LMFDB - Embedded newform 2001.4.a.a.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2001,4,Mod(1,2001)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2001, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("2001.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2001 = 3 \cdot 23 \cdot 29 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2001.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:	$118.062821921$
Analytic rank:	$1$
Dimension:	$31$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Character	$\chi$	$=$	2001.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.36157 q^{2} +3.00000 q^{3} +3.30018 q^{4} +17.6011 q^{5} -10.0847 q^{6} +3.08404 q^{7} +15.7988 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q-3.36157 q^{2} +3.00000 q^{3} +3.30018 q^{4} +17.6011 q^{5} -10.0847 q^{6} +3.08404 q^{7} +15.7988 q^{8} +9.00000 q^{9} -59.1674 q^{10} -5.14153 q^{11} +9.90053 q^{12} -16.2143 q^{13} -10.3672 q^{14} +52.8033 q^{15} -79.5102 q^{16} -77.3052 q^{17} -30.2542 q^{18} -55.8110 q^{19} +58.0868 q^{20} +9.25212 q^{21} +17.2836 q^{22} +23.0000 q^{23} +47.3964 q^{24} +184.799 q^{25} +54.5054 q^{26} +27.0000 q^{27} +10.1779 q^{28} +29.0000 q^{29} -177.502 q^{30} +314.679 q^{31} +140.889 q^{32} -15.4246 q^{33} +259.867 q^{34} +54.2825 q^{35} +29.7016 q^{36} -286.386 q^{37} +187.613 q^{38} -48.6428 q^{39} +278.077 q^{40} -300.623 q^{41} -31.1017 q^{42} +21.0086 q^{43} -16.9679 q^{44} +158.410 q^{45} -77.3162 q^{46} -560.353 q^{47} -238.531 q^{48} -333.489 q^{49} -621.216 q^{50} -231.916 q^{51} -53.5099 q^{52} +199.189 q^{53} -90.7625 q^{54} -90.4966 q^{55} +48.7241 q^{56} -167.433 q^{57} -97.4856 q^{58} -671.319 q^{59} +174.260 q^{60} -686.061 q^{61} -1057.82 q^{62} +27.7564 q^{63} +162.473 q^{64} -285.389 q^{65} +51.8509 q^{66} -644.829 q^{67} -255.121 q^{68} +69.0000 q^{69} -182.475 q^{70} +365.645 q^{71} +142.189 q^{72} +1082.99 q^{73} +962.707 q^{74} +554.398 q^{75} -184.186 q^{76} -15.8567 q^{77} +163.516 q^{78} +455.825 q^{79} -1399.47 q^{80} +81.0000 q^{81} +1010.57 q^{82} -446.961 q^{83} +30.5336 q^{84} -1360.66 q^{85} -70.6219 q^{86} +87.0000 q^{87} -81.2300 q^{88} -242.655 q^{89} -532.507 q^{90} -50.0054 q^{91} +75.9041 q^{92} +944.037 q^{93} +1883.67 q^{94} -982.335 q^{95} +422.667 q^{96} +944.694 q^{97} +1121.05 q^{98} -46.2737 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 31 q - 10 q^{2} + 93 q^{3} + 74 q^{4} - 25 q^{5} - 30 q^{6} - 76 q^{7} - 117 q^{8} + 279 q^{9}+O(q^{10}) $ 31 * q - 10 * q^2 + 93 * q^3 + 74 * q^4 - 25 * q^5 - 30 * q^6 - 76 * q^7 - 117 * q^8 + 279 * q^9	\( 31 q - 10 q^{2} + 93 q^{3} + 74 q^{4} - 25 q^{5} - 30 q^{6} - 76 q^{7} - 117 q^{8} + 279 q^{9} - 6 q^{10} - 109 q^{11} + 222 q^{12} - 317 q^{13} - 159 q^{14} - 75 q^{15} + 54 q^{16} - 180 q^{17} - 90 q^{18} - 217 q^{19} - 193 q^{20} - 228 q^{21} - 160 q^{22} + 713 q^{23} - 351 q^{24} + 208 q^{25} + 211 q^{26} + 837 q^{27} + 5 q^{28} + 899 q^{29} - 18 q^{30} - 622 q^{31} - 696 q^{32} - 327 q^{33} + 126 q^{34} - 627 q^{35} + 666 q^{36} - 539 q^{37} + 35 q^{38} - 951 q^{39} + 371 q^{40} - 459 q^{41} - 477 q^{42} - 719 q^{43} - 876 q^{44} - 225 q^{45} - 230 q^{46} - 1384 q^{47} + 162 q^{48} - 747 q^{49} - 2005 q^{50} - 540 q^{51} - 2190 q^{52} - 730 q^{53} - 270 q^{54} - 1408 q^{55} - 1107 q^{56} - 651 q^{57} - 290 q^{58} - 3765 q^{59} - 579 q^{60} - 1300 q^{61} - 70 q^{62} - 684 q^{63} - 813 q^{64} - 2490 q^{65} - 480 q^{66} - 2735 q^{67} - 2031 q^{68} + 2139 q^{69} - 2581 q^{70} - 4800 q^{71} - 1053 q^{72} - 1858 q^{73} - 801 q^{74} + 624 q^{75} - 2699 q^{76} - 1661 q^{77} + 633 q^{78} - 4466 q^{79} + 3587 q^{80} + 2511 q^{81} - 5615 q^{82} - 1234 q^{83} + 15 q^{84} - 3623 q^{85} - 1873 q^{86} + 2697 q^{87} - 6893 q^{88} - 455 q^{89} - 54 q^{90} - 2425 q^{91} + 1702 q^{92} - 1866 q^{93} - 3912 q^{94} - 1851 q^{95} - 2088 q^{96} - 2114 q^{97} + 1769 q^{98} - 981 q^{99}+O(q^{100}) \) 31 * q - 10 * q^2 + 93 * q^3 + 74 * q^4 - 25 * q^5 - 30 * q^6 - 76 * q^7 - 117 * q^8 + 279 * q^9 - 6 * q^10 - 109 * q^11 + 222 * q^12 - 317 * q^13 - 159 * q^14 - 75 * q^15 + 54 * q^16 - 180 * q^17 - 90 * q^18 - 217 * q^19 - 193 * q^20 - 228 * q^21 - 160 * q^22 + 713 * q^23 - 351 * q^24 + 208 * q^25 + 211 * q^26 + 837 * q^27 + 5 * q^28 + 899 * q^29 - 18 * q^30 - 622 * q^31 - 696 * q^32 - 327 * q^33 + 126 * q^34 - 627 * q^35 + 666 * q^36 - 539 * q^37 + 35 * q^38 - 951 * q^39 + 371 * q^40 - 459 * q^41 - 477 * q^42 - 719 * q^43 - 876 * q^44 - 225 * q^45 - 230 * q^46 - 1384 * q^47 + 162 * q^48 - 747 * q^49 - 2005 * q^50 - 540 * q^51 - 2190 * q^52 - 730 * q^53 - 270 * q^54 - 1408 * q^55 - 1107 * q^56 - 651 * q^57 - 290 * q^58 - 3765 * q^59 - 579 * q^60 - 1300 * q^61 - 70 * q^62 - 684 * q^63 - 813 * q^64 - 2490 * q^65 - 480 * q^66 - 2735 * q^67 - 2031 * q^68 + 2139 * q^69 - 2581 * q^70 - 4800 * q^71 - 1053 * q^72 - 1858 * q^73 - 801 * q^74 + 624 * q^75 - 2699 * q^76 - 1661 * q^77 + 633 * q^78 - 4466 * q^79 + 3587 * q^80 + 2511 * q^81 - 5615 * q^82 - 1234 * q^83 + 15 * q^84 - 3623 * q^85 - 1873 * q^86 + 2697 * q^87 - 6893 * q^88 - 455 * q^89 - 54 * q^90 - 2425 * q^91 + 1702 * q^92 - 1866 * q^93 - 3912 * q^94 - 1851 * q^95 - 2088 * q^96 - 2114 * q^97 + 1769 * q^98 - 981 * q^99

Display 100 coefficients 10 coefficients

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 $

$2$	−3.36157	−1.18850	−0.594248	−	0.804282i	$-0.702550\pi$
$2$	−3.36157	−1.18850	−0.594248	+	0.804282i	$0.702550\pi$
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	3.30018	0.412522
$5$	17.6011	1.57429	0.787146	−	0.616767i	$-0.211558\pi$
$5$	17.6011	1.57429	0.787146	+	0.616767i	$0.211558\pi$
$6$	−10.0847	−0.686178
$7$	3.08404	0.166522	0.0832612	−	0.996528i	$-0.473466\pi$
$7$	3.08404	0.166522	0.0832612	+	0.996528i	$0.473466\pi$
$8$	15.7988	0.698215
$9$	9.00000	0.333333
$10$	−59.1674	−1.87104
$11$	−5.14153	−0.140930	−0.0704650	−	0.997514i	$-0.522448\pi$
$11$	−5.14153	−0.140930	−0.0704650	+	0.997514i	$0.522448\pi$
$12$	9.90053	0.238170
$13$	−16.2143	−0.345925	−0.172963	−	0.984928i	$-0.555334\pi$
$13$	−16.2143	−0.345925	−0.172963	+	0.984928i	$0.555334\pi$
$14$	−10.3672	−0.197911
$15$	52.8033	0.908918
$16$	−79.5102	−1.24235
$17$	−77.3052	−1.10290	−0.551449	−	0.834209i	$-0.685925\pi$
$17$	−77.3052	−1.10290	−0.551449	+	0.834209i	$0.685925\pi$
$18$	−30.2542	−0.396165
