# Properties

 Label 49.6.a.g.1.2 Level $49$ Weight $6$ Character 49.1 Self dual yes Analytic conductor $7.859$ Analytic rank $1$ Dimension $4$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [49,6,Mod(1,49)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(49, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("49.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$49 = 7^{2}$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 49.a (trivial)

## Newform invariants

comment: select newform

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

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

 Self dual: yes Analytic conductor: $$7.85880717084$$ Analytic rank: $$1$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{2}, \sqrt{113})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - 2x^{3} - 59x^{2} + 60x + 674$$ x^4 - 2*x^3 - 59*x^2 + 60*x + 674 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$7$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$-6.22929$$ of defining polynomial Character $$\chi$$ $$=$$ 49.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-7.81507 q^{2} +23.5186 q^{3} +29.0754 q^{4} -74.2753 q^{5} -183.799 q^{6} +22.8562 q^{8} +310.123 q^{9} +O(q^{10})$$ $$q-7.81507 q^{2} +23.5186 q^{3} +29.0754 q^{4} -74.2753 q^{5} -183.799 q^{6} +22.8562 q^{8} +310.123 q^{9} +580.467 q^{10} -424.219 q^{11} +683.811 q^{12} -252.233 q^{13} -1746.85 q^{15} -1109.03 q^{16} -1104.35 q^{17} -2423.64 q^{18} -6.47100 q^{19} -2159.58 q^{20} +3315.30 q^{22} -3612.39 q^{23} +537.546 q^{24} +2391.82 q^{25} +1971.22 q^{26} +1578.65 q^{27} -5005.02 q^{29} +13651.8 q^{30} +2821.69 q^{31} +7935.79 q^{32} -9977.03 q^{33} +8630.55 q^{34} +9016.95 q^{36} -2046.88 q^{37} +50.5713 q^{38} -5932.17 q^{39} -1697.65 q^{40} +9393.81 q^{41} +10320.8 q^{43} -12334.3 q^{44} -23034.5 q^{45} +28231.1 q^{46} +17035.6 q^{47} -26082.9 q^{48} -18692.3 q^{50} -25972.7 q^{51} -7333.78 q^{52} -39506.7 q^{53} -12337.3 q^{54} +31509.0 q^{55} -152.189 q^{57} +39114.6 q^{58} +33949.8 q^{59} -50790.3 q^{60} +28295.2 q^{61} -22051.7 q^{62} -26529.7 q^{64} +18734.7 q^{65} +77971.2 q^{66} +56100.9 q^{67} -32109.3 q^{68} -84958.2 q^{69} -15537.4 q^{71} +7088.26 q^{72} -78219.5 q^{73} +15996.5 q^{74} +56252.3 q^{75} -188.147 q^{76} +46360.3 q^{78} -45335.5 q^{79} +82373.9 q^{80} -38232.4 q^{81} -73413.3 q^{82} +1381.82 q^{83} +82025.7 q^{85} -80657.6 q^{86} -117711. q^{87} -9696.05 q^{88} -68879.4 q^{89} +180016. q^{90} -105031. q^{92} +66362.1 q^{93} -133134. q^{94} +480.635 q^{95} +186638. q^{96} +108857. q^{97} -131560. q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 10 q^{2} + 10 q^{4} - 270 q^{8} + 220 q^{9}+O(q^{10})$$ 4 * q - 10 * q^2 + 10 * q^4 - 270 * q^8 + 220 * q^9 $$4 q - 10 q^{2} + 10 q^{4} - 270 q^{8} + 220 q^{9} - 1952 q^{11} - 4096 q^{15} - 1566 q^{16} - 5974 q^{18} + 3524 q^{22} - 7136 q^{23} + 2764 q^{25} - 3352 q^{29} + 25608 q^{30} + 27810 q^{32} + 27670 q^{36} - 9208 q^{37} + 2464 q^{39} + 20448 q^{43} + 1900 q^{44} + 56712 q^{46} - 43070 q^{50} - 67408 q^{51} - 102920 q^{53} - 15576 q^{57} + 96972 q^{58} - 87080 q^{60} - 40318 q^{64} - 63168 q^{65} - 22896 q^{67} - 153824 q^{71} + 77358 q^{72} + 17596 q^{74} + 133056 q^{78} - 90688 q^{79} - 17204 q^{81} + 272656 q^{85} - 161860 q^{86} + 154812 q^{88} - 212200 q^{92} + 247760 q^{93} + 108224 q^{95} - 42272 q^{99}+O(q^{100})$$ 4 * q - 10 * q^2 + 10 * q^4 - 270 * q^8 + 220 * q^9 - 1952 * q^11 - 4096 * q^15 - 1566 * q^16 - 5974 * q^18 + 3524 * q^22 - 7136 * q^23 + 2764 * q^25 - 3352 * q^29 + 25608 * q^30 + 27810 * q^32 + 27670 * q^36 - 9208 * q^37 + 2464 * q^39 + 20448 * q^43 + 1900 * q^44 + 56712 * q^46 - 43070 * q^50 - 67408 * q^51 - 102920 * q^53 - 15576 * q^57 + 96972 * q^58 - 87080 * q^60 - 40318 * q^64 - 63168 * q^65 - 22896 * q^67 - 153824 * q^71 + 77358 * q^72 + 17596 * q^74 + 133056 * q^78 - 90688 * q^79 - 17204 * q^81 + 272656 * q^85 - 161860 * q^86 + 154812 * q^88 - 212200 * q^92 + 247760 * q^93 + 108224 * q^95 - 42272 * q^99

## Coefficient data

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

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −7.81507 −1.38152 −0.690761 0.723083i $$-0.742724\pi$$
−0.690761 + 0.723083i $$0.742724\pi$$
$$3$$ 23.5186 1.50872 0.754359 0.656462i $$-0.227948\pi$$
0.754359 + 0.656462i $$0.227948\pi$$
$$4$$ 29.0754 0.908605
$$5$$ −74.2753 −1.32868 −0.664339 0.747432i $$-0.731287\pi$$
−0.664339 + 0.747432i $$0.731287\pi$$
$$6$$ −183.799 −2.08433
$$7$$ 0 0
$$8$$ 22.8562 0.126264
$$9$$ 310.123 1.27623
$$10$$ 580.467 1.83560
$$11$$ −424.219 −1.05708 −0.528541 0.848908i $$-0.677261\pi$$
−0.528541 + 0.848908i $$0.677261\pi$$
$$12$$ 683.811 1.37083
$$13$$ −252.233 −0.413946 −0.206973 0.978347i $$-0.566361\pi$$
−0.206973 + 0.978347i $$0.566361\pi$$
$$14$$ 0 0
$$15$$ −1746.85 −2.00460
$$16$$ −1109.03 −1.08304
$$17$$ −1104.35 −0.926794 −0.463397 0.886151i $$-0.653370\pi$$
−0.463397 + 0.886151i $$0.653370\pi$$
$$18$$ −2423.64 −1.76314
$$19$$ −6.47100 −0.00411232 −0.00205616 0.999998i $$-0.500654\pi$$
−0.00205616 + 0.999998i $$0.500654\pi$$
$$20$$ −2159.58 −1.20724
$$21$$ 0 0
$$22$$ 3315.30 1.46038
$$23$$ −3612.39 −1.42388 −0.711942 0.702239i $$-0.752184\pi$$
−0.711942 + 0.702239i $$0.752184\pi$$
$$24$$ 537.546 0.190497
$$25$$ 2391.82 0.765383
