# Properties

 Label 2025.2.b.n.649.4 Level $2025$ Weight $2$ Character 2025.649 Analytic conductor $16.170$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2025,2,Mod(649,2025)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2025, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

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

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

chi := DirichletCharacter("2025.649");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2025 = 3^{4} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2025.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: $$16.1697064093$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.34810603776.6 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{8} + 12x^{6} + 42x^{4} + 49x^{2} + 9$$ x^8 + 12*x^6 + 42*x^4 + 49*x^2 + 9 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 225) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 649.4 Root $$-0.473255i$$ of defining polynomial Character $$\chi$$ $$=$$ 2025.649 Dual form 2025.2.b.n.649.5

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-0.473255i q^{2} +1.77603 q^{4} -2.56305i q^{7} -1.78702i q^{8} +O(q^{10})$$ $$q-0.473255i q^{2} +1.77603 q^{4} -2.56305i q^{7} -1.78702i q^{8} -6.16860 q^{11} +2.13230i q^{13} -1.21298 q^{14} +2.70634 q^{16} -3.16860i q^{17} -0.356267 q^{19} +2.91932i q^{22} -4.21298i q^{23} +1.00912 q^{26} -4.55206i q^{28} -1.68623 q^{29} -8.25840 q^{31} -4.85484i q^{32} -1.49956 q^{34} -3.63274i q^{37} +0.168605i q^{38} -2.73353 q^{41} -7.67817i q^{43} -10.9556 q^{44} -1.99381 q^{46} +11.4289i q^{47} +0.430757 q^{49} +3.78702i q^{52} -9.43507i q^{53} -4.58024 q^{56} +0.798017i q^{58} -10.2159 q^{59} -0.0109932 q^{61} +3.90833i q^{62} +3.11511 q^{64} -0.982817i q^{67} -5.62754i q^{68} -6.43507 q^{71} +6.61467i q^{73} -1.71921 q^{74} -0.632740 q^{76} +15.8105i q^{77} +9.47138 q^{79} +1.29366i q^{82} -10.4198i q^{83} -3.63373 q^{86} +11.0234i q^{88} +6.26940 q^{89} +5.46519 q^{91} -7.48237i q^{92} +5.40877 q^{94} -7.20679i q^{97} -0.203858i q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q - 8 q^{4}+O(q^{10})$$ 8 * q - 8 * q^4 $$8 q - 8 q^{4} - 2 q^{11} - 6 q^{14} + 8 q^{16} - 4 q^{19} - 20 q^{26} - 2 q^{29} - 8 q^{31} - 18 q^{34} - 10 q^{41} - 44 q^{44} + 6 q^{49} - 60 q^{56} - 34 q^{59} - 26 q^{61} - 38 q^{64} - 16 q^{71} - 80 q^{74} + 22 q^{76} + 14 q^{79} - 68 q^{86} + 18 q^{89} - 34 q^{91} - 6 q^{94}+O(q^{100})$$ 8 * q - 8 * q^4 - 2 * q^11 - 6 * q^14 + 8 * q^16 - 4 * q^19 - 20 * q^26 - 2 * q^29 - 8 * q^31 - 18 * q^34 - 10 * q^41 - 44 * q^44 + 6 * q^49 - 60 * q^56 - 34 * q^59 - 26 * q^61 - 38 * q^64 - 16 * q^71 - 80 * q^74 + 22 * q^76 + 14 * q^79 - 68 * q^86 + 18 * q^89 - 34 * q^91 - 6 * q^94

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2025\mathbb{Z}\right)^\times$$.

 $$n$$ $$326$$ $$1702$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ − 0.473255i − 0.334641i −0.985903 0.167321i $$-0.946488\pi$$
0.985903 0.167321i $$-0.0535115\pi$$
$$3$$ 0 0
$$4$$ 1.77603 0.888015
$$5$$ 0 0
$$6$$ 0 0
$$7$$ − 2.56305i − 0.968743i −0.874862 0.484372i $$-0.839048\pi$$
0.874862 0.484372i $$-0.160952\pi$$
$$8$$ − 1.78702i − 0.631808i
$$9$$ 0 0
$$10$$ 0 0
$$11$$ −6.16860 −1.85990 −0.929952 0.367681i $$-0.880152\pi$$
−0.929952 + 0.367681i $$0.880152\pi$$
$$12$$ 0 0
$$13$$ 2.13230i 0.591393i 0.955282 + 0.295696i $$0.0955517\pi$$
−0.955282 + 0.295696i $$0.904448\pi$$
$$14$$ −1.21298 −0.324182
$$15$$ 0 0
$$16$$ 2.70634 0.676586
$$17$$ − 3.16860i − 0.768500i −0.923229 0.384250i $$-0.874460\pi$$
0.923229 0.384250i $$-0.125540\pi$$
$$18$$ 0 0
$$19$$ −0.356267 −0.0817332 −0.0408666 0.999165i $$-0.513012\pi$$
−0.0408666 + 0.999165i $$0.513012\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 2.91932i 0.622401i
$$23$$ − 4.21298i − 0.878466i −0.898373 0.439233i $$-0.855250\pi$$
0.898373 0.439233i $$-0.144750\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 1.00912 0.197905
$$27$$ 0 0
$$28$$ − 4.55206i − 0.860259i
$$29$$ −1.68623 −0.313125 −0.156563 0.987668i $$-0.550041\pi$$
−0.156563 + 0.987668i $$0.550041\pi$$
$$30$$ 0 0
$$31$$ −8.25840 −1.48325 −0.741627 0.670813i $$-0.765945\pi$$
−0.741627 + 0.670813i $$0.765945\pi$$
$$32$$ − 4.85484i − 0.858222i
$$33$$ 0 0
$$34$$ −1.49956 −0.257172
$$35$$ 0 0
$$36$$ 0 0
$$37$$ − 3.63274i − 0.597219i −0.954375 0.298609i $$-0.903477\pi$$
0.954375 0.298609i $$-0.0965228\pi$$
$$38$$ 0.168605i 0.0273513i
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −2.73353 −0.426906 −0.213453 0.976953i $$-0.568471\pi$$
−0.213453 + 0.976953i $$0.568471\pi$$
$$42$$ 0 0
$$43$$ − 7.67817i − 1.17091i −0.810705 0.585455i $$-0.800916\pi$$
0.810705 0.585455i $$-0.199084\pi$$
$$44$$ −10.9556 −1.65162
$$45$$ 0 0
$$46$$ −1.99381 −0.293971
$$47$$ 11.4289i 1.66707i 0.552464 + 0.833537i $$0.313688\pi$$
−0.552464 + 0.833537i $$0.686312\pi$$
$$48$$ 0 0
$$49$$ 0.430757 0.0615367
