# Properties

 Label 9405.2.a.v.1.2 Level $9405$ Weight $2$ Character 9405.1 Self dual yes Analytic conductor $75.099$ Analytic rank $0$ Dimension $5$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9405,2,Mod(1,9405)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9405, 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("9405.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$9405 = 3^{2} \cdot 5 \cdot 11 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 9405.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: $$75.0993031010$$ Analytic rank: $$0$$ Dimension: $$5$$ Coefficient field: $$\Q(\zeta_{22})^+$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{5} - x^{4} - 4x^{3} + 3x^{2} + 3x - 1$$ x^5 - x^4 - 4*x^3 + 3*x^2 + 3*x - 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1045) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$-1.68251$$ of defining polynomial Character $$\chi$$ $$=$$ 9405.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-0.0881559 q^{2} -1.99223 q^{4} -1.00000 q^{5} -1.95185 q^{7} +0.351939 q^{8} +O(q^{10})$$ $$q-0.0881559 q^{2} -1.99223 q^{4} -1.00000 q^{5} -1.95185 q^{7} +0.351939 q^{8} +0.0881559 q^{10} +1.00000 q^{11} +3.20786 q^{13} +0.172067 q^{14} +3.95343 q^{16} -0.503983 q^{17} +1.00000 q^{19} +1.99223 q^{20} -0.0881559 q^{22} -1.05529 q^{23} +1.00000 q^{25} -0.282792 q^{26} +3.88853 q^{28} +7.91058 q^{29} +9.52800 q^{31} -1.05240 q^{32} +0.0444291 q^{34} +1.95185 q^{35} -5.22650 q^{37} -0.0881559 q^{38} -0.351939 q^{40} -8.32622 q^{41} -8.04835 q^{43} -1.99223 q^{44} +0.0930303 q^{46} +9.39522 q^{47} -3.19028 q^{49} -0.0881559 q^{50} -6.39079 q^{52} -2.37178 q^{53} -1.00000 q^{55} -0.686931 q^{56} -0.697365 q^{58} +2.80063 q^{59} -1.53640 q^{61} -0.839949 q^{62} -7.81409 q^{64} -3.20786 q^{65} -9.79354 q^{67} +1.00405 q^{68} -0.172067 q^{70} +4.45027 q^{71} -1.55991 q^{73} +0.460747 q^{74} -1.99223 q^{76} -1.95185 q^{77} +3.00849 q^{79} -3.95343 q^{80} +0.734005 q^{82} +2.22730 q^{83} +0.503983 q^{85} +0.709509 q^{86} +0.351939 q^{88} -6.10788 q^{89} -6.26126 q^{91} +2.10238 q^{92} -0.828244 q^{94} -1.00000 q^{95} -4.69521 q^{97} +0.281242 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$5 q + 3 q^{2} + 5 q^{4} - 5 q^{5} - 11 q^{7} - 3 q^{8}+O(q^{10})$$ 5 * q + 3 * q^2 + 5 * q^4 - 5 * q^5 - 11 * q^7 - 3 * q^8 $$5 q + 3 q^{2} + 5 q^{4} - 5 q^{5} - 11 q^{7} - 3 q^{8} - 3 q^{10} + 5 q^{11} + q^{13} - 3 q^{16} + 3 q^{17} + 5 q^{19} - 5 q^{20} + 3 q^{22} + 8 q^{23} + 5 q^{25} + 16 q^{26} - 22 q^{28} - 11 q^{29} - 5 q^{31} + 2 q^{32} + 4 q^{34} + 11 q^{35} - 9 q^{37} + 3 q^{38} + 3 q^{40} - 15 q^{41} - 13 q^{43} + 5 q^{44} + 18 q^{46} + 20 q^{47} + 20 q^{49} + 3 q^{50} + q^{52} + 5 q^{53} - 5 q^{55} - 33 q^{58} + 17 q^{59} + 3 q^{61} - 14 q^{62} - 17 q^{64} - q^{65} - 28 q^{67} + 25 q^{68} + 6 q^{71} - 16 q^{73} + 21 q^{74} + 5 q^{76} - 11 q^{77} + 3 q^{79} + 3 q^{80} + 2 q^{82} + 33 q^{83} - 3 q^{85} - 10 q^{86} - 3 q^{88} + 16 q^{89} - 22 q^{91} + 19 q^{92} - 10 q^{94} - 5 q^{95} - 14 q^{97} - 10 q^{98}+O(q^{100})$$ 5 * q + 3 * q^2 + 5 * q^4 - 5 * q^5 - 11 * q^7 - 3 * q^8 - 3 * q^10 + 5 * q^11 + q^13 - 3 * q^16 + 3 * q^17 + 5 * q^19 - 5 * q^20 + 3 * q^22 + 8 * q^23 + 5 * q^25 + 16 * q^26 - 22 * q^28 - 11 * q^29 - 5 * q^31 + 2 * q^32 + 4 * q^34 + 11 * q^35 - 9 * q^37 + 3 * q^38 + 3 * q^40 - 15 * q^41 - 13 * q^43 + 5 * q^44 + 18 * q^46 + 20 * q^47 + 20 * q^49 + 3 * q^50 + q^52 + 5 * q^53 - 5 * q^55 - 33 * q^58 + 17 * q^59 + 3 * q^61 - 14 * q^62 - 17 * q^64 - q^65 - 28 * q^67 + 25 * q^68 + 6 * q^71 - 16 * q^73 + 21 * q^74 + 5 * q^76 - 11 * q^77 + 3 * q^79 + 3 * q^80 + 2 * q^82 + 33 * q^83 - 3 * q^85 - 10 * q^86 - 3 * q^88 + 16 * q^89 - 22 * q^91 + 19 * q^92 - 10 * q^94 - 5 * q^95 - 14 * q^97 - 10 * q^98

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −0.0881559 −0.0623356 −0.0311678 0.999514i $$-0.509923\pi$$
−0.0311678 + 0.999514i $$0.509923\pi$$
$$3$$ 0 0
$$4$$ −1.99223 −0.996114
$$5$$ −1.00000 −0.447214
$$6$$ 0 0
$$7$$ −1.95185 −0.737730 −0.368865 0.929483i $$-0.620253\pi$$
−0.368865 + 0.929483i $$0.620253\pi$$
$$8$$ 0.351939 0.124429
$$9$$ 0 0
$$10$$ 0.0881559 0.0278774
$$11$$ 1.00000 0.301511
$$12$$ 0 0
$$13$$ 3.20786 0.889700 0.444850 0.895605i $$-0.353257\pi$$
0.444850 + 0.895605i $$0.353257\pi$$
$$14$$ 0.172067 0.0459869
$$15$$ 0 0
$$16$$ 3.95343 0.988358
$$17$$ −0.503983 −0.122234 −0.0611169 0.998131i $$-0.519466\pi$$
−0.0611169 + 0.998131i $$0.519466\pi$$
$$18$$ 0 0
$$19$$ 1.00000 0.229416
$$20$$ 1.99223 0.445476
$$21$$ 0 0
$$22$$ −0.0881559 −0.0187949
$$23$$ −1.05529 −0.220044 −0.110022 0.993929i $$-0.535092\pi$$
−0.110022 + 0.993929i $$0.535092\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ −0.282792 −0.0554601
$$27$$ 0 0
$$28$$ 3.88853 0.734863