$$26$$ 1971.22 0.571876
$$27$$ 1578.65 0.416751
$$28$$ 0 0
$$29$$ −5005.02 −1.10512 −0.552561 0.833472i $$-0.686350\pi$$
−0.552561 + 0.833472i $$0.686350\pi$$
$$30$$ 13651.8 2.76940
$$31$$ 2821.69 0.527357 0.263679 0.964611i $$-0.415064\pi$$
0.263679 + 0.964611i $$0.415064\pi$$
$$32$$ 7935.79 1.36998
$$33$$ −9977.03 −1.59484
$$34$$ 8630.55 1.28039
$$35$$ 0 0
$$36$$ 9016.95 1.15959
$$37$$ −2046.88 −0.245803 −0.122902 0.992419i $$-0.539220\pi$$
−0.122902 + 0.992419i $$0.539220\pi$$
$$38$$ 50.5713 0.00568127
$$39$$ −5932.17 −0.624528
$$40$$ −1697.65 −0.167764
$$41$$ 9393.81 0.872734 0.436367 0.899769i $$-0.356265\pi$$
0.436367 + 0.899769i $$0.356265\pi$$
$$42$$ 0 0
$$43$$ 10320.8 0.851218 0.425609 0.904907i $$-0.360060\pi$$
0.425609 + 0.904907i $$0.360060\pi$$
$$44$$ −12334.3 −0.960470
$$45$$ −23034.5 −1.69570
$$46$$ 28231.1 1.96713
$$47$$ 17035.6 1.12490 0.562448 0.826833i $$-0.309860\pi$$
0.562448 + 0.826833i $$0.309860\pi$$
$$48$$ −26082.9 −1.63400
$$49$$ 0 0
$$50$$ −18692.3 −1.05739
$$51$$ −25972.7 −1.39827
$$52$$ −7333.78 −0.376114
$$53$$ −39506.7 −1.93188 −0.965941 0.258761i $$-0.916686\pi$$
−0.965941 + 0.258761i $$0.916686\pi$$
$$54$$ −12337.3 −0.575750
$$55$$ 31509.0 1.40452
$$56$$ 0 0
$$57$$ −152.189 −0.00620433
$$58$$ 39114.6 1.52675
$$59$$ 33949.8 1.26972 0.634859 0.772628i $$-0.281058\pi$$
0.634859 + 0.772628i $$0.281058\pi$$
$$60$$ −50790.3 −1.82139
$$61$$ 28295.2 0.973618 0.486809 0.873508i $$-0.338161\pi$$
0.486809 + 0.873508i $$0.338161\pi$$
$$62$$ −22051.7 −0.728556
$$63$$ 0 0
$$64$$ −26529.7 −0.809621
$$65$$ 18734.7 0.550001
$$66$$ 77971.2 2.20330
$$67$$ 56100.9 1.52680 0.763402 0.645924i $$-0.223528\pi$$
0.763402 + 0.645924i $$0.223528\pi$$
$$68$$ −32109.3 −0.842090
$$69$$ −84958.2 −2.14824
$$70$$ 0 0
$$71$$ −15537.4 −0.365791 −0.182895 0.983132i $$-0.558547\pi$$
−0.182895 + 0.983132i $$0.558547\pi$$
$$72$$ 7088.26 0.161142
$$73$$ −78219.5 −1.71794 −0.858970 0.512025i $$-0.828895\pi$$
−0.858970 + 0.512025i $$0.828895\pi$$
$$74$$ 15996.5 0.339583
$$75$$ 56252.3 1.15475
$$76$$ −188.147 −0.00373648
$$77$$ 0 0
$$78$$ 46360.3 0.862800
$$79$$ −45335.5 −0.817279 −0.408640 0.912696i $$-0.633997\pi$$
−0.408640 + 0.912696i $$0.633997\pi$$
$$80$$ 82373.9 1.43901
$$81$$ −38232.4 −0.647469
$$82$$ −73413.3 −1.20570
$$83$$ 1381.82 0.0220169 0.0110085 0.999939i $$-0.496496\pi$$
0.0110085 + 0.999939i $$0.496496\pi$$
$$84$$ 0 0
$$85$$ 82025.7 1.23141
$$86$$ −80657.6 −1.17598
$$87$$ −117711. −1.66732
$$88$$ −9696.05 −0.133471
$$89$$ −68879.4 −0.921753 −0.460876 0.887464i $$-0.652465\pi$$
−0.460876 + 0.887464i $$0.652465\pi$$
$$90$$ 180016. 2.34264
$$91$$ 0 0
$$92$$ −105031. −1.29375
$$93$$ 66362.1 0.795633
$$94$$ −133134. −1.55407
$$95$$ 480.635 0.00546395
$$96$$ 186638. 2.06692
$$97$$ 108857. 1.17470 0.587351 0.809332i $$-0.300171\pi$$
0.587351 + 0.809332i $$0.300171\pi$$
$$98$$ 0 0
$$99$$ −131560. −1.34908
$$100$$ 69543.1 0.695431
$$101$$ 17972.3 0.175307 0.0876535 0.996151i $$-0.472063\pi$$
0.0876535 + 0.996151i $$0.472063\pi$$
$$102$$ 202978. 1.93174
$$103$$ 31773.7 0.295103 0.147552 0.989054i $$-0.452861\pi$$
0.147552 + 0.989054i $$0.452861\pi$$
$$104$$ −5765.11 −0.0522666
$$105$$ 0 0
$$106$$ 308747. 2.66894
$$107$$ 8229.36 0.0694875 0.0347438 0.999396i $$-0.488938\pi$$
0.0347438 + 0.999396i $$0.488938\pi$$
$$108$$ 45899.8 0.378662
$$109$$ −11068.7 −0.0892338 −0.0446169 0.999004i $$-0.514207\pi$$
−0.0446169 + 0.999004i $$0.514207\pi$$
$$110$$ −246245. −1.94038
$$111$$ −48139.6 −0.370847
$$112$$ 0 0
$$113$$ 65184.3 0.480228 0.240114 0.970745i $$-0.422815\pi$$
0.240114 + 0.970745i $$0.422815\pi$$
$$114$$ 1189.37 0.00857143
$$115$$ 268311. 1.89188
$$116$$ −145523. −1.00412
$$117$$ −78223.5 −0.528290
$$118$$ −265320. −1.75414
$$119$$ 0 0
$$120$$ −39926.4 −0.253109
$$121$$ 18910.9 0.117422
$$122$$ −221129. −1.34508
$$123$$ 220929. 1.31671
$$124$$ 82041.7 0.479160
$$125$$ 54456.9 0.311730
$$126$$ 0 0
$$127$$ −194777. −1.07159 −0.535796 0.844348i $$-0.679988\pi$$
−0.535796 + 0.844348i $$0.679988\pi$$
$$128$$ −46614.1 −0.251473
$$129$$ 242730. 1.28425
$$130$$ −146413. −0.759839
$$131$$ −236503. −1.20409 −0.602046 0.798462i $$-0.705647\pi$$
−0.602046 + 0.798462i $$0.705647\pi$$
$$132$$ −290086. −1.44908
$$133$$ 0 0
$$134$$ −438433. −2.10931
$$135$$ −117255. −0.553727
$$136$$ −25241.2 −0.117021
$$137$$ −200903. −0.914503 −0.457252 0.889337i $$-0.651166\pi$$
−0.457252 + 0.889337i $$0.651166\pi$$
$$138$$ 663954. 2.96784
$$139$$ 52985.2 0.232604 0.116302 0.993214i $$-0.462896\pi$$
0.116302 + 0.993214i $$0.462896\pi$$
$$140$$ 0 0
$$141$$ 400653. 1.69715
$$142$$ 121426. 0.505348
$$143$$ 107002. 0.437575
$$144$$ −343938. −1.38221
$$145$$ 371749. 1.46835
$$146$$ 611291. 2.37337
$$147$$ 0 0
$$148$$ −59513.7 −0.223338
$$149$$ −100770. −0.371849 −0.185925 0.982564i $$-0.559528\pi$$
−0.185925 + 0.982564i $$0.559528\pi$$
$$150$$ −439616. −1.59531
$$151$$ −457904. −1.63430 −0.817150 0.576425i $$-0.804447\pi$$
−0.817150 + 0.576425i $$0.804447\pi$$
$$152$$ −147.903 −0.000519239 0
$$153$$ −342484. −1.18280
$$154$$ 0 0
$$155$$ −209582. −0.700688
$$156$$ −172480. −0.567450
$$157$$ −179037. −0.579688 −0.289844 0.957074i $$-0.593603\pi$$
−0.289844 + 0.957074i $$0.593603\pi$$
$$158$$ 354300. 1.12909
$$159$$ −929141. −2.91467
$$160$$ −589433. −1.82027
$$161$$ 0 0
