Properties

 Label 4275.2.a.bm.1.2 Level $4275$ Weight $2$ Character 4275.1 Self dual yes Analytic conductor $34.136$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

Related objects

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4275,2,Mod(1,4275)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4275, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("4275.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$4275 = 3^{2} \cdot 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4275.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: $$34.1360468641$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.169.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 4x - 1$$ x^3 - x^2 - 4*x - 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 475) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

 Embedding label 1.2 Root $$-0.273891$$ of defining polynomial Character $$\chi$$ $$=$$ 4275.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.27389 q^{2} -0.377203 q^{4} -3.65109 q^{7} -3.02830 q^{8} +O(q^{10})$$ $$q+1.27389 q^{2} -0.377203 q^{4} -3.65109 q^{7} -3.02830 q^{8} -2.65109 q^{11} -6.13161 q^{13} -4.65109 q^{14} -3.10331 q^{16} +2.34891 q^{17} +1.00000 q^{19} -3.37720 q^{22} +5.48052 q^{23} -7.81100 q^{26} +1.37720 q^{28} -0.651093 q^{29} -6.67939 q^{31} +2.10331 q^{32} +2.99225 q^{34} +8.70769 q^{37} +1.27389 q^{38} -1.93273 q^{41} +2.65884 q^{43} +1.00000 q^{44} +6.98158 q^{46} +3.71836 q^{47} +6.33048 q^{49} +2.31286 q^{52} +13.7544 q^{53} +11.0566 q^{56} -0.829422 q^{58} +7.84997 q^{59} -1.92498 q^{61} -8.50881 q^{62} +8.88601 q^{64} +4.44447 q^{67} -0.886014 q^{68} -3.54778 q^{71} +2.48052 q^{73} +11.0926 q^{74} -0.377203 q^{76} +9.67939 q^{77} -15.1599 q^{79} -2.46209 q^{82} -14.7282 q^{83} +3.38708 q^{86} +8.02830 q^{88} +5.06727 q^{89} +22.3871 q^{91} -2.06727 q^{92} +4.73678 q^{94} -3.22717 q^{97} +8.06434 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 2 q^{2} + 4 q^{4} - 4 q^{7} + 3 q^{8}+O(q^{10})$$ 3 * q + 2 * q^2 + 4 * q^4 - 4 * q^7 + 3 * q^8 $$3 q + 2 q^{2} + 4 q^{4} - 4 q^{7} + 3 q^{8} - q^{11} - 3 q^{13} - 7 q^{14} - 6 q^{16} + 14 q^{17} + 3 q^{19} - 5 q^{22} + 8 q^{23} + 11 q^{26} - q^{28} + 5 q^{29} - q^{31} + 3 q^{32} + 5 q^{34} - 5 q^{37} + 2 q^{38} - q^{41} + 5 q^{43} + 3 q^{44} - 12 q^{46} + 9 q^{47} - 7 q^{49} + 22 q^{52} + 31 q^{53} + 9 q^{56} - q^{58} + 6 q^{59} + 3 q^{61} - 5 q^{62} + q^{64} + 13 q^{67} + 23 q^{68} - 7 q^{71} - q^{73} + q^{74} + 4 q^{76} + 10 q^{77} - 18 q^{79} + 34 q^{82} + 3 q^{83} - 40 q^{86} + 12 q^{88} + 20 q^{89} + 17 q^{91} - 11 q^{92} + 45 q^{94} + 13 q^{97} + 4 q^{98}+O(q^{100})$$ 3 * q + 2 * q^2 + 4 * q^4 - 4 * q^7 + 3 * q^8 - q^11 - 3 * q^13 - 7 * q^14 - 6 * q^16 + 14 * q^17 + 3 * q^19 - 5 * q^22 + 8 * q^23 + 11 * q^26 - q^28 + 5 * q^29 - q^31 + 3 * q^32 + 5 * q^34 - 5 * q^37 + 2 * q^38 - q^41 + 5 * q^43 + 3 * q^44 - 12 * q^46 + 9 * q^47 - 7 * q^49 + 22 * q^52 + 31 * q^53 + 9 * q^56 - q^58 + 6 * q^59 + 3 * q^61 - 5 * q^62 + q^64 + 13 * q^67 + 23 * q^68 - 7 * q^71 - q^73 + q^74 + 4 * q^76 + 10 * q^77 - 18 * q^79 + 34 * q^82 + 3 * q^83 - 40 * q^86 + 12 * q^88 + 20 * q^89 + 17 * q^91 - 11 * q^92 + 45 * q^94 + 13 * q^97 + 4 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.27389 0.900777 0.450388 0.892833i $$-0.351286\pi$$
0.450388 + 0.892833i $$0.351286\pi$$
$$3$$ 0 0
$$4$$ −0.377203 −0.188601
$$5$$ 0 0
$$6$$ 0 0
$$7$$ −3.65109 −1.37998 −0.689992 0.723817i $$-0.742386\pi$$
−0.689992 + 0.723817i $$0.742386\pi$$
$$8$$ −3.02830 −1.07066
$$9$$ 0 0
$$10$$ 0 0
$$11$$ −2.65109 −0.799335 −0.399667 0.916660i $$-0.630874\pi$$
−0.399667 + 0.916660i $$0.630874\pi$$
$$12$$ 0 0
$$13$$ −6.13161 −1.70060 −0.850301 0.526297i $$-0.823580\pi$$
−0.850301 + 0.526297i $$0.823580\pi$$
$$14$$ −4.65109 −1.24306
$$15$$ 0 0
$$16$$ −3.10331 −0.775828
$$17$$ 2.34891 0.569694 0.284847 0.958573i $$-0.408057\pi$$
0.284847 + 0.958573i $$0.408057\pi$$
$$18$$ 0 0
$$19$$ 1.00000 0.229416
$$20$$ 0 0
$$21$$ 0 0
$$22$$ −3.37720 −0.720022
$$23$$ 5.48052 1.14277 0.571383 0.820683i $$-0.306407\pi$$
0.571383 + 0.820683i $$0.306407\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ −7.81100 −1.53186
$$27$$ 0 0
$$28$$ 1.37720 0.260267
$$29$$ −0.651093 −0.120905 −0.0604525 0.998171i $$-0.519254\pi$$
−0.0604525 + 0.998171i $$0.519254\pi$$
$$30$$ 0 0
$$31$$ −6.67939 −1.19965 −0.599827 0.800130i $$-0.704764\pi$$
−0.599827 + 0.800130i $$0.704764\pi$$
$$32$$ 2.10331 0.371817
$$33$$ 0 0
$$34$$ 2.99225 0.513167
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 8.70769 1.43153 0.715767 0.698339i $$-0.246077\pi$$
0.715767 + 0.698339i $$0.246077\pi$$
$$38$$ 1.27389 0.206652
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −1.93273 −0.301842 −0.150921 0.988546i $$-0.548224\pi$$
