# Properties

 Label 425.4.b.f.324.6 Level $425$ Weight $4$ Character 425.324 Analytic conductor $25.076$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$425 = 5^{2} \cdot 17$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 425.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$25.0758117524$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: 6.0.27793984.1 Defining polynomial: $$x^{6} - 2x^{3} + 49x^{2} - 14x + 2$$ x^6 - 2*x^3 + 49*x^2 - 14*x + 2 Coefficient ring: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 17) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 324.6 Root $$1.79483 - 1.79483i$$ of defining polynomial Character $$\chi$$ $$=$$ 425.324 Dual form 425.4.b.f.324.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+5.03251i q^{2} +8.47535i q^{3} -17.3261 q^{4} -42.6523 q^{6} -3.81828i q^{7} -46.9339i q^{8} -44.8316 q^{9} +O(q^{10})$$ $$q+5.03251i q^{2} +8.47535i q^{3} -17.3261 q^{4} -42.6523 q^{6} -3.81828i q^{7} -46.9339i q^{8} -44.8316 q^{9} -52.3720 q^{11} -146.845i q^{12} -8.06025i q^{13} +19.2156 q^{14} +97.5862 q^{16} +17.0000i q^{17} -225.616i q^{18} +66.5154 q^{19} +32.3613 q^{21} -263.563i q^{22} +180.226i q^{23} +397.782 q^{24} +40.5633 q^{26} -151.129i q^{27} +66.1562i q^{28} +41.2800 q^{29} -34.9114 q^{31} +115.632i q^{32} -443.871i q^{33} -85.5527 q^{34} +776.759 q^{36} -130.368i q^{37} +334.739i q^{38} +68.3134 q^{39} -17.9081 q^{41} +162.859i q^{42} +277.620i q^{43} +907.405 q^{44} -906.987 q^{46} -463.789i q^{47} +827.078i q^{48} +328.421 q^{49} -144.081 q^{51} +139.653i q^{52} -329.944i q^{53} +760.560 q^{54} -179.207 q^{56} +563.741i q^{57} +207.742i q^{58} -678.656 q^{59} +340.280 q^{61} -175.692i q^{62} +171.180i q^{63} +198.770 q^{64} +2233.79 q^{66} -15.3925i q^{67} -294.545i q^{68} -1527.48 q^{69} -670.203 q^{71} +2104.12i q^{72} +193.480i q^{73} +656.080 q^{74} -1152.46 q^{76} +199.971i q^{77} +343.788i q^{78} -1080.15 q^{79} +70.4207 q^{81} -90.1229i q^{82} -865.668i q^{83} -560.697 q^{84} -1397.13 q^{86} +349.863i q^{87} +2458.02i q^{88} -1129.46 q^{89} -30.7763 q^{91} -3122.61i q^{92} -295.886i q^{93} +2334.02 q^{94} -980.023 q^{96} +379.412i q^{97} +1652.78i q^{98} +2347.92 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 50 q^{4} - 148 q^{6} - 118 q^{9}+O(q^{10})$$ 6 * q - 50 * q^4 - 148 * q^6 - 118 * q^9 $$6 q - 50 q^{4} - 148 q^{6} - 118 q^{9} - 56 q^{11} - 184 q^{14} + 274 q^{16} - 160 q^{19} - 384 q^{21} + 1332 q^{24} + 52 q^{26} + 912 q^{29} + 460 q^{31} + 34 q^{34} + 2626 q^{36} - 536 q^{39} - 588 q^{41} + 2244 q^{44} - 1408 q^{46} + 538 q^{49} - 136 q^{51} + 2200 q^{54} + 1368 q^{56} - 1272 q^{59} - 168 q^{61} + 1838 q^{64} + 4936 q^{66} - 1152 q^{69} - 804 q^{71} - 1672 q^{74} - 1816 q^{76} + 1188 q^{79} - 1010 q^{81} + 4080 q^{84} - 2528 q^{86} + 340 q^{89} - 2032 q^{91} + 4032 q^{94} + 1356 q^{96} + 5840 q^{99}+O(q^{100})$$ 6 * q - 50 * q^4 - 148 * q^6 - 118 * q^9 - 56 * q^11 - 184 * q^14 + 274 * q^16 - 160 * q^19 - 384 * q^21 + 1332 * q^24 + 52 * q^26 + 912 * q^29 + 460 * q^31 + 34 * q^34 + 2626 * q^36 - 536 * q^39 - 588 * q^41 + 2244 * q^44 - 1408 * q^46 + 538 * q^49 - 136 * q^51 + 2200 * q^54 + 1368 * q^56 - 1272 * q^59 - 168 * q^61 + 1838 * q^64 + 4936 * q^66 - 1152 * q^69 - 804 * q^71 - 1672 * q^74 - 1816 * q^76 + 1188 * q^79 - 1010 * q^81 + 4080 * q^84 - 2528 * q^86 + 340 * q^89 - 2032 * q^91 + 4032 * q^94 + 1356 * q^96 + 5840 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/425\mathbb{Z}\right)^\times$$.

 $$n$$ $$52$$ $$326$$ $$\chi(n)$$ $$-1$$ $$1$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 5.03251i 1.77926i 0.456681 + 0.889630i $$0.349038\pi$$
−0.456681 + 0.889630i $$0.650962\pi$$
$$3$$ 8.47535i 1.63108i 0.578699 + 0.815541i $$0.303561\pi$$
−0.578699 + 0.815541i $$0.696439\pi$$
$$4$$ −17.3261 −2.16577
$$5$$ 0 0
$$6$$ −42.6523 −2.90212
$$7$$ − 3.81828i − 0.206168i −0.994673 0.103084i $$-0.967129\pi$$
0.994673 0.103084i $$-0.0328711\pi$$
$$8$$ − 46.9339i − 2.07421i
$$9$$ −44.8316 −1.66043
$$10$$ 0 0
$$11$$ −52.3720 −1.43552 −0.717761 0.696289i $$-0.754833\pi$$
−0.717761 + 0.696289i $$0.754833\pi$$
$$12$$ − 146.845i − 3.53255i
$$13$$ − 8.06025i − 0.171962i −0.996297 0.0859811i $$-0.972598\pi$$
0.996297 0.0859811i $$-0.0274025\pi$$
$$14$$ 19.2156 0.366827
$$15$$ 0 0
$$16$$ 97.5862 1.52478
$$17$$ 17.0000i 0.242536i
$$18$$ − 225.616i − 2.95434i
$$19$$ 66.5154 0.803141 0.401570 0.915828i $$-0.368465\pi$$
0.401570 + 0.915828i $$0.368465\pi$$
$$20$$ 0 0
$$21$$ 32.3613 0.336277
$$22$$ − 263.563i − 2.55417i
$$23$$ 180.226i 1.63390i 0.576711 + 0.816948i $$0.304336\pi$$
−0.576711 + 0.816948i $$0.695664\pi$$
$$24$$ 397.782 3.38320
$$25$$ 0 0
$$26$$ 40.5633 0.305966
$$27$$ − 151.129i − 1.07722i
$$28$$ 66.1562i 0.446512i
$$29$$ 41.2800 0.264328 0.132164 0.991228i $$-0.457807\pi$$
0.132164 + 0.991228i $$0.457807\pi$$
$$30$$ 0 0
$$31$$ −34.9114 −0.202267 −0.101133 0.994873i $$-0.532247\pi$$
−0.101133 + 0.994873i $$0.532247\pi$$
$$32$$ 115.632i 0.638783i
$$33$$ − 443.871i − 2.34146i
$$34$$ −85.5527 −0.431534
$$35$$ 0 0
$$36$$ 776.759 3.59611
$$37$$ − 130.368i − 0.579255i −0.957139 0.289627i $$-0.906469\pi$$
0.957139 0.289627i $$-0.0935314\pi$$
