# Properties

 Label 7056.2.b.x.1567.8 Level $7056$ Weight $2$ Character 7056.1567 Analytic conductor $56.342$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$7056 = 2^{4} \cdot 3^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 7056.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$56.3424436662$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.339738624.1 Defining polynomial: $$x^{8} - 4 x^{6} + 14 x^{4} - 8 x^{2} + 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{6}$$ Twist minimal: no (minimal twist has level 2352) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 1567.8 Root $$-0.662827 + 0.382683i$$ of defining polynomial Character $$\chi$$ $$=$$ 7056.1567 Dual form 7056.2.b.x.1567.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+4.29725i q^{5} +O(q^{10})$$ $$q+4.29725i q^{5} -5.06262i q^{11} -3.37849i q^{13} +2.76652i q^{17} -4.70319 q^{19} +4.83120i q^{23} -13.4663 q^{25} -2.46054 q^{29} +5.69764 q^{31} -2.33530 q^{37} -5.14822i q^{41} -13.0199i q^{43} +5.35449 q^{47} +4.22371 q^{53} +21.7553 q^{55} +9.60498 q^{59} +3.87698i q^{61} +14.5182 q^{65} -4.76342i q^{67} -12.3181i q^{71} -11.5102i q^{73} +13.9166i q^{79} +7.32191 q^{83} -11.8884 q^{85} -14.0448i q^{89} -20.2108i q^{95} +14.5716i q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + O(q^{10})$$ $$8q - 24q^{25} - 16q^{29} + 32q^{31} - 16q^{47} - 16q^{53} + 64q^{55} + 48q^{59} + 16q^{65} - 64q^{85} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/7056\mathbb{Z}\right)^\times$$.

 $$n$$ $$785$$ $$1765$$ $$4609$$ $$6175$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$ $$-1$$

## 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 0
$$4$$ 0 0
$$5$$ 4.29725i 1.92179i 0.276916 + 0.960894i $$0.410688\pi$$
−0.276916 + 0.960894i $$0.589312\pi$$
$$6$$ 0 0
$$7$$ 0 0
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ − 5.06262i − 1.52644i −0.646141 0.763218i $$-0.723618\pi$$
0.646141 0.763218i $$-0.276382\pi$$
$$12$$ 0 0
$$13$$ − 3.37849i − 0.937025i −0.883457 0.468513i $$-0.844790\pi$$
0.883457 0.468513i $$-0.155210\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 2.76652i 0.670978i 0.942044 + 0.335489i $$0.108902\pi$$
−0.942044 + 0.335489i $$0.891098\pi$$
$$18$$ 0 0
$$19$$ −4.70319 −1.07898 −0.539492 0.841990i $$-0.681384\pi$$
−0.539492 + 0.841990i $$0.681384\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 4.83120i 1.00737i 0.863886 + 0.503687i $$0.168024\pi$$
−0.863886 + 0.503687i $$0.831976\pi$$
$$24$$ 0 0
$$25$$ −13.4663 −2.69327
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ −2.46054 −0.456912 −0.228456 0.973554i $$-0.573368\pi$$
−0.228456 + 0.973554i $$0.573368\pi$$
$$30$$ 0 0
$$31$$ 5.69764 1.02333 0.511663 0.859186i $$-0.329030\pi$$
0.511663 + 0.859186i $$0.329030\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −2.33530 −0.383921 −0.191961 0.981403i $$-0.561485\pi$$
−0.191961 + 0.981403i $$0.561485\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ − 5.14822i − 0.804018i −0.915636 0.402009i $$-0.868312\pi$$
0.915636 0.402009i $$-0.131688\pi$$
$$42$$ 0 0
$$43$$ − 13.0199i − 1.98552i −0.120117 0.992760i $$-0.538327\pi$$
0.120117 0.992760i $$-0.461673\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 5.35449 0.781033 0.390517 0.920596i $$-0.372296\pi$$
0.390517 + 0.920596i $$0.372296\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ 4.22371 0.580171 0.290085 0.957001i $$-0.406316\pi$$
0.290085 + 0.957001i $$0.406316\pi$$
$$54$$ 0 0
$$55$$ 21.7553 2.93349
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ 9.60498 1.25046 0.625231 0.780440i $$-0.285005\pi$$
0.625231 + 0.780440i $$0.285005\pi$$
$$60$$ 0 0
$$61$$ 3.87698i 0.496397i 0.968709 + 0.248198i $$0.0798385\pi$$
−0.968709 + 0.248198i $$0.920162\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ 14.5182 1.80076
$$66$$ 0 0
$$67$$ − 4.76342i − 0.581944i −0.956732 0.290972i $$-0.906021\pi$$
0.956732 0.290972i $$-0.0939787\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ − 12.3181i − 1.46189i −0.682437 0.730944i $$-0.739080\pi$$