−0.150921 + 0.988546i $$0.548224\pi$$
$$42$$ 0 0
$$43$$ 2.65884 0.405470 0.202735 0.979234i $$-0.435017\pi$$
0.202735 + 0.979234i $$0.435017\pi$$
$$44$$ 1.00000 0.150756
$$45$$ 0 0
$$46$$ 6.98158 1.02938
$$47$$ 3.71836 0.542378 0.271189 0.962526i $$-0.412583\pi$$
0.271189 + 0.962526i $$0.412583\pi$$
$$48$$ 0 0
$$49$$ 6.33048 0.904355
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 2.31286 0.320736
$$53$$ 13.7544 1.88931 0.944656 0.328061i $$-0.106395\pi$$
0.944656 + 0.328061i $$0.106395\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 11.0566 1.47750
$$57$$ 0 0
$$58$$ −0.829422 −0.108908
$$59$$ 7.84997 1.02198 0.510989 0.859587i $$-0.329279\pi$$
0.510989 + 0.859587i $$0.329279\pi$$
$$60$$ 0 0
$$61$$ −1.92498 −0.246469 −0.123234 0.992378i $$-0.539327\pi$$
−0.123234 + 0.992378i $$0.539327\pi$$
$$62$$ −8.50881 −1.08062
$$63$$ 0 0
$$64$$ 8.88601 1.11075
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 4.44447 0.542978 0.271489 0.962442i $$-0.412484\pi$$
0.271489 + 0.962442i $$0.412484\pi$$
$$68$$ −0.886014 −0.107445
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −3.54778 −0.421044 −0.210522 0.977589i $$-0.567516\pi$$
−0.210522 + 0.977589i $$0.567516\pi$$
$$72$$ 0 0
$$73$$ 2.48052 0.290322 0.145161 0.989408i $$-0.453630\pi$$
0.145161 + 0.989408i $$0.453630\pi$$
$$74$$ 11.0926 1.28949
$$75$$ 0 0
$$76$$ −0.377203 −0.0432681
$$77$$ 9.67939 1.10307
$$78$$ 0 0
$$79$$ −15.1599 −1.70562 −0.852811 0.522219i $$-0.825104\pi$$
−0.852811 + 0.522219i $$0.825104\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ −2.46209 −0.271893
$$83$$ −14.7282 −1.61663 −0.808317 0.588748i $$-0.799621\pi$$
−0.808317 + 0.588748i $$0.799621\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 3.38708 0.365238
$$87$$ 0 0
$$88$$ 8.02830 0.855819
$$89$$ 5.06727 0.537129 0.268565 0.963262i $$-0.413451\pi$$
0.268565 + 0.963262i $$0.413451\pi$$
$$90$$ 0 0
$$91$$ 22.3871 2.34680
$$92$$ −2.06727 −0.215527
$$93$$ 0 0
$$94$$ 4.73678 0.488562
$$95$$ 0 0
$$96$$ 0 0
$$97$$ −3.22717 −0.327670 −0.163835 0.986488i $$-0.552386\pi$$
−0.163835 + 0.986488i $$0.552386\pi$$
$$98$$ 8.06434 0.814622
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −16.8032 −1.67199 −0.835993 0.548740i $$-0.815108\pi$$
−0.835993 + 0.548740i $$0.815108\pi$$
$$102$$ 0 0
$$103$$ 9.85772 0.971310 0.485655 0.874151i $$-0.338581\pi$$
0.485655 + 0.874151i $$0.338581\pi$$
$$104$$ 18.5683 1.82077
$$105$$ 0 0
$$106$$ 17.5216 1.70185
$$107$$ −5.13936 −0.496841 −0.248420 0.968652i $$-0.579911\pi$$
−0.248420 + 0.968652i $$0.579911\pi$$
$$108$$ 0 0
$$109$$ 13.9738 1.33845 0.669225 0.743060i $$-0.266626\pi$$
0.669225 + 0.743060i $$0.266626\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 11.3305 1.07063
$$113$$ −0.527235 −0.0495981 −0.0247990 0.999692i $$-0.507895\pi$$
−0.0247990 + 0.999692i $$0.507895\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0.245594 0.0228029
$$117$$ 0 0
$$118$$ 10.0000 0.920575
$$119$$ −8.57608 −0.786168
$$120$$ 0 0
$$121$$ −3.97170 −0.361064
$$122$$ −2.45222 −0.222013
$$123$$ 0 0
$$124$$ 2.51948 0.226256
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 11.8217 1.04900 0.524502 0.851409i $$-0.324252\pi$$
0.524502 + 0.851409i $$0.324252\pi$$
$$128$$ 7.11319 0.628723
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −8.12386 −0.709785 −0.354892 0.934907i $$-0.615483\pi$$
−0.354892 + 0.934907i $$0.615483\pi$$
$$132$$ 0 0
$$133$$ −3.65109 −0.316590
$$134$$ 5.66177 0.489102
$$135$$ 0 0
$$136$$ −7.11319 −0.609951
$$137$$ 17.5761 1.50163 0.750813 0.660515i $$-0.229662\pi$$
0.750813 + 0.660515i $$0.229662\pi$$
$$138$$ 0 0
$$139$$ 18.4154 1.56197 0.780986 0.624549i $$-0.214717\pi$$
0.780986 + 0.624549i $$0.214717\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ −4.51948 −0.379267
$$143$$ 16.2555 1.35935
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 3.15990 0.261516
$$147$$ 0 0
$$148$$ −3.28456 −0.269989
$$149$$ 17.2915 1.41658 0.708288 0.705924i $$-0.249468\pi$$
0.708288 + 0.705924i $$0.249468\pi$$
$$150$$ 0 0
$$151$$ 2.58383 0.210269 0.105134 0.994458i $$-0.466473\pi$$
0.105134 + 0.994458i $$0.466473\pi$$
$$152$$ −3.02830 −0.245627
$$153$$ 0 0
$$154$$ 12.3305 0.993619
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 10.9738 0.875807 0.437903 0.899022i $$-0.355721\pi$$
0.437903 + 0.899022i $$0.355721\pi$$
$$158$$ −19.3121 −1.53638
$$159$$ 0 0
$$160$$ 0 0
$$161$$ −20.0099 −1.57700
$$162$$ 0 0
$$163$$ −22.6794 −1.77639 −0.888193 0.459470i $$-0.848039\pi$$
−0.888193 + 0.459470i $$0.848039\pi$$
$$164$$ 0.729033 0.0569279
$$165$$ 0 0
$$166$$ −18.7622 −1.45623
$$167$$ −5.16283 −0.399512 −0.199756 0.979846i $$-0.564015\pi$$
−0.199756 + 0.979846i $$0.564015\pi$$
$$168$$ 0 0
