# Properties

 Label 2025.2.a.p.1.3 Level $2025$ Weight $2$ Character 2025.1 Self dual yes Analytic conductor $16.170$ Analytic rank $1$ Dimension $4$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2025,2,Mod(1,2025)]

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, 0]))

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

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

chi := DirichletCharacter("2025.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2025 = 3^{4} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2025.a (trivial)

## 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: yes Analytic conductor: $$16.1697064093$$ Analytic rank: $$1$$ Dimension: $$4$$ Coefficient field: 4.4.11661.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - 2x^{3} - 4x^{2} + 5x + 3$$ x^4 - 2*x^3 - 4*x^2 + 5*x + 3 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 225) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.3 Root $$1.47325$$ of defining polynomial Character $$\chi$$ $$=$$ 2025.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+0.473255 q^{2} -1.77603 q^{4} -2.56305 q^{7} -1.78702 q^{8} +O(q^{10})$$ $$q+0.473255 q^{2} -1.77603 q^{4} -2.56305 q^{7} -1.78702 q^{8} +6.16860 q^{11} -2.13230 q^{13} -1.21298 q^{14} +2.70634 q^{16} +3.16860 q^{17} +0.356267 q^{19} +2.91932 q^{22} -4.21298 q^{23} -1.00912 q^{26} +4.55206 q^{28} -1.68623 q^{29} -8.25840 q^{31} +4.85484 q^{32} +1.49956 q^{34} -3.63274 q^{37} +0.168605 q^{38} +2.73353 q^{41} +7.67817 q^{43} -10.9556 q^{44} -1.99381 q^{46} -11.4289 q^{47} -0.430757 q^{49} +3.78702 q^{52} -9.43507 q^{53} +4.58024 q^{56} -0.798017 q^{58} -10.2159 q^{59} -0.0109932 q^{61} -3.90833 q^{62} -3.11511 q^{64} -0.982817 q^{67} -5.62754 q^{68} +6.43507 q^{71} -6.61467 q^{73} -1.71921 q^{74} -0.632740 q^{76} -15.8105 q^{77} -9.47138 q^{79} +1.29366 q^{82} -10.4198 q^{83} +3.63373 q^{86} -11.0234 q^{88} +6.26940 q^{89} +5.46519 q^{91} +7.48237 q^{92} -5.40877 q^{94} -7.20679 q^{97} -0.203858 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 2 q^{2} + 4 q^{4} - q^{7} - 9 q^{8}+O(q^{10})$$ 4 * q - 2 * q^2 + 4 * q^4 - q^7 - 9 * q^8 $$4 q - 2 q^{2} + 4 q^{4} - q^{7} - 9 q^{8} + q^{11} + 2 q^{13} - 3 q^{14} + 4 q^{16} - 11 q^{17} + 2 q^{19} + 3 q^{22} - 15 q^{23} + 10 q^{26} - 4 q^{28} - q^{29} - 4 q^{31} - 10 q^{32} + 9 q^{34} - q^{37} - 23 q^{38} + 5 q^{41} - 10 q^{43} - 22 q^{44} - 20 q^{47} - 3 q^{49} + 17 q^{52} - 20 q^{53} + 30 q^{56} - 18 q^{58} - 17 q^{59} - 13 q^{61} + 6 q^{62} + 19 q^{64} + 17 q^{67} - 34 q^{68} + 8 q^{71} + 2 q^{73} - 40 q^{74} + 11 q^{76} - 12 q^{77} - 7 q^{79} + 12 q^{82} - 30 q^{83} + 34 q^{86} + 9 q^{88} + 9 q^{89} - 17 q^{91} + 12 q^{92} + 3 q^{94} - 19 q^{97} - 13 q^{98}+O(q^{100})$$ 4 * q - 2 * q^2 + 4 * q^4 - q^7 - 9 * q^8 + q^11 + 2 * q^13 - 3 * q^14 + 4 * q^16 - 11 * q^17 + 2 * q^19 + 3 * q^22 - 15 * q^23 + 10 * q^26 - 4 * q^28 - q^29 - 4 * q^31 - 10 * q^32 + 9 * q^34 - q^37 - 23 * q^38 + 5 * q^41 - 10 * q^43 - 22 * q^44 - 20 * q^47 - 3 * q^49 + 17 * q^52 - 20 * q^53 + 30 * q^56 - 18 * q^58 - 17 * q^59 - 13 * q^61 + 6 * q^62 + 19 * q^64 + 17 * q^67 - 34 * q^68 + 8 * q^71 + 2 * q^73 - 40 * q^74 + 11 * q^76 - 12 * q^77 - 7 * q^79 + 12 * q^82 - 30 * q^83 + 34 * q^86 + 9 * q^88 + 9 * q^89 - 17 * q^91 + 12 * q^92 + 3 * q^94 - 19 * q^97 - 13 * q^98

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0.473255 0.334641 0.167321 0.985903i $$-0.446488\pi$$
0.167321 + 0.985903i $$0.446488\pi$$
$$3$$ 0 0
$$4$$ −1.77603 −0.888015
$$5$$ 0 0
$$6$$ 0 0
$$7$$ −2.56305 −0.968743 −0.484372 0.874862i $$-0.660952\pi$$
−0.484372 + 0.874862i $$0.660952\pi$$
$$8$$ −1.78702 −0.631808
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 6.16860 1.85990 0.929952 0.367681i $$-0.119848\pi$$
0.929952 + 0.367681i $$0.119848\pi$$
$$12$$ 0 0
$$13$$ −2.13230 −0.591393 −0.295696 0.955282i $$-0.595552\pi$$
−0.295696 + 0.955282i $$0.595552\pi$$
$$14$$ −1.21298 −0.324182
$$15$$ 0 0
$$16$$ 2.70634 0.676586
$$17$$ 3.16860 0.768500 0.384250 0.923229i $$-0.374460\pi$$
0.384250 + 0.923229i $$0.374460\pi$$
$$18$$ 0 0
$$19$$ 0.356267 0.0817332 0.0408666 0.999165i $$-0.486988\pi$$
0.0408666 + 0.999165i $$0.486988\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 2.91932 0.622401
$$23$$ −4.21298 −0.878466 −0.439233 0.898373i $$-0.644750\pi$$
−0.439233 + 0.898373i $$0.644750\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ −1.00912 −0.197905
$$27$$ 0 0
$$28$$ 4.55206 0.860259
$$29$$ −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 $$0.765945\pi$$
