# Properties

 Label 825.2.c.g.199.5 Level $825$ Weight $2$ Character 825.199 Analytic conductor $6.588$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [825,2,Mod(199,825)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(825, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("825.199");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$825 = 3 \cdot 5^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 825.c (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: $$6.58765816676$$ 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: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{3}$$ Twist minimal: no (minimal twist has level 165) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 199.5 Root $$1.45161 - 1.45161i$$ of defining polynomial Character $$\chi$$ $$=$$ 825.199 Dual form 825.2.c.g.199.2

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.90321i q^{2} -1.00000i q^{3} -1.62222 q^{4} +1.90321 q^{6} -4.42864i q^{7} +0.719004i q^{8} -1.00000 q^{9} +O(q^{10})$$ $$q+1.90321i q^{2} -1.00000i q^{3} -1.62222 q^{4} +1.90321 q^{6} -4.42864i q^{7} +0.719004i q^{8} -1.00000 q^{9} +1.00000 q^{11} +1.62222i q^{12} +0.622216i q^{13} +8.42864 q^{14} -4.61285 q^{16} -5.18421i q^{17} -1.90321i q^{18} -7.05086 q^{19} -4.42864 q^{21} +1.90321i q^{22} -8.85728i q^{23} +0.719004 q^{24} -1.18421 q^{26} +1.00000i q^{27} +7.18421i q^{28} +7.80642 q^{29} +2.75557 q^{31} -7.34122i q^{32} -1.00000i q^{33} +9.86665 q^{34} +1.62222 q^{36} -2.00000i q^{37} -13.4193i q^{38} +0.622216 q^{39} -0.193576 q^{41} -8.42864i q^{42} -5.67307i q^{43} -1.62222 q^{44} +16.8573 q^{46} -2.75557i q^{47} +4.61285i q^{48} -12.6128 q^{49} -5.18421 q^{51} -1.00937i q^{52} +10.8573i q^{53} -1.90321 q^{54} +3.18421 q^{56} +7.05086i q^{57} +14.8573i q^{58} +4.85728 q^{59} +6.85728 q^{61} +5.24443i q^{62} +4.42864i q^{63} +4.74620 q^{64} +1.90321 q^{66} -1.24443i q^{67} +8.40990i q^{68} -8.85728 q^{69} +2.75557 q^{71} -0.719004i q^{72} -4.23506i q^{73} +3.80642 q^{74} +11.4380 q^{76} -4.42864i q^{77} +1.18421i q^{78} -8.56199 q^{79} +1.00000 q^{81} -0.368416i q^{82} -0.133353i q^{83} +7.18421 q^{84} +10.7971 q^{86} -7.80642i q^{87} +0.719004i q^{88} -5.61285 q^{89} +2.75557 q^{91} +14.3684i q^{92} -2.75557i q^{93} +5.24443 q^{94} -7.34122 q^{96} +7.24443i q^{97} -24.0049i q^{98} -1.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 10 q^{4} - 2 q^{6} - 6 q^{9}+O(q^{10})$$ 6 * q - 10 * q^4 - 2 * q^6 - 6 * q^9 $$6 q - 10 q^{4} - 2 q^{6} - 6 q^{9} + 6 q^{11} + 24 q^{14} + 26 q^{16} - 16 q^{19} + 18 q^{24} + 20 q^{26} + 20 q^{29} + 16 q^{31} + 60 q^{34} + 10 q^{36} + 4 q^{39} - 28 q^{41} - 10 q^{44} + 48 q^{46} - 22 q^{49} - 4 q^{51} + 2 q^{54} - 8 q^{56} - 24 q^{59} - 12 q^{61} - 26 q^{64} - 2 q^{66} + 16 q^{71} - 4 q^{74} + 96 q^{76} - 24 q^{79} + 6 q^{81} + 16 q^{84} - 16 q^{86} + 20 q^{89} + 16 q^{91} + 32 q^{94} - 58 q^{96} - 6 q^{99}+O(q^{100})$$ 6 * q - 10 * q^4 - 2 * q^6 - 6 * q^9 + 6 * q^11 + 24 * q^14 + 26 * q^16 - 16 * q^19 + 18 * q^24 + 20 * q^26 + 20 * q^29 + 16 * q^31 + 60 * q^34 + 10 * q^36 + 4 * q^39 - 28 * q^41 - 10 * q^44 + 48 * q^46 - 22 * q^49 - 4 * q^51 + 2 * q^54 - 8 * q^56 - 24 * q^59 - 12 * q^61 - 26 * q^64 - 2 * q^66 + 16 * q^71 - 4 * q^74 + 96 * q^76 - 24 * q^79 + 6 * q^81 + 16 * q^84 - 16 * q^86 + 20 * q^89 + 16 * q^91 + 32 * q^94 - 58 * q^96 - 6 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/825\mathbb{Z}\right)^\times$$.

 $$n$$ $$376$$ $$551$$ $$727$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.90321i 1.34577i 0.739745 + 0.672887i $$0.234946\pi$$
−0.739745 + 0.672887i $$0.765054\pi$$
$$3$$ − 1.00000i − 0.577350i
$$4$$ −1.62222 −0.811108
$$5$$ 0 0
$$6$$ 1.90321 0.776983
$$7$$ − 4.42864i − 1.67387i −0.547304 0.836934i $$-0.684346\pi$$
0.547304 0.836934i $$-0.315654\pi$$
$$8$$ 0.719004i 0.254206i
$$9$$ −1.00000 −0.333333
$$10$$ 0 0
$$11$$ 1.00000 0.301511
$$12$$ 1.62222i 0.468293i
$$13$$ 0.622216i 0.172572i 0.996270 + 0.0862858i $$0.0274998\pi$$
−0.996270 + 0.0862858i $$0.972500\pi$$
$$14$$ 8.42864 2.25265
$$15$$ 0 0
$$16$$ −4.61285 −1.15321
$$17$$ − 5.18421i − 1.25736i −0.777666 0.628678i $$-0.783597\pi$$
0.777666 0.628678i $$-0.216403\pi$$
$$18$$ − 1.90321i − 0.448591i
$$19$$ −7.05086 −1.61758 −0.808789 0.588100i $$-0.799876\pi$$
−0.808789 + 0.588100i $$0.799876\pi$$
$$20$$ 0 0
$$21$$ −4.42864 −0.966408
$$22$$ 1.90321i 0.405766i
$$23$$ − 8.85728i − 1.84687i −0.383754 0.923435i $$-0.625369\pi$$
0.383754 0.923435i $$-0.374631\pi$$
$$24$$ 0.719004 0.146766
$$25$$ 0 0
$$26$$ −1.18421 −0.232242
$$27$$ 1.00000i 0.192450i
$$28$$ 7.18421i 1.35769i
$$29$$ 7.80642 1.44962 0.724808 0.688951i $$-0.241928\pi$$
0.724808 + 0.688951i $$0.241928\pi$$
$$30$$ 0 0
$$31$$ 2.75557 0.494915 0.247457 0.968899i $$-0.420405\pi$$
0.247457 + 0.968899i $$0.420405\pi$$
$$32$$ − 7.34122i − 1.29776i
$$33$$ − 1.00000i − 0.174078i
$$34$$ 9.86665 1.69212
$$35$$ 0 0
$$36$$ 1.62222 0.270369
$$37$$ − 2.00000i − 0.328798i −0.986394 0.164399i $$-0.947432\pi$$
0.986394 0.164399i $$-0.0525685\pi$$
$$38$$ − 13.4193i − 2.17689i
$$39$$ 0.622216 0.0996342
$$40$$ 0 0
$$41$$ −0.193576 −0.0302315 −0.0151158 0.999886i $$-0.504812\pi$$
