# Properties

 Label 380.2.c.a.229.2 Level $380$ Weight $2$ Character 380.229 Analytic conductor $3.034$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$380 = 2^{2} \cdot 5 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 380.c (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.03431527681$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{-2}, \sqrt{5})$$ Defining polynomial: $$x^{4} + 6x^{2} + 4$$ x^4 + 6*x^2 + 4 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 229.2 Root $$-0.874032i$$ of defining polynomial Character $$\chi$$ $$=$$ 380.229 Dual form 380.2.c.a.229.3

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-0.874032i q^{3} +2.23607 q^{5} +2.82843i q^{7} +2.23607 q^{9} +O(q^{10})$$ $$q-0.874032i q^{3} +2.23607 q^{5} +2.82843i q^{7} +2.23607 q^{9} -0.763932 q^{11} +5.45052i q^{13} -1.95440i q^{15} -7.40492i q^{17} +1.00000 q^{19} +2.47214 q^{21} +1.08036i q^{23} +5.00000 q^{25} -4.57649i q^{27} +4.47214 q^{29} -4.00000 q^{31} +0.667701i q^{33} +6.32456i q^{35} -2.62210i q^{37} +4.76393 q^{39} -6.00000 q^{41} -8.48528i q^{43} +5.00000 q^{45} +8.48528i q^{47} -1.00000 q^{49} -6.47214 q^{51} -2.62210i q^{53} -1.70820 q^{55} -0.874032i q^{57} +1.52786 q^{59} -11.7082 q^{61} +6.32456i q^{63} +12.1877i q^{65} +11.1074i q^{67} +0.944272 q^{69} -10.4721 q^{71} -5.24419i q^{73} -4.37016i q^{75} -2.16073i q^{77} -15.4164 q^{79} +2.70820 q^{81} -13.7295i q^{83} -16.5579i q^{85} -3.90879i q^{87} -2.94427 q^{89} -15.4164 q^{91} +3.49613i q^{93} +2.23607 q^{95} +13.9358i q^{97} -1.70820 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q+O(q^{10})$$ 4 * q $$4 q - 12 q^{11} + 4 q^{19} - 8 q^{21} + 20 q^{25} - 16 q^{31} + 28 q^{39} - 24 q^{41} + 20 q^{45} - 4 q^{49} - 8 q^{51} + 20 q^{55} + 24 q^{59} - 20 q^{61} - 32 q^{69} - 24 q^{71} - 8 q^{79} - 16 q^{81} + 24 q^{89} - 8 q^{91} + 20 q^{99}+O(q^{100})$$ 4 * q - 12 * q^11 + 4 * q^19 - 8 * q^21 + 20 * q^25 - 16 * q^31 + 28 * q^39 - 24 * q^41 + 20 * q^45 - 4 * q^49 - 8 * q^51 + 20 * q^55 + 24 * q^59 - 20 * q^61 - 32 * q^69 - 24 * q^71 - 8 * q^79 - 16 * q^81 + 24 * q^89 - 8 * q^91 + 20 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/380\mathbb{Z}\right)^\times$$.

 $$n$$ $$21$$ $$77$$ $$191$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ − 0.874032i − 0.504623i −0.967646 0.252311i $$-0.918809\pi$$
0.967646 0.252311i $$-0.0811907\pi$$
$$4$$ 0 0
$$5$$ 2.23607 1.00000
$$6$$ 0 0
$$7$$ 2.82843i 1.06904i 0.845154 + 0.534522i $$0.179509\pi$$
−0.845154 + 0.534522i $$0.820491\pi$$
$$8$$ 0 0
$$9$$ 2.23607 0.745356
$$10$$ 0 0
$$11$$ −0.763932 −0.230334 −0.115167 0.993346i $$-0.536740\pi$$
−0.115167 + 0.993346i $$0.536740\pi$$
$$12$$ 0 0
$$13$$ 5.45052i 1.51170i 0.654743 + 0.755852i $$0.272777\pi$$
−0.654743 + 0.755852i $$0.727223\pi$$
$$14$$ 0 0
$$15$$ − 1.95440i − 0.504623i
$$16$$ 0 0
$$17$$ − 7.40492i − 1.79596i −0.440040 0.897978i $$-0.645036\pi$$
0.440040 0.897978i $$-0.354964\pi$$
$$18$$ 0 0
$$19$$ 1.00000 0.229416
$$20$$ 0 0
$$21$$ 2.47214 0.539464
$$22$$ 0 0
$$23$$ 1.08036i 0.225271i 0.993636 + 0.112636i $$0.0359293\pi$$
−0.993636 + 0.112636i $$0.964071\pi$$
$$24$$ 0 0
$$25$$ 5.00000 1.00000
$$26$$ 0 0
$$27$$ − 4.57649i − 0.880746i
$$28$$ 0 0
$$29$$ 4.47214 0.830455 0.415227 0.909718i $$-0.363702\pi$$
0.415227 + 0.909718i $$0.363702\pi$$
$$30$$ 0 0
$$31$$ −4.00000 −0.718421 −0.359211 0.933257i $$-0.616954\pi$$
−0.359211 + 0.933257i $$0.616954\pi$$
$$32$$ 0 0
$$33$$ 0.667701i 0.116232i
$$34$$ 0 0
$$35$$ 6.32456i 1.06904i
$$36$$ 0 0
$$37$$ − 2.62210i − 0.431070i −0.976496 0.215535i $$-0.930850\pi$$
0.976496 0.215535i $$-0.0691495\pi$$
$$38$$ 0 0
$$39$$ 4.76393 0.762840
$$40$$ 0 0
$$41$$ −6.00000 −0.937043 −0.468521 0.883452i $$-0.655213\pi$$
−0.468521 + 0.883452i $$0.655213\pi$$
$$42$$ 0 0
$$43$$ − 8.48528i − 1.29399i −0.762493 0.646997i $$-0.776025\pi$$
0.762493 0.646997i $$-0.223975\pi$$
$$44$$ 0 0
$$45$$ 5.00000 0.745356
$$46$$ 0 0
$$47$$ 8.48528i 1.23771i 0.785507 + 0.618853i $$0.212402\pi$$
−0.785507 + 0.618853i $$0.787598\pi$$
$$48$$ 0 0
$$49$$ −1.00000 −0.142857
$$50$$ 0 0
$$51$$ −6.47214 −0.906280
$$52$$ 0 0
$$53$$ − 2.62210i − 0.360173i −0.983651 0.180086i $$-0.942362\pi$$
0.983651 0.180086i $$-0.0576377\pi$$
$$54$$ 0 0
$$55$$ −1.70820 −0.230334
$$56$$ 0 0
$$57$$ − 0.874032i − 0.115768i
$$58$$ 0 0
$$59$$ 1.52786 0.198911 0.0994555 0.995042i $$-0.468290\pi$$
