# Properties

 Label 175.6.b.a.99.2 Level $175$ Weight $6$ Character 175.99 Analytic conductor $28.067$ Analytic rank $0$ Dimension $2$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [175,6,Mod(99,175)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(175, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("175.99");

S:= CuspForms(chi, 6);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$175 = 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 175.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: $$28.0671684673$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 7) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 99.2 Root $$1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 175.99 Dual form 175.6.b.a.99.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+10.0000i q^{2} -14.0000i q^{3} -68.0000 q^{4} +140.000 q^{6} +49.0000i q^{7} -360.000i q^{8} +47.0000 q^{9} +O(q^{10})$$ $$q+10.0000i q^{2} -14.0000i q^{3} -68.0000 q^{4} +140.000 q^{6} +49.0000i q^{7} -360.000i q^{8} +47.0000 q^{9} +232.000 q^{11} +952.000i q^{12} -140.000i q^{13} -490.000 q^{14} +1424.00 q^{16} +1722.00i q^{17} +470.000i q^{18} +98.0000 q^{19} +686.000 q^{21} +2320.00i q^{22} +1824.00i q^{23} -5040.00 q^{24} +1400.00 q^{26} -4060.00i q^{27} -3332.00i q^{28} -3418.00 q^{29} -7644.00 q^{31} +2720.00i q^{32} -3248.00i q^{33} -17220.0 q^{34} -3196.00 q^{36} +10398.0i q^{37} +980.000i q^{38} -1960.00 q^{39} -17962.0 q^{41} +6860.00i q^{42} +10880.0i q^{43} -15776.0 q^{44} -18240.0 q^{46} -9324.00i q^{47} -19936.0i q^{48} -2401.00 q^{49} +24108.0 q^{51} +9520.00i q^{52} +2262.00i q^{53} +40600.0 q^{54} +17640.0 q^{56} -1372.00i q^{57} -34180.0i q^{58} +2730.00 q^{59} +25648.0 q^{61} -76440.0i q^{62} +2303.00i q^{63} +18368.0 q^{64} +32480.0 q^{66} +48404.0i q^{67} -117096. i q^{68} +25536.0 q^{69} -58560.0 q^{71} -16920.0i q^{72} +68082.0i q^{73} -103980. q^{74} -6664.00 q^{76} +11368.0i q^{77} -19600.0i q^{78} -31784.0 q^{79} -45419.0 q^{81} -179620. i q^{82} -20538.0i q^{83} -46648.0 q^{84} -108800. q^{86} +47852.0i q^{87} -83520.0i q^{88} +50582.0 q^{89} +6860.00 q^{91} -124032. i q^{92} +107016. i q^{93} +93240.0 q^{94} +38080.0 q^{96} +58506.0i q^{97} -24010.0i q^{98} +10904.0 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 136 q^{4} + 280 q^{6} + 94 q^{9}+O(q^{10})$$ 2 * q - 136 * q^4 + 280 * q^6 + 94 * q^9 $$2 q - 136 q^{4} + 280 q^{6} + 94 q^{9} + 464 q^{11} - 980 q^{14} + 2848 q^{16} + 196 q^{19} + 1372 q^{21} - 10080 q^{24} + 2800 q^{26} - 6836 q^{29} - 15288 q^{31} - 34440 q^{34} - 6392 q^{36} - 3920 q^{39} - 35924 q^{41} - 31552 q^{44} - 36480 q^{46} - 4802 q^{49} + 48216 q^{51} + 81200 q^{54} + 35280 q^{56} + 5460 q^{59} + 51296 q^{61} + 36736 q^{64} + 64960 q^{66} + 51072 q^{69} - 117120 q^{71} - 207960 q^{74} - 13328 q^{76} - 63568 q^{79} - 90838 q^{81} - 93296 q^{84} - 217600 q^{86} + 101164 q^{89} + 13720 q^{91} + 186480 q^{94} + 76160 q^{96} + 21808 q^{99}+O(q^{100})$$ 2 * q - 136 * q^4 + 280 * q^6 + 94 * q^9 + 464 * q^11 - 980 * q^14 + 2848 * q^16 + 196 * q^19 + 1372 * q^21 - 10080 * q^24 + 2800 * q^26 - 6836 * q^29 - 15288 * q^31 - 34440 * q^34 - 6392 * q^36 - 3920 * q^39 - 35924 * q^41 - 31552 * q^44 - 36480 * q^46 - 4802 * q^49 + 48216 * q^51 + 81200 * q^54 + 35280 * q^56 + 5460 * q^59 + 51296 * q^61 + 36736 * q^64 + 64960 * q^66 + 51072 * q^69 - 117120 * q^71 - 207960 * q^74 - 13328 * q^76 - 63568 * q^79 - 90838 * q^81 - 93296 * q^84 - 217600 * q^86 + 101164 * q^89 + 13720 * q^91 + 186480 * q^94 + 76160 * q^96 + 21808 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/175\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$127$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 10.0000i 1.76777i 0.467707 + 0.883883i $$0.345080\pi$$
−0.467707 + 0.883883i $$0.654920\pi$$
$$3$$ − 14.0000i − 0.898100i −0.893507 0.449050i $$-0.851762\pi$$
0.893507 0.449050i $$-0.148238\pi$$
$$4$$ −68.0000 −2.12500
$$5$$ 0 0
$$6$$ 140.000 1.58763
$$7$$ 49.0000i 0.377964i
$$8$$ − 360.000i − 1.98874i
$$9$$ 47.0000 0.193416
$$10$$ 0 0
$$11$$ 232.000 0.578104 0.289052 0.957313i $$-0.406660\pi$$
0.289052 + 0.957313i $$0.406660\pi$$
$$12$$ 952.000i 1.90846i
$$13$$ − 140.000i − 0.229757i −0.993380 0.114879i $$-0.963352\pi$$
0.993380 0.114879i $$-0.0366479\pi$$
$$14$$ −490.000 −0.668153
$$15$$ 0 0
$$16$$ 1424.00 1.39062
$$17$$ 1722.00i 1.44514i 0.691296 + 0.722572i $$0.257040\pi$$
−0.691296 + 0.722572i $$0.742960\pi$$
$$18$$ 470.000i 0.341914i
$$19$$ 98.0000 0.0622791 0.0311395 0.999515i $$-0.490086\pi$$
0.0311395 + 0.999515i $$0.490086\pi$$
$$20$$ 0 0
$$21$$ 686.000 0.339450
$$22$$ 2320.00i 1.02195i
$$23$$ 1824.00i 0.718961i 0.933153 + 0.359480i $$0.117046\pi$$
−0.933153 + 0.359480i $$0.882954\pi$$
$$24$$ −5040.00 −1.78609
$$25$$ 0 0
$$26$$ 1400.00 0.406158
$$27$$ − 4060.00i − 1.07181i
$$28$$ − 3332.00i − 0.803175i
$$29$$ −3418.00 −0.754705 −0.377352 0.926070i $$-0.623165\pi$$
−0.377352 + 0.926070i $$0.623165\pi$$
$$30$$ 0 0
$$31$$ −7644.00 −1.42862 −0.714310 0.699830i $$-0.753259\pi$$
−0.714310 + 0.699830i $$0.753259\pi$$
$$32$$ 2720.00i 0.469563i
$$33$$ − 3248.00i − 0.519196i
$$34$$ −17220.0 −2.55468
$$35$$ 0 0
$$36$$ −3196.00 −0.411008
$$37$$ 10398.0i 1.24866i 0.781159 + 0.624332i $$0.214629\pi$$
−0.781159 + 0.624332i $$0.785371\pi$$
$$38$$ 980.000i 0.110095i
