LMFDB - Embedded newform 4027.2.a.b.1.11

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4027,2,Mod(1,4027)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4027, 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("4027.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4027 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4027.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:	$32.1557568940$
Analytic rank:	$1$
Dimension:	$159$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.11
Character	$\chi$	$=$	4027.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.59401 q^{2} -2.55671 q^{3} +4.72888 q^{4} +1.35267 q^{5} +6.63213 q^{6} -3.48116 q^{7} -7.07874 q^{8} +3.53677 q^{9} +O(q^{10})$	\(q-2.59401 q^{2} -2.55671 q^{3} +4.72888 q^{4} +1.35267 q^{5} +6.63213 q^{6} -3.48116 q^{7} -7.07874 q^{8} +3.53677 q^{9} -3.50885 q^{10} +1.80644 q^{11} -12.0904 q^{12} -6.45851 q^{13} +9.03015 q^{14} -3.45840 q^{15} +8.90456 q^{16} +1.32477 q^{17} -9.17441 q^{18} -4.79710 q^{19} +6.39664 q^{20} +8.90031 q^{21} -4.68592 q^{22} +0.714751 q^{23} +18.0983 q^{24} -3.17027 q^{25} +16.7534 q^{26} -1.37236 q^{27} -16.4620 q^{28} +5.17571 q^{29} +8.97111 q^{30} +8.01732 q^{31} -8.94102 q^{32} -4.61854 q^{33} -3.43646 q^{34} -4.70887 q^{35} +16.7250 q^{36} +4.44250 q^{37} +12.4437 q^{38} +16.5125 q^{39} -9.57523 q^{40} -11.7039 q^{41} -23.0875 q^{42} +2.71557 q^{43} +8.54244 q^{44} +4.78409 q^{45} -1.85407 q^{46} +0.881464 q^{47} -22.7664 q^{48} +5.11845 q^{49} +8.22371 q^{50} -3.38705 q^{51} -30.5415 q^{52} +4.32494 q^{53} +3.55991 q^{54} +2.44352 q^{55} +24.6422 q^{56} +12.2648 q^{57} -13.4258 q^{58} +5.14368 q^{59} -16.3543 q^{60} +5.68491 q^{61} -20.7970 q^{62} -12.3120 q^{63} +5.38396 q^{64} -8.73626 q^{65} +11.9805 q^{66} +9.07780 q^{67} +6.26467 q^{68} -1.82741 q^{69} +12.2149 q^{70} -0.995263 q^{71} -25.0359 q^{72} -13.8183 q^{73} -11.5239 q^{74} +8.10547 q^{75} -22.6849 q^{76} -6.28850 q^{77} -42.8336 q^{78} +2.05191 q^{79} +12.0450 q^{80} -7.10158 q^{81} +30.3600 q^{82} -1.66483 q^{83} +42.0885 q^{84} +1.79198 q^{85} -7.04421 q^{86} -13.2328 q^{87} -12.7873 q^{88} -6.79529 q^{89} -12.4100 q^{90} +22.4831 q^{91} +3.37997 q^{92} -20.4980 q^{93} -2.28653 q^{94} -6.48892 q^{95} +22.8596 q^{96} -3.22502 q^{97} -13.2773 q^{98} +6.38896 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 159 q - 22 q^{2} - 19 q^{3} + 148 q^{4} - 