# Properties

 Label 3525.2.a.bd.1.8 Level $3525$ Weight $2$ Character 3525.1 Self dual yes Analytic conductor $28.147$ Analytic rank $1$ Dimension $8$ CM no Inner twists $1$

# Learn more

## Newspace parameters

 Level: $$N$$ $$=$$ $$3525 = 3 \cdot 5^{2} \cdot 47$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3525.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$28.1472667125$$ Analytic rank: $$1$$ Dimension: $$8$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ Defining polynomial: $$x^{8} - 3 x^{7} - 7 x^{6} + 24 x^{5} + 8 x^{4} - 47 x^{3} + 8 x^{2} + 13 x + 2$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.8 Root $$-2.25864$$ of defining polynomial Character $$\chi$$ $$=$$ 3525.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+2.25864 q^{2} +1.00000 q^{3} +3.10144 q^{4} +2.25864 q^{6} -3.65257 q^{7} +2.48774 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q+2.25864 q^{2} +1.00000 q^{3} +3.10144 q^{4} +2.25864 q^{6} -3.65257 q^{7} +2.48774 q^{8} +1.00000 q^{9} -5.39045 q^{11} +3.10144 q^{12} -3.76717 q^{13} -8.24983 q^{14} -0.583969 q^{16} +6.40426 q^{17} +2.25864 q^{18} -8.07640 q^{19} -3.65257 q^{21} -12.1751 q^{22} +4.76354 q^{23} +2.48774 q^{24} -8.50866 q^{26} +1.00000 q^{27} -11.3282 q^{28} -3.78248 q^{29} -5.26069 q^{31} -6.29446 q^{32} -5.39045 q^{33} +14.4649 q^{34} +3.10144 q^{36} -3.48931 q^{37} -18.2416 q^{38} -3.76717 q^{39} +6.70344 q^{41} -8.24983 q^{42} -5.28901 q^{43} -16.7181 q^{44} +10.7591 q^{46} -1.00000 q^{47} -0.583969 q^{48} +6.34130 q^{49} +6.40426 q^{51} -11.6836 q^{52} +9.62076 q^{53} +2.25864 q^{54} -9.08666 q^{56} -8.07640 q^{57} -8.54325 q^{58} +10.0449 q^{59} -4.68111 q^{61} -11.8820 q^{62} -3.65257 q^{63} -13.0489 q^{64} -12.1751 q^{66} -14.1087 q^{67} +19.8624 q^{68} +4.76354 q^{69} +12.0784 q^{71} +2.48774 q^{72} +11.5541 q^{73} -7.88109 q^{74} -25.0484 q^{76} +19.6890 q^{77} -8.50866 q^{78} +1.64874 q^{79} +1.00000 q^{81} +15.1406 q^{82} -10.7120 q^{83} -11.3282 q^{84} -11.9459 q^{86} -3.78248 q^{87} -13.4100 q^{88} -7.50610 q^{89} +13.7599 q^{91} +14.7738 q^{92} -5.26069 q^{93} -2.25864 q^{94} -6.29446 q^{96} -11.7481 q^{97} +14.3227 q^{98} -5.39045 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - 3q^{2} + 8q^{3} + 7q^{4} - 3q^{6} - 8q^{7} - 6q^{8} + 8q^{9} + O(q^{10})$$ $$8q - 3q^{2} + 8q^{3} + 7q^{4} - 3q^{6} - 8q^{7} - 6q^{8} + 8q^{9} - 8q^{11} + 7q^{12} - 10q^{13} + q^{14} + 5q^{16} - 6q^{17} - 3q^{18} - 2q^{19} - 8q^{21} - 10q^{23} - 6q^{24} - 14q^{26} + 8q^{27} - 44q^{28} - 13q^{29} - 10q^{32} - 8q^{33} + 28q^{34} + 7q^{36} - 3q^{37} - 36q^{38} - 10q^{39} - 16q^{41} + q^{42} - 25q^{43} - 17q^{44} - 5q^{46} - 8q^{47} + 5q^{48} + 16q^{49} - 6q^{51} + 17q^{52} - 4q^{53} - 3q^{54} + 37q^{56} - 2q^{57} - 15q^{58} - 8q^{59} + 15q^{61} - 6q^{62} - 8q^{63} - 14q^{64} - 27q^{67} - 14q^{68} - 10q^{69} + 14q^{71} - 6q^{72} - 28q^{73} - 21q^{74} + 6q^{76} - 4q^{77} - 14q^{78} + 7q^{79} + 8q^{81} + 53q^{82} - 60q^{83} - 44q^{84} - 3q^{86} - 13q^{87} - 54q^{88} - 34q^{89} + 23q^{91} + 43q^{92} + 3q^{94} - 10q^{96} - 7q^{97} - 40q^{98} - 8q^{99} + O(q^{100})$$

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 2.25864 1.59710 0.798548 0.601931i $$-0.205602\pi$$
0.798548 + 0.601931i $$0.205602\pi$$
$$3$$ 1.00000 0.577350
$$4$$ 3.10144 1.55072
$$5$$ 0 0
$$6$$ 2.25864 0.922084
$$7$$ −3.65257 −1.38054 −0.690272 0.723550i $$-0.742509\pi$$
−0.690272 + 0.723550i $$0.742509\pi$$
$$8$$ 2.48774 0.879549
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ −5.39045 −1.62528 −0.812640 0.582766i $$-0.801971\pi$$
−0.812640 + 0.582766i $$0.801971\pi$$
$$12$$ 3.10144 0.895307
$$13$$ −3.76717 −1.04482 −0.522412 0.852693i $$-0.674968\pi$$
−0.522412 + 0.852693i $$0.674968\pi$$
$$14$$ −8.24983 −2.20486
$$15$$ 0 0
$$16$$ −0.583969 −0.145992
$$17$$ 6.40426 1.55326 0.776631 0.629956i $$-0.216927\pi$$
0.776631 + 0.629956i $$0.216927\pi$$
$$18$$ 2.25864 0.532366
$$19$$ −8.07640 −1.85285 −0.926427 0.376475i $$-0.877136\pi$$
−0.926427 + 0.376475i $$0.877136\pi$$
$$20$$ 0 0
$$21$$ −3.65257 −0.797057
$$22$$ −12.1751 −2.59573
$$23$$ 4.76354 0.993268 0.496634 0.867960i $$-0.334569\pi$$
0.496634 + 0.867960i $$0.334569\pi$$
$$24$$ 2.48774 0.507808
$$25$$ 0 0
$$26$$ −8.50866 −1.66869
$$27$$ 1.00000 0.192450
$$28$$ −11.3282 −2.14083
$$29$$ −3.78248 −0.702390 −0.351195 0.936302i $$-0.614225\pi$$
−0.351195 + 0.936302i $$0.614225\pi$$
$$30$$ 0 0
$$31$$ −5.26069 −0.944847 −0.472423 0.881372i $$-0.656621\pi$$
−0.472423 + 0.881372i $$0.656621\pi$$
$$32$$ −6.29446 −1.11271
$$33$$ −5.39045 −0.938356
$$34$$ 14.4649 2.48071
$$35$$ 0 0
$$36$$ 3.10144 0.516906
$$37$$ −3.48931 −0.573640 −0.286820 0.957985i $$-0.592598\pi$$
−0.286820 + 0.957985i $$0.592598\pi$$
$$38$$ −18.2416 −2.95919
$$39$$ −3.76717 −0.603230
$$40$$ 0 0
$$41$$ 6.70344 1.04690 0.523451 0.852056i $$-0.324644\pi$$
