LMFDB - Embedded newform 1859.4.a.k.1.15

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1859,4,Mod(1,1859)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1859, 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("1859.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$N$	$=$	$1859 = 11 \cdot 13^{2}$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	1859.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:	$109.684550701$
Analytic rank:	$0$
Dimension:	$18$
Coefficient field:	$\mathbb{Q}[x]/(x^{18} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{18} - 2 x^{17} - 108 x^{16} + 212 x^{15} + 4721 x^{14} - 8963 x^{13} - 107626 x^{12} + 194656 x^{11} + \cdots + 9847296$ x^18 - 2x^17 - 108x^16 + 212x^15 + 4721x^14 - 8963x^13 - 107626x^12 + 194656x^11 + 1375700x^10 - 2336289x^9 - 9875440x^8 + 15599166x^7 + 37652624x^6 - 55787484x^5 - 64147728x^4 + 92949968x^3 + 22620800x^2 - 43697856*x + 9847296
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$2^{5}\cdot 3^{2}$
Twist minimal:	no (minimal twist has level 143)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.15
Root			$3.97618$ of defining polynomial
Character	$\chi$	$=$	1859.1

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.97618 q^{2} -1.00306 q^{3} +7.81003 q^{4} -0.345299 q^{5} -3.98837 q^{6} -17.7538 q^{7} -0.755338 q^{8} -25.9939 q^{9} -1.37297 q^{10} +11.0000 q^{11} -7.83397 q^{12} -70.5923 q^{14} +0.346357 q^{15} -65.4836 q^{16} -18.9261 q^{17} -103.356 q^{18} -17.8720 q^{19} -2.69679 q^{20} +17.8082 q^{21} +43.7380 q^{22} +196.640 q^{23} +0.757652 q^{24} -124.881 q^{25} +53.1562 q^{27} -138.658 q^{28} +94.8199 q^{29} +1.37718 q^{30} +115.933 q^{31} -254.332 q^{32} -11.0337 q^{33} -75.2535 q^{34} +6.13035 q^{35} -203.013 q^{36} -215.254 q^{37} -71.0624 q^{38} +0.260817 q^{40} +338.126 q^{41} +70.8086 q^{42} +390.863 q^{43} +85.9104 q^{44} +8.97565 q^{45} +781.876 q^{46} +439.810 q^{47} +65.6843 q^{48} -27.8036 q^{49} -496.549 q^{50} +18.9840 q^{51} -585.925 q^{53} +211.359 q^{54} -3.79829 q^{55} +13.4101 q^{56} +17.9268 q^{57} +377.021 q^{58} -796.004 q^{59} +2.70506 q^{60} +347.654 q^{61} +460.970 q^{62} +461.489 q^{63} -487.403 q^{64} -43.8720 q^{66} +868.735 q^{67} -147.813 q^{68} -197.242 q^{69} +24.3754 q^{70} +792.078 q^{71} +19.6341 q^{72} +342.330 q^{73} -855.888 q^{74} +125.263 q^{75} -139.581 q^{76} -195.291 q^{77} +868.378 q^{79} +22.6114 q^{80} +648.515 q^{81} +1344.45 q^{82} +1019.35 q^{83} +139.082 q^{84} +6.53514 q^{85} +1554.14 q^{86} -95.1105 q^{87} -8.30872 q^{88} -472.117 q^{89} +35.6888 q^{90} +1535.76 q^{92} -116.288 q^{93} +1748.76 q^{94} +6.17118 q^{95} +255.112 q^{96} -1031.67 q^{97} -110.552 q^{98} -285.932 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$18 q + 2 q^{2} + 76 q^{4} + 20 q^{5} + 49 q^{6} + 28 q^{7} - 12 q^{8} + 180 q^{9} + 56 q^{10} + 198 q^{11} + 54 q^{12} + 4 q^{14} + 60 q^{15} + 364 q^{16} - 138 q^{17} + 298 q^{18} + 24 q^{19} + 160 q^{20}+ \cdots + 1980 q^{99}+O(q^{100})$ 18 * q + 2 * q^2 + 76 * q^4 + 20 * q^5 + 49 * q^6 + 28 * q^7 - 12 * q^8 + 180 * q^9 + 56 * q^10 + 198 * q^11 + 54 * q^12 + 4 * q^14 + 60 * q^15 + 364 * q^16 - 138 * q^17 + 298 * q^18 + 24 * q^19 + 160 * q^20 + 352 * q^21 + 22 * q^22 + 236 * q^23 + 124 * q^24 + 586 * q^25 - 6 * q^27 + 691 * q^28 - 286 * q^29 - 356 * q^30 + 642 * q^31 - 379 * q^32 + 2068 * q^34 + 34 * q^35 + 215 * q^36 + 530 * q^37 + 25 * q^38 - 108 * q^40 + 608 * q^41 + 563 * q^42 - 460 * q^43 + 836 * q^44 + 452 * q^45 - 580 * q^46 + 986 * q^47 + 837 * q^48 + 1082 * q^49 - 2578 * q^50 + 170 * q^51 + 1216 * q^53 + 1539 * q^54 + 220 * q^55 + 1137 * q^56 + 1282 * q^57 + 90 * q^58 + 1366 * q^59 + 5104 * q^60 - 922 * q^61 - 1398 * q^62 - 3236 * q^63 + 1296 * q^64 + 539 * q^66 + 1118 * q^67 - 2274 * q^68 + 1644 * q^69 + 416 * q^70 + 1118 * q^71 + 3295 * q^72 + 2444 * q^73 - 2018 * q^74 - 410 * q^75 - 1326 * q^76 + 308 * q^77 + 180 * q^79 + 5446 * q^80 + 426 * q^81 - 974 * q^82 - 696 * q^83 + 1061 * q^84 + 4580 * q^85 + 4652 * q^86 - 1240 * q^87 - 132 * q^88 - 3664 * q^89 + 248 * q^90 + 227 * q^92 + 3696 * q^93 - 244 * q^94 - 476 * q^95 - 5426 * q^96 + 2660 * q^97 + 6003 * q^98 + 1980 * 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

)

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

$2$	3.97618	1.40579	0.702897	−	0.711292i	$-0.251890\pi$
$2$	3.97618	1.40579	0.702897	+	0.711292i	$0.251890\pi$
$3$	−1.00306	−0.193040	−0.0965199	−	0.995331i	$-0.530771\pi$
$3$	−1.00306	−0.193040	−0.0965199	+	0.995331i	$0.530771\pi$
$4$	7.81003	0.976254
$5$	−0.345299	−0.0308845	−0.0154422	−	0.999881i	$-0.504916\pi$
$5$	−0.345299	−0.0308845	−0.0154422	+	0.999881i	$0.504916\pi$
$6$	−3.98837	−0.271374
$7$	−17.7538	−0.958614	−0.479307	−	0.877647i	$-0.659112\pi$
$7$	−17.7538	−0.958614	−0.479307	+	0.877647i	$0.659112\pi$
$8$	−0.755338	−0.0333815
$9$	−25.9939	−0.962736
$10$	−1.37297	−0.0434172
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	−7.83397	−0.188456
$13$	0	0
$13$	0	0
$14$	−70.5923	−1.34761
$15$	0.346357	0.00596193
$16$	−65.4836	−1.02318
$17$	−18.9261	−0.270014	−0.135007	−	0.990845i	$-0.543106\pi$
$17$	−18.9261	−0.270014	−0.135007	+	0.990845i	$0.543106\pi$
$18$	−103.356	−1.35341
$19$	−17.8720	−0.215796	−0.107898	−	0.994162i	$-0.534412\pi$
$19$	−17.8720	−0.215796	−0.107898	+	0.994162i	$0.534412\pi$
$20$	−2.69679	−0.0301511
$21$	17.8082	0.185051
$22$	43.7380	0.423863
$23$	196.640	1.78271	0.891353	−	0.453310i	$-0.149757\pi$
$23$	196.640	1.78271	0.891353	+	0.453310i	$0.149757\pi$
$24$	0.757652	0.00644396
