LMFDB - Embedded newform 4015.2.a.g.1.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4015,2,Mod(1,4015)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4015, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("4015.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4015 = 5 \cdot 11 \cdot 73 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4015.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:	$32.0599364115$
Analytic rank:	$1$
Dimension:	$32$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Character	$\chi$	$=$	4015.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.75190 q^{2} -0.429918 q^{3} +1.06914 q^{4} -1.00000 q^{5} +0.753172 q^{6} +3.79529 q^{7} +1.63077 q^{8} -2.81517 q^{9} +O(q^{10})$	\(q-1.75190 q^{2} -0.429918 q^{3} +1.06914 q^{4} -1.00000 q^{5} +0.753172 q^{6} +3.79529 q^{7} +1.63077 q^{8} -2.81517 q^{9} +1.75190 q^{10} -1.00000 q^{11} -0.459642 q^{12} +2.05520 q^{13} -6.64896 q^{14} +0.429918 q^{15} -4.99522 q^{16} +7.44337 q^{17} +4.93188 q^{18} -4.12238 q^{19} -1.06914 q^{20} -1.63167 q^{21} +1.75190 q^{22} -2.75423 q^{23} -0.701099 q^{24} +1.00000 q^{25} -3.60050 q^{26} +2.50005 q^{27} +4.05769 q^{28} -8.70488 q^{29} -0.753172 q^{30} +4.84064 q^{31} +5.48956 q^{32} +0.429918 q^{33} -13.0400 q^{34} -3.79529 q^{35} -3.00981 q^{36} -1.42395 q^{37} +7.22197 q^{38} -0.883568 q^{39} -1.63077 q^{40} -11.1940 q^{41} +2.85851 q^{42} +3.21471 q^{43} -1.06914 q^{44} +2.81517 q^{45} +4.82512 q^{46} -9.44758 q^{47} +2.14754 q^{48} +7.40426 q^{49} -1.75190 q^{50} -3.20004 q^{51} +2.19729 q^{52} +10.2656 q^{53} -4.37982 q^{54} +1.00000 q^{55} +6.18926 q^{56} +1.77228 q^{57} +15.2500 q^{58} -8.30534 q^{59} +0.459642 q^{60} -2.62164 q^{61} -8.48030 q^{62} -10.6844 q^{63} +0.373304 q^{64} -2.05520 q^{65} -0.753172 q^{66} +1.50213 q^{67} +7.95799 q^{68} +1.18409 q^{69} +6.64896 q^{70} +6.84288 q^{71} -4.59090 q^{72} -1.00000 q^{73} +2.49461 q^{74} -0.429918 q^{75} -4.40739 q^{76} -3.79529 q^{77} +1.54792 q^{78} +1.81457 q^{79} +4.99522 q^{80} +7.37070 q^{81} +19.6108 q^{82} -8.27854 q^{83} -1.74448 q^{84} -7.44337 q^{85} -5.63184 q^{86} +3.74239 q^{87} -1.63077 q^{88} -7.32143 q^{89} -4.93188 q^{90} +7.80009 q^{91} -2.94465 q^{92} -2.08108 q^{93} +16.5512 q^{94} +4.12238 q^{95} -2.36006 q^{96} -15.3235 q^{97} -12.9715 q^{98} +2.81517 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q - 5 q^{2} - 7 q^{3} + 37 q^{4} - 32 q^{5} - 3 q^{6} - 18 q^{8} + 29 q^{9}+O(q^{10}) $ 32 * q - 5 * q^2 - 7 * q^3 + 37 * q^4 - 32 * q^5 - 3 * q^6 - 18 * q^8 + 29 * q^9	\( 32 q - 5 q^{2} - 7 q^{3} + 37 q^{4} - 32 q^{5} - 3 q^{6} - 18 q^{8} + 29 q^{9} + 5 q^{10} - 32 q^{11} - 24 q^{12} - q^{13} - 5 q^{14} + 7 q^{15} + 47 q^{16} - 30 q^{17} - 11 q^{18} + 16 q^{19} - 37 q^{20} + q^{21} + 5 q^{22} - 26 q^{23} - 21 q^{24} + 32 q^{25} - q^{26} - 31 q^{27} - 24 q^{28} - 10 q^{29} + 3 q^{30} - 2 q^{31} - 31 q^{32} + 7 q^{33} - 14 q^{34} + 38 q^{36} - 28 q^{37} - 63 q^{38} - 2 q^{39} + 18 q^{40} - 62 q^{41} - 9 q^{42} + 8 q^{43} - 37 q^{44} - 29 q^{45} + 19 q^{46} - 21 q^{47} - 79 q^{48} + 34 q^{49} - 5 q^{50} + 17 q^{51} + 15 q^{52} - 32 q^{53} + 5 q^{54} + 32 q^{55} - 52 q^{56} - 57 q^{57} + 4 q^{58} - 37 q^{59} + 24 q^{60} + 15 q^{61} - 22 q^{62} + 5 q^{63} + 70 q^{64} + q^{65} + 3 q^{66} - 42 q^{67} - 81 q^{68} - 8 q^{69} + 5 q^{70} - 40 q^{71} - 27 q^{72} - 32 q^{73} - 17 q^{74} - 7 q^{75} + 21 q^{76} - 105 q^{78} + 18 q^{79} - 47 q^{80} + 12 q^{81} - 70 q^{82} - 26 q^{83} + 22 q^{84} + 30 q^{85} - 45 q^{86} - 18 q^{87} + 18 q^{88} - 83 q^{89} + 11 q^{90} - 18 q^{91} - 73 q^{92} - 68 q^{93} + 56 q^{94} - 16 q^{95} - 35 q^{96} - 99 q^{97} - 61 q^{98} - 29 q^{99}+O(q^{100}) \) 32 * q - 5 * q^2 - 7 * q^3 + 37 * q^4 - 32 * q^5 - 3 * q^6 - 18 * q^8 + 29 * q^9 + 5 * q^10 - 32 * q^11 - 24 * q^12 - q^13 - 5 * q^14 + 7 * q^15 + 47 * q^16 - 30 * q^17 - 11 * q^18 + 16 * q^19 - 37 * q^20 + q^21 + 5 * q^22 - 26 * q^23 - 21 * q^24 + 32 * q^25 - q^26 - 31 * q^27 - 24 * q^28 - 10 * q^29 + 3 * q^30 - 2 * q^31 - 31 * q^32 + 7 * q^33 - 14 * q^34 + 38 * q^36 - 28 * q^37 - 63 * q^38 - 2 * q^39 + 18 * q^40 - 62 * q^41 - 9 * q^42 + 8 * q^43 - 37 * q^44 - 29 * q^45 + 19 * q^46 - 21 * q^47 - 79 * q^48 + 34 * q^49 - 5 * q^50 + 17 * q^51 + 15 * q^52 - 32 * q^53 + 5 * q^54 + 32 * q^55 - 52 * q^56 - 57 * q^57 + 4 * q^58 - 37 * q^59 + 24 * q^60 + 15 * q^61 - 22 * q^62 + 5 * q^63 + 70 * q^64 + q^65 + 3 * q^66 - 42 * q^67 - 81 * q^68 - 8 * q^69 + 5 * q^70 - 40 * q^71 - 27 * q^72 - 32 * q^73 - 17 * q^74 - 7 * q^75 + 21 * q^76 - 105 * q^78 + 18 * q^79 - 47 * q^80 + 12 * q^81 - 70 * q^82 - 26 * q^83 + 22 * q^84 + 30 * q^85 - 45 * q^86 - 18 * q^87 + 18 * q^88 - 83 * q^89 + 11 * q^90 - 18 * q^91 - 73 * q^92 - 68 * q^93 + 56 * q^94 - 16 * q^95 - 35 * q^96 - 99 * q^97 - 61 * q^98 - 29 * 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 $

$2$	−1.75190	−1.23878	−0.619389	−	0.785085i	$-0.712619\pi$
$2$	−1.75190	−1.23878	−0.619389	+	0.785085i	$0.712619\pi$
$3$	−0.429918	−0.248213	−0.124107	−	0.992269i	$-0.539606\pi$
$3$	−0.429918	−0.248213	−0.124107	+	0.992269i	$0.539606\pi$
$4$	1.06914	0.534569
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0.753172	0.307481
$7$	3.79529	1.43449	0.717243	−	0.696823i	$-0.245404\pi$
$7$	3.79529	1.43449	0.717243	+	0.696823i	$0.245404\pi$
$8$	1.63077	0.576565
$9$	−2.81517	−0.938390
$10$	1.75190	0.553998
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	−0.459642	−0.132687
$13$	2.05520	0.570010	0.285005	−	0.958526i	$-0.408005\pi$
$13$	2.05520	0.570010	0.285005	+	0.958526i	$0.408005\pi$
$14$	−6.64896	−1.77701
$15$	0.429918	0.111004
$16$	−4.99522	−1.24880
$17$	7.44337	1.80528	0.902641	−	0.430394i	$-0.141625\pi$
$17$	7.44337	1.80528	0.902641	+	0.430394i	$0.141625\pi$
$18$	4.93188	1.16246
$19$	−4.12238	−0.945738	−0.472869	−	0.881133i	$-0.656782\pi$
$19$	−4.12238	−0.945738	−0.472869	+	0.881133i	$0.656782\pi$
$20$	−1.06914	−0.239067
$21$	−1.63167	−0.356059
$22$	1.75190	0.373505
$23$	−2.75423	−0.574296	−0.287148	−	0.957886i	$-0.592707\pi$
$23$	−2.75423	−0.574296	−0.287148	+	0.957886i	$0.592707\pi$
$24$	−0.701099	−0.143111
$25$	1.00000	0.200000
$26$	−3.60050	−0.706115
$27$	2.50005	0.481134
$28$	4.05769	0.766832
