LMFDB - Embedded newform 85.4.a.e.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [85,4,Mod(1,85)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(85, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("85.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$N$	$=$	$85 = 5 \cdot 17$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	85.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:	$5.01516235049$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.1304.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{3} - 11x - 2$ x^3 - 11*x - 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$2$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-3.22168$ of defining polynomial
Character	$\chi$	$=$	85.1

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.22168 q^{2} -1.15753 q^{3} -6.50750 q^{4} -5.00000 q^{5} -1.41413 q^{6} +14.6743 q^{7} -17.7235 q^{8} -25.6601 q^{9} -6.10839 q^{10} -71.3743 q^{11} +7.53262 q^{12} -28.4035 q^{13} +17.9273 q^{14} +5.78764 q^{15} +30.4076 q^{16} +17.0000 q^{17} -31.3484 q^{18} +85.9592 q^{19} +32.5375 q^{20} -16.9860 q^{21} -87.1964 q^{22} -7.17872 q^{23} +20.5154 q^{24} +25.0000 q^{25} -34.7000 q^{26} +60.9556 q^{27} -95.4934 q^{28} -27.1610 q^{29} +7.07063 q^{30} -15.6170 q^{31} +178.936 q^{32} +82.6178 q^{33} +20.7685 q^{34} -73.3717 q^{35} +166.983 q^{36} -67.0036 q^{37} +105.014 q^{38} +32.8779 q^{39} +88.6175 q^{40} -38.5701 q^{41} -20.7514 q^{42} -251.495 q^{43} +464.469 q^{44} +128.301 q^{45} -8.77008 q^{46} +57.9121 q^{47} -35.1977 q^{48} -127.663 q^{49} +30.5419 q^{50} -19.6780 q^{51} +184.836 q^{52} -677.807 q^{53} +74.4680 q^{54} +356.872 q^{55} -260.081 q^{56} -99.5002 q^{57} -33.1819 q^{58} +598.645 q^{59} -37.6631 q^{60} +346.063 q^{61} -19.0789 q^{62} -376.546 q^{63} -24.6588 q^{64} +142.018 q^{65} +100.932 q^{66} -849.923 q^{67} -110.628 q^{68} +8.30957 q^{69} -89.6366 q^{70} -911.836 q^{71} +454.787 q^{72} +704.352 q^{73} -81.8568 q^{74} -28.9382 q^{75} -559.380 q^{76} -1047.37 q^{77} +40.1662 q^{78} +55.8414 q^{79} -152.038 q^{80} +622.266 q^{81} -47.1202 q^{82} -1235.19 q^{83} +110.536 q^{84} -85.0000 q^{85} -307.245 q^{86} +31.4396 q^{87} +1265.00 q^{88} -72.8069 q^{89} +156.742 q^{90} -416.803 q^{91} +46.7155 q^{92} +18.0771 q^{93} +70.7499 q^{94} -429.796 q^{95} -207.124 q^{96} +972.939 q^{97} -155.964 q^{98} +1831.47 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$3 q - 6 q^{2} - 4 q^{3} + 10 q^{4} - 15 q^{5} + 36 q^{6} + 8 q^{7} - 66 q^{8} + 39 q^{9} + 30 q^{10} - 118 q^{11} - 212 q^{12} - 10 q^{13} - 68 q^{14} + 20 q^{15} + 242 q^{16} + 51 q^{17} - 342 q^{18} - 160 q^{19}+ \cdots + 770 q^{99}+O(q^{100})$ 3 * q - 6 * q^2 - 4 * q^3 + 10 * q^4 - 15 * q^5 + 36 * q^6 + 8 * q^7 - 66 * q^8 + 39 * q^9 + 30 * q^10 - 118 * q^11 - 212 * q^12 - 10 * q^13 - 68 * q^14 + 20 * q^15 + 242 * q^16 + 51 * q^17 - 342 * q^18 - 160 * q^19 - 50 * q^20 - 472 * q^21 + 44 * q^22 - 92 * q^23 + 804 * q^24 + 75 * q^25 - 44 * q^26 - 280 * q^27 + 644 * q^28 + 374 * q^29 - 180 * q^30 + 70 * q^31 - 594 * q^32 - 8 * q^33 - 102 * q^34 - 40 * q^35 + 1258 * q^36 - 246 * q^37 + 896 * q^38 + 248 * q^39 + 330 * q^40 - 354 * q^41 + 1688 * q^42 - 286 * q^43 + 348 * q^44 - 195 * q^45 - 4 * q^46 - 390 * q^47 - 2004 * q^48 + 1091 * q^49 - 150 * q^50 - 68 * q^51 - 76 * q^52 - 1282 * q^53 + 1560 * q^54 + 590 * q^55 - 2996 * q^56 - 160 * q^57 - 1356 * q^58 - 304 * q^59 + 1060 * q^60 + 246 * q^61 + 532 * q^62 + 728 * q^63 + 1826 * q^64 + 50 * q^65 - 152 * q^66 - 718 * q^67 + 170 * q^68 - 1144 * q^69 + 340 * q^70 - 602 * q^71 - 3138 * q^72 + 282 * q^73 + 916 * q^74 - 100 * q^75 - 1888 * q^76 - 256 * q^77 - 568 * q^78 + 206 * q^79 - 1210 * q^80 + 435 * q^81 + 1572 * q^82 - 2414 * q^83 - 4120 * q^84 - 255 * q^85 + 256 * q^86 - 8 * q^87 + 1412 * q^88 - 870 * q^89 + 1710 * q^90 - 1456 * q^91 + 1524 * q^92 + 3528 * q^93 + 2224 * q^94 + 800 * q^95 + 4068 * q^96 + 2230 * q^97 - 3822 * q^98 + 770 * 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 (See

a_n

instead) (See

a_n

instead) (See

a_n

instead) Display

a_n

with

n

up to: 50 250 1000 (See only

a_p

) (See only

a_p

) (See only

a_p

)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$\alpha_n$			$\theta_n$
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$\alpha_p$			$\theta_p$

