LMFDB - Embedded newform 2013.4.a.a.1.18

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2013,4,Mod(1,2013)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2013, 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("2013.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2013 = 3 \cdot 11 \cdot 61 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2013.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:	$118.770844842$
Analytic rank:	$1$
Dimension:	$36$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.18
Character	$\chi$	$=$	2013.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.942036 q^{2} +3.00000 q^{3} -7.11257 q^{4} +1.29514 q^{5} -2.82611 q^{6} +21.4949 q^{7} +14.2366 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q-0.942036 q^{2} +3.00000 q^{3} -7.11257 q^{4} +1.29514 q^{5} -2.82611 q^{6} +21.4949 q^{7} +14.2366 q^{8} +9.00000 q^{9} -1.22007 q^{10} +11.0000 q^{11} -21.3377 q^{12} +2.72743 q^{13} -20.2490 q^{14} +3.88542 q^{15} +43.4892 q^{16} +79.3106 q^{17} -8.47832 q^{18} -127.760 q^{19} -9.21177 q^{20} +64.4847 q^{21} -10.3624 q^{22} +5.59949 q^{23} +42.7097 q^{24} -123.323 q^{25} -2.56934 q^{26} +27.0000 q^{27} -152.884 q^{28} -271.435 q^{29} -3.66020 q^{30} -190.020 q^{31} -154.861 q^{32} +33.0000 q^{33} -74.7134 q^{34} +27.8389 q^{35} -64.0131 q^{36} -346.311 q^{37} +120.355 q^{38} +8.18230 q^{39} +18.4384 q^{40} +392.154 q^{41} -60.7469 q^{42} +21.8247 q^{43} -78.2383 q^{44} +11.6563 q^{45} -5.27492 q^{46} +127.601 q^{47} +130.468 q^{48} +119.031 q^{49} +116.174 q^{50} +237.932 q^{51} -19.3991 q^{52} -298.825 q^{53} -25.4350 q^{54} +14.2465 q^{55} +306.014 q^{56} -383.281 q^{57} +255.702 q^{58} -306.061 q^{59} -27.6353 q^{60} +61.0000 q^{61} +179.006 q^{62} +193.454 q^{63} -202.029 q^{64} +3.53241 q^{65} -31.0872 q^{66} +548.900 q^{67} -564.102 q^{68} +16.7985 q^{69} -26.2252 q^{70} +372.399 q^{71} +128.129 q^{72} -121.718 q^{73} +326.238 q^{74} -369.968 q^{75} +908.705 q^{76} +236.444 q^{77} -7.70802 q^{78} +820.585 q^{79} +56.3246 q^{80} +81.0000 q^{81} -369.423 q^{82} -557.247 q^{83} -458.652 q^{84} +102.718 q^{85} -20.5597 q^{86} -814.306 q^{87} +156.602 q^{88} +55.2264 q^{89} -10.9806 q^{90} +58.6259 q^{91} -39.8268 q^{92} -570.061 q^{93} -120.205 q^{94} -165.468 q^{95} -464.583 q^{96} -1100.58 q^{97} -112.131 q^{98} +99.0000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 36 q - 14 q^{2} + 108 q^{3} + 146 q^{4} - 65 q^{5} - 42 q^{6} - 105 q^{7} - 189 q^{8} + 324 q^{9}+O(q^{10}) $ 36 * q - 14 * q^2 + 108 * q^3 + 146 * q^4 - 65 * q^5 - 42 * q^6 - 105 * q^7 - 189 * q^8 + 324 * q^9	\( 36 q - 14 q^{2} + 108 q^{3} + 146 q^{4} - 65 q^{5} - 42 q^{6} - 105 q^{7} - 189 q^{8} + 324 q^{9} - 161 q^{10} + 396 q^{11} + 438 q^{12} - 233 q^{13} - 264 q^{14} - 195 q^{15} + 574 q^{16} - 556 q^{17} - 126 q^{18} - 615 q^{19} - 136 q^{20} - 315 q^{21} - 154 q^{22} - 457 q^{23} - 567 q^{24} + 863 q^{25} + 115 q^{26} + 972 q^{27} - 424 q^{28} - 754 q^{29} - 483 q^{30} - 508 q^{31} - 1511 q^{32} + 1188 q^{33} - 860 q^{34} - 826 q^{35} + 1314 q^{36} - 412 q^{37} - 599 q^{38} - 699 q^{39} - 2791 q^{40} - 2066 q^{41} - 792 q^{42} - 2063 q^{43} + 1606 q^{44} - 585 q^{45} - 787 q^{46} - 1815 q^{47} + 1722 q^{48} + 2825 q^{49} + 808 q^{50} - 1668 q^{51} - 2882 q^{52} - 759 q^{53} - 378 q^{54} - 715 q^{55} - 1749 q^{56} - 1845 q^{57} - 335 q^{58} - 2337 q^{59} - 408 q^{60} + 2196 q^{61} - 1689 q^{62} - 945 q^{63} + 4723 q^{64} - 3550 q^{65} - 462 q^{66} - 1331 q^{67} - 6166 q^{68} - 1371 q^{69} - 1750 q^{70} - 361 q^{71} - 1701 q^{72} - 4627 q^{73} - 3394 q^{74} + 2589 q^{75} - 7214 q^{76} - 1155 q^{77} + 345 q^{78} - 2583 q^{79} - 2643 q^{80} + 2916 q^{81} + 1090 q^{82} - 6123 q^{83} - 1272 q^{84} + 295 q^{85} + 613 q^{86} - 2262 q^{87} - 2079 q^{88} - 2485 q^{89} - 1449 q^{90} - 3156 q^{91} - 6291 q^{92} - 1524 q^{93} - 1744 q^{94} - 5572 q^{95} - 4533 q^{96} - 2558 q^{97} - 2314 q^{98} + 3564 q^{99}+O(q^{100}) \) 36 * q - 14 * q^2 + 108 * q^3 + 146 * q^4 - 65 * q^5 - 42 * q^6 - 105 * q^7 - 189 * q^8 + 324 * q^9 - 161 * q^10 + 396 * q^11 + 438 * q^12 - 233 * q^13 - 264 * q^14 - 195 * q^15 + 574 * q^16 - 556 * q^17 - 126 * q^18 - 615 * q^19 - 136 * q^20 - 315 * q^21 - 154 * q^22 - 457 * q^23 - 567 * q^24 + 863 * q^25 + 115 * q^26 + 972 * q^27 - 424 * q^28 - 754 * q^29 - 483 * q^30 - 508 * q^31 - 1511 * q^32 + 1188 * q^33 - 860 * q^34 - 826 * q^35 + 1314 * q^36 - 412 * q^37 - 599 * q^38 - 699 * q^39 - 2791 * q^40 - 2066 * q^41 - 792 * q^42 - 2063 * q^43 + 1606 * q^44 - 585 * q^45 - 787 * q^46 - 1815 * q^47 + 1722 * q^48 + 2825 * q^49 + 808 * q^50 - 1668 * q^51 - 2882 * q^52 - 759 * q^53 - 378 * q^54 - 715 * q^55 - 1749 * q^56 - 1845 * q^57 - 335 * q^58 - 2337 * q^59 - 408 * q^60 + 2196 * q^61 - 1689 * q^62 - 945 * q^63 + 4723 * q^64 - 3550 * q^65 - 462 * q^66 - 1331 * q^67 - 6166 * q^68 - 1371 * q^69 - 1750 * q^70 - 361 * q^71 - 1701 * q^72 - 4627 * q^73 - 3394 * q^74 + 2589 * q^75 - 7214 * q^76 - 1155 * q^77 + 345 * q^78 - 2583 * q^79 - 2643 * q^80 + 2916 * q^81 + 1090 * q^82 - 6123 * q^83 - 1272 * q^84 + 295 * q^85 + 613 * q^86 - 2262 * q^87 - 2079 * q^88 - 2485 * q^89 - 1449 * q^90 - 3156 * q^91 - 6291 * q^92 - 1524 * q^93 - 1744 * q^94 - 5572 * q^95 - 4533 * q^96 - 2558 * q^97 - 2314 * q^98 + 3564 * 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 $

$2$	−0.942036	−0.333060	−0.166530	−	0.986036i	$-0.553256\pi$
$2$	−0.942036	−0.333060	−0.166530	+	0.986036i	$0.553256\pi$
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	−7.11257	−0.889071
$5$	1.29514	0.115841	0.0579204	−	0.998321i	$-0.481553\pi$
$5$	1.29514	0.115841	0.0579204	+	0.998321i	$0.481553\pi$
$6$	−2.82611	−0.192292
$7$	21.4949	1.16062	0.580308	−	0.814397i	$-0.302932\pi$
$7$	21.4949	1.16062	0.580308	+	0.814397i	$0.302932\pi$
$8$	14.2366	0.629174
$9$	9.00000	0.333333
$10$	−1.22007	−0.0385819
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	−21.3377	−0.513305
$13$	2.72743	0.0581888	0.0290944	−	0.999577i	$-0.490738\pi$
$13$	2.72743	0.0581888	0.0290944	+	0.999577i	$0.490738\pi$
$14$	−20.2490	−0.386555
$15$	3.88542	0.0668807
$16$	43.4892	0.679518
$17$	79.3106	1.13151	0.565754	−	0.824574i	$-0.308585\pi$
$17$	79.3106	1.13151	0.565754	+	0.824574i	$0.308585\pi$
$18$	−8.47832	−0.111020
$19$	−127.760	−1.54264	−0.771322	−	0.636445i	$-0.780404\pi$
