# Properties

 Label 147.6.a.j.1.1 Level $147$ Weight $6$ Character 147.1 Self dual yes Analytic conductor $23.576$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [147,6,Mod(1,147)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(147, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("147.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$147 = 3 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 147.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: $$23.5764215125$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{193})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 48$$ x^2 - x - 48 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$7.44622$$ of defining polynomial Character $$\chi$$ $$=$$ 147.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-8.44622 q^{2} +9.00000 q^{3} +39.3387 q^{4} -36.0000 q^{5} -76.0160 q^{6} -61.9840 q^{8} +81.0000 q^{9} +O(q^{10})$$ $$q-8.44622 q^{2} +9.00000 q^{3} +39.3387 q^{4} -36.0000 q^{5} -76.0160 q^{6} -61.9840 q^{8} +81.0000 q^{9} +304.064 q^{10} +295.570 q^{11} +354.048 q^{12} -1148.13 q^{13} -324.000 q^{15} -735.307 q^{16} +1032.38 q^{17} -684.144 q^{18} +2108.51 q^{19} -1416.19 q^{20} -2496.45 q^{22} -640.988 q^{23} -557.856 q^{24} -1829.00 q^{25} +9697.34 q^{26} +729.000 q^{27} +7631.58 q^{29} +2736.58 q^{30} -966.976 q^{31} +8194.05 q^{32} +2660.13 q^{33} -8719.74 q^{34} +3186.43 q^{36} -1773.21 q^{37} -17809.0 q^{38} -10333.2 q^{39} +2231.42 q^{40} -11976.4 q^{41} -19802.9 q^{43} +11627.3 q^{44} -2916.00 q^{45} +5413.93 q^{46} -27966.1 q^{47} -6617.76 q^{48} +15448.1 q^{50} +9291.46 q^{51} -45165.8 q^{52} -7114.33 q^{53} -6157.30 q^{54} -10640.5 q^{55} +18976.6 q^{57} -64458.0 q^{58} -20869.5 q^{59} -12745.7 q^{60} -23868.3 q^{61} +8167.30 q^{62} -45679.0 q^{64} +41332.6 q^{65} -22468.0 q^{66} +34671.5 q^{67} +40612.6 q^{68} -5768.90 q^{69} -28413.2 q^{71} -5020.70 q^{72} -15292.7 q^{73} +14976.9 q^{74} -16461.0 q^{75} +82946.0 q^{76} +87276.1 q^{78} -73059.5 q^{79} +26471.0 q^{80} +6561.00 q^{81} +101155. q^{82} -30340.9 q^{83} -37165.8 q^{85} +167260. q^{86} +68684.2 q^{87} -18320.6 q^{88} -36089.5 q^{89} +24629.2 q^{90} -25215.6 q^{92} -8702.79 q^{93} +236208. q^{94} -75906.4 q^{95} +73746.5 q^{96} -153963. q^{97} +23941.2 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 3 q^{2} + 18 q^{3} + 37 q^{4} - 72 q^{5} - 27 q^{6} - 249 q^{8} + 162 q^{9}+O(q^{10})$$ 2 * q - 3 * q^2 + 18 * q^3 + 37 * q^4 - 72 * q^5 - 27 * q^6 - 249 * q^8 + 162 * q^9 $$2 q - 3 q^{2} + 18 q^{3} + 37 q^{4} - 72 q^{5} - 27 q^{6} - 249 q^{8} + 162 q^{9} + 108 q^{10} + 480 q^{11} + 333 q^{12} - 1296 q^{13} - 648 q^{15} - 1679 q^{16} - 936 q^{17} - 243 q^{18} + 216 q^{19} - 1332 q^{20} - 1492 q^{22} - 504 q^{23} - 2241 q^{24} - 3658 q^{25} + 8892 q^{26} + 1458 q^{27} + 6372 q^{29} + 972 q^{30} - 9936 q^{31} + 9039 q^{32} + 4320 q^{33} - 19440 q^{34} + 2997 q^{36} + 11124 q^{37} - 28116 q^{38} - 11664 q^{39} + 8964 q^{40} - 20952 q^{41} - 6264 q^{43} + 11196 q^{44} - 5832 q^{45} + 6160 q^{46} - 7920 q^{47} - 15111 q^{48} + 5487 q^{50} - 8424 q^{51} - 44820 q^{52} + 2220 q^{53} - 2187 q^{54} - 17280 q^{55} + 1944 q^{57} - 71318 q^{58} - 29736 q^{59} - 11988 q^{60} + 17280 q^{61} - 40680 q^{62} - 10879 q^{64} + 46656 q^{65} - 13428 q^{66} - 20680 q^{67} + 45216 q^{68} - 4536 q^{69} - 92280 q^{71} - 20169 q^{72} - 56592 q^{73} + 85218 q^{74} - 32922 q^{75} + 87372 q^{76} + 80028 q^{78} - 56096 q^{79} + 60444 q^{80} + 13122 q^{81} + 52272 q^{82} + 71352 q^{83} + 33696 q^{85} + 240996 q^{86} + 57348 q^{87} - 52812 q^{88} - 123192 q^{89} + 8748 q^{90} - 25536 q^{92} - 89424 q^{93} + 345384 q^{94} - 7776 q^{95} + 81351 q^{96} - 35856 q^{97} + 38880 q^{99}+O(q^{100})$$ 2 * q - 3 * q^2 + 18 * q^3 + 37 * q^4 - 72 * q^5 - 27 * q^6 - 249 * q^8 + 162 * q^9 + 108 * q^10 + 480 * q^11 + 333 * q^12 - 1296 * q^13 - 648 * q^15 - 1679 * q^16 - 936 * q^17 - 243 * q^18 + 216 * q^19 - 1332 * q^20 - 1492 * q^22 - 504 * q^23 - 2241 * q^24 - 3658 * q^25 + 8892 * q^26 + 1458 * q^27 + 6372 * q^29 + 972 * q^30 - 9936 * q^31 + 9039 * q^32 + 4320 * q^33 - 19440 * q^34 + 2997 * q^36 + 11124 * q^37 - 28116 * q^38 - 11664 * q^39 + 8964 * q^40 - 20952 * q^41 - 6264 * q^43 + 11196 * q^44 - 5832 * q^45 + 6160 * q^46 - 7920 * q^47 - 15111 * q^48 + 5487 * q^50 - 8424 * q^51 - 44820 * q^52 + 2220 * q^53 - 2187 * q^54 - 17280 * q^55 + 1944 * q^57 - 71318 * q^58 - 29736 * q^59 - 11988 * q^60 + 17280 * q^61 - 40680 * q^62 - 10879 * q^64 + 46656 * q^65 - 13428 * q^66 - 20680 * q^67 + 45216 * q^68 - 4536 * q^69 - 92280 * q^71 - 20169 * q^72 - 56592 * q^73 + 85218 * q^74 - 32922 * q^75 + 87372 * q^76 + 80028 * q^78 - 56096 * q^79 + 60444 * q^80 + 13122 * q^81 + 52272 * q^82 + 71352 * q^83 + 33696 * q^85 + 240996 * q^86 + 57348 * q^87 - 52812 * q^88 - 123192 * q^89 + 8748 * q^90 - 25536 * q^92 - 89424 * q^93 + 345384 * q^94 - 7776 * q^95 + 81351 * q^96 - 35856 * q^97 + 38880 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −8.44622 −1.49310 −0.746548 0.665332i $$-0.768290\pi$$
−0.746548 + 0.665332i $$0.768290\pi$$
$$3$$ 9.00000 0.577350
$$4$$ 39.3387 1.22933
$$5$$ −36.0000 −0.643988 −0.321994 0.946742i $$-0.604353\pi$$
−0.321994 + 0.946742i $$0.604353\pi$$
$$6$$ −76.0160 −0.862039
$$7$$ 0 0
$$8$$ −61.9840 −0.342416
$$9$$ 81.0000 0.333333
$$10$$ 304.064 0.961535
$$11$$ 295.570 0.736509 0.368255 0.929725i $$-0.379955\pi$$