0.0994555 + 0.995042i $$0.468290\pi$$
$$60$$ 0 0
$$61$$ −11.7082 −1.49908 −0.749541 0.661958i $$-0.769726\pi$$
−0.749541 + 0.661958i $$0.769726\pi$$
$$62$$ 0 0
$$63$$ 6.32456i 0.796819i
$$64$$ 0 0
$$65$$ 12.1877i 1.51170i
$$66$$ 0 0
$$67$$ 11.1074i 1.35698i 0.734609 + 0.678491i $$0.237366\pi$$
−0.734609 + 0.678491i $$0.762634\pi$$
$$68$$ 0 0
$$69$$ 0.944272 0.113677
$$70$$ 0 0
$$71$$ −10.4721 −1.24281 −0.621407 0.783488i $$-0.713439\pi$$
−0.621407 + 0.783488i $$0.713439\pi$$
$$72$$ 0 0
$$73$$ − 5.24419i − 0.613786i −0.951744 0.306893i $$-0.900711\pi$$
0.951744 0.306893i $$-0.0992894\pi$$
$$74$$ 0 0
$$75$$ − 4.37016i − 0.504623i
$$76$$ 0 0
$$77$$ − 2.16073i − 0.246238i
$$78$$ 0 0
$$79$$ −15.4164 −1.73448 −0.867241 0.497889i $$-0.834109\pi$$
−0.867241 + 0.497889i $$0.834109\pi$$
$$80$$ 0 0
$$81$$ 2.70820 0.300912
$$82$$ 0 0
$$83$$ − 13.7295i − 1.50701i −0.657445 0.753503i $$-0.728363\pi$$
0.657445 0.753503i $$-0.271637\pi$$
$$84$$ 0 0
$$85$$ − 16.5579i − 1.79596i
$$86$$ 0 0
$$87$$ − 3.90879i − 0.419066i
$$88$$ 0 0
$$89$$ −2.94427 −0.312092 −0.156046 0.987750i $$-0.549875\pi$$
−0.156046 + 0.987750i $$0.549875\pi$$
$$90$$ 0 0
$$91$$ −15.4164 −1.61608
$$92$$ 0 0
$$93$$ 3.49613i 0.362532i
$$94$$ 0 0
$$95$$ 2.23607 0.229416
$$96$$ 0 0
$$97$$ 13.9358i 1.41497i 0.706730 + 0.707483i $$0.250170\pi$$
−0.706730 + 0.707483i $$0.749830\pi$$
$$98$$ 0 0
$$99$$ −1.70820 −0.171681
$$100$$ 0 0
$$101$$ 5.23607 0.521008 0.260504 0.965473i $$-0.416111\pi$$
0.260504 + 0.965473i $$0.416111\pi$$
$$102$$ 0 0
$$103$$ 5.86319i 0.577717i 0.957372 + 0.288858i $$0.0932757\pi$$
−0.957372 + 0.288858i $$0.906724\pi$$
$$104$$ 0 0
$$105$$ 5.52786 0.539464
$$106$$ 0 0
$$107$$ − 13.2681i − 1.28268i −0.767258 0.641338i $$-0.778380\pi$$
0.767258 0.641338i $$-0.221620\pi$$
$$108$$ 0 0
$$109$$ −2.00000 −0.191565 −0.0957826 0.995402i $$-0.530535\pi$$
−0.0957826 + 0.995402i $$0.530535\pi$$
$$110$$ 0 0
$$111$$ −2.29180 −0.217528
$$112$$ 0 0
$$113$$ 16.3516i 1.53823i 0.639113 + 0.769113i $$0.279302\pi$$
−0.639113 + 0.769113i $$0.720698\pi$$
$$114$$ 0 0
$$115$$ 2.41577i 0.225271i
$$116$$ 0 0
$$117$$ 12.1877i 1.12676i
$$118$$ 0 0
$$119$$ 20.9443 1.91996
$$120$$ 0 0
$$121$$ −10.4164 −0.946946
$$122$$ 0 0
$$123$$ 5.24419i 0.472853i
$$124$$ 0 0
$$125$$ 11.1803 1.00000
$$126$$ 0 0
$$127$$ − 19.5927i − 1.73857i −0.494314 0.869284i $$-0.664581\pi$$
0.494314 0.869284i $$-0.335419\pi$$
$$128$$ 0 0
$$129$$ −7.41641 −0.652978
$$130$$ 0 0
$$131$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$132$$ 0 0
$$133$$ 2.82843i 0.245256i
$$134$$ 0 0
$$135$$ − 10.2333i − 0.880746i
$$136$$ 0 0
$$137$$ − 7.40492i − 0.632645i −0.948652 0.316322i $$-0.897552\pi$$
0.948652 0.316322i $$-0.102448\pi$$
$$138$$ 0 0
$$139$$ −2.29180 −0.194388 −0.0971938 0.995265i $$-0.530987\pi$$
−0.0971938 + 0.995265i $$0.530987\pi$$
$$140$$ 0 0
$$141$$ 7.41641 0.624574
$$142$$ 0 0
$$143$$ − 4.16383i − 0.348197i
$$144$$ 0 0
$$145$$ 10.0000 0.830455
$$146$$ 0 0
$$147$$ 0.874032i 0.0720889i
$$148$$ 0 0
$$149$$ 12.6525 1.03653 0.518266 0.855220i $$-0.326578\pi$$
0.518266 + 0.855220i $$0.326578\pi$$
$$150$$ 0 0
$$151$$ −8.00000 −0.651031 −0.325515 0.945537i $$-0.605538\pi$$
−0.325515 + 0.945537i $$0.605538\pi$$
$$152$$ 0 0
$$153$$ − 16.5579i − 1.33863i
$$154$$ 0 0
$$155$$ −8.94427 −0.718421
$$156$$ 0 0
$$157$$ − 0.412662i − 0.0329340i −0.999864 0.0164670i $$-0.994758\pi$$
0.999864 0.0164670i $$-0.00524185\pi$$
$$158$$ 0 0
$$159$$ −2.29180 −0.181751
$$160$$ 0 0
$$161$$ −3.05573 −0.240825
$$162$$ 0 0
$$163$$ 8.48528i 0.664619i 0.943170 + 0.332309i $$0.107828\pi$$
−0.943170 + 0.332309i $$0.892172\pi$$
$$164$$ 0 0
$$165$$ 1.49302i 0.116232i
$$166$$ 0 0
$$167$$ 17.4319i 1.34892i 0.738310 + 0.674462i $$0.235624\pi$$
−0.738310 + 0.674462i $$0.764376\pi$$
$$168$$ 0 0
$$169$$ −16.7082 −1.28525
$$170$$ 0 0
$$171$$ 2.23607 0.170996
$$172$$ 0 0
$$173$$ − 8.94665i − 0.680201i −0.940389 0.340101i $$-0.889539\pi$$
0.940389 0.340101i $$-0.110461\pi$$
$$174$$ 0 0
$$175$$ 14.1421i 1.06904i
$$176$$ 0 0
$$177$$ − 1.33540i − 0.100375i
$$178$$ 0 0
$$179$$ 22.4721 1.67965 0.839823 0.542860i $$-0.182659\pi$$
0.839823 + 0.542860i $$0.182659\pi$$
$$180$$ 0 0
$$181$$ 6.00000 0.445976 0.222988 0.974821i $$-0.428419\pi$$
0.222988 + 0.974821i $$0.428419\pi$$
$$182$$ 0 0
$$183$$ 10.2333i 0.756471i
$$184$$ 0 0
$$185$$ − 5.86319i − 0.431070i
$$186$$ 0 0
$$187$$ 5.65685i 0.413670i
$$188$$ 0 0
