# Properties

 Label 1035.6.a.b.1.2 Level $1035$ Weight $6$ Character 1035.1 Self dual yes Analytic conductor $165.997$ Analytic rank $1$ Dimension $7$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1035,6,Mod(1,1035)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1035, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1035.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1035 = 3^{2} \cdot 5 \cdot 23$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 1035.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: $$165.997253506$$ Analytic rank: $$1$$ Dimension: $$7$$ Coefficient field: $$\mathbb{Q}[x]/(x^{7} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{7} - 3x^{6} - 196x^{5} + 464x^{4} + 11003x^{3} - 21041x^{2} - 142416x + 243340$$ x^7 - 3*x^6 - 196*x^5 + 464*x^4 + 11003*x^3 - 21041*x^2 - 142416*x + 243340 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 115) Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$-8.71037$$ of defining polynomial Character $$\chi$$ $$=$$ 1035.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-9.71037 q^{2} +62.2913 q^{4} +25.0000 q^{5} +84.0805 q^{7} -294.140 q^{8} +O(q^{10})$$ $$q-9.71037 q^{2} +62.2913 q^{4} +25.0000 q^{5} +84.0805 q^{7} -294.140 q^{8} -242.759 q^{10} -219.102 q^{11} -248.743 q^{13} -816.453 q^{14} +862.884 q^{16} -246.199 q^{17} +31.8413 q^{19} +1557.28 q^{20} +2127.56 q^{22} -529.000 q^{23} +625.000 q^{25} +2415.39 q^{26} +5237.49 q^{28} +5198.89 q^{29} -2426.03 q^{31} +1033.55 q^{32} +2390.68 q^{34} +2102.01 q^{35} -1218.62 q^{37} -309.191 q^{38} -7353.49 q^{40} +7062.12 q^{41} +8934.89 q^{43} -13648.1 q^{44} +5136.79 q^{46} -10262.7 q^{47} -9737.46 q^{49} -6068.98 q^{50} -15494.5 q^{52} +10911.4 q^{53} -5477.54 q^{55} -24731.4 q^{56} -50483.1 q^{58} -3809.83 q^{59} +31290.1 q^{61} +23557.6 q^{62} -37648.4 q^{64} -6218.58 q^{65} +7696.34 q^{67} -15336.1 q^{68} -20411.3 q^{70} -71989.2 q^{71} -60862.6 q^{73} +11833.2 q^{74} +1983.44 q^{76} -18422.2 q^{77} +53155.3 q^{79} +21572.1 q^{80} -68575.8 q^{82} +56773.0 q^{83} -6154.98 q^{85} -86761.1 q^{86} +64446.5 q^{88} -116077. q^{89} -20914.5 q^{91} -32952.1 q^{92} +99655.0 q^{94} +796.032 q^{95} -96388.0 q^{97} +94554.4 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$7 q - 4 q^{2} + 178 q^{4} + 175 q^{5} + 33 q^{7} - 546 q^{8}+O(q^{10})$$ 7 * q - 4 * q^2 + 178 * q^4 + 175 * q^5 + 33 * q^7 - 546 * q^8 $$7 q - 4 q^{2} + 178 q^{4} + 175 q^{5} + 33 q^{7} - 546 q^{8} - 100 q^{10} - 1373 q^{11} + 605 q^{13} - 1317 q^{14} + 3770 q^{16} - 2505 q^{17} - 115 q^{19} + 4450 q^{20} + 2977 q^{22} - 3703 q^{23} + 4375 q^{25} - 9379 q^{26} + 5777 q^{28} - 2440 q^{29} + 13565 q^{31} - 14086 q^{32} + 26997 q^{34} + 825 q^{35} + 9414 q^{37} - 28717 q^{38} - 13650 q^{40} - 13725 q^{41} + 76694 q^{43} - 55203 q^{44} + 2116 q^{46} - 59692 q^{47} - 53608 q^{49} - 2500 q^{50} + 61195 q^{52} - 49536 q^{53} - 34325 q^{55} + 54461 q^{56} - 95562 q^{58} - 44536 q^{59} - 49097 q^{61} + 25763 q^{62} - 18654 q^{64} + 15125 q^{65} + 788 q^{67} - 163845 q^{68} - 32925 q^{70} - 49521 q^{71} - 3760 q^{73} - 88170 q^{74} - 411465 q^{76} - 77728 q^{77} + 918 q^{79} + 94250 q^{80} - 227459 q^{82} - 99202 q^{83} - 62625 q^{85} - 24584 q^{86} - 201275 q^{88} + 141676 q^{89} - 223605 q^{91} - 94162 q^{92} - 354292 q^{94} - 2875 q^{95} + 28731 q^{97} + 149557 q^{98}+O(q^{100})$$ 7 * q - 4 * q^2 + 178 * q^4 + 175 * q^5 + 33 * q^7 - 546 * q^8 - 100 * q^10 - 1373 * q^11 + 605 * q^13 - 1317 * q^14 + 3770 * q^16 - 2505 * q^17 - 115 * q^19 + 4450 * q^20 + 2977 * q^22 - 3703 * q^23 + 4375 * q^25 - 9379 * q^26 + 5777 * q^28 - 2440 * q^29 + 13565 * q^31 - 14086 * q^32 + 26997 * q^34 + 825 * q^35 + 9414 * q^37 - 28717 * q^38 - 13650 * q^40 - 13725 * q^41 + 76694 * q^43 - 55203 * q^44 + 2116 * q^46 - 59692 * q^47 - 53608 * q^49 - 2500 * q^50 + 61195 * q^52 - 49536 * q^53 - 34325 * q^55 + 54461 * q^56 - 95562 * q^58 - 44536 * q^59 - 49097 * q^61 + 25763 * q^62 - 18654 * q^64 + 15125 * q^65 + 788 * q^67 - 163845 * q^68 - 32925 * q^70 - 49521 * q^71 - 3760 * q^73 - 88170 * q^74 - 411465 * q^76 - 77728 * q^77 + 918 * q^79 + 94250 * q^80 - 227459 * q^82 - 99202 * q^83 - 62625 * q^85 - 24584 * q^86 - 201275 * q^88 + 141676 * q^89 - 223605 * q^91 - 94162 * q^92 - 354292 * q^94 - 2875 * q^95 + 28731 * q^97 + 149557 * q^98

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −9.71037 −1.71657 −0.858284 0.513176i $$-0.828469\pi$$
−0.858284 + 0.513176i $$0.828469\pi$$
$$3$$ 0 0
$$4$$ 62.2913 1.94660
$$5$$ 25.0000 0.447214
$$6$$ 0 0
$$7$$ 84.0805 0.648560 0.324280 0.945961i $$-0.394878\pi$$
0.324280 + 0.945961i $$0.394878\pi$$
$$8$$ −294.140 −1.62491
$$9$$ 0 0
$$10$$ −242.759 −0.767672
$$11$$ −219.102 −0.545964 −0.272982 0.962019i $$-0.588010\pi$$
−0.272982 + 0.962019i $$0.588010\pi$$
$$12$$ 0 0
$$13$$ −248.743 −0.408219 −0.204109 0.978948i $$-0.565430\pi$$
