# Properties

 Label 525.4.d.c.274.2 Level $525$ Weight $4$ Character 525.274 Analytic conductor $30.976$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

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

## $q$-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 $$0.177932\pi$$
−0.847791 + 0.530330i $$0.822068\pi$$
$$3$$ − 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 $$-0.881859\pi$$
0.931910 0.362689i $$-0.118141\pi$$
$$14$$ 21.0000 0.400892
$$15$$ 0 0
$$16$$ −71.0000 −1.10938
$$17$$ − 42.0000i − 0.599206i −0.954064 0.299603i $$-0.903146\pi$$
0.954064 0.299603i $$-0.0968542\pi$$
$$18$$ − 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 $$-0.654704\pi$$
0.467109 0.884200i $$-0.345296\pi$$
$$38$$ 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 $$-0.842366\pi$$
0.879863 0.475228i $$-0.157634\pi$$
$$44$$ 36.0000 0.123346
$$45$$ 0 0
$$46$$ 0 0
$$47$$ − 240.000i − 0.744843i −0.928064 0.372421i $$-0.878528\pi$$
0.928064 0.372421i $$-0.121472\pi$$
$$48$$ 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 $$-0.776718\pi$$
0.763899 0.645335i $$-0.223282\pi$$
$$54$$ −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 $$-0.973270\pi$$
0.996476 0.0838775i $$-0.0267305\pi$$
$$68$$ 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 $$-0.868166\pi$$
0.915451 0.402429i $$-0.131834\pi$$
$$74$$ 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 $$-0.956932\pi$$
0.990860 0.134891i $$-0.0430684\pi$$
$$84$$ 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 $$0.0478260\pi$$
−0.988734 + 0.149685i $$0.952174\pi$$
$$98$$ − 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 $$0.00852716\pi$$
−0.999641 + 0.0267857i $$0.991473\pi$$
$$104$$ 714.000 0.673206
$$105$$ 0 0
$$106$$ 1494.00 1.36896
$$107$$ − 12.0000i − 0.0108419i −0.999985 0.00542095i $$-0.998274\pi$$
0.999985 0.00542095i $$-0.00172555\pi$$
$$108$$ − 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 $$0.0535151\pi$$
−0.985901 + 0.167332i $$0.946485\pi$$
$$114$$ 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 $$-0.852431\pi$$
0.894448 0.447172i $$-0.147569\pi$$
$$128$$ − 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 $$0.262934\pi$$
−0.677800 + 0.735246i $$0.737066\pi$$
$$138$$ 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 $$-0.858316\pi$$
0.902562 0.430560i $$-0.141684\pi$$
$$158$$ 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 $$-0.751192\pi$$
0.709750 0.704454i $$-0.248808\pi$$
$$164$$ 318.000 0.151412
$$165$$ 0 0
$$166$$ 612.000 0.286147
$$167$$ − 1176.00i − 0.544920i −0.962167 0.272460i $$-0.912163\pi$$
0.962167 0.272460i $$-0.0878372\pi$$
$$168$$ − 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 $$0.0612282\pi$$
−0.981557 + 0.191170i $$0.938772\pi$$
$$174$$ −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 $$0.271038\pi$$
−0.658862 + 0.752263i $$0.728962\pi$$
$$194$$ −858.000 −0.317530
$$195$$ 0 0
$$196$$ 49.0000 0.0178571
$$197$$ 1314.00i 0.475221i 0.971360 + 0.237611i $$0.0763643\pi$$
−0.971360 + 0.237611i $$0.923636\pi$$
$$198$$ 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 $$-0.908513\pi$$
0.958980 0.283475i $$-0.0914873\pi$$
$$224$$ −315.000 −0.0939590
$$225$$ 0 0
$$226$$ −1206.00 −0.354964
$$227$$ 4716.00i 1.37891i 0.724330 + 0.689454i $$0.242149\pi$$
−0.724330 + 0.689454i $$0.757851\pi$$
$$228$$ 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 $$0.00617579\pi$$
−0.999812 + 0.0194006i $$0.993824\pi$$
$$234$$ −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 $$0.310469\pi$$
−0.560864 + 0.827908i $$0.689531\pi$$
$$258$$ − 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 $$0.0982750\pi$$
−0.952717 + 0.303858i $$0.901725\pi$$
$$264$$ −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 $$0.249717\pi$$
−0.707736 + 0.706477i $$0.750283\pi$$
$$278$$ 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 $$0.111233\pi$$
−0.939562 + 0.342380i $$0.888767\pi$$
$$284$$ 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 $$0.170441\pi$$
−0.860036 + 0.510233i $$0.829559\pi$$
$$294$$ −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 $$-0.986622\pi$$
