# Properties

 Label 1792.2.b.k.897.3 Level $1792$ Weight $2$ Character 1792.897 Analytic conductor $14.309$ 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] = [1792,2,Mod(897,1792)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1792, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1792.897");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1792 = 2^{8} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1792.b (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: $$14.3091920422$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{5})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 3x^{2} + 1$$ x^4 + 3*x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: no (minimal twist has level 224) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 897.3 Root $$0.618034i$$ of defining polynomial Character $$\chi$$ $$=$$ 1792.897 Dual form 1792.2.b.k.897.2

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.23607i q^{3} -3.23607i q^{5} -1.00000 q^{7} +1.47214 q^{9} +O(q^{10})$$ $$q+1.23607i q^{3} -3.23607i q^{5} -1.00000 q^{7} +1.47214 q^{9} +6.47214i q^{11} +0.763932i q^{13} +4.00000 q^{15} +4.47214 q^{17} -1.23607i q^{19} -1.23607i q^{21} -4.00000 q^{23} -5.47214 q^{25} +5.52786i q^{27} -4.47214i q^{29} -2.47214 q^{31} -8.00000 q^{33} +3.23607i q^{35} +4.47214i q^{37} -0.944272 q^{39} +8.47214 q^{41} -6.47214i q^{43} -4.76393i q^{45} +10.4721 q^{47} +1.00000 q^{49} +5.52786i q^{51} +10.0000i q^{53} +20.9443 q^{55} +1.52786 q^{57} +9.23607i q^{59} +11.2361i q^{61} -1.47214 q^{63} +2.47214 q^{65} +4.00000i q^{67} -4.94427i q^{69} +4.94427 q^{71} +2.94427 q^{73} -6.76393i q^{75} -6.47214i q^{77} +12.9443 q^{79} -2.41641 q^{81} -9.23607i q^{83} -14.4721i q^{85} +5.52786 q^{87} +6.00000 q^{89} -0.763932i q^{91} -3.05573i q^{93} -4.00000 q^{95} +12.4721 q^{97} +9.52786i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{7} - 12 q^{9}+O(q^{10})$$ 4 * q - 4 * q^7 - 12 * q^9 $$4 q - 4 q^{7} - 12 q^{9} + 16 q^{15} - 16 q^{23} - 4 q^{25} + 8 q^{31} - 32 q^{33} + 32 q^{39} + 16 q^{41} + 24 q^{47} + 4 q^{49} + 48 q^{55} + 24 q^{57} + 12 q^{63} - 8 q^{65} - 16 q^{71} - 24 q^{73} + 16 q^{79} + 44 q^{81} + 40 q^{87} + 24 q^{89} - 16 q^{95} + 32 q^{97}+O(q^{100})$$ 4 * q - 4 * q^7 - 12 * q^9 + 16 * q^15 - 16 * q^23 - 4 * q^25 + 8 * q^31 - 32 * q^33 + 32 * q^39 + 16 * q^41 + 24 * q^47 + 4 * q^49 + 48 * q^55 + 24 * q^57 + 12 * q^63 - 8 * q^65 - 16 * q^71 - 24 * q^73 + 16 * q^79 + 44 * q^81 + 40 * q^87 + 24 * q^89 - 16 * q^95 + 32 * q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1792\mathbb{Z}\right)^\times$$.

 $$n$$ $$1023$$ $$1025$$ $$1541$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## 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
$$3$$ 1.23607i 0.713644i 0.934172 + 0.356822i $$0.116140\pi$$
−0.934172 + 0.356822i $$0.883860\pi$$
$$4$$ 0 0
$$5$$ − 3.23607i − 1.44721i −0.690212 0.723607i $$-0.742483\pi$$
0.690212 0.723607i $$-0.257517\pi$$
$$6$$ 0 0
$$7$$ −1.00000 −0.377964
$$8$$ 0 0
$$9$$ 1.47214 0.490712
$$10$$ 0 0
$$11$$ 6.47214i 1.95142i 0.219061 + 0.975711i $$0.429701\pi$$
−0.219061 + 0.975711i $$0.570299\pi$$
$$12$$ 0 0
$$13$$ 0.763932i 0.211877i 0.994373 + 0.105938i $$0.0337846\pi$$
−0.994373 + 0.105938i $$0.966215\pi$$
$$14$$ 0 0
$$15$$ 4.00000 1.03280
$$16$$ 0 0
$$17$$ 4.47214 1.08465 0.542326 0.840168i $$-0.317544\pi$$
0.542326 + 0.840168i $$0.317544\pi$$
$$18$$ 0 0
$$19$$ − 1.23607i − 0.283573i −0.989897 0.141787i $$-0.954715\pi$$
0.989897 0.141787i $$-0.0452847\pi$$
$$20$$ 0 0
$$21$$ − 1.23607i − 0.269732i
$$22$$ 0 0
$$23$$ −4.00000 −0.834058 −0.417029 0.908893i $$-0.636929\pi$$
−0.417029 + 0.908893i $$0.636929\pi$$
$$24$$ 0 0
$$25$$ −5.47214 −1.09443
$$26$$ 0 0
$$27$$ 5.52786i 1.06384i
$$28$$ 0 0
$$29$$ − 4.47214i − 0.830455i −0.909718 0.415227i $$-0.863702\pi$$
0.909718 0.415227i $$-0.136298\pi$$
$$30$$ 0 0
$$31$$ −2.47214 −0.444009 −0.222004 0.975046i $$-0.571260\pi$$
−0.222004 + 0.975046i $$0.571260\pi$$
$$32$$ 0 0
$$33$$ −8.00000 −1.39262
$$34$$ 0 0
$$35$$ 3.23607i 0.546995i
$$36$$ 0 0
$$37$$ 4.47214i 0.735215i 0.929981 + 0.367607i $$0.119823\pi$$
−0.929981 + 0.367607i $$0.880177\pi$$
$$38$$ 0 0
$$39$$ −0.944272 −0.151205
$$40$$ 0 0
$$41$$ 8.47214 1.32313 0.661563 0.749890i $$-0.269894\pi$$
0.661563 + 0.749890i $$0.269894\pi$$
$$42$$ 0 0
$$43$$ − 6.47214i − 0.986991i −0.869748 0.493496i $$-0.835719\pi$$
0.869748 0.493496i $$-0.164281\pi$$
$$44$$ 0 0
$$45$$ − 4.76393i − 0.710165i
$$46$$ 0 0
$$47$$ 10.4721 1.52752 0.763759 0.645501i $$-0.223352\pi$$
0.763759 + 0.645501i $$0.223352\pi$$
$$48$$ 0 0
