# Properties

 Label 441.6.a.r.1.2 Level $441$ Weight $6$ Character 441.1 Self dual yes Analytic conductor $70.729$ Analytic rank $1$ 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] = [441,6,Mod(1,441)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(441, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("441.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$441 = 3^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 441.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: $$70.7292645375$$ Analytic rank: $$1$$ 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: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 147) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$7.44622$$ of defining polynomial Character $$\chi$$ $$=$$ 441.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+8.44622 q^{2} +39.3387 q^{4} +36.0000 q^{5} +61.9840 q^{8} +O(q^{10})$$ $$q+8.44622 q^{2} +39.3387 q^{4} +36.0000 q^{5} +61.9840 q^{8} +304.064 q^{10} -295.570 q^{11} -1148.13 q^{13} -735.307 q^{16} -1032.38 q^{17} +2108.51 q^{19} +1416.19 q^{20} -2496.45 q^{22} +640.988 q^{23} -1829.00 q^{25} -9697.34 q^{26} -7631.58 q^{29} -966.976 q^{31} -8194.05 q^{32} -8719.74 q^{34} -1773.21 q^{37} +17809.0 q^{38} +2231.42 q^{40} +11976.4 q^{41} -19802.9 q^{43} -11627.3 q^{44} +5413.93 q^{46} +27966.1 q^{47} -15448.1 q^{50} -45165.8 q^{52} +7114.33 q^{53} -10640.5 q^{55} -64458.0 q^{58} +20869.5 q^{59} -23868.3 q^{61} -8167.30 q^{62} -45679.0 q^{64} -41332.6 q^{65} +34671.5 q^{67} -40612.6 q^{68} +28413.2 q^{71} -15292.7 q^{73} -14976.9 q^{74} +82946.0 q^{76} -73059.5 q^{79} -26471.0 q^{80} +101155. q^{82} +30340.9 q^{83} -37165.8 q^{85} -167260. q^{86} -18320.6 q^{88} +36089.5 q^{89} +25215.6 q^{92} +236208. q^{94} +75906.4 q^{95} -153963. q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 3 q^{2} + 37 q^{4} + 72 q^{5} + 249 q^{8}+O(q^{10})$$ 2 * q + 3 * q^2 + 37 * q^4 + 72 * q^5 + 249 * q^8 $$2 q + 3 q^{2} + 37 q^{4} + 72 q^{5} + 249 q^{8} + 108 q^{10} - 480 q^{11} - 1296 q^{13} - 1679 q^{16} + 936 q^{17} + 216 q^{19} + 1332 q^{20} - 1492 q^{22} + 504 q^{23} - 3658 q^{25} - 8892 q^{26} - 6372 q^{29} - 9936 q^{31} - 9039 q^{32} - 19440 q^{34} + 11124 q^{37} + 28116 q^{38} + 8964 q^{40} + 20952 q^{41} - 6264 q^{43} - 11196 q^{44} + 6160 q^{46} + 7920 q^{47} - 5487 q^{50} - 44820 q^{52} - 2220 q^{53} - 17280 q^{55} - 71318 q^{58} + 29736 q^{59} + 17280 q^{61} + 40680 q^{62} - 10879 q^{64} - 46656 q^{65} - 20680 q^{67} - 45216 q^{68} + 92280 q^{71} - 56592 q^{73} - 85218 q^{74} + 87372 q^{76} - 56096 q^{79} - 60444 q^{80} + 52272 q^{82} - 71352 q^{83} + 33696 q^{85} - 240996 q^{86} - 52812 q^{88} + 123192 q^{89} + 25536 q^{92} + 345384 q^{94} + 7776 q^{95} - 35856 q^{97}+O(q^{100})$$ 2 * q + 3 * q^2 + 37 * q^4 + 72 * q^5 + 249 * q^8 + 108 * q^10 - 480 * q^11 - 1296 * q^13 - 1679 * q^16 + 936 * q^17 + 216 * q^19 + 1332 * q^20 - 1492 * q^22 + 504 * q^23 - 3658 * q^25 - 8892 * q^26 - 6372 * q^29 - 9936 * q^31 - 9039 * q^32 - 19440 * q^34 + 11124 * q^37 + 28116 * q^38 + 8964 * q^40 + 20952 * q^41 - 6264 * q^43 - 11196 * q^44 + 6160 * q^46 + 7920 * q^47 - 5487 * q^50 - 44820 * q^52 - 2220 * q^53 - 17280 * q^55 - 71318 * q^58 + 29736 * q^59 + 17280 * q^61 + 40680 * q^62 - 10879 * q^64 - 46656 * q^65 - 20680 * q^67 - 45216 * q^68 + 92280 * q^71 - 56592 * q^73 - 85218 * q^74 + 87372 * q^76 - 56096 * q^79 - 60444 * q^80 + 52272 * q^82 - 71352 * q^83 + 33696 * q^85 - 240996 * q^86 - 52812 * q^88 + 123192 * q^89 + 25536 * q^92 + 345384 * q^94 + 7776 * q^95 - 35856 * q^97

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 8.44622 1.49310 0.746548 0.665332i $$-0.231710\pi$$
0.746548 + 0.665332i $$0.231710\pi$$
$$3$$ 0 0
$$4$$ 39.3387 1.22933
$$5$$ 36.0000 0.643988 0.321994 0.946742i $$-0.395647\pi$$
0.321994 + 0.946742i $$0.395647\pi$$
$$6$$ 0 0
$$7$$ 0 0
$$8$$ 61.9840 0.342416
$$9$$ 0 0
$$10$$ 304.064 0.961535
$$11$$ −295.570 −0.736509 −0.368255 0.929725i $$-0.620045\pi$$
−0.368255 + 0.929725i $$0.620045\pi$$
$$12$$ 0 0
$$13$$ −1148.13 −1.88422 −0.942111 0.335302i $$-0.891162\pi$$
−0.942111 + 0.335302i $$0.891162\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ −735.307 −0.718073
$$17$$ −1032.38 −0.866401 −0.433200 0.901298i $$-0.642616\pi$$
−0.433200 + 0.901298i $$0.642616\pi$$
$$18$$ 0 0
$$19$$ 2108.51 1.33996 0.669980 0.742379i $$-0.266303\pi$$
0.669980 + 0.742379i $$0.266303\pi$$
$$20$$ 1416.19 0.791675
$$21$$ 0 0
$$22$$ −2496.45 −1.09968
$$23$$ 640.988 0.252657 0.126328 0.991988i $$-0.459681\pi$$
0.126328 + 0.991988i $$0.459681\pi$$
$$24$$ 0 0
$$25$$ −1829.00 −0.585280
$$26$$ −9697.34 −2.81332
$$27$$ 0 0
$$28$$ 0 0
$$29$$ −7631.58 −1.68508 −0.842538 0.538637i $$-0.818940\pi$$
−0.842538 + 0.538637i $$0.818940\pi$$
$$30$$ 0 0
$$31$$ −966.976 −0.180722 −0.0903611 0.995909i $$-0.528802\pi$$