$19$	−127.760	−1.54264	−0.771322	+	0.636445i	$0.780404\pi$
$20$	−9.21177	−0.102991
$21$	64.4847	0.670082
$22$	−10.3624	−0.100421
$23$	5.59949	0.0507641	0.0253821	−	0.999678i	$-0.491920\pi$
$23$	5.59949	0.0507641	0.0253821	+	0.999678i	$0.491920\pi$
$24$	42.7097	0.363254
$25$	−123.323	−0.986581
$26$	−2.56934	−0.0193804
$27$	27.0000	0.192450
$28$	−152.884	−1.03187
$29$	−271.435	−1.73808	−0.869039	−	0.494744i	$-0.835262\pi$
$29$	−271.435	−1.73808	−0.869039	+	0.494744i	$0.835262\pi$
$30$	−3.66020	−0.0222753
$31$	−190.020	−1.10092	−0.550462	−	0.834860i	$-0.685548\pi$
$31$	−190.020	−1.10092	−0.550462	+	0.834860i	$0.685548\pi$
$32$	−154.861	−0.855494
$33$	33.0000	0.174078
$34$	−74.7134	−0.376860
$35$	27.8389	0.134447
$36$	−64.0131	−0.296357
$37$	−346.311	−1.53874	−0.769368	−	0.638806i	$-0.779429\pi$
$37$	−346.311	−1.53874	−0.769368	+	0.638806i	$0.779429\pi$
$38$	120.355	0.513793
$39$	8.18230	0.0335953
$40$	18.4384	0.0728840
$41$	392.154	1.49376	0.746880	−	0.664958i	$-0.231551\pi$
$41$	392.154	1.49376	0.746880	+	0.664958i	$0.231551\pi$
$42$	−60.7469	−0.223177
$43$	21.8247	0.0774008	0.0387004	−	0.999251i	$-0.487678\pi$
$43$	21.8247	0.0774008	0.0387004	+	0.999251i	$0.487678\pi$
$44$	−78.2383	−0.268065
$45$	11.6563	0.0386136
$46$	−5.27492	−0.0169075
$47$	127.601	0.396012	0.198006	−	0.980201i	$-0.436553\pi$
$47$	127.601	0.396012	0.198006	+	0.980201i	$0.436553\pi$
$48$	130.468	0.392320
$49$	119.031	0.347028
$50$	116.174	0.328591
$51$	237.932	0.653277
$52$	−19.3991	−0.0517340
$53$	−298.825	−0.774467	−0.387234	−	0.921982i	$-0.626569\pi$
$53$	−298.825	−0.774467	−0.387234	+	0.921982i	$0.626569\pi$
$54$	−25.4350	−0.0640974
$55$	14.2465	0.0349273
$56$	306.014	0.730229
$57$	−383.281	−0.890646
$58$	255.702	0.578884
$59$	−306.061	−0.675351	−0.337675	−	0.941263i	$-0.609641\pi$
$59$	−306.061	−0.675351	−0.337675	+	0.941263i	$0.609641\pi$
$60$	−27.6353	−0.0594617
$61$	61.0000	0.128037
$61$	61.0000	0.128037
$62$	179.006	0.366674
$63$	193.454	0.386872
$64$	−202.029	−0.394587
$65$	3.53241	0.00674063
$66$	−31.0872	−0.0579783
$67$	548.900	1.00088	0.500439	−	0.865772i	$-0.333172\pi$
$67$	548.900	1.00088	0.500439	+	0.865772i	$0.333172\pi$
$68$	−564.102	−1.00599
$69$	16.7985	0.0293087
$70$	−26.2252	−0.0447788
$71$	372.399	0.622474	0.311237	−	0.950332i	$-0.399257\pi$
$71$	372.399	0.622474	0.311237	+	0.950332i	$0.399257\pi$
$72$	128.129	0.209725
$73$	−121.718	−0.195151	−0.0975755	−	0.995228i	$-0.531109\pi$
$73$	−121.718	−0.195151	−0.0975755	+	0.995228i	$0.531109\pi$
$74$	326.238	0.512491
$75$	−369.968	−0.569603
$76$	908.705	1.37152
$77$	236.444	0.349939
$78$	−7.70802	−0.0111893
$79$	820.585	1.16865	0.584323	−	0.811521i	$-0.301360\pi$
$79$	820.585	1.16865	0.584323	+	0.811521i	$0.301360\pi$
$80$	56.3246	0.0787160
$81$	81.0000	0.111111
$82$	−369.423	−0.497512
$83$	−557.247	−0.736938	−0.368469	−	0.929640i	$-0.620118\pi$
$83$	−557.247	−0.736938	−0.368469	+	0.929640i	$0.620118\pi$
$84$	−458.652	−0.595750
$85$	102.718	0.131075
$86$	−20.5597	−0.0257791
$87$	−814.306	−1.00348
$88$	156.602	0.189703
$89$	55.2264	0.0657751	0.0328875	−	0.999459i	$-0.489530\pi$
$89$	55.2264	0.0657751	0.0328875	+	0.999459i	$0.489530\pi$
$90$	−10.9806	−0.0128606
$91$	58.6259	0.0675348
$92$	−39.8268	−0.0451329
$93$	−570.061	−0.635619
$94$	−120.205	−0.131896
$95$	−165.468	−0.178701
$96$	−464.583	−0.493920
$97$	−1100.58	−1.15203	−0.576016	−	0.817439i	$-0.695393\pi$
$97$	−1100.58	−1.15203	−0.576016	+	0.817439i	$0.695393\pi$
$98$	−112.131	−0.115581
$99$	99.0000	0.100504
$100$	877.141	0.877141
$101$	87.8217	0.0865207	0.0432603	−	0.999064i	$-0.486226\pi$
$101$	87.8217	0.0865207	0.0432603	+	0.999064i	$0.486226\pi$
$102$	−224.140	−0.217580
$103$	−1173.47	−1.12257	−0.561286	−	0.827622i	$-0.689693\pi$
$103$	−1173.47	−1.12257	−0.561286	+	0.827622i	$0.689693\pi$
$104$	38.8293	0.0366109
$105$	83.5167	0.0776228
$106$	281.504	0.257944
$107$	−1746.19	−1.57767	−0.788836	−	0.614603i	$-0.789316\pi$
$107$	−1746.19	−1.57767	−0.788836	+	0.614603i	$0.789316\pi$
$108$	−192.039	−0.171102
$109$	−669.754	−0.588539	−0.294270	−	0.955722i	$-0.595076\pi$
$109$	−669.754	−0.588539	−0.294270	+	0.955722i	$0.595076\pi$
$110$	−13.4207	−0.0116329
$111$	−1038.93	−0.888390
$112$	934.795	0.788659
$113$	200.432	0.166859	0.0834293	−	0.996514i	$-0.473413\pi$
$113$	200.432	0.166859	0.0834293	+	0.996514i	$0.473413\pi$
$114$	361.065	0.296639
$115$	7.25212	0.00588056
$116$	1930.60	1.54527
$117$	24.5469	0.0193963
$118$	288.320	0.224932
$119$	1704.77	1.31325
$120$	55.3151	0.0420796
$121$	121.000	0.0909091
$122$	−57.4642	−0.0426440
$123$	1176.46	0.862423
$124$	1351.53	0.978800
$125$	−321.612	−0.230127
$126$	−182.241	−0.128852
$127$	2553.85	1.78439	0.892194	−	0.451651i	$-0.149165\pi$
$127$	2553.85	1.78439	0.892194	+	0.451651i	$0.149165\pi$
$128$	1429.21	0.986916
$129$	65.4741	0.0446874
$130$	−3.32766	−0.00224504
$131$	−1463.07	−0.975796	−0.487898	−	0.872901i	$-0.662236\pi$
$131$	−1463.07	−0.975796	−0.487898	+	0.872901i	$0.662236\pi$
$132$	−234.715	−0.154767
$133$	−2746.20	−1.79042
$134$	−517.083	−0.333352
$135$	34.9688	0.0222936
$136$	1129.11	0.711916
$137$	2573.27	1.60474	0.802370	−	0.596827i	$-0.203572\pi$
$137$	2573.27	1.60474	0.802370	+	0.596827i	$0.203572\pi$
$138$	−15.8248	−0.00976155
$139$	−1601.43	−0.977207	−0.488603	−	0.872506i	$-0.662493\pi$
$139$	−1601.43	−0.977207	−0.488603	+	0.872506i	$0.662493\pi$
$140$	−198.006	−0.119533
$141$	382.804	0.228638
$142$	−350.813	−0.207321
$143$	30.0018	0.0175446
$144$	391.403	0.226506
$145$	−351.547	−0.201340
$146$	114.663	0.0649970
$147$	357.092	0.200357
$148$	2463.16	1.36805
$149$	481.654	0.264823	0.132411	−	0.991195i	$-0.457728\pi$
$149$	481.654	0.264823	0.132411	+	0.991195i	$0.457728\pi$
$150$	348.523	0.189712
$151$	578.253	0.311639	0.155820	−	0.987786i	$-0.450198\pi$
$151$	578.253	0.311639	0.155820	+	0.987786i	$0.450198\pi$
$152$	−1818.87	−0.970592
$153$	713.796	0.377170
$154$	−222.739	−0.116551
$155$	−246.103	−0.127532
$156$	−58.1972	−0.0298686
$157$	−3620.54	−1.84045	−0.920223	−	0.391393i	$-0.871993\pi$
$157$	−3620.54	−1.84045	−0.920223	+	0.391393i	$0.871993\pi$
$158$	−773.020	−0.389229
$159$	−896.475	−0.447139
$160$	−200.567	−0.0991012
$161$	120.360	0.0589176
$162$	−76.3049	−0.0370067
$163$	3160.84	1.51887	0.759436	−	0.650582i	$-0.225475\pi$
$163$	3160.84	1.51887	0.759436	+	0.650582i	$0.225475\pi$
$164$	−2789.22	−1.32806
$165$	42.7396	0.0201653