$$49$$ 1.00000 0.142857
$$50$$ 0 0
$$51$$ 5.52786i 0.774056i
$$52$$ 0 0
$$53$$ 10.0000i 1.37361i 0.726844 + 0.686803i $$0.240986\pi$$
−0.726844 + 0.686803i $$0.759014\pi$$
$$54$$ 0 0
$$55$$ 20.9443 2.82413
$$56$$ 0 0
$$57$$ 1.52786 0.202371
$$58$$ 0 0
$$59$$ 9.23607i 1.20243i 0.799086 + 0.601217i $$0.205317\pi$$
−0.799086 + 0.601217i $$0.794683\pi$$
$$60$$ 0 0
$$61$$ 11.2361i 1.43863i 0.694683 + 0.719316i $$0.255544\pi$$
−0.694683 + 0.719316i $$0.744456\pi$$
$$62$$ 0 0
$$63$$ −1.47214 −0.185472
$$64$$ 0 0
$$65$$ 2.47214 0.306631
$$66$$ 0 0
$$67$$ 4.00000i 0.488678i 0.969690 + 0.244339i $$0.0785709\pi$$
−0.969690 + 0.244339i $$0.921429\pi$$
$$68$$ 0 0
$$69$$ − 4.94427i − 0.595220i
$$70$$ 0 0
$$71$$ 4.94427 0.586777 0.293389 0.955993i $$-0.405217\pi$$
0.293389 + 0.955993i $$0.405217\pi$$
$$72$$ 0 0
$$73$$ 2.94427 0.344601 0.172300 0.985044i $$-0.444880\pi$$
0.172300 + 0.985044i $$0.444880\pi$$
$$74$$ 0 0
$$75$$ − 6.76393i − 0.781032i
$$76$$ 0 0
$$77$$ − 6.47214i − 0.737568i
$$78$$ 0 0
$$79$$ 12.9443 1.45634 0.728172 0.685394i $$-0.240370\pi$$
0.728172 + 0.685394i $$0.240370\pi$$
$$80$$ 0 0
$$81$$ −2.41641 −0.268490
$$82$$ 0 0
$$83$$ − 9.23607i − 1.01379i −0.862008 0.506895i $$-0.830793\pi$$
0.862008 0.506895i $$-0.169207\pi$$
$$84$$ 0 0
$$85$$ − 14.4721i − 1.56972i
$$86$$ 0 0
$$87$$ 5.52786 0.592649
$$88$$ 0 0
$$89$$ 6.00000 0.635999 0.317999 0.948091i $$-0.396989\pi$$
0.317999 + 0.948091i $$0.396989\pi$$
$$90$$ 0 0
$$91$$ − 0.763932i − 0.0800818i
$$92$$ 0 0
$$93$$ − 3.05573i − 0.316864i
$$94$$ 0 0
$$95$$ −4.00000 −0.410391
$$96$$ 0 0
$$97$$ 12.4721 1.26635 0.633177 0.774007i $$-0.281751\pi$$
0.633177 + 0.774007i $$0.281751\pi$$
$$98$$ 0 0
$$99$$ 9.52786i 0.957586i
$$100$$ 0 0
$$101$$ 1.70820i 0.169973i 0.996382 + 0.0849863i $$0.0270847\pi$$
−0.996382 + 0.0849863i $$0.972915\pi$$
$$102$$ 0 0
$$103$$ 5.52786 0.544677 0.272338 0.962202i $$-0.412203\pi$$
0.272338 + 0.962202i $$0.412203\pi$$
$$104$$ 0 0
$$105$$ −4.00000 −0.390360
$$106$$ 0 0
$$107$$ − 8.94427i − 0.864675i −0.901712 0.432338i $$-0.857689\pi$$
0.901712 0.432338i $$-0.142311\pi$$
$$108$$ 0 0
$$109$$ 8.47214i 0.811483i 0.913988 + 0.405742i $$0.132987\pi$$
−0.913988 + 0.405742i $$0.867013\pi$$
$$110$$ 0 0
$$111$$ −5.52786 −0.524682
$$112$$ 0 0
$$113$$ −12.4721 −1.17328 −0.586640 0.809848i $$-0.699550\pi$$
−0.586640 + 0.809848i $$0.699550\pi$$
$$114$$ 0 0
$$115$$ 12.9443i 1.20706i
$$116$$ 0 0
$$117$$ 1.12461i 0.103970i
$$118$$ 0 0
$$119$$ −4.47214 −0.409960
$$120$$ 0 0
$$121$$ −30.8885 −2.80805
$$122$$ 0 0
$$123$$ 10.4721i 0.944241i
$$124$$ 0 0
$$125$$ 1.52786i 0.136656i
$$126$$ 0 0
$$127$$ 8.94427 0.793676 0.396838 0.917889i $$-0.370108\pi$$
0.396838 + 0.917889i $$0.370108\pi$$
$$128$$ 0 0
$$129$$ 8.00000 0.704361
$$130$$ 0 0
$$131$$ 11.7082i 1.02295i 0.859298 + 0.511475i $$0.170901\pi$$
−0.859298 + 0.511475i $$0.829099\pi$$
$$132$$ 0 0
$$133$$ 1.23607i 0.107181i
$$134$$ 0 0
$$135$$ 17.8885 1.53960
$$136$$ 0 0
$$137$$ −14.9443 −1.27678 −0.638388 0.769715i $$-0.720398\pi$$
−0.638388 + 0.769715i $$0.720398\pi$$
$$138$$ 0 0
$$139$$ − 1.23607i − 0.104842i −0.998625 0.0524210i $$-0.983306\pi$$
0.998625 0.0524210i $$-0.0166938\pi$$
$$140$$ 0 0
$$141$$ 12.9443i 1.09010i
$$142$$ 0 0
$$143$$ −4.94427 −0.413461
$$144$$ 0 0
$$145$$ −14.4721 −1.20185
$$146$$ 0 0
$$147$$ 1.23607i 0.101949i
$$148$$ 0 0
$$149$$ − 2.94427i − 0.241204i −0.992701 0.120602i $$-0.961517\pi$$
0.992701 0.120602i $$-0.0384825\pi$$
$$150$$ 0 0
$$151$$ 8.94427 0.727875 0.363937 0.931423i $$-0.381432\pi$$
0.363937 + 0.931423i $$0.381432\pi$$
$$152$$ 0 0
$$153$$ 6.58359 0.532252
$$154$$ 0 0
$$155$$ 8.00000i 0.642575i
$$156$$ 0 0
$$157$$ 0.763932i 0.0609684i 0.999535 + 0.0304842i $$0.00970493\pi$$
−0.999535 + 0.0304842i $$0.990295\pi$$
$$158$$ 0 0
$$159$$ −12.3607 −0.980266
$$160$$ 0 0
$$161$$ 4.00000 0.315244
$$162$$ 0 0
$$163$$ − 3.41641i − 0.267594i −0.991009 0.133797i $$-0.957283\pi$$
0.991009 0.133797i $$-0.0427170\pi$$
$$164$$ 0 0
$$165$$ 25.8885i 2.01542i
$$166$$ 0 0
$$167$$ −23.4164 −1.81202 −0.906008 0.423261i $$-0.860886\pi$$
−0.906008 + 0.423261i $$0.860886\pi$$
$$168$$ 0 0
$$169$$ 12.4164 0.955108
$$170$$ 0 0
$$171$$ − 1.81966i − 0.139153i
$$172$$ 0 0
$$173$$ 5.70820i 0.433987i 0.976173 + 0.216993i $$0.0696250\pi$$
−0.976173 + 0.216993i $$0.930375\pi$$
$$174$$ 0 0
$$175$$ 5.47214 0.413655
$$176$$ 0 0
$$177$$ −11.4164 −0.858110
$$178$$ 0 0
$$179$$ − 7.05573i − 0.527370i −0.964609 0.263685i $$-0.915062\pi$$
0.964609 0.263685i $$-0.0849379\pi$$
$$180$$ 0 0
$$181$$ 12.1803i 0.905358i 0.891674 + 0.452679i $$0.149532\pi$$
−0.891674 + 0.452679i $$0.850468\pi$$
