# Properties

 Label 4400.2.b.bb.4049.5 Level $4400$ Weight $2$ Character 4400.4049 Analytic conductor $35.134$ 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] = [4400,2,Mod(4049,4400)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4400, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("4400.4049");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$4400 = 2^{4} \cdot 5^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4400.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: $$35.1341768894$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: 6.0.96668224.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} + 15x^{4} + 61x^{2} + 36$$ x^6 + 15*x^4 + 61*x^2 + 36 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 2200) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 4049.5 Root $$2.59261i$$ of defining polynomial Character $$\chi$$ $$=$$ 4400.4049 Dual form 4400.2.b.bb.4049.2

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+2.59261i q^{3} +2.72165i q^{7} -3.72165 q^{9} +O(q^{10})$$ $$q+2.59261i q^{3} +2.72165i q^{7} -3.72165 q^{9} -1.00000 q^{11} +1.00000i q^{13} +4.59261i q^{17} +8.18523 q^{19} -7.05619 q^{21} -0.407385i q^{23} -1.87096i q^{27} -7.46358 q^{29} +4.44330 q^{31} -2.59261i q^{33} +7.31427i q^{37} -2.59261 q^{39} -3.31427 q^{41} +7.49950i q^{43} +7.05619i q^{47} -0.407385 q^{49} -11.9069 q^{51} +0.979724i q^{53} +21.2211i q^{57} +7.05619 q^{59} -4.46358 q^{61} -10.1290i q^{63} -2.25807i q^{67} +1.05619 q^{69} -10.3143 q^{71} -12.1650i q^{73} -2.72165i q^{77} +3.14931 q^{79} -6.31427 q^{81} -16.6488i q^{83} -19.3502i q^{87} -8.27835 q^{89} -2.72165 q^{91} +11.5198i q^{93} -3.03592i q^{97} +3.72165 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 12 q^{9}+O(q^{10})$$ 6 * q - 12 * q^9 $$6 q - 12 q^{9} - 6 q^{11} + 14 q^{19} - 20 q^{29} + 6 q^{31} + 2 q^{39} + 8 q^{41} - 20 q^{49} - 26 q^{51} - 2 q^{61} - 36 q^{69} - 34 q^{71} + 22 q^{79} - 10 q^{81} - 60 q^{89} - 6 q^{91} + 12 q^{99}+O(q^{100})$$ 6 * q - 12 * q^9 - 6 * q^11 + 14 * q^19 - 20 * q^29 + 6 * q^31 + 2 * q^39 + 8 * q^41 - 20 * q^49 - 26 * q^51 - 2 * q^61 - 36 * q^69 - 34 * q^71 + 22 * q^79 - 10 * q^81 - 60 * q^89 - 6 * q^91 + 12 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/4400\mathbb{Z}\right)^\times$$.

 $$n$$ $$177$$ $$1201$$ $$2751$$ $$3301$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$ $$1$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ 2.59261i 1.49685i 0.663221 + 0.748423i $$0.269189\pi$$
−0.663221 + 0.748423i $$0.730811\pi$$
$$4$$ 0 0
$$5$$ 0 0
$$6$$ 0 0
$$7$$ 2.72165i 1.02869i 0.857584 + 0.514344i $$0.171964\pi$$
−0.857584 + 0.514344i $$0.828036\pi$$
$$8$$ 0 0
$$9$$ −3.72165 −1.24055
$$10$$ 0 0
$$11$$ −1.00000 −0.301511
$$12$$ 0 0
$$13$$ 1.00000i 0.277350i 0.990338 + 0.138675i $$0.0442844\pi$$
−0.990338 + 0.138675i $$0.955716\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 4.59261i 1.11387i 0.830555 + 0.556936i $$0.188023\pi$$
−0.830555 + 0.556936i $$0.811977\pi$$
$$18$$ 0 0
$$19$$ 8.18523 1.87782 0.938910 0.344162i $$-0.111837\pi$$
0.938910 + 0.344162i $$0.111837\pi$$
$$20$$ 0 0
$$21$$ −7.05619 −1.53979
$$22$$ 0 0
$$23$$ − 0.407385i − 0.0849457i −0.999098 0.0424728i $$-0.986476\pi$$
0.999098 0.0424728i $$-0.0135236\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ − 1.87096i − 0.360067i
$$28$$ 0 0
$$29$$ −7.46358 −1.38595 −0.692976 0.720961i $$-0.743701\pi$$
−0.692976 + 0.720961i $$0.743701\pi$$
$$30$$ 0 0
$$31$$ 4.44330 0.798041 0.399020 0.916942i $$-0.369350\pi$$
0.399020 + 0.916942i $$0.369350\pi$$
$$32$$ 0 0
$$33$$ − 2.59261i − 0.451316i
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 7.31427i 1.20246i 0.799077 + 0.601229i $$0.205322\pi$$
−0.799077 + 0.601229i $$0.794678\pi$$
$$38$$ 0 0
$$39$$ −2.59261 −0.415151
$$40$$ 0 0
$$41$$ −3.31427 −0.517601 −0.258801 0.965931i $$-0.583327\pi$$
−0.258801 + 0.965931i $$0.583327\pi$$
$$42$$ 0 0
$$43$$ 7.49950i 1.14366i 0.820371 + 0.571831i $$0.193767\pi$$
−0.820371 + 0.571831i $$0.806233\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 7.05619i 1.02925i 0.857415 + 0.514626i $$0.172069\pi$$
−0.857415 + 0.514626i $$0.827931\pi$$
$$48$$ 0 0
$$49$$ −0.407385 −0.0581979
$$50$$ 0 0
$$51$$ −11.9069 −1.66730
$$52$$ 0 0
$$53$$ 0.979724i 0.134575i 0.997734 + 0.0672877i $$0.0214345\pi$$
