# Properties

 Label 1400.2.g.j.449.2 Level $1400$ Weight $2$ Character 1400.449 Analytic conductor $11.179$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1400,2,Mod(449,1400)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1400, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1400.449");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1400 = 2^{3} \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1400.g (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: no Analytic conductor: $$11.1790562830$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{17})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 9x^{2} + 16$$ x^4 + 9*x^2 + 16 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 449.2 Root $$-1.56155i$$ of defining polynomial Character $$\chi$$ $$=$$ 1400.449 Dual form 1400.2.g.j.449.3

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.56155i q^{3} -1.00000i q^{7} +0.561553 q^{9} +O(q^{10})$$ $$q-1.56155i q^{3} -1.00000i q^{7} +0.561553 q^{9} -6.12311 q^{11} +2.00000i q^{13} -1.56155i q^{17} -3.56155 q^{19} -1.56155 q^{21} +1.43845i q^{23} -5.56155i q^{27} -3.43845 q^{29} -9.12311 q^{31} +9.56155i q^{33} -8.80776i q^{37} +3.12311 q^{39} -2.43845 q^{41} +6.56155i q^{43} +8.24621i q^{47} -1.00000 q^{49} -2.43845 q^{51} -1.12311i q^{53} +5.56155i q^{57} -11.3693 q^{59} +11.1231 q^{61} -0.561553i q^{63} -7.87689i q^{67} +2.24621 q^{69} +1.68466 q^{71} +6.43845i q^{73} +6.12311i q^{77} +5.68466 q^{79} -7.00000 q^{81} -1.31534i q^{83} +5.36932i q^{87} -9.80776 q^{89} +2.00000 q^{91} +14.2462i q^{93} -6.00000i q^{97} -3.43845 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 6 q^{9}+O(q^{10})$$ 4 * q - 6 * q^9 $$4 q - 6 q^{9} - 8 q^{11} - 6 q^{19} + 2 q^{21} - 22 q^{29} - 20 q^{31} - 4 q^{39} - 18 q^{41} - 4 q^{49} - 18 q^{51} + 4 q^{59} + 28 q^{61} - 24 q^{69} - 18 q^{71} - 2 q^{79} - 28 q^{81} + 2 q^{89} + 8 q^{91} - 22 q^{99}+O(q^{100})$$ 4 * q - 6 * q^9 - 8 * q^11 - 6 * q^19 + 2 * q^21 - 22 * q^29 - 20 * q^31 - 4 * q^39 - 18 * q^41 - 4 * q^49 - 18 * q^51 + 4 * q^59 + 28 * q^61 - 24 * q^69 - 18 * q^71 - 2 * q^79 - 28 * q^81 + 2 * q^89 + 8 * q^91 - 22 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1400\mathbb{Z}\right)^\times$$.

 $$n$$ $$351$$ $$701$$ $$801$$ $$1177$$ $$\chi(n)$$ $$1$$ $$1$$ $$1$$ $$-1$$

## 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
$$3$$ − 1.56155i − 0.901563i −0.892634 0.450781i $$-0.851145\pi$$
0.892634 0.450781i $$-0.148855\pi$$
$$4$$ 0 0
$$5$$ 0 0
$$6$$ 0 0
$$7$$ − 1.00000i − 0.377964i
$$8$$ 0 0
$$9$$ 0.561553 0.187184
$$10$$ 0 0
$$11$$ −6.12311 −1.84619 −0.923093 0.384577i $$-0.874347\pi$$
−0.923093 + 0.384577i $$0.874347\pi$$
$$12$$ 0 0
$$13$$ 2.00000i 0.554700i 0.960769 + 0.277350i $$0.0894562\pi$$
−0.960769 + 0.277350i $$0.910544\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ − 1.56155i − 0.378732i −0.981907 0.189366i $$-0.939357\pi$$
0.981907 0.189366i $$-0.0606433\pi$$
$$18$$ 0 0
$$19$$ −3.56155 −0.817076 −0.408538 0.912741i $$-0.633961\pi$$
−0.408538 + 0.912741i $$0.633961\pi$$
$$20$$ 0 0
$$21$$ −1.56155 −0.340759
$$22$$ 0 0
$$23$$ 1.43845i 0.299937i 0.988691 + 0.149968i $$0.0479172\pi$$
−0.988691 + 0.149968i $$0.952083\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ − 5.56155i − 1.07032i
$$28$$ 0 0
$$29$$ −3.43845 −0.638504 −0.319252 0.947670i $$-0.603432\pi$$
−0.319252 + 0.947670i $$0.603432\pi$$
$$30$$ 0 0
$$31$$ −9.12311 −1.63856 −0.819279 0.573395i $$-0.805626\pi$$
−0.819279 + 0.573395i $$0.805626\pi$$
$$32$$ 0 0
$$33$$ 9.56155i 1.66445i
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ − 8.80776i − 1.44799i −0.689807 0.723994i $$-0.742304\pi$$
0.689807 0.723994i $$-0.257696\pi$$
$$38$$ 0 0
$$39$$ 3.12311 0.500097
$$40$$ 0 0
$$41$$ −2.43845 −0.380821 −0.190411 0.981705i $$-0.560982\pi$$
−0.190411 + 0.981705i $$0.560982\pi$$
$$42$$ 0 0
$$43$$ 6.56155i 1.00063i 0.865844 + 0.500314i $$0.166782\pi$$
−0.865844 + 0.500314i $$0.833218\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 8.24621i 1.20283i 0.798935 + 0.601417i $$0.205397\pi$$
−0.798935 + 0.601417i $$0.794603\pi$$
$$48$$ 0 0
$$49$$ −1.00000 −0.142857
$$50$$ 0 0
$$51$$ −2.43845 −0.341451
$$52$$ 0 0
$$53$$ − 1.12311i − 0.154270i −0.997021 0.0771352i $$-0.975423\pi$$