$$32$$ 4.85484 0.858222
$$33$$ 0 0
$$34$$ 1.49956 0.257172
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −3.63274 −0.597219 −0.298609 0.954375i $$-0.596523\pi$$
−0.298609 + 0.954375i $$0.596523\pi$$
$$38$$ 0.168605 0.0273513
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 2.73353 0.426906 0.213453 0.976953i $$-0.431529\pi$$
0.213453 + 0.976953i $$0.431529\pi$$
$$42$$ 0 0
$$43$$ 7.67817 1.17091 0.585455 0.810705i $$-0.300916\pi$$
0.585455 + 0.810705i $$0.300916\pi$$
$$44$$ −10.9556 −1.65162
$$45$$ 0 0
$$46$$ −1.99381 −0.293971
$$47$$ −11.4289 −1.66707 −0.833537 0.552464i $$-0.813688\pi$$
−0.833537 + 0.552464i $$0.813688\pi$$
$$48$$ 0 0
$$49$$ −0.430757 −0.0615367
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 3.78702 0.525166
$$53$$ −9.43507 −1.29601 −0.648003 0.761637i $$-0.724396\pi$$
−0.648003 + 0.761637i $$0.724396\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 4.58024 0.612060
$$57$$ 0 0
$$58$$ −0.798017 −0.104785
$$59$$ −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 $$0.500224\pi$$
$$62$$ −3.90833 −0.496358
$$63$$ 0 0
$$64$$ −3.11511 −0.389389
$$65$$ 0 0
$$66$$ 0 0
$$67$$ −0.982817 −0.120070 −0.0600351 0.998196i $$-0.519121\pi$$
−0.0600351 + 0.998196i $$0.519121\pi$$
$$68$$ −5.62754 −0.682439
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 6.43507 0.763703 0.381851 0.924224i $$-0.375287\pi$$
0.381851 + 0.924224i $$0.375287\pi$$
$$72$$ 0 0
$$73$$ −6.61467 −0.774189 −0.387094 0.922040i $$-0.626521\pi$$
−0.387094 + 0.922040i $$0.626521\pi$$
$$74$$ −1.71921 −0.199854
$$75$$ 0 0
$$76$$ −0.632740 −0.0725803
$$77$$ −15.8105 −1.80177
$$78$$ 0 0
$$79$$ −9.47138 −1.06561 −0.532807 0.846237i $$-0.678863\pi$$
−0.532807 + 0.846237i $$0.678863\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 1.29366 0.142860
$$83$$ −10.4198 −1.14372 −0.571859 0.820352i $$-0.693777\pi$$
−0.571859 + 0.820352i $$0.693777\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 3.63373 0.391835
$$87$$ 0 0
$$88$$ −11.0234 −1.17510
$$89$$ 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
$$92$$ 7.48237 0.780091
$$93$$ 0 0
$$94$$ −5.40877 −0.557872
$$95$$ 0 0
$$96$$ 0 0
$$97$$ −7.20679 −0.731738 −0.365869 0.930666i $$-0.619228\pi$$
−0.365869 + 0.930666i $$0.619228\pi$$
$$98$$ −0.203858 −0.0205927
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −6.97094 −0.693634 −0.346817 0.937933i $$-0.612738\pi$$
−0.346817 + 0.937933i $$0.612738\pi$$
$$102$$ 0 0
$$103$$ 6.11511 0.602540 0.301270 0.953539i $$-0.402589\pi$$
0.301270 + 0.953539i $$0.402589\pi$$
$$104$$ 3.81046 0.373647
$$105$$ 0 0
$$106$$ −4.46519 −0.433698
$$107$$ −14.5349 −1.40514 −0.702570 0.711615i $$-0.747964\pi$$
−0.702570 + 0.711615i $$0.747964\pi$$
$$108$$ 0 0
$$109$$ 1.90214 0.182192 0.0910958 0.995842i $$-0.470963\pi$$
0.0910958 + 0.995842i $$0.470963\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ −6.93650 −0.655438
$$113$$ −6.57925 −0.618924 −0.309462 0.950912i $$-0.600149\pi$$
−0.309462 + 0.950912i $$0.600149\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 2.99480 0.278060
$$117$$ 0 0
$$118$$ −4.83472 −0.445072
$$119$$ −8.12130 −0.744479
$$120$$ 0 0
$$121$$ 27.0517 2.45924
$$122$$ −0.00520257 −0.000471019 0
$$123$$ 0 0
$$124$$ 14.6672 1.31715
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 9.25840 0.821550 0.410775 0.911737i $$-0.365258\pi$$
0.410775 + 0.911737i $$0.365258\pi$$
$$128$$ −11.1839 −0.988528
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −0.269397 −0.0235373 −0.0117687 0.999931i $$-0.503746\pi$$
−0.0117687 + 0.999931i $$0.503746\pi$$
$$132$$ 0 0
$$133$$ −0.913130 −0.0791784
$$134$$ −0.465123 −0.0401805
$$135$$ 0 0
$$136$$ −5.66237 −0.485544
$$137$$ 3.47618 0.296990 0.148495 0.988913i $$-0.452557\pi$$
0.148495 + 0.988913i $$0.452557\pi$$
$$138$$ 0 0
$$139$$ 14.7479 1.25090 0.625448 0.780266i $$-0.284916\pi$$
0.625448 + 0.780266i $$0.284916\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 3.04543 0.255567
$$143$$ −13.1533 −1.09993
$$144$$ 0 0
$$145$$ 0 0
$$146$$ −3.13042 −0.259076
$$147$$ 0 0
$$148$$ 6.45186 0.530339
$$149$$ 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 $$0.637886\pi$$
$$152$$ −0.636657 −0.0516397
$$153$$ 0 0
$$154$$ −7.48237 −0.602947
$$155$$ 0 0
$$156$$ 0 0
$$157$$ −1.06261 −0.0848055 −0.0424028 0.999101i $$-0.513501\pi$$
−0.0424028 + 0.999101i $$0.513501\pi$$
$$158$$ −4.48237 −0.356598
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 10.7981 0.851008
$$162$$ 0 0
$$163$$ 17.1386 1.34240 0.671198 0.741278i $$-0.265780\pi$$