0.523451 + 0.852056i $$0.324644\pi$$
$$42$$ −8.24983 −1.27298
$$43$$ −5.28901 −0.806566 −0.403283 0.915075i $$-0.632131\pi$$
−0.403283 + 0.915075i $$0.632131\pi$$
$$44$$ −16.7181 −2.52035
$$45$$ 0 0
$$46$$ 10.7591 1.58634
$$47$$ −1.00000 −0.145865
$$48$$ −0.583969 −0.0842887
$$49$$ 6.34130 0.905899
$$50$$ 0 0
$$51$$ 6.40426 0.896776
$$52$$ −11.6836 −1.62023
$$53$$ 9.62076 1.32151 0.660756 0.750601i $$-0.270236\pi$$
0.660756 + 0.750601i $$0.270236\pi$$
$$54$$ 2.25864 0.307361
$$55$$ 0 0
$$56$$ −9.08666 −1.21426
$$57$$ −8.07640 −1.06975
$$58$$ −8.54325 −1.12178
$$59$$ 10.0449 1.30774 0.653870 0.756607i $$-0.273144\pi$$
0.653870 + 0.756607i $$0.273144\pi$$
$$60$$ 0 0
$$61$$ −4.68111 −0.599355 −0.299677 0.954041i $$-0.596879\pi$$
−0.299677 + 0.954041i $$0.596879\pi$$
$$62$$ −11.8820 −1.50901
$$63$$ −3.65257 −0.460181
$$64$$ −13.0489 −1.63112
$$65$$ 0 0
$$66$$ −12.1751 −1.49865
$$67$$ −14.1087 −1.72365 −0.861824 0.507207i $$-0.830678\pi$$
−0.861824 + 0.507207i $$0.830678\pi$$
$$68$$ 19.8624 2.40867
$$69$$ 4.76354 0.573463
$$70$$ 0 0
$$71$$ 12.0784 1.43344 0.716721 0.697360i $$-0.245642\pi$$
0.716721 + 0.697360i $$0.245642\pi$$
$$72$$ 2.48774 0.293183
$$73$$ 11.5541 1.35231 0.676155 0.736759i $$-0.263645\pi$$
0.676155 + 0.736759i $$0.263645\pi$$
$$74$$ −7.88109 −0.916158
$$75$$ 0 0
$$76$$ −25.0484 −2.87325
$$77$$ 19.6890 2.24377
$$78$$ −8.50866 −0.963417
$$79$$ 1.64874 0.185498 0.0927491 0.995690i $$-0.470435\pi$$
0.0927491 + 0.995690i $$0.470435\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 15.1406 1.67200
$$83$$ −10.7120 −1.17579 −0.587895 0.808937i $$-0.700043\pi$$
−0.587895 + 0.808937i $$0.700043\pi$$
$$84$$ −11.3282 −1.23601
$$85$$ 0 0
$$86$$ −11.9459 −1.28816
$$87$$ −3.78248 −0.405525
$$88$$ −13.4100 −1.42951
$$89$$ −7.50610 −0.795645 −0.397822 0.917462i $$-0.630234\pi$$
−0.397822 + 0.917462i $$0.630234\pi$$
$$90$$ 0 0
$$91$$ 13.7599 1.44243
$$92$$ 14.7738 1.54028
$$93$$ −5.26069 −0.545508
$$94$$ −2.25864 −0.232960
$$95$$ 0 0
$$96$$ −6.29446 −0.642425
$$97$$ −11.7481 −1.19284 −0.596420 0.802673i $$-0.703411\pi$$
−0.596420 + 0.802673i $$0.703411\pi$$
$$98$$ 14.3227 1.44681
$$99$$ −5.39045 −0.541760
$$100$$ 0 0
$$101$$ 9.79905 0.975042 0.487521 0.873111i $$-0.337901\pi$$
0.487521 + 0.873111i $$0.337901\pi$$
$$102$$ 14.4649 1.43224
$$103$$ 7.91135 0.779528 0.389764 0.920915i $$-0.372557\pi$$
0.389764 + 0.920915i $$0.372557\pi$$
$$104$$ −9.37174 −0.918975
$$105$$ 0 0
$$106$$ 21.7298 2.11058
$$107$$ −16.0171 −1.54843 −0.774217 0.632921i $$-0.781856\pi$$
−0.774217 + 0.632921i $$0.781856\pi$$
$$108$$ 3.10144 0.298436
$$109$$ −6.27573 −0.601106 −0.300553 0.953765i $$-0.597171\pi$$
−0.300553 + 0.953765i $$0.597171\pi$$
$$110$$ 0 0
$$111$$ −3.48931 −0.331191
$$112$$ 2.13299 0.201549
$$113$$ −15.5200 −1.45999 −0.729997 0.683450i $$-0.760479\pi$$
−0.729997 + 0.683450i $$0.760479\pi$$
$$114$$ −18.2416 −1.70849
$$115$$ 0 0
$$116$$ −11.7311 −1.08921
$$117$$ −3.76717 −0.348275
$$118$$ 22.6879 2.08859
$$119$$ −23.3920 −2.14434
$$120$$ 0 0
$$121$$ 18.0569 1.64154
$$122$$ −10.5729 −0.957227
$$123$$ 6.70344 0.604429
$$124$$ −16.3157 −1.46519
$$125$$ 0 0
$$126$$ −8.24983 −0.734954
$$127$$ −16.1501 −1.43309 −0.716543 0.697543i $$-0.754277\pi$$
−0.716543 + 0.697543i $$0.754277\pi$$
$$128$$ −16.8839 −1.49234
$$129$$ −5.28901 −0.465671
$$130$$ 0 0
$$131$$ 14.0586 1.22831 0.614155 0.789186i $$-0.289497\pi$$
0.614155 + 0.789186i $$0.289497\pi$$
$$132$$ −16.7181 −1.45513
$$133$$ 29.4997 2.55794
$$134$$ −31.8664 −2.75283
$$135$$ 0 0
$$136$$ 15.9321 1.36617
$$137$$ 20.6494 1.76419 0.882097 0.471069i $$-0.156132\pi$$
0.882097 + 0.471069i $$0.156132\pi$$
$$138$$ 10.7591 0.915876
$$139$$ −3.85467 −0.326949 −0.163474 0.986548i $$-0.552270\pi$$
−0.163474 + 0.986548i $$0.552270\pi$$
$$140$$ 0 0
$$141$$ −1.00000 −0.0842152
$$142$$ 27.2807 2.28935
$$143$$ 20.3067 1.69813
$$144$$ −0.583969 −0.0486641
$$145$$ 0 0
$$146$$ 26.0966 2.15977
$$147$$ 6.34130 0.523021
$$148$$ −10.8219 −0.889553
$$149$$ −9.31652 −0.763239 −0.381620 0.924319i $$-0.624634\pi$$
−0.381620 + 0.924319i $$0.624634\pi$$
$$150$$ 0 0
$$151$$ 10.9014 0.887146 0.443573 0.896238i $$-0.353711\pi$$
0.443573 + 0.896238i $$0.353711\pi$$
$$152$$ −20.0920 −1.62968
$$153$$ 6.40426 0.517754
$$154$$ 44.4703 3.58352
$$155$$ 0 0
$$156$$ −11.6836 −0.935439
$$157$$ −8.68544 −0.693174 −0.346587 0.938018i $$-0.612659\pi$$
−0.346587 + 0.938018i $$0.612659\pi$$
$$158$$ 3.72391 0.296258
$$159$$ 9.62076 0.762976
$$160$$ 0 0
$$161$$ −17.3992 −1.37125
$$162$$ 2.25864 0.177455
$$163$$ 14.4978 1.13556 0.567778 0.823182i $$-0.307803\pi$$
0.567778 + 0.823182i $$0.307803\pi$$
$$164$$ 20.7903 1.62345
$$165$$ 0 0
$$166$$ −24.1944 −1.87785
$$167$$ −14.1218 −1.09278 −0.546388 0.837532i $$-0.683998\pi$$
−0.546388 + 0.837532i $$0.683998\pi$$
$$168$$ −9.08666 −0.701051
$$169$$ 1.19157 0.0916591
$$170$$ 0 0
$$171$$ −8.07640 −0.617618