−0.0151158 + 0.999886i $$0.504812\pi$$
$$42$$ − 8.42864i − 1.30057i
$$43$$ − 5.67307i − 0.865135i −0.901602 0.432568i $$-0.857608\pi$$
0.901602 0.432568i $$-0.142392\pi$$
$$44$$ −1.62222 −0.244558
$$45$$ 0 0
$$46$$ 16.8573 2.48547
$$47$$ − 2.75557i − 0.401941i −0.979597 0.200971i $$-0.935590\pi$$
0.979597 0.200971i $$-0.0644095\pi$$
$$48$$ 4.61285i 0.665807i
$$49$$ −12.6128 −1.80184
$$50$$ 0 0
$$51$$ −5.18421 −0.725934
$$52$$ − 1.00937i − 0.139974i
$$53$$ 10.8573i 1.49136i 0.666303 + 0.745681i $$0.267876\pi$$
−0.666303 + 0.745681i $$0.732124\pi$$
$$54$$ −1.90321 −0.258994
$$55$$ 0 0
$$56$$ 3.18421 0.425508
$$57$$ 7.05086i 0.933909i
$$58$$ 14.8573i 1.95086i
$$59$$ 4.85728 0.632364 0.316182 0.948699i $$-0.397599\pi$$
0.316182 + 0.948699i $$0.397599\pi$$
$$60$$ 0 0
$$61$$ 6.85728 0.877985 0.438992 0.898491i $$-0.355336\pi$$
0.438992 + 0.898491i $$0.355336\pi$$
$$62$$ 5.24443i 0.666043i
$$63$$ 4.42864i 0.557956i
$$64$$ 4.74620 0.593275
$$65$$ 0 0
$$66$$ 1.90321 0.234269
$$67$$ − 1.24443i − 0.152031i −0.997107 0.0760157i $$-0.975780\pi$$
0.997107 0.0760157i $$-0.0242199\pi$$
$$68$$ 8.40990i 1.01985i
$$69$$ −8.85728 −1.06629
$$70$$ 0 0
$$71$$ 2.75557 0.327026 0.163513 0.986541i $$-0.447717\pi$$
0.163513 + 0.986541i $$0.447717\pi$$
$$72$$ − 0.719004i − 0.0847354i
$$73$$ − 4.23506i − 0.495677i −0.968801 0.247838i $$-0.920280\pi$$
0.968801 0.247838i $$-0.0797202\pi$$
$$74$$ 3.80642 0.442488
$$75$$ 0 0
$$76$$ 11.4380 1.31203
$$77$$ − 4.42864i − 0.504690i
$$78$$ 1.18421i 0.134085i
$$79$$ −8.56199 −0.963299 −0.481650 0.876364i $$-0.659962\pi$$
−0.481650 + 0.876364i $$0.659962\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ − 0.368416i − 0.0406848i
$$83$$ − 0.133353i − 0.0146374i −0.999973 0.00731870i $$-0.997670\pi$$
0.999973 0.00731870i $$-0.00232964\pi$$
$$84$$ 7.18421 0.783861
$$85$$ 0 0
$$86$$ 10.7971 1.16428
$$87$$ − 7.80642i − 0.836936i
$$88$$ 0.719004i 0.0766461i
$$89$$ −5.61285 −0.594961 −0.297480 0.954728i $$-0.596146\pi$$
−0.297480 + 0.954728i $$0.596146\pi$$
$$90$$ 0 0
$$91$$ 2.75557 0.288862
$$92$$ 14.3684i 1.49801i
$$93$$ − 2.75557i − 0.285739i
$$94$$ 5.24443 0.540922
$$95$$ 0 0
$$96$$ −7.34122 −0.749260
$$97$$ 7.24443i 0.735561i 0.929913 + 0.367780i $$0.119882\pi$$
−0.929913 + 0.367780i $$0.880118\pi$$
$$98$$ − 24.0049i − 2.42486i
$$99$$ −1.00000 −0.100504
$$100$$ 0 0
$$101$$ 4.66370 0.464056 0.232028 0.972709i $$-0.425464\pi$$
0.232028 + 0.972709i $$0.425464\pi$$
$$102$$ − 9.86665i − 0.976944i
$$103$$ 11.6128i 1.14425i 0.820167 + 0.572124i $$0.193880\pi$$
−0.820167 + 0.572124i $$0.806120\pi$$
$$104$$ −0.447375 −0.0438688
$$105$$ 0 0
$$106$$ −20.6637 −2.00704
$$107$$ 2.62222i 0.253499i 0.991935 + 0.126750i $$0.0404545\pi$$
−0.991935 + 0.126750i $$0.959546\pi$$
$$108$$ − 1.62222i − 0.156098i
$$109$$ 19.7146 1.88831 0.944156 0.329499i $$-0.106880\pi$$
0.944156 + 0.329499i $$0.106880\pi$$
$$110$$ 0 0
$$111$$ −2.00000 −0.189832
$$112$$ 20.4286i 1.93032i
$$113$$ 6.00000i 0.564433i 0.959351 + 0.282216i $$0.0910696\pi$$
−0.959351 + 0.282216i $$0.908930\pi$$
$$114$$ −13.4193 −1.25683
$$115$$ 0 0
$$116$$ −12.6637 −1.17580
$$117$$ − 0.622216i − 0.0575239i
$$118$$ 9.24443i 0.851019i
$$119$$ −22.9590 −2.10465
$$120$$ 0 0
$$121$$ 1.00000 0.0909091
$$122$$ 13.0509i 1.18157i
$$123$$ 0.193576i 0.0174542i
$$124$$ −4.47013 −0.401429
$$125$$ 0 0
$$126$$ −8.42864 −0.750883
$$127$$ − 15.1842i − 1.34738i −0.739014 0.673690i $$-0.764708\pi$$
0.739014 0.673690i $$-0.235292\pi$$
$$128$$ − 5.64941i − 0.499342i
$$129$$ −5.67307 −0.499486
$$130$$ 0 0
$$131$$ 1.24443 0.108726 0.0543632 0.998521i $$-0.482687\pi$$
0.0543632 + 0.998521i $$0.482687\pi$$
$$132$$ 1.62222i 0.141196i
$$133$$ 31.2257i 2.70761i
$$134$$ 2.36842 0.204600
$$135$$ 0 0
$$136$$ 3.72746 0.319627
$$137$$ − 0.488863i − 0.0417663i −0.999782 0.0208832i $$-0.993352\pi$$
0.999782 0.0208832i $$-0.00664780\pi$$
$$138$$ − 16.8573i − 1.43499i
$$139$$ −17.8064 −1.51032 −0.755161 0.655540i $$-0.772441\pi$$
−0.755161 + 0.655540i $$0.772441\pi$$
$$140$$ 0 0
$$141$$ −2.75557 −0.232061
$$142$$ 5.24443i 0.440103i
$$143$$ 0.622216i 0.0520323i
$$144$$ 4.61285 0.384404
$$145$$ 0 0
$$146$$ 8.06022 0.667069
$$147$$ 12.6128i 1.04029i
$$148$$ 3.24443i 0.266691i
$$149$$ 1.43801 0.117806 0.0589031 0.998264i $$-0.481240\pi$$
0.0589031 + 0.998264i $$0.481240\pi$$
$$150$$ 0 0
$$151$$ −12.1748 −0.990774 −0.495387 0.868672i $$-0.664974\pi$$
−0.495387 + 0.868672i $$0.664974\pi$$
$$152$$ − 5.06959i − 0.411198i
$$153$$ 5.18421i 0.419118i
$$154$$ 8.42864 0.679199
$$155$$ 0 0
$$156$$ −1.00937 −0.0808141
$$157$$ 18.4701i 1.47408i 0.675851 + 0.737038i $$0.263776\pi$$
−0.675851 + 0.737038i $$0.736224\pi$$
$$158$$ − 16.2953i − 1.29638i
$$159$$ 10.8573 0.861038
$$160$$ 0 0
$$161$$ −39.2257 −3.09142
$$162$$ 1.90321i 0.149530i
$$163$$ 10.1017i 0.791227i 0.918417 + 0.395614i $$0.129468\pi$$
−0.918417 + 0.395614i $$0.870532\pi$$
$$164$$ 0.314022 0.0245210
$$165$$ 0 0
$$166$$ 0.253799 0.0196986
$$167$$ − 16.3368i − 1.26418i −0.774896 0.632089i $$-0.782198\pi$$
0.774896 0.632089i $$-0.217802\pi$$
$$168$$ − 3.18421i − 0.245667i
$$169$$ 12.6128 0.970219
$$170$$ 0 0
$$171$$ 7.05086 0.539192
$$172$$ 9.20294i 0.701718i
