# Properties

 Label 495.6.a.e.1.1 Level $495$ Weight $6$ Character 495.1 Self dual yes Analytic conductor $79.390$ Analytic rank $1$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$495 = 3^{2} \cdot 5 \cdot 11$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 495.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$79.3899908074$$ Analytic rank: $$1$$ Dimension: $$3$$ Coefficient field: 3.3.34253.1 Defining polynomial: $$x^{3} - x^{2} - 52x + 48$$ x^3 - x^2 - 52*x + 48 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 165) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-7.17710$$ of defining polynomial Character $$\chi$$ $$=$$ 495.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-5.17710 q^{2} -5.19759 q^{4} -25.0000 q^{5} -123.437 q^{7} +192.576 q^{8} +O(q^{10})$$ $$q-5.17710 q^{2} -5.19759 q^{4} -25.0000 q^{5} -123.437 q^{7} +192.576 q^{8} +129.428 q^{10} +121.000 q^{11} -500.053 q^{13} +639.048 q^{14} -830.662 q^{16} +422.631 q^{17} -932.948 q^{19} +129.940 q^{20} -626.430 q^{22} +1225.18 q^{23} +625.000 q^{25} +2588.83 q^{26} +641.576 q^{28} +2111.62 q^{29} -159.612 q^{31} -1862.00 q^{32} -2188.00 q^{34} +3085.93 q^{35} +5414.46 q^{37} +4829.97 q^{38} -4814.40 q^{40} +18066.7 q^{41} +6815.47 q^{43} -628.908 q^{44} -6342.88 q^{46} +15098.9 q^{47} -1570.23 q^{49} -3235.69 q^{50} +2599.07 q^{52} -15367.7 q^{53} -3025.00 q^{55} -23771.0 q^{56} -10932.1 q^{58} -23400.5 q^{59} +10768.4 q^{61} +826.328 q^{62} +36221.0 q^{64} +12501.3 q^{65} +14507.1 q^{67} -2196.66 q^{68} -15976.2 q^{70} +28114.0 q^{71} -28836.7 q^{73} -28031.2 q^{74} +4849.08 q^{76} -14935.9 q^{77} -8150.52 q^{79} +20766.6 q^{80} -93533.0 q^{82} +109864. q^{83} -10565.8 q^{85} -35284.4 q^{86} +23301.7 q^{88} -69673.6 q^{89} +61725.2 q^{91} -6367.98 q^{92} -78168.5 q^{94} +23323.7 q^{95} +91551.4 q^{97} +8129.23 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 7 q^{2} + 25 q^{4} - 75 q^{5} - 172 q^{7} + 231 q^{8}+O(q^{10})$$ 3 * q + 7 * q^2 + 25 * q^4 - 75 * q^5 - 172 * q^7 + 231 * q^8 $$3 q + 7 q^{2} + 25 q^{4} - 75 q^{5} - 172 q^{7} + 231 q^{8} - 175 q^{10} + 363 q^{11} - 654 q^{13} + 728 q^{14} - 415 q^{16} + 2366 q^{17} - 2872 q^{19} - 625 q^{20} + 847 q^{22} - 2272 q^{23} + 1875 q^{25} - 3422 q^{26} + 4592 q^{28} + 7738 q^{29} + 568 q^{31} - 1001 q^{32} + 2506 q^{34} + 4300 q^{35} - 9126 q^{37} - 13076 q^{38} - 5775 q^{40} + 8758 q^{41} - 14672 q^{43} + 3025 q^{44} - 28768 q^{46} + 19392 q^{47} - 26629 q^{49} + 4375 q^{50} - 61506 q^{52} + 4598 q^{53} - 9075 q^{55} - 2688 q^{56} + 8550 q^{58} + 9348 q^{59} - 60078 q^{61} + 14096 q^{62} - 7087 q^{64} + 16350 q^{65} - 38468 q^{67} - 59778 q^{68} - 18200 q^{70} + 74032 q^{71} - 44442 q^{73} - 82542 q^{74} - 98708 q^{76} - 20812 q^{77} - 108116 q^{79} + 10375 q^{80} - 92230 q^{82} + 81892 q^{83} - 59150 q^{85} - 126412 q^{86} + 27951 q^{88} - 167342 q^{89} - 31832 q^{91} - 72960 q^{92} + 12728 q^{94} + 71800 q^{95} + 159702 q^{97} - 163121 q^{98}+O(q^{100})$$ 3 * q + 7 * q^2 + 25 * q^4 - 75 * q^5 - 172 * q^7 + 231 * q^8 - 175 * q^10 + 363 * q^11 - 654 * q^13 + 728 * q^14 - 415 * q^16 + 2366 * q^17 - 2872 * q^19 - 625 * q^20 + 847 * q^22 - 2272 * q^23 + 1875 * q^25 - 3422 * q^26 + 4592 * q^28 + 7738 * q^29 + 568 * q^31 - 1001 * q^32 + 2506 * q^34 + 4300 * q^35 - 9126 * q^37 - 13076 * q^38 - 5775 * q^40 + 8758 * q^41 - 14672 * q^43 + 3025 * q^44 - 28768 * q^46 + 19392 * q^47 - 26629 * q^49 + 4375 * q^50 - 61506 * q^52 + 4598 * q^53 - 9075 * q^55 - 2688 * q^56 + 8550 * q^58 + 9348 * q^59 - 60078 * q^61 + 14096 * q^62 - 7087 * q^64 + 16350 * q^65 - 38468 * q^67 - 59778 * q^68 - 18200 * q^70 + 74032 * q^71 - 44442 * q^73 - 82542 * q^74 - 98708 * q^76 - 20812 * q^77 - 108116 * q^79 + 10375 * q^80 - 92230 * q^82 + 81892 * q^83 - 59150 * q^85 - 126412 * q^86 + 27951 * q^88 - 167342 * q^89 - 31832 * q^91 - 72960 * q^92 + 12728 * q^94 + 71800 * q^95 + 159702 * q^97 - 163121 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −5.17710 −0.915191 −0.457596 0.889160i $$-0.651289\pi$$
−0.457596 + 0.889160i $$0.651289\pi$$
$$3$$ 0 0
$$4$$ −5.19759 −0.162425
$$5$$ −25.0000 −0.447214
$$6$$ 0 0
$$7$$ −123.437 −0.952141 −0.476071 0.879407i $$-0.657939\pi$$
−0.476071 + 0.879407i $$0.657939\pi$$
$$8$$ 192.576 1.06384
$$9$$ 0 0
$$10$$ 129.428 0.409286
$$11$$ 121.000 0.301511
$$12$$ 0 0
$$13$$ −500.053 −0.820649 −0.410324 0.911940i $$-0.634585\pi$$
−0.410324 + 0.911940i $$0.634585\pi$$
$$14$$ 639.048 0.871392
$$15$$ 0 0
$$16$$ −830.662 −0.811194
$$17$$ 422.631 0.354682 0.177341 0.984150i $$-0.443251\pi$$
0.177341 + 0.984150i $$0.443251\pi$$
$$18$$ 0 0
$$19$$ −932.948 −0.592889 −0.296445 0.955050i $$-0.595801\pi$$
−0.296445 + 0.955050i $$0.595801\pi$$
$$20$$ 129.940 0.0726385
$$21$$ 0 0
$$22$$ −626.430 −0.275941
$$23$$ 1225.18 0.482926 0.241463 0.970410i $$-0.422373\pi$$