$$172$$ −16.4035 −1.25076
$$173$$ −6.32710 −0.481041 −0.240520 0.970644i $$-0.577318\pi$$
−0.240520 + 0.970644i $$0.577318\pi$$
$$174$$ −8.54325 −0.647662
$$175$$ 0 0
$$176$$ 3.14785 0.237278
$$177$$ 10.0449 0.755024
$$178$$ −16.9535 −1.27072
$$179$$ 8.85078 0.661538 0.330769 0.943712i $$-0.392692\pi$$
0.330769 + 0.943712i $$0.392692\pi$$
$$180$$ 0 0
$$181$$ 13.6795 1.01679 0.508395 0.861124i $$-0.330239\pi$$
0.508395 + 0.861124i $$0.330239\pi$$
$$182$$ 31.0785 2.30369
$$183$$ −4.68111 −0.346038
$$184$$ 11.8505 0.873628
$$185$$ 0 0
$$186$$ −11.8820 −0.871228
$$187$$ −34.5218 −2.52449
$$188$$ −3.10144 −0.226195
$$189$$ −3.65257 −0.265686
$$190$$ 0 0
$$191$$ 15.7719 1.14121 0.570607 0.821223i $$-0.306708\pi$$
0.570607 + 0.821223i $$0.306708\pi$$
$$192$$ −13.0489 −0.941727
$$193$$ 7.82923 0.563561 0.281780 0.959479i $$-0.409075\pi$$
0.281780 + 0.959479i $$0.409075\pi$$
$$194$$ −26.5347 −1.90508
$$195$$ 0 0
$$196$$ 19.6671 1.40479
$$197$$ 10.6587 0.759401 0.379701 0.925109i $$-0.376027\pi$$
0.379701 + 0.925109i $$0.376027\pi$$
$$198$$ −12.1751 −0.865243
$$199$$ −6.40218 −0.453839 −0.226919 0.973914i $$-0.572865\pi$$
−0.226919 + 0.973914i $$0.572865\pi$$
$$200$$ 0 0
$$201$$ −14.1087 −0.995149
$$202$$ 22.1325 1.55724
$$203$$ 13.8158 0.969679
$$204$$ 19.8624 1.39065
$$205$$ 0 0
$$206$$ 17.8689 1.24498
$$207$$ 4.76354 0.331089
$$208$$ 2.19991 0.152536
$$209$$ 43.5354 3.01141
$$210$$ 0 0
$$211$$ 7.13642 0.491291 0.245646 0.969360i $$-0.421000\pi$$
0.245646 + 0.969360i $$0.421000\pi$$
$$212$$ 29.8382 2.04929
$$213$$ 12.0784 0.827599
$$214$$ −36.1768 −2.47300
$$215$$ 0 0
$$216$$ 2.48774 0.169269
$$217$$ 19.2150 1.30440
$$218$$ −14.1746 −0.960024
$$219$$ 11.5541 0.780756
$$220$$ 0 0
$$221$$ −24.1259 −1.62289
$$222$$ −7.88109 −0.528944
$$223$$ −5.41572 −0.362664 −0.181332 0.983422i $$-0.558041\pi$$
−0.181332 + 0.983422i $$0.558041\pi$$
$$224$$ 22.9910 1.53615
$$225$$ 0 0
$$226$$ −35.0539 −2.33175
$$227$$ −22.8812 −1.51868 −0.759339 0.650695i $$-0.774478\pi$$
−0.759339 + 0.650695i $$0.774478\pi$$
$$228$$ −25.0484 −1.65887
$$229$$ 25.9203 1.71286 0.856431 0.516261i $$-0.172677\pi$$
0.856431 + 0.516261i $$0.172677\pi$$
$$230$$ 0 0
$$231$$ 19.6890 1.29544
$$232$$ −9.40984 −0.617786
$$233$$ 1.38698 0.0908638 0.0454319 0.998967i $$-0.485534\pi$$
0.0454319 + 0.998967i $$0.485534\pi$$
$$234$$ −8.50866 −0.556229
$$235$$ 0 0
$$236$$ 31.1538 2.02794
$$237$$ 1.64874 0.107097
$$238$$ −52.8341 −3.42472
$$239$$ −20.8507 −1.34872 −0.674361 0.738402i $$-0.735581\pi$$
−0.674361 + 0.738402i $$0.735581\pi$$
$$240$$ 0 0
$$241$$ −19.2621 −1.24078 −0.620389 0.784294i $$-0.713025\pi$$
−0.620389 + 0.784294i $$0.713025\pi$$
$$242$$ 40.7840 2.62169
$$243$$ 1.00000 0.0641500
$$244$$ −14.5182 −0.929430
$$245$$ 0 0
$$246$$ 15.1406 0.965332
$$247$$ 30.4252 1.93591
$$248$$ −13.0872 −0.831040
$$249$$ −10.7120 −0.678843
$$250$$ 0 0
$$251$$ 10.0252 0.632784 0.316392 0.948629i $$-0.397529\pi$$
0.316392 + 0.948629i $$0.397529\pi$$
$$252$$ −11.3282 −0.713611
$$253$$ −25.6776 −1.61434
$$254$$ −36.4771 −2.28878
$$255$$ 0 0
$$256$$ −12.0367 −0.752293
$$257$$ 11.1976 0.698488 0.349244 0.937032i $$-0.386438\pi$$
0.349244 + 0.937032i $$0.386438\pi$$
$$258$$ −11.9459 −0.743722
$$259$$ 12.7450 0.791935
$$260$$ 0 0
$$261$$ −3.78248 −0.234130
$$262$$ 31.7534 1.96173
$$263$$ −0.764798 −0.0471595 −0.0235797 0.999722i $$-0.507506\pi$$
−0.0235797 + 0.999722i $$0.507506\pi$$
$$264$$ −13.4100 −0.825331
$$265$$ 0 0
$$266$$ 66.6290 4.08528
$$267$$ −7.50610 −0.459366
$$268$$ −43.7571 −2.67289
$$269$$ 2.34969 0.143263 0.0716316 0.997431i $$-0.477179\pi$$
0.0716316 + 0.997431i $$0.477179\pi$$
$$270$$ 0 0
$$271$$ −15.8548 −0.963111 −0.481556 0.876416i $$-0.659928\pi$$
−0.481556 + 0.876416i $$0.659928\pi$$
$$272$$ −3.73989 −0.226764
$$273$$ 13.7599 0.832785
$$274$$ 46.6394 2.81759
$$275$$ 0 0
$$276$$ 14.7738 0.889280
$$277$$ −17.8932 −1.07510 −0.537548 0.843233i $$-0.680649\pi$$
−0.537548 + 0.843233i $$0.680649\pi$$
$$278$$ −8.70630 −0.522169
$$279$$ −5.26069 −0.314949
$$280$$ 0 0
$$281$$ −21.7136 −1.29533 −0.647663 0.761927i $$-0.724253\pi$$
−0.647663 + 0.761927i $$0.724253\pi$$
$$282$$ −2.25864 −0.134500
$$283$$ −14.5961 −0.867648 −0.433824 0.900998i $$-0.642836\pi$$
−0.433824 + 0.900998i $$0.642836\pi$$
$$284$$ 37.4604 2.22286
$$285$$ 0 0
$$286$$ 45.8655 2.71208
$$287$$ −24.4848 −1.44529
$$288$$ −6.29446 −0.370904
$$289$$ 24.0145 1.41262
$$290$$ 0 0
$$291$$ −11.7481 −0.688686
$$292$$ 35.8344 2.09705
$$293$$ 6.51441 0.380576 0.190288 0.981728i $$-0.439058\pi$$
0.190288 + 0.981728i $$0.439058\pi$$
$$294$$ 14.3227 0.835316
$$295$$ 0 0
$$296$$ −8.68051 −0.504545
$$297$$ −5.39045 −0.312785
$$298$$ −21.0426 −1.21897
$$299$$ −17.9451 −1.03779
$$300$$ 0 0
$$301$$ 19.3185 1.11350
$$302$$ 24.6224 1.41686
$$303$$ 9.79905 0.562941
$$304$$ 4.71637 0.270502
$$305$$ 0 0
$$306$$ 14.4649 0.826903
