# Properties

 Label 1850.2.a.q.1.2 Level $1850$ Weight $2$ Character 1850.1 Self dual yes Analytic conductor $14.772$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1850,2,Mod(1,1850)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1850, 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("1850.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1850 = 2 \cdot 5^{2} \cdot 37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1850.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: $$14.7723243739$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{33})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 8$$ x^2 - x - 8 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 370) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$3.37228$$ of defining polynomial Character $$\chi$$ $$=$$ 1850.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} -2.00000 q^{3} +1.00000 q^{4} +2.00000 q^{6} +4.37228 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{2} -2.00000 q^{3} +1.00000 q^{4} +2.00000 q^{6} +4.37228 q^{7} -1.00000 q^{8} +1.00000 q^{9} +2.37228 q^{11} -2.00000 q^{12} -6.74456 q^{13} -4.37228 q^{14} +1.00000 q^{16} -0.372281 q^{17} -1.00000 q^{18} -2.00000 q^{19} -8.74456 q^{21} -2.37228 q^{22} +4.74456 q^{23} +2.00000 q^{24} +6.74456 q^{26} +4.00000 q^{27} +4.37228 q^{28} -9.11684 q^{29} -8.37228 q^{31} -1.00000 q^{32} -4.74456 q^{33} +0.372281 q^{34} +1.00000 q^{36} -1.00000 q^{37} +2.00000 q^{38} +13.4891 q^{39} -0.372281 q^{41} +8.74456 q^{42} -1.62772 q^{43} +2.37228 q^{44} -4.74456 q^{46} -2.74456 q^{47} -2.00000 q^{48} +12.1168 q^{49} +0.744563 q^{51} -6.74456 q^{52} +4.37228 q^{53} -4.00000 q^{54} -4.37228 q^{56} +4.00000 q^{57} +9.11684 q^{58} +1.25544 q^{59} +0.372281 q^{61} +8.37228 q^{62} +4.37228 q^{63} +1.00000 q^{64} +4.74456 q^{66} +6.74456 q^{67} -0.372281 q^{68} -9.48913 q^{69} +4.74456 q^{71} -1.00000 q^{72} +2.74456 q^{73} +1.00000 q^{74} -2.00000 q^{76} +10.3723 q^{77} -13.4891 q^{78} +6.74456 q^{79} -11.0000 q^{81} +0.372281 q^{82} -10.7446 q^{83} -8.74456 q^{84} +1.62772 q^{86} +18.2337 q^{87} -2.37228 q^{88} +10.0000 q^{89} -29.4891 q^{91} +4.74456 q^{92} +16.7446 q^{93} +2.74456 q^{94} +2.00000 q^{96} -17.1168 q^{97} -12.1168 q^{98} +2.37228 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} - 4 q^{3} + 2 q^{4} + 4 q^{6} + 3 q^{7} - 2 q^{8} + 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^2 - 4 * q^3 + 2 * q^4 + 4 * q^6 + 3 * q^7 - 2 * q^8 + 2 * q^9 $$2 q - 2 q^{2} - 4 q^{3} + 2 q^{4} + 4 q^{6} + 3 q^{7} - 2 q^{8} + 2 q^{9} - q^{11} - 4 q^{12} - 2 q^{13} - 3 q^{14} + 2 q^{16} + 5 q^{17} - 2 q^{18} - 4 q^{19} - 6 q^{21} + q^{22} - 2 q^{23} + 4 q^{24} + 2 q^{26} + 8 q^{27} + 3 q^{28} - q^{29} - 11 q^{31} - 2 q^{32} + 2 q^{33} - 5 q^{34} + 2 q^{36} - 2 q^{37} + 4 q^{38} + 4 q^{39} + 5 q^{41} + 6 q^{42} - 9 q^{43} - q^{44} + 2 q^{46} + 6 q^{47} - 4 q^{48} + 7 q^{49} - 10 q^{51} - 2 q^{52} + 3 q^{53} - 8 q^{54} - 3 q^{56} + 8 q^{57} + q^{58} + 14 q^{59} - 5 q^{61} + 11 q^{62} + 3 q^{63} + 2 q^{64} - 2 q^{66} + 2 q^{67} + 5 q^{68} + 4 q^{69} - 2 q^{71} - 2 q^{72} - 6 q^{73} + 2 q^{74} - 4 q^{76} + 15 q^{77} - 4 q^{78} + 2 q^{79} - 22 q^{81} - 5 q^{82} - 10 q^{83} - 6 q^{84} + 9 q^{86} + 2 q^{87} + q^{88} + 20 q^{89} - 36 q^{91} - 2 q^{92} + 22 q^{93} - 6 q^{94} + 4 q^{96} - 17 q^{97} - 7 q^{98} - q^{99}+O(q^{100})$$ 2 * q - 2 * q^2 - 4 * q^3 + 2 * q^4 + 4 * q^6 + 3 * q^7 - 2 * q^8 + 2 * q^9 - q^11 - 4 * q^12 - 2 * q^13 - 3 * q^14 + 2 * q^16 + 5 * q^17 - 2 * q^18 - 4 * q^19 - 6 * q^21 + q^22 - 2 * q^23 + 4 * q^24 + 2 * q^26 + 8 * q^27 + 3 * q^28 - q^29 - 11 * q^31 - 2 * q^32 + 2 * q^33 - 5 * q^34 + 2 * q^36 - 2 * q^37 + 4 * q^38 + 4 * q^39 + 5 * q^41 + 6 * q^42 - 9 * q^43 - q^44 + 2 * q^46 + 6 * q^47 - 4 * q^48 + 7 * q^49 - 10 * q^51 - 2 * q^52 + 3 * q^53 - 8 * q^54 - 3 * q^56 + 8 * q^57 + q^58 + 14 * q^59 - 5 * q^61 + 11 * q^62 + 3 * q^63 + 2 * q^64 - 2 * q^66 + 2 * q^67 + 5 * q^68 + 4 * q^69 - 2 * q^71 - 2 * q^72 - 6 * q^73 + 2 * q^74 - 4 * q^76 + 15 * q^77 - 4 * q^78 + 2 * q^79 - 22 * q^81 - 5 * q^82 - 10 * q^83 - 6 * q^84 + 9 * q^86 + 2 * q^87 + q^88 + 20 * q^89 - 36 * q^91 - 2 * q^92 + 22 * q^93 - 6 * q^94 + 4 * q^96 - 17 * q^97 - 7 * q^98 - q^99

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −1.00000 −0.707107
$$3$$ −2.00000 −1.15470 −0.577350 0.816497i $$-0.695913\pi$$
−0.577350 + 0.816497i $$0.695913\pi$$
$$4$$ 1.00000 0.500000
$$5$$ 0 0
$$6$$ 2.00000 0.816497
$$7$$ 4.37228 1.65257 0.826284 0.563254i $$-0.190451\pi$$
0.826284 + 0.563254i $$0.190451\pi$$
$$8$$ −1.00000 −0.353553
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ 2.37228 0.715270 0.357635 0.933862i $$-0.383583\pi$$
0.357635 + 0.933862i $$0.383583\pi$$
$$12$$ −2.00000 −0.577350
$$13$$ −6.74456 −1.87061 −0.935303 0.353849i $$-0.884873\pi$$
−0.935303 + 0.353849i $$0.884873\pi$$
$$14$$ −4.37228 −1.16854
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ −0.372281 −0.0902915 −0.0451457 0.998980i $$-0.514375\pi$$
−0.0451457 + 0.998980i $$0.514375\pi$$
$$18$$ −1.00000 −0.235702
