# Properties

 Label 825.2.c.d.199.1 Level $825$ Weight $2$ Character 825.199 Analytic conductor $6.588$ Analytic rank $0$ Dimension $4$ 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: $$4$$ Coefficient field: $$\Q(\zeta_{8})$$ Defining polynomial: $$x^{4} + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 199.1 Root $$0.707107 - 0.707107i$$ of defining polynomial Character $$\chi$$ $$=$$ 825.199 Dual form 825.2.c.d.199.4

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-2.41421i q^{2} +1.00000i q^{3} -3.82843 q^{4} +2.41421 q^{6} -0.414214i q^{7} +4.41421i q^{8} -1.00000 q^{9} +O(q^{10})$$ $$q-2.41421i q^{2} +1.00000i q^{3} -3.82843 q^{4} +2.41421 q^{6} -0.414214i q^{7} +4.41421i q^{8} -1.00000 q^{9} -1.00000 q^{11} -3.82843i q^{12} -2.82843i q^{13} -1.00000 q^{14} +3.00000 q^{16} -2.41421i q^{17} +2.41421i q^{18} -6.41421 q^{19} +0.414214 q^{21} +2.41421i q^{22} +1.00000i q^{23} -4.41421 q^{24} -6.82843 q^{26} -1.00000i q^{27} +1.58579i q^{28} -1.17157 q^{29} -8.48528 q^{31} +1.58579i q^{32} -1.00000i q^{33} -5.82843 q^{34} +3.82843 q^{36} +0.171573i q^{37} +15.4853i q^{38} +2.82843 q^{39} -10.8995 q^{41} -1.00000i q^{42} -11.6569i q^{43} +3.82843 q^{44} +2.41421 q^{46} +7.48528i q^{47} +3.00000i q^{48} +6.82843 q^{49} +2.41421 q^{51} +10.8284i q^{52} +7.65685i q^{53} -2.41421 q^{54} +1.82843 q^{56} -6.41421i q^{57} +2.82843i q^{58} -11.0000 q^{59} +8.82843 q^{61} +20.4853i q^{62} +0.414214i q^{63} +9.82843 q^{64} -2.41421 q^{66} +0.343146i q^{67} +9.24264i q^{68} -1.00000 q^{69} +7.82843 q^{71} -4.41421i q^{72} -8.82843i q^{73} +0.414214 q^{74} +24.5563 q^{76} +0.414214i q^{77} -6.82843i q^{78} -13.2426 q^{79} +1.00000 q^{81} +26.3137i q^{82} -4.48528i q^{83} -1.58579 q^{84} -28.1421 q^{86} -1.17157i q^{87} -4.41421i q^{88} -3.65685 q^{89} -1.17157 q^{91} -3.82843i q^{92} -8.48528i q^{93} +18.0711 q^{94} -1.58579 q^{96} -5.82843i q^{97} -16.4853i q^{98} +1.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{4} + 4 q^{6} - 4 q^{9} + O(q^{10})$$ $$4 q - 4 q^{4} + 4 q^{6} - 4 q^{9} - 4 q^{11} - 4 q^{14} + 12 q^{16} - 20 q^{19} - 4 q^{21} - 12 q^{24} - 16 q^{26} - 16 q^{29} - 12 q^{34} + 4 q^{36} - 4 q^{41} + 4 q^{44} + 4 q^{46} + 16 q^{49} + 4 q^{51} - 4 q^{54} - 4 q^{56} - 44 q^{59} + 24 q^{61} + 28 q^{64} - 4 q^{66} - 4 q^{69} + 20 q^{71} - 4 q^{74} + 36 q^{76} - 36 q^{79} + 4 q^{81} - 12 q^{84} - 56 q^{86} + 8 q^{89} - 16 q^{91} + 44 q^{94} - 12 q^{96} + 4 q^{99} + O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ − 2.41421i − 1.70711i −0.521005 0.853553i $$-0.674443\pi$$
0.521005 0.853553i $$-0.325557\pi$$
$$3$$ 1.00000i 0.577350i
$$4$$ −3.82843 −1.91421
$$5$$ 0 0
$$6$$ 2.41421 0.985599
$$7$$ − 0.414214i − 0.156558i −0.996931 0.0782790i $$-0.975058\pi$$
0.996931 0.0782790i $$-0.0249425\pi$$
$$8$$ 4.41421i 1.56066i
$$9$$ −1.00000 −0.333333
$$10$$ 0 0
$$11$$ −1.00000 −0.301511
$$12$$ − 3.82843i − 1.10517i
$$13$$ − 2.82843i − 0.784465i −0.919866 0.392232i $$-0.871703\pi$$
0.919866 0.392232i $$-0.128297\pi$$
$$14$$ −1.00000 −0.267261
$$15$$ 0 0
$$16$$ 3.00000 0.750000
$$17$$ − 2.41421i − 0.585533i −0.956184 0.292766i $$-0.905424\pi$$
0.956184 0.292766i $$-0.0945758\pi$$
$$18$$ 2.41421i 0.569036i
$$19$$ −6.41421 −1.47152 −0.735761 0.677242i $$-0.763175\pi$$
−0.735761 + 0.677242i $$0.763175\pi$$
$$20$$ 0 0
$$21$$ 0.414214 0.0903888
$$22$$ 2.41421i 0.514712i
$$23$$ 1.00000i 0.208514i 0.994550 + 0.104257i $$0.0332465\pi$$
−0.994550 + 0.104257i $$0.966753\pi$$
$$24$$ −4.41421 −0.901048
$$25$$ 0 0
$$26$$ −6.82843 −1.33916
$$27$$ − 1.00000i − 0.192450i
$$28$$ 1.58579i 0.299685i
$$29$$ −1.17157 −0.217556 −0.108778 0.994066i $$-0.534694\pi$$
−0.108778 + 0.994066i $$0.534694\pi$$
$$30$$ 0 0
$$31$$ −8.48528 −1.52400 −0.762001 0.647576i $$-0.775783\pi$$
−0.762001 + 0.647576i $$0.775783\pi$$
$$32$$ 1.58579i 0.280330i
$$33$$ − 1.00000i − 0.174078i
$$34$$ −5.82843 −0.999567
$$35$$ 0 0
$$36$$ 3.82843 0.638071
$$37$$ 0.171573i 0.0282064i 0.999901 + 0.0141032i $$0.00448934\pi$$
−0.999901 + 0.0141032i $$0.995511\pi$$
$$38$$ 15.4853i 2.51204i
$$39$$ 2.82843 0.452911
$$40$$ 0 0
$$41$$ −10.8995 −1.70222 −0.851108 0.524991i $$-0.824069\pi$$
−0.851108 + 0.524991i $$0.824069\pi$$
$$42$$ − 1.00000i − 0.154303i
$$43$$ − 11.6569i − 1.77765i −0.458243 0.888827i $$-0.651521\pi$$
0.458243 0.888827i $$-0.348479\pi$$
$$44$$ 3.82843 0.577157
$$45$$ 0 0
$$46$$ 2.41421 0.355956
$$47$$ 7.48528i 1.09184i 0.837837 + 0.545920i $$0.183820\pi$$
−0.837837 + 0.545920i $$0.816180\pi$$
$$48$$ 3.00000i 0.433013i
$$49$$ 6.82843 0.975490
$$50$$ 0 0
$$51$$ 2.41421 0.338058
$$52$$ 10.8284i 1.50163i
$$53$$ 7.65685i 1.05175i 0.850562 + 0.525875i $$0.176262\pi$$
−0.850562 + 0.525875i $$0.823738\pi$$
$$54$$ −2.41421 −0.328533
$$55$$ 0 0
$$56$$ 1.82843 0.244334
$$57$$ − 6.41421i − 0.849583i
$$58$$ 2.82843i 0.371391i
$$59$$ −11.0000 −1.43208 −0.716039 0.698060i $$-0.754047\pi$$
−0.716039 + 0.698060i $$0.754047\pi$$
$$60$$ 0 0
$$61$$ 8.82843 1.13036 0.565182 0.824966i $$-0.308806\pi$$