0.368255 + 0.929725i $$0.379955\pi$$
$$12$$ 354.048 0.709756
$$13$$ −1148.13 −1.88422 −0.942111 0.335302i $$-0.891162\pi$$
−0.942111 + 0.335302i $$0.891162\pi$$
$$14$$ 0 0
$$15$$ −324.000 −0.371806
$$16$$ −735.307 −0.718073
$$17$$ 1032.38 0.866401 0.433200 0.901298i $$-0.357384\pi$$
0.433200 + 0.901298i $$0.357384\pi$$
$$18$$ −684.144 −0.497698
$$19$$ 2108.51 1.33996 0.669980 0.742379i $$-0.266303\pi$$
0.669980 + 0.742379i $$0.266303\pi$$
$$20$$ −1416.19 −0.791675
$$21$$ 0 0
$$22$$ −2496.45 −1.09968
$$23$$ −640.988 −0.252657 −0.126328 0.991988i $$-0.540319\pi$$
−0.126328 + 0.991988i $$0.540319\pi$$
$$24$$ −557.856 −0.197694
$$25$$ −1829.00 −0.585280
$$26$$ 9697.34 2.81332
$$27$$ 729.000 0.192450
$$28$$ 0 0
$$29$$ 7631.58 1.68508 0.842538 0.538637i $$-0.181060\pi$$
0.842538 + 0.538637i $$0.181060\pi$$
$$30$$ 2736.58 0.555142
$$31$$ −966.976 −0.180722 −0.0903611 0.995909i $$-0.528802\pi$$
−0.0903611 + 0.995909i $$0.528802\pi$$
$$32$$ 8194.05 1.41457
$$33$$ 2660.13 0.425224
$$34$$ −8719.74 −1.29362
$$35$$ 0 0
$$36$$ 3186.43 0.409778
$$37$$ −1773.21 −0.212939 −0.106470 0.994316i $$-0.533955\pi$$
−0.106470 + 0.994316i $$0.533955\pi$$
$$38$$ −17809.0 −2.00069
$$39$$ −10333.2 −1.08786
$$40$$ 2231.42 0.220512
$$41$$ −11976.4 −1.11267 −0.556335 0.830958i $$-0.687793\pi$$
−0.556335 + 0.830958i $$0.687793\pi$$
$$42$$ 0 0
$$43$$ −19802.9 −1.63327 −0.816636 0.577153i $$-0.804163\pi$$
−0.816636 + 0.577153i $$0.804163\pi$$
$$44$$ 11627.3 0.905416
$$45$$ −2916.00 −0.214663
$$46$$ 5413.93 0.377240
$$47$$ −27966.1 −1.84666 −0.923332 0.384002i $$-0.874545\pi$$
−0.923332 + 0.384002i $$0.874545\pi$$
$$48$$ −6617.76 −0.414580
$$49$$ 0 0
$$50$$ 15448.1 0.873879
$$51$$ 9291.46 0.500217
$$52$$ −45165.8 −2.31634
$$53$$ −7114.33 −0.347892 −0.173946 0.984755i $$-0.555652\pi$$
−0.173946 + 0.984755i $$0.555652\pi$$
$$54$$ −6157.30 −0.287346
$$55$$ −10640.5 −0.474303
$$56$$ 0 0
$$57$$ 18976.6 0.773627
$$58$$ −64458.0 −2.51598
$$59$$ −20869.5 −0.780518 −0.390259 0.920705i $$-0.627614\pi$$
−0.390259 + 0.920705i $$0.627614\pi$$
$$60$$ −12745.7 −0.457074
$$61$$ −23868.3 −0.821291 −0.410646 0.911795i $$-0.634697\pi$$
−0.410646 + 0.911795i $$0.634697\pi$$
$$62$$ 8167.30 0.269835
$$63$$ 0 0
$$64$$ −45679.0 −1.39401
$$65$$ 41332.6 1.21342
$$66$$ −22468.0 −0.634900
$$67$$ 34671.5 0.943595 0.471798 0.881707i $$-0.343605\pi$$
0.471798 + 0.881707i $$0.343605\pi$$
$$68$$ 40612.6 1.06510
$$69$$ −5768.90 −0.145871
$$70$$ 0 0
$$71$$ −28413.2 −0.668921 −0.334461 0.942410i $$-0.608554\pi$$
−0.334461 + 0.942410i $$0.608554\pi$$
$$72$$ −5020.70 −0.114139
$$73$$ −15292.7 −0.335874 −0.167937 0.985798i $$-0.553711\pi$$
−0.167937 + 0.985798i $$0.553711\pi$$
$$74$$ 14976.9 0.317939
$$75$$ −16461.0 −0.337912
$$76$$ 82946.0 1.64726
$$77$$ 0 0
$$78$$ 87276.1 1.62427
$$79$$ −73059.5 −1.31707 −0.658535 0.752550i $$-0.728824\pi$$
−0.658535 + 0.752550i $$0.728824\pi$$
$$80$$ 26471.0 0.462430
$$81$$ 6561.00 0.111111
$$82$$ 101155. 1.66132
$$83$$ −30340.9 −0.483429 −0.241715 0.970347i $$-0.577710\pi$$
−0.241715 + 0.970347i $$0.577710\pi$$
$$84$$ 0 0
$$85$$ −37165.8 −0.557951
$$86$$ 167260. 2.43863
$$87$$ 68684.2 0.972879
$$88$$ −18320.6 −0.252193
$$89$$ −36089.5 −0.482954 −0.241477 0.970407i $$-0.577632\pi$$
−0.241477 + 0.970407i $$0.577632\pi$$
$$90$$ 24629.2 0.320512
$$91$$ 0 0
$$92$$ −25215.6 −0.310599
$$93$$ −8702.79 −0.104340
$$94$$ 236208. 2.75725
$$95$$ −75906.4 −0.862918
$$96$$ 73746.5 0.816701
$$97$$ −153963. −1.66145 −0.830724 0.556685i $$-0.812073\pi$$
−0.830724 + 0.556685i $$0.812073\pi$$
$$98$$ 0 0
$$99$$ 23941.2 0.245503
$$100$$ −71950.4 −0.719504
$$101$$ 139809. 1.36374 0.681869 0.731474i $$-0.261167\pi$$
0.681869 + 0.731474i $$0.261167\pi$$
$$102$$ −78477.7 −0.746871
$$103$$ −115925. −1.07668 −0.538339 0.842728i $$-0.680948\pi$$
−0.538339 + 0.842728i $$0.680948\pi$$
$$104$$ 71165.6 0.645188
$$105$$ 0 0
$$106$$ 60089.2 0.519436
$$107$$ 83061.8 0.701361 0.350681 0.936495i $$-0.385950\pi$$
0.350681 + 0.936495i $$0.385950\pi$$
$$108$$ 28677.9 0.236585
$$109$$ 45356.2 0.365654 0.182827 0.983145i $$-0.441475\pi$$
0.182827 + 0.983145i $$0.441475\pi$$
$$110$$ 89872.1 0.708179
$$111$$ −15958.9 −0.122941
$$112$$ 0 0
$$113$$ −355.533 −0.00261929 −0.00130965 0.999999i $$-0.500417\pi$$
−0.00130965 + 0.999999i $$0.500417\pi$$
$$114$$ −160281. −1.15510
$$115$$ 23075.6 0.162708
$$116$$ 300216. 2.07152
$$117$$ −92998.4 −0.628074
$$118$$ 176269. 1.16539
$$119$$ 0 0
$$120$$ 20082.8 0.127313
$$121$$ −73689.5 −0.457554
$$122$$ 201597. 1.22627
$$123$$ −107787. −0.642400
$$124$$ −38039.6 −0.222168
$$125$$ 178344. 1.02090
$$126$$ 0 0
$$127$$ 168967. 0.929593 0.464797 0.885417i $$-0.346127\pi$$
0.464797 + 0.885417i $$0.346127\pi$$
$$128$$ 123605. 0.666824
$$129$$ −178226. −0.942970
$$130$$ −349104. −1.81174
$$131$$ −173969. −0.885715 −0.442858 0.896592i $$-0.646035\pi$$
−0.442858 + 0.896592i $$0.646035\pi$$
$$132$$ 104646. 0.522742
$$133$$ 0 0
$$134$$ −292843. −1.40888
$$135$$ −26244.0 −0.123935
$$136$$ −63991.3 −0.296670
$$137$$ 367723. 1.67386 0.836931 0.547308i $$-0.184347\pi$$
0.836931 + 0.547308i $$0.184347\pi$$
$$138$$ 48725.4 0.217800
$$139$$ 217967. 0.956870 0.478435 0.878123i $$-0.341204\pi$$
0.478435 + 0.878123i $$0.341204\pi$$
$$140$$ 0 0
$$141$$ −251695. −1.06617
$$142$$ 239985. 0.998763
$$143$$ −339352. −1.38775
$$144$$ −59559.8 −0.239358
$$145$$ −274737. −1.08517
