# Properties

 Label 1850.2.b.o.149.6 Level $1850$ Weight $2$ Character 1850.149 Analytic conductor $14.772$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1850,2,Mod(149,1850)]

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([1, 0]))

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

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

chi := DirichletCharacter("1850.149");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1850 = 2 \cdot 5^{2} \cdot 37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1850.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: $$14.7723243739$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: 6.0.3182656.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} - 2x^{3} + 25x^{2} - 10x + 2$$ x^6 - 2*x^3 + 25*x^2 - 10*x + 2 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 370) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 149.6 Root $$-1.67298 - 1.67298i$$ of defining polynomial Character $$\chi$$ $$=$$ 1850.149 Dual form 1850.2.b.o.149.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000i q^{2} +3.34596i q^{3} -1.00000 q^{4} -3.34596 q^{6} -2.59774i q^{7} -1.00000i q^{8} -8.19547 q^{9} +O(q^{10})$$ $$q+1.00000i q^{2} +3.34596i q^{3} -1.00000 q^{4} -3.34596 q^{6} -2.59774i q^{7} -1.00000i q^{8} -8.19547 q^{9} +4.74823 q^{11} -3.34596i q^{12} -6.69193i q^{13} +2.59774 q^{14} +1.00000 q^{16} -0.748228i q^{17} -8.19547i q^{18} +3.34596 q^{19} +8.69193 q^{21} +4.74823i q^{22} -1.49646i q^{23} +3.34596 q^{24} +6.69193 q^{26} -17.3839i q^{27} +2.59774i q^{28} -3.94370 q^{29} +7.79321 q^{31} +1.00000i q^{32} +15.8874i q^{33} +0.748228 q^{34} +8.19547 q^{36} -1.00000i q^{37} +3.34596i q^{38} +22.3909 q^{39} -6.44724 q^{41} +8.69193i q^{42} -1.94370i q^{43} -4.74823 q^{44} +1.49646 q^{46} +1.84951i q^{47} +3.34596i q^{48} +0.251772 q^{49} +2.50354 q^{51} +6.69193i q^{52} -10.4472i q^{53} +17.3839 q^{54} -2.59774 q^{56} +11.1955i q^{57} -3.94370i q^{58} -5.84951 q^{59} +7.94370 q^{61} +7.79321i q^{62} +21.2897i q^{63} -1.00000 q^{64} -15.8874 q^{66} +1.84951i q^{67} +0.748228i q^{68} +5.00709 q^{69} -3.88740 q^{71} +8.19547i q^{72} +7.49646i q^{73} +1.00000 q^{74} -3.34596 q^{76} -12.3346i q^{77} +22.3909i q^{78} +16.5414 q^{79} +33.5793 q^{81} -6.44724i q^{82} -15.2334i q^{83} -8.69193 q^{84} +1.94370 q^{86} -13.1955i q^{87} -4.74823i q^{88} -6.00000 q^{89} -17.3839 q^{91} +1.49646i q^{92} +26.0758i q^{93} -1.84951 q^{94} -3.34596 q^{96} -10.4472i q^{97} +0.251772i q^{98} -38.9140 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 6 q^{4} - 22 q^{9}+O(q^{10})$$ 6 * q - 6 * q^4 - 22 * q^9 $$6 q - 6 q^{4} - 22 q^{9} + 22 q^{11} + 2 q^{14} + 6 q^{16} + 12 q^{21} + 10 q^{29} + 6 q^{31} - 2 q^{34} + 22 q^{36} + 80 q^{39} - 18 q^{41} - 22 q^{44} - 4 q^{46} + 8 q^{49} + 28 q^{51} + 24 q^{54} - 2 q^{56} - 28 q^{59} + 14 q^{61} - 6 q^{64} - 28 q^{66} + 56 q^{69} + 44 q^{71} + 6 q^{74} + 52 q^{79} + 94 q^{81} - 12 q^{84} - 22 q^{86} - 36 q^{89} - 24 q^{91} - 4 q^{94} - 38 q^{99}+O(q^{100})$$ 6 * q - 6 * q^4 - 22 * q^9 + 22 * q^11 + 2 * q^14 + 6 * q^16 + 12 * q^21 + 10 * q^29 + 6 * q^31 - 2 * q^34 + 22 * q^36 + 80 * q^39 - 18 * q^41 - 22 * q^44 - 4 * q^46 + 8 * q^49 + 28 * q^51 + 24 * q^54 - 2 * q^56 - 28 * q^59 + 14 * q^61 - 6 * q^64 - 28 * q^66 + 56 * q^69 + 44 * q^71 + 6 * q^74 + 52 * q^79 + 94 * q^81 - 12 * q^84 - 22 * q^86 - 36 * q^89 - 24 * q^91 - 4 * q^94 - 38 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1850\mathbb{Z}\right)^\times$$.

 $$n$$ $$1001$$ $$1777$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.00000i 0.707107i
$$3$$ 3.34596i 1.93179i 0.258929 + 0.965896i $$0.416630\pi$$
−0.258929 + 0.965896i $$0.583370\pi$$
$$4$$ −1.00000 −0.500000
$$5$$ 0 0
$$6$$ −3.34596 −1.36598
$$7$$ − 2.59774i − 0.981852i −0.871201 0.490926i $$-0.836659\pi$$
0.871201 0.490926i $$-0.163341\pi$$
$$8$$ − 1.00000i − 0.353553i
$$9$$ −8.19547 −2.73182
$$10$$ 0 0
$$11$$ 4.74823 1.43164 0.715822 0.698282i $$-0.246052\pi$$
0.715822 + 0.698282i $$0.246052\pi$$
$$12$$ − 3.34596i − 0.965896i
$$13$$ − 6.69193i − 1.85601i −0.372572 0.928003i $$-0.621524\pi$$
0.372572 0.928003i $$-0.378476\pi$$
$$14$$ 2.59774 0.694274
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ − 0.748228i − 0.181472i −0.995875 0.0907360i $$-0.971078\pi$$
0.995875 0.0907360i $$-0.0289219\pi$$
$$18$$ − 8.19547i − 1.93169i
$$19$$ 3.34596 0.767617 0.383808 0.923413i $$-0.374612\pi$$
0.383808 + 0.923413i $$0.374612\pi$$
$$20$$ 0 0
$$21$$ 8.69193 1.89673
$$22$$ 4.74823i 1.01233i
$$23$$ − 1.49646i − 0.312033i −0.987754 0.156016i $$-0.950135\pi$$
0.987754 0.156016i $$-0.0498653\pi$$
$$24$$ 3.34596 0.682992
$$25$$ 0 0
$$26$$ 6.69193 1.31239
$$27$$ − 17.3839i − 3.34552i
$$28$$ 2.59774i 0.490926i
$$29$$ −3.94370 −0.732326 −0.366163 0.930551i $$-0.619329\pi$$
−0.366163 + 0.930551i $$0.619329\pi$$
$$30$$ 0 0
$$31$$ 7.79321 1.39970 0.699851 0.714289i $$-0.253250\pi$$
0.699851 + 0.714289i $$0.253250\pi$$
$$32$$ 1.00000i 0.176777i
$$33$$ 15.8874i 2.76564i
$$34$$ 0.748228 0.128320
$$35$$ 0 0
$$36$$ 8.19547 1.36591
$$37$$ − 1.00000i − 0.164399i
