Properties

 Label 507.4.b.h.337.1 Level $507$ Weight $4$ Character 507.337 Analytic conductor $29.914$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

Related objects

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [507,4,Mod(337,507)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(507, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

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

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

chi := DirichletCharacter("507.337");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$507 = 3 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 507.b (of order $$2$$, degree $$1$$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: no Analytic conductor: $$29.9139683729$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} + \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{8} + 54x^{6} + 889x^{4} + 4584x^{2} + 5776$$ x^8 + 54*x^6 + 889*x^4 + 4584*x^2 + 5776 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{6}\cdot 13^{2}$$ Twist minimal: no (minimal twist has level 39) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

 Embedding label 337.1 Root $$-4.33039i$$ of defining polynomial Character $$\chi$$ $$=$$ 507.337 Dual form 507.4.b.h.337.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-5.33039i q^{2} -3.00000 q^{3} -20.4131 q^{4} +16.4131i q^{5} +15.9912i q^{6} +9.67968i q^{7} +66.1667i q^{8} +9.00000 q^{9} +O(q^{10})$$ $$q-5.33039i q^{2} -3.00000 q^{3} -20.4131 q^{4} +16.4131i q^{5} +15.9912i q^{6} +9.67968i q^{7} +66.1667i q^{8} +9.00000 q^{9} +87.4882 q^{10} +27.5882i q^{11} +61.2393 q^{12} +51.5965 q^{14} -49.2393i q^{15} +189.390 q^{16} -107.928 q^{17} -47.9735i q^{18} +2.24723i q^{19} -335.042i q^{20} -29.0391i q^{21} +147.056 q^{22} -41.8090 q^{23} -198.500i q^{24} -144.390 q^{25} -27.0000 q^{27} -197.592i q^{28} +61.6213 q^{29} -262.465 q^{30} -191.932i q^{31} -480.187i q^{32} -82.7645i q^{33} +575.300i q^{34} -158.874 q^{35} -183.718 q^{36} +98.4236i q^{37} +11.9786 q^{38} -1086.00 q^{40} +30.7452i q^{41} -154.790 q^{42} -238.325 q^{43} -563.160i q^{44} +147.718i q^{45} +222.858i q^{46} -511.482i q^{47} -568.169 q^{48} +249.304 q^{49} +769.653i q^{50} +323.785 q^{51} +492.825 q^{53} +143.921i q^{54} -452.807 q^{55} -640.472 q^{56} -6.74170i q^{57} -328.466i q^{58} +484.179i q^{59} +1005.13i q^{60} -444.021 q^{61} -1023.07 q^{62} +87.1172i q^{63} -1044.47 q^{64} -441.167 q^{66} -190.114i q^{67} +2203.15 q^{68} +125.427 q^{69} +846.858i q^{70} -484.785i q^{71} +595.500i q^{72} -957.780i q^{73} +524.636 q^{74} +433.169 q^{75} -45.8729i q^{76} -267.045 q^{77} -375.216 q^{79} +3108.47i q^{80} +81.0000 q^{81} +163.884 q^{82} +715.765i q^{83} +592.777i q^{84} -1771.43i q^{85} +1270.37i q^{86} -184.864 q^{87} -1825.42 q^{88} -1038.15i q^{89} +787.394 q^{90} +853.451 q^{92} +575.796i q^{93} -2726.40 q^{94} -36.8840 q^{95} +1440.56i q^{96} -65.5636i q^{97} -1328.89i q^{98} +248.293i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q - 24 q^{3} - 44 q^{4} + 72 q^{9}+O(q^{10})$$ 8 * q - 24 * q^3 - 44 * q^4 + 72 * q^9 $$8 q - 24 q^{3} - 44 q^{4} + 72 q^{9} + 124 q^{10} + 132 q^{12} + 80 q^{14} + 244 q^{16} - 196 q^{17} + 440 q^{22} - 208 q^{23} + 116 q^{25} - 216 q^{27} + 388 q^{29} - 372 q^{30} + 176 q^{35} - 396 q^{36} - 664 q^{38} - 1996 q^{40} - 240 q^{42} - 900 q^{43} - 732 q^{48} - 2140 q^{49} + 588 q^{51} + 524 q^{53} + 408 q^{55} - 4328 q^{56} - 1856 q^{61} - 5560 q^{62} - 2052 q^{64} - 1320 q^{66} + 3572 q^{68} + 624 q^{69} + 2316 q^{74} - 348 q^{75} - 5016 q^{77} - 1492 q^{79} + 648 q^{81} - 3468 q^{82} - 1164 q^{87} - 6120 q^{88} + 1116 q^{90} + 664 q^{92} - 1544 q^{94} - 4408 q^{95}+O(q^{100})$$ 8 * q - 24 * q^3 - 44 * q^4 + 72 * q^9 + 124 * q^10 + 132 * q^12 + 80 * q^14 + 244 * q^16 - 196 * q^17 + 440 * q^22 - 208 * q^23 + 116 * q^25 - 216 * q^27 + 388 * q^29 - 372 * q^30 + 176 * q^35 - 396 * q^36 - 664 * q^38 - 1996 * q^40 - 240 * q^42 - 900 * q^43 - 732 * q^48 - 2140 * q^49 + 588 * q^51 + 524 * q^53 + 408 * q^55 - 4328 * q^56 - 1856 * q^61 - 5560 * q^62 - 2052 * q^64 - 1320 * q^66 + 3572 * q^68 + 624 * q^69 + 2316 * q^74 - 348 * q^75 - 5016 * q^77 - 1492 * q^79 + 648 * q^81 - 3468 * q^82 - 1164 * q^87 - 6120 * q^88 + 1116 * q^90 + 664 * q^92 - 1544 * q^94 - 4408 * q^95

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/507\mathbb{Z}\right)^\times$$.

 $$n$$ $$170$$ $$340$$ $$\chi(n)$$ $$1$$ $$-1$$

Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ − 5.33039i − 1.88458i −0.334800 0.942289i $$-0.608669\pi$$
0.334800 0.942289i $$-0.391331\pi$$
$$3$$ −3.00000 −0.577350
$$4$$ −20.4131 −2.55164
$$5$$ 16.4131i 1.46803i 0.679132 + 0.734016i $$0.262356\pi$$
−0.679132 + 0.734016i $$0.737644\pi$$
$$6$$ 15.9912i 1.08806i
$$7$$ 9.67968i 0.522654i 0.965250 + 0.261327i $$0.0841601\pi$$
−0.965250 + 0.261327i $$0.915840\pi$$
$$8$$ 66.1667i 2.92418i
$$9$$ 9.00000 0.333333
$$10$$ 87.4882 2.76662
$$11$$ 27.5882i 0.756195i 0.925766 + 0.378098i $$0.123422\pi$$
−0.925766 + 0.378098i $$0.876578\pi$$
$$12$$ 61.2393 1.47319
$$13$$ 0 0
$$14$$ 51.5965 0.984982
$$15$$ − 49.2393i − 0.847568i
$$16$$ 189.390 2.95921
$$17$$ −107.928 −1.53979 −0.769895 0.638171i $$-0.779691\pi$$
−0.769895 + 0.638171i $$0.779691\pi$$
$$18$$ − 47.9735i − 0.628193i
$$19$$ 2.24723i 0.0271342i 0.999908 + 0.0135671i $$0.00431868\pi$$
−0.999908 + 0.0135671i $$0.995681\pi$$
$$20$$ − 335.042i − 3.74588i
$$21$$ − 29.0391i − 0.301754i
$$22$$ 147.056 1.42511
$$23$$ −41.8090 −0.379034 −0.189517 0.981877i $$-0.560692\pi$$