0.997021 0.0771352i $$-0.0245773\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 5.56155i 0.736646i
$$58$$ 0 0
$$59$$ −11.3693 −1.48016 −0.740079 0.672519i $$-0.765212\pi$$
−0.740079 + 0.672519i $$0.765212\pi$$
$$60$$ 0 0
$$61$$ 11.1231 1.42417 0.712084 0.702094i $$-0.247752\pi$$
0.712084 + 0.702094i $$0.247752\pi$$
$$62$$ 0 0
$$63$$ − 0.561553i − 0.0707490i
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ − 7.87689i − 0.962316i −0.876634 0.481158i $$-0.840216\pi$$
0.876634 0.481158i $$-0.159784\pi$$
$$68$$ 0 0
$$69$$ 2.24621 0.270412
$$70$$ 0 0
$$71$$ 1.68466 0.199932 0.0999661 0.994991i $$-0.468127\pi$$
0.0999661 + 0.994991i $$0.468127\pi$$
$$72$$ 0 0
$$73$$ 6.43845i 0.753563i 0.926302 + 0.376782i $$0.122969\pi$$
−0.926302 + 0.376782i $$0.877031\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 6.12311i 0.697793i
$$78$$ 0 0
$$79$$ 5.68466 0.639574 0.319787 0.947489i $$-0.396389\pi$$
0.319787 + 0.947489i $$0.396389\pi$$
$$80$$ 0 0
$$81$$ −7.00000 −0.777778
$$82$$ 0 0
$$83$$ − 1.31534i − 0.144377i −0.997391 0.0721887i $$-0.977002\pi$$
0.997391 0.0721887i $$-0.0229984\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 5.36932i 0.575651i
$$88$$ 0 0
$$89$$ −9.80776 −1.03962 −0.519810 0.854282i $$-0.673997\pi$$
−0.519810 + 0.854282i $$0.673997\pi$$
$$90$$ 0 0
$$91$$ 2.00000 0.209657
$$92$$ 0 0
$$93$$ 14.2462i 1.47726i
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ − 6.00000i − 0.609208i −0.952479 0.304604i $$-0.901476\pi$$
0.952479 0.304604i $$-0.0985241\pi$$
$$98$$ 0 0
$$99$$ −3.43845 −0.345577
$$100$$ 0 0
$$101$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$102$$ 0 0
$$103$$ − 1.12311i − 0.110663i −0.998468 0.0553314i $$-0.982378\pi$$
0.998468 0.0553314i $$-0.0176215\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ − 7.80776i − 0.754805i −0.926049 0.377403i $$-0.876817\pi$$
0.926049 0.377403i $$-0.123183\pi$$
$$108$$ 0 0
$$109$$ −19.9309 −1.90903 −0.954516 0.298161i $$-0.903627\pi$$
−0.954516 + 0.298161i $$0.903627\pi$$
$$110$$ 0 0
$$111$$ −13.7538 −1.30545
$$112$$ 0 0
$$113$$ 17.2462i 1.62239i 0.584778 + 0.811194i $$0.301182\pi$$
−0.584778 + 0.811194i $$0.698818\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ 1.12311i 0.103831i
$$118$$ 0 0
$$119$$ −1.56155 −0.143147
$$120$$ 0 0
$$121$$ 26.4924 2.40840
$$122$$ 0 0
$$123$$ 3.80776i 0.343335i
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ − 10.8078i − 0.959034i −0.877533 0.479517i $$-0.840812\pi$$
0.877533 0.479517i $$-0.159188\pi$$
$$128$$ 0 0
$$129$$ 10.2462 0.902129
$$130$$ 0 0
$$131$$ −18.2462 −1.59418 −0.797089 0.603861i $$-0.793628\pi$$
−0.797089 + 0.603861i $$0.793628\pi$$
$$132$$ 0 0
$$133$$ 3.56155i 0.308826i
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 14.6847i 1.25460i 0.778780 + 0.627298i $$0.215839\pi$$
−0.778780 + 0.627298i $$0.784161\pi$$
$$138$$ 0 0
$$139$$ 7.31534 0.620479 0.310240 0.950658i $$-0.399591\pi$$
0.310240 + 0.950658i $$0.399591\pi$$
$$140$$ 0 0
$$141$$ 12.8769 1.08443
$$142$$ 0 0
$$143$$ − 12.2462i − 1.02408i
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ 1.56155i 0.128795i
$$148$$ 0 0
$$149$$ −13.4384 −1.10092 −0.550460 0.834861i $$-0.685548\pi$$
−0.550460 + 0.834861i $$0.685548\pi$$
$$150$$ 0 0
$$151$$ −1.43845 −0.117059 −0.0585296 0.998286i $$-0.518641\pi$$
−0.0585296 + 0.998286i $$0.518641\pi$$
$$152$$ 0 0
$$153$$ − 0.876894i − 0.0708927i
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 8.49242i 0.677769i 0.940828 + 0.338885i $$0.110050\pi$$
−0.940828 + 0.338885i $$0.889950\pi$$
$$158$$ 0 0
$$159$$ −1.75379 −0.139084
$$160$$ 0 0
$$161$$ 1.43845 0.113366
$$162$$ 0 0
$$163$$ 14.9309i 1.16948i 0.811222 + 0.584738i $$0.198803\pi$$
−0.811222 + 0.584738i $$0.801197\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ − 21.6155i − 1.67266i −0.548227 0.836330i $$-0.684697\pi$$
0.548227 0.836330i $$-0.315303\pi$$
$$168$$ 0 0
$$169$$ 9.00000 0.692308
$$170$$ 0 0
$$171$$ −2.00000 −0.152944
$$172$$ 0 0
$$173$$ − 18.4924i − 1.40595i −0.711213 0.702976i $$-0.751854\pi$$
0.711213 0.702976i $$-0.248146\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 17.7538i 1.33446i
$$178$$ 0 0
$$179$$ 20.6847 1.54604 0.773022 0.634379i $$-0.218744\pi$$
0.773022 + 0.634379i $$0.218744\pi$$
$$180$$ 0 0
$$181$$ −11.1231 −0.826774 −0.413387 0.910555i $$-0.635654\pi$$
−0.413387 + 0.910555i $$0.635654\pi$$