−0.997734 + 0.0672877i $$0.978565\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 21.2211i 2.81081i
$$58$$ 0 0
$$59$$ 7.05619 0.918638 0.459319 0.888271i $$-0.348093\pi$$
0.459319 + 0.888271i $$0.348093\pi$$
$$60$$ 0 0
$$61$$ −4.46358 −0.571503 −0.285751 0.958304i $$-0.592243\pi$$
−0.285751 + 0.958304i $$0.592243\pi$$
$$62$$ 0 0
$$63$$ − 10.1290i − 1.27614i
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ − 2.25807i − 0.275868i −0.990441 0.137934i $$-0.955954\pi$$
0.990441 0.137934i $$-0.0440461\pi$$
$$68$$ 0 0
$$69$$ 1.05619 0.127151
$$70$$ 0 0
$$71$$ −10.3143 −1.22408 −0.612039 0.790828i $$-0.709650\pi$$
−0.612039 + 0.790828i $$0.709650\pi$$
$$72$$ 0 0
$$73$$ − 12.1650i − 1.42380i −0.702281 0.711900i $$-0.747835\pi$$
0.702281 0.711900i $$-0.252165\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ − 2.72165i − 0.310161i
$$78$$ 0 0
$$79$$ 3.14931 0.354325 0.177163 0.984182i $$-0.443308\pi$$
0.177163 + 0.984182i $$0.443308\pi$$
$$80$$ 0 0
$$81$$ −6.31427 −0.701585
$$82$$ 0 0
$$83$$ − 16.6488i − 1.82744i −0.406340 0.913722i $$-0.633195\pi$$
0.406340 0.913722i $$-0.366805\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ − 19.3502i − 2.07456i
$$88$$ 0 0
$$89$$ −8.27835 −0.877503 −0.438752 0.898608i $$-0.644579\pi$$
−0.438752 + 0.898608i $$0.644579\pi$$
$$90$$ 0 0
$$91$$ −2.72165 −0.285307
$$92$$ 0 0
$$93$$ 11.5198i 1.19454i
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ − 3.03592i − 0.308251i −0.988051 0.154125i $$-0.950744\pi$$
0.988051 0.154125i $$-0.0492560\pi$$
$$98$$ 0 0
$$99$$ 3.72165 0.374040
$$100$$ 0 0
$$101$$ 5.59261 0.556486 0.278243 0.960511i $$-0.410248\pi$$
0.278243 + 0.960511i $$0.410248\pi$$
$$102$$ 0 0
$$103$$ 4.64881i 0.458061i 0.973419 + 0.229030i $$0.0735555\pi$$
−0.973419 + 0.229030i $$0.926445\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 0.701375i 0.0678045i 0.999425 + 0.0339023i $$0.0107935\pi$$
−0.999425 + 0.0339023i $$0.989207\pi$$
$$108$$ 0 0
$$109$$ −4.27835 −0.409791 −0.204896 0.978784i $$-0.565686\pi$$
−0.204896 + 0.978784i $$0.565686\pi$$
$$110$$ 0 0
$$111$$ −18.9631 −1.79990
$$112$$ 0 0
$$113$$ 11.6847i 1.09921i 0.835426 + 0.549603i $$0.185221\pi$$
−0.835426 + 0.549603i $$0.814779\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ − 3.72165i − 0.344067i
$$118$$ 0 0
$$119$$ −12.4995 −1.14583
$$120$$ 0 0
$$121$$ 1.00000 0.0909091
$$122$$ 0 0
$$123$$ − 8.59261i − 0.774770i
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 9.51977i 0.844743i 0.906423 + 0.422372i $$0.138802\pi$$
−0.906423 + 0.422372i $$0.861198\pi$$
$$128$$ 0 0
$$129$$ −19.4433 −1.71189
$$130$$ 0 0
$$131$$ −22.4792 −1.96402 −0.982009 0.188833i $$-0.939530\pi$$
−0.982009 + 0.188833i $$0.939530\pi$$
$$132$$ 0 0
$$133$$ 22.2773i 1.93169i
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ − 15.5354i − 1.32728i −0.748052 0.663640i $$-0.769011\pi$$
0.748052 0.663640i $$-0.230989\pi$$
$$138$$ 0 0
$$139$$ 13.3299 1.13063 0.565314 0.824876i $$-0.308755\pi$$
0.565314 + 0.824876i $$0.308755\pi$$
$$140$$ 0 0
$$141$$ −18.2940 −1.54063
$$142$$ 0 0
$$143$$ − 1.00000i − 0.0836242i
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ − 1.05619i − 0.0871133i
$$148$$ 0 0
$$149$$ 0.628532 0.0514913 0.0257457 0.999669i $$-0.491804\pi$$
0.0257457 + 0.999669i $$0.491804\pi$$
$$150$$ 0 0
$$151$$ 19.1280 1.55662 0.778308 0.627882i $$-0.216078\pi$$
0.778308 + 0.627882i $$0.216078\pi$$
$$152$$ 0 0
$$153$$ − 17.0921i − 1.38182i
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ − 8.88660i − 0.709228i −0.935013 0.354614i $$-0.884612\pi$$
0.935013 0.354614i $$-0.115388\pi$$
$$158$$ 0 0
$$159$$ −2.54005 −0.201439
$$160$$ 0 0
$$161$$ 1.10876 0.0873826
$$162$$ 0 0
$$163$$ − 20.8340i − 1.63185i −0.578159 0.815924i $$-0.696229\pi$$
0.578159 0.815924i $$-0.303771\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 17.4267i 1.34851i 0.738496 + 0.674257i $$0.235536\pi$$
−0.738496 + 0.674257i $$0.764464\pi$$
$$168$$ 0 0
$$169$$ 12.0000 0.923077
$$170$$ 0 0
$$171$$ −30.4626 −2.32953
$$172$$ 0 0
$$173$$ − 8.18523i − 0.622311i −0.950359 0.311156i $$-0.899284\pi$$
0.950359 0.311156i $$-0.100716\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 18.2940i 1.37506i
$$178$$ 0 0
$$179$$ −10.7217 −0.801374 −0.400687 0.916215i $$-0.631228\pi$$
−0.400687 + 0.916215i $$0.631228\pi$$