$$146$$ 129165. 0.501492
$$147$$ 0 0
$$148$$ −69755.7 −0.261773
$$149$$ −64906.1 −0.239508 −0.119754 0.992804i $$-0.538211\pi$$
−0.119754 + 0.992804i $$0.538211\pi$$
$$150$$ 139033. 0.504534
$$151$$ −223777. −0.798681 −0.399341 0.916803i $$-0.630761\pi$$
−0.399341 + 0.916803i $$0.630761\pi$$
$$152$$ −130694. −0.458825
$$153$$ 83623.1 0.288800
$$154$$ 0 0
$$155$$ 34811.1 0.116383
$$156$$ −406492. −1.33734
$$157$$ −459973. −1.48930 −0.744652 0.667453i $$-0.767384\pi$$
−0.744652 + 0.667453i $$0.767384\pi$$
$$158$$ 617077. 1.96651
$$159$$ −64028.9 −0.200855
$$160$$ −294986. −0.910964
$$161$$ 0 0
$$162$$ −55415.7 −0.165899
$$163$$ 91068.6 0.268472 0.134236 0.990949i $$-0.457142\pi$$
0.134236 + 0.990949i $$0.457142\pi$$
$$164$$ −471135. −1.36784
$$165$$ −95764.6 −0.273839
$$166$$ 256266. 0.721806
$$167$$ −314772. −0.873384 −0.436692 0.899611i $$-0.643850\pi$$
−0.436692 + 0.899611i $$0.643850\pi$$
$$168$$ 0 0
$$169$$ 946905. 2.55029
$$170$$ 313911. 0.833075
$$171$$ 170789. 0.446654
$$172$$ −779021. −2.00784
$$173$$ −362143. −0.919951 −0.459975 0.887932i $$-0.652142\pi$$
−0.459975 + 0.887932i $$0.652142\pi$$
$$174$$ −580122. −1.45260
$$175$$ 0 0
$$176$$ −217334. −0.528867
$$177$$ −187826. −0.450632
$$178$$ 304820. 0.721096
$$179$$ −173896. −0.405656 −0.202828 0.979214i $$-0.565013\pi$$
−0.202828 + 0.979214i $$0.565013\pi$$
$$180$$ −114712. −0.263892
$$181$$ 134973. 0.306233 0.153116 0.988208i $$-0.451069\pi$$
0.153116 + 0.988208i $$0.451069\pi$$
$$182$$ 0 0
$$183$$ −214815. −0.474173
$$184$$ 39731.0 0.0865138
$$185$$ 63835.6 0.137130
$$186$$ 73505.7 0.155790
$$187$$ 305141. 0.638112
$$188$$ −1.10015e6 −2.27017
$$189$$ 0 0
$$190$$ 641123. 1.28842
$$191$$ 181413. 0.359821 0.179910 0.983683i $$-0.442419\pi$$
0.179910 + 0.983683i $$0.442419\pi$$
$$192$$ −411111. −0.804833
$$193$$ 965999. 1.86674 0.933369 0.358919i $$-0.116855\pi$$
0.933369 + 0.358919i $$0.116855\pi$$
$$194$$ 1.30040e6 2.48070
$$195$$ 371993. 0.700566
$$196$$ 0 0
$$197$$ −699058. −1.28336 −0.641679 0.766974i $$-0.721762\pi$$
−0.641679 + 0.766974i $$0.721762\pi$$
$$198$$ −202212. −0.366560
$$199$$ −416191. −0.745006 −0.372503 0.928031i $$-0.621500\pi$$
−0.372503 + 0.928031i $$0.621500\pi$$
$$200$$ 113369. 0.200410
$$201$$ 312044. 0.544785
$$202$$ −1.18086e6 −2.03619
$$203$$ 0 0
$$204$$ 365513. 0.614933
$$205$$ 431150. 0.716545
$$206$$ 979132. 1.60758
$$207$$ −51920.1 −0.0842189
$$208$$ 844226. 1.35301
$$209$$ 623212. 0.986894
$$210$$ 0 0
$$211$$ −407152. −0.629580 −0.314790 0.949161i $$-0.601934\pi$$
−0.314790 + 0.949161i $$0.601934\pi$$
$$212$$ −279868. −0.427675
$$213$$ −255719. −0.386202
$$214$$ −701558. −1.04720
$$215$$ 712906. 1.05181
$$216$$ −45186.3 −0.0658981
$$217$$ 0 0
$$218$$ −383089. −0.545957
$$219$$ −137634. −0.193917
$$220$$ −418584. −0.583076
$$221$$ −1.18531e6 −1.63249
$$222$$ 134792. 0.183562
$$223$$ 882022. 1.18773 0.593865 0.804565i $$-0.297602\pi$$
0.593865 + 0.804565i $$0.297602\pi$$
$$224$$ 0 0
$$225$$ −148149. −0.195093
$$226$$ 3002.91 0.00391085
$$227$$ −1.12650e6 −1.45100 −0.725499 0.688223i $$-0.758391\pi$$
−0.725499 + 0.688223i $$0.758391\pi$$
$$228$$ 746514. 0.951045
$$229$$ 310084. 0.390743 0.195371 0.980729i $$-0.437409\pi$$
0.195371 + 0.980729i $$0.437409\pi$$
$$230$$ −194902. −0.242938
$$231$$ 0 0
$$232$$ −473036. −0.576998
$$233$$ 1.13654e6 1.37149 0.685746 0.727841i $$-0.259476\pi$$
0.685746 + 0.727841i $$0.259476\pi$$
$$234$$ 785485. 0.937774
$$235$$ 1.00678e6 1.18923
$$236$$ −820980. −0.959516
$$237$$ −657536. −0.760411
$$238$$ 0 0
$$239$$ 87506.8 0.0990940 0.0495470 0.998772i $$-0.484222\pi$$
0.0495470 + 0.998772i $$0.484222\pi$$
$$240$$ 238239. 0.266984
$$241$$ 537768. 0.596421 0.298210 0.954500i $$-0.403610\pi$$
0.298210 + 0.954500i $$0.403610\pi$$
$$242$$ 622398. 0.683171
$$243$$ 59049.0 0.0641500
$$244$$ −938948. −1.00964
$$245$$ 0 0
$$246$$ 910397. 0.959164
$$247$$ −2.42084e6 −2.52478
$$248$$ 59937.1 0.0618823
$$249$$ −273068. −0.279108
$$250$$ −1.50633e6 −1.52430
$$251$$ 1.35353e6 1.35607 0.678036 0.735028i $$-0.262831\pi$$
0.678036 + 0.735028i $$0.262831\pi$$
$$252$$ 0 0
$$253$$ −189457. −0.186084
$$254$$ −1.42713e6 −1.38797
$$255$$ −334492. −0.322133
$$256$$ 417731. 0.398380
$$257$$ −976900. −0.922608 −0.461304 0.887242i $$-0.652618\pi$$
−0.461304 + 0.887242i $$0.652618\pi$$
$$258$$ 1.50534e6 1.40794
$$259$$ 0 0
$$260$$ 1.62597e6 1.49169
$$261$$ 618158. 0.561692
$$262$$ 1.46938e6 1.32246
$$263$$ −1.24375e6 −1.10877 −0.554387 0.832259i $$-0.687047\pi$$
−0.554387 + 0.832259i $$0.687047\pi$$
$$264$$ −164885. −0.145604
$$265$$ 256116. 0.224038
$$266$$ 0 0
$$267$$ −324805. −0.278833
$$268$$ 1.36393e6 1.15999
$$269$$ −1.08408e6 −0.913445 −0.456722 0.889609i $$-0.650977\pi$$
−0.456722 + 0.889609i $$0.650977\pi$$
$$270$$ 221663. 0.185047
$$271$$ −2.16627e6 −1.79180 −0.895900 0.444256i $$-0.853468\pi$$
−0.895900 + 0.444256i $$0.853468\pi$$
$$272$$ −759119. −0.622139
$$273$$ 0 0
$$274$$ −3.10587e6 −2.49924
$$275$$ −540597. −0.431064
$$276$$ −226941. −0.179324
$$277$$ 253859. 0.198789 0.0993946 0.995048i $$-0.468309\pi$$
0.0993946 + 0.995048i $$0.468309\pi$$
$$278$$ −1.84099e6 −1.42870
$$279$$ −78325.1 −0.0602407
$$280$$ 0 0
$$281$$ 1.14116e6 0.862143 0.431072 0.902318i $$-0.358136\pi$$
0.431072 + 0.902318i $$0.358136\pi$$
$$282$$ 2.12587e6 1.59190
$$283$$ −609918. −0.452694 −0.226347 0.974047i $$-0.572678\pi$$
−0.226347 + 0.974047i $$0.572678\pi$$