$$189$$ 12.9443 0.941557
$$190$$ 0 0
$$191$$ −3.05573 −0.221105 −0.110552 0.993870i $$-0.535262\pi$$
−0.110552 + 0.993870i $$0.535262\pi$$
$$192$$ 0 0
$$193$$ − 13.9358i − 1.00312i −0.865123 0.501561i $$-0.832759\pi$$
0.865123 0.501561i $$-0.167241\pi$$
$$194$$ 0 0
$$195$$ 10.6525 0.762840
$$196$$ 0 0
$$197$$ 12.6491i 0.901212i 0.892723 + 0.450606i $$0.148792\pi$$
−0.892723 + 0.450606i $$0.851208\pi$$
$$198$$ 0 0
$$199$$ 16.0000 1.13421 0.567105 0.823646i $$-0.308063\pi$$
0.567105 + 0.823646i $$0.308063\pi$$
$$200$$ 0 0
$$201$$ 9.70820 0.684764
$$202$$ 0 0
$$203$$ 12.6491i 0.887794i
$$204$$ 0 0
$$205$$ −13.4164 −0.937043
$$206$$ 0 0
$$207$$ 2.41577i 0.167907i
$$208$$ 0 0
$$209$$ −0.763932 −0.0528423
$$210$$ 0 0
$$211$$ 15.4164 1.06131 0.530655 0.847588i $$-0.321946\pi$$
0.530655 + 0.847588i $$0.321946\pi$$
$$212$$ 0 0
$$213$$ 9.15298i 0.627152i
$$214$$ 0 0
$$215$$ − 18.9737i − 1.29399i
$$216$$ 0 0
$$217$$ − 11.3137i − 0.768025i
$$218$$ 0 0
$$219$$ −4.58359 −0.309730
$$220$$ 0 0
$$221$$ 40.3607 2.71495
$$222$$ 0 0
$$223$$ − 18.7673i − 1.25675i −0.777909 0.628377i $$-0.783720\pi$$
0.777909 0.628377i $$-0.216280\pi$$
$$224$$ 0 0
$$225$$ 11.1803 0.745356
$$226$$ 0 0
$$227$$ 5.86319i 0.389153i 0.980887 + 0.194577i $$0.0623333\pi$$
−0.980887 + 0.194577i $$0.937667\pi$$
$$228$$ 0 0
$$229$$ −7.70820 −0.509372 −0.254686 0.967024i $$-0.581972\pi$$
−0.254686 + 0.967024i $$0.581972\pi$$
$$230$$ 0 0
$$231$$ −1.88854 −0.124257
$$232$$ 0 0
$$233$$ 12.6491i 0.828671i 0.910124 + 0.414335i $$0.135986\pi$$
−0.910124 + 0.414335i $$0.864014\pi$$
$$234$$ 0 0
$$235$$ 18.9737i 1.23771i
$$236$$ 0 0
$$237$$ 13.4744i 0.875259i
$$238$$ 0 0
$$239$$ 29.8885 1.93333 0.966665 0.256046i $$-0.0824199\pi$$
0.966665 + 0.256046i $$0.0824199\pi$$
$$240$$ 0 0
$$241$$ −28.8328 −1.85728 −0.928642 0.370976i $$-0.879023\pi$$
−0.928642 + 0.370976i $$0.879023\pi$$
$$242$$ 0 0
$$243$$ − 16.0965i − 1.03259i
$$244$$ 0 0
$$245$$ −2.23607 −0.142857
$$246$$ 0 0
$$247$$ 5.45052i 0.346808i
$$248$$ 0 0
$$249$$ −12.0000 −0.760469
$$250$$ 0 0
$$251$$ −5.88854 −0.371682 −0.185841 0.982580i $$-0.559501\pi$$
−0.185841 + 0.982580i $$0.559501\pi$$
$$252$$ 0 0
$$253$$ − 0.825324i − 0.0518877i
$$254$$ 0 0
$$255$$ −14.4721 −0.906280
$$256$$ 0 0
$$257$$ − 17.4319i − 1.08737i −0.839288 0.543687i $$-0.817028\pi$$
0.839288 0.543687i $$-0.182972\pi$$
$$258$$ 0 0
$$259$$ 7.41641 0.460833
$$260$$ 0 0
$$261$$ 10.0000 0.618984
$$262$$ 0 0
$$263$$ − 15.8902i − 0.979832i −0.871770 0.489916i $$-0.837027\pi$$
0.871770 0.489916i $$-0.162973\pi$$
$$264$$ 0 0
$$265$$ − 5.86319i − 0.360173i
$$266$$ 0 0
$$267$$ 2.57339i 0.157489i
$$268$$ 0 0
$$269$$ −14.9443 −0.911168 −0.455584 0.890193i $$-0.650570\pi$$
−0.455584 + 0.890193i $$0.650570\pi$$
$$270$$ 0 0
$$271$$ 21.1246 1.28323 0.641614 0.767027i $$-0.278265\pi$$
0.641614 + 0.767027i $$0.278265\pi$$
$$272$$ 0 0
$$273$$ 13.4744i 0.815510i
$$274$$ 0 0
$$275$$ −3.81966 −0.230334
$$276$$ 0 0
$$277$$ − 22.2148i − 1.33476i −0.744719 0.667378i $$-0.767417\pi$$
0.744719 0.667378i $$-0.232583\pi$$
$$278$$ 0 0
$$279$$ −8.94427 −0.535480
$$280$$ 0 0
$$281$$ −25.4164 −1.51622 −0.758108 0.652129i $$-0.773876\pi$$
−0.758108 + 0.652129i $$0.773876\pi$$
$$282$$ 0 0
$$283$$ 2.41577i 0.143602i 0.997419 + 0.0718012i $$0.0228747\pi$$
−0.997419 + 0.0718012i $$0.977125\pi$$
$$284$$ 0 0
$$285$$ − 1.95440i − 0.115768i
$$286$$ 0 0
$$287$$ − 16.9706i − 1.00174i
$$288$$ 0 0
$$289$$ −37.8328 −2.22546
$$290$$ 0 0
$$291$$ 12.1803 0.714024
$$292$$ 0 0
$$293$$ − 2.62210i − 0.153184i −0.997062 0.0765922i $$-0.975596\pi$$
0.997062 0.0765922i $$-0.0244040\pi$$
$$294$$ 0 0
$$295$$ 3.41641 0.198911
$$296$$ 0 0
$$297$$ 3.49613i 0.202866i
$$298$$ 0 0
$$299$$ −5.88854 −0.340543
$$300$$ 0 0
$$301$$ 24.0000 1.38334
$$302$$ 0 0
$$303$$ − 4.57649i − 0.262913i
$$304$$ 0 0
$$305$$ −26.1803 −1.49908
$$306$$ 0 0
$$307$$ − 5.45052i − 0.311078i −0.987830 0.155539i $$-0.950289\pi$$
0.987830 0.155539i $$-0.0497114\pi$$
$$308$$ 0 0
$$309$$ 5.12461 0.291529
$$310$$ 0 0
$$311$$ −9.70820 −0.550502 −0.275251 0.961372i $$-0.588761\pi$$
−0.275251 + 0.961372i $$0.588761\pi$$
$$312$$ 0 0
$$313$$ 10.4884i 0.592839i 0.955058 + 0.296419i $$0.0957926\pi$$
−0.955058 + 0.296419i $$0.904207\pi$$
$$314$$ 0 0
$$315$$ 14.1421i 0.796819i
$$316$$ 0 0
$$317$$ − 8.94665i − 0.502494i −0.967923 0.251247i $$-0.919159\pi$$
0.967923 0.251247i $$-0.0808406\pi$$
