# Properties

 Label 7942.2.a.bi.1.1 Level $7942$ Weight $2$ Character 7942.1 Self dual yes Analytic conductor $63.417$ Analytic rank $1$ Dimension $3$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7942,2,Mod(1,7942)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7942, 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("7942.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$7942 = 2 \cdot 11 \cdot 19^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 7942.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: $$63.4171892853$$ Analytic rank: $$1$$ Dimension: $$3$$ Coefficient field: 3.3.621.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - 6x - 3$$ x^3 - 6*x - 3 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 418) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$2.66908$$ of defining polynomial Character $$\chi$$ $$=$$ 7942.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} -2.66908 q^{3} +1.00000 q^{4} -4.12398 q^{5} -2.66908 q^{6} -4.21417 q^{7} +1.00000 q^{8} +4.12398 q^{9} +O(q^{10})$$ $$q+1.00000 q^{2} -2.66908 q^{3} +1.00000 q^{4} -4.12398 q^{5} -2.66908 q^{6} -4.21417 q^{7} +1.00000 q^{8} +4.12398 q^{9} -4.12398 q^{10} -1.00000 q^{11} -2.66908 q^{12} +2.21417 q^{13} -4.21417 q^{14} +11.0072 q^{15} +1.00000 q^{16} -3.45490 q^{17} +4.12398 q^{18} -4.12398 q^{20} +11.2480 q^{21} -1.00000 q^{22} -5.45490 q^{23} -2.66908 q^{24} +12.0072 q^{25} +2.21417 q^{26} -3.00000 q^{27} -4.21417 q^{28} -5.57889 q^{29} +11.0072 q^{30} -7.00724 q^{31} +1.00000 q^{32} +2.66908 q^{33} -3.45490 q^{34} +17.3792 q^{35} +4.12398 q^{36} +2.90981 q^{37} -5.90981 q^{39} -4.12398 q^{40} +11.9170 q^{41} +11.2480 q^{42} +1.46214 q^{43} -1.00000 q^{44} -17.0072 q^{45} -5.45490 q^{46} +7.58612 q^{47} -2.66908 q^{48} +10.7593 q^{49} +12.0072 q^{50} +9.22141 q^{51} +2.21417 q^{52} +13.2214 q^{53} -3.00000 q^{54} +4.12398 q^{55} -4.21417 q^{56} -5.57889 q^{58} -4.79306 q^{59} +11.0072 q^{60} +8.90981 q^{61} -7.00724 q^{62} -17.3792 q^{63} +1.00000 q^{64} -9.13122 q^{65} +2.66908 q^{66} -1.30437 q^{67} -3.45490 q^{68} +14.5596 q^{69} +17.3792 q^{70} +6.80030 q^{71} +4.12398 q^{72} -1.45490 q^{73} +2.90981 q^{74} -32.0483 q^{75} +4.21417 q^{77} -5.90981 q^{78} +9.15777 q^{79} -4.12398 q^{80} -4.36471 q^{81} +11.9170 q^{82} -13.2142 q^{83} +11.2480 q^{84} +14.2480 q^{85} +1.46214 q^{86} +14.8905 q^{87} -1.00000 q^{88} -8.24797 q^{89} -17.0072 q^{90} -9.33092 q^{91} -5.45490 q^{92} +18.7029 q^{93} +7.58612 q^{94} -2.66908 q^{96} -2.18038 q^{97} +10.7593 q^{98} -4.12398 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 3 q^{2} + 3 q^{4} - 3 q^{5} - 6 q^{7} + 3 q^{8} + 3 q^{9}+O(q^{10})$$ 3 * q + 3 * q^2 + 3 * q^4 - 3 * q^5 - 6 * q^7 + 3 * q^8 + 3 * q^9 $$3 q + 3 q^{2} + 3 q^{4} - 3 q^{5} - 6 q^{7} + 3 q^{8} + 3 q^{9} - 3 q^{10} - 3 q^{11} - 6 q^{14} + 9 q^{15} + 3 q^{16} - 9 q^{17} + 3 q^{18} - 3 q^{20} + 15 q^{21} - 3 q^{22} - 15 q^{23} + 12 q^{25} - 9 q^{27} - 6 q^{28} - 6 q^{29} + 9 q^{30} + 3 q^{31} + 3 q^{32} - 9 q^{34} + 3 q^{36} + 6 q^{37} - 15 q^{39} - 3 q^{40} + 9 q^{41} + 15 q^{42} - 21 q^{43} - 3 q^{44} - 27 q^{45} - 15 q^{46} - 12 q^{47} + 27 q^{49} + 12 q^{50} - 3 q^{51} + 9 q^{53} - 9 q^{54} + 3 q^{55} - 6 q^{56} - 6 q^{58} + 3 q^{59} + 9 q^{60} + 24 q^{61} + 3 q^{62} + 3 q^{64} + 6 q^{65} - 9 q^{68} - 3 q^{69} - 21 q^{71} + 3 q^{72} - 3 q^{73} + 6 q^{74} - 36 q^{75} + 6 q^{77} - 15 q^{78} + 6 q^{79} - 3 q^{80} - 9 q^{81} + 9 q^{82} - 33 q^{83} + 15 q^{84} + 24 q^{85} - 21 q^{86} + 6 q^{87} - 3 q^{88} - 6 q^{89} - 27 q^{90} - 36 q^{91} - 15 q^{92} + 36 q^{93} - 12 q^{94} - 12 q^{97} + 27 q^{98} - 3 q^{99}+O(q^{100})$$ 3 * q + 3 * q^2 + 3 * q^4 - 3 * q^5 - 6 * q^7 + 3 * q^8 + 3 * q^9 - 3 * q^10 - 3 * q^11 - 6 * q^14 + 9 * q^15 + 3 * q^16 - 9 * q^17 + 3 * q^18 - 3 * q^20 + 15 * q^21 - 3 * q^22 - 15 * q^23 + 12 * q^25 - 9 * q^27 - 6 * q^28 - 6 * q^29 + 9 * q^30 + 3 * q^31 + 3 * q^32 - 9 * q^34 + 3 * q^36 + 6 * q^37 - 15 * q^39 - 3 * q^40 + 9 * q^41 + 15 * q^42 - 21 * q^43 - 3 * q^44 - 27 * q^45 - 15 * q^46 - 12 * q^47 + 27 * q^49 + 12 * q^50 - 3 * q^51 + 9 * q^53 - 9 * q^54 + 3 * q^55 - 6 * q^56 - 6 * q^58 + 3 * q^59 + 9 * q^60 + 24 * q^61 + 3 * q^62 + 3 * q^64 + 6 * q^65 - 9 * q^68 - 3 * q^69 - 21 * q^71 + 3 * q^72 - 3 * q^73 + 6 * q^74 - 36 * q^75 + 6 * q^77 - 15 * q^78 + 6 * q^79 - 3 * q^80 - 9 * q^81 + 9 * q^82 - 33 * q^83 + 15 * q^84 + 24 * q^85 - 21 * q^86 + 6 * q^87 - 3 * q^88 - 6 * q^89 - 27 * q^90 - 36 * q^91 - 15 * q^92 + 36 * q^93 - 12 * q^94 - 12 * q^97 + 27 * q^98 - 3 * 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
$$3$$ −2.66908 −1.54099 −0.770497 0.637444i $$-0.779992\pi$$
−0.770497 + 0.637444i $$0.779992\pi$$
$$4$$ 1.00000 0.500000
$$5$$ −4.12398 −1.84430 −0.922151 0.386831i $$-0.873570\pi$$
−0.922151 + 0.386831i $$0.873570\pi$$
$$6$$ −2.66908 −1.08965
$$7$$ −4.21417 −1.59281 −0.796404 0.604765i $$-0.793267\pi$$
−0.796404 + 0.604765i $$0.793267\pi$$
$$8$$ 1.00000 0.353553
$$9$$ 4.12398 1.37466
$$10$$ −4.12398 −1.30412
$$11$$ −1.00000 −0.301511
$$12$$ −2.66908 −0.770497
$$13$$ 2.21417 0.614102 0.307051 0.951693i $$-0.400658\pi$$
0.307051 + 0.951693i $$0.400658\pi$$
$$14$$ −4.21417 −1.12629
$$15$$ 11.0072 2.84206
$$16$$ 1.00000 0.250000
$$17$$ −3.45490 −0.837937 −0.418969 0.908001i $$-0.637608\pi$$
