# Properties

 Label 525.4.d.c.274.1 Level $525$ Weight $4$ Character 525.274 Analytic conductor $30.976$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Learn more about

## Newspace parameters

 Level: $$N$$ $$=$$ $$525 = 3 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 525.d (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$30.9760027530$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 21) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 274.1 Root $$-1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 525.274 Dual form 525.4.d.c.274.2

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-3.00000i q^{2} +3.00000i q^{3} -1.00000 q^{4} +9.00000 q^{6} +7.00000i q^{7} -21.0000i q^{8} -9.00000 q^{9} +O(q^{10})$$ $$q-3.00000i q^{2} +3.00000i q^{3} -1.00000 q^{4} +9.00000 q^{6} +7.00000i q^{7} -21.0000i q^{8} -9.00000 q^{9} -36.0000 q^{11} -3.00000i q^{12} +34.0000i q^{13} +21.0000 q^{14} -71.0000 q^{16} +42.0000i q^{17} +27.0000i q^{18} +124.000 q^{19} -21.0000 q^{21} +108.000i q^{22} +63.0000 q^{24} +102.000 q^{26} -27.0000i q^{27} -7.00000i q^{28} -102.000 q^{29} -160.000 q^{31} +45.0000i q^{32} -108.000i q^{33} +126.000 q^{34} +9.00000 q^{36} +398.000i q^{37} -372.000i q^{38} -102.000 q^{39} -318.000 q^{41} +63.0000i q^{42} +268.000i q^{43} +36.0000 q^{44} +240.000i q^{47} -213.000i q^{48} -49.0000 q^{49} -126.000 q^{51} -34.0000i q^{52} +498.000i q^{53} -81.0000 q^{54} +147.000 q^{56} +372.000i q^{57} +306.000i q^{58} +132.000 q^{59} +398.000 q^{61} +480.000i q^{62} -63.0000i q^{63} -433.000 q^{64} -324.000 q^{66} +92.0000i q^{67} -42.0000i q^{68} -720.000 q^{71} +189.000i q^{72} +502.000i q^{73} +1194.00 q^{74} -124.000 q^{76} -252.000i q^{77} +306.000i q^{78} +1024.00 q^{79} +81.0000 q^{81} +954.000i q^{82} +204.000i q^{83} +21.0000 q^{84} +804.000 q^{86} -306.000i q^{87} +756.000i q^{88} -354.000 q^{89} -238.000 q^{91} -480.000i q^{93} +720.000 q^{94} -135.000 q^{96} -286.000i q^{97} +147.000i q^{98} +324.000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{4} + 18q^{6} - 18q^{9} + O(q^{10})$$ $$2q - 2q^{4} + 18q^{6} - 18q^{9} - 72q^{11} + 42q^{14} - 142q^{16} + 248q^{19} - 42q^{21} + 126q^{24} + 204q^{26} - 204q^{29} - 320q^{31} + 252q^{34} + 18q^{36} - 204q^{39} - 636q^{41} + 72q^{44} - 98q^{49} - 252q^{51} - 162q^{54} + 294q^{56} + 264q^{59} + 796q^{61} - 866q^{64} - 648q^{66} - 1440q^{71} + 2388q^{74} - 248q^{76} + 2048q^{79} + 162q^{81} + 42q^{84} + 1608q^{86} - 708q^{89} - 476q^{91} + 1440q^{94} - 270q^{96} + 648q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/525\mathbb{Z}\right)^\times$$.

 $$n$$ $$127$$ $$176$$ $$451$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ − 3.00000i − 1.06066i −0.847791 0.530330i $$-0.822068\pi$$
0.847791 0.530330i $$-0.177932\pi$$
$$3$$ 3.00000i 0.577350i
$$4$$ −1.00000 −0.125000
$$5$$ 0 0
$$6$$ 9.00000 0.612372
$$7$$ 7.00000i 0.377964i
$$8$$ − 21.0000i − 0.928078i
$$9$$ −9.00000 −0.333333
$$10$$ 0 0
$$11$$ −36.0000 −0.986764 −0.493382 0.869813i $$-0.664240\pi$$
−0.493382 + 0.869813i $$0.664240\pi$$
$$12$$ − 3.00000i − 0.0721688i
$$13$$ 34.0000i 0.725377i 0.931910 + 0.362689i $$0.118141\pi$$
−0.931910 + 0.362689i $$0.881859\pi$$
$$14$$ 21.0000 0.400892
$$15$$ 0 0
$$16$$ −71.0000 −1.10938
$$17$$ 42.0000i 0.599206i 0.954064 + 0.299603i $$0.0968542\pi$$
−0.954064 + 0.299603i $$0.903146\pi$$
$$18$$ 27.0000i 0.353553i
$$19$$ 124.000 1.49724 0.748620 0.663000i $$-0.230717\pi$$
0.748620 + 0.663000i $$0.230717\pi$$
$$20$$ 0 0
$$21$$ −21.0000 −0.218218
$$22$$ 108.000i 1.04662i
$$23$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$24$$ 63.0000 0.535826
$$25$$ 0 0
$$26$$ 102.000 0.769379
$$27$$ − 27.0000i − 0.192450i
$$28$$ − 7.00000i − 0.0472456i
$$29$$ −102.000 −0.653135 −0.326568 0.945174i $$-0.605892\pi$$
−0.326568 + 0.945174i $$0.605892\pi$$
$$30$$ 0 0
$$31$$ −160.000 −0.926995 −0.463498 0.886098i $$-0.653406\pi$$
−0.463498 + 0.886098i $$0.653406\pi$$
$$32$$ 45.0000i 0.248592i
$$33$$ − 108.000i − 0.569709i
$$34$$ 126.000 0.635554
$$35$$ 0 0
$$36$$ 9.00000 0.0416667
$$37$$ 398.000i 1.76840i 0.467109 + 0.884200i $$0.345296\pi$$
−0.467109 + 0.884200i $$0.654704\pi$$
$$38$$ − 372.000i − 1.58806i
$$39$$ −102.000 −0.418797
$$40$$ 0 0
$$41$$ −318.000 −1.21130 −0.605649 0.795732i $$-0.707087\pi$$
−0.605649 + 0.795732i $$0.707087\pi$$
$$42$$ 63.0000i 0.231455i
$$43$$ 268.000i 0.950456i 0.879863 + 0.475228i $$0.157634\pi$$
−0.879863 + 0.475228i $$0.842366\pi$$
$$44$$ 36.0000 0.123346
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 240.000i 0.744843i 0.928064 + 0.372421i $$0.121472\pi$$
−0.928064 + 0.372421i $$0.878528\pi$$
$$48$$ − 213.000i − 0.640498i
$$49$$ −49.0000 −0.142857
$$50$$ 0 0
$$51$$ −126.000 −0.345952
$$52$$ − 34.0000i − 0.0906721i
$$53$$ 498.000i 1.29067i 0.763899 + 0.645335i $$0.223282\pi$$
−0.763899 + 0.645335i $$0.776718\pi$$
$$54$$ −81.0000 −0.204124
$$55$$ 0 0
$$56$$ 147.000 0.350780
$$57$$ 372.000i 0.864432i
$$58$$ 306.000i 0.692755i
$$59$$ 132.000 0.291270 0.145635 0.989338i $$-0.453477\pi$$
0.145635 + 0.989338i $$0.453477\pi$$
$$60$$ 0 0