−0.189517 + 0.981877i $$0.560692\pi$$
$$24$$ − 198.500i − 1.68828i
$$25$$ −144.390 −1.15512
$$26$$ 0 0
$$27$$ −27.0000 −0.192450
$$28$$ − 197.592i − 1.33362i
$$29$$ 61.6213 0.394579 0.197289 0.980345i $$-0.436786\pi$$
0.197289 + 0.980345i $$0.436786\pi$$
$$30$$ −262.465 −1.59731
$$31$$ − 191.932i − 1.11200i −0.831182 0.556000i $$-0.812335\pi$$
0.831182 0.556000i $$-0.187665\pi$$
$$32$$ − 480.187i − 2.65269i
$$33$$ − 82.7645i − 0.436589i
$$34$$ 575.300i 2.90185i
$$35$$ −158.874 −0.767272
$$36$$ −183.718 −0.850545
$$37$$ 98.4236i 0.437317i 0.975801 + 0.218659i $$0.0701681\pi$$
−0.975801 + 0.218659i $$0.929832\pi$$
$$38$$ 11.9786 0.0511366
$$39$$ 0 0
$$40$$ −1086.00 −4.29279
$$41$$ 30.7452i 0.117112i 0.998284 + 0.0585561i $$0.0186496\pi$$
−0.998284 + 0.0585561i $$0.981350\pi$$
$$42$$ −154.790 −0.568680
$$43$$ −238.325 −0.845216 −0.422608 0.906313i $$-0.638885\pi$$
−0.422608 + 0.906313i $$0.638885\pi$$
$$44$$ − 563.160i − 1.92953i
$$45$$ 147.718i 0.489344i
$$46$$ 222.858i 0.714319i
$$47$$ − 511.482i − 1.58739i −0.608316 0.793695i $$-0.708155\pi$$
0.608316 0.793695i $$-0.291845\pi$$
$$48$$ −568.169 −1.70850
$$49$$ 249.304 0.726833
$$50$$ 769.653i 2.17691i
$$51$$ 323.785 0.888998
$$52$$ 0 0
$$53$$ 492.825 1.27726 0.638630 0.769514i $$-0.279502\pi$$
0.638630 + 0.769514i $$0.279502\pi$$
$$54$$ 143.921i 0.362687i
$$55$$ −452.807 −1.11012
$$56$$ −640.472 −1.52833
$$57$$ − 6.74170i − 0.0156660i
$$58$$ − 328.466i − 0.743615i
$$59$$ 484.179i 1.06838i 0.845363 + 0.534192i $$0.179384\pi$$
−0.845363 + 0.534192i $$0.820616\pi$$
$$60$$ 1005.13i 2.16269i
$$61$$ −444.021 −0.931985 −0.465993 0.884789i $$-0.654303\pi$$
−0.465993 + 0.884789i $$0.654303\pi$$
$$62$$ −1023.07 −2.09565
$$63$$ 87.1172i 0.174218i
$$64$$ −1044.47 −2.03998
$$65$$ 0 0
$$66$$ −441.167 −0.822787
$$67$$ − 190.114i − 0.346658i −0.984864 0.173329i $$-0.944548\pi$$
0.984864 0.173329i $$-0.0554524\pi$$
$$68$$ 2203.15 3.92898
$$69$$ 125.427 0.218835
$$70$$ 846.858i 1.44598i
$$71$$ − 484.785i − 0.810329i −0.914244 0.405164i $$-0.867214\pi$$
0.914244 0.405164i $$-0.132786\pi$$
$$72$$ 595.500i 0.974727i
$$73$$ − 957.780i − 1.53561i −0.640683 0.767806i $$-0.721349\pi$$
0.640683 0.767806i $$-0.278651\pi$$
$$74$$ 524.636 0.824159
$$75$$ 433.169 0.666907
$$76$$ − 45.8729i − 0.0692367i
$$77$$ −267.045 −0.395228
$$78$$ 0 0
$$79$$ −375.216 −0.534368 −0.267184 0.963646i $$-0.586093\pi$$
−0.267184 + 0.963646i $$0.586093\pi$$
$$80$$ 3108.47i 4.34422i
$$81$$ 81.0000 0.111111
$$82$$ 163.884 0.220707
$$83$$ 715.765i 0.946571i 0.880909 + 0.473286i $$0.156932\pi$$
−0.880909 + 0.473286i $$0.843068\pi$$
$$84$$ 592.777i 0.769967i
$$85$$ − 1771.43i − 2.26046i
$$86$$ 1270.37i 1.59288i
$$87$$ −184.864 −0.227810
$$88$$ −1825.42 −2.21125
$$89$$ − 1038.15i − 1.23645i −0.786002 0.618224i $$-0.787852\pi$$
0.786002 0.618224i $$-0.212148\pi$$
$$90$$ 787.394 0.922207
$$91$$ 0 0
$$92$$ 853.451 0.967157
$$93$$ 575.796i 0.642013i
$$94$$ −2726.40 −2.99156
$$95$$ −36.8840 −0.0398339
$$96$$ 1440.56i 1.53153i
$$97$$ − 65.5636i − 0.0686286i −0.999411 0.0343143i $$-0.989075\pi$$
0.999411 0.0343143i $$-0.0109247\pi$$
$$98$$ − 1328.89i − 1.36977i
$$99$$ 248.293i 0.252065i
$$100$$ 2947.44 2.94744
$$101$$ −531.798 −0.523920 −0.261960 0.965079i $$-0.584369\pi$$
−0.261960 + 0.965079i $$0.584369\pi$$
$$102$$ − 1725.90i − 1.67539i
$$103$$ 735.984 0.704064 0.352032 0.935988i $$-0.385491\pi$$
0.352032 + 0.935988i $$0.385491\pi$$
$$104$$ 0 0
$$105$$ 476.621 0.442985
$$106$$ − 2626.95i − 2.40710i
$$107$$ −783.265 −0.707673 −0.353837 0.935307i $$-0.615123\pi$$
−0.353837 + 0.935307i $$0.615123\pi$$
$$108$$ 551.153 0.491063
$$109$$ 532.339i 0.467788i 0.972262 + 0.233894i $$0.0751468\pi$$
−0.972262 + 0.233894i $$0.924853\pi$$
$$110$$ 2413.64i 2.09210i
$$111$$ − 295.271i − 0.252485i
$$112$$ 1833.23i 1.54664i
$$113$$ −180.589 −0.150340 −0.0751699 0.997171i $$-0.523950\pi$$
−0.0751699 + 0.997171i $$0.523950\pi$$
$$114$$ −35.9359 −0.0295237
$$115$$ − 686.215i − 0.556434i
$$116$$ −1257.88 −1.00682
$$117$$ 0 0
$$118$$ 2580.86 2.01346
$$119$$ − 1044.71i − 0.804777i
$$120$$ 3258.00 2.47844
$$121$$ 569.893 0.428169
$$122$$ 2366.81i 1.75640i
$$123$$ − 92.2357i − 0.0676147i
$$124$$ 3917.92i 2.83742i
$$125$$ − 318.242i − 0.227716i
$$126$$ 464.369 0.328327
$$127$$ −1431.63 −1.00029 −0.500146 0.865941i $$-0.666720\pi$$
−0.500146 + 0.865941i $$0.666720\pi$$
$$128$$ 1725.94i 1.19182i
$$129$$ 714.976 0.487986
$$130$$ 0 0
$$131$$ 2067.32 1.37880 0.689400 0.724381i $$-0.257874\pi$$
0.689400 + 0.724381i $$0.257874\pi$$
$$132$$ 1689.48i 1.11402i
$$133$$ −21.7525 −0.0141818
$$134$$ −1013.38 −0.653304
$$135$$ − 443.153i − 0.282523i
$$136$$ − 7141.25i − 4.50262i
$$137$$ − 387.512i − 0.241660i −0.992673 0.120830i $$-0.961444\pi$$
0.992673 0.120830i $$-0.0385555\pi$$
$$138$$ − 668.575i − 0.412412i
$$139$$ 752.568 0.459223 0.229611 0.973282i $$-0.426254\pi$$
0.229611 + 0.973282i $$0.426254\pi$$
$$140$$ 3243.10 1.95780
$$141$$ 1534.45i 0.916480i
$$142$$ −2584.09 −1.52713
$$143$$ 0 0
$$144$$ 1704.51 0.986404
$$145$$ 1011.40i 0.579254i
$$146$$ −5105.34 −2.89398
$$147$$ −747.911 −0.419637
$$148$$ − 2009.13i − 1.11587i
$$149$$ 2636.72i 1.44972i 0.688895 + 0.724862i $$0.258096\pi$$
−0.688895 + 0.724862i $$0.741904\pi$$
$$150$$ − 2308.96i − 1.25684i
$$151$$ − 3332.42i − 1.79595i −0.440046 0.897975i $$-0.645038\pi$$
0.440046 0.897975i $$-0.354962\pi$$
$$152$$ −148.692 −0.0793454
$$153$$ −971.354 −0.513263
$$154$$ 1423.45i 0.744839i
$$155$$ 3150.20 1.63245
