LMFDB - Embedded newform 4021.2.a.b.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4021,2,Mod(1,4021)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4021, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4021.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4021 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4021.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:	$32.1078466528$
Analytic rank:	$1$
Dimension:	$151$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Character	$\chi$	$=$	4021.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.64544 q^{2} -1.40778 q^{3} +4.99834 q^{4} -2.43306 q^{5} +3.72420 q^{6} +1.18791 q^{7} -7.93191 q^{8} -1.01815 q^{9} +O(q^{10})$	\(q-2.64544 q^{2} -1.40778 q^{3} +4.99834 q^{4} -2.43306 q^{5} +3.72420 q^{6} +1.18791 q^{7} -7.93191 q^{8} -1.01815 q^{9} +6.43651 q^{10} +2.95254 q^{11} -7.03657 q^{12} +4.35719 q^{13} -3.14254 q^{14} +3.42522 q^{15} +10.9867 q^{16} -4.48809 q^{17} +2.69344 q^{18} +1.43895 q^{19} -12.1613 q^{20} -1.67232 q^{21} -7.81075 q^{22} -0.349728 q^{23} +11.1664 q^{24} +0.919793 q^{25} -11.5267 q^{26} +5.65668 q^{27} +5.93758 q^{28} -0.220410 q^{29} -9.06121 q^{30} -3.47987 q^{31} -13.2008 q^{32} -4.15653 q^{33} +11.8730 q^{34} -2.89026 q^{35} -5.08904 q^{36} -1.64270 q^{37} -3.80664 q^{38} -6.13397 q^{39} +19.2988 q^{40} -7.37983 q^{41} +4.42402 q^{42} +2.64501 q^{43} +14.7578 q^{44} +2.47721 q^{45} +0.925182 q^{46} -4.07869 q^{47} -15.4669 q^{48} -5.58887 q^{49} -2.43325 q^{50} +6.31826 q^{51} +21.7787 q^{52} +10.2465 q^{53} -14.9644 q^{54} -7.18371 q^{55} -9.42240 q^{56} -2.02573 q^{57} +0.583082 q^{58} +4.96669 q^{59} +17.1204 q^{60} -3.21229 q^{61} +9.20577 q^{62} -1.20947 q^{63} +12.9484 q^{64} -10.6013 q^{65} +10.9958 q^{66} +8.12584 q^{67} -22.4330 q^{68} +0.492341 q^{69} +7.64600 q^{70} -13.4118 q^{71} +8.07584 q^{72} +8.79812 q^{73} +4.34566 q^{74} -1.29487 q^{75} +7.19234 q^{76} +3.50735 q^{77} +16.2270 q^{78} -1.18668 q^{79} -26.7313 q^{80} -4.90894 q^{81} +19.5229 q^{82} +17.4229 q^{83} -8.35882 q^{84} +10.9198 q^{85} -6.99720 q^{86} +0.310290 q^{87} -23.4193 q^{88} -13.6348 q^{89} -6.55331 q^{90} +5.17595 q^{91} -1.74806 q^{92} +4.89890 q^{93} +10.7899 q^{94} -3.50105 q^{95} +18.5838 q^{96} -5.94610 q^{97} +14.7850 q^{98} -3.00612 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 151 q - 16 q^{2} - 29 q^{3} + 120 q^{4} - 27 q^{5} - 18 q^{6} - 18 q^{7} - 48 q^{8} + 98 q^{9}+O(q^{10}) $ 151 * q - 16 * q^2 - 29 * q^3 + 120 * q^4 - 27 * q^5 - 18 * q^6 - 18 * q^7 - 48 * q^8 + 98 * q^9	\( 151 q - 16 q^{2} - 29 q^{3} + 120 q^{4} - 27 q^{5} - 18 q^{6} - 18 q^{7} - 48 q^{8} + 98 q^{9} - 7 q^{10} - 126 q^{11} - 47 q^{12} - 10 q^{13} - 58 q^{14} - 50 q^{15} + 74 q^{16} - 39 q^{17} - 47 q^{18} - 48 q^{19} - 64 q^{20} - 50 q^{21} - 7 q^{22} - 91 q^{23} - 44 q^{24} + 102 q^{25} - 94 q^{26} - 92 q^{27} - 27 q^{28} - 79 q^{29} - 12 q^{30} - 33 q^{31} - 102 q^{32} - 7 q^{33} - 17 q^{34} - 183 q^{35} - 6 q^{36} - 24 q^{37} - 77 q^{38} - 81 q^{39} - 37 q^{40} - 62 q^{41} + 4 q^{42} - 74 q^{43} - 209 q^{44} - 35 q^{45} - 37 q^{46} - 127 q^{47} - 59 q^{48} + 75 q^{49} - 79 q^{50} - 138 q^{51} - 12 q^{52} - 104 q^{53} - 43 q^{54} - 49 q^{55} - 167 q^{56} - 19 q^{57} - 32 q^{58} - 207 q^{59} - 94 q^{60} - 57 q^{61} - 69 q^{62} - 49 q^{63} - 4 q^{64} - 78 q^{65} - 51 q^{66} - 92 q^{67} - 80 q^{68} - 84 q^{69} + 5 q^{70} - 217 q^{71} - 79 q^{72} + 2 q^{73} - 113 q^{74} - 173 q^{75} - 70 q^{76} - 85 q^{77} - 46 q^{78} - 70 q^{79} - 95 q^{80} - 49 q^{81} - 12 q^{82} - 109 q^{83} - 114 q^{84} - 25 q^{85} - 136 q^{86} - 60 q^{87} - 2 q^{88} - 144 q^{89} - 55 q^{90} - 62 q^{91} - 175 q^{92} - 42 q^{93} - 15 q^{94} - 189 q^{95} - 64 q^{96} - 24 q^{97} - 37 q^{98} - 293 q^{99}+O(q^{100}) \) 151 * q - 16 * q^2 - 29 * q^3 + 120 * q^4 - 27 * q^5 - 18 * q^6 - 18 * q^7 - 48 * q^8 + 98 * q^9 - 7 * q^10 - 126 * q^11 - 47 * q^12 - 10 * q^13 - 58 * q^14 - 50 * q^15 + 74 * q^16 - 39 * q^17 - 47 * q^18 - 48 * q^19 - 64 * q^20 - 50 * q^21 - 7 * q^22 - 91 * q^23 - 44 * q^24 + 102 * q^25 - 94 * q^26 - 92 * q^27 - 27 * q^28 - 79 * q^29 - 12 * q^30 - 33 * q^31 - 102 * q^32 - 7 * q^33 - 17 * q^34 - 183 * q^35 - 6 * q^36 - 24 * q^37 - 77 * q^38 - 81 * q^39 - 37 * q^40 - 62 * q^41 + 4 * q^42 - 74 * q^43 - 209 * q^44 - 35 * q^45 - 37 * q^46 - 127 * q^47 - 59 * q^48 + 75 * q^49 - 79 * q^50 - 138 * q^51 - 12 * q^52 - 104 * q^53 - 43 * q^54 - 49 * q^55 - 167 * q^56 - 19 * q^57 - 32 * q^58 - 207 * q^59 - 94 * q^60 - 57 * q^61 - 69 * q^62 - 49 * q^63 - 4 * q^64 - 78 * q^65 - 51 * q^66 - 92 * q^67 - 80 * q^68 - 84 * q^69 + 5 * q^70 - 217 * q^71 - 79 * q^72 + 2 * q^73 - 113 * q^74 - 173 * q^75 - 70 * q^76 - 85 * q^77 - 46 * q^78 - 70 * q^79 - 95 * q^80 - 49 * q^81 - 12 * q^82 - 109 * q^83 - 114 * q^84 - 25 * q^85 - 136 * q^86 - 60 * q^87 - 2 * q^88 - 144 * q^89 - 55 * q^90 - 62 * q^91 - 175 * q^92 - 42 * q^93 - 15 * q^94 - 189 * q^95 - 64 * q^96 - 24 * q^97 - 37 * q^98 - 293 * 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 $

$2$	−2.64544	−1.87061	−0.935303	−	0.353847i	$-0.884873\pi$
$2$	−2.64544	−1.87061	−0.935303	+	0.353847i	$0.884873\pi$
$3$	−1.40778	−0.812784	−0.406392	−	0.913699i	$-0.633213\pi$
$3$	−1.40778	−0.812784	−0.406392	+	0.913699i	$0.633213\pi$
$4$	4.99834	2.49917
$5$	−2.43306	−1.08810	−0.544049	−	0.839053i	$-0.683110\pi$
$5$	−2.43306	−1.08810	−0.544049	+	0.839053i	$0.683110\pi$
$6$	3.72420	1.52040
$7$	1.18791	0.448988	0.224494	−	0.974475i	$-0.427927\pi$
$7$	1.18791	0.448988	0.224494	+	0.974475i	$0.427927\pi$
$8$	−7.93191	−2.80435
$9$	−1.01815	−0.339382
$10$	6.43651	2.03540
$11$	2.95254	0.890224	0.445112	−	0.895475i	$-0.353164\pi$
$11$	2.95254	0.890224	0.445112	+	0.895475i	$0.353164\pi$
$12$	−7.03657	−2.03128
$13$	4.35719	1.20847	0.604233	−	0.796808i	$-0.293480\pi$
$13$	4.35719	1.20847	0.604233	+	0.796808i	$0.293480\pi$
$14$	−3.14254	−0.839880
$15$	3.42522	0.884389
$16$	10.9867	2.74667
$17$	−4.48809	−1.08852	−0.544261	−	0.838916i	$-0.683190\pi$
$17$	−4.48809	−1.08852	−0.544261	+	0.838916i	$0.683190\pi$
$18$	2.69344	0.634850
$19$	1.43895	0.330117	0.165059	−	0.986284i	$-0.447219\pi$
$19$	1.43895	0.330117	0.165059	+	0.986284i	$0.447219\pi$
$20$	−12.1613	−2.71934
$21$	−1.67232	−0.364930
$22$	−7.81075	−1.66526