$$182$$ 0 0
$$183$$ −13.8885 −1.02667
$$184$$ 0 0
$$185$$ 14.4721 1.06401
$$186$$ 0 0
$$187$$ 28.9443i 2.11661i
$$188$$ 0 0
$$189$$ − 5.52786i − 0.402093i
$$190$$ 0 0
$$191$$ 4.94427 0.357755 0.178877 0.983871i $$-0.442753\pi$$
0.178877 + 0.983871i $$0.442753\pi$$
$$192$$ 0 0
$$193$$ 0.472136 0.0339851 0.0169925 0.999856i $$-0.494591\pi$$
0.0169925 + 0.999856i $$0.494591\pi$$
$$194$$ 0 0
$$195$$ 3.05573i 0.218825i
$$196$$ 0 0
$$197$$ − 10.9443i − 0.779747i −0.920868 0.389874i $$-0.872519\pi$$
0.920868 0.389874i $$-0.127481\pi$$
$$198$$ 0 0
$$199$$ −15.4164 −1.09284 −0.546420 0.837511i $$-0.684010\pi$$
−0.546420 + 0.837511i $$0.684010\pi$$
$$200$$ 0 0
$$201$$ −4.94427 −0.348742
$$202$$ 0 0
$$203$$ 4.47214i 0.313882i
$$204$$ 0 0
$$205$$ − 27.4164i − 1.91484i
$$206$$ 0 0
$$207$$ −5.88854 −0.409282
$$208$$ 0 0
$$209$$ 8.00000 0.553372
$$210$$ 0 0
$$211$$ − 12.0000i − 0.826114i −0.910705 0.413057i $$-0.864461\pi$$
0.910705 0.413057i $$-0.135539\pi$$
$$212$$ 0 0
$$213$$ 6.11146i 0.418750i
$$214$$ 0 0
$$215$$ −20.9443 −1.42839
$$216$$ 0 0
$$217$$ 2.47214 0.167820
$$218$$ 0 0
$$219$$ 3.63932i 0.245922i
$$220$$ 0 0
$$221$$ 3.41641i 0.229812i
$$222$$ 0 0
$$223$$ 12.9443 0.866813 0.433406 0.901199i $$-0.357312\pi$$
0.433406 + 0.901199i $$0.357312\pi$$
$$224$$ 0 0
$$225$$ −8.05573 −0.537049
$$226$$ 0 0
$$227$$ − 17.2361i − 1.14400i −0.820254 0.571999i $$-0.806168\pi$$
0.820254 0.571999i $$-0.193832\pi$$
$$228$$ 0 0
$$229$$ − 23.5967i − 1.55932i −0.626205 0.779658i $$-0.715393\pi$$
0.626205 0.779658i $$-0.284607\pi$$
$$230$$ 0 0
$$231$$ 8.00000 0.526361
$$232$$ 0 0
$$233$$ 15.8885 1.04089 0.520447 0.853894i $$-0.325765\pi$$
0.520447 + 0.853894i $$0.325765\pi$$
$$234$$ 0 0
$$235$$ − 33.8885i − 2.21064i
$$236$$ 0 0
$$237$$ 16.0000i 1.03931i
$$238$$ 0 0
$$239$$ −13.8885 −0.898375 −0.449188 0.893437i $$-0.648287\pi$$
−0.449188 + 0.893437i $$0.648287\pi$$
$$240$$ 0 0
$$241$$ 12.4721 0.803401 0.401700 0.915771i $$-0.368419\pi$$
0.401700 + 0.915771i $$0.368419\pi$$
$$242$$ 0 0
$$243$$ 13.5967i 0.872232i
$$244$$ 0 0
$$245$$ − 3.23607i − 0.206745i
$$246$$ 0 0
$$247$$ 0.944272 0.0600826
$$248$$ 0 0
$$249$$ 11.4164 0.723485
$$250$$ 0 0
$$251$$ − 4.29180i − 0.270896i −0.990784 0.135448i $$-0.956753\pi$$
0.990784 0.135448i $$-0.0432473\pi$$
$$252$$ 0 0
$$253$$ − 25.8885i − 1.62760i
$$254$$ 0 0
$$255$$ 17.8885 1.12022
$$256$$ 0 0
$$257$$ −14.0000 −0.873296 −0.436648 0.899632i $$-0.643834\pi$$
−0.436648 + 0.899632i $$0.643834\pi$$
$$258$$ 0 0
$$259$$ − 4.47214i − 0.277885i
$$260$$ 0 0
$$261$$ − 6.58359i − 0.407514i
$$262$$ 0 0
$$263$$ −11.0557 −0.681725 −0.340863 0.940113i $$-0.610719\pi$$
−0.340863 + 0.940113i $$0.610719\pi$$
$$264$$ 0 0
$$265$$ 32.3607 1.98790
$$266$$ 0 0
$$267$$ 7.41641i 0.453877i
$$268$$ 0 0
$$269$$ − 4.18034i − 0.254880i −0.991846 0.127440i $$-0.959324\pi$$
0.991846 0.127440i $$-0.0406760\pi$$
$$270$$ 0 0
$$271$$ −24.0000 −1.45790 −0.728948 0.684569i $$-0.759990\pi$$
−0.728948 + 0.684569i $$0.759990\pi$$
$$272$$ 0 0
$$273$$ 0.944272 0.0571499
$$274$$ 0 0
$$275$$ − 35.4164i − 2.13569i
$$276$$ 0 0
$$277$$ − 7.88854i − 0.473977i −0.971512 0.236988i $$-0.923840\pi$$
0.971512 0.236988i $$-0.0761603\pi$$
$$278$$ 0 0
$$279$$ −3.63932 −0.217880
$$280$$ 0 0
$$281$$ −26.0000 −1.55103 −0.775515 0.631329i $$-0.782510\pi$$
−0.775515 + 0.631329i $$0.782510\pi$$
$$282$$ 0 0
$$283$$ 6.18034i 0.367383i 0.982984 + 0.183692i $$0.0588048\pi$$
−0.982984 + 0.183692i $$0.941195\pi$$
$$284$$ 0 0
$$285$$ − 4.94427i − 0.292873i
$$286$$ 0 0
$$287$$ −8.47214 −0.500094
$$288$$ 0 0
$$289$$ 3.00000 0.176471
$$290$$ 0 0
$$291$$ 15.4164i 0.903726i
$$292$$ 0 0
$$293$$ 12.7639i 0.745677i 0.927896 + 0.372838i $$0.121615\pi$$
−0.927896 + 0.372838i $$0.878385\pi$$
$$294$$ 0 0
$$295$$ 29.8885 1.74018
$$296$$ 0 0
$$297$$ −35.7771 −2.07600
$$298$$ 0 0
$$299$$ − 3.05573i − 0.176717i
$$300$$ 0 0
$$301$$ 6.47214i 0.373048i
$$302$$ 0 0
$$303$$ −2.11146 −0.121300
$$304$$ 0 0
$$305$$ 36.3607 2.08201
$$306$$ 0 0
$$307$$ 1.81966i 0.103853i 0.998651 + 0.0519267i $$0.0165362\pi$$
−0.998651 + 0.0519267i $$0.983464\pi$$
$$308$$ 0 0
$$309$$ 6.83282i 0.388705i
$$310$$ 0 0
$$311$$ −8.00000 −0.453638 −0.226819 0.973937i $$-0.572833\pi$$
−0.226819 + 0.973937i $$0.572833\pi$$
$$312$$ 0 0
$$313$$ 8.47214 0.478873 0.239437 0.970912i $$-0.423037\pi$$
0.239437 + 0.970912i $$0.423037\pi$$
$$314$$ 0 0
$$315$$ 4.76393i 0.268417i
$$316$$ 0 0
$$317$$ 9.05573i 0.508620i 0.967123 + 0.254310i $$0.0818484\pi$$
−0.967123 + 0.254310i $$0.918152\pi$$
$$318$$ 0 0
$$319$$ 28.9443 1.62057
$$320$$ 0 0
$$321$$ 11.0557 0.617071