−0.418969 + 0.908001i $$0.637608\pi$$
$$18$$ 4.12398 0.972032
$$19$$ 0 0
$$20$$ −4.12398 −0.922151
$$21$$ 11.2480 2.45451
$$22$$ −1.00000 −0.213201
$$23$$ −5.45490 −1.13743 −0.568713 0.822536i $$-0.692559\pi$$
−0.568713 + 0.822536i $$0.692559\pi$$
$$24$$ −2.66908 −0.544823
$$25$$ 12.0072 2.40145
$$26$$ 2.21417 0.434235
$$27$$ −3.00000 −0.577350
$$28$$ −4.21417 −0.796404
$$29$$ −5.57889 −1.03597 −0.517987 0.855389i $$-0.673318\pi$$
−0.517987 + 0.855389i $$0.673318\pi$$
$$30$$ 11.0072 2.00964
$$31$$ −7.00724 −1.25854 −0.629268 0.777188i $$-0.716645\pi$$
−0.629268 + 0.777188i $$0.716645\pi$$
$$32$$ 1.00000 0.176777
$$33$$ 2.66908 0.464627
$$34$$ −3.45490 −0.592511
$$35$$ 17.3792 2.93762
$$36$$ 4.12398 0.687331
$$37$$ 2.90981 0.478370 0.239185 0.970974i $$-0.423120\pi$$
0.239185 + 0.970974i $$0.423120\pi$$
$$38$$ 0 0
$$39$$ −5.90981 −0.946327
$$40$$ −4.12398 −0.652059
$$41$$ 11.9170 1.86113 0.930565 0.366127i $$-0.119316\pi$$
0.930565 + 0.366127i $$0.119316\pi$$
$$42$$ 11.2480 1.73560
$$43$$ 1.46214 0.222974 0.111487 0.993766i $$-0.464439\pi$$
0.111487 + 0.993766i $$0.464439\pi$$
$$44$$ −1.00000 −0.150756
$$45$$ −17.0072 −2.53529
$$46$$ −5.45490 −0.804282
$$47$$ 7.58612 1.10655 0.553275 0.832999i $$-0.313378\pi$$
0.553275 + 0.832999i $$0.313378\pi$$
$$48$$ −2.66908 −0.385248
$$49$$ 10.7593 1.53704
$$50$$ 12.0072 1.69808
$$51$$ 9.22141 1.29126
$$52$$ 2.21417 0.307051
$$53$$ 13.2214 1.81610 0.908050 0.418861i $$-0.137571\pi$$
0.908050 + 0.418861i $$0.137571\pi$$
$$54$$ −3.00000 −0.408248
$$55$$ 4.12398 0.556078
$$56$$ −4.21417 −0.563143
$$57$$ 0 0
$$58$$ −5.57889 −0.732544
$$59$$ −4.79306 −0.624004 −0.312002 0.950082i $$-0.600999\pi$$
−0.312002 + 0.950082i $$0.600999\pi$$
$$60$$ 11.0072 1.42103
$$61$$ 8.90981 1.14078 0.570392 0.821373i $$-0.306791\pi$$
0.570392 + 0.821373i $$0.306791\pi$$
$$62$$ −7.00724 −0.889920
$$63$$ −17.3792 −2.18957
$$64$$ 1.00000 0.125000
$$65$$ −9.13122 −1.13259
$$66$$ 2.66908 0.328541
$$67$$ −1.30437 −0.159354 −0.0796769 0.996821i $$-0.525389\pi$$
−0.0796769 + 0.996821i $$0.525389\pi$$
$$68$$ −3.45490 −0.418969
$$69$$ 14.5596 1.75277
$$70$$ 17.3792 2.07721
$$71$$ 6.80030 0.807047 0.403524 0.914969i $$-0.367785\pi$$
0.403524 + 0.914969i $$0.367785\pi$$
$$72$$ 4.12398 0.486016
$$73$$ −1.45490 −0.170284 −0.0851418 0.996369i $$-0.527134\pi$$
−0.0851418 + 0.996369i $$0.527134\pi$$
$$74$$ 2.90981 0.338258
$$75$$ −32.0483 −3.70061
$$76$$ 0 0
$$77$$ 4.21417 0.480250
$$78$$ −5.90981 −0.669154
$$79$$ 9.15777 1.03033 0.515165 0.857091i $$-0.327731\pi$$
0.515165 + 0.857091i $$0.327731\pi$$
$$80$$ −4.12398 −0.461075
$$81$$ −4.36471 −0.484968
$$82$$ 11.9170 1.31602
$$83$$ −13.2142 −1.45044 −0.725222 0.688515i $$-0.758263\pi$$
−0.725222 + 0.688515i $$0.758263\pi$$
$$84$$ 11.2480 1.22725
$$85$$ 14.2480 1.54541
$$86$$ 1.46214 0.157667
$$87$$ 14.8905 1.59643
$$88$$ −1.00000 −0.106600
$$89$$ −8.24797 −0.874283 −0.437141 0.899393i $$-0.644009\pi$$
−0.437141 + 0.899393i $$0.644009\pi$$
$$90$$ −17.0072 −1.79272
$$91$$ −9.33092 −0.978146
$$92$$ −5.45490 −0.568713
$$93$$ 18.7029 1.93940
$$94$$ 7.58612 0.782449
$$95$$ 0 0
$$96$$ −2.66908 −0.272412
$$97$$ −2.18038 −0.221384 −0.110692 0.993855i $$-0.535307\pi$$
−0.110692 + 0.993855i $$0.535307\pi$$
$$98$$ 10.7593 1.08685
$$99$$ −4.12398 −0.414476
$$100$$ 12.0072 1.20072
$$101$$ −6.24797 −0.621696 −0.310848 0.950460i $$-0.600613\pi$$
−0.310848 + 0.950460i $$0.600613\pi$$
$$102$$ 9.22141 0.913056
$$103$$ −0.605441 −0.0596559 −0.0298280 0.999555i $$-0.509496\pi$$
−0.0298280 + 0.999555i $$0.509496\pi$$
$$104$$ 2.21417 0.217118
$$105$$ −46.3864 −4.52685
$$106$$ 13.2214 1.28418
$$107$$ −3.88325 −0.375408 −0.187704 0.982226i $$-0.560105\pi$$
−0.187704 + 0.982226i $$0.560105\pi$$
$$108$$ −3.00000 −0.288675
$$109$$ 4.97345 0.476370 0.238185 0.971220i $$-0.423448\pi$$
0.238185 + 0.971220i $$0.423448\pi$$
$$110$$ 4.12398 0.393206
$$111$$ −7.76651 −0.737164
$$112$$ −4.21417 −0.398202
$$113$$ 8.00000 0.752577 0.376288 0.926503i $$-0.377200\pi$$
0.376288 + 0.926503i $$0.377200\pi$$
$$114$$ 0 0
$$115$$ 22.4959 2.09776
$$116$$ −5.57889 −0.517987
$$117$$ 9.13122 0.844182
$$118$$ −4.79306 −0.441237
$$119$$ 14.5596 1.33467
$$120$$ 11.0072 1.00482
$$121$$ 1.00000 0.0909091
$$122$$ 8.90981 0.806656
$$123$$ −31.8075 −2.86799
$$124$$ −7.00724 −0.629268
$$125$$ −28.8977 −2.58469
$$126$$ −17.3792 −1.54826
$$127$$ 2.48146 0.220194 0.110097 0.993921i $$-0.464884\pi$$
0.110097 + 0.993921i $$0.464884\pi$$
$$128$$ 1.00000 0.0883883
$$129$$ −3.90257 −0.343602
$$130$$ −9.13122 −0.800861
$$131$$ 11.0072 0.961707 0.480853 0.876801i $$-0.340327\pi$$
0.480853 + 0.876801i $$0.340327\pi$$
$$132$$ 2.66908 0.232314
$$133$$ 0 0
$$134$$ −1.30437 −0.112680
$$135$$ 12.3719 1.06481
$$136$$ −3.45490 −0.296256
$$137$$ 6.94360 0.593232 0.296616 0.954997i $$-0.404142\pi$$
0.296616 + 0.954997i $$0.404142\pi$$
$$138$$ 14.5596 1.23939
$$139$$ −4.37195 −0.370824 −0.185412 0.982661i $$-0.559362\pi$$
−0.185412 + 0.982661i $$0.559362\pi$$
$$140$$ 17.3792 1.46881
$$141$$ −20.2480 −1.70519
$$142$$ 6.80030 0.570668
$$143$$ −2.21417 −0.185159
$$144$$ 4.12398 0.343665
$$145$$ 23.0072 1.91065
$$146$$ −1.45490 −0.120409
$$147$$ −28.7173 −2.36857
$$148$$ 2.90981 0.239185
$$149$$ 11.7665 0.963950 0.481975 0.876185i $$-0.339920\pi$$
0.481975 + 0.876185i $$0.339920\pi$$
$$150$$ −32.0483 −2.61673