$19$	−55.8110	−0.673890	−0.336945	−	0.941524i	$-0.609394\pi$
$19$	−55.8110	−0.673890	−0.336945	+	0.941524i	$0.609394\pi$
$20$	58.0868	0.649430
$21$	9.25212	0.0961418
$22$	17.2836	0.167495
$23$	23.0000	0.208514
$23$	23.0000	0.208514
$24$	47.3964	0.403115
$25$	184.799	1.47839
$26$	54.5054	0.411130
$27$	27.0000	0.192450
$28$	10.1779	0.0686942
$29$	29.0000	0.185695
$29$	29.0000	0.185695
$30$	−177.502	−1.08024
$31$	314.679	1.82316	0.911581	−	0.411121i	$-0.134863\pi$
$31$	314.679	1.82316	0.911581	+	0.411121i	$0.134863\pi$
$32$	140.889	0.778310
$33$	−15.4246	−0.0813659
$34$	259.867	1.31079
$35$	54.2825	0.262155
$36$	29.7016	0.137507
$37$	−286.386	−1.27247	−0.636237	−	0.771493i	$-0.719510\pi$
$37$	−286.386	−1.27247	−0.636237	+	0.771493i	$0.719510\pi$
$38$	187.613	0.800916
$39$	−48.6428	−0.199720
$40$	278.077	1.09919
$41$	−300.623	−1.14511	−0.572554	−	0.819867i	$-0.694047\pi$
$41$	−300.623	−1.14511	−0.572554	+	0.819867i	$0.694047\pi$
$42$	−31.1017	−0.114264
$43$	21.0086	0.0745065	0.0372532	−	0.999306i	$-0.488139\pi$
$43$	21.0086	0.0745065	0.0372532	+	0.999306i	$0.488139\pi$
$44$	−16.9679	−0.0581367
$45$	158.410	0.524764
$46$	−77.3162	−0.247818
$47$	−560.353	−1.73906	−0.869531	−	0.493879i	$-0.835579\pi$
$47$	−560.353	−1.73906	−0.869531	+	0.493879i	$0.835579\pi$
$48$	−238.531	−0.717270
$49$	−333.489	−0.972270
$50$	−621.216	−1.75706
$51$	−231.916	−0.636759
$52$	−53.5099	−0.142702
$53$	199.189	0.516240	0.258120	−	0.966113i	$-0.416897\pi$
$53$	199.189	0.516240	0.258120	+	0.966113i	$0.416897\pi$
$54$	−90.7625	−0.228726
$55$	−90.4966	−0.221865
$56$	48.7241	0.116268
$57$	−167.433	−0.389071
$58$	−97.4856	−0.220698
$59$	−671.319	−1.48133	−0.740663	−	0.671877i	$-0.765488\pi$
$59$	−671.319	−1.48133	−0.740663	+	0.671877i	$0.765488\pi$
$60$	174.260	0.374949
$61$	−686.061	−1.44002	−0.720009	−	0.693965i	$-0.755862\pi$
$61$	−686.061	−1.44002	−0.720009	+	0.693965i	$0.755862\pi$
$62$	−1057.82	−2.16682
$63$	27.7564	0.0555075
$64$	162.473	0.317330
$65$	−285.389	−0.544587
$66$	51.8509	0.0967030
$67$	−644.829	−1.17580	−0.587899	−	0.808935i	$-0.700045\pi$
$67$	−644.829	−1.17580	−0.587899	+	0.808935i	$0.700045\pi$
$68$	−255.121	−0.454970
$69$	69.0000	0.120386
$70$	−182.475	−0.311570
$71$	365.645	0.611184	0.305592	−	0.952163i	$-0.401146\pi$
$71$	365.645	0.611184	0.305592	+	0.952163i	$0.401146\pi$
$72$	142.189	0.232738
$73$	1082.99	1.73637	0.868184	−	0.496242i	$-0.165287\pi$
$73$	1082.99	1.73637	0.868184	+	0.496242i	$0.165287\pi$
$74$	962.707	1.51233
$75$	554.398	0.853551
$76$	−184.186	−0.277995
$77$	−15.8567	−0.0234680
$78$	163.516	0.237366
$79$	455.825	0.649168	0.324584	−	0.945857i	$-0.394776\pi$
$79$	455.825	0.649168	0.324584	+	0.945857i	$0.394776\pi$
$80$	−1399.47	−1.95582
$81$	81.0000	0.111111
$82$	1010.57	1.36096
$83$	−446.961	−0.591088	−0.295544	−	0.955329i	$-0.595501\pi$
$83$	−446.961	−0.591088	−0.295544	+	0.955329i	$0.595501\pi$
$84$	30.5336	0.0396606
$85$	−1360.66	−1.73628
$86$	−70.6219	−0.0885506
$87$	87.0000	0.107211
$88$	−81.2300	−0.0983994
$89$	−242.655	−0.289004	−0.144502	−	0.989505i	$-0.546158\pi$
$89$	−242.655	−0.289004	−0.144502	+	0.989505i	$0.546158\pi$
$90$	−532.507	−0.623680
$91$	−50.0054	−0.0576043
$92$	75.9041	0.0860168
$93$	944.037	1.05260
$94$	1883.67	2.06687
$95$	−982.335	−1.06090
$96$	422.667	0.449357
$97$	944.694	0.988857	0.494429	−	0.869218i	$-0.335377\pi$
$97$	944.694	0.988857	0.494429	+	0.869218i	$0.335377\pi$
$98$	1121.05	1.15554
$99$	−46.2737	−0.0469766
$100$	609.870	0.609870
$101$	−1815.90	−1.78899	−0.894497	−	0.447074i	$-0.852466\pi$
$101$	−1815.90	−1.78899	−0.894497	+	0.447074i	$0.852466\pi$
$102$	779.602	0.756785
$103$	−1700.82	−1.62705	−0.813526	−	0.581529i	$-0.802455\pi$
$103$	−1700.82	−1.62705	−0.813526	+	0.581529i	$0.802455\pi$
$104$	−256.166	−0.241530
$105$	162.848	0.151355
$106$	−669.589	−0.613549
$107$	−952.617	−0.860682	−0.430341	−	0.902667i	$-0.641607\pi$
$107$	−952.617	−0.860682	−0.430341	+	0.902667i	$0.641607\pi$
$108$	89.1048	0.0793899
$109$	−829.057	−0.728525	−0.364263	−	0.931296i	$-0.618679\pi$
$109$	−829.057	−0.728525	−0.364263	+	0.931296i	$0.618679\pi$
$110$	304.211	0.263685
$111$	−859.158	−0.734664
$112$	−245.213	−0.206879
$113$	117.158	0.0975337	0.0487669	−	0.998810i	$-0.484471\pi$
$113$	117.158	0.0975337	0.0487669	+	0.998810i	$0.484471\pi$
$114$	562.838	0.462409
$115$	404.826	0.328262
$116$	95.7051	0.0766034
$117$	−145.928	−0.115308
$118$	2256.69	1.76055
$119$	−238.412	−0.183657
$120$	834.230	0.634620
$121$	−1304.56	−0.980139
$122$	2306.24	1.71146
$123$	−901.869	−0.661128
$124$	1038.50	0.752095
$125$	1052.53	0.753131
$126$	−93.3050	−0.0659704
$127$	460.045	0.321436	0.160718	−	0.987000i	$-0.448619\pi$
$127$	460.045	0.321436	0.160718	+	0.987000i	$0.448619\pi$
$128$	−1673.28	−1.15545
$129$	63.0258	0.0430163
$130$	959.356	0.647239
$131$	1605.44	1.07075	0.535375	−	0.844614i	$-0.320170\pi$
$131$	1605.44	1.07075	0.535375	+	0.844614i	$0.320170\pi$
$132$	−50.9038	−0.0335652
$133$	−172.123	−0.112218
$134$	2167.64	1.39743
$135$	475.230	0.302973
$136$	−1221.33	−0.770060
$137$	−532.645	−0.332167	−0.166084	−	0.986112i	$-0.553112\pi$
$137$	−532.645	−0.332167	−0.166084	+	0.986112i	$0.553112\pi$
$138$	−231.949	−0.143078
$139$	1540.08	0.939768	0.469884	−	0.882728i	$-0.344296\pi$
$139$	1540.08	0.939768	0.469884	+	0.882728i	$0.344296\pi$
$140$	179.142	0.108145
$141$	−1681.06	−1.00405
$142$	−1229.14	−0.726389
$143$	83.3661	0.0487512
$144$	−715.592	−0.414116
$145$	510.432	0.292339
$146$	−3640.56	−2.06367
$147$	−1000.47	−0.561341
$148$	−945.124	−0.524924
$149$	1422.05	0.781870	0.390935	−	0.920418i	$-0.372152\pi$
$149$	1422.05	0.781870	0.390935	+	0.920418i	$0.372152\pi$
$150$	−1863.65	−1.01444
$151$	193.789	0.104439	0.0522195	−	0.998636i	$-0.483370\pi$
$151$	193.789	0.104439	0.0522195	+	0.998636i	$0.483370\pi$
$152$	−881.747	−0.470520
$153$	−695.747	−0.367633
$154$	53.3034	0.0278916
$155$	5538.70	2.87019
$156$	−160.530	−0.0823889
$157$	1855.65	0.943291	0.471646	−	0.881788i	$-0.343660\pi$
$157$	1855.65	0.943291	0.471646	+	0.881788i	$0.343660\pi$
$158$	−1532.29	−0.771533
$159$	597.567	0.298051
$160$	2479.81	1.22529
$161$	70.9329	0.0347223
$162$	−272.287	−0.132055
$163$	−757.642	−0.364068	−0.182034	−	0.983292i	$-0.558268\pi$
$163$	−757.642	−0.364068	−0.182034	+	0.983292i	$0.558268\pi$
$164$	−992.109	−0.472382