$$318$$ 0 0
$$319$$ −3.41641 −0.191282
$$320$$ 0 0
$$321$$ −11.5967 −0.647267
$$322$$ 0 0
$$323$$ − 7.40492i − 0.412021i
$$324$$ 0 0
$$325$$ 27.2526i 1.51170i
$$326$$ 0 0
$$327$$ 1.74806i 0.0966682i
$$328$$ 0 0
$$329$$ −24.0000 −1.32316
$$330$$ 0 0
$$331$$ 7.41641 0.407643 0.203821 0.979008i $$-0.434664\pi$$
0.203821 + 0.979008i $$0.434664\pi$$
$$332$$ 0 0
$$333$$ − 5.86319i − 0.321301i
$$334$$ 0 0
$$335$$ 24.8369i 1.35698i
$$336$$ 0 0
$$337$$ 16.7642i 0.913206i 0.889671 + 0.456603i $$0.150934\pi$$
−0.889671 + 0.456603i $$0.849066\pi$$
$$338$$ 0 0
$$339$$ 14.2918 0.776224
$$340$$ 0 0
$$341$$ 3.05573 0.165477
$$342$$ 0 0
$$343$$ 16.9706i 0.916324i
$$344$$ 0 0
$$345$$ 2.11146 0.113677
$$346$$ 0 0
$$347$$ − 9.40802i − 0.505049i −0.967590 0.252525i $$-0.918739\pi$$
0.967590 0.252525i $$-0.0812608\pi$$
$$348$$ 0 0
$$349$$ −25.4164 −1.36051 −0.680255 0.732976i $$-0.738131\pi$$
−0.680255 + 0.732976i $$0.738131\pi$$
$$350$$ 0 0
$$351$$ 24.9443 1.33143
$$352$$ 0 0
$$353$$ − 9.56564i − 0.509128i −0.967056 0.254564i $$-0.918068\pi$$
0.967056 0.254564i $$-0.0819319\pi$$
$$354$$ 0 0
$$355$$ −23.4164 −1.24281
$$356$$ 0 0
$$357$$ − 18.3060i − 0.968854i
$$358$$ 0 0
$$359$$ −29.1246 −1.53714 −0.768569 0.639767i $$-0.779031\pi$$
−0.768569 + 0.639767i $$0.779031\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ 0 0
$$363$$ 9.10427i 0.477850i
$$364$$ 0 0
$$365$$ − 11.7264i − 0.613786i
$$366$$ 0 0
$$367$$ 8.48528i 0.442928i 0.975169 + 0.221464i $$0.0710835\pi$$
−0.975169 + 0.221464i $$0.928916\pi$$
$$368$$ 0 0
$$369$$ −13.4164 −0.698430
$$370$$ 0 0
$$371$$ 7.41641 0.385041
$$372$$ 0 0
$$373$$ − 13.1105i − 0.678835i −0.940636 0.339417i $$-0.889770\pi$$
0.940636 0.339417i $$-0.110230\pi$$
$$374$$ 0 0
$$375$$ − 9.77198i − 0.504623i
$$376$$ 0 0
$$377$$ 24.3755i 1.25540i
$$378$$ 0 0
$$379$$ 20.0000 1.02733 0.513665 0.857991i $$-0.328287\pi$$
0.513665 + 0.857991i $$0.328287\pi$$
$$380$$ 0 0
$$381$$ −17.1246 −0.877320
$$382$$ 0 0
$$383$$ 36.4056i 1.86024i 0.367257 + 0.930120i $$0.380297\pi$$
−0.367257 + 0.930120i $$0.619703\pi$$
$$384$$ 0 0
$$385$$ − 4.83153i − 0.246238i
$$386$$ 0 0
$$387$$ − 18.9737i − 0.964486i
$$388$$ 0 0
$$389$$ 14.9443 0.757705 0.378852 0.925457i $$-0.376319\pi$$
0.378852 + 0.925457i $$0.376319\pi$$
$$390$$ 0 0
$$391$$ 8.00000 0.404577
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ −34.4721 −1.73448
$$396$$ 0 0
$$397$$ 11.7264i 0.588530i 0.955724 + 0.294265i $$0.0950748\pi$$
−0.955724 + 0.294265i $$0.904925\pi$$
$$398$$ 0 0
$$399$$ 2.47214 0.123762
$$400$$ 0 0
$$401$$ −4.47214 −0.223328 −0.111664 0.993746i $$-0.535618\pi$$
−0.111664 + 0.993746i $$0.535618\pi$$
$$402$$ 0 0
$$403$$ − 21.8021i − 1.08604i
$$404$$ 0 0
$$405$$ 6.05573 0.300912
$$406$$ 0 0
$$407$$ 2.00310i 0.0992901i
$$408$$ 0 0
$$409$$ −17.4164 −0.861186 −0.430593 0.902546i $$-0.641696\pi$$
−0.430593 + 0.902546i $$0.641696\pi$$
$$410$$ 0 0
$$411$$ −6.47214 −0.319247
$$412$$ 0 0
$$413$$ 4.32145i 0.212645i
$$414$$ 0 0
$$415$$ − 30.7000i − 1.50701i
$$416$$ 0 0
$$417$$ 2.00310i 0.0980924i
$$418$$ 0 0
$$419$$ 20.9443 1.02319 0.511597 0.859225i $$-0.329054\pi$$
0.511597 + 0.859225i $$0.329054\pi$$
$$420$$ 0 0
$$421$$ 14.0000 0.682318 0.341159 0.940006i $$-0.389181\pi$$
0.341159 + 0.940006i $$0.389181\pi$$
$$422$$ 0 0
$$423$$ 18.9737i 0.922531i
$$424$$ 0 0
$$425$$ − 37.0246i − 1.79596i
$$426$$ 0 0
$$427$$ − 33.1158i − 1.60259i
$$428$$ 0 0
$$429$$ −3.63932 −0.175708
$$430$$ 0 0
$$431$$ 1.52786 0.0735946 0.0367973 0.999323i $$-0.488284\pi$$
0.0367973 + 0.999323i $$0.488284\pi$$
$$432$$ 0 0
$$433$$ 28.4906i 1.36917i 0.728933 + 0.684585i $$0.240017\pi$$
−0.728933 + 0.684585i $$0.759983\pi$$
$$434$$ 0 0
$$435$$ − 8.74032i − 0.419066i
$$436$$ 0 0
$$437$$ 1.08036i 0.0516808i
$$438$$ 0 0
$$439$$ 18.8328 0.898841 0.449421 0.893320i $$-0.351630\pi$$
0.449421 + 0.893320i $$0.351630\pi$$
$$440$$ 0 0
$$441$$ −2.23607 −0.106479
$$442$$ 0 0
$$443$$ 13.7295i 0.652307i 0.945317 + 0.326153i $$0.105753\pi$$
−0.945317 + 0.326153i $$0.894247\pi$$
$$444$$ 0 0
$$445$$ −6.58359 −0.312092
$$446$$ 0 0
$$447$$ − 11.0587i − 0.523057i
$$448$$ 0 0
$$449$$ 16.4721 0.777368 0.388684 0.921371i $$-0.372930\pi$$
0.388684 + 0.921371i $$0.372930\pi$$
$$450$$ 0 0
$$451$$ 4.58359 0.215833
$$452$$ 0 0
$$453$$ 6.99226i 0.328525i
$$454$$ 0 0
$$455$$ −34.4721 −1.61608
$$456$$ 0 0
$$457$$ 37.9473i 1.77510i 0.460710 + 0.887551i $$0.347595\pi$$