$$19$$ −2.00000 −0.458831 −0.229416 0.973329i $$-0.573682\pi$$
−0.229416 + 0.973329i $$0.573682\pi$$
$$20$$ 0 0
$$21$$ −8.74456 −1.90822
$$22$$ −2.37228 −0.505772
$$23$$ 4.74456 0.989310 0.494655 0.869090i $$-0.335294\pi$$
0.494655 + 0.869090i $$0.335294\pi$$
$$24$$ 2.00000 0.408248
$$25$$ 0 0
$$26$$ 6.74456 1.32272
$$27$$ 4.00000 0.769800
$$28$$ 4.37228 0.826284
$$29$$ −9.11684 −1.69296 −0.846478 0.532424i $$-0.821281\pi$$
−0.846478 + 0.532424i $$0.821281\pi$$
$$30$$ 0 0
$$31$$ −8.37228 −1.50371 −0.751853 0.659331i $$-0.770840\pi$$
−0.751853 + 0.659331i $$0.770840\pi$$
$$32$$ −1.00000 −0.176777
$$33$$ −4.74456 −0.825922
$$34$$ 0.372281 0.0638457
$$35$$ 0 0
$$36$$ 1.00000 0.166667
$$37$$ −1.00000 −0.164399
$$38$$ 2.00000 0.324443
$$39$$ 13.4891 2.15999
$$40$$ 0 0
$$41$$ −0.372281 −0.0581406 −0.0290703 0.999577i $$-0.509255\pi$$
−0.0290703 + 0.999577i $$0.509255\pi$$
$$42$$ 8.74456 1.34932
$$43$$ −1.62772 −0.248225 −0.124112 0.992268i $$-0.539608\pi$$
−0.124112 + 0.992268i $$0.539608\pi$$
$$44$$ 2.37228 0.357635
$$45$$ 0 0
$$46$$ −4.74456 −0.699548
$$47$$ −2.74456 −0.400336 −0.200168 0.979762i $$-0.564149\pi$$
−0.200168 + 0.979762i $$0.564149\pi$$
$$48$$ −2.00000 −0.288675
$$49$$ 12.1168 1.73098
$$50$$ 0 0
$$51$$ 0.744563 0.104260
$$52$$ −6.74456 −0.935303
$$53$$ 4.37228 0.600579 0.300290 0.953848i $$-0.402917\pi$$
0.300290 + 0.953848i $$0.402917\pi$$
$$54$$ −4.00000 −0.544331
$$55$$ 0 0
$$56$$ −4.37228 −0.584271
$$57$$ 4.00000 0.529813
$$58$$ 9.11684 1.19710
$$59$$ 1.25544 0.163444 0.0817220 0.996655i $$-0.473958\pi$$
0.0817220 + 0.996655i $$0.473958\pi$$
$$60$$ 0 0
$$61$$ 0.372281 0.0476657 0.0238329 0.999716i $$-0.492413\pi$$
0.0238329 + 0.999716i $$0.492413\pi$$
$$62$$ 8.37228 1.06328
$$63$$ 4.37228 0.550856
$$64$$ 1.00000 0.125000
$$65$$ 0 0
$$66$$ 4.74456 0.584015
$$67$$ 6.74456 0.823979 0.411990 0.911188i $$-0.364834\pi$$
0.411990 + 0.911188i $$0.364834\pi$$
$$68$$ −0.372281 −0.0451457
$$69$$ −9.48913 −1.14236
$$70$$ 0 0
$$71$$ 4.74456 0.563076 0.281538 0.959550i $$-0.409155\pi$$
0.281538 + 0.959550i $$0.409155\pi$$
$$72$$ −1.00000 −0.117851
$$73$$ 2.74456 0.321227 0.160613 0.987017i $$-0.448653\pi$$
0.160613 + 0.987017i $$0.448653\pi$$
$$74$$ 1.00000 0.116248
$$75$$ 0 0
$$76$$ −2.00000 −0.229416
$$77$$ 10.3723 1.18203
$$78$$ −13.4891 −1.52734
$$79$$ 6.74456 0.758823 0.379411 0.925228i $$-0.376127\pi$$
0.379411 + 0.925228i $$0.376127\pi$$
$$80$$ 0 0
$$81$$ −11.0000 −1.22222
$$82$$ 0.372281 0.0411116
$$83$$ −10.7446 −1.17937 −0.589684 0.807634i $$-0.700748\pi$$
−0.589684 + 0.807634i $$0.700748\pi$$
$$84$$ −8.74456 −0.954110
$$85$$ 0 0
$$86$$ 1.62772 0.175521
$$87$$ 18.2337 1.95486
$$88$$ −2.37228 −0.252886
$$89$$ 10.0000 1.06000 0.529999 0.847998i $$-0.322192\pi$$
0.529999 + 0.847998i $$0.322192\pi$$
$$90$$ 0 0
$$91$$ −29.4891 −3.09130
$$92$$ 4.74456 0.494655
$$93$$ 16.7446 1.73633
$$94$$ 2.74456 0.283080
$$95$$ 0 0
$$96$$ 2.00000 0.204124
$$97$$ −17.1168 −1.73795 −0.868976 0.494854i $$-0.835222\pi$$
−0.868976 + 0.494854i $$0.835222\pi$$
$$98$$ −12.1168 −1.22399
$$99$$ 2.37228 0.238423
$$100$$ 0 0
$$101$$ 11.4891 1.14321 0.571605 0.820529i $$-0.306321\pi$$
0.571605 + 0.820529i $$0.306321\pi$$
$$102$$ −0.744563 −0.0737227
$$103$$ −13.4891 −1.32912 −0.664562 0.747234i $$-0.731382\pi$$
−0.664562 + 0.747234i $$0.731382\pi$$
$$104$$ 6.74456 0.661359
$$105$$ 0 0
$$106$$ −4.37228 −0.424674
$$107$$ −19.4891 −1.88408 −0.942042 0.335494i $$-0.891097\pi$$
−0.942042 + 0.335494i $$0.891097\pi$$
$$108$$ 4.00000 0.384900
$$109$$ −17.1168 −1.63950 −0.819748 0.572724i $$-0.805887\pi$$
−0.819748 + 0.572724i $$0.805887\pi$$
$$110$$ 0 0
$$111$$ 2.00000 0.189832
$$112$$ 4.37228 0.413142
$$113$$ −11.6277 −1.09384 −0.546922 0.837184i $$-0.684201\pi$$
−0.546922 + 0.837184i $$0.684201\pi$$
$$114$$ −4.00000 −0.374634
$$115$$ 0 0
$$116$$ −9.11684 −0.846478
$$117$$ −6.74456 −0.623535
$$118$$ −1.25544 −0.115572
$$119$$ −1.62772 −0.149213
$$120$$ 0 0
$$121$$ −5.37228 −0.488389
$$122$$ −0.372281 −0.0337048
$$123$$ 0.744563 0.0671350
$$124$$ −8.37228 −0.751853
$$125$$ 0 0
$$126$$ −4.37228 −0.389514
$$127$$ 5.25544 0.466345 0.233172 0.972435i $$-0.425089\pi$$
0.233172 + 0.972435i $$0.425089\pi$$
$$128$$ −1.00000 −0.0883883
$$129$$ 3.25544 0.286625
$$130$$ 0 0
$$131$$ −14.7446 −1.28824 −0.644119 0.764925i $$-0.722776\pi$$
−0.644119 + 0.764925i $$0.722776\pi$$
$$132$$ −4.74456 −0.412961
$$133$$ −8.74456 −0.758250
$$134$$ −6.74456 −0.582641
$$135$$ 0 0
$$136$$ 0.372281 0.0319229
$$137$$ −5.25544 −0.449002 −0.224501 0.974474i $$-0.572075\pi$$
−0.224501 + 0.974474i $$0.572075\pi$$
$$138$$ 9.48913 0.807768
$$139$$ −19.1168 −1.62147 −0.810735 0.585414i $$-0.800932\pi$$
−0.810735 + 0.585414i $$0.800932\pi$$
$$140$$ 0 0
$$141$$ 5.48913 0.462268
$$142$$ −4.74456 −0.398155
$$143$$ −16.0000 −1.33799
$$144$$ 1.00000 0.0833333
$$145$$ 0 0
$$146$$ −2.74456 −0.227142
$$147$$ −24.2337 −1.99876
$$148$$ −1.00000 −0.0821995
$$149$$ 11.4891 0.941226 0.470613 0.882340i $$-0.344033\pi$$
0.470613 + 0.882340i $$0.344033\pi$$
$$150$$ 0 0