$$38$$ 334.739i 1.42900i
$$39$$ 68.3134 0.280485
$$40$$ 0 0
$$41$$ −17.9081 −0.0682142 −0.0341071 0.999418i $$-0.510859\pi$$
−0.0341071 + 0.999418i $$0.510859\pi$$
$$42$$ 162.859i 0.598325i
$$43$$ 277.620i 0.984573i 0.870433 + 0.492287i $$0.163839\pi$$
−0.870433 + 0.492287i $$0.836161\pi$$
$$44$$ 907.405 3.10901
$$45$$ 0 0
$$46$$ −906.987 −2.90713
$$47$$ − 463.789i − 1.43937i −0.694299 0.719687i $$-0.744285\pi$$
0.694299 0.719687i $$-0.255715\pi$$
$$48$$ 827.078i 2.48705i
$$49$$ 328.421 0.957495
$$50$$ 0 0
$$51$$ −144.081 −0.395596
$$52$$ 139.653i 0.372431i
$$53$$ − 329.944i − 0.855118i −0.903987 0.427559i $$-0.859374\pi$$
0.903987 0.427559i $$-0.140626\pi$$
$$54$$ 760.560 1.91665
$$55$$ 0 0
$$56$$ −179.207 −0.427635
$$57$$ 563.741i 1.30999i
$$58$$ 207.742i 0.470308i
$$59$$ −678.656 −1.49752 −0.748759 0.662843i $$-0.769350\pi$$
−0.748759 + 0.662843i $$0.769350\pi$$
$$60$$ 0 0
$$61$$ 340.280 0.714237 0.357118 0.934059i $$-0.383759\pi$$
0.357118 + 0.934059i $$0.383759\pi$$
$$62$$ − 175.692i − 0.359885i
$$63$$ 171.180i 0.342328i
$$64$$ 198.770 0.388223
$$65$$ 0 0
$$66$$ 2233.79 4.16606
$$67$$ − 15.3925i − 0.0280671i −0.999902 0.0140336i $$-0.995533\pi$$
0.999902 0.0140336i $$-0.00446717\pi$$
$$68$$ − 294.545i − 0.525276i
$$69$$ −1527.48 −2.66502
$$70$$ 0 0
$$71$$ −670.203 −1.12026 −0.560130 0.828405i $$-0.689249\pi$$
−0.560130 + 0.828405i $$0.689249\pi$$
$$72$$ 2104.12i 3.44408i
$$73$$ 193.480i 0.310207i 0.987898 + 0.155103i $$0.0495711\pi$$
−0.987898 + 0.155103i $$0.950429\pi$$
$$74$$ 656.080 1.03065
$$75$$ 0 0
$$76$$ −1152.46 −1.73942
$$77$$ 199.971i 0.295959i
$$78$$ 343.788i 0.499055i
$$79$$ −1080.15 −1.53831 −0.769156 0.639061i $$-0.779323\pi$$
−0.769156 + 0.639061i $$0.779323\pi$$
$$80$$ 0 0
$$81$$ 70.4207 0.0965990
$$82$$ − 90.1229i − 0.121371i
$$83$$ − 865.668i − 1.14481i −0.819970 0.572406i $$-0.806010\pi$$
0.819970 0.572406i $$-0.193990\pi$$
$$84$$ −560.697 −0.728298
$$85$$ 0 0
$$86$$ −1397.13 −1.75181
$$87$$ 349.863i 0.431141i
$$88$$ 2458.02i 2.97757i
$$89$$ −1129.46 −1.34520 −0.672599 0.740008i $$-0.734822\pi$$
−0.672599 + 0.740008i $$0.734822\pi$$
$$90$$ 0 0
$$91$$ −30.7763 −0.0354531
$$92$$ − 3122.61i − 3.53864i
$$93$$ − 295.886i − 0.329914i
$$94$$ 2334.02 2.56102
$$95$$ 0 0
$$96$$ −980.023 −1.04191
$$97$$ 379.412i 0.397149i 0.980086 + 0.198574i $$0.0636311\pi$$
−0.980086 + 0.198574i $$0.936369\pi$$
$$98$$ 1652.78i 1.70363i
$$99$$ 2347.92 2.38359
$$100$$ 0 0
$$101$$ 131.732 0.129780 0.0648902 0.997892i $$-0.479330\pi$$
0.0648902 + 0.997892i $$0.479330\pi$$
$$102$$ − 725.089i − 0.703868i
$$103$$ 195.988i 0.187488i 0.995596 + 0.0937442i $$0.0298836\pi$$
−0.995596 + 0.0937442i $$0.970116\pi$$
$$104$$ −378.299 −0.356685
$$105$$ 0 0
$$106$$ 1660.45 1.52148
$$107$$ 485.147i 0.438326i 0.975688 + 0.219163i $$0.0703327\pi$$
−0.975688 + 0.219163i $$0.929667\pi$$
$$108$$ 2618.49i 2.33300i
$$109$$ 1255.12 1.10292 0.551460 0.834201i $$-0.314071\pi$$
0.551460 + 0.834201i $$0.314071\pi$$
$$110$$ 0 0
$$111$$ 1104.92 0.944812
$$112$$ − 372.612i − 0.314362i
$$113$$ − 1013.35i − 0.843612i −0.906686 0.421806i $$-0.861396\pi$$
0.906686 0.421806i $$-0.138604\pi$$
$$114$$ −2837.03 −2.33081
$$115$$ 0 0
$$116$$ −715.224 −0.572473
$$117$$ 361.354i 0.285531i
$$118$$ − 3415.34i − 2.66447i
$$119$$ 64.9108 0.0500031
$$120$$ 0 0
$$121$$ 1411.83 1.06073
$$122$$ 1712.46i 1.27081i
$$123$$ − 151.778i − 0.111263i
$$124$$ 604.880 0.438063
$$125$$ 0 0
$$126$$ −861.464 −0.609090
$$127$$ − 1927.72i − 1.34691i −0.739227 0.673456i $$-0.764809\pi$$
0.739227 0.673456i $$-0.235191\pi$$
$$128$$ 1925.37i 1.32953i
$$129$$ −2352.93 −1.60592
$$130$$ 0 0
$$131$$ −406.738 −0.271274 −0.135637 0.990759i $$-0.543308\pi$$
−0.135637 + 0.990759i $$0.543308\pi$$
$$132$$ 7690.58i 5.07105i
$$133$$ − 253.975i − 0.165582i
$$134$$ 77.4631 0.0499387
$$135$$ 0 0
$$136$$ 797.877 0.503069
$$137$$ 130.552i 0.0814149i 0.999171 + 0.0407074i $$0.0129612\pi$$
−0.999171 + 0.0407074i $$0.987039\pi$$
$$138$$ − 7687.03i − 4.74177i
$$139$$ −2073.54 −1.26529 −0.632644 0.774443i $$-0.718030\pi$$
−0.632644 + 0.774443i $$0.718030\pi$$
$$140$$ 0 0
$$141$$ 3930.78 2.34774
$$142$$ − 3372.80i − 1.99323i
$$143$$ 422.131i 0.246856i
$$144$$ −4374.95 −2.53180
$$145$$ 0 0
$$146$$ −973.689 −0.551939
$$147$$ 2783.48i 1.56175i
$$148$$ 2258.78i 1.25453i
$$149$$ 1852.73 1.01867 0.509334 0.860569i $$-0.329892\pi$$
0.509334 + 0.860569i $$0.329892\pi$$
$$150$$ 0 0
$$151$$ 2050.86 1.10527 0.552637 0.833422i $$-0.313622\pi$$
0.552637 + 0.833422i $$0.313622\pi$$
$$152$$ − 3121.83i − 1.66588i
$$153$$ − 762.138i − 0.402714i
$$154$$ −1006.36 −0.526588
$$155$$ 0 0
$$156$$ −1183.61 −0.607465
$$157$$ 262.991i 0.133688i 0.997763 + 0.0668438i $$0.0212929\pi$$
−0.997763 + 0.0668438i $$0.978707\pi$$
$$158$$ − 5435.88i − 2.73706i
$$159$$ 2796.39 1.39477
$$160$$ 0 0
$$161$$ 688.152 0.336857
$$162$$ 354.393i 0.171875i
$$163$$ − 1444.98i − 0.694354i −0.937800 0.347177i $$-0.887140\pi$$
0.937800 0.347177i $$-0.112860\pi$$
$$164$$ 310.279 0.147736
$$165$$ 0 0
$$166$$ 4356.48 2.03692
$$167$$ 501.565i 0.232409i 0.993225 + 0.116204i $$0.0370728\pi$$
−0.993225 + 0.116204i $$0.962927\pi$$
$$168$$ − 1518.84i − 0.697508i