$$182$$ 0 0
$$183$$ − 17.3693i − 1.28398i
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 9.56155i 0.699210i
$$188$$ 0 0
$$189$$ −5.56155 −0.404543
$$190$$ 0 0
$$191$$ 25.3693 1.83566 0.917830 0.396974i $$-0.129940\pi$$
0.917830 + 0.396974i $$0.129940\pi$$
$$192$$ 0 0
$$193$$ − 20.6155i − 1.48394i −0.670434 0.741969i $$-0.733892\pi$$
0.670434 0.741969i $$-0.266108\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ − 0.561553i − 0.0400090i −0.999800 0.0200045i $$-0.993632\pi$$
0.999800 0.0200045i $$-0.00636805\pi$$
$$198$$ 0 0
$$199$$ 4.87689 0.345714 0.172857 0.984947i $$-0.444700\pi$$
0.172857 + 0.984947i $$0.444700\pi$$
$$200$$ 0 0
$$201$$ −12.3002 −0.867588
$$202$$ 0 0
$$203$$ 3.43845i 0.241332i
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 0.807764i 0.0561435i
$$208$$ 0 0
$$209$$ 21.8078 1.50847
$$210$$ 0 0
$$211$$ 6.43845 0.443241 0.221620 0.975133i $$-0.428865\pi$$
0.221620 + 0.975133i $$0.428865\pi$$
$$212$$ 0 0
$$213$$ − 2.63068i − 0.180251i
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 9.12311i 0.619317i
$$218$$ 0 0
$$219$$ 10.0540 0.679385
$$220$$ 0 0
$$221$$ 3.12311 0.210083
$$222$$ 0 0
$$223$$ − 25.3693i − 1.69886i −0.527705 0.849428i $$-0.676947\pi$$
0.527705 0.849428i $$-0.323053\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ − 11.3693i − 0.754608i −0.926089 0.377304i $$-0.876851\pi$$
0.926089 0.377304i $$-0.123149\pi$$
$$228$$ 0 0
$$229$$ 14.0000 0.925146 0.462573 0.886581i $$-0.346926\pi$$
0.462573 + 0.886581i $$0.346926\pi$$
$$230$$ 0 0
$$231$$ 9.56155 0.629104
$$232$$ 0 0
$$233$$ − 0.561553i − 0.0367885i −0.999831 0.0183943i $$-0.994145\pi$$
0.999831 0.0183943i $$-0.00585541\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ − 8.87689i − 0.576616i
$$238$$ 0 0
$$239$$ 4.00000 0.258738 0.129369 0.991596i $$-0.458705\pi$$
0.129369 + 0.991596i $$0.458705\pi$$
$$240$$ 0 0
$$241$$ −12.0540 −0.776465 −0.388232 0.921562i $$-0.626914\pi$$
−0.388232 + 0.921562i $$0.626914\pi$$
$$242$$ 0 0
$$243$$ − 5.75379i − 0.369106i
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ − 7.12311i − 0.453232i
$$248$$ 0 0
$$249$$ −2.05398 −0.130165
$$250$$ 0 0
$$251$$ −24.6847 −1.55808 −0.779041 0.626973i $$-0.784294\pi$$
−0.779041 + 0.626973i $$0.784294\pi$$
$$252$$ 0 0
$$253$$ − 8.80776i − 0.553739i
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ − 12.2462i − 0.763898i −0.924183 0.381949i $$-0.875253\pi$$
0.924183 0.381949i $$-0.124747\pi$$
$$258$$ 0 0
$$259$$ −8.80776 −0.547288
$$260$$ 0 0
$$261$$ −1.93087 −0.119518
$$262$$ 0 0
$$263$$ 1.68466i 0.103880i 0.998650 + 0.0519402i $$0.0165405\pi$$
−0.998650 + 0.0519402i $$0.983459\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 15.3153i 0.937284i
$$268$$ 0 0
$$269$$ −28.2462 −1.72220 −0.861101 0.508434i $$-0.830225\pi$$
−0.861101 + 0.508434i $$0.830225\pi$$
$$270$$ 0 0
$$271$$ −0.246211 −0.0149563 −0.00747813 0.999972i $$-0.502380\pi$$
−0.00747813 + 0.999972i $$0.502380\pi$$
$$272$$ 0 0
$$273$$ − 3.12311i − 0.189019i
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ − 22.4924i − 1.35144i −0.737159 0.675719i $$-0.763833\pi$$
0.737159 0.675719i $$-0.236167\pi$$
$$278$$ 0 0
$$279$$ −5.12311 −0.306712
$$280$$ 0 0
$$281$$ 7.43845 0.443741 0.221870 0.975076i $$-0.428784\pi$$
0.221870 + 0.975076i $$0.428784\pi$$
$$282$$ 0 0
$$283$$ − 1.31534i − 0.0781889i −0.999236 0.0390945i $$-0.987553\pi$$
0.999236 0.0390945i $$-0.0124473\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 2.43845i 0.143937i
$$288$$ 0 0
$$289$$ 14.5616 0.856562
$$290$$ 0 0
$$291$$ −9.36932 −0.549239
$$292$$ 0 0
$$293$$ − 30.0000i − 1.75262i −0.481749 0.876309i $$-0.659998\pi$$
0.481749 0.876309i $$-0.340002\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 34.0540i 1.97601i
$$298$$ 0 0
$$299$$ −2.87689 −0.166375
$$300$$ 0 0
$$301$$ 6.56155 0.378202
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ − 16.6847i − 0.952244i −0.879379 0.476122i $$-0.842042\pi$$
0.879379 0.476122i $$-0.157958\pi$$
$$308$$ 0 0
$$309$$ −1.75379 −0.0997696
$$310$$ 0 0
$$311$$ 11.1231 0.630733 0.315367 0.948970i $$-0.397872\pi$$
0.315367 + 0.948970i $$0.397872\pi$$
$$312$$ 0 0
$$313$$ 22.0000i 1.24351i 0.783210 + 0.621757i $$0.213581\pi$$
−0.783210 + 0.621757i $$0.786419\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 8.31534i 0.467036i 0.972352 + 0.233518i $$0.0750238\pi$$
