# Properties

 Label 112.6.a.j.1.2 Level $112$ Weight $6$ Character 112.1 Self dual yes Analytic conductor $17.963$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [112,6,Mod(1,112)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(112, 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("112.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$112 = 2^{4} \cdot 7$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 112.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: $$17.9629878191$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{193})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 48$$ x^2 - x - 48 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 56) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$-6.44622$$ of defining polynomial Character $$\chi$$ $$=$$ 112.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+20.8924 q^{3} +90.4622 q^{5} +49.0000 q^{7} +193.494 q^{9} +O(q^{10})$$ $$q+20.8924 q^{3} +90.4622 q^{5} +49.0000 q^{7} +193.494 q^{9} +552.494 q^{11} -593.172 q^{13} +1889.98 q^{15} -1422.19 q^{17} -318.997 q^{19} +1023.73 q^{21} -659.954 q^{23} +5058.41 q^{25} -1034.30 q^{27} -8185.27 q^{29} +9598.03 q^{31} +11543.0 q^{33} +4432.65 q^{35} +5180.56 q^{37} -12392.8 q^{39} -2192.46 q^{41} -7458.29 q^{43} +17503.9 q^{45} +19561.8 q^{47} +2401.00 q^{49} -29713.1 q^{51} +36569.6 q^{53} +49979.9 q^{55} -6664.63 q^{57} -16361.7 q^{59} -10893.1 q^{61} +9481.22 q^{63} -53659.6 q^{65} -8035.91 q^{67} -13788.0 q^{69} -55983.3 q^{71} -77752.5 q^{73} +105683. q^{75} +27072.2 q^{77} +3208.43 q^{79} -68628.1 q^{81} -76626.9 q^{83} -128655. q^{85} -171010. q^{87} -84288.6 q^{89} -29065.4 q^{91} +200526. q^{93} -28857.2 q^{95} +101273. q^{97} +106904. q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 14 q^{3} + 42 q^{5} + 98 q^{7} - 2 q^{9}+O(q^{10})$$ 2 * q + 14 * q^3 + 42 * q^5 + 98 * q^7 - 2 * q^9 $$2 q + 14 q^{3} + 42 q^{5} + 98 q^{7} - 2 q^{9} + 716 q^{11} - 714 q^{13} + 2224 q^{15} - 1344 q^{17} + 1946 q^{19} + 686 q^{21} + 1792 q^{23} + 4282 q^{25} + 1988 q^{27} - 1200 q^{29} + 6804 q^{31} + 10416 q^{33} + 2058 q^{35} + 14640 q^{37} - 11560 q^{39} + 7896 q^{41} - 524 q^{43} + 26978 q^{45} + 18396 q^{47} + 4802 q^{49} - 30252 q^{51} + 45132 q^{53} + 42056 q^{55} - 22276 q^{57} - 22582 q^{59} - 52822 q^{61} - 98 q^{63} - 47804 q^{65} - 9848 q^{67} - 30688 q^{69} + 840 q^{71} - 122052 q^{73} + 111034 q^{75} + 35084 q^{77} - 31704 q^{79} - 41954 q^{81} - 36974 q^{83} - 132444 q^{85} - 219156 q^{87} - 210588 q^{89} - 34986 q^{91} + 219784 q^{93} - 138624 q^{95} - 44240 q^{97} + 74940 q^{99}+O(q^{100})$$ 2 * q + 14 * q^3 + 42 * q^5 + 98 * q^7 - 2 * q^9 + 716 * q^11 - 714 * q^13 + 2224 * q^15 - 1344 * q^17 + 1946 * q^19 + 686 * q^21 + 1792 * q^23 + 4282 * q^25 + 1988 * q^27 - 1200 * q^29 + 6804 * q^31 + 10416 * q^33 + 2058 * q^35 + 14640 * q^37 - 11560 * q^39 + 7896 * q^41 - 524 * q^43 + 26978 * q^45 + 18396 * q^47 + 4802 * q^49 - 30252 * q^51 + 45132 * q^53 + 42056 * q^55 - 22276 * q^57 - 22582 * q^59 - 52822 * q^61 - 98 * q^63 - 47804 * q^65 - 9848 * q^67 - 30688 * q^69 + 840 * q^71 - 122052 * q^73 + 111034 * q^75 + 35084 * q^77 - 31704 * q^79 - 41954 * q^81 - 36974 * q^83 - 132444 * q^85 - 219156 * q^87 - 210588 * q^89 - 34986 * q^91 + 219784 * q^93 - 138624 * q^95 - 44240 * q^97 + 74940 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ 20.8924 1.34025 0.670125 0.742248i $$-0.266240\pi$$
0.670125 + 0.742248i $$0.266240\pi$$
$$4$$ 0 0
$$5$$ 90.4622 1.61824 0.809119 0.587645i $$-0.199945\pi$$
0.809119 + 0.587645i $$0.199945\pi$$
$$6$$ 0 0
$$7$$ 49.0000 0.377964
$$8$$ 0 0
$$9$$ 193.494 0.796272
$$10$$ 0 0
$$11$$ 552.494 1.37672 0.688361 0.725369i $$-0.258331\pi$$
0.688361 + 0.725369i $$0.258331\pi$$
$$12$$ 0 0
$$13$$ −593.172 −0.973469 −0.486734 0.873550i $$-0.661812\pi$$
−0.486734 + 0.873550i $$0.661812\pi$$
$$14$$ 0 0
$$15$$ 1889.98 2.16884
$$16$$ 0 0
$$17$$ −1422.19 −1.19354 −0.596769 0.802413i $$-0.703549\pi$$
−0.596769 + 0.802413i $$0.703549\pi$$
$$18$$ 0 0
$$19$$ −318.997 −0.202723 −0.101361 0.994850i $$-0.532320\pi$$
−0.101361 + 0.994850i $$0.532320\pi$$
$$20$$ 0 0
$$21$$ 1023.73 0.506567
$$22$$ 0 0
$$23$$ −659.954 −0.260132 −0.130066 0.991505i $$-0.541519\pi$$
−0.130066 + 0.991505i $$0.541519\pi$$
$$24$$ 0 0
$$25$$ 5058.41 1.61869
$$26$$ 0 0
$$27$$ −1034.30 −0.273046
$$28$$ 0 0
$$29$$ −8185.27 −1.80733 −0.903667 0.428237i $$-0.859135\pi$$
−0.903667 + 0.428237i $$0.859135\pi$$
$$30$$ 0 0
$$31$$ 9598.03 1.79382 0.896908 0.442217i $$-0.145808\pi$$