$$169$$ 2132.03 0.970429
$$170$$ 0 0
$$171$$ −2981.99 −1.33356
$$172$$ − 4810.08i − 2.13236i
$$173$$ − 2590.14i − 1.13829i −0.822237 0.569146i $$-0.807274\pi$$
0.822237 0.569146i $$-0.192726\pi$$
$$174$$ −1760.69 −0.767112
$$175$$ 0 0
$$176$$ −5110.79 −2.18886
$$177$$ − 5751.85i − 2.44257i
$$178$$ − 5684.02i − 2.39346i
$$179$$ −2165.65 −0.904294 −0.452147 0.891943i $$-0.649342\pi$$
−0.452147 + 0.891943i $$0.649342\pi$$
$$180$$ 0 0
$$181$$ −1925.56 −0.790750 −0.395375 0.918520i $$-0.629385\pi$$
−0.395375 + 0.918520i $$0.629385\pi$$
$$182$$ − 154.882i − 0.0630803i
$$183$$ 2884.00i 1.16498i
$$184$$ 8458.69 3.38904
$$185$$ 0 0
$$186$$ 1489.05 0.587003
$$187$$ − 890.324i − 0.348165i
$$188$$ 8035.68i 3.11735i
$$189$$ −577.055 −0.222088
$$190$$ 0 0
$$191$$ −2783.52 −1.05449 −0.527247 0.849712i $$-0.676776\pi$$
−0.527247 + 0.849712i $$0.676776\pi$$
$$192$$ 1684.65i 0.633223i
$$193$$ 2258.27i 0.842246i 0.907004 + 0.421123i $$0.138364\pi$$
−0.907004 + 0.421123i $$0.861636\pi$$
$$194$$ −1909.39 −0.706631
$$195$$ 0 0
$$196$$ −5690.27 −2.07371
$$197$$ 1270.70i 0.459560i 0.973243 + 0.229780i $$0.0738007\pi$$
−0.973243 + 0.229780i $$0.926199\pi$$
$$198$$ 11815.9i 4.24102i
$$199$$ 4794.36 1.70786 0.853928 0.520392i $$-0.174214\pi$$
0.853928 + 0.520392i $$0.174214\pi$$
$$200$$ 0 0
$$201$$ 130.457 0.0457798
$$202$$ 662.942i 0.230913i
$$203$$ − 157.619i − 0.0544960i
$$204$$ 2496.37 0.856769
$$205$$ 0 0
$$206$$ −986.313 −0.333591
$$207$$ − 8079.80i − 2.71297i
$$208$$ − 786.569i − 0.262205i
$$209$$ −3483.54 −1.15293
$$210$$ 0 0
$$211$$ −2807.00 −0.915837 −0.457918 0.888994i $$-0.651405\pi$$
−0.457918 + 0.888994i $$0.651405\pi$$
$$212$$ 5716.66i 1.85199i
$$213$$ − 5680.21i − 1.82724i
$$214$$ −2441.50 −0.779896
$$215$$ 0 0
$$216$$ −7093.09 −2.23437
$$217$$ 133.302i 0.0417009i
$$218$$ 6316.38i 1.96238i
$$219$$ −1639.81 −0.505973
$$220$$ 0 0
$$221$$ 137.024 0.0417070
$$222$$ 5560.51i 1.68107i
$$223$$ 4684.30i 1.40665i 0.710866 + 0.703327i $$0.248303\pi$$
−0.710866 + 0.703327i $$0.751697\pi$$
$$224$$ 441.516 0.131697
$$225$$ 0 0
$$226$$ 5099.70 1.50101
$$227$$ 1395.72i 0.408095i 0.978961 + 0.204047i $$0.0654096\pi$$
−0.978961 + 0.204047i $$0.934590\pi$$
$$228$$ − 9767.47i − 2.83713i
$$229$$ −894.638 −0.258163 −0.129082 0.991634i $$-0.541203\pi$$
−0.129082 + 0.991634i $$0.541203\pi$$
$$230$$ 0 0
$$231$$ −1694.83 −0.482733
$$232$$ − 1937.43i − 0.548270i
$$233$$ 1196.13i 0.336313i 0.985760 + 0.168156i $$0.0537814\pi$$
−0.985760 + 0.168156i $$0.946219\pi$$
$$234$$ −1818.52 −0.508035
$$235$$ 0 0
$$236$$ 11758.5 3.24328
$$237$$ − 9154.67i − 2.50911i
$$238$$ 326.664i 0.0889685i
$$239$$ −4948.82 −1.33938 −0.669691 0.742639i $$-0.733574\pi$$
−0.669691 + 0.742639i $$0.733574\pi$$
$$240$$ 0 0
$$241$$ −6702.73 −1.79154 −0.895770 0.444518i $$-0.853375\pi$$
−0.895770 + 0.444518i $$0.853375\pi$$
$$242$$ 7105.03i 1.88731i
$$243$$ − 3483.65i − 0.919656i
$$244$$ −5895.75 −1.54687
$$245$$ 0 0
$$246$$ 763.824 0.197966
$$247$$ − 536.130i − 0.138110i
$$248$$ 1638.53i 0.419543i
$$249$$ 7336.85 1.86728
$$250$$ 0 0
$$251$$ −4756.08 −1.19602 −0.598010 0.801489i $$-0.704042\pi$$
−0.598010 + 0.801489i $$0.704042\pi$$
$$252$$ − 2965.89i − 0.741402i
$$253$$ − 9438.77i − 2.34550i
$$254$$ 9701.29 2.39651
$$255$$ 0 0
$$256$$ −8099.28 −1.97736
$$257$$ − 2892.84i − 0.702143i −0.936349 0.351071i $$-0.885817\pi$$
0.936349 0.351071i $$-0.114183\pi$$
$$258$$ − 11841.1i − 2.85735i
$$259$$ −497.784 −0.119424
$$260$$ 0 0
$$261$$ −1850.65 −0.438898
$$262$$ − 2046.92i − 0.482667i
$$263$$ 5415.48i 1.26971i 0.772633 + 0.634853i $$0.218939\pi$$
−0.772633 + 0.634853i $$0.781061\pi$$
$$264$$ −20832.6 −4.85666
$$265$$ 0 0
$$266$$ 1278.13 0.294613
$$267$$ − 9572.58i − 2.19413i
$$268$$ 266.693i 0.0607869i
$$269$$ −5787.00 −1.31167 −0.655835 0.754904i $$-0.727683\pi$$
−0.655835 + 0.754904i $$0.727683\pi$$
$$270$$ 0 0
$$271$$ 5465.13 1.22503 0.612515 0.790459i $$-0.290158\pi$$
0.612515 + 0.790459i $$0.290158\pi$$
$$272$$ 1658.97i 0.369815i
$$273$$ − 260.840i − 0.0578270i
$$274$$ −657.006 −0.144858
$$275$$ 0 0
$$276$$ 26465.3 5.77182
$$277$$ 1207.65i 0.261952i 0.991386 + 0.130976i $$0.0418111\pi$$
−0.991386 + 0.130976i $$0.958189\pi$$
$$278$$ − 10435.1i − 2.25128i
$$279$$ 1565.13 0.335850
$$280$$ 0 0
$$281$$ −1197.18 −0.254155 −0.127077 0.991893i $$-0.540560\pi$$
−0.127077 + 0.991893i $$0.540560\pi$$
$$282$$ 19781.7i 4.17724i
$$283$$ 3164.73i 0.664748i 0.943148 + 0.332374i $$0.107850\pi$$
−0.943148 + 0.332374i $$0.892150\pi$$
$$284$$ 11612.0 2.42622
$$285$$ 0 0
$$286$$ −2124.38 −0.439221
$$287$$ 68.3784i 0.0140636i
$$288$$ − 5183.98i − 1.06066i
$$289$$ −289.000 −0.0588235
$$290$$ 0 0
$$291$$ −3215.65 −0.647782
$$292$$ − 3352.26i − 0.671836i
$$293$$ 7456.21i 1.48668i 0.668915 + 0.743339i $$0.266759\pi$$
−0.668915 + 0.743339i $$0.733241\pi$$
$$294$$ −14007.9 −2.77877
$$295$$ 0 0
$$296$$ −6118.70 −1.20149
$$297$$ 7914.94i 1.54637i
$$298$$ 9323.89i 1.81248i
$$299$$ 1452.66 0.280969
$$300$$ 0 0
$$301$$ 1060.03 0.202988
$$302$$ 10321.0i 1.96657i
$$303$$ 1116.47i 0.211683i
$$304$$ 6490.98 1.22462
$$305$$ 0 0
$$306$$ 3835.46 0.716532
$$307$$ 6535.48i 1.21498i 0.794327 + 0.607491i $$0.207824\pi$$