$$156$$ 0 0
$$157$$ −1625.26 −0.826179 −0.413089 0.910690i $$-0.635550\pi$$
−0.413089 + 0.910690i $$0.635550\pi$$
$$158$$ 2000.05i 1.00706i
$$159$$ −1478.48 −0.737426
$$160$$ 7881.36 3.89423
$$161$$ − 404.698i − 0.198104i
$$162$$ − 431.762i − 0.209398i
$$163$$ − 1835.37i − 0.881944i −0.897521 0.440972i $$-0.854634\pi$$
0.897521 0.440972i $$-0.145366\pi$$
$$164$$ − 627.605i − 0.298828i
$$165$$ 1358.42 0.640927
$$166$$ 3815.31 1.78389
$$167$$ 1945.00i 0.901248i 0.892714 + 0.450624i $$0.148798\pi$$
−0.892714 + 0.450624i $$0.851202\pi$$
$$168$$ 1921.42 0.882384
$$169$$ 0 0
$$170$$ −9442.44 −4.26001
$$171$$ 20.2251i 0.00904474i
$$172$$ 4864.96 2.15668
$$173$$ −2531.63 −1.11258 −0.556289 0.830989i $$-0.687775\pi$$
−0.556289 + 0.830989i $$0.687775\pi$$
$$174$$ 985.397i 0.429326i
$$175$$ − 1397.65i − 0.603726i
$$176$$ 5224.91i 2.23774i
$$177$$ − 1452.54i − 0.616832i
$$178$$ −5533.76 −2.33018
$$179$$ −4263.01 −1.78007 −0.890035 0.455892i $$-0.849320\pi$$
−0.890035 + 0.455892i $$0.849320\pi$$
$$180$$ − 3015.38i − 1.24863i
$$181$$ −3944.61 −1.61989 −0.809946 0.586504i $$-0.800504\pi$$
−0.809946 + 0.586504i $$0.800504\pi$$
$$182$$ 0 0
$$183$$ 1332.06 0.538082
$$184$$ − 2766.36i − 1.10836i
$$185$$ −1615.43 −0.641995
$$186$$ 3069.22 1.20992
$$187$$ − 2977.54i − 1.16438i
$$188$$ 10440.9i 4.05044i
$$189$$ − 261.351i − 0.100585i
$$190$$ 196.606i 0.0750701i
$$191$$ −214.109 −0.0811119 −0.0405559 0.999177i $$-0.512913\pi$$
−0.0405559 + 0.999177i $$0.512913\pi$$
$$192$$ 3133.42 1.17779
$$193$$ − 1207.19i − 0.450234i −0.974332 0.225117i $$-0.927724\pi$$
0.974332 0.225117i $$-0.0722764\pi$$
$$194$$ −349.480 −0.129336
$$195$$ 0 0
$$196$$ −5089.06 −1.85461
$$197$$ 927.631i 0.335487i 0.985831 + 0.167744i $$0.0536481\pi$$
−0.985831 + 0.167744i $$0.946352\pi$$
$$198$$ 1323.50 0.475036
$$199$$ −478.951 −0.170613 −0.0853064 0.996355i $$-0.527187\pi$$
−0.0853064 + 0.996355i $$0.527187\pi$$
$$200$$ − 9553.77i − 3.37777i
$$201$$ 570.341i 0.200143i
$$202$$ 2834.69i 0.987368i
$$203$$ 596.474i 0.206228i
$$204$$ −6609.44 −2.26840
$$205$$ −504.624 −0.171924
$$206$$ − 3923.08i − 1.32686i
$$207$$ −376.281 −0.126345
$$208$$ 0 0
$$209$$ −61.9970 −0.0205188
$$210$$ − 2540.58i − 0.834840i
$$211$$ −1450.95 −0.473402 −0.236701 0.971583i $$-0.576066\pi$$
−0.236701 + 0.971583i $$0.576066\pi$$
$$212$$ −10060.1 −3.25910
$$213$$ 1454.35i 0.467843i
$$214$$ 4175.11i 1.33367i
$$215$$ − 3911.66i − 1.24080i
$$216$$ − 1786.50i − 0.562759i
$$217$$ 1857.84 0.581191
$$218$$ 2837.58 0.881583
$$219$$ 2873.34i 0.886586i
$$220$$ 9243.19 2.83262
$$221$$ 0 0
$$222$$ −1573.91 −0.475828
$$223$$ 2059.79i 0.618536i 0.950975 + 0.309268i $$0.100084\pi$$
−0.950975 + 0.309268i $$0.899916\pi$$
$$224$$ 4648.06 1.38644
$$225$$ −1299.51 −0.385039
$$226$$ 962.612i 0.283327i
$$227$$ − 4482.46i − 1.31062i −0.755359 0.655311i $$-0.772537\pi$$
0.755359 0.655311i $$-0.227463\pi$$
$$228$$ 137.619i 0.0399738i
$$229$$ − 1630.39i − 0.470477i −0.971938 0.235239i $$-0.924413\pi$$
0.971938 0.235239i $$-0.0755872\pi$$
$$230$$ −3657.80 −1.04864
$$231$$ 801.134 0.228185
$$232$$ 4077.27i 1.15382i
$$233$$ 1903.69 0.535258 0.267629 0.963522i $$-0.413760\pi$$
0.267629 + 0.963522i $$0.413760\pi$$
$$234$$ 0 0
$$235$$ 8395.00 2.33034
$$236$$ − 9883.58i − 2.72613i
$$237$$ 1125.65 0.308517
$$238$$ −5568.72 −1.51667
$$239$$ − 3763.79i − 1.01866i −0.860572 0.509328i $$-0.829894\pi$$
0.860572 0.509328i $$-0.170106\pi$$
$$240$$ − 9325.40i − 2.50813i
$$241$$ − 3614.74i − 0.966166i −0.875575 0.483083i $$-0.839517\pi$$
0.875575 0.483083i $$-0.160483\pi$$
$$242$$ − 3037.75i − 0.806918i
$$243$$ −243.000 −0.0641500
$$244$$ 9063.85 2.37809
$$245$$ 4091.84i 1.06701i
$$246$$ −491.652 −0.127425
$$247$$ 0 0
$$248$$ 12699.5 3.25169
$$249$$ − 2147.30i − 0.546503i
$$250$$ −1696.36 −0.429148
$$251$$ 5729.77 1.44088 0.720438 0.693520i $$-0.243941\pi$$
0.720438 + 0.693520i $$0.243941\pi$$
$$252$$ − 1778.33i − 0.444541i
$$253$$ − 1153.43i − 0.286624i
$$254$$ 7631.18i 1.88513i
$$255$$ 5314.30i 1.30508i
$$256$$ 844.191 0.206101
$$257$$ 5525.79 1.34120 0.670602 0.741818i $$-0.266036\pi$$
0.670602 + 0.741818i $$0.266036\pi$$
$$258$$ − 3811.10i − 0.919647i
$$259$$ −952.709 −0.228565
$$260$$ 0 0
$$261$$ 554.591 0.131526
$$262$$ − 11019.6i − 2.59846i
$$263$$ 5223.21 1.22463 0.612313 0.790615i $$-0.290239\pi$$
0.612313 + 0.790615i $$0.290239\pi$$
$$264$$ 5476.25 1.27667
$$265$$ 8088.78i 1.87506i
$$266$$ 115.949i 0.0267267i
$$267$$ 3114.46i 0.713864i
$$268$$ 3880.81i 0.884545i
$$269$$ 7203.88 1.63282 0.816410 0.577473i $$-0.195961\pi$$
0.816410 + 0.577473i $$0.195961\pi$$
$$270$$ −2362.18 −0.532436
$$271$$ − 8577.69i − 1.92272i −0.275293 0.961360i $$-0.588775\pi$$
0.275293 0.961360i $$-0.411225\pi$$
$$272$$ −20440.5 −4.55656
$$273$$ 0 0
$$274$$ −2065.59 −0.455427
$$275$$ − 3983.44i − 0.873493i
$$276$$ −2560.35 −0.558388
$$277$$ −7169.19 −1.55507 −0.777536 0.628838i $$-0.783531\pi$$
−0.777536 + 0.628838i $$0.783531\pi$$
$$278$$ − 4011.48i − 0.865442i
$$279$$ − 1727.39i − 0.370667i
$$280$$ − 10512.1i − 2.24364i
$$281$$ 849.157i 0.180272i 0.995929 + 0.0901360i $$0.0287302\pi$$
−0.995929 + 0.0901360i $$0.971270\pi$$
$$282$$ 8179.20 1.72718
$$283$$ −1115.37 −0.234283 −0.117141 0.993115i $$-0.537373\pi$$
−0.117141 + 0.993115i $$0.537373\pi$$
$$284$$ 9895.95i 2.06766i
$$285$$ 110.652 0.0229981
$$286$$ 0 0
$$287$$ −297.604 −0.0612091
$$288$$ − 4321.69i − 0.884229i
$$289$$ 6735.49 1.37095
$$290$$ 5391.14 1.09165
$$291$$ 196.691i 0.0396227i
$$292$$ 19551.2i 3.91832i
