# Properties

 Label 175.4.b.b.99.1 Level $175$ Weight $4$ Character 175.99 Analytic conductor $10.325$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$175 = 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 175.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$10.3253342510$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 7) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 99.1 Root $$-1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 175.99 Dual form 175.4.b.b.99.2

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000i q^{2} +2.00000i q^{3} +7.00000 q^{4} +2.00000 q^{6} -7.00000i q^{7} -15.0000i q^{8} +23.0000 q^{9} +O(q^{10})$$ $$q-1.00000i q^{2} +2.00000i q^{3} +7.00000 q^{4} +2.00000 q^{6} -7.00000i q^{7} -15.0000i q^{8} +23.0000 q^{9} -8.00000 q^{11} +14.0000i q^{12} -28.0000i q^{13} -7.00000 q^{14} +41.0000 q^{16} +54.0000i q^{17} -23.0000i q^{18} +110.000 q^{19} +14.0000 q^{21} +8.00000i q^{22} -48.0000i q^{23} +30.0000 q^{24} -28.0000 q^{26} +100.000i q^{27} -49.0000i q^{28} +110.000 q^{29} +12.0000 q^{31} -161.000i q^{32} -16.0000i q^{33} +54.0000 q^{34} +161.000 q^{36} -246.000i q^{37} -110.000i q^{38} +56.0000 q^{39} +182.000 q^{41} -14.0000i q^{42} -128.000i q^{43} -56.0000 q^{44} -48.0000 q^{46} +324.000i q^{47} +82.0000i q^{48} -49.0000 q^{49} -108.000 q^{51} -196.000i q^{52} +162.000i q^{53} +100.000 q^{54} -105.000 q^{56} +220.000i q^{57} -110.000i q^{58} -810.000 q^{59} -488.000 q^{61} -12.0000i q^{62} -161.000i q^{63} +167.000 q^{64} -16.0000 q^{66} +244.000i q^{67} +378.000i q^{68} +96.0000 q^{69} -768.000 q^{71} -345.000i q^{72} +702.000i q^{73} -246.000 q^{74} +770.000 q^{76} +56.0000i q^{77} -56.0000i q^{78} -440.000 q^{79} +421.000 q^{81} -182.000i q^{82} +1302.00i q^{83} +98.0000 q^{84} -128.000 q^{86} +220.000i q^{87} +120.000i q^{88} -730.000 q^{89} -196.000 q^{91} -336.000i q^{92} +24.0000i q^{93} +324.000 q^{94} +322.000 q^{96} +294.000i q^{97} +49.0000i q^{98} -184.000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 14q^{4} + 4q^{6} + 46q^{9} + O(q^{10})$$ $$2q + 14q^{4} + 4q^{6} + 46q^{9} - 16q^{11} - 14q^{14} + 82q^{16} + 220q^{19} + 28q^{21} + 60q^{24} - 56q^{26} + 220q^{29} + 24q^{31} + 108q^{34} + 322q^{36} + 112q^{39} + 364q^{41} - 112q^{44} - 96q^{46} - 98q^{49} - 216q^{51} + 200q^{54} - 210q^{56} - 1620q^{59} - 976q^{61} + 334q^{64} - 32q^{66} + 192q^{69} - 1536q^{71} - 492q^{74} + 1540q^{76} - 880q^{79} + 842q^{81} + 196q^{84} - 256q^{86} - 1460q^{89} - 392q^{91} + 648q^{94} + 644q^{96} - 368q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/175\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$127$$ $$\chi(n)$$ $$1$$ $$-1$$

## Coefficient data

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

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ − 1.00000i − 0.353553i −0.984251 0.176777i $$-0.943433\pi$$
0.984251 0.176777i $$-0.0565670\pi$$
$$3$$ 2.00000i 0.384900i 0.981307 + 0.192450i $$0.0616434\pi$$
−0.981307 + 0.192450i $$0.938357\pi$$
$$4$$ 7.00000 0.875000
$$5$$ 0 0
$$6$$ 2.00000 0.136083
$$7$$ − 7.00000i − 0.377964i
$$8$$ − 15.0000i − 0.662913i
$$9$$ 23.0000 0.851852
$$10$$ 0 0
$$11$$ −8.00000 −0.219281 −0.109640 0.993971i $$-0.534970\pi$$
−0.109640 + 0.993971i $$0.534970\pi$$
$$12$$ 14.0000i 0.336788i
$$13$$ − 28.0000i − 0.597369i −0.954352 0.298685i $$-0.903452\pi$$
0.954352 0.298685i $$-0.0965479\pi$$
$$14$$ −7.00000 −0.133631
$$15$$ 0 0
$$16$$ 41.0000 0.640625
$$17$$ 54.0000i 0.770407i 0.922832 + 0.385204i $$0.125869\pi$$
−0.922832 + 0.385204i $$0.874131\pi$$
$$18$$ − 23.0000i − 0.301175i
$$19$$ 110.000 1.32820 0.664098 0.747645i $$-0.268816\pi$$
0.664098 + 0.747645i $$0.268816\pi$$
$$20$$ 0 0
$$21$$ 14.0000 0.145479
$$22$$ 8.00000i 0.0775275i
$$23$$ − 48.0000i − 0.435161i −0.976042 0.217580i $$-0.930184\pi$$
0.976042 0.217580i $$-0.0698164\pi$$
$$24$$ 30.0000 0.255155
$$25$$ 0 0
$$26$$ −28.0000 −0.211202
$$27$$ 100.000i 0.712778i
$$28$$ − 49.0000i − 0.330719i
$$29$$ 110.000 0.704362 0.352181 0.935932i $$-0.385440\pi$$
0.352181 + 0.935932i $$0.385440\pi$$
$$30$$ 0 0
$$31$$ 12.0000 0.0695246 0.0347623 0.999396i $$-0.488933\pi$$
0.0347623 + 0.999396i $$0.488933\pi$$
$$32$$ − 161.000i − 0.889408i
$$33$$ − 16.0000i − 0.0844013i
$$34$$ 54.0000 0.272380
$$35$$ 0 0
$$36$$ 161.000 0.745370
$$37$$ − 246.000i − 1.09303i −0.837449 0.546516i $$-0.815954\pi$$
0.837449 0.546516i $$-0.184046\pi$$
$$38$$ − 110.000i − 0.469588i
$$39$$ 56.0000 0.229928
$$40$$ 0 0
$$41$$ 182.000 0.693259 0.346630 0.938002i $$-0.387326\pi$$
0.346630 + 0.938002i $$0.387326\pi$$
$$42$$ − 14.0000i − 0.0514344i
$$43$$ − 128.000i − 0.453949i −0.973901 0.226975i $$-0.927117\pi$$
0.973901 0.226975i $$-0.0728834\pi$$
$$44$$ −56.0000 −0.191871
$$45$$ 0 0
$$46$$ −48.0000 −0.153852
$$47$$ 324.000i 1.00554i 0.864421 + 0.502769i $$0.167685\pi$$
−0.864421 + 0.502769i $$0.832315\pi$$
$$48$$ 82.0000i 0.246577i
$$49$$ −49.0000 −0.142857
$$50$$ 0 0
$$51$$ −108.000 −0.296530
$$52$$ − 196.000i − 0.522698i
$$53$$ 162.000i 0.419857i 0.977717 + 0.209928i $$0.0673231\pi$$
−0.977717 + 0.209928i $$0.932677\pi$$
$$54$$ 100.000 0.252005
$$55$$ 0 0
$$56$$ −105.000 −0.250557
$$57$$ 220.000i 0.511223i