$$29$$ 7.91058 1.46896 0.734479 0.678631i $$-0.237426\pi$$
0.734479 + 0.678631i $$0.237426\pi$$
$$30$$ 0 0
$$31$$ 9.52800 1.71128 0.855639 0.517573i $$-0.173164\pi$$
0.855639 + 0.517573i $$0.173164\pi$$
$$32$$ −1.05240 −0.186039
$$33$$ 0 0
$$34$$ 0.0444291 0.00761953
$$35$$ 1.95185 0.329923
$$36$$ 0 0
$$37$$ −5.22650 −0.859231 −0.429615 0.903012i $$-0.641351\pi$$
−0.429615 + 0.903012i $$0.641351\pi$$
$$38$$ −0.0881559 −0.0143008
$$39$$ 0 0
$$40$$ −0.351939 −0.0556464
$$41$$ −8.32622 −1.30034 −0.650168 0.759790i $$-0.725302\pi$$
−0.650168 + 0.759790i $$0.725302\pi$$
$$42$$ 0 0
$$43$$ −8.04835 −1.22736 −0.613681 0.789554i $$-0.710312\pi$$
−0.613681 + 0.789554i $$0.710312\pi$$
$$44$$ −1.99223 −0.300340
$$45$$ 0 0
$$46$$ 0.0930303 0.0137166
$$47$$ 9.39522 1.37043 0.685216 0.728339i $$-0.259708\pi$$
0.685216 + 0.728339i $$0.259708\pi$$
$$48$$ 0 0
$$49$$ −3.19028 −0.455755
$$50$$ −0.0881559 −0.0124671
$$51$$ 0 0
$$52$$ −6.39079 −0.886243
$$53$$ −2.37178 −0.325789 −0.162895 0.986643i $$-0.552083\pi$$
−0.162895 + 0.986643i $$0.552083\pi$$
$$54$$ 0 0
$$55$$ −1.00000 −0.134840
$$56$$ −0.686931 −0.0917950
$$57$$ 0 0
$$58$$ −0.697365 −0.0915685
$$59$$ 2.80063 0.364611 0.182305 0.983242i $$-0.441644\pi$$
0.182305 + 0.983242i $$0.441644\pi$$
$$60$$ 0 0
$$61$$ −1.53640 −0.196715 −0.0983577 0.995151i $$-0.531359\pi$$
−0.0983577 + 0.995151i $$0.531359\pi$$
$$62$$ −0.839949 −0.106674
$$63$$ 0 0
$$64$$ −7.81409 −0.976761
$$65$$ −3.20786 −0.397886
$$66$$ 0 0
$$67$$ −9.79354 −1.19647 −0.598236 0.801320i $$-0.704131\pi$$
−0.598236 + 0.801320i $$0.704131\pi$$
$$68$$ 1.00405 0.121759
$$69$$ 0 0
$$70$$ −0.172067 −0.0205659
$$71$$ 4.45027 0.528150 0.264075 0.964502i $$-0.414933\pi$$
0.264075 + 0.964502i $$0.414933\pi$$
$$72$$ 0 0
$$73$$ −1.55991 −0.182573 −0.0912866 0.995825i $$-0.529098\pi$$
−0.0912866 + 0.995825i $$0.529098\pi$$
$$74$$ 0.460747 0.0535607
$$75$$ 0 0
$$76$$ −1.99223 −0.228524
$$77$$ −1.95185 −0.222434
$$78$$ 0 0
$$79$$ 3.00849 0.338482 0.169241 0.985575i $$-0.445868\pi$$
0.169241 + 0.985575i $$0.445868\pi$$
$$80$$ −3.95343 −0.442007
$$81$$ 0 0
$$82$$ 0.734005 0.0810573
$$83$$ 2.22730 0.244478 0.122239 0.992501i $$-0.460992\pi$$
0.122239 + 0.992501i $$0.460992\pi$$
$$84$$ 0 0
$$85$$ 0.503983 0.0546647
$$86$$ 0.709509 0.0765084
$$87$$ 0 0
$$88$$ 0.351939 0.0375168
$$89$$ −6.10788 −0.647434 −0.323717 0.946154i $$-0.604933\pi$$
−0.323717 + 0.946154i $$0.604933\pi$$
$$90$$ 0 0
$$91$$ −6.26126 −0.656358
$$92$$ 2.10238 0.219189
$$93$$ 0 0
$$94$$ −0.828244 −0.0854268
$$95$$ −1.00000 −0.102598
$$96$$ 0 0
$$97$$ −4.69521 −0.476726 −0.238363 0.971176i $$-0.576611\pi$$
−0.238363 + 0.971176i $$0.576611\pi$$
$$98$$ 0.281242 0.0284098
$$99$$ 0 0
$$100$$ −1.99223 −0.199223
$$101$$ −0.160574 −0.0159777 −0.00798885 0.999968i $$-0.502543\pi$$
−0.00798885 + 0.999968i $$0.502543\pi$$
$$102$$ 0 0
$$103$$ −12.1749 −1.19962 −0.599812 0.800141i $$-0.704758\pi$$
−0.599812 + 0.800141i $$0.704758\pi$$
$$104$$ 1.12897 0.110705
$$105$$ 0 0
$$106$$ 0.209087 0.0203083
$$107$$ −16.2082 −1.56690 −0.783452 0.621452i $$-0.786543\pi$$
−0.783452 + 0.621452i $$0.786543\pi$$
$$108$$ 0 0
$$109$$ 17.2276 1.65010 0.825050 0.565059i $$-0.191147\pi$$
0.825050 + 0.565059i $$0.191147\pi$$
$$110$$ 0.0881559 0.00840534
$$111$$ 0 0
$$112$$ −7.71650 −0.729141
$$113$$ 9.74136 0.916390 0.458195 0.888852i $$-0.348496\pi$$
0.458195 + 0.888852i $$0.348496\pi$$
$$114$$ 0 0
$$115$$ 1.05529 0.0984065
$$116$$ −15.7597 −1.46325
$$117$$ 0 0
$$118$$ −0.246892 −0.0227283
$$119$$ 0.983699 0.0901756
$$120$$ 0 0
$$121$$ 1.00000 0.0909091
$$122$$ 0.135442 0.0122624
$$123$$ 0 0
$$124$$ −18.9819 −1.70463
$$125$$ −1.00000 −0.0894427
$$126$$ 0 0
$$127$$ −6.14600 −0.545370 −0.272685 0.962103i $$-0.587912\pi$$
−0.272685 + 0.962103i $$0.587912\pi$$
$$128$$ 2.79365 0.246926
$$129$$ 0 0
$$130$$ 0.282792 0.0248025
$$131$$ 13.4596 1.17597 0.587987 0.808871i $$-0.299921\pi$$
0.587987 + 0.808871i $$0.299921\pi$$
$$132$$ 0 0
$$133$$ −1.95185 −0.169247
$$134$$ 0.863359 0.0745828
$$135$$ 0 0
$$136$$ −0.177371 −0.0152095
$$137$$ −7.45968 −0.637324 −0.318662 0.947868i $$-0.603233\pi$$
−0.318662 + 0.947868i $$0.603233\pi$$
$$138$$ 0 0
$$139$$ −18.3550 −1.55685 −0.778425 0.627738i $$-0.783981\pi$$
−0.778425 + 0.627738i $$0.783981\pi$$
$$140$$ −3.88853 −0.328641
$$141$$ 0 0
$$142$$ −0.392318 −0.0329226
$$143$$ 3.20786 0.268255
$$144$$ 0 0
$$145$$ −7.91058 −0.656938
$$146$$ 0.137515 0.0113808
$$147$$ 0 0
$$148$$ 10.4124 0.855892
$$149$$ 19.2387 1.57610 0.788048 0.615614i $$-0.211092\pi$$
0.788048 + 0.615614i $$0.211092\pi$$
$$150$$ 0 0
$$151$$ 7.76410 0.631833 0.315917 0.948787i $$-0.397688\pi$$
0.315917 + 0.948787i $$0.397688\pi$$
$$152$$ 0.351939 0.0285460
$$153$$ 0 0
$$154$$ 0.172067 0.0138656
$$155$$ −9.52800 −0.765307
$$156$$ 0 0