−0.972352 + 0.233518i $$0.924976\pi$$
$$318$$ 0 0
$$319$$ 21.0540 1.17880
$$320$$ 0 0
$$321$$ −12.1922 −0.680504
$$322$$ 0 0
$$323$$ 5.56155i 0.309453i
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ 31.1231i 1.72111i
$$328$$ 0 0
$$329$$ 8.24621 0.454628
$$330$$ 0 0
$$331$$ 27.0000 1.48405 0.742027 0.670370i $$-0.233865\pi$$
0.742027 + 0.670370i $$0.233865\pi$$
$$332$$ 0 0
$$333$$ − 4.94602i − 0.271040i
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ − 6.68466i − 0.364137i −0.983286 0.182068i $$-0.941721\pi$$
0.983286 0.182068i $$-0.0582792\pi$$
$$338$$ 0 0
$$339$$ 26.9309 1.46268
$$340$$ 0 0
$$341$$ 55.8617 3.02508
$$342$$ 0 0
$$343$$ 1.00000i 0.0539949i
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 18.6155i 0.999334i 0.866218 + 0.499667i $$0.166544\pi$$
−0.866218 + 0.499667i $$0.833456\pi$$
$$348$$ 0 0
$$349$$ 21.1231 1.13069 0.565347 0.824853i $$-0.308742\pi$$
0.565347 + 0.824853i $$0.308742\pi$$
$$350$$ 0 0
$$351$$ 11.1231 0.593707
$$352$$ 0 0
$$353$$ − 28.2462i − 1.50339i −0.659509 0.751697i $$-0.729236\pi$$
0.659509 0.751697i $$-0.270764\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 2.43845i 0.129056i
$$358$$ 0 0
$$359$$ 2.31534 0.122199 0.0610995 0.998132i $$-0.480539\pi$$
0.0610995 + 0.998132i $$0.480539\pi$$
$$360$$ 0 0
$$361$$ −6.31534 −0.332386
$$362$$ 0 0
$$363$$ − 41.3693i − 2.17133i
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ − 34.2462i − 1.78764i −0.448428 0.893819i $$-0.648016\pi$$
0.448428 0.893819i $$-0.351984\pi$$
$$368$$ 0 0
$$369$$ −1.36932 −0.0712838
$$370$$ 0 0
$$371$$ −1.12311 −0.0583087
$$372$$ 0 0
$$373$$ − 23.0540i − 1.19369i −0.802357 0.596845i $$-0.796421\pi$$
0.802357 0.596845i $$-0.203579\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ − 6.87689i − 0.354178i
$$378$$ 0 0
$$379$$ −32.8617 −1.68799 −0.843997 0.536348i $$-0.819804\pi$$
−0.843997 + 0.536348i $$0.819804\pi$$
$$380$$ 0 0
$$381$$ −16.8769 −0.864629
$$382$$ 0 0
$$383$$ − 3.75379i − 0.191810i −0.995391 0.0959048i $$-0.969426\pi$$
0.995391 0.0959048i $$-0.0305744\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 3.68466i 0.187302i
$$388$$ 0 0
$$389$$ −6.56155 −0.332684 −0.166342 0.986068i $$-0.553196\pi$$
−0.166342 + 0.986068i $$0.553196\pi$$
$$390$$ 0 0
$$391$$ 2.24621 0.113596
$$392$$ 0 0
$$393$$ 28.4924i 1.43725i
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ − 5.50758i − 0.276417i −0.990403 0.138209i $$-0.955866\pi$$
0.990403 0.138209i $$-0.0441345\pi$$
$$398$$ 0 0
$$399$$ 5.56155 0.278426
$$400$$ 0 0
$$401$$ 1.49242 0.0745280 0.0372640 0.999305i $$-0.488136\pi$$
0.0372640 + 0.999305i $$0.488136\pi$$
$$402$$ 0 0
$$403$$ − 18.2462i − 0.908909i
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 53.9309i 2.67325i
$$408$$ 0 0
$$409$$ 25.8078 1.27611 0.638056 0.769990i $$-0.279739\pi$$
0.638056 + 0.769990i $$0.279739\pi$$
$$410$$ 0 0
$$411$$ 22.9309 1.13110
$$412$$ 0 0
$$413$$ 11.3693i 0.559448i
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ − 11.4233i − 0.559401i
$$418$$ 0 0
$$419$$ −1.80776 −0.0883151 −0.0441575 0.999025i $$-0.514060\pi$$
−0.0441575 + 0.999025i $$0.514060\pi$$
$$420$$ 0 0
$$421$$ 33.9309 1.65369 0.826845 0.562430i $$-0.190134\pi$$
0.826845 + 0.562430i $$0.190134\pi$$
$$422$$ 0 0
$$423$$ 4.63068i 0.225152i
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ − 11.1231i − 0.538285i
$$428$$ 0 0
$$429$$ −19.1231 −0.923272
$$430$$ 0 0
$$431$$ −30.2462 −1.45691 −0.728454 0.685094i $$-0.759761\pi$$
−0.728454 + 0.685094i $$0.759761\pi$$
$$432$$ 0 0
$$433$$ 34.5464i 1.66019i 0.557619 + 0.830097i $$0.311715\pi$$
−0.557619 + 0.830097i $$0.688285\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ − 5.12311i − 0.245071i
$$438$$ 0 0
$$439$$ −1.12311 −0.0536029 −0.0268015 0.999641i $$-0.508532\pi$$
−0.0268015 + 0.999641i $$0.508532\pi$$
$$440$$ 0 0
$$441$$ −0.561553 −0.0267406
$$442$$ 0 0
$$443$$ 18.4384i 0.876037i 0.898966 + 0.438019i $$0.144320\pi$$
−0.898966 + 0.438019i $$0.855680\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 20.9848i 0.992549i
$$448$$ 0 0
$$449$$ 14.3693 0.678130 0.339065 0.940763i $$-0.389889\pi$$
0.339065 + 0.940763i $$0.389889\pi$$
$$450$$ 0 0
$$451$$ 14.9309 0.703067
$$452$$ 0 0
$$453$$ 2.24621i 0.105536i