−0.0903611 + 0.995909i $$0.528802\pi$$
$$32$$ −8194.05 −1.41457
$$33$$ 0 0
$$34$$ −8719.74 −1.29362
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −1773.21 −0.212939 −0.106470 0.994316i $$-0.533955\pi$$
−0.106470 + 0.994316i $$0.533955\pi$$
$$38$$ 17809.0 2.00069
$$39$$ 0 0
$$40$$ 2231.42 0.220512
$$41$$ 11976.4 1.11267 0.556335 0.830958i $$-0.312207\pi$$
0.556335 + 0.830958i $$0.312207\pi$$
$$42$$ 0 0
$$43$$ −19802.9 −1.63327 −0.816636 0.577153i $$-0.804163\pi$$
−0.816636 + 0.577153i $$0.804163\pi$$
$$44$$ −11627.3 −0.905416
$$45$$ 0 0
$$46$$ 5413.93 0.377240
$$47$$ 27966.1 1.84666 0.923332 0.384002i $$-0.125455\pi$$
0.923332 + 0.384002i $$0.125455\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ −15448.1 −0.873879
$$51$$ 0 0
$$52$$ −45165.8 −2.31634
$$53$$ 7114.33 0.347892 0.173946 0.984755i $$-0.444348\pi$$
0.173946 + 0.984755i $$0.444348\pi$$
$$54$$ 0 0
$$55$$ −10640.5 −0.474303
$$56$$ 0 0
$$57$$ 0 0
$$58$$ −64458.0 −2.51598
$$59$$ 20869.5 0.780518 0.390259 0.920705i $$-0.372386\pi$$
0.390259 + 0.920705i $$0.372386\pi$$
$$60$$ 0 0
$$61$$ −23868.3 −0.821291 −0.410646 0.911795i $$-0.634697\pi$$
−0.410646 + 0.911795i $$0.634697\pi$$
$$62$$ −8167.30 −0.269835
$$63$$ 0 0
$$64$$ −45679.0 −1.39401
$$65$$ −41332.6 −1.21342
$$66$$ 0 0
$$67$$ 34671.5 0.943595 0.471798 0.881707i $$-0.343605\pi$$
0.471798 + 0.881707i $$0.343605\pi$$
$$68$$ −40612.6 −1.06510
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 28413.2 0.668921 0.334461 0.942410i $$-0.391446\pi$$
0.334461 + 0.942410i $$0.391446\pi$$
$$72$$ 0 0
$$73$$ −15292.7 −0.335874 −0.167937 0.985798i $$-0.553711\pi$$
−0.167937 + 0.985798i $$0.553711\pi$$
$$74$$ −14976.9 −0.317939
$$75$$ 0 0
$$76$$ 82946.0 1.64726
$$77$$ 0 0
$$78$$ 0 0
$$79$$ −73059.5 −1.31707 −0.658535 0.752550i $$-0.728824\pi$$
−0.658535 + 0.752550i $$0.728824\pi$$
$$80$$ −26471.0 −0.462430
$$81$$ 0 0
$$82$$ 101155. 1.66132
$$83$$ 30340.9 0.483429 0.241715 0.970347i $$-0.422290\pi$$
0.241715 + 0.970347i $$0.422290\pi$$
$$84$$ 0 0
$$85$$ −37165.8 −0.557951
$$86$$ −167260. −2.43863
$$87$$ 0 0
$$88$$ −18320.6 −0.252193
$$89$$ 36089.5 0.482954 0.241477 0.970407i $$-0.422368\pi$$
0.241477 + 0.970407i $$0.422368\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 25215.6 0.310599
$$93$$ 0 0
$$94$$ 236208. 2.75725
$$95$$ 75906.4 0.862918
$$96$$ 0 0
$$97$$ −153963. −1.66145 −0.830724 0.556685i $$-0.812073\pi$$
−0.830724 + 0.556685i $$0.812073\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ −71950.4 −0.719504
$$101$$ −139809. −1.36374 −0.681869 0.731474i $$-0.738833\pi$$
−0.681869 + 0.731474i $$0.738833\pi$$
$$102$$ 0 0
$$103$$ −115925. −1.07668 −0.538339 0.842728i $$-0.680948\pi$$
−0.538339 + 0.842728i $$0.680948\pi$$
$$104$$ −71165.6 −0.645188
$$105$$ 0 0
$$106$$ 60089.2 0.519436
$$107$$ −83061.8 −0.701361 −0.350681 0.936495i $$-0.614050\pi$$
−0.350681 + 0.936495i $$0.614050\pi$$
$$108$$ 0 0
$$109$$ 45356.2 0.365654 0.182827 0.983145i $$-0.441475\pi$$
0.182827 + 0.983145i $$0.441475\pi$$
$$110$$ −89872.1 −0.708179
$$111$$ 0 0
$$112$$ 0 0
$$113$$ 355.533 0.00261929 0.00130965 0.999999i $$-0.499583\pi$$
0.00130965 + 0.999999i $$0.499583\pi$$
$$114$$ 0 0
$$115$$ 23075.6 0.162708
$$116$$ −300216. −2.07152
$$117$$ 0 0
$$118$$ 176269. 1.16539
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −73689.5 −0.457554
$$122$$ −201597. −1.22627
$$123$$ 0 0
$$124$$ −38039.6 −0.222168
$$125$$ −178344. −1.02090
$$126$$ 0 0
$$127$$ 168967. 0.929593 0.464797 0.885417i $$-0.346127\pi$$
0.464797 + 0.885417i $$0.346127\pi$$
$$128$$ −123605. −0.666824
$$129$$ 0 0
$$130$$ −349104. −1.81174
$$131$$ 173969. 0.885715 0.442858 0.896592i $$-0.353965\pi$$
0.442858 + 0.896592i $$0.353965\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 292843. 1.40888
$$135$$ 0 0
$$136$$ −63991.3 −0.296670
$$137$$ −367723. −1.67386 −0.836931 0.547308i $$-0.815653\pi$$
−0.836931 + 0.547308i $$0.815653\pi$$
$$138$$ 0 0
$$139$$ 217967. 0.956870 0.478435 0.878123i $$-0.341204\pi$$
0.478435 + 0.878123i $$0.341204\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 239985. 0.998763
$$143$$ 339352. 1.38775
$$144$$ 0 0
$$145$$ −274737. −1.08517
$$146$$ −129165. −0.501492
$$147$$ 0 0
$$148$$ −69755.7 −0.261773
$$149$$ 64906.1 0.239508 0.119754 0.992804i $$-0.461789\pi$$
0.119754 + 0.992804i $$0.461789\pi$$
$$150$$ 0 0
$$151$$ −223777. −0.798681 −0.399341 0.916803i $$-0.630761\pi$$
−0.399341 + 0.916803i $$0.630761\pi$$
$$152$$ 130694. 0.458825
$$153$$ 0 0
$$154$$ 0 0
$$155$$ −34811.1 −0.116383
$$156$$ 0 0
$$157$$ −459973. −1.48930 −0.744652 0.667453i $$-0.767384\pi$$
−0.744652 + 0.667453i $$0.767384\pi$$
$$158$$ −617077. −1.96651
$$159$$ 0 0
$$160$$ −294986. −0.910964
$$161$$ 0 0
$$162$$ 0 0
$$163$$ 91068.6 0.268472 0.134236 0.990949i $$-0.457142\pi$$