$$39$$ −1960.00 −0.206345
$$40$$ 0 0
$$41$$ −17962.0 −1.66876 −0.834382 0.551186i $$-0.814175\pi$$
−0.834382 + 0.551186i $$0.814175\pi$$
$$42$$ 6860.00i 0.600069i
$$43$$ 10880.0i 0.897342i 0.893697 + 0.448671i $$0.148102\pi$$
−0.893697 + 0.448671i $$0.851898\pi$$
$$44$$ −15776.0 −1.22847
$$45$$ 0 0
$$46$$ −18240.0 −1.27096
$$47$$ − 9324.00i − 0.615684i −0.951438 0.307842i $$-0.900393\pi$$
0.951438 0.307842i $$-0.0996068\pi$$
$$48$$ − 19936.0i − 1.24892i
$$49$$ −2401.00 −0.142857
$$50$$ 0 0
$$51$$ 24108.0 1.29788
$$52$$ 9520.00i 0.488235i
$$53$$ 2262.00i 0.110612i 0.998469 + 0.0553061i $$0.0176135\pi$$
−0.998469 + 0.0553061i $$0.982387\pi$$
$$54$$ 40600.0 1.89471
$$55$$ 0 0
$$56$$ 17640.0 0.751672
$$57$$ − 1372.00i − 0.0559329i
$$58$$ − 34180.0i − 1.33414i
$$59$$ 2730.00 0.102102 0.0510508 0.998696i $$-0.483743\pi$$
0.0510508 + 0.998696i $$0.483743\pi$$
$$60$$ 0 0
$$61$$ 25648.0 0.882529 0.441264 0.897377i $$-0.354530\pi$$
0.441264 + 0.897377i $$0.354530\pi$$
$$62$$ − 76440.0i − 2.52547i
$$63$$ 2303.00i 0.0731042i
$$64$$ 18368.0 0.560547
$$65$$ 0 0
$$66$$ 32480.0 0.917817
$$67$$ 48404.0i 1.31733i 0.752437 + 0.658664i $$0.228878\pi$$
−0.752437 + 0.658664i $$0.771122\pi$$
$$68$$ − 117096.i − 3.07093i
$$69$$ 25536.0 0.645699
$$70$$ 0 0
$$71$$ −58560.0 −1.37865 −0.689327 0.724450i $$-0.742094\pi$$
−0.689327 + 0.724450i $$0.742094\pi$$
$$72$$ − 16920.0i − 0.384653i
$$73$$ 68082.0i 1.49529i 0.664099 + 0.747645i $$0.268815\pi$$
−0.664099 + 0.747645i $$0.731185\pi$$
$$74$$ −103980. −2.20735
$$75$$ 0 0
$$76$$ −6664.00 −0.132343
$$77$$ 11368.0i 0.218503i
$$78$$ − 19600.0i − 0.364770i
$$79$$ −31784.0 −0.572982 −0.286491 0.958083i $$-0.592489\pi$$
−0.286491 + 0.958083i $$0.592489\pi$$
$$80$$ 0 0
$$81$$ −45419.0 −0.769175
$$82$$ − 179620.i − 2.94999i
$$83$$ − 20538.0i − 0.327237i −0.986524 0.163619i $$-0.947683\pi$$
0.986524 0.163619i $$-0.0523167\pi$$
$$84$$ −46648.0 −0.721331
$$85$$ 0 0
$$86$$ −108800. −1.58629
$$87$$ 47852.0i 0.677801i
$$88$$ − 83520.0i − 1.14970i
$$89$$ 50582.0 0.676894 0.338447 0.940985i $$-0.390098\pi$$
0.338447 + 0.940985i $$0.390098\pi$$
$$90$$ 0 0
$$91$$ 6860.00 0.0868402
$$92$$ − 124032.i − 1.52779i
$$93$$ 107016.i 1.28304i
$$94$$ 93240.0 1.08839
$$95$$ 0 0
$$96$$ 38080.0 0.421715
$$97$$ 58506.0i 0.631351i 0.948867 + 0.315676i $$0.102231\pi$$
−0.948867 + 0.315676i $$0.897769\pi$$
$$98$$ − 24010.0i − 0.252538i
$$99$$ 10904.0 0.111814
$$100$$ 0 0
$$101$$ 38696.0 0.377453 0.188726 0.982030i $$-0.439564\pi$$
0.188726 + 0.982030i $$0.439564\pi$$
$$102$$ 241080.i 2.29436i
$$103$$ 53060.0i 0.492804i 0.969168 + 0.246402i $$0.0792483\pi$$
−0.969168 + 0.246402i $$0.920752\pi$$
$$104$$ −50400.0 −0.456927
$$105$$ 0 0
$$106$$ −22620.0 −0.195537
$$107$$ 146324.i 1.23554i 0.786360 + 0.617769i $$0.211963\pi$$
−0.786360 + 0.617769i $$0.788037\pi$$
$$108$$ 276080.i 2.27759i
$$109$$ −92898.0 −0.748928 −0.374464 0.927241i $$-0.622173\pi$$
−0.374464 + 0.927241i $$0.622173\pi$$
$$110$$ 0 0
$$111$$ 145572. 1.12143
$$112$$ 69776.0i 0.525607i
$$113$$ − 83354.0i − 0.614088i −0.951695 0.307044i $$-0.900660\pi$$
0.951695 0.307044i $$-0.0993398\pi$$
$$114$$ 13720.0 0.0988762
$$115$$ 0 0
$$116$$ 232424. 1.60375
$$117$$ − 6580.00i − 0.0444387i
$$118$$ 27300.0i 0.180492i
$$119$$ −84378.0 −0.546213
$$120$$ 0 0
$$121$$ −107227. −0.665795
$$122$$ 256480.i 1.56011i
$$123$$ 251468.i 1.49872i
$$124$$ 519792. 3.03582
$$125$$ 0 0
$$126$$ −23030.0 −0.129231
$$127$$ − 60384.0i − 0.332210i −0.986108 0.166105i $$-0.946881\pi$$
0.986108 0.166105i $$-0.0531191\pi$$
$$128$$ 270720.i 1.46048i
$$129$$ 152320. 0.805903
$$130$$ 0 0
$$131$$ −61586.0 −0.313548 −0.156774 0.987635i $$-0.550109\pi$$
−0.156774 + 0.987635i $$0.550109\pi$$
$$132$$ 220864.i 1.10329i
$$133$$ 4802.00i 0.0235393i
$$134$$ −484040. −2.32873
$$135$$ 0 0
$$136$$ 619920. 2.87401
$$137$$ 204462.i 0.930703i 0.885126 + 0.465352i $$0.154072\pi$$
−0.885126 + 0.465352i $$0.845928\pi$$
$$138$$ 255360.i 1.14145i
$$139$$ 35406.0 0.155432 0.0777159 0.996976i $$-0.475237\pi$$
0.0777159 + 0.996976i $$0.475237\pi$$
$$140$$ 0 0
$$141$$ −130536. −0.552946
$$142$$ − 585600.i − 2.43714i
$$143$$ − 32480.0i − 0.132824i
$$144$$ 66928.0 0.268969
$$145$$ 0 0
$$146$$ −680820. −2.64332
$$147$$ 33614.0i 0.128300i
$$148$$ − 707064.i − 2.65341i
$$149$$ 20226.0 0.0746353 0.0373177 0.999303i $$-0.488119\pi$$
0.0373177 + 0.999303i $$0.488119\pi$$
$$150$$ 0 0
$$151$$ 70904.0 0.253063 0.126531 0.991963i $$-0.459616\pi$$
0.126531 + 0.991963i $$0.459616\pi$$
$$152$$ − 35280.0i − 0.123857i
$$153$$ 80934.0i 0.279513i
$$154$$ −113680. −0.386262
$$155$$ 0 0
$$156$$ 133280. 0.438484
$$157$$ − 293524.i − 0.950374i −0.879885 0.475187i $$-0.842380\pi$$
0.879885 0.475187i $$-0.157620\pi$$
$$158$$ − 317840.i − 1.01290i
$$159$$ 31668.0 0.0993408
$$160$$ 0 0
$$161$$ −89376.0 −0.271742
$$162$$ − 454190.i − 1.35972i
$$163$$ 13192.0i 0.0388903i 0.999811 + 0.0194452i $$0.00618998\pi$$
−0.999811 + 0.0194452i $$0.993810\pi$$
$$164$$ 1.22142e6 3.54612
$$165$$ 0 0
$$166$$ 205380. 0.578479
$$167$$ − 493612.i − 1.36960i −0.728730 0.684801i $$-0.759889\pi$$
0.728730 0.684801i $$-0.240111\pi$$
$$168$$ − 246960.i − 0.675077i
$$169$$ 351693. 0.947212
$$170$$ 0 0
$$171$$ 4606.00 0.0120457
$$172$$ − 739840.i − 1.90685i
$$173$$ 240716.i 0.611490i 0.952113 + 0.305745i $$0.0989056\pi$$
−0.952113 + 0.305745i $$0.901094\pi$$