$29$	−8.70488	−1.61646	−0.808228	−	0.588870i	$-0.799573\pi$
$29$	−8.70488	−1.61646	−0.808228	+	0.588870i	$0.799573\pi$
$30$	−0.753172	−0.137510
$31$	4.84064	0.869405	0.434702	−	0.900574i	$-0.356854\pi$
$31$	4.84064	0.869405	0.434702	+	0.900574i	$0.356854\pi$
$32$	5.48956	0.970426
$33$	0.429918	0.0748391
$34$	−13.0400	−2.23634
$35$	−3.79529	−0.641522
$36$	−3.00981	−0.501635
$37$	−1.42395	−0.234096	−0.117048	−	0.993126i	$-0.537343\pi$
$37$	−1.42395	−0.234096	−0.117048	+	0.993126i	$0.537343\pi$
$38$	7.22197	1.17156
$39$	−0.883568	−0.141484
$40$	−1.63077	−0.257848
$41$	−11.1940	−1.74821	−0.874107	−	0.485733i	$-0.838553\pi$
$41$	−11.1940	−1.74821	−0.874107	+	0.485733i	$0.838553\pi$
$42$	2.85851	0.441077
$43$	3.21471	0.490239	0.245119	−	0.969493i	$-0.421173\pi$
$43$	3.21471	0.490239	0.245119	+	0.969493i	$0.421173\pi$
$44$	−1.06914	−0.161179
$45$	2.81517	0.419661
$46$	4.82512	0.711425
$47$	−9.44758	−1.37807	−0.689036	−	0.724727i	$-0.741966\pi$
$47$	−9.44758	−1.37807	−0.689036	+	0.724727i	$0.741966\pi$
$48$	2.14754	0.309970
$49$	7.40426	1.05775
$50$	−1.75190	−0.247755
$51$	−3.20004	−0.448095
$52$	2.19729	0.304710
$53$	10.2656	1.41009	0.705044	−	0.709164i	$-0.250927\pi$
$53$	10.2656	1.41009	0.705044	+	0.709164i	$0.250927\pi$
$54$	−4.37982	−0.596018
$55$	1.00000	0.134840
$56$	6.18926	0.827075
$57$	1.77228	0.234745
$58$	15.2500	2.00243
$59$	−8.30534	−1.08126	−0.540632	−	0.841259i	$-0.681815\pi$
$59$	−8.30534	−1.08126	−0.540632	+	0.841259i	$0.681815\pi$
$60$	0.459642	0.0593395
$61$	−2.62164	−0.335666	−0.167833	−	0.985815i	$-0.553677\pi$
$61$	−2.62164	−0.335666	−0.167833	+	0.985815i	$0.553677\pi$
$62$	−8.48030	−1.07700
$63$	−10.6844	−1.34611
$64$	0.373304	0.0466630
$65$	−2.05520	−0.254916
$66$	−0.753172	−0.0927090
$67$	1.50213	0.183515	0.0917575	−	0.995781i	$-0.470752\pi$
$67$	1.50213	0.183515	0.0917575	+	0.995781i	$0.470752\pi$
$68$	7.95799	0.965048
$69$	1.18409	0.142548
$70$	6.64896	0.794703
$71$	6.84288	0.812101	0.406051	−	0.913851i	$-0.366906\pi$
$71$	6.84288	0.812101	0.406051	+	0.913851i	$0.366906\pi$
$72$	−4.59090	−0.541043
$73$	−1.00000	−0.117041
$73$	−1.00000	−0.117041
$74$	2.49461	0.289992
$75$	−0.429918	−0.0496427
$76$	−4.40739	−0.505562
$77$	−3.79529	−0.432514
$78$	1.54792	0.175267
$79$	1.81457	0.204155	0.102077	−	0.994776i	$-0.467451\pi$
$79$	1.81457	0.204155	0.102077	+	0.994776i	$0.467451\pi$
$80$	4.99522	0.558483
$81$	7.37070	0.818966
$82$	19.6108	2.16565
$83$	−8.27854	−0.908688	−0.454344	−	0.890826i	$-0.650126\pi$
$83$	−8.27854	−0.908688	−0.454344	+	0.890826i	$0.650126\pi$
$84$	−1.74448	−0.190338
$85$	−7.44337	−0.807347
$86$	−5.63184	−0.607297
$87$	3.74239	0.401226
$88$	−1.63077	−0.173841
$89$	−7.32143	−0.776070	−0.388035	−	0.921645i	$-0.626846\pi$
$89$	−7.32143	−0.776070	−0.388035	+	0.921645i	$0.626846\pi$
$90$	−4.93188	−0.519866
$91$	7.80009	0.817672
$92$	−2.94465	−0.307001
$93$	−2.08108	−0.215798
$94$	16.5512	1.70712
$95$	4.12238	0.422947
$96$	−2.36006	−0.240873
$97$	−15.3235	−1.55587	−0.777933	−	0.628347i	$-0.783732\pi$
$97$	−15.3235	−1.55587	−0.777933	+	0.628347i	$0.783732\pi$
$98$	−12.9715	−1.31032
$99$	2.81517	0.282935
$100$	1.06914	0.106914
$101$	2.27501	0.226372	0.113186	−	0.993574i	$-0.463894\pi$
$101$	2.27501	0.226372	0.113186	+	0.993574i	$0.463894\pi$
$102$	5.60614	0.555090
$103$	−5.97525	−0.588759	−0.294379	−	0.955689i	$-0.595113\pi$
$103$	−5.97525	−0.588759	−0.294379	+	0.955689i	$0.595113\pi$
$104$	3.35156	0.328648
$105$	1.63167	0.159234
$106$	−17.9843	−1.74678
$107$	10.9582	1.05937	0.529686	−	0.848194i	$-0.322310\pi$
$107$	10.9582	1.05937	0.529686	+	0.848194i	$0.322310\pi$
$108$	2.67290	0.257200
$109$	−6.87679	−0.658677	−0.329339	−	0.944212i	$-0.606826\pi$
$109$	−6.87679	−0.658677	−0.329339	+	0.944212i	$0.606826\pi$
$110$	−1.75190	−0.167037
$111$	0.612181	0.0581057
$112$	−18.9583	−1.79139
$113$	13.0954	1.23191	0.615957	−	0.787780i	$-0.288770\pi$
$113$	13.0954	1.23191	0.615957	+	0.787780i	$0.288770\pi$
$114$	−3.10486	−0.290796
$115$	2.75423	0.256833
$116$	−9.30673	−0.864108
$117$	−5.78574	−0.534892
$118$	14.5501	1.33944
$119$	28.2498	2.58965
$120$	0.701099	0.0640012
$121$	1.00000	0.0909091
$122$	4.59283	0.415815
$123$	4.81252	0.433930
$124$	5.17532	0.464757
$125$	−1.00000	−0.0894427
$126$	18.7180	1.66753
$127$	−13.2524	−1.17596	−0.587981	−	0.808875i	$-0.700077\pi$
$127$	−13.2524	−1.17596	−0.587981	+	0.808875i	$0.700077\pi$
$128$	−11.6331	−1.02823
$129$	−1.38206	−0.121684
$130$	3.60050	0.315784
$131$	1.39821	0.122162	0.0610809	−	0.998133i	$-0.480545\pi$
$131$	1.39821	0.122162	0.0610809	+	0.998133i	$0.480545\pi$
$132$	0.459642	0.0400067
$133$	−15.6456	−1.35665
$134$	−2.63158	−0.227334
$135$	−2.50005	−0.215170
$136$	12.1384	1.04086
$137$	5.93212	0.506816	0.253408	−	0.967360i	$-0.418449\pi$
$137$	5.93212	0.506816	0.253408	+	0.967360i	$0.418449\pi$
$138$	−2.07441	−0.176585
$139$	18.7476	1.59015	0.795074	−	0.606513i	$-0.207432\pi$
$139$	18.7476	1.59015	0.795074	+	0.606513i	$0.207432\pi$
$140$	−4.05769	−0.342938
$141$	4.06169	0.342056
$142$	−11.9880	−1.00601
$143$	−2.05520	−0.171864
$144$	14.0624	1.17187
$145$	8.70488	0.722901
$146$	1.75190	0.144988
$147$	−3.18322	−0.262548
$148$	−1.52240	−0.125140
$149$	−22.0781	−1.80871	−0.904353	−	0.426784i	$-0.859646\pi$
$149$	−22.0781	−1.80871	−0.904353	+	0.426784i	$0.859646\pi$
$150$	0.753172	0.0614962
$151$	3.64640	0.296740	0.148370	−	0.988932i	$-0.452597\pi$
$151$	3.64640	0.296740	0.148370	+	0.988932i	$0.452597\pi$
$152$	−6.72266	−0.545279
$153$	−20.9544	−1.69406
$154$	6.64896	0.535788
$155$	−4.84064	−0.388810
$156$	−0.944656	−0.0756330
$157$	23.3158	1.86080	0.930400	−	0.366546i	$-0.119460\pi$
$157$	23.3158	1.86080	0.930400	+	0.366546i	$0.119460\pi$
$158$	−3.17894	−0.252903
$159$	−4.41336	−0.350003
$160$	−5.48956	−0.433988
$161$	−10.4531	−0.823820
$162$	−12.9127	−1.01452
$163$	21.9224	1.71710	0.858549	−	0.512732i	$-0.171366\pi$
$163$	21.9224	1.71710	0.858549	+	0.512732i	$0.171366\pi$
$164$	−11.9680	−0.934542
$165$	−0.429918	−0.0334691
$166$	14.5031	1.12566
$167$	6.22225	0.481492	0.240746	−	0.970588i	$-0.422608\pi$
$167$	6.22225	0.481492	0.240746	+	0.970588i	$0.422608\pi$
$168$	−2.66087	−0.205291
$169$	−8.77615	−0.675089
$170$	13.0400	1.00012
$171$	11.6052	0.887471