$$38$$ 3.34596i 0.542787i
$$39$$ 22.3909 3.58542
$$40$$ 0 0
$$41$$ −6.44724 −1.00689 −0.503445 0.864027i $$-0.667934\pi$$
−0.503445 + 0.864027i $$0.667934\pi$$
$$42$$ 8.69193i 1.34119i
$$43$$ − 1.94370i − 0.296411i −0.988957 0.148206i $$-0.952650\pi$$
0.988957 0.148206i $$-0.0473497\pi$$
$$44$$ −4.74823 −0.715822
$$45$$ 0 0
$$46$$ 1.49646 0.220640
$$47$$ 1.84951i 0.269778i 0.990861 + 0.134889i $$0.0430678\pi$$
−0.990861 + 0.134889i $$0.956932\pi$$
$$48$$ 3.34596i 0.482948i
$$49$$ 0.251772 0.0359674
$$50$$ 0 0
$$51$$ 2.50354 0.350566
$$52$$ 6.69193i 0.928003i
$$53$$ − 10.4472i − 1.43504i −0.696538 0.717520i $$-0.745277\pi$$
0.696538 0.717520i $$-0.254723\pi$$
$$54$$ 17.3839 2.36564
$$55$$ 0 0
$$56$$ −2.59774 −0.347137
$$57$$ 11.1955i 1.48288i
$$58$$ − 3.94370i − 0.517833i
$$59$$ −5.84951 −0.761541 −0.380770 0.924670i $$-0.624341\pi$$
−0.380770 + 0.924670i $$0.624341\pi$$
$$60$$ 0 0
$$61$$ 7.94370 1.01709 0.508543 0.861036i $$-0.330184\pi$$
0.508543 + 0.861036i $$0.330184\pi$$
$$62$$ 7.79321i 0.989738i
$$63$$ 21.2897i 2.68225i
$$64$$ −1.00000 −0.125000
$$65$$ 0 0
$$66$$ −15.8874 −1.95560
$$67$$ 1.84951i 0.225953i 0.993598 + 0.112977i $$0.0360385\pi$$
−0.993598 + 0.112977i $$0.963961\pi$$
$$68$$ 0.748228i 0.0907360i
$$69$$ 5.00709 0.602783
$$70$$ 0 0
$$71$$ −3.88740 −0.461349 −0.230675 0.973031i $$-0.574093\pi$$
−0.230675 + 0.973031i $$0.574093\pi$$
$$72$$ 8.19547i 0.965845i
$$73$$ 7.49646i 0.877394i 0.898635 + 0.438697i $$0.144560\pi$$
−0.898635 + 0.438697i $$0.855440\pi$$
$$74$$ 1.00000 0.116248
$$75$$ 0 0
$$76$$ −3.34596 −0.383808
$$77$$ − 12.3346i − 1.40566i
$$78$$ 22.3909i 2.53527i
$$79$$ 16.5414 1.86106 0.930528 0.366220i $$-0.119348\pi$$
0.930528 + 0.366220i $$0.119348\pi$$
$$80$$ 0 0
$$81$$ 33.5793 3.73104
$$82$$ − 6.44724i − 0.711979i
$$83$$ − 15.2334i − 1.67208i −0.548670 0.836039i $$-0.684866\pi$$
0.548670 0.836039i $$-0.315134\pi$$
$$84$$ −8.69193 −0.948367
$$85$$ 0 0
$$86$$ 1.94370 0.209594
$$87$$ − 13.1955i − 1.41470i
$$88$$ − 4.74823i − 0.506163i
$$89$$ −6.00000 −0.635999 −0.317999 0.948091i $$-0.603011\pi$$
−0.317999 + 0.948091i $$0.603011\pi$$
$$90$$ 0 0
$$91$$ −17.3839 −1.82232
$$92$$ 1.49646i 0.156016i
$$93$$ 26.0758i 2.70393i
$$94$$ −1.84951 −0.190762
$$95$$ 0 0
$$96$$ −3.34596 −0.341496
$$97$$ − 10.4472i − 1.06076i −0.847761 0.530378i $$-0.822050\pi$$
0.847761 0.530378i $$-0.177950\pi$$
$$98$$ 0.251772i 0.0254328i
$$99$$ −38.9140 −3.91100
$$100$$ 0 0
$$101$$ −12.1884 −1.21279 −0.606395 0.795164i $$-0.707385\pi$$
−0.606395 + 0.795164i $$0.707385\pi$$
$$102$$ 2.50354i 0.247888i
$$103$$ 1.30807i 0.128888i 0.997921 + 0.0644442i $$0.0205274\pi$$
−0.997921 + 0.0644442i $$0.979473\pi$$
$$104$$ −6.69193 −0.656197
$$105$$ 0 0
$$106$$ 10.4472 1.01473
$$107$$ − 3.04498i − 0.294369i −0.989109 0.147185i $$-0.952979\pi$$
0.989109 0.147185i $$-0.0470211\pi$$
$$108$$ 17.3839i 1.67276i
$$109$$ 1.44015 0.137942 0.0689709 0.997619i $$-0.478028\pi$$
0.0689709 + 0.997619i $$0.478028\pi$$
$$110$$ 0 0
$$111$$ 3.34596 0.317585
$$112$$ − 2.59774i − 0.245463i
$$113$$ − 11.1392i − 1.04788i −0.851754 0.523942i $$-0.824461\pi$$
0.851754 0.523942i $$-0.175539\pi$$
$$114$$ −11.1955 −1.04855
$$115$$ 0 0
$$116$$ 3.94370 0.366163
$$117$$ 54.8435i 5.07028i
$$118$$ − 5.84951i − 0.538491i
$$119$$ −1.94370 −0.178179
$$120$$ 0 0
$$121$$ 11.5457 1.04961
$$122$$ 7.94370i 0.719189i
$$123$$ − 21.5722i − 1.94510i
$$124$$ −7.79321 −0.699851
$$125$$ 0 0
$$126$$ −21.2897 −1.89663
$$127$$ − 8.84242i − 0.784638i −0.919829 0.392319i $$-0.871673\pi$$
0.919829 0.392319i $$-0.128327\pi$$
$$128$$ − 1.00000i − 0.0883883i
$$129$$ 6.50354 0.572605
$$130$$ 0 0
$$131$$ 6.15049 0.537371 0.268686 0.963228i $$-0.413411\pi$$
0.268686 + 0.963228i $$0.413411\pi$$
$$132$$ − 15.8874i − 1.38282i
$$133$$ − 8.69193i − 0.753686i
$$134$$ −1.84951 −0.159773
$$135$$ 0 0
$$136$$ −0.748228 −0.0641600
$$137$$ 10.8945i 0.930779i 0.885106 + 0.465389i $$0.154086\pi$$
−0.885106 + 0.465389i $$0.845914\pi$$
$$138$$ 5.00709i 0.426232i
$$139$$ −1.04921 −0.0889932 −0.0444966 0.999010i $$-0.514168\pi$$
−0.0444966 + 0.999010i $$0.514168\pi$$
$$140$$ 0 0
$$141$$ −6.18838 −0.521156
$$142$$ − 3.88740i − 0.326223i
$$143$$ − 31.7748i − 2.65714i
$$144$$ −8.19547 −0.682956
$$145$$ 0 0
$$146$$ −7.49646 −0.620411
$$147$$ 0.842420i 0.0694816i
$$148$$ 1.00000i 0.0821995i
$$149$$ 4.18838 0.343126 0.171563 0.985173i $$-0.445118\pi$$
0.171563 + 0.985173i $$0.445118\pi$$
$$150$$ 0 0
$$151$$ −6.69193 −0.544581 −0.272291 0.962215i $$-0.587781\pi$$
−0.272291 + 0.962215i $$0.587781\pi$$
$$152$$ − 3.34596i − 0.271393i
$$153$$ 6.13208i 0.495749i
$$154$$ 12.3346 0.993954
$$155$$ 0 0
$$156$$ −22.3909 −1.79271
$$157$$ − 3.94370i − 0.314741i −0.987540 0.157371i $$-0.949698\pi$$
0.987540 0.157371i $$-0.0503018\pi$$
$$158$$ 16.5414i 1.31597i
$$159$$ 34.9561 2.77220
$$160$$ 0 0
$$161$$ −3.88740 −0.306370
$$162$$ 33.5793i 2.63824i
$$163$$ 12.1463i 0.951368i 0.879616 + 0.475684i $$0.157799\pi$$
−0.879616 + 0.475684i $$0.842201\pi$$
$$164$$ 6.44724 0.503445
$$165$$ 0 0
$$166$$ 15.2334 1.18234
$$167$$ 17.0829i 1.32191i 0.750425 + 0.660956i $$0.229849\pi$$
