# Properties

 Label 1050.6.g.i.799.2 Level $1050$ Weight $6$ Character 1050.799 Analytic conductor $168.403$ 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] = [1050,6,Mod(799,1050)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1050, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("1050.799");

S:= CuspForms(chi, 6);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 1050.g (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: $$168.403010804$$ 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: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 42) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 799.2 Root $$1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 1050.799 Dual form 1050.6.g.i.799.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+4.00000i q^{2} +9.00000i q^{3} -16.0000 q^{4} -36.0000 q^{6} +49.0000i q^{7} -64.0000i q^{8} -81.0000 q^{9} +O(q^{10})$$ $$q+4.00000i q^{2} +9.00000i q^{3} -16.0000 q^{4} -36.0000 q^{6} +49.0000i q^{7} -64.0000i q^{8} -81.0000 q^{9} +664.000 q^{11} -144.000i q^{12} +318.000i q^{13} -196.000 q^{14} +256.000 q^{16} -1582.00i q^{17} -324.000i q^{18} -236.000 q^{19} -441.000 q^{21} +2656.00i q^{22} +2212.00i q^{23} +576.000 q^{24} -1272.00 q^{26} -729.000i q^{27} -784.000i q^{28} +4954.00 q^{29} -7128.00 q^{31} +1024.00i q^{32} +5976.00i q^{33} +6328.00 q^{34} +1296.00 q^{36} -4358.00i q^{37} -944.000i q^{38} -2862.00 q^{39} +10542.0 q^{41} -1764.00i q^{42} -8452.00i q^{43} -10624.0 q^{44} -8848.00 q^{46} -5352.00i q^{47} +2304.00i q^{48} -2401.00 q^{49} +14238.0 q^{51} -5088.00i q^{52} -33354.0i q^{53} +2916.00 q^{54} +3136.00 q^{56} -2124.00i q^{57} +19816.0i q^{58} +15436.0 q^{59} -36762.0 q^{61} -28512.0i q^{62} -3969.00i q^{63} -4096.00 q^{64} -23904.0 q^{66} -40972.0i q^{67} +25312.0i q^{68} -19908.0 q^{69} -9092.00 q^{71} +5184.00i q^{72} -73454.0i q^{73} +17432.0 q^{74} +3776.00 q^{76} +32536.0i q^{77} -11448.0i q^{78} -89400.0 q^{79} +6561.00 q^{81} +42168.0i q^{82} -6428.00i q^{83} +7056.00 q^{84} +33808.0 q^{86} +44586.0i q^{87} -42496.0i q^{88} +122658. q^{89} -15582.0 q^{91} -35392.0i q^{92} -64152.0i q^{93} +21408.0 q^{94} -9216.00 q^{96} -21370.0i q^{97} -9604.00i q^{98} -53784.0 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 32 q^{4} - 72 q^{6} - 162 q^{9}+O(q^{10})$$ 2 * q - 32 * q^4 - 72 * q^6 - 162 * q^9 $$2 q - 32 q^{4} - 72 q^{6} - 162 q^{9} + 1328 q^{11} - 392 q^{14} + 512 q^{16} - 472 q^{19} - 882 q^{21} + 1152 q^{24} - 2544 q^{26} + 9908 q^{29} - 14256 q^{31} + 12656 q^{34} + 2592 q^{36} - 5724 q^{39} + 21084 q^{41} - 21248 q^{44} - 17696 q^{46} - 4802 q^{49} + 28476 q^{51} + 5832 q^{54} + 6272 q^{56} + 30872 q^{59} - 73524 q^{61} - 8192 q^{64} - 47808 q^{66} - 39816 q^{69} - 18184 q^{71} + 34864 q^{74} + 7552 q^{76} - 178800 q^{79} + 13122 q^{81} + 14112 q^{84} + 67616 q^{86} + 245316 q^{89} - 31164 q^{91} + 42816 q^{94} - 18432 q^{96} - 107568 q^{99}+O(q^{100})$$ 2 * q - 32 * q^4 - 72 * q^6 - 162 * q^9 + 1328 * q^11 - 392 * q^14 + 512 * q^16 - 472 * q^19 - 882 * q^21 + 1152 * q^24 - 2544 * q^26 + 9908 * q^29 - 14256 * q^31 + 12656 * q^34 + 2592 * q^36 - 5724 * q^39 + 21084 * q^41 - 21248 * q^44 - 17696 * q^46 - 4802 * q^49 + 28476 * q^51 + 5832 * q^54 + 6272 * q^56 + 30872 * q^59 - 73524 * q^61 - 8192 * q^64 - 47808 * q^66 - 39816 * q^69 - 18184 * q^71 + 34864 * q^74 + 7552 * q^76 - 178800 * q^79 + 13122 * q^81 + 14112 * q^84 + 67616 * q^86 + 245316 * q^89 - 31164 * q^91 + 42816 * q^94 - 18432 * q^96 - 107568 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1050\mathbb{Z}\right)^\times$$.

 $$n$$ $$127$$ $$451$$ $$701$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 4.00000i 0.707107i
$$3$$ 9.00000i 0.577350i
$$4$$ −16.0000 −0.500000
$$5$$ 0 0
$$6$$ −36.0000 −0.408248
$$7$$ 49.0000i 0.377964i
$$8$$ − 64.0000i − 0.353553i
$$9$$ −81.0000 −0.333333
$$10$$ 0 0
$$11$$ 664.000 1.65457 0.827287 0.561779i $$-0.189883\pi$$
0.827287 + 0.561779i $$0.189883\pi$$
$$12$$ − 144.000i − 0.288675i
$$13$$ 318.000i 0.521878i 0.965355 + 0.260939i $$0.0840321\pi$$
−0.965355 + 0.260939i $$0.915968\pi$$
$$14$$ −196.000 −0.267261
$$15$$ 0 0
$$16$$ 256.000 0.250000
$$17$$ − 1582.00i − 1.32765i −0.747887 0.663826i $$-0.768932\pi$$
0.747887 0.663826i $$-0.231068\pi$$
$$18$$ − 324.000i − 0.235702i
$$19$$ −236.000 −0.149978 −0.0749891 0.997184i $$-0.523892\pi$$
−0.0749891 + 0.997184i $$0.523892\pi$$
$$20$$ 0 0
$$21$$ −441.000 −0.218218
$$22$$ 2656.00i 1.16996i
$$23$$ 2212.00i 0.871898i 0.899971 + 0.435949i $$0.143587\pi$$
−0.899971 + 0.435949i $$0.856413\pi$$
$$24$$ 576.000 0.204124
$$25$$ 0 0
$$26$$ −1272.00 −0.369023
$$27$$ − 729.000i − 0.192450i
$$28$$ − 784.000i − 0.188982i
$$29$$ 4954.00 1.09386 0.546929 0.837179i $$-0.315797\pi$$
0.546929 + 0.837179i $$0.315797\pi$$
$$30$$ 0 0
$$31$$ −7128.00 −1.33218 −0.666091 0.745871i $$-0.732034\pi$$
−0.666091 + 0.745871i $$0.732034\pi$$
$$32$$ 1024.00i 0.176777i
$$33$$ 5976.00i 0.955269i
$$34$$ 6328.00 0.938792
$$35$$ 0 0
$$36$$ 1296.00 0.166667
$$37$$ − 4358.00i − 0.523339i −0.965158 0.261669i $$-0.915727\pi$$
0.965158 0.261669i $$-0.0842730\pi$$
$$38$$ − 944.000i − 0.106051i
$$39$$ −2862.00 −0.301306
$$40$$ 0 0
$$41$$ 10542.0 0.979407 0.489704 0.871889i $$-0.337105\pi$$
0.489704 + 0.871889i $$0.337105\pi$$