$$151$$ −20.0000 −1.62758 −0.813788 0.581161i $$-0.802599\pi$$
−0.813788 + 0.581161i $$0.802599\pi$$
$$152$$ 2.00000 0.162221
$$153$$ −0.372281 −0.0300972
$$154$$ −10.3723 −0.835822
$$155$$ 0 0
$$156$$ 13.4891 1.07999
$$157$$ −11.6277 −0.927993 −0.463996 0.885837i $$-0.653585\pi$$
−0.463996 + 0.885837i $$0.653585\pi$$
$$158$$ −6.74456 −0.536569
$$159$$ −8.74456 −0.693489
$$160$$ 0 0
$$161$$ 20.7446 1.63490
$$162$$ 11.0000 0.864242
$$163$$ −13.6277 −1.06741 −0.533703 0.845672i $$-0.679200\pi$$
−0.533703 + 0.845672i $$0.679200\pi$$
$$164$$ −0.372281 −0.0290703
$$165$$ 0 0
$$166$$ 10.7446 0.833940
$$167$$ 21.4891 1.66288 0.831439 0.555616i $$-0.187517\pi$$
0.831439 + 0.555616i $$0.187517\pi$$
$$168$$ 8.74456 0.674658
$$169$$ 32.4891 2.49916
$$170$$ 0 0
$$171$$ −2.00000 −0.152944
$$172$$ −1.62772 −0.124112
$$173$$ 5.86141 0.445634 0.222817 0.974860i $$-0.428475\pi$$
0.222817 + 0.974860i $$0.428475\pi$$
$$174$$ −18.2337 −1.38229
$$175$$ 0 0
$$176$$ 2.37228 0.178817
$$177$$ −2.51087 −0.188729
$$178$$ −10.0000 −0.749532
$$179$$ 8.23369 0.615415 0.307707 0.951481i $$-0.400438\pi$$
0.307707 + 0.951481i $$0.400438\pi$$
$$180$$ 0 0
$$181$$ −15.4891 −1.15130 −0.575649 0.817697i $$-0.695250\pi$$
−0.575649 + 0.817697i $$0.695250\pi$$
$$182$$ 29.4891 2.18588
$$183$$ −0.744563 −0.0550397
$$184$$ −4.74456 −0.349774
$$185$$ 0 0
$$186$$ −16.7446 −1.22777
$$187$$ −0.883156 −0.0645828
$$188$$ −2.74456 −0.200168
$$189$$ 17.4891 1.27215
$$190$$ 0 0
$$191$$ 24.3723 1.76352 0.881758 0.471702i $$-0.156360\pi$$
0.881758 + 0.471702i $$0.156360\pi$$
$$192$$ −2.00000 −0.144338
$$193$$ −2.00000 −0.143963 −0.0719816 0.997406i $$-0.522932\pi$$
−0.0719816 + 0.997406i $$0.522932\pi$$
$$194$$ 17.1168 1.22892
$$195$$ 0 0
$$196$$ 12.1168 0.865489
$$197$$ −4.51087 −0.321387 −0.160693 0.987004i $$-0.551373\pi$$
−0.160693 + 0.987004i $$0.551373\pi$$
$$198$$ −2.37228 −0.168591
$$199$$ −22.7446 −1.61232 −0.806160 0.591698i $$-0.798458\pi$$
−0.806160 + 0.591698i $$0.798458\pi$$
$$200$$ 0 0
$$201$$ −13.4891 −0.951450
$$202$$ −11.4891 −0.808372
$$203$$ −39.8614 −2.79772
$$204$$ 0.744563 0.0521298
$$205$$ 0 0
$$206$$ 13.4891 0.939832
$$207$$ 4.74456 0.329770
$$208$$ −6.74456 −0.467651
$$209$$ −4.74456 −0.328188
$$210$$ 0 0
$$211$$ 3.11684 0.214572 0.107286 0.994228i $$-0.465784\pi$$
0.107286 + 0.994228i $$0.465784\pi$$
$$212$$ 4.37228 0.300290
$$213$$ −9.48913 −0.650184
$$214$$ 19.4891 1.33225
$$215$$ 0 0
$$216$$ −4.00000 −0.272166
$$217$$ −36.6060 −2.48498
$$218$$ 17.1168 1.15930
$$219$$ −5.48913 −0.370921
$$220$$ 0 0
$$221$$ 2.51087 0.168900
$$222$$ −2.00000 −0.134231
$$223$$ −4.37228 −0.292790 −0.146395 0.989226i $$-0.546767\pi$$
−0.146395 + 0.989226i $$0.546767\pi$$
$$224$$ −4.37228 −0.292135
$$225$$ 0 0
$$226$$ 11.6277 0.773464
$$227$$ 5.62772 0.373525 0.186762 0.982405i $$-0.440201\pi$$
0.186762 + 0.982405i $$0.440201\pi$$
$$228$$ 4.00000 0.264906
$$229$$ −10.0000 −0.660819 −0.330409 0.943838i $$-0.607187\pi$$
−0.330409 + 0.943838i $$0.607187\pi$$
$$230$$ 0 0
$$231$$ −20.7446 −1.36489
$$232$$ 9.11684 0.598550
$$233$$ 28.2337 1.84965 0.924825 0.380392i $$-0.124211\pi$$
0.924825 + 0.380392i $$0.124211\pi$$
$$234$$ 6.74456 0.440906
$$235$$ 0 0
$$236$$ 1.25544 0.0817220
$$237$$ −13.4891 −0.876213
$$238$$ 1.62772 0.105509
$$239$$ −22.6060 −1.46226 −0.731129 0.682239i $$-0.761006\pi$$
−0.731129 + 0.682239i $$0.761006\pi$$
$$240$$ 0 0
$$241$$ 24.2337 1.56103 0.780515 0.625138i $$-0.214957\pi$$
0.780515 + 0.625138i $$0.214957\pi$$
$$242$$ 5.37228 0.345343
$$243$$ 10.0000 0.641500
$$244$$ 0.372281 0.0238329
$$245$$ 0 0
$$246$$ −0.744563 −0.0474716
$$247$$ 13.4891 0.858292
$$248$$ 8.37228 0.531640
$$249$$ 21.4891 1.36182
$$250$$ 0 0
$$251$$ −11.4891 −0.725187 −0.362594 0.931947i $$-0.618109\pi$$
−0.362594 + 0.931947i $$0.618109\pi$$
$$252$$ 4.37228 0.275428
$$253$$ 11.2554 0.707623
$$254$$ −5.25544 −0.329755
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ −24.9783 −1.55810 −0.779050 0.626962i $$-0.784298\pi$$
−0.779050 + 0.626962i $$0.784298\pi$$
$$258$$ −3.25544 −0.202675
$$259$$ −4.37228 −0.271680
$$260$$ 0 0
$$261$$ −9.11684 −0.564318
$$262$$ 14.7446 0.910922
$$263$$ 17.1168 1.05547 0.527735 0.849409i $$-0.323041\pi$$
0.527735 + 0.849409i $$0.323041\pi$$
$$264$$ 4.74456 0.292008
$$265$$ 0 0
$$266$$ 8.74456 0.536164
$$267$$ −20.0000 −1.22398
$$268$$ 6.74456 0.411990
$$269$$ 10.0000 0.609711 0.304855 0.952399i $$-0.401392\pi$$
0.304855 + 0.952399i $$0.401392\pi$$
$$270$$ 0 0
$$271$$ 4.74456 0.288212 0.144106 0.989562i $$-0.453969\pi$$
0.144106 + 0.989562i $$0.453969\pi$$
$$272$$ −0.372281 −0.0225729
$$273$$ 58.9783 3.56953
$$274$$ 5.25544 0.317493
$$275$$ 0 0
$$276$$ −9.48913 −0.571178
$$277$$ 10.7446 0.645578 0.322789 0.946471i $$-0.395380\pi$$
0.322789 + 0.946471i $$0.395380\pi$$
$$278$$ 19.1168 1.14655
$$279$$ −8.37228 −0.501235
$$280$$ 0 0
$$281$$ 13.2554 0.790753 0.395377 0.918519i $$-0.370614\pi$$
0.395377 + 0.918519i $$0.370614\pi$$
$$282$$ −5.48913 −0.326873
$$283$$ 17.4891 1.03962 0.519810 0.854282i $$-0.326003\pi$$
0.519810 + 0.854282i $$0.326003\pi$$