−0.204109 + 0.978948i $$0.565430\pi$$
$$14$$ −816.453 −1.11330
$$15$$ 0 0
$$16$$ 862.884 0.842660
$$17$$ −246.199 −0.206616 −0.103308 0.994649i $$-0.532943\pi$$
−0.103308 + 0.994649i $$0.532943\pi$$
$$18$$ 0 0
$$19$$ 31.8413 0.0202352 0.0101176 0.999949i $$-0.496779\pi$$
0.0101176 + 0.999949i $$0.496779\pi$$
$$20$$ 1557.28 0.870547
$$21$$ 0 0
$$22$$ 2127.56 0.937183
$$23$$ −529.000 −0.208514
$$24$$ 0 0
$$25$$ 625.000 0.200000
$$26$$ 2415.39 0.700735
$$27$$ 0 0
$$28$$ 5237.49 1.26249
$$29$$ 5198.89 1.14793 0.573965 0.818880i $$-0.305405\pi$$
0.573965 + 0.818880i $$0.305405\pi$$
$$30$$ 0 0
$$31$$ −2426.03 −0.453410 −0.226705 0.973963i $$-0.572795\pi$$
−0.226705 + 0.973963i $$0.572795\pi$$
$$32$$ 1033.55 0.178425
$$33$$ 0 0
$$34$$ 2390.68 0.354670
$$35$$ 2102.01 0.290045
$$36$$ 0 0
$$37$$ −1218.62 −0.146340 −0.0731700 0.997319i $$-0.523312\pi$$
−0.0731700 + 0.997319i $$0.523312\pi$$
$$38$$ −309.191 −0.0347350
$$39$$ 0 0
$$40$$ −7353.49 −0.726681
$$41$$ 7062.12 0.656108 0.328054 0.944659i $$-0.393607\pi$$
0.328054 + 0.944659i $$0.393607\pi$$
$$42$$ 0 0
$$43$$ 8934.89 0.736917 0.368458 0.929644i $$-0.379886\pi$$
0.368458 + 0.929644i $$0.379886\pi$$
$$44$$ −13648.1 −1.06277
$$45$$ 0 0
$$46$$ 5136.79 0.357929
$$47$$ −10262.7 −0.677671 −0.338835 0.940846i $$-0.610033\pi$$
−0.338835 + 0.940846i $$0.610033\pi$$
$$48$$ 0 0
$$49$$ −9737.46 −0.579369
$$50$$ −6068.98 −0.343313
$$51$$ 0 0
$$52$$ −15494.5 −0.794640
$$53$$ 10911.4 0.533571 0.266785 0.963756i $$-0.414039\pi$$
0.266785 + 0.963756i $$0.414039\pi$$
$$54$$ 0 0
$$55$$ −5477.54 −0.244162
$$56$$ −24731.4 −1.05385
$$57$$ 0 0
$$58$$ −50483.1 −1.97050
$$59$$ −3809.83 −0.142487 −0.0712435 0.997459i $$-0.522697\pi$$
−0.0712435 + 0.997459i $$0.522697\pi$$
$$60$$ 0 0
$$61$$ 31290.1 1.07667 0.538334 0.842732i $$-0.319054\pi$$
0.538334 + 0.842732i $$0.319054\pi$$
$$62$$ 23557.6 0.778309
$$63$$ 0 0
$$64$$ −37648.4 −1.14894
$$65$$ −6218.58 −0.182561
$$66$$ 0 0
$$67$$ 7696.34 0.209458 0.104729 0.994501i $$-0.466602\pi$$
0.104729 + 0.994501i $$0.466602\pi$$
$$68$$ −15336.1 −0.402199
$$69$$ 0 0
$$70$$ −20411.3 −0.497882
$$71$$ −71989.2 −1.69481 −0.847405 0.530946i $$-0.821837\pi$$
−0.847405 + 0.530946i $$0.821837\pi$$
$$72$$ 0 0
$$73$$ −60862.6 −1.33673 −0.668365 0.743834i $$-0.733005\pi$$
−0.668365 + 0.743834i $$0.733005\pi$$
$$74$$ 11833.2 0.251203
$$75$$ 0 0
$$76$$ 1983.44 0.0393898
$$77$$ −18422.2 −0.354090
$$78$$ 0 0
$$79$$ 53155.3 0.958250 0.479125 0.877747i $$-0.340954\pi$$
0.479125 + 0.877747i $$0.340954\pi$$
$$80$$ 21572.1 0.376849
$$81$$ 0 0
$$82$$ −68575.8 −1.12625
$$83$$ 56773.0 0.904579 0.452290 0.891871i $$-0.350607\pi$$
0.452290 + 0.891871i $$0.350607\pi$$
$$84$$ 0 0
$$85$$ −6154.98 −0.0924015
$$86$$ −86761.1 −1.26497
$$87$$ 0 0
$$88$$ 64446.5 0.887141
$$89$$ −116077. −1.55335 −0.776677 0.629900i $$-0.783096\pi$$
−0.776677 + 0.629900i $$0.783096\pi$$
$$90$$ 0 0
$$91$$ −20914.5 −0.264754
$$92$$ −32952.1 −0.405895
$$93$$ 0 0
$$94$$ 99655.0 1.16327
$$95$$ 796.032 0.00904944
$$96$$ 0 0
$$97$$ −96388.0 −1.04014 −0.520072 0.854122i $$-0.674095\pi$$
−0.520072 + 0.854122i $$0.674095\pi$$
$$98$$ 94554.4 0.994527
$$99$$ 0 0
$$100$$ 38932.1 0.389321
$$101$$ 18650.2 0.181920 0.0909602 0.995855i $$-0.471006\pi$$
0.0909602 + 0.995855i $$0.471006\pi$$
$$102$$ 0 0
$$103$$ −57213.0 −0.531376 −0.265688 0.964059i $$-0.585599\pi$$
−0.265688 + 0.964059i $$0.585599\pi$$
$$104$$ 73165.2 0.663317
$$105$$ 0 0
$$106$$ −105954. −0.915910
$$107$$ −109572. −0.925206 −0.462603 0.886566i $$-0.653084\pi$$
−0.462603 + 0.886566i $$0.653084\pi$$
$$108$$ 0 0
$$109$$ −61327.9 −0.494415 −0.247208 0.968963i $$-0.579513\pi$$
−0.247208 + 0.968963i $$0.579513\pi$$
$$110$$ 53188.9 0.419121
$$111$$ 0 0
$$112$$ 72551.7 0.546516
$$113$$ 247962. 1.82679 0.913397 0.407071i $$-0.133450\pi$$
0.913397 + 0.407071i $$0.133450\pi$$
$$114$$ 0 0
$$115$$ −13225.0 −0.0932505
$$116$$ 323845. 2.23456
$$117$$ 0 0
$$118$$ 36994.8 0.244589
$$119$$ −20700.6 −0.134003
$$120$$ 0 0
$$121$$ −113045. −0.701924
$$122$$ −303838. −1.84817
$$123$$ 0 0
$$124$$ −151120. −0.882609
$$125$$ 15625.0 0.0894427
$$126$$ 0 0
$$127$$ 302231. 1.66276 0.831380 0.555704i $$-0.187551\pi$$
0.831380 + 0.555704i $$0.187551\pi$$
$$128$$ 332507. 1.79381
$$129$$ 0 0
$$130$$ 60384.7 0.313378
$$131$$ −97654.9 −0.497183 −0.248591 0.968608i $$-0.579968\pi$$
−0.248591 + 0.968608i $$0.579968\pi$$
$$132$$ 0 0
$$133$$ 2677.23 0.0131237
$$134$$ −74734.3 −0.359549
$$135$$ 0 0
$$136$$ 72416.9 0.335732
$$137$$ −403988. −1.83894 −0.919470 0.393160i $$-0.871382\pi$$
−0.919470 + 0.393160i $$0.871382\pi$$
$$138$$ 0 0
$$139$$ 347715. 1.52646 0.763232 0.646125i $$-0.223611\pi$$
0.763232 + 0.646125i $$0.223611\pi$$
$$140$$ 130937. 0.564602
$$141$$ 0 0
$$142$$ 699041. 2.90926
$$143$$ 54500.0 0.222873
$$144$$ 0 0
$$145$$ 129972. 0.513370
$$146$$ 590999. 2.29459
$$147$$ 0 0
$$148$$ −75909.3 −0.284866
$$149$$ −101121. −0.373144 −0.186572 0.982441i $$-0.559738\pi$$
