# Properties

 Label 3971.2.a.i.1.6 Level $3971$ Weight $2$ Character 3971.1 Self dual yes Analytic conductor $31.709$ Analytic rank $0$ Dimension $7$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3971,2,Mod(1,3971)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3971, 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("3971.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3971 = 11 \cdot 19^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3971.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: $$31.7085946427$$ Analytic rank: $$0$$ Dimension: $$7$$ Coefficient field: $$\mathbb{Q}[x]/(x^{7} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{7} - x^{6} - 14x^{5} + 10x^{4} + 59x^{3} - 27x^{2} - 66x + 30$$ x^7 - x^6 - 14*x^5 + 10*x^4 + 59*x^3 - 27*x^2 - 66*x + 30 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 209) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.6 Root $$2.61330$$ of defining polynomial Character $$\chi$$ $$=$$ 3971.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+2.61330 q^{2} -1.19599 q^{3} +4.82936 q^{4} +4.07680 q^{5} -3.12549 q^{6} +3.61829 q^{7} +7.39397 q^{8} -1.56960 q^{9} +O(q^{10})$$ $$q+2.61330 q^{2} -1.19599 q^{3} +4.82936 q^{4} +4.07680 q^{5} -3.12549 q^{6} +3.61829 q^{7} +7.39397 q^{8} -1.56960 q^{9} +10.6539 q^{10} -1.00000 q^{11} -5.77587 q^{12} +1.47857 q^{13} +9.45570 q^{14} -4.87582 q^{15} +9.66398 q^{16} -3.27003 q^{17} -4.10185 q^{18} +19.6883 q^{20} -4.32745 q^{21} -2.61330 q^{22} -7.45793 q^{23} -8.84313 q^{24} +11.6203 q^{25} +3.86395 q^{26} +5.46521 q^{27} +17.4740 q^{28} -1.02535 q^{29} -12.7420 q^{30} -1.64921 q^{31} +10.4670 q^{32} +1.19599 q^{33} -8.54558 q^{34} +14.7511 q^{35} -7.58018 q^{36} +6.71293 q^{37} -1.76836 q^{39} +30.1438 q^{40} +3.92451 q^{41} -11.3089 q^{42} +5.38113 q^{43} -4.82936 q^{44} -6.39896 q^{45} -19.4898 q^{46} -3.71597 q^{47} -11.5580 q^{48} +6.09205 q^{49} +30.3674 q^{50} +3.91093 q^{51} +7.14054 q^{52} +0.102902 q^{53} +14.2823 q^{54} -4.07680 q^{55} +26.7536 q^{56} -2.67955 q^{58} -13.2986 q^{59} -23.5471 q^{60} -6.49664 q^{61} -4.30989 q^{62} -5.67929 q^{63} +8.02543 q^{64} +6.02783 q^{65} +3.12549 q^{66} +3.70989 q^{67} -15.7921 q^{68} +8.91962 q^{69} +38.5490 q^{70} -6.32968 q^{71} -11.6056 q^{72} -1.37759 q^{73} +17.5429 q^{74} -13.8978 q^{75} -3.61829 q^{77} -4.62125 q^{78} -13.6725 q^{79} +39.3981 q^{80} -1.82753 q^{81} +10.2559 q^{82} +5.44061 q^{83} -20.8988 q^{84} -13.3313 q^{85} +14.0625 q^{86} +1.22631 q^{87} -7.39397 q^{88} -12.1357 q^{89} -16.7224 q^{90} +5.34990 q^{91} -36.0170 q^{92} +1.97244 q^{93} -9.71096 q^{94} -12.5184 q^{96} +13.7910 q^{97} +15.9204 q^{98} +1.56960 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$7 q + q^{2} - 2 q^{3} + 15 q^{4} + 2 q^{5} - 2 q^{6} + 10 q^{7} + 9 q^{8} + 11 q^{9}+O(q^{10})$$ 7 * q + q^2 - 2 * q^3 + 15 * q^4 + 2 * q^5 - 2 * q^6 + 10 * q^7 + 9 * q^8 + 11 * q^9 $$7 q + q^{2} - 2 q^{3} + 15 q^{4} + 2 q^{5} - 2 q^{6} + 10 q^{7} + 9 q^{8} + 11 q^{9} + 6 q^{10} - 7 q^{11} + 16 q^{12} + 4 q^{13} - 6 q^{14} - 12 q^{15} + 27 q^{16} + 2 q^{17} - 9 q^{18} - 4 q^{20} + 14 q^{21} - q^{22} + 10 q^{23} - 2 q^{24} + 9 q^{25} - 8 q^{26} + 4 q^{27} + 26 q^{28} + 18 q^{29} - 42 q^{30} - 24 q^{31} + 49 q^{32} + 2 q^{33} + 6 q^{34} + 8 q^{35} + 29 q^{36} + 24 q^{39} + 2 q^{40} + 12 q^{41} - 44 q^{42} + 2 q^{43} - 15 q^{44} - 4 q^{45} + 4 q^{46} + 8 q^{47} + 72 q^{48} + 17 q^{49} + 33 q^{50} + 24 q^{51} + 60 q^{52} - 2 q^{53} - 52 q^{54} - 2 q^{55} - 26 q^{56} - 8 q^{58} + 10 q^{59} - 42 q^{60} + 14 q^{61} + 14 q^{62} + 55 q^{64} + 14 q^{65} + 2 q^{66} - 8 q^{67} - 18 q^{68} + 6 q^{69} + 66 q^{70} - 10 q^{71} - 53 q^{72} - 6 q^{73} + 26 q^{74} - 26 q^{75} - 10 q^{77} - 22 q^{78} - 52 q^{79} - 12 q^{80} - q^{81} + 24 q^{82} - 10 q^{83} + 6 q^{84} - 12 q^{85} - 8 q^{86} + 6 q^{87} - 9 q^{88} - 20 q^{90} - 12 q^{91} + 2 q^{93} - 24 q^{94} + 6 q^{96} + 24 q^{97} - 19 q^{98} - 11 q^{99}+O(q^{100})$$ 7 * q + q^2 - 2 * q^3 + 15 * q^4 + 2 * q^5 - 2 * q^6 + 10 * q^7 + 9 * q^8 + 11 * q^9 + 6 * q^10 - 7 * q^11 + 16 * q^12 + 4 * q^13 - 6 * q^14 - 12 * q^15 + 27 * q^16 + 2 * q^17 - 9 * q^18 - 4 * q^20 + 14 * q^21 - q^22 + 10 * q^23 - 2 * q^24 + 9 * q^25 - 8 * q^26 + 4 * q^27 + 26 * q^28 + 18 * q^29 - 42 * q^30 - 24 * q^31 + 49 * q^32 + 2 * q^33 + 6 * q^34 + 8 * q^35 + 29 * q^36 + 24 * q^39 + 2 * q^40 + 12 * q^41 - 44 * q^42 + 2 * q^43 - 15 * q^44 - 4 * q^45 + 4 * q^46 + 8 * q^47 + 72 * q^48 + 17 * q^49 + 33 * q^50 + 24 * q^51 + 60 * q^52 - 2 * q^53 - 52 * q^54 - 2 * q^55 - 26 * q^56 - 8 * q^58 + 10 * q^59 - 42 * q^60 + 14 * q^61 + 14 * q^62 + 55 * q^64 + 14 * q^65 + 2 * q^66 - 8 * q^67 - 18 * q^68 + 6 * q^69 + 66 * q^70 - 10 * q^71 - 53 * q^72 - 6 * q^73 + 26 * q^74 - 26 * q^75 - 10 * q^77 - 22 * q^78 - 52 * q^79 - 12 * q^80 - q^81 + 24 * q^82 - 10 * q^83 + 6 * q^84 - 12 * q^85 - 8 * q^86 + 6 * q^87 - 9 * q^88 - 20 * q^90 - 12 * q^91 + 2 * q^93 - 24 * q^94 + 6 * q^96 + 24 * q^97 - 19 * q^98 - 11 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 2.61330 1.84789 0.923943 0.382531i $$-0.124948\pi$$
0.923943 + 0.382531i $$0.124948\pi$$
$$3$$ −1.19599 −0.690506 −0.345253 0.938510i $$-0.612207\pi$$
−0.345253 + 0.938510i $$0.612207\pi$$
$$4$$ 4.82936 2.41468
$$5$$ 4.07680 1.82320 0.911600 0.411078i $$-0.134847\pi$$
0.911600 + 0.411078i $$0.134847\pi$$
$$6$$ −3.12549 −1.27598