$$173$$ 9.18421i 0.698262i 0.937074 + 0.349131i $$0.113523\pi$$
−0.937074 + 0.349131i $$0.886477\pi$$
$$174$$ 14.8573 1.12633
$$175$$ 0 0
$$176$$ −4.61285 −0.347706
$$177$$ − 4.85728i − 0.365095i
$$178$$ − 10.6824i − 0.800683i
$$179$$ 25.3274 1.89306 0.946530 0.322617i $$-0.104563\pi$$
0.946530 + 0.322617i $$0.104563\pi$$
$$180$$ 0 0
$$181$$ −13.6128 −1.01184 −0.505918 0.862582i $$-0.668846\pi$$
−0.505918 + 0.862582i $$0.668846\pi$$
$$182$$ 5.24443i 0.388743i
$$183$$ − 6.85728i − 0.506905i
$$184$$ 6.36842 0.469486
$$185$$ 0 0
$$186$$ 5.24443 0.384540
$$187$$ − 5.18421i − 0.379107i
$$188$$ 4.47013i 0.326017i
$$189$$ 4.42864 0.322136
$$190$$ 0 0
$$191$$ 6.10171 0.441504 0.220752 0.975330i $$-0.429149\pi$$
0.220752 + 0.975330i $$0.429149\pi$$
$$192$$ − 4.74620i − 0.342528i
$$193$$ − 18.3368i − 1.31991i −0.751305 0.659955i $$-0.770575\pi$$
0.751305 0.659955i $$-0.229425\pi$$
$$194$$ −13.7877 −0.989898
$$195$$ 0 0
$$196$$ 20.4608 1.46148
$$197$$ − 6.69535i − 0.477024i −0.971140 0.238512i $$-0.923340\pi$$
0.971140 0.238512i $$-0.0766596\pi$$
$$198$$ − 1.90321i − 0.135255i
$$199$$ −14.1017 −0.999644 −0.499822 0.866128i $$-0.666601\pi$$
−0.499822 + 0.866128i $$0.666601\pi$$
$$200$$ 0 0
$$201$$ −1.24443 −0.0877754
$$202$$ 8.87601i 0.624514i
$$203$$ − 34.5718i − 2.42647i
$$204$$ 8.40990 0.588811
$$205$$ 0 0
$$206$$ −22.1017 −1.53990
$$207$$ 8.85728i 0.615623i
$$208$$ − 2.87019i − 0.199012i
$$209$$ −7.05086 −0.487718
$$210$$ 0 0
$$211$$ −10.6637 −0.734120 −0.367060 0.930197i $$-0.619636\pi$$
−0.367060 + 0.930197i $$0.619636\pi$$
$$212$$ − 17.6128i − 1.20966i
$$213$$ − 2.75557i − 0.188808i
$$214$$ −4.99063 −0.341153
$$215$$ 0 0
$$216$$ −0.719004 −0.0489220
$$217$$ − 12.2034i − 0.828422i
$$218$$ 37.5210i 2.54124i
$$219$$ −4.23506 −0.286179
$$220$$ 0 0
$$221$$ 3.22570 0.216984
$$222$$ − 3.80642i − 0.255470i
$$223$$ − 8.85728i − 0.593127i −0.955013 0.296564i $$-0.904159\pi$$
0.955013 0.296564i $$-0.0958407\pi$$
$$224$$ −32.5116 −2.17227
$$225$$ 0 0
$$226$$ −11.4193 −0.759599
$$227$$ 13.3778i 0.887915i 0.896048 + 0.443957i $$0.146426\pi$$
−0.896048 + 0.443957i $$0.853574\pi$$
$$228$$ − 11.4380i − 0.757501i
$$229$$ −11.5111 −0.760677 −0.380339 0.924847i $$-0.624193\pi$$
−0.380339 + 0.924847i $$0.624193\pi$$
$$230$$ 0 0
$$231$$ −4.42864 −0.291383
$$232$$ 5.61285i 0.368502i
$$233$$ 4.32693i 0.283467i 0.989905 + 0.141733i $$0.0452675\pi$$
−0.989905 + 0.141733i $$0.954732\pi$$
$$234$$ 1.18421 0.0774141
$$235$$ 0 0
$$236$$ −7.87955 −0.512915
$$237$$ 8.56199i 0.556161i
$$238$$ − 43.6958i − 2.83238i
$$239$$ 3.34614 0.216444 0.108222 0.994127i $$-0.465484\pi$$
0.108222 + 0.994127i $$0.465484\pi$$
$$240$$ 0 0
$$241$$ −1.34614 −0.0867126 −0.0433563 0.999060i $$-0.513805\pi$$
−0.0433563 + 0.999060i $$0.513805\pi$$
$$242$$ 1.90321i 0.122343i
$$243$$ − 1.00000i − 0.0641500i
$$244$$ −11.1240 −0.712140
$$245$$ 0 0
$$246$$ −0.368416 −0.0234894
$$247$$ − 4.38715i − 0.279148i
$$248$$ 1.98126i 0.125810i
$$249$$ −0.133353 −0.00845091
$$250$$ 0 0
$$251$$ 22.7556 1.43632 0.718159 0.695879i $$-0.244985\pi$$
0.718159 + 0.695879i $$0.244985\pi$$
$$252$$ − 7.18421i − 0.452563i
$$253$$ − 8.85728i − 0.556852i
$$254$$ 28.8988 1.81327
$$255$$ 0 0
$$256$$ 20.2444 1.26528
$$257$$ − 6.85728i − 0.427745i −0.976862 0.213873i $$-0.931392\pi$$
0.976862 0.213873i $$-0.0686078\pi$$
$$258$$ − 10.7971i − 0.672195i
$$259$$ −8.85728 −0.550365
$$260$$ 0 0
$$261$$ −7.80642 −0.483206
$$262$$ 2.36842i 0.146321i
$$263$$ 29.5812i 1.82406i 0.410129 + 0.912028i $$0.365484\pi$$
−0.410129 + 0.912028i $$0.634516\pi$$
$$264$$ 0.719004 0.0442516
$$265$$ 0 0
$$266$$ −59.4291 −3.64383
$$267$$ 5.61285i 0.343501i
$$268$$ 2.01874i 0.123314i
$$269$$ −8.48886 −0.517575 −0.258788 0.965934i $$-0.583323\pi$$
−0.258788 + 0.965934i $$0.583323\pi$$
$$270$$ 0 0
$$271$$ 14.6637 0.890757 0.445378 0.895343i $$-0.353069\pi$$
0.445378 + 0.895343i $$0.353069\pi$$
$$272$$ 23.9140i 1.45000i
$$273$$ − 2.75557i − 0.166775i
$$274$$ 0.930409 0.0562081
$$275$$ 0 0
$$276$$ 14.3684 0.864877
$$277$$ 14.6035i 0.877438i 0.898624 + 0.438719i $$0.144568\pi$$
−0.898624 + 0.438719i $$0.855432\pi$$
$$278$$ − 33.8894i − 2.03255i
$$279$$ −2.75557 −0.164972
$$280$$ 0 0
$$281$$ −0.193576 −0.0115478 −0.00577389 0.999983i $$-0.501838\pi$$
−0.00577389 + 0.999983i $$0.501838\pi$$
$$282$$ − 5.24443i − 0.312301i
$$283$$ − 27.1842i − 1.61593i −0.589228 0.807967i $$-0.700568\pi$$
0.589228 0.807967i $$-0.299432\pi$$
$$284$$ −4.47013 −0.265253
$$285$$ 0 0
$$286$$ −1.18421 −0.0700237
$$287$$ 0.857279i 0.0506036i
$$288$$ 7.34122i 0.432585i
$$289$$ −9.87601 −0.580942
$$290$$ 0 0
$$291$$ 7.24443 0.424676
$$292$$ 6.87019i 0.402047i
$$293$$ 2.81579i 0.164500i 0.996612 + 0.0822502i $$0.0262106\pi$$
−0.996612 + 0.0822502i $$0.973789\pi$$
$$294$$ −24.0049 −1.40000
$$295$$ 0 0
$$296$$ 1.43801 0.0835825
$$297$$ 1.00000i 0.0580259i
$$298$$ 2.73683i 0.158540i
$$299$$ 5.51114 0.318717
$$300$$ 0 0
$$301$$ −25.1240 −1.44812
$$302$$ − 23.1713i − 1.33336i
$$303$$ − 4.66370i − 0.267923i
$$304$$ 32.5245 1.86541
$$305$$ 0 0
$$306$$ −9.86665 −0.564039
$$307$$ − 24.4286i − 1.39422i −0.716966 0.697108i $$-0.754470\pi$$
0.716966 0.697108i $$-0.245530\pi$$