$2$	1.22168	0.431928	0.215964	−	0.976401i	$-0.430711\pi$
$2$	1.22168	0.431928	0.215964	+	0.976401i	$0.430711\pi$
$3$	−1.15753	−0.222766	−0.111383	−	0.993778i	$-0.535528\pi$
$3$	−1.15753	−0.222766	−0.111383	+	0.993778i	$0.535528\pi$
$4$	−6.50750	−0.813438
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	−1.41413	−0.0962191
$7$	14.6743	0.792340	0.396170	−	0.918177i	$-0.370339\pi$
$7$	14.6743	0.792340	0.396170	+	0.918177i	$0.370339\pi$
$8$	−17.7235	−0.783275
$9$	−25.6601	−0.950375
$10$	−6.10839	−0.193164
$11$	−71.3743	−1.95638	−0.978189	−	0.207716i	$-0.933397\pi$
$11$	−71.3743	−1.95638	−0.978189	+	0.207716i	$0.933397\pi$
$12$	7.53262	0.181207
$13$	−28.4035	−0.605979	−0.302989	−	0.952994i	$-0.597985\pi$
$13$	−28.4035	−0.605979	−0.302989	+	0.952994i	$0.597985\pi$
$14$	17.9273	0.342234
$15$	5.78764	0.0996241
$16$	30.4076	0.475119
$17$	17.0000	0.242536
$17$	17.0000	0.242536
$18$	−31.3484	−0.410494
$19$	85.9592	1.03792	0.518958	−	0.854800i	$-0.326320\pi$
$19$	85.9592	1.03792	0.518958	+	0.854800i	$0.326320\pi$
$20$	32.5375	0.363781
$21$	−16.9860	−0.176507
$22$	−87.1964	−0.845015
$23$	−7.17872	−0.0650811	−0.0325406	−	0.999470i	$-0.510360\pi$
$23$	−7.17872	−0.0650811	−0.0325406	+	0.999470i	$0.510360\pi$
$24$	20.5154	0.174487
$25$	25.0000	0.200000
$26$	−34.7000	−0.261739
$27$	60.9556	0.434478
$28$	−95.4934	−0.644520
$29$	−27.1610	−0.173919	−0.0869597	−	0.996212i	$-0.527715\pi$
$29$	−27.1610	−0.173919	−0.0869597	+	0.996212i	$0.527715\pi$
$30$	7.07063	0.0430305
$31$	−15.6170	−0.0904803	−0.0452401	−	0.998976i	$-0.514405\pi$
$31$	−15.6170	−0.0904803	−0.0452401	+	0.998976i	$0.514405\pi$
$32$	178.936	0.988493
$33$	82.6178	0.435815
$34$	20.7685	0.104758
$35$	−73.3717	−0.354345
$36$	166.983	0.773071
$37$	−67.0036	−0.297712	−0.148856	−	0.988859i	$-0.547559\pi$
$37$	−67.0036	−0.297712	−0.148856	+	0.988859i	$0.547559\pi$
$38$	105.014	0.448305
$39$	32.8779	0.134992
$40$	88.6175	0.350291
$41$	−38.5701	−0.146918	−0.0734590	−	0.997298i	$-0.523404\pi$
$41$	−38.5701	−0.146918	−0.0734590	+	0.997298i	$0.523404\pi$
$42$	−20.7514	−0.0762383
$43$	−251.495	−0.891920	−0.445960	−	0.895053i	$-0.647138\pi$
$43$	−251.495	−0.891920	−0.445960	+	0.895053i	$0.647138\pi$
$44$	464.469	1.59139
$45$	128.301	0.425021
$46$	−8.77008	−0.0281104
$47$	57.9121	0.179731	0.0898654	−	0.995954i	$-0.471356\pi$
$47$	57.9121	0.179731	0.0898654	+	0.995954i	$0.471356\pi$
$48$	−35.1977	−0.105841
$49$	−127.663	−0.372197
$50$	30.5419	0.0863856
$51$	−19.6780	−0.0540288
$52$	184.836	0.492926
$53$	−677.807	−1.75668	−0.878339	−	0.478039i	$-0.841348\pi$
$53$	−677.807	−1.75668	−0.878339	+	0.478039i	$0.841348\pi$
$54$	74.4680	0.187663
$55$	356.872	0.874919
$56$	−260.081	−0.620620
$57$	−99.5002	−0.231213
$58$	−33.1819	−0.0751207
$59$	598.645	1.32097	0.660483	−	0.750841i	$-0.270352\pi$
$59$	598.645	1.32097	0.660483	+	0.750841i	$0.270352\pi$
$60$	−37.6631	−0.0810381
$61$	346.063	0.726375	0.363187	−	0.931716i	$-0.381689\pi$
$61$	346.063	0.726375	0.363187	+	0.931716i	$0.381689\pi$
$62$	−19.0789	−0.0390810
$63$	−376.546	−0.753021
$64$	−24.6588	−0.0481617
$65$	142.018	0.271002
$66$	100.932	0.188241
$67$	−849.923	−1.54977	−0.774885	−	0.632102i	$-0.782192\pi$
$67$	−849.923	−1.54977	−0.774885	+	0.632102i	$0.782192\pi$
$68$	−110.628	−0.197288
$69$	8.30957	0.0144979
$70$	−89.6366	−0.153052
$71$	−911.836	−1.52415	−0.762077	−	0.647486i	$-0.775820\pi$
$71$	−911.836	−1.52415	−0.762077	+	0.647486i	$0.775820\pi$
$72$	454.787	0.744405
$73$	704.352	1.12929	0.564645	−	0.825334i	$-0.309013\pi$
$73$	704.352	1.12929	0.564645	+	0.825334i	$0.309013\pi$
$74$	−81.8568	−0.128590
$75$	−28.9382	−0.0445533
$76$	−559.380	−0.844280
$77$	−1047.37	−1.55012
$78$	40.1662	0.0583067
$79$	55.8414	0.0795272	0.0397636	−	0.999209i	$-0.487340\pi$
$79$	55.8414	0.0795272	0.0397636	+	0.999209i	$0.487340\pi$
$80$	−152.038	−0.212480
$81$	622.266	0.853588
$82$	−47.1202	−0.0634580
$83$	−1235.19	−1.63350	−0.816748	−	0.576995i	$-0.804225\pi$
$83$	−1235.19	−1.63350	−0.816748	+	0.576995i	$0.804225\pi$
$84$	110.536	0.143577
$85$	−85.0000	−0.108465
$86$	−307.245	−0.385246
$87$	31.4396	0.0387434
$88$	1265.00	1.53238
$89$	−72.8069	−0.0867136	−0.0433568	−	0.999060i	$-0.513805\pi$
$89$	−72.8069	−0.0867136	−0.0433568	+	0.999060i	$0.513805\pi$
$90$	156.742	0.183578
$91$	−416.803	−0.480141
$92$	46.7155	0.0529395
$93$	18.0771	0.0201560
$94$	70.7499	0.0776308
$95$	−429.796	−0.464170
$96$	−207.124	−0.220203
$97$	972.939	1.01842	0.509211	−	0.860642i	$-0.329937\pi$
$97$	972.939	1.01842	0.509211	+	0.860642i	$0.329937\pi$
$98$	−155.964	−0.160762
$99$	1831.47	1.85929
$100$	−162.688	−0.162688
$101$	−658.119	−0.648369	−0.324185	−	0.945994i	$-0.605090\pi$
$101$	−658.119	−0.648369	−0.324185	+	0.945994i	$0.605090\pi$
$102$	−24.0401	−0.0233366
$103$	−1727.68	−1.65275	−0.826375	−	0.563120i	$-0.809601\pi$
$103$	−1727.68	−1.65275	−0.826375	+	0.563120i	$0.809601\pi$
$104$	503.410	0.474648
$105$	84.9298	0.0789362
$106$	−828.061	−0.758758
$107$	−903.004	−0.815857	−0.407929	−	0.913014i	$-0.633749\pi$
$107$	−903.004	−0.815857	−0.407929	+	0.913014i	$0.633749\pi$
$108$	−396.669	−0.353421
$109$	833.557	0.732480	0.366240	−	0.930520i	$-0.380645\pi$
$109$	833.557	0.732480	0.366240	+	0.930520i	$0.380645\pi$
$110$	435.982	0.377902
$111$	77.5586	0.0663201
$112$	446.212	0.376456
$113$	1006.49	0.837900	0.418950	−	0.908009i	$-0.362398\pi$
$113$	1006.49	0.837900	0.418950	+	0.908009i	$0.362398\pi$
$114$	−121.557	−0.0998673
$115$	35.8936	0.0291052
$116$	176.750	0.141473
$117$	728.838	0.575907
$118$	731.352	0.570562
$119$	249.464	0.192171
$120$	−102.577	−0.0780331
$121$	3763.29	2.82742
$122$	422.778	0.313742
$123$	44.6459	0.0327284
$124$	101.627	0.0736001
$125$	−125.000	−0.0894427
$126$	−460.017	−0.325251
$127$	1114.87	0.778970	0.389485	−	0.921033i	$-0.372653\pi$
$127$	1114.87	0.778970	0.389485	+	0.921033i	$0.372653\pi$
$128$	−1461.62	−1.00929
$129$	291.112	0.198690
$130$	173.500	0.117053
$131$	1558.43	1.03939	0.519697	−	0.854351i	$-0.326045\pi$