$$7$$ 3.61829 1.36759 0.683793 0.729676i $$-0.260329\pi$$
0.683793 + 0.729676i $$0.260329\pi$$
$$8$$ 7.39397 2.61416
$$9$$ −1.56960 −0.523201
$$10$$ 10.6539 3.36907
$$11$$ −1.00000 −0.301511
$$12$$ −5.77587 −1.66735
$$13$$ 1.47857 0.410081 0.205041 0.978753i $$-0.434267\pi$$
0.205041 + 0.978753i $$0.434267\pi$$
$$14$$ 9.45570 2.52714
$$15$$ −4.87582 −1.25893
$$16$$ 9.66398 2.41600
$$17$$ −3.27003 −0.793099 −0.396549 0.918013i $$-0.629792\pi$$
−0.396549 + 0.918013i $$0.629792\pi$$
$$18$$ −4.10185 −0.966816
$$19$$ 0 0
$$20$$ 19.6883 4.40244
$$21$$ −4.32745 −0.944327
$$22$$ −2.61330 −0.557158
$$23$$ −7.45793 −1.55509 −0.777543 0.628830i $$-0.783534\pi$$
−0.777543 + 0.628830i $$0.783534\pi$$
$$24$$ −8.84313 −1.80510
$$25$$ 11.6203 2.32406
$$26$$ 3.86395 0.757783
$$27$$ 5.46521 1.05178
$$28$$ 17.4740 3.30228
$$29$$ −1.02535 −0.190403 −0.0952013 0.995458i $$-0.530349\pi$$
−0.0952013 + 0.995458i $$0.530349\pi$$
$$30$$ −12.7420 −2.32636
$$31$$ −1.64921 −0.296207 −0.148104 0.988972i $$-0.547317\pi$$
−0.148104 + 0.988972i $$0.547317\pi$$
$$32$$ 10.4670 1.85032
$$33$$ 1.19599 0.208195
$$34$$ −8.54558 −1.46556
$$35$$ 14.7511 2.49338
$$36$$ −7.58018 −1.26336
$$37$$ 6.71293 1.10360 0.551799 0.833977i $$-0.313941\pi$$
0.551799 + 0.833977i $$0.313941\pi$$
$$38$$ 0 0
$$39$$ −1.76836 −0.283164
$$40$$ 30.1438 4.76615
$$41$$ 3.92451 0.612905 0.306453 0.951886i $$-0.400858\pi$$
0.306453 + 0.951886i $$0.400858\pi$$
$$42$$ −11.3089 −1.74501
$$43$$ 5.38113 0.820614 0.410307 0.911947i $$-0.365422\pi$$
0.410307 + 0.911947i $$0.365422\pi$$
$$44$$ −4.82936 −0.728053
$$45$$ −6.39896 −0.953901
$$46$$ −19.4898 −2.87362
$$47$$ −3.71597 −0.542030 −0.271015 0.962575i $$-0.587359\pi$$
−0.271015 + 0.962575i $$0.587359\pi$$
$$48$$ −11.5580 −1.66826
$$49$$ 6.09205 0.870292
$$50$$ 30.3674 4.29460
$$51$$ 3.91093 0.547640
$$52$$ 7.14054 0.990215
$$53$$ 0.102902 0.0141347 0.00706733 0.999975i $$-0.497750\pi$$
0.00706733 + 0.999975i $$0.497750\pi$$
$$54$$ 14.2823 1.94357
$$55$$ −4.07680 −0.549716
$$56$$ 26.7536 3.57509
$$57$$ 0 0
$$58$$ −2.67955 −0.351842
$$59$$ −13.2986 −1.73134 −0.865668 0.500619i $$-0.833106\pi$$
−0.865668 + 0.500619i $$0.833106\pi$$
$$60$$ −23.5471 −3.03991
$$61$$ −6.49664 −0.831809 −0.415905 0.909408i $$-0.636535\pi$$
−0.415905 + 0.909408i $$0.636535\pi$$
$$62$$ −4.30989 −0.547357
$$63$$ −5.67929 −0.715523
$$64$$ 8.02543 1.00318
$$65$$ 6.02783 0.747661
$$66$$ 3.12549 0.384721
$$67$$ 3.70989 0.453235 0.226618 0.973984i $$-0.427233\pi$$
0.226618 + 0.973984i $$0.427233\pi$$
$$68$$ −15.7921 −1.91508
$$69$$ 8.91962 1.07380
$$70$$ 38.5490 4.60749
$$71$$ −6.32968 −0.751194 −0.375597 0.926783i $$-0.622562\pi$$
−0.375597 + 0.926783i $$0.622562\pi$$
$$72$$ −11.6056 −1.36773
$$73$$ −1.37759 −0.161235 −0.0806173 0.996745i $$-0.525689\pi$$
−0.0806173 + 0.996745i $$0.525689\pi$$
$$74$$ 17.5429 2.03932
$$75$$ −13.8978 −1.60478
$$76$$ 0 0
$$77$$ −3.61829 −0.412343
$$78$$ −4.62125 −0.523254
$$79$$ −13.6725 −1.53828 −0.769141 0.639079i $$-0.779316\pi$$
−0.769141 + 0.639079i $$0.779316\pi$$
$$80$$ 39.3981 4.40484
$$81$$ −1.82753 −0.203059
$$82$$ 10.2559 1.13258
$$83$$ 5.44061 0.597184 0.298592 0.954381i $$-0.403483\pi$$
0.298592 + 0.954381i $$0.403483\pi$$
$$84$$ −20.8988 −2.28025
$$85$$ −13.3313 −1.44598
$$86$$ 14.0625 1.51640
$$87$$ 1.22631 0.131474
$$88$$ −7.39397 −0.788200
$$89$$ −12.1357 −1.28638 −0.643191 0.765706i $$-0.722390\pi$$
−0.643191 + 0.765706i $$0.722390\pi$$
$$90$$ −16.7224 −1.76270
$$91$$ 5.34990 0.560822
$$92$$ −36.0170 −3.75503
$$93$$ 1.97244 0.204533
$$94$$ −9.71096 −1.00161
$$95$$ 0 0
$$96$$ −12.5184 −1.27766
$$97$$ 13.7910 1.40026 0.700131 0.714014i $$-0.253125\pi$$
0.700131 + 0.714014i $$0.253125\pi$$
$$98$$ 15.9204 1.60820
$$99$$ 1.56960 0.157751
$$100$$ 56.1186 5.61186
$$101$$ −11.0029 −1.09483 −0.547413 0.836863i $$-0.684387\pi$$
−0.547413 + 0.836863i $$0.684387\pi$$
$$102$$ 10.2204 1.01198
$$103$$ 4.99191 0.491867 0.245934 0.969287i $$-0.420906\pi$$
0.245934 + 0.969287i $$0.420906\pi$$
$$104$$ 10.9325 1.07202
$$105$$ −17.6421 −1.72170
$$106$$ 0.268914 0.0261192
$$107$$ −7.31345 −0.707018 −0.353509 0.935431i $$-0.615012\pi$$
−0.353509 + 0.935431i $$0.615012\pi$$
$$108$$ 26.3934 2.53971
$$109$$ 1.44482 0.138389 0.0691944 0.997603i $$-0.477957\pi$$
0.0691944 + 0.997603i $$0.477957\pi$$
$$110$$ −10.6539 −1.01581
$$111$$ −8.02861 −0.762042
$$112$$ 34.9671 3.30408
$$113$$ 12.0369 1.13234 0.566169 0.824289i $$-0.308425\pi$$
0.566169 + 0.824289i $$0.308425\pi$$
$$114$$ 0 0
$$115$$ −30.4045 −2.83523
$$116$$ −4.95178 −0.459761
$$117$$ −2.32077 −0.214555
$$118$$ −34.7534 −3.19931
$$119$$ −11.8319 −1.08463
$$120$$ −36.0517 −3.29105
$$121$$ 1.00000 0.0909091
$$122$$ −16.9777 −1.53709
$$123$$ −4.69368 −0.423215
$$124$$ −7.96463 −0.715245
$$125$$ 26.9897 2.41403
$$126$$ −14.8417 −1.32220
$$127$$ 4.69692 0.416784 0.208392 0.978045i $$-0.433177\pi$$
0.208392 + 0.978045i $$0.433177\pi$$
$$128$$ 0.0389415 0.00344197
$$129$$ −6.43578 −0.566639
$$130$$ 15.7526 1.38159
$$131$$ 3.74466 0.327173 0.163586 0.986529i $$-0.447694\pi$$
0.163586 + 0.986529i $$0.447694\pi$$
$$132$$ 5.77587 0.502725
$$133$$ 0 0
$$134$$ 9.69507 0.837526
$$135$$ 22.2806 1.91761
$$136$$ −24.1785 −2.07329
$$137$$ −15.7595 −1.34643 −0.673213 0.739449i $$-0.735086\pi$$
−0.673213 + 0.739449i $$0.735086\pi$$
$$138$$ 23.3097 1.98425
$$139$$ −2.52822 −0.214440 −0.107220 0.994235i $$-0.534195\pi$$