$$157$$ −4.82520 −0.385093 −0.192546 0.981288i $$-0.561675\pi$$
−0.192546 + 0.981288i $$0.561675\pi$$
$$158$$ −0.265216 −0.0210995
$$159$$ 0 0
$$160$$ 1.05240 0.0831992
$$161$$ 2.05977 0.162333
$$162$$ 0 0
$$163$$ 16.0681 1.25855 0.629276 0.777182i $$-0.283352\pi$$
0.629276 + 0.777182i $$0.283352\pi$$
$$164$$ 16.5877 1.29528
$$165$$ 0 0
$$166$$ −0.196350 −0.0152397
$$167$$ 19.8521 1.53620 0.768102 0.640327i $$-0.221201\pi$$
0.768102 + 0.640327i $$0.221201\pi$$
$$168$$ 0 0
$$169$$ −2.70963 −0.208433
$$170$$ −0.0444291 −0.00340756
$$171$$ 0 0
$$172$$ 16.0341 1.22259
$$173$$ 23.1362 1.75901 0.879507 0.475886i $$-0.157873\pi$$
0.879507 + 0.475886i $$0.157873\pi$$
$$174$$ 0 0
$$175$$ −1.95185 −0.147546
$$176$$ 3.95343 0.298001
$$177$$ 0 0
$$178$$ 0.538446 0.0403582
$$179$$ −4.66377 −0.348586 −0.174293 0.984694i $$-0.555764\pi$$
−0.174293 + 0.984694i $$0.555764\pi$$
$$180$$ 0 0
$$181$$ 23.7883 1.76817 0.884084 0.467328i $$-0.154783\pi$$
0.884084 + 0.467328i $$0.154783\pi$$
$$182$$ 0.551967 0.0409145
$$183$$ 0 0
$$184$$ −0.371398 −0.0273798
$$185$$ 5.22650 0.384260
$$186$$ 0 0
$$187$$ −0.503983 −0.0368549
$$188$$ −18.7174 −1.36511
$$189$$ 0 0
$$190$$ 0.0881559 0.00639550
$$191$$ 4.12335 0.298355 0.149178 0.988810i $$-0.452337\pi$$
0.149178 + 0.988810i $$0.452337\pi$$
$$192$$ 0 0
$$193$$ 16.6564 1.19895 0.599477 0.800392i $$-0.295375\pi$$
0.599477 + 0.800392i $$0.295375\pi$$
$$194$$ 0.413910 0.0297170
$$195$$ 0 0
$$196$$ 6.35578 0.453984
$$197$$ −12.7284 −0.906862 −0.453431 0.891291i $$-0.649800\pi$$
−0.453431 + 0.891291i $$0.649800\pi$$
$$198$$ 0 0
$$199$$ −17.0542 −1.20894 −0.604471 0.796627i $$-0.706615\pi$$
−0.604471 + 0.796627i $$0.706615\pi$$
$$200$$ 0.351939 0.0248858
$$201$$ 0 0
$$202$$ 0.0141555 0.000995980 0
$$203$$ −15.4403 −1.08369
$$204$$ 0 0
$$205$$ 8.32622 0.581528
$$206$$ 1.07329 0.0747794
$$207$$ 0 0
$$208$$ 12.6821 0.879342
$$209$$ 1.00000 0.0691714
$$210$$ 0 0
$$211$$ −15.0057 −1.03304 −0.516518 0.856276i $$-0.672772\pi$$
−0.516518 + 0.856276i $$0.672772\pi$$
$$212$$ 4.72513 0.324523
$$213$$ 0 0
$$214$$ 1.42885 0.0976740
$$215$$ 8.04835 0.548893
$$216$$ 0 0
$$217$$ −18.5972 −1.26246
$$218$$ −1.51871 −0.102860
$$219$$ 0 0
$$220$$ 1.99223 0.134316
$$221$$ −1.61671 −0.108752
$$222$$ 0 0
$$223$$ 4.21760 0.282432 0.141216 0.989979i $$-0.454899\pi$$
0.141216 + 0.989979i $$0.454899\pi$$
$$224$$ 2.05412 0.137246
$$225$$ 0 0
$$226$$ −0.858759 −0.0571238
$$227$$ −18.5091 −1.22849 −0.614245 0.789115i $$-0.710539\pi$$
−0.614245 + 0.789115i $$0.710539\pi$$
$$228$$ 0 0
$$229$$ 10.0978 0.667282 0.333641 0.942700i $$-0.391723\pi$$
0.333641 + 0.942700i $$0.391723\pi$$
$$230$$ −0.0930303 −0.00613424
$$231$$ 0 0
$$232$$ 2.78404 0.182781
$$233$$ 5.28276 0.346085 0.173043 0.984914i $$-0.444640\pi$$
0.173043 + 0.984914i $$0.444640\pi$$
$$234$$ 0 0
$$235$$ −9.39522 −0.612876
$$236$$ −5.57949 −0.363194
$$237$$ 0 0
$$238$$ −0.0867189 −0.00562115
$$239$$ −6.01161 −0.388859 −0.194429 0.980917i $$-0.562286\pi$$
−0.194429 + 0.980917i $$0.562286\pi$$
$$240$$ 0 0
$$241$$ −24.1747 −1.55723 −0.778613 0.627504i $$-0.784076\pi$$
−0.778613 + 0.627504i $$0.784076\pi$$
$$242$$ −0.0881559 −0.00566688
$$243$$ 0 0
$$244$$ 3.06085 0.195951
$$245$$ 3.19028 0.203820
$$246$$ 0 0
$$247$$ 3.20786 0.204111
$$248$$ 3.35327 0.212933
$$249$$ 0 0
$$250$$ 0.0881559 0.00557547
$$251$$ 21.5063 1.35747 0.678734 0.734385i $$-0.262529\pi$$
0.678734 + 0.734385i $$0.262529\pi$$
$$252$$ 0 0
$$253$$ −1.05529 −0.0663457
$$254$$ 0.541807 0.0339960
$$255$$ 0 0
$$256$$ 15.3819 0.961369
$$257$$ 8.04652 0.501928 0.250964 0.967996i $$-0.419252\pi$$
0.250964 + 0.967996i $$0.419252\pi$$
$$258$$ 0 0
$$259$$ 10.2013 0.633880
$$260$$ 6.39079 0.396340
$$261$$ 0 0
$$262$$ −1.18655 −0.0733050
$$263$$ 19.5827 1.20752 0.603762 0.797165i $$-0.293668\pi$$
0.603762 + 0.797165i $$0.293668\pi$$
$$264$$ 0 0
$$265$$ 2.37178 0.145697
$$266$$ 0.172067 0.0105501
$$267$$ 0 0
$$268$$ 19.5110 1.19182
$$269$$ 27.2287 1.66017 0.830083 0.557640i $$-0.188293\pi$$
0.830083 + 0.557640i $$0.188293\pi$$
$$270$$ 0 0
$$271$$ 16.4952 1.00201 0.501006 0.865444i $$-0.332963\pi$$
0.501006 + 0.865444i $$0.332963\pi$$
$$272$$ −1.99246 −0.120811
$$273$$ 0 0
$$274$$ 0.657615 0.0397280
$$275$$ 1.00000 0.0603023
$$276$$ 0 0
$$277$$ −14.0482 −0.844077 −0.422039 0.906578i $$-0.638685\pi$$
−0.422039 + 0.906578i $$0.638685\pi$$
$$278$$ 1.61810 0.0970472
$$279$$ 0 0
$$280$$ 0.686931 0.0410520
$$281$$ 4.14119 0.247043 0.123521 0.992342i $$-0.460581\pi$$
0.123521 + 0.992342i $$0.460581\pi$$
$$282$$ 0 0
$$283$$ 4.15687 0.247100 0.123550 0.992338i $$-0.460572\pi$$
0.123550 + 0.992338i $$0.460572\pi$$
$$284$$ −8.86596 −0.526098
$$285$$ 0 0
$$286$$ −0.282792 −0.0167218
$$287$$ 16.2515 0.959297
$$288$$ 0 0
$$289$$ −16.7460 −0.985059
$$290$$ 0.697365 0.0409507