0.241463 + 0.970410i $$0.422373\pi$$
$$24$$ 0 0
$$25$$ 625.000 0.200000
$$26$$ 2588.83 0.751051
$$27$$ 0 0
$$28$$ 641.576 0.154651
$$29$$ 2111.62 0.466251 0.233126 0.972447i $$-0.425105\pi$$
0.233126 + 0.972447i $$0.425105\pi$$
$$30$$ 0 0
$$31$$ −159.612 −0.0298306 −0.0149153 0.999889i $$-0.504748\pi$$
−0.0149153 + 0.999889i $$0.504748\pi$$
$$32$$ −1862.00 −0.321444
$$33$$ 0 0
$$34$$ −2188.00 −0.324601
$$35$$ 3085.93 0.425811
$$36$$ 0 0
$$37$$ 5414.46 0.650205 0.325103 0.945679i $$-0.394601\pi$$
0.325103 + 0.945679i $$0.394601\pi$$
$$38$$ 4829.97 0.542607
$$39$$ 0 0
$$40$$ −4814.40 −0.475764
$$41$$ 18066.7 1.67849 0.839244 0.543756i $$-0.182998\pi$$
0.839244 + 0.543756i $$0.182998\pi$$
$$42$$ 0 0
$$43$$ 6815.47 0.562114 0.281057 0.959691i $$-0.409315\pi$$
0.281057 + 0.959691i $$0.409315\pi$$
$$44$$ −628.908 −0.0489729
$$45$$ 0 0
$$46$$ −6342.88 −0.441969
$$47$$ 15098.9 0.997012 0.498506 0.866886i $$-0.333882\pi$$
0.498506 + 0.866886i $$0.333882\pi$$
$$48$$ 0 0
$$49$$ −1570.23 −0.0934270
$$50$$ −3235.69 −0.183038
$$51$$ 0 0
$$52$$ 2599.07 0.133294
$$53$$ −15367.7 −0.751484 −0.375742 0.926724i $$-0.622612\pi$$
−0.375742 + 0.926724i $$0.622612\pi$$
$$54$$ 0 0
$$55$$ −3025.00 −0.134840
$$56$$ −23771.0 −1.01293
$$57$$ 0 0
$$58$$ −10932.1 −0.426709
$$59$$ −23400.5 −0.875177 −0.437588 0.899175i $$-0.644167\pi$$
−0.437588 + 0.899175i $$0.644167\pi$$
$$60$$ 0 0
$$61$$ 10768.4 0.370532 0.185266 0.982688i $$-0.440685\pi$$
0.185266 + 0.982688i $$0.440685\pi$$
$$62$$ 826.328 0.0273007
$$63$$ 0 0
$$64$$ 36221.0 1.10538
$$65$$ 12501.3 0.367005
$$66$$ 0 0
$$67$$ 14507.1 0.394814 0.197407 0.980322i $$-0.436748\pi$$
0.197407 + 0.980322i $$0.436748\pi$$
$$68$$ −2196.66 −0.0576090
$$69$$ 0 0
$$70$$ −15976.2 −0.389698
$$71$$ 28114.0 0.661876 0.330938 0.943653i $$-0.392635\pi$$
0.330938 + 0.943653i $$0.392635\pi$$
$$72$$ 0 0
$$73$$ −28836.7 −0.633342 −0.316671 0.948536i $$-0.602565\pi$$
−0.316671 + 0.948536i $$0.602565\pi$$
$$74$$ −28031.2 −0.595062
$$75$$ 0 0
$$76$$ 4849.08 0.0962998
$$77$$ −14935.9 −0.287081
$$78$$ 0 0
$$79$$ −8150.52 −0.146932 −0.0734662 0.997298i $$-0.523406\pi$$
−0.0734662 + 0.997298i $$0.523406\pi$$
$$80$$ 20766.6 0.362777
$$81$$ 0 0
$$82$$ −93533.0 −1.53614
$$83$$ 109864. 1.75049 0.875246 0.483679i $$-0.160700\pi$$
0.875246 + 0.483679i $$0.160700\pi$$
$$84$$ 0 0
$$85$$ −10565.8 −0.158618
$$86$$ −35284.4 −0.514442
$$87$$ 0 0
$$88$$ 23301.7 0.320760
$$89$$ −69673.6 −0.932380 −0.466190 0.884685i $$-0.654374\pi$$
−0.466190 + 0.884685i $$0.654374\pi$$
$$90$$ 0 0
$$91$$ 61725.2 0.781374
$$92$$ −6367.98 −0.0784390
$$93$$ 0 0
$$94$$ −78168.5 −0.912457
$$95$$ 23323.7 0.265148
$$96$$ 0 0
$$97$$ 91551.4 0.987952 0.493976 0.869476i $$-0.335543\pi$$
0.493976 + 0.869476i $$0.335543\pi$$
$$98$$ 8129.23 0.0855036
$$99$$ 0 0
$$100$$ −3248.49 −0.0324849
$$101$$ −21299.3 −0.207760 −0.103880 0.994590i $$-0.533126\pi$$
−0.103880 + 0.994590i $$0.533126\pi$$
$$102$$ 0 0
$$103$$ −6548.06 −0.0608163 −0.0304081 0.999538i $$-0.509681\pi$$
−0.0304081 + 0.999538i $$0.509681\pi$$
$$104$$ −96298.1 −0.873040
$$105$$ 0 0
$$106$$ 79560.3 0.687752
$$107$$ −127171. −1.07381 −0.536905 0.843643i $$-0.680407\pi$$
−0.536905 + 0.843643i $$0.680407\pi$$
$$108$$ 0 0
$$109$$ 57285.2 0.461823 0.230912 0.972975i $$-0.425829\pi$$
0.230912 + 0.972975i $$0.425829\pi$$
$$110$$ 15660.7 0.123404
$$111$$ 0 0
$$112$$ 102535. 0.772371
$$113$$ 67774.2 0.499308 0.249654 0.968335i $$-0.419683\pi$$
0.249654 + 0.968335i $$0.419683\pi$$
$$114$$ 0 0
$$115$$ −30629.5 −0.215971
$$116$$ −10975.3 −0.0757307
$$117$$ 0 0
$$118$$ 121147. 0.800954
$$119$$ −52168.4 −0.337707
$$120$$ 0 0
$$121$$ 14641.0 0.0909091
$$122$$ −55749.1 −0.339108
$$123$$ 0 0
$$124$$ 829.598 0.00484522
$$125$$ −15625.0 −0.0894427
$$126$$ 0 0
$$127$$ −115352. −0.634622 −0.317311 0.948322i $$-0.602780\pi$$
−0.317311 + 0.948322i $$0.602780\pi$$
$$128$$ −127936. −0.690187
$$129$$ 0 0
$$130$$ −64720.6 −0.335880
$$131$$ −43276.4 −0.220329 −0.110165 0.993913i $$-0.535138\pi$$
−0.110165 + 0.993913i $$0.535138\pi$$
$$132$$ 0 0
$$133$$ 115161. 0.564514
$$134$$ −75104.6 −0.361331
$$135$$ 0 0
$$136$$ 81388.4 0.377325
$$137$$ −401439. −1.82734 −0.913668 0.406461i $$-0.866763\pi$$
−0.913668 + 0.406461i $$0.866763\pi$$
$$138$$ 0 0
$$139$$ −302018. −1.32586 −0.662928 0.748683i $$-0.730686\pi$$
−0.662928 + 0.748683i $$0.730686\pi$$
$$140$$ −16039.4 −0.0691621
$$141$$ 0 0
$$142$$ −145549. −0.605743
$$143$$ −60506.4 −0.247435
$$144$$ 0 0
$$145$$ −52790.4 −0.208514
$$146$$ 149290. 0.579629
$$147$$ 0 0
$$148$$ −28142.1 −0.105609
$$149$$ −326942. −1.20644 −0.603218 0.797576i $$-0.706115\pi$$
−0.603218 + 0.797576i $$0.706115\pi$$
$$150$$ 0 0
$$151$$ 164681. 0.587761 0.293881 0.955842i $$-0.405053\pi$$
0.293881 + 0.955842i $$0.405053\pi$$
$$152$$ −179663. −0.630740
$$153$$ 0 0
$$154$$ 77324.8 0.262734
$$155$$ 3990.30 0.0133406
$$156$$ 0 0