$25$	−124.881	−0.999046
$26$	0	0
$27$	53.1562	0.378886
$28$	−138.658	−0.935851
$29$	94.8199	0.607159	0.303580	−	0.952806i	$-0.401818\pi$
$29$	94.8199	0.607159	0.303580	+	0.952806i	$0.401818\pi$
$30$	1.37718	0.00838124
$31$	115.933	0.671683	0.335841	−	0.941919i	$-0.390979\pi$
$31$	115.933	0.671683	0.335841	+	0.941919i	$0.390979\pi$
$32$	−254.332	−1.40500
$33$	−11.0337	−0.0582037
$34$	−75.2535	−0.379584
$35$	6.13035	0.0296063
$36$	−203.013	−0.939875
$37$	−215.254	−0.956419	−0.478209	−	0.878246i	$-0.658714\pi$
$37$	−215.254	−0.956419	−0.478209	+	0.878246i	$0.658714\pi$
$38$	−71.0624	−0.303364
$39$	0	0
$40$	0.260817	0.00103097
$41$	338.126	1.28796	0.643980	−	0.765042i	$-0.277282\pi$
$41$	338.126	1.28796	0.643980	+	0.765042i	$0.277282\pi$
$42$	70.8086	0.260143
$43$	390.863	1.38619	0.693094	−	0.720848i	$-0.256247\pi$
$43$	390.863	1.38619	0.693094	+	0.720848i	$0.256247\pi$
$44$	85.9104	0.294352
$45$	8.97565	0.0297336
$46$	781.876	2.50612
$47$	439.810	1.36495	0.682477	−	0.730907i	$-0.260903\pi$
$47$	439.810	1.36495	0.682477	+	0.730907i	$0.260903\pi$
$48$	65.6843	0.197515
$49$	−27.8036	−0.0810601
$50$	−496.549	−1.40445
$51$	18.9840	0.0521235
$52$	0	0
$53$	−585.925	−1.51855	−0.759273	−	0.650772i	$-0.774445\pi$
$53$	−585.925	−1.51855	−0.759273	+	0.650772i	$0.774445\pi$
$54$	211.359	0.532635
$55$	−3.79829	−0.00931201
$56$	13.4101	0.0320000
$57$	17.9268	0.0416572
$58$	377.021	0.853540
$59$	−796.004	−1.75646	−0.878228	−	0.478242i	$-0.841274\pi$
$59$	−796.004	−1.75646	−0.878228	+	0.478242i	$0.841274\pi$
$60$	2.70506	0.00582036
$61$	347.654	0.729713	0.364856	−	0.931064i	$-0.381118\pi$
$61$	347.654	0.729713	0.364856	+	0.931064i	$0.381118\pi$
$62$	460.970	0.944247
$63$	461.489	0.922891
$64$	−487.403	−0.951958
$65$	0	0
$66$	−43.8720	−0.0818223
$67$	868.735	1.58407	0.792037	−	0.610474i	$-0.209021\pi$
$67$	868.735	1.58407	0.792037	+	0.610474i	$0.209021\pi$
$68$	−147.813	−0.263603
$69$	−197.242	−0.344133
$70$	24.3754	0.0416203
$71$	792.078	1.32398	0.661989	−	0.749514i	$-0.269713\pi$
$71$	792.078	1.32398	0.661989	+	0.749514i	$0.269713\pi$
$72$	19.6341	0.0321376
$73$	342.330	0.548859	0.274430	−	0.961607i	$-0.411511\pi$
$73$	342.330	0.548859	0.274430	+	0.961607i	$0.411511\pi$
$74$	−855.888	−1.34453
$75$	125.263	0.192856
$76$	−139.581	−0.210672
$77$	−195.291	−0.289033
$78$	0	0
$79$	868.378	1.23671	0.618355	−	0.785899i	$-0.287799\pi$
$79$	868.378	1.23671	0.618355	+	0.785899i	$0.287799\pi$
$80$	22.6114	0.0316004
$81$	648.515	0.889596
$82$	1344.45	1.81061
$83$	1019.35	1.34805	0.674024	−	0.738710i	$-0.264565\pi$
$83$	1019.35	1.34805	0.674024	+	0.738710i	$0.264565\pi$
$84$	139.082	0.180656
$85$	6.53514	0.00833924
$86$	1554.14	1.94869
$87$	−95.1105	−0.117206
$88$	−8.30872	−0.0100649
$89$	−472.117	−0.562296	−0.281148	−	0.959664i	$-0.590715\pi$
$89$	−472.117	−0.562296	−0.281148	+	0.959664i	$0.590715\pi$
$90$	35.6888	0.0417992
$91$	0	0
$92$	1535.76	1.74037
$93$	−116.288	−0.129661
$94$	1748.76	1.91884
$95$	6.17118	0.00666474
$96$	255.112	0.271221
$97$	−1031.67	−1.07990	−0.539952	−	0.841696i	$-0.681558\pi$
$97$	−1031.67	−1.07990	−0.539952	+	0.841696i	$0.681558\pi$
$98$	−110.552	−0.113954
$99$	−285.932	−0.290276
$100$	−975.323	−0.975323
$101$	1555.46	1.53241	0.766207	−	0.642594i	$-0.222142\pi$
$101$	1555.46	1.53241	0.766207	+	0.642594i	$0.222142\pi$
$102$	75.4840	0.0732748
$103$	−1667.47	−1.59516	−0.797579	−	0.603215i	$-0.793886\pi$
$103$	−1667.47	−1.59516	−0.797579	+	0.603215i	$0.793886\pi$
$104$	0	0
$105$	−6.14914	−0.00571519
$106$	−2329.74	−2.13476
$107$	813.150	0.734674	0.367337	−	0.930088i	$-0.380269\pi$
$107$	813.150	0.734674	0.367337	+	0.930088i	$0.380269\pi$
$108$	415.152	0.369889
$109$	−1557.11	−1.36829	−0.684145	−	0.729346i	$-0.739825\pi$
$109$	−1557.11	−1.36829	−0.684145	+	0.729346i	$0.739825\pi$
$110$	−15.1027	−0.0130908
$111$	215.913	0.184627
$112$	1162.58	0.980836
$113$	−853.103	−0.710205	−0.355103	−	0.934827i	$-0.615554\pi$
$113$	−853.103	−0.710205	−0.355103	+	0.934827i	$0.615554\pi$
$114$	71.2801	0.0585614
$115$	−67.8995	−0.0550579
$116$	740.547	0.592742
$117$	0	0
$118$	−3165.06	−2.46921
$119$	336.009	0.258839
$120$	−0.261616	−0.000199018 0
$121$	121.000	0.0909091
$122$	1382.33	1.02583
$123$	−339.162	−0.248628
$124$	905.440	0.655733
$125$	86.2835	0.0617395
$126$	1834.97	1.29739
$127$	284.900	0.199061	0.0995305	−	0.995035i	$-0.468266\pi$
$127$	284.900	0.199061	0.0995305	+	0.995035i	$0.468266\pi$
$128$	96.6560	0.0667442
$129$	−392.061	−0.267589
$130$	0	0
$131$	1646.75	1.09830	0.549148	−	0.835725i	$-0.314952\pi$
$131$	1646.75	1.09830	0.549148	+	0.835725i	$0.314952\pi$
$132$	−86.1736	−0.0568216
$133$	317.296	0.206865
$134$	3454.25	2.22688
$135$	−18.3548	−0.0117017
$136$	14.2956	0.00901349
$137$	−211.431	−0.131852	−0.0659261	−	0.997825i	$-0.521000\pi$
$137$	−211.431	−0.131852	−0.0659261	+	0.997825i	$0.521000\pi$
$138$	−784.272	−0.483780
$139$	−1882.82	−1.14891	−0.574455	−	0.818536i	$-0.694786\pi$
$139$	−1882.82	−1.14891	−0.574455	+	0.818536i	$0.694786\pi$
$140$	47.8783	0.0289032
$141$	−441.157	−0.263490
$142$	3149.45	1.86124
$143$	0	0
$144$	1702.17	0.985054
$145$	−32.7412	−0.0187518
$146$	1361.17	0.771582
$147$	27.8888	0.0156478
$148$	−1681.14	−0.933708
$149$	1132.07	0.622433	0.311216	−	0.950339i	$-0.399264\pi$
$149$	1132.07	0.622433	0.311216	+	0.950339i	$0.399264\pi$
$150$	498.070	0.271115
$151$	19.4066	0.0104589	0.00522943	−	0.999986i	$-0.498335\pi$
$151$	19.4066	0.0104589	0.00522943	+	0.999986i	$0.498335\pi$
$152$	13.4994	0.00720359
$153$	491.961	0.259952
$154$	−776.515	−0.406320
$155$	−40.0315	−0.0207446