$172$	3.43697	0.262067
$173$	9.20019	0.699477	0.349739	−	0.936847i	$-0.386270\pi$
$173$	9.20019	0.699477	0.349739	+	0.936847i	$0.386270\pi$
$174$	−6.55627	−0.497030
$175$	3.79529	0.286897
$176$	4.99522	0.376529
$177$	3.57062	0.268384
$178$	12.8264	0.961378
$179$	−14.4830	−1.08251	−0.541256	−	0.840858i	$-0.682051\pi$
$179$	−14.4830	−1.08251	−0.541256	+	0.840858i	$0.682051\pi$
$180$	3.00981	0.224338
$181$	21.8106	1.62117	0.810585	−	0.585621i	$-0.199149\pi$
$181$	21.8106	1.62117	0.810585	+	0.585621i	$0.199149\pi$
$182$	−13.6649	−1.01291
$183$	1.12709	0.0833168
$184$	−4.49152	−0.331119
$185$	1.42395	0.104691
$186$	3.64583	0.267325
$187$	−7.44337	−0.544313
$188$	−10.1008	−0.736675
$189$	9.48841	0.690181
$190$	−7.22197	−0.523937
$191$	−15.3522	−1.11084	−0.555422	−	0.831569i	$-0.687443\pi$
$191$	−15.3522	−1.11084	−0.555422	+	0.831569i	$0.687443\pi$
$192$	−0.160490	−0.0115824
$193$	−25.5185	−1.83686	−0.918430	−	0.395583i	$-0.870543\pi$
$193$	−25.5185	−1.83686	−0.918430	+	0.395583i	$0.870543\pi$
$194$	26.8452	1.92737
$195$	0.883568	0.0632736
$196$	7.91618	0.565441
$197$	12.8034	0.912206	0.456103	−	0.889927i	$-0.349245\pi$
$197$	12.8034	0.912206	0.456103	+	0.889927i	$0.349245\pi$
$198$	−4.93188	−0.350494
$199$	−2.63578	−0.186846	−0.0934228	−	0.995627i	$-0.529781\pi$
$199$	−2.63578	−0.186846	−0.0934228	+	0.995627i	$0.529781\pi$
$200$	1.63077	0.115313
$201$	−0.645795	−0.0455509
$202$	−3.98559	−0.280425
$203$	−33.0376	−2.31878
$204$	−3.42129	−0.239538
$205$	11.1940	0.781825
$206$	10.4680	0.729341
$207$	7.75362	0.538914
$208$	−10.2662	−0.711831
$209$	4.12238	0.285151
$210$	−2.85851	−0.197256
$211$	18.7958	1.29396	0.646979	−	0.762508i	$-0.276032\pi$
$211$	18.7958	1.29396	0.646979	+	0.762508i	$0.276032\pi$
$212$	10.9753	0.753790
$213$	−2.94188	−0.201574
$214$	−19.1977	−1.31233
$215$	−3.21471	−0.219241
$216$	4.07701	0.277405
$217$	18.3717	1.24715
$218$	12.0474	0.815955
$219$	0.429918	0.0290512
$220$	1.06914	0.0720813
$221$	15.2976	1.02903
$222$	−1.07248	−0.0719800
$223$	−25.0190	−1.67540	−0.837698	−	0.546134i	$-0.816099\pi$
$223$	−25.0190	−1.67540	−0.837698	+	0.546134i	$0.816099\pi$
$224$	20.8345	1.39206
$225$	−2.81517	−0.187678
$226$	−22.9418	−1.52607
$227$	−23.8560	−1.58338	−0.791688	−	0.610926i	$-0.790797\pi$
$227$	−23.8560	−1.58338	−0.791688	+	0.610926i	$0.790797\pi$
$228$	1.89482	0.125487
$229$	−20.3816	−1.34685	−0.673425	−	0.739255i	$-0.735178\pi$
$229$	−20.3816	−1.34685	−0.673425	+	0.739255i	$0.735178\pi$
$230$	−4.82512	−0.318159
$231$	1.63167	0.107356
$232$	−14.1957	−0.931992
$233$	−18.0927	−1.18529	−0.592645	−	0.805464i	$-0.701916\pi$
$233$	−18.0927	−1.18529	−0.592645	+	0.805464i	$0.701916\pi$
$234$	10.1360	0.662612
$235$	9.44758	0.616292
$236$	−8.87956	−0.578010
$237$	−0.780116	−0.0506740
$238$	−49.4907	−3.20800
$239$	−6.31921	−0.408756	−0.204378	−	0.978892i	$-0.565517\pi$
$239$	−6.31921	−0.408756	−0.204378	+	0.978892i	$0.565517\pi$
$240$	−2.14754	−0.138623
$241$	−13.1436	−0.846655	−0.423327	−	0.905977i	$-0.639138\pi$
$241$	−13.1436	−0.846655	−0.423327	+	0.905977i	$0.639138\pi$
$242$	−1.75190	−0.112616
$243$	−10.6689	−0.684413
$244$	−2.80289	−0.179437
$245$	−7.40426	−0.473041
$246$	−8.43103	−0.537543
$247$	−8.47231	−0.539080
$248$	7.89398	0.501268
$249$	3.55909	0.225548
$250$	1.75190	0.110800
$251$	−25.7730	−1.62677	−0.813387	−	0.581722i	$-0.802379\pi$
$251$	−25.7730	−1.62677	−0.813387	+	0.581722i	$0.802379\pi$
$252$	−11.4231	−0.719588
$253$	2.75423	0.173157
$254$	23.2168	1.45675
$255$	3.20004	0.200394
$256$	19.6334	1.22709
$257$	16.0395	1.00052	0.500260	−	0.865875i	$-0.333238\pi$
$257$	16.0395	1.00052	0.500260	+	0.865875i	$0.333238\pi$
$258$	2.42123	0.150739
$259$	−5.40430	−0.335807
$260$	−2.19729	−0.136270
$261$	24.5057	1.51687
$262$	−2.44951	−0.151331
$263$	−2.08118	−0.128331	−0.0641655	−	0.997939i	$-0.520439\pi$
$263$	−2.08118	−0.128331	−0.0641655	+	0.997939i	$0.520439\pi$
$264$	0.701099	0.0431496
$265$	−10.2656	−0.630610
$266$	27.4095	1.68058
$267$	3.14761	0.192631
$268$	1.60599	0.0981015
$269$	−3.77127	−0.229938	−0.114969	−	0.993369i	$-0.536677\pi$
$269$	−3.77127	−0.229938	−0.114969	+	0.993369i	$0.536677\pi$
$270$	4.37982	0.266547
$271$	−26.8879	−1.63332	−0.816662	−	0.577116i	$-0.804178\pi$
$271$	−26.8879	−1.63332	−0.816662	+	0.577116i	$0.804178\pi$
$272$	−37.1813	−2.25445
$273$	−3.35340	−0.202957
$274$	−10.3925	−0.627832
$275$	−1.00000	−0.0603023
$276$	1.26596	0.0762017
$277$	28.2598	1.69796	0.848982	−	0.528421i	$-0.177216\pi$
$277$	28.2598	1.69796	0.848982	+	0.528421i	$0.177216\pi$
$278$	−32.8438	−1.96984
$279$	−13.6272	−0.815841
$280$	−6.18926	−0.369879
$281$	−28.5803	−1.70496	−0.852478	−	0.522763i	$-0.824901\pi$
$281$	−28.5803	−1.70496	−0.852478	+	0.522763i	$0.824901\pi$
$282$	−7.11565	−0.423731
$283$	17.0621	1.01424	0.507119	−	0.861876i	$-0.330710\pi$
$283$	17.0621	1.01424	0.507119	+	0.861876i	$0.330710\pi$
$284$	7.31599	0.434124
$285$	−1.77228	−0.104981
$286$	3.60050	0.212902
$287$	−42.4847	−2.50779
$288$	−15.4540	−0.910638
$289$	38.4038	2.25904
$290$	−15.2500	−0.895514
$291$	6.58785	0.386187
$292$	−1.06914	−0.0625666
$293$	0.775653	0.0453141	0.0226571	−	0.999743i	$-0.492787\pi$
$293$	0.775653	0.0453141	0.0226571	+	0.999743i	$0.492787\pi$
$294$	5.57668	0.325238
$295$	8.30534	0.483556
$296$	−2.32214	−0.134971
$297$	−2.50005	−0.145067
$298$	38.6785	2.24059
$299$	−5.66049	−0.327354
$300$	−0.459642	−0.0265374
$301$	12.2008	0.703241
$302$	−6.38811	−0.367595
$303$	−0.978070	−0.0561887
$304$	20.5922	1.18104
$305$	2.62164	0.150114
$306$	36.7098	2.09856
$307$	−1.98565	−0.113327	−0.0566635	−	0.998393i	$-0.518046\pi$
$307$	−1.98565	−0.113327	−0.0566635	+	0.998393i	$0.518046\pi$
$308$	−4.05769	−0.231209
$309$	2.56887	0.146138
$310$	8.48030	0.481649
$311$	−7.53165	−0.427081	−0.213540	−	0.976934i	$-0.568500\pi$
$311$	−7.53165	−0.427081	−0.213540	+	0.976934i	$0.568500\pi$
$312$	−1.44090	−0.0815748
$313$	−8.36974	−0.473086	−0.236543	−	0.971621i	$-0.576014\pi$
$313$	−8.36974	−0.473086	−0.236543	+	0.971621i	$0.576014\pi$
$314$	−40.8468	−2.30512
$315$	10.6844	0.601998
$316$	1.94003	0.109135
$317$	2.60030	0.146048	0.0730238	−	0.997330i	$-0.476735\pi$
$317$	2.60030	0.146048	0.0730238	+	0.997330i	$0.476735\pi$
$318$	7.73175	0.433575