−0.107220 + 0.994235i $$0.534195\pi$$
$$140$$ 71.2381 6.02072
$$141$$ 4.44427 0.374275
$$142$$ −16.5414 −1.38812
$$143$$ −1.47857 −0.123644
$$144$$ −15.1686 −1.26405
$$145$$ −4.18015 −0.347142
$$146$$ −3.60006 −0.297943
$$147$$ −7.28604 −0.600942
$$148$$ 32.4191 2.66484
$$149$$ 1.84902 0.151477 0.0757387 0.997128i $$-0.475869\pi$$
0.0757387 + 0.997128i $$0.475869\pi$$
$$150$$ −36.3191 −2.96545
$$151$$ 15.3184 1.24659 0.623296 0.781986i $$-0.285793\pi$$
0.623296 + 0.781986i $$0.285793\pi$$
$$152$$ 0 0
$$153$$ 5.13265 0.414950
$$154$$ −9.45570 −0.761962
$$155$$ −6.72351 −0.540045
$$156$$ −8.54003 −0.683750
$$157$$ 24.6631 1.96833 0.984165 0.177254i $$-0.0567215\pi$$
0.984165 + 0.177254i $$0.0567215\pi$$
$$158$$ −35.7305 −2.84257
$$159$$ −0.123070 −0.00976006
$$160$$ 42.6718 3.37350
$$161$$ −26.9850 −2.12671
$$162$$ −4.77590 −0.375230
$$163$$ −3.72149 −0.291490 −0.145745 0.989322i $$-0.546558\pi$$
−0.145745 + 0.989322i $$0.546558\pi$$
$$164$$ 18.9529 1.47997
$$165$$ 4.87582 0.379582
$$166$$ 14.2180 1.10353
$$167$$ −2.64758 −0.204876 −0.102438 0.994739i $$-0.532664\pi$$
−0.102438 + 0.994739i $$0.532664\pi$$
$$168$$ −31.9970 −2.46863
$$169$$ −10.8138 −0.831833
$$170$$ −34.8386 −2.67200
$$171$$ 0 0
$$172$$ 25.9874 1.98152
$$173$$ −7.34552 −0.558469 −0.279235 0.960223i $$-0.590081\pi$$
−0.279235 + 0.960223i $$0.590081\pi$$
$$174$$ 3.20472 0.242949
$$175$$ 42.0457 3.17835
$$176$$ −9.66398 −0.728450
$$177$$ 15.9051 1.19550
$$178$$ −31.7143 −2.37708
$$179$$ −9.55394 −0.714095 −0.357047 0.934086i $$-0.616217\pi$$
−0.357047 + 0.934086i $$0.616217\pi$$
$$180$$ −30.9029 −2.30336
$$181$$ −6.02638 −0.447937 −0.223969 0.974596i $$-0.571901\pi$$
−0.223969 + 0.974596i $$0.571901\pi$$
$$182$$ 13.9809 1.03633
$$183$$ 7.76993 0.574369
$$184$$ −55.1437 −4.06525
$$185$$ 27.3673 2.01208
$$186$$ 5.15459 0.377953
$$187$$ 3.27003 0.239128
$$188$$ −17.9458 −1.30883
$$189$$ 19.7747 1.43840
$$190$$ 0 0
$$191$$ 17.7069 1.28123 0.640613 0.767864i $$-0.278680\pi$$
0.640613 + 0.767864i $$0.278680\pi$$
$$192$$ −9.59835 −0.692701
$$193$$ −3.69348 −0.265863 −0.132931 0.991125i $$-0.542439\pi$$
−0.132931 + 0.991125i $$0.542439\pi$$
$$194$$ 36.0400 2.58752
$$195$$ −7.20924 −0.516264
$$196$$ 29.4207 2.10148
$$197$$ −25.7789 −1.83667 −0.918336 0.395802i $$-0.870467\pi$$
−0.918336 + 0.395802i $$0.870467\pi$$
$$198$$ 4.10185 0.291506
$$199$$ 18.8953 1.33945 0.669726 0.742608i $$-0.266411\pi$$
0.669726 + 0.742608i $$0.266411\pi$$
$$200$$ 85.9202 6.07548
$$201$$ −4.43700 −0.312962
$$202$$ −28.7538 −2.02311
$$203$$ −3.71002 −0.260392
$$204$$ 18.8873 1.32237
$$205$$ 15.9994 1.11745
$$206$$ 13.0454 0.908914
$$207$$ 11.7060 0.813623
$$208$$ 14.2889 0.990755
$$209$$ 0 0
$$210$$ −46.1043 −3.18150
$$211$$ 10.1993 0.702150 0.351075 0.936347i $$-0.385816\pi$$
0.351075 + 0.936347i $$0.385816\pi$$
$$212$$ 0.496950 0.0341306
$$213$$ 7.57024 0.518704
$$214$$ −19.1123 −1.30649
$$215$$ 21.9378 1.49614
$$216$$ 40.4096 2.74953
$$217$$ −5.96733 −0.405089
$$218$$ 3.77576 0.255727
$$219$$ 1.64759 0.111334
$$220$$ −19.6883 −1.32739
$$221$$ −4.83497 −0.325235
$$222$$ −20.9812 −1.40817
$$223$$ 0.262700 0.0175917 0.00879584 0.999961i $$-0.497200\pi$$
0.00879584 + 0.999961i $$0.497200\pi$$
$$224$$ 37.8726 2.53047
$$225$$ −18.2393 −1.21595
$$226$$ 31.4561 2.09243
$$227$$ −27.3256 −1.81367 −0.906834 0.421489i $$-0.861508\pi$$
−0.906834 + 0.421489i $$0.861508\pi$$
$$228$$ 0 0
$$229$$ −5.53171 −0.365546 −0.182773 0.983155i $$-0.558507\pi$$
−0.182773 + 0.983155i $$0.558507\pi$$
$$230$$ −79.4562 −5.23918
$$231$$ 4.32745 0.284725
$$232$$ −7.58141 −0.497744
$$233$$ 27.4733 1.79984 0.899918 0.436059i $$-0.143626\pi$$
0.899918 + 0.436059i $$0.143626\pi$$
$$234$$ −6.06487 −0.396473
$$235$$ −15.1493 −0.988229
$$236$$ −64.2239 −4.18062
$$237$$ 16.3523 1.06219
$$238$$ −30.9204 −2.00427
$$239$$ 1.40339 0.0907781 0.0453890 0.998969i $$-0.485547\pi$$
0.0453890 + 0.998969i $$0.485547\pi$$
$$240$$ −47.1198 −3.04157
$$241$$ 20.7696 1.33789 0.668944 0.743312i $$-0.266746\pi$$
0.668944 + 0.743312i $$0.266746\pi$$
$$242$$ 2.61330 0.167990
$$243$$ −14.2099 −0.911566
$$244$$ −31.3746 −2.00855
$$245$$ 24.8361 1.58672
$$246$$ −12.2660 −0.782052
$$247$$ 0 0
$$248$$ −12.1942 −0.774334
$$249$$ −6.50692 −0.412359
$$250$$ 70.5322 4.46085
$$251$$ −1.29936 −0.0820148 −0.0410074 0.999159i $$-0.513057\pi$$
−0.0410074 + 0.999159i $$0.513057\pi$$
$$252$$ −27.4273 −1.72776
$$253$$ 7.45793 0.468876
$$254$$ 12.2745 0.770169
$$255$$ 15.9441 0.998457
$$256$$ −15.9491 −0.996819
$$257$$ −3.41219 −0.212847 −0.106423 0.994321i $$-0.533940\pi$$
−0.106423 + 0.994321i $$0.533940\pi$$
$$258$$ −16.8187 −1.04708
$$259$$ 24.2893 1.50927
$$260$$ 29.1106 1.80536
$$261$$ 1.60939 0.0996189
$$262$$ 9.78594 0.604577
$$263$$ 14.2418 0.878189 0.439094 0.898441i $$-0.355299\pi$$
0.439094 + 0.898441i $$0.355299\pi$$
$$264$$ 8.84313 0.544257
$$265$$ 0.419510 0.0257703
$$266$$ 0 0
$$267$$ 14.5142 0.888254
$$268$$ 17.9164 1.09442
$$269$$ −5.39477 −0.328925 −0.164462 0.986383i $$-0.552589\pi$$
−0.164462 + 0.986383i $$0.552589\pi$$
$$270$$ 58.2259 3.54351
$$271$$ −11.0624 −0.671995 −0.335998 0.941863i $$-0.609073\pi$$
−0.335998 + 0.941863i $$0.609073\pi$$
$$272$$ −31.6015 −1.91612
$$273$$ −6.39843 −0.387251
$$274$$ −41.1844 −2.48804
$$275$$ −11.6203 −0.700731
$$276$$ 43.0760 2.59287
$$277$$ 14.0808 0.846036 0.423018 0.906121i $$-0.360971\pi$$