$$169$$ 24.5966 1.89205
$$170$$ 0 0
$$171$$ 0 0
$$172$$ −1.00292 −0.0764722
$$173$$ −17.2165 −1.30895 −0.654473 0.756085i $$-0.727109\pi$$
−0.654473 + 0.756085i $$0.727109\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 8.22717 0.620146
$$177$$ 0 0
$$178$$ 6.45514 0.483833
$$179$$ −10.6999 −0.799751 −0.399875 0.916570i $$-0.630947\pi$$
−0.399875 + 0.916570i $$0.630947\pi$$
$$180$$ 0 0
$$181$$ −16.7720 −1.24666 −0.623328 0.781961i $$-0.714220\pi$$
−0.623328 + 0.781961i $$0.714220\pi$$
$$182$$ 28.5187 2.11395
$$183$$ 0 0
$$184$$ −16.5966 −1.22352
$$185$$ 0 0
$$186$$ 0 0
$$187$$ −6.22717 −0.455376
$$188$$ −1.40258 −0.102293
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 13.8598 1.00286 0.501431 0.865197i $$-0.332807\pi$$
0.501431 + 0.865197i $$0.332807\pi$$
$$192$$ 0 0
$$193$$ −21.0694 −1.51661 −0.758304 0.651901i $$-0.773972\pi$$
−0.758304 + 0.651901i $$0.773972\pi$$
$$194$$ −4.11106 −0.295157
$$195$$ 0 0
$$196$$ −2.38788 −0.170563
$$197$$ 21.2555 1.51439 0.757195 0.653189i $$-0.226569\pi$$
0.757195 + 0.653189i $$0.226569\pi$$
$$198$$ 0 0
$$199$$ −7.69006 −0.545134 −0.272567 0.962137i $$-0.587873\pi$$
−0.272567 + 0.962137i $$0.587873\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ −21.4055 −1.50609
$$203$$ 2.37720 0.166847
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 12.5577 0.874933
$$207$$ 0 0
$$208$$ 19.0283 1.31937
$$209$$ −2.65109 −0.183380
$$210$$ 0 0
$$211$$ −1.69781 −0.116882 −0.0584411 0.998291i $$-0.518613\pi$$
−0.0584411 + 0.998291i $$0.518613\pi$$
$$212$$ −5.18820 −0.356327
$$213$$ 0 0
$$214$$ −6.54698 −0.447542
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 24.3871 1.65550
$$218$$ 17.8011 1.20564
$$219$$ 0 0
$$220$$ 0 0
$$221$$ −14.4026 −0.968822
$$222$$ 0 0
$$223$$ 25.2632 1.69175 0.845875 0.533381i $$-0.179079\pi$$
0.845875 + 0.533381i $$0.179079\pi$$
$$224$$ −7.67939 −0.513101
$$225$$ 0 0
$$226$$ −0.671640 −0.0446768
$$227$$ −16.8217 −1.11649 −0.558247 0.829675i $$-0.688526\pi$$
−0.558247 + 0.829675i $$0.688526\pi$$
$$228$$ 0 0
$$229$$ −14.6249 −0.966442 −0.483221 0.875498i $$-0.660533\pi$$
−0.483221 + 0.875498i $$0.660533\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 1.97170 0.129449
$$233$$ 8.91431 0.583996 0.291998 0.956419i $$-0.405680\pi$$
0.291998 + 0.956419i $$0.405680\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ −2.96103 −0.192747
$$237$$ 0 0
$$238$$ −10.9250 −0.708162
$$239$$ 24.6015 1.59134 0.795668 0.605733i $$-0.207120\pi$$
0.795668 + 0.605733i $$0.207120\pi$$
$$240$$ 0 0
$$241$$ −14.6150 −0.941438 −0.470719 0.882283i $$-0.656005\pi$$
−0.470719 + 0.882283i $$0.656005\pi$$
$$242$$ −5.05952 −0.325238
$$243$$ 0 0
$$244$$ 0.726109 0.0464844
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −6.13161 −0.390145
$$248$$ 20.2272 1.28443
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 13.3382 0.841902 0.420951 0.907083i $$-0.361696\pi$$
0.420951 + 0.907083i $$0.361696\pi$$
$$252$$ 0 0
$$253$$ −14.5294 −0.913453
$$254$$ 15.0595 0.944918
$$255$$ 0 0
$$256$$ −8.71061 −0.544413
$$257$$ −3.35103 −0.209031 −0.104516 0.994523i $$-0.533329\pi$$
−0.104516 + 0.994523i $$0.533329\pi$$
$$258$$ 0 0
$$259$$ −31.7926 −1.97549
$$260$$ 0 0
$$261$$ 0 0
$$262$$ −10.3489 −0.639358
$$263$$ 10.8860 0.671260 0.335630 0.941994i $$-0.391051\pi$$
0.335630 + 0.941994i $$0.391051\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ −4.65109 −0.285177
$$267$$ 0 0
$$268$$ −1.67647 −0.102406
$$269$$ −18.4338 −1.12393 −0.561964 0.827162i $$-0.689954\pi$$
−0.561964 + 0.827162i $$0.689954\pi$$
$$270$$ 0 0
$$271$$ −20.4076 −1.23967 −0.619837 0.784730i $$-0.712801\pi$$
−0.619837 + 0.784730i $$0.712801\pi$$
$$272$$ −7.28939 −0.441984
$$273$$ 0 0
$$274$$ 22.3900 1.35263
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 2.69994 0.162223 0.0811117 0.996705i $$-0.474153\pi$$
0.0811117 + 0.996705i $$0.474153\pi$$
$$278$$ 23.4592 1.40699
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 15.2242 0.908202 0.454101 0.890950i $$-0.349960\pi$$
0.454101 + 0.890950i $$0.349960\pi$$
$$282$$ 0 0
$$283$$ −6.18045 −0.367390 −0.183695 0.982983i $$-0.558806\pi$$
−0.183695 + 0.982983i $$0.558806\pi$$
$$284$$ 1.33823 0.0794095
$$285$$ 0 0
$$286$$ 20.7077 1.22447
$$287$$ 7.05659 0.416537
$$288$$ 0 0
$$289$$ −11.4826 −0.675449
$$290$$ 0 0
$$291$$ 0 0
$$292$$ −0.935657 −0.0547552
$$293$$ 30.6893 1.79289 0.896443 0.443159i $$-0.146142\pi$$
0.896443 + 0.443159i $$0.146142\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ −26.3695 −1.53269
$$297$$ 0 0
$$298$$ 22.0275 1.27602
$$299$$ −33.6044 −1.94339
$$300$$ 0 0
$$301$$ −9.70769 −0.559542
$$302$$ 3.29151 0.189405
$$303$$ 0 0
$$304$$ −3.10331 −0.177987
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 14.7827 0.843693 0.421847 0.906667i $$-0.361382\pi$$
