# Properties

 Label 7616.2.a.bp.1.1 Level $7616$ Weight $2$ Character 7616.1 Self dual yes Analytic conductor $60.814$ Analytic rank $0$ Dimension $4$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7616,2,Mod(1,7616)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7616, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("7616.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$7616 = 2^{6} \cdot 7 \cdot 17$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 7616.a (trivial)

## 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: yes Analytic conductor: $$60.8140661794$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: 4.4.5225.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{3} - 8x^{2} + x + 11$$ x^4 - x^3 - 8*x^2 + x + 11 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 952) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$3.10522$$ of defining polynomial Character $$\chi$$ $$=$$ 7616.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-2.10522 q^{3} +3.40632 q^{5} +1.00000 q^{7} +1.43195 q^{9} +O(q^{10})$$ $$q-2.10522 q^{3} +3.40632 q^{5} +1.00000 q^{7} +1.43195 q^{9} -1.23607 q^{11} +6.47214 q^{13} -7.17104 q^{15} +1.00000 q^{17} -2.21044 q^{19} -2.10522 q^{21} -1.39781 q^{23} +6.60299 q^{25} +3.30110 q^{27} +0.633874 q^{29} -0.965861 q^{31} +2.60219 q^{33} +3.40632 q^{35} +8.31040 q^{37} -13.6253 q^{39} -5.60299 q^{41} +11.0796 q^{43} +4.87766 q^{45} -7.67652 q^{47} +1.00000 q^{49} -2.10522 q^{51} -11.7167 q^{53} -4.21044 q^{55} +4.65345 q^{57} -0.602193 q^{59} +9.84352 q^{61} +1.43195 q^{63} +22.0461 q^{65} +5.30636 q^{67} +2.94269 q^{69} +3.33603 q^{71} +14.0239 q^{73} -13.9007 q^{75} -1.23607 q^{77} +0.323477 q^{79} -11.2454 q^{81} +13.8870 q^{83} +3.40632 q^{85} -1.33444 q^{87} -14.7509 q^{89} +6.47214 q^{91} +2.03335 q^{93} -7.52945 q^{95} -5.90855 q^{97} -1.76998 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 3 q^{3} + q^{5} + 4 q^{7} + 7 q^{9}+O(q^{10})$$ 4 * q + 3 * q^3 + q^5 + 4 * q^7 + 7 * q^9 $$4 q + 3 q^{3} + q^{5} + 4 q^{7} + 7 q^{9} + 4 q^{11} + 8 q^{13} - 12 q^{15} + 4 q^{17} + 14 q^{19} + 3 q^{21} - 8 q^{23} + 11 q^{25} + 12 q^{27} - 4 q^{29} - 5 q^{31} + 8 q^{33} + q^{35} + 4 q^{37} - 4 q^{39} - 7 q^{41} + 19 q^{43} + 2 q^{45} - 8 q^{47} + 4 q^{49} + 3 q^{51} - 5 q^{53} + 6 q^{55} + 44 q^{57} + 23 q^{61} + 7 q^{63} - 8 q^{65} + 15 q^{67} + 2 q^{69} - 2 q^{71} - 5 q^{73} + 10 q^{75} + 4 q^{77} + 24 q^{79} - 8 q^{81} + 10 q^{83} + q^{85} - 16 q^{87} - 16 q^{89} + 8 q^{91} + 20 q^{93} - 22 q^{95} - 15 q^{97} + 2 q^{99}+O(q^{100})$$ 4 * q + 3 * q^3 + q^5 + 4 * q^7 + 7 * q^9 + 4 * q^11 + 8 * q^13 - 12 * q^15 + 4 * q^17 + 14 * q^19 + 3 * q^21 - 8 * q^23 + 11 * q^25 + 12 * q^27 - 4 * q^29 - 5 * q^31 + 8 * q^33 + q^35 + 4 * q^37 - 4 * q^39 - 7 * q^41 + 19 * q^43 + 2 * q^45 - 8 * q^47 + 4 * q^49 + 3 * q^51 - 5 * q^53 + 6 * q^55 + 44 * q^57 + 23 * q^61 + 7 * q^63 - 8 * q^65 + 15 * q^67 + 2 * q^69 - 2 * q^71 - 5 * q^73 + 10 * q^75 + 4 * q^77 + 24 * q^79 - 8 * q^81 + 10 * q^83 + q^85 - 16 * q^87 - 16 * q^89 + 8 * q^91 + 20 * q^93 - 22 * q^95 - 15 * q^97 + 2 * q^99

## 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
$$3$$ −2.10522 −1.21545 −0.607724 0.794148i $$-0.707917\pi$$
−0.607724 + 0.794148i $$0.707917\pi$$
$$4$$ 0 0
$$5$$ 3.40632 1.52335 0.761675 0.647959i $$-0.224377\pi$$
0.761675 + 0.647959i $$0.224377\pi$$
$$6$$ 0 0
$$7$$ 1.00000 0.377964
$$8$$ 0 0
$$9$$ 1.43195 0.477315
$$10$$ 0 0
$$11$$ −1.23607 −0.372689 −0.186344 0.982485i $$-0.559664\pi$$
−0.186344 + 0.982485i $$0.559664\pi$$
$$12$$ 0 0
$$13$$ 6.47214 1.79505 0.897524 0.440966i $$-0.145364\pi$$
0.897524 + 0.440966i $$0.145364\pi$$
$$14$$ 0 0
$$15$$ −7.17104 −1.85155
$$16$$ 0 0
$$17$$ 1.00000 0.242536
$$18$$ 0 0
$$19$$ −2.21044 −0.507109 −0.253555 0.967321i $$-0.581600\pi$$
−0.253555 + 0.967321i $$0.581600\pi$$
$$20$$ 0 0
$$21$$ −2.10522 −0.459396
$$22$$ 0 0
$$23$$ −1.39781 −0.291463 −0.145731 0.989324i $$-0.546554\pi$$
−0.145731 + 0.989324i $$0.546554\pi$$
$$24$$ 0 0
$$25$$ 6.60299 1.32060
$$26$$ 0 0
$$27$$ 3.30110 0.635296
$$28$$ 0 0
$$29$$ 0.633874 0.117708 0.0588538 0.998267i $$-0.481255\pi$$
0.0588538 + 0.998267i $$0.481255\pi$$
$$30$$ 0 0
$$31$$ −0.965861 −0.173474 −0.0867368 0.996231i $$-0.527644\pi$$
−0.0867368 + 0.996231i $$0.527644\pi$$
$$32$$ 0 0
$$33$$ 2.60219 0.452984
$$34$$ 0 0
$$35$$ 3.40632 0.575772