−0.750425 + 0.660956i $$0.770151\pi$$
$$168$$ − 8.69193i − 0.670597i
$$169$$ −31.7819 −2.44476
$$170$$ 0 0
$$171$$ −27.4217 −2.09699
$$172$$ 1.94370i 0.148206i
$$173$$ 15.3417i 1.16641i 0.812325 + 0.583205i $$0.198202\pi$$
−0.812325 + 0.583205i $$0.801798\pi$$
$$174$$ 13.1955 1.00035
$$175$$ 0 0
$$176$$ 4.74823 0.357911
$$177$$ − 19.5722i − 1.47114i
$$178$$ − 6.00000i − 0.449719i
$$179$$ −2.45148 −0.183232 −0.0916161 0.995794i $$-0.529203\pi$$
−0.0916161 + 0.995794i $$0.529203\pi$$
$$180$$ 0 0
$$181$$ 13.1955 0.980812 0.490406 0.871494i $$-0.336849\pi$$
0.490406 + 0.871494i $$0.336849\pi$$
$$182$$ − 17.3839i − 1.28858i
$$183$$ 26.5793i 1.96480i
$$184$$ −1.49646 −0.110320
$$185$$ 0 0
$$186$$ −26.0758 −1.91197
$$187$$ − 3.55276i − 0.259803i
$$188$$ − 1.84951i − 0.134889i
$$189$$ −45.1586 −3.28481
$$190$$ 0 0
$$191$$ −21.7790 −1.57588 −0.787938 0.615755i $$-0.788851\pi$$
−0.787938 + 0.615755i $$0.788851\pi$$
$$192$$ − 3.34596i − 0.241474i
$$193$$ 12.9929i 0.935250i 0.883927 + 0.467625i $$0.154890\pi$$
−0.883927 + 0.467625i $$0.845110\pi$$
$$194$$ 10.4472 0.750068
$$195$$ 0 0
$$196$$ −0.251772 −0.0179837
$$197$$ 0.616147i 0.0438986i 0.999759 + 0.0219493i $$0.00698725\pi$$
−0.999759 + 0.0219493i $$0.993013\pi$$
$$198$$ − 38.9140i − 2.76549i
$$199$$ 1.54852 0.109772 0.0548859 0.998493i $$-0.482520\pi$$
0.0548859 + 0.998493i $$0.482520\pi$$
$$200$$ 0 0
$$201$$ −6.18838 −0.436495
$$202$$ − 12.1884i − 0.857572i
$$203$$ 10.2447i 0.719036i
$$204$$ −2.50354 −0.175283
$$205$$ 0 0
$$206$$ −1.30807 −0.0911378
$$207$$ 12.2642i 0.852418i
$$208$$ − 6.69193i − 0.464002i
$$209$$ 15.8874 1.09895
$$210$$ 0 0
$$211$$ 20.9366 1.44134 0.720668 0.693280i $$-0.243835\pi$$
0.720668 + 0.693280i $$0.243835\pi$$
$$212$$ 10.4472i 0.717520i
$$213$$ − 13.0071i − 0.891231i
$$214$$ 3.04498 0.208150
$$215$$ 0 0
$$216$$ −17.3839 −1.18282
$$217$$ − 20.2447i − 1.37430i
$$218$$ 1.44015i 0.0975396i
$$219$$ −25.0829 −1.69494
$$220$$ 0 0
$$221$$ −5.00709 −0.336813
$$222$$ 3.34596i 0.224566i
$$223$$ 2.71034i 0.181498i 0.995874 + 0.0907488i $$0.0289260\pi$$
−0.995874 + 0.0907488i $$0.971074\pi$$
$$224$$ 2.59774 0.173568
$$225$$ 0 0
$$226$$ 11.1392 0.740966
$$227$$ 26.1321i 1.73445i 0.497919 + 0.867224i $$0.334098\pi$$
−0.497919 + 0.867224i $$0.665902\pi$$
$$228$$ − 11.1955i − 0.741438i
$$229$$ 24.7677 1.63670 0.818348 0.574723i $$-0.194890\pi$$
0.818348 + 0.574723i $$0.194890\pi$$
$$230$$ 0 0
$$231$$ 41.2713 2.71545
$$232$$ 3.94370i 0.258916i
$$233$$ 12.2783i 0.804381i 0.915556 + 0.402190i $$0.131751\pi$$
−0.915556 + 0.402190i $$0.868249\pi$$
$$234$$ −54.8435 −3.58523
$$235$$ 0 0
$$236$$ 5.84951 0.380770
$$237$$ 55.3470i 3.59518i
$$238$$ − 1.94370i − 0.125991i
$$239$$ 17.1771 1.11109 0.555546 0.831486i $$-0.312509\pi$$
0.555546 + 0.831486i $$0.312509\pi$$
$$240$$ 0 0
$$241$$ −15.2713 −0.983708 −0.491854 0.870678i $$-0.663681\pi$$
−0.491854 + 0.870678i $$0.663681\pi$$
$$242$$ 11.5457i 0.742184i
$$243$$ 60.2036i 3.86206i
$$244$$ −7.94370 −0.508543
$$245$$ 0 0
$$246$$ 21.5722 1.37540
$$247$$ − 22.3909i − 1.42470i
$$248$$ − 7.79321i − 0.494869i
$$249$$ 50.9703 3.23011
$$250$$ 0 0
$$251$$ 10.0379 0.633586 0.316793 0.948495i $$-0.397394\pi$$
0.316793 + 0.948495i $$0.397394\pi$$
$$252$$ − 21.2897i − 1.34112i
$$253$$ − 7.10552i − 0.446720i
$$254$$ 8.84242 0.554823
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ − 20.9703i − 1.30809i −0.756456 0.654045i $$-0.773071\pi$$
0.756456 0.654045i $$-0.226929\pi$$
$$258$$ 6.50354i 0.404893i
$$259$$ −2.59774 −0.161415
$$260$$ 0 0
$$261$$ 32.3205 2.00059
$$262$$ 6.15049i 0.379979i
$$263$$ − 22.3725i − 1.37955i −0.724024 0.689775i $$-0.757710\pi$$
0.724024 0.689775i $$-0.242290\pi$$
$$264$$ 15.8874 0.977802
$$265$$ 0 0
$$266$$ 8.69193 0.532936
$$267$$ − 20.0758i − 1.22862i
$$268$$ − 1.84951i − 0.112977i
$$269$$ 4.18838 0.255370 0.127685 0.991815i $$-0.459245\pi$$
0.127685 + 0.991815i $$0.459245\pi$$
$$270$$ 0 0
$$271$$ −12.1126 −0.735788 −0.367894 0.929868i $$-0.619921\pi$$
−0.367894 + 0.929868i $$0.619921\pi$$
$$272$$ − 0.748228i − 0.0453680i
$$273$$ − 58.1657i − 3.52035i
$$274$$ −10.8945 −0.658160
$$275$$ 0 0
$$276$$ −5.00709 −0.301391
$$277$$ − 9.30807i − 0.559268i −0.960107 0.279634i $$-0.909787\pi$$
0.960107 0.279634i $$-0.0902131\pi$$
$$278$$ − 1.04921i − 0.0629277i
$$279$$ −63.8690 −3.82374
$$280$$ 0 0
$$281$$ 19.2713 1.14963 0.574813 0.818285i $$-0.305075\pi$$
0.574813 + 0.818285i $$0.305075\pi$$
$$282$$ − 6.18838i − 0.368513i
$$283$$ 13.6848i 0.813479i 0.913544 + 0.406740i $$0.133334\pi$$
−0.913544 + 0.406740i $$0.866666\pi$$
$$284$$ 3.88740 0.230675
$$285$$ 0 0
$$286$$ 31.7748 1.87888
$$287$$ 16.7482i 0.988617i
$$288$$ − 8.19547i − 0.482923i
$$289$$ 16.4402 0.967068
$$290$$ 0 0
$$291$$ 34.9561 2.04916
$$292$$ − 7.49646i − 0.438697i
$$293$$ 10.4472i 0.610334i 0.952299 + 0.305167i $$0.0987124\pi$$
−0.952299 + 0.305167i $$0.901288\pi$$
$$294$$ −0.842420 −0.0491309
$$295$$ 0 0
$$296$$ −1.00000 −0.0581238
$$297$$ − 82.5425i − 4.78960i
$$298$$ 4.18838i 0.242627i
$$299$$ −10.0142 −0.579135
$$300$$ 0 0
$$301$$ −5.04921 −0.291032
$$302$$ − 6.69193i − 0.385077i
$$303$$ − 40.7819i − 2.34286i