$$293$$ − 1863.53i − 0.371565i −0.982591 0.185782i $$-0.940518\pi$$
0.982591 0.185782i $$-0.0594819\pi$$
$$294$$ 3986.66i 0.790839i
$$295$$ −7946.87 −1.56842
$$296$$ −6512.36 −1.27879
$$297$$ − 744.880i − 0.145530i
$$298$$ 14054.8 2.73212
$$299$$ 0 0
$$300$$ −8842.31 −1.70170
$$301$$ − 2306.91i − 0.441755i
$$302$$ −17763.1 −3.38461
$$303$$ 1595.39 0.302485
$$304$$ 425.602i 0.0802959i
$$305$$ − 7287.76i − 1.36818i
$$306$$ 5177.70i 0.967285i
$$307$$ − 6387.50i − 1.18747i −0.804660 0.593736i $$-0.797652\pi$$
0.804660 0.593736i $$-0.202348\pi$$
$$308$$ 5451.21 1.00848
$$309$$ −2207.95 −0.406492
$$310$$ − 16791.8i − 3.07648i
$$311$$ 3492.59 0.636806 0.318403 0.947955i $$-0.396853\pi$$
0.318403 + 0.947955i $$0.396853\pi$$
$$312$$ 0 0
$$313$$ −5912.01 −1.06762 −0.533812 0.845603i $$-0.679241\pi$$
−0.533812 + 0.845603i $$0.679241\pi$$
$$314$$ 8663.29i 1.55700i
$$315$$ −1429.86 −0.255757
$$316$$ 7659.31 1.36351
$$317$$ 1677.54i 0.297224i 0.988896 + 0.148612i $$0.0474805\pi$$
−0.988896 + 0.148612i $$0.952519\pi$$
$$318$$ 7880.86i 1.38974i
$$319$$ 1700.02i 0.298378i
$$320$$ − 17143.0i − 2.99476i
$$321$$ 2349.79 0.408575
$$322$$ −2157.20 −0.373342
$$323$$ − 242.540i − 0.0417810i
$$324$$ −1653.46 −0.283515
$$325$$ 0 0
$$326$$ −9783.22 −1.66209
$$327$$ − 1597.02i − 0.270077i
$$328$$ −2034.31 −0.342457
$$329$$ 4950.98 0.829655
$$330$$ − 7240.92i − 1.20788i
$$331$$ − 2010.31i − 0.333827i −0.985972 0.166913i $$-0.946620\pi$$
0.985972 0.166913i $$-0.0533800\pi$$
$$332$$ − 14611.0i − 2.41531i
$$333$$ 885.812i 0.145772i
$$334$$ 10367.6 1.69847
$$335$$ 3120.35 0.508905
$$336$$ − 5499.69i − 0.892955i
$$337$$ −7139.24 −1.15400 −0.577002 0.816743i $$-0.695778\pi$$
−0.577002 + 0.816743i $$0.695778\pi$$
$$338$$ 0 0
$$339$$ 541.768 0.0867988
$$340$$ 36160.5i 5.76787i
$$341$$ 5295.05 0.840889
$$342$$ 107.808 0.0170455
$$343$$ 5733.31i 0.902536i
$$344$$ − 15769.2i − 2.47156i
$$345$$ 2058.64i 0.321257i
$$346$$ 13494.6i 2.09674i
$$347$$ 1.13990 0.000176349 0 8.81743e−5 1.00000i $$-0.499972\pi$$
8.81743e−5 1.00000i $$0.499972\pi$$
$$348$$ 3773.64 0.581289
$$349$$ 12199.1i 1.87107i 0.353235 + 0.935535i $$0.385082\pi$$
−0.353235 + 0.935535i $$0.614918\pi$$
$$350$$ −7450.00 −1.13777
$$351$$ 0 0
$$352$$ 13247.5 2.00595
$$353$$ 10892.3i 1.64232i 0.570698 + 0.821160i $$0.306673\pi$$
−0.570698 + 0.821160i $$0.693327\pi$$
$$354$$ −7742.59 −1.16247
$$355$$ 7956.81 1.18959
$$356$$ 21191.9i 3.15497i
$$357$$ 3134.13i 0.464638i
$$358$$ 22723.5i 3.35468i
$$359$$ − 3525.78i − 0.518339i −0.965832 0.259169i $$-0.916551\pi$$
0.965832 0.259169i $$-0.0834488\pi$$
$$360$$ −9773.99 −1.43093
$$361$$ 6853.95 0.999264
$$362$$ 21026.3i 3.05281i
$$363$$ −1709.68 −0.247204
$$364$$ 0 0
$$365$$ 15720.1 2.25433
$$366$$ − 7100.42i − 1.01406i
$$367$$ 2383.75 0.339049 0.169525 0.985526i $$-0.445777\pi$$
0.169525 + 0.985526i $$0.445777\pi$$
$$368$$ −7918.19 −1.12164
$$369$$ 276.707i 0.0390374i
$$370$$ 8610.90i 1.20989i
$$371$$ 4770.39i 0.667564i
$$372$$ − 11753.8i − 1.63818i
$$373$$ −13282.2 −1.84377 −0.921885 0.387463i $$-0.873352\pi$$
−0.921885 + 0.387463i $$0.873352\pi$$
$$374$$ −15871.5 −2.19437
$$375$$ 954.727i 0.131472i
$$376$$ 33843.1 4.64181
$$377$$ 0 0
$$378$$ −1393.11 −0.189560
$$379$$ 4436.73i 0.601318i 0.953732 + 0.300659i $$0.0972065\pi$$
−0.953732 + 0.300659i $$0.902793\pi$$
$$380$$ 752.917 0.101642
$$381$$ 4294.90 0.577519
$$382$$ 1141.28i 0.152862i
$$383$$ 810.412i 0.108120i 0.998538 + 0.0540602i $$0.0172163\pi$$
−0.998538 + 0.0540602i $$0.982784\pi$$
$$384$$ − 5177.83i − 0.688100i
$$385$$ − 4383.03i − 0.580207i
$$386$$ −6434.78 −0.848501
$$387$$ −2144.93 −0.281739
$$388$$ 1338.35i 0.175115i
$$389$$ −3463.79 −0.451469 −0.225734 0.974189i $$-0.572478\pi$$
−0.225734 + 0.974189i $$0.572478\pi$$
$$390$$ 0 0
$$391$$ 4512.37 0.583633
$$392$$ 16495.6i 2.12539i
$$393$$ −6201.96 −0.796050
$$394$$ 4944.64 0.632252
$$395$$ − 6158.45i − 0.784469i
$$396$$ − 5068.44i − 0.643178i
$$397$$ 425.405i 0.0537796i 0.999638 + 0.0268898i $$0.00856031\pi$$
−0.999638 + 0.0268898i $$0.991440\pi$$
$$398$$ 2553.00i 0.321533i
$$399$$ 65.2575 0.00818787
$$400$$ −27345.9 −3.41823
$$401$$ − 1186.85i − 0.147801i −0.997266 0.0739007i $$-0.976455\pi$$
0.997266 0.0739007i $$-0.0235448\pi$$
$$402$$ 3040.14 0.377185
$$403$$ 0 0
$$404$$ 10855.6 1.33685
$$405$$ 1329.46i 0.163115i
$$406$$ 3179.44 0.388653
$$407$$ −2715.33 −0.330697
$$408$$ 21423.7i 2.59959i
$$409$$ − 8007.42i − 0.968071i −0.875048 0.484036i $$-0.839170\pi$$
0.875048 0.484036i $$-0.160830\pi$$
$$410$$ 2689.84i 0.324005i
$$411$$ 1162.54i 0.139522i
$$412$$ −15023.7 −1.79652
$$413$$ −4686.70 −0.558395
$$414$$ 2005.73i 0.238106i
$$415$$ −11747.9 −1.38960
$$416$$ 0 0
$$417$$ −2257.70 −0.265132
$$418$$ 330.468i 0.0386692i
$$419$$ 6832.46 0.796629 0.398314 0.917249i $$-0.369595\pi$$
0.398314 + 0.917249i $$0.369595\pi$$
$$420$$ −9729.30 −1.13034
$$421$$ − 10739.6i − 1.24326i −0.783309 0.621632i $$-0.786470\pi$$
0.783309 0.621632i $$-0.213530\pi$$
$$422$$ 7734.16i 0.892163i
$$423$$ − 4603.34i − 0.529130i
$$424$$ 32608.6i 3.73494i
$$425$$ 15583.7 1.77864
$$426$$ 7752.28 0.881688
$$427$$ − 4297.99i − 0.487106i
$$428$$ 15988.9 1.80573
$$429$$ 0 0
$$430$$ −20850.7 −2.33839
$$431$$ − 5214.45i − 0.582763i −0.956607 0.291382i $$-0.905885\pi$$
0.956607 0.291382i $$-0.0941150\pi$$
$$432$$ −5113.52 −0.569501
$$433$$ −8642.24 −0.959168 −0.479584 0.877496i $$-0.659212\pi$$
−0.479584 + 0.877496i $$0.659212\pi$$
$$434$$ − 9903.02i − 1.09530i
$$435$$ − 3034.19i − 0.334432i
$$436$$ − 10866.7i − 1.19362i
