# Properties

 Label 825.6.a.j.1.2 Level $825$ Weight $6$ Character 825.1 Self dual yes Analytic conductor $132.317$ Analytic rank $1$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$825 = 3 \cdot 5^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 825.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$132.316651346$$ 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 1.2 Root $$0.921799$$ of defining polynomial Character $$\chi$$ $$=$$ 825.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+2.92180 q^{2} -9.00000 q^{3} -23.4631 q^{4} -26.2962 q^{6} +85.0105 q^{7} -162.052 q^{8} +81.0000 q^{9} +O(q^{10})$$ $$q+2.92180 q^{2} -9.00000 q^{3} -23.4631 q^{4} -26.2962 q^{6} +85.0105 q^{7} -162.052 q^{8} +81.0000 q^{9} -121.000 q^{11} +211.168 q^{12} -724.085 q^{13} +248.384 q^{14} +277.335 q^{16} +2098.75 q^{17} +236.666 q^{18} -6.40223 q^{19} -765.094 q^{21} -353.538 q^{22} -1569.80 q^{23} +1458.47 q^{24} -2115.63 q^{26} -729.000 q^{27} -1994.61 q^{28} -5145.94 q^{29} -1031.88 q^{31} +5995.98 q^{32} +1089.00 q^{33} +6132.14 q^{34} -1900.51 q^{36} +12641.6 q^{37} -18.7060 q^{38} +6516.76 q^{39} +13808.7 q^{41} -2235.45 q^{42} +17012.0 q^{43} +2839.03 q^{44} -4586.64 q^{46} -8078.06 q^{47} -2496.02 q^{48} -9580.22 q^{49} -18888.8 q^{51} +16989.3 q^{52} +22110.8 q^{53} -2129.99 q^{54} -13776.1 q^{56} +57.6201 q^{57} -15035.4 q^{58} -16890.6 q^{59} -34398.6 q^{61} -3014.94 q^{62} +6885.85 q^{63} +8644.32 q^{64} +3181.84 q^{66} +37306.5 q^{67} -49243.3 q^{68} +14128.2 q^{69} -56607.5 q^{71} -13126.2 q^{72} +27777.9 q^{73} +36936.2 q^{74} +150.216 q^{76} -10286.3 q^{77} +19040.7 q^{78} -12759.9 q^{79} +6561.00 q^{81} +40346.2 q^{82} +69258.6 q^{83} +17951.5 q^{84} +49705.6 q^{86} +46313.5 q^{87} +19608.3 q^{88} +59029.2 q^{89} -61554.8 q^{91} +36832.4 q^{92} +9286.89 q^{93} -23602.5 q^{94} -53963.8 q^{96} -104905. q^{97} -27991.5 q^{98} -9801.00 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 7 q^{2} - 27 q^{3} + 25 q^{4} - 63 q^{6} + 172 q^{7} + 231 q^{8} + 243 q^{9}+O(q^{10})$$ 3 * q + 7 * q^2 - 27 * q^3 + 25 * q^4 - 63 * q^6 + 172 * q^7 + 231 * q^8 + 243 * q^9 $$3 q + 7 q^{2} - 27 q^{3} + 25 q^{4} - 63 q^{6} + 172 q^{7} + 231 q^{8} + 243 q^{9} - 363 q^{11} - 225 q^{12} + 654 q^{13} - 728 q^{14} - 415 q^{16} + 2366 q^{17} + 567 q^{18} - 2872 q^{19} - 1548 q^{21} - 847 q^{22} - 2272 q^{23} - 2079 q^{24} + 3422 q^{26} - 2187 q^{27} - 4592 q^{28} - 7738 q^{29} + 568 q^{31} - 1001 q^{32} + 3267 q^{33} + 2506 q^{34} + 2025 q^{36} + 9126 q^{37} - 13076 q^{38} - 5886 q^{39} - 8758 q^{41} + 6552 q^{42} + 14672 q^{43} - 3025 q^{44} - 28768 q^{46} + 19392 q^{47} + 3735 q^{48} - 26629 q^{49} - 21294 q^{51} + 61506 q^{52} + 4598 q^{53} - 5103 q^{54} + 2688 q^{56} + 25848 q^{57} - 8550 q^{58} - 9348 q^{59} - 60078 q^{61} + 14096 q^{62} + 13932 q^{63} - 7087 q^{64} + 7623 q^{66} + 38468 q^{67} - 59778 q^{68} + 20448 q^{69} - 74032 q^{71} + 18711 q^{72} + 44442 q^{73} + 82542 q^{74} - 98708 q^{76} - 20812 q^{77} - 30798 q^{78} - 108116 q^{79} + 19683 q^{81} + 92230 q^{82} + 81892 q^{83} + 41328 q^{84} + 126412 q^{86} + 69642 q^{87} - 27951 q^{88} + 167342 q^{89} - 31832 q^{91} - 72960 q^{92} - 5112 q^{93} + 12728 q^{94} + 9009 q^{96} - 159702 q^{97} - 163121 q^{98} - 29403 q^{99}+O(q^{100})$$ 3 * q + 7 * q^2 - 27 * q^3 + 25 * q^4 - 63 * q^6 + 172 * q^7 + 231 * q^8 + 243 * q^9 - 363 * q^11 - 225 * q^12 + 654 * q^13 - 728 * q^14 - 415 * q^16 + 2366 * q^17 + 567 * q^18 - 2872 * q^19 - 1548 * q^21 - 847 * q^22 - 2272 * q^23 - 2079 * q^24 + 3422 * q^26 - 2187 * q^27 - 4592 * q^28 - 7738 * q^29 + 568 * q^31 - 1001 * q^32 + 3267 * q^33 + 2506 * q^34 + 2025 * q^36 + 9126 * q^37 - 13076 * q^38 - 5886 * q^39 - 8758 * q^41 + 6552 * q^42 + 14672 * q^43 - 3025 * q^44 - 28768 * q^46 + 19392 * q^47 + 3735 * q^48 - 26629 * q^49 - 21294 * q^51 + 61506 * q^52 + 4598 * q^53 - 5103 * q^54 + 2688 * q^56 + 25848 * q^57 - 8550 * q^58 - 9348 * q^59 - 60078 * q^61 + 14096 * q^62 + 13932 * q^63 - 7087 * q^64 + 7623 * q^66 + 38468 * q^67 - 59778 * q^68 + 20448 * q^69 - 74032 * q^71 + 18711 * q^72 + 44442 * q^73 + 82542 * q^74 - 98708 * q^76 - 20812 * q^77 - 30798 * q^78 - 108116 * q^79 + 19683 * q^81 + 92230 * q^82 + 81892 * q^83 + 41328 * q^84 + 126412 * q^86 + 69642 * q^87 - 27951 * q^88 + 167342 * q^89 - 31832 * q^91 - 72960 * q^92 - 5112 * q^93 + 12728 * q^94 + 9009 * q^96 - 159702 * q^97 - 163121 * q^98 - 29403 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 2.92180 0.516506 0.258253 0.966077i $$-0.416853\pi$$
0.258253 + 0.966077i $$0.416853\pi$$
$$3$$ −9.00000 −0.577350
$$4$$ −23.4631 −0.733222
$$5$$ 0 0
$$6$$ −26.2962 −0.298205
$$7$$ 85.0105 0.655734 0.327867 0.944724i $$-0.393670\pi$$
0.327867 + 0.944724i $$0.393670\pi$$
$$8$$ −162.052 −0.895219
$$9$$ 81.0000 0.333333
$$10$$ 0 0
$$11$$ −121.000 −0.301511
$$12$$ 211.168 0.423326
$$13$$ −724.085 −1.18831 −0.594157 0.804349i $$-0.702514\pi$$
−0.594157 + 0.804349i $$0.702514\pi$$
$$14$$ 248.384 0.338690
$$15$$ 0 0
$$16$$ 277.335 0.270835
$$17$$ 2098.75 1.76132 0.880662 0.473745i $$-0.157098\pi$$
0.880662 + 0.473745i $$0.157098\pi$$
$$18$$ 236.666 0.172169
$$19$$ −6.40223 −0.00406862 −0.00203431 0.999998i $$-0.500648\pi$$