$165$	−271.490	−0.128094
$166$	1502.49	0.702505
$167$	3116.00	1.44385	0.721927	−	0.691969i	$-0.243257\pi$
$167$	3116.00	1.44385	0.721927	+	0.691969i	$0.243257\pi$
$168$	146.172	0.0671276
$169$	−1934.10	−0.880336
$170$	4573.95	2.06357
$171$	−502.299	−0.224630
$172$	69.3320	0.0307356
$173$	−1093.54	−0.480580	−0.240290	−	0.970701i	$-0.577242\pi$
$173$	−1093.54	−0.480580	−0.240290	+	0.970701i	$0.577242\pi$
$174$	−292.457	−0.127420
$175$	569.928	0.246186
$176$	408.804	0.175084
$177$	−2013.96	−0.855244
$178$	815.701	0.343480
$179$	51.5050	0.0215065	0.0107533	−	0.999942i	$-0.496577\pi$
$179$	51.5050	0.0215065	0.0107533	+	0.999942i	$0.496577\pi$
$180$	522.781	0.216477
$181$	−485.068	−0.199198	−0.0995990	−	0.995028i	$-0.531756\pi$
$181$	−485.068	−0.199198	−0.0995990	+	0.995028i	$0.531756\pi$
$182$	168.097	0.0684625
$183$	−2058.18	−0.831395
$184$	363.372	0.145588
$185$	−5040.71	−2.00325
$186$	−3173.45	−1.25101
$187$	397.467	0.155431
$188$	−1849.26	−0.717401
$189$	83.2691	0.0320473
$190$	3302.19	1.26087
$191$	2409.24	0.912705	0.456352	−	0.889799i	$-0.349156\pi$
$191$	2409.24	0.912705	0.456352	+	0.889799i	$0.349156\pi$
$192$	487.418	0.183210
$193$	1267.02	0.472550	0.236275	−	0.971686i	$-0.424073\pi$
$193$	1267.02	0.472550	0.236275	+	0.971686i	$0.424073\pi$
$194$	−3175.66	−1.17525
$195$	−856.167	−0.314417
$196$	−1100.57	−0.401083
$197$	4181.03	1.51211	0.756057	−	0.654506i	$-0.227123\pi$
$197$	4181.03	1.51211	0.756057	+	0.654506i	$0.227123\pi$
$198$	155.553	0.0558315
$199$	−5383.48	−1.91771	−0.958857	−	0.283890i	$-0.908375\pi$
$199$	−5383.48	−1.91771	−0.958857	+	0.283890i	$0.908375\pi$
$200$	2919.61	1.03224
$201$	−1934.49	−0.678847
$202$	6104.27	2.12621
$203$	89.4371	0.0309224
$204$	−765.363	−0.262677
$205$	−5291.30	−1.80273
$206$	5717.41	1.93374
$207$	207.000	0.0695048
$208$	1289.20	0.429759
$209$	286.954	0.0949713
$210$	−547.424	−0.179885
$211$	1540.32	0.502560	0.251280	−	0.967914i	$-0.419148\pi$
$211$	1540.32	0.502560	0.251280	+	0.967914i	$0.419148\pi$
$212$	657.359	0.212960
$213$	1096.93	0.352867
$214$	3202.29	1.02292
$215$	369.775	0.117295
$216$	426.568	0.134372
$217$	970.482	0.303597
$218$	2786.94	0.865849
$219$	3248.98	1.00249
$220$	−298.655	−0.0915241
$221$	1253.45	0.381520
$222$	2888.12	0.873144
$223$	−2099.72	−0.630529	−0.315264	−	0.949004i	$-0.602093\pi$
$223$	−2099.72	−0.630529	−0.315264	+	0.949004i	$0.602093\pi$
$224$	434.508	0.129606
$225$	1663.19	0.492798
$226$	−393.836	−0.115918
$227$	−5058.76	−1.47913	−0.739563	−	0.673087i	$-0.764968\pi$
$227$	−5058.76	−1.47913	−0.739563	+	0.673087i	$0.764968\pi$
$228$	−552.558	−0.160500
$229$	1590.36	0.458925	0.229463	−	0.973317i	$-0.426303\pi$
$229$	1590.36	0.458925	0.229463	+	0.973317i	$0.426303\pi$
$230$	−1360.85	−0.390139
$231$	−47.5700	−0.0135493
$232$	458.165	0.129655
$233$	−1337.35	−0.376020	−0.188010	−	0.982167i	$-0.560204\pi$
$233$	−1337.35	−0.376020	−0.188010	+	0.982167i	$0.560204\pi$
$234$	490.549	0.137043
$235$	−9862.84	−2.73779
$236$	−2215.47	−0.611080
$237$	1367.47	0.374797
$238$	801.441	0.218276
$239$	1515.89	0.410270	0.205135	−	0.978734i	$-0.434237\pi$
$239$	1515.89	0.410270	0.205135	+	0.978734i	$0.434237\pi$
$240$	−4198.41	−1.12919
$241$	−3556.60	−0.950627	−0.475313	−	0.879817i	$-0.657665\pi$
$241$	−3556.60	−0.950627	−0.475313	+	0.879817i	$0.657665\pi$
$242$	4385.39	1.16489
$243$	243.000	0.0641500
$244$	−2264.12	−0.594039
$245$	−5869.77	−1.53064
$246$	3031.70	0.785748
$247$	904.934	0.233116
$248$	4971.55	1.27296
$249$	−1340.88	−0.341265
$250$	−3538.17	−0.895093
$251$	3351.92	0.842914	0.421457	−	0.906848i	$-0.361519\pi$
$251$	3351.92	0.842914	0.421457	+	0.906848i	$0.361519\pi$
$252$	91.6009	0.0228981
$253$	−118.255	−0.0293859
$254$	−1546.47	−0.382025
$255$	−4081.97	−1.00244
$256$	4325.06	1.05592
$257$	−3216.31	−0.780654	−0.390327	−	0.920676i	$-0.627638\pi$
$257$	−3216.31	−0.780654	−0.390327	+	0.920676i	$0.627638\pi$
$258$	−211.866	−0.0511247
$259$	−883.226	−0.211896
$260$	−941.834	−0.224654
$261$	261.000	0.0618984
$262$	−5396.82	−1.27258
$263$	6313.86	1.48034	0.740169	−	0.672420i	$-0.234745\pi$
$263$	6313.86	1.48034	0.740169	+	0.672420i	$0.234745\pi$
$264$	−243.690	−0.0568109
$265$	3505.95	0.812713
$266$	578.605	0.133370
$267$	−727.964	−0.166856
$268$	−2128.05	−0.485042
$269$	5077.72	1.15091	0.575454	−	0.817834i	$-0.304826\pi$
$269$	5077.72	1.15091	0.575454	+	0.817834i	$0.304826\pi$
$270$	−1597.52	−0.360082
$271$	−1770.58	−0.396883	−0.198441	−	0.980113i	$-0.563588\pi$
$271$	−1770.58	−0.396883	−0.198441	+	0.980113i	$0.563588\pi$
$272$	6146.56	1.37018
$273$	−150.016	−0.0332579
$274$	1790.52	0.394779
$275$	−950.150	−0.208350
$276$	227.712	0.0496618
$277$	−5189.43	−1.12564	−0.562821	−	0.826579i	$-0.690284\pi$
$277$	−5189.43	−1.12564	−0.562821	+	0.826579i	$0.690284\pi$
$278$	−5177.09	−1.11691
$279$	2832.11	0.607721
$280$	857.599	0.183040
$281$	6833.07	1.45063	0.725315	−	0.688417i	$-0.241694\pi$
$281$	6833.07	1.45063	0.725315	+	0.688417i	$0.241694\pi$
$282$	5651.00	1.19331
$283$	8166.69	1.71541	0.857703	−	0.514146i	$-0.171891\pi$
$283$	8166.69	1.71541	0.857703	+	0.514146i	$0.171891\pi$
$284$	1206.69	0.252127
$285$	−2947.01	−0.612511
$286$	−280.241	−0.0579406
$287$	−927.133	−0.190686
$288$	1268.00	0.259437
$289$	1063.10	0.216385
$290$	−1715.86	−0.347443
$291$	2834.08	0.570917
$292$	3574.07	0.716290
$293$	484.183	0.0965401	0.0482701	−	0.998834i	$-0.484629\pi$
$293$	484.183	0.0965401	0.0482701	+	0.998834i	$0.484629\pi$
$294$	3363.14	0.667151
$295$	−11816.0	−2.33204
$296$	−4524.55	−0.888461
$297$	−138.821	−0.0271220
$298$	−4780.32	−0.929249
$299$	−372.928	−0.0721304
$300$	1829.61	0.352109
$301$	64.7913	0.0124070
$302$	−651.435	−0.124125
$303$	−5447.69	−1.03288
$304$	4437.54	0.837206
$305$	−12075.4	−2.26701
$306$	2338.80	0.436930
$307$	−2382.30	−0.442883	−0.221441	−	0.975174i	$-0.571076\pi$
$307$	−2382.30	−0.442883	−0.221441	+	0.975174i	$0.571076\pi$
$308$	−52.3298	−0.00968107
$309$	−5102.45	−0.939379
$310$	−18618.8	−3.41121
$311$	−5875.78	−1.07133	−0.535667	−	0.844430i	$-0.679940\pi$
$311$	−5875.78	−1.07133	−0.535667	+	0.844430i	$0.679940\pi$
$312$	−768.498	−0.139447
$313$	4671.66	0.843636	0.421818	−	0.906681i	$-0.361392\pi$
$313$	4671.66	0.843636	0.421818	+	0.906681i	$0.361392\pi$
$314$	−6237.89	−1.12110
$315$	488.543	0.0873850
$316$	1504.30	0.267796
$317$	2781.96	0.492904	0.246452	−	0.969155i	$-0.420735\pi$