0.423018 + 0.906121i $$0.360971\pi$$
$$278$$ −6.60700 −0.396261
$$279$$ 2.58861 0.154976
$$280$$ 109.069 6.51812
$$281$$ 10.8199 0.645459 0.322729 0.946491i $$-0.395400\pi$$
0.322729 + 0.946491i $$0.395400\pi$$
$$282$$ 11.6142 0.691617
$$283$$ 1.90947 0.113506 0.0567532 0.998388i $$-0.481925\pi$$
0.0567532 + 0.998388i $$0.481925\pi$$
$$284$$ −30.5683 −1.81389
$$285$$ 0 0
$$286$$ −3.86395 −0.228480
$$287$$ 14.2000 0.838201
$$288$$ −16.4290 −0.968089
$$289$$ −6.30690 −0.370994
$$290$$ −10.9240 −0.641479
$$291$$ −16.4939 −0.966890
$$292$$ −6.65287 −0.389330
$$293$$ −3.63550 −0.212388 −0.106194 0.994345i $$-0.533867\pi$$
−0.106194 + 0.994345i $$0.533867\pi$$
$$294$$ −19.0406 −1.11047
$$295$$ −54.2159 −3.15657
$$296$$ 49.6352 2.88499
$$297$$ −5.46521 −0.317124
$$298$$ 4.83205 0.279913
$$299$$ −11.0271 −0.637712
$$300$$ −67.1174 −3.87502
$$301$$ 19.4705 1.12226
$$302$$ 40.0316 2.30356
$$303$$ 13.1593 0.755984
$$304$$ 0 0
$$305$$ −26.4855 −1.51656
$$306$$ 13.4132 0.766781
$$307$$ 23.5329 1.34310 0.671548 0.740961i $$-0.265630\pi$$
0.671548 + 0.740961i $$0.265630\pi$$
$$308$$ −17.4740 −0.995675
$$309$$ −5.97028 −0.339637
$$310$$ −17.5706 −0.997941
$$311$$ −16.0026 −0.907425 −0.453713 0.891148i $$-0.649901\pi$$
−0.453713 + 0.891148i $$0.649901\pi$$
$$312$$ −13.0752 −0.740237
$$313$$ 16.0034 0.904566 0.452283 0.891875i $$-0.350610\pi$$
0.452283 + 0.891875i $$0.350610\pi$$
$$314$$ 64.4522 3.63725
$$315$$ −23.1533 −1.30454
$$316$$ −66.0296 −3.71446
$$317$$ 20.8766 1.17255 0.586273 0.810113i $$-0.300595\pi$$
0.586273 + 0.810113i $$0.300595\pi$$
$$318$$ −0.321619 −0.0180355
$$319$$ 1.02535 0.0574086
$$320$$ 32.7181 1.82900
$$321$$ 8.74683 0.488200
$$322$$ −70.5199 −3.92992
$$323$$ 0 0
$$324$$ −8.82581 −0.490323
$$325$$ 17.1814 0.953054
$$326$$ −9.72539 −0.538640
$$327$$ −1.72800 −0.0955584
$$328$$ 29.0177 1.60224
$$329$$ −13.4455 −0.741273
$$330$$ 12.7420 0.701424
$$331$$ −15.1136 −0.830721 −0.415360 0.909657i $$-0.636345\pi$$
−0.415360 + 0.909657i $$0.636345\pi$$
$$332$$ 26.2746 1.44201
$$333$$ −10.5366 −0.577404
$$334$$ −6.91893 −0.378587
$$335$$ 15.1245 0.826339
$$336$$ −41.8204 −2.28149
$$337$$ 12.2766 0.668751 0.334376 0.942440i $$-0.391475\pi$$
0.334376 + 0.942440i $$0.391475\pi$$
$$338$$ −28.2598 −1.53713
$$339$$ −14.3961 −0.781886
$$340$$ −64.3814 −3.49157
$$341$$ 1.64921 0.0893098
$$342$$ 0 0
$$343$$ −3.28525 −0.177387
$$344$$ 39.7879 2.14522
$$345$$ 36.3635 1.95775
$$346$$ −19.1961 −1.03199
$$347$$ −30.9067 −1.65916 −0.829580 0.558387i $$-0.811420\pi$$
−0.829580 + 0.558387i $$0.811420\pi$$
$$348$$ 5.92229 0.317468
$$349$$ −23.9024 −1.27947 −0.639733 0.768597i $$-0.720955\pi$$
−0.639733 + 0.768597i $$0.720955\pi$$
$$350$$ 109.878 5.87323
$$351$$ 8.08069 0.431315
$$352$$ −10.4670 −0.557892
$$353$$ 15.0158 0.799210 0.399605 0.916687i $$-0.369147\pi$$
0.399605 + 0.916687i $$0.369147\pi$$
$$354$$ 41.5648 2.20914
$$355$$ −25.8048 −1.36958
$$356$$ −58.6076 −3.10620
$$357$$ 14.1509 0.748945
$$358$$ −24.9673 −1.31957
$$359$$ 14.0826 0.743251 0.371626 0.928383i $$-0.378800\pi$$
0.371626 + 0.928383i $$0.378800\pi$$
$$360$$ −47.3137 −2.49365
$$361$$ 0 0
$$362$$ −15.7488 −0.827737
$$363$$ −1.19599 −0.0627733
$$364$$ 25.8366 1.35420
$$365$$ −5.61616 −0.293963
$$366$$ 20.3052 1.06137
$$367$$ 21.3142 1.11259 0.556296 0.830984i $$-0.312222\pi$$
0.556296 + 0.830984i $$0.312222\pi$$
$$368$$ −72.0733 −3.75708
$$369$$ −6.15992 −0.320673
$$370$$ 71.5190 3.71810
$$371$$ 0.372329 0.0193304
$$372$$ 9.52564 0.493881
$$373$$ 2.42088 0.125349 0.0626743 0.998034i $$-0.480037\pi$$
0.0626743 + 0.998034i $$0.480037\pi$$
$$374$$ 8.54558 0.441882
$$375$$ −32.2794 −1.66690
$$376$$ −27.4758 −1.41696
$$377$$ −1.51605 −0.0780806
$$378$$ 51.6774 2.65800
$$379$$ −8.87535 −0.455896 −0.227948 0.973673i $$-0.573202\pi$$
−0.227948 + 0.973673i $$0.573202\pi$$
$$380$$ 0 0
$$381$$ −5.61748 −0.287792
$$382$$ 46.2735 2.36756
$$383$$ 3.54065 0.180919 0.0904595 0.995900i $$-0.471166\pi$$
0.0904595 + 0.995900i $$0.471166\pi$$
$$384$$ −0.0465737 −0.00237670
$$385$$ −14.7511 −0.751784
$$386$$ −9.65219 −0.491284
$$387$$ −8.44624 −0.429346
$$388$$ 66.6016 3.38118
$$389$$ −16.7041 −0.846933 −0.423466 0.905912i $$-0.639187\pi$$
−0.423466 + 0.905912i $$0.639187\pi$$
$$390$$ −18.8399 −0.953997
$$391$$ 24.3877 1.23334
$$392$$ 45.0444 2.27509
$$393$$ −4.47858 −0.225915
$$394$$ −67.3681 −3.39396
$$395$$ −55.7403 −2.80460
$$396$$ 7.58018 0.380918
$$397$$ 28.2073 1.41568 0.707841 0.706372i $$-0.249669\pi$$
0.707841 + 0.706372i $$0.249669\pi$$
$$398$$ 49.3792 2.47515
$$399$$ 0 0
$$400$$ 112.298 5.61492
$$401$$ 36.2078 1.80813 0.904065 0.427395i $$-0.140569\pi$$
0.904065 + 0.427395i $$0.140569\pi$$
$$402$$ −11.5952 −0.578317
$$403$$ −2.43847 −0.121469
$$404$$ −53.1368 −2.64365
$$405$$ −7.45049 −0.370218
$$406$$ −9.69540 −0.481175
$$407$$ −6.71293 −0.332748
$$408$$ 28.9173 1.43162
$$409$$ −36.9236 −1.82576 −0.912878 0.408233i $$-0.866145\pi$$
−0.912878 + 0.408233i $$0.866145\pi$$
$$410$$ 41.8114 2.06492
$$411$$ 18.8482 0.929715
$$412$$ 24.1077 1.18770
$$413$$ −48.1184 −2.36775
$$414$$ 30.5913 1.50348
$$415$$ 22.1803 1.08879
$$416$$ 15.4762 0.758781
$$417$$ 3.02373 0.148072
$$418$$ 0 0
$$419$$ 18.0690 0.882726 0.441363 0.897329i $$-0.354495\pi$$
0.441363 + 0.897329i $$0.354495\pi$$
$$420$$ −85.2002 −4.15735
$$421$$ 8.57629 0.417983 0.208991 0.977918i $$-0.432982\pi$$
0.208991 + 0.977918i $$0.432982\pi$$