$$174$$ −478520. −1.19819
$$175$$ 0 0
$$176$$ 330368. 0.803926
$$177$$ − 38220.0i − 0.0916975i
$$178$$ 505820.i 1.19659i
$$179$$ −294932. −0.688001 −0.344001 0.938969i $$-0.611782\pi$$
−0.344001 + 0.938969i $$0.611782\pi$$
$$180$$ 0 0
$$181$$ −336980. −0.764553 −0.382277 0.924048i $$-0.624860\pi$$
−0.382277 + 0.924048i $$0.624860\pi$$
$$182$$ 68600.0i 0.153513i
$$183$$ − 359072.i − 0.792600i
$$184$$ 656640. 1.42982
$$185$$ 0 0
$$186$$ −1.07016e6 −2.26812
$$187$$ 399504.i 0.835444i
$$188$$ 634032.i 1.30833i
$$189$$ 198940. 0.405105
$$190$$ 0 0
$$191$$ 358264. 0.710591 0.355296 0.934754i $$-0.384380\pi$$
0.355296 + 0.934754i $$0.384380\pi$$
$$192$$ − 257152.i − 0.503427i
$$193$$ − 989554.i − 1.91226i −0.292948 0.956128i $$-0.594636\pi$$
0.292948 0.956128i $$-0.405364\pi$$
$$194$$ −585060. −1.11608
$$195$$ 0 0
$$196$$ 163268. 0.303571
$$197$$ 990050.i 1.81757i 0.417263 + 0.908786i $$0.362989\pi$$
−0.417263 + 0.908786i $$0.637011\pi$$
$$198$$ 109040.i 0.197662i
$$199$$ 840756. 1.50500 0.752501 0.658591i $$-0.228847\pi$$
0.752501 + 0.658591i $$0.228847\pi$$
$$200$$ 0 0
$$201$$ 677656. 1.18309
$$202$$ 386960.i 0.667249i
$$203$$ − 167482.i − 0.285252i
$$204$$ −1.63934e6 −2.75800
$$205$$ 0 0
$$206$$ −530600. −0.871163
$$207$$ 85728.0i 0.139058i
$$208$$ − 199360.i − 0.319506i
$$209$$ 22736.0 0.0360038
$$210$$ 0 0
$$211$$ 1.15073e6 1.77938 0.889689 0.456568i $$-0.150921\pi$$
0.889689 + 0.456568i $$0.150921\pi$$
$$212$$ − 153816.i − 0.235051i
$$213$$ 819840.i 1.23817i
$$214$$ −1.46324e6 −2.18414
$$215$$ 0 0
$$216$$ −1.46160e6 −2.13154
$$217$$ − 374556.i − 0.539967i
$$218$$ − 928980.i − 1.32393i
$$219$$ 953148. 1.34292
$$220$$ 0 0
$$221$$ 241080. 0.332032
$$222$$ 1.45572e6i 1.98242i
$$223$$ − 824264.i − 1.10995i −0.831866 0.554976i $$-0.812727\pi$$
0.831866 0.554976i $$-0.187273\pi$$
$$224$$ −133280. −0.177478
$$225$$ 0 0
$$226$$ 833540. 1.08556
$$227$$ − 74382.0i − 0.0958083i −0.998852 0.0479042i $$-0.984746\pi$$
0.998852 0.0479042i $$-0.0152542\pi$$
$$228$$ 93296.0i 0.118857i
$$229$$ −1.13196e6 −1.42640 −0.713199 0.700961i $$-0.752755\pi$$
−0.713199 + 0.700961i $$0.752755\pi$$
$$230$$ 0 0
$$231$$ 159152. 0.196238
$$232$$ 1.23048e6i 1.50091i
$$233$$ − 198726.i − 0.239809i −0.992785 0.119904i $$-0.961741\pi$$
0.992785 0.119904i $$-0.0382588\pi$$
$$234$$ 65800.0 0.0785572
$$235$$ 0 0
$$236$$ −185640. −0.216966
$$237$$ 444976.i 0.514595i
$$238$$ − 843780.i − 0.965577i
$$239$$ −482904. −0.546847 −0.273424 0.961894i $$-0.588156\pi$$
−0.273424 + 0.961894i $$0.588156\pi$$
$$240$$ 0 0
$$241$$ 805910. 0.893807 0.446904 0.894582i $$-0.352527\pi$$
0.446904 + 0.894582i $$0.352527\pi$$
$$242$$ − 1.07227e6i − 1.17697i
$$243$$ − 350714.i − 0.381011i
$$244$$ −1.74406e6 −1.87537
$$245$$ 0 0
$$246$$ −2.51468e6 −2.64938
$$247$$ − 13720.0i − 0.0143091i
$$248$$ 2.75184e6i 2.84115i
$$249$$ −287532. −0.293892
$$250$$ 0 0
$$251$$ 430738. 0.431548 0.215774 0.976443i $$-0.430773\pi$$
0.215774 + 0.976443i $$0.430773\pi$$
$$252$$ − 156604.i − 0.155347i
$$253$$ 423168.i 0.415634i
$$254$$ 603840. 0.587270
$$255$$ 0 0
$$256$$ −2.11942e6 −2.02124
$$257$$ 1.17691e6i 1.11150i 0.831349 + 0.555751i $$0.187569\pi$$
−0.831349 + 0.555751i $$0.812431\pi$$
$$258$$ 1.52320e6i 1.42465i
$$259$$ −509502. −0.471951
$$260$$ 0 0
$$261$$ −160646. −0.145972
$$262$$ − 615860.i − 0.554279i
$$263$$ 1.29098e6i 1.15088i 0.817845 + 0.575438i $$0.195169\pi$$
−0.817845 + 0.575438i $$0.804831\pi$$
$$264$$ −1.16928e6 −1.03254
$$265$$ 0 0
$$266$$ −48020.0 −0.0416119
$$267$$ − 708148.i − 0.607919i
$$268$$ − 3.29147e6i − 2.79932i
$$269$$ 1.27756e6 1.07646 0.538232 0.842797i $$-0.319093\pi$$
0.538232 + 0.842797i $$0.319093\pi$$
$$270$$ 0 0
$$271$$ 1.65054e6 1.36522 0.682612 0.730781i $$-0.260844\pi$$
0.682612 + 0.730781i $$0.260844\pi$$
$$272$$ 2.45213e6i 2.00965i
$$273$$ − 96040.0i − 0.0779912i
$$274$$ −2.04462e6 −1.64527
$$275$$ 0 0
$$276$$ −1.73645e6 −1.37211
$$277$$ 1.06409e6i 0.833257i 0.909077 + 0.416628i $$0.136788\pi$$
−0.909077 + 0.416628i $$0.863212\pi$$
$$278$$ 354060.i 0.274767i
$$279$$ −359268. −0.276317
$$280$$ 0 0
$$281$$ −22342.0 −0.0168794 −0.00843969 0.999964i $$-0.502686\pi$$
−0.00843969 + 0.999964i $$0.502686\pi$$
$$282$$ − 1.30536e6i − 0.977479i
$$283$$ − 2.49574e6i − 1.85239i −0.377042 0.926196i $$-0.623059\pi$$
0.377042 0.926196i $$-0.376941\pi$$
$$284$$ 3.98208e6 2.92964
$$285$$ 0 0
$$286$$ 324800. 0.234802
$$287$$ − 880138.i − 0.630734i
$$288$$ 127840.i 0.0908208i
$$289$$ −1.54543e6 −1.08844
$$290$$ 0 0
$$291$$ 819084. 0.567017
$$292$$ − 4.62958e6i − 3.17749i
$$293$$ − 1.93178e6i − 1.31458i −0.753637 0.657291i $$-0.771702\pi$$
0.753637 0.657291i $$-0.228298\pi$$
$$294$$ −336140. −0.226805
$$295$$ 0 0
$$296$$ 3.74328e6 2.48326
$$297$$ − 941920.i − 0.619616i
$$298$$ 202260.i 0.131938i
$$299$$ 255360. 0.165187
$$300$$ 0 0
$$301$$ −533120. −0.339163
$$302$$ 709040.i 0.447356i
$$303$$ − 541744.i − 0.338991i
$$304$$ 139552. 0.0866068
$$305$$ 0 0
$$306$$ −809340. −0.494114
$$307$$ 459074.i 0.277995i 0.990293 + 0.138997i $$0.0443880\pi$$
−0.990293 + 0.138997i $$0.955612\pi$$
$$308$$ − 773024.i − 0.464319i
$$309$$ 742840. 0.442587
$$310$$ 0 0
$$311$$ 667128. 0.391118 0.195559 0.980692i $$-0.437348\pi$$
0.195559 + 0.980692i $$0.437348\pi$$
$$312$$ 705600.i 0.410367i
$$313$$ − 111034.i − 0.0640612i −0.999487 0.0320306i $$-0.989803\pi$$