−0.00203431 + 0.999998i $$0.500648\pi$$
$$20$$ 0 0
$$21$$ −765.094 −0.378588
$$22$$ −353.538 −0.155732
$$23$$ −1569.80 −0.618764 −0.309382 0.950938i $$-0.600122\pi$$
−0.309382 + 0.950938i $$0.600122\pi$$
$$24$$ 1458.47 0.516855
$$25$$ 0 0
$$26$$ −2115.63 −0.613771
$$27$$ −729.000 −0.192450
$$28$$ −1994.61 −0.480798
$$29$$ −5145.94 −1.13624 −0.568119 0.822946i $$-0.692329\pi$$
−0.568119 + 0.822946i $$0.692329\pi$$
$$30$$ 0 0
$$31$$ −1031.88 −0.192852 −0.0964259 0.995340i $$-0.530741\pi$$
−0.0964259 + 0.995340i $$0.530741\pi$$
$$32$$ 5995.98 1.03511
$$33$$ 1089.00 0.174078
$$34$$ 6132.14 0.909735
$$35$$ 0 0
$$36$$ −1900.51 −0.244407
$$37$$ 12641.6 1.51809 0.759045 0.651039i $$-0.225666\pi$$
0.759045 + 0.651039i $$0.225666\pi$$
$$38$$ −18.7060 −0.00210147
$$39$$ 6516.76 0.686073
$$40$$ 0 0
$$41$$ 13808.7 1.28290 0.641450 0.767165i $$-0.278333\pi$$
0.641450 + 0.767165i $$0.278333\pi$$
$$42$$ −2235.45 −0.195543
$$43$$ 17012.0 1.40308 0.701542 0.712628i $$-0.252495\pi$$
0.701542 + 0.712628i $$0.252495\pi$$
$$44$$ 2839.03 0.221075
$$45$$ 0 0
$$46$$ −4586.64 −0.319595
$$47$$ −8078.06 −0.533412 −0.266706 0.963778i $$-0.585935\pi$$
−0.266706 + 0.963778i $$0.585935\pi$$
$$48$$ −2496.02 −0.156367
$$49$$ −9580.22 −0.570013
$$50$$ 0 0
$$51$$ −18888.8 −1.01690
$$52$$ 16989.3 0.871297
$$53$$ 22110.8 1.08122 0.540612 0.841272i $$-0.318193\pi$$
0.540612 + 0.841272i $$0.318193\pi$$
$$54$$ −2129.99 −0.0994016
$$55$$ 0 0
$$56$$ −13776.1 −0.587025
$$57$$ 57.6201 0.00234902
$$58$$ −15035.4 −0.586874
$$59$$ −16890.6 −0.631706 −0.315853 0.948808i $$-0.602291\pi$$
−0.315853 + 0.948808i $$0.602291\pi$$
$$60$$ 0 0
$$61$$ −34398.6 −1.18363 −0.591816 0.806073i $$-0.701589\pi$$
−0.591816 + 0.806073i $$0.701589\pi$$
$$62$$ −3014.94 −0.0996091
$$63$$ 6885.85 0.218578
$$64$$ 8644.32 0.263804
$$65$$ 0 0
$$66$$ 3181.84 0.0899122
$$67$$ 37306.5 1.01531 0.507654 0.861561i $$-0.330513\pi$$
0.507654 + 0.861561i $$0.330513\pi$$
$$68$$ −49243.3 −1.29144
$$69$$ 14128.2 0.357244
$$70$$ 0 0
$$71$$ −56607.5 −1.33269 −0.666344 0.745645i $$-0.732142\pi$$
−0.666344 + 0.745645i $$0.732142\pi$$
$$72$$ −13126.2 −0.298406
$$73$$ 27777.9 0.610088 0.305044 0.952338i $$-0.401329\pi$$
0.305044 + 0.952338i $$0.401329\pi$$
$$74$$ 36936.2 0.784102
$$75$$ 0 0
$$76$$ 150.216 0.00298320
$$77$$ −10286.3 −0.197711
$$78$$ 19040.7 0.354361
$$79$$ −12759.9 −0.230028 −0.115014 0.993364i $$-0.536691\pi$$
−0.115014 + 0.993364i $$0.536691\pi$$
$$80$$ 0 0
$$81$$ 6561.00 0.111111
$$82$$ 40346.2 0.662625
$$83$$ 69258.6 1.10352 0.551758 0.834004i $$-0.313957\pi$$
0.551758 + 0.834004i $$0.313957\pi$$
$$84$$ 17951.5 0.277589
$$85$$ 0 0
$$86$$ 49705.6 0.724702
$$87$$ 46313.5 0.656008
$$88$$ 19608.3 0.269919
$$89$$ 59029.2 0.789936 0.394968 0.918695i $$-0.370756\pi$$
0.394968 + 0.918695i $$0.370756\pi$$
$$90$$ 0 0
$$91$$ −61554.8 −0.779217
$$92$$ 36832.4 0.453691
$$93$$ 9286.89 0.111343
$$94$$ −23602.5 −0.275510
$$95$$ 0 0
$$96$$ −53963.8 −0.597620
$$97$$ −104905. −1.13205 −0.566027 0.824387i $$-0.691520\pi$$
−0.566027 + 0.824387i $$0.691520\pi$$
$$98$$ −27991.5 −0.294415
$$99$$ −9801.00 −0.100504
$$100$$ 0 0
$$101$$ 135417. 1.32090 0.660451 0.750869i $$-0.270365\pi$$
0.660451 + 0.750869i $$0.270365\pi$$
$$102$$ −55189.3 −0.525236
$$103$$ −167505. −1.55573 −0.777867 0.628428i $$-0.783698\pi$$
−0.777867 + 0.628428i $$0.783698\pi$$
$$104$$ 117339. 1.06380
$$105$$ 0 0
$$106$$ 64603.4 0.558458
$$107$$ −57989.6 −0.489656 −0.244828 0.969567i $$-0.578731\pi$$
−0.244828 + 0.969567i $$0.578731\pi$$
$$108$$ 17104.6 0.141109
$$109$$ −14956.1 −0.120574 −0.0602869 0.998181i $$-0.519202\pi$$
−0.0602869 + 0.998181i $$0.519202\pi$$
$$110$$ 0 0
$$111$$ −113774. −0.876469
$$112$$ 23576.4 0.177596
$$113$$ −198323. −1.46109 −0.730544 0.682866i $$-0.760733\pi$$
−0.730544 + 0.682866i $$0.760733\pi$$
$$114$$ 168.354 0.00121328
$$115$$ 0 0
$$116$$ 120740. 0.833115
$$117$$ −58650.9 −0.396105
$$118$$ −49351.0 −0.326280
$$119$$ 178416. 1.15496
$$120$$ 0 0
$$121$$ 14641.0 0.0909091
$$122$$ −100506. −0.611353
$$123$$ −124278. −0.740682
$$124$$ 24211.0 0.141403
$$125$$ 0 0
$$126$$ 20119.1 0.112897
$$127$$ 40986.5 0.225492 0.112746 0.993624i $$-0.464035\pi$$
0.112746 + 0.993624i $$0.464035\pi$$
$$128$$ −166614. −0.898851
$$129$$ −153108. −0.810071
$$130$$ 0 0
$$131$$ −238406. −1.21378 −0.606889 0.794787i $$-0.707583\pi$$
−0.606889 + 0.794787i $$0.707583\pi$$
$$132$$ −25551.3 −0.127637
$$133$$ −544.257 −0.00266793
$$134$$ 109002. 0.524413
$$135$$ 0 0
$$136$$ −340107. −1.57677
$$137$$ −149534. −0.680673 −0.340337 0.940304i $$-0.610541\pi$$
−0.340337 + 0.940304i $$0.610541\pi$$
$$138$$ 41279.8 0.184518
$$139$$ 167700. 0.736202 0.368101 0.929786i $$-0.380008\pi$$
0.368101 + 0.929786i $$0.380008\pi$$
$$140$$ 0 0
$$141$$ 72702.5 0.307965
$$142$$ −165396. −0.688341
$$143$$ 87614.3 0.358290
$$144$$ 22464.2 0.0902785
$$145$$ 0 0
$$146$$ 81161.5 0.315114
$$147$$ 86221.9 0.329097
$$148$$ −296611. −1.11310
$$149$$ −272698. −1.00627 −0.503136 0.864207i $$-0.667821\pi$$
−0.503136 + 0.864207i $$0.667821\pi$$
$$150$$ 0 0
$$151$$ −448320. −1.60009 −0.800047 0.599937i $$-0.795192\pi$$
−0.800047 + 0.599937i $$0.795192\pi$$