$131$	1558.43	1.03939	0.519697	+	0.854351i	$0.326045\pi$
$132$	−537.635	−0.354509
$133$	1261.40	0.822382
$134$	−1038.33	−0.669389
$135$	−304.778	−0.194304
$136$	−301.299	−0.189972
$137$	235.190	0.146669	0.0733346	−	0.997307i	$-0.476636\pi$
$137$	235.190	0.146669	0.0733346	+	0.997307i	$0.476636\pi$
$138$	10.1516	0.00626205
$139$	−2138.46	−1.30490	−0.652452	−	0.757830i	$-0.726260\pi$
$139$	−2138.46	−1.30490	−0.652452	+	0.757830i	$0.726260\pi$
$140$	477.467	0.288238
$141$	−67.0349	−0.0400380
$142$	−1113.97	−0.658325
$143$	2027.28	1.18552
$144$	−780.264	−0.451542
$145$	135.805	0.0777791
$146$	860.491	0.487772
$147$	147.774	0.0829129
$148$	436.026	0.242170
$149$	−3317.71	−1.82414	−0.912072	−	0.410030i	$-0.865518\pi$
$149$	−3317.71	−1.82414	−0.912072	+	0.410030i	$0.865518\pi$
$150$	−35.3531	−0.0192438
$151$	848.486	0.457277	0.228638	−	0.973511i	$-0.426573\pi$
$151$	848.486	0.457277	0.228638	+	0.973511i	$0.426573\pi$
$152$	−1523.50	−0.812973
$153$	−436.222	−0.230500
$154$	−1279.55	−0.669539
$155$	78.0848	0.0404640
$156$	−213.953	−0.109807
$157$	949.821	0.482828	0.241414	−	0.970422i	$-0.422389\pi$
$157$	949.821	0.482828	0.241414	+	0.970422i	$0.422389\pi$
$158$	68.2202	0.0343500
$159$	784.580	0.391329
$160$	−894.681	−0.442067
$161$	−105.343	−0.0515664
$162$	760.208	0.368689
$163$	1942.09	0.933229	0.466615	−	0.884461i	$-0.345473\pi$
$163$	1942.09	0.933229	0.466615	+	0.884461i	$0.345473\pi$
$164$	250.995	0.119509
$165$	−413.089	−0.194903
$166$	−1509.01	−0.705553
$167$	−3358.46	−1.55620	−0.778100	−	0.628141i	$-0.783816\pi$
$167$	−3358.46	−1.55620	−0.778100	+	0.628141i	$0.783816\pi$
$168$	301.051	0.138253
$169$	−1390.24	−0.632790
$170$	−103.843	−0.0468492
$171$	−2205.72	−0.986409
$172$	1636.60	0.725522
$173$	216.466	0.0951308	0.0475654	−	0.998868i	$-0.484854\pi$
$173$	216.466	0.0951308	0.0475654	+	0.998868i	$0.484854\pi$
$174$	38.4090	0.0167344
$175$	366.859	0.158468
$176$	−2170.32	−0.929513
$177$	−692.949	−0.294267
$178$	−88.9465	−0.0374541
$179$	−1142.30	−0.476981	−0.238490	−	0.971145i	$-0.576653\pi$
$179$	−1142.30	−0.476981	−0.238490	+	0.971145i	$0.576653\pi$
$180$	−834.917	−0.345728
$181$	1884.85	0.774031	0.387015	−	0.922073i	$-0.373506\pi$
$181$	1884.85	0.774031	0.387015	+	0.922073i	$0.373506\pi$
$182$	−509.199	−0.207387
$183$	−400.578	−0.161812
$184$	127.232	0.0509764
$185$	335.018	0.133141
$186$	22.0843	0.00870593
$187$	−1213.36	−0.474491
$188$	−376.863	−0.146200
$189$	894.483	0.344254
$190$	−525.072	−0.200488
$191$	2274.84	0.861790	0.430895	−	0.902402i	$-0.358198\pi$
$191$	2274.84	0.861790	0.430895	+	0.902402i	$0.358198\pi$
$192$	28.5432	0.0107288
$193$	−2905.75	−1.08373	−0.541866	−	0.840465i	$-0.682282\pi$
$193$	−2905.75	−1.08373	−0.541866	+	0.840465i	$0.682282\pi$
$194$	1188.62	0.439885
$195$	−164.389	−0.0603701
$196$	830.771	0.302759
$197$	4837.99	1.74971	0.874853	−	0.484388i	$-0.160958\pi$
$197$	4837.99	1.74971	0.874853	+	0.484388i	$0.160958\pi$
$198$	2237.47	0.803081
$199$	−2365.26	−0.842558	−0.421279	−	0.906931i	$-0.638419\pi$
$199$	−2365.26	−0.842558	−0.421279	+	0.906931i	$0.638419\pi$
$200$	−443.087	−0.156655
$201$	983.809	0.345237
$202$	−804.009	−0.280049
$203$	−398.569	−0.137803
$204$	128.055	0.0439491
$205$	192.850	0.0657037
$206$	−2110.67	−0.713869
$207$	184.207	0.0618515
$208$	−863.685	−0.287912
$209$	−6135.28	−2.03056
$210$	103.757	0.0340948
$211$	594.164	0.193858	0.0969288	−	0.995291i	$-0.469098\pi$
$211$	594.164	0.193858	0.0969288	+	0.995291i	$0.469098\pi$
$212$	4410.83	1.42895
$213$	1055.48	0.339530
$214$	−1103.18	−0.352392
$215$	1257.47	0.398879
$216$	−1080.35	−0.340316
$217$	−229.169	−0.0716912
$218$	1018.34	0.316379
$219$	−815.307	−0.251568
$220$	−2322.34	−0.711692
$221$	−482.860	−0.146971
$222$	94.7516	0.0286455
$223$	3040.15	0.912931	0.456466	−	0.889741i	$-0.349115\pi$
$223$	3040.15	0.912931	0.456466	+	0.889741i	$0.349115\pi$
$224$	2625.77	0.783223
$225$	−641.503	−0.190075
$226$	1229.61	0.361912
$227$	−459.136	−0.134246	−0.0671232	−	0.997745i	$-0.521382\pi$
$227$	−459.136	−0.134246	−0.0671232	+	0.997745i	$0.521382\pi$
$228$	647.498	0.188077
$229$	2153.49	0.621427	0.310713	−	0.950504i	$-0.399432\pi$
$229$	2153.49	0.621427	0.310713	+	0.950504i	$0.399432\pi$
$230$	43.8504	0.0125713
$231$	1212.36	0.345314
$232$	481.387	0.136227
$233$	5456.45	1.53418	0.767090	−	0.641540i	$-0.221704\pi$
$233$	5456.45	1.53418	0.767090	+	0.641540i	$0.221704\pi$
$234$	890.405	0.248750
$235$	−289.561	−0.0803781
$236$	−3895.69	−1.07452
$237$	−64.6380	−0.0177160
$238$	304.764	0.0830040
$239$	−6073.58	−1.64380	−0.821898	−	0.569634i	$-0.807085\pi$
$239$	−6073.58	−1.64380	−0.821898	+	0.569634i	$0.807085\pi$
$240$	175.989	0.0473334
$241$	7191.69	1.92223	0.961115	−	0.276148i	$-0.0890581\pi$
$241$	7191.69	1.92223	0.961115	+	0.276148i	$0.0890581\pi$
$242$	4597.53	1.22124
$243$	−2366.09	−0.624629
$244$	−2252.01	−0.590861
$245$	638.317	0.166451
$246$	54.5429	0.0141363
$247$	−2441.54	−0.628955
$248$	276.787	0.0708710
$249$	1429.77	0.363888
$250$	−152.710	−0.0386328
$251$	−3778.49	−0.950185	−0.475092	−	0.879936i	$-0.657585\pi$
$251$	−3778.49	−0.950185	−0.475092	+	0.879936i	$0.657585\pi$
$252$	2450.37	0.612536
$253$	512.376	0.127323
$254$	1362.02	0.336459
$255$	98.3899	0.0241624
$256$	−1588.35	−0.387781
$257$	−5002.13	−1.21410	−0.607051	−	0.794663i	$-0.707648\pi$
$257$	−5002.13	−1.21410	−0.607051	+	0.794663i	$0.707648\pi$
$258$	355.645	0.0858197
$259$	−983.235	−0.235889
$260$	−924.181	−0.220443
$261$	696.954	0.165289
$262$	1903.90	0.448943
$263$	4745.06	1.11252	0.556260	−	0.831008i	$-0.312236\pi$
$263$	4745.06	1.11252	0.556260	+	0.831008i	$0.312236\pi$
$264$	−1464.27	−0.341363
$265$	3389.03	0.785610
$266$	1541.02	0.355210
$267$	84.2760	0.0193169
$268$	5530.88	1.26064
$269$	7844.29	1.77797	0.888987	−	0.457932i	$-0.151410\pi$
$269$	7844.29	1.77797	0.888987	+	0.457932i	$0.151410\pi$
$270$	−372.340	−0.0839256
$271$	3730.24	0.836148	0.418074	−	0.908413i	$-0.362705\pi$
$271$	3730.24	0.836148	0.418074	+	0.908413i	$0.362705\pi$
$272$	516.930	0.115233