$$180$$ 0 0
$$181$$ 23.4626 1.74396 0.871980 0.489542i $$-0.162836\pi$$
0.871980 + 0.489542i $$0.162836\pi$$
$$182$$ 0 0
$$183$$ − 11.5723i − 0.855452i
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ − 4.59261i − 0.335845i
$$188$$ 0 0
$$189$$ 5.09211 0.370397
$$190$$ 0 0
$$191$$ 15.5354 1.12410 0.562052 0.827102i $$-0.310012\pi$$
0.562052 + 0.827102i $$0.310012\pi$$
$$192$$ 0 0
$$193$$ − 1.62954i − 0.117297i −0.998279 0.0586485i $$-0.981321\pi$$
0.998279 0.0586485i $$-0.0186791\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 2.77784i 0.197913i 0.995092 + 0.0989566i $$0.0315505\pi$$
−0.995092 + 0.0989566i $$0.968449\pi$$
$$198$$ 0 0
$$199$$ −26.1639 −1.85471 −0.927356 0.374179i $$-0.877924\pi$$
−0.927356 + 0.374179i $$0.877924\pi$$
$$200$$ 0 0
$$201$$ 5.85431 0.412931
$$202$$ 0 0
$$203$$ − 20.3133i − 1.42571i
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 1.51615i 0.105379i
$$208$$ 0 0
$$209$$ −8.18523 −0.566184
$$210$$ 0 0
$$211$$ 2.38711 0.164335 0.0821677 0.996619i $$-0.473816\pi$$
0.0821677 + 0.996619i $$0.473816\pi$$
$$212$$ 0 0
$$213$$ − 26.7409i − 1.83226i
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 12.0931i 0.820934i
$$218$$ 0 0
$$219$$ 31.5390 2.13121
$$220$$ 0 0
$$221$$ −4.59261 −0.308933
$$222$$ 0 0
$$223$$ − 25.4267i − 1.70269i −0.524603 0.851347i $$-0.675786\pi$$
0.524603 0.851347i $$-0.324214\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 10.2783i 0.682198i 0.940027 + 0.341099i $$0.110799\pi$$
−0.940027 + 0.341099i $$0.889201\pi$$
$$228$$ 0 0
$$229$$ −29.0349 −1.91868 −0.959340 0.282252i $$-0.908919\pi$$
−0.959340 + 0.282252i $$0.908919\pi$$
$$230$$ 0 0
$$231$$ 7.05619 0.464263
$$232$$ 0 0
$$233$$ 5.90688i 0.386973i 0.981103 + 0.193486i $$0.0619795\pi$$
−0.981103 + 0.193486i $$0.938020\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 8.16495i 0.530371i
$$238$$ 0 0
$$239$$ 3.79449 0.245445 0.122723 0.992441i $$-0.460837\pi$$
0.122723 + 0.992441i $$0.460837\pi$$
$$240$$ 0 0
$$241$$ −23.5916 −1.51967 −0.759834 0.650117i $$-0.774720\pi$$
−0.759834 + 0.650117i $$0.774720\pi$$
$$242$$ 0 0
$$243$$ − 21.9833i − 1.41023i
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 8.18523i 0.520814i
$$248$$ 0 0
$$249$$ 43.1639 2.73540
$$250$$ 0 0
$$251$$ 21.5916 1.36285 0.681425 0.731888i $$-0.261361\pi$$
0.681425 + 0.731888i $$0.261361\pi$$
$$252$$ 0 0
$$253$$ 0.407385i 0.0256121i
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 4.01564i 0.250489i 0.992126 + 0.125244i $$0.0399715\pi$$
−0.992126 + 0.125244i $$0.960028\pi$$
$$258$$ 0 0
$$259$$ −19.9069 −1.23695
$$260$$ 0 0
$$261$$ 27.7768 1.71934
$$262$$ 0 0
$$263$$ 29.2404i 1.80304i 0.432736 + 0.901521i $$0.357548\pi$$
−0.432736 + 0.901521i $$0.642452\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ − 21.4626i − 1.31349i
$$268$$ 0 0
$$269$$ 8.03592 0.489959 0.244979 0.969528i $$-0.421219\pi$$
0.244979 + 0.969528i $$0.421219\pi$$
$$270$$ 0 0
$$271$$ 19.4267 1.18009 0.590043 0.807372i $$-0.299111\pi$$
0.590043 + 0.807372i $$0.299111\pi$$
$$272$$ 0 0
$$273$$ − 7.05619i − 0.427060i
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 1.83041i 0.109979i 0.998487 + 0.0549894i $$0.0175125\pi$$
−0.998487 + 0.0549894i $$0.982487\pi$$
$$278$$ 0 0
$$279$$ −16.5364 −0.990010
$$280$$ 0 0
$$281$$ 17.6847 1.05498 0.527491 0.849561i $$-0.323133\pi$$
0.527491 + 0.849561i $$0.323133\pi$$
$$282$$ 0 0
$$283$$ − 16.8507i − 1.00167i −0.865543 0.500835i $$-0.833026\pi$$
0.865543 0.500835i $$-0.166974\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ − 9.02028i − 0.532450i
$$288$$ 0 0
$$289$$ −4.09211 −0.240712
$$290$$ 0 0
$$291$$ 7.87096 0.461404
$$292$$ 0 0
$$293$$ − 28.3705i − 1.65742i −0.559678 0.828710i $$-0.689075\pi$$
0.559678 0.828710i $$-0.310925\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 1.87096i 0.108564i
$$298$$ 0 0
$$299$$ 0.407385 0.0235597
$$300$$ 0 0
$$301$$ −20.4110 −1.17647
$$302$$ 0 0
$$303$$ 14.4995i 0.832974i
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ − 2.29399i − 0.130925i −0.997855 0.0654625i $$-0.979148\pi$$
0.997855 0.0654625i $$-0.0208523\pi$$
$$308$$ 0 0
$$309$$ −12.0526 −0.685647
$$310$$ 0 0
$$311$$ −11.2581 −0.638387 −0.319193 0.947690i $$-0.603412\pi$$
−0.319193 + 0.947690i $$0.603412\pi$$
$$312$$ 0 0
$$313$$ 7.18523i 0.406133i 0.979165 + 0.203067i $$0.0650908\pi$$