$$304$$ 3.34596 0.191904
$$305$$ 0 0
$$306$$ −6.13208 −0.350548
$$307$$ − 4.95502i − 0.282798i −0.989953 0.141399i $$-0.954840\pi$$
0.989953 0.141399i $$-0.0451601\pi$$
$$308$$ 12.3346i 0.702831i
$$309$$ −4.37677 −0.248985
$$310$$ 0 0
$$311$$ 3.68060 0.208708 0.104354 0.994540i $$-0.466723\pi$$
0.104354 + 0.994540i $$0.466723\pi$$
$$312$$ − 22.3909i − 1.26764i
$$313$$ 0.992912i 0.0561227i 0.999606 + 0.0280614i $$0.00893338\pi$$
−0.999606 + 0.0280614i $$0.991067\pi$$
$$314$$ 3.94370 0.222556
$$315$$ 0 0
$$316$$ −16.5414 −0.930528
$$317$$ − 6.55985i − 0.368438i −0.982885 0.184219i $$-0.941024\pi$$
0.982885 0.184219i $$-0.0589755\pi$$
$$318$$ 34.9561i 1.96024i
$$319$$ −18.7256 −1.04843
$$320$$ 0 0
$$321$$ 10.1884 0.568660
$$322$$ − 3.88740i − 0.216636i
$$323$$ − 2.50354i − 0.139301i
$$324$$ −33.5793 −1.86552
$$325$$ 0 0
$$326$$ −12.1463 −0.672719
$$327$$ 4.81870i 0.266475i
$$328$$ 6.44724i 0.355989i
$$329$$ 4.80453 0.264882
$$330$$ 0 0
$$331$$ −18.0379 −0.991452 −0.495726 0.868479i $$-0.665098\pi$$
−0.495726 + 0.868479i $$0.665098\pi$$
$$332$$ 15.2334i 0.836039i
$$333$$ 8.19547i 0.449109i
$$334$$ −17.0829 −0.934733
$$335$$ 0 0
$$336$$ 8.69193 0.474183
$$337$$ − 14.8945i − 0.811354i −0.914016 0.405677i $$-0.867036\pi$$
0.914016 0.405677i $$-0.132964\pi$$
$$338$$ − 31.7819i − 1.72871i
$$339$$ 37.2713 2.02430
$$340$$ 0 0
$$341$$ 37.0039 2.00387
$$342$$ − 27.4217i − 1.48280i
$$343$$ − 18.8382i − 1.01717i
$$344$$ −1.94370 −0.104797
$$345$$ 0 0
$$346$$ −15.3417 −0.824776
$$347$$ − 9.38385i − 0.503752i −0.967760 0.251876i $$-0.918953\pi$$
0.967760 0.251876i $$-0.0810475\pi$$
$$348$$ 13.1955i 0.707351i
$$349$$ 32.9703 1.76486 0.882429 0.470446i $$-0.155907\pi$$
0.882429 + 0.470446i $$0.155907\pi$$
$$350$$ 0 0
$$351$$ −116.331 −6.20931
$$352$$ 4.74823i 0.253081i
$$353$$ − 27.4260i − 1.45974i −0.683587 0.729869i $$-0.739581\pi$$
0.683587 0.729869i $$-0.260419\pi$$
$$354$$ 19.5722 1.04025
$$355$$ 0 0
$$356$$ 6.00000 0.317999
$$357$$ − 6.50354i − 0.344204i
$$358$$ − 2.45148i − 0.129565i
$$359$$ 25.0829 1.32382 0.661912 0.749582i $$-0.269745\pi$$
0.661912 + 0.749582i $$0.269745\pi$$
$$360$$ 0 0
$$361$$ −7.80453 −0.410765
$$362$$ 13.1955i 0.693539i
$$363$$ 38.6314i 2.02762i
$$364$$ 17.3839 0.911161
$$365$$ 0 0
$$366$$ −26.5793 −1.38932
$$367$$ 2.48513i 0.129723i 0.997894 + 0.0648614i $$0.0206605\pi$$
−0.997894 + 0.0648614i $$0.979339\pi$$
$$368$$ − 1.49646i − 0.0780082i
$$369$$ 52.8382 2.75065
$$370$$ 0 0
$$371$$ −27.1392 −1.40900
$$372$$ − 26.0758i − 1.35197i
$$373$$ 35.1586i 1.82045i 0.414119 + 0.910223i $$0.364090\pi$$
−0.414119 + 0.910223i $$0.635910\pi$$
$$374$$ 3.55276 0.183709
$$375$$ 0 0
$$376$$ 1.84951 0.0953810
$$377$$ 26.3909i 1.35920i
$$378$$ − 45.1586i − 2.32271i
$$379$$ −24.6778 −1.26761 −0.633805 0.773492i $$-0.718508\pi$$
−0.633805 + 0.773492i $$0.718508\pi$$
$$380$$ 0 0
$$381$$ 29.5864 1.51576
$$382$$ − 21.7790i − 1.11431i
$$383$$ 13.3839i 0.683883i 0.939721 + 0.341941i $$0.111084\pi$$
−0.939721 + 0.341941i $$0.888916\pi$$
$$384$$ 3.34596 0.170748
$$385$$ 0 0
$$386$$ −12.9929 −0.661322
$$387$$ 15.9295i 0.809743i
$$388$$ 10.4472i 0.530378i
$$389$$ −18.2220 −0.923894 −0.461947 0.886908i $$-0.652849\pi$$
−0.461947 + 0.886908i $$0.652849\pi$$
$$390$$ 0 0
$$391$$ −1.11969 −0.0566252
$$392$$ − 0.251772i − 0.0127164i
$$393$$ 20.5793i 1.03809i
$$394$$ −0.616147 −0.0310410
$$395$$ 0 0
$$396$$ 38.9140 1.95550
$$397$$ − 2.22521i − 0.111680i −0.998440 0.0558399i $$-0.982216\pi$$
0.998440 0.0558399i $$-0.0177837\pi$$
$$398$$ 1.54852i 0.0776204i
$$399$$ 29.0829 1.45596
$$400$$ 0 0
$$401$$ 12.3909 0.618774 0.309387 0.950936i $$-0.399876\pi$$
0.309387 + 0.950936i $$0.399876\pi$$
$$402$$ − 6.18838i − 0.308648i
$$403$$ − 52.1516i − 2.59785i
$$404$$ 12.1884 0.606395
$$405$$ 0 0
$$406$$ −10.2447 −0.508435
$$407$$ − 4.74823i − 0.235361i
$$408$$ − 2.50354i − 0.123944i
$$409$$ −7.27125 −0.359540 −0.179770 0.983709i $$-0.557535\pi$$
−0.179770 + 0.983709i $$0.557535\pi$$
$$410$$ 0 0
$$411$$ −36.4525 −1.79807
$$412$$ − 1.30807i − 0.0644442i
$$413$$ 15.1955i 0.747720i
$$414$$ −12.2642 −0.602751
$$415$$ 0 0
$$416$$ 6.69193 0.328099
$$417$$ − 3.51063i − 0.171916i
$$418$$ 15.8874i 0.777078i
$$419$$ −28.4809 −1.39138 −0.695691 0.718341i $$-0.744902\pi$$
−0.695691 + 0.718341i $$0.744902\pi$$
$$420$$ 0 0
$$421$$ 22.0000 1.07221 0.536107 0.844150i $$-0.319894\pi$$
0.536107 + 0.844150i $$0.319894\pi$$
$$422$$ 20.9366i 1.01918i
$$423$$ − 15.1576i − 0.736987i
$$424$$ −10.4472 −0.507363
$$425$$ 0 0
$$426$$ 13.0071 0.630195
$$427$$ − 20.6356i − 0.998628i
$$428$$ 3.04498i 0.147185i
$$429$$ 106.317 5.13305
$$430$$ 0 0
$$431$$ 1.51487 0.0729686 0.0364843 0.999334i $$-0.488384\pi$$
0.0364843 + 0.999334i $$0.488384\pi$$
$$432$$ − 17.3839i − 0.836381i
$$433$$ − 24.9929i − 1.20108i −0.799594 0.600541i $$-0.794952\pi$$
0.799594 0.600541i $$-0.205048\pi$$
$$434$$ 20.2447 0.971776
$$435$$ 0 0
$$436$$ −1.44015 −0.0689709
$$437$$ − 5.00709i − 0.239521i
$$438$$ − 25.0829i − 1.19851i
$$439$$ 5.10128 0.243471 0.121735 0.992563i $$-0.461154\pi$$
0.121735 + 0.992563i $$0.461154\pi$$
$$440$$ 0 0