$317$	2781.96	0.492904	0.246452	+	0.969155i	$0.420735\pi$
$318$	−2008.77	−0.354233
$319$	−149.104	−0.0261700
$320$	2859.70	0.499570
$321$	−2857.85	−0.496915
$322$	−238.446	−0.0412673
$323$	4314.48	0.743233
$324$	267.314	0.0458358
$325$	−2996.38	−0.511413
$326$	2546.87	0.432694
$327$	−2487.17	−0.420614
$328$	−4749.48	−0.799531
$329$	−1728.15	−0.289593
$330$	912.633	0.152239
$331$	−10516.2	−1.74629	−0.873144	−	0.487462i	$-0.837923\pi$
$331$	−10516.2	−1.74629	−0.873144	+	0.487462i	$0.837923\pi$
$332$	−1475.05	−0.243837
$333$	−2577.47	−0.424158
$334$	−10474.7	−1.71601
$335$	−11349.7	−1.85105
$336$	−735.638	−0.119442
$337$	−4074.72	−0.658647	−0.329324	−	0.944217i	$-0.606821\pi$
$337$	−4074.72	−0.658647	−0.329324	+	0.944217i	$0.606821\pi$
$338$	6501.61	1.04628
$339$	351.474	0.0563111
$340$	−4490.41	−0.716255
$341$	−1617.93	−0.256938
$342$	1688.51	0.266972
$343$	−2086.32	−0.328427
$344$	331.910	0.0520216
$345$	1214.48	0.189522
$346$	3676.01	0.571167
$347$	−12093.4	−1.87092	−0.935459	−	0.353435i	$-0.885013\pi$
$347$	−12093.4	−1.87092	−0.935459	+	0.353435i	$0.885013\pi$
$348$	287.115	0.0442270
$349$	−2503.11	−0.383922	−0.191961	−	0.981403i	$-0.561485\pi$
$349$	−2503.11	−0.383922	−0.191961	+	0.981403i	$0.561485\pi$
$350$	−1915.86	−0.292591
$351$	−437.785	−0.0665733
$352$	−724.385	−0.109687
$353$	−123.180	−0.0185728	−0.00928639	−	0.999957i	$-0.502956\pi$
$353$	−123.180	−0.0185728	−0.00928639	+	0.999957i	$0.502956\pi$
$354$	6770.06	1.01645
$355$	6435.76	0.962182
$356$	−800.803	−0.119220
$357$	−715.237	−0.106035
$358$	−173.138	−0.0255604
$359$	−181.429	−0.0266726	−0.0133363	−	0.999911i	$-0.504245\pi$
$359$	−181.429	−0.0266726	−0.0133363	+	0.999911i	$0.504245\pi$
$360$	2502.69	0.366398
$361$	−3744.13	−0.545872
$362$	1630.59	0.236746
$363$	−3913.69	−0.565883
$364$	−165.027	−0.0237630
$365$	19061.9	2.73355
$366$	6918.73	0.988109
$367$	−6091.87	−0.866467	−0.433233	−	0.901282i	$-0.642627\pi$
$367$	−6091.87	−0.866467	−0.433233	+	0.901282i	$0.642627\pi$
$368$	−1828.74	−0.259047
$369$	−2705.61	−0.381703
$370$	16944.7	2.38085
$371$	614.307	0.0859656
$372$	3115.49	0.434222
$373$	11588.8	1.60871	0.804353	−	0.594152i	$-0.202512\pi$
$373$	11588.8	1.60871	0.804353	+	0.594152i	$0.202512\pi$
$374$	−1336.11	−0.184729
$375$	3157.60	0.434821
$376$	−8852.90	−1.21424
$377$	−470.213	−0.0642367
$378$	−279.915	−0.0380880
$379$	5176.71	0.701609	0.350804	−	0.936449i	$-0.385908\pi$
$379$	5176.71	0.701609	0.350804	+	0.936449i	$0.385908\pi$
$380$	−3241.88	−0.437645
$381$	1380.13	0.185581
$382$	−8098.84	−1.08475
$383$	4570.08	0.609713	0.304857	−	0.952398i	$-0.401391\pi$
$383$	4570.08	0.609713	0.304857	+	0.952398i	$0.401391\pi$
$384$	−5019.83	−0.667102
$385$	−279.095	−0.0369455
$386$	−4259.18	−0.561624
$387$	189.077	0.0248355
$388$	3117.66	0.407925
$389$	−2141.04	−0.279062	−0.139531	−	0.990218i	$-0.544559\pi$
$389$	−2141.04	−0.279062	−0.139531	+	0.990218i	$0.544559\pi$
$390$	2878.07	0.373684
$391$	−1778.02	−0.229970
$392$	−5268.72	−0.678854
$393$	4816.33	0.618198
$394$	−14054.9	−1.79714
$395$	8023.02	1.02198
$396$	−152.712	−0.0193789
$397$	−15660.9	−1.97984	−0.989920	−	0.141630i	$-0.954766\pi$
$397$	−15660.9	−1.97984	−0.989920	+	0.141630i	$0.954766\pi$
$398$	18097.0	2.27919
$399$	−516.370	−0.0647890
$400$	−14693.4	−1.83668
$401$	475.237	0.0591825	0.0295913	−	0.999562i	$-0.490579\pi$
$401$	475.237	0.0591825	0.0295913	+	0.999562i	$0.490579\pi$
$402$	6502.92	0.806807
$403$	−5102.29	−0.630677
$404$	−5992.78	−0.738000
$405$	1425.69	0.174921
$406$	−300.650	−0.0367512
$407$	1472.46	0.179330
$408$	−3663.99	−0.444594
$409$	5995.97	0.724893	0.362447	−	0.932005i	$-0.381941\pi$
$409$	5995.97	0.724893	0.362447	+	0.932005i	$0.381941\pi$
$410$	17787.1	2.14254
$411$	−1597.93	−0.191777
$412$	−5612.99	−0.671195
$413$	−2070.37	−0.246674
$414$	−695.846	−0.0826062
$415$	−7867.00	−0.930545
$416$	−2284.41	−0.269237
$417$	4620.23	0.542575
$418$	−964.616	−0.112873
$419$	4713.72	0.549595	0.274797	−	0.961502i	$-0.411389\pi$
$419$	4713.72	0.549595	0.274797	+	0.961502i	$0.411389\pi$
$420$	537.426	0.0624374
$421$	−1969.83	−0.228037	−0.114019	−	0.993479i	$-0.536372\pi$
$421$	−1969.83	−0.228037	−0.114019	+	0.993479i	$0.536372\pi$
$422$	−5177.91	−0.597291
$423$	−5043.18	−0.579687
$424$	3146.95	0.360447
$425$	−14285.9	−1.63052
$426$	−3687.43	−0.419381
$427$	−2115.84	−0.239795
$428$	−3143.80	−0.355050
$429$	250.098	0.0281465
$430$	−1243.02	−0.139405
$431$	6070.55	0.678442	0.339221	−	0.940707i	$-0.389837\pi$
$431$	6070.55	0.678442	0.339221	+	0.940707i	$0.389837\pi$
$432$	−2146.78	−0.239090
$433$	−14757.4	−1.63786	−0.818931	−	0.573892i	$-0.805433\pi$
$433$	−14757.4	−1.63786	−0.818931	+	0.573892i	$0.805433\pi$
$434$	−3262.35	−0.360824
$435$	1531.30	0.168782
$436$	−2736.04	−0.300533
$437$	−1283.65	−0.140516
$438$	−10921.7	−1.19146
$439$	9184.20	0.998492	0.499246	−	0.866460i	$-0.333610\pi$
$439$	9184.20	0.998492	0.499246	+	0.866460i	$0.333610\pi$
$440$	−1429.74	−0.154909
$441$	−3001.40	−0.324090
$442$	−4213.55	−0.453435
$443$	−4089.21	−0.438565	−0.219283	−	0.975661i	$-0.570372\pi$
$443$	−4089.21	−0.438565	−0.219283	+	0.975661i	$0.570372\pi$
$444$	−2835.37	−0.303065
$445$	−4270.99	−0.454976
$446$	7058.38	0.749381
$447$	4266.14	0.451413
$448$	501.073	0.0528425
$449$	−16835.8	−1.76956	−0.884780	−	0.466009i	$-0.845691\pi$
$449$	−16835.8	−1.76956	−0.884780	+	0.466009i	$0.845691\pi$
$450$	−5590.95	−0.585688
$451$	1545.66	0.161380
$452$	386.642	0.0402348
$453$	581.366	0.0602979
$454$	17005.4	1.75794
$455$	−880.151	−0.0906860
$456$	−2645.24	−0.271655
$457$	−3791.92	−0.388137	−0.194069	−	0.980988i	$-0.562168\pi$
$457$	−3791.92	−0.388137	−0.194069	+	0.980988i	$0.562168\pi$
$458$	−5346.11	−0.545431
$459$	−2087.24	−0.212253
$460$	1336.00	0.135416
$461$	−2400.55	−0.242527	−0.121263	−	0.992620i	$-0.538695\pi$
$461$	−2400.55	−0.242527	−0.121263	+	0.992620i	$0.538695\pi$
$462$	159.910	0.0161032
$463$	11828.9	1.18734	0.593669	−	0.804709i	$-0.297679\pi$
$463$	11828.9	1.18734	0.593669	+	0.804709i	$0.297679\pi$
$464$	−2305.80	−0.230698
$465$	16616.1	1.65710
$466$	4495.60	0.446899
$467$	1010.81	0.100160	0.0500802	−	0.998745i	$-0.484052\pi$
$467$	1010.81	0.100160	0.0500802	+	0.998745i	$0.484052\pi$
$468$	−481.589	−0.0475672
$469$	−1988.68	−0.195797
$470$	33154.6	3.25385
$471$	5566.94	0.544609