0.421847 + 0.906667i $$0.361382\pi$$
$$308$$ −3.65109 −0.208040
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 26.9992 1.53098 0.765492 0.643445i $$-0.222496\pi$$
0.765492 + 0.643445i $$0.222496\pi$$
$$312$$ 0 0
$$313$$ 9.74666 0.550914 0.275457 0.961313i $$-0.411171\pi$$
0.275457 + 0.961313i $$0.411171\pi$$
$$314$$ 13.9795 0.788906
$$315$$ 0 0
$$316$$ 5.71836 0.321683
$$317$$ 19.7261 1.10793 0.553964 0.832540i $$-0.313114\pi$$
0.553964 + 0.832540i $$0.313114\pi$$
$$318$$ 0 0
$$319$$ 1.72611 0.0966436
$$320$$ 0 0
$$321$$ 0 0
$$322$$ −25.4904 −1.42052
$$323$$ 2.34891 0.130697
$$324$$ 0 0
$$325$$ 0 0
$$326$$ −28.8911 −1.60013
$$327$$ 0 0
$$328$$ 5.85289 0.323172
$$329$$ −13.5761 −0.748473
$$330$$ 0 0
$$331$$ 19.9426 1.09614 0.548072 0.836431i $$-0.315362\pi$$
0.548072 + 0.836431i $$0.315362\pi$$
$$332$$ 5.55553 0.304899
$$333$$ 0 0
$$334$$ −6.57688 −0.359871
$$335$$ 0 0
$$336$$ 0 0
$$337$$ −2.05952 −0.112189 −0.0560945 0.998425i $$-0.517865\pi$$
−0.0560945 + 0.998425i $$0.517865\pi$$
$$338$$ 31.3334 1.70431
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 17.7077 0.958925
$$342$$ 0 0
$$343$$ 2.44447 0.131989
$$344$$ −8.05177 −0.434122
$$345$$ 0 0
$$346$$ −21.9319 −1.17907
$$347$$ −10.5011 −0.563727 −0.281863 0.959455i $$-0.590952\pi$$
−0.281863 + 0.959455i $$0.590952\pi$$
$$348$$ 0 0
$$349$$ 16.5059 0.883540 0.441770 0.897128i $$-0.354351\pi$$
0.441770 + 0.897128i $$0.354351\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ −5.57608 −0.297206
$$353$$ 15.8860 0.845527 0.422764 0.906240i $$-0.361060\pi$$
0.422764 + 0.906240i $$0.361060\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ −1.91139 −0.101303
$$357$$ 0 0
$$358$$ −13.6305 −0.720397
$$359$$ −1.28376 −0.0677544 −0.0338772 0.999426i $$-0.510786\pi$$
−0.0338772 + 0.999426i $$0.510786\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ −21.3657 −1.12296
$$363$$ 0 0
$$364$$ −8.44447 −0.442610
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 19.8804 1.03775 0.518874 0.854851i $$-0.326351\pi$$
0.518874 + 0.854851i $$0.326351\pi$$
$$368$$ −17.0078 −0.886590
$$369$$ 0 0
$$370$$ 0 0
$$371$$ −50.2186 −2.60722
$$372$$ 0 0
$$373$$ 16.4359 0.851020 0.425510 0.904954i $$-0.360095\pi$$
0.425510 + 0.904954i $$0.360095\pi$$
$$374$$ −7.93273 −0.410192
$$375$$ 0 0
$$376$$ −11.2603 −0.580705
$$377$$ 3.99225 0.205611
$$378$$ 0 0
$$379$$ 0.338233 0.0173739 0.00868694 0.999962i $$-0.497235\pi$$
0.00868694 + 0.999962i $$0.497235\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 17.6559 0.903355
$$383$$ −13.7339 −0.701767 −0.350884 0.936419i $$-0.614119\pi$$
−0.350884 + 0.936419i $$0.614119\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ −26.8401 −1.36612
$$387$$ 0 0
$$388$$ 1.21730 0.0617989
$$389$$ 2.26109 0.114642 0.0573210 0.998356i $$-0.481744\pi$$
0.0573210 + 0.998356i $$0.481744\pi$$
$$390$$ 0 0
$$391$$ 12.8732 0.651027
$$392$$ −19.1706 −0.968260
$$393$$ 0 0
$$394$$ 27.0771 1.36413
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 7.82942 0.392947 0.196474 0.980509i $$-0.437051\pi$$
0.196474 + 0.980509i $$0.437051\pi$$
$$398$$ −9.79630 −0.491044
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 35.0510 1.75036 0.875181 0.483796i $$-0.160742\pi$$
0.875181 + 0.483796i $$0.160742\pi$$
$$402$$ 0 0
$$403$$ 40.9554 2.04013
$$404$$ 6.33823 0.315339
$$405$$ 0 0
$$406$$ 3.02830 0.150292
$$407$$ −23.0849 −1.14428
$$408$$ 0 0
$$409$$ −8.34811 −0.412787 −0.206394 0.978469i $$-0.566173\pi$$
−0.206394 + 0.978469i $$0.566173\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ −3.71836 −0.183190
$$413$$ −28.6610 −1.41031
$$414$$ 0 0
$$415$$ 0 0
$$416$$ −12.8967 −0.632312
$$417$$ 0 0
$$418$$ −3.37720 −0.165184
$$419$$ 19.8705 0.970738 0.485369 0.874309i $$-0.338685\pi$$
0.485369 + 0.874309i $$0.338685\pi$$
$$420$$ 0 0
$$421$$ −0.400672 −0.0195276 −0.00976379 0.999952i $$-0.503108\pi$$
−0.00976379 + 0.999952i $$0.503108\pi$$
$$422$$ −2.16283 −0.105285
$$423$$ 0 0
$$424$$ −41.6524 −2.02282
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 7.02830 0.340123
$$428$$ 1.93858 0.0937048
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 14.2533 0.686559 0.343280 0.939233i $$-0.388462\pi$$
0.343280 + 0.939233i $$0.388462\pi$$
$$432$$ 0 0
$$433$$ −12.8393 −0.617017 −0.308509 0.951222i $$-0.599830\pi$$
−0.308509 + 0.951222i $$0.599830\pi$$
$$434$$ 31.0665 1.49124
$$435$$ 0 0
$$436$$ −5.27097 −0.252434
$$437$$ 5.48052 0.262169
$$438$$ 0 0
$$439$$ −13.2555 −0.632649 −0.316324 0.948651i $$-0.602449\pi$$
−0.316324 + 0.948651i $$0.602449\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ −18.3473 −0.872692
$$443$$ 5.72611 0.272056 0.136028 0.990705i $$-0.456566\pi$$