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 3.78702i 0.525166i
$$53$$ − 9.43507i − 1.29601i −0.761637 0.648003i $$-0.775604\pi$$
0.761637 0.648003i $$-0.224396\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ −4.58024 −0.612060
$$57$$ 0 0
$$58$$ 0.798017i 0.104785i
$$59$$ −10.2159 −1.33000 −0.664999 0.746844i $$-0.731568\pi$$
−0.664999 + 0.746844i $$0.731568\pi$$
$$60$$ 0 0
$$61$$ −0.0109932 −0.00140753 −0.000703767 1.00000i $$-0.500224\pi$$
−0.000703767 1.00000i $$0.500224\pi$$
$$62$$ 3.90833i 0.496358i
$$63$$ 0 0
$$64$$ 3.11511 0.389389
$$65$$ 0 0
$$66$$ 0 0
$$67$$ − 0.982817i − 0.120070i −0.998196 0.0600351i $$-0.980879\pi$$
0.998196 0.0600351i $$-0.0191213\pi$$
$$68$$ − 5.62754i − 0.682439i
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −6.43507 −0.763703 −0.381851 0.924224i $$-0.624713\pi$$
−0.381851 + 0.924224i $$0.624713\pi$$
$$72$$ 0 0
$$73$$ 6.61467i 0.774189i 0.922040 + 0.387094i $$0.126521\pi$$
−0.922040 + 0.387094i $$0.873479\pi$$
$$74$$ −1.71921 −0.199854
$$75$$ 0 0
$$76$$ −0.632740 −0.0725803
$$77$$ 15.8105i 1.80177i
$$78$$ 0 0
$$79$$ 9.47138 1.06561 0.532807 0.846237i $$-0.321137\pi$$
0.532807 + 0.846237i $$0.321137\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 1.29366i 0.142860i
$$83$$ − 10.4198i − 1.14372i −0.820352 0.571859i $$-0.806223\pi$$
0.820352 0.571859i $$-0.193777\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ −3.63373 −0.391835
$$87$$ 0 0
$$88$$ 11.0234i 1.17510i
$$89$$ 6.26940 0.664555 0.332277 0.943182i $$-0.392183\pi$$
0.332277 + 0.943182i $$0.392183\pi$$
$$90$$ 0 0
$$91$$ 5.46519 0.572908
$$92$$ − 7.48237i − 0.780091i
$$93$$ 0 0
$$94$$ 5.40877 0.557872
$$95$$ 0 0
$$96$$ 0 0
$$97$$ − 7.20679i − 0.731738i −0.930666 0.365869i $$-0.880772\pi$$
0.930666 0.365869i $$-0.119228\pi$$
$$98$$ − 0.203858i − 0.0205927i
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 6.97094 0.693634 0.346817 0.937933i $$-0.387262\pi$$
0.346817 + 0.937933i $$0.387262\pi$$
$$102$$ 0 0
$$103$$ − 6.11511i − 0.602540i −0.953539 0.301270i $$-0.902589\pi$$
0.953539 0.301270i $$-0.0974106\pi$$
$$104$$ 3.81046 0.373647
$$105$$ 0 0
$$106$$ −4.46519 −0.433698
$$107$$ 14.5349i 1.40514i 0.711615 + 0.702570i $$0.247964\pi$$
−0.711615 + 0.702570i $$0.752036\pi$$
$$108$$ 0 0
$$109$$ −1.90214 −0.182192 −0.0910958 0.995842i $$-0.529037\pi$$
−0.0910958 + 0.995842i $$0.529037\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ − 6.93650i − 0.655438i
$$113$$ − 6.57925i − 0.618924i −0.950912 0.309462i $$-0.899851\pi$$
0.950912 0.309462i $$-0.100149\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ −2.99480 −0.278060
$$117$$ 0 0
$$118$$ 4.83472i 0.445072i
$$119$$ −8.12130 −0.744479
$$120$$ 0 0
$$121$$ 27.0517 2.45924
$$122$$ 0.00520257i 0 0.000471019i
$$123$$ 0 0
$$124$$ −14.6672 −1.31715
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 9.25840i 0.821550i 0.911737 + 0.410775i $$0.134742\pi$$
−0.911737 + 0.410775i $$0.865258\pi$$
$$128$$ − 11.1839i − 0.988528i
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 0.269397 0.0235373 0.0117687 0.999931i $$-0.496254\pi$$
0.0117687 + 0.999931i $$0.496254\pi$$
$$132$$ 0 0
$$133$$ 0.913130i 0.0791784i
$$134$$ −0.465123 −0.0401805
$$135$$ 0 0
$$136$$ −5.66237 −0.485544
$$137$$ − 3.47618i − 0.296990i −0.988913 0.148495i $$-0.952557\pi$$
0.988913 0.148495i $$-0.0474430\pi$$
$$138$$ 0 0
$$139$$ −14.7479 −1.25090 −0.625448 0.780266i $$-0.715084\pi$$
−0.625448 + 0.780266i $$0.715084\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 3.04543i 0.255567i
$$143$$ − 13.1533i − 1.09993i
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 3.13042 0.259076
$$147$$ 0 0
$$148$$ − 6.45186i − 0.530339i
$$149$$ 10.1533 0.831790 0.415895 0.909413i $$-0.363468\pi$$
0.415895 + 0.909413i $$0.363468\pi$$
$$150$$ 0 0
$$151$$ −10.3162 −0.839521 −0.419761 0.907635i $$-0.637886\pi$$
−0.419761 + 0.907635i $$0.637886\pi$$
$$152$$ 0.636657i 0.0516397i
$$153$$ 0 0
$$154$$ 7.48237 0.602947
$$155$$ 0 0
$$156$$ 0 0
$$157$$ − 1.06261i − 0.0848055i −0.999101 0.0424028i $$-0.986499\pi$$
0.999101 0.0424028i $$-0.0135013\pi$$
$$158$$ − 4.48237i − 0.356598i
$$159$$ 0 0
$$160$$ 0 0
$$161$$ −10.7981 −0.851008
$$162$$ 0 0
$$163$$ − 17.1386i − 1.34240i −0.741278 0.671198i $$-0.765780\pi$$
0.741278 0.671198i $$-0.234220\pi$$
$$164$$ −4.85484 −0.379099
$$165$$ 0 0
$$166$$ −4.93120 −0.382735
$$167$$ 4.37345i 0.338428i 0.985579 + 0.169214i $$0.0541229\pi$$
−0.985579 + 0.169214i $$0.945877\pi$$
$$168$$ 0 0
$$169$$ 8.45331 0.650255
$$170$$ 0 0
$$171$$ 0 0
$$172$$ − 13.6367i − 1.03979i
$$173$$ − 14.6601i − 1.11459i −0.830316 0.557293i $$-0.811840\pi$$
0.830316 0.557293i $$-0.188160\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ −16.6944 −1.25838
$$177$$ 0 0
$$178$$ − 2.96702i − 0.222388i
$$179$$ 6.87014 0.513499 0.256749 0.966478i $$-0.417349\pi$$
0.256749 + 0.966478i $$0.417349\pi$$
$$180$$ 0 0
$$181$$ −10.9709 −0.815463 −0.407732 0.913102i $$-0.633680\pi$$