$$58$$ − 110.000i − 0.249029i
$$59$$ −810.000 −1.78734 −0.893670 0.448725i $$-0.851878\pi$$
−0.893670 + 0.448725i $$0.851878\pi$$
$$60$$ 0 0
$$61$$ −488.000 −1.02430 −0.512148 0.858898i $$-0.671150\pi$$
−0.512148 + 0.858898i $$0.671150\pi$$
$$62$$ − 12.0000i − 0.0245807i
$$63$$ − 161.000i − 0.321970i
$$64$$ 167.000 0.326172
$$65$$ 0 0
$$66$$ −16.0000 −0.0298404
$$67$$ 244.000i 0.444916i 0.974942 + 0.222458i $$0.0714080\pi$$
−0.974942 + 0.222458i $$0.928592\pi$$
$$68$$ 378.000i 0.674106i
$$69$$ 96.0000 0.167493
$$70$$ 0 0
$$71$$ −768.000 −1.28373 −0.641865 0.766818i $$-0.721839\pi$$
−0.641865 + 0.766818i $$0.721839\pi$$
$$72$$ − 345.000i − 0.564703i
$$73$$ 702.000i 1.12552i 0.826621 + 0.562759i $$0.190260\pi$$
−0.826621 + 0.562759i $$0.809740\pi$$
$$74$$ −246.000 −0.386445
$$75$$ 0 0
$$76$$ 770.000 1.16217
$$77$$ 56.0000i 0.0828804i
$$78$$ − 56.0000i − 0.0812917i
$$79$$ −440.000 −0.626631 −0.313316 0.949649i $$-0.601440\pi$$
−0.313316 + 0.949649i $$0.601440\pi$$
$$80$$ 0 0
$$81$$ 421.000 0.577503
$$82$$ − 182.000i − 0.245104i
$$83$$ 1302.00i 1.72184i 0.508737 + 0.860922i $$0.330113\pi$$
−0.508737 + 0.860922i $$0.669887\pi$$
$$84$$ 98.0000 0.127294
$$85$$ 0 0
$$86$$ −128.000 −0.160495
$$87$$ 220.000i 0.271109i
$$88$$ 120.000i 0.145364i
$$89$$ −730.000 −0.869436 −0.434718 0.900567i $$-0.643152\pi$$
−0.434718 + 0.900567i $$0.643152\pi$$
$$90$$ 0 0
$$91$$ −196.000 −0.225784
$$92$$ − 336.000i − 0.380765i
$$93$$ 24.0000i 0.0267600i
$$94$$ 324.000 0.355511
$$95$$ 0 0
$$96$$ 322.000 0.342333
$$97$$ 294.000i 0.307744i 0.988091 + 0.153872i $$0.0491744\pi$$
−0.988091 + 0.153872i $$0.950826\pi$$
$$98$$ 49.0000i 0.0505076i
$$99$$ −184.000 −0.186795
$$100$$ 0 0
$$101$$ −688.000 −0.677808 −0.338904 0.940821i $$-0.610056\pi$$
−0.338904 + 0.940821i $$0.610056\pi$$
$$102$$ 108.000i 0.104839i
$$103$$ − 1388.00i − 1.32780i −0.747820 0.663901i $$-0.768899\pi$$
0.747820 0.663901i $$-0.231101\pi$$
$$104$$ −420.000 −0.396004
$$105$$ 0 0
$$106$$ 162.000 0.148442
$$107$$ 244.000i 0.220452i 0.993907 + 0.110226i $$0.0351575\pi$$
−0.993907 + 0.110226i $$0.964843\pi$$
$$108$$ 700.000i 0.623681i
$$109$$ −90.0000 −0.0790866 −0.0395433 0.999218i $$-0.512590\pi$$
−0.0395433 + 0.999218i $$0.512590\pi$$
$$110$$ 0 0
$$111$$ 492.000 0.420708
$$112$$ − 287.000i − 0.242133i
$$113$$ − 1318.00i − 1.09723i −0.836075 0.548615i $$-0.815155\pi$$
0.836075 0.548615i $$-0.184845\pi$$
$$114$$ 220.000 0.180745
$$115$$ 0 0
$$116$$ 770.000 0.616316
$$117$$ − 644.000i − 0.508870i
$$118$$ 810.000i 0.631920i
$$119$$ 378.000 0.291187
$$120$$ 0 0
$$121$$ −1267.00 −0.951916
$$122$$ 488.000i 0.362143i
$$123$$ 364.000i 0.266836i
$$124$$ 84.0000 0.0608341
$$125$$ 0 0
$$126$$ −161.000 −0.113833
$$127$$ − 1776.00i − 1.24090i −0.784245 0.620451i $$-0.786950\pi$$
0.784245 0.620451i $$-0.213050\pi$$
$$128$$ − 1455.00i − 1.00473i
$$129$$ 256.000 0.174725
$$130$$ 0 0
$$131$$ −1118.00 −0.745650 −0.372825 0.927902i $$-0.621611\pi$$
−0.372825 + 0.927902i $$0.621611\pi$$
$$132$$ − 112.000i − 0.0738511i
$$133$$ − 770.000i − 0.502011i
$$134$$ 244.000 0.157301
$$135$$ 0 0
$$136$$ 810.000 0.510713
$$137$$ 2274.00i 1.41811i 0.705154 + 0.709054i $$0.250878\pi$$
−0.705154 + 0.709054i $$0.749122\pi$$
$$138$$ − 96.0000i − 0.0592178i
$$139$$ 210.000 0.128144 0.0640718 0.997945i $$-0.479591\pi$$
0.0640718 + 0.997945i $$0.479591\pi$$
$$140$$ 0 0
$$141$$ −648.000 −0.387032
$$142$$ 768.000i 0.453867i
$$143$$ 224.000i 0.130992i
$$144$$ 943.000 0.545718
$$145$$ 0 0
$$146$$ 702.000 0.397931
$$147$$ − 98.0000i − 0.0549857i
$$148$$ − 1722.00i − 0.956402i
$$149$$ 2010.00 1.10514 0.552569 0.833467i $$-0.313648\pi$$
0.552569 + 0.833467i $$0.313648\pi$$
$$150$$ 0 0
$$151$$ 1112.00 0.599293 0.299647 0.954050i $$-0.403131\pi$$
0.299647 + 0.954050i $$0.403131\pi$$
$$152$$ − 1650.00i − 0.880478i
$$153$$ 1242.00i 0.656273i
$$154$$ 56.0000 0.0293027
$$155$$ 0 0
$$156$$ 392.000 0.201187
$$157$$ 124.000i 0.0630336i 0.999503 + 0.0315168i $$0.0100338\pi$$
−0.999503 + 0.0315168i $$0.989966\pi$$
$$158$$ 440.000i 0.221548i
$$159$$ −324.000 −0.161603
$$160$$ 0 0
$$161$$ −336.000 −0.164475
$$162$$ − 421.000i − 0.204178i
$$163$$ − 2008.00i − 0.964900i −0.875924 0.482450i $$-0.839747\pi$$
0.875924 0.482450i $$-0.160253\pi$$
$$164$$ 1274.00 0.606602
$$165$$ 0 0
$$166$$ 1302.00 0.608764
$$167$$ 2884.00i 1.33635i 0.744004 + 0.668176i $$0.232924\pi$$
−0.744004 + 0.668176i $$0.767076\pi$$
$$168$$ − 210.000i − 0.0964396i
$$169$$ 1413.00 0.643150
$$170$$ 0 0
$$171$$ 2530.00 1.13143
$$172$$ − 896.000i − 0.397206i
$$173$$ − 2228.00i − 0.979143i −0.871963 0.489571i $$-0.837153\pi$$
0.871963 0.489571i $$-0.162847\pi$$
$$174$$ 220.000 0.0958515
$$175$$ 0 0
$$176$$ −328.000 −0.140477
$$177$$ − 1620.00i − 0.687947i
$$178$$ 730.000i 0.307392i
$$179$$ 820.000 0.342400 0.171200 0.985236i $$-0.445236\pi$$
0.171200 + 0.985236i $$0.445236\pi$$
$$180$$ 0 0
$$181$$ 3892.00 1.59829 0.799144 0.601140i $$-0.205287\pi$$
0.799144 + 0.601140i $$0.205287\pi$$
$$182$$ 196.000i 0.0798268i
$$183$$ − 976.000i − 0.394251i
$$184$$ −720.000 −0.288473
$$185$$ 0 0
$$186$$ 24.0000 0.00946110
$$187$$ − 432.000i − 0.168936i
$$188$$ 2268.00i 0.879845i
$$189$$ 700.000 0.269405