$$42$$ − 1764.00i − 0.154303i
$$43$$ − 8452.00i − 0.697089i −0.937292 0.348545i $$-0.886676\pi$$
0.937292 0.348545i $$-0.113324\pi$$
$$44$$ −10624.0 −0.827287
$$45$$ 0 0
$$46$$ −8848.00 −0.616525
$$47$$ − 5352.00i − 0.353404i −0.984264 0.176702i $$-0.943457\pi$$
0.984264 0.176702i $$-0.0565429\pi$$
$$48$$ 2304.00i 0.144338i
$$49$$ −2401.00 −0.142857
$$50$$ 0 0
$$51$$ 14238.0 0.766520
$$52$$ − 5088.00i − 0.260939i
$$53$$ − 33354.0i − 1.63102i −0.578746 0.815508i $$-0.696458\pi$$
0.578746 0.815508i $$-0.303542\pi$$
$$54$$ 2916.00 0.136083
$$55$$ 0 0
$$56$$ 3136.00 0.133631
$$57$$ − 2124.00i − 0.0865899i
$$58$$ 19816.0i 0.773475i
$$59$$ 15436.0 0.577304 0.288652 0.957434i $$-0.406793\pi$$
0.288652 + 0.957434i $$0.406793\pi$$
$$60$$ 0 0
$$61$$ −36762.0 −1.26495 −0.632477 0.774579i $$-0.717962\pi$$
−0.632477 + 0.774579i $$0.717962\pi$$
$$62$$ − 28512.0i − 0.941995i
$$63$$ − 3969.00i − 0.125988i
$$64$$ −4096.00 −0.125000
$$65$$ 0 0
$$66$$ −23904.0 −0.675477
$$67$$ − 40972.0i − 1.11506i −0.830155 0.557532i $$-0.811748\pi$$
0.830155 0.557532i $$-0.188252\pi$$
$$68$$ 25312.0i 0.663826i
$$69$$ −19908.0 −0.503390
$$70$$ 0 0
$$71$$ −9092.00 −0.214049 −0.107025 0.994256i $$-0.534132\pi$$
−0.107025 + 0.994256i $$0.534132\pi$$
$$72$$ 5184.00i 0.117851i
$$73$$ − 73454.0i − 1.61327i −0.591047 0.806637i $$-0.701285\pi$$
0.591047 0.806637i $$-0.298715\pi$$
$$74$$ 17432.0 0.370056
$$75$$ 0 0
$$76$$ 3776.00 0.0749891
$$77$$ 32536.0i 0.625370i
$$78$$ − 11448.0i − 0.213056i
$$79$$ −89400.0 −1.61165 −0.805823 0.592156i $$-0.798277\pi$$
−0.805823 + 0.592156i $$0.798277\pi$$
$$80$$ 0 0
$$81$$ 6561.00 0.111111
$$82$$ 42168.0i 0.692546i
$$83$$ − 6428.00i − 0.102419i −0.998688 0.0512095i $$-0.983692\pi$$
0.998688 0.0512095i $$-0.0163076\pi$$
$$84$$ 7056.00 0.109109
$$85$$ 0 0
$$86$$ 33808.0 0.492916
$$87$$ 44586.0i 0.631539i
$$88$$ − 42496.0i − 0.584980i
$$89$$ 122658. 1.64142 0.820712 0.571342i $$-0.193577\pi$$
0.820712 + 0.571342i $$0.193577\pi$$
$$90$$ 0 0
$$91$$ −15582.0 −0.197251
$$92$$ − 35392.0i − 0.435949i
$$93$$ − 64152.0i − 0.769135i
$$94$$ 21408.0 0.249894
$$95$$ 0 0
$$96$$ −9216.00 −0.102062
$$97$$ − 21370.0i − 0.230608i −0.993330 0.115304i $$-0.963216\pi$$
0.993330 0.115304i $$-0.0367843\pi$$
$$98$$ − 9604.00i − 0.101015i
$$99$$ −53784.0 −0.551525
$$100$$ 0 0
$$101$$ −36814.0 −0.359095 −0.179548 0.983749i $$-0.557463\pi$$
−0.179548 + 0.983749i $$0.557463\pi$$
$$102$$ 56952.0i 0.542012i
$$103$$ 104528.i 0.970822i 0.874286 + 0.485411i $$0.161330\pi$$
−0.874286 + 0.485411i $$0.838670\pi$$
$$104$$ 20352.0 0.184512
$$105$$ 0 0
$$106$$ 133416. 1.15330
$$107$$ − 214440.i − 1.81070i −0.424667 0.905350i $$-0.639609\pi$$
0.424667 0.905350i $$-0.360391\pi$$
$$108$$ 11664.0i 0.0962250i
$$109$$ −28798.0 −0.232165 −0.116082 0.993240i $$-0.537034\pi$$
−0.116082 + 0.993240i $$0.537034\pi$$
$$110$$ 0 0
$$111$$ 39222.0 0.302150
$$112$$ 12544.0i 0.0944911i
$$113$$ − 56014.0i − 0.412668i −0.978482 0.206334i $$-0.933847\pi$$
0.978482 0.206334i $$-0.0661533\pi$$
$$114$$ 8496.00 0.0612283
$$115$$ 0 0
$$116$$ −79264.0 −0.546929
$$117$$ − 25758.0i − 0.173959i
$$118$$ 61744.0i 0.408216i
$$119$$ 77518.0 0.501805
$$120$$ 0 0
$$121$$ 279845. 1.73762
$$122$$ − 147048.i − 0.894457i
$$123$$ 94878.0i 0.565461i
$$124$$ 114048. 0.666091
$$125$$ 0 0
$$126$$ 15876.0 0.0890871
$$127$$ − 185400.i − 1.02000i −0.860174 0.510000i $$-0.829645\pi$$
0.860174 0.510000i $$-0.170355\pi$$
$$128$$ − 16384.0i − 0.0883883i
$$129$$ 76068.0 0.402465
$$130$$ 0 0
$$131$$ 64532.0 0.328547 0.164273 0.986415i $$-0.447472\pi$$
0.164273 + 0.986415i $$0.447472\pi$$
$$132$$ − 95616.0i − 0.477635i
$$133$$ − 11564.0i − 0.0566864i
$$134$$ 163888. 0.788470
$$135$$ 0 0
$$136$$ −101248. −0.469396
$$137$$ − 152930.i − 0.696131i −0.937470 0.348066i $$-0.886839\pi$$
0.937470 0.348066i $$-0.113161\pi$$
$$138$$ − 79632.0i − 0.355951i
$$139$$ 343460. 1.50778 0.753892 0.656998i $$-0.228174\pi$$
0.753892 + 0.656998i $$0.228174\pi$$
$$140$$ 0 0
$$141$$ 48168.0 0.204038
$$142$$ − 36368.0i − 0.151356i
$$143$$ 211152.i 0.863486i
$$144$$ −20736.0 −0.0833333
$$145$$ 0 0
$$146$$ 293816. 1.14076
$$147$$ − 21609.0i − 0.0824786i
$$148$$ 69728.0i 0.261669i
$$149$$ 174858. 0.645238 0.322619 0.946529i $$-0.395437\pi$$
0.322619 + 0.946529i $$0.395437\pi$$
$$150$$ 0 0
$$151$$ −452552. −1.61520 −0.807600 0.589731i $$-0.799234\pi$$
−0.807600 + 0.589731i $$0.799234\pi$$
$$152$$ 15104.0i 0.0530253i
$$153$$ 128142.i 0.442551i
$$154$$ −130144. −0.442204
$$155$$ 0 0
$$156$$ 45792.0 0.150653
$$157$$ 499066.i 1.61588i 0.589265 + 0.807940i $$0.299417\pi$$
−0.589265 + 0.807940i $$0.700583\pi$$
$$158$$ − 357600.i − 1.13961i
$$159$$ 300186. 0.941668
$$160$$ 0 0
$$161$$ −108388. −0.329546
$$162$$ 26244.0i 0.0785674i
$$163$$ − 475588.i − 1.40204i −0.713139 0.701022i $$-0.752727\pi$$
0.713139 0.701022i $$-0.247273\pi$$
$$164$$ −168672. −0.489704
$$165$$ 0 0
$$166$$ 25712.0 0.0724212
$$167$$ − 120224.i − 0.333580i −0.985992 0.166790i $$-0.946660\pi$$
0.985992 0.166790i $$-0.0533402\pi$$
$$168$$ 28224.0i 0.0771517i
$$169$$ 270169. 0.727644
$$170$$ 0 0
$$171$$ 19116.0 0.0499927
$$172$$ 135232.i 0.348545i
$$173$$ 508874.i 1.29269i 0.763045 + 0.646346i $$0.223704\pi$$
−0.763045 + 0.646346i $$0.776296\pi$$
$$174$$ −178344. −0.446566
$$175$$ 0 0
