Properties

 Label 17.4.a.b.1.1 Level $17$ Weight $4$ Character 17.1 Self dual yes Analytic conductor $1.003$ Analytic rank $0$ 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] = [17,4,Mod(1,17)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(17, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("17.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$17$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 17.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: $$1.00303247010$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.2636.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - 14x - 4$$ x^3 - 14*x - 4 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

 Embedding label 1.1 Root $$-3.58966$$ of defining polynomial Character $$\chi$$ $$=$$ 17.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-5.03251 q^{2} +8.47535 q^{3} +17.3261 q^{4} +0.885690 q^{5} -42.6523 q^{6} +3.81828 q^{7} -46.9339 q^{8} +44.8316 q^{9} +O(q^{10})$$ $$q-5.03251 q^{2} +8.47535 q^{3} +17.3261 q^{4} +0.885690 q^{5} -42.6523 q^{6} +3.81828 q^{7} -46.9339 q^{8} +44.8316 q^{9} -4.45724 q^{10} -52.3720 q^{11} +146.845 q^{12} -8.06025 q^{13} -19.2156 q^{14} +7.50653 q^{15} +97.5862 q^{16} -17.0000 q^{17} -225.616 q^{18} -66.5154 q^{19} +15.3456 q^{20} +32.3613 q^{21} +263.563 q^{22} +180.226 q^{23} -397.782 q^{24} -124.216 q^{25} +40.5633 q^{26} +151.129 q^{27} +66.1562 q^{28} -41.2800 q^{29} -37.7767 q^{30} -34.9114 q^{31} -115.632 q^{32} -443.871 q^{33} +85.5527 q^{34} +3.38182 q^{35} +776.759 q^{36} +130.368 q^{37} +334.739 q^{38} -68.3134 q^{39} -41.5689 q^{40} -17.9081 q^{41} -162.859 q^{42} +277.620 q^{43} -907.405 q^{44} +39.7069 q^{45} -906.987 q^{46} +463.789 q^{47} +827.078 q^{48} -328.421 q^{49} +625.116 q^{50} -144.081 q^{51} -139.653 q^{52} -329.944 q^{53} -760.560 q^{54} -46.3853 q^{55} -179.207 q^{56} -563.741 q^{57} +207.742 q^{58} +678.656 q^{59} +130.059 q^{60} +340.280 q^{61} +175.692 q^{62} +171.180 q^{63} -198.770 q^{64} -7.13888 q^{65} +2233.79 q^{66} +15.3925 q^{67} -294.545 q^{68} +1527.48 q^{69} -17.0190 q^{70} -670.203 q^{71} -2104.12 q^{72} +193.480 q^{73} -656.080 q^{74} -1052.77 q^{75} -1152.46 q^{76} -199.971 q^{77} +343.788 q^{78} +1080.15 q^{79} +86.4311 q^{80} +70.4207 q^{81} +90.1229 q^{82} -865.668 q^{83} +560.697 q^{84} -15.0567 q^{85} -1397.13 q^{86} -349.863 q^{87} +2458.02 q^{88} +1129.46 q^{89} -199.825 q^{90} -30.7763 q^{91} +3122.61 q^{92} -295.886 q^{93} -2334.02 q^{94} -58.9120 q^{95} -980.023 q^{96} -379.412 q^{97} +1652.78 q^{98} -2347.92 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + q^{2} + 4 q^{3} + 25 q^{4} - 8 q^{5} - 74 q^{6} + 22 q^{7} - 39 q^{8} + 59 q^{9}+O(q^{10})$$ 3 * q + q^2 + 4 * q^3 + 25 * q^4 - 8 * q^5 - 74 * q^6 + 22 * q^7 - 39 * q^8 + 59 * q^9 $$3 q + q^{2} + 4 q^{3} + 25 q^{4} - 8 q^{5} - 74 q^{6} + 22 q^{7} - 39 q^{8} + 59 q^{9} - 56 q^{10} - 28 q^{11} + 22 q^{12} + 30 q^{13} + 92 q^{14} + 108 q^{15} + 137 q^{16} - 51 q^{17} - 103 q^{18} + 80 q^{19} - 168 q^{20} - 192 q^{21} + 286 q^{22} + 142 q^{23} - 666 q^{24} - 223 q^{25} + 26 q^{26} - 20 q^{27} + 476 q^{28} - 456 q^{29} + 400 q^{30} + 230 q^{31} - 71 q^{32} - 332 q^{33} - 17 q^{34} - 332 q^{35} + 1313 q^{36} + 356 q^{37} + 724 q^{38} + 268 q^{39} - 424 q^{40} - 294 q^{41} - 1128 q^{42} + 556 q^{43} - 1122 q^{44} - 384 q^{45} - 704 q^{46} + 640 q^{47} + 774 q^{48} - 269 q^{49} + 547 q^{50} - 68 q^{51} - 774 q^{52} + 302 q^{53} - 1100 q^{54} + 76 q^{55} + 684 q^{56} - 720 q^{57} - 1304 q^{58} + 636 q^{59} + 1328 q^{60} - 84 q^{61} + 508 q^{62} + 1122 q^{63} - 919 q^{64} + 408 q^{65} + 2468 q^{66} + 1008 q^{67} - 425 q^{68} + 576 q^{69} - 1504 q^{70} - 402 q^{71} - 927 q^{72} + 838 q^{73} + 836 q^{74} - 1548 q^{75} - 908 q^{76} - 504 q^{77} + 1308 q^{78} - 594 q^{79} - 40 q^{80} - 505 q^{81} + 358 q^{82} - 2396 q^{83} - 2040 q^{84} + 136 q^{85} - 1264 q^{86} + 1428 q^{87} + 1838 q^{88} - 170 q^{89} - 2008 q^{90} - 1016 q^{91} + 4896 q^{92} + 632 q^{93} - 2016 q^{94} - 472 q^{95} + 678 q^{96} - 270 q^{97} + 2857 q^{98} - 2920 q^{99}+O(q^{100})$$ 3 * q + q^2 + 4 * q^3 + 25 * q^4 - 8 * q^5 - 74 * q^6 + 22 * q^7 - 39 * q^8 + 59 * q^9 - 56 * q^10 - 28 * q^11 + 22 * q^12 + 30 * q^13 + 92 * q^14 + 108 * q^15 + 137 * q^16 - 51 * q^17 - 103 * q^18 + 80 * q^19 - 168 * q^20 - 192 * q^21 + 286 * q^22 + 142 * q^23 - 666 * q^24 - 223 * q^25 + 26 * q^26 - 20 * q^27 + 476 * q^28 - 456 * q^29 + 400 * q^30 + 230 * q^31 - 71 * q^32 - 332 * q^33 - 17 * q^34 - 332 * q^35 + 1313 * q^36 + 356 * q^37 + 724 * q^38 + 268 * q^39 - 424 * q^40 - 294 * q^41 - 1128 * q^42 + 556 * q^43 - 1122 * q^44 - 384 * q^45 - 704 * q^46 + 640 * q^47 + 774 * q^48 - 269 * q^49 + 547 * q^50 - 68 * q^51 - 774 * q^52 + 302 * q^53 - 1100 * q^54 + 76 * q^55 + 684 * q^56 - 720 * q^57 - 1304 * q^58 + 636 * q^59 + 1328 * q^60 - 84 * q^61 + 508 * q^62 + 1122 * q^63 - 919 * q^64 + 408 * q^65 + 2468 * q^66 + 1008 * q^67 - 425 * q^68 + 576 * q^69 - 1504 * q^70 - 402 * q^71 - 927 * q^72 + 838 * q^73 + 836 * q^74 - 1548 * q^75 - 908 * q^76 - 504 * q^77 + 1308 * q^78 - 594 * q^79 - 40 * q^80 - 505 * q^81 + 358 * q^82 - 2396 * q^83 - 2040 * q^84 + 136 * q^85 - 1264 * q^86 + 1428 * q^87 + 1838 * q^88 - 170 * q^89 - 2008 * q^90 - 1016 * q^91 + 4896 * q^92 + 632 * q^93 - 2016 * q^94 - 472 * q^95 + 678 * q^96 - 270 * q^97 + 2857 * q^98 - 2920 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −5.03251 −1.77926 −0.889630 0.456681i $$-0.849038\pi$$
−0.889630 + 0.456681i $$0.849038\pi$$
$$3$$ 8.47535 1.63108 0.815541 0.578699i $$-0.196439\pi$$
0.815541 + 0.578699i $$0.196439\pi$$