$$61$$ 398.000 0.835388 0.417694 0.908588i $$-0.362838\pi$$
0.417694 + 0.908588i $$0.362838\pi$$
$$62$$ 480.000i 0.983227i
$$63$$ − 63.0000i − 0.125988i
$$64$$ −433.000 −0.845703
$$65$$ 0 0
$$66$$ −324.000 −0.604267
$$67$$ 92.0000i 0.167755i 0.996476 + 0.0838775i $$0.0267305\pi$$
−0.996476 + 0.0838775i $$0.973270\pi$$
$$68$$ − 42.0000i − 0.0749007i
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −720.000 −1.20350 −0.601748 0.798686i $$-0.705529\pi$$
−0.601748 + 0.798686i $$0.705529\pi$$
$$72$$ 189.000i 0.309359i
$$73$$ 502.000i 0.804858i 0.915451 + 0.402429i $$0.131834\pi$$
−0.915451 + 0.402429i $$0.868166\pi$$
$$74$$ 1194.00 1.87567
$$75$$ 0 0
$$76$$ −124.000 −0.187155
$$77$$ − 252.000i − 0.372962i
$$78$$ 306.000i 0.444201i
$$79$$ 1024.00 1.45834 0.729171 0.684332i $$-0.239906\pi$$
0.729171 + 0.684332i $$0.239906\pi$$
$$80$$ 0 0
$$81$$ 81.0000 0.111111
$$82$$ 954.000i 1.28478i
$$83$$ 204.000i 0.269782i 0.990860 + 0.134891i $$0.0430684\pi$$
−0.990860 + 0.134891i $$0.956932\pi$$
$$84$$ 21.0000 0.0272772
$$85$$ 0 0
$$86$$ 804.000 1.00811
$$87$$ − 306.000i − 0.377088i
$$88$$ 756.000i 0.915794i
$$89$$ −354.000 −0.421617 −0.210809 0.977527i $$-0.567610\pi$$
−0.210809 + 0.977527i $$0.567610\pi$$
$$90$$ 0 0
$$91$$ −238.000 −0.274167
$$92$$ 0 0
$$93$$ − 480.000i − 0.535201i
$$94$$ 720.000 0.790025
$$95$$ 0 0
$$96$$ −135.000 −0.143525
$$97$$ − 286.000i − 0.299370i −0.988734 0.149685i $$-0.952174\pi$$
0.988734 0.149685i $$-0.0478260\pi$$
$$98$$ 147.000i 0.151523i
$$99$$ 324.000 0.328921
$$100$$ 0 0
$$101$$ 414.000 0.407867 0.203933 0.978985i $$-0.434627\pi$$
0.203933 + 0.978985i $$0.434627\pi$$
$$102$$ 378.000i 0.366937i
$$103$$ − 56.0000i − 0.0535713i −0.999641 0.0267857i $$-0.991473\pi$$
0.999641 0.0267857i $$-0.00852716\pi$$
$$104$$ 714.000 0.673206
$$105$$ 0 0
$$106$$ 1494.00 1.36896
$$107$$ 12.0000i 0.0108419i 0.999985 + 0.00542095i $$0.00172555\pi$$
−0.999985 + 0.00542095i $$0.998274\pi$$
$$108$$ 27.0000i 0.0240563i
$$109$$ −1478.00 −1.29878 −0.649389 0.760457i $$-0.724975\pi$$
−0.649389 + 0.760457i $$0.724975\pi$$
$$110$$ 0 0
$$111$$ −1194.00 −1.02099
$$112$$ − 497.000i − 0.419304i
$$113$$ − 402.000i − 0.334664i −0.985901 0.167332i $$-0.946485\pi$$
0.985901 0.167332i $$-0.0535151\pi$$
$$114$$ 1116.00 0.916868
$$115$$ 0 0
$$116$$ 102.000 0.0816419
$$117$$ − 306.000i − 0.241792i
$$118$$ − 396.000i − 0.308939i
$$119$$ −294.000 −0.226478
$$120$$ 0 0
$$121$$ −35.0000 −0.0262960
$$122$$ − 1194.00i − 0.886063i
$$123$$ − 954.000i − 0.699344i
$$124$$ 160.000 0.115874
$$125$$ 0 0
$$126$$ −189.000 −0.133631
$$127$$ 1280.00i 0.894344i 0.894448 + 0.447172i $$0.147569\pi$$
−0.894448 + 0.447172i $$0.852431\pi$$
$$128$$ 1659.00i 1.14560i
$$129$$ −804.000 −0.548746
$$130$$ 0 0
$$131$$ 1764.00 1.17650 0.588250 0.808679i $$-0.299817\pi$$
0.588250 + 0.808679i $$0.299817\pi$$
$$132$$ 108.000i 0.0712136i
$$133$$ 868.000i 0.565903i
$$134$$ 276.000 0.177931
$$135$$ 0 0
$$136$$ 882.000 0.556109
$$137$$ − 2358.00i − 1.47049i −0.677800 0.735246i $$-0.737066\pi$$
0.677800 0.735246i $$-0.262934\pi$$
$$138$$ 0 0
$$139$$ 52.0000 0.0317308 0.0158654 0.999874i $$-0.494950\pi$$
0.0158654 + 0.999874i $$0.494950\pi$$
$$140$$ 0 0
$$141$$ −720.000 −0.430035
$$142$$ 2160.00i 1.27650i
$$143$$ − 1224.00i − 0.715776i
$$144$$ 639.000 0.369792
$$145$$ 0 0
$$146$$ 1506.00 0.853681
$$147$$ − 147.000i − 0.0824786i
$$148$$ − 398.000i − 0.221050i
$$149$$ 1746.00 0.959986 0.479993 0.877272i $$-0.340639\pi$$
0.479993 + 0.877272i $$0.340639\pi$$
$$150$$ 0 0
$$151$$ −232.000 −0.125032 −0.0625162 0.998044i $$-0.519913\pi$$
−0.0625162 + 0.998044i $$0.519913\pi$$
$$152$$ − 2604.00i − 1.38955i
$$153$$ − 378.000i − 0.199735i
$$154$$ −756.000 −0.395586
$$155$$ 0 0
$$156$$ 102.000 0.0523496
$$157$$ 1694.00i 0.861120i 0.902562 + 0.430560i $$0.141684\pi$$
−0.902562 + 0.430560i $$0.858316\pi$$
$$158$$ − 3072.00i − 1.54681i
$$159$$ −1494.00 −0.745169
$$160$$ 0 0
$$161$$ 0 0
$$162$$ − 243.000i − 0.117851i
$$163$$ 2932.00i 1.40891i 0.709750 + 0.704454i $$0.248808\pi$$
−0.709750 + 0.704454i $$0.751192\pi$$
$$164$$ 318.000 0.151412
$$165$$ 0 0
$$166$$ 612.000 0.286147
$$167$$ 1176.00i 0.544920i 0.962167 + 0.272460i $$0.0878372\pi$$
−0.962167 + 0.272460i $$0.912163\pi$$
$$168$$ 441.000i 0.202523i
$$169$$ 1041.00 0.473828
$$170$$ 0 0
$$171$$ −1116.00 −0.499080
$$172$$ − 268.000i − 0.118807i
$$173$$ − 870.000i − 0.382340i −0.981557 0.191170i $$-0.938772\pi$$
0.981557 0.191170i $$-0.0612282\pi$$
$$174$$ −918.000 −0.399962
$$175$$ 0 0
$$176$$ 2556.00 1.09469
$$177$$ 396.000i 0.168165i
$$178$$ 1062.00i 0.447193i
$$179$$ 2316.00 0.967072 0.483536 0.875324i $$-0.339352\pi$$
0.483536 + 0.875324i $$0.339352\pi$$
$$180$$ 0 0
$$181$$ −106.000 −0.0435299 −0.0217650 0.999763i $$-0.506929\pi$$
−0.0217650 + 0.999763i $$0.506929\pi$$
$$182$$ 714.000i 0.290798i
$$183$$ 1194.00i 0.482312i
$$184$$ 0 0
$$185$$ 0 0
$$186$$ −1440.00 −0.567666
$$187$$ − 1512.00i − 0.591275i
$$188$$ − 240.000i − 0.0931053i
$$189$$ 189.000 0.0727393
$$190$$ 0 0
$$191$$ −1128.00 −0.427326 −0.213663 0.976907i $$-0.568539\pi$$