$$176$$ 169984. 0.413644
$$177$$ 138924.i 0.333307i
$$178$$ 490632.i 1.16066i
$$179$$ −487560. −1.13735 −0.568677 0.822561i $$-0.692544\pi$$
−0.568677 + 0.822561i $$0.692544\pi$$
$$180$$ 0 0
$$181$$ −544410. −1.23518 −0.617589 0.786501i $$-0.711891\pi$$
−0.617589 + 0.786501i $$0.711891\pi$$
$$182$$ − 62328.0i − 0.139478i
$$183$$ − 330858.i − 0.730321i
$$184$$ 141568. 0.308262
$$185$$ 0 0
$$186$$ 256608. 0.543861
$$187$$ − 1.05045e6i − 2.19670i
$$188$$ 85632.0i 0.176702i
$$189$$ 35721.0 0.0727393
$$190$$ 0 0
$$191$$ 376404. 0.746570 0.373285 0.927717i $$-0.378231\pi$$
0.373285 + 0.927717i $$0.378231\pi$$
$$192$$ − 36864.0i − 0.0721688i
$$193$$ 844946.i 1.63281i 0.577480 + 0.816405i $$0.304036\pi$$
−0.577480 + 0.816405i $$0.695964\pi$$
$$194$$ 85480.0 0.163065
$$195$$ 0 0
$$196$$ 38416.0 0.0714286
$$197$$ 492794.i 0.904690i 0.891843 + 0.452345i $$0.149412\pi$$
−0.891843 + 0.452345i $$0.850588\pi$$
$$198$$ − 215136.i − 0.389987i
$$199$$ 914776. 1.63750 0.818751 0.574148i $$-0.194667\pi$$
0.818751 + 0.574148i $$0.194667\pi$$
$$200$$ 0 0
$$201$$ 368748. 0.643783
$$202$$ − 147256.i − 0.253919i
$$203$$ 242746.i 0.413440i
$$204$$ −227808. −0.383260
$$205$$ 0 0
$$206$$ −418112. −0.686475
$$207$$ − 179172.i − 0.290633i
$$208$$ 81408.0i 0.130469i
$$209$$ −156704. −0.248150
$$210$$ 0 0
$$211$$ 311780. 0.482106 0.241053 0.970512i $$-0.422507\pi$$
0.241053 + 0.970512i $$0.422507\pi$$
$$212$$ 533664.i 0.815508i
$$213$$ − 81828.0i − 0.123581i
$$214$$ 857760. 1.28036
$$215$$ 0 0
$$216$$ −46656.0 −0.0680414
$$217$$ − 349272.i − 0.503517i
$$218$$ − 115192.i − 0.164165i
$$219$$ 661086. 0.931425
$$220$$ 0 0
$$221$$ 503076. 0.692872
$$222$$ 156888.i 0.213652i
$$223$$ − 1.28776e6i − 1.73409i −0.498226 0.867047i $$-0.666015\pi$$
0.498226 0.867047i $$-0.333985\pi$$
$$224$$ −50176.0 −0.0668153
$$225$$ 0 0
$$226$$ 224056. 0.291800
$$227$$ − 1.28905e6i − 1.66037i −0.557485 0.830187i $$-0.688234\pi$$
0.557485 0.830187i $$-0.311766\pi$$
$$228$$ 33984.0i 0.0432950i
$$229$$ −678214. −0.854630 −0.427315 0.904103i $$-0.640540\pi$$
−0.427315 + 0.904103i $$0.640540\pi$$
$$230$$ 0 0
$$231$$ −292824. −0.361058
$$232$$ − 317056.i − 0.386737i
$$233$$ − 1.11731e6i − 1.34829i −0.738598 0.674146i $$-0.764512\pi$$
0.738598 0.674146i $$-0.235488\pi$$
$$234$$ 103032. 0.123008
$$235$$ 0 0
$$236$$ −246976. −0.288652
$$237$$ − 804600.i − 0.930485i
$$238$$ 310072.i 0.354830i
$$239$$ 1.26196e6 1.42906 0.714528 0.699606i $$-0.246641\pi$$
0.714528 + 0.699606i $$0.246641\pi$$
$$240$$ 0 0
$$241$$ 948218. 1.05164 0.525818 0.850597i $$-0.323759\pi$$
0.525818 + 0.850597i $$0.323759\pi$$
$$242$$ 1.11938e6i 1.22868i
$$243$$ 59049.0i 0.0641500i
$$244$$ 588192. 0.632477
$$245$$ 0 0
$$246$$ −379512. −0.399841
$$247$$ − 75048.0i − 0.0782703i
$$248$$ 456192.i 0.470997i
$$249$$ 57852.0 0.0591317
$$250$$ 0 0
$$251$$ −486396. −0.487310 −0.243655 0.969862i $$-0.578347\pi$$
−0.243655 + 0.969862i $$0.578347\pi$$
$$252$$ 63504.0i 0.0629941i
$$253$$ 1.46877e6i 1.44262i
$$254$$ 741600. 0.721249
$$255$$ 0 0
$$256$$ 65536.0 0.0625000
$$257$$ 1.03910e6i 0.981349i 0.871343 + 0.490675i $$0.163250\pi$$
−0.871343 + 0.490675i $$0.836750\pi$$
$$258$$ 304272.i 0.284585i
$$259$$ 213542. 0.197803
$$260$$ 0 0
$$261$$ −401274. −0.364619
$$262$$ 258128.i 0.232317i
$$263$$ 1.35104e6i 1.20443i 0.798335 + 0.602213i $$0.205714\pi$$
−0.798335 + 0.602213i $$0.794286\pi$$
$$264$$ 382464. 0.337739
$$265$$ 0 0
$$266$$ 46256.0 0.0400833
$$267$$ 1.10392e6i 0.947677i
$$268$$ 655552.i 0.557532i
$$269$$ 1.11811e6 0.942115 0.471057 0.882103i $$-0.343872\pi$$
0.471057 + 0.882103i $$0.343872\pi$$
$$270$$ 0 0
$$271$$ −190104. −0.157242 −0.0786209 0.996905i $$-0.525052\pi$$
−0.0786209 + 0.996905i $$0.525052\pi$$
$$272$$ − 404992.i − 0.331913i
$$273$$ − 140238.i − 0.113883i
$$274$$ 611720. 0.492239
$$275$$ 0 0
$$276$$ 318528. 0.251695
$$277$$ 200506.i 0.157010i 0.996914 + 0.0785051i $$0.0250147\pi$$
−0.996914 + 0.0785051i $$0.974985\pi$$
$$278$$ 1.37384e6i 1.06616i
$$279$$ 577368. 0.444061
$$280$$ 0 0
$$281$$ 1.09237e6 0.825285 0.412643 0.910893i $$-0.364606\pi$$
0.412643 + 0.910893i $$0.364606\pi$$
$$282$$ 192672.i 0.144277i
$$283$$ 1.81258e6i 1.34534i 0.739944 + 0.672669i $$0.234852\pi$$
−0.739944 + 0.672669i $$0.765148\pi$$
$$284$$ 145472. 0.107025
$$285$$ 0 0
$$286$$ −844608. −0.610577
$$287$$ 516558.i 0.370181i
$$288$$ − 82944.0i − 0.0589256i
$$289$$ −1.08287e6 −0.762659
$$290$$ 0 0
$$291$$ 192330. 0.133142
$$292$$ 1.17526e6i 0.806637i
$$293$$ 2.10031e6i 1.42927i 0.699499 + 0.714634i $$0.253407\pi$$
−0.699499 + 0.714634i $$0.746593\pi$$
$$294$$ 86436.0 0.0583212
$$295$$ 0 0
$$296$$ −278912. −0.185028
$$297$$ − 484056.i − 0.318423i
$$298$$ 699432.i 0.456252i
$$299$$ −703416. −0.455024
$$300$$ 0 0
$$301$$ 414148. 0.263475
$$302$$ − 1.81021e6i − 1.14212i
$$303$$ − 331326.i − 0.207324i
$$304$$ −60416.0 −0.0374945
$$305$$ 0 0
$$306$$ −512568. −0.312931
$$307$$ 1.64104e6i 0.993743i 0.867824 + 0.496872i $$0.165518\pi$$
−0.867824 + 0.496872i $$0.834482\pi$$
$$308$$ − 520576.i − 0.312685i
$$309$$ −940752. −0.560504
$$310$$ 0 0
$$311$$ −945232. −0.554163 −0.277081 0.960846i $$-0.589367\pi$$
−0.277081 + 0.960846i $$0.589367\pi$$
$$312$$ 183168.i 0.106528i
$$313$$ 415354.i 0.239639i 0.992796 + 0.119820i $$0.0382316\pi$$
−0.992796 + 0.119820i $$0.961768\pi$$