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 37.1080i 1.73584i 0.496707 + 0.867918i $$0.334542\pi$$
−0.496707 + 0.867918i $$0.665458\pi$$
$$458$$ 0 0
$$459$$ −8.68466 −0.405365
$$460$$ 0 0
$$461$$ −26.4924 −1.23388 −0.616938 0.787012i $$-0.711627\pi$$
−0.616938 + 0.787012i $$0.711627\pi$$
$$462$$ 0 0
$$463$$ 12.4924i 0.580572i 0.956940 + 0.290286i $$0.0937505\pi$$
−0.956940 + 0.290286i $$0.906250\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 33.1231i 1.53275i 0.642391 + 0.766377i $$0.277943\pi$$
−0.642391 + 0.766377i $$0.722057\pi$$
$$468$$ 0 0
$$469$$ −7.87689 −0.363721
$$470$$ 0 0
$$471$$ 13.2614 0.611052
$$472$$ 0 0
$$473$$ − 40.1771i − 1.84734i
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ − 0.630683i − 0.0288770i
$$478$$ 0 0
$$479$$ −8.73863 −0.399278 −0.199639 0.979869i $$-0.563977\pi$$
−0.199639 + 0.979869i $$0.563977\pi$$
$$480$$ 0 0
$$481$$ 17.6155 0.803199
$$482$$ 0 0
$$483$$ − 2.24621i − 0.102206i
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ − 31.4384i − 1.42461i −0.701869 0.712306i $$-0.747651\pi$$
0.701869 0.712306i $$-0.252349\pi$$
$$488$$ 0 0
$$489$$ 23.3153 1.05436
$$490$$ 0 0
$$491$$ 8.31534 0.375266 0.187633 0.982239i $$-0.439918\pi$$
0.187633 + 0.982239i $$0.439918\pi$$
$$492$$ 0 0
$$493$$ 5.36932i 0.241822i
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ − 1.68466i − 0.0755673i
$$498$$ 0 0
$$499$$ −16.4924 −0.738302 −0.369151 0.929369i $$-0.620352\pi$$
−0.369151 + 0.929369i $$0.620352\pi$$
$$500$$ 0 0
$$501$$ −33.7538 −1.50801
$$502$$ 0 0
$$503$$ 36.2462i 1.61614i 0.589087 + 0.808069i $$0.299487\pi$$
−0.589087 + 0.808069i $$0.700513\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ − 14.0540i − 0.624159i
$$508$$ 0 0
$$509$$ −26.0000 −1.15243 −0.576215 0.817298i $$-0.695471\pi$$
−0.576215 + 0.817298i $$0.695471\pi$$
$$510$$ 0 0
$$511$$ 6.43845 0.284820
$$512$$ 0 0
$$513$$ 19.8078i 0.874534i
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ − 50.4924i − 2.22065i
$$518$$ 0 0
$$519$$ −28.8769 −1.26755
$$520$$ 0 0
$$521$$ −31.5616 −1.38274 −0.691368 0.722502i $$-0.742992\pi$$
−0.691368 + 0.722502i $$0.742992\pi$$
$$522$$ 0 0
$$523$$ 40.6847i 1.77902i 0.456920 + 0.889508i $$0.348953\pi$$
−0.456920 + 0.889508i $$0.651047\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 14.2462i 0.620575i
$$528$$ 0 0
$$529$$ 20.9309 0.910038
$$530$$ 0 0
$$531$$ −6.38447 −0.277062
$$532$$ 0 0
$$533$$ − 4.87689i − 0.211242i
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ − 32.3002i − 1.39386i
$$538$$ 0 0
$$539$$ 6.12311 0.263741
$$540$$ 0 0
$$541$$ −20.4233 −0.878066 −0.439033 0.898471i $$-0.644679\pi$$
−0.439033 + 0.898471i $$0.644679\pi$$
$$542$$ 0 0
$$543$$ 17.3693i 0.745389i
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 15.7386i 0.672935i 0.941695 + 0.336468i $$0.109232\pi$$
−0.941695 + 0.336468i $$0.890768\pi$$
$$548$$ 0 0
$$549$$ 6.24621 0.266582
$$550$$ 0 0
$$551$$ 12.2462 0.521706
$$552$$ 0 0
$$553$$ − 5.68466i − 0.241736i
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ − 5.43845i − 0.230434i −0.993340 0.115217i $$-0.963244\pi$$
0.993340 0.115217i $$-0.0367564\pi$$
$$558$$ 0 0
$$559$$ −13.1231 −0.555048
$$560$$ 0 0
$$561$$ 14.9309 0.630382
$$562$$ 0 0
$$563$$ 1.12311i 0.0473333i 0.999720 + 0.0236666i $$0.00753403\pi$$
−0.999720 + 0.0236666i $$0.992466\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ 7.00000i 0.293972i
$$568$$ 0 0
$$569$$ −45.9848 −1.92778 −0.963892 0.266292i $$-0.914202\pi$$
−0.963892 + 0.266292i $$0.914202\pi$$
$$570$$ 0 0
$$571$$ 39.6847 1.66075 0.830376 0.557204i $$-0.188126\pi$$
0.830376 + 0.557204i $$0.188126\pi$$
$$572$$ 0 0
$$573$$ − 39.6155i − 1.65496i
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ − 24.5464i − 1.02188i −0.859616 0.510940i $$-0.829297\pi$$
0.859616 0.510940i $$-0.170703\pi$$
$$578$$ 0 0
$$579$$ −32.1922 −1.33786
$$580$$ 0 0
$$581$$ −1.31534 −0.0545696
$$582$$ 0 0
$$583$$ 6.87689i 0.284812i
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 37.1771i 1.53446i 0.641371 + 0.767231i $$0.278366\pi$$
−0.641371 + 0.767231i $$0.721634\pi$$
$$588$$ 0 0
$$589$$ 32.4924 1.33883
$$590$$ 0 0
$$591$$ −0.876894 −0.0360706
$$592$$ 0 0
$$593$$ 24.6847i 1.01368i 0.862041 + 0.506839i $$0.169186\pi$$
−0.862041 + 0.506839i $$0.830814\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ − 7.61553i − 0.311683i
$$598$$ 0 0