0.682437 0.730944i $$-0.260920\pi$$
$$72$$ 0 0
$$73$$ − 11.5102i − 1.34716i −0.739113 0.673581i $$-0.764755\pi$$
0.739113 0.673581i $$-0.235245\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 13.9166i 1.56574i 0.622185 + 0.782870i $$0.286245\pi$$
−0.622185 + 0.782870i $$0.713755\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ 7.32191 0.803685 0.401842 0.915709i $$-0.368370\pi$$
0.401842 + 0.915709i $$0.368370\pi$$
$$84$$ 0 0
$$85$$ −11.8884 −1.28948
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ − 14.0448i − 1.48874i −0.667765 0.744372i $$-0.732749\pi$$
0.667765 0.744372i $$-0.267251\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ − 20.2108i − 2.07358i
$$96$$ 0 0
$$97$$ 14.5716i 1.47952i 0.672868 + 0.739762i $$0.265062\pi$$
−0.672868 + 0.739762i $$0.734938\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 13.1087i 1.30436i 0.758063 + 0.652181i $$0.226146\pi$$
−0.758063 + 0.652181i $$0.773854\pi$$
$$102$$ 0 0
$$103$$ −0.209698 −0.0206622 −0.0103311 0.999947i $$-0.503289\pi$$
−0.0103311 + 0.999947i $$0.503289\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ − 6.42657i − 0.621280i −0.950528 0.310640i $$-0.899457\pi$$
0.950528 0.310640i $$-0.100543\pi$$
$$108$$ 0 0
$$109$$ 0.940024 0.0900379 0.0450190 0.998986i $$-0.485665\pi$$
0.0450190 + 0.998986i $$0.485665\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ −1.09821 −0.103311 −0.0516554 0.998665i $$-0.516450\pi$$
−0.0516554 + 0.998665i $$0.516450\pi$$
$$114$$ 0 0
$$115$$ −20.7609 −1.93596
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −14.6301 −1.33001
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ − 36.3820i − 3.25411i
$$126$$ 0 0
$$127$$ 5.22625i 0.463755i 0.972745 + 0.231877i $$0.0744868\pi$$
−0.972745 + 0.231877i $$0.925513\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 0.875015 0.0764504 0.0382252 0.999269i $$-0.487830\pi$$
0.0382252 + 0.999269i $$0.487830\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ −0.236838 −0.0202344 −0.0101172 0.999949i $$-0.503220\pi$$
−0.0101172 + 0.999949i $$0.503220\pi$$
$$138$$ 0 0
$$139$$ 3.00555 0.254927 0.127464 0.991843i $$-0.459316\pi$$
0.127464 + 0.991843i $$0.459316\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ −17.1040 −1.43031
$$144$$ 0 0
$$145$$ − 10.5736i − 0.878087i
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ 14.8017 1.21260 0.606299 0.795237i $$-0.292653\pi$$
0.606299 + 0.795237i $$0.292653\pi$$
$$150$$ 0 0
$$151$$ − 12.3654i − 1.00628i −0.864205 0.503140i $$-0.832178\pi$$
0.864205 0.503140i $$-0.167822\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 24.4842i 1.96662i
$$156$$ 0 0
$$157$$ − 10.8906i − 0.869164i −0.900632 0.434582i $$-0.856896\pi$$
0.900632 0.434582i $$-0.143104\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ 1.51024i 0.118291i 0.998249 + 0.0591455i $$0.0188376\pi$$
−0.998249 + 0.0591455i $$0.981162\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 10.3333 0.799612 0.399806 0.916600i $$-0.369078\pi$$
0.399806 + 0.916600i $$0.369078\pi$$
$$168$$ 0 0
$$169$$ 1.58579 0.121984
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ − 3.35642i − 0.255184i −0.991827 0.127592i $$-0.959275\pi$$
0.991827 0.127592i $$-0.0407248\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ − 13.2589i − 0.991018i −0.868603 0.495509i $$-0.834982\pi$$
0.868603 0.495509i $$-0.165018\pi$$
$$180$$ 0 0
$$181$$ − 11.8519i − 0.880946i −0.897766 0.440473i $$-0.854811\pi$$
0.897766 0.440473i $$-0.145189\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ − 10.0354i − 0.737816i
$$186$$ 0 0
$$187$$ 14.0058 1.02421
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 1.34660i 0.0974368i 0.998813 + 0.0487184i $$0.0155137\pi$$
−0.998813 + 0.0487184i $$0.984486\pi$$
$$192$$ 0 0
$$193$$ −26.1154 −1.87982 −0.939912 0.341416i $$-0.889094\pi$$
−0.939912 + 0.341416i $$0.889094\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 7.49083 0.533699 0.266850 0.963738i $$-0.414017\pi$$