$166$	524.947	0.245444
$167$	−1411.12	−0.653867	−0.326933	−	0.945047i	$-0.606015\pi$
$167$	−1411.12	−0.653867	−0.326933	+	0.945047i	$0.606015\pi$
$168$	918.042	0.421598
$169$	−2189.56	−0.996614
$170$	−96.7644	−0.0436558
$171$	−1149.84	−0.514215
$172$	−155.230	−0.0688148
$173$	3525.85	1.54951	0.774756	−	0.632260i	$-0.217872\pi$
$173$	3525.85	1.54951	0.774756	+	0.632260i	$0.217872\pi$
$174$	767.105	0.334219
$175$	−2650.81	−1.14504
$176$	478.381	0.204882
$177$	−918.182	−0.389914
$178$	−52.0252	−0.0219070
$179$	−3824.19	−1.59683	−0.798417	−	0.602105i	$-0.794329\pi$
$179$	−3824.19	−1.59683	−0.798417	+	0.602105i	$0.794329\pi$
$180$	−82.9059	−0.0343302
$181$	−1057.90	−0.434436	−0.217218	−	0.976123i	$-0.569698\pi$
$181$	−1057.90	−0.434436	−0.217218	+	0.976123i	$0.569698\pi$
$182$	−55.2277	−0.0224931
$183$	183.000	0.0739221
$184$	79.7176	0.0319395
$185$	−448.522	−0.178248
$186$	537.018	0.211699
$187$	872.417	0.341163
$188$	−907.573	−0.352083
$189$	580.362	0.223361
$190$	155.876	0.0595182
$191$	2101.06	0.795956	0.397978	−	0.917395i	$-0.369712\pi$
$191$	2101.06	0.795956	0.397978	+	0.917395i	$0.369712\pi$
$192$	−606.086	−0.227815
$193$	155.390	0.0579545	0.0289773	−	0.999580i	$-0.490775\pi$
$193$	155.390	0.0579545	0.0289773	+	0.999580i	$0.490775\pi$
$194$	1036.79	0.383696
$195$	10.5972	0.00389171
$196$	−846.613	−0.308532
$197$	−4579.12	−1.65608	−0.828042	−	0.560666i	$-0.810545\pi$
$197$	−4579.12	−1.65608	−0.828042	+	0.560666i	$0.810545\pi$
$198$	−93.2616	−0.0334738
$199$	−3811.10	−1.35760	−0.678799	−	0.734324i	$-0.737499\pi$
$199$	−3811.10	−1.35760	−0.678799	+	0.734324i	$0.737499\pi$
$200$	−1755.69	−0.620731
$201$	1646.70	0.577857
$202$	−82.7312	−0.0288166
$203$	−5834.47	−2.01724
$204$	−1692.31	−0.580810
$205$	507.894	0.173038
$206$	1105.45	0.373884
$207$	50.3954	0.0169214
$208$	118.614	0.0395403
$209$	−1405.36	−0.465125
$210$	−78.6757	−0.0258530
$211$	−689.770	−0.225051	−0.112525	−	0.993649i	$-0.535894\pi$
$211$	−689.770	−0.225051	−0.112525	+	0.993649i	$0.535894\pi$
$212$	2125.41	0.688556
$213$	1117.20	0.359385
$214$	1644.98	0.525460
$215$	28.2660	0.00896617
$216$	384.388	0.121085
$217$	−4084.47	−1.27775
$218$	630.932	0.196019
$219$	−365.154	−0.112670
$220$	−101.329	−0.0310529
$221$	216.314	0.0658411
$222$	978.713	0.295887
$223$	−2316.11	−0.695509	−0.347754	−	0.937586i	$-0.613056\pi$
$223$	−2316.11	−0.695509	−0.347754	+	0.937586i	$0.613056\pi$
$224$	−3328.72	−0.992900
$225$	−1109.90	−0.328860
$226$	−188.814	−0.0555739
$227$	345.719	0.101085	0.0505423	−	0.998722i	$-0.483905\pi$
$227$	345.719	0.101085	0.0505423	+	0.998722i	$0.483905\pi$
$228$	2726.11	0.791848
$229$	−316.283	−0.0912688	−0.0456344	−	0.998958i	$-0.514531\pi$
$229$	−316.283	−0.0912688	−0.0456344	+	0.998958i	$0.514531\pi$
$230$	−6.83176	−0.00195858
$231$	709.332	0.202037
$232$	−3864.31	−1.09355
$233$	−4514.84	−1.26943	−0.634715	−	0.772746i	$-0.718883\pi$
$233$	−4514.84	−1.26943	−0.634715	+	0.772746i	$0.718883\pi$
$234$	−23.1241	−0.00646012
$235$	165.262	0.0458744
$236$	2176.88	0.600435
$237$	2461.75	0.674718
$238$	−1605.96	−0.437390
$239$	−961.948	−0.260348	−0.130174	−	0.991491i	$-0.541554\pi$
$239$	−961.948	−0.260348	−0.130174	+	0.991491i	$0.541554\pi$
$240$	168.974	0.0454467
$241$	5759.07	1.53931	0.769656	−	0.638459i	$-0.220428\pi$
$241$	5759.07	1.53931	0.769656	+	0.638459i	$0.220428\pi$
$242$	−113.986	−0.0302782
$243$	243.000	0.0641500
$244$	−433.867	−0.113834
$245$	154.161	0.0402000
$246$	−1108.27	−0.287239
$247$	−348.458	−0.0897646
$248$	−2705.24	−0.692673
$249$	−1671.74	−0.425471
$250$	302.971	0.0766462
$251$	26.6978	0.00671374	0.00335687	−	0.999994i	$-0.498931\pi$
$251$	26.6978	0.00671374	0.00335687	+	0.999994i	$0.498931\pi$
$252$	−1375.96	−0.343956
$253$	61.5944	0.0153060
$254$	−2405.82	−0.594309
$255$	308.155	0.0756761
$256$	269.866	0.0658852
$257$	264.197	0.0641252	0.0320626	−	0.999486i	$-0.489792\pi$
$257$	264.197	0.0641252	0.0320626	+	0.999486i	$0.489792\pi$
$258$	−61.6790	−0.0148836
$259$	−7443.93	−1.78588
$260$	−25.1245	−0.00599290
$261$	−2442.92	−0.579359
$262$	1378.27	0.324999
$263$	2498.99	0.585911	0.292955	−	0.956126i	$-0.405361\pi$
$263$	2498.99	0.585911	0.292955	+	0.956126i	$0.405361\pi$
$264$	469.807	0.109525
$265$	−387.020	−0.0897149
$266$	2587.02	0.596316
$267$	165.679	0.0379753
$268$	−3904.09	−0.889851
$269$	−5533.05	−1.25411	−0.627056	−	0.778974i	$-0.715740\pi$
$269$	−5533.05	−1.25411	−0.627056	+	0.778974i	$0.715740\pi$
$270$	−32.9418	−0.00742510
$271$	−1667.05	−0.373676	−0.186838	−	0.982391i	$-0.559824\pi$
$271$	−1667.05	−0.373676	−0.186838	+	0.982391i	$0.559824\pi$
$272$	3449.15	0.768881
$273$	175.878	0.0389912
$274$	−2424.11	−0.534475
$275$	−1356.55	−0.297465
$276$	−119.480	−0.0260575
$277$	4750.37	1.03041	0.515203	−	0.857068i	$-0.327717\pi$
$277$	4750.37	1.03041	0.515203	+	0.857068i	$0.327717\pi$
$278$	1508.61	0.325469
$279$	−1710.18	−0.366975
$280$	396.331	0.0845903
$281$	−2359.48	−0.500907	−0.250453	−	0.968129i	$-0.580580\pi$
$281$	−2359.48	−0.500907	−0.250453	+	0.968129i	$0.580580\pi$
$282$	−360.615	−0.0761501
$283$	−4706.17	−0.988526	−0.494263	−	0.869312i	$-0.664562\pi$
$283$	−4706.17	−0.988526	−0.494263	+	0.869312i	$0.664562\pi$
$284$	−2648.71	−0.553423
$285$	−496.403	−0.103173
$286$	−28.2627	−0.00584340
$287$	8429.31	1.73368
$288$	−1393.75	−0.285165
$289$	1377.17	0.280312
$290$	331.169	0.0670584
$291$	−3301.74	−0.665126
$292$	865.728	0.173503
$293$	5929.36	1.18224	0.591121	−	0.806583i	$-0.298685\pi$
$293$	5929.36	1.18224	0.591121	+	0.806583i	$0.298685\pi$
$294$	−336.393	−0.0667308
$295$	−396.391	−0.0782332
$296$	−4930.29	−0.968133
$297$	297.000	0.0580259
$298$	−453.735	−0.0882019
$299$	15.2722	0.00295390
$300$	2631.42	0.506417
$301$	469.120	0.0898326
$302$	−544.735	−0.103795
$303$	263.465	0.0499527
$304$	−5556.19	−1.04826
$305$	79.0035	0.0148319
$306$	−672.421	−0.125620
$307$	9524.87	1.77073	0.885364	−	0.464899i	$-0.153909\pi$
$307$	9524.87	1.77073	0.885364	+	0.464899i	$0.153909\pi$
$308$	−1681.72	−0.311120
$309$	−3520.40	−0.648117
$310$	231.838	0.0424758
$311$	5884.59	1.07294	0.536470	−	0.843919i	$-0.319757\pi$
$311$	5884.59	1.07294	0.536470	+	0.843919i	$0.319757\pi$
$312$	116.488	0.0211373
$313$	6661.16	1.20291	0.601456	−	0.798906i	$-0.294588\pi$
$313$	6661.16	1.20291	0.601456	+	0.798906i	$0.294588\pi$
$314$	3410.67	0.612979
$315$	250.550	0.0448155
$316$	−5836.47	−1.03901