$$4$$ 17.3261 2.16577
$$5$$ 0.885690 0.0792185 0.0396092 0.999215i $$-0.487389\pi$$
0.0396092 + 0.999215i $$0.487389\pi$$
$$6$$ −42.6523 −2.90212
$$7$$ 3.81828 0.206168 0.103084 0.994673i $$-0.467129\pi$$
0.103084 + 0.994673i $$0.467129\pi$$
$$8$$ −46.9339 −2.07421
$$9$$ 44.8316 1.66043
$$10$$ −4.45724 −0.140950
$$11$$ −52.3720 −1.43552 −0.717761 0.696289i $$-0.754833\pi$$
−0.717761 + 0.696289i $$0.754833\pi$$
$$12$$ 146.845 3.53255
$$13$$ −8.06025 −0.171962 −0.0859811 0.996297i $$-0.527402\pi$$
−0.0859811 + 0.996297i $$0.527402\pi$$
$$14$$ −19.2156 −0.366827
$$15$$ 7.50653 0.129212
$$16$$ 97.5862 1.52478
$$17$$ −17.0000 −0.242536
$$18$$ −225.616 −2.95434
$$19$$ −66.5154 −0.803141 −0.401570 0.915828i $$-0.631535\pi$$
−0.401570 + 0.915828i $$0.631535\pi$$
$$20$$ 15.3456 0.171569
$$21$$ 32.3613 0.336277
$$22$$ 263.563 2.55417
$$23$$ 180.226 1.63390 0.816948 0.576711i $$-0.195664\pi$$
0.816948 + 0.576711i $$0.195664\pi$$
$$24$$ −397.782 −3.38320
$$25$$ −124.216 −0.993724
$$26$$ 40.5633 0.305966
$$27$$ 151.129 1.07722
$$28$$ 66.1562 0.446512
$$29$$ −41.2800 −0.264328 −0.132164 0.991228i $$-0.542193\pi$$
−0.132164 + 0.991228i $$0.542193\pi$$
$$30$$ −37.7767 −0.229902
$$31$$ −34.9114 −0.202267 −0.101133 0.994873i $$-0.532247\pi$$
−0.101133 + 0.994873i $$0.532247\pi$$
$$32$$ −115.632 −0.638783
$$33$$ −443.871 −2.34146
$$34$$ 85.5527 0.431534
$$35$$ 3.38182 0.0163323
$$36$$ 776.759 3.59611
$$37$$ 130.368 0.579255 0.289627 0.957139i $$-0.406469\pi$$
0.289627 + 0.957139i $$0.406469\pi$$
$$38$$ 334.739 1.42900
$$39$$ −68.3134 −0.280485
$$40$$ −41.5689 −0.164315
$$41$$ −17.9081 −0.0682142 −0.0341071 0.999418i $$-0.510859\pi$$
−0.0341071 + 0.999418i $$0.510859\pi$$
$$42$$ −162.859 −0.598325
$$43$$ 277.620 0.984573 0.492287 0.870433i $$-0.336161\pi$$
0.492287 + 0.870433i $$0.336161\pi$$
$$44$$ −907.405 −3.10901
$$45$$ 39.7069 0.131537
$$46$$ −906.987 −2.90713
$$47$$ 463.789 1.43937 0.719687 0.694299i $$-0.244285\pi$$
0.719687 + 0.694299i $$0.244285\pi$$
$$48$$ 827.078 2.48705
$$49$$ −328.421 −0.957495
$$50$$ 625.116 1.76809
$$51$$ −144.081 −0.395596
$$52$$ −139.653 −0.372431
$$53$$ −329.944 −0.855118 −0.427559 0.903987i $$-0.640626\pi$$
−0.427559 + 0.903987i $$0.640626\pi$$
$$54$$ −760.560 −1.91665
$$55$$ −46.3853 −0.113720
$$56$$ −179.207 −0.427635
$$57$$ −563.741 −1.30999
$$58$$ 207.742 0.470308
$$59$$ 678.656 1.49752 0.748759 0.662843i $$-0.230650\pi$$
0.748759 + 0.662843i $$0.230650\pi$$
$$60$$ 130.059 0.279843
$$61$$ 340.280 0.714237 0.357118 0.934059i $$-0.383759\pi$$
0.357118 + 0.934059i $$0.383759\pi$$
$$62$$ 175.692 0.359885
$$63$$ 171.180 0.342328
$$64$$ −198.770 −0.388223
$$65$$ −7.13888 −0.0136226
$$66$$ 2233.79 4.16606
$$67$$ 15.3925 0.0280671 0.0140336 0.999902i $$-0.495533\pi$$
0.0140336 + 0.999902i $$0.495533\pi$$
$$68$$ −294.545 −0.525276
$$69$$ 1527.48 2.66502
$$70$$ −17.0190 −0.0290595
$$71$$ −670.203 −1.12026 −0.560130 0.828405i $$-0.689249\pi$$
−0.560130 + 0.828405i $$0.689249\pi$$
$$72$$ −2104.12 −3.44408
$$73$$ 193.480 0.310207 0.155103 0.987898i $$-0.450429\pi$$
0.155103 + 0.987898i $$0.450429\pi$$
$$74$$ −656.080 −1.03065
$$75$$ −1052.77 −1.62085
$$76$$ −1152.46 −1.73942
$$77$$ −199.971 −0.295959
$$78$$ 343.788 0.499055
$$79$$ 1080.15 1.53831 0.769156 0.639061i $$-0.220677\pi$$
0.769156 + 0.639061i $$0.220677\pi$$
$$80$$ 86.4311 0.120791
$$81$$ 70.4207 0.0965990
$$82$$ 90.1229 0.121371
$$83$$ −865.668 −1.14481 −0.572406 0.819970i $$-0.693990\pi$$
−0.572406 + 0.819970i $$0.693990\pi$$
$$84$$ 560.697 0.728298
$$85$$ −15.0567 −0.0192133
$$86$$ −1397.13 −1.75181
$$87$$ −349.863 −0.431141
$$88$$ 2458.02 2.97757
$$89$$ 1129.46 1.34520 0.672599 0.740008i $$-0.265178\pi$$
0.672599 + 0.740008i $$0.265178\pi$$
$$90$$ −199.825 −0.234038
$$91$$ −30.7763 −0.0354531
$$92$$ 3122.61 3.53864
$$93$$ −295.886 −0.329914
$$94$$ −2334.02 −2.56102
$$95$$ −58.9120 −0.0636236
$$96$$ −980.023 −1.04191
$$97$$ −379.412 −0.397149 −0.198574 0.980086i $$-0.563631\pi$$
−0.198574 + 0.980086i $$0.563631\pi$$
$$98$$ 1652.78 1.70363
$$99$$ −2347.92 −2.38359
$$100$$ −2152.18 −2.15218
$$101$$ 131.732 0.129780 0.0648902 0.997892i $$-0.479330\pi$$
0.0648902 + 0.997892i $$0.479330\pi$$
$$102$$ 725.089 0.703868
$$103$$ 195.988 0.187488 0.0937442 0.995596i $$-0.470116\pi$$
0.0937442 + 0.995596i $$0.470116\pi$$
$$104$$ 378.299 0.356685
$$105$$ 28.6621 0.0266394
$$106$$ 1660.45 1.52148
$$107$$ −485.147 −0.438326 −0.219163 0.975688i $$-0.570333\pi$$
−0.219163 + 0.975688i $$0.570333\pi$$
$$108$$ 2618.49 2.33300
$$109$$ −1255.12 −1.10292 −0.551460 0.834201i $$-0.685929\pi$$
−0.551460 + 0.834201i $$0.685929\pi$$
$$110$$ 233.435 0.202337
$$111$$ 1104.92 0.944812
$$112$$ 372.612 0.314362
$$113$$ −1013.35 −0.843612 −0.421806 0.906686i $$-0.638604\pi$$
−0.421806 + 0.906686i $$0.638604\pi$$
$$114$$ 2837.03 2.33081
$$115$$ 159.624 0.129435
$$116$$ −715.224 −0.572473
$$117$$ −361.354 −0.285531
$$118$$ −3415.34 −2.66447
$$119$$ −64.9108 −0.0500031
$$120$$ −352.311 −0.268012
$$121$$ 1411.83 1.06073
$$122$$ −1712.46 −1.27081
$$123$$ −151.778 −0.111263
$$124$$ −604.880 −0.438063
$$125$$ −220.728 −0.157940
$$126$$ −861.464 −0.609090
$$127$$ 1927.72 1.34691 0.673456 0.739227i $$-0.264809\pi$$
0.673456 + 0.739227i $$0.264809\pi$$
$$128$$ 1925.37 1.32953
$$129$$ 2352.93 1.60592
$$130$$ 35.9265 0.0242381
$$131$$ −406.738 −0.271274 −0.135637 0.990759i $$-0.543308\pi$$
−0.135637 + 0.990759i $$0.543308\pi$$
$$132$$ −7690.58 −5.07105
$$133$$ −253.975 −0.165582
$$134$$ −77.4631 −0.0499387
$$135$$ 133.854 0.0853355
$$136$$ 797.877 0.503069
$$137$$ −130.552 −0.0814149 −0.0407074 0.999171i $$-0.512961\pi$$