0.134236 + 0.990949i $$0.457142\pi$$
$$164$$ 471135. 1.36784
$$165$$ 0 0
$$166$$ 256266. 0.721806
$$167$$ 314772. 0.873384 0.436692 0.899611i $$-0.356150\pi$$
0.436692 + 0.899611i $$0.356150\pi$$
$$168$$ 0 0
$$169$$ 946905. 2.55029
$$170$$ −313911. −0.833075
$$171$$ 0 0
$$172$$ −779021. −2.00784
$$173$$ 362143. 0.919951 0.459975 0.887932i $$-0.347858\pi$$
0.459975 + 0.887932i $$0.347858\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 217334. 0.528867
$$177$$ 0 0
$$178$$ 304820. 0.721096
$$179$$ 173896. 0.405656 0.202828 0.979214i $$-0.434987\pi$$
0.202828 + 0.979214i $$0.434987\pi$$
$$180$$ 0 0
$$181$$ 134973. 0.306233 0.153116 0.988208i $$-0.451069\pi$$
0.153116 + 0.988208i $$0.451069\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 39731.0 0.0865138
$$185$$ −63835.6 −0.137130
$$186$$ 0 0
$$187$$ 305141. 0.638112
$$188$$ 1.10015e6 2.27017
$$189$$ 0 0
$$190$$ 641123. 1.28842
$$191$$ −181413. −0.359821 −0.179910 0.983683i $$-0.557581\pi$$
−0.179910 + 0.983683i $$0.557581\pi$$
$$192$$ 0 0
$$193$$ 965999. 1.86674 0.933369 0.358919i $$-0.116855\pi$$
0.933369 + 0.358919i $$0.116855\pi$$
$$194$$ −1.30040e6 −2.48070
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 699058. 1.28336 0.641679 0.766974i $$-0.278238\pi$$
0.641679 + 0.766974i $$0.278238\pi$$
$$198$$ 0 0
$$199$$ −416191. −0.745006 −0.372503 0.928031i $$-0.621500\pi$$
−0.372503 + 0.928031i $$0.621500\pi$$
$$200$$ −113369. −0.200410
$$201$$ 0 0
$$202$$ −1.18086e6 −2.03619
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 431150. 0.716545
$$206$$ −979132. −1.60758
$$207$$ 0 0
$$208$$ 844226. 1.35301
$$209$$ −623212. −0.986894
$$210$$ 0 0
$$211$$ −407152. −0.629580 −0.314790 0.949161i $$-0.601934\pi$$
−0.314790 + 0.949161i $$0.601934\pi$$
$$212$$ 279868. 0.427675
$$213$$ 0 0
$$214$$ −701558. −1.04720
$$215$$ −712906. −1.05181
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 383089. 0.545957
$$219$$ 0 0
$$220$$ −418584. −0.583076
$$221$$ 1.18531e6 1.63249
$$222$$ 0 0
$$223$$ 882022. 1.18773 0.593865 0.804565i $$-0.297602\pi$$
0.593865 + 0.804565i $$0.297602\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 3002.91 0.00391085
$$227$$ 1.12650e6 1.45100 0.725499 0.688223i $$-0.241609\pi$$
0.725499 + 0.688223i $$0.241609\pi$$
$$228$$ 0 0
$$229$$ 310084. 0.390743 0.195371 0.980729i $$-0.437409\pi$$
0.195371 + 0.980729i $$0.437409\pi$$
$$230$$ 194902. 0.242938
$$231$$ 0 0
$$232$$ −473036. −0.576998
$$233$$ −1.13654e6 −1.37149 −0.685746 0.727841i $$-0.740524\pi$$
−0.685746 + 0.727841i $$0.740524\pi$$
$$234$$ 0 0
$$235$$ 1.00678e6 1.18923
$$236$$ 820980. 0.959516
$$237$$ 0 0
$$238$$ 0 0
$$239$$ −87506.8 −0.0990940 −0.0495470 0.998772i $$-0.515778\pi$$
−0.0495470 + 0.998772i $$0.515778\pi$$
$$240$$ 0 0
$$241$$ 537768. 0.596421 0.298210 0.954500i $$-0.403610\pi$$
0.298210 + 0.954500i $$0.403610\pi$$
$$242$$ −622398. −0.683171
$$243$$ 0 0
$$244$$ −938948. −1.00964
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −2.42084e6 −2.52478
$$248$$ −59937.1 −0.0618823
$$249$$ 0 0
$$250$$ −1.50633e6 −1.52430
$$251$$ −1.35353e6 −1.35607 −0.678036 0.735028i $$-0.737169\pi$$
−0.678036 + 0.735028i $$0.737169\pi$$
$$252$$ 0 0
$$253$$ −189457. −0.186084
$$254$$ 1.42713e6 1.38797
$$255$$ 0 0
$$256$$ 417731. 0.398380
$$257$$ 976900. 0.922608 0.461304 0.887242i $$-0.347382\pi$$
0.461304 + 0.887242i $$0.347382\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ −1.62597e6 −1.49169
$$261$$ 0 0
$$262$$ 1.46938e6 1.32246
$$263$$ 1.24375e6 1.10877 0.554387 0.832259i $$-0.312953\pi$$
0.554387 + 0.832259i $$0.312953\pi$$
$$264$$ 0 0
$$265$$ 256116. 0.224038
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 1.36393e6 1.15999
$$269$$ 1.08408e6 0.913445 0.456722 0.889609i $$-0.349023\pi$$
0.456722 + 0.889609i $$0.349023\pi$$
$$270$$ 0 0
$$271$$ −2.16627e6 −1.79180 −0.895900 0.444256i $$-0.853468\pi$$
−0.895900 + 0.444256i $$0.853468\pi$$
$$272$$ 759119. 0.622139
$$273$$ 0 0
$$274$$ −3.10587e6 −2.49924
$$275$$ 540597. 0.431064
$$276$$ 0 0
$$277$$ 253859. 0.198789 0.0993946 0.995048i $$-0.468309\pi$$
0.0993946 + 0.995048i $$0.468309\pi$$
$$278$$ 1.84099e6 1.42870
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −1.14116e6 −0.862143 −0.431072 0.902318i $$-0.641864\pi$$
−0.431072 + 0.902318i $$0.641864\pi$$
$$282$$ 0 0
$$283$$ −609918. −0.452694 −0.226347 0.974047i $$-0.572678\pi$$
−0.226347 + 0.974047i $$0.572678\pi$$
$$284$$ 1.11774e6 0.822327
$$285$$ 0 0
$$286$$ 2.86624e6 2.07204
$$287$$ 0 0
$$288$$ 0 0
$$289$$ −354040. −0.249349
$$290$$ −2.32049e6 −1.62026
$$291$$ 0 0
$$292$$ −601593. −0.412901
$$293$$ −156438. −0.106457 −0.0532283 0.998582i $$-0.516951\pi$$
−0.0532283 + 0.998582i $$0.516951\pi$$
$$294$$ 0 0
$$295$$ 751303. 0.502644
$$296$$ −109911. −0.0729139
$$297$$ 0 0
$$298$$ 548211. 0.357608
$$299$$ −735937. −0.476061
$$300$$ 0 0