0.565182 + 0.824966i $$0.308806\pi$$
$$62$$ 20.4853i 2.60163i
$$63$$ 0.414214i 0.0521860i
$$64$$ 9.82843 1.22855
$$65$$ 0 0
$$66$$ −2.41421 −0.297169
$$67$$ 0.343146i 0.0419219i 0.999780 + 0.0209610i $$0.00667257\pi$$
−0.999780 + 0.0209610i $$0.993327\pi$$
$$68$$ 9.24264i 1.12083i
$$69$$ −1.00000 −0.120386
$$70$$ 0 0
$$71$$ 7.82843 0.929063 0.464532 0.885556i $$-0.346223\pi$$
0.464532 + 0.885556i $$0.346223\pi$$
$$72$$ − 4.41421i − 0.520220i
$$73$$ − 8.82843i − 1.03329i −0.856200 0.516645i $$-0.827181\pi$$
0.856200 0.516645i $$-0.172819\pi$$
$$74$$ 0.414214 0.0481513
$$75$$ 0 0
$$76$$ 24.5563 2.81681
$$77$$ 0.414214i 0.0472040i
$$78$$ − 6.82843i − 0.773167i
$$79$$ −13.2426 −1.48991 −0.744957 0.667113i $$-0.767530\pi$$
−0.744957 + 0.667113i $$0.767530\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 26.3137i 2.90586i
$$83$$ − 4.48528i − 0.492324i −0.969229 0.246162i $$-0.920831\pi$$
0.969229 0.246162i $$-0.0791695\pi$$
$$84$$ −1.58579 −0.173023
$$85$$ 0 0
$$86$$ −28.1421 −3.03464
$$87$$ − 1.17157i − 0.125606i
$$88$$ − 4.41421i − 0.470557i
$$89$$ −3.65685 −0.387626 −0.193813 0.981039i $$-0.562085\pi$$
−0.193813 + 0.981039i $$0.562085\pi$$
$$90$$ 0 0
$$91$$ −1.17157 −0.122814
$$92$$ − 3.82843i − 0.399141i
$$93$$ − 8.48528i − 0.879883i
$$94$$ 18.0711 1.86389
$$95$$ 0 0
$$96$$ −1.58579 −0.161849
$$97$$ − 5.82843i − 0.591787i −0.955221 0.295894i $$-0.904383\pi$$
0.955221 0.295894i $$-0.0956174\pi$$
$$98$$ − 16.4853i − 1.66526i
$$99$$ 1.00000 0.100504
$$100$$ 0 0
$$101$$ 14.8995 1.48256 0.741278 0.671199i $$-0.234220\pi$$
0.741278 + 0.671199i $$0.234220\pi$$
$$102$$ − 5.82843i − 0.577100i
$$103$$ − 13.6569i − 1.34565i −0.739802 0.672825i $$-0.765081\pi$$
0.739802 0.672825i $$-0.234919\pi$$
$$104$$ 12.4853 1.22428
$$105$$ 0 0
$$106$$ 18.4853 1.79545
$$107$$ 5.31371i 0.513696i 0.966452 + 0.256848i $$0.0826839\pi$$
−0.966452 + 0.256848i $$0.917316\pi$$
$$108$$ 3.82843i 0.368391i
$$109$$ −5.31371 −0.508961 −0.254480 0.967078i $$-0.581904\pi$$
−0.254480 + 0.967078i $$0.581904\pi$$
$$110$$ 0 0
$$111$$ −0.171573 −0.0162850
$$112$$ − 1.24264i − 0.117419i
$$113$$ − 10.0000i − 0.940721i −0.882474 0.470360i $$-0.844124\pi$$
0.882474 0.470360i $$-0.155876\pi$$
$$114$$ −15.4853 −1.45033
$$115$$ 0 0
$$116$$ 4.48528 0.416448
$$117$$ 2.82843i 0.261488i
$$118$$ 26.5563i 2.44471i
$$119$$ −1.00000 −0.0916698
$$120$$ 0 0
$$121$$ 1.00000 0.0909091
$$122$$ − 21.3137i − 1.92965i
$$123$$ − 10.8995i − 0.982774i
$$124$$ 32.4853 2.91726
$$125$$ 0 0
$$126$$ 1.00000 0.0890871
$$127$$ − 7.24264i − 0.642680i −0.946964 0.321340i $$-0.895867\pi$$
0.946964 0.321340i $$-0.104133\pi$$
$$128$$ − 20.5563i − 1.81694i
$$129$$ 11.6569 1.02633
$$130$$ 0 0
$$131$$ 6.82843 0.596602 0.298301 0.954472i $$-0.403580\pi$$
0.298301 + 0.954472i $$0.403580\pi$$
$$132$$ 3.82843i 0.333222i
$$133$$ 2.65685i 0.230378i
$$134$$ 0.828427 0.0715652
$$135$$ 0 0
$$136$$ 10.6569 0.913818
$$137$$ 12.1421i 1.03737i 0.854965 + 0.518686i $$0.173579\pi$$
−0.854965 + 0.518686i $$0.826421\pi$$
$$138$$ 2.41421i 0.205512i
$$139$$ 18.9706 1.60906 0.804531 0.593911i $$-0.202417\pi$$
0.804531 + 0.593911i $$0.202417\pi$$
$$140$$ 0 0
$$141$$ −7.48528 −0.630374
$$142$$ − 18.8995i − 1.58601i
$$143$$ 2.82843i 0.236525i
$$144$$ −3.00000 −0.250000
$$145$$ 0 0
$$146$$ −21.3137 −1.76394
$$147$$ 6.82843i 0.563199i
$$148$$ − 0.656854i − 0.0539931i
$$149$$ −7.72792 −0.633096 −0.316548 0.948576i $$-0.602524\pi$$
−0.316548 + 0.948576i $$0.602524\pi$$
$$150$$ 0 0
$$151$$ 14.0000 1.13930 0.569652 0.821886i $$-0.307078\pi$$
0.569652 + 0.821886i $$0.307078\pi$$
$$152$$ − 28.3137i − 2.29655i
$$153$$ 2.41421i 0.195178i
$$154$$ 1.00000 0.0805823
$$155$$ 0 0
$$156$$ −10.8284 −0.866968
$$157$$ − 6.00000i − 0.478852i −0.970915 0.239426i $$-0.923041\pi$$
0.970915 0.239426i $$-0.0769593\pi$$
$$158$$ 31.9706i 2.54344i
$$159$$ −7.65685 −0.607228
$$160$$ 0 0
$$161$$ 0.414214 0.0326446
$$162$$ − 2.41421i − 0.189679i
$$163$$ 15.7990i 1.23747i 0.785599 + 0.618736i $$0.212355\pi$$
−0.785599 + 0.618736i $$0.787645\pi$$
$$164$$ 41.7279 3.25840
$$165$$ 0 0
$$166$$ −10.8284 −0.840449
$$167$$ − 21.7990i − 1.68686i −0.537242 0.843428i $$-0.680534\pi$$
0.537242 0.843428i $$-0.319466\pi$$
$$168$$ 1.82843i 0.141066i
$$169$$ 5.00000 0.384615
$$170$$ 0 0
$$171$$ 6.41421 0.490507
$$172$$ 44.6274i 3.40281i
$$173$$ − 12.5563i − 0.954642i −0.878729 0.477321i $$-0.841608\pi$$
0.878729 0.477321i $$-0.158392\pi$$
$$174$$ −2.82843 −0.214423
$$175$$ 0 0
$$176$$ −3.00000 −0.226134
$$177$$ − 11.0000i − 0.826811i
$$178$$ 8.82843i 0.661719i
$$179$$ −16.7990 −1.25562 −0.627808 0.778368i $$-0.716048\pi$$
−0.627808 + 0.778368i $$0.716048\pi$$
$$180$$ 0 0
$$181$$ −21.9706 −1.63306 −0.816530 0.577304i $$-0.804105\pi$$
−0.816530 + 0.577304i $$0.804105\pi$$
$$182$$ 2.82843i 0.209657i
$$183$$ 8.82843i 0.652616i
$$184$$ −4.41421 −0.325420
$$185$$ 0 0
$$186$$ −20.4853 −1.50205
$$187$$ 2.41421i 0.176545i
$$188$$ − 28.6569i − 2.09002i
$$189$$ −0.414214 −0.0301296
$$190$$ 0 0
$$191$$ 6.17157 0.446559 0.223280 0.974754i $$-0.428324\pi$$