70 q^{5} - 23 q^{6} - 19 q^{7} - 66 q^{8} + 126 q^{9}+O(q^{10}) $ 159 * q - 22 * q^2 - 19 * q^3 + 148 * q^4 - 70 * q^5 - 23 * q^6 - 19 * q^7 - 66 * q^8 + 126 * q^9	\( 159 q - 22 q^{2} - 19 q^{3} + 148 q^{4} - 70 q^{5} - 23 q^{6} - 19 q^{7} - 66 q^{8} + 126 q^{9} - 23 q^{10} - 33 q^{11} - 57 q^{12} - 90 q^{13} - 28 q^{14} - 22 q^{15} + 130 q^{16} - 145 q^{17} - 50 q^{18} - 28 q^{19} - 121 q^{20} - 69 q^{21} - 26 q^{22} - 79 q^{23} - 62 q^{24} + 123 q^{25} - 40 q^{26} - 70 q^{27} - 43 q^{28} - 109 q^{29} - 43 q^{30} - 21 q^{31} - 139 q^{32} - 83 q^{33} - 93 q^{35} + 75 q^{36} - 65 q^{37} - 122 q^{38} - 18 q^{39} - 43 q^{40} - 71 q^{41} - 88 q^{42} - 72 q^{43} - 79 q^{44} - 181 q^{45} - 11 q^{46} - 114 q^{47} - 118 q^{48} + 118 q^{49} - 77 q^{50} - 29 q^{51} - 169 q^{52} - 220 q^{53} - 80 q^{54} - 37 q^{55} - 72 q^{56} - 90 q^{57} - 8 q^{58} - 60 q^{59} - 42 q^{60} - 108 q^{61} - 152 q^{62} - 65 q^{63} + 114 q^{64} - 81 q^{65} - 40 q^{66} - 50 q^{67} - 319 q^{68} - 103 q^{69} + 4 q^{70} - 7 q^{71} - 129 q^{72} - 94 q^{73} - 79 q^{74} - 59 q^{75} - 46 q^{76} - 329 q^{77} + 8 q^{78} - 18 q^{79} - 190 q^{80} + 59 q^{81} - 56 q^{82} - 201 q^{83} - 71 q^{84} - 26 q^{85} - 52 q^{86} - 126 q^{87} - 66 q^{88} - 114 q^{89} - 33 q^{90} - 30 q^{91} - 204 q^{92} - 125 q^{93} + 9 q^{94} - 84 q^{95} - 88 q^{96} - 56 q^{97} - 110 q^{98} - 46 q^{99}+O(q^{100}) \) 159 * q - 22 * q^2 - 19 * q^3 + 148 * q^4 - 70 * q^5 - 23 * q^6 - 19 * q^7 - 66 * q^8 + 126 * q^9 - 23 * q^10 - 33 * q^11 - 57 * q^12 - 90 * q^13 - 28 * q^14 - 22 * q^15 + 130 * q^16 - 145 * q^17 - 50 * q^18 - 28 * q^19 - 121 * q^20 - 69 * q^21 - 26 * q^22 - 79 * q^23 - 62 * q^24 + 123 * q^25 - 40 * q^26 - 70 * q^27 - 43 * q^28 - 109 * q^29 - 43 * q^30 - 21 * q^31 - 139 * q^32 - 83 * q^33 - 93 * q^35 + 75 * q^36 - 65 * q^37 - 122 * q^38 - 18 * q^39 - 43 * q^40 - 71 * q^41 - 88 * q^42 - 72 * q^43 - 79 * q^44 - 181 * q^45 - 11 * q^46 - 114 * q^47 - 118 * q^48 + 118 * q^49 - 77 * q^50 - 29 * q^51 - 169 * q^52 - 220 * q^53 - 80 * q^54 - 37 * q^55 - 72 * q^56 - 90 * q^57 - 8 * q^58 - 60 * q^59 - 42 * q^60 - 108 * q^61 - 152 * q^62 - 65 * q^63 + 114 * q^64 - 81 * q^65 - 40 * q^66 - 50 * q^67 - 319 * q^68 - 103 * q^69 + 4 * q^70 - 7 * q^71 - 129 * q^72 - 94 * q^73 - 79 * q^74 - 59 * q^75 - 46 * q^76 - 329 * q^77 + 8 * q^78 - 18 * q^79 - 190 * q^80 + 59 * q^81 - 56 * q^82 - 201 * q^83 - 71 * q^84 - 26 * q^85 - 52 * q^86 - 126 * q^87 - 66 * q^88 - 114 * q^89 - 33 * q^90 - 30 * q^91 - 204 * q^92 - 125 * q^93 + 9 * q^94 - 84 * q^95 - 88 * q^96 - 56 * q^97 - 110 * q^98 - 46 * 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 $