−0.979165 + 0.203067i $$0.934909\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ − 29.5198i − 1.65800i −0.559252 0.828998i $$-0.688912\pi$$
0.559252 0.828998i $$-0.311088\pi$$
$$318$$ 0 0
$$319$$ 7.46358 0.417880
$$320$$ 0 0
$$321$$ −1.81840 −0.101493
$$322$$ 0 0
$$323$$ 37.5916i 2.09165i
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ − 11.0921i − 0.613395i
$$328$$ 0 0
$$329$$ −19.2045 −1.05878
$$330$$ 0 0
$$331$$ 6.12904 0.336882 0.168441 0.985712i $$-0.446127\pi$$
0.168441 + 0.985712i $$0.446127\pi$$
$$332$$ 0 0
$$333$$ − 27.2211i − 1.49171i
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 10.0000i 0.544735i 0.962193 + 0.272367i $$0.0878066\pi$$
−0.962193 + 0.272367i $$0.912193\pi$$
$$338$$ 0 0
$$339$$ −30.2940 −1.64534
$$340$$ 0 0
$$341$$ −4.44330 −0.240618
$$342$$ 0 0
$$343$$ 17.9428i 0.968820i
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 14.7217i 0.790300i 0.918617 + 0.395150i $$0.129307\pi$$
−0.918617 + 0.395150i $$0.870693\pi$$
$$348$$ 0 0
$$349$$ −32.2976 −1.72885 −0.864426 0.502760i $$-0.832318\pi$$
−0.864426 + 0.502760i $$0.832318\pi$$
$$350$$ 0 0
$$351$$ 1.87096 0.0998646
$$352$$ 0 0
$$353$$ 3.37046i 0.179391i 0.995969 + 0.0896957i $$0.0285895\pi$$
−0.995969 + 0.0896957i $$0.971411\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ − 32.4064i − 1.71513i
$$358$$ 0 0
$$359$$ −19.7419 −1.04194 −0.520970 0.853575i $$-0.674429\pi$$
−0.520970 + 0.853575i $$0.674429\pi$$
$$360$$ 0 0
$$361$$ 47.9980 2.52621
$$362$$ 0 0
$$363$$ 2.59261i 0.136077i
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 30.4792i 1.59100i 0.605952 + 0.795501i $$0.292792\pi$$
−0.605952 + 0.795501i $$0.707208\pi$$
$$368$$ 0 0
$$369$$ 12.3345 0.642111
$$370$$ 0 0
$$371$$ −2.66647 −0.138436
$$372$$ 0 0
$$373$$ − 27.4985i − 1.42382i −0.702272 0.711909i $$-0.747831\pi$$
0.702272 0.711909i $$-0.252169\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ − 7.46358i − 0.384394i
$$378$$ 0 0
$$379$$ 3.87096 0.198838 0.0994190 0.995046i $$-0.468302\pi$$
0.0994190 + 0.995046i $$0.468302\pi$$
$$380$$ 0 0
$$381$$ −24.6811 −1.26445
$$382$$ 0 0
$$383$$ 33.2414i 1.69856i 0.527945 + 0.849279i $$0.322963\pi$$
−0.527945 + 0.849279i $$0.677037\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ − 27.9105i − 1.41877i
$$388$$ 0 0
$$389$$ −33.9980 −1.72377 −0.861883 0.507107i $$-0.830715\pi$$
−0.861883 + 0.507107i $$0.830715\pi$$
$$390$$ 0 0
$$391$$ 1.87096 0.0946187
$$392$$ 0 0
$$393$$ − 58.2800i − 2.93983i
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ − 2.85069i − 0.143072i −0.997438 0.0715360i $$-0.977210\pi$$
0.997438 0.0715360i $$-0.0227901\pi$$
$$398$$ 0 0
$$399$$ −57.7566 −2.89144
$$400$$ 0 0
$$401$$ 16.0967 0.803833 0.401917 0.915676i $$-0.368344\pi$$
0.401917 + 0.915676i $$0.368344\pi$$
$$402$$ 0 0
$$403$$ 4.44330i 0.221337i
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ − 7.31427i − 0.362555i
$$408$$ 0 0
$$409$$ −0.757568 −0.0374593 −0.0187297 0.999825i $$-0.505962\pi$$
−0.0187297 + 0.999825i $$0.505962\pi$$
$$410$$ 0 0
$$411$$ 40.2773 1.98673
$$412$$ 0 0
$$413$$ 19.2045i 0.944991i
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ 34.5593i 1.69238i
$$418$$ 0 0
$$419$$ 36.8700 1.80122 0.900608 0.434633i $$-0.143122\pi$$
0.900608 + 0.434633i $$0.143122\pi$$
$$420$$ 0 0
$$421$$ −32.2055 −1.56960 −0.784800 0.619749i $$-0.787234\pi$$
−0.784800 + 0.619749i $$0.787234\pi$$
$$422$$ 0 0
$$423$$ − 26.2607i − 1.27684i
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ − 12.1483i − 0.587898i
$$428$$ 0 0
$$429$$ 2.59261 0.125173
$$430$$ 0 0
$$431$$ −17.9428 −0.864274 −0.432137 0.901808i $$-0.642240\pi$$
−0.432137 + 0.901808i $$0.642240\pi$$
$$432$$ 0 0
$$433$$ − 14.2258i − 0.683647i −0.939764 0.341824i $$-0.888955\pi$$
0.939764 0.341824i $$-0.111045\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ − 3.33454i − 0.159513i
$$438$$ 0 0
$$439$$ −9.70601 −0.463243 −0.231621 0.972806i $$-0.574403\pi$$
−0.231621 + 0.972806i $$0.574403\pi$$
$$440$$ 0 0
$$441$$ 1.51615 0.0721974
$$442$$ 0 0
$$443$$ 4.44431i 0.211156i 0.994411 + 0.105578i $$0.0336692\pi$$
−0.994411 + 0.105578i $$0.966331\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 1.62954i 0.0770746i
$$448$$ 0 0
$$449$$ 12.4479 0.587454 0.293727 0.955889i $$-0.405104\pi$$
0.293727 + 0.955889i $$0.405104\pi$$