$$307$$ −30.2614 −1.72711 −0.863555 0.504254i $$-0.831767\pi$$
−0.863555 + 0.504254i $$0.831767\pi$$
$$308$$ 61.0642 3.47945
$$309$$ 7.91135 0.450061
$$310$$ 0 0
$$311$$ −14.8690 −0.843145 −0.421572 0.906795i $$-0.638522\pi$$
−0.421572 + 0.906795i $$0.638522\pi$$
$$312$$ −9.37174 −0.530571
$$313$$ −19.0799 −1.07846 −0.539231 0.842158i $$-0.681285\pi$$
−0.539231 + 0.842158i $$0.681285\pi$$
$$314$$ −19.6172 −1.10707
$$315$$ 0 0
$$316$$ 5.11347 0.287655
$$317$$ 11.9688 0.672234 0.336117 0.941820i $$-0.390886\pi$$
0.336117 + 0.941820i $$0.390886\pi$$
$$318$$ 21.7298 1.21855
$$319$$ 20.3893 1.14158
$$320$$ 0 0
$$321$$ −16.0171 −0.893989
$$322$$ −39.2984 −2.19002
$$323$$ −51.7234 −2.87797
$$324$$ 3.10144 0.172302
$$325$$ 0 0
$$326$$ 32.7453 1.81359
$$327$$ −6.27573 −0.347049
$$328$$ 16.6764 0.920802
$$329$$ 3.65257 0.201373
$$330$$ 0 0
$$331$$ −16.0901 −0.884390 −0.442195 0.896919i $$-0.645800\pi$$
−0.442195 + 0.896919i $$0.645800\pi$$
$$332$$ −33.2224 −1.82332
$$333$$ −3.48931 −0.191213
$$334$$ −31.8960 −1.74527
$$335$$ 0 0
$$336$$ 2.13299 0.116364
$$337$$ 9.68906 0.527797 0.263898 0.964550i $$-0.414992\pi$$
0.263898 + 0.964550i $$0.414992\pi$$
$$338$$ 2.69132 0.146388
$$339$$ −15.5200 −0.842928
$$340$$ 0 0
$$341$$ 28.3574 1.53564
$$342$$ −18.2416 −0.986395
$$343$$ 2.40596 0.129910
$$344$$ −13.1577 −0.709415
$$345$$ 0 0
$$346$$ −14.2906 −0.768268
$$347$$ −3.87840 −0.208203 −0.104102 0.994567i $$-0.533197\pi$$
−0.104102 + 0.994567i $$0.533197\pi$$
$$348$$ −11.7311 −0.628855
$$349$$ −26.0674 −1.39536 −0.697679 0.716411i $$-0.745784\pi$$
−0.697679 + 0.716411i $$0.745784\pi$$
$$350$$ 0 0
$$351$$ −3.76717 −0.201077
$$352$$ 33.9299 1.80847
$$353$$ −32.9876 −1.75575 −0.877877 0.478886i $$-0.841041\pi$$
−0.877877 + 0.478886i $$0.841041\pi$$
$$354$$ 22.6879 1.20585
$$355$$ 0 0
$$356$$ −23.2797 −1.23382
$$357$$ −23.3920 −1.23804
$$358$$ 19.9907 1.05654
$$359$$ −0.425406 −0.0224520 −0.0112260 0.999937i $$-0.503573\pi$$
−0.0112260 + 0.999937i $$0.503573\pi$$
$$360$$ 0 0
$$361$$ 46.2283 2.43307
$$362$$ 30.8970 1.62391
$$363$$ 18.0569 0.947742
$$364$$ 42.6753 2.23680
$$365$$ 0 0
$$366$$ −10.5729 −0.552655
$$367$$ 1.97076 0.102873 0.0514365 0.998676i $$-0.483620\pi$$
0.0514365 + 0.998676i $$0.483620\pi$$
$$368$$ −2.78176 −0.145009
$$369$$ 6.70344 0.348967
$$370$$ 0 0
$$371$$ −35.1405 −1.82441
$$372$$ −16.3157 −0.845928
$$373$$ −3.91613 −0.202770 −0.101385 0.994847i $$-0.532327\pi$$
−0.101385 + 0.994847i $$0.532327\pi$$
$$374$$ −77.9722 −4.03185
$$375$$ 0 0
$$376$$ −2.48774 −0.128295
$$377$$ 14.2493 0.733874
$$378$$ −8.24983 −0.424326
$$379$$ −6.09374 −0.313014 −0.156507 0.987677i $$-0.550023\pi$$
−0.156507 + 0.987677i $$0.550023\pi$$
$$380$$ 0 0
$$381$$ −16.1501 −0.827393
$$382$$ 35.6230 1.82263
$$383$$ 27.9056 1.42591 0.712956 0.701209i $$-0.247356\pi$$
0.712956 + 0.701209i $$0.247356\pi$$
$$384$$ −16.8839 −0.861603
$$385$$ 0 0
$$386$$ 17.6834 0.900061
$$387$$ −5.28901 −0.268855
$$388$$ −36.4360 −1.84976
$$389$$ 7.08100 0.359021 0.179511 0.983756i $$-0.442549\pi$$
0.179511 + 0.983756i $$0.442549\pi$$
$$390$$ 0 0
$$391$$ 30.5070 1.54280
$$392$$ 15.7755 0.796783
$$393$$ 14.0586 0.709165
$$394$$ 24.0741 1.21284
$$395$$ 0 0
$$396$$ −16.7181 −0.840117
$$397$$ −12.8045 −0.642637 −0.321319 0.946971i $$-0.604126\pi$$
−0.321319 + 0.946971i $$0.604126\pi$$
$$398$$ −14.4602 −0.724824
$$399$$ 29.4997 1.47683
$$400$$ 0 0
$$401$$ 17.9986 0.898808 0.449404 0.893329i $$-0.351636\pi$$
0.449404 + 0.893329i $$0.351636\pi$$
$$402$$ −31.8664 −1.58935
$$403$$ 19.8179 0.987200
$$404$$ 30.3911 1.51202
$$405$$ 0 0
$$406$$ 31.2049 1.54867
$$407$$ 18.8090 0.932326
$$408$$ 15.9321 0.788759
$$409$$ 28.9900 1.43346 0.716731 0.697350i $$-0.245637\pi$$
0.716731 + 0.697350i $$0.245637\pi$$
$$410$$ 0 0
$$411$$ 20.6494 1.01856
$$412$$ 24.5365 1.20883
$$413$$ −36.6899 −1.80539
$$414$$ 10.7591 0.528781
$$415$$ 0 0
$$416$$ 23.7123 1.16259
$$417$$ −3.85467 −0.188764
$$418$$ 98.3306 4.80951
$$419$$ −0.494504 −0.0241581 −0.0120791 0.999927i $$-0.503845\pi$$
−0.0120791 + 0.999927i $$0.503845\pi$$
$$420$$ 0 0
$$421$$ −24.4814 −1.19315 −0.596575 0.802557i $$-0.703472\pi$$
−0.596575 + 0.802557i $$0.703472\pi$$
$$422$$ 16.1186 0.784640
$$423$$ −1.00000 −0.0486217
$$424$$ 23.9340 1.16234
$$425$$ 0 0
$$426$$ 27.2807 1.32175
$$427$$ 17.0981 0.827435
$$428$$ −49.6761 −2.40118
$$429$$ 20.3067 0.980418
$$430$$ 0 0
$$431$$ 20.5802 0.991311 0.495656 0.868519i $$-0.334928\pi$$
0.495656 + 0.868519i $$0.334928\pi$$
$$432$$ −0.583969 −0.0280962
$$433$$ 10.8674 0.522254 0.261127 0.965304i $$-0.415906\pi$$
0.261127 + 0.965304i $$0.415906\pi$$
$$434$$ 43.3998 2.08326
$$435$$ 0 0
$$436$$ −19.4638 −0.932145
$$437$$ −38.4723 −1.84038
$$438$$ 26.0966 1.24694
$$439$$ −4.05108 −0.193347 −0.0966737 0.995316i $$-0.530820\pi$$
−0.0966737 + 0.995316i $$0.530820\pi$$
$$440$$ 0 0