0.671198 + 0.741278i $$0.265780\pi$$
$$164$$ −4.85484 −0.379099
$$165$$ 0 0
$$166$$ −4.93120 −0.382735
$$167$$ −4.37345 −0.338428 −0.169214 0.985579i $$-0.554123\pi$$
−0.169214 + 0.985579i $$0.554123\pi$$
$$168$$ 0 0
$$169$$ −8.45331 −0.650255
$$170$$ 0 0
$$171$$ 0 0
$$172$$ −13.6367 −1.03979
$$173$$ −14.6601 −1.11459 −0.557293 0.830316i $$-0.688160\pi$$
−0.557293 + 0.830316i $$0.688160\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 16.6944 1.25838
$$177$$ 0 0
$$178$$ 2.96702 0.222388
$$179$$ 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 $$0.633680\pi$$
$$182$$ 2.58643 0.191719
$$183$$ 0 0
$$184$$ 7.52869 0.555022
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 19.5459 1.42934
$$188$$ 20.2980 1.48039
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −13.7325 −0.993652 −0.496826 0.867850i $$-0.665501\pi$$
−0.496826 + 0.867850i $$0.665501\pi$$
$$192$$ 0 0
$$193$$ 0.482374 0.0347220 0.0173610 0.999849i $$-0.494474\pi$$
0.0173610 + 0.999849i $$0.494474\pi$$
$$194$$ −3.41064 −0.244870
$$195$$ 0 0
$$196$$ 0.765037 0.0546455
$$197$$ 5.53488 0.394344 0.197172 0.980369i $$-0.436824\pi$$
0.197172 + 0.980369i $$0.436824\pi$$
$$198$$ 0 0
$$199$$ 17.4590 1.23764 0.618818 0.785534i $$-0.287612\pi$$
0.618818 + 0.785534i $$0.287612\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ −3.29903 −0.232119
$$203$$ 4.32190 0.303338
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 2.89401 0.201635
$$207$$ 0 0
$$208$$ −5.77073 −0.400128
$$209$$ 2.19767 0.152016
$$210$$ 0 0
$$211$$ −1.63666 −0.112672 −0.0563360 0.998412i $$-0.517942\pi$$
−0.0563360 + 0.998412i $$0.517942\pi$$
$$212$$ 16.7570 1.15087
$$213$$ 0 0
$$214$$ −6.87870 −0.470218
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 21.1667 1.43689
$$218$$ 0.900195 0.0609689
$$219$$ 0 0
$$220$$ 0 0
$$221$$ −6.75641 −0.454485
$$222$$ 0 0
$$223$$ −7.74785 −0.518835 −0.259417 0.965765i $$-0.583531\pi$$
−0.259417 + 0.965765i $$0.583531\pi$$
$$224$$ −12.4432 −0.831397
$$225$$ 0 0
$$226$$ −3.11366 −0.207118
$$227$$ −11.2603 −0.747371 −0.373685 0.927555i $$-0.621906\pi$$
−0.373685 + 0.927555i $$0.621906\pi$$
$$228$$ 0 0
$$229$$ −10.4776 −0.692377 −0.346189 0.938165i $$-0.612524\pi$$
−0.346189 + 0.938165i $$0.612524\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 3.01333 0.197835
$$233$$ −2.90214 −0.190125 −0.0950627 0.995471i $$-0.530305\pi$$
−0.0950627 + 0.995471i $$0.530305\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 18.1438 1.18106
$$237$$ 0 0
$$238$$ −3.84344 −0.249133
$$239$$ −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 $$0.309105\pi$$
$$242$$ 12.8023 0.822965
$$243$$ 0 0
$$244$$ 0.0195242 0.00124991
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −0.759666 −0.0483364
$$248$$ 14.7580 0.937131
$$249$$ 0 0
$$250$$ 0 0
$$251$$ −8.46999 −0.534621 −0.267311 0.963610i $$-0.586135\pi$$
−0.267311 + 0.963610i $$0.586135\pi$$
$$252$$ 0 0
$$253$$ −25.9882 −1.63386
$$254$$ 4.38158 0.274925
$$255$$ 0 0
$$256$$ 0.937390 0.0585869
$$257$$ −2.87251 −0.179182 −0.0895910 0.995979i $$-0.528556\pi$$
−0.0895910 + 0.995979i $$0.528556\pi$$
$$258$$ 0 0
$$259$$ 9.31091 0.578552
$$260$$ 0 0
$$261$$ 0 0
$$262$$ −0.127493 −0.00787656
$$263$$ 25.6239 1.58003 0.790017 0.613084i $$-0.210071\pi$$
0.790017 + 0.613084i $$0.210071\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ −0.432143 −0.0264964
$$267$$ 0 0
$$268$$ 1.74551 0.106624
$$269$$ −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 $$0.272318\pi$$
$$272$$ 8.57533 0.519956
$$273$$ 0 0
$$274$$ 1.64512 0.0993853
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −24.1338 −1.45006 −0.725028 0.688719i $$-0.758173\pi$$
−0.725028 + 0.688719i $$0.758173\pi$$
$$278$$ 6.97949 0.418602
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −3.36726 −0.200874 −0.100437 0.994943i $$-0.532024\pi$$
−0.100437 + 0.994943i $$0.532024\pi$$
$$282$$ 0 0
$$283$$ −21.8497 −1.29883 −0.649415 0.760434i $$-0.724986\pi$$
−0.649415 + 0.760434i $$0.724986\pi$$
$$284$$ −11.4289 −0.678179
$$285$$ 0 0
$$286$$ −6.22486 −0.368083
$$287$$ −7.00619 −0.413562
$$288$$ 0 0
$$289$$ −6.95994 −0.409408
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 11.7479 0.687491
$$293$$ 13.7540 0.803520 0.401760 0.915745i $$-0.368399\pi$$
0.401760 + 0.915745i $$0.368399\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 6.49179 0.377328
$$297$$ 0 0
$$298$$ 4.80509 0.278352
$$299$$ 8.98332 0.519519
$$300$$ 0 0
$$301$$ −19.6796 −1.13431
$$302$$ −4.88219 −0.280939
$$303$$ 0 0
$$304$$ 0.964180 0.0552995