$$190$$ 0 0
$$191$$ −5048.00 −1.91236 −0.956179 0.292782i $$-0.905419\pi$$
−0.956179 + 0.292782i $$0.905419\pi$$
$$192$$ 334.000i 0.125544i
$$193$$ 2962.00i 1.10471i 0.833608 + 0.552356i $$0.186271\pi$$
−0.833608 + 0.552356i $$0.813729\pi$$
$$194$$ 294.000 0.108804
$$195$$ 0 0
$$196$$ −343.000 −0.125000
$$197$$ 3334.00i 1.20577i 0.797826 + 0.602887i $$0.205983\pi$$
−0.797826 + 0.602887i $$0.794017\pi$$
$$198$$ 184.000i 0.0660420i
$$199$$ −1860.00 −0.662572 −0.331286 0.943530i $$-0.607483\pi$$
−0.331286 + 0.943530i $$0.607483\pi$$
$$200$$ 0 0
$$201$$ −488.000 −0.171248
$$202$$ 688.000i 0.239641i
$$203$$ − 770.000i − 0.266224i
$$204$$ −756.000 −0.259464
$$205$$ 0 0
$$206$$ −1388.00 −0.469449
$$207$$ − 1104.00i − 0.370692i
$$208$$ − 1148.00i − 0.382690i
$$209$$ −880.000 −0.291248
$$210$$ 0 0
$$211$$ −4268.00 −1.39252 −0.696259 0.717791i $$-0.745153\pi$$
−0.696259 + 0.717791i $$0.745153\pi$$
$$212$$ 1134.00i 0.367375i
$$213$$ − 1536.00i − 0.494108i
$$214$$ 244.000 0.0779416
$$215$$ 0 0
$$216$$ 1500.00 0.472510
$$217$$ − 84.0000i − 0.0262778i
$$218$$ 90.0000i 0.0279613i
$$219$$ −1404.00 −0.433212
$$220$$ 0 0
$$221$$ 1512.00 0.460218
$$222$$ − 492.000i − 0.148743i
$$223$$ 5432.00i 1.63118i 0.578629 + 0.815591i $$0.303588\pi$$
−0.578629 + 0.815591i $$0.696412\pi$$
$$224$$ −1127.00 −0.336165
$$225$$ 0 0
$$226$$ −1318.00 −0.387929
$$227$$ − 2046.00i − 0.598228i −0.954217 0.299114i $$-0.903309\pi$$
0.954217 0.299114i $$-0.0966911\pi$$
$$228$$ 1540.00i 0.447320i
$$229$$ 2980.00 0.859930 0.429965 0.902846i $$-0.358526\pi$$
0.429965 + 0.902846i $$0.358526\pi$$
$$230$$ 0 0
$$231$$ −112.000 −0.0319007
$$232$$ − 1650.00i − 0.466930i
$$233$$ − 4458.00i − 1.25345i −0.779241 0.626724i $$-0.784395\pi$$
0.779241 0.626724i $$-0.215605\pi$$
$$234$$ −644.000 −0.179913
$$235$$ 0 0
$$236$$ −5670.00 −1.56392
$$237$$ − 880.000i − 0.241190i
$$238$$ − 378.000i − 0.102950i
$$239$$ −4440.00 −1.20167 −0.600836 0.799372i $$-0.705166\pi$$
−0.600836 + 0.799372i $$0.705166\pi$$
$$240$$ 0 0
$$241$$ 3302.00 0.882575 0.441287 0.897366i $$-0.354522\pi$$
0.441287 + 0.897366i $$0.354522\pi$$
$$242$$ 1267.00i 0.336553i
$$243$$ 3542.00i 0.935059i
$$244$$ −3416.00 −0.896258
$$245$$ 0 0
$$246$$ 364.000 0.0943406
$$247$$ − 3080.00i − 0.793424i
$$248$$ − 180.000i − 0.0460888i
$$249$$ −2604.00 −0.662738
$$250$$ 0 0
$$251$$ 1582.00 0.397829 0.198914 0.980017i $$-0.436258\pi$$
0.198914 + 0.980017i $$0.436258\pi$$
$$252$$ − 1127.00i − 0.281724i
$$253$$ 384.000i 0.0954224i
$$254$$ −1776.00 −0.438725
$$255$$ 0 0
$$256$$ −119.000 −0.0290527
$$257$$ 2354.00i 0.571356i 0.958326 + 0.285678i $$0.0922188\pi$$
−0.958326 + 0.285678i $$0.907781\pi$$
$$258$$ − 256.000i − 0.0617747i
$$259$$ −1722.00 −0.413127
$$260$$ 0 0
$$261$$ 2530.00 0.600012
$$262$$ 1118.00i 0.263627i
$$263$$ 3872.00i 0.907824i 0.891046 + 0.453912i $$0.149972\pi$$
−0.891046 + 0.453912i $$0.850028\pi$$
$$264$$ −240.000 −0.0559507
$$265$$ 0 0
$$266$$ −770.000 −0.177488
$$267$$ − 1460.00i − 0.334646i
$$268$$ 1708.00i 0.389301i
$$269$$ −180.000 −0.0407985 −0.0203992 0.999792i $$-0.506494\pi$$
−0.0203992 + 0.999792i $$0.506494\pi$$
$$270$$ 0 0
$$271$$ 2032.00 0.455480 0.227740 0.973722i $$-0.426866\pi$$
0.227740 + 0.973722i $$0.426866\pi$$
$$272$$ 2214.00i 0.493542i
$$273$$ − 392.000i − 0.0869045i
$$274$$ 2274.00 0.501377
$$275$$ 0 0
$$276$$ 672.000 0.146557
$$277$$ − 5426.00i − 1.17696i −0.808513 0.588478i $$-0.799727\pi$$
0.808513 0.588478i $$-0.200273\pi$$
$$278$$ − 210.000i − 0.0453056i
$$279$$ 276.000 0.0592247
$$280$$ 0 0
$$281$$ 842.000 0.178753 0.0893764 0.995998i $$-0.471513\pi$$
0.0893764 + 0.995998i $$0.471513\pi$$
$$282$$ 648.000i 0.136836i
$$283$$ 3782.00i 0.794405i 0.917731 + 0.397202i $$0.130019\pi$$
−0.917731 + 0.397202i $$0.869981\pi$$
$$284$$ −5376.00 −1.12326
$$285$$ 0 0
$$286$$ 224.000 0.0463126
$$287$$ − 1274.00i − 0.262027i
$$288$$ − 3703.00i − 0.757644i
$$289$$ 1997.00 0.406473
$$290$$ 0 0
$$291$$ −588.000 −0.118451
$$292$$ 4914.00i 0.984829i
$$293$$ 4312.00i 0.859760i 0.902886 + 0.429880i $$0.141444\pi$$
−0.902886 + 0.429880i $$0.858556\pi$$
$$294$$ −98.0000 −0.0194404
$$295$$ 0 0
$$296$$ −3690.00 −0.724584
$$297$$ − 800.000i − 0.156299i
$$298$$ − 2010.00i − 0.390725i
$$299$$ −1344.00 −0.259952
$$300$$ 0 0
$$301$$ −896.000 −0.171577
$$302$$ − 1112.00i − 0.211882i
$$303$$ − 1376.00i − 0.260888i
$$304$$ 4510.00 0.850876
$$305$$ 0 0
$$306$$ 1242.00 0.232027
$$307$$ 2674.00i 0.497112i 0.968618 + 0.248556i $$0.0799559\pi$$
−0.968618 + 0.248556i $$0.920044\pi$$
$$308$$ 392.000i 0.0725204i
$$309$$ 2776.00 0.511072
$$310$$ 0 0
$$311$$ −3768.00 −0.687021 −0.343511 0.939149i $$-0.611616\pi$$
−0.343511 + 0.939149i $$0.611616\pi$$
$$312$$ − 840.000i − 0.152422i
$$313$$ − 2438.00i − 0.440268i −0.975470 0.220134i $$-0.929351\pi$$
0.975470 0.220134i $$-0.0706495\pi$$
$$314$$ 124.000 0.0222857
$$315$$ 0 0
$$316$$ −3080.00 −0.548302
$$317$$ − 3186.00i − 0.564491i −0.959342 0.282245i $$-0.908921\pi$$
0.959342 0.282245i $$-0.0910792\pi$$
$$318$$ 324.000i 0.0571353i
$$319$$ −880.000 −0.154453
$$320$$ 0 0
$$321$$ −488.000 −0.0848520
$$322$$ 336.000i 0.0581508i
$$323$$ 5940.00i 1.02325i
$$324$$ 2947.00 0.505316
$$325$$ 0 0