$$308$$ 7.18421i 0.409358i
$$309$$ 11.6128 0.660632
$$310$$ 0 0
$$311$$ 19.8796 1.12727 0.563633 0.826025i $$-0.309403\pi$$
0.563633 + 0.826025i $$0.309403\pi$$
$$312$$ 0.447375i 0.0253276i
$$313$$ 15.7146i 0.888239i 0.895967 + 0.444120i $$0.146483\pi$$
−0.895967 + 0.444120i $$0.853517\pi$$
$$314$$ −35.1526 −1.98377
$$315$$ 0 0
$$316$$ 13.8894 0.781340
$$317$$ 16.4889i 0.926107i 0.886330 + 0.463053i $$0.153246\pi$$
−0.886330 + 0.463053i $$0.846754\pi$$
$$318$$ 20.6637i 1.15876i
$$319$$ 7.80642 0.437076
$$320$$ 0 0
$$321$$ 2.62222 0.146358
$$322$$ − 74.6548i − 4.16035i
$$323$$ 36.5531i 2.03387i
$$324$$ −1.62222 −0.0901231
$$325$$ 0 0
$$326$$ −19.2257 −1.06481
$$327$$ − 19.7146i − 1.09022i
$$328$$ − 0.139182i − 0.00768504i
$$329$$ −12.2034 −0.672796
$$330$$ 0 0
$$331$$ 15.3461 0.843500 0.421750 0.906712i $$-0.361416\pi$$
0.421750 + 0.906712i $$0.361416\pi$$
$$332$$ 0.216327i 0.0118725i
$$333$$ 2.00000i 0.109599i
$$334$$ 31.0923 1.70130
$$335$$ 0 0
$$336$$ 20.4286 1.11447
$$337$$ 28.2351i 1.53806i 0.639212 + 0.769031i $$0.279261\pi$$
−0.639212 + 0.769031i $$0.720739\pi$$
$$338$$ 24.0049i 1.30570i
$$339$$ 6.00000 0.325875
$$340$$ 0 0
$$341$$ 2.75557 0.149222
$$342$$ 13.4193i 0.725631i
$$343$$ 24.8573i 1.34217i
$$344$$ 4.07896 0.219923
$$345$$ 0 0
$$346$$ −17.4795 −0.939703
$$347$$ 2.62222i 0.140768i 0.997520 + 0.0703840i $$0.0224224\pi$$
−0.997520 + 0.0703840i $$0.977578\pi$$
$$348$$ 12.6637i 0.678846i
$$349$$ −5.14272 −0.275284 −0.137642 0.990482i $$-0.543952\pi$$
−0.137642 + 0.990482i $$0.543952\pi$$
$$350$$ 0 0
$$351$$ −0.622216 −0.0332114
$$352$$ − 7.34122i − 0.391288i
$$353$$ 9.34614i 0.497445i 0.968575 + 0.248722i $$0.0800107\pi$$
−0.968575 + 0.248722i $$0.919989\pi$$
$$354$$ 9.24443 0.491336
$$355$$ 0 0
$$356$$ 9.10525 0.482577
$$357$$ 22.9590i 1.21512i
$$358$$ 48.2034i 2.54763i
$$359$$ −10.7556 −0.567657 −0.283829 0.958875i $$-0.591605\pi$$
−0.283829 + 0.958875i $$0.591605\pi$$
$$360$$ 0 0
$$361$$ 30.7146 1.61656
$$362$$ − 25.9081i − 1.36170i
$$363$$ − 1.00000i − 0.0524864i
$$364$$ −4.47013 −0.234298
$$365$$ 0 0
$$366$$ 13.0509 0.682179
$$367$$ 33.7975i 1.76422i 0.471046 + 0.882108i $$0.343876\pi$$
−0.471046 + 0.882108i $$0.656124\pi$$
$$368$$ 40.8573i 2.12983i
$$369$$ 0.193576 0.0100772
$$370$$ 0 0
$$371$$ 48.0830 2.49634
$$372$$ 4.47013i 0.231765i
$$373$$ − 33.9496i − 1.75784i −0.476965 0.878922i $$-0.658263\pi$$
0.476965 0.878922i $$-0.341737\pi$$
$$374$$ 9.86665 0.510192
$$375$$ 0 0
$$376$$ 1.98126 0.102176
$$377$$ 4.85728i 0.250163i
$$378$$ 8.42864i 0.433522i
$$379$$ 20.0000 1.02733 0.513665 0.857991i $$-0.328287\pi$$
0.513665 + 0.857991i $$0.328287\pi$$
$$380$$ 0 0
$$381$$ −15.1842 −0.777911
$$382$$ 11.6128i 0.594165i
$$383$$ 14.6351i 0.747820i 0.927465 + 0.373910i $$0.121983\pi$$
−0.927465 + 0.373910i $$0.878017\pi$$
$$384$$ −5.64941 −0.288295
$$385$$ 0 0
$$386$$ 34.8988 1.77630
$$387$$ 5.67307i 0.288378i
$$388$$ − 11.7520i − 0.596619i
$$389$$ 5.61285 0.284583 0.142291 0.989825i $$-0.454553\pi$$
0.142291 + 0.989825i $$0.454553\pi$$
$$390$$ 0 0
$$391$$ −45.9180 −2.32217
$$392$$ − 9.06868i − 0.458038i
$$393$$ − 1.24443i − 0.0627733i
$$394$$ 12.7427 0.641966
$$395$$ 0 0
$$396$$ 1.62222 0.0815194
$$397$$ − 12.7556i − 0.640184i −0.947387 0.320092i $$-0.896286\pi$$
0.947387 0.320092i $$-0.103714\pi$$
$$398$$ − 26.8385i − 1.34529i
$$399$$ 31.2257 1.56324
$$400$$ 0 0
$$401$$ 2.00000 0.0998752 0.0499376 0.998752i $$-0.484098\pi$$
0.0499376 + 0.998752i $$0.484098\pi$$
$$402$$ − 2.36842i − 0.118126i
$$403$$ 1.71456i 0.0854082i
$$404$$ −7.56553 −0.376399
$$405$$ 0 0
$$406$$ 65.7975 3.26548
$$407$$ − 2.00000i − 0.0991363i
$$408$$ − 3.72746i − 0.184537i
$$409$$ 7.12399 0.352258 0.176129 0.984367i $$-0.443642\pi$$
0.176129 + 0.984367i $$0.443642\pi$$
$$410$$ 0 0
$$411$$ −0.488863 −0.0241138
$$412$$ − 18.8385i − 0.928108i
$$413$$ − 21.5111i − 1.05849i
$$414$$ −16.8573 −0.828490
$$415$$ 0 0
$$416$$ 4.56782 0.223956
$$417$$ 17.8064i 0.871984i
$$418$$ − 13.4193i − 0.656358i
$$419$$ −15.6128 −0.762738 −0.381369 0.924423i $$-0.624547\pi$$
−0.381369 + 0.924423i $$0.624547\pi$$
$$420$$ 0 0
$$421$$ 7.89829 0.384939 0.192470 0.981303i $$-0.438350\pi$$
0.192470 + 0.981303i $$0.438350\pi$$
$$422$$ − 20.2953i − 0.987959i
$$423$$ 2.75557i 0.133980i
$$424$$ −7.80642 −0.379113
$$425$$ 0 0
$$426$$ 5.24443 0.254094
$$427$$ − 30.3684i − 1.46963i
$$428$$ − 4.25380i − 0.205615i
$$429$$ 0.622216 0.0300409
$$430$$ 0 0
$$431$$ 34.3051 1.65242 0.826210 0.563362i $$-0.190492\pi$$
0.826210 + 0.563362i $$0.190492\pi$$
$$432$$ − 4.61285i − 0.221936i
$$433$$ − 14.4701i − 0.695390i −0.937608 0.347695i $$-0.886964\pi$$
0.937608 0.347695i $$-0.113036\pi$$
$$434$$ 23.2257 1.11487
$$435$$ 0 0
$$436$$ −31.9813 −1.53162
$$437$$ 62.4514i 2.98746i
$$438$$ − 8.06022i − 0.385132i
$$439$$ 19.3176 0.921977 0.460988 0.887406i $$-0.347495\pi$$
0.460988 + 0.887406i $$0.347495\pi$$
$$440$$ 0 0
$$441$$ 12.6128 0.600612
$$442$$ 6.13918i 0.292011i
$$443$$ 13.1240i 0.623539i 0.950158 + 0.311770i $$0.100922\pi$$
−0.950158 + 0.311770i $$0.899078\pi$$
$$444$$ 3.24443 0.153974
$$445$$ 0 0
$$446$$ 16.8573 0.798215
$$447$$ − 1.43801i − 0.0680154i
$$448$$ − 21.0192i − 0.993064i