$$36$$ 0 0
$$37$$ 8.31040 1.36622 0.683110 0.730315i $$-0.260627\pi$$
0.683110 + 0.730315i $$0.260627\pi$$
$$38$$ 0 0
$$39$$ −13.6253 −2.18179
$$40$$ 0 0
$$41$$ −5.60299 −0.875039 −0.437520 0.899209i $$-0.644143\pi$$
−0.437520 + 0.899209i $$0.644143\pi$$
$$42$$ 0 0
$$43$$ 11.0796 1.68962 0.844811 0.535065i $$-0.179713\pi$$
0.844811 + 0.535065i $$0.179713\pi$$
$$44$$ 0 0
$$45$$ 4.87766 0.727119
$$46$$ 0 0
$$47$$ −7.67652 −1.11974 −0.559868 0.828582i $$-0.689148\pi$$
−0.559868 + 0.828582i $$0.689148\pi$$
$$48$$ 0 0
$$49$$ 1.00000 0.142857
$$50$$ 0 0
$$51$$ −2.10522 −0.294790
$$52$$ 0 0
$$53$$ −11.7167 −1.60941 −0.804707 0.593672i $$-0.797678\pi$$
−0.804707 + 0.593672i $$0.797678\pi$$
$$54$$ 0 0
$$55$$ −4.21044 −0.567735
$$56$$ 0 0
$$57$$ 4.65345 0.616365
$$58$$ 0 0
$$59$$ −0.602193 −0.0783989 −0.0391995 0.999231i $$-0.512481\pi$$
−0.0391995 + 0.999231i $$0.512481\pi$$
$$60$$ 0 0
$$61$$ 9.84352 1.26033 0.630167 0.776460i $$-0.282986\pi$$
0.630167 + 0.776460i $$0.282986\pi$$
$$62$$ 0 0
$$63$$ 1.43195 0.180408
$$64$$ 0 0
$$65$$ 22.0461 2.73449
$$66$$ 0 0
$$67$$ 5.30636 0.648275 0.324137 0.946010i $$-0.394926\pi$$
0.324137 + 0.946010i $$0.394926\pi$$
$$68$$ 0 0
$$69$$ 2.94269 0.354258
$$70$$ 0 0
$$71$$ 3.33603 0.395914 0.197957 0.980211i $$-0.436569\pi$$
0.197957 + 0.980211i $$0.436569\pi$$
$$72$$ 0 0
$$73$$ 14.0239 1.64137 0.820684 0.571382i $$-0.193592\pi$$
0.820684 + 0.571382i $$0.193592\pi$$
$$74$$ 0 0
$$75$$ −13.9007 −1.60512
$$76$$ 0 0
$$77$$ −1.23607 −0.140863
$$78$$ 0 0
$$79$$ 0.323477 0.0363940 0.0181970 0.999834i $$-0.494207\pi$$
0.0181970 + 0.999834i $$0.494207\pi$$
$$80$$ 0 0
$$81$$ −11.2454 −1.24949
$$82$$ 0 0
$$83$$ 13.8870 1.52429 0.762146 0.647405i $$-0.224146\pi$$
0.762146 + 0.647405i $$0.224146\pi$$
$$84$$ 0 0
$$85$$ 3.40632 0.369467
$$86$$ 0 0
$$87$$ −1.33444 −0.143067
$$88$$ 0 0
$$89$$ −14.7509 −1.56359 −0.781794 0.623537i $$-0.785695\pi$$
−0.781794 + 0.623537i $$0.785695\pi$$
$$90$$ 0 0
$$91$$ 6.47214 0.678464
$$92$$ 0 0
$$93$$ 2.03335 0.210848
$$94$$ 0 0
$$95$$ −7.52945 −0.772505
$$96$$ 0 0
$$97$$ −5.90855 −0.599922 −0.299961 0.953951i $$-0.596974\pi$$
−0.299961 + 0.953951i $$0.596974\pi$$
$$98$$ 0 0
$$99$$ −1.76998 −0.177890
$$100$$ 0 0
$$101$$ −5.08485 −0.505961 −0.252981 0.967471i $$-0.581411\pi$$
−0.252981 + 0.967471i $$0.581411\pi$$
$$102$$ 0 0
$$103$$ 5.28477 0.520724 0.260362 0.965511i $$-0.416158\pi$$
0.260362 + 0.965511i $$0.416158\pi$$
$$104$$ 0 0
$$105$$ −7.17104 −0.699822
$$106$$ 0 0
$$107$$ −9.25309 −0.894530 −0.447265 0.894402i $$-0.647602\pi$$
−0.447265 + 0.894402i $$0.647602\pi$$
$$108$$ 0 0
$$109$$ 16.1000 1.54210 0.771048 0.636777i $$-0.219733\pi$$
0.771048 + 0.636777i $$0.219733\pi$$
$$110$$ 0 0
$$111$$ −17.4952 −1.66057
$$112$$ 0 0
$$113$$ 10.5509 0.992548 0.496274 0.868166i $$-0.334701\pi$$
0.496274 + 0.868166i $$0.334701\pi$$
$$114$$ 0 0
$$115$$ −4.76137 −0.444000
$$116$$ 0 0
$$117$$ 9.26775 0.856804
$$118$$ 0 0
$$119$$ 1.00000 0.0916698
$$120$$ 0 0
$$121$$ −9.47214 −0.861103
$$122$$ 0 0
$$123$$ 11.7955 1.06357
$$124$$ 0 0
$$125$$ 5.46027 0.488382
$$126$$ 0 0
$$127$$ 17.9272 1.59078 0.795389 0.606100i $$-0.207267\pi$$
0.795389 + 0.606100i $$0.207267\pi$$
$$128$$ 0 0
$$129$$ −23.3250 −2.05365
$$130$$ 0 0
$$131$$ 1.49777 0.130860 0.0654302 0.997857i $$-0.479158\pi$$
0.0654302 + 0.997857i $$0.479158\pi$$
$$132$$ 0 0
$$133$$ −2.21044 −0.191669
$$134$$ 0 0
$$135$$ 11.2446 0.967779
$$136$$ 0 0
$$137$$ 13.6030 1.16218 0.581091 0.813839i $$-0.302626\pi$$
0.581091 + 0.813839i $$0.302626\pi$$
$$138$$ 0 0
$$139$$ −6.28073 −0.532724 −0.266362 0.963873i $$-0.585822\pi$$
−0.266362 + 0.963873i $$0.585822\pi$$
$$140$$ 0 0
$$141$$ 16.1608 1.36098
$$142$$ 0 0
$$143$$ −8.00000 −0.668994
$$144$$ 0 0
$$145$$ 2.15918 0.179310
$$146$$ 0 0
$$147$$ −2.10522 −0.173636
$$148$$ 0 0
$$149$$ −21.7004 −1.77776 −0.888882 0.458136i $$-0.848517\pi$$
−0.888882 + 0.458136i $$0.848517\pi$$
$$150$$ 0 0
$$151$$ −4.60745 −0.374949 −0.187475 0.982269i $$-0.560030\pi$$
−0.187475 + 0.982269i $$0.560030\pi$$
$$152$$ 0 0
$$153$$ 1.43195 0.115766
$$154$$ 0 0
$$155$$ −3.29003 −0.264261
$$156$$ 0 0
$$157$$ 4.53391 0.361846 0.180923 0.983497i $$-0.442092\pi$$
0.180923 + 0.983497i $$0.442092\pi$$
$$158$$ 0 0
$$159$$ 24.6662 1.95616
$$160$$ 0 0
$$161$$ −1.39781 −0.110163
$$162$$ 0 0
$$163$$ −12.2421 −0.958877 −0.479438 0.877576i $$-0.659160\pi$$