0.136028 + 0.990705i $$0.456566\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 32.1826 1.52389
$$447$$ 0 0
$$448$$ −32.4437 −1.53282
$$449$$ −3.11399 −0.146958 −0.0734790 0.997297i $$-0.523410\pi$$
−0.0734790 + 0.997297i $$0.523410\pi$$
$$450$$ 0 0
$$451$$ 5.12386 0.241273
$$452$$ 0.198875 0.00935427
$$453$$ 0 0
$$454$$ −21.4290 −1.00571
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −27.4535 −1.28422 −0.642111 0.766611i $$-0.721941\pi$$
−0.642111 + 0.766611i $$0.721941\pi$$
$$458$$ −18.6305 −0.870548
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −13.9837 −0.651286 −0.325643 0.945493i $$-0.605581\pi$$
−0.325643 + 0.945493i $$0.605581\pi$$
$$462$$ 0 0
$$463$$ −17.9992 −0.836494 −0.418247 0.908333i $$-0.637355\pi$$
−0.418247 + 0.908333i $$0.637355\pi$$
$$464$$ 2.02055 0.0938015
$$465$$ 0 0
$$466$$ 11.3559 0.526050
$$467$$ 12.6871 0.587091 0.293545 0.955945i $$-0.405165\pi$$
0.293545 + 0.955945i $$0.405165\pi$$
$$468$$ 0 0
$$469$$ −16.2272 −0.749301
$$470$$ 0 0
$$471$$ 0 0
$$472$$ −23.7720 −1.09420
$$473$$ −7.04884 −0.324106
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 3.23492 0.148272
$$477$$ 0 0
$$478$$ 31.3396 1.43344
$$479$$ −24.7544 −1.13106 −0.565529 0.824729i $$-0.691328\pi$$
−0.565529 + 0.824729i $$0.691328\pi$$
$$480$$ 0 0
$$481$$ −53.3921 −2.43447
$$482$$ −18.6180 −0.848025
$$483$$ 0 0
$$484$$ 1.49814 0.0680972
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 4.08277 0.185008 0.0925039 0.995712i $$-0.470513\pi$$
0.0925039 + 0.995712i $$0.470513\pi$$
$$488$$ 5.82942 0.263886
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −29.2547 −1.32024 −0.660122 0.751158i $$-0.729496\pi$$
−0.660122 + 0.751158i $$0.729496\pi$$
$$492$$ 0 0
$$493$$ −1.52936 −0.0688788
$$494$$ −7.81100 −0.351433
$$495$$ 0 0
$$496$$ 20.7282 0.930725
$$497$$ 12.9533 0.581034
$$498$$ 0 0
$$499$$ 1.57315 0.0704240 0.0352120 0.999380i $$-0.488789\pi$$
0.0352120 + 0.999380i $$0.488789\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 16.9914 0.758365
$$503$$ 20.7819 0.926619 0.463310 0.886196i $$-0.346662\pi$$
0.463310 + 0.886196i $$0.346662\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ −18.5088 −0.822817
$$507$$ 0 0
$$508$$ −4.45917 −0.197844
$$509$$ 31.8238 1.41056 0.705282 0.708926i $$-0.250820\pi$$
0.705282 + 0.708926i $$0.250820\pi$$
$$510$$ 0 0
$$511$$ −9.05659 −0.400640
$$512$$ −25.3227 −1.11912
$$513$$ 0 0
$$514$$ −4.26884 −0.188291
$$515$$ 0 0
$$516$$ 0 0
$$517$$ −9.85772 −0.433542
$$518$$ −40.5003 −1.77948
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 27.4720 1.20357 0.601784 0.798659i $$-0.294457\pi$$
0.601784 + 0.798659i $$0.294457\pi$$
$$522$$ 0 0
$$523$$ 10.4466 0.456798 0.228399 0.973568i $$-0.426651\pi$$
0.228399 + 0.973568i $$0.426651\pi$$
$$524$$ 3.06434 0.133866
$$525$$ 0 0
$$526$$ 13.8676 0.604656
$$527$$ −15.6893 −0.683435
$$528$$ 0 0
$$529$$ 7.03605 0.305915
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 1.37720 0.0597093
$$533$$ 11.8508 0.513314
$$534$$ 0 0
$$535$$ 0 0
$$536$$ −13.4592 −0.581348
$$537$$ 0 0
$$538$$ −23.4826 −1.01241
$$539$$ −16.7827 −0.722882
$$540$$ 0 0
$$541$$ 13.4989 0.580365 0.290182 0.956971i $$-0.406284\pi$$
0.290182 + 0.956971i $$0.406284\pi$$
$$542$$ −25.9971 −1.11667
$$543$$ 0 0
$$544$$ 4.94048 0.211822
$$545$$ 0 0
$$546$$ 0 0
$$547$$ −8.54698 −0.365442 −0.182721 0.983165i $$-0.558491\pi$$
−0.182721 + 0.983165i $$0.558491\pi$$
$$548$$ −6.62975 −0.283209
$$549$$ 0 0
$$550$$ 0 0
$$551$$ −0.651093 −0.0277375
$$552$$ 0 0
$$553$$ 55.3502 2.35373
$$554$$ 3.43942 0.146127
$$555$$ 0 0
$$556$$ −6.94633 −0.294590
$$557$$ 27.2378 1.15410 0.577052 0.816707i $$-0.304203\pi$$
0.577052 + 0.816707i $$0.304203\pi$$
$$558$$ 0 0
$$559$$ −16.3030 −0.689543
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 19.3940 0.818088
$$563$$ −1.44235 −0.0607876 −0.0303938 0.999538i $$-0.509676\pi$$
−0.0303938 + 0.999538i $$0.509676\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ −7.87322 −0.330936
$$567$$ 0 0
$$568$$ 10.7437 0.450797
$$569$$ 14.2632 0.597945 0.298973 0.954262i $$-0.403356\pi$$
0.298973 + 0.954262i $$0.403356\pi$$
$$570$$ 0 0
$$571$$ 9.91723 0.415023 0.207512 0.978233i $$-0.433464\pi$$
0.207512 + 0.978233i $$0.433464\pi$$
$$572$$ −6.13161 −0.256375
$$573$$ 0 0
$$574$$ 8.98933 0.375207
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 8.23997 0.343034 0.171517 0.985181i $$-0.445133\pi$$
0.171517 + 0.985181i $$0.445133\pi$$
$$578$$ −14.6276 −0.608429
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 53.7742 2.23093
$$582$$ 0 0
$$583$$ −36.4642 −1.51019
$$584$$ −7.51173 −0.310838
$$585$$ 0 0
$$586$$ 39.0948 1.61499
$$587$$ 36.9554 1.52531 0.762656 0.646804i $$-0.223895\pi$$
0.762656 + 0.646804i $$0.223895\pi$$
$$588$$ 0 0
$$589$$ −6.67939 −0.275219
