# Properties

 Label 825.2.c.g.199.1 Level $825$ Weight $2$ Character 825.199 Analytic conductor $6.588$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$825 = 3 \cdot 5^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 825.c (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.58765816676$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: 6.0.350464.1 Defining polynomial: $$x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2$$ x^6 - 2*x^5 + 2*x^4 + 2*x^3 + 4*x^2 - 4*x + 2 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{3}$$ Twist minimal: no (minimal twist has level 165) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 199.1 Root $$-0.854638 + 0.854638i$$ of defining polynomial Character $$\chi$$ $$=$$ 825.199 Dual form 825.2.c.g.199.6

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-2.70928i q^{2} -1.00000i q^{3} -5.34017 q^{4} -2.70928 q^{6} +1.07838i q^{7} +9.04945i q^{8} -1.00000 q^{9} +O(q^{10})$$ $$q-2.70928i q^{2} -1.00000i q^{3} -5.34017 q^{4} -2.70928 q^{6} +1.07838i q^{7} +9.04945i q^{8} -1.00000 q^{9} +1.00000 q^{11} +5.34017i q^{12} +4.34017i q^{13} +2.92162 q^{14} +13.8371 q^{16} +7.75872i q^{17} +2.70928i q^{18} -5.26180 q^{19} +1.07838 q^{21} -2.70928i q^{22} +2.15676i q^{23} +9.04945 q^{24} +11.7587 q^{26} +1.00000i q^{27} -5.75872i q^{28} -1.41855 q^{29} -4.68035 q^{31} -19.3896i q^{32} -1.00000i q^{33} +21.0205 q^{34} +5.34017 q^{36} -2.00000i q^{37} +14.2557i q^{38} +4.34017 q^{39} -9.41855 q^{41} -2.92162i q^{42} -7.60197i q^{43} -5.34017 q^{44} +5.84324 q^{46} +4.68035i q^{47} -13.8371i q^{48} +5.83710 q^{49} +7.75872 q^{51} -23.1773i q^{52} -0.156755i q^{53} +2.70928 q^{54} -9.75872 q^{56} +5.26180i q^{57} +3.84324i q^{58} -6.15676 q^{59} -4.15676 q^{61} +12.6803i q^{62} -1.07838i q^{63} -24.8576 q^{64} -2.70928 q^{66} -8.68035i q^{67} -41.4329i q^{68} +2.15676 q^{69} -4.68035 q^{71} -9.04945i q^{72} +10.4969i q^{73} -5.41855 q^{74} +28.0989 q^{76} +1.07838i q^{77} -11.7587i q^{78} +8.09890 q^{79} +1.00000 q^{81} +25.5174i q^{82} +11.0205i q^{83} -5.75872 q^{84} -20.5958 q^{86} +1.41855i q^{87} +9.04945i q^{88} +12.8371 q^{89} -4.68035 q^{91} -11.5174i q^{92} +4.68035i q^{93} +12.6803 q^{94} -19.3896 q^{96} +14.6803i q^{97} -15.8143i q^{98} -1.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 10 q^{4} - 2 q^{6} - 6 q^{9}+O(q^{10})$$ 6 * q - 10 * q^4 - 2 * q^6 - 6 * q^9 $$6 q - 10 q^{4} - 2 q^{6} - 6 q^{9} + 6 q^{11} + 24 q^{14} + 26 q^{16} - 16 q^{19} + 18 q^{24} + 20 q^{26} + 20 q^{29} + 16 q^{31} + 60 q^{34} + 10 q^{36} + 4 q^{39} - 28 q^{41} - 10 q^{44} + 48 q^{46} - 22 q^{49} - 4 q^{51} + 2 q^{54} - 8 q^{56} - 24 q^{59} - 12 q^{61} - 26 q^{64} - 2 q^{66} + 16 q^{71} - 4 q^{74} + 96 q^{76} - 24 q^{79} + 6 q^{81} + 16 q^{84} - 16 q^{86} + 20 q^{89} + 16 q^{91} + 32 q^{94} - 58 q^{96} - 6 q^{99}+O(q^{100})$$ 6 * q - 10 * q^4 - 2 * q^6 - 6 * q^9 + 6 * q^11 + 24 * q^14 + 26 * q^16 - 16 * q^19 + 18 * q^24 + 20 * q^26 + 20 * q^29 + 16 * q^31 + 60 * q^34 + 10 * q^36 + 4 * q^39 - 28 * q^41 - 10 * q^44 + 48 * q^46 - 22 * q^49 - 4 * q^51 + 2 * q^54 - 8 * q^56 - 24 * q^59 - 12 * q^61 - 26 * q^64 - 2 * q^66 + 16 * q^71 - 4 * q^74 + 96 * q^76 - 24 * q^79 + 6 * q^81 + 16 * q^84 - 16 * q^86 + 20 * q^89 + 16 * q^91 + 32 * q^94 - 58 * q^96 - 6 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/825\mathbb{Z}\right)^\times$$.

 $$n$$ $$376$$ $$551$$ $$727$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ − 2.70928i − 1.91575i −0.287190 0.957873i $$-0.592721\pi$$
0.287190 0.957873i $$-0.407279\pi$$
$$3$$ − 1.00000i − 0.577350i
$$4$$ −5.34017 −2.67009
$$5$$ 0 0
$$6$$ −2.70928 −1.10606
$$7$$ 1.07838i 0.407588i 0.979014 + 0.203794i $$0.0653274\pi$$
−0.979014 + 0.203794i $$0.934673\pi$$
$$8$$ 9.04945i 3.19946i
$$9$$ −1.00000 −0.333333
$$10$$ 0 0
$$11$$ 1.00000 0.301511
$$12$$ 5.34017i 1.54158i
$$13$$ 4.34017i 1.20375i 0.798591 + 0.601874i $$0.205579\pi$$
−0.798591 + 0.601874i $$0.794421\pi$$
$$14$$ 2.92162 0.780836
$$15$$ 0 0
$$16$$ 13.8371 3.45928
$$17$$ 7.75872i 1.88177i 0.338730 + 0.940883i $$0.390003\pi$$
−0.338730 + 0.940883i $$0.609997\pi$$
$$18$$ 2.70928i 0.638582i
$$19$$ −5.26180 −1.20714 −0.603569 0.797311i $$-0.706255\pi$$
−0.603569 + 0.797311i $$0.706255\pi$$
$$20$$ 0 0
$$21$$ 1.07838 0.235321
$$22$$ − 2.70928i − 0.577619i
$$23$$ 2.15676i 0.449715i 0.974392 + 0.224857i $$0.0721916\pi$$
−0.974392 + 0.224857i $$0.927808\pi$$
$$24$$ 9.04945 1.84721
$$25$$ 0 0
$$26$$ 11.7587 2.30608
$$27$$ 1.00000i 0.192450i
$$28$$ − 5.75872i − 1.08830i
$$29$$ −1.41855 −0.263418 −0.131709 0.991288i $$-0.542046\pi$$
−0.131709 + 0.991288i $$0.542046\pi$$
$$30$$ 0 0
$$31$$ −4.68035 −0.840615 −0.420307 0.907382i $$-0.638078\pi$$
−0.420307 + 0.907382i $$0.638078\pi$$
$$32$$ − 19.3896i − 3.42763i
$$33$$ − 1.00000i − 0.174078i
$$34$$ 21.0205 3.60499
$$35$$ 0 0
$$36$$ 5.34017 0.890029
$$37$$ − 2.00000i − 0.328798i −0.986394 0.164399i $$-0.947432\pi$$
0.986394 0.164399i $$-0.0525685\pi$$
$$38$$ 14.2557i 2.31257i
$$39$$ 4.34017 0.694984
$$40$$ 0 0
$$41$$ −9.41855 −1.47093 −0.735465 0.677562i $$-0.763036\pi$$
−0.735465 + 0.677562i $$0.763036\pi$$
$$42$$ − 2.92162i − 0.450816i