0.266850 + 0.963738i $$0.414017\pi$$
$$198$$ 0 0
$$199$$ −4.49349 −0.318535 −0.159267 0.987235i $$-0.550913\pi$$
−0.159267 + 0.987235i $$0.550913\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 22.1232 1.54515
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ 23.8104i 1.64700i
$$210$$ 0 0
$$211$$ 5.93122i 0.408322i 0.978937 + 0.204161i $$0.0654466\pi$$
−0.978937 + 0.204161i $$0.934553\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ 55.9498 3.81575
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 9.34665 0.628724
$$222$$ 0 0
$$223$$ −20.6163 −1.38057 −0.690286 0.723537i $$-0.742515\pi$$
−0.690286 + 0.723537i $$0.742515\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 20.4911 1.36004 0.680021 0.733193i $$-0.261971\pi$$
0.680021 + 0.733193i $$0.261971\pi$$
$$228$$ 0 0
$$229$$ − 29.0409i − 1.91908i −0.281576 0.959539i $$-0.590857\pi$$
0.281576 0.959539i $$-0.409143\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −17.7364 −1.16195 −0.580975 0.813922i $$-0.697328\pi$$
−0.580975 + 0.813922i $$0.697328\pi$$
$$234$$ 0 0
$$235$$ 23.0096i 1.50098i
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 20.4248i 1.32117i 0.750750 + 0.660586i $$0.229692\pi$$
−0.750750 + 0.660586i $$0.770308\pi$$
$$240$$ 0 0
$$241$$ 3.25742i 0.209829i 0.994481 + 0.104915i $$0.0334569\pi$$
−0.994481 + 0.104915i $$0.966543\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 15.8897i 1.01104i
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 3.95633 0.249721 0.124861 0.992174i $$-0.460152\pi$$
0.124861 + 0.992174i $$0.460152\pi$$
$$252$$ 0 0
$$253$$ 24.4585 1.53769
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 5.14822i 0.321137i 0.987025 + 0.160569i $$0.0513328\pi$$
−0.987025 + 0.160569i $$0.948667\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ − 0.0785021i − 0.00484065i −0.999997 0.00242032i $$-0.999230\pi$$
0.999997 0.00242032i $$-0.000770414\pi$$
$$264$$ 0 0
$$265$$ 18.1503i 1.11497i
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ 0.883296i 0.0538555i 0.999637 + 0.0269277i $$0.00857240\pi$$
−0.999637 + 0.0269277i $$0.991428\pi$$
$$270$$ 0 0
$$271$$ 19.3084 1.17290 0.586451 0.809984i $$-0.300524\pi$$
0.586451 + 0.809984i $$0.300524\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 68.1749i 4.11110i
$$276$$ 0 0
$$277$$ 16.9327 1.01739 0.508694 0.860948i $$-0.330129\pi$$
0.508694 + 0.860948i $$0.330129\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −5.05417 −0.301507 −0.150753 0.988571i $$-0.548170\pi$$
−0.150753 + 0.988571i $$0.548170\pi$$
$$282$$ 0 0
$$283$$ 19.1156 1.13631 0.568153 0.822923i $$-0.307658\pi$$
0.568153 + 0.822923i $$0.307658\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ 9.34639 0.549788
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ − 2.37837i − 0.138946i −0.997584 0.0694730i $$-0.977868\pi$$
0.997584 0.0694730i $$-0.0221318\pi$$
$$294$$ 0 0
$$295$$ 41.2750i 2.40312i
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 16.3222 0.943936
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ −16.6604 −0.953969
$$306$$ 0 0
$$307$$ −4.89599 −0.279429 −0.139714 0.990192i $$-0.544618\pi$$
−0.139714 + 0.990192i $$0.544618\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 10.4800 0.594266 0.297133 0.954836i $$-0.403970\pi$$
0.297133 + 0.954836i $$0.403970\pi$$
$$312$$ 0 0
$$313$$ − 9.60172i − 0.542722i −0.962478 0.271361i $$-0.912526\pi$$
0.962478 0.271361i $$-0.0874737\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 16.7934 0.943214 0.471607 0.881809i $$-0.343674\pi$$
0.471607 + 0.881809i $$0.343674\pi$$
$$318$$ 0 0
$$319$$ 12.4568i 0.697446i
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ − 13.0114i − 0.723976i
$$324$$ 0 0
$$325$$ 45.4960i 2.52366i
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ − 26.8362i − 1.47505i −0.675318 0.737526i $$-0.735994\pi$$
0.675318 0.737526i $$-0.264006\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ 20.4696 1.11837