$$437$$ − 93.9545i − 0.0102848i
$$438$$ 15316.0 1.67084
$$439$$ 13026.2 1.41619 0.708097 0.706116i $$-0.249554\pi$$
0.708097 + 0.706116i $$0.249554\pi$$
$$440$$ − 29960.7i − 3.24619i
$$441$$ 2243.73 0.242278
$$442$$ 0 0
$$443$$ −11533.0 −1.23690 −0.618450 0.785824i $$-0.712239\pi$$
−0.618450 + 0.785824i $$0.712239\pi$$
$$444$$ 6027.39i 0.644250i
$$445$$ 17039.3 1.81515
$$446$$ 10979.5 1.16568
$$447$$ − 7910.17i − 0.836998i
$$448$$ − 10110.2i − 1.06621i
$$449$$ 9882.75i 1.03874i 0.854548 + 0.519372i $$0.173834\pi$$
−0.854548 + 0.519372i $$0.826166\pi$$
$$450$$ 6926.88i 0.725636i
$$451$$ −848.204 −0.0885596
$$452$$ 3686.38 0.383613
$$453$$ 9997.26i 1.03689i
$$454$$ −23893.3 −2.46997
$$455$$ 0 0
$$456$$ 446.075 0.0458101
$$457$$ 15628.1i 1.59967i 0.600218 + 0.799836i $$0.295080\pi$$
−0.600218 + 0.799836i $$0.704920\pi$$
$$458$$ −8690.63 −0.886652
$$459$$ 2914.06 0.296333
$$460$$ 14007.8i 1.41982i
$$461$$ 7747.46i 0.782723i 0.920237 + 0.391361i $$0.127996\pi$$
−0.920237 + 0.391361i $$0.872004\pi$$
$$462$$ − 4270.36i − 0.430033i
$$463$$ − 333.422i − 0.0334675i −0.999860 0.0167337i $$-0.994673\pi$$
0.999860 0.0167337i $$-0.00532676\pi$$
$$464$$ 11670.4 1.16764
$$465$$ −9450.59 −0.942496
$$466$$ − 10147.4i − 1.00874i
$$467$$ −8198.33 −0.812363 −0.406182 0.913792i $$-0.633140\pi$$
−0.406182 + 0.913792i $$0.633140\pi$$
$$468$$ 0 0
$$469$$ 1840.24 0.181182
$$470$$ − 44748.7i − 4.39171i
$$471$$ 4875.79 0.476995
$$472$$ −32036.5 −3.12415
$$473$$ − 6574.96i − 0.639148i
$$474$$ − 6000.14i − 0.581425i
$$475$$ − 324.477i − 0.0313432i
$$476$$ 21325.8i 2.05350i
$$477$$ 4435.43 0.425753
$$478$$ −20062.5 −1.91974
$$479$$ − 6435.88i − 0.613910i −0.951724 0.306955i $$-0.900690\pi$$
0.951724 0.306955i $$-0.0993101\pi$$
$$480$$ −23644.1 −2.24833
$$481$$ 0 0
$$482$$ −19268.0 −1.82082
$$483$$ 1214.09i 0.114375i
$$484$$ −11633.3 −1.09253
$$485$$ 1076.10 0.100749
$$486$$ 1295.29i 0.120896i
$$487$$ − 8095.37i − 0.753257i −0.926364 0.376629i $$-0.877083\pi$$
0.926364 0.376629i $$-0.122917\pi$$
$$488$$ − 29379.4i − 2.72529i
$$489$$ 5506.10i 0.509191i
$$490$$ 21811.1 2.01087
$$491$$ −5116.46 −0.470270 −0.235135 0.971963i $$-0.575553\pi$$
−0.235135 + 0.971963i $$0.575553\pi$$
$$492$$ 1882.82i 0.172528i
$$493$$ −6650.67 −0.607568
$$494$$ 0 0
$$495$$ −4075.26 −0.370039
$$496$$ − 36349.9i − 3.29064i
$$497$$ 4692.56 0.423521
$$498$$ −11445.9 −1.02993
$$499$$ 18050.7i 1.61936i 0.586870 + 0.809682i $$0.300360\pi$$
−0.586870 + 0.809682i $$0.699640\pi$$
$$500$$ 6496.31i 0.581047i
$$501$$ − 5834.99i − 0.520336i
$$502$$ − 30541.9i − 2.71544i
$$503$$ 10531.1 0.933512 0.466756 0.884386i $$-0.345423\pi$$
0.466756 + 0.884386i $$0.345423\pi$$
$$504$$ −5764.25 −0.509445
$$505$$ − 8728.45i − 0.769131i
$$506$$ −6148.25 −0.540165
$$507$$ 0 0
$$508$$ 29224.1 2.55238
$$509$$ 1963.31i 0.170967i 0.996340 + 0.0854834i $$0.0272435\pi$$
−0.996340 + 0.0854834i $$0.972757\pi$$
$$510$$ 28327.3 2.45952
$$511$$ 9271.00 0.802593
$$512$$ 9307.69i 0.803409i
$$513$$ − 60.6753i − 0.00522198i
$$514$$ − 29454.6i − 2.52760i
$$515$$ 12079.8i 1.03359i
$$516$$ −14594.9 −1.24516
$$517$$ 14110.9 1.20038
$$518$$ 5078.31i 0.430750i
$$519$$ 7594.89 0.642348
$$520$$ 0 0
$$521$$ −7044.93 −0.592407 −0.296203 0.955125i $$-0.595721\pi$$
−0.296203 + 0.955125i $$0.595721\pi$$
$$522$$ − 2956.19i − 0.247872i
$$523$$ −3213.29 −0.268657 −0.134328 0.990937i $$-0.542888\pi$$
−0.134328 + 0.990937i $$0.542888\pi$$
$$524$$ −42200.4 −3.51819
$$525$$ 4192.94i 0.348561i
$$526$$ − 27841.7i − 2.30790i
$$527$$ 20714.9i 1.71225i
$$528$$ − 15674.7i − 1.29196i
$$529$$ −10419.0 −0.856333
$$530$$ 43116.4 3.53369
$$531$$ 4357.61i 0.356128i
$$532$$ 444.036 0.0361868
$$533$$ 0 0
$$534$$ 16601.3 1.34533
$$535$$ − 12855.8i − 1.03889i
$$536$$ 12579.2 1.01369
$$537$$ 12789.0 1.02772
$$538$$ − 38399.5i − 3.07718i
$$539$$ 6877.83i 0.549627i
$$540$$ 9046.13i 0.720895i
$$541$$ 11251.4i 0.894150i 0.894497 + 0.447075i $$0.147534\pi$$
−0.894497 + 0.447075i $$0.852466\pi$$
$$542$$ −45722.4 −3.62352
$$543$$ 11833.8 0.935245
$$544$$ 51825.8i 4.08458i
$$545$$ −8737.33 −0.686727
$$546$$ 0 0
$$547$$ 1533.54 0.119871 0.0599353 0.998202i $$-0.480911\pi$$
0.0599353 + 0.998202i $$0.480911\pi$$
$$548$$ 7910.31i 0.616628i
$$549$$ −3996.19 −0.310662
$$550$$ −21233.3 −1.64617
$$551$$ 138.477i 0.0107066i
$$552$$ 8299.08i 0.639914i
$$553$$ − 3631.97i − 0.279289i
$$554$$ 38214.6i 2.93066i
$$555$$ 4846.30 0.370656
$$556$$ −15362.2 −1.17177
$$557$$ 16845.7i 1.28146i 0.767766 + 0.640731i $$0.221369\pi$$
−0.767766 + 0.640731i $$0.778631\pi$$
$$558$$ −9207.65 −0.698550
$$559$$ 0 0
$$560$$ −30089.0 −2.27052
$$561$$ 8932.62i 0.672256i
$$562$$ 4526.34 0.339737
$$563$$ 20820.1 1.55855 0.779273 0.626685i $$-0.215589\pi$$
0.779273 + 0.626685i $$0.215589\pi$$
$$564$$ − 31322.8i − 2.33852i
$$565$$ − 2964.03i − 0.220704i
$$566$$ 5945.37i 0.441524i
$$567$$ 784.054i 0.0580726i
$$568$$ 32076.6 2.36955
$$569$$ −23636.6 −1.74147 −0.870735 0.491752i $$-0.836357\pi$$
−0.870735 + 0.491752i $$0.836357\pi$$
$$570$$ − 589.819i − 0.0433417i
$$571$$ 26955.1 1.97554 0.987771 0.155913i $$-0.0498319\pi$$
0.987771 + 0.155913i $$0.0498319\pi$$
$$572$$ 0 0
$$573$$ 642.326 0.0468300
$$574$$ 1586.35i 0.115353i
$$575$$ 6036.78 0.437828
$$576$$ −9400.25 −0.679995
$$577$$ − 23499.8i − 1.69551i −0.530388 0.847755i $$-0.677954\pi$$
0.530388 0.847755i $$-0.322046\pi$$
$$578$$ − 35902.8i − 2.58367i
$$579$$ 3621.56i 0.259943i
$$580$$ − 20645.7i − 1.47805i
$$581$$ −6928.38 −0.494729
$$582$$ 1048.44 0.0746721
$$583$$ 13596.1i 0.965857i