−0.0407074 + 0.999171i $$0.512961\pi$$
$$138$$ −7687.03 −4.74177
$$139$$ 2073.54 1.26529 0.632644 0.774443i $$-0.281970\pi$$
0.632644 + 0.774443i $$0.281970\pi$$
$$140$$ 58.5938 0.0353720
$$141$$ 3930.78 2.34774
$$142$$ 3372.80 1.99323
$$143$$ 422.131 0.246856
$$144$$ 4374.95 2.53180
$$145$$ −36.5613 −0.0209397
$$146$$ −973.689 −0.551939
$$147$$ −2783.48 −1.56175
$$148$$ 2258.78 1.25453
$$149$$ −1852.73 −1.01867 −0.509334 0.860569i $$-0.670108\pi$$
−0.509334 + 0.860569i $$0.670108\pi$$
$$150$$ 5298.08 2.88391
$$151$$ 2050.86 1.10527 0.552637 0.833422i $$-0.313622\pi$$
0.552637 + 0.833422i $$0.313622\pi$$
$$152$$ 3121.83 1.66588
$$153$$ −762.138 −0.402714
$$154$$ 1006.36 0.526588
$$155$$ −30.9207 −0.0160233
$$156$$ −1183.61 −0.607465
$$157$$ −262.991 −0.133688 −0.0668438 0.997763i $$-0.521293\pi$$
−0.0668438 + 0.997763i $$0.521293\pi$$
$$158$$ −5435.88 −2.73706
$$159$$ −2796.39 −1.39477
$$160$$ −102.414 −0.0506035
$$161$$ 688.152 0.336857
$$162$$ −354.393 −0.171875
$$163$$ −1444.98 −0.694354 −0.347177 0.937800i $$-0.612860\pi$$
−0.347177 + 0.937800i $$0.612860\pi$$
$$164$$ −310.279 −0.147736
$$165$$ −393.132 −0.185487
$$166$$ 4356.48 2.03692
$$167$$ −501.565 −0.232409 −0.116204 0.993225i $$-0.537073\pi$$
−0.116204 + 0.993225i $$0.537073\pi$$
$$168$$ −1518.84 −0.697508
$$169$$ −2132.03 −0.970429
$$170$$ 75.7731 0.0341855
$$171$$ −2981.99 −1.33356
$$172$$ 4810.08 2.13236
$$173$$ −2590.14 −1.13829 −0.569146 0.822237i $$-0.692726\pi$$
−0.569146 + 0.822237i $$0.692726\pi$$
$$174$$ 1760.69 0.767112
$$175$$ −474.290 −0.204874
$$176$$ −5110.79 −2.18886
$$177$$ 5751.85 2.44257
$$178$$ −5684.02 −2.39346
$$179$$ 2165.65 0.904294 0.452147 0.891943i $$-0.350658\pi$$
0.452147 + 0.891943i $$0.350658\pi$$
$$180$$ 687.968 0.284878
$$181$$ −1925.56 −0.790750 −0.395375 0.918520i $$-0.629385\pi$$
−0.395375 + 0.918520i $$0.629385\pi$$
$$182$$ 154.882 0.0630803
$$183$$ 2884.00 1.16498
$$184$$ −8458.69 −3.38904
$$185$$ 115.466 0.0458877
$$186$$ 1489.05 0.587003
$$187$$ 890.324 0.348165
$$188$$ 8035.68 3.11735
$$189$$ 577.055 0.222088
$$190$$ 296.475 0.113203
$$191$$ −2783.52 −1.05449 −0.527247 0.849712i $$-0.676776\pi$$
−0.527247 + 0.849712i $$0.676776\pi$$
$$192$$ −1684.65 −0.633223
$$193$$ 2258.27 0.842246 0.421123 0.907004i $$-0.361636\pi$$
0.421123 + 0.907004i $$0.361636\pi$$
$$194$$ 1909.39 0.706631
$$195$$ −60.5045 −0.0222196
$$196$$ −5690.27 −2.07371
$$197$$ −1270.70 −0.459560 −0.229780 0.973243i $$-0.573801\pi$$
−0.229780 + 0.973243i $$0.573801\pi$$
$$198$$ 11815.9 4.24102
$$199$$ −4794.36 −1.70786 −0.853928 0.520392i $$-0.825786\pi$$
−0.853928 + 0.520392i $$0.825786\pi$$
$$200$$ 5829.92 2.06119
$$201$$ 130.457 0.0457798
$$202$$ −662.942 −0.230913
$$203$$ −157.619 −0.0544960
$$204$$ −2496.37 −0.856769
$$205$$ −15.8611 −0.00540383
$$206$$ −986.313 −0.333591
$$207$$ 8079.80 2.71297
$$208$$ −786.569 −0.262205
$$209$$ 3483.54 1.15293
$$210$$ −144.242 −0.0473984
$$211$$ −2807.00 −0.915837 −0.457918 0.888994i $$-0.651405\pi$$
−0.457918 + 0.888994i $$0.651405\pi$$
$$212$$ −5716.66 −1.85199
$$213$$ −5680.21 −1.82724
$$214$$ 2441.50 0.779896
$$215$$ 245.885 0.0779964
$$216$$ −7093.09 −2.23437
$$217$$ −133.302 −0.0417009
$$218$$ 6316.38 1.96238
$$219$$ 1639.81 0.505973
$$220$$ −803.679 −0.246291
$$221$$ 137.024 0.0417070
$$222$$ −5560.51 −1.68107
$$223$$ 4684.30 1.40665 0.703327 0.710866i $$-0.251697\pi$$
0.703327 + 0.710866i $$0.251697\pi$$
$$224$$ −441.516 −0.131697
$$225$$ −5568.79 −1.65001
$$226$$ 5099.70 1.50101
$$227$$ −1395.72 −0.408095 −0.204047 0.978961i $$-0.565410\pi$$
−0.204047 + 0.978961i $$0.565410\pi$$
$$228$$ −9767.47 −2.83713
$$229$$ 894.638 0.258163 0.129082 0.991634i $$-0.458797\pi$$
0.129082 + 0.991634i $$0.458797\pi$$
$$230$$ −803.309 −0.230298
$$231$$ −1694.83 −0.482733
$$232$$ 1937.43 0.548270
$$233$$ 1196.13 0.336313 0.168156 0.985760i $$-0.446219\pi$$
0.168156 + 0.985760i $$0.446219\pi$$
$$234$$ 1818.52 0.508035
$$235$$ 410.773 0.114025
$$236$$ 11758.5 3.24328
$$237$$ 9154.67 2.50911
$$238$$ 326.664 0.0889685
$$239$$ 4948.82 1.33938 0.669691 0.742639i $$-0.266426\pi$$
0.669691 + 0.742639i $$0.266426\pi$$
$$240$$ 732.534 0.197020
$$241$$ −6702.73 −1.79154 −0.895770 0.444518i $$-0.853375\pi$$
−0.895770 + 0.444518i $$0.853375\pi$$
$$242$$ −7105.03 −1.88731
$$243$$ −3483.65 −0.919656
$$244$$ 5895.75 1.54687
$$245$$ −290.879 −0.0758513
$$246$$ 763.824 0.197966
$$247$$ 536.130 0.138110
$$248$$ 1638.53 0.419543
$$249$$ −7336.85 −1.86728
$$250$$ 1110.81 0.281016
$$251$$ −4756.08 −1.19602 −0.598010 0.801489i $$-0.704042\pi$$
−0.598010 + 0.801489i $$0.704042\pi$$
$$252$$ 2965.89 0.741402
$$253$$ −9438.77 −2.34550
$$254$$ −9701.29 −2.39651
$$255$$ −127.611 −0.0313385
$$256$$ −8099.28 −1.97736
$$257$$ 2892.84 0.702143 0.351071 0.936349i $$-0.385817\pi$$
0.351071 + 0.936349i $$0.385817\pi$$
$$258$$ −11841.1 −2.85735
$$259$$ 497.784 0.119424
$$260$$ −123.689 −0.0295034
$$261$$ −1850.65 −0.438898
$$262$$ 2046.92 0.482667
$$263$$ 5415.48 1.26971 0.634853 0.772633i $$-0.281061\pi$$
0.634853 + 0.772633i $$0.281061\pi$$
$$264$$ 20832.6 4.85666
$$265$$ −292.228 −0.0677412
$$266$$ 1278.13 0.294613
$$267$$ 9572.58 2.19413
$$268$$ 266.693 0.0607869
$$269$$ 5787.00 1.31167 0.655835 0.754904i $$-0.272317\pi$$
0.655835 + 0.754904i $$0.272317\pi$$
$$270$$ −673.620 −0.151834
$$271$$ 5465.13 1.22503 0.612515 0.790459i $$-0.290158\pi$$
0.612515 + 0.790459i $$0.290158\pi$$
$$272$$ −1658.97 −0.369815
$$273$$ −260.840 −0.0578270
$$274$$ 657.006 0.144858
$$275$$ 6505.42 1.42651
$$276$$ 26465.3 5.77182
$$277$$ −1207.65 −0.261952 −0.130976 0.991386i $$-0.541811\pi$$