−0.213663 + 0.976907i $$0.568539\pi$$
$$192$$ − 1299.00i − 0.488267i
$$193$$ − 4034.00i − 1.50453i −0.658862 0.752263i $$-0.728962\pi$$
0.658862 0.752263i $$-0.271038\pi$$
$$194$$ −858.000 −0.317530
$$195$$ 0 0
$$196$$ 49.0000 0.0178571
$$197$$ − 1314.00i − 0.475221i −0.971360 0.237611i $$-0.923636\pi$$
0.971360 0.237611i $$-0.0763643\pi$$
$$198$$ − 972.000i − 0.348874i
$$199$$ −5096.00 −1.81531 −0.907653 0.419722i $$-0.862128\pi$$
−0.907653 + 0.419722i $$0.862128\pi$$
$$200$$ 0 0
$$201$$ −276.000 −0.0968534
$$202$$ − 1242.00i − 0.432608i
$$203$$ − 714.000i − 0.246862i
$$204$$ 126.000 0.0432439
$$205$$ 0 0
$$206$$ −168.000 −0.0568209
$$207$$ 0 0
$$208$$ − 2414.00i − 0.804715i
$$209$$ −4464.00 −1.47742
$$210$$ 0 0
$$211$$ −3076.00 −1.00360 −0.501802 0.864982i $$-0.667330\pi$$
−0.501802 + 0.864982i $$0.667330\pi$$
$$212$$ − 498.000i − 0.161334i
$$213$$ − 2160.00i − 0.694839i
$$214$$ 36.0000 0.0114996
$$215$$ 0 0
$$216$$ −567.000 −0.178609
$$217$$ − 1120.00i − 0.350371i
$$218$$ 4434.00i 1.37756i
$$219$$ −1506.00 −0.464685
$$220$$ 0 0
$$221$$ −1428.00 −0.434650
$$222$$ 3582.00i 1.08292i
$$223$$ 1888.00i 0.566950i 0.958980 + 0.283475i $$0.0914873\pi$$
−0.958980 + 0.283475i $$0.908513\pi$$
$$224$$ −315.000 −0.0939590
$$225$$ 0 0
$$226$$ −1206.00 −0.354964
$$227$$ − 4716.00i − 1.37891i −0.724330 0.689454i $$-0.757851\pi$$
0.724330 0.689454i $$-0.242149\pi$$
$$228$$ − 372.000i − 0.108054i
$$229$$ 1690.00 0.487678 0.243839 0.969816i $$-0.421593\pi$$
0.243839 + 0.969816i $$0.421593\pi$$
$$230$$ 0 0
$$231$$ 756.000 0.215330
$$232$$ 2142.00i 0.606160i
$$233$$ − 138.000i − 0.0388012i −0.999812 0.0194006i $$-0.993824\pi$$
0.999812 0.0194006i $$-0.00617579\pi$$
$$234$$ −918.000 −0.256460
$$235$$ 0 0
$$236$$ −132.000 −0.0364088
$$237$$ 3072.00i 0.841974i
$$238$$ 882.000i 0.240217i
$$239$$ −1896.00 −0.513147 −0.256573 0.966525i $$-0.582594\pi$$
−0.256573 + 0.966525i $$0.582594\pi$$
$$240$$ 0 0
$$241$$ −3598.00 −0.961691 −0.480846 0.876805i $$-0.659670\pi$$
−0.480846 + 0.876805i $$0.659670\pi$$
$$242$$ 105.000i 0.0278911i
$$243$$ 243.000i 0.0641500i
$$244$$ −398.000 −0.104424
$$245$$ 0 0
$$246$$ −2862.00 −0.741766
$$247$$ 4216.00i 1.08606i
$$248$$ 3360.00i 0.860323i
$$249$$ −612.000 −0.155759
$$250$$ 0 0
$$251$$ −3060.00 −0.769504 −0.384752 0.923020i $$-0.625713\pi$$
−0.384752 + 0.923020i $$0.625713\pi$$
$$252$$ 63.0000i 0.0157485i
$$253$$ 0 0
$$254$$ 3840.00 0.948595
$$255$$ 0 0
$$256$$ 1513.00 0.369385
$$257$$ − 6822.00i − 1.65582i −0.560864 0.827908i $$-0.689531\pi$$
0.560864 0.827908i $$-0.310469\pi$$
$$258$$ 2412.00i 0.582033i
$$259$$ −2786.00 −0.668392
$$260$$ 0 0
$$261$$ 918.000 0.217712
$$262$$ − 5292.00i − 1.24787i
$$263$$ − 2592.00i − 0.607717i −0.952717 0.303858i $$-0.901725\pi$$
0.952717 0.303858i $$-0.0982750\pi$$
$$264$$ −2268.00 −0.528734
$$265$$ 0 0
$$266$$ 2604.00 0.600231
$$267$$ − 1062.00i − 0.243421i
$$268$$ − 92.0000i − 0.0209694i
$$269$$ −8214.00 −1.86177 −0.930886 0.365311i $$-0.880963\pi$$
−0.930886 + 0.365311i $$0.880963\pi$$
$$270$$ 0 0
$$271$$ −5344.00 −1.19788 −0.598939 0.800795i $$-0.704411\pi$$
−0.598939 + 0.800795i $$0.704411\pi$$
$$272$$ − 2982.00i − 0.664744i
$$273$$ − 714.000i − 0.158290i
$$274$$ −7074.00 −1.55969
$$275$$ 0 0
$$276$$ 0 0
$$277$$ − 6514.00i − 1.41295i −0.707736 0.706477i $$-0.750283\pi$$
0.707736 0.706477i $$-0.249717\pi$$
$$278$$ − 156.000i − 0.0336556i
$$279$$ 1440.00 0.308998
$$280$$ 0 0
$$281$$ 6618.00 1.40497 0.702485 0.711698i $$-0.252074\pi$$
0.702485 + 0.711698i $$0.252074\pi$$
$$282$$ 2160.00i 0.456121i
$$283$$ − 3260.00i − 0.684759i −0.939562 0.342380i $$-0.888767\pi$$
0.939562 0.342380i $$-0.111233\pi$$
$$284$$ 720.000 0.150437
$$285$$ 0 0
$$286$$ −3672.00 −0.759195
$$287$$ − 2226.00i − 0.457828i
$$288$$ − 405.000i − 0.0828641i
$$289$$ 3149.00 0.640953
$$290$$ 0 0
$$291$$ 858.000 0.172841
$$292$$ − 502.000i − 0.100607i
$$293$$ − 5118.00i − 1.02047i −0.860036 0.510233i $$-0.829559\pi$$
0.860036 0.510233i $$-0.170441\pi$$
$$294$$ −441.000 −0.0874818
$$295$$ 0 0
$$296$$ 8358.00 1.64121
$$297$$ 972.000i 0.189903i
$$298$$ − 5238.00i − 1.01822i
$$299$$ 0 0
$$300$$ 0 0
$$301$$ −1876.00 −0.359239
$$302$$ 696.000i 0.132617i
$$303$$ 1242.00i 0.235482i
$$304$$ −8804.00 −1.66100
$$305$$ 0 0
$$306$$ −1134.00 −0.211851
$$307$$ 452.000i 0.0840293i 0.999117 + 0.0420147i $$0.0133776\pi$$
−0.999117 + 0.0420147i $$0.986622\pi$$
$$308$$ 252.000i 0.0466202i
$$309$$ 168.000 0.0309294
$$310$$ 0 0
$$311$$ 5016.00 0.914570 0.457285 0.889320i $$-0.348822\pi$$
0.457285 + 0.889320i $$0.348822\pi$$
$$312$$ 2142.00i 0.388676i
$$313$$ − 5402.00i − 0.975524i −0.872977 0.487762i $$-0.837813\pi$$
0.872977 0.487762i $$-0.162187\pi$$
$$314$$ 5082.00 0.913356
$$315$$ 0 0
$$316$$ −1024.00 −0.182293
$$317$$ 10086.0i 1.78702i 0.449041 + 0.893511i $$0.351766\pi$$
−0.449041 + 0.893511i $$0.648234\pi$$
$$318$$ 4482.00i 0.790371i
$$319$$ 3672.00 0.644491
$$320$$ 0 0
$$321$$ −36.0000 −0.00625958
$$322$$ 0 0
$$323$$ 5208.00i 0.897154i
$$324$$ −81.0000 −0.0138889
$$325$$ 0 0
$$326$$ 8796.00 1.49437
$$327$$ − 4434.00i − 0.749849i
$$328$$ 6678.00i 1.12418i