$156$	0	0
$157$	−78.6951	−0.0400035	−0.0200018	−	0.999800i	$-0.506367\pi$
$157$	−78.6951	−0.0400035	−0.0200018	+	0.999800i	$0.506367\pi$
$158$	3452.83	1.73856
$159$	587.720	0.293140
$160$	87.8206	0.0433927
$161$	−3491.10	−1.70893
$162$	2578.62	1.25059
$163$	3139.19	1.50847	0.754234	−	0.656605i	$-0.228008\pi$
$163$	3139.19	1.50847	0.754234	+	0.656605i	$0.228008\pi$
$164$	2640.78	1.25738
$165$	3.80992	0.00179759
$166$	4053.11	1.89508
$167$	−546.889	−0.253411	−0.126705	−	0.991940i	$-0.540440\pi$
$167$	−546.889	−0.253411	−0.126705	+	0.991940i	$0.540440\pi$
$168$	−13.4512	−0.00617727
$169$	0	0
$170$	25.9849	0.0117232
$171$	464.563	0.207754
$172$	3052.65	1.35327
$173$	3936.60	1.73002	0.865011	−	0.501753i	$-0.167311\pi$
$173$	3936.60	1.73002	0.865011	+	0.501753i	$0.167311\pi$
$174$	−378.177	−0.164767
$175$	2217.10	0.957699
$176$	−720.320	−0.308501
$177$	798.443	0.339066
$178$	−1877.22	−0.790472
$179$	1017.48	0.424859	0.212429	−	0.977176i	$-0.431862\pi$
$179$	1017.48	0.424859	0.212429	+	0.977176i	$0.431862\pi$
$180$	70.1001	0.0290275
$181$	−2418.67	−0.993252	−0.496626	−	0.867965i	$-0.665428\pi$
$181$	−2418.67	−0.993252	−0.496626	+	0.867965i	$0.665428\pi$
$182$	0	0
$183$	−348.719	−0.140864
$184$	−148.530	−0.0595095
$185$	74.3268	0.0295385
$186$	−462.383	−0.182277
$187$	−208.187	−0.0814124
$188$	3434.93	1.33254
$189$	−943.724	−0.363205
$190$	24.5377	0.00936924
$191$	3292.32	1.24725	0.623624	−	0.781725i	$-0.285660\pi$
$191$	3292.32	1.24725	0.623624	+	0.781725i	$0.285660\pi$
$192$	488.896	0.183766
$193$	597.292	0.222767	0.111383	−	0.993778i	$-0.464472\pi$
$193$	597.292	0.222767	0.111383	+	0.993778i	$0.464472\pi$
$194$	−4102.13	−1.51812
$195$	0	0
$196$	−217.147	−0.0791353
$197$	−3280.26	−1.18634	−0.593170	−	0.805077i	$-0.702124\pi$
$197$	−3280.26	−1.18634	−0.593170	+	0.805077i	$0.702124\pi$
$198$	−1136.92	−0.408068
$199$	−1680.17	−0.598512	−0.299256	−	0.954173i	$-0.596739\pi$
$199$	−1680.17	−0.598512	−0.299256	+	0.954173i	$0.596739\pi$
$200$	94.3272	0.0333497
$201$	−871.397	−0.305789
$202$	6184.79	2.15426
$203$	−1683.41	−0.582031
$204$	148.266	0.0508858
$205$	−116.754	−0.0397780
$206$	−6630.19	−2.24246
$207$	−5111.43	−1.71627
$208$	0	0
$209$	−196.592	−0.0650649
$210$	−24.4501	−0.00803437
$211$	−2179.99	−0.711264	−0.355632	−	0.934626i	$-0.615734\pi$
$211$	−2179.99	−0.711264	−0.355632	+	0.934626i	$0.615734\pi$
$212$	−4576.09	−1.48249
$213$	−794.505	−0.255580
$214$	3233.23	1.03280
$215$	−134.964	−0.0428116
$216$	−40.1509	−0.0126478
$217$	−2058.25	−0.643884
$218$	−6191.34	−1.92353
$219$	−343.379	−0.105952
$220$	−29.6647	−0.00909089
$221$	0	0
$222$	858.511	0.259547
$223$	−168.248	−0.0505233	−0.0252616	−	0.999681i	$-0.508042\pi$
$223$	−168.248	−0.0505233	−0.0252616	+	0.999681i	$0.508042\pi$
$224$	4515.36	1.34685
$225$	3246.13	0.961817
$226$	−3392.09	−0.998402
$227$	2882.86	0.842917	0.421459	−	0.906848i	$-0.361518\pi$
$227$	2882.86	0.842917	0.421459	+	0.906848i	$0.361518\pi$
$228$	140.009	0.0406680
$229$	−191.646	−0.0553027	−0.0276513	−	0.999618i	$-0.508803\pi$
$229$	−191.646	−0.0553027	−0.0276513	+	0.999618i	$0.508803\pi$
$230$	−269.981	−0.0774000
$231$	195.890	0.0557948
$232$	−71.6211	−0.0202679
$233$	1425.45	0.400792	0.200396	−	0.979715i	$-0.435777\pi$
$233$	1425.45	0.400792	0.200396	+	0.979715i	$0.435777\pi$
$234$	0	0
$235$	−151.866	−0.0421559
$236$	−6216.82	−1.71475
$237$	−871.039	−0.238734
$238$	1336.03	0.363874
$239$	388.599	0.105173	0.0525866	−	0.998616i	$-0.483253\pi$
$239$	388.599	0.105173	0.0525866	+	0.998616i	$0.483253\pi$
$240$	−22.6807	−0.00610014
$241$	1048.03	0.280123	0.140062	−	0.990143i	$-0.455270\pi$
$241$	1048.03	0.280123	0.140062	+	0.990143i	$0.455270\pi$
$242$	481.118	0.127799
$243$	−2085.72	−0.550613
$244$	2715.19	0.712385
$245$	9.60055	0.00250350
$246$	−1348.57	−0.349519
$247$	0	0
$248$	−87.5685	−0.0224218
$249$	−1022.47	−0.260227
$250$	343.079	0.0867929
$251$	2720.68	0.684175	0.342087	−	0.939668i	$-0.388866\pi$
$251$	2720.68	0.684175	0.342087	+	0.939668i	$0.388866\pi$
$252$	3604.25	0.900977
$253$	2163.04	0.537506
$254$	1132.81	0.279839
$255$	−6.55517	−0.00160981
$256$	4283.54	1.04579
$257$	2172.82	0.527381	0.263690	−	0.964607i	$-0.415060\pi$
$257$	2172.82	0.527381	0.263690	+	0.964607i	$0.415060\pi$
$258$	−1558.90	−0.376175
$259$	3821.57	0.916836
$260$	0	0
$261$	−2464.74	−0.584534
$262$	6547.76	1.54398
$263$	−5714.65	−1.33985	−0.669925	−	0.742429i	$-0.733674\pi$
$263$	−5714.65	−1.33985	−0.669925	+	0.742429i	$0.733674\pi$
$264$	8.33417	0.00194293
$265$	202.319	0.0468995
$266$	1261.63	0.290809
$267$	473.564	0.108545
$268$	6784.85	1.54646
$269$	1390.82	0.315241	0.157621	−	0.987500i	$-0.449618\pi$
$269$	1390.82	0.315241	0.157621	+	0.987500i	$0.449618\pi$
$270$	−72.9820	−0.0164502
$271$	205.434	0.0460488	0.0230244	−	0.999735i	$-0.492670\pi$
$271$	205.434	0.0460488	0.0230244	+	0.999735i	$0.492670\pi$
$272$	1239.35	0.276274
$273$	0	0
$274$	−840.688	−0.185357
$275$	−1373.69	−0.301224
$276$	−1540.47	−0.335962
$277$	−4377.91	−0.949614	−0.474807	−	0.880090i	$-0.657482\pi$
$277$	−4377.91	−0.949614	−0.474807	+	0.880090i	$0.657482\pi$
$278$	−7486.43	−1.61513
$279$	−3013.54	−0.646653
$280$	−4.63049	−0.000988302 0
$281$	7993.25	1.69693	0.848465	−	0.529251i	$-0.177527\pi$
$281$	7993.25	1.69693	0.848465	+	0.529251i	$0.177527\pi$
$282$	−1754.12	−0.370413
$283$	1568.54	0.329470	0.164735	−	0.986338i	$-0.447323\pi$
$283$	1568.54	0.329470	0.164735	+	0.986338i	$0.447323\pi$
$284$	6186.16	1.29254
$285$	−6.19009	−0.00128656
$286$	0	0
$287$	−6003.01	−1.23466
$288$	6611.08	1.35264