$$157$$ 248620. 0.804982 0.402491 0.915424i $$-0.368144\pi$$
0.402491 + 0.915424i $$0.368144\pi$$
$$158$$ 42196.1 0.134471
$$159$$ 0 0
$$160$$ 46550.0 0.143754
$$161$$ −151233. −0.459813
$$162$$ 0 0
$$163$$ −417656. −1.23126 −0.615629 0.788036i $$-0.711098\pi$$
−0.615629 + 0.788036i $$0.711098\pi$$
$$164$$ −93903.0 −0.272628
$$165$$ 0 0
$$166$$ −568777. −1.60203
$$167$$ −704955. −1.95601 −0.978004 0.208588i $$-0.933113\pi$$
−0.978004 + 0.208588i $$0.933113\pi$$
$$168$$ 0 0
$$169$$ −121240. −0.326535
$$170$$ 54700.1 0.145166
$$171$$ 0 0
$$172$$ −35424.0 −0.0913012
$$173$$ 171060. 0.434545 0.217272 0.976111i $$-0.430284\pi$$
0.217272 + 0.976111i $$0.430284\pi$$
$$174$$ 0 0
$$175$$ −77148.3 −0.190428
$$176$$ −100510. −0.244584
$$177$$ 0 0
$$178$$ 360707. 0.853306
$$179$$ 166327. 0.387999 0.193999 0.981002i $$-0.437854\pi$$
0.193999 + 0.981002i $$0.437854\pi$$
$$180$$ 0 0
$$181$$ −584391. −1.32589 −0.662945 0.748668i $$-0.730694\pi$$
−0.662945 + 0.748668i $$0.730694\pi$$
$$182$$ −319558. −0.715107
$$183$$ 0 0
$$184$$ 235940. 0.513756
$$185$$ −135361. −0.290781
$$186$$ 0 0
$$187$$ 51138.3 0.106941
$$188$$ −78477.8 −0.161939
$$189$$ 0 0
$$190$$ −120749. −0.242661
$$191$$ 715510. 1.41916 0.709581 0.704624i $$-0.248884\pi$$
0.709581 + 0.704624i $$0.248884\pi$$
$$192$$ 0 0
$$193$$ −922088. −1.78188 −0.890941 0.454118i $$-0.849954\pi$$
−0.890941 + 0.454118i $$0.849954\pi$$
$$194$$ −473971. −0.904165
$$195$$ 0 0
$$196$$ 8161.40 0.0151749
$$197$$ 613632. 1.12653 0.563265 0.826276i $$-0.309545\pi$$
0.563265 + 0.826276i $$0.309545\pi$$
$$198$$ 0 0
$$199$$ −378985. −0.678406 −0.339203 0.940713i $$-0.610157\pi$$
−0.339203 + 0.940713i $$0.610157\pi$$
$$200$$ 120360. 0.212768
$$201$$ 0 0
$$202$$ 110269. 0.190140
$$203$$ −260652. −0.443937
$$204$$ 0 0
$$205$$ −451666. −0.750642
$$206$$ 33900.0 0.0556585
$$207$$ 0 0
$$208$$ 415375. 0.665705
$$209$$ −112887. −0.178763
$$210$$ 0 0
$$211$$ −473721. −0.732515 −0.366257 0.930514i $$-0.619361\pi$$
−0.366257 + 0.930514i $$0.619361\pi$$
$$212$$ 79875.1 0.122060
$$213$$ 0 0
$$214$$ 658376. 0.982741
$$215$$ −170387. −0.251385
$$216$$ 0 0
$$217$$ 19702.1 0.0284029
$$218$$ −296571. −0.422657
$$219$$ 0 0
$$220$$ 15722.7 0.0219013
$$221$$ −211338. −0.291069
$$222$$ 0 0
$$223$$ −822747. −1.10791 −0.553954 0.832547i $$-0.686882\pi$$
−0.553954 + 0.832547i $$0.686882\pi$$
$$224$$ 229840. 0.306060
$$225$$ 0 0
$$226$$ −350874. −0.456962
$$227$$ 556097. 0.716285 0.358143 0.933667i $$-0.383410\pi$$
0.358143 + 0.933667i $$0.383410\pi$$
$$228$$ 0 0
$$229$$ −634919. −0.800073 −0.400036 0.916499i $$-0.631003\pi$$
−0.400036 + 0.916499i $$0.631003\pi$$
$$230$$ 158572. 0.197655
$$231$$ 0 0
$$232$$ 406646. 0.496017
$$233$$ 906561. 1.09397 0.546987 0.837141i $$-0.315775\pi$$
0.546987 + 0.837141i $$0.315775\pi$$
$$234$$ 0 0
$$235$$ −377472. −0.445877
$$236$$ 121626. 0.142150
$$237$$ 0 0
$$238$$ 270081. 0.309066
$$239$$ −662586. −0.750322 −0.375161 0.926960i $$-0.622413\pi$$
−0.375161 + 0.926960i $$0.622413\pi$$
$$240$$ 0 0
$$241$$ −1.31823e6 −1.46200 −0.731001 0.682376i $$-0.760947\pi$$
−0.731001 + 0.682376i $$0.760947\pi$$
$$242$$ −75798.0 −0.0831992
$$243$$ 0 0
$$244$$ −55969.6 −0.0601836
$$245$$ 39255.7 0.0417818
$$246$$ 0 0
$$247$$ 466523. 0.486554
$$248$$ −30737.4 −0.0317350
$$249$$ 0 0
$$250$$ 80892.3 0.0818572
$$251$$ 11073.9 0.0110947 0.00554737 0.999985i $$-0.498234\pi$$
0.00554737 + 0.999985i $$0.498234\pi$$
$$252$$ 0 0
$$253$$ 148247. 0.145608
$$254$$ 597188. 0.580800
$$255$$ 0 0
$$256$$ −496734. −0.473723
$$257$$ 788596. 0.744769 0.372384 0.928079i $$-0.378540\pi$$
0.372384 + 0.928079i $$0.378540\pi$$
$$258$$ 0 0
$$259$$ −668346. −0.619087
$$260$$ −64976.7 −0.0596107
$$261$$ 0 0
$$262$$ 224046. 0.201644
$$263$$ −211325. −0.188391 −0.0941956 0.995554i $$-0.530028\pi$$
−0.0941956 + 0.995554i $$0.530028\pi$$
$$264$$ 0 0
$$265$$ 384193. 0.336074
$$266$$ −596198. −0.516639
$$267$$ 0 0
$$268$$ −75401.8 −0.0641276
$$269$$ −552342. −0.465401 −0.232701 0.972548i $$-0.574756\pi$$
−0.232701 + 0.972548i $$0.574756\pi$$
$$270$$ 0 0
$$271$$ −1.49255e6 −1.23454 −0.617271 0.786751i $$-0.711762\pi$$
−0.617271 + 0.786751i $$0.711762\pi$$
$$272$$ −351063. −0.287715
$$273$$ 0 0
$$274$$ 2.07829e6 1.67236
$$275$$ 75625.0 0.0603023
$$276$$ 0 0
$$277$$ 1.01431e6 0.794275 0.397138 0.917759i $$-0.370004\pi$$
0.397138 + 0.917759i $$0.370004\pi$$
$$278$$ 1.56358e6 1.21341
$$279$$ 0 0
$$280$$ 594276. 0.452995
$$281$$ 658216. 0.497282 0.248641 0.968596i $$-0.420016\pi$$
0.248641 + 0.968596i $$0.420016\pi$$
$$282$$ 0 0
$$283$$ 1.65981e6 1.23195 0.615974 0.787767i $$-0.288763\pi$$
0.615974 + 0.787767i $$0.288763\pi$$
$$284$$ −146125. −0.107505
$$285$$ 0 0
$$286$$ 313248. 0.226450
$$287$$ −2.23010e6 −1.59816
$$288$$ 0 0
$$289$$ −1.24124e6 −0.874201
$$290$$ 273301. 0.190830
$$291$$ 0 0
$$292$$ 149881. 0.102870