0.999487 0.0320306i $$-0.0101974\pi$$
$$314$$ 2.93524e6 1.68004
$$315$$ 0 0
$$316$$ 2.16131e6 1.21759
$$317$$ 68778.0i 0.0384416i 0.999815 + 0.0192208i $$0.00611855\pi$$
−0.999815 + 0.0192208i $$0.993881\pi$$
$$318$$ 316680.i 0.175611i
$$319$$ −792976. −0.436298
$$320$$ 0 0
$$321$$ 2.04854e6 1.10964
$$322$$ − 893760.i − 0.480376i
$$323$$ 168756.i 0.0900022i
$$324$$ 3.08849e6 1.63450
$$325$$ 0 0
$$326$$ −131920. −0.0687490
$$327$$ 1.30057e6i 0.672613i
$$328$$ 6.46632e6i 3.31874i
$$329$$ 456876. 0.232707
$$330$$ 0 0
$$331$$ −564448. −0.283174 −0.141587 0.989926i $$-0.545221\pi$$
−0.141587 + 0.989926i $$0.545221\pi$$
$$332$$ 1.39658e6i 0.695379i
$$333$$ 488706.i 0.241511i
$$334$$ 4.93612e6 2.42114
$$335$$ 0 0
$$336$$ 976864. 0.472048
$$337$$ − 2.07729e6i − 0.996376i −0.867069 0.498188i $$-0.833999\pi$$
0.867069 0.498188i $$-0.166001\pi$$
$$338$$ 3.51693e6i 1.67445i
$$339$$ −1.16696e6 −0.551512
$$340$$ 0 0
$$341$$ −1.77341e6 −0.825891
$$342$$ 46060.0i 0.0212941i
$$343$$ − 117649.i − 0.0539949i
$$344$$ 3.91680e6 1.78458
$$345$$ 0 0
$$346$$ −2.40716e6 −1.08097
$$347$$ 53248.0i 0.0237399i 0.999930 + 0.0118700i $$0.00377842\pi$$
−0.999930 + 0.0118700i $$0.996222\pi$$
$$348$$ − 3.25394e6i − 1.44033i
$$349$$ 2.27200e6 0.998494 0.499247 0.866460i $$-0.333610\pi$$
0.499247 + 0.866460i $$0.333610\pi$$
$$350$$ 0 0
$$351$$ −568400. −0.246256
$$352$$ 631040.i 0.271456i
$$353$$ 4.00645e6i 1.71129i 0.517565 + 0.855644i $$0.326838\pi$$
−0.517565 + 0.855644i $$0.673162\pi$$
$$354$$ 382200. 0.162100
$$355$$ 0 0
$$356$$ −3.43958e6 −1.43840
$$357$$ 1.18129e6i 0.490554i
$$358$$ − 2.94932e6i − 1.21623i
$$359$$ −73784.0 −0.0302152 −0.0151076 0.999886i $$-0.504809\pi$$
−0.0151076 + 0.999886i $$0.504809\pi$$
$$360$$ 0 0
$$361$$ −2.46650e6 −0.996121
$$362$$ − 3.36980e6i − 1.35155i
$$363$$ 1.50118e6i 0.597951i
$$364$$ −466480. −0.184535
$$365$$ 0 0
$$366$$ 3.59072e6 1.40113
$$367$$ − 1.40431e6i − 0.544250i −0.962262 0.272125i $$-0.912274\pi$$
0.962262 0.272125i $$-0.0877264\pi$$
$$368$$ 2.59738e6i 0.999805i
$$369$$ −844214. −0.322765
$$370$$ 0 0
$$371$$ −110838. −0.0418075
$$372$$ − 7.27709e6i − 2.72647i
$$373$$ − 1.60323e6i − 0.596657i −0.954463 0.298329i $$-0.903571\pi$$
0.954463 0.298329i $$-0.0964291\pi$$
$$374$$ −3.99504e6 −1.47687
$$375$$ 0 0
$$376$$ −3.35664e6 −1.22443
$$377$$ 478520.i 0.173399i
$$378$$ 1.98940e6i 0.716131i
$$379$$ 4.77012e6 1.70581 0.852906 0.522064i $$-0.174838\pi$$
0.852906 + 0.522064i $$0.174838\pi$$
$$380$$ 0 0
$$381$$ −845376. −0.298358
$$382$$ 3.58264e6i 1.25616i
$$383$$ − 2.23079e6i − 0.777072i −0.921434 0.388536i $$-0.872981\pi$$
0.921434 0.388536i $$-0.127019\pi$$
$$384$$ 3.79008e6 1.31166
$$385$$ 0 0
$$386$$ 9.89554e6 3.38042
$$387$$ 511360.i 0.173560i
$$388$$ − 3.97841e6i − 1.34162i
$$389$$ −4.84024e6 −1.62178 −0.810892 0.585196i $$-0.801018\pi$$
−0.810892 + 0.585196i $$0.801018\pi$$
$$390$$ 0 0
$$391$$ −3.14093e6 −1.03900
$$392$$ 864360.i 0.284105i
$$393$$ 862204.i 0.281597i
$$394$$ −9.90050e6 −3.21304
$$395$$ 0 0
$$396$$ −741472. −0.237606
$$397$$ − 995820.i − 0.317106i −0.987350 0.158553i $$-0.949317\pi$$
0.987350 0.158553i $$-0.0506829\pi$$
$$398$$ 8.40756e6i 2.66049i
$$399$$ 67228.0 0.0211406
$$400$$ 0 0
$$401$$ −3.31605e6 −1.02982 −0.514909 0.857245i $$-0.672174\pi$$
−0.514909 + 0.857245i $$0.672174\pi$$
$$402$$ 6.77656e6i 2.09143i
$$403$$ 1.07016e6i 0.328236i
$$404$$ −2.63133e6 −0.802087
$$405$$ 0 0
$$406$$ 1.67482e6 0.504258
$$407$$ 2.41234e6i 0.721858i
$$408$$ − 8.67888e6i − 2.58115i
$$409$$ −3.07273e6 −0.908274 −0.454137 0.890932i $$-0.650052\pi$$
−0.454137 + 0.890932i $$0.650052\pi$$
$$410$$ 0 0
$$411$$ 2.86247e6 0.835865
$$412$$ − 3.60808e6i − 1.04721i
$$413$$ 133770.i 0.0385908i
$$414$$ −857280. −0.245823
$$415$$ 0 0
$$416$$ 380800. 0.107886
$$417$$ − 495684.i − 0.139593i
$$418$$ 227360.i 0.0636463i
$$419$$ −2.81438e6 −0.783154 −0.391577 0.920145i $$-0.628070\pi$$
−0.391577 + 0.920145i $$0.628070\pi$$
$$420$$ 0 0
$$421$$ 3.05802e6 0.840883 0.420441 0.907320i $$-0.361875\pi$$
0.420441 + 0.907320i $$0.361875\pi$$
$$422$$ 1.15073e7i 3.14552i
$$423$$ − 438228.i − 0.119083i
$$424$$ 814320. 0.219979
$$425$$ 0 0
$$426$$ −8.19840e6 −2.18880
$$427$$ 1.25675e6i 0.333565i
$$428$$ − 9.95003e6i − 2.62552i
$$429$$ −454720. −0.119289
$$430$$ 0 0
$$431$$ 1.93750e6 0.502398 0.251199 0.967936i $$-0.419175\pi$$
0.251199 + 0.967936i $$0.419175\pi$$
$$432$$ − 5.78144e6i − 1.49048i
$$433$$ 3.94790e6i 1.01192i 0.862557 + 0.505961i $$0.168862\pi$$
−0.862557 + 0.505961i $$0.831138\pi$$
$$434$$ 3.74556e6 0.954536
$$435$$ 0 0
$$436$$ 6.31706e6 1.59147
$$437$$ 178752.i 0.0447762i
$$438$$ 9.53148e6i 2.37397i
$$439$$ 7.41770e6 1.83700 0.918498 0.395426i $$-0.129403\pi$$
0.918498 + 0.395426i $$0.129403\pi$$
$$440$$ 0 0
$$441$$ −112847. −0.0276308
$$442$$ 2.41080e6i 0.586956i
$$443$$ 1.40269e6i 0.339589i 0.985480 + 0.169794i $$0.0543103\pi$$
−0.985480 + 0.169794i $$0.945690\pi$$
$$444$$ −9.89890e6 −2.38303
$$445$$ 0 0
$$446$$ 8.24264e6 1.96214
$$447$$ − 283164.i − 0.0670300i
$$448$$ 900032.i 0.211867i
$$449$$ 590574. 0.138248 0.0691239 0.997608i $$-0.477980\pi$$
0.0691239 + 0.997608i $$0.477980\pi$$
$$450$$ 0 0
$$451$$ −4.16718e6 −0.964720
$$452$$ 5.66807e6i 1.30494i
$$453$$ − 992656.i − 0.227276i
$$454$$ 743820. 0.169367
$$455$$ 0 0
$$456$$ −493920. −0.111236