$$441$$ −2.06339 −0.0982566
$$442$$ − 5.00709i − 0.238163i
$$443$$ − 25.7369i − 1.22280i −0.791323 0.611399i $$-0.790607\pi$$
0.791323 0.611399i $$-0.209393\pi$$
$$444$$ −3.34596 −0.158792
$$445$$ 0 0
$$446$$ −2.71034 −0.128338
$$447$$ 14.0142i 0.662848i
$$448$$ 2.59774i 0.122731i
$$449$$ 17.7748 0.838844 0.419422 0.907791i $$-0.362233\pi$$
0.419422 + 0.907791i $$0.362233\pi$$
$$450$$ 0 0
$$451$$ −30.6130 −1.44151
$$452$$ 11.1392i 0.523942i
$$453$$ − 22.3909i − 1.05202i
$$454$$ −26.1321 −1.22644
$$455$$ 0 0
$$456$$ 11.1955 0.524276
$$457$$ 34.3346i 1.60611i 0.595907 + 0.803053i $$0.296793\pi$$
−0.595907 + 0.803053i $$0.703207\pi$$
$$458$$ 24.7677i 1.15732i
$$459$$ −13.0071 −0.607119
$$460$$ 0 0
$$461$$ −16.9508 −0.789477 −0.394738 0.918794i $$-0.629165\pi$$
−0.394738 + 0.918794i $$0.629165\pi$$
$$462$$ 41.2713i 1.92011i
$$463$$ − 10.9929i − 0.510884i −0.966824 0.255442i $$-0.917779\pi$$
0.966824 0.255442i $$-0.0822210\pi$$
$$464$$ −3.94370 −0.183082
$$465$$ 0 0
$$466$$ −12.2783 −0.568783
$$467$$ 37.4175i 1.73148i 0.500498 + 0.865738i $$0.333150\pi$$
−0.500498 + 0.865738i $$0.666850\pi$$
$$468$$ − 54.8435i − 2.53514i
$$469$$ 4.80453 0.221853
$$470$$ 0 0
$$471$$ 13.1955 0.608015
$$472$$ 5.84951i 0.269245i
$$473$$ − 9.22912i − 0.424356i
$$474$$ −55.3470 −2.54217
$$475$$ 0 0
$$476$$ 1.94370 0.0890893
$$477$$ 85.6201i 3.92027i
$$478$$ 17.1771i 0.785660i
$$479$$ 0.465654 0.0212763 0.0106381 0.999943i $$-0.496614\pi$$
0.0106381 + 0.999943i $$0.496614\pi$$
$$480$$ 0 0
$$481$$ −6.69193 −0.305126
$$482$$ − 15.2713i − 0.695586i
$$483$$ − 13.0071i − 0.591843i
$$484$$ −11.5457 −0.524803
$$485$$ 0 0
$$486$$ −60.2036 −2.73089
$$487$$ 34.9561i 1.58401i 0.610514 + 0.792006i $$0.290963\pi$$
−0.610514 + 0.792006i $$0.709037\pi$$
$$488$$ − 7.94370i − 0.359594i
$$489$$ −40.6409 −1.83785
$$490$$ 0 0
$$491$$ −24.7819 −1.11839 −0.559195 0.829036i $$-0.688890\pi$$
−0.559195 + 0.829036i $$0.688890\pi$$
$$492$$ 21.5722i 0.972552i
$$493$$ 2.95079i 0.132897i
$$494$$ 22.3909 1.00742
$$495$$ 0 0
$$496$$ 7.79321 0.349925
$$497$$ 10.0984i 0.452976i
$$498$$ 50.9703i 2.28403i
$$499$$ 2.33888 0.104702 0.0523512 0.998629i $$-0.483328\pi$$
0.0523512 + 0.998629i $$0.483328\pi$$
$$500$$ 0 0
$$501$$ −57.1586 −2.55366
$$502$$ 10.0379i 0.448013i
$$503$$ 18.6161i 0.830053i 0.909809 + 0.415026i $$0.136228\pi$$
−0.909809 + 0.415026i $$0.863772\pi$$
$$504$$ 21.2897 0.948317
$$505$$ 0 0
$$506$$ 7.10552 0.315879
$$507$$ − 106.341i − 4.72277i
$$508$$ 8.84242i 0.392319i
$$509$$ −6.89448 −0.305593 −0.152796 0.988258i $$-0.548828\pi$$
−0.152796 + 0.988258i $$0.548828\pi$$
$$510$$ 0 0
$$511$$ 19.4738 0.861471
$$512$$ 1.00000i 0.0441942i
$$513$$ − 58.1657i − 2.56808i
$$514$$ 20.9703 0.924959
$$515$$ 0 0
$$516$$ −6.50354 −0.286303
$$517$$ 8.78188i 0.386227i
$$518$$ − 2.59774i − 0.114138i
$$519$$ −51.3329 −2.25326
$$520$$ 0 0
$$521$$ 4.86083 0.212957 0.106478 0.994315i $$-0.466042\pi$$
0.106478 + 0.994315i $$0.466042\pi$$
$$522$$ 32.3205i 1.41463i
$$523$$ − 22.7677i − 0.995562i −0.867303 0.497781i $$-0.834148\pi$$
0.867303 0.497781i $$-0.165852\pi$$
$$524$$ −6.15049 −0.268686
$$525$$ 0 0
$$526$$ 22.3725 0.975489
$$527$$ − 5.83110i − 0.254007i
$$528$$ 15.8874i 0.691410i
$$529$$ 20.7606 0.902636
$$530$$ 0 0
$$531$$ 47.9395 2.08040
$$532$$ 8.69193i 0.376843i
$$533$$ 43.1445i 1.86879i
$$534$$ 20.0758 0.868764
$$535$$ 0 0
$$536$$ 1.84951 0.0798865
$$537$$ − 8.20256i − 0.353967i
$$538$$ 4.18838i 0.180574i
$$539$$ 1.19547 0.0514926
$$540$$ 0 0
$$541$$ 8.99291 0.386636 0.193318 0.981136i $$-0.438075\pi$$
0.193318 + 0.981136i $$0.438075\pi$$
$$542$$ − 12.1126i − 0.520281i
$$543$$ 44.1516i 1.89472i
$$544$$ 0.748228 0.0320800
$$545$$ 0 0
$$546$$ 58.1657 2.48926
$$547$$ − 18.8382i − 0.805463i −0.915318 0.402731i $$-0.868061\pi$$
0.915318 0.402731i $$-0.131939\pi$$
$$548$$ − 10.8945i − 0.465389i
$$549$$ −65.1023 −2.77850
$$550$$ 0 0
$$551$$ −13.1955 −0.562146
$$552$$ − 5.00709i − 0.213116i
$$553$$ − 42.9703i − 1.82728i
$$554$$ 9.30807 0.395462
$$555$$ 0 0
$$556$$ 1.04921 0.0444966
$$557$$ 3.69901i 0.156732i 0.996925 + 0.0783661i $$0.0249703\pi$$
−0.996925 + 0.0783661i $$0.975030\pi$$
$$558$$ − 63.8690i − 2.70379i
$$559$$ −13.0071 −0.550141
$$560$$ 0 0
$$561$$ 11.8874 0.501886
$$562$$ 19.2713i 0.812909i
$$563$$ 14.3488i 0.604730i 0.953192 + 0.302365i $$0.0977762\pi$$
−0.953192 + 0.302365i $$0.902224\pi$$
$$564$$ 6.18838 0.260578
$$565$$ 0 0
$$566$$ −13.6848 −0.575217
$$567$$ − 87.2302i − 3.66332i
$$568$$ 3.88740i 0.163112i
$$569$$ −15.9858 −0.670161 −0.335080 0.942190i $$-0.608763\pi$$
−0.335080 + 0.942190i $$0.608763\pi$$
$$570$$ 0 0
$$571$$ −30.2079 −1.26416 −0.632080 0.774903i $$-0.717799\pi$$
−0.632080 + 0.774903i $$0.717799\pi$$
$$572$$ 31.7748i 1.32857i
$$573$$ − 72.8718i − 3.04426i
$$574$$ −16.7482 −0.699058
$$575$$ 0 0
$$576$$ 8.19547 0.341478
$$577$$ − 19.1813i − 0.798528i −0.916836 0.399264i $$-0.869266\pi$$
0.916836 0.399264i $$-0.130734\pi$$
$$578$$ 16.4402i 0.683820i
$$579$$ −43.4738 −1.80671
$$580$$ 0 0
$$581$$ −39.5722 −1.64173
$$582$$ 34.9561i 1.44898i
$$583$$ − 49.6059i − 2.05447i
$$584$$ 7.49646 0.310206
$$585$$ 0 0