$$441$$ 6.34130 0.301966
$$442$$ −54.4917 −2.59191
$$443$$ −19.8347 −0.942376 −0.471188 0.882033i $$-0.656175\pi$$
−0.471188 + 0.882033i $$0.656175\pi$$
$$444$$ −10.8219 −0.513584
$$445$$ 0 0
$$446$$ −12.2321 −0.579209
$$447$$ −9.31652 −0.440656
$$448$$ 47.6622 2.25183
$$449$$ −12.3888 −0.584663 −0.292331 0.956317i $$-0.594431\pi$$
−0.292331 + 0.956317i $$0.594431\pi$$
$$450$$ 0 0
$$451$$ −36.1346 −1.70151
$$452$$ −48.1341 −2.26404
$$453$$ 10.9014 0.512194
$$454$$ −51.6803 −2.42548
$$455$$ 0 0
$$456$$ −20.0920 −0.940894
$$457$$ 0.726902 0.0340030 0.0170015 0.999855i $$-0.494588\pi$$
0.0170015 + 0.999855i $$0.494588\pi$$
$$458$$ 58.5445 2.73561
$$459$$ 6.40426 0.298925
$$460$$ 0 0
$$461$$ −5.78513 −0.269440 −0.134720 0.990884i $$-0.543014\pi$$
−0.134720 + 0.990884i $$0.543014\pi$$
$$462$$ 44.4703 2.06894
$$463$$ −23.2529 −1.08066 −0.540328 0.841455i $$-0.681700\pi$$
−0.540328 + 0.841455i $$0.681700\pi$$
$$464$$ 2.20885 0.102544
$$465$$ 0 0
$$466$$ 3.13267 0.145118
$$467$$ 19.5028 0.902481 0.451241 0.892402i $$-0.350982\pi$$
0.451241 + 0.892402i $$0.350982\pi$$
$$468$$ −11.6836 −0.540076
$$469$$ 51.5330 2.37957
$$470$$ 0 0
$$471$$ −8.68544 −0.400204
$$472$$ 24.9892 1.15022
$$473$$ 28.5101 1.31090
$$474$$ 3.72391 0.171045
$$475$$ 0 0
$$476$$ −72.5489 −3.32527
$$477$$ 9.62076 0.440504
$$478$$ −47.0942 −2.15404
$$479$$ 15.9371 0.728185 0.364093 0.931363i $$-0.381379\pi$$
0.364093 + 0.931363i $$0.381379\pi$$
$$480$$ 0 0
$$481$$ 13.1448 0.599353
$$482$$ −43.5060 −1.98164
$$483$$ −17.3992 −0.791691
$$484$$ 56.0023 2.54556
$$485$$ 0 0
$$486$$ 2.25864 0.102454
$$487$$ −21.5979 −0.978695 −0.489347 0.872089i $$-0.662765\pi$$
−0.489347 + 0.872089i $$0.662765\pi$$
$$488$$ −11.6454 −0.527162
$$489$$ 14.4978 0.655613
$$490$$ 0 0
$$491$$ −19.5726 −0.883298 −0.441649 0.897188i $$-0.645606\pi$$
−0.441649 + 0.897188i $$0.645606\pi$$
$$492$$ 20.7903 0.937299
$$493$$ −24.2240 −1.09099
$$494$$ 68.7194 3.09183
$$495$$ 0 0
$$496$$ 3.07208 0.137940
$$497$$ −44.1173 −1.97893
$$498$$ −24.1944 −1.08418
$$499$$ −30.3032 −1.35656 −0.678278 0.734805i $$-0.737274\pi$$
−0.678278 + 0.734805i $$0.737274\pi$$
$$500$$ 0 0
$$501$$ −14.1218 −0.630915
$$502$$ 22.6432 1.01062
$$503$$ 23.6646 1.05515 0.527577 0.849507i $$-0.323101\pi$$
0.527577 + 0.849507i $$0.323101\pi$$
$$504$$ −9.08666 −0.404752
$$505$$ 0 0
$$506$$ −57.9964 −2.57825
$$507$$ 1.19157 0.0529194
$$508$$ −50.0884 −2.22231
$$509$$ 12.2650 0.543638 0.271819 0.962348i $$-0.412375\pi$$
0.271819 + 0.962348i $$0.412375\pi$$
$$510$$ 0 0
$$511$$ −42.2024 −1.86692
$$512$$ 6.58130 0.290855
$$513$$ −8.07640 −0.356582
$$514$$ 25.2913 1.11555
$$515$$ 0 0
$$516$$ −16.4035 −0.722125
$$517$$ 5.39045 0.237072
$$518$$ 28.7863 1.26480
$$519$$ −6.32710 −0.277729
$$520$$ 0 0
$$521$$ −29.7739 −1.30442 −0.652210 0.758038i $$-0.726158\pi$$
−0.652210 + 0.758038i $$0.726158\pi$$
$$522$$ −8.54325 −0.373928
$$523$$ −36.9743 −1.61677 −0.808386 0.588653i $$-0.799658\pi$$
−0.808386 + 0.588653i $$0.799658\pi$$
$$524$$ 43.6020 1.90476
$$525$$ 0 0
$$526$$ −1.72740 −0.0753182
$$527$$ −33.6908 −1.46759
$$528$$ 3.14785 0.136993
$$529$$ −0.308648 −0.0134195
$$530$$ 0 0
$$531$$ 10.0449 0.435914
$$532$$ 91.4913 3.96665
$$533$$ −25.2530 −1.09383
$$534$$ −16.9535 −0.733652
$$535$$ 0 0
$$536$$ −35.0987 −1.51603
$$537$$ 8.85078 0.381939
$$538$$ 5.30710 0.228805
$$539$$ −34.1824 −1.47234
$$540$$ 0 0
$$541$$ 17.3797 0.747212 0.373606 0.927587i $$-0.378121\pi$$
0.373606 + 0.927587i $$0.378121\pi$$
$$542$$ −35.8102 −1.53818
$$543$$ 13.6795 0.587044
$$544$$ −40.3113 −1.72833
$$545$$ 0 0
$$546$$ 31.0785 1.33004
$$547$$ −3.14180 −0.134334 −0.0671668 0.997742i $$-0.521396\pi$$
−0.0671668 + 0.997742i $$0.521396\pi$$
$$548$$ 64.0426 2.73577
$$549$$ −4.68111 −0.199785
$$550$$ 0 0
$$551$$ 30.5489 1.30143
$$552$$ 11.8505 0.504389
$$553$$ −6.02216 −0.256088
$$554$$ −40.4141 −1.71703
$$555$$ 0 0
$$556$$ −11.9550 −0.507005
$$557$$ 4.70436 0.199330 0.0996650 0.995021i $$-0.468223\pi$$
0.0996650 + 0.995021i $$0.468223\pi$$
$$558$$ −11.8820 −0.503004
$$559$$ 19.9246 0.842721
$$560$$ 0 0
$$561$$ −34.5218 −1.45751
$$562$$ −49.0432 −2.06876
$$563$$ −9.19564 −0.387550 −0.193775 0.981046i $$-0.562073\pi$$
−0.193775 + 0.981046i $$0.562073\pi$$
$$564$$ −3.10144 −0.130594
$$565$$ 0 0
$$566$$ −32.9673 −1.38572
$$567$$ −3.65257 −0.153394
$$568$$ 30.0479 1.26078
$$569$$ 4.54858 0.190686 0.0953432 0.995444i $$-0.469605\pi$$
0.0953432 + 0.995444i $$0.469605\pi$$
$$570$$ 0 0
$$571$$ −8.85447 −0.370548 −0.185274 0.982687i $$-0.559317\pi$$
−0.185274 + 0.982687i $$0.559317\pi$$
$$572$$ 62.9800 2.63333
$$573$$ 15.7719 0.658881
$$574$$ −55.3023 −2.30827
$$575$$ 0 0
$$576$$ −13.0489 −0.543706
$$577$$ 8.34516 0.347414 0.173707 0.984797i $$-0.444425\pi$$
0.173707 + 0.984797i $$0.444425\pi$$
$$578$$ 54.2401 2.25609
$$579$$ 7.82923 0.325372
$$580$$ 0 0
$$581$$ 39.1262 1.62323