−0.460710 + 0.887551i $$0.652405\pi$$
$$458$$ 0 0
$$459$$ −33.8885 −1.58178
$$460$$ 0 0
$$461$$ 23.8885 1.11260 0.556300 0.830981i $$-0.312220\pi$$
0.556300 + 0.830981i $$0.312220\pi$$
$$462$$ 0 0
$$463$$ − 14.5548i − 0.676419i −0.941071 0.338209i $$-0.890179\pi$$
0.941071 0.338209i $$-0.109821\pi$$
$$464$$ 0 0
$$465$$ 7.81758i 0.362532i
$$466$$ 0 0
$$467$$ 30.7000i 1.42063i 0.703885 + 0.710314i $$0.251447\pi$$
−0.703885 + 0.710314i $$0.748553\pi$$
$$468$$ 0 0
$$469$$ −31.4164 −1.45067
$$470$$ 0 0
$$471$$ −0.360680 −0.0166192
$$472$$ 0 0
$$473$$ 6.48218i 0.298051i
$$474$$ 0 0
$$475$$ 5.00000 0.229416
$$476$$ 0 0
$$477$$ − 5.86319i − 0.268457i
$$478$$ 0 0
$$479$$ −11.2361 −0.513389 −0.256695 0.966493i $$-0.582633\pi$$
−0.256695 + 0.966493i $$0.582633\pi$$
$$480$$ 0 0
$$481$$ 14.2918 0.651650
$$482$$ 0 0
$$483$$ 2.67080i 0.121526i
$$484$$ 0 0
$$485$$ 31.1614i 1.41497i
$$486$$ 0 0
$$487$$ − 10.6947i − 0.484624i −0.970198 0.242312i $$-0.922094\pi$$
0.970198 0.242312i $$-0.0779057\pi$$
$$488$$ 0 0
$$489$$ 7.41641 0.335382
$$490$$ 0 0
$$491$$ 5.88854 0.265746 0.132873 0.991133i $$-0.457580\pi$$
0.132873 + 0.991133i $$0.457580\pi$$
$$492$$ 0 0
$$493$$ − 33.1158i − 1.49146i
$$494$$ 0 0
$$495$$ −3.81966 −0.171681
$$496$$ 0 0
$$497$$ − 29.6197i − 1.32862i
$$498$$ 0 0
$$499$$ 17.1246 0.766603 0.383301 0.923623i $$-0.374787\pi$$
0.383301 + 0.923623i $$0.374787\pi$$
$$500$$ 0 0
$$501$$ 15.2361 0.680697
$$502$$ 0 0
$$503$$ − 20.2117i − 0.901193i −0.892728 0.450597i $$-0.851211\pi$$
0.892728 0.450597i $$-0.148789\pi$$
$$504$$ 0 0
$$505$$ 11.7082 0.521008
$$506$$ 0 0
$$507$$ 14.6035i 0.648564i
$$508$$ 0 0
$$509$$ 10.5836 0.469109 0.234555 0.972103i $$-0.424637\pi$$
0.234555 + 0.972103i $$0.424637\pi$$
$$510$$ 0 0
$$511$$ 14.8328 0.656165
$$512$$ 0 0
$$513$$ − 4.57649i − 0.202057i
$$514$$ 0 0
$$515$$ 13.1105i 0.577717i
$$516$$ 0 0
$$517$$ − 6.48218i − 0.285086i
$$518$$ 0 0
$$519$$ −7.81966 −0.343245
$$520$$ 0 0
$$521$$ 0.111456 0.00488298 0.00244149 0.999997i $$-0.499223\pi$$
0.00244149 + 0.999997i $$0.499223\pi$$
$$522$$ 0 0
$$523$$ 24.8369i 1.08604i 0.839720 + 0.543020i $$0.182719\pi$$
−0.839720 + 0.543020i $$0.817281\pi$$
$$524$$ 0 0
$$525$$ 12.3607 0.539464
$$526$$ 0 0
$$527$$ 29.6197i 1.29025i
$$528$$ 0 0
$$529$$ 21.8328 0.949253
$$530$$ 0 0
$$531$$ 3.41641 0.148259
$$532$$ 0 0
$$533$$ − 32.7031i − 1.41653i
$$534$$ 0 0
$$535$$ − 29.6684i − 1.28268i
$$536$$ 0 0
$$537$$ − 19.6414i − 0.847588i
$$538$$ 0 0
$$539$$ 0.763932 0.0329049
$$540$$ 0 0
$$541$$ 11.7082 0.503375 0.251688 0.967809i $$-0.419014\pi$$
0.251688 + 0.967809i $$0.419014\pi$$
$$542$$ 0 0
$$543$$ − 5.24419i − 0.225050i
$$544$$ 0 0
$$545$$ −4.47214 −0.191565
$$546$$ 0 0
$$547$$ − 8.27895i − 0.353982i −0.984212 0.176991i $$-0.943364\pi$$
0.984212 0.176991i $$-0.0566364\pi$$
$$548$$ 0 0
$$549$$ −26.1803 −1.11735
$$550$$ 0 0
$$551$$ 4.47214 0.190519
$$552$$ 0 0
$$553$$ − 43.6042i − 1.85424i
$$554$$ 0 0
$$555$$ −5.12461 −0.217528
$$556$$ 0 0
$$557$$ 20.0540i 0.849716i 0.905260 + 0.424858i $$0.139676\pi$$
−0.905260 + 0.424858i $$0.860324\pi$$
$$558$$ 0 0
$$559$$ 46.2492 1.95613
$$560$$ 0 0
$$561$$ 4.94427 0.208747
$$562$$ 0 0
$$563$$ − 30.0810i − 1.26776i −0.773429 0.633882i $$-0.781460\pi$$
0.773429 0.633882i $$-0.218540\pi$$
$$564$$ 0 0
$$565$$ 36.5632i 1.53823i
$$566$$ 0 0
$$567$$ 7.65996i 0.321688i
$$568$$ 0 0
$$569$$ 38.9443 1.63263 0.816314 0.577608i $$-0.196014\pi$$
0.816314 + 0.577608i $$0.196014\pi$$
$$570$$ 0 0
$$571$$ 25.1246 1.05143 0.525716 0.850660i $$-0.323797\pi$$
0.525716 + 0.850660i $$0.323797\pi$$
$$572$$ 0 0
$$573$$ 2.67080i 0.111574i
$$574$$ 0 0
$$575$$ 5.40182i 0.225271i
$$576$$ 0 0
$$577$$ − 5.65685i − 0.235498i −0.993043 0.117749i $$-0.962432\pi$$
0.993043 0.117749i $$-0.0375678\pi$$
$$578$$ 0 0
$$579$$ −12.1803 −0.506198
$$580$$ 0 0
$$581$$ 38.8328 1.61106
$$582$$ 0 0
$$583$$ 2.00310i 0.0829601i
$$584$$ 0 0
$$585$$ 27.2526i 1.12676i
$$586$$ 0 0
$$587$$ 5.40182i 0.222957i 0.993767 + 0.111478i $$0.0355586\pi$$
−0.993767 + 0.111478i $$0.964441\pi$$
$$588$$ 0 0
$$589$$ −4.00000 −0.164817
$$590$$ 0 0
$$591$$ 11.0557 0.454772
$$592$$ 0 0
$$593$$ 2.16073i 0.0887304i 0.999015 + 0.0443652i $$0.0141265\pi$$
−0.999015 + 0.0443652i $$0.985873\pi$$
$$594$$ 0 0
$$595$$ 46.8328 1.91996
$$596$$ 0 0
$$597$$ − 13.9845i − 0.572348i
$$598$$ 0 0
$$599$$ 36.0000 1.47092 0.735460 0.677568i $$-0.236966\pi$$