$$449$$ 32.3051 1.52457 0.762287 0.647240i $$-0.224077\pi$$
0.762287 + 0.647240i $$0.224077\pi$$
$$450$$ 0 0
$$451$$ −0.193576 −0.00911514
$$452$$ − 9.73329i − 0.457816i
$$453$$ 12.1748i 0.572024i
$$454$$ −25.4608 −1.19493
$$455$$ 0 0
$$456$$ −5.06959 −0.237405
$$457$$ − 23.4608i − 1.09745i −0.836004 0.548724i $$-0.815114\pi$$
0.836004 0.548724i $$-0.184886\pi$$
$$458$$ − 21.9081i − 1.02370i
$$459$$ 5.18421 0.241978
$$460$$ 0 0
$$461$$ −28.8671 −1.34448 −0.672238 0.740335i $$-0.734667\pi$$
−0.672238 + 0.740335i $$0.734667\pi$$
$$462$$ − 8.42864i − 0.392136i
$$463$$ 19.3461i 0.899091i 0.893257 + 0.449546i $$0.148414\pi$$
−0.893257 + 0.449546i $$0.851586\pi$$
$$464$$ −36.0098 −1.67172
$$465$$ 0 0
$$466$$ −8.23506 −0.381482
$$467$$ 3.14272i 0.145428i 0.997353 + 0.0727139i $$0.0231660\pi$$
−0.997353 + 0.0727139i $$0.976834\pi$$
$$468$$ 1.00937i 0.0466580i
$$469$$ −5.51114 −0.254481
$$470$$ 0 0
$$471$$ 18.4701 0.851059
$$472$$ 3.49240i 0.160751i
$$473$$ − 5.67307i − 0.260848i
$$474$$ −16.2953 −0.748467
$$475$$ 0 0
$$476$$ 37.2444 1.70710
$$477$$ − 10.8573i − 0.497121i
$$478$$ 6.36842i 0.291285i
$$479$$ 24.8573 1.13576 0.567879 0.823112i $$-0.307764\pi$$
0.567879 + 0.823112i $$0.307764\pi$$
$$480$$ 0 0
$$481$$ 1.24443 0.0567412
$$482$$ − 2.56199i − 0.116696i
$$483$$ 39.2257i 1.78483i
$$484$$ −1.62222 −0.0737371
$$485$$ 0 0
$$486$$ 1.90321 0.0863314
$$487$$ − 11.5299i − 0.522468i −0.965275 0.261234i $$-0.915871\pi$$
0.965275 0.261234i $$-0.0841295\pi$$
$$488$$ 4.93041i 0.223189i
$$489$$ 10.1017 0.456815
$$490$$ 0 0
$$491$$ 16.3872 0.739542 0.369771 0.929123i $$-0.379436\pi$$
0.369771 + 0.929123i $$0.379436\pi$$
$$492$$ − 0.314022i − 0.0141572i
$$493$$ − 40.4701i − 1.82268i
$$494$$ 8.34968 0.375670
$$495$$ 0 0
$$496$$ −12.7110 −0.570742
$$497$$ − 12.2034i − 0.547398i
$$498$$ − 0.253799i − 0.0113730i
$$499$$ 25.3274 1.13381 0.566905 0.823783i $$-0.308141\pi$$
0.566905 + 0.823783i $$0.308141\pi$$
$$500$$ 0 0
$$501$$ −16.3368 −0.729873
$$502$$ 43.3087i 1.93296i
$$503$$ − 19.0923i − 0.851285i −0.904891 0.425643i $$-0.860048\pi$$
0.904891 0.425643i $$-0.139952\pi$$
$$504$$ −3.18421 −0.141836
$$505$$ 0 0
$$506$$ 16.8573 0.749397
$$507$$ − 12.6128i − 0.560156i
$$508$$ 24.6321i 1.09287i
$$509$$ 32.4514 1.43838 0.719191 0.694812i $$-0.244512\pi$$
0.719191 + 0.694812i $$0.244512\pi$$
$$510$$ 0 0
$$511$$ −18.7556 −0.829698
$$512$$ 27.2306i 1.20343i
$$513$$ − 7.05086i − 0.311303i
$$514$$ 13.0509 0.575649
$$515$$ 0 0
$$516$$ 9.20294 0.405137
$$517$$ − 2.75557i − 0.121190i
$$518$$ − 16.8573i − 0.740666i
$$519$$ 9.18421 0.403142
$$520$$ 0 0
$$521$$ −29.2257 −1.28040 −0.640200 0.768208i $$-0.721149\pi$$
−0.640200 + 0.768208i $$0.721149\pi$$
$$522$$ − 14.8573i − 0.650285i
$$523$$ − 6.71408i − 0.293586i −0.989167 0.146793i $$-0.953105\pi$$
0.989167 0.146793i $$-0.0468952\pi$$
$$524$$ −2.01874 −0.0881889
$$525$$ 0 0
$$526$$ −56.2993 −2.45477
$$527$$ − 14.2854i − 0.622284i
$$528$$ 4.61285i 0.200748i
$$529$$ −55.4514 −2.41093
$$530$$ 0 0
$$531$$ −4.85728 −0.210788
$$532$$ − 50.6548i − 2.19616i
$$533$$ − 0.120446i − 0.00521710i
$$534$$ −10.6824 −0.462274
$$535$$ 0 0
$$536$$ 0.894751 0.0386473
$$537$$ − 25.3274i − 1.09296i
$$538$$ − 16.1561i − 0.696539i
$$539$$ −12.6128 −0.543274
$$540$$ 0 0
$$541$$ −18.0000 −0.773880 −0.386940 0.922105i $$-0.626468\pi$$
−0.386940 + 0.922105i $$0.626468\pi$$
$$542$$ 27.9081i 1.19876i
$$543$$ 13.6128i 0.584183i
$$544$$ −38.0584 −1.63174
$$545$$ 0 0
$$546$$ 5.24443 0.224441
$$547$$ 41.3689i 1.76881i 0.466724 + 0.884403i $$0.345434\pi$$
−0.466724 + 0.884403i $$0.654566\pi$$
$$548$$ 0.793040i 0.0338770i
$$549$$ −6.85728 −0.292662
$$550$$ 0 0
$$551$$ −55.0420 −2.34487
$$552$$ − 6.36842i − 0.271058i
$$553$$ 37.9180i 1.61244i
$$554$$ −27.7935 −1.18083
$$555$$ 0 0
$$556$$ 28.8859 1.22503
$$557$$ − 20.7971i − 0.881200i −0.897704 0.440600i $$-0.854766\pi$$
0.897704 0.440600i $$-0.145234\pi$$
$$558$$ − 5.24443i − 0.222014i
$$559$$ 3.52987 0.149298
$$560$$ 0 0
$$561$$ −5.18421 −0.218877
$$562$$ − 0.368416i − 0.0155407i
$$563$$ − 37.7275i − 1.59002i −0.606594 0.795012i $$-0.707465\pi$$
0.606594 0.795012i $$-0.292535\pi$$
$$564$$ 4.47013 0.188226
$$565$$ 0 0
$$566$$ 51.7373 2.17468
$$567$$ − 4.42864i − 0.185985i
$$568$$ 1.98126i 0.0831320i
$$569$$ 7.33630 0.307554 0.153777 0.988106i $$-0.450856\pi$$
0.153777 + 0.988106i $$0.450856\pi$$
$$570$$ 0 0
$$571$$ 36.6450 1.53354 0.766772 0.641919i $$-0.221862\pi$$
0.766772 + 0.641919i $$0.221862\pi$$
$$572$$ − 1.00937i − 0.0422038i
$$573$$ − 6.10171i − 0.254903i
$$574$$ −1.63158 −0.0681010
$$575$$ 0 0
$$576$$ −4.74620 −0.197758
$$577$$ 4.22216i 0.175771i 0.996131 + 0.0878853i $$0.0280109\pi$$
−0.996131 + 0.0878853i $$0.971989\pi$$
$$578$$ − 18.7961i − 0.781817i
$$579$$ −18.3368 −0.762050
$$580$$ 0 0
$$581$$ −0.590573 −0.0245011
$$582$$ 13.7877i 0.571518i
$$583$$ 10.8573i 0.449663i
$$584$$ 3.04503 0.126004
$$585$$ 0 0
$$586$$ −5.35905 −0.221380
$$587$$ 34.3684i 1.41854i 0.704939 + 0.709268i $$0.250974\pi$$
−0.704939 + 0.709268i $$0.749026\pi$$
$$588$$ − 20.4608i − 0.843787i
$$589$$ −19.4291 −0.800563
$$590$$ 0 0
$$591$$ −6.69535 −0.275410
$$592$$ 9.22570i 0.379174i