0.223280 + 0.974754i $$0.428324\pi$$
$$192$$ 9.82843i 0.709306i
$$193$$ 3.31371i 0.238526i 0.992863 + 0.119263i $$0.0380532\pi$$
−0.992863 + 0.119263i $$0.961947\pi$$
$$194$$ −14.0711 −1.01024
$$195$$ 0 0
$$196$$ −26.1421 −1.86730
$$197$$ − 4.75736i − 0.338948i −0.985535 0.169474i $$-0.945793\pi$$
0.985535 0.169474i $$-0.0542068\pi$$
$$198$$ − 2.41421i − 0.171571i
$$199$$ −10.8284 −0.767607 −0.383803 0.923415i $$-0.625386\pi$$
−0.383803 + 0.923415i $$0.625386\pi$$
$$200$$ 0 0
$$201$$ −0.343146 −0.0242036
$$202$$ − 35.9706i − 2.53088i
$$203$$ 0.485281i 0.0340601i
$$204$$ −9.24264 −0.647114
$$205$$ 0 0
$$206$$ −32.9706 −2.29717
$$207$$ − 1.00000i − 0.0695048i
$$208$$ − 8.48528i − 0.588348i
$$209$$ 6.41421 0.443680
$$210$$ 0 0
$$211$$ −13.3137 −0.916553 −0.458277 0.888810i $$-0.651533\pi$$
−0.458277 + 0.888810i $$0.651533\pi$$
$$212$$ − 29.3137i − 2.01327i
$$213$$ 7.82843i 0.536395i
$$214$$ 12.8284 0.876933
$$215$$ 0 0
$$216$$ 4.41421 0.300349
$$217$$ 3.51472i 0.238595i
$$218$$ 12.8284i 0.868851i
$$219$$ 8.82843 0.596570
$$220$$ 0 0
$$221$$ −6.82843 −0.459330
$$222$$ 0.414214i 0.0278002i
$$223$$ − 21.1716i − 1.41775i −0.705332 0.708877i $$-0.749202\pi$$
0.705332 0.708877i $$-0.250798\pi$$
$$224$$ 0.656854 0.0438879
$$225$$ 0 0
$$226$$ −24.1421 −1.60591
$$227$$ 18.4853i 1.22691i 0.789729 + 0.613456i $$0.210221\pi$$
−0.789729 + 0.613456i $$0.789779\pi$$
$$228$$ 24.5563i 1.62628i
$$229$$ 2.51472 0.166177 0.0830886 0.996542i $$-0.473522\pi$$
0.0830886 + 0.996542i $$0.473522\pi$$
$$230$$ 0 0
$$231$$ −0.414214 −0.0272533
$$232$$ − 5.17157i − 0.339530i
$$233$$ 16.5563i 1.08464i 0.840171 + 0.542321i $$0.182454\pi$$
−0.840171 + 0.542321i $$0.817546\pi$$
$$234$$ 6.82843 0.446388
$$235$$ 0 0
$$236$$ 42.1127 2.74130
$$237$$ − 13.2426i − 0.860202i
$$238$$ 2.41421i 0.156490i
$$239$$ −23.6569 −1.53023 −0.765117 0.643891i $$-0.777319\pi$$
−0.765117 + 0.643891i $$0.777319\pi$$
$$240$$ 0 0
$$241$$ 14.1421 0.910975 0.455488 0.890242i $$-0.349465\pi$$
0.455488 + 0.890242i $$0.349465\pi$$
$$242$$ − 2.41421i − 0.155192i
$$243$$ 1.00000i 0.0641500i
$$244$$ −33.7990 −2.16376
$$245$$ 0 0
$$246$$ −26.3137 −1.67770
$$247$$ 18.1421i 1.15436i
$$248$$ − 37.4558i − 2.37845i
$$249$$ 4.48528 0.284243
$$250$$ 0 0
$$251$$ 8.97056 0.566217 0.283108 0.959088i $$-0.408634\pi$$
0.283108 + 0.959088i $$0.408634\pi$$
$$252$$ − 1.58579i − 0.0998952i
$$253$$ − 1.00000i − 0.0628695i
$$254$$ −17.4853 −1.09712
$$255$$ 0 0
$$256$$ −29.9706 −1.87316
$$257$$ − 9.31371i − 0.580973i −0.956879 0.290487i $$-0.906183\pi$$
0.956879 0.290487i $$-0.0938172\pi$$
$$258$$ − 28.1421i − 1.75205i
$$259$$ 0.0710678 0.00441594
$$260$$ 0 0
$$261$$ 1.17157 0.0725185
$$262$$ − 16.4853i − 1.01846i
$$263$$ 22.9706i 1.41643i 0.705999 + 0.708213i $$0.250498\pi$$
−0.705999 + 0.708213i $$0.749502\pi$$
$$264$$ 4.41421 0.271676
$$265$$ 0 0
$$266$$ 6.41421 0.393281
$$267$$ − 3.65685i − 0.223796i
$$268$$ − 1.31371i − 0.0802475i
$$269$$ 15.7990 0.963281 0.481641 0.876369i $$-0.340041\pi$$
0.481641 + 0.876369i $$0.340041\pi$$
$$270$$ 0 0
$$271$$ 8.89949 0.540606 0.270303 0.962775i $$-0.412876\pi$$
0.270303 + 0.962775i $$0.412876\pi$$
$$272$$ − 7.24264i − 0.439150i
$$273$$ − 1.17157i − 0.0709068i
$$274$$ 29.3137 1.77091
$$275$$ 0 0
$$276$$ 3.82843 0.230444
$$277$$ − 4.82843i − 0.290112i −0.989423 0.145056i $$-0.953664\pi$$
0.989423 0.145056i $$-0.0463362\pi$$
$$278$$ − 45.7990i − 2.74684i
$$279$$ 8.48528 0.508001
$$280$$ 0 0
$$281$$ 32.0711 1.91320 0.956600 0.291405i $$-0.0941227\pi$$
0.956600 + 0.291405i $$0.0941227\pi$$
$$282$$ 18.0711i 1.07612i
$$283$$ − 0.899495i − 0.0534694i −0.999643 0.0267347i $$-0.991489\pi$$
0.999643 0.0267347i $$-0.00851094\pi$$
$$284$$ −29.9706 −1.77843
$$285$$ 0 0
$$286$$ 6.82843 0.403773
$$287$$ 4.51472i 0.266495i
$$288$$ − 1.58579i − 0.0934434i
$$289$$ 11.1716 0.657151
$$290$$ 0 0
$$291$$ 5.82843 0.341668
$$292$$ 33.7990i 1.97794i
$$293$$ 17.5858i 1.02737i 0.857978 + 0.513686i $$0.171720\pi$$
−0.857978 + 0.513686i $$0.828280\pi$$
$$294$$ 16.4853 0.961441
$$295$$ 0 0
$$296$$ −0.757359 −0.0440206
$$297$$ 1.00000i 0.0580259i
$$298$$ 18.6569i 1.08076i
$$299$$ 2.82843 0.163572
$$300$$ 0 0
$$301$$ −4.82843 −0.278306
$$302$$ − 33.7990i − 1.94491i
$$303$$ 14.8995i 0.855954i
$$304$$ −19.2426 −1.10364
$$305$$ 0 0
$$306$$ 5.82843 0.333189
$$307$$ − 6.68629i − 0.381607i −0.981628 0.190803i $$-0.938891\pi$$
0.981628 0.190803i $$-0.0611093\pi$$
$$308$$ − 1.58579i − 0.0903586i
$$309$$ 13.6569 0.776911
$$310$$ 0 0
$$311$$ 13.6569 0.774409 0.387205 0.921994i $$-0.373441\pi$$
0.387205 + 0.921994i $$0.373441\pi$$
$$312$$ 12.4853i 0.706840i
$$313$$ − 27.1421i − 1.53416i −0.641549 0.767082i $$-0.721708\pi$$
0.641549 0.767082i $$-0.278292\pi$$
$$314$$ −14.4853 −0.817452
$$315$$ 0 0
$$316$$ 50.6985 2.85201
$$317$$ − 30.8284i − 1.73150i −0.500479 0.865748i $$-0.666843\pi$$
0.500479 0.865748i $$-0.333157\pi$$
$$318$$ 18.4853i 1.03660i
$$319$$ 1.17157 0.0655955
$$320$$ 0 0
$$321$$ −5.31371 −0.296582
$$322$$ − 1.00000i − 0.0557278i