$$151$$ 19.5861 1.59390 0.796948 0.604048i $$-0.206446\pi$$
0.796948 + 0.604048i $$0.206446\pi$$
$$152$$ 0 0
$$153$$ −14.2480 −1.15188
$$154$$ 4.21417 0.339588
$$155$$ 28.8977 2.32112
$$156$$ −5.90981 −0.473163
$$157$$ 20.6199 1.64565 0.822824 0.568296i $$-0.192397\pi$$
0.822824 + 0.568296i $$0.192397\pi$$
$$158$$ 9.15777 0.728553
$$159$$ −35.2890 −2.79860
$$160$$ −4.12398 −0.326029
$$161$$ 22.9879 1.81170
$$162$$ −4.36471 −0.342924
$$163$$ −17.8341 −1.39687 −0.698437 0.715672i $$-0.746121\pi$$
−0.698437 + 0.715672i $$0.746121\pi$$
$$164$$ 11.9170 0.930565
$$165$$ −11.0072 −0.856912
$$166$$ −13.2142 −1.02562
$$167$$ −4.18038 −0.323488 −0.161744 0.986833i $$-0.551712\pi$$
−0.161744 + 0.986833i $$0.551712\pi$$
$$168$$ 11.2480 0.867799
$$169$$ −8.09743 −0.622879
$$170$$ 14.2480 1.09277
$$171$$ 0 0
$$172$$ 1.46214 0.111487
$$173$$ −23.7101 −1.80265 −0.901323 0.433147i $$-0.857403\pi$$
−0.901323 + 0.433147i $$0.857403\pi$$
$$174$$ 14.8905 1.12885
$$175$$ −50.6006 −3.82505
$$176$$ −1.00000 −0.0753778
$$177$$ 12.7931 0.961585
$$178$$ −8.24797 −0.618211
$$179$$ 10.8269 0.809237 0.404619 0.914486i $$-0.367404\pi$$
0.404619 + 0.914486i $$0.367404\pi$$
$$180$$ −17.0072 −1.26764
$$181$$ 17.8341 1.32560 0.662799 0.748798i $$-0.269368\pi$$
0.662799 + 0.748798i $$0.269368\pi$$
$$182$$ −9.33092 −0.691654
$$183$$ −23.7810 −1.75794
$$184$$ −5.45490 −0.402141
$$185$$ −12.0000 −0.882258
$$186$$ 18.7029 1.37136
$$187$$ 3.45490 0.252648
$$188$$ 7.58612 0.553275
$$189$$ 12.6425 0.919608
$$190$$ 0 0
$$191$$ 0.612679 0.0443319 0.0221659 0.999754i $$-0.492944\pi$$
0.0221659 + 0.999754i $$0.492944\pi$$
$$192$$ −2.66908 −0.192624
$$193$$ −11.6425 −0.838047 −0.419024 0.907975i $$-0.637628\pi$$
−0.419024 + 0.907975i $$0.637628\pi$$
$$194$$ −2.18038 −0.156542
$$195$$ 24.3719 1.74531
$$196$$ 10.7593 0.768519
$$197$$ −12.9098 −0.919786 −0.459893 0.887974i $$-0.652112\pi$$
−0.459893 + 0.887974i $$0.652112\pi$$
$$198$$ −4.12398 −0.293079
$$199$$ −18.3647 −1.30184 −0.650920 0.759146i $$-0.725617\pi$$
−0.650920 + 0.759146i $$0.725617\pi$$
$$200$$ 12.0072 0.849040
$$201$$ 3.48146 0.245563
$$202$$ −6.24797 −0.439605
$$203$$ 23.5104 1.65011
$$204$$ 9.22141 0.645628
$$205$$ −49.1457 −3.43248
$$206$$ −0.605441 −0.0421831
$$207$$ −22.4959 −1.56358
$$208$$ 2.21417 0.153525
$$209$$ 0 0
$$210$$ −46.3864 −3.20097
$$211$$ −2.13122 −0.146719 −0.0733596 0.997306i $$-0.523372\pi$$
−0.0733596 + 0.997306i $$0.523372\pi$$
$$212$$ 13.2214 0.908050
$$213$$ −18.1505 −1.24365
$$214$$ −3.88325 −0.265454
$$215$$ −6.02985 −0.411232
$$216$$ −3.00000 −0.204124
$$217$$ 29.5297 2.00461
$$218$$ 4.97345 0.336844
$$219$$ 3.88325 0.262406
$$220$$ 4.12398 0.278039
$$221$$ −7.64976 −0.514579
$$222$$ −7.76651 −0.521254
$$223$$ −6.90981 −0.462715 −0.231357 0.972869i $$-0.574317\pi$$
−0.231357 + 0.972869i $$0.574317\pi$$
$$224$$ −4.21417 −0.281571
$$225$$ 49.5176 3.30118
$$226$$ 8.00000 0.532152
$$227$$ 2.72548 0.180896 0.0904482 0.995901i $$-0.471170\pi$$
0.0904482 + 0.995901i $$0.471170\pi$$
$$228$$ 0 0
$$229$$ 3.48870 0.230539 0.115270 0.993334i $$-0.463227\pi$$
0.115270 + 0.993334i $$0.463227\pi$$
$$230$$ 22.4959 1.48334
$$231$$ −11.2480 −0.740062
$$232$$ −5.57889 −0.366272
$$233$$ −17.0902 −1.11962 −0.559808 0.828622i $$-0.689125\pi$$
−0.559808 + 0.828622i $$0.689125\pi$$
$$234$$ 9.13122 0.596927
$$235$$ −31.2850 −2.04081
$$236$$ −4.79306 −0.312002
$$237$$ −24.4428 −1.58773
$$238$$ 14.5596 0.943757
$$239$$ 6.06035 0.392011 0.196006 0.980603i $$-0.437203\pi$$
0.196006 + 0.980603i $$0.437203\pi$$
$$240$$ 11.0072 0.710514
$$241$$ −4.85341 −0.312635 −0.156318 0.987707i $$-0.549962\pi$$
−0.156318 + 0.987707i $$0.549962\pi$$
$$242$$ 1.00000 0.0642824
$$243$$ 20.6498 1.32468
$$244$$ 8.90981 0.570392
$$245$$ −44.3711 −2.83476
$$246$$ −31.8075 −2.02797
$$247$$ 0 0
$$248$$ −7.00724 −0.444960
$$249$$ 35.2697 2.23513
$$250$$ −28.8977 −1.82765
$$251$$ −1.58612 −0.100115 −0.0500576 0.998746i $$-0.515941\pi$$
−0.0500576 + 0.998746i $$0.515941\pi$$
$$252$$ −17.3792 −1.09479
$$253$$ 5.45490 0.342947
$$254$$ 2.48146 0.155701
$$255$$ −38.0289 −2.38147
$$256$$ 1.00000 0.0625000
$$257$$ −19.5185 −1.21753 −0.608767 0.793349i $$-0.708335\pi$$
−0.608767 + 0.793349i $$0.708335\pi$$
$$258$$ −3.90257 −0.242963
$$259$$ −12.2624 −0.761951
$$260$$ −9.13122 −0.566294
$$261$$ −23.0072 −1.42411
$$262$$ 11.0072 0.680029
$$263$$ −13.0483 −0.804591 −0.402295 0.915510i $$-0.631787\pi$$
−0.402295 + 0.915510i $$0.631787\pi$$
$$264$$ 2.66908 0.164270
$$265$$ −54.5249 −3.34944
$$266$$ 0 0
$$267$$ 22.0145 1.34726
$$268$$ −1.30437 −0.0796769
$$269$$ −23.6006 −1.43895 −0.719477 0.694516i $$-0.755618\pi$$
−0.719477 + 0.694516i $$0.755618\pi$$
$$270$$ 12.3719 0.752933
$$271$$ −16.4090 −0.996778 −0.498389 0.866954i $$-0.666075\pi$$
−0.498389 + 0.866954i $$0.666075\pi$$
$$272$$ −3.45490 −0.209484
$$273$$ 24.9050 1.50732
$$274$$ 6.94360 0.419478
$$275$$ −12.0072 −0.724064
$$276$$ 14.5596 0.876383
$$277$$ −11.5861 −0.696143 −0.348071 0.937468i $$-0.613163\pi$$
−0.348071 + 0.937468i $$0.613163\pi$$
$$278$$ −4.37195 −0.262212
$$279$$ −28.8977 −1.73006
$$280$$ 17.3792 1.03861
$$281$$ −7.03379 −0.419601 −0.209800 0.977744i $$-0.567281\pi$$
−0.209800 + 0.977744i $$0.567281\pi$$
$$282$$ −20.2480 −1.20575
$$283$$ 8.33092 0.495222 0.247611 0.968860i $$-0.420355\pi$$
0.247611 + 0.968860i $$0.420355\pi$$