$$326$$ −2008.00 −0.341144
$$327$$ − 180.000i − 0.0304404i
$$328$$ − 2730.00i − 0.459570i
$$329$$ 2268.00 0.380057
$$330$$ 0 0
$$331$$ 8672.00 1.44005 0.720025 0.693949i $$-0.244131\pi$$
0.720025 + 0.693949i $$0.244131\pi$$
$$332$$ 9114.00i 1.50661i
$$333$$ − 5658.00i − 0.931101i
$$334$$ 2884.00 0.472471
$$335$$ 0 0
$$336$$ 574.000 0.0931972
$$337$$ 814.000i 0.131577i 0.997834 + 0.0657884i $$0.0209562\pi$$
−0.997834 + 0.0657884i $$0.979044\pi$$
$$338$$ − 1413.00i − 0.227388i
$$339$$ 2636.00 0.422324
$$340$$ 0 0
$$341$$ −96.0000 −0.0152454
$$342$$ − 2530.00i − 0.400020i
$$343$$ 343.000i 0.0539949i
$$344$$ −1920.00 −0.300929
$$345$$ 0 0
$$346$$ −2228.00 −0.346179
$$347$$ 9344.00i 1.44557i 0.691074 + 0.722784i $$0.257138\pi$$
−0.691074 + 0.722784i $$0.742862\pi$$
$$348$$ 1540.00i 0.237220i
$$349$$ 5180.00 0.794496 0.397248 0.917711i $$-0.369965\pi$$
0.397248 + 0.917711i $$0.369965\pi$$
$$350$$ 0 0
$$351$$ 2800.00 0.425792
$$352$$ 1288.00i 0.195030i
$$353$$ − 12178.0i − 1.83617i −0.396379 0.918087i $$-0.629733\pi$$
0.396379 0.918087i $$-0.370267\pi$$
$$354$$ −1620.00 −0.243226
$$355$$ 0 0
$$356$$ −5110.00 −0.760757
$$357$$ 756.000i 0.112078i
$$358$$ − 820.000i − 0.121057i
$$359$$ −440.000 −0.0646861 −0.0323431 0.999477i $$-0.510297\pi$$
−0.0323431 + 0.999477i $$0.510297\pi$$
$$360$$ 0 0
$$361$$ 5241.00 0.764106
$$362$$ − 3892.00i − 0.565080i
$$363$$ − 2534.00i − 0.366393i
$$364$$ −1372.00 −0.197561
$$365$$ 0 0
$$366$$ −976.000 −0.139389
$$367$$ − 9816.00i − 1.39616i −0.716019 0.698080i $$-0.754038\pi$$
0.716019 0.698080i $$-0.245962\pi$$
$$368$$ − 1968.00i − 0.278775i
$$369$$ 4186.00 0.590554
$$370$$ 0 0
$$371$$ 1134.00 0.158691
$$372$$ 168.000i 0.0234150i
$$373$$ 442.000i 0.0613563i 0.999529 + 0.0306781i $$0.00976669\pi$$
−0.999529 + 0.0306781i $$0.990233\pi$$
$$374$$ −432.000 −0.0597278
$$375$$ 0 0
$$376$$ 4860.00 0.666583
$$377$$ − 3080.00i − 0.420764i
$$378$$ − 700.000i − 0.0952490i
$$379$$ 3960.00 0.536706 0.268353 0.963321i $$-0.413521\pi$$
0.268353 + 0.963321i $$0.413521\pi$$
$$380$$ 0 0
$$381$$ 3552.00 0.477623
$$382$$ 5048.00i 0.676121i
$$383$$ − 6708.00i − 0.894942i −0.894298 0.447471i $$-0.852325\pi$$
0.894298 0.447471i $$-0.147675\pi$$
$$384$$ 2910.00 0.386720
$$385$$ 0 0
$$386$$ 2962.00 0.390575
$$387$$ − 2944.00i − 0.386697i
$$388$$ 2058.00i 0.269276i
$$389$$ 13350.0 1.74003 0.870015 0.493025i $$-0.164109\pi$$
0.870015 + 0.493025i $$0.164109\pi$$
$$390$$ 0 0
$$391$$ 2592.00 0.335251
$$392$$ 735.000i 0.0947018i
$$393$$ − 2236.00i − 0.287001i
$$394$$ 3334.00 0.426306
$$395$$ 0 0
$$396$$ −1288.00 −0.163446
$$397$$ − 1356.00i − 0.171425i −0.996320 0.0857125i $$-0.972683\pi$$
0.996320 0.0857125i $$-0.0273166\pi$$
$$398$$ 1860.00i 0.234255i
$$399$$ 1540.00 0.193224
$$400$$ 0 0
$$401$$ 6222.00 0.774843 0.387421 0.921903i $$-0.373366\pi$$
0.387421 + 0.921903i $$0.373366\pi$$
$$402$$ 488.000i 0.0605453i
$$403$$ − 336.000i − 0.0415319i
$$404$$ −4816.00 −0.593082
$$405$$ 0 0
$$406$$ −770.000 −0.0941243
$$407$$ 1968.00i 0.239681i
$$408$$ 1620.00i 0.196573i
$$409$$ −5150.00 −0.622619 −0.311309 0.950309i $$-0.600768\pi$$
−0.311309 + 0.950309i $$0.600768\pi$$
$$410$$ 0 0
$$411$$ −4548.00 −0.545830
$$412$$ − 9716.00i − 1.16183i
$$413$$ 5670.00i 0.675551i
$$414$$ −1104.00 −0.131060
$$415$$ 0 0
$$416$$ −4508.00 −0.531305
$$417$$ 420.000i 0.0493225i
$$418$$ 880.000i 0.102972i
$$419$$ −2310.00 −0.269334 −0.134667 0.990891i $$-0.542996\pi$$
−0.134667 + 0.990891i $$0.542996\pi$$
$$420$$ 0 0
$$421$$ 1262.00 0.146095 0.0730476 0.997328i $$-0.476727\pi$$
0.0730476 + 0.997328i $$0.476727\pi$$
$$422$$ 4268.00i 0.492329i
$$423$$ 7452.00i 0.856569i
$$424$$ 2430.00 0.278328
$$425$$ 0 0
$$426$$ −1536.00 −0.174694
$$427$$ 3416.00i 0.387147i
$$428$$ 1708.00i 0.192896i
$$429$$ −448.000 −0.0504188
$$430$$ 0 0
$$431$$ −4488.00 −0.501576 −0.250788 0.968042i $$-0.580690\pi$$
−0.250788 + 0.968042i $$0.580690\pi$$
$$432$$ 4100.00i 0.456623i
$$433$$ − 17038.0i − 1.89098i −0.325652 0.945490i $$-0.605584\pi$$
0.325652 0.945490i $$-0.394416\pi$$
$$434$$ −84.0000 −0.00929062
$$435$$ 0 0
$$436$$ −630.000 −0.0692008
$$437$$ − 5280.00i − 0.577979i
$$438$$ 1404.00i 0.153164i
$$439$$ −16200.0 −1.76124 −0.880619 0.473824i $$-0.842873\pi$$
−0.880619 + 0.473824i $$0.842873\pi$$
$$440$$ 0 0
$$441$$ −1127.00 −0.121693
$$442$$ − 1512.00i − 0.162712i
$$443$$ 8772.00i 0.940791i 0.882456 + 0.470395i $$0.155889\pi$$
−0.882456 + 0.470395i $$0.844111\pi$$
$$444$$ 3444.00 0.368119
$$445$$ 0 0
$$446$$ 5432.00 0.576710
$$447$$ 4020.00i 0.425368i
$$448$$ − 1169.00i − 0.123281i
$$449$$ −2130.00 −0.223877 −0.111939 0.993715i $$-0.535706\pi$$
−0.111939 + 0.993715i $$0.535706\pi$$
$$450$$ 0 0
$$451$$ −1456.00 −0.152019
$$452$$ − 9226.00i − 0.960076i
$$453$$ 2224.00i 0.230668i
$$454$$ −2046.00 −0.211506
$$455$$ 0 0
$$456$$ 3300.00 0.338896
$$457$$ 10534.0i 1.07825i 0.842226 + 0.539124i $$0.181245\pi$$
−0.842226 + 0.539124i $$0.818755\pi$$
$$458$$ − 2980.00i − 0.304031i
$$459$$ −5400.00 −0.549129
$$460$$ 0 0
$$461$$ −9268.00 −0.936342 −0.468171 0.883638i $$-0.655087\pi$$
−0.468171 + 0.883638i $$0.655087\pi$$
$$462$$ 112.000i 0.0112786i
$$463$$ 9392.00i 0.942728i 0.881939 + 0.471364i $$0.156238\pi$$