$$284$$ −1.11774e6 −0.822327
$$285$$ −683158. −0.498206
$$286$$ 2.86624e6 2.07204
$$287$$ 0 0
$$288$$ 663718. 0.471523
$$289$$ −354040. −0.249349
$$290$$ 2.32049e6 1.62026
$$291$$ −1.38567e6 −0.959237
$$292$$ −601593. −0.412901
$$293$$ 156438. 0.106457 0.0532283 0.998582i $$-0.483049\pi$$
0.0532283 + 0.998582i $$0.483049\pi$$
$$294$$ 0 0
$$295$$ 751303. 0.502644
$$296$$ 109911. 0.0729139
$$297$$ 215470. 0.141741
$$298$$ 548211. 0.357608
$$299$$ 735937. 0.476061
$$300$$ −647554. −0.415406
$$301$$ 0 0
$$302$$ 1.89007e6 1.19251
$$303$$ 1.25828e6 0.787354
$$304$$ −1.55040e6 −0.962189
$$305$$ 859259. 0.528901
$$306$$ −706299. −0.431206
$$307$$ 293229. 0.177566 0.0887831 0.996051i $$-0.471702\pi$$
0.0887831 + 0.996051i $$0.471702\pi$$
$$308$$ 0 0
$$309$$ −1.04333e6 −0.621620
$$310$$ −294023. −0.173771
$$311$$ 2.45216e6 1.43763 0.718816 0.695200i $$-0.244684\pi$$
0.718816 + 0.695200i $$0.244684\pi$$
$$312$$ 640490. 0.372500
$$313$$ 1.83541e6 1.05894 0.529471 0.848328i $$-0.322390\pi$$
0.529471 + 0.848328i $$0.322390\pi$$
$$314$$ 3.88503e6 2.22367
$$315$$ 0 0
$$316$$ −2.87406e6 −1.61912
$$317$$ 589960. 0.329742 0.164871 0.986315i $$-0.447279\pi$$
0.164871 + 0.986315i $$0.447279\pi$$
$$318$$ 540803. 0.299896
$$319$$ 2.25567e6 1.24107
$$320$$ 1.64444e6 0.897726
$$321$$ 747556. 0.404931
$$322$$ 0 0
$$323$$ 2.17679e6 1.16094
$$324$$ 258101. 0.136593
$$325$$ 2.09993e6 1.10280
$$326$$ −769186. −0.400855
$$327$$ 408206. 0.211111
$$328$$ 742344. 0.380996
$$329$$ 0 0
$$330$$ 808849. 0.408868
$$331$$ 177318. 0.0889577 0.0444789 0.999010i $$-0.485837\pi$$
0.0444789 + 0.999010i $$0.485837\pi$$
$$332$$ −1.19357e6 −0.594296
$$333$$ −143630. −0.0709798
$$334$$ 2.65864e6 1.30405
$$335$$ −1.24817e6 −0.607664
$$336$$ 0 0
$$337$$ −3.04781e6 −1.46189 −0.730943 0.682438i $$-0.760920\pi$$
−0.730943 + 0.682438i $$0.760920\pi$$
$$338$$ −7.99777e6 −3.80783
$$339$$ −3199.80 −0.00151225
$$340$$ −1.46205e6 −0.685908
$$341$$ −285809. −0.133104
$$342$$ −1.44253e6 −0.666896
$$343$$ 0 0
$$344$$ 1.22747e6 0.559259
$$345$$ 207680. 0.0939393
$$346$$ 3.05874e6 1.37357
$$347$$ −2.42361e6 −1.08054 −0.540268 0.841493i $$-0.681677\pi$$
−0.540268 + 0.841493i $$0.681677\pi$$
$$348$$ 2.70195e6 1.19599
$$349$$ −2.67690e6 −1.17644 −0.588218 0.808702i $$-0.700170\pi$$
−0.588218 + 0.808702i $$0.700170\pi$$
$$350$$ 0 0
$$351$$ −836985. −0.362619
$$352$$ 2.42191e6 1.04184
$$353$$ 950412. 0.405953 0.202976 0.979184i $$-0.434939\pi$$
0.202976 + 0.979184i $$0.434939\pi$$
$$354$$ 1.58642e6 0.672837
$$355$$ 1.02288e6 0.430777
$$356$$ −1.41971e6 −0.593711
$$357$$ 0 0
$$358$$ 1.46877e6 0.605683
$$359$$ 2.78881e6 1.14204 0.571022 0.820935i $$-0.306547\pi$$
0.571022 + 0.820935i $$0.306547\pi$$
$$360$$ 180745. 0.0735040
$$361$$ 1.96972e6 0.795495
$$362$$ −1.14001e6 −0.457235
$$363$$ −663206. −0.264169
$$364$$ 0 0
$$365$$ 550536. 0.216299
$$366$$ 1.81437e6 0.707985
$$367$$ 153881. 0.0596377 0.0298189 0.999555i $$-0.490507\pi$$
0.0298189 + 0.999555i $$0.490507\pi$$
$$368$$ 471323. 0.181426
$$369$$ −970087. −0.370890
$$370$$ −539169. −0.204749
$$371$$ 0 0
$$372$$ −342356. −0.128269
$$373$$ −2.38381e6 −0.887156 −0.443578 0.896236i $$-0.646291\pi$$
−0.443578 + 0.896236i $$0.646291\pi$$
$$374$$ −2.57729e6 −0.952763
$$375$$ 1.60510e6 0.589417
$$376$$ 1.73345e6 0.632328
$$377$$ −8.76203e6 −3.17506
$$378$$ 0 0
$$379$$ 3.65191e6 1.30594 0.652969 0.757385i $$-0.273523\pi$$
0.652969 + 0.757385i $$0.273523\pi$$
$$380$$ −2.98606e6 −1.06081
$$381$$ 1.52070e6 0.536701
$$382$$ −1.53226e6 −0.537246
$$383$$ 2.15730e6 0.751472 0.375736 0.926727i $$-0.377390\pi$$
0.375736 + 0.926727i $$0.377390\pi$$
$$384$$ 1.11245e6 0.384991
$$385$$ 0 0
$$386$$ −8.15904e6 −2.78722
$$387$$ −1.60404e6 −0.544424
$$388$$ −6.05669e6 −2.04247
$$389$$ −3.66471e6 −1.22791 −0.613954 0.789342i $$-0.710422\pi$$
−0.613954 + 0.789342i $$0.710422\pi$$
$$390$$ −3.14194e6 −1.04601
$$391$$ −661746. −0.218902
$$392$$ 0 0
$$393$$ −1.56572e6 −0.511368
$$394$$ 5.90440e6 1.91617
$$395$$ 2.63014e6 0.848177
$$396$$ 941813. 0.301805
$$397$$ −3.94648e6 −1.25671 −0.628353 0.777928i $$-0.716271\pi$$
−0.628353 + 0.777928i $$0.716271\pi$$
$$398$$ 3.51524e6 1.11236
$$399$$ 0 0
$$400$$ 1.34488e6 0.420274
$$401$$ 25016.1 0.00776887 0.00388444 0.999992i $$-0.498764\pi$$
0.00388444 + 0.999992i $$0.498764\pi$$
$$402$$ −2.63559e6 −0.813416
$$403$$ 1.11021e6 0.340521
$$404$$ 5.49989e6 1.67649
$$405$$ −236196. −0.0715542
$$406$$ 0 0
$$407$$ −524107. −0.156832
$$408$$ −575922. −0.171282
$$409$$ 832700. 0.246139 0.123069 0.992398i $$-0.460726\pi$$
0.123069 + 0.992398i $$0.460726\pi$$
$$410$$ −3.64159e6 −1.06987
$$411$$ 3.30951e6 0.966405
$$412$$ −4.56035e6 −1.32360
$$413$$ 0 0
$$414$$ 438528. 0.125747
$$415$$ 1.09227e6 0.311323
$$416$$ −9.40782e6 −2.66536
$$417$$ 1.96170e6 0.552449
$$418$$ −5.26379e6 −1.47353
$$419$$ 3.95178e6 1.09966 0.549828 0.835278i $$-0.314693\pi$$
0.549828 + 0.835278i $$0.314693\pi$$
$$420$$ 0 0
$$421$$ 4.72285e6 1.29867 0.649336 0.760502i $$-0.275047\pi$$
0.649336 + 0.760502i $$0.275047\pi$$
$$422$$ 3.43890e6 0.940023
$$423$$ −2.26526e6 −0.615555
$$424$$ 440974. 0.119124
$$425$$ −1.88823e6 −0.507087
$$426$$ 2.15986e6 0.576636
$$427$$ 0 0
$$428$$ 3.26754e6 0.862207
$$429$$ −3.05417e6 −0.801216
$$430$$ −6.02136e6 −1.57045
$$431$$ 4.07810e6 1.05746 0.528731 0.848790i $$-0.322668\pi$$
0.528731 + 0.848790i $$0.322668\pi$$