$$336$$ 0 0
$$337$$ 23.0827 1.25739 0.628697 0.777651i $$-0.283589\pi$$
0.628697 + 0.777651i $$0.283589\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ − 28.8450i − 1.56204i
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ − 1.30690i − 0.0701580i −0.999385 0.0350790i $$-0.988832\pi$$
0.999385 0.0350790i $$-0.0111683\pi$$
$$348$$ 0 0
$$349$$ − 26.5489i − 1.42113i −0.703633 0.710564i $$-0.748440\pi$$
0.703633 0.710564i $$-0.251560\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ 16.8893i 0.898928i 0.893298 + 0.449464i $$0.148385\pi$$
−0.893298 + 0.449464i $$0.851615\pi$$
$$354$$ 0 0
$$355$$ 52.9339 2.80944
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ − 33.5901i − 1.77282i −0.462904 0.886408i $$-0.653193\pi$$
0.462904 0.886408i $$-0.346807\pi$$
$$360$$ 0 0
$$361$$ 3.11995 0.164208
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 49.4620 2.58896
$$366$$ 0 0
$$367$$ 7.35978 0.384178 0.192089 0.981378i $$-0.438474\pi$$
0.192089 + 0.981378i $$0.438474\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ 3.63008 0.187958 0.0939791 0.995574i $$-0.470041\pi$$
0.0939791 + 0.995574i $$0.470041\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 8.31293i 0.428138i
$$378$$ 0 0
$$379$$ 13.6647i 0.701908i 0.936393 + 0.350954i $$0.114143\pi$$
−0.936393 + 0.350954i $$0.885857\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ −13.6569 −0.697833 −0.348916 0.937154i $$-0.613450\pi$$
−0.348916 + 0.937154i $$0.613450\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ −26.9080 −1.36429 −0.682144 0.731218i $$-0.738952\pi$$
−0.682144 + 0.731218i $$0.738952\pi$$
$$390$$ 0 0
$$391$$ −13.3656 −0.675927
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ −59.8031 −3.00902
$$396$$ 0 0
$$397$$ 36.7242i 1.84314i 0.388216 + 0.921568i $$0.373091\pi$$
−0.388216 + 0.921568i $$0.626909\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 27.6816 1.38235 0.691176 0.722686i $$-0.257093\pi$$
0.691176 + 0.722686i $$0.257093\pi$$
$$402$$ 0 0
$$403$$ − 19.2494i − 0.958883i
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 11.8227i 0.586032i
$$408$$ 0 0
$$409$$ 8.42820i 0.416747i 0.978049 + 0.208374i $$0.0668170\pi$$
−0.978049 + 0.208374i $$0.933183\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 31.4641i 1.54451i
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ −18.3029 −0.894154 −0.447077 0.894495i $$-0.647535\pi$$
−0.447077 + 0.894495i $$0.647535\pi$$
$$420$$ 0 0
$$421$$ 22.6274 1.10279 0.551396 0.834243i $$-0.314095\pi$$
0.551396 + 0.834243i $$0.314095\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ − 37.2549i − 1.80713i
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 4.44906i 0.214304i 0.994243 + 0.107152i $$0.0341731\pi$$
−0.994243 + 0.107152i $$0.965827\pi$$
$$432$$ 0 0
$$433$$ 1.66205i 0.0798730i 0.999202 + 0.0399365i $$0.0127156\pi$$
−0.999202 + 0.0399365i $$0.987284\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ − 22.7220i − 1.08694i
$$438$$ 0 0
$$439$$ −18.7551 −0.895130 −0.447565 0.894251i $$-0.647709\pi$$
−0.447565 + 0.894251i $$0.647709\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 33.0314i 1.56937i 0.619896 + 0.784684i $$0.287175\pi$$
−0.619896 + 0.784684i $$0.712825\pi$$
$$444$$ 0 0
$$445$$ 60.3539 2.86105
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ −9.29441 −0.438630 −0.219315 0.975654i $$-0.570382\pi$$
−0.219315 + 0.975654i $$0.570382\pi$$
$$450$$ 0 0
$$451$$ −26.0635 −1.22728
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −27.4448 −1.28381 −0.641906 0.766784i $$-0.721856\pi$$
−0.641906 + 0.766784i $$0.721856\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ − 4.28209i − 0.199437i −0.995016 0.0997183i $$-0.968206\pi$$
0.995016 0.0997183i $$-0.0317942\pi$$
$$462$$ 0 0
$$463$$ − 16.1278i − 0.749521i −0.927122 0.374760i $$-0.877725\pi$$
0.927122 0.374760i $$-0.122275\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 27.5403 1.27441 0.637207 0.770692i $$-0.280090\pi$$