$$322$$ 0 0
$$323$$ − 5.52786i − 0.307579i
$$324$$ 0 0
$$325$$ − 4.18034i − 0.231884i
$$326$$ 0 0
$$327$$ −10.4721 −0.579110
$$328$$ 0 0
$$329$$ −10.4721 −0.577348
$$330$$ 0 0
$$331$$ − 22.4721i − 1.23518i −0.786500 0.617590i $$-0.788109\pi$$
0.786500 0.617590i $$-0.211891\pi$$
$$332$$ 0 0
$$333$$ 6.58359i 0.360779i
$$334$$ 0 0
$$335$$ 12.9443 0.707221
$$336$$ 0 0
$$337$$ 10.3607 0.564382 0.282191 0.959358i $$-0.408939\pi$$
0.282191 + 0.959358i $$0.408939\pi$$
$$338$$ 0 0
$$339$$ − 15.4164i − 0.837304i
$$340$$ 0 0
$$341$$ − 16.0000i − 0.866449i
$$342$$ 0 0
$$343$$ −1.00000 −0.0539949
$$344$$ 0 0
$$345$$ −16.0000 −0.861411
$$346$$ 0 0
$$347$$ − 6.47214i − 0.347442i −0.984795 0.173721i $$-0.944421\pi$$
0.984795 0.173721i $$-0.0555792\pi$$
$$348$$ 0 0
$$349$$ 26.6525i 1.42667i 0.700821 + 0.713337i $$0.252817\pi$$
−0.700821 + 0.713337i $$0.747183\pi$$
$$350$$ 0 0
$$351$$ −4.22291 −0.225402
$$352$$ 0 0
$$353$$ −15.8885 −0.845662 −0.422831 0.906209i $$-0.638964\pi$$
−0.422831 + 0.906209i $$0.638964\pi$$
$$354$$ 0 0
$$355$$ − 16.0000i − 0.849192i
$$356$$ 0 0
$$357$$ − 5.52786i − 0.292566i
$$358$$ 0 0
$$359$$ 16.9443 0.894284 0.447142 0.894463i $$-0.352442\pi$$
0.447142 + 0.894463i $$0.352442\pi$$
$$360$$ 0 0
$$361$$ 17.4721 0.919586
$$362$$ 0 0
$$363$$ − 38.1803i − 2.00395i
$$364$$ 0 0
$$365$$ − 9.52786i − 0.498711i
$$366$$ 0 0
$$367$$ −22.8328 −1.19186 −0.595932 0.803035i $$-0.703217\pi$$
−0.595932 + 0.803035i $$0.703217\pi$$
$$368$$ 0 0
$$369$$ 12.4721 0.649273
$$370$$ 0 0
$$371$$ − 10.0000i − 0.519174i
$$372$$ 0 0
$$373$$ − 2.94427i − 0.152449i −0.997091 0.0762243i $$-0.975713\pi$$
0.997091 0.0762243i $$-0.0242865\pi$$
$$374$$ 0 0
$$375$$ −1.88854 −0.0975240
$$376$$ 0 0
$$377$$ 3.41641 0.175954
$$378$$ 0 0
$$379$$ 4.58359i 0.235443i 0.993047 + 0.117722i $$0.0375591\pi$$
−0.993047 + 0.117722i $$0.962441\pi$$
$$380$$ 0 0
$$381$$ 11.0557i 0.566402i
$$382$$ 0 0
$$383$$ −15.4164 −0.787742 −0.393871 0.919166i $$-0.628864\pi$$
−0.393871 + 0.919166i $$0.628864\pi$$
$$384$$ 0 0
$$385$$ −20.9443 −1.06742
$$386$$ 0 0
$$387$$ − 9.52786i − 0.484329i
$$388$$ 0 0
$$389$$ 4.47214i 0.226746i 0.993552 + 0.113373i $$0.0361656\pi$$
−0.993552 + 0.113373i $$0.963834\pi$$
$$390$$ 0 0
$$391$$ −17.8885 −0.904663
$$392$$ 0 0
$$393$$ −14.4721 −0.730023
$$394$$ 0 0
$$395$$ − 41.8885i − 2.10764i
$$396$$ 0 0
$$397$$ − 15.2361i − 0.764676i −0.924022 0.382338i $$-0.875119\pi$$
0.924022 0.382338i $$-0.124881\pi$$
$$398$$ 0 0
$$399$$ −1.52786 −0.0764889
$$400$$ 0 0
$$401$$ −23.5279 −1.17493 −0.587463 0.809251i $$-0.699873\pi$$
−0.587463 + 0.809251i $$0.699873\pi$$
$$402$$ 0 0
$$403$$ − 1.88854i − 0.0940751i
$$404$$ 0 0
$$405$$ 7.81966i 0.388562i
$$406$$ 0 0
$$407$$ −28.9443 −1.43471
$$408$$ 0 0
$$409$$ 21.4164 1.05897 0.529487 0.848318i $$-0.322385\pi$$
0.529487 + 0.848318i $$0.322385\pi$$
$$410$$ 0 0
$$411$$ − 18.4721i − 0.911163i
$$412$$ 0 0
$$413$$ − 9.23607i − 0.454477i
$$414$$ 0 0
$$415$$ −29.8885 −1.46717
$$416$$ 0 0
$$417$$ 1.52786 0.0748198
$$418$$ 0 0
$$419$$ 22.1803i 1.08358i 0.840514 + 0.541790i $$0.182253\pi$$
−0.840514 + 0.541790i $$0.817747\pi$$
$$420$$ 0 0
$$421$$ 2.00000i 0.0974740i 0.998812 + 0.0487370i $$0.0155196\pi$$
−0.998812 + 0.0487370i $$0.984480\pi$$
$$422$$ 0 0
$$423$$ 15.4164 0.749571
$$424$$ 0 0
$$425$$ −24.4721 −1.18707
$$426$$ 0 0
$$427$$ − 11.2361i − 0.543751i
$$428$$ 0 0
$$429$$ − 6.11146i − 0.295064i
$$430$$ 0 0
$$431$$ −28.0000 −1.34871 −0.674356 0.738406i $$-0.735579\pi$$
−0.674356 + 0.738406i $$0.735579\pi$$
$$432$$ 0 0
$$433$$ 9.41641 0.452524 0.226262 0.974067i $$-0.427349\pi$$
0.226262 + 0.974067i $$0.427349\pi$$
$$434$$ 0 0
$$435$$ − 17.8885i − 0.857690i
$$436$$ 0 0
$$437$$ 4.94427i 0.236517i
$$438$$ 0 0
$$439$$ 32.0000 1.52728 0.763638 0.645644i $$-0.223411\pi$$
0.763638 + 0.645644i $$0.223411\pi$$
$$440$$ 0 0
$$441$$ 1.47214 0.0701017
$$442$$ 0 0
$$443$$ 13.8885i 0.659865i 0.944005 + 0.329932i $$0.107026\pi$$
−0.944005 + 0.329932i $$0.892974\pi$$
$$444$$ 0 0
$$445$$ − 19.4164i − 0.920426i
$$446$$ 0 0
$$447$$ 3.63932 0.172134
$$448$$ 0 0
$$449$$ −7.88854 −0.372283 −0.186142 0.982523i $$-0.559598\pi$$
−0.186142 + 0.982523i $$0.559598\pi$$
$$450$$ 0 0
$$451$$ 54.8328i 2.58198i
$$452$$ 0 0
$$453$$ 11.0557i 0.519443i
$$454$$ 0 0
$$455$$ −2.47214 −0.115896
$$456$$ 0 0
$$457$$ 7.52786 0.352139 0.176069 0.984378i $$-0.443662\pi$$
0.176069 + 0.984378i $$0.443662\pi$$
$$458$$ 0 0
$$459$$ 24.7214i 1.15389i
$$460$$ 0 0
$$461$$ 21.7082i 1.01105i 0.862811 + 0.505526i $$0.168702\pi$$
−0.862811 + 0.505526i $$0.831298\pi$$
$$462$$ 0 0