0.896908 + 0.442217i $$0.145808\pi$$
$$32$$ 0 0
$$33$$ 11543.0 1.84515
$$34$$ 0 0
$$35$$ 4432.65 0.611636
$$36$$ 0 0
$$37$$ 5180.56 0.622118 0.311059 0.950391i $$-0.399316\pi$$
0.311059 + 0.950391i $$0.399316\pi$$
$$38$$ 0 0
$$39$$ −12392.8 −1.30469
$$40$$ 0 0
$$41$$ −2192.46 −0.203691 −0.101846 0.994800i $$-0.532475\pi$$
−0.101846 + 0.994800i $$0.532475\pi$$
$$42$$ 0 0
$$43$$ −7458.29 −0.615131 −0.307566 0.951527i $$-0.599514\pi$$
−0.307566 + 0.951527i $$0.599514\pi$$
$$44$$ 0 0
$$45$$ 17503.9 1.28856
$$46$$ 0 0
$$47$$ 19561.8 1.29171 0.645853 0.763462i $$-0.276502\pi$$
0.645853 + 0.763462i $$0.276502\pi$$
$$48$$ 0 0
$$49$$ 2401.00 0.142857
$$50$$ 0 0
$$51$$ −29713.1 −1.59964
$$52$$ 0 0
$$53$$ 36569.6 1.78826 0.894129 0.447809i $$-0.147795\pi$$
0.894129 + 0.447809i $$0.147795\pi$$
$$54$$ 0 0
$$55$$ 49979.9 2.22786
$$56$$ 0 0
$$57$$ −6664.63 −0.271700
$$58$$ 0 0
$$59$$ −16361.7 −0.611927 −0.305963 0.952043i $$-0.598979\pi$$
−0.305963 + 0.952043i $$0.598979\pi$$
$$60$$ 0 0
$$61$$ −10893.1 −0.374825 −0.187412 0.982281i $$-0.560010\pi$$
−0.187412 + 0.982281i $$0.560010\pi$$
$$62$$ 0 0
$$63$$ 9481.22 0.300963
$$64$$ 0 0
$$65$$ −53659.6 −1.57530
$$66$$ 0 0
$$67$$ −8035.91 −0.218700 −0.109350 0.994003i $$-0.534877\pi$$
−0.109350 + 0.994003i $$0.534877\pi$$
$$68$$ 0 0
$$69$$ −13788.0 −0.348642
$$70$$ 0 0
$$71$$ −55983.3 −1.31799 −0.658996 0.752146i $$-0.729019\pi$$
−0.658996 + 0.752146i $$0.729019\pi$$
$$72$$ 0 0
$$73$$ −77752.5 −1.70768 −0.853841 0.520533i $$-0.825733\pi$$
−0.853841 + 0.520533i $$0.825733\pi$$
$$74$$ 0 0
$$75$$ 105683. 2.16945
$$76$$ 0 0
$$77$$ 27072.2 0.520352
$$78$$ 0 0
$$79$$ 3208.43 0.0578396 0.0289198 0.999582i $$-0.490793\pi$$
0.0289198 + 0.999582i $$0.490793\pi$$
$$80$$ 0 0
$$81$$ −68628.1 −1.16222
$$82$$ 0 0
$$83$$ −76626.9 −1.22092 −0.610458 0.792049i $$-0.709015\pi$$
−0.610458 + 0.792049i $$0.709015\pi$$
$$84$$ 0 0
$$85$$ −128655. −1.93143
$$86$$ 0 0
$$87$$ −171010. −2.42228
$$88$$ 0 0
$$89$$ −84288.6 −1.12796 −0.563980 0.825788i $$-0.690731\pi$$
−0.563980 + 0.825788i $$0.690731\pi$$
$$90$$ 0 0
$$91$$ −29065.4 −0.367937
$$92$$ 0 0
$$93$$ 200526. 2.40416
$$94$$ 0 0
$$95$$ −28857.2 −0.328054
$$96$$ 0 0
$$97$$ 101273. 1.09286 0.546428 0.837506i $$-0.315987\pi$$
0.546428 + 0.837506i $$0.315987\pi$$
$$98$$ 0 0
$$99$$ 106904. 1.09625
$$100$$ 0 0
$$101$$ 77456.5 0.755535 0.377767 0.925901i $$-0.376692\pi$$
0.377767 + 0.925901i $$0.376692\pi$$
$$102$$ 0 0
$$103$$ 56716.9 0.526768 0.263384 0.964691i $$-0.415161\pi$$
0.263384 + 0.964691i $$0.415161\pi$$
$$104$$ 0 0
$$105$$ 92608.9 0.819746
$$106$$ 0 0
$$107$$ 38699.2 0.326770 0.163385 0.986562i $$-0.447759\pi$$
0.163385 + 0.986562i $$0.447759\pi$$
$$108$$ 0 0
$$109$$ −16595.3 −0.133788 −0.0668941 0.997760i $$-0.521309\pi$$
−0.0668941 + 0.997760i $$0.521309\pi$$
$$110$$ 0 0
$$111$$ 108235. 0.833794
$$112$$ 0 0
$$113$$ −76723.9 −0.565242 −0.282621 0.959232i $$-0.591204\pi$$
−0.282621 + 0.959232i $$0.591204\pi$$
$$114$$ 0 0
$$115$$ −59700.9 −0.420955
$$116$$ 0 0
$$117$$ −114775. −0.775146
$$118$$ 0 0
$$119$$ −69687.4 −0.451115
$$120$$ 0 0
$$121$$ 144199. 0.895361
$$122$$ 0 0
$$123$$ −45805.9 −0.272997
$$124$$ 0 0
$$125$$ 174901. 1.00119
$$126$$ 0 0
$$127$$ −68539.3 −0.377077 −0.188539 0.982066i $$-0.560375\pi$$
−0.188539 + 0.982066i $$0.560375\pi$$
$$128$$ 0 0
$$129$$ −155822. −0.824430
$$130$$ 0 0
$$131$$ −95936.2 −0.488432 −0.244216 0.969721i $$-0.578531\pi$$
−0.244216 + 0.969721i $$0.578531\pi$$
$$132$$ 0 0
$$133$$ −15630.9 −0.0766221
$$134$$ 0 0
$$135$$ −93564.8 −0.441853
$$136$$ 0 0
$$137$$ 291295. 1.32597 0.662983 0.748635i $$-0.269290\pi$$
0.662983 + 0.748635i $$0.269290\pi$$
$$138$$ 0 0
$$139$$ 281554. 1.23602 0.618009 0.786171i $$-0.287939\pi$$
0.618009 + 0.786171i $$0.287939\pi$$
$$140$$ 0 0
$$141$$ 408693. 1.73121
$$142$$ 0 0
$$143$$ −327724. −1.34019
$$144$$ 0 0
$$145$$ −740458. −2.92469
$$146$$ 0 0
$$147$$ 50162.8 0.191464
$$148$$ 0 0
$$149$$ −128283. −0.473372 −0.236686 0.971586i $$-0.576061\pi$$
−0.236686 + 0.971586i $$0.576061\pi$$
$$150$$ 0 0
$$151$$ −19017.7 −0.0678760 −0.0339380 0.999424i $$-0.510805\pi$$
−0.0339380 + 0.999424i $$0.510805\pi$$
$$152$$ 0 0
$$153$$ −275186. −0.950381
$$154$$ 0 0
$$155$$ 868259. 2.90282
$$156$$ 0 0
$$157$$ −272988. −0.883883 −0.441941 0.897044i $$-0.645710\pi$$
−0.441941 + 0.897044i $$0.645710\pi$$
$$158$$ 0 0
$$159$$ 764028. 2.39672
$$160$$ 0 0
$$161$$ −32337.7 −0.0983207
$$162$$ 0 0
$$163$$ −315849. −0.931131 −0.465565 0.885013i $$-0.654149\pi$$
−0.465565 + 0.885013i $$0.654149\pi$$