0.637207 + 0.770692i $$0.280090\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ −65.9149 −3.03077
$$474$$ 0 0
$$475$$ 63.3347 2.90600
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ −19.5509 −0.893304 −0.446652 0.894708i $$-0.647384\pi$$
−0.446652 + 0.894708i $$0.647384\pi$$
$$480$$ 0 0
$$481$$ 7.88980i 0.359744i
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ −62.6179 −2.84333
$$486$$ 0 0
$$487$$ − 37.7917i − 1.71250i −0.516558 0.856252i $$-0.672787\pi$$
0.516558 0.856252i $$-0.327213\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 29.2605i 1.32051i 0.751042 + 0.660254i $$0.229551\pi$$
−0.751042 + 0.660254i $$0.770449\pi$$
$$492$$ 0 0
$$493$$ − 6.80713i − 0.306578i
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ − 12.6422i − 0.565944i −0.959128 0.282972i $$-0.908680\pi$$
0.959128 0.282972i $$-0.0913203\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ 27.0714 1.20706 0.603528 0.797342i $$-0.293761\pi$$
0.603528 + 0.797342i $$0.293761\pi$$
$$504$$ 0 0
$$505$$ −56.3312 −2.50671
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ − 11.6983i − 0.518520i −0.965808 0.259260i $$-0.916521\pi$$
0.965808 0.259260i $$-0.0834786\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ − 0.901124i − 0.0397083i
$$516$$ 0 0
$$517$$ − 27.1077i − 1.19220i
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 7.20643i 0.315719i 0.987462 + 0.157860i $$0.0504594\pi$$
−0.987462 + 0.157860i $$0.949541\pi$$
$$522$$ 0 0
$$523$$ 17.7145 0.774602 0.387301 0.921953i $$-0.373407\pi$$
0.387301 + 0.921953i $$0.373407\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 15.7626i 0.686630i
$$528$$ 0 0
$$529$$ −0.340485 −0.0148037
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ −17.3932 −0.753385
$$534$$ 0 0
$$535$$ 27.6166 1.19397
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ 23.1180 0.993921 0.496961 0.867773i $$-0.334449\pi$$
0.496961 + 0.867773i $$0.334449\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ 4.03951i 0.173034i
$$546$$ 0 0
$$547$$ 16.0524i 0.686351i 0.939271 + 0.343176i $$0.111503\pi$$
−0.939271 + 0.343176i $$0.888497\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 11.5724 0.493001
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 12.5808 0.533067 0.266533 0.963826i $$-0.414122\pi$$
0.266533 + 0.963826i $$0.414122\pi$$
$$558$$ 0 0
$$559$$ −43.9877 −1.86048
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 36.5948 1.54229 0.771144 0.636660i $$-0.219685\pi$$
0.771144 + 0.636660i $$0.219685\pi$$
$$564$$ 0 0
$$565$$ − 4.71927i − 0.198541i
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 24.4989 1.02705 0.513524 0.858075i $$-0.328340\pi$$
0.513524 + 0.858075i $$0.328340\pi$$
$$570$$ 0 0
$$571$$ − 32.7924i − 1.37232i −0.727451 0.686159i $$-0.759295\pi$$
0.727451 0.686159i $$-0.240705\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ − 65.0586i − 2.71313i
$$576$$ 0 0
$$577$$ − 19.2140i − 0.799888i −0.916540 0.399944i $$-0.869030\pi$$
0.916540 0.399944i $$-0.130970\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ − 21.3830i − 0.885594i
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 24.1451 0.996573 0.498286 0.867012i $$-0.333963\pi$$
0.498286 + 0.867012i $$0.333963\pi$$
$$588$$ 0 0
$$589$$ −26.7971 −1.10415
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ − 2.54582i − 0.104544i −0.998633 0.0522722i $$-0.983354\pi$$
0.998633 0.0522722i $$-0.0166463\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ 14.9899i 0.612469i 0.951956 + 0.306234i $$0.0990691\pi$$
−0.951956 + 0.306234i $$0.900931\pi$$
$$600$$ 0 0
$$601$$ − 24.3343i − 0.992618i −0.868146 0.496309i $$-0.834688\pi$$
0.868146 0.496309i $$-0.165312\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ − 62.8691i − 2.55599i
$$606$$ 0 0
$$607$$ −8.91288 −0.361763 −0.180881 0.983505i $$-0.557895\pi$$
−0.180881 + 0.983505i $$0.557895\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ − 18.0901i − 0.731848i
$$612$$ 0 0
$$613$$ −17.1748 −0.693685 −0.346842 0.937923i $$-0.612746\pi$$