$$293$$ 410864. 0.279594 0.139797 0.990180i $$-0.455355\pi$$
0.139797 + 0.990180i $$0.455355\pi$$
$$294$$ 0 0
$$295$$ 585013. 0.391391
$$296$$ 1.04269e6 0.691715
$$297$$ 0 0
$$298$$ 1.69261e6 1.10412
$$299$$ −612654. −0.396312
$$300$$ 0 0
$$301$$ −841283. −0.535212
$$302$$ −852570. −0.537914
$$303$$ 0 0
$$304$$ 774965. 0.480948
$$305$$ −269210. −0.165707
$$306$$ 0 0
$$307$$ 1.34831e6 0.816477 0.408238 0.912875i $$-0.366143\pi$$
0.408238 + 0.912875i $$0.366143\pi$$
$$308$$ 77630.7 0.0466291
$$309$$ 0 0
$$310$$ −20658.2 −0.0122092
$$311$$ 2.52585e6 1.48083 0.740417 0.672148i $$-0.234628\pi$$
0.740417 + 0.672148i $$0.234628\pi$$
$$312$$ 0 0
$$313$$ 2.23061e6 1.28695 0.643476 0.765466i $$-0.277491\pi$$
0.643476 + 0.765466i $$0.277491\pi$$
$$314$$ −1.28713e6 −0.736713
$$315$$ 0 0
$$316$$ 42363.0 0.0238654
$$317$$ 120908. 0.0675780 0.0337890 0.999429i $$-0.489243\pi$$
0.0337890 + 0.999429i $$0.489243\pi$$
$$318$$ 0 0
$$319$$ 255506. 0.140580
$$320$$ −905524. −0.494339
$$321$$ 0 0
$$322$$ 782948. 0.420817
$$323$$ −394292. −0.210287
$$324$$ 0 0
$$325$$ −312533. −0.164130
$$326$$ 2.16225e6 1.12684
$$327$$ 0 0
$$328$$ 3.47920e6 1.78564
$$329$$ −1.86377e6 −0.949296
$$330$$ 0 0
$$331$$ 578480. 0.290214 0.145107 0.989416i $$-0.453647\pi$$
0.145107 + 0.989416i $$0.453647\pi$$
$$332$$ −571028. −0.284323
$$333$$ 0 0
$$334$$ 3.64963e6 1.79012
$$335$$ −362677. −0.176566
$$336$$ 0 0
$$337$$ −1.78710e6 −0.857186 −0.428593 0.903498i $$-0.640991\pi$$
−0.428593 + 0.903498i $$0.640991\pi$$
$$338$$ 627674. 0.298842
$$339$$ 0 0
$$340$$ 54916.5 0.0257635
$$341$$ −19313.1 −0.00899425
$$342$$ 0 0
$$343$$ 2.26844e6 1.04110
$$344$$ 1.31249e6 0.598000
$$345$$ 0 0
$$346$$ −885598. −0.397692
$$347$$ 1.46282e6 0.652179 0.326089 0.945339i $$-0.394269\pi$$
0.326089 + 0.945339i $$0.394269\pi$$
$$348$$ 0 0
$$349$$ 1.48501e6 0.652627 0.326314 0.945262i $$-0.394193\pi$$
0.326314 + 0.945262i $$0.394193\pi$$
$$350$$ 399405. 0.174278
$$351$$ 0 0
$$352$$ −225302. −0.0969189
$$353$$ −1.34528e6 −0.574616 −0.287308 0.957838i $$-0.592760\pi$$
−0.287308 + 0.957838i $$0.592760\pi$$
$$354$$ 0 0
$$355$$ −702850. −0.296000
$$356$$ 362135. 0.151442
$$357$$ 0 0
$$358$$ −861092. −0.355093
$$359$$ 1.83956e6 0.753316 0.376658 0.926352i $$-0.377073\pi$$
0.376658 + 0.926352i $$0.377073\pi$$
$$360$$ 0 0
$$361$$ −1.60571e6 −0.648483
$$362$$ 3.02546e6 1.21344
$$363$$ 0 0
$$364$$ −320822. −0.126914
$$365$$ 720917. 0.283239
$$366$$ 0 0
$$367$$ −312079. −0.120948 −0.0604741 0.998170i $$-0.519261\pi$$
−0.0604741 + 0.998170i $$0.519261\pi$$
$$368$$ −1.01771e6 −0.391746
$$369$$ 0 0
$$370$$ 700780. 0.266120
$$371$$ 1.89695e6 0.715519
$$372$$ 0 0
$$373$$ 3.38658e6 1.26034 0.630172 0.776455i $$-0.282984\pi$$
0.630172 + 0.776455i $$0.282984\pi$$
$$374$$ −264748. −0.0978710
$$375$$ 0 0
$$376$$ 2.90768e6 1.06066
$$377$$ −1.05592e6 −0.382629
$$378$$ 0 0
$$379$$ −1.62370e6 −0.580641 −0.290321 0.956929i $$-0.593762\pi$$
−0.290321 + 0.956929i $$0.593762\pi$$
$$380$$ −121227. −0.0430666
$$381$$ 0 0
$$382$$ −3.70427e6 −1.29881
$$383$$ 3.06187e6 1.06657 0.533285 0.845935i $$-0.320957\pi$$
0.533285 + 0.845935i $$0.320957\pi$$
$$384$$ 0 0
$$385$$ 373398. 0.128387
$$386$$ 4.77375e6 1.63076
$$387$$ 0 0
$$388$$ −475847. −0.160468
$$389$$ −4.21840e6 −1.41343 −0.706715 0.707499i $$-0.749824\pi$$
−0.706715 + 0.707499i $$0.749824\pi$$
$$390$$ 0 0
$$391$$ 517798. 0.171285
$$392$$ −302388. −0.0993915
$$393$$ 0 0
$$394$$ −3.17684e6 −1.03099
$$395$$ 203763. 0.0657101
$$396$$ 0 0
$$397$$ 513185. 0.163417 0.0817085 0.996656i $$-0.473962\pi$$
0.0817085 + 0.996656i $$0.473962\pi$$
$$398$$ 1.96205e6 0.620871
$$399$$ 0 0
$$400$$ −519164. −0.162239
$$401$$ 2.18458e6 0.678432 0.339216 0.940709i $$-0.389838\pi$$
0.339216 + 0.940709i $$0.389838\pi$$
$$402$$ 0 0
$$403$$ 79814.5 0.0244804
$$404$$ 110705. 0.0337453
$$405$$ 0 0
$$406$$ 1.34942e6 0.406287
$$407$$ 655149. 0.196044
$$408$$ 0 0
$$409$$ 1.22307e6 0.361529 0.180764 0.983526i $$-0.442143\pi$$
0.180764 + 0.983526i $$0.442143\pi$$
$$410$$ 2.33832e6 0.686981
$$411$$ 0 0
$$412$$ 34034.1 0.00987806
$$413$$ 2.88850e6 0.833292
$$414$$ 0 0
$$415$$ −2.74660e6 −0.782844
$$416$$ 931098. 0.263792
$$417$$ 0 0
$$418$$ 584426. 0.163602
$$419$$ −2.42879e6 −0.675858 −0.337929 0.941172i $$-0.609726\pi$$
−0.337929 + 0.941172i $$0.609726\pi$$
$$420$$ 0 0
$$421$$ −727467. −0.200036 −0.100018 0.994986i $$-0.531890\pi$$
−0.100018 + 0.994986i $$0.531890\pi$$
$$422$$ 2.45250e6 0.670391
$$423$$ 0 0
$$424$$ −2.95945e6 −0.799460
$$425$$ 264144. 0.0709363
$$426$$ 0 0
$$427$$ −1.32922e6 −0.352799
$$428$$ 660981. 0.174413
$$429$$ 0 0
$$430$$ 882110. 0.230066
$$431$$ −6.18223e6 −1.60307 −0.801534 0.597949i $$-0.795982\pi$$
−0.801534 + 0.597949i $$0.795982\pi$$
$$432$$ 0 0
$$433$$ 2.42337e6 0.621154 0.310577 0.950548i $$-0.399478\pi$$
0.310577 + 0.950548i $$0.399478\pi$$
$$434$$ −102000. −0.0259941