$$284$$ 4.74456 0.281538
$$285$$ 0 0
$$286$$ 16.0000 0.946100
$$287$$ −1.62772 −0.0960812
$$288$$ −1.00000 −0.0589256
$$289$$ −16.8614 −0.991847
$$290$$ 0 0
$$291$$ 34.2337 2.00681
$$292$$ 2.74456 0.160613
$$293$$ −9.86141 −0.576110 −0.288055 0.957614i $$-0.593009\pi$$
−0.288055 + 0.957614i $$0.593009\pi$$
$$294$$ 24.2337 1.41334
$$295$$ 0 0
$$296$$ 1.00000 0.0581238
$$297$$ 9.48913 0.550615
$$298$$ −11.4891 −0.665547
$$299$$ −32.0000 −1.85061
$$300$$ 0 0
$$301$$ −7.11684 −0.410208
$$302$$ 20.0000 1.15087
$$303$$ −22.9783 −1.32007
$$304$$ −2.00000 −0.114708
$$305$$ 0 0
$$306$$ 0.372281 0.0212819
$$307$$ 0.510875 0.0291572 0.0145786 0.999894i $$-0.495359\pi$$
0.0145786 + 0.999894i $$0.495359\pi$$
$$308$$ 10.3723 0.591016
$$309$$ 26.9783 1.53474
$$310$$ 0 0
$$311$$ 4.37228 0.247929 0.123965 0.992287i $$-0.460439\pi$$
0.123965 + 0.992287i $$0.460439\pi$$
$$312$$ −13.4891 −0.763671
$$313$$ −19.4891 −1.10159 −0.550795 0.834640i $$-0.685675\pi$$
−0.550795 + 0.834640i $$0.685675\pi$$
$$314$$ 11.6277 0.656190
$$315$$ 0 0
$$316$$ 6.74456 0.379411
$$317$$ −19.6277 −1.10240 −0.551201 0.834372i $$-0.685830\pi$$
−0.551201 + 0.834372i $$0.685830\pi$$
$$318$$ 8.74456 0.490371
$$319$$ −21.6277 −1.21092
$$320$$ 0 0
$$321$$ 38.9783 2.17555
$$322$$ −20.7446 −1.15605
$$323$$ 0.744563 0.0414286
$$324$$ −11.0000 −0.611111
$$325$$ 0 0
$$326$$ 13.6277 0.754770
$$327$$ 34.2337 1.89313
$$328$$ 0.372281 0.0205558
$$329$$ −12.0000 −0.661581
$$330$$ 0 0
$$331$$ 0.510875 0.0280802 0.0140401 0.999901i $$-0.495531\pi$$
0.0140401 + 0.999901i $$0.495531\pi$$
$$332$$ −10.7446 −0.589684
$$333$$ −1.00000 −0.0547997
$$334$$ −21.4891 −1.17583
$$335$$ 0 0
$$336$$ −8.74456 −0.477055
$$337$$ 18.7446 1.02108 0.510541 0.859854i $$-0.329445\pi$$
0.510541 + 0.859854i $$0.329445\pi$$
$$338$$ −32.4891 −1.76718
$$339$$ 23.2554 1.26306
$$340$$ 0 0
$$341$$ −19.8614 −1.07556
$$342$$ 2.00000 0.108148
$$343$$ 22.3723 1.20799
$$344$$ 1.62772 0.0877607
$$345$$ 0 0
$$346$$ −5.86141 −0.315111
$$347$$ −22.9783 −1.23354 −0.616769 0.787145i $$-0.711559\pi$$
−0.616769 + 0.787145i $$0.711559\pi$$
$$348$$ 18.2337 0.977428
$$349$$ −22.0000 −1.17763 −0.588817 0.808267i $$-0.700406\pi$$
−0.588817 + 0.808267i $$0.700406\pi$$
$$350$$ 0 0
$$351$$ −26.9783 −1.43999
$$352$$ −2.37228 −0.126443
$$353$$ −5.86141 −0.311971 −0.155986 0.987759i $$-0.549855\pi$$
−0.155986 + 0.987759i $$0.549855\pi$$
$$354$$ 2.51087 0.133451
$$355$$ 0 0
$$356$$ 10.0000 0.529999
$$357$$ 3.25544 0.172296
$$358$$ −8.23369 −0.435164
$$359$$ 14.9783 0.790522 0.395261 0.918569i $$-0.370654\pi$$
0.395261 + 0.918569i $$0.370654\pi$$
$$360$$ 0 0
$$361$$ −15.0000 −0.789474
$$362$$ 15.4891 0.814090
$$363$$ 10.7446 0.563943
$$364$$ −29.4891 −1.54565
$$365$$ 0 0
$$366$$ 0.744563 0.0389189
$$367$$ −21.1168 −1.10229 −0.551145 0.834409i $$-0.685809\pi$$
−0.551145 + 0.834409i $$0.685809\pi$$
$$368$$ 4.74456 0.247327
$$369$$ −0.372281 −0.0193802
$$370$$ 0 0
$$371$$ 19.1168 0.992497
$$372$$ 16.7446 0.868165
$$373$$ 8.51087 0.440676 0.220338 0.975424i $$-0.429284\pi$$
0.220338 + 0.975424i $$0.429284\pi$$
$$374$$ 0.883156 0.0456669
$$375$$ 0 0
$$376$$ 2.74456 0.141540
$$377$$ 61.4891 3.16685
$$378$$ −17.4891 −0.899544
$$379$$ −8.00000 −0.410932 −0.205466 0.978664i $$-0.565871\pi$$
−0.205466 + 0.978664i $$0.565871\pi$$
$$380$$ 0 0
$$381$$ −10.5109 −0.538488
$$382$$ −24.3723 −1.24699
$$383$$ −9.48913 −0.484872 −0.242436 0.970167i $$-0.577946\pi$$
−0.242436 + 0.970167i $$0.577946\pi$$
$$384$$ 2.00000 0.102062
$$385$$ 0 0
$$386$$ 2.00000 0.101797
$$387$$ −1.62772 −0.0827416
$$388$$ −17.1168 −0.868976
$$389$$ −13.8614 −0.702801 −0.351401 0.936225i $$-0.614294\pi$$
−0.351401 + 0.936225i $$0.614294\pi$$
$$390$$ 0 0
$$391$$ −1.76631 −0.0893262
$$392$$ −12.1168 −0.611993
$$393$$ 29.4891 1.48753
$$394$$ 4.51087 0.227255
$$395$$ 0 0
$$396$$ 2.37228 0.119212
$$397$$ 20.9783 1.05287 0.526434 0.850216i $$-0.323529\pi$$
0.526434 + 0.850216i $$0.323529\pi$$
$$398$$ 22.7446 1.14008
$$399$$ 17.4891 0.875551
$$400$$ 0 0
$$401$$ 10.0000 0.499376 0.249688 0.968326i $$-0.419672\pi$$
0.249688 + 0.968326i $$0.419672\pi$$
$$402$$ 13.4891 0.672776
$$403$$ 56.4674 2.81284
$$404$$ 11.4891 0.571605
$$405$$ 0 0
$$406$$ 39.8614 1.97829
$$407$$ −2.37228 −0.117590
$$408$$ −0.744563 −0.0368613
$$409$$ −28.2337 −1.39607 −0.698033 0.716066i $$-0.745941\pi$$
−0.698033 + 0.716066i $$0.745941\pi$$
$$410$$ 0 0
$$411$$ 10.5109 0.518463
$$412$$ −13.4891 −0.664562
$$413$$ 5.48913 0.270102
$$414$$ −4.74456 −0.233183
$$415$$ 0 0
$$416$$ 6.74456 0.330679
$$417$$ 38.2337 1.87231
$$418$$ 4.74456 0.232064
$$419$$ −9.48913 −0.463574 −0.231787 0.972767i $$-0.574457\pi$$
−0.231787 + 0.972767i $$0.574457\pi$$
$$420$$ 0 0
$$421$$ 15.4891 0.754894 0.377447 0.926031i $$-0.376802\pi$$
0.377447 + 0.926031i $$0.376802\pi$$
$$422$$ −3.11684 −0.151726
$$423$$ −2.74456 −0.133445
$$424$$ −4.37228 −0.212337
$$425$$ 0 0
$$426$$ 9.48913 0.459750
$$427$$ 1.62772 0.0787708
$$428$$ −19.4891 −0.942042
$$429$$ 32.0000 1.54497
$$430$$ 0 0
$$431$$ 4.37228 0.210605 0.105303 0.994440i $$-0.466419\pi$$