$$432$$ −536039. −0.138193
$$433$$ −1.79927e6 −0.461186 −0.230593 0.973050i $$-0.574067\pi$$
−0.230593 + 0.973050i $$0.574067\pi$$
$$434$$ 0 0
$$435$$ −2.47263e6 −0.626522
$$436$$ 1.78425e6 0.449511
$$437$$ −1.35153e6 −0.338550
$$438$$ 1.16249e6 0.289536
$$439$$ 4.51827e6 1.11895 0.559475 0.828847i $$-0.311003\pi$$
0.559475 + 0.828847i $$0.311003\pi$$
$$440$$ 659542. 0.162409
$$441$$ 0 0
$$442$$ 1.00114e7 2.43746
$$443$$ −2.85256e6 −0.690597 −0.345299 0.938493i $$-0.612222\pi$$
−0.345299 + 0.938493i $$0.612222\pi$$
$$444$$ −627802. −0.151135
$$445$$ 1.29922e6 0.311016
$$446$$ −7.44976e6 −1.77339
$$447$$ −584155. −0.138280
$$448$$ 0 0
$$449$$ −1.90246e6 −0.445348 −0.222674 0.974893i $$-0.571478\pi$$
−0.222674 + 0.974893i $$0.571478\pi$$
$$450$$ 1.25130e6 0.291293
$$451$$ −3.53986e6 −0.819491
$$452$$ −13986.2 −0.00321998
$$453$$ −2.01400e6 −0.461119
$$454$$ 9.51468e6 2.16648
$$455$$ 0 0
$$456$$ −1.17625e6 −0.264903
$$457$$ 2.64834e6 0.593176 0.296588 0.955006i $$-0.404151\pi$$
0.296588 + 0.955006i $$0.404151\pi$$
$$458$$ −2.61904e6 −0.583416
$$459$$ 752608. 0.166739
$$460$$ 907763. 0.200022
$$461$$ 1.09031e6 0.238944 0.119472 0.992838i $$-0.461880\pi$$
0.119472 + 0.992838i $$0.461880\pi$$
$$462$$ 0 0
$$463$$ −2.50851e6 −0.543831 −0.271916 0.962321i $$-0.587657\pi$$
−0.271916 + 0.962321i $$0.587657\pi$$
$$464$$ −5.61155e6 −1.21001
$$465$$ 313300. 0.0671937
$$466$$ −9.59943e6 −2.04777
$$467$$ −3.20935e6 −0.680966 −0.340483 0.940251i $$-0.610591\pi$$
−0.340483 + 0.940251i $$0.610591\pi$$
$$468$$ −3.65843e6 −0.772112
$$469$$ 0 0
$$470$$ −8.50350e6 −1.77563
$$471$$ −4.13976e6 −0.859850
$$472$$ 1.29358e6 0.267262
$$473$$ −5.85315e6 −1.20292
$$474$$ 5.55369e6 1.13537
$$475$$ −3.85647e6 −0.784252
$$476$$ 0 0
$$477$$ −576260. −0.115964
$$478$$ −739102. −0.147957
$$479$$ 2.31462e6 0.460936 0.230468 0.973080i $$-0.425974\pi$$
0.230468 + 0.973080i $$0.425974\pi$$
$$480$$ −2.65487e6 −0.525945
$$481$$ 2.03587e6 0.401225
$$482$$ −4.54211e6 −0.890513
$$483$$ 0 0
$$484$$ −2.89885e6 −0.562486
$$485$$ 5.54266e6 1.06995
$$486$$ −498741. −0.0957821
$$487$$ −4.63735e6 −0.886028 −0.443014 0.896515i $$-0.646091\pi$$
−0.443014 + 0.896515i $$0.646091\pi$$
$$488$$ 1.47945e6 0.281224
$$489$$ 819618. 0.155003
$$490$$ 0 0
$$491$$ 5.02151e6 0.940007 0.470003 0.882665i $$-0.344253\pi$$
0.470003 + 0.882665i $$0.344253\pi$$
$$492$$ −4.24021e6 −0.789724
$$493$$ 7.87872e6 1.45995
$$494$$ 2.04470e7 3.76974
$$495$$ −861881. −0.158101
$$496$$ 711024. 0.129772
$$497$$ 0 0
$$498$$ 2.30639e6 0.416735
$$499$$ 3.37822e6 0.607347 0.303673 0.952776i $$-0.401787\pi$$
0.303673 + 0.952776i $$0.401787\pi$$
$$500$$ 7.01582e6 1.25503
$$501$$ −2.83295e6 −0.504249
$$502$$ −1.14322e7 −2.02475
$$503$$ 5.03743e6 0.887747 0.443873 0.896090i $$-0.353604\pi$$
0.443873 + 0.896090i $$0.353604\pi$$
$$504$$ 0 0
$$505$$ −5.03311e6 −0.878230
$$506$$ 1.60019e6 0.277841
$$507$$ 8.52214e6 1.47241
$$508$$ 6.64694e6 1.14278
$$509$$ 6.72466e6 1.15047 0.575236 0.817988i $$-0.304910\pi$$
0.575236 + 0.817988i $$0.304910\pi$$
$$510$$ 2.82520e6 0.480976
$$511$$ 0 0
$$512$$ −7.48361e6 −1.26164
$$513$$ 1.53711e6 0.257876
$$514$$ 8.25112e6 1.37754
$$515$$ 4.17332e6 0.693367
$$516$$ −7.01119e6 −1.15922
$$517$$ −8.26595e6 −1.36009
$$518$$ 0 0
$$519$$ −3.25929e6 −0.531134
$$520$$ −2.56196e6 −0.415493
$$521$$ 4.42770e6 0.714635 0.357317 0.933983i $$-0.383691\pi$$
0.357317 + 0.933983i $$0.383691\pi$$
$$522$$ −5.22110e6 −0.838660
$$523$$ 8.95911e6 1.43222 0.716111 0.697986i $$-0.245920\pi$$
0.716111 + 0.697986i $$0.245920\pi$$
$$524$$ −6.84372e6 −1.08884
$$525$$ 0 0
$$526$$ 1.05050e7 1.65551
$$527$$ −998291. −0.156578
$$528$$ −1.95601e6 −0.305342
$$529$$ −6.02548e6 −0.936165
$$530$$ −2.16321e6 −0.334510
$$531$$ −1.69043e6 −0.260173
$$532$$ 0 0
$$533$$ 1.37504e7 2.09652
$$534$$ 2.74338e6 0.416325
$$535$$ −2.99022e6 −0.451668
$$536$$ −2.14908e6 −0.323103
$$537$$ −1.56507e6 −0.234205
$$538$$ 9.15641e6 1.36386
$$539$$ 0 0
$$540$$ −1.03240e6 −0.152358
$$541$$ −1.00467e7 −1.47581 −0.737907 0.674902i $$-0.764186\pi$$
−0.737907 + 0.674902i $$0.764186\pi$$
$$542$$ 1.82968e7 2.67533
$$543$$ 1.21476e6 0.176804
$$544$$ 8.45941e6 1.22558
$$545$$ −1.63282e6 −0.235477
$$546$$ 0 0
$$547$$ −1.31426e7 −1.87808 −0.939039 0.343811i $$-0.888282\pi$$
−0.939039 + 0.343811i $$0.888282\pi$$
$$548$$ 1.44657e7 2.05774
$$549$$ −1.93333e6 −0.273764
$$550$$ 4.56600e6 0.643620
$$551$$ 1.60913e7 2.25794
$$552$$ 357579. 0.0499487
$$553$$ 0 0
$$554$$ −2.14415e6 −0.296811
$$555$$ 574520. 0.0791722
$$556$$ 8.57452e6 1.17631
$$557$$ 9.06752e6 1.23837 0.619185 0.785245i $$-0.287463\pi$$
0.619185 + 0.785245i $$0.287463\pi$$
$$558$$ 661551. 0.0899452
$$559$$ 2.27363e7 3.07744
$$560$$ 0 0
$$561$$ 2.74627e6 0.368414
$$562$$ −9.63846e6 −1.28726
$$563$$ −1.05180e7 −1.39849 −0.699247 0.714880i $$-0.746481\pi$$
−0.699247 + 0.714880i $$0.746481\pi$$
$$564$$ −9.90136e6 −1.31068
$$565$$ 12799.2 0.00168679
$$566$$ 5.15150e6 0.675916
$$567$$ 0 0
$$568$$ 1.76117e6 0.229050
$$569$$ 7.32307e6 0.948227 0.474114 0.880464i $$-0.342769\pi$$
0.474114 + 0.880464i $$0.342769\pi$$
$$570$$ 5.77010e6 0.743869
$$571$$ −6.97981e6 −0.895887 −0.447943 0.894062i $$-0.647843\pi$$
−0.447943 + 0.894062i $$0.647843\pi$$
$$572$$ −1.33497e7 −1.70600
$$573$$ 1.63272e6 0.207743
$$574$$ 0 0
$$575$$ 1.17237e6 0.147875
$$576$$ −3.70000e6 −0.464670
$$577$$ 5.81210e6 0.726765 0.363382 0.931640i $$-0.381622\pi$$