$289$	−4554.80	−0.927092
$290$	−130.185	−0.0263611
$291$	1034.84	0.208464
$292$	2673.61	0.535826
$293$	−3535.82	−0.705000	−0.352500	−	0.935812i	$-0.614668\pi$
$293$	−3535.82	−0.705000	−0.352500	+	0.935812i	$0.614668\pi$
$294$	110.891	0.0219976
$295$	274.859	0.0542472
$296$	162.589	0.0319267
$297$	584.719	0.114238
$298$	4501.30	0.875012
$299$	0	0
$300$	978.312	0.188276
$301$	−6939.29	−1.32882
$302$	77.1642	0.0147030
$303$	−1560.22	−0.295817
$304$	1170.32	0.220798
$305$	−120.044	−0.0225368
$306$	1956.13	0.365439
$307$	−7614.99	−1.41567	−0.707835	−	0.706378i	$-0.750328\pi$
$307$	−7614.99	−1.41567	−0.707835	+	0.706378i	$0.750328\pi$
$308$	−1525.23	−0.282170
$309$	1672.58	0.307929
$310$	−159.173	−0.0291626
$311$	−3281.61	−0.598338	−0.299169	−	0.954200i	$-0.596709\pi$
$311$	−3281.61	−0.598338	−0.299169	+	0.954200i	$0.596709\pi$
$312$	0	0
$313$	−3592.43	−0.648741	−0.324371	−	0.945930i	$-0.605152\pi$
$313$	−3592.43	−0.648741	−0.324371	+	0.945930i	$0.605152\pi$
$314$	−312.906	−0.0562366
$315$	−159.352	−0.0285030
$316$	6782.06	1.20734
$317$	−1410.44	−0.249900	−0.124950	−	0.992163i	$-0.539877\pi$
$317$	−1410.44	−0.249900	−0.124950	+	0.992163i	$0.539877\pi$
$318$	2336.88	0.412094
$319$	1043.02	0.183065
$320$	168.299	0.0294007
$321$	−815.641	−0.141821
$322$	−13881.2	−2.40240
$323$	338.247	0.0582679
$324$	5064.93	0.868471
$325$	0	0
$326$	12482.0	2.12059
$327$	1561.88	0.264135
$328$	−255.399	−0.0429941
$329$	−7808.28	−1.30846
$330$	15.1490	0.00252704
$331$	1366.57	0.226929	0.113465	−	0.993542i	$-0.463805\pi$
$331$	1366.57	0.226929	0.113465	+	0.993542i	$0.463805\pi$
$332$	7961.14	1.31604
$333$	5595.28	0.920779
$334$	−2174.53	−0.356243
$335$	−299.973	−0.0489232
$336$	−1166.14	−0.189340
$337$	−2821.22	−0.456029	−0.228015	−	0.973658i	$-0.573223\pi$
$337$	−2821.22	−0.456029	−0.228015	+	0.973658i	$0.573223\pi$
$338$	0	0
$339$	855.717	0.137098
$340$	51.0397	0.00814122
$341$	1275.26	0.202520
$342$	1847.19	0.292060
$343$	6583.16	1.03632
$344$	−295.234	−0.0462730
$345$	68.1075	0.0106284
$346$	15652.6	2.43205
$347$	9796.45	1.51556	0.757782	−	0.652508i	$-0.226283\pi$
$347$	9796.45	1.51556	0.757782	+	0.652508i	$0.226283\pi$
$348$	−742.816	−0.114423
$349$	8589.64	1.31746	0.658729	−	0.752380i	$-0.271094\pi$
$349$	8589.64	1.31746	0.658729	+	0.752380i	$0.271094\pi$
$350$	8815.61	1.34633
$351$	0	0
$352$	−2797.65	−0.423624
$353$	6999.51	1.05537	0.527686	−	0.849440i	$-0.323060\pi$
$353$	6999.51	1.05537	0.527686	+	0.849440i	$0.323060\pi$
$354$	3174.76	0.476656
$355$	−273.504	−0.0408903
$356$	−3687.25	−0.548944
$357$	−337.038	−0.0499663
$358$	4045.67	0.597264
$359$	9944.85	1.46203	0.731015	−	0.682361i	$-0.239047\pi$
$359$	9944.85	1.46203	0.731015	+	0.682361i	$0.239047\pi$
$360$	−6.77965	−0.000992552 0
$361$	−6539.59	−0.953432
$362$	−9617.09	−1.39631
$363$	−121.371	−0.0175491
$364$	0	0
$365$	−118.206	−0.0169512
$366$	−1386.57	−0.198025
$367$	1135.70	0.161534	0.0807669	−	0.996733i	$-0.474263\pi$
$367$	1135.70	0.161534	0.0807669	+	0.996733i	$0.474263\pi$
$368$	−12876.7	−1.82403
$369$	−8789.20	−1.23997
$370$	295.537	0.0415250
$371$	10402.4	1.45570
$372$	−908.214	−0.126583
$373$	3712.67	0.515375	0.257688	−	0.966228i	$-0.417039\pi$
$373$	3712.67	0.515375	0.257688	+	0.966228i	$0.417039\pi$
$374$	−827.788	−0.114449
$375$	−86.5479	−0.0119182
$376$	−332.205	−0.0455642
$377$	0	0
$378$	−3752.42	−0.510592
$379$	10945.5	1.48347	0.741734	−	0.670694i	$-0.234003\pi$
$379$	10945.5	1.48347	0.741734	+	0.670694i	$0.234003\pi$
$380$	48.1971	0.00650648
$381$	−285.773	−0.0384267
$382$	13090.9	1.75337
$383$	8159.93	1.08865	0.544325	−	0.838875i	$-0.316786\pi$
$383$	8159.93	1.08865	0.544325	+	0.838875i	$0.316786\pi$
$384$	−96.9521	−0.0128843
$385$	67.4339	0.00892662
$386$	2374.94	0.313164
$387$	−10160.0	−1.33453
$388$	−8057.41	−1.05426
$389$	5139.56	0.669887	0.334943	−	0.942238i	$-0.391283\pi$
$389$	5139.56	0.669887	0.334943	+	0.942238i	$0.391283\pi$
$390$	0	0
$391$	−3721.62	−0.481356
$392$	21.0011	0.00270591
$393$	−1651.79	−0.212015
$394$	−13042.9	−1.66775
$395$	−299.850	−0.0381951
$396$	−2233.14	−0.283383
$397$	13809.5	1.74579	0.872896	−	0.487907i	$-0.162240\pi$
$397$	13809.5	1.74579	0.872896	+	0.487907i	$0.162240\pi$
$398$	−6680.66	−0.841385
$399$	−318.268	−0.0399331
$400$	8177.65	1.02221
$401$	−7503.51	−0.934433	−0.467216	−	0.884143i	$-0.654743\pi$
$401$	−7503.51	−0.934433	−0.467216	+	0.884143i	$0.654743\pi$
$402$	−3464.84	−0.429876
$403$	0	0
$404$	12148.2	1.49603
$405$	−223.931	−0.0274747
$406$	−6693.55	−0.818215
$407$	−2367.79	−0.288371
$408$	−14.3394	−0.00173996
$409$	9398.53	1.13625	0.568126	−	0.822941i	$-0.307669\pi$
$409$	9398.53	1.13625	0.568126	+	0.822941i	$0.307669\pi$
$410$	−464.237	−0.0559196
$411$	212.079	0.0254527
$412$	−13023.0	−1.55728
$413$	14132.1	1.68376
$414$	−20324.0	−2.41273
$415$	−351.979	−0.0416337
$416$	0	0
$417$	1888.59	0.221785
$418$	−781.686	−0.0914678
$419$	−13391.1	−1.56133	−0.780664	−	0.624951i	$-0.785119\pi$
$419$	−13391.1	−1.56133	−0.780664	+	0.624951i	$0.785119\pi$
$420$	−48.0250	−0.00557947
$421$	6154.68	0.712495	0.356248	−	0.934392i	$-0.384056\pi$
$421$	6154.68	0.712495	0.356248	+	0.934392i	$0.384056\pi$
$422$	−8668.04	−0.999890
$423$	−11432.4	−1.31409
$424$	442.571	0.0506914
$425$	2363.50	0.269757
$426$	−3159.10	−0.359293
$427$	−6172.16	−0.699512
$428$	6350.73	0.717229
$429$	0	0
$430$	−536.644	−0.0601843
$431$	3945.66	0.440965	0.220482	−	0.975391i	$-0.429237\pi$
$431$	3945.66	0.440965	0.220482	+	0.975391i	$0.429237\pi$
$432$	−3480.86	−0.387669
$433$	−9096.00	−1.00953	−0.504764	−	0.863257i	$-0.668421\pi$