0.735460 + 0.677568i $$0.236966\pi$$
$$600$$ 0 0
$$601$$ 16.8328 0.686625 0.343312 0.939221i $$-0.388451\pi$$
0.343312 + 0.939221i $$0.388451\pi$$
$$602$$ 0 0
$$603$$ 24.8369i 1.01143i
$$604$$ 0 0
$$605$$ −23.2918 −0.946946
$$606$$ 0 0
$$607$$ 22.4211i 0.910044i 0.890480 + 0.455022i $$0.150369\pi$$
−0.890480 + 0.455022i $$0.849631\pi$$
$$608$$ 0 0
$$609$$ 11.0557 0.448001
$$610$$ 0 0
$$611$$ −46.2492 −1.87104
$$612$$ 0 0
$$613$$ 39.1853i 1.58268i 0.611376 + 0.791340i $$0.290616\pi$$
−0.611376 + 0.791340i $$0.709384\pi$$
$$614$$ 0 0
$$615$$ 11.7264i 0.472853i
$$616$$ 0 0
$$617$$ 3.08347i 0.124136i 0.998072 + 0.0620678i $$0.0197695\pi$$
−0.998072 + 0.0620678i $$0.980230\pi$$
$$618$$ 0 0
$$619$$ −41.1246 −1.65294 −0.826469 0.562982i $$-0.809654\pi$$
−0.826469 + 0.562982i $$0.809654\pi$$
$$620$$ 0 0
$$621$$ 4.94427 0.198407
$$622$$ 0 0
$$623$$ − 8.32766i − 0.333641i
$$624$$ 0 0
$$625$$ 25.0000 1.00000
$$626$$ 0 0
$$627$$ 0.667701i 0.0266654i
$$628$$ 0 0
$$629$$ −19.4164 −0.774183
$$630$$ 0 0
$$631$$ −1.70820 −0.0680025 −0.0340013 0.999422i $$-0.510825\pi$$
−0.0340013 + 0.999422i $$0.510825\pi$$
$$632$$ 0 0
$$633$$ − 13.4744i − 0.535561i
$$634$$ 0 0
$$635$$ − 43.8105i − 1.73857i
$$636$$ 0 0
$$637$$ − 5.45052i − 0.215958i
$$638$$ 0 0
$$639$$ −23.4164 −0.926339
$$640$$ 0 0
$$641$$ −12.1115 −0.478374 −0.239187 0.970974i $$-0.576881\pi$$
−0.239187 + 0.970974i $$0.576881\pi$$
$$642$$ 0 0
$$643$$ 30.7000i 1.21069i 0.795963 + 0.605346i $$0.206965\pi$$
−0.795963 + 0.605346i $$0.793035\pi$$
$$644$$ 0 0
$$645$$ −16.5836 −0.652978
$$646$$ 0 0
$$647$$ − 9.40802i − 0.369867i −0.982751 0.184934i $$-0.940793\pi$$
0.982751 0.184934i $$-0.0592071\pi$$
$$648$$ 0 0
$$649$$ −1.16718 −0.0458160
$$650$$ 0 0
$$651$$ −9.88854 −0.387563
$$652$$ 0 0
$$653$$ 22.2148i 0.869331i 0.900592 + 0.434665i $$0.143133\pi$$
−0.900592 + 0.434665i $$0.856867\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ − 11.7264i − 0.457489i
$$658$$ 0 0
$$659$$ 44.9443 1.75078 0.875390 0.483417i $$-0.160605\pi$$
0.875390 + 0.483417i $$0.160605\pi$$
$$660$$ 0 0
$$661$$ −26.0000 −1.01128 −0.505641 0.862744i $$-0.668744\pi$$
−0.505641 + 0.862744i $$0.668744\pi$$
$$662$$ 0 0
$$663$$ − 35.2765i − 1.37003i
$$664$$ 0 0
$$665$$ 6.32456i 0.245256i
$$666$$ 0 0
$$667$$ 4.83153i 0.187078i
$$668$$ 0 0
$$669$$ −16.4033 −0.634186
$$670$$ 0 0
$$671$$ 8.94427 0.345290
$$672$$ 0 0
$$673$$ 4.62520i 0.178288i 0.996019 + 0.0891442i $$0.0284132\pi$$
−0.996019 + 0.0891442i $$0.971587\pi$$
$$674$$ 0 0
$$675$$ − 22.8825i − 0.880746i
$$676$$ 0 0
$$677$$ 42.7302i 1.64225i 0.570746 + 0.821127i $$0.306654\pi$$
−0.570746 + 0.821127i $$0.693346\pi$$
$$678$$ 0 0
$$679$$ −39.4164 −1.51266
$$680$$ 0 0
$$681$$ 5.12461 0.196376
$$682$$ 0 0
$$683$$ − 19.5927i − 0.749692i −0.927087 0.374846i $$-0.877696\pi$$
0.927087 0.374846i $$-0.122304\pi$$
$$684$$ 0 0
$$685$$ − 16.5579i − 0.632645i
$$686$$ 0 0
$$687$$ 6.73722i 0.257041i
$$688$$ 0 0
$$689$$ 14.2918 0.544474
$$690$$ 0 0
$$691$$ 25.1246 0.955785 0.477893 0.878418i $$-0.341401\pi$$
0.477893 + 0.878418i $$0.341401\pi$$
$$692$$ 0 0
$$693$$ − 4.83153i − 0.183535i
$$694$$ 0 0
$$695$$ −5.12461 −0.194388
$$696$$ 0 0
$$697$$ 44.4295i 1.68289i
$$698$$ 0 0
$$699$$ 11.0557 0.418166
$$700$$ 0 0
$$701$$ 0.875388 0.0330630 0.0165315 0.999863i $$-0.494738\pi$$
0.0165315 + 0.999863i $$0.494738\pi$$
$$702$$ 0 0
$$703$$ − 2.62210i − 0.0988942i
$$704$$ 0 0
$$705$$ 16.5836 0.624574
$$706$$ 0 0
$$707$$ 14.8098i 0.556981i
$$708$$ 0 0
$$709$$ −33.4164 −1.25498 −0.627490 0.778625i $$-0.715918\pi$$
−0.627490 + 0.778625i $$0.715918\pi$$
$$710$$ 0 0
$$711$$ −34.4721 −1.29281
$$712$$ 0 0
$$713$$ − 4.32145i − 0.161840i
$$714$$ 0 0
$$715$$ − 9.31061i − 0.348197i
$$716$$ 0 0
$$717$$ − 26.1235i − 0.975602i
$$718$$ 0 0
$$719$$ −27.5967 −1.02919 −0.514593 0.857435i $$-0.672057\pi$$
−0.514593 + 0.857435i $$0.672057\pi$$
$$720$$ 0 0
$$721$$ −16.5836 −0.617605
$$722$$ 0 0
$$723$$ 25.2008i 0.937228i
$$724$$ 0 0
$$725$$ 22.3607 0.830455
$$726$$ 0 0
$$727$$ 8.48528i 0.314702i 0.987543 + 0.157351i $$0.0502953\pi$$
−0.987543 + 0.157351i $$0.949705\pi$$
$$728$$ 0 0
$$729$$ −5.94427 −0.220158
$$730$$ 0 0
$$731$$ −62.8328 −2.32396
$$732$$ 0 0
$$733$$ 33.9411i 1.25364i 0.779162 + 0.626822i $$0.215645\pi$$
−0.779162 + 0.626822i $$0.784355\pi$$
$$734$$ 0 0
$$735$$ 1.95440i 0.0720889i
$$736$$ 0 0
$$737$$ − 8.48528i − 0.312559i
$$738$$ 0 0