$$43$$ − 7.60197i − 1.15929i −0.814869 0.579645i $$-0.803191\pi$$
0.814869 0.579645i $$-0.196809\pi$$
$$44$$ −5.34017 −0.805061
$$45$$ 0 0
$$46$$ 5.84324 0.861539
$$47$$ 4.68035i 0.682699i 0.939937 + 0.341349i $$0.110884\pi$$
−0.939937 + 0.341349i $$0.889116\pi$$
$$48$$ − 13.8371i − 1.99721i
$$49$$ 5.83710 0.833872
$$50$$ 0 0
$$51$$ 7.75872 1.08644
$$52$$ − 23.1773i − 3.21411i
$$53$$ − 0.156755i − 0.0215320i −0.999942 0.0107660i $$-0.996573\pi$$
0.999942 0.0107660i $$-0.00342699\pi$$
$$54$$ 2.70928 0.368686
$$55$$ 0 0
$$56$$ −9.75872 −1.30406
$$57$$ 5.26180i 0.696942i
$$58$$ 3.84324i 0.504643i
$$59$$ −6.15676 −0.801541 −0.400771 0.916178i $$-0.631258\pi$$
−0.400771 + 0.916178i $$0.631258\pi$$
$$60$$ 0 0
$$61$$ −4.15676 −0.532218 −0.266109 0.963943i $$-0.585738\pi$$
−0.266109 + 0.963943i $$0.585738\pi$$
$$62$$ 12.6803i 1.61041i
$$63$$ − 1.07838i − 0.135863i
$$64$$ −24.8576 −3.10720
$$65$$ 0 0
$$66$$ −2.70928 −0.333489
$$67$$ − 8.68035i − 1.06047i −0.847850 0.530237i $$-0.822103\pi$$
0.847850 0.530237i $$-0.177897\pi$$
$$68$$ − 41.4329i − 5.02448i
$$69$$ 2.15676 0.259643
$$70$$ 0 0
$$71$$ −4.68035 −0.555455 −0.277727 0.960660i $$-0.589581\pi$$
−0.277727 + 0.960660i $$0.589581\pi$$
$$72$$ − 9.04945i − 1.06649i
$$73$$ 10.4969i 1.22857i 0.789083 + 0.614286i $$0.210556\pi$$
−0.789083 + 0.614286i $$0.789444\pi$$
$$74$$ −5.41855 −0.629894
$$75$$ 0 0
$$76$$ 28.0989 3.22316
$$77$$ 1.07838i 0.122893i
$$78$$ − 11.7587i − 1.33141i
$$79$$ 8.09890 0.911197 0.455599 0.890185i $$-0.349425\pi$$
0.455599 + 0.890185i $$0.349425\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 25.5174i 2.81793i
$$83$$ 11.0205i 1.20966i 0.796355 + 0.604830i $$0.206759\pi$$
−0.796355 + 0.604830i $$0.793241\pi$$
$$84$$ −5.75872 −0.628328
$$85$$ 0 0
$$86$$ −20.5958 −2.22090
$$87$$ 1.41855i 0.152085i
$$88$$ 9.04945i 0.964674i
$$89$$ 12.8371 1.36073 0.680365 0.732873i $$-0.261821\pi$$
0.680365 + 0.732873i $$0.261821\pi$$
$$90$$ 0 0
$$91$$ −4.68035 −0.490634
$$92$$ − 11.5174i − 1.20078i
$$93$$ 4.68035i 0.485329i
$$94$$ 12.6803 1.30788
$$95$$ 0 0
$$96$$ −19.3896 −1.97894
$$97$$ 14.6803i 1.49056i 0.666750 + 0.745282i $$0.267685\pi$$
−0.666750 + 0.745282i $$0.732315\pi$$
$$98$$ − 15.8143i − 1.59749i
$$99$$ −1.00000 −0.100504
$$100$$ 0 0
$$101$$ −15.5753 −1.54980 −0.774900 0.632083i $$-0.782200\pi$$
−0.774900 + 0.632083i $$0.782200\pi$$
$$102$$ − 21.0205i − 2.08134i
$$103$$ − 6.83710i − 0.673680i −0.941562 0.336840i $$-0.890642\pi$$
0.941562 0.336840i $$-0.109358\pi$$
$$104$$ −39.2762 −3.85135
$$105$$ 0 0
$$106$$ −0.424694 −0.0412499
$$107$$ 6.34017i 0.612928i 0.951882 + 0.306464i $$0.0991458\pi$$
−0.951882 + 0.306464i $$0.900854\pi$$
$$108$$ − 5.34017i − 0.513858i
$$109$$ −2.31351 −0.221594 −0.110797 0.993843i $$-0.535340\pi$$
−0.110797 + 0.993843i $$0.535340\pi$$
$$110$$ 0 0
$$111$$ −2.00000 −0.189832
$$112$$ 14.9216i 1.40996i
$$113$$ 6.00000i 0.564433i 0.959351 + 0.282216i $$0.0910696\pi$$
−0.959351 + 0.282216i $$0.908930\pi$$
$$114$$ 14.2557 1.33516
$$115$$ 0 0
$$116$$ 7.57531 0.703350
$$117$$ − 4.34017i − 0.401249i
$$118$$ 16.6803i 1.53555i
$$119$$ −8.36683 −0.766987
$$120$$ 0 0
$$121$$ 1.00000 0.0909091
$$122$$ 11.2618i 1.01960i
$$123$$ 9.41855i 0.849242i
$$124$$ 24.9939 2.24451
$$125$$ 0 0
$$126$$ −2.92162 −0.260279
$$127$$ − 2.24128i − 0.198881i −0.995044 0.0994406i $$-0.968295\pi$$
0.995044 0.0994406i $$-0.0317053\pi$$
$$128$$ 28.5669i 2.52498i
$$129$$ −7.60197 −0.669316
$$130$$ 0 0
$$131$$ 8.68035 0.758405 0.379203 0.925314i $$-0.376198\pi$$
0.379203 + 0.925314i $$0.376198\pi$$
$$132$$ 5.34017i 0.464802i
$$133$$ − 5.67420i − 0.492016i
$$134$$ −23.5174 −2.03160
$$135$$ 0 0
$$136$$ −70.2122 −6.02064
$$137$$ − 15.3607i − 1.31235i −0.754608 0.656176i $$-0.772173\pi$$
0.754608 0.656176i $$-0.227827\pi$$
$$138$$ − 5.84324i − 0.497410i
$$139$$ −8.58145 −0.727869 −0.363935 0.931425i $$-0.618567\pi$$
−0.363935 + 0.931425i $$0.618567\pi$$
$$140$$ 0 0
$$141$$ 4.68035 0.394156
$$142$$ 12.6803i 1.06411i
$$143$$ 4.34017i 0.362943i
$$144$$ −13.8371 −1.15309
$$145$$ 0 0
$$146$$ 28.4391 2.35363
$$147$$ − 5.83710i − 0.481436i
$$148$$ 10.6803i 0.877919i
$$149$$ 18.0989 1.48272 0.741360 0.671108i $$-0.234181\pi$$
0.741360 + 0.671108i $$0.234181\pi$$
$$150$$ 0 0
$$151$$ 22.9360 1.86651 0.933253 0.359221i $$-0.116958\pi$$
0.933253 + 0.359221i $$0.116958\pi$$
$$152$$ − 47.6163i − 3.86220i
$$153$$ − 7.75872i − 0.627256i
$$154$$ 2.92162 0.235431
$$155$$ 0 0
$$156$$ −23.1773 −1.85567
$$157$$ − 10.9939i − 0.877405i −0.898632 0.438703i $$-0.855438\pi$$
0.898632 0.438703i $$-0.144562\pi$$
$$158$$ − 21.9421i − 1.74562i
$$159$$ −0.156755 −0.0124315
$$160$$ 0 0
$$161$$ −2.32580 −0.183298
$$162$$ − 2.70928i − 0.212861i
$$163$$ 6.52359i 0.510967i 0.966813 + 0.255484i $$0.0822347\pi$$
−0.966813 + 0.255484i $$0.917765\pi$$
$$164$$ 50.2967 3.92751
$$165$$ 0 0
$$166$$ 29.8576 2.31740
$$167$$ 1.97334i 0.152701i 0.997081 + 0.0763507i $$0.0243269\pi$$
−0.997081 + 0.0763507i $$0.975673\pi$$
$$168$$ 9.75872i 0.752902i
$$169$$ −5.83710 −0.449008
$$170$$ 0 0
$$171$$ 5.26180 0.402380
$$172$$ 40.5958i 3.09540i
$$173$$ − 3.75872i − 0.285770i −0.989739 0.142885i $$-0.954362\pi$$
