# Properties

 Label 825.6.a.y.1.3 Level $825$ Weight $6$ Character 825.1 Self dual yes Analytic conductor $132.317$ Analytic rank $0$ Dimension $13$ CM no Inner twists $1$

# Learn more

## 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: $$0$$ Dimension: $$13$$ Coefficient field: $$\mathbb{Q}[x]/(x^{13} - \cdots)$$ Defining polynomial: $$x^{13} - 306 x^{11} - 206 x^{10} + 34574 x^{9} + 39928 x^{8} - 1788312 x^{7} - 2591628 x^{6} + 42852537 x^{5} + 63733360 x^{4} - 448113518 x^{3} + \cdots + 522579400$$ x^13 - 306*x^11 - 206*x^10 + 34574*x^9 + 39928*x^8 - 1788312*x^7 - 2591628*x^6 + 42852537*x^5 + 63733360*x^4 - 448113518*x^3 - 549984598*x^2 + 1518551280*x + 522579400 Coefficient ring: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$2^{9}\cdot 3^{2}\cdot 5^{7}$$ Twist minimal: no (minimal twist has level 165) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.3 Root $$8.39855$$ of defining polynomial Character $$\chi$$ $$=$$ 825.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-7.39855 q^{2} +9.00000 q^{3} +22.7385 q^{4} -66.5869 q^{6} +150.852 q^{7} +68.5217 q^{8} +81.0000 q^{9} +O(q^{10})$$ $$q-7.39855 q^{2} +9.00000 q^{3} +22.7385 q^{4} -66.5869 q^{6} +150.852 q^{7} +68.5217 q^{8} +81.0000 q^{9} +121.000 q^{11} +204.646 q^{12} -868.426 q^{13} -1116.08 q^{14} -1234.59 q^{16} -2317.87 q^{17} -599.282 q^{18} -2655.24 q^{19} +1357.67 q^{21} -895.224 q^{22} -2537.58 q^{23} +616.695 q^{24} +6425.09 q^{26} +729.000 q^{27} +3430.14 q^{28} -819.917 q^{29} +7303.36 q^{31} +6941.50 q^{32} +1089.00 q^{33} +17148.9 q^{34} +1841.82 q^{36} +2993.84 q^{37} +19644.9 q^{38} -7815.84 q^{39} -4567.72 q^{41} -10044.8 q^{42} -1022.10 q^{43} +2751.36 q^{44} +18774.4 q^{46} +24499.3 q^{47} -11111.3 q^{48} +5949.25 q^{49} -20860.8 q^{51} -19746.7 q^{52} +13318.9 q^{53} -5393.54 q^{54} +10336.6 q^{56} -23897.2 q^{57} +6066.20 q^{58} -29760.6 q^{59} -17257.1 q^{61} -54034.2 q^{62} +12219.0 q^{63} -11850.0 q^{64} -8057.02 q^{66} +18645.0 q^{67} -52704.9 q^{68} -22838.2 q^{69} +47722.0 q^{71} +5550.26 q^{72} +19156.3 q^{73} -22150.1 q^{74} -60376.2 q^{76} +18253.1 q^{77} +57825.8 q^{78} -3713.15 q^{79} +6561.00 q^{81} +33794.5 q^{82} +51919.0 q^{83} +30871.3 q^{84} +7562.07 q^{86} -7379.26 q^{87} +8291.13 q^{88} -52966.5 q^{89} -131004. q^{91} -57700.8 q^{92} +65730.2 q^{93} -181259. q^{94} +62473.5 q^{96} +114526. q^{97} -44015.8 q^{98} +9801.00 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$13 q + 13 q^{2} + 117 q^{3} + 209 q^{4} + 117 q^{6} + 304 q^{7} + 399 q^{8} + 1053 q^{9}+O(q^{10})$$ 13 * q + 13 * q^2 + 117 * q^3 + 209 * q^4 + 117 * q^6 + 304 * q^7 + 399 * q^8 + 1053 * q^9 $$13 q + 13 q^{2} + 117 q^{3} + 209 q^{4} + 117 q^{6} + 304 q^{7} + 399 q^{8} + 1053 q^{9} + 1573 q^{11} + 1881 q^{12} + 986 q^{13} - 610 q^{14} + 3501 q^{16} + 1476 q^{17} + 1053 q^{18} + 270 q^{19} + 2736 q^{21} + 1573 q^{22} + 9084 q^{23} + 3591 q^{24} + 2652 q^{26} + 9477 q^{27} + 10920 q^{28} + 11952 q^{29} + 19096 q^{31} + 11661 q^{32} + 14157 q^{33} - 1302 q^{34} + 16929 q^{36} + 39964 q^{37} + 1574 q^{38} + 8874 q^{39} + 35184 q^{41} - 5490 q^{42} - 96 q^{43} + 25289 q^{44} - 4120 q^{46} + 34984 q^{47} + 31509 q^{48} + 14557 q^{49} + 13284 q^{51} + 39002 q^{52} + 22984 q^{53} + 9477 q^{54} + 59802 q^{56} + 2430 q^{57} + 18896 q^{58} - 9192 q^{59} + 5438 q^{61} + 272 q^{62} + 24624 q^{63} + 106557 q^{64} + 14157 q^{66} + 71508 q^{67} + 127948 q^{68} + 81756 q^{69} + 101700 q^{71} + 32319 q^{72} + 77390 q^{73} + 13676 q^{74} + 139966 q^{76} + 36784 q^{77} + 23868 q^{78} + 93954 q^{79} + 85293 q^{81} + 53284 q^{82} + 185918 q^{83} + 98280 q^{84} + 370930 q^{86} + 107568 q^{87} + 48279 q^{88} - 18418 q^{89} + 174536 q^{91} + 274264 q^{92} + 171864 q^{93} + 64520 q^{94} + 104949 q^{96} + 94312 q^{97} + 145677 q^{98} + 127413 q^{99}+O(q^{100})$$ 13 * q + 13 * q^2 + 117 * q^3 + 209 * q^4 + 117 * q^6 + 304 * q^7 + 399 * q^8 + 1053 * q^9 + 1573 * q^11 + 1881 * q^12 + 986 * q^13 - 610 * q^14 + 3501 * q^16 + 1476 * q^17 + 1053 * q^18 + 270 * q^19 + 2736 * q^21 + 1573 * q^22 + 9084 * q^23 + 3591 * q^24 + 2652 * q^26 + 9477 * q^27 + 10920 * q^28 + 11952 * q^29 + 19096 * q^31 + 11661 * q^32 + 14157 * q^33 - 1302 * q^34 + 16929 * q^36 + 39964 * q^37 + 1574 * q^38 + 8874 * q^39 + 35184 * q^41 - 5490 * q^42 - 96 * q^43 + 25289 * q^44 - 4120 * q^46 + 34984 * q^47 + 31509 * q^48 + 14557 * q^49 + 13284 * q^51 + 39002 * q^52 + 22984 * q^53 + 9477 * q^54 + 59802 * q^56 + 2430 * q^57 + 18896 * q^58 - 9192 * q^59 + 5438 * q^61 + 272 * q^62 + 24624 * q^63 + 106557 * q^64 + 14157 * q^66 + 71508 * q^67 + 127948 * q^68 + 81756 * q^69 + 101700 * q^71 + 32319 * q^72 + 77390 * q^73 + 13676 * q^74 + 139966 * q^76 + 36784 * q^77 + 23868 * q^78 + 93954 * q^79 + 85293 * q^81 + 53284 * q^82 + 185918 * q^83 + 98280 * q^84 + 370930 * q^86 + 107568 * q^87 + 48279 * q^88 - 18418 * q^89 + 174536 * q^91 + 274264 * q^92 + 171864 * q^93 + 64520 * q^94 + 104949 * q^96 + 94312 * q^97 + 145677 * q^98 + 127413 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −7.39855 −1.30789 −0.653945 0.756542i $$-0.726887\pi$$
−0.653945 + 0.756542i $$0.726887\pi$$
$$3$$ 9.00000 0.577350
$$4$$ 22.7385 0.710578
$$5$$ 0 0
$$6$$ −66.5869 −0.755111