$$291$$ 0 0
$$292$$ 3.10769 0.181864
$$293$$ 5.34865 0.312471 0.156236 0.987720i $$-0.450064\pi$$
0.156236 + 0.987720i $$0.450064\pi$$
$$294$$ 0 0
$$295$$ −2.80063 −0.163059
$$296$$ −1.83941 −0.106913
$$297$$ 0 0
$$298$$ −1.69601 −0.0982470
$$299$$ −3.38523 −0.195773
$$300$$ 0 0
$$301$$ 15.7092 0.905461
$$302$$ −0.684451 −0.0393858
$$303$$ 0 0
$$304$$ 3.95343 0.226745
$$305$$ 1.53640 0.0879738
$$306$$ 0 0
$$307$$ −24.7343 −1.41166 −0.705830 0.708381i $$-0.749426\pi$$
−0.705830 + 0.708381i $$0.749426\pi$$
$$308$$ 3.88853 0.221570
$$309$$ 0 0
$$310$$ 0.839949 0.0477059
$$311$$ −17.9991 −1.02063 −0.510317 0.859986i $$-0.670472\pi$$
−0.510317 + 0.859986i $$0.670472\pi$$
$$312$$ 0 0
$$313$$ −34.3706 −1.94274 −0.971372 0.237564i $$-0.923651\pi$$
−0.971372 + 0.237564i $$0.923651\pi$$
$$314$$ 0.425370 0.0240050
$$315$$ 0 0
$$316$$ −5.99360 −0.337166
$$317$$ 13.1196 0.736870 0.368435 0.929654i $$-0.379894\pi$$
0.368435 + 0.929654i $$0.379894\pi$$
$$318$$ 0 0
$$319$$ 7.91058 0.442908
$$320$$ 7.81409 0.436821
$$321$$ 0 0
$$322$$ −0.181581 −0.0101191
$$323$$ −0.503983 −0.0280424
$$324$$ 0 0
$$325$$ 3.20786 0.177940
$$326$$ −1.41650 −0.0784527
$$327$$ 0 0
$$328$$ −2.93032 −0.161800
$$329$$ −18.3380 −1.01101
$$330$$ 0 0
$$331$$ −0.911287 −0.0500889 −0.0250444 0.999686i $$-0.507973\pi$$
−0.0250444 + 0.999686i $$0.507973\pi$$
$$332$$ −4.43730 −0.243528
$$333$$ 0 0
$$334$$ −1.75008 −0.0957603
$$335$$ 9.79354 0.535078
$$336$$ 0 0
$$337$$ 12.9894 0.707577 0.353789 0.935325i $$-0.384893\pi$$
0.353789 + 0.935325i $$0.384893\pi$$
$$338$$ 0.238870 0.0129928
$$339$$ 0 0
$$340$$ −1.00405 −0.0544522
$$341$$ 9.52800 0.515970
$$342$$ 0 0
$$343$$ 19.8899 1.07395
$$344$$ −2.83252 −0.152719
$$345$$ 0 0
$$346$$ −2.03959 −0.109649
$$347$$ 8.50005 0.456307 0.228153 0.973625i $$-0.426731\pi$$
0.228153 + 0.973625i $$0.426731\pi$$
$$348$$ 0 0
$$349$$ −22.9281 −1.22731 −0.613657 0.789573i $$-0.710302\pi$$
−0.613657 + 0.789573i $$0.710302\pi$$
$$350$$ 0.172067 0.00919737
$$351$$ 0 0
$$352$$ −1.05240 −0.0560929
$$353$$ 32.5037 1.73000 0.864998 0.501775i $$-0.167319\pi$$
0.864998 + 0.501775i $$0.167319\pi$$
$$354$$ 0 0
$$355$$ −4.45027 −0.236196
$$356$$ 12.1683 0.644919
$$357$$ 0 0
$$358$$ 0.411139 0.0217293
$$359$$ 7.05497 0.372347 0.186174 0.982517i $$-0.440391\pi$$
0.186174 + 0.982517i $$0.440391\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ −2.09708 −0.110220
$$363$$ 0 0
$$364$$ 12.4739 0.653808
$$365$$ 1.55991 0.0816492
$$366$$ 0 0
$$367$$ 14.8301 0.774127 0.387063 0.922053i $$-0.373489\pi$$
0.387063 + 0.922053i $$0.373489\pi$$
$$368$$ −4.17203 −0.217482
$$369$$ 0 0
$$370$$ −0.460747 −0.0239531
$$371$$ 4.62936 0.240344
$$372$$ 0 0
$$373$$ 9.69631 0.502056 0.251028 0.967980i $$-0.419231\pi$$
0.251028 + 0.967980i $$0.419231\pi$$
$$374$$ 0.0444291 0.00229737
$$375$$ 0 0
$$376$$ 3.30654 0.170522
$$377$$ 25.3761 1.30693
$$378$$ 0 0
$$379$$ 1.03558 0.0531941 0.0265970 0.999646i $$-0.491533\pi$$
0.0265970 + 0.999646i $$0.491533\pi$$
$$380$$ 1.99223 0.102199
$$381$$ 0 0
$$382$$ −0.363498 −0.0185982
$$383$$ −1.35607 −0.0692922 −0.0346461 0.999400i $$-0.511030\pi$$
−0.0346461 + 0.999400i $$0.511030\pi$$
$$384$$ 0 0
$$385$$ 1.95185 0.0994754
$$386$$ −1.46836 −0.0747376
$$387$$ 0 0
$$388$$ 9.35393 0.474874
$$389$$ 34.2838 1.73826 0.869129 0.494585i $$-0.164680\pi$$
0.869129 + 0.494585i $$0.164680\pi$$
$$390$$ 0 0
$$391$$ 0.531850 0.0268968
$$392$$ −1.12278 −0.0567092
$$393$$ 0 0
$$394$$ 1.12208 0.0565298
$$395$$ −3.00849 −0.151374
$$396$$ 0 0
$$397$$ −35.4936 −1.78137 −0.890687 0.454617i $$-0.849776\pi$$
−0.890687 + 0.454617i $$0.849776\pi$$
$$398$$ 1.50343 0.0753601
$$399$$ 0 0
$$400$$ 3.95343 0.197672
$$401$$ −28.4914 −1.42279 −0.711396 0.702791i $$-0.751937\pi$$
−0.711396 + 0.702791i $$0.751937\pi$$
$$402$$ 0 0
$$403$$ 30.5645 1.52253
$$404$$ 0.319900 0.0159156
$$405$$ 0 0
$$406$$ 1.36115 0.0675528
$$407$$ −5.22650 −0.259068
$$408$$ 0 0
$$409$$ 31.1180 1.53869 0.769343 0.638836i $$-0.220584\pi$$
0.769343 + 0.638836i $$0.220584\pi$$
$$410$$ −0.734005 −0.0362499
$$411$$ 0 0
$$412$$ 24.2551 1.19496
$$413$$ −5.46641 −0.268984
$$414$$ 0 0
$$415$$ −2.22730 −0.109334
$$416$$ −3.37594 −0.165519
$$417$$ 0 0
$$418$$ −0.0881559 −0.00431185
$$419$$ 23.8806 1.16664 0.583322 0.812241i $$-0.301753\pi$$
0.583322 + 0.812241i $$0.301753\pi$$
$$420$$ 0 0
$$421$$ −1.13683 −0.0554059 −0.0277029 0.999616i $$-0.508819\pi$$
−0.0277029 + 0.999616i $$0.508819\pi$$
$$422$$ 1.32284 0.0643950
$$423$$ 0 0
$$424$$ −0.834722 −0.0405377
$$425$$ −0.503983 −0.0244468
$$426$$ 0 0
$$427$$ 2.99881 0.145123
$$428$$ 32.2904 1.56082
$$429$$ 0 0
$$430$$ −0.709509 −0.0342156
$$431$$ 0.598886 0.0288473 0.0144237 0.999896i $$-0.495409\pi$$
0.0144237 + 0.999896i $$0.495409\pi$$
$$432$$ 0 0
$$433$$ 32.3456 1.55443 0.777216 0.629234i $$-0.216631\pi$$