−0.407732 + 0.913102i $$0.633680\pi$$
$$182$$ − 2.58643i − 0.191719i
$$183$$ 0 0
$$184$$ −7.52869 −0.555022
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 19.5459i 1.42934i
$$188$$ 20.2980i 1.48039i
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 13.7325 0.993652 0.496826 0.867850i $$-0.334499\pi$$
0.496826 + 0.867850i $$0.334499\pi$$
$$192$$ 0 0
$$193$$ − 0.482374i − 0.0347220i −0.999849 0.0173610i $$-0.994474\pi$$
0.999849 0.0173610i $$-0.00552646\pi$$
$$194$$ −3.41064 −0.244870
$$195$$ 0 0
$$196$$ 0.765037 0.0546455
$$197$$ − 5.53488i − 0.394344i −0.980369 0.197172i $$-0.936824\pi$$
0.980369 0.197172i $$-0.0631757\pi$$
$$198$$ 0 0
$$199$$ −17.4590 −1.23764 −0.618818 0.785534i $$-0.712388\pi$$
−0.618818 + 0.785534i $$0.712388\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ − 3.29903i − 0.232119i
$$203$$ 4.32190i 0.303338i
$$204$$ 0 0
$$205$$ 0 0
$$206$$ −2.89401 −0.201635
$$207$$ 0 0
$$208$$ 5.77073i 0.400128i
$$209$$ 2.19767 0.152016
$$210$$ 0 0
$$211$$ −1.63666 −0.112672 −0.0563360 0.998412i $$-0.517942\pi$$
−0.0563360 + 0.998412i $$0.517942\pi$$
$$212$$ − 16.7570i − 1.15087i
$$213$$ 0 0
$$214$$ 6.87870 0.470218
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 21.1667i 1.43689i
$$218$$ 0.900195i 0.0609689i
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 6.75641 0.454485
$$222$$ 0 0
$$223$$ 7.74785i 0.518835i 0.965765 + 0.259417i $$0.0835305\pi$$
−0.965765 + 0.259417i $$0.916469\pi$$
$$224$$ −12.4432 −0.831397
$$225$$ 0 0
$$226$$ −3.11366 −0.207118
$$227$$ 11.2603i 0.747371i 0.927555 + 0.373685i $$0.121906\pi$$
−0.927555 + 0.373685i $$0.878094\pi$$
$$228$$ 0 0
$$229$$ 10.4776 0.692377 0.346189 0.938165i $$-0.387476\pi$$
0.346189 + 0.938165i $$0.387476\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 3.01333i 0.197835i
$$233$$ − 2.90214i − 0.190125i −0.995471 0.0950627i $$-0.969695\pi$$
0.995471 0.0950627i $$-0.0303051\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ −18.1438 −1.18106
$$237$$ 0 0
$$238$$ 3.84344i 0.249133i
$$239$$ −16.3545 −1.05788 −0.528941 0.848659i $$-0.677411\pi$$
−0.528941 + 0.848659i $$0.677411\pi$$
$$240$$ 0 0
$$241$$ 17.5239 1.12881 0.564406 0.825497i $$-0.309105\pi$$
0.564406 + 0.825497i $$0.309105\pi$$
$$242$$ − 12.8023i − 0.822965i
$$243$$ 0 0
$$244$$ −0.0195242 −0.00124991
$$245$$ 0 0
$$246$$ 0 0
$$247$$ − 0.759666i − 0.0483364i
$$248$$ 14.7580i 0.937131i
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 8.46999 0.534621 0.267311 0.963610i $$-0.413865\pi$$
0.267311 + 0.963610i $$0.413865\pi$$
$$252$$ 0 0
$$253$$ 25.9882i 1.63386i
$$254$$ 4.38158 0.274925
$$255$$ 0 0
$$256$$ 0.937390 0.0585869
$$257$$ 2.87251i 0.179182i 0.995979 + 0.0895910i $$0.0285560\pi$$
−0.995979 + 0.0895910i $$0.971444\pi$$
$$258$$ 0 0
$$259$$ −9.31091 −0.578552
$$260$$ 0 0
$$261$$ 0 0
$$262$$ − 0.127493i − 0.00787656i
$$263$$ 25.6239i 1.58003i 0.613084 + 0.790017i $$0.289929\pi$$
−0.613084 + 0.790017i $$0.710071\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0.432143 0.0264964
$$267$$ 0 0
$$268$$ − 1.74551i − 0.106624i
$$269$$ −0.337210 −0.0205600 −0.0102800 0.999947i $$-0.503272\pi$$
−0.0102800 + 0.999947i $$0.503272\pi$$
$$270$$ 0 0
$$271$$ 21.5927 1.31166 0.655831 0.754908i $$-0.272318\pi$$
0.655831 + 0.754908i $$0.272318\pi$$
$$272$$ − 8.57533i − 0.519956i
$$273$$ 0 0
$$274$$ −1.64512 −0.0993853
$$275$$ 0 0
$$276$$ 0 0
$$277$$ − 24.1338i − 1.45006i −0.688719 0.725028i $$-0.741827\pi$$
0.688719 0.725028i $$-0.258173\pi$$
$$278$$ 6.97949i 0.418602i
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 3.36726 0.200874 0.100437 0.994943i $$-0.467976\pi$$
0.100437 + 0.994943i $$0.467976\pi$$
$$282$$ 0 0
$$283$$ 21.8497i 1.29883i 0.760434 + 0.649415i $$0.224986\pi$$
−0.760434 + 0.649415i $$0.775014\pi$$
$$284$$ −11.4289 −0.678179
$$285$$ 0 0
$$286$$ −6.22486 −0.368083
$$287$$ 7.00619i 0.413562i
$$288$$ 0 0
$$289$$ 6.95994 0.409408
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 11.7479i 0.687491i
$$293$$ 13.7540i 0.803520i 0.915745 + 0.401760i $$0.131601\pi$$
−0.915745 + 0.401760i $$0.868399\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ −6.49179 −0.377328
$$297$$ 0 0
$$298$$ − 4.80509i − 0.278352i
$$299$$ 8.98332 0.519519
$$300$$ 0 0
$$301$$ −19.6796 −1.13431
$$302$$ 4.88219i 0.280939i
$$303$$ 0 0
$$304$$ −0.964180 −0.0552995
$$305$$ 0 0
$$306$$ 0 0
$$307$$ − 34.2183i − 1.95294i −0.215644 0.976472i $$-0.569185\pi$$
0.215644 0.976472i $$-0.430815\pi$$
$$308$$ 28.0799i 1.60000i
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 23.0397 1.30646 0.653232 0.757158i $$-0.273413\pi$$
0.653232 + 0.757158i $$0.273413\pi$$
$$312$$ 0 0
$$313$$ − 3.59130i − 0.202992i −0.994836 0.101496i $$-0.967637\pi$$
0.994836 0.101496i $$-0.0323629\pi$$
$$314$$ −0.502885 −0.0283794
$$315$$ 0 0
$$316$$ 16.8215 0.946281
$$317$$ − 13.1807i − 0.740299i −0.928972 0.370150i $$-0.879306\pi$$
0.928972 0.370150i $$-0.120694\pi$$
$$318$$ 0 0
$$319$$ 10.4017 0.582383