$$599$$ 29.3002 1.19717 0.598587 0.801058i $$-0.295729\pi$$
0.598587 + 0.801058i $$0.295729\pi$$
$$600$$ 0 0
$$601$$ −42.0540 −1.71542 −0.857709 0.514136i $$-0.828113\pi$$
−0.857709 + 0.514136i $$0.828113\pi$$
$$602$$ 0 0
$$603$$ − 4.42329i − 0.180130i
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 12.6307i 0.512664i 0.966589 + 0.256332i $$0.0825140\pi$$
−0.966589 + 0.256332i $$0.917486\pi$$
$$608$$ 0 0
$$609$$ 5.36932 0.217576
$$610$$ 0 0
$$611$$ −16.4924 −0.667212
$$612$$ 0 0
$$613$$ 5.68466i 0.229601i 0.993389 + 0.114801i $$0.0366229\pi$$
−0.993389 + 0.114801i $$0.963377\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 2.80776i 0.113036i 0.998402 + 0.0565182i $$0.0179999\pi$$
−0.998402 + 0.0565182i $$0.982000\pi$$
$$618$$ 0 0
$$619$$ −4.63068 −0.186123 −0.0930614 0.995660i $$-0.529665\pi$$
−0.0930614 + 0.995660i $$0.529665\pi$$
$$620$$ 0 0
$$621$$ 8.00000 0.321029
$$622$$ 0 0
$$623$$ 9.80776i 0.392940i
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ − 34.0540i − 1.35998i
$$628$$ 0 0
$$629$$ −13.7538 −0.548399
$$630$$ 0 0
$$631$$ −3.05398 −0.121577 −0.0607884 0.998151i $$-0.519361\pi$$
−0.0607884 + 0.998151i $$0.519361\pi$$
$$632$$ 0 0
$$633$$ − 10.0540i − 0.399610i
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ − 2.00000i − 0.0792429i
$$638$$ 0 0
$$639$$ 0.946025 0.0374242
$$640$$ 0 0
$$641$$ −20.5616 −0.812133 −0.406066 0.913844i $$-0.633100\pi$$
−0.406066 + 0.913844i $$0.633100\pi$$
$$642$$ 0 0
$$643$$ − 46.7386i − 1.84319i −0.388151 0.921596i $$-0.626886\pi$$
0.388151 0.921596i $$-0.373114\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 0.630683i 0.0247947i 0.999923 + 0.0123974i $$0.00394630\pi$$
−0.999923 + 0.0123974i $$0.996054\pi$$
$$648$$ 0 0
$$649$$ 69.6155 2.73265
$$650$$ 0 0
$$651$$ 14.2462 0.558353
$$652$$ 0 0
$$653$$ − 23.8617i − 0.933782i −0.884315 0.466891i $$-0.845374\pi$$
0.884315 0.466891i $$-0.154626\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 3.61553i 0.141055i
$$658$$ 0 0
$$659$$ −14.4384 −0.562442 −0.281221 0.959643i $$-0.590739\pi$$
−0.281221 + 0.959643i $$0.590739\pi$$
$$660$$ 0 0
$$661$$ 10.4924 0.408108 0.204054 0.978960i $$-0.434588\pi$$
0.204054 + 0.978960i $$0.434588\pi$$
$$662$$ 0 0
$$663$$ − 4.87689i − 0.189403i
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ − 4.94602i − 0.191511i
$$668$$ 0 0
$$669$$ −39.6155 −1.53162
$$670$$ 0 0
$$671$$ −68.1080 −2.62928
$$672$$ 0 0
$$673$$ 13.5076i 0.520679i 0.965517 + 0.260339i $$0.0838345\pi$$
−0.965517 + 0.260339i $$0.916166\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ − 39.3693i − 1.51309i −0.653944 0.756543i $$-0.726887\pi$$
0.653944 0.756543i $$-0.273113\pi$$
$$678$$ 0 0
$$679$$ −6.00000 −0.230259
$$680$$ 0 0
$$681$$ −17.7538 −0.680327
$$682$$ 0 0
$$683$$ 25.8769i 0.990152i 0.868850 + 0.495076i $$0.164860\pi$$
−0.868850 + 0.495076i $$0.835140\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ − 21.8617i − 0.834077i
$$688$$ 0 0
$$689$$ 2.24621 0.0855738
$$690$$ 0 0
$$691$$ −21.5616 −0.820240 −0.410120 0.912032i $$-0.634513\pi$$
−0.410120 + 0.912032i $$0.634513\pi$$
$$692$$ 0 0
$$693$$ 3.43845i 0.130616i
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 3.80776i 0.144229i
$$698$$ 0 0
$$699$$ −0.876894 −0.0331672
$$700$$ 0 0
$$701$$ −30.1080 −1.13716 −0.568581 0.822627i $$-0.692507\pi$$
−0.568581 + 0.822627i $$0.692507\pi$$
$$702$$ 0 0
$$703$$ 31.3693i 1.18312i
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ −12.6307 −0.474355 −0.237178 0.971466i $$-0.576222\pi$$
−0.237178 + 0.971466i $$0.576222\pi$$
$$710$$ 0 0
$$711$$ 3.19224 0.119718
$$712$$ 0 0
$$713$$ − 13.1231i − 0.491464i
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ − 6.24621i − 0.233269i
$$718$$ 0 0
$$719$$ −21.7538 −0.811279 −0.405640 0.914033i $$-0.632951\pi$$
−0.405640 + 0.914033i $$0.632951\pi$$
$$720$$ 0 0
$$721$$ −1.12311 −0.0418266
$$722$$ 0 0
$$723$$ 18.8229i 0.700032i
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ − 39.6155i − 1.46926i −0.678469 0.734629i $$-0.737356\pi$$
0.678469 0.734629i $$-0.262644\pi$$
$$728$$ 0 0
$$729$$ −29.9848 −1.11055
$$730$$ 0 0
$$731$$ 10.2462 0.378970
$$732$$ 0 0
$$733$$ 5.75379i 0.212521i 0.994338 + 0.106261i $$0.0338878\pi$$
−0.994338 + 0.106261i $$0.966112\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 48.2311i 1.77661i