−0.186572 + 0.982441i $$0.559738\pi$$
$$150$$ 0 0
$$151$$ 15842.0 0.0565416 0.0282708 0.999600i $$-0.491000\pi$$
0.0282708 + 0.999600i $$0.491000\pi$$
$$152$$ −9365.79 −0.0328803
$$153$$ 0 0
$$154$$ 178886. 0.607820
$$155$$ −60650.6 −0.202771
$$156$$ 0 0
$$157$$ 219032. 0.709184 0.354592 0.935021i $$-0.384620\pi$$
0.354592 + 0.935021i $$0.384620\pi$$
$$158$$ −516158. −1.64490
$$159$$ 0 0
$$160$$ 25838.7 0.0797941
$$161$$ −44478.6 −0.135234
$$162$$ 0 0
$$163$$ 541493. 1.59634 0.798168 0.602435i $$-0.205803\pi$$
0.798168 + 0.602435i $$0.205803\pi$$
$$164$$ 439909. 1.27718
$$165$$ 0 0
$$166$$ −551287. −1.55277
$$167$$ −360046. −0.999003 −0.499502 0.866313i $$-0.666484\pi$$
−0.499502 + 0.866313i $$0.666484\pi$$
$$168$$ 0 0
$$169$$ −309420. −0.833358
$$170$$ 59767.1 0.158613
$$171$$ 0 0
$$172$$ 556566. 1.43448
$$173$$ 274433. 0.697143 0.348571 0.937282i $$-0.386667\pi$$
0.348571 + 0.937282i $$0.386667\pi$$
$$174$$ 0 0
$$175$$ 52550.3 0.129712
$$176$$ −189059. −0.460062
$$177$$ 0 0
$$178$$ 1.12715e6 2.66644
$$179$$ −261807. −0.610730 −0.305365 0.952235i $$-0.598778\pi$$
−0.305365 + 0.952235i $$0.598778\pi$$
$$180$$ 0 0
$$181$$ 359209. 0.814987 0.407493 0.913208i $$-0.366403\pi$$
0.407493 + 0.913208i $$0.366403\pi$$
$$182$$ 203087. 0.454469
$$183$$ 0 0
$$184$$ 155600. 0.338817
$$185$$ −30465.4 −0.0654453
$$186$$ 0 0
$$187$$ 53942.6 0.112805
$$188$$ −639279. −1.31916
$$189$$ 0 0
$$190$$ −7729.77 −0.0155340
$$191$$ −640678. −1.27074 −0.635370 0.772208i $$-0.719152\pi$$
−0.635370 + 0.772208i $$0.719152\pi$$
$$192$$ 0 0
$$193$$ 99379.5 0.192045 0.0960226 0.995379i $$-0.469388\pi$$
0.0960226 + 0.995379i $$0.469388\pi$$
$$194$$ 935963. 1.78548
$$195$$ 0 0
$$196$$ −606559. −1.12780
$$197$$ 378446. 0.694766 0.347383 0.937723i $$-0.387070\pi$$
0.347383 + 0.937723i $$0.387070\pi$$
$$198$$ 0 0
$$199$$ 763816. 1.36728 0.683638 0.729821i $$-0.260397\pi$$
0.683638 + 0.729821i $$0.260397\pi$$
$$200$$ −183837. −0.324982
$$201$$ 0 0
$$202$$ −181101. −0.312278
$$203$$ 437125. 0.744502
$$204$$ 0 0
$$205$$ 176553. 0.293421
$$206$$ 555559. 0.912142
$$207$$ 0 0
$$208$$ −214636. −0.343990
$$209$$ −6976.48 −0.0110477
$$210$$ 0 0
$$211$$ 255573. 0.395192 0.197596 0.980284i $$-0.436687\pi$$
0.197596 + 0.980284i $$0.436687\pi$$
$$212$$ 679687. 1.03865
$$213$$ 0 0
$$214$$ 1.06398e6 1.58818
$$215$$ 223372. 0.329559
$$216$$ 0 0
$$217$$ −203982. −0.294064
$$218$$ 595516. 0.848697
$$219$$ 0 0
$$220$$ −341203. −0.475287
$$221$$ 61240.3 0.0843445
$$222$$ 0 0
$$223$$ −449097. −0.604752 −0.302376 0.953189i $$-0.597780\pi$$
−0.302376 + 0.953189i $$0.597780\pi$$
$$224$$ 86901.2 0.115719
$$225$$ 0 0
$$226$$ −2.40780e6 −3.13581
$$227$$ 381030. 0.490789 0.245394 0.969423i $$-0.421083\pi$$
0.245394 + 0.969423i $$0.421083\pi$$
$$228$$ 0 0
$$229$$ −1.20514e6 −1.51862 −0.759311 0.650728i $$-0.774464\pi$$
−0.759311 + 0.650728i $$0.774464\pi$$
$$230$$ 128420. 0.160071
$$231$$ 0 0
$$232$$ −1.52920e6 −1.86528
$$233$$ 792519. 0.956356 0.478178 0.878263i $$-0.341297\pi$$
0.478178 + 0.878263i $$0.341297\pi$$
$$234$$ 0 0
$$235$$ −256569. −0.303064
$$236$$ −237319. −0.277366
$$237$$ 0 0
$$238$$ 201010. 0.230025
$$239$$ 185603. 0.210180 0.105090 0.994463i $$-0.466487\pi$$
0.105090 + 0.994463i $$0.466487\pi$$
$$240$$ 0 0
$$241$$ −510987. −0.566718 −0.283359 0.959014i $$-0.591449\pi$$
−0.283359 + 0.959014i $$0.591449\pi$$
$$242$$ 1.09771e6 1.20490
$$243$$ 0 0
$$244$$ 1.94910e6 2.09585
$$245$$ −243437. −0.259102
$$246$$ 0 0
$$247$$ −7920.31 −0.00826037
$$248$$ 713590. 0.736749
$$249$$ 0 0
$$250$$ −151725. −0.153534
$$251$$ −651645. −0.652870 −0.326435 0.945220i $$-0.605847\pi$$
−0.326435 + 0.945220i $$0.605847\pi$$
$$252$$ 0 0
$$253$$ 115905. 0.113841
$$254$$ −2.93478e6 −2.85424
$$255$$ 0 0
$$256$$ −2.02401e6 −1.93025
$$257$$ −803251. −0.758610 −0.379305 0.925272i $$-0.623837\pi$$
−0.379305 + 0.925272i $$0.623837\pi$$
$$258$$ 0 0
$$259$$ −102462. −0.0949104
$$260$$ −387363. −0.355374
$$261$$ 0 0
$$262$$ 948266. 0.853447
$$263$$ 189088. 0.168568 0.0842841 0.996442i $$-0.473140\pi$$
0.0842841 + 0.996442i $$0.473140\pi$$
$$264$$ 0 0
$$265$$ 272786. 0.238620
$$266$$ −25996.9 −0.0225278
$$267$$ 0 0
$$268$$ 479415. 0.407732
$$269$$ 898302. 0.756905 0.378453 0.925621i $$-0.376456\pi$$
0.378453 + 0.925621i $$0.376456\pi$$
$$270$$ 0 0
$$271$$ −2.06767e6 −1.71024 −0.855122 0.518427i $$-0.826518\pi$$
−0.855122 + 0.518427i $$0.826518\pi$$
$$272$$ −212441. −0.174107
$$273$$ 0 0
$$274$$ 3.92288e6 3.15666
$$275$$ −136938. −0.109193
$$276$$ 0 0
$$277$$ 1.80297e6 1.41185 0.705927 0.708284i $$-0.250530\pi$$
0.705927 + 0.708284i $$0.250530\pi$$
$$278$$ −3.37644e6 −2.62028
$$279$$ 0 0
$$280$$ −618286. −0.471296
$$281$$ −1.05239e6 −0.795081 −0.397540 0.917585i $$-0.630136\pi$$
−0.397540 + 0.917585i $$0.630136\pi$$
$$282$$ 0 0
$$283$$ −1.44912e6 −1.07557 −0.537784 0.843082i $$-0.680739\pi$$
−0.537784 + 0.843082i $$0.680739\pi$$