$$329$$ −1680.00 −0.281524
$$330$$ 0 0
$$331$$ −8044.00 −1.33577 −0.667883 0.744267i $$-0.732799\pi$$
−0.667883 + 0.744267i $$0.732799\pi$$
$$332$$ − 204.000i − 0.0337228i
$$333$$ − 3582.00i − 0.589467i
$$334$$ 3528.00 0.577975
$$335$$ 0 0
$$336$$ 1491.00 0.242085
$$337$$ 4178.00i 0.675342i 0.941264 + 0.337671i $$0.109639\pi$$
−0.941264 + 0.337671i $$0.890361\pi$$
$$338$$ − 3123.00i − 0.502570i
$$339$$ 1206.00 0.193218
$$340$$ 0 0
$$341$$ 5760.00 0.914726
$$342$$ 3348.00i 0.529354i
$$343$$ − 343.000i − 0.0539949i
$$344$$ 5628.00 0.882097
$$345$$ 0 0
$$346$$ −2610.00 −0.405533
$$347$$ 156.000i 0.0241341i 0.999927 + 0.0120670i $$0.00384115\pi$$
−0.999927 + 0.0120670i $$0.996159\pi$$
$$348$$ 306.000i 0.0471360i
$$349$$ 12418.0 1.90464 0.952321 0.305097i $$-0.0986888\pi$$
0.952321 + 0.305097i $$0.0986888\pi$$
$$350$$ 0 0
$$351$$ 918.000 0.139599
$$352$$ − 1620.00i − 0.245302i
$$353$$ 7830.00i 1.18059i 0.807187 + 0.590296i $$0.200989\pi$$
−0.807187 + 0.590296i $$0.799011\pi$$
$$354$$ 1188.00 0.178366
$$355$$ 0 0
$$356$$ 354.000 0.0527021
$$357$$ − 882.000i − 0.130757i
$$358$$ − 6948.00i − 1.02574i
$$359$$ 9312.00 1.36899 0.684497 0.729016i $$-0.260022\pi$$
0.684497 + 0.729016i $$0.260022\pi$$
$$360$$ 0 0
$$361$$ 8517.00 1.24173
$$362$$ 318.000i 0.0461705i
$$363$$ − 105.000i − 0.0151820i
$$364$$ 238.000 0.0342709
$$365$$ 0 0
$$366$$ 3582.00 0.511569
$$367$$ − 3760.00i − 0.534797i −0.963586 0.267398i $$-0.913836\pi$$
0.963586 0.267398i $$-0.0861640\pi$$
$$368$$ 0 0
$$369$$ 2862.00 0.403766
$$370$$ 0 0
$$371$$ −3486.00 −0.487828
$$372$$ 480.000i 0.0669001i
$$373$$ − 5870.00i − 0.814845i −0.913240 0.407422i $$-0.866428\pi$$
0.913240 0.407422i $$-0.133572\pi$$
$$374$$ −4536.00 −0.627142
$$375$$ 0 0
$$376$$ 5040.00 0.691272
$$377$$ − 3468.00i − 0.473769i
$$378$$ − 567.000i − 0.0771517i
$$379$$ 1852.00 0.251005 0.125502 0.992093i $$-0.459946\pi$$
0.125502 + 0.992093i $$0.459946\pi$$
$$380$$ 0 0
$$381$$ −3840.00 −0.516350
$$382$$ 3384.00i 0.453247i
$$383$$ − 2160.00i − 0.288175i −0.989565 0.144087i $$-0.953975\pi$$
0.989565 0.144087i $$-0.0460246\pi$$
$$384$$ −4977.00 −0.661410
$$385$$ 0 0
$$386$$ −12102.0 −1.59579
$$387$$ − 2412.00i − 0.316819i
$$388$$ 286.000i 0.0374213i
$$389$$ 6786.00 0.884483 0.442241 0.896896i $$-0.354183\pi$$
0.442241 + 0.896896i $$0.354183\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ 1029.00i 0.132583i
$$393$$ 5292.00i 0.679252i
$$394$$ −3942.00 −0.504048
$$395$$ 0 0
$$396$$ −324.000 −0.0411152
$$397$$ − 6514.00i − 0.823497i −0.911298 0.411748i $$-0.864918\pi$$
0.911298 0.411748i $$-0.135082\pi$$
$$398$$ 15288.0i 1.92542i
$$399$$ −2604.00 −0.326724
$$400$$ 0 0
$$401$$ 3330.00 0.414694 0.207347 0.978267i $$-0.433517\pi$$
0.207347 + 0.978267i $$0.433517\pi$$
$$402$$ 828.000i 0.102729i
$$403$$ − 5440.00i − 0.672421i
$$404$$ −414.000 −0.0509833
$$405$$ 0 0
$$406$$ −2142.00 −0.261837
$$407$$ − 14328.0i − 1.74499i
$$408$$ 2646.00i 0.321070i
$$409$$ 5398.00 0.652601 0.326301 0.945266i $$-0.394198\pi$$
0.326301 + 0.945266i $$0.394198\pi$$
$$410$$ 0 0
$$411$$ 7074.00 0.848990
$$412$$ 56.0000i 0.00669641i
$$413$$ 924.000i 0.110090i
$$414$$ 0 0
$$415$$ 0 0
$$416$$ −1530.00 −0.180323
$$417$$ 156.000i 0.0183198i
$$418$$ 13392.0i 1.56704i
$$419$$ −13092.0 −1.52646 −0.763229 0.646128i $$-0.776387\pi$$
−0.763229 + 0.646128i $$0.776387\pi$$
$$420$$ 0 0
$$421$$ −322.000 −0.0372763 −0.0186381 0.999826i $$-0.505933\pi$$
−0.0186381 + 0.999826i $$0.505933\pi$$
$$422$$ 9228.00i 1.06448i
$$423$$ − 2160.00i − 0.248281i
$$424$$ 10458.0 1.19784
$$425$$ 0 0
$$426$$ −6480.00 −0.736988
$$427$$ 2786.00i 0.315747i
$$428$$ − 12.0000i − 0.00135524i
$$429$$ 3672.00 0.413254
$$430$$ 0 0
$$431$$ 2616.00 0.292363 0.146181 0.989258i $$-0.453302\pi$$
0.146181 + 0.989258i $$0.453302\pi$$
$$432$$ 1917.00i 0.213499i
$$433$$ − 4322.00i − 0.479681i −0.970812 0.239841i $$-0.922905\pi$$
0.970812 0.239841i $$-0.0770952\pi$$
$$434$$ −3360.00 −0.371625
$$435$$ 0 0
$$436$$ 1478.00 0.162347
$$437$$ 0 0
$$438$$ 4518.00i 0.492873i
$$439$$ 9016.00 0.980205 0.490103 0.871665i $$-0.336959\pi$$
0.490103 + 0.871665i $$0.336959\pi$$
$$440$$ 0 0
$$441$$ 441.000 0.0476190
$$442$$ 4284.00i 0.461016i
$$443$$ 5268.00i 0.564989i 0.959269 + 0.282495i $$0.0911619\pi$$
−0.959269 + 0.282495i $$0.908838\pi$$
$$444$$ 1194.00 0.127623
$$445$$ 0 0
$$446$$ 5664.00 0.601341
$$447$$ 5238.00i 0.554248i
$$448$$ − 3031.00i − 0.319646i
$$449$$ 5310.00 0.558117 0.279058 0.960274i $$-0.409978\pi$$
0.279058 + 0.960274i $$0.409978\pi$$
$$450$$ 0 0
$$451$$ 11448.0 1.19527
$$452$$ 402.000i 0.0418329i
$$453$$ − 696.000i − 0.0721875i
$$454$$ −14148.0 −1.46255
$$455$$ 0 0
$$456$$ 7812.00 0.802260
$$457$$ 15770.0i 1.61420i 0.590415 + 0.807100i $$0.298964\pi$$
−0.590415 + 0.807100i $$0.701036\pi$$
$$458$$ − 5070.00i − 0.517261i
$$459$$ 1134.00 0.115317
$$460$$ 0 0
$$461$$ −5370.00 −0.542529 −0.271264 0.962505i $$-0.587442\pi$$
−0.271264 + 0.962505i $$0.587442\pi$$
$$462$$ − 2268.00i − 0.228392i
$$463$$ 3328.00i 0.334050i 0.985953 + 0.167025i $$0.0534161\pi$$
−0.985953 + 0.167025i $$0.946584\pi$$