$$284$$ 6.80030 0.403524
$$285$$ 0 0
$$286$$ −2.21417 −0.130927
$$287$$ −50.2205 −2.96442
$$288$$ 4.12398 0.243008
$$289$$ −5.06364 −0.297861
$$290$$ 23.0072 1.35103
$$291$$ 5.81962 0.341152
$$292$$ −1.45490 −0.0851418
$$293$$ −10.2142 −0.596718 −0.298359 0.954454i $$-0.596439\pi$$
−0.298359 + 0.954454i $$0.596439\pi$$
$$294$$ −28.7173 −1.67483
$$295$$ 19.7665 1.15085
$$296$$ 2.90981 0.169129
$$297$$ 3.00000 0.174078
$$298$$ 11.7665 0.681616
$$299$$ −12.0781 −0.698495
$$300$$ −32.0483 −1.85031
$$301$$ −6.16172 −0.355156
$$302$$ 19.5861 1.12705
$$303$$ 16.6763 0.958029
$$304$$ 0 0
$$305$$ −36.7439 −2.10395
$$306$$ −14.2480 −0.814502
$$307$$ 1.93242 0.110289 0.0551444 0.998478i $$-0.482438\pi$$
0.0551444 + 0.998478i $$0.482438\pi$$
$$308$$ 4.21417 0.240125
$$309$$ 1.61597 0.0919294
$$310$$ 28.8977 1.64128
$$311$$ −27.7173 −1.57171 −0.785853 0.618413i $$-0.787776\pi$$
−0.785853 + 0.618413i $$0.787776\pi$$
$$312$$ −5.90981 −0.334577
$$313$$ 11.5258 0.651476 0.325738 0.945460i $$-0.394387\pi$$
0.325738 + 0.945460i $$0.394387\pi$$
$$314$$ 20.6199 1.16365
$$315$$ 71.6715 4.03823
$$316$$ 9.15777 0.515165
$$317$$ 33.8977 1.90389 0.951943 0.306275i $$-0.0990827\pi$$
0.951943 + 0.306275i $$0.0990827\pi$$
$$318$$ −35.2890 −1.97891
$$319$$ 5.57889 0.312358
$$320$$ −4.12398 −0.230538
$$321$$ 10.3647 0.578502
$$322$$ 22.9879 1.28107
$$323$$ 0 0
$$324$$ −4.36471 −0.242484
$$325$$ 26.5861 1.47473
$$326$$ −17.8341 −0.987739
$$327$$ −13.2745 −0.734083
$$328$$ 11.9170 0.658009
$$329$$ −31.9693 −1.76252
$$330$$ −11.0072 −0.605928
$$331$$ 16.2697 0.894262 0.447131 0.894468i $$-0.352446\pi$$
0.447131 + 0.894468i $$0.352446\pi$$
$$332$$ −13.2142 −0.725222
$$333$$ 12.0000 0.657596
$$334$$ −4.18038 −0.228740
$$335$$ 5.37919 0.293896
$$336$$ 11.2480 0.613627
$$337$$ 6.82685 0.371882 0.185941 0.982561i $$-0.440467\pi$$
0.185941 + 0.982561i $$0.440467\pi$$
$$338$$ −8.09743 −0.440442
$$339$$ −21.3526 −1.15972
$$340$$ 14.2480 0.772704
$$341$$ 7.00724 0.379463
$$342$$ 0 0
$$343$$ −15.8422 −0.855400
$$344$$ 1.46214 0.0788334
$$345$$ −60.0434 −3.23263
$$346$$ −23.7101 −1.27466
$$347$$ −5.09019 −0.273256 −0.136628 0.990622i $$-0.543626\pi$$
−0.136628 + 0.990622i $$0.543626\pi$$
$$348$$ 14.8905 0.798214
$$349$$ 30.9919 1.65896 0.829478 0.558539i $$-0.188638\pi$$
0.829478 + 0.558539i $$0.188638\pi$$
$$350$$ −50.6006 −2.70472
$$351$$ −6.64252 −0.354552
$$352$$ −1.00000 −0.0533002
$$353$$ −17.8301 −0.949003 −0.474501 0.880255i $$-0.657372\pi$$
−0.474501 + 0.880255i $$0.657372\pi$$
$$354$$ 12.7931 0.679944
$$355$$ −28.0443 −1.48844
$$356$$ −8.24797 −0.437141
$$357$$ −38.8606 −2.05672
$$358$$ 10.8269 0.572217
$$359$$ 24.4090 1.28826 0.644130 0.764916i $$-0.277220\pi$$
0.644130 + 0.764916i $$0.277220\pi$$
$$360$$ −17.0072 −0.896360
$$361$$ 0 0
$$362$$ 17.8341 0.937339
$$363$$ −2.66908 −0.140090
$$364$$ −9.33092 −0.489073
$$365$$ 6.00000 0.314054
$$366$$ −23.7810 −1.24305
$$367$$ −26.6232 −1.38972 −0.694860 0.719145i $$-0.744534\pi$$
−0.694860 + 0.719145i $$0.744534\pi$$
$$368$$ −5.45490 −0.284357
$$369$$ 49.1457 2.55842
$$370$$ −12.0000 −0.623850
$$371$$ −55.7173 −2.89270
$$372$$ 18.7029 0.969699
$$373$$ 12.6160 0.653230 0.326615 0.945157i $$-0.394092\pi$$
0.326615 + 0.945157i $$0.394092\pi$$
$$374$$ 3.45490 0.178649
$$375$$ 77.1303 3.98299
$$376$$ 7.58612 0.391225
$$377$$ −12.3526 −0.636193
$$378$$ 12.6425 0.650261
$$379$$ 22.8905 1.17581 0.587903 0.808932i $$-0.299954\pi$$
0.587903 + 0.808932i $$0.299954\pi$$
$$380$$ 0 0
$$381$$ −6.62321 −0.339317
$$382$$ 0.612679 0.0313474
$$383$$ 0.0298464 0.00152508 0.000762539 1.00000i $$-0.499757\pi$$
0.000762539 1.00000i $$0.499757\pi$$
$$384$$ −2.66908 −0.136206
$$385$$ −17.3792 −0.885725
$$386$$ −11.6425 −0.592589
$$387$$ 6.02985 0.306514
$$388$$ −2.18038 −0.110692
$$389$$ 9.22865 0.467911 0.233956 0.972247i $$-0.424833\pi$$
0.233956 + 0.972247i $$0.424833\pi$$
$$390$$ 24.3719 1.23412
$$391$$ 18.8462 0.953092
$$392$$ 10.7593 0.543425
$$393$$ −29.3792 −1.48198
$$394$$ −12.9098 −0.650387
$$395$$ −37.7665 −1.90024
$$396$$ −4.12398 −0.207238
$$397$$ 17.2697 0.866740 0.433370 0.901216i $$-0.357324\pi$$
0.433370 + 0.901216i $$0.357324\pi$$
$$398$$ −18.3647 −0.920540
$$399$$ 0 0
$$400$$ 12.0072 0.600362
$$401$$ 13.2850 0.663424 0.331712 0.943381i $$-0.392374\pi$$
0.331712 + 0.943381i $$0.392374\pi$$
$$402$$ 3.48146 0.173639
$$403$$ −15.5152 −0.772870
$$404$$ −6.24797 −0.310848
$$405$$ 18.0000 0.894427
$$406$$ 23.5104 1.16680
$$407$$ −2.90981 −0.144234
$$408$$ 9.22141 0.456528
$$409$$ −1.94360 −0.0961048 −0.0480524 0.998845i $$-0.515301\pi$$
−0.0480524 + 0.998845i $$0.515301\pi$$
$$410$$ −49.1457 −2.42713
$$411$$ −18.5330 −0.914166
$$412$$ −0.605441 −0.0298280
$$413$$ 20.1988 0.993918
$$414$$ −22.4959 −1.10561
$$415$$ 54.4950 2.67506
$$416$$ 2.21417 0.108559
$$417$$ 11.6691 0.571437
$$418$$ 0 0
$$419$$ −2.49593 −0.121934 −0.0609671 0.998140i $$-0.519418\pi$$
−0.0609671 + 0.998140i $$0.519418\pi$$
$$420$$ −46.3864 −2.26343
$$421$$ −15.6353 −0.762017 −0.381009 0.924571i $$-0.624423\pi$$
−0.381009 + 0.924571i $$0.624423\pi$$
$$422$$ −2.13122 −0.103746
$$423$$ 31.2850 1.52113
$$424$$ 13.2214 0.642089
$$425$$ −41.4839 −2.01226
$$426$$ −18.1505 −0.879396
$$427$$ −37.5475 −1.81705
$$428$$ −3.88325 −0.187704
$$429$$ 5.90981 0.285328
$$430$$ −6.02985 −0.290785