$$593$$ − 27.9398i − 1.14735i −0.819083 0.573675i $$-0.805517\pi$$
0.819083 0.573675i $$-0.194483\pi$$
$$594$$ −1.90321 −0.0780897
$$595$$ 0 0
$$596$$ −2.33276 −0.0955535
$$597$$ 14.1017i 0.577145i
$$598$$ 10.4889i 0.428921i
$$599$$ 31.2257 1.27585 0.637924 0.770100i $$-0.279794\pi$$
0.637924 + 0.770100i $$0.279794\pi$$
$$600$$ 0 0
$$601$$ −8.75557 −0.357147 −0.178574 0.983927i $$-0.557148\pi$$
−0.178574 + 0.983927i $$0.557148\pi$$
$$602$$ − 47.8163i − 1.94885i
$$603$$ 1.24443i 0.0506772i
$$604$$ 19.7502 0.803625
$$605$$ 0 0
$$606$$ 8.87601 0.360563
$$607$$ − 15.1842i − 0.616308i −0.951336 0.308154i $$-0.900289\pi$$
0.951336 0.308154i $$-0.0997112\pi$$
$$608$$ 51.7619i 2.09922i
$$609$$ −34.5718 −1.40092
$$610$$ 0 0
$$611$$ 1.71456 0.0693636
$$612$$ − 8.40990i − 0.339950i
$$613$$ − 42.7239i − 1.72560i −0.505543 0.862802i $$-0.668708\pi$$
0.505543 0.862802i $$-0.331292\pi$$
$$614$$ 46.4929 1.87630
$$615$$ 0 0
$$616$$ 3.18421 0.128295
$$617$$ − 3.51114i − 0.141353i −0.997499 0.0706765i $$-0.977484\pi$$
0.997499 0.0706765i $$-0.0225158\pi$$
$$618$$ 22.1017i 0.889061i
$$619$$ −17.5941 −0.707167 −0.353584 0.935403i $$-0.615037\pi$$
−0.353584 + 0.935403i $$0.615037\pi$$
$$620$$ 0 0
$$621$$ 8.85728 0.355430
$$622$$ 37.8350i 1.51705i
$$623$$ 24.8573i 0.995886i
$$624$$ −2.87019 −0.114899
$$625$$ 0 0
$$626$$ −29.9081 −1.19537
$$627$$ 7.05086i 0.281584i
$$628$$ − 29.9625i − 1.19564i
$$629$$ −10.3684 −0.413416
$$630$$ 0 0
$$631$$ 15.8163 0.629636 0.314818 0.949152i $$-0.398057\pi$$
0.314818 + 0.949152i $$0.398057\pi$$
$$632$$ − 6.15610i − 0.244877i
$$633$$ 10.6637i 0.423844i
$$634$$ −31.3818 −1.24633
$$635$$ 0 0
$$636$$ −17.6128 −0.698395
$$637$$ − 7.84791i − 0.310946i
$$638$$ 14.8573i 0.588205i
$$639$$ −2.75557 −0.109009
$$640$$ 0 0
$$641$$ 25.8163 1.01968 0.509841 0.860269i $$-0.329704\pi$$
0.509841 + 0.860269i $$0.329704\pi$$
$$642$$ 4.99063i 0.196965i
$$643$$ − 18.1017i − 0.713862i −0.934131 0.356931i $$-0.883823\pi$$
0.934131 0.356931i $$-0.116177\pi$$
$$644$$ 63.6325 2.50747
$$645$$ 0 0
$$646$$ −69.5683 −2.73713
$$647$$ − 47.0420i − 1.84941i −0.380684 0.924705i $$-0.624311\pi$$
0.380684 0.924705i $$-0.375689\pi$$
$$648$$ 0.719004i 0.0282451i
$$649$$ 4.85728 0.190665
$$650$$ 0 0
$$651$$ −12.2034 −0.478290
$$652$$ − 16.3872i − 0.641770i
$$653$$ − 30.0830i − 1.17724i −0.808411 0.588619i $$-0.799672\pi$$
0.808411 0.588619i $$-0.200328\pi$$
$$654$$ 37.5210 1.46719
$$655$$ 0 0
$$656$$ 0.892937 0.0348633
$$657$$ 4.23506i 0.165226i
$$658$$ − 23.2257i − 0.905432i
$$659$$ 10.2854 0.400664 0.200332 0.979728i $$-0.435798\pi$$
0.200332 + 0.979728i $$0.435798\pi$$
$$660$$ 0 0
$$661$$ −27.7146 −1.07797 −0.538986 0.842315i $$-0.681192\pi$$
−0.538986 + 0.842315i $$0.681192\pi$$
$$662$$ 29.2070i 1.13516i
$$663$$ − 3.22570i − 0.125276i
$$664$$ 0.0958814 0.00372092
$$665$$ 0 0
$$666$$ −3.80642 −0.147496
$$667$$ − 69.1437i − 2.67725i
$$668$$ 26.5018i 1.02538i
$$669$$ −8.85728 −0.342442
$$670$$ 0 0
$$671$$ 6.85728 0.264722
$$672$$ 32.5116i 1.25416i
$$673$$ 9.86665i 0.380331i 0.981752 + 0.190166i $$0.0609025\pi$$
−0.981752 + 0.190166i $$0.939097\pi$$
$$674$$ −53.7373 −2.06988
$$675$$ 0 0
$$676$$ −20.4608 −0.786952
$$677$$ − 5.65433i − 0.217314i −0.994079 0.108657i $$-0.965345\pi$$
0.994079 0.108657i $$-0.0346550\pi$$
$$678$$ 11.4193i 0.438554i
$$679$$ 32.0830 1.23123
$$680$$ 0 0
$$681$$ 13.3778 0.512638
$$682$$ 5.24443i 0.200820i
$$683$$ − 34.1847i − 1.30804i −0.756477 0.654020i $$-0.773081\pi$$
0.756477 0.654020i $$-0.226919\pi$$
$$684$$ −11.4380 −0.437343
$$685$$ 0 0
$$686$$ −47.3087 −1.80625
$$687$$ 11.5111i 0.439177i
$$688$$ 26.1690i 0.997684i
$$689$$ −6.75557 −0.257367
$$690$$ 0 0
$$691$$ −19.2257 −0.731380 −0.365690 0.930737i $$-0.619167\pi$$
−0.365690 + 0.930737i $$0.619167\pi$$
$$692$$ − 14.8988i − 0.566366i
$$693$$ 4.42864i 0.168230i
$$694$$ −4.99063 −0.189442
$$695$$ 0 0
$$696$$ 5.61285 0.212754
$$697$$ 1.00354i 0.0380118i
$$698$$ − 9.78769i − 0.370469i
$$699$$ 4.32693 0.163659
$$700$$ 0 0
$$701$$ −29.9081 −1.12961 −0.564807 0.825223i $$-0.691050\pi$$
−0.564807 + 0.825223i $$0.691050\pi$$
$$702$$ − 1.18421i − 0.0446951i
$$703$$ 14.1017i 0.531856i
$$704$$ 4.74620 0.178879
$$705$$ 0 0
$$706$$ −17.7877 −0.669448
$$707$$ − 20.6539i − 0.776768i
$$708$$ 7.87955i 0.296132i
$$709$$ 15.3274 0.575633 0.287816 0.957686i $$-0.407071\pi$$
0.287816 + 0.957686i $$0.407071\pi$$
$$710$$ 0 0
$$711$$ 8.56199 0.321100
$$712$$ − 4.03566i − 0.151243i
$$713$$ − 24.4068i − 0.914043i
$$714$$ −43.6958 −1.63528
$$715$$ 0 0
$$716$$ −41.0865 −1.53548
$$717$$ − 3.34614i − 0.124964i
$$718$$ − 20.4701i − 0.763938i
$$719$$ −23.8163 −0.888197 −0.444098 0.895978i $$-0.646476\pi$$
−0.444098 + 0.895978i $$0.646476\pi$$
$$720$$ 0 0
$$721$$ 51.4291 1.91532
$$722$$ 58.4563i 2.17552i
$$723$$ 1.34614i 0.0500635i
$$724$$ 22.0830 0.820707
$$725$$ 0 0
$$726$$ 1.90321 0.0706348
$$727$$ − 32.9403i − 1.22169i −0.791752 0.610843i $$-0.790831\pi$$
0.791752 0.610843i $$-0.209169\pi$$
$$728$$ 1.98126i 0.0734305i
$$729$$ −1.00000 −0.0370370
$$730$$ 0 0
$$731$$ −29.4104 −1.08778
$$732$$ 11.1240i 0.411154i
$$733$$ 29.8666i 1.10315i 0.834125 + 0.551575i $$0.185973\pi$$
−0.834125 + 0.551575i $$0.814027\pi$$