$$457$$ 2.90484e6i 0.650627i 0.945606 + 0.325313i $$0.105470\pi$$
−0.945606 + 0.325313i $$0.894530\pi$$
$$458$$ − 1.13196e7i − 2.52154i
$$459$$ 6.99132e6 1.54891
$$460$$ 0 0
$$461$$ −922684. −0.202209 −0.101105 0.994876i $$-0.532238\pi$$
−0.101105 + 0.994876i $$0.532238\pi$$
$$462$$ 1.59152e6i 0.346902i
$$463$$ 7.18235e6i 1.55709i 0.627588 + 0.778546i $$0.284042\pi$$
−0.627588 + 0.778546i $$0.715958\pi$$
$$464$$ −4.86723e6 −1.04951
$$465$$ 0 0
$$466$$ 1.98726e6 0.423926
$$467$$ 612570.i 0.129976i 0.997886 + 0.0649881i $$0.0207009\pi$$
−0.997886 + 0.0649881i $$0.979299\pi$$
$$468$$ 447440.i 0.0944322i
$$469$$ −2.37180e6 −0.497904
$$470$$ 0 0
$$471$$ −4.10934e6 −0.853531
$$472$$ − 982800.i − 0.203053i
$$473$$ 2.52416e6i 0.518757i
$$474$$ −4.44976e6 −0.909684
$$475$$ 0 0
$$476$$ 5.73770e6 1.16070
$$477$$ 106314.i 0.0213941i
$$478$$ − 4.82904e6i − 0.966699i
$$479$$ −2.60330e6 −0.518424 −0.259212 0.965820i $$-0.583463\pi$$
−0.259212 + 0.965820i $$0.583463\pi$$
$$480$$ 0 0
$$481$$ 1.45572e6 0.286890
$$482$$ 8.05910e6i 1.58004i
$$483$$ 1.25126e6i 0.244051i
$$484$$ 7.29144e6 1.41482
$$485$$ 0 0
$$486$$ 3.50714e6 0.673539
$$487$$ − 5.46309e6i − 1.04380i −0.853008 0.521898i $$-0.825224\pi$$
0.853008 0.521898i $$-0.174776\pi$$
$$488$$ − 9.23328e6i − 1.75512i
$$489$$ 184688. 0.0349274
$$490$$ 0 0
$$491$$ 1.64090e6 0.307170 0.153585 0.988135i $$-0.450918\pi$$
0.153585 + 0.988135i $$0.450918\pi$$
$$492$$ − 1.70998e7i − 3.18478i
$$493$$ − 5.88580e6i − 1.09066i
$$494$$ 137200. 0.0252951
$$495$$ 0 0
$$496$$ −1.08851e7 −1.98667
$$497$$ − 2.86944e6i − 0.521082i
$$498$$ − 2.87532e6i − 0.519533i
$$499$$ −2.99796e6 −0.538983 −0.269491 0.963003i $$-0.586856\pi$$
−0.269491 + 0.963003i $$0.586856\pi$$
$$500$$ 0 0
$$501$$ −6.91057e6 −1.23004
$$502$$ 4.30738e6i 0.762876i
$$503$$ − 6.89405e6i − 1.21494i −0.794343 0.607469i $$-0.792185\pi$$
0.794343 0.607469i $$-0.207815\pi$$
$$504$$ 829080. 0.145385
$$505$$ 0 0
$$506$$ −4.23168e6 −0.734745
$$507$$ − 4.92370e6i − 0.850691i
$$508$$ 4.10611e6i 0.705946i
$$509$$ −2.30476e6 −0.394305 −0.197152 0.980373i $$-0.563169\pi$$
−0.197152 + 0.980373i $$0.563169\pi$$
$$510$$ 0 0
$$511$$ −3.33602e6 −0.565166
$$512$$ − 1.25312e7i − 2.11260i
$$513$$ − 397880.i − 0.0667511i
$$514$$ −1.17691e7 −1.96488
$$515$$ 0 0
$$516$$ −1.03578e7 −1.71254
$$517$$ − 2.16317e6i − 0.355929i
$$518$$ − 5.09502e6i − 0.834299i
$$519$$ 3.37002e6 0.549180
$$520$$ 0 0
$$521$$ −1.20960e7 −1.95231 −0.976155 0.217073i $$-0.930349\pi$$
−0.976155 + 0.217073i $$0.930349\pi$$
$$522$$ − 1.60646e6i − 0.258044i
$$523$$ 5.48443e6i 0.876753i 0.898791 + 0.438377i $$0.144446\pi$$
−0.898791 + 0.438377i $$0.855554\pi$$
$$524$$ 4.18785e6 0.666289
$$525$$ 0 0
$$526$$ −1.29098e7 −2.03448
$$527$$ − 1.31630e7i − 2.06456i
$$528$$ − 4.62515e6i − 0.722007i
$$529$$ 3.10937e6 0.483095
$$530$$ 0 0
$$531$$ 128310. 0.0197480
$$532$$ − 326536.i − 0.0500210i
$$533$$ 2.51468e6i 0.383411i
$$534$$ 7.08148e6 1.07466
$$535$$ 0 0
$$536$$ 1.74254e7 2.61982
$$537$$ 4.12905e6i 0.617894i
$$538$$ 1.27756e7i 1.90294i
$$539$$ −557032. −0.0825863
$$540$$ 0 0
$$541$$ −6.71799e6 −0.986839 −0.493420 0.869791i $$-0.664253\pi$$
−0.493420 + 0.869791i $$0.664253\pi$$
$$542$$ 1.65054e7i 2.41340i
$$543$$ 4.71772e6i 0.686646i
$$544$$ −4.68384e6 −0.678586
$$545$$ 0 0
$$546$$ 960400. 0.137870
$$547$$ 5.00235e6i 0.714835i 0.933945 + 0.357418i $$0.116343\pi$$
−0.933945 + 0.357418i $$0.883657\pi$$
$$548$$ − 1.39034e7i − 1.97774i
$$549$$ 1.20546e6 0.170695
$$550$$ 0 0
$$551$$ −334964. −0.0470023
$$552$$ − 9.19296e6i − 1.28413i
$$553$$ − 1.55742e6i − 0.216567i
$$554$$ −1.06409e7 −1.47300
$$555$$ 0 0
$$556$$ −2.40761e6 −0.330293
$$557$$ − 9.01961e6i − 1.23183i −0.787814 0.615913i $$-0.788787\pi$$
0.787814 0.615913i $$-0.211213\pi$$
$$558$$ − 3.59268e6i − 0.488465i
$$559$$ 1.52320e6 0.206171
$$560$$ 0 0
$$561$$ 5.59306e6 0.750312
$$562$$ − 223420.i − 0.0298388i
$$563$$ 1.24051e7i 1.64941i 0.565561 + 0.824707i $$0.308660\pi$$
−0.565561 + 0.824707i $$0.691340\pi$$
$$564$$ 8.87645e6 1.17501
$$565$$ 0 0
$$566$$ 2.49574e7 3.27460
$$567$$ − 2.22553e6i − 0.290721i
$$568$$ 2.10816e7i 2.74178i
$$569$$ −6.48804e6 −0.840103 −0.420052 0.907500i $$-0.637988\pi$$
−0.420052 + 0.907500i $$0.637988\pi$$
$$570$$ 0 0
$$571$$ −1.02285e7 −1.31287 −0.656435 0.754382i $$-0.727936\pi$$
−0.656435 + 0.754382i $$0.727936\pi$$
$$572$$ 2.20864e6i 0.282251i
$$573$$ − 5.01570e6i − 0.638182i
$$574$$ 8.80138e6 1.11499
$$575$$ 0 0
$$576$$ 863296. 0.108419
$$577$$ − 2.65338e6i − 0.331787i −0.986144 0.165894i $$-0.946949\pi$$
0.986144 0.165894i $$-0.0530508\pi$$
$$578$$ − 1.54543e7i − 1.92411i
$$579$$ −1.38538e7 −1.71740
$$580$$ 0 0
$$581$$ 1.00636e6 0.123684
$$582$$ 8.19084e6i 1.00235i
$$583$$ 524784.i 0.0639454i
$$584$$ 2.45095e7 2.97374
$$585$$ 0 0
$$586$$ 1.93178e7 2.32387
$$587$$ 1.43044e7i 1.71346i 0.515766 + 0.856729i $$0.327507\pi$$
−0.515766 + 0.856729i $$0.672493\pi$$
$$588$$ − 2.28575e6i − 0.272638i
$$589$$ −749112. −0.0889731
$$590$$ 0 0
$$591$$ 1.38607e7 1.63236
$$592$$ 1.48068e7i 1.73642i
$$593$$ − 1.00265e7i − 1.17088i −0.810714 0.585442i $$-0.800921\pi$$
0.810714 0.585442i $$-0.199079\pi$$
$$594$$ 9.41920e6 1.09534
$$595$$ 0 0
$$596$$ −1.37537e6 −0.158600
$$597$$ − 1.17706e7i − 1.35164i
$$598$$ 2.55360e6i 0.292011i
$$599$$ 7.52292e6 0.856681 0.428341 0.903617i $$-0.359098\pi$$