$$164$$ 0 0
$$165$$ 1.04420e6 2.98589
$$166$$ 0 0
$$167$$ 588655. 1.63331 0.816657 0.577123i $$-0.195825\pi$$
0.816657 + 0.577123i $$0.195825\pi$$
$$168$$ 0 0
$$169$$ −19440.5 −0.0523590
$$170$$ 0 0
$$171$$ −61724.1 −0.161423
$$172$$ 0 0
$$173$$ 718256. 1.82458 0.912292 0.409540i $$-0.134311\pi$$
0.912292 + 0.409540i $$0.134311\pi$$
$$174$$ 0 0
$$175$$ 247862. 0.611808
$$176$$ 0 0
$$177$$ −341837. −0.820136
$$178$$ 0 0
$$179$$ −53637.2 −0.125122 −0.0625610 0.998041i $$-0.519927\pi$$
−0.0625610 + 0.998041i $$0.519927\pi$$
$$180$$ 0 0
$$181$$ −392213. −0.889867 −0.444934 0.895564i $$-0.646773\pi$$
−0.444934 + 0.895564i $$0.646773\pi$$
$$182$$ 0 0
$$183$$ −227584. −0.502359
$$184$$ 0 0
$$185$$ 468645. 1.00673
$$186$$ 0 0
$$187$$ −785753. −1.64317
$$188$$ 0 0
$$189$$ −50680.5 −0.103202
$$190$$ 0 0
$$191$$ −395509. −0.784463 −0.392232 0.919866i $$-0.628297\pi$$
−0.392232 + 0.919866i $$0.628297\pi$$
$$192$$ 0 0
$$193$$ −562892. −1.08776 −0.543878 0.839164i $$-0.683045\pi$$
−0.543878 + 0.839164i $$0.683045\pi$$
$$194$$ 0 0
$$195$$ −1.12108e6 −2.11130
$$196$$ 0 0
$$197$$ 402202. 0.738378 0.369189 0.929354i $$-0.379635\pi$$
0.369189 + 0.929354i $$0.379635\pi$$
$$198$$ 0 0
$$199$$ 455975. 0.816222 0.408111 0.912932i $$-0.366188\pi$$
0.408111 + 0.912932i $$0.366188\pi$$
$$200$$ 0 0
$$201$$ −167890. −0.293112
$$202$$ 0 0
$$203$$ −401078. −0.683108
$$204$$ 0 0
$$205$$ −198335. −0.329621
$$206$$ 0 0
$$207$$ −127697. −0.207136
$$208$$ 0 0
$$209$$ −176244. −0.279093
$$210$$ 0 0
$$211$$ 1.18264e6 1.82871 0.914356 0.404911i $$-0.132698\pi$$
0.914356 + 0.404911i $$0.132698\pi$$
$$212$$ 0 0
$$213$$ −1.16963e6 −1.76644
$$214$$ 0 0
$$215$$ −674693. −0.995429
$$216$$ 0 0
$$217$$ 470303. 0.677999
$$218$$ 0 0
$$219$$ −1.62444e6 −2.28872
$$220$$ 0 0
$$221$$ 843604. 1.16187
$$222$$ 0 0
$$223$$ −931112. −1.25383 −0.626917 0.779086i $$-0.715683\pi$$
−0.626917 + 0.779086i $$0.715683\pi$$
$$224$$ 0 0
$$225$$ 978774. 1.28892
$$226$$ 0 0
$$227$$ −192653. −0.248149 −0.124074 0.992273i $$-0.539596\pi$$
−0.124074 + 0.992273i $$0.539596\pi$$
$$228$$ 0 0
$$229$$ −783934. −0.987850 −0.493925 0.869505i $$-0.664438\pi$$
−0.493925 + 0.869505i $$0.664438\pi$$
$$230$$ 0 0
$$231$$ 565605. 0.697402
$$232$$ 0 0
$$233$$ −1.39976e6 −1.68914 −0.844568 0.535448i $$-0.820143\pi$$
−0.844568 + 0.535448i $$0.820143\pi$$
$$234$$ 0 0
$$235$$ 1.76960e6 2.09029
$$236$$ 0 0
$$237$$ 67032.0 0.0775196
$$238$$ 0 0
$$239$$ 643631. 0.728857 0.364429 0.931231i $$-0.381264\pi$$
0.364429 + 0.931231i $$0.381264\pi$$
$$240$$ 0 0
$$241$$ −58756.4 −0.0651647 −0.0325824 0.999469i $$-0.510373\pi$$
−0.0325824 + 0.999469i $$0.510373\pi$$
$$242$$ 0 0
$$243$$ −1.18247e6 −1.28462
$$244$$ 0 0
$$245$$ 217200. 0.231177
$$246$$ 0 0
$$247$$ 189220. 0.197344
$$248$$ 0 0
$$249$$ −1.60092e6 −1.63633
$$250$$ 0 0
$$251$$ 641680. 0.642886 0.321443 0.946929i $$-0.395832\pi$$
0.321443 + 0.946929i $$0.395832\pi$$
$$252$$ 0 0
$$253$$ −364621. −0.358129
$$254$$ 0 0
$$255$$ −2.68791e6 −2.58860
$$256$$ 0 0
$$257$$ 810671. 0.765617 0.382809 0.923828i $$-0.374957\pi$$
0.382809 + 0.923828i $$0.374957\pi$$
$$258$$ 0 0
$$259$$ 253848. 0.235138
$$260$$ 0 0
$$261$$ −1.58380e6 −1.43913
$$262$$ 0 0
$$263$$ 2.04061e6 1.81916 0.909581 0.415526i $$-0.136403\pi$$
0.909581 + 0.415526i $$0.136403\pi$$
$$264$$ 0 0
$$265$$ 3.30817e6 2.89383
$$266$$ 0 0
$$267$$ −1.76100e6 −1.51175
$$268$$ 0 0
$$269$$ −1.28300e6 −1.08105 −0.540524 0.841328i $$-0.681774\pi$$
−0.540524 + 0.841328i $$0.681774\pi$$
$$270$$ 0 0
$$271$$ 22511.3 0.0186199 0.00930994 0.999957i $$-0.497037\pi$$
0.00930994 + 0.999957i $$0.497037\pi$$
$$272$$ 0 0
$$273$$ −607247. −0.493127
$$274$$ 0 0
$$275$$ 2.79474e6 2.22849
$$276$$ 0 0
$$277$$ 368210. 0.288334 0.144167 0.989553i $$-0.453950\pi$$
0.144167 + 0.989553i $$0.453950\pi$$
$$278$$ 0 0
$$279$$ 1.85716e6 1.42837
$$280$$ 0 0
$$281$$ 2.03709e6 1.53902 0.769511 0.638634i $$-0.220500\pi$$
0.769511 + 0.638634i $$0.220500\pi$$
$$282$$ 0 0
$$283$$ 656410. 0.487202 0.243601 0.969876i $$-0.421671\pi$$
0.243601 + 0.969876i $$0.421671\pi$$
$$284$$ 0 0
$$285$$ −602897. −0.439675
$$286$$ 0 0
$$287$$ −107431. −0.0769880
$$288$$ 0 0
$$289$$ 602773. 0.424531
$$290$$ 0 0
$$291$$ 2.11583e6 1.46470
$$292$$ 0 0
$$293$$ −962295. −0.654846 −0.327423 0.944878i $$-0.606180\pi$$
−0.327423 + 0.944878i $$0.606180\pi$$
$$294$$ 0 0
$$295$$ −1.48012e6 −0.990243
$$296$$ 0 0
$$297$$ −571443. −0.375908
$$298$$ 0 0
$$299$$ 391466. 0.253230
$$300$$ 0 0
$$301$$ −365456. −0.232498
$$302$$ 0 0
$$303$$ 1.61826e6 1.01261
$$304$$ 0 0