$317$	−6223.04	−1.10259	−0.551294	−	0.834311i	$-0.685866\pi$
$317$	−6223.04	−1.10259	−0.551294	+	0.834311i	$0.685866\pi$
$318$	844.512	0.148924
$319$	−2985.79	−0.524050
$320$	−261.655	−0.0457093
$321$	−5238.58	−0.910870
$322$	−113.384	−0.0196231
$323$	−10132.8	−1.74552
$324$	−576.118	−0.0987857
$325$	−336.354	−0.0574079
$326$	−2977.62	−0.505875
$327$	−2009.26	−0.339793
$328$	5582.94	0.939835
$329$	2742.78	0.459618
$330$	−40.2622	−0.00671625
$331$	−9194.43	−1.52680	−0.763401	−	0.645925i	$-0.776472\pi$
$331$	−9194.43	−1.52680	−0.763401	+	0.645925i	$0.776472\pi$
$332$	3963.46	0.655190
$333$	−3116.80	−0.512912
$334$	1329.33	0.217777
$335$	710.902	0.115942
$336$	2804.39	0.455333
$337$	−2066.63	−0.334055	−0.167027	−	0.985952i	$-0.553417\pi$
$337$	−2066.63	−0.334055	−0.167027	+	0.985952i	$0.553417\pi$
$338$	2062.65	0.331932
$339$	601.295	0.0963359
$340$	−730.591	−0.116535
$341$	−2090.22	−0.331941
$342$	1083.19	0.171264
$343$	−4814.20	−0.757849
$344$	310.709	0.0486986
$345$	21.7564	0.00339514
$346$	−3321.48	−0.516081
$347$	−4363.74	−0.675095	−0.337547	−	0.941309i	$-0.609597\pi$
$347$	−4363.74	−0.675095	−0.337547	+	0.941309i	$0.609597\pi$
$348$	5791.81	0.892165
$349$	1622.29	0.248823	0.124411	−	0.992231i	$-0.460296\pi$
$349$	1622.29	0.248823	0.124411	+	0.992231i	$0.460296\pi$
$350$	2497.16	0.381367
$351$	73.6407	0.0111984
$352$	−1703.47	−0.257941
$353$	5849.26	0.881940	0.440970	−	0.897522i	$-0.354634\pi$
$353$	5849.26	0.881940	0.440970	+	0.897522i	$0.354634\pi$
$354$	864.960	0.129865
$355$	482.309	0.0721079
$356$	−392.801	−0.0584787
$357$	5114.32	0.758203
$358$	3602.52	0.531841
$359$	4163.05	0.612026	0.306013	−	0.952027i	$-0.401005\pi$
$359$	4163.05	0.612026	0.306013	+	0.952027i	$0.401005\pi$
$360$	165.945	0.0242947
$361$	9463.72	1.37975
$362$	996.578	0.144693
$363$	363.000	0.0524864
$364$	−416.981	−0.0600432
$365$	−157.642	−0.0226064
$366$	−172.393	−0.0246205
$367$	−12364.2	−1.75859	−0.879296	−	0.476275i	$-0.841987\pi$
$367$	−12364.2	−1.75859	−0.879296	+	0.476275i	$0.841987\pi$
$368$	243.517	0.0344951
$369$	3529.39	0.497920
$370$	422.523	0.0593674
$371$	−6423.21	−0.898859
$372$	4054.60	0.565110
$373$	13785.2	1.91359	0.956797	−	0.290755i	$-0.0939065\pi$
$373$	13785.2	1.91359	0.956797	+	0.290755i	$0.0939065\pi$
$374$	−821.848	−0.113628
$375$	−964.837	−0.132864
$376$	1816.61	0.249160
$377$	−740.322	−0.101137
$378$	−546.722	−0.0743925
$379$	7650.00	1.03682	0.518409	−	0.855133i	$-0.326525\pi$
$379$	7650.00	1.03682	0.518409	+	0.855133i	$0.326525\pi$
$380$	1176.90	0.158878
$381$	7661.55	1.03022
$382$	−1979.28	−0.265101
$383$	3708.56	0.494774	0.247387	−	0.968917i	$-0.420428\pi$
$383$	3708.56	0.494774	0.247387	+	0.968917i	$0.420428\pi$
$384$	4287.62	0.569796
$385$	306.228	0.0405372
$386$	−146.383	−0.0193023
$387$	196.422	0.0258003
$388$	7827.96	1.02424
$389$	−9967.89	−1.29921	−0.649604	−	0.760272i	$-0.725065\pi$
$389$	−9967.89	−1.29921	−0.649604	+	0.760272i	$0.725065\pi$
$390$	−9.98297	−0.00129617
$391$	444.099	0.0574400
$392$	1694.59	0.218341
$393$	−4389.22	−0.563376
$394$	4313.69	0.551575
$395$	1062.77	0.135377
$396$	−704.144	−0.0893550
$397$	−6672.18	−0.843494	−0.421747	−	0.906714i	$-0.638583\pi$
$397$	−6672.18	−0.843494	−0.421747	+	0.906714i	$0.638583\pi$
$398$	3590.20	0.452162
$399$	−8238.59	−1.03370
$400$	−5363.20	−0.670400
$401$	−3733.25	−0.464911	−0.232456	−	0.972607i	$-0.574676\pi$
$401$	−3733.25	−0.464911	−0.232456	+	0.972607i	$0.574676\pi$
$402$	−1551.25	−0.192461
$403$	−518.268	−0.0640614
$404$	−624.638	−0.0769230
$405$	104.906	0.0128712
$406$	5496.28	0.671862
$407$	−3809.42	−0.463946
$408$	3387.34	0.411025
$409$	2855.96	0.345277	0.172639	−	0.984985i	$-0.444771\pi$
$409$	2855.96	0.345277	0.172639	+	0.984985i	$0.444771\pi$
$410$	−478.455	−0.0576322
$411$	7719.82	0.926497
$412$	8346.35	0.998047
$413$	−6578.74	−0.783822
$414$	−47.4743	−0.00563583
$415$	−721.713	−0.0853674
$416$	−422.373	−0.0497802
$417$	−4804.30	−0.564191
$418$	1323.90	0.154914
$419$	−7182.48	−0.837440	−0.418720	−	0.908115i	$-0.637521\pi$
$419$	−7182.48	−0.837440	−0.418720	+	0.908115i	$0.637521\pi$
$420$	−594.018	−0.0690122
$421$	−14876.8	−1.72221	−0.861104	−	0.508430i	$-0.830226\pi$
$421$	−14876.8	−1.72221	−0.861104	+	0.508430i	$0.830226\pi$
$422$	649.788	0.0749554
$423$	1148.41	0.132004
$424$	−4254.25	−0.487275
$425$	−9780.79	−1.11632
$426$	−1052.44	−0.119697
$427$	1311.19	0.148602
$428$	12419.9	1.40266
$429$	90.0053	0.0101294
$430$	−26.6276	−0.00298627
$431$	5784.40	0.646461	0.323231	−	0.946320i	$-0.395231\pi$
$431$	5784.40	0.646461	0.323231	+	0.946320i	$0.395231\pi$
$432$	1174.21	0.130773
$433$	−2075.84	−0.230389	−0.115194	−	0.993343i	$-0.536749\pi$
$433$	−2075.84	−0.230389	−0.115194	+	0.993343i	$0.536749\pi$
$434$	3847.71	0.425567
$435$	−1054.64	−0.116244
$436$	4763.67	0.523253
$437$	−715.393	−0.0783110
$438$	343.988	0.0375260
$439$	14023.8	1.52465	0.762325	−	0.647194i	$-0.224058\pi$
$439$	14023.8	1.52465	0.762325	+	0.647194i	$0.224058\pi$
$440$	202.822	0.0219754
$441$	1071.28	0.115676
$442$	−203.776	−0.0219290
$443$	−5090.51	−0.545954	−0.272977	−	0.962021i	$-0.588008\pi$
$443$	−5090.51	−0.545954	−0.272977	+	0.962021i	$0.588008\pi$
$444$	7389.49	0.789842
$445$	71.5258	0.00761944
$446$	2181.86	0.231646
$447$	1444.96	0.152895
$448$	−4342.59	−0.457964
$449$	−8826.99	−0.927776	−0.463888	−	0.885894i	$-0.653546\pi$
$449$	−8826.99	−0.927776	−0.463888	+	0.885894i	$0.653546\pi$
$450$	1045.57	0.109530
$451$	4313.70	0.450386
$452$	−1425.58	−0.148349
$453$	1734.76	0.179925
$454$	−325.680	−0.0336672
$455$	75.9287	0.00782328
$456$	−5456.61	−0.560371
$457$	−9116.65	−0.933170	−0.466585	−	0.884476i	$-0.654516\pi$
$457$	−9116.65	−0.933170	−0.466585	+	0.884476i	$0.654516\pi$
$458$	297.950	0.0303980
$459$	2141.39	0.217759
$460$	−51.5812	−0.00522823
$461$	564.760	0.0570574	0.0285287	−	0.999593i	$-0.490918\pi$
$461$	564.760	0.0570574	0.0285287	+	0.999593i	$0.490918\pi$
$462$	−668.216	−0.0672905
$463$	18852.1	1.89230	0.946148	−	0.323736i	$-0.104939\pi$
$463$	18852.1	1.89230	0.946148	+	0.323736i	$0.104939\pi$
$464$	−11804.5	−1.18106
$465$	−738.308	−0.0736306
$466$	4253.15	0.422797
$467$	−15031.3	−1.48943	−0.744716	−	0.667382i	$-0.767415\pi$
$467$	−15031.3	−1.48943	−0.744716	+	0.667382i	$0.767415\pi$
$468$	−174.592	−0.0172447
$469$	11798.5	1.16163
$470$	−155.682	−0.0152789
$471$	−10861.6	−1.06258
$472$	−4357.26	−0.424913
$473$	240.072	0.0233372