$$422$$ 26.6539 1.29749
$$423$$ 5.83260 0.283591
$$424$$ 0.760853 0.0369503
$$425$$ −37.9987 −1.84321
$$426$$ 19.7833 0.958506
$$427$$ −23.5067 −1.13757
$$428$$ −35.3193 −1.70722
$$429$$ 1.76836 0.0853771
$$430$$ 57.3301 2.76470
$$431$$ −4.28147 −0.206231 −0.103116 0.994669i $$-0.532881\pi$$
−0.103116 + 0.994669i $$0.532881\pi$$
$$432$$ 52.8157 2.54110
$$433$$ −18.2035 −0.874804 −0.437402 0.899266i $$-0.644101\pi$$
−0.437402 + 0.899266i $$0.644101\pi$$
$$434$$ −15.5945 −0.748558
$$435$$ 4.99942 0.239704
$$436$$ 6.97756 0.334165
$$437$$ 0 0
$$438$$ 4.30564 0.205732
$$439$$ −29.4442 −1.40529 −0.702647 0.711538i $$-0.747999\pi$$
−0.702647 + 0.711538i $$0.747999\pi$$
$$440$$ −30.1438 −1.43705
$$441$$ −9.56210 −0.455338
$$442$$ −12.6352 −0.600997
$$443$$ −24.3633 −1.15753 −0.578767 0.815493i $$-0.696466\pi$$
−0.578767 + 0.815493i $$0.696466\pi$$
$$444$$ −38.7730 −1.84009
$$445$$ −49.4748 −2.34533
$$446$$ 0.686514 0.0325074
$$447$$ −2.21141 −0.104596
$$448$$ 29.0384 1.37193
$$449$$ −38.7776 −1.83003 −0.915015 0.403421i $$-0.867821\pi$$
−0.915015 + 0.403421i $$0.867821\pi$$
$$450$$ −47.6648 −2.24694
$$451$$ −3.92451 −0.184798
$$452$$ 58.1306 2.73423
$$453$$ −18.3206 −0.860779
$$454$$ −71.4102 −3.35145
$$455$$ 21.8105 1.02249
$$456$$ 0 0
$$457$$ −7.47672 −0.349746 −0.174873 0.984591i $$-0.555952\pi$$
−0.174873 + 0.984591i $$0.555952\pi$$
$$458$$ −14.4560 −0.675487
$$459$$ −17.8714 −0.834165
$$460$$ −146.834 −6.84618
$$461$$ 15.9782 0.744181 0.372091 0.928196i $$-0.378641\pi$$
0.372091 + 0.928196i $$0.378641\pi$$
$$462$$ 11.3089 0.526139
$$463$$ −6.10221 −0.283594 −0.141797 0.989896i $$-0.545288\pi$$
−0.141797 + 0.989896i $$0.545288\pi$$
$$464$$ −9.90896 −0.460012
$$465$$ 8.04126 0.372904
$$466$$ 71.7961 3.32589
$$467$$ −26.6373 −1.23263 −0.616313 0.787502i $$-0.711374\pi$$
−0.616313 + 0.787502i $$0.711374\pi$$
$$468$$ −11.2078 −0.518082
$$469$$ 13.4235 0.619838
$$470$$ −39.5896 −1.82613
$$471$$ −29.4969 −1.35914
$$472$$ −98.3298 −4.52599
$$473$$ −5.38113 −0.247424
$$474$$ 42.7334 1.96281
$$475$$ 0 0
$$476$$ −57.1406 −2.61904
$$477$$ −0.161515 −0.00739527
$$478$$ 3.66750 0.167747
$$479$$ −16.2094 −0.740626 −0.370313 0.928907i $$-0.620750\pi$$
−0.370313 + 0.928907i $$0.620750\pi$$
$$480$$ −51.0351 −2.32942
$$481$$ 9.92553 0.452565
$$482$$ 54.2773 2.47226
$$483$$ 32.2738 1.46851
$$484$$ 4.82936 0.219516
$$485$$ 56.2231 2.55296
$$486$$ −37.1348 −1.68447
$$487$$ 4.82448 0.218618 0.109309 0.994008i $$-0.465136\pi$$
0.109309 + 0.994008i $$0.465136\pi$$
$$488$$ −48.0360 −2.17449
$$489$$ 4.45088 0.201276
$$490$$ 64.9042 2.93207
$$491$$ −8.53579 −0.385215 −0.192607 0.981276i $$-0.561694\pi$$
−0.192607 + 0.981276i $$0.561694\pi$$
$$492$$ −22.6675 −1.02193
$$493$$ 3.35293 0.151008
$$494$$ 0 0
$$495$$ 6.39896 0.287612
$$496$$ −15.9380 −0.715635
$$497$$ −22.9026 −1.02732
$$498$$ −17.0046 −0.761993
$$499$$ −13.4325 −0.601319 −0.300660 0.953732i $$-0.597207\pi$$
−0.300660 + 0.953732i $$0.597207\pi$$
$$500$$ 130.343 5.82910
$$501$$ 3.16648 0.141468
$$502$$ −3.39562 −0.151554
$$503$$ −1.66487 −0.0742327 −0.0371163 0.999311i $$-0.511817\pi$$
−0.0371163 + 0.999311i $$0.511817\pi$$
$$504$$ −41.9925 −1.87049
$$505$$ −44.8565 −1.99609
$$506$$ 19.4898 0.866429
$$507$$ 12.9333 0.574386
$$508$$ 22.6831 1.00640
$$509$$ 14.9684 0.663463 0.331731 0.943374i $$-0.392367\pi$$
0.331731 + 0.943374i $$0.392367\pi$$
$$510$$ 41.6667 1.84503
$$511$$ −4.98452 −0.220502
$$512$$ −41.7577 −1.84545
$$513$$ 0 0
$$514$$ −8.91709 −0.393316
$$515$$ 20.3510 0.896772
$$516$$ −31.0807 −1.36825
$$517$$ 3.71597 0.163428
$$518$$ 63.4754 2.78895
$$519$$ 8.78518 0.385627
$$520$$ 44.5696 1.95451
$$521$$ 7.89123 0.345721 0.172861 0.984946i $$-0.444699\pi$$
0.172861 + 0.984946i $$0.444699\pi$$
$$522$$ 4.20583 0.184084
$$523$$ 22.3062 0.975381 0.487690 0.873017i $$-0.337840\pi$$
0.487690 + 0.873017i $$0.337840\pi$$
$$524$$ 18.0843 0.790017
$$525$$ −50.2863 −2.19467
$$526$$ 37.2182 1.62279
$$527$$ 5.39297 0.234922
$$528$$ 11.5580 0.502999
$$529$$ 32.6207 1.41829
$$530$$ 1.09631 0.0476206
$$531$$ 20.8736 0.905837
$$532$$ 0 0
$$533$$ 5.80266 0.251341
$$534$$ 37.9300 1.64139
$$535$$ −29.8155 −1.28904
$$536$$ 27.4308 1.18483
$$537$$ 11.4264 0.493087
$$538$$ −14.0982 −0.607815
$$539$$ −6.09205 −0.262403
$$540$$ 107.601 4.63040
$$541$$ 38.9694 1.67542 0.837712 0.546112i $$-0.183893\pi$$
0.837712 + 0.546112i $$0.183893\pi$$
$$542$$ −28.9095 −1.24177
$$543$$ 7.20750 0.309304
$$544$$ −34.2273 −1.46749
$$545$$ 5.89025 0.252311
$$546$$ −16.7211 −0.715595
$$547$$ −37.7503 −1.61409 −0.807043 0.590492i $$-0.798934\pi$$
−0.807043 + 0.590492i $$0.798934\pi$$
$$548$$ −76.1083 −3.25119
$$549$$ 10.1971 0.435204
$$550$$ −30.3674 −1.29487
$$551$$ 0 0
$$552$$ 65.9514 2.80708
$$553$$ −49.4713 −2.10373
$$554$$ 36.7975 1.56338
$$555$$ −32.7310 −1.38935
$$556$$ −12.2097 −0.517805
$$557$$ −3.17436 −0.134502 −0.0672511 0.997736i $$-0.521423\pi$$
−0.0672511 + 0.997736i $$0.521423\pi$$
$$558$$ 6.76482 0.286378
$$559$$ 7.95637 0.336519
$$560$$ 142.554 6.02401
$$561$$ −3.91093 −0.165120
$$562$$ 28.2756 1.19273
$$563$$ 19.9431 0.840503 0.420252 0.907408i $$-0.361942\pi$$
0.420252 + 0.907408i $$0.361942\pi$$
$$564$$ 21.4630 0.903754
$$565$$ 49.0721 2.06448
$$566$$ 4.99004 0.209747
$$567$$ −6.61255 −0.277701
$$568$$ −46.8015 −1.96375
$$569$$ −36.6424 −1.53613 −0.768064 0.640374i $$-0.778780\pi$$