$$305$$ −985418. −0.606556
$$306$$ 0 0
$$307$$ 296468. 0.179528 0.0897638 0.995963i $$-0.471389\pi$$
0.0897638 + 0.995963i $$0.471389\pi$$
$$308$$ 0 0
$$309$$ 1.18495e6 0.706001
$$310$$ 0 0
$$311$$ −1.12063e6 −0.656996 −0.328498 0.944505i $$-0.606542\pi$$
−0.328498 + 0.944505i $$0.606542\pi$$
$$312$$ 0 0
$$313$$ 1.74908e6 1.00913 0.504567 0.863373i $$-0.331652\pi$$
0.504567 + 0.863373i $$0.331652\pi$$
$$314$$ 0 0
$$315$$ 857692. 0.487029
$$316$$ 0 0
$$317$$ 3.17498e6 1.77457 0.887284 0.461223i $$-0.152589\pi$$
0.887284 + 0.461223i $$0.152589\pi$$
$$318$$ 0 0
$$319$$ −4.52232e6 −2.48819
$$320$$ 0 0
$$321$$ 808520. 0.437954
$$322$$ 0 0
$$323$$ 453675. 0.241957
$$324$$ 0 0
$$325$$ −3.00051e6 −1.57575
$$326$$ 0 0
$$327$$ −346715. −0.179310
$$328$$ 0 0
$$329$$ 958526. 0.488219
$$330$$ 0 0
$$331$$ −878405. −0.440681 −0.220341 0.975423i $$-0.570717\pi$$
−0.220341 + 0.975423i $$0.570717\pi$$
$$332$$ 0 0
$$333$$ 1.00241e6 0.495375
$$334$$ 0 0
$$335$$ −726946. −0.353908
$$336$$ 0 0
$$337$$ 2.05261e6 0.984536 0.492268 0.870444i $$-0.336168\pi$$
0.492268 + 0.870444i $$0.336168\pi$$
$$338$$ 0 0
$$339$$ −1.60295e6 −0.757566
$$340$$ 0 0
$$341$$ 5.30286e6 2.46958
$$342$$ 0 0
$$343$$ 117649. 0.0539949
$$344$$ 0 0
$$345$$ −1.24730e6 −0.564186
$$346$$ 0 0
$$347$$ −476182. −0.212300 −0.106150 0.994350i $$-0.533852\pi$$
−0.106150 + 0.994350i $$0.533852\pi$$
$$348$$ 0 0
$$349$$ 351681. 0.154556 0.0772780 0.997010i $$-0.475377\pi$$
0.0772780 + 0.997010i $$0.475377\pi$$
$$350$$ 0 0
$$351$$ 613515. 0.265802
$$352$$ 0 0
$$353$$ −1.73722e6 −0.742023 −0.371011 0.928628i $$-0.620989\pi$$
−0.371011 + 0.928628i $$0.620989\pi$$
$$354$$ 0 0
$$355$$ −5.06438e6 −2.13282
$$356$$ 0 0
$$357$$ −1.45594e6 −0.604607
$$358$$ 0 0
$$359$$ 1.69731e6 0.695063 0.347531 0.937668i $$-0.387020\pi$$
0.347531 + 0.937668i $$0.387020\pi$$
$$360$$ 0 0
$$361$$ −2.37434e6 −0.958903
$$362$$ 0 0
$$363$$ 3.01267e6 1.20001
$$364$$ 0 0
$$365$$ −7.03366e6 −2.76344
$$366$$ 0 0
$$367$$ 1.11920e6 0.433752 0.216876 0.976199i $$-0.430413\pi$$
0.216876 + 0.976199i $$0.430413\pi$$
$$368$$ 0 0
$$369$$ −424228. −0.162194
$$370$$ 0 0
$$371$$ 1.79191e6 0.675898
$$372$$ 0 0
$$373$$ −837822. −0.311803 −0.155901 0.987773i $$-0.549828\pi$$
−0.155901 + 0.987773i $$0.549828\pi$$
$$374$$ 0 0
$$375$$ 3.65411e6 1.34185
$$376$$ 0 0
$$377$$ 4.85527e6 1.75938
$$378$$ 0 0
$$379$$ 1.94713e6 0.696300 0.348150 0.937439i $$-0.386810\pi$$
0.348150 + 0.937439i $$0.386810\pi$$
$$380$$ 0 0
$$381$$ −1.43195e6 −0.505378
$$382$$ 0 0
$$383$$ 202063. 0.0703866 0.0351933 0.999381i $$-0.488795\pi$$
0.0351933 + 0.999381i $$0.488795\pi$$
$$384$$ 0 0
$$385$$ 2.44901e6 0.842053
$$386$$ 0 0
$$387$$ −1.44314e6 −0.489812
$$388$$ 0 0
$$389$$ 3.23741e6 1.08473 0.542367 0.840141i $$-0.317528\pi$$
0.542367 + 0.840141i $$0.317528\pi$$
$$390$$ 0 0
$$391$$ 938581. 0.310477
$$392$$ 0 0
$$393$$ −2.00434e6 −0.654622
$$394$$ 0 0
$$395$$ 290242. 0.0935982
$$396$$ 0 0
$$397$$ 823921. 0.262367 0.131183 0.991358i $$-0.458122\pi$$
0.131183 + 0.991358i $$0.458122\pi$$
$$398$$ 0 0
$$399$$ −326567. −0.102693
$$400$$ 0 0
$$401$$ 2.39337e6 0.743273 0.371637 0.928378i $$-0.378797\pi$$
0.371637 + 0.928378i $$0.378797\pi$$
$$402$$ 0 0
$$403$$ −5.69328e6 −1.74622
$$404$$ 0 0
$$405$$ −6.20825e6 −1.88075
$$406$$ 0 0
$$407$$ 2.86223e6 0.856483
$$408$$ 0 0
$$409$$ −581801. −0.171975 −0.0859876 0.996296i $$-0.527405\pi$$
−0.0859876 + 0.996296i $$0.527405\pi$$
$$410$$ 0 0
$$411$$ 6.08587e6 1.77713
$$412$$ 0 0
$$413$$ −801725. −0.231287
$$414$$ 0 0
$$415$$ −6.93184e6 −1.97573
$$416$$ 0 0
$$417$$ 5.88236e6 1.65658
$$418$$ 0 0
$$419$$ −3.11898e6 −0.867915 −0.433957 0.900933i $$-0.642883\pi$$
−0.433957 + 0.900933i $$0.642883\pi$$
$$420$$ 0 0
$$421$$ −1.14481e6 −0.314796 −0.157398 0.987535i $$-0.550311\pi$$
−0.157398 + 0.987535i $$0.550311\pi$$
$$422$$ 0 0
$$423$$ 3.78509e6 1.02855
$$424$$ 0 0
$$425$$ −7.19403e6 −1.93197
$$426$$ 0 0
$$427$$ −533764. −0.141671
$$428$$ 0 0
$$429$$ −6.84695e6 −1.79620
$$430$$ 0 0
$$431$$ 4.31012e6 1.11762 0.558812 0.829294i $$-0.311257\pi$$
0.558812 + 0.829294i $$0.311257\pi$$
$$432$$ 0 0
$$433$$ −2.79301e6 −0.715901 −0.357950 0.933741i $$-0.616524\pi$$
−0.357950 + 0.933741i $$0.616524\pi$$
$$434$$ 0 0
$$435$$ −1.54700e7 −3.91982
$$436$$ 0 0
$$437$$ 210523. 0.0527347
$$438$$ 0 0
$$439$$ −252467. −0.0625236 −0.0312618 0.999511i $$-0.509953\pi$$
−0.0312618 + 0.999511i $$0.509953\pi$$
$$440$$ 0 0
$$441$$ 464580. 0.113753
$$442$$ 0 0
$$443$$ 3.43282e6 0.831077 0.415539 0.909576i $$-0.363593\pi$$