$319$	8.70488	0.487380
$320$	−0.373304	−0.0208683
$321$	−4.71114	−0.262950
$322$	18.3127	1.02053
$323$	−30.6844	−1.70732
$324$	7.88029	0.437794
$325$	2.05520	0.114002
$326$	−38.4058	−2.12710
$327$	2.95646	0.163492
$328$	−18.2549	−1.00796
$329$	−35.8564	−1.97682
$330$	0.753172	0.0414607
$331$	23.9153	1.31450	0.657252	−	0.753671i	$-0.271719\pi$
$331$	23.9153	1.31450	0.657252	+	0.753671i	$0.271719\pi$
$332$	−8.85091	−0.485757
$333$	4.00866	0.219673
$334$	−10.9007	−0.596461
$335$	−1.50213	−0.0820704
$336$	8.15053	0.444648
$337$	1.00440	0.0547131	0.0273565	−	0.999626i	$-0.491291\pi$
$337$	1.00440	0.0547131	0.0273565	+	0.999626i	$0.491291\pi$
$338$	15.3749	0.836284
$339$	−5.62996	−0.305777
$340$	−7.95799	−0.431583
$341$	−4.84064	−0.262135
$342$	−20.3311	−1.09938
$343$	1.53427	0.0828429
$344$	5.24246	0.282654
$345$	−1.18409	−0.0637494
$346$	−16.1178	−0.866497
$347$	−15.2719	−0.819837	−0.409919	−	0.912122i	$-0.634443\pi$
$347$	−15.2719	−0.819837	−0.409919	+	0.912122i	$0.634443\pi$
$348$	4.00113	0.214483
$349$	−3.30916	−0.177135	−0.0885675	−	0.996070i	$-0.528229\pi$
$349$	−3.30916	−0.177135	−0.0885675	+	0.996070i	$0.528229\pi$
$350$	−6.64896	−0.355402
$351$	5.13810	0.274251
$352$	−5.48956	−0.292595
$353$	−18.1228	−0.964579	−0.482290	−	0.876012i	$-0.660195\pi$
$353$	−18.1228	−0.964579	−0.482290	+	0.876012i	$0.660195\pi$
$354$	−6.25535	−0.332468
$355$	−6.84288	−0.363183
$356$	−7.82762	−0.414863
$357$	−12.1451	−0.642786
$358$	25.3727	1.34099
$359$	−22.5641	−1.19089	−0.595444	−	0.803397i	$-0.703024\pi$
$359$	−22.5641	−1.19089	−0.595444	+	0.803397i	$0.703024\pi$
$360$	4.59090	0.241962
$361$	−2.00602	−0.105580
$362$	−38.2099	−2.00827
$363$	−0.429918	−0.0225648
$364$	8.33938	0.437102
$365$	1.00000	0.0523424
$366$	−1.97454	−0.103211
$367$	−2.16141	−0.112825	−0.0564123	−	0.998408i	$-0.517966\pi$
$367$	−2.16141	−0.112825	−0.0564123	+	0.998408i	$0.517966\pi$
$368$	13.7580	0.717184
$369$	31.5131	1.64051
$370$	−2.49461	−0.129689
$371$	38.9609	2.02275
$372$	−2.22496	−0.115359
$373$	−11.1553	−0.577597	−0.288798	−	0.957390i	$-0.593256\pi$
$373$	−11.1553	−0.577597	−0.288798	+	0.957390i	$0.593256\pi$
$374$	13.0400	0.674283
$375$	0.429918	0.0222009
$376$	−15.4069	−0.794548
$377$	−17.8903	−0.921396
$378$	−16.6227	−0.854980
$379$	13.4291	0.689809	0.344904	−	0.938638i	$-0.387911\pi$
$379$	13.4291	0.689809	0.344904	+	0.938638i	$0.387911\pi$
$380$	4.40739	0.226094
$381$	5.69745	0.291889
$382$	26.8954	1.37609
$383$	−3.61368	−0.184650	−0.0923252	−	0.995729i	$-0.529430\pi$
$383$	−3.61368	−0.184650	−0.0923252	+	0.995729i	$0.529430\pi$
$384$	5.00128	0.255221
$385$	3.79529	0.193426
$386$	44.7057	2.27546
$387$	−9.04996	−0.460035
$388$	−16.3830	−0.831718
$389$	−13.0708	−0.662715	−0.331358	−	0.943505i	$-0.607507\pi$
$389$	−13.0708	−0.662715	−0.331358	+	0.943505i	$0.607507\pi$
$390$	−1.54792	−0.0783819
$391$	−20.5007	−1.03677
$392$	12.0747	0.609862
$393$	−0.601114	−0.0303222
$394$	−22.4303	−1.13002
$395$	−1.81457	−0.0913009
$396$	3.00981	0.151248
$397$	−12.1056	−0.607563	−0.303781	−	0.952742i	$-0.598249\pi$
$397$	−12.1056	−0.607563	−0.303781	+	0.952742i	$0.598249\pi$
$398$	4.61762	0.231460
$399$	6.72634	0.336738
$400$	−4.99522	−0.249761
$401$	−36.3714	−1.81630	−0.908151	−	0.418643i	$-0.862506\pi$
$401$	−36.3714	−1.81630	−0.908151	+	0.418643i	$0.862506\pi$
$402$	1.13137	0.0564274
$403$	9.94849	0.495569
$404$	2.43231	0.121012
$405$	−7.37070	−0.366253
$406$	57.8784	2.87246
$407$	1.42395	0.0705825
$408$	−5.21854	−0.258356
$409$	−6.79529	−0.336006	−0.168003	−	0.985787i	$-0.553732\pi$
$409$	−6.79529	−0.336006	−0.168003	+	0.985787i	$0.553732\pi$
$410$	−19.6108	−0.968507
$411$	−2.55033	−0.125798
$412$	−6.38837	−0.314732
$413$	−31.5212	−1.55106
$414$	−13.5835	−0.667594
$415$	8.27854	0.406378
$416$	11.2821	0.553153
$417$	−8.05992	−0.394696
$418$	−7.22197	−0.353238
$419$	−18.2003	−0.889143	−0.444571	−	0.895743i	$-0.646644\pi$
$419$	−18.2003	−0.889143	−0.444571	+	0.895743i	$0.646644\pi$
$420$	1.74448	0.0851217
$421$	−15.3469	−0.747960	−0.373980	−	0.927437i	$-0.622007\pi$
$421$	−15.3469	−0.747960	−0.373980	+	0.927437i	$0.622007\pi$
$422$	−32.9283	−1.60293
$423$	26.5966	1.29317
$424$	16.7408	0.813007
$425$	7.44337	0.361056
$426$	5.15387	0.249706
$427$	−9.94988	−0.481508
$428$	11.7159	0.566308
$429$	0.883568	0.0426591
$430$	5.63184	0.271591
$431$	−27.0675	−1.30380	−0.651898	−	0.758307i	$-0.726027\pi$
$431$	−27.0675	−1.30380	−0.651898	+	0.758307i	$0.726027\pi$
$432$	−12.4883	−0.600843
$433$	4.88813	0.234908	0.117454	−	0.993078i	$-0.462527\pi$
$433$	4.88813	0.234908	0.117454	+	0.993078i	$0.462527\pi$
$434$	−32.1852	−1.54494
$435$	−3.74239	−0.179434
$436$	−7.35224	−0.352109
$437$	11.3540	0.543133
$438$	−0.753172	−0.0359879
$439$	−6.54448	−0.312351	−0.156176	−	0.987729i	$-0.549917\pi$
$439$	−6.54448	−0.312351	−0.156176	+	0.987729i	$0.549917\pi$
$440$	1.63077	0.0777440
$441$	−20.8442	−0.992583
$442$	−26.7998	−1.27474
$443$	−26.7445	−1.27067	−0.635334	−	0.772238i	$-0.719138\pi$
$443$	−26.7445	−1.27067	−0.635334	+	0.772238i	$0.719138\pi$
$444$	0.654507	0.0310615
$445$	7.32143	0.347069
$446$	43.8306	2.07544
$447$	9.49177	0.448945
$448$	1.41680	0.0669374
$449$	33.2630	1.56978	0.784889	−	0.619636i	$-0.212720\pi$
$449$	33.2630	1.56978	0.784889	+	0.619636i	$0.212720\pi$
$450$	4.93188	0.232491
$451$	11.1940	0.527107
$452$	14.0008	0.658543
$453$	−1.56765	−0.0736548
$454$	41.7931	1.96145
$455$	−7.80009	−0.365674
$456$	2.89019	0.135346
$457$	16.3819	0.766313	0.383156	−	0.923684i	$-0.374837\pi$
$457$	16.3819	0.766313	0.383156	+	0.923684i	$0.374837\pi$
$458$	35.7064	1.66845
$459$	18.6088	0.868583
$460$	2.94465	0.137295
$461$	8.88147	0.413651	0.206826	−	0.978378i	$-0.433687\pi$
$461$	8.88147	0.413651	0.206826	+	0.978378i	$0.433687\pi$
$462$	−2.85851	−0.132990
$463$	−29.4506	−1.36869	−0.684343	−	0.729160i	$-0.739911\pi$
$463$	−29.4506	−1.36869	−0.684343	+	0.729160i	$0.739911\pi$
$464$	43.4828	2.01864
$465$	2.08108	0.0965077
$466$	31.6965	1.46831
$467$	7.07755	0.327510	0.163755	−	0.986501i	$-0.447639\pi$
$467$	7.07755	0.327510	0.163755	+	0.986501i	$0.447639\pi$
$468$	−6.18576	−0.285937
$469$	5.70104	0.263250
$470$	−16.5512	−0.763449
$471$	−10.0239	−0.461875
$472$	−13.5441	−0.623419