$472$	−10606.0	−1.03428
$473$	−108.016	−0.0105002
$474$	−4596.86	−0.445445
$475$	−10313.8	−0.996275
$476$	−786.803	−0.0757627
$477$	1792.70	0.172080
$478$	−5095.76	−0.487604
$479$	17547.1	1.67380	0.836899	−	0.547358i	$-0.184366\pi$
$479$	17547.1	1.67380	0.836899	+	0.547358i	$0.184366\pi$
$480$	7439.42	0.707420
$481$	4643.54	0.440181
$482$	11955.8	1.12982
$483$	212.799	0.0200469
$484$	−4305.29	−0.404329
$485$	16627.7	1.55675
$486$	−816.862	−0.0762420
$487$	−3861.57	−0.359311	−0.179655	−	0.983730i	$-0.557498\pi$
$487$	−3861.57	−0.359311	−0.179655	+	0.983730i	$0.557498\pi$
$488$	−10838.9	−1.00544
$489$	−2272.93	−0.210195
$490$	19731.7	1.81916
$491$	10952.4	1.00667	0.503333	−	0.864093i	$-0.332107\pi$
$491$	10952.4	1.00667	0.503333	+	0.864093i	$0.332107\pi$
$492$	−2976.33	−0.272730
$493$	−2241.85	−0.204803
$494$	−3042.00	−0.277057
$495$	−814.470	−0.0739549
$496$	−25020.2	−2.26500
$497$	1127.66	0.101776
$498$	4507.47	0.405592
$499$	−21290.0	−1.90996	−0.954982	−	0.296663i	$-0.904126\pi$
$499$	−21290.0	−1.90996	−0.954982	+	0.296663i	$0.904126\pi$
$500$	3473.55	0.310683
$501$	9348.01	0.833609
$502$	−11267.7	−1.00180
$503$	−22027.9	−1.95263	−0.976316	−	0.216348i	$-0.930585\pi$
$503$	−22027.9	−1.95263	−0.976316	+	0.216348i	$0.930585\pi$
$504$	438.517	0.0387562
$505$	−31961.8	−2.81640
$506$	397.523	0.0349250
$507$	−5802.29	−0.508262
$508$	1518.23	0.132599
$509$	−8486.23	−0.738989	−0.369495	−	0.929233i	$-0.620469\pi$
$509$	−8486.23	−0.738989	−0.369495	+	0.929233i	$0.620469\pi$
$510$	13721.9	1.19140
$511$	3340.00	0.289144
$512$	−1152.80	−0.0995056
$513$	−1506.90	−0.129690
$514$	10811.9	0.927804
$515$	−29936.2	−2.56145
$516$	207.996	0.0177452
$517$	2881.07	0.245086
$518$	2969.03	0.251837
$519$	−3280.62	−0.277463
$520$	−4508.80	−0.380239
$521$	12228.5	1.02829	0.514146	−	0.857703i	$-0.328109\pi$
$521$	12228.5	1.02829	0.514146	+	0.857703i	$0.328109\pi$
$522$	−877.371	−0.0735660
$523$	12831.5	1.07282	0.536408	−	0.843959i	$-0.319781\pi$
$523$	12831.5	1.07282	0.536408	+	0.843959i	$0.319781\pi$
$524$	5298.25	0.441708
$525$	1709.78	0.142135
$526$	−21224.5	−1.75938
$527$	−24326.3	−2.01076
$528$	1226.41	0.101085
$529$	529.000	0.0434783
$530$	−11785.5	−0.965905
$531$	−6041.87	−0.493775
$532$	−568.037	−0.0462924
$533$	4874.38	0.396121
$534$	2447.10	0.198308
$535$	−16767.1	−1.35496
$536$	−10187.5	−0.820959
$537$	154.515	0.0124168
$538$	−17069.1	−1.36785
$539$	1714.64	0.137022
$540$	1568.34	0.124983
$541$	19421.6	1.54344	0.771720	−	0.635962i	$-0.219397\pi$
$541$	19421.6	1.54344	0.771720	+	0.635962i	$0.219397\pi$
$542$	5951.94	0.471693
$543$	−1455.20	−0.115007
$544$	−10891.5	−0.858397
$545$	−14592.3	−1.14691
$546$	504.291	0.0395268
$547$	−15713.2	−1.22824	−0.614121	−	0.789212i	$-0.710489\pi$
$547$	−15713.2	−1.22824	−0.614121	+	0.789212i	$0.710489\pi$
$548$	−1757.82	−0.137026
$549$	−6174.55	−0.480006
$550$	3194.00	0.247623
$551$	−1618.52	−0.125138
$552$	1090.12	0.0840552
$553$	1405.78	0.108101
$554$	17444.6	1.33782
$555$	−15122.1	−1.15657
$556$	5082.53	0.387675
$557$	19074.7	1.45102	0.725511	−	0.688210i	$-0.241603\pi$
$557$	19074.7	1.45102	0.725511	+	0.688210i	$0.241603\pi$
$558$	−9520.35	−0.722273
$559$	−340.639	−0.0257737
$560$	−4316.02	−0.325688
$561$	1192.40	0.0897383
$562$	−22969.9	−1.72407
$563$	52.4547	0.00392665	0.00196332	−	0.999998i	$-0.499375\pi$
$563$	52.4547	0.00392665	0.00196332	+	0.999998i	$0.499375\pi$
$564$	−5547.79	−0.414192
$565$	2062.11	0.153546
$566$	−27452.9	−2.03875
$567$	249.807	0.0185025
$568$	5776.75	0.426738
$569$	7095.98	0.522810	0.261405	−	0.965229i	$-0.415814\pi$
$569$	7095.98	0.522810	0.261405	+	0.965229i	$0.415814\pi$
$570$	9906.58	0.727966
$571$	−3486.00	−0.255490	−0.127745	−	0.991807i	$-0.540774\pi$
$571$	−3486.00	−0.255490	−0.127745	+	0.991807i	$0.540774\pi$
$572$	275.123	0.0201109
$573$	7227.72	0.526950
$574$	3116.63	0.226630
$575$	4250.38	0.308266
$576$	1462.26	0.105777
$577$	7649.20	0.551890	0.275945	−	0.961173i	$-0.411009\pi$
$577$	7649.20	0.551890	0.275945	+	0.961173i	$0.411009\pi$
$578$	−3573.68	−0.257172
$579$	3801.06	0.272827
$580$	1684.52	0.120596
$581$	−1378.44	−0.0984294
$582$	−9526.98	−0.678532
$583$	−1024.14	−0.0727537
$584$	17110.0	1.21236
$585$	−2568.50	−0.181529
$586$	−1627.62	−0.114738
$587$	−15516.1	−1.09100	−0.545500	−	0.838111i	$-0.683660\pi$
$587$	−15516.1	−1.09100	−0.545500	+	0.838111i	$0.683660\pi$
$588$	−3301.71	−0.231565
$589$	−17562.5	−1.22861
$590$	39720.2	2.77162
$591$	12543.1	0.873019
$592$	22770.6	1.58086
$593$	−11854.2	−0.820898	−0.410449	−	0.911884i	$-0.634628\pi$
$593$	−11854.2	−0.820898	−0.410449	+	0.911884i	$0.634628\pi$
$594$	466.658	0.0322343
$595$	−4196.32	−0.289130
$596$	4693.01	0.322539
$597$	−16150.5	−1.10719
$598$	1253.62	0.0857266
$599$	−8769.89	−0.598211	−0.299105	−	0.954220i	$-0.596688\pi$
$599$	−8769.89	−0.598211	−0.299105	+	0.954220i	$0.596688\pi$
$600$	8758.82	0.595962
$601$	−6794.22	−0.461134	−0.230567	−	0.973056i	$-0.574058\pi$
$601$	−6794.22	−0.461134	−0.230567	+	0.973056i	$0.574058\pi$
$602$	−217.801	−0.0147457
$603$	−5803.46	−0.391932
$604$	639.537	0.0430834
$605$	−22961.8	−1.54302
$606$	18312.8	1.22757
$607$	−4583.13	−0.306464	−0.153232	−	0.988190i	$-0.548968\pi$
$607$	−4583.13	−0.306464	−0.153232	+	0.988190i	$0.548968\pi$
$608$	−7863.16	−0.524495
$609$	268.311	0.0178531
$610$	40592.5	2.69433
$611$	9085.71	0.601585
$612$	−2296.09	−0.151657
$613$	−12725.0	−0.838431	−0.419216	−	0.907887i	$-0.637695\pi$
$613$	−12725.0	−0.838431	−0.419216	+	0.907887i	$0.637695\pi$
$614$	8008.28	0.526364
$615$	−15873.9	−1.04081
$616$	−250.516	−0.0163857
$617$	3279.16	0.213961	0.106981	−	0.994261i	$-0.465882\pi$
$617$	3279.16	0.213961	0.106981	+	0.994261i	$0.465882\pi$
$618$	17152.2	1.11645
$619$	−4471.88	−0.290372	−0.145186	−	0.989404i	$-0.546378\pi$
$619$	−4471.88	−0.290372	−0.145186	+	0.989404i	$0.546378\pi$
$620$	18278.7	1.18402
$621$	621.000	0.0401286
$622$	19751.9	1.27327
$623$	−748.356	−0.0481256
$624$	3867.60	0.248122
$625$	−4574.15	−0.292745
$626$	−15704.1	−1.00266
$627$	860.861	0.0548317
$628$	6123.96	0.389128
$629$	22139.1	1.40341
$630$	−1642.27	−0.103857
$631$	8411.70	0.530689	0.265344	−	0.964154i	$-0.414514\pi$
$631$	8411.70	0.530689	0.265344	+	0.964154i	$0.414514\pi$
$632$	7201.48	0.453259
$633$	4620.97	0.290153
$634$	−9351.76	−0.585814
$635$	8097.30	0.506034