$$152$$ 1037.49 0.00364231
$$153$$ 169999. 0.587108
$$154$$ −30054.4 −0.102119
$$155$$ 0 0
$$156$$ −152903. −0.503044
$$157$$ 366634. 1.18709 0.593545 0.804801i $$-0.297728\pi$$
0.593545 + 0.804801i $$0.297728\pi$$
$$158$$ −37281.9 −0.118811
$$159$$ −198998. −0.624245
$$160$$ 0 0
$$161$$ −133450. −0.405744
$$162$$ 19169.9 0.0573896
$$163$$ −501806. −1.47934 −0.739668 0.672972i $$-0.765018\pi$$
−0.739668 + 0.672972i $$0.765018\pi$$
$$164$$ −323994. −0.940650
$$165$$ 0 0
$$166$$ 202360. 0.569973
$$167$$ 195231. 0.541700 0.270850 0.962622i $$-0.412695\pi$$
0.270850 + 0.962622i $$0.412695\pi$$
$$168$$ 123985. 0.338919
$$169$$ 153006. 0.412089
$$170$$ 0 0
$$171$$ −518.581 −0.00135621
$$172$$ −399154. −1.02877
$$173$$ −618180. −1.57036 −0.785181 0.619266i $$-0.787430\pi$$
−0.785181 + 0.619266i $$0.787430\pi$$
$$174$$ 135319. 0.338832
$$175$$ 0 0
$$176$$ −33557.6 −0.0816600
$$177$$ 152015. 0.364716
$$178$$ 172471. 0.408007
$$179$$ −88898.6 −0.207378 −0.103689 0.994610i $$-0.533065\pi$$
−0.103689 + 0.994610i $$0.533065\pi$$
$$180$$ 0 0
$$181$$ −378128. −0.857911 −0.428956 0.903326i $$-0.641118\pi$$
−0.428956 + 0.903326i $$0.641118\pi$$
$$182$$ −179851. −0.402470
$$183$$ 309588. 0.683370
$$184$$ 254389. 0.553930
$$185$$ 0 0
$$186$$ 27134.4 0.0575093
$$187$$ −253949. −0.531059
$$188$$ 189536. 0.391109
$$189$$ −61972.7 −0.126196
$$190$$ 0 0
$$191$$ 125442. 0.248806 0.124403 0.992232i $$-0.460299\pi$$
0.124403 + 0.992232i $$0.460299\pi$$
$$192$$ −77798.9 −0.152307
$$193$$ 928398. 1.79408 0.897038 0.441953i $$-0.145714\pi$$
0.897038 + 0.441953i $$0.145714\pi$$
$$194$$ −306512. −0.584713
$$195$$ 0 0
$$196$$ 224781. 0.417946
$$197$$ −1.03233e6 −1.89519 −0.947595 0.319473i $$-0.896494\pi$$
−0.947595 + 0.319473i $$0.896494\pi$$
$$198$$ −28636.6 −0.0519108
$$199$$ −155017. −0.277489 −0.138744 0.990328i $$-0.544307\pi$$
−0.138744 + 0.990328i $$0.544307\pi$$
$$200$$ 0 0
$$201$$ −335759. −0.586188
$$202$$ 395662. 0.682254
$$203$$ −437459. −0.745070
$$204$$ 443189. 0.745614
$$205$$ 0 0
$$206$$ −489417. −0.803546
$$207$$ −127154. −0.206255
$$208$$ −200814. −0.321837
$$209$$ 774.670 0.00122674
$$210$$ 0 0
$$211$$ −364263. −0.563259 −0.281630 0.959523i $$-0.590875\pi$$
−0.281630 + 0.959523i $$0.590875\pi$$
$$212$$ −518789. −0.792777
$$213$$ 509468. 0.769428
$$214$$ −169434. −0.252910
$$215$$ 0 0
$$216$$ 118136. 0.172285
$$217$$ −87720.4 −0.126459
$$218$$ −43698.8 −0.0622771
$$219$$ −250001. −0.352234
$$220$$ 0 0
$$221$$ −1.51968e6 −2.09301
$$222$$ −332426. −0.452702
$$223$$ 74806.0 0.100734 0.0503668 0.998731i $$-0.483961\pi$$
0.0503668 + 0.998731i $$0.483961\pi$$
$$224$$ 509721. 0.678755
$$225$$ 0 0
$$226$$ −579459. −0.754661
$$227$$ −1.22677e6 −1.58015 −0.790075 0.613010i $$-0.789958\pi$$
−0.790075 + 0.613010i $$0.789958\pi$$
$$228$$ −1351.94 −0.00172235
$$229$$ −440852. −0.555526 −0.277763 0.960650i $$-0.589593\pi$$
−0.277763 + 0.960650i $$0.589593\pi$$
$$230$$ 0 0
$$231$$ 92576.4 0.114149
$$232$$ 833910. 1.01718
$$233$$ 514549. 0.620922 0.310461 0.950586i $$-0.399517\pi$$
0.310461 + 0.950586i $$0.399517\pi$$
$$234$$ −171366. −0.204590
$$235$$ 0 0
$$236$$ 396306. 0.463181
$$237$$ 114839. 0.132807
$$238$$ 521296. 0.596544
$$239$$ −984315. −1.11465 −0.557326 0.830294i $$-0.688173\pi$$
−0.557326 + 0.830294i $$0.688173\pi$$
$$240$$ 0 0
$$241$$ 284794. 0.315855 0.157927 0.987451i $$-0.449519\pi$$
0.157927 + 0.987451i $$0.449519\pi$$
$$242$$ 42778.1 0.0469551
$$243$$ −59049.0 −0.0641500
$$244$$ 807098. 0.867865
$$245$$ 0 0
$$246$$ −363116. −0.382567
$$247$$ 4635.76 0.00483480
$$248$$ 167218. 0.172645
$$249$$ −623328. −0.637115
$$250$$ 0 0
$$251$$ 134529. 0.134782 0.0673911 0.997727i $$-0.478532\pi$$
0.0673911 + 0.997727i $$0.478532\pi$$
$$252$$ −161563. −0.160266
$$253$$ 189946. 0.186564
$$254$$ 119754. 0.116468
$$255$$ 0 0
$$256$$ −763432. −0.728066
$$257$$ −2.06732e6 −1.95242 −0.976212 0.216817i $$-0.930433\pi$$
−0.976212 + 0.216817i $$0.930433\pi$$
$$258$$ −447350. −0.418407
$$259$$ 1.07467e6 0.995462
$$260$$ 0 0
$$261$$ −416821. −0.378746
$$262$$ −696575. −0.626924
$$263$$ 661739. 0.589926 0.294963 0.955509i $$-0.404693\pi$$
0.294963 + 0.955509i $$0.404693\pi$$
$$264$$ −176475. −0.155838
$$265$$ 0 0
$$266$$ −1590.21 −0.00137800
$$267$$ −531263. −0.456070
$$268$$ −875327. −0.744446
$$269$$ 703008. 0.592351 0.296176 0.955133i $$-0.404289\pi$$
0.296176 + 0.955133i $$0.404289\pi$$
$$270$$ 0 0
$$271$$ −1.33847e6 −1.10710 −0.553549 0.832817i $$-0.686727\pi$$
−0.553549 + 0.832817i $$0.686727\pi$$
$$272$$ 582059. 0.477029
$$273$$ 553993. 0.449881
$$274$$ −436908. −0.351572
$$275$$ 0 0
$$276$$ −331492. −0.261939
$$277$$ 33171.1 0.0259753 0.0129877 0.999916i $$-0.495866\pi$$
0.0129877 + 0.999916i $$0.495866\pi$$
$$278$$ 489987. 0.380253
$$279$$ −83582.0 −0.0642839
$$280$$ 0 0
$$281$$ −321114. −0.242601 −0.121301 0.992616i $$-0.538707\pi$$
−0.121301 + 0.992616i $$0.538707\pi$$
$$282$$ 212422. 0.159066
$$283$$ 1.90591e6 1.41461 0.707305 0.706908i $$-0.249911\pi$$
0.707305 + 0.706908i $$0.249911\pi$$
$$284$$ 1.32819e6 0.977155
$$285$$ 0 0
$$286$$ 255991. 0.185059
$$287$$ 1.17388e6 0.841240
$$288$$ 485675. 0.345036
$$289$$ 2.98491e6 2.10226
$$290$$ 0 0
$$291$$ 944146. 0.653592