$$450$$ 0 0
$$451$$ 3.31427 0.156063
$$452$$ 0 0
$$453$$ 49.5916i 2.33002i
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ − 16.2368i − 0.759525i −0.925084 0.379762i $$-0.876006\pi$$
0.925084 0.379762i $$-0.123994\pi$$
$$458$$ 0 0
$$459$$ 8.59261 0.401069
$$460$$ 0 0
$$461$$ −41.8689 −1.95003 −0.975016 0.222136i $$-0.928697\pi$$
−0.975016 + 0.222136i $$0.928697\pi$$
$$462$$ 0 0
$$463$$ − 18.7014i − 0.869127i −0.900641 0.434563i $$-0.856903\pi$$
0.900641 0.434563i $$-0.143097\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 12.2820i 0.568342i 0.958774 + 0.284171i $$0.0917183\pi$$
−0.958774 + 0.284171i $$0.908282\pi$$
$$468$$ 0 0
$$469$$ 6.14569 0.283781
$$470$$ 0 0
$$471$$ 23.0395 1.06161
$$472$$ 0 0
$$473$$ − 7.49950i − 0.344827i
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ − 3.64619i − 0.166948i
$$478$$ 0 0
$$479$$ 6.75757 0.308761 0.154381 0.988011i $$-0.450662\pi$$
0.154381 + 0.988011i $$0.450662\pi$$
$$480$$ 0 0
$$481$$ −7.31427 −0.333502
$$482$$ 0 0
$$483$$ 2.87459i 0.130798i
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 34.0395i 1.54248i 0.636545 + 0.771239i $$0.280363\pi$$
−0.636545 + 0.771239i $$0.719637\pi$$
$$488$$ 0 0
$$489$$ 54.0146 2.44263
$$490$$ 0 0
$$491$$ 17.3299 0.782088 0.391044 0.920372i $$-0.372114\pi$$
0.391044 + 0.920372i $$0.372114\pi$$
$$492$$ 0 0
$$493$$ − 34.2773i − 1.54377i
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ − 28.0718i − 1.25919i
$$498$$ 0 0
$$499$$ −14.0931 −0.630895 −0.315447 0.948943i $$-0.602155\pi$$
−0.315447 + 0.948943i $$0.602155\pi$$
$$500$$ 0 0
$$501$$ −45.1806 −2.01852
$$502$$ 0 0
$$503$$ − 15.7825i − 0.703706i −0.936055 0.351853i $$-0.885552\pi$$
0.936055 0.351853i $$-0.114448\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 31.1114i 1.38170i
$$508$$ 0 0
$$509$$ 13.3907 0.593534 0.296767 0.954950i $$-0.404092\pi$$
0.296767 + 0.954950i $$0.404092\pi$$
$$510$$ 0 0
$$511$$ 33.1088 1.46465
$$512$$ 0 0
$$513$$ − 15.3143i − 0.676141i
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ − 7.05619i − 0.310331i
$$518$$ 0 0
$$519$$ 21.2211 0.931505
$$520$$ 0 0
$$521$$ 11.6857 0.511961 0.255981 0.966682i $$-0.417602\pi$$
0.255981 + 0.966682i $$0.417602\pi$$
$$522$$ 0 0
$$523$$ − 18.9022i − 0.826538i −0.910609 0.413269i $$-0.864387\pi$$
0.910609 0.413269i $$-0.135613\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 20.4064i 0.888916i
$$528$$ 0 0
$$529$$ 22.8340 0.992784
$$530$$ 0 0
$$531$$ −26.2607 −1.13962
$$532$$ 0 0
$$533$$ − 3.31427i − 0.143557i
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ − 27.7971i − 1.19953i
$$538$$ 0 0
$$539$$ 0.407385 0.0175473
$$540$$ 0 0
$$541$$ −10.8497 −0.466464 −0.233232 0.972421i $$-0.574930\pi$$
−0.233232 + 0.972421i $$0.574930\pi$$
$$542$$ 0 0
$$543$$ 60.8294i 2.61044i
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ − 44.9621i − 1.92244i −0.275784 0.961220i $$-0.588938\pi$$
0.275784 0.961220i $$-0.411062\pi$$
$$548$$ 0 0
$$549$$ 16.6119 0.708978
$$550$$ 0 0
$$551$$ −61.0911 −2.60257
$$552$$ 0 0
$$553$$ 8.57133i 0.364490i
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 1.99899i 0.0847000i 0.999103 + 0.0423500i $$0.0134845\pi$$
−0.999103 + 0.0423500i $$0.986516\pi$$
$$558$$ 0 0
$$559$$ −7.49950 −0.317195
$$560$$ 0 0
$$561$$ 11.9069 0.502709
$$562$$ 0 0
$$563$$ 28.8340i 1.21521i 0.794239 + 0.607605i $$0.207870\pi$$
−0.794239 + 0.607605i $$0.792130\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ − 17.1852i − 0.721712i
$$568$$ 0 0
$$569$$ −25.6894 −1.07695 −0.538477 0.842640i $$-0.681000\pi$$
−0.538477 + 0.842640i $$0.681000\pi$$
$$570$$ 0 0
$$571$$ 36.4792 1.52661 0.763304 0.646040i $$-0.223576\pi$$
0.763304 + 0.646040i $$0.223576\pi$$
$$572$$ 0 0
$$573$$ 40.2773i 1.68261i
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 36.1078i 1.50319i 0.659628 + 0.751593i $$0.270714\pi$$
−0.659628 + 0.751593i $$0.729286\pi$$
$$578$$ 0 0
$$579$$ 4.22477 0.175576
$$580$$ 0 0
$$581$$ 45.3122 1.87987
$$582$$ 0 0
$$583$$ − 0.979724i − 0.0405760i
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ − 5.77784i − 0.238477i −0.992866 0.119239i $$-0.961955\pi$$
0.992866 0.119239i $$-0.0380453\pi$$
$$588$$ 0 0
$$589$$ 36.3694 1.49858
$$590$$ 0 0
$$591$$ −7.20188 −0.296246
$$592$$ 0 0
$$593$$ 32.9418i 1.35276i 0.736554 + 0.676379i $$0.236452\pi$$