$$582$$ −26.5347 −1.09990
$$583$$ −51.8602 −2.14783
$$584$$ 28.7437 1.18942
$$585$$ 0 0
$$586$$ 14.7137 0.607817
$$587$$ 18.7043 0.772010 0.386005 0.922497i $$-0.373855\pi$$
0.386005 + 0.922497i $$0.373855\pi$$
$$588$$ 19.6671 0.811058
$$589$$ 42.4874 1.75066
$$590$$ 0 0
$$591$$ 10.6587 0.438440
$$592$$ 2.03765 0.0837470
$$593$$ 19.5188 0.801542 0.400771 0.916178i $$-0.368742\pi$$
0.400771 + 0.916178i $$0.368742\pi$$
$$594$$ −12.1751 −0.499548
$$595$$ 0 0
$$596$$ −28.8946 −1.18357
$$597$$ −6.40218 −0.262024
$$598$$ −40.5314 −1.65745
$$599$$ −10.2208 −0.417610 −0.208805 0.977957i $$-0.566957\pi$$
−0.208805 + 0.977957i $$0.566957\pi$$
$$600$$ 0 0
$$601$$ −23.3384 −0.951992 −0.475996 0.879448i $$-0.657912\pi$$
−0.475996 + 0.879448i $$0.657912\pi$$
$$602$$ 43.6335 1.77837
$$603$$ −14.1087 −0.574550
$$604$$ 33.8101 1.37571
$$605$$ 0 0
$$606$$ 22.1325 0.899071
$$607$$ 31.6937 1.28641 0.643204 0.765695i $$-0.277605\pi$$
0.643204 + 0.765695i $$0.277605\pi$$
$$608$$ 50.8366 2.06169
$$609$$ 13.8158 0.559845
$$610$$ 0 0
$$611$$ 3.76717 0.152403
$$612$$ 19.8624 0.802890
$$613$$ 34.6718 1.40038 0.700191 0.713955i $$-0.253098\pi$$
0.700191 + 0.713955i $$0.253098\pi$$
$$614$$ −68.3495 −2.75836
$$615$$ 0 0
$$616$$ 48.9811 1.97351
$$617$$ −45.0825 −1.81495 −0.907476 0.420104i $$-0.861994\pi$$
−0.907476 + 0.420104i $$0.861994\pi$$
$$618$$ 17.8689 0.718791
$$619$$ 15.5285 0.624141 0.312071 0.950059i $$-0.398977\pi$$
0.312071 + 0.950059i $$0.398977\pi$$
$$620$$ 0 0
$$621$$ 4.76354 0.191154
$$622$$ −33.5837 −1.34658
$$623$$ 27.4166 1.09842
$$624$$ 2.19991 0.0880669
$$625$$ 0 0
$$626$$ −43.0947 −1.72241
$$627$$ 43.5354 1.73864
$$628$$ −26.9373 −1.07492
$$629$$ −22.3465 −0.891012
$$630$$ 0 0
$$631$$ 11.8534 0.471878 0.235939 0.971768i $$-0.424183\pi$$
0.235939 + 0.971768i $$0.424183\pi$$
$$632$$ 4.10165 0.163155
$$633$$ 7.13642 0.283647
$$634$$ 27.0331 1.07362
$$635$$ 0 0
$$636$$ 29.8382 1.18316
$$637$$ −23.8887 −0.946506
$$638$$ 46.0519 1.82321
$$639$$ 12.0784 0.477814
$$640$$ 0 0
$$641$$ 11.3123 0.446807 0.223404 0.974726i $$-0.428283\pi$$
0.223404 + 0.974726i $$0.428283\pi$$
$$642$$ −36.1768 −1.42779
$$643$$ 3.71679 0.146576 0.0732879 0.997311i $$-0.476651\pi$$
0.0732879 + 0.997311i $$0.476651\pi$$
$$644$$ −53.9625 −2.12642
$$645$$ 0 0
$$646$$ −116.824 −4.59639
$$647$$ −20.9097 −0.822047 −0.411023 0.911625i $$-0.634829\pi$$
−0.411023 + 0.911625i $$0.634829\pi$$
$$648$$ 2.48774 0.0977277
$$649$$ −54.1467 −2.12545
$$650$$ 0 0
$$651$$ 19.2150 0.753097
$$652$$ 44.9640 1.76093
$$653$$ 31.2980 1.22479 0.612393 0.790553i $$-0.290207\pi$$
0.612393 + 0.790553i $$0.290207\pi$$
$$654$$ −14.1746 −0.554270
$$655$$ 0 0
$$656$$ −3.91461 −0.152840
$$657$$ 11.5541 0.450770
$$658$$ 8.24983 0.321612
$$659$$ 11.6158 0.452487 0.226243 0.974071i $$-0.427356\pi$$
0.226243 + 0.974071i $$0.427356\pi$$
$$660$$ 0 0
$$661$$ −9.93633 −0.386478 −0.193239 0.981152i $$-0.561899\pi$$
−0.193239 + 0.981152i $$0.561899\pi$$
$$662$$ −36.3416 −1.41246
$$663$$ −24.1259 −0.936974
$$664$$ −26.6486 −1.03417
$$665$$ 0 0
$$666$$ −7.88109 −0.305386
$$667$$ −18.0180 −0.697661
$$668$$ −43.7978 −1.69459
$$669$$ −5.41572 −0.209384
$$670$$ 0 0
$$671$$ 25.2333 0.974119
$$672$$ 22.9910 0.886896
$$673$$ 22.5946 0.870957 0.435479 0.900199i $$-0.356579\pi$$
0.435479 + 0.900199i $$0.356579\pi$$
$$674$$ 21.8841 0.842943
$$675$$ 0 0
$$676$$ 3.69557 0.142137
$$677$$ −30.0102 −1.15338 −0.576692 0.816962i $$-0.695657\pi$$
−0.576692 + 0.816962i $$0.695657\pi$$
$$678$$ −35.0539 −1.34624
$$679$$ 42.9108 1.64677
$$680$$ 0 0
$$681$$ −22.8812 −0.876809
$$682$$ 64.0491 2.45257
$$683$$ 2.71242 0.103788 0.0518939 0.998653i $$-0.483474\pi$$
0.0518939 + 0.998653i $$0.483474\pi$$
$$684$$ −25.0484 −0.957751
$$685$$ 0 0
$$686$$ 5.43419 0.207479
$$687$$ 25.9203 0.988921
$$688$$ 3.08862 0.117753
$$689$$ −36.2430 −1.38075
$$690$$ 0 0
$$691$$ 12.4120 0.472174 0.236087 0.971732i $$-0.424135\pi$$
0.236087 + 0.971732i $$0.424135\pi$$
$$692$$ −19.6231 −0.745958
$$693$$ 19.6890 0.747923
$$694$$ −8.75989 −0.332521
$$695$$ 0 0
$$696$$ −9.40984 −0.356679
$$697$$ 42.9306 1.62611
$$698$$ −58.8768 −2.22852
$$699$$ 1.38698 0.0524603
$$700$$ 0 0
$$701$$ 38.1760 1.44189 0.720943 0.692994i $$-0.243709\pi$$
0.720943 + 0.692994i $$0.243709\pi$$
$$702$$ −8.50866 −0.321139
$$703$$ 28.1811 1.06287
$$704$$ 70.3396 2.65102
$$705$$ 0 0
$$706$$ −74.5070 −2.80411
$$707$$ −35.7918 −1.34609
$$708$$ 31.1538 1.17083
$$709$$ −21.2535 −0.798194 −0.399097 0.916909i $$-0.630676\pi$$
−0.399097 + 0.916909i $$0.630676\pi$$
$$710$$ 0 0
$$711$$ 1.64874 0.0618327
$$712$$ −18.6732 −0.699809
$$713$$ −25.0595 −0.938486
$$714$$ −52.8341 −1.97727
$$715$$ 0 0
$$716$$ 27.4501 1.02586
$$717$$ −20.8507 −0.778685
$$718$$ −0.960836 −0.0358581
$$719$$ 25.3271 0.944543 0.472272 0.881453i $$-0.343434\pi$$
0.472272 + 0.881453i $$0.343434\pi$$
$$720$$ 0 0
$$721$$ −28.8968 −1.07617