$$162$$ 298789. 0.894494
$$163$$ 243610. 0.718168 0.359084 0.933305i $$-0.383089\pi$$
0.359084 + 0.933305i $$0.383089\pi$$
$$164$$ 273128. 0.792971
$$165$$ 741047. 2.11902
$$166$$ −10799.0 −0.0304169
$$167$$ 117033. 0.324725 0.162362 0.986731i $$-0.448089\pi$$
0.162362 + 0.986731i $$0.448089\pi$$
$$168$$ 0 0
$$169$$ −307671. −0.828648
$$170$$ −641037. −1.70122
$$171$$ −2006.81 −0.00524826
$$172$$ 300080. 0.773421
$$173$$ 269733. 0.685203 0.342602 0.939481i $$-0.388692\pi$$
0.342602 + 0.939481i $$0.388692\pi$$
$$174$$ 919919. 2.30344
$$175$$ 0 0
$$176$$ 470474. 1.14486
$$177$$ 798451. 1.91564
$$178$$ 538298. 1.27342
$$179$$ 376525. 0.878336 0.439168 0.898405i $$-0.355273\pi$$
0.439168 + 0.898405i $$0.355273\pi$$
$$180$$ −669737. −1.54072
$$181$$ −434641. −0.986131 −0.493065 0.869992i $$-0.664124\pi$$
−0.493065 + 0.869992i $$0.664124\pi$$
$$182$$ 0 0
$$183$$ 665464. 1.46891
$$184$$ −82565.5 −0.179785
$$185$$ 152032. 0.326593
$$186$$ −518625. −1.09919
$$187$$ 468485. 0.979697
$$188$$ 495316. 1.02209
$$189$$ 0 0
$$190$$ −3756.20 −0.00754857
$$191$$ 565940. 1.12250 0.561250 0.827646i $$-0.310320\pi$$
0.561250 + 0.827646i $$0.310320\pi$$
$$192$$ −623940. −1.22149
$$193$$ 514461. 0.994167 0.497084 0.867703i $$-0.334404\pi$$
0.497084 + 0.867703i $$0.334404\pi$$
$$194$$ −850727. −1.62288
$$195$$ 440614. 0.829796
$$196$$ 0 0
$$197$$ −298541. −0.548073 −0.274037 0.961719i $$-0.588359\pi$$
−0.274037 + 0.961719i $$0.588359\pi$$
$$198$$ 1.02815e6 1.86378
$$199$$ −591919. −1.05957 −0.529785 0.848132i $$-0.677727\pi$$
−0.529785 + 0.848132i $$0.677727\pi$$
$$200$$ 54668.1 0.0966404
$$201$$ 1.31941e6 2.30351
$$202$$ −140455. −0.242191
$$203$$ 0 0
$$204$$ −755165. −1.27048
$$205$$ −697728. −1.15958
$$206$$ −248313. −0.407692
$$207$$ −1.12029e6 −1.81720
$$208$$ 279736. 0.448321
$$209$$ 2745.12 0.00434706
$$210$$ 0 0
$$211$$ −140535. −0.217309 −0.108655 0.994080i $$-0.534654\pi$$
−0.108655 + 0.994080i $$0.534654\pi$$
$$212$$ −1.14867e6 −1.75532
$$213$$ −365418. −0.551875
$$214$$ −64313.1 −0.0959986
$$215$$ −766579. −1.13099
$$216$$ 36082.0 0.0526206
$$217$$ 0 0
$$218$$ 86502.5 0.123278
$$219$$ −1.83961e6 −2.59189
$$220$$ 916136. 1.27615
$$221$$ 278553. 0.383643
$$222$$ 376215. 0.512334
$$223$$ 490.526 0.000660541 0 0.000330271 1.00000i $$-0.499895\pi$$
0.000330271 1.00000i $$0.499895\pi$$
$$224$$ 0 0
$$225$$ 741761. 0.976804
$$226$$ −509420. −0.663445
$$227$$ 593898. 0.764975 0.382488 0.923961i $$-0.375067\pi$$
0.382488 + 0.923961i $$0.375067\pi$$
$$228$$ −4424.94 −0.00563729
$$229$$ −35880.3 −0.0452135 −0.0226067 0.999744i $$-0.507197\pi$$
−0.0226067 + 0.999744i $$0.507197\pi$$
$$230$$ −2.09687e6 −2.61368
$$231$$ 0 0
$$232$$ −114396. −0.139537
$$233$$ 1.24822e6 1.50626 0.753131 0.657871i $$-0.228543\pi$$
0.753131 + 0.657871i $$0.228543\pi$$
$$234$$ 611322. 0.729845
$$235$$ −1.26532e6 −1.49462
$$236$$ 987102. 1.15367
$$237$$ −1.06623e6 −1.23304
$$238$$ 0 0
$$239$$ −576943. −0.653339 −0.326669 0.945139i $$-0.605926\pi$$
−0.326669 + 0.945139i $$0.605926\pi$$
$$240$$ 1.93732e6 2.17106
$$241$$ −1.38241e6 −1.53318 −0.766592 0.642135i $$-0.778049\pi$$
−0.766592 + 0.642135i $$0.778049\pi$$
$$242$$ −147790. −0.162221
$$243$$ −1.28278e6 −1.39360
$$244$$ 822694. 0.884634
$$245$$ 0 0
$$246$$ −1.72658e6 −1.81906
$$247$$ 1632.20 0.00170228
$$248$$ 64493.2 0.0665863
$$249$$ 32498.5 0.0332173
$$250$$ −425585. −0.430662
$$251$$ 323217. 0.323824 0.161912 0.986805i $$-0.448234\pi$$
0.161912 + 0.986805i $$0.448234\pi$$
$$252$$ 0 0
$$253$$ 1.53244e6 1.50516
$$254$$ 1.52220e6 1.48043
$$255$$ 1.92913e6 1.85785
$$256$$ 1.21324e6 1.15704
$$257$$ 1.84601e6 1.74342 0.871711 0.490021i $$-0.163011\pi$$
0.871711 + 0.490021i $$0.163011\pi$$
$$258$$ −1.89695e6 −1.77422
$$259$$ 0 0
$$260$$ 544719. 0.499734
$$261$$ −1.55217e6 −1.41039
$$262$$ 1.84829e6 1.66348
$$263$$ −458222. −0.408495 −0.204248 0.978919i $$-0.565475\pi$$
−0.204248 + 0.978919i $$0.565475\pi$$
$$264$$ −228037. −0.201371
$$265$$ 2.93437e6 2.56685
$$266$$ 0 0
$$267$$ −1.61995e6 −1.39066
$$268$$ 1.63115e6 1.38726
$$269$$ 416958. 0.351327 0.175663 0.984450i $$-0.443793\pi$$
0.175663 + 0.984450i $$0.443793\pi$$
$$270$$ 916354. 0.764987
$$271$$ −900379. −0.744735 −0.372368 0.928085i $$-0.621454\pi$$
−0.372368 + 0.928085i $$0.621454\pi$$
$$272$$ 1.22476e6 1.00376
$$273$$ 0 0
$$274$$ 1.57007e6 1.26341
$$275$$ −1.01466e6 −0.809073
$$276$$ −2.47019e6 −1.95190
$$277$$ −447641. −0.350535 −0.175267 0.984521i $$-0.556079\pi$$
−0.175267 + 0.984521i $$0.556079\pi$$
$$278$$ −414083. −0.321348
$$279$$ 875072. 0.673029
$$280$$ 0 0
$$281$$ −768521. −0.580617 −0.290309 0.956933i $$-0.593758\pi$$
−0.290309 + 0.956933i $$0.593758\pi$$
$$282$$ −3.13113e6 −2.34465
$$283$$ −2.13220e6 −1.58256 −0.791282 0.611452i $$-0.790586\pi$$
−0.791282 + 0.611452i $$0.790586\pi$$
$$284$$ −451756. −0.332359
$$285$$ 11303.9 0.00824356
$$286$$ −836230. −0.604520
$$287$$ 0 0
$$288$$ 2.46107e6 1.74841
$$289$$ −200275. −0.141053
$$290$$ −2.90525e6 −2.02856
$$291$$ 2.56017e6 1.77229
$$292$$ −2.27426e6 −1.56093
$$293$$ −2.42669e6 −1.65138 −0.825688 0.564128i $$-0.809213\pi$$
−0.825688 + 0.564128i $$0.809213\pi$$
$$294$$ 0 0
$$295$$ −2.52163e6 −1.68704
$$296$$ −46783.9 −0.0310361
$$297$$ −669693. −0.440539
$$298$$ 787528. 0.513719
$$299$$ 911164. 0.589411
$$300$$ 1.63556e6 1.04921