−0.346842 + 0.937923i $$0.612746\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −35.9980 −1.44922 −0.724612 0.689157i $$-0.757981\pi$$
−0.724612 + 0.689157i $$0.757981\pi$$
$$618$$ 0 0
$$619$$ 4.11123 0.165244 0.0826222 0.996581i $$-0.473671\pi$$
0.0826222 + 0.996581i $$0.473671\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 89.0108 3.56043
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ − 6.46065i − 0.257603i
$$630$$ 0 0
$$631$$ − 34.8970i − 1.38923i −0.719383 0.694613i $$-0.755575\pi$$
0.719383 0.694613i $$-0.244425\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ −22.4585 −0.891239
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ −7.32735 −0.289413 −0.144707 0.989475i $$-0.546224\pi$$
−0.144707 + 0.989475i $$0.546224\pi$$
$$642$$ 0 0
$$643$$ −9.24275 −0.364498 −0.182249 0.983252i $$-0.558338\pi$$
−0.182249 + 0.983252i $$0.558338\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ −15.1803 −0.596798 −0.298399 0.954441i $$-0.596453\pi$$
−0.298399 + 0.954441i $$0.596453\pi$$
$$648$$ 0 0
$$649$$ − 48.6263i − 1.90875i
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 2.12849 0.0832943 0.0416471 0.999132i $$-0.486739\pi$$
0.0416471 + 0.999132i $$0.486739\pi$$
$$654$$ 0 0
$$655$$ 3.76016i 0.146922i
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ − 13.0520i − 0.508436i −0.967147 0.254218i $$-0.918182\pi$$
0.967147 0.254218i $$-0.0818180\pi$$
$$660$$ 0 0
$$661$$ − 19.2786i − 0.749851i −0.927055 0.374926i $$-0.877668\pi$$
0.927055 0.374926i $$-0.122332\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ − 11.8874i − 0.460281i
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 19.6277 0.757718
$$672$$ 0 0
$$673$$ −7.08216 −0.272997 −0.136499 0.990640i $$-0.543585\pi$$
−0.136499 + 0.990640i $$0.543585\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 36.5042i 1.40297i 0.712685 + 0.701485i $$0.247479\pi$$
−0.712685 + 0.701485i $$0.752521\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ − 30.1206i − 1.15253i −0.817261 0.576267i $$-0.804509\pi$$
0.817261 0.576267i $$-0.195491\pi$$
$$684$$ 0 0
$$685$$ − 1.01775i − 0.0388863i
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ − 14.2698i − 0.543635i
$$690$$ 0 0
$$691$$ −39.8113 −1.51449 −0.757247 0.653129i $$-0.773456\pi$$
−0.757247 + 0.653129i $$0.773456\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 12.9156i 0.489916i
$$696$$ 0 0
$$697$$ 14.2426 0.539478
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −28.0795 −1.06055 −0.530275 0.847826i $$-0.677911\pi$$
−0.530275 + 0.847826i $$0.677911\pi$$
$$702$$ 0 0
$$703$$ 10.9834 0.414245
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ −36.9475 −1.38759 −0.693797 0.720170i $$-0.744064\pi$$
−0.693797 + 0.720170i $$0.744064\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ 27.5264i 1.03087i
$$714$$ 0 0
$$715$$ − 73.5002i − 2.74875i
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ 18.7090 0.697728 0.348864 0.937173i $$-0.386568\pi$$
0.348864 + 0.937173i $$0.386568\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ 33.1345 1.23059
$$726$$ 0 0
$$727$$ 31.2078 1.15743 0.578716 0.815529i $$-0.303554\pi$$
0.578716 + 0.815529i $$0.303554\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 36.0198 1.33224
$$732$$ 0 0
$$733$$ 4.06897i 0.150291i 0.997173 + 0.0751455i $$0.0239421\pi$$
−0.997173 + 0.0751455i $$0.976058\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ −24.1154 −0.888301
$$738$$ 0 0
$$739$$ − 39.9025i − 1.46784i −0.679237 0.733919i $$-0.737689\pi$$
0.679237 0.733919i $$-0.262311\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 41.7530i 1.53177i 0.642979 + 0.765884i $$0.277698\pi$$
−0.642979 + 0.765884i $$0.722302\pi$$
$$744$$ 0 0
$$745$$ 63.6064i 2.33036i
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 9.09927i 0.332037i 0.986123 + 0.166019i $$0.0530911\pi$$
−0.986123 + 0.166019i $$0.946909\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 53.1371 1.93386