$$435$$ 0 0
$$436$$ −297745. −0.0750115
$$437$$ −1.14303e6 −0.286321
$$438$$ 0 0
$$439$$ −3.79993e6 −0.941054 −0.470527 0.882385i $$-0.655936\pi$$
−0.470527 + 0.882385i $$0.655936\pi$$
$$440$$ −582542. −0.143448
$$441$$ 0 0
$$442$$ 1.09412e6 0.266384
$$443$$ 5.58044e6 1.35101 0.675506 0.737354i $$-0.263925\pi$$
0.675506 + 0.737354i $$0.263925\pi$$
$$444$$ 0 0
$$445$$ 1.74184e6 0.416973
$$446$$ 4.25945e6 1.01395
$$447$$ 0 0
$$448$$ −4.47102e6 −1.05247
$$449$$ −2.69609e6 −0.631130 −0.315565 0.948904i $$-0.602194\pi$$
−0.315565 + 0.948904i $$0.602194\pi$$
$$450$$ 0 0
$$451$$ 2.18607e6 0.506083
$$452$$ −352263. −0.0810999
$$453$$ 0 0
$$454$$ −2.87897e6 −0.655538
$$455$$ −1.54313e6 −0.349441
$$456$$ 0 0
$$457$$ 1.14257e6 0.255912 0.127956 0.991780i $$-0.459158\pi$$
0.127956 + 0.991780i $$0.459158\pi$$
$$458$$ 3.28704e6 0.732220
$$459$$ 0 0
$$460$$ 159199. 0.0350790
$$461$$ −5.37091e6 −1.17705 −0.588526 0.808478i $$-0.700292\pi$$
−0.588526 + 0.808478i $$0.700292\pi$$
$$462$$ 0 0
$$463$$ −3.80227e6 −0.824311 −0.412155 0.911114i $$-0.635224\pi$$
−0.412155 + 0.911114i $$0.635224\pi$$
$$464$$ −1.75404e6 −0.378220
$$465$$ 0 0
$$466$$ −4.69336e6 −1.00120
$$467$$ −9.41694e6 −1.99810 −0.999051 0.0435609i $$-0.986130\pi$$
−0.999051 + 0.0435609i $$0.986130\pi$$
$$468$$ 0 0
$$469$$ −1.79071e6 −0.375919
$$470$$ 1.95421e6 0.408063
$$471$$ 0 0
$$472$$ −4.50638e6 −0.931049
$$473$$ 824672. 0.169484
$$474$$ 0 0
$$475$$ −583093. −0.118578
$$476$$ 271150. 0.0548519
$$477$$ 0 0
$$478$$ 3.43028e6 0.686688
$$479$$ 4.69047e6 0.934065 0.467033 0.884240i $$-0.345323\pi$$
0.467033 + 0.884240i $$0.345323\pi$$
$$480$$ 0 0
$$481$$ −2.70751e6 −0.533590
$$482$$ 6.82461e6 1.33801
$$483$$ 0 0
$$484$$ −76097.9 −0.0147659
$$485$$ −2.28879e6 −0.441825
$$486$$ 0 0
$$487$$ 7.07177e6 1.35116 0.675579 0.737288i $$-0.263894\pi$$
0.675579 + 0.737288i $$0.263894\pi$$
$$488$$ 2.07373e6 0.394187
$$489$$ 0 0
$$490$$ −203231. −0.0382384
$$491$$ −906051. −0.169609 −0.0848045 0.996398i $$-0.527027\pi$$
−0.0848045 + 0.996398i $$0.527027\pi$$
$$492$$ 0 0
$$493$$ 892434. 0.165371
$$494$$ −2.41524e6 −0.445290
$$495$$ 0 0
$$496$$ 132584. 0.0241984
$$497$$ −3.47031e6 −0.630199
$$498$$ 0 0
$$499$$ −7.13323e6 −1.28243 −0.641217 0.767360i $$-0.721570\pi$$
−0.641217 + 0.767360i $$0.721570\pi$$
$$500$$ 81212.3 0.0145277
$$501$$ 0 0
$$502$$ −57330.9 −0.0101538
$$503$$ 2.34927e6 0.414013 0.207006 0.978340i $$-0.433628\pi$$
0.207006 + 0.978340i $$0.433628\pi$$
$$504$$ 0 0
$$505$$ 532482. 0.0929131
$$506$$ −767489. −0.133259
$$507$$ 0 0
$$508$$ 599551. 0.103078
$$509$$ −1.97148e6 −0.337286 −0.168643 0.985677i $$-0.553938\pi$$
−0.168643 + 0.985677i $$0.553938\pi$$
$$510$$ 0 0
$$511$$ 3.55952e6 0.603031
$$512$$ 6.66559e6 1.12373
$$513$$ 0 0
$$514$$ −4.08264e6 −0.681606
$$515$$ 163702. 0.0271979
$$516$$ 0 0
$$517$$ 1.82697e6 0.300610
$$518$$ 3.46010e6 0.566583
$$519$$ 0 0
$$520$$ 2.40745e6 0.390435
$$521$$ −1.20074e7 −1.93801 −0.969004 0.247046i $$-0.920540\pi$$
−0.969004 + 0.247046i $$0.920540\pi$$
$$522$$ 0 0
$$523$$ 4.16772e6 0.666260 0.333130 0.942881i $$-0.391895\pi$$
0.333130 + 0.942881i $$0.391895\pi$$
$$524$$ 224933. 0.0357869
$$525$$ 0 0
$$526$$ 1.09405e6 0.172414
$$527$$ −67456.9 −0.0105804
$$528$$ 0 0
$$529$$ −4.93528e6 −0.766783
$$530$$ −1.98901e6 −0.307572
$$531$$ 0 0
$$532$$ −598557. −0.0916910
$$533$$ −9.03428e6 −1.37745
$$534$$ 0 0
$$535$$ 3.17927e6 0.480222
$$536$$ 2.79371e6 0.420020
$$537$$ 0 0
$$538$$ 2.85953e6 0.425931
$$539$$ −189998. −0.0281693
$$540$$ 0 0
$$541$$ 1.01106e7 1.48519 0.742594 0.669741i $$-0.233595\pi$$
0.742594 + 0.669741i $$0.233595\pi$$
$$542$$ 7.72709e6 1.12984
$$543$$ 0 0
$$544$$ −786938. −0.114010
$$545$$ −1.43213e6 −0.206534
$$546$$ 0 0
$$547$$ −7.39087e6 −1.05615 −0.528077 0.849196i $$-0.677087\pi$$
−0.528077 + 0.849196i $$0.677087\pi$$
$$548$$ 2.08652e6 0.296805
$$549$$ 0 0
$$550$$ −391519. −0.0551881
$$551$$ −1.97003e6 −0.276435
$$552$$ 0 0
$$553$$ 1.00608e6 0.139900
$$554$$ −5.25119e6 −0.726914
$$555$$ 0 0
$$556$$ 1.56977e6 0.215352
$$557$$ −1.52459e6 −0.208217 −0.104108 0.994566i $$-0.533199\pi$$
−0.104108 + 0.994566i $$0.533199\pi$$
$$558$$ 0 0
$$559$$ −3.40809e6 −0.461299
$$560$$ −2.56337e6 −0.345415
$$561$$ 0 0
$$562$$ −3.40765e6 −0.455108
$$563$$ 1.40845e7 1.87271 0.936353 0.351059i $$-0.114178\pi$$
0.936353 + 0.351059i $$0.114178\pi$$
$$564$$ 0 0
$$565$$ −1.69436e6 −0.223297
$$566$$ −8.59301e6 −1.12747
$$567$$ 0 0
$$568$$ 5.41407e6 0.704131
$$569$$ 1.30473e7 1.68942 0.844712 0.535221i $$-0.179772\pi$$
0.844712 + 0.535221i $$0.179772\pi$$
$$570$$ 0 0
$$571$$ −1.19873e7 −1.53862 −0.769312 0.638873i $$-0.779401\pi$$
−0.769312 + 0.638873i $$0.779401\pi$$
$$572$$ 314487. 0.0401895
$$573$$ 0 0
$$574$$ 1.15455e7 1.46262
$$575$$ 765737. 0.0965851
$$576$$ 0 0
$$577$$ 568904. 0.0711376 0.0355688 0.999367i $$-0.488676\pi$$