$$284$$ −4.48430e6 −3.29912
$$285$$ 0 0
$$286$$ −529215. −0.382576
$$287$$ 593787. 0.425526
$$288$$ 0 0
$$289$$ −1.35924e6 −0.957310
$$290$$ −1.26208e6 −0.881234
$$291$$ 0 0
$$292$$ −3.79121e6 −2.60208
$$293$$ −1.44959e6 −0.986450 −0.493225 0.869902i $$-0.664182\pi$$
−0.493225 + 0.869902i $$0.664182\pi$$
$$294$$ 0 0
$$295$$ −95245.7 −0.0637221
$$296$$ 358444. 0.237789
$$297$$ 0 0
$$298$$ 981925. 0.640527
$$299$$ 131585. 0.0851195
$$300$$ 0 0
$$301$$ 751251. 0.477935
$$302$$ −153832. −0.0970574
$$303$$ 0 0
$$304$$ 27475.3 0.0170514
$$305$$ 782252. 0.481501
$$306$$ 0 0
$$307$$ 2.03959e6 1.23508 0.617542 0.786538i $$-0.288129\pi$$
0.617542 + 0.786538i $$0.288129\pi$$
$$308$$ −1.14754e6 −0.689274
$$309$$ 0 0
$$310$$ 588940. 0.348070
$$311$$ −650243. −0.381219 −0.190610 0.981666i $$-0.561046\pi$$
−0.190610 + 0.981666i $$0.561046\pi$$
$$312$$ 0 0
$$313$$ −716801. −0.413559 −0.206780 0.978388i $$-0.566298\pi$$
−0.206780 + 0.978388i $$0.566298\pi$$
$$314$$ −2.12689e6 −1.21736
$$315$$ 0 0
$$316$$ 3.31111e6 1.86533
$$317$$ 869542. 0.486007 0.243004 0.970025i $$-0.421867\pi$$
0.243004 + 0.970025i $$0.421867\pi$$
$$318$$ 0 0
$$319$$ −1.13908e6 −0.626728
$$320$$ −941210. −0.513821
$$321$$ 0 0
$$322$$ 431904. 0.232139
$$323$$ −7839.30 −0.00418091
$$324$$ 0 0
$$325$$ −155464. −0.0816437
$$326$$ −5.25810e6 −2.74022
$$327$$ 0 0
$$328$$ −2.07725e6 −1.06612
$$329$$ −862897. −0.439510
$$330$$ 0 0
$$331$$ −1.68564e6 −0.845659 −0.422829 0.906209i $$-0.638963\pi$$
−0.422829 + 0.906209i $$0.638963\pi$$
$$332$$ 3.53646e6 1.76086
$$333$$ 0 0
$$334$$ 3.49618e6 1.71486
$$335$$ 192408. 0.0936725
$$336$$ 0 0
$$337$$ −1.64864e6 −0.790771 −0.395385 0.918515i $$-0.629389\pi$$
−0.395385 + 0.918515i $$0.629389\pi$$
$$338$$ 3.00458e6 1.43051
$$339$$ 0 0
$$340$$ −383401. −0.179869
$$341$$ 531546. 0.247545
$$342$$ 0 0
$$343$$ −2.23187e6 −1.02432
$$344$$ −2.62811e6 −1.19742
$$345$$ 0 0
$$346$$ −2.66485e6 −1.19669
$$347$$ −1.65489e6 −0.737810 −0.368905 0.929467i $$-0.620267\pi$$
−0.368905 + 0.929467i $$0.620267\pi$$
$$348$$ 0 0
$$349$$ −2.77967e6 −1.22160 −0.610801 0.791784i $$-0.709153\pi$$
−0.610801 + 0.791784i $$0.709153\pi$$
$$350$$ −510283. −0.222659
$$351$$ 0 0
$$352$$ −226452. −0.0974135
$$353$$ 2.23694e6 0.955471 0.477735 0.878504i $$-0.341458\pi$$
0.477735 + 0.878504i $$0.341458\pi$$
$$354$$ 0 0
$$355$$ −1.79973e6 −0.757942
$$356$$ −7.23057e6 −3.02376
$$357$$ 0 0
$$358$$ 2.54225e6 1.04836
$$359$$ −1.93471e6 −0.792282 −0.396141 0.918190i $$-0.629651\pi$$
−0.396141 + 0.918190i $$0.629651\pi$$
$$360$$ 0 0
$$361$$ −2.47509e6 −0.999591
$$362$$ −3.48805e6 −1.39898
$$363$$ 0 0
$$364$$ −1.30279e6 −0.515372
$$365$$ −1.52157e6 −0.597804
$$366$$ 0 0
$$367$$ −3.42337e6 −1.32675 −0.663374 0.748288i $$-0.730876\pi$$
−0.663374 + 0.748288i $$0.730876\pi$$
$$368$$ −456466. −0.175707
$$369$$ 0 0
$$370$$ 295831. 0.112341
$$371$$ 917439. 0.346053
$$372$$ 0 0
$$373$$ 1.54365e6 0.574483 0.287242 0.957858i $$-0.407262\pi$$
0.287242 + 0.957858i $$0.407262\pi$$
$$374$$ −523803. −0.193637
$$375$$ 0 0
$$376$$ 3.01868e6 1.10115
$$377$$ −1.29319e6 −0.468606
$$378$$ 0 0
$$379$$ 517292. 0.184985 0.0924927 0.995713i $$-0.470517\pi$$
0.0924927 + 0.995713i $$0.470517\pi$$
$$380$$ 49585.9 0.0176157
$$381$$ 0 0
$$382$$ 6.22123e6 2.18131
$$383$$ 1.12499e6 0.391877 0.195939 0.980616i $$-0.437225\pi$$
0.195939 + 0.980616i $$0.437225\pi$$
$$384$$ 0 0
$$385$$ −460555. −0.158354
$$386$$ −965012. −0.329659
$$387$$ 0 0
$$388$$ −6.00413e6 −2.02475
$$389$$ −1.85887e6 −0.622838 −0.311419 0.950273i $$-0.600804\pi$$
−0.311419 + 0.950273i $$0.600804\pi$$
$$390$$ 0 0
$$391$$ 130239. 0.0430824
$$392$$ 2.86417e6 0.941422
$$393$$ 0 0
$$394$$ −3.67485e6 −1.19261
$$395$$ 1.32888e6 0.428542
$$396$$ 0 0
$$397$$ 3.27369e6 1.04246 0.521232 0.853415i $$-0.325473\pi$$
0.521232 + 0.853415i $$0.325473\pi$$
$$398$$ −7.41694e6 −2.34702
$$399$$ 0 0
$$400$$ 539302. 0.168532
$$401$$ 92661.9 0.0287766 0.0143883 0.999896i $$-0.495420\pi$$
0.0143883 + 0.999896i $$0.495420\pi$$
$$402$$ 0 0
$$403$$ 603457. 0.185090
$$404$$ 1.16175e6 0.354127
$$405$$ 0 0
$$406$$ −4.24465e6 −1.27799
$$407$$ 267001. 0.0798964
$$408$$ 0 0
$$409$$ −4.09923e6 −1.21170 −0.605849 0.795580i $$-0.707166\pi$$
−0.605849 + 0.795580i $$0.707166\pi$$
$$410$$ −1.71440e6 −0.503676
$$411$$ 0 0
$$412$$ −3.56387e6 −1.03438
$$413$$ −320332. −0.0924114
$$414$$ 0 0
$$415$$ 1.41933e6 0.404540
$$416$$ −257088. −0.0728364
$$417$$ 0 0
$$418$$ 67744.2 0.0189641
$$419$$ 6.11636e6 1.70199 0.850996 0.525171i $$-0.175999\pi$$
0.850996 + 0.525171i $$0.175999\pi$$
$$420$$ 0 0
$$421$$ −3.46820e6 −0.953671 −0.476836 0.878992i $$-0.658216\pi$$
−0.476836 + 0.878992i $$0.658216\pi$$
$$422$$ −2.48171e6 −0.678374
$$423$$ 0 0
$$424$$ −3.20948e6 −0.867003
$$425$$ −153874. −0.0413232
$$426$$ 0 0
$$427$$ 2.63089e6 0.698284
$$428$$ −6.82535e6 −1.80101
$$429$$ 0 0
$$430$$ −2.16903e6 −0.565710