$$323$$ 15.4853i 0.861624i
$$324$$ −3.82843 −0.212690
$$325$$ 0 0
$$326$$ 38.1421 2.11250
$$327$$ − 5.31371i − 0.293849i
$$328$$ − 48.1127i − 2.65658i
$$329$$ 3.10051 0.170936
$$330$$ 0 0
$$331$$ −32.1421 −1.76669 −0.883346 0.468722i $$-0.844715\pi$$
−0.883346 + 0.468722i $$0.844715\pi$$
$$332$$ 17.1716i 0.942412i
$$333$$ − 0.171573i − 0.00940214i
$$334$$ −52.6274 −2.87964
$$335$$ 0 0
$$336$$ 1.24264 0.0677916
$$337$$ − 4.14214i − 0.225637i −0.993616 0.112818i $$-0.964012\pi$$
0.993616 0.112818i $$-0.0359878\pi$$
$$338$$ − 12.0711i − 0.656580i
$$339$$ 10.0000 0.543125
$$340$$ 0 0
$$341$$ 8.48528 0.459504
$$342$$ − 15.4853i − 0.837348i
$$343$$ − 5.72792i − 0.309279i
$$344$$ 51.4558 2.77431
$$345$$ 0 0
$$346$$ −30.3137 −1.62968
$$347$$ 21.1716i 1.13655i 0.822839 + 0.568275i $$0.192389\pi$$
−0.822839 + 0.568275i $$0.807611\pi$$
$$348$$ 4.48528i 0.240436i
$$349$$ 2.48528 0.133034 0.0665170 0.997785i $$-0.478811\pi$$
0.0665170 + 0.997785i $$0.478811\pi$$
$$350$$ 0 0
$$351$$ −2.82843 −0.150970
$$352$$ − 1.58579i − 0.0845227i
$$353$$ − 4.48528i − 0.238727i −0.992851 0.119364i $$-0.961915\pi$$
0.992851 0.119364i $$-0.0380855\pi$$
$$354$$ −26.5563 −1.41145
$$355$$ 0 0
$$356$$ 14.0000 0.741999
$$357$$ − 1.00000i − 0.0529256i
$$358$$ 40.5563i 2.14347i
$$359$$ −15.5147 −0.818836 −0.409418 0.912347i $$-0.634268\pi$$
−0.409418 + 0.912347i $$0.634268\pi$$
$$360$$ 0 0
$$361$$ 22.1421 1.16538
$$362$$ 53.0416i 2.78781i
$$363$$ 1.00000i 0.0524864i
$$364$$ 4.48528 0.235093
$$365$$ 0 0
$$366$$ 21.3137 1.11409
$$367$$ − 1.31371i − 0.0685750i −0.999412 0.0342875i $$-0.989084\pi$$
0.999412 0.0342875i $$-0.0109162\pi$$
$$368$$ 3.00000i 0.156386i
$$369$$ 10.8995 0.567405
$$370$$ 0 0
$$371$$ 3.17157 0.164660
$$372$$ 32.4853i 1.68428i
$$373$$ − 23.6569i − 1.22491i −0.790507 0.612453i $$-0.790183\pi$$
0.790507 0.612453i $$-0.209817\pi$$
$$374$$ 5.82843 0.301381
$$375$$ 0 0
$$376$$ −33.0416 −1.70399
$$377$$ 3.31371i 0.170665i
$$378$$ 1.00000i 0.0514344i
$$379$$ −9.17157 −0.471112 −0.235556 0.971861i $$-0.575691\pi$$
−0.235556 + 0.971861i $$0.575691\pi$$
$$380$$ 0 0
$$381$$ 7.24264 0.371052
$$382$$ − 14.8995i − 0.762324i
$$383$$ 20.0000i 1.02195i 0.859595 + 0.510976i $$0.170716\pi$$
−0.859595 + 0.510976i $$0.829284\pi$$
$$384$$ 20.5563 1.04901
$$385$$ 0 0
$$386$$ 8.00000 0.407189
$$387$$ 11.6569i 0.592551i
$$388$$ 22.3137i 1.13281i
$$389$$ 17.6569 0.895238 0.447619 0.894224i $$-0.352272\pi$$
0.447619 + 0.894224i $$0.352272\pi$$
$$390$$ 0 0
$$391$$ 2.41421 0.122092
$$392$$ 30.1421i 1.52241i
$$393$$ 6.82843i 0.344449i
$$394$$ −11.4853 −0.578620
$$395$$ 0 0
$$396$$ −3.82843 −0.192386
$$397$$ 35.9411i 1.80383i 0.431910 + 0.901917i $$0.357840\pi$$
−0.431910 + 0.901917i $$0.642160\pi$$
$$398$$ 26.1421i 1.31039i
$$399$$ −2.65685 −0.133009
$$400$$ 0 0
$$401$$ 31.7990 1.58797 0.793983 0.607940i $$-0.208004\pi$$
0.793983 + 0.607940i $$0.208004\pi$$
$$402$$ 0.828427i 0.0413182i
$$403$$ 24.0000i 1.19553i
$$404$$ −57.0416 −2.83793
$$405$$ 0 0
$$406$$ 1.17157 0.0581442
$$407$$ − 0.171573i − 0.00850455i
$$408$$ 10.6569i 0.527593i
$$409$$ 4.14214 0.204815 0.102408 0.994743i $$-0.467345\pi$$
0.102408 + 0.994743i $$0.467345\pi$$
$$410$$ 0 0
$$411$$ −12.1421 −0.598927
$$412$$ 52.2843i 2.57586i
$$413$$ 4.55635i 0.224203i
$$414$$ −2.41421 −0.118652
$$415$$ 0 0
$$416$$ 4.48528 0.219909
$$417$$ 18.9706i 0.928992i
$$418$$ − 15.4853i − 0.757410i
$$419$$ 25.4853 1.24504 0.622519 0.782605i $$-0.286109\pi$$
0.622519 + 0.782605i $$0.286109\pi$$
$$420$$ 0 0
$$421$$ −27.0000 −1.31590 −0.657950 0.753062i $$-0.728576\pi$$
−0.657950 + 0.753062i $$0.728576\pi$$
$$422$$ 32.1421i 1.56465i
$$423$$ − 7.48528i − 0.363947i
$$424$$ −33.7990 −1.64142
$$425$$ 0 0
$$426$$ 18.8995 0.915684
$$427$$ − 3.65685i − 0.176968i
$$428$$ − 20.3431i − 0.983323i
$$429$$ −2.82843 −0.136558
$$430$$ 0 0
$$431$$ 1.17157 0.0564327 0.0282163 0.999602i $$-0.491017\pi$$
0.0282163 + 0.999602i $$0.491017\pi$$
$$432$$ − 3.00000i − 0.144338i
$$433$$ − 13.3137i − 0.639816i −0.947449 0.319908i $$-0.896348\pi$$
0.947449 0.319908i $$-0.103652\pi$$
$$434$$ 8.48528 0.407307
$$435$$ 0 0
$$436$$ 20.3431 0.974260
$$437$$ − 6.41421i − 0.306833i
$$438$$ − 21.3137i − 1.01841i
$$439$$ −2.27208 −0.108440 −0.0542202 0.998529i $$-0.517267\pi$$
−0.0542202 + 0.998529i $$0.517267\pi$$
$$440$$ 0 0
$$441$$ −6.82843 −0.325163
$$442$$ 16.4853i 0.784125i
$$443$$ − 7.97056i − 0.378693i −0.981910 0.189346i $$-0.939363\pi$$
0.981910 0.189346i $$-0.0606369\pi$$
$$444$$ 0.656854 0.0311729
$$445$$ 0 0
$$446$$ −51.1127 −2.42026
$$447$$ − 7.72792i − 0.365518i
$$448$$ − 4.07107i − 0.192340i
$$449$$ 6.48528 0.306059 0.153030 0.988222i $$-0.451097\pi$$
0.153030 + 0.988222i $$0.451097\pi$$
$$450$$ 0 0
$$451$$ 10.8995 0.513237
$$452$$ 38.2843i 1.80074i
$$453$$ 14.0000i 0.657777i
$$454$$ 44.6274 2.09447
$$455$$ 0 0
$$456$$ 28.3137 1.32591
$$457$$ − 3.85786i − 0.180463i −0.995921 0.0902316i $$-0.971239\pi$$
0.995921 0.0902316i $$-0.0287607\pi$$
$$458$$ − 6.07107i − 0.283682i
$$459$$ −2.41421 −0.112686
$$460$$ 0 0