0.999117 0.0420147i $$-0.0133776\pi$$
$$308$$ − 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 $$0.162187\pi$$
−0.872977 + 0.487762i $$0.837813\pi$$
$$314$$ 5082.00 0.913356
$$315$$ 0 0
$$316$$ −1024.00 −0.182293
$$317$$ − 10086.0i − 1.78702i −0.449041 0.893511i $$-0.648234\pi$$
0.449041 0.893511i $$-0.351766\pi$$
$$318$$ − 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 $$-0.890361\pi$$
0.941264 0.337671i $$-0.109639\pi$$
$$338$$ 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 $$-0.996159\pi$$
0.999927 0.0120670i $$-0.00384115\pi$$
$$348$$ − 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 $$-0.799011\pi$$
0.807187 0.590296i $$-0.200989\pi$$
$$354$$ 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 $$0.0861640\pi$$
−0.963586 + 0.267398i $$0.913836\pi$$
$$368$$ 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 $$0.133572\pi$$
−0.913240 + 0.407422i $$0.866428\pi$$
$$374$$ −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 $$0.0460246\pi$$
−0.989565 + 0.144087i $$0.953975\pi$$
$$384$$ −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 $$0.135082\pi$$
−0.911298 + 0.411748i $$0.864918\pi$$
$$398$$ − 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 $$0.0770952\pi$$
−0.970812 + 0.239841i $$0.922905\pi$$
$$434$$ −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 $$-0.908838\pi$$
0.959269 0.282495i $$-0.0911619\pi$$
$$444$$ 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 $$-0.701036\pi$$
0.590415 0.807100i $$-0.298964\pi$$
$$458$$ 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 $$-0.946584\pi$$
0.985953 0.167025i $$-0.0534161\pi$$
$$464$$ 7242.00 0.724572
$$465$$ 0 0
$$466$$ −414.000 −0.0411549
$$467$$ − 4548.00i − 0.450656i −0.974283 0.225328i $$-0.927655\pi$$
0.974283 0.225328i $$-0.0723454\pi$$
$$468$$ − 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 $$-0.718734\pi$$
0.634355 0.773042i $$-0.281266\pi$$
$$488$$ 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 $$-0.906213\pi$$
0.956906 0.290397i $$-0.0937873\pi$$
$$504$$ −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 $$0.264351\pi$$
−0.674518 + 0.738258i $$0.735649\pi$$
$$524$$ −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 $$-0.694167\pi$$
0.572863 0.819651i $$-0.305833\pi$$
$$548$$ − 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 $$-0.701972\pi$$
0.592786 0.805360i $$-0.298028\pi$$
$$558$$ 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 $$-0.768352\pi$$
0.746677 0.665187i $$-0.231648\pi$$
$$564$$ 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 $$-0.999977\pi$$
1.00000 7.21500e-5i $$-2.29661e-5\pi$$
$$578$$ 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 $$-0.622479\pi$$
0.375354 0.926881i $$-0.377521\pi$$
$$588$$ − 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 $$0.0253541\pi$$
−0.996829 + 0.0795679i $$0.974646\pi$$
$$594$$ 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 $$-0.809740\pi$$
0.826621 0.562759i $$-0.190260\pi$$
$$608$$ − 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 $$-0.973943\pi$$
0.996651 0.0817676i $$-0.0260565\pi$$
$$614$$ 1356.00 0.0891266
$$615$$ 0 0
$$616$$ −5292.00 −0.346138
$$617$$ 15798.0i 1.03080i 0.856950 + 0.515400i $$0.172357\pi$$
−0.856950 + 0.515400i $$0.827643\pi$$
$$618$$ 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 $$-0.877734\pi$$
0.927132 0.374735i $$-0.122266\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 15624.0 0.951576
$$647$$ − 13560.0i − 0.823955i −0.911194 0.411977i $$-0.864838\pi$$
0.911194 0.411977i $$-0.135162\pi$$
$$648$$ 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 $$0.243265\pi$$
−0.721908 + 0.691989i $$0.756735\pi$$
$$654$$ −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 $$-0.777576\pi$$
0.765636 0.643274i $$-0.222424\pi$$
$$674$$ 12534.0 0.716308
$$675$$ 0 0
$$676$$ −1041.00 −0.0592285
$$677$$ 25554.0i 1.45069i 0.688383 + 0.725347i $$0.258321\pi$$
−0.688383 + 0.725347i $$0.741679\pi$$
$$678$$ 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 $$0.0836686\pi$$
−0.965653 + 0.259836i $$0.916331\pi$$
$$684$$ 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 $$0.317274\pi$$