$$463$$ −35.7771 −1.66270 −0.831351 0.555748i $$-0.812432\pi$$
−0.831351 + 0.555748i $$0.812432\pi$$
$$464$$ 0 0
$$465$$ −9.88854 −0.458570
$$466$$ 0 0
$$467$$ − 32.0689i − 1.48397i −0.670416 0.741985i $$-0.733884\pi$$
0.670416 0.741985i $$-0.266116\pi$$
$$468$$ 0 0
$$469$$ − 4.00000i − 0.184703i
$$470$$ 0 0
$$471$$ −0.944272 −0.0435098
$$472$$ 0 0
$$473$$ 41.8885 1.92604
$$474$$ 0 0
$$475$$ 6.76393i 0.310350i
$$476$$ 0 0
$$477$$ 14.7214i 0.674045i
$$478$$ 0 0
$$479$$ −8.58359 −0.392194 −0.196097 0.980584i $$-0.562827\pi$$
−0.196097 + 0.980584i $$0.562827\pi$$
$$480$$ 0 0
$$481$$ −3.41641 −0.155775
$$482$$ 0 0
$$483$$ 4.94427i 0.224972i
$$484$$ 0 0
$$485$$ − 40.3607i − 1.83268i
$$486$$ 0 0
$$487$$ −20.0000 −0.906287 −0.453143 0.891438i $$-0.649697\pi$$
−0.453143 + 0.891438i $$0.649697\pi$$
$$488$$ 0 0
$$489$$ 4.22291 0.190967
$$490$$ 0 0
$$491$$ − 37.8885i − 1.70989i −0.518722 0.854943i $$-0.673592\pi$$
0.518722 0.854943i $$-0.326408\pi$$
$$492$$ 0 0
$$493$$ − 20.0000i − 0.900755i
$$494$$ 0 0
$$495$$ 30.8328 1.38583
$$496$$ 0 0
$$497$$ −4.94427 −0.221781
$$498$$ 0 0
$$499$$ 21.8885i 0.979866i 0.871760 + 0.489933i $$0.162979\pi$$
−0.871760 + 0.489933i $$0.837021\pi$$
$$500$$ 0 0
$$501$$ − 28.9443i − 1.29313i
$$502$$ 0 0
$$503$$ 4.94427 0.220454 0.110227 0.993906i $$-0.464842\pi$$
0.110227 + 0.993906i $$0.464842\pi$$
$$504$$ 0 0
$$505$$ 5.52786 0.245987
$$506$$ 0 0
$$507$$ 15.3475i 0.681607i
$$508$$ 0 0
$$509$$ − 41.1246i − 1.82282i −0.411503 0.911408i $$-0.634996\pi$$
0.411503 0.911408i $$-0.365004\pi$$
$$510$$ 0 0
$$511$$ −2.94427 −0.130247
$$512$$ 0 0
$$513$$ 6.83282 0.301676
$$514$$ 0 0
$$515$$ − 17.8885i − 0.788263i
$$516$$ 0 0
$$517$$ 67.7771i 2.98083i
$$518$$ 0 0
$$519$$ −7.05573 −0.309712
$$520$$ 0 0
$$521$$ 6.58359 0.288432 0.144216 0.989546i $$-0.453934\pi$$
0.144216 + 0.989546i $$0.453934\pi$$
$$522$$ 0 0
$$523$$ 4.29180i 0.187667i 0.995588 + 0.0938336i $$0.0299122\pi$$
−0.995588 + 0.0938336i $$0.970088\pi$$
$$524$$ 0 0
$$525$$ 6.76393i 0.295202i
$$526$$ 0 0
$$527$$ −11.0557 −0.481595
$$528$$ 0 0
$$529$$ −7.00000 −0.304348
$$530$$ 0 0
$$531$$ 13.5967i 0.590049i
$$532$$ 0 0
$$533$$ 6.47214i 0.280339i
$$534$$ 0 0
$$535$$ −28.9443 −1.25137
$$536$$ 0 0
$$537$$ 8.72136 0.376354
$$538$$ 0 0
$$539$$ 6.47214i 0.278775i
$$540$$ 0 0
$$541$$ − 5.05573i − 0.217363i −0.994077 0.108681i $$-0.965337\pi$$
0.994077 0.108681i $$-0.0346628\pi$$
$$542$$ 0 0
$$543$$ −15.0557 −0.646103
$$544$$ 0 0
$$545$$ 27.4164 1.17439
$$546$$ 0 0
$$547$$ − 4.58359i − 0.195980i −0.995187 0.0979901i $$-0.968759\pi$$
0.995187 0.0979901i $$-0.0312414\pi$$
$$548$$ 0 0
$$549$$ 16.5410i 0.705954i
$$550$$ 0 0
$$551$$ −5.52786 −0.235495
$$552$$ 0 0
$$553$$ −12.9443 −0.550446
$$554$$ 0 0
$$555$$ 17.8885i 0.759326i
$$556$$ 0 0
$$557$$ 9.05573i 0.383704i 0.981424 + 0.191852i $$0.0614493\pi$$
−0.981424 + 0.191852i $$0.938551\pi$$
$$558$$ 0 0
$$559$$ 4.94427 0.209120
$$560$$ 0 0
$$561$$ −35.7771 −1.51051
$$562$$ 0 0
$$563$$ − 17.8197i − 0.751009i −0.926821 0.375505i $$-0.877469\pi$$
0.926821 0.375505i $$-0.122531\pi$$
$$564$$ 0 0
$$565$$ 40.3607i 1.69799i
$$566$$ 0 0
$$567$$ 2.41641 0.101480
$$568$$ 0 0
$$569$$ −26.3607 −1.10510 −0.552549 0.833481i $$-0.686345\pi$$
−0.552549 + 0.833481i $$0.686345\pi$$
$$570$$ 0 0
$$571$$ 14.4721i 0.605640i 0.953048 + 0.302820i $$0.0979281\pi$$
−0.953048 + 0.302820i $$0.902072\pi$$
$$572$$ 0 0
$$573$$ 6.11146i 0.255310i
$$574$$ 0 0
$$575$$ 21.8885 0.912815
$$576$$ 0 0
$$577$$ −6.00000 −0.249783 −0.124892 0.992170i $$-0.539858\pi$$
−0.124892 + 0.992170i $$0.539858\pi$$
$$578$$ 0 0
$$579$$ 0.583592i 0.0242533i
$$580$$ 0 0
$$581$$ 9.23607i 0.383177i
$$582$$ 0 0
$$583$$ −64.7214 −2.68048
$$584$$ 0 0
$$585$$ 3.63932 0.150467
$$586$$ 0 0
$$587$$ 3.70820i 0.153054i 0.997068 + 0.0765270i $$0.0243831\pi$$
−0.997068 + 0.0765270i $$0.975617\pi$$
$$588$$ 0 0
$$589$$ 3.05573i 0.125909i
$$590$$ 0 0
$$591$$ 13.5279 0.556462
$$592$$ 0 0
$$593$$ 32.8328 1.34828 0.674141 0.738603i $$-0.264514\pi$$
0.674141 + 0.738603i $$0.264514\pi$$
$$594$$ 0 0
$$595$$ 14.4721i 0.593300i
$$596$$ 0 0
$$597$$ − 19.0557i − 0.779899i
$$598$$ 0 0
$$599$$ −17.8885 −0.730906 −0.365453 0.930830i $$-0.619086\pi$$
−0.365453 + 0.930830i $$0.619086\pi$$
$$600$$ 0 0
$$601$$ −29.7771 −1.21463 −0.607316 0.794460i $$-0.707754\pi$$
−0.607316 + 0.794460i $$0.707754\pi$$
$$602$$ 0 0
$$603$$ 5.88854i 0.239800i
$$604$$ 0 0
$$605$$ 99.9574i 4.06385i
$$606$$ 0 0
$$607$$ 9.88854 0.401364 0.200682 0.979656i $$-0.435684\pi$$
0.200682 + 0.979656i $$0.435684\pi$$
$$608$$ 0 0
$$609$$ −5.52786 −0.224000
$$610$$ 0 0
$$611$$ 8.00000i 0.323645i