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 5.11024i 0.284783i
$$323$$ 1.12887i 0.0628119i
$$324$$ 0 0
$$325$$ 0 0
$$326$$ −8.11090 −0.449221
$$327$$ 0 0
$$328$$ 4.88489i 0.269723i
$$329$$ 29.2928 1.61497
$$330$$ 0 0
$$331$$ 1.18253 0.0649976 0.0324988 0.999472i $$-0.489653\pi$$
0.0324988 + 0.999472i $$0.489653\pi$$
$$332$$ − 18.5058i − 1.01564i
$$333$$ 0 0
$$334$$ 2.06975 0.113252
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 24.7995i 1.35091i 0.737400 + 0.675457i $$0.236053\pi$$
−0.737400 + 0.675457i $$0.763947\pi$$
$$338$$ − 4.00057i − 0.217602i
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 50.9428 2.75871
$$342$$ 0 0
$$343$$ − 19.0454i − 1.02836i
$$344$$ −13.7211 −0.739790
$$345$$ 0 0
$$346$$ −6.93796 −0.372987
$$347$$ − 22.1692i − 1.19010i −0.803687 0.595052i $$-0.797132\pi$$
0.803687 0.595052i $$-0.202868\pi$$
$$348$$ 0 0
$$349$$ −14.9185 −0.798569 −0.399285 0.916827i $$-0.630741\pi$$
−0.399285 + 0.916827i $$0.630741\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 29.9476i 1.59621i
$$353$$ 16.9145i 0.900269i 0.892961 + 0.450134i $$0.148624\pi$$
−0.892961 + 0.450134i $$0.851376\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 11.1346 0.590135
$$357$$ 0 0
$$358$$ − 3.25133i − 0.171838i
$$359$$ −0.636657 −0.0336015 −0.0168007 0.999859i $$-0.505348\pi$$
−0.0168007 + 0.999859i $$0.505348\pi$$
$$360$$ 0 0
$$361$$ −18.8731 −0.993320
$$362$$ 5.19205i 0.272888i
$$363$$ 0 0
$$364$$ 9.70634 0.508751
$$365$$ 0 0
$$366$$ 0 0
$$367$$ − 20.0979i − 1.04910i −0.851379 0.524552i $$-0.824233\pi$$
0.851379 0.524552i $$-0.175767\pi$$
$$368$$ − 11.4018i − 0.594358i
$$369$$ 0 0
$$370$$ 0 0
$$371$$ −24.1826 −1.25550
$$372$$ 0 0
$$373$$ − 19.6429i − 1.01707i −0.861041 0.508536i $$-0.830187\pi$$
0.861041 0.508536i $$-0.169813\pi$$
$$374$$ 9.25017 0.478315
$$375$$ 0 0
$$376$$ 20.4237 1.05327
$$377$$ − 3.59555i − 0.185180i
$$378$$ 0 0
$$379$$ 7.94219 0.407963 0.203982 0.978975i $$-0.434612\pi$$
0.203982 + 0.978975i $$0.434612\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ − 6.49899i − 0.332517i
$$383$$ − 31.1888i − 1.59367i −0.604195 0.796836i $$-0.706505\pi$$
0.604195 0.796836i $$-0.293495\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ −0.228285 −0.0116194
$$387$$ 0 0
$$388$$ − 12.7995i − 0.649795i
$$389$$ 31.4494 1.59455 0.797274 0.603618i $$-0.206275\pi$$
0.797274 + 0.603618i $$0.206275\pi$$
$$390$$ 0 0
$$391$$ −13.3493 −0.675101
$$392$$ − 0.769772i − 0.0388794i
$$393$$ 0 0
$$394$$ −2.61941 −0.131964
$$395$$ 0 0
$$396$$ 0 0
$$397$$ − 17.7174i − 0.889211i −0.895726 0.444606i $$-0.853344\pi$$
0.895726 0.444606i $$-0.146656\pi$$
$$398$$ 8.26255i 0.414164i
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −7.14247 −0.356678 −0.178339 0.983969i $$-0.557072\pi$$
−0.178339 + 0.983969i $$0.557072\pi$$
$$402$$ 0 0
$$403$$ − 17.6094i − 0.877185i
$$404$$ 12.3806 0.615958
$$405$$ 0 0
$$406$$ 2.04536 0.101509
$$407$$ 22.4089i 1.11077i
$$408$$ 0 0
$$409$$ −24.7518 −1.22390 −0.611948 0.790898i $$-0.709614\pi$$
−0.611948 + 0.790898i $$0.709614\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ − 10.8606i − 0.535065i
$$413$$ 26.1839i 1.28843i
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 10.3520 0.507546
$$417$$ 0 0
$$418$$ − 1.04006i − 0.0508708i
$$419$$ 10.6591 0.520732 0.260366 0.965510i $$-0.416157\pi$$
0.260366 + 0.965510i $$0.416157\pi$$
$$420$$ 0 0
$$421$$ −8.17861 −0.398601 −0.199301 0.979938i $$-0.563867\pi$$
−0.199301 + 0.979938i $$0.563867\pi$$
$$422$$ 0.774555i 0.0377048i
$$423$$ 0 0
$$424$$ −16.8607 −0.818828
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 0.0281761i 0.00136354i
$$428$$ 25.8144i 1.24779i
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 1.67248 0.0805604 0.0402802 0.999188i $$-0.487175\pi$$
0.0402802 + 0.999188i $$0.487175\pi$$
$$432$$ 0 0
$$433$$ − 9.95994i − 0.478644i −0.970940 0.239322i $$-0.923075\pi$$
0.970940 0.239322i $$-0.0769252\pi$$
$$434$$ 10.0173 0.480843
$$435$$ 0 0
$$436$$ −3.37825 −0.161789
$$437$$ 1.50094i 0.0717998i
$$438$$ 0 0
$$439$$ −12.8158 −0.611663 −0.305832 0.952086i $$-0.598934\pi$$
−0.305832 + 0.952086i $$0.598934\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ − 3.19750i − 0.152090i
$$443$$ − 7.75404i − 0.368406i −0.982888 0.184203i $$-0.941030\pi$$
0.982888 0.184203i $$-0.0589703\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 3.66671 0.173624
$$447$$ 0 0
$$448$$ − 7.98420i − 0.377218i
$$449$$ 33.3401 1.57342 0.786709 0.617324i $$-0.211783\pi$$
0.786709 + 0.617324i $$0.211783\pi$$
$$450$$ 0 0
$$451$$ 16.8621 0.794004
$$452$$ − 11.6849i − 0.549614i
$$453$$ 0 0
$$454$$ 5.32898 0.250101
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 38.2192i 1.78782i 0.448246 + 0.893910i $$0.352049\pi$$
−0.448246 + 0.893910i $$0.647951\pi$$
$$458$$ − 4.95856i − 0.231698i
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 31.3033 1.45794 0.728971 0.684545i $$-0.239999\pi$$