−0.794327 + 0.607491i $$0.792176\pi$$
$$308$$ − 3464.73i − 0.640978i
$$309$$ −1661.07 −0.305809
$$310$$ 0 0
$$311$$ −8935.89 −1.62928 −0.814642 0.579963i $$-0.803067\pi$$
−0.814642 + 0.579963i $$0.803067\pi$$
$$312$$ − 3206.22i − 0.581783i
$$313$$ − 2628.71i − 0.474707i −0.971423 0.237353i $$-0.923720\pi$$
0.971423 0.237353i $$-0.0762799\pi$$
$$314$$ −1323.50 −0.237865
$$315$$ 0 0
$$316$$ 18714.9 3.33163
$$317$$ − 4268.54i − 0.756293i −0.925746 0.378147i $$-0.876562\pi$$
0.925746 0.378147i $$-0.123438\pi$$
$$318$$ 14072.9i 2.48166i
$$319$$ −2161.92 −0.379449
$$320$$ 0 0
$$321$$ −4111.79 −0.714946
$$322$$ 3463.13i 0.599357i
$$323$$ 1130.76i 0.194790i
$$324$$ −1220.12 −0.209211
$$325$$ 0 0
$$326$$ 7271.89 1.23544
$$327$$ 10637.5i 1.79895i
$$328$$ 840.500i 0.141490i
$$329$$ −1770.88 −0.296753
$$330$$ 0 0
$$331$$ 992.298 0.164778 0.0823892 0.996600i $$-0.473745\pi$$
0.0823892 + 0.996600i $$0.473745\pi$$
$$332$$ 14998.7i 2.47940i
$$333$$ 5844.63i 0.961812i
$$334$$ −2524.13 −0.413516
$$335$$ 0 0
$$336$$ 3158.02 0.512750
$$337$$ − 8042.26i − 1.29997i −0.759947 0.649985i $$-0.774775\pi$$
0.759947 0.649985i $$-0.225225\pi$$
$$338$$ 10729.5i 1.72665i
$$339$$ 8588.52 1.37600
$$340$$ 0 0
$$341$$ 1828.38 0.290359
$$342$$ − 15006.9i − 2.37275i
$$343$$ − 2563.68i − 0.403573i
$$344$$ 13029.8 2.04221
$$345$$ 0 0
$$346$$ 13034.9 2.02532
$$347$$ 7414.16i 1.14701i 0.819202 + 0.573506i $$0.194417\pi$$
−0.819202 + 0.573506i $$0.805583\pi$$
$$348$$ − 6061.78i − 0.933751i
$$349$$ 859.194 0.131781 0.0658905 0.997827i $$-0.479011\pi$$
0.0658905 + 0.997827i $$0.479011\pi$$
$$350$$ 0 0
$$351$$ −1218.14 −0.185241
$$352$$ − 6055.89i − 0.916988i
$$353$$ 569.084i 0.0858053i 0.999079 + 0.0429027i $$0.0136605\pi$$
−0.999079 + 0.0429027i $$0.986339\pi$$
$$354$$ 28946.3 4.34598
$$355$$ 0 0
$$356$$ 19569.2 2.91339
$$357$$ 550.142i 0.0815592i
$$358$$ − 10898.7i − 1.60897i
$$359$$ 5005.21 0.735835 0.367918 0.929858i $$-0.380071\pi$$
0.367918 + 0.929858i $$0.380071\pi$$
$$360$$ 0 0
$$361$$ −2434.71 −0.354965
$$362$$ − 9690.40i − 1.40695i
$$363$$ 11965.7i 1.73013i
$$364$$ 533.235 0.0767833
$$365$$ 0 0
$$366$$ −14513.7 −2.07280
$$367$$ 10975.3i 1.56105i 0.625127 + 0.780523i $$0.285047\pi$$
−0.625127 + 0.780523i $$0.714953\pi$$
$$368$$ 17587.5i 2.49134i
$$369$$ 802.851 0.113265
$$370$$ 0 0
$$371$$ −1259.82 −0.176298
$$372$$ 5126.57i 0.714517i
$$373$$ − 3211.72i − 0.445835i −0.974837 0.222918i $$-0.928442\pi$$
0.974837 0.222918i $$-0.0715581\pi$$
$$374$$ 4480.56 0.619477
$$375$$ 0 0
$$376$$ −21767.4 −2.98556
$$377$$ − 332.727i − 0.0454544i
$$378$$ − 2904.03i − 0.395152i
$$379$$ −8051.48 −1.09123 −0.545616 0.838035i $$-0.683704\pi$$
−0.545616 + 0.838035i $$0.683704\pi$$
$$380$$ 0 0
$$381$$ 16338.1 2.19692
$$382$$ − 14008.1i − 1.87622i
$$383$$ − 2584.16i − 0.344763i −0.985030 0.172382i $$-0.944854\pi$$
0.985030 0.172382i $$-0.0551462\pi$$
$$384$$ −16318.2 −2.16858
$$385$$ 0 0
$$386$$ −11364.7 −1.49858
$$387$$ − 12446.2i − 1.63482i
$$388$$ − 6573.74i − 0.860132i
$$389$$ 5174.31 0.674417 0.337208 0.941430i $$-0.390517\pi$$
0.337208 + 0.941430i $$0.390517\pi$$
$$390$$ 0 0
$$391$$ −3063.83 −0.396278
$$392$$ − 15414.1i − 1.98604i
$$393$$ − 3447.25i − 0.442470i
$$394$$ −6394.79 −0.817677
$$395$$ 0 0
$$396$$ −40680.4 −5.16230
$$397$$ 5149.36i 0.650980i 0.945545 + 0.325490i $$0.105529\pi$$
−0.945545 + 0.325490i $$0.894471\pi$$
$$398$$ 24127.7i 3.03872i
$$399$$ 2152.53 0.270078
$$400$$ 0 0
$$401$$ 8700.49 1.08350 0.541748 0.840541i $$-0.317763\pi$$
0.541748 + 0.840541i $$0.317763\pi$$
$$402$$ 656.527i 0.0814542i
$$403$$ 281.394i 0.0347823i
$$404$$ −2282.41 −0.281074
$$405$$ 0 0
$$406$$ 793.219 0.0969625
$$407$$ 6827.65i 0.831533i
$$408$$ 6762.29i 0.820547i
$$409$$ −12346.0 −1.49260 −0.746299 0.665611i $$-0.768171\pi$$
−0.746299 + 0.665611i $$0.768171\pi$$
$$410$$ 0 0
$$411$$ −1106.48 −0.132794
$$412$$ − 3395.72i − 0.406056i
$$413$$ 2591.30i 0.308740i
$$414$$ 40661.7 4.82708
$$415$$ 0 0
$$416$$ 932.023 0.109847
$$417$$ − 17574.0i − 2.06379i
$$418$$ − 17531.0i − 2.05136i
$$419$$ 5763.33 0.671974 0.335987 0.941867i $$-0.390930\pi$$
0.335987 + 0.941867i $$0.390930\pi$$
$$420$$ 0 0
$$421$$ −1876.12 −0.217188 −0.108594 0.994086i $$-0.534635\pi$$
−0.108594 + 0.994086i $$0.534635\pi$$
$$422$$ − 14126.2i − 1.62951i
$$423$$ 20792.4i 2.38998i
$$424$$ −15485.6 −1.77369
$$425$$ 0 0
$$426$$ 28585.7 3.25113
$$427$$ − 1299.29i − 0.147253i
$$428$$ − 8405.72i − 0.949313i
$$429$$ −3577.71 −0.402642
$$430$$ 0 0
$$431$$ 83.9299 0.00937996 0.00468998 0.999989i $$-0.498507\pi$$
0.00468998 + 0.999989i $$0.498507\pi$$
$$432$$ − 14748.1i − 1.64252i
$$433$$ − 15345.0i − 1.70308i −0.524291 0.851539i $$-0.675669\pi$$
0.524291 0.851539i $$-0.324331\pi$$
$$434$$ −670.842 −0.0741968
$$435$$ 0 0
$$436$$ −21746.3 −2.38867
$$437$$ 11987.8i 1.31225i
$$438$$ − 8252.36i − 0.900258i
$$439$$ −3064.74 −0.333194 −0.166597 0.986025i $$-0.553278\pi$$
−0.166597 + 0.986025i $$0.553278\pi$$
$$440$$ 0 0
$$441$$ −14723.6 −1.58985
$$442$$ 689.575i 0.0742076i
$$443$$ − 1792.97i − 0.192295i −0.995367 0.0961474i $$-0.969348\pi$$
0.995367 0.0961474i $$-0.0306520\pi$$
$$444$$ −19144.0 −2.04624
$$445$$ 0 0
$$446$$ −23573.8 −2.50281