−0.130976 + 0.991386i $$0.541811\pi$$
$$278$$ −10435.1 −2.25128
$$279$$ −1565.13 −0.335850
$$280$$ −158.722 −0.0338766
$$281$$ −1197.18 −0.254155 −0.127077 0.991893i $$-0.540560\pi$$
−0.127077 + 0.991893i $$0.540560\pi$$
$$282$$ −19781.7 −4.17724
$$283$$ 3164.73 0.664748 0.332374 0.943148i $$-0.392150\pi$$
0.332374 + 0.943148i $$0.392150\pi$$
$$284$$ −11612.0 −2.42622
$$285$$ −499.300 −0.103775
$$286$$ −2124.38 −0.439221
$$287$$ −68.3784 −0.0140636
$$288$$ −5183.98 −1.06066
$$289$$ 289.000 0.0588235
$$290$$ 183.995 0.0372571
$$291$$ −3215.65 −0.647782
$$292$$ 3352.26 0.671836
$$293$$ 7456.21 1.48668 0.743339 0.668915i $$-0.233241\pi$$
0.743339 + 0.668915i $$0.233241\pi$$
$$294$$ 14007.9 2.77877
$$295$$ 601.079 0.118631
$$296$$ −6118.70 −1.20149
$$297$$ −7914.94 −1.54637
$$298$$ 9323.89 1.81248
$$299$$ −1452.66 −0.280969
$$300$$ −18240.5 −3.51038
$$301$$ 1060.03 0.202988
$$302$$ −10321.0 −1.96657
$$303$$ 1116.47 0.211683
$$304$$ −6490.98 −1.22462
$$305$$ 301.383 0.0565808
$$306$$ 3835.46 0.716532
$$307$$ −6535.48 −1.21498 −0.607491 0.794327i $$-0.707824\pi$$
−0.607491 + 0.794327i $$0.707824\pi$$
$$308$$ −3464.73 −0.640978
$$309$$ 1661.07 0.305809
$$310$$ 155.608 0.0285096
$$311$$ −8935.89 −1.62928 −0.814642 0.579963i $$-0.803067\pi$$
−0.814642 + 0.579963i $$0.803067\pi$$
$$312$$ 3206.22 0.581783
$$313$$ −2628.71 −0.474707 −0.237353 0.971423i $$-0.576280\pi$$
−0.237353 + 0.971423i $$0.576280\pi$$
$$314$$ 1323.50 0.237865
$$315$$ 151.612 0.0271187
$$316$$ 18714.9 3.33163
$$317$$ 4268.54 0.756293 0.378147 0.925746i $$-0.376562\pi$$
0.378147 + 0.925746i $$0.376562\pi$$
$$318$$ 14072.9 2.48166
$$319$$ 2161.92 0.379449
$$320$$ −176.048 −0.0307544
$$321$$ −4111.79 −0.714946
$$322$$ −3463.13 −0.599357
$$323$$ 1130.76 0.194790
$$324$$ 1220.12 0.209211
$$325$$ 1001.21 0.170883
$$326$$ 7271.89 1.23544
$$327$$ −10637.5 −1.79895
$$328$$ 840.500 0.141490
$$329$$ 1770.88 0.296753
$$330$$ 1978.44 0.330029
$$331$$ 992.298 0.164778 0.0823892 0.996600i $$-0.473745\pi$$
0.0823892 + 0.996600i $$0.473745\pi$$
$$332$$ −14998.7 −2.47940
$$333$$ 5844.63 0.961812
$$334$$ 2524.13 0.413516
$$335$$ 13.6330 0.00222344
$$336$$ 3158.02 0.512750
$$337$$ 8042.26 1.29997 0.649985 0.759947i $$-0.274775\pi$$
0.649985 + 0.759947i $$0.274775\pi$$
$$338$$ 10729.5 1.72665
$$339$$ −8588.52 −1.37600
$$340$$ −260.875 −0.0416116
$$341$$ 1828.38 0.290359
$$342$$ 15006.9 2.37275
$$343$$ −2563.68 −0.403573
$$344$$ −13029.8 −2.04221
$$345$$ 1352.87 0.211119
$$346$$ 13034.9 2.02532
$$347$$ −7414.16 −1.14701 −0.573506 0.819202i $$-0.694417\pi$$
−0.573506 + 0.819202i $$0.694417\pi$$
$$348$$ −6061.78 −0.933751
$$349$$ −859.194 −0.131781 −0.0658905 0.997827i $$-0.520989\pi$$
−0.0658905 + 0.997827i $$0.520989\pi$$
$$350$$ 2386.87 0.364525
$$351$$ −1218.14 −0.185241
$$352$$ 6055.89 0.916988
$$353$$ 569.084 0.0858053 0.0429027 0.999079i $$-0.486339\pi$$
0.0429027 + 0.999079i $$0.486339\pi$$
$$354$$ −28946.3 −4.34598
$$355$$ −593.592 −0.0887453
$$356$$ 19569.2 2.91339
$$357$$ −550.142 −0.0815592
$$358$$ −10898.7 −1.60897
$$359$$ −5005.21 −0.735835 −0.367918 0.929858i $$-0.619929\pi$$
−0.367918 + 0.929858i $$0.619929\pi$$
$$360$$ −1863.60 −0.272834
$$361$$ −2434.71 −0.354965
$$362$$ 9690.40 1.40695
$$363$$ 11965.7 1.73013
$$364$$ −533.235 −0.0767833
$$365$$ 171.363 0.0245741
$$366$$ −14513.7 −2.07280
$$367$$ −10975.3 −1.56105 −0.780523 0.625127i $$-0.785047\pi$$
−0.780523 + 0.625127i $$0.785047\pi$$
$$368$$ 17587.5 2.49134
$$369$$ −802.851 −0.113265
$$370$$ −581.083 −0.0816462
$$371$$ −1259.82 −0.176298
$$372$$ −5126.57 −0.714517
$$373$$ −3211.72 −0.445835 −0.222918 0.974837i $$-0.571558\pi$$
−0.222918 + 0.974837i $$0.571558\pi$$
$$374$$ −4480.56 −0.619477
$$375$$ −1870.74 −0.257613
$$376$$ −21767.4 −2.98556
$$377$$ 332.727 0.0454544
$$378$$ −2904.03 −0.395152
$$379$$ 8051.48 1.09123 0.545616 0.838035i $$-0.316296\pi$$
0.545616 + 0.838035i $$0.316296\pi$$
$$380$$ −1020.72 −0.137794
$$381$$ 16338.1 2.19692
$$382$$ 14008.1 1.87622
$$383$$ −2584.16 −0.344763 −0.172382 0.985030i $$-0.555146\pi$$
−0.172382 + 0.985030i $$0.555146\pi$$
$$384$$ 16318.2 2.16858
$$385$$ −177.112 −0.0234454
$$386$$ −11364.7 −1.49858
$$387$$ 12446.2 1.63482
$$388$$ −6573.74 −0.860132
$$389$$ −5174.31 −0.674417 −0.337208 0.941430i $$-0.609483\pi$$
−0.337208 + 0.941430i $$0.609483\pi$$
$$390$$ 304.489 0.0395344
$$391$$ −3063.83 −0.396278
$$392$$ 15414.1 1.98604
$$393$$ −3447.25 −0.442470
$$394$$ 6394.79 0.817677
$$395$$ 956.680 0.121863
$$396$$ −40680.4 −5.16230
$$397$$ −5149.36 −0.650980 −0.325490 0.945545i $$-0.605529\pi$$
−0.325490 + 0.945545i $$0.605529\pi$$
$$398$$ 24127.7 3.03872
$$399$$ −2152.53 −0.270078
$$400$$ −12121.7 −1.51522
$$401$$ 8700.49 1.08350 0.541748 0.840541i $$-0.317763\pi$$
0.541748 + 0.840541i $$0.317763\pi$$
$$402$$ −656.527 −0.0814542
$$403$$ 281.394 0.0347823
$$404$$ 2282.41 0.281074
$$405$$ 62.3709 0.00765243
$$406$$ 793.219 0.0969625
$$407$$ −6827.65 −0.831533
$$408$$ 6762.29 0.820547
$$409$$ 12346.0 1.49260 0.746299 0.665611i $$-0.231829\pi$$
0.746299 + 0.665611i $$0.231829\pi$$
$$410$$ 79.8209 0.00961482
$$411$$ −1106.48 −0.132794
$$412$$ 3395.72 0.406056
$$413$$ 2591.30 0.308740
$$414$$ −40661.7 −4.82708
$$415$$ −766.713 −0.0906903
$$416$$ 932.023 0.109847
$$417$$ 17574.0 2.06379
$$418$$ −17531.0 −2.05136
$$419$$ −5763.33 −0.671974 −0.335987 0.941867i $$-0.609070\pi$$
−0.335987 + 0.941867i $$0.609070\pi$$
$$420$$ 496.604 0.0576947
$$421$$ −1876.12 −0.217188 −0.108594 0.994086i $$-0.534635\pi$$
−0.108594 + 0.994086i $$0.534635\pi$$
$$422$$ 14126.2 1.62951
$$423$$ 20792.4 2.38998