0.728971 + 0.684545i $$0.239999\pi$$
$$462$$ 0 0
$$463$$ 12.0852i 0.561645i 0.959760 + 0.280823i $$0.0906073\pi$$
−0.959760 + 0.280823i $$0.909393\pi$$
$$464$$ −4.56352 −0.211856
$$465$$ 0 0
$$466$$ −1.37345 −0.0636238
$$467$$ 7.60466i 0.351902i 0.984399 + 0.175951i $$0.0563000\pi$$
−0.984399 + 0.175951i $$0.943700\pi$$
$$468$$ 0 0
$$469$$ −2.51901 −0.116317
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 18.2561i 0.840303i
$$473$$ 47.3636i 2.17778i
$$474$$ 0 0
$$475$$ 0 0
$$476$$ −14.4237 −0.661108
$$477$$ 0 0
$$478$$ 7.73982i 0.354011i
$$479$$ 32.4833 1.48420 0.742101 0.670289i $$-0.233830\pi$$
0.742101 + 0.670289i $$0.233830\pi$$
$$480$$ 0 0
$$481$$ 7.74608 0.353191
$$482$$ − 8.29326i − 0.377748i
$$483$$ 0 0
$$484$$ 48.0446 2.18385
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 4.46121i 0.202157i 0.994878 + 0.101078i $$0.0322293\pi$$
−0.994878 + 0.101078i $$0.967771\pi$$
$$488$$ 0.0196451i 0 0.000889291i
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 32.8420 1.48214 0.741070 0.671428i $$-0.234319\pi$$
0.741070 + 0.671428i $$0.234319\pi$$
$$492$$ 0 0
$$493$$ 5.34300i 0.240637i
$$494$$ −0.359515 −0.0161754
$$495$$ 0 0
$$496$$ −22.3501 −1.00355
$$497$$ 16.4934i 0.739832i
$$498$$ 0 0
$$499$$ 34.2021 1.53109 0.765547 0.643380i $$-0.222468\pi$$
0.765547 + 0.643380i $$0.222468\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ − 4.00846i − 0.178906i
$$503$$ 22.1773i 0.988837i 0.869224 + 0.494419i $$0.164619\pi$$
−0.869224 + 0.494419i $$0.835381\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 12.2990 0.546758
$$507$$ 0 0
$$508$$ 16.4432i 0.729549i
$$509$$ 21.5632 0.955773 0.477887 0.878422i $$-0.341403\pi$$
0.477887 + 0.878422i $$0.341403\pi$$
$$510$$ 0 0
$$511$$ 16.9538 0.749990
$$512$$ − 22.8115i − 1.00813i
$$513$$ 0 0
$$514$$ 1.35943 0.0599617
$$515$$ 0 0
$$516$$ 0 0
$$517$$ − 70.5003i − 3.10060i
$$518$$ 4.40643i 0.193607i
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 20.2626 0.887718 0.443859 0.896097i $$-0.353609\pi$$
0.443859 + 0.896097i $$0.353609\pi$$
$$522$$ 0 0
$$523$$ 31.8114i 1.39101i 0.718520 + 0.695507i $$0.244820\pi$$
−0.718520 + 0.695507i $$0.755180\pi$$
$$524$$ 0.478457 0.0209015
$$525$$ 0 0
$$526$$ 12.1266 0.528745
$$527$$ 26.1676i 1.13988i
$$528$$ 0 0
$$529$$ 5.25083 0.228297
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 1.62175i 0.0703117i
$$533$$ − 5.82870i − 0.252469i
$$534$$ 0 0
$$535$$ 0 0
$$536$$ −1.75632 −0.0758613
$$537$$ 0 0
$$538$$ 0.159586i 0.00688024i
$$539$$ −2.65717 −0.114452
$$540$$ 0 0
$$541$$ −15.1315 −0.650553 −0.325277 0.945619i $$-0.605457\pi$$
−0.325277 + 0.945619i $$0.605457\pi$$
$$542$$ − 10.2188i − 0.438937i
$$543$$ 0 0
$$544$$ −15.3831 −0.659543
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 4.08744i 0.174766i 0.996175 + 0.0873831i $$0.0278504\pi$$
−0.996175 + 0.0873831i $$0.972150\pi$$
$$548$$ − 6.17381i − 0.263732i
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 0.600748 0.0255927
$$552$$ 0 0
$$553$$ − 24.2757i − 1.03231i
$$554$$ −11.4214 −0.485249
$$555$$ 0 0
$$556$$ −26.1926 −1.11082
$$557$$ 13.1425i 0.556864i 0.960456 + 0.278432i $$0.0898147\pi$$
−0.960456 + 0.278432i $$0.910185\pi$$
$$558$$ 0 0
$$559$$ 16.3721 0.692467
$$560$$ 0 0
$$561$$ 0 0
$$562$$ − 1.59357i − 0.0672207i
$$563$$ − 24.5221i − 1.03348i −0.856141 0.516742i $$-0.827145\pi$$
0.856141 0.516742i $$-0.172855\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 10.3405 0.434642
$$567$$ 0 0
$$568$$ 11.4996i 0.482514i
$$569$$ −22.7299 −0.952885 −0.476442 0.879206i $$-0.658074\pi$$
−0.476442 + 0.879206i $$0.658074\pi$$
$$570$$ 0 0
$$571$$ −0.494186 −0.0206810 −0.0103405 0.999947i $$-0.503292\pi$$
−0.0103405 + 0.999947i $$0.503292\pi$$
$$572$$ − 23.3607i − 0.976758i
$$573$$ 0 0
$$574$$ 3.31571 0.138395
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 9.41187i 0.391821i 0.980622 + 0.195911i $$0.0627662\pi$$
−0.980622 + 0.195911i $$0.937234\pi$$
$$578$$ − 3.29382i − 0.137005i
$$579$$ 0 0
$$580$$ 0 0
$$581$$ −26.7064 −1.10797
$$582$$ 0 0
$$583$$ 58.2012i 2.41045i
$$584$$ 11.8206 0.489139
$$585$$ 0 0
$$586$$ 6.50916 0.268891
$$587$$ − 9.97321i − 0.411638i −0.978590 0.205819i $$-0.934014\pi$$
0.978590 0.205819i $$-0.0659859\pi$$
$$588$$ 0 0
$$589$$ 2.94219 0.121231
$$590$$ 0 0
$$591$$ 0 0
$$592$$ − 9.83144i − 0.404070i
$$593$$ 38.3421i 1.57452i 0.616621 + 0.787260i $$0.288501\pi$$
−0.616621 + 0.787260i $$0.711499\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 18.0326 0.738642
$$597$$ 0 0
$$598$$ − 4.25140i − 0.173852i
$$599$$ 10.1533 0.414852 0.207426 0.978251i $$-0.433491\pi$$
0.207426 + 0.978251i $$0.433491\pi$$
$$600$$ 0 0
$$601$$ −21.2743 −0.867796 −0.433898 0.900962i $$-0.642862\pi$$
−0.433898 + 0.900962i $$0.642862\pi$$
$$602$$ 9.31344i 0.379587i
$$603$$ 0 0
$$604$$ −18.3219 −0.745508
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 37.7355i 1.53164i 0.643056 + 0.765819i $$0.277666\pi$$
−0.643056 + 0.765819i $$0.722334\pi$$