$$7$$ 150.852 1.16360 0.581802 0.813330i $$-0.302348\pi$$
0.581802 + 0.813330i $$0.302348\pi$$
$$8$$ 68.5217 0.378533
$$9$$ 81.0000 0.333333
$$10$$ 0 0
$$11$$ 121.000 0.301511
$$12$$ 204.646 0.410252
$$13$$ −868.426 −1.42520 −0.712598 0.701573i $$-0.752482\pi$$
−0.712598 + 0.701573i $$0.752482\pi$$
$$14$$ −1116.08 −1.52187
$$15$$ 0 0
$$16$$ −1234.59 −1.20566
$$17$$ −2317.87 −1.94521 −0.972606 0.232458i $$-0.925323\pi$$
−0.972606 + 0.232458i $$0.925323\pi$$
$$18$$ −599.282 −0.435964
$$19$$ −2655.24 −1.68741 −0.843705 0.536808i $$-0.819630\pi$$
−0.843705 + 0.536808i $$0.819630\pi$$
$$20$$ 0 0
$$21$$ 1357.67 0.671807
$$22$$ −895.224 −0.394344
$$23$$ −2537.58 −1.00023 −0.500116 0.865959i $$-0.666709\pi$$
−0.500116 + 0.865959i $$0.666709\pi$$
$$24$$ 616.695 0.218546
$$25$$ 0 0
$$26$$ 6425.09 1.86400
$$27$$ 729.000 0.192450
$$28$$ 3430.14 0.826831
$$29$$ −819.917 −0.181040 −0.0905201 0.995895i $$-0.528853\pi$$
−0.0905201 + 0.995895i $$0.528853\pi$$
$$30$$ 0 0
$$31$$ 7303.36 1.36495 0.682477 0.730907i $$-0.260903\pi$$
0.682477 + 0.730907i $$0.260903\pi$$
$$32$$ 6941.50 1.19833
$$33$$ 1089.00 0.174078
$$34$$ 17148.9 2.54413
$$35$$ 0 0
$$36$$ 1841.82 0.236859
$$37$$ 2993.84 0.359521 0.179761 0.983710i $$-0.442468\pi$$
0.179761 + 0.983710i $$0.442468\pi$$
$$38$$ 19644.9 2.20695
$$39$$ −7815.84 −0.822837
$$40$$ 0 0
$$41$$ −4567.72 −0.424365 −0.212182 0.977230i $$-0.568057\pi$$
−0.212182 + 0.977230i $$0.568057\pi$$
$$42$$ −10044.8 −0.878650
$$43$$ −1022.10 −0.0842992 −0.0421496 0.999111i $$-0.513421\pi$$
−0.0421496 + 0.999111i $$0.513421\pi$$
$$44$$ 2751.36 0.214247
$$45$$ 0 0
$$46$$ 18774.4 1.30819
$$47$$ 24499.3 1.61774 0.808871 0.587987i $$-0.200079\pi$$
0.808871 + 0.587987i $$0.200079\pi$$
$$48$$ −11111.3 −0.696086
$$49$$ 5949.25 0.353974
$$50$$ 0 0
$$51$$ −20860.8 −1.12307
$$52$$ −19746.7 −1.01271
$$53$$ 13318.9 0.651297 0.325649 0.945491i $$-0.394417\pi$$
0.325649 + 0.945491i $$0.394417\pi$$
$$54$$ −5393.54 −0.251704
$$55$$ 0 0
$$56$$ 10336.6 0.440462
$$57$$ −23897.2 −0.974226
$$58$$ 6066.20 0.236781
$$59$$ −29760.6 −1.11304 −0.556522 0.830833i $$-0.687864\pi$$
−0.556522 + 0.830833i $$0.687864\pi$$
$$60$$ 0 0
$$61$$ −17257.1 −0.593805 −0.296903 0.954908i $$-0.595954\pi$$
−0.296903 + 0.954908i $$0.595954\pi$$
$$62$$ −54034.2 −1.78521
$$63$$ 12219.0 0.387868
$$64$$ −11850.0 −0.361634
$$65$$ 0 0
$$66$$ −8057.02 −0.227675
$$67$$ 18645.0 0.507429 0.253715 0.967279i $$-0.418348\pi$$
0.253715 + 0.967279i $$0.418348\pi$$
$$68$$ −52704.9 −1.38223
$$69$$ −22838.2 −0.577484
$$70$$ 0 0
$$71$$ 47722.0 1.12350 0.561750 0.827307i $$-0.310128\pi$$
0.561750 + 0.827307i $$0.310128\pi$$
$$72$$ 5550.26 0.126178
$$73$$ 19156.3 0.420730 0.210365 0.977623i $$-0.432535\pi$$
0.210365 + 0.977623i $$0.432535\pi$$
$$74$$ −22150.1 −0.470214
$$75$$ 0 0
$$76$$ −60376.2 −1.19904
$$77$$ 18253.1 0.350840
$$78$$ 57825.8 1.07618
$$79$$ −3713.15 −0.0669383 −0.0334692 0.999440i $$-0.510656\pi$$
−0.0334692 + 0.999440i $$0.510656\pi$$
$$80$$ 0 0
$$81$$ 6561.00 0.111111
$$82$$ 33794.5 0.555023
$$83$$ 51919.0 0.827239 0.413619 0.910450i $$-0.364264\pi$$
0.413619 + 0.910450i $$0.364264\pi$$
$$84$$ 30871.3 0.477371
$$85$$ 0 0
$$86$$ 7562.07 0.110254
$$87$$ −7379.26 −0.104524
$$88$$ 8291.13 0.114132
$$89$$ −52966.5 −0.708803 −0.354402 0.935093i $$-0.615315\pi$$
−0.354402 + 0.935093i $$0.615315\pi$$
$$90$$ 0 0
$$91$$ −131004. −1.65836
$$92$$ −57700.8 −0.710742
$$93$$ 65730.2 0.788057
$$94$$ −181259. −2.11583
$$95$$ 0 0
$$96$$ 62473.5 0.691859
$$97$$ 114526. 1.23588 0.617940 0.786225i $$-0.287968\pi$$
0.617940 + 0.786225i $$0.287968\pi$$
$$98$$ −44015.8 −0.462960
$$99$$ 9801.00 0.100504
$$100$$ 0 0
$$101$$ 154646. 1.50846 0.754232 0.656608i $$-0.228009\pi$$
0.754232 + 0.656608i $$0.228009\pi$$
$$102$$ 154340. 1.46885
$$103$$ −117955. −1.09552 −0.547762 0.836634i $$-0.684520\pi$$
−0.547762 + 0.836634i $$0.684520\pi$$
$$104$$ −59506.1 −0.539483
$$105$$ 0 0
$$106$$ −98540.6 −0.851826
$$107$$ 233086. 1.96815 0.984073 0.177763i $$-0.0568862\pi$$
0.984073 + 0.177763i $$0.0568862\pi$$
$$108$$ 16576.4 0.136751
$$109$$ −18774.3 −0.151355 −0.0756775 0.997132i $$-0.524112\pi$$
−0.0756775 + 0.997132i $$0.524112\pi$$
$$110$$ 0 0
$$111$$ 26944.6 0.207570
$$112$$ −186240. −1.40291
$$113$$ 165056. 1.21600 0.608001 0.793936i $$-0.291971\pi$$
0.608001 + 0.793936i $$0.291971\pi$$
$$114$$ 176804. 1.27418
$$115$$ 0 0
$$116$$ −18643.7 −0.128643
$$117$$ −70342.5 −0.475065
$$118$$ 220186. 1.45574
$$119$$ −349655. −2.26346
$$120$$ 0 0
$$121$$ 14641.0 0.0909091
$$122$$ 127678. 0.776632
$$123$$ −41109.4 −0.245007
$$124$$ 166067. 0.969907
$$125$$ 0 0
$$126$$ −90402.8 −0.507289
$$127$$ 203866. 1.12159 0.560797 0.827953i $$-0.310495\pi$$
0.560797 + 0.827953i $$0.310495\pi$$
$$128$$ −134455. −0.725357
$$129$$ −9198.92 −0.0486701
$$130$$ 0 0
$$131$$ 82311.6 0.419067 0.209533 0.977802i $$-0.432806\pi$$
0.209533 + 0.977802i $$0.432806\pi$$
$$132$$ 24762.2 0.123696
$$133$$ −400548. −1.96348
$$134$$ −137946. −0.663662
$$135$$ 0 0
$$136$$ −158825. −0.736326
$$137$$ 202042. 0.919687 0.459844 0.888000i $$-0.347905\pi$$
0.459844 + 0.888000i $$0.347905\pi$$
$$138$$ 168970. 0.755286
$$139$$ −200312. −0.879367 −0.439683 0.898153i $$-0.644909\pi$$
−0.439683 + 0.898153i $$0.644909\pi$$
$$140$$ 0 0