−0.736554 + 0.676379i $$0.763548\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ − 67.8330i − 2.77622i
$$598$$ 0 0
$$599$$ −8.14830 −0.332931 −0.166465 0.986047i $$-0.553235\pi$$
−0.166465 + 0.986047i $$0.553235\pi$$
$$600$$ 0 0
$$601$$ 7.39073 0.301474 0.150737 0.988574i $$-0.451835\pi$$
0.150737 + 0.988574i $$0.451835\pi$$
$$602$$ 0 0
$$603$$ 8.40376i 0.342228i
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 27.9980i 1.13640i 0.822889 + 0.568202i $$0.192361\pi$$
−0.822889 + 0.568202i $$0.807639\pi$$
$$608$$ 0 0
$$609$$ 52.6644 2.13407
$$610$$ 0 0
$$611$$ −7.05619 −0.285463
$$612$$ 0 0
$$613$$ 39.0036i 1.57534i 0.616096 + 0.787671i $$0.288713\pi$$
−0.616096 + 0.787671i $$0.711287\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 26.1686i 1.05351i 0.850018 + 0.526754i $$0.176591\pi$$
−0.850018 + 0.526754i $$0.823409\pi$$
$$618$$ 0 0
$$619$$ −22.2820 −0.895588 −0.447794 0.894137i $$-0.647790\pi$$
−0.447794 + 0.894137i $$0.647790\pi$$
$$620$$ 0 0
$$621$$ −0.762203 −0.0305862
$$622$$ 0 0
$$623$$ − 22.5308i − 0.902677i
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ − 21.2211i − 0.847491i
$$628$$ 0 0
$$629$$ −33.5916 −1.33939
$$630$$ 0 0
$$631$$ −34.1629 −1.36000 −0.680002 0.733210i $$-0.738021\pi$$
−0.680002 + 0.733210i $$0.738021\pi$$
$$632$$ 0 0
$$633$$ 6.18885i 0.245985i
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ − 0.407385i − 0.0161412i
$$638$$ 0 0
$$639$$ 38.3861 1.51853
$$640$$ 0 0
$$641$$ −5.37046 −0.212120 −0.106060 0.994360i $$-0.533824\pi$$
−0.106060 + 0.994360i $$0.533824\pi$$
$$642$$ 0 0
$$643$$ − 15.8856i − 0.626467i −0.949676 0.313233i $$-0.898588\pi$$
0.949676 0.313233i $$-0.101412\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 4.87197i 0.191537i 0.995404 + 0.0957685i $$0.0305309\pi$$
−0.995404 + 0.0957685i $$0.969469\pi$$
$$648$$ 0 0
$$649$$ −7.05619 −0.276980
$$650$$ 0 0
$$651$$ −31.3528 −1.22881
$$652$$ 0 0
$$653$$ − 16.8101i − 0.657831i −0.944359 0.328916i $$-0.893317\pi$$
0.944359 0.328916i $$-0.106683\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 45.2737i 1.76630i
$$658$$ 0 0
$$659$$ 2.79449 0.108858 0.0544290 0.998518i $$-0.482666\pi$$
0.0544290 + 0.998518i $$0.482666\pi$$
$$660$$ 0 0
$$661$$ 33.4267 1.30015 0.650073 0.759872i $$-0.274738\pi$$
0.650073 + 0.759872i $$0.274738\pi$$
$$662$$ 0 0
$$663$$ − 11.9069i − 0.462425i
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 3.04055i 0.117731i
$$668$$ 0 0
$$669$$ 65.9215 2.54867
$$670$$ 0 0
$$671$$ 4.46358 0.172315
$$672$$ 0 0
$$673$$ 11.9115i 0.459155i 0.973290 + 0.229578i $$0.0737345\pi$$
−0.973290 + 0.229578i $$0.926266\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ − 13.4277i − 0.516067i −0.966136 0.258033i $$-0.916926\pi$$
0.966136 0.258033i $$-0.0830745\pi$$
$$678$$ 0 0
$$679$$ 8.26271 0.317094
$$680$$ 0 0
$$681$$ −26.6478 −1.02115
$$682$$ 0 0
$$683$$ 47.7160i 1.82580i 0.408181 + 0.912901i $$0.366163\pi$$
−0.408181 + 0.912901i $$0.633837\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ − 75.2763i − 2.87197i
$$688$$ 0 0
$$689$$ −0.979724 −0.0373245
$$690$$ 0 0
$$691$$ 29.2450 1.11253 0.556267 0.831004i $$-0.312233\pi$$
0.556267 + 0.831004i $$0.312233\pi$$
$$692$$ 0 0
$$693$$ 10.1290i 0.384770i
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ − 15.2211i − 0.576542i
$$698$$ 0 0
$$699$$ −15.3143 −0.579239
$$700$$ 0 0
$$701$$ 18.6442 0.704181 0.352090 0.935966i $$-0.385471\pi$$
0.352090 + 0.935966i $$0.385471\pi$$
$$702$$ 0 0
$$703$$ 59.8689i 2.25800i
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 15.2211i 0.572450i
$$708$$ 0 0
$$709$$ 30.3705 1.14059 0.570293 0.821441i $$-0.306830\pi$$
0.570293 + 0.821441i $$0.306830\pi$$
$$710$$ 0 0
$$711$$ −11.7206 −0.439558
$$712$$ 0 0
$$713$$ − 1.81014i − 0.0677901i
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 9.83766i 0.367394i
$$718$$ 0 0
$$719$$ 4.49950 0.167803 0.0839014 0.996474i $$-0.473262\pi$$
0.0839014 + 0.996474i $$0.473262\pi$$
$$720$$ 0 0
$$721$$ −12.6524 −0.471201
$$722$$ 0 0
$$723$$ − 61.1639i − 2.27471i
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 38.7925i 1.43873i 0.694631 + 0.719367i $$0.255568\pi$$
−0.694631 + 0.719367i $$0.744432\pi$$
$$728$$ 0 0
$$729$$ 38.0516 1.40932
$$730$$ 0 0
$$731$$ −34.4423 −1.27389
$$732$$ 0 0
$$733$$ 29.5547i 1.09163i 0.837907 + 0.545813i $$0.183779\pi$$