$$305$$ 0 0
$$306$$ 0 0
$$307$$ −34.2183 −1.95294 −0.976472 0.215644i $$-0.930815\pi$$
−0.976472 + 0.215644i $$0.930815\pi$$
$$308$$ 28.0799 1.60000
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −23.0397 −1.30646 −0.653232 0.757158i $$-0.726587\pi$$
−0.653232 + 0.757158i $$0.726587\pi$$
$$312$$ 0 0
$$313$$ 3.59130 0.202992 0.101496 0.994836i $$-0.467637\pi$$
0.101496 + 0.994836i $$0.467637\pi$$
$$314$$ −0.502885 −0.0283794
$$315$$ 0 0
$$316$$ 16.8215 0.946281
$$317$$ 13.1807 0.740299 0.370150 0.928972i $$-0.379306\pi$$
0.370150 + 0.928972i $$0.379306\pi$$
$$318$$ 0 0
$$319$$ −10.4017 −0.582383
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 5.11024 0.284783
$$323$$ 1.12887 0.0628119
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 8.11090 0.449221
$$327$$ 0 0
$$328$$ −4.88489 −0.269723
$$329$$ 29.2928 1.61497
$$330$$ 0 0
$$331$$ 1.18253 0.0649976 0.0324988 0.999472i $$-0.489653\pi$$
0.0324988 + 0.999472i $$0.489653\pi$$
$$332$$ 18.5058 1.01564
$$333$$ 0 0
$$334$$ −2.06975 −0.113252
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 24.7995 1.35091 0.675457 0.737400i $$-0.263947\pi$$
0.675457 + 0.737400i $$0.263947\pi$$
$$338$$ −4.00057 −0.217602
$$339$$ 0 0
$$340$$ 0 0
$$341$$ −50.9428 −2.75871
$$342$$ 0 0
$$343$$ 19.0454 1.02836
$$344$$ −13.7211 −0.739790
$$345$$ 0 0
$$346$$ −6.93796 −0.372987
$$347$$ 22.1692 1.19010 0.595052 0.803687i $$-0.297132\pi$$
0.595052 + 0.803687i $$0.297132\pi$$
$$348$$ 0 0
$$349$$ 14.9185 0.798569 0.399285 0.916827i $$-0.369259\pi$$
0.399285 + 0.916827i $$0.369259\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 29.9476 1.59621
$$353$$ 16.9145 0.900269 0.450134 0.892961i $$-0.351376\pi$$
0.450134 + 0.892961i $$0.351376\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ −11.1346 −0.590135
$$357$$ 0 0
$$358$$ 3.25133 0.171838
$$359$$ −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
$$362$$ −5.19205 −0.272888
$$363$$ 0 0
$$364$$ −9.70634 −0.508751
$$365$$ 0 0
$$366$$ 0 0
$$367$$ −20.0979 −1.04910 −0.524552 0.851379i $$-0.675767\pi$$
−0.524552 + 0.851379i $$0.675767\pi$$
$$368$$ −11.4018 −0.594358
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 24.1826 1.25550
$$372$$ 0 0
$$373$$ 19.6429 1.01707 0.508536 0.861041i $$-0.330187\pi$$
0.508536 + 0.861041i $$0.330187\pi$$
$$374$$ 9.25017 0.478315
$$375$$ 0 0
$$376$$ 20.4237 1.05327
$$377$$ 3.59555 0.185180
$$378$$ 0 0
$$379$$ −7.94219 −0.407963 −0.203982 0.978975i $$-0.565388\pi$$
−0.203982 + 0.978975i $$0.565388\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ −6.49899 −0.332517
$$383$$ −31.1888 −1.59367 −0.796836 0.604195i $$-0.793495\pi$$
−0.796836 + 0.604195i $$0.793495\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0.228285 0.0116194
$$387$$ 0 0
$$388$$ 12.7995 0.649795
$$389$$ 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
$$392$$ 0.769772 0.0388794
$$393$$ 0 0
$$394$$ 2.61941 0.131964
$$395$$ 0 0
$$396$$ 0 0
$$397$$ −17.7174 −0.889211 −0.444606 0.895726i $$-0.646656\pi$$
−0.444606 + 0.895726i $$0.646656\pi$$
$$398$$ 8.26255 0.414164
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 7.14247 0.356678 0.178339 0.983969i $$-0.442928\pi$$
0.178339 + 0.983969i $$0.442928\pi$$
$$402$$ 0 0
$$403$$ 17.6094 0.877185
$$404$$ 12.3806 0.615958
$$405$$ 0 0
$$406$$ 2.04536 0.101509
$$407$$ −22.4089 −1.11077
$$408$$ 0 0
$$409$$ 24.7518 1.22390 0.611948 0.790898i $$-0.290386\pi$$
0.611948 + 0.790898i $$0.290386\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ −10.8606 −0.535065
$$413$$ 26.1839 1.28843
$$414$$ 0 0
$$415$$ 0 0
$$416$$ −10.3520 −0.507546
$$417$$ 0 0
$$418$$ 1.04006 0.0508708
$$419$$ 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 $$0.563867\pi$$
$$422$$ −0.774555 −0.0377048
$$423$$ 0 0
$$424$$ 16.8607 0.818828
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 0.0281761 0.00136354
$$428$$ 25.8144 1.24779
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −1.67248 −0.0805604 −0.0402802 0.999188i $$-0.512825\pi$$
−0.0402802 + 0.999188i $$0.512825\pi$$
$$432$$ 0 0
$$433$$ 9.95994 0.478644 0.239322 0.970940i $$-0.423075\pi$$
0.239322 + 0.970940i $$0.423075\pi$$
$$434$$ 10.0173 0.480843
$$435$$ 0 0
$$436$$ −3.37825 −0.161789
$$437$$ −1.50094 −0.0717998
$$438$$ 0 0
$$439$$ 12.8158 0.611663 0.305832 0.952086i $$-0.401066\pi$$
0.305832 + 0.952086i $$0.401066\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ −3.19750 −0.152090
$$443$$ −7.75404 −0.368406 −0.184203 0.982888i $$-0.558970\pi$$