$$292$$ −651756. −0.447330
$$293$$ −272957. −0.185748 −0.0928742 0.995678i $$-0.529605\pi$$
−0.0928742 + 0.995678i $$0.529605\pi$$
$$294$$ 251923. 0.169981
$$295$$ 0 0
$$296$$ −2.04859e6 −1.35902
$$297$$ 88209.0 0.0580259
$$298$$ −796768. −0.519746
$$299$$ 1.13667e6 0.735286
$$300$$ 0 0
$$301$$ 1.44620e6 0.920050
$$302$$ −1.30990e6 −0.826459
$$303$$ −1.21876e6 −0.762624
$$304$$ −1775.57 −0.00110193
$$305$$ 0 0
$$306$$ 496703. 0.303245
$$307$$ 843806. 0.510971 0.255486 0.966813i $$-0.417765\pi$$
0.255486 + 0.966813i $$0.417765\pi$$
$$308$$ 241348. 0.144966
$$309$$ 1.50755e6 0.898204
$$310$$ 0 0
$$311$$ 2.06909e6 1.21305 0.606523 0.795066i $$-0.292564\pi$$
0.606523 + 0.795066i $$0.292564\pi$$
$$312$$ −1.05605e6 −0.614186
$$313$$ −603113. −0.347967 −0.173983 0.984749i $$-0.555664\pi$$
−0.173983 + 0.984749i $$0.555664\pi$$
$$314$$ 1.07123e6 0.613139
$$315$$ 0 0
$$316$$ 299387. 0.168661
$$317$$ 1.67334e6 0.935268 0.467634 0.883922i $$-0.345107\pi$$
0.467634 + 0.883922i $$0.345107\pi$$
$$318$$ −581431. −0.322426
$$319$$ 622659. 0.342589
$$320$$ 0 0
$$321$$ 521907. 0.282703
$$322$$ −389913. −0.209569
$$323$$ −13436.7 −0.00716616
$$324$$ −153941. −0.0814691
$$325$$ 0 0
$$326$$ −1.46618e6 −0.764086
$$327$$ 134605. 0.0696133
$$328$$ −2.23772e6 −1.14848
$$329$$ −686720. −0.349776
$$330$$ 0 0
$$331$$ −2.20421e6 −1.10582 −0.552909 0.833242i $$-0.686482\pi$$
−0.552909 + 0.833242i $$0.686482\pi$$
$$332$$ −1.62502e6 −0.809122
$$333$$ 1.02397e6 0.506030
$$334$$ 570427. 0.279791
$$335$$ 0 0
$$336$$ −212188. −0.102535
$$337$$ −1.07377e6 −0.515036 −0.257518 0.966273i $$-0.582905\pi$$
−0.257518 + 0.966273i $$0.582905\pi$$
$$338$$ 447052. 0.212847
$$339$$ 1.78490e6 0.843559
$$340$$ 0 0
$$341$$ 124857. 0.0581470
$$342$$ −1515.19 −0.000700489 0
$$343$$ −2.24319e6 −1.02951
$$344$$ −2.75683e6 −1.25607
$$345$$ 0 0
$$346$$ −1.80620e6 −0.811101
$$347$$ 1.39783e6 0.623203 0.311601 0.950213i $$-0.399135\pi$$
0.311601 + 0.950213i $$0.399135\pi$$
$$348$$ −1.08666e6 −0.480999
$$349$$ −2.66674e6 −1.17197 −0.585985 0.810322i $$-0.699292\pi$$
−0.585985 + 0.810322i $$0.699292\pi$$
$$350$$ 0 0
$$351$$ 527858. 0.228691
$$352$$ −725514. −0.312097
$$353$$ 594048. 0.253738 0.126869 0.991920i $$-0.459507\pi$$
0.126869 + 0.991920i $$0.459507\pi$$
$$354$$ 444159. 0.188378
$$355$$ 0 0
$$356$$ −1.38501e6 −0.579198
$$357$$ −1.60575e6 −0.666816
$$358$$ −259744. −0.107112
$$359$$ −3.35774e6 −1.37503 −0.687513 0.726172i $$-0.741298\pi$$
−0.687513 + 0.726172i $$0.741298\pi$$
$$360$$ 0 0
$$361$$ −2.47606e6 −0.999983
$$362$$ −1.10481e6 −0.443116
$$363$$ −131769. −0.0524864
$$364$$ 1.44427e6 0.571339
$$365$$ 0 0
$$366$$ 904553. 0.352965
$$367$$ −2.58775e6 −1.00290 −0.501450 0.865187i $$-0.667200\pi$$
−0.501450 + 0.865187i $$0.667200\pi$$
$$368$$ −435362. −0.167583
$$369$$ 1.11850e6 0.427633
$$370$$ 0 0
$$371$$ 1.87965e6 0.708995
$$372$$ −217899. −0.0816391
$$373$$ 1.07376e6 0.399608 0.199804 0.979836i $$-0.435969\pi$$
0.199804 + 0.979836i $$0.435969\pi$$
$$374$$ −741989. −0.274295
$$375$$ 0 0
$$376$$ 1.30907e6 0.477520
$$377$$ 3.72610e6 1.35021
$$378$$ −181072. −0.0651810
$$379$$ 5.16745e6 1.84790 0.923949 0.382516i $$-0.124942\pi$$
0.923949 + 0.382516i $$0.124942\pi$$
$$380$$ 0 0
$$381$$ −368878. −0.130188
$$382$$ 366517. 0.128510
$$383$$ −2.42092e6 −0.843302 −0.421651 0.906758i $$-0.638549\pi$$
−0.421651 + 0.906758i $$0.638549\pi$$
$$384$$ 1.49953e6 0.518952
$$385$$ 0 0
$$386$$ 2.71259e6 0.926651
$$387$$ 1.37797e6 0.467695
$$388$$ 2.46140e6 0.830047
$$389$$ 1.92154e6 0.643835 0.321918 0.946768i $$-0.395673\pi$$
0.321918 + 0.946768i $$0.395673\pi$$
$$390$$ 0 0
$$391$$ −3.29463e6 −1.08984
$$392$$ 1.55249e6 0.510287
$$393$$ 2.14566e6 0.700775
$$394$$ −3.01626e6 −0.978877
$$395$$ 0 0
$$396$$ 229962. 0.0736915
$$397$$ −4.09572e6 −1.30423 −0.652114 0.758121i $$-0.726118\pi$$
−0.652114 + 0.758121i $$0.726118\pi$$
$$398$$ −452928. −0.143325
$$399$$ 4898.31 0.00154033
$$400$$ 0 0
$$401$$ 5.97525e6 1.85565 0.927823 0.373022i $$-0.121678\pi$$
0.927823 + 0.373022i $$0.121678\pi$$
$$402$$ −981020. −0.302770
$$403$$ 747167. 0.229168
$$404$$ −3.17731e6 −0.968514
$$405$$ 0 0
$$406$$ −1.27817e6 −0.384833
$$407$$ −1.52963e6 −0.457721
$$408$$ 3.06097e6 0.910350
$$409$$ −1.92665e6 −0.569500 −0.284750 0.958602i $$-0.591911\pi$$
−0.284750 + 0.958602i $$0.591911\pi$$
$$410$$ 0 0
$$411$$ 1.34581e6 0.392987
$$412$$ 3.93019e6 1.14070
$$413$$ −1.43588e6 −0.414231
$$414$$ −371518. −0.106532
$$415$$ 0 0
$$416$$ −4.34160e6 −1.23003
$$417$$ −1.50930e6 −0.425047
$$418$$ 2263.43 0.000633616 0
$$419$$ −1.47857e6 −0.411441 −0.205720 0.978611i $$-0.565954\pi$$
−0.205720 + 0.978611i $$0.565954\pi$$
$$420$$ 0 0
$$421$$ 4.82644e6 1.32715 0.663577 0.748108i $$-0.269037\pi$$
0.663577 + 0.748108i $$0.269037\pi$$
$$422$$ −1.06430e6 −0.290927
$$423$$ −654323. −0.177804
$$424$$ −3.58311e6 −0.967932
$$425$$ 0 0
$$426$$ 1.48856e6 0.397414
$$427$$ −2.92425e6 −0.776147
$$428$$ 1.36062e6 0.359026
$$429$$ −788528. −0.206859
$$430$$ 0 0
$$431$$ 2.89229e6 0.749978 0.374989 0.927029i $$-0.377647\pi$$
0.374989 + 0.927029i $$0.377647\pi$$
$$432$$ −202178. −0.0521223
$$433$$ −2.36700e6 −0.606706 −0.303353 0.952878i $$-0.598106\pi$$