$$431$$ 2.18327e6 0.566127 0.283064 0.959101i $$-0.408649\pi$$
0.283064 + 0.959101i $$0.408649\pi$$
$$432$$ 0 0
$$433$$ 4.15213e6 1.06427 0.532135 0.846660i $$-0.321390\pi$$
0.532135 + 0.846660i $$0.321390\pi$$
$$434$$ 1.98074e6 0.504780
$$435$$ 0 0
$$436$$ −3.82019e6 −0.962430
$$437$$ −16844.0 −0.00421932
$$438$$ 0 0
$$439$$ −3.98317e6 −0.986432 −0.493216 0.869907i $$-0.664179\pi$$
−0.493216 + 0.869907i $$0.664179\pi$$
$$440$$ 1.61116e6 0.396741
$$441$$ 0 0
$$442$$ −594666. −0.144783
$$443$$ −5.08171e6 −1.23027 −0.615135 0.788422i $$-0.710898\pi$$
−0.615135 + 0.788422i $$0.710898\pi$$
$$444$$ 0 0
$$445$$ −2.90192e6 −0.694681
$$446$$ 4.36089e6 1.03810
$$447$$ 0 0
$$448$$ −3.16550e6 −0.745156
$$449$$ −6.32873e6 −1.48150 −0.740749 0.671782i $$-0.765529\pi$$
−0.740749 + 0.671782i $$0.765529\pi$$
$$450$$ 0 0
$$451$$ −1.54732e6 −0.358211
$$452$$ 1.54459e7 3.55604
$$453$$ 0 0
$$454$$ −3.69994e6 −0.842472
$$455$$ −522862. −0.118402
$$456$$ 0 0
$$457$$ 2.82211e6 0.632097 0.316048 0.948743i $$-0.397644\pi$$
0.316048 + 0.948743i $$0.397644\pi$$
$$458$$ 1.17024e7 2.60682
$$459$$ 0 0
$$460$$ −823802. −0.181522
$$461$$ −76172.5 −0.0166935 −0.00834673 0.999965i $$-0.502657\pi$$
−0.00834673 + 0.999965i $$0.502657\pi$$
$$462$$ 0 0
$$463$$ −8.27628e6 −1.79425 −0.897125 0.441777i $$-0.854348\pi$$
−0.897125 + 0.441777i $$0.854348\pi$$
$$464$$ 4.48604e6 0.967315
$$465$$ 0 0
$$466$$ −7.69565e6 −1.64165
$$467$$ 6.55871e6 1.39164 0.695819 0.718217i $$-0.255042\pi$$
0.695819 + 0.718217i $$0.255042\pi$$
$$468$$ 0 0
$$469$$ 647112. 0.135846
$$470$$ 2.49138e6 0.520229
$$471$$ 0 0
$$472$$ 1.12062e6 0.231528
$$473$$ −1.95765e6 −0.402330
$$474$$ 0 0
$$475$$ 19900.8 0.00404703
$$476$$ −1.28946e6 −0.260851
$$477$$ 0 0
$$478$$ −1.80228e6 −0.360788
$$479$$ −6.69118e6 −1.33249 −0.666245 0.745733i $$-0.732099\pi$$
−0.666245 + 0.745733i $$0.732099\pi$$
$$480$$ 0 0
$$481$$ 303123. 0.0597387
$$482$$ 4.96187e6 0.972810
$$483$$ 0 0
$$484$$ −7.04175e6 −1.36637
$$485$$ −2.40970e6 −0.465167
$$486$$ 0 0
$$487$$ 3.76173e6 0.718729 0.359364 0.933197i $$-0.382994\pi$$
0.359364 + 0.933197i $$0.382994\pi$$
$$488$$ −9.20365e6 −1.74949
$$489$$ 0 0
$$490$$ 2.36386e6 0.444766
$$491$$ 2.46265e6 0.460997 0.230499 0.973073i $$-0.425964\pi$$
0.230499 + 0.973073i $$0.425964\pi$$
$$492$$ 0 0
$$493$$ −1.27996e6 −0.237181
$$494$$ 76909.1 0.0141795
$$495$$ 0 0
$$496$$ −2.09338e6 −0.382070
$$497$$ −6.05289e6 −1.09919
$$498$$ 0 0
$$499$$ 2.03788e6 0.366375 0.183188 0.983078i $$-0.441358\pi$$
0.183188 + 0.983078i $$0.441358\pi$$
$$500$$ 973301. 0.174109
$$501$$ 0 0
$$502$$ 6.32771e6 1.12069
$$503$$ −4.22596e6 −0.744742 −0.372371 0.928084i $$-0.621455\pi$$
−0.372371 + 0.928084i $$0.621455\pi$$
$$504$$ 0 0
$$505$$ 466256. 0.0813572
$$506$$ −1.12548e6 −0.195416
$$507$$ 0 0
$$508$$ 1.88264e7 3.23674
$$509$$ 3.53097e6 0.604087 0.302044 0.953294i $$-0.402331\pi$$
0.302044 + 0.953294i $$0.402331\pi$$
$$510$$ 0 0
$$511$$ −5.11736e6 −0.866950
$$512$$ 9.01370e6 1.51960
$$513$$ 0 0
$$514$$ 7.79986e6 1.30220
$$515$$ −1.43032e6 −0.237638
$$516$$ 0 0
$$517$$ 2.24858e6 0.369984
$$518$$ 994944. 0.162920
$$519$$ 0 0
$$520$$ 1.82913e6 0.296645
$$521$$ 6.66348e6 1.07549 0.537746 0.843107i $$-0.319276\pi$$
0.537746 + 0.843107i $$0.319276\pi$$
$$522$$ 0 0
$$523$$ −6.77352e6 −1.08283 −0.541415 0.840755i $$-0.682111\pi$$
−0.541415 + 0.840755i $$0.682111\pi$$
$$524$$ −6.08305e6 −0.967817
$$525$$ 0 0
$$526$$ −1.83612e6 −0.289359
$$527$$ 597285. 0.0936818
$$528$$ 0 0
$$529$$ 279841. 0.0434783
$$530$$ −2.64885e6 −0.409607
$$531$$ 0 0
$$532$$ 166768. 0.0255467
$$533$$ −1.75665e6 −0.267836
$$534$$ 0 0
$$535$$ −2.73929e6 −0.413765
$$536$$ −2.26380e6 −0.340350
$$537$$ 0 0
$$538$$ −8.72284e6 −1.29928
$$539$$ 2.13349e6 0.316315
$$540$$ 0 0
$$541$$ 5.07069e6 0.744859 0.372429 0.928061i $$-0.378525\pi$$
0.372429 + 0.928061i $$0.378525\pi$$
$$542$$ 2.00778e7 2.93575
$$543$$ 0 0
$$544$$ −254458. −0.0368655
$$545$$ −1.53320e6 −0.221109
$$546$$ 0 0
$$547$$ −1.25719e7 −1.79652 −0.898259 0.439466i $$-0.855168\pi$$
−0.898259 + 0.439466i $$0.855168\pi$$
$$548$$ −2.51650e7 −3.57969
$$549$$ 0 0
$$550$$ 1.32972e6 0.187437
$$551$$ 165539. 0.0232286
$$552$$ 0 0
$$553$$ 4.46933e6 0.621483
$$554$$ −1.75075e7 −2.42354
$$555$$ 0 0
$$556$$ 2.16596e7 2.97142
$$557$$ 1.21618e7 1.66097 0.830484 0.557042i $$-0.188064\pi$$
0.830484 + 0.557042i $$0.188064\pi$$
$$558$$ 0 0
$$559$$ −2.22249e6 −0.300823
$$560$$ 1.81379e6 0.244409
$$561$$ 0 0
$$562$$ 1.02191e7 1.36481
$$563$$ 4.47399e6 0.594873 0.297436 0.954742i $$-0.403868\pi$$
0.297436 + 0.954742i $$0.403868\pi$$
$$564$$ 0 0
$$565$$ 6.19905e6 0.816967
$$566$$ 1.40715e7 1.84629
$$567$$ 0 0
$$568$$ 2.11749e7 2.75391
$$569$$ 1.20581e7 1.56135 0.780674 0.624938i $$-0.214876\pi$$
0.780674 + 0.624938i $$0.214876\pi$$
$$570$$ 0 0
$$571$$ −1.18705e7 −1.52363 −0.761814 0.647796i $$-0.775691\pi$$
−0.761814 + 0.647796i $$0.775691\pi$$