$23$	−0.349728	−0.0729233	−0.0364616	−	0.999335i	$-0.511609\pi$
$23$	−0.349728	−0.0729233	−0.0364616	+	0.999335i	$0.511609\pi$
$24$	11.1664	2.27933
$25$	0.919793	0.183959
$26$	−11.5267	−2.26056
$27$	5.65668	1.08863
$28$	5.93758	1.12210
$29$	−0.220410	−0.0409292	−0.0204646	−	0.999791i	$-0.506515\pi$
$29$	−0.220410	−0.0409292	−0.0204646	+	0.999791i	$0.506515\pi$
$30$	−9.06121	−1.65434
$31$	−3.47987	−0.625003	−0.312501	−	0.949917i	$-0.601167\pi$
$31$	−3.47987	−0.625003	−0.312501	+	0.949917i	$0.601167\pi$
$32$	−13.2008	−2.33359
$33$	−4.15653	−0.723560
$34$	11.8730	2.03620
$35$	−2.89026	−0.488543
$36$	−5.08904	−0.848173
$37$	−1.64270	−0.270058	−0.135029	−	0.990842i	$-0.543113\pi$
$37$	−1.64270	−0.270058	−0.135029	+	0.990842i	$0.543113\pi$
$38$	−3.80664	−0.617519
$39$	−6.13397	−0.982222
$40$	19.2988	3.05141
$41$	−7.37983	−1.15254	−0.576268	−	0.817261i	$-0.695492\pi$
$41$	−7.37983	−1.15254	−0.576268	+	0.817261i	$0.695492\pi$
$42$	4.42402	0.682641
$43$	2.64501	0.403360	0.201680	−	0.979451i	$-0.435360\pi$
$43$	2.64501	0.403360	0.201680	+	0.979451i	$0.435360\pi$
$44$	14.7578	2.22482
$45$	2.47721	0.369281
$46$	0.925182	0.136411
$47$	−4.07869	−0.594938	−0.297469	−	0.954732i	$-0.596142\pi$
$47$	−4.07869	−0.594938	−0.297469	+	0.954732i	$0.596142\pi$
$48$	−15.4669	−2.23245
$49$	−5.58887	−0.798410
$50$	−2.43325	−0.344114
$51$	6.31826	0.884734
$52$	21.7787	3.02016
$53$	10.2465	1.40747	0.703734	−	0.710464i	$-0.251515\pi$
$53$	10.2465	1.40747	0.703734	+	0.710464i	$0.251515\pi$
$54$	−14.9644	−2.03640
$55$	−7.18371	−0.968651
$56$	−9.42240	−1.25912
$57$	−2.02573	−0.268314
$58$	0.583082	0.0765624
$59$	4.96669	0.646608	0.323304	−	0.946295i	$-0.395206\pi$
$59$	4.96669	0.646608	0.323304	+	0.946295i	$0.395206\pi$
$60$	17.1204	2.21024
$61$	−3.21229	−0.411292	−0.205646	−	0.978626i	$-0.565930\pi$
$61$	−3.21229	−0.411292	−0.205646	+	0.978626i	$0.565930\pi$
$62$	9.20577	1.16913
$63$	−1.20947	−0.152379
$64$	12.9484	1.61856
$65$	−10.6013	−1.31493
$66$	10.9958	1.35350
$67$	8.12584	0.992730	0.496365	−	0.868114i	$-0.334668\pi$
$67$	8.12584	0.992730	0.496365	+	0.868114i	$0.334668\pi$
$68$	−22.4330	−2.72040
$69$	0.492341	0.0592709
$70$	7.64600	0.913872
$71$	−13.4118	−1.59168	−0.795842	−	0.605505i	$-0.792971\pi$
$71$	−13.4118	−1.59168	−0.795842	+	0.605505i	$0.792971\pi$
$72$	8.07584	0.951747
$73$	8.79812	1.02974	0.514871	−	0.857268i	$-0.327840\pi$
$73$	8.79812	1.02974	0.514871	+	0.857268i	$0.327840\pi$
$74$	4.34566	0.505172
$75$	−1.29487	−0.149519
$76$	7.19234	0.825018
$77$	3.50735	0.399700
$78$	16.2270	1.83735
$79$	−1.18668	−0.133512	−0.0667561	−	0.997769i	$-0.521265\pi$
$79$	−1.18668	−0.133512	−0.0667561	+	0.997769i	$0.521265\pi$
$80$	−26.7313	−2.98865
$81$	−4.90894	−0.545438
$82$	19.5229	2.15594
$83$	17.4229	1.91241	0.956205	−	0.292699i	$-0.0945534\pi$
$83$	17.4229	1.91241	0.956205	+	0.292699i	$0.0945534\pi$
$84$	−8.35882	−0.912022
$85$	10.9198	1.18442
$86$	−6.99720	−0.754528
$87$	0.310290	0.0332666
$88$	−23.4193	−2.49650
$89$	−13.6348	−1.44528	−0.722642	−	0.691222i	$-0.757072\pi$
$89$	−13.6348	−1.44528	−0.722642	+	0.691222i	$0.757072\pi$
$90$	−6.55331	−0.690780
$91$	5.17595	0.542587
$92$	−1.74806	−0.182247
$93$	4.89890	0.507992
$94$	10.7899	1.11289
$95$	−3.50105	−0.359200
$96$	18.5838	1.89670
$97$	−5.94610	−0.603735	−0.301867	−	0.953350i	$-0.597610\pi$
$97$	−5.94610	−0.603735	−0.301867	+	0.953350i	$0.597610\pi$
$98$	14.7850	1.49351
$99$	−3.00612	−0.302126
$100$	4.59743	0.459743
$101$	−0.228212	−0.0227079	−0.0113540	−	0.999936i	$-0.503614\pi$
$101$	−0.228212	−0.0227079	−0.0113540	+	0.999936i	$0.503614\pi$
$102$	−16.7146	−1.65499
$103$	12.5677	1.23833	0.619165	−	0.785261i	$-0.287471\pi$
$103$	12.5677	1.23833	0.619165	+	0.785261i	$0.287471\pi$
$104$	−34.5608	−3.38897
$105$	4.06886	0.397080
$106$	−27.1065	−2.63282
$107$	−8.90296	−0.860682	−0.430341	−	0.902666i	$-0.641607\pi$
$107$	−8.90296	−0.860682	−0.430341	+	0.902666i	$0.641607\pi$
$108$	28.2740	2.72067
$109$	1.50692	0.144337	0.0721683	−	0.997392i	$-0.477008\pi$
$109$	1.50692	0.144337	0.0721683	+	0.997392i	$0.477008\pi$
$110$	19.0040	1.81196
$111$	2.31256	0.219499
$112$	13.0512	1.23322
$113$	8.42016	0.792102	0.396051	−	0.918228i	$-0.370380\pi$
$113$	8.42016	0.792102	0.396051	+	0.918228i	$0.370380\pi$
$114$	5.35893	0.501910
$115$	0.850909	0.0793477
$116$	−1.10168	−0.102289
$117$	−4.43625	−0.410132
$118$	−13.1391	−1.20955
$119$	−5.33145	−0.488734
$120$	−27.1686	−2.48014
$121$	−2.28252	−0.207502
$122$	8.49792	0.769365
$123$	10.3892	0.936763
$124$	−17.3936	−1.56199
$125$	9.92740	0.887934
$126$	3.19957	0.285040
$127$	−11.4069	−1.01220	−0.506099	−	0.862476i	$-0.668913\pi$
$127$	−11.4069	−1.01220	−0.506099	+	0.862476i	$0.668913\pi$
$128$	−7.85273	−0.694090
$129$	−3.72360	−0.327845
$130$	28.0451	2.45972
$131$	−1.42392	−0.124408	−0.0622042	−	0.998063i	$-0.519813\pi$
$131$	−1.42392	−0.124408	−0.0622042	+	0.998063i	$0.519813\pi$
$132$	−20.7757	−1.80830
$133$	1.70934	0.148219
$134$	−21.4964	−1.85701
$135$	−13.7631	−1.18454
$136$	35.5991	3.05260
$137$	−10.6475	−0.909674	−0.454837	−	0.890575i	$-0.650302\pi$
$137$	−10.6475	−0.909674	−0.454837	+	0.890575i	$0.650302\pi$
$138$	−1.30246	−0.110872
$139$	15.6648	1.32867	0.664335	−	0.747435i	$-0.268715\pi$
$139$	15.6648	1.32867	0.664335	+	0.747435i	$0.268715\pi$
$140$	−14.4465	−1.22095
$141$	5.74191	0.483556
$142$	35.4800	2.97741
$143$	12.8648	1.07581
$144$	−11.1861	−0.932172
$145$	0.536272	0.0445350
$146$	−23.2749	−1.92624
$147$	7.86791	0.648935
$148$	−8.21076	−0.674920
$149$	−8.36824	−0.685553	−0.342776	−	0.939417i	$-0.611367\pi$
$149$	−8.36824	−0.685553	−0.342776	+	0.939417i	$0.611367\pi$
$150$	3.42549	0.279690
$151$	15.1525	1.23309	0.616545	−	0.787320i	$-0.288532\pi$
$151$	15.1525	1.23309	0.616545	+	0.787320i	$0.288532\pi$
$152$	−11.4136	−0.925765
$153$	4.56954	0.369425
$154$	−9.27848	−0.747681
$155$	8.46674	0.680065
$156$	−30.6597	−2.45474
$157$	−3.54591	−0.282994	−0.141497	−	0.989939i	$-0.545192\pi$
$157$	−3.54591	−0.282994	−0.141497	+	0.989939i	$0.545192\pi$
$158$	3.13930	0.249749
$159$	−14.4249	−1.14397
$160$	32.1183	2.53918
$161$	−0.415445	−0.0327417
$162$	12.9863	1.02030
$163$	10.7081	0.838720	0.419360	−	0.907820i	$-0.362255\pi$
$163$	10.7081	0.838720	0.419360	+	0.907820i	$0.362255\pi$
$164$	−36.8869	−2.88038