−0.837907 + 0.545813i $$0.816221\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 2.25807i 0.0831772i
$$738$$ 0 0
$$739$$ 10.9079 0.401253 0.200627 0.979668i $$-0.435702\pi$$
0.200627 + 0.979668i $$0.435702\pi$$
$$740$$ 0 0
$$741$$ −21.2211 −0.779578
$$742$$ 0 0
$$743$$ − 7.94743i − 0.291563i −0.989317 0.145782i $$-0.953430\pi$$
0.989317 0.145782i $$-0.0465697\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ 61.9611i 2.26704i
$$748$$ 0 0
$$749$$ −1.90890 −0.0697497
$$750$$ 0 0
$$751$$ 18.2783 0.666986 0.333493 0.942753i $$-0.391773\pi$$
0.333493 + 0.942753i $$0.391773\pi$$
$$752$$ 0 0
$$753$$ 55.9787i 2.03998i
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 30.2727i 1.10028i 0.835072 + 0.550140i $$0.185426\pi$$
−0.835072 + 0.550140i $$0.814574\pi$$
$$758$$ 0 0
$$759$$ −1.05619 −0.0383374
$$760$$ 0 0
$$761$$ 14.7429 0.534431 0.267216 0.963637i $$-0.413896\pi$$
0.267216 + 0.963637i $$0.413896\pi$$
$$762$$ 0 0
$$763$$ − 11.6442i − 0.421547i
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 7.05619i 0.254784i
$$768$$ 0 0
$$769$$ −3.94280 −0.142181 −0.0710905 0.997470i $$-0.522648\pi$$
−0.0710905 + 0.997470i $$0.522648\pi$$
$$770$$ 0 0
$$771$$ −10.4110 −0.374943
$$772$$ 0 0
$$773$$ − 52.6478i − 1.89361i −0.321808 0.946805i $$-0.604291\pi$$
0.321808 0.946805i $$-0.395709\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ − 51.6109i − 1.85153i
$$778$$ 0 0
$$779$$ −27.1280 −0.971962
$$780$$ 0 0
$$781$$ 10.3143 0.369073
$$782$$ 0 0
$$783$$ 13.9641i 0.499036i
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 44.5713i 1.58880i 0.607397 + 0.794398i $$0.292214\pi$$
−0.607397 + 0.794398i $$0.707786\pi$$
$$788$$ 0 0
$$789$$ −75.8091 −2.69888
$$790$$ 0 0
$$791$$ −31.8017 −1.13074
$$792$$ 0 0
$$793$$ − 4.46358i − 0.158506i
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ − 9.03693i − 0.320104i −0.987109 0.160052i $$-0.948834\pi$$
0.987109 0.160052i $$-0.0511662\pi$$
$$798$$ 0 0
$$799$$ −32.4064 −1.14646
$$800$$ 0 0
$$801$$ 30.8091 1.08859
$$802$$ 0 0
$$803$$ 12.1650i 0.429292i
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 20.8340i 0.733393i
$$808$$ 0 0
$$809$$ −9.14468 −0.321510 −0.160755 0.986994i $$-0.551393\pi$$
−0.160755 + 0.986994i $$0.551393\pi$$
$$810$$ 0 0
$$811$$ 1.35482 0.0475741 0.0237870 0.999717i $$-0.492428\pi$$
0.0237870 + 0.999717i $$0.492428\pi$$
$$812$$ 0 0
$$813$$ 50.3658i 1.76641i
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 61.3851i 2.14759i
$$818$$ 0 0
$$819$$ 10.1290 0.353937
$$820$$ 0 0
$$821$$ −27.2414 −0.950732 −0.475366 0.879788i $$-0.657684\pi$$
−0.475366 + 0.879788i $$0.657684\pi$$
$$822$$ 0 0
$$823$$ − 6.96771i − 0.242879i −0.992599 0.121440i $$-0.961249\pi$$
0.992599 0.121440i $$-0.0387510\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 2.68472i 0.0933570i 0.998910 + 0.0466785i $$0.0148636\pi$$
−0.998910 + 0.0466785i $$0.985136\pi$$
$$828$$ 0 0
$$829$$ 20.2222 0.702345 0.351172 0.936311i $$-0.385783\pi$$
0.351172 + 0.936311i $$0.385783\pi$$
$$830$$ 0 0
$$831$$ −4.74555 −0.164621
$$832$$ 0 0
$$833$$ − 1.87096i − 0.0648250i
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ − 8.31326i − 0.287348i
$$838$$ 0 0
$$839$$ 35.5344 1.22678 0.613392 0.789779i $$-0.289805\pi$$
0.613392 + 0.789779i $$0.289805\pi$$
$$840$$ 0 0
$$841$$ 26.7050 0.920862
$$842$$ 0 0
$$843$$ 45.8497i 1.57915i
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 2.72165i 0.0935170i
$$848$$ 0 0
$$849$$ 43.6873 1.49935
$$850$$ 0 0
$$851$$ 2.97972 0.102144
$$852$$ 0 0
$$853$$ 26.3253i 0.901360i 0.892686 + 0.450680i $$0.148818\pi$$
−0.892686 + 0.450680i $$0.851182\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 48.1270i 1.64399i 0.569496 + 0.821994i $$0.307138\pi$$
−0.569496 + 0.821994i $$0.692862\pi$$
$$858$$ 0 0
$$859$$ 40.8148 1.39258 0.696291 0.717760i $$-0.254832\pi$$
0.696291 + 0.717760i $$0.254832\pi$$
$$860$$ 0 0
$$861$$ 23.3861 0.796996
$$862$$ 0 0
$$863$$ 39.8866i 1.35776i 0.734251 + 0.678878i $$0.237533\pi$$
−0.734251 + 0.678878i $$0.762467\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ − 10.6093i − 0.360310i
$$868$$ 0 0
$$869$$ −3.14931 −0.106833
$$870$$ 0 0
$$871$$ 2.25807 0.0765119
$$872$$ 0 0
$$873$$ 11.2986i 0.382401i
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 11.7263i 0.395969i 0.980205 + 0.197984i $$0.0634395\pi$$