−0.768064 + 0.640374i $$0.778780\pi$$
$$570$$ 0 0
$$571$$ 11.5300 0.482515 0.241258 0.970461i $$-0.422440\pi$$
0.241258 + 0.970461i $$0.422440\pi$$
$$572$$ −7.14054 −0.298561
$$573$$ −21.1773 −0.884694
$$574$$ 37.1090 1.54890
$$575$$ −86.6634 −3.61411
$$576$$ −12.5968 −0.524865
$$577$$ 28.5590 1.18893 0.594463 0.804123i $$-0.297365\pi$$
0.594463 + 0.804123i $$0.297365\pi$$
$$578$$ −16.4818 −0.685554
$$579$$ 4.41737 0.183580
$$580$$ −20.1874 −0.838237
$$581$$ 19.6857 0.816701
$$582$$ −43.1036 −1.78670
$$583$$ −0.102902 −0.00426176
$$584$$ −10.1859 −0.421494
$$585$$ −9.46131 −0.391177
$$586$$ −9.50067 −0.392469
$$587$$ 18.1461 0.748969 0.374484 0.927233i $$-0.377820\pi$$
0.374484 + 0.927233i $$0.377820\pi$$
$$588$$ −35.1869 −1.45108
$$589$$ 0 0
$$590$$ −141.683 −5.83298
$$591$$ 30.8314 1.26823
$$592$$ 64.8736 2.66629
$$593$$ 15.3085 0.628644 0.314322 0.949316i $$-0.398223\pi$$
0.314322 + 0.949316i $$0.398223\pi$$
$$594$$ −14.2823 −0.586008
$$595$$ −48.2364 −1.97750
$$596$$ 8.92957 0.365769
$$597$$ −22.5986 −0.924900
$$598$$ −28.8171 −1.17842
$$599$$ −17.8806 −0.730580 −0.365290 0.930894i $$-0.619030\pi$$
−0.365290 + 0.930894i $$0.619030\pi$$
$$600$$ −102.760 −4.19515
$$601$$ 32.5803 1.32898 0.664489 0.747298i $$-0.268649\pi$$
0.664489 + 0.747298i $$0.268649\pi$$
$$602$$ 50.8823 2.07381
$$603$$ −5.82306 −0.237133
$$604$$ 73.9779 3.01012
$$605$$ 4.07680 0.165746
$$606$$ 34.3893 1.39697
$$607$$ −43.6494 −1.77167 −0.885837 0.463997i $$-0.846415\pi$$
−0.885837 + 0.463997i $$0.846415\pi$$
$$608$$ 0 0
$$609$$ 4.43715 0.179802
$$610$$ −69.2147 −2.80242
$$611$$ −5.49432 −0.222276
$$612$$ 24.7874 1.00197
$$613$$ 0.843061 0.0340509 0.0170254 0.999855i $$-0.494580\pi$$
0.0170254 + 0.999855i $$0.494580\pi$$
$$614$$ 61.4987 2.48189
$$615$$ −19.1352 −0.771606
$$616$$ −26.7536 −1.07793
$$617$$ 4.89882 0.197219 0.0986094 0.995126i $$-0.468561\pi$$
0.0986094 + 0.995126i $$0.468561\pi$$
$$618$$ −15.6021 −0.627611
$$619$$ −14.8704 −0.597691 −0.298846 0.954301i $$-0.596602\pi$$
−0.298846 + 0.954301i $$0.596602\pi$$
$$620$$ −32.4702 −1.30404
$$621$$ −40.7591 −1.63561
$$622$$ −41.8197 −1.67682
$$623$$ −43.9105 −1.75924
$$624$$ −17.0894 −0.684122
$$625$$ 51.9299 2.07720
$$626$$ 41.8217 1.67153
$$627$$ 0 0
$$628$$ 119.107 4.75289
$$629$$ −21.9515 −0.875263
$$630$$ −60.5067 −2.41064
$$631$$ 2.83922 0.113027 0.0565137 0.998402i $$-0.482002\pi$$
0.0565137 + 0.998402i $$0.482002\pi$$
$$632$$ −101.094 −4.02132
$$633$$ −12.1983 −0.484839
$$634$$ 54.5569 2.16673
$$635$$ 19.1484 0.759881
$$636$$ −0.594348 −0.0235674
$$637$$ 9.00751 0.356891
$$638$$ 2.67955 0.106084
$$639$$ 9.93508 0.393026
$$640$$ 0.158757 0.00627541
$$641$$ −20.6746 −0.816599 −0.408299 0.912848i $$-0.633878\pi$$
−0.408299 + 0.912848i $$0.633878\pi$$
$$642$$ 22.8581 0.902138
$$643$$ −24.6254 −0.971130 −0.485565 0.874201i $$-0.661386\pi$$
−0.485565 + 0.874201i $$0.661386\pi$$
$$644$$ −130.320 −5.13533
$$645$$ −26.2374 −1.03310
$$646$$ 0 0
$$647$$ 45.3626 1.78339 0.891693 0.452640i $$-0.149518\pi$$
0.891693 + 0.452640i $$0.149518\pi$$
$$648$$ −13.5127 −0.530830
$$649$$ 13.2986 0.522017
$$650$$ 44.9003 1.76113
$$651$$ 7.13688 0.279716
$$652$$ −17.9724 −0.703854
$$653$$ 26.1012 1.02142 0.510710 0.859753i $$-0.329383\pi$$
0.510710 + 0.859753i $$0.329383\pi$$
$$654$$ −4.51578 −0.176581
$$655$$ 15.2662 0.596501
$$656$$ 37.9264 1.48078
$$657$$ 2.16227 0.0843582
$$658$$ −35.1371 −1.36979
$$659$$ 45.8507 1.78609 0.893045 0.449967i $$-0.148564\pi$$
0.893045 + 0.449967i $$0.148564\pi$$
$$660$$ 23.5471 0.916569
$$661$$ 15.4225 0.599864 0.299932 0.953961i $$-0.403036\pi$$
0.299932 + 0.953961i $$0.403036\pi$$
$$662$$ −39.4965 −1.53508
$$663$$ 5.78258 0.224577
$$664$$ 40.2277 1.56114
$$665$$ 0 0
$$666$$ −27.5354 −1.06698
$$667$$ 7.64698 0.296092
$$668$$ −12.7861 −0.494709
$$669$$ −0.314187 −0.0121472
$$670$$ 39.5249 1.52698
$$671$$ 6.49664 0.250800
$$672$$ −45.2953 −1.74730
$$673$$ 37.8633 1.45952 0.729762 0.683702i $$-0.239631\pi$$
0.729762 + 0.683702i $$0.239631\pi$$
$$674$$ 32.0826 1.23578
$$675$$ 63.5074 2.44440
$$676$$ −52.2239 −2.00861
$$677$$ 25.3510 0.974320 0.487160 0.873313i $$-0.338033\pi$$
0.487160 + 0.873313i $$0.338033\pi$$
$$678$$ −37.6213 −1.44484
$$679$$ 49.8998 1.91498
$$680$$ −98.5710 −3.78003
$$681$$ 32.6812 1.25235
$$682$$ 4.30989 0.165034
$$683$$ 21.7513 0.832290 0.416145 0.909298i $$-0.363381\pi$$
0.416145 + 0.909298i $$0.363381\pi$$
$$684$$ 0 0
$$685$$ −64.2484 −2.45480
$$686$$ −8.58534 −0.327790
$$687$$ 6.61588 0.252412
$$688$$ 52.0031 1.98260
$$689$$ 0.152147 0.00579636
$$690$$ 95.0289 3.61769
$$691$$ 17.3051 0.658318 0.329159 0.944275i $$-0.393235\pi$$
0.329159 + 0.944275i $$0.393235\pi$$
$$692$$ −35.4741 −1.34852
$$693$$ 5.67929 0.215738
$$694$$ −80.7687 −3.06594
$$695$$ −10.3070 −0.390968
$$696$$ 9.06730 0.343695
$$697$$ −12.8333 −0.486095
$$698$$ −62.4643 −2.36431
$$699$$ −32.8578 −1.24280
$$700$$ 203.054 7.67470
$$701$$ −29.6923 −1.12146 −0.560732 0.827997i $$-0.689480\pi$$
−0.560732 + 0.827997i $$0.689480\pi$$
$$702$$ 21.1173 0.797021
$$703$$ 0 0
$$704$$ −8.02543 −0.302470
$$705$$ 18.1184 0.682378
$$706$$ 39.2409 1.47685
$$707$$ −39.8116 −1.49727
$$708$$ 76.8112 2.88674
$$709$$ −21.4898 −0.807067 −0.403534 0.914965i $$-0.632218\pi$$
−0.403534 + 0.914965i $$0.632218\pi$$
$$710$$ −67.4359 −2.53082
$$711$$ 21.4605 0.804831
$$712$$ −89.7310 −3.36281
$$713$$ 12.2997 0.460627
$$714$$ 36.9806 1.38396