0.989739 0.142885i $$-0.0456380\pi$$
$$174$$ 3.84324 0.291356
$$175$$ 0 0
$$176$$ 13.8371 1.04301
$$177$$ 6.15676i 0.462770i
$$178$$ − 34.7792i − 2.60681i
$$179$$ −15.1506 −1.13241 −0.566205 0.824264i $$-0.691589\pi$$
−0.566205 + 0.824264i $$0.691589\pi$$
$$180$$ 0 0
$$181$$ 4.83710 0.359539 0.179769 0.983709i $$-0.442465\pi$$
0.179769 + 0.983709i $$0.442465\pi$$
$$182$$ 12.6803i 0.939930i
$$183$$ 4.15676i 0.307276i
$$184$$ −19.5174 −1.43885
$$185$$ 0 0
$$186$$ 12.6803 0.929768
$$187$$ 7.75872i 0.567374i
$$188$$ − 24.9939i − 1.82286i
$$189$$ −1.07838 −0.0784404
$$190$$ 0 0
$$191$$ 2.52359 0.182601 0.0913003 0.995823i $$-0.470898\pi$$
0.0913003 + 0.995823i $$0.470898\pi$$
$$192$$ 24.8576i 1.79394i
$$193$$ − 0.0266620i − 0.00191917i −1.00000 0.000959586i $$-0.999695\pi$$
1.00000 0.000959586i $$-0.000305446\pi$$
$$194$$ 39.7731 2.85554
$$195$$ 0 0
$$196$$ −31.1711 −2.22651
$$197$$ 21.1194i 1.50470i 0.658766 + 0.752348i $$0.271079\pi$$
−0.658766 + 0.752348i $$0.728921\pi$$
$$198$$ 2.70928i 0.192540i
$$199$$ −10.5236 −0.745998 −0.372999 0.927832i $$-0.621670\pi$$
−0.372999 + 0.927832i $$0.621670\pi$$
$$200$$ 0 0
$$201$$ −8.68035 −0.612264
$$202$$ 42.1978i 2.96903i
$$203$$ − 1.52973i − 0.107366i
$$204$$ −41.4329 −2.90089
$$205$$ 0 0
$$206$$ −18.5236 −1.29060
$$207$$ − 2.15676i − 0.149905i
$$208$$ 60.0554i 4.16409i
$$209$$ −5.26180 −0.363966
$$210$$ 0 0
$$211$$ 9.57531 0.659191 0.329596 0.944122i $$-0.393088\pi$$
0.329596 + 0.944122i $$0.393088\pi$$
$$212$$ 0.837101i 0.0574924i
$$213$$ 4.68035i 0.320692i
$$214$$ 17.1773 1.17421
$$215$$ 0 0
$$216$$ −9.04945 −0.615737
$$217$$ − 5.04718i − 0.342625i
$$218$$ 6.26794i 0.424518i
$$219$$ 10.4969 0.709317
$$220$$ 0 0
$$221$$ −33.6742 −2.26517
$$222$$ 5.41855i 0.363669i
$$223$$ 2.15676i 0.144427i 0.997389 + 0.0722135i $$0.0230063\pi$$
−0.997389 + 0.0722135i $$0.976994\pi$$
$$224$$ 20.9093 1.39706
$$225$$ 0 0
$$226$$ 16.2557 1.08131
$$227$$ 9.65983i 0.641145i 0.947224 + 0.320573i $$0.103875\pi$$
−0.947224 + 0.320573i $$0.896125\pi$$
$$228$$ − 28.0989i − 1.86089i
$$229$$ 3.36069 0.222081 0.111040 0.993816i $$-0.464582\pi$$
0.111040 + 0.993816i $$0.464582\pi$$
$$230$$ 0 0
$$231$$ 1.07838 0.0709520
$$232$$ − 12.8371i − 0.842797i
$$233$$ 2.39803i 0.157100i 0.996910 + 0.0785501i $$0.0250291\pi$$
−0.996910 + 0.0785501i $$0.974971\pi$$
$$234$$ −11.7587 −0.768692
$$235$$ 0 0
$$236$$ 32.8781 2.14018
$$237$$ − 8.09890i − 0.526080i
$$238$$ 22.6681i 1.46935i
$$239$$ 7.20394 0.465984 0.232992 0.972479i $$-0.425148\pi$$
0.232992 + 0.972479i $$0.425148\pi$$
$$240$$ 0 0
$$241$$ −5.20394 −0.335215 −0.167608 0.985854i $$-0.553604\pi$$
−0.167608 + 0.985854i $$0.553604\pi$$
$$242$$ − 2.70928i − 0.174159i
$$243$$ − 1.00000i − 0.0641500i
$$244$$ 22.1978 1.42107
$$245$$ 0 0
$$246$$ 25.5174 1.62693
$$247$$ − 22.8371i − 1.45309i
$$248$$ − 42.3545i − 2.68952i
$$249$$ 11.0205 0.698397
$$250$$ 0 0
$$251$$ 15.3197 0.966968 0.483484 0.875353i $$-0.339371\pi$$
0.483484 + 0.875353i $$0.339371\pi$$
$$252$$ 5.75872i 0.362765i
$$253$$ 2.15676i 0.135594i
$$254$$ −6.07223 −0.381006
$$255$$ 0 0
$$256$$ 27.6803 1.73002
$$257$$ 4.15676i 0.259291i 0.991560 + 0.129646i $$0.0413840\pi$$
−0.991560 + 0.129646i $$0.958616\pi$$
$$258$$ 20.5958i 1.28224i
$$259$$ 2.15676 0.134014
$$260$$ 0 0
$$261$$ 1.41855 0.0878061
$$262$$ − 23.5174i − 1.45291i
$$263$$ 18.7070i 1.15352i 0.816912 + 0.576762i $$0.195684\pi$$
−0.816912 + 0.576762i $$0.804316\pi$$
$$264$$ 9.04945 0.556955
$$265$$ 0 0
$$266$$ −15.3730 −0.942578
$$267$$ − 12.8371i − 0.785618i
$$268$$ 46.3545i 2.83155i
$$269$$ −23.3607 −1.42433 −0.712163 0.702014i $$-0.752284\pi$$
−0.712163 + 0.702014i $$0.752284\pi$$
$$270$$ 0 0
$$271$$ −5.57531 −0.338676 −0.169338 0.985558i $$-0.554163\pi$$
−0.169338 + 0.985558i $$0.554163\pi$$
$$272$$ 107.358i 6.50955i
$$273$$ 4.68035i 0.283267i
$$274$$ −41.6163 −2.51414
$$275$$ 0 0
$$276$$ −11.5174 −0.693269
$$277$$ − 26.0144i − 1.56305i −0.623872 0.781526i $$-0.714442\pi$$
0.623872 0.781526i $$-0.285558\pi$$
$$278$$ 23.2495i 1.39441i
$$279$$ 4.68035 0.280205
$$280$$ 0 0
$$281$$ −9.41855 −0.561864 −0.280932 0.959728i $$-0.590643\pi$$
−0.280932 + 0.959728i $$0.590643\pi$$
$$282$$ − 12.6803i − 0.755104i
$$283$$ − 14.2413i − 0.846556i −0.906000 0.423278i $$-0.860879\pi$$
0.906000 0.423278i $$-0.139121\pi$$
$$284$$ 24.9939 1.48311
$$285$$ 0 0
$$286$$ 11.7587 0.695308
$$287$$ − 10.1568i − 0.599534i
$$288$$ 19.3896i 1.14254i
$$289$$ −43.1978 −2.54105
$$290$$ 0 0
$$291$$ 14.6803 0.860577
$$292$$ − 56.0554i − 3.28039i
$$293$$ 15.7587i 0.920634i 0.887754 + 0.460317i $$0.152264\pi$$
−0.887754 + 0.460317i $$0.847736\pi$$
$$294$$ −15.8143 −0.922310
$$295$$ 0 0
$$296$$ 18.0989 1.05198
$$297$$ 1.00000i 0.0580259i
$$298$$ − 49.0349i − 2.84052i
$$299$$ −9.36069 −0.541343
$$300$$ 0 0
$$301$$ 8.19779 0.472513
$$302$$ − 62.1399i − 3.57575i
$$303$$ 15.5753i 0.894778i
$$304$$ −72.8080 −4.17582
$$305$$ 0 0
$$306$$ −21.0205 −1.20166
$$307$$ − 18.9216i − 1.07991i −0.841693 0.539957i $$-0.818440\pi$$
0.841693 0.539957i $$-0.181560\pi$$
$$308$$ − 5.75872i − 0.328134i
$$309$$ −6.83710 −0.388949
$$310$$ 0 0