−0.303353 + 0.952878i $$0.598106\pi$$
$$434$$ −256301. −0.0653170
$$435$$ 0 0
$$436$$ 350917. 0.0884073
$$437$$ 10050.2 0.00251752
$$438$$ −730453. −0.181931
$$439$$ 3.22037e6 0.797525 0.398762 0.917054i $$-0.369440\pi$$
0.398762 + 0.917054i $$0.369440\pi$$
$$440$$ 0 0
$$441$$ −775997. −0.190004
$$442$$ −4.44019e6 −1.08105
$$443$$ 3.80335e6 0.920782 0.460391 0.887716i $$-0.347709\pi$$
0.460391 + 0.887716i $$0.347709\pi$$
$$444$$ 2.66950e6 0.642646
$$445$$ 0 0
$$446$$ 218568. 0.0520295
$$447$$ 2.45428e6 0.580972
$$448$$ 734858. 0.172985
$$449$$ −6.55303e6 −1.53400 −0.767001 0.641646i $$-0.778252\pi$$
−0.767001 + 0.641646i $$0.778252\pi$$
$$450$$ 0 0
$$451$$ −1.67085e6 −0.386809
$$452$$ 4.65326e6 1.07130
$$453$$ 4.03488e6 0.923815
$$454$$ −3.58437e6 −0.816157
$$455$$ 0 0
$$456$$ −9337.45 −0.00210289
$$457$$ 7.41470e6 1.66074 0.830372 0.557209i $$-0.188128\pi$$
0.830372 + 0.557209i $$0.188128\pi$$
$$458$$ −1.28808e6 −0.286932
$$459$$ −1.52999e6 −0.338967
$$460$$ 0 0
$$461$$ 409525. 0.0897487 0.0448743 0.998993i $$-0.485711\pi$$
0.0448743 + 0.998993i $$0.485711\pi$$
$$462$$ 270490. 0.0589584
$$463$$ −7.07192e6 −1.53315 −0.766575 0.642154i $$-0.778041\pi$$
−0.766575 + 0.642154i $$0.778041\pi$$
$$464$$ −1.42715e6 −0.307734
$$465$$ 0 0
$$466$$ 1.50341e6 0.320710
$$467$$ 6.35423e6 1.34825 0.674125 0.738617i $$-0.264521\pi$$
0.674125 + 0.738617i $$0.264521\pi$$
$$468$$ 1.37613e6 0.290432
$$469$$ 3.17145e6 0.665772
$$470$$ 0 0
$$471$$ −3.29970e6 −0.685366
$$472$$ 2.73716e6 0.565516
$$473$$ −2.05845e6 −0.423046
$$474$$ 335537. 0.0685954
$$475$$ 0 0
$$476$$ −4.18619e6 −0.846841
$$477$$ 1.79098e6 0.360408
$$478$$ −2.87597e6 −0.575725
$$479$$ 6.36751e6 1.26803 0.634017 0.773319i $$-0.281405\pi$$
0.634017 + 0.773319i $$0.281405\pi$$
$$480$$ 0 0
$$481$$ −9.15358e6 −1.80397
$$482$$ 832109. 0.163141
$$483$$ 1.20105e6 0.234257
$$484$$ −343523. −0.0666565
$$485$$ 0 0
$$486$$ −172529. −0.0331339
$$487$$ 2.94280e6 0.562261 0.281130 0.959670i $$-0.409291\pi$$
0.281130 + 0.959670i $$0.409291\pi$$
$$488$$ 5.57437e6 1.05961
$$489$$ 4.51626e6 0.854095
$$490$$ 0 0
$$491$$ −501628. −0.0939027 −0.0469513 0.998897i $$-0.514951\pi$$
−0.0469513 + 0.998897i $$0.514951\pi$$
$$492$$ 2.91595e6 0.543084
$$493$$ −1.08001e7 −2.00129
$$494$$ 13544.8 0.00249720
$$495$$ 0 0
$$496$$ −286176. −0.0522311
$$497$$ −4.81224e6 −0.873888
$$498$$ −1.82124e6 −0.329074
$$499$$ −9.19784e6 −1.65362 −0.826808 0.562484i $$-0.809846\pi$$
−0.826808 + 0.562484i $$0.809846\pi$$
$$500$$ 0 0
$$501$$ −1.75708e6 −0.312750
$$502$$ 393068. 0.0696158
$$503$$ 6.53811e6 1.15221 0.576106 0.817375i $$-0.304572\pi$$
0.576106 + 0.817375i $$0.304572\pi$$
$$504$$ −1.11587e6 −0.195675
$$505$$ 0 0
$$506$$ 554984. 0.0963616
$$507$$ −1.37705e6 −0.237920
$$508$$ −961669. −0.165336
$$509$$ −1.49017e6 −0.254942 −0.127471 0.991842i $$-0.540686\pi$$
−0.127471 + 0.991842i $$0.540686\pi$$
$$510$$ 0 0
$$511$$ 2.36141e6 0.400055
$$512$$ 3.10107e6 0.522801
$$513$$ 4667.23 0.000783007 0
$$514$$ −6.04029e6 −1.00844
$$515$$ 0 0
$$516$$ 3.59238e6 0.593962
$$517$$ 977445. 0.160830
$$518$$ 3.13996e6 0.514162
$$519$$ 5.56362e6 0.906649
$$520$$ 0 0
$$521$$ 3.82163e6 0.616814 0.308407 0.951255i $$-0.400204\pi$$
0.308407 + 0.951255i $$0.400204\pi$$
$$522$$ −1.21787e6 −0.195625
$$523$$ 4.18273e6 0.668660 0.334330 0.942456i $$-0.391490\pi$$
0.334330 + 0.942456i $$0.391490\pi$$
$$524$$ 5.59375e6 0.889968
$$525$$ 0 0
$$526$$ 1.93347e6 0.304700
$$527$$ −2.16566e6 −0.339675
$$528$$ 302018. 0.0471464
$$529$$ −3.97207e6 −0.617131
$$530$$ 0 0
$$531$$ −1.36814e6 −0.210569
$$532$$ 12769.9 0.00195619
$$533$$ −9.99866e6 −1.52449
$$534$$ −1.55224e6 −0.235563
$$535$$ 0 0
$$536$$ −6.04560e6 −0.908923
$$537$$ 800088. 0.119730
$$538$$ 2.05405e6 0.305953
$$539$$ 1.15921e6 0.171866
$$540$$ 0 0
$$541$$ 2.80424e6 0.411928 0.205964 0.978560i $$-0.433967\pi$$
0.205964 + 0.978560i $$0.433967\pi$$
$$542$$ −3.91075e6 −0.571823
$$543$$ 3.40315e6 0.495315
$$544$$ 1.25841e7 1.82316
$$545$$ 0 0
$$546$$ 1.61866e6 0.232366
$$547$$ 8.42403e6 1.20379 0.601896 0.798574i $$-0.294412\pi$$
0.601896 + 0.798574i $$0.294412\pi$$
$$548$$ 3.50853e6 0.499084
$$549$$ −2.78629e6 −0.394544
$$550$$ 0 0
$$551$$ 32945.5 0.00462293
$$552$$ −2.28951e6 −0.319811
$$553$$ −1.08473e6 −0.150837
$$554$$ 96919.4 0.0134164
$$555$$ 0 0
$$556$$ −3.93477e6 −0.539799
$$557$$ 3.56702e6 0.487155 0.243577 0.969881i $$-0.421679\pi$$
0.243577 + 0.969881i $$0.421679\pi$$
$$558$$ −244210. −0.0332030
$$559$$ −1.23181e7 −1.66730
$$560$$ 0 0
$$561$$ 2.28554e6 0.306607
$$562$$ −938231. −0.125305
$$563$$ −4.70552e6 −0.625658 −0.312829 0.949810i $$-0.601277\pi$$
−0.312829 + 0.949810i $$0.601277\pi$$
$$564$$ −1.70583e6 −0.225807
$$565$$ 0 0
$$566$$ 5.56869e6 0.730655
$$567$$ 557754. 0.0728593
$$568$$ 9.17337e6 1.19305
$$569$$ 2.60879e6 0.337799 0.168900 0.985633i $$-0.445979\pi$$
0.168900 + 0.985633i $$0.445979\pi$$
$$570$$ 0 0
$$571$$ 1.13036e6 0.145086 0.0725431 0.997365i $$-0.476889\pi$$
0.0725431 + 0.997365i $$0.476889\pi$$
$$572$$ −2.05570e6 −0.262706
$$573$$ −1.12898e6 −0.143648
$$574$$ 3.42985e6 0.434506
$$575$$ 0 0
$$576$$ 700190. 0.0879346
$$577$$ −1.12075e7 −1.40142 −0.700710 0.713446i $$-0.747133\pi$$