−0.543038 + 0.839708i $$0.682726\pi$$
$$728$$ − 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 $$-0.944006\pi$$
0.984568 0.175004i $$-0.0559939\pi$$
$$734$$ −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 $$-0.802793\pi$$
0.814144 0.580663i $$-0.197207\pi$$
$$744$$ −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 $$0.166157\pi$$
−0.866825 + 0.498612i $$0.833843\pi$$
$$758$$ 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 $$-0.729105\pi$$
0.659200 0.751967i $$-0.270895\pi$$
$$774$$ −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 $$-0.797555\pi$$
0.804479 0.593982i $$-0.202445\pi$$
$$788$$ − 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 $$0.413207\pi$$
−0.269302 + 0.963056i $$0.586793\pi$$
$$798$$ − 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 $$-0.959499\pi$$
0.991916 0.126894i $$-0.0405009\pi$$
$$824$$ −1176.00 −0.0497183
$$825$$ 0 0
$$826$$ 2772.00 0.116768
$$827$$ − 25884.0i − 1.08836i −0.838968 0.544181i $$-0.816841\pi$$
0.838968 0.544181i $$-0.183159\pi$$
$$828$$ 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 $$0.00760300\pi$$
−0.999715 + 0.0238832i $$0.992397\pi$$
$$854$$ 8358.00 0.334900
$$855$$ 0 0
$$856$$ 252.000 0.0100621
$$857$$ − 34578.0i − 1.37825i −0.724642 0.689126i $$-0.757995\pi$$
0.724642 0.689126i $$-0.242005\pi$$
$$858$$ 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 $$-0.727195\pi$$
0.654676 0.755910i $$-0.272805\pi$$
$$864$$ 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 $$0.268879\pi$$
−0.663950 + 0.747777i $$0.731121\pi$$
$$878$$ 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 $$0.0872451\pi$$
−0.962672 + 0.270670i $$0.912755\pi$$
$$884$$ 1428.00 0.0543313
$$885$$ 0 0
$$886$$ 15804.0 0.599262
$$887$$ 26136.0i 0.989359i 0.869076 + 0.494679i $$0.164714\pi$$
−0.869076 + 0.494679i $$0.835286\pi$$
$$888$$ − 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 $$0.0529860\pi$$
−0.986177 + 0.165693i $$0.947014\pi$$
$$908$$ − 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 $$0.0778715\pi$$
−0.970224 + 0.242208i $$0.922128\pi$$
$$938$$ − 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 $$-0.923844\pi$$
0.971516 0.236974i $$-0.0761558\pi$$
$$948$$ 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 $$-0.533293\pi$$
0.104402 0.994535i $$-0.466707\pi$$
$$954$$ −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 $$-0.894108\pi$$
0.945174 0.326567i $$-0.105892\pi$$
$$968$$ − 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 $$0.126000\pi$$
−0.922673 + 0.385584i $$0.874000\pi$$
$$978$$ − 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 $$0.0823419\pi$$
−0.966727 + 0.255809i $$0.917658\pi$$
$$984$$ −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 $$0.154294\pi$$
−0.884801 + 0.465970i $$0.845706\pi$$
$$998$$ 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.274.2 2
5.2 odd 4 21.4.a.a.1.1 1
5.3 odd 4 525.4.a.g.1.1 1
5.4 even 2 inner 525.4.d.c.274.1 2
15.2 even 4 63.4.a.c.1.1 1
15.8 even 4 1575.4.a.b.1.1 1
20.7 even 4 336.4.a.f.1.1 1
35.2 odd 12 147.4.e.i.67.1 2
35.12 even 12 147.4.e.g.67.1 2
35.17 even 12 147.4.e.g.79.1 2
35.27 even 4 147.4.a.c.1.1 1
35.32 odd 12 147.4.e.i.79.1 2
40.27 even 4 1344.4.a.n.1.1 1
40.37 odd 4 1344.4.a.ba.1.1 1
60.47 odd 4 1008.4.a.v.1.1 1
105.2 even 12 441.4.e.b.361.1 2
105.17 odd 12 441.4.e.d.226.1 2
105.32 even 12 441.4.e.b.226.1 2
105.47 odd 12 441.4.e.d.361.1 2
105.62 odd 4 441.4.a.j.1.1 1
140.27 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 5.2 odd 4
63.4.a.c.1.1 1 15.2 even 4
147.4.a.c.1.1 1 35.27 even 4
147.4.e.g.67.1 2 35.12 even 12
147.4.e.g.79.1 2 35.17 even 12
147.4.e.i.67.1 2 35.2 odd 12
147.4.e.i.79.1 2 35.32 odd 12
336.4.a.f.1.1 1 20.7 even 4
441.4.a.j.1.1 1 105.62 odd 4
441.4.e.b.226.1 2 105.32 even 12
441.4.e.b.361.1 2 105.2 even 12
441.4.e.d.226.1 2 105.17 odd 12
441.4.e.d.361.1 2 105.47 odd 12
525.4.a.g.1.1 1 5.3 odd 4
525.4.d.c.274.1 2 5.4 even 2 inner
525.4.d.c.274.2 2 1.1 even 1 trivial
1008.4.a.v.1.1 1 60.47 odd 4
1344.4.a.n.1.1 1 40.27 even 4
1344.4.a.ba.1.1 1 40.37 odd 4
1575.4.a.b.1.1 1 15.8 even 4
2352.4.a.r.1.1 1 140.27 odd 4