0.777216 + 0.629234i $$0.216631\pi$$
$$434$$ 1.63945 0.0786963
$$435$$ 0 0
$$436$$ −34.3212 −1.64369
$$437$$ −1.05529 −0.0504815
$$438$$ 0 0
$$439$$ 9.36506 0.446970 0.223485 0.974707i $$-0.428257\pi$$
0.223485 + 0.974707i $$0.428257\pi$$
$$440$$ −0.351939 −0.0167780
$$441$$ 0 0
$$442$$ 0.142522 0.00677910
$$443$$ −10.5557 −0.501517 −0.250759 0.968050i $$-0.580680\pi$$
−0.250759 + 0.968050i $$0.580680\pi$$
$$444$$ 0 0
$$445$$ 6.10788 0.289541
$$446$$ −0.371807 −0.0176056
$$447$$ 0 0
$$448$$ 15.2519 0.720586
$$449$$ 13.4399 0.634269 0.317135 0.948381i $$-0.397279\pi$$
0.317135 + 0.948381i $$0.397279\pi$$
$$450$$ 0 0
$$451$$ −8.32622 −0.392066
$$452$$ −19.4070 −0.912829
$$453$$ 0 0
$$454$$ 1.63168 0.0765787
$$455$$ 6.26126 0.293532
$$456$$ 0 0
$$457$$ 3.88536 0.181750 0.0908748 0.995862i $$-0.471034\pi$$
0.0908748 + 0.995862i $$0.471034\pi$$
$$458$$ −0.890182 −0.0415955
$$459$$ 0 0
$$460$$ −2.10238 −0.0980242
$$461$$ −15.7252 −0.732396 −0.366198 0.930537i $$-0.619341\pi$$
−0.366198 + 0.930537i $$0.619341\pi$$
$$462$$ 0 0
$$463$$ 35.4757 1.64869 0.824347 0.566084i $$-0.191542\pi$$
0.824347 + 0.566084i $$0.191542\pi$$
$$464$$ 31.2740 1.45186
$$465$$ 0 0
$$466$$ −0.465707 −0.0215734
$$467$$ −13.5528 −0.627148 −0.313574 0.949564i $$-0.601526\pi$$
−0.313574 + 0.949564i $$0.601526\pi$$
$$468$$ 0 0
$$469$$ 19.1155 0.882673
$$470$$ 0.828244 0.0382040
$$471$$ 0 0
$$472$$ 0.985649 0.0453682
$$473$$ −8.04835 −0.370063
$$474$$ 0 0
$$475$$ 1.00000 0.0458831
$$476$$ −1.95975 −0.0898252
$$477$$ 0 0
$$478$$ 0.529959 0.0242398
$$479$$ −39.8445 −1.82054 −0.910271 0.414012i $$-0.864127\pi$$
−0.910271 + 0.414012i $$0.864127\pi$$
$$480$$ 0 0
$$481$$ −16.7659 −0.764458
$$482$$ 2.13114 0.0970707
$$483$$ 0 0
$$484$$ −1.99223 −0.0905558
$$485$$ 4.69521 0.213198
$$486$$ 0 0
$$487$$ 35.9781 1.63032 0.815161 0.579234i $$-0.196648\pi$$
0.815161 + 0.579234i $$0.196648\pi$$
$$488$$ −0.540717 −0.0244771
$$489$$ 0 0
$$490$$ −0.281242 −0.0127052
$$491$$ 37.4470 1.68996 0.844980 0.534798i $$-0.179612\pi$$
0.844980 + 0.534798i $$0.179612\pi$$
$$492$$ 0 0
$$493$$ −3.98680 −0.179557
$$494$$ −0.282792 −0.0127234
$$495$$ 0 0
$$496$$ 37.6683 1.69136
$$497$$ −8.68626 −0.389632
$$498$$ 0 0
$$499$$ 28.2460 1.26446 0.632232 0.774779i $$-0.282139\pi$$
0.632232 + 0.774779i $$0.282139\pi$$
$$500$$ 1.99223 0.0890952
$$501$$ 0 0
$$502$$ −1.89591 −0.0846186
$$503$$ 15.8937 0.708667 0.354333 0.935119i $$-0.384708\pi$$
0.354333 + 0.935119i $$0.384708\pi$$
$$504$$ 0 0
$$505$$ 0.160574 0.00714545
$$506$$ 0.0930303 0.00413570
$$507$$ 0 0
$$508$$ 12.2442 0.543251
$$509$$ −1.13300 −0.0502195 −0.0251097 0.999685i $$-0.507994\pi$$
−0.0251097 + 0.999685i $$0.507994\pi$$
$$510$$ 0 0
$$511$$ 3.04470 0.134690
$$512$$ −6.94330 −0.306854
$$513$$ 0 0
$$514$$ −0.709348 −0.0312880
$$515$$ 12.1749 0.536488
$$516$$ 0 0
$$517$$ 9.39522 0.413201
$$518$$ −0.899308 −0.0395133
$$519$$ 0 0
$$520$$ −1.12897 −0.0495086
$$521$$ 35.7749 1.56732 0.783662 0.621187i $$-0.213349\pi$$
0.783662 + 0.621187i $$0.213349\pi$$
$$522$$ 0 0
$$523$$ −11.7973 −0.515862 −0.257931 0.966163i $$-0.583041\pi$$
−0.257931 + 0.966163i $$0.583041\pi$$
$$524$$ −26.8147 −1.17140
$$525$$ 0 0
$$526$$ −1.72633 −0.0752718
$$527$$ −4.80195 −0.209176
$$528$$ 0 0
$$529$$ −21.8864 −0.951581
$$530$$ −0.209087 −0.00908214
$$531$$ 0 0
$$532$$ 3.88853 0.168589
$$533$$ −26.7093 −1.15691
$$534$$ 0 0
$$535$$ 16.2082 0.700741
$$536$$ −3.44673 −0.148876
$$537$$ 0 0
$$538$$ −2.40037 −0.103488
$$539$$ −3.19028 −0.137415
$$540$$ 0 0
$$541$$ −29.1292 −1.25236 −0.626181 0.779678i $$-0.715383\pi$$
−0.626181 + 0.779678i $$0.715383\pi$$
$$542$$ −1.45415 −0.0624611
$$543$$ 0 0
$$544$$ 0.530390 0.0227403
$$545$$ −17.2276 −0.737947
$$546$$ 0 0
$$547$$ 25.3904 1.08561 0.542807 0.839857i $$-0.317361\pi$$
0.542807 + 0.839857i $$0.317361\pi$$
$$548$$ 14.8614 0.634847
$$549$$ 0 0
$$550$$ −0.0881559 −0.00375898
$$551$$ 7.91058 0.337002
$$552$$ 0 0
$$553$$ −5.87212 −0.249708
$$554$$ 1.23844 0.0526161
$$555$$ 0 0
$$556$$ 36.5673 1.55080
$$557$$ 2.80423 0.118819 0.0594096 0.998234i $$-0.481078\pi$$
0.0594096 + 0.998234i $$0.481078\pi$$
$$558$$ 0 0
$$559$$ −25.8180 −1.09198
$$560$$ 7.71650 0.326082
$$561$$ 0 0
$$562$$ −0.365070 −0.0153996
$$563$$ −14.6693 −0.618239 −0.309119 0.951023i $$-0.600034\pi$$
−0.309119 + 0.951023i $$0.600034\pi$$
$$564$$ 0 0
$$565$$ −9.74136 −0.409822
$$566$$ −0.366453 −0.0154031
$$567$$ 0 0
$$568$$ 1.56622 0.0657172
$$569$$ 38.3728 1.60867 0.804336 0.594175i $$-0.202521\pi$$
0.804336 + 0.594175i $$0.202521\pi$$
$$570$$ 0 0
$$571$$ 25.0213 1.04711 0.523554 0.851993i $$-0.324606\pi$$
0.523554 + 0.851993i $$0.324606\pi$$
$$572$$ −6.39079 −0.267212
$$573$$ 0 0
$$574$$ −1.43267 −0.0597984
$$575$$ −1.05529 −0.0440087
$$576$$ 0 0