$$584$$ 63373.1 4.49040
$$585$$ 0 0
$$586$$ −9933.33 −0.700243
$$587$$ − 4637.50i − 0.326082i −0.986619 0.163041i $$-0.947870\pi$$
0.986619 0.163041i $$-0.0521303\pi$$
$$588$$ 15267.2 1.07076
$$589$$ 431.316 0.0301733
$$590$$ 42359.9i 2.95582i
$$591$$ − 2782.89i − 0.193694i
$$592$$ 18640.4i 1.29411i
$$593$$ 12633.5i 0.874869i 0.899250 + 0.437434i $$0.144113\pi$$
−0.899250 + 0.437434i $$0.855887\pi$$
$$594$$ −3970.51 −0.274262
$$595$$ 17146.9 1.18144
$$596$$ − 53823.7i − 3.69917i
$$597$$ 1436.85 0.0985033
$$598$$ 0 0
$$599$$ −18757.1 −1.27946 −0.639730 0.768600i $$-0.720954\pi$$
−0.639730 + 0.768600i $$0.720954\pi$$
$$600$$ 28661.3i 1.95016i
$$601$$ −3632.98 −0.246576 −0.123288 0.992371i $$-0.539344\pi$$
−0.123288 + 0.992371i $$0.539344\pi$$
$$602$$ −12296.8 −0.832522
$$603$$ − 1711.02i − 0.115553i
$$604$$ 68025.0i 4.58261i
$$605$$ 9353.71i 0.628566i
$$606$$ − 8504.08i − 0.570057i
$$607$$ −12700.0 −0.849219 −0.424610 0.905377i $$-0.639589\pi$$
−0.424610 + 0.905377i $$0.639589\pi$$
$$608$$ 1079.09 0.0719786
$$609$$ − 1789.42i − 0.119066i
$$610$$ −38846.6 −2.57845
$$611$$ 0 0
$$612$$ 19828.3 1.30966
$$613$$ − 21640.1i − 1.42584i −0.701248 0.712918i $$-0.747373\pi$$
0.701248 0.712918i $$-0.252627\pi$$
$$614$$ −34047.9 −2.23788
$$615$$ 1513.87 0.0992605
$$616$$ − 17669.5i − 1.15572i
$$617$$ 16541.7i 1.07933i 0.841881 + 0.539663i $$0.181448\pi$$
−0.841881 + 0.539663i $$0.818552\pi$$
$$618$$ 11769.2i 0.766066i
$$619$$ − 21138.9i − 1.37261i −0.727316 0.686303i $$-0.759233\pi$$
0.727316 0.686303i $$-0.240767\pi$$
$$620$$ −64305.2 −4.16542
$$621$$ 1128.84 0.0729451
$$622$$ − 18616.9i − 1.20011i
$$623$$ 10049.0 0.646235
$$624$$ 0 0
$$625$$ −12825.4 −0.820823
$$626$$ 31513.3i 2.01202i
$$627$$ 185.991 0.0118465
$$628$$ 33176.6 2.10811
$$629$$ − 10622.7i − 0.673377i
$$630$$ 7621.73i 0.481995i
$$631$$ 5489.80i 0.346348i 0.984891 + 0.173174i $$0.0554023\pi$$
−0.984891 + 0.173174i $$0.944598\pi$$
$$632$$ − 24826.8i − 1.56259i
$$633$$ 4352.86 0.273319
$$634$$ 8941.94 0.560141
$$635$$ − 23497.5i − 1.46846i
$$636$$ 30180.3 1.88164
$$637$$ 0 0
$$638$$ 9061.76 0.562318
$$639$$ − 4363.06i − 0.270110i
$$640$$ −28328.1 −1.74963
$$641$$ −4297.04 −0.264778 −0.132389 0.991198i $$-0.542265\pi$$
−0.132389 + 0.991198i $$0.542265\pi$$
$$642$$ − 12525.3i − 0.769993i
$$643$$ − 25696.9i − 1.57603i −0.615655 0.788016i $$-0.711108\pi$$
0.615655 0.788016i $$-0.288892\pi$$
$$644$$ 8261.14i 0.505488i
$$645$$ 11735.0i 0.716378i
$$646$$ −1292.83 −0.0787396
$$647$$ −2174.98 −0.132160 −0.0660798 0.997814i $$-0.521049\pi$$
−0.0660798 + 0.997814i $$0.521049\pi$$
$$648$$ 5359.50i 0.324909i
$$649$$ −13357.6 −0.807907
$$650$$ 0 0
$$651$$ −5573.52 −0.335551
$$652$$ 37465.5i 2.25040i
$$653$$ −15454.5 −0.926160 −0.463080 0.886316i $$-0.653256\pi$$
−0.463080 + 0.886316i $$0.653256\pi$$
$$654$$ −8512.73 −0.508982
$$655$$ 33931.1i 2.02412i
$$656$$ 5822.82i 0.346560i
$$657$$ − 8620.02i − 0.511870i
$$658$$ − 26390.7i − 1.56355i
$$659$$ −3148.77 −0.186129 −0.0930643 0.995660i $$-0.529666\pi$$
−0.0930643 + 0.995660i $$0.529666\pi$$
$$660$$ −27729.6 −1.63541
$$661$$ − 2099.70i − 0.123553i −0.998090 0.0617767i $$-0.980323\pi$$
0.998090 0.0617767i $$-0.0196767\pi$$
$$662$$ −10715.7 −0.629122
$$663$$ 0 0
$$664$$ −47359.8 −2.76795
$$665$$ − 357.026i − 0.0208193i
$$666$$ 4721.73 0.274720
$$667$$ −2576.32 −0.149559
$$668$$ − 39703.4i − 2.29966i
$$669$$ − 6179.36i − 0.357112i
$$670$$ − 16632.7i − 0.959071i
$$671$$ − 12249.7i − 0.704763i
$$672$$ −13944.2 −0.800460
$$673$$ −30970.8 −1.77390 −0.886950 0.461865i $$-0.847181\pi$$
−0.886950 + 0.461865i $$0.847181\pi$$
$$674$$ 38055.0i 2.17481i
$$675$$ 3898.52 0.222302
$$676$$ 0 0
$$677$$ 14640.6 0.831141 0.415570 0.909561i $$-0.363582\pi$$
0.415570 + 0.909561i $$0.363582\pi$$
$$678$$ − 2887.83i − 0.163579i
$$679$$ 634.635 0.0358690
$$680$$ 117210. 6.60999
$$681$$ 13447.4i 0.756688i
$$682$$ − 28224.7i − 1.58472i
$$683$$ − 6685.83i − 0.374563i −0.982306 0.187281i $$-0.940032\pi$$
0.982306 0.187281i $$-0.0599676\pi$$
$$684$$ − 412.856i − 0.0230789i
$$685$$ 6360.27 0.354764
$$686$$ 30560.8 1.70090
$$687$$ 4891.18i 0.271630i
$$688$$ −45136.3 −2.50117
$$689$$ 0 0
$$690$$ 10973.4 0.605434
$$691$$ 30194.1i 1.66228i 0.556062 + 0.831141i $$0.312312\pi$$
−0.556062 + 0.831141i $$0.687688\pi$$
$$692$$ 51678.4 2.83890
$$693$$ −2403.40 −0.131743
$$694$$ − 6.07611i 0 0.000332343i
$$695$$ 12352.0i 0.674154i
$$696$$ − 12231.8i − 0.666158i
$$697$$ − 3318.28i − 0.180328i
$$698$$ 65026.0 3.52618
$$699$$ −5711.08 −0.309031
$$700$$ 28530.3i 1.54049i
$$701$$ 30300.9 1.63260 0.816298 0.577631i $$-0.196023\pi$$
0.816298 + 0.577631i $$0.196023\pi$$
$$702$$ 0 0
$$703$$ −221.181 −0.0118663
$$704$$ − 28815.1i − 1.54263i
$$705$$ −25185.0 −1.34542
$$706$$ 58060.3 3.09508
$$707$$ − 5147.64i − 0.273829i
$$708$$ 29650.8i 1.57393i
$$709$$ 26123.2i 1.38375i 0.722017 + 0.691875i $$0.243215\pi$$
−0.722017 + 0.691875i $$0.756785\pi$$
$$710$$ − 42412.9i − 2.24187i
$$711$$ −3376.94 −0.178123
$$712$$ 68691.1 3.61560
$$713$$ 8024.48i 0.421486i
$$714$$ 16706.2 0.875647
$$715$$ 0 0
$$716$$ 87021.3 4.54209
$$717$$ 11291.4i 0.588122i
$$718$$ −18793.8 −0.976850
$$719$$ −19325.7 −1.00240 −0.501200 0.865331i $$-0.667108\pi$$
−0.501200 + 0.865331i $$0.667108\pi$$
$$720$$ 27976.2i 1.44807i
$$721$$ 7124.09i 0.367982i
$$722$$ − 36534.2i − 1.88319i
$$723$$ 10844.2i 0.557816i
$$724$$ 80521.7 4.13338
$$725$$ −8897.47 −0.455784
$$726$$ 9113.26i 0.465875i
$$727$$ −26065.8 −1.32975 −0.664875 0.746954i $$-0.731515\pi$$