$273$	482.462	0.106959
$274$	287.327	0.0633505
$275$	−1784.36	−0.391276
$276$	−54.0745	−0.0117931
$277$	−1888.55	−0.409646	−0.204823	−	0.978799i	$-0.565662\pi$
$277$	−1888.55	−0.409646	−0.204823	+	0.978799i	$0.565662\pi$
$278$	−2612.51	−0.563625
$279$	400.733	0.0859902
$280$	1300.40	0.277550
$281$	−3340.77	−0.709231	−0.354615	−	0.935012i	$-0.615388\pi$
$281$	−3340.77	−0.709231	−0.354615	+	0.935012i	$0.615388\pi$
$282$	−81.8950	−0.0172935
$283$	−7783.21	−1.63485	−0.817427	−	0.576032i	$-0.804600\pi$
$283$	−7783.21	−1.63485	−0.817427	+	0.576032i	$0.804600\pi$
$284$	5933.77	1.23981
$285$	497.501	0.103401
$286$	2476.68	0.512061
$287$	−565.991	−0.116409
$288$	−4591.53	−0.939439
$289$	289.000	0.0588235
$290$	165.910	0.0335950
$291$	−1126.20	−0.226870
$292$	−4583.57	−0.918608
$293$	−8514.61	−1.69771	−0.848855	−	0.528626i	$-0.822707\pi$
$293$	−8514.61	−1.69771	−0.848855	+	0.528626i	$0.822707\pi$
$294$	180.532	0.0358124
$295$	−2993.23	−0.590754
$296$	1187.54	0.233190
$297$	−4350.66	−0.850003
$298$	−4053.17	−0.787899
$299$	203.901	0.0394378
$300$	188.315	0.0362413
$301$	−3690.52	−0.706705
$302$	1036.58	0.197511
$303$	761.792	0.144435
$304$	2613.82	0.493134
$305$	−1730.32	−0.324845
$306$	−532.923	−0.0995594
$307$	−8954.07	−1.66461	−0.832306	−	0.554317i	$-0.812980\pi$
$307$	−8954.07	−1.66461	−0.832306	+	0.554317i	$0.812980\pi$
$308$	6815.77	1.26092
$309$	1999.84	0.368177
$310$	95.3945	0.0174776
$311$	−3842.82	−0.700664	−0.350332	−	0.936626i	$-0.613931\pi$
$311$	−3842.82	−0.700664	−0.350332	+	0.936626i	$0.613931\pi$
$312$	−582.711	−0.105736
$313$	−4432.84	−0.800507	−0.400254	−	0.916404i	$-0.631078\pi$
$313$	−4432.84	−0.800507	−0.400254	+	0.916404i	$0.631078\pi$
$314$	1160.37	0.208547
$315$	1882.73	0.336761
$316$	−363.388	−0.0646904
$317$	7694.21	1.36325	0.681625	−	0.731702i	$-0.261274\pi$
$317$	7694.21	1.36325	0.681625	+	0.731702i	$0.261274\pi$
$318$	958.504	0.169026
$319$	1938.59	0.340252
$320$	123.294	0.0215386
$321$	1045.25	0.181746
$322$	−128.695	−0.0222730
$323$	1461.31	0.251731
$324$	−4049.40	−0.694341
$325$	−710.088	−0.121196
$326$	2372.61	0.403088
$327$	−964.866	−0.163172
$328$	683.596	0.115077
$329$	849.823	0.142408
$330$	−504.661	−0.0841839
$331$	7276.70	1.20835	0.604175	−	0.796852i	$-0.293503\pi$
$331$	7276.70	1.20835	0.604175	+	0.796852i	$0.293503\pi$
$332$	8038.03	1.32875
$333$	1719.32	0.282938
$334$	−4102.95	−0.672166
$335$	4249.61	0.693078
$336$	−516.503	−0.0838618
$337$	−3300.91	−0.533566	−0.266783	−	0.963757i	$-0.585961\pi$
$337$	−3300.91	−0.533566	−0.266783	+	0.963757i	$0.585961\pi$
$338$	−1698.42	−0.273320
$339$	−1165.04	−0.186656
$340$	553.138	0.0882297
$341$	1114.65	0.177014
$342$	−2694.68	−0.426058
$343$	−6906.68	−1.08725
$344$	4457.36	0.698619
$345$	−41.5478	−0.00648365
$346$	264.452	0.0410897
$347$	−7998.41	−1.23740	−0.618699	−	0.785628i	$-0.712340\pi$
$347$	−7998.41	−1.23740	−0.618699	+	0.785628i	$0.712340\pi$
$348$	−204.593	−0.0315154
$349$	−4378.96	−0.671634	−0.335817	−	0.941927i	$-0.609012\pi$
$349$	−4378.96	−0.671634	−0.335817	+	0.941927i	$0.609012\pi$
$350$	448.183	0.0684468
$351$	−1731.35	−0.263284
$352$	−12771.5	−1.93387
$353$	1508.37	0.227430	0.113715	−	0.993513i	$-0.463725\pi$
$353$	1508.37	0.227430	0.113715	+	0.993513i	$0.463725\pi$
$354$	−846.560	−0.127102
$355$	4559.18	0.681623
$356$	473.791	0.0705362
$357$	−288.761	−0.0428092
$358$	−1395.52	−0.206021
$359$	3436.61	0.505229	0.252614	−	0.967567i	$-0.418710\pi$
$359$	3436.61	0.505229	0.252614	+	0.967567i	$0.418710\pi$
$360$	−2273.94	−0.332908
$361$	529.984	0.0772684
$362$	2302.67	0.334326
$363$	−4356.11	−0.629853
$364$	2712.35	0.390565
$365$	−3521.76	−0.505034
$366$	−489.377	−0.0698911
$367$	10656.6	1.51573	0.757863	−	0.652413i	$-0.226243\pi$
$367$	10656.6	1.51573	0.757863	+	0.652413i	$0.226243\pi$
$368$	−218.288	−0.0309213
$369$	989.713	0.139627
$370$	409.284	0.0575072
$371$	−9946.37	−1.39189
$372$	−117.637	−0.0163956
$373$	−9062.32	−1.25799	−0.628994	−	0.777410i	$-0.716533\pi$
$373$	−9062.32	−1.25799	−0.628994	+	0.777410i	$0.716533\pi$
$374$	−1482.34	−0.204946
$375$	144.691	0.0199248
$376$	−1026.40	−0.140779
$377$	771.467	0.105391
$378$	1092.77	0.148693
$379$	1080.44	0.146434	0.0732169	−	0.997316i	$-0.476673\pi$
$379$	1080.44	0.146434	0.0732169	+	0.997316i	$0.476673\pi$
$380$	2796.90	0.377573
$381$	−1290.50	−0.173528
$382$	2779.13	0.372232
$383$	6920.65	0.923313	0.461656	−	0.887059i	$-0.347255\pi$
$383$	6920.65	0.923313	0.461656	+	0.887059i	$0.347255\pi$
$384$	1691.86	0.224837
$385$	5236.86	0.693234
$386$	−3549.88	−0.468094
$387$	6453.39	0.847659
$388$	−6331.40	−0.828423
$389$	9664.52	1.25967	0.629833	−	0.776730i	$-0.283123\pi$
$389$	9664.52	1.25967	0.629833	+	0.776730i	$0.283123\pi$
$390$	−200.831	−0.0260755
$391$	−122.038	−0.0157845
$392$	2262.64	0.291532
$393$	−1803.92	−0.231542
$394$	5910.46	0.755748
$395$	−279.207	−0.0355656
$396$	−11918.3	−1.51242
$397$	−7464.80	−0.943697	−0.471848	−	0.881680i	$-0.656413\pi$
$397$	−7464.80	−0.943697	−0.471848	+	0.881680i	$0.656413\pi$
$398$	−2889.59	−0.363924
$399$	−1460.10	−0.183199
$400$	760.191	0.0950239
$401$	2948.70	0.367209	0.183605	−	0.983000i	$-0.441223\pi$
$401$	2948.70	0.367209	0.183605	+	0.983000i	$0.441223\pi$
$402$	1201.90	0.149117
$403$	443.577	0.0548291
$404$	4282.71	0.527408
$405$	−3111.33	−0.381736
$406$	−486.923	−0.0595212
$407$	4782.34	0.582436
$408$	348.762	0.0423194
$409$	5546.04	0.670499	0.335250	−	0.942129i	$-0.391179\pi$
$409$	5546.04	0.670499	0.335250	+	0.942129i	$0.391179\pi$
$410$	235.601	0.0283793
$411$	−272.239	−0.0326730
$412$	11242.9	1.34441
$413$	8784.73	1.04665
$414$	225.041	0.0267154
$415$	6175.97	0.730521
$416$	−5082.42	−0.599005
$417$	2475.33	0.290689
$418$	−7495.33	−0.877054
$419$	8245.75	0.961411	0.480706	−	0.876882i	$-0.340381\pi$
$419$	8245.75	0.961411	0.480706	+	0.876882i	$0.340381\pi$
$420$	−552.681	−0.0642097
$421$	−9972.70	−1.15449	−0.577244	−	0.816571i	$-0.695872\pi$
$421$	−9972.70	−1.15449	−0.577244	+	0.816571i	$0.695872\pi$