$$586$$ −10.4472 −0.431572
$$587$$ − 26.6130i − 1.09844i −0.835679 0.549218i $$-0.814926\pi$$
0.835679 0.549218i $$-0.185074\pi$$
$$588$$ − 0.842420i − 0.0347408i
$$589$$ 26.0758 1.07443
$$590$$ 0 0
$$591$$ −2.06160 −0.0848031
$$592$$ − 1.00000i − 0.0410997i
$$593$$ − 26.2642i − 1.07854i −0.842133 0.539270i $$-0.818700\pi$$
0.842133 0.539270i $$-0.181300\pi$$
$$594$$ 82.5425 3.38676
$$595$$ 0 0
$$596$$ −4.18838 −0.171563
$$597$$ 5.18130i 0.212056i
$$598$$ − 10.0142i − 0.409510i
$$599$$ 38.2415 1.56251 0.781253 0.624215i $$-0.214581\pi$$
0.781253 + 0.624215i $$0.214581\pi$$
$$600$$ 0 0
$$601$$ 30.2447 1.23371 0.616853 0.787078i $$-0.288407\pi$$
0.616853 + 0.787078i $$0.288407\pi$$
$$602$$ − 5.04921i − 0.205791i
$$603$$ − 15.1576i − 0.617264i
$$604$$ 6.69193 0.272291
$$605$$ 0 0
$$606$$ 40.7819 1.65665
$$607$$ − 5.90157i − 0.239537i −0.992802 0.119769i $$-0.961785\pi$$
0.992802 0.119769i $$-0.0382153\pi$$
$$608$$ 3.34596i 0.135697i
$$609$$ −34.2783 −1.38903
$$610$$ 0 0
$$611$$ 12.3768 0.500710
$$612$$ − 6.13208i − 0.247875i
$$613$$ − 6.15473i − 0.248587i −0.992245 0.124294i $$-0.960334\pi$$
0.992245 0.124294i $$-0.0396665\pi$$
$$614$$ 4.95502 0.199968
$$615$$ 0 0
$$616$$ −12.3346 −0.496977
$$617$$ 16.5035i 0.664408i 0.943208 + 0.332204i $$0.107792\pi$$
−0.943208 + 0.332204i $$0.892208\pi$$
$$618$$ − 4.37677i − 0.176059i
$$619$$ −9.04921 −0.363719 −0.181859 0.983325i $$-0.558212\pi$$
−0.181859 + 0.983325i $$0.558212\pi$$
$$620$$ 0 0
$$621$$ −26.0142 −1.04391
$$622$$ 3.68060i 0.147579i
$$623$$ 15.5864i 0.624456i
$$624$$ 22.3909 0.896355
$$625$$ 0 0
$$626$$ −0.992912 −0.0396848
$$627$$ 53.1586i 2.12295i
$$628$$ 3.94370i 0.157371i
$$629$$ −0.748228 −0.0298338
$$630$$ 0 0
$$631$$ −0.507780 −0.0202144 −0.0101072 0.999949i $$-0.503217\pi$$
−0.0101072 + 0.999949i $$0.503217\pi$$
$$632$$ − 16.5414i − 0.657983i
$$633$$ 70.0531i 2.78436i
$$634$$ 6.55985 0.260525
$$635$$ 0 0
$$636$$ −34.9561 −1.38610
$$637$$ − 1.68484i − 0.0667558i
$$638$$ − 18.7256i − 0.741353i
$$639$$ 31.8590 1.26032
$$640$$ 0 0
$$641$$ −40.6356 −1.60501 −0.802505 0.596645i $$-0.796500\pi$$
−0.802505 + 0.596645i $$0.796500\pi$$
$$642$$ 10.1884i 0.402103i
$$643$$ 9.53011i 0.375831i 0.982185 + 0.187915i $$0.0601731\pi$$
−0.982185 + 0.187915i $$0.939827\pi$$
$$644$$ 3.88740 0.153185
$$645$$ 0 0
$$646$$ 2.50354 0.0985006
$$647$$ − 21.0071i − 0.825874i −0.910760 0.412937i $$-0.864503\pi$$
0.910760 0.412937i $$-0.135497\pi$$
$$648$$ − 33.5793i − 1.31912i
$$649$$ −27.7748 −1.09026
$$650$$ 0 0
$$651$$ 67.7380 2.65486
$$652$$ − 12.1463i − 0.475684i
$$653$$ 45.5496i 1.78249i 0.453519 + 0.891247i $$0.350168\pi$$
−0.453519 + 0.891247i $$0.649832\pi$$
$$654$$ −4.81870 −0.188426
$$655$$ 0 0
$$656$$ −6.44724 −0.251723
$$657$$ − 61.4370i − 2.39689i
$$658$$ 4.80453i 0.187300i
$$659$$ −9.38385 −0.365543 −0.182772 0.983155i $$-0.558507\pi$$
−0.182772 + 0.983155i $$0.558507\pi$$
$$660$$ 0 0
$$661$$ 4.05630 0.157772 0.0788859 0.996884i $$-0.474864\pi$$
0.0788859 + 0.996884i $$0.474864\pi$$
$$662$$ − 18.0379i − 0.701062i
$$663$$ − 16.7535i − 0.650653i
$$664$$ −15.2334 −0.591169
$$665$$ 0 0
$$666$$ −8.19547 −0.317568
$$667$$ 5.90157i 0.228510i
$$668$$ − 17.0829i − 0.660956i
$$669$$ −9.06869 −0.350616
$$670$$ 0 0
$$671$$ 37.7185 1.45611
$$672$$ 8.69193i 0.335298i
$$673$$ − 7.60906i − 0.293308i −0.989188 0.146654i $$-0.953150\pi$$
0.989188 0.146654i $$-0.0468503\pi$$
$$674$$ 14.8945 0.573714
$$675$$ 0 0
$$676$$ 31.7819 1.22238
$$677$$ 28.7677i 1.10563i 0.833303 + 0.552816i $$0.186447\pi$$
−0.833303 + 0.552816i $$0.813553\pi$$
$$678$$ 37.2713i 1.43139i
$$679$$ −27.1392 −1.04151
$$680$$ 0 0
$$681$$ −87.4370 −3.35059
$$682$$ 37.0039i 1.41695i
$$683$$ − 0.0704767i − 0.00269671i −0.999999 0.00134836i $$-0.999571\pi$$
0.999999 0.00134836i $$-0.000429196\pi$$
$$684$$ 27.4217 1.04850
$$685$$ 0 0
$$686$$ 18.8382 0.719245
$$687$$ 82.8718i 3.16176i
$$688$$ − 1.94370i − 0.0741028i
$$689$$ −69.9122 −2.66344
$$690$$ 0 0
$$691$$ 10.6498 0.405137 0.202569 0.979268i $$-0.435071\pi$$
0.202569 + 0.979268i $$0.435071\pi$$
$$692$$ − 15.3417i − 0.583205i
$$693$$ 101.088i 3.84002i
$$694$$ 9.38385 0.356206
$$695$$ 0 0
$$696$$ −13.1955 −0.500173
$$697$$ 4.82401i 0.182722i
$$698$$ 32.9703i 1.24794i
$$699$$ −41.0829 −1.55390
$$700$$ 0 0
$$701$$ −7.98582 −0.301620 −0.150810 0.988563i $$-0.548188\pi$$
−0.150810 + 0.988563i $$0.548188\pi$$
$$702$$ − 116.331i − 4.39065i
$$703$$ − 3.34596i − 0.126195i
$$704$$ −4.74823 −0.178956
$$705$$ 0 0
$$706$$ 27.4260 1.03219
$$707$$ 31.6622i 1.19078i
$$708$$ 19.5722i 0.735570i
$$709$$ 37.2149 1.39764 0.698818 0.715299i $$-0.253710\pi$$
0.698818 + 0.715299i $$0.253710\pi$$
$$710$$ 0 0
$$711$$ −135.565 −5.08408
$$712$$ 6.00000i 0.224860i
$$713$$ − 11.6622i − 0.436752i
$$714$$ 6.50354 0.243389
$$715$$ 0 0
$$716$$ 2.45148 0.0916161
$$717$$ 57.4738i 2.14640i
$$718$$ 25.0829i 0.936084i
$$719$$ −30.1742 −1.12531 −0.562654 0.826692i $$-0.690220\pi$$
−0.562654 + 0.826692i $$0.690220\pi$$
$$720$$ 0 0
$$721$$ 3.39803 0.126549
$$722$$ − 7.80453i − 0.290455i
$$723$$ − 51.0970i − 1.90032i
$$724$$ −13.1955 −0.490406
$$725$$ 0 0