$$738$$ 0 0
$$739$$ 28.1771 1.03651 0.518255 0.855226i $$-0.326582\pi$$
0.518255 + 0.855226i $$0.326582\pi$$
$$740$$ 0 0
$$741$$ −11.1231 −0.408617
$$742$$ 0 0
$$743$$ 30.7386i 1.12769i 0.825880 + 0.563846i $$0.190679\pi$$
−0.825880 + 0.563846i $$0.809321\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ − 0.738634i − 0.0270252i
$$748$$ 0 0
$$749$$ −7.80776 −0.285289
$$750$$ 0 0
$$751$$ 2.63068 0.0959950 0.0479975 0.998847i $$-0.484716\pi$$
0.0479975 + 0.998847i $$0.484716\pi$$
$$752$$ 0 0
$$753$$ 38.5464i 1.40471i
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ − 25.6847i − 0.933525i −0.884383 0.466762i $$-0.845420\pi$$
0.884383 0.466762i $$-0.154580\pi$$
$$758$$ 0 0
$$759$$ −13.7538 −0.499231
$$760$$ 0 0
$$761$$ −27.4233 −0.994094 −0.497047 0.867724i $$-0.665582\pi$$
−0.497047 + 0.867724i $$0.665582\pi$$
$$762$$ 0 0
$$763$$ 19.9309i 0.721546i
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ − 22.7386i − 0.821044i
$$768$$ 0 0
$$769$$ −20.3002 −0.732043 −0.366022 0.930606i $$-0.619280\pi$$
−0.366022 + 0.930606i $$0.619280\pi$$
$$770$$ 0 0
$$771$$ −19.1231 −0.688702
$$772$$ 0 0
$$773$$ 29.6155i 1.06520i 0.846368 + 0.532598i $$0.178784\pi$$
−0.846368 + 0.532598i $$0.821216\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ 13.7538i 0.493414i
$$778$$ 0 0
$$779$$ 8.68466 0.311160
$$780$$ 0 0
$$781$$ −10.3153 −0.369112
$$782$$ 0 0
$$783$$ 19.1231i 0.683404i
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ − 9.75379i − 0.347685i −0.984773 0.173843i $$-0.944382\pi$$
0.984773 0.173843i $$-0.0556184\pi$$
$$788$$ 0 0
$$789$$ 2.63068 0.0936548
$$790$$ 0 0
$$791$$ 17.2462 0.613205
$$792$$ 0 0
$$793$$ 22.2462i 0.789986i
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ − 0.384472i − 0.0136187i −0.999977 0.00680935i $$-0.997833\pi$$
0.999977 0.00680935i $$-0.00216750\pi$$
$$798$$ 0 0
$$799$$ 12.8769 0.455552
$$800$$ 0 0
$$801$$ −5.50758 −0.194601
$$802$$ 0 0
$$803$$ − 39.4233i − 1.39122i
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 44.1080i 1.55267i
$$808$$ 0 0
$$809$$ 12.4233 0.436780 0.218390 0.975862i $$-0.429920\pi$$
0.218390 + 0.975862i $$0.429920\pi$$
$$810$$ 0 0
$$811$$ 2.38447 0.0837301 0.0418651 0.999123i $$-0.486670\pi$$
0.0418651 + 0.999123i $$0.486670\pi$$
$$812$$ 0 0
$$813$$ 0.384472i 0.0134840i
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ − 23.3693i − 0.817589i
$$818$$ 0 0
$$819$$ 1.12311 0.0392445
$$820$$ 0 0
$$821$$ −11.8617 −0.413978 −0.206989 0.978343i $$-0.566366\pi$$
−0.206989 + 0.978343i $$0.566366\pi$$
$$822$$ 0 0
$$823$$ 16.5616i 0.577299i 0.957435 + 0.288650i $$0.0932063\pi$$
−0.957435 + 0.288650i $$0.906794\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ − 7.24621i − 0.251975i −0.992032 0.125988i $$-0.959790\pi$$
0.992032 0.125988i $$-0.0402100\pi$$
$$828$$ 0 0
$$829$$ −14.2462 −0.494791 −0.247396 0.968915i $$-0.579575\pi$$
−0.247396 + 0.968915i $$0.579575\pi$$
$$830$$ 0 0
$$831$$ −35.1231 −1.21841
$$832$$ 0 0
$$833$$ 1.56155i 0.0541046i
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 50.7386i 1.75378i
$$838$$ 0 0
$$839$$ −7.12311 −0.245917 −0.122958 0.992412i $$-0.539238\pi$$
−0.122958 + 0.992412i $$0.539238\pi$$
$$840$$ 0 0
$$841$$ −17.1771 −0.592313
$$842$$ 0 0
$$843$$ − 11.6155i − 0.400060i
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ − 26.4924i − 0.910290i
$$848$$ 0 0
$$849$$ −2.05398 −0.0704923
$$850$$ 0 0
$$851$$ 12.6695 0.434305
$$852$$ 0 0
$$853$$ 19.1231i 0.654763i 0.944892 + 0.327381i $$0.106166\pi$$
−0.944892 + 0.327381i $$0.893834\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ − 42.0540i − 1.43654i −0.695766 0.718268i $$-0.744935\pi$$
0.695766 0.718268i $$-0.255065\pi$$
$$858$$ 0 0
$$859$$ 11.1771 0.381357 0.190679 0.981653i $$-0.438931\pi$$
0.190679 + 0.981653i $$0.438931\pi$$
$$860$$ 0 0
$$861$$ 3.80776 0.129768
$$862$$ 0 0
$$863$$ − 3.19224i − 0.108665i −0.998523 0.0543325i $$-0.982697\pi$$
0.998523 0.0543325i $$-0.0173031\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ − 22.7386i − 0.772244i
$$868$$ 0 0
$$869$$ −34.8078 −1.18077
$$870$$ 0 0
$$871$$ 15.7538 0.533797
$$872$$ 0 0
$$873$$ − 3.36932i − 0.114034i
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 17.6155i 0.594834i 0.954748 + 0.297417i $$0.0961252\pi$$
−0.954748 + 0.297417i $$0.903875\pi$$
$$878$$ 0 0