−0.664875 + 0.746954i $$0.731515\pi$$
$$728$$ 0 0
$$729$$ 729.000 0.0370370
$$730$$ − 83794.4i − 4.24845i
$$731$$ 25722.0 1.30145
$$732$$ −27191.5 −1.37299
$$733$$ − 1055.45i − 0.0531843i −0.999646 0.0265921i $$-0.991534\pi$$
0.999646 0.0265921i $$-0.00846554\pi$$
$$734$$ − 12706.4i − 0.638965i
$$735$$ − 12275.5i − 0.616041i
$$736$$ 20076.2i 1.00546i
$$737$$ 5244.89 0.262141
$$738$$ 1474.96 0.0735690
$$739$$ 9410.40i 0.468426i 0.972185 + 0.234213i $$0.0752514\pi$$
−0.972185 + 0.234213i $$0.924749\pi$$
$$740$$ 32976.0 1.63814
$$741$$ 0 0
$$742$$ 25428.1 1.25808
$$743$$ 7523.70i 0.371491i 0.982598 + 0.185746i $$0.0594700\pi$$
−0.982598 + 0.185746i $$0.940530\pi$$
$$744$$ −38098.5 −1.87736
$$745$$ −43276.8 −2.12824
$$746$$ 70799.4i 3.47473i
$$747$$ 6441.89i 0.315524i
$$748$$ 60780.8i 2.97108i
$$749$$ − 7581.76i − 0.369868i
$$750$$ 5089.07 0.247769
$$751$$ 12984.1 0.630886 0.315443 0.948945i $$-0.397847\pi$$
0.315443 + 0.948945i $$0.397847\pi$$
$$752$$ − 96869.3i − 4.69742i
$$753$$ −17189.3 −0.831890
$$754$$ 0 0
$$755$$ 54695.3 2.63651
$$756$$ 5334.99i 0.256656i
$$757$$ −27934.6 −1.34122 −0.670609 0.741811i $$-0.733967\pi$$
−0.670609 + 0.741811i $$0.733967\pi$$
$$758$$ 23649.5 1.13323
$$759$$ 3460.30i 0.165482i
$$760$$ − 2440.49i − 0.116482i
$$761$$ − 15519.3i − 0.739255i −0.929180 0.369627i $$-0.879485\pi$$
0.929180 0.369627i $$-0.120515\pi$$
$$762$$ − 22893.5i − 1.08838i
$$763$$ −5152.88 −0.244491
$$764$$ 4370.62 0.206968
$$765$$ − 15942.9i − 0.753487i
$$766$$ 4319.82 0.203761
$$767$$ 0 0
$$768$$ −2532.57 −0.118993
$$769$$ − 12885.2i − 0.604228i −0.953272 0.302114i $$-0.902308\pi$$
0.953272 0.302114i $$-0.0976924\pi$$
$$770$$ −23363.3 −1.09345
$$771$$ −16577.4 −0.774344
$$772$$ 24642.4i 1.14883i
$$773$$ − 5892.04i − 0.274155i −0.990560 0.137078i $$-0.956229\pi$$
0.990560 0.137078i $$-0.0437710\pi$$
$$774$$ 11433.3i 0.530958i
$$775$$ 27713.0i 1.28449i
$$776$$ 4338.12 0.200682
$$777$$ 2858.13 0.131962
$$778$$ 18463.4i 0.850828i
$$779$$ −69.0916 −0.00317775
$$780$$ 0 0
$$781$$ 13374.3 0.612766
$$782$$ − 24052.7i − 1.09990i
$$783$$ −1663.77 −0.0759367
$$784$$ 47215.5 2.15085
$$785$$ − 26675.6i − 1.21286i
$$786$$ 33058.9i 1.50022i
$$787$$ − 21020.4i − 0.952091i −0.879421 0.476045i $$-0.842070\pi$$
0.879421 0.476045i $$-0.157930\pi$$
$$788$$ − 18935.8i − 0.856042i
$$789$$ −15669.6 −0.707038
$$790$$ −32827.0 −1.47839
$$791$$ − 1748.05i − 0.0785757i
$$792$$ −16428.7 −0.737084
$$793$$ 0 0
$$794$$ 2267.58 0.101352
$$795$$ − 24266.4i − 1.08256i
$$796$$ 9776.87 0.435342
$$797$$ −31355.6 −1.39356 −0.696782 0.717283i $$-0.745386\pi$$
−0.696782 + 0.717283i $$0.745386\pi$$
$$798$$ − 347.848i − 0.0154307i
$$799$$ 55203.3i 2.44425i
$$800$$ 69334.0i 3.06416i
$$801$$ − 9343.37i − 0.412150i
$$802$$ −6326.37 −0.278543
$$803$$ 26423.4 1.16122
$$804$$ − 11642.4i − 0.510692i
$$805$$ 6642.34 0.290822
$$806$$ 0 0
$$807$$ −21611.7 −0.942709
$$808$$ − 35187.3i − 1.53204i
$$809$$ −18132.5 −0.788017 −0.394009 0.919107i $$-0.628912\pi$$
−0.394009 + 0.919107i $$0.628912\pi$$
$$810$$ 7086.55 0.307402
$$811$$ 24755.3i 1.07186i 0.844263 + 0.535928i $$0.180038\pi$$
−0.844263 + 0.535928i $$0.819962\pi$$
$$812$$ − 12175.9i − 0.526219i
$$813$$ 25733.1i 1.11008i
$$814$$ 14473.8i 0.623225i
$$815$$ 30124.0 1.29472
$$816$$ 61321.4 2.63073
$$817$$ − 535.572i − 0.0229343i
$$818$$ −42682.7 −1.82441
$$819$$ 0 0
$$820$$ 10300.9 0.438688
$$821$$ − 4082.65i − 0.173551i −0.996228 0.0867755i $$-0.972344\pi$$
0.996228 0.0867755i $$-0.0276563\pi$$
$$822$$ 6196.77 0.262941
$$823$$ 34327.0 1.45391 0.726954 0.686687i $$-0.240935\pi$$
0.726954 + 0.686687i $$0.240935\pi$$
$$824$$ 48697.6i 2.05881i
$$825$$ 11950.3i 0.504312i
$$826$$ 24981.9i 1.05234i
$$827$$ 3228.87i 0.135767i 0.997693 + 0.0678833i $$0.0216245\pi$$
−0.997693 + 0.0678833i $$0.978375\pi$$
$$828$$ 7681.06 0.322386
$$829$$ 10452.4 0.437908 0.218954 0.975735i $$-0.429735\pi$$
0.218954 + 0.975735i $$0.429735\pi$$
$$830$$ 62621.0i 2.61880i
$$831$$ 21507.6 0.897821
$$832$$ 0 0
$$833$$ −26906.9 −1.11917
$$834$$ 12034.5i 0.499663i
$$835$$ −31923.4 −1.32306
$$836$$ 1265.55 0.0523564
$$837$$ 5182.16i 0.214004i
$$838$$ − 36419.7i − 1.50131i
$$839$$ − 28289.0i − 1.16406i −0.813169 0.582028i $$-0.802259\pi$$
0.813169 0.582028i $$-0.197741\pi$$
$$840$$ 31536.4i 1.29537i
$$841$$ −20591.8 −0.844308
$$842$$ −57246.1 −2.34303
$$843$$ − 2547.47i − 0.104080i
$$844$$ 29618.5 1.20795
$$845$$ 0 0
$$846$$ −24537.6 −0.997187
$$847$$ 5516.39i 0.223784i
$$848$$ 93335.9 3.77968
$$849$$ 3346.12 0.135263
$$850$$ − 83067.2i − 3.35198i
$$851$$ − 4114.99i − 0.165758i
$$852$$ − 29687.8i − 1.19377i
$$853$$ − 26631.8i − 1.06900i −0.845170 0.534498i $$-0.820501\pi$$
0.845170 0.534498i $$-0.179499\pi$$
$$854$$ −22910.0 −0.917989
$$855$$ −331.956 −0.0132780
$$856$$ − 51826.0i − 2.06936i
$$857$$ −11796.7 −0.470209 −0.235104 0.971970i $$-0.575543\pi$$
−0.235104 + 0.971970i $$0.575543\pi$$
$$858$$ 0 0
$$859$$ −22672.8 −0.900567 −0.450283 0.892886i $$-0.648677\pi$$
−0.450283 + 0.892886i $$0.648677\pi$$
$$860$$ 79849.0i 3.16608i
$$861$$ 892.812 0.0353391
$$862$$ −27795.0 −1.09826
$$863$$ − 21421.1i − 0.844940i −0.906377 0.422470i $$-0.861163\pi$$
0.906377 0.422470i $$-0.138837\pi$$
$$864$$ 12965.1i 0.510510i
$$865$$ − 41551.9i − 1.63330i
$$866$$ 46066.6i 1.80763i
$$867$$ −20206.5 −0.791520
$$868$$ −37924.3 −1.48299
$$869$$ − 10351.5i − 0.404086i
$$870$$ −16173.4 −0.630264
$$871$$ 0 0
$$872$$ −35223.1 −1.36790
$$873$$ − 590.072i − 0.0228762i
$$874$$ −500.814 −0.0193825
$$875$$ 3080.48 0.119016