−0.980205 + 0.197984i $$0.936561\pi$$
$$878$$ 0 0
$$879$$ 73.5537 2.48090
$$880$$ 0 0
$$881$$ −11.7363 −0.395405 −0.197703 0.980262i $$-0.563348\pi$$
−0.197703 + 0.980262i $$0.563348\pi$$
$$882$$ 0 0
$$883$$ 14.0146i 0.471630i 0.971798 + 0.235815i $$0.0757759\pi$$
−0.971798 + 0.235815i $$0.924224\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ − 4.12904i − 0.138639i −0.997594 0.0693197i $$-0.977917\pi$$
0.997594 0.0693197i $$-0.0220829\pi$$
$$888$$ 0 0
$$889$$ −25.9095 −0.868977
$$890$$ 0 0
$$891$$ 6.31427 0.211536
$$892$$ 0 0
$$893$$ 57.7566i 1.93275i
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 1.05619i 0.0352653i
$$898$$ 0 0
$$899$$ −33.1629 −1.10605
$$900$$ 0 0
$$901$$ −4.49950 −0.149900
$$902$$ 0 0
$$903$$ − 52.9179i − 1.76100i
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ − 25.2258i − 0.837608i −0.908077 0.418804i $$-0.862449\pi$$
0.908077 0.418804i $$-0.137551\pi$$
$$908$$ 0 0
$$909$$ −20.8138 −0.690349
$$910$$ 0 0
$$911$$ 19.7419 0.654079 0.327040 0.945011i $$-0.393949\pi$$
0.327040 + 0.945011i $$0.393949\pi$$
$$912$$ 0 0
$$913$$ 16.6488i 0.550995i
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ − 61.1806i − 2.02036i
$$918$$ 0 0
$$919$$ 39.1114 1.29017 0.645083 0.764113i $$-0.276823\pi$$
0.645083 + 0.764113i $$0.276823\pi$$
$$920$$ 0 0
$$921$$ 5.94743 0.195975
$$922$$ 0 0
$$923$$ − 10.3143i − 0.339498i
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ − 17.3012i − 0.568247i
$$928$$ 0 0
$$929$$ 0.670093 0.0219850 0.0109925 0.999940i $$-0.496501\pi$$
0.0109925 + 0.999940i $$0.496501\pi$$
$$930$$ 0 0
$$931$$ −3.33454 −0.109285
$$932$$ 0 0
$$933$$ − 29.1878i − 0.955567i
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 31.3715i 1.02486i 0.858729 + 0.512431i $$0.171255\pi$$
−0.858729 + 0.512431i $$0.828745\pi$$
$$938$$ 0 0
$$939$$ −18.6285 −0.607919
$$940$$ 0 0
$$941$$ −39.5703 −1.28996 −0.644978 0.764201i $$-0.723133\pi$$
−0.644978 + 0.764201i $$0.723133\pi$$
$$942$$ 0 0
$$943$$ 1.35018i 0.0439680i
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ − 9.31427i − 0.302673i −0.988482 0.151336i $$-0.951642\pi$$
0.988482 0.151336i $$-0.0483577\pi$$
$$948$$ 0 0
$$949$$ 12.1650 0.394891
$$950$$ 0 0
$$951$$ 76.5334 2.48177
$$952$$ 0 0
$$953$$ − 26.7409i − 0.866223i −0.901340 0.433112i $$-0.857416\pi$$
0.901340 0.433112i $$-0.142584\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ 19.3502i 0.625503i
$$958$$ 0 0
$$959$$ 42.2820 1.36536
$$960$$ 0 0
$$961$$ −11.2571 −0.363131
$$962$$ 0 0
$$963$$ − 2.61027i − 0.0841149i
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ − 29.0442i − 0.933998i −0.884258 0.466999i $$-0.845335\pi$$
0.884258 0.466999i $$-0.154665\pi$$
$$968$$ 0 0
$$969$$ −97.4606 −3.13088
$$970$$ 0 0
$$971$$ 41.3482 1.32693 0.663463 0.748209i $$-0.269086\pi$$
0.663463 + 0.748209i $$0.269086\pi$$
$$972$$ 0 0
$$973$$ 36.2794i 1.16306i
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 14.9105i 0.477029i 0.971139 + 0.238515i $$0.0766605\pi$$
−0.971139 + 0.238515i $$0.923339\pi$$
$$978$$ 0 0
$$979$$ 8.27835 0.264577
$$980$$ 0 0
$$981$$ 15.9225 0.508367
$$982$$ 0 0
$$983$$ 21.4682i 0.684730i 0.939567 + 0.342365i $$0.111228\pi$$
−0.939567 + 0.342365i $$0.888772\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ − 49.7899i − 1.58483i
$$988$$ 0 0
$$989$$ 3.05518 0.0971492
$$990$$ 0 0
$$991$$ −3.47922 −0.110521 −0.0552605 0.998472i $$-0.517599\pi$$
−0.0552605 + 0.998472i $$0.517599\pi$$
$$992$$ 0 0
$$993$$ 15.8902i 0.504261i
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 23.4054i 0.741255i 0.928782 + 0.370628i $$0.120857\pi$$
−0.928782 + 0.370628i $$0.879143\pi$$
$$998$$ 0 0
$$999$$ 13.6847 0.432966
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 4400.2.b.bb.4049.5 6
4.3 odd 2 2200.2.b.m.1849.2 6
5.2 odd 4 4400.2.a.by.1.3 3
5.3 odd 4 4400.2.a.bz.1.1 3
5.4 even 2 inner 4400.2.b.bb.4049.2 6
20.3 even 4 2200.2.a.u.1.3 3
20.7 even 4 2200.2.a.v.1.1 yes 3
20.19 odd 2 2200.2.b.m.1849.5 6

By twisted newform
Twist Min Dim Char Parity Ord Type
2200.2.a.u.1.3 3 20.3 even 4
2200.2.a.v.1.1 yes 3 20.7 even 4
2200.2.b.m.1849.2 6 4.3 odd 2
2200.2.b.m.1849.5 6 20.19 odd 2
4400.2.a.by.1.3 3 5.2 odd 4
4400.2.a.bz.1.1 3 5.3 odd 4
4400.2.b.bb.4049.2 6 5.4 even 2 inner
4400.2.b.bb.4049.5 6 1.1 even 1 trivial