−0.881939 + 0.471364i $$0.843762\pi$$
$$464$$ 4510.00 0.451232
$$465$$ 0 0
$$466$$ −4458.00 −0.443161
$$467$$ − 10806.0i − 1.07075i −0.844613 0.535377i $$-0.820170\pi$$
0.844613 0.535377i $$-0.179830\pi$$
$$468$$ − 4508.00i − 0.445261i
$$469$$ 1708.00 0.168162
$$470$$ 0 0
$$471$$ −248.000 −0.0242616
$$472$$ 12150.0i 1.18485i
$$473$$ 1024.00i 0.0995424i
$$474$$ −880.000 −0.0852737
$$475$$ 0 0
$$476$$ 2646.00 0.254788
$$477$$ 3726.00i 0.357656i
$$478$$ 4440.00i 0.424855i
$$479$$ −4940.00 −0.471220 −0.235610 0.971848i $$-0.575709\pi$$
−0.235610 + 0.971848i $$0.575709\pi$$
$$480$$ 0 0
$$481$$ −6888.00 −0.652943
$$482$$ − 3302.00i − 0.312037i
$$483$$ − 672.000i − 0.0633065i
$$484$$ −8869.00 −0.832926
$$485$$ 0 0
$$486$$ 3542.00 0.330593
$$487$$ − 5216.00i − 0.485338i −0.970109 0.242669i $$-0.921977\pi$$
0.970109 0.242669i $$-0.0780229\pi$$
$$488$$ 7320.00i 0.679018i
$$489$$ 4016.00 0.371390
$$490$$ 0 0
$$491$$ 4412.00 0.405521 0.202760 0.979228i $$-0.435009\pi$$
0.202760 + 0.979228i $$0.435009\pi$$
$$492$$ 2548.00i 0.233481i
$$493$$ 5940.00i 0.542645i
$$494$$ −3080.00 −0.280518
$$495$$ 0 0
$$496$$ 492.000 0.0445392
$$497$$ 5376.00i 0.485204i
$$498$$ 2604.00i 0.234313i
$$499$$ −19060.0 −1.70991 −0.854953 0.518706i $$-0.826414\pi$$
−0.854953 + 0.518706i $$0.826414\pi$$
$$500$$ 0 0
$$501$$ −5768.00 −0.514362
$$502$$ − 1582.00i − 0.140654i
$$503$$ − 12768.0i − 1.13180i −0.824473 0.565902i $$-0.808528\pi$$
0.824473 0.565902i $$-0.191472\pi$$
$$504$$ −2415.00 −0.213438
$$505$$ 0 0
$$506$$ 384.000 0.0337369
$$507$$ 2826.00i 0.247548i
$$508$$ − 12432.0i − 1.08579i
$$509$$ 5500.00 0.478945 0.239473 0.970903i $$-0.423025\pi$$
0.239473 + 0.970903i $$0.423025\pi$$
$$510$$ 0 0
$$511$$ 4914.00 0.425406
$$512$$ − 11521.0i − 0.994455i
$$513$$ 11000.0i 0.946709i
$$514$$ 2354.00 0.202005
$$515$$ 0 0
$$516$$ 1792.00 0.152884
$$517$$ − 2592.00i − 0.220495i
$$518$$ 1722.00i 0.146062i
$$519$$ 4456.00 0.376872
$$520$$ 0 0
$$521$$ −7338.00 −0.617051 −0.308526 0.951216i $$-0.599836\pi$$
−0.308526 + 0.951216i $$0.599836\pi$$
$$522$$ − 2530.00i − 0.212136i
$$523$$ 17582.0i 1.46999i 0.678070 + 0.734997i $$0.262817\pi$$
−0.678070 + 0.734997i $$0.737183\pi$$
$$524$$ −7826.00 −0.652444
$$525$$ 0 0
$$526$$ 3872.00 0.320964
$$527$$ 648.000i 0.0535623i
$$528$$ − 656.000i − 0.0540696i
$$529$$ 9863.00 0.810635
$$530$$ 0 0
$$531$$ −18630.0 −1.52255
$$532$$ − 5390.00i − 0.439260i
$$533$$ − 5096.00i − 0.414132i
$$534$$ −1460.00 −0.118315
$$535$$ 0 0
$$536$$ 3660.00 0.294940
$$537$$ 1640.00i 0.131790i
$$538$$ 180.000i 0.0144244i
$$539$$ 392.000 0.0313259
$$540$$ 0 0
$$541$$ −1618.00 −0.128583 −0.0642914 0.997931i $$-0.520479\pi$$
−0.0642914 + 0.997931i $$0.520479\pi$$
$$542$$ − 2032.00i − 0.161037i
$$543$$ 7784.00i 0.615181i
$$544$$ 8694.00 0.685206
$$545$$ 0 0
$$546$$ −392.000 −0.0307254
$$547$$ 16144.0i 1.26192i 0.775817 + 0.630958i $$0.217338\pi$$
−0.775817 + 0.630958i $$0.782662\pi$$
$$548$$ 15918.0i 1.24085i
$$549$$ −11224.0 −0.872548
$$550$$ 0 0
$$551$$ 12100.0 0.935531
$$552$$ − 1440.00i − 0.111033i
$$553$$ 3080.00i 0.236844i
$$554$$ −5426.00 −0.416117
$$555$$ 0 0
$$556$$ 1470.00 0.112126
$$557$$ 4654.00i 0.354033i 0.984208 + 0.177016i $$0.0566446\pi$$
−0.984208 + 0.177016i $$0.943355\pi$$
$$558$$ − 276.000i − 0.0209391i
$$559$$ −3584.00 −0.271175
$$560$$ 0 0
$$561$$ 864.000 0.0650234
$$562$$ − 842.000i − 0.0631986i
$$563$$ − 10078.0i − 0.754418i −0.926128 0.377209i $$-0.876884\pi$$
0.926128 0.377209i $$-0.123116\pi$$
$$564$$ −4536.00 −0.338653
$$565$$ 0 0
$$566$$ 3782.00 0.280865
$$567$$ − 2947.00i − 0.218276i
$$568$$ 11520.0i 0.851001i
$$569$$ 5930.00 0.436904 0.218452 0.975848i $$-0.429899\pi$$
0.218452 + 0.975848i $$0.429899\pi$$
$$570$$ 0 0
$$571$$ −19048.0 −1.39603 −0.698016 0.716082i $$-0.745933\pi$$
−0.698016 + 0.716082i $$0.745933\pi$$
$$572$$ 1568.00i 0.114618i
$$573$$ − 10096.0i − 0.736067i
$$574$$ −1274.00 −0.0926406
$$575$$ 0 0
$$576$$ 3841.00 0.277850
$$577$$ − 14366.0i − 1.03651i −0.855227 0.518253i $$-0.826582\pi$$
0.855227 0.518253i $$-0.173418\pi$$
$$578$$ − 1997.00i − 0.143710i
$$579$$ −5924.00 −0.425204
$$580$$ 0 0
$$581$$ 9114.00 0.650796
$$582$$ 588.000i 0.0418787i
$$583$$ − 1296.00i − 0.0920666i
$$584$$ 10530.0 0.746121
$$585$$ 0 0
$$586$$ 4312.00 0.303971
$$587$$ − 3626.00i − 0.254959i −0.991841 0.127480i $$-0.959311\pi$$
0.991841 0.127480i $$-0.0406887\pi$$
$$588$$ − 686.000i − 0.0481125i
$$589$$ 1320.00 0.0923424
$$590$$ 0 0
$$591$$ −6668.00 −0.464103
$$592$$ − 10086.0i − 0.700223i
$$593$$ 1062.00i 0.0735432i 0.999324 + 0.0367716i $$0.0117074\pi$$
−0.999324 + 0.0367716i $$0.988293\pi$$
$$594$$ −800.000 −0.0552599
$$595$$ 0 0
$$596$$ 14070.0 0.966996
$$597$$ − 3720.00i − 0.255024i
$$598$$ 1344.00i 0.0919068i
$$599$$ 10200.0 0.695761 0.347880 0.937539i $$-0.386902\pi$$
0.347880 + 0.937539i $$0.386902\pi$$
$$600$$ 0 0
$$601$$ −25158.0 −1.70751 −0.853757 0.520671i $$-0.825682\pi$$
−0.853757 + 0.520671i $$0.825682\pi$$
$$602$$ 896.000i 0.0606615i
$$603$$ 5612.00i 0.379002i
$$604$$ 7784.00 0.524382
$$605$$ 0 0
$$606$$ −1376.00 −0.0922379
$$607$$ 25664.0i 1.71609i 0.513570 + 0.858047i $$0.328323\pi$$
−0.513570 + 0.858047i $$0.671677\pi$$
$$608$$ − 17710.0i − 1.18131i