$$301$$ 0 0
$$302$$ −1.89007e6 −1.19251
$$303$$ 0 0
$$304$$ −1.55040e6 −0.962189
$$305$$ −859259. −0.528901
$$306$$ 0 0
$$307$$ 293229. 0.177566 0.0887831 0.996051i $$-0.471702\pi$$
0.0887831 + 0.996051i $$0.471702\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ −294023. −0.173771
$$311$$ −2.45216e6 −1.43763 −0.718816 0.695200i $$-0.755316\pi$$
−0.718816 + 0.695200i $$0.755316\pi$$
$$312$$ 0 0
$$313$$ 1.83541e6 1.05894 0.529471 0.848328i $$-0.322390\pi$$
0.529471 + 0.848328i $$0.322390\pi$$
$$314$$ −3.88503e6 −2.22367
$$315$$ 0 0
$$316$$ −2.87406e6 −1.61912
$$317$$ −589960. −0.329742 −0.164871 0.986315i $$-0.552721\pi$$
−0.164871 + 0.986315i $$0.552721\pi$$
$$318$$ 0 0
$$319$$ 2.25567e6 1.24107
$$320$$ −1.64444e6 −0.897726
$$321$$ 0 0
$$322$$ 0 0
$$323$$ −2.17679e6 −1.16094
$$324$$ 0 0
$$325$$ 2.09993e6 1.10280
$$326$$ 769186. 0.400855
$$327$$ 0 0
$$328$$ 742344. 0.380996
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 177318. 0.0889577 0.0444789 0.999010i $$-0.485837\pi$$
0.0444789 + 0.999010i $$0.485837\pi$$
$$332$$ 1.19357e6 0.594296
$$333$$ 0 0
$$334$$ 2.65864e6 1.30405
$$335$$ 1.24817e6 0.607664
$$336$$ 0 0
$$337$$ −3.04781e6 −1.46189 −0.730943 0.682438i $$-0.760920\pi$$
−0.730943 + 0.682438i $$0.760920\pi$$
$$338$$ 7.99777e6 3.80783
$$339$$ 0 0
$$340$$ −1.46205e6 −0.685908
$$341$$ 285809. 0.133104
$$342$$ 0 0
$$343$$ 0 0
$$344$$ −1.22747e6 −0.559259
$$345$$ 0 0
$$346$$ 3.05874e6 1.37357
$$347$$ 2.42361e6 1.08054 0.540268 0.841493i $$-0.318323\pi$$
0.540268 + 0.841493i $$0.318323\pi$$
$$348$$ 0 0
$$349$$ −2.67690e6 −1.17644 −0.588218 0.808702i $$-0.700170\pi$$
−0.588218 + 0.808702i $$0.700170\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 2.42191e6 1.04184
$$353$$ −950412. −0.405953 −0.202976 0.979184i $$-0.565061\pi$$
−0.202976 + 0.979184i $$0.565061\pi$$
$$354$$ 0 0
$$355$$ 1.02288e6 0.430777
$$356$$ 1.41971e6 0.593711
$$357$$ 0 0
$$358$$ 1.46877e6 0.605683
$$359$$ −2.78881e6 −1.14204 −0.571022 0.820935i $$-0.693453\pi$$
−0.571022 + 0.820935i $$0.693453\pi$$
$$360$$ 0 0
$$361$$ 1.96972e6 0.795495
$$362$$ 1.14001e6 0.457235
$$363$$ 0 0
$$364$$ 0 0
$$365$$ −550536. −0.216299
$$366$$ 0 0
$$367$$ 153881. 0.0596377 0.0298189 0.999555i $$-0.490507\pi$$
0.0298189 + 0.999555i $$0.490507\pi$$
$$368$$ −471323. −0.181426
$$369$$ 0 0
$$370$$ −539169. −0.204749
$$371$$ 0 0
$$372$$ 0 0
$$373$$ −2.38381e6 −0.887156 −0.443578 0.896236i $$-0.646291\pi$$
−0.443578 + 0.896236i $$0.646291\pi$$
$$374$$ 2.57729e6 0.952763
$$375$$ 0 0
$$376$$ 1.73345e6 0.632328
$$377$$ 8.76203e6 3.17506
$$378$$ 0 0
$$379$$ 3.65191e6 1.30594 0.652969 0.757385i $$-0.273523\pi$$
0.652969 + 0.757385i $$0.273523\pi$$
$$380$$ 2.98606e6 1.06081
$$381$$ 0 0
$$382$$ −1.53226e6 −0.537246
$$383$$ −2.15730e6 −0.751472 −0.375736 0.926727i $$-0.622610\pi$$
−0.375736 + 0.926727i $$0.622610\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 8.15904e6 2.78722
$$387$$ 0 0
$$388$$ −6.05669e6 −2.04247
$$389$$ 3.66471e6 1.22791 0.613954 0.789342i $$-0.289578\pi$$
0.613954 + 0.789342i $$0.289578\pi$$
$$390$$ 0 0
$$391$$ −661746. −0.218902
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 5.90440e6 1.91617
$$395$$ −2.63014e6 −0.848177
$$396$$ 0 0
$$397$$ −3.94648e6 −1.25671 −0.628353 0.777928i $$-0.716271\pi$$
−0.628353 + 0.777928i $$0.716271\pi$$
$$398$$ −3.51524e6 −1.11236
$$399$$ 0 0
$$400$$ 1.34488e6 0.420274
$$401$$ −25016.1 −0.00776887 −0.00388444 0.999992i $$-0.501236\pi$$
−0.00388444 + 0.999992i $$0.501236\pi$$
$$402$$ 0 0
$$403$$ 1.11021e6 0.340521
$$404$$ −5.49989e6 −1.67649
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 524107. 0.156832
$$408$$ 0 0
$$409$$ 832700. 0.246139 0.123069 0.992398i $$-0.460726\pi$$
0.123069 + 0.992398i $$0.460726\pi$$
$$410$$ 3.64159e6 1.06987
$$411$$ 0 0
$$412$$ −4.56035e6 −1.32360
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 1.09227e6 0.311323
$$416$$ 9.40782e6 2.66536
$$417$$ 0 0
$$418$$ −5.26379e6 −1.47353
$$419$$ −3.95178e6 −1.09966 −0.549828 0.835278i $$-0.685307\pi$$
−0.549828 + 0.835278i $$0.685307\pi$$
$$420$$ 0 0
$$421$$ 4.72285e6 1.29867 0.649336 0.760502i $$-0.275047\pi$$
0.649336 + 0.760502i $$0.275047\pi$$
$$422$$ −3.43890e6 −0.940023
$$423$$ 0 0
$$424$$ 440974. 0.119124
$$425$$ 1.88823e6 0.507087
$$426$$ 0 0
$$427$$ 0 0
$$428$$ −3.26754e6 −0.862207
$$429$$ 0 0
$$430$$ −6.02136e6 −1.57045
$$431$$ −4.07810e6 −1.05746 −0.528731 0.848790i $$-0.677332\pi$$
−0.528731 + 0.848790i $$0.677332\pi$$
$$432$$ 0 0
$$433$$ −1.79927e6 −0.461186 −0.230593 0.973050i $$-0.574067\pi$$
−0.230593 + 0.973050i $$0.574067\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 1.78425e6 0.449511
$$437$$ 1.35153e6 0.338550
$$438$$ 0 0
$$439$$ 4.51827e6 1.11895 0.559475 0.828847i $$-0.311003\pi$$
0.559475 + 0.828847i $$0.311003\pi$$
$$440$$ −659542. −0.162409