$$577$$ −28.9574 −1.20551 −0.602756 0.797925i $$-0.705931\pi$$
−0.602756 + 0.797925i $$0.705931\pi$$
$$578$$ 1.47626 0.0614043
$$579$$ 0 0
$$580$$ 15.7597 0.654386
$$581$$ −4.34736 −0.180359
$$582$$ 0 0
$$583$$ −2.37178 −0.0982292
$$584$$ −0.548991 −0.0227174
$$585$$ 0 0
$$586$$ −0.471515 −0.0194781
$$587$$ 45.1372 1.86301 0.931506 0.363726i $$-0.118496\pi$$
0.931506 + 0.363726i $$0.118496\pi$$
$$588$$ 0 0
$$589$$ 9.52800 0.392594
$$590$$ 0.246892 0.0101644
$$591$$ 0 0
$$592$$ −20.6626 −0.849227
$$593$$ −8.57775 −0.352246 −0.176123 0.984368i $$-0.556356\pi$$
−0.176123 + 0.984368i $$0.556356\pi$$
$$594$$ 0 0
$$595$$ −0.983699 −0.0403277
$$596$$ −38.3279 −1.56997
$$597$$ 0 0
$$598$$ 0.298428 0.0122036
$$599$$ −6.04473 −0.246981 −0.123490 0.992346i $$-0.539409\pi$$
−0.123490 + 0.992346i $$0.539409\pi$$
$$600$$ 0 0
$$601$$ 22.5599 0.920237 0.460118 0.887858i $$-0.347807\pi$$
0.460118 + 0.887858i $$0.347807\pi$$
$$602$$ −1.38486 −0.0564425
$$603$$ 0 0
$$604$$ −15.4679 −0.629378
$$605$$ −1.00000 −0.0406558
$$606$$ 0 0
$$607$$ 38.8854 1.57831 0.789155 0.614194i $$-0.210519\pi$$
0.789155 + 0.614194i $$0.210519\pi$$
$$608$$ −1.05240 −0.0426803
$$609$$ 0 0
$$610$$ −0.135442 −0.00548390
$$611$$ 30.1385 1.21927
$$612$$ 0 0
$$613$$ −26.0557 −1.05238 −0.526191 0.850367i $$-0.676380\pi$$
−0.526191 + 0.850367i $$0.676380\pi$$
$$614$$ 2.18047 0.0879967
$$615$$ 0 0
$$616$$ −0.686931 −0.0276772
$$617$$ −1.39432 −0.0561331 −0.0280665 0.999606i $$-0.508935\pi$$
−0.0280665 + 0.999606i $$0.508935\pi$$
$$618$$ 0 0
$$619$$ 9.82493 0.394897 0.197449 0.980313i $$-0.436734\pi$$
0.197449 + 0.980313i $$0.436734\pi$$
$$620$$ 18.9819 0.762333
$$621$$ 0 0
$$622$$ 1.58673 0.0636219
$$623$$ 11.9217 0.477632
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 3.02998 0.121102
$$627$$ 0 0
$$628$$ 9.61290 0.383596
$$629$$ 2.63407 0.105027
$$630$$ 0 0
$$631$$ −39.0163 −1.55321 −0.776607 0.629986i $$-0.783061\pi$$
−0.776607 + 0.629986i $$0.783061\pi$$
$$632$$ 1.05880 0.0421169
$$633$$ 0 0
$$634$$ −1.15657 −0.0459333
$$635$$ 6.14600 0.243897
$$636$$ 0 0
$$637$$ −10.2340 −0.405485
$$638$$ −0.697365 −0.0276089
$$639$$ 0 0
$$640$$ −2.79365 −0.110429
$$641$$ −24.6377 −0.973130 −0.486565 0.873644i $$-0.661750\pi$$
−0.486565 + 0.873644i $$0.661750\pi$$
$$642$$ 0 0
$$643$$ 31.8076 1.25437 0.627185 0.778871i $$-0.284207\pi$$
0.627185 + 0.778871i $$0.284207\pi$$
$$644$$ −4.10354 −0.161702
$$645$$ 0 0
$$646$$ 0.0444291 0.00174804
$$647$$ −18.5631 −0.729790 −0.364895 0.931049i $$-0.618895\pi$$
−0.364895 + 0.931049i $$0.618895\pi$$
$$648$$ 0 0
$$649$$ 2.80063 0.109934
$$650$$ −0.282792 −0.0110920
$$651$$ 0 0
$$652$$ −32.0114 −1.25366
$$653$$ −11.2747 −0.441212 −0.220606 0.975363i $$-0.570804\pi$$
−0.220606 + 0.975363i $$0.570804\pi$$
$$654$$ 0 0
$$655$$ −13.4596 −0.525911
$$656$$ −32.9171 −1.28520
$$657$$ 0 0
$$658$$ 1.61661 0.0630219
$$659$$ 39.1971 1.52690 0.763451 0.645866i $$-0.223503\pi$$
0.763451 + 0.645866i $$0.223503\pi$$
$$660$$ 0 0
$$661$$ −42.1179 −1.63820 −0.819099 0.573652i $$-0.805526\pi$$
−0.819099 + 0.573652i $$0.805526\pi$$
$$662$$ 0.0803354 0.00312232
$$663$$ 0 0
$$664$$ 0.783874 0.0304202
$$665$$ 1.95185 0.0756895
$$666$$ 0 0
$$667$$ −8.34798 −0.323235
$$668$$ −39.5500 −1.53024
$$669$$ 0 0
$$670$$ −0.863359 −0.0333545
$$671$$ −1.53640 −0.0593119
$$672$$ 0 0
$$673$$ 25.5042 0.983116 0.491558 0.870845i $$-0.336428\pi$$
0.491558 + 0.870845i $$0.336428\pi$$
$$674$$ −1.14509 −0.0441073
$$675$$ 0 0
$$676$$ 5.39821 0.207623
$$677$$ −48.5446 −1.86572 −0.932861 0.360237i $$-0.882696\pi$$
−0.932861 + 0.360237i $$0.882696\pi$$
$$678$$ 0 0
$$679$$ 9.16434 0.351695
$$680$$ 0.177371 0.00680187
$$681$$ 0 0
$$682$$ −0.839949 −0.0321633
$$683$$ −22.0305 −0.842975 −0.421488 0.906834i $$-0.638492\pi$$
−0.421488 + 0.906834i $$0.638492\pi$$
$$684$$ 0 0
$$685$$ 7.45968 0.285020
$$686$$ −1.75341 −0.0669456
$$687$$ 0 0
$$688$$ −31.8186 −1.21307
$$689$$ −7.60834 −0.289855
$$690$$ 0 0
$$691$$ −23.7336 −0.902870 −0.451435 0.892304i $$-0.649088\pi$$
−0.451435 + 0.892304i $$0.649088\pi$$
$$692$$ −46.0926 −1.75218
$$693$$ 0 0
$$694$$ −0.749330 −0.0284442
$$695$$ 18.3550 0.696244
$$696$$ 0 0
$$697$$ 4.19627 0.158945
$$698$$ 2.02125 0.0765054
$$699$$ 0 0
$$700$$ 3.88853 0.146973
$$701$$ −3.46600 −0.130909 −0.0654546 0.997856i $$-0.520850\pi$$
−0.0654546 + 0.997856i $$0.520850\pi$$
$$702$$ 0 0
$$703$$ −5.22650 −0.197121
$$704$$ −7.81409 −0.294505
$$705$$ 0 0
$$706$$ −2.86539 −0.107840
$$707$$ 0.313416 0.0117872
$$708$$ 0 0
$$709$$ 11.4552 0.430208 0.215104 0.976591i $$-0.430991\pi$$
0.215104 + 0.976591i $$0.430991\pi$$
$$710$$ 0.392318 0.0147234
$$711$$ 0 0
$$712$$ −2.14960 −0.0805597
$$713$$ −10.0548 −0.376556
$$714$$ 0 0
$$715$$ −3.20786 −0.119967
$$716$$ 9.29129 0.347232
$$717$$ 0 0
$$718$$ −0.621938 −0.0232105