$$431$$ −0.262441 −0.0126413 −0.00632067 0.999980i $$-0.502012\pi$$
−0.00632067 + 0.999980i $$0.502012\pi$$
$$432$$ −3.00000 −0.144338
$$433$$ 32.2769 1.55113 0.775565 0.631268i $$-0.217465\pi$$
0.775565 + 0.631268i $$0.217465\pi$$
$$434$$ 29.5297 1.41747
$$435$$ −61.4081 −2.94429
$$436$$ 4.97345 0.238185
$$437$$ 0 0
$$438$$ 3.88325 0.185549
$$439$$ 17.0902 0.815670 0.407835 0.913056i $$-0.366284\pi$$
0.407835 + 0.913056i $$0.366284\pi$$
$$440$$ 4.12398 0.196603
$$441$$ 44.3711 2.11291
$$442$$ −7.64976 −0.363862
$$443$$ 16.2624 0.772652 0.386326 0.922362i $$-0.373744\pi$$
0.386326 + 0.922362i $$0.373744\pi$$
$$444$$ −7.76651 −0.368582
$$445$$ 34.0145 1.61244
$$446$$ −6.90981 −0.327189
$$447$$ −31.4057 −1.48544
$$448$$ −4.21417 −0.199101
$$449$$ 27.3382 1.29017 0.645084 0.764112i $$-0.276822\pi$$
0.645084 + 0.764112i $$0.276822\pi$$
$$450$$ 49.5176 2.33428
$$451$$ −11.9170 −0.561152
$$452$$ 8.00000 0.376288
$$453$$ −52.2769 −2.45618
$$454$$ 2.72548 0.127913
$$455$$ 38.4806 1.80400
$$456$$ 0 0
$$457$$ 10.4920 0.490794 0.245397 0.969423i $$-0.421082\pi$$
0.245397 + 0.969423i $$0.421082\pi$$
$$458$$ 3.48870 0.163016
$$459$$ 10.3647 0.483783
$$460$$ 22.4959 1.04888
$$461$$ 6.18038 0.287849 0.143925 0.989589i $$-0.454028\pi$$
0.143925 + 0.989589i $$0.454028\pi$$
$$462$$ −11.2480 −0.523303
$$463$$ −11.1433 −0.517873 −0.258937 0.965894i $$-0.583372\pi$$
−0.258937 + 0.965894i $$0.583372\pi$$
$$464$$ −5.57889 −0.258993
$$465$$ −77.1303 −3.57683
$$466$$ −17.0902 −0.791688
$$467$$ −34.5104 −1.59695 −0.798476 0.602027i $$-0.794360\pi$$
−0.798476 + 0.602027i $$0.794360\pi$$
$$468$$ 9.13122 0.422091
$$469$$ 5.49683 0.253820
$$470$$ −31.2850 −1.44307
$$471$$ −55.0362 −2.53593
$$472$$ −4.79306 −0.220619
$$473$$ −1.46214 −0.0672293
$$474$$ −24.4428 −1.12270
$$475$$ 0 0
$$476$$ 14.5596 0.667337
$$477$$ 54.5249 2.49652
$$478$$ 6.06035 0.277194
$$479$$ 41.6980 1.90523 0.952616 0.304176i $$-0.0983812\pi$$
0.952616 + 0.304176i $$0.0983812\pi$$
$$480$$ 11.0072 0.502409
$$481$$ 6.44282 0.293768
$$482$$ −4.85341 −0.221067
$$483$$ −61.3566 −2.79182
$$484$$ 1.00000 0.0454545
$$485$$ 8.99187 0.408300
$$486$$ 20.6498 0.936692
$$487$$ −9.10137 −0.412423 −0.206211 0.978507i $$-0.566113\pi$$
−0.206211 + 0.978507i $$0.566113\pi$$
$$488$$ 8.90981 0.403328
$$489$$ 47.6006 2.15257
$$490$$ −44.3711 −2.00448
$$491$$ −30.8413 −1.39185 −0.695925 0.718115i $$-0.745005\pi$$
−0.695925 + 0.718115i $$0.745005\pi$$
$$492$$ −31.8075 −1.43399
$$493$$ 19.2745 0.868081
$$494$$ 0 0
$$495$$ 17.0072 0.764418
$$496$$ −7.00724 −0.314634
$$497$$ −28.6577 −1.28547
$$498$$ 35.2697 1.58047
$$499$$ 21.7665 0.974403 0.487201 0.873290i $$-0.338018\pi$$
0.487201 + 0.873290i $$0.338018\pi$$
$$500$$ −28.8977 −1.29235
$$501$$ 11.1578 0.498493
$$502$$ −1.58612 −0.0707922
$$503$$ 38.0893 1.69832 0.849159 0.528138i $$-0.177109\pi$$
0.849159 + 0.528138i $$0.177109\pi$$
$$504$$ −17.3792 −0.774131
$$505$$ 25.7665 1.14659
$$506$$ 5.45490 0.242500
$$507$$ 21.6127 0.959853
$$508$$ 2.48146 0.110097
$$509$$ −14.9388 −0.662149 −0.331074 0.943605i $$-0.607411\pi$$
−0.331074 + 0.943605i $$0.607411\pi$$
$$510$$ −38.0289 −1.68395
$$511$$ 6.13122 0.271229
$$512$$ 1.00000 0.0441942
$$513$$ 0 0
$$514$$ −19.5185 −0.860926
$$515$$ 2.49683 0.110023
$$516$$ −3.90257 −0.171801
$$517$$ −7.58612 −0.333637
$$518$$ −12.2624 −0.538781
$$519$$ 63.2842 2.77787
$$520$$ −9.13122 −0.400431
$$521$$ −7.22536 −0.316549 −0.158274 0.987395i $$-0.550593\pi$$
−0.158274 + 0.987395i $$0.550593\pi$$
$$522$$ −23.0072 −1.00700
$$523$$ −4.90586 −0.214518 −0.107259 0.994231i $$-0.534207\pi$$
−0.107259 + 0.994231i $$0.534207\pi$$
$$524$$ 11.0072 0.480853
$$525$$ 135.057 5.89437
$$526$$ −13.0483 −0.568931
$$527$$ 24.2093 1.05458
$$528$$ 2.66908 0.116157
$$529$$ 6.75598 0.293738
$$530$$ −54.5249 −2.36841
$$531$$ −19.7665 −0.857793
$$532$$ 0 0
$$533$$ 26.3864 1.14292
$$534$$ 22.0145 0.952659
$$535$$ 16.0145 0.692366
$$536$$ −1.30437 −0.0563401
$$537$$ −28.8977 −1.24703
$$538$$ −23.6006 −1.01749
$$539$$ −10.7593 −0.463435
$$540$$ 12.3719 0.532404
$$541$$ 30.8712 1.32726 0.663628 0.748063i $$-0.269016\pi$$
0.663628 + 0.748063i $$0.269016\pi$$
$$542$$ −16.4090 −0.704828
$$543$$ −47.6006 −2.04274
$$544$$ −3.45490 −0.148128
$$545$$ −20.5104 −0.878569
$$546$$ 24.9050 1.06583
$$547$$ −2.54115 −0.108652 −0.0543259 0.998523i $$-0.517301\pi$$
−0.0543259 + 0.998523i $$0.517301\pi$$
$$548$$ 6.94360 0.296616
$$549$$ 36.7439 1.56819
$$550$$ −12.0072 −0.511990
$$551$$ 0 0
$$552$$ 14.5596 0.619696
$$553$$ −38.5925 −1.64112
$$554$$ −11.5861 −0.492247
$$555$$ 32.0289 1.35955
$$556$$ −4.37195 −0.185412
$$557$$ 23.9017 1.01275 0.506373 0.862314i $$-0.330986\pi$$
0.506373 + 0.862314i $$0.330986\pi$$
$$558$$ −28.8977 −1.22334
$$559$$ 3.23744 0.136929
$$560$$ 17.3792 0.734405
$$561$$ −9.22141 −0.389328
$$562$$ −7.03379 −0.296703
$$563$$ 10.6232 0.447715 0.223857 0.974622i $$-0.428135\pi$$
0.223857 + 0.974622i $$0.428135\pi$$
$$564$$ −20.2480 −0.852593
$$565$$ −32.9919 −1.38798
$$566$$ 8.33092 0.350175
$$567$$ 18.3937 0.772461
$$568$$ 6.80030 0.285334
$$569$$ −29.3373 −1.22988 −0.614941 0.788573i $$-0.710820\pi$$
−0.614941 + 0.788573i $$0.710820\pi$$
$$570$$ 0 0
$$571$$ 19.8639 0.831280 0.415640 0.909529i $$-0.363558\pi$$
0.415640 + 0.909529i $$0.363558\pi$$
$$572$$ −2.21417 −0.0925793