$$572$$ 3.39488e6 0.433844
$$573$$ 0 0
$$574$$ −5.76589e6 −0.730444
$$575$$ −330625. −0.0417029
$$576$$ 0 0
$$577$$ 3.10897e6 0.388756 0.194378 0.980927i $$-0.437731\pi$$
0.194378 + 0.980927i $$0.437731\pi$$
$$578$$ 1.31988e7 1.64329
$$579$$ 0 0
$$580$$ 8.09614e6 0.999327
$$581$$ 4.77351e6 0.586674
$$582$$ 0 0
$$583$$ −2.39071e6 −0.291310
$$584$$ 1.79021e7 2.17206
$$585$$ 0 0
$$586$$ 1.40760e7 1.69331
$$587$$ 7.66593e6 0.918269 0.459134 0.888367i $$-0.348160\pi$$
0.459134 + 0.888367i $$0.348160\pi$$
$$588$$ 0 0
$$589$$ −77247.8 −0.00917483
$$590$$ 924871. 0.109383
$$591$$ 0 0
$$592$$ −1.05153e6 −0.123315
$$593$$ 1.04472e6 0.122001 0.0610004 0.998138i $$-0.480571\pi$$
0.0610004 + 0.998138i $$0.480571\pi$$
$$594$$ 0 0
$$595$$ −517514. −0.0599280
$$596$$ −6.29897e6 −0.726364
$$597$$ 0 0
$$598$$ −1.27774e6 −0.146113
$$599$$ 4.05765e6 0.462070 0.231035 0.972945i $$-0.425789\pi$$
0.231035 + 0.972945i $$0.425789\pi$$
$$600$$ 0 0
$$601$$ −8.30225e6 −0.937582 −0.468791 0.883309i $$-0.655310\pi$$
−0.468791 + 0.883309i $$0.655310\pi$$
$$602$$ −7.29492e6 −0.820407
$$603$$ 0 0
$$604$$ 986820. 0.110064
$$605$$ −2.82614e6 −0.313910
$$606$$ 0 0
$$607$$ 7.44097e6 0.819705 0.409853 0.912152i $$-0.365580\pi$$
0.409853 + 0.912152i $$0.365580\pi$$
$$608$$ 32909.5 0.00361046
$$609$$ 0 0
$$610$$ −7.59595e6 −0.826528
$$611$$ 2.55279e6 0.276638
$$612$$ 0 0
$$613$$ 7.25848e6 0.780179 0.390090 0.920777i $$-0.372444\pi$$
0.390090 + 0.920777i $$0.372444\pi$$
$$614$$ −1.98052e7 −2.12010
$$615$$ 0 0
$$616$$ 5.41869e6 0.575364
$$617$$ −3.29613e6 −0.348571 −0.174286 0.984695i $$-0.555762\pi$$
−0.174286 + 0.984695i $$0.555762\pi$$
$$618$$ 0 0
$$619$$ −3.57956e6 −0.375494 −0.187747 0.982217i $$-0.560119\pi$$
−0.187747 + 0.982217i $$0.560119\pi$$
$$620$$ −3.77801e6 −0.394715
$$621$$ 0 0
$$622$$ 6.31410e6 0.654388
$$623$$ −9.75980e6 −1.00744
$$624$$ 0 0
$$625$$ 390625. 0.0400000
$$626$$ 6.96040e6 0.709902
$$627$$ 0 0
$$628$$ 1.36438e7 1.38050
$$629$$ 300023. 0.0302362
$$630$$ 0 0
$$631$$ 1.45987e7 1.45963 0.729814 0.683646i $$-0.239607\pi$$
0.729814 + 0.683646i $$0.239607\pi$$
$$632$$ −1.56351e7 −1.55707
$$633$$ 0 0
$$634$$ −8.44358e6 −0.834264
$$635$$ 7.55578e6 0.743609
$$636$$ 0 0
$$637$$ 2.42213e6 0.236509
$$638$$ 1.10609e7 1.07582
$$639$$ 0 0
$$640$$ 8.31266e6 0.802214
$$641$$ −1.63500e7 −1.57171 −0.785857 0.618409i $$-0.787778\pi$$
−0.785857 + 0.618409i $$0.787778\pi$$
$$642$$ 0 0
$$643$$ 3.67043e6 0.350098 0.175049 0.984560i $$-0.443992\pi$$
0.175049 + 0.984560i $$0.443992\pi$$
$$644$$ −2.77063e6 −0.263247
$$645$$ 0 0
$$646$$ 76122.5 0.00717681
$$647$$ −972667. −0.0913490 −0.0456745 0.998956i $$-0.514544\pi$$
−0.0456745 + 0.998956i $$0.514544\pi$$
$$648$$ 0 0
$$649$$ 834739. 0.0777928
$$650$$ 1.50962e6 0.140147
$$651$$ 0 0
$$652$$ 3.37303e7 3.10743
$$653$$ −1.36330e7 −1.25115 −0.625575 0.780164i $$-0.715136\pi$$
−0.625575 + 0.780164i $$0.715136\pi$$
$$654$$ 0 0
$$655$$ −2.44137e6 −0.222347
$$656$$ 6.09379e6 0.552876
$$657$$ 0 0
$$658$$ 8.37905e6 0.754449
$$659$$ 6.41148e6 0.575102 0.287551 0.957765i $$-0.407159\pi$$
0.287551 + 0.957765i $$0.407159\pi$$
$$660$$ 0 0
$$661$$ −1.50140e7 −1.33657 −0.668285 0.743905i $$-0.732971\pi$$
−0.668285 + 0.743905i $$0.732971\pi$$
$$662$$ 1.63682e7 1.45163
$$663$$ 0 0
$$664$$ −1.66992e7 −1.46986
$$665$$ 66930.8 0.00586911
$$666$$ 0 0
$$667$$ −2.75021e6 −0.239360
$$668$$ −2.24277e7 −1.94466
$$669$$ 0 0
$$670$$ −1.86836e6 −0.160795
$$671$$ −6.85570e6 −0.587822
$$672$$ 0 0
$$673$$ −4.45498e6 −0.379148 −0.189574 0.981866i $$-0.560711\pi$$
−0.189574 + 0.981866i $$0.560711\pi$$
$$674$$ 1.60089e7 1.35741
$$675$$ 0 0
$$676$$ −1.92742e7 −1.62222
$$677$$ −2.22147e7 −1.86281 −0.931406 0.363981i $$-0.881417\pi$$
−0.931406 + 0.363981i $$0.881417\pi$$
$$678$$ 0 0
$$679$$ −8.10435e6 −0.674596
$$680$$ 1.81042e6 0.150144
$$681$$ 0 0
$$682$$ −5.16151e6 −0.424928
$$683$$ −1.84077e7 −1.50990 −0.754948 0.655785i $$-0.772338\pi$$
−0.754948 + 0.655785i $$0.772338\pi$$
$$684$$ 0 0
$$685$$ −1.00997e7 −0.822399
$$686$$ 2.16723e7 1.75831
$$687$$ 0 0
$$688$$ 7.70978e6 0.620970
$$689$$ −2.71414e6 −0.217813
$$690$$ 0 0
$$691$$ −1.19728e6 −0.0953894 −0.0476947 0.998862i $$-0.515187\pi$$
−0.0476947 + 0.998862i $$0.515187\pi$$
$$692$$ 1.70948e7 1.35706
$$693$$ 0 0
$$694$$ 1.60696e7 1.26650
$$695$$ 8.69288e6 0.682655
$$696$$ 0 0
$$697$$ −1.73869e6 −0.135563
$$698$$ 2.69917e7 2.09696
$$699$$ 0 0
$$700$$ 3.27343e6 0.252498
$$701$$ −1.74390e7 −1.34038 −0.670188 0.742191i $$-0.733787\pi$$
−0.670188 + 0.742191i $$0.733787\pi$$
$$702$$ 0 0
$$703$$ −38802.4 −0.00296121
$$704$$ 8.24883e6 0.627279
$$705$$ 0 0
$$706$$ −2.17215e7 −1.64013
$$707$$ 1.56812e6 0.117986
$$708$$ 0 0
$$709$$ −1.84583e7 −1.37904 −0.689520 0.724267i $$-0.742178\pi$$
−0.689520 + 0.724267i $$0.742178\pi$$
$$710$$ 1.74760e7 1.30106
$$711$$ 0 0
$$712$$ 3.41428e7 2.52406
$$713$$ 1.28337e6 0.0945425
$$714$$ 0 0
$$715$$ 1.36250e6 0.0996716