$433$	−9096.00	−1.00953	−0.504764	+	0.863257i	$0.668421\pi$
$434$	−8183.96	−0.905168
$435$	32.8415	0.00361984
$436$	−12161.1	−1.33580
$437$	−3514.35	−0.384700
$438$	−1365.34	−0.148946
$439$	−17691.5	−1.92339	−0.961696	−	0.274119i	$-0.911614\pi$
$439$	−17691.5	−1.92339	−0.961696	+	0.274119i	$0.911614\pi$
$440$	2.86899	0.000310849 0
$441$	722.723	0.0780394
$442$	0	0
$443$	7394.32	0.793035	0.396518	−	0.918027i	$-0.370219\pi$
$443$	7394.32	0.793035	0.396518	+	0.918027i	$0.370219\pi$
$444$	1686.29	0.180243
$445$	163.021	0.0173662
$446$	−668.983	−0.0710253
$447$	−1135.54	−0.120154
$448$	8653.23	0.912560
$449$	−4963.11	−0.521656	−0.260828	−	0.965385i	$-0.583996\pi$
$449$	−4963.11	−0.521656	−0.260828	+	0.965385i	$0.583996\pi$
$450$	12907.2	1.35212
$451$	3719.39	0.388335
$452$	−6662.76	−0.693341
$453$	−19.4661	−0.00201897
$454$	11462.8	1.18497
$455$	0	0
$456$	−13.5408	−0.00139058
$457$	1719.90	0.176048	0.0880238	−	0.996118i	$-0.471945\pi$
$457$	1719.90	0.176048	0.0880238	+	0.996118i	$0.471945\pi$
$458$	−762.019	−0.0777441
$459$	−1006.04	−0.102305
$460$	−530.297	−0.0537505
$461$	−3452.06	−0.348760	−0.174380	−	0.984678i	$-0.555792\pi$
$461$	−3452.06	−0.348760	−0.174380	+	0.984678i	$0.555792\pi$
$462$	778.894	0.0784360
$463$	−6323.57	−0.634733	−0.317366	−	0.948303i	$-0.602798\pi$
$463$	−6323.57	−0.634733	−0.317366	+	0.948303i	$0.602798\pi$
$464$	−6209.15	−0.621234
$465$	40.1541	0.00400452
$466$	5667.86	0.563430
$467$	2084.12	0.206513	0.103256	−	0.994655i	$-0.467074\pi$
$467$	2084.12	0.206513	0.103256	+	0.994655i	$0.467074\pi$
$468$	0	0
$469$	−15423.3	−1.51851
$470$	−603.846	−0.0592624
$471$	78.9362	0.00772227
$472$	601.252	0.0586332
$473$	4299.49	0.417951
$474$	−3463.41	−0.335611
$475$	2231.87	0.215590
$476$	2624.24	0.252693
$477$	15230.4	1.46196
$478$	1545.14	0.147852
$479$	4579.51	0.436833	0.218416	−	0.975856i	$-0.429911\pi$
$479$	4579.51	0.436833	0.218416	+	0.975856i	$0.429911\pi$
$480$	−88.0897	−0.00837651
$481$	0	0
$482$	4167.17	0.393795
$483$	3501.80	0.329891
$484$	945.014	0.0887504
$485$	356.236	0.0333522
$486$	−8293.21	−0.774049
$487$	6734.31	0.626613	0.313307	−	0.949652i	$-0.398563\pi$
$487$	6734.31	0.626613	0.313307	+	0.949652i	$0.398563\pi$
$488$	−262.596	−0.0243589
$489$	−3148.81	−0.291194
$490$	38.1736	0.00351940
$491$	−8339.41	−0.766501	−0.383251	−	0.923644i	$-0.625195\pi$
$491$	−8339.41	−0.766501	−0.383251	+	0.923644i	$0.625195\pi$
$492$	−2648.87	−0.242724
$493$	−1794.57	−0.163942
$494$	0	0
$495$	98.7321	0.00896501
$496$	−7591.71	−0.687254
$497$	−14062.4	−1.26918
$498$	−4065.53	−0.365825
$499$	−10769.9	−0.966189	−0.483094	−	0.875568i	$-0.660487\pi$
$499$	−10769.9	−0.966189	−0.483094	+	0.875568i	$0.660487\pi$
$500$	673.877	0.0602734
$501$	548.565	0.0489183
$502$	10817.9	0.961808
$503$	564.189	0.0500118	0.0250059	−	0.999687i	$-0.492040\pi$
$503$	564.189	0.0500118	0.0250059	+	0.999687i	$0.492040\pi$
$504$	−348.580	−0.0308075
$505$	−537.098	−0.0473278
$506$	8600.64	0.755622
$507$	0	0
$508$	2225.08	0.194334
$509$	6357.58	0.553625	0.276812	−	0.960924i	$-0.410722\pi$
$509$	6357.58	0.553625	0.276812	+	0.960924i	$0.410722\pi$
$510$	−26.0645	−0.00226305
$511$	−6077.65	−0.526144
$512$	16258.9	1.40342
$513$	−950.009	−0.0817620
$514$	8639.53	0.741388
$515$	575.777	0.0492656
$516$	−3062.01	−0.261235
$517$	4837.91	0.411549
$518$	15195.2	1.28888
$519$	−3948.66	−0.333963
$520$	0	0
$521$	−13026.1	−1.09537	−0.547683	−	0.836686i	$-0.684490\pi$
$521$	−13026.1	−1.09537	−0.547683	+	0.836686i	$0.684490\pi$
$522$	−9800.24	−0.821734
$523$	−1935.30	−0.161806	−0.0809031	−	0.996722i	$-0.525780\pi$
$523$	−1935.30	−0.161806	−0.0809031	+	0.996722i	$0.525780\pi$
$524$	12861.1	1.07222
$525$	−2223.90	−0.184874
$526$	−22722.5	−1.88355
$527$	−2194.15	−0.181364
$528$	722.527	0.0595530
$529$	26500.2	2.17804
$530$	804.458	0.0659310
$531$	20691.2	1.69100
$532$	2478.09	0.201953
$533$	0	0
$534$	1882.98	0.152592
$535$	−280.780	−0.0226900
$536$	−656.189	−0.0528788
$537$	−1020.59	−0.0820147
$538$	5530.16	0.443164
$539$	−305.840	−0.0244405
$540$	−143.351	−0.0114238
$541$	−4020.24	−0.319489	−0.159745	−	0.987158i	$-0.551067\pi$
$541$	−4020.24	−0.319489	−0.159745	+	0.987158i	$0.551067\pi$
$542$	816.844	0.0647351
$543$	2426.09	0.191737
$544$	4813.51	0.379370
$545$	537.667	0.0422589
$546$	0	0
$547$	−4795.87	−0.374875	−0.187438	−	0.982277i	$-0.560018\pi$
$547$	−4795.87	−0.374875	−0.187438	+	0.982277i	$0.560018\pi$
$548$	−1651.28	−0.128721
$549$	−9036.86	−0.702520
$550$	−5462.04	−0.423458
$551$	−1694.62	−0.131022
$552$	148.985	0.0114877
$553$	−15417.0	−1.18553
$554$	−17407.4	−1.33496
$555$	−74.5546	−0.00570210
$556$	−14704.9	−1.12163
$557$	950.719	0.0723218	0.0361609	−	0.999346i	$-0.488487\pi$
$557$	950.719	0.0723218	0.0361609	+	0.999346i	$0.488487\pi$
$558$	−11982.4	−0.909060
$559$	0	0
$560$	−401.438	−0.0302926
$561$	208.824	0.0157158
$562$	31782.6	2.38553
$563$	2622.84	0.196340	0.0981702	−	0.995170i	$-0.468701\pi$
$563$	2622.84	0.196340	0.0981702	+	0.995170i	$0.468701\pi$
$564$	−3445.45	−0.257234
$565$	294.575	0.0219343
$566$	6236.80	0.463167
$567$	−11513.6	−0.852778
$568$	−598.286	−0.0441964
$569$	6080.78	0.448013	0.224007	−	0.974588i	$-0.428086\pi$
$569$	6080.78	0.448013	0.224007	+	0.974588i	$0.428086\pi$
$570$	−24.6129	−0.00180864
$571$	−15913.8	−1.16632	−0.583161	−	0.812356i	$-0.698184\pi$
$571$	−15913.8	−1.16632	−0.583161	+	0.812356i	$0.698184\pi$
$572$	0	0
$573$	−3302.41	−0.240768
$574$	−23869.1	−1.73567
$575$	−24556.5	−1.78101
$576$	12669.5	0.916484
$577$	18578.9	1.34047	0.670233	−	0.742150i	$-0.266194\pi$