$$424$$ 15485.6 1.77369
$$425$$ 2111.66 0.241014
$$426$$ 28585.7 3.25113
$$427$$ 1299.29 0.147253
$$428$$ −8405.72 −0.949313
$$429$$ 3577.71 0.402642
$$430$$ −1237.42 −0.138776
$$431$$ 83.9299 0.00937996 0.00468998 0.999989i $$-0.498507\pi$$
0.00468998 + 0.999989i $$0.498507\pi$$
$$432$$ 14748.1 1.64252
$$433$$ −15345.0 −1.70308 −0.851539 0.524291i $$-0.824331\pi$$
−0.851539 + 0.524291i $$0.824331\pi$$
$$434$$ 670.842 0.0741968
$$435$$ −309.870 −0.0341543
$$436$$ −21746.3 −2.38867
$$437$$ −11987.8 −1.31225
$$438$$ −8252.36 −0.900258
$$439$$ 3064.74 0.333194 0.166597 0.986025i $$-0.446722\pi$$
0.166597 + 0.986025i $$0.446722\pi$$
$$440$$ 2177.05 0.235879
$$441$$ −14723.6 −1.58985
$$442$$ −689.575 −0.0742076
$$443$$ −1792.97 −0.192295 −0.0961474 0.995367i $$-0.530652\pi$$
−0.0961474 + 0.995367i $$0.530652\pi$$
$$444$$ 19144.0 2.04624
$$445$$ 1000.35 0.106564
$$446$$ −23573.8 −2.50281
$$447$$ −15702.6 −1.66153
$$448$$ −758.960 −0.0800391
$$449$$ 2499.19 0.262681 0.131341 0.991337i $$-0.458072\pi$$
0.131341 + 0.991337i $$0.458072\pi$$
$$450$$ 28025.0 2.93580
$$451$$ 937.885 0.0979231
$$452$$ −17557.5 −1.82707
$$453$$ 17381.7 1.80279
$$454$$ 7024.00 0.726107
$$455$$ −27.2583 −0.00280854
$$456$$ 26458.6 2.71719
$$457$$ 14784.4 1.51331 0.756656 0.653813i $$-0.226832\pi$$
0.756656 + 0.653813i $$0.226832\pi$$
$$458$$ −4502.28 −0.459340
$$459$$ −2569.20 −0.261263
$$460$$ 2765.67 0.280326
$$461$$ −17746.9 −1.79297 −0.896483 0.443078i $$-0.853887\pi$$
−0.896483 + 0.443078i $$0.853887\pi$$
$$462$$ 8529.23 0.858909
$$463$$ 18486.4 1.85559 0.927793 0.373096i $$-0.121704\pi$$
0.927793 + 0.373096i $$0.121704\pi$$
$$464$$ −4028.36 −0.403043
$$465$$ −262.064 −0.0261353
$$466$$ −6019.52 −0.598388
$$467$$ 7406.57 0.733908 0.366954 0.930239i $$-0.380401\pi$$
0.366954 + 0.930239i $$0.380401\pi$$
$$468$$ −6260.87 −0.618395
$$469$$ 58.7731 0.00578655
$$470$$ −2067.22 −0.202880
$$471$$ −2228.94 −0.218055
$$472$$ −31852.0 −3.10616
$$473$$ −14539.5 −1.41338
$$474$$ −46071.0 −4.46437
$$475$$ 8262.24 0.798101
$$476$$ −1124.65 −0.108295
$$477$$ −14791.9 −1.41986
$$478$$ −24905.0 −2.38311
$$479$$ −18550.9 −1.76955 −0.884775 0.466019i $$-0.845688\pi$$
−0.884775 + 0.466019i $$0.845688\pi$$
$$480$$ −867.997 −0.0825384
$$481$$ −1050.80 −0.0996100
$$482$$ 33731.6 3.18762
$$483$$ 5832.34 0.549442
$$484$$ 24461.5 2.29729
$$485$$ −336.041 −0.0314615
$$486$$ 17531.5 1.63631
$$487$$ 10203.4 0.949406 0.474703 0.880146i $$-0.342556\pi$$
0.474703 + 0.880146i $$0.342556\pi$$
$$488$$ −15970.7 −1.48147
$$489$$ −12246.7 −1.13255
$$490$$ 1463.85 0.134959
$$491$$ −1247.46 −0.114658 −0.0573290 0.998355i $$-0.518258\pi$$
−0.0573290 + 0.998355i $$0.518258\pi$$
$$492$$ −2629.73 −0.240970
$$493$$ 701.760 0.0641089
$$494$$ −2698.08 −0.245734
$$495$$ −2079.53 −0.188824
$$496$$ −3406.87 −0.308413
$$497$$ −2559.03 −0.230962
$$498$$ 36922.7 3.32238
$$499$$ 70.0303 0.00628254 0.00314127 0.999995i $$-0.499000\pi$$
0.00314127 + 0.999995i $$0.499000\pi$$
$$500$$ −3824.36 −0.342061
$$501$$ −4250.94 −0.379078
$$502$$ 23935.0 2.12803
$$503$$ 1444.29 0.128028 0.0640138 0.997949i $$-0.479610\pi$$
0.0640138 + 0.997949i $$0.479610\pi$$
$$504$$ −8034.15 −0.710058
$$505$$ 116.674 0.0102810
$$506$$ 47500.7 4.17325
$$507$$ −18069.7 −1.58285
$$508$$ 33400.0 2.91710
$$509$$ 14272.8 1.24289 0.621445 0.783458i $$-0.286546\pi$$
0.621445 + 0.783458i $$0.286546\pi$$
$$510$$ 642.204 0.0557593
$$511$$ 738.761 0.0639547
$$512$$ 25356.7 2.18871
$$513$$ −10052.4 −0.865157
$$514$$ −14558.3 −1.24929
$$515$$ 173.585 0.0148525
$$516$$ 40767.2 3.47805
$$517$$ −24289.6 −2.06625
$$518$$ −2505.10 −0.212486
$$519$$ −21952.3 −1.85665
$$520$$ 335.055 0.0282561
$$521$$ 14874.0 1.25075 0.625376 0.780324i $$-0.284946\pi$$
0.625376 + 0.780324i $$0.284946\pi$$
$$522$$ 9313.42 0.780914
$$523$$ −8142.90 −0.680811 −0.340406 0.940279i $$-0.610564\pi$$
−0.340406 + 0.940279i $$0.610564\pi$$
$$524$$ −7047.21 −0.587517
$$525$$ −4019.78 −0.334167
$$526$$ −27253.4 −2.25914
$$527$$ 593.494 0.0490569
$$528$$ −43315.7 −3.57022
$$529$$ 20314.2 1.66962
$$530$$ 1470.64 0.120529
$$531$$ 30425.3 2.48652
$$532$$ −4400.40 −0.358612
$$533$$ 144.344 0.0117303
$$534$$ −48174.1 −3.90393
$$535$$ −429.689 −0.0347235
$$536$$ −722.432 −0.0582170
$$537$$ 18354.7 1.47498
$$538$$ −29123.1 −2.33380
$$539$$ 17200.0 1.37451
$$540$$ 2319.17 0.184817
$$541$$ 3179.67 0.252689 0.126344 0.991986i $$-0.459676\pi$$
0.126344 + 0.991986i $$0.459676\pi$$
$$542$$ −27503.3 −2.17965
$$543$$ −16319.8 −1.28978
$$544$$ 1965.75 0.154928
$$545$$ −1111.64 −0.0873716
$$546$$ 1312.68 0.102889
$$547$$ 2107.07 0.164702 0.0823509 0.996603i $$-0.473757\pi$$
0.0823509 + 0.996603i $$0.473757\pi$$
$$548$$ −2261.97 −0.176326
$$549$$ 15255.3 1.18594
$$550$$ −32738.6 −2.53814
$$551$$ 2745.76 0.212292
$$552$$ −71690.4 −5.52780
$$553$$ 4124.33 0.317151
$$554$$ 6077.51 0.466081
$$555$$ 978.614 0.0748466
$$556$$ 35926.4 2.74032
$$557$$ 467.382 0.0355540 0.0177770 0.999842i $$-0.494341\pi$$
0.0177770 + 0.999842i $$0.494341\pi$$
$$558$$ 7876.55 0.597565
$$559$$ −2237.69 −0.169309
$$560$$ 330.019 0.0249033
$$561$$ 7545.81 0.567887
$$562$$ 6024.80 0.452208
$$563$$ 14612.6 1.09387 0.546935 0.837175i $$-0.315794\pi$$
0.546935 + 0.837175i $$0.315794\pi$$
$$564$$ 68105.2 5.08466
$$565$$ −897.515 −0.0668297
$$566$$ −15926.5 −1.18276
$$567$$ 268.886 0.0199156
$$568$$ 31455.3 2.32365
$$569$$ 11602.3 0.854821 0.427410 0.904058i $$-0.359426\pi$$
0.427410 + 0.904058i $$0.359426\pi$$
$$570$$ 2512.73 0.184643
$$571$$ −10534.9 −0.772104 −0.386052 0.922477i $$-0.626161\pi$$
−0.386052 + 0.922477i $$0.626161\pi$$