−0.479438 + 0.877576i $$0.659160\pi$$
$$164$$ 0 0
$$165$$ 8.86389 0.690053
$$166$$ 0 0
$$167$$ −1.34129 −0.103792 −0.0518959 0.998652i $$-0.516526\pi$$
−0.0518959 + 0.998652i $$0.516526\pi$$
$$168$$ 0 0
$$169$$ 28.8885 2.22220
$$170$$ 0 0
$$171$$ −3.16523 −0.242051
$$172$$ 0 0
$$173$$ −1.83300 −0.139361 −0.0696803 0.997569i $$-0.522198\pi$$
−0.0696803 + 0.997569i $$0.522198\pi$$
$$174$$ 0 0
$$175$$ 6.60299 0.499139
$$176$$ 0 0
$$177$$ 1.26775 0.0952898
$$178$$ 0 0
$$179$$ 20.2343 1.51238 0.756191 0.654351i $$-0.227058\pi$$
0.756191 + 0.654351i $$0.227058\pi$$
$$180$$ 0 0
$$181$$ 6.03010 0.448214 0.224107 0.974565i $$-0.428054\pi$$
0.224107 + 0.974565i $$0.428054\pi$$
$$182$$ 0 0
$$183$$ −20.7228 −1.53187
$$184$$ 0 0
$$185$$ 28.3078 2.08123
$$186$$ 0 0
$$187$$ −1.23607 −0.0903902
$$188$$ 0 0
$$189$$ 3.30110 0.240119
$$190$$ 0 0
$$191$$ 10.9146 0.789753 0.394876 0.918734i $$-0.370787\pi$$
0.394876 + 0.918734i $$0.370787\pi$$
$$192$$ 0 0
$$193$$ −19.1717 −1.38001 −0.690006 0.723804i $$-0.742392\pi$$
−0.690006 + 0.723804i $$0.742392\pi$$
$$194$$ 0 0
$$195$$ −46.4119 −3.32363
$$196$$ 0 0
$$197$$ −2.89557 −0.206301 −0.103151 0.994666i $$-0.532892\pi$$
−0.103151 + 0.994666i $$0.532892\pi$$
$$198$$ 0 0
$$199$$ −26.1830 −1.85607 −0.928033 0.372498i $$-0.878501\pi$$
−0.928033 + 0.372498i $$0.878501\pi$$
$$200$$ 0 0
$$201$$ −11.1710 −0.787944
$$202$$ 0 0
$$203$$ 0.633874 0.0444893
$$204$$ 0 0
$$205$$ −19.0855 −1.33299
$$206$$ 0 0
$$207$$ −2.00158 −0.139120
$$208$$ 0 0
$$209$$ 2.73225 0.188994
$$210$$ 0 0
$$211$$ −2.11048 −0.145291 −0.0726456 0.997358i $$-0.523144\pi$$
−0.0726456 + 0.997358i $$0.523144\pi$$
$$212$$ 0 0
$$213$$ −7.02307 −0.481213
$$214$$ 0 0
$$215$$ 37.7406 2.57389
$$216$$ 0 0
$$217$$ −0.965861 −0.0655669
$$218$$ 0 0
$$219$$ −29.5233 −1.99500
$$220$$ 0 0
$$221$$ 6.47214 0.435363
$$222$$ 0 0
$$223$$ 18.8126 1.25979 0.629893 0.776682i $$-0.283099\pi$$
0.629893 + 0.776682i $$0.283099\pi$$
$$224$$ 0 0
$$225$$ 9.45512 0.630341
$$226$$ 0 0
$$227$$ −17.3737 −1.15313 −0.576565 0.817051i $$-0.695607\pi$$
−0.576565 + 0.817051i $$0.695607\pi$$
$$228$$ 0 0
$$229$$ 2.71523 0.179428 0.0897138 0.995968i $$-0.471405\pi$$
0.0897138 + 0.995968i $$0.471405\pi$$
$$230$$ 0 0
$$231$$ 2.60219 0.171212
$$232$$ 0 0
$$233$$ 0.715233 0.0468565 0.0234282 0.999726i $$-0.492542\pi$$
0.0234282 + 0.999726i $$0.492542\pi$$
$$234$$ 0 0
$$235$$ −26.1487 −1.70575
$$236$$ 0 0
$$237$$ −0.680990 −0.0442351
$$238$$ 0 0
$$239$$ −8.60745 −0.556770 −0.278385 0.960470i $$-0.589799\pi$$
−0.278385 + 0.960470i $$0.589799\pi$$
$$240$$ 0 0
$$241$$ 6.31241 0.406618 0.203309 0.979115i $$-0.434830\pi$$
0.203309 + 0.979115i $$0.434830\pi$$
$$242$$ 0 0
$$243$$ 13.7707 0.883389
$$244$$ 0 0
$$245$$ 3.40632 0.217622
$$246$$ 0 0
$$247$$ −14.3063 −0.910285
$$248$$ 0 0
$$249$$ −29.2351 −1.85270
$$250$$ 0 0
$$251$$ −26.5711 −1.67715 −0.838577 0.544783i $$-0.816612\pi$$
−0.838577 + 0.544783i $$0.816612\pi$$
$$252$$ 0 0
$$253$$ 1.72778 0.108625
$$254$$ 0 0
$$255$$ −7.17104 −0.449068
$$256$$ 0 0
$$257$$ 6.75085 0.421107 0.210553 0.977582i $$-0.432473\pi$$
0.210553 + 0.977582i $$0.432473\pi$$
$$258$$ 0 0
$$259$$ 8.31040 0.516383
$$260$$ 0 0
$$261$$ 0.907674 0.0561836
$$262$$ 0 0
$$263$$ −1.20439 −0.0742657 −0.0371328 0.999310i $$-0.511822\pi$$
−0.0371328 + 0.999310i $$0.511822\pi$$
$$264$$ 0 0
$$265$$ −39.9108 −2.45170
$$266$$ 0 0
$$267$$ 31.0538 1.90046
$$268$$ 0 0
$$269$$ −18.0673 −1.10158 −0.550791 0.834643i $$-0.685674\pi$$
−0.550791 + 0.834643i $$0.685674\pi$$
$$270$$ 0 0
$$271$$ −19.7464 −1.19951 −0.599754 0.800185i $$-0.704735\pi$$
−0.599754 + 0.800185i $$0.704735\pi$$
$$272$$ 0 0
$$273$$ −13.6253 −0.824638
$$274$$ 0 0
$$275$$ −8.16174 −0.492171
$$276$$ 0 0
$$277$$ 7.99744 0.480519 0.240260 0.970709i $$-0.422767\pi$$
0.240260 + 0.970709i $$0.422767\pi$$
$$278$$ 0 0
$$279$$ −1.38306 −0.0828016
$$280$$ 0 0
$$281$$ 3.07890 0.183672 0.0918359 0.995774i $$-0.470726\pi$$
0.0918359 + 0.995774i $$0.470726\pi$$
$$282$$ 0 0
$$283$$ 28.6927 1.70560 0.852801 0.522236i $$-0.174902\pi$$
0.852801 + 0.522236i $$0.174902\pi$$
$$284$$ 0 0
$$285$$ 15.8511 0.938940
$$286$$ 0 0
$$287$$ −5.60299 −0.330734
$$288$$ 0 0
$$289$$ 1.00000 0.0588235
$$290$$ 0 0
$$291$$ 12.4388 0.729175
$$292$$ 0 0
$$293$$ 2.14866 0.125526 0.0627630 0.998028i $$-0.480009\pi$$
0.0627630 + 0.998028i $$0.480009\pi$$
$$294$$ 0 0
$$295$$ −2.05126 −0.119429