$577$	18578.9	1.34047	0.670233	+	0.742150i	$0.266194\pi$
$578$	−18110.7	−1.30330
$579$	−599.122	−0.0430029
$580$	−255.710	−0.0183065
$581$	−18097.3	−1.29226
$582$	4114.70	0.293058
$583$	−6445.17	−0.457859
$584$	−258.575	−0.0183218
$585$	0	0
$586$	−14059.1	−0.991084
$587$	27182.1	1.91129	0.955643	−	0.294526i	$-0.0951619\pi$
$587$	27182.1	1.91129	0.955643	+	0.294526i	$0.0951619\pi$
$588$	217.813	0.0152763
$589$	−2071.95	−0.144946
$590$	1092.89	0.0762603
$591$	3290.31	0.229011
$592$	14095.6	0.978590
$593$	23344.7	1.61661	0.808305	−	0.588764i	$-0.200385\pi$
$593$	23344.7	1.61661	0.808305	+	0.588764i	$0.200385\pi$
$594$	2324.95	0.160596
$595$	−116.023	−0.00799411
$596$	8841.48	0.607653
$597$	1685.32	0.115537
$598$	0	0
$599$	4661.33	0.317958	0.158979	−	0.987282i	$-0.449180\pi$
$599$	4661.33	0.317958	0.158979	+	0.987282i	$0.449180\pi$
$600$	−94.6162	−0.00643782
$601$	−12321.2	−0.836257	−0.418128	−	0.908388i	$-0.637314\pi$
$601$	−12321.2	−0.836257	−0.418128	+	0.908388i	$0.637314\pi$
$602$	−27591.9	−1.86804
$603$	−22581.8	−1.52504
$604$	151.566	0.0102105
$605$	−41.7811	−0.00280768
$606$	−6203.74	−0.415857
$607$	−15215.8	−1.01745	−0.508725	−	0.860929i	$-0.669883\pi$
$607$	−15215.8	−1.01745	−0.508725	+	0.860929i	$0.669883\pi$
$608$	4545.43	0.303193
$609$	1688.57	0.112355
$610$	−477.318	−0.0316821
$611$	0	0
$612$	3842.23	0.253780
$613$	−29935.3	−1.97239	−0.986194	−	0.165596i	$-0.947045\pi$
$613$	−29935.3	−1.97239	−0.986194	+	0.165596i	$0.947045\pi$
$614$	−30278.6	−1.99014
$615$	117.112	0.00767873
$616$	147.511	0.00964836
$617$	22842.6	1.49045	0.745227	−	0.666811i	$-0.232341\pi$
$617$	22842.6	1.49045	0.745227	+	0.666811i	$0.232341\pi$
$618$	6650.50	0.432884
$619$	−9191.08	−0.596802	−0.298401	−	0.954441i	$-0.596453\pi$
$619$	−9191.08	−0.596802	−0.298401	+	0.954441i	$0.596453\pi$
$620$	−312.647	−0.0202520
$621$	10452.6	0.675443
$622$	−13048.3	−0.841140
$623$	8381.86	0.539024
$624$	0	0
$625$	15580.3	0.997139
$626$	−14284.1	−0.911996
$627$	197.194	0.0125601
$628$	−614.611	−0.0390536
$629$	4073.90	0.258247
$630$	−633.611	−0.0400693
$631$	−18134.4	−1.14409	−0.572043	−	0.820224i	$-0.693849\pi$
$631$	−18134.4	−1.14409	−0.572043	+	0.820224i	$0.693849\pi$
$632$	−655.918	−0.0412833
$633$	2186.67	0.137302
$634$	−5608.17	−0.351308
$635$	−98.3755	−0.00614789
$636$	4590.11	0.286179
$637$	0	0
$638$	4147.24	0.257352
$639$	−20589.2	−1.27464
$640$	−33.3752	−0.00206136
$641$	12696.7	0.782355	0.391178	−	0.920315i	$-0.372068\pi$
$641$	12696.7	0.782355	0.391178	+	0.920315i	$0.372068\pi$
$642$	−3243.14	−0.199372
$643$	8874.40	0.544280	0.272140	−	0.962258i	$-0.412269\pi$
$643$	8874.40	0.544280	0.272140	+	0.962258i	$0.412269\pi$
$644$	−27265.6	−1.66835
$645$	135.378	0.00826435
$646$	1344.93	0.0819127
$647$	30325.4	1.84268	0.921342	−	0.388754i	$-0.127094\pi$
$647$	30325.4	1.84268	0.921342	+	0.388754i	$0.127094\pi$
$648$	−489.848	−0.0296961
$649$	−8756.04	−0.529591
$650$	0	0
$651$	2064.55	0.124295
$652$	24517.2	1.47265
$653$	29334.9	1.75798	0.878992	−	0.476836i	$-0.158217\pi$
$653$	29334.9	1.75798	0.878992	+	0.476836i	$0.158217\pi$
$654$	6210.31	0.371319
$655$	−568.619	−0.0339203
$656$	−22141.7	−1.31782
$657$	−8898.48	−0.528406
$658$	−31047.2	−1.83943
$659$	−7966.06	−0.470886	−0.235443	−	0.971888i	$-0.575654\pi$
$659$	−7966.06	−0.470886	−0.235443	+	0.971888i	$0.575654\pi$
$660$	29.7556	0.00175490
$661$	1963.72	0.115552	0.0577759	−	0.998330i	$-0.481599\pi$
$661$	1963.72	0.115552	0.0577759	+	0.998330i	$0.481599\pi$
$662$	5433.74	0.319016
$663$	0	0
$664$	−769.951	−0.0449999
$665$	−109.562	−0.00638891
$666$	22247.8	1.29442
$667$	18645.4	1.08239
$668$	−4271.22	−0.247393
$669$	168.763	0.00975301
$670$	−1192.75	−0.0687760
$671$	3824.19	0.220017
$672$	−4529.19	−0.259996
$673$	−23351.7	−1.33751	−0.668754	−	0.743484i	$-0.733172\pi$
$673$	−23351.7	−1.33751	−0.668754	+	0.743484i	$0.733172\pi$
$674$	−11217.7	−0.641083
$675$	−6638.19	−0.378525
$676$	0	0
$677$	−29559.6	−1.67809	−0.839045	−	0.544061i	$-0.816886\pi$
$677$	−29559.6	−1.67809	−0.839045	+	0.544061i	$0.816886\pi$
$678$	3402.49	0.192731
$679$	18316.1	1.03521
$680$	−4.93624	−0.000278377 0
$681$	−2891.70	−0.162717
$682$	5070.68	0.284701
$683$	−4530.76	−0.253828	−0.126914	−	0.991914i	$-0.540507\pi$
$683$	−4530.76	−0.253828	−0.126914	+	0.991914i	$0.540507\pi$
$684$	3628.25	0.202821
$685$	73.0068	0.00407219
$686$	26175.9	1.45685
$687$	192.233	0.0106756
$688$	−25595.1	−1.41832
$689$	0	0
$690$	270.808	0.0149413
$691$	10222.7	0.562791	0.281395	−	0.959592i	$-0.409203\pi$
$691$	10222.7	0.562791	0.281395	+	0.959592i	$0.409203\pi$
$692$	30744.9	1.68894
$693$	5076.38	0.278262
$694$	38952.5	2.13057
$695$	650.135	0.0354835
$696$	71.8405	0.00391251
$697$	−6399.39	−0.347768
$698$	34154.0	1.85207
$699$	−1429.82	−0.0773688
$700$	17315.7	0.934958
$701$	−10683.6	−0.575625	−0.287812	−	0.957687i	$-0.592928\pi$
$701$	−10683.6	−0.575625	−0.287812	+	0.957687i	$0.592928\pi$
$702$	0	0
$703$	3847.02	0.206391
$704$	−5361.43	−0.287026
$705$	152.331	0.00813776
$706$	27831.3	1.48363
$707$	−27615.2	−1.46899
$708$	6235.87	0.331014
$709$	−7556.09	−0.400247	−0.200123	−	0.979771i	$-0.564134\pi$
$709$	−7556.09	−0.400247	−0.200123	+	0.979771i	$0.564134\pi$
$710$	−1087.50	−0.0574833
$711$	−22572.5	−1.19063
$712$	356.608	0.0187703
$713$	22797.0	1.19741
$714$	−1340.13	−0.0702423
$715$	0	0
$716$	7946.52	0.414770
$717$	−389.790	−0.0203026
$718$	39542.5	2.05531
$719$	−17562.2	−0.910929	−0.455464	−	0.890254i	$-0.650527\pi$
$719$	−17562.2	−0.910929	−0.455464	+	0.890254i	$0.650527\pi$
$720$	−587.758	−0.0304228