$$612$$ 0 0
$$613$$ − 29.4164i − 1.18812i −0.804422 0.594059i $$-0.797525\pi$$
0.804422 0.594059i $$-0.202475\pi$$
$$614$$ 0 0
$$615$$ 33.8885 1.36652
$$616$$ 0 0
$$617$$ −34.3607 −1.38331 −0.691654 0.722229i $$-0.743118\pi$$
−0.691654 + 0.722229i $$0.743118\pi$$
$$618$$ 0 0
$$619$$ 48.0689i 1.93205i 0.258446 + 0.966026i $$0.416790\pi$$
−0.258446 + 0.966026i $$0.583210\pi$$
$$620$$ 0 0
$$621$$ − 22.1115i − 0.887302i
$$622$$ 0 0
$$623$$ −6.00000 −0.240385
$$624$$ 0 0
$$625$$ −22.4164 −0.896656
$$626$$ 0 0
$$627$$ 9.88854i 0.394910i
$$628$$ 0 0
$$629$$ 20.0000i 0.797452i
$$630$$ 0 0
$$631$$ 44.9443 1.78920 0.894602 0.446865i $$-0.147459\pi$$
0.894602 + 0.446865i $$0.147459\pi$$
$$632$$ 0 0
$$633$$ 14.8328 0.589551
$$634$$ 0 0
$$635$$ − 28.9443i − 1.14862i
$$636$$ 0 0
$$637$$ 0.763932i 0.0302681i
$$638$$ 0 0
$$639$$ 7.27864 0.287939
$$640$$ 0 0
$$641$$ 14.5836 0.576017 0.288009 0.957628i $$-0.407007\pi$$
0.288009 + 0.957628i $$0.407007\pi$$
$$642$$ 0 0
$$643$$ − 43.7082i − 1.72368i −0.507177 0.861842i $$-0.669311\pi$$
0.507177 0.861842i $$-0.330689\pi$$
$$644$$ 0 0
$$645$$ − 25.8885i − 1.01936i
$$646$$ 0 0
$$647$$ 20.3607 0.800461 0.400230 0.916415i $$-0.368930\pi$$
0.400230 + 0.916415i $$0.368930\pi$$
$$648$$ 0 0
$$649$$ −59.7771 −2.34646
$$650$$ 0 0
$$651$$ 3.05573i 0.119763i
$$652$$ 0 0
$$653$$ − 9.41641i − 0.368493i −0.982880 0.184246i $$-0.941016\pi$$
0.982880 0.184246i $$-0.0589844\pi$$
$$654$$ 0 0
$$655$$ 37.8885 1.48043
$$656$$ 0 0
$$657$$ 4.33437 0.169100
$$658$$ 0 0
$$659$$ 37.3050i 1.45319i 0.687064 + 0.726597i $$0.258899\pi$$
−0.687064 + 0.726597i $$0.741101\pi$$
$$660$$ 0 0
$$661$$ 22.6525i 0.881079i 0.897733 + 0.440540i $$0.145213\pi$$
−0.897733 + 0.440540i $$0.854787\pi$$
$$662$$ 0 0
$$663$$ −4.22291 −0.164004
$$664$$ 0 0
$$665$$ 4.00000 0.155113
$$666$$ 0 0
$$667$$ 17.8885i 0.692647i
$$668$$ 0 0
$$669$$ 16.0000i 0.618596i
$$670$$ 0 0
$$671$$ −72.7214 −2.80738
$$672$$ 0 0
$$673$$ −2.94427 −0.113493 −0.0567467 0.998389i $$-0.518073\pi$$
−0.0567467 + 0.998389i $$0.518073\pi$$
$$674$$ 0 0
$$675$$ − 30.2492i − 1.16429i
$$676$$ 0 0
$$677$$ − 19.8197i − 0.761731i −0.924630 0.380866i $$-0.875626\pi$$
0.924630 0.380866i $$-0.124374\pi$$
$$678$$ 0 0
$$679$$ −12.4721 −0.478637
$$680$$ 0 0
$$681$$ 21.3050 0.816408
$$682$$ 0 0
$$683$$ − 23.7771i − 0.909805i −0.890541 0.454902i $$-0.849674\pi$$
0.890541 0.454902i $$-0.150326\pi$$
$$684$$ 0 0
$$685$$ 48.3607i 1.84777i
$$686$$ 0 0
$$687$$ 29.1672 1.11280
$$688$$ 0 0
$$689$$ −7.63932 −0.291035
$$690$$ 0 0
$$691$$ − 14.1803i − 0.539446i −0.962938 0.269723i $$-0.913068\pi$$
0.962938 0.269723i $$-0.0869321\pi$$
$$692$$ 0 0
$$693$$ − 9.52786i − 0.361934i
$$694$$ 0 0
$$695$$ −4.00000 −0.151729
$$696$$ 0 0
$$697$$ 37.8885 1.43513
$$698$$ 0 0
$$699$$ 19.6393i 0.742827i
$$700$$ 0 0
$$701$$ − 41.4164i − 1.56428i −0.623105 0.782138i $$-0.714129\pi$$
0.623105 0.782138i $$-0.285871\pi$$
$$702$$ 0 0
$$703$$ 5.52786 0.208487
$$704$$ 0 0
$$705$$ 41.8885 1.57761
$$706$$ 0 0
$$707$$ − 1.70820i − 0.0642436i
$$708$$ 0 0
$$709$$ − 1.63932i − 0.0615660i −0.999526 0.0307830i $$-0.990200\pi$$
0.999526 0.0307830i $$-0.00980008\pi$$
$$710$$ 0 0
$$711$$ 19.0557 0.714646
$$712$$ 0 0
$$713$$ 9.88854 0.370329
$$714$$ 0 0
$$715$$ 16.0000i 0.598366i
$$716$$ 0 0
$$717$$ − 17.1672i − 0.641120i
$$718$$ 0 0
$$719$$ 51.1935 1.90920 0.954598 0.297898i $$-0.0962856\pi$$
0.954598 + 0.297898i $$0.0962856\pi$$
$$720$$ 0 0
$$721$$ −5.52786 −0.205868
$$722$$ 0 0
$$723$$ 15.4164i 0.573342i
$$724$$ 0 0
$$725$$ 24.4721i 0.908872i
$$726$$ 0 0
$$727$$ −12.3607 −0.458432 −0.229216 0.973376i $$-0.573616\pi$$
−0.229216 + 0.973376i $$0.573616\pi$$
$$728$$ 0 0
$$729$$ −24.0557 −0.890953
$$730$$ 0 0
$$731$$ − 28.9443i − 1.07054i
$$732$$ 0 0
$$733$$ − 4.76393i − 0.175960i −0.996122 0.0879799i $$-0.971959\pi$$
0.996122 0.0879799i $$-0.0280411\pi$$
$$734$$ 0 0
$$735$$ 4.00000 0.147542
$$736$$ 0 0
$$737$$ −25.8885 −0.953617
$$738$$ 0 0
$$739$$ − 3.41641i − 0.125675i −0.998024 0.0628373i $$-0.979985\pi$$
0.998024 0.0628373i $$-0.0200149\pi$$
$$740$$ 0 0
$$741$$ 1.16718i 0.0428776i
$$742$$ 0 0
$$743$$ −24.9443 −0.915117 −0.457558 0.889180i $$-0.651276\pi$$
−0.457558 + 0.889180i $$0.651276\pi$$
$$744$$ 0 0
$$745$$ −9.52786 −0.349074
$$746$$ 0 0
$$747$$ − 13.5967i − 0.497479i
$$748$$ 0 0
$$749$$ 8.94427i 0.326817i
$$750$$ 0 0
$$751$$ 36.0000 1.31366 0.656829 0.754039i $$-0.271897\pi$$
0.656829 + 0.754039i $$0.271897\pi$$
$$752$$ 0 0
$$753$$ 5.30495 0.193323
$$754$$ 0 0
$$755$$ − 28.9443i − 1.05339i
$$756$$ 0 0
$$757$$ − 39.3050i − 1.42856i −0.699859 0.714281i $$-0.746754\pi$$