$2$	−2.59401	−1.83424	−0.917121	−	0.398610i	$-0.869493\pi$
$2$	−2.59401	−1.83424	−0.917121	+	0.398610i	$0.869493\pi$
$3$	−2.55671	−1.47612	−0.738059	−	0.674737i	$-0.764257\pi$
$3$	−2.55671	−1.47612	−0.738059	+	0.674737i	$0.764257\pi$
$4$	4.72888	2.36444
$5$	1.35267	0.604934	0.302467	−	0.953160i	$-0.402190\pi$
$5$	1.35267	0.604934	0.302467	+	0.953160i	$0.402190\pi$
$6$	6.63213	2.70756
$7$	−3.48116	−1.31575	−0.657877	−	0.753126i	$-0.728545\pi$
$7$	−3.48116	−1.31575	−0.657877	+	0.753126i	$0.728545\pi$
$8$	−7.07874	−2.50271
$9$	3.53677	1.17892
$10$	−3.50885	−1.10960
$11$	1.80644	0.544662	0.272331	−	0.962204i	$-0.412205\pi$
$11$	1.80644	0.544662	0.272331	+	0.962204i	$0.412205\pi$
$12$	−12.0904	−3.49019
$13$	−6.45851	−1.79127	−0.895634	−	0.444792i	$-0.853277\pi$
$13$	−6.45851	−1.79127	−0.895634	+	0.444792i	$0.853277\pi$
$14$	9.03015	2.41341
$15$	−3.45840	−0.892954
$16$	8.90456	2.22614
$17$	1.32477	0.321303	0.160652	−	0.987011i	$-0.448640\pi$
$17$	1.32477	0.321303	0.160652	+	0.987011i	$0.448640\pi$
$18$	−9.17441	−2.16243
$19$	−4.79710	−1.10053	−0.550265	−	0.834990i	$-0.685473\pi$
$19$	−4.79710	−1.10053	−0.550265	+	0.834990i	$0.685473\pi$
$20$	6.39664	1.43033
$21$	8.90031	1.94221
$22$	−4.68592	−0.999042
$23$	0.714751	0.149036	0.0745179	−	0.997220i	$-0.476258\pi$
$23$	0.714751	0.149036	0.0745179	+	0.997220i	$0.476258\pi$
$24$	18.0983	3.69430
$25$	−3.17027	−0.634055
$26$	16.7534	3.28562
$27$	−1.37236	−0.264111
$28$	−16.4620	−3.11102
$29$	5.17571	0.961105	0.480553	−	0.876966i	$-0.340436\pi$
$29$	5.17571	0.961105	0.480553	+	0.876966i	$0.340436\pi$
$30$	8.97111	1.63789
$31$	8.01732	1.43995	0.719977	−	0.693998i	$-0.244152\pi$
$31$	8.01732	1.43995	0.719977	+	0.693998i	$0.244152\pi$
$32$	−8.94102	−1.58056
$33$	−4.61854	−0.803985
$34$	−3.43646	−0.589348
$35$	−4.70887	−0.795945
$36$	16.7250	2.78749
$37$	4.44250	0.730343	0.365172	−	0.930940i	$-0.381010\pi$
$37$	4.44250	0.730343	0.365172	+	0.930940i	$0.381010\pi$
$38$	12.4437	2.01864
$39$	16.5125	2.64412
$40$	−9.57523	−1.51398
$41$	−11.7039	−1.82784	−0.913921	−	0.405893i	$-0.866961\pi$
$41$	−11.7039	−1.82784	−0.913921	+	0.405893i	$0.866961\pi$
$42$	−23.0875	−3.56248