$$608$$ 1.72962i 0.0701452i
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −24.3698 −0.985895
$$612$$ 0 0
$$613$$ − 32.2633i − 1.30310i −0.758605 0.651551i $$-0.774119\pi$$
0.758605 0.651551i $$-0.225881\pi$$
$$614$$ −16.1940 −0.653536
$$615$$ 0 0
$$616$$ 28.2537 1.13837
$$617$$ 26.0178i 1.04744i 0.851891 + 0.523719i $$0.175456\pi$$
−0.851891 + 0.523719i $$0.824544\pi$$
$$618$$ 0 0
$$619$$ 11.8815 0.477559 0.238780 0.971074i $$-0.423253\pi$$
0.238780 + 0.971074i $$0.423253\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ − 10.9037i − 0.437197i
$$623$$ − 16.0688i − 0.643783i
$$624$$ 0 0
$$625$$ 0 0
$$626$$ −1.69960 −0.0679296
$$627$$ 0 0
$$628$$ − 1.88723i − 0.0753086i
$$629$$ −11.5107 −0.458962
$$630$$ 0 0
$$631$$ −13.2726 −0.528372 −0.264186 0.964472i $$-0.585103\pi$$
−0.264186 + 0.964472i $$0.585103\pi$$
$$632$$ − 16.9256i − 0.673263i
$$633$$ 0 0
$$634$$ −6.23780 −0.247735
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 0.918501i 0.0363923i
$$638$$ − 4.92265i − 0.194890i
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 44.8149 1.77008 0.885042 0.465511i $$-0.154129\pi$$
0.885042 + 0.465511i $$0.154129\pi$$
$$642$$ 0 0
$$643$$ − 14.9255i − 0.588605i −0.955712 0.294302i $$-0.904913\pi$$
0.955712 0.294302i $$-0.0950873\pi$$
$$644$$ −19.1777 −0.755708
$$645$$ 0 0
$$646$$ 0.534242 0.0210195
$$647$$ 41.2684i 1.62243i 0.584749 + 0.811214i $$0.301193\pi$$
−0.584749 + 0.811214i $$0.698807\pi$$
$$648$$ 0 0
$$649$$ 63.0179 2.47367
$$650$$ 0 0
$$651$$ 0 0
$$652$$ − 30.4386i − 1.19207i
$$653$$ 27.6252i 1.08106i 0.841326 + 0.540528i $$0.181776\pi$$
−0.841326 + 0.540528i $$0.818224\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ −7.39788 −0.288839
$$657$$ 0 0
$$658$$ − 13.8630i − 0.540435i
$$659$$ −40.0225 −1.55905 −0.779527 0.626369i $$-0.784540\pi$$
−0.779527 + 0.626369i $$0.784540\pi$$
$$660$$ 0 0
$$661$$ 24.9929 0.972112 0.486056 0.873928i $$-0.338435\pi$$
0.486056 + 0.873928i $$0.338435\pi$$
$$662$$ − 0.559636i − 0.0217509i
$$663$$ 0 0
$$664$$ −18.6204 −0.722610
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 7.10405i 0.275070i
$$668$$ 7.76738i 0.300529i
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 0.0678126 0.00261788
$$672$$ 0 0
$$673$$ − 40.8048i − 1.57291i −0.617649 0.786454i $$-0.711915\pi$$
0.617649 0.786454i $$-0.288085\pi$$
$$674$$ 11.7365 0.452072
$$675$$ 0 0
$$676$$ 15.0133 0.577436
$$677$$ − 41.1894i − 1.58304i −0.611146 0.791518i $$-0.709291\pi$$
0.611146 0.791518i $$-0.290709\pi$$
$$678$$ 0 0
$$679$$ −18.4714 −0.708867
$$680$$ 0 0
$$681$$ 0 0
$$682$$ − 24.1089i − 0.923178i
$$683$$ 1.33820i 0.0512047i 0.999672 + 0.0256023i $$0.00815037\pi$$
−0.999672 + 0.0256023i $$0.991850\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ −9.01333 −0.344131
$$687$$ 0 0
$$688$$ − 20.7798i − 0.792221i
$$689$$ 20.1184 0.766449
$$690$$ 0 0
$$691$$ −25.2813 −0.961748 −0.480874 0.876790i $$-0.659680\pi$$
−0.480874 + 0.876790i $$0.659680\pi$$
$$692$$ − 26.0368i − 0.989770i
$$693$$ 0 0
$$694$$ −10.4917 −0.398258
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 8.66148i 0.328077i
$$698$$ 7.06025i 0.267234i
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −18.2064 −0.687645 −0.343822 0.939035i $$-0.611722\pi$$
−0.343822 + 0.939035i $$0.611722\pi$$
$$702$$ 0 0
$$703$$ 1.29422i 0.0488126i
$$704$$ −19.2159 −0.724227
$$705$$ 0 0
$$706$$ 8.00487 0.301267
$$707$$ − 17.8669i − 0.671953i
$$708$$ 0 0
$$709$$ −41.8206 −1.57061 −0.785304 0.619111i $$-0.787493\pi$$
−0.785304 + 0.619111i $$0.787493\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ − 11.2036i − 0.419871i
$$713$$ 34.7925i 1.30299i
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 12.2016 0.455995
$$717$$ 0 0
$$718$$ 0.301301i 0.0112444i
$$719$$ −48.9786 −1.82660 −0.913298 0.407293i $$-0.866473\pi$$
−0.913298 + 0.407293i $$0.866473\pi$$
$$720$$ 0 0
$$721$$ −15.6734 −0.583707
$$722$$ 8.93177i 0.332406i
$$723$$ 0 0
$$724$$ −19.4847 −0.724144
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 43.8009i 1.62449i 0.583319 + 0.812243i $$0.301754\pi$$
−0.583319 + 0.812243i $$0.698246\pi$$
$$728$$ − 9.76642i − 0.361968i
$$729$$ 0 0
$$730$$ 0 0
$$731$$ −24.3291 −0.899843
$$732$$ 0 0
$$733$$ − 6.78664i − 0.250670i −0.992114 0.125335i $$-0.959999\pi$$
0.992114 0.125335i $$-0.0400006\pi$$
$$734$$ −9.51144 −0.351074
$$735$$ 0 0
$$736$$ −20.4533 −0.753919
$$737$$ 6.06261i 0.223319i
$$738$$ 0 0
$$739$$ −28.7245 −1.05665 −0.528324 0.849043i $$-0.677179\pi$$
−0.528324 + 0.849043i $$0.677179\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 11.4445i 0.420142i
$$743$$ − 31.4523i − 1.15387i −0.816789 0.576937i $$-0.804248\pi$$
0.816789 0.576937i $$-0.195752\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ −9.29610 −0.340354
$$747$$ 0 0
$$748$$ 34.7141i 1.26927i
$$749$$ 37.2537 1.36122
$$750$$ 0 0
$$751$$ 10.9532 0.399687 0.199844 0.979828i $$-0.435957\pi$$
0.199844 + 0.979828i $$0.435957\pi$$