$$716$$ −1.63083e7 −1.18885
$$717$$ 0 0
$$718$$ 1.87868e7 1.36001
$$719$$ −2.12129e7 −1.53030 −0.765152 0.643849i $$-0.777336\pi$$
−0.765152 + 0.643849i $$0.777336\pi$$
$$720$$ 0 0
$$721$$ −4.81050e6 −0.344629
$$722$$ 2.40340e7 1.71586
$$723$$ 0 0
$$724$$ 2.23756e7 1.58646
$$725$$ 3.24930e6 0.229586
$$726$$ 0 0
$$727$$ 1.69692e7 1.19077 0.595383 0.803442i $$-0.297000\pi$$
0.595383 + 0.803442i $$0.297000\pi$$
$$728$$ 6.15177e6 0.430201
$$729$$ 0 0
$$730$$ 1.47750e7 1.02617
$$731$$ −2.19976e6 −0.152259
$$732$$ 0 0
$$733$$ 903452. 0.0621077 0.0310538 0.999518i $$-0.490114\pi$$
0.0310538 + 0.999518i $$0.490114\pi$$
$$734$$ 3.32422e7 2.27745
$$735$$ 0 0
$$736$$ −546747. −0.0372042
$$737$$ −1.68628e6 −0.114357
$$738$$ 0 0
$$739$$ 386739. 0.0260499 0.0130250 0.999915i $$-0.495854\pi$$
0.0130250 + 0.999915i $$0.495854\pi$$
$$740$$ −1.89773e6 −0.127396
$$741$$ 0 0
$$742$$ −8.90867e6 −0.594023
$$743$$ −1.90773e7 −1.26779 −0.633893 0.773421i $$-0.718544\pi$$
−0.633893 + 0.773421i $$0.718544\pi$$
$$744$$ 0 0
$$745$$ −2.52803e6 −0.166875
$$746$$ −1.49894e7 −0.986139
$$747$$ 0 0
$$748$$ 3.36016e6 0.219586
$$749$$ −9.21284e6 −0.600052
$$750$$ 0 0
$$751$$ 1.92271e7 1.24398 0.621990 0.783025i $$-0.286324\pi$$
0.621990 + 0.783025i $$0.286324\pi$$
$$752$$ −8.85555e6 −0.571046
$$753$$ 0 0
$$754$$ 1.25573e7 0.804394
$$755$$ 396050. 0.0252862
$$756$$ 0 0
$$757$$ 5.23395e6 0.331963 0.165982 0.986129i $$-0.446921\pi$$
0.165982 + 0.986129i $$0.446921\pi$$
$$758$$ −5.02309e6 −0.317540
$$759$$ 0 0
$$760$$ −234145. −0.0147045
$$761$$ 1.96724e7 1.23139 0.615694 0.787985i $$-0.288876\pi$$
0.615694 + 0.787985i $$0.288876\pi$$
$$762$$ 0 0
$$763$$ −5.15648e6 −0.320658
$$764$$ −3.99087e7 −2.47363
$$765$$ 0 0
$$766$$ −1.09240e7 −0.672683
$$767$$ 947669. 0.0581659
$$768$$ 0 0
$$769$$ −2.05543e7 −1.25339 −0.626695 0.779264i $$-0.715593\pi$$
−0.626695 + 0.779264i $$0.715593\pi$$
$$770$$ 4.47215e6 0.271825
$$771$$ 0 0
$$772$$ 6.19048e6 0.373836
$$773$$ −2.02610e7 −1.21959 −0.609793 0.792561i $$-0.708747\pi$$
−0.609793 + 0.792561i $$0.708747\pi$$
$$774$$ 0 0
$$775$$ −1.51627e6 −0.0906820
$$776$$ 2.83515e7 1.69014
$$777$$ 0 0
$$778$$ 1.80503e7 1.06914
$$779$$ 224867. 0.0132765
$$780$$ 0 0
$$781$$ 1.57729e7 0.925305
$$782$$ −1.26467e6 −0.0739539
$$783$$ 0 0
$$784$$ −8.40230e6 −0.488211
$$785$$ 5.47581e6 0.317157
$$786$$ 0 0
$$787$$ −2.01612e7 −1.16032 −0.580162 0.814501i $$-0.697011\pi$$
−0.580162 + 0.814501i $$0.697011\pi$$
$$788$$ 2.35739e7 1.35243
$$789$$ 0 0
$$790$$ −1.29039e7 −0.735622
$$791$$ 2.08488e7 1.18479
$$792$$ 0 0
$$793$$ −7.78319e6 −0.439516
$$794$$ −3.17887e7 −1.78946
$$795$$ 0 0
$$796$$ 4.75791e7 2.66154
$$797$$ 9.93780e6 0.554172 0.277086 0.960845i $$-0.410631\pi$$
0.277086 + 0.960845i $$0.410631\pi$$
$$798$$ 0 0
$$799$$ 2.52668e6 0.140018
$$800$$ 645967. 0.0356850
$$801$$ 0 0
$$802$$ −899781. −0.0493970
$$803$$ 1.33351e7 0.729806
$$804$$ 0 0
$$805$$ −1.11197e6 −0.0604786
$$806$$ −5.85979e6 −0.317720
$$807$$ 0 0
$$808$$ −5.48578e6 −0.295604
$$809$$ −3.48213e7 −1.87057 −0.935284 0.353898i $$-0.884856\pi$$
−0.935284 + 0.353898i $$0.884856\pi$$
$$810$$ 0 0
$$811$$ 7.77806e6 0.415259 0.207630 0.978208i $$-0.433425\pi$$
0.207630 + 0.978208i $$0.433425\pi$$
$$812$$ 2.72291e7 1.44925
$$813$$ 0 0
$$814$$ −2.59268e6 −0.137147
$$815$$ 1.35373e7 0.713903
$$816$$ 0 0
$$817$$ 284499. 0.0149116
$$818$$ 3.98051e7 2.07996
$$819$$ 0 0
$$820$$ 1.09977e7 0.571173
$$821$$ −2.03211e7 −1.05218 −0.526089 0.850430i $$-0.676342\pi$$
−0.526089 + 0.850430i $$0.676342\pi$$
$$822$$ 0 0
$$823$$ 2.14456e7 1.10367 0.551834 0.833954i $$-0.313928\pi$$
0.551834 + 0.833954i $$0.313928\pi$$
$$824$$ 1.68286e7 0.863436
$$825$$ 0 0
$$826$$ 3.11055e6 0.158630
$$827$$ −2.88309e7 −1.46587 −0.732934 0.680300i $$-0.761850\pi$$
−0.732934 + 0.680300i $$0.761850\pi$$
$$828$$ 0 0
$$829$$ −5.70528e6 −0.288330 −0.144165 0.989554i $$-0.546050\pi$$
−0.144165 + 0.989554i $$0.546050\pi$$
$$830$$ −1.37822e7 −0.694421
$$831$$ 0 0
$$832$$ 9.36479e6 0.469018
$$833$$ 2.39735e6 0.119707
$$834$$ 0 0
$$835$$ −9.00115e6 −0.446768
$$836$$ −434574. −0.0215054
$$837$$ 0 0
$$838$$ −5.93921e7 −2.92159
$$839$$ 952844. 0.0467323 0.0233661 0.999727i $$-0.492562\pi$$
0.0233661 + 0.999727i $$0.492562\pi$$
$$840$$ 0 0
$$841$$ 6.51728e6 0.317743
$$842$$ 3.36775e7 1.63704
$$843$$ 0 0
$$844$$ 1.59200e7 0.769282
$$845$$ −7.73550e6 −0.372689
$$846$$ 0 0
$$847$$ −9.50493e6 −0.455240
$$848$$ 9.41529e6 0.449619
$$849$$ 0 0
$$850$$ 1.49418e6 0.0709341
$$851$$ 644649. 0.0305140
$$852$$ 0 0
$$853$$ −4.02321e7 −1.89321 −0.946607 0.322389i $$-0.895514\pi$$
−0.946607 + 0.322389i $$0.895514\pi$$
$$854$$ −2.55469e7 −1.19865
$$855$$ 0 0
$$856$$ 3.22293e7 1.50337
$$857$$ 2.86224e7 1.33123 0.665617 0.746294i $$-0.268169\pi$$
0.665617 + 0.746294i $$0.268169\pi$$
$$858$$ 0 0
$$859$$ −3.61703e7 −1.67251 −0.836256 0.548339i $$-0.815260\pi$$