$$141$$ 220494. 0.934003
$$142$$ −353074. −1.46942
$$143$$ −105080. −0.429713
$$144$$ −100002. −0.401886
$$145$$ 0 0
$$146$$ −141728. −0.550269
$$147$$ 53543.2 0.204367
$$148$$ 68075.4 0.255468
$$149$$ 80425.4 0.296775 0.148388 0.988929i $$-0.452592\pi$$
0.148388 + 0.988929i $$0.452592\pi$$
$$150$$ 0 0
$$151$$ 395216. 1.41056 0.705280 0.708928i $$-0.250821\pi$$
0.705280 + 0.708928i $$0.250821\pi$$
$$152$$ −181942. −0.638739
$$153$$ −187748. −0.648404
$$154$$ −135046. −0.458860
$$155$$ 0 0
$$156$$ −177720. −0.584690
$$157$$ −510056. −1.65146 −0.825732 0.564063i $$-0.809237\pi$$
−0.825732 + 0.564063i $$0.809237\pi$$
$$158$$ 27471.9 0.0875480
$$159$$ 119870. 0.376027
$$160$$ 0 0
$$161$$ −382799. −1.16387
$$162$$ −48541.9 −0.145321
$$163$$ 307324. 0.905998 0.452999 0.891511i $$-0.350354\pi$$
0.452999 + 0.891511i $$0.350354\pi$$
$$164$$ −103863. −0.301544
$$165$$ 0 0
$$166$$ −384125. −1.08194
$$167$$ 214345. 0.594733 0.297367 0.954763i $$-0.403892\pi$$
0.297367 + 0.954763i $$0.403892\pi$$
$$168$$ 93029.6 0.254301
$$169$$ 382871. 1.03118
$$170$$ 0 0
$$171$$ −215075. −0.562470
$$172$$ −23241.1 −0.0599011
$$173$$ 366833. 0.931865 0.465933 0.884820i $$-0.345719\pi$$
0.465933 + 0.884820i $$0.345719\pi$$
$$174$$ 54595.8 0.136705
$$175$$ 0 0
$$176$$ −149386. −0.363519
$$177$$ −267846. −0.642616
$$178$$ 391875. 0.927037
$$179$$ 678162. 1.58198 0.790990 0.611829i $$-0.209566\pi$$
0.790990 + 0.611829i $$0.209566\pi$$
$$180$$ 0 0
$$181$$ −270856. −0.614529 −0.307265 0.951624i $$-0.599414\pi$$
−0.307265 + 0.951624i $$0.599414\pi$$
$$182$$ 969236. 2.16896
$$183$$ −155314. −0.342834
$$184$$ −173879. −0.378620
$$185$$ 0 0
$$186$$ −486308. −1.03069
$$187$$ −280462. −0.586504
$$188$$ 557077. 1.14953
$$189$$ 109971. 0.223936
$$190$$ 0 0
$$191$$ 81645.4 0.161938 0.0809689 0.996717i $$-0.474199\pi$$
0.0809689 + 0.996717i $$0.474199\pi$$
$$192$$ −106650. −0.208789
$$193$$ −170728. −0.329922 −0.164961 0.986300i $$-0.552750\pi$$
−0.164961 + 0.986300i $$0.552750\pi$$
$$194$$ −847329. −1.61640
$$195$$ 0 0
$$196$$ 135277. 0.251526
$$197$$ −173263. −0.318082 −0.159041 0.987272i $$-0.550840\pi$$
−0.159041 + 0.987272i $$0.550840\pi$$
$$198$$ −72513.2 −0.131448
$$199$$ 42534.0 0.0761384 0.0380692 0.999275i $$-0.487879\pi$$
0.0380692 + 0.999275i $$0.487879\pi$$
$$200$$ 0 0
$$201$$ 167805. 0.292964
$$202$$ −1.14416e6 −1.97291
$$203$$ −123686. −0.210659
$$204$$ −474344. −0.798028
$$205$$ 0 0
$$206$$ 872693. 1.43283
$$207$$ −205544. −0.333410
$$208$$ 1.07215e6 1.71830
$$209$$ −321284. −0.508773
$$210$$ 0 0
$$211$$ −625694. −0.967511 −0.483755 0.875203i $$-0.660728\pi$$
−0.483755 + 0.875203i $$0.660728\pi$$
$$212$$ 302852. 0.462797
$$213$$ 429498. 0.648653
$$214$$ −1.72450e6 −2.57412
$$215$$ 0 0
$$216$$ 49952.3 0.0728486
$$217$$ 1.10172e6 1.58827
$$218$$ 138902. 0.197956
$$219$$ 172406. 0.242909
$$220$$ 0 0
$$221$$ 2.01290e6 2.77231
$$222$$ −199351. −0.271478
$$223$$ −42469.9 −0.0571899 −0.0285950 0.999591i $$-0.509103\pi$$
−0.0285950 + 0.999591i $$0.509103\pi$$
$$224$$ 1.04714e6 1.39439
$$225$$ 0 0
$$226$$ −1.22117e6 −1.59040
$$227$$ 21815.5 0.0280996 0.0140498 0.999901i $$-0.495528\pi$$
0.0140498 + 0.999901i $$0.495528\pi$$
$$228$$ −543386. −0.692263
$$229$$ 58861.1 0.0741720 0.0370860 0.999312i $$-0.488192\pi$$
0.0370860 + 0.999312i $$0.488192\pi$$
$$230$$ 0 0
$$231$$ 164278. 0.202557
$$232$$ −56182.1 −0.0685296
$$233$$ 418005. 0.504419 0.252210 0.967673i $$-0.418843\pi$$
0.252210 + 0.967673i $$0.418843\pi$$
$$234$$ 520432. 0.621333
$$235$$ 0 0
$$236$$ −676712. −0.790904
$$237$$ −33418.3 −0.0386468
$$238$$ 2.58694e6 2.96035
$$239$$ −1.29252e6 −1.46367 −0.731834 0.681483i $$-0.761335\pi$$
−0.731834 + 0.681483i $$0.761335\pi$$
$$240$$ 0 0
$$241$$ 496148. 0.550261 0.275131 0.961407i $$-0.411279\pi$$
0.275131 + 0.961407i $$0.411279\pi$$
$$242$$ −108322. −0.118899
$$243$$ 59049.0 0.0641500
$$244$$ −392401. −0.421945
$$245$$ 0 0
$$246$$ 304150. 0.320443
$$247$$ 2.30588e6 2.40489
$$248$$ 500439. 0.516680
$$249$$ 467271. 0.477606
$$250$$ 0 0
$$251$$ −1.58588e6 −1.58886 −0.794431 0.607354i $$-0.792231\pi$$
−0.794431 + 0.607354i $$0.792231\pi$$
$$252$$ 277841. 0.275610
$$253$$ −307047. −0.301581
$$254$$ −1.50831e6 −1.46692
$$255$$ 0 0
$$256$$ 1.37397e6 1.31032
$$257$$ 1.41881e6 1.33995 0.669977 0.742381i $$-0.266304\pi$$
0.669977 + 0.742381i $$0.266304\pi$$
$$258$$ 68058.6 0.0636552
$$259$$ 451626. 0.418340
$$260$$ 0 0
$$261$$ −66413.3 −0.0603467
$$262$$ −608987. −0.548093
$$263$$ 1.62046e6 1.44461 0.722304 0.691575i $$-0.243083\pi$$
0.722304 + 0.691575i $$0.243083\pi$$
$$264$$ 74620.1 0.0658941
$$265$$ 0 0
$$266$$ 2.96347e6 2.56801
$$267$$ −476698. −0.409228
$$268$$ 423959. 0.360568
$$269$$ 1.25821e6 1.06017 0.530083 0.847946i $$-0.322161\pi$$
0.530083 + 0.847946i $$0.322161\pi$$
$$270$$ 0 0
$$271$$ −2.33387e6 −1.93043 −0.965214 0.261462i $$-0.915796\pi$$
−0.965214 + 0.261462i $$0.915796\pi$$
$$272$$ 2.86163e6 2.34526
$$273$$ −1.17903e6 −0.957457
$$274$$ −1.49482e6 −1.20285
$$275$$ 0 0
$$276$$ −519307. −0.410347
$$277$$ −952978. −0.746248 −0.373124 0.927781i $$-0.621713\pi$$
−0.373124 + 0.927781i $$0.621713\pi$$
$$278$$ 1.48202e6 1.15012
$$279$$ 591572. 0.454985
$$280$$ 0 0
$$281$$ −194035. −0.146593 −0.0732966 0.997310i $$-0.523352\pi$$