$$573$$ −1.63529 −0.0683151
$$574$$ −50.2205 −2.09616
$$575$$ −65.4983 −2.73147
$$576$$ 4.12398 0.171833
$$577$$ −3.17709 −0.132264 −0.0661320 0.997811i $$-0.521066\pi$$
−0.0661320 + 0.997811i $$0.521066\pi$$
$$578$$ −5.06364 −0.210620
$$579$$ 31.0748 1.29143
$$580$$ 23.0072 0.955324
$$581$$ 55.6868 2.31028
$$582$$ 5.81962 0.241231
$$583$$ −13.2214 −0.547575
$$584$$ −1.45490 −0.0602044
$$585$$ −37.6570 −1.55693
$$586$$ −10.2142 −0.421944
$$587$$ 4.00000 0.165098 0.0825488 0.996587i $$-0.473694\pi$$
0.0825488 + 0.996587i $$0.473694\pi$$
$$588$$ −28.7173 −1.18428
$$589$$ 0 0
$$590$$ 19.7665 0.813774
$$591$$ 34.4573 1.41738
$$592$$ 2.90981 0.119592
$$593$$ −3.45885 −0.142038 −0.0710190 0.997475i $$-0.522625\pi$$
−0.0710190 + 0.997475i $$0.522625\pi$$
$$594$$ 3.00000 0.123091
$$595$$ −60.0434 −2.46154
$$596$$ 11.7665 0.481975
$$597$$ 49.0169 2.00613
$$598$$ −12.0781 −0.493911
$$599$$ −40.3864 −1.65014 −0.825072 0.565027i $$-0.808866\pi$$
−0.825072 + 0.565027i $$0.808866\pi$$
$$600$$ −32.0483 −1.30836
$$601$$ −18.3309 −0.747734 −0.373867 0.927482i $$-0.621968\pi$$
−0.373867 + 0.927482i $$0.621968\pi$$
$$602$$ −6.16172 −0.251133
$$603$$ −5.37919 −0.219057
$$604$$ 19.5861 0.796948
$$605$$ −4.12398 −0.167664
$$606$$ 16.6763 0.677429
$$607$$ −12.4283 −0.504451 −0.252226 0.967668i $$-0.581162\pi$$
−0.252226 + 0.967668i $$0.581162\pi$$
$$608$$ 0 0
$$609$$ −62.7511 −2.54280
$$610$$ −36.7439 −1.48772
$$611$$ 16.7970 0.679534
$$612$$ −14.2480 −0.575940
$$613$$ −0.0531082 −0.00214502 −0.00107251 0.999999i $$-0.500341\pi$$
−0.00107251 + 0.999999i $$0.500341\pi$$
$$614$$ 1.93242 0.0779860
$$615$$ 131.174 5.28944
$$616$$ 4.21417 0.169794
$$617$$ −17.4887 −0.704068 −0.352034 0.935987i $$-0.614510\pi$$
−0.352034 + 0.935987i $$0.614510\pi$$
$$618$$ 1.61597 0.0650039
$$619$$ 25.2995 1.01687 0.508437 0.861099i $$-0.330224\pi$$
0.508437 + 0.861099i $$0.330224\pi$$
$$620$$ 28.8977 1.16056
$$621$$ 16.3647 0.656693
$$622$$ −27.7173 −1.11136
$$623$$ 34.7584 1.39256
$$624$$ −5.90981 −0.236582
$$625$$ 59.1376 2.36550
$$626$$ 11.5258 0.460663
$$627$$ 0 0
$$628$$ 20.6199 0.822824
$$629$$ −10.0531 −0.400844
$$630$$ 71.6715 2.85546
$$631$$ 9.02261 0.359184 0.179592 0.983741i $$-0.442522\pi$$
0.179592 + 0.983741i $$0.442522\pi$$
$$632$$ 9.15777 0.364277
$$633$$ 5.68840 0.226093
$$634$$ 33.8977 1.34625
$$635$$ −10.2335 −0.406104
$$636$$ −35.2890 −1.39930
$$637$$ 23.8229 0.943898
$$638$$ 5.57889 0.220870
$$639$$ 28.0443 1.10942
$$640$$ −4.12398 −0.163015
$$641$$ −30.4573 −1.20299 −0.601495 0.798876i $$-0.705428\pi$$
−0.601495 + 0.798876i $$0.705428\pi$$
$$642$$ 10.3647 0.409063
$$643$$ −36.7584 −1.44961 −0.724804 0.688955i $$-0.758070\pi$$
−0.724804 + 0.688955i $$0.758070\pi$$
$$644$$ 22.9879 0.905851
$$645$$ 16.0941 0.633706
$$646$$ 0 0
$$647$$ −41.0700 −1.61463 −0.807314 0.590122i $$-0.799079\pi$$
−0.807314 + 0.590122i $$0.799079\pi$$
$$648$$ −4.36471 −0.171462
$$649$$ 4.79306 0.188144
$$650$$ 26.5861 1.04279
$$651$$ −78.8172 −3.08909
$$652$$ −17.8341 −0.698437
$$653$$ 28.4806 1.11453 0.557265 0.830335i $$-0.311851\pi$$
0.557265 + 0.830335i $$0.311851\pi$$
$$654$$ −13.2745 −0.519075
$$655$$ −45.3937 −1.77368
$$656$$ 11.9170 0.465282
$$657$$ −6.00000 −0.234082
$$658$$ −31.9693 −1.24629
$$659$$ 10.5740 0.411906 0.205953 0.978562i $$-0.433971\pi$$
0.205953 + 0.978562i $$0.433971\pi$$
$$660$$ −11.0072 −0.428456
$$661$$ −7.70287 −0.299607 −0.149803 0.988716i $$-0.547864\pi$$
−0.149803 + 0.988716i $$0.547864\pi$$
$$662$$ 16.2697 0.632339
$$663$$ 20.4178 0.792962
$$664$$ −13.2142 −0.512809
$$665$$ 0 0
$$666$$ 12.0000 0.464991
$$667$$ 30.4323 1.17834
$$668$$ −4.18038 −0.161744
$$669$$ 18.4428 0.713041
$$670$$ 5.37919 0.207816
$$671$$ −8.90981 −0.343959
$$672$$ 11.2480 0.433900
$$673$$ 31.5176 1.21492 0.607458 0.794352i $$-0.292189\pi$$
0.607458 + 0.794352i $$0.292189\pi$$
$$674$$ 6.82685 0.262961
$$675$$ −36.0217 −1.38648
$$676$$ −8.09743 −0.311440
$$677$$ −11.9928 −0.460919 −0.230460 0.973082i $$-0.574023\pi$$
−0.230460 + 0.973082i $$0.574023\pi$$
$$678$$ −21.3526 −0.820043
$$679$$ 9.18852 0.352623
$$680$$ 14.2480 0.546385
$$681$$ −7.27452 −0.278760
$$682$$ 7.00724 0.268321
$$683$$ −39.1722 −1.49888 −0.749442 0.662070i $$-0.769678\pi$$
−0.749442 + 0.662070i $$0.769678\pi$$
$$684$$ 0 0
$$685$$ −28.6353 −1.09410
$$686$$ −15.8422 −0.604859
$$687$$ −9.31160 −0.355260
$$688$$ 1.46214 0.0557436
$$689$$ 29.2745 1.11527
$$690$$ −60.0434 −2.28581
$$691$$ −49.1722 −1.87060 −0.935300 0.353855i $$-0.884871\pi$$
−0.935300 + 0.353855i $$0.884871\pi$$
$$692$$ −23.7101 −0.901323
$$693$$ 17.3792 0.660181
$$694$$ −5.09019 −0.193221
$$695$$ 18.0298 0.683911
$$696$$ 14.8905 0.564423
$$697$$ −41.1722 −1.55951
$$698$$ 30.9919 1.17306
$$699$$ 45.6151 1.72532
$$700$$ −50.6006 −1.91252
$$701$$ 37.8486 1.42952 0.714760 0.699370i $$-0.246536\pi$$
0.714760 + 0.699370i $$0.246536\pi$$
$$702$$ −6.64252 −0.250706
$$703$$ 0 0
$$704$$ −1.00000 −0.0376889
$$705$$ 83.5023 3.14488
$$706$$ −17.8301 −0.671046
$$707$$ 26.3300 0.990242
$$708$$ 12.7931 0.480793
$$709$$ 27.7511 1.04222 0.521108 0.853491i $$-0.325519\pi$$
0.521108 + 0.853491i $$0.325519\pi$$
$$710$$ −28.0443 −1.05248
$$711$$ 37.7665 1.41635
$$712$$ −8.24797 −0.309106
$$713$$ 38.2238 1.43149
$$714$$ −38.8606 −1.45432
$$715$$ 9.13122 0.341488
$$716$$ 10.8269 0.404619
$$717$$ −16.1755 −0.604087