−0.700710 + 0.713446i $$0.747133\pi$$
$$578$$ 8.72132e6 1.08583
$$579$$ −8.35558e6 −1.03581
$$580$$ 0 0
$$581$$ 5.88771e6 0.723613
$$582$$ 2.75861e6 0.337584
$$583$$ −2.67541e6 −0.326001
$$584$$ −4.50147e6 −0.546162
$$585$$ 0 0
$$586$$ −797526. −0.0959402
$$587$$ −9.05798e6 −1.08502 −0.542508 0.840051i $$-0.682525\pi$$
−0.542508 + 0.840051i $$0.682525\pi$$
$$588$$ −2.02303e6 −0.241301
$$589$$ 6606.31 0.000784641 0
$$590$$ 0 0
$$591$$ 9.29097e6 1.09419
$$592$$ 3.50596e6 0.411152
$$593$$ −1.00907e7 −1.17837 −0.589186 0.807997i $$-0.700552\pi$$
−0.589186 + 0.807997i $$0.700552\pi$$
$$594$$ 257729. 0.0299707
$$595$$ 0 0
$$596$$ 6.39833e6 0.737821
$$597$$ 1.39515e6 0.160208
$$598$$ 3.32112e6 0.379780
$$599$$ −1.29126e7 −1.47044 −0.735222 0.677827i $$-0.762922\pi$$
−0.735222 + 0.677827i $$0.762922\pi$$
$$600$$ 0 0
$$601$$ 1.82591e6 0.206202 0.103101 0.994671i $$-0.467124\pi$$
0.103101 + 0.994671i $$0.467124\pi$$
$$602$$ 4.22550e6 0.475211
$$603$$ 3.02183e6 0.338436
$$604$$ 1.05190e7 1.17322
$$605$$ 0 0
$$606$$ −3.56096e6 −0.393900
$$607$$ 1.26409e6 0.139253 0.0696267 0.997573i $$-0.477819\pi$$
0.0696267 + 0.997573i $$0.477819\pi$$
$$608$$ −38387.7 −0.00421146
$$609$$ 3.93713e6 0.430166
$$610$$ 0 0
$$611$$ 5.84920e6 0.633860
$$612$$ −3.98870e6 −0.430480
$$613$$ −1.59125e7 −1.71036 −0.855180 0.518332i $$-0.826553\pi$$
−0.855180 + 0.518332i $$0.826553\pi$$
$$614$$ 2.46543e6 0.263920
$$615$$ 0 0
$$616$$ 1.66691e6 0.176995
$$617$$ −85225.6 −0.00901274 −0.00450637 0.999990i $$-0.501434\pi$$
−0.00450637 + 0.999990i $$0.501434\pi$$
$$618$$ 4.40475e6 0.463928
$$619$$ −1.20387e7 −1.26285 −0.631425 0.775437i $$-0.717530\pi$$
−0.631425 + 0.775437i $$0.717530\pi$$
$$620$$ 0 0
$$621$$ 1.14439e6 0.119081
$$622$$ 6.04545e6 0.626546
$$623$$ 5.01810e6 0.517987
$$624$$ 1.80733e6 0.185813
$$625$$ 0 0
$$626$$ −1.76218e6 −0.179727
$$627$$ −6972.03 −0.000708256 0
$$628$$ −8.60236e6 −0.870400
$$629$$ 2.65316e7 2.67385
$$630$$ 0 0
$$631$$ 121417. 0.0121396 0.00606981 0.999982i $$-0.498068\pi$$
0.00606981 + 0.999982i $$0.498068\pi$$
$$632$$ 2.06777e6 0.205925
$$633$$ 3.27836e6 0.325198
$$634$$ 4.88916e6 0.483071
$$635$$ 0 0
$$636$$ 4.66910e6 0.457710
$$637$$ 6.93689e6 0.677355
$$638$$ 1.81928e6 0.176949
$$639$$ −4.58521e6 −0.444229
$$640$$ 0 0
$$641$$ 1.25328e7 1.20477 0.602385 0.798206i $$-0.294217\pi$$
0.602385 + 0.798206i $$0.294217\pi$$
$$642$$ 1.52491e6 0.146018
$$643$$ 3.45380e6 0.329435 0.164717 0.986341i $$-0.447329\pi$$
0.164717 + 0.986341i $$0.447329\pi$$
$$644$$ 3.13114e6 0.297501
$$645$$ 0 0
$$646$$ −39259.4 −0.00370137
$$647$$ −1.71115e7 −1.60704 −0.803521 0.595276i $$-0.797043\pi$$
−0.803521 + 0.595276i $$0.797043\pi$$
$$648$$ −1.06322e6 −0.0994688
$$649$$ 2.04376e6 0.190467
$$650$$ 0 0
$$651$$ 789483. 0.0730114
$$652$$ 1.17739e7 1.08468
$$653$$ −1.61622e7 −1.48326 −0.741630 0.670809i $$-0.765947\pi$$
−0.741630 + 0.670809i $$0.765947\pi$$
$$654$$ 393289. 0.0359557
$$655$$ 0 0
$$656$$ 3.82964e6 0.347455
$$657$$ 2.25001e6 0.203363
$$658$$ −2.00646e6 −0.180661
$$659$$ 9.54805e6 0.856449 0.428224 0.903672i $$-0.359139\pi$$
0.428224 + 0.903672i $$0.359139\pi$$
$$660$$ 0 0
$$661$$ −1.24374e7 −1.10720 −0.553599 0.832784i $$-0.686746\pi$$
−0.553599 + 0.832784i $$0.686746\pi$$
$$662$$ −6.44027e6 −0.571162
$$663$$ 1.36771e7 1.20840
$$664$$ −1.12235e7 −0.987889
$$665$$ 0 0
$$666$$ 2.99183e6 0.261367
$$667$$ 8.07810e6 0.703064
$$668$$ −4.58073e6 −0.397186
$$669$$ −673254. −0.0581586
$$670$$ 0 0
$$671$$ 4.16224e6 0.356878
$$672$$ −4.58749e6 −0.391879
$$673$$ −1.37772e7 −1.17253 −0.586266 0.810119i $$-0.699403\pi$$
−0.586266 + 0.810119i $$0.699403\pi$$
$$674$$ −3.13735e6 −0.266019
$$675$$ 0 0
$$676$$ −3.58999e6 −0.302153
$$677$$ 1.06456e6 0.0892683 0.0446341 0.999003i $$-0.485788\pi$$
0.0446341 + 0.999003i $$0.485788\pi$$
$$678$$ 5.21513e6 0.435703
$$679$$ −8.91804e6 −0.742326
$$680$$ 0 0
$$681$$ 1.10409e7 0.912300
$$682$$ 364807. 0.0300333
$$683$$ 1.48070e7 1.21455 0.607275 0.794492i $$-0.292263\pi$$
0.607275 + 0.794492i $$0.292263\pi$$
$$684$$ 12167.5 0.000994400 0
$$685$$ 0 0
$$686$$ −6.55415e6 −0.531748
$$687$$ 3.96767e6 0.320733
$$688$$ 4.71803e6 0.380005
$$689$$ −1.60101e7 −1.28483
$$690$$ 0 0
$$691$$ 1.56554e7 1.24729 0.623646 0.781707i $$-0.285651\pi$$
0.623646 + 0.781707i $$0.285651\pi$$
$$692$$ 1.45044e7 1.15142
$$693$$ −833188. −0.0659037
$$694$$ 4.08417e6 0.321888
$$695$$ 0 0
$$696$$ −7.50519e6 −0.587271
$$697$$ 2.89810e7 2.25960
$$698$$ −7.79167e6 −0.605330
$$699$$ −4.63094e6 −0.358490
$$700$$ 0 0
$$701$$ 1.56474e7 1.20267 0.601334 0.798998i $$-0.294636\pi$$
0.601334 + 0.798998i $$0.294636\pi$$
$$702$$ 1.54229e6 0.118120
$$703$$ −80934.4 −0.00617653
$$704$$ −1.04596e6 −0.0795398
$$705$$ 0 0
$$706$$ 1.73569e6 0.131057
$$707$$ 1.15119e7 0.866160
$$708$$ −3.56675e6 −0.267417
$$709$$ −635875. −0.0475068 −0.0237534 0.999718i $$-0.507562\pi$$
−0.0237534 + 0.999718i $$0.507562\pi$$
$$710$$ 0 0
$$711$$ −1.03355e6 −0.0766759
$$712$$ −9.56580e6 −0.707166
$$713$$ 1.61984e6 0.119330
$$714$$ −4.69167e6 −0.344415
$$715$$ 0 0
$$716$$ 2.08584e6 0.152054
$$717$$ 8.85884e6 0.643545
$$718$$ −9.81064e6 −0.710209
$$719$$ −2.50359e6 −0.180610 −0.0903048 0.995914i $$-0.528784\pi$$