$$301$$ 0 0
$$302$$ 3.57855e6 2.25782
$$303$$ 422682. 0.264489
$$304$$ 7176.56 0.00445382
$$305$$ −2.10164e6 −1.29362
$$306$$ 2.67654e6 1.63407
$$307$$ 2.44328e6 1.47954 0.739772 0.672857i $$-0.234933\pi$$
0.739772 + 0.672857i $$0.234933\pi$$
$$308$$ 0 0
$$309$$ 747271. 0.445228
$$310$$ 1.63790e6 0.968016
$$311$$ −1.15465e6 −0.676938 −0.338469 0.940978i $$-0.609909\pi$$
−0.338469 + 0.940978i $$0.609909\pi$$
$$312$$ −135587. −0.0788555
$$313$$ −1.65706e6 −0.956044 −0.478022 0.878348i $$-0.658646\pi$$
−0.478022 + 0.878348i $$0.658646\pi$$
$$314$$ 1.39919e6 0.800852
$$315$$ 0 0
$$316$$ −1.31815e6 −0.742584
$$317$$ 821361. 0.459077 0.229539 0.973300i $$-0.426278\pi$$
0.229539 + 0.973300i $$0.426278\pi$$
$$318$$ 7.26130e6 4.02668
$$319$$ 2.12322e6 1.16821
$$320$$ 1.97050e6 1.07572
$$321$$ 193543. 0.104837
$$322$$ 0 0
$$323$$ 7146.23 0.00381128
$$324$$ −1.11162e6 −0.588294
$$325$$ −603298. −0.316828
$$326$$ −1.90383e6 −0.992166
$$327$$ −260319. −0.134629
$$328$$ 214707. 0.110195
$$329$$ 0 0
$$330$$ −5.79134e6 −2.92748
$$331$$ −95670.7 −0.0479964 −0.0239982 0.999712i $$-0.507640\pi$$
−0.0239982 + 0.999712i $$0.507640\pi$$
$$332$$ 40177.0 0.0200047
$$333$$ −634785. −0.313701
$$334$$ −914618. −0.448615
$$335$$ −4.16691e6 −2.02863
$$336$$ 0 0
$$337$$ 2.37020e6 1.13687 0.568435 0.822728i $$-0.307549\pi$$
0.568435 + 0.822728i $$0.307549\pi$$
$$338$$ 2.40447e6 1.14480
$$339$$ 1.53304e6 0.724528
$$340$$ 2.38493e6 1.11887
$$341$$ −1.19701e6 −0.557460
$$342$$ 15683.4 0.00725060
$$343$$ 0 0
$$344$$ 235894. 0.107478
$$345$$ 6.31029e6 2.85431
$$346$$ −2.10799e6 −0.946624
$$347$$ 490571. 0.218715 0.109358 0.994002i $$-0.465121\pi$$
0.109358 + 0.994002i $$0.465121\pi$$
$$348$$ −3.42249e6 −1.51493
$$349$$ −4.21208e6 −1.85111 −0.925557 0.378607i $$-0.876403\pi$$
−0.925557 + 0.378607i $$0.876403\pi$$
$$350$$ 0 0
$$351$$ −398188. −0.172512
$$352$$ −3.36651e6 −1.44818
$$353$$ 3.17378e6 1.35563 0.677814 0.735234i $$-0.262928\pi$$
0.677814 + 0.735234i $$0.262928\pi$$
$$354$$ −6.23995e6 −2.64651
$$355$$ 1.15405e6 0.486018
$$356$$ −2.00269e6 −0.837509
$$357$$ 0 0
$$358$$ −2.94257e6 −1.21344
$$359$$ −3.40098e6 −1.39273 −0.696366 0.717687i $$-0.745201\pi$$
−0.696366 + 0.717687i $$0.745201\pi$$
$$360$$ −526483. −0.214105
$$361$$ −2.47606e6 −0.999983
$$362$$ 3.39675e6 1.36236
$$363$$ 444757. 0.177156
$$364$$ 0 0
$$365$$ 5.80978e6 2.28259
$$366$$ −5.20065e6 −2.02934
$$367$$ 1.96872e6 0.762988 0.381494 0.924371i $$-0.375410\pi$$
0.381494 + 0.924371i $$0.375410\pi$$
$$368$$ 4.00626e6 1.54213
$$369$$ 2.91324e6 1.11381
$$370$$ −1.18814e6 −0.451196
$$371$$ 0 0
$$372$$ 1.92950e6 0.722917
$$373$$ −3.47889e6 −1.29470 −0.647349 0.762194i $$-0.724122\pi$$
−0.647349 + 0.762194i $$0.724122\pi$$
$$374$$ −3.66124e6 −1.35347
$$375$$ 1.28075e6 0.470312
$$376$$ 389369. 0.142034
$$377$$ 1.26243e6 0.457462
$$378$$ 0 0
$$379$$ −421294. −0.150656 −0.0753281 0.997159i $$-0.524000\pi$$
−0.0753281 + 0.997159i $$0.524000\pi$$
$$380$$ 13974.7 0.00496457
$$381$$ −4.58089e6 −1.61673
$$382$$ −4.42286e6 −1.55076
$$383$$ 2.66910e6 0.929754 0.464877 0.885375i $$-0.346099\pi$$
0.464877 + 0.885375i $$0.346099\pi$$
$$384$$ −1.09630e6 −0.379402
$$385$$ 0 0
$$386$$ −4.02055e6 −1.37346
$$387$$ 3.20071e6 1.08635
$$388$$ 3.16506e6 1.06734
$$389$$ 3.10178e6 1.03929 0.519645 0.854382i $$-0.326064\pi$$
0.519645 + 0.854382i $$0.326064\pi$$
$$390$$ −3.44343e6 −1.14638
$$391$$ 3.98933e6 1.31965
$$392$$ 0 0
$$393$$ −5.56223e6 −1.81663
$$394$$ 2.33312e6 0.757176
$$395$$ 3.36731e6 1.08590
$$396$$ −3.82516e6 −1.22578
$$397$$ 613257. 0.195284 0.0976419 0.995222i $$-0.468870\pi$$
0.0976419 + 0.995222i $$0.468870\pi$$
$$398$$ 4.62589e6 1.46382
$$399$$ 0 0
$$400$$ −2.65262e6 −0.828942
$$401$$ 2.82223e6 0.876459 0.438229 0.898863i $$-0.355606\pi$$
0.438229 + 0.898863i $$0.355606\pi$$
$$402$$ −1.03113e7 −3.18236
$$403$$ −711724. −0.218298
$$404$$ 522550. 0.159285
$$405$$ 2.83973e6 0.860278
$$406$$ 0 0
$$407$$ 868324. 0.259834
$$408$$ −593637. −0.176551
$$409$$ 2.28350e6 0.674983 0.337492 0.941329i $$-0.390422\pi$$
0.337492 + 0.941329i $$0.390422\pi$$
$$410$$ 5.45279e6 1.60199
$$411$$ −4.72496e6 −1.37973
$$412$$ 923831. 0.268132
$$413$$ 0 0
$$414$$ 8.75511e6 2.51050
$$415$$ −102635. −0.0292534
$$416$$ −2.00167e6 −0.567100
$$417$$ 1.24614e6 0.350934
$$418$$ −21453.3 −0.00600556
$$419$$ −2.65270e6 −0.738163 −0.369082 0.929397i $$-0.620328\pi$$
−0.369082 + 0.929397i $$0.620328\pi$$
$$420$$ 0 0
$$421$$ 2.93674e6 0.807532 0.403766 0.914862i $$-0.367701\pi$$
0.403766 + 0.914862i $$0.367701\pi$$
$$422$$ 1.09829e6 0.300218
$$423$$ 5.28314e6 1.43562
$$424$$ −902974. −0.243927
$$425$$ −2.64140e6 −0.709353
$$426$$ 2.85577e6 0.762428
$$427$$ 0 0
$$428$$ 239272. 0.0631367
$$429$$ 2.51654e6 0.660177
$$430$$ 5.99087e6 1.56249
$$431$$ −2.44565e6 −0.634164 −0.317082 0.948398i $$-0.602703\pi$$
−0.317082 + 0.948398i $$0.602703\pi$$
$$432$$ −1.75078e6 −0.451358
$$433$$ −2.11718e6 −0.542673 −0.271336 0.962485i $$-0.587466\pi$$
−0.271336 + 0.962485i $$0.587466\pi$$
$$434$$ 0 0
$$435$$ 8.74301e6 2.21533
$$436$$ −321826. −0.0810783
$$437$$ 23375.7 0.00585547
$$438$$ 1.43767e7 3.58075
$$439$$ 4.64764e6 1.15099 0.575495 0.817806i $$-0.304810\pi$$
0.575495 + 0.817806i $$0.304810\pi$$
$$440$$ 720178. 0.177340
$$441$$ 0 0
$$442$$ −2.17691e6 −0.530012