$$734$$ −64.3239 −2.37424
$$735$$ 0 0
$$736$$ −65.0232 −2.39679
$$737$$ − 1.24443i − 0.0458392i
$$738$$ 0.368416i 0.0135616i
$$739$$ 5.06959 0.186488 0.0932440 0.995643i $$-0.470276\pi$$
0.0932440 + 0.995643i $$0.470276\pi$$
$$740$$ 0 0
$$741$$ −4.38715 −0.161166
$$742$$ 91.5121i 3.35951i
$$743$$ − 22.4385i − 0.823188i −0.911367 0.411594i $$-0.864972\pi$$
0.911367 0.411594i $$-0.135028\pi$$
$$744$$ 1.98126 0.0726367
$$745$$ 0 0
$$746$$ 64.6133 2.36566
$$747$$ 0.133353i 0.00487913i
$$748$$ 8.40990i 0.307497i
$$749$$ 11.6128 0.424324
$$750$$ 0 0
$$751$$ −6.63512 −0.242119 −0.121060 0.992645i $$-0.538629\pi$$
−0.121060 + 0.992645i $$0.538629\pi$$
$$752$$ 12.7110i 0.463523i
$$753$$ − 22.7556i − 0.829259i
$$754$$ −9.24443 −0.336662
$$755$$ 0 0
$$756$$ −7.18421 −0.261287
$$757$$ 8.75557i 0.318227i 0.987260 + 0.159113i $$0.0508635\pi$$
−0.987260 + 0.159113i $$0.949136\pi$$
$$758$$ 38.0642i 1.38256i
$$759$$ −8.85728 −0.321499
$$760$$ 0 0
$$761$$ 3.15257 0.114280 0.0571402 0.998366i $$-0.481802\pi$$
0.0571402 + 0.998366i $$0.481802\pi$$
$$762$$ − 28.8988i − 1.04689i
$$763$$ − 87.3087i − 3.16079i
$$764$$ −9.89829 −0.358108
$$765$$ 0 0
$$766$$ −27.8537 −1.00640
$$767$$ 3.02227i 0.109128i
$$768$$ − 20.2444i − 0.730508i
$$769$$ 28.9590 1.04429 0.522144 0.852857i $$-0.325132\pi$$
0.522144 + 0.852857i $$0.325132\pi$$
$$770$$ 0 0
$$771$$ −6.85728 −0.246959
$$772$$ 29.7462i 1.07059i
$$773$$ − 29.1427i − 1.04819i −0.851660 0.524095i $$-0.824404\pi$$
0.851660 0.524095i $$-0.175596\pi$$
$$774$$ −10.7971 −0.388092
$$775$$ 0 0
$$776$$ −5.20877 −0.186984
$$777$$ 8.85728i 0.317753i
$$778$$ 10.6824i 0.382984i
$$779$$ 1.36488 0.0489018
$$780$$ 0 0
$$781$$ 2.75557 0.0986020
$$782$$ − 87.3916i − 3.12512i
$$783$$ 7.80642i 0.278979i
$$784$$ 58.1811 2.07790
$$785$$ 0 0
$$786$$ 2.36842 0.0844786
$$787$$ − 11.2672i − 0.401632i −0.979629 0.200816i $$-0.935641\pi$$
0.979629 0.200816i $$-0.0643593\pi$$
$$788$$ 10.8613i 0.386918i
$$789$$ 29.5812 1.05312
$$790$$ 0 0
$$791$$ 26.5718 0.944786
$$792$$ − 0.719004i − 0.0255487i
$$793$$ 4.26671i 0.151515i
$$794$$ 24.2766 0.861543
$$795$$ 0 0
$$796$$ 22.8760 0.810819
$$797$$ 41.9625i 1.48639i 0.669075 + 0.743195i $$0.266690\pi$$
−0.669075 + 0.743195i $$0.733310\pi$$
$$798$$ 59.4291i 2.10377i
$$799$$ −14.2854 −0.505383
$$800$$ 0 0
$$801$$ 5.61285 0.198320
$$802$$ 3.80642i 0.134409i
$$803$$ − 4.23506i − 0.149452i
$$804$$ 2.01874 0.0711953
$$805$$ 0 0
$$806$$ −3.26317 −0.114940
$$807$$ 8.48886i 0.298822i
$$808$$ 3.35322i 0.117966i
$$809$$ 27.8064 0.977622 0.488811 0.872390i $$-0.337431\pi$$
0.488811 + 0.872390i $$0.337431\pi$$
$$810$$ 0 0
$$811$$ 6.78415 0.238224 0.119112 0.992881i $$-0.461995\pi$$
0.119112 + 0.992881i $$0.461995\pi$$
$$812$$ 56.0830i 1.96813i
$$813$$ − 14.6637i − 0.514279i
$$814$$ 3.80642 0.133415
$$815$$ 0 0
$$816$$ 23.9140 0.837156
$$817$$ 40.0000i 1.39942i
$$818$$ 13.5585i 0.474060i
$$819$$ −2.75557 −0.0962874
$$820$$ 0 0
$$821$$ 3.62269 0.126433 0.0632164 0.998000i $$-0.479864\pi$$
0.0632164 + 0.998000i $$0.479864\pi$$
$$822$$ − 0.930409i − 0.0324517i
$$823$$ − 42.0642i − 1.46627i −0.680085 0.733134i $$-0.738057\pi$$
0.680085 0.733134i $$-0.261943\pi$$
$$824$$ −8.34968 −0.290875
$$825$$ 0 0
$$826$$ 40.9403 1.42449
$$827$$ 30.8256i 1.07191i 0.844246 + 0.535956i $$0.180049\pi$$
−0.844246 + 0.535956i $$0.819951\pi$$
$$828$$ − 14.3684i − 0.499337i
$$829$$ −7.12399 −0.247426 −0.123713 0.992318i $$-0.539480\pi$$
−0.123713 + 0.992318i $$0.539480\pi$$
$$830$$ 0 0
$$831$$ 14.6035 0.506589
$$832$$ 2.95316i 0.102382i
$$833$$ 65.3876i 2.26555i
$$834$$ −33.8894 −1.17349
$$835$$ 0 0
$$836$$ 11.4380 0.395592
$$837$$ 2.75557i 0.0952464i
$$838$$ − 29.7146i − 1.02647i
$$839$$ −3.34614 −0.115522 −0.0577608 0.998330i $$-0.518396\pi$$
−0.0577608 + 0.998330i $$0.518396\pi$$
$$840$$ 0 0
$$841$$ 31.9403 1.10139
$$842$$ 15.0321i 0.518041i
$$843$$ 0.193576i 0.00666712i
$$844$$ 17.2988 0.595450
$$845$$ 0 0
$$846$$ −5.24443 −0.180307
$$847$$ − 4.42864i − 0.152170i
$$848$$ − 50.0830i − 1.71986i
$$849$$ −27.1842 −0.932960
$$850$$ 0 0
$$851$$ −17.7146 −0.607247
$$852$$ 4.47013i 0.153144i
$$853$$ 26.4197i 0.904595i 0.891867 + 0.452297i $$0.149395\pi$$
−0.891867 + 0.452297i $$0.850605\pi$$
$$854$$ 57.7975 1.97779
$$855$$ 0 0
$$856$$ −1.88538 −0.0644411
$$857$$ 38.7783i 1.32464i 0.749220 + 0.662321i $$0.230429\pi$$
−0.749220 + 0.662321i $$0.769571\pi$$
$$858$$ 1.18421i 0.0404282i
$$859$$ 27.3087 0.931760 0.465880 0.884848i $$-0.345738\pi$$
0.465880 + 0.884848i $$0.345738\pi$$
$$860$$ 0 0
$$861$$ 0.857279 0.0292160
$$862$$ 65.2899i 2.22378i
$$863$$ 49.5308i 1.68605i 0.537875 + 0.843024i $$0.319227\pi$$
−0.537875 + 0.843024i $$0.680773\pi$$
$$864$$ 7.34122 0.249753
$$865$$ 0 0
$$866$$ 27.5397 0.935838
$$867$$ 9.87601i 0.335407i
$$868$$ 19.7966i 0.671940i
$$869$$ −8.56199 −0.290446
$$870$$ 0 0
$$871$$ 0.774305 0.0262363
$$872$$ 14.1748i 0.480021i
$$873$$ − 7.24443i − 0.245187i
$$874$$ −118.858 −4.02044
$$875$$ 0 0
$$876$$ 6.87019 0.232122
$$877$$ − 4.50177i − 0.152014i −0.997107 0.0760070i $$-0.975783\pi$$
0.997107 0.0760070i $$-0.0242171\pi$$
$$878$$ 36.7654i 1.24077i
$$879$$ 2.81579 0.0949743
$$880$$ 0 0