0.105303 + 0.994440i $$0.466419\pi$$
$$432$$ 4.00000 0.192450
$$433$$ 32.9783 1.58483 0.792417 0.609980i $$-0.208823\pi$$
0.792417 + 0.609980i $$0.208823\pi$$
$$434$$ 36.6060 1.75714
$$435$$ 0 0
$$436$$ −17.1168 −0.819748
$$437$$ −9.48913 −0.453926
$$438$$ 5.48913 0.262281
$$439$$ −7.62772 −0.364051 −0.182026 0.983294i $$-0.558265\pi$$
−0.182026 + 0.983294i $$0.558265\pi$$
$$440$$ 0 0
$$441$$ 12.1168 0.576993
$$442$$ −2.51087 −0.119430
$$443$$ −28.9783 −1.37680 −0.688399 0.725332i $$-0.741686\pi$$
−0.688399 + 0.725332i $$0.741686\pi$$
$$444$$ 2.00000 0.0949158
$$445$$ 0 0
$$446$$ 4.37228 0.207034
$$447$$ −22.9783 −1.08683
$$448$$ 4.37228 0.206571
$$449$$ 18.0000 0.849473 0.424736 0.905317i $$-0.360367\pi$$
0.424736 + 0.905317i $$0.360367\pi$$
$$450$$ 0 0
$$451$$ −0.883156 −0.0415862
$$452$$ −11.6277 −0.546922
$$453$$ 40.0000 1.87936
$$454$$ −5.62772 −0.264122
$$455$$ 0 0
$$456$$ −4.00000 −0.187317
$$457$$ 9.86141 0.461297 0.230649 0.973037i $$-0.425915\pi$$
0.230649 + 0.973037i $$0.425915\pi$$
$$458$$ 10.0000 0.467269
$$459$$ −1.48913 −0.0695064
$$460$$ 0 0
$$461$$ 24.0951 1.12222 0.561110 0.827741i $$-0.310374\pi$$
0.561110 + 0.827741i $$0.310374\pi$$
$$462$$ 20.7446 0.965124
$$463$$ −17.4891 −0.812789 −0.406394 0.913698i $$-0.633214\pi$$
−0.406394 + 0.913698i $$0.633214\pi$$
$$464$$ −9.11684 −0.423239
$$465$$ 0 0
$$466$$ −28.2337 −1.30790
$$467$$ 21.3505 0.987985 0.493992 0.869466i $$-0.335537\pi$$
0.493992 + 0.869466i $$0.335537\pi$$
$$468$$ −6.74456 −0.311768
$$469$$ 29.4891 1.36168
$$470$$ 0 0
$$471$$ 23.2554 1.07155
$$472$$ −1.25544 −0.0577862
$$473$$ −3.86141 −0.177548
$$474$$ 13.4891 0.619576
$$475$$ 0 0
$$476$$ −1.62772 −0.0746064
$$477$$ 4.37228 0.200193
$$478$$ 22.6060 1.03397
$$479$$ 14.7446 0.673696 0.336848 0.941559i $$-0.390639\pi$$
0.336848 + 0.941559i $$0.390639\pi$$
$$480$$ 0 0
$$481$$ 6.74456 0.307526
$$482$$ −24.2337 −1.10381
$$483$$ −41.4891 −1.88782
$$484$$ −5.37228 −0.244195
$$485$$ 0 0
$$486$$ −10.0000 −0.453609
$$487$$ −19.7228 −0.893726 −0.446863 0.894602i $$-0.647459\pi$$
−0.446863 + 0.894602i $$0.647459\pi$$
$$488$$ −0.372281 −0.0168524
$$489$$ 27.2554 1.23253
$$490$$ 0 0
$$491$$ −30.9783 −1.39803 −0.699014 0.715108i $$-0.746378\pi$$
−0.699014 + 0.715108i $$0.746378\pi$$
$$492$$ 0.744563 0.0335675
$$493$$ 3.39403 0.152859
$$494$$ −13.4891 −0.606904
$$495$$ 0 0
$$496$$ −8.37228 −0.375927
$$497$$ 20.7446 0.930521
$$498$$ −21.4891 −0.962951
$$499$$ 20.9783 0.939115 0.469558 0.882902i $$-0.344413\pi$$
0.469558 + 0.882902i $$0.344413\pi$$
$$500$$ 0 0
$$501$$ −42.9783 −1.92013
$$502$$ 11.4891 0.512785
$$503$$ −6.51087 −0.290306 −0.145153 0.989409i $$-0.546367\pi$$
−0.145153 + 0.989409i $$0.546367\pi$$
$$504$$ −4.37228 −0.194757
$$505$$ 0 0
$$506$$ −11.2554 −0.500365
$$507$$ −64.9783 −2.88579
$$508$$ 5.25544 0.233172
$$509$$ −22.7446 −1.00814 −0.504068 0.863664i $$-0.668164\pi$$
−0.504068 + 0.863664i $$0.668164\pi$$
$$510$$ 0 0
$$511$$ 12.0000 0.530849
$$512$$ −1.00000 −0.0441942
$$513$$ −8.00000 −0.353209
$$514$$ 24.9783 1.10174
$$515$$ 0 0
$$516$$ 3.25544 0.143313
$$517$$ −6.51087 −0.286348
$$518$$ 4.37228 0.192107
$$519$$ −11.7228 −0.514574
$$520$$ 0 0
$$521$$ −36.0951 −1.58135 −0.790677 0.612233i $$-0.790271\pi$$
−0.790677 + 0.612233i $$0.790271\pi$$
$$522$$ 9.11684 0.399033
$$523$$ −28.0000 −1.22435 −0.612177 0.790721i $$-0.709706\pi$$
−0.612177 + 0.790721i $$0.709706\pi$$
$$524$$ −14.7446 −0.644119
$$525$$ 0 0
$$526$$ −17.1168 −0.746330
$$527$$ 3.11684 0.135772
$$528$$ −4.74456 −0.206481
$$529$$ −0.489125 −0.0212663
$$530$$ 0 0
$$531$$ 1.25544 0.0544813
$$532$$ −8.74456 −0.379125
$$533$$ 2.51087 0.108758
$$534$$ 20.0000 0.865485
$$535$$ 0 0
$$536$$ −6.74456 −0.291321
$$537$$ −16.4674 −0.710620
$$538$$ −10.0000 −0.431131
$$539$$ 28.7446 1.23812
$$540$$ 0 0
$$541$$ 30.0000 1.28980 0.644900 0.764267i $$-0.276899\pi$$
0.644900 + 0.764267i $$0.276899\pi$$
$$542$$ −4.74456 −0.203796
$$543$$ 30.9783 1.32940
$$544$$ 0.372281 0.0159614
$$545$$ 0 0
$$546$$ −58.9783 −2.52404
$$547$$ −33.6277 −1.43782 −0.718909 0.695104i $$-0.755358\pi$$
−0.718909 + 0.695104i $$0.755358\pi$$
$$548$$ −5.25544 −0.224501
$$549$$ 0.372281 0.0158886
$$550$$ 0 0
$$551$$ 18.2337 0.776781
$$552$$ 9.48913 0.403884
$$553$$ 29.4891 1.25401
$$554$$ −10.7446 −0.456493
$$555$$ 0 0
$$556$$ −19.1168 −0.810735
$$557$$ 2.74456 0.116291 0.0581454 0.998308i $$-0.481481\pi$$
0.0581454 + 0.998308i $$0.481481\pi$$
$$558$$ 8.37228 0.354427
$$559$$ 10.9783 0.464331
$$560$$ 0 0
$$561$$ 1.76631 0.0745738
$$562$$ −13.2554 −0.559147
$$563$$ 30.0951 1.26836 0.634179 0.773187i $$-0.281338\pi$$
0.634179 + 0.773187i $$0.281338\pi$$
$$564$$ 5.48913 0.231134
$$565$$ 0 0
$$566$$ −17.4891 −0.735123
$$567$$ −48.0951 −2.01980
$$568$$ −4.74456 −0.199077
$$569$$ 12.9783 0.544077 0.272038 0.962286i $$-0.412302\pi$$
0.272038 + 0.962286i $$0.412302\pi$$
$$570$$ 0 0
$$571$$ 12.6060 0.527543 0.263772 0.964585i $$-0.415033\pi$$
0.263772 + 0.964585i $$0.415033\pi$$
$$572$$ −16.0000 −0.668994
$$573$$ −48.7446 −2.03633
$$574$$ 1.62772 0.0679397