$$756$$ 0 0
$$757$$ −0.902155 −0.0327894 −0.0163947 0.999866i $$-0.505219\pi$$
−0.0163947 + 0.999866i $$0.505219\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 23.7781i 0.861955i 0.902363 + 0.430978i $$0.141831\pi$$
−0.902363 + 0.430978i $$0.858169\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ − 32.4503i − 1.17171i
$$768$$ 0 0
$$769$$ − 16.3028i − 0.587895i −0.955822 0.293948i $$-0.905031\pi$$
0.955822 0.293948i $$-0.0949691\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ 25.0881i 0.902358i 0.892434 + 0.451179i $$0.148996\pi$$
−0.892434 + 0.451179i $$0.851004\pi$$
$$774$$ 0 0
$$775$$ −76.7264 −2.75609
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 24.2131i 0.867523i
$$780$$ 0 0
$$781$$ −62.3618 −2.23148
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 46.7996 1.67035
$$786$$ 0 0
$$787$$ 4.47130 0.159385 0.0796924 0.996820i $$-0.474606\pi$$
0.0796924 + 0.996820i $$0.474606\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ 13.0984 0.465136
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 6.62320i 0.234606i 0.993096 + 0.117303i $$0.0374248\pi$$
−0.993096 + 0.117303i $$0.962575\pi$$
$$798$$ 0 0
$$799$$ 14.8133i 0.524056i
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ −58.2715 −2.05636
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ 8.37044 0.294289 0.147144 0.989115i $$-0.452992\pi$$
0.147144 + 0.989115i $$0.452992\pi$$
$$810$$ 0 0
$$811$$ 1.20944 0.0424692 0.0212346 0.999775i $$-0.493240\pi$$
0.0212346 + 0.999775i $$0.493240\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ −6.48988 −0.227330
$$816$$ 0 0
$$817$$ 61.2351i 2.14235i
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −51.2519 −1.78871 −0.894353 0.447363i $$-0.852363\pi$$
−0.894353 + 0.447363i $$0.852363\pi$$
$$822$$ 0 0
$$823$$ 31.5546i 1.09992i 0.835189 + 0.549962i $$0.185358\pi$$
−0.835189 + 0.549962i $$0.814642\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 23.4800i 0.816480i 0.912875 + 0.408240i $$0.133857\pi$$
−0.912875 + 0.408240i $$0.866143\pi$$
$$828$$ 0 0
$$829$$ − 17.2639i − 0.599599i −0.954002 0.299800i $$-0.903080\pi$$
0.954002 0.299800i $$-0.0969199\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ 0 0
$$834$$ 0 0
$$835$$ 44.4046i 1.53668i
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ −37.4781 −1.29389 −0.646943 0.762538i $$-0.723953\pi$$
−0.646943 + 0.762538i $$0.723953\pi$$
$$840$$ 0 0
$$841$$ −22.9457 −0.791232
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ 6.81452i 0.234427i
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ − 11.2823i − 0.386753i
$$852$$ 0 0
$$853$$ − 9.50115i − 0.325313i −0.986683 0.162657i $$-0.947994\pi$$
0.986683 0.162657i $$-0.0520063\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 32.7797i 1.11973i 0.828583 + 0.559866i $$0.189147\pi$$
−0.828583 + 0.559866i $$0.810853\pi$$
$$858$$ 0 0
$$859$$ −5.37306 −0.183327 −0.0916633 0.995790i $$-0.529218\pi$$
−0.0916633 + 0.995790i $$0.529218\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ − 29.6824i − 1.01040i −0.863003 0.505200i $$-0.831419\pi$$
0.863003 0.505200i $$-0.168581\pi$$
$$864$$ 0 0
$$865$$ 14.4234 0.490409
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ 70.4544 2.39000
$$870$$ 0 0
$$871$$ −16.0932 −0.545296
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −30.3242 −1.02398 −0.511988 0.858993i $$-0.671091\pi$$
−0.511988 + 0.858993i $$0.671091\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 16.5012i 0.555939i 0.960590 + 0.277969i $$0.0896614\pi$$
−0.960590 + 0.277969i $$0.910339\pi$$
$$882$$ 0 0
$$883$$ − 22.5909i − 0.760245i −0.924936 0.380122i $$-0.875882\pi$$
0.924936 0.380122i $$-0.124118\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ −38.6036 −1.29618 −0.648090 0.761563i $$-0.724432\pi$$
−0.648090 + 0.761563i $$0.724432\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ −25.1832 −0.842723
$$894$$ 0 0
$$895$$ 56.9769 1.90453
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ −14.0193 −0.467570