−0.0732966 + 0.997310i $$0.523352\pi$$
$$282$$ −1.63133e6 −1.22157
$$283$$ −589829. −0.437784 −0.218892 0.975749i $$-0.570244\pi$$
−0.218892 + 0.975749i $$0.570244\pi$$
$$284$$ 1.08513e6 0.798334
$$285$$ 0 0
$$286$$ 777436. 0.562017
$$287$$ −689048. −0.493793
$$288$$ 562261. 0.399445
$$289$$ 3.95267e6 2.78385
$$290$$ 0 0
$$291$$ 1.03074e6 0.713535
$$292$$ 435584. 0.298961
$$293$$ −2.18674e6 −1.48809 −0.744045 0.668130i $$-0.767095\pi$$
−0.744045 + 0.668130i $$0.767095\pi$$
$$294$$ −396142. −0.267290
$$295$$ 0 0
$$296$$ 205143. 0.136090
$$297$$ 88209.0 0.0580259
$$298$$ −595031. −0.388150
$$299$$ 2.20370e6 1.42553
$$300$$ 0 0
$$301$$ −154186. −0.0980908
$$302$$ −2.92402e6 −1.84486
$$303$$ 1.39181e6 0.870913
$$304$$ 3.27814e6 2.03444
$$305$$ 0 0
$$306$$ 1.38906e6 0.848042
$$307$$ 2.25874e6 1.36780 0.683898 0.729578i $$-0.260283\pi$$
0.683898 + 0.729578i $$0.260283\pi$$
$$308$$ 415047. 0.249299
$$309$$ −1.06159e6 −0.632501
$$310$$ 0 0
$$311$$ −3.00436e6 −1.76137 −0.880687 0.473699i $$-0.842918\pi$$
−0.880687 + 0.473699i $$0.842918\pi$$
$$312$$ −535554. −0.311471
$$313$$ 2.76965e6 1.59795 0.798976 0.601363i $$-0.205376\pi$$
0.798976 + 0.601363i $$0.205376\pi$$
$$314$$ 3.77367e6 2.15993
$$315$$ 0 0
$$316$$ −84431.4 −0.0475649
$$317$$ 2.00320e6 1.11963 0.559817 0.828616i $$-0.310871\pi$$
0.559817 + 0.828616i $$0.310871\pi$$
$$318$$ −886866. −0.491802
$$319$$ −99210.0 −0.0545857
$$320$$ 0 0
$$321$$ 2.09778e6 1.13631
$$322$$ 2.83215e6 1.52222
$$323$$ 6.15451e6 3.28237
$$324$$ 149187. 0.0789531
$$325$$ 0 0
$$326$$ −2.27375e6 −1.18495
$$327$$ −168968. −0.0873849
$$328$$ −312988. −0.160636
$$329$$ 3.69576e6 1.88241
$$330$$ 0 0
$$331$$ 1.09643e6 0.550059 0.275029 0.961436i $$-0.411312\pi$$
0.275029 + 0.961436i $$0.411312\pi$$
$$332$$ 1.18056e6 0.587817
$$333$$ 242501. 0.119840
$$334$$ −1.58584e6 −0.777846
$$335$$ 0 0
$$336$$ −1.67616e6 −0.809969
$$337$$ 594378. 0.285094 0.142547 0.989788i $$-0.454471\pi$$
0.142547 + 0.989788i $$0.454471\pi$$
$$338$$ −2.83269e6 −1.34868
$$339$$ 1.48550e6 0.702060
$$340$$ 0 0
$$341$$ 883706. 0.411549
$$342$$ 1.59124e6 0.735649
$$343$$ −1.63791e6 −0.751718
$$344$$ −70036.2 −0.0319100
$$345$$ 0 0
$$346$$ −2.71403e6 −1.21878
$$347$$ −822560. −0.366728 −0.183364 0.983045i $$-0.558699\pi$$
−0.183364 + 0.983045i $$0.558699\pi$$
$$348$$ −167793. −0.0742722
$$349$$ 956928. 0.420548 0.210274 0.977642i $$-0.432564\pi$$
0.210274 + 0.977642i $$0.432564\pi$$
$$350$$ 0 0
$$351$$ −633083. −0.274279
$$352$$ 839921. 0.361312
$$353$$ 218804. 0.0934583 0.0467291 0.998908i $$-0.485120\pi$$
0.0467291 + 0.998908i $$0.485120\pi$$
$$354$$ 1.98167e6 0.840472
$$355$$ 0 0
$$356$$ −1.20438e6 −0.503660
$$357$$ −3.14689e6 −1.30681
$$358$$ −5.01741e6 −2.06906
$$359$$ −2.72858e6 −1.11738 −0.558689 0.829378i $$-0.688695\pi$$
−0.558689 + 0.829378i $$0.688695\pi$$
$$360$$ 0 0
$$361$$ 4.57422e6 1.84735
$$362$$ 2.00394e6 0.803737
$$363$$ 131769. 0.0524864
$$364$$ −2.97882e6 −1.17840
$$365$$ 0 0
$$366$$ 1.14910e6 0.448389
$$367$$ −1.94374e6 −0.753307 −0.376654 0.926354i $$-0.622925\pi$$
−0.376654 + 0.926354i $$0.622925\pi$$
$$368$$ 3.13288e6 1.20594
$$369$$ −369985. −0.141455
$$370$$ 0 0
$$371$$ 2.00918e6 0.757852
$$372$$ 1.49461e6 0.559976
$$373$$ −4.88061e6 −1.81636 −0.908180 0.418580i $$-0.862528\pi$$
−0.908180 + 0.418580i $$0.862528\pi$$
$$374$$ 2.07501e6 0.767083
$$375$$ 0 0
$$376$$ 1.67873e6 0.612368
$$377$$ 712038. 0.258018
$$378$$ −813625. −0.292883
$$379$$ −4.08830e6 −1.46199 −0.730995 0.682383i $$-0.760944\pi$$
−0.730995 + 0.682383i $$0.760944\pi$$
$$380$$ 0 0
$$381$$ 1.83479e6 0.647553
$$382$$ −604057. −0.211797
$$383$$ 3.04558e6 1.06090 0.530448 0.847717i $$-0.322024\pi$$
0.530448 + 0.847717i $$0.322024\pi$$
$$384$$ −1.21009e6 −0.418785
$$385$$ 0 0
$$386$$ 1.26314e6 0.431502
$$387$$ −82790.3 −0.0280997
$$388$$ 2.60416e6 0.878189
$$389$$ 2.13646e6 0.715847 0.357923 0.933751i $$-0.383485\pi$$
0.357923 + 0.933751i $$0.383485\pi$$
$$390$$ 0 0
$$391$$ 5.88179e6 1.94566
$$392$$ 407653. 0.133991
$$393$$ 740805. 0.241948
$$394$$ 1.28189e6 0.416017
$$395$$ 0 0
$$396$$ 222860. 0.0714158
$$397$$ −2.27058e6 −0.723038 −0.361519 0.932365i $$-0.617742\pi$$
−0.361519 + 0.932365i $$0.617742\pi$$
$$398$$ −314690. −0.0995807
$$399$$ −3.60493e6 −1.13361
$$400$$ 0 0
$$401$$ 981140. 0.304698 0.152349 0.988327i $$-0.451316\pi$$
0.152349 + 0.988327i $$0.451316\pi$$
$$402$$ −1.24151e6 −0.383165
$$403$$ −6.34243e6 −1.94533
$$404$$ 3.51642e6 1.07188
$$405$$ 0 0
$$406$$ 915096. 0.275519
$$407$$ 362255. 0.108400
$$408$$ −1.42942e6 −0.425118
$$409$$ −1.98079e6 −0.585505 −0.292752 0.956188i $$-0.594571\pi$$
−0.292752 + 0.956188i $$0.594571\pi$$
$$410$$ 0 0
$$411$$ 1.81838e6 0.530982
$$412$$ −2.68211e6 −0.778456
$$413$$ −4.48945e6 −1.29514
$$414$$ 1.52073e6 0.436064
$$415$$ 0 0
$$416$$ −6.02818e6 −1.70786
$$417$$ −1.80281e6 −0.507703
$$418$$ 2.37704e6 0.665419
$$419$$ 153079. 0.0425972 0.0212986 0.999773i $$-0.493220\pi$$
0.0212986 + 0.999773i $$0.493220\pi$$
$$420$$ 0 0
$$421$$ 848256. 0.233250 0.116625 0.993176i $$-0.462792\pi$$
0.116625 + 0.993176i $$0.462792\pi$$
$$422$$ 4.62923e6 1.26540
$$423$$ 1.98444e6 0.539247
$$424$$ 912635. 0.246537
$$425$$ 0 0
$$426$$ −3.17766e6 −0.848367
$$427$$ −2.60327e6 −0.690954