0.428341 + 0.903617i $$0.359098\pi$$
$$600$$ 0 0
$$601$$ 3.38625e6 0.382413 0.191207 0.981550i $$-0.438760\pi$$
0.191207 + 0.981550i $$0.438760\pi$$
$$602$$ − 5.33120e6i − 0.599562i
$$603$$ 2.27499e6i 0.254792i
$$604$$ −4.82147e6 −0.537759
$$605$$ 0 0
$$606$$ 5.41744e6 0.599256
$$607$$ 6.90861e6i 0.761060i 0.924769 + 0.380530i $$0.124258\pi$$
−0.924769 + 0.380530i $$0.875742\pi$$
$$608$$ 266560.i 0.0292439i
$$609$$ −2.34475e6 −0.256185
$$610$$ 0 0
$$611$$ −1.30536e6 −0.141458
$$612$$ − 5.50351e6i − 0.593966i
$$613$$ − 9.68896e6i − 1.04142i −0.853734 0.520710i $$-0.825667\pi$$
0.853734 0.520710i $$-0.174333\pi$$
$$614$$ −4.59074e6 −0.491430
$$615$$ 0 0
$$616$$ 4.09248e6 0.434545
$$617$$ 7.84742e6i 0.829877i 0.909849 + 0.414939i $$0.136197\pi$$
−0.909849 + 0.414939i $$0.863803\pi$$
$$618$$ 7.42840e6i 0.782391i
$$619$$ 1.01972e7 1.06968 0.534840 0.844953i $$-0.320372\pi$$
0.534840 + 0.844953i $$0.320372\pi$$
$$620$$ 0 0
$$621$$ 7.40544e6 0.770587
$$622$$ 6.67128e6i 0.691406i
$$623$$ 2.47852e6i 0.255842i
$$624$$ −2.79104e6 −0.286949
$$625$$ 0 0
$$626$$ 1.11034e6 0.113245
$$627$$ − 318304.i − 0.0323350i
$$628$$ 1.99596e7i 2.01954i
$$629$$ −1.79054e7 −1.80450
$$630$$ 0 0
$$631$$ −8.36258e6 −0.836116 −0.418058 0.908420i $$-0.637289\pi$$
−0.418058 + 0.908420i $$0.637289\pi$$
$$632$$ 1.14422e7i 1.13951i
$$633$$ − 1.61102e7i − 1.59806i
$$634$$ −687780. −0.0679558
$$635$$ 0 0
$$636$$ −2.15342e6 −0.211099
$$637$$ 336140.i 0.0328225i
$$638$$ − 7.92976e6i − 0.771273i
$$639$$ −2.75232e6 −0.266653
$$640$$ 0 0
$$641$$ 1.10283e6 0.106014 0.0530070 0.998594i $$-0.483119\pi$$
0.0530070 + 0.998594i $$0.483119\pi$$
$$642$$ 2.04854e7i 1.96158i
$$643$$ 1.71354e7i 1.63443i 0.576330 + 0.817217i $$0.304484\pi$$
−0.576330 + 0.817217i $$0.695516\pi$$
$$644$$ 6.07757e6 0.577451
$$645$$ 0 0
$$646$$ −1.68756e6 −0.159103
$$647$$ 54964.0i 0.00516200i 0.999997 + 0.00258100i $$0.000821558\pi$$
−0.999997 + 0.00258100i $$0.999178\pi$$
$$648$$ 1.63508e7i 1.52969i
$$649$$ 633360. 0.0590254
$$650$$ 0 0
$$651$$ −5.24378e6 −0.484945
$$652$$ − 897056.i − 0.0826420i
$$653$$ − 485166.i − 0.0445254i −0.999752 0.0222627i $$-0.992913\pi$$
0.999752 0.0222627i $$-0.00708702\pi$$
$$654$$ −1.30057e7 −1.18902
$$655$$ 0 0
$$656$$ −2.55779e7 −2.32063
$$657$$ 3.19985e6i 0.289212i
$$658$$ 4.56876e6i 0.411371i
$$659$$ 2.72136e6 0.244103 0.122051 0.992524i $$-0.461053\pi$$
0.122051 + 0.992524i $$0.461053\pi$$
$$660$$ 0 0
$$661$$ −2.14525e6 −0.190974 −0.0954869 0.995431i $$-0.530441\pi$$
−0.0954869 + 0.995431i $$0.530441\pi$$
$$662$$ − 5.64448e6i − 0.500586i
$$663$$ − 3.37512e6i − 0.298198i
$$664$$ −7.39368e6 −0.650789
$$665$$ 0 0
$$666$$ −4.88706e6 −0.426935
$$667$$ − 6.23443e6i − 0.542603i
$$668$$ 3.35656e7i 2.91041i
$$669$$ −1.15397e7 −0.996848
$$670$$ 0 0
$$671$$ 5.95034e6 0.510194
$$672$$ 1.86592e6i 0.159393i
$$673$$ 2.92796e6i 0.249188i 0.992208 + 0.124594i $$0.0397629\pi$$
−0.992208 + 0.124594i $$0.960237\pi$$
$$674$$ 2.07729e7 1.76136
$$675$$ 0 0
$$676$$ −2.39151e7 −2.01282
$$677$$ 1.34992e7i 1.13198i 0.824414 + 0.565988i $$0.191505\pi$$
−0.824414 + 0.565988i $$0.808495\pi$$
$$678$$ − 1.16696e7i − 0.974945i
$$679$$ −2.86679e6 −0.238628
$$680$$ 0 0
$$681$$ −1.04135e6 −0.0860455
$$682$$ − 1.77341e7i − 1.45998i
$$683$$ − 5.42972e6i − 0.445375i −0.974890 0.222688i $$-0.928517\pi$$
0.974890 0.222688i $$-0.0714830\pi$$
$$684$$ −313208. −0.0255972
$$685$$ 0 0
$$686$$ 1.17649e6 0.0954504
$$687$$ 1.58474e7i 1.28105i
$$688$$ 1.54931e7i 1.24787i
$$689$$ 316680. 0.0254140
$$690$$ 0 0
$$691$$ 2.08280e7 1.65940 0.829702 0.558207i $$-0.188510\pi$$
0.829702 + 0.558207i $$0.188510\pi$$
$$692$$ − 1.63687e7i − 1.29942i
$$693$$ 534296.i 0.0422619i
$$694$$ −532480. −0.0419667
$$695$$ 0 0
$$696$$ 1.72267e7 1.34797
$$697$$ − 3.09306e7i − 2.41160i
$$698$$ 2.27200e7i 1.76510i
$$699$$ −2.78216e6 −0.215372
$$700$$ 0 0
$$701$$ 2.35141e7 1.80731 0.903655 0.428261i $$-0.140874\pi$$
0.903655 + 0.428261i $$0.140874\pi$$
$$702$$ − 5.68400e6i − 0.435323i
$$703$$ 1.01900e6i 0.0777656i
$$704$$ 4.26138e6 0.324055
$$705$$ 0 0
$$706$$ −4.00645e7 −3.02516
$$707$$ 1.89610e6i 0.142664i
$$708$$ 2.59896e6i 0.194857i
$$709$$ 1.95747e7 1.46244 0.731221 0.682140i $$-0.238951\pi$$
0.731221 + 0.682140i $$0.238951\pi$$
$$710$$ 0 0
$$711$$ −1.49385e6 −0.110824
$$712$$ − 1.82095e7i − 1.34617i
$$713$$ − 1.39427e7i − 1.02712i
$$714$$ −1.18129e7 −0.867185
$$715$$ 0 0
$$716$$ 2.00554e7 1.46200
$$717$$ 6.76066e6i 0.491124i
$$718$$ − 737840.i − 0.0534135i
$$719$$ 2.61152e7 1.88396 0.941978 0.335674i $$-0.108964\pi$$
0.941978 + 0.335674i $$0.108964\pi$$
$$720$$ 0 0
$$721$$ −2.59994e6 −0.186262
$$722$$ − 2.46650e7i − 1.76091i
$$723$$ − 1.12827e7i − 0.802729i
$$724$$ 2.29146e7 1.62468
$$725$$ 0 0
$$726$$ −1.50118e7 −1.05704
$$727$$ − 1.54126e7i − 1.08154i −0.841172 0.540768i $$-0.818134\pi$$
0.841172 0.540768i $$-0.181866\pi$$
$$728$$ − 2.46960e6i − 0.172702i
$$729$$ −1.59468e7 −1.11136
$$730$$ 0 0
$$731$$ −1.87354e7 −1.29679
$$732$$ 2.44169e7i 1.68427i
$$733$$ − 1.69868e7i − 1.16776i −0.811841 0.583878i $$-0.801535\pi$$
0.811841 0.583878i $$-0.198465\pi$$
$$734$$ 1.40431e7 0.962107
$$735$$ 0 0
$$736$$ −4.96128e6 −0.337597
$$737$$ 1.12297e7i 0.761554i
$$738$$ − 8.44214e6i − 0.570574i
$$739$$ −2.01511e6 −0.135734 −0.0678669 0.997694i $$-0.521619\pi$$
−0.0678669 + 0.997694i $$0.521619\pi$$