$$572$$ 7313.91 0.534633
$$573$$ −23591.3 −1.71997
$$574$$ 344.115 0.0250228
$$575$$ −22386.8 −1.62364
$$576$$ −8911.18 −0.644617
$$577$$ 14404.7 1.03930 0.519650 0.854379i $$-0.326062\pi$$
0.519650 + 0.854379i $$0.326062\pi$$
$$578$$ −1454.40 −0.104662
$$579$$ 19139.6 1.37377
$$580$$ −633.466 −0.0453504
$$581$$ −3305.37 −0.236024
$$582$$ 16182.8 1.15257
$$583$$ 17279.8 1.22754
$$584$$ −9080.77 −0.643433
$$585$$ −320.047 −0.0226194
$$586$$ −37523.5 −2.64519
$$587$$ −11004.9 −0.773799 −0.386900 0.922122i $$-0.626454\pi$$
−0.386900 + 0.922122i $$0.626454\pi$$
$$588$$ −48227.0 −3.38240
$$589$$ 2322.14 0.162449
$$590$$ −3024.94 −0.211076
$$591$$ −10769.6 −0.749581
$$592$$ 12722.2 0.883239
$$593$$ 1853.59 0.128361 0.0641804 0.997938i $$-0.479557\pi$$
0.0641804 + 0.997938i $$0.479557\pi$$
$$594$$ 39832.0 2.75139
$$595$$ −57.4909 −0.00396117
$$596$$ −32100.7 −2.20620
$$597$$ −40633.9 −2.78565
$$598$$ 7310.53 0.499916
$$599$$ 19074.7 1.30112 0.650559 0.759456i $$-0.274535\pi$$
0.650559 + 0.759456i $$0.274535\pi$$
$$600$$ 49410.7 3.36197
$$601$$ −27776.0 −1.88520 −0.942600 0.333923i $$-0.891627\pi$$
−0.942600 + 0.333923i $$0.891627\pi$$
$$602$$ −5334.62 −0.361168
$$603$$ 690.073 0.0466035
$$604$$ 35533.4 2.39377
$$605$$ 1250.44 0.0840291
$$606$$ −5618.67 −0.376638
$$607$$ 18728.3 1.25232 0.626159 0.779695i $$-0.284626\pi$$
0.626159 + 0.779695i $$0.284626\pi$$
$$608$$ 7691.32 0.513033
$$609$$ −1335.88 −0.0888874
$$610$$ −1516.71 −0.100672
$$611$$ −3738.25 −0.247518
$$612$$ −13204.9 −0.872184
$$613$$ −24405.3 −1.60802 −0.804012 0.594613i $$-0.797305\pi$$
−0.804012 + 0.594613i $$0.797305\pi$$
$$614$$ 32889.8 2.16177
$$615$$ −134.428 −0.00881409
$$616$$ 9385.43 0.613880
$$617$$ −22516.4 −1.46917 −0.734584 0.678518i $$-0.762623\pi$$
−0.734584 + 0.678518i $$0.762623\pi$$
$$618$$ −8359.35 −0.544114
$$619$$ −5146.53 −0.334179 −0.167089 0.985942i $$-0.553437\pi$$
−0.167089 + 0.985942i $$0.553437\pi$$
$$620$$ −535.736 −0.0347027
$$621$$ 27237.4 1.76006
$$622$$ 44969.9 2.89892
$$623$$ 4312.60 0.277337
$$624$$ −6666.45 −0.427679
$$625$$ 15331.4 0.981213
$$626$$ 13229.0 0.844627
$$627$$ 29524.3 1.88052
$$628$$ −4556.62 −0.289536
$$629$$ −2216.26 −0.140490
$$630$$ −762.990 −0.0482512
$$631$$ −3858.77 −0.243447 −0.121724 0.992564i $$-0.538842\pi$$
−0.121724 + 0.992564i $$0.538842\pi$$
$$632$$ −50695.8 −3.19078
$$633$$ −23790.3 −1.49381
$$634$$ −21481.5 −1.34564
$$635$$ 1707.36 0.106700
$$636$$ −48450.7 −3.02075
$$637$$ 2647.15 0.164653
$$638$$ −10879.9 −0.675138
$$639$$ −30046.3 −1.86011
$$640$$ 1705.28 0.105324
$$641$$ 18689.3 1.15161 0.575805 0.817587i $$-0.304689\pi$$
0.575805 + 0.817587i $$0.304689\pi$$
$$642$$ 20692.6 1.27208
$$643$$ 26473.5 1.62366 0.811831 0.583893i $$-0.198471\pi$$
0.811831 + 0.583893i $$0.198471\pi$$
$$644$$ 11923.0 0.729555
$$645$$ 2083.96 0.127219
$$646$$ −5690.57 −0.346583
$$647$$ 14397.7 0.874855 0.437427 0.899254i $$-0.355890\pi$$
0.437427 + 0.899254i $$0.355890\pi$$
$$648$$ −3305.12 −0.200366
$$649$$ −35542.6 −2.14972
$$650$$ −5038.59 −0.304046
$$651$$ −1129.78 −0.0680177
$$652$$ −25036.0 −1.50381
$$653$$ 20939.5 1.25486 0.627431 0.778672i $$-0.284107\pi$$
0.627431 + 0.778672i $$0.284107\pi$$
$$654$$ 53533.6 3.20081
$$655$$ −360.244 −0.0214899
$$656$$ −1747.59 −0.104012
$$657$$ 8674.02 0.515077
$$658$$ −8911.96 −0.528001
$$659$$ 4031.76 0.238323 0.119162 0.992875i $$-0.461979\pi$$
0.119162 + 0.992875i $$0.461979\pi$$
$$660$$ −6811.47 −0.401721
$$661$$ 6691.52 0.393752 0.196876 0.980428i $$-0.436920\pi$$
0.196876 + 0.980428i $$0.436920\pi$$
$$662$$ −4993.75 −0.293184
$$663$$ 1161.33 0.0680275
$$664$$ 40629.2 2.37458
$$665$$ −224.943 −0.0131171
$$666$$ −29413.1 −1.71131
$$667$$ −7439.71 −0.431884
$$668$$ −8690.19 −0.503344
$$669$$ 39701.1 2.29437
$$670$$ −68.6083 −0.00395607
$$671$$ −17821.2 −1.02530
$$672$$ −3742.01 −0.214808
$$673$$ 10319.2 0.591048 0.295524 0.955335i $$-0.404506\pi$$
0.295524 + 0.955335i $$0.404506\pi$$
$$674$$ −40472.7 −2.31298
$$675$$ −18772.6 −1.07046
$$676$$ −36939.9 −2.10172
$$677$$ −19813.3 −1.12480 −0.562398 0.826866i $$-0.690121\pi$$
−0.562398 + 0.826866i $$0.690121\pi$$
$$678$$ 43221.8 2.44826
$$679$$ −1448.70 −0.0818793
$$680$$ 706.671 0.0398524
$$681$$ −11829.3 −0.665636
$$682$$ −9201.33 −0.516624
$$683$$ 5924.61 0.331916 0.165958 0.986133i $$-0.446928\pi$$
0.165958 + 0.986133i $$0.446928\pi$$
$$684$$ −51666.4 −2.88818
$$685$$ −115.629 −0.00644957
$$686$$ 12901.7 0.718061
$$687$$ 7582.38 0.421085
$$688$$ 27091.9 1.50126
$$689$$ 2659.43 0.147048
$$690$$ −6808.33 −0.375636
$$691$$ 1973.16 0.108629 0.0543143 0.998524i $$-0.482703\pi$$
0.0543143 + 0.998524i $$0.482703\pi$$
$$692$$ −44877.1 −2.46528
$$693$$ −8965.03 −0.491419
$$694$$ 37311.8 2.04083
$$695$$ 1836.51 0.100234
$$696$$ 16420.4 0.894274
$$697$$ 304.439 0.0165444
$$698$$ 4323.90 0.234473
$$699$$ 10137.6 0.548554
$$700$$ −8217.63 −0.443710
$$701$$ −12840.1 −0.691815 −0.345907 0.938269i $$-0.612429\pi$$
−0.345907 + 0.938269i $$0.612429\pi$$
$$702$$ 6130.30 0.329591
$$703$$ −8671.50 −0.465223
$$704$$ 10410.0 0.557302
$$705$$ 3481.45 0.185984
$$706$$ −2863.92 −0.152670
$$707$$ 502.990 0.0267566
$$708$$ 99657.5 5.29005
$$709$$ −27749.7 −1.46990 −0.734952 0.678119i $$-0.762796\pi$$
−0.734952 + 0.678119i $$0.762796\pi$$
$$710$$ 2987.26 0.157901
$$711$$ 48425.0 2.55426
$$712$$ −53010.0 −2.79022
$$713$$ −6291.92 −0.330483
$$714$$ 2768.60 0.145115
$$715$$ 373.877 0.0195555
$$716$$ 37522.4 1.95849
$$717$$ 41943.0 2.18464
$$718$$ 25188.8 1.30924
$$719$$ 16888.3 0.875979 0.437989 0.898980i $$-0.355691\pi$$
0.437989 + 0.898980i $$0.355691\pi$$
$$720$$ 3874.85 0.200565