$$464$$ 7242.00 0.724572
$$465$$ 0 0
$$466$$ −414.000 −0.0411549
$$467$$ 4548.00i 0.450656i 0.974283 + 0.225328i $$0.0723454\pi$$
−0.974283 + 0.225328i $$0.927655\pi$$
$$468$$ 306.000i 0.0302240i
$$469$$ −644.000 −0.0634055
$$470$$ 0 0
$$471$$ −5082.00 −0.497168
$$472$$ − 2772.00i − 0.270321i
$$473$$ − 9648.00i − 0.937876i
$$474$$ 9216.00 0.893048
$$475$$ 0 0
$$476$$ 294.000 0.0283098
$$477$$ − 4482.00i − 0.430224i
$$478$$ 5688.00i 0.544274i
$$479$$ 8064.00 0.769214 0.384607 0.923080i $$-0.374337\pi$$
0.384607 + 0.923080i $$0.374337\pi$$
$$480$$ 0 0
$$481$$ −13532.0 −1.28276
$$482$$ 10794.0i 1.02003i
$$483$$ 0 0
$$484$$ 35.0000 0.00328700
$$485$$ 0 0
$$486$$ 729.000 0.0680414
$$487$$ 16616.0i 1.54608i 0.634355 + 0.773042i $$0.281266\pi$$
−0.634355 + 0.773042i $$0.718734\pi$$
$$488$$ − 8358.00i − 0.775305i
$$489$$ −8796.00 −0.813433
$$490$$ 0 0
$$491$$ −7140.00 −0.656260 −0.328130 0.944633i $$-0.606418\pi$$
−0.328130 + 0.944633i $$0.606418\pi$$
$$492$$ 954.000i 0.0874180i
$$493$$ − 4284.00i − 0.391362i
$$494$$ 12648.0 1.15194
$$495$$ 0 0
$$496$$ 11360.0 1.02839
$$497$$ − 5040.00i − 0.454879i
$$498$$ 1836.00i 0.165207i
$$499$$ 9124.00 0.818530 0.409265 0.912416i $$-0.365785\pi$$
0.409265 + 0.912416i $$0.365785\pi$$
$$500$$ 0 0
$$501$$ −3528.00 −0.314610
$$502$$ 9180.00i 0.816182i
$$503$$ 6552.00i 0.580794i 0.956906 + 0.290397i $$0.0937873\pi$$
−0.956906 + 0.290397i $$0.906213\pi$$
$$504$$ −1323.00 −0.116927
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 3123.00i 0.273565i
$$508$$ − 1280.00i − 0.111793i
$$509$$ −2790.00 −0.242956 −0.121478 0.992594i $$-0.538763\pi$$
−0.121478 + 0.992594i $$0.538763\pi$$
$$510$$ 0 0
$$511$$ −3514.00 −0.304208
$$512$$ 8733.00i 0.753804i
$$513$$ − 3348.00i − 0.288144i
$$514$$ −20466.0 −1.75626
$$515$$ 0 0
$$516$$ 804.000 0.0685933
$$517$$ − 8640.00i − 0.734984i
$$518$$ 8358.00i 0.708937i
$$519$$ 2610.00 0.220744
$$520$$ 0 0
$$521$$ −14862.0 −1.24974 −0.624871 0.780728i $$-0.714849\pi$$
−0.624871 + 0.780728i $$0.714849\pi$$
$$522$$ − 2754.00i − 0.230918i
$$523$$ − 17660.0i − 1.47652i −0.674518 0.738258i $$-0.735649\pi$$
0.674518 0.738258i $$-0.264351\pi$$
$$524$$ −1764.00 −0.147062
$$525$$ 0 0
$$526$$ −7776.00 −0.644581
$$527$$ − 6720.00i − 0.555461i
$$528$$ 7668.00i 0.632021i
$$529$$ 12167.0 1.00000
$$530$$ 0 0
$$531$$ −1188.00 −0.0970900
$$532$$ − 868.000i − 0.0707379i
$$533$$ − 10812.0i − 0.878649i
$$534$$ −3186.00 −0.258187
$$535$$ 0 0
$$536$$ 1932.00 0.155690
$$537$$ 6948.00i 0.558340i
$$538$$ 24642.0i 1.97471i
$$539$$ 1764.00 0.140966
$$540$$ 0 0
$$541$$ −19834.0 −1.57621 −0.788106 0.615540i $$-0.788938\pi$$
−0.788106 + 0.615540i $$0.788938\pi$$
$$542$$ 16032.0i 1.27054i
$$543$$ − 318.000i − 0.0251320i
$$544$$ −1890.00 −0.148958
$$545$$ 0 0
$$546$$ −2142.00 −0.167892
$$547$$ 20972.0i 1.63930i 0.572863 + 0.819651i $$0.305833\pi$$
−0.572863 + 0.819651i $$0.694167\pi$$
$$548$$ 2358.00i 0.183812i
$$549$$ −3582.00 −0.278463
$$550$$ 0 0
$$551$$ −12648.0 −0.977900
$$552$$ 0 0
$$553$$ 7168.00i 0.551201i
$$554$$ −19542.0 −1.49866
$$555$$ 0 0
$$556$$ −52.0000 −0.00396635
$$557$$ 21174.0i 1.61072i 0.592786 + 0.805360i $$0.298028\pi$$
−0.592786 + 0.805360i $$0.701972\pi$$
$$558$$ − 4320.00i − 0.327742i
$$559$$ −9112.00 −0.689439
$$560$$ 0 0
$$561$$ 4536.00 0.341373
$$562$$ − 19854.0i − 1.49020i
$$563$$ 17772.0i 1.33037i 0.746677 + 0.665187i $$0.231648\pi$$
−0.746677 + 0.665187i $$0.768352\pi$$
$$564$$ 720.000 0.0537544
$$565$$ 0 0
$$566$$ −9780.00 −0.726297
$$567$$ 567.000i 0.0419961i
$$568$$ 15120.0i 1.11694i
$$569$$ −8250.00 −0.607835 −0.303917 0.952698i $$-0.598295\pi$$
−0.303917 + 0.952698i $$0.598295\pi$$
$$570$$ 0 0
$$571$$ 20756.0 1.52121 0.760606 0.649214i $$-0.224902\pi$$
0.760606 + 0.649214i $$0.224902\pi$$
$$572$$ 1224.00i 0.0894720i
$$573$$ − 3384.00i − 0.246717i
$$574$$ −6678.00 −0.485600
$$575$$ 0 0
$$576$$ 3897.00 0.281901
$$577$$ 2.00000i 0 0.000144300i 1.00000 7.21500e-5i $$2.29661e-5\pi$$
−1.00000 7.21500e-5i $$0.999977\pi$$
$$578$$ − 9447.00i − 0.679833i
$$579$$ 12102.0 0.868639
$$580$$ 0 0
$$581$$ −1428.00 −0.101968
$$582$$ − 2574.00i − 0.183326i
$$583$$ − 17928.0i − 1.27359i
$$584$$ 10542.0 0.746971
$$585$$ 0 0
$$586$$ −15354.0 −1.08237
$$587$$ 26364.0i 1.85376i 0.375354 + 0.926881i $$0.377521\pi$$
−0.375354 + 0.926881i $$0.622479\pi$$
$$588$$ 147.000i 0.0103098i
$$589$$ −19840.0 −1.38793
$$590$$ 0 0
$$591$$ 3942.00 0.274369
$$592$$ − 28258.0i − 1.96182i
$$593$$ − 2298.00i − 0.159136i −0.996829 0.0795679i $$-0.974646\pi$$
0.996829 0.0795679i $$-0.0253541\pi$$
$$594$$ 2916.00 0.201422
$$595$$ 0 0
$$596$$ −1746.00 −0.119998
$$597$$ − 15288.0i − 1.04807i
$$598$$ 0 0
$$599$$ −3072.00 −0.209547 −0.104773 0.994496i $$-0.533412\pi$$
−0.104773 + 0.994496i $$0.533412\pi$$
$$600$$ 0 0
$$601$$ 24554.0 1.66652 0.833260 0.552881i $$-0.186472\pi$$
0.833260 + 0.552881i $$0.186472\pi$$
$$602$$ 5628.00i 0.381030i
$$603$$ − 828.000i − 0.0559184i
$$604$$ 232.000 0.0156290
$$605$$ 0 0
$$606$$ 3726.00 0.249766
$$607$$ 16832.0i 1.12552i 0.826621 + 0.562759i $$0.190260\pi$$
−0.826621 + 0.562759i $$0.809740\pi$$