$$461$$ −32.7696 −1.52623 −0.763115 0.646263i $$-0.776331\pi$$
−0.763115 + 0.646263i $$0.776331\pi$$
$$462$$ 1.00000i 0.0465242i
$$463$$ 34.9706i 1.62522i 0.582808 + 0.812610i $$0.301954\pi$$
−0.582808 + 0.812610i $$0.698046\pi$$
$$464$$ −3.51472 −0.163167
$$465$$ 0 0
$$466$$ 39.9706 1.85160
$$467$$ − 10.6274i − 0.491778i −0.969298 0.245889i $$-0.920920\pi$$
0.969298 0.245889i $$-0.0790799\pi$$
$$468$$ − 10.8284i − 0.500544i
$$469$$ 0.142136 0.00656321
$$470$$ 0 0
$$471$$ 6.00000 0.276465
$$472$$ − 48.5563i − 2.23499i
$$473$$ 11.6569i 0.535983i
$$474$$ −31.9706 −1.46846
$$475$$ 0 0
$$476$$ 3.82843 0.175476
$$477$$ − 7.65685i − 0.350583i
$$478$$ 57.1127i 2.61227i
$$479$$ 24.4853 1.11876 0.559381 0.828911i $$-0.311039\pi$$
0.559381 + 0.828911i $$0.311039\pi$$
$$480$$ 0 0
$$481$$ 0.485281 0.0221269
$$482$$ − 34.1421i − 1.55513i
$$483$$ 0.414214i 0.0188474i
$$484$$ −3.82843 −0.174019
$$485$$ 0 0
$$486$$ 2.41421 0.109511
$$487$$ 6.48528i 0.293876i 0.989146 + 0.146938i $$0.0469418\pi$$
−0.989146 + 0.146938i $$0.953058\pi$$
$$488$$ 38.9706i 1.76411i
$$489$$ −15.7990 −0.714455
$$490$$ 0 0
$$491$$ −24.1421 −1.08952 −0.544760 0.838592i $$-0.683379\pi$$
−0.544760 + 0.838592i $$0.683379\pi$$
$$492$$ 41.7279i 1.88124i
$$493$$ 2.82843i 0.127386i
$$494$$ 43.7990 1.97061
$$495$$ 0 0
$$496$$ −25.4558 −1.14300
$$497$$ − 3.24264i − 0.145452i
$$498$$ − 10.8284i − 0.485233i
$$499$$ −35.1716 −1.57450 −0.787248 0.616637i $$-0.788495\pi$$
−0.787248 + 0.616637i $$0.788495\pi$$
$$500$$ 0 0
$$501$$ 21.7990 0.973907
$$502$$ − 21.6569i − 0.966593i
$$503$$ − 34.2843i − 1.52866i −0.644825 0.764330i $$-0.723070\pi$$
0.644825 0.764330i $$-0.276930\pi$$
$$504$$ −1.82843 −0.0814446
$$505$$ 0 0
$$506$$ −2.41421 −0.107325
$$507$$ 5.00000i 0.222058i
$$508$$ 27.7279i 1.23023i
$$509$$ −4.62742 −0.205107 −0.102553 0.994728i $$-0.532701\pi$$
−0.102553 + 0.994728i $$0.532701\pi$$
$$510$$ 0 0
$$511$$ −3.65685 −0.161770
$$512$$ 31.2426i 1.38074i
$$513$$ 6.41421i 0.283194i
$$514$$ −22.4853 −0.991783
$$515$$ 0 0
$$516$$ −44.6274 −1.96461
$$517$$ − 7.48528i − 0.329202i
$$518$$ − 0.171573i − 0.00753848i
$$519$$ 12.5563 0.551163
$$520$$ 0 0
$$521$$ −36.1421 −1.58342 −0.791708 0.610900i $$-0.790808\pi$$
−0.791708 + 0.610900i $$0.790808\pi$$
$$522$$ − 2.82843i − 0.123797i
$$523$$ 42.2132i 1.84585i 0.384974 + 0.922927i $$0.374210\pi$$
−0.384974 + 0.922927i $$0.625790\pi$$
$$524$$ −26.1421 −1.14202
$$525$$ 0 0
$$526$$ 55.4558 2.41799
$$527$$ 20.4853i 0.892353i
$$528$$ − 3.00000i − 0.130558i
$$529$$ 22.0000 0.956522
$$530$$ 0 0
$$531$$ 11.0000 0.477359
$$532$$ − 10.1716i − 0.440994i
$$533$$ 30.8284i 1.33533i
$$534$$ −8.82843 −0.382043
$$535$$ 0 0
$$536$$ −1.51472 −0.0654259
$$537$$ − 16.7990i − 0.724930i
$$538$$ − 38.1421i − 1.64442i
$$539$$ −6.82843 −0.294121
$$540$$ 0 0
$$541$$ 11.3137 0.486414 0.243207 0.969974i $$-0.421801\pi$$
0.243207 + 0.969974i $$0.421801\pi$$
$$542$$ − 21.4853i − 0.922872i
$$543$$ − 21.9706i − 0.942847i
$$544$$ 3.82843 0.164142
$$545$$ 0 0
$$546$$ −2.82843 −0.121046
$$547$$ − 35.8701i − 1.53369i −0.641831 0.766846i $$-0.721825\pi$$
0.641831 0.766846i $$-0.278175\pi$$
$$548$$ − 46.4853i − 1.98575i
$$549$$ −8.82843 −0.376788
$$550$$ 0 0
$$551$$ 7.51472 0.320138
$$552$$ − 4.41421i − 0.187881i
$$553$$ 5.48528i 0.233258i
$$554$$ −11.6569 −0.495252
$$555$$ 0 0
$$556$$ −72.6274 −3.08009
$$557$$ 5.17157i 0.219127i 0.993980 + 0.109563i $$0.0349452\pi$$
−0.993980 + 0.109563i $$0.965055\pi$$
$$558$$ − 20.4853i − 0.867211i
$$559$$ −32.9706 −1.39451
$$560$$ 0 0
$$561$$ −2.41421 −0.101928
$$562$$ − 77.4264i − 3.26604i
$$563$$ 15.3137i 0.645396i 0.946502 + 0.322698i $$0.104590\pi$$
−0.946502 + 0.322698i $$0.895410\pi$$
$$564$$ 28.6569 1.20667
$$565$$ 0 0
$$566$$ −2.17157 −0.0912780
$$567$$ − 0.414214i − 0.0173953i
$$568$$ 34.5563i 1.44995i
$$569$$ −15.2426 −0.639005 −0.319502 0.947585i $$-0.603516\pi$$
−0.319502 + 0.947585i $$0.603516\pi$$
$$570$$ 0 0
$$571$$ −9.02944 −0.377870 −0.188935 0.981990i $$-0.560504\pi$$
−0.188935 + 0.981990i $$0.560504\pi$$
$$572$$ − 10.8284i − 0.452759i
$$573$$ 6.17157i 0.257821i
$$574$$ 10.8995 0.454936
$$575$$ 0 0
$$576$$ −9.82843 −0.409518
$$577$$ − 23.9706i − 0.997908i −0.866629 0.498954i $$-0.833718\pi$$
0.866629 0.498954i $$-0.166282\pi$$
$$578$$ − 26.9706i − 1.12183i
$$579$$ −3.31371 −0.137713
$$580$$ 0 0
$$581$$ −1.85786 −0.0770772
$$582$$ − 14.0711i − 0.583265i
$$583$$ − 7.65685i − 0.317115i
$$584$$ 38.9706 1.61261
$$585$$ 0 0
$$586$$ 42.4558 1.75383
$$587$$ 36.6569i 1.51299i 0.653999 + 0.756495i $$0.273090\pi$$
−0.653999 + 0.756495i $$0.726910\pi$$
$$588$$ − 26.1421i − 1.07808i
$$589$$ 54.4264 2.24260
$$590$$ 0 0
$$591$$ 4.75736 0.195692
$$592$$ 0.514719i 0.0211548i
$$593$$ 3.79899i 0.156006i 0.996953 + 0.0780029i $$0.0248543\pi$$
−0.996953 + 0.0780029i $$0.975146\pi$$
$$594$$ 2.41421 0.0990564
$$595$$ 0 0
$$596$$ 29.5858 1.21188
$$597$$ − 10.8284i − 0.443178i
$$598$$ − 6.82843i − 0.279235i
$$599$$ −36.3137 −1.48374 −0.741869 0.670545i $$-0.766060\pi$$
−0.741869 + 0.670545i $$0.766060\pi$$