−0.836256 + 0.548339i $$0.815260\pi$$
$$860$$ 1.39142e7 0.641521
$$861$$ 0 0
$$862$$ −2.12003e7 −0.971795
$$863$$ −6.70416e6 −0.306420 −0.153210 0.988194i $$-0.548961\pi$$
−0.153210 + 0.988194i $$0.548961\pi$$
$$864$$ 0 0
$$865$$ 6.86083e6 0.311772
$$866$$ −4.03188e7 −1.82689
$$867$$ 0 0
$$868$$ −1.27063e7 −0.572425
$$869$$ −1.16464e7 −0.523170
$$870$$ 0 0
$$871$$ −1.91441e6 −0.0855047
$$872$$ 1.80390e7 0.803379
$$873$$ 0 0
$$874$$ 163562. 0.00724275
$$875$$ 1.31376e6 0.0580090
$$876$$ 0 0
$$877$$ 2.69344e7 1.18252 0.591261 0.806480i $$-0.298630\pi$$
0.591261 + 0.806480i $$0.298630\pi$$
$$878$$ 3.86780e7 1.69328
$$879$$ 0 0
$$880$$ −4.72648e6 −0.205746
$$881$$ −8.29228e6 −0.359943 −0.179972 0.983672i $$-0.557601\pi$$
−0.179972 + 0.983672i $$0.557601\pi$$
$$882$$ 0 0
$$883$$ 2.48054e7 1.07064 0.535322 0.844648i $$-0.320190\pi$$
0.535322 + 0.844648i $$0.320190\pi$$
$$884$$ 3.81474e6 0.164185
$$885$$ 0 0
$$886$$ 4.93452e7 2.11184
$$887$$ −3.33079e6 −0.142147 −0.0710736 0.997471i $$-0.522643\pi$$
−0.0710736 + 0.997471i $$0.522643\pi$$
$$888$$ 0 0
$$889$$ 2.54118e7 1.07840
$$890$$ 2.81787e7 1.19247
$$891$$ 0 0
$$892$$ −2.79748e7 −1.17721
$$893$$ −326779. −0.0137128
$$894$$ 0 0
$$895$$ −6.54518e6 −0.273127
$$896$$ 2.79573e7 1.16339
$$897$$ 0 0
$$898$$ 6.14543e7 2.54309
$$899$$ −1.26126e7 −0.520483
$$900$$ 0 0
$$901$$ −2.68638e6 −0.110244
$$902$$ 1.50251e7 0.614894
$$903$$ 0 0
$$904$$ −7.29355e7 −2.96837
$$905$$ 8.98022e6 0.364473
$$906$$ 0 0
$$907$$ −3.87276e7 −1.56316 −0.781578 0.623807i $$-0.785585\pi$$
−0.781578 + 0.623807i $$0.785585\pi$$
$$908$$ 2.37348e7 0.955371
$$909$$ 0 0
$$910$$ 5.07718e6 0.203245
$$911$$ −1.56365e7 −0.624229 −0.312114 0.950045i $$-0.601037\pi$$
−0.312114 + 0.950045i $$0.601037\pi$$
$$912$$ 0 0
$$913$$ −1.24391e7 −0.493868
$$914$$ −2.74037e7 −1.08504
$$915$$ 0 0
$$916$$ −7.50698e7 −2.95615
$$917$$ −8.21088e6 −0.322453
$$918$$ 0 0
$$919$$ 6.11633e6 0.238892 0.119446 0.992841i $$-0.461888\pi$$
0.119446 + 0.992841i $$0.461888\pi$$
$$920$$ 3.89000e6 0.151523
$$921$$ 0 0
$$922$$ 739664. 0.0286554
$$923$$ 1.79068e7 0.691853
$$924$$ 0 0
$$925$$ −761636. −0.0292680
$$926$$ 8.03658e7 3.07995
$$927$$ 0 0
$$928$$ 5.37330e6 0.204819
$$929$$ 1.46888e7 0.558403 0.279202 0.960232i $$-0.409930\pi$$
0.279202 + 0.960232i $$0.409930\pi$$
$$930$$ 0 0
$$931$$ −310053. −0.0117236
$$932$$ 4.93670e7 1.86165
$$933$$ 0 0
$$934$$ −6.36875e7 −2.38884
$$935$$ 1.34857e6 0.0504479
$$936$$ 0 0
$$937$$ 1.85682e7 0.690910 0.345455 0.938435i $$-0.387725\pi$$
0.345455 + 0.938435i $$0.387725\pi$$
$$938$$ −6.28370e6 −0.233189
$$939$$ 0 0
$$940$$ −1.59820e7 −0.589944
$$941$$ 5.08610e7 1.87245 0.936227 0.351397i $$-0.114293\pi$$
0.936227 + 0.351397i $$0.114293\pi$$
$$942$$ 0 0
$$943$$ −3.73586e6 −0.136808
$$944$$ −3.28744e6 −0.120068
$$945$$ 0 0
$$946$$ 1.90095e7 0.690626
$$947$$ 2.18050e7 0.790098 0.395049 0.918660i $$-0.370728\pi$$
0.395049 + 0.918660i $$0.370728\pi$$
$$948$$ 0 0
$$949$$ 1.51392e7 0.545678
$$950$$ −193244. −0.00694700
$$951$$ 0 0
$$952$$ 6.08885e6 0.217743
$$953$$ −2.63434e7 −0.939592 −0.469796 0.882775i $$-0.655673\pi$$
−0.469796 + 0.882775i $$0.655673\pi$$
$$954$$ 0 0
$$955$$ −1.60170e7 −0.568292
$$956$$ 1.15615e7 0.409137
$$957$$ 0 0
$$958$$ 6.49738e7 2.28731
$$959$$ −3.39676e7 −1.19266
$$960$$ 0 0
$$961$$ −2.27436e7 −0.794419
$$962$$ −2.94344e6 −0.102546
$$963$$ 0 0
$$964$$ −3.18301e7 −1.10318
$$965$$ 2.48449e6 0.0858852
$$966$$ 0 0
$$967$$ −6.88107e6 −0.236641 −0.118320 0.992975i $$-0.537751\pi$$
−0.118320 + 0.992975i $$0.537751\pi$$
$$968$$ 3.32512e7 1.14056
$$969$$ 0 0
$$970$$ 2.33991e7 0.798490
$$971$$ 8.39882e6 0.285871 0.142936 0.989732i $$-0.454346\pi$$
0.142936 + 0.989732i $$0.454346\pi$$
$$972$$ 0 0
$$973$$ 2.92361e7 0.990004
$$974$$ −3.65278e7 −1.23375
$$975$$ 0 0
$$976$$ 2.69997e7 0.907265
$$977$$ −3.21445e7 −1.07738 −0.538692 0.842503i $$-0.681081\pi$$
−0.538692 + 0.842503i $$0.681081\pi$$
$$978$$ 0 0
$$979$$ 2.54326e7 0.848075
$$980$$ −1.51640e7 −0.504368
$$981$$ 0 0
$$982$$ −2.39132e7 −0.791333
$$983$$ 1.00759e7 0.332583 0.166292 0.986077i $$-0.446821\pi$$
0.166292 + 0.986077i $$0.446821\pi$$
$$984$$ 0 0
$$985$$ 9.46115e6 0.310709
$$986$$ 1.24289e7 0.407137
$$987$$ 0 0
$$988$$ −493366. −0.0160797
$$989$$ −4.72656e6 −0.153658
$$990$$ 0 0
$$991$$ −1.13729e7 −0.367864 −0.183932 0.982939i $$-0.558883\pi$$
−0.183932 + 0.982939i $$0.558883\pi$$
$$992$$ −2.50741e6 −0.0808996
$$993$$ 0 0
$$994$$ 5.87758e7 1.88683
$$995$$ 1.90954e7 0.611465
$$996$$ 0 0
$$997$$ −2.73242e7 −0.870582 −0.435291 0.900290i $$-0.643354\pi$$
−0.435291 + 0.900290i $$0.643354\pi$$
$$998$$ −1.97885e7 −0.628908
$$999$$ 0 0
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 1035.6.a.b.1.2 7
3.2 odd 2 115.6.a.c.1.6 7
15.14 odd 2 575.6.a.d.1.2 7

By twisted newform
Twist Min Dim Char Parity Ord Type
115.6.a.c.1.6 7 3.2 odd 2
575.6.a.d.1.2 7 15.14 odd 2
1035.6.a.b.1.2 7 1.1 even 1 trivial