$165$	10.1131	0.787304
$166$	−46.0911	−3.57736
$167$	8.01794	0.620447	0.310223	−	0.950664i	$-0.399596\pi$
$167$	8.01794	0.620447	0.310223	+	0.950664i	$0.399596\pi$
$168$	13.2647	1.02339
$169$	5.98507	0.460390
$170$	−28.8877	−2.21558
$171$	−1.46506	−0.112036
$172$	13.2206	1.00806
$173$	−22.5497	−1.71442	−0.857211	−	0.514965i	$-0.827805\pi$
$173$	−22.5497	−1.71442	−0.857211	+	0.514965i	$0.827805\pi$
$174$	−0.820853	−0.0622287
$175$	1.09263	0.0825952
$176$	32.4386	2.44515
$177$	−6.99203	−0.525553
$178$	36.0700	2.70356
$179$	16.0552	1.20002	0.600011	−	0.799992i	$-0.295163\pi$
$179$	16.0552	1.20002	0.600011	+	0.799992i	$0.295163\pi$
$180$	12.3819	0.922896
$181$	10.0656	0.748172	0.374086	−	0.927394i	$-0.377956\pi$
$181$	10.0656	0.748172	0.374086	+	0.927394i	$0.377956\pi$
$182$	−13.6926	−1.01497
$183$	4.52221	0.334292
$184$	2.77401	0.204503
$185$	3.99679	0.293850
$186$	−12.9597	−0.950254
$187$	−13.2513	−0.969028
$188$	−20.3866	−1.48685
$189$	6.71963	0.488781
$190$	9.26180	0.671922
$191$	−3.90187	−0.282329	−0.141165	−	0.989986i	$-0.545085\pi$
$191$	−3.90187	−0.282329	−0.141165	+	0.989986i	$0.545085\pi$
$192$	−18.2286	−1.31554
$193$	−26.5989	−1.91463	−0.957316	−	0.289042i	$-0.906663\pi$
$193$	−26.5989	−1.91463	−0.957316	+	0.289042i	$0.906663\pi$
$194$	15.7300	1.12935
$195$	14.9243	1.06875
$196$	−27.9350	−1.99536
$197$	24.1294	1.71915	0.859574	−	0.511011i	$-0.170729\pi$
$197$	24.1294	1.71915	0.859574	+	0.511011i	$0.170729\pi$
$198$	7.95249	0.565159
$199$	15.4989	1.09868	0.549342	−	0.835597i	$-0.314878\pi$
$199$	15.4989	1.09868	0.549342	+	0.835597i	$0.314878\pi$
$200$	−7.29571	−0.515885
$201$	−11.4394	−0.806875
$202$	0.603719	0.0424775
$203$	−0.261828	−0.0183767
$204$	31.5808	2.21110
$205$	17.9556	1.25407
$206$	−33.2470	−2.31643
$207$	0.356074	0.0247489
$208$	47.8711	3.31926
$209$	4.24855	0.293878
$210$	−10.7639	−0.742781
$211$	−8.54871	−0.588517	−0.294259	−	0.955726i	$-0.595073\pi$
$211$	−8.54871	−0.588517	−0.294259	+	0.955726i	$0.595073\pi$
$212$	51.2155	3.51750
$213$	18.8809	1.29369
$214$	23.5522	1.61000
$215$	−6.43547	−0.438895
$216$	−44.8683	−3.05290
$217$	−4.13377	−0.280619
$218$	−3.98646	−0.269997
$219$	−12.3858	−0.836958
$220$	−35.9066	−2.42082
$221$	−19.5555	−1.31544
$222$	−6.11774	−0.410596
$223$	23.8624	1.59794	0.798971	−	0.601370i	$-0.205378\pi$
$223$	23.8624	1.59794	0.798971	+	0.601370i	$0.205378\pi$
$224$	−15.6813	−1.04775
$225$	−0.936484	−0.0624323
$226$	−22.2750	−1.48171
$227$	4.80165	0.318697	0.159348	−	0.987222i	$-0.449061\pi$
$227$	4.80165	0.318697	0.159348	+	0.987222i	$0.449061\pi$
$228$	−10.1253	−0.670561
$229$	2.22454	0.147002	0.0735008	−	0.997295i	$-0.476583\pi$
$229$	2.22454	0.147002	0.0735008	+	0.997295i	$0.476583\pi$
$230$	−2.25103	−0.148428
$231$	−4.93759	−0.324870
$232$	1.74827	0.114780
$233$	18.1181	1.18696	0.593478	−	0.804850i	$-0.297755\pi$
$233$	18.1181	1.18696	0.593478	+	0.804850i	$0.297755\pi$
$234$	11.7358	0.767195
$235$	9.92370	0.647351
$236$	24.8252	1.61598
$237$	1.67059	0.108517
$238$	14.1040	0.914228
$239$	14.7706	0.955428	0.477714	−	0.878515i	$-0.341466\pi$
$239$	14.7706	0.955428	0.477714	+	0.878515i	$0.341466\pi$
$240$	37.6319	2.42913
$241$	−6.57987	−0.423847	−0.211924	−	0.977286i	$-0.567973\pi$
$241$	−6.57987	−0.423847	−0.211924	+	0.977286i	$0.567973\pi$
$242$	6.03826	0.388154
$243$	−10.0593	−0.645305
$244$	−16.0561	−1.02789
$245$	13.5981	0.868748
$246$	−27.4840	−1.75231
$247$	6.26976	0.398935
$248$	27.6020	1.75273
$249$	−24.5276	−1.55438
$250$	−26.2623	−1.66097
$251$	3.60112	0.227301	0.113650	−	0.993521i	$-0.463746\pi$
$251$	3.60112	0.227301	0.113650	+	0.993521i	$0.463746\pi$
$252$	−6.04532	−0.380820
$253$	−1.03258	−0.0649180
$254$	30.1762	1.89342
$255$	−15.3727	−0.962677
$256$	−5.12298	−0.320186
$257$	17.9979	1.12268	0.561341	−	0.827585i	$-0.310286\pi$
$257$	17.9979	1.12268	0.561341	+	0.827585i	$0.310286\pi$
$258$	9.85054	0.613268
$259$	−1.95138	−0.121253
$260$	−52.9889	−3.28623
$261$	0.224410	0.0138906
$262$	3.76689	0.232719
$263$	−5.29842	−0.326714	−0.163357	−	0.986567i	$-0.552232\pi$
$263$	−5.29842	−0.326714	−0.163357	+	0.986567i	$0.552232\pi$
$264$	32.9692	2.02912
$265$	−24.9304	−1.53146
$266$	−4.52195	−0.277259
$267$	19.1948	1.17470
$268$	40.6157	2.48100
$269$	3.21608	0.196088	0.0980441	−	0.995182i	$-0.468741\pi$
$269$	3.21608	0.196088	0.0980441	+	0.995182i	$0.468741\pi$
$270$	36.4093	2.21580
$271$	0.398723	0.0242207	0.0121104	−	0.999927i	$-0.496145\pi$
$271$	0.398723	0.0242207	0.0121104	+	0.999927i	$0.496145\pi$
$272$	−49.3093	−2.98981
$273$	−7.28661	−0.441006
$274$	28.1672	1.70164
$275$	2.71572	0.163764
$276$	2.46088	0.148128
$277$	−13.1463	−0.789884	−0.394942	−	0.918706i	$-0.629235\pi$
$277$	−13.1463	−0.789884	−0.394942	+	0.918706i	$0.629235\pi$
$278$	−41.4402	−2.48542
$279$	3.54302	0.212115
$280$	22.9253	1.37005
$281$	−10.2163	−0.609454	−0.304727	−	0.952440i	$-0.598565\pi$
$281$	−10.2163	−0.609454	−0.304727	+	0.952440i	$0.598565\pi$
$282$	−15.1898	−0.904542
$283$	−30.0026	−1.78347	−0.891734	−	0.452561i	$-0.850511\pi$
$283$	−30.0026	−1.78347	−0.891734	+	0.452561i	$0.850511\pi$
$284$	−67.0365	−3.97788
$285$	4.92872	0.291952
$286$	−34.0329	−2.01241
$287$	−8.76658	−0.517475
$288$	13.4403	0.791979
$289$	3.14298	0.184881
$290$	−1.41867	−0.0833074
$291$	8.37082	0.490706
$292$	43.9760	2.57350
$293$	−16.0297	−0.936466	−0.468233	−	0.883605i	$-0.655109\pi$
$293$	−16.0297	−0.936466	−0.468233	+	0.883605i	$0.655109\pi$
$294$	−20.8141	−1.21390
$295$	−12.0843	−0.703574
$296$	13.0297	0.757338
$297$	16.7016	0.969123
$298$	22.1376	1.28240
$299$	−1.52383	−0.0881253
$300$	−6.47219	−0.373672
$301$	3.14203	0.181104
$302$	−40.0849	−2.30663
$303$	0.321272	0.0184566
$304$	15.8093	0.906723
$305$	7.81571	0.447526
$306$	−12.0884	−0.691049
$307$	−8.61525	−0.491699	−0.245849	−	0.969308i	$-0.579067\pi$
$307$	−8.61525	−0.491699	−0.245849	+	0.969308i	$0.579067\pi$
$308$	17.5309	0.998917
$309$	−17.6926	−1.00650
$310$	−22.3982	−1.27213
$311$	−1.64826	−0.0934640	−0.0467320	−	0.998907i	$-0.514881\pi$
$311$	−1.64826	−0.0934640	−0.0467320	+	0.998907i	$0.514881\pi$
$312$	48.6541	2.75450
$313$	−10.8948	−0.615813	−0.307906	−	0.951417i	$-0.599628\pi$
$313$	−10.8948	−0.615813	−0.307906	+	0.951417i	$0.599628\pi$
$314$	9.38047	0.529371
$315$	2.94271	0.165803