$$722$$ 104.413 3.88584
$$723$$ −19.2621 −0.716364
$$724$$ 42.4261 1.57675
$$725$$ 0 0
$$726$$ 40.7840 1.51363
$$727$$ 38.4277 1.42521 0.712603 0.701567i $$-0.247516\pi$$
0.712603 + 0.701567i $$0.247516\pi$$
$$728$$ 34.2310 1.26868
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ −33.8722 −1.25281
$$732$$ −14.5182 −0.536607
$$733$$ −39.9589 −1.47592 −0.737959 0.674846i $$-0.764210\pi$$
−0.737959 + 0.674846i $$0.764210\pi$$
$$734$$ 4.45123 0.164298
$$735$$ 0 0
$$736$$ −29.9839 −1.10522
$$737$$ 76.0520 2.80141
$$738$$ 15.1406 0.557335
$$739$$ 6.27640 0.230881 0.115441 0.993314i $$-0.463172\pi$$
0.115441 + 0.993314i $$0.463172\pi$$
$$740$$ 0 0
$$741$$ 30.4252 1.11770
$$742$$ −79.3696 −2.91375
$$743$$ −18.5595 −0.680882 −0.340441 0.940266i $$-0.610576\pi$$
−0.340441 + 0.940266i $$0.610576\pi$$
$$744$$ −13.0872 −0.479801
$$745$$ 0 0
$$746$$ −8.84511 −0.323843
$$747$$ −10.7120 −0.391930
$$748$$ −107.067 −3.91476
$$749$$ 58.5037 2.13768
$$750$$ 0 0
$$751$$ 46.3145 1.69004 0.845019 0.534736i $$-0.179589\pi$$
0.845019 + 0.534736i $$0.179589\pi$$
$$752$$ 0.583969 0.0212952
$$753$$ 10.0252 0.365338
$$754$$ 32.1839 1.17207
$$755$$ 0 0
$$756$$ −11.3282 −0.412003
$$757$$ −28.8540 −1.04872 −0.524358 0.851498i $$-0.675695\pi$$
−0.524358 + 0.851498i $$0.675695\pi$$
$$758$$ −13.7635 −0.499914
$$759$$ −25.6776 −0.932039
$$760$$ 0 0
$$761$$ 2.16914 0.0786313 0.0393156 0.999227i $$-0.487482\pi$$
0.0393156 + 0.999227i $$0.487482\pi$$
$$762$$ −36.4771 −1.32143
$$763$$ 22.9226 0.829852
$$764$$ 48.9155 1.76970
$$765$$ 0 0
$$766$$ 63.0287 2.27732
$$767$$ −37.8410 −1.36636
$$768$$ −12.0367 −0.434337
$$769$$ −30.4385 −1.09764 −0.548821 0.835940i $$-0.684923\pi$$
−0.548821 + 0.835940i $$0.684923\pi$$
$$770$$ 0 0
$$771$$ 11.1976 0.403272
$$772$$ 24.2819 0.873923
$$773$$ 0.878615 0.0316016 0.0158008 0.999875i $$-0.494970\pi$$
0.0158008 + 0.999875i $$0.494970\pi$$
$$774$$ −11.9459 −0.429388
$$775$$ 0 0
$$776$$ −29.2262 −1.04916
$$777$$ 12.7450 0.457224
$$778$$ 15.9934 0.573391
$$779$$ −54.1397 −1.93976
$$780$$ 0 0
$$781$$ −65.1080 −2.32975
$$782$$ 68.9041 2.46401
$$783$$ −3.78248 −0.135175
$$784$$ −3.70312 −0.132254
$$785$$ 0 0
$$786$$ 31.7534 1.13260
$$787$$ −17.9038 −0.638202 −0.319101 0.947721i $$-0.603381\pi$$
−0.319101 + 0.947721i $$0.603381\pi$$
$$788$$ 33.0573 1.17762
$$789$$ −0.764798 −0.0272275
$$790$$ 0 0
$$791$$ 56.6878 2.01559
$$792$$ −13.4100 −0.476505
$$793$$ 17.6345 0.626221
$$794$$ −28.9206 −1.02635
$$795$$ 0 0
$$796$$ −19.8560 −0.703776
$$797$$ −2.03099 −0.0719413 −0.0359707 0.999353i $$-0.511452\pi$$
−0.0359707 + 0.999353i $$0.511452\pi$$
$$798$$ 66.6290 2.35864
$$799$$ −6.40426 −0.226566
$$800$$ 0 0
$$801$$ −7.50610 −0.265215
$$802$$ 40.6523 1.43548
$$803$$ −62.2820 −2.19788
$$804$$ −43.7571 −1.54320
$$805$$ 0 0
$$806$$ 44.7614 1.57665
$$807$$ 2.34969 0.0827130
$$808$$ 24.3775 0.857598
$$809$$ −3.21240 −0.112942 −0.0564710 0.998404i $$-0.517985\pi$$
−0.0564710 + 0.998404i $$0.517985\pi$$
$$810$$ 0 0
$$811$$ 2.93628 0.103107 0.0515533 0.998670i $$-0.483583\pi$$
0.0515533 + 0.998670i $$0.483583\pi$$
$$812$$ 42.8488 1.50370
$$813$$ −15.8548 −0.556053
$$814$$ 42.4826 1.48901
$$815$$ 0 0
$$816$$ −3.73989 −0.130922
$$817$$ 42.7162 1.49445
$$818$$ 65.4778 2.28938
$$819$$ 13.7599 0.480809
$$820$$ 0 0
$$821$$ −13.8856 −0.484611 −0.242305 0.970200i $$-0.577904\pi$$
−0.242305 + 0.970200i $$0.577904\pi$$
$$822$$ 46.6394 1.62673
$$823$$ 10.2213 0.356294 0.178147 0.984004i $$-0.442990\pi$$
0.178147 + 0.984004i $$0.442990\pi$$
$$824$$ 19.6814 0.685634
$$825$$ 0 0
$$826$$ −82.8691 −2.88339
$$827$$ −27.1692 −0.944765 −0.472383 0.881394i $$-0.656606\pi$$
−0.472383 + 0.881394i $$0.656606\pi$$
$$828$$ 14.7738 0.513426
$$829$$ −7.41476 −0.257525 −0.128763 0.991675i $$-0.541101\pi$$
−0.128763 + 0.991675i $$0.541101\pi$$
$$830$$ 0 0
$$831$$ −17.8932 −0.620707
$$832$$ 49.1576 1.70423
$$833$$ 40.6113 1.40710
$$834$$ −8.70630 −0.301474
$$835$$ 0 0
$$836$$ 135.022 4.66984
$$837$$ −5.26069 −0.181836
$$838$$ −1.11691 −0.0385829
$$839$$ 6.99598 0.241528 0.120764 0.992681i $$-0.461466\pi$$
0.120764 + 0.992681i $$0.461466\pi$$
$$840$$ 0 0
$$841$$ −14.6928 −0.506649
$$842$$ −55.2946 −1.90558
$$843$$ −21.7136 −0.747857
$$844$$ 22.1331 0.761854
$$845$$ 0 0
$$846$$ −2.25864 −0.0776535
$$847$$ −65.9542 −2.26621
$$848$$ −5.61823 −0.192931
$$849$$ −14.5961 −0.500937
$$850$$ 0 0
$$851$$ −16.6215 −0.569778
$$852$$ 37.4604 1.28337
$$853$$ 25.3316 0.867338 0.433669 0.901072i $$-0.357219\pi$$
0.433669 + 0.901072i $$0.357219\pi$$
$$854$$ 38.6184 1.32149
$$855$$ 0 0
$$856$$ −39.8465 −1.36192
$$857$$ −11.2384 −0.383898 −0.191949 0.981405i $$-0.561481\pi$$
−0.191949 + 0.981405i $$0.561481\pi$$
$$858$$ 45.8655 1.56582
$$859$$ −41.7711 −1.42521 −0.712605 0.701566i $$-0.752485\pi$$
−0.712605 + 0.701566i $$0.752485\pi$$
$$860$$ 0 0
$$861$$ −24.4848 −0.834441