0.0355688 + 0.999367i $$0.488676\pi$$
$$578$$ 6.42603e6 0.800061
$$579$$ 0 0
$$580$$ 274383. 0.0338678
$$581$$ −1.35613e7 −1.66671
$$582$$ 0 0
$$583$$ −1.85949e6 −0.226581
$$584$$ −5.55325e6 −0.673775
$$585$$ 0 0
$$586$$ −2.12708e6 −0.255882
$$587$$ −2.93323e6 −0.351358 −0.175679 0.984447i $$-0.556212\pi$$
−0.175679 + 0.984447i $$0.556212\pi$$
$$588$$ 0 0
$$589$$ 148910. 0.0176862
$$590$$ −3.02868e6 −0.358198
$$591$$ 0 0
$$592$$ −4.49758e6 −0.527442
$$593$$ −5.90557e6 −0.689644 −0.344822 0.938668i $$-0.612061\pi$$
−0.344822 + 0.938668i $$0.612061\pi$$
$$594$$ 0 0
$$595$$ 1.30421e6 0.151027
$$596$$ 1.69931e6 0.195955
$$597$$ 0 0
$$598$$ 3.17178e6 0.362702
$$599$$ −1.07665e7 −1.22605 −0.613025 0.790064i $$-0.710047\pi$$
−0.613025 + 0.790064i $$0.710047\pi$$
$$600$$ 0 0
$$601$$ −1.19746e7 −1.35231 −0.676154 0.736760i $$-0.736355\pi$$
−0.676154 + 0.736760i $$0.736355\pi$$
$$602$$ 4.35541e6 0.489822
$$603$$ 0 0
$$604$$ −855944. −0.0954669
$$605$$ −366025. −0.0406558
$$606$$ 0 0
$$607$$ −3.45506e6 −0.380613 −0.190307 0.981725i $$-0.560948\pi$$
−0.190307 + 0.981725i $$0.560948\pi$$
$$608$$ 1.73715e6 0.190580
$$609$$ 0 0
$$610$$ 1.39373e6 0.151654
$$611$$ −7.55024e6 −0.818197
$$612$$ 0 0
$$613$$ −8.22419e6 −0.883979 −0.441990 0.897020i $$-0.645727\pi$$
−0.441990 + 0.897020i $$0.645727\pi$$
$$614$$ −6.98034e6 −0.747233
$$615$$ 0 0
$$616$$ −2.87630e6 −0.305409
$$617$$ −9.83567e6 −1.04014 −0.520069 0.854124i $$-0.674094\pi$$
−0.520069 + 0.854124i $$0.674094\pi$$
$$618$$ 0 0
$$619$$ 267648. 0.0280762 0.0140381 0.999901i $$-0.495531\pi$$
0.0140381 + 0.999901i $$0.495531\pi$$
$$620$$ −20739.9 −0.00216685
$$621$$ 0 0
$$622$$ −1.30766e7 −1.35525
$$623$$ 8.60032e6 0.887758
$$624$$ 0 0
$$625$$ 390625. 0.0400000
$$626$$ −1.15481e7 −1.17781
$$627$$ 0 0
$$628$$ −1.29222e6 −0.130749
$$629$$ 2.28831e6 0.230616
$$630$$ 0 0
$$631$$ −4.17135e6 −0.417065 −0.208532 0.978015i $$-0.566869\pi$$
−0.208532 + 0.978015i $$0.566869\pi$$
$$632$$ −1.56959e6 −0.156313
$$633$$ 0 0
$$634$$ −625951. −0.0618468
$$635$$ 2.88379e6 0.283811
$$636$$ 0 0
$$637$$ 785197. 0.0766708
$$638$$ −1.32278e6 −0.128658
$$639$$ 0 0
$$640$$ 3.19839e6 0.308661
$$641$$ −8.24673e6 −0.792751 −0.396376 0.918088i $$-0.629732\pi$$
−0.396376 + 0.918088i $$0.629732\pi$$
$$642$$ 0 0
$$643$$ 1.07773e7 1.02797 0.513986 0.857799i $$-0.328168\pi$$
0.513986 + 0.857799i $$0.328168\pi$$
$$644$$ 786046. 0.0746850
$$645$$ 0 0
$$646$$ 2.04129e6 0.192453
$$647$$ −7.89194e6 −0.741179 −0.370590 0.928797i $$-0.620844\pi$$
−0.370590 + 0.928797i $$0.620844\pi$$
$$648$$ 0 0
$$649$$ −2.83147e6 −0.263876
$$650$$ 1.61802e6 0.150210
$$651$$ 0 0
$$652$$ 2.17080e6 0.199987
$$653$$ −1.47858e7 −1.35695 −0.678473 0.734625i $$-0.737358\pi$$
−0.678473 + 0.734625i $$0.737358\pi$$
$$654$$ 0 0
$$655$$ 1.08191e6 0.0985343
$$656$$ −1.50073e7 −1.36158
$$657$$ 0 0
$$658$$ 9.64891e6 0.868788
$$659$$ 475088. 0.0426148 0.0213074 0.999773i $$-0.493217\pi$$
0.0213074 + 0.999773i $$0.493217\pi$$
$$660$$ 0 0
$$661$$ 2.73604e6 0.243567 0.121784 0.992557i $$-0.461139\pi$$
0.121784 + 0.992557i $$0.461139\pi$$
$$662$$ −2.99485e6 −0.265601
$$663$$ 0 0
$$664$$ 2.11571e7 1.86224
$$665$$ −2.87902e6 −0.252458
$$666$$ 0 0
$$667$$ 2.58711e6 0.225165
$$668$$ 3.66407e6 0.317704
$$669$$ 0 0
$$670$$ 1.87762e6 0.161592
$$671$$ 1.30297e6 0.111720
$$672$$ 0 0
$$673$$ 1.52861e7 1.30094 0.650471 0.759531i $$-0.274572\pi$$
0.650471 + 0.759531i $$0.274572\pi$$
$$674$$ 9.25202e6 0.784489
$$675$$ 0 0
$$676$$ 630157. 0.0530374
$$677$$ 3.93225e6 0.329739 0.164869 0.986315i $$-0.447280\pi$$
0.164869 + 0.986315i $$0.447280\pi$$
$$678$$ 0 0
$$679$$ −1.13009e7 −0.940670
$$680$$ −2.03471e6 −0.168745
$$681$$ 0 0
$$682$$ 99985.7 0.00823146
$$683$$ −3.42591e6 −0.281011 −0.140506 0.990080i $$-0.544873\pi$$
−0.140506 + 0.990080i $$0.544873\pi$$
$$684$$ 0 0
$$685$$ 1.00360e7 0.817210
$$686$$ −1.17439e7 −0.952803
$$687$$ 0 0
$$688$$ −5.66135e6 −0.455984
$$689$$ 7.68467e6 0.616705
$$690$$ 0 0
$$691$$ 1.81896e7 1.44920 0.724599 0.689171i $$-0.242025\pi$$
0.724599 + 0.689171i $$0.242025\pi$$
$$692$$ −889102. −0.0705808
$$693$$ 0 0
$$694$$ −7.57316e6 −0.596868
$$695$$ 7.55046e6 0.592941
$$696$$ 0 0
$$697$$ 7.63552e6 0.595328
$$698$$ −7.68804e6 −0.597279
$$699$$ 0 0
$$700$$ 400985. 0.0309302
$$701$$ 1.15598e6 0.0888494 0.0444247 0.999013i $$-0.485855\pi$$
0.0444247 + 0.999013i $$0.485855\pi$$
$$702$$ 0 0
$$703$$ −5.05141e6 −0.385500
$$704$$ 4.38274e6 0.333283
$$705$$ 0 0
$$706$$ 6.96468e6 0.525883
$$707$$ 2.62913e6 0.197817
$$708$$ 0 0
$$709$$ 2.42664e7 1.81296 0.906481 0.422246i $$-0.138758\pi$$
0.906481 + 0.422246i $$0.138758\pi$$
$$710$$ 3.63873e6 0.270897
$$711$$ 0 0
$$712$$ −1.34174e7 −0.991904
$$713$$ −195553. −0.0144059
$$714$$ 0 0
$$715$$ 1.51266e6 0.110656
$$716$$ −864499. −0.0630205
$$717$$ 0 0
$$718$$ −9.52359e6 −0.689429
$$719$$ 2.25503e7 1.62679 0.813394 0.581714i $$-0.197618\pi$$