$474$	−2319.06	−0.224722
$475$	15755.7	1.52194
$476$	−12125.3	−1.16757
$477$	−2689.42	−0.258156
$478$	906.190	0.0867116
$479$	−14710.8	−1.40324	−0.701622	−	0.712550i	$-0.747540\pi$
$479$	−14710.8	−1.40324	−0.701622	+	0.712550i	$0.747540\pi$
$480$	−601.700	−0.0572161
$481$	−944.541	−0.0895372
$482$	−5425.25	−0.512683
$483$	361.081	0.0340161
$484$	−860.621	−0.0808246
$485$	−1425.41	−0.133452
$486$	−228.915	−0.0213658
$487$	11769.7	1.09515	0.547575	−	0.836757i	$-0.315551\pi$
$487$	11769.7	1.09515	0.547575	+	0.836757i	$0.315551\pi$
$488$	868.432	0.0805575
$489$	9482.52	0.876921
$490$	−145.225	−0.0133890
$491$	16072.2	1.47725	0.738624	−	0.674117i	$-0.235476\pi$
$491$	16072.2	1.47725	0.738624	+	0.674117i	$0.235476\pi$
$492$	−8367.67	−0.766755
$493$	−21527.7	−1.96665
$494$	328.260	0.0298970
$495$	128.219	0.0116424
$496$	−8263.82	−0.748098
$497$	8004.68	0.722452
$498$	1574.84	0.141707
$499$	−13745.0	−1.23309	−0.616546	−	0.787319i	$-0.711469\pi$
$499$	−13745.0	−1.23309	−0.616546	+	0.787319i	$0.711469\pi$
$500$	2287.49	0.204599
$501$	−4233.36	−0.377510
$502$	−25.1503	−0.00223608
$503$	−21938.9	−1.94474	−0.972372	−	0.233435i	$-0.925003\pi$
$503$	−21938.9	−1.94474	−0.972372	+	0.233435i	$0.925003\pi$
$504$	2754.12	0.243410
$505$	113.741	0.0100226
$506$	−58.0241	−0.00509780
$507$	−6568.68	−0.575395
$508$	−18164.4	−1.58645
$509$	−4312.09	−0.375501	−0.187751	−	0.982217i	$-0.560120\pi$
$509$	−4312.09	−0.375501	−0.187751	+	0.982217i	$0.560120\pi$
$510$	−290.293	−0.0252047
$511$	−2616.32	−0.226495
$512$	−11687.9	−1.00886
$513$	−3449.53	−0.296882
$514$	−248.884	−0.0213576
$515$	−1519.80	−0.130040
$516$	−465.689	−0.0397303
$517$	1403.61	0.119402
$518$	7012.45	0.594805
$519$	10577.6	0.894612
$520$	50.2894	0.00424103
$521$	−8223.57	−0.691519	−0.345759	−	0.938323i	$-0.612379\pi$
$521$	−8223.57	−0.691519	−0.345759	+	0.938323i	$0.612379\pi$
$522$	2301.32	0.192961
$523$	−1499.65	−0.125382	−0.0626912	−	0.998033i	$-0.519968\pi$
$523$	−1499.65	−0.125382	−0.0626912	+	0.998033i	$0.519968\pi$
$524$	10406.2	0.867552
$525$	−7952.42	−0.661090
$526$	−2354.14	−0.195143
$527$	−15070.6	−1.24571
$528$	1435.14	0.118289
$529$	−12135.6	−0.997423
$530$	364.587	0.0298805
$531$	−2754.55	−0.225117
$532$	19532.5	1.59181
$533$	1069.57	0.0869201
$534$	−156.076	−0.0126480
$535$	−2261.57	−0.182759
$536$	7814.45	0.629726
$537$	−11472.6	−0.921932
$538$	5212.34	0.417695
$539$	1309.34	0.104633
$540$	−248.718	−0.0198206
$541$	6445.91	0.512257	0.256129	−	0.966643i	$-0.417553\pi$
$541$	6445.91	0.512257	0.256129	+	0.966643i	$0.417553\pi$
$542$	1570.42	0.124457
$543$	−3173.69	−0.250822
$544$	−12282.1	−0.967999
$545$	−867.424	−0.0681768
$546$	−165.683	−0.0129864
$547$	−20254.5	−1.58322	−0.791610	−	0.611027i	$-0.790757\pi$
$547$	−20254.5	−1.58322	−0.791610	+	0.611027i	$0.790757\pi$
$548$	−18302.6	−1.42673
$549$	549.000	0.0426790
$550$	1277.92	0.0990738
$551$	34678.7	2.68124
$552$	239.153	0.0184403
$553$	17638.4	1.35635
$554$	−4475.02	−0.343187
$555$	−1345.56	−0.102912
$556$	11390.3	0.868806
$557$	19049.6	1.44912	0.724559	−	0.689212i	$-0.242043\pi$
$557$	19049.6	1.44912	0.724559	+	0.689212i	$0.242043\pi$
$558$	1611.05	0.122225
$559$	59.5254	0.00450386
$560$	1210.69	0.0913589
$561$	2617.25	0.196970
$562$	2222.72	0.166832
$563$	−11977.1	−0.896578	−0.448289	−	0.893889i	$-0.647966\pi$
$563$	−11977.1	−0.896578	−0.448289	+	0.893889i	$0.647966\pi$
$564$	−2722.72	−0.203275
$565$	259.587	0.0193290
$566$	4433.38	0.329239
$567$	1741.09	0.128957
$568$	5301.69	0.391644
$569$	−3738.11	−0.275413	−0.137706	−	0.990473i	$-0.543973\pi$
$569$	−3738.11	−0.275413	−0.137706	+	0.990473i	$0.543973\pi$
$570$	467.629	0.0343629
$571$	7135.47	0.522960	0.261480	−	0.965209i	$-0.415789\pi$
$571$	7135.47	0.522960	0.261480	+	0.965209i	$0.415789\pi$
$572$	−213.390	−0.0155984
$573$	6303.19	0.459545
$574$	−7940.72	−0.577420
$575$	−690.544	−0.0500829
$576$	−1818.26	−0.131529
$577$	−3559.26	−0.256800	−0.128400	−	0.991722i	$-0.540984\pi$
$577$	−3559.26	−0.256800	−0.128400	+	0.991722i	$0.540984\pi$
$578$	−1297.35	−0.0933607
$579$	466.170	0.0334601
$580$	2500.40	0.179006
$581$	−11978.0	−0.855301
$582$	3110.36	0.221527
$583$	−3287.07	−0.233511
$584$	−1732.85	−0.122784
$585$	31.7917	0.00224688
$586$	−5585.67	−0.393758
$587$	3880.88	0.272881	0.136440	−	0.990648i	$-0.456434\pi$
$587$	3880.88	0.272881	0.136440	+	0.990648i	$0.456434\pi$
$588$	−2539.84	−0.178131
$589$	24277.1	1.69833
$590$	373.415	0.0260563
$591$	−13737.3	−0.956140
$592$	−15060.8	−1.04560
$593$	5107.33	0.353681	0.176841	−	0.984239i	$-0.443412\pi$
$593$	5107.33	0.353681	0.176841	+	0.984239i	$0.443412\pi$
$594$	−279.785	−0.0193261
$595$	2207.92	0.152128
$596$	−3425.79	−0.235446
$597$	−11433.3	−0.783809
$598$	−14.3870	−0.000983826 0
$599$	−1445.49	−0.0985993	−0.0492996	−	0.998784i	$-0.515699\pi$
$599$	−1445.49	−0.0985993	−0.0492996	+	0.998784i	$0.515699\pi$
$600$	−5267.08	−0.358379
$601$	−26802.9	−1.81916	−0.909578	−	0.415533i	$-0.863595\pi$
$601$	−26802.9	−1.81916	−0.909578	+	0.415533i	$0.863595\pi$
$602$	−441.928	−0.0299196
$603$	4940.10	0.333626
$604$	−4112.86	−0.277069
$605$	156.712	0.0105310
$606$	−248.194	−0.0166373
$607$	20385.3	1.36312	0.681560	−	0.731762i	$-0.261302\pi$
$607$	20385.3	1.36312	0.681560	+	0.731762i	$0.261302\pi$
$608$	19785.1	1.31972
$609$	−17503.4	−1.16465
$610$	−74.4242	−0.00493991
$611$	348.024	0.0230435
$612$	−5076.92	−0.335331
$613$	14308.7	0.942776	0.471388	−	0.881926i	$-0.343753\pi$
$613$	14308.7	0.942776	0.471388	+	0.881926i	$0.343753\pi$
$614$	−8972.77	−0.589758
$615$	1523.68	0.0999038
$616$	3366.15	0.220172
$617$	15298.4	0.998202	0.499101	−	0.866544i	$-0.333664\pi$
$617$	15298.4	0.998202	0.499101	+	0.866544i	$0.333664\pi$
$618$	3316.34	0.215862
$619$	23502.3	1.52607	0.763034	−	0.646359i	$-0.223709\pi$
$619$	23502.3	1.52607	0.763034	+	0.646359i	$0.223709\pi$
$620$	1750.42	0.113385
$621$	151.186	0.00976956
$622$	−5543.49	−0.357353
$623$	1187.08	0.0763396
$624$	355.842	0.0228286
$625$	14998.8	0.959923
$626$	−6275.05	−0.400642
$627$	−4216.09	−0.268540
$628$	25751.3	1.63629
$629$	−27466.2	−1.74109
$630$	−236.027	−0.0149263
$631$	6379.09	0.402453	0.201226	−	0.979545i	$-0.435507\pi$
$631$	6379.09	0.402453	0.201226	+	0.979545i	$0.435507\pi$
$632$	11682.3	0.735282
$633$	−2069.31	−0.129933
$634$	5862.33	0.367228
$635$	3307.59	0.206705
$636$	6376.24	0.397538
$637$	324.648	0.0201931