$$862$$ 46.4831 1.58322
$$863$$ 30.0343 1.02238 0.511190 0.859468i $$-0.329205\pi$$
0.511190 + 0.859468i $$0.329205\pi$$
$$864$$ −6.29446 −0.214142
$$865$$ 0 0
$$866$$ 24.5455 0.834090
$$867$$ 24.0145 0.815577
$$868$$ 59.5942 2.02276
$$869$$ −8.88746 −0.301487
$$870$$ 0 0
$$871$$ 53.1498 1.80091
$$872$$ −15.6124 −0.528702
$$873$$ −11.7481 −0.397613
$$874$$ −86.8949 −2.93926
$$875$$ 0 0
$$876$$ 35.8344 1.21073
$$877$$ 8.46607 0.285879 0.142939 0.989731i $$-0.454345\pi$$
0.142939 + 0.989731i $$0.454345\pi$$
$$878$$ −9.14991 −0.308794
$$879$$ 6.51441 0.219726
$$880$$ 0 0
$$881$$ −12.4949 −0.420965 −0.210482 0.977598i $$-0.567503\pi$$
−0.210482 + 0.977598i $$0.567503\pi$$
$$882$$ 14.3227 0.482270
$$883$$ 8.80299 0.296244 0.148122 0.988969i $$-0.452677\pi$$
0.148122 + 0.988969i $$0.452677\pi$$
$$884$$ −74.8250 −2.51664
$$885$$ 0 0
$$886$$ −44.7994 −1.50507
$$887$$ 56.7788 1.90645 0.953223 0.302269i $$-0.0977441\pi$$
0.953223 + 0.302269i $$0.0977441\pi$$
$$888$$ −8.68051 −0.291299
$$889$$ 58.9893 1.97844
$$890$$ 0 0
$$891$$ −5.39045 −0.180587
$$892$$ −16.7965 −0.562389
$$893$$ 8.07640 0.270266
$$894$$ −21.0426 −0.703771
$$895$$ 0 0
$$896$$ 61.6697 2.06024
$$897$$ −17.9451 −0.599169
$$898$$ −27.9817 −0.933763
$$899$$ 19.8985 0.663651
$$900$$ 0 0
$$901$$ 61.6138 2.05265
$$902$$ −81.6148 −2.71748
$$903$$ 19.3185 0.642879
$$904$$ −38.6096 −1.28414
$$905$$ 0 0
$$906$$ 24.6224 0.818024
$$907$$ 39.1164 1.29884 0.649419 0.760430i $$-0.275012\pi$$
0.649419 + 0.760430i $$0.275012\pi$$
$$908$$ −70.9645 −2.35504
$$909$$ 9.79905 0.325014
$$910$$ 0 0
$$911$$ 20.2215 0.669969 0.334984 0.942224i $$-0.391269\pi$$
0.334984 + 0.942224i $$0.391269\pi$$
$$912$$ 4.71637 0.156175
$$913$$ 57.7422 1.91099
$$914$$ 1.64181 0.0543061
$$915$$ 0 0
$$916$$ 80.3902 2.65617
$$917$$ −51.3502 −1.69573
$$918$$ 14.4649 0.477413
$$919$$ −30.6670 −1.01161 −0.505806 0.862647i $$-0.668805\pi$$
−0.505806 + 0.862647i $$0.668805\pi$$
$$920$$ 0 0
$$921$$ −30.2614 −0.997148
$$922$$ −13.0665 −0.430322
$$923$$ −45.5014 −1.49770
$$924$$ 61.0642 2.00886
$$925$$ 0 0
$$926$$ −52.5199 −1.72591
$$927$$ 7.91135 0.259843
$$928$$ 23.8087 0.781558
$$929$$ 17.3908 0.570573 0.285286 0.958442i $$-0.407911\pi$$
0.285286 + 0.958442i $$0.407911\pi$$
$$930$$ 0 0
$$931$$ −51.2149 −1.67850
$$932$$ 4.30162 0.140904
$$933$$ −14.8690 −0.486790
$$934$$ 44.0497 1.44135
$$935$$ 0 0
$$936$$ −9.37174 −0.306325
$$937$$ 25.7353 0.840736 0.420368 0.907354i $$-0.361901\pi$$
0.420368 + 0.907354i $$0.361901\pi$$
$$938$$ 116.394 3.80041
$$939$$ −19.0799 −0.622651
$$940$$ 0 0
$$941$$ 5.20220 0.169587 0.0847934 0.996399i $$-0.472977\pi$$
0.0847934 + 0.996399i $$0.472977\pi$$
$$942$$ −19.6172 −0.639164
$$943$$ 31.9322 1.03985
$$944$$ −5.86594 −0.190920
$$945$$ 0 0
$$946$$ 64.3940 2.09363
$$947$$ −15.3966 −0.500323 −0.250162 0.968204i $$-0.580484\pi$$
−0.250162 + 0.968204i $$0.580484\pi$$
$$948$$ 5.11347 0.166078
$$949$$ −43.5264 −1.41293
$$950$$ 0 0
$$951$$ 11.9688 0.388115
$$952$$ −58.1933 −1.88606
$$953$$ 25.8939 0.838785 0.419393 0.907805i $$-0.362243\pi$$
0.419393 + 0.907805i $$0.362243\pi$$
$$954$$ 21.7298 0.703528
$$955$$ 0 0
$$956$$ −64.6672 −2.09149
$$957$$ 20.3893 0.659092
$$958$$ 35.9961 1.16298
$$959$$ −75.4233 −2.43554
$$960$$ 0 0
$$961$$ −3.32519 −0.107264
$$962$$ 29.6894 0.957225
$$963$$ −16.0171 −0.516145
$$964$$ −59.7400 −1.92410
$$965$$ 0 0
$$966$$ −39.2984 −1.26441
$$967$$ 59.1183 1.90112 0.950559 0.310545i $$-0.100512\pi$$
0.950559 + 0.310545i $$0.100512\pi$$
$$968$$ 44.9209 1.44381
$$969$$ −51.7234 −1.66159
$$970$$ 0 0
$$971$$ −6.55439 −0.210340 −0.105170 0.994454i $$-0.533539\pi$$
−0.105170 + 0.994454i $$0.533539\pi$$
$$972$$ 3.10144 0.0994786
$$973$$ 14.0795 0.451367
$$974$$ −48.7818 −1.56307
$$975$$ 0 0
$$976$$ 2.73362 0.0875012
$$977$$ −12.4405 −0.398007 −0.199004 0.979999i $$-0.563771\pi$$
−0.199004 + 0.979999i $$0.563771\pi$$
$$978$$ 32.7453 1.04708
$$979$$ 40.4612 1.29315
$$980$$ 0 0
$$981$$ −6.27573 −0.200369
$$982$$ −44.2073 −1.41071
$$983$$ −11.5633 −0.368812 −0.184406 0.982850i $$-0.559036\pi$$
−0.184406 + 0.982850i $$0.559036\pi$$
$$984$$ 16.6764 0.531625
$$985$$ 0 0
$$986$$ −54.7132 −1.74242
$$987$$ 3.65257 0.116263
$$988$$ 94.3617 3.00205
$$989$$ −25.1944 −0.801136
$$990$$ 0 0
$$991$$ −10.5006 −0.333562 −0.166781 0.985994i $$-0.553337\pi$$
−0.166781 + 0.985994i $$0.553337\pi$$
$$992$$ 33.1132 1.05134
$$993$$ −16.0901 −0.510603
$$994$$ −99.6448 −3.16054
$$995$$ 0 0
$$996$$ −33.2224 −1.05269
$$997$$ −31.0404 −0.983059 −0.491529 0.870861i $$-0.663562\pi$$
−0.491529 + 0.870861i $$0.663562\pi$$
$$998$$ −68.4438 −2.16655
$$999$$ −3.48931 −0.110397
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 3525.2.a.bd.1.8 8
5.4 even 2 3525.2.a.be.1.1 yes 8

By twisted newform
Twist Min Dim Char Parity Ord Type
3525.2.a.bd.1.8 8 1.1 even 1 trivial
3525.2.a.be.1.1 yes 8 5.4 even 2