$$715$$ −6.02783 −0.225428
$$716$$ −46.1394 −1.72431
$$717$$ −1.67845 −0.0626828
$$718$$ 36.8021 1.37344
$$719$$ 9.61388 0.358537 0.179269 0.983800i $$-0.442627\pi$$
0.179269 + 0.983800i $$0.442627\pi$$
$$720$$ −61.8395 −2.30462
$$721$$ 18.0622 0.672671
$$722$$ 0 0
$$723$$ −24.8403 −0.923821
$$724$$ −29.1036 −1.08163
$$725$$ −11.9149 −0.442507
$$726$$ −3.12549 −0.115998
$$727$$ 6.84046 0.253699 0.126849 0.991922i $$-0.459514\pi$$
0.126849 + 0.991922i $$0.459514\pi$$
$$728$$ 39.5570 1.46608
$$729$$ 22.4775 0.832501
$$730$$ −14.6767 −0.543210
$$731$$ −17.5965 −0.650828
$$732$$ 37.5238 1.38692
$$733$$ 38.2277 1.41197 0.705987 0.708225i $$-0.250504\pi$$
0.705987 + 0.708225i $$0.250504\pi$$
$$734$$ 55.7005 2.05594
$$735$$ −29.7037 −1.09564
$$736$$ −78.0620 −2.87740
$$737$$ −3.70989 −0.136656
$$738$$ −16.0978 −0.592567
$$739$$ −17.7157 −0.651682 −0.325841 0.945425i $$-0.605647\pi$$
−0.325841 + 0.945425i $$0.605647\pi$$
$$740$$ 132.166 4.85853
$$741$$ 0 0
$$742$$ 0.973009 0.0357203
$$743$$ −21.3028 −0.781525 −0.390763 0.920491i $$-0.627789\pi$$
−0.390763 + 0.920491i $$0.627789\pi$$
$$744$$ 14.5842 0.534682
$$745$$ 7.53808 0.276174
$$746$$ 6.32650 0.231630
$$747$$ −8.53960 −0.312447
$$748$$ 15.7921 0.577418
$$749$$ −26.4622 −0.966908
$$750$$ −84.3559 −3.08024
$$751$$ −25.1552 −0.917926 −0.458963 0.888455i $$-0.651779\pi$$
−0.458963 + 0.888455i $$0.651779\pi$$
$$752$$ −35.9111 −1.30954
$$753$$ 1.55402 0.0566317
$$754$$ −3.96190 −0.144284
$$755$$ 62.4499 2.27279
$$756$$ 95.4992 3.47327
$$757$$ 15.5116 0.563780 0.281890 0.959447i $$-0.409039\pi$$
0.281890 + 0.959447i $$0.409039\pi$$
$$758$$ −23.1940 −0.842443
$$759$$ −8.91962 −0.323762
$$760$$ 0 0
$$761$$ 35.2085 1.27631 0.638154 0.769908i $$-0.279698\pi$$
0.638154 + 0.769908i $$0.279698\pi$$
$$762$$ −14.6802 −0.531807
$$763$$ 5.22779 0.189259
$$764$$ 85.5129 3.09375
$$765$$ 20.9248 0.756538
$$766$$ 9.25280 0.334317
$$767$$ −19.6630 −0.709988
$$768$$ 19.0750 0.688310
$$769$$ −5.57667 −0.201100 −0.100550 0.994932i $$-0.532060\pi$$
−0.100550 + 0.994932i $$0.532060\pi$$
$$770$$ −38.5490 −1.38921
$$771$$ 4.08095 0.146972
$$772$$ −17.8371 −0.641973
$$773$$ −35.9175 −1.29186 −0.645931 0.763396i $$-0.723531\pi$$
−0.645931 + 0.763396i $$0.723531\pi$$
$$774$$ −22.0726 −0.793383
$$775$$ −19.1643 −0.688403
$$776$$ 101.970 3.66052
$$777$$ −29.0499 −1.04216
$$778$$ −43.6530 −1.56503
$$779$$ 0 0
$$780$$ −34.8160 −1.24661
$$781$$ 6.32968 0.226494
$$782$$ 63.7324 2.27906
$$783$$ −5.60375 −0.200262
$$784$$ 58.8734 2.10262
$$785$$ 100.547 3.58866
$$786$$ −11.7039 −0.417464
$$787$$ −7.53242 −0.268502 −0.134251 0.990947i $$-0.542863\pi$$
−0.134251 + 0.990947i $$0.542863\pi$$
$$788$$ −124.496 −4.43497
$$789$$ −17.0331 −0.606395
$$790$$ −145.666 −5.18257
$$791$$ 43.5531 1.54857
$$792$$ 11.6056 0.412387
$$793$$ −9.60573 −0.341110
$$794$$ 73.7141 2.61602
$$795$$ −0.501731 −0.0177946
$$796$$ 91.2522 3.23435
$$797$$ 49.3837 1.74926 0.874629 0.484792i $$-0.161105\pi$$
0.874629 + 0.484792i $$0.161105\pi$$
$$798$$ 0 0
$$799$$ 12.1513 0.429883
$$800$$ 121.629 4.30025
$$801$$ 19.0482 0.673036
$$802$$ 94.6219 3.34122
$$803$$ 1.37759 0.0486141
$$804$$ −21.4278 −0.755702
$$805$$ −110.012 −3.87743
$$806$$ −6.37247 −0.224461
$$807$$ 6.45210 0.227125
$$808$$ −81.3549 −2.86205
$$809$$ 34.4637 1.21168 0.605840 0.795587i $$-0.292837\pi$$
0.605840 + 0.795587i $$0.292837\pi$$
$$810$$ −19.4704 −0.684120
$$811$$ −20.0278 −0.703272 −0.351636 0.936137i $$-0.614375\pi$$
−0.351636 + 0.936137i $$0.614375\pi$$
$$812$$ −17.9170 −0.628763
$$813$$ 13.2306 0.464017
$$814$$ −17.5429 −0.614879
$$815$$ −15.1718 −0.531444
$$816$$ 37.7952 1.32310
$$817$$ 0 0
$$818$$ −96.4926 −3.37379
$$819$$ −8.39722 −0.293423
$$820$$ 77.2670 2.69828
$$821$$ 6.45245 0.225192 0.112596 0.993641i $$-0.464083\pi$$
0.112596 + 0.993641i $$0.464083\pi$$
$$822$$ 49.2562 1.71801
$$823$$ −43.0190 −1.49955 −0.749774 0.661694i $$-0.769838\pi$$
−0.749774 + 0.661694i $$0.769838\pi$$
$$824$$ 36.9100 1.28582
$$825$$ 13.8978 0.483859
$$826$$ −125.748 −4.37533
$$827$$ 43.1557 1.50067 0.750335 0.661058i $$-0.229892\pi$$
0.750335 + 0.661058i $$0.229892\pi$$
$$828$$ 56.5324 1.96464
$$829$$ −13.4937 −0.468656 −0.234328 0.972158i $$-0.575289\pi$$
−0.234328 + 0.972158i $$0.575289\pi$$
$$830$$ 57.9638 2.01195
$$831$$ −16.8406 −0.584193
$$832$$ 11.8662 0.411385
$$833$$ −19.9212 −0.690228
$$834$$ 7.90191 0.273621
$$835$$ −10.7937 −0.373530
$$836$$ 0 0
$$837$$ −9.01329 −0.311545
$$838$$ 47.2197 1.63118
$$839$$ 14.6851 0.506988 0.253494 0.967337i $$-0.418420\pi$$
0.253494 + 0.967337i $$0.418420\pi$$
$$840$$ −130.446 −4.50080
$$841$$ −27.9487 −0.963747
$$842$$ 22.4124 0.772384
$$843$$ −12.9405 −0.445693
$$844$$ 49.2562 1.69547
$$845$$ −44.0858 −1.51660
$$846$$ 15.2424 0.524043
$$847$$ 3.61829 0.124326
$$848$$ 0.994441 0.0341493
$$849$$ −2.28372 −0.0783769
$$850$$ −99.3023 −3.40604
$$851$$ −50.0645 −1.71619
$$852$$ 36.5594 1.25250
$$853$$ −51.5775 −1.76598 −0.882990 0.469392i $$-0.844473\pi$$
−0.882990 + 0.469392i $$0.844473\pi$$
$$854$$ −61.4303 −2.10210
$$855$$ 0 0
$$856$$ −54.0755 −1.84826
$$857$$ 46.6355 1.59304 0.796520 0.604612i $$-0.206672\pi$$
0.796520 + 0.604612i $$0.206672\pi$$
$$858$$ 4.62125 0.157767
$$859$$ −18.6711 −0.637048 −0.318524 0.947915i $$-0.603187\pi$$
−0.318524 + 0.947915i $$0.603187\pi$$
$$860$$ 105.945 3.61271
$$861$$ −16.9831 −0.578783
$$862$$ −11.1888 −0.381091