0.363382 + 0.931640i $$0.381622\pi$$
$$578$$ 2.99030e6 0.372302
$$579$$ 8.69399e6 1.07776
$$580$$ −1.08078e7 −1.33403
$$581$$ 0 0
$$582$$ 1.17036e7 1.43223
$$583$$ −2.10278e6 −0.256226
$$584$$ 947901. 0.115009
$$585$$ 3.34794e6 0.404472
$$586$$ −1.32131e6 −0.158950
$$587$$ 7.37446e6 0.883355 0.441677 0.897174i $$-0.354384\pi$$
0.441677 + 0.897174i $$0.354384\pi$$
$$588$$ 0 0
$$589$$ −2.03888e6 −0.242161
$$590$$ −6.34567e6 −0.750495
$$591$$ −6.29152e6 −0.740947
$$592$$ 1.30385e6 0.152906
$$593$$ −9.46528e6 −1.10534 −0.552671 0.833399i $$-0.686391\pi$$
−0.552671 + 0.833399i $$0.686391\pi$$
$$594$$ −1.81991e6 −0.211633
$$595$$ 0 0
$$596$$ −2.55332e6 −0.294435
$$597$$ −3.74571e6 −0.430129
$$598$$ −6.21589e6 −0.710804
$$599$$ −8.52195e6 −0.970448 −0.485224 0.874390i $$-0.661262\pi$$
−0.485224 + 0.874390i $$0.661262\pi$$
$$600$$ 1.02032e6 0.115706
$$601$$ 657065. 0.0742031 0.0371016 0.999311i $$-0.488187\pi$$
0.0371016 + 0.999311i $$0.488187\pi$$
$$602$$ 0 0
$$603$$ 2.80839e6 0.314532
$$604$$ −8.80310e6 −0.981846
$$605$$ 2.65282e6 0.294659
$$606$$ −1.06277e7 −1.17560
$$607$$ −5.86885e6 −0.646519 −0.323260 0.946310i $$-0.604779\pi$$
−0.323260 + 0.946310i $$0.604779\pi$$
$$608$$ 1.72773e7 1.89547
$$609$$ 0 0
$$610$$ −7.25750e6 −0.789700
$$611$$ 3.21087e7 3.47952
$$612$$ 3.28962e6 0.355032
$$613$$ 3.84402e6 0.413175 0.206588 0.978428i $$-0.433764\pi$$
0.206588 + 0.978428i $$0.433764\pi$$
$$614$$ −2.47667e6 −0.265123
$$615$$ 3.88035e6 0.413698
$$616$$ 0 0
$$617$$ 6.44660e6 0.681739 0.340869 0.940111i $$-0.389279\pi$$
0.340869 + 0.940111i $$0.389279\pi$$
$$618$$ 8.81219e6 0.928138
$$619$$ −6.73740e6 −0.706749 −0.353375 0.935482i $$-0.614966\pi$$
−0.353375 + 0.935482i $$0.614966\pi$$
$$620$$ 1.36942e6 0.143073
$$621$$ −467281. −0.0486238
$$622$$ −2.07115e7 −2.14652
$$623$$ 0 0
$$624$$ 7.59804e6 0.781160
$$625$$ −704759. −0.0721673
$$626$$ −1.55023e7 −1.58110
$$627$$ 5.60891e6 0.569783
$$628$$ −1.80947e7 −1.83085
$$629$$ −1.83063e6 −0.184491
$$630$$ 0 0
$$631$$ −9.14514e6 −0.914360 −0.457180 0.889374i $$-0.651140\pi$$
−0.457180 + 0.889374i $$0.651140\pi$$
$$632$$ 4.52852e6 0.450987
$$633$$ −3.66437e6 −0.363488
$$634$$ −4.98293e6 −0.492336
$$635$$ −6.08282e6 −0.598647
$$636$$ −2.51881e6 −0.246918
$$637$$ 0 0
$$638$$ −1.90518e7 −1.85304
$$639$$ −2.30147e6 −0.222974
$$640$$ −4.44978e6 −0.429426
$$641$$ −1.04088e6 −0.100059 −0.0500296 0.998748i $$-0.515932\pi$$
−0.0500296 + 0.998748i $$0.515932\pi$$
$$642$$ −6.31403e6 −0.604601
$$643$$ 9.08713e6 0.866761 0.433381 0.901211i $$-0.357321\pi$$
0.433381 + 0.901211i $$0.357321\pi$$
$$644$$ 0 0
$$645$$ 6.41615e6 0.607261
$$646$$ −1.83857e7 −1.73340
$$647$$ −2.20711e6 −0.207283 −0.103641 0.994615i $$-0.533049\pi$$
−0.103641 + 0.994615i $$0.533049\pi$$
$$648$$ −406677. −0.0380463
$$649$$ −6.16840e6 −0.574859
$$650$$ −1.77364e7 −1.64658
$$651$$ 0 0
$$652$$ 3.58252e6 0.330042
$$653$$ −1.83610e7 −1.68505 −0.842524 0.538658i $$-0.818931\pi$$
−0.842524 + 0.538658i $$0.818931\pi$$
$$654$$ −3.44780e6 −0.315208
$$655$$ 6.26289e6 0.570390
$$656$$ 8.80632e6 0.798978
$$657$$ −1.23871e6 −0.111958
$$658$$ 0 0
$$659$$ 6.21208e6 0.557216 0.278608 0.960405i $$-0.410127\pi$$
0.278608 + 0.960405i $$0.410127\pi$$
$$660$$ −3.76725e6 −0.336639
$$661$$ −1.54230e7 −1.37298 −0.686491 0.727138i $$-0.740850\pi$$
−0.686491 + 0.727138i $$0.740850\pi$$
$$662$$ −1.49767e6 −0.132822
$$663$$ −1.06678e7 −0.942519
$$664$$ 1.88065e6 0.165534
$$665$$ 0 0
$$666$$ 1.21313e6 0.105980
$$667$$ −4.89176e6 −0.425746
$$668$$ −1.23827e7 −1.07368
$$669$$ 7.93820e6 0.685736
$$670$$ 1.05424e7 0.907300
$$671$$ −7.05475e6 −0.604889
$$672$$ 0 0
$$673$$ −2.27201e7 −1.93362 −0.966811 0.255491i $$-0.917763\pi$$
−0.966811 + 0.255491i $$0.917763\pi$$
$$674$$ 2.57425e7 2.18274
$$675$$ −1.33334e6 −0.112637
$$676$$ 3.72500e7 3.13516
$$677$$ −1.36173e7 −1.14188 −0.570940 0.820992i $$-0.693421\pi$$
−0.570940 + 0.820992i $$0.693421\pi$$
$$678$$ 27026.2 0.00225793
$$679$$ 0 0
$$680$$ 2.30369e6 0.191052
$$681$$ −1.01385e7 −0.837734
$$682$$ 2.41401e6 0.198736
$$683$$ 2.39985e6 0.196849 0.0984245 0.995145i $$-0.468620\pi$$
0.0984245 + 0.995145i $$0.468620\pi$$
$$684$$ 6.71863e6 0.549086
$$685$$ −1.32380e7 −1.07795
$$686$$ 0 0
$$687$$ 2.79076e6 0.225595
$$688$$ 1.45612e7 1.17281
$$689$$ 8.16816e6 0.655505
$$690$$ −1.75411e6 −0.140260
$$691$$ 1.40365e6 0.111831 0.0559156 0.998435i $$-0.482192\pi$$
0.0559156 + 0.998435i $$0.482192\pi$$
$$692$$ −1.42462e7 −1.13093
$$693$$ 0 0
$$694$$ 2.04703e7 1.61334
$$695$$ −7.84680e6 −0.616212
$$696$$ −4.25732e6 −0.333130
$$697$$ −1.23642e7 −0.964018
$$698$$ 2.26097e7 1.75653
$$699$$ 1.02288e7 0.791831
$$700$$ 0 0
$$701$$ −5.78991e6 −0.445017 −0.222509 0.974931i $$-0.571425\pi$$
−0.222509 + 0.974931i $$0.571425\pi$$
$$702$$ 7.06936e6 0.541424
$$703$$ −3.73884e6 −0.285330
$$704$$ −1.35013e7 −1.02670
$$705$$ 9.06103e6 0.686602
$$706$$ −8.02739e6 −0.606126
$$707$$ 0 0
$$708$$ −7.38882e6 −0.553977
$$709$$ −1.13143e7 −0.845304 −0.422652 0.906292i $$-0.638901\pi$$
−0.422652 + 0.906292i $$0.638901\pi$$
$$710$$ −8.63944e6 −0.643191
$$711$$ −5.91782e6 −0.439024
$$712$$ 2.23697e6 0.165371
$$713$$ 619821. 0.0456607
$$714$$ 0 0
$$715$$ 1.22167e7 0.893692
$$716$$ −6.84085e6 −0.498686
$$717$$ 787562. 0.0572119
$$718$$ −2.35549e7 −1.70518
$$719$$ 2.73780e7 1.97506 0.987529 0.157437i $$-0.0503231\pi$$
0.987529 + 0.157437i $$0.0503231\pi$$