−0.0903048 + 0.995914i $$0.528784\pi$$
$$720$$ 0 0
$$721$$ −1.42397e7 −1.02015
$$722$$ −7.23454e6 −0.516497
$$723$$ −2.56314e6 −0.182359
$$724$$ 8.87205e6 0.629039
$$725$$ 0 0
$$726$$ −385003. −0.0271095
$$727$$ −8.82888e6 −0.619540 −0.309770 0.950811i $$-0.600252\pi$$
−0.309770 + 0.950811i $$0.600252\pi$$
$$728$$ 9.97508e6 0.697570
$$729$$ 531441. 0.0370370
$$730$$ 0 0
$$731$$ 3.57040e7 2.47129
$$732$$ −7.26389e6 −0.501062
$$733$$ 2.93960e6 0.202082 0.101041 0.994882i $$-0.467783\pi$$
0.101041 + 0.994882i $$0.467783\pi$$
$$734$$ −7.56089e6 −0.518004
$$735$$ 0 0
$$736$$ −9.41250e6 −0.640487
$$737$$ −4.51409e6 −0.306127
$$738$$ 3.26804e6 0.220875
$$739$$ −1.51762e6 −0.102224 −0.0511120 0.998693i $$-0.516277\pi$$
−0.0511120 + 0.998693i $$0.516277\pi$$
$$740$$ 0 0
$$741$$ −41721.8 −0.00279137
$$742$$ 5.49197e6 0.366200
$$743$$ 1.07165e7 0.712168 0.356084 0.934454i $$-0.384112\pi$$
0.356084 + 0.934454i $$0.384112\pi$$
$$744$$ −1.50496e6 −0.0996764
$$745$$ 0 0
$$746$$ 3.13731e6 0.206400
$$747$$ 5.60995e6 0.367839
$$748$$ 5.95844e6 0.389384
$$749$$ −4.92973e6 −0.321084
$$750$$ 0 0
$$751$$ −2.69694e7 −1.74490 −0.872452 0.488700i $$-0.837471\pi$$
−0.872452 + 0.488700i $$0.837471\pi$$
$$752$$ −2.24033e6 −0.144467
$$753$$ −1.21076e6 −0.0778165
$$754$$ 1.08869e7 0.697391
$$755$$ 0 0
$$756$$ 1.45407e6 0.0925296
$$757$$ 2.81995e7 1.78855 0.894275 0.447518i $$-0.147692\pi$$
0.894275 + 0.447518i $$0.147692\pi$$
$$758$$ 1.50982e7 0.954450
$$759$$ −1.70951e6 −0.107713
$$760$$ 0 0
$$761$$ 5.29643e6 0.331529 0.165764 0.986165i $$-0.446991\pi$$
0.165764 + 0.986165i $$0.446991\pi$$
$$762$$ −1.07779e6 −0.0672428
$$763$$ −1.27143e6 −0.0790643
$$764$$ −2.94326e6 −0.182430
$$765$$ 0 0
$$766$$ −7.07344e6 −0.435571
$$767$$ 1.22302e7 0.750665
$$768$$ 6.87089e6 0.420349
$$769$$ 6.09571e6 0.371714 0.185857 0.982577i $$-0.440494\pi$$
0.185857 + 0.982577i $$0.440494\pi$$
$$770$$ 0 0
$$771$$ 1.86059e7 1.12723
$$772$$ −2.17831e7 −1.31546
$$773$$ −1.02486e7 −0.616901 −0.308451 0.951240i $$-0.599810\pi$$
−0.308451 + 0.951240i $$0.599810\pi$$
$$774$$ 4.02615e6 0.241567
$$775$$ 0 0
$$776$$ 1.70001e7 1.01344
$$777$$ −9.67201e6 −0.574730
$$778$$ 5.61434e6 0.332545
$$779$$ −88406.4 −0.00521963
$$780$$ 0 0
$$781$$ 6.84951e6 0.401820
$$782$$ −9.62624e6 −0.562911
$$783$$ 3.75139e6 0.218669
$$784$$ −2.65693e6 −0.154380
$$785$$ 0 0
$$786$$ 6.26917e6 0.361954
$$787$$ 1.62333e7 0.934267 0.467133 0.884187i $$-0.345287\pi$$
0.467133 + 0.884187i $$0.345287\pi$$
$$788$$ 2.42217e7 1.38959
$$789$$ −5.95565e6 −0.340594
$$790$$ 0 0
$$791$$ −1.68595e7 −0.958084
$$792$$ 1.58827e6 0.0899729
$$793$$ 2.49075e7 1.40653
$$794$$ −1.19669e7 −0.673642
$$795$$ 0 0
$$796$$ 3.63717e6 0.203461
$$797$$ −1.29153e7 −0.720211 −0.360105 0.932912i $$-0.617259\pi$$
−0.360105 + 0.932912i $$0.617259\pi$$
$$798$$ 14311.9 0.000795590 0
$$799$$ −1.69539e7 −0.939511
$$800$$ 0 0
$$801$$ 4.78136e6 0.263312
$$802$$ 1.74585e7 0.958452
$$803$$ −3.36113e6 −0.183948
$$804$$ 7.87794e6 0.429806
$$805$$ 0 0
$$806$$ 2.18307e6 0.118367
$$807$$ −6.32707e6 −0.341994
$$808$$ −2.19447e7 −1.18250
$$809$$ 1.40361e6 0.0754007 0.0377004 0.999289i $$-0.487997\pi$$
0.0377004 + 0.999289i $$0.487997\pi$$
$$810$$ 0 0
$$811$$ −1.27909e7 −0.682887 −0.341444 0.939902i $$-0.610916\pi$$
−0.341444 + 0.939902i $$0.610916\pi$$
$$812$$ 1.02641e7 0.546301
$$813$$ 1.20462e7 0.639183
$$814$$ −4.46928e6 −0.236416
$$815$$ 0 0
$$816$$ −5.23853e6 −0.275413
$$817$$ −108915. −0.00570862
$$818$$ −5.62928e6 −0.294150
$$819$$ −4.98594e6 −0.259739
$$820$$ 0 0
$$821$$ −5.20603e6 −0.269556 −0.134778 0.990876i $$-0.543032\pi$$
−0.134778 + 0.990876i $$0.543032\pi$$
$$822$$ 3.93217e6 0.202980
$$823$$ −3.77702e7 −1.94379 −0.971896 0.235409i $$-0.924357\pi$$
−0.971896 + 0.235409i $$0.924357\pi$$
$$824$$ 2.71446e7 1.39272
$$825$$ 0 0
$$826$$ −4.19535e6 −0.213953
$$827$$ 2.54890e7 1.29595 0.647976 0.761661i $$-0.275616\pi$$
0.647976 + 0.761661i $$0.275616\pi$$
$$828$$ 2.98342e6 0.151230
$$829$$ 1.71410e7 0.866262 0.433131 0.901331i $$-0.357409\pi$$
0.433131 + 0.901331i $$0.357409\pi$$
$$830$$ 0 0
$$831$$ −298540. −0.0149969
$$832$$ −6.25922e6 −0.313482
$$833$$ −2.01065e7 −1.00398
$$834$$ −4.40988e6 −0.219539
$$835$$ 0 0
$$836$$ −18176.1 −0.000899469 0
$$837$$ 752238. 0.0371143
$$838$$ −4.32009e6 −0.212512
$$839$$ −1.01094e7 −0.495818 −0.247909 0.968783i $$-0.579743\pi$$
−0.247909 + 0.968783i $$0.579743\pi$$
$$840$$ 0 0
$$841$$ 5.96954e6 0.291039
$$842$$ 1.41019e7 0.685483
$$843$$ 2.89003e6 0.140066
$$844$$ 8.54672e6 0.412994
$$845$$ 0 0
$$846$$ −1.91180e6 −0.0918368
$$847$$ 1.24464e6 0.0596121
$$848$$ 6.13212e6 0.292834
$$849$$ −1.71532e7 −0.816726
$$850$$ 0 0
$$851$$ −1.98448e7 −0.939339
$$852$$ −1.19537e7 −0.564161
$$853$$ −3.29141e7 −1.54885 −0.774425 0.632666i $$-0.781961\pi$$
−0.774425 + 0.632666i $$0.781961\pi$$
$$854$$ −8.54406e6 −0.400885
$$855$$ 0 0
$$856$$ 9.39734e6 0.438349
$$857$$ −6.94241e6 −0.322893 −0.161446 0.986881i $$-0.551616\pi$$
−0.161446 + 0.986881i $$0.551616\pi$$
$$858$$ −2.30392e6 −0.106844
$$859$$ −1.84275e7 −0.852085 −0.426042 0.904703i $$-0.640093\pi$$
−0.426042 + 0.904703i $$0.640093\pi$$
$$860$$ 0 0
$$861$$ −1.05649e7 −0.485690