$$314$$ −1.99626e6 −1.14260
$$315$$ 0 0
$$316$$ 1.43040e6 0.805823
$$317$$ − 1.18481e6i − 0.662220i −0.943592 0.331110i $$-0.892577\pi$$
0.943592 0.331110i $$-0.107423\pi$$
$$318$$ 1.20074e6i 0.665860i
$$319$$ 3.28946e6 1.80987
$$320$$ 0 0
$$321$$ 1.92996e6 1.04541
$$322$$ − 433552.i − 0.233024i
$$323$$ 373352.i 0.199119i
$$324$$ −104976. −0.0555556
$$325$$ 0 0
$$326$$ 1.90235e6 0.991395
$$327$$ − 259182.i − 0.134040i
$$328$$ − 674688.i − 0.346273i
$$329$$ 262248. 0.133574
$$330$$ 0 0
$$331$$ 1.37155e6 0.688083 0.344042 0.938954i $$-0.388204\pi$$
0.344042 + 0.938954i $$0.388204\pi$$
$$332$$ 102848.i 0.0512095i
$$333$$ 352998.i 0.174446i
$$334$$ 480896. 0.235877
$$335$$ 0 0
$$336$$ −112896. −0.0545545
$$337$$ − 963522.i − 0.462154i −0.972935 0.231077i $$-0.925775\pi$$
0.972935 0.231077i $$-0.0742250\pi$$
$$338$$ 1.08068e6i 0.514522i
$$339$$ 504126. 0.238254
$$340$$ 0 0
$$341$$ −4.73299e6 −2.20419
$$342$$ 76464.0i 0.0353502i
$$343$$ − 117649.i − 0.0539949i
$$344$$ −540928. −0.246458
$$345$$ 0 0
$$346$$ −2.03550e6 −0.914071
$$347$$ − 2.57731e6i − 1.14906i −0.818483 0.574531i $$-0.805185\pi$$
0.818483 0.574531i $$-0.194815\pi$$
$$348$$ − 713376.i − 0.315770i
$$349$$ 3.06751e6 1.34810 0.674051 0.738684i $$-0.264553\pi$$
0.674051 + 0.738684i $$0.264553\pi$$
$$350$$ 0 0
$$351$$ 231822. 0.100435
$$352$$ 679936.i 0.292490i
$$353$$ − 3.10144e6i − 1.32473i −0.749182 0.662364i $$-0.769553\pi$$
0.749182 0.662364i $$-0.230447\pi$$
$$354$$ −555696. −0.235683
$$355$$ 0 0
$$356$$ −1.96253e6 −0.820712
$$357$$ 697662.i 0.289717i
$$358$$ − 1.95024e6i − 0.804230i
$$359$$ 327508. 0.134118 0.0670588 0.997749i $$-0.478638\pi$$
0.0670588 + 0.997749i $$0.478638\pi$$
$$360$$ 0 0
$$361$$ −2.42040e6 −0.977507
$$362$$ − 2.17764e6i − 0.873403i
$$363$$ 2.51860e6i 1.00321i
$$364$$ 249312. 0.0986256
$$365$$ 0 0
$$366$$ 1.32343e6 0.516415
$$367$$ 2.86739e6i 1.11128i 0.831424 + 0.555638i $$0.187526\pi$$
−0.831424 + 0.555638i $$0.812474\pi$$
$$368$$ 566272.i 0.217974i
$$369$$ −853902. −0.326469
$$370$$ 0 0
$$371$$ 1.63435e6 0.616466
$$372$$ 1.02643e6i 0.384568i
$$373$$ 3.58029e6i 1.33244i 0.745757 + 0.666218i $$0.232088\pi$$
−0.745757 + 0.666218i $$0.767912\pi$$
$$374$$ 4.20179e6 1.55330
$$375$$ 0 0
$$376$$ −342528. −0.124947
$$377$$ 1.57537e6i 0.570860i
$$378$$ 142884.i 0.0514344i
$$379$$ −1.64235e6 −0.587310 −0.293655 0.955912i $$-0.594872\pi$$
−0.293655 + 0.955912i $$0.594872\pi$$
$$380$$ 0 0
$$381$$ 1.66860e6 0.588898
$$382$$ 1.50562e6i 0.527905i
$$383$$ − 2.05698e6i − 0.716527i −0.933621 0.358263i $$-0.883369\pi$$
0.933621 0.358263i $$-0.116631\pi$$
$$384$$ 147456. 0.0510310
$$385$$ 0 0
$$386$$ −3.37978e6 −1.15457
$$387$$ 684612.i 0.232363i
$$388$$ 341920.i 0.115304i
$$389$$ −616142. −0.206446 −0.103223 0.994658i $$-0.532916\pi$$
−0.103223 + 0.994658i $$0.532916\pi$$
$$390$$ 0 0
$$391$$ 3.49938e6 1.15758
$$392$$ 153664.i 0.0505076i
$$393$$ 580788.i 0.189686i
$$394$$ −1.97118e6 −0.639713
$$395$$ 0 0
$$396$$ 860544. 0.275762
$$397$$ − 2.19212e6i − 0.698052i −0.937113 0.349026i $$-0.886513\pi$$
0.937113 0.349026i $$-0.113487\pi$$
$$398$$ 3.65910e6i 1.15789i
$$399$$ 104076. 0.0327279
$$400$$ 0 0
$$401$$ 3.28454e6 1.02003 0.510015 0.860165i $$-0.329640\pi$$
0.510015 + 0.860165i $$0.329640\pi$$
$$402$$ 1.47499e6i 0.455223i
$$403$$ − 2.26670e6i − 0.695236i
$$404$$ 589024. 0.179548
$$405$$ 0 0
$$406$$ −970984. −0.292346
$$407$$ − 2.89371e6i − 0.865903i
$$408$$ − 911232.i − 0.271006i
$$409$$ 3.61219e6 1.06773 0.533866 0.845569i $$-0.320739\pi$$
0.533866 + 0.845569i $$0.320739\pi$$
$$410$$ 0 0
$$411$$ 1.37637e6 0.401912
$$412$$ − 1.67245e6i − 0.485411i
$$413$$ 756364.i 0.218200i
$$414$$ 716688. 0.205508
$$415$$ 0 0
$$416$$ −325632. −0.0922558
$$417$$ 3.09114e6i 0.870520i
$$418$$ − 626816.i − 0.175469i
$$419$$ −5.41489e6 −1.50680 −0.753398 0.657564i $$-0.771587\pi$$
−0.753398 + 0.657564i $$0.771587\pi$$
$$420$$ 0 0
$$421$$ 3.60629e6 0.991644 0.495822 0.868424i $$-0.334867\pi$$
0.495822 + 0.868424i $$0.334867\pi$$
$$422$$ 1.24712e6i 0.340900i
$$423$$ 433512.i 0.117801i
$$424$$ −2.13466e6 −0.576651
$$425$$ 0 0
$$426$$ 327312. 0.0873852
$$427$$ − 1.80134e6i − 0.478107i
$$428$$ 3.43104e6i 0.905350i
$$429$$ −1.90037e6 −0.498534
$$430$$ 0 0
$$431$$ −2.78214e6 −0.721416 −0.360708 0.932679i $$-0.617465\pi$$
−0.360708 + 0.932679i $$0.617465\pi$$
$$432$$ − 186624.i − 0.0481125i
$$433$$ 6.27619e6i 1.60871i 0.594152 + 0.804353i $$0.297488\pi$$
−0.594152 + 0.804353i $$0.702512\pi$$
$$434$$ 1.39709e6 0.356041
$$435$$ 0 0
$$436$$ 460768. 0.116082
$$437$$ − 522032.i − 0.130766i
$$438$$ 2.64434e6i 0.658617i
$$439$$ −641592. −0.158890 −0.0794452 0.996839i $$-0.525315\pi$$
−0.0794452 + 0.996839i $$0.525315\pi$$
$$440$$ 0 0
$$441$$ 194481. 0.0476190
$$442$$ 2.01230e6i 0.489934i
$$443$$ 6.05546e6i 1.46601i 0.680222 + 0.733006i $$0.261883\pi$$
−0.680222 + 0.733006i $$0.738117\pi$$
$$444$$ −627552. −0.151075
$$445$$ 0 0
$$446$$ 5.15104e6 1.22619
$$447$$ 1.57372e6i 0.372528i
$$448$$ − 200704.i − 0.0472456i
$$449$$ 5.16681e6 1.20950 0.604752 0.796414i $$-0.293272\pi$$
0.604752 + 0.796414i $$0.293272\pi$$
$$450$$ 0 0
$$451$$ 6.99989e6 1.62050
$$452$$ 896224.i 0.206334i
$$453$$ − 4.07297e6i − 0.932536i
$$454$$ 5.15621e6 1.17406
$$455$$ 0 0
$$456$$ −135936. −0.0306142