$$879$$ −46.8466 −1.58010
$$880$$ 0 0
$$881$$ 8.73863 0.294412 0.147206 0.989106i $$-0.452972\pi$$
0.147206 + 0.989106i $$0.452972\pi$$
$$882$$ 0 0
$$883$$ 31.4924i 1.05980i 0.848059 + 0.529902i $$0.177771\pi$$
−0.848059 + 0.529902i $$0.822229\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 8.38447i 0.281523i 0.990044 + 0.140762i $$0.0449551\pi$$
−0.990044 + 0.140762i $$0.955045\pi$$
$$888$$ 0 0
$$889$$ −10.8078 −0.362481
$$890$$ 0 0
$$891$$ 42.8617 1.43592
$$892$$ 0 0
$$893$$ − 29.3693i − 0.982807i
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 4.49242i 0.149998i
$$898$$ 0 0
$$899$$ 31.3693 1.04623
$$900$$ 0 0
$$901$$ −1.75379 −0.0584272
$$902$$ 0 0
$$903$$ − 10.2462i − 0.340973i
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ − 4.00000i − 0.132818i −0.997792 0.0664089i $$-0.978846\pi$$
0.997792 0.0664089i $$-0.0211542\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 22.4233 0.742917 0.371458 0.928450i $$-0.378858\pi$$
0.371458 + 0.928450i $$0.378858\pi$$
$$912$$ 0 0
$$913$$ 8.05398i 0.266548i
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 18.2462i 0.602543i
$$918$$ 0 0
$$919$$ −38.8078 −1.28015 −0.640075 0.768312i $$-0.721097\pi$$
−0.640075 + 0.768312i $$0.721097\pi$$
$$920$$ 0 0
$$921$$ −26.0540 −0.858508
$$922$$ 0 0
$$923$$ 3.36932i 0.110902i
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ − 0.630683i − 0.0207144i
$$928$$ 0 0
$$929$$ 34.3542 1.12712 0.563562 0.826074i $$-0.309431\pi$$
0.563562 + 0.826074i $$0.309431\pi$$
$$930$$ 0 0
$$931$$ 3.56155 0.116725
$$932$$ 0 0
$$933$$ − 17.3693i − 0.568646i
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 32.3002i 1.05520i 0.849493 + 0.527601i $$0.176908\pi$$
−0.849493 + 0.527601i $$0.823092\pi$$
$$938$$ 0 0
$$939$$ 34.3542 1.12111
$$940$$ 0 0
$$941$$ 23.3693 0.761818 0.380909 0.924613i $$-0.375611\pi$$
0.380909 + 0.924613i $$0.375611\pi$$
$$942$$ 0 0
$$943$$ − 3.50758i − 0.114222i
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 16.4924i 0.535932i 0.963428 + 0.267966i $$0.0863514\pi$$
−0.963428 + 0.267966i $$0.913649\pi$$
$$948$$ 0 0
$$949$$ −12.8769 −0.418002
$$950$$ 0 0
$$951$$ 12.9848 0.421062
$$952$$ 0 0
$$953$$ − 36.1231i − 1.17014i −0.810982 0.585071i $$-0.801067\pi$$
0.810982 0.585071i $$-0.198933\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ − 32.8769i − 1.06276i
$$958$$ 0 0
$$959$$ 14.6847 0.474192
$$960$$ 0 0
$$961$$ 52.2311 1.68487
$$962$$ 0 0
$$963$$ − 4.38447i − 0.141288i
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 41.3693i 1.33035i 0.746689 + 0.665174i $$0.231643\pi$$
−0.746689 + 0.665174i $$0.768357\pi$$
$$968$$ 0 0
$$969$$ 8.68466 0.278991
$$970$$ 0 0
$$971$$ −14.6847 −0.471253 −0.235627 0.971844i $$-0.575714\pi$$
−0.235627 + 0.971844i $$0.575714\pi$$
$$972$$ 0 0
$$973$$ − 7.31534i − 0.234519i
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 16.7538i 0.536001i 0.963419 + 0.268001i $$0.0863629\pi$$
−0.963419 + 0.268001i $$0.913637\pi$$
$$978$$ 0 0
$$979$$ 60.0540 1.91933
$$980$$ 0 0
$$981$$ −11.1922 −0.357341
$$982$$ 0 0
$$983$$ 28.3542i 0.904357i 0.891927 + 0.452179i $$0.149353\pi$$
−0.891927 + 0.452179i $$0.850647\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ − 12.8769i − 0.409876i
$$988$$ 0 0
$$989$$ −9.43845 −0.300125
$$990$$ 0 0
$$991$$ 48.6695 1.54604 0.773019 0.634383i $$-0.218746\pi$$
0.773019 + 0.634383i $$0.218746\pi$$
$$992$$ 0 0
$$993$$ − 42.1619i − 1.33797i
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 4.24621i 0.134479i 0.997737 + 0.0672394i $$0.0214191\pi$$
−0.997737 + 0.0672394i $$0.978581\pi$$
$$998$$ 0 0
$$999$$ −48.9848 −1.54981
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 1400.2.g.j.449.2 4
4.3 odd 2 2800.2.g.v.449.3 4
5.2 odd 4 1400.2.a.q.1.1 yes 2
5.3 odd 4 1400.2.a.o.1.2 2
5.4 even 2 inner 1400.2.g.j.449.3 4
20.3 even 4 2800.2.a.bo.1.1 2
20.7 even 4 2800.2.a.bj.1.2 2
20.19 odd 2 2800.2.g.v.449.2 4
35.13 even 4 9800.2.a.bx.1.1 2
35.27 even 4 9800.2.a.bt.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
1400.2.a.o.1.2 2 5.3 odd 4
1400.2.a.q.1.1 yes 2 5.2 odd 4
1400.2.g.j.449.2 4 1.1 even 1 trivial
1400.2.g.j.449.3 4 5.4 even 2 inner
2800.2.a.bj.1.2 2 20.7 even 4
2800.2.a.bo.1.1 2 20.3 even 4
2800.2.g.v.449.2 4 20.19 odd 2
2800.2.g.v.449.3 4 4.3 odd 2
9800.2.a.bt.1.2 2 35.27 even 4
9800.2.a.bx.1.1 2 35.13 even 4