$$311$$ −20.8781 −1.18389 −0.591945 0.805978i $$-0.701640\pi$$
−0.591945 + 0.805978i $$0.701640\pi$$
$$312$$ 39.2762i 2.22358i
$$313$$ − 6.31351i − 0.356861i −0.983953 0.178430i $$-0.942898\pi$$
0.983953 0.178430i $$-0.0571019\pi$$
$$314$$ −29.7854 −1.68089
$$315$$ 0 0
$$316$$ −43.2495 −2.43297
$$317$$ 31.3607i 1.76139i 0.473682 + 0.880696i $$0.342925\pi$$
−0.473682 + 0.880696i $$0.657075\pi$$
$$318$$ 0.424694i 0.0238156i
$$319$$ −1.41855 −0.0794236
$$320$$ 0 0
$$321$$ 6.34017 0.353874
$$322$$ 6.30122i 0.351154i
$$323$$ − 40.8248i − 2.27155i
$$324$$ −5.34017 −0.296676
$$325$$ 0 0
$$326$$ 17.6742 0.978884
$$327$$ 2.31351i 0.127937i
$$328$$ − 85.2327i − 4.70619i
$$329$$ −5.04718 −0.278260
$$330$$ 0 0
$$331$$ 19.2039 1.05554 0.527772 0.849386i $$-0.323028\pi$$
0.527772 + 0.849386i $$0.323028\pi$$
$$332$$ − 58.8515i − 3.22989i
$$333$$ 2.00000i 0.109599i
$$334$$ 5.34632 0.292537
$$335$$ 0 0
$$336$$ 14.9216 0.814041
$$337$$ 13.5031i 0.735559i 0.929913 + 0.367780i $$0.119882\pi$$
−0.929913 + 0.367780i $$0.880118\pi$$
$$338$$ 15.8143i 0.860185i
$$339$$ 6.00000 0.325875
$$340$$ 0 0
$$341$$ −4.68035 −0.253455
$$342$$ − 14.2557i − 0.770857i
$$343$$ 13.8432i 0.747465i
$$344$$ 68.7936 3.70910
$$345$$ 0 0
$$346$$ −10.1834 −0.547464
$$347$$ 6.34017i 0.340358i 0.985413 + 0.170179i $$0.0544346\pi$$
−0.985413 + 0.170179i $$0.945565\pi$$
$$348$$ − 7.57531i − 0.406079i
$$349$$ −16.1568 −0.864851 −0.432426 0.901670i $$-0.642342\pi$$
−0.432426 + 0.901670i $$0.642342\pi$$
$$350$$ 0 0
$$351$$ −4.34017 −0.231661
$$352$$ − 19.3896i − 1.03347i
$$353$$ 13.2039i 0.702775i 0.936230 + 0.351387i $$0.114290\pi$$
−0.936230 + 0.351387i $$0.885710\pi$$
$$354$$ 16.6803 0.886550
$$355$$ 0 0
$$356$$ −68.5523 −3.63327
$$357$$ 8.36683i 0.442820i
$$358$$ 41.0472i 2.16941i
$$359$$ −3.31965 −0.175205 −0.0876023 0.996156i $$-0.527920\pi$$
−0.0876023 + 0.996156i $$0.527920\pi$$
$$360$$ 0 0
$$361$$ 8.68649 0.457184
$$362$$ − 13.1050i − 0.688786i
$$363$$ − 1.00000i − 0.0524864i
$$364$$ 24.9939 1.31003
$$365$$ 0 0
$$366$$ 11.2618 0.588663
$$367$$ − 36.1445i − 1.88673i −0.331762 0.943363i $$-0.607643\pi$$
0.331762 0.943363i $$-0.392357\pi$$
$$368$$ 29.8432i 1.55569i
$$369$$ 9.41855 0.490310
$$370$$ 0 0
$$371$$ 0.169042 0.00877620
$$372$$ − 24.9939i − 1.29587i
$$373$$ 2.81044i 0.145519i 0.997350 + 0.0727595i $$0.0231806\pi$$
−0.997350 + 0.0727595i $$0.976819\pi$$
$$374$$ 21.0205 1.08695
$$375$$ 0 0
$$376$$ −42.3545 −2.18427
$$377$$ − 6.15676i − 0.317089i
$$378$$ 2.92162i 0.150272i
$$379$$ 20.0000 1.02733 0.513665 0.857991i $$-0.328287\pi$$
0.513665 + 0.857991i $$0.328287\pi$$
$$380$$ 0 0
$$381$$ −2.24128 −0.114824
$$382$$ − 6.83710i − 0.349817i
$$383$$ − 33.5585i − 1.71476i −0.514685 0.857379i $$-0.672091\pi$$
0.514685 0.857379i $$-0.327909\pi$$
$$384$$ 28.5669 1.45780
$$385$$ 0 0
$$386$$ −0.0722347 −0.00367665
$$387$$ 7.60197i 0.386430i
$$388$$ − 78.3956i − 3.97993i
$$389$$ −12.8371 −0.650867 −0.325433 0.945565i $$-0.605510\pi$$
−0.325433 + 0.945565i $$0.605510\pi$$
$$390$$ 0 0
$$391$$ −16.7337 −0.846258
$$392$$ 52.8225i 2.66794i
$$393$$ − 8.68035i − 0.437866i
$$394$$ 57.2183 2.88262
$$395$$ 0 0
$$396$$ 5.34017 0.268354
$$397$$ − 5.31965i − 0.266986i −0.991050 0.133493i $$-0.957381\pi$$
0.991050 0.133493i $$-0.0426193\pi$$
$$398$$ 28.5113i 1.42914i
$$399$$ −5.67420 −0.284065
$$400$$ 0 0
$$401$$ 2.00000 0.0998752 0.0499376 0.998752i $$-0.484098\pi$$
0.0499376 + 0.998752i $$0.484098\pi$$
$$402$$ 23.5174i 1.17294i
$$403$$ − 20.3135i − 1.01189i
$$404$$ 83.1748 4.13810
$$405$$ 0 0
$$406$$ −4.14447 −0.205687
$$407$$ − 2.00000i − 0.0991363i
$$408$$ 70.2122i 3.47602i
$$409$$ −26.1978 −1.29540 −0.647699 0.761897i $$-0.724268\pi$$
−0.647699 + 0.761897i $$0.724268\pi$$
$$410$$ 0 0
$$411$$ −15.3607 −0.757687
$$412$$ 36.5113i 1.79878i
$$413$$ − 6.63931i − 0.326699i
$$414$$ −5.84324 −0.287180
$$415$$ 0 0
$$416$$ 84.1543 4.12600
$$417$$ 8.58145i 0.420235i
$$418$$ 14.2557i 0.697267i
$$419$$ 2.83710 0.138601 0.0693007 0.997596i $$-0.477923\pi$$
0.0693007 + 0.997596i $$0.477923\pi$$
$$420$$ 0 0
$$421$$ 11.4764 0.559326 0.279663 0.960098i $$-0.409777\pi$$
0.279663 + 0.960098i $$0.409777\pi$$
$$422$$ − 25.9421i − 1.26284i
$$423$$ − 4.68035i − 0.227566i
$$424$$ 1.41855 0.0688909
$$425$$ 0 0
$$426$$ 12.6803 0.614365
$$427$$ − 4.48255i − 0.216926i
$$428$$ − 33.8576i − 1.63657i
$$429$$ 4.34017 0.209546
$$430$$ 0 0
$$431$$ 23.5708 1.13536 0.567682 0.823248i $$-0.307840\pi$$
0.567682 + 0.823248i $$0.307840\pi$$
$$432$$ 13.8371i 0.665738i
$$433$$ 14.9939i 0.720559i 0.932844 + 0.360279i $$0.117319\pi$$
−0.932844 + 0.360279i $$0.882681\pi$$
$$434$$ −13.6742 −0.656383
$$435$$ 0 0
$$436$$ 12.3545 0.591676
$$437$$ − 11.3484i − 0.542868i
$$438$$ − 28.4391i − 1.35887i
$$439$$ −4.77924 −0.228101 −0.114050 0.993475i $$-0.536383\pi$$
−0.114050 + 0.993475i $$0.536383\pi$$
$$440$$ 0 0
$$441$$ −5.83710 −0.277957
$$442$$ 91.2327i 4.33950i
$$443$$ − 20.1978i − 0.959626i −0.877371 0.479813i $$-0.840704\pi$$
0.877371 0.479813i $$-0.159296\pi$$
$$444$$ 10.6803 0.506867
$$445$$ 0 0
$$446$$ 5.84324 0.276686
$$447$$ − 18.0989i − 0.856048i
$$448$$ − 26.8059i − 1.26646i
$$449$$ 21.5708 1.01799 0.508994 0.860770i $$-0.330018\pi$$