$$296$$ 0 0
$$297$$ −4.08038 −0.236768
$$298$$ 0 0
$$299$$ −9.04679 −0.523190
$$300$$ 0 0
$$301$$ 11.0796 0.638617
$$302$$ 0 0
$$303$$ 10.7047 0.614970
$$304$$ 0 0
$$305$$ 33.5301 1.91993
$$306$$ 0 0
$$307$$ 16.0291 0.914830 0.457415 0.889253i $$-0.348775\pi$$
0.457415 + 0.889253i $$0.348775\pi$$
$$308$$ 0 0
$$309$$ −11.1256 −0.632913
$$310$$ 0 0
$$311$$ −34.4505 −1.95351 −0.976756 0.214356i $$-0.931235\pi$$
−0.976756 + 0.214356i $$0.931235\pi$$
$$312$$ 0 0
$$313$$ 32.5585 1.84031 0.920157 0.391551i $$-0.128061\pi$$
0.920157 + 0.391551i $$0.128061\pi$$
$$314$$ 0 0
$$315$$ 4.87766 0.274825
$$316$$ 0 0
$$317$$ −2.82571 −0.158708 −0.0793539 0.996847i $$-0.525286\pi$$
−0.0793539 + 0.996847i $$0.525286\pi$$
$$318$$ 0 0
$$319$$ −0.783512 −0.0438682
$$320$$ 0 0
$$321$$ 19.4798 1.08725
$$322$$ 0 0
$$323$$ −2.21044 −0.122992
$$324$$ 0 0
$$325$$ 42.7354 2.37053
$$326$$ 0 0
$$327$$ −33.8939 −1.87434
$$328$$ 0 0
$$329$$ −7.67652 −0.423220
$$330$$ 0 0
$$331$$ 16.5822 0.911439 0.455720 0.890123i $$-0.349382\pi$$
0.455720 + 0.890123i $$0.349382\pi$$
$$332$$ 0 0
$$333$$ 11.9000 0.652118
$$334$$ 0 0
$$335$$ 18.0751 0.987549
$$336$$ 0 0
$$337$$ 15.7553 0.858247 0.429123 0.903246i $$-0.358823\pi$$
0.429123 + 0.903246i $$0.358823\pi$$
$$338$$ 0 0
$$339$$ −22.2120 −1.20639
$$340$$ 0 0
$$341$$ 1.19387 0.0646516
$$342$$ 0 0
$$343$$ 1.00000 0.0539949
$$344$$ 0 0
$$345$$ 10.0237 0.539659
$$346$$ 0 0
$$347$$ −12.6630 −0.679785 −0.339893 0.940464i $$-0.610391\pi$$
−0.339893 + 0.940464i $$0.610391\pi$$
$$348$$ 0 0
$$349$$ 25.1547 1.34650 0.673250 0.739415i $$-0.264898\pi$$
0.673250 + 0.739415i $$0.264898\pi$$
$$350$$ 0 0
$$351$$ 21.3651 1.14039
$$352$$ 0 0
$$353$$ 28.5801 1.52116 0.760581 0.649243i $$-0.224914\pi$$
0.760581 + 0.649243i $$0.224914\pi$$
$$354$$ 0 0
$$355$$ 11.3636 0.603115
$$356$$ 0 0
$$357$$ −2.10522 −0.111420
$$358$$ 0 0
$$359$$ 25.4038 1.34076 0.670379 0.742018i $$-0.266131\pi$$
0.670379 + 0.742018i $$0.266131\pi$$
$$360$$ 0 0
$$361$$ −14.1140 −0.742840
$$362$$ 0 0
$$363$$ 19.9409 1.04663
$$364$$ 0 0
$$365$$ 47.7697 2.50038
$$366$$ 0 0
$$367$$ 4.87073 0.254250 0.127125 0.991887i $$-0.459425\pi$$
0.127125 + 0.991887i $$0.459425\pi$$
$$368$$ 0 0
$$369$$ −8.02317 −0.417670
$$370$$ 0 0
$$371$$ −11.7167 −0.608301
$$372$$ 0 0
$$373$$ −23.2388 −1.20326 −0.601629 0.798776i $$-0.705481\pi$$
−0.601629 + 0.798776i $$0.705481\pi$$
$$374$$ 0 0
$$375$$ −11.4951 −0.593603
$$376$$ 0 0
$$377$$ 4.10252 0.211291
$$378$$ 0 0
$$379$$ 15.8658 0.814971 0.407486 0.913212i $$-0.366406\pi$$
0.407486 + 0.913212i $$0.366406\pi$$
$$380$$ 0 0
$$381$$ −37.7406 −1.93351
$$382$$ 0 0
$$383$$ 2.06828 0.105684 0.0528421 0.998603i $$-0.483172\pi$$
0.0528421 + 0.998603i $$0.483172\pi$$
$$384$$ 0 0
$$385$$ −4.21044 −0.214584
$$386$$ 0 0
$$387$$ 15.8654 0.806482
$$388$$ 0 0
$$389$$ −3.51679 −0.178308 −0.0891542 0.996018i $$-0.528416\pi$$
−0.0891542 + 0.996018i $$0.528416\pi$$
$$390$$ 0 0
$$391$$ −1.39781 −0.0706901
$$392$$ 0 0
$$393$$ −3.15313 −0.159054
$$394$$ 0 0
$$395$$ 1.10187 0.0554408
$$396$$ 0 0
$$397$$ −5.47459 −0.274762 −0.137381 0.990518i $$-0.543868\pi$$
−0.137381 + 0.990518i $$0.543868\pi$$
$$398$$ 0 0
$$399$$ 4.65345 0.232964
$$400$$ 0 0
$$401$$ −6.76137 −0.337647 −0.168823 0.985646i $$-0.553997\pi$$
−0.168823 + 0.985646i $$0.553997\pi$$
$$402$$ 0 0
$$403$$ −6.25118 −0.311393
$$404$$ 0 0
$$405$$ −38.3053 −1.90340
$$406$$ 0 0
$$407$$ −10.2722 −0.509175
$$408$$ 0 0
$$409$$ −14.6378 −0.723793 −0.361897 0.932218i $$-0.617871\pi$$
−0.361897 + 0.932218i $$0.617871\pi$$
$$410$$ 0 0
$$411$$ −28.6373 −1.41257
$$412$$ 0 0
$$413$$ −0.602193 −0.0296320
$$414$$ 0 0
$$415$$ 47.3034 2.32203
$$416$$ 0 0
$$417$$ 13.2223 0.647499
$$418$$ 0 0
$$419$$ 29.5736 1.44476 0.722382 0.691494i $$-0.243047\pi$$
0.722382 + 0.691494i $$0.243047\pi$$
$$420$$ 0 0
$$421$$ −12.0223 −0.585930 −0.292965 0.956123i $$-0.594642\pi$$
−0.292965 + 0.956123i $$0.594642\pi$$
$$422$$ 0 0
$$423$$ −10.9924 −0.534467
$$424$$ 0 0
$$425$$ 6.60299 0.320292
$$426$$ 0 0
$$427$$ 9.84352 0.476361
$$428$$ 0 0
$$429$$ 16.8417 0.813127
$$430$$ 0 0
$$431$$ 11.1034 0.534834 0.267417 0.963581i $$-0.413830\pi$$
0.267417 + 0.963581i $$0.413830\pi$$
$$432$$ 0 0
$$433$$ 17.2165 0.827372 0.413686 0.910420i $$-0.364241\pi$$
0.413686 + 0.910420i $$0.364241\pi$$
$$434$$ 0 0
$$435$$ −4.54554 −0.217942
$$436$$ 0 0
$$437$$ 3.08976 0.147803