−0.184203 + 0.982888i $$0.558970\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ −3.66671 −0.173624
$$447$$ 0 0
$$448$$ 7.98420 0.377218
$$449$$ 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
$$452$$ 11.6849 0.549614
$$453$$ 0 0
$$454$$ −5.32898 −0.250101
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 38.2192 1.78782 0.893910 0.448246i $$-0.147951\pi$$
0.893910 + 0.448246i $$0.147951\pi$$
$$458$$ −4.95856 −0.231698
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −31.3033 −1.45794 −0.728971 0.684545i $$-0.760001\pi$$
−0.728971 + 0.684545i $$0.760001\pi$$
$$462$$ 0 0
$$463$$ −12.0852 −0.561645 −0.280823 0.959760i $$-0.590607\pi$$
−0.280823 + 0.959760i $$0.590607\pi$$
$$464$$ −4.56352 −0.211856
$$465$$ 0 0
$$466$$ −1.37345 −0.0636238
$$467$$ −7.60466 −0.351902 −0.175951 0.984399i $$-0.556300\pi$$
−0.175951 + 0.984399i $$0.556300\pi$$
$$468$$ 0 0
$$469$$ 2.51901 0.116317
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 18.2561 0.840303
$$473$$ 47.3636 2.17778
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 14.4237 0.661108
$$477$$ 0 0
$$478$$ −7.73982 −0.354011
$$479$$ 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
$$482$$ 8.29326 0.377748
$$483$$ 0 0
$$484$$ −48.0446 −2.18385
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 4.46121 0.202157 0.101078 0.994878i $$-0.467771\pi$$
0.101078 + 0.994878i $$0.467771\pi$$
$$488$$ 0.0196451 0.000889291 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −32.8420 −1.48214 −0.741070 0.671428i $$-0.765681\pi$$
−0.741070 + 0.671428i $$0.765681\pi$$
$$492$$ 0 0
$$493$$ −5.34300 −0.240637
$$494$$ −0.359515 −0.0161754
$$495$$ 0 0
$$496$$ −22.3501 −1.00355
$$497$$ −16.4934 −0.739832
$$498$$ 0 0
$$499$$ −34.2021 −1.53109 −0.765547 0.643380i $$-0.777532\pi$$
−0.765547 + 0.643380i $$0.777532\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ −4.00846 −0.178906
$$503$$ 22.1773 0.988837 0.494419 0.869224i $$-0.335381\pi$$
0.494419 + 0.869224i $$0.335381\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ −12.2990 −0.546758
$$507$$ 0 0
$$508$$ −16.4432 −0.729549
$$509$$ 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
$$512$$ 22.8115 1.00813
$$513$$ 0 0
$$514$$ −1.35943 −0.0599617
$$515$$ 0 0
$$516$$ 0 0
$$517$$ −70.5003 −3.10060
$$518$$ 4.40643 0.193607
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −20.2626 −0.887718 −0.443859 0.896097i $$-0.646391\pi$$
−0.443859 + 0.896097i $$0.646391\pi$$
$$522$$ 0 0
$$523$$ −31.8114 −1.39101 −0.695507 0.718520i $$-0.744820\pi$$
−0.695507 + 0.718520i $$0.744820\pi$$
$$524$$ 0.478457 0.0209015
$$525$$ 0 0
$$526$$ 12.1266 0.528745
$$527$$ −26.1676 −1.13988
$$528$$ 0 0
$$529$$ −5.25083 −0.228297
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 1.62175 0.0703117
$$533$$ −5.82870 −0.252469
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 1.75632 0.0758613
$$537$$ 0 0
$$538$$ −0.159586 −0.00688024
$$539$$ −2.65717 −0.114452
$$540$$ 0 0
$$541$$ −15.1315 −0.650553 −0.325277 0.945619i $$-0.605457\pi$$
−0.325277 + 0.945619i $$0.605457\pi$$
$$542$$ 10.2188 0.438937
$$543$$ 0 0
$$544$$ 15.3831 0.659543
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 4.08744 0.174766 0.0873831 0.996175i $$-0.472150\pi$$
0.0873831 + 0.996175i $$0.472150\pi$$
$$548$$ −6.17381 −0.263732
$$549$$ 0 0
$$550$$ 0 0
$$551$$ −0.600748 −0.0255927
$$552$$ 0 0
$$553$$ 24.2757 1.03231
$$554$$ −11.4214 −0.485249
$$555$$ 0 0
$$556$$ −26.1926 −1.11082
$$557$$ −13.1425 −0.556864 −0.278432 0.960456i $$-0.589815\pi$$
−0.278432 + 0.960456i $$0.589815\pi$$
$$558$$ 0 0
$$559$$ −16.3721 −0.692467
$$560$$ 0 0
$$561$$ 0 0
$$562$$ −1.59357 −0.0672207
$$563$$ −24.5221 −1.03348 −0.516742 0.856141i $$-0.672855\pi$$
−0.516742 + 0.856141i $$0.672855\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ −10.3405 −0.434642
$$567$$ 0 0
$$568$$ −11.4996 −0.482514
$$569$$ −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 $$0.503292\pi$$
$$572$$ 23.3607 0.976758
$$573$$ 0 0
$$574$$ −3.31571 −0.138395
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 9.41187 0.391821 0.195911 0.980622i $$-0.437234\pi$$
0.195911 + 0.980622i $$0.437234\pi$$
$$578$$ −3.29382 −0.137005
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 26.7064 1.10797
$$582$$ 0 0
$$583$$ −58.2012 −2.41045
$$584$$ 11.8206 0.489139
$$585$$ 0 0
$$586$$ 6.50916 0.268891
$$587$$ 9.97321 0.411638 0.205819 0.978590i $$-0.434014\pi$$
0.205819 + 0.978590i $$0.434014\pi$$
$$588$$ 0 0
$$589$$ −2.94219 −0.121231
$$590$$ 0 0
$$591$$ 0 0
$$592$$ −9.83144 −0.404070
$$593$$ 38.3421 1.57452 0.787260 0.616621i $$-0.211499\pi$$