$$608$$ 5580.00i 0.372202i
$$609$$ 2142.00 0.142526
$$610$$ 0 0
$$611$$ −8160.00 −0.540292
$$612$$ 378.000i 0.0249669i
$$613$$ 2482.00i 0.163535i 0.996651 + 0.0817676i $$0.0260565\pi$$
−0.996651 + 0.0817676i $$0.973943\pi$$
$$614$$ 1356.00 0.0891266
$$615$$ 0 0
$$616$$ −5292.00 −0.346138
$$617$$ − 15798.0i − 1.03080i −0.856950 0.515400i $$-0.827643\pi$$
0.856950 0.515400i $$-0.172357\pi$$
$$618$$ − 504.000i − 0.0328056i
$$619$$ 15460.0 1.00386 0.501930 0.864908i $$-0.332623\pi$$
0.501930 + 0.864908i $$0.332623\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ − 15048.0i − 0.970048i
$$623$$ − 2478.00i − 0.159356i
$$624$$ 7242.00 0.464603
$$625$$ 0 0
$$626$$ −16206.0 −1.03470
$$627$$ − 13392.0i − 0.852990i
$$628$$ − 1694.00i − 0.107640i
$$629$$ −16716.0 −1.05964
$$630$$ 0 0
$$631$$ −7720.00 −0.487050 −0.243525 0.969895i $$-0.578304\pi$$
−0.243525 + 0.969895i $$0.578304\pi$$
$$632$$ − 21504.0i − 1.35345i
$$633$$ − 9228.00i − 0.579431i
$$634$$ 30258.0 1.89542
$$635$$ 0 0
$$636$$ 1494.00 0.0931462
$$637$$ − 1666.00i − 0.103625i
$$638$$ − 11016.0i − 0.683586i
$$639$$ 6480.00 0.401166
$$640$$ 0 0
$$641$$ −17262.0 −1.06366 −0.531832 0.846850i $$-0.678496\pi$$
−0.531832 + 0.846850i $$0.678496\pi$$
$$642$$ 108.000i 0.00663928i
$$643$$ 12220.0i 0.749471i 0.927132 + 0.374735i $$0.122266\pi$$
−0.927132 + 0.374735i $$0.877734\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 15624.0 0.951576
$$647$$ 13560.0i 0.823955i 0.911194 + 0.411977i $$0.135162\pi$$
−0.911194 + 0.411977i $$0.864838\pi$$
$$648$$ − 1701.00i − 0.103120i
$$649$$ −4752.00 −0.287415
$$650$$ 0 0
$$651$$ 3360.00 0.202287
$$652$$ − 2932.00i − 0.176113i
$$653$$ − 23094.0i − 1.38398i −0.721908 0.691989i $$-0.756735\pi$$
0.721908 0.691989i $$-0.243265\pi$$
$$654$$ −13302.0 −0.795335
$$655$$ 0 0
$$656$$ 22578.0 1.34378
$$657$$ − 4518.00i − 0.268286i
$$658$$ 5040.00i 0.298601i
$$659$$ −22548.0 −1.33285 −0.666423 0.745574i $$-0.732175\pi$$
−0.666423 + 0.745574i $$0.732175\pi$$
$$660$$ 0 0
$$661$$ 17462.0 1.02752 0.513762 0.857933i $$-0.328252\pi$$
0.513762 + 0.857933i $$0.328252\pi$$
$$662$$ 24132.0i 1.41679i
$$663$$ − 4284.00i − 0.250945i
$$664$$ 4284.00 0.250379
$$665$$ 0 0
$$666$$ −10746.0 −0.625224
$$667$$ 0 0
$$668$$ − 1176.00i − 0.0681150i
$$669$$ −5664.00 −0.327329
$$670$$ 0 0
$$671$$ −14328.0 −0.824331
$$672$$ − 945.000i − 0.0542473i
$$673$$ 22462.0i 1.28655i 0.765636 + 0.643274i $$0.222424\pi$$
−0.765636 + 0.643274i $$0.777576\pi$$
$$674$$ 12534.0 0.716308
$$675$$ 0 0
$$676$$ −1041.00 −0.0592285
$$677$$ − 25554.0i − 1.45069i −0.688383 0.725347i $$-0.741679\pi$$
0.688383 0.725347i $$-0.258321\pi$$
$$678$$ − 3618.00i − 0.204939i
$$679$$ 2002.00 0.113151
$$680$$ 0 0
$$681$$ 14148.0 0.796112
$$682$$ − 17280.0i − 0.970213i
$$683$$ − 9276.00i − 0.519672i −0.965653 0.259836i $$-0.916331\pi$$
0.965653 0.259836i $$-0.0836686\pi$$
$$684$$ 1116.00 0.0623850
$$685$$ 0 0
$$686$$ −1029.00 −0.0572703
$$687$$ 5070.00i 0.281561i
$$688$$ − 19028.0i − 1.05441i
$$689$$ −16932.0 −0.936223
$$690$$ 0 0
$$691$$ 27380.0 1.50736 0.753679 0.657243i $$-0.228277\pi$$
0.753679 + 0.657243i $$0.228277\pi$$
$$692$$ 870.000i 0.0477925i
$$693$$ 2268.00i 0.124321i
$$694$$ 468.000 0.0255980
$$695$$ 0 0
$$696$$ −6426.00 −0.349967
$$697$$ − 13356.0i − 0.725817i
$$698$$ − 37254.0i − 2.02018i
$$699$$ 414.000 0.0224019
$$700$$ 0 0
$$701$$ 25830.0 1.39171 0.695853 0.718184i $$-0.255027\pi$$
0.695853 + 0.718184i $$0.255027\pi$$
$$702$$ − 2754.00i − 0.148067i
$$703$$ 49352.0i 2.64772i
$$704$$ 15588.0 0.834510
$$705$$ 0 0
$$706$$ 23490.0 1.25221
$$707$$ 2898.00i 0.154159i
$$708$$ − 396.000i − 0.0210206i
$$709$$ 6226.00 0.329792 0.164896 0.986311i $$-0.447271\pi$$
0.164896 + 0.986311i $$0.447271\pi$$
$$710$$ 0 0
$$711$$ −9216.00 −0.486114
$$712$$ 7434.00i 0.391293i
$$713$$ 0 0
$$714$$ −2646.00 −0.138689
$$715$$ 0 0
$$716$$ −2316.00 −0.120884
$$717$$ − 5688.00i − 0.296265i
$$718$$ − 27936.0i − 1.45204i
$$719$$ 15072.0 0.781767 0.390884 0.920440i $$-0.372169\pi$$
0.390884 + 0.920440i $$0.372169\pi$$
$$720$$ 0 0
$$721$$ 392.000 0.0202480
$$722$$ − 25551.0i − 1.31705i
$$723$$ − 10794.0i − 0.555233i
$$724$$ 106.000 0.00544124
$$725$$ 0 0
$$726$$ −315.000 −0.0161030
$$727$$ − 32920.0i − 1.67942i −0.543038 0.839708i $$-0.682726\pi$$
0.543038 0.839708i $$-0.317274\pi$$
$$728$$ 4998.00i 0.254448i
$$729$$ −729.000 −0.0370370
$$730$$ 0 0
$$731$$ −11256.0 −0.569519
$$732$$ − 1194.00i − 0.0602889i
$$733$$ 6946.00i 0.350009i 0.984568 + 0.175004i $$0.0559939\pi$$
−0.984568 + 0.175004i $$0.944006\pi$$
$$734$$ −11280.0 −0.567238
$$735$$ 0 0
$$736$$ 0 0
$$737$$ − 3312.00i − 0.165535i
$$738$$ − 8586.00i − 0.428259i
$$739$$ 2356.00 0.117276 0.0586379 0.998279i $$-0.481324\pi$$
0.0586379 + 0.998279i $$0.481324\pi$$
$$740$$ 0 0
$$741$$ −12648.0 −0.627039
$$742$$ 10458.0i 0.517419i
$$743$$ 23520.0i 1.16133i 0.814144 + 0.580663i $$0.197207\pi$$
−0.814144 + 0.580663i $$0.802793\pi$$
$$744$$ −10080.0 −0.496708
$$745$$ 0 0
$$746$$ −17610.0 −0.864273
$$747$$ − 1836.00i − 0.0899273i
$$748$$ 1512.00i 0.0739094i
$$749$$ −84.0000 −0.00409785
$$750$$ 0 0