0.813394 + 0.581714i $$0.197618\pi$$
$$720$$ 0 0
$$721$$ 808276. 0.0579057
$$722$$ 8.31291e6 0.593486
$$723$$ 0 0
$$724$$ 3.03743e6 0.215357
$$725$$ 1.31976e6 0.0932503
$$726$$ 0 0
$$727$$ −2.17941e7 −1.52933 −0.764667 0.644425i $$-0.777097\pi$$
−0.764667 + 0.644425i $$0.777097\pi$$
$$728$$ 1.18868e7 0.831257
$$729$$ 0 0
$$730$$ −3.73226e6 −0.259218
$$731$$ 2.88043e6 0.199372
$$732$$ 0 0
$$733$$ 1.73565e7 1.19317 0.596583 0.802551i $$-0.296524\pi$$
0.596583 + 0.802551i $$0.296524\pi$$
$$734$$ 1.61567e6 0.110691
$$735$$ 0 0
$$736$$ −2.28129e6 −0.155233
$$737$$ 1.75536e6 0.119041
$$738$$ 0 0
$$739$$ 1.16961e6 0.0787825 0.0393912 0.999224i $$-0.487458\pi$$
0.0393912 + 0.999224i $$0.487458\pi$$
$$740$$ 703553. 0.0472300
$$741$$ 0 0
$$742$$ −9.82071e6 −0.654837
$$743$$ −2.98047e6 −0.198067 −0.0990336 0.995084i $$-0.531575\pi$$
−0.0990336 + 0.995084i $$0.531575\pi$$
$$744$$ 0 0
$$745$$ 8.17354e6 0.539535
$$746$$ −1.75327e7 −1.15346
$$747$$ 0 0
$$748$$ −265796. −0.0173698
$$749$$ 1.56976e7 1.02242
$$750$$ 0 0
$$751$$ −1.83578e7 −1.18774 −0.593868 0.804563i $$-0.702400\pi$$
−0.593868 + 0.804563i $$0.702400\pi$$
$$752$$ −1.25421e7 −0.808770
$$753$$ 0 0
$$754$$ 5.46661e6 0.350178
$$755$$ −4.11702e6 −0.262855
$$756$$ 0 0
$$757$$ 1.60050e7 1.01511 0.507557 0.861618i $$-0.330549\pi$$
0.507557 + 0.861618i $$0.330549\pi$$
$$758$$ 8.40607e6 0.531398
$$759$$ 0 0
$$760$$ 4.49158e6 0.282075
$$761$$ 1.67436e7 1.04807 0.524033 0.851698i $$-0.324427\pi$$
0.524033 + 0.851698i $$0.324427\pi$$
$$762$$ 0 0
$$763$$ −7.07113e6 −0.439721
$$764$$ −3.71892e6 −0.230507
$$765$$ 0 0
$$766$$ −1.58516e7 −0.976116
$$767$$ 1.17015e7 0.718213
$$768$$ 0 0
$$769$$ −1.73863e7 −1.06021 −0.530104 0.847933i $$-0.677847\pi$$
−0.530104 + 0.847933i $$0.677847\pi$$
$$770$$ −1.93312e6 −0.117498
$$771$$ 0 0
$$772$$ 4.79263e6 0.289422
$$773$$ −1.79095e7 −1.07804 −0.539021 0.842292i $$-0.681206\pi$$
−0.539021 + 0.842292i $$0.681206\pi$$
$$774$$ 0 0
$$775$$ −99757.5 −0.00596611
$$776$$ 1.76306e7 1.05102
$$777$$ 0 0
$$778$$ 2.18391e7 1.29356
$$779$$ −1.68552e7 −0.995157
$$780$$ 0 0
$$781$$ 3.40179e6 0.199563
$$782$$ −2.68070e6 −0.156758
$$783$$ 0 0
$$784$$ 1.30433e6 0.0757874
$$785$$ −6.21549e6 −0.359999
$$786$$ 0 0
$$787$$ −3.01291e7 −1.73400 −0.867000 0.498308i $$-0.833955\pi$$
−0.867000 + 0.498308i $$0.833955\pi$$
$$788$$ −3.18941e6 −0.182976
$$789$$ 0 0
$$790$$ −1.05490e6 −0.0601374
$$791$$ −8.36587e6 −0.475412
$$792$$ 0 0
$$793$$ −5.38476e6 −0.304077
$$794$$ −2.65681e6 −0.149558
$$795$$ 0 0
$$796$$ 1.96981e6 0.110190
$$797$$ 5.51251e6 0.307400 0.153700 0.988118i $$-0.450881\pi$$
0.153700 + 0.988118i $$0.450881\pi$$
$$798$$ 0 0
$$799$$ 6.38125e6 0.353622
$$800$$ −1.16375e6 −0.0642887
$$801$$ 0 0
$$802$$ −1.13098e7 −0.620895
$$803$$ −3.48924e6 −0.190960
$$804$$ 0 0
$$805$$ 3.78082e6 0.205635
$$806$$ −413208. −0.0224043
$$807$$ 0 0
$$808$$ −4.10173e6 −0.221024
$$809$$ −3.11564e7 −1.67369 −0.836847 0.547436i $$-0.815604\pi$$
−0.836847 + 0.547436i $$0.815604\pi$$
$$810$$ 0 0
$$811$$ −1.11336e7 −0.594406 −0.297203 0.954814i $$-0.596054\pi$$
−0.297203 + 0.954814i $$0.596054\pi$$
$$812$$ 1.35476e6 0.0721063
$$813$$ 0 0
$$814$$ −3.39178e6 −0.179418
$$815$$ 1.04414e7 0.550636
$$816$$ 0 0
$$817$$ −6.35848e6 −0.333271
$$818$$ −6.33195e6 −0.330868
$$819$$ 0 0
$$820$$ 2.34758e6 0.121923
$$821$$ 3.44603e7 1.78427 0.892136 0.451767i $$-0.149206\pi$$
0.892136 + 0.451767i $$0.149206\pi$$
$$822$$ 0 0
$$823$$ −9.74417e6 −0.501470 −0.250735 0.968056i $$-0.580672\pi$$
−0.250735 + 0.968056i $$0.580672\pi$$
$$824$$ −1.26100e6 −0.0646989
$$825$$ 0 0
$$826$$ −1.49541e7 −0.762622
$$827$$ 9.63214e6 0.489733 0.244866 0.969557i $$-0.421256\pi$$
0.244866 + 0.969557i $$0.421256\pi$$
$$828$$ 0 0
$$829$$ −2.79310e7 −1.41156 −0.705780 0.708431i $$-0.749403\pi$$
−0.705780 + 0.708431i $$0.749403\pi$$
$$830$$ 1.42194e7 0.716452
$$831$$ 0 0
$$832$$ −1.81124e7 −0.907126
$$833$$ −663626. −0.0331368
$$834$$ 0 0
$$835$$ 1.76239e7 0.874753
$$836$$ 586739. 0.0290355
$$837$$ 0 0
$$838$$ 1.25741e7 0.618540
$$839$$ 3.51475e7 1.72381 0.861906 0.507068i $$-0.169271\pi$$
0.861906 + 0.507068i $$0.169271\pi$$
$$840$$ 0 0
$$841$$ −1.60522e7 −0.782610
$$842$$ 3.76617e6 0.183071
$$843$$ 0 0
$$844$$ 2.46221e6 0.118978
$$845$$ 3.03101e6 0.146031
$$846$$ 0 0
$$847$$ −1.80725e6 −0.0865583
$$848$$ 1.27654e7 0.609599
$$849$$ 0 0
$$850$$ −1.36750e6 −0.0649203
$$851$$ 6.63368e6 0.314001
$$852$$ 0 0
$$853$$ 2.50498e7 1.17878 0.589389 0.807849i $$-0.299369\pi$$
0.589389 + 0.807849i $$0.299369\pi$$
$$854$$ 6.88152e6 0.322879
$$855$$ 0 0
$$856$$ −2.44900e7 −1.14236
$$857$$ 1.89631e7 0.881975 0.440987 0.897513i $$-0.354628\pi$$
0.440987 + 0.897513i $$0.354628\pi$$
$$858$$ 0 0
$$859$$ −3.04605e7 −1.40849 −0.704245 0.709957i $$-0.748714\pi$$
−0.704245 + 0.709957i $$0.748714\pi$$
$$860$$ 885600. 0.0408312