0.787260 + 0.616621i $$0.211499\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ −18.0326 −0.738642
$$597$$ 0 0
$$598$$ 4.25140 0.173852
$$599$$ 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 $$0.642862\pi$$
$$602$$ −9.31344 −0.379587
$$603$$ 0 0
$$604$$ 18.3219 0.745508
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 37.7355 1.53164 0.765819 0.643056i $$-0.222334\pi$$
0.765819 + 0.643056i $$0.222334\pi$$
$$608$$ 1.72962 0.0701452
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 24.3698 0.985895
$$612$$ 0 0
$$613$$ 32.2633 1.30310 0.651551 0.758605i $$-0.274119\pi$$
0.651551 + 0.758605i $$0.274119\pi$$
$$614$$ −16.1940 −0.653536
$$615$$ 0 0
$$616$$ 28.2537 1.13837
$$617$$ −26.0178 −1.04744 −0.523719 0.851891i $$-0.675456\pi$$
−0.523719 + 0.851891i $$0.675456\pi$$
$$618$$ 0 0
$$619$$ −11.8815 −0.477559 −0.238780 0.971074i $$-0.576747\pi$$
−0.238780 + 0.971074i $$0.576747\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ −10.9037 −0.437197
$$623$$ −16.0688 −0.643783
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 1.69960 0.0679296
$$627$$ 0 0
$$628$$ 1.88723 0.0753086
$$629$$ −11.5107 −0.458962
$$630$$ 0 0
$$631$$ −13.2726 −0.528372 −0.264186 0.964472i $$-0.585103\pi$$
−0.264186 + 0.964472i $$0.585103\pi$$
$$632$$ 16.9256 0.673263
$$633$$ 0 0
$$634$$ 6.23780 0.247735
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 0.918501 0.0363923
$$638$$ −4.92265 −0.194890
$$639$$ 0 0
$$640$$ 0 0
$$641$$ −44.8149 −1.77008 −0.885042 0.465511i $$-0.845871\pi$$
−0.885042 + 0.465511i $$0.845871\pi$$
$$642$$ 0 0
$$643$$ 14.9255 0.588605 0.294302 0.955712i $$-0.404913\pi$$
0.294302 + 0.955712i $$0.404913\pi$$
$$644$$ −19.1777 −0.755708
$$645$$ 0 0
$$646$$ 0.534242 0.0210195
$$647$$ −41.2684 −1.62243 −0.811214 0.584749i $$-0.801193\pi$$
−0.811214 + 0.584749i $$0.801193\pi$$
$$648$$ 0 0
$$649$$ −63.0179 −2.47367
$$650$$ 0 0
$$651$$ 0 0
$$652$$ −30.4386 −1.19207
$$653$$ 27.6252 1.08106 0.540528 0.841326i $$-0.318224\pi$$
0.540528 + 0.841326i $$0.318224\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 7.39788 0.288839
$$657$$ 0 0
$$658$$ 13.8630 0.540435
$$659$$ −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 $$0.338435\pi$$
$$662$$ 0.559636 0.0217509
$$663$$ 0 0
$$664$$ 18.6204 0.722610
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 7.10405 0.275070
$$668$$ 7.76738 0.300529
$$669$$ 0 0
$$670$$ 0 0
$$671$$ −0.0678126 −0.00261788
$$672$$ 0 0
$$673$$ 40.8048 1.57291 0.786454 0.617649i $$-0.211915\pi$$
0.786454 + 0.617649i $$0.211915\pi$$
$$674$$ 11.7365 0.452072
$$675$$ 0 0
$$676$$ 15.0133 0.577436
$$677$$ 41.1894 1.58304 0.791518 0.611146i $$-0.209291\pi$$
0.791518 + 0.611146i $$0.209291\pi$$
$$678$$ 0 0
$$679$$ 18.4714 0.708867
$$680$$ 0 0
$$681$$ 0 0
$$682$$ −24.1089 −0.923178
$$683$$ 1.33820 0.0512047 0.0256023 0.999672i $$-0.491850\pi$$
0.0256023 + 0.999672i $$0.491850\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 9.01333 0.344131
$$687$$ 0 0
$$688$$ 20.7798 0.792221
$$689$$ 20.1184 0.766449
$$690$$ 0 0
$$691$$ −25.2813 −0.961748 −0.480874 0.876790i $$-0.659680\pi$$
−0.480874 + 0.876790i $$0.659680\pi$$
$$692$$ 26.0368 0.989770
$$693$$ 0 0
$$694$$ 10.4917 0.398258
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 8.66148 0.328077
$$698$$ 7.06025 0.267234
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 18.2064 0.687645 0.343822 0.939035i $$-0.388278\pi$$
0.343822 + 0.939035i $$0.388278\pi$$
$$702$$ 0 0
$$703$$ −1.29422 −0.0488126
$$704$$ −19.2159 −0.724227
$$705$$ 0 0
$$706$$ 8.00487 0.301267
$$707$$ 17.8669 0.671953
$$708$$ 0 0
$$709$$ 41.8206 1.57061 0.785304 0.619111i $$-0.212507\pi$$
0.785304 + 0.619111i $$0.212507\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ −11.2036 −0.419871
$$713$$ 34.7925 1.30299
$$714$$ 0 0
$$715$$ 0 0
$$716$$ −12.2016 −0.455995
$$717$$ 0 0
$$718$$ −0.301301 −0.0112444
$$719$$ −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
$$722$$ −8.93177 −0.332406
$$723$$ 0 0
$$724$$ 19.4847 0.724144
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 43.8009 1.62449 0.812243 0.583319i $$-0.198246\pi$$
0.812243 + 0.583319i $$0.198246\pi$$
$$728$$ −9.76642 −0.361968
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 24.3291 0.899843
$$732$$ 0 0
$$733$$ 6.78664 0.250670 0.125335 0.992114i $$-0.459999\pi$$
0.125335 + 0.992114i $$0.459999\pi$$
$$734$$ −9.51144 −0.351074
$$735$$ 0 0
$$736$$ −20.4533 −0.753919
$$737$$ −6.06261 −0.223319
$$738$$ 0 0
$$739$$ 28.7245 1.05665 0.528324 0.849043i $$-0.322821\pi$$