$$441$$ 0 0
$$442$$ 1.00114e7 2.43746
$$443$$ 2.85256e6 0.690597 0.345299 0.938493i $$-0.387778\pi$$
0.345299 + 0.938493i $$0.387778\pi$$
$$444$$ 0 0
$$445$$ 1.29922e6 0.311016
$$446$$ 7.44976e6 1.77339
$$447$$ 0 0
$$448$$ 0 0
$$449$$ 1.90246e6 0.445348 0.222674 0.974893i $$-0.428522\pi$$
0.222674 + 0.974893i $$0.428522\pi$$
$$450$$ 0 0
$$451$$ −3.53986e6 −0.819491
$$452$$ 13986.2 0.00321998
$$453$$ 0 0
$$454$$ 9.51468e6 2.16648
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 2.64834e6 0.593176 0.296588 0.955006i $$-0.404151\pi$$
0.296588 + 0.955006i $$0.404151\pi$$
$$458$$ 2.61904e6 0.583416
$$459$$ 0 0
$$460$$ 907763. 0.200022
$$461$$ −1.09031e6 −0.238944 −0.119472 0.992838i $$-0.538120\pi$$
−0.119472 + 0.992838i $$0.538120\pi$$
$$462$$ 0 0
$$463$$ −2.50851e6 −0.543831 −0.271916 0.962321i $$-0.587657\pi$$
−0.271916 + 0.962321i $$0.587657\pi$$
$$464$$ 5.61155e6 1.21001
$$465$$ 0 0
$$466$$ −9.59943e6 −2.04777
$$467$$ 3.20935e6 0.680966 0.340483 0.940251i $$-0.389409\pi$$
0.340483 + 0.940251i $$0.389409\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 8.50350e6 1.77563
$$471$$ 0 0
$$472$$ 1.29358e6 0.267262
$$473$$ 5.85315e6 1.20292
$$474$$ 0 0
$$475$$ −3.85647e6 −0.784252
$$476$$ 0 0
$$477$$ 0 0
$$478$$ −739102. −0.147957
$$479$$ −2.31462e6 −0.460936 −0.230468 0.973080i $$-0.574026\pi$$
−0.230468 + 0.973080i $$0.574026\pi$$
$$480$$ 0 0
$$481$$ 2.03587e6 0.401225
$$482$$ 4.54211e6 0.890513
$$483$$ 0 0
$$484$$ −2.89885e6 −0.562486
$$485$$ −5.54266e6 −1.06995
$$486$$ 0 0
$$487$$ −4.63735e6 −0.886028 −0.443014 0.896515i $$-0.646091\pi$$
−0.443014 + 0.896515i $$0.646091\pi$$
$$488$$ −1.47945e6 −0.281224
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −5.02151e6 −0.940007 −0.470003 0.882665i $$-0.655747\pi$$
−0.470003 + 0.882665i $$0.655747\pi$$
$$492$$ 0 0
$$493$$ 7.87872e6 1.45995
$$494$$ −2.04470e7 −3.76974
$$495$$ 0 0
$$496$$ 711024. 0.129772
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 3.37822e6 0.607347 0.303673 0.952776i $$-0.401787\pi$$
0.303673 + 0.952776i $$0.401787\pi$$
$$500$$ −7.01582e6 −1.25503
$$501$$ 0 0
$$502$$ −1.14322e7 −2.02475
$$503$$ −5.03743e6 −0.887747 −0.443873 0.896090i $$-0.646396\pi$$
−0.443873 + 0.896090i $$0.646396\pi$$
$$504$$ 0 0
$$505$$ −5.03311e6 −0.878230
$$506$$ −1.60019e6 −0.277841
$$507$$ 0 0
$$508$$ 6.64694e6 1.14278
$$509$$ −6.72466e6 −1.15047 −0.575236 0.817988i $$-0.695090\pi$$
−0.575236 + 0.817988i $$0.695090\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 7.48361e6 1.26164
$$513$$ 0 0
$$514$$ 8.25112e6 1.37754
$$515$$ −4.17332e6 −0.693367
$$516$$ 0 0
$$517$$ −8.26595e6 −1.36009
$$518$$ 0 0
$$519$$ 0 0
$$520$$ −2.56196e6 −0.415493
$$521$$ −4.42770e6 −0.714635 −0.357317 0.933983i $$-0.616309\pi$$
−0.357317 + 0.933983i $$0.616309\pi$$
$$522$$ 0 0
$$523$$ 8.95911e6 1.43222 0.716111 0.697986i $$-0.245920\pi$$
0.716111 + 0.697986i $$0.245920\pi$$
$$524$$ 6.84372e6 1.08884
$$525$$ 0 0
$$526$$ 1.05050e7 1.65551
$$527$$ 998291. 0.156578
$$528$$ 0 0
$$529$$ −6.02548e6 −0.936165
$$530$$ 2.16321e6 0.334510
$$531$$ 0 0
$$532$$ 0 0
$$533$$ −1.37504e7 −2.09652
$$534$$ 0 0
$$535$$ −2.99022e6 −0.451668
$$536$$ 2.14908e6 0.323103
$$537$$ 0 0
$$538$$ 9.15641e6 1.36386
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −1.00467e7 −1.47581 −0.737907 0.674902i $$-0.764186\pi$$
−0.737907 + 0.674902i $$0.764186\pi$$
$$542$$ −1.82968e7 −2.67533
$$543$$ 0 0
$$544$$ 8.45941e6 1.22558
$$545$$ 1.63282e6 0.235477
$$546$$ 0 0
$$547$$ −1.31426e7 −1.87808 −0.939039 0.343811i $$-0.888282\pi$$
−0.939039 + 0.343811i $$0.888282\pi$$
$$548$$ −1.44657e7 −2.05774
$$549$$ 0 0
$$550$$ 4.56600e6 0.643620
$$551$$ −1.60913e7 −2.25794
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 2.14415e6 0.296811
$$555$$ 0 0
$$556$$ 8.57452e6 1.17631
$$557$$ −9.06752e6 −1.23837 −0.619185 0.785245i $$-0.712537\pi$$
−0.619185 + 0.785245i $$0.712537\pi$$
$$558$$ 0 0
$$559$$ 2.27363e7 3.07744
$$560$$ 0 0
$$561$$ 0 0
$$562$$ −9.63846e6 −1.28726
$$563$$ 1.05180e7 1.39849 0.699247 0.714880i $$-0.253519\pi$$
0.699247 + 0.714880i $$0.253519\pi$$
$$564$$ 0 0
$$565$$ 12799.2 0.00168679
$$566$$ −5.15150e6 −0.675916
$$567$$ 0 0
$$568$$ 1.76117e6 0.229050
$$569$$ −7.32307e6 −0.948227 −0.474114 0.880464i $$-0.657231\pi$$
−0.474114 + 0.880464i $$0.657231\pi$$
$$570$$ 0 0
$$571$$ −6.97981e6 −0.895887 −0.447943 0.894062i $$-0.647843\pi$$
−0.447943 + 0.894062i $$0.647843\pi$$
$$572$$ 1.33497e7 1.70600
$$573$$ 0 0
$$574$$ 0 0
$$575$$ −1.17237e6 −0.147875
$$576$$ 0 0
$$577$$ 5.81210e6 0.726765 0.363382 0.931640i $$-0.381622\pi$$
0.363382 + 0.931640i $$0.381622\pi$$
$$578$$ −2.99030e6 −0.372302
$$579$$ 0 0
$$580$$ −1.08078e7 −1.33403
$$581$$ 0 0
$$582$$ 0 0
$$583$$ −2.10278e6 −0.256226
$$584$$ −947901. −0.115009
$$585$$ 0 0
$$586$$ −1.32131e6 −0.158950