0.415539 + 0.909576i $$0.363593\pi$$
$$444$$ 0 0
$$445$$ −7.62494e6 −1.82531
$$446$$ 0 0
$$447$$ −2.68014e6 −0.634437
$$448$$ 0 0
$$449$$ −1.68031e6 −0.393345 −0.196673 0.980469i $$-0.563014\pi$$
−0.196673 + 0.980469i $$0.563014\pi$$
$$450$$ 0 0
$$451$$ −1.21132e6 −0.280426
$$452$$ 0 0
$$453$$ −397327. −0.0909709
$$454$$ 0 0
$$455$$ −2.62932e6 −0.595409
$$456$$ 0 0
$$457$$ −2.94585e6 −0.659812 −0.329906 0.944014i $$-0.607017\pi$$
−0.329906 + 0.944014i $$0.607017\pi$$
$$458$$ 0 0
$$459$$ 1.47097e6 0.325890
$$460$$ 0 0
$$461$$ −1.91583e6 −0.419860 −0.209930 0.977716i $$-0.567324\pi$$
−0.209930 + 0.977716i $$0.567324\pi$$
$$462$$ 0 0
$$463$$ 3.75401e6 0.813847 0.406924 0.913462i $$-0.366602\pi$$
0.406924 + 0.913462i $$0.366602\pi$$
$$464$$ 0 0
$$465$$ 1.81401e7 3.89051
$$466$$ 0 0
$$467$$ 2.10594e6 0.446842 0.223421 0.974722i $$-0.428278\pi$$
0.223421 + 0.974722i $$0.428278\pi$$
$$468$$ 0 0
$$469$$ −393759. −0.0826607
$$470$$ 0 0
$$471$$ −5.70339e6 −1.18462
$$472$$ 0 0
$$473$$ −4.12066e6 −0.846864
$$474$$ 0 0
$$475$$ −1.61362e6 −0.328146
$$476$$ 0 0
$$477$$ 7.07600e6 1.42394
$$478$$ 0 0
$$479$$ 274926. 0.0547491 0.0273745 0.999625i $$-0.491285\pi$$
0.0273745 + 0.999625i $$0.491285\pi$$
$$480$$ 0 0
$$481$$ −3.07296e6 −0.605612
$$482$$ 0 0
$$483$$ −675614. −0.131774
$$484$$ 0 0
$$485$$ 9.16135e6 1.76850
$$486$$ 0 0
$$487$$ −7.67128e6 −1.46570 −0.732851 0.680389i $$-0.761811\pi$$
−0.732851 + 0.680389i $$0.761811\pi$$
$$488$$ 0 0
$$489$$ −6.59886e6 −1.24795
$$490$$ 0 0
$$491$$ 2.42352e6 0.453673 0.226837 0.973933i $$-0.427162\pi$$
0.226837 + 0.973933i $$0.427162\pi$$
$$492$$ 0 0
$$493$$ 1.16410e7 2.15712
$$494$$ 0 0
$$495$$ 9.67081e6 1.77399
$$496$$ 0 0
$$497$$ −2.74318e6 −0.498154
$$498$$ 0 0
$$499$$ 8.22317e6 1.47839 0.739193 0.673494i $$-0.235207\pi$$
0.739193 + 0.673494i $$0.235207\pi$$
$$500$$ 0 0
$$501$$ 1.22984e7 2.18905
$$502$$ 0 0
$$503$$ −2.98186e6 −0.525493 −0.262746 0.964865i $$-0.584628\pi$$
−0.262746 + 0.964865i $$0.584628\pi$$
$$504$$ 0 0
$$505$$ 7.00689e6 1.22263
$$506$$ 0 0
$$507$$ −406160. −0.0701741
$$508$$ 0 0
$$509$$ −6.26867e6 −1.07246 −0.536230 0.844072i $$-0.680152\pi$$
−0.536230 + 0.844072i $$0.680152\pi$$
$$510$$ 0 0
$$511$$ −3.80987e6 −0.645443
$$512$$ 0 0
$$513$$ 329938. 0.0553527
$$514$$ 0 0
$$515$$ 5.13073e6 0.852435
$$516$$ 0 0
$$517$$ 1.08078e7 1.77832
$$518$$ 0 0
$$519$$ 1.50061e7 2.44540
$$520$$ 0 0
$$521$$ −4.12134e6 −0.665188 −0.332594 0.943070i $$-0.607924\pi$$
−0.332594 + 0.943070i $$0.607924\pi$$
$$522$$ 0 0
$$523$$ 4.70469e6 0.752102 0.376051 0.926599i $$-0.377282\pi$$
0.376051 + 0.926599i $$0.377282\pi$$
$$524$$ 0 0
$$525$$ 5.17845e6 0.819976
$$526$$ 0 0
$$527$$ −1.36502e7 −2.14099
$$528$$ 0 0
$$529$$ −6.00080e6 −0.932331
$$530$$ 0 0
$$531$$ −3.16590e6 −0.487261
$$532$$ 0 0
$$533$$ 1.30051e6 0.198287
$$534$$ 0 0
$$535$$ 3.50081e6 0.528791
$$536$$ 0 0
$$537$$ −1.12061e6 −0.167695
$$538$$ 0 0
$$539$$ 1.32654e6 0.196674
$$540$$ 0 0
$$541$$ 4.47840e6 0.657855 0.328927 0.944355i $$-0.393313\pi$$
0.328927 + 0.944355i $$0.393313\pi$$
$$542$$ 0 0
$$543$$ −8.19428e6 −1.19265
$$544$$ 0 0
$$545$$ −1.50124e6 −0.216501
$$546$$ 0 0
$$547$$ 6.49187e6 0.927687 0.463843 0.885917i $$-0.346470\pi$$
0.463843 + 0.885917i $$0.346470\pi$$
$$548$$ 0 0
$$549$$ −2.10776e6 −0.298463
$$550$$ 0 0
$$551$$ 2.61108e6 0.366388
$$552$$ 0 0
$$553$$ 157213. 0.0218613
$$554$$ 0 0
$$555$$ 9.79115e6 1.34928
$$556$$ 0 0
$$557$$ −2.50858e6 −0.342602 −0.171301 0.985219i $$-0.554797\pi$$
−0.171301 + 0.985219i $$0.554797\pi$$
$$558$$ 0 0
$$559$$ 4.42404e6 0.598811
$$560$$ 0 0
$$561$$ −1.64163e7 −2.20226
$$562$$ 0 0
$$563$$ 3.52702e6 0.468961 0.234480 0.972121i $$-0.424661\pi$$
0.234480 + 0.972121i $$0.424661\pi$$
$$564$$ 0 0
$$565$$ −6.94061e6 −0.914696
$$566$$ 0 0
$$567$$ −3.36278e6 −0.439279
$$568$$ 0 0
$$569$$ 9.75283e6 1.26284 0.631422 0.775439i $$-0.282471\pi$$
0.631422 + 0.775439i $$0.282471\pi$$
$$570$$ 0 0
$$571$$ −1.70085e6 −0.218312 −0.109156 0.994025i $$-0.534815\pi$$
−0.109156 + 0.994025i $$0.534815\pi$$
$$572$$ 0 0
$$573$$ −8.26314e6 −1.05138
$$574$$ 0 0
$$575$$ −3.33832e6 −0.421074
$$576$$ 0 0
$$577$$ −1.17066e7 −1.46383 −0.731914 0.681397i $$-0.761373\pi$$
−0.731914 + 0.681397i $$0.761373\pi$$
$$578$$ 0 0
$$579$$ −1.17602e7 −1.45787
$$580$$ 0 0
$$581$$ −3.75472e6 −0.461463
$$582$$ 0 0
$$583$$ 2.02045e7 2.46193
$$584$$ 0 0
$$585$$ −1.03828e7 −1.25437
$$586$$ 0 0
$$587$$ 3.75683e6 0.450015 0.225007 0.974357i $$-0.427759\pi$$
0.225007 + 0.974357i $$0.427759\pi$$
$$588$$ 0 0
$$589$$ −3.06175e6 −0.363648
$$590$$ 0 0