$721$	29604.0	1.52914
$722$	−26002.6	−1.34033
$723$	−1051.24	−0.0540749
$724$	−18889.9	−0.969667
$725$	−11841.2	−0.606580
$726$	−482.592	−0.0246704
$727$	−9943.83	−0.507285	−0.253642	−	0.967298i	$-0.581629\pi$
$727$	−9943.83	−0.507285	−0.253642	+	0.967298i	$0.581629\pi$
$728$	0	0
$729$	−15417.8	−0.783305
$730$	−470.009	−0.0238299
$731$	−7397.49	−0.374290
$732$	−2723.51	−0.137519
$733$	−23341.4	−1.17617	−0.588086	−	0.808798i	$-0.700118\pi$
$733$	−23341.4	−1.17617	−0.588086	+	0.808798i	$0.700118\pi$
$734$	4515.74	0.227083
$735$	−9.62997	−0.000483274 0
$736$	−50011.9	−2.50470
$737$	9556.09	0.477616
$738$	−34947.5	−1.74314
$739$	25326.3	1.26068	0.630340	−	0.776319i	$-0.282916\pi$
$739$	25326.3	1.26068	0.630340	+	0.776319i	$0.282916\pi$
$740$	580.495	0.0288371
$741$	0	0
$742$	41361.7	2.04641
$743$	−2963.26	−0.146314	−0.0731571	−	0.997320i	$-0.523307\pi$
$743$	−2963.26	−0.146314	−0.0731571	+	0.997320i	$0.523307\pi$
$744$	87.8368	0.00432830
$745$	−390.901	−0.0192235
$746$	14762.3	0.724511
$747$	−26496.8	−1.29781
$748$	−1625.94	−0.0794792
$749$	−14436.5	−0.704269
$750$	−344.130	−0.0167545
$751$	13183.0	0.640551	0.320276	−	0.947324i	$-0.396225\pi$
$751$	13183.0	0.640551	0.320276	+	0.947324i	$0.396225\pi$
$752$	−28800.3	−1.39660
$753$	−2729.02	−0.132073
$754$	0	0
$755$	−6.70107	−0.000323016 0
$756$	−7370.52	−0.354581
$757$	−15734.2	−0.755440	−0.377720	−	0.925920i	$-0.623292\pi$
$757$	−15734.2	−0.755440	−0.377720	+	0.925920i	$0.623292\pi$
$758$	43521.5	2.08545
$759$	−2169.67	−0.103760
$760$	−4.66133	−0.000222479 0
$761$	12567.4	0.598644	0.299322	−	0.954152i	$-0.403240\pi$
$761$	12567.4	0.598644	0.299322	+	0.954152i	$0.403240\pi$
$762$	−1136.28	−0.0540200
$763$	27644.5	1.31166
$764$	25713.2	1.21763
$765$	−169.874	−0.00802849
$766$	32445.4	1.53042
$767$	0	0
$768$	−4296.67	−0.201878
$769$	−4592.72	−0.215367	−0.107684	−	0.994185i	$-0.534343\pi$
$769$	−4592.72	−0.215367	−0.107684	+	0.994185i	$0.534343\pi$
$770$	268.130	0.0125490
$771$	−2179.48	−0.101805
$772$	4664.87	0.217477
$773$	−41775.9	−1.94382	−0.971910	−	0.235351i	$-0.924376\pi$
$773$	−41775.9	−1.94382	−0.971910	+	0.235351i	$0.924376\pi$
$774$	−40398.2	−1.87608
$775$	−14477.8	−0.671042
$776$	779.262	0.0360488
$777$	−3833.28	−0.176986
$778$	20435.8	0.941722
$779$	−6042.99	−0.277937
$780$	0	0
$781$	8712.86	0.399194
$782$	−14797.8	−0.676687
$783$	5040.27	0.230044
$784$	1820.68	0.0829392
$785$	27.1733	0.00123549
$786$	−6567.83	−0.298049
$787$	9118.11	0.412993	0.206497	−	0.978447i	$-0.433794\pi$
$787$	9118.11	0.412993	0.206497	+	0.978447i	$0.433794\pi$
$788$	−25619.0	−1.15817
$789$	5732.16	0.258644
$790$	−1192.26	−0.0536945
$791$	15145.8	0.680812
$792$	215.976	0.00968985
$793$	0	0
$794$	54909.1	2.45422
$795$	−202.939	−0.00905347
$796$	−13122.2	−0.584300
$797$	4843.97	0.215285	0.107643	−	0.994190i	$-0.465670\pi$
$797$	4843.97	0.215285	0.107643	+	0.994190i	$0.465670\pi$
$798$	−1265.49	−0.0561377
$799$	−8323.86	−0.368557
$800$	31761.2	1.40366
$801$	12272.2	0.541342
$802$	−29835.3	−1.31362
$803$	3765.63	0.165487
$804$	−6805.64	−0.298528
$805$	1205.47	0.0527793
$806$	0	0
$807$	−1395.08	−0.0608541
$808$	−1174.90	−0.0511543
$809$	36278.8	1.57663	0.788317	−	0.615270i	$-0.210953\pi$
$809$	36278.8	1.57663	0.788317	+	0.615270i	$0.210953\pi$
$810$	−890.393	−0.0386237
$811$	36038.7	1.56041	0.780203	−	0.625527i	$-0.215116\pi$
$811$	36038.7	1.56041	0.780203	+	0.625527i	$0.215116\pi$
$812$	−13147.5	−0.568210
$813$	−206.064	−0.00888926
$814$	−9414.77	−0.405390
$815$	−1083.96	−0.0465882
$816$	−1243.14	−0.0533318
$817$	−6985.51	−0.299133
$818$	37370.3	1.59734
$819$	0	0
$820$	−911.856	−0.0388334
$821$	2356.50	0.100174	0.0500868	−	0.998745i	$-0.484050\pi$
$821$	2356.50	0.100174	0.0500868	+	0.998745i	$0.484050\pi$
$822$	843.264	0.0357813
$823$	32938.1	1.39508	0.697539	−	0.716546i	$-0.254278\pi$
$823$	32938.1	1.39508	0.697539	+	0.716546i	$0.254278\pi$
$824$	1259.51	0.0532488
$825$	1377.90	0.0581482
$826$	56191.7	2.36702
$827$	−11298.0	−0.475053	−0.237526	−	0.971381i	$-0.576337\pi$
$827$	−11298.0	−0.475053	−0.237526	+	0.971381i	$0.576337\pi$
$828$	−39920.4	−1.67552
$829$	37028.9	1.55135	0.775673	−	0.631134i	$-0.217410\pi$
$829$	37028.9	1.55135	0.775673	+	0.631134i	$0.217410\pi$
$830$	−1399.53	−0.0585284
$831$	4391.32	0.183313
$832$	0	0
$833$	526.213	0.0218874
$834$	7509.37	0.311784
$835$	188.840	0.00782645
$836$	−1535.39	−0.0635199
$837$	6162.56	0.254491
$838$	−53245.3	−2.19490
$839$	−23970.0	−0.986336	−0.493168	−	0.869934i	$-0.664161\pi$
$839$	−23970.0	−0.986336	−0.493168	+	0.869934i	$0.664161\pi$
$840$	4.64468	0.000190782 0
$841$	−15398.2	−0.631358
$842$	24472.1	1.00162
$843$	−8017.74	−0.327575
$844$	−17025.8	−0.694374
$845$	0	0
$846$	−45457.1	−1.84734
$847$	−2148.21	−0.0871467
$848$	38368.5	1.55375
$849$	−1573.35	−0.0636008
$850$	9397.71	0.379222
$851$	−42327.5	−1.70501
$852$	−6205.11	−0.249511
$853$	4473.25	0.179556	0.0897780	−	0.995962i	$-0.471384\pi$
$853$	4473.25	0.179556	0.0897780	+	0.995962i	$0.471384\pi$
$854$	−24541.6	−0.983370
$855$	−160.413	−0.00641638
$856$	−614.203	−0.0245246
$857$	999.977	0.0398583	0.0199292	−	0.999801i	$-0.493656\pi$
$857$	999.977	0.0398583	0.0199292	+	0.999801i	$0.493656\pi$
$858$	0	0
$859$	−22246.9	−0.883649	−0.441824	−	0.897102i	$-0.645669\pi$
$859$	−22246.9	−0.883649	−0.441824	+	0.897102i	$0.645669\pi$
$860$	−1054.08	−0.0417950
$861$	6021.40	0.238338
$862$	15688.7	0.619905
$863$	−15088.3	−0.595146	−0.297573	−	0.954699i	$-0.596177\pi$
$863$	−15088.3	−0.595146	−0.297573	+	0.954699i	$0.596177\pi$