$$752$$ 30.9305i 1.12792i
$$753$$ 0 0
$$754$$ −1.70161 −0.0619689
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 45.7942i 1.66442i 0.554461 + 0.832210i $$0.312925\pi$$
−0.554461 + 0.832210i $$0.687075\pi$$
$$758$$ − 3.75868i − 0.136521i
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −33.9138 −1.22937 −0.614687 0.788771i $$-0.710717\pi$$
−0.614687 + 0.788771i $$0.710717\pi$$
$$762$$ 0 0
$$763$$ 4.87528i 0.176497i
$$764$$ 24.3894 0.882378
$$765$$ 0 0
$$766$$ −14.7602 −0.533309
$$767$$ − 21.7833i − 0.786551i
$$768$$ 0 0
$$769$$ −7.15972 −0.258186 −0.129093 0.991632i $$-0.541207\pi$$
−0.129093 + 0.991632i $$0.541207\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ − 0.856710i − 0.0308337i
$$773$$ − 14.5998i − 0.525117i −0.964916 0.262558i $$-0.915434\pi$$
0.964916 0.262558i $$-0.0845663\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ −12.8787 −0.462318
$$777$$ 0 0
$$778$$ − 14.8836i − 0.533602i
$$779$$ 0.973866 0.0348924
$$780$$ 0 0
$$781$$ 39.6954 1.42041
$$782$$ 6.31760i 0.225917i
$$783$$ 0 0
$$784$$ 1.16578 0.0416348
$$785$$ 0 0
$$786$$ 0 0
$$787$$ − 18.4728i − 0.658483i −0.944246 0.329242i $$-0.893207\pi$$
0.944246 0.329242i $$-0.106793\pi$$
$$788$$ − 9.83011i − 0.350183i
$$789$$ 0 0
$$790$$ 0 0
$$791$$ −16.8630 −0.599578
$$792$$ 0 0
$$793$$ − 0.0234407i 0 0.000832405i
$$794$$ −8.38484 −0.297567
$$795$$ 0 0
$$796$$ −31.0077 −1.09904
$$797$$ − 41.0374i − 1.45362i −0.686838 0.726810i $$-0.741002\pi$$
0.686838 0.726810i $$-0.258998\pi$$
$$798$$ 0 0
$$799$$ 36.2136 1.28115
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 3.38021i 0.119359i
$$803$$ − 40.8033i − 1.43992i
$$804$$ 0 0
$$805$$ 0 0
$$806$$ −8.33371 −0.293543
$$807$$ 0 0
$$808$$ − 12.4572i − 0.438244i
$$809$$ −7.19375 −0.252919 −0.126459 0.991972i $$-0.540361\pi$$
−0.126459 + 0.991972i $$0.540361\pi$$
$$810$$ 0 0
$$811$$ 38.2183 1.34203 0.671014 0.741445i $$-0.265859\pi$$
0.671014 + 0.741445i $$0.265859\pi$$
$$812$$ 7.67583i 0.269369i
$$813$$ 0 0
$$814$$ 10.6051 0.371710
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 2.73547i 0.0957021i
$$818$$ 11.7139i 0.409566i
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −0.668560 −0.0233329 −0.0116665 0.999932i $$-0.503714\pi$$
−0.0116665 + 0.999932i $$0.503714\pi$$
$$822$$ 0 0
$$823$$ − 1.42033i − 0.0495096i −0.999694 0.0247548i $$-0.992119\pi$$
0.999694 0.0247548i $$-0.00788051\pi$$
$$824$$ −10.9279 −0.380690
$$825$$ 0 0
$$826$$ 12.3917 0.431161
$$827$$ − 49.8169i − 1.73230i −0.499782 0.866152i $$-0.666586\pi$$
0.499782 0.866152i $$-0.333414\pi$$
$$828$$ 0 0
$$829$$ −36.4150 −1.26475 −0.632373 0.774664i $$-0.717919\pi$$
−0.632373 + 0.774664i $$0.717919\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 6.64235i 0.230282i
$$833$$ − 1.36490i − 0.0472909i
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 3.90312 0.134992
$$837$$ 0 0
$$838$$ − 5.04447i − 0.174258i
$$839$$ 20.0890 0.693549 0.346774 0.937949i $$-0.387277\pi$$
0.346774 + 0.937949i $$0.387277\pi$$
$$840$$ 0 0
$$841$$ −26.1566 −0.901953
$$842$$ 3.87056i 0.133388i
$$843$$ 0 0
$$844$$ −2.90675 −0.100055
$$845$$ 0 0
$$846$$ 0 0
$$847$$ − 69.3349i − 2.38238i
$$848$$ − 25.5345i − 0.876860i
$$849$$ 0 0
$$850$$ 0 0
$$851$$ −15.3046 −0.524637
$$852$$ 0 0
$$853$$ 26.9084i 0.921326i 0.887575 + 0.460663i $$0.152388\pi$$
−0.887575 + 0.460663i $$0.847612\pi$$
$$854$$ 0.0133345 0.000456296 0
$$855$$ 0 0
$$856$$ 25.9742 0.887779
$$857$$ 48.8408i 1.66837i 0.551486 + 0.834184i $$0.314061\pi$$
−0.551486 + 0.834184i $$0.685939\pi$$
$$858$$ 0 0
$$859$$ 41.4094 1.41287 0.706435 0.707778i $$-0.250302\pi$$
0.706435 + 0.707778i $$0.250302\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ − 0.791507i − 0.0269588i
$$863$$ − 50.8101i − 1.72960i −0.502119 0.864799i $$-0.667446\pi$$
0.502119 0.864799i $$-0.332554\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ −4.71359 −0.160174
$$867$$ 0 0
$$868$$ 37.5928i 1.27598i
$$869$$ −58.4252 −1.98194
$$870$$ 0 0
$$871$$ 2.09566 0.0710087
$$872$$ 3.39916i 0.115110i
$$873$$ 0 0
$$874$$ 0.710328 0.0240272
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 0.409589i 0.0138308i 0.999976 + 0.00691542i $$0.00220127\pi$$
−0.999976 + 0.00691542i $$0.997799\pi$$
$$878$$ 6.06512i 0.204688i
$$879$$ 0 0
$$880$$ 0 0
$$881$$ −5.32851 −0.179522 −0.0897610 0.995963i $$-0.528610\pi$$
−0.0897610 + 0.995963i $$0.528610\pi$$
$$882$$ 0 0
$$883$$ − 14.2064i − 0.478083i −0.971009 0.239042i $$-0.923167\pi$$
0.971009 0.239042i $$-0.0768333\pi$$
$$884$$ 11.9996 0.403590
$$885$$ 0 0
$$886$$ −3.66964 −0.123284
$$887$$ 7.23193i 0.242825i 0.992602 + 0.121412i $$0.0387423\pi$$
−0.992602 + 0.121412i $$0.961258\pi$$
$$888$$ 0 0
$$889$$ 23.7298 0.795871
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 13.7604i 0.460733i
$$893$$ − 4.07173i − 0.136255i
$$894$$ 0 0
$$895$$ 0 0
$$896$$ −28.6650 −0.957629
$$897$$ 0 0
$$898$$ − 15.7784i − 0.526531i
$$899$$ 13.9256 0.464444
$$900$$ 0 0