$$726$$ −38.6314 −1.43375
$$727$$ − 18.7677i − 0.696056i −0.937484 0.348028i $$-0.886851\pi$$
0.937484 0.348028i $$-0.113149\pi$$
$$728$$ 17.3839i 0.644288i
$$729$$ −100.701 −3.72967
$$730$$ 0 0
$$731$$ −1.45433 −0.0537903
$$732$$ − 26.5793i − 0.982400i
$$733$$ − 26.1094i − 0.964374i −0.876068 0.482187i $$-0.839843\pi$$
0.876068 0.482187i $$-0.160157\pi$$
$$734$$ −2.48513 −0.0917279
$$735$$ 0 0
$$736$$ 1.49646 0.0551601
$$737$$ 8.78188i 0.323485i
$$738$$ 52.8382i 1.94500i
$$739$$ −42.6130 −1.56754 −0.783772 0.621049i $$-0.786707\pi$$
−0.783772 + 0.621049i $$0.786707\pi$$
$$740$$ 0 0
$$741$$ 74.9193 2.75223
$$742$$ − 27.1392i − 0.996310i
$$743$$ 35.4922i 1.30208i 0.759042 + 0.651042i $$0.225668\pi$$
−0.759042 + 0.651042i $$0.774332\pi$$
$$744$$ 26.0758 0.955984
$$745$$ 0 0
$$746$$ −35.1586 −1.28725
$$747$$ 124.845i 4.56782i
$$748$$ 3.55276i 0.129902i
$$749$$ −7.91005 −0.289027
$$750$$ 0 0
$$751$$ 21.5722 0.787182 0.393591 0.919286i $$-0.371233\pi$$
0.393591 + 0.919286i $$0.371233\pi$$
$$752$$ 1.84951i 0.0674446i
$$753$$ 33.5864i 1.22396i
$$754$$ −26.3909 −0.961101
$$755$$ 0 0
$$756$$ 45.1586 1.64240
$$757$$ − 23.7890i − 0.864625i −0.901724 0.432312i $$-0.857698\pi$$
0.901724 0.432312i $$-0.142302\pi$$
$$758$$ − 24.6778i − 0.896336i
$$759$$ 23.7748 0.862970
$$760$$ 0 0
$$761$$ −18.6724 −0.676876 −0.338438 0.940989i $$-0.609898\pi$$
−0.338438 + 0.940989i $$0.609898\pi$$
$$762$$ 29.5864i 1.07180i
$$763$$ − 3.74114i − 0.135438i
$$764$$ 21.7790 0.787938
$$765$$ 0 0
$$766$$ −13.3839 −0.483578
$$767$$ 39.1445i 1.41342i
$$768$$ 3.34596i 0.120737i
$$769$$ 2.48937 0.0897689 0.0448845 0.998992i $$-0.485708\pi$$
0.0448845 + 0.998992i $$0.485708\pi$$
$$770$$ 0 0
$$771$$ 70.1657 2.52696
$$772$$ − 12.9929i − 0.467625i
$$773$$ − 0.950786i − 0.0341974i −0.999854 0.0170987i $$-0.994557\pi$$
0.999854 0.0170987i $$-0.00544295\pi$$
$$774$$ −15.9295 −0.572575
$$775$$ 0 0
$$776$$ −10.4472 −0.375034
$$777$$ − 8.69193i − 0.311821i
$$778$$ − 18.2220i − 0.653292i
$$779$$ −21.5722 −0.772906
$$780$$ 0 0
$$781$$ −18.4582 −0.660488
$$782$$ − 1.11969i − 0.0400401i
$$783$$ 68.5567i 2.45002i
$$784$$ 0.251772 0.00899185
$$785$$ 0 0
$$786$$ −20.5793 −0.734040
$$787$$ − 29.4359i − 1.04928i −0.851325 0.524639i $$-0.824200\pi$$
0.851325 0.524639i $$-0.175800\pi$$
$$788$$ − 0.616147i − 0.0219493i
$$789$$ 74.8577 2.66500
$$790$$ 0 0
$$791$$ −28.9366 −1.02887
$$792$$ 38.9140i 1.38275i
$$793$$ − 53.1586i − 1.88772i
$$794$$ 2.22521 0.0789696
$$795$$ 0 0
$$796$$ −1.54852 −0.0548859
$$797$$ − 14.1742i − 0.502076i −0.967977 0.251038i $$-0.919228\pi$$
0.967977 0.251038i $$-0.0807719\pi$$
$$798$$ 29.0829i 1.02952i
$$799$$ 1.38385 0.0489572
$$800$$ 0 0
$$801$$ 49.1728 1.73744
$$802$$ 12.3909i 0.437539i
$$803$$ 35.5949i 1.25612i
$$804$$ 6.18838 0.218247
$$805$$ 0 0
$$806$$ 52.1516 1.83696
$$807$$ 14.0142i 0.493322i
$$808$$ 12.1884i 0.428786i
$$809$$ 32.9929 1.15997 0.579985 0.814627i $$-0.303059\pi$$
0.579985 + 0.814627i $$0.303059\pi$$
$$810$$ 0 0
$$811$$ 40.5567 1.42414 0.712069 0.702110i $$-0.247758\pi$$
0.712069 + 0.702110i $$0.247758\pi$$
$$812$$ − 10.2447i − 0.359518i
$$813$$ − 40.5283i − 1.42139i
$$814$$ 4.74823 0.166425
$$815$$ 0 0
$$816$$ 2.50354 0.0876416
$$817$$ − 6.50354i − 0.227530i
$$818$$ − 7.27125i − 0.254233i
$$819$$ 142.469 4.97826
$$820$$ 0 0
$$821$$ −38.8661 −1.35644 −0.678219 0.734860i $$-0.737248\pi$$
−0.678219 + 0.734860i $$0.737248\pi$$
$$822$$ − 36.4525i − 1.27143i
$$823$$ 49.6243i 1.72979i 0.501949 + 0.864897i $$0.332616\pi$$
−0.501949 + 0.864897i $$0.667384\pi$$
$$824$$ 1.30807 0.0455689
$$825$$ 0 0
$$826$$ −15.1955 −0.528718
$$827$$ − 30.0195i − 1.04388i −0.852982 0.521940i $$-0.825209\pi$$
0.852982 0.521940i $$-0.174791\pi$$
$$828$$ − 12.2642i − 0.426209i
$$829$$ −10.5598 −0.366759 −0.183379 0.983042i $$-0.558704\pi$$
−0.183379 + 0.983042i $$0.558704\pi$$
$$830$$ 0 0
$$831$$ 31.1445 1.08039
$$832$$ 6.69193i 0.232001i
$$833$$ − 0.188383i − 0.00652708i
$$834$$ 3.51063 0.121563
$$835$$ 0 0
$$836$$ −15.8874 −0.549477
$$837$$ − 135.476i − 4.68273i
$$838$$ − 28.4809i − 0.983856i
$$839$$ −29.3839 −1.01444 −0.507222 0.861816i $$-0.669327\pi$$
−0.507222 + 0.861816i $$0.669327\pi$$
$$840$$ 0 0
$$841$$ −13.4472 −0.463698
$$842$$ 22.0000i 0.758170i
$$843$$ 64.4809i 2.22084i
$$844$$ −20.9366 −0.720668
$$845$$ 0 0
$$846$$ 15.1576 0.521128
$$847$$ − 29.9926i − 1.03056i
$$848$$ − 10.4472i − 0.358760i
$$849$$ −45.7890 −1.57147
$$850$$ 0 0
$$851$$ −1.49646 −0.0512979
$$852$$ 13.0071i 0.445615i
$$853$$ − 6.22521i − 0.213147i −0.994305 0.106573i $$-0.966012\pi$$
0.994305 0.106573i $$-0.0339879\pi$$
$$854$$ 20.6356 0.706137
$$855$$ 0 0
$$856$$ −3.04498 −0.104075
$$857$$ − 3.65119i − 0.124722i −0.998054 0.0623611i $$-0.980137\pi$$
0.998054 0.0623611i $$-0.0198630\pi$$
$$858$$ 106.317i 3.62961i
$$859$$ 36.3162 1.23909 0.619547 0.784960i $$-0.287316\pi$$
0.619547 + 0.784960i $$0.287316\pi$$
$$860$$ 0 0
$$861$$ −56.0390 −1.90980
$$862$$ 1.51487i 0.0515966i
$$863$$ − 24.5751i − 0.836546i −0.908321 0.418273i $$-0.862636\pi$$
0.908321 0.418273i $$-0.137364\pi$$
$$864$$ 17.3839 0.591411
$$865$$ 0 0
$$866$$ 24.9929 0.849294