$$609$$ 1540.00 0.102470
$$610$$ 0 0
$$611$$ 9072.00 0.600677
$$612$$ 8694.00i 0.574239i
$$613$$ − 19018.0i − 1.25307i −0.779395 0.626533i $$-0.784473\pi$$
0.779395 0.626533i $$-0.215527\pi$$
$$614$$ 2674.00 0.175755
$$615$$ 0 0
$$616$$ 840.000 0.0549425
$$617$$ 17334.0i 1.13102i 0.824741 + 0.565511i $$0.191321\pi$$
−0.824741 + 0.565511i $$0.808679\pi$$
$$618$$ − 2776.00i − 0.180691i
$$619$$ −18730.0 −1.21619 −0.608096 0.793864i $$-0.708066\pi$$
−0.608096 + 0.793864i $$0.708066\pi$$
$$620$$ 0 0
$$621$$ 4800.00 0.310173
$$622$$ 3768.00i 0.242899i
$$623$$ 5110.00i 0.328616i
$$624$$ 2296.00 0.147297
$$625$$ 0 0
$$626$$ −2438.00 −0.155658
$$627$$ − 1760.00i − 0.112101i
$$628$$ 868.000i 0.0551544i
$$629$$ 13284.0 0.842079
$$630$$ 0 0
$$631$$ −6928.00 −0.437083 −0.218541 0.975828i $$-0.570130\pi$$
−0.218541 + 0.975828i $$0.570130\pi$$
$$632$$ 6600.00i 0.415402i
$$633$$ − 8536.00i − 0.535980i
$$634$$ −3186.00 −0.199578
$$635$$ 0 0
$$636$$ −2268.00 −0.141403
$$637$$ 1372.00i 0.0853385i
$$638$$ 880.000i 0.0546074i
$$639$$ −17664.0 −1.09355
$$640$$ 0 0
$$641$$ 16302.0 1.00451 0.502255 0.864720i $$-0.332504\pi$$
0.502255 + 0.864720i $$0.332504\pi$$
$$642$$ 488.000i 0.0299997i
$$643$$ − 4718.00i − 0.289362i −0.989478 0.144681i $$-0.953784\pi$$
0.989478 0.144681i $$-0.0462156\pi$$
$$644$$ −2352.00 −0.143916
$$645$$ 0 0
$$646$$ 5940.00 0.361774
$$647$$ − 21436.0i − 1.30253i −0.758851 0.651264i $$-0.774239\pi$$
0.758851 0.651264i $$-0.225761\pi$$
$$648$$ − 6315.00i − 0.382834i
$$649$$ 6480.00 0.391930
$$650$$ 0 0
$$651$$ 168.000 0.0101143
$$652$$ − 14056.0i − 0.844287i
$$653$$ − 4458.00i − 0.267159i −0.991038 0.133580i $$-0.957353\pi$$
0.991038 0.133580i $$-0.0426472\pi$$
$$654$$ −180.000 −0.0107623
$$655$$ 0 0
$$656$$ 7462.00 0.444119
$$657$$ 16146.0i 0.958775i
$$658$$ − 2268.00i − 0.134371i
$$659$$ 26640.0 1.57473 0.787365 0.616487i $$-0.211445\pi$$
0.787365 + 0.616487i $$0.211445\pi$$
$$660$$ 0 0
$$661$$ 7432.00 0.437324 0.218662 0.975801i $$-0.429831\pi$$
0.218662 + 0.975801i $$0.429831\pi$$
$$662$$ − 8672.00i − 0.509134i
$$663$$ 3024.00i 0.177138i
$$664$$ 19530.0 1.14143
$$665$$ 0 0
$$666$$ −5658.00 −0.329194
$$667$$ − 5280.00i − 0.306510i
$$668$$ 20188.0i 1.16931i
$$669$$ −10864.0 −0.627842
$$670$$ 0 0
$$671$$ 3904.00 0.224608
$$672$$ − 2254.00i − 0.129390i
$$673$$ − 58.0000i − 0.00332204i −0.999999 0.00166102i $$-0.999471\pi$$
0.999999 0.00166102i $$-0.000528720\pi$$
$$674$$ 814.000 0.0465194
$$675$$ 0 0
$$676$$ 9891.00 0.562756
$$677$$ − 21516.0i − 1.22146i −0.791840 0.610729i $$-0.790876\pi$$
0.791840 0.610729i $$-0.209124\pi$$
$$678$$ − 2636.00i − 0.149314i
$$679$$ 2058.00 0.116316
$$680$$ 0 0
$$681$$ 4092.00 0.230258
$$682$$ 96.0000i 0.00539007i
$$683$$ − 18108.0i − 1.01447i −0.861808 0.507235i $$-0.830668\pi$$
0.861808 0.507235i $$-0.169332\pi$$
$$684$$ 17710.0 0.989998
$$685$$ 0 0
$$686$$ 343.000 0.0190901
$$687$$ 5960.00i 0.330987i
$$688$$ − 5248.00i − 0.290811i
$$689$$ 4536.00 0.250810
$$690$$ 0 0
$$691$$ −10078.0 −0.554827 −0.277413 0.960751i $$-0.589477\pi$$
−0.277413 + 0.960751i $$0.589477\pi$$
$$692$$ − 15596.0i − 0.856750i
$$693$$ 1288.00i 0.0706018i
$$694$$ 9344.00 0.511086
$$695$$ 0 0
$$696$$ 3300.00 0.179722
$$697$$ 9828.00i 0.534092i
$$698$$ − 5180.00i − 0.280897i
$$699$$ 8916.00 0.482452
$$700$$ 0 0
$$701$$ 18762.0 1.01089 0.505443 0.862860i $$-0.331329\pi$$
0.505443 + 0.862860i $$0.331329\pi$$
$$702$$ − 2800.00i − 0.150540i
$$703$$ − 27060.0i − 1.45176i
$$704$$ −1336.00 −0.0715233
$$705$$ 0 0
$$706$$ −12178.0 −0.649186
$$707$$ 4816.00i 0.256187i
$$708$$ − 11340.0i − 0.601954i
$$709$$ −6810.00 −0.360726 −0.180363 0.983600i $$-0.557727\pi$$
−0.180363 + 0.983600i $$0.557727\pi$$
$$710$$ 0 0
$$711$$ −10120.0 −0.533797
$$712$$ 10950.0i 0.576360i
$$713$$ − 576.000i − 0.0302544i
$$714$$ 756.000 0.0396255
$$715$$ 0 0
$$716$$ 5740.00 0.299600
$$717$$ − 8880.00i − 0.462524i
$$718$$ 440.000i 0.0228700i
$$719$$ −4860.00 −0.252083 −0.126041 0.992025i $$-0.540227\pi$$
−0.126041 + 0.992025i $$0.540227\pi$$
$$720$$ 0 0
$$721$$ −9716.00 −0.501862
$$722$$ − 5241.00i − 0.270152i
$$723$$ 6604.00i 0.339703i
$$724$$ 27244.0 1.39850
$$725$$ 0 0
$$726$$ −2534.00 −0.129539
$$727$$ − 13636.0i − 0.695641i −0.937561 0.347821i $$-0.886922\pi$$
0.937561 0.347821i $$-0.113078\pi$$
$$728$$ 2940.00i 0.149675i
$$729$$ 4283.00 0.217599
$$730$$ 0 0
$$731$$ 6912.00 0.349726
$$732$$ − 6832.00i − 0.344970i
$$733$$ − 2088.00i − 0.105214i −0.998615 0.0526071i $$-0.983247\pi$$
0.998615 0.0526071i $$-0.0167531\pi$$
$$734$$ −9816.00 −0.493617
$$735$$ 0 0
$$736$$ −7728.00 −0.387035
$$737$$ − 1952.00i − 0.0975615i
$$738$$ − 4186.00i − 0.208792i
$$739$$ 5160.00 0.256852 0.128426 0.991719i $$-0.459008\pi$$
0.128426 + 0.991719i $$0.459008\pi$$
$$740$$ 0 0
$$741$$ 6160.00 0.305389
$$742$$ − 1134.00i − 0.0561057i
$$743$$ 28152.0i 1.39004i 0.718992 + 0.695018i $$0.244604\pi$$
−0.718992 + 0.695018i $$0.755396\pi$$
$$744$$ 360.000 0.0177396
$$745$$ 0 0
$$746$$ 442.000 0.0216927
$$747$$ 29946.0i 1.46676i
$$748$$ − 3024.00i − 0.147819i
$$749$$ 1708.00 0.0833230
$$750$$ 0 0
$$751$$ −16808.0 −0.816688 −0.408344 0.912828i $$-0.633894\pi$$
−0.408344 + 0.912828i $$0.633894\pi$$