$$587$$ −7.37446e6 −0.883355 −0.441677 0.897174i $$-0.645616\pi$$
−0.441677 + 0.897174i $$0.645616\pi$$
$$588$$ 0 0
$$589$$ −2.03888e6 −0.242161
$$590$$ 6.34567e6 0.750495
$$591$$ 0 0
$$592$$ 1.30385e6 0.152906
$$593$$ 9.46528e6 1.10534 0.552671 0.833399i $$-0.313609\pi$$
0.552671 + 0.833399i $$0.313609\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 2.55332e6 0.294435
$$597$$ 0 0
$$598$$ −6.21589e6 −0.710804
$$599$$ 8.52195e6 0.970448 0.485224 0.874390i $$-0.338738\pi$$
0.485224 + 0.874390i $$0.338738\pi$$
$$600$$ 0 0
$$601$$ 657065. 0.0742031 0.0371016 0.999311i $$-0.488187\pi$$
0.0371016 + 0.999311i $$0.488187\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ −8.80310e6 −0.981846
$$605$$ −2.65282e6 −0.294659
$$606$$ 0 0
$$607$$ −5.86885e6 −0.646519 −0.323260 0.946310i $$-0.604779\pi$$
−0.323260 + 0.946310i $$0.604779\pi$$
$$608$$ −1.72773e7 −1.89547
$$609$$ 0 0
$$610$$ −7.25750e6 −0.789700
$$611$$ −3.21087e7 −3.47952
$$612$$ 0 0
$$613$$ 3.84402e6 0.413175 0.206588 0.978428i $$-0.433764\pi$$
0.206588 + 0.978428i $$0.433764\pi$$
$$614$$ 2.47667e6 0.265123
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −6.44660e6 −0.681739 −0.340869 0.940111i $$-0.610721\pi$$
−0.340869 + 0.940111i $$0.610721\pi$$
$$618$$ 0 0
$$619$$ −6.73740e6 −0.706749 −0.353375 0.935482i $$-0.614966\pi$$
−0.353375 + 0.935482i $$0.614966\pi$$
$$620$$ −1.36942e6 −0.143073
$$621$$ 0 0
$$622$$ −2.07115e7 −2.14652
$$623$$ 0 0
$$624$$ 0 0
$$625$$ −704759. −0.0721673
$$626$$ 1.55023e7 1.58110
$$627$$ 0 0
$$628$$ −1.80947e7 −1.83085
$$629$$ 1.83063e6 0.184491
$$630$$ 0 0
$$631$$ −9.14514e6 −0.914360 −0.457180 0.889374i $$-0.651140\pi$$
−0.457180 + 0.889374i $$0.651140\pi$$
$$632$$ −4.52852e6 −0.450987
$$633$$ 0 0
$$634$$ −4.98293e6 −0.492336
$$635$$ 6.08282e6 0.598647
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 1.90518e7 1.85304
$$639$$ 0 0
$$640$$ −4.44978e6 −0.429426
$$641$$ 1.04088e6 0.100059 0.0500296 0.998748i $$-0.484068\pi$$
0.0500296 + 0.998748i $$0.484068\pi$$
$$642$$ 0 0
$$643$$ 9.08713e6 0.866761 0.433381 0.901211i $$-0.357321\pi$$
0.433381 + 0.901211i $$0.357321\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ −1.83857e7 −1.73340
$$647$$ 2.20711e6 0.207283 0.103641 0.994615i $$-0.466951\pi$$
0.103641 + 0.994615i $$0.466951\pi$$
$$648$$ 0 0
$$649$$ −6.16840e6 −0.574859
$$650$$ 1.77364e7 1.64658
$$651$$ 0 0
$$652$$ 3.58252e6 0.330042
$$653$$ 1.83610e7 1.68505 0.842524 0.538658i $$-0.181069\pi$$
0.842524 + 0.538658i $$0.181069\pi$$
$$654$$ 0 0
$$655$$ 6.26289e6 0.570390
$$656$$ −8.80632e6 −0.798978
$$657$$ 0 0
$$658$$ 0 0
$$659$$ −6.21208e6 −0.557216 −0.278608 0.960405i $$-0.589873\pi$$
−0.278608 + 0.960405i $$0.589873\pi$$
$$660$$ 0 0
$$661$$ −1.54230e7 −1.37298 −0.686491 0.727138i $$-0.740850\pi$$
−0.686491 + 0.727138i $$0.740850\pi$$
$$662$$ 1.49767e6 0.132822
$$663$$ 0 0
$$664$$ 1.88065e6 0.165534
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −4.89176e6 −0.425746
$$668$$ 1.23827e7 1.07368
$$669$$ 0 0
$$670$$ 1.05424e7 0.907300
$$671$$ 7.05475e6 0.604889
$$672$$ 0 0
$$673$$ −2.27201e7 −1.93362 −0.966811 0.255491i $$-0.917763\pi$$
−0.966811 + 0.255491i $$0.917763\pi$$
$$674$$ −2.57425e7 −2.18274
$$675$$ 0 0
$$676$$ 3.72500e7 3.13516
$$677$$ 1.36173e7 1.14188 0.570940 0.820992i $$-0.306579\pi$$
0.570940 + 0.820992i $$0.306579\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ −2.30369e6 −0.191052
$$681$$ 0 0
$$682$$ 2.41401e6 0.198736
$$683$$ −2.39985e6 −0.196849 −0.0984245 0.995145i $$-0.531380\pi$$
−0.0984245 + 0.995145i $$0.531380\pi$$
$$684$$ 0 0
$$685$$ −1.32380e7 −1.07795
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 1.45612e7 1.17281
$$689$$ −8.16816e6 −0.655505
$$690$$ 0 0
$$691$$ 1.40365e6 0.111831 0.0559156 0.998435i $$-0.482192\pi$$
0.0559156 + 0.998435i $$0.482192\pi$$
$$692$$ 1.42462e7 1.13093
$$693$$ 0 0
$$694$$ 2.04703e7 1.61334
$$695$$ 7.84680e6 0.616212
$$696$$ 0 0
$$697$$ −1.23642e7 −0.964018
$$698$$ −2.26097e7 −1.75653
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 5.78991e6 0.445017 0.222509 0.974931i $$-0.428575\pi$$
0.222509 + 0.974931i $$0.428575\pi$$
$$702$$ 0 0
$$703$$ −3.73884e6 −0.285330
$$704$$ 1.35013e7 1.02670
$$705$$ 0 0
$$706$$ −8.02739e6 −0.606126
$$707$$ 0 0
$$708$$ 0 0
$$709$$ −1.13143e7 −0.845304 −0.422652 0.906292i $$-0.638901\pi$$
−0.422652 + 0.906292i $$0.638901\pi$$
$$710$$ 8.63944e6 0.643191
$$711$$ 0 0
$$712$$ 2.23697e6 0.165371
$$713$$ −619821. −0.0456607
$$714$$ 0 0
$$715$$ 1.22167e7 0.893692
$$716$$ 6.84085e6 0.498686
$$717$$ 0 0
$$718$$ −2.35549e7 −1.70518
$$719$$ −2.73780e7 −1.97506 −0.987529 0.157437i $$-0.949677\pi$$
−0.987529 + 0.157437i $$0.949677\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 1.66367e7 1.18775
$$723$$ 0 0
$$724$$ 5.30967e6 0.376462
$$725$$ 1.39582e7 0.986241
$$726$$ 0 0