$$443$$ 4.42925e6 1.07231 0.536155 0.844119i $$-0.319876\pi$$
0.536155 + 0.844119i $$0.319876\pi$$
$$444$$ −1.39968e6 −0.336954
$$445$$ 5.11604e6 1.22471
$$446$$ −3833.50 −0.000912553 0
$$447$$ −2.36998e6 −0.561016
$$448$$ 0 0
$$449$$ −6.70171e6 −1.56881 −0.784404 0.620250i $$-0.787031\pi$$
−0.784404 + 0.620250i $$0.787031\pi$$
$$450$$ −5.79691e6 −1.34948
$$451$$ −3.98503e6 −0.922551
$$452$$ 1.89526e6 0.436337
$$453$$ −1.07692e7 −2.46570
$$454$$ −4.64136e6 −1.05683
$$455$$ 0 0
$$456$$ −3478.46 −0.000783385 0
$$457$$ −5.88344e6 −1.31777 −0.658887 0.752242i $$-0.728972\pi$$
−0.658887 + 0.752242i $$0.728972\pi$$
$$458$$ 280407. 0.0624634
$$459$$ −1.74338e6 −0.386242
$$460$$ 7.80124e6 1.71897
$$461$$ 1.54764e6 0.339171 0.169585 0.985515i $$-0.445757\pi$$
0.169585 + 0.985515i $$0.445757\pi$$
$$462$$ 0 0
$$463$$ −3.93764e6 −0.853656 −0.426828 0.904333i $$-0.640369\pi$$
−0.426828 + 0.904333i $$0.640369\pi$$
$$464$$ 5.55074e6 1.19689
$$465$$ −4.92907e6 −1.05714
$$466$$ −9.75491e6 −2.08093
$$467$$ −6.81586e6 −1.44620 −0.723100 0.690743i $$-0.757284\pi$$
−0.723100 + 0.690743i $$0.757284\pi$$
$$468$$ −2.27438e6 −0.480007
$$469$$ 0 0
$$470$$ 9.88860e6 2.06486
$$471$$ −4.21070e6 −0.874585
$$472$$ 775964. 0.160320
$$473$$ −4.37827e6 −0.899807
$$474$$ 8.33263e6 1.70348
$$475$$ −15477.5 −0.00314750
$$476$$ 0 0
$$477$$ −1.22519e7 −2.46552
$$478$$ 4.50885e6 0.902602
$$479$$ −5.20406e6 −1.03634 −0.518171 0.855277i $$-0.673387\pi$$
−0.518171 + 0.855277i $$0.673387\pi$$
$$480$$ −1.38626e7 −2.74627
$$481$$ 516291. 0.101749
$$482$$ 1.08036e7 2.11813
$$483$$ 0 0
$$484$$ 549840. 0.106690
$$485$$ −8.08540e6 −1.56080
$$486$$ 1.00251e7 1.92529
$$487$$ −154998. −0.0296145 −0.0148073 0.999890i $$-0.504713\pi$$
−0.0148073 + 0.999890i $$0.504713\pi$$
$$488$$ 646723. 0.122933
$$489$$ 5.72936e6 1.08351
$$490$$ 0 0
$$491$$ −1.61951e6 −0.303165 −0.151583 0.988445i $$-0.548437\pi$$
−0.151583 + 0.988445i $$0.548437\pi$$
$$492$$ 6.42359e6 1.19637
$$493$$ 5.52727e6 1.02422
$$494$$ −12755.8 −0.00235174
$$495$$ 9.77168e6 1.79249
$$496$$ −3.12935e6 −0.571150
$$497$$ 0 0
$$498$$ −253978. −0.0458905
$$499$$ 4.10674e6 0.738322 0.369161 0.929366i $$-0.379645\pi$$
0.369161 + 0.929366i $$0.379645\pi$$
$$500$$ 1.58336e6 0.283239
$$501$$ 2.75244e6 0.489918
$$502$$ −2.52596e6 −0.447371
$$503$$ −3.40748e6 −0.600501 −0.300250 0.953860i $$-0.597070\pi$$
−0.300250 + 0.953860i $$0.597070\pi$$
$$504$$ 0 0
$$505$$ −1.33490e6 −0.232927
$$506$$ −1.19762e7 −2.07941
$$507$$ −7.23599e6 −1.25020
$$508$$ −5.66322e6 −0.973654
$$509$$ 1.07091e7 1.83213 0.916067 0.401025i $$-0.131346\pi$$
0.916067 + 0.401025i $$0.131346\pi$$
$$510$$ −1.50763e7 −2.56666
$$511$$ 0 0
$$512$$ −7.98992e6 −1.34700
$$513$$ −10215.4 −0.00171381
$$514$$ −1.44267e7 −2.40858
$$515$$ −2.36000e6 −0.392097
$$516$$ 7.05746e6 1.16687
$$517$$ −7.22682e6 −1.18911
$$518$$ 0 0
$$519$$ 6.34374e6 1.03378
$$520$$ 428205. 0.0694454
$$521$$ 7.92001e6 1.27830 0.639148 0.769084i $$-0.279287\pi$$
0.639148 + 0.769084i $$0.279287\pi$$
$$522$$ 1.21303e7 1.94848
$$523$$ −8.32746e6 −1.33125 −0.665623 0.746288i $$-0.731834\pi$$
−0.665623 + 0.746288i $$0.731834\pi$$
$$524$$ −6.87643e6 −1.09404
$$525$$ 0 0
$$526$$ 3.58104e6 0.564345
$$527$$ −3.11612e6 −0.488752
$$528$$ 1.10649e7 1.72728
$$529$$ 6.61298e6 1.02744
$$530$$ −2.29323e7 −3.54616
$$531$$ 1.05286e7 1.62045
$$532$$ 0 0
$$533$$ −2.36943e6 −0.361265
$$534$$ 1.26600e7 1.92124
$$535$$ −611239. −0.0923265
$$536$$ 1.28226e6 0.192780
$$537$$ 8.85532e6 1.32516
$$538$$ −3.25856e6 −0.485366
$$539$$ 0 0
$$540$$ −3.40922e6 −0.503119
$$541$$ 623261. 0.0915539 0.0457770 0.998952i $$-0.485424\pi$$
0.0457770 + 0.998952i $$0.485424\pi$$
$$542$$ 7.03653e6 1.02887
$$543$$ −1.02221e7 −1.48779
$$544$$ −8.76386e6 −1.26969
$$545$$ 822129. 0.118563
$$546$$ 0 0
$$547$$ 1.05691e7 1.51032 0.755159 0.655541i $$-0.227559\pi$$
0.755159 + 0.655541i $$0.227559\pi$$
$$548$$ −5.84133e6 −0.830922
$$549$$ 8.77502e6 1.24256
$$550$$ 7.92962e6 1.11775
$$551$$ 32387.5 0.00454462
$$552$$ −1.94182e6 −0.271245
$$553$$ 0 0
$$554$$ 3.49835e6 0.484272
$$555$$ 3.57559e6 0.492737
$$556$$ 1.54056e6 0.211345
$$557$$ −1.35398e7 −1.84916 −0.924579 0.380991i $$-0.875583\pi$$
−0.924579 + 0.380991i $$0.875583\pi$$
$$558$$ −6.83875e6 −0.929804
$$559$$ −2.60324e6 −0.352359
$$560$$ 0 0
$$561$$ 1.10181e7 1.47809
$$562$$ 6.00605e6 0.802136
$$563$$ −1.39757e7 −1.85824 −0.929122 0.369774i $$-0.879435\pi$$
−0.929122 + 0.369774i $$0.879435\pi$$
$$564$$ 1.16491e7 1.54204
$$565$$ −4.84159e6 −0.638068
$$566$$ 1.66633e7 2.18635
$$567$$ 0 0
$$568$$ −355127. −0.0461862
$$569$$ −5.61993e6 −0.727697 −0.363848 0.931458i $$-0.618537\pi$$
−0.363848 + 0.931458i $$0.618537\pi$$
$$570$$ −88340.5 −0.0113887
$$571$$ 8.70790e6 1.11769 0.558847 0.829271i $$-0.311244\pi$$
0.558847 + 0.829271i $$0.311244\pi$$
$$572$$ 3.11113e6 0.397583
$$573$$ 1.33101e7 1.69354
$$574$$ 0 0
$$575$$ −8.64019e6 −1.08982
$$576$$ −8.22747e6 −1.03326
$$577$$ −6.63992e6 −0.830278 −0.415139 0.909758i $$-0.636267\pi$$
−0.415139 + 0.909758i $$0.636267\pi$$
$$578$$ 1.56517e6 0.194868
$$579$$ 1.20994e7 1.49992
$$580$$ 1.08087e7 1.33415
$$581$$ 0 0
$$582$$ −2.00079e7 −2.44846
$$583$$ 1.67595e7 2.04216
$$584$$ −1.78780e6 −0.216914
$$585$$ 5.81007e6 0.701927
$$586$$ 1.89648e7 2.28141