$$719$$ −12.8364 −0.478718 −0.239359 0.970931i $$-0.576937\pi$$
−0.239359 + 0.970931i $$0.576937\pi$$
$$720$$ 0 0
$$721$$ 23.7635 0.884998
$$722$$ −0.0881559 −0.00328082
$$723$$ 0 0
$$724$$ −47.3917 −1.76130
$$725$$ 7.91058 0.293792
$$726$$ 0 0
$$727$$ −22.3762 −0.829887 −0.414943 0.909847i $$-0.636199\pi$$
−0.414943 + 0.909847i $$0.636199\pi$$
$$728$$ −2.20358 −0.0816701
$$729$$ 0 0
$$730$$ −0.137515 −0.00508966
$$731$$ 4.05623 0.150025
$$732$$ 0 0
$$733$$ 35.6014 1.31497 0.657483 0.753469i $$-0.271621\pi$$
0.657483 + 0.753469i $$0.271621\pi$$
$$734$$ −1.30736 −0.0482557
$$735$$ 0 0
$$736$$ 1.11059 0.0409367
$$737$$ −9.79354 −0.360750
$$738$$ 0 0
$$739$$ 24.3156 0.894465 0.447233 0.894418i $$-0.352410\pi$$
0.447233 + 0.894418i $$0.352410\pi$$
$$740$$ −10.4124 −0.382766
$$741$$ 0 0
$$742$$ −0.408106 −0.0149820
$$743$$ −34.2814 −1.25766 −0.628831 0.777542i $$-0.716466\pi$$
−0.628831 + 0.777542i $$0.716466\pi$$
$$744$$ 0 0
$$745$$ −19.2387 −0.704852
$$746$$ −0.854787 −0.0312960
$$747$$ 0 0
$$748$$ 1.00405 0.0367117
$$749$$ 31.6359 1.15595
$$750$$ 0 0
$$751$$ 32.2464 1.17669 0.588345 0.808610i $$-0.299780\pi$$
0.588345 + 0.808610i $$0.299780\pi$$
$$752$$ 37.1433 1.35448
$$753$$ 0 0
$$754$$ −2.23705 −0.0814685
$$755$$ −7.76410 −0.282565
$$756$$ 0 0
$$757$$ 2.94119 0.106899 0.0534497 0.998571i $$-0.482978\pi$$
0.0534497 + 0.998571i $$0.482978\pi$$
$$758$$ −0.0912923 −0.00331589
$$759$$ 0 0
$$760$$ −0.351939 −0.0127662
$$761$$ −26.6626 −0.966519 −0.483260 0.875477i $$-0.660547\pi$$
−0.483260 + 0.875477i $$0.660547\pi$$
$$762$$ 0 0
$$763$$ −33.6256 −1.21733
$$764$$ −8.21466 −0.297196
$$765$$ 0 0
$$766$$ 0.119546 0.00431937
$$767$$ 8.98403 0.324394
$$768$$ 0 0
$$769$$ −7.23123 −0.260765 −0.130382 0.991464i $$-0.541621\pi$$
−0.130382 + 0.991464i $$0.541621\pi$$
$$770$$ −0.172067 −0.00620087
$$771$$ 0 0
$$772$$ −33.1834 −1.19430
$$773$$ 8.64003 0.310760 0.155380 0.987855i $$-0.450340\pi$$
0.155380 + 0.987855i $$0.450340\pi$$
$$774$$ 0 0
$$775$$ 9.52800 0.342256
$$776$$ −1.65243 −0.0593186
$$777$$ 0 0
$$778$$ −3.02232 −0.108355
$$779$$ −8.32622 −0.298318
$$780$$ 0 0
$$781$$ 4.45027 0.159243
$$782$$ −0.0468857 −0.00167663
$$783$$ 0 0
$$784$$ −12.6126 −0.450449
$$785$$ 4.82520 0.172219
$$786$$ 0 0
$$787$$ 16.4120 0.585025 0.292512 0.956262i $$-0.405509\pi$$
0.292512 + 0.956262i $$0.405509\pi$$
$$788$$ 25.3579 0.903338
$$789$$ 0 0
$$790$$ 0.265216 0.00943597
$$791$$ −19.0137 −0.676048
$$792$$ 0 0
$$793$$ −4.92854 −0.175018
$$794$$ 3.12897 0.111043
$$795$$ 0 0
$$796$$ 33.9759 1.20424
$$797$$ 54.6736 1.93664 0.968319 0.249716i $$-0.0803372\pi$$
0.968319 + 0.249716i $$0.0803372\pi$$
$$798$$ 0 0
$$799$$ −4.73503 −0.167513
$$800$$ −1.05240 −0.0372078
$$801$$ 0 0
$$802$$ 2.51168 0.0886907
$$803$$ −1.55991 −0.0550479
$$804$$ 0 0
$$805$$ −2.05977 −0.0725974
$$806$$ −2.69444 −0.0949076
$$807$$ 0 0
$$808$$ −0.0565122 −0.00198809
$$809$$ 20.6416 0.725719 0.362859 0.931844i $$-0.381801\pi$$
0.362859 + 0.931844i $$0.381801\pi$$
$$810$$ 0 0
$$811$$ 0.728429 0.0255786 0.0127893 0.999918i $$-0.495929\pi$$
0.0127893 + 0.999918i $$0.495929\pi$$
$$812$$ 30.7605 1.07948
$$813$$ 0 0
$$814$$ 0.460747 0.0161492
$$815$$ −16.0681 −0.562842
$$816$$ 0 0
$$817$$ −8.04835 −0.281576
$$818$$ −2.74324 −0.0959150
$$819$$ 0 0
$$820$$ −16.5877 −0.579269
$$821$$ −7.88685 −0.275253 −0.137626 0.990484i $$-0.543947\pi$$
−0.137626 + 0.990484i $$0.543947\pi$$
$$822$$ 0 0
$$823$$ 21.3819 0.745327 0.372663 0.927967i $$-0.378445\pi$$
0.372663 + 0.927967i $$0.378445\pi$$
$$824$$ −4.28480 −0.149268
$$825$$ 0 0
$$826$$ 0.481896 0.0167673
$$827$$ 18.0869 0.628943 0.314471 0.949267i $$-0.398173\pi$$
0.314471 + 0.949267i $$0.398173\pi$$
$$828$$ 0 0
$$829$$ 38.7216 1.34486 0.672429 0.740162i $$-0.265251\pi$$
0.672429 + 0.740162i $$0.265251\pi$$
$$830$$ 0.196350 0.00681541
$$831$$ 0 0
$$832$$ −25.0665 −0.869025
$$833$$ 1.60785 0.0557087
$$834$$ 0 0
$$835$$ −19.8521 −0.687012
$$836$$ −1.99223 −0.0689027
$$837$$ 0 0
$$838$$ −2.10522 −0.0727235
$$839$$ −11.1336 −0.384375 −0.192187 0.981358i $$-0.561558\pi$$
−0.192187 + 0.981358i $$0.561558\pi$$
$$840$$ 0 0
$$841$$ 33.5774 1.15784
$$842$$ 0.100219 0.00345376
$$843$$ 0 0
$$844$$ 29.8948 1.02902
$$845$$ 2.70963 0.0932142
$$846$$ 0 0
$$847$$ −1.95185 −0.0670663
$$848$$ −9.37668 −0.321996
$$849$$ 0 0
$$850$$ 0.0444291 0.00152391
$$851$$ 5.51548 0.189068
$$852$$ 0 0
$$853$$ −37.4831 −1.28340 −0.641698 0.766957i $$-0.721770\pi$$
−0.641698 + 0.766957i $$0.721770\pi$$
$$854$$ −0.264363 −0.00904632
$$855$$ 0 0
$$856$$ −5.70429 −0.194968
$$857$$ −38.7649 −1.32418 −0.662091 0.749423i $$-0.730331\pi$$
−0.662091 + 0.749423i $$0.730331\pi$$
$$858$$ 0 0
$$859$$ 14.9492 0.510060 0.255030 0.966933i $$-0.417915\pi$$
0.255030 + 0.966933i $$0.417915\pi$$
$$860$$ −16.0341 −0.546760