$$727$$ 9.86471e6 0.692226 0.346113 0.938193i $$-0.387501\pi$$
0.346113 + 0.938193i $$0.387501\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ −4.64995e6 −0.322954
$$731$$ 2.04442e7 1.41507
$$732$$ 0 0
$$733$$ −3.87876e6 −0.266645 −0.133322 0.991073i $$-0.542565\pi$$
−0.133322 + 0.991073i $$0.542565\pi$$
$$734$$ 1.29972e6 0.0890448
$$735$$ 0 0
$$736$$ −5.25229e6 −0.357400
$$737$$ −1.02479e7 −0.694967
$$738$$ 0 0
$$739$$ −7.95498e6 −0.535831 −0.267916 0.963442i $$-0.586335\pi$$
−0.267916 + 0.963442i $$0.586335\pi$$
$$740$$ −2.51121e6 −0.168579
$$741$$ 0 0
$$742$$ 0 0
$$743$$ −1.65977e7 −1.10300 −0.551500 0.834175i $$-0.685944\pi$$
−0.551500 + 0.834175i $$0.685944\pi$$
$$744$$ 0 0
$$745$$ 2.33662e6 0.154240
$$746$$ −2.01342e7 −1.32461
$$747$$ 0 0
$$748$$ 1.20039e7 0.784453
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 1.51072e7 0.977426 0.488713 0.872445i $$-0.337467\pi$$
0.488713 + 0.872445i $$0.337467\pi$$
$$752$$ −2.05637e7 −1.32604
$$753$$ 0 0
$$754$$ 7.40061e7 4.74066
$$755$$ −8.05598e6 −0.514341
$$756$$ 0 0
$$757$$ 5.80923e6 0.368450 0.184225 0.982884i $$-0.441022\pi$$
0.184225 + 0.982884i $$0.441022\pi$$
$$758$$ 3.08449e7 1.94989
$$759$$ 0 0
$$760$$ 4.70498e6 0.295477
$$761$$ 2.54270e7 1.59160 0.795799 0.605561i $$-0.207051\pi$$
0.795799 + 0.605561i $$0.207051\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ −7.13656e6 −0.442340
$$765$$ 0 0
$$766$$ −1.82210e7 −1.12202
$$767$$ −2.39609e7 −1.47067
$$768$$ 0 0
$$769$$ −1.53909e7 −0.938532 −0.469266 0.883057i $$-0.655481\pi$$
−0.469266 + 0.883057i $$0.655481\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 3.80011e7 2.29484
$$773$$ 905393. 0.0544990 0.0272495 0.999629i $$-0.491325\pi$$
0.0272495 + 0.999629i $$0.491325\pi$$
$$774$$ 0 0
$$775$$ 1.76860e6 0.105773
$$776$$ −9.54323e6 −0.568907
$$777$$ 0 0
$$778$$ 3.09530e7 1.83338
$$779$$ 2.52523e7 1.49093
$$780$$ 0 0
$$781$$ −8.39810e6 −0.492667
$$782$$ −5.58926e6 −0.326841
$$783$$ 0 0
$$784$$ 0 0
$$785$$ −1.65590e7 −0.959093
$$786$$ 0 0
$$787$$ 2.79334e7 1.60763 0.803817 0.594877i $$-0.202799\pi$$
0.803817 + 0.594877i $$0.202799\pi$$
$$788$$ 2.75000e7 1.57767
$$789$$ 0 0
$$790$$ −2.22148e7 −1.26641
$$791$$ 0 0
$$792$$ 0 0
$$793$$ 2.74039e7 1.54749
$$794$$ −3.33328e7 −1.87638
$$795$$ 0 0
$$796$$ −1.63724e7 −0.915860
$$797$$ 2.18824e7 1.22025 0.610126 0.792304i $$-0.291119\pi$$
0.610126 + 0.792304i $$0.291119\pi$$
$$798$$ 0 0
$$799$$ −2.88718e7 −1.59995
$$800$$ 1.49869e7 0.827918
$$801$$ 0 0
$$802$$ −211291. −0.0115997
$$803$$ 4.52005e6 0.247374
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 9.37710e6 0.508430
$$807$$ 0 0
$$808$$ −8.66590e6 −0.466966
$$809$$ −2.44194e7 −1.31179 −0.655893 0.754854i $$-0.727708\pi$$
−0.655893 + 0.754854i $$0.727708\pi$$
$$810$$ 0 0
$$811$$ −4.46711e6 −0.238492 −0.119246 0.992865i $$-0.538048\pi$$
−0.119246 + 0.992865i $$0.538048\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 4.42673e6 0.234165
$$815$$ 3.27847e6 0.172893
$$816$$ 0 0
$$817$$ −4.17547e7 −2.18852
$$818$$ 7.03317e6 0.367509
$$819$$ 0 0
$$820$$ 1.69609e7 0.880873
$$821$$ −1.34708e7 −0.697485 −0.348743 0.937219i $$-0.613391\pi$$
−0.348743 + 0.937219i $$0.613391\pi$$
$$822$$ 0 0
$$823$$ −4.03958e6 −0.207891 −0.103946 0.994583i $$-0.533147\pi$$
−0.103946 + 0.994583i $$0.533147\pi$$
$$824$$ −7.18552e6 −0.368672
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −1.24927e7 −0.635175 −0.317588 0.948229i $$-0.602873\pi$$
−0.317588 + 0.948229i $$0.602873\pi$$
$$828$$ 0 0
$$829$$ 1.45980e7 0.737749 0.368874 0.929479i $$-0.379743\pi$$
0.368874 + 0.929479i $$0.379743\pi$$
$$830$$ 9.22557e6 0.464834
$$831$$ 0 0
$$832$$ 5.24453e7 2.62663
$$833$$ 0 0
$$834$$ 0 0
$$835$$ 1.13318e7 0.562449
$$836$$ −2.45163e7 −1.21322
$$837$$ 0 0
$$838$$ −3.33776e7 −1.64189
$$839$$ −9.15983e6 −0.449244 −0.224622 0.974446i $$-0.572115\pi$$
−0.224622 + 0.974446i $$0.572115\pi$$
$$840$$ 0 0
$$841$$ 3.77299e7 1.83948
$$842$$ 3.98903e7 1.93904
$$843$$ 0 0
$$844$$ −1.60168e7 −0.773964
$$845$$ 3.40886e7 1.64236
$$846$$ 0 0
$$847$$ 0 0
$$848$$ −5.23121e6 −0.249812
$$849$$ 0 0
$$850$$ 1.59484e7 0.757129
$$851$$ −1.13661e6 −0.0538005
$$852$$ 0 0
$$853$$ 8.68253e6 0.408577 0.204289 0.978911i $$-0.434512\pi$$
0.204289 + 0.978911i $$0.434512\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ −5.14850e6 −0.240158
$$857$$ 1.04988e7 0.488302 0.244151 0.969737i $$-0.421491\pi$$
0.244151 + 0.969737i $$0.421491\pi$$
$$858$$ 0 0
$$859$$ −9.03780e6 −0.417907 −0.208954 0.977926i $$-0.567006\pi$$
−0.208954 + 0.977926i $$0.567006\pi$$
$$860$$ −2.80448e7 −1.29302
$$861$$ 0 0
$$862$$ −3.44445e7 −1.57889
$$863$$ 1.59858e7 0.730645 0.365322 0.930881i $$-0.380959\pi$$
0.365322 + 0.930881i $$0.380959\pi$$
$$864$$ 0 0
$$865$$ 1.30371e7 0.592437
$$866$$ −1.51970e7 −0.688594
$$867$$ 0 0
$$868$$ 0 0