$$428$$ 5.30003e6 1.39852
$$429$$ −945716. −0.248095
$$430$$ 0 0
$$431$$ −492966. −0.127827 −0.0639137 0.997955i $$-0.520358\pi$$
−0.0639137 + 0.997955i $$0.520358\pi$$
$$432$$ −900018. −0.232029
$$433$$ 5.12774e6 1.31434 0.657168 0.753744i $$-0.271754\pi$$
0.657168 + 0.753744i $$0.271754\pi$$
$$434$$ −8.15116e6 −2.07728
$$435$$ 0 0
$$436$$ −426899. −0.107550
$$437$$ 6.73790e6 1.68780
$$438$$ −1.27556e6 −0.317698
$$439$$ −3.98100e6 −0.985895 −0.492947 0.870059i $$-0.664081\pi$$
−0.492947 + 0.870059i $$0.664081\pi$$
$$440$$ 0 0
$$441$$ 481889. 0.117991
$$442$$ −1.48925e7 −3.62588
$$443$$ −3.09112e6 −0.748353 −0.374177 0.927357i $$-0.622075\pi$$
−0.374177 + 0.927357i $$0.622075\pi$$
$$444$$ 612679. 0.147494
$$445$$ 0 0
$$446$$ 314216. 0.0747981
$$447$$ 723829. 0.171343
$$448$$ −1.78760e6 −0.420799
$$449$$ −1.51168e6 −0.353870 −0.176935 0.984223i $$-0.556618\pi$$
−0.176935 + 0.984223i $$0.556618\pi$$
$$450$$ 0 0
$$451$$ −552694. −0.127951
$$452$$ 3.75312e6 0.864065
$$453$$ 3.55694e6 0.814388
$$454$$ −161403. −0.0367512
$$455$$ 0 0
$$456$$ −1.63748e6 −0.368776
$$457$$ −5.58801e6 −1.25160 −0.625802 0.779982i $$-0.715228\pi$$
−0.625802 + 0.779982i $$0.715228\pi$$
$$458$$ −435487. −0.0970088
$$459$$ −1.68973e6 −0.374356
$$460$$ 0 0
$$461$$ −819486. −0.179593 −0.0897964 0.995960i $$-0.528622\pi$$
−0.0897964 + 0.995960i $$0.528622\pi$$
$$462$$ −1.21542e6 −0.264923
$$463$$ −987740. −0.214136 −0.107068 0.994252i $$-0.534146\pi$$
−0.107068 + 0.994252i $$0.534146\pi$$
$$464$$ 1.01226e6 0.218272
$$465$$ 0 0
$$466$$ −3.09263e6 −0.659725
$$467$$ −3.26318e6 −0.692387 −0.346193 0.938163i $$-0.612526\pi$$
−0.346193 + 0.938163i $$0.612526\pi$$
$$468$$ −1.59948e6 −0.337571
$$469$$ 2.81263e6 0.590447
$$470$$ 0 0
$$471$$ −4.59051e6 −0.953473
$$472$$ −2.03925e6 −0.421323
$$473$$ −123674. −0.0254172
$$474$$ 247247. 0.0505459
$$475$$ 0 0
$$476$$ −7.95063e6 −1.60836
$$477$$ 1.07883e6 0.217099
$$478$$ 9.56277e6 1.91432
$$479$$ 5.61817e6 1.11881 0.559405 0.828895i $$-0.311030\pi$$
0.559405 + 0.828895i $$0.311030\pi$$
$$480$$ 0 0
$$481$$ −2.59993e6 −0.512388
$$482$$ −3.67078e6 −0.719681
$$483$$ −3.44519e6 −0.671962
$$484$$ 332914. 0.0645980
$$485$$ 0 0
$$486$$ −436877. −0.0839012
$$487$$ −4.31761e6 −0.824938 −0.412469 0.910972i $$-0.635334\pi$$
−0.412469 + 0.910972i $$0.635334\pi$$
$$488$$ −1.18249e6 −0.224775
$$489$$ 2.76591e6 0.523078
$$490$$ 0 0
$$491$$ 8.77622e6 1.64287 0.821436 0.570301i $$-0.193173\pi$$
0.821436 + 0.570301i $$0.193173\pi$$
$$492$$ −934767. −0.174097
$$493$$ 1.90046e6 0.352162
$$494$$ −1.70602e7 −3.14533
$$495$$ 0 0
$$496$$ −9.01667e6 −1.64567
$$497$$ 7.19895e6 1.30731
$$498$$ −3.45712e6 −0.624657
$$499$$ −9.08110e6 −1.63263 −0.816313 0.577609i $$-0.803986\pi$$
−0.816313 + 0.577609i $$0.803986\pi$$
$$500$$ 0 0
$$501$$ 1.92911e6 0.343370
$$502$$ 1.17332e7 2.07806
$$503$$ 4.35992e6 0.768350 0.384175 0.923260i $$-0.374486\pi$$
0.384175 + 0.923260i $$0.374486\pi$$
$$504$$ 837266. 0.146821
$$505$$ 0 0
$$506$$ 2.27170e6 0.394435
$$507$$ 3.44584e6 0.595354
$$508$$ 4.63561e6 0.796980
$$509$$ 5.30215e6 0.907105 0.453552 0.891230i $$-0.350156\pi$$
0.453552 + 0.891230i $$0.350156\pi$$
$$510$$ 0 0
$$511$$ 2.88975e6 0.489563
$$512$$ −5.86284e6 −0.988400
$$513$$ −1.93567e6 −0.324742
$$514$$ −1.04971e7 −1.75251
$$515$$ 0 0
$$516$$ −209170. −0.0345839
$$517$$ 2.96442e6 0.487767
$$518$$ −3.34138e6 −0.547143
$$519$$ 3.30150e6 0.538013
$$520$$ 0 0
$$521$$ 1.80284e6 0.290981 0.145490 0.989360i $$-0.453524\pi$$
0.145490 + 0.989360i $$0.453524\pi$$
$$522$$ 491362. 0.0789269
$$523$$ 1.48512e6 0.237414 0.118707 0.992929i $$-0.462125\pi$$
0.118707 + 0.992929i $$0.462125\pi$$
$$524$$ 1.87164e6 0.297779
$$525$$ 0 0
$$526$$ −1.19891e7 −1.88939
$$527$$ −1.69282e7 −2.65513
$$528$$ −1.34447e6 −0.209878
$$529$$ 2976.47 0.000462448 0
$$530$$ 0 0
$$531$$ −2.41061e6 −0.371015
$$532$$ −9.10786e6 −1.39520
$$533$$ 3.96672e6 0.604803
$$534$$ 3.52687e6 0.535225
$$535$$ 0 0
$$536$$ 1.27759e6 0.192078
$$537$$ 6.10346e6 0.913356
$$538$$ −9.30896e6 −1.38658
$$539$$ 719859. 0.106727
$$540$$ 0 0
$$541$$ −2.18504e6 −0.320972 −0.160486 0.987038i $$-0.551306\pi$$
−0.160486 + 0.987038i $$0.551306\pi$$
$$542$$ 1.72673e7 2.52479
$$543$$ −2.43771e6 −0.354799
$$544$$ −1.60895e7 −2.33102
$$545$$ 0 0
$$546$$ 8.72313e6 1.25225
$$547$$ 6.00853e6 0.858618 0.429309 0.903158i $$-0.358757\pi$$
0.429309 + 0.903158i $$0.358757\pi$$
$$548$$ 4.59413e6 0.653509
$$549$$ −1.39783e6 −0.197935
$$550$$ 0 0
$$551$$ 2.17708e6 0.305489
$$552$$ −1.56491e6 −0.218596
$$553$$ −560135. −0.0778897
$$554$$ 7.05065e6 0.976011
$$555$$ 0 0
$$556$$ −4.55479e6 −0.624858
$$557$$ 8.40252e6 1.14755 0.573775 0.819013i $$-0.305478\pi$$
0.573775 + 0.819013i $$0.305478\pi$$
$$558$$ −4.37677e6 −0.595071
$$559$$ 887620. 0.120143
$$560$$ 0 0
$$561$$ −2.52416e6 −0.338618
$$562$$ 1.43558e6 0.191728
$$563$$ −1.68040e6 −0.223430 −0.111715 0.993740i $$-0.535634\pi$$
−0.111715 + 0.993740i $$0.535634\pi$$
$$564$$ 5.01369e6 0.663682
$$565$$ 0 0
$$566$$ 4.36387e6 0.572573
$$567$$ 989738. 0.129289
$$568$$ 3.27000e6 0.425281
$$569$$ 1.43427e7 1.85716 0.928580 0.371131i $$-0.121030\pi$$
0.928580 + 0.371131i $$0.121030\pi$$
$$570$$ 0 0
$$571$$ 9.83718e6 1.26264 0.631321 0.775521i $$-0.282513\pi$$