$$457$$ 227798.i 0.0510222i 0.999675 + 0.0255111i $$0.00812132\pi$$
−0.999675 + 0.0255111i $$0.991879\pi$$
$$458$$ − 2.71286e6i − 0.604315i
$$459$$ −1.15328e6 −0.255507
$$460$$ 0 0
$$461$$ 585146. 0.128237 0.0641183 0.997942i $$-0.479577\pi$$
0.0641183 + 0.997942i $$0.479577\pi$$
$$462$$ − 1.17130e6i − 0.255306i
$$463$$ − 3.41454e6i − 0.740251i −0.928982 0.370126i $$-0.879315\pi$$
0.928982 0.370126i $$-0.120685\pi$$
$$464$$ 1.26822e6 0.273465
$$465$$ 0 0
$$466$$ 4.46924e6 0.953386
$$467$$ − 716300.i − 0.151986i −0.997108 0.0759929i $$-0.975787\pi$$
0.997108 0.0759929i $$-0.0242126\pi$$
$$468$$ 412128.i 0.0869796i
$$469$$ 2.00763e6 0.421455
$$470$$ 0 0
$$471$$ −4.49159e6 −0.932928
$$472$$ − 987904.i − 0.204108i
$$473$$ − 5.61213e6i − 1.15339i
$$474$$ 3.21840e6 0.657952
$$475$$ 0 0
$$476$$ −1.24029e6 −0.250903
$$477$$ 2.70167e6i 0.543672i
$$478$$ 5.04782e6i 1.01050i
$$479$$ −5.24092e6 −1.04368 −0.521842 0.853042i $$-0.674755\pi$$
−0.521842 + 0.853042i $$0.674755\pi$$
$$480$$ 0 0
$$481$$ 1.38584e6 0.273119
$$482$$ 3.79287e6i 0.743619i
$$483$$ − 975492.i − 0.190264i
$$484$$ −4.47752e6 −0.868809
$$485$$ 0 0
$$486$$ −236196. −0.0453609
$$487$$ − 1.11702e6i − 0.213421i −0.994290 0.106710i $$-0.965968\pi$$
0.994290 0.106710i $$-0.0340318\pi$$
$$488$$ 2.35277e6i 0.447229i
$$489$$ 4.28029e6 0.809471
$$490$$ 0 0
$$491$$ 1.34458e6 0.251699 0.125850 0.992049i $$-0.459834\pi$$
0.125850 + 0.992049i $$0.459834\pi$$
$$492$$ − 1.51805e6i − 0.282731i
$$493$$ − 7.83723e6i − 1.45226i
$$494$$ 300192. 0.0553454
$$495$$ 0 0
$$496$$ −1.82477e6 −0.333045
$$497$$ − 445508.i − 0.0809030i
$$498$$ 231408.i 0.0418124i
$$499$$ 6.54648e6 1.17695 0.588473 0.808517i $$-0.299729\pi$$
0.588473 + 0.808517i $$0.299729\pi$$
$$500$$ 0 0
$$501$$ 1.08202e6 0.192592
$$502$$ − 1.94558e6i − 0.344580i
$$503$$ − 8.22050e6i − 1.44870i −0.689432 0.724350i $$-0.742140\pi$$
0.689432 0.724350i $$-0.257860\pi$$
$$504$$ −254016. −0.0445435
$$505$$ 0 0
$$506$$ −5.87507e6 −1.02009
$$507$$ 2.43152e6i 0.420105i
$$508$$ 2.96640e6i 0.510000i
$$509$$ 5.11045e6 0.874308 0.437154 0.899387i $$-0.355987\pi$$
0.437154 + 0.899387i $$0.355987\pi$$
$$510$$ 0 0
$$511$$ 3.59925e6 0.609760
$$512$$ 262144.i 0.0441942i
$$513$$ 172044.i 0.0288633i
$$514$$ −4.15639e6 −0.693919
$$515$$ 0 0
$$516$$ −1.21709e6 −0.201232
$$517$$ − 3.55373e6i − 0.584733i
$$518$$ 854168.i 0.139868i
$$519$$ −4.57987e6 −0.746336
$$520$$ 0 0
$$521$$ 9.69999e6 1.56559 0.782793 0.622282i $$-0.213794\pi$$
0.782793 + 0.622282i $$0.213794\pi$$
$$522$$ − 1.60510e6i − 0.257825i
$$523$$ − 3.17295e6i − 0.507234i −0.967305 0.253617i $$-0.918380\pi$$
0.967305 0.253617i $$-0.0816204\pi$$
$$524$$ −1.03251e6 −0.164273
$$525$$ 0 0
$$526$$ −5.40418e6 −0.851658
$$527$$ 1.12765e7i 1.76867i
$$528$$ 1.52986e6i 0.238817i
$$529$$ 1.54340e6 0.239794
$$530$$ 0 0
$$531$$ −1.25032e6 −0.192435
$$532$$ 185024.i 0.0283432i
$$533$$ 3.35236e6i 0.511131i
$$534$$ −4.41569e6 −0.670109
$$535$$ 0 0
$$536$$ −2.62221e6 −0.394235
$$537$$ − 4.38804e6i − 0.656651i
$$538$$ 4.47244e6i 0.666176i
$$539$$ −1.59426e6 −0.236368
$$540$$ 0 0
$$541$$ −6.62575e6 −0.973289 −0.486644 0.873600i $$-0.661779\pi$$
−0.486644 + 0.873600i $$0.661779\pi$$
$$542$$ − 760416.i − 0.111187i
$$543$$ − 4.89969e6i − 0.713131i
$$544$$ 1.61997e6 0.234698
$$545$$ 0 0
$$546$$ 560952. 0.0805275
$$547$$ − 3.84707e6i − 0.549745i −0.961481 0.274873i $$-0.911364\pi$$
0.961481 0.274873i $$-0.0886357\pi$$
$$548$$ 2.44688e6i 0.348066i
$$549$$ 2.97772e6 0.421651
$$550$$ 0 0
$$551$$ −1.16914e6 −0.164055
$$552$$ 1.27411e6i 0.177975i
$$553$$ − 4.38060e6i − 0.609145i
$$554$$ −802024. −0.111023
$$555$$ 0 0
$$556$$ −5.49536e6 −0.753892
$$557$$ − 5.00176e6i − 0.683101i −0.939863 0.341550i $$-0.889048\pi$$
0.939863 0.341550i $$-0.110952\pi$$
$$558$$ 2.30947e6i 0.313998i
$$559$$ 2.68774e6 0.363795
$$560$$ 0 0
$$561$$ 9.45403e6 1.26826
$$562$$ 4.36948e6i 0.583565i
$$563$$ 2.27772e6i 0.302852i 0.988469 + 0.151426i $$0.0483865\pi$$
−0.988469 + 0.151426i $$0.951614\pi$$
$$564$$ −770688. −0.102019
$$565$$ 0 0
$$566$$ −7.25032e6 −0.951297
$$567$$ 321489.i 0.0419961i
$$568$$ 581888.i 0.0756778i
$$569$$ −8.86979e6 −1.14850 −0.574252 0.818678i $$-0.694707\pi$$
−0.574252 + 0.818678i $$0.694707\pi$$
$$570$$ 0 0
$$571$$ 1.40102e7 1.79826 0.899132 0.437678i $$-0.144199\pi$$
0.899132 + 0.437678i $$0.144199\pi$$
$$572$$ − 3.37843e6i − 0.431743i
$$573$$ 3.38764e6i 0.431033i
$$574$$ −2.06623e6 −0.261758
$$575$$ 0 0
$$576$$ 331776. 0.0416667
$$577$$ − 8.75327e6i − 1.09454i −0.836957 0.547269i $$-0.815668\pi$$
0.836957 0.547269i $$-0.184332\pi$$
$$578$$ − 4.33147e6i − 0.539281i
$$579$$ −7.60451e6 −0.942703
$$580$$ 0 0
$$581$$ 314972. 0.0387108
$$582$$ 769320.i 0.0941455i
$$583$$ − 2.21471e7i − 2.69864i
$$584$$ −4.70106e6 −0.570379
$$585$$ 0 0
$$586$$ −8.40122e6 −1.01064
$$587$$ 1.06117e7i 1.27113i 0.772048 + 0.635564i $$0.219232\pi$$
−0.772048 + 0.635564i $$0.780768\pi$$
$$588$$ 345744.i 0.0412393i
$$589$$ 1.68221e6 0.199798
$$590$$ 0 0
$$591$$ −4.43515e6 −0.522323
$$592$$ − 1.11565e6i − 0.130835i
$$593$$ 1.88552e6i 0.220188i 0.993921 + 0.110094i $$0.0351152\pi$$
−0.993921 + 0.110094i $$0.964885\pi$$
$$594$$ 1.93622e6 0.225159
$$595$$ 0 0
$$596$$ −2.79773e6 −0.322619
$$597$$ 8.23298e6i 0.945413i
$$598$$ − 2.81366e6i − 0.321751i