$$447$$ 15702.6i 1.66153i
$$448$$ − 758.960i − 0.0800391i
$$449$$ −2499.19 −0.262681 −0.131341 0.991337i $$-0.541928\pi$$
−0.131341 + 0.991337i $$0.541928\pi$$
$$450$$ 0 0
$$451$$ 937.885 0.0979231
$$452$$ 17557.5i 1.82707i
$$453$$ 17381.7i 1.80279i
$$454$$ −7024.00 −0.726107
$$455$$ 0 0
$$456$$ 26458.6 2.71719
$$457$$ − 14784.4i − 1.51331i −0.653813 0.756656i $$-0.726832\pi$$
0.653813 0.756656i $$-0.273168\pi$$
$$458$$ − 4502.28i − 0.459340i
$$459$$ 2569.20 0.261263
$$460$$ 0 0
$$461$$ −17746.9 −1.79297 −0.896483 0.443078i $$-0.853887\pi$$
−0.896483 + 0.443078i $$0.853887\pi$$
$$462$$ − 8529.23i − 0.858909i
$$463$$ 18486.4i 1.85559i 0.373096 + 0.927793i $$0.378296\pi$$
−0.373096 + 0.927793i $$0.621704\pi$$
$$464$$ 4028.36 0.403043
$$465$$ 0 0
$$466$$ −6019.52 −0.598388
$$467$$ − 7406.57i − 0.733908i −0.930239 0.366954i $$-0.880401\pi$$
0.930239 0.366954i $$-0.119599\pi$$
$$468$$ − 6260.87i − 0.618395i
$$469$$ −58.7731 −0.00578655
$$470$$ 0 0
$$471$$ −2228.94 −0.218055
$$472$$ 31852.0i 3.10616i
$$473$$ − 14539.5i − 1.41338i
$$474$$ 46071.0 4.46437
$$475$$ 0 0
$$476$$ −1124.65 −0.108295
$$477$$ 14791.9i 1.41986i
$$478$$ − 24905.0i − 2.38311i
$$479$$ 18550.9 1.76955 0.884775 0.466019i $$-0.154312\pi$$
0.884775 + 0.466019i $$0.154312\pi$$
$$480$$ 0 0
$$481$$ −1050.80 −0.0996100
$$482$$ − 33731.6i − 3.18762i
$$483$$ 5832.34i 0.549442i
$$484$$ −24461.5 −2.29729
$$485$$ 0 0
$$486$$ 17531.5 1.63631
$$487$$ − 10203.4i − 0.949406i −0.880146 0.474703i $$-0.842556\pi$$
0.880146 0.474703i $$-0.157444\pi$$
$$488$$ − 15970.7i − 1.48147i
$$489$$ 12246.7 1.13255
$$490$$ 0 0
$$491$$ −1247.46 −0.114658 −0.0573290 0.998355i $$-0.518258\pi$$
−0.0573290 + 0.998355i $$0.518258\pi$$
$$492$$ 2629.73i 0.240970i
$$493$$ 701.760i 0.0641089i
$$494$$ 2698.08 0.245734
$$495$$ 0 0
$$496$$ −3406.87 −0.308413
$$497$$ 2559.03i 0.230962i
$$498$$ 36922.7i 3.32238i
$$499$$ −70.0303 −0.00628254 −0.00314127 0.999995i $$-0.501000\pi$$
−0.00314127 + 0.999995i $$0.501000\pi$$
$$500$$ 0 0
$$501$$ −4250.94 −0.379078
$$502$$ − 23935.0i − 2.12803i
$$503$$ 1444.29i 0.128028i 0.997949 + 0.0640138i $$0.0203902\pi$$
−0.997949 + 0.0640138i $$0.979610\pi$$
$$504$$ 8034.15 0.710058
$$505$$ 0 0
$$506$$ 47500.7 4.17325
$$507$$ 18069.7i 1.58285i
$$508$$ 33400.0i 2.91710i
$$509$$ −14272.8 −1.24289 −0.621445 0.783458i $$-0.713454\pi$$
−0.621445 + 0.783458i $$0.713454\pi$$
$$510$$ 0 0
$$511$$ 738.761 0.0639547
$$512$$ − 25356.7i − 2.18871i
$$513$$ − 10052.4i − 0.865157i
$$514$$ 14558.3 1.24929
$$515$$ 0 0
$$516$$ 40767.2 3.47805
$$517$$ 24289.6i 2.06625i
$$518$$ − 2505.10i − 0.212486i
$$519$$ 21952.3 1.85665
$$520$$ 0 0
$$521$$ 14874.0 1.25075 0.625376 0.780324i $$-0.284946\pi$$
0.625376 + 0.780324i $$0.284946\pi$$
$$522$$ − 9313.42i − 0.780914i
$$523$$ − 8142.90i − 0.680811i −0.940279 0.340406i $$-0.889436\pi$$
0.940279 0.340406i $$-0.110564\pi$$
$$524$$ 7047.21 0.587517
$$525$$ 0 0
$$526$$ −27253.4 −2.25914
$$527$$ − 593.494i − 0.0490569i
$$528$$ − 43315.7i − 3.57022i
$$529$$ −20314.2 −1.66962
$$530$$ 0 0
$$531$$ 30425.3 2.48652
$$532$$ 4400.40i 0.358612i
$$533$$ 144.344i 0.0117303i
$$534$$ 48174.1 3.90393
$$535$$ 0 0
$$536$$ −722.432 −0.0582170
$$537$$ − 18354.7i − 1.47498i
$$538$$ − 29123.1i − 2.33380i
$$539$$ −17200.0 −1.37451
$$540$$ 0 0
$$541$$ 3179.67 0.252689 0.126344 0.991986i $$-0.459676\pi$$
0.126344 + 0.991986i $$0.459676\pi$$
$$542$$ 27503.3i 2.17965i
$$543$$ − 16319.8i − 1.28978i
$$544$$ −1965.75 −0.154928
$$545$$ 0 0
$$546$$ 1312.68 0.102889
$$547$$ − 2107.07i − 0.164702i −0.996603 0.0823509i $$-0.973757\pi$$
0.996603 0.0823509i $$-0.0262428\pi$$
$$548$$ − 2261.97i − 0.176326i
$$549$$ −15255.3 −1.18594
$$550$$ 0 0
$$551$$ 2745.76 0.212292
$$552$$ 71690.4i 5.52780i
$$553$$ 4124.33i 0.317151i
$$554$$ −6077.51 −0.466081
$$555$$ 0 0
$$556$$ 35926.4 2.74032
$$557$$ − 467.382i − 0.0355540i −0.999842 0.0177770i $$-0.994341\pi$$
0.999842 0.0177770i $$-0.00565890\pi$$
$$558$$ 7876.55i 0.597565i
$$559$$ 2237.69 0.169309
$$560$$ 0 0
$$561$$ 7545.81 0.567887
$$562$$ − 6024.80i − 0.452208i
$$563$$ 14612.6i 1.09387i 0.837175 + 0.546935i $$0.184206\pi$$
−0.837175 + 0.546935i $$0.815794\pi$$
$$564$$ −68105.2 −5.08466
$$565$$ 0 0
$$566$$ −15926.5 −1.18276
$$567$$ − 268.886i − 0.0199156i
$$568$$ 31455.3i 2.32365i
$$569$$ −11602.3 −0.854821 −0.427410 0.904058i $$-0.640574\pi$$
−0.427410 + 0.904058i $$0.640574\pi$$
$$570$$ 0 0
$$571$$ −10534.9 −0.772104 −0.386052 0.922477i $$-0.626161\pi$$
−0.386052 + 0.922477i $$0.626161\pi$$
$$572$$ − 7313.91i − 0.534633i
$$573$$ − 23591.3i − 1.71997i
$$574$$ −344.115 −0.0250228
$$575$$ 0 0
$$576$$ −8911.18 −0.644617
$$577$$ − 14404.7i − 1.03930i −0.854379 0.519650i $$-0.826062\pi$$
0.854379 0.519650i $$-0.173938\pi$$
$$578$$ − 1454.40i − 0.104662i
$$579$$ −19139.6 −1.37377
$$580$$ 0 0
$$581$$ −3305.37 −0.236024
$$582$$ − 16182.8i − 1.15257i
$$583$$ 17279.8i 1.22754i
$$584$$ 9080.77 0.643433
$$585$$ 0 0
$$586$$ −37523.5 −2.64519
$$587$$ 11004.9i 0.773799i 0.922122 + 0.386900i $$0.126454\pi$$
−0.922122 + 0.386900i $$0.873546\pi$$
$$588$$ − 48227.0i − 3.38240i
$$589$$ −2322.14 −0.162449
$$590$$ 0 0
$$591$$ −10769.6 −0.749581