$316$	−5.93144	−0.333670
$317$	−6.66222	−0.374188	−0.187094	−	0.982342i	$-0.559907\pi$
$317$	−6.66222	−0.374188	−0.187094	+	0.982342i	$0.559907\pi$
$318$	38.1601	2.13991
$319$	−0.650770	−0.0364361
$320$	−31.5044	−1.76115
$321$	12.5334	0.699548
$322$	1.09903	0.0612468
$323$	−6.45813	−0.359340
$324$	−24.5365	−1.36314
$325$	4.00771	0.222308
$326$	−28.3275	−1.56891
$327$	−2.12142	−0.117315
$328$	58.5361	3.23212
$329$	−4.84512	−0.267120
$330$	−26.7536	−1.47274
$331$	−6.21261	−0.341476	−0.170738	−	0.985316i	$-0.554615\pi$
$331$	−6.21261	−0.341476	−0.170738	+	0.985316i	$0.554615\pi$
$332$	87.0854	4.77943
$333$	1.67251	0.0916529
$334$	−21.2110	−1.16061
$335$	−19.7707	−1.08019
$336$	−18.3733	−1.00234
$337$	5.50457	0.299853	0.149927	−	0.988697i	$-0.452096\pi$
$337$	5.50457	0.299853	0.149927	+	0.988697i	$0.452096\pi$
$338$	−15.8331	−0.861209
$339$	−11.8538	−0.643808
$340$	54.5809	2.96006
$341$	−10.2744	−0.556392
$342$	3.87572	0.209575
$343$	−14.9545	−0.807465
$344$	−20.9800	−1.13116
$345$	−1.19790	−0.0644925
$346$	59.6538	3.20701
$347$	16.6222	0.892328	0.446164	−	0.894951i	$-0.352790\pi$
$347$	16.6222	0.892328	0.446164	+	0.894951i	$0.352790\pi$
$348$	1.55093	0.0831388
$349$	−16.7655	−0.897436	−0.448718	−	0.893673i	$-0.648119\pi$
$349$	−16.7655	−0.897436	−0.448718	+	0.893673i	$0.648119\pi$
$350$	−2.89049	−0.154503
$351$	24.6472	1.31557
$352$	−38.9758	−2.07742
$353$	−14.4186	−0.767427	−0.383713	−	0.923452i	$-0.625355\pi$
$353$	−14.4186	−0.767427	−0.383713	+	0.923452i	$0.625355\pi$
$354$	18.4970	0.983102
$355$	32.6317	1.73191
$356$	−68.1512	−3.61201
$357$	7.50553	0.397235
$358$	−42.4730	−2.24477
$359$	−33.2220	−1.75339	−0.876694	−	0.481048i	$-0.840256\pi$
$359$	−33.2220	−1.75339	−0.876694	+	0.481048i	$0.840256\pi$
$360$	−19.6490	−1.03559
$361$	−16.9294	−0.891023
$362$	−26.6280	−1.39954
$363$	3.21329	0.168654
$364$	25.8711	1.35602
$365$	−21.4064	−1.12046
$366$	−11.9632	−0.625328
$367$	4.25331	0.222021	0.111011	−	0.993819i	$-0.464591\pi$
$367$	4.25331	0.222021	0.111011	+	0.993819i	$0.464591\pi$
$368$	−3.84235	−0.200296
$369$	7.51375	0.391150
$370$	−10.5733	−0.549677
$371$	12.1719	0.631936
$372$	24.4863	1.26956
$373$	3.08856	0.159919	0.0799597	−	0.996798i	$-0.474521\pi$
$373$	3.08856	0.159919	0.0799597	+	0.996798i	$0.474521\pi$
$374$	35.0554	1.81267
$375$	−13.9756	−0.721698
$376$	32.3518	1.66841
$377$	−0.960369	−0.0494615
$378$	−17.7764	−0.914317
$379$	−4.66656	−0.239705	−0.119852	−	0.992792i	$-0.538242\pi$
$379$	−4.66656	−0.239705	−0.119852	+	0.992792i	$0.538242\pi$
$380$	−17.4994	−0.897701
$381$	16.0584	0.822698
$382$	10.3222	0.528127
$383$	−21.2994	−1.08835	−0.544175	−	0.838972i	$-0.683157\pi$
$383$	−21.2994	−1.08835	−0.544175	+	0.838972i	$0.683157\pi$
$384$	11.0549	0.564145
$385$	−8.53361	−0.434913
$386$	70.3658	3.58152
$387$	−2.69301	−0.136893
$388$	−29.7206	−1.50883
$389$	4.95187	0.251070	0.125535	−	0.992089i	$-0.459935\pi$
$389$	4.95187	0.251070	0.125535	+	0.992089i	$0.459935\pi$
$390$	−39.4814	−1.99922
$391$	1.56961	0.0793786
$392$	44.3304	2.23902
$393$	2.00457	0.101117
$394$	−63.8328	−3.21585
$395$	2.88727	0.145275
$396$	−15.0256	−0.755064
$397$	−7.05067	−0.353863	−0.176931	−	0.984223i	$-0.556617\pi$
$397$	−7.05067	−0.353863	−0.176931	+	0.984223i	$0.556617\pi$
$398$	−41.0012	−2.05521
$399$	−2.40638	−0.120470
$400$	10.1055	0.505274
$401$	3.84079	0.191800	0.0958999	−	0.995391i	$-0.469427\pi$
$401$	3.84079	0.191800	0.0958999	+	0.995391i	$0.469427\pi$
$402$	30.2623	1.50935
$403$	−15.1624	−0.755295
$404$	−1.14068	−0.0567509
$405$	11.9438	0.593490
$406$	0.692649	0.0343756
$407$	−4.85013	−0.240412
$408$	−50.1159	−2.48111
$409$	−32.6368	−1.61379	−0.806893	−	0.590697i	$-0.798853\pi$
$409$	−32.6368	−1.61379	−0.806893	+	0.590697i	$0.798853\pi$
$410$	−47.5004	−2.34588
$411$	14.9893	0.739368
$412$	62.8175	3.09480
$413$	5.89999	0.290319
$414$	−0.941971	−0.0462954
$415$	−42.3909	−2.08089
$416$	−57.5182	−2.82006
$417$	−22.0526	−1.07992
$418$	−11.2393	−0.549730
$419$	−13.0822	−0.639110	−0.319555	−	0.947568i	$-0.603533\pi$
$419$	−13.0822	−0.639110	−0.319555	+	0.947568i	$0.603533\pi$
$420$	20.3375	0.992370
$421$	−8.54498	−0.416457	−0.208229	−	0.978080i	$-0.566770\pi$
$421$	−8.54498	−0.416457	−0.208229	+	0.978080i	$0.566770\pi$
$422$	22.6151	1.10088
$423$	4.15270	0.201911
$424$	−81.2744	−3.94704
$425$	−4.12812	−0.200243
$426$	−49.9481	−2.41999
$427$	−3.81592	−0.184665
$428$	−44.5000	−2.15099
$429$	−18.1108	−0.874397
$430$	17.0246	0.821001
$431$	−19.4060	−0.934753	−0.467376	−	0.884058i	$-0.654801\pi$
$431$	−19.4060	−0.934753	−0.467376	+	0.884058i	$0.654801\pi$
$432$	62.1482	2.99011
$433$	−5.56285	−0.267334	−0.133667	−	0.991026i	$-0.542675\pi$
$433$	−5.56285	−0.267334	−0.133667	+	0.991026i	$0.542675\pi$
$434$	10.9356	0.524927
$435$	−0.754955	−0.0361973
$436$	7.53209	0.360722
$437$	−0.503240	−0.0240732
$438$	32.7660	1.56562
$439$	−11.4846	−0.548128	−0.274064	−	0.961711i	$-0.588368\pi$
$439$	−11.4846	−0.548128	−0.274064	+	0.961711i	$0.588368\pi$
$440$	56.9805	2.71644
$441$	5.69029	0.270966
$442$	51.7327	2.46067
$443$	−21.7652	−1.03410	−0.517048	−	0.855956i	$-0.672969\pi$
$443$	−21.7652	−1.03410	−0.517048	+	0.855956i	$0.672969\pi$
$444$	11.5590	0.548564
$445$	33.1743	1.57261
$446$	−63.1263	−2.98912
$447$	11.7807	0.557206
$448$	15.3816	0.726712
$449$	11.4113	0.538534	0.269267	−	0.963066i	$-0.413219\pi$
$449$	11.4113	0.538534	0.269267	+	0.963066i	$0.413219\pi$
$450$	2.47741	0.116786
$451$	−21.7892	−1.02601
$452$	42.0868	1.97960
$453$	−21.3314	−1.00224
$454$	−12.7025	−0.596156
$455$	−12.5934	−0.590388
$456$	16.0679	0.752447
$457$	−10.5272	−0.492443	−0.246222	−	0.969214i	$-0.579189\pi$
$457$	−10.5272	−0.492443	−0.246222	+	0.969214i	$0.579189\pi$
$458$	−5.88488	−0.274982
$459$	−25.3877	−1.18500
$460$	4.25313	0.198303
$461$	−28.6716	−1.33537	−0.667685	−	0.744444i	$-0.732715\pi$
$461$	−28.6716	−1.33537	−0.667685	+	0.744444i	$0.732715\pi$
$462$	13.0621	0.607703
$463$	−26.2910	−1.22185	−0.610923	−	0.791690i	$-0.709201\pi$
$463$	−26.2910	−1.22185	−0.610923	+	0.791690i	$0.709201\pi$
$464$	−2.42158	−0.112419
$465$	−11.9193	−0.552746
$466$	−47.9303	−2.22033
$467$	−17.4334	−0.806722	−0.403361	−	0.915041i	$-0.632158\pi$
$467$	−17.4334	−0.806722	−0.403361	+	0.915041i	$0.632158\pi$
$468$	−22.1739	−1.02499