$422$	725.877	0.0837325
$423$	−1486.03	−0.170812
$424$	12013.1	1.37596
$425$	425.000	0.0485071
$426$	1289.45	0.146653
$427$	5078.25	0.575536
$428$	5876.30	0.663649
$429$	−2346.64	−0.264095
$430$	1536.23	0.172287
$431$	10506.0	1.17415	0.587073	−	0.809534i	$-0.300280\pi$
$431$	10506.0	1.17415	0.587073	+	0.809534i	$0.300280\pi$
$432$	1853.52	0.206429
$433$	13866.9	1.53903	0.769514	−	0.638631i	$-0.220499\pi$
$433$	13866.9	1.53903	0.769514	+	0.638631i	$0.220499\pi$
$434$	−279.970	−0.0309654
$435$	−157.198	−0.0173266
$436$	−5424.38	−0.595827
$437$	−617.077	−0.0675487
$438$	−996.043	−0.108659
$439$	270.717	0.0294319	0.0147160	−	0.999892i	$-0.495316\pi$
$439$	270.717	0.0294319	0.0147160	+	0.999892i	$0.495316\pi$
$440$	−6325.01	−0.685302
$441$	3275.86	0.353726
$442$	−589.899	−0.0634811
$443$	−6809.29	−0.730291	−0.365146	−	0.930950i	$-0.618981\pi$
$443$	−6809.29	−0.730291	−0.365146	+	0.930950i	$0.618981\pi$
$444$	−504.713	−0.0539473
$445$	364.034	0.0387795
$446$	3714.09	0.394321
$447$	3840.34	0.406358
$448$	−361.852	−0.0381605
$449$	−9130.09	−0.959633	−0.479817	−	0.877369i	$-0.659297\pi$
$449$	−9130.09	−0.959633	−0.479817	+	0.877369i	$0.659297\pi$
$450$	−783.710	−0.0820988
$451$	2752.91	0.287427
$452$	−6549.74	−0.681579
$453$	−982.146	−0.101866
$454$	−560.916	−0.0579848
$455$	2084.02	0.214726
$456$	1763.49	0.181103
$457$	−2524.80	−0.258436	−0.129218	−	0.991616i	$-0.541247\pi$
$457$	−2524.80	−0.258436	−0.129218	+	0.991616i	$0.541247\pi$
$458$	2630.87	0.268412
$459$	1036.24	0.105376
$460$	−233.578	−0.0236752
$461$	11102.1	1.12164	0.560819	−	0.827938i	$-0.310486\pi$
$461$	11102.1	1.12164	0.560819	+	0.827938i	$0.310486\pi$
$462$	1481.11	0.149151
$463$	−3045.92	−0.305736	−0.152868	−	0.988247i	$-0.548851\pi$
$463$	−3045.92	−0.305736	−0.152868	+	0.988247i	$0.548851\pi$
$464$	−825.901	−0.0826325
$465$	−90.3854	−0.00901402
$466$	6666.02	0.662655
$467$	−14515.5	−1.43832	−0.719161	−	0.694844i	$-0.755473\pi$
$467$	−14515.5	−1.43832	−0.719161	+	0.694844i	$0.755473\pi$
$468$	−4742.92	−0.468465
$469$	−12472.1	−1.22795
$470$	−353.750	−0.0347176
$471$	−1099.44	−0.107558
$472$	−10610.1	−1.03468
$473$	17950.3	1.74493
$474$	−78.9668	−0.00765203
$475$	2148.98	0.207583
$476$	−1623.39	−0.156319
$477$	17392.6	1.66950
$478$	−7419.96	−0.710002
$479$	−5511.93	−0.525775	−0.262888	−	0.964826i	$-0.584675\pi$
$479$	−5511.93	−0.525775	−0.262888	+	0.964826i	$0.584675\pi$
$480$	1035.62	0.0984777
$481$	1903.14	0.180407
$482$	8785.92	0.830265
$483$	121.937	0.0114873
$484$	−24489.6	−2.29993
$485$	−4864.69	−0.455452
$486$	−2890.60	−0.269795
$487$	−10132.2	−0.942777	−0.471388	−	0.881926i	$-0.656247\pi$
$487$	−10132.2	−0.942777	−0.471388	+	0.881926i	$0.656247\pi$
$488$	−6133.45	−0.568951
$489$	−2248.03	−0.207892
$490$	779.818	0.0718951
$491$	−5965.90	−0.548345	−0.274172	−	0.961681i	$-0.588404\pi$
$491$	−5965.90	−0.548345	−0.274172	+	0.961681i	$0.588404\pi$
$492$	−290.534	−0.0266225
$493$	−461.736	−0.0421817
$494$	−2982.78	−0.271663
$495$	−9157.37	−0.831501
$496$	−474.875	−0.0429890
$497$	−13380.6	−1.20765
$498$	1746.72	0.157173
$499$	19690.2	1.76644	0.883220	−	0.468959i	$-0.155371\pi$
$499$	19690.2	1.76644	0.883220	+	0.468959i	$0.155371\pi$
$500$	813.438	0.0727561
$501$	3887.51	0.346669
$502$	−4616.10	−0.410412
$503$	−18780.3	−1.66476	−0.832378	−	0.554209i	$-0.813021\pi$
$503$	−18780.3	−1.66476	−0.832378	+	0.554209i	$0.813021\pi$
$504$	6673.70	0.589822
$505$	3290.60	0.289960
$506$	625.958	0.0549945
$507$	1609.24	0.140964
$508$	−7255.05	−0.633644
$509$	−4447.70	−0.387310	−0.193655	−	0.981070i	$-0.562034\pi$
$509$	−4447.70	−0.387310	−0.193655	+	0.981070i	$0.562034\pi$
$510$	120.201	0.0104364
$511$	10335.9	0.894782
$512$	9752.47	0.841801
$513$	5239.69	0.450951
$514$	−6110.99	−0.524405
$515$	8638.39	0.739132
$516$	−1894.41	−0.161622
$517$	−4133.44	−0.351622
$518$	−1201.20	−0.101887
$519$	−250.566	−0.0211920
$520$	−2517.05	−0.212269
$521$	−9380.60	−0.788813	−0.394407	−	0.918936i	$-0.629050\pi$
$521$	−9380.60	−0.788813	−0.394407	+	0.918936i	$0.629050\pi$
$522$	851.453	0.0713929
$523$	−2039.00	−0.170477	−0.0852384	−	0.996361i	$-0.527165\pi$
$523$	−2039.00	−0.170477	−0.0852384	+	0.996361i	$0.527165\pi$
$524$	−10141.5	−0.845482
$525$	−424.649	−0.0353014
$526$	5796.93	0.480529
$527$	−265.488	−0.0219447
$528$	2512.21	0.207064
$529$	−12115.5	−0.995764
$530$	4140.31	0.339327
$531$	−15361.3	−1.25541
$532$	−8208.54	−0.668957
$533$	1095.53	0.0890291
$534$	102.958	0.00834351
$535$	4515.02	0.364862
$536$	15063.6	1.21390
$537$	1322.24	0.106255
$538$	9583.20	0.767957
$539$	9111.89	0.728158
$540$	1983.34	0.158055
$541$	−10362.5	−0.823510	−0.411755	−	0.911295i	$-0.635084\pi$
$541$	−10362.5	−0.823510	−0.411755	+	0.911295i	$0.635084\pi$
$542$	4557.15	0.361156
$543$	−2181.76	−0.172428
$544$	3041.92	0.239745
$545$	−4167.79	−0.327575
$546$	589.412	0.0461988
$547$	1466.22	0.114609	0.0573044	−	0.998357i	$-0.481749\pi$
$547$	1466.22	0.114609	0.0573044	+	0.998357i	$0.481749\pi$
$548$	−1530.50	−0.119306
$549$	−8880.03	−0.690328
$550$	−2179.91	−0.169003
$551$	−2334.73	−0.180514
$552$	−147.275	−0.0113558
$553$	819.436	0.0630126
$554$	−2307.20	−0.176938
$555$	−387.793	−0.0296593
$556$	13916.0	1.06146
$557$	2400.26	0.182590	0.0912949	−	0.995824i	$-0.470899\pi$
$557$	2400.26	0.182590	0.0912949	+	0.995824i	$0.470899\pi$
$558$	489.567	0.0371416
$559$	7143.34	0.540485
$560$	−2231.06	−0.168356
$561$	1404.50	0.105701
$562$	−4081.35	−0.306337
$563$	−14902.0	−1.11553	−0.557766	−	0.829998i	$-0.688341\pi$
$563$	−14902.0	−1.11553	−0.557766	+	0.829998i	$0.688341\pi$
$564$	436.230	0.0325684
$565$	−5032.45	−0.374720
$566$	−9508.57	−0.706140
$567$	9131.34	0.676332
$568$	16160.9	1.19383
$569$	−9568.31	−0.704964	−0.352482	−	0.935819i	$-0.614662\pi$
$569$	−9568.31	−0.704964	−0.352482	+	0.935819i	$0.614662\pi$
$570$	607.786	0.0446620
$571$	10645.8	0.780235	0.390117	−	0.920765i	$-0.372434\pi$
$571$	10645.8	0.780235	0.390117	+	0.920765i	$0.372434\pi$
$572$	−13192.5	−0.964350
$573$	−2633.20	−0.191978