$$599$$ −1.27256e7 −1.44915 −0.724573 0.689198i $$-0.757963\pi$$
−0.724573 + 0.689198i $$0.757963\pi$$
$$600$$ 0 0
$$601$$ 7.18846e6 0.811801 0.405900 0.913917i $$-0.366958\pi$$
0.405900 + 0.913917i $$0.366958\pi$$
$$602$$ 1.65659e6i 0.186305i
$$603$$ 3.31873e6i 0.371688i
$$604$$ 7.24083e6 0.807600
$$605$$ 0 0
$$606$$ 1.32530e6 0.146600
$$607$$ − 1.08494e7i − 1.19519i −0.801800 0.597593i $$-0.796124\pi$$
0.801800 0.597593i $$-0.203876\pi$$
$$608$$ − 241664.i − 0.0265126i
$$609$$ −2.18471e6 −0.238699
$$610$$ 0 0
$$611$$ 1.70194e6 0.184434
$$612$$ − 2.05027e6i − 0.221275i
$$613$$ − 4.90511e6i − 0.527227i −0.964628 0.263614i $$-0.915086\pi$$
0.964628 0.263614i $$-0.0849144\pi$$
$$614$$ −6.56418e6 −0.702683
$$615$$ 0 0
$$616$$ 2.08230e6 0.221102
$$617$$ − 2.58445e6i − 0.273310i −0.990619 0.136655i $$-0.956365\pi$$
0.990619 0.136655i $$-0.0436351\pi$$
$$618$$ − 3.76301e6i − 0.396336i
$$619$$ 4.99336e6 0.523801 0.261901 0.965095i $$-0.415651\pi$$
0.261901 + 0.965095i $$0.415651\pi$$
$$620$$ 0 0
$$621$$ 1.61255e6 0.167797
$$622$$ − 3.78093e6i − 0.391852i
$$623$$ 6.01024e6i 0.620400i
$$624$$ −732672. −0.0753266
$$625$$ 0 0
$$626$$ −1.66142e6 −0.169450
$$627$$ − 1.41034e6i − 0.143269i
$$628$$ − 7.98506e6i − 0.807940i
$$629$$ −6.89436e6 −0.694812
$$630$$ 0 0
$$631$$ −1.18219e7 −1.18199 −0.590997 0.806674i $$-0.701265\pi$$
−0.590997 + 0.806674i $$0.701265\pi$$
$$632$$ 5.72160e6i 0.569803i
$$633$$ 2.80602e6i 0.278344i
$$634$$ 4.73926e6 0.468260
$$635$$ 0 0
$$636$$ −4.80298e6 −0.470834
$$637$$ − 763518.i − 0.0745540i
$$638$$ 1.31578e7i 1.27977i
$$639$$ 736452. 0.0713497
$$640$$ 0 0
$$641$$ −5.47007e6 −0.525833 −0.262916 0.964819i $$-0.584684\pi$$
−0.262916 + 0.964819i $$0.584684\pi$$
$$642$$ 7.71984e6i 0.739215i
$$643$$ 9.64934e6i 0.920386i 0.887819 + 0.460193i $$0.152220\pi$$
−0.887819 + 0.460193i $$0.847780\pi$$
$$644$$ 1.73421e6 0.164773
$$645$$ 0 0
$$646$$ −1.49341e6 −0.140798
$$647$$ − 292368.i − 0.0274580i −0.999906 0.0137290i $$-0.995630\pi$$
0.999906 0.0137290i $$-0.00437022\pi$$
$$648$$ − 419904.i − 0.0392837i
$$649$$ 1.02495e7 0.955193
$$650$$ 0 0
$$651$$ 3.14345e6 0.290706
$$652$$ 7.60941e6i 0.701022i
$$653$$ 6.94081e6i 0.636982i 0.947926 + 0.318491i $$0.103176\pi$$
−0.947926 + 0.318491i $$0.896824\pi$$
$$654$$ 1.03673e6 0.0947808
$$655$$ 0 0
$$656$$ 2.69875e6 0.244852
$$657$$ 5.94977e6i 0.537758i
$$658$$ 1.04899e6i 0.0944512i
$$659$$ 1.32912e7 1.19221 0.596104 0.802908i $$-0.296715\pi$$
0.596104 + 0.802908i $$0.296715\pi$$
$$660$$ 0 0
$$661$$ 2.05219e6 0.182690 0.0913448 0.995819i $$-0.470883\pi$$
0.0913448 + 0.995819i $$0.470883\pi$$
$$662$$ 5.48619e6i 0.486548i
$$663$$ 4.52768e6i 0.400030i
$$664$$ −411392. −0.0362106
$$665$$ 0 0
$$666$$ −1.41199e6 −0.123352
$$667$$ 1.09582e7i 0.953732i
$$668$$ 1.92358e6i 0.166790i
$$669$$ 1.15898e7 1.00118
$$670$$ 0 0
$$671$$ −2.44100e7 −2.09296
$$672$$ − 451584.i − 0.0385758i
$$673$$ − 1.57039e7i − 1.33650i −0.743935 0.668252i $$-0.767043\pi$$
0.743935 0.668252i $$-0.232957\pi$$
$$674$$ 3.85409e6 0.326792
$$675$$ 0 0
$$676$$ −4.32270e6 −0.363822
$$677$$ 969534.i 0.0813002i 0.999173 + 0.0406501i $$0.0129429\pi$$
−0.999173 + 0.0406501i $$0.987057\pi$$
$$678$$ 2.01650e6i 0.168471i
$$679$$ 1.04713e6 0.0871618
$$680$$ 0 0
$$681$$ 1.16015e7 0.958617
$$682$$ − 1.89320e7i − 1.55860i
$$683$$ − 1.49908e7i − 1.22962i −0.788673 0.614812i $$-0.789232\pi$$
0.788673 0.614812i $$-0.210768\pi$$
$$684$$ −305856. −0.0249964
$$685$$ 0 0
$$686$$ 470596. 0.0381802
$$687$$ − 6.10393e6i − 0.493421i
$$688$$ − 2.16371e6i − 0.174272i
$$689$$ 1.06066e7 0.851191
$$690$$ 0 0
$$691$$ −7.16038e6 −0.570481 −0.285240 0.958456i $$-0.592073\pi$$
−0.285240 + 0.958456i $$0.592073\pi$$
$$692$$ − 8.14198e6i − 0.646346i
$$693$$ − 2.63542e6i − 0.208457i
$$694$$ 1.03092e7 0.812509
$$695$$ 0 0
$$696$$ 2.85350e6 0.223283
$$697$$ − 1.66774e7i − 1.30031i
$$698$$ 1.22701e7i 0.953253i
$$699$$ 1.00558e7 0.778437
$$700$$ 0 0
$$701$$ −91834.0 −0.00705844 −0.00352922 0.999994i $$-0.501123\pi$$
−0.00352922 + 0.999994i $$0.501123\pi$$
$$702$$ 927288.i 0.0710186i
$$703$$ 1.02849e6i 0.0784894i
$$704$$ −2.71974e6 −0.206822
$$705$$ 0 0
$$706$$ 1.24058e7 0.936725
$$707$$ − 1.80389e6i − 0.135725i
$$708$$ − 2.22278e6i − 0.166653i
$$709$$ −2.20981e7 −1.65097 −0.825487 0.564422i $$-0.809099\pi$$
−0.825487 + 0.564422i $$0.809099\pi$$
$$710$$ 0 0
$$711$$ 7.24140e6 0.537216
$$712$$ − 7.85011e6i − 0.580331i
$$713$$ − 1.57671e7i − 1.16153i
$$714$$ −2.79065e6 −0.204861
$$715$$ 0 0
$$716$$ 7.80096e6 0.568677
$$717$$ 1.13576e7i 0.825066i
$$718$$ 1.31003e6i 0.0948355i
$$719$$ −1.58388e7 −1.14262 −0.571308 0.820736i $$-0.693564\pi$$
−0.571308 + 0.820736i $$0.693564\pi$$
$$720$$ 0 0
$$721$$ −5.12187e6 −0.366936
$$722$$ − 9.68161e6i − 0.691202i
$$723$$ 8.53396e6i 0.607163i
$$724$$ 8.71056e6 0.617589
$$725$$ 0 0
$$726$$ −1.00744e7 −0.709379
$$727$$ − 6.31418e6i − 0.443078i −0.975151 0.221539i $$-0.928892\pi$$
0.975151 0.221539i $$-0.0711081\pi$$
$$728$$ 997248.i 0.0697388i
$$729$$ −531441. −0.0370370
$$730$$ 0 0
$$731$$ −1.33711e7 −0.925492
$$732$$ 5.29373e6i 0.365161i
$$733$$ 6.93003e6i 0.476404i 0.971216 + 0.238202i $$0.0765580\pi$$
−0.971216 + 0.238202i $$0.923442\pi$$
$$734$$ −1.14696e7 −0.785791
$$735$$ 0 0
$$736$$ −2.26509e6 −0.154131
$$737$$ − 2.72054e7i − 1.84496i
$$738$$ − 3.41561e6i − 0.230849i