$$740$$ 0 0
$$741$$ −192080. −0.0128510
$$742$$ − 1.10838e6i − 0.0739059i
$$743$$ − 1.51381e7i − 1.00600i −0.864286 0.503001i $$-0.832229\pi$$
0.864286 0.503001i $$-0.167771\pi$$
$$744$$ 3.85258e7 2.55164
$$745$$ 0 0
$$746$$ 1.60323e7 1.05475
$$747$$ − 965286.i − 0.0632928i
$$748$$ − 2.71663e7i − 1.77532i
$$749$$ −7.16988e6 −0.466989
$$750$$ 0 0
$$751$$ 7.21401e6 0.466742 0.233371 0.972388i $$-0.425024\pi$$
0.233371 + 0.972388i $$0.425024\pi$$
$$752$$ − 1.32774e7i − 0.856185i
$$753$$ − 6.03033e6i − 0.387573i
$$754$$ −4.78520e6 −0.306529
$$755$$ 0 0
$$756$$ −1.35279e7 −0.860848
$$757$$ 1.09697e7i 0.695755i 0.937540 + 0.347877i $$0.113097\pi$$
−0.937540 + 0.347877i $$0.886903\pi$$
$$758$$ 4.77012e7i 3.01548i
$$759$$ 5.92435e6 0.373281
$$760$$ 0 0
$$761$$ 1.92442e7 1.20459 0.602293 0.798275i $$-0.294254\pi$$
0.602293 + 0.798275i $$0.294254\pi$$
$$762$$ − 8.45376e6i − 0.527427i
$$763$$ − 4.55200e6i − 0.283068i
$$764$$ −2.43620e7 −1.51001
$$765$$ 0 0
$$766$$ 2.23079e7 1.37368
$$767$$ − 382200.i − 0.0234586i
$$768$$ 2.96719e7i 1.81528i
$$769$$ −8.21185e6 −0.500755 −0.250378 0.968148i $$-0.580555\pi$$
−0.250378 + 0.968148i $$0.580555\pi$$
$$770$$ 0 0
$$771$$ 1.64767e7 0.998241
$$772$$ 6.72897e7i 4.06355i
$$773$$ 1.86187e7i 1.12073i 0.828247 + 0.560363i $$0.189338\pi$$
−0.828247 + 0.560363i $$0.810662\pi$$
$$774$$ −5.11360e6 −0.306813
$$775$$ 0 0
$$776$$ 2.10622e7 1.25559
$$777$$ 7.13303e6i 0.423859i
$$778$$ − 4.84024e7i − 2.86694i
$$779$$ −1.76028e6 −0.103929
$$780$$ 0 0
$$781$$ −1.35859e7 −0.797006
$$782$$ − 3.14093e7i − 1.83671i
$$783$$ 1.38771e7i 0.808898i
$$784$$ −3.41902e6 −0.198661
$$785$$ 0 0
$$786$$ −8.62204e6 −0.497799
$$787$$ − 2.62501e7i − 1.51075i −0.655291 0.755377i $$-0.727454\pi$$
0.655291 0.755377i $$-0.272546\pi$$
$$788$$ − 6.73234e7i − 3.86234i
$$789$$ 1.80737e7 1.03360
$$790$$ 0 0
$$791$$ 4.08435e6 0.232103
$$792$$ − 3.92544e6i − 0.222370i
$$793$$ − 3.59072e6i − 0.202768i
$$794$$ 9.95820e6 0.560570
$$795$$ 0 0
$$796$$ −5.71714e7 −3.19813
$$797$$ 1.00373e7i 0.559720i 0.960041 + 0.279860i $$0.0902881\pi$$
−0.960041 + 0.279860i $$0.909712\pi$$
$$798$$ 672280.i 0.0373717i
$$799$$ 1.60559e7 0.889751
$$800$$ 0 0
$$801$$ 2.37735e6 0.130922
$$802$$ − 3.31605e7i − 1.82048i
$$803$$ 1.57950e7i 0.864433i
$$804$$ −4.60806e7 −2.51407
$$805$$ 0 0
$$806$$ −1.07016e7 −0.580245
$$807$$ − 1.78858e7i − 0.966772i
$$808$$ − 1.39306e7i − 0.750655i
$$809$$ −1.40884e7 −0.756816 −0.378408 0.925639i $$-0.623528\pi$$
−0.378408 + 0.925639i $$0.623528\pi$$
$$810$$ 0 0
$$811$$ 1.81433e7 0.968646 0.484323 0.874889i $$-0.339066\pi$$
0.484323 + 0.874889i $$0.339066\pi$$
$$812$$ 1.13888e7i 0.606160i
$$813$$ − 2.31076e7i − 1.22611i
$$814$$ −2.41234e7 −1.27608
$$815$$ 0 0
$$816$$ 3.43298e7 1.80487
$$817$$ 1.06624e6i 0.0558856i
$$818$$ − 3.07273e7i − 1.60562i
$$819$$ 322420. 0.0167962
$$820$$ 0 0
$$821$$ −2.13669e7 −1.10633 −0.553164 0.833072i $$-0.686580\pi$$
−0.553164 + 0.833072i $$0.686580\pi$$
$$822$$ 2.86247e7i 1.47761i
$$823$$ 1.78017e7i 0.916142i 0.888916 + 0.458071i $$0.151459\pi$$
−0.888916 + 0.458071i $$0.848541\pi$$
$$824$$ 1.91016e7 0.980058
$$825$$ 0 0
$$826$$ −1.33770e6 −0.0682195
$$827$$ − 1.62921e7i − 0.828350i −0.910197 0.414175i $$-0.864070\pi$$
0.910197 0.414175i $$-0.135930\pi$$
$$828$$ − 5.82950e6i − 0.295499i
$$829$$ 2.08499e6 0.105370 0.0526851 0.998611i $$-0.483222\pi$$
0.0526851 + 0.998611i $$0.483222\pi$$
$$830$$ 0 0
$$831$$ 1.48973e7 0.748348
$$832$$ − 2.57152e6i − 0.128790i
$$833$$ − 4.13452e6i − 0.206449i
$$834$$ 4.95684e6 0.246769
$$835$$ 0 0
$$836$$ −1.54605e6 −0.0765081
$$837$$ 3.10346e7i 1.53120i
$$838$$ − 2.81438e7i − 1.38443i
$$839$$ 2.27850e7 1.11749 0.558745 0.829340i $$-0.311283\pi$$
0.558745 + 0.829340i $$0.311283\pi$$
$$840$$ 0 0
$$841$$ −8.82842e6 −0.430421
$$842$$ 3.05802e7i 1.48648i
$$843$$ 312788.i 0.0151594i
$$844$$ −7.82498e7 −3.78118
$$845$$ 0 0
$$846$$ 4.38228e6 0.210511
$$847$$ − 5.25412e6i − 0.251647i
$$848$$ 3.22109e6i 0.153820i
$$849$$ −3.49403e7 −1.66363
$$850$$ 0 0
$$851$$ −1.89660e7 −0.897740
$$852$$ − 5.57491e7i − 2.63111i
$$853$$ − 2.26975e7i − 1.06808i −0.845458 0.534042i $$-0.820672\pi$$
0.845458 0.534042i $$-0.179328\pi$$
$$854$$ −1.25675e7 −0.589664
$$855$$ 0 0
$$856$$ 5.26766e7 2.45716
$$857$$ − 2.52900e7i − 1.17624i −0.808774 0.588120i $$-0.799868\pi$$
0.808774 0.588120i $$-0.200132\pi$$
$$858$$ − 4.54720e6i − 0.210875i
$$859$$ 1.03947e7 0.480652 0.240326 0.970692i $$-0.422746\pi$$
0.240326 + 0.970692i $$0.422746\pi$$
$$860$$ 0 0
$$861$$ −1.23219e7 −0.566462
$$862$$ 1.93750e7i 0.888122i
$$863$$ 4.33399e7i 1.98089i 0.137892 + 0.990447i $$0.455967\pi$$
−0.137892 + 0.990447i $$0.544033\pi$$
$$864$$ 1.10432e7 0.503281
$$865$$ 0 0
$$866$$ −3.94790e7 −1.78884
$$867$$ 2.16360e7i 0.977527i
$$868$$ 2.54698e7i 1.14743i
$$869$$ −7.37389e6 −0.331243
$$870$$ 0 0
$$871$$ 6.77656e6 0.302666
$$872$$ 3.34433e7i 1.48942i
$$873$$ 2.74978e6i 0.122113i
$$874$$ −1.78752e6 −0.0791539
$$875$$ 0 0
$$876$$ −6.48141e7 −2.85370
$$877$$ − 3.71659e7i − 1.63172i −0.578248 0.815861i $$-0.696264\pi$$
0.578248 0.815861i $$-0.303736\pi$$
$$878$$ 7.41770e7i 3.24738i
$$879$$ −2.70449e7 −1.18063
$$880$$ 0 0
$$881$$ 9.04785e6 0.392740 0.196370 0.980530i $$-0.437085\pi$$
0.196370 + 0.980530i $$0.437085\pi$$
$$882$$ − 1.12847e6i − 0.0488448i