0.508994 + 0.860770i $$0.330018\pi$$
$$450$$ 0 0
$$451$$ −9.41855 −0.443502
$$452$$ − 32.0410i − 1.50708i
$$453$$ − 22.9360i − 1.07763i
$$454$$ 26.1711 1.22827
$$455$$ 0 0
$$456$$ −47.6163 −2.22984
$$457$$ 28.1711i 1.31779i 0.752235 + 0.658895i $$0.228976\pi$$
−0.752235 + 0.658895i $$0.771024\pi$$
$$458$$ − 9.10504i − 0.425451i
$$459$$ −7.75872 −0.362146
$$460$$ 0 0
$$461$$ −1.47187 −0.0685520 −0.0342760 0.999412i $$-0.510913\pi$$
−0.0342760 + 0.999412i $$0.510913\pi$$
$$462$$ − 2.92162i − 0.135926i
$$463$$ 23.2039i 1.07838i 0.842185 + 0.539189i $$0.181269\pi$$
−0.842185 + 0.539189i $$0.818731\pi$$
$$464$$ −19.6286 −0.911236
$$465$$ 0 0
$$466$$ 6.49693 0.300964
$$467$$ 14.1568i 0.655097i 0.944834 + 0.327548i $$0.106222\pi$$
−0.944834 + 0.327548i $$0.893778\pi$$
$$468$$ 23.1773i 1.07137i
$$469$$ 9.36069 0.432237
$$470$$ 0 0
$$471$$ −10.9939 −0.506570
$$472$$ − 55.7152i − 2.56450i
$$473$$ − 7.60197i − 0.349539i
$$474$$ −21.9421 −1.00784
$$475$$ 0 0
$$476$$ 44.6803 2.04792
$$477$$ 0.156755i 0.00717734i
$$478$$ − 19.5174i − 0.892707i
$$479$$ 13.8432 0.632514 0.316257 0.948674i $$-0.397574\pi$$
0.316257 + 0.948674i $$0.397574\pi$$
$$480$$ 0 0
$$481$$ 8.68035 0.395790
$$482$$ 14.0989i 0.642187i
$$483$$ 2.32580i 0.105827i
$$484$$ −5.34017 −0.242735
$$485$$ 0 0
$$486$$ −2.70928 −0.122895
$$487$$ − 40.9939i − 1.85761i −0.370570 0.928804i $$-0.620838\pi$$
0.370570 0.928804i $$-0.379162\pi$$
$$488$$ − 37.6163i − 1.70281i
$$489$$ 6.52359 0.295007
$$490$$ 0 0
$$491$$ 34.8371 1.57218 0.786088 0.618114i $$-0.212103\pi$$
0.786088 + 0.618114i $$0.212103\pi$$
$$492$$ − 50.2967i − 2.26755i
$$493$$ − 11.0061i − 0.495692i
$$494$$ −61.8720 −2.78375
$$495$$ 0 0
$$496$$ −64.7624 −2.90792
$$497$$ − 5.04718i − 0.226397i
$$498$$ − 29.8576i − 1.33795i
$$499$$ −15.1506 −0.678235 −0.339117 0.940744i $$-0.610128\pi$$
−0.339117 + 0.940744i $$0.610128\pi$$
$$500$$ 0 0
$$501$$ 1.97334 0.0881622
$$502$$ − 41.5052i − 1.85247i
$$503$$ 6.65368i 0.296673i 0.988937 + 0.148337i $$0.0473919\pi$$
−0.988937 + 0.148337i $$0.952608\pi$$
$$504$$ 9.75872 0.434688
$$505$$ 0 0
$$506$$ 5.84324 0.259764
$$507$$ 5.83710i 0.259235i
$$508$$ 11.9688i 0.531030i
$$509$$ −41.3484 −1.83274 −0.916368 0.400337i $$-0.868893\pi$$
−0.916368 + 0.400337i $$0.868893\pi$$
$$510$$ 0 0
$$511$$ −11.3197 −0.500752
$$512$$ − 17.8599i − 0.789303i
$$513$$ − 5.26180i − 0.232314i
$$514$$ 11.2618 0.496736
$$515$$ 0 0
$$516$$ 40.5958 1.78713
$$517$$ 4.68035i 0.205841i
$$518$$ − 5.84324i − 0.256737i
$$519$$ −3.75872 −0.164990
$$520$$ 0 0
$$521$$ 7.67420 0.336213 0.168106 0.985769i $$-0.446235\pi$$
0.168106 + 0.985769i $$0.446235\pi$$
$$522$$ − 3.84324i − 0.168214i
$$523$$ − 23.2351i − 1.01600i −0.861357 0.508001i $$-0.830385\pi$$
0.861357 0.508001i $$-0.169615\pi$$
$$524$$ −46.3545 −2.02501
$$525$$ 0 0
$$526$$ 50.6824 2.20986
$$527$$ − 36.3135i − 1.58184i
$$528$$ − 13.8371i − 0.602183i
$$529$$ 18.3484 0.797757
$$530$$ 0 0
$$531$$ 6.15676 0.267180
$$532$$ 30.3012i 1.31372i
$$533$$ − 40.8781i − 1.77063i
$$534$$ −34.7792 −1.50505
$$535$$ 0 0
$$536$$ 78.5523 3.39294
$$537$$ 15.1506i 0.653797i
$$538$$ 63.2905i 2.72865i
$$539$$ 5.83710 0.251422
$$540$$ 0 0
$$541$$ −18.0000 −0.773880 −0.386940 0.922105i $$-0.626468\pi$$
−0.386940 + 0.922105i $$0.626468\pi$$
$$542$$ 15.1050i 0.648817i
$$543$$ − 4.83710i − 0.207580i
$$544$$ 150.439 6.45001
$$545$$ 0 0
$$546$$ 12.6803 0.542669
$$547$$ − 23.0661i − 0.986235i −0.869963 0.493117i $$-0.835857\pi$$
0.869963 0.493117i $$-0.164143\pi$$
$$548$$ 82.0288i 3.50409i
$$549$$ 4.15676 0.177406
$$550$$ 0 0
$$551$$ 7.46412 0.317982
$$552$$ 19.5174i 0.830718i
$$553$$ 8.73367i 0.371393i
$$554$$ −70.4801 −2.99441
$$555$$ 0 0
$$556$$ 45.8264 1.94347
$$557$$ 10.5958i 0.448960i 0.974479 + 0.224480i $$0.0720683\pi$$
−0.974479 + 0.224480i $$0.927932\pi$$
$$558$$ − 12.6803i − 0.536802i
$$559$$ 32.9939 1.39549
$$560$$ 0 0
$$561$$ 7.75872 0.327574
$$562$$ 25.5174i 1.07639i
$$563$$ 36.2122i 1.52616i 0.646303 + 0.763080i $$0.276314\pi$$
−0.646303 + 0.763080i $$0.723686\pi$$
$$564$$ −24.9939 −1.05243
$$565$$ 0 0
$$566$$ −38.5835 −1.62179
$$567$$ 1.07838i 0.0452876i
$$568$$ − 42.3545i − 1.77716i
$$569$$ 27.5753 1.15602 0.578008 0.816031i $$-0.303830\pi$$
0.578008 + 0.816031i $$0.303830\pi$$
$$570$$ 0 0
$$571$$ −27.9299 −1.16883 −0.584414 0.811456i $$-0.698676\pi$$
−0.584414 + 0.811456i $$0.698676\pi$$
$$572$$ − 23.1773i − 0.969091i
$$573$$ − 2.52359i − 0.105425i
$$574$$ −27.5174 −1.14856
$$575$$ 0 0
$$576$$ 24.8576 1.03573
$$577$$ 41.4017i 1.72358i 0.507268 + 0.861788i $$0.330655\pi$$
−0.507268 + 0.861788i $$0.669345\pi$$
$$578$$ 117.035i 4.86800i
$$579$$ −0.0266620 −0.00110803
$$580$$ 0 0
$$581$$ −11.8843 −0.493043
$$582$$ − 39.7731i − 1.64865i
$$583$$ − 0.156755i − 0.00649215i
$$584$$ −94.9914 −3.93077
$$585$$ 0 0
$$586$$ 42.6947 1.76370
$$587$$ 8.48255i 0.350112i 0.984558 + 0.175056i $$0.0560107\pi$$
−0.984558 + 0.175056i $$0.943989\pi$$
$$588$$ 31.1711i 1.28548i
$$589$$ 24.6270 1.01474
$$590$$ 0 0
$$591$$ 21.1194 0.868737
$$592$$ − 27.6742i − 1.13740i
$$593$$ − 7.56093i − 0.310490i −0.987876 0.155245i $$-0.950383\pi$$