$636$	1972.08	0.122953
$637$	5407.27	0.336333
$638$	501.225	0.0311030
$639$	3290.80	0.203728
$640$	−29451.5	−1.81902
$641$	−20349.9	−1.25393	−0.626967	−	0.779046i	$-0.715704\pi$
$641$	−20349.9	−1.25393	−0.626967	+	0.779046i	$0.715704\pi$
$642$	9606.87	0.590581
$643$	−20634.1	−1.26552	−0.632759	−	0.774349i	$-0.718078\pi$
$643$	−20634.1	−1.26552	−0.632759	+	0.774349i	$0.718078\pi$
$644$	234.091	0.0143237
$645$	1109.32	0.0677203
$646$	−14503.4	−0.883329
$647$	−7147.89	−0.434332	−0.217166	−	0.976135i	$-0.569681\pi$
$647$	−7147.89	−0.434332	−0.217166	+	0.976135i	$0.569681\pi$
$648$	1279.70	0.0775794
$649$	3451.60	0.208763
$650$	10072.6	0.607813
$651$	2911.45	0.175282
$652$	−2500.35	−0.150186
$653$	32056.9	1.92111	0.960555	−	0.278091i	$-0.0897016\pi$
$653$	32056.9	1.92111	0.960555	+	0.278091i	$0.0897016\pi$
$654$	8360.81	0.499898
$655$	28257.6	1.68567
$656$	23902.6	1.42262
$657$	9746.95	0.578789
$658$	5809.30	0.344180
$659$	−19616.9	−1.15958	−0.579792	−	0.814765i	$-0.696866\pi$
$659$	−19616.9	−1.15958	−0.579792	+	0.814765i	$0.696866\pi$
$660$	−895.964	−0.0528415
$661$	22936.9	1.34969	0.674843	−	0.737961i	$-0.264211\pi$
$661$	22936.9	1.34969	0.674843	+	0.737961i	$0.264211\pi$
$662$	35350.9	2.07546
$663$	3760.34	0.220271
$664$	−7061.44	−0.412706
$665$	−3029.56	−0.176664
$666$	8664.37	0.504110
$667$	667.000	0.0387202
$668$	10283.4	0.595622
$669$	−6299.17	−0.364036
$670$	38152.9	2.19996
$671$	3527.40	0.202942
$672$	1303.52	0.0748281
$673$	4751.95	0.272176	0.136088	−	0.990697i	$-0.456547\pi$
$673$	4751.95	0.272176	0.136088	+	0.990697i	$0.456547\pi$
$674$	13697.5	0.782799
$675$	4989.58	0.284517
$676$	−6382.86	−0.363158
$677$	−28515.6	−1.61882	−0.809411	−	0.587243i	$-0.800213\pi$
$677$	−28515.6	−1.61882	−0.809411	+	0.587243i	$0.800213\pi$
$678$	−1181.51	−0.0669255
$679$	2913.47	0.164667
$680$	−21496.8	−1.21230
$681$	−15176.3	−0.853974
$682$	5438.79	0.305370
$683$	26693.1	1.49544	0.747718	−	0.664017i	$-0.231150\pi$
$683$	26693.1	1.49544	0.747718	+	0.664017i	$0.231150\pi$
$684$	−1657.67	−0.0926649
$685$	−9375.14	−0.522928
$686$	7013.31	0.390334
$687$	4771.08	0.264961
$688$	−1670.40	−0.0925630
$689$	−3229.70	−0.178580
$690$	−4082.55	−0.225247
$691$	−24877.1	−1.36957	−0.684783	−	0.728747i	$-0.740103\pi$
$691$	−24877.1	−1.36957	−0.684783	+	0.728747i	$0.740103\pi$
$692$	−3608.87	−0.198250
$693$	−142.710	−0.00782266
$694$	40652.9	2.22358
$695$	27107.1	1.47947
$696$	1374.50	0.0748565
$697$	23239.7	1.26294
$698$	8414.40	0.456289
$699$	−4012.05	−0.217096
$700$	1880.86	0.101557
$701$	18653.2	1.00502	0.502511	−	0.864571i	$-0.332410\pi$
$701$	18653.2	1.00502	0.502511	+	0.864571i	$0.332410\pi$
$702$	1471.65	0.0791221
$703$	15983.5	0.857508
$704$	−835.359	−0.0447212
$705$	−29588.5	−1.58066
$706$	414.077	0.0220737
$707$	−5600.30	−0.297908
$708$	−6646.41	−0.352807
$709$	9247.78	0.489856	0.244928	−	0.969541i	$-0.421236\pi$
$709$	9247.78	0.489856	0.244928	+	0.969541i	$0.421236\pi$
$710$	−21634.3	−1.14355
$711$	4102.42	0.216389
$712$	−3833.65	−0.201787
$713$	7237.62	0.380156
$714$	2404.32	0.126022
$715$	1467.34	0.0767486
$716$	169.976	0.00887192
$717$	4547.66	0.236869
$718$	609.888	0.0317003
$719$	−15040.6	−0.780141	−0.390071	−	0.920785i	$-0.627549\pi$
$719$	−15040.6	−0.780141	−0.390071	+	0.920785i	$0.627549\pi$
$720$	−12595.2	−0.651939
$721$	−5245.38	−0.270941
$722$	12586.2	0.648766
$723$	−10669.8	−0.548845
$724$	−1600.81	−0.0821736
$725$	5359.18	0.274531
$726$	13156.2	0.672550
$727$	−12509.9	−0.638195	−0.319097	−	0.947722i	$-0.603380\pi$
$727$	−12509.9	−0.638195	−0.319097	+	0.947722i	$0.603380\pi$
$728$	−790.026	−0.0402202
$729$	729.000	0.0370370
$730$	−64078.0	−3.24881
$731$	−1624.07	−0.0821731
$732$	−6792.37	−0.342969
$733$	13779.0	0.694326	0.347163	−	0.937805i	$-0.387145\pi$
$733$	13779.0	0.694326	0.347163	+	0.937805i	$0.387145\pi$
$734$	20478.3	1.02979
$735$	−17609.3	−0.883714
$736$	3240.45	0.162289
$737$	3315.41	0.165705
$738$	9095.10	0.453652
$739$	25800.1	1.28427	0.642133	−	0.766593i	$-0.278050\pi$
$739$	25800.1	1.28427	0.642133	+	0.766593i	$0.278050\pi$
$740$	−16635.2	−0.826383
$741$	2714.80	0.134589
$742$	−2065.04	−0.102170
$743$	23521.9	1.16142	0.580710	−	0.814111i	$-0.302775\pi$
$743$	23521.9	1.16142	0.580710	+	0.814111i	$0.302775\pi$
$744$	14914.7	0.734943
$745$	25029.6	1.23089
$746$	−38956.7	−1.91194
$747$	−4022.64	−0.197029
$748$	1311.71	0.0641189
$749$	−2937.91	−0.143323
$750$	−10614.5	−0.516782
$751$	9585.84	0.465769	0.232884	−	0.972504i	$-0.425184\pi$
$751$	9585.84	0.465769	0.232884	+	0.972504i	$0.425184\pi$
$752$	44553.8	2.16052
$753$	10055.8	0.486657
$754$	1580.66	0.0763450
$755$	3410.90	0.164418
$756$	274.803	0.0132202
$757$	−10502.3	−0.504244	−0.252122	−	0.967695i	$-0.581128\pi$
$757$	−10502.3	−0.504244	−0.252122	+	0.967695i	$0.581128\pi$
$758$	−17401.9	−0.833859
$759$	−354.765	−0.0169660
$760$	−15519.7	−0.740736
$761$	−13762.8	−0.655588	−0.327794	−	0.944749i	$-0.606305\pi$
$761$	−13762.8	−0.655588	−0.327794	+	0.944749i	$0.606305\pi$
$762$	−4639.42	−0.220562
$763$	−2556.84	−0.121316
$764$	7950.92	0.376511
$765$	−12245.9	−0.578761
$766$	−15362.7	−0.724642
$767$	10884.9	0.512428
$768$	12975.2	0.609638
$769$	−2401.68	−0.112623	−0.0563113	−	0.998413i	$-0.517934\pi$
$769$	−2401.68	−0.112623	−0.0563113	+	0.998413i	$0.517934\pi$
$770$	938.199	0.0439095
$771$	−9648.94	−0.450711
$772$	4181.39	0.194937
$773$	39709.4	1.84767	0.923833	−	0.382795i	$-0.125038\pi$
$773$	39709.4	1.84767	0.923833	+	0.382795i	$0.125038\pi$
$774$	−635.597	−0.0295169
$775$	58152.4	2.69535
$776$	14925.0	0.690435
$777$	−2649.68	−0.122338
$778$	7197.27	0.331664
$779$	16778.1	0.771677
$780$	−2825.50	−0.129704
$781$	−1879.97	−0.0861341
$782$	5976.95	0.273319
$783$	783.000	0.0357371
$784$	26515.8	1.20790
$785$	32661.4	1.48502
$786$	−16190.5	−0.734726
$787$	−15446.5	−0.699631	−0.349815	−	0.936819i	$-0.613756\pi$
$787$	−15446.5	−0.699631	−0.349815	+	0.936819i	$0.613756\pi$
$788$	13798.1	0.623780
$789$	18941.6	0.854674
$790$	−26970.0	−1.21462
$791$	361.320	0.0162416
$792$	−731.070	−0.0327998
$793$	11124.0	0.498138
$794$	52645.1	2.35303
$795$	10517.9	0.469220
$796$	−17766.4	−0.791099
$797$	7014.17	0.311738	0.155869	−	0.987778i	$-0.450182\pi$
$797$	7014.17	0.311738	0.155869	+	0.987778i	$0.450182\pi$
$798$	1735.81	0.0770015
$799$	43318.2	1.91801
$800$	26036.2	1.15065
$801$	−2183.89	−0.0963346