$$718$$ 24.4090 0.910937
$$719$$ −17.0410 −0.635523 −0.317762 0.948171i $$-0.602931\pi$$
−0.317762 + 0.948171i $$0.602931\pi$$
$$720$$ −17.0072 −0.633822
$$721$$ 2.55144 0.0950204
$$722$$ 0 0
$$723$$ 12.9541 0.481769
$$724$$ 17.8341 0.662799
$$725$$ −66.9870 −2.48784
$$726$$ −2.66908 −0.0990588
$$727$$ 13.5225 0.501521 0.250761 0.968049i $$-0.419319\pi$$
0.250761 + 0.968049i $$0.419319\pi$$
$$728$$ −9.33092 −0.345827
$$729$$ −42.0217 −1.55636
$$730$$ 6.00000 0.222070
$$731$$ −5.05156 −0.186839
$$732$$ −23.7810 −0.878970
$$733$$ −25.6682 −0.948076 −0.474038 0.880504i $$-0.657204\pi$$
−0.474038 + 0.880504i $$0.657204\pi$$
$$734$$ −26.6232 −0.982681
$$735$$ 118.430 4.36835
$$736$$ −5.45490 −0.201070
$$737$$ 1.30437 0.0480470
$$738$$ 49.1457 1.80908
$$739$$ 21.3493 0.785348 0.392674 0.919678i $$-0.371550\pi$$
0.392674 + 0.919678i $$0.371550\pi$$
$$740$$ −12.0000 −0.441129
$$741$$ 0 0
$$742$$ −55.7173 −2.04545
$$743$$ 7.55718 0.277246 0.138623 0.990345i $$-0.455732\pi$$
0.138623 + 0.990345i $$0.455732\pi$$
$$744$$ 18.7029 0.685680
$$745$$ −48.5249 −1.77781
$$746$$ 12.6160 0.461904
$$747$$ −54.4950 −1.99387
$$748$$ 3.45490 0.126324
$$749$$ 16.3647 0.597954
$$750$$ 77.1303 2.81640
$$751$$ 53.5861 1.95539 0.977693 0.210040i $$-0.0673595\pi$$
0.977693 + 0.210040i $$0.0673595\pi$$
$$752$$ 7.58612 0.276637
$$753$$ 4.23349 0.154277
$$754$$ −12.3526 −0.449856
$$755$$ −80.7728 −2.93962
$$756$$ 12.6425 0.459804
$$757$$ −13.9436 −0.506789 −0.253394 0.967363i $$-0.581547\pi$$
−0.253394 + 0.967363i $$0.581547\pi$$
$$758$$ 22.8905 0.831420
$$759$$ −14.5596 −0.528479
$$760$$ 0 0
$$761$$ −45.7994 −1.66023 −0.830114 0.557594i $$-0.811724\pi$$
−0.830114 + 0.557594i $$0.811724\pi$$
$$762$$ −6.62321 −0.239934
$$763$$ −20.9590 −0.758766
$$764$$ 0.612679 0.0221659
$$765$$ 58.7584 2.12441
$$766$$ 0.0298464 0.00107839
$$767$$ −10.6127 −0.383202
$$768$$ −2.66908 −0.0963121
$$769$$ −40.8751 −1.47399 −0.736997 0.675896i $$-0.763757\pi$$
−0.736997 + 0.675896i $$0.763757\pi$$
$$770$$ −17.3792 −0.626302
$$771$$ 52.0965 1.87621
$$772$$ −11.6425 −0.419024
$$773$$ −37.4983 −1.34872 −0.674361 0.738402i $$-0.735581\pi$$
−0.674361 + 0.738402i $$0.735581\pi$$
$$774$$ 6.02985 0.216738
$$775$$ −84.1376 −3.02231
$$776$$ −2.18038 −0.0782712
$$777$$ 32.7294 1.17416
$$778$$ 9.22865 0.330863
$$779$$ 0 0
$$780$$ 24.3719 0.872656
$$781$$ −6.80030 −0.243334
$$782$$ 18.8462 0.673938
$$783$$ 16.7367 0.598119
$$784$$ 10.7593 0.384260
$$785$$ −85.0362 −3.03507
$$786$$ −29.3792 −1.04792
$$787$$ −8.80754 −0.313955 −0.156977 0.987602i $$-0.550175\pi$$
−0.156977 + 0.987602i $$0.550175\pi$$
$$788$$ −12.9098 −0.459893
$$789$$ 34.8269 1.23987
$$790$$ −37.7665 −1.34367
$$791$$ −33.7134 −1.19871
$$792$$ −4.12398 −0.146539
$$793$$ 19.7279 0.700557
$$794$$ 17.2697 0.612878
$$795$$ 145.531 5.16146
$$796$$ −18.3647 −0.650920
$$797$$ 26.1312 0.925615 0.462808 0.886459i $$-0.346842\pi$$
0.462808 + 0.886459i $$0.346842\pi$$
$$798$$ 0 0
$$799$$ −26.2093 −0.927220
$$800$$ 12.0072 0.424520
$$801$$ −34.0145 −1.20184
$$802$$ 13.2850 0.469111
$$803$$ 1.45490 0.0513425
$$804$$ 3.48146 0.122782
$$805$$ −94.8018 −3.34132
$$806$$ −15.5152 −0.546501
$$807$$ 62.9919 2.21742
$$808$$ −6.24797 −0.219803
$$809$$ −6.01053 −0.211319 −0.105659 0.994402i $$-0.533695\pi$$
−0.105659 + 0.994402i $$0.533695\pi$$
$$810$$ 18.0000 0.632456
$$811$$ 40.2809 1.41445 0.707226 0.706987i $$-0.249946\pi$$
0.707226 + 0.706987i $$0.249946\pi$$
$$812$$ 23.5104 0.825054
$$813$$ 43.7970 1.53603
$$814$$ −2.90981 −0.101989
$$815$$ 73.5475 2.57626
$$816$$ 9.22141 0.322814
$$817$$ 0 0
$$818$$ −1.94360 −0.0679564
$$819$$ −38.4806 −1.34462
$$820$$ −49.1457 −1.71624
$$821$$ 50.1496 1.75023 0.875117 0.483911i $$-0.160784\pi$$
0.875117 + 0.483911i $$0.160784\pi$$
$$822$$ −18.5330 −0.646413
$$823$$ −42.7400 −1.48982 −0.744911 0.667164i $$-0.767508\pi$$
−0.744911 + 0.667164i $$0.767508\pi$$
$$824$$ −0.605441 −0.0210915
$$825$$ 32.0483 1.11578
$$826$$ 20.1988 0.702806
$$827$$ −49.0555 −1.70583 −0.852913 0.522052i $$-0.825167\pi$$
−0.852913 + 0.522052i $$0.825167\pi$$
$$828$$ −22.4959 −0.781788
$$829$$ −49.9653 −1.73537 −0.867683 0.497117i $$-0.834392\pi$$
−0.867683 + 0.497117i $$0.834392\pi$$
$$830$$ 54.4950 1.89155
$$831$$ 30.9243 1.07275
$$832$$ 2.21417 0.0767627
$$833$$ −37.1722 −1.28794
$$834$$ 11.6691 0.404067
$$835$$ 17.2398 0.596609
$$836$$ 0 0
$$837$$ 21.0217 0.726617
$$838$$ −2.49593 −0.0862206
$$839$$ 52.0257 1.79613 0.898063 0.439868i $$-0.144975\pi$$
0.898063 + 0.439868i $$0.144975\pi$$
$$840$$ −46.3864 −1.60048
$$841$$ 2.12398 0.0732408
$$842$$ −15.6353 −0.538828
$$843$$ 18.7737 0.646602
$$844$$ −2.13122 −0.0733596
$$845$$ 33.3937 1.14878
$$846$$ 31.2850 1.07560
$$847$$ −4.21417 −0.144801
$$848$$ 13.2214 0.454025
$$849$$ −22.2359 −0.763134
$$850$$ −41.4839 −1.42288
$$851$$ −15.8727 −0.544110
$$852$$ −18.1505 −0.621827
$$853$$ 16.2335 0.555824 0.277912 0.960607i $$-0.410358\pi$$
0.277912 + 0.960607i $$0.410358\pi$$
$$854$$ −37.5475 −1.28485
$$855$$ 0 0
$$856$$ −3.88325 −0.132727
$$857$$ −9.60150 −0.327981 −0.163990 0.986462i $$-0.552437\pi$$
−0.163990 + 0.986462i $$0.552437\pi$$
$$858$$ 5.90981 0.201758
$$859$$ 19.4057 0.662115 0.331058 0.943611i $$-0.392595\pi$$
0.331058 + 0.943611i $$0.392595\pi$$
$$860$$ −6.02985 −0.205616
$$861$$ 134.043 4.56816