$$862$$ 8.45068e6 0.387368
$$863$$ 2.60648e7 1.19131 0.595657 0.803239i $$-0.296892\pi$$
0.595657 + 0.803239i $$0.296892\pi$$
$$864$$ −4.37107e6 −0.199207
$$865$$ 0 0
$$866$$ −6.91589e6 −0.313367
$$867$$ −2.68642e7 −1.21374
$$868$$ 2.05819e6 0.0927228
$$869$$ 1.54395e6 0.0693559
$$870$$ 0 0
$$871$$ −2.70131e7 −1.20650
$$872$$ 2.42367e6 0.107940
$$873$$ −8.49731e6 −0.377352
$$874$$ 29364.8 0.00130031
$$875$$ 0 0
$$876$$ 5.86580e6 0.258266
$$877$$ −1.86288e6 −0.0817875 −0.0408937 0.999164i $$-0.513021\pi$$
−0.0408937 + 0.999164i $$0.513021\pi$$
$$878$$ 9.40927e6 0.411926
$$879$$ 2.45661e6 0.107242
$$880$$ 0 0
$$881$$ 1.01838e7 0.442051 0.221025 0.975268i $$-0.429060\pi$$
0.221025 + 0.975268i $$0.429060\pi$$
$$882$$ −2.26731e6 −0.0981384
$$883$$ −6.25954e6 −0.270172 −0.135086 0.990834i $$-0.543131\pi$$
−0.135086 + 0.990834i $$0.543131\pi$$
$$884$$ 3.56563e7 1.53464
$$885$$ 0 0
$$886$$ 1.11126e7 0.475589
$$887$$ −1.80319e7 −0.769540 −0.384770 0.923012i $$-0.625719\pi$$
−0.384770 + 0.923012i $$0.625719\pi$$
$$888$$ 1.84374e7 0.784632
$$889$$ 3.48428e6 0.147863
$$890$$ 0 0
$$891$$ −793881. −0.0335013
$$892$$ −1.75518e6 −0.0738600
$$893$$ 51717.6 0.00217025
$$894$$ 7.17091e6 0.300075
$$895$$ 0 0
$$896$$ −1.41640e7 −0.589407
$$897$$ −1.02300e7 −0.424518
$$898$$ −1.91466e7 −0.792321
$$899$$ 5.30998e6 0.219126
$$900$$ 0 0
$$901$$ 4.64052e7 1.90439
$$902$$ −4.88189e6 −0.199789
$$903$$ −1.30158e7 −0.531191
$$904$$ 3.21386e7 1.30799
$$905$$ 0 0
$$906$$ 1.17891e7 0.477156
$$907$$ −2.69733e7 −1.08872 −0.544360 0.838852i $$-0.683228\pi$$
−0.544360 + 0.838852i $$0.683228\pi$$
$$908$$ 2.87838e7 1.15860
$$909$$ 1.09688e7 0.440301
$$910$$ 0 0
$$911$$ −4.31660e7 −1.72324 −0.861621 0.507553i $$-0.830550\pi$$
−0.861621 + 0.507553i $$0.830550\pi$$
$$912$$ 15980.1 0.000636198 0
$$913$$ −8.38029e6 −0.332723
$$914$$ 2.16643e7 0.857784
$$915$$ 0 0
$$916$$ 1.03438e7 0.407323
$$917$$ −2.02670e7 −0.795915
$$918$$ −4.47033e6 −0.175079
$$919$$ −3.69944e7 −1.44493 −0.722465 0.691407i $$-0.756991\pi$$
−0.722465 + 0.691407i $$0.756991\pi$$
$$920$$ 0 0
$$921$$ −7.59425e6 −0.295010
$$922$$ 1.19655e6 0.0463557
$$923$$ 4.09887e7 1.58365
$$924$$ −2.17213e6 −0.0836962
$$925$$ 0 0
$$926$$ −2.06627e7 −0.791882
$$927$$ −1.35679e7 −0.518578
$$928$$ −3.08550e7 −1.17613
$$929$$ 1.63853e6 0.0622895 0.0311447 0.999515i $$-0.490085\pi$$
0.0311447 + 0.999515i $$0.490085\pi$$
$$930$$ 0 0
$$931$$ 61334.7 0.00231917
$$932$$ −1.20729e7 −0.455273
$$933$$ −1.86218e7 −0.700353
$$934$$ 1.85658e7 0.696380
$$935$$ 0 0
$$936$$ 9.50449e6 0.354600
$$937$$ 2.02831e7 0.754719 0.377360 0.926067i $$-0.376832\pi$$
0.377360 + 0.926067i $$0.376832\pi$$
$$938$$ 9.26633e6 0.343875
$$939$$ 5.42802e6 0.200899
$$940$$ 0 0
$$941$$ −1.83342e7 −0.674975 −0.337487 0.941330i $$-0.609577\pi$$
−0.337487 + 0.941330i $$0.609577\pi$$
$$942$$ −9.64107e6 −0.353996
$$943$$ −2.16769e7 −0.793812
$$944$$ −4.68436e6 −0.171088
$$945$$ 0 0
$$946$$ −6.01438e6 −0.218506
$$947$$ 6.09095e6 0.220704 0.110352 0.993893i $$-0.464802\pi$$
0.110352 + 0.993893i $$0.464802\pi$$
$$948$$ −2.69448e6 −0.0973766
$$949$$ −2.01136e7 −0.724976
$$950$$ 0 0
$$951$$ −1.50601e7 −0.539977
$$952$$ −2.89127e7 −1.03394
$$953$$ 5.31211e7 1.89467 0.947337 0.320238i $$-0.103763\pi$$
0.947337 + 0.320238i $$0.103763\pi$$
$$954$$ 5.23288e6 0.186153
$$955$$ 0 0
$$956$$ 2.30951e7 0.817287
$$957$$ −5.60393e6 −0.197794
$$958$$ 1.86046e7 0.654947
$$959$$ −1.27120e7 −0.446340
$$960$$ 0 0
$$961$$ −2.75644e7 −0.962808
$$962$$ −2.67449e7 −0.931759
$$963$$ −4.69716e6 −0.163219
$$964$$ −6.68214e6 −0.231592
$$965$$ 0 0
$$966$$ 3.50922e6 0.120995
$$967$$ 2.17670e6 0.0748570 0.0374285 0.999299i $$-0.488083\pi$$
0.0374285 + 0.999299i $$0.488083\pi$$
$$968$$ −2.37260e6 −0.0813836
$$969$$ 120930. 0.00413739
$$970$$ 0 0
$$971$$ 4.83305e7 1.64503 0.822514 0.568745i $$-0.192571\pi$$
0.822514 + 0.568745i $$0.192571\pi$$
$$972$$ 1.38547e6 0.0470362
$$973$$ 1.42563e7 0.482753
$$974$$ 8.59826e6 0.290411
$$975$$ 0 0
$$976$$ −9.53996e6 −0.320569
$$977$$ 3.84816e7 1.28978 0.644892 0.764274i $$-0.276902\pi$$
0.644892 + 0.764274i $$0.276902\pi$$
$$978$$ 1.31956e7 0.441145
$$979$$ −7.14253e6 −0.238175
$$980$$ 0 0
$$981$$ −1.21145e6 −0.0401913
$$982$$ −1.46566e6 −0.0485013
$$983$$ 1.53385e7 0.506288 0.253144 0.967429i $$-0.418535\pi$$
0.253144 + 0.967429i $$0.418535\pi$$
$$984$$ 2.01395e7 0.663073
$$985$$ 0 0
$$986$$ −3.15556e7 −1.03368
$$987$$ 6.18048e6 0.201943
$$988$$ −108769. −0.00354498
$$989$$ −2.67054e7 −0.868178
$$990$$ 0 0
$$991$$ −4.29143e7 −1.38809 −0.694046 0.719931i $$-0.744174\pi$$
−0.694046 + 0.719931i $$0.744174\pi$$
$$992$$ −6.18712e6 −0.199622
$$993$$ 1.98379e7 0.638444
$$994$$ −1.40604e7 −0.451368
$$995$$ 0 0
$$996$$ 1.46252e7 0.467147
$$997$$ −6.10887e7 −1.94636 −0.973180 0.230044i $$-0.926113\pi$$
−0.973180 + 0.230044i $$0.926113\pi$$
$$998$$ −2.68742e7 −0.854102
$$999$$ −9.21572e6 −0.292156
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 825.6.a.j.1.2 3
5.4 even 2 165.6.a.a.1.2 3
15.14 odd 2 495.6.a.e.1.2 3

By twisted newform
Twist Min Dim Char Parity Ord Type
165.6.a.a.1.2 3 5.4 even 2
495.6.a.e.1.2 3 15.14 odd 2
825.6.a.j.1.2 3 1.1 even 1 trivial