$864$	−13519.3	−0.532335
$865$	−1359.30	−0.0534308
$866$	−36167.4	−1.41919
$867$	4568.76	0.178966
$868$	−16075.0	−0.628595
$869$	9552.16	0.372882
$870$	130.584	0.00508875
$871$	0	0
$872$	1176.14	0.0456756
$873$	26817.2	1.03966
$874$	−13973.7	−0.540809
$875$	−1531.86	−0.0591843
$876$	−2681.80	−0.103436
$877$	246.007	0.00947213	0.00473606	−	0.999989i	$-0.498492\pi$
$877$	246.007	0.00947213	0.00473606	+	0.999989i	$0.498492\pi$
$878$	−70344.6	−2.70389
$879$	3546.66	0.136093
$880$	248.726	0.00952788
$881$	41960.1	1.60462	0.802312	−	0.596906i	$-0.203603\pi$
$881$	41960.1	1.60462	0.802312	+	0.596906i	$0.203603\pi$
$882$	2873.68	0.109707
$883$	−42534.3	−1.62106	−0.810529	−	0.585699i	$-0.800820\pi$
$883$	−42534.3	−1.62106	−0.810529	+	0.585699i	$0.800820\pi$
$884$	0	0
$885$	−275.701	−0.0104719
$886$	29401.2	1.11484
$887$	−22011.5	−0.833229	−0.416614	−	0.909083i	$-0.636783\pi$
$887$	−22011.5	−0.833229	−0.416614	+	0.909083i	$0.636783\pi$
$888$	−163.087	−0.00616313
$889$	−5058.04	−0.190823
$890$	648.203	0.0244133
$891$	7133.67	0.268223
$892$	−1314.02	−0.0493236
$893$	−7860.28	−0.294551
$894$	−4515.10	−0.168912
$895$	−351.333	−0.0131215
$896$	−1716.01	−0.0639819
$897$	0	0
$898$	−19734.2	−0.733340
$899$	10992.7	0.407818
$900$	25352.4	0.938978
$901$	11089.2	0.410029
$902$	14789.0	0.545918
$903$	6960.55	0.256515
$904$	644.381	0.0237077
$905$	835.165	0.0306761
$906$	−77.4006	−0.00283826
$907$	21613.0	0.791234	0.395617	−	0.918416i	$-0.370531\pi$
$907$	21613.0	0.791234	0.395617	+	0.918416i	$0.370531\pi$
$908$	22515.2	0.822902
$909$	−40432.4	−1.47531
$910$	0	0
$911$	28072.1	1.02093	0.510466	−	0.859898i	$-0.329473\pi$
$911$	28072.1	1.02093	0.510466	+	0.859898i	$0.329473\pi$
$912$	−1173.91	−0.0426229
$913$	11212.8	0.406451
$914$	6838.66	0.247487
$915$	120.412	0.00435050
$916$	−1496.76	−0.0539895
$917$	−29236.0	−1.05284
$918$	−4000.19	−0.143819
$919$	1135.13	0.0407450	0.0203725	−	0.999792i	$-0.493515\pi$
$919$	1135.13	0.0407450	0.0203725	+	0.999792i	$0.493515\pi$
$920$	51.2870	0.00183792
$921$	7638.33	0.273281
$922$	−13726.0	−0.490285
$923$	0	0
$924$	1529.91	0.0544700
$925$	26881.1	0.955507
$926$	−25143.7	−0.892303
$927$	43344.1	1.53571
$928$	−24115.8	−0.853059
$929$	7945.48	0.280606	0.140303	−	0.990109i	$-0.455192\pi$
$929$	7945.48	0.280606	0.140303	+	0.990109i	$0.455192\pi$
$930$	159.660	0.00562953
$931$	496.906	0.0174924
$932$	11132.8	0.391275
$933$	3291.67	0.115503
$934$	8286.83	0.290314
$935$	71.8866	0.00251438
$936$	0	0
$937$	−5455.25	−0.190198	−0.0950988	−	0.995468i	$-0.530317\pi$
$937$	−5455.25	−0.190198	−0.0950988	+	0.995468i	$0.530317\pi$
$938$	−61326.0	−2.13472
$939$	3603.43	0.125233
$940$	−1186.08	−0.0411548
$941$	−23398.9	−0.810608	−0.405304	−	0.914182i	$-0.632834\pi$
$941$	−23398.9	−0.810608	−0.405304	+	0.914182i	$0.632834\pi$
$942$	313.865	0.0108559
$943$	66489.0	2.29606
$944$	52125.2	1.79717
$945$	325.867	0.0112174
$946$	17095.6	0.587553
$947$	22362.6	0.767357	0.383679	−	0.923467i	$-0.374657\pi$
$947$	22362.6	0.767357	0.383679	+	0.923467i	$0.374657\pi$
$948$	−6802.84	−0.233065
$949$	0	0
$950$	8874.32	0.303075
$951$	1414.76	0.0482406
$952$	−253.800	−0.00864045
$953$	−41159.0	−1.39903	−0.699513	−	0.714620i	$-0.746600\pi$
$953$	−41159.0	−1.39903	−0.699513	+	0.714620i	$0.746600\pi$
$954$	60559.1	2.05521
$955$	−1136.84	−0.0385206
$956$	3034.97	0.102676
$957$	−1046.22	−0.0353389
$958$	18209.0	0.614097
$959$	3753.70	0.126395
$960$	−168.815	−0.00567551
$961$	−16350.6	−0.548842
$962$	0	0
$963$	−21136.9	−0.707297
$964$	8185.17	0.273471
$965$	−206.244	−0.00688003
$966$	13923.8	0.463758
$967$	16577.3	0.551284	0.275642	−	0.961260i	$-0.411110\pi$
$967$	16577.3	0.551284	0.275642	+	0.961260i	$0.411110\pi$
$968$	−91.3959	−0.00303468
$969$	−339.283	−0.0112480
$970$	1416.46	0.0468863
$971$	−1715.58	−0.0567000	−0.0283500	−	0.999598i	$-0.509025\pi$
$971$	−1715.58	−0.0567000	−0.0283500	+	0.999598i	$0.509025\pi$
$972$	−16289.6	−0.537539
$973$	33427.1	1.10136
$974$	26776.8	0.880889
$975$	0	0
$976$	−22765.6	−0.746629
$977$	29160.5	0.954890	0.477445	−	0.878662i	$-0.341563\pi$
$977$	29160.5	0.954890	0.477445	+	0.878662i	$0.341563\pi$
$978$	−12520.2	−0.409359
$979$	−5193.29	−0.169539
$980$	74.9806	0.00244405
$981$	40475.2	1.31730
$982$	−33159.0	−1.07754
$983$	53729.0	1.74332	0.871662	−	0.490107i	$-0.163042\pi$
$983$	53729.0	1.74332	0.871662	+	0.490107i	$0.163042\pi$
$984$	256.182	0.00829957
$985$	1132.67	0.0366395
$986$	−7135.53	−0.230468
$987$	7832.21	0.252585
$988$	0	0
$989$	76859.2	2.47116
$990$	392.577	0.0126029
$991$	4991.12	0.159988	0.0799940	−	0.996795i	$-0.474510\pi$
$991$	4991.12	0.159988	0.0799940	+	0.996795i	$0.474510\pi$
$992$	−29485.5	−0.943714
$993$	−1370.76	−0.0438064
$994$	−55914.6	−1.78421
$995$	580.160	0.0184847
$996$	−7985.53	−0.254047
$997$	11498.7	0.365264	0.182632	−	0.983181i	$-0.441538\pi$
$997$	11498.7	0.365264	0.182632	+	0.983181i	$0.441538\pi$
$998$	−42823.2	−1.35826
$999$	−11442.1	−0.362374

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1859.4.a.k.1.15		18
13.5	odd	4		143.4.b.a.12.8	✓	36
13.8	odd	4		143.4.b.a.12.29	yes	36
13.12	even	2		1859.4.a.j.1.4		18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
143.4.b.a.12.8	✓	36	13.5	odd	4
143.4.b.a.12.29	yes	36	13.8	odd	4
1859.4.a.j.1.4		18	13.12	even	2
1859.4.a.k.1.15		18	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

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	1859.4.a.k.1.15

Level	$1859$
Weight	$4$
Character	1859.1
Self dual	yes
Analytic conductor	$109.685$
Analytic rank	$0$
Dimension	$18$
CM	no
Inner twists	$1$