$$590$$ 0 0
$$591$$ 0 0
$$592$$ −27.0227 −1.11062
$$593$$ −1.36170 −0.0559184 −0.0279592 0.999609i $$-0.508901\pi$$
−0.0279592 + 0.999609i $$0.508901\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ −6.52241 −0.267168
$$597$$ 0 0
$$598$$ −42.8083 −1.75056
$$599$$ −33.5753 −1.37185 −0.685924 0.727673i $$-0.740602\pi$$
−0.685924 + 0.727673i $$0.740602\pi$$
$$600$$ 0 0
$$601$$ 12.6561 0.516255 0.258127 0.966111i $$-0.416895\pi$$
0.258127 + 0.966111i $$0.416895\pi$$
$$602$$ −12.3665 −0.504022
$$603$$ 0 0
$$604$$ −0.974627 −0.0396570
$$605$$ 0 0
$$606$$ 0 0
$$607$$ −17.4047 −0.706435 −0.353217 0.935541i $$-0.614912\pi$$
−0.353217 + 0.935541i $$0.614912\pi$$
$$608$$ 2.10331 0.0853006
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −22.7995 −0.922370
$$612$$ 0 0
$$613$$ 37.2603 1.50493 0.752465 0.658633i $$-0.228865\pi$$
0.752465 + 0.658633i $$0.228865\pi$$
$$614$$ 18.8315 0.759979
$$615$$ 0 0
$$616$$ −29.3121 −1.18102
$$617$$ −18.3764 −0.739806 −0.369903 0.929070i $$-0.620609\pi$$
−0.369903 + 0.929070i $$0.620609\pi$$
$$618$$ 0 0
$$619$$ 14.5526 0.584919 0.292459 0.956278i $$-0.405526\pi$$
0.292459 + 0.956278i $$0.405526\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 34.3940 1.37907
$$623$$ −18.5011 −0.741229
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 12.4162 0.496250
$$627$$ 0 0
$$628$$ −4.13936 −0.165178
$$629$$ 20.4535 0.815536
$$630$$ 0 0
$$631$$ 22.0304 0.877017 0.438509 0.898727i $$-0.355507\pi$$
0.438509 + 0.898727i $$0.355507\pi$$
$$632$$ 45.9087 1.82615
$$633$$ 0 0
$$634$$ 25.1289 0.997996
$$635$$ 0 0
$$636$$ 0 0
$$637$$ −38.8160 −1.53795
$$638$$ 2.19887 0.0870543
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 43.6765 1.72512 0.862558 0.505958i $$-0.168861\pi$$
0.862558 + 0.505958i $$0.168861\pi$$
$$642$$ 0 0
$$643$$ 25.9263 1.02243 0.511217 0.859452i $$-0.329195\pi$$
0.511217 + 0.859452i $$0.329195\pi$$
$$644$$ 7.54778 0.297424
$$645$$ 0 0
$$646$$ 2.99225 0.117728
$$647$$ −42.1046 −1.65530 −0.827652 0.561242i $$-0.810324\pi$$
−0.827652 + 0.561242i $$0.810324\pi$$
$$648$$ 0 0
$$649$$ −20.8110 −0.816903
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 8.55473 0.335029
$$653$$ 40.4671 1.58360 0.791801 0.610779i $$-0.209144\pi$$
0.791801 + 0.610779i $$0.209144\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 5.99788 0.234178
$$657$$ 0 0
$$658$$ −17.2944 −0.674207
$$659$$ −33.4204 −1.30187 −0.650937 0.759131i $$-0.725624\pi$$
−0.650937 + 0.759131i $$0.725624\pi$$
$$660$$ 0 0
$$661$$ 19.4883 0.758006 0.379003 0.925396i $$-0.376267\pi$$
0.379003 + 0.925396i $$0.376267\pi$$
$$662$$ 25.4047 0.987382
$$663$$ 0 0
$$664$$ 44.6015 1.73087
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −3.56833 −0.138166
$$668$$ 1.94743 0.0753485
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 5.10331 0.197011
$$672$$ 0 0
$$673$$ 0.747456 0.0288123 0.0144062 0.999896i $$-0.495414\pi$$
0.0144062 + 0.999896i $$0.495414\pi$$
$$674$$ −2.62360 −0.101057
$$675$$ 0 0
$$676$$ −9.27792 −0.356843
$$677$$ 25.0275 0.961885 0.480942 0.876752i $$-0.340295\pi$$
0.480942 + 0.876752i $$0.340295\pi$$
$$678$$ 0 0
$$679$$ 11.7827 0.452179
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 22.5577 0.863777
$$683$$ −36.7253 −1.40525 −0.702627 0.711558i $$-0.747990\pi$$
−0.702627 + 0.711558i $$0.747990\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 3.11399 0.118893
$$687$$ 0 0
$$688$$ −8.25122 −0.314575
$$689$$ −84.3366 −3.21297
$$690$$ 0 0
$$691$$ −21.3177 −0.810963 −0.405482 0.914103i $$-0.632896\pi$$
−0.405482 + 0.914103i $$0.632896\pi$$
$$692$$ 6.49411 0.246869
$$693$$ 0 0
$$694$$ −13.3772 −0.507792
$$695$$ 0 0
$$696$$ 0 0
$$697$$ −4.53981 −0.171958
$$698$$ 21.0267 0.795872
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −27.1161 −1.02416 −0.512081 0.858937i $$-0.671125\pi$$
−0.512081 + 0.858937i $$0.671125\pi$$
$$702$$ 0 0
$$703$$ 8.70769 0.328417
$$704$$ −23.5577 −0.887862
$$705$$ 0 0
$$706$$ 20.2370 0.761631
$$707$$ 61.3502 2.30731
$$708$$ 0 0
$$709$$ −10.1161 −0.379918 −0.189959 0.981792i $$-0.560836\pi$$
−0.189959 + 0.981792i $$0.560836\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ −15.3452 −0.575085
$$713$$ −36.6065 −1.37092
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 4.03605 0.150834
$$717$$ 0 0
$$718$$ −1.63537 −0.0610316
$$719$$ −24.1471 −0.900535 −0.450268 0.892894i $$-0.648671\pi$$
−0.450268 + 0.892894i $$0.648671\pi$$
$$720$$ 0 0
$$721$$ −35.9914 −1.34039
$$722$$ 1.27389 0.0474093
$$723$$ 0 0
$$724$$ 6.32646 0.235121
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 19.8988 0.738006 0.369003 0.929428i $$-0.379699\pi$$
0.369003 + 0.929428i $$0.379699\pi$$
$$728$$ −67.7947 −2.51264
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 6.24537 0.230994