$$600$$ 0 0
$$601$$ −14.8284 −0.604864 −0.302432 0.953171i $$-0.597799\pi$$
−0.302432 + 0.953171i $$0.597799\pi$$
$$602$$ 11.6569i 0.475098i
$$603$$ − 0.343146i − 0.0139740i
$$604$$ −53.5980 −2.18087
$$605$$ 0 0
$$606$$ 35.9706 1.46120
$$607$$ 2.97056i 0.120571i 0.998181 + 0.0602857i $$0.0192012\pi$$
−0.998181 + 0.0602857i $$0.980799\pi$$
$$608$$ − 10.1716i − 0.412512i
$$609$$ −0.485281 −0.0196646
$$610$$ 0 0
$$611$$ 21.1716 0.856510
$$612$$ − 9.24264i − 0.373612i
$$613$$ − 42.0000i − 1.69636i −0.529705 0.848182i $$-0.677697\pi$$
0.529705 0.848182i $$-0.322303\pi$$
$$614$$ −16.1421 −0.651444
$$615$$ 0 0
$$616$$ −1.82843 −0.0736694
$$617$$ 9.85786i 0.396863i 0.980115 + 0.198431i $$0.0635847\pi$$
−0.980115 + 0.198431i $$0.936415\pi$$
$$618$$ − 32.9706i − 1.32627i
$$619$$ −38.6274 −1.55257 −0.776283 0.630384i $$-0.782897\pi$$
−0.776283 + 0.630384i $$0.782897\pi$$
$$620$$ 0 0
$$621$$ 1.00000 0.0401286
$$622$$ − 32.9706i − 1.32200i
$$623$$ 1.51472i 0.0606859i
$$624$$ 8.48528 0.339683
$$625$$ 0 0
$$626$$ −65.5269 −2.61898
$$627$$ 6.41421i 0.256159i
$$628$$ 22.9706i 0.916625i
$$629$$ 0.414214 0.0165158
$$630$$ 0 0
$$631$$ 26.6274 1.06002 0.530010 0.847991i $$-0.322188\pi$$
0.530010 + 0.847991i $$0.322188\pi$$
$$632$$ − 58.4558i − 2.32525i
$$633$$ − 13.3137i − 0.529172i
$$634$$ −74.4264 −2.95585
$$635$$ 0 0
$$636$$ 29.3137 1.16236
$$637$$ − 19.3137i − 0.765237i
$$638$$ − 2.82843i − 0.111979i
$$639$$ −7.82843 −0.309688
$$640$$ 0 0
$$641$$ −42.4853 −1.67807 −0.839034 0.544079i $$-0.816879\pi$$
−0.839034 + 0.544079i $$0.816879\pi$$
$$642$$ 12.8284i 0.506298i
$$643$$ − 32.9706i − 1.30023i −0.759835 0.650116i $$-0.774720\pi$$
0.759835 0.650116i $$-0.225280\pi$$
$$644$$ −1.58579 −0.0624887
$$645$$ 0 0
$$646$$ 37.3848 1.47088
$$647$$ − 17.3431i − 0.681829i −0.940094 0.340915i $$-0.889263\pi$$
0.940094 0.340915i $$-0.110737\pi$$
$$648$$ 4.41421i 0.173407i
$$649$$ 11.0000 0.431788
$$650$$ 0 0
$$651$$ −3.51472 −0.137753
$$652$$ − 60.4853i − 2.36879i
$$653$$ − 40.4853i − 1.58431i −0.610319 0.792156i $$-0.708959\pi$$
0.610319 0.792156i $$-0.291041\pi$$
$$654$$ −12.8284 −0.501631
$$655$$ 0 0
$$656$$ −32.6985 −1.27666
$$657$$ 8.82843i 0.344430i
$$658$$ − 7.48528i − 0.291807i
$$659$$ −15.1127 −0.588707 −0.294354 0.955697i $$-0.595104\pi$$
−0.294354 + 0.955697i $$0.595104\pi$$
$$660$$ 0 0
$$661$$ −34.6569 −1.34800 −0.673998 0.738733i $$-0.735424\pi$$
−0.673998 + 0.738733i $$0.735424\pi$$
$$662$$ 77.5980i 3.01593i
$$663$$ − 6.82843i − 0.265194i
$$664$$ 19.7990 0.768350
$$665$$ 0 0
$$666$$ −0.414214 −0.0160504
$$667$$ − 1.17157i − 0.0453635i
$$668$$ 83.4558i 3.22900i
$$669$$ 21.1716 0.818540
$$670$$ 0 0
$$671$$ −8.82843 −0.340818
$$672$$ 0.656854i 0.0253387i
$$673$$ 11.6569i 0.449339i 0.974435 + 0.224669i $$0.0721302\pi$$
−0.974435 + 0.224669i $$0.927870\pi$$
$$674$$ −10.0000 −0.385186
$$675$$ 0 0
$$676$$ −19.1421 −0.736236
$$677$$ − 40.4853i − 1.55598i −0.628279 0.777988i $$-0.716240\pi$$
0.628279 0.777988i $$-0.283760\pi$$
$$678$$ − 24.1421i − 0.927173i
$$679$$ −2.41421 −0.0926490
$$680$$ 0 0
$$681$$ −18.4853 −0.708358
$$682$$ − 20.4853i − 0.784422i
$$683$$ 29.4853i 1.12822i 0.825699 + 0.564111i $$0.190781\pi$$
−0.825699 + 0.564111i $$0.809219\pi$$
$$684$$ −24.5563 −0.938935
$$685$$ 0 0
$$686$$ −13.8284 −0.527972
$$687$$ 2.51472i 0.0959425i
$$688$$ − 34.9706i − 1.33324i
$$689$$ 21.6569 0.825060
$$690$$ 0 0
$$691$$ 15.4558 0.587968 0.293984 0.955810i $$-0.405019\pi$$
0.293984 + 0.955810i $$0.405019\pi$$
$$692$$ 48.0711i 1.82739i
$$693$$ − 0.414214i − 0.0157347i
$$694$$ 51.1127 1.94021
$$695$$ 0 0
$$696$$ 5.17157 0.196028
$$697$$ 26.3137i 0.996703i
$$698$$ − 6.00000i − 0.227103i
$$699$$ −16.5563 −0.626219
$$700$$ 0 0
$$701$$ −4.61522 −0.174315 −0.0871573 0.996195i $$-0.527778\pi$$
−0.0871573 + 0.996195i $$0.527778\pi$$
$$702$$ 6.82843i 0.257722i
$$703$$ − 1.10051i − 0.0415063i
$$704$$ −9.82843 −0.370423
$$705$$ 0 0
$$706$$ −10.8284 −0.407533
$$707$$ − 6.17157i − 0.232106i
$$708$$ 42.1127i 1.58269i
$$709$$ 0.857864 0.0322178 0.0161089 0.999870i $$-0.494872\pi$$
0.0161089 + 0.999870i $$0.494872\pi$$
$$710$$ 0 0
$$711$$ 13.2426 0.496638
$$712$$ − 16.1421i − 0.604952i
$$713$$ − 8.48528i − 0.317776i
$$714$$ −2.41421 −0.0903497
$$715$$ 0 0
$$716$$ 64.3137 2.40352
$$717$$ − 23.6569i − 0.883481i
$$718$$ 37.4558i 1.39784i
$$719$$ 1.65685 0.0617902 0.0308951 0.999523i $$-0.490164\pi$$
0.0308951 + 0.999523i $$0.490164\pi$$
$$720$$ 0 0
$$721$$ −5.65685 −0.210672
$$722$$ − 53.4558i − 1.98942i
$$723$$ 14.1421i 0.525952i
$$724$$ 84.1127 3.12602
$$725$$ 0 0
$$726$$ 2.41421 0.0895999
$$727$$ 16.9706i 0.629403i 0.949191 + 0.314702i $$0.101904\pi$$
−0.949191 + 0.314702i $$0.898096\pi$$
$$728$$ − 5.17157i − 0.191671i
$$729$$ −1.00000 −0.0370370
$$730$$ 0 0
$$731$$ −28.1421 −1.04087
$$732$$ − 33.7990i − 1.24925i
$$733$$ 3.85786i 0.142493i 0.997459 + 0.0712467i $$0.0226978\pi$$
−0.997459 + 0.0712467i $$0.977302\pi$$
$$734$$ −3.17157 −0.117065
$$735$$ 0 0
$$736$$ −1.58579 −0.0584529
$$737$$ − 0.343146i − 0.0126399i