$$592$$ − 12722.2i − 0.883239i
$$593$$ 1853.59i 0.128361i 0.997938 + 0.0641804i $$0.0204433\pi$$
−0.997938 + 0.0641804i $$0.979557\pi$$
$$594$$ −39832.0 −2.75139
$$595$$ 0 0
$$596$$ −32100.7 −2.20620
$$597$$ 40633.9i 2.78565i
$$598$$ 7310.53i 0.499916i
$$599$$ −19074.7 −1.30112 −0.650559 0.759456i $$-0.725465\pi$$
−0.650559 + 0.759456i $$0.725465\pi$$
$$600$$ 0 0
$$601$$ −27776.0 −1.88520 −0.942600 0.333923i $$-0.891627\pi$$
−0.942600 + 0.333923i $$0.891627\pi$$
$$602$$ 5334.62i 0.361168i
$$603$$ 690.073i 0.0466035i
$$604$$ −35533.4 −2.39377
$$605$$ 0 0
$$606$$ −5618.67 −0.376638
$$607$$ − 18728.3i − 1.25232i −0.779695 0.626159i $$-0.784626\pi$$
0.779695 0.626159i $$-0.215374\pi$$
$$608$$ 7691.32i 0.513033i
$$609$$ 1335.88 0.0888874
$$610$$ 0 0
$$611$$ −3738.25 −0.247518
$$612$$ 13204.9i 0.872184i
$$613$$ − 24405.3i − 1.60802i −0.594613 0.804012i $$-0.702695\pi$$
0.594613 0.804012i $$-0.297305\pi$$
$$614$$ −32889.8 −2.16177
$$615$$ 0 0
$$616$$ 9385.43 0.613880
$$617$$ 22516.4i 1.46917i 0.678518 + 0.734584i $$0.262623\pi$$
−0.678518 + 0.734584i $$0.737377\pi$$
$$618$$ − 8359.35i − 0.544114i
$$619$$ 5146.53 0.334179 0.167089 0.985942i $$-0.446563\pi$$
0.167089 + 0.985942i $$0.446563\pi$$
$$620$$ 0 0
$$621$$ 27237.4 1.76006
$$622$$ − 44969.9i − 2.89892i
$$623$$ 4312.60i 0.277337i
$$624$$ 6666.45 0.427679
$$625$$ 0 0
$$626$$ 13229.0 0.844627
$$627$$ − 29524.3i − 1.88052i
$$628$$ − 4556.62i − 0.289536i
$$629$$ 2216.26 0.140490
$$630$$ 0 0
$$631$$ −3858.77 −0.243447 −0.121724 0.992564i $$-0.538842\pi$$
−0.121724 + 0.992564i $$0.538842\pi$$
$$632$$ 50695.8i 3.19078i
$$633$$ − 23790.3i − 1.49381i
$$634$$ 21481.5 1.34564
$$635$$ 0 0
$$636$$ −48450.7 −3.02075
$$637$$ − 2647.15i − 0.164653i
$$638$$ − 10879.9i − 0.675138i
$$639$$ 30046.3 1.86011
$$640$$ 0 0
$$641$$ 18689.3 1.15161 0.575805 0.817587i $$-0.304689\pi$$
0.575805 + 0.817587i $$0.304689\pi$$
$$642$$ − 20692.6i − 1.27208i
$$643$$ 26473.5i 1.62366i 0.583893 + 0.811831i $$0.301529\pi$$
−0.583893 + 0.811831i $$0.698471\pi$$
$$644$$ −11923.0 −0.729555
$$645$$ 0 0
$$646$$ −5690.57 −0.346583
$$647$$ − 14397.7i − 0.874855i −0.899254 0.437427i $$-0.855890\pi$$
0.899254 0.437427i $$-0.144110\pi$$
$$648$$ − 3305.12i − 0.200366i
$$649$$ 35542.6 2.14972
$$650$$ 0 0
$$651$$ −1129.78 −0.0680177
$$652$$ 25036.0i 1.50381i
$$653$$ 20939.5i 1.25486i 0.778672 + 0.627431i $$0.215893\pi$$
−0.778672 + 0.627431i $$0.784107\pi$$
$$654$$ −53533.6 −3.20081
$$655$$ 0 0
$$656$$ −1747.59 −0.104012
$$657$$ − 8674.02i − 0.515077i
$$658$$ − 8911.96i − 0.528001i
$$659$$ −4031.76 −0.238323 −0.119162 0.992875i $$-0.538021\pi$$
−0.119162 + 0.992875i $$0.538021\pi$$
$$660$$ 0 0
$$661$$ 6691.52 0.393752 0.196876 0.980428i $$-0.436920\pi$$
0.196876 + 0.980428i $$0.436920\pi$$
$$662$$ 4993.75i 0.293184i
$$663$$ 1161.33i 0.0680275i
$$664$$ −40629.2 −2.37458
$$665$$ 0 0
$$666$$ −29413.1 −1.71131
$$667$$ 7439.71i 0.431884i
$$668$$ − 8690.19i − 0.503344i
$$669$$ −39701.1 −2.29437
$$670$$ 0 0
$$671$$ −17821.2 −1.02530
$$672$$ 3742.01i 0.214808i
$$673$$ 10319.2i 0.591048i 0.955335 + 0.295524i $$0.0954942\pi$$
−0.955335 + 0.295524i $$0.904506\pi$$
$$674$$ 40472.7 2.31298
$$675$$ 0 0
$$676$$ −36939.9 −2.10172
$$677$$ 19813.3i 1.12480i 0.826866 + 0.562398i $$0.190121\pi$$
−0.826866 + 0.562398i $$0.809879\pi$$
$$678$$ 43221.8i 2.44826i
$$679$$ 1448.70 0.0818793
$$680$$ 0 0
$$681$$ −11829.3 −0.665636
$$682$$ 9201.33i 0.516624i
$$683$$ 5924.61i 0.331916i 0.986133 + 0.165958i $$0.0530717\pi$$
−0.986133 + 0.165958i $$0.946928\pi$$
$$684$$ 51666.4 2.88818
$$685$$ 0 0
$$686$$ 12901.7 0.718061
$$687$$ − 7582.38i − 0.421085i
$$688$$ 27091.9i 1.50126i
$$689$$ −2659.43 −0.147048
$$690$$ 0 0
$$691$$ 1973.16 0.108629 0.0543143 0.998524i $$-0.482703\pi$$
0.0543143 + 0.998524i $$0.482703\pi$$
$$692$$ 44877.1i 2.46528i
$$693$$ − 8965.03i − 0.491419i
$$694$$ −37311.8 −2.04083
$$695$$ 0 0
$$696$$ 16420.4 0.894274
$$697$$ − 304.439i − 0.0165444i
$$698$$ 4323.90i 0.234473i
$$699$$ −10137.6 −0.548554
$$700$$ 0 0
$$701$$ −12840.1 −0.691815 −0.345907 0.938269i $$-0.612429\pi$$
−0.345907 + 0.938269i $$0.612429\pi$$
$$702$$ − 6130.30i − 0.329591i
$$703$$ − 8671.50i − 0.465223i
$$704$$ −10410.0 −0.557302
$$705$$ 0 0
$$706$$ −2863.92 −0.152670
$$707$$ − 502.990i − 0.0267566i
$$708$$ 99657.5i 5.29005i
$$709$$ 27749.7 1.46990 0.734952 0.678119i $$-0.237204\pi$$
0.734952 + 0.678119i $$0.237204\pi$$
$$710$$ 0 0
$$711$$ 48425.0 2.55426
$$712$$ 53010.0i 2.79022i
$$713$$ − 6291.92i − 0.330483i
$$714$$ −2768.60 −0.145115
$$715$$ 0 0
$$716$$ 37522.4 1.95849
$$717$$ − 41943.0i − 2.18464i
$$718$$ 25188.8i 1.30924i
$$719$$ −16888.3 −0.875979 −0.437989 0.898980i $$-0.644309\pi$$
−0.437989 + 0.898980i $$0.644309\pi$$
$$720$$ 0 0
$$721$$ 748.339 0.0386541
$$722$$ − 12252.7i − 0.631575i
$$723$$ − 56808.0i − 2.92215i
$$724$$ 33362.6 1.71258
$$725$$ 0 0
$$726$$ −60217.6 −3.07835
$$727$$ − 2135.25i − 0.108930i −0.998516 0.0544649i $$-0.982655\pi$$
0.998516 0.0544649i $$-0.0173453\pi$$
$$728$$ 1444.45i 0.0735371i
$$729$$ 31426.5 1.59663
$$730$$ 0 0
$$731$$ −4719.54 −0.238794
$$732$$ − 49968.6i − 2.52308i
$$733$$ 4795.27i 0.241633i 0.992675 + 0.120817i $$0.0385513\pi$$