$802$	−1597.54	−0.0703382
$803$	−5568.24	−0.244706
$804$	−6384.15	−0.280039
$805$	1248.50	0.0546631
$806$	17151.7	0.749557
$807$	15233.2	0.664477
$808$	−28689.0	−1.24910
$809$	13728.3	0.596613	0.298307	−	0.954470i	$-0.403578\pi$
$809$	13728.3	0.596613	0.298307	+	0.954470i	$0.403578\pi$
$810$	−4792.56	−0.207893
$811$	−19728.7	−0.854215	−0.427108	−	0.904201i	$-0.640467\pi$
$811$	−19728.7	−0.854215	−0.427108	+	0.904201i	$0.640467\pi$
$812$	295.158	0.0127562
$813$	−5311.75	−0.229140
$814$	−4949.79	−0.213133
$815$	−13335.3	−0.573150
$816$	18439.7	0.791076
$817$	−1172.51	−0.0502092
$818$	−20155.9	−0.861533
$819$	−450.049	−0.0192014
$820$	−17462.2	−0.743667
$821$	−9887.17	−0.420298	−0.210149	−	0.977669i	$-0.567395\pi$
$821$	−9887.17	−0.420298	−0.210149	+	0.977669i	$0.567395\pi$
$822$	5371.57	0.227926
$823$	−38571.7	−1.63369	−0.816843	−	0.576860i	$-0.804278\pi$
$823$	−38571.7	−1.63369	−0.816843	+	0.576860i	$0.804278\pi$
$824$	−26870.8	−1.13603
$825$	−2850.45	−0.120291
$826$	6959.71	0.293171
$827$	−9664.63	−0.406375	−0.203187	−	0.979140i	$-0.565130\pi$
$827$	−9664.63	−0.406375	−0.203187	+	0.979140i	$0.565130\pi$
$828$	683.137	0.0286723
$829$	−16819.5	−0.704664	−0.352332	−	0.935875i	$-0.614611\pi$
$829$	−16819.5	−0.704664	−0.352332	+	0.935875i	$0.614611\pi$
$830$	26445.5	1.10595
$831$	−15568.3	−0.649889
$832$	−2634.38	−0.109772
$833$	25780.4	1.07232
$834$	−15531.3	−0.644848
$835$	54845.1	2.27305
$836$	946.998	0.0391778
$837$	8496.33	0.350868
$838$	−15845.5	−0.653191
$839$	40253.9	1.65640	0.828199	−	0.560433i	$-0.189365\pi$
$839$	40253.9	1.65640	0.828199	+	0.560433i	$0.189365\pi$
$840$	2572.80	0.105678
$841$	841.000	0.0344828
$842$	6621.73	0.271021
$843$	20499.2	0.837522
$844$	5083.34	0.207317
$845$	−34042.3	−1.38591
$846$	16953.0	0.688956
$847$	−4023.33	−0.163215
$848$	−15837.6	−0.641350
$849$	24500.1	0.990390
$850$	48023.3	1.93786
$851$	−6586.88	−0.265329
$852$	3620.08	0.145565
$853$	−21475.4	−0.862021	−0.431010	−	0.902347i	$-0.641843\pi$
$853$	−21475.4	−0.862021	−0.431010	+	0.902347i	$0.641843\pi$
$854$	7112.55	0.284996
$855$	−8841.02	−0.353633
$856$	−15050.2	−0.600941
$857$	45194.6	1.80142	0.900710	−	0.434421i	$-0.143047\pi$
$857$	45194.6	1.80142	0.900710	+	0.434421i	$0.143047\pi$
$858$	−840.723	−0.0334520
$859$	13411.1	0.532691	0.266345	−	0.963878i	$-0.414184\pi$
$859$	13411.1	0.532691	0.266345	+	0.963878i	$0.414184\pi$
$860$	1220.32	0.0483868
$861$	−2781.40	−0.110093
$862$	−20406.6	−0.806325
$863$	−26443.7	−1.04305	−0.521526	−	0.853235i	$-0.674637\pi$
$863$	−26443.7	−1.04305	−0.521526	+	0.853235i	$0.674637\pi$
$864$	3804.01	0.149786
$865$	−19247.5	−0.756572
$866$	49608.0	1.94659
$867$	3189.29	0.124930
$868$	3202.76	0.125241
$869$	−2343.63	−0.0914872
$870$	−5147.57	−0.200596
$871$	10455.4	0.406738
$872$	−13098.1	−0.508667
$873$	8502.25	0.329619
$874$	4315.09	0.167002
$875$	3246.05	0.125413
$876$	10722.2	0.413550
$877$	−15055.5	−0.579691	−0.289845	−	0.957073i	$-0.593604\pi$
$877$	−15055.5	−0.579691	−0.289845	+	0.957073i	$0.593604\pi$
$878$	−30873.4	−1.18670
$879$	1452.55	0.0557375
$880$	7195.41	0.275633
$881$	−30853.3	−1.17988	−0.589939	−	0.807448i	$-0.700848\pi$
$881$	−30853.3	−1.17988	−0.589939	+	0.807448i	$0.700848\pi$
$882$	10089.4	0.385180
$883$	19472.2	0.742119	0.371060	−	0.928609i	$-0.378995\pi$
$883$	19472.2	0.742119	0.371060	+	0.928609i	$0.378995\pi$
$884$	4136.60	0.157385
$885$	−35447.9	−1.34640
$886$	13746.2	0.521233
$887$	−22638.6	−0.856969	−0.428484	−	0.903549i	$-0.640952\pi$
$887$	−22638.6	−0.856969	−0.428484	+	0.903549i	$0.640952\pi$
$888$	−13573.7	−0.512953
$889$	1418.80	0.0535263
$890$	14357.2	0.540737
$891$	−416.464	−0.0156589
$892$	−6929.46	−0.260107
$893$	31273.8	1.17194
$894$	−14341.0	−0.536502
$895$	906.546	0.0338575
$896$	−5160.45	−0.192409
$897$	−1118.78	−0.0416445
$898$	56594.9	2.10311
$899$	9125.69	0.338553
$900$	5488.83	0.203290
$901$	−15398.4	−0.569360
$902$	−5195.85	−0.191799
$903$	194.374	0.00716319
$904$	1850.96	0.0680995
$905$	−8537.74	−0.313596
$906$	−1954.30	−0.0716638
$907$	−36150.2	−1.32343	−0.661713	−	0.749757i	$-0.730170\pi$
$907$	−36150.2	−1.32343	−0.661713	+	0.749757i	$0.730170\pi$
$908$	−16694.8	−0.610172
$909$	−16343.1	−0.596331
$910$	2958.69	0.107780
$911$	−36785.4	−1.33782	−0.668910	−	0.743343i	$-0.733239\pi$
$911$	−36785.4	−1.33782	−0.668910	+	0.743343i	$0.733239\pi$
$912$	13312.6	0.483361
$913$	2298.06	0.0833020
$914$	12746.8	0.461299
$915$	−36226.3	−1.30886
$916$	5248.46	0.189317
$917$	4951.25	0.178304
$918$	7016.41	0.252262
$919$	−19831.9	−0.711854	−0.355927	−	0.934514i	$-0.615835\pi$
$919$	−19831.9	−0.711854	−0.355927	+	0.934514i	$0.615835\pi$
$920$	6395.76	0.229198
$921$	−7146.90	−0.255699
$922$	8069.64	0.288242
$923$	−5928.66	−0.211424
$924$	−156.989	−0.00558937
$925$	−52923.9	−1.88122
$926$	−39763.8	−1.41115
$927$	−15307.3	−0.542350
$928$	4085.78	0.144529
$929$	−13953.0	−0.492768	−0.246384	−	0.969172i	$-0.579243\pi$
$929$	−13953.0	−0.492768	−0.246384	+	0.969172i	$0.579243\pi$
$930$	−55856.3	−1.96946
$931$	18612.3	0.655204
$932$	−4413.49	−0.155117
$933$	−17627.3	−0.618535
$934$	−3397.93	−0.119040
$935$	6995.86	0.244694
$936$	−2305.49	−0.0805100
$937$	44605.8	1.55518	0.777592	−	0.628770i	$-0.216441\pi$
$937$	44605.8	1.55518	0.777592	+	0.628770i	$0.216441\pi$
$938$	6685.09	0.232703
$939$	14015.0	0.487073
$940$	−32549.1	−1.12940
$941$	55122.4	1.90961	0.954803	−	0.297240i	$-0.0960661\pi$
$941$	55122.4	1.90961	0.954803	+	0.297240i	$0.0960661\pi$
$942$	−18713.7	−0.647266
$943$	−6914.33	−0.238771
$944$	53376.7	1.84032
$945$	1465.63	0.0504517
$946$	363.104	0.0124794
$947$	22620.4	0.776202	0.388101	−	0.921617i	$-0.373131\pi$
$947$	22620.4	0.776202	0.388101	+	0.921617i	$0.373131\pi$
$948$	4512.91	0.154612
$949$	−17559.9	−0.600653
$950$	34670.7	1.18407
$951$	8345.88	0.284578
$952$	−3766.63	−0.128232
$953$	−43256.9	−1.47034	−0.735168	−	0.677885i	$-0.762897\pi$
$953$	−43256.9	−1.47034	−0.735168	+	0.677885i	$0.762897\pi$
$954$	−6026.30	−0.204516
$955$	42405.3	1.43686
$956$	5002.69	0.169245
$957$	−447.313	−0.0151093
$958$	−58986.0	−1.98930
$959$	−1642.70	−0.0553133
$960$	8579.11	0.288427
$961$	69231.9	2.32392
$962$	−15609.6	−0.523153
$963$	−8573.55	−0.286894
$964$	−11737.4	−0.392154
$965$	22301.0	0.743931
$966$	−715.339	−0.0238257
$967$	−54656.9	−1.81763	−0.908815	−	0.417200i	$-0.863011\pi$