$469$	9.65278	0.445724
$470$	−26.2525	−1.21094
$471$	4.99187	0.230013
$472$	−39.3953	−1.81332
$473$	7.80949	0.359081
$474$	−4.41945	−0.202992
$475$	1.32353	0.0607279
$476$	−26.6484	−1.22143
$477$	−10.4325	−0.477669
$478$	−39.0746	−1.78723
$479$	14.3898	0.657486	0.328743	−	0.944419i	$-0.393375\pi$
$479$	14.3898	0.657486	0.328743	+	0.944419i	$0.393375\pi$
$480$	−45.2156	−2.06380
$481$	−7.15754	−0.326356
$482$	17.4066	0.792851
$483$	0.584857	0.0266119
$484$	−11.4088	−0.518582
$485$	14.4672	0.656923
$486$	26.6113	1.20711
$487$	−21.0997	−0.956120	−0.478060	−	0.878327i	$-0.658660\pi$
$487$	−21.0997	−0.956120	−0.478060	+	0.878327i	$0.658660\pi$
$488$	25.4796	1.15341
$489$	−15.0746	−0.681698
$490$	−35.9728	−1.62509
$491$	−17.6696	−0.797420	−0.398710	−	0.917077i	$-0.630542\pi$
$491$	−17.6696	−0.797420	−0.398710	+	0.917077i	$0.630542\pi$
$492$	51.9287	2.34113
$493$	0.989222	0.0445523
$494$	−16.5863	−0.746251
$495$	7.31407	0.328743
$496$	−38.2322	−1.71668
$497$	−15.9320	−0.714647
$498$	64.8863	2.90762
$499$	−6.72555	−0.301077	−0.150539	−	0.988604i	$-0.548101\pi$
$499$	−6.72555	−0.301077	−0.150539	+	0.988604i	$0.548101\pi$
$500$	49.6205	2.21910
$501$	−11.2875	−0.504289
$502$	−9.52653	−0.425190
$503$	19.3116	0.861062	0.430531	−	0.902576i	$-0.358326\pi$
$503$	19.3116	0.861062	0.430531	+	0.902576i	$0.358326\pi$
$504$	9.59338	0.427323
$505$	0.555253	0.0247084
$506$	2.73164	0.121436
$507$	−8.42569	−0.374198
$508$	−57.0154	−2.52965
$509$	−25.3999	−1.12583	−0.562915	−	0.826515i	$-0.690320\pi$
$509$	−25.3999	−1.12583	−0.562915	+	0.826515i	$0.690320\pi$
$510$	40.6676	1.80079
$511$	10.4514	0.462342
$512$	29.2580	1.29303
$513$	8.13966	0.359375
$514$	−47.6124	−2.10009
$515$	−30.5780	−1.34743
$516$	−18.6118	−0.819339
$517$	−12.0425	−0.529627
$518$	5.16225	0.226816
$519$	31.7451	1.39346
$520$	84.0886	3.68753
$521$	4.22181	0.184961	0.0924805	−	0.995714i	$-0.470520\pi$
$521$	4.22181	0.184961	0.0924805	+	0.995714i	$0.470520\pi$
$522$	−0.593663	−0.0259839
$523$	−18.2296	−0.797123	−0.398561	−	0.917142i	$-0.630490\pi$
$523$	−18.2296	−0.797123	−0.398561	+	0.917142i	$0.630490\pi$
$524$	−7.11723	−0.310918
$525$	−1.53819	−0.0671321
$526$	14.0166	0.611154
$527$	15.6180	0.680330
$528$	−45.6665	−1.98738
$529$	−22.8777	−0.994682
$530$	65.9518	2.86476
$531$	−5.05682	−0.219447
$532$	8.54386	0.370423
$533$	−32.1553	−1.39280
$534$	−50.7787	−2.19741
$535$	21.6615	0.936507
$536$	−64.4534	−2.78396
$537$	−22.6022	−0.975359
$538$	−8.50795	−0.366804
$539$	−16.5013	−0.710763
$540$	−68.7924	−2.96035
$541$	12.2807	0.527990	0.263995	−	0.964524i	$-0.414960\pi$
$541$	12.2807	0.527990	0.263995	+	0.964524i	$0.414960\pi$
$542$	−1.05480	−0.0453074
$543$	−14.1702	−0.608102
$544$	59.2463	2.54016
$545$	−3.66643	−0.157053
$546$	19.2763	0.824948
$547$	−39.9817	−1.70949	−0.854747	−	0.519046i	$-0.826287\pi$
$547$	−39.9817	−1.70949	−0.854747	+	0.519046i	$0.826287\pi$
$548$	−53.2196	−2.27343
$549$	3.27059	0.139585
$550$	−7.18427	−0.306338
$551$	−0.317159	−0.0135114
$552$	−3.90520	−0.166216
$553$	−1.40967	−0.0599454
$554$	34.7777	1.47756
$555$	−5.62661	−0.238836
$556$	78.2979	3.32057
$557$	14.9734	0.634442	0.317221	−	0.948352i	$-0.397250\pi$
$557$	14.9734	0.634442	0.317221	+	0.948352i	$0.397250\pi$
$558$	−9.37282	−0.396783
$559$	11.5248	0.487447
$560$	−31.7544	−1.34187
$561$	18.6549	0.787611
$562$	27.0266	1.14005
$563$	−3.18844	−0.134377	−0.0671884	−	0.997740i	$-0.521403\pi$
$563$	−3.18844	−0.134377	−0.0671884	+	0.997740i	$0.521403\pi$
$564$	28.7000	1.20849
$565$	−20.4868	−0.861885
$566$	79.3699	3.33616
$567$	−5.83138	−0.244895
$568$	106.381	4.46364
$569$	33.5576	1.40681	0.703404	−	0.710791i	$-0.251663\pi$
$569$	33.5576	1.40681	0.703404	+	0.710791i	$0.251663\pi$
$570$	−13.0386	−0.546127
$571$	12.2852	0.514118	0.257059	−	0.966396i	$-0.417247\pi$
$571$	12.2852	0.514118	0.257059	+	0.966396i	$0.417247\pi$
$572$	64.3024	2.68862
$573$	5.49299	0.229473
$574$	23.1914	0.967992
$575$	−0.321677	−0.0134149
$576$	−13.1834	−0.549309
$577$	30.3603	1.26392	0.631958	−	0.775003i	$-0.282252\pi$
$577$	30.3603	1.26392	0.631958	+	0.775003i	$0.282252\pi$
$578$	−8.31455	−0.345840
$579$	37.4455	1.55618
$580$	2.68047	0.111300
$581$	20.6968	0.858649
$582$	−22.1445	−0.917918
$583$	30.2532	1.25296
$584$	−69.7859	−2.88776
$585$	10.7937	0.446264
$586$	42.4056	1.75176
$587$	28.0309	1.15696	0.578480	−	0.815696i	$-0.303646\pi$
$587$	28.0309	1.15696	0.578480	+	0.815696i	$0.303646\pi$
$588$	39.3265	1.62180
$589$	−5.00735	−0.206324
$590$	31.9682	1.31611
$591$	−33.9690	−1.39730
$592$	−18.0478	−0.741761
$593$	−11.4692	−0.470982	−0.235491	−	0.971877i	$-0.575670\pi$
$593$	−11.4692	−0.470982	−0.235491	+	0.971877i	$0.575670\pi$
$594$	−44.1829	−1.81285
$595$	12.9718	0.531790
$596$	−41.8273	−1.71331
$597$	−21.8190	−0.892994
$598$	4.03119	0.164848
$599$	−13.9254	−0.568975	−0.284488	−	0.958680i	$-0.591823\pi$
$599$	−13.9254	−0.568975	−0.284488	+	0.958680i	$0.591823\pi$
$600$	10.2708	0.419303
$601$	−8.99325	−0.366842	−0.183421	−	0.983034i	$-0.558717\pi$
$601$	−8.99325	−0.366842	−0.183421	+	0.983034i	$0.558717\pi$
$602$	−8.31205	−0.338774
$603$	−8.27330	−0.336915
$604$	75.7371	3.08170
$605$	5.55351	0.225782
$606$	−0.849906	−0.0345251
$607$	41.4986	1.68437	0.842187	−	0.539185i	$-0.181268\pi$
$607$	41.4986	1.68437	0.842187	+	0.539185i	$0.181268\pi$
$608$	−18.9952	−0.770358
$609$	0.368597	0.0149363
$610$	−20.6760	−0.837145
$611$	−17.7716	−0.718962
$612$	22.8401	0.923255
$613$	−6.44765	−0.260418	−0.130209	−	0.991487i	$-0.541565\pi$
$613$	−6.44765	−0.260418	−0.130209	+	0.991487i	$0.541565\pi$
$614$	22.7911	0.919774
$615$	−25.2776	−1.01929
$616$	−27.8200	−1.12090
$617$	38.4459	1.54777	0.773887	−	0.633324i	$-0.218310\pi$
$617$	38.4459	1.54777	0.773887	+	0.633324i	$0.218310\pi$
$618$	46.8046	1.88276
$619$	17.8117	0.715914	0.357957	−	0.933738i	$-0.383473\pi$
$619$	17.8117	0.715914	0.357957	+	0.933738i	$0.383473\pi$
$620$	42.3196	1.69960
$621$	−1.97830	−0.0793863
$622$	4.36036	0.174834
$623$	−16.1969	−0.648915
$624$	−67.3921	−2.69784
$625$	−28.7529	−1.15012
$626$	28.8216	1.15194
$627$	−5.98103	−0.238859
$628$	−17.7236	−0.707250
$629$	7.37258	0.293964
$630$	−7.78475	−0.310152
$631$	−0.565117	−0.0224970	−0.0112485	−	0.999937i	$-0.503581\pi$
$631$	−0.565117	−0.0224970	−0.0112485	+	0.999937i	$0.503581\pi$