$$587$$ −1.36652e7 −1.63689 −0.818446 0.574583i $$-0.805164\pi$$
−0.818446 + 0.574583i $$0.805164\pi$$
$$588$$ 0 0
$$589$$ −18259.2 −0.00216866
$$590$$ 1.97067e7 2.33069
$$591$$ −7.02126e6 −0.826888
$$592$$ 2.27006e6 0.266215
$$593$$ −7.02589e6 −0.820474 −0.410237 0.911979i $$-0.634554\pi$$
−0.410237 + 0.911979i $$0.634554\pi$$
$$594$$ 5.23370e6 0.608615
$$595$$ 0 0
$$596$$ −2.92994e6 −0.337864
$$597$$ −1.39211e7 −1.59859
$$598$$ −7.12081e6 −0.814285
$$599$$ −3.58663e6 −0.408432 −0.204216 0.978926i $$-0.565464\pi$$
−0.204216 + 0.978926i $$0.565464\pi$$
$$600$$ 1.28572e6 0.145803
$$601$$ 1.58600e7 1.79108 0.895542 0.444977i $$-0.146788\pi$$
0.895542 + 0.444977i $$0.146788\pi$$
$$602$$ 0 0
$$603$$ 1.73982e7 1.94855
$$604$$ −1.33137e7 −1.48493
$$605$$ −1.40461e6 −0.156015
$$606$$ −3.30329e6 −0.365397
$$607$$ 6.44170e6 0.709625 0.354812 0.934938i $$-0.384545\pi$$
0.354812 + 0.934938i $$0.384545\pi$$
$$608$$ −51352.5 −0.00563381
$$609$$ 0 0
$$610$$ 1.64244e7 1.78717
$$611$$ −4.29694e6 −0.465647
$$612$$ −9.95784e6 −1.07470
$$613$$ −4.58865e6 −0.493212 −0.246606 0.969116i $$-0.579315\pi$$
−0.246606 + 0.969116i $$0.579315\pi$$
$$614$$ −1.90944e7 −2.04402
$$615$$ −1.64096e7 −1.74948
$$616$$ 0 0
$$617$$ 1.47104e6 0.155565 0.0777825 0.996970i $$-0.475216\pi$$
0.0777825 + 0.996970i $$0.475216\pi$$
$$618$$ −5.83998e6 −0.615092
$$619$$ −3.13569e6 −0.328932 −0.164466 0.986383i $$-0.552590\pi$$
−0.164466 + 0.986383i $$0.552590\pi$$
$$620$$ −6.09367e6 −0.636649
$$621$$ −5.70269e6 −0.593404
$$622$$ 9.02366e6 0.935205
$$623$$ 0 0
$$624$$ 6.57898e6 0.676390
$$625$$ −1.15193e7 −1.17957
$$626$$ 1.29501e7 1.32080
$$627$$ 64561.3 0.00655849
$$628$$ −5.20557e6 −0.526707
$$629$$ 2.26046e6 0.227809
$$630$$ 0 0
$$631$$ 484547. 0.0484465 0.0242233 0.999707i $$-0.492289\pi$$
0.0242233 + 0.999707i $$0.492289\pi$$
$$632$$ −1.03620e6 −0.103193
$$633$$ −3.30518e6 −0.327858
$$634$$ −6.41899e6 −0.634226
$$635$$ 1.44672e7 1.42380
$$636$$ −2.70151e7 −2.64828
$$637$$ 0 0
$$638$$ −1.65932e7 −1.61390
$$639$$ −4.81851e6 −0.466832
$$640$$ 3.46228e6 0.334127
$$641$$ 3.04085e6 0.292314 0.146157 0.989261i $$-0.453310\pi$$
0.146157 + 0.989261i $$0.453310\pi$$
$$642$$ −1.51255e6 −0.144835
$$643$$ 5.25888e6 0.501609 0.250805 0.968038i $$-0.419305\pi$$
0.250805 + 0.968038i $$0.419305\pi$$
$$644$$ 0 0
$$645$$ −1.80288e7 −1.70635
$$646$$ −55848.3 −0.00526536
$$647$$ 2.11970e7 1.99074 0.995368 0.0961386i $$-0.0306492\pi$$
0.995368 + 0.0961386i $$0.0306492\pi$$
$$648$$ −873849. −0.0817521
$$649$$ −1.44021e7 −1.34219
$$650$$ 4.71481e6 0.437705
$$651$$ 0 0
$$652$$ 7.08305e6 0.652531
$$653$$ 1.30106e7 1.19403 0.597013 0.802232i $$-0.296354\pi$$
0.597013 + 0.802232i $$0.296354\pi$$
$$654$$ 2.03442e6 0.185992
$$655$$ 1.75664e7 1.59985
$$656$$ −1.04181e7 −0.945208
$$657$$ −2.42577e7 −2.19248
$$658$$ 0 0
$$659$$ −1.59874e7 −1.43405 −0.717024 0.697049i $$-0.754496\pi$$
−0.717024 + 0.697049i $$0.754496\pi$$
$$660$$ 2.15462e7 1.92536
$$661$$ 4.03142e6 0.358884 0.179442 0.983769i $$-0.442571\pi$$
0.179442 + 0.983769i $$0.442571\pi$$
$$662$$ 747673. 0.0663082
$$663$$ 6.55117e6 0.578809
$$664$$ 31583.3 0.00277995
$$665$$ 0 0
$$666$$ 4.96089e6 0.433385
$$667$$ 1.80800e7 1.57357
$$668$$ 3.40276e6 0.295047
$$669$$ 11536.5 0.000996570 0
$$670$$ 3.25647e7 2.80260
$$671$$ −1.20034e7 −1.02919
$$672$$ 0 0
$$673$$ 2.98234e6 0.253816 0.126908 0.991914i $$-0.459495\pi$$
0.126908 + 0.991914i $$0.459495\pi$$
$$674$$ −1.85233e7 −1.57061
$$675$$ 3.77585e6 0.318974
$$676$$ −8.94566e6 −0.752914
$$677$$ 1.94696e6 0.163262 0.0816309 0.996663i $$-0.473987\pi$$
0.0816309 + 0.996663i $$0.473987\pi$$
$$678$$ −1.19808e7 −1.00095
$$679$$ 0 0
$$680$$ 1.87480e6 0.155483
$$681$$ 1.39676e7 1.15413
$$682$$ 9.35476e6 0.770144
$$683$$ 1.19940e7 0.983812 0.491906 0.870648i $$-0.336300\pi$$
0.491906 + 0.870648i $$0.336300\pi$$
$$684$$ −58348.7 −0.00476860
$$685$$ 1.49221e7 1.21508
$$686$$ 0 0
$$687$$ −843854. −0.0682143
$$688$$ −1.14461e7 −0.921905
$$689$$ 9.96490e6 0.799696
$$690$$ −4.93154e7 −3.94330
$$691$$ −8.66304e6 −0.690201 −0.345100 0.938566i $$-0.612155\pi$$
−0.345100 + 0.938566i $$0.612155\pi$$
$$692$$ 7.84260e6 0.622579
$$693$$ 0 0
$$694$$ −3.83385e6 −0.302160
$$695$$ −3.93549e6 −0.309056
$$696$$ −2.69043e6 −0.210522
$$697$$ −1.03740e7 −0.808845
$$698$$ 3.29177e7 2.55736
$$699$$ 2.93563e7 2.27252
$$700$$ 0 0
$$701$$ −8.13382e6 −0.625172 −0.312586 0.949889i $$-0.601195\pi$$
−0.312586 + 0.949889i $$0.601195\pi$$
$$702$$ 3.11187e6 0.238330
$$703$$ 13245.3 0.00101082
$$704$$ 1.12544e7 0.855835
$$705$$ −2.97586e7 −2.25497
$$706$$ −2.48033e7 −1.87283
$$707$$ 0 0
$$708$$ 2.32152e7 1.74056
$$709$$ 2.21326e7 1.65355 0.826773 0.562535i $$-0.190174\pi$$
0.826773 + 0.562535i $$0.190174\pi$$
$$710$$ −9.01895e6 −0.671445
$$711$$ −1.40596e7 −1.04303
$$712$$ −1.57432e6 −0.116384
$$713$$ −1.01930e7 −0.750896
$$714$$ 0 0
$$715$$ −7.94762e6 −0.581396
$$716$$ 1.09476e7 0.798061
$$717$$ −1.35689e7 −0.985704
$$718$$ 2.65789e7 1.92409
$$719$$ −7.23196e6 −0.521716 −0.260858 0.965377i $$-0.584005\pi$$
−0.260858 + 0.965377i $$0.584005\pi$$
$$720$$ 2.55461e7 1.83651
$$721$$ 0 0
$$722$$ 1.93506e7 1.38150
$$723$$ −3.25123e7 −2.31314
$$724$$ −1.26374e7 −0.896004
$$725$$ −1.19711e7 −0.845843
$$726$$ −3.47581e6 −0.244745
$$727$$ −1.70200e7 −1.19433 −0.597163 0.802120i $$-0.703705\pi$$