$574$	−691.458	−0.0502803
$575$	−179.468	−0.0130162
$576$	632.748	0.0457717
$577$	335.528	0.0242084	0.0121042	−	0.999927i	$-0.496147\pi$
$577$	335.528	0.0242084	0.0121042	+	0.999927i	$0.496147\pi$
$578$	353.065	0.0254075
$579$	3363.48	0.241419
$580$	−883.750	−0.0632685
$581$	−18125.7	−1.29428
$582$	−1375.86	−0.0979916
$583$	48378.0	3.43673
$584$	−12483.6	−0.884545
$585$	−3644.19	−0.257553
$586$	−10402.1	−0.733289
$587$	12512.7	0.879817	0.439908	−	0.898043i	$-0.355011\pi$
$587$	12512.7	0.879817	0.439908	+	0.898043i	$0.355011\pi$
$588$	−961.640	−0.0674445
$589$	−1342.42	−0.0939109
$590$	−3656.76	−0.255163
$591$	−5600.10	−0.389776
$592$	−2037.42	−0.141449
$593$	−11061.3	−0.765994	−0.382997	−	0.923749i	$-0.625108\pi$
$593$	−11061.3	−0.765994	−0.382997	+	0.923749i	$0.625108\pi$
$594$	−5315.10	−0.367140
$595$	−1247.32	−0.0859414
$596$	21590.0	1.48383
$597$	2737.86	0.187694
$598$	249.101	0.0170343
$599$	4672.09	0.318692	0.159346	−	0.987223i	$-0.449062\pi$
$599$	4672.09	0.318692	0.159346	+	0.987223i	$0.449062\pi$
$600$	512.886	0.0348975
$601$	26076.4	1.76985	0.884924	−	0.465736i	$-0.154210\pi$
$601$	26076.4	1.76985	0.884924	+	0.465736i	$0.154210\pi$
$602$	−4508.63	−0.305246
$603$	21809.1	1.47286
$604$	−5521.53	−0.371966
$605$	−18816.5	−1.26446
$606$	930.663	0.0623855
$607$	11437.6	0.764810	0.382405	−	0.923995i	$-0.375096\pi$
$607$	11437.6	0.764810	0.382405	+	0.923995i	$0.375096\pi$
$608$	15381.2	1.02597
$609$	461.355	0.0306980
$610$	−2113.89	−0.140310
$611$	−1644.91	−0.108913
$612$	2838.72	0.187497
$613$	28634.7	1.88670	0.943348	−	0.331806i	$-0.107658\pi$
$613$	28634.7	1.88670	0.943348	+	0.331806i	$0.107658\pi$
$614$	−10939.0	−0.718993
$615$	−223.230	−0.0146366
$616$	18563.1	1.21417
$617$	−3394.42	−0.221482	−0.110741	−	0.993849i	$-0.535322\pi$
$617$	−3394.42	−0.221482	−0.110741	+	0.993849i	$0.535322\pi$
$618$	2443.15	0.159026
$619$	10197.3	0.662139	0.331069	−	0.943606i	$-0.392591\pi$
$619$	10197.3	0.662139	0.331069	+	0.943606i	$0.392591\pi$
$620$	−508.137	−0.0329150
$621$	−437.583	−0.0282763
$622$	−4694.69	−0.302637
$623$	−1068.39	−0.0687067
$624$	999.739	0.0641372
$625$	625.000	0.0400000
$626$	−5415.50	−0.345762
$627$	7101.76	0.452339
$628$	−6180.96	−0.392751
$629$	−1139.06	−0.0722057
$630$	2300.09	0.145457
$631$	13195.9	0.832522	0.416261	−	0.909245i	$-0.363340\pi$
$631$	13195.9	0.832522	0.416261	+	0.909245i	$0.363340\pi$
$632$	−989.705	−0.0622917
$633$	−687.761	−0.0431849
$634$	9399.85	0.588826
$635$	−5574.37	−0.348366
$636$	−5105.66	−0.318322
$637$	3626.09	0.225543
$638$	2368.34	0.146965
$639$	23397.8	1.44852
$640$	7308.08	0.451370
$641$	−10280.5	−0.633469	−0.316734	−	0.948514i	$-0.602586\pi$
$641$	−10280.5	−0.633469	−0.316734	+	0.948514i	$0.602586\pi$
$642$	1276.96	0.0785010
$643$	−16401.3	−1.00592	−0.502959	−	0.864311i	$-0.667755\pi$
$643$	−16401.3	−1.00592	−0.502959	+	0.864311i	$0.667755\pi$
$644$	685.520	0.0419461
$645$	−1455.56	−0.0888568
$646$	1785.24	0.108730
$647$	18960.3	1.15210	0.576048	−	0.817416i	$-0.304594\pi$
$647$	18960.3	1.15210	0.576048	+	0.817416i	$0.304594\pi$
$648$	−11028.7	−0.668594
$649$	−42727.9	−2.58431
$650$	−867.499	−0.0523479
$651$	265.269	0.0159704
$652$	−12638.2	−0.759124
$653$	1539.55	0.0922623	0.0461311	−	0.998935i	$-0.485311\pi$
$653$	1539.55	0.0922623	0.0461311	+	0.998935i	$0.485311\pi$
$654$	−1178.76	−0.0704785
$655$	−7792.14	−0.464831
$656$	−1172.83	−0.0698036
$657$	−18073.8	−1.07325
$658$	1038.21	0.0615100
$659$	−9245.33	−0.546505	−0.273253	−	0.961942i	$-0.588099\pi$
$659$	−9245.33	−0.546505	−0.273253	+	0.961942i	$0.588099\pi$
$660$	2688.18	0.158541
$661$	28073.9	1.65197	0.825983	−	0.563696i	$-0.190621\pi$
$661$	28073.9	1.65197	0.825983	+	0.563696i	$0.190621\pi$
$662$	8889.78	0.521920
$663$	558.924	0.0327403
$664$	21891.9	1.27948
$665$	−6306.98	−0.367781
$666$	2100.46	0.122209
$667$	194.981	0.0113189
$668$	21855.2	1.26587
$669$	−3519.06	−0.203370
$670$	5191.66	0.299360
$671$	−24700.0	−1.42106
$672$	−3039.41	−0.174476
$673$	−19690.2	−1.12779	−0.563893	−	0.825848i	$-0.690697\pi$
$673$	−19690.2	−1.12779	−0.563893	+	0.825848i	$0.690697\pi$
$674$	−4032.64	−0.230462
$675$	1523.89	0.0868956
$676$	9046.99	0.514735
$677$	−12023.4	−0.682566	−0.341283	−	0.939961i	$-0.610861\pi$
$677$	−12023.4	−0.682566	−0.341283	+	0.939961i	$0.610861\pi$
$678$	−1423.30	−0.0806219
$679$	14277.2	0.806937
$680$	1506.50	0.0849581
$681$	531.463	0.0299056
$682$	1361.74	0.0764572
$683$	−6517.76	−0.365147	−0.182573	−	0.983192i	$-0.558443\pi$
$683$	−6517.76	−0.365147	−0.182573	+	0.983192i	$0.558443\pi$
$684$	14353.8	0.802383
$685$	−1175.95	−0.0655924
$686$	−8437.74	−0.469613
$687$	−2492.73	−0.138433
$688$	−7647.36	−0.423769
$689$	19252.1	1.06451
$690$	−50.7580	−0.00280047
$691$	−25926.3	−1.42733	−0.713664	−	0.700488i	$-0.752966\pi$
$691$	−25926.3	−1.42733	−0.713664	+	0.700488i	$0.752966\pi$
$692$	−1408.66	−0.0773830
$693$	26875.7	1.47319
$694$	−9771.47	−0.534467
$695$	10692.3	0.583571
$696$	−557.219	−0.0303467
$697$	−655.691	−0.0356328
$698$	−5349.68	−0.290098
$699$	−6315.99	−0.341764
$700$	−2387.33	−0.128904
$701$	1635.68	0.0881294	0.0440647	−	0.999029i	$-0.485969\pi$
$701$	1635.68	0.0881294	0.0440647	+	0.999029i	$0.485969\pi$
$702$	−2115.16	−0.113720
$703$	−5759.58	−0.308999
$704$	1760.00	0.0942225
$705$	335.175	0.0179055
$706$	1842.75	0.0982333
$707$	−9657.47	−0.513729
$708$	4509.37	0.239368
$709$	−26057.0	−1.38024	−0.690122	−	0.723693i	$-0.742443\pi$
$709$	−26057.0	−1.38024	−0.690122	+	0.723693i	$0.742443\pi$
$710$	5569.84	0.294412
$711$	−1432.90	−0.0755807
$712$	1290.39	0.0679206
$713$	112.110	0.00588856
$714$	−352.773	−0.0184905
$715$	−10136.4	−0.530182
$716$	7433.53	0.387994
$717$	7030.34	0.366183
$718$	4198.42	0.218223
$719$	106.968	0.00554828	0.00277414	−	0.999996i	$-0.499117\pi$
$719$	106.968	0.00554828	0.00277414	+	0.999996i	$0.499117\pi$
$720$	3901.32	0.201936
$721$	−25352.6	−1.30954
$722$	647.469	0.0333744
$723$	−8324.58	−0.428208
$724$	−12265.6	−0.629626
$725$	−679.024	−0.0347839
$726$	−5321.77	−0.272051