0.987876 0.155245i $$-0.0496167\pi$$
$$594$$ 2.70928 0.111163
$$595$$ 0 0
$$596$$ −96.6512 −3.95899
$$597$$ 10.5236i 0.430702i
$$598$$ 25.3607i 1.03708i
$$599$$ −5.67420 −0.231842 −0.115921 0.993258i $$-0.536982\pi$$
−0.115921 + 0.993258i $$0.536982\pi$$
$$600$$ 0 0
$$601$$ −1.31965 −0.0538298 −0.0269149 0.999638i $$-0.508568\pi$$
−0.0269149 + 0.999638i $$0.508568\pi$$
$$602$$ − 22.2101i − 0.905215i
$$603$$ 8.68035i 0.353491i
$$604$$ −122.482 −4.98373
$$605$$ 0 0
$$606$$ 42.1978 1.71417
$$607$$ − 2.24128i − 0.0909706i −0.998965 0.0454853i $$-0.985517\pi$$
0.998965 0.0454853i $$-0.0144834\pi$$
$$608$$ 102.024i 4.13763i
$$609$$ −1.52973 −0.0619879
$$610$$ 0 0
$$611$$ −20.3135 −0.821797
$$612$$ 41.4329i 1.67483i
$$613$$ − 42.8638i − 1.73125i −0.500692 0.865626i $$-0.666921\pi$$
0.500692 0.865626i $$-0.333079\pi$$
$$614$$ −51.2639 −2.06884
$$615$$ 0 0
$$616$$ −9.75872 −0.393190
$$617$$ 11.3607i 0.457364i 0.973501 + 0.228682i $$0.0734416\pi$$
−0.973501 + 0.228682i $$0.926558\pi$$
$$618$$ 18.5236i 0.745128i
$$619$$ 45.1917 1.81641 0.908203 0.418530i $$-0.137455\pi$$
0.908203 + 0.418530i $$0.137455\pi$$
$$620$$ 0 0
$$621$$ −2.15676 −0.0865476
$$622$$ 56.5646i 2.26803i
$$623$$ 13.8432i 0.554618i
$$624$$ 60.0554 2.40414
$$625$$ 0 0
$$626$$ −17.1050 −0.683655
$$627$$ 5.26180i 0.210136i
$$628$$ 58.7091i 2.34275i
$$629$$ 15.5174 0.618721
$$630$$ 0 0
$$631$$ −9.78992 −0.389731 −0.194865 0.980830i $$-0.562427\pi$$
−0.194865 + 0.980830i $$0.562427\pi$$
$$632$$ 73.2905i 2.91534i
$$633$$ − 9.57531i − 0.380584i
$$634$$ 84.9647 3.37438
$$635$$ 0 0
$$636$$ 0.837101 0.0331932
$$637$$ 25.3340i 1.00377i
$$638$$ 3.84324i 0.152156i
$$639$$ 4.68035 0.185152
$$640$$ 0 0
$$641$$ 0.210079 0.00829764 0.00414882 0.999991i $$-0.498679\pi$$
0.00414882 + 0.999991i $$0.498679\pi$$
$$642$$ − 17.1773i − 0.677933i
$$643$$ − 14.5236i − 0.572754i −0.958117 0.286377i $$-0.907549\pi$$
0.958117 0.286377i $$-0.0924511\pi$$
$$644$$ 12.4202 0.489423
$$645$$ 0 0
$$646$$ −110.606 −4.35172
$$647$$ 15.4641i 0.607957i 0.952679 + 0.303979i $$0.0983152\pi$$
−0.952679 + 0.303979i $$0.901685\pi$$
$$648$$ 9.04945i 0.355496i
$$649$$ −6.15676 −0.241674
$$650$$ 0 0
$$651$$ −5.04718 −0.197815
$$652$$ − 34.8371i − 1.36433i
$$653$$ 17.8310i 0.697779i 0.937164 + 0.348890i $$0.113441\pi$$
−0.937164 + 0.348890i $$0.886559\pi$$
$$654$$ 6.26794 0.245096
$$655$$ 0 0
$$656$$ −130.325 −5.08835
$$657$$ − 10.4969i − 0.409524i
$$658$$ 13.6742i 0.533076i
$$659$$ 32.3135 1.25876 0.629378 0.777099i $$-0.283310\pi$$
0.629378 + 0.777099i $$0.283310\pi$$
$$660$$ 0 0
$$661$$ −5.68649 −0.221179 −0.110589 0.993866i $$-0.535274\pi$$
−0.110589 + 0.993866i $$0.535274\pi$$
$$662$$ − 52.0288i − 2.02215i
$$663$$ 33.6742i 1.30780i
$$664$$ −99.7296 −3.87026
$$665$$ 0 0
$$666$$ 5.41855 0.209965
$$667$$ − 3.05947i − 0.118463i
$$668$$ − 10.5380i − 0.407726i
$$669$$ 2.15676 0.0833850
$$670$$ 0 0
$$671$$ −4.15676 −0.160470
$$672$$ − 20.9093i − 0.806595i
$$673$$ 21.0205i 0.810281i 0.914254 + 0.405141i $$0.132777\pi$$
−0.914254 + 0.405141i $$0.867223\pi$$
$$674$$ 36.5835 1.40915
$$675$$ 0 0
$$676$$ 31.1711 1.19889
$$677$$ 36.7526i 1.41252i 0.707954 + 0.706258i $$0.249618\pi$$
−0.707954 + 0.706258i $$0.750382\pi$$
$$678$$ − 16.2557i − 0.624295i
$$679$$ −15.8310 −0.607536
$$680$$ 0 0
$$681$$ 9.65983 0.370165
$$682$$ 12.6803i 0.485556i
$$683$$ 17.3074i 0.662248i 0.943587 + 0.331124i $$0.107428\pi$$
−0.943587 + 0.331124i $$0.892572\pi$$
$$684$$ −28.0989 −1.07439
$$685$$ 0 0
$$686$$ 37.5052 1.43195
$$687$$ − 3.36069i − 0.128218i
$$688$$ − 105.189i − 4.01030i
$$689$$ 0.680346 0.0259191
$$690$$ 0 0
$$691$$ 17.6742 0.672358 0.336179 0.941798i $$-0.390865\pi$$
0.336179 + 0.941798i $$0.390865\pi$$
$$692$$ 20.0722i 0.763032i
$$693$$ − 1.07838i − 0.0409642i
$$694$$ 17.1773 0.652040
$$695$$ 0 0
$$696$$ −12.8371 −0.486589
$$697$$ − 73.0759i − 2.76795i
$$698$$ 43.7731i 1.65684i
$$699$$ 2.39803 0.0907019
$$700$$ 0 0
$$701$$ −17.1050 −0.646048 −0.323024 0.946391i $$-0.604700\pi$$
−0.323024 + 0.946391i $$0.604700\pi$$
$$702$$ 11.7587i 0.443804i
$$703$$ 10.5236i 0.396905i
$$704$$ −24.8576 −0.936857
$$705$$ 0 0
$$706$$ 35.7731 1.34634
$$707$$ − 16.7961i − 0.631681i
$$708$$ − 32.8781i − 1.23564i
$$709$$ −25.1506 −0.944551 −0.472276 0.881451i $$-0.656567\pi$$
−0.472276 + 0.881451i $$0.656567\pi$$
$$710$$ 0 0
$$711$$ −8.09890 −0.303732
$$712$$ 116.169i 4.35361i
$$713$$ − 10.0944i − 0.378037i
$$714$$ 22.6681 0.848331
$$715$$ 0 0
$$716$$ 80.9069 3.02363
$$717$$ − 7.20394i − 0.269036i
$$718$$ 8.99386i 0.335648i
$$719$$ 1.78992 0.0667528 0.0333764 0.999443i $$-0.489374\pi$$
0.0333764 + 0.999443i $$0.489374\pi$$
$$720$$ 0 0
$$721$$ 7.37298 0.274584
$$722$$ − 23.5341i − 0.875848i
$$723$$ 5.20394i 0.193536i
$$724$$ −25.8310 −0.960000
$$725$$ 0 0
$$726$$ −2.70928 −0.100551
$$727$$ 25.9877i 0.963831i 0.876218 + 0.481915i $$0.160059\pi$$
−0.876218 + 0.481915i $$0.839941\pi$$
$$728$$ − 42.3545i − 1.56976i
$$729$$ −1.00000 −0.0370370
$$730$$ 0 0
$$731$$ 58.9816 2.18151
$$732$$ − 22.1978i − 0.820454i
$$733$$ 41.0205i 1.51513i 0.652761 + 0.757564i $$0.273610\pi$$
−0.652761 + 0.757564i $$0.726390\pi$$
$$734$$ −97.9253 −3.61449
$$735$$ 0 0