$632$	9.41266	0.374416
$633$	12.0347	0.478337
$634$	17.6245	0.699958
$635$	27.7537	1.10137
$636$	−72.1004	−2.85897
$637$	−24.3517	−0.964851
$638$	1.72157	0.0681576
$639$	13.6551	0.540189
$640$	19.1062	0.755238
$641$	−14.2094	−0.561236	−0.280618	−	0.959819i	$-0.590539\pi$
$641$	−14.2094	−0.561236	−0.280618	+	0.959819i	$0.590539\pi$
$642$	−33.1564	−1.30858
$643$	0.205041	0.00808603	0.00404302	−	0.999992i	$-0.498713\pi$
$643$	0.205041	0.00808603	0.00404302	+	0.999992i	$0.498713\pi$
$644$	−2.07654	−0.0818269
$645$	9.05975	0.356727
$646$	17.0846	0.672183
$647$	−28.2123	−1.10914	−0.554570	−	0.832137i	$-0.687117\pi$
$647$	−28.2123	−1.10914	−0.554570	+	0.832137i	$0.687117\pi$
$648$	38.9372	1.52960
$649$	14.6643	0.575626
$650$	−10.6021	−0.415850
$651$	5.81946	0.228082
$652$	53.5225	2.09610
$653$	24.0668	0.941808	0.470904	−	0.882184i	$-0.343928\pi$
$653$	24.0668	0.941808	0.470904	+	0.882184i	$0.343928\pi$
$654$	5.61207	0.219449
$655$	3.46449	0.135369
$656$	−81.0799	−3.16564
$657$	−8.95778	−0.349476
$658$	12.8174	0.499676
$659$	32.4898	1.26562	0.632812	−	0.774305i	$-0.281901\pi$
$659$	32.4898	1.26562	0.632812	+	0.774305i	$0.281901\pi$
$660$	50.5487	1.96761
$661$	36.8003	1.43137	0.715683	−	0.698426i	$-0.246116\pi$
$661$	36.8003	1.43137	0.715683	+	0.698426i	$0.246116\pi$
$662$	16.4351	0.638767
$663$	27.5298	1.06917
$664$	−138.197	−5.36307
$665$	−4.15893	−0.161276
$666$	−4.42451	−0.171446
$667$	0.0770836	0.00298469
$668$	40.0764	1.55060
$669$	−33.5930	−1.29878
$670$	52.3021	2.02061
$671$	−9.48442	−0.366142
$672$	22.0759	0.851598
$673$	20.1739	0.777646	0.388823	−	0.921312i	$-0.372882\pi$
$673$	20.1739	0.777646	0.388823	+	0.921312i	$0.372882\pi$
$674$	−14.5620	−0.560907
$675$	5.20297	0.200263
$676$	29.9154	1.15059
$677$	2.39552	0.0920674	0.0460337	−	0.998940i	$-0.485342\pi$
$677$	2.39552	0.0920674	0.0460337	+	0.998940i	$0.485342\pi$
$678$	31.3584	1.20431
$679$	−7.06344	−0.271070
$680$	−86.6149	−3.32153
$681$	−6.75968	−0.259031
$682$	27.1804	1.04079
$683$	−27.1680	−1.03955	−0.519777	−	0.854302i	$-0.673985\pi$
$683$	−27.1680	−1.03955	−0.519777	+	0.854302i	$0.673985\pi$
$684$	−7.32286	−0.279996
$685$	25.9059	0.989815
$686$	39.5611	1.51045
$687$	−3.13167	−0.119481
$688$	29.0599	1.10790
$689$	44.6460	1.70088
$690$	3.16896	0.120640
$691$	17.8549	0.679231	0.339616	−	0.940564i	$-0.389703\pi$
$691$	17.8549	0.679231	0.339616	+	0.940564i	$0.389703\pi$
$692$	−112.711	−4.28463
$693$	−3.57100	−0.135651
$694$	−43.9730	−1.66919
$695$	−38.1134	−1.44572
$696$	−2.46119	−0.0932912
$697$	33.1214	1.25456
$698$	44.3520	1.67875
$699$	−25.5063	−0.964738
$700$	5.46134	0.206419
$701$	−22.8911	−0.864584	−0.432292	−	0.901734i	$-0.642295\pi$
$701$	−22.8911	−0.864584	−0.432292	+	0.901734i	$0.642295\pi$
$702$	−65.2026	−2.46091
$703$	−2.36376	−0.0891508
$704$	38.2308	1.44088
$705$	−13.9704	−0.526156
$706$	38.1436	1.43555
$707$	−0.271095	−0.0101956
$708$	−34.9485	−1.31344
$709$	−27.7316	−1.04148	−0.520742	−	0.853714i	$-0.674344\pi$
$709$	−27.7316	−1.04148	−0.520742	+	0.853714i	$0.674344\pi$
$710$	−86.3250	−3.23972
$711$	1.20822	0.0453117
$712$	108.150	4.05309
$713$	1.21701	0.0455772
$714$	−19.8554	−0.743070
$715$	−31.3008	−1.17058
$716$	80.2493	2.99906
$717$	−20.7937	−0.776556
$718$	87.8866	3.27990
$719$	−26.7917	−0.999161	−0.499581	−	0.866267i	$-0.666513\pi$
$719$	−26.7917	−0.999161	−0.499581	+	0.866267i	$0.666513\pi$
$720$	27.2164	1.01429
$721$	14.9293	0.555996
$722$	44.7857	1.66675
$723$	9.26304	0.344496
$724$	50.3114	1.86981
$725$	−0.202732	−0.00752927
$726$	−8.50057	−0.315486
$727$	−20.9994	−0.778824	−0.389412	−	0.921064i	$-0.627322\pi$
$727$	−20.9994	−0.778824	−0.389412	+	0.921064i	$0.627322\pi$
$728$	−41.0552	−1.52161
$729$	28.8882	1.06993
$730$	56.6292	2.09594
$731$	−11.8710	−0.439066
$732$	22.6035	0.835451
$733$	−22.4206	−0.828126	−0.414063	−	0.910248i	$-0.635891\pi$
$733$	−22.4206	−0.828126	−0.414063	+	0.910248i	$0.635891\pi$
$734$	−11.2519	−0.415314
$735$	−19.1431	−0.706105
$736$	4.61668	0.170173
$737$	23.9919	0.883751
$738$	−19.8771	−0.731688
$739$	−4.91573	−0.180828	−0.0904140	−	0.995904i	$-0.528819\pi$
$739$	−4.91573	−0.180828	−0.0904140	+	0.995904i	$0.528819\pi$
$740$	19.9773	0.734380
$741$	−8.82646	−0.324248
$742$	−32.2001	−1.18210
$743$	−1.01477	−0.0372284	−0.0186142	−	0.999827i	$-0.505925\pi$
$743$	−1.01477	−0.0372284	−0.0186142	+	0.999827i	$0.505925\pi$
$744$	−38.8576	−1.42459
$745$	20.3604	0.745949
$746$	−8.17058	−0.299146
$747$	−17.7390	−0.649038
$748$	−66.2343	−2.42176
$749$	−10.5759	−0.386436
$750$	36.9716	1.35001
$751$	−53.3845	−1.94803	−0.974014	−	0.226489i	$-0.927275\pi$
$751$	−53.3845	−1.94803	−0.974014	+	0.226489i	$0.927275\pi$
$752$	−44.8113	−1.63410
$753$	−5.06960	−0.184746
$754$	2.54060	0.0925230
$755$	−36.8669	−1.34172
$756$	33.5870	1.22155
$757$	−35.7934	−1.30093	−0.650467	−	0.759535i	$-0.725427\pi$
$757$	−35.7934	−1.30093	−0.650467	+	0.759535i	$0.725427\pi$
$758$	12.3451	0.448394
$759$	1.45365	0.0527643
$760$	27.7700	1.00732
$761$	−30.2504	−1.09658	−0.548288	−	0.836290i	$-0.684720\pi$
$761$	−30.2504	−1.09658	−0.548288	+	0.836290i	$0.684720\pi$
$762$	−42.4815	−1.53894
$763$	1.79009	0.0648055
$764$	−19.5029	−0.705589
$765$	−11.1180	−0.401971
$766$	56.3463	2.03588
$767$	21.6408	0.781404
$768$	7.21205	0.260242
$769$	40.6756	1.46680	0.733400	−	0.679797i	$-0.237932\pi$
$769$	40.6756	1.46680	0.733400	+	0.679797i	$0.237932\pi$
$770$	22.5751	0.813551
$771$	−25.3372	−0.912497
$772$	−132.950	−4.78499
$773$	−27.5166	−0.989704	−0.494852	−	0.868977i	$-0.664778\pi$
$773$	−27.5166	−0.989704	−0.494852	+	0.868977i	$0.664778\pi$
$774$	7.12418	0.256073
$775$	−3.20076	−0.114975
$776$	47.1639	1.69309
$777$	2.74712	0.0985524
$778$	−13.0999	−0.469653
$779$	−10.6192	−0.380472
$780$	74.5969	2.67100
$781$	−39.5987	−1.41695
$782$	−4.15230	−0.148486
$783$	−1.24679	−0.0445567
$784$	−61.4032	−2.19297
$785$	8.62741	0.307926
$786$	−5.30297	−0.189151
$787$	8.59548	0.306396	0.153198	−	0.988196i	$-0.451043\pi$
$787$	8.59548	0.306396	0.153198	+	0.988196i	$0.451043\pi$
$788$	120.607	4.29644
$789$	7.45902	0.265548
$790$	−7.63810	−0.271751
$791$	10.0024	0.355644
$792$	23.8442	0.847268
$793$	−13.9966	−0.497032
$794$	18.6521	0.661938
$795$	35.0966	1.24475
$796$	77.4685	2.74580
$797$	−46.4441	−1.64513	−0.822567	−	0.568669i	$-0.807459\pi$