$$738$$ − 26.3137i − 0.968621i
$$739$$ −17.5858 −0.646904 −0.323452 0.946245i $$-0.604843\pi$$
−0.323452 + 0.946245i $$0.604843\pi$$
$$740$$ 0 0
$$741$$ −18.1421 −0.666468
$$742$$ − 7.65685i − 0.281092i
$$743$$ 31.1127i 1.14141i 0.821154 + 0.570707i $$0.193331\pi$$
−0.821154 + 0.570707i $$0.806669\pi$$
$$744$$ 37.4558 1.37320
$$745$$ 0 0
$$746$$ −57.1127 −2.09104
$$747$$ 4.48528i 0.164108i
$$748$$ − 9.24264i − 0.337944i
$$749$$ 2.20101 0.0804232
$$750$$ 0 0
$$751$$ −47.5980 −1.73687 −0.868437 0.495799i $$-0.834875\pi$$
−0.868437 + 0.495799i $$0.834875\pi$$
$$752$$ 22.4558i 0.818880i
$$753$$ 8.97056i 0.326905i
$$754$$ 8.00000 0.291343
$$755$$ 0 0
$$756$$ 1.58579 0.0576745
$$757$$ 33.3137i 1.21081i 0.795919 + 0.605404i $$0.206988\pi$$
−0.795919 + 0.605404i $$0.793012\pi$$
$$758$$ 22.1421i 0.804239i
$$759$$ 1.00000 0.0362977
$$760$$ 0 0
$$761$$ −10.8284 −0.392530 −0.196265 0.980551i $$-0.562881\pi$$
−0.196265 + 0.980551i $$0.562881\pi$$
$$762$$ − 17.4853i − 0.633425i
$$763$$ 2.20101i 0.0796819i
$$764$$ −23.6274 −0.854810
$$765$$ 0 0
$$766$$ 48.2843 1.74458
$$767$$ 31.1127i 1.12341i
$$768$$ − 29.9706i − 1.08147i
$$769$$ −3.65685 −0.131870 −0.0659348 0.997824i $$-0.521003\pi$$
−0.0659348 + 0.997824i $$0.521003\pi$$
$$770$$ 0 0
$$771$$ 9.31371 0.335425
$$772$$ − 12.6863i − 0.456590i
$$773$$ 4.82843i 0.173666i 0.996223 + 0.0868332i $$0.0276747\pi$$
−0.996223 + 0.0868332i $$0.972325\pi$$
$$774$$ 28.1421 1.01155
$$775$$ 0 0
$$776$$ 25.7279 0.923579
$$777$$ 0.0710678i 0.00254954i
$$778$$ − 42.6274i − 1.52827i
$$779$$ 69.9117 2.50485
$$780$$ 0 0
$$781$$ −7.82843 −0.280123
$$782$$ − 5.82843i − 0.208424i
$$783$$ 1.17157i 0.0418686i
$$784$$ 20.4853 0.731617
$$785$$ 0 0
$$786$$ 16.4853 0.588011
$$787$$ 22.0711i 0.786749i 0.919378 + 0.393374i $$0.128692\pi$$
−0.919378 + 0.393374i $$0.871308\pi$$
$$788$$ 18.2132i 0.648819i
$$789$$ −22.9706 −0.817774
$$790$$ 0 0
$$791$$ −4.14214 −0.147277
$$792$$ 4.41421i 0.156852i
$$793$$ − 24.9706i − 0.886731i
$$794$$ 86.7696 3.07934
$$795$$ 0 0
$$796$$ 41.4558 1.46936
$$797$$ − 7.02944i − 0.248995i −0.992220 0.124498i $$-0.960268\pi$$
0.992220 0.124498i $$-0.0397319\pi$$
$$798$$ 6.41421i 0.227061i
$$799$$ 18.0711 0.639308
$$800$$ 0 0
$$801$$ 3.65685 0.129209
$$802$$ − 76.7696i − 2.71083i
$$803$$ 8.82843i 0.311548i
$$804$$ 1.31371 0.0463309
$$805$$ 0 0
$$806$$ 57.9411 2.04089
$$807$$ 15.7990i 0.556151i
$$808$$ 65.7696i 2.31376i
$$809$$ 17.7279 0.623281 0.311640 0.950200i $$-0.399122\pi$$
0.311640 + 0.950200i $$0.399122\pi$$
$$810$$ 0 0
$$811$$ −32.2132 −1.13116 −0.565579 0.824694i $$-0.691347\pi$$
−0.565579 + 0.824694i $$0.691347\pi$$
$$812$$ − 1.85786i − 0.0651983i
$$813$$ 8.89949i 0.312119i
$$814$$ −0.414214 −0.0145182
$$815$$ 0 0
$$816$$ 7.24264 0.253543
$$817$$ 74.7696i 2.61586i
$$818$$ − 10.0000i − 0.349642i
$$819$$ 1.17157 0.0409381
$$820$$ 0 0
$$821$$ −23.7990 −0.830590 −0.415295 0.909687i $$-0.636322\pi$$
−0.415295 + 0.909687i $$0.636322\pi$$
$$822$$ 29.3137i 1.02243i
$$823$$ 18.9706i 0.661272i 0.943758 + 0.330636i $$0.107263\pi$$
−0.943758 + 0.330636i $$0.892737\pi$$
$$824$$ 60.2843 2.10010
$$825$$ 0 0
$$826$$ 11.0000 0.382739
$$827$$ 35.3137i 1.22798i 0.789315 + 0.613989i $$0.210436\pi$$
−0.789315 + 0.613989i $$0.789564\pi$$
$$828$$ 3.82843i 0.133047i
$$829$$ −19.9411 −0.692584 −0.346292 0.938127i $$-0.612559\pi$$
−0.346292 + 0.938127i $$0.612559\pi$$
$$830$$ 0 0
$$831$$ 4.82843 0.167496
$$832$$ − 27.7990i − 0.963757i
$$833$$ − 16.4853i − 0.571181i
$$834$$ 45.7990 1.58589
$$835$$ 0 0
$$836$$ −24.5563 −0.849299
$$837$$ 8.48528i 0.293294i
$$838$$ − 61.5269i − 2.12541i
$$839$$ 24.6863 0.852265 0.426133 0.904661i $$-0.359876\pi$$
0.426133 + 0.904661i $$0.359876\pi$$
$$840$$ 0 0
$$841$$ −27.6274 −0.952670
$$842$$ 65.1838i 2.24638i
$$843$$ 32.0711i 1.10459i
$$844$$ 50.9706 1.75448
$$845$$ 0 0
$$846$$ −18.0711 −0.621296
$$847$$ − 0.414214i − 0.0142325i
$$848$$ 22.9706i 0.788812i
$$849$$ 0.899495 0.0308706
$$850$$ 0 0
$$851$$ −0.171573 −0.00588144
$$852$$ − 29.9706i − 1.02677i
$$853$$ − 24.8284i − 0.850109i −0.905168 0.425055i $$-0.860255\pi$$
0.905168 0.425055i $$-0.139745\pi$$
$$854$$ −8.82843 −0.302103
$$855$$ 0 0
$$856$$ −23.4558 −0.801704
$$857$$ − 22.6985i − 0.775365i −0.921793 0.387683i $$-0.873276\pi$$
0.921793 0.387683i $$-0.126724\pi$$
$$858$$ 6.82843i 0.233119i
$$859$$ 3.51472 0.119921 0.0599603 0.998201i $$-0.480903\pi$$
0.0599603 + 0.998201i $$0.480903\pi$$
$$860$$ 0 0
$$861$$ −4.51472 −0.153861
$$862$$ − 2.82843i − 0.0963366i
$$863$$ − 51.3137i − 1.74674i −0.487058 0.873369i $$-0.661930\pi$$
0.487058 0.873369i $$-0.338070\pi$$
$$864$$ 1.58579 0.0539496
$$865$$ 0 0
$$866$$ −32.1421 −1.09223
$$867$$ 11.1716i 0.379407i
$$868$$ − 13.4558i − 0.456721i
$$869$$ 13.2426 0.449226
$$870$$ 0 0
$$871$$ 0.970563 0.0328863
$$872$$ − 23.4558i − 0.794315i
$$873$$ 5.82843i 0.197262i
$$874$$ −15.4853 −0.523797
$$875$$ 0 0
$$876$$ −33.7990 −1.14196
$$877$$ 47.1127i 1.59088i 0.606031 + 0.795441i $$0.292761\pi$$
−0.606031 + 0.795441i $$0.707239\pi$$