$$591$$ 8.40299e6 0.989612
$$592$$ 0 0
$$593$$ 8.64338e6 1.00936 0.504681 0.863306i $$-0.331610\pi$$
0.504681 + 0.863306i $$0.331610\pi$$
$$594$$ 0 0
$$595$$ −6.30408e6 −0.730011
$$596$$ 0 0
$$597$$ 9.52643e6 1.09394
$$598$$ 0 0
$$599$$ −9.31930e6 −1.06125 −0.530623 0.847608i $$-0.678042\pi$$
−0.530623 + 0.847608i $$0.678042\pi$$
$$600$$ 0 0
$$601$$ 910438. 0.102817 0.0514084 0.998678i $$-0.483629\pi$$
0.0514084 + 0.998678i $$0.483629\pi$$
$$602$$ 0 0
$$603$$ −1.55490e6 −0.174144
$$604$$ 0 0
$$605$$ 1.30445e7 1.44891
$$606$$ 0 0
$$607$$ −5.70173e6 −0.628109 −0.314054 0.949405i $$-0.601687\pi$$
−0.314054 + 0.949405i $$0.601687\pi$$
$$608$$ 0 0
$$609$$ −8.37951e6 −0.915536
$$610$$ 0 0
$$611$$ −1.16035e7 −1.25743
$$612$$ 0 0
$$613$$ 1.61327e7 1.73403 0.867016 0.498280i $$-0.166035\pi$$
0.867016 + 0.498280i $$0.166035\pi$$
$$614$$ 0 0
$$615$$ −4.14370e6 −0.441774
$$616$$ 0 0
$$617$$ −5.53575e6 −0.585415 −0.292708 0.956202i $$-0.594556\pi$$
−0.292708 + 0.956202i $$0.594556\pi$$
$$618$$ 0 0
$$619$$ 1.88448e7 1.97681 0.988407 0.151828i $$-0.0485161\pi$$
0.988407 + 0.151828i $$0.0485161\pi$$
$$620$$ 0 0
$$621$$ 682588. 0.0710280
$$622$$ 0 0
$$623$$ −4.13014e6 −0.426329
$$624$$ 0 0
$$625$$ 14378.1 0.00147232
$$626$$ 0 0
$$627$$ −3.68217e6 −0.374055
$$628$$ 0 0
$$629$$ −7.36776e6 −0.742521
$$630$$ 0 0
$$631$$ −2.63269e6 −0.263224 −0.131612 0.991301i $$-0.542015\pi$$
−0.131612 + 0.991301i $$0.542015\pi$$
$$632$$ 0 0
$$633$$ 2.47082e7 2.45093
$$634$$ 0 0
$$635$$ −6.20021e6 −0.610200
$$636$$ 0 0
$$637$$ −1.42420e6 −0.139067
$$638$$ 0 0
$$639$$ −1.08324e7 −1.04948
$$640$$ 0 0
$$641$$ 1.42939e7 1.37406 0.687028 0.726631i $$-0.258915\pi$$
0.687028 + 0.726631i $$0.258915\pi$$
$$642$$ 0 0
$$643$$ 1.63874e7 1.56308 0.781542 0.623853i $$-0.214434\pi$$
0.781542 + 0.623853i $$0.214434\pi$$
$$644$$ 0 0
$$645$$ −1.40960e7 −1.33412
$$646$$ 0 0
$$647$$ −1.19303e7 −1.12045 −0.560224 0.828341i $$-0.689285\pi$$
−0.560224 + 0.828341i $$0.689285\pi$$
$$648$$ 0 0
$$649$$ −9.03977e6 −0.842453
$$650$$ 0 0
$$651$$ 9.82579e6 0.908688
$$652$$ 0 0
$$653$$ 3.47862e6 0.319245 0.159622 0.987178i $$-0.448972\pi$$
0.159622 + 0.987178i $$0.448972\pi$$
$$654$$ 0 0
$$655$$ −8.67860e6 −0.790399
$$656$$ 0 0
$$657$$ −1.50447e7 −1.35978
$$658$$ 0 0
$$659$$ −2.09397e7 −1.87826 −0.939131 0.343560i $$-0.888367\pi$$
−0.939131 + 0.343560i $$0.888367\pi$$
$$660$$ 0 0
$$661$$ −1.02566e7 −0.913059 −0.456530 0.889708i $$-0.650908\pi$$
−0.456530 + 0.889708i $$0.650908\pi$$
$$662$$ 0 0
$$663$$ 1.76249e7 1.55720
$$664$$ 0 0
$$665$$ −1.41400e6 −0.123993
$$666$$ 0 0
$$667$$ 5.40190e6 0.470145
$$668$$ 0 0
$$669$$ −1.94532e7 −1.68045
$$670$$ 0 0
$$671$$ −6.01840e6 −0.516029
$$672$$ 0 0
$$673$$ 716398. 0.0609701 0.0304851 0.999535i $$-0.490295\pi$$
0.0304851 + 0.999535i $$0.490295\pi$$
$$674$$ 0 0
$$675$$ −5.23190e6 −0.441977
$$676$$ 0 0
$$677$$ 945625. 0.0792953 0.0396476 0.999214i $$-0.487376\pi$$
0.0396476 + 0.999214i $$0.487376\pi$$
$$678$$ 0 0
$$679$$ 4.96236e6 0.413061
$$680$$ 0 0
$$681$$ −4.02500e6 −0.332582
$$682$$ 0 0
$$683$$ −3.56992e6 −0.292824 −0.146412 0.989224i $$-0.546773\pi$$
−0.146412 + 0.989224i $$0.546773\pi$$
$$684$$ 0 0
$$685$$ 2.63512e7 2.14573
$$686$$ 0 0
$$687$$ −1.63783e7 −1.32397
$$688$$ 0 0
$$689$$ −2.16920e7 −1.74081
$$690$$ 0 0
$$691$$ −9.67811e6 −0.771073 −0.385536 0.922693i $$-0.625984\pi$$
−0.385536 + 0.922693i $$0.625984\pi$$
$$692$$ 0 0
$$693$$ 5.23832e6 0.414342
$$694$$ 0 0
$$695$$ 2.54700e7 2.00017
$$696$$ 0 0
$$697$$ 3.11810e6 0.243113
$$698$$ 0 0
$$699$$ −2.92445e7 −2.26387
$$700$$ 0 0
$$701$$ 2.52404e7 1.94000 0.970000 0.243105i $$-0.0781659\pi$$
0.970000 + 0.243105i $$0.0781659\pi$$
$$702$$ 0 0
$$703$$ −1.65259e6 −0.126118
$$704$$ 0 0
$$705$$ 3.69713e7 2.80151
$$706$$ 0 0
$$707$$ 3.79537e6 0.285565
$$708$$ 0 0
$$709$$ −1.74744e7 −1.30553 −0.652765 0.757560i $$-0.726391\pi$$
−0.652765 + 0.757560i $$0.726391\pi$$
$$710$$ 0 0
$$711$$ 620813. 0.0460561
$$712$$ 0 0
$$713$$ −6.33426e6 −0.466629
$$714$$ 0 0
$$715$$ −2.96466e7 −2.16875
$$716$$ 0 0
$$717$$ 1.34470e7 0.976852
$$718$$ 0 0
$$719$$ 9.73361e6 0.702185 0.351093 0.936341i $$-0.385810\pi$$
0.351093 + 0.936341i $$0.385810\pi$$
$$720$$ 0 0
$$721$$ 2.77913e6 0.199100
$$722$$ 0 0
$$723$$ −1.22756e6 −0.0873371
$$724$$ 0 0
$$725$$ −4.14045e7 −2.92552
$$726$$ 0 0
$$727$$ 1.60454e7 1.12594 0.562971 0.826477i $$-0.309658\pi$$
0.562971 + 0.826477i $$0.309658\pi$$
$$728$$ 0 0
$$729$$ −8.02815e6 −0.559496
$$730$$ 0 0
$$731$$ 1.06071e7 0.734182
$$732$$ 0 0