−0.992675 + 0.120817i $$0.961449\pi$$
$$734$$ −55233.1 −2.77751
$$735$$ 0 0
$$736$$ −20839.9 −1.04371
$$737$$ 806.138i 0.0402910i
$$738$$ 4040.36i 0.201528i
$$739$$ 32747.6 1.63010 0.815048 0.579393i $$-0.196710\pi$$
0.815048 + 0.579393i $$0.196710\pi$$
$$740$$ 0 0
$$741$$ 4543.89 0.225269
$$742$$ − 6340.05i − 0.313680i
$$743$$ 12299.4i 0.607298i 0.952784 + 0.303649i $$0.0982050\pi$$
−0.952784 + 0.303649i $$0.901795\pi$$
$$744$$ −13887.1 −0.684309
$$745$$ 0 0
$$746$$ 16163.0 0.793257
$$747$$ 38809.3i 1.90088i
$$748$$ 15425.9i 0.754046i
$$749$$ 1852.43 0.0903688
$$750$$ 0 0
$$751$$ 30102.6 1.46266 0.731332 0.682021i $$-0.238899\pi$$
0.731332 + 0.682021i $$0.238899\pi$$
$$752$$ − 45259.4i − 2.19474i
$$753$$ − 40309.4i − 1.95081i
$$754$$ 1674.45 0.0808753
$$755$$ 0 0
$$756$$ 9998.14 0.480990
$$757$$ − 38826.3i − 1.86416i −0.362257 0.932078i $$-0.617994\pi$$
0.362257 0.932078i $$-0.382006\pi$$
$$758$$ − 40519.2i − 1.94159i
$$759$$ 79996.9 3.82570
$$760$$ 0 0
$$761$$ 19981.6 0.951815 0.475907 0.879495i $$-0.342120\pi$$
0.475907 + 0.879495i $$0.342120\pi$$
$$762$$ 82221.8i 3.90890i
$$763$$ − 4792.39i − 0.227387i
$$764$$ 48227.7 2.28379
$$765$$ 0 0
$$766$$ 13004.8 0.613424
$$767$$ 5470.14i 0.257517i
$$768$$ − 68644.2i − 3.22524i
$$769$$ 22407.7 1.05077 0.525384 0.850865i $$-0.323922\pi$$
0.525384 + 0.850865i $$0.323922\pi$$
$$770$$ 0 0
$$771$$ 24517.9 1.14525
$$772$$ − 39127.0i − 1.82411i
$$773$$ − 6902.77i − 0.321184i −0.987021 0.160592i $$-0.948660\pi$$
0.987021 0.160592i $$-0.0513403\pi$$
$$774$$ 62635.4 2.90876
$$775$$ 0 0
$$776$$ 17807.3 0.823768
$$777$$ − 4218.89i − 0.194790i
$$778$$ 26039.8i 1.19996i
$$779$$ −1191.17 −0.0547856
$$780$$ 0 0
$$781$$ 35099.9 1.60816
$$782$$ − 15418.8i − 0.705082i
$$783$$ − 6238.62i − 0.284738i
$$784$$ 32049.3 1.45997
$$785$$ 0 0
$$786$$ 17348.3 0.787270
$$787$$ 22185.9i 1.00488i 0.864611 + 0.502442i $$0.167565\pi$$
−0.864611 + 0.502442i $$0.832435\pi$$
$$788$$ − 22016.3i − 0.995301i
$$789$$ −45898.1 −2.07099
$$790$$ 0 0
$$791$$ −3869.27 −0.173926
$$792$$ − 110197.i − 4.94405i
$$793$$ − 2742.74i − 0.122822i
$$794$$ −25914.2 −1.15826
$$795$$ 0 0
$$796$$ −83067.8 −3.69882
$$797$$ 16291.1i 0.724040i 0.932170 + 0.362020i $$0.117913\pi$$
−0.932170 + 0.362020i $$0.882087\pi$$
$$798$$ 10832.6i 0.480539i
$$799$$ 7884.41 0.349100
$$800$$ 0 0
$$801$$ 50635.5 2.23361
$$802$$ 43785.3i 1.92782i
$$803$$ − 10132.9i − 0.445309i
$$804$$ −2260.32 −0.0991485
$$805$$ 0 0
$$806$$ −1416.12 −0.0618867
$$807$$ − 49046.8i − 2.13944i
$$808$$ − 6182.70i − 0.269191i
$$809$$ −17696.8 −0.769082 −0.384541 0.923108i $$-0.625640\pi$$
−0.384541 + 0.923108i $$0.625640\pi$$
$$810$$ 0 0
$$811$$ −3095.34 −0.134022 −0.0670111 0.997752i $$-0.521346\pi$$
−0.0670111 + 0.997752i $$0.521346\pi$$
$$812$$ 2730.93i 0.118026i
$$813$$ 46318.9i 1.99812i
$$814$$ −34360.2 −1.47951
$$815$$ 0 0
$$816$$ −14060.3 −0.603198
$$817$$ 18466.0i 0.790751i
$$818$$ − 62131.5i − 2.65572i
$$819$$ 1379.75 0.0588674
$$820$$ 0 0
$$821$$ 12323.5 0.523864 0.261932 0.965086i $$-0.415640\pi$$
0.261932 + 0.965086i $$0.415640\pi$$
$$822$$ − 5568.36i − 0.236276i
$$823$$ − 34436.5i − 1.45854i −0.684225 0.729271i $$-0.739860\pi$$
0.684225 0.729271i $$-0.260140\pi$$
$$824$$ 9198.50 0.388889
$$825$$ 0 0
$$826$$ −13040.8 −0.549329
$$827$$ − 18761.6i − 0.788880i −0.918922 0.394440i $$-0.870939\pi$$
0.918922 0.394440i $$-0.129061\pi$$
$$828$$ 139992.i 5.87567i
$$829$$ −22423.8 −0.939457 −0.469728 0.882811i $$-0.655648\pi$$
−0.469728 + 0.882811i $$0.655648\pi$$
$$830$$ 0 0
$$831$$ −10235.3 −0.427265
$$832$$ − 1602.13i − 0.0667596i
$$833$$ 5583.15i 0.232227i
$$834$$ 88441.1 3.67202
$$835$$ 0 0
$$836$$ 60356.4 2.49697
$$837$$ 5276.13i 0.217885i
$$838$$ 29004.0i 1.19562i
$$839$$ −9128.63 −0.375632 −0.187816 0.982204i $$-0.560141\pi$$
−0.187816 + 0.982204i $$0.560141\pi$$
$$840$$ 0 0
$$841$$ −22685.0 −0.930131
$$842$$ − 9441.58i − 0.386435i
$$843$$ − 10146.5i − 0.414547i
$$844$$ 48634.4 1.98349
$$845$$ 0 0
$$846$$ −104638. −4.25240
$$847$$ − 5390.75i − 0.218688i
$$848$$ − 32198.0i − 1.30387i
$$849$$ −26822.2 −1.08426
$$850$$ 0 0
$$851$$ 23495.7 0.946442
$$852$$ 98416.1i 3.95737i
$$853$$ 27204.8i 1.09200i 0.837786 + 0.545999i $$0.183850\pi$$
−0.837786 + 0.545999i $$0.816150\pi$$
$$854$$ 6538.68 0.262001
$$855$$ 0 0
$$856$$ 22769.8 0.909179
$$857$$ 38060.0i 1.51704i 0.651649 + 0.758520i $$0.274077\pi$$
−0.651649 + 0.758520i $$0.725923\pi$$
$$858$$ − 18004.9i − 0.716405i
$$859$$ 33326.2 1.32372 0.661860 0.749627i $$-0.269767\pi$$
0.661860 + 0.749627i $$0.269767\pi$$
$$860$$ 0 0
$$861$$ −579.531 −0.0229389
$$862$$ 422.378i 0.0166894i
$$863$$ − 41724.2i − 1.64578i −0.568201 0.822890i $$-0.692360\pi$$
0.568201 0.822890i $$-0.307640\pi$$
$$864$$ 17475.4 0.688108
$$865$$ 0 0
$$866$$ 77223.8 3.03022
$$867$$ − 2449.38i − 0.0959460i
$$868$$ − 2309.60i − 0.0903146i
$$869$$ 56569.7 2.20828
$$870$$ 0 0
$$871$$ −124.068 −0.00482649
$$872$$ − 58907.5i − 2.28768i
$$873$$ − 17009.6i − 0.659438i
$$874$$ −60328.6 −2.33483
$$875$$ 0 0
$$876$$ 28411.6 1.09582
$$877$$ 49337.3i 1.89966i 0.312767 + 0.949830i $$0.398744\pi$$
−0.312767 + 0.949830i $$0.601256\pi$$
$$878$$ − 15423.3i − 0.592838i