$$900$$ 0 0
$$901$$ 11.6849i 0.389282i
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 50.9307 1.69299
$$906$$ 0 0
$$907$$ − 36.6743i − 1.21775i −0.793266 0.608875i $$-0.791621\pi$$
0.793266 0.608875i $$-0.208379\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ − 52.6835i − 1.74548i −0.488185 0.872740i $$-0.662341\pi$$
0.488185 0.872740i $$-0.337659\pi$$
$$912$$ 0 0
$$913$$ − 37.0680i − 1.22677i
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 0 0
$$918$$ 0 0
$$919$$ − 5.11618i − 0.168767i −0.996433 0.0843836i $$-0.973108\pi$$
0.996433 0.0843836i $$-0.0268921\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ −41.6166 −1.36983
$$924$$ 0 0
$$925$$ 31.4480 1.03400
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ 35.3694i 1.16043i 0.814462 + 0.580217i $$0.197032\pi$$
−0.814462 + 0.580217i $$0.802968\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ 60.1864i 1.96831i
$$936$$ 0 0
$$937$$ 23.6290i 0.771924i 0.922515 + 0.385962i $$0.126130\pi$$
−0.922515 + 0.385962i $$0.873870\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ − 19.2187i − 0.626510i −0.949669 0.313255i $$-0.898581\pi$$
0.949669 0.313255i $$-0.101419\pi$$
$$942$$ 0 0
$$943$$ 24.8721 0.809947
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ − 27.8159i − 0.903894i −0.892045 0.451947i $$-0.850730\pi$$
0.892045 0.451947i $$-0.149270\pi$$
$$948$$ 0 0
$$949$$ −38.8870 −1.26232
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ −56.8087 −1.84021 −0.920107 0.391668i $$-0.871898\pi$$
−0.920107 + 0.391668i $$0.871898\pi$$
$$954$$ 0 0
$$955$$ −5.78669 −0.187253
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ 0 0
$$960$$ 0 0
$$961$$ 1.46310 0.0471967
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ − 112.224i − 3.61262i
$$966$$ 0 0
$$967$$ − 20.3860i − 0.655570i −0.944752 0.327785i $$-0.893698\pi$$
0.944752 0.327785i $$-0.106302\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 11.2287 0.360347 0.180174 0.983635i $$-0.442334\pi$$
0.180174 + 0.983635i $$0.442334\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 12.1833 0.389778 0.194889 0.980825i $$-0.437565\pi$$
0.194889 + 0.980825i $$0.437565\pi$$
$$978$$ 0 0
$$979$$ −71.1033 −2.27247
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0 0
$$983$$ −9.06374 −0.289088 −0.144544 0.989498i $$-0.546172\pi$$
−0.144544 + 0.989498i $$0.546172\pi$$
$$984$$ 0 0
$$985$$ 32.1899i 1.02566i
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 62.9018 2.00016
$$990$$ 0 0
$$991$$ 5.37158i 0.170634i 0.996354 + 0.0853170i $$0.0271903\pi$$
−0.996354 + 0.0853170i $$0.972810\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ − 19.3096i − 0.612157i
$$996$$ 0 0
$$997$$ 14.3440i 0.454278i 0.973862 + 0.227139i $$0.0729372\pi$$
−0.973862 + 0.227139i $$0.927063\pi$$
$$998$$ 0 0
$$999$$ 0 0
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 7056.2.b.x.1567.8 8
3.2 odd 2 2352.2.b.k.1567.1 8
4.3 odd 2 7056.2.b.w.1567.8 8
7.6 odd 2 7056.2.b.w.1567.1 8
12.11 even 2 2352.2.b.l.1567.1 yes 8
21.2 odd 6 2352.2.bl.r.31.1 8
21.5 even 6 2352.2.bl.o.31.4 8
21.11 odd 6 2352.2.bl.t.607.4 8
21.17 even 6 2352.2.bl.q.607.1 8
21.20 even 2 2352.2.b.l.1567.8 yes 8
28.27 even 2 inner 7056.2.b.x.1567.1 8
84.11 even 6 2352.2.bl.o.607.4 8
84.23 even 6 2352.2.bl.q.31.1 8
84.47 odd 6 2352.2.bl.t.31.4 8
84.59 odd 6 2352.2.bl.r.607.1 8
84.83 odd 2 2352.2.b.k.1567.8 yes 8

By twisted newform
Twist Min Dim Char Parity Ord Type
2352.2.b.k.1567.1 8 3.2 odd 2
2352.2.b.k.1567.8 yes 8 84.83 odd 2
2352.2.b.l.1567.1 yes 8 12.11 even 2
2352.2.b.l.1567.8 yes 8 21.20 even 2
2352.2.bl.o.31.4 8 21.5 even 6
2352.2.bl.o.607.4 8 84.11 even 6
2352.2.bl.q.31.1 8 84.23 even 6
2352.2.bl.q.607.1 8 21.17 even 6
2352.2.bl.r.31.1 8 21.2 odd 6
2352.2.bl.r.607.1 8 84.59 odd 6
2352.2.bl.t.31.4 8 84.47 odd 6
2352.2.bl.t.607.4 8 21.11 odd 6
7056.2.b.w.1567.1 8 7.6 odd 2
7056.2.b.w.1567.8 8 4.3 odd 2
7056.2.b.x.1567.1 8 28.27 even 2 inner
7056.2.b.x.1567.8 8 1.1 even 1 trivial