$$733$$ 1.04534e7 0.718615 0.359307 0.933219i $$-0.383013\pi$$
0.359307 + 0.933219i $$0.383013\pi$$
$$734$$ 0 0
$$735$$ 4.53783e6 0.309835
$$736$$ 0 0
$$737$$ −4.43979e6 −0.301088
$$738$$ 0 0
$$739$$ 7.47638e6 0.503594 0.251797 0.967780i $$-0.418978\pi$$
0.251797 + 0.967780i $$0.418978\pi$$
$$740$$ 0 0
$$741$$ 3.95327e6 0.264491
$$742$$ 0 0
$$743$$ 1.33998e7 0.890483 0.445241 0.895411i $$-0.353118\pi$$
0.445241 + 0.895411i $$0.353118\pi$$
$$744$$ 0 0
$$745$$ −1.16047e7 −0.766028
$$746$$ 0 0
$$747$$ −1.48269e7 −0.972182
$$748$$ 0 0
$$749$$ 1.89626e6 0.123507
$$750$$ 0 0
$$751$$ −9.13903e6 −0.591290 −0.295645 0.955298i $$-0.595535\pi$$
−0.295645 + 0.955298i $$0.595535\pi$$
$$752$$ 0 0
$$753$$ 1.34063e7 0.861629
$$754$$ 0 0
$$755$$ −1.72039e6 −0.109840
$$756$$ 0 0
$$757$$ −1.16376e7 −0.738117 −0.369059 0.929406i $$-0.620320\pi$$
−0.369059 + 0.929406i $$0.620320\pi$$
$$758$$ 0 0
$$759$$ −7.61782e6 −0.479983
$$760$$ 0 0
$$761$$ −8.07072e6 −0.505186 −0.252593 0.967573i $$-0.581283\pi$$
−0.252593 + 0.967573i $$0.581283\pi$$
$$762$$ 0 0
$$763$$ −813167. −0.0505672
$$764$$ 0 0
$$765$$ −2.48939e7 −1.53794
$$766$$ 0 0
$$767$$ 9.70532e6 0.595692
$$768$$ 0 0
$$769$$ −1.30085e7 −0.793255 −0.396628 0.917980i $$-0.629820\pi$$
−0.396628 + 0.917980i $$0.629820\pi$$
$$770$$ 0 0
$$771$$ 1.69369e7 1.02612
$$772$$ 0 0
$$773$$ 2.37637e7 1.43043 0.715214 0.698906i $$-0.246329\pi$$
0.715214 + 0.698906i $$0.246329\pi$$
$$774$$ 0 0
$$775$$ 4.85508e7 2.90364
$$776$$ 0 0
$$777$$ 5.30350e6 0.315145
$$778$$ 0 0
$$779$$ 699389. 0.0412929
$$780$$ 0 0
$$781$$ −3.09305e7 −1.81451
$$782$$ 0 0
$$783$$ 8.46600e6 0.493485
$$784$$ 0 0
$$785$$ −2.46951e7 −1.43033
$$786$$ 0 0
$$787$$ 2.84940e7 1.63989 0.819947 0.572439i $$-0.194003\pi$$
0.819947 + 0.572439i $$0.194003\pi$$
$$788$$ 0 0
$$789$$ 4.26334e7 2.43813
$$790$$ 0 0
$$791$$ −3.75947e6 −0.213641
$$792$$ 0 0
$$793$$ 6.46150e6 0.364880
$$794$$ 0 0
$$795$$ 6.91157e7 3.87845
$$796$$ 0 0
$$797$$ 1.43450e7 0.799938 0.399969 0.916529i $$-0.369021\pi$$
0.399969 + 0.916529i $$0.369021\pi$$
$$798$$ 0 0
$$799$$ −2.78206e7 −1.54170
$$800$$ 0 0
$$801$$ −1.63094e7 −0.898164
$$802$$ 0 0
$$803$$ −4.29578e7 −2.35100
$$804$$ 0 0
$$805$$ −2.92534e6 −0.159106
$$806$$ 0 0
$$807$$ −2.68050e7 −1.44888
$$808$$ 0 0
$$809$$ −3.59284e6 −0.193004 −0.0965020 0.995333i $$-0.530765\pi$$
−0.0965020 + 0.995333i $$0.530765\pi$$
$$810$$ 0 0
$$811$$ −3.52968e7 −1.88445 −0.942223 0.334988i $$-0.891268\pi$$
−0.942223 + 0.334988i $$0.891268\pi$$
$$812$$ 0 0
$$813$$ 470316. 0.0249553
$$814$$ 0 0
$$815$$ −2.85724e7 −1.50679
$$816$$ 0 0
$$817$$ 2.37917e6 0.124701
$$818$$ 0 0
$$819$$ −5.62399e6 −0.292978
$$820$$ 0 0
$$821$$ 1.04558e7 0.541378 0.270689 0.962667i $$-0.412748\pi$$
0.270689 + 0.962667i $$0.412748\pi$$
$$822$$ 0 0
$$823$$ −4.99190e6 −0.256901 −0.128451 0.991716i $$-0.541000\pi$$
−0.128451 + 0.991716i $$0.541000\pi$$
$$824$$ 0 0
$$825$$ 5.83890e7 2.98673
$$826$$ 0 0
$$827$$ 2.06198e7 1.04839 0.524193 0.851599i $$-0.324367\pi$$
0.524193 + 0.851599i $$0.324367\pi$$
$$828$$ 0 0
$$829$$ −506843. −0.0256146 −0.0128073 0.999918i $$-0.504077\pi$$
−0.0128073 + 0.999918i $$0.504077\pi$$
$$830$$ 0 0
$$831$$ 7.69281e6 0.386440
$$832$$ 0 0
$$833$$ −3.41468e6 −0.170505
$$834$$ 0 0
$$835$$ 5.32511e7 2.64309
$$836$$ 0 0
$$837$$ −9.92721e6 −0.489794
$$838$$ 0 0
$$839$$ 4.41851e6 0.216706 0.108353 0.994112i $$-0.465442\pi$$
0.108353 + 0.994112i $$0.465442\pi$$
$$840$$ 0 0
$$841$$ 4.64876e7 2.26645
$$842$$ 0 0
$$843$$ 4.25598e7 2.06267
$$844$$ 0 0
$$845$$ −1.75863e6 −0.0847292
$$846$$ 0 0
$$847$$ 7.06574e6 0.338415
$$848$$ 0 0
$$849$$ 1.37140e7 0.652973
$$850$$ 0 0
$$851$$ −3.41893e6 −0.161833
$$852$$ 0 0
$$853$$ −1.11737e7 −0.525806 −0.262903 0.964822i $$-0.584680\pi$$
−0.262903 + 0.964822i $$0.584680\pi$$
$$854$$ 0 0
$$855$$ −5.58370e6 −0.261220
$$856$$ 0 0
$$857$$ 2.12308e7 0.987448 0.493724 0.869619i $$-0.335635\pi$$
0.493724 + 0.869619i $$0.335635\pi$$
$$858$$ 0 0
$$859$$ −3.64229e7 −1.68419 −0.842096 0.539328i $$-0.818678\pi$$
−0.842096 + 0.539328i $$0.818678\pi$$
$$860$$ 0 0
$$861$$ −2.24449e6 −0.103183
$$862$$ 0 0
$$863$$ 2.53298e7 1.15772 0.578861 0.815426i $$-0.303497\pi$$
0.578861 + 0.815426i $$0.303497\pi$$
$$864$$ 0 0
$$865$$ 6.49750e7 2.95261
$$866$$ 0 0
$$867$$ 1.25934e7 0.568978
$$868$$ 0 0
$$869$$ 1.77264e6 0.0796290
$$870$$ 0 0
$$871$$ 4.76667e6 0.212897
$$872$$ 0 0
$$873$$ 1.95957e7 0.870211
$$874$$ 0 0
$$875$$ 8.57014e6 0.378415
$$876$$ 0 0
$$877$$ −577138. −0.0253385 −0.0126692 0.999920i $$-0.504033\pi$$
−0.0126692 + 0.999920i $$0.504033\pi$$