$$862$$ −0.262441 −0.00893877
$$863$$ −1.54839 −0.0527077 −0.0263539 0.999653i $$-0.508390\pi$$
−0.0263539 + 0.999653i $$0.508390\pi$$
$$864$$ −3.00000 −0.102062
$$865$$ 97.7801 3.32462
$$866$$ 32.2769 1.09681
$$867$$ 13.5152 0.459002
$$868$$ 29.5297 1.00230
$$869$$ −9.15777 −0.310656
$$870$$ −61.4081 −2.08193
$$871$$ −2.88810 −0.0978594
$$872$$ 4.97345 0.168422
$$873$$ −8.99187 −0.304329
$$874$$ 0 0
$$875$$ 121.780 4.11692
$$876$$ 3.88325 0.131203
$$877$$ 12.1200 0.409265 0.204632 0.978839i $$-0.434400\pi$$
0.204632 + 0.978839i $$0.434400\pi$$
$$878$$ 17.0902 0.576766
$$879$$ 27.2624 0.919539
$$880$$ 4.12398 0.139019
$$881$$ 53.6836 1.80864 0.904322 0.426850i $$-0.140377\pi$$
0.904322 + 0.426850i $$0.140377\pi$$
$$882$$ 44.3711 1.49405
$$883$$ 34.4283 1.15861 0.579303 0.815112i $$-0.303325\pi$$
0.579303 + 0.815112i $$0.303325\pi$$
$$884$$ −7.64976 −0.257289
$$885$$ −52.7584 −1.77345
$$886$$ 16.2624 0.546347
$$887$$ 27.5330 0.924468 0.462234 0.886758i $$-0.347048\pi$$
0.462234 + 0.886758i $$0.347048\pi$$
$$888$$ −7.76651 −0.260627
$$889$$ −10.4573 −0.350727
$$890$$ 34.0145 1.14017
$$891$$ 4.36471 0.146223
$$892$$ −6.90981 −0.231357
$$893$$ 0 0
$$894$$ −31.4057 −1.05037
$$895$$ −44.6498 −1.49248
$$896$$ −4.21417 −0.140786
$$897$$ 32.2374 1.07638
$$898$$ 27.3382 0.912286
$$899$$ 39.0926 1.30381
$$900$$ 49.5176 1.65059
$$901$$ −45.6787 −1.52178
$$902$$ −11.9170 −0.396794
$$903$$ 16.4461 0.547292
$$904$$ 8.00000 0.266076
$$905$$ −73.5475 −2.44480
$$906$$ −52.2769 −1.73678
$$907$$ −15.5677 −0.516917 −0.258459 0.966022i $$-0.583215\pi$$
−0.258459 + 0.966022i $$0.583215\pi$$
$$908$$ 2.72548 0.0904482
$$909$$ −25.7665 −0.854621
$$910$$ 38.4806 1.27562
$$911$$ 0.623208 0.0206478 0.0103239 0.999947i $$-0.496714\pi$$
0.0103239 + 0.999947i $$0.496714\pi$$
$$912$$ 0 0
$$913$$ 13.2142 0.437325
$$914$$ 10.4920 0.347044
$$915$$ 98.0724 3.24217
$$916$$ 3.48870 0.115270
$$917$$ −46.3864 −1.53181
$$918$$ 10.3647 0.342086
$$919$$ −44.4887 −1.46755 −0.733773 0.679394i $$-0.762243\pi$$
−0.733773 + 0.679394i $$0.762243\pi$$
$$920$$ 22.4959 0.741669
$$921$$ −5.15777 −0.169954
$$922$$ 6.18038 0.203540
$$923$$ 15.0571 0.495609
$$924$$ −11.2480 −0.370031
$$925$$ 34.9388 1.14878
$$926$$ −11.1433 −0.366192
$$927$$ −2.49683 −0.0820067
$$928$$ −5.57889 −0.183136
$$929$$ 44.1224 1.44761 0.723805 0.690005i $$-0.242391\pi$$
0.723805 + 0.690005i $$0.242391\pi$$
$$930$$ −77.1303 −2.52920
$$931$$ 0 0
$$932$$ −17.0902 −0.559808
$$933$$ 73.9798 2.42199
$$934$$ −34.5104 −1.12922
$$935$$ −14.2480 −0.465958
$$936$$ 9.13122 0.298463
$$937$$ 24.8672 0.812377 0.406188 0.913789i $$-0.366858\pi$$
0.406188 + 0.913789i $$0.366858\pi$$
$$938$$ 5.49683 0.179478
$$939$$ −30.7632 −1.00392
$$940$$ −31.2850 −1.02041
$$941$$ −29.8301 −0.972435 −0.486217 0.873838i $$-0.661624\pi$$
−0.486217 + 0.873838i $$0.661624\pi$$
$$942$$ −55.0362 −1.79318
$$943$$ −65.0063 −2.11690
$$944$$ −4.79306 −0.156001
$$945$$ −52.1376 −1.69603
$$946$$ −1.46214 −0.0475383
$$947$$ 9.07572 0.294921 0.147461 0.989068i $$-0.452890\pi$$
0.147461 + 0.989068i $$0.452890\pi$$
$$948$$ −24.4428 −0.793866
$$949$$ −3.22141 −0.104571
$$950$$ 0 0
$$951$$ −90.4757 −2.93388
$$952$$ 14.5596 0.471878
$$953$$ −1.27058 −0.0411580 −0.0205790 0.999788i $$-0.506551\pi$$
−0.0205790 + 0.999788i $$0.506551\pi$$
$$954$$ 54.5249 1.76531
$$955$$ −2.52668 −0.0817613
$$956$$ 6.06035 0.196006
$$957$$ −14.8905 −0.481341
$$958$$ 41.6980 1.34720
$$959$$ −29.2615 −0.944905
$$960$$ 11.0072 0.355257
$$961$$ 18.1014 0.583915
$$962$$ 6.44282 0.207725
$$963$$ −16.0145 −0.516059
$$964$$ −4.85341 −0.156318
$$965$$ 48.0136 1.54561
$$966$$ −61.3566 −1.97412
$$967$$ 6.67632 0.214696 0.107348 0.994222i $$-0.465764\pi$$
0.107348 + 0.994222i $$0.465764\pi$$
$$968$$ 1.00000 0.0321412
$$969$$ 0 0
$$970$$ 8.99187 0.288711
$$971$$ 0.191566 0.00614764 0.00307382 0.999995i $$-0.499022\pi$$
0.00307382 + 0.999995i $$0.499022\pi$$
$$972$$ 20.6498 0.662341
$$973$$ 18.4242 0.590651
$$974$$ −9.10137 −0.291627
$$975$$ −70.9605 −2.27255
$$976$$ 8.90981 0.285196
$$977$$ −16.4959 −0.527752 −0.263876 0.964557i $$-0.585001\pi$$
−0.263876 + 0.964557i $$0.585001\pi$$
$$978$$ 47.6006 1.52210
$$979$$ 8.24797 0.263606
$$980$$ −44.3711 −1.41738
$$981$$ 20.5104 0.654847
$$982$$ −30.8413 −0.984186
$$983$$ 35.0072 1.11656 0.558279 0.829653i $$-0.311462\pi$$
0.558279 + 0.829653i $$0.311462\pi$$
$$984$$ −31.8075 −1.01399
$$985$$ 53.2398 1.69636
$$986$$ 19.2745 0.613826
$$987$$ 85.3285 2.71604
$$988$$ 0 0
$$989$$ −7.97584 −0.253617
$$990$$ 17.0072 0.540525
$$991$$ 11.3759 0.361367 0.180684 0.983541i $$-0.442169\pi$$
0.180684 + 0.983541i $$0.442169\pi$$
$$992$$ −7.00724 −0.222480
$$993$$ −43.4251 −1.37805
$$994$$ −28.6577 −0.908966
$$995$$ 75.7358 2.40099
$$996$$ 35.2697 1.11756
$$997$$ 8.89533 0.281718 0.140859 0.990030i $$-0.455014\pi$$
0.140859 + 0.990030i $$0.455014\pi$$
$$998$$ 21.7665 0.689007
$$999$$ −8.72942 −0.276187
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 7942.2.a.bi.1.1 3
19.18 odd 2 418.2.a.g.1.3 3
57.56 even 2 3762.2.a.bg.1.3 3
76.75 even 2 3344.2.a.q.1.1 3
209.208 even 2 4598.2.a.bo.1.3 3

By twisted newform
Twist Min Dim Char Parity Ord Type
418.2.a.g.1.3 3 19.18 odd 2
3344.2.a.q.1.1 3 76.75 even 2
3762.2.a.bg.1.3 3 57.56 even 2
4598.2.a.bo.1.3 3 209.208 even 2
7942.2.a.bi.1.1 3 1.1 even 1 trivial