$$575$$ 0 0
$$576$$ 1.00000 0.0416667
$$577$$ −15.4891 −0.644821 −0.322410 0.946600i $$-0.604493\pi$$
−0.322410 + 0.946600i $$0.604493\pi$$
$$578$$ 16.8614 0.701342
$$579$$ 4.00000 0.166234
$$580$$ 0 0
$$581$$ −46.9783 −1.94899
$$582$$ −34.2337 −1.41903
$$583$$ 10.3723 0.429576
$$584$$ −2.74456 −0.113571
$$585$$ 0 0
$$586$$ 9.86141 0.407371
$$587$$ 26.8397 1.10779 0.553896 0.832586i $$-0.313141\pi$$
0.553896 + 0.832586i $$0.313141\pi$$
$$588$$ −24.2337 −0.999380
$$589$$ 16.7446 0.689948
$$590$$ 0 0
$$591$$ 9.02175 0.371105
$$592$$ −1.00000 −0.0410997
$$593$$ −24.2337 −0.995158 −0.497579 0.867419i $$-0.665778\pi$$
−0.497579 + 0.867419i $$0.665778\pi$$
$$594$$ −9.48913 −0.389344
$$595$$ 0 0
$$596$$ 11.4891 0.470613
$$597$$ 45.4891 1.86175
$$598$$ 32.0000 1.30858
$$599$$ −5.48913 −0.224280 −0.112140 0.993692i $$-0.535770\pi$$
−0.112140 + 0.993692i $$0.535770\pi$$
$$600$$ 0 0
$$601$$ 0.372281 0.0151857 0.00759284 0.999971i $$-0.497583\pi$$
0.00759284 + 0.999971i $$0.497583\pi$$
$$602$$ 7.11684 0.290061
$$603$$ 6.74456 0.274660
$$604$$ −20.0000 −0.813788
$$605$$ 0 0
$$606$$ 22.9783 0.933428
$$607$$ 39.7228 1.61230 0.806150 0.591712i $$-0.201548\pi$$
0.806150 + 0.591712i $$0.201548\pi$$
$$608$$ 2.00000 0.0811107
$$609$$ 79.7228 3.23053
$$610$$ 0 0
$$611$$ 18.5109 0.748870
$$612$$ −0.372281 −0.0150486
$$613$$ 20.0951 0.811633 0.405817 0.913955i $$-0.366987\pi$$
0.405817 + 0.913955i $$0.366987\pi$$
$$614$$ −0.510875 −0.0206172
$$615$$ 0 0
$$616$$ −10.3723 −0.417911
$$617$$ −25.7228 −1.03556 −0.517781 0.855513i $$-0.673242\pi$$
−0.517781 + 0.855513i $$0.673242\pi$$
$$618$$ −26.9783 −1.08522
$$619$$ 12.8832 0.517818 0.258909 0.965902i $$-0.416637\pi$$
0.258909 + 0.965902i $$0.416637\pi$$
$$620$$ 0 0
$$621$$ 18.9783 0.761571
$$622$$ −4.37228 −0.175313
$$623$$ 43.7228 1.75172
$$624$$ 13.4891 0.539997
$$625$$ 0 0
$$626$$ 19.4891 0.778942
$$627$$ 9.48913 0.378959
$$628$$ −11.6277 −0.463996
$$629$$ 0.372281 0.0148438
$$630$$ 0 0
$$631$$ 44.0951 1.75540 0.877699 0.479212i $$-0.159078\pi$$
0.877699 + 0.479212i $$0.159078\pi$$
$$632$$ −6.74456 −0.268284
$$633$$ −6.23369 −0.247767
$$634$$ 19.6277 0.779516
$$635$$ 0 0
$$636$$ −8.74456 −0.346744
$$637$$ −81.7228 −3.23798
$$638$$ 21.6277 0.856250
$$639$$ 4.74456 0.187692
$$640$$ 0 0
$$641$$ −9.39403 −0.371042 −0.185521 0.982640i $$-0.559397\pi$$
−0.185521 + 0.982640i $$0.559397\pi$$
$$642$$ −38.9783 −1.53835
$$643$$ 2.37228 0.0935536 0.0467768 0.998905i $$-0.485105\pi$$
0.0467768 + 0.998905i $$0.485105\pi$$
$$644$$ 20.7446 0.817450
$$645$$ 0 0
$$646$$ −0.744563 −0.0292944
$$647$$ −9.48913 −0.373056 −0.186528 0.982450i $$-0.559724\pi$$
−0.186528 + 0.982450i $$0.559724\pi$$
$$648$$ 11.0000 0.432121
$$649$$ 2.97825 0.116907
$$650$$ 0 0
$$651$$ 73.2119 2.86940
$$652$$ −13.6277 −0.533703
$$653$$ −23.4891 −0.919201 −0.459600 0.888126i $$-0.652007\pi$$
−0.459600 + 0.888126i $$0.652007\pi$$
$$654$$ −34.2337 −1.33864
$$655$$ 0 0
$$656$$ −0.372281 −0.0145351
$$657$$ 2.74456 0.107076
$$658$$ 12.0000 0.467809
$$659$$ 12.0000 0.467454 0.233727 0.972302i $$-0.424908\pi$$
0.233727 + 0.972302i $$0.424908\pi$$
$$660$$ 0 0
$$661$$ −47.3505 −1.84172 −0.920861 0.389891i $$-0.872513\pi$$
−0.920861 + 0.389891i $$0.872513\pi$$
$$662$$ −0.510875 −0.0198557
$$663$$ −5.02175 −0.195029
$$664$$ 10.7446 0.416970
$$665$$ 0 0
$$666$$ 1.00000 0.0387492
$$667$$ −43.2554 −1.67486
$$668$$ 21.4891 0.831439
$$669$$ 8.74456 0.338084
$$670$$ 0 0
$$671$$ 0.883156 0.0340939
$$672$$ 8.74456 0.337329
$$673$$ 31.4891 1.21382 0.606908 0.794772i $$-0.292410\pi$$
0.606908 + 0.794772i $$0.292410\pi$$
$$674$$ −18.7446 −0.722014
$$675$$ 0 0
$$676$$ 32.4891 1.24958
$$677$$ 42.0000 1.61419 0.807096 0.590421i $$-0.201038\pi$$
0.807096 + 0.590421i $$0.201038\pi$$
$$678$$ −23.2554 −0.893120
$$679$$ −74.8397 −2.87208
$$680$$ 0 0
$$681$$ −11.2554 −0.431309
$$682$$ 19.8614 0.760533
$$683$$ −14.3723 −0.549940 −0.274970 0.961453i $$-0.588668\pi$$
−0.274970 + 0.961453i $$0.588668\pi$$
$$684$$ −2.00000 −0.0764719
$$685$$ 0 0
$$686$$ −22.3723 −0.854178
$$687$$ 20.0000 0.763048
$$688$$ −1.62772 −0.0620562
$$689$$ −29.4891 −1.12345
$$690$$ 0 0
$$691$$ −29.3505 −1.11655 −0.558273 0.829657i $$-0.688536\pi$$
−0.558273 + 0.829657i $$0.688536\pi$$
$$692$$ 5.86141 0.222817
$$693$$ 10.3723 0.394010
$$694$$ 22.9783 0.872242
$$695$$ 0 0
$$696$$ −18.2337 −0.691146
$$697$$ 0.138593 0.00524960
$$698$$ 22.0000 0.832712
$$699$$ −56.4674 −2.13579
$$700$$ 0 0
$$701$$ 42.4674 1.60397 0.801985 0.597344i $$-0.203777\pi$$
0.801985 + 0.597344i $$0.203777\pi$$
$$702$$ 26.9783 1.01823
$$703$$ 2.00000 0.0754314
$$704$$ 2.37228 0.0894087
$$705$$ 0 0
$$706$$ 5.86141 0.220597
$$707$$ 50.2337 1.88923
$$708$$ −2.51087 −0.0943645
$$709$$ 22.8832 0.859395 0.429697 0.902973i $$-0.358620\pi$$
0.429697 + 0.902973i $$0.358620\pi$$
$$710$$ 0 0
$$711$$ 6.74456 0.252941
$$712$$ −10.0000 −0.374766
$$713$$ −39.7228 −1.48763
$$714$$ −3.25544 −0.121832
$$715$$ 0 0
$$716$$ 8.23369 0.307707
$$717$$ 45.2119 1.68847
$$718$$ −14.9783 −0.558983
$$719$$ −23.2554 −0.867281 −0.433641 0.901086i $$-0.642771\pi$$