$$739$$ −1.42331e7 −0.958714 −0.479357 0.877620i $$-0.659130\pi$$
−0.479357 + 0.877620i $$0.659130\pi$$
$$740$$ 0 0
$$741$$ 675432. 0.0451894
$$742$$ 6.53738e6i 0.435907i
$$743$$ − 5.94460e6i − 0.395048i −0.980298 0.197524i $$-0.936710\pi$$
0.980298 0.197524i $$-0.0632901\pi$$
$$744$$ −4.10573e6 −0.271930
$$745$$ 0 0
$$746$$ −1.43212e7 −0.942175
$$747$$ 520668.i 0.0341397i
$$748$$ 1.68072e7i 1.09835i
$$749$$ 1.05076e7 0.684380
$$750$$ 0 0
$$751$$ −682752. −0.0441736 −0.0220868 0.999756i $$-0.507031\pi$$
−0.0220868 + 0.999756i $$0.507031\pi$$
$$752$$ − 1.37011e6i − 0.0883510i
$$753$$ − 4.37756e6i − 0.281349i
$$754$$ −6.30149e6 −0.403659
$$755$$ 0 0
$$756$$ −571536. −0.0363696
$$757$$ − 1.46333e7i − 0.928116i −0.885805 0.464058i $$-0.846393\pi$$
0.885805 0.464058i $$-0.153607\pi$$
$$758$$ − 6.56939e6i − 0.415291i
$$759$$ −1.32189e7 −0.832897
$$760$$ 0 0
$$761$$ −1.16367e7 −0.728399 −0.364200 0.931321i $$-0.618657\pi$$
−0.364200 + 0.931321i $$0.618657\pi$$
$$762$$ 6.67440e6i 0.416414i
$$763$$ − 1.41110e6i − 0.0877500i
$$764$$ −6.02246e6 −0.373285
$$765$$ 0 0
$$766$$ 8.22790e6 0.506661
$$767$$ 4.90865e6i 0.301282i
$$768$$ 589824.i 0.0360844i
$$769$$ −1.91472e7 −1.16759 −0.583793 0.811902i $$-0.698432\pi$$
−0.583793 + 0.811902i $$0.698432\pi$$
$$770$$ 0 0
$$771$$ −9.35188e6 −0.566582
$$772$$ − 1.35191e7i − 0.816405i
$$773$$ − 5.39261e6i − 0.324601i −0.986741 0.162301i $$-0.948109\pi$$
0.986741 0.162301i $$-0.0518914\pi$$
$$774$$ −2.73845e6 −0.164305
$$775$$ 0 0
$$776$$ −1.36768e6 −0.0815324
$$777$$ 1.92188e6i 0.114202i
$$778$$ − 2.46457e6i − 0.145979i
$$779$$ −2.48791e6 −0.146890
$$780$$ 0 0
$$781$$ −6.03709e6 −0.354160
$$782$$ 1.39975e7i 0.818530i
$$783$$ − 3.61147e6i − 0.210513i
$$784$$ −614656. −0.0357143
$$785$$ 0 0
$$786$$ −2.32315e6 −0.134129
$$787$$ − 3.04348e6i − 0.175159i −0.996158 0.0875796i $$-0.972087\pi$$
0.996158 0.0875796i $$-0.0279132\pi$$
$$788$$ − 7.88470e6i − 0.452345i
$$789$$ −1.21594e7 −0.695376
$$790$$ 0 0
$$791$$ 2.74469e6 0.155974
$$792$$ 3.44218e6i 0.194993i
$$793$$ − 1.16903e7i − 0.660151i
$$794$$ 8.76847e6 0.493597
$$795$$ 0 0
$$796$$ −1.46364e7 −0.818751
$$797$$ − 2.29652e7i − 1.28063i −0.768111 0.640316i $$-0.778803\pi$$
0.768111 0.640316i $$-0.221197\pi$$
$$798$$ 416304.i 0.0231421i
$$799$$ −8.46686e6 −0.469197
$$800$$ 0 0
$$801$$ −9.93530e6 −0.547141
$$802$$ 1.31382e7i 0.721271i
$$803$$ − 4.87735e7i − 2.66928i
$$804$$ −5.89997e6 −0.321892
$$805$$ 0 0
$$806$$ 9.06682e6 0.491606
$$807$$ 1.00630e7i 0.543930i
$$808$$ 2.35610e6i 0.126959i
$$809$$ −1.90787e7 −1.02489 −0.512445 0.858720i $$-0.671260\pi$$
−0.512445 + 0.858720i $$0.671260\pi$$
$$810$$ 0 0
$$811$$ 1.09414e7 0.584147 0.292074 0.956396i $$-0.405655\pi$$
0.292074 + 0.956396i $$0.405655\pi$$
$$812$$ − 3.88394e6i − 0.206720i
$$813$$ − 1.71094e6i − 0.0907836i
$$814$$ 1.15748e7 0.612286
$$815$$ 0 0
$$816$$ 3.64493e6 0.191630
$$817$$ 1.99467e6i 0.104548i
$$818$$ 1.44488e7i 0.755001i
$$819$$ 1.26214e6 0.0657504
$$820$$ 0 0
$$821$$ 2.12594e7 1.10076 0.550380 0.834914i $$-0.314483\pi$$
0.550380 + 0.834914i $$0.314483\pi$$
$$822$$ 5.50548e6i 0.284194i
$$823$$ − 1.42256e7i − 0.732103i −0.930595 0.366052i $$-0.880709\pi$$
0.930595 0.366052i $$-0.119291\pi$$
$$824$$ 6.68979e6 0.343237
$$825$$ 0 0
$$826$$ −3.02546e6 −0.154291
$$827$$ − 2.76103e6i − 0.140381i −0.997534 0.0701904i $$-0.977639\pi$$
0.997534 0.0701904i $$-0.0223607\pi$$
$$828$$ 2.86675e6i 0.145316i
$$829$$ 3.82147e7 1.93127 0.965637 0.259895i $$-0.0836880\pi$$
0.965637 + 0.259895i $$0.0836880\pi$$
$$830$$ 0 0
$$831$$ −1.80455e6 −0.0906499
$$832$$ − 1.30253e6i − 0.0652347i
$$833$$ 3.79838e6i 0.189665i
$$834$$ −1.23646e7 −0.615550
$$835$$ 0 0
$$836$$ 2.50726e6 0.124075
$$837$$ 5.19631e6i 0.256378i
$$838$$ − 2.16596e7i − 1.06547i
$$839$$ −1.06044e7 −0.520094 −0.260047 0.965596i $$-0.583738\pi$$
−0.260047 + 0.965596i $$0.583738\pi$$
$$840$$ 0 0
$$841$$ 4.03097e6 0.196526
$$842$$ 1.44252e7i 0.701198i
$$843$$ 9.83133e6i 0.476479i
$$844$$ −4.98848e6 −0.241053
$$845$$ 0 0
$$846$$ −1.73405e6 −0.0832981
$$847$$ 1.37124e7i 0.656758i
$$848$$ − 8.53862e6i − 0.407754i
$$849$$ −1.63132e7 −0.776731
$$850$$ 0 0
$$851$$ 9.63990e6 0.456298
$$852$$ 1.30925e6i 0.0617907i
$$853$$ − 4.07009e7i − 1.91527i −0.287977 0.957637i $$-0.592983\pi$$
0.287977 0.957637i $$-0.407017\pi$$
$$854$$ 7.20535e6 0.338073
$$855$$ 0 0
$$856$$ −1.37242e7 −0.640179
$$857$$ 3.10120e7i 1.44237i 0.692741 + 0.721187i $$0.256403\pi$$
−0.692741 + 0.721187i $$0.743597\pi$$
$$858$$ − 7.60147e6i − 0.352517i
$$859$$ −1.09104e7 −0.504495 −0.252247 0.967663i $$-0.581170\pi$$
−0.252247 + 0.967663i $$0.581170\pi$$
$$860$$ 0 0
$$861$$ −4.64902e6 −0.213724
$$862$$ − 1.11286e7i − 0.510118i
$$863$$ 1.04089e7i 0.475751i 0.971296 + 0.237875i $$0.0764510\pi$$
−0.971296 + 0.237875i $$0.923549\pi$$
$$864$$ 746496. 0.0340207
$$865$$ 0 0
$$866$$ −2.51048e7 −1.13753
$$867$$ − 9.74580e6i − 0.440321i
$$868$$ 5.58835e6i 0.251759i
$$869$$ −5.93616e7 −2.66659
$$870$$ 0 0
$$871$$ 1.30291e7 0.581928
$$872$$ 1.84307e6i 0.0820826i
$$873$$ 1.73097e6i 0.0768695i
$$874$$ 2.08813e6 0.0924652
$$875$$ 0 0
$$876$$ −1.05774e7 −0.465712
$$877$$ − 1.64064e7i − 0.720299i −0.932895 0.360150i $$-0.882726\pi$$
0.932895 0.360150i $$-0.117274\pi$$
$$878$$ − 2.56637e6i − 0.112352i