$$878$$ 5.48528i 0.185119i
$$879$$ −17.5858 −0.593154
$$880$$ 0 0
$$881$$ −24.0000 −0.808581 −0.404290 0.914631i $$-0.632481\pi$$
−0.404290 + 0.914631i $$0.632481\pi$$
$$882$$ 16.4853i 0.555088i
$$883$$ − 6.62742i − 0.223030i −0.993763 0.111515i $$-0.964430\pi$$
0.993763 0.111515i $$-0.0355704\pi$$
$$884$$ 26.1421 0.879255
$$885$$ 0 0
$$886$$ −19.2426 −0.646469
$$887$$ − 6.14214i − 0.206233i −0.994669 0.103116i $$-0.967119\pi$$
0.994669 0.103116i $$-0.0328814\pi$$
$$888$$ − 0.757359i − 0.0254153i
$$889$$ −3.00000 −0.100617
$$890$$ 0 0
$$891$$ −1.00000 −0.0335013
$$892$$ 81.0538i 2.71388i
$$893$$ − 48.0122i − 1.60667i
$$894$$ −18.6569 −0.623979
$$895$$ 0 0
$$896$$ −8.51472 −0.284457
$$897$$ 2.82843i 0.0944384i
$$898$$ − 15.6569i − 0.522476i
$$899$$ 9.94113 0.331555
$$900$$ 0 0
$$901$$ 18.4853 0.615834
$$902$$ − 26.3137i − 0.876151i
$$903$$ − 4.82843i − 0.160680i
$$904$$ 44.1421 1.46815
$$905$$ 0 0
$$906$$ 33.7990 1.12290
$$907$$ − 14.4853i − 0.480976i −0.970652 0.240488i $$-0.922693\pi$$
0.970652 0.240488i $$-0.0773074\pi$$
$$908$$ − 70.7696i − 2.34857i
$$909$$ −14.8995 −0.494185
$$910$$ 0 0
$$911$$ 45.4853 1.50699 0.753497 0.657451i $$-0.228365\pi$$
0.753497 + 0.657451i $$0.228365\pi$$
$$912$$ − 19.2426i − 0.637188i
$$913$$ 4.48528i 0.148441i
$$914$$ −9.31371 −0.308070
$$915$$ 0 0
$$916$$ −9.62742 −0.318099
$$917$$ − 2.82843i − 0.0934029i
$$918$$ 5.82843i 0.192367i
$$919$$ −44.2132 −1.45846 −0.729230 0.684269i $$-0.760121\pi$$
−0.729230 + 0.684269i $$0.760121\pi$$
$$920$$ 0 0
$$921$$ 6.68629 0.220321
$$922$$ 79.1127i 2.60544i
$$923$$ − 22.1421i − 0.728817i
$$924$$ 1.58579 0.0521685
$$925$$ 0 0
$$926$$ 84.4264 2.77442
$$927$$ 13.6569i 0.448550i
$$928$$ − 1.85786i − 0.0609874i
$$929$$ −25.7990 −0.846437 −0.423219 0.906028i $$-0.639100\pi$$
−0.423219 + 0.906028i $$0.639100\pi$$
$$930$$ 0 0
$$931$$ −43.7990 −1.43545
$$932$$ − 63.3848i − 2.07624i
$$933$$ 13.6569i 0.447105i
$$934$$ −25.6569 −0.839518
$$935$$ 0 0
$$936$$ −12.4853 −0.408094
$$937$$ 16.0000i 0.522697i 0.965244 + 0.261349i $$0.0841672\pi$$
−0.965244 + 0.261349i $$0.915833\pi$$
$$938$$ − 0.343146i − 0.0112041i
$$939$$ 27.1421 0.885750
$$940$$ 0 0
$$941$$ −3.10051 −0.101074 −0.0505368 0.998722i $$-0.516093\pi$$
−0.0505368 + 0.998722i $$0.516093\pi$$
$$942$$ − 14.4853i − 0.471956i
$$943$$ − 10.8995i − 0.354936i
$$944$$ −33.0000 −1.07406
$$945$$ 0 0
$$946$$ 28.1421 0.914980
$$947$$ − 2.79899i − 0.0909549i −0.998965 0.0454775i $$-0.985519\pi$$
0.998965 0.0454775i $$-0.0144809\pi$$
$$948$$ 50.6985i 1.64661i
$$949$$ −24.9706 −0.810579
$$950$$ 0 0
$$951$$ 30.8284 0.999680
$$952$$ − 4.41421i − 0.143065i
$$953$$ 19.0416i 0.616819i 0.951254 + 0.308409i $$0.0997967\pi$$
−0.951254 + 0.308409i $$0.900203\pi$$
$$954$$ −18.4853 −0.598483
$$955$$ 0 0
$$956$$ 90.5685 2.92920
$$957$$ 1.17157i 0.0378716i
$$958$$ − 59.1127i − 1.90984i
$$959$$ 5.02944 0.162409
$$960$$ 0 0
$$961$$ 41.0000 1.32258
$$962$$ − 1.17157i − 0.0377730i
$$963$$ − 5.31371i − 0.171232i
$$964$$ −54.1421 −1.74380
$$965$$ 0 0
$$966$$ 1.00000 0.0321745
$$967$$ − 38.0000i − 1.22200i −0.791632 0.610999i $$-0.790768\pi$$
0.791632 0.610999i $$-0.209232\pi$$
$$968$$ 4.41421i 0.141878i
$$969$$ −15.4853 −0.497459
$$970$$ 0 0
$$971$$ −27.6863 −0.888495 −0.444248 0.895904i $$-0.646529\pi$$
−0.444248 + 0.895904i $$0.646529\pi$$
$$972$$ − 3.82843i − 0.122797i
$$973$$ − 7.85786i − 0.251912i
$$974$$ 15.6569 0.501678
$$975$$ 0 0
$$976$$ 26.4853 0.847773
$$977$$ 52.5685i 1.68182i 0.541178 + 0.840908i $$0.317979\pi$$
−0.541178 + 0.840908i $$0.682021\pi$$
$$978$$ 38.1421i 1.21965i
$$979$$ 3.65685 0.116874
$$980$$ 0 0
$$981$$ 5.31371 0.169654
$$982$$ 58.2843i 1.85993i
$$983$$ − 5.28427i − 0.168542i −0.996443 0.0842710i $$-0.973144\pi$$
0.996443 0.0842710i $$-0.0268562\pi$$
$$984$$ 48.1127 1.53378
$$985$$ 0 0
$$986$$ 6.82843 0.217461
$$987$$ 3.10051i 0.0986902i
$$988$$ − 69.4558i − 2.20968i
$$989$$ 11.6569 0.370666
$$990$$ 0 0
$$991$$ 14.2843 0.453755 0.226877 0.973923i $$-0.427148\pi$$
0.226877 + 0.973923i $$0.427148\pi$$
$$992$$ − 13.4558i − 0.427223i
$$993$$ − 32.1421i − 1.02000i
$$994$$ −7.82843 −0.248303
$$995$$ 0 0
$$996$$ −17.1716 −0.544102
$$997$$ − 43.2548i − 1.36989i −0.728593 0.684947i $$-0.759825\pi$$
0.728593 0.684947i $$-0.240175\pi$$
$$998$$ 84.9117i 2.68783i
$$999$$ 0.171573 0.00542833
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.d.199.1 4
3.2 odd 2 2475.2.c.o.199.4 4
5.2 odd 4 825.2.a.f.1.2 yes 2
5.3 odd 4 825.2.a.d.1.1 2
5.4 even 2 inner 825.2.c.d.199.4 4
15.2 even 4 2475.2.a.l.1.1 2
15.8 even 4 2475.2.a.w.1.2 2
15.14 odd 2 2475.2.c.o.199.1 4
55.32 even 4 9075.2.a.w.1.1 2
55.43 even 4 9075.2.a.ca.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
825.2.a.d.1.1 2 5.3 odd 4
825.2.a.f.1.2 yes 2 5.2 odd 4
825.2.c.d.199.1 4 1.1 even 1 trivial
825.2.c.d.199.4 4 5.4 even 2 inner
2475.2.a.l.1.1 2 15.2 even 4
2475.2.a.w.1.2 2 15.8 even 4
2475.2.c.o.199.1 4 15.14 odd 2
2475.2.c.o.199.4 4 3.2 odd 2
9075.2.a.w.1.1 2 55.32 even 4
9075.2.a.ca.1.2 2 55.43 even 4