−0.597163 + 0.802120i $$0.703705\pi$$
$$728$$ 0 0
$$729$$ −2.08788e7 −1.45508
$$730$$ −4.54039e7 −3.15345
$$731$$ −1.13977e7 −0.788904
$$732$$ 1.93486e7 1.33466
$$733$$ 1.00011e6 0.0687525 0.0343763 0.999409i $$-0.489056\pi$$
0.0343763 + 0.999409i $$0.489056\pi$$
$$734$$ −1.53857e7 −1.05409
$$735$$ 0 0
$$736$$ −2.86671e7 −1.95070
$$737$$ −2.37991e7 −1.61396
$$738$$ −2.27672e7 −1.53875
$$739$$ 3.25979e6 0.219572 0.109786 0.993955i $$-0.464983\pi$$
0.109786 + 0.993955i $$0.464983\pi$$
$$740$$ 4.42040e6 0.296744
$$741$$ 38387.1 0.00256826
$$742$$ 0 0
$$743$$ 1.36125e7 0.904617 0.452309 0.891861i $$-0.350601\pi$$
0.452309 + 0.891861i $$0.350601\pi$$
$$744$$ 1.51679e6 0.100460
$$745$$ 7.48475e6 0.494068
$$746$$ 2.71878e7 1.78865
$$747$$ 428536. 0.0280986
$$748$$ 1.36214e7 0.890158
$$749$$ 0 0
$$750$$ −1.00092e7 −0.649747
$$751$$ 6.56544e6 0.424780 0.212390 0.977185i $$-0.431875\pi$$
0.212390 + 0.977185i $$0.431875\pi$$
$$752$$ −1.88931e7 −1.21831
$$753$$ 7.60160e6 0.488560
$$754$$ −9.86600e6 −0.631994
$$755$$ 3.40110e7 2.17146
$$756$$ 0 0
$$757$$ 2.62531e7 1.66510 0.832551 0.553948i $$-0.186879\pi$$
0.832551 + 0.553948i $$0.186879\pi$$
$$758$$ 3.29244e6 0.208135
$$759$$ 3.60409e7 2.27086
$$760$$ 10985.5 0.000689901 0
$$761$$ −5.25111e6 −0.328692 −0.164346 0.986403i $$-0.552551\pi$$
−0.164346 + 0.986403i $$0.552551\pi$$
$$762$$ 3.58000e7 2.23355
$$763$$ 0 0
$$764$$ 1.64549e7 1.01991
$$765$$ 2.54381e7 1.57156
$$766$$ −2.08592e7 −1.28448
$$767$$ −8.56327e6 −0.525595
$$768$$ 2.85337e7 1.74564
$$769$$ −1.77307e7 −1.08121 −0.540605 0.841277i $$-0.681805\pi$$
−0.540605 + 0.841277i $$0.681805\pi$$
$$770$$ 0 0
$$771$$ 4.34156e7 2.63033
$$772$$ 1.49581e7 0.903305
$$773$$ 3.82592e6 0.230296 0.115148 0.993348i $$-0.463266\pi$$
0.115148 + 0.993348i $$0.463266\pi$$
$$774$$ −2.50138e7 −1.50082
$$775$$ 6.74898e6 0.403631
$$776$$ 2.48807e6 0.148323
$$777$$ 0 0
$$778$$ −2.42406e7 −1.43580
$$779$$ −60787.3 −0.00358896
$$780$$ 1.28110e7 0.753957
$$781$$ 6.59126e6 0.386671
$$782$$ −3.11769e7 −1.82312
$$783$$ −7.90117e6 −0.460561
$$784$$ 0 0
$$785$$ 1.32980e7 0.770218
$$786$$ 4.34692e7 2.50972
$$787$$ 2.94263e7 1.69355 0.846777 0.531948i $$-0.178540\pi$$
0.846777 + 0.531948i $$0.178540\pi$$
$$788$$ −8.68019e6 −0.497982
$$789$$ −1.07767e7 −0.616304
$$790$$ −2.63157e7 −1.50020
$$791$$ 0 0
$$792$$ −3.00697e6 −0.170340
$$793$$ −7.13700e6 −0.403026
$$794$$ −4.79265e6 −0.269789
$$795$$ 6.90122e7 3.87265
$$796$$ −1.72102e7 −0.962730
$$797$$ 1.35805e7 0.757305 0.378653 0.925539i $$-0.376387\pi$$
0.378653 + 0.925539i $$0.376387\pi$$
$$798$$ 0 0
$$799$$ −1.88132e7 −1.04255
$$800$$ 1.89810e7 1.04856
$$801$$ −2.13611e7 −1.17637
$$802$$ −2.20559e7 −1.21085
$$803$$ 3.31822e7 1.81600
$$804$$ 3.83624e7 2.09299
$$805$$ 0 0
$$806$$ 5.56218e6 0.301583
$$807$$ 9.80625e6 0.530053
$$808$$ 410778. 0.0221350
$$809$$ −1.15714e7 −0.621604 −0.310802 0.950475i $$-0.600598\pi$$
−0.310802 + 0.950475i $$0.600598\pi$$
$$810$$ −2.21927e7 −1.18849
$$811$$ −3.52530e6 −0.188210 −0.0941052 0.995562i $$-0.529999\pi$$
−0.0941052 + 0.995562i $$0.529999\pi$$
$$812$$ 0 0
$$813$$ −2.11756e7 −1.12360
$$814$$ −6.78602e6 −0.358966
$$815$$ −1.80942e7 −0.954214
$$816$$ 2.88046e7 1.51439
$$817$$ −66785.7 −0.00350049
$$818$$ −1.78457e7 −0.932505
$$819$$ 0 0
$$820$$ −2.02867e7 −1.05360
$$821$$ 1.78951e7 0.926567 0.463283 0.886210i $$-0.346671\pi$$
0.463283 + 0.886210i $$0.346671\pi$$
$$822$$ 3.69259e7 1.90612
$$823$$ −3.61421e7 −1.86000 −0.930001 0.367557i $$-0.880194\pi$$
−0.930001 + 0.367557i $$0.880194\pi$$
$$824$$ 726226. 0.0372610
$$825$$ −2.38633e7 −1.22066
$$826$$ 0 0
$$827$$ −1.00605e7 −0.511512 −0.255756 0.966741i $$-0.582324\pi$$
−0.255756 + 0.966741i $$0.582324\pi$$
$$828$$ −3.25727e7 −1.65112
$$829$$ 2.03654e7 1.02921 0.514607 0.857426i $$-0.327938\pi$$
0.514607 + 0.857426i $$0.327938\pi$$
$$830$$ 802102. 0.0404142
$$831$$ −1.05279e7 −0.528858
$$832$$ 6.69166e6 0.335140
$$833$$ 0 0
$$834$$ −9.73865e6 −0.484823
$$835$$ −8.69263e6 −0.431454
$$836$$ 79815.4 0.00394976
$$837$$ 4.45446e6 0.219777
$$838$$ 2.07310e7 1.01979
$$839$$ −5.95014e6 −0.291825 −0.145912 0.989298i $$-0.546612\pi$$
−0.145912 + 0.989298i $$0.546612\pi$$
$$840$$ 0 0
$$841$$ 4.53905e6 0.221297
$$842$$ −2.29508e7 −1.11562
$$843$$ −1.80745e7 −0.875987
$$844$$ −4.08611e6 −0.197448
$$845$$ 2.28524e7 1.10101
$$846$$ −4.12881e7 −1.98335
$$847$$ 0 0
$$848$$ 4.38143e7 2.09231
$$849$$ −5.01462e7 −2.38764
$$850$$ 2.06428e7 0.979987
$$851$$ 7.39411e6 0.349995
$$852$$ −1.06247e7 −0.501436
$$853$$ 1.59836e7 0.752146 0.376073 0.926590i $$-0.377274\pi$$
0.376073 + 0.926590i $$0.377274\pi$$
$$854$$ 0 0
$$855$$ 149056. 0.00697325
$$856$$ 188092. 0.00877378
$$857$$ −2.34591e6 −0.109109 −0.0545544 0.998511i $$-0.517374\pi$$
−0.0545544 + 0.998511i $$0.517374\pi$$
$$858$$ −1.96669e7 −0.912050
$$859$$ −1.30223e7 −0.602152 −0.301076 0.953600i $$-0.597346\pi$$
−0.301076 + 0.953600i $$0.597346\pi$$
$$860$$ −2.22886e7 −1.02763
$$861$$ 0 0
$$862$$ 1.91130e7 0.876112
$$863$$ 1.96688e7 0.898983 0.449492 0.893285i $$-0.351605\pi$$
0.449492 + 0.893285i $$0.351605\pi$$
$$864$$ 1.25278e7 0.570941
$$865$$ −2.00345e7 −0.910414
$$866$$ 1.65459e7 0.749715
$$867$$ −4.71019e6 −0.212809
$$868$$ 0 0
$$869$$ 1.92322e7 0.863931
$$870$$ −6.83273e7 −3.06053
$$871$$ −1.41505e7 −0.632015