$$879$$ −1.89028e7 −0.825188
$$880$$ 0 0
$$881$$ 1.48577e7 0.644927 0.322464 0.946582i $$-0.395489\pi$$
0.322464 + 0.946582i $$0.395489\pi$$
$$882$$ 777924.i 0.0336718i
$$883$$ − 2.72018e7i − 1.17407i −0.809560 0.587037i $$-0.800294\pi$$
0.809560 0.587037i $$-0.199706\pi$$
$$884$$ −8.04922e6 −0.346436
$$885$$ 0 0
$$886$$ −2.42218e7 −1.03663
$$887$$ − 2.71242e7i − 1.15757i −0.815480 0.578785i $$-0.803527\pi$$
0.815480 0.578785i $$-0.196473\pi$$
$$888$$ − 2.51021e6i − 0.106826i
$$889$$ 9.08460e6 0.385524
$$890$$ 0 0
$$891$$ 4.35650e6 0.183842
$$892$$ 2.06042e7i 0.867047i
$$893$$ 1.26307e6i 0.0530029i
$$894$$ −6.29489e6 −0.263417
$$895$$ 0 0
$$896$$ 802816. 0.0334077
$$897$$ − 6.33074e6i − 0.262708i
$$898$$ 2.06673e7i 0.855248i
$$899$$ −3.53121e7 −1.45722
$$900$$ 0 0
$$901$$ −5.27660e7 −2.16542
$$902$$ 2.79996e7i 1.14587i
$$903$$ 3.72733e6i 0.152117i
$$904$$ −3.58490e6 −0.145900
$$905$$ 0 0
$$906$$ 1.62919e7 0.659402
$$907$$ 8.42269e6i 0.339964i 0.985447 + 0.169982i $$0.0543709\pi$$
−0.985447 + 0.169982i $$0.945629\pi$$
$$908$$ 2.06248e7i 0.830187i
$$909$$ 2.98193e6 0.119698
$$910$$ 0 0
$$911$$ 3.08637e7 1.23212 0.616060 0.787700i $$-0.288728\pi$$
0.616060 + 0.787700i $$0.288728\pi$$
$$912$$ − 543744.i − 0.0216475i
$$913$$ − 4.26819e6i − 0.169460i
$$914$$ −911192. −0.0360782
$$915$$ 0 0
$$916$$ 1.08514e7 0.427315
$$917$$ 3.16207e6i 0.124179i
$$918$$ − 4.61311e6i − 0.180671i
$$919$$ −4.93895e6 −0.192906 −0.0964531 0.995338i $$-0.530750\pi$$
−0.0964531 + 0.995338i $$0.530750\pi$$
$$920$$ 0 0
$$921$$ −1.47694e7 −0.573738
$$922$$ 2.34058e6i 0.0906770i
$$923$$ − 2.89126e6i − 0.111707i
$$924$$ 4.68518e6 0.180529
$$925$$ 0 0
$$926$$ 1.36581e7 0.523437
$$927$$ − 8.46677e6i − 0.323607i
$$928$$ 5.07290e6i 0.193369i
$$929$$ −5.62575e6 −0.213866 −0.106933 0.994266i $$-0.534103\pi$$
−0.106933 + 0.994266i $$0.534103\pi$$
$$930$$ 0 0
$$931$$ 566636. 0.0214255
$$932$$ 1.78770e7i 0.674146i
$$933$$ − 8.50709e6i − 0.319946i
$$934$$ 2.86520e6 0.107470
$$935$$ 0 0
$$936$$ −1.64851e6 −0.0615039
$$937$$ − 2.60073e7i − 0.967714i −0.875147 0.483857i $$-0.839236\pi$$
0.875147 0.483857i $$-0.160764\pi$$
$$938$$ 8.03051e6i 0.298014i
$$939$$ −3.73819e6 −0.138356
$$940$$ 0 0
$$941$$ 3.02160e6 0.111241 0.0556203 0.998452i $$-0.482286\pi$$
0.0556203 + 0.998452i $$0.482286\pi$$
$$942$$ − 1.79664e7i − 0.659680i
$$943$$ 2.33189e7i 0.853943i
$$944$$ 3.95162e6 0.144326
$$945$$ 0 0
$$946$$ 2.24485e7 0.815567
$$947$$ 3.48282e7i 1.26199i 0.775787 + 0.630995i $$0.217353\pi$$
−0.775787 + 0.630995i $$0.782647\pi$$
$$948$$ 1.28736e7i 0.465242i
$$949$$ 2.33584e7 0.841932
$$950$$ 0 0
$$951$$ 1.06633e7 0.382333
$$952$$ − 4.96115e6i − 0.177415i
$$953$$ − 9.39009e6i − 0.334917i −0.985879 0.167459i $$-0.946444\pi$$
0.985879 0.167459i $$-0.0535560\pi$$
$$954$$ −1.08067e7 −0.384434
$$955$$ 0 0
$$956$$ −2.01913e7 −0.714528
$$957$$ 2.96051e7i 1.04493i
$$958$$ − 2.09637e7i − 0.737996i
$$959$$ 7.49357e6 0.263113
$$960$$ 0 0
$$961$$ 2.21792e7 0.774708
$$962$$ 5.54338e6i 0.193124i
$$963$$ 1.73696e7i 0.603566i
$$964$$ −1.51715e7 −0.525818
$$965$$ 0 0
$$966$$ 3.90197e6 0.134537
$$967$$ − 1.44768e7i − 0.497860i −0.968521 0.248930i $$-0.919921\pi$$
0.968521 0.248930i $$-0.0800789\pi$$
$$968$$ − 1.79101e7i − 0.614340i
$$969$$ −3.36017e6 −0.114961
$$970$$ 0 0
$$971$$ 9.24976e6 0.314834 0.157417 0.987532i $$-0.449683\pi$$
0.157417 + 0.987532i $$0.449683\pi$$
$$972$$ − 944784.i − 0.0320750i
$$973$$ 1.68295e7i 0.569889i
$$974$$ 4.46806e6 0.150911
$$975$$ 0 0
$$976$$ −9.41107e6 −0.316238
$$977$$ 4.97780e7i 1.66840i 0.551459 + 0.834202i $$0.314071\pi$$
−0.551459 + 0.834202i $$0.685929\pi$$
$$978$$ 1.71212e7i 0.572382i
$$979$$ 8.14449e7 2.71586
$$980$$ 0 0
$$981$$ 2.33264e6 0.0773882
$$982$$ 5.37830e6i 0.177978i
$$983$$ − 8.95601e6i − 0.295618i −0.989016 0.147809i $$-0.952778\pi$$
0.989016 0.147809i $$-0.0472221\pi$$
$$984$$ 6.07219e6 0.199921
$$985$$ 0 0
$$986$$ 3.13489e7 1.02690
$$987$$ 2.36023e6i 0.0771191i
$$988$$ 1.20077e6i 0.0391351i
$$989$$ 1.86958e7 0.607790
$$990$$ 0 0
$$991$$ 2.62400e7 0.848751 0.424376 0.905486i $$-0.360494\pi$$
0.424376 + 0.905486i $$0.360494\pi$$
$$992$$ − 7.29907e6i − 0.235499i
$$993$$ 1.23439e7i 0.397265i
$$994$$ 1.78203e6 0.0572070
$$995$$ 0 0
$$996$$ −925632. −0.0295658
$$997$$ − 2.80506e7i − 0.893727i −0.894602 0.446863i $$-0.852541\pi$$
0.894602 0.446863i $$-0.147459\pi$$
$$998$$ 2.61859e7i 0.832226i
$$999$$ −3.17698e6 −0.100717
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 1050.6.g.i.799.2 2
5.2 odd 4 42.6.a.d.1.1 1
5.3 odd 4 1050.6.a.k.1.1 1
5.4 even 2 inner 1050.6.g.i.799.1 2
15.2 even 4 126.6.a.i.1.1 1
20.7 even 4 336.6.a.h.1.1 1
35.2 odd 12 294.6.e.i.67.1 2
35.12 even 12 294.6.e.p.67.1 2
35.17 even 12 294.6.e.p.79.1 2
35.27 even 4 294.6.a.b.1.1 1
35.32 odd 12 294.6.e.i.79.1 2
60.47 odd 4 1008.6.a.j.1.1 1
105.62 odd 4 882.6.a.s.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
42.6.a.d.1.1 1 5.2 odd 4
126.6.a.i.1.1 1 15.2 even 4
294.6.a.b.1.1 1 35.27 even 4
294.6.e.i.67.1 2 35.2 odd 12
294.6.e.i.79.1 2 35.32 odd 12
294.6.e.p.67.1 2 35.12 even 12
294.6.e.p.79.1 2 35.17 even 12
336.6.a.h.1.1 1 20.7 even 4
882.6.a.s.1.1 1 105.62 odd 4
1008.6.a.j.1.1 1 60.47 odd 4
1050.6.a.k.1.1 1 5.3 odd 4
1050.6.g.i.799.1 2 5.4 even 2 inner
1050.6.g.i.799.2 2 1.1 even 1 trivial