$$720$$ 2.14415e6 0.154143
$$721$$ 0 0
$$722$$ −1.66367e7 −1.18775
$$723$$ 4.83992e6 0.344344
$$724$$ 5.30967e6 0.376462
$$725$$ −1.39582e7 −0.986241
$$726$$ 5.60158e6 0.394429
$$727$$ 9.86471e6 0.692226 0.346113 0.938193i $$-0.387501\pi$$
0.346113 + 0.938193i $$0.387501\pi$$
$$728$$ 0 0
$$729$$ 531441. 0.0370370
$$730$$ −4.64995e6 −0.322954
$$731$$ −2.04442e7 −1.41507
$$732$$ −8.45053e6 −0.582916
$$733$$ −3.87876e6 −0.266645 −0.133322 0.991073i $$-0.542565\pi$$
−0.133322 + 0.991073i $$0.542565\pi$$
$$734$$ −1.29972e6 −0.0890448
$$735$$ 0 0
$$736$$ −5.25229e6 −0.357400
$$737$$ 1.02479e7 0.694967
$$738$$ 8.19357e6 0.553774
$$739$$ −7.95498e6 −0.535831 −0.267916 0.963442i $$-0.586335\pi$$
−0.267916 + 0.963442i $$0.586335\pi$$
$$740$$ 2.51121e6 0.168579
$$741$$ −2.17876e7 −1.45768
$$742$$ 0 0
$$743$$ 1.65977e7 1.10300 0.551500 0.834175i $$-0.314056\pi$$
0.551500 + 0.834175i $$0.314056\pi$$
$$744$$ 539433. 0.0357277
$$745$$ 2.33662e6 0.154240
$$746$$ 2.01342e7 1.32461
$$747$$ −2.45761e6 −0.161143
$$748$$ 1.20039e7 0.784453
$$749$$ 0 0
$$750$$ −1.35570e7 −0.880056
$$751$$ 1.51072e7 0.977426 0.488713 0.872445i $$-0.337467\pi$$
0.488713 + 0.872445i $$0.337467\pi$$
$$752$$ 2.05637e7 1.32604
$$753$$ 1.21818e7 0.782929
$$754$$ 7.40061e7 4.74066
$$755$$ 8.05598e6 0.514341
$$756$$ 0 0
$$757$$ 5.80923e6 0.368450 0.184225 0.982884i $$-0.441022\pi$$
0.184225 + 0.982884i $$0.441022\pi$$
$$758$$ −3.08449e7 −1.94989
$$759$$ −1.70511e6 −0.107436
$$760$$ 4.70498e6 0.295477
$$761$$ −2.54270e7 −1.59160 −0.795799 0.605561i $$-0.792949\pi$$
−0.795799 + 0.605561i $$0.792949\pi$$
$$762$$ −1.28442e7 −0.801346
$$763$$ 0 0
$$764$$ 7.13656e6 0.442340
$$765$$ −3.01043e6 −0.185984
$$766$$ −1.82210e7 −1.12202
$$767$$ 2.39609e7 1.47067
$$768$$ 3.75958e6 0.230005
$$769$$ −1.53909e7 −0.938532 −0.469266 0.883057i $$-0.655481\pi$$
−0.469266 + 0.883057i $$0.655481\pi$$
$$770$$ 0 0
$$771$$ −8.79210e6 −0.532668
$$772$$ 3.80011e7 2.29484
$$773$$ −905393. −0.0544990 −0.0272495 0.999629i $$-0.508675\pi$$
−0.0272495 + 0.999629i $$0.508675\pi$$
$$774$$ 1.35481e7 0.812877
$$775$$ 1.76860e6 0.105773
$$776$$ 9.54323e6 0.568907
$$777$$ 0 0
$$778$$ 3.09530e7 1.83338
$$779$$ −2.52523e7 −1.49093
$$780$$ 1.46337e7 0.861229
$$781$$ −8.39810e6 −0.492667
$$782$$ 5.58926e6 0.326841
$$783$$ 5.56342e6 0.324293
$$784$$ 0 0
$$785$$ 1.65590e7 0.959093
$$786$$ 1.32244e7 0.763521
$$787$$ 2.79334e7 1.60763 0.803817 0.594877i $$-0.202799\pi$$
0.803817 + 0.594877i $$0.202799\pi$$
$$788$$ −2.75000e7 −1.57767
$$789$$ −1.11937e7 −0.640151
$$790$$ −2.22148e7 −1.26641
$$791$$ 0 0
$$792$$ −1.48397e6 −0.0840643
$$793$$ 2.74039e7 1.54749
$$794$$ 3.33328e7 1.87638
$$795$$ 2.30504e6 0.129348
$$796$$ −1.63724e7 −0.915860
$$797$$ −2.18824e7 −1.22025 −0.610126 0.792304i $$-0.708881\pi$$
−0.610126 + 0.792304i $$0.708881\pi$$
$$798$$ 0 0
$$799$$ −2.88718e7 −1.59995
$$800$$ −1.49869e7 −0.827918
$$801$$ −2.92325e6 −0.160985
$$802$$ −211291. −0.0115997
$$803$$ −4.52005e6 −0.247374
$$804$$ 1.22754e7 0.669722
$$805$$ 0 0
$$806$$ −9.37710e6 −0.508430
$$807$$ −9.75676e6 −0.527377
$$808$$ −8.66590e6 −0.466966
$$809$$ 2.44194e7 1.31179 0.655893 0.754854i $$-0.272292\pi$$
0.655893 + 0.754854i $$0.272292\pi$$
$$810$$ 1.99496e6 0.106837
$$811$$ −4.46711e6 −0.238492 −0.119246 0.992865i $$-0.538048\pi$$
−0.119246 + 0.992865i $$0.538048\pi$$
$$812$$ 0 0
$$813$$ −1.94964e7 −1.03450
$$814$$ 4.42673e6 0.234165
$$815$$ −3.27847e6 −0.172893
$$816$$ −6.83207e6 −0.359192
$$817$$ −4.17547e7 −2.18852
$$818$$ −7.03317e6 −0.367509
$$819$$ 0 0
$$820$$ 1.69609e7 0.880873
$$821$$ 1.34708e7 0.697485 0.348743 0.937219i $$-0.386609\pi$$
0.348743 + 0.937219i $$0.386609\pi$$
$$822$$ −2.79529e7 −1.44293
$$823$$ −4.03958e6 −0.207891 −0.103946 0.994583i $$-0.533147\pi$$
−0.103946 + 0.994583i $$0.533147\pi$$
$$824$$ 7.18552e6 0.368672
$$825$$ −4.86537e6 −0.248875
$$826$$ 0 0
$$827$$ 1.24927e7 0.635175 0.317588 0.948229i $$-0.397127\pi$$
0.317588 + 0.948229i $$0.397127\pi$$
$$828$$ −2.04247e6 −0.103533
$$829$$ 1.45980e7 0.737749 0.368874 0.929479i $$-0.379743\pi$$
0.368874 + 0.929479i $$0.379743\pi$$
$$830$$ −9.22557e6 −0.464834
$$831$$ 2.28473e6 0.114771
$$832$$ 5.24453e7 2.62663
$$833$$ 0 0
$$834$$ −1.65690e7 −0.824859
$$835$$ 1.13318e7 0.562449
$$836$$ 2.45163e7 1.21322
$$837$$ −704926. −0.0347800
$$838$$ −3.33776e7 −1.64189
$$839$$ 9.15983e6 0.449244 0.224622 0.974446i $$-0.427885\pi$$
0.224622 + 0.974446i $$0.427885\pi$$
$$840$$ 0 0
$$841$$ 3.77299e7 1.83948
$$842$$ −3.98903e7 −1.93904
$$843$$ 1.02704e7 0.497759
$$844$$ −1.60168e7 −0.773964
$$845$$ −3.40886e7 −1.64236
$$846$$ 1.91329e7 0.919082
$$847$$ 0 0
$$848$$ 5.23121e6 0.249812
$$849$$ −5.48926e6 −0.261363
$$850$$ 1.59484e7 0.757129
$$851$$ 1.13661e6 0.0538005
$$852$$ −1.00597e7 −0.474771
$$853$$ 8.68253e6 0.408577 0.204289 0.978911i $$-0.434512\pi$$
0.204289 + 0.978911i $$0.434512\pi$$
$$854$$ 0 0
$$855$$ −6.14842e6 −0.287639
$$856$$ −5.14850e6 −0.240158
$$857$$ −1.04988e7 −0.488302 −0.244151 0.969737i $$-0.578509\pi$$
−0.244151 + 0.969737i $$0.578509\pi$$
$$858$$ 2.57962e7 1.19629
$$859$$ −9.03780e6 −0.417907 −0.208954 0.977926i $$-0.567006\pi$$
−0.208954 + 0.977926i $$0.567006\pi$$
$$860$$ 2.80448e7 1.29302
$$861$$ 0 0
$$862$$ −3.44445e7 −1.57889
$$863$$ −1.59858e7 −0.730645 −0.365322 0.930881i $$-0.619041\pi$$
−0.365322 + 0.930881i $$0.619041\pi$$
$$864$$ 5.97346e6 0.272234