−0.433641 + 0.901086i $$0.642771\pi$$
$$720$$ 0 0
$$721$$ −58.9783 −2.19646
$$722$$ 15.0000 0.558242
$$723$$ −48.4674 −1.80252
$$724$$ −15.4891 −0.575649
$$725$$ 0 0
$$726$$ −10.7446 −0.398768
$$727$$ 48.0000 1.78022 0.890111 0.455744i $$-0.150627\pi$$
0.890111 + 0.455744i $$0.150627\pi$$
$$728$$ 29.4891 1.09294
$$729$$ 13.0000 0.481481
$$730$$ 0 0
$$731$$ 0.605969 0.0224126
$$732$$ −0.744563 −0.0275198
$$733$$ 23.6277 0.872710 0.436355 0.899775i $$-0.356269\pi$$
0.436355 + 0.899775i $$0.356269\pi$$
$$734$$ 21.1168 0.779437
$$735$$ 0 0
$$736$$ −4.74456 −0.174887
$$737$$ 16.0000 0.589368
$$738$$ 0.372281 0.0137039
$$739$$ −8.13859 −0.299383 −0.149691 0.988733i $$-0.547828\pi$$
−0.149691 + 0.988733i $$0.547828\pi$$
$$740$$ 0 0
$$741$$ −26.9783 −0.991071
$$742$$ −19.1168 −0.701801
$$743$$ −0.372281 −0.0136577 −0.00682884 0.999977i $$-0.502174\pi$$
−0.00682884 + 0.999977i $$0.502174\pi$$
$$744$$ −16.7446 −0.613885
$$745$$ 0 0
$$746$$ −8.51087 −0.311605
$$747$$ −10.7446 −0.393123
$$748$$ −0.883156 −0.0322914
$$749$$ −85.2119 −3.11358
$$750$$ 0 0
$$751$$ −0.744563 −0.0271695 −0.0135847 0.999908i $$-0.504324\pi$$
−0.0135847 + 0.999908i $$0.504324\pi$$
$$752$$ −2.74456 −0.100084
$$753$$ 22.9783 0.837374
$$754$$ −61.4891 −2.23930
$$755$$ 0 0
$$756$$ 17.4891 0.636073
$$757$$ 47.9565 1.74301 0.871504 0.490388i $$-0.163145\pi$$
0.871504 + 0.490388i $$0.163145\pi$$
$$758$$ 8.00000 0.290573
$$759$$ −22.5109 −0.817093
$$760$$ 0 0
$$761$$ −0.372281 −0.0134952 −0.00674759 0.999977i $$-0.502148\pi$$
−0.00674759 + 0.999977i $$0.502148\pi$$
$$762$$ 10.5109 0.380769
$$763$$ −74.8397 −2.70938
$$764$$ 24.3723 0.881758
$$765$$ 0 0
$$766$$ 9.48913 0.342856
$$767$$ −8.46738 −0.305739
$$768$$ −2.00000 −0.0721688
$$769$$ 11.7663 0.424304 0.212152 0.977237i $$-0.431953\pi$$
0.212152 + 0.977237i $$0.431953\pi$$
$$770$$ 0 0
$$771$$ 49.9565 1.79914
$$772$$ −2.00000 −0.0719816
$$773$$ −27.3505 −0.983730 −0.491865 0.870671i $$-0.663685\pi$$
−0.491865 + 0.870671i $$0.663685\pi$$
$$774$$ 1.62772 0.0585071
$$775$$ 0 0
$$776$$ 17.1168 0.614459
$$777$$ 8.74456 0.313709
$$778$$ 13.8614 0.496956
$$779$$ 0.744563 0.0266767
$$780$$ 0 0
$$781$$ 11.2554 0.402751
$$782$$ 1.76631 0.0631632
$$783$$ −36.4674 −1.30324
$$784$$ 12.1168 0.432744
$$785$$ 0 0
$$786$$ −29.4891 −1.05184
$$787$$ 22.0000 0.784215 0.392108 0.919919i $$-0.371746\pi$$
0.392108 + 0.919919i $$0.371746\pi$$
$$788$$ −4.51087 −0.160693
$$789$$ −34.2337 −1.21875
$$790$$ 0 0
$$791$$ −50.8397 −1.80765
$$792$$ −2.37228 −0.0842953
$$793$$ −2.51087 −0.0891638
$$794$$ −20.9783 −0.744490
$$795$$ 0 0
$$796$$ −22.7446 −0.806160
$$797$$ 4.51087 0.159783 0.0798917 0.996804i $$-0.474543\pi$$
0.0798917 + 0.996804i $$0.474543\pi$$
$$798$$ −17.4891 −0.619108
$$799$$ 1.02175 0.0361469
$$800$$ 0 0
$$801$$ 10.0000 0.353333
$$802$$ −10.0000 −0.353112
$$803$$ 6.51087 0.229764
$$804$$ −13.4891 −0.475725
$$805$$ 0 0
$$806$$ −56.4674 −1.98898
$$807$$ −20.0000 −0.704033
$$808$$ −11.4891 −0.404186
$$809$$ −28.5109 −1.00239 −0.501194 0.865335i $$-0.667106\pi$$
−0.501194 + 0.865335i $$0.667106\pi$$
$$810$$ 0 0
$$811$$ 26.5109 0.930923 0.465461 0.885068i $$-0.345888\pi$$
0.465461 + 0.885068i $$0.345888\pi$$
$$812$$ −39.8614 −1.39886
$$813$$ −9.48913 −0.332798
$$814$$ 2.37228 0.0831484
$$815$$ 0 0
$$816$$ 0.744563 0.0260649
$$817$$ 3.25544 0.113893
$$818$$ 28.2337 0.987168
$$819$$ −29.4891 −1.03043
$$820$$ 0 0
$$821$$ −21.2554 −0.741820 −0.370910 0.928669i $$-0.620954\pi$$
−0.370910 + 0.928669i $$0.620954\pi$$
$$822$$ −10.5109 −0.366609
$$823$$ −17.7228 −0.617778 −0.308889 0.951098i $$-0.599957\pi$$
−0.308889 + 0.951098i $$0.599957\pi$$
$$824$$ 13.4891 0.469916
$$825$$ 0 0
$$826$$ −5.48913 −0.190991
$$827$$ 2.64947 0.0921310 0.0460655 0.998938i $$-0.485332\pi$$
0.0460655 + 0.998938i $$0.485332\pi$$
$$828$$ 4.74456 0.164885
$$829$$ 11.3505 0.394220 0.197110 0.980381i $$-0.436844\pi$$
0.197110 + 0.980381i $$0.436844\pi$$
$$830$$ 0 0
$$831$$ −21.4891 −0.745449
$$832$$ −6.74456 −0.233826
$$833$$ −4.51087 −0.156293
$$834$$ −38.2337 −1.32392
$$835$$ 0 0
$$836$$ −4.74456 −0.164094
$$837$$ −33.4891 −1.15755
$$838$$ 9.48913 0.327796
$$839$$ −6.51087 −0.224780 −0.112390 0.993664i $$-0.535851\pi$$
−0.112390 + 0.993664i $$0.535851\pi$$
$$840$$ 0 0
$$841$$ 54.1168 1.86610
$$842$$ −15.4891 −0.533791
$$843$$ −26.5109 −0.913083
$$844$$ 3.11684 0.107286
$$845$$ 0 0
$$846$$ 2.74456 0.0943600
$$847$$ −23.4891 −0.807096
$$848$$ 4.37228 0.150145
$$849$$ −34.9783 −1.20045
$$850$$ 0 0
$$851$$ −4.74456 −0.162642
$$852$$ −9.48913 −0.325092
$$853$$ −12.5109 −0.428364 −0.214182 0.976794i $$-0.568709\pi$$
−0.214182 + 0.976794i $$0.568709\pi$$
$$854$$ −1.62772 −0.0556994
$$855$$ 0 0
$$856$$ 19.4891 0.666125
$$857$$ 57.8614 1.97651 0.988254 0.152820i $$-0.0488356\pi$$
0.988254 + 0.152820i $$0.0488356\pi$$
$$858$$ −32.0000 −1.09246
$$859$$ 37.7228 1.28709 0.643543 0.765410i $$-0.277464\pi$$
0.643543 + 0.765410i $$0.277464\pi$$
$$860$$ 0 0
$$861$$ 3.25544 0.110945
$$862$$ −4.37228 −0.148920
$$863$$ 20.8397 0.709390 0.354695 0.934982i $$-0.384585\pi$$