$$863$$ −43.0160 −1.46428 −0.732141 0.681153i $$-0.761479\pi$$
−0.732141 + 0.681153i $$0.761479\pi$$
$$864$$ 57.2042 1.94613
$$865$$ −29.9462 −1.01820
$$866$$ −47.5712 −1.61654
$$867$$ 7.54300 0.256174
$$868$$ −28.8184 −0.978160
$$869$$ 13.6725 0.463809
$$870$$ 13.0650 0.442945
$$871$$ 5.48533 0.185863
$$872$$ 10.6830 0.361771
$$873$$ −21.6464 −0.732619
$$874$$ 0 0
$$875$$ 97.6565 3.30139
$$876$$ 7.95678 0.268835
$$877$$ −56.0429 −1.89243 −0.946217 0.323534i $$-0.895129\pi$$
−0.946217 + 0.323534i $$0.895129\pi$$
$$878$$ −76.9466 −2.59682
$$879$$ 4.34803 0.146655
$$880$$ −39.3981 −1.32811
$$881$$ 17.9947 0.606257 0.303128 0.952950i $$-0.401969\pi$$
0.303128 + 0.952950i $$0.401969\pi$$
$$882$$ −24.9887 −0.841412
$$883$$ 1.99550 0.0671540 0.0335770 0.999436i $$-0.489310\pi$$
0.0335770 + 0.999436i $$0.489310\pi$$
$$884$$ −23.3498 −0.785339
$$885$$ 64.8418 2.17963
$$886$$ −63.6687 −2.13899
$$887$$ −7.20927 −0.242063 −0.121032 0.992649i $$-0.538620\pi$$
−0.121032 + 0.992649i $$0.538620\pi$$
$$888$$ −59.3633 −1.99210
$$889$$ 16.9948 0.569988
$$890$$ −129.293 −4.33390
$$891$$ 1.82753 0.0612246
$$892$$ 1.26867 0.0424782
$$893$$ 0 0
$$894$$ −5.77909 −0.193282
$$895$$ −38.9495 −1.30194
$$896$$ 0.140902 0.00470720
$$897$$ 13.1883 0.440344
$$898$$ −101.338 −3.38168
$$899$$ 1.69102 0.0563986
$$900$$ −88.0840 −2.93613
$$901$$ −0.336492 −0.0112102
$$902$$ −10.2559 −0.341485
$$903$$ −23.2866 −0.774928
$$904$$ 89.0006 2.96012
$$905$$ −24.5684 −0.816680
$$906$$ −47.8774 −1.59062
$$907$$ 25.7832 0.856116 0.428058 0.903751i $$-0.359198\pi$$
0.428058 + 0.903751i $$0.359198\pi$$
$$908$$ −131.965 −4.37942
$$909$$ 17.2701 0.572814
$$910$$ 56.9974 1.88945
$$911$$ −31.1334 −1.03149 −0.515747 0.856741i $$-0.672486\pi$$
−0.515747 + 0.856741i $$0.672486\pi$$
$$912$$ 0 0
$$913$$ −5.44061 −0.180058
$$914$$ −19.5389 −0.646291
$$915$$ 31.6764 1.04719
$$916$$ −26.7146 −0.882676
$$917$$ 13.5493 0.447437
$$918$$ −46.7034 −1.54144
$$919$$ 5.42725 0.179029 0.0895143 0.995986i $$-0.471469\pi$$
0.0895143 + 0.995986i $$0.471469\pi$$
$$920$$ −224.810 −7.41176
$$921$$ −28.1452 −0.927416
$$922$$ 41.7560 1.37516
$$923$$ −9.35887 −0.308051
$$924$$ 20.8988 0.687520
$$925$$ 78.0063 2.56483
$$926$$ −15.9469 −0.524049
$$927$$ −7.83531 −0.257345
$$928$$ −10.7323 −0.352305
$$929$$ −21.6025 −0.708756 −0.354378 0.935102i $$-0.615307\pi$$
−0.354378 + 0.935102i $$0.615307\pi$$
$$930$$ 21.0143 0.689085
$$931$$ 0 0
$$932$$ 132.678 4.34603
$$933$$ 19.1390 0.626583
$$934$$ −69.6113 −2.27775
$$935$$ 13.3313 0.435979
$$936$$ −17.1597 −0.560882
$$937$$ 31.6840 1.03507 0.517536 0.855661i $$-0.326849\pi$$
0.517536 + 0.855661i $$0.326849\pi$$
$$938$$ 35.0796 1.14539
$$939$$ −19.1399 −0.624608
$$940$$ −73.1612 −2.38626
$$941$$ 5.63987 0.183854 0.0919272 0.995766i $$-0.470697\pi$$
0.0919272 + 0.995766i $$0.470697\pi$$
$$942$$ −77.0843 −2.51154
$$943$$ −29.2687 −0.953120
$$944$$ −128.518 −4.18290
$$945$$ 80.6176 2.62249
$$946$$ −14.0625 −0.457212
$$947$$ −51.7369 −1.68122 −0.840611 0.541639i $$-0.817804\pi$$
−0.840611 + 0.541639i $$0.817804\pi$$
$$948$$ 78.9709 2.56486
$$949$$ −2.03686 −0.0661194
$$950$$ 0 0
$$951$$ −24.9682 −0.809650
$$952$$ −87.4850 −2.83540
$$953$$ 16.2553 0.526561 0.263281 0.964719i $$-0.415196\pi$$
0.263281 + 0.964719i $$0.415196\pi$$
$$954$$ −0.422088 −0.0136656
$$955$$ 72.1874 2.33593
$$956$$ 6.77750 0.219200
$$957$$ −1.22631 −0.0396410
$$958$$ −42.3601 −1.36859
$$959$$ −57.0225 −1.84135
$$960$$ −39.1306 −1.26293
$$961$$ −28.2801 −0.912261
$$962$$ 25.9384 0.836289
$$963$$ 11.4792 0.369913
$$964$$ 100.304 3.23057
$$965$$ −15.0576 −0.484721
$$966$$ 84.3413 2.71364
$$967$$ −54.5004 −1.75261 −0.876307 0.481754i $$-0.840000\pi$$
−0.876307 + 0.481754i $$0.840000\pi$$
$$968$$ 7.39397 0.237651
$$969$$ 0 0
$$970$$ 146.928 4.71758
$$971$$ 34.2679 1.09971 0.549855 0.835260i $$-0.314683\pi$$
0.549855 + 0.835260i $$0.314683\pi$$
$$972$$ −68.6247 −2.20114
$$973$$ −9.14783 −0.293266
$$974$$ 12.6078 0.403981
$$975$$ −20.5488 −0.658090
$$976$$ −62.7834 −2.00965
$$977$$ −55.5644 −1.77766 −0.888831 0.458234i $$-0.848482\pi$$
−0.888831 + 0.458234i $$0.848482\pi$$
$$978$$ 11.6315 0.371934
$$979$$ 12.1357 0.387858
$$980$$ 119.942 3.83141
$$981$$ −2.26780 −0.0724052
$$982$$ −22.3066 −0.711832
$$983$$ 41.3971 1.32036 0.660181 0.751106i $$-0.270479\pi$$
0.660181 + 0.751106i $$0.270479\pi$$
$$984$$ −34.7049 −1.10635
$$985$$ −105.095 −3.34862
$$986$$ 8.76221 0.279046
$$987$$ 16.0807 0.511853
$$988$$ 0 0
$$989$$ −40.1321 −1.27613
$$990$$ 16.7224 0.531474
$$991$$ −60.8219 −1.93207 −0.966036 0.258406i $$-0.916803\pi$$
−0.966036 + 0.258406i $$0.916803\pi$$
$$992$$ −17.2623 −0.548077
$$993$$ 18.0758 0.573618
$$994$$ −59.8515 −1.89837
$$995$$ 77.0324 2.44209
$$996$$ −31.4242 −0.995715
$$997$$ 26.0627 0.825414 0.412707 0.910864i $$-0.364583\pi$$
0.412707 + 0.910864i $$0.364583\pi$$
$$998$$ −35.1031 −1.11117
$$999$$ 36.6876 1.16074
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 3971.2.a.i.1.6 7
19.18 odd 2 209.2.a.d.1.2 7
57.56 even 2 1881.2.a.p.1.6 7
76.75 even 2 3344.2.a.ba.1.4 7
95.94 odd 2 5225.2.a.n.1.6 7
209.208 even 2 2299.2.a.q.1.6 7

By twisted newform
Twist Min Dim Char Parity Ord Type
209.2.a.d.1.2 7 19.18 odd 2
1881.2.a.p.1.6 7 57.56 even 2
2299.2.a.q.1.6 7 209.208 even 2
3344.2.a.ba.1.4 7 76.75 even 2
3971.2.a.i.1.6 7 1.1 even 1 trivial
5225.2.a.n.1.6 7 95.94 odd 2