$$438$$ 0 0
$$439$$ −25.1681 −1.20121 −0.600603 0.799548i $$-0.705073\pi$$
−0.600603 + 0.799548i $$0.705073\pi$$
$$440$$ 0 0
$$441$$ 1.43195 0.0681879
$$442$$ 0 0
$$443$$ 31.1531 1.48013 0.740065 0.672536i $$-0.234795\pi$$
0.740065 + 0.672536i $$0.234795\pi$$
$$444$$ 0 0
$$445$$ −50.2461 −2.38189
$$446$$ 0 0
$$447$$ 45.6841 2.16078
$$448$$ 0 0
$$449$$ 10.7444 0.507057 0.253529 0.967328i $$-0.418409\pi$$
0.253529 + 0.967328i $$0.418409\pi$$
$$450$$ 0 0
$$451$$ 6.92567 0.326117
$$452$$ 0 0
$$453$$ 9.69969 0.455731
$$454$$ 0 0
$$455$$ 22.0461 1.03354
$$456$$ 0 0
$$457$$ −30.0863 −1.40738 −0.703690 0.710508i $$-0.748465\pi$$
−0.703690 + 0.710508i $$0.748465\pi$$
$$458$$ 0 0
$$459$$ 3.30110 0.154082
$$460$$ 0 0
$$461$$ −8.29082 −0.386142 −0.193071 0.981185i $$-0.561845\pi$$
−0.193071 + 0.981185i $$0.561845\pi$$
$$462$$ 0 0
$$463$$ −11.2795 −0.524203 −0.262102 0.965040i $$-0.584416\pi$$
−0.262102 + 0.965040i $$0.584416\pi$$
$$464$$ 0 0
$$465$$ 6.92622 0.321196
$$466$$ 0 0
$$467$$ 38.2582 1.77038 0.885188 0.465233i $$-0.154029\pi$$
0.885188 + 0.465233i $$0.154029\pi$$
$$468$$ 0 0
$$469$$ 5.30636 0.245025
$$470$$ 0 0
$$471$$ −9.54488 −0.439805
$$472$$ 0 0
$$473$$ −13.6951 −0.629702
$$474$$ 0 0
$$475$$ −14.5955 −0.669687
$$476$$ 0 0
$$477$$ −16.7777 −0.768198
$$478$$ 0 0
$$479$$ 33.0812 1.51152 0.755759 0.654850i $$-0.227268\pi$$
0.755759 + 0.654850i $$0.227268\pi$$
$$480$$ 0 0
$$481$$ 53.7860 2.45243
$$482$$ 0 0
$$483$$ 2.94269 0.133897
$$484$$ 0 0
$$485$$ −20.1264 −0.913892
$$486$$ 0 0
$$487$$ 25.3757 1.14988 0.574941 0.818195i $$-0.305025\pi$$
0.574941 + 0.818195i $$0.305025\pi$$
$$488$$ 0 0
$$489$$ 25.7723 1.16547
$$490$$ 0 0
$$491$$ 30.4505 1.37421 0.687107 0.726556i $$-0.258880\pi$$
0.687107 + 0.726556i $$0.258880\pi$$
$$492$$ 0 0
$$493$$ 0.633874 0.0285483
$$494$$ 0 0
$$495$$ −6.02912 −0.270989
$$496$$ 0 0
$$497$$ 3.33603 0.149641
$$498$$ 0 0
$$499$$ 11.9016 0.532790 0.266395 0.963864i $$-0.414167\pi$$
0.266395 + 0.963864i $$0.414167\pi$$
$$500$$ 0 0
$$501$$ 2.82370 0.126154
$$502$$ 0 0
$$503$$ −8.90567 −0.397084 −0.198542 0.980092i $$-0.563621\pi$$
−0.198542 + 0.980092i $$0.563621\pi$$
$$504$$ 0 0
$$505$$ −17.3206 −0.770756
$$506$$ 0 0
$$507$$ −60.8167 −2.70096
$$508$$ 0 0
$$509$$ 27.1838 1.20490 0.602451 0.798156i $$-0.294191\pi$$
0.602451 + 0.798156i $$0.294191\pi$$
$$510$$ 0 0
$$511$$ 14.0239 0.620379
$$512$$ 0 0
$$513$$ −7.29687 −0.322165
$$514$$ 0 0
$$515$$ 18.0016 0.793245
$$516$$ 0 0
$$517$$ 9.48870 0.417313
$$518$$ 0 0
$$519$$ 3.85887 0.169386
$$520$$ 0 0
$$521$$ 7.10981 0.311487 0.155743 0.987798i $$-0.450223\pi$$
0.155743 + 0.987798i $$0.450223\pi$$
$$522$$ 0 0
$$523$$ −30.4581 −1.33184 −0.665919 0.746024i $$-0.731961\pi$$
−0.665919 + 0.746024i $$0.731961\pi$$
$$524$$ 0 0
$$525$$ −13.9007 −0.606678
$$526$$ 0 0
$$527$$ −0.965861 −0.0420735
$$528$$ 0 0
$$529$$ −21.0461 −0.915049
$$530$$ 0 0
$$531$$ −0.862309 −0.0374210
$$532$$ 0 0
$$533$$ −36.2633 −1.57074
$$534$$ 0 0
$$535$$ −31.5189 −1.36268
$$536$$ 0 0
$$537$$ −42.5976 −1.83822
$$538$$ 0 0
$$539$$ −1.23607 −0.0532412
$$540$$ 0 0
$$541$$ 1.88794 0.0811688 0.0405844 0.999176i $$-0.487078\pi$$
0.0405844 + 0.999176i $$0.487078\pi$$
$$542$$ 0 0
$$543$$ −12.6947 −0.544781
$$544$$ 0 0
$$545$$ 54.8415 2.34915
$$546$$ 0 0
$$547$$ 13.8844 0.593654 0.296827 0.954931i $$-0.404072\pi$$
0.296827 + 0.954931i $$0.404072\pi$$
$$548$$ 0 0
$$549$$ 14.0954 0.601577
$$550$$ 0 0
$$551$$ −1.40114 −0.0596906
$$552$$ 0 0
$$553$$ 0.323477 0.0137556
$$554$$ 0 0
$$555$$ −59.5942 −2.52963
$$556$$ 0 0
$$557$$ 18.5183 0.784644 0.392322 0.919828i $$-0.371672\pi$$
0.392322 + 0.919828i $$0.371672\pi$$
$$558$$ 0 0
$$559$$ 71.7086 3.03295
$$560$$ 0 0
$$561$$ 2.60219 0.109865
$$562$$ 0 0
$$563$$ −36.1932 −1.52536 −0.762681 0.646775i $$-0.776117\pi$$
−0.762681 + 0.646775i $$0.776117\pi$$
$$564$$ 0 0
$$565$$ 35.9398 1.51200
$$566$$ 0 0
$$567$$ −11.2454 −0.472261
$$568$$ 0 0
$$569$$ 28.9137 1.21213 0.606063 0.795417i $$-0.292748\pi$$
0.606063 + 0.795417i $$0.292748\pi$$
$$570$$ 0 0
$$571$$ 15.1214 0.632813 0.316406 0.948624i $$-0.397524\pi$$
0.316406 + 0.948624i $$0.397524\pi$$
$$572$$ 0 0
$$573$$ −22.9776 −0.959904
$$574$$ 0 0
$$575$$ −9.22970 −0.384905
$$576$$ 0 0
$$577$$ −18.1608 −0.756042 −0.378021 0.925797i $$-0.623395\pi$$
−0.378021 + 0.925797i $$0.623395\pi$$
$$578$$ 0 0
$$579$$ 40.3607 1.67733
$$580$$ 0 0