$$872$$ −252988. −0.0112670
$$873$$ 3.37592e7 1.49919
$$874$$ −182683. −0.00808946
$$875$$ 0 0
$$876$$ −5.34874e7 −2.35500
$$877$$ 2.34581e7 1.02990 0.514950 0.857221i $$-0.327811\pi$$
0.514950 + 0.857221i $$0.327811\pi$$
$$878$$ −3.63217e7 −1.59012
$$879$$ −5.70724e7 −2.49146
$$880$$ −3.49446e7 −1.52115
$$881$$ 4.59257e6 0.199350 0.0996750 0.995020i $$-0.468220\pi$$
0.0996750 + 0.995020i $$0.468220\pi$$
$$882$$ 0 0
$$883$$ −1.23402e7 −0.532622 −0.266311 0.963887i $$-0.585805\pi$$
−0.266311 + 0.963887i $$0.585805\pi$$
$$884$$ 8.09903e6 0.348580
$$885$$ −5.93052e7 −2.54527
$$886$$ −3.46149e7 −1.48142
$$887$$ 1.36554e7 0.582769 0.291384 0.956606i $$-0.405884\pi$$
0.291384 + 0.956606i $$0.405884\pi$$
$$888$$ −1.10029e6 −0.0468247
$$889$$ 0 0
$$890$$ −3.99822e7 −1.69197
$$891$$ 1.62189e7 0.684428
$$892$$ 14262.2 0.000600171 0
$$893$$ −110237. −0.00462594
$$894$$ 1.85215e7 0.775056
$$895$$ −2.79665e7 −1.16703
$$896$$ 0 0
$$897$$ 2.14293e7 0.889255
$$898$$ 5.23744e7 2.16735
$$899$$ −1.41226e7 −0.582795
$$900$$ 2.15670e7 0.887529
$$901$$ 4.36291e7 1.79046
$$902$$ 3.11433e7 1.27453
$$903$$ 0 0
$$904$$ 1.48987e6 0.0606355
$$905$$ 3.22831e7 1.31025
$$906$$ 8.41624e7 3.40642
$$907$$ 7.39599e6 0.298523 0.149262 0.988798i $$-0.452310\pi$$
0.149262 + 0.988798i $$0.452310\pi$$
$$908$$ 1.72678e7 0.695060
$$909$$ 5.57362e6 0.223732
$$910$$ 0 0
$$911$$ −3.51041e7 −1.40140 −0.700699 0.713457i $$-0.747129\pi$$
−0.700699 + 0.713457i $$0.747129\pi$$
$$912$$ 168783. 0.00671955
$$913$$ −586195. −0.0232737
$$914$$ 4.59795e7 1.82053
$$915$$ −4.94275e7 −1.95171
$$916$$ −1.04323e6 −0.0410812
$$917$$ 0 0
$$918$$ 1.36246e7 0.533602
$$919$$ −4.81337e7 −1.88001 −0.940005 0.341159i $$-0.889180\pi$$
−0.940005 + 0.341159i $$0.889180\pi$$
$$920$$ 6.13258e6 0.238877
$$921$$ 5.74626e7 2.23221
$$922$$ −1.20949e7 −0.468572
$$923$$ 3.91905e6 0.151418
$$924$$ 0 0
$$925$$ −4.89577e6 −0.188134
$$926$$ 3.07729e7 1.17935
$$927$$ 9.85376e6 0.376619
$$928$$ −3.97188e7 −1.51400
$$929$$ −1.83602e7 −0.697971 −0.348986 0.937128i $$-0.613474\pi$$
−0.348986 + 0.937128i $$0.613474\pi$$
$$930$$ 3.85210e7 1.46046
$$931$$ 0 0
$$932$$ 3.62924e7 1.36860
$$933$$ −2.71557e7 −1.02131
$$934$$ 5.32665e7 1.99796
$$935$$ −3.47969e7 −1.30170
$$936$$ −1.78789e6 −0.0667041
$$937$$ 2.27081e7 0.844951 0.422475 0.906374i $$-0.361161\pi$$
0.422475 + 0.906374i $$0.361161\pi$$
$$938$$ 0 0
$$939$$ −3.89718e7 −1.44240
$$940$$ −3.67898e7 −1.35802
$$941$$ 4.15801e7 1.53078 0.765389 0.643568i $$-0.222547\pi$$
0.765389 + 0.643568i $$0.222547\pi$$
$$942$$ 3.29069e7 1.20826
$$943$$ −3.39340e7 −1.24267
$$944$$ −3.76515e7 −1.37516
$$945$$ 0 0
$$946$$ 3.42165e7 1.24310
$$947$$ 1.55341e7 0.562874 0.281437 0.959580i $$-0.409189\pi$$
0.281437 + 0.959580i $$0.409189\pi$$
$$948$$ −3.10009e7 −1.12035
$$949$$ 1.97296e7 0.711135
$$950$$ 120958. 0.00434835
$$951$$ 1.93172e7 0.692618
$$952$$ 0 0
$$953$$ 3.94908e7 1.40852 0.704262 0.709940i $$-0.251278\pi$$
0.704262 + 0.709940i $$0.251278\pi$$
$$954$$ 9.57499e7 3.40618
$$955$$ −4.20354e7 −1.49144
$$956$$ −1.67748e7 −0.593627
$$957$$ 4.99352e7 1.76249
$$958$$ 4.06701e7 1.43173
$$959$$ 0 0
$$960$$ 4.63433e7 1.62296
$$961$$ −2.06672e7 −0.721894
$$962$$ −4.03485e6 −0.140569
$$963$$ 2.55212e6 0.0886819
$$964$$ −4.01941e7 −1.39306
$$965$$ −3.82118e7 −1.32093
$$966$$ 0 0
$$967$$ −1.87472e7 −0.644718 −0.322359 0.946617i $$-0.604476\pi$$
−0.322359 + 0.946617i $$0.604476\pi$$
$$968$$ 432231. 0.0148261
$$969$$ 168069. 0.00575014
$$970$$ 6.31880e7 2.15628
$$971$$ −5.35423e7 −1.82242 −0.911211 0.411940i $$-0.864851\pi$$
−0.911211 + 0.411940i $$0.864851\pi$$
$$972$$ −3.72974e7 −1.26623
$$973$$ 0 0
$$974$$ 1.21132e6 0.0409131
$$975$$ −1.41887e7 −0.478004
$$976$$ −3.13804e7 −1.05447
$$977$$ 4.46104e7 1.49520 0.747600 0.664149i $$-0.231206\pi$$
0.747600 + 0.664149i $$0.231206\pi$$
$$978$$ −4.47754e7 −1.49690
$$979$$ 2.92200e7 0.974368
$$980$$ 0 0
$$981$$ −3.43265e6 −0.113883
$$982$$ 1.26566e7 0.418829
$$983$$ −2.00068e7 −0.660380 −0.330190 0.943914i $$-0.607113\pi$$
−0.330190 + 0.943914i $$0.607113\pi$$
$$984$$ 5.04960e6 0.166253
$$985$$ 2.21742e7 0.728213
$$986$$ −4.31961e7 −1.41498
$$987$$ 0 0
$$988$$ 47456.9 0.00154670
$$989$$ −3.72826e7 −1.21204
$$990$$ −7.63664e7 −2.47636
$$991$$ 3.24635e7 1.05005 0.525026 0.851086i $$-0.324056\pi$$
0.525026 + 0.851086i $$0.324056\pi$$
$$992$$ 2.23923e7 0.722471
$$993$$ −2.25004e6 −0.0724130
$$994$$ 0 0
$$995$$ 4.39649e7 1.40783
$$996$$ 944905. 0.0301814
$$997$$ 2.78940e7 0.888736 0.444368 0.895844i $$-0.353428\pi$$
0.444368 + 0.895844i $$0.353428\pi$$
$$998$$ −3.20945e7 −1.02001
$$999$$ −3.23130e6 −0.102439
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 49.6.a.g.1.2 yes 4
3.2 odd 2 441.6.a.z.1.4 4
4.3 odd 2 784.6.a.bf.1.1 4
7.2 even 3 49.6.c.h.18.3 8
7.3 odd 6 49.6.c.h.30.4 8
7.4 even 3 49.6.c.h.30.3 8
7.5 odd 6 49.6.c.h.18.4 8
7.6 odd 2 inner 49.6.a.g.1.1 4
21.20 even 2 441.6.a.z.1.3 4
28.27 even 2 784.6.a.bf.1.4 4

By twisted newform
Twist Min Dim Char Parity Ord Type
49.6.a.g.1.1 4 7.6 odd 2 inner
49.6.a.g.1.2 yes 4 1.1 even 1 trivial
49.6.c.h.18.3 8 7.2 even 3
49.6.c.h.18.4 8 7.5 odd 6
49.6.c.h.30.3 8 7.4 even 3
49.6.c.h.30.4 8 7.3 odd 6
441.6.a.z.1.3 4 21.20 even 2
441.6.a.z.1.4 4 3.2 odd 2
784.6.a.bf.1.1 4 4.3 odd 2
784.6.a.bf.1.4 4 28.27 even 2