0.631321 + 0.775521i $$0.282513\pi$$
$$572$$ −2.38935e6 −0.305344
$$573$$ 734809. 0.0934949
$$574$$ 5.09795e6 0.645827
$$575$$ 0 0
$$576$$ −959852. −0.120545
$$577$$ 4.88454e6 0.610779 0.305390 0.952228i $$-0.401213\pi$$
0.305390 + 0.952228i $$0.401213\pi$$
$$578$$ −2.92440e7 −3.64097
$$579$$ −1.53655e6 −0.190481
$$580$$ 0 0
$$581$$ 7.83207e6 0.962578
$$582$$ −7.62596e6 −0.933226
$$583$$ 1.61159e6 0.196374
$$584$$ 1.31262e6 0.159260
$$585$$ 0 0
$$586$$ 1.61787e7 1.94626
$$587$$ 3.75376e6 0.449646 0.224823 0.974400i $$-0.427820\pi$$
0.224823 + 0.974400i $$0.427820\pi$$
$$588$$ 1.21749e6 0.145219
$$589$$ −1.93922e7 −2.30324
$$590$$ 0 0
$$591$$ −1.55936e6 −0.183645
$$592$$ −3.69617e6 −0.433459
$$593$$ 7.63599e6 0.891720 0.445860 0.895103i $$-0.352898\pi$$
0.445860 + 0.895103i $$0.352898\pi$$
$$594$$ −652618. −0.0758915
$$595$$ 0 0
$$596$$ 1.82875e6 0.210882
$$597$$ 382806. 0.0439585
$$598$$ −1.63042e7 −1.86443
$$599$$ −1.06753e6 −0.121567 −0.0607834 0.998151i $$-0.519360\pi$$
−0.0607834 + 0.998151i $$0.519360\pi$$
$$600$$ 0 0
$$601$$ −4.83197e6 −0.545680 −0.272840 0.962059i $$-0.587963\pi$$
−0.272840 + 0.962059i $$0.587963\pi$$
$$602$$ 1.14075e6 0.128292
$$603$$ 1.51025e6 0.169143
$$604$$ 8.98661e6 1.00231
$$605$$ 0 0
$$606$$ −1.02974e7 −1.13906
$$607$$ −8.95712e6 −0.986726 −0.493363 0.869824i $$-0.664233\pi$$
−0.493363 + 0.869824i $$0.664233\pi$$
$$608$$ −1.84314e7 −2.02208
$$609$$ −1.11317e6 −0.121624
$$610$$ 0 0
$$611$$ −2.12758e7 −2.30560
$$612$$ −4.26910e6 −0.460742
$$613$$ −8.42569e6 −0.905637 −0.452818 0.891603i $$-0.649581\pi$$
−0.452818 + 0.891603i $$0.649581\pi$$
$$614$$ −1.67114e7 −1.78893
$$615$$ 0 0
$$616$$ 1.25073e6 0.132804
$$617$$ −4.84242e6 −0.512094 −0.256047 0.966664i $$-0.582420\pi$$
−0.256047 + 0.966664i $$0.582420\pi$$
$$618$$ 7.85424e6 0.827243
$$619$$ −4.22402e6 −0.443097 −0.221549 0.975149i $$-0.571111\pi$$
−0.221549 + 0.975149i $$0.571111\pi$$
$$620$$ 0 0
$$621$$ −1.84990e6 −0.192495
$$622$$ 2.22279e7 2.30368
$$623$$ −7.99008e6 −0.824767
$$624$$ 9.64938e6 0.992059
$$625$$ 0 0
$$626$$ −2.04914e7 −2.08995
$$627$$ −2.89156e6 −0.293740
$$628$$ −1.15979e7 −1.17349
$$629$$ −6.93934e6 −0.699345
$$630$$ 0 0
$$631$$ 1.18298e7 1.18278 0.591392 0.806384i $$-0.298579\pi$$
0.591392 + 0.806384i $$0.298579\pi$$
$$632$$ −254431. −0.0253383
$$633$$ −5.63125e6 −0.558593
$$634$$ −1.48208e7 −1.46436
$$635$$ 0 0
$$636$$ 2.72567e6 0.267196
$$637$$ −5.16648e6 −0.504483
$$638$$ 734010. 0.0713921
$$639$$ 3.86549e6 0.374500
$$640$$ 0 0
$$641$$ −2.19467e6 −0.210972 −0.105486 0.994421i $$-0.533640\pi$$
−0.105486 + 0.994421i $$0.533640\pi$$
$$642$$ −1.55205e7 −1.48617
$$643$$ 467202. 0.0445633 0.0222816 0.999752i $$-0.492907\pi$$
0.0222816 + 0.999752i $$0.492907\pi$$
$$644$$ −8.70426e6 −0.827022
$$645$$ 0 0
$$646$$ −4.55345e7 −4.29298
$$647$$ 944645. 0.0887172 0.0443586 0.999016i $$-0.485876\pi$$
0.0443586 + 0.999016i $$0.485876\pi$$
$$648$$ 449571. 0.0420592
$$649$$ −3.60104e6 −0.335595
$$650$$ 0 0
$$651$$ 9.91552e6 0.916986
$$652$$ 6.98808e6 0.643782
$$653$$ 917543. 0.0842061 0.0421030 0.999113i $$-0.486594\pi$$
0.0421030 + 0.999113i $$0.486594\pi$$
$$654$$ 1.25012e6 0.114290
$$655$$ 0 0
$$656$$ 5.63927e6 0.511638
$$657$$ 1.55166e6 0.140243
$$658$$ −2.73433e7 −2.46199
$$659$$ −5.17061e6 −0.463798 −0.231899 0.972740i $$-0.574494\pi$$
−0.231899 + 0.972740i $$0.574494\pi$$
$$660$$ 0 0
$$661$$ −1.78425e6 −0.158837 −0.0794185 0.996841i $$-0.525306\pi$$
−0.0794185 + 0.996841i $$0.525306\pi$$
$$662$$ −8.11195e6 −0.719417
$$663$$ 1.81161e7 1.60059
$$664$$ 3.55758e6 0.313137
$$665$$ 0 0
$$666$$ −1.79416e6 −0.156738
$$667$$ 2.08061e6 0.181082
$$668$$ 4.87388e6 0.422604
$$669$$ −382229. −0.0330186
$$670$$ 0 0
$$671$$ −2.08811e6 −0.179039
$$672$$ 9.42423e6 0.805050
$$673$$ 9.10656e6 0.775026 0.387513 0.921864i $$-0.373334\pi$$
0.387513 + 0.921864i $$0.373334\pi$$
$$674$$ −4.39753e6 −0.372872
$$675$$ 0 0
$$676$$ 8.70591e6 0.732736
$$677$$ 657297. 0.0551175 0.0275588 0.999620i $$-0.491227\pi$$
0.0275588 + 0.999620i $$0.491227\pi$$
$$678$$ −1.09906e7 −0.918217
$$679$$ 1.72765e7 1.43807
$$680$$ 0 0
$$681$$ 196339. 0.0162233
$$682$$ −6.53814e6 −0.538262
$$683$$ 1.92866e7 1.58199 0.790997 0.611821i $$-0.209563\pi$$
0.790997 + 0.611821i $$0.209563\pi$$
$$684$$ −4.89047e6 −0.399678
$$685$$ 0 0
$$686$$ 1.21182e7 0.983165
$$687$$ 529750. 0.0428232
$$688$$ 1.26188e6 0.101636
$$689$$ −1.15665e7 −0.928226
$$690$$ 0 0
$$691$$ 7.09653e6 0.565394 0.282697 0.959209i $$-0.408771\pi$$
0.282697 + 0.959209i $$0.408771\pi$$
$$692$$ 8.34123e6 0.662163
$$693$$ 1.47850e6 0.116947
$$694$$ 6.08575e6 0.479640
$$695$$ 0 0
$$696$$ −505639. −0.0395656
$$697$$ 1.05874e7 0.825480
$$698$$ −7.07987e6 −0.550031
$$699$$ 3.76204e6 0.291227
$$700$$ 0 0
$$701$$ 1.57310e7 1.20910 0.604550 0.796567i $$-0.293353\pi$$
0.604550 + 0.796567i $$0.293353\pi$$
$$702$$ 4.68389e6 0.358727
$$703$$ −7.94938e6 −0.606659
$$704$$ −1.43385e6 −0.109037
$$705$$ 0 0
$$706$$ −1.61883e6 −0.122233
$$707$$ 2.33286e7 1.75526
$$708$$ −6.09041e6 −0.456629
$$709$$ 2.46071e7 1.83842 0.919210 0.393767i $$-0.128828\pi$$
0.919210 + 0.393767i $$0.128828\pi$$
$$710$$ 0 0
$$711$$ −300765. −0.0223128
$$712$$ −3.62935e6 −0.268305
$$713$$ −1.85329e7 −1.36527
$$714$$ 2.32824e7 1.70916