$$751$$ 3008.00 0.146156 0.0730782 0.997326i $$-0.476718\pi$$
0.0730782 + 0.997326i $$0.476718\pi$$
$$752$$ − 17040.0i − 0.826310i
$$753$$ − 9180.00i − 0.444273i
$$754$$ −10404.0 −0.502508
$$755$$ 0 0
$$756$$ −189.000 −0.00909241
$$757$$ − 20770.0i − 0.997224i −0.866825 0.498612i $$-0.833843\pi$$
0.866825 0.498612i $$-0.166157\pi$$
$$758$$ − 5556.00i − 0.266231i
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 11538.0 0.549609 0.274804 0.961500i $$-0.411387\pi$$
0.274804 + 0.961500i $$0.411387\pi$$
$$762$$ 11520.0i 0.547671i
$$763$$ − 10346.0i − 0.490892i
$$764$$ 1128.00 0.0534157
$$765$$ 0 0
$$766$$ −6480.00 −0.305655
$$767$$ 4488.00i 0.211281i
$$768$$ 4539.00i 0.213264i
$$769$$ −8498.00 −0.398499 −0.199249 0.979949i $$-0.563850\pi$$
−0.199249 + 0.979949i $$0.563850\pi$$
$$770$$ 0 0
$$771$$ 20466.0 0.955986
$$772$$ 4034.00i 0.188066i
$$773$$ 32322.0i 1.50393i 0.659200 + 0.751967i $$0.270895\pi$$
−0.659200 + 0.751967i $$0.729105\pi$$
$$774$$ −7236.00 −0.336037
$$775$$ 0 0
$$776$$ −6006.00 −0.277839
$$777$$ − 8358.00i − 0.385896i
$$778$$ − 20358.0i − 0.938136i
$$779$$ −39432.0 −1.81360
$$780$$ 0 0
$$781$$ 25920.0 1.18757
$$782$$ 0 0
$$783$$ 2754.00i 0.125696i
$$784$$ 3479.00 0.158482
$$785$$ 0 0
$$786$$ 15876.0 0.720456
$$787$$ 26228.0i 1.18796i 0.804479 + 0.593982i $$0.202445\pi$$
−0.804479 + 0.593982i $$0.797555\pi$$
$$788$$ 1314.00i 0.0594027i
$$789$$ 7776.00 0.350866
$$790$$ 0 0
$$791$$ 2814.00 0.126491
$$792$$ − 6804.00i − 0.305265i
$$793$$ 13532.0i 0.605972i
$$794$$ −19542.0 −0.873450
$$795$$ 0 0
$$796$$ 5096.00 0.226913
$$797$$ − 43338.0i − 1.92611i −0.269302 0.963056i $$-0.586793\pi$$
0.269302 0.963056i $$-0.413207\pi$$
$$798$$ 7812.00i 0.346544i
$$799$$ −10080.0 −0.446314
$$800$$ 0 0
$$801$$ 3186.00 0.140539
$$802$$ − 9990.00i − 0.439849i
$$803$$ − 18072.0i − 0.794206i
$$804$$ 276.000 0.0121067
$$805$$ 0 0
$$806$$ −16320.0 −0.713210
$$807$$ − 24642.0i − 1.07489i
$$808$$ − 8694.00i − 0.378532i
$$809$$ 28902.0 1.25604 0.628022 0.778195i $$-0.283865\pi$$
0.628022 + 0.778195i $$0.283865\pi$$
$$810$$ 0 0
$$811$$ 27164.0 1.17615 0.588075 0.808807i $$-0.299886\pi$$
0.588075 + 0.808807i $$0.299886\pi$$
$$812$$ 714.000i 0.0308577i
$$813$$ − 16032.0i − 0.691595i
$$814$$ −42984.0 −1.85085
$$815$$ 0 0
$$816$$ 8946.00 0.383790
$$817$$ 33232.0i 1.42306i
$$818$$ − 16194.0i − 0.692188i
$$819$$ 2142.00 0.0913889
$$820$$ 0 0
$$821$$ −17202.0 −0.731247 −0.365624 0.930763i $$-0.619144\pi$$
−0.365624 + 0.930763i $$0.619144\pi$$
$$822$$ − 21222.0i − 0.900489i
$$823$$ 5992.00i 0.253789i 0.991916 + 0.126894i $$0.0405009\pi$$
−0.991916 + 0.126894i $$0.959499\pi$$
$$824$$ −1176.00 −0.0497183
$$825$$ 0 0
$$826$$ 2772.00 0.116768
$$827$$ 25884.0i 1.08836i 0.838968 + 0.544181i $$0.183159\pi$$
−0.838968 + 0.544181i $$0.816841\pi$$
$$828$$ 0 0
$$829$$ 1474.00 0.0617541 0.0308770 0.999523i $$-0.490170\pi$$
0.0308770 + 0.999523i $$0.490170\pi$$
$$830$$ 0 0
$$831$$ 19542.0 0.815770
$$832$$ − 14722.0i − 0.613454i
$$833$$ − 2058.00i − 0.0856008i
$$834$$ 468.000 0.0194311
$$835$$ 0 0
$$836$$ 4464.00 0.184678
$$837$$ 4320.00i 0.178400i
$$838$$ 39276.0i 1.61905i
$$839$$ −33528.0 −1.37964 −0.689818 0.723983i $$-0.742310\pi$$
−0.689818 + 0.723983i $$0.742310\pi$$
$$840$$ 0 0
$$841$$ −13985.0 −0.573414
$$842$$ 966.000i 0.0395375i
$$843$$ 19854.0i 0.811160i
$$844$$ 3076.00 0.125451
$$845$$ 0 0
$$846$$ −6480.00 −0.263342
$$847$$ − 245.000i − 0.00993896i
$$848$$ − 35358.0i − 1.43184i
$$849$$ 9780.00 0.395346
$$850$$ 0 0
$$851$$ 0 0
$$852$$ 2160.00i 0.0868549i
$$853$$ − 1190.00i − 0.0477665i −0.999715 0.0238832i $$-0.992397\pi$$
0.999715 0.0238832i $$-0.00760300\pi$$
$$854$$ 8358.00 0.334900
$$855$$ 0 0
$$856$$ 252.000 0.0100621
$$857$$ 34578.0i 1.37825i 0.724642 + 0.689126i $$0.242005\pi$$
−0.724642 + 0.689126i $$0.757995\pi$$
$$858$$ − 11016.0i − 0.438322i
$$859$$ 44404.0 1.76373 0.881865 0.471501i $$-0.156288\pi$$
0.881865 + 0.471501i $$0.156288\pi$$
$$860$$ 0 0
$$861$$ 6678.00 0.264327
$$862$$ − 7848.00i − 0.310097i
$$863$$ 38328.0i 1.51182i 0.654676 + 0.755910i $$0.272805\pi$$
−0.654676 + 0.755910i $$0.727195\pi$$
$$864$$ 1215.00 0.0478416
$$865$$ 0 0
$$866$$ −12966.0 −0.508779
$$867$$ 9447.00i 0.370054i
$$868$$ 1120.00i 0.0437964i
$$869$$ −36864.0 −1.43904
$$870$$ 0 0
$$871$$ −3128.00 −0.121686
$$872$$ 31038.0i 1.20537i
$$873$$ 2574.00i 0.0997900i
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 1506.00 0.0580856
$$877$$ − 38842.0i − 1.49555i −0.663950 0.747777i $$-0.731121\pi$$
0.663950 0.747777i $$-0.268879\pi$$
$$878$$ − 27048.0i − 1.03966i
$$879$$ 15354.0 0.589167
$$880$$ 0 0
$$881$$ −35046.0 −1.34022 −0.670108 0.742264i $$-0.733752\pi$$
−0.670108 + 0.742264i $$0.733752\pi$$
$$882$$ − 1323.00i − 0.0505076i
$$883$$ − 14204.0i − 0.541339i −0.962672 0.270670i $$-0.912755\pi$$
0.962672 0.270670i $$-0.0872451\pi$$
$$884$$ 1428.00 0.0543313
$$885$$ 0 0
$$886$$ 15804.0 0.599262
$$887$$ − 26136.0i − 0.989359i −0.869076 0.494679i $$-0.835286\pi$$
0.869076 0.494679i $$-0.164714\pi$$
$$888$$ 25074.0i 0.947554i
$$889$$ −8960.00 −0.338030
$$890$$ 0 0
$$891$$ −2916.00 −0.109640