0.699859 0.714281i $$-0.253246\pi$$
$$758$$ 0 0
$$759$$ 32.0000 1.16153
$$760$$ 0 0
$$761$$ 3.52786 0.127885 0.0639425 0.997954i $$-0.479633\pi$$
0.0639425 + 0.997954i $$0.479633\pi$$
$$762$$ 0 0
$$763$$ − 8.47214i − 0.306712i
$$764$$ 0 0
$$765$$ − 21.3050i − 0.770282i
$$766$$ 0 0
$$767$$ −7.05573 −0.254768
$$768$$ 0 0
$$769$$ −18.3607 −0.662103 −0.331052 0.943613i $$-0.607403\pi$$
−0.331052 + 0.943613i $$0.607403\pi$$
$$770$$ 0 0
$$771$$ − 17.3050i − 0.623223i
$$772$$ 0 0
$$773$$ − 40.1803i − 1.44519i −0.691274 0.722593i $$-0.742950\pi$$
0.691274 0.722593i $$-0.257050\pi$$
$$774$$ 0 0
$$775$$ 13.5279 0.485935
$$776$$ 0 0
$$777$$ 5.52786 0.198311
$$778$$ 0 0
$$779$$ − 10.4721i − 0.375203i
$$780$$ 0 0
$$781$$ 32.0000i 1.14505i
$$782$$ 0 0
$$783$$ 24.7214 0.883469
$$784$$ 0 0
$$785$$ 2.47214 0.0882343
$$786$$ 0 0
$$787$$ − 12.2918i − 0.438155i −0.975707 0.219078i $$-0.929695\pi$$
0.975707 0.219078i $$-0.0703048\pi$$
$$788$$ 0 0
$$789$$ − 13.6656i − 0.486509i
$$790$$ 0 0
$$791$$ 12.4721 0.443458
$$792$$ 0 0
$$793$$ −8.58359 −0.304812
$$794$$ 0 0
$$795$$ 40.0000i 1.41865i
$$796$$ 0 0
$$797$$ − 52.1803i − 1.84832i −0.382003 0.924161i $$-0.624765\pi$$
0.382003 0.924161i $$-0.375235\pi$$
$$798$$ 0 0
$$799$$ 46.8328 1.65683
$$800$$ 0 0
$$801$$ 8.83282 0.312092
$$802$$ 0 0
$$803$$ 19.0557i 0.672462i
$$804$$ 0 0
$$805$$ − 12.9443i − 0.456226i
$$806$$ 0 0
$$807$$ 5.16718 0.181894
$$808$$ 0 0
$$809$$ 17.4164 0.612328 0.306164 0.951979i $$-0.400954\pi$$
0.306164 + 0.951979i $$0.400954\pi$$
$$810$$ 0 0
$$811$$ 53.0132i 1.86154i 0.365601 + 0.930772i $$0.380864\pi$$
−0.365601 + 0.930772i $$0.619136\pi$$
$$812$$ 0 0
$$813$$ − 29.6656i − 1.04042i
$$814$$ 0 0
$$815$$ −11.0557 −0.387265
$$816$$ 0 0
$$817$$ −8.00000 −0.279885
$$818$$ 0 0
$$819$$ − 1.12461i − 0.0392971i
$$820$$ 0 0
$$821$$ − 4.11146i − 0.143491i −0.997423 0.0717454i $$-0.977143\pi$$
0.997423 0.0717454i $$-0.0228569\pi$$
$$822$$ 0 0
$$823$$ 19.7771 0.689386 0.344693 0.938715i $$-0.387983\pi$$
0.344693 + 0.938715i $$0.387983\pi$$
$$824$$ 0 0
$$825$$ 43.7771 1.52412
$$826$$ 0 0
$$827$$ − 15.0557i − 0.523539i −0.965130 0.261769i $$-0.915694\pi$$
0.965130 0.261769i $$-0.0843060\pi$$
$$828$$ 0 0
$$829$$ 11.2361i 0.390245i 0.980779 + 0.195122i $$0.0625104\pi$$
−0.980779 + 0.195122i $$0.937490\pi$$
$$830$$ 0 0
$$831$$ 9.75078 0.338251
$$832$$ 0 0
$$833$$ 4.47214 0.154950
$$834$$ 0 0
$$835$$ 75.7771i 2.62237i
$$836$$ 0 0
$$837$$ − 13.6656i − 0.472353i
$$838$$ 0 0
$$839$$ 21.5279 0.743224 0.371612 0.928388i $$-0.378805\pi$$
0.371612 + 0.928388i $$0.378805\pi$$
$$840$$ 0 0
$$841$$ 9.00000 0.310345
$$842$$ 0 0
$$843$$ − 32.1378i − 1.10688i
$$844$$ 0 0
$$845$$ − 40.1803i − 1.38225i
$$846$$ 0 0
$$847$$ 30.8885 1.06134
$$848$$ 0 0
$$849$$ −7.63932 −0.262181
$$850$$ 0 0
$$851$$ − 17.8885i − 0.613211i
$$852$$ 0 0
$$853$$ − 13.7082i − 0.469360i −0.972073 0.234680i $$-0.924596\pi$$
0.972073 0.234680i $$-0.0754042\pi$$
$$854$$ 0 0
$$855$$ −5.88854 −0.201384
$$856$$ 0 0
$$857$$ 27.5279 0.940334 0.470167 0.882577i $$-0.344194\pi$$
0.470167 + 0.882577i $$0.344194\pi$$
$$858$$ 0 0
$$859$$ 33.8197i 1.15391i 0.816775 + 0.576956i $$0.195760\pi$$
−0.816775 + 0.576956i $$0.804240\pi$$
$$860$$ 0 0
$$861$$ − 10.4721i − 0.356889i
$$862$$ 0 0
$$863$$ −20.9443 −0.712951 −0.356476 0.934305i $$-0.616022\pi$$
−0.356476 + 0.934305i $$0.616022\pi$$
$$864$$ 0 0
$$865$$ 18.4721 0.628071
$$866$$ 0 0
$$867$$ 3.70820i 0.125937i
$$868$$ 0 0
$$869$$ 83.7771i 2.84194i
$$870$$ 0 0
$$871$$ −3.05573 −0.103539
$$872$$ 0 0
$$873$$ 18.3607 0.621415
$$874$$ 0 0
$$875$$ − 1.52786i − 0.0516512i
$$876$$ 0 0
$$877$$ 13.4164i 0.453040i 0.974007 + 0.226520i $$0.0727348\pi$$
−0.974007 + 0.226520i $$0.927265\pi$$
$$878$$ 0 0
$$879$$ −15.7771 −0.532148
$$880$$ 0 0
$$881$$ 24.8328 0.836639 0.418319 0.908300i $$-0.362619\pi$$
0.418319 + 0.908300i $$0.362619\pi$$
$$882$$ 0 0
$$883$$ 23.0557i 0.775887i 0.921683 + 0.387944i $$0.126814\pi$$
−0.921683 + 0.387944i $$0.873186\pi$$
$$884$$ 0 0
$$885$$ 36.9443i 1.24187i
$$886$$ 0 0
$$887$$ 33.3050 1.11827 0.559135 0.829076i $$-0.311133\pi$$
0.559135 + 0.829076i $$0.311133\pi$$
$$888$$ 0 0
$$889$$ −8.94427 −0.299981
$$890$$ 0 0
$$891$$ − 15.6393i − 0.523937i
$$892$$ 0 0
$$893$$ − 12.9443i − 0.433164i
$$894$$ 0 0
$$895$$ −22.8328 −0.763217
$$896$$ 0 0
$$897$$ 3.77709 0.126113
$$898$$ 0 0
$$899$$ 11.0557i 0.368729i
$$900$$ 0 0
$$901$$ 44.7214i 1.48988i
$$902$$ 0 0
$$903$$ −8.00000 −0.266223
$$904$$ 0 0
$$905$$ 39.4164 1.31025
$$906$$ 0 0
$$907$$ − 0.944272i − 0.0313540i −0.999877 0.0156770i $$-0.995010\pi$$
0.999877 0.0156770i $$-0.00499035\pi$$