0.528324 + 0.849043i $$0.322821\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 11.4445 0.420142
$$743$$ −31.4523 −1.15387 −0.576937 0.816789i $$-0.695752\pi$$
−0.576937 + 0.816789i $$0.695752\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 9.29610 0.340354
$$747$$ 0 0
$$748$$ −34.7141 −1.26927
$$749$$ 37.2537 1.36122
$$750$$ 0 0
$$751$$ 10.9532 0.399687 0.199844 0.979828i $$-0.435957\pi$$
0.199844 + 0.979828i $$0.435957\pi$$
$$752$$ −30.9305 −1.12792
$$753$$ 0 0
$$754$$ 1.70161 0.0619689
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 45.7942 1.66442 0.832210 0.554461i $$-0.187075\pi$$
0.832210 + 0.554461i $$0.187075\pi$$
$$758$$ −3.75868 −0.136521
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 33.9138 1.22937 0.614687 0.788771i $$-0.289283\pi$$
0.614687 + 0.788771i $$0.289283\pi$$
$$762$$ 0 0
$$763$$ −4.87528 −0.176497
$$764$$ 24.3894 0.882378
$$765$$ 0 0
$$766$$ −14.7602 −0.533309
$$767$$ 21.7833 0.786551
$$768$$ 0 0
$$769$$ 7.15972 0.258186 0.129093 0.991632i $$-0.458793\pi$$
0.129093 + 0.991632i $$0.458793\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ −0.856710 −0.0308337
$$773$$ −14.5998 −0.525117 −0.262558 0.964916i $$-0.584566\pi$$
−0.262558 + 0.964916i $$0.584566\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 12.8787 0.462318
$$777$$ 0 0
$$778$$ 14.8836 0.533602
$$779$$ 0.973866 0.0348924
$$780$$ 0 0
$$781$$ 39.6954 1.42041
$$782$$ −6.31760 −0.225917
$$783$$ 0 0
$$784$$ −1.16578 −0.0416348
$$785$$ 0 0
$$786$$ 0 0
$$787$$ −18.4728 −0.658483 −0.329242 0.944246i $$-0.606793\pi$$
−0.329242 + 0.944246i $$0.606793\pi$$
$$788$$ −9.83011 −0.350183
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 16.8630 0.599578
$$792$$ 0 0
$$793$$ 0.0234407 0.000832405 0
$$794$$ −8.38484 −0.297567
$$795$$ 0 0
$$796$$ −31.0077 −1.09904
$$797$$ 41.0374 1.45362 0.726810 0.686838i $$-0.241002\pi$$
0.726810 + 0.686838i $$0.241002\pi$$
$$798$$ 0 0
$$799$$ −36.2136 −1.28115
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 3.38021 0.119359
$$803$$ −40.8033 −1.43992
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 8.33371 0.293543
$$807$$ 0 0
$$808$$ 12.4572 0.438244
$$809$$ −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 $$0.265859\pi$$
$$812$$ −7.67583 −0.269369
$$813$$ 0 0
$$814$$ −10.6051 −0.371710
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 2.73547 0.0957021
$$818$$ 11.7139 0.409566
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 0.668560 0.0233329 0.0116665 0.999932i $$-0.496286\pi$$
0.0116665 + 0.999932i $$0.496286\pi$$
$$822$$ 0 0
$$823$$ 1.42033 0.0495096 0.0247548 0.999694i $$-0.492119\pi$$
0.0247548 + 0.999694i $$0.492119\pi$$
$$824$$ −10.9279 −0.380690
$$825$$ 0 0
$$826$$ 12.3917 0.431161
$$827$$ 49.8169 1.73230 0.866152 0.499782i $$-0.166586\pi$$
0.866152 + 0.499782i $$0.166586\pi$$
$$828$$ 0 0
$$829$$ 36.4150 1.26475 0.632373 0.774664i $$-0.282081\pi$$
0.632373 + 0.774664i $$0.282081\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 6.64235 0.230282
$$833$$ −1.36490 −0.0472909
$$834$$ 0 0
$$835$$ 0 0
$$836$$ −3.90312 −0.134992
$$837$$ 0 0
$$838$$ 5.04447 0.174258
$$839$$ 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
$$842$$ −3.87056 −0.133388
$$843$$ 0 0
$$844$$ 2.90675 0.100055
$$845$$ 0 0
$$846$$ 0 0
$$847$$ −69.3349 −2.38238
$$848$$ −25.5345 −0.876860
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 15.3046 0.524637
$$852$$ 0 0
$$853$$ −26.9084 −0.921326 −0.460663 0.887575i $$-0.652388\pi$$
−0.460663 + 0.887575i $$0.652388\pi$$
$$854$$ 0.0133345 0.000456296 0
$$855$$ 0 0
$$856$$ 25.9742 0.887779
$$857$$ −48.8408 −1.66837 −0.834184 0.551486i $$-0.814061\pi$$
−0.834184 + 0.551486i $$0.814061\pi$$
$$858$$ 0 0
$$859$$ −41.4094 −1.41287 −0.706435 0.707778i $$-0.749698\pi$$
−0.706435 + 0.707778i $$0.749698\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ −0.791507 −0.0269588
$$863$$ −50.8101 −1.72960 −0.864799 0.502119i $$-0.832554\pi$$
−0.864799 + 0.502119i $$0.832554\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 4.71359 0.160174
$$867$$ 0 0
$$868$$ −37.5928 −1.27598
$$869$$ −58.4252 −1.98194
$$870$$ 0 0
$$871$$ 2.09566 0.0710087
$$872$$ −3.39916 −0.115110
$$873$$ 0 0
$$874$$ −0.710328 −0.0240272
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 0.409589 0.0138308 0.00691542 0.999976i $$-0.497799\pi$$
0.00691542 + 0.999976i $$0.497799\pi$$
$$878$$ 6.06512 0.204688
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 5.32851 0.179522 0.0897610 0.995963i $$-0.471390\pi$$