$638$	2812.72	0.174540
$639$	3351.59	0.207491
$640$	1851.02	0.114325
$641$	−25161.1	−1.55040	−0.775198	−	0.631719i	$-0.782350\pi$
$641$	−25161.1	−1.55040	−0.775198	+	0.631719i	$0.782350\pi$
$642$	4934.93	0.303374
$643$	−5314.89	−0.325970	−0.162985	−	0.986629i	$-0.552112\pi$
$643$	−5314.89	−0.325970	−0.162985	+	0.986629i	$0.552112\pi$
$644$	−856.072	−0.0523819
$645$	84.7981	0.00517662
$646$	9545.42	0.581362
$647$	19289.5	1.17210	0.586049	−	0.810276i	$-0.300683\pi$
$647$	19289.5	1.17210	0.586049	+	0.810276i	$0.300683\pi$
$648$	1153.16	0.0699082
$649$	−3366.67	−0.203626
$650$	316.858	0.0191203
$651$	−12253.4	−0.737709
$652$	−22481.7	−1.35038
$653$	−4545.97	−0.272431	−0.136216	−	0.990679i	$-0.543494\pi$
$653$	−4545.97	−0.272431	−0.136216	+	0.990679i	$0.543494\pi$
$654$	1892.80	0.113172
$655$	−1894.88	−0.113037
$656$	17054.5	1.01504
$657$	−1095.46	−0.0650503
$658$	−2583.79	−0.153080
$659$	3621.20	0.214055	0.107027	−	0.994256i	$-0.465867\pi$
$659$	3621.20	0.214055	0.107027	+	0.994256i	$0.465867\pi$
$660$	−303.988	−0.0179284
$661$	−13855.9	−0.815329	−0.407665	−	0.913132i	$-0.633657\pi$
$661$	−13855.9	−0.815329	−0.407665	+	0.913132i	$0.633657\pi$
$662$	8661.48	0.508517
$663$	648.943	0.0380134
$664$	−7933.29	−0.463662
$665$	−3556.71	−0.207403
$666$	2936.14	0.170830
$667$	−1519.90	−0.0882320
$668$	10036.7	0.581334
$669$	−6948.34	−0.401552
$670$	−669.695	−0.0386158
$671$	671.000	0.0386046
$672$	−9986.17	−0.573251
$673$	2012.11	0.115247	0.0576233	−	0.998338i	$-0.481648\pi$
$673$	2012.11	0.115247	0.0576233	+	0.998338i	$0.481648\pi$
$674$	1946.84	0.111260
$675$	−3329.71	−0.189868
$676$	15573.4	0.886061
$677$	1699.88	0.0965016	0.0482508	−	0.998835i	$-0.484635\pi$
$677$	1699.88	0.0965016	0.0482508	+	0.998835i	$0.484635\pi$
$678$	−566.441	−0.0320856
$679$	−23656.9	−1.33707
$680$	1462.36	0.0824689
$681$	1037.16	0.0583612
$682$	1969.07	0.110556
$683$	−15659.6	−0.877302	−0.438651	−	0.898657i	$-0.644544\pi$
$683$	−15659.6	−0.877302	−0.438651	+	0.898657i	$0.644544\pi$
$684$	8178.34	0.457174
$685$	3332.75	0.185894
$686$	4535.15	0.252409
$687$	−948.848	−0.0526941
$688$	949.138	0.0525953
$689$	−815.025	−0.0450653
$690$	−20.4953	−0.00113079
$691$	−11827.0	−0.651114	−0.325557	−	0.945522i	$-0.605552\pi$
$691$	−11827.0	−0.651114	−0.325557	+	0.945522i	$0.605552\pi$
$692$	−25077.9	−1.37763
$693$	2127.99	0.116646
$694$	4110.80	0.224847
$695$	−2074.08	−0.113200
$696$	−11592.9	−0.631363
$697$	31102.0	1.69020
$698$	−1528.26	−0.0828730
$699$	−13544.5	−0.732906
$700$	18854.0	1.01802
$701$	−23700.6	−1.27697	−0.638487	−	0.769633i	$-0.720439\pi$
$701$	−23700.6	−1.27697	−0.638487	+	0.769633i	$0.720439\pi$
$702$	−69.3722	−0.00372975
$703$	44244.9	2.37372
$704$	−2222.32	−0.118973
$705$	495.785	0.0264856
$706$	−5510.22	−0.293739
$707$	1887.72	0.100417
$708$	6530.63	0.346661
$709$	14239.7	0.754277	0.377138	−	0.926157i	$-0.376908\pi$
$709$	14239.7	0.754277	0.377138	+	0.926157i	$0.376908\pi$
$710$	−454.352	−0.0240162
$711$	7385.26	0.389549
$712$	786.235	0.0413840
$713$	−1064.02	−0.0558874
$714$	−4817.87	−0.252527
$715$	38.8565	0.00203238
$716$	27199.8	1.41970
$717$	−2885.85	−0.150312
$718$	−3921.74	−0.203841
$719$	−17537.0	−0.909622	−0.454811	−	0.890588i	$-0.650293\pi$
$719$	−17537.0	−0.909622	−0.454811	+	0.890588i	$0.650293\pi$
$720$	506.921	0.0262387
$721$	−25223.5	−1.30287
$722$	−8915.17	−0.459540
$723$	17277.2	0.888722
$724$	7524.37	0.386245
$725$	33474.1	1.71475
$726$	−341.959	−0.0174811
$727$	9282.37	0.473541	0.236770	−	0.971566i	$-0.423911\pi$
$727$	9282.37	0.473541	0.236770	+	0.971566i	$0.423911\pi$
$728$	834.633	0.0424911
$729$	729.000	0.0370370
$730$	148.504	0.00752930
$731$	1730.93	0.0875797
$732$	−1301.60	−0.0657220
$733$	18964.2	0.955604	0.477802	−	0.878468i	$-0.341434\pi$
$733$	18964.2	0.955604	0.477802	+	0.878468i	$0.341434\pi$
$734$	11647.5	0.585717
$735$	462.484	0.0232095
$736$	−867.143	−0.0434284
$737$	6037.90	0.301776
$738$	−3324.81	−0.165837
$739$	−8539.53	−0.425077	−0.212538	−	0.977153i	$-0.568173\pi$
$739$	−8539.53	−0.425077	−0.212538	+	0.977153i	$0.568173\pi$
$740$	3190.14	0.158476
$741$	−1045.37	−0.0518256
$742$	6050.90	0.299374
$743$	−35202.6	−1.73817	−0.869083	−	0.494667i	$-0.835290\pi$
$743$	−35202.6	−1.73817	−0.869083	+	0.494667i	$0.835290\pi$
$744$	−8115.72	−0.399915
$745$	623.809	0.0306773
$746$	−12986.2	−0.637342
$747$	−5015.22	−0.245646
$748$	−6205.12	−0.303318
$749$	−37534.3	−1.83107
$750$	908.912	0.0442517
$751$	35786.0	1.73882	0.869408	−	0.494095i	$-0.164501\pi$
$751$	35786.0	1.73882	0.869408	+	0.494095i	$0.164501\pi$
$752$	5549.28	0.269097
$753$	80.0933	0.00387618
$754$	697.410	0.0336846
$755$	748.918	0.0361005
$756$	−4127.87	−0.198583
$757$	8715.81	0.418469	0.209235	−	0.977865i	$-0.432903\pi$
$757$	8715.81	0.418469	0.209235	+	0.977865i	$0.432903\pi$
$758$	−7206.57	−0.345323
$759$	184.783	0.00883690
$760$	−2355.69	−0.112434
$761$	−25246.9	−1.20263	−0.601313	−	0.799014i	$-0.705355\pi$
$761$	−25246.9	−1.20263	−0.601313	+	0.799014i	$0.705355\pi$
$762$	−7217.45	−0.343124
$763$	−14396.3	−0.683067
$764$	−14944.0	−0.707661
$765$	924.465	0.0436916
$766$	−3493.59	−0.164789
$767$	−834.760	−0.0392978
$768$	809.597	0.0380388
$769$	−2393.43	−0.112236	−0.0561178	−	0.998424i	$-0.517872\pi$
$769$	−2393.43	−0.112236	−0.0561178	+	0.998424i	$0.517872\pi$
$770$	−288.478	−0.0135013
$771$	792.592	0.0370227
$772$	−1105.22	−0.0515257
$773$	28311.2	1.31732	0.658658	−	0.752443i	$-0.271125\pi$
$773$	28311.2	1.31732	0.658658	+	0.752443i	$0.271125\pi$
$774$	−185.037	−0.00859304
$775$	23433.8	1.08615
$776$	−15668.5	−0.724828
$777$	−22331.8	−1.03108
$778$	9390.11	0.432715
$779$	−50101.8	−2.30434
$780$	−75.3735	−0.00346000
$781$	4096.39	0.187683
$782$	−418.357	−0.0191310
$783$	−7328.75	−0.334493
$784$	5176.54	0.235812
$785$	−4689.10	−0.213199
$786$	4134.80	0.187638
$787$	447.771	0.0202812	0.0101406	−	0.999949i	$-0.496772\pi$
$787$	447.771	0.0202812	0.0101406	+	0.999949i	$0.496772\pi$
$788$	32569.3	1.47238
$789$	7496.98	0.338276
$790$	−1001.17	−0.0450886
$791$	4308.26	0.193659
$792$	1409.42	0.0632344
$793$	166.373	0.00745031
$794$	6285.43	0.280934
$795$	−1161.06	−0.0517969
$796$	27106.7	1.20700
$797$	−28139.9	−1.25065	−0.625325	−	0.780364i	$-0.715034\pi$
$797$	−28139.9	−1.25065	−0.625325	+	0.780364i	$0.715034\pi$
$798$	7761.05	0.344283
$799$	10120.1	0.448091
$800$	19097.9	0.844014
$801$	497.037	0.0219250