$$892$$ − 1888.00i − 0.0708687i
$$893$$ 29760.0i 1.11521i
$$894$$ 15714.0 0.587869
$$895$$ 0 0
$$896$$ −11613.0 −0.432995
$$897$$ 0 0
$$898$$ − 15930.0i − 0.591972i
$$899$$ 16320.0 0.605453
$$900$$ 0 0
$$901$$ −20916.0 −0.773377
$$902$$ − 34344.0i − 1.26777i
$$903$$ − 5628.00i − 0.207407i
$$904$$ −8442.00 −0.310594
$$905$$ 0 0
$$906$$ −2088.00 −0.0765664
$$907$$ − 9052.00i − 0.331386i −0.986177 0.165693i $$-0.947014\pi$$
0.986177 0.165693i $$-0.0529860\pi$$
$$908$$ 4716.00i 0.172363i
$$909$$ −3726.00 −0.135956
$$910$$ 0 0
$$911$$ 5016.00 0.182423 0.0912116 0.995832i $$-0.470926\pi$$
0.0912116 + 0.995832i $$0.470926\pi$$
$$912$$ − 26412.0i − 0.958979i
$$913$$ − 7344.00i − 0.266211i
$$914$$ 47310.0 1.71212
$$915$$ 0 0
$$916$$ −1690.00 −0.0609598
$$917$$ 12348.0i 0.444675i
$$918$$ − 3402.00i − 0.122312i
$$919$$ −44552.0 −1.59917 −0.799584 0.600555i $$-0.794946\pi$$
−0.799584 + 0.600555i $$0.794946\pi$$
$$920$$ 0 0
$$921$$ −1356.00 −0.0485144
$$922$$ 16110.0i 0.575439i
$$923$$ − 24480.0i − 0.872989i
$$924$$ −756.000 −0.0269162
$$925$$ 0 0
$$926$$ 9984.00 0.354314
$$927$$ 504.000i 0.0178571i
$$928$$ − 4590.00i − 0.162364i
$$929$$ −24234.0 −0.855858 −0.427929 0.903812i $$-0.640757\pi$$
−0.427929 + 0.903812i $$0.640757\pi$$
$$930$$ 0 0
$$931$$ −6076.00 −0.213891
$$932$$ 138.000i 0.00485015i
$$933$$ 15048.0i 0.528027i
$$934$$ 13644.0 0.477993
$$935$$ 0 0
$$936$$ −6426.00 −0.224402
$$937$$ − 13894.0i − 0.484415i −0.970224 0.242208i $$-0.922128\pi$$
0.970224 0.242208i $$-0.0778715\pi$$
$$938$$ 1932.00i 0.0672516i
$$939$$ 16206.0 0.563219
$$940$$ 0 0
$$941$$ 46758.0 1.61984 0.809919 0.586542i $$-0.199511\pi$$
0.809919 + 0.586542i $$0.199511\pi$$
$$942$$ 15246.0i 0.527326i
$$943$$ 0 0
$$944$$ −9372.00 −0.323128
$$945$$ 0 0
$$946$$ −28944.0 −0.994768
$$947$$ 13812.0i 0.473949i 0.971516 + 0.236974i $$0.0761558\pi$$
−0.971516 + 0.236974i $$0.923844\pi$$
$$948$$ − 3072.00i − 0.105247i
$$949$$ −17068.0 −0.583826
$$950$$ 0 0
$$951$$ −30258.0 −1.03174
$$952$$ 6174.00i 0.210190i
$$953$$ 58518.0i 1.98907i 0.104402 + 0.994535i $$0.466707\pi$$
−0.104402 + 0.994535i $$0.533293\pi$$
$$954$$ −13446.0 −0.456321
$$955$$ 0 0
$$956$$ 1896.00 0.0641433
$$957$$ 11016.0i 0.372097i
$$958$$ − 24192.0i − 0.815875i
$$959$$ 16506.0 0.555794
$$960$$ 0 0
$$961$$ −4191.00 −0.140680
$$962$$ 40596.0i 1.36057i
$$963$$ − 108.000i − 0.00361397i
$$964$$ 3598.00 0.120211
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 19640.0i 0.653133i 0.945174 + 0.326567i $$0.105892\pi$$
−0.945174 + 0.326567i $$0.894108\pi$$
$$968$$ 735.000i 0.0244047i
$$969$$ −15624.0 −0.517972
$$970$$ 0 0
$$971$$ −58308.0 −1.92708 −0.963539 0.267568i $$-0.913780\pi$$
−0.963539 + 0.267568i $$0.913780\pi$$
$$972$$ − 243.000i − 0.00801875i
$$973$$ 364.000i 0.0119931i
$$974$$ 49848.0 1.63987
$$975$$ 0 0
$$976$$ −28258.0 −0.926759
$$977$$ − 23550.0i − 0.771168i −0.922673 0.385584i $$-0.874000\pi$$
0.922673 0.385584i $$-0.126000\pi$$
$$978$$ 26388.0i 0.862776i
$$979$$ 12744.0 0.416037
$$980$$ 0 0
$$981$$ 13302.0 0.432926
$$982$$ 21420.0i 0.696069i
$$983$$ − 15768.0i − 0.511619i −0.966727 0.255809i $$-0.917658\pi$$
0.966727 0.255809i $$-0.0823419\pi$$
$$984$$ −20034.0 −0.649045
$$985$$ 0 0
$$986$$ −12852.0 −0.415102
$$987$$ − 5040.00i − 0.162538i
$$988$$ − 4216.00i − 0.135758i
$$989$$ 0 0
$$990$$ 0 0
$$991$$ 35264.0 1.13037 0.565186 0.824964i $$-0.308805\pi$$
0.565186 + 0.824964i $$0.308805\pi$$
$$992$$ − 7200.00i − 0.230444i
$$993$$ − 24132.0i − 0.771204i
$$994$$ −15120.0 −0.482472
$$995$$ 0 0
$$996$$ 612.000 0.0194698
$$997$$ − 29338.0i − 0.931940i −0.884801 0.465970i $$-0.845706\pi$$
0.884801 0.465970i $$-0.154294\pi$$
$$998$$ − 27372.0i − 0.868182i
$$999$$ 10746.0 0.340329
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 525.4.d.c.274.1 2
5.2 odd 4 525.4.a.g.1.1 1
5.3 odd 4 21.4.a.a.1.1 1
5.4 even 2 inner 525.4.d.c.274.2 2
15.2 even 4 1575.4.a.b.1.1 1
15.8 even 4 63.4.a.c.1.1 1
20.3 even 4 336.4.a.f.1.1 1
35.3 even 12 147.4.e.g.79.1 2
35.13 even 4 147.4.a.c.1.1 1
35.18 odd 12 147.4.e.i.79.1 2
35.23 odd 12 147.4.e.i.67.1 2
35.33 even 12 147.4.e.g.67.1 2
40.3 even 4 1344.4.a.n.1.1 1
40.13 odd 4 1344.4.a.ba.1.1 1
60.23 odd 4 1008.4.a.v.1.1 1
105.23 even 12 441.4.e.b.361.1 2
105.38 odd 12 441.4.e.d.226.1 2
105.53 even 12 441.4.e.b.226.1 2
105.68 odd 12 441.4.e.d.361.1 2
105.83 odd 4 441.4.a.j.1.1 1
140.83 odd 4 2352.4.a.r.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
21.4.a.a.1.1 1 5.3 odd 4
63.4.a.c.1.1 1 15.8 even 4
147.4.a.c.1.1 1 35.13 even 4
147.4.e.g.67.1 2 35.33 even 12
147.4.e.g.79.1 2 35.3 even 12
147.4.e.i.67.1 2 35.23 odd 12
147.4.e.i.79.1 2 35.18 odd 12
336.4.a.f.1.1 1 20.3 even 4
441.4.a.j.1.1 1 105.83 odd 4
441.4.e.b.226.1 2 105.53 even 12
441.4.e.b.361.1 2 105.23 even 12
441.4.e.d.226.1 2 105.38 odd 12
441.4.e.d.361.1 2 105.68 odd 12
525.4.a.g.1.1 1 5.2 odd 4
525.4.d.c.274.1 2 1.1 even 1 trivial
525.4.d.c.274.2 2 5.4 even 2 inner
1008.4.a.v.1.1 1 60.23 odd 4
1344.4.a.n.1.1 1 40.3 even 4
1344.4.a.ba.1.1 1 40.13 odd 4
1575.4.a.b.1.1 1 15.2 even 4
2352.4.a.r.1.1 1 140.83 odd 4