$$867$$ 55.0082i 1.86817i
$$868$$ 20.2447i 0.687149i
$$869$$ 78.5425 2.66437
$$870$$ 0 0
$$871$$ 12.3768 0.419371
$$872$$ − 1.44015i − 0.0487698i
$$873$$ 85.6201i 2.89780i
$$874$$ 5.00709 0.169367
$$875$$ 0 0
$$876$$ 25.0829 0.847472
$$877$$ 20.0563i 0.677253i 0.940921 + 0.338627i $$0.109962\pi$$
−0.940921 + 0.338627i $$0.890038\pi$$
$$878$$ 5.10128i 0.172160i
$$879$$ −34.9561 −1.17904
$$880$$ 0 0
$$881$$ −15.0266 −0.506258 −0.253129 0.967433i $$-0.581460\pi$$
−0.253129 + 0.967433i $$0.581460\pi$$
$$882$$ − 2.06339i − 0.0694779i
$$883$$ 6.62145i 0.222830i 0.993774 + 0.111415i $$0.0355382\pi$$
−0.993774 + 0.111415i $$0.964462\pi$$
$$884$$ 5.00709 0.168407
$$885$$ 0 0
$$886$$ 25.7369 0.864648
$$887$$ − 37.5538i − 1.26093i −0.776216 0.630467i $$-0.782863\pi$$
0.776216 0.630467i $$-0.217137\pi$$
$$888$$ − 3.34596i − 0.112283i
$$889$$ −22.9703 −0.770398
$$890$$ 0 0
$$891$$ 159.442 5.34152
$$892$$ − 2.71034i − 0.0907488i
$$893$$ 6.18838i 0.207086i
$$894$$ −14.0142 −0.468704
$$895$$ 0 0
$$896$$ −2.59774 −0.0867842
$$897$$ − 33.5071i − 1.11877i
$$898$$ 17.7748i 0.593153i
$$899$$ −30.7341 −1.02504
$$900$$ 0 0
$$901$$ −7.81692 −0.260419
$$902$$ − 30.6130i − 1.01930i
$$903$$ − 16.8945i − 0.562213i
$$904$$ −11.1392 −0.370483
$$905$$ 0 0
$$906$$ 22.3909 0.743889
$$907$$ 50.4667i 1.67572i 0.545885 + 0.837860i $$0.316193\pi$$
−0.545885 + 0.837860i $$0.683807\pi$$
$$908$$ − 26.1321i − 0.867224i
$$909$$ 99.8895 3.31313
$$910$$ 0 0
$$911$$ 15.9395 0.528098 0.264049 0.964509i $$-0.414942\pi$$
0.264049 + 0.964509i $$0.414942\pi$$
$$912$$ 11.1955i 0.370719i
$$913$$ − 72.3315i − 2.39382i
$$914$$ −34.3346 −1.13569
$$915$$ 0 0
$$916$$ −24.7677 −0.818348
$$917$$ − 15.9774i − 0.527619i
$$918$$ − 13.0071i − 0.429298i
$$919$$ −32.5414 −1.07344 −0.536721 0.843760i $$-0.680337\pi$$
−0.536721 + 0.843760i $$0.680337\pi$$
$$920$$ 0 0
$$921$$ 16.5793 0.546307
$$922$$ − 16.9508i − 0.558244i
$$923$$ 26.0142i 0.856267i
$$924$$ −41.2713 −1.35772
$$925$$ 0 0
$$926$$ 10.9929 0.361250
$$927$$ − 10.7203i − 0.352100i
$$928$$ − 3.94370i − 0.129458i
$$929$$ 34.1094 1.11909 0.559547 0.828799i $$-0.310975\pi$$
0.559547 + 0.828799i $$0.310975\pi$$
$$930$$ 0 0
$$931$$ 0.842420 0.0276092
$$932$$ − 12.2783i − 0.402190i
$$933$$ 12.3152i 0.403180i
$$934$$ −37.4175 −1.22434
$$935$$ 0 0
$$936$$ 54.8435 1.79262
$$937$$ 0.0141751i 0 0.000463082i 1.00000 0.000231541i $$7.37017e-5\pi$$
−1.00000 0.000231541i $$0.999926\pi$$
$$938$$ 4.80453i 0.156873i
$$939$$ −3.32225 −0.108417
$$940$$ 0 0
$$941$$ 11.1813 0.364500 0.182250 0.983252i $$-0.441662\pi$$
0.182250 + 0.983252i $$0.441662\pi$$
$$942$$ 13.1955i 0.429932i
$$943$$ 9.64802i 0.314183i
$$944$$ −5.84951 −0.190385
$$945$$ 0 0
$$946$$ 9.22912 0.300065
$$947$$ − 53.6427i − 1.74315i −0.490259 0.871577i $$-0.663098\pi$$
0.490259 0.871577i $$-0.336902\pi$$
$$948$$ − 55.3470i − 1.79759i
$$949$$ 50.1657 1.62845
$$950$$ 0 0
$$951$$ 21.9490 0.711745
$$952$$ 1.94370i 0.0629956i
$$953$$ 9.88740i 0.320284i 0.987094 + 0.160142i $$0.0511952\pi$$
−0.987094 + 0.160142i $$0.948805\pi$$
$$954$$ −85.6201 −2.77205
$$955$$ 0 0
$$956$$ −17.1771 −0.555546
$$957$$ − 62.6551i − 2.02535i
$$958$$ 0.465654i 0.0150446i
$$959$$ 28.3010 0.913886
$$960$$ 0 0
$$961$$ 29.7341 0.959163
$$962$$ − 6.69193i − 0.215756i
$$963$$ 24.9550i 0.804164i
$$964$$ 15.2713 0.491854
$$965$$ 0 0
$$966$$ 13.0071 0.418496
$$967$$ 51.6254i 1.66016i 0.557644 + 0.830080i $$0.311705\pi$$
−0.557644 + 0.830080i $$0.688295\pi$$
$$968$$ − 11.5457i − 0.371092i
$$969$$ 8.37677 0.269100
$$970$$ 0 0
$$971$$ 20.5598 0.659797 0.329898 0.944016i $$-0.392986\pi$$
0.329898 + 0.944016i $$0.392986\pi$$
$$972$$ − 60.2036i − 1.93103i
$$973$$ 2.72558i 0.0873781i
$$974$$ −34.9561 −1.12007
$$975$$ 0 0
$$976$$ 7.94370 0.254272
$$977$$ 38.4472i 1.23004i 0.788513 + 0.615018i $$0.210851\pi$$
−0.788513 + 0.615018i $$0.789149\pi$$
$$978$$ − 40.6409i − 1.29955i
$$979$$ −28.4894 −0.910524
$$980$$ 0 0
$$981$$ −11.8027 −0.376833
$$982$$ − 24.7819i − 0.790822i
$$983$$ − 11.8973i − 0.379466i −0.981836 0.189733i $$-0.939238\pi$$
0.981836 0.189733i $$-0.0607622\pi$$
$$984$$ −21.5722 −0.687698
$$985$$ 0 0
$$986$$ −2.95079 −0.0939722
$$987$$ 16.0758i 0.511698i
$$988$$ 22.3909i 0.712351i
$$989$$ −2.90866 −0.0924900
$$990$$ 0 0
$$991$$ −19.7932 −0.628752 −0.314376 0.949299i $$-0.601795\pi$$
−0.314376 + 0.949299i $$0.601795\pi$$
$$992$$ 7.79321i 0.247435i
$$993$$ − 60.3541i − 1.91528i
$$994$$ −10.0984 −0.320303
$$995$$ 0 0
$$996$$ −50.9703 −1.61505
$$997$$ 5.88740i 0.186456i 0.995645 + 0.0932279i $$0.0297185\pi$$
−0.995645 + 0.0932279i $$0.970281\pi$$
$$998$$ 2.33888i 0.0740358i
$$999$$ −17.3839 −0.550001
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.b.o.149.6 6
5.2 odd 4 1850.2.a.z.1.3 3
5.3 odd 4 370.2.a.g.1.1 3
5.4 even 2 inner 1850.2.b.o.149.1 6
15.8 even 4 3330.2.a.bg.1.1 3
20.3 even 4 2960.2.a.u.1.3 3

By twisted newform
Twist Min Dim Char Parity Ord Type
370.2.a.g.1.1 3 5.3 odd 4
1850.2.a.z.1.3 3 5.2 odd 4
1850.2.b.o.149.1 6 5.4 even 2 inner
1850.2.b.o.149.6 6 1.1 even 1 trivial
2960.2.a.u.1.3 3 20.3 even 4
3330.2.a.bg.1.1 3 15.8 even 4