$802$	3516.85	0.154843
$803$	−1338.90	−0.0588402
$804$	−11712.3	−0.513756
$805$	155.884	0.00682506
$806$	488.227	0.0213363
$807$	−16599.2	−0.724062
$808$	1250.28	0.0544366
$809$	23949.1	1.04080	0.520400	−	0.853923i	$-0.325783\pi$
$809$	23949.1	1.04080	0.520400	+	0.853923i	$0.325783\pi$
$810$	−98.8255	−0.00428688
$811$	12707.5	0.550209	0.275105	−	0.961414i	$-0.411287\pi$
$811$	12707.5	0.550209	0.275105	+	0.961414i	$0.411287\pi$
$812$	41498.1	1.79347
$813$	−5001.16	−0.215742
$814$	3588.62	0.154522
$815$	4093.73	0.175947
$816$	10347.5	0.443914
$817$	−2788.33	−0.119402
$818$	−2690.42	−0.114998
$819$	527.633	0.0225116
$820$	−3612.43	−0.153843
$821$	−36089.2	−1.53413	−0.767065	−	0.641570i	$-0.778284\pi$
$821$	−36089.2	−1.53413	−0.767065	+	0.641570i	$0.778284\pi$
$822$	−7272.34	−0.308579
$823$	16154.4	0.684211	0.342106	−	0.939661i	$-0.388860\pi$
$823$	16154.4	0.684211	0.342106	+	0.939661i	$0.388860\pi$
$824$	−16706.1	−0.706293
$825$	−4069.65	−0.171742
$826$	6197.41	0.261060
$827$	−18940.7	−0.796412	−0.398206	−	0.917296i	$-0.630367\pi$
$827$	−18940.7	−0.796412	−0.398206	+	0.917296i	$0.630367\pi$
$828$	−358.441	−0.0150443
$829$	7117.33	0.298185	0.149092	−	0.988823i	$-0.452365\pi$
$829$	7117.33	0.298185	0.149092	+	0.988823i	$0.452365\pi$
$830$	679.879	0.0284325
$831$	14251.1	0.594905
$832$	−551.020	−0.0229606
$833$	9440.39	0.392665
$834$	4525.82	0.187909
$835$	−1827.60	−0.0757444
$836$	9995.75	0.413529
$837$	−5130.55	−0.211873
$838$	6766.16	0.278918
$839$	45871.7	1.88756	0.943782	−	0.330570i	$-0.107241\pi$
$839$	45871.7	1.88756	0.943782	+	0.330570i	$0.107241\pi$
$840$	1188.99	0.0488382
$841$	49288.1	2.02091
$842$	14014.4	0.573598
$843$	−7078.44	−0.289199
$844$	4906.04	0.200086
$845$	−2835.79	−0.115449
$846$	−1081.85	−0.0439653
$847$	2600.88	0.105510
$848$	−12995.7	−0.526265
$849$	−14118.5	−0.570726
$850$	9213.86	0.371803
$851$	−1939.17	−0.0781126
$852$	−7946.14	−0.319519
$853$	−40736.5	−1.63516	−0.817580	−	0.575814i	$-0.804685\pi$
$853$	−40736.5	−1.63516	−0.817580	+	0.575814i	$0.804685\pi$
$854$	−1235.19	−0.0494932
$855$	−1489.21	−0.0595671
$856$	−24859.8	−0.992631
$857$	6443.64	0.256838	0.128419	−	0.991720i	$-0.459010\pi$
$857$	6443.64	0.256838	0.128419	+	0.991720i	$0.459010\pi$
$858$	−84.7882	−0.00337369
$859$	41161.4	1.63493	0.817467	−	0.575975i	$-0.195377\pi$
$859$	41161.4	1.63493	0.817467	+	0.575975i	$0.195377\pi$
$860$	−201.044	−0.00797157
$861$	25287.9	1.00094
$862$	−5449.12	−0.215310
$863$	28476.1	1.12322	0.561610	−	0.827402i	$-0.310182\pi$
$863$	28476.1	1.12322	0.561610	+	0.827402i	$0.310182\pi$
$864$	−4181.25	−0.164640
$865$	4566.47	0.179497
$866$	1955.51	0.0767333
$867$	4131.52	0.161838
$868$	29051.0	1.13601
$869$	9026.43	0.352360
$870$	993.508	0.0387162
$871$	1497.09	0.0582398
$872$	−9535.00	−0.370293
$873$	−9905.23	−0.384011
$874$	673.926	0.0260823
$875$	−6913.03	−0.267089
$876$	2597.18	0.100172
$877$	27025.7	1.04059	0.520293	−	0.853988i	$-0.325823\pi$
$877$	27025.7	1.04059	0.520293	+	0.853988i	$0.325823\pi$
$878$	−13211.0	−0.507800
$879$	17788.1	0.682568
$880$	619.570	0.0237338
$881$	−7511.13	−0.287238	−0.143619	−	0.989633i	$-0.545874\pi$
$881$	−7511.13	−0.287238	−0.143619	+	0.989633i	$0.545874\pi$
$882$	−1009.18	−0.0385270
$883$	32044.4	1.22127	0.610634	−	0.791913i	$-0.290915\pi$
$883$	32044.4	1.22127	0.610634	+	0.791913i	$0.290915\pi$
$884$	−1538.55	−0.0585374
$885$	−1189.17	−0.0451679
$886$	4795.45	0.181835
$887$	48492.8	1.83566	0.917829	−	0.396975i	$-0.129940\pi$
$887$	48492.8	1.83566	0.917829	+	0.396975i	$0.129940\pi$
$888$	−14790.9	−0.558952
$889$	54894.7	2.07099
$890$	−67.3799	−0.00253773
$891$	891.000	0.0335013
$892$	16473.5	0.618357
$893$	−16302.4	−0.610906
$894$	−1361.20	−0.0509234
$895$	−4952.86	−0.184978
$896$	30720.6	1.14543
$897$	45.8167	0.00170544
$898$	8315.34	0.309005
$899$	51578.2	1.91349
$900$	7894.26	0.292380
$901$	−23700.0	−0.876317
$902$	−4063.66	−0.150005
$903$	1407.36	0.0518649
$904$	2853.46	0.104983
$905$	−1370.13	−0.0503254
$906$	−1634.20	−0.0599258
$907$	48575.6	1.77831	0.889154	−	0.457608i	$-0.151294\pi$
$907$	48575.6	1.77831	0.889154	+	0.457608i	$0.151294\pi$
$908$	−2458.95	−0.0898714
$909$	790.396	0.0288402
$910$	−71.5276	−0.00260562
$911$	8866.70	0.322466	0.161233	−	0.986916i	$-0.448453\pi$
$911$	8866.70	0.322466	0.161233	+	0.986916i	$0.448453\pi$
$912$	−16668.6	−0.605210
$913$	−6129.72	−0.222195
$914$	8588.21	0.310802
$915$	237.011	0.00856320
$916$	2249.58	0.0811444
$917$	−31448.6	−1.13252
$918$	−2017.26	−0.0725268
$919$	−15606.2	−0.560176	−0.280088	−	0.959974i	$-0.590364\pi$
$919$	−15606.2	−0.560176	−0.280088	+	0.959974i	$0.590364\pi$
$920$	103.245	0.00369989
$921$	28574.6	1.02233
$922$	−532.024	−0.0190035
$923$	1015.69	0.0362210
$924$	−5045.17	−0.179625
$925$	42708.0	1.51809
$926$	−17759.4	−0.630248
$927$	−10561.2	−0.374191
$928$	42034.7	1.48692
$929$	1232.32	0.0435209	0.0217605	−	0.999763i	$-0.493073\pi$
$929$	1232.32	0.0435209	0.0217605	+	0.999763i	$0.493073\pi$
$930$	695.513	0.0245234
$931$	−15207.4	−0.535341
$932$	32112.1	1.12861
$933$	17653.8	0.619462
$934$	14160.0	0.496070
$935$	1129.90	0.0395206
$936$	349.464	0.0122036
$937$	−21620.7	−0.753809	−0.376904	−	0.926252i	$-0.623011\pi$
$937$	−21620.7	−0.753809	−0.376904	+	0.926252i	$0.623011\pi$
$938$	−11114.6	−0.386894
$939$	19983.5	0.694501
$940$	−1175.43	−0.0407856
$941$	−52777.4	−1.82837	−0.914183	−	0.405301i	$-0.867167\pi$
$941$	−52777.4	−1.82837	−0.914183	+	0.405301i	$0.867167\pi$
$942$	10232.0	0.353904
$943$	2195.86	0.0758294
$944$	−13310.3	−0.458913
$945$	751.650	0.0258743
$946$	−226.156	−0.00777270
$947$	−31591.7	−1.08405	−0.542023	−	0.840364i	$-0.682341\pi$
$947$	−31591.7	−1.08405	−0.542023	+	0.840364i	$0.682341\pi$
$948$	−17509.4	−0.599872
$949$	−331.978	−0.0113556
$950$	−14842.5	−0.506899
$951$	−18669.1	−0.636580
$952$	24270.1	0.826261
$953$	−25276.9	−0.859182	−0.429591	−	0.903024i	$-0.641342\pi$
$953$	−25276.9	−0.859182	−0.429591	+	0.903024i	$0.641342\pi$
$954$	2533.54	0.0859814
$955$	2721.17	0.0922042
$956$	6841.92	0.231468
$957$	−8957.36	−0.302561
$958$	13858.1	0.467364
$959$	55312.2	1.86249
$960$	−784.966	−0.0263903
$961$	6316.70	0.212034
$962$	889.792	0.0298212
$963$	−15715.8	−0.525891
$964$	−40961.8	−1.36856
$965$	201.252	0.00671350
$966$	−340.152	−0.0113294
$967$	41535.7	1.38128	0.690640	−	0.723199i	$-0.257329\pi$
$967$	41535.7	1.38128	0.690640	+	0.723199i	$0.257329\pi$