$$861$$ 0 0
$$862$$ −0.0527954 −0.00179822
$$863$$ 45.1615 1.53732 0.768658 0.639660i $$-0.220925\pi$$
0.768658 + 0.639660i $$0.220925\pi$$
$$864$$ 0 0
$$865$$ −23.1362 −0.786655
$$866$$ −2.85146 −0.0968965
$$867$$ 0 0
$$868$$ 37.0499 1.25756
$$869$$ 3.00849 0.102056
$$870$$ 0 0
$$871$$ −31.4163 −1.06450
$$872$$ 6.06304 0.205320
$$873$$ 0 0
$$874$$ 0.0930303 0.00314680
$$875$$ 1.95185 0.0659845
$$876$$ 0 0
$$877$$ 13.1538 0.444171 0.222086 0.975027i $$-0.428714\pi$$
0.222086 + 0.975027i $$0.428714\pi$$
$$878$$ −0.825585 −0.0278622
$$879$$ 0 0
$$880$$ −3.95343 −0.133270
$$881$$ 19.8386 0.668380 0.334190 0.942506i $$-0.391537\pi$$
0.334190 + 0.942506i $$0.391537\pi$$
$$882$$ 0 0
$$883$$ 18.2493 0.614138 0.307069 0.951687i $$-0.400652\pi$$
0.307069 + 0.951687i $$0.400652\pi$$
$$884$$ 3.22085 0.108329
$$885$$ 0 0
$$886$$ 0.930549 0.0312624
$$887$$ 19.6448 0.659606 0.329803 0.944050i $$-0.393018\pi$$
0.329803 + 0.944050i $$0.393018\pi$$
$$888$$ 0 0
$$889$$ 11.9961 0.402335
$$890$$ −0.538446 −0.0180488
$$891$$ 0 0
$$892$$ −8.40243 −0.281334
$$893$$ 9.39522 0.314399
$$894$$ 0 0
$$895$$ 4.66377 0.155892
$$896$$ −5.45278 −0.182165
$$897$$ 0 0
$$898$$ −1.18481 −0.0395376
$$899$$ 75.3720 2.51380
$$900$$ 0 0
$$901$$ 1.19534 0.0398225
$$902$$ 0.734005 0.0244397
$$903$$ 0 0
$$904$$ 3.42836 0.114026
$$905$$ −23.7883 −0.790749
$$906$$ 0 0
$$907$$ −36.7829 −1.22136 −0.610678 0.791879i $$-0.709103\pi$$
−0.610678 + 0.791879i $$0.709103\pi$$
$$908$$ 36.8743 1.22372
$$909$$ 0 0
$$910$$ −0.551967 −0.0182975
$$911$$ 35.8447 1.18759 0.593794 0.804617i $$-0.297630\pi$$
0.593794 + 0.804617i $$0.297630\pi$$
$$912$$ 0 0
$$913$$ 2.22730 0.0737130
$$914$$ −0.342518 −0.0113295
$$915$$ 0 0
$$916$$ −20.1171 −0.664689
$$917$$ −26.2712 −0.867550
$$918$$ 0 0
$$919$$ −2.43716 −0.0803943 −0.0401972 0.999192i $$-0.512799\pi$$
−0.0401972 + 0.999192i $$0.512799\pi$$
$$920$$ 0.371398 0.0122446
$$921$$ 0 0
$$922$$ 1.38627 0.0456544
$$923$$ 14.2759 0.469895
$$924$$ 0 0
$$925$$ −5.22650 −0.171846
$$926$$ −3.12739 −0.102772
$$927$$ 0 0
$$928$$ −8.32506 −0.273284
$$929$$ −50.7417 −1.66478 −0.832391 0.554189i $$-0.813029\pi$$
−0.832391 + 0.554189i $$0.813029\pi$$
$$930$$ 0 0
$$931$$ −3.19028 −0.104557
$$932$$ −10.5245 −0.344740
$$933$$ 0 0
$$934$$ 1.19476 0.0390937
$$935$$ 0.503983 0.0164820
$$936$$ 0 0
$$937$$ −34.9861 −1.14295 −0.571473 0.820621i $$-0.693628\pi$$
−0.571473 + 0.820621i $$0.693628\pi$$
$$938$$ −1.68515 −0.0550220
$$939$$ 0 0
$$940$$ 18.7174 0.610495
$$941$$ 38.7579 1.26347 0.631736 0.775183i $$-0.282342\pi$$
0.631736 + 0.775183i $$0.282342\pi$$
$$942$$ 0 0
$$943$$ 8.78660 0.286131
$$944$$ 11.0721 0.360366
$$945$$ 0 0
$$946$$ 0.709509 0.0230681
$$947$$ −33.8957 −1.10146 −0.550732 0.834682i $$-0.685651\pi$$
−0.550732 + 0.834682i $$0.685651\pi$$
$$948$$ 0 0
$$949$$ −5.00396 −0.162435
$$950$$ −0.0881559 −0.00286016
$$951$$ 0 0
$$952$$ 0.346202 0.0112205
$$953$$ −56.6907 −1.83639 −0.918195 0.396128i $$-0.870354\pi$$
−0.918195 + 0.396128i $$0.870354\pi$$
$$954$$ 0 0
$$955$$ −4.12335 −0.133428
$$956$$ 11.9765 0.387348
$$957$$ 0 0
$$958$$ 3.51253 0.113485
$$959$$ 14.5602 0.470173
$$960$$ 0 0
$$961$$ 59.7827 1.92847
$$962$$ 1.47801 0.0476530
$$963$$ 0 0
$$964$$ 48.1614 1.55118
$$965$$ −16.6564 −0.536189
$$966$$ 0 0
$$967$$ 45.0600 1.44903 0.724517 0.689257i $$-0.242063\pi$$
0.724517 + 0.689257i $$0.242063\pi$$
$$968$$ 0.351939 0.0113117
$$969$$ 0 0
$$970$$ −0.413910 −0.0132899
$$971$$ 24.1262 0.774247 0.387124 0.922028i $$-0.373469\pi$$
0.387124 + 0.922028i $$0.373469\pi$$
$$972$$ 0 0
$$973$$ 35.8262 1.14853
$$974$$ −3.17168 −0.101627
$$975$$ 0 0
$$976$$ −6.07404 −0.194425
$$977$$ 12.1551 0.388875 0.194437 0.980915i $$-0.437712\pi$$
0.194437 + 0.980915i $$0.437712\pi$$
$$978$$ 0 0
$$979$$ −6.10788 −0.195209
$$980$$ −6.35578 −0.203028
$$981$$ 0 0
$$982$$ −3.30118 −0.105345
$$983$$ 26.5655 0.847308 0.423654 0.905824i $$-0.360747\pi$$
0.423654 + 0.905824i $$0.360747\pi$$
$$984$$ 0 0
$$985$$ 12.7284 0.405561
$$986$$ 0.351460 0.0111928
$$987$$ 0 0
$$988$$ −6.39079 −0.203318
$$989$$ 8.49336 0.270073
$$990$$ 0 0
$$991$$ 6.91424 0.219638 0.109819 0.993952i $$-0.464973\pi$$
0.109819 + 0.993952i $$0.464973\pi$$
$$992$$ −10.0272 −0.318365
$$993$$ 0 0
$$994$$ 0.765745 0.0242880
$$995$$ 17.0542 0.540655
$$996$$ 0 0
$$997$$ 42.2221 1.33719 0.668593 0.743628i $$-0.266897\pi$$
0.668593 + 0.743628i $$0.266897\pi$$
$$998$$ −2.49005 −0.0788212
$$999$$ 0 0
Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000

## Twists

By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 9405.2.a.v.1.2 5
3.2 odd 2 1045.2.a.d.1.4 5
15.14 odd 2 5225.2.a.j.1.2 5

By twisted newform
Twist Min Dim Char Parity Ord Type
1045.2.a.d.1.4 5 3.2 odd 2
5225.2.a.j.1.2 5 15.14 odd 2
9405.2.a.v.1.2 5 1.1 even 1 trivial