$$876$$ − 58653.7i − 2.26224i
$$877$$ 5155.20i 0.198493i 0.995063 + 0.0992466i $$0.0316433\pi$$
−0.995063 + 0.0992466i $$0.968357\pi$$
$$878$$ − 69435.0i − 2.66893i
$$879$$ 5590.58i 0.214523i
$$880$$ −85756.9 −3.28507
$$881$$ −23692.2 −0.906027 −0.453013 0.891504i $$-0.649651\pi$$
−0.453013 + 0.891504i $$0.649651\pi$$
$$882$$ − 11960.0i − 0.456591i
$$883$$ 14591.5 0.556108 0.278054 0.960565i $$-0.410311\pi$$
0.278054 + 0.960565i $$0.410311\pi$$
$$884$$ 0 0
$$885$$ 23840.6 0.905529
$$886$$ 61475.2i 2.33104i
$$887$$ −9722.26 −0.368029 −0.184014 0.982924i $$-0.558909\pi$$
−0.184014 + 0.982924i $$0.558909\pi$$
$$888$$ 19537.1 0.738312
$$889$$ − 13857.8i − 0.522806i
$$890$$ − 90826.1i − 3.42078i
$$891$$ 2234.64i 0.0840217i
$$892$$ − 42046.6i − 1.57828i
$$893$$ 1149.42 0.0430726
$$894$$ −42164.3 −1.57739
$$895$$ − 69969.2i − 2.61320i
$$896$$ −16706.6 −0.622911
$$897$$ 0 0
$$898$$ 52678.9 1.95759
$$899$$ − 11827.1i − 0.438771i
$$900$$ 26526.9 0.982479
$$901$$ −53189.7 −1.96671
$$902$$ 4521.26i 0.166898i
$$903$$ 6920.74i 0.255048i
$$904$$ − 11949.0i − 0.439621i
$$905$$ − 64743.2i − 2.37805i
$$906$$ 53289.3 1.95411
$$907$$ −11799.0 −0.431951 −0.215975 0.976399i $$-0.569293\pi$$
−0.215975 + 0.976399i $$0.569293\pi$$
$$908$$ 91500.9i 3.34423i
$$909$$ −4786.18 −0.174640
$$910$$ 0 0
$$911$$ −43012.4 −1.56429 −0.782143 0.623099i $$-0.785873\pi$$
−0.782143 + 0.623099i $$0.785873\pi$$
$$912$$ − 1276.81i − 0.0463589i
$$913$$ −19746.6 −0.715793
$$914$$ 83303.8 3.01471
$$915$$ 21863.3i 0.789921i
$$916$$ 33281.3i 1.20049i
$$917$$ 20011.0i 0.720635i
$$918$$ − 15533.1i − 0.558462i
$$919$$ −4951.41 −0.177728 −0.0888639 0.996044i $$-0.528324\pi$$
−0.0888639 + 0.996044i $$0.528324\pi$$
$$920$$ 45404.5 1.62711
$$921$$ 19162.5i 0.685587i
$$922$$ 41297.0 1.47510
$$923$$ 0 0
$$924$$ −16353.6 −0.582245
$$925$$ − 14211.3i − 0.505152i
$$926$$ −1777.27 −0.0630721
$$927$$ 6623.85 0.234688
$$928$$ − 29589.8i − 1.04669i
$$929$$ 8934.86i 0.315547i 0.987475 + 0.157774i $$0.0504316\pi$$
−0.987475 + 0.157774i $$0.949568\pi$$
$$930$$ 50375.4i 1.77621i
$$931$$ 560.243i 0.0197221i
$$932$$ −38860.2 −1.36578
$$933$$ −10477.8 −0.367660
$$934$$ 43700.3i 1.53096i
$$935$$ 48870.6 1.70935
$$936$$ 0 0
$$937$$ −13182.8 −0.459620 −0.229810 0.973235i $$-0.573811\pi$$
−0.229810 + 0.973235i $$0.573811\pi$$
$$938$$ − 9809.20i − 0.341452i
$$939$$ 17736.0 0.616393
$$940$$ −171368. −5.94618
$$941$$ 21693.7i 0.751536i 0.926714 + 0.375768i $$0.122621\pi$$
−0.926714 + 0.375768i $$0.877379\pi$$
$$942$$ − 25989.9i − 0.898934i
$$943$$ − 1285.43i − 0.0443895i
$$944$$ 91698.4i 3.16158i
$$945$$ 4289.59 0.147662
$$946$$ −35047.1 −1.20452
$$947$$ − 49790.0i − 1.70851i −0.519856 0.854254i $$-0.674014\pi$$
0.519856 0.854254i $$-0.325986\pi$$
$$948$$ −22977.9 −0.787224
$$949$$ 0 0
$$950$$ −1729.59 −0.0590687
$$951$$ − 5032.61i − 0.171602i
$$952$$ 69125.0 2.35331
$$953$$ −4217.93 −0.143371 −0.0716853 0.997427i $$-0.522838\pi$$
−0.0716853 + 0.997427i $$0.522838\pi$$
$$954$$ − 23642.6i − 0.802365i
$$955$$ − 3514.19i − 0.119075i
$$956$$ 76830.5i 2.59924i
$$957$$ − 5100.05i − 0.172269i
$$958$$ −34305.8 −1.15696
$$959$$ 3750.99 0.126304
$$960$$ 51429.0i 1.72903i
$$961$$ −7046.87 −0.236544
$$962$$ 0 0
$$963$$ −7049.38 −0.235891
$$964$$ 73788.1i 2.46530i
$$965$$ 19813.7 0.660958
$$966$$ 6471.60 0.215549
$$967$$ 40927.9i 1.36107i 0.732717 + 0.680534i $$0.238252\pi$$
−0.732717 + 0.680534i $$0.761748\pi$$
$$968$$ 37707.9i 1.25204i
$$969$$ 727.619i 0.0241223i
$$970$$ − 5736.04i − 0.189869i
$$971$$ −17114.8 −0.565645 −0.282822 0.959172i $$-0.591271\pi$$
−0.282822 + 0.959172i $$0.591271\pi$$
$$972$$ 4960.38 0.163688
$$973$$ 7284.62i 0.240015i
$$974$$ −43151.5 −1.41957
$$975$$ 0 0
$$976$$ −84093.0 −2.75794
$$977$$ 118.470i 0.00387940i 0.999998 + 0.00193970i $$0.000617427\pi$$
−0.999998 + 0.00193970i $$0.999383\pi$$
$$978$$ 29349.7 0.959610
$$979$$ 28640.7 0.934996
$$980$$ − 83527.2i − 2.72263i
$$981$$ 4791.05i 0.155929i
$$982$$ 27272.8i 0.886261i
$$983$$ 26002.8i 0.843705i 0.906665 + 0.421852i $$0.138620\pi$$
−0.906665 + 0.421852i $$0.861380\pi$$
$$984$$ 6102.93 0.197718
$$985$$ −15225.3 −0.492506
$$986$$ 35450.7i 1.14501i
$$987$$ −14853.0 −0.479002
$$988$$ 0 0
$$989$$ 9964.14 0.320365
$$990$$ 21722.8i 0.697368i
$$991$$ −16062.0 −0.514860 −0.257430 0.966297i $$-0.582876\pi$$
−0.257430 + 0.966297i $$0.582876\pi$$
$$992$$ −92163.3 −2.94979
$$993$$ 6030.93i 0.192735i
$$994$$ − 25013.2i − 0.798159i
$$995$$ − 7861.07i − 0.250465i
$$996$$ 43832.9i 1.39448i
$$997$$ 1361.54 0.0432503 0.0216251 0.999766i $$-0.493116\pi$$
0.0216251 + 0.999766i $$0.493116\pi$$
$$998$$ 96217.5 3.05182
$$999$$ − 2657.44i − 0.0841617i
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 507.4.b.h.337.1 8
13.5 odd 4 507.4.a.i.1.1 4
13.7 odd 12 39.4.e.c.16.1 8
13.8 odd 4 507.4.a.m.1.4 4
13.11 odd 12 39.4.e.c.22.1 yes 8
13.12 even 2 inner 507.4.b.h.337.8 8
39.5 even 4 1521.4.a.bb.1.4 4
39.8 even 4 1521.4.a.v.1.1 4
39.11 even 12 117.4.g.e.100.4 8
39.20 even 12 117.4.g.e.55.4 8
52.7 even 12 624.4.q.i.289.1 8
52.11 even 12 624.4.q.i.529.1 8

By twisted newform
Twist Min Dim Char Parity Ord Type
39.4.e.c.16.1 8 13.7 odd 12
39.4.e.c.22.1 yes 8 13.11 odd 12
117.4.g.e.55.4 8 39.20 even 12
117.4.g.e.100.4 8 39.11 even 12
507.4.a.i.1.1 4 13.5 odd 4
507.4.a.m.1.4 4 13.8 odd 4
507.4.b.h.337.1 8 1.1 even 1 trivial
507.4.b.h.337.8 8 13.12 even 2 inner
624.4.q.i.289.1 8 52.7 even 12
624.4.q.i.529.1 8 52.11 even 12
1521.4.a.v.1.1 4 39.8 even 4
1521.4.a.bb.1.4 4 39.5 even 4