$$861$$ 0 0
$$862$$ 3.20060e7 1.46711
$$863$$ −3.46176e7 −1.58223 −0.791115 0.611667i $$-0.790499\pi$$
−0.791115 + 0.611667i $$0.790499\pi$$
$$864$$ 0 0
$$865$$ −4.27651e6 −0.194334
$$866$$ −1.25460e7 −0.568475
$$867$$ 0 0
$$868$$ −102403. −0.00461333
$$869$$ −986213. −0.0443018
$$870$$ 0 0
$$871$$ −7.25430e6 −0.324004
$$872$$ 1.10317e7 0.491307
$$873$$ 0 0
$$874$$ 5.91758e6 0.262039
$$875$$ 1.92871e6 0.0851621
$$876$$ 0 0
$$877$$ −3.61979e7 −1.58922 −0.794611 0.607119i $$-0.792325\pi$$
−0.794611 + 0.607119i $$0.792325\pi$$
$$878$$ 1.96727e7 0.861245
$$879$$ 0 0
$$880$$ 2.51275e6 0.109381
$$881$$ −2.40371e7 −1.04338 −0.521690 0.853135i $$-0.674698\pi$$
−0.521690 + 0.853135i $$0.674698\pi$$
$$882$$ 0 0
$$883$$ 3.64938e7 1.57513 0.787567 0.616229i $$-0.211341\pi$$
0.787567 + 0.616229i $$0.211341\pi$$
$$884$$ 1.09845e6 0.0472768
$$885$$ 0 0
$$886$$ −2.88905e7 −1.23644
$$887$$ −2.29197e7 −0.978137 −0.489069 0.872245i $$-0.662663\pi$$
−0.489069 + 0.872245i $$0.662663\pi$$
$$888$$ 0 0
$$889$$ 1.42387e7 0.604250
$$890$$ −9.01768e6 −0.381610
$$891$$ 0 0
$$892$$ 4.27630e6 0.179952
$$893$$ −1.40865e7 −0.591117
$$894$$ 0 0
$$895$$ −4.15817e6 −0.173518
$$896$$ 1.57920e7 0.657156
$$897$$ 0 0
$$898$$ 1.39579e7 0.577605
$$899$$ −337039. −0.0139085
$$900$$ 0 0
$$901$$ −6.49487e6 −0.266538
$$902$$ −1.13175e7 −0.463163
$$903$$ 0 0
$$904$$ 1.30517e7 0.531184
$$905$$ 1.46098e7 0.592956
$$906$$ 0 0
$$907$$ −2.66664e7 −1.07633 −0.538167 0.842838i $$-0.680883\pi$$
−0.538167 + 0.842838i $$0.680883\pi$$
$$908$$ −2.89036e6 −0.116342
$$909$$ 0 0
$$910$$ 7.98894e6 0.319805
$$911$$ −1.73286e7 −0.691778 −0.345889 0.938276i $$-0.612423\pi$$
−0.345889 + 0.938276i $$0.612423\pi$$
$$912$$ 0 0
$$913$$ 1.32935e7 0.527793
$$914$$ −5.91518e6 −0.234208
$$915$$ 0 0
$$916$$ 3.30005e6 0.129952
$$917$$ 5.34192e6 0.209785
$$918$$ 0 0
$$919$$ −1.72198e7 −0.672573 −0.336286 0.941760i $$-0.609171\pi$$
−0.336286 + 0.941760i $$0.609171\pi$$
$$920$$ −5.89850e6 −0.229759
$$921$$ 0 0
$$922$$ 2.78058e7 1.07723
$$923$$ −1.40585e7 −0.543168
$$924$$ 0 0
$$925$$ 3.38404e6 0.130041
$$926$$ 1.96848e7 0.754402
$$927$$ 0 0
$$928$$ −3.93183e6 −0.149874
$$929$$ −5.47137e6 −0.207997 −0.103998 0.994577i $$-0.533164\pi$$
−0.103998 + 0.994577i $$0.533164\pi$$
$$930$$ 0 0
$$931$$ 1.46494e6 0.0553919
$$932$$ −4.71193e6 −0.177688
$$933$$ 0 0
$$934$$ 4.87525e7 1.82865
$$935$$ −1.27846e6 −0.0478252
$$936$$ 0 0
$$937$$ 8.04892e6 0.299494 0.149747 0.988724i $$-0.452154\pi$$
0.149747 + 0.988724i $$0.452154\pi$$
$$938$$ 9.27072e6 0.344038
$$939$$ 0 0
$$940$$ 1.96195e6 0.0724215
$$941$$ 1.13721e7 0.418665 0.209333 0.977844i $$-0.432871\pi$$
0.209333 + 0.977844i $$0.432871\pi$$
$$942$$ 0 0
$$943$$ 2.21349e7 0.810584
$$944$$ 1.94379e7 0.709938
$$945$$ 0 0
$$946$$ −4.26941e6 −0.155110
$$947$$ −4.97017e7 −1.80093 −0.900463 0.434932i $$-0.856772\pi$$
−0.900463 + 0.434932i $$0.856772\pi$$
$$948$$ 0 0
$$949$$ 1.44199e7 0.519751
$$950$$ 3.01873e6 0.108521
$$951$$ 0 0
$$952$$ −1.00464e7 −0.359266
$$953$$ −2.95809e7 −1.05507 −0.527533 0.849534i $$-0.676883\pi$$
−0.527533 + 0.849534i $$0.676883\pi$$
$$954$$ 0 0
$$955$$ −1.78877e7 −0.634669
$$956$$ 3.44385e6 0.121871
$$957$$ 0 0
$$958$$ −2.42830e7 −0.854849
$$959$$ 4.95526e7 1.73988
$$960$$ 0 0
$$961$$ −2.86037e7 −0.999110
$$962$$ 1.40171e7 0.488337
$$963$$ 0 0
$$964$$ 6.85161e6 0.237465
$$965$$ 2.30522e7 0.796882
$$966$$ 0 0
$$967$$ 3.10475e7 1.06773 0.533863 0.845571i $$-0.320740\pi$$
0.533863 + 0.845571i $$0.320740\pi$$
$$968$$ 2.81950e6 0.0967128
$$969$$ 0 0
$$970$$ 1.18493e7 0.404355
$$971$$ 1.89426e7 0.644751 0.322376 0.946612i $$-0.395519\pi$$
0.322376 + 0.946612i $$0.395519\pi$$
$$972$$ 0 0
$$973$$ 3.72803e7 1.26240
$$974$$ −3.66113e7 −1.23657
$$975$$ 0 0
$$976$$ −8.94489e6 −0.300573
$$977$$ −5.49466e7 −1.84164 −0.920819 0.389989i $$-0.872479\pi$$
−0.920819 + 0.389989i $$0.872479\pi$$
$$978$$ 0 0
$$979$$ −8.43050e6 −0.281123
$$980$$ −204035. −0.00678640
$$981$$ 0 0
$$982$$ 4.69072e6 0.155225
$$983$$ 4.72203e7 1.55864 0.779318 0.626628i $$-0.215565\pi$$
0.779318 + 0.626628i $$0.215565\pi$$
$$984$$ 0 0
$$985$$ −1.53408e7 −0.503799
$$986$$ −4.62022e6 −0.151346
$$987$$ 0 0
$$988$$ −2.42480e6 −0.0790283
$$989$$ 8.35018e6 0.271459
$$990$$ 0 0
$$991$$ 4.48844e7 1.45182 0.725908 0.687792i $$-0.241420\pi$$
0.725908 + 0.687792i $$0.241420\pi$$
$$992$$ 297198. 0.00958885
$$993$$ 0 0
$$994$$ 1.79662e7 0.576753
$$995$$ 9.47463e6 0.303392
$$996$$ 0 0
$$997$$ 5.08797e7 1.62109 0.810543 0.585679i $$-0.199172\pi$$
0.810543 + 0.585679i $$0.199172\pi$$
$$998$$ 3.69295e7 1.17367
$$999$$ 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 495.6.a.e.1.1 3
3.2 odd 2 165.6.a.a.1.3 3
15.14 odd 2 825.6.a.j.1.1 3

By twisted newform
Twist Min Dim Char Parity Ord Type
165.6.a.a.1.3 3 3.2 odd 2
495.6.a.e.1.1 3 1.1 even 1 trivial
825.6.a.j.1.1 3 15.14 odd 2