$$736$$ 41.8187 1.54146
$$737$$ − 8.68035i − 0.319745i
$$738$$ − 25.5174i − 0.939310i
$$739$$ 47.6163 1.75160 0.875798 0.482678i $$-0.160336\pi$$
0.875798 + 0.482678i $$0.160336\pi$$
$$740$$ 0 0
$$741$$ −22.8371 −0.838942
$$742$$ − 0.457980i − 0.0168130i
$$743$$ − 0.550252i − 0.0201868i −0.999949 0.0100934i $$-0.996787\pi$$
0.999949 0.0100934i $$-0.00321288\pi$$
$$744$$ −42.3545 −1.55279
$$745$$ 0 0
$$746$$ 7.61425 0.278778
$$747$$ − 11.0205i − 0.403220i
$$748$$ − 41.4329i − 1.51494i
$$749$$ −6.83710 −0.249822
$$750$$ 0 0
$$751$$ 41.5585 1.51649 0.758245 0.651969i $$-0.226057\pi$$
0.758245 + 0.651969i $$0.226057\pi$$
$$752$$ 64.7624i 2.36164i
$$753$$ − 15.3197i − 0.558279i
$$754$$ −16.6803 −0.607462
$$755$$ 0 0
$$756$$ 5.75872 0.209443
$$757$$ 1.31965i 0.0479636i 0.999712 + 0.0239818i $$0.00763438\pi$$
−0.999712 + 0.0239818i $$0.992366\pi$$
$$758$$ − 54.1855i − 1.96811i
$$759$$ 2.15676 0.0782853
$$760$$ 0 0
$$761$$ −2.21461 −0.0802797 −0.0401399 0.999194i $$-0.512780\pi$$
−0.0401399 + 0.999194i $$0.512780\pi$$
$$762$$ 6.07223i 0.219974i
$$763$$ − 2.49484i − 0.0903192i
$$764$$ −13.4764 −0.487559
$$765$$ 0 0
$$766$$ −90.9192 −3.28504
$$767$$ − 26.7214i − 0.964853i
$$768$$ − 27.6803i − 0.998828i
$$769$$ 14.3668 0.518081 0.259041 0.965866i $$-0.416594\pi$$
0.259041 + 0.965866i $$0.416594\pi$$
$$770$$ 0 0
$$771$$ 4.15676 0.149702
$$772$$ 0.142380i 0.00512436i
$$773$$ − 40.1568i − 1.44434i −0.691717 0.722169i $$-0.743145\pi$$
0.691717 0.722169i $$-0.256855\pi$$
$$774$$ 20.5958 0.740302
$$775$$ 0 0
$$776$$ −132.849 −4.76900
$$777$$ − 2.15676i − 0.0773732i
$$778$$ 34.7792i 1.24690i
$$779$$ 49.5585 1.77562
$$780$$ 0 0
$$781$$ −4.68035 −0.167476
$$782$$ 45.3361i 1.62122i
$$783$$ − 1.41855i − 0.0506949i
$$784$$ 80.7686 2.88459
$$785$$ 0 0
$$786$$ −23.5174 −0.838840
$$787$$ 49.5897i 1.76768i 0.467788 + 0.883841i $$0.345051\pi$$
−0.467788 + 0.883841i $$0.654949\pi$$
$$788$$ − 112.781i − 4.01767i
$$789$$ 18.7070 0.665987
$$790$$ 0 0
$$791$$ −6.47027 −0.230056
$$792$$ − 9.04945i − 0.321558i
$$793$$ − 18.0410i − 0.640656i
$$794$$ −14.4124 −0.511477
$$795$$ 0 0
$$796$$ 56.1978 1.99188
$$797$$ − 46.7091i − 1.65452i −0.561818 0.827261i $$-0.689898\pi$$
0.561818 0.827261i $$-0.310102\pi$$
$$798$$ 15.3730i 0.544198i
$$799$$ −36.3135 −1.28468
$$800$$ 0 0
$$801$$ −12.8371 −0.453577
$$802$$ − 5.41855i − 0.191336i
$$803$$ 10.4969i 0.370429i
$$804$$ 46.3545 1.63480
$$805$$ 0 0
$$806$$ −55.0349 −1.93852
$$807$$ 23.3607i 0.822335i
$$808$$ − 140.948i − 4.95853i
$$809$$ 18.5814 0.653289 0.326644 0.945147i $$-0.394082\pi$$
0.326644 + 0.945147i $$0.394082\pi$$
$$810$$ 0 0
$$811$$ 27.3028 0.958732 0.479366 0.877615i $$-0.340867\pi$$
0.479366 + 0.877615i $$0.340867\pi$$
$$812$$ 8.16904i 0.286677i
$$813$$ 5.57531i 0.195535i
$$814$$ −5.41855 −0.189920
$$815$$ 0 0
$$816$$ 107.358 3.75829
$$817$$ 40.0000i 1.39942i
$$818$$ 70.9770i 2.48165i
$$819$$ 4.68035 0.163545
$$820$$ 0 0
$$821$$ −31.2085 −1.08918 −0.544592 0.838701i $$-0.683315\pi$$
−0.544592 + 0.838701i $$0.683315\pi$$
$$822$$ 41.6163i 1.45154i
$$823$$ 50.1855i 1.74936i 0.484704 + 0.874678i $$0.338927\pi$$
−0.484704 + 0.874678i $$0.661073\pi$$
$$824$$ 61.8720 2.15541
$$825$$ 0 0
$$826$$ −17.9877 −0.625873
$$827$$ 27.3874i 0.952352i 0.879350 + 0.476176i $$0.157977\pi$$
−0.879350 + 0.476176i $$0.842023\pi$$
$$828$$ 11.5174i 0.400259i
$$829$$ 26.1978 0.909887 0.454943 0.890520i $$-0.349659\pi$$
0.454943 + 0.890520i $$0.349659\pi$$
$$830$$ 0 0
$$831$$ −26.0144 −0.902429
$$832$$ − 107.886i − 3.74029i
$$833$$ 45.2885i 1.56915i
$$834$$ 23.2495 0.805065
$$835$$ 0 0
$$836$$ 28.0989 0.971821
$$837$$ − 4.68035i − 0.161776i
$$838$$ − 7.68649i − 0.265525i
$$839$$ −7.20394 −0.248708 −0.124354 0.992238i $$-0.539686\pi$$
−0.124354 + 0.992238i $$0.539686\pi$$
$$840$$ 0 0
$$841$$ −26.9877 −0.930611
$$842$$ − 31.0928i − 1.07153i
$$843$$ 9.41855i 0.324392i
$$844$$ −51.1338 −1.76010
$$845$$ 0 0
$$846$$ −12.6803 −0.435959
$$847$$ 1.07838i 0.0370535i
$$848$$ − 2.16904i − 0.0744852i
$$849$$ −14.2413 −0.488759
$$850$$ 0 0
$$851$$ 4.31351 0.147865
$$852$$ − 24.9939i − 0.856275i
$$853$$ − 39.8043i − 1.36287i −0.731877 0.681437i $$-0.761356\pi$$
0.731877 0.681437i $$-0.238644\pi$$
$$854$$ −12.1445 −0.415575
$$855$$ 0 0
$$856$$ −57.3751 −1.96104
$$857$$ − 36.9504i − 1.26220i −0.775701 0.631100i $$-0.782604\pi$$
0.775701 0.631100i $$-0.217396\pi$$
$$858$$ − 11.7587i − 0.401436i
$$859$$ −57.5052 −1.96205 −0.981025 0.193879i $$-0.937893\pi$$
−0.981025 + 0.193879i $$0.937893\pi$$
$$860$$ 0 0
$$861$$ −10.1568 −0.346141
$$862$$ − 63.8597i − 2.17507i
$$863$$ 1.89657i 0.0645599i 0.999479 + 0.0322800i $$0.0102768\pi$$
−0.999479 + 0.0322800i $$0.989723\pi$$
$$864$$ 19.3896 0.659648
$$865$$ 0 0
$$866$$ 40.6225 1.38041
$$867$$ 43.1978i 1.46707i
$$868$$ 26.9528i 0.914838i
$$869$$ 8.09890 0.274736
$$870$$ 0 0
$$871$$ 37.6742 1.27654
$$872$$ − 20.9360i − 0.708982i
$$873$$ − 14.6803i − 0.496854i
$$874$$ −30.7460 −1.04000
$$875$$ 0 0
$$876$$ −56.0554 −1.89394
$$877$$ 32.5380i 1.09873i 0.835583 + 0.549365i $$0.185130\pi$$
−0.835583 + 0.549365i $$0.814870\pi$$
$$878$$ 12.9483i 0.436983i
$$879$$ 15.7587 0.531529