$$881$$ −15.1240 −0.509540 −0.254770 0.967002i $$-0.582000\pi$$
−0.254770 + 0.967002i $$0.582000\pi$$
$$882$$ 24.0049i 0.808288i
$$883$$ 30.2480i 1.01793i 0.860789 + 0.508963i $$0.169971\pi$$
−0.860789 + 0.508963i $$0.830029\pi$$
$$884$$ −5.23277 −0.175997
$$885$$ 0 0
$$886$$ −24.9777 −0.839143
$$887$$ 57.1941i 1.92039i 0.279333 + 0.960194i $$0.409887\pi$$
−0.279333 + 0.960194i $$0.590113\pi$$
$$888$$ − 1.43801i − 0.0482564i
$$889$$ −67.2454 −2.25534
$$890$$ 0 0
$$891$$ 1.00000 0.0335013
$$892$$ 14.3684i 0.481090i
$$893$$ 19.4291i 0.650171i
$$894$$ 2.73683 0.0915334
$$895$$ 0 0
$$896$$ −25.0192 −0.835833
$$897$$ − 5.51114i − 0.184012i
$$898$$ 61.4835i 2.05173i
$$899$$ 21.5111 0.717437
$$900$$ 0 0
$$901$$ 56.2864 1.87517
$$902$$ − 0.368416i − 0.0122669i
$$903$$ 25.1240i 0.836074i
$$904$$ −4.31402 −0.143482
$$905$$ 0 0
$$906$$ −23.1713 −0.769815
$$907$$ − 53.2641i − 1.76861i −0.466913 0.884303i $$-0.654634\pi$$
0.466913 0.884303i $$-0.345366\pi$$
$$908$$ − 21.7017i − 0.720195i
$$909$$ −4.66370 −0.154685
$$910$$ 0 0
$$911$$ −0.590573 −0.0195665 −0.00978327 0.999952i $$-0.503114\pi$$
−0.00978327 + 0.999952i $$0.503114\pi$$
$$912$$ − 32.5245i − 1.07699i
$$913$$ − 0.133353i − 0.00441334i
$$914$$ 44.6508 1.47692
$$915$$ 0 0
$$916$$ 18.6735 0.616991
$$917$$ − 5.51114i − 0.181994i
$$918$$ 9.86665i 0.325648i
$$919$$ −55.8707 −1.84300 −0.921502 0.388375i $$-0.873037\pi$$
−0.921502 + 0.388375i $$0.873037\pi$$
$$920$$ 0 0
$$921$$ −24.4286 −0.804951
$$922$$ − 54.9403i − 1.80936i
$$923$$ 1.71456i 0.0564354i
$$924$$ 7.18421 0.236343
$$925$$ 0 0
$$926$$ −36.8198 −1.20997
$$927$$ − 11.6128i − 0.381416i
$$928$$ − 57.3087i − 1.88125i
$$929$$ −15.3274 −0.502876 −0.251438 0.967873i $$-0.580903\pi$$
−0.251438 + 0.967873i $$0.580903\pi$$
$$930$$ 0 0
$$931$$ 88.9314 2.91461
$$932$$ − 7.01921i − 0.229922i
$$933$$ − 19.8796i − 0.650827i
$$934$$ −5.98126 −0.195713
$$935$$ 0 0
$$936$$ 0.447375 0.0146229
$$937$$ − 27.8479i − 0.909752i −0.890555 0.454876i $$-0.849684\pi$$
0.890555 0.454876i $$-0.150316\pi$$
$$938$$ − 10.4889i − 0.342474i
$$939$$ 15.7146 0.512825
$$940$$ 0 0
$$941$$ 10.4157 0.339543 0.169772 0.985483i $$-0.445697\pi$$
0.169772 + 0.985483i $$0.445697\pi$$
$$942$$ 35.1526i 1.14533i
$$943$$ 1.71456i 0.0558337i
$$944$$ −22.4059 −0.729250
$$945$$ 0 0
$$946$$ 10.7971 0.351043
$$947$$ 8.47013i 0.275242i 0.990485 + 0.137621i $$0.0439456\pi$$
−0.990485 + 0.137621i $$0.956054\pi$$
$$948$$ − 13.8894i − 0.451107i
$$949$$ 2.63512 0.0855397
$$950$$ 0 0
$$951$$ 16.4889 0.534688
$$952$$ − 16.5076i − 0.535014i
$$953$$ 8.71408i 0.282277i 0.989990 + 0.141138i $$0.0450763\pi$$
−0.989990 + 0.141138i $$0.954924\pi$$
$$954$$ 20.6637 0.669012
$$955$$ 0 0
$$956$$ −5.42816 −0.175559
$$957$$ − 7.80642i − 0.252346i
$$958$$ 47.3087i 1.52847i
$$959$$ −2.16500 −0.0699114
$$960$$ 0 0
$$961$$ −23.4068 −0.755059
$$962$$ 2.36842i 0.0763608i
$$963$$ − 2.62222i − 0.0844997i
$$964$$ 2.18373 0.0703333
$$965$$ 0 0
$$966$$ −74.6548 −2.40198
$$967$$ 44.2449i 1.42282i 0.702777 + 0.711410i $$0.251943\pi$$
−0.702777 + 0.711410i $$0.748057\pi$$
$$968$$ 0.719004i 0.0231097i
$$969$$ 36.5531 1.17425
$$970$$ 0 0
$$971$$ −57.1437 −1.83383 −0.916914 0.399085i $$-0.869328\pi$$
−0.916914 + 0.399085i $$0.869328\pi$$
$$972$$ 1.62222i 0.0520326i
$$973$$ 78.8582i 2.52808i
$$974$$ 21.9438 0.703124
$$975$$ 0 0
$$976$$ −31.6316 −1.01250
$$977$$ − 16.2480i − 0.519819i −0.965633 0.259909i $$-0.916307\pi$$
0.965633 0.259909i $$-0.0836927\pi$$
$$978$$ 19.2257i 0.614770i
$$979$$ −5.61285 −0.179387
$$980$$ 0 0
$$981$$ −19.7146 −0.629437
$$982$$ 31.1882i 0.995256i
$$983$$ − 1.12399i − 0.0358496i −0.999839 0.0179248i $$-0.994294\pi$$
0.999839 0.0179248i $$-0.00570594\pi$$
$$984$$ −0.139182 −0.00443696
$$985$$ 0 0
$$986$$ 77.0232 2.45292
$$987$$ 12.2034i 0.388439i
$$988$$ 7.11691i 0.226419i
$$989$$ −50.2480 −1.59779
$$990$$ 0 0
$$991$$ −53.6513 −1.70429 −0.852144 0.523307i $$-0.824698\pi$$
−0.852144 + 0.523307i $$0.824698\pi$$
$$992$$ − 20.2292i − 0.642279i
$$993$$ − 15.3461i − 0.486995i
$$994$$ 23.2257 0.736674
$$995$$ 0 0
$$996$$ 0.216327 0.00685460
$$997$$ − 35.7275i − 1.13150i −0.824577 0.565750i $$-0.808587\pi$$
0.824577 0.565750i $$-0.191413\pi$$
$$998$$ 48.2034i 1.52585i
$$999$$ 2.00000 0.0632772
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 825.2.c.g.199.5 6
3.2 odd 2 2475.2.c.r.199.2 6
5.2 odd 4 825.2.a.k.1.1 3
5.3 odd 4 165.2.a.c.1.3 3
5.4 even 2 inner 825.2.c.g.199.2 6
15.2 even 4 2475.2.a.bb.1.3 3
15.8 even 4 495.2.a.e.1.1 3
15.14 odd 2 2475.2.c.r.199.5 6
20.3 even 4 2640.2.a.be.1.3 3
35.13 even 4 8085.2.a.bk.1.3 3
55.32 even 4 9075.2.a.cf.1.3 3
55.43 even 4 1815.2.a.m.1.1 3
60.23 odd 4 7920.2.a.cj.1.3 3
165.98 odd 4 5445.2.a.z.1.3 3

By twisted newform
Twist Min Dim Char Parity Ord Type
165.2.a.c.1.3 3 5.3 odd 4
495.2.a.e.1.1 3 15.8 even 4
825.2.a.k.1.1 3 5.2 odd 4
825.2.c.g.199.2 6 5.4 even 2 inner
825.2.c.g.199.5 6 1.1 even 1 trivial
1815.2.a.m.1.1 3 55.43 even 4
2475.2.a.bb.1.3 3 15.2 even 4
2475.2.c.r.199.2 6 3.2 odd 2
2475.2.c.r.199.5 6 15.14 odd 2
2640.2.a.be.1.3 3 20.3 even 4
5445.2.a.z.1.3 3 165.98 odd 4
7920.2.a.cj.1.3 3 60.23 odd 4
8085.2.a.bk.1.3 3 35.13 even 4
9075.2.a.cf.1.3 3 55.32 even 4