$$721$$ 748.339 0.0386541
$$722$$ 12252.7 0.631575
$$723$$ −56808.0 −2.92215
$$724$$ −33362.6 −1.71258
$$725$$ 5127.62 0.262669
$$726$$ −60217.6 −3.07835
$$727$$ 2135.25 0.108930 0.0544649 0.998516i $$-0.482655\pi$$
0.0544649 + 0.998516i $$0.482655\pi$$
$$728$$ 1444.45 0.0735371
$$729$$ −31426.5 −1.59663
$$730$$ −862.386 −0.0437238
$$731$$ −4719.54 −0.238794
$$732$$ 49968.6 2.52308
$$733$$ 4795.27 0.241633 0.120817 0.992675i $$-0.461449\pi$$
0.120817 + 0.992675i $$0.461449\pi$$
$$734$$ 55233.1 2.77751
$$735$$ −2465.30 −0.123720
$$736$$ −20839.9 −1.04371
$$737$$ −806.138 −0.0402910
$$738$$ 4040.36 0.201528
$$739$$ −32747.6 −1.63010 −0.815048 0.579393i $$-0.803290\pi$$
−0.815048 + 0.579393i $$0.803290\pi$$
$$740$$ 2000.58 0.0993821
$$741$$ 4543.89 0.225269
$$742$$ 6340.05 0.313680
$$743$$ 12299.4 0.607298 0.303649 0.952784i $$-0.401795\pi$$
0.303649 + 0.952784i $$0.401795\pi$$
$$744$$ 13887.1 0.684309
$$745$$ −1640.94 −0.0806974
$$746$$ 16163.0 0.793257
$$747$$ −38809.3 −1.90088
$$748$$ 15425.9 0.754046
$$749$$ −1852.43 −0.0903688
$$750$$ 9414.54 0.458361
$$751$$ 30102.6 1.46266 0.731332 0.682021i $$-0.238899\pi$$
0.731332 + 0.682021i $$0.238899\pi$$
$$752$$ 45259.4 2.19474
$$753$$ −40309.4 −1.95081
$$754$$ −1674.45 −0.0808753
$$755$$ 1816.42 0.0875581
$$756$$ 9998.14 0.480990
$$757$$ 38826.3 1.86416 0.932078 0.362257i $$-0.117994\pi$$
0.932078 + 0.362257i $$0.117994\pi$$
$$758$$ −40519.2 −1.94159
$$759$$ −79996.9 −3.82570
$$760$$ 2764.97 0.131968
$$761$$ 19981.6 0.951815 0.475907 0.879495i $$-0.342120\pi$$
0.475907 + 0.879495i $$0.342120\pi$$
$$762$$ −82221.8 −3.90890
$$763$$ −4792.39 −0.227387
$$764$$ −48227.7 −2.28379
$$765$$ −675.017 −0.0319024
$$766$$ 13004.8 0.613424
$$767$$ −5470.14 −0.257517
$$768$$ −68644.2 −3.22524
$$769$$ −22407.7 −1.05077 −0.525384 0.850865i $$-0.676078\pi$$
−0.525384 + 0.850865i $$0.676078\pi$$
$$770$$ 891.320 0.0417155
$$771$$ 24517.9 1.14525
$$772$$ 39127.0 1.82411
$$773$$ −6902.77 −0.321184 −0.160592 0.987021i $$-0.551340\pi$$
−0.160592 + 0.987021i $$0.551340\pi$$
$$774$$ −62635.4 −2.90876
$$775$$ 4336.54 0.200997
$$776$$ 17807.3 0.823768
$$777$$ 4218.89 0.194790
$$778$$ 26039.8 1.19996
$$779$$ 1191.17 0.0547856
$$780$$ −1048.31 −0.0481225
$$781$$ 35099.9 1.60816
$$782$$ 15418.8 0.705082
$$783$$ −6238.62 −0.284738
$$784$$ −32049.3 −1.45997
$$785$$ −232.928 −0.0105905
$$786$$ 17348.3 0.787270
$$787$$ −22185.9 −1.00488 −0.502442 0.864611i $$-0.667565\pi$$
−0.502442 + 0.864611i $$0.667565\pi$$
$$788$$ −22016.3 −0.995301
$$789$$ 45898.1 2.07099
$$790$$ −4814.50 −0.216826
$$791$$ −3869.27 −0.173926
$$792$$ 110197. 4.94405
$$793$$ −2742.74 −0.122822
$$794$$ 25914.2 1.15826
$$795$$ −2476.73 −0.110491
$$796$$ −83067.8 −3.69882
$$797$$ −16291.1 −0.724040 −0.362020 0.932170i $$-0.617913\pi$$
−0.362020 + 0.932170i $$0.617913\pi$$
$$798$$ 10832.6 0.480539
$$799$$ −7884.41 −0.349100
$$800$$ 14363.3 0.634775
$$801$$ 50635.5 2.23361
$$802$$ −43785.3 −1.92782
$$803$$ −10132.9 −0.445309
$$804$$ 2260.32 0.0991485
$$805$$ 609.489 0.0266853
$$806$$ −1416.12 −0.0618867
$$807$$ 49046.8 2.13944
$$808$$ −6182.70 −0.269191
$$809$$ 17696.8 0.769082 0.384541 0.923108i $$-0.374360\pi$$
0.384541 + 0.923108i $$0.374360\pi$$
$$810$$ −313.882 −0.0136157
$$811$$ −3095.34 −0.134022 −0.0670111 0.997752i $$-0.521346\pi$$
−0.0670111 + 0.997752i $$0.521346\pi$$
$$812$$ −2730.93 −0.118026
$$813$$ 46318.9 1.99812
$$814$$ 34360.2 1.47951
$$815$$ −1279.81 −0.0550057
$$816$$ −14060.3 −0.603198
$$817$$ −18466.0 −0.790751
$$818$$ −62131.5 −2.65572
$$819$$ −1379.75 −0.0588674
$$820$$ −274.811 −0.0117034
$$821$$ 12323.5 0.523864 0.261932 0.965086i $$-0.415640\pi$$
0.261932 + 0.965086i $$0.415640\pi$$
$$822$$ 5568.36 0.236276
$$823$$ −34436.5 −1.45854 −0.729271 0.684225i $$-0.760140\pi$$
−0.729271 + 0.684225i $$0.760140\pi$$
$$824$$ −9198.50 −0.388889
$$825$$ 55135.7 2.32676
$$826$$ −13040.8 −0.549329
$$827$$ 18761.6 0.788880 0.394440 0.918922i $$-0.370939\pi$$
0.394440 + 0.918922i $$0.370939\pi$$
$$828$$ 139992. 5.87567
$$829$$ 22423.8 0.939457 0.469728 0.882811i $$-0.344352\pi$$
0.469728 + 0.882811i $$0.344352\pi$$
$$830$$ 3858.49 0.161362
$$831$$ −10235.3 −0.427265
$$832$$ 1602.13 0.0667596
$$833$$ 5583.15 0.232227
$$834$$ −88441.1 −3.67202
$$835$$ −444.231 −0.0184111
$$836$$ 60356.4 2.49697
$$837$$ −5276.13 −0.217885
$$838$$ 29004.0 1.19562
$$839$$ 9128.63 0.375632 0.187816 0.982204i $$-0.439859\pi$$
0.187816 + 0.982204i $$0.439859\pi$$
$$840$$ −1345.22 −0.0552555
$$841$$ −22685.0 −0.930131
$$842$$ 9441.58 0.386435
$$843$$ −10146.5 −0.414547
$$844$$ −48634.4 −1.98349
$$845$$ −1888.32 −0.0768759
$$846$$ −104638. −4.25240
$$847$$ 5390.75 0.218688
$$848$$ −32198.0 −1.30387
$$849$$ 26822.2 1.08426
$$850$$ −10627.0 −0.428826
$$851$$ 23495.7 0.946442
$$852$$ −98416.1 −3.95737
$$853$$ 27204.8 1.09200 0.545999 0.837786i $$-0.316150\pi$$
0.545999 + 0.837786i $$0.316150\pi$$
$$854$$ −6538.68 −0.262001
$$855$$ −2641.12 −0.105643
$$856$$ 22769.8 0.909179
$$857$$ −38060.0 −1.51704 −0.758520 0.651649i $$-0.774077\pi$$
−0.758520 + 0.651649i $$0.774077\pi$$
$$858$$ −18004.9 −0.716405
$$859$$ −33326.2 −1.32372 −0.661860 0.749627i $$-0.730233\pi$$
−0.661860 + 0.749627i $$0.730233\pi$$
$$860$$ 4260.24 0.168922
$$861$$ −579.531 −0.0229389
$$862$$ −422.378 −0.0166894
$$863$$ −41724.2 −1.64578 −0.822890 0.568201i $$-0.807640\pi$$
−0.822890 + 0.568201i $$0.807640\pi$$
$$864$$ −17475.4 −0.688108
$$865$$ −2294.06 −0.0901737
$$866$$ 77223.8 3.03022
$$867$$ 2449.38 0.0959460
$$868$$ −2309.60 −0.0903146
$$869$$ −56569.7 −2.20828
$$870$$ 1559.42 0.0607694