$$883$$ 3.29679e7i 1.42295i 0.702712 + 0.711474i $$0.251972\pi$$
−0.702712 + 0.711474i $$0.748028\pi$$
$$884$$ −1.63934e7 −0.705569
$$885$$ 0 0
$$886$$ −1.40269e7 −0.600313
$$887$$ 1.61099e7i 0.687517i 0.939058 + 0.343758i $$0.111700\pi$$
−0.939058 + 0.343758i $$0.888300\pi$$
$$888$$ − 5.24059e7i − 2.23022i
$$889$$ 2.95882e6 0.125564
$$890$$ 0 0
$$891$$ −1.05372e7 −0.444663
$$892$$ 5.60500e7i 2.35865i
$$893$$ − 913752.i − 0.0383442i
$$894$$ 2.83164e6 0.118493
$$895$$ 0 0
$$896$$ −1.32653e7 −0.552009
$$897$$ − 3.57504e6i − 0.148354i
$$898$$ 5.90574e6i 0.244390i
$$899$$ 2.61272e7 1.07819
$$900$$ 0 0
$$901$$ −3.89516e6 −0.159850
$$902$$ − 4.16718e7i − 1.70540i
$$903$$ 7.46368e6i 0.304603i
$$904$$ −3.00074e7 −1.22126
$$905$$ 0 0
$$906$$ 9.92656e6 0.401771
$$907$$ 4.47286e7i 1.80537i 0.430300 + 0.902686i $$0.358408\pi$$
−0.430300 + 0.902686i $$0.641592\pi$$
$$908$$ 5.05798e6i 0.203593i
$$909$$ 1.81871e6 0.0730053
$$910$$ 0 0
$$911$$ −6.60518e6 −0.263687 −0.131844 0.991271i $$-0.542090\pi$$
−0.131844 + 0.991271i $$0.542090\pi$$
$$912$$ − 1.95373e6i − 0.0777816i
$$913$$ − 4.76482e6i − 0.189177i
$$914$$ −2.90484e7 −1.15016
$$915$$ 0 0
$$916$$ 7.69730e7 3.03110
$$917$$ − 3.01771e6i − 0.118510i
$$918$$ 6.99132e7i 2.73812i
$$919$$ 3.08930e7 1.20662 0.603311 0.797506i $$-0.293848\pi$$
0.603311 + 0.797506i $$0.293848\pi$$
$$920$$ 0 0
$$921$$ 6.42704e6 0.249667
$$922$$ − 9.22684e6i − 0.357459i
$$923$$ 8.19840e6i 0.316756i
$$924$$ −1.08223e7 −0.417005
$$925$$ 0 0
$$926$$ −7.18235e7 −2.75258
$$927$$ 2.49382e6i 0.0953160i
$$928$$ − 9.29696e6i − 0.354381i
$$929$$ 4.87215e6 0.185217 0.0926087 0.995703i $$-0.470479\pi$$
0.0926087 + 0.995703i $$0.470479\pi$$
$$930$$ 0 0
$$931$$ −235298. −0.00889701
$$932$$ 1.35134e7i 0.509593i
$$933$$ − 9.33979e6i − 0.351264i
$$934$$ −6.12570e6 −0.229767
$$935$$ 0 0
$$936$$ −2.36880e6 −0.0883769
$$937$$ 3.25004e7i 1.20932i 0.796485 + 0.604658i $$0.206690\pi$$
−0.796485 + 0.604658i $$0.793310\pi$$
$$938$$ − 2.37180e7i − 0.880177i
$$939$$ −1.55448e6 −0.0575334
$$940$$ 0 0
$$941$$ −2.64040e6 −0.0972066 −0.0486033 0.998818i $$-0.515477\pi$$
−0.0486033 + 0.998818i $$0.515477\pi$$
$$942$$ − 4.10934e7i − 1.50884i
$$943$$ − 3.27627e7i − 1.19978i
$$944$$ 3.88752e6 0.141985
$$945$$ 0 0
$$946$$ −2.52416e7 −0.917042
$$947$$ 4.08179e7i 1.47903i 0.673142 + 0.739513i $$0.264944\pi$$
−0.673142 + 0.739513i $$0.735056\pi$$
$$948$$ − 3.02584e7i − 1.09351i
$$949$$ 9.53148e6 0.343554
$$950$$ 0 0
$$951$$ 962892. 0.0345244
$$952$$ 3.03761e7i 1.08627i
$$953$$ − 6.71983e6i − 0.239677i −0.992793 0.119838i $$-0.961762\pi$$
0.992793 0.119838i $$-0.0382376\pi$$
$$954$$ −1.06314e6 −0.0378198
$$955$$ 0 0
$$956$$ 3.28375e7 1.16205
$$957$$ 1.11017e7i 0.391840i
$$958$$ − 2.60330e7i − 0.916454i
$$959$$ −1.00186e7 −0.351773
$$960$$ 0 0
$$961$$ 2.98016e7 1.04095
$$962$$ 1.45572e7i 0.507154i
$$963$$ 6.87723e6i 0.238972i
$$964$$ −5.48019e7 −1.89934
$$965$$ 0 0
$$966$$ −1.25126e7 −0.431426
$$967$$ 2.78979e6i 0.0959413i 0.998849 + 0.0479707i $$0.0152754\pi$$
−0.998849 + 0.0479707i $$0.984725\pi$$
$$968$$ 3.86017e7i 1.32409i
$$969$$ 2.36258e6 0.0808310
$$970$$ 0 0
$$971$$ 3.33594e7 1.13545 0.567727 0.823217i $$-0.307823\pi$$
0.567727 + 0.823217i $$0.307823\pi$$
$$972$$ 2.38486e7i 0.809648i
$$973$$ 1.73489e6i 0.0587477i
$$974$$ 5.46309e7 1.84519
$$975$$ 0 0
$$976$$ 3.65228e7 1.22727
$$977$$ 7.60033e6i 0.254739i 0.991855 + 0.127370i $$0.0406534\pi$$
−0.991855 + 0.127370i $$0.959347\pi$$
$$978$$ 1.84688e6i 0.0617435i
$$979$$ 1.17350e7 0.391316
$$980$$ 0 0
$$981$$ −4.36621e6 −0.144854
$$982$$ 1.64090e7i 0.543004i
$$983$$ − 5.79760e6i − 0.191366i −0.995412 0.0956829i $$-0.969497\pi$$
0.995412 0.0956829i $$-0.0305035\pi$$
$$984$$ 9.05285e7 2.98056
$$985$$ 0 0
$$986$$ 5.88580e7 1.92803
$$987$$ − 6.39626e6i − 0.208994i
$$988$$ 932960.i 0.0304068i
$$989$$ −1.98451e7 −0.645153
$$990$$ 0 0
$$991$$ 1.26825e7 0.410224 0.205112 0.978739i $$-0.434244\pi$$
0.205112 + 0.978739i $$0.434244\pi$$
$$992$$ − 2.07917e7i − 0.670827i
$$993$$ 7.90227e6i 0.254319i
$$994$$ 2.86944e7 0.921152
$$995$$ 0 0
$$996$$ 1.95522e7 0.624521
$$997$$ − 1.44400e7i − 0.460077i −0.973182 0.230039i $$-0.926115\pi$$
0.973182 0.230039i $$-0.0738853\pi$$
$$998$$ − 2.99796e7i − 0.952796i
$$999$$ 4.22159e7 1.33833
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 175.6.b.a.99.2 2
5.2 odd 4 7.6.a.a.1.1 1
5.3 odd 4 175.6.a.b.1.1 1
5.4 even 2 inner 175.6.b.a.99.1 2
15.2 even 4 63.6.a.e.1.1 1
20.7 even 4 112.6.a.g.1.1 1
35.2 odd 12 49.6.c.c.18.1 2
35.12 even 12 49.6.c.b.18.1 2
35.17 even 12 49.6.c.b.30.1 2
35.27 even 4 49.6.a.a.1.1 1
35.32 odd 12 49.6.c.c.30.1 2
40.27 even 4 448.6.a.c.1.1 1
40.37 odd 4 448.6.a.m.1.1 1
55.32 even 4 847.6.a.b.1.1 1
60.47 odd 4 1008.6.a.y.1.1 1
105.62 odd 4 441.6.a.k.1.1 1
140.27 odd 4 784.6.a.c.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
7.6.a.a.1.1 1 5.2 odd 4
49.6.a.a.1.1 1 35.27 even 4
49.6.c.b.18.1 2 35.12 even 12
49.6.c.b.30.1 2 35.17 even 12
49.6.c.c.18.1 2 35.2 odd 12
49.6.c.c.30.1 2 35.32 odd 12
63.6.a.e.1.1 1 15.2 even 4
112.6.a.g.1.1 1 20.7 even 4
175.6.a.b.1.1 1 5.3 odd 4
175.6.b.a.99.1 2 5.4 even 2 inner
175.6.b.a.99.2 2 1.1 even 1 trivial
441.6.a.k.1.1 1 105.62 odd 4
448.6.a.c.1.1 1 40.27 even 4
448.6.a.m.1.1 1 40.37 odd 4
784.6.a.c.1.1 1 140.27 odd 4
847.6.a.b.1.1 1 55.32 even 4
1008.6.a.y.1.1 1 60.47 odd 4