$$901$$ −29.8960 −0.995981
$$902$$ − 7.98006i − 0.265707i
$$903$$ 0 0
$$904$$ −11.7573 −0.391041
$$905$$ 0 0
$$906$$ 0 0
$$907$$ − 38.8101i − 1.28867i −0.764743 0.644335i $$-0.777134\pi$$
0.764743 0.644335i $$-0.222866\pi$$
$$908$$ 19.9986i 0.663677i
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −40.2781 −1.33447 −0.667236 0.744846i $$-0.732523\pi$$
−0.667236 + 0.744846i $$0.732523\pi$$
$$912$$ 0 0
$$913$$ 64.2754i 2.12721i
$$914$$ 18.0874 0.598279
$$915$$ 0 0
$$916$$ 18.6085 0.614842
$$917$$ − 0.690479i − 0.0228016i
$$918$$ 0 0
$$919$$ 13.0468 0.430375 0.215187 0.976573i $$-0.430964\pi$$
0.215187 + 0.976573i $$0.430964\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ − 14.8144i − 0.487888i
$$923$$ − 13.7215i − 0.451648i
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 5.71936 0.187950
$$927$$ 0 0
$$928$$ 8.18638i 0.268731i
$$929$$ −0.293825 −0.00964008 −0.00482004 0.999988i $$-0.501534\pi$$
−0.00482004 + 0.999988i $$0.501534\pi$$
$$930$$ 0 0
$$931$$ −0.153464 −0.00502959
$$932$$ − 5.15428i − 0.168834i
$$933$$ 0 0
$$934$$ 3.59894 0.117761
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 16.9141i 0.552559i 0.961077 + 0.276280i $$0.0891016\pi$$
−0.961077 + 0.276280i $$0.910898\pi$$
$$938$$ 1.19213i 0.0389246i
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 57.2093 1.86497 0.932485 0.361209i $$-0.117636\pi$$
0.932485 + 0.361209i $$0.117636\pi$$
$$942$$ 0 0
$$943$$ 11.5163i 0.375023i
$$944$$ −27.6478 −0.899858
$$945$$ 0 0
$$946$$ 22.4150 0.728775
$$947$$ 38.8746i 1.26325i 0.775272 + 0.631627i $$0.217613\pi$$
−0.775272 + 0.631627i $$0.782387\pi$$
$$948$$ 0 0
$$949$$ −14.1044 −0.457850
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 14.5130i 0.470368i
$$953$$ − 54.4516i − 1.76386i −0.471381 0.881930i $$-0.656244\pi$$
0.471381 0.881930i $$-0.343756\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ −29.0460 −0.939415
$$957$$ 0 0
$$958$$ − 15.3729i − 0.496675i
$$959$$ −8.90965 −0.287707
$$960$$ 0 0
$$961$$ 37.2012 1.20004
$$962$$ − 3.66587i − 0.118192i
$$963$$ 0 0
$$964$$ 31.1229 1.00240
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 19.5701i 0.629333i 0.949202 + 0.314666i $$0.101893\pi$$
−0.949202 + 0.314666i $$0.898107\pi$$
$$968$$ − 48.3420i − 1.55377i
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 6.31009 0.202500 0.101250 0.994861i $$-0.467716\pi$$
0.101250 + 0.994861i $$0.467716\pi$$
$$972$$ 0 0
$$973$$ 37.7995i 1.21180i
$$974$$ 2.11129 0.0676500
$$975$$ 0 0
$$976$$ −0.0297513 −0.000952317 0
$$977$$ − 7.41911i − 0.237358i −0.992933 0.118679i $$-0.962134\pi$$
0.992933 0.118679i $$-0.0378660\pi$$
$$978$$ 0 0
$$979$$ −38.6734 −1.23601
$$980$$ 0 0
$$981$$ 0 0
$$982$$ − 15.5426i − 0.495986i
$$983$$ − 10.9827i − 0.350295i −0.984542 0.175148i $$-0.943960\pi$$
0.984542 0.175148i $$-0.0560403\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 2.52860 0.0805270
$$987$$ 0 0
$$988$$ − 1.34919i − 0.0429234i
$$989$$ −32.3479 −1.02860
$$990$$ 0 0
$$991$$ −21.3721 −0.678908 −0.339454 0.940623i $$-0.610242\pi$$
−0.339454 + 0.940623i $$0.610242\pi$$
$$992$$ 40.0932i 1.27296i
$$993$$ 0 0
$$994$$ 7.80559 0.247578
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 30.5348i 0.967047i 0.875331 + 0.483524i $$0.160643\pi$$
−0.875331 + 0.483524i $$0.839357\pi$$
$$998$$ − 16.1863i − 0.512368i
$$999$$ 0 0
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 2025.2.b.n.649.4 8
3.2 odd 2 2025.2.b.o.649.5 8
5.2 odd 4 2025.2.a.q.1.3 4
5.3 odd 4 2025.2.a.y.1.2 4
5.4 even 2 inner 2025.2.b.n.649.5 8
9.2 odd 6 675.2.k.c.199.5 16
9.4 even 3 225.2.k.c.124.5 16
9.5 odd 6 675.2.k.c.424.4 16
9.7 even 3 225.2.k.c.49.4 16
15.2 even 4 2025.2.a.z.1.2 4
15.8 even 4 2025.2.a.p.1.3 4
15.14 odd 2 2025.2.b.o.649.4 8
45.2 even 12 675.2.e.c.226.3 8
45.4 even 6 225.2.k.c.124.4 16
45.7 odd 12 225.2.e.e.76.2 yes 8
45.13 odd 12 225.2.e.c.151.3 yes 8
45.14 odd 6 675.2.k.c.424.5 16
45.22 odd 12 225.2.e.e.151.2 yes 8
45.23 even 12 675.2.e.e.451.2 8
45.29 odd 6 675.2.k.c.199.4 16
45.32 even 12 675.2.e.c.451.3 8
45.34 even 6 225.2.k.c.49.5 16
45.38 even 12 675.2.e.e.226.2 8
45.43 odd 12 225.2.e.c.76.3 8

By twisted newform
Twist Min Dim Char Parity Ord Type
225.2.e.c.76.3 8 45.43 odd 12
225.2.e.c.151.3 yes 8 45.13 odd 12
225.2.e.e.76.2 yes 8 45.7 odd 12
225.2.e.e.151.2 yes 8 45.22 odd 12
225.2.k.c.49.4 16 9.7 even 3
225.2.k.c.49.5 16 45.34 even 6
225.2.k.c.124.4 16 45.4 even 6
225.2.k.c.124.5 16 9.4 even 3
675.2.e.c.226.3 8 45.2 even 12
675.2.e.c.451.3 8 45.32 even 12
675.2.e.e.226.2 8 45.38 even 12
675.2.e.e.451.2 8 45.23 even 12
675.2.k.c.199.4 16 45.29 odd 6
675.2.k.c.199.5 16 9.2 odd 6
675.2.k.c.424.4 16 9.5 odd 6
675.2.k.c.424.5 16 45.14 odd 6
2025.2.a.p.1.3 4 15.8 even 4
2025.2.a.q.1.3 4 5.2 odd 4
2025.2.a.y.1.2 4 5.3 odd 4
2025.2.a.z.1.2 4 15.2 even 4
2025.2.b.n.649.4 8 1.1 even 1 trivial
2025.2.b.n.649.5 8 5.4 even 2 inner
2025.2.b.o.649.4 8 15.14 odd 2
2025.2.b.o.649.5 8 3.2 odd 2