$$715$$ 0 0
$$716$$ 1.54204e7 1.12412
$$717$$ −1.16327e7 −0.845049
$$718$$ 2.01875e7 1.46141
$$719$$ −8.76856e6 −0.632566 −0.316283 0.948665i $$-0.602435\pi$$
−0.316283 + 0.948665i $$0.602435\pi$$
$$720$$ 0 0
$$721$$ −1.77937e7 −1.27476
$$722$$ −3.38426e7 −2.41613
$$723$$ 4.46533e6 0.317693
$$724$$ −6.15887e6 −0.436671
$$725$$ 0 0
$$726$$ −974899. −0.0686465
$$727$$ −1.62173e7 −1.13800 −0.568999 0.822338i $$-0.692669\pi$$
−0.568999 + 0.822338i $$0.692669\pi$$
$$728$$ −8.97659e6 −0.627745
$$729$$ 531441. 0.0370370
$$730$$ 0 0
$$731$$ 2.36910e6 0.163980
$$732$$ −3.53161e6 −0.243610
$$733$$ −1.12288e7 −0.771919 −0.385960 0.922516i $$-0.626130\pi$$
−0.385960 + 0.922516i $$0.626130\pi$$
$$734$$ 1.43808e7 0.985243
$$735$$ 0 0
$$736$$ −1.76146e7 −1.19861
$$737$$ 2.25605e6 0.152996
$$738$$ 2.73735e6 0.185008
$$739$$ 8.22290e6 0.553878 0.276939 0.960888i $$-0.410680\pi$$
0.276939 + 0.960888i $$0.410680\pi$$
$$740$$ 0 0
$$741$$ 2.07529e7 1.38846
$$742$$ −1.48650e7 −0.991188
$$743$$ −2.24984e6 −0.149513 −0.0747566 0.997202i $$-0.523818\pi$$
−0.0747566 + 0.997202i $$0.523818\pi$$
$$744$$ 4.50395e6 0.298305
$$745$$ 0 0
$$746$$ 3.61094e7 2.37560
$$747$$ 4.20544e6 0.275746
$$748$$ −6.37729e6 −0.416757
$$749$$ 3.51615e7 2.29014
$$750$$ 0 0
$$751$$ −171752. −0.0111122 −0.00555612 0.999985i $$-0.501769\pi$$
−0.00555612 + 0.999985i $$0.501769\pi$$
$$752$$ −3.02467e7 −1.95044
$$753$$ −1.42729e7 −0.917330
$$754$$ −5.26804e6 −0.337459
$$755$$ 0 0
$$756$$ 2.50057e6 0.159124
$$757$$ 2.62239e7 1.66325 0.831624 0.555339i $$-0.187411\pi$$
0.831624 + 0.555339i $$0.187411\pi$$
$$758$$ 3.02475e7 1.91212
$$759$$ −2.76343e6 −0.174118
$$760$$ 0 0
$$761$$ −2.29774e7 −1.43827 −0.719134 0.694871i $$-0.755461\pi$$
−0.719134 + 0.694871i $$0.755461\pi$$
$$762$$ −1.35748e7 −0.846928
$$763$$ −2.83213e6 −0.176117
$$764$$ 1.85649e6 0.115069
$$765$$ 0 0
$$766$$ −2.25329e7 −1.38754
$$767$$ 2.58449e7 1.58631
$$768$$ 1.23657e7 0.756515
$$769$$ −2.94748e6 −0.179736 −0.0898680 0.995954i $$-0.528645\pi$$
−0.0898680 + 0.995954i $$0.528645\pi$$
$$770$$ 0 0
$$771$$ 1.27693e7 0.773623
$$772$$ −3.88209e6 −0.234435
$$773$$ −1.27928e7 −0.770049 −0.385024 0.922906i $$-0.625807\pi$$
−0.385024 + 0.922906i $$0.625807\pi$$
$$774$$ 612528. 0.0367514
$$775$$ 0 0
$$776$$ 7.84754e6 0.467821
$$777$$ 4.06463e6 0.241529
$$778$$ −1.58067e7 −0.936249
$$779$$ 1.21284e7 0.716077
$$780$$ 0 0
$$781$$ 5.77437e6 0.338748
$$782$$ −4.35167e7 −2.54471
$$783$$ −597720. −0.0348412
$$784$$ −7.34490e6 −0.426772
$$785$$ 0 0
$$786$$ −5.48088e6 −0.316442
$$787$$ −1.94779e6 −0.112100 −0.0560499 0.998428i $$-0.517851\pi$$
−0.0560499 + 0.998428i $$0.517851\pi$$
$$788$$ −3.93973e6 −0.226022
$$789$$ 1.45842e7 0.834045
$$790$$ 0 0
$$791$$ 2.48989e7 1.41495
$$792$$ 671581. 0.0380440
$$793$$ 1.49865e7 0.846289
$$794$$ 1.67990e7 0.945655
$$795$$ 0 0
$$796$$ 967160. 0.0541023
$$797$$ −2.72445e7 −1.51926 −0.759630 0.650355i $$-0.774620\pi$$
−0.759630 + 0.650355i $$0.774620\pi$$
$$798$$ 2.66713e7 1.48264
$$799$$ −5.67862e7 −3.14685
$$800$$ 0 0
$$801$$ −4.29028e6 −0.236268
$$802$$ −7.25901e6 −0.398512
$$803$$ 2.31791e6 0.126855
$$804$$ 3.81563e6 0.208174
$$805$$ 0 0
$$806$$ 4.69247e7 2.54428
$$807$$ 1.13239e7 0.612087
$$808$$ 1.05966e7 0.571003
$$809$$ 1.56700e7 0.841780 0.420890 0.907112i $$-0.361718\pi$$
0.420890 + 0.907112i $$0.361718\pi$$
$$810$$ 0 0
$$811$$ 1.41672e7 0.756365 0.378182 0.925731i $$-0.376549\pi$$
0.378182 + 0.925731i $$0.376549\pi$$
$$812$$ −2.81243e6 −0.149690
$$813$$ −2.10048e7 −1.11453
$$814$$ −2.68016e6 −0.141775
$$815$$ 0 0
$$816$$ 2.57547e7 1.35404
$$817$$ 2.71393e6 0.142247
$$818$$ 1.46550e7 0.765776
$$819$$ −1.06113e7 −0.552788
$$820$$ 0 0
$$821$$ 6.51336e6 0.337246 0.168623 0.985681i $$-0.446068\pi$$
0.168623 + 0.985681i $$0.446068\pi$$
$$822$$ −1.34533e7 −0.694466
$$823$$ 2.92352e7 1.50455 0.752275 0.658849i $$-0.228956\pi$$
0.752275 + 0.658849i $$0.228956\pi$$
$$824$$ −8.08246e6 −0.414692
$$825$$ 0 0
$$826$$ 3.32154e7 1.69390
$$827$$ −2.56735e7 −1.30533 −0.652666 0.757646i $$-0.726349\pi$$
−0.652666 + 0.757646i $$0.726349\pi$$
$$828$$ −4.67376e6 −0.236914
$$829$$ −9.02846e6 −0.456276 −0.228138 0.973629i $$-0.573264\pi$$
−0.228138 + 0.973629i $$0.573264\pi$$
$$830$$ 0 0
$$831$$ −8.57680e6 −0.430847
$$832$$ 1.02909e7 0.515399
$$833$$ −1.37896e7 −0.688555
$$834$$ 1.33382e7 0.664019
$$835$$ 0 0
$$836$$ −7.30552e6 −0.361523
$$837$$ 5.32415e6 0.262686
$$838$$ −1.13256e6 −0.0557124
$$839$$ 3.00826e7 1.47540 0.737701 0.675128i $$-0.235912\pi$$
0.737701 + 0.675128i $$0.235912\pi$$
$$840$$ 0 0
$$841$$ −1.98389e7 −0.967224
$$842$$ −6.27586e6 −0.305065
$$843$$ −1.74631e6 −0.0846356
$$844$$ −1.42273e7 −0.687492
$$845$$ 0 0
$$846$$ −1.46820e7 −0.705276
$$847$$ 2.20862e6 0.105782
$$848$$ −1.64434e7 −0.785241
$$849$$ −5.30846e6 −0.252755
$$850$$ 0 0
$$851$$ −7.59711e6 −0.359604
$$852$$ 9.76614e6 0.460919
$$853$$ 2.48619e7 1.16994 0.584968 0.811057i $$-0.301107\pi$$
0.584968 + 0.811057i $$0.301107\pi$$
$$854$$ 1.92604e7 0.903692
$$855$$ 0 0
$$856$$ 1.59715e7 0.745008
$$857$$ 8.27535e6 0.384888 0.192444 0.981308i $$-0.438359\pi$$
0.192444 + 0.981308i $$0.438359\pi$$
$$858$$ 6.99693e6 0.324481
$$859$$ 1.89682e7 0.877088 0.438544 0.898710i $$-0.355494\pi$$
0.438544 + 0.898710i $$0.355494\pi$$