$$865$$ 1.30371e7 0.592437
$$866$$ 1.51970e7 0.688594
$$867$$ −3.18636e6 −0.143962
$$868$$ 0 0
$$869$$ −2.15942e7 −0.970035
$$870$$ 2.08844e7 0.935457
$$871$$ −3.98073e7 −1.77794
$$872$$ −2.81136e6 −0.125206
$$873$$ −1.24710e7 −0.553816
$$874$$ 1.14153e7 0.505487
$$875$$ 0 0
$$876$$ −5.41434e6 −0.238388
$$877$$ 2.67453e7 1.17422 0.587109 0.809508i $$-0.300266\pi$$
0.587109 + 0.809508i $$0.300266\pi$$
$$878$$ −3.81623e7 −1.67070
$$879$$ 1.40794e6 0.0614628
$$880$$ 7.82404e6 0.340584
$$881$$ 6.40715e6 0.278115 0.139058 0.990284i $$-0.455593\pi$$
0.139058 + 0.990284i $$0.455593\pi$$
$$882$$ 0 0
$$883$$ 4.96462e6 0.214281 0.107141 0.994244i $$-0.465831\pi$$
0.107141 + 0.994244i $$0.465831\pi$$
$$884$$ −4.66285e7 −2.00688
$$885$$ 6.76173e6 0.290201
$$886$$ 2.40933e7 1.03113
$$887$$ −8.71625e6 −0.371981 −0.185990 0.982552i $$-0.559549\pi$$
−0.185990 + 0.982552i $$0.559549\pi$$
$$888$$ 989196. 0.0420969
$$889$$ 0 0
$$890$$ −1.09735e7 −0.464377
$$891$$ 1.93923e6 0.0818344
$$892$$ 3.46976e7 1.46011
$$893$$ −5.89669e7 −2.47446
$$894$$ 4.93390e6 0.206465
$$895$$ 6.26026e6 0.261237
$$896$$ 0 0
$$897$$ 6.62343e6 0.274854
$$898$$ 1.60686e7 0.664946
$$899$$ −7.37956e6 −0.304531
$$900$$ −5.82798e6 −0.239835
$$901$$ −7.34472e6 −0.301414
$$902$$ 2.98984e7 1.22358
$$903$$ 0 0
$$904$$ 22037.4 0.000896888 0
$$905$$ −4.85904e6 −0.197210
$$906$$ 1.70107e7 0.688494
$$907$$ −2.36255e7 −0.953594 −0.476797 0.879014i $$-0.658202\pi$$
−0.476797 + 0.879014i $$0.658202\pi$$
$$908$$ −4.43151e7 −1.78376
$$909$$ 1.13245e7 0.454579
$$910$$ 0 0
$$911$$ −1.69360e7 −0.676105 −0.338053 0.941127i $$-0.609768\pi$$
−0.338053 + 0.941127i $$0.609768\pi$$
$$912$$ −1.39536e7 −0.555520
$$913$$ −8.96785e6 −0.356050
$$914$$ −2.23685e7 −0.885668
$$915$$ 7.73334e6 0.305361
$$916$$ 1.21983e7 0.480353
$$917$$ 0 0
$$918$$ −6.35669e6 −0.248957
$$919$$ −3.13804e7 −1.22566 −0.612829 0.790216i $$-0.709968\pi$$
−0.612829 + 0.790216i $$0.709968\pi$$
$$920$$ −1.43032e6 −0.0557138
$$921$$ 2.63906e6 0.102518
$$922$$ −9.20896e6 −0.356766
$$923$$ 3.26220e7 1.26040
$$924$$ 0 0
$$925$$ 3.24320e6 0.124629
$$926$$ 2.11875e7 0.811992
$$927$$ −9.38996e6 −0.358893
$$928$$ 6.25336e7 2.38365
$$929$$ 1.43089e7 0.543959 0.271980 0.962303i $$-0.412322\pi$$
0.271980 + 0.962303i $$0.412322\pi$$
$$930$$ −2.64620e6 −0.100327
$$931$$ 0 0
$$932$$ 4.47098e7 1.68602
$$933$$ 2.20694e7 0.830017
$$934$$ 2.71069e7 1.01675
$$935$$ −1.09851e7 −0.410937
$$936$$ 5.76441e6 0.215063
$$937$$ −2.81206e7 −1.04635 −0.523173 0.852227i $$-0.675252\pi$$
−0.523173 + 0.852227i $$0.675252\pi$$
$$938$$ 0 0
$$939$$ 1.65187e7 0.611381
$$940$$ 3.96054e7 1.46196
$$941$$ 3.29569e7 1.21331 0.606656 0.794964i $$-0.292510\pi$$
0.606656 + 0.794964i $$0.292510\pi$$
$$942$$ 3.49653e7 1.28384
$$943$$ 7.67672e6 0.281123
$$944$$ 1.53455e7 0.560469
$$945$$ 0 0
$$946$$ 4.94370e7 1.79607
$$947$$ 1.62975e7 0.590535 0.295268 0.955415i $$-0.404591\pi$$
0.295268 + 0.955415i $$0.404591\pi$$
$$948$$ −2.58666e7 −0.934799
$$949$$ 1.75579e7 0.632861
$$950$$ 3.25726e7 1.17096
$$951$$ 5.30964e6 0.190377
$$952$$ 0 0
$$953$$ −3.03230e7 −1.08153 −0.540767 0.841172i $$-0.681866\pi$$
−0.540767 + 0.841172i $$0.681866\pi$$
$$954$$ 4.86722e6 0.173145
$$955$$ −6.53088e6 −0.231720
$$956$$ 3.44240e6 0.121820
$$957$$ 2.03010e7 0.716535
$$958$$ −1.95498e7 −0.688221
$$959$$ 0 0
$$960$$ 1.48000e7 0.518302
$$961$$ −2.76941e7 −0.967339
$$962$$ −1.71954e7 −0.599067
$$963$$ 6.72801e6 0.233787
$$964$$ 2.11551e7 0.733200
$$965$$ −3.47759e7 −1.20216
$$966$$ 0 0
$$967$$ −1.49059e7 −0.512616 −0.256308 0.966595i $$-0.582506\pi$$
−0.256308 + 0.966595i $$0.582506\pi$$
$$968$$ 4.56757e6 0.156674
$$969$$ 1.95911e7 0.670271
$$970$$ −4.68145e7 −1.59754
$$971$$ −1.65608e7 −0.563679 −0.281840 0.959462i $$-0.590945\pi$$
−0.281840 + 0.959462i $$0.590945\pi$$
$$972$$ 2.32291e6 0.0788618
$$973$$ 0 0
$$974$$ 3.91681e7 1.32292
$$975$$ 1.88993e7 0.636700
$$976$$ 1.75505e7 0.589747
$$977$$ 7.56862e6 0.253676 0.126838 0.991923i $$-0.459517\pi$$
0.126838 + 0.991923i $$0.459517\pi$$
$$978$$ −6.92267e6 −0.231434
$$979$$ −1.06670e7 −0.355700
$$980$$ 0 0
$$981$$ 3.67386e6 0.121885
$$982$$ −4.24128e7 −1.40352
$$983$$ 3.32042e7 1.09600 0.547998 0.836480i $$-0.315390\pi$$
0.547998 + 0.836480i $$0.315390\pi$$
$$984$$ 6.68110e6 0.219968
$$985$$ 2.51661e7 0.826466
$$986$$ −6.65454e7 −2.17985
$$987$$ 0 0
$$988$$ −9.52327e7 −3.10380
$$989$$ 1.26935e7 0.412657
$$990$$ 7.27964e6 0.236060
$$991$$ 3.48003e7 1.12564 0.562819 0.826580i $$-0.309717\pi$$
0.562819 + 0.826580i $$0.309717\pi$$
$$992$$ −7.92345e6 −0.255644
$$993$$ 1.59586e6 0.0513598
$$994$$ 0 0
$$995$$ 1.49829e7 0.479774
$$996$$ −1.07421e7 −0.343117
$$997$$ −9.40852e6 −0.299767 −0.149883 0.988704i $$-0.547890\pi$$
−0.149883 + 0.988704i $$0.547890\pi$$
$$998$$ −2.85332e7 −0.906826
$$999$$ −1.29267e6 −0.0409802
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 147.6.a.j.1.1 yes 2
3.2 odd 2 441.6.a.r.1.2 2
7.2 even 3 147.6.e.m.67.2 4
7.3 odd 6 147.6.e.n.79.2 4
7.4 even 3 147.6.e.m.79.2 4
7.5 odd 6 147.6.e.n.67.2 4
7.6 odd 2 147.6.a.h.1.1 2
21.20 even 2 441.6.a.q.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
147.6.a.h.1.1 2 7.6 odd 2
147.6.a.j.1.1 yes 2 1.1 even 1 trivial
147.6.e.m.67.2 4 7.2 even 3
147.6.e.m.79.2 4 7.4 even 3
147.6.e.n.67.2 4 7.5 odd 6
147.6.e.n.79.2 4 7.3 odd 6
441.6.a.q.1.2 2 21.20 even 2
441.6.a.r.1.2 2 3.2 odd 2