$473$	−3.21471	−0.147813
$474$	1.36668	0.0627738
$475$	−4.12238	−0.189148
$476$	30.2029	1.38435
$477$	−28.8994	−1.32321
$478$	11.0706	0.506358
$479$	9.88783	0.451787	0.225893	−	0.974152i	$-0.427470\pi$
$479$	9.88783	0.451787	0.225893	+	0.974152i	$0.427470\pi$
$480$	2.36006	0.107722
$481$	−2.92650	−0.133437
$482$	23.0262	1.04882
$483$	4.49398	0.204483
$484$	1.06914	0.0485972
$485$	15.3235	0.695805
$486$	18.6909	0.847835
$487$	−5.46584	−0.247681	−0.123840	−	0.992302i	$-0.539521\pi$
$487$	−5.46584	−0.247681	−0.123840	+	0.992302i	$0.539521\pi$
$488$	−4.27529	−0.193533
$489$	−9.42485	−0.426207
$490$	12.9715	0.585992
$491$	16.0968	0.726437	0.363218	−	0.931704i	$-0.381678\pi$
$491$	16.0968	0.726437	0.363218	+	0.931704i	$0.381678\pi$
$492$	5.14525	0.231966
$493$	−64.7937	−2.91816
$494$	14.8426	0.667800
$495$	−2.81517	−0.126533
$496$	−24.1801	−1.08572
$497$	25.9708	1.16495
$498$	−6.23516	−0.279404
$499$	−4.65306	−0.208299	−0.104150	−	0.994562i	$-0.533212\pi$
$499$	−4.65306	−0.208299	−0.104150	+	0.994562i	$0.533212\pi$
$500$	−1.06914	−0.0478133
$501$	−2.67506	−0.119513
$502$	45.1515	2.01521
$503$	22.2724	0.993079	0.496540	−	0.868014i	$-0.334604\pi$
$503$	22.2724	0.993079	0.496540	+	0.868014i	$0.334604\pi$
$504$	−17.4238	−0.776119
$505$	−2.27501	−0.101237
$506$	−4.82512	−0.214503
$507$	3.77303	0.167566
$508$	−14.1687	−0.628633
$509$	−11.5218	−0.510696	−0.255348	−	0.966849i	$-0.582190\pi$
$509$	−11.5218	−0.510696	−0.255348	+	0.966849i	$0.582190\pi$
$510$	−5.60614	−0.248244
$511$	−3.79529	−0.167894
$512$	−11.1294	−0.491856
$513$	−10.3061	−0.455027
$514$	−28.0996	−1.23942
$515$	5.97525	0.263301
$516$	−1.47762	−0.0650484
$517$	9.44758	0.415504
$518$	9.46778	0.415990
$519$	−3.95533	−0.173620
$520$	−3.35156	−0.146976
$521$	−9.77652	−0.428317	−0.214159	−	0.976799i	$-0.568701\pi$
$521$	−9.77652	−0.428317	−0.214159	+	0.976799i	$0.568701\pi$
$522$	−42.9315	−1.87906
$523$	−13.4697	−0.588987	−0.294494	−	0.955653i	$-0.595151\pi$
$523$	−13.4697	−0.588987	−0.294494	+	0.955653i	$0.595151\pi$
$524$	1.49488	0.0653039
$525$	−1.63167	−0.0712117
$526$	3.64601	0.158973
$527$	36.0307	1.56952
$528$	−2.14754	−0.0934595
$529$	−15.4142	−0.670184
$530$	17.9843	0.781186
$531$	23.3810	1.01465
$532$	−16.7273	−0.725222
$533$	−23.0060	−0.996500
$534$	−5.51429	−0.238627
$535$	−10.9582	−0.473766
$536$	2.44964	0.105808
$537$	6.22651	0.268694
$538$	6.60687	0.284842
$539$	−7.40426	−0.318924
$540$	−2.67290	−0.115023
$541$	22.7450	0.977885	0.488943	−	0.872316i	$-0.337383\pi$
$541$	22.7450	0.977885	0.488943	+	0.872316i	$0.337383\pi$
$542$	47.1048	2.02332
$543$	−9.37678	−0.402396
$544$	40.8608	1.75189
$545$	6.87679	0.294569
$546$	5.87481	0.251419
$547$	−41.0151	−1.75368	−0.876839	−	0.480785i	$-0.840352\pi$
$547$	−41.0151	−1.75368	−0.876839	+	0.480785i	$0.840352\pi$
$548$	6.34226	0.270928
$549$	7.38035	0.314986
$550$	1.75190	0.0747011
$551$	35.8848	1.52874
$552$	1.93098	0.0821881
$553$	6.88682	0.292857
$554$	−49.5082	−2.10340
$555$	−0.612181	−0.0259857
$556$	20.0437	0.850044
$557$	16.6723	0.706430	0.353215	−	0.935542i	$-0.385088\pi$
$557$	16.6723	0.706430	0.353215	+	0.935542i	$0.385088\pi$
$558$	23.8735	1.01065
$559$	6.60687	0.279441
$560$	18.9583	0.801136
$561$	3.20004	0.135106
$562$	50.0696	2.11206
$563$	−5.97078	−0.251638	−0.125819	−	0.992053i	$-0.540156\pi$
$563$	−5.97078	−0.251638	−0.125819	+	0.992053i	$0.540156\pi$
$564$	4.34251	0.182852
$565$	−13.0954	−0.550928
$566$	−29.8911	−1.25641
$567$	27.9740	1.17480
$568$	11.1592	0.468229
$569$	−46.4888	−1.94891	−0.974455	−	0.224581i	$-0.927899\pi$
$569$	−46.4888	−1.94891	−0.974455	+	0.224581i	$0.927899\pi$
$570$	3.10486	0.130048
$571$	10.2268	0.427977	0.213988	−	0.976836i	$-0.431355\pi$
$571$	10.2268	0.427977	0.213988	+	0.976836i	$0.431355\pi$
$572$	−2.19729	−0.0918735
$573$	6.60018	0.275726
$574$	74.4287	3.10659
$575$	−2.75423	−0.114859
$576$	−1.05091	−0.0437881
$577$	21.2515	0.884712	0.442356	−	0.896840i	$-0.354143\pi$
$577$	21.2515	0.884712	0.442356	+	0.896840i	$0.354143\pi$
$578$	−67.2794	−2.79845
$579$	10.9709	0.455933
$580$	9.30673	0.386441
$581$	−31.4195	−1.30350
$582$	−11.5412	−0.478400
$583$	−10.2656	−0.425157
$584$	−1.63077	−0.0674818
$585$	5.78574	0.239211
$586$	−1.35886	−0.0561341
$587$	−8.90747	−0.367651	−0.183825	−	0.982959i	$-0.558848\pi$
$587$	−8.90747	−0.367651	−0.183825	+	0.982959i	$0.558848\pi$
$588$	−3.40331	−0.140350
$589$	−19.9549	−0.822229
$590$	−14.5501	−0.599018
$591$	−5.50442	−0.226422
$592$	7.11294	0.292340
$593$	4.21687	0.173166	0.0865831	−	0.996245i	$-0.472405\pi$
$593$	4.21687	0.173166	0.0865831	+	0.996245i	$0.472405\pi$
$594$	4.37982	0.179706
$595$	−28.2498	−1.15813
$596$	−23.6045	−0.966879
$597$	1.13317	0.0463776
$598$	9.91658	0.405519
$599$	−7.66765	−0.313292	−0.156646	−	0.987655i	$-0.550068\pi$
$599$	−7.66765	−0.313292	−0.156646	+	0.987655i	$0.550068\pi$
$600$	−0.701099	−0.0286222
$601$	23.3340	0.951813	0.475907	−	0.879496i	$-0.342120\pi$
$601$	23.3340	0.951813	0.475907	+	0.879496i	$0.342120\pi$
$602$	−21.3745	−0.871159
$603$	−4.22877	−0.172209
$604$	3.89851	0.158628
$605$	−1.00000	−0.0406558
$606$	1.71348	0.0696052
$607$	20.6632	0.838693	0.419347	−	0.907826i	$-0.362259\pi$
$607$	20.6632	0.838693	0.419347	+	0.907826i	$0.362259\pi$
$608$	−22.6300	−0.917769
$609$	14.2035	0.575553
$610$	−4.59283	−0.185958
$611$	−19.4167	−0.785515
$612$	−22.4031	−0.905592
$613$	−11.8552	−0.478827	−0.239414	−	0.970918i	$-0.576955\pi$
$613$	−11.8552	−0.478827	−0.239414	+	0.970918i	$0.576955\pi$
$614$	3.47865	0.140387
$615$	−4.81252	−0.194059
$616$	−6.18926	−0.249372
$617$	−9.27583	−0.373431	−0.186715	−	0.982414i	$-0.559784\pi$
$617$	−9.27583	−0.373431	−0.186715	+	0.982414i	$0.559784\pi$
$618$	−4.50039	−0.181032
$619$	19.8005	0.795850	0.397925	−	0.917418i	$-0.369730\pi$
$619$	19.8005	0.795850	0.397925	+	0.917418i	$0.369730\pi$
$620$	−5.17532	−0.207846
$621$	−6.88570	−0.276313
$622$	13.1947	0.529058
$623$	−27.7870	−1.11326
$624$	4.41362	0.176686
$625$	1.00000	0.0400000
$626$	14.6629	0.586048
$627$	−1.77228	−0.0707782
$628$	24.9278	0.994726
$629$	−10.5990	−0.422609
$630$	−18.7180	−0.745741
$631$	−18.7000	−0.744434	−0.372217	−	0.928146i	$-0.621402\pi$
$631$	−18.7000	−0.744434	−0.372217	+	0.928146i	$0.621402\pi$