$$860$$ 0 0
$$861$$ −6.20143e6 −0.285091
$$862$$ 3.64723e6 0.167184
$$863$$ 1.70264e7 0.778209 0.389104 0.921194i $$-0.372785\pi$$
0.389104 + 0.921194i $$0.372785\pi$$
$$864$$ 5.06035e6 0.230620
$$865$$ 0 0
$$866$$ −3.79378e7 −1.71901
$$867$$ 3.55741e7 1.60726
$$868$$ 2.50515e7 1.12859
$$869$$ −449291. −0.0201827
$$870$$ 0 0
$$871$$ −1.61918e7 −0.723186
$$872$$ −1.28645e6 −0.0572928
$$873$$ 9.27664e6 0.411960
$$874$$ −4.98506e7 −2.20746
$$875$$ 0 0
$$876$$ 3.92026e6 0.172605
$$877$$ 2.69746e7 1.18428 0.592142 0.805834i $$-0.298283\pi$$
0.592142 + 0.805834i $$0.298283\pi$$
$$878$$ 2.94536e7 1.28944
$$879$$ −1.96807e7 −0.859149
$$880$$ 0 0
$$881$$ 2.13740e7 0.927781 0.463890 0.885893i $$-0.346453\pi$$
0.463890 + 0.885893i $$0.346453\pi$$
$$882$$ −3.56528e6 −0.154320
$$883$$ −3.26501e6 −0.140923 −0.0704616 0.997514i $$-0.522447\pi$$
−0.0704616 + 0.997514i $$0.522447\pi$$
$$884$$ 4.57703e7 1.96994
$$885$$ 0 0
$$886$$ 2.28698e7 0.978764
$$887$$ −6.68667e6 −0.285365 −0.142682 0.989769i $$-0.545573\pi$$
−0.142682 + 0.989769i $$0.545573\pi$$
$$888$$ 1.84629e6 0.0785718
$$889$$ 3.07536e7 1.30509
$$890$$ 0 0
$$891$$ 793881. 0.0335013
$$892$$ −965702. −0.0406379
$$893$$ −6.50516e7 −2.72979
$$894$$ −5.35528e6 −0.224098
$$895$$ 0 0
$$896$$ −2.02828e7 −0.844029
$$897$$ 1.98333e7 0.823027
$$898$$ 1.11842e7 0.462824
$$899$$ −5.98815e6 −0.247112
$$900$$ 0 0
$$901$$ −3.08715e7 −1.26691
$$902$$ 4.08913e6 0.167346
$$903$$ −1.38767e6 −0.0566328
$$904$$ 1.13099e7 0.460297
$$905$$ 0 0
$$906$$ −2.63162e7 −1.06513
$$907$$ 2.82586e7 1.14060 0.570299 0.821437i $$-0.306827\pi$$
0.570299 + 0.821437i $$0.306827\pi$$
$$908$$ 496050. 0.0199669
$$909$$ 1.25263e7 0.502822
$$910$$ 0 0
$$911$$ −9.75713e6 −0.389517 −0.194758 0.980851i $$-0.562392\pi$$
−0.194758 + 0.980851i $$0.562392\pi$$
$$912$$ 2.95033e7 1.17458
$$913$$ 6.28219e6 0.249422
$$914$$ 4.13432e7 1.63696
$$915$$ 0 0
$$916$$ 1.33841e6 0.0527050
$$917$$ 1.24169e7 0.487628
$$918$$ 1.25015e7 0.489617
$$919$$ −3.69755e7 −1.44419 −0.722097 0.691792i $$-0.756821\pi$$
−0.722097 + 0.691792i $$0.756821\pi$$
$$920$$ 0 0
$$921$$ 2.03287e7 0.789697
$$922$$ 6.06300e6 0.234888
$$923$$ −4.14431e7 −1.60121
$$924$$ 3.73542e6 0.143933
$$925$$ 0 0
$$926$$ 7.30784e6 0.280067
$$927$$ −9.55433e6 −0.365175
$$928$$ −5.69145e6 −0.216947
$$929$$ −5.05419e7 −1.92137 −0.960687 0.277633i $$-0.910450\pi$$
−0.960687 + 0.277633i $$0.910450\pi$$
$$930$$ 0 0
$$931$$ −1.57967e7 −0.597299
$$932$$ 9.50480e6 0.358429
$$933$$ −2.70393e7 −1.01693
$$934$$ 2.41428e7 0.905566
$$935$$ 0 0
$$936$$ −4.81999e6 −0.179828
$$937$$ 1.08317e7 0.403040 0.201520 0.979484i $$-0.435412\pi$$
0.201520 + 0.979484i $$0.435412\pi$$
$$938$$ −2.08094e7 −0.772240
$$939$$ 2.49268e7 0.922578
$$940$$ 0 0
$$941$$ 1.45659e6 0.0536246 0.0268123 0.999640i $$-0.491464\pi$$
0.0268123 + 0.999640i $$0.491464\pi$$
$$942$$ 3.39631e7 1.24704
$$943$$ 1.15909e7 0.424463
$$944$$ 3.67423e7 1.34195
$$945$$ 0 0
$$946$$ 915011. 0.0332429
$$947$$ −4.32433e7 −1.56691 −0.783455 0.621449i $$-0.786544\pi$$
−0.783455 + 0.621449i $$0.786544\pi$$
$$948$$ −759883. −0.0274616
$$949$$ −1.66358e7 −0.599623
$$950$$ 0 0
$$951$$ 1.80288e7 0.646421
$$952$$ −2.39590e7 −0.856792
$$953$$ 1.69154e7 0.603322 0.301661 0.953415i $$-0.402459\pi$$
0.301661 + 0.953415i $$0.402459\pi$$
$$954$$ −7.98179e6 −0.283942
$$955$$ 0 0
$$956$$ −2.93900e7 −1.04005
$$957$$ −892890. −0.0315151
$$958$$ −4.15663e7 −1.46328
$$959$$ 3.04784e7 1.07015
$$960$$ 0 0
$$961$$ 2.47099e7 0.863102
$$962$$ 1.92357e7 0.670147
$$963$$ 1.88800e7 0.656049
$$964$$ 1.12817e7 0.391003
$$965$$ 0 0
$$966$$ 2.54894e7 0.878853
$$967$$ 2.51066e7 0.863421 0.431710 0.902012i $$-0.357910\pi$$
0.431710 + 0.902012i $$0.357910\pi$$
$$968$$ 1.00323e6 0.0344121
$$969$$ 5.53906e7 1.89508
$$970$$ 0 0
$$971$$ −3.10806e7 −1.05789 −0.528947 0.848655i $$-0.677413\pi$$
−0.528947 + 0.848655i $$0.677413\pi$$
$$972$$ 1.34269e6 0.0455836
$$973$$ −3.02174e7 −1.02323
$$974$$ 3.19441e7 1.07893
$$975$$ 0 0
$$976$$ 2.13055e7 0.715925
$$977$$ −1.72964e7 −0.579722 −0.289861 0.957069i $$-0.593609\pi$$
−0.289861 + 0.957069i $$0.593609\pi$$
$$978$$ −2.04637e7 −0.684129
$$979$$ −6.40894e6 −0.213712
$$980$$ 0 0
$$981$$ −1.52072e6 −0.0504517
$$982$$ −6.49313e7 −2.14870
$$983$$ 2.02617e7 0.668793 0.334396 0.942433i $$-0.391468\pi$$
0.334396 + 0.942433i $$0.391468\pi$$
$$984$$ −2.81689e6 −0.0927432
$$985$$ 0 0
$$986$$ −1.40607e7 −0.460589
$$987$$ 3.32619e7 1.08681
$$988$$ 5.24323e7 1.70886
$$989$$ 2.59367e6 0.0843186
$$990$$ 0 0
$$991$$ 1.64795e7 0.533041 0.266520 0.963829i $$-0.414126\pi$$
0.266520 + 0.963829i $$0.414126\pi$$
$$992$$ 5.06962e7 1.63567
$$993$$ 9.86783e6 0.317577
$$994$$ −5.32618e7 −1.70982
$$995$$ 0 0
$$996$$ 1.06250e7 0.339377
$$997$$ 364567. 0.0116155 0.00580777 0.999983i $$-0.498151\pi$$
0.00580777 + 0.999983i $$0.498151\pi$$
$$998$$ 6.71869e7 2.13530
$$999$$ 2.18251e6 0.0691899
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.y.1.3 13
5.2 odd 4 165.6.c.b.34.6 26
5.3 odd 4 165.6.c.b.34.21 yes 26
5.4 even 2 825.6.a.v.1.11 13

By twisted newform
Twist Min Dim Char Parity Ord Type
165.6.c.b.34.6 26 5.2 odd 4
165.6.c.b.34.21 yes 26 5.3 odd 4
825.6.a.v.1.11 13 5.4 even 2
825.6.a.y.1.3 13 1.1 even 1 trivial