$797$	−46.4441	−1.64513	−0.822567	+	0.568669i	$0.807459\pi$
$798$	6.36593	0.225351
$799$	18.3055	0.647603
$800$	−12.1420	−0.429284
$801$	13.8822	0.490504
$802$	−10.1606	−0.358782
$803$	25.9768	0.916701
$804$	−57.1781	−2.01652
$805$	1.01080	0.0356262
$806$	40.1113	1.41286
$807$	−4.52755	−0.159377
$808$	1.81015	0.0636810
$809$	1.80706	0.0635327	0.0317664	−	0.999495i	$-0.489887\pi$
$809$	1.80706	0.0635327	0.0317664	+	0.999495i	$0.489887\pi$
$810$	−31.5964	−1.11019
$811$	16.3774	0.575089	0.287545	−	0.957767i	$-0.407161\pi$
$811$	16.3774	0.575089	0.287545	+	0.957767i	$0.407161\pi$
$812$	−1.30870	−0.0459265
$813$	−0.561316	−0.0196862
$814$	12.8307	0.449716
$815$	−26.0534	−0.912610
$816$	69.4168	2.43007
$817$	3.80603	0.133156
$818$	86.3386	3.01876
$819$	−5.26987	−0.184144
$820$	89.7481	3.13414
$821$	33.6089	1.17296	0.586480	−	0.809964i	$-0.300513\pi$
$821$	33.6089	1.17296	0.586480	+	0.809964i	$0.300513\pi$
$822$	−39.6533	−1.38307
$823$	−16.7520	−0.583938	−0.291969	−	0.956428i	$-0.594310\pi$
$823$	−16.7520	−0.583938	−0.291969	+	0.956428i	$0.594310\pi$
$824$	−99.6857	−3.47272
$825$	−3.82315	−0.133105
$826$	−15.6080	−0.543073
$827$	−29.4959	−1.02567	−0.512837	−	0.858486i	$-0.671405\pi$
$827$	−29.4959	−1.02567	−0.512837	+	0.858486i	$0.671405\pi$
$828$	1.77978	0.0618515
$829$	55.4206	1.92484	0.962418	−	0.271571i	$-0.0875432\pi$
$829$	55.4206	1.92484	0.962418	+	0.271571i	$0.0875432\pi$
$830$	112.143	3.89253
$831$	18.5071	0.642005
$832$	56.4188	1.95597
$833$	25.0834	0.869087
$834$	58.3389	2.02011
$835$	−19.5082	−0.675107
$836$	21.2357	0.734451
$837$	−19.6845	−0.680396
$838$	34.6083	1.19552
$839$	−32.7433	−1.13042	−0.565212	−	0.824946i	$-0.691205\pi$
$839$	−32.7433	−1.13042	−0.565212	+	0.824946i	$0.691205\pi$
$840$	−32.2738	−1.11355
$841$	−28.9514	−0.998325
$842$	22.6052	0.779027
$843$	14.3824	0.495355
$844$	−42.7293	−1.47080
$845$	−14.5621	−0.500950
$846$	−10.9857	−0.377696
$847$	−2.71143	−0.0931659
$848$	112.575	3.86585
$849$	42.2371	1.44957
$850$	10.9207	0.374576
$851$	0.574497	0.0196935
$852$	94.3728	3.23316
$853$	−36.9916	−1.26657	−0.633284	−	0.773920i	$-0.718293\pi$
$853$	−36.9916	−1.26657	−0.633284	+	0.773920i	$0.718293\pi$
$854$	10.0948	0.345436
$855$	3.56458	0.121906
$856$	70.6175	2.41366
$857$	6.46996	0.221010	0.110505	−	0.993876i	$-0.464753\pi$
$857$	6.46996	0.221010	0.110505	+	0.993876i	$0.464753\pi$
$858$	47.9109	1.63565
$859$	−4.91978	−0.167861	−0.0839303	−	0.996472i	$-0.526747\pi$
$859$	−4.91978	−0.167861	−0.0839303	+	0.996472i	$0.526747\pi$
$860$	−32.1666	−1.09687
$861$	12.3414	0.420595
$862$	51.3373	1.74855
$863$	2.96312	0.100866	0.0504330	−	0.998727i	$-0.483940\pi$
$863$	2.96312	0.100866	0.0504330	+	0.998727i	$0.483940\pi$
$864$	−74.6726	−2.54041
$865$	54.8649	1.86546
$866$	14.7162	0.500076
$867$	−4.42463	−0.150268
$868$	−20.6620	−0.701314
$869$	−3.50373	−0.118856
$870$	1.99719	0.0677109
$871$	35.4058	1.19968
$872$	−11.9527	−0.404771
$873$	6.05400	0.204897
$874$	1.33129	0.0450315
$875$	11.7929	0.398672
$876$	−61.9086	−2.09170
$877$	39.9622	1.34943	0.674713	−	0.738080i	$-0.264267\pi$
$877$	39.9622	1.34943	0.674713	+	0.738080i	$0.264267\pi$
$878$	30.3817	1.02533
$879$	22.5664	0.761145
$880$	−78.9252	−2.66057
$881$	−16.2290	−0.546769	−0.273384	−	0.961905i	$-0.588143\pi$
$881$	−16.2290	−0.546769	−0.273384	+	0.961905i	$0.588143\pi$
$882$	−15.0533	−0.506871
$883$	−13.6522	−0.459433	−0.229717	−	0.973258i	$-0.573780\pi$
$883$	−13.6522	−0.459433	−0.229717	+	0.973258i	$0.573780\pi$
$884$	−97.7447	−3.28751
$885$	17.0120	0.571853
$886$	57.5785	1.93439
$887$	15.8266	0.531407	0.265703	−	0.964055i	$-0.414396\pi$
$887$	15.8266	0.531407	0.265703	+	0.964055i	$0.414396\pi$
$888$	−18.3430	−0.615552
$889$	−13.5504	−0.454465
$890$	−87.7605	−2.94174
$891$	−14.4938	−0.485561
$892$	119.272	3.99352
$893$	−5.86901	−0.196399
$894$	−31.1650	−1.04231
$895$	−39.0633	−1.30574
$896$	−9.32834	−0.311638
$897$	2.14522	0.0716268
$898$	−30.1880	−1.00739
$899$	0.766999	0.0255808
$900$	−4.68086	−0.156029
$901$	−45.9873	−1.53206
$902$	57.6420	1.91927
$903$	−4.42330	−0.147198
$904$	−66.7879	−2.22133
$905$	−24.4903	−0.814085
$906$	56.4308	1.87479
$907$	32.7037	1.08591	0.542954	−	0.839763i	$-0.317306\pi$
$907$	32.7037	1.08591	0.542954	+	0.839763i	$0.317306\pi$
$908$	24.0002	0.796476
$909$	0.232353	0.00770666
$910$	33.3151	1.10438
$911$	−27.0399	−0.895871	−0.447936	−	0.894066i	$-0.647841\pi$
$911$	−27.0399	−0.895871	−0.447936	+	0.894066i	$0.647841\pi$
$912$	−22.2560	−0.736970
$913$	51.4417	1.70247
$914$	27.8491	0.921168
$915$	−11.0028	−0.363742
$916$	11.1190	0.367382
$917$	−1.69149	−0.0558579
$918$	67.1616	2.21666
$919$	4.76147	0.157066	0.0785331	−	0.996912i	$-0.474976\pi$
$919$	4.76147	0.157066	0.0785331	+	0.996912i	$0.474976\pi$
$920$	−6.74933	−0.222519
$921$	12.1284	0.399645
$922$	75.8490	2.49795
$923$	−58.4375	−1.92350
$924$	−24.6797	−0.811904
$925$	−1.51094	−0.0496795
$926$	69.5511	2.28559
$927$	−12.7957	−0.420267
$928$	2.90959	0.0955119
$929$	−20.8870	−0.685279	−0.342639	−	0.939467i	$-0.611321\pi$
$929$	−20.8870	−0.685279	−0.342639	+	0.939467i	$0.611321\pi$
$930$	31.5318	1.03397
$931$	−8.04208	−0.263569
$932$	90.5603	2.96640
$933$	2.32039	0.0759660
$934$	46.1190	1.50906
$935$	32.2412	1.05440
$936$	35.1880	1.15015
$937$	−18.5968	−0.607532	−0.303766	−	0.952747i	$-0.598244\pi$
$937$	−18.5968	−0.607532	−0.303766	+	0.952747i	$0.598244\pi$
$938$	−25.5358	−0.833774
$939$	15.3376	0.500523
$940$	49.6020	1.61784
$941$	−14.8466	−0.483987	−0.241993	−	0.970278i	$-0.577801\pi$
$941$	−14.8466	−0.483987	−0.241993	+	0.970278i	$0.577801\pi$
$942$	−13.2057	−0.430264
$943$	2.58093	0.0840467
$944$	54.5675	1.77602
$945$	−16.3493	−0.531842
$946$	−20.6595	−0.671698
$947$	25.3658	0.824279	0.412139	−	0.911121i	$-0.364782\pi$
$947$	25.3658	0.824279	0.412139	+	0.911121i	$0.364782\pi$
$948$	8.35018	0.271201
$949$	38.3351	1.24441
$950$	−3.50132	−0.113598
$951$	9.37896	0.304134
$952$	42.2886	1.37058
$953$	−22.8145	−0.739034	−0.369517	−	0.929224i	$-0.620477\pi$
$953$	−22.8145	−0.739034	−0.369517	+	0.929224i	$0.620477\pi$
$954$	27.5984	0.893531
$955$	9.49349	0.307202
$956$	73.8282	2.38777
$957$	0.916143	0.0296147
$958$	−38.0673	−1.22990
$959$	−12.6482	−0.408433
$960$	44.3513	1.43143
$961$	−18.8905	−0.609371