$632$	2.95915	0.117709
$633$	−8.08067	−0.321178
$634$	−4.55546	−0.180921
$635$	13.2524	0.525906
$636$	−4.71850	−0.187101
$637$	15.2172	0.602929
$638$	−15.2500	−0.603755
$639$	−19.2639	−0.762068
$640$	11.6331	0.459839
$641$	−27.1831	−1.07367	−0.536833	−	0.843688i	$-0.680380\pi$
$641$	−27.1831	−1.07367	−0.536833	+	0.843688i	$0.680380\pi$
$642$	8.25343	0.325737
$643$	7.68728	0.303157	0.151578	−	0.988445i	$-0.451564\pi$
$643$	7.68728	0.303157	0.151578	+	0.988445i	$0.451564\pi$
$644$	−11.1758	−0.440389
$645$	1.38206	0.0544186
$646$	53.7558	2.11499
$647$	7.75794	0.304996	0.152498	−	0.988304i	$-0.451268\pi$
$647$	7.75794	0.304996	0.152498	+	0.988304i	$0.451268\pi$
$648$	12.0199	0.472187
$649$	8.30534	0.326013
$650$	−3.60050	−0.141223
$651$	−7.89831	−0.309559
$652$	23.4381	0.917908
$653$	47.4525	1.85696	0.928480	−	0.371382i	$-0.121116\pi$
$653$	47.4525	1.85696	0.928480	+	0.371382i	$0.121116\pi$
$654$	−5.17941	−0.202531
$655$	−1.39821	−0.0546324
$656$	55.9167	2.18318
$657$	2.81517	0.109830
$658$	62.8166	2.44885
$659$	1.76339	0.0686920	0.0343460	−	0.999410i	$-0.489065\pi$
$659$	1.76339	0.0686920	0.0343460	+	0.999410i	$0.489065\pi$
$660$	−0.459642	−0.0178915
$661$	9.85750	0.383412	0.191706	−	0.981452i	$-0.438598\pi$
$661$	9.85750	0.383412	0.191706	+	0.981452i	$0.438598\pi$
$662$	−41.8971	−1.62838
$663$	−6.57672	−0.255419
$664$	−13.5004	−0.523918
$665$	15.6456	0.606711
$666$	−7.02275	−0.272126
$667$	23.9752	0.928324
$668$	6.65245	0.257391
$669$	10.7561	0.415855
$670$	2.63158	0.101667
$671$	2.62164	0.101207
$672$	−8.95713	−0.345529
$673$	3.28682	0.126698	0.0633488	−	0.997991i	$-0.479822\pi$
$673$	3.28682	0.126698	0.0633488	+	0.997991i	$0.479822\pi$
$674$	−1.75960	−0.0677773
$675$	2.50005	0.0962269
$676$	−9.38292	−0.360882
$677$	−3.37426	−0.129683	−0.0648417	−	0.997896i	$-0.520654\pi$
$677$	−3.37426	−0.129683	−0.0648417	+	0.997896i	$0.520654\pi$
$678$	9.86310	0.378790
$679$	−58.1572	−2.23187
$680$	−12.1384	−0.465488
$681$	10.2561	0.393015
$682$	8.48030	0.324727
$683$	−32.5143	−1.24413	−0.622063	−	0.782967i	$-0.713705\pi$
$683$	−32.5143	−1.24413	−0.622063	+	0.782967i	$0.713705\pi$
$684$	12.4076	0.474415
$685$	−5.93212	−0.226655
$686$	−2.68788	−0.102624
$687$	8.76240	0.334306
$688$	−16.0582	−0.612213
$689$	21.0979	0.803764
$690$	2.07441	0.0789713
$691$	30.1185	1.14576	0.572880	−	0.819639i	$-0.305826\pi$
$691$	30.1185	1.14576	0.572880	+	0.819639i	$0.305826\pi$
$692$	9.83628	0.373919
$693$	10.6844	0.405867
$694$	26.7547	1.01560
$695$	−18.7476	−0.711136
$696$	6.10298	0.231333
$697$	−83.3213	−3.15602
$698$	5.79730	0.219431
$699$	7.77837	0.294205
$700$	4.05769	0.153366
$701$	−50.8780	−1.92164	−0.960818	−	0.277179i	$-0.910601\pi$
$701$	−50.8780	−1.92164	−0.960818	+	0.277179i	$0.910601\pi$
$702$	−9.00141	−0.339736
$703$	5.87005	0.221393
$704$	−0.373304	−0.0140694
$705$	−4.06169	−0.152972
$706$	31.7493	1.19490
$707$	8.63435	0.324728
$708$	3.81748	0.143470
$709$	14.7197	0.552809	0.276405	−	0.961041i	$-0.410857\pi$
$709$	14.7197	0.552809	0.276405	+	0.961041i	$0.410857\pi$
$710$	11.9880	0.449902
$711$	−5.10832	−0.191577
$712$	−11.9396	−0.447455
$713$	−13.3322	−0.499296
$714$	21.2769	0.796269
$715$	2.05520	0.0768601
$716$	−15.4843	−0.578677
$717$	2.71674	0.101459
$718$	39.5300	1.47525
$719$	−27.1331	−1.01189	−0.505947	−	0.862565i	$-0.668857\pi$
$719$	−27.1331	−1.01189	−0.505947	+	0.862565i	$0.668857\pi$
$720$	−14.0624	−0.524075
$721$	−22.6778	−0.844567
$722$	3.51434	0.130790
$723$	5.65068	0.210151
$724$	23.3186	0.866628
$725$	−8.70488	−0.323291
$726$	0.753172	0.0279528
$727$	−45.5654	−1.68993	−0.844963	−	0.534825i	$-0.820377\pi$
$727$	−45.5654	−1.68993	−0.844963	+	0.534825i	$0.820377\pi$
$728$	12.7202	0.471441
$729$	−17.5253	−0.649086
$730$	−1.75190	−0.0648406
$731$	23.9283	0.885019
$732$	1.20501	0.0445386
$733$	−38.3359	−1.41597	−0.707984	−	0.706229i	$-0.750395\pi$
$733$	−38.3359	−1.41597	−0.707984	+	0.706229i	$0.750395\pi$
$734$	3.78656	0.139765
$735$	3.18322	0.117415
$736$	−15.1195	−0.557312
$737$	−1.50213	−0.0553319
$738$	−55.2077	−2.03222
$739$	16.0143	0.589095	0.294547	−	0.955637i	$-0.404831\pi$
$739$	16.0143	0.589095	0.294547	+	0.955637i	$0.404831\pi$
$740$	1.52240	0.0559645
$741$	3.64240	0.133807
$742$	−68.2555	−2.50574
$743$	19.8343	0.727650	0.363825	−	0.931467i	$-0.381471\pi$
$743$	19.8343	0.727650	0.363825	+	0.931467i	$0.381471\pi$
$744$	−3.39377	−0.124421
$745$	22.0781	0.808878
$746$	19.5428	0.715514
$747$	23.3055	0.852704
$748$	−7.95799	−0.290973
$749$	41.5897	1.51966
$750$	−0.753172	−0.0275019
$751$	20.2385	0.738514	0.369257	−	0.929327i	$-0.379612\pi$
$751$	20.2385	0.738514	0.369257	+	0.929327i	$0.379612\pi$
$752$	47.1928	1.72094
$753$	11.0803	0.403787
$754$	31.3419	1.14140
$755$	−3.64640	−0.132706
$756$	10.1444	0.368949
$757$	−20.1363	−0.731866	−0.365933	−	0.930641i	$-0.619250\pi$
$757$	−20.1363	−0.731866	−0.365933	+	0.930641i	$0.619250\pi$
$758$	−23.5265	−0.854519
$759$	−1.18409	−0.0429798
$760$	6.72266	0.243856
$761$	27.6396	1.00194	0.500968	−	0.865466i	$-0.332977\pi$
$761$	27.6396	1.00194	0.500968	+	0.865466i	$0.332977\pi$
$762$	−9.98134	−0.361586
$763$	−26.0995	−0.944864
$764$	−16.4136	−0.593823
$765$	20.9544	0.757606
$766$	6.33079	0.228741
$767$	−17.0691	−0.616331
$768$	−8.44075	−0.304579
$769$	−33.3025	−1.20092	−0.600459	−	0.799656i	$-0.705015\pi$
$769$	−33.3025	−1.20092	−0.600459	+	0.799656i	$0.705015\pi$
$770$	−6.64896	−0.239612
$771$	−6.89569	−0.248342
$772$	−27.2828	−0.981929
$773$	−39.1461	−1.40799	−0.703994	−	0.710206i	$-0.748602\pi$
$773$	−39.1461	−1.40799	−0.703994	+	0.710206i	$0.748602\pi$
$774$	15.8546	0.569881
$775$	4.84064	0.173881
$776$	−24.9892	−0.897058
$777$	2.32341	0.0833518
$778$	22.8987	0.820957
$779$	46.1460	1.65335
$780$	0.944656	0.0338241
$781$	−6.84288	−0.244858
$782$	35.9151	1.28432
$783$	−21.7626	−0.777732
$784$	−36.9859	−1.32092
$785$	−23.3158	−0.832175
$786$	1.05309	0.0375624
$787$	50.4394	1.79797	0.898985	−	0.437980i	$-0.144306\pi$
$787$	50.4394	1.79797	0.898985	+	0.437980i	$0.144306\pi$
$788$	13.6886	0.487637
$789$	0.894736	0.0318535
$790$	3.17894	0.113101
$791$	49.7010	1.76716
$792$	4.59090	0.163131
$793$	−5.38799	−0.191333
$794$	21.2078	0.752635
$795$	4.41336	0.156526
$796$	−2.81802	−0.0998820