$727$	−17019.9	−0.868273	−0.434137	−	0.900847i	$-0.642947\pi$
$727$	−17019.9	−0.868273	−0.434137	+	0.900847i	$0.642947\pi$
$728$	7387.21	0.376083
$729$	−14062.4	−0.714442
$730$	−4302.46	−0.218138
$731$	−4275.41	−0.216322
$732$	2606.76	0.131624
$733$	−27064.0	−1.36375	−0.681877	−	0.731467i	$-0.738836\pi$
$733$	−27064.0	−1.36375	−0.681877	+	0.731467i	$0.738836\pi$
$734$	13019.0	0.654685
$735$	−738.870	−0.0370798
$736$	−1284.53	−0.0643322
$737$	60662.6	3.03194
$738$	1209.11	0.0603089
$739$	−13177.0	−0.655919	−0.327960	−	0.944692i	$-0.606361\pi$
$739$	−13177.0	−0.655919	−0.327960	+	0.944692i	$0.606361\pi$
$740$	−2180.13	−0.108302
$741$	2826.16	0.140110
$742$	−12151.3	−0.601195
$743$	1164.23	0.0574854	0.0287427	−	0.999587i	$-0.490850\pi$
$743$	1164.23	0.0574854	0.0287427	+	0.999587i	$0.490850\pi$
$744$	−320.389	−0.0157877
$745$	16588.6	0.815782
$746$	−11071.2	−0.543360
$747$	31695.2	1.55243
$748$	7895.97	0.385969
$749$	−13251.0	−0.646437
$750$	176.766	0.00860610
$751$	5632.54	0.273681	0.136840	−	0.990593i	$-0.456305\pi$
$751$	5632.54	0.273681	0.136840	+	0.990593i	$0.456305\pi$
$752$	1760.97	0.0853936
$753$	4373.71	0.211669
$754$	942.484	0.0455215
$755$	−4242.43	−0.204500
$756$	−5820.85	−0.280030
$757$	−40260.8	−1.93303	−0.966515	−	0.256611i	$-0.917394\pi$
$757$	−40260.8	−1.93303	−0.966515	+	0.256611i	$0.917394\pi$
$758$	1319.95	0.0632489
$759$	−593.089	−0.0283634
$760$	7617.49	0.363573
$761$	−22060.7	−1.05085	−0.525427	−	0.850839i	$-0.676094\pi$
$761$	−22060.7	−1.05085	−0.525427	+	0.850839i	$0.676094\pi$
$762$	−1576.57	−0.0749517
$763$	12231.9	0.580374
$764$	−14803.6	−0.701013
$765$	2181.11	0.103083
$766$	8454.80	0.398805
$767$	−17003.6	−0.800477
$768$	1838.56	0.0863846
$769$	8923.07	0.418432	0.209216	−	0.977869i	$-0.432909\pi$
$769$	8923.07	0.418432	0.209216	+	0.977869i	$0.432909\pi$
$770$	6397.75	0.299427
$771$	5790.10	0.270461
$772$	18909.2	0.881548
$773$	3076.31	0.143140	0.0715699	−	0.997436i	$-0.477199\pi$
$773$	3076.31	0.143140	0.0715699	+	0.997436i	$0.477199\pi$
$774$	7883.96	0.366128
$775$	−390.424	−0.0180961
$776$	−17243.9	−0.797705
$777$	1138.12	0.0525481
$778$	11806.9	0.544086
$779$	−3315.45	−0.152488
$780$	1069.76	0.0491073
$781$	65081.6	2.98182
$782$	−149.091	−0.00681777
$783$	−1655.61	−0.0755642
$784$	−3881.95	−0.176838
$785$	−4749.11	−0.215927
$786$	−2203.81	−0.100009
$787$	−20627.3	−0.934286	−0.467143	−	0.884182i	$-0.654717\pi$
$787$	−20627.3	−0.934286	−0.467143	+	0.884182i	$0.654717\pi$
$788$	−31483.2	−1.42328
$789$	−5492.54	−0.247832
$790$	−341.101	−0.0153618
$791$	14769.6	0.663902
$792$	−32460.1	−1.45634
$793$	−9829.42	−0.440167
$794$	−9119.58	−0.407609
$795$	−3922.90	−0.175007
$796$	15392.0	0.685369
$797$	13976.5	0.621172	0.310586	−	0.950545i	$-0.399475\pi$
$797$	13976.5	0.621172	0.310586	+	0.950545i	$0.399475\pi$
$798$	−1783.77	−0.0791289
$799$	984.506	0.0435911
$800$	4473.41	0.197699
$801$	1868.23	0.0824105
$802$	3602.36	0.158608
$803$	−50272.6	−2.20932
$804$	−6402.14	−0.280829
$805$	526.715	0.0230612
$806$	541.908	0.0236822
$807$	−9079.99	−0.396073
$808$	11664.2	0.507852
$809$	5425.25	0.235774	0.117887	−	0.993027i	$-0.462388\pi$
$809$	5425.25	0.235774	0.117887	+	0.993027i	$0.462388\pi$
$810$	−3801.04	−0.164883
$811$	−36180.8	−1.56656	−0.783279	−	0.621670i	$-0.786455\pi$
$811$	−36180.8	−1.56656	−0.783279	+	0.621670i	$0.786455\pi$
$812$	2593.69	0.112095
$813$	−4317.86	−0.186266
$814$	5842.47	0.251571
$815$	−9710.46	−0.417353
$816$	−598.361	−0.0256701
$817$	−21618.3	−0.925738
$818$	6775.48	0.289608
$819$	10695.2	0.456314
$820$	−1254.97	−0.0534459
$821$	−17712.7	−0.752955	−0.376478	−	0.926426i	$-0.622865\pi$
$821$	−17712.7	−0.752955	−0.376478	+	0.926426i	$0.622865\pi$
$822$	−332.589	−0.0141124
$823$	−15244.9	−0.645691	−0.322845	−	0.946452i	$-0.604639\pi$
$823$	−15244.9	−0.645691	−0.322845	+	0.946452i	$0.604639\pi$
$824$	30620.5	1.29456
$825$	2065.44	0.0871631
$826$	10732.1	0.452080
$827$	4079.89	0.171550	0.0857750	−	0.996315i	$-0.472663\pi$
$827$	4079.89	0.171550	0.0857750	+	0.996315i	$0.472663\pi$
$828$	−1198.73	−0.0503124
$829$	13959.9	0.584857	0.292429	−	0.956287i	$-0.405537\pi$
$829$	13959.9	0.584857	0.292429	+	0.956287i	$0.405537\pi$
$830$	7545.04	0.315533
$831$	2186.05	0.0912554
$832$	700.397	0.0291850
$833$	−2170.28	−0.0902710
$834$	3024.05	0.125557
$835$	16792.3	0.695954
$836$	39925.3	1.65173
$837$	−951.941	−0.0393117
$838$	10073.6	0.415261
$839$	45111.6	1.85629	0.928144	−	0.372221i	$-0.121404\pi$
$839$	45111.6	1.85629	0.928144	+	0.372221i	$0.121404\pi$
$840$	−1505.25	−0.0618288
$841$	−23651.3	−0.969752
$842$	−12183.4	−0.498656
$843$	3867.04	0.157993
$844$	−3866.52	−0.157691
$845$	6951.20	0.282992
$846$	−1815.45	−0.0737784
$847$	55223.8	2.24028
$848$	−20610.5	−0.834632
$849$	9009.28	0.364191
$850$	519.213	0.0209516
$851$	481.000	0.0193754
$852$	−6868.51	−0.276187
$853$	−37653.2	−1.51140	−0.755698	−	0.654920i	$-0.772702\pi$
$853$	−37653.2	−1.51140	−0.755698	+	0.654920i	$0.772702\pi$
$854$	6203.99	0.248590
$855$	11028.6	0.441136
$856$	16004.4	0.639041
$857$	7916.64	0.315551	0.157776	−	0.987475i	$-0.449568\pi$
$857$	7916.64	0.315551	0.157776	+	0.987475i	$0.449568\pi$
$858$	−2866.83	−0.114070
$859$	4012.33	0.159370	0.0796850	−	0.996820i	$-0.474609\pi$
$859$	4012.33	0.159370	0.0796850	+	0.996820i	$0.474609\pi$
$860$	−8183.01	−0.324463
$861$	655.150	0.0259320
$862$	12835.0	0.507147
$863$	22547.6	0.889374	0.444687	−	0.895686i	$-0.353315\pi$
$863$	22547.6	0.889374	0.444687	+	0.895686i	$0.353315\pi$
$864$	10907.2	0.429478
$865$	−1082.33	−0.0425438
$866$	16940.8	0.664749
$867$	−334.526	−0.0131039
$868$	1491.32	0.0583163
$869$	−3985.64	−0.155585
$870$	−192.045	−0.00748384
$871$	24140.8	0.939127
$872$	−14773.5	−0.573733
$873$	−24965.7	−0.967883
$874$	−753.869	−0.0291762
$875$	−1834.29	−0.0708691
$876$	5305.62	0.204635
$877$	16964.9	0.653210	0.326605	−	0.945161i	$-0.394095\pi$
$877$	16964.9	0.653210	0.326605	+	0.945161i	$0.394095\pi$
$878$	330.729	0.0127125
$879$	9855.90	0.378193