$$581$$ 13.8870 0.576128
$$582$$ 0 0
$$583$$ 14.4827 0.599810
$$584$$ 0 0
$$585$$ 31.5689 1.30521
$$586$$ 0 0
$$587$$ 45.0538 1.85957 0.929784 0.368105i $$-0.119993\pi$$
0.929784 + 0.368105i $$0.119993\pi$$
$$588$$ 0 0
$$589$$ 2.13497 0.0879701
$$590$$ 0 0
$$591$$ 6.09581 0.250748
$$592$$ 0 0
$$593$$ 6.43139 0.264106 0.132053 0.991243i $$-0.457843\pi$$
0.132053 + 0.991243i $$0.457843\pi$$
$$594$$ 0 0
$$595$$ 3.40632 0.139645
$$596$$ 0 0
$$597$$ 55.1210 2.25595
$$598$$ 0 0
$$599$$ 41.0655 1.67789 0.838946 0.544215i $$-0.183172\pi$$
0.838946 + 0.544215i $$0.183172\pi$$
$$600$$ 0 0
$$601$$ 39.2626 1.60156 0.800778 0.598961i $$-0.204420\pi$$
0.800778 + 0.598961i $$0.204420\pi$$
$$602$$ 0 0
$$603$$ 7.59841 0.309431
$$604$$ 0 0
$$605$$ −32.2651 −1.31176
$$606$$ 0 0
$$607$$ 11.9954 0.486879 0.243440 0.969916i $$-0.421724\pi$$
0.243440 + 0.969916i $$0.421724\pi$$
$$608$$ 0 0
$$609$$ −1.33444 −0.0540744
$$610$$ 0 0
$$611$$ −49.6835 −2.00998
$$612$$ 0 0
$$613$$ −28.0774 −1.13404 −0.567018 0.823706i $$-0.691903\pi$$
−0.567018 + 0.823706i $$0.691903\pi$$
$$614$$ 0 0
$$615$$ 40.1792 1.62018
$$616$$ 0 0
$$617$$ −6.08688 −0.245049 −0.122524 0.992466i $$-0.539099\pi$$
−0.122524 + 0.992466i $$0.539099\pi$$
$$618$$ 0 0
$$619$$ 3.24659 0.130491 0.0652456 0.997869i $$-0.479217\pi$$
0.0652456 + 0.997869i $$0.479217\pi$$
$$620$$ 0 0
$$621$$ −4.61429 −0.185165
$$622$$ 0 0
$$623$$ −14.7509 −0.590980
$$624$$ 0 0
$$625$$ −14.4155 −0.576621
$$626$$ 0 0
$$627$$ −5.75199 −0.229712
$$628$$ 0 0
$$629$$ 8.31040 0.331357
$$630$$ 0 0
$$631$$ 6.53093 0.259992 0.129996 0.991515i $$-0.458504\pi$$
0.129996 + 0.991515i $$0.458504\pi$$
$$632$$ 0 0
$$633$$ 4.44302 0.176594
$$634$$ 0 0
$$635$$ 61.0655 2.42331
$$636$$ 0 0
$$637$$ 6.47214 0.256435
$$638$$ 0 0
$$639$$ 4.77701 0.188976
$$640$$ 0 0
$$641$$ −6.98745 −0.275988 −0.137994 0.990433i $$-0.544065\pi$$
−0.137994 + 0.990433i $$0.544065\pi$$
$$642$$ 0 0
$$643$$ 21.1348 0.833475 0.416737 0.909027i $$-0.363173\pi$$
0.416737 + 0.909027i $$0.363173\pi$$
$$644$$ 0 0
$$645$$ −79.4522 −3.12843
$$646$$ 0 0
$$647$$ −34.2892 −1.34805 −0.674024 0.738709i $$-0.735436\pi$$
−0.674024 + 0.738709i $$0.735436\pi$$
$$648$$ 0 0
$$649$$ 0.744352 0.0292184
$$650$$ 0 0
$$651$$ 2.03335 0.0796932
$$652$$ 0 0
$$653$$ −20.4234 −0.799231 −0.399615 0.916683i $$-0.630856\pi$$
−0.399615 + 0.916683i $$0.630856\pi$$
$$654$$ 0 0
$$655$$ 5.10187 0.199346
$$656$$ 0 0
$$657$$ 20.0814 0.783450
$$658$$ 0 0
$$659$$ −9.16578 −0.357048 −0.178524 0.983936i $$-0.557132\pi$$
−0.178524 + 0.983936i $$0.557132\pi$$
$$660$$ 0 0
$$661$$ −33.1919 −1.29102 −0.645508 0.763754i $$-0.723354\pi$$
−0.645508 + 0.763754i $$0.723354\pi$$
$$662$$ 0 0
$$663$$ −13.6253 −0.529161
$$664$$ 0 0
$$665$$ −7.52945 −0.291979
$$666$$ 0 0
$$667$$ −0.886034 −0.0343074
$$668$$ 0 0
$$669$$ −39.6047 −1.53121
$$670$$ 0 0
$$671$$ −12.1673 −0.469712
$$672$$ 0 0
$$673$$ 44.8277 1.72798 0.863990 0.503509i $$-0.167958\pi$$
0.863990 + 0.503509i $$0.167958\pi$$
$$674$$ 0 0
$$675$$ 21.7971 0.838970
$$676$$ 0 0
$$677$$ 17.9276 0.689013 0.344506 0.938784i $$-0.388046\pi$$
0.344506 + 0.938784i $$0.388046\pi$$
$$678$$ 0 0
$$679$$ −5.90855 −0.226749
$$680$$ 0 0
$$681$$ 36.5753 1.40157
$$682$$ 0 0
$$683$$ −35.3742 −1.35356 −0.676778 0.736187i $$-0.736624\pi$$
−0.676778 + 0.736187i $$0.736624\pi$$
$$684$$ 0 0
$$685$$ 46.3361 1.77041
$$686$$ 0 0
$$687$$ −5.71616 −0.218085
$$688$$ 0 0
$$689$$ −75.8322 −2.88898
$$690$$ 0 0
$$691$$ −32.6537 −1.24221 −0.621103 0.783729i $$-0.713315\pi$$
−0.621103 + 0.783729i $$0.713315\pi$$
$$692$$ 0 0
$$693$$ −1.76998 −0.0672361
$$694$$ 0 0
$$695$$ −21.3941 −0.811526
$$696$$ 0 0
$$697$$ −5.60299 −0.212228
$$698$$ 0 0
$$699$$ −1.50572 −0.0569516
$$700$$ 0 0
$$701$$ 12.3696 0.467194 0.233597 0.972334i $$-0.424950\pi$$
0.233597 + 0.972334i $$0.424950\pi$$
$$702$$ 0 0
$$703$$ −18.3696 −0.692823
$$704$$ 0 0
$$705$$ 55.0486 2.07325
$$706$$ 0 0
$$707$$ −5.08485 −0.191235
$$708$$ 0 0
$$709$$ −20.9482 −0.786727 −0.393363 0.919383i $$-0.628689\pi$$
−0.393363 + 0.919383i $$0.628689\pi$$
$$710$$ 0 0
$$711$$ 0.463202 0.0173714
$$712$$ 0 0
$$713$$ 1.35009 0.0505611
$$714$$ 0 0
$$715$$ −27.2505 −1.01911
$$716$$ 0 0
$$717$$ 18.1206 0.676725
$$718$$ 0 0
$$719$$ 28.0239 1.04511 0.522557 0.852604i $$-0.324978\pi$$
0.522557 + 0.852604i $$0.324978\pi$$
$$720$$ 0 0
$$721$$ 5.28477 0.196815
$$722$$ 0 0
$$723$$ −13.2890 −0.494223