$$880$$ 0 0
$$881$$ 18.1978 0.613099 0.306550 0.951855i $$-0.400825\pi$$
0.306550 + 0.951855i $$0.400825\pi$$
$$882$$ 15.8143i 0.532496i
$$883$$ − 36.3956i − 1.22481i −0.790545 0.612405i $$-0.790202\pi$$
0.790545 0.612405i $$-0.209798\pi$$
$$884$$ 179.826 6.04821
$$885$$ 0 0
$$886$$ −54.7214 −1.83840
$$887$$ 27.8699i 0.935780i 0.883787 + 0.467890i $$0.154986\pi$$
−0.883787 + 0.467890i $$0.845014\pi$$
$$888$$ − 18.0989i − 0.607359i
$$889$$ 2.41694 0.0810616
$$890$$ 0 0
$$891$$ 1.00000 0.0335013
$$892$$ − 11.5174i − 0.385633i
$$893$$ − 24.6270i − 0.824112i
$$894$$ −49.0349 −1.63997
$$895$$ 0 0
$$896$$ −30.8059 −1.02915
$$897$$ 9.36069i 0.312544i
$$898$$ − 58.4412i − 1.95021i
$$899$$ 6.63931 0.221433
$$900$$ 0 0
$$901$$ 1.21622 0.0405182
$$902$$ 25.5174i 0.849638i
$$903$$ − 8.19779i − 0.272805i
$$904$$ −54.2967 −1.80588
$$905$$ 0 0
$$906$$ −62.1399 −2.06446
$$907$$ − 27.9376i − 0.927653i −0.885926 0.463826i $$-0.846476\pi$$
0.885926 0.463826i $$-0.153524\pi$$
$$908$$ − 51.5851i − 1.71191i
$$909$$ 15.5753 0.516600
$$910$$ 0 0
$$911$$ −11.8843 −0.393744 −0.196872 0.980429i $$-0.563078\pi$$
−0.196872 + 0.980429i $$0.563078\pi$$
$$912$$ 72.8080i 2.41091i
$$913$$ 11.0205i 0.364726i
$$914$$ 76.3234 2.52455
$$915$$ 0 0
$$916$$ −17.9467 −0.592975
$$917$$ 9.36069i 0.309117i
$$918$$ 21.0205i 0.693781i
$$919$$ 45.6041 1.50434 0.752170 0.658970i $$-0.229007\pi$$
0.752170 + 0.658970i $$0.229007\pi$$
$$920$$ 0 0
$$921$$ −18.9216 −0.623489
$$922$$ 3.98771i 0.131328i
$$923$$ − 20.3135i − 0.668627i
$$924$$ −5.75872 −0.189448
$$925$$ 0 0
$$926$$ 62.8659 2.06590
$$927$$ 6.83710i 0.224560i
$$928$$ 27.5052i 0.902901i
$$929$$ 25.1506 0.825165 0.412582 0.910920i $$-0.364627\pi$$
0.412582 + 0.910920i $$0.364627\pi$$
$$930$$ 0 0
$$931$$ −30.7136 −1.00660
$$932$$ − 12.8059i − 0.419471i
$$933$$ 20.8781i 0.683520i
$$934$$ 38.3545 1.25500
$$935$$ 0 0
$$936$$ 39.2762 1.28378
$$937$$ 5.33403i 0.174255i 0.996197 + 0.0871276i $$0.0277688\pi$$
−0.996197 + 0.0871276i $$0.972231\pi$$
$$938$$ − 25.3607i − 0.828056i
$$939$$ −6.31351 −0.206034
$$940$$ 0 0
$$941$$ 56.8203 1.85229 0.926144 0.377170i $$-0.123103\pi$$
0.926144 + 0.377170i $$0.123103\pi$$
$$942$$ 29.7854i 0.970460i
$$943$$ − 20.3135i − 0.661499i
$$944$$ −85.1917 −2.77275
$$945$$ 0 0
$$946$$ −20.5958 −0.669628
$$947$$ − 20.9939i − 0.682209i −0.940025 0.341104i $$-0.889199\pi$$
0.940025 0.341104i $$-0.110801\pi$$
$$948$$ 43.2495i 1.40468i
$$949$$ −45.5585 −1.47889
$$950$$ 0 0
$$951$$ 31.3607 1.01694
$$952$$ − 75.7152i − 2.45395i
$$953$$ 25.2351i 0.817446i 0.912658 + 0.408723i $$0.134026\pi$$
−0.912658 + 0.408723i $$0.865974\pi$$
$$954$$ 0.424694 0.0137500
$$955$$ 0 0
$$956$$ −38.4703 −1.24422
$$957$$ 1.41855i 0.0458552i
$$958$$ − 37.5052i − 1.21174i
$$959$$ 16.5646 0.534900
$$960$$ 0 0
$$961$$ −9.09436 −0.293367
$$962$$ − 23.5174i − 0.758233i
$$963$$ − 6.34017i − 0.204309i
$$964$$ 27.7899 0.895053
$$965$$ 0 0
$$966$$ 6.30122 0.202739
$$967$$ 13.1317i 0.422287i 0.977455 + 0.211144i $$0.0677187\pi$$
−0.977455 + 0.211144i $$0.932281\pi$$
$$968$$ 9.04945i 0.290860i
$$969$$ −40.8248 −1.31148
$$970$$ 0 0
$$971$$ 8.94053 0.286915 0.143458 0.989656i $$-0.454178\pi$$
0.143458 + 0.989656i $$0.454178\pi$$
$$972$$ 5.34017i 0.171286i
$$973$$ − 9.25404i − 0.296671i
$$974$$ −111.064 −3.55871
$$975$$ 0 0
$$976$$ −57.5174 −1.84109
$$977$$ 50.3956i 1.61230i 0.591713 + 0.806149i $$0.298452\pi$$
−0.591713 + 0.806149i $$0.701548\pi$$
$$978$$ − 17.6742i − 0.565159i
$$979$$ 12.8371 0.410276
$$980$$ 0 0
$$981$$ 2.31351 0.0738647
$$982$$ − 94.3833i − 3.01189i
$$983$$ 32.1978i 1.02695i 0.858105 + 0.513475i $$0.171642\pi$$
−0.858105 + 0.513475i $$0.828358\pi$$
$$984$$ −85.2327 −2.71712
$$985$$ 0 0
$$986$$ −29.8187 −0.949620
$$987$$ 5.04718i 0.160654i
$$988$$ 121.954i 3.87988i
$$989$$ 16.3956 0.521349
$$990$$ 0 0
$$991$$ −46.7747 −1.48585 −0.742924 0.669376i $$-0.766562\pi$$
−0.742924 + 0.669376i $$0.766562\pi$$
$$992$$ 90.7501i 2.88132i
$$993$$ − 19.2039i − 0.609419i
$$994$$ −13.6742 −0.433719
$$995$$ 0 0
$$996$$ −58.8515 −1.86478
$$997$$ 38.2122i 1.21019i 0.796153 + 0.605096i $$0.206865\pi$$
−0.796153 + 0.605096i $$0.793135\pi$$
$$998$$ 41.0472i 1.29933i
$$999$$ 2.00000 0.0632772
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 825.2.c.g.199.1 6
3.2 odd 2 2475.2.c.r.199.6 6
5.2 odd 4 825.2.a.k.1.3 3
5.3 odd 4 165.2.a.c.1.1 3
5.4 even 2 inner 825.2.c.g.199.6 6
15.2 even 4 2475.2.a.bb.1.1 3
15.8 even 4 495.2.a.e.1.3 3
15.14 odd 2 2475.2.c.r.199.1 6
20.3 even 4 2640.2.a.be.1.2 3
35.13 even 4 8085.2.a.bk.1.1 3
55.32 even 4 9075.2.a.cf.1.1 3
55.43 even 4 1815.2.a.m.1.3 3
60.23 odd 4 7920.2.a.cj.1.2 3
165.98 odd 4 5445.2.a.z.1.1 3

By twisted newform
Twist Min Dim Char Parity Ord Type
165.2.a.c.1.1 3 5.3 odd 4
495.2.a.e.1.3 3 15.8 even 4
825.2.a.k.1.3 3 5.2 odd 4
825.2.c.g.199.1 6 1.1 even 1 trivial
825.2.c.g.199.6 6 5.4 even 2 inner
1815.2.a.m.1.3 3 55.43 even 4
2475.2.a.bb.1.1 3 15.2 even 4
2475.2.c.r.199.1 6 15.14 odd 2
2475.2.c.r.199.6 6 3.2 odd 2
2640.2.a.be.1.2 3 20.3 even 4
5445.2.a.z.1.1 3 165.98 odd 4
7920.2.a.cj.1.2 3 60.23 odd 4
8085.2.a.bk.1.1 3 35.13 even 4
9075.2.a.cf.1.1 3 55.32 even 4