$880$	10851.6	0.415691
$881$	−28213.2	−1.07892	−0.539459	−	0.842012i	$-0.681371\pi$
$881$	−28213.2	−1.07892	−0.539459	+	0.842012i	$0.681371\pi$
$882$	4002.05	0.152784
$883$	−44019.1	−1.67764	−0.838822	−	0.544405i	$-0.816755\pi$
$883$	−44019.1	−1.67764	−0.838822	+	0.544405i	$0.816755\pi$
$884$	3142.21	0.119552
$885$	3464.74	0.131600
$886$	−8318.75	−0.315433
$887$	14078.8	0.532941	0.266471	−	0.963843i	$-0.414142\pi$
$887$	14078.8	0.532941	0.266471	+	0.963843i	$0.414142\pi$
$888$	−1374.61	−0.0519469
$889$	16360.1	0.617209
$890$	444.733	0.0167500
$891$	−44413.8	−1.66994
$892$	−19783.8	−0.742613
$893$	4978.08	0.186545
$894$	4691.66	0.175517
$895$	5711.50	0.213312
$896$	−21448.3	−0.799705
$897$	−236.021	−0.00878541
$898$	−11154.0	−0.414493
$899$	424.172	0.0157363
$900$	4174.58	0.154614
$901$	−11522.7	−0.426057
$902$	3363.17	0.124148
$903$	4271.88	0.157430
$904$	−17838.5	−0.656306
$905$	−9424.23	−0.346157
$906$	−1199.87	−0.0439988
$907$	−12670.6	−0.463858	−0.231929	−	0.972733i	$-0.574504\pi$
$907$	−12670.6	−0.463858	−0.231929	+	0.972733i	$0.574504\pi$
$908$	2987.83	0.109201
$909$	16887.4	0.616194
$910$	2546.00	0.0927461
$911$	15336.4	0.557760	0.278880	−	0.960326i	$-0.410037\pi$
$911$	15336.4	0.557760	0.278880	+	0.960326i	$0.410037\pi$
$912$	−3025.57	−0.109854
$913$	88161.1	3.19574
$914$	−3084.49	−0.111626
$915$	2002.89	0.0723644
$916$	−14013.9	−0.505492
$917$	22868.9	0.823553
$918$	1265.96	0.0455150
$919$	43929.1	1.57681	0.788404	−	0.615158i	$-0.210908\pi$
$919$	43929.1	1.57681	0.788404	+	0.615158i	$0.210908\pi$
$920$	−636.160	−0.0227973
$921$	10364.6	0.370819
$922$	13563.2	0.484467
$923$	25899.3	0.923605
$924$	−7889.45	−0.280892
$925$	−1675.09	−0.0595423
$926$	−3721.13	−0.132056
$927$	44332.5	1.57073
$928$	−4860.08	−0.171918
$929$	28646.0	1.01167	0.505836	−	0.862629i	$-0.331184\pi$
$929$	28646.0	1.01167	0.505836	+	0.862629i	$0.331184\pi$
$930$	−110.422	−0.00389341
$931$	−10973.8	−0.386309
$932$	−35507.9	−1.24796
$933$	4448.17	0.156084
$934$	−17733.2	−0.621252
$935$	6066.82	0.212199
$936$	−12917.6	−0.451094
$937$	25892.8	0.902753	0.451376	−	0.892334i	$-0.350933\pi$
$937$	25892.8	0.902753	0.451376	+	0.892334i	$0.350933\pi$
$938$	−15236.8	−0.530384
$939$	5131.13	0.178326
$940$	1884.32	0.0653826
$941$	−12650.8	−0.438263	−0.219131	−	0.975695i	$-0.570322\pi$
$941$	−12650.8	−0.438263	−0.219131	+	0.975695i	$0.570322\pi$
$942$	−1343.17	−0.0464572
$943$	276.884	0.00956158
$944$	18203.4	0.627617
$945$	−4472.42	−0.153955
$946$	21929.4	0.753686
$947$	−19766.7	−0.678280	−0.339140	−	0.940736i	$-0.610136\pi$
$947$	−19766.7	−0.678280	−0.339140	+	0.940736i	$0.610136\pi$
$948$	420.632	0.0144109
$949$	−20006.1	−0.684326
$950$	2625.36	0.0896610
$951$	−8906.27	−0.303686
$952$	−4421.37	−0.150523
$953$	−6066.08	−0.206190	−0.103095	−	0.994671i	$-0.532875\pi$
$953$	−6066.08	−0.206190	−0.103095	+	0.994671i	$0.532875\pi$
$954$	21248.2	0.721105
$955$	−11374.2	−0.385404
$956$	39523.9	1.33713
$957$	−2243.98	−0.0757968
$958$	−6733.80	−0.227097
$959$	3451.27	0.116212
$960$	−142.716	−0.00479807
$961$	−29547.1	−0.991813
$962$	2325.02	0.0779228
$963$	23171.2	0.775370
$964$	−46799.9	−1.56361
$965$	14528.7	0.484659
$966$	148.968	0.00496167
$967$	−31669.2	−1.05317	−0.526584	−	0.850123i	$-0.676527\pi$
$967$	−31669.2	−1.05317	−0.526584	+	0.850123i	$0.676527\pi$
$968$	−66698.7	−2.21464
$969$	−1691.50	−0.0560773
$970$	−5943.09	−0.196723
$971$	−24982.5	−0.825671	−0.412835	−	0.910806i	$-0.635462\pi$
$971$	−24982.5	−0.825671	−0.412835	+	0.910806i	$0.635462\pi$
$972$	15397.3	0.508097
$973$	−31380.5	−1.03393
$974$	−12378.2	−0.407212
$975$	821.947	0.0269983
$976$	10523.0	0.345115
$977$	26574.1	0.870194	0.435097	−	0.900384i	$-0.356714\pi$
$977$	26574.1	0.870194	0.435097	+	0.900384i	$0.356714\pi$
$978$	−2746.36	−0.0897945
$979$	5196.54	0.169645
$980$	−4153.85	−0.135398
$981$	−21389.2	−0.696131
$982$	−7288.41	−0.236846
$983$	−19222.4	−0.623703	−0.311852	−	0.950131i	$-0.600949\pi$
$983$	−19222.4	−0.623703	−0.311852	+	0.950131i	$0.600949\pi$
$984$	−791.282	−0.0256353
$985$	−24189.9	−0.782493
$986$	−564.093	−0.0182195
$987$	−983.694	−0.0317237
$988$	15888.4	0.511616
$989$	1805.41	0.0580472
$990$	−11187.4	−0.359149
$991$	−22396.5	−0.717909	−0.358954	−	0.933355i	$-0.616867\pi$
$991$	−22396.5	−0.717909	−0.358954	+	0.933355i	$0.616867\pi$
$992$	−2794.44	−0.0894391
$993$	−8422.99	−0.269180
$994$	−16346.8	−0.521618
$995$	11826.3	0.376803
$996$	−9304.24	−0.296000
$997$	26548.7	0.843337	0.421669	−	0.906750i	$-0.361445\pi$
$997$	26548.7	0.843337	0.421669	+	0.906750i	$0.361445\pi$
$998$	24055.0	0.762975
$999$	−4084.24	−0.129349

Display

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a_n

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a_n

instead) (See

a_n

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a_n

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up to: 50 250 1000 (See only

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a_p

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Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	85.4.a.e.1.3	✓	3
3.2	odd	2		765.4.a.l.1.1		3
4.3	odd	2		1360.4.a.s.1.2		3
5.2	odd	4		425.4.b.g.324.4		6
5.3	odd	4		425.4.b.g.324.3		6
5.4	even	2		425.4.a.h.1.1		3
17.16	even	2		1445.4.a.j.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
85.4.a.e.1.3	✓	3	1.1	even	1	trivial
425.4.a.h.1.1		3	5.4	even	2
425.4.b.g.324.3		6	5.3	odd	4
425.4.b.g.324.4		6	5.2	odd	4
765.4.a.l.1.1		3	3.2	odd	2
1360.4.a.s.1.2		3	4.3	odd	2
1445.4.a.j.1.3		3	17.16	even	2

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Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

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Varieties

Fields

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Database

Properties

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	85.4.a.e.1.3

Level	$85$
Weight	$4$
Character	85.1
Self dual	yes
Analytic conductor	$5.015$
Analytic rank	$1$
Dimension	$3$
CM	no
Inner twists	$1$