$$739$$ 18.8328 0.692776 0.346388 0.938091i $$-0.387408\pi$$
0.346388 + 0.938091i $$0.387408\pi$$
$$740$$ 0 0
$$741$$ 4.76393 0.175007
$$742$$ 0 0
$$743$$ 2.62210i 0.0961954i 0.998843 + 0.0480977i $$0.0153159\pi$$
−0.998843 + 0.0480977i $$0.984684\pi$$
$$744$$ 0 0
$$745$$ 28.2918 1.03653
$$746$$ 0 0
$$747$$ − 30.7000i − 1.12326i
$$748$$ 0 0
$$749$$ 37.5279 1.37124
$$750$$ 0 0
$$751$$ 20.5836 0.751106 0.375553 0.926801i $$-0.377453\pi$$
0.375553 + 0.926801i $$0.377453\pi$$
$$752$$ 0 0
$$753$$ 5.14678i 0.187559i
$$754$$ 0 0
$$755$$ −17.8885 −0.651031
$$756$$ 0 0
$$757$$ − 38.7727i − 1.40922i −0.709597 0.704608i $$-0.751123\pi$$
0.709597 0.704608i $$-0.248877\pi$$
$$758$$ 0 0
$$759$$ −0.721360 −0.0261837
$$760$$ 0 0
$$761$$ −42.5410 −1.54211 −0.771055 0.636768i $$-0.780271\pi$$
−0.771055 + 0.636768i $$0.780271\pi$$
$$762$$ 0 0
$$763$$ − 5.65685i − 0.204792i
$$764$$ 0 0
$$765$$ − 37.0246i − 1.33863i
$$766$$ 0 0
$$767$$ 8.32766i 0.300694i
$$768$$ 0 0
$$769$$ 38.5410 1.38982 0.694912 0.719094i $$-0.255443\pi$$
0.694912 + 0.719094i $$0.255443\pi$$
$$770$$ 0 0
$$771$$ −15.2361 −0.548714
$$772$$ 0 0
$$773$$ − 5.86319i − 0.210884i −0.994425 0.105442i $$-0.966374\pi$$
0.994425 0.105442i $$-0.0336258\pi$$
$$774$$ 0 0
$$775$$ −20.0000 −0.718421
$$776$$ 0 0
$$777$$ − 6.48218i − 0.232547i
$$778$$ 0 0
$$779$$ −6.00000 −0.214972
$$780$$ 0 0
$$781$$ 8.00000 0.286263
$$782$$ 0 0
$$783$$ − 20.4667i − 0.731420i
$$784$$ 0 0
$$785$$ − 0.922740i − 0.0329340i
$$786$$ 0 0
$$787$$ 24.8369i 0.885338i 0.896685 + 0.442669i $$0.145968\pi$$
−0.896685 + 0.442669i $$0.854032\pi$$
$$788$$ 0 0
$$789$$ −13.8885 −0.494445
$$790$$ 0 0
$$791$$ −46.2492 −1.64443
$$792$$ 0 0
$$793$$ − 63.8158i − 2.26617i
$$794$$ 0 0
$$795$$ −5.12461 −0.181751
$$796$$ 0 0
$$797$$ − 52.2958i − 1.85241i −0.377018 0.926206i $$-0.623050\pi$$
0.377018 0.926206i $$-0.376950\pi$$
$$798$$ 0 0
$$799$$ 62.8328 2.22287
$$800$$ 0 0
$$801$$ −6.58359 −0.232620
$$802$$ 0 0
$$803$$ 4.00621i 0.141376i
$$804$$ 0 0
$$805$$ −6.83282 −0.240825
$$806$$ 0 0
$$807$$ 13.0618i 0.459796i
$$808$$ 0 0
$$809$$ 7.52786 0.264666 0.132333 0.991205i $$-0.457753\pi$$
0.132333 + 0.991205i $$0.457753\pi$$
$$810$$ 0 0
$$811$$ 8.00000 0.280918 0.140459 0.990086i $$-0.455142\pi$$
0.140459 + 0.990086i $$0.455142\pi$$
$$812$$ 0 0
$$813$$ − 18.4636i − 0.647546i
$$814$$ 0 0
$$815$$ 18.9737i 0.664619i
$$816$$ 0 0
$$817$$ − 8.48528i − 0.296862i
$$818$$ 0 0
$$819$$ −34.4721 −1.20455
$$820$$ 0 0
$$821$$ 38.9443 1.35916 0.679582 0.733599i $$-0.262161\pi$$
0.679582 + 0.733599i $$0.262161\pi$$
$$822$$ 0 0
$$823$$ 35.9442i 1.25294i 0.779447 + 0.626469i $$0.215500\pi$$
−0.779447 + 0.626469i $$0.784500\pi$$
$$824$$ 0 0
$$825$$ 3.33851i 0.116232i
$$826$$ 0 0
$$827$$ − 7.86629i − 0.273538i −0.990603 0.136769i $$-0.956328\pi$$
0.990603 0.136769i $$-0.0436717\pi$$
$$828$$ 0 0
$$829$$ −18.0000 −0.625166 −0.312583 0.949890i $$-0.601194\pi$$
−0.312583 + 0.949890i $$0.601194\pi$$
$$830$$ 0 0
$$831$$ −19.4164 −0.673548
$$832$$ 0 0
$$833$$ 7.40492i 0.256565i
$$834$$ 0 0
$$835$$ 38.9790i 1.34892i
$$836$$ 0 0
$$837$$ 18.3060i 0.632747i
$$838$$ 0 0
$$839$$ −49.3050 −1.70220 −0.851098 0.525007i $$-0.824063\pi$$
−0.851098 + 0.525007i $$0.824063\pi$$
$$840$$ 0 0
$$841$$ −9.00000 −0.310345
$$842$$ 0 0
$$843$$ 22.2148i 0.765117i
$$844$$ 0 0
$$845$$ −37.3607 −1.28525
$$846$$ 0 0
$$847$$ − 29.4621i − 1.01233i
$$848$$ 0 0
$$849$$ 2.11146 0.0724650
$$850$$ 0 0
$$851$$ 2.83282 0.0971077
$$852$$ 0 0
$$853$$ 27.4589i 0.940176i 0.882619 + 0.470088i $$0.155778\pi$$
−0.882619 + 0.470088i $$0.844222\pi$$
$$854$$ 0 0
$$855$$ 5.00000 0.170996
$$856$$ 0 0
$$857$$ 31.3190i 1.06984i 0.844903 + 0.534919i $$0.179658\pi$$
−0.844903 + 0.534919i $$0.820342\pi$$
$$858$$ 0 0
$$859$$ −40.0000 −1.36478 −0.682391 0.730987i $$-0.739060\pi$$
−0.682391 + 0.730987i $$0.739060\pi$$
$$860$$ 0 0
$$861$$ −14.8328 −0.505501
$$862$$ 0 0
$$863$$ − 7.86629i − 0.267772i −0.990997 0.133886i $$-0.957254\pi$$
0.990997 0.133886i $$-0.0427455\pi$$
$$864$$ 0 0
$$865$$ − 20.0053i − 0.680201i
$$866$$ 0 0
$$867$$ 33.0671i 1.12302i
$$868$$ 0 0
$$869$$ 11.7771 0.399510
$$870$$ 0 0
$$871$$ −60.5410 −2.05135
$$872$$ 0 0
$$873$$ 31.1614i 1.05465i
$$874$$ 0 0
$$875$$ 31.6228i 1.06904i
$$876$$ 0 0
$$877$$ 13.9358i 0.470579i 0.971925 + 0.235289i $$0.0756038\pi$$
−0.971925 + 0.235289i $$0.924396\pi$$
$$878$$ 0 0
$$879$$ −2.29180 −0.0773004