$$732$$ 0 0
$$733$$ −19.9455 −0.736705 −0.368352 0.929686i $$-0.620078\pi$$
−0.368352 + 0.929686i $$0.620078\pi$$
$$734$$ 25.3254 0.934779
$$735$$ 0 0
$$736$$ 11.5272 0.424900
$$737$$ −11.7827 −0.434021
$$738$$ 0 0
$$739$$ −38.2624 −1.40751 −0.703753 0.710445i $$-0.748494\pi$$
−0.703753 + 0.710445i $$0.748494\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ −63.9730 −2.34852
$$743$$ −12.5547 −0.460588 −0.230294 0.973121i $$-0.573969\pi$$
−0.230294 + 0.973121i $$0.573969\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 20.9376 0.766579
$$747$$ 0 0
$$748$$ 2.34891 0.0858845
$$749$$ 18.7643 0.685632
$$750$$ 0 0
$$751$$ 23.2215 0.847366 0.423683 0.905810i $$-0.360737\pi$$
0.423683 + 0.905810i $$0.360737\pi$$
$$752$$ −11.5392 −0.420792
$$753$$ 0 0
$$754$$ 5.08569 0.185210
$$755$$ 0 0
$$756$$ 0 0
$$757$$ −18.4105 −0.669143 −0.334571 0.942370i $$-0.608592\pi$$
−0.334571 + 0.942370i $$0.608592\pi$$
$$758$$ 0.430872 0.0156500
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −11.7982 −0.427684 −0.213842 0.976868i $$-0.568598\pi$$
−0.213842 + 0.976868i $$0.568598\pi$$
$$762$$ 0 0
$$763$$ −51.0197 −1.84704
$$764$$ −5.22797 −0.189141
$$765$$ 0 0
$$766$$ −17.4954 −0.632136
$$767$$ −48.1329 −1.73798
$$768$$ 0 0
$$769$$ 41.2653 1.48807 0.744033 0.668143i $$-0.232910\pi$$
0.744033 + 0.668143i $$0.232910\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 7.94743 0.286034
$$773$$ 5.51736 0.198446 0.0992229 0.995065i $$-0.468364\pi$$
0.0992229 + 0.995065i $$0.468364\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 9.77283 0.350824
$$777$$ 0 0
$$778$$ 2.88039 0.103267
$$779$$ −1.93273 −0.0692474
$$780$$ 0 0
$$781$$ 9.40550 0.336555
$$782$$ 16.3991 0.586430
$$783$$ 0 0
$$784$$ −19.6455 −0.701624
$$785$$ 0 0
$$786$$ 0 0
$$787$$ −16.7771 −0.598038 −0.299019 0.954247i $$-0.596659\pi$$
−0.299019 + 0.954247i $$0.596659\pi$$
$$788$$ −8.01762 −0.285616
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 1.92498 0.0684446
$$792$$ 0 0
$$793$$ 11.8032 0.419146
$$794$$ 9.97383 0.353958
$$795$$ 0 0
$$796$$ 2.90071 0.102813
$$797$$ −29.4260 −1.04232 −0.521162 0.853458i $$-0.674501\pi$$
−0.521162 + 0.853458i $$0.674501\pi$$
$$798$$ 0 0
$$799$$ 8.73408 0.308989
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 44.6511 1.57668
$$803$$ −6.57608 −0.232065
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 52.1727 1.83771
$$807$$ 0 0
$$808$$ 50.8852 1.79014
$$809$$ −5.48614 −0.192883 −0.0964413 0.995339i $$-0.530746\pi$$
−0.0964413 + 0.995339i $$0.530746\pi$$
$$810$$ 0 0
$$811$$ 16.3927 0.575626 0.287813 0.957687i $$-0.407072\pi$$
0.287813 + 0.957687i $$0.407072\pi$$
$$812$$ −0.896688 −0.0314676
$$813$$ 0 0
$$814$$ −29.4076 −1.03074
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 2.65884 0.0930212
$$818$$ −10.6346 −0.371829
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −15.1628 −0.529186 −0.264593 0.964360i $$-0.585238\pi$$
−0.264593 + 0.964360i $$0.585238\pi$$
$$822$$ 0 0
$$823$$ −16.6094 −0.578968 −0.289484 0.957183i $$-0.593484\pi$$
−0.289484 + 0.957183i $$0.593484\pi$$
$$824$$ −29.8521 −1.03995
$$825$$ 0 0
$$826$$ −36.5109 −1.27038
$$827$$ 15.2293 0.529574 0.264787 0.964307i $$-0.414698\pi$$
0.264787 + 0.964307i $$0.414698\pi$$
$$828$$ 0 0
$$829$$ 47.4565 1.64823 0.824116 0.566422i $$-0.191673\pi$$
0.824116 + 0.566422i $$0.191673\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ −54.4856 −1.88895
$$833$$ 14.8697 0.515205
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 1.00000 0.0345857
$$837$$ 0 0
$$838$$ 25.3129 0.874418
$$839$$ −2.26614 −0.0782359 −0.0391179 0.999235i $$-0.512455\pi$$
−0.0391179 + 0.999235i $$0.512455\pi$$
$$840$$ 0 0
$$841$$ −28.5761 −0.985382
$$842$$ −0.510413 −0.0175900
$$843$$ 0 0
$$844$$ 0.640420 0.0220442
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 14.5011 0.498262
$$848$$ −42.6842 −1.46578
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 47.7226 1.63591
$$852$$ 0 0
$$853$$ −51.8753 −1.77618 −0.888089 0.459672i $$-0.847967\pi$$
−0.888089 + 0.459672i $$0.847967\pi$$
$$854$$ 8.95328 0.306375
$$855$$ 0 0
$$856$$ 15.5635 0.531949
$$857$$ −6.17058 −0.210783 −0.105391 0.994431i $$-0.533610\pi$$
−0.105391 + 0.994431i $$0.533610\pi$$
$$858$$ 0 0
$$859$$ −31.0635 −1.05987 −0.529937 0.848037i $$-0.677785\pi$$
−0.529937 + 0.848037i $$0.677785\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 18.1572 0.618437
$$863$$ −3.24772 −0.110554 −0.0552768 0.998471i $$-0.517604\pi$$
−0.0552768 + 0.998471i $$0.517604\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ −16.3559 −0.555795
$$867$$ 0 0
$$868$$ −9.19887 −0.312230
$$869$$ 40.1903 1.36336
$$870$$ 0 0
$$871$$ −27.2517 −0.923390
$$872$$ −42.3169 −1.43303
$$873$$ 0 0