0.354695 + 0.934982i $$0.384585\pi$$
$$864$$ −4.00000 −0.136083
$$865$$ 0 0
$$866$$ −32.9783 −1.12065
$$867$$ 33.7228 1.14529
$$868$$ −36.6060 −1.24249
$$869$$ 16.0000 0.542763
$$870$$ 0 0
$$871$$ −45.4891 −1.54134
$$872$$ 17.1168 0.579649
$$873$$ −17.1168 −0.579317
$$874$$ 9.48913 0.320974
$$875$$ 0 0
$$876$$ −5.48913 −0.185460
$$877$$ −43.6277 −1.47320 −0.736602 0.676327i $$-0.763571\pi$$
−0.736602 + 0.676327i $$0.763571\pi$$
$$878$$ 7.62772 0.257423
$$879$$ 19.7228 0.665234
$$880$$ 0 0
$$881$$ −4.37228 −0.147306 −0.0736530 0.997284i $$-0.523466\pi$$
−0.0736530 + 0.997284i $$0.523466\pi$$
$$882$$ −12.1168 −0.407995
$$883$$ 5.62772 0.189388 0.0946939 0.995506i $$-0.469813\pi$$
0.0946939 + 0.995506i $$0.469813\pi$$
$$884$$ 2.51087 0.0844499
$$885$$ 0 0
$$886$$ 28.9783 0.973543
$$887$$ 19.6277 0.659034 0.329517 0.944150i $$-0.393114\pi$$
0.329517 + 0.944150i $$0.393114\pi$$
$$888$$ −2.00000 −0.0671156
$$889$$ 22.9783 0.770666
$$890$$ 0 0
$$891$$ −26.0951 −0.874219
$$892$$ −4.37228 −0.146395
$$893$$ 5.48913 0.183687
$$894$$ 22.9783 0.768508
$$895$$ 0 0
$$896$$ −4.37228 −0.146068
$$897$$ 64.0000 2.13690
$$898$$ −18.0000 −0.600668
$$899$$ 76.3288 2.54571
$$900$$ 0 0
$$901$$ −1.62772 −0.0542272
$$902$$ 0.883156 0.0294059
$$903$$ 14.2337 0.473667
$$904$$ 11.6277 0.386732
$$905$$ 0 0
$$906$$ −40.0000 −1.32891
$$907$$ 28.4674 0.945244 0.472622 0.881265i $$-0.343308\pi$$
0.472622 + 0.881265i $$0.343308\pi$$
$$908$$ 5.62772 0.186762
$$909$$ 11.4891 0.381070
$$910$$ 0 0
$$911$$ 52.2337 1.73058 0.865290 0.501272i $$-0.167134\pi$$
0.865290 + 0.501272i $$0.167134\pi$$
$$912$$ 4.00000 0.132453
$$913$$ −25.4891 −0.843567
$$914$$ −9.86141 −0.326186
$$915$$ 0 0
$$916$$ −10.0000 −0.330409
$$917$$ −64.4674 −2.12890
$$918$$ 1.48913 0.0491485
$$919$$ −49.7228 −1.64020 −0.820102 0.572217i $$-0.806083\pi$$
−0.820102 + 0.572217i $$0.806083\pi$$
$$920$$ 0 0
$$921$$ −1.02175 −0.0336678
$$922$$ −24.0951 −0.793530
$$923$$ −32.0000 −1.05329
$$924$$ −20.7446 −0.682446
$$925$$ 0 0
$$926$$ 17.4891 0.574728
$$927$$ −13.4891 −0.443041
$$928$$ 9.11684 0.299275
$$929$$ −32.0951 −1.05301 −0.526503 0.850173i $$-0.676497\pi$$
−0.526503 + 0.850173i $$0.676497\pi$$
$$930$$ 0 0
$$931$$ −24.2337 −0.794227
$$932$$ 28.2337 0.924825
$$933$$ −8.74456 −0.286284
$$934$$ −21.3505 −0.698611
$$935$$ 0 0
$$936$$ 6.74456 0.220453
$$937$$ 8.97825 0.293307 0.146653 0.989188i $$-0.453150\pi$$
0.146653 + 0.989188i $$0.453150\pi$$
$$938$$ −29.4891 −0.962854
$$939$$ 38.9783 1.27201
$$940$$ 0 0
$$941$$ 18.0000 0.586783 0.293392 0.955992i $$-0.405216\pi$$
0.293392 + 0.955992i $$0.405216\pi$$
$$942$$ −23.2554 −0.757703
$$943$$ −1.76631 −0.0575190
$$944$$ 1.25544 0.0408610
$$945$$ 0 0
$$946$$ 3.86141 0.125545
$$947$$ 26.0951 0.847977 0.423988 0.905668i $$-0.360630\pi$$
0.423988 + 0.905668i $$0.360630\pi$$
$$948$$ −13.4891 −0.438106
$$949$$ −18.5109 −0.600888
$$950$$ 0 0
$$951$$ 39.2554 1.27294
$$952$$ 1.62772 0.0527547
$$953$$ −11.7663 −0.381148 −0.190574 0.981673i $$-0.561035\pi$$
−0.190574 + 0.981673i $$0.561035\pi$$
$$954$$ −4.37228 −0.141558
$$955$$ 0 0
$$956$$ −22.6060 −0.731129
$$957$$ 43.2554 1.39825
$$958$$ −14.7446 −0.476375
$$959$$ −22.9783 −0.742006
$$960$$ 0 0
$$961$$ 39.0951 1.26113
$$962$$ −6.74456 −0.217453
$$963$$ −19.4891 −0.628028
$$964$$ 24.2337 0.780515
$$965$$ 0 0
$$966$$ 41.4891 1.33489
$$967$$ 28.0000 0.900419 0.450210 0.892923i $$-0.351349\pi$$
0.450210 + 0.892923i $$0.351349\pi$$
$$968$$ 5.37228 0.172672
$$969$$ −1.48913 −0.0478376
$$970$$ 0 0
$$971$$ 20.6060 0.661277 0.330639 0.943757i $$-0.392736\pi$$
0.330639 + 0.943757i $$0.392736\pi$$
$$972$$ 10.0000 0.320750
$$973$$ −83.5842 −2.67959
$$974$$ 19.7228 0.631960
$$975$$ 0 0
$$976$$ 0.372281 0.0119164
$$977$$ 29.1168 0.931530 0.465765 0.884908i $$-0.345779\pi$$
0.465765 + 0.884908i $$0.345779\pi$$
$$978$$ −27.2554 −0.871533
$$979$$ 23.7228 0.758184
$$980$$ 0 0
$$981$$ −17.1168 −0.546499
$$982$$ 30.9783 0.988556
$$983$$ −2.13859 −0.0682105 −0.0341053 0.999418i $$-0.510858\pi$$
−0.0341053 + 0.999418i $$0.510858\pi$$
$$984$$ −0.744563 −0.0237358
$$985$$ 0 0
$$986$$ −3.39403 −0.108088
$$987$$ 24.0000 0.763928
$$988$$ 13.4891 0.429146
$$989$$ −7.72281 −0.245571
$$990$$ 0 0
$$991$$ −14.6060 −0.463974 −0.231987 0.972719i $$-0.574523\pi$$
−0.231987 + 0.972719i $$0.574523\pi$$
$$992$$ 8.37228 0.265820
$$993$$ −1.02175 −0.0324242
$$994$$ −20.7446 −0.657978
$$995$$ 0 0
$$996$$ 21.4891 0.680909
$$997$$ 8.23369 0.260764 0.130382 0.991464i $$-0.458380\pi$$
0.130382 + 0.991464i $$0.458380\pi$$
$$998$$ −20.9783 −0.664055
$$999$$ −4.00000 −0.126554
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 1850.2.a.q.1.2 2
5.2 odd 4 1850.2.b.m.149.2 4
5.3 odd 4 1850.2.b.m.149.3 4
5.4 even 2 370.2.a.f.1.1 2
15.14 odd 2 3330.2.a.bb.1.1 2
20.19 odd 2 2960.2.a.o.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
370.2.a.f.1.1 2 5.4 even 2
1850.2.a.q.1.2 2 1.1 even 1 trivial
1850.2.b.m.149.2 4 5.2 odd 4
1850.2.b.m.149.3 4 5.3 odd 4
2960.2.a.o.1.2 2 20.19 odd 2
3330.2.a.bb.1.1 2 15.14 odd 2