$$724$$ 0 0
$$725$$ 4.18546 0.155444
$$726$$ 0 0
$$727$$ 23.9844 0.889531 0.444765 0.895647i $$-0.353287\pi$$
0.444765 + 0.895647i $$0.353287\pi$$
$$728$$ 0 0
$$729$$ 4.74583 0.175772
$$730$$ 0 0
$$731$$ 11.0796 0.409793
$$732$$ 0 0
$$733$$ 31.0864 1.14820 0.574102 0.818784i $$-0.305351\pi$$
0.574102 + 0.818784i $$0.305351\pi$$
$$734$$ 0 0
$$735$$ −7.17104 −0.264508
$$736$$ 0 0
$$737$$ −6.55902 −0.241604
$$738$$ 0 0
$$739$$ −12.0841 −0.444519 −0.222260 0.974988i $$-0.571343\pi$$
−0.222260 + 0.974988i $$0.571343\pi$$
$$740$$ 0 0
$$741$$ 30.1178 1.10640
$$742$$ 0 0
$$743$$ −52.8588 −1.93920 −0.969600 0.244695i $$-0.921312\pi$$
−0.969600 + 0.244695i $$0.921312\pi$$
$$744$$ 0 0
$$745$$ −73.9184 −2.70816
$$746$$ 0 0
$$747$$ 19.8854 0.727568
$$748$$ 0 0
$$749$$ −9.25309 −0.338100
$$750$$ 0 0
$$751$$ −30.6499 −1.11843 −0.559216 0.829022i $$-0.688898\pi$$
−0.559216 + 0.829022i $$0.688898\pi$$
$$752$$ 0 0
$$753$$ 55.9380 2.03849
$$754$$ 0 0
$$755$$ −15.6944 −0.571179
$$756$$ 0 0
$$757$$ 3.57276 0.129854 0.0649271 0.997890i $$-0.479319\pi$$
0.0649271 + 0.997890i $$0.479319\pi$$
$$758$$ 0 0
$$759$$ −3.63736 −0.132028
$$760$$ 0 0
$$761$$ −4.64537 −0.168395 −0.0841973 0.996449i $$-0.526833\pi$$
−0.0841973 + 0.996449i $$0.526833\pi$$
$$762$$ 0 0
$$763$$ 16.1000 0.582858
$$764$$ 0 0
$$765$$ 4.87766 0.176352
$$766$$ 0 0
$$767$$ −3.89748 −0.140730
$$768$$ 0 0
$$769$$ −13.1361 −0.473700 −0.236850 0.971546i $$-0.576115\pi$$
−0.236850 + 0.971546i $$0.576115\pi$$
$$770$$ 0 0
$$771$$ −14.2120 −0.511833
$$772$$ 0 0
$$773$$ 44.0372 1.58391 0.791954 0.610581i $$-0.209064\pi$$
0.791954 + 0.610581i $$0.209064\pi$$
$$774$$ 0 0
$$775$$ −6.37756 −0.229089
$$776$$ 0 0
$$777$$ −17.4952 −0.627637
$$778$$ 0 0
$$779$$ 12.3850 0.443740
$$780$$ 0 0
$$781$$ −4.12356 −0.147552
$$782$$ 0 0
$$783$$ 2.09248 0.0747792
$$784$$ 0 0
$$785$$ 15.4439 0.551218
$$786$$ 0 0
$$787$$ 39.3178 1.40153 0.700765 0.713393i $$-0.252842\pi$$
0.700765 + 0.713393i $$0.252842\pi$$
$$788$$ 0 0
$$789$$ 2.53550 0.0902661
$$790$$ 0 0
$$791$$ 10.5509 0.375148
$$792$$ 0 0
$$793$$ 63.7086 2.26236
$$794$$ 0 0
$$795$$ 84.0210 2.97992
$$796$$ 0 0
$$797$$ 32.5987 1.15470 0.577352 0.816496i $$-0.304086\pi$$
0.577352 + 0.816496i $$0.304086\pi$$
$$798$$ 0 0
$$799$$ −7.67652 −0.271576
$$800$$ 0 0
$$801$$ −21.1224 −0.746324
$$802$$ 0 0
$$803$$ −17.3344 −0.611719
$$804$$ 0 0
$$805$$ −4.76137 −0.167816
$$806$$ 0 0
$$807$$ 38.0356 1.33892
$$808$$ 0 0
$$809$$ −56.0461 −1.97048 −0.985239 0.171187i $$-0.945240\pi$$
−0.985239 + 0.171187i $$0.945240\pi$$
$$810$$ 0 0
$$811$$ 9.39490 0.329900 0.164950 0.986302i $$-0.447254\pi$$
0.164950 + 0.986302i $$0.447254\pi$$
$$812$$ 0 0
$$813$$ 41.5705 1.45794
$$814$$ 0 0
$$815$$ −41.7005 −1.46071
$$816$$ 0 0
$$817$$ −24.4907 −0.856822
$$818$$ 0 0
$$819$$ 9.26775 0.323841
$$820$$ 0 0
$$821$$ 7.40016 0.258267 0.129134 0.991627i $$-0.458780\pi$$
0.129134 + 0.991627i $$0.458780\pi$$
$$822$$ 0 0
$$823$$ 39.2675 1.36878 0.684390 0.729116i $$-0.260068\pi$$
0.684390 + 0.729116i $$0.260068\pi$$
$$824$$ 0 0
$$825$$ 17.1822 0.598209
$$826$$ 0 0
$$827$$ 30.5052 1.06077 0.530385 0.847757i $$-0.322047\pi$$
0.530385 + 0.847757i $$0.322047\pi$$
$$828$$ 0 0
$$829$$ 46.0461 1.59925 0.799624 0.600501i $$-0.205032\pi$$
0.799624 + 0.600501i $$0.205032\pi$$
$$830$$ 0 0
$$831$$ −16.8364 −0.584047
$$832$$ 0 0
$$833$$ 1.00000 0.0346479
$$834$$ 0 0
$$835$$ −4.56885 −0.158111
$$836$$ 0 0
$$837$$ −3.18840 −0.110207
$$838$$ 0 0
$$839$$ −43.1903 −1.49110 −0.745548 0.666452i $$-0.767812\pi$$
−0.745548 + 0.666452i $$0.767812\pi$$
$$840$$ 0 0
$$841$$ −28.5982 −0.986145
$$842$$ 0 0
$$843$$ −6.48176 −0.223244
$$844$$ 0 0
$$845$$ 98.4035 3.38518
$$846$$ 0 0
$$847$$ −9.47214 −0.325466
$$848$$ 0 0
$$849$$ −60.4043 −2.07307
$$850$$ 0 0
$$851$$ −11.6163 −0.398203
$$852$$ 0 0
$$853$$ −31.9186 −1.09287 −0.546437 0.837500i $$-0.684016\pi$$
−0.546437 + 0.837500i $$0.684016\pi$$
$$854$$ 0 0
$$855$$ −10.7818 −0.368728
$$856$$ 0 0
$$857$$ 11.6363 0.397490 0.198745 0.980051i $$-0.436314\pi$$
0.198745 + 0.980051i $$0.436314\pi$$
$$858$$ 0 0
$$859$$ 44.9205 1.53267 0.766335 0.642442i $$-0.222078\pi$$
0.766335 + 0.642442i $$0.222078\pi$$
$$860$$ 0 0
$$861$$ 11.7955 0.401990
$$862$$ 0 0
$$863$$ −22.2676 −0.757999 −0.379000 0.925397i $$-0.623732\pi$$
−0.379000 + 0.925397i $$0.623732\pi$$
$$864$$ 0 0
$$865$$ −6.24379 −0.212295