$797$	33.9378	1.20214	0.601070	−	0.799196i	$-0.294741\pi$
$797$	33.9378	1.20214	0.601070	+	0.799196i	$0.294741\pi$
$798$	−11.7838	−0.417143
$799$	−70.3219	−2.48781
$800$	5.48956	0.194085
$801$	20.6111	0.728256
$802$	63.7189	2.24999
$803$	1.00000	0.0352892
$804$	−0.690444	−0.0243501
$805$	10.4531	0.368423
$806$	−17.4287	−0.613900
$807$	1.62134	0.0570738
$808$	3.71003	0.130518
$809$	−47.9067	−1.68431	−0.842155	−	0.539236i	$-0.818713\pi$
$809$	−47.9067	−1.68431	−0.842155	+	0.539236i	$0.818713\pi$
$810$	12.9127	0.453706
$811$	−28.0595	−0.985302	−0.492651	−	0.870227i	$-0.663972\pi$
$811$	−28.0595	−0.985302	−0.492651	+	0.870227i	$0.663972\pi$
$812$	−35.3218	−1.23955
$813$	11.5596	0.405413
$814$	−2.49461	−0.0874360
$815$	−21.9224	−0.767909
$816$	15.9849	0.559583
$817$	−13.2522	−0.463637
$818$	11.9046	0.416236
$819$	−21.9586	−0.767295
$820$	11.9680	0.417940
$821$	34.1649	1.19236	0.596182	−	0.802849i	$-0.296684\pi$
$821$	34.1649	1.19236	0.596182	+	0.802849i	$0.296684\pi$
$822$	4.46791	0.155836
$823$	2.35723	0.0821679	0.0410839	−	0.999156i	$-0.486919\pi$
$823$	2.35723	0.0821679	0.0410839	+	0.999156i	$0.486919\pi$
$824$	−9.74427	−0.339458
$825$	0.429918	0.0149678
$826$	55.2219	1.92141
$827$	19.4764	0.677262	0.338631	−	0.940919i	$-0.390036\pi$
$827$	19.4764	0.677262	0.338631	+	0.940919i	$0.390036\pi$
$828$	8.28969	0.288087
$829$	−34.5517	−1.20003	−0.600015	−	0.799989i	$-0.704839\pi$
$829$	−34.5517	−1.20003	−0.600015	+	0.799989i	$0.704839\pi$
$830$	−14.5031	−0.503411
$831$	−12.1494	−0.421458
$832$	0.767214	0.0265984
$833$	55.1126	1.90954
$834$	14.1201	0.488940
$835$	−6.22225	−0.215330
$836$	4.40739	0.152433
$837$	12.1018	0.418300
$838$	31.8850	1.10145
$839$	−15.4554	−0.533580	−0.266790	−	0.963755i	$-0.585963\pi$
$839$	−15.4554	−0.533580	−0.266790	+	0.963755i	$0.585963\pi$
$840$	2.66087	0.0918089
$841$	46.7750	1.61293
$842$	26.8861	0.926556
$843$	12.2872	0.423193
$844$	20.0953	0.691710
$845$	8.77615	0.301909
$846$	−46.5944	−1.60195
$847$	3.79529	0.130408
$848$	−51.2789	−1.76092
$849$	−7.33532	−0.251747
$850$	−13.0400	−0.447269
$851$	3.92188	0.134440
$852$	−3.14528	−0.107755
$853$	37.7516	1.29259	0.646295	−	0.763088i	$-0.276318\pi$
$853$	37.7516	1.29259	0.646295	+	0.763088i	$0.276318\pi$
$854$	17.4311	0.596481
$855$	−11.6052	−0.396889
$856$	17.8704	0.610797
$857$	52.0692	1.77865	0.889325	−	0.457276i	$-0.151175\pi$
$857$	52.0692	1.77865	0.889325	+	0.457276i	$0.151175\pi$
$858$	−1.54792	−0.0528451
$859$	8.22106	0.280499	0.140249	−	0.990116i	$-0.455210\pi$
$859$	8.22106	0.280499	0.140249	+	0.990116i	$0.455210\pi$
$860$	−3.43697	−0.117200
$861$	18.2649	0.622467
$862$	47.4195	1.61511
$863$	18.1868	0.619086	0.309543	−	0.950885i	$-0.399824\pi$
$863$	18.1868	0.619086	0.309543	+	0.950885i	$0.399824\pi$
$864$	13.7242	0.466905
$865$	−9.20019	−0.312816
$866$	−8.56349	−0.290999
$867$	−16.5105	−0.560725
$868$	19.6418	0.666688
$869$	−1.81457	−0.0615550
$870$	6.55627	0.222278
$871$	3.08719	0.104605
$872$	−11.2145	−0.379770
$873$	43.1383	1.46001
$874$	−19.8909	−0.672821
$875$	−3.79529	−0.128304
$876$	0.459642	0.0155299
$877$	32.1276	1.08487	0.542435	−	0.840098i	$-0.317502\pi$
$877$	32.1276	1.08487	0.542435	+	0.840098i	$0.317502\pi$
$878$	11.4652	0.386933
$879$	−0.333467	−0.0112476
$880$	−4.99522	−0.168389
$881$	54.0228	1.82008	0.910038	−	0.414525i	$-0.136052\pi$
$881$	54.0228	1.82008	0.910038	+	0.414525i	$0.136052\pi$
$882$	36.5169	1.22959
$883$	56.8673	1.91374	0.956869	−	0.290521i	$-0.0938287\pi$
$883$	56.8673	1.91374	0.956869	+	0.290521i	$0.0938287\pi$
$884$	16.3553	0.550087
$885$	−3.57062	−0.120025
$886$	46.8535	1.57407
$887$	26.2590	0.881693	0.440846	−	0.897583i	$-0.354678\pi$
$887$	26.2590	0.881693	0.440846	+	0.897583i	$0.354678\pi$
$888$	0.998328	0.0335017
$889$	−50.2968	−1.68690
$890$	−12.8264	−0.429941
$891$	−7.37070	−0.246928
$892$	−26.7488	−0.895615
$893$	38.9465	1.30329
$894$	−16.6286	−0.556143
$895$	14.4830	0.484114
$896$	−44.1511	−1.47498
$897$	2.43355	0.0812537
$898$	−58.2733	−1.94461
$899$	−42.1372	−1.40535
$900$	−3.00981	−0.100327
$901$	76.4106	2.54561
$902$	−19.6108	−0.652968
$903$	−5.24533	−0.174554
$904$	21.3556	0.710278
$905$	−21.8106	−0.725009
$906$	2.74637	0.0912419
$907$	−17.5506	−0.582760	−0.291380	−	0.956607i	$-0.594114\pi$
$907$	−17.5506	−0.582760	−0.291380	+	0.956607i	$0.594114\pi$
$908$	−25.5053	−0.846424
$909$	−6.40455	−0.212426
$910$	13.6649	0.452988
$911$	−15.6622	−0.518911	−0.259455	−	0.965755i	$-0.583543\pi$
$911$	−15.6622	−0.518911	−0.259455	+	0.965755i	$0.583543\pi$
$912$	−8.85295	−0.293150
$913$	8.27854	0.273980
$914$	−28.6994	−0.949291
$915$	−1.12709	−0.0372604
$916$	−21.7907	−0.719985
$917$	5.30660	0.175239
$918$	−32.6006	−1.07598
$919$	−46.0381	−1.51866	−0.759328	−	0.650707i	$-0.774472\pi$
$919$	−46.0381	−1.51866	−0.759328	+	0.650707i	$0.774472\pi$
$920$	4.49152	0.148081
$921$	0.853667	0.0281293
$922$	−15.5594	−0.512422
$923$	14.0635	0.462906
$924$	1.74448	0.0573891
$925$	−1.42395	−0.0468191
$926$	51.5944	1.69550
$927$	16.8213	0.552486
$928$	−47.7860	−1.56865
$929$	−2.86249	−0.0939153	−0.0469577	−	0.998897i	$-0.514953\pi$
$929$	−2.86249	−0.0939153	−0.0469577	+	0.998897i	$0.514953\pi$
$930$	−3.64583	−0.119552
$931$	−30.5231	−1.00035
$932$	−19.3436	−0.633620
$933$	3.23799	0.106007
$934$	−12.3991	−0.405712
$935$	7.44337	0.243424
$936$	−9.43522	−0.308400
$937$	44.9914	1.46980	0.734902	−	0.678173i	$-0.237228\pi$
$937$	44.9914	1.46980	0.734902	+	0.678173i	$0.237228\pi$
$938$	−9.98763	−0.326108
$939$	3.59830	0.117426
$940$	10.1008	0.329451
$941$	−46.2115	−1.50645	−0.753226	−	0.657761i	$-0.771504\pi$
$941$	−46.2115	−1.50645	−0.753226	+	0.657761i	$0.771504\pi$
$942$	17.5608	0.572161
$943$	30.8309	1.00399
$944$	41.4870	1.35029
$945$	−9.48841	−0.308658
$946$	5.63184	0.183107
$947$	−28.8865	−0.938685	−0.469342	−	0.883016i	$-0.655509\pi$
$947$	−28.8865	−0.938685	−0.469342	+	0.883016i	$0.655509\pi$
$948$	−0.834052	−0.0270887
$949$	−2.05520	−0.0667146
$950$	7.22197	0.234312
$951$	−1.11792	−0.0362510
$952$	46.0690	1.49310
$953$	39.6647	1.28487	0.642433	−	0.766342i	$-0.277925\pi$
$953$	39.6647	1.28487	0.642433	+	0.766342i	$0.277925\pi$
$954$	50.6287	1.63917
$955$	15.3522	0.496785
$956$	−6.75612	−0.218508