$$869$$ 2.15942e7 0.970035
$$870$$ 0 0
$$871$$ −3.98073e7 −1.77794
$$872$$ 2.81136e6 0.125206
$$873$$ 0 0
$$874$$ 1.14153e7 0.505487
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 2.67453e7 1.17422 0.587109 0.809508i $$-0.300266\pi$$
0.587109 + 0.809508i $$0.300266\pi$$
$$878$$ 3.81623e7 1.67070
$$879$$ 0 0
$$880$$ 7.82404e6 0.340584
$$881$$ −6.40715e6 −0.278115 −0.139058 0.990284i $$-0.544407\pi$$
−0.139058 + 0.990284i $$0.544407\pi$$
$$882$$ 0 0
$$883$$ 4.96462e6 0.214281 0.107141 0.994244i $$-0.465831\pi$$
0.107141 + 0.994244i $$0.465831\pi$$
$$884$$ 4.66285e7 2.00688
$$885$$ 0 0
$$886$$ 2.40933e7 1.03113
$$887$$ 8.71625e6 0.371981 0.185990 0.982552i $$-0.440451\pi$$
0.185990 + 0.982552i $$0.440451\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 1.09735e7 0.464377
$$891$$ 0 0
$$892$$ 3.46976e7 1.46011
$$893$$ 5.89669e7 2.47446
$$894$$ 0 0
$$895$$ 6.26026e6 0.261237
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 1.60686e7 0.664946
$$899$$ 7.37956e6 0.304531
$$900$$ 0 0
$$901$$ −7.34472e6 −0.301414
$$902$$ −2.98984e7 −1.22358
$$903$$ 0 0
$$904$$ 22037.4 0.000896888 0
$$905$$ 4.85904e6 0.197210
$$906$$ 0 0
$$907$$ −2.36255e7 −0.953594 −0.476797 0.879014i $$-0.658202\pi$$
−0.476797 + 0.879014i $$0.658202\pi$$
$$908$$ 4.43151e7 1.78376
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 1.69360e7 0.676105 0.338053 0.941127i $$-0.390232\pi$$
0.338053 + 0.941127i $$0.390232\pi$$
$$912$$ 0 0
$$913$$ −8.96785e6 −0.356050
$$914$$ 2.23685e7 0.885668
$$915$$ 0 0
$$916$$ 1.21983e7 0.480353
$$917$$ 0 0
$$918$$ 0 0
$$919$$ −3.13804e7 −1.22566 −0.612829 0.790216i $$-0.709968\pi$$
−0.612829 + 0.790216i $$0.709968\pi$$
$$920$$ 1.43032e6 0.0557138
$$921$$ 0 0
$$922$$ −9.20896e6 −0.356766
$$923$$ −3.26220e7 −1.26040
$$924$$ 0 0
$$925$$ 3.24320e6 0.124629
$$926$$ −2.11875e7 −0.811992
$$927$$ 0 0
$$928$$ 6.25336e7 2.38365
$$929$$ −1.43089e7 −0.543959 −0.271980 0.962303i $$-0.587678\pi$$
−0.271980 + 0.962303i $$0.587678\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ −4.47098e7 −1.68602
$$933$$ 0 0
$$934$$ 2.71069e7 1.01675
$$935$$ 1.09851e7 0.410937
$$936$$ 0 0
$$937$$ −2.81206e7 −1.04635 −0.523173 0.852227i $$-0.675252\pi$$
−0.523173 + 0.852227i $$0.675252\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 3.96054e7 1.46196
$$941$$ −3.29569e7 −1.21331 −0.606656 0.794964i $$-0.707490\pi$$
−0.606656 + 0.794964i $$0.707490\pi$$
$$942$$ 0 0
$$943$$ 7.67672e6 0.281123
$$944$$ −1.53455e7 −0.560469
$$945$$ 0 0
$$946$$ 4.94370e7 1.79607
$$947$$ −1.62975e7 −0.590535 −0.295268 0.955415i $$-0.595409\pi$$
−0.295268 + 0.955415i $$0.595409\pi$$
$$948$$ 0 0
$$949$$ 1.75579e7 0.632861
$$950$$ −3.25726e7 −1.17096
$$951$$ 0 0
$$952$$ 0 0
$$953$$ 3.03230e7 1.08153 0.540767 0.841172i $$-0.318134\pi$$
0.540767 + 0.841172i $$0.318134\pi$$
$$954$$ 0 0
$$955$$ −6.53088e6 −0.231720
$$956$$ −3.44240e6 −0.121820
$$957$$ 0 0
$$958$$ −1.95498e7 −0.688221
$$959$$ 0 0
$$960$$ 0 0
$$961$$ −2.76941e7 −0.967339
$$962$$ 1.71954e7 0.599067
$$963$$ 0 0
$$964$$ 2.11551e7 0.733200
$$965$$ 3.47759e7 1.20216
$$966$$ 0 0
$$967$$ −1.49059e7 −0.512616 −0.256308 0.966595i $$-0.582506\pi$$
−0.256308 + 0.966595i $$0.582506\pi$$
$$968$$ −4.56757e6 −0.156674
$$969$$ 0 0
$$970$$ −4.68145e7 −1.59754
$$971$$ 1.65608e7 0.563679 0.281840 0.959462i $$-0.409055\pi$$
0.281840 + 0.959462i $$0.409055\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ −3.91681e7 −1.32292
$$975$$ 0 0
$$976$$ 1.75505e7 0.589747
$$977$$ −7.56862e6 −0.253676 −0.126838 0.991923i $$-0.540483\pi$$
−0.126838 + 0.991923i $$0.540483\pi$$
$$978$$ 0 0
$$979$$ −1.06670e7 −0.355700
$$980$$ 0 0
$$981$$ 0 0
$$982$$ −4.24128e7 −1.40352
$$983$$ −3.32042e7 −1.09600 −0.547998 0.836480i $$-0.684610\pi$$
−0.547998 + 0.836480i $$0.684610\pi$$
$$984$$ 0 0
$$985$$ 2.51661e7 0.826466
$$986$$ 6.65454e7 2.17985
$$987$$ 0 0
$$988$$ −9.52327e7 −3.10380
$$989$$ −1.26935e7 −0.412657
$$990$$ 0 0
$$991$$ 3.48003e7 1.12564 0.562819 0.826580i $$-0.309717\pi$$
0.562819 + 0.826580i $$0.309717\pi$$
$$992$$ 7.92345e6 0.255644
$$993$$ 0 0
$$994$$ 0 0
$$995$$ −1.49829e7 −0.479774
$$996$$ 0 0
$$997$$ −9.40852e6 −0.299767 −0.149883 0.988704i $$-0.547890\pi$$
−0.149883 + 0.988704i $$0.547890\pi$$
$$998$$ 2.85332e7 0.906826
$$999$$ 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 441.6.a.r.1.2 2
3.2 odd 2 147.6.a.j.1.1 yes 2
7.6 odd 2 441.6.a.q.1.2 2
21.2 odd 6 147.6.e.m.67.2 4
21.5 even 6 147.6.e.n.67.2 4
21.11 odd 6 147.6.e.m.79.2 4
21.17 even 6 147.6.e.n.79.2 4
21.20 even 2 147.6.a.h.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
147.6.a.h.1.1 2 21.20 even 2
147.6.a.j.1.1 yes 2 3.2 odd 2
147.6.e.m.67.2 4 21.2 odd 6
147.6.e.m.79.2 4 21.11 odd 6
147.6.e.n.67.2 4 21.5 even 6
147.6.e.n.79.2 4 21.17 even 6
441.6.a.q.1.2 2 7.6 odd 2
441.6.a.r.1.2 2 1.1 even 1 trivial