$$752$$ 13284.0i 0.644172i
$$753$$ 3164.00i 0.153124i
$$754$$ −3080.00 −0.148763
$$755$$ 0 0
$$756$$ 4900.00 0.235729
$$757$$ 21674.0i 1.04063i 0.853975 + 0.520314i $$0.174185\pi$$
−0.853975 + 0.520314i $$0.825815\pi$$
$$758$$ − 3960.00i − 0.189754i
$$759$$ −768.000 −0.0367281
$$760$$ 0 0
$$761$$ 7422.00 0.353544 0.176772 0.984252i $$-0.443434\pi$$
0.176772 + 0.984252i $$0.443434\pi$$
$$762$$ − 3552.00i − 0.168865i
$$763$$ 630.000i 0.0298919i
$$764$$ −35336.0 −1.67331
$$765$$ 0 0
$$766$$ −6708.00 −0.316410
$$767$$ 22680.0i 1.06770i
$$768$$ − 238.000i − 0.0111824i
$$769$$ −13790.0 −0.646658 −0.323329 0.946287i $$-0.604802\pi$$
−0.323329 + 0.946287i $$0.604802\pi$$
$$770$$ 0 0
$$771$$ −4708.00 −0.219915
$$772$$ 20734.0i 0.966623i
$$773$$ 6232.00i 0.289973i 0.989434 + 0.144987i $$0.0463139\pi$$
−0.989434 + 0.144987i $$0.953686\pi$$
$$774$$ −2944.00 −0.136718
$$775$$ 0 0
$$776$$ 4410.00 0.204007
$$777$$ − 3444.00i − 0.159013i
$$778$$ − 13350.0i − 0.615194i
$$779$$ 20020.0 0.920784
$$780$$ 0 0
$$781$$ 6144.00 0.281498
$$782$$ − 2592.00i − 0.118529i
$$783$$ 11000.0i 0.502054i
$$784$$ −2009.00 −0.0915179
$$785$$ 0 0
$$786$$ −2236.00 −0.101470
$$787$$ − 1766.00i − 0.0799887i −0.999200 0.0399943i $$-0.987266\pi$$
0.999200 0.0399943i $$-0.0127340\pi$$
$$788$$ 23338.0i 1.05505i
$$789$$ −7744.00 −0.349422
$$790$$ 0 0
$$791$$ −9226.00 −0.414714
$$792$$ 2760.00i 0.123829i
$$793$$ 13664.0i 0.611883i
$$794$$ −1356.00 −0.0606079
$$795$$ 0 0
$$796$$ −13020.0 −0.579751
$$797$$ 1204.00i 0.0535105i 0.999642 + 0.0267552i $$0.00851748\pi$$
−0.999642 + 0.0267552i $$0.991483\pi$$
$$798$$ − 1540.00i − 0.0683150i
$$799$$ −17496.0 −0.774673
$$800$$ 0 0
$$801$$ −16790.0 −0.740631
$$802$$ − 6222.00i − 0.273948i
$$803$$ − 5616.00i − 0.246805i
$$804$$ −3416.00 −0.149842
$$805$$ 0 0
$$806$$ −336.000 −0.0146837
$$807$$ − 360.000i − 0.0157033i
$$808$$ 10320.0i 0.449327i
$$809$$ 7050.00 0.306384 0.153192 0.988196i $$-0.451045\pi$$
0.153192 + 0.988196i $$0.451045\pi$$
$$810$$ 0 0
$$811$$ 23282.0 1.00807 0.504033 0.863684i $$-0.331849\pi$$
0.504033 + 0.863684i $$0.331849\pi$$
$$812$$ − 5390.00i − 0.232946i
$$813$$ 4064.00i 0.175315i
$$814$$ 1968.00 0.0847400
$$815$$ 0 0
$$816$$ −4428.00 −0.189964
$$817$$ − 14080.0i − 0.602934i
$$818$$ 5150.00i 0.220129i
$$819$$ −4508.00 −0.192335
$$820$$ 0 0
$$821$$ 10142.0 0.431131 0.215565 0.976489i $$-0.430841\pi$$
0.215565 + 0.976489i $$0.430841\pi$$
$$822$$ 4548.00i 0.192980i
$$823$$ 9192.00i 0.389323i 0.980870 + 0.194662i $$0.0623609\pi$$
−0.980870 + 0.194662i $$0.937639\pi$$
$$824$$ −20820.0 −0.880217
$$825$$ 0 0
$$826$$ 5670.00 0.238843
$$827$$ − 46716.0i − 1.96430i −0.188104 0.982149i $$-0.560234\pi$$
0.188104 0.982149i $$-0.439766\pi$$
$$828$$ − 7728.00i − 0.324356i
$$829$$ −11240.0 −0.470906 −0.235453 0.971886i $$-0.575657\pi$$
−0.235453 + 0.971886i $$0.575657\pi$$
$$830$$ 0 0
$$831$$ 10852.0 0.453010
$$832$$ − 4676.00i − 0.194845i
$$833$$ − 2646.00i − 0.110058i
$$834$$ 420.000 0.0174381
$$835$$ 0 0
$$836$$ −6160.00 −0.254842
$$837$$ 1200.00i 0.0495556i
$$838$$ 2310.00i 0.0952239i
$$839$$ −700.000 −0.0288042 −0.0144021 0.999896i $$-0.504584\pi$$
−0.0144021 + 0.999896i $$0.504584\pi$$
$$840$$ 0 0
$$841$$ −12289.0 −0.503875
$$842$$ − 1262.00i − 0.0516525i
$$843$$ 1684.00i 0.0688019i
$$844$$ −29876.0 −1.21845
$$845$$ 0 0
$$846$$ 7452.00 0.302843
$$847$$ 8869.00i 0.359790i
$$848$$ 6642.00i 0.268971i
$$849$$ −7564.00 −0.305767
$$850$$ 0 0
$$851$$ −11808.0 −0.475644
$$852$$ − 10752.0i − 0.432344i
$$853$$ 37492.0i 1.50493i 0.658635 + 0.752463i $$0.271134\pi$$
−0.658635 + 0.752463i $$0.728866\pi$$
$$854$$ 3416.00 0.136877
$$855$$ 0 0
$$856$$ 3660.00 0.146140
$$857$$ 28894.0i 1.15169i 0.817558 + 0.575846i $$0.195327\pi$$
−0.817558 + 0.575846i $$0.804673\pi$$
$$858$$ 448.000i 0.0178257i
$$859$$ 2770.00 0.110025 0.0550123 0.998486i $$-0.482480\pi$$
0.0550123 + 0.998486i $$0.482480\pi$$
$$860$$ 0 0
$$861$$ 2548.00 0.100854
$$862$$ 4488.00i 0.177334i
$$863$$ − 17688.0i − 0.697690i −0.937180 0.348845i $$-0.886574\pi$$
0.937180 0.348845i $$-0.113426\pi$$
$$864$$ 16100.0 0.633950
$$865$$ 0 0
$$866$$ −17038.0 −0.668562
$$867$$ 3994.00i 0.156451i
$$868$$ − 588.000i − 0.0229931i
$$869$$ 3520.00 0.137408
$$870$$ 0 0
$$871$$ 6832.00 0.265779
$$872$$ 1350.00i 0.0524275i
$$873$$ 6762.00i 0.262152i
$$874$$ −5280.00 −0.204346
$$875$$ 0 0
$$876$$ −9828.00 −0.379061
$$877$$ − 33566.0i − 1.29241i −0.763164 0.646205i $$-0.776355\pi$$
0.763164 0.646205i $$-0.223645\pi$$
$$878$$ 16200.0i 0.622692i
$$879$$ −8624.00 −0.330922
$$880$$ 0 0
$$881$$ −16758.0 −0.640853 −0.320426 0.947273i $$-0.603826\pi$$
−0.320426 + 0.947273i $$0.603826\pi$$
$$882$$ 1127.00i 0.0430250i
$$883$$ − 11468.0i − 0.437066i −0.975830 0.218533i $$-0.929873\pi$$
0.975830 0.218533i $$-0.0701271\pi$$
$$884$$ 10584.0 0.402691
$$885$$ 0 0
$$886$$ 8772.00 0.332620
$$887$$ − 50356.0i − 1.90619i −0.302674 0.953094i $$-0.597879\pi$$
0.302674 0.953094i $$-0.402121\pi$$
$$888$$ − 7380.00i − 0.278893i
$$889$$ −12432.0 −0.469017
$$890$$ 0 0
$$891$$ −3368.00 −0.126636
$$892$$ 38024.0i 1.42728i
$$893$$ 35640.0i 1.33555i
$$894$$ 4020.00 0.150390
$$895$$ 0 0
$$896$$ −10185.0 −0.379751