$$866$$ 0 0
$$867$$ −2.10522 −0.0714970
$$868$$ 0 0
$$869$$ −0.399840 −0.0135636
$$870$$ 0 0
$$871$$ 34.3435 1.16368
$$872$$ 0 0
$$873$$ −8.46072 −0.286352
$$874$$ 0 0
$$875$$ 5.46027 0.184591
$$876$$ 0 0
$$877$$ −18.2095 −0.614890 −0.307445 0.951566i $$-0.599474\pi$$
−0.307445 + 0.951566i $$0.599474\pi$$
$$878$$ 0 0
$$879$$ −4.52340 −0.152570
$$880$$ 0 0
$$881$$ 31.7548 1.06985 0.534923 0.844901i $$-0.320341\pi$$
0.534923 + 0.844901i $$0.320341\pi$$
$$882$$ 0 0
$$883$$ −57.7644 −1.94393 −0.971964 0.235130i $$-0.924448\pi$$
−0.971964 + 0.235130i $$0.924448\pi$$
$$884$$ 0 0
$$885$$ 4.31835 0.145160
$$886$$ 0 0
$$887$$ −21.9442 −0.736813 −0.368407 0.929665i $$-0.620097\pi$$
−0.368407 + 0.929665i $$0.620097\pi$$
$$888$$ 0 0
$$889$$ 17.9272 0.601257
$$890$$ 0 0
$$891$$ 13.9000 0.465669
$$892$$ 0 0
$$893$$ 16.9685 0.567828
$$894$$ 0 0
$$895$$ 68.9244 2.30389
$$896$$ 0 0
$$897$$ 19.0455 0.635910
$$898$$ 0 0
$$899$$ −0.612234 −0.0204192
$$900$$ 0 0
$$901$$ −11.7167 −0.390340
$$902$$ 0 0
$$903$$ −23.3250 −0.776206
$$904$$ 0 0
$$905$$ 20.5404 0.682786
$$906$$ 0 0
$$907$$ −10.0883 −0.334978 −0.167489 0.985874i $$-0.553566\pi$$
−0.167489 + 0.985874i $$0.553566\pi$$
$$908$$ 0 0
$$909$$ −7.28123 −0.241503
$$910$$ 0 0
$$911$$ 53.3155 1.76642 0.883210 0.468977i $$-0.155377\pi$$
0.883210 + 0.468977i $$0.155377\pi$$
$$912$$ 0 0
$$913$$ −17.1652 −0.568086
$$914$$ 0 0
$$915$$ −70.5883 −2.33358
$$916$$ 0 0
$$917$$ 1.49777 0.0494606
$$918$$ 0 0
$$919$$ −14.3247 −0.472529 −0.236264 0.971689i $$-0.575923\pi$$
−0.236264 + 0.971689i $$0.575923\pi$$
$$920$$ 0 0
$$921$$ −33.7448 −1.11193
$$922$$ 0 0
$$923$$ 21.5912 0.710684
$$924$$ 0 0
$$925$$ 54.8734 1.80423
$$926$$ 0 0
$$927$$ 7.56750 0.248549
$$928$$ 0 0
$$929$$ 30.0091 0.984567 0.492284 0.870435i $$-0.336162\pi$$
0.492284 + 0.870435i $$0.336162\pi$$
$$930$$ 0 0
$$931$$ −2.21044 −0.0724442
$$932$$ 0 0
$$933$$ 72.5259 2.37439
$$934$$ 0 0
$$935$$ −4.21044 −0.137696
$$936$$ 0 0
$$937$$ −45.1807 −1.47599 −0.737994 0.674807i $$-0.764227\pi$$
−0.737994 + 0.674807i $$0.764227\pi$$
$$938$$ 0 0
$$939$$ −68.5427 −2.23681
$$940$$ 0 0
$$941$$ 37.4598 1.22116 0.610578 0.791956i $$-0.290937\pi$$
0.610578 + 0.791956i $$0.290937\pi$$
$$942$$ 0 0
$$943$$ 7.83189 0.255041
$$944$$ 0 0
$$945$$ 11.2446 0.365786
$$946$$ 0 0
$$947$$ −1.33347 −0.0433318 −0.0216659 0.999765i $$-0.506897\pi$$
−0.0216659 + 0.999765i $$0.506897\pi$$
$$948$$ 0 0
$$949$$ 90.7643 2.94633
$$950$$ 0 0
$$951$$ 5.94874 0.192901
$$952$$ 0 0
$$953$$ 4.14271 0.134196 0.0670978 0.997746i $$-0.478626\pi$$
0.0670978 + 0.997746i $$0.478626\pi$$
$$954$$ 0 0
$$955$$ 37.1786 1.20307
$$956$$ 0 0
$$957$$ 1.64946 0.0533196
$$958$$ 0 0
$$959$$ 13.6030 0.439263
$$960$$ 0 0
$$961$$ −30.0671 −0.969907
$$962$$ 0 0
$$963$$ −13.2499 −0.426973
$$964$$ 0 0
$$965$$ −65.3050 −2.10224
$$966$$ 0 0
$$967$$ −23.9265 −0.769423 −0.384712 0.923037i $$-0.625699\pi$$
−0.384712 + 0.923037i $$0.625699\pi$$
$$968$$ 0 0
$$969$$ 4.65345 0.149490
$$970$$ 0 0
$$971$$ 41.7634 1.34025 0.670126 0.742248i $$-0.266240\pi$$
0.670126 + 0.742248i $$0.266240\pi$$
$$972$$ 0 0
$$973$$ −6.28073 −0.201351
$$974$$ 0 0
$$975$$ −89.9674 −2.88126
$$976$$ 0 0
$$977$$ −35.4143 −1.13300 −0.566501 0.824061i $$-0.691703\pi$$
−0.566501 + 0.824061i $$0.691703\pi$$
$$978$$ 0 0
$$979$$ 18.2331 0.582731
$$980$$ 0 0
$$981$$ 23.0543 0.736066
$$982$$ 0 0
$$983$$ 45.7062 1.45780 0.728901 0.684620i $$-0.240032\pi$$
0.728901 + 0.684620i $$0.240032\pi$$
$$984$$ 0 0
$$985$$ −9.86324 −0.314269
$$986$$ 0 0
$$987$$ 16.1608 0.514403
$$988$$ 0 0
$$989$$ −15.4871 −0.492462
$$990$$ 0 0
$$991$$ −13.3586 −0.424351 −0.212176 0.977232i $$-0.568055\pi$$
−0.212176 + 0.977232i $$0.568055\pi$$
$$992$$ 0 0
$$993$$ −34.9091 −1.10781
$$994$$ 0 0
$$995$$ −89.1877 −2.82744
$$996$$ 0 0
$$997$$ 31.8346 1.00821 0.504106 0.863642i $$-0.331822\pi$$
0.504106 + 0.863642i $$0.331822\pi$$
$$998$$ 0 0
$$999$$ 27.4334 0.867955
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 7616.2.a.bp.1.1 4
4.3 odd 2 7616.2.a.bj.1.4 4
8.3 odd 2 952.2.a.g.1.1 4
8.5 even 2 1904.2.a.q.1.4 4
24.11 even 2 8568.2.a.bj.1.4 4
56.27 even 2 6664.2.a.o.1.4 4

By twisted newform
Twist Min Dim Char Parity Ord Type
952.2.a.g.1.1 4 8.3 odd 2
1904.2.a.q.1.4 4 8.5 even 2
6664.2.a.o.1.4 4 56.27 even 2
7616.2.a.bj.1.4 4 4.3 odd 2
7616.2.a.bp.1.1 4 1.1 even 1 trivial
8568.2.a.bj.1.4 4 24.11 even 2