$967$	−54656.9	−1.81763	−0.908815	+	0.417200i	$0.863011\pi$
$968$	−20610.6	−0.684348
$969$	12943.4	0.429105
$970$	−55895.1	−1.85019
$971$	40402.2	1.33529	0.667647	−	0.744478i	$-0.267302\pi$
$971$	40402.2	1.33529	0.667647	+	0.744478i	$0.267302\pi$
$972$	801.943	0.0264633
$973$	4749.66	0.156492
$974$	12980.9	0.427039
$975$	−8989.15	−0.295265
$976$	54548.9	1.78900
$977$	−35515.9	−1.16300	−0.581501	−	0.813545i	$-0.697534\pi$
$977$	−35515.9	−1.16300	−0.581501	+	0.813545i	$0.697534\pi$
$978$	7640.61	0.249816
$979$	1247.62	0.0407293
$980$	−19371.3	−0.631421
$981$	−7461.51	−0.242842
$982$	−36817.1	−1.19642
$983$	−4746.08	−0.153994	−0.0769972	−	0.997031i	$-0.524533\pi$
$983$	−4746.08	−0.153994	−0.0769972	+	0.997031i	$0.524533\pi$
$984$	−14248.4	−0.461610
$985$	73590.8	2.38051
$986$	7536.15	0.243408
$987$	−5184.45	−0.167196
$988$	2986.44	0.0961653
$989$	483.197	0.0155357
$990$	2737.90	0.0878951
$991$	−42542.5	−1.36368	−0.681840	−	0.731502i	$-0.738820\pi$
$991$	−42542.5	−1.36368	−0.681840	+	0.731502i	$0.738820\pi$
$992$	44334.9	1.41898
$993$	−31548.5	−1.00822
$994$	−3790.72	−0.120960
$995$	−94755.3	−3.01904
$996$	−4425.15	−0.140779
$997$	43847.6	1.39284	0.696422	−	0.717632i	$-0.254774\pi$
$997$	43847.6	1.39284	0.696422	+	0.717632i	$0.254774\pi$
$998$	71568.0	2.26998
$999$	−7732.42	−0.244888

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2001.4.a.a.1.8	✓	31

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2001.4.a.a.1.8	✓	31	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 2001 = 3 \cdot 23 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2001.a (trivial)

\(f(q)\)	\(=\)	\(q-3.36157 q^{2} +3.00000 q^{3} +3.30018 q^{4} +17.6011 q^{5} -10.0847 q^{6} +3.08404 q^{7} +15.7988 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q-3.36157 q^{2} +3.00000 q^{3} +3.30018 q^{4} +17.6011 q^{5} -10.0847 q^{6} +3.08404 q^{7} +15.7988 q^{8} +9.00000 q^{9} -59.1674 q^{10} -5.14153 q^{11} +9.90053 q^{12} -16.2143 q^{13} -10.3672 q^{14} +52.8033 q^{15} -79.5102 q^{16} -77.3052 q^{17} -30.2542 q^{18} -55.8110 q^{19} +58.0868 q^{20} +9.25212 q^{21} +17.2836 q^{22} +23.0000 q^{23} +47.3964 q^{24} +184.799 q^{25} +54.5054 q^{26} +27.0000 q^{27} +10.1779 q^{28} +29.0000 q^{29} -177.502 q^{30} +314.679 q^{31} +140.889 q^{32} -15.4246 q^{33} +259.867 q^{34} +54.2825 q^{35} +29.7016 q^{36} -286.386 q^{37} +187.613 q^{38} -48.6428 q^{39} +278.077 q^{40} -300.623 q^{41} -31.1017 q^{42} +21.0086 q^{43} -16.9679 q^{44} +158.410 q^{45} -77.3162 q^{46} -560.353 q^{47} -238.531 q^{48} -333.489 q^{49} -621.216 q^{50} -231.916 q^{51} -53.5099 q^{52} +199.189 q^{53} -90.7625 q^{54} -90.4966 q^{55} +48.7241 q^{56} -167.433 q^{57} -97.4856 q^{58} -671.319 q^{59} +174.260 q^{60} -686.061 q^{61} -1057.82 q^{62} +27.7564 q^{63} +162.473 q^{64} -285.389 q^{65} +51.8509 q^{66} -644.829 q^{67} -255.121 q^{68} +69.0000 q^{69} -182.475 q^{70} +365.645 q^{71} +142.189 q^{72} +1082.99 q^{73} +962.707 q^{74} +554.398 q^{75} -184.186 q^{76} -15.8567 q^{77} +163.516 q^{78} +455.825 q^{79} -1399.47 q^{80} +81.0000 q^{81} +1010.57 q^{82} -446.961 q^{83} +30.5336 q^{84} -1360.66 q^{85} -70.6219 q^{86} +87.0000 q^{87} -81.2300 q^{88} -242.655 q^{89} -532.507 q^{90} -50.0054 q^{91} +75.9041 q^{92} +944.037 q^{93} +1883.67 q^{94} -982.335 q^{95} +422.667 q^{96} +944.694 q^{97} +1121.05 q^{98} -46.2737 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 31 q - 10 q^{2} + 93 q^{3} + 74 q^{4} - 25 q^{5} - 30 q^{6} - 76 q^{7} - 117 q^{8} + 279 q^{9}+O(q^{10}) \) 31 * q - 10 * q^2 + 93 * q^3 + 74 * q^4 - 25 * q^5 - 30 * q^6 - 76 * q^7 - 117 * q^8 + 279 * q^9	\( 31 q - 10 q^{2} + 93 q^{3} + 74 q^{4} - 25 q^{5} - 30 q^{6} - 76 q^{7} - 117 q^{8} + 279 q^{9} - 6 q^{10} - 109 q^{11} + 222 q^{12} - 317 q^{13} - 159 q^{14} - 75 q^{15} + 54 q^{16} - 180 q^{17} - 90 q^{18} - 217 q^{19} - 193 q^{20} - 228 q^{21} - 160 q^{22} + 713 q^{23} - 351 q^{24} + 208 q^{25} + 211 q^{26} + 837 q^{27} + 5 q^{28} + 899 q^{29} - 18 q^{30} - 622 q^{31} - 696 q^{32} - 327 q^{33} + 126 q^{34} - 627 q^{35} + 666 q^{36} - 539 q^{37} + 35 q^{38} - 951 q^{39} + 371 q^{40} - 459 q^{41} - 477 q^{42} - 719 q^{43} - 876 q^{44} - 225 q^{45} - 230 q^{46} - 1384 q^{47} + 162 q^{48} - 747 q^{49} - 2005 q^{50} - 540 q^{51} - 2190 q^{52} - 730 q^{53} - 270 q^{54} - 1408 q^{55} - 1107 q^{56} - 651 q^{57} - 290 q^{58} - 3765 q^{59} - 579 q^{60} - 1300 q^{61} - 70 q^{62} - 684 q^{63} - 813 q^{64} - 2490 q^{65} - 480 q^{66} - 2735 q^{67} - 2031 q^{68} + 2139 q^{69} - 2581 q^{70} - 4800 q^{71} - 1053 q^{72} - 1858 q^{73} - 801 q^{74} + 624 q^{75} - 2699 q^{76} - 1661 q^{77} + 633 q^{78} - 4466 q^{79} + 3587 q^{80} + 2511 q^{81} - 5615 q^{82} - 1234 q^{83} + 15 q^{84} - 3623 q^{85} - 1873 q^{86} + 2697 q^{87} - 6893 q^{88} - 455 q^{89} - 54 q^{90} - 2425 q^{91} + 1702 q^{92} - 1866 q^{93} - 3912 q^{94} - 1851 q^{95} - 2088 q^{96} - 2114 q^{97} + 1769 q^{98} - 981 q^{99}+O(q^{100}) \) 31 * q - 10 * q^2 + 93 * q^3 + 74 * q^4 - 25 * q^5 - 30 * q^6 - 76 * q^7 - 117 * q^8 + 279 * q^9 - 6 * q^10 - 109 * q^11 + 222 * q^12 - 317 * q^13 - 159 * q^14 - 75 * q^15 + 54 * q^16 - 180 * q^17 - 90 * q^18 - 217 * q^19 - 193 * q^20 - 228 * q^21 - 160 * q^22 + 713 * q^23 - 351 * q^24 + 208 * q^25 + 211 * q^26 + 837 * q^27 + 5 * q^28 + 899 * q^29 - 18 * q^30 - 622 * q^31 - 696 * q^32 - 327 * q^33 + 126 * q^34 - 627 * q^35 + 666 * q^36 - 539 * q^37 + 35 * q^38 - 951 * q^39 + 371 * q^40 - 459 * q^41 - 477 * q^42 - 719 * q^43 - 876 * q^44 - 225 * q^45 - 230 * q^46 - 1384 * q^47 + 162 * q^48 - 747 * q^49 - 2005 * q^50 - 540 * q^51 - 2190 * q^52 - 730 * q^53 - 270 * q^54 - 1408 * q^55 - 1107 * q^56 - 651 * q^57 - 290 * q^58 - 3765 * q^59 - 579 * q^60 - 1300 * q^61 - 70 * q^62 - 684 * q^63 - 813 * q^64 - 2490 * q^65 - 480 * q^66 - 2735 * q^67 - 2031 * q^68 + 2139 * q^69 - 2581 * q^70 - 4800 * q^71 - 1053 * q^72 - 1858 * q^73 - 801 * q^74 + 624 * q^75 - 2699 * q^76 - 1661 * q^77 + 633 * q^78 - 4466 * q^79 + 3587 * q^80 + 2511 * q^81 - 5615 * q^82 - 1234 * q^83 + 15 * q^84 - 3623 * q^85 - 1873 * q^86 + 2697 * q^87 - 6893 * q^88 - 455 * q^89 - 54 * q^90 - 2425 * q^91 + 1702 * q^92 - 1866 * q^93 - 3912 * q^94 - 1851 * q^95 - 2088 * q^96 - 2114 * q^97 + 1769 * q^98 - 981 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more