$$879$$ −63194.0 −2.42489
$$880$$ 0 0
$$881$$ 8845.46 0.338265 0.169132 0.985593i $$-0.445903\pi$$
0.169132 + 0.985593i $$0.445903\pi$$
$$882$$ − 74096.8i − 2.82876i
$$883$$ 14724.2i 0.561165i 0.959830 + 0.280582i $$0.0905276\pi$$
−0.959830 + 0.280582i $$0.909472\pi$$
$$884$$ −2374.10 −0.0903277
$$885$$ 0 0
$$886$$ 9023.14 0.342143
$$887$$ − 3864.38i − 0.146283i −0.997322 0.0731415i $$-0.976698\pi$$
0.997322 0.0731415i $$-0.0233025\pi$$
$$888$$ − 51858.1i − 1.95974i
$$889$$ −7360.60 −0.277690
$$890$$ 0 0
$$891$$ −3688.07 −0.138670
$$892$$ − 81160.9i − 3.04649i
$$893$$ − 30849.1i − 1.15602i
$$894$$ −79023.2 −2.95630
$$895$$ 0 0
$$896$$ 7351.61 0.274107
$$897$$ 12311.8i 0.458283i
$$898$$ − 12577.2i − 0.467379i
$$899$$ −1441.14 −0.0534648
$$900$$ 0 0
$$901$$ 5609.04 0.207397
$$902$$ 4719.92i 0.174231i
$$903$$ 8984.15i 0.331089i
$$904$$ −47560.6 −1.74982
$$905$$ 0 0
$$906$$ −87473.7 −3.20764
$$907$$ 743.409i 0.0272155i 0.999907 + 0.0136078i $$0.00433162\pi$$
−0.999907 + 0.0136078i $$0.995668\pi$$
$$908$$ − 24182.5i − 0.883839i
$$909$$ −5905.76 −0.215491
$$910$$ 0 0
$$911$$ 16291.0 0.592475 0.296238 0.955114i $$-0.404268\pi$$
0.296238 + 0.955114i $$0.404268\pi$$
$$912$$ 55013.4i 1.99745i
$$913$$ 45336.8i 1.64340i
$$914$$ 74402.5 2.69258
$$915$$ 0 0
$$916$$ 15500.6 0.559122
$$917$$ 1553.04i 0.0559280i
$$918$$ 12929.5i 0.464856i
$$919$$ 6188.99 0.222150 0.111075 0.993812i $$-0.464571\pi$$
0.111075 + 0.993812i $$0.464571\pi$$
$$920$$ 0 0
$$921$$ −55390.5 −1.98174
$$922$$ − 89311.6i − 3.19015i
$$923$$ 5402.00i 0.192643i
$$924$$ 29364.8 1.04549
$$925$$ 0 0
$$926$$ −93033.0 −3.30157
$$927$$ − 8786.47i − 0.311311i
$$928$$ 4773.30i 0.168848i
$$929$$ 31661.7 1.11818 0.559089 0.829108i $$-0.311151\pi$$
0.559089 + 0.829108i $$0.311151\pi$$
$$930$$ 0 0
$$931$$ 21845.0 0.769003
$$932$$ − 20724.3i − 0.728376i
$$933$$ − 75734.8i − 2.65750i
$$934$$ 37273.6 1.30581
$$935$$ 0 0
$$936$$ 16959.8 0.592251
$$937$$ − 35010.5i − 1.22064i −0.792153 0.610322i $$-0.791040\pi$$
0.792153 0.610322i $$-0.208960\pi$$
$$938$$ − 295.776i − 0.0102958i
$$939$$ 22279.2 0.774286
$$940$$ 0 0
$$941$$ −45625.8 −1.58061 −0.790307 0.612711i $$-0.790079\pi$$
−0.790307 + 0.612711i $$0.790079\pi$$
$$942$$ − 11217.2i − 0.387977i
$$943$$ − 3227.51i − 0.111455i
$$944$$ −66227.5 −2.28339
$$945$$ 0 0
$$946$$ 73170.2 2.51477
$$947$$ 21508.4i 0.738044i 0.929421 + 0.369022i $$0.120307\pi$$
−0.929421 + 0.369022i $$0.879693\pi$$
$$948$$ 158615.i 5.43416i
$$949$$ 1559.50 0.0533439
$$950$$ 0 0
$$951$$ 36177.4 1.23358
$$952$$ − 3046.52i − 0.103717i
$$953$$ 35686.7i 1.21302i 0.795076 + 0.606509i $$0.207431\pi$$
−0.795076 + 0.606509i $$0.792569\pi$$
$$954$$ −74440.5 −2.52631
$$955$$ 0 0
$$956$$ 85744.0 2.90079
$$957$$ − 18323.0i − 0.618912i
$$958$$ 93357.8i 3.14849i
$$959$$ 498.486 0.0167851
$$960$$ 0 0
$$961$$ −28572.2 −0.959088
$$962$$ − 5288.17i − 0.177232i
$$963$$ − 21749.9i − 0.727810i
$$964$$ 116133. 3.88006
$$965$$ 0 0
$$966$$ −29351.3 −0.977600
$$967$$ 3731.33i 0.124086i 0.998073 + 0.0620432i $$0.0197617\pi$$
−0.998073 + 0.0620432i $$0.980238\pi$$
$$968$$ − 66262.5i − 2.20016i
$$969$$ −9583.60 −0.317719
$$970$$ 0 0
$$971$$ 17645.1 0.583171 0.291585 0.956545i $$-0.405817\pi$$
0.291585 + 0.956545i $$0.405817\pi$$
$$972$$ 60358.3i 1.99176i
$$973$$ 7917.35i 0.260862i
$$974$$ 51348.7 1.68924
$$975$$ 0 0
$$976$$ 33206.7 1.08906
$$977$$ − 24941.2i − 0.816723i −0.912820 0.408362i $$-0.866100\pi$$
0.912820 0.408362i $$-0.133900\pi$$
$$978$$ 61631.8i 2.01510i
$$979$$ 59152.1 1.93106
$$980$$ 0 0
$$981$$ −56268.9 −1.83132
$$982$$ − 6277.85i − 0.204006i
$$983$$ − 22506.2i − 0.730252i −0.930958 0.365126i $$-0.881026\pi$$
0.930958 0.365126i $$-0.118974\pi$$
$$984$$ −7123.53 −0.230782
$$985$$ 0 0
$$986$$ −3531.62 −0.114066
$$987$$ − 15008.8i − 0.484029i
$$988$$ 9289.07i 0.299114i
$$989$$ −50034.2 −1.60869
$$990$$ 0 0
$$991$$ 32694.1 1.04799 0.523997 0.851720i $$-0.324440\pi$$
0.523997 + 0.851720i $$0.324440\pi$$
$$992$$ − 4036.88i − 0.129205i
$$993$$ 8410.08i 0.268767i
$$994$$ −12878.3 −0.410941
$$995$$ 0 0
$$996$$ −127119. −4.04410
$$997$$ − 18248.8i − 0.579686i −0.957074 0.289843i $$-0.906397\pi$$
0.957074 0.289843i $$-0.0936030\pi$$
$$998$$ − 352.428i − 0.0111783i
$$999$$ −19702.5 −0.623983
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 425.4.b.f.324.6 6
5.2 odd 4 17.4.a.b.1.1 3
5.3 odd 4 425.4.a.g.1.3 3
5.4 even 2 inner 425.4.b.f.324.1 6
15.2 even 4 153.4.a.g.1.3 3
20.7 even 4 272.4.a.h.1.1 3
35.27 even 4 833.4.a.d.1.1 3
40.27 even 4 1088.4.a.x.1.3 3
40.37 odd 4 1088.4.a.v.1.1 3
55.32 even 4 2057.4.a.e.1.3 3
60.47 odd 4 2448.4.a.bi.1.2 3
85.47 odd 4 289.4.b.b.288.6 6
85.67 odd 4 289.4.a.b.1.1 3
85.72 odd 4 289.4.b.b.288.5 6

By twisted newform
Twist Min Dim Char Parity Ord Type
17.4.a.b.1.1 3 5.2 odd 4
153.4.a.g.1.3 3 15.2 even 4
272.4.a.h.1.1 3 20.7 even 4
289.4.a.b.1.1 3 85.67 odd 4
289.4.b.b.288.5 6 85.72 odd 4
289.4.b.b.288.6 6 85.47 odd 4
425.4.a.g.1.3 3 5.3 odd 4
425.4.b.f.324.1 6 5.4 even 2 inner
425.4.b.f.324.6 6 1.1 even 1 trivial
833.4.a.d.1.1 3 35.27 even 4
1088.4.a.v.1.1 3 40.37 odd 4
1088.4.a.x.1.3 3 40.27 even 4
2057.4.a.e.1.3 3 55.32 even 4
2448.4.a.bi.1.2 3 60.47 odd 4