$$878$$ 0 0
$$879$$ −2.01047e7 −0.877658
$$880$$ 0 0
$$881$$ −2.15071e7 −0.933561 −0.466781 0.884373i $$-0.654586\pi$$
−0.466781 + 0.884373i $$0.654586\pi$$
$$882$$ 0 0
$$883$$ 3.71948e6 0.160539 0.0802695 0.996773i $$-0.474422\pi$$
0.0802695 + 0.996773i $$0.474422\pi$$
$$884$$ 0 0
$$885$$ −3.09233e7 −1.32717
$$886$$ 0 0
$$887$$ −1.81760e7 −0.775694 −0.387847 0.921724i $$-0.626781\pi$$
−0.387847 + 0.921724i $$0.626781\pi$$
$$888$$ 0 0
$$889$$ −3.35842e6 −0.142522
$$890$$ 0 0
$$891$$ −3.79166e7 −1.60006
$$892$$ 0 0
$$893$$ −6.24015e6 −0.261858
$$894$$ 0 0
$$895$$ −4.85214e6 −0.202477
$$896$$ 0 0
$$897$$ 8.17868e6 0.339392
$$898$$ 0 0
$$899$$ −7.85625e7 −3.24202
$$900$$ 0 0
$$901$$ −5.20090e7 −2.13435
$$902$$ 0 0
$$903$$ −7.63527e6 −0.311605
$$904$$ 0 0
$$905$$ −3.54804e7 −1.44002
$$906$$ 0 0
$$907$$ 1.35430e7 0.546635 0.273317 0.961924i $$-0.411879\pi$$
0.273317 + 0.961924i $$0.411879\pi$$
$$908$$ 0 0
$$909$$ 1.49874e7 0.601612
$$910$$ 0 0
$$911$$ 412746. 0.0164773 0.00823866 0.999966i $$-0.497378\pi$$
0.00823866 + 0.999966i $$0.497378\pi$$
$$912$$ 0 0
$$913$$ −4.23359e7 −1.68086
$$914$$ 0 0
$$915$$ −2.05878e7 −0.812937
$$916$$ 0 0
$$917$$ −4.70087e6 −0.184610
$$918$$ 0 0
$$919$$ −2.79059e7 −1.08995 −0.544976 0.838452i $$-0.683461\pi$$
−0.544976 + 0.838452i $$0.683461\pi$$
$$920$$ 0 0
$$921$$ 6.19394e6 0.240612
$$922$$ 0 0
$$923$$ 3.32077e7 1.28302
$$924$$ 0 0
$$925$$ 2.62054e7 1.00702
$$926$$ 0 0
$$927$$ 1.09744e7 0.419451
$$928$$ 0 0
$$929$$ −1.94840e7 −0.740695 −0.370348 0.928893i $$-0.620761\pi$$
−0.370348 + 0.928893i $$0.620761\pi$$
$$930$$ 0 0
$$931$$ −765912. −0.0289604
$$932$$ 0 0
$$933$$ −2.34128e7 −0.880539
$$934$$ 0 0
$$935$$ −7.10809e7 −2.65904
$$936$$ 0 0
$$937$$ −1.89631e7 −0.705602 −0.352801 0.935698i $$-0.614771\pi$$
−0.352801 + 0.935698i $$0.614771\pi$$
$$938$$ 0 0
$$939$$ 3.65425e7 1.35249
$$940$$ 0 0
$$941$$ −2.30498e7 −0.848579 −0.424290 0.905526i $$-0.639476\pi$$
−0.424290 + 0.905526i $$0.639476\pi$$
$$942$$ 0 0
$$943$$ 1.44692e6 0.0529866
$$944$$ 0 0
$$945$$ −4.58467e6 −0.167005
$$946$$ 0 0
$$947$$ 3.49310e7 1.26572 0.632858 0.774268i $$-0.281882\pi$$
0.632858 + 0.774268i $$0.281882\pi$$
$$948$$ 0 0
$$949$$ 4.61206e7 1.66238
$$950$$ 0 0
$$951$$ 6.63331e7 2.37837
$$952$$ 0 0
$$953$$ 2.74501e7 0.979066 0.489533 0.871985i $$-0.337167\pi$$
0.489533 + 0.871985i $$0.337167\pi$$
$$954$$ 0 0
$$955$$ −3.57786e7 −1.26945
$$956$$ 0 0
$$957$$ −9.44822e7 −3.33480
$$958$$ 0 0
$$959$$ 1.42735e7 0.501168
$$960$$ 0 0
$$961$$ 6.34930e7 2.21778
$$962$$ 0 0
$$963$$ 7.48807e6 0.260198
$$964$$ 0 0
$$965$$ −5.09204e7 −1.76025
$$966$$ 0 0
$$967$$ 9.37608e6 0.322445 0.161222 0.986918i $$-0.448456\pi$$
0.161222 + 0.986918i $$0.448456\pi$$
$$968$$ 0 0
$$969$$ 9.47839e6 0.324284
$$970$$ 0 0
$$971$$ −7.23359e6 −0.246210 −0.123105 0.992394i $$-0.539285\pi$$
−0.123105 + 0.992394i $$0.539285\pi$$
$$972$$ 0 0
$$973$$ 1.37962e7 0.467171
$$974$$ 0 0
$$975$$ −6.26879e7 −2.11190
$$976$$ 0 0
$$977$$ −2.88022e7 −0.965360 −0.482680 0.875797i $$-0.660337\pi$$
−0.482680 + 0.875797i $$0.660337\pi$$
$$978$$ 0 0
$$979$$ −4.65690e7 −1.55289
$$980$$ 0 0
$$981$$ −3.21109e6 −0.106532
$$982$$ 0 0
$$983$$ −5.21848e7 −1.72250 −0.861252 0.508178i $$-0.830319\pi$$
−0.861252 + 0.508178i $$0.830319\pi$$
$$984$$ 0 0
$$985$$ 3.63841e7 1.19487
$$986$$ 0 0
$$987$$ 2.00260e7 0.654335
$$988$$ 0 0
$$989$$ 4.92212e6 0.160015
$$990$$ 0 0
$$991$$ 3.89578e7 1.26012 0.630058 0.776548i $$-0.283031\pi$$
0.630058 + 0.776548i $$0.283031\pi$$
$$992$$ 0 0
$$993$$ −1.83520e7 −0.590624
$$994$$ 0 0
$$995$$ 4.12485e7 1.32084
$$996$$ 0 0
$$997$$ −3.01716e7 −0.961305 −0.480652 0.876911i $$-0.659600\pi$$
−0.480652 + 0.876911i $$0.659600\pi$$
$$998$$ 0 0
$$999$$ −5.35824e6 −0.169867
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 112.6.a.j.1.2 2
3.2 odd 2 1008.6.a.bi.1.1 2
4.3 odd 2 56.6.a.d.1.1 2
7.6 odd 2 784.6.a.q.1.1 2
8.3 odd 2 448.6.a.x.1.2 2
8.5 even 2 448.6.a.r.1.1 2
12.11 even 2 504.6.a.m.1.1 2
28.3 even 6 392.6.i.h.177.1 4
28.11 odd 6 392.6.i.k.177.2 4
28.19 even 6 392.6.i.h.361.1 4
28.23 odd 6 392.6.i.k.361.2 4
28.27 even 2 392.6.a.e.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
56.6.a.d.1.1 2 4.3 odd 2
112.6.a.j.1.2 2 1.1 even 1 trivial
392.6.a.e.1.2 2 28.27 even 2
392.6.i.h.177.1 4 28.3 even 6
392.6.i.h.361.1 4 28.19 even 6
392.6.i.k.177.2 4 28.11 odd 6
392.6.i.k.361.2 4 28.23 odd 6
448.6.a.r.1.1 2 8.5 even 2
448.6.a.x.1.2 2 8.3 odd 2
504.6.a.m.1.1 2 12.11 even 2
784.6.a.q.1.1 2 7.6 odd 2
1008.6.a.bi.1.1 2 3.2 odd 2