$$874$$ 6.98158 0.236155
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 41.7042 1.40825 0.704125 0.710076i $$-0.251339\pi$$
0.704125 + 0.710076i $$0.251339\pi$$
$$878$$ −16.8860 −0.569875
$$879$$ 0 0
$$880$$ 0 0
$$881$$ −7.42392 −0.250118 −0.125059 0.992149i $$-0.539912\pi$$
−0.125059 + 0.992149i $$0.539912\pi$$
$$882$$ 0 0
$$883$$ −32.0333 −1.07801 −0.539004 0.842303i $$-0.681199\pi$$
−0.539004 + 0.842303i $$0.681199\pi$$
$$884$$ 5.43269 0.182721
$$885$$ 0 0
$$886$$ 7.29444 0.245061
$$887$$ 20.0275 0.672457 0.336229 0.941780i $$-0.390848\pi$$
0.336229 + 0.941780i $$0.390848\pi$$
$$888$$ 0 0
$$889$$ −43.1620 −1.44761
$$890$$ 0 0
$$891$$ 0 0
$$892$$ −9.52936 −0.319066
$$893$$ 3.71836 0.124430
$$894$$ 0 0
$$895$$ 0 0
$$896$$ −25.9709 −0.867627
$$897$$ 0 0
$$898$$ −3.96688 −0.132376
$$899$$ 4.34891 0.145044
$$900$$ 0 0
$$901$$ 32.3078 1.07633
$$902$$ 6.52723 0.217333
$$903$$ 0 0
$$904$$ 1.59662 0.0531029
$$905$$ 0 0
$$906$$ 0 0
$$907$$ −4.94823 −0.164303 −0.0821517 0.996620i $$-0.526179\pi$$
−0.0821517 + 0.996620i $$0.526179\pi$$
$$908$$ 6.34518 0.210572
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −32.3014 −1.07019 −0.535096 0.844791i $$-0.679725\pi$$
−0.535096 + 0.844791i $$0.679725\pi$$
$$912$$ 0 0
$$913$$ 39.0459 1.29223
$$914$$ −34.9728 −1.15680
$$915$$ 0 0
$$916$$ 5.51656 0.182272
$$917$$ 29.6610 0.979491
$$918$$ 0 0
$$919$$ 21.3072 0.702861 0.351430 0.936214i $$-0.385695\pi$$
0.351430 + 0.936214i $$0.385695\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ −17.8137 −0.586663
$$923$$ 21.7536 0.716029
$$924$$ 0 0
$$925$$ 0 0
$$926$$ −22.9290 −0.753494
$$927$$ 0 0
$$928$$ −1.36945 −0.0449545
$$929$$ −24.7848 −0.813164 −0.406582 0.913614i $$-0.633279\pi$$
−0.406582 + 0.913614i $$0.633279\pi$$
$$930$$ 0 0
$$931$$ 6.33048 0.207473
$$932$$ −3.36250 −0.110142
$$933$$ 0 0
$$934$$ 16.1620 0.528838
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 32.3687 1.05744 0.528719 0.848797i $$-0.322673\pi$$
0.528719 + 0.848797i $$0.322673\pi$$
$$938$$ −20.6716 −0.674953
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 1.48264 0.0483326 0.0241663 0.999708i $$-0.492307\pi$$
0.0241663 + 0.999708i $$0.492307\pi$$
$$942$$ 0 0
$$943$$ −10.5924 −0.344935
$$944$$ −24.3609 −0.792880
$$945$$ 0 0
$$946$$ −8.97945 −0.291947
$$947$$ −8.86064 −0.287932 −0.143966 0.989583i $$-0.545986\pi$$
−0.143966 + 0.989583i $$0.545986\pi$$
$$948$$ 0 0
$$949$$ −15.2095 −0.493723
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 25.9709 0.841722
$$953$$ 10.1239 0.327944 0.163972 0.986465i $$-0.447569\pi$$
0.163972 + 0.986465i $$0.447569\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ −9.27974 −0.300128
$$957$$ 0 0
$$958$$ −31.5344 −1.01883
$$959$$ −64.1719 −2.07222
$$960$$ 0 0
$$961$$ 13.6142 0.439169
$$962$$ −68.0157 −2.19291
$$963$$ 0 0
$$964$$ 5.51284 0.177557
$$965$$ 0 0
$$966$$ 0 0
$$967$$ −41.8139 −1.34465 −0.672323 0.740258i $$-0.734703\pi$$
−0.672323 + 0.740258i $$0.734703\pi$$
$$968$$ 12.0275 0.386578
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 5.53016 0.177471 0.0887356 0.996055i $$-0.471717\pi$$
0.0887356 + 0.996055i $$0.471717\pi$$
$$972$$ 0 0
$$973$$ −67.2362 −2.15549
$$974$$ 5.20100 0.166651
$$975$$ 0 0
$$976$$ 5.97383 0.191218
$$977$$ −31.9447 −1.02200 −0.511001 0.859580i $$-0.670725\pi$$
−0.511001 + 0.859580i $$0.670725\pi$$
$$978$$ 0 0
$$979$$ −13.4338 −0.429346
$$980$$ 0 0
$$981$$ 0 0
$$982$$ −37.2672 −1.18925
$$983$$ 8.82862 0.281589 0.140795 0.990039i $$-0.455034\pi$$
0.140795 + 0.990039i $$0.455034\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ −1.94823 −0.0620444
$$987$$ 0 0
$$988$$ 2.31286 0.0735819
$$989$$ 14.5718 0.463357
$$990$$ 0 0
$$991$$ −43.7304 −1.38914 −0.694570 0.719425i $$-0.744405\pi$$
−0.694570 + 0.719425i $$0.744405\pi$$
$$992$$ −14.0488 −0.446051
$$993$$ 0 0
$$994$$ 16.5011 0.523382
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 46.6738 1.47817 0.739086 0.673611i $$-0.235257\pi$$
0.739086 + 0.673611i $$0.235257\pi$$
$$998$$ 2.00403 0.0634363
$$999$$ 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 4275.2.a.bm.1.2 3
3.2 odd 2 475.2.a.e.1.2 3
5.4 even 2 4275.2.a.ba.1.2 3
12.11 even 2 7600.2.a.cc.1.1 3
15.2 even 4 475.2.b.b.324.3 6
15.8 even 4 475.2.b.b.324.4 6
15.14 odd 2 475.2.a.g.1.2 yes 3
57.56 even 2 9025.2.a.bc.1.2 3
60.59 even 2 7600.2.a.bh.1.3 3
285.284 even 2 9025.2.a.y.1.2 3

By twisted newform
Twist Min Dim Char Parity Ord Type
475.2.a.e.1.2 3 3.2 odd 2
475.2.a.g.1.2 yes 3 15.14 odd 2
475.2.b.b.324.3 6 15.2 even 4
475.2.b.b.324.4 6 15.8 even 4
4275.2.a.ba.1.2 3 5.4 even 2
4275.2.a.bm.1.2 3 1.1 even 1 trivial
7600.2.a.bh.1.3 3 60.59 even 2
7600.2.a.cc.1.1 3 12.11 even 2
9025.2.a.y.1.2 3 285.284 even 2
9025.2.a.bc.1.2 3 57.56 even 2