$957$	−3.74239	−0.120974
$958$	−17.3225	−0.559663
$959$	22.5142	0.727020
$960$	0.160490	0.00517980
$961$	−7.56820	−0.244135
$962$	5.12692	0.165299
$963$	−30.8493	−0.994105
$964$	−14.0523	−0.452596
$965$	25.5185	0.821469
$966$	−7.87298	−0.253309
$967$	−1.53336	−0.0493095	−0.0246547	−	0.999696i	$-0.507849\pi$
$967$	−1.53336	−0.0493095	−0.0246547	+	0.999696i	$0.507849\pi$
$968$	1.63077	0.0524150
$969$	13.1918	0.423781
$970$	−26.8452	−0.861947
$971$	25.2177	0.809273	0.404637	−	0.914478i	$-0.367398\pi$
$971$	25.2177	0.809273	0.404637	+	0.914478i	$0.367398\pi$
$972$	−11.4066	−0.365866
$973$	71.1525	2.28104
$974$	9.57558	0.306821
$975$	−0.883568	−0.0282968
$976$	13.0956	0.419181
$977$	−46.6415	−1.49219	−0.746097	−	0.665837i	$-0.768075\pi$
$977$	−46.6415	−1.49219	−0.746097	+	0.665837i	$0.768075\pi$
$978$	16.5114	0.527975
$979$	7.32143	0.233994
$980$	−7.91618	−0.252873
$981$	19.3593	0.618096
$982$	−28.1998	−0.899893
$983$	4.57128	0.145801	0.0729006	−	0.997339i	$-0.476774\pi$
$983$	4.57128	0.145801	0.0729006	+	0.997339i	$0.476774\pi$
$984$	7.84812	0.250189
$985$	−12.8034	−0.407951
$986$	113.512	3.61495
$987$	15.4153	0.490674
$988$	−9.05807	−0.288176
$989$	−8.85404	−0.281542
$990$	4.93188	0.156746
$991$	43.9720	1.39682	0.698408	−	0.715700i	$-0.253892\pi$
$991$	43.9720	1.39682	0.698408	+	0.715700i	$0.253892\pi$
$992$	26.5730	0.843693
$993$	−10.2816	−0.326277
$994$	−45.4981	−1.44311
$995$	2.63578	0.0835599
$996$	3.80516	0.120571
$997$	−24.8824	−0.788035	−0.394017	−	0.919103i	$-0.628915\pi$
$997$	−24.8824	−0.788035	−0.394017	+	0.919103i	$0.628915\pi$
$998$	8.15167	0.258037
$999$	−3.55994	−0.112631

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4015.2.a.g.1.9	✓	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4015.2.a.g.1.9	✓	32	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:	\( N \)	\(=\)	\( 4015 = 5 \cdot 11 \cdot 73 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4015.a (trivial)

\(f(q)\)	\(=\)	\(q-1.75190 q^{2} -0.429918 q^{3} +1.06914 q^{4} -1.00000 q^{5} +0.753172 q^{6} +3.79529 q^{7} +1.63077 q^{8} -2.81517 q^{9} +O(q^{10})\)	\(q-1.75190 q^{2} -0.429918 q^{3} +1.06914 q^{4} -1.00000 q^{5} +0.753172 q^{6} +3.79529 q^{7} +1.63077 q^{8} -2.81517 q^{9} +1.75190 q^{10} -1.00000 q^{11} -0.459642 q^{12} +2.05520 q^{13} -6.64896 q^{14} +0.429918 q^{15} -4.99522 q^{16} +7.44337 q^{17} +4.93188 q^{18} -4.12238 q^{19} -1.06914 q^{20} -1.63167 q^{21} +1.75190 q^{22} -2.75423 q^{23} -0.701099 q^{24} +1.00000 q^{25} -3.60050 q^{26} +2.50005 q^{27} +4.05769 q^{28} -8.70488 q^{29} -0.753172 q^{30} +4.84064 q^{31} +5.48956 q^{32} +0.429918 q^{33} -13.0400 q^{34} -3.79529 q^{35} -3.00981 q^{36} -1.42395 q^{37} +7.22197 q^{38} -0.883568 q^{39} -1.63077 q^{40} -11.1940 q^{41} +2.85851 q^{42} +3.21471 q^{43} -1.06914 q^{44} +2.81517 q^{45} +4.82512 q^{46} -9.44758 q^{47} +2.14754 q^{48} +7.40426 q^{49} -1.75190 q^{50} -3.20004 q^{51} +2.19729 q^{52} +10.2656 q^{53} -4.37982 q^{54} +1.00000 q^{55} +6.18926 q^{56} +1.77228 q^{57} +15.2500 q^{58} -8.30534 q^{59} +0.459642 q^{60} -2.62164 q^{61} -8.48030 q^{62} -10.6844 q^{63} +0.373304 q^{64} -2.05520 q^{65} -0.753172 q^{66} +1.50213 q^{67} +7.95799 q^{68} +1.18409 q^{69} +6.64896 q^{70} +6.84288 q^{71} -4.59090 q^{72} -1.00000 q^{73} +2.49461 q^{74} -0.429918 q^{75} -4.40739 q^{76} -3.79529 q^{77} +1.54792 q^{78} +1.81457 q^{79} +4.99522 q^{80} +7.37070 q^{81} +19.6108 q^{82} -8.27854 q^{83} -1.74448 q^{84} -7.44337 q^{85} -5.63184 q^{86} +3.74239 q^{87} -1.63077 q^{88} -7.32143 q^{89} -4.93188 q^{90} +7.80009 q^{91} -2.94465 q^{92} -2.08108 q^{93} +16.5512 q^{94} +4.12238 q^{95} -2.36006 q^{96} -15.3235 q^{97} -12.9715 q^{98} +2.81517 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 5 q^{2} - 7 q^{3} + 37 q^{4} - 32 q^{5} - 3 q^{6} - 18 q^{8} + 29 q^{9}+O(q^{10}) \) 32 * q - 5 * q^2 - 7 * q^3 + 37 * q^4 - 32 * q^5 - 3 * q^6 - 18 * q^8 + 29 * q^9	\( 32 q - 5 q^{2} - 7 q^{3} + 37 q^{4} - 32 q^{5} - 3 q^{6} - 18 q^{8} + 29 q^{9} + 5 q^{10} - 32 q^{11} - 24 q^{12} - q^{13} - 5 q^{14} + 7 q^{15} + 47 q^{16} - 30 q^{17} - 11 q^{18} + 16 q^{19} - 37 q^{20} + q^{21} + 5 q^{22} - 26 q^{23} - 21 q^{24} + 32 q^{25} - q^{26} - 31 q^{27} - 24 q^{28} - 10 q^{29} + 3 q^{30} - 2 q^{31} - 31 q^{32} + 7 q^{33} - 14 q^{34} + 38 q^{36} - 28 q^{37} - 63 q^{38} - 2 q^{39} + 18 q^{40} - 62 q^{41} - 9 q^{42} + 8 q^{43} - 37 q^{44} - 29 q^{45} + 19 q^{46} - 21 q^{47} - 79 q^{48} + 34 q^{49} - 5 q^{50} + 17 q^{51} + 15 q^{52} - 32 q^{53} + 5 q^{54} + 32 q^{55} - 52 q^{56} - 57 q^{57} + 4 q^{58} - 37 q^{59} + 24 q^{60} + 15 q^{61} - 22 q^{62} + 5 q^{63} + 70 q^{64} + q^{65} + 3 q^{66} - 42 q^{67} - 81 q^{68} - 8 q^{69} + 5 q^{70} - 40 q^{71} - 27 q^{72} - 32 q^{73} - 17 q^{74} - 7 q^{75} + 21 q^{76} - 105 q^{78} + 18 q^{79} - 47 q^{80} + 12 q^{81} - 70 q^{82} - 26 q^{83} + 22 q^{84} + 30 q^{85} - 45 q^{86} - 18 q^{87} + 18 q^{88} - 83 q^{89} + 11 q^{90} - 18 q^{91} - 73 q^{92} - 68 q^{93} + 56 q^{94} - 16 q^{95} - 35 q^{96} - 99 q^{97} - 61 q^{98} - 29 q^{99}+O(q^{100}) \) 32 * q - 5 * q^2 - 7 * q^3 + 37 * q^4 - 32 * q^5 - 3 * q^6 - 18 * q^8 + 29 * q^9 + 5 * q^10 - 32 * q^11 - 24 * q^12 - q^13 - 5 * q^14 + 7 * q^15 + 47 * q^16 - 30 * q^17 - 11 * q^18 + 16 * q^19 - 37 * q^20 + q^21 + 5 * q^22 - 26 * q^23 - 21 * q^24 + 32 * q^25 - q^26 - 31 * q^27 - 24 * q^28 - 10 * q^29 + 3 * q^30 - 2 * q^31 - 31 * q^32 + 7 * q^33 - 14 * q^34 + 38 * q^36 - 28 * q^37 - 63 * q^38 - 2 * q^39 + 18 * q^40 - 62 * q^41 - 9 * q^42 + 8 * q^43 - 37 * q^44 - 29 * q^45 + 19 * q^46 - 21 * q^47 - 79 * q^48 + 34 * q^49 - 5 * q^50 + 17 * q^51 + 15 * q^52 - 32 * q^53 + 5 * q^54 + 32 * q^55 - 52 * q^56 - 57 * q^57 + 4 * q^58 - 37 * q^59 + 24 * q^60 + 15 * q^61 - 22 * q^62 + 5 * q^63 + 70 * q^64 + q^65 + 3 * q^66 - 42 * q^67 - 81 * q^68 - 8 * q^69 + 5 * q^70 - 40 * q^71 - 27 * q^72 - 32 * q^73 - 17 * q^74 - 7 * q^75 + 21 * q^76 - 105 * q^78 + 18 * q^79 - 47 * q^80 + 12 * q^81 - 70 * q^82 - 26 * q^83 + 22 * q^84 + 30 * q^85 - 45 * q^86 - 18 * q^87 + 18 * q^88 - 83 * q^89 + 11 * q^90 - 18 * q^91 - 73 * q^92 - 68 * q^93 + 56 * q^94 - 16 * q^95 - 35 * q^96 - 99 * q^97 - 61 * q^98 - 29 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more