0.0897610 + 0.995963i $$0.471390\pi$$
$$882$$ 0 0
$$883$$ 14.2064 0.478083 0.239042 0.971009i $$-0.423167\pi$$
0.239042 + 0.971009i $$0.423167\pi$$
$$884$$ 11.9996 0.403590
$$885$$ 0 0
$$886$$ −3.66964 −0.123284
$$887$$ −7.23193 −0.242825 −0.121412 0.992602i $$-0.538742\pi$$
−0.121412 + 0.992602i $$0.538742\pi$$
$$888$$ 0 0
$$889$$ −23.7298 −0.795871
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 13.7604 0.460733
$$893$$ −4.07173 −0.136255
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 28.6650 0.957629
$$897$$ 0 0
$$898$$ 15.7784 0.526531
$$899$$ 13.9256 0.464444
$$900$$ 0 0
$$901$$ −29.8960 −0.995981
$$902$$ 7.98006 0.265707
$$903$$ 0 0
$$904$$ 11.7573 0.391041
$$905$$ 0 0
$$906$$ 0 0
$$907$$ −38.8101 −1.28867 −0.644335 0.764743i $$-0.722866\pi$$
−0.644335 + 0.764743i $$0.722866\pi$$
$$908$$ 19.9986 0.663677
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 40.2781 1.33447 0.667236 0.744846i $$-0.267477\pi$$
0.667236 + 0.744846i $$0.267477\pi$$
$$912$$ 0 0
$$913$$ −64.2754 −2.12721
$$914$$ 18.0874 0.598279
$$915$$ 0 0
$$916$$ 18.6085 0.614842
$$917$$ 0.690479 0.0228016
$$918$$ 0 0
$$919$$ −13.0468 −0.430375 −0.215187 0.976573i $$-0.569036\pi$$
−0.215187 + 0.976573i $$0.569036\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ −14.8144 −0.487888
$$923$$ −13.7215 −0.451648
$$924$$ 0 0
$$925$$ 0 0
$$926$$ −5.71936 −0.187950
$$927$$ 0 0
$$928$$ −8.18638 −0.268731
$$929$$ −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
$$932$$ 5.15428 0.168834
$$933$$ 0 0
$$934$$ −3.59894 −0.117761
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 16.9141 0.552559 0.276280 0.961077i $$-0.410898\pi$$
0.276280 + 0.961077i $$0.410898\pi$$
$$938$$ 1.19213 0.0389246
$$939$$ 0 0
$$940$$ 0 0
$$941$$ −57.2093 −1.86497 −0.932485 0.361209i $$-0.882364\pi$$
−0.932485 + 0.361209i $$0.882364\pi$$
$$942$$ 0 0
$$943$$ −11.5163 −0.375023
$$944$$ −27.6478 −0.899858
$$945$$ 0 0
$$946$$ 22.4150 0.728775
$$947$$ −38.8746 −1.26325 −0.631627 0.775272i $$-0.717613\pi$$
−0.631627 + 0.775272i $$0.717613\pi$$
$$948$$ 0 0
$$949$$ 14.1044 0.457850
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 14.5130 0.470368
$$953$$ −54.4516 −1.76386 −0.881930 0.471381i $$-0.843756\pi$$
−0.881930 + 0.471381i $$0.843756\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 29.0460 0.939415
$$957$$ 0 0
$$958$$ 15.3729 0.496675
$$959$$ −8.90965 −0.287707
$$960$$ 0 0
$$961$$ 37.2012 1.20004
$$962$$ 3.66587 0.118192
$$963$$ 0 0
$$964$$ −31.1229 −1.00240
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 19.5701 0.629333 0.314666 0.949202i $$-0.398107\pi$$
0.314666 + 0.949202i $$0.398107\pi$$
$$968$$ −48.3420 −1.55377
$$969$$ 0 0
$$970$$ 0 0
$$971$$ −6.31009 −0.202500 −0.101250 0.994861i $$-0.532284\pi$$
−0.101250 + 0.994861i $$0.532284\pi$$
$$972$$ 0 0
$$973$$ −37.7995 −1.21180
$$974$$ 2.11129 0.0676500
$$975$$ 0 0
$$976$$ −0.0297513 −0.000952317 0
$$977$$ 7.41911 0.237358 0.118679 0.992933i $$-0.462134\pi$$
0.118679 + 0.992933i $$0.462134\pi$$
$$978$$ 0 0
$$979$$ 38.6734 1.23601
$$980$$ 0 0
$$981$$ 0 0
$$982$$ −15.5426 −0.495986
$$983$$ −10.9827 −0.350295 −0.175148 0.984542i $$-0.556040\pi$$
−0.175148 + 0.984542i $$0.556040\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ −2.52860 −0.0805270
$$987$$ 0 0
$$988$$ 1.34919 0.0429234
$$989$$ −32.3479 −1.02860
$$990$$ 0 0
$$991$$ −21.3721 −0.678908 −0.339454 0.940623i $$-0.610242\pi$$
−0.339454 + 0.940623i $$0.610242\pi$$
$$992$$ −40.0932 −1.27296
$$993$$ 0 0
$$994$$ −7.80559 −0.247578
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 30.5348 0.967047 0.483524 0.875331i $$-0.339357\pi$$
0.483524 + 0.875331i $$0.339357\pi$$
$$998$$ −16.1863 −0.512368
$$999$$ 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.a.p.1.3 4
3.2 odd 2 2025.2.a.y.1.2 4
5.2 odd 4 2025.2.b.o.649.5 8
5.3 odd 4 2025.2.b.o.649.4 8
5.4 even 2 2025.2.a.z.1.2 4
9.2 odd 6 225.2.e.c.76.3 8
9.4 even 3 675.2.e.e.451.2 8
9.5 odd 6 225.2.e.c.151.3 yes 8
9.7 even 3 675.2.e.e.226.2 8
15.2 even 4 2025.2.b.n.649.4 8
15.8 even 4 2025.2.b.n.649.5 8
15.14 odd 2 2025.2.a.q.1.3 4
45.2 even 12 225.2.k.c.49.4 16
45.4 even 6 675.2.e.c.451.3 8
45.7 odd 12 675.2.k.c.199.5 16
45.13 odd 12 675.2.k.c.424.5 16
45.14 odd 6 225.2.e.e.151.2 yes 8
45.22 odd 12 675.2.k.c.424.4 16
45.23 even 12 225.2.k.c.124.4 16
45.29 odd 6 225.2.e.e.76.2 yes 8
45.32 even 12 225.2.k.c.124.5 16
45.34 even 6 675.2.e.c.226.3 8
45.38 even 12 225.2.k.c.49.5 16
45.43 odd 12 675.2.k.c.199.4 16

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