$968$	1722.63	0.0571976
$969$	−30398.3	−1.00777
$970$	1342.78	0.0444476
$971$	26480.2	0.875169	0.437584	−	0.899177i	$-0.355834\pi$
$971$	26480.2	0.875169	0.437584	+	0.899177i	$0.355834\pi$
$972$	−1728.35	−0.0570339
$973$	−34422.6	−1.13416
$974$	−11087.5	−0.364751
$975$	−1009.06	−0.0331445
$976$	2652.84	0.0870034
$977$	41284.9	1.35191	0.675957	−	0.736941i	$-0.263731\pi$
$977$	41284.9	1.35191	0.675957	+	0.736941i	$0.263731\pi$
$978$	−8932.87	−0.292067
$979$	607.490	0.0198319
$980$	−1096.48	−0.0357407
$981$	−6027.78	−0.196180
$982$	−15140.6	−0.492012
$983$	8678.92	0.281602	0.140801	−	0.990038i	$-0.455032\pi$
$983$	8678.92	0.281602	0.140801	+	0.990038i	$0.455032\pi$
$984$	16748.8	0.542614
$985$	−5930.59	−0.191842
$986$	20279.9	0.655013
$987$	8228.33	0.265360
$988$	2478.43	0.0798071
$989$	122.207	0.00392918
$990$	−120.787	−0.00387763
$991$	−21063.8	−0.675190	−0.337595	−	0.941291i	$-0.609613\pi$
$991$	−21063.8	−0.675190	−0.337595	+	0.941291i	$0.609613\pi$
$992$	29426.7	0.941834
$993$	−27583.3	−0.881499
$994$	−7540.69	−0.240620
$995$	−4935.91	−0.157265
$996$	11890.4	0.378274
$997$	30828.4	0.979283	0.489641	−	0.871924i	$-0.337128\pi$
$997$	30828.4	0.979283	0.489641	+	0.871924i	$0.337128\pi$
$998$	12948.3	0.410694
$999$	−9350.41	−0.296130

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2013.4.a.a.1.18	✓	36

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2013.4.a.a.1.18	✓	36	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:	\( N \)	\(=\)	\( 2013 = 3 \cdot 11 \cdot 61 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2013.a (trivial)

\(f(q)\)	\(=\)	\(q-0.942036 q^{2} +3.00000 q^{3} -7.11257 q^{4} +1.29514 q^{5} -2.82611 q^{6} +21.4949 q^{7} +14.2366 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q-0.942036 q^{2} +3.00000 q^{3} -7.11257 q^{4} +1.29514 q^{5} -2.82611 q^{6} +21.4949 q^{7} +14.2366 q^{8} +9.00000 q^{9} -1.22007 q^{10} +11.0000 q^{11} -21.3377 q^{12} +2.72743 q^{13} -20.2490 q^{14} +3.88542 q^{15} +43.4892 q^{16} +79.3106 q^{17} -8.47832 q^{18} -127.760 q^{19} -9.21177 q^{20} +64.4847 q^{21} -10.3624 q^{22} +5.59949 q^{23} +42.7097 q^{24} -123.323 q^{25} -2.56934 q^{26} +27.0000 q^{27} -152.884 q^{28} -271.435 q^{29} -3.66020 q^{30} -190.020 q^{31} -154.861 q^{32} +33.0000 q^{33} -74.7134 q^{34} +27.8389 q^{35} -64.0131 q^{36} -346.311 q^{37} +120.355 q^{38} +8.18230 q^{39} +18.4384 q^{40} +392.154 q^{41} -60.7469 q^{42} +21.8247 q^{43} -78.2383 q^{44} +11.6563 q^{45} -5.27492 q^{46} +127.601 q^{47} +130.468 q^{48} +119.031 q^{49} +116.174 q^{50} +237.932 q^{51} -19.3991 q^{52} -298.825 q^{53} -25.4350 q^{54} +14.2465 q^{55} +306.014 q^{56} -383.281 q^{57} +255.702 q^{58} -306.061 q^{59} -27.6353 q^{60} +61.0000 q^{61} +179.006 q^{62} +193.454 q^{63} -202.029 q^{64} +3.53241 q^{65} -31.0872 q^{66} +548.900 q^{67} -564.102 q^{68} +16.7985 q^{69} -26.2252 q^{70} +372.399 q^{71} +128.129 q^{72} -121.718 q^{73} +326.238 q^{74} -369.968 q^{75} +908.705 q^{76} +236.444 q^{77} -7.70802 q^{78} +820.585 q^{79} +56.3246 q^{80} +81.0000 q^{81} -369.423 q^{82} -557.247 q^{83} -458.652 q^{84} +102.718 q^{85} -20.5597 q^{86} -814.306 q^{87} +156.602 q^{88} +55.2264 q^{89} -10.9806 q^{90} +58.6259 q^{91} -39.8268 q^{92} -570.061 q^{93} -120.205 q^{94} -165.468 q^{95} -464.583 q^{96} -1100.58 q^{97} -112.131 q^{98} +99.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q - 14 q^{2} + 108 q^{3} + 146 q^{4} - 65 q^{5} - 42 q^{6} - 105 q^{7} - 189 q^{8} + 324 q^{9}+O(q^{10}) \) 36 * q - 14 * q^2 + 108 * q^3 + 146 * q^4 - 65 * q^5 - 42 * q^6 - 105 * q^7 - 189 * q^8 + 324 * q^9	\( 36 q - 14 q^{2} + 108 q^{3} + 146 q^{4} - 65 q^{5} - 42 q^{6} - 105 q^{7} - 189 q^{8} + 324 q^{9} - 161 q^{10} + 396 q^{11} + 438 q^{12} - 233 q^{13} - 264 q^{14} - 195 q^{15} + 574 q^{16} - 556 q^{17} - 126 q^{18} - 615 q^{19} - 136 q^{20} - 315 q^{21} - 154 q^{22} - 457 q^{23} - 567 q^{24} + 863 q^{25} + 115 q^{26} + 972 q^{27} - 424 q^{28} - 754 q^{29} - 483 q^{30} - 508 q^{31} - 1511 q^{32} + 1188 q^{33} - 860 q^{34} - 826 q^{35} + 1314 q^{36} - 412 q^{37} - 599 q^{38} - 699 q^{39} - 2791 q^{40} - 2066 q^{41} - 792 q^{42} - 2063 q^{43} + 1606 q^{44} - 585 q^{45} - 787 q^{46} - 1815 q^{47} + 1722 q^{48} + 2825 q^{49} + 808 q^{50} - 1668 q^{51} - 2882 q^{52} - 759 q^{53} - 378 q^{54} - 715 q^{55} - 1749 q^{56} - 1845 q^{57} - 335 q^{58} - 2337 q^{59} - 408 q^{60} + 2196 q^{61} - 1689 q^{62} - 945 q^{63} + 4723 q^{64} - 3550 q^{65} - 462 q^{66} - 1331 q^{67} - 6166 q^{68} - 1371 q^{69} - 1750 q^{70} - 361 q^{71} - 1701 q^{72} - 4627 q^{73} - 3394 q^{74} + 2589 q^{75} - 7214 q^{76} - 1155 q^{77} + 345 q^{78} - 2583 q^{79} - 2643 q^{80} + 2916 q^{81} + 1090 q^{82} - 6123 q^{83} - 1272 q^{84} + 295 q^{85} + 613 q^{86} - 2262 q^{87} - 2079 q^{88} - 2485 q^{89} - 1449 q^{90} - 3156 q^{91} - 6291 q^{92} - 1524 q^{93} - 1744 q^{94} - 5572 q^{95} - 4533 q^{96} - 2558 q^{97} - 2314 q^{98} + 3564 q^{99}+O(q^{100}) \) 36 * q - 14 * q^2 + 108 * q^3 + 146 * q^4 - 65 * q^5 - 42 * q^6 - 105 * q^7 - 189 * q^8 + 324 * q^9 - 161 * q^10 + 396 * q^11 + 438 * q^12 - 233 * q^13 - 264 * q^14 - 195 * q^15 + 574 * q^16 - 556 * q^17 - 126 * q^18 - 615 * q^19 - 136 * q^20 - 315 * q^21 - 154 * q^22 - 457 * q^23 - 567 * q^24 + 863 * q^25 + 115 * q^26 + 972 * q^27 - 424 * q^28 - 754 * q^29 - 483 * q^30 - 508 * q^31 - 1511 * q^32 + 1188 * q^33 - 860 * q^34 - 826 * q^35 + 1314 * q^36 - 412 * q^37 - 599 * q^38 - 699 * q^39 - 2791 * q^40 - 2066 * q^41 - 792 * q^42 - 2063 * q^43 + 1606 * q^44 - 585 * q^45 - 787 * q^46 - 1815 * q^47 + 1722 * q^48 + 2825 * q^49 + 808 * q^50 - 1668 * q^51 - 2882 * q^52 - 759 * q^53 - 378 * q^54 - 715 * q^55 - 1749 * q^56 - 1845 * q^57 - 335 * q^58 - 2337 * q^59 - 408 * q^60 + 2196 * q^61 - 1689 * q^62 - 945 * q^63 + 4723 * q^64 - 3550 * q^65 - 462 * q^66 - 1331 * q^67 - 6166 * q^68 - 1371 * q^69 - 1750 * q^70 - 361 * q^71 - 1701 * q^72 - 4627 * q^73 - 3394 * q^74 + 2589 * q^75 - 7214 * q^76 - 1155 * q^77 + 345 * q^78 - 2583 * q^79 - 2643 * q^80 + 2916 * q^81 + 1090 * q^82 - 6123 * q^83 - 1272 * q^84 + 295 * q^85 + 613 * q^86 - 2262 * q^87 - 2079 * q^88 - 2485 * q^89 - 1449 * q^90 - 3156 * q^91 - 6291 * q^92 - 1524 * q^93 - 1744 * q^94 - 5572 * q^95 - 4533 * q^96 - 2558 * q^97 - 2314 * q^98 + 3564 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more