$962$	18.9348	0.610484
$963$	9.06452	0.292100
$964$	−32.8884	−1.05926
$965$	64.7169	2.08331
$966$	−1.54720	−0.0497804
$967$	−58.1942	−1.87140	−0.935700	−	0.352797i	$-0.885231\pi$
$967$	−58.1942	−1.87140	−0.935700	+	0.352797i	$0.885231\pi$
$968$	18.1047	0.581908
$969$	9.09164	0.292066
$970$	−38.2721	−1.22884
$971$	−53.3732	−1.71283	−0.856414	−	0.516290i	$-0.827313\pi$
$971$	−53.3732	−1.71283	−0.856414	+	0.516290i	$0.827313\pi$
$972$	−50.2798	−1.61273
$973$	18.6084	0.596557
$974$	55.8180	1.78852
$975$	−5.64199	−0.180688
$976$	−35.2925	−1.12968
$977$	32.8019	1.04943	0.524713	−	0.851279i	$-0.324173\pi$
$977$	32.8019	1.04943	0.524713	+	0.851279i	$0.324173\pi$
$978$	39.8790	1.27519
$979$	−40.2572	−1.28663
$980$	67.9677	2.17115
$981$	−1.53426	−0.0489853
$982$	46.7439	1.49166
$983$	44.3271	1.41382	0.706908	−	0.707306i	$-0.250090\pi$
$983$	44.3271	1.41382	0.706908	+	0.707306i	$0.250090\pi$
$984$	−82.4062	−2.62701
$985$	−58.7083	−1.87060
$986$	−2.61692	−0.0833399
$987$	6.82087	0.217111
$988$	31.3384	0.997006
$989$	−0.925033	−0.0294143
$990$	−19.3489	−0.614949
$991$	28.5674	0.907474	0.453737	−	0.891136i	$-0.350091\pi$
$991$	28.5674	0.907474	0.453737	+	0.891136i	$0.350091\pi$
$992$	45.9370	1.45850
$993$	8.74601	0.277546
$994$	42.1470	1.33682
$995$	−37.7097	−1.19548
$996$	−122.597	−3.88465
$997$	31.3651	0.993344	0.496672	−	0.867938i	$-0.334555\pi$
$997$	31.3651	0.993344	0.496672	+	0.867938i	$0.334555\pi$
$998$	17.7920	0.563197
$999$	−9.29222	−0.293993

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4021.2.a.b.1.7	✓	151

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4021.2.a.b.1.7	✓	151	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:	\( N \)	\(=\)	\( 4021 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4021.a (trivial)

\(f(q)\)	\(=\)	\(q-2.64544 q^{2} -1.40778 q^{3} +4.99834 q^{4} -2.43306 q^{5} +3.72420 q^{6} +1.18791 q^{7} -7.93191 q^{8} -1.01815 q^{9} +O(q^{10})\)	\(q-2.64544 q^{2} -1.40778 q^{3} +4.99834 q^{4} -2.43306 q^{5} +3.72420 q^{6} +1.18791 q^{7} -7.93191 q^{8} -1.01815 q^{9} +6.43651 q^{10} +2.95254 q^{11} -7.03657 q^{12} +4.35719 q^{13} -3.14254 q^{14} +3.42522 q^{15} +10.9867 q^{16} -4.48809 q^{17} +2.69344 q^{18} +1.43895 q^{19} -12.1613 q^{20} -1.67232 q^{21} -7.81075 q^{22} -0.349728 q^{23} +11.1664 q^{24} +0.919793 q^{25} -11.5267 q^{26} +5.65668 q^{27} +5.93758 q^{28} -0.220410 q^{29} -9.06121 q^{30} -3.47987 q^{31} -13.2008 q^{32} -4.15653 q^{33} +11.8730 q^{34} -2.89026 q^{35} -5.08904 q^{36} -1.64270 q^{37} -3.80664 q^{38} -6.13397 q^{39} +19.2988 q^{40} -7.37983 q^{41} +4.42402 q^{42} +2.64501 q^{43} +14.7578 q^{44} +2.47721 q^{45} +0.925182 q^{46} -4.07869 q^{47} -15.4669 q^{48} -5.58887 q^{49} -2.43325 q^{50} +6.31826 q^{51} +21.7787 q^{52} +10.2465 q^{53} -14.9644 q^{54} -7.18371 q^{55} -9.42240 q^{56} -2.02573 q^{57} +0.583082 q^{58} +4.96669 q^{59} +17.1204 q^{60} -3.21229 q^{61} +9.20577 q^{62} -1.20947 q^{63} +12.9484 q^{64} -10.6013 q^{65} +10.9958 q^{66} +8.12584 q^{67} -22.4330 q^{68} +0.492341 q^{69} +7.64600 q^{70} -13.4118 q^{71} +8.07584 q^{72} +8.79812 q^{73} +4.34566 q^{74} -1.29487 q^{75} +7.19234 q^{76} +3.50735 q^{77} +16.2270 q^{78} -1.18668 q^{79} -26.7313 q^{80} -4.90894 q^{81} +19.5229 q^{82} +17.4229 q^{83} -8.35882 q^{84} +10.9198 q^{85} -6.99720 q^{86} +0.310290 q^{87} -23.4193 q^{88} -13.6348 q^{89} -6.55331 q^{90} +5.17595 q^{91} -1.74806 q^{92} +4.89890 q^{93} +10.7899 q^{94} -3.50105 q^{95} +18.5838 q^{96} -5.94610 q^{97} +14.7850 q^{98} -3.00612 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 151 q - 16 q^{2} - 29 q^{3} + 120 q^{4} - 27 q^{5} - 18 q^{6} - 18 q^{7} - 48 q^{8} + 98 q^{9}+O(q^{10}) \) 151 * q - 16 * q^2 - 29 * q^3 + 120 * q^4 - 27 * q^5 - 18 * q^6 - 18 * q^7 - 48 * q^8 + 98 * q^9	\( 151 q - 16 q^{2} - 29 q^{3} + 120 q^{4} - 27 q^{5} - 18 q^{6} - 18 q^{7} - 48 q^{8} + 98 q^{9} - 7 q^{10} - 126 q^{11} - 47 q^{12} - 10 q^{13} - 58 q^{14} - 50 q^{15} + 74 q^{16} - 39 q^{17} - 47 q^{18} - 48 q^{19} - 64 q^{20} - 50 q^{21} - 7 q^{22} - 91 q^{23} - 44 q^{24} + 102 q^{25} - 94 q^{26} - 92 q^{27} - 27 q^{28} - 79 q^{29} - 12 q^{30} - 33 q^{31} - 102 q^{32} - 7 q^{33} - 17 q^{34} - 183 q^{35} - 6 q^{36} - 24 q^{37} - 77 q^{38} - 81 q^{39} - 37 q^{40} - 62 q^{41} + 4 q^{42} - 74 q^{43} - 209 q^{44} - 35 q^{45} - 37 q^{46} - 127 q^{47} - 59 q^{48} + 75 q^{49} - 79 q^{50} - 138 q^{51} - 12 q^{52} - 104 q^{53} - 43 q^{54} - 49 q^{55} - 167 q^{56} - 19 q^{57} - 32 q^{58} - 207 q^{59} - 94 q^{60} - 57 q^{61} - 69 q^{62} - 49 q^{63} - 4 q^{64} - 78 q^{65} - 51 q^{66} - 92 q^{67} - 80 q^{68} - 84 q^{69} + 5 q^{70} - 217 q^{71} - 79 q^{72} + 2 q^{73} - 113 q^{74} - 173 q^{75} - 70 q^{76} - 85 q^{77} - 46 q^{78} - 70 q^{79} - 95 q^{80} - 49 q^{81} - 12 q^{82} - 109 q^{83} - 114 q^{84} - 25 q^{85} - 136 q^{86} - 60 q^{87} - 2 q^{88} - 144 q^{89} - 55 q^{90} - 62 q^{91} - 175 q^{92} - 42 q^{93} - 15 q^{94} - 189 q^{95} - 64 q^{96} - 24 q^{97} - 37 q^{98} - 293 q^{99}+O(q^{100}) \) 151 * q - 16 * q^2 - 29 * q^3 + 120 * q^4 - 27 * q^5 - 18 * q^6 - 18 * q^7 - 48 * q^8 + 98 * q^9 - 7 * q^10 - 126 * q^11 - 47 * q^12 - 10 * q^13 - 58 * q^14 - 50 * q^15 + 74 * q^16 - 39 * q^17 - 47 * q^18 - 48 * q^19 - 64 * q^20 - 50 * q^21 - 7 * q^22 - 91 * q^23 - 44 * q^24 + 102 * q^25 - 94 * q^26 - 92 * q^27 - 27 * q^28 - 79 * q^29 - 12 * q^30 - 33 * q^31 - 102 * q^32 - 7 * q^33 - 17 * q^34 - 183 * q^35 - 6 * q^36 - 24 * q^37 - 77 * q^38 - 81 * q^39 - 37 * q^40 - 62 * q^41 + 4 * q^42 - 74 * q^43 - 209 * q^44 - 35 * q^45 - 37 * q^46 - 127 * q^47 - 59 * q^48 + 75 * q^49 - 79 * q^50 - 138 * q^51 - 12 * q^52 - 104 * q^53 - 43 * q^54 - 49 * q^55 - 167 * q^56 - 19 * q^57 - 32 * q^58 - 207 * q^59 - 94 * q^60 - 57 * q^61 - 69 * q^62 - 49 * q^63 - 4 * q^64 - 78 * q^65 - 51 * q^66 - 92 * q^67 - 80 * q^68 - 84 * q^69 + 5 * q^70 - 217 * q^71 - 79 * q^72 + 2 * q^73 - 113 * q^74 - 173 * q^75 - 70 * q^76 - 85 * q^77 - 46 * q^78 - 70 * q^79 - 95 * q^80 - 49 * q^81 - 12 * q^82 - 109 * q^83 - 114 * q^84 - 25 * q^85 - 136 * q^86 - 60 * q^87 - 2 * q^88 - 144 * q^89 - 55 * q^90 - 62 * q^91 - 175 * q^92 - 42 * q^93 - 15 * q^94 - 189 * q^95 - 64 * q^96 - 24 * q^97 - 37 * q^98 - 293 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more