$$908$$ 0 0
$$909$$ 2.51471i 0.0834076i
$$910$$ 0 0
$$911$$ −34.8328 −1.15406 −0.577031 0.816722i $$-0.695789\pi$$
−0.577031 + 0.816722i $$0.695789\pi$$
$$912$$ 0 0
$$913$$ 59.7771 1.97833
$$914$$ 0 0
$$915$$ 44.9443i 1.48581i
$$916$$ 0 0
$$917$$ − 11.7082i − 0.386639i
$$918$$ 0 0
$$919$$ 35.7771 1.18018 0.590089 0.807338i $$-0.299093\pi$$
0.590089 + 0.807338i $$0.299093\pi$$
$$920$$ 0 0
$$921$$ −2.24922 −0.0741144
$$922$$ 0 0
$$923$$ 3.77709i 0.124324i
$$924$$ 0 0
$$925$$ − 24.4721i − 0.804639i
$$926$$ 0 0
$$927$$ 8.13777 0.267279
$$928$$ 0 0
$$929$$ −47.3050 −1.55203 −0.776013 0.630717i $$-0.782761\pi$$
−0.776013 + 0.630717i $$0.782761\pi$$
$$930$$ 0 0
$$931$$ − 1.23607i − 0.0405105i
$$932$$ 0 0
$$933$$ − 9.88854i − 0.323736i
$$934$$ 0 0
$$935$$ 93.6656 3.06319
$$936$$ 0 0
$$937$$ 9.05573 0.295838 0.147919 0.988999i $$-0.452743\pi$$
0.147919 + 0.988999i $$0.452743\pi$$
$$938$$ 0 0
$$939$$ 10.4721i 0.341745i
$$940$$ 0 0
$$941$$ − 35.5967i − 1.16042i −0.814467 0.580210i $$-0.802970\pi$$
0.814467 0.580210i $$-0.197030\pi$$
$$942$$ 0 0
$$943$$ −33.8885 −1.10356
$$944$$ 0 0
$$945$$ −17.8885 −0.581914
$$946$$ 0 0
$$947$$ − 4.58359i − 0.148947i −0.997223 0.0744734i $$-0.976272\pi$$
0.997223 0.0744734i $$-0.0237276\pi$$
$$948$$ 0 0
$$949$$ 2.24922i 0.0730129i
$$950$$ 0 0
$$951$$ −11.1935 −0.362974
$$952$$ 0 0
$$953$$ −51.8885 −1.68083 −0.840417 0.541940i $$-0.817690\pi$$
−0.840417 + 0.541940i $$0.817690\pi$$
$$954$$ 0 0
$$955$$ − 16.0000i − 0.517748i
$$956$$ 0 0
$$957$$ 35.7771i 1.15651i
$$958$$ 0 0
$$959$$ 14.9443 0.482576
$$960$$ 0 0
$$961$$ −24.8885 −0.802856
$$962$$ 0 0
$$963$$ − 13.1672i − 0.424307i
$$964$$ 0 0
$$965$$ − 1.52786i − 0.0491837i
$$966$$ 0 0
$$967$$ −29.8885 −0.961151 −0.480575 0.876953i $$-0.659572\pi$$
−0.480575 + 0.876953i $$0.659572\pi$$
$$968$$ 0 0
$$969$$ 6.83282 0.219502
$$970$$ 0 0
$$971$$ − 22.7639i − 0.730529i −0.930904 0.365265i $$-0.880978\pi$$
0.930904 0.365265i $$-0.119022\pi$$
$$972$$ 0 0
$$973$$ 1.23607i 0.0396265i
$$974$$ 0 0
$$975$$ 5.16718 0.165482
$$976$$ 0 0
$$977$$ −12.8328 −0.410558 −0.205279 0.978703i $$-0.565810\pi$$
−0.205279 + 0.978703i $$0.565810\pi$$
$$978$$ 0 0
$$979$$ 38.8328i 1.24110i
$$980$$ 0 0
$$981$$ 12.4721i 0.398205i
$$982$$ 0 0
$$983$$ 5.52786 0.176311 0.0881557 0.996107i $$-0.471903\pi$$
0.0881557 + 0.996107i $$0.471903\pi$$
$$984$$ 0 0
$$985$$ −35.4164 −1.12846
$$986$$ 0 0
$$987$$ − 12.9443i − 0.412021i
$$988$$ 0 0
$$989$$ 25.8885i 0.823208i
$$990$$ 0 0
$$991$$ −24.0000 −0.762385 −0.381193 0.924496i $$-0.624487\pi$$
−0.381193 + 0.924496i $$0.624487\pi$$
$$992$$ 0 0
$$993$$ 27.7771 0.881479
$$994$$ 0 0
$$995$$ 49.8885i 1.58157i
$$996$$ 0 0
$$997$$ − 18.0689i − 0.572247i −0.958193 0.286124i $$-0.907633\pi$$
0.958193 0.286124i $$-0.0923668\pi$$
$$998$$ 0 0
$$999$$ −24.7214 −0.782149
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 1792.2.b.k.897.3 4
4.3 odd 2 1792.2.b.m.897.2 4
8.3 odd 2 1792.2.b.m.897.3 4
8.5 even 2 inner 1792.2.b.k.897.2 4
16.3 odd 4 448.2.a.i.1.2 2
16.5 even 4 224.2.a.c.1.2 2
16.11 odd 4 224.2.a.d.1.1 yes 2
16.13 even 4 448.2.a.j.1.1 2
48.5 odd 4 2016.2.a.r.1.1 2
48.11 even 4 2016.2.a.o.1.1 2
48.29 odd 4 4032.2.a.bw.1.2 2
48.35 even 4 4032.2.a.bv.1.2 2
80.59 odd 4 5600.2.a.z.1.2 2
80.69 even 4 5600.2.a.bk.1.1 2
112.5 odd 12 1568.2.i.n.1537.2 4
112.11 odd 12 1568.2.i.m.961.2 4
112.13 odd 4 3136.2.a.bf.1.2 2
112.27 even 4 1568.2.a.k.1.2 2
112.37 even 12 1568.2.i.v.1537.1 4
112.53 even 12 1568.2.i.v.961.1 4
112.59 even 12 1568.2.i.w.961.1 4
112.69 odd 4 1568.2.a.v.1.1 2
112.75 even 12 1568.2.i.w.1537.1 4
112.83 even 4 3136.2.a.by.1.1 2
112.101 odd 12 1568.2.i.n.961.2 4
112.107 odd 12 1568.2.i.m.1537.2 4

By twisted newform
Twist Min Dim Char Parity Ord Type
224.2.a.c.1.2 2 16.5 even 4
224.2.a.d.1.1 yes 2 16.11 odd 4
448.2.a.i.1.2 2 16.3 odd 4
448.2.a.j.1.1 2 16.13 even 4
1568.2.a.k.1.2 2 112.27 even 4
1568.2.a.v.1.1 2 112.69 odd 4
1568.2.i.m.961.2 4 112.11 odd 12
1568.2.i.m.1537.2 4 112.107 odd 12
1568.2.i.n.961.2 4 112.101 odd 12
1568.2.i.n.1537.2 4 112.5 odd 12
1568.2.i.v.961.1 4 112.53 even 12
1568.2.i.v.1537.1 4 112.37 even 12
1568.2.i.w.961.1 4 112.59 even 12
1568.2.i.w.1537.1 4 112.75 even 12
1792.2.b.k.897.2 4 8.5 even 2 inner
1792.2.b.k.897.3 4 1.1 even 1 trivial
1792.2.b.m.897.2 4 4.3 odd 2
1792.2.b.m.897.3 4 8.3 odd 2
2016.2.a.o.1.1 2 48.11 even 4
2016.2.a.r.1.1 2 48.5 odd 4
3136.2.a.bf.1.2 2 112.13 odd 4
3136.2.a.by.1.1 2 112.83 even 4
4032.2.a.bv.1.2 2 48.35 even 4
4032.2.a.bw.1.2 2 48.29 odd 4
5600.2.a.z.1.2 2 80.59 odd 4
5600.2.a.bk.1.1 2 80.69 even 4