$$871$$ −124.068 −0.00482649
$$872$$ 58907.5 2.28768
$$873$$ −17009.6 −0.659438
$$874$$ 60328.6 2.33483
$$875$$ −842.801 −0.0325621
$$876$$ 28411.6 1.09582
$$877$$ −49337.3 −1.89966 −0.949830 0.312767i $$-0.898744\pi$$
−0.949830 + 0.312767i $$0.898744\pi$$
$$878$$ −15423.3 −0.592838
$$879$$ 63194.0 2.42489
$$880$$ −4526.57 −0.173398
$$881$$ 8845.46 0.338265 0.169132 0.985593i $$-0.445903\pi$$
0.169132 + 0.985593i $$0.445903\pi$$
$$882$$ 74096.8 2.82876
$$883$$ 14724.2 0.561165 0.280582 0.959830i $$-0.409472\pi$$
0.280582 + 0.959830i $$0.409472\pi$$
$$884$$ 2374.10 0.0903277
$$885$$ 5094.36 0.193497
$$886$$ 9023.14 0.342143
$$887$$ 3864.38 0.146283 0.0731415 0.997322i $$-0.476698\pi$$
0.0731415 + 0.997322i $$0.476698\pi$$
$$888$$ −51858.1 −1.95974
$$889$$ 7360.60 0.277690
$$890$$ −5034.28 −0.189606
$$891$$ −3688.07 −0.138670
$$892$$ 81160.9 3.04649
$$893$$ −30849.1 −1.15602
$$894$$ 79023.2 2.95630
$$895$$ 1918.10 0.0716368
$$896$$ 7351.61 0.274107
$$897$$ −12311.8 −0.458283
$$898$$ −12577.2 −0.467379
$$899$$ 1441.14 0.0534648
$$900$$ −96485.6 −3.57354
$$901$$ 5609.04 0.207397
$$902$$ −4719.92 −0.174231
$$903$$ 8984.15 0.331089
$$904$$ 47560.6 1.74982
$$905$$ −1705.45 −0.0626421
$$906$$ −87473.7 −3.20764
$$907$$ −743.409 −0.0272155 −0.0136078 0.999907i $$-0.504332\pi$$
−0.0136078 + 0.999907i $$0.504332\pi$$
$$908$$ −24182.5 −0.883839
$$909$$ 5905.76 0.215491
$$910$$ 137.177 0.00499713
$$911$$ 16291.0 0.592475 0.296238 0.955114i $$-0.404268\pi$$
0.296238 + 0.955114i $$0.404268\pi$$
$$912$$ −55013.4 −1.99745
$$913$$ 45336.8 1.64340
$$914$$ −74402.5 −2.69258
$$915$$ 2554.33 0.0922879
$$916$$ 15500.6 0.559122
$$917$$ −1553.04 −0.0559280
$$918$$ 12929.5 0.464856
$$919$$ −6188.99 −0.222150 −0.111075 0.993812i $$-0.535429\pi$$
−0.111075 + 0.993812i $$0.535429\pi$$
$$920$$ −7491.78 −0.268474
$$921$$ −55390.5 −1.98174
$$922$$ 89311.6 3.19015
$$923$$ 5402.00 0.192643
$$924$$ −29364.8 −1.04549
$$925$$ −16193.8 −0.575620
$$926$$ −93033.0 −3.30157
$$927$$ 8786.47 0.311311
$$928$$ 4773.30 0.168848
$$929$$ −31661.7 −1.11818 −0.559089 0.829108i $$-0.688849\pi$$
−0.559089 + 0.829108i $$0.688849\pi$$
$$930$$ 1318.84 0.0465015
$$931$$ 21845.0 0.769003
$$932$$ 20724.3 0.728376
$$933$$ −75734.8 −2.65750
$$934$$ −37273.6 −1.30581
$$935$$ 788.551 0.0275811
$$936$$ 16959.8 0.592251
$$937$$ 35010.5 1.22064 0.610322 0.792153i $$-0.291040\pi$$
0.610322 + 0.792153i $$0.291040\pi$$
$$938$$ −295.776 −0.0102958
$$939$$ −22279.2 −0.774286
$$940$$ 7117.12 0.246952
$$941$$ −45625.8 −1.58061 −0.790307 0.612711i $$-0.790079\pi$$
−0.790307 + 0.612711i $$0.790079\pi$$
$$942$$ 11217.2 0.387977
$$943$$ −3227.51 −0.111455
$$944$$ 66227.5 2.28339
$$945$$ 511.092 0.0175934
$$946$$ 73170.2 2.51477
$$947$$ −21508.4 −0.738044 −0.369022 0.929421i $$-0.620307\pi$$
−0.369022 + 0.929421i $$0.620307\pi$$
$$948$$ 158615. 5.43416
$$949$$ −1559.50 −0.0533439
$$950$$ −41579.8 −1.42003
$$951$$ 36177.4 1.23358
$$952$$ 3046.52 0.103717
$$953$$ 35686.7 1.21302 0.606509 0.795076i $$-0.292569\pi$$
0.606509 + 0.795076i $$0.292569\pi$$
$$954$$ 74440.5 2.52631
$$955$$ −2465.34 −0.0835355
$$956$$ 85744.0 2.90079
$$957$$ 18323.0 0.618912
$$958$$ 93357.8 3.14849
$$959$$ −498.486 −0.0167851
$$960$$ −1492.07 −0.0501630
$$961$$ −28572.2 −0.959088
$$962$$ 5288.17 0.177232
$$963$$ −21749.9 −0.727810
$$964$$ −116133. −3.88006
$$965$$ 2000.12 0.0667215
$$966$$ −29351.3 −0.977600
$$967$$ −3731.33 −0.124086 −0.0620432 0.998073i $$-0.519762\pi$$
−0.0620432 + 0.998073i $$0.519762\pi$$
$$968$$ −66262.5 −2.20016
$$969$$ 9583.60 0.317719
$$970$$ 1691.13 0.0559782
$$971$$ 17645.1 0.583171 0.291585 0.956545i $$-0.405817\pi$$
0.291585 + 0.956545i $$0.405817\pi$$
$$972$$ −60358.3 −1.99176
$$973$$ 7917.35 0.260862
$$974$$ −51348.7 −1.68924
$$975$$ 8485.59 0.278725
$$976$$ 33206.7 1.08906
$$977$$ 24941.2 0.816723 0.408362 0.912820i $$-0.366100\pi$$
0.408362 + 0.912820i $$0.366100\pi$$
$$978$$ 61631.8 2.01510
$$979$$ −59152.1 −1.93106
$$980$$ −5039.81 −0.164276
$$981$$ −56268.9 −1.83132
$$982$$ 6277.85 0.204006
$$983$$ −22506.2 −0.730252 −0.365126 0.930958i $$-0.618974\pi$$
−0.365126 + 0.930958i $$0.618974\pi$$
$$984$$ 7123.53 0.230782
$$985$$ −1125.44 −0.0364057
$$986$$ −3531.62 −0.114066
$$987$$ 15008.8 0.484029
$$988$$ 9289.07 0.299114
$$989$$ 50034.2 1.60869
$$990$$ 10465.3 0.335967
$$991$$ 32694.1 1.04799 0.523997 0.851720i $$-0.324440\pi$$
0.523997 + 0.851720i $$0.324440\pi$$
$$992$$ 4036.88 0.129205
$$993$$ 8410.08 0.268767
$$994$$ 12878.3 0.410941
$$995$$ −4246.32 −0.135294
$$996$$ −127119. −4.04410
$$997$$ 18248.8 0.579686 0.289843 0.957074i $$-0.406397\pi$$
0.289843 + 0.957074i $$0.406397\pi$$
$$998$$ −352.428 −0.0111783
$$999$$ 19702.5 0.623983
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 17.4.a.b.1.1 3
3.2 odd 2 153.4.a.g.1.3 3
4.3 odd 2 272.4.a.h.1.1 3
5.2 odd 4 425.4.b.f.324.1 6
5.3 odd 4 425.4.b.f.324.6 6
5.4 even 2 425.4.a.g.1.3 3
7.6 odd 2 833.4.a.d.1.1 3
8.3 odd 2 1088.4.a.x.1.3 3
8.5 even 2 1088.4.a.v.1.1 3
11.10 odd 2 2057.4.a.e.1.3 3
12.11 even 2 2448.4.a.bi.1.2 3
17.4 even 4 289.4.b.b.288.5 6
17.13 even 4 289.4.b.b.288.6 6
17.16 even 2 289.4.a.b.1.1 3

By twisted newform
Twist Min Dim Char Parity Ord Type
17.4.a.b.1.1 3 1.1 even 1 trivial
153.4.a.g.1.3 3 3.2 odd 2
272.4.a.h.1.1 3 4.3 odd 2
289.4.a.b.1.1 3 17.16 even 2
289.4.b.b.288.5 6 17.4 even 4
289.4.b.b.288.6 6 17.13 even 4
425.4.a.g.1.3 3 5.4 even 2
425.4.b.f.324.1 6 5.2 odd 4
425.4.b.f.324.6 6 5.3 odd 4
833.4.a.d.1.1 3 7.6 odd 2
1088.4.a.v.1.1 3 8.5 even 2
1088.4.a.x.1.3 3 8.3 odd 2
2057.4.a.e.1.3 3 11.10 odd 2
2448.4.a.bi.1.2 3 12.11 even 2