$$880$$ 0 0
$$881$$ 0.652476 0.0219825 0.0109912 0.999940i $$-0.496501\pi$$
0.0109912 + 0.999940i $$0.496501\pi$$
$$882$$ 0 0
$$883$$ 52.9148i 1.78072i 0.455253 + 0.890362i $$0.349549\pi$$
−0.455253 + 0.890362i $$0.650451\pi$$
$$884$$ 0 0
$$885$$ − 2.98605i − 0.100375i
$$886$$ 0 0
$$887$$ − 49.0547i − 1.64710i −0.567247 0.823548i $$-0.691991\pi$$
0.567247 0.823548i $$-0.308009\pi$$
$$888$$ 0 0
$$889$$ 55.4164 1.85861
$$890$$ 0 0
$$891$$ −2.06888 −0.0693102
$$892$$ 0 0
$$893$$ 8.48528i 0.283949i
$$894$$ 0 0
$$895$$ 50.2492 1.67965
$$896$$ 0 0
$$897$$ 5.14678i 0.171846i
$$898$$ 0 0
$$899$$ −17.8885 −0.596616
$$900$$ 0 0
$$901$$ −19.4164 −0.646854
$$902$$ 0 0
$$903$$ − 20.9768i − 0.698063i
$$904$$ 0 0
$$905$$ 13.4164 0.445976
$$906$$ 0 0
$$907$$ − 38.5663i − 1.28057i −0.768136 0.640287i $$-0.778815\pi$$
0.768136 0.640287i $$-0.221185\pi$$
$$908$$ 0 0
$$909$$ 11.7082 0.388337
$$910$$ 0 0
$$911$$ −46.4721 −1.53969 −0.769845 0.638231i $$-0.779667\pi$$
−0.769845 + 0.638231i $$0.779667\pi$$
$$912$$ 0 0
$$913$$ 10.4884i 0.347115i
$$914$$ 0 0
$$915$$ 22.8825i 0.756471i
$$916$$ 0 0
$$917$$ 0 0
$$918$$ 0 0
$$919$$ −26.8328 −0.885133 −0.442566 0.896736i $$-0.645932\pi$$
−0.442566 + 0.896736i $$0.645932\pi$$
$$920$$ 0 0
$$921$$ −4.76393 −0.156977
$$922$$ 0 0
$$923$$ − 57.0786i − 1.87877i
$$924$$ 0 0
$$925$$ − 13.1105i − 0.431070i
$$926$$ 0 0
$$927$$ 13.1105i 0.430605i
$$928$$ 0 0
$$929$$ −7.52786 −0.246981 −0.123491 0.992346i $$-0.539409\pi$$
−0.123491 + 0.992346i $$0.539409\pi$$
$$930$$ 0 0
$$931$$ −1.00000 −0.0327737
$$932$$ 0 0
$$933$$ 8.48528i 0.277796i
$$934$$ 0 0
$$935$$ 12.6491i 0.413670i
$$936$$ 0 0
$$937$$ − 43.1915i − 1.41101i −0.708707 0.705503i $$-0.750721\pi$$
0.708707 0.705503i $$-0.249279\pi$$
$$938$$ 0 0
$$939$$ 9.16718 0.299160
$$940$$ 0 0
$$941$$ −38.9443 −1.26955 −0.634773 0.772698i $$-0.718907\pi$$
−0.634773 + 0.772698i $$0.718907\pi$$
$$942$$ 0 0
$$943$$ − 6.48218i − 0.211089i
$$944$$ 0 0
$$945$$ 28.9443 0.941557
$$946$$ 0 0
$$947$$ − 47.6706i − 1.54909i −0.632521 0.774543i $$-0.717980\pi$$
0.632521 0.774543i $$-0.282020\pi$$
$$948$$ 0 0
$$949$$ 28.5836 0.927863
$$950$$ 0 0
$$951$$ −7.81966 −0.253570
$$952$$ 0 0
$$953$$ 36.5632i 1.18440i 0.805791 + 0.592199i $$0.201740\pi$$
−0.805791 + 0.592199i $$0.798260\pi$$
$$954$$ 0 0
$$955$$ −6.83282 −0.221105
$$956$$ 0 0
$$957$$ 2.98605i 0.0965253i
$$958$$ 0 0
$$959$$ 20.9443 0.676326
$$960$$ 0 0
$$961$$ −15.0000 −0.483871
$$962$$ 0 0
$$963$$ − 29.6684i − 0.956050i
$$964$$ 0 0
$$965$$ − 31.1614i − 1.00312i
$$966$$ 0 0
$$967$$ − 2.41577i − 0.0776858i −0.999245 0.0388429i $$-0.987633\pi$$
0.999245 0.0388429i $$-0.0123672\pi$$
$$968$$ 0 0
$$969$$ −6.47214 −0.207915
$$970$$ 0 0
$$971$$ −40.3607 −1.29524 −0.647618 0.761965i $$-0.724235\pi$$
−0.647618 + 0.761965i $$0.724235\pi$$
$$972$$ 0 0
$$973$$ − 6.48218i − 0.207809i
$$974$$ 0 0
$$975$$ 23.8197 0.762840
$$976$$ 0 0
$$977$$ 24.9945i 0.799644i 0.916593 + 0.399822i $$0.130928\pi$$
−0.916593 + 0.399822i $$0.869072\pi$$
$$978$$ 0 0
$$979$$ 2.24922 0.0718855
$$980$$ 0 0
$$981$$ −4.47214 −0.142784
$$982$$ 0 0
$$983$$ 30.2387i 0.964464i 0.876044 + 0.482232i $$0.160174\pi$$
−0.876044 + 0.482232i $$0.839826\pi$$
$$984$$ 0 0
$$985$$ 28.2843i 0.901212i
$$986$$ 0 0
$$987$$ 20.9768i 0.667698i
$$988$$ 0 0
$$989$$ 9.16718 0.291500
$$990$$ 0 0
$$991$$ 38.8328 1.23357 0.616783 0.787134i $$-0.288436\pi$$
0.616783 + 0.787134i $$0.288436\pi$$
$$992$$ 0 0
$$993$$ − 6.48218i − 0.205706i
$$994$$ 0 0
$$995$$ 35.7771 1.13421
$$996$$ 0 0
$$997$$ − 11.7264i − 0.371378i −0.982609 0.185689i $$-0.940548\pi$$
0.982609 0.185689i $$-0.0594517\pi$$
$$998$$ 0 0
$$999$$ −12.0000 −0.379663
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 380.2.c.a.229.2 4
3.2 odd 2 3420.2.f.a.1369.2 4
4.3 odd 2 1520.2.d.f.609.3 4
5.2 odd 4 1900.2.a.j.1.2 4
5.3 odd 4 1900.2.a.j.1.3 4
5.4 even 2 inner 380.2.c.a.229.3 yes 4
15.14 odd 2 3420.2.f.a.1369.1 4
20.3 even 4 7600.2.a.ce.1.2 4
20.7 even 4 7600.2.a.ce.1.3 4
20.19 odd 2 1520.2.d.f.609.2 4

By twisted newform
Twist Min Dim Char Parity Ord Type
380.2.c.a.229.2 4 1.1 even 1 trivial
380.2.c.a.229.3 yes 4 5.4 even 2 inner
1520.2.d.f.609.2 4 20.19 odd 2
1520.2.d.f.609.3 4 4.3 odd 2
1900.2.a.j.1.2 4 5.2 odd 4
1900.2.a.j.1.3 4 5.3 odd 4
3420.2.f.a.1369.1 4 15.14 odd 2
3420.2.f.a.1369.2 4 3.2 odd 2
7600.2.a.ce.1.2 4 20.3 even 4
7600.2.a.ce.1.3 4 20.7 even 4