$$897$$ − 2688.00i − 0.100055i
$$898$$ 2130.00i 0.0791526i
$$899$$ 1320.00 0.0489705
$$900$$ 0 0
$$901$$ −8748.00 −0.323461
$$902$$ 1456.00i 0.0537467i
$$903$$ − 1792.00i − 0.0660399i
$$904$$ −19770.0 −0.727368
$$905$$ 0 0
$$906$$ 2224.00 0.0815535
$$907$$ − 8716.00i − 0.319085i −0.987191 0.159542i $$-0.948998\pi$$
0.987191 0.159542i $$-0.0510019\pi$$
$$908$$ − 14322.0i − 0.523450i
$$909$$ −15824.0 −0.577392
$$910$$ 0 0
$$911$$ 7632.00 0.277563 0.138781 0.990323i $$-0.455682\pi$$
0.138781 + 0.990323i $$0.455682\pi$$
$$912$$ 9020.00i 0.327502i
$$913$$ − 10416.0i − 0.377568i
$$914$$ 10534.0 0.381219
$$915$$ 0 0
$$916$$ 20860.0 0.752439
$$917$$ 7826.00i 0.281829i
$$918$$ 5400.00i 0.194147i
$$919$$ 23080.0 0.828443 0.414221 0.910176i $$-0.364054\pi$$
0.414221 + 0.910176i $$0.364054\pi$$
$$920$$ 0 0
$$921$$ −5348.00 −0.191338
$$922$$ 9268.00i 0.331047i
$$923$$ 21504.0i 0.766861i
$$924$$ −784.000 −0.0279131
$$925$$ 0 0
$$926$$ 9392.00 0.333305
$$927$$ − 31924.0i − 1.13109i
$$928$$ − 17710.0i − 0.626465i
$$929$$ −45110.0 −1.59312 −0.796561 0.604558i $$-0.793350\pi$$
−0.796561 + 0.604558i $$0.793350\pi$$
$$930$$ 0 0
$$931$$ −5390.00 −0.189742
$$932$$ − 31206.0i − 1.09677i
$$933$$ − 7536.00i − 0.264435i
$$934$$ −10806.0 −0.378569
$$935$$ 0 0
$$936$$ −9660.00 −0.337337
$$937$$ 16674.0i 0.581340i 0.956823 + 0.290670i $$0.0938782\pi$$
−0.956823 + 0.290670i $$0.906122\pi$$
$$938$$ − 1708.00i − 0.0594543i
$$939$$ 4876.00 0.169459
$$940$$ 0 0
$$941$$ 43832.0 1.51847 0.759236 0.650815i $$-0.225573\pi$$
0.759236 + 0.650815i $$0.225573\pi$$
$$942$$ 248.000i 0.00857779i
$$943$$ − 8736.00i − 0.301679i
$$944$$ −33210.0 −1.14501
$$945$$ 0 0
$$946$$ 1024.00 0.0351936
$$947$$ − 736.000i − 0.0252553i −0.999920 0.0126277i $$-0.995980\pi$$
0.999920 0.0126277i $$-0.00401962\pi$$
$$948$$ − 6160.00i − 0.211042i
$$949$$ 19656.0 0.672351
$$950$$ 0 0
$$951$$ 6372.00 0.217273
$$952$$ − 5670.00i − 0.193031i
$$953$$ − 38138.0i − 1.29634i −0.761496 0.648169i $$-0.775535\pi$$
0.761496 0.648169i $$-0.224465\pi$$
$$954$$ 3726.00 0.126450
$$955$$ 0 0
$$956$$ −31080.0 −1.05146
$$957$$ − 1760.00i − 0.0594490i
$$958$$ 4940.00i 0.166601i
$$959$$ 15918.0 0.535995
$$960$$ 0 0
$$961$$ −29647.0 −0.995166
$$962$$ 6888.00i 0.230850i
$$963$$ 5612.00i 0.187792i
$$964$$ 23114.0 0.772253
$$965$$ 0 0
$$966$$ −672.000 −0.0223822
$$967$$ 26224.0i 0.872086i 0.899926 + 0.436043i $$0.143620\pi$$
−0.899926 + 0.436043i $$0.856380\pi$$
$$968$$ 19005.0i 0.631037i
$$969$$ −11880.0 −0.393850
$$970$$ 0 0
$$971$$ 18762.0 0.620084 0.310042 0.950723i $$-0.399657\pi$$
0.310042 + 0.950723i $$0.399657\pi$$
$$972$$ 24794.0i 0.818177i
$$973$$ − 1470.00i − 0.0484337i
$$974$$ −5216.00 −0.171593
$$975$$ 0 0
$$976$$ −20008.0 −0.656189
$$977$$ 38394.0i 1.25725i 0.777709 + 0.628625i $$0.216382\pi$$
−0.777709 + 0.628625i $$0.783618\pi$$
$$978$$ − 4016.00i − 0.131306i
$$979$$ 5840.00 0.190651
$$980$$ 0 0
$$981$$ −2070.00 −0.0673700
$$982$$ − 4412.00i − 0.143373i
$$983$$ − 5388.00i − 0.174822i −0.996172 0.0874112i $$-0.972141\pi$$
0.996172 0.0874112i $$-0.0278594\pi$$
$$984$$ 5460.00 0.176889
$$985$$ 0 0
$$986$$ 5940.00 0.191854
$$987$$ 4536.00i 0.146284i
$$988$$ − 21560.0i − 0.694246i
$$989$$ −6144.00 −0.197541
$$990$$ 0 0
$$991$$ 25472.0 0.816493 0.408247 0.912872i $$-0.366140\pi$$
0.408247 + 0.912872i $$0.366140\pi$$
$$992$$ − 1932.00i − 0.0618357i
$$993$$ 17344.0i 0.554275i
$$994$$ 5376.00 0.171546
$$995$$ 0 0
$$996$$ −18228.0 −0.579896
$$997$$ − 17096.0i − 0.543065i −0.962429 0.271532i $$-0.912470\pi$$
0.962429 0.271532i $$-0.0875304\pi$$
$$998$$ 19060.0i 0.604543i
$$999$$ 24600.0 0.779089
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 175.4.b.b.99.1 2
5.2 odd 4 175.4.a.b.1.1 1
5.3 odd 4 7.4.a.a.1.1 1
5.4 even 2 inner 175.4.b.b.99.2 2
15.2 even 4 1575.4.a.e.1.1 1
15.8 even 4 63.4.a.b.1.1 1
20.3 even 4 112.4.a.f.1.1 1
35.3 even 12 49.4.c.b.30.1 2
35.13 even 4 49.4.a.b.1.1 1
35.18 odd 12 49.4.c.c.30.1 2
35.23 odd 12 49.4.c.c.18.1 2
35.27 even 4 1225.4.a.j.1.1 1
35.33 even 12 49.4.c.b.18.1 2
40.3 even 4 448.4.a.e.1.1 1
40.13 odd 4 448.4.a.i.1.1 1
55.43 even 4 847.4.a.b.1.1 1
60.23 odd 4 1008.4.a.c.1.1 1
65.38 odd 4 1183.4.a.b.1.1 1
85.33 odd 4 2023.4.a.a.1.1 1
105.23 even 12 441.4.e.h.361.1 2
105.38 odd 12 441.4.e.e.226.1 2
105.53 even 12 441.4.e.h.226.1 2
105.68 odd 12 441.4.e.e.361.1 2
105.83 odd 4 441.4.a.i.1.1 1
140.83 odd 4 784.4.a.g.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
7.4.a.a.1.1 1 5.3 odd 4
49.4.a.b.1.1 1 35.13 even 4
49.4.c.b.18.1 2 35.33 even 12
49.4.c.b.30.1 2 35.3 even 12
49.4.c.c.18.1 2 35.23 odd 12
49.4.c.c.30.1 2 35.18 odd 12
63.4.a.b.1.1 1 15.8 even 4
112.4.a.f.1.1 1 20.3 even 4
175.4.a.b.1.1 1 5.2 odd 4
175.4.b.b.99.1 2 1.1 even 1 trivial
175.4.b.b.99.2 2 5.4 even 2 inner
441.4.a.i.1.1 1 105.83 odd 4
441.4.e.e.226.1 2 105.38 odd 12
441.4.e.e.361.1 2 105.68 odd 12
441.4.e.h.226.1 2 105.53 even 12
441.4.e.h.361.1 2 105.23 even 12
448.4.a.e.1.1 1 40.3 even 4
448.4.a.i.1.1 1 40.13 odd 4
784.4.a.g.1.1 1 140.83 odd 4
847.4.a.b.1.1 1 55.43 even 4
1008.4.a.c.1.1 1 60.23 odd 4
1183.4.a.b.1.1 1 65.38 odd 4
1225.4.a.j.1.1 1 35.27 even 4
1575.4.a.e.1.1 1 15.2 even 4
2023.4.a.a.1.1 1 85.33 odd 4