$43$	2.71557	0.414120	0.207060	−	0.978328i	$-0.433610\pi$
$43$	2.71557	0.414120	0.207060	+	0.978328i	$0.433610\pi$
$44$	8.54244	1.28782
$45$	4.78409	0.713171
$46$	−1.85407	−0.273368
$47$	0.881464	0.128575	0.0642874	−	0.997931i	$-0.479523\pi$
$47$	0.881464	0.128575	0.0642874	+	0.997931i	$0.479523\pi$
$48$	−22.7664	−3.28604
$49$	5.11845	0.731208
$50$	8.22371	1.16301
$51$	−3.38705	−0.474281
$52$	−30.5415	−4.23535
$53$	4.32494	0.594076	0.297038	−	0.954866i	$-0.404001\pi$
$53$	4.32494	0.594076	0.297038	+	0.954866i	$0.404001\pi$
$54$	3.55991	0.484442
$55$	2.44352	0.329485
$56$	24.6422	3.29295
$57$	12.2648	1.62451
$58$	−13.4258	−1.76290
$59$	5.14368	0.669650	0.334825	−	0.942280i	$-0.391323\pi$
$59$	5.14368	0.669650	0.334825	+	0.942280i	$0.391323\pi$
$60$	−16.3543	−2.11134
$61$	5.68491	0.727878	0.363939	−	0.931423i	$-0.381432\pi$
$61$	5.68491	0.727878	0.363939	+	0.931423i	$0.381432\pi$
$62$	−20.7970	−2.64122
$63$	−12.3120	−1.55117
$64$	5.38396	0.672995
$65$	−8.73626	−1.08360
$66$	11.9805	1.47470
$67$	9.07780	1.10903	0.554515	−	0.832174i	$-0.312904\pi$
$67$	9.07780	1.10903	0.554515	+	0.832174i	$0.312904\pi$
$68$	6.26467	0.759702
$69$	−1.82741	−0.219994
$70$	12.2149	1.45995
$71$	−0.995263	−0.118116	−0.0590580	−	0.998255i	$-0.518810\pi$
$71$	−0.995263	−0.118116	−0.0590580	+	0.998255i	$0.518810\pi$
$72$	−25.0359	−2.95051
$73$	−13.8183	−1.61731	−0.808655	−	0.588284i	$-0.799804\pi$
$73$	−13.8183	−1.61731	−0.808655	+	0.588284i	$0.799804\pi$
$74$	−11.5239	−1.33963
$75$	8.10547	0.935939
$76$	−22.6849	−2.60214
$77$	−6.28850	−0.716641
$78$	−42.8336	−4.84996
$79$	2.05191	0.230858	0.115429	−	0.993316i	$-0.463176\pi$
$79$	2.05191	0.230858	0.115429	+	0.993316i	$0.463176\pi$
$80$	12.0450	1.34667
$81$	−7.10158	−0.789064
$82$	30.3600	3.35270
$83$	−1.66483	−0.182739	−0.0913693	−	0.995817i	$-0.529124\pi$
$83$	−1.66483	−0.182739	−0.0913693	+	0.995817i	$0.529124\pi$
$84$	42.0885	4.59223
$85$	1.79198	0.194367
$86$	−7.04421	−0.759597
$87$	−13.2328	−1.41870
$88$	−12.7873	−1.36313
$89$	−6.79529	−0.720300	−0.360150	−	0.932894i	$-0.617274\pi$
$89$	−6.79529	−0.720300	−0.360150	+	0.932894i	$0.617274\pi$
$90$	−12.4100	−1.30813
$91$	22.4831	2.35687
$92$	3.37997	0.352386
$93$	−20.4980	−2.12554
$94$	−2.28653	−0.235837
$95$	−6.48892	−0.665749
$96$	22.8596	2.33310
$97$	−3.22502	−0.327451	−0.163726	−	0.986506i	$-0.552351\pi$
$97$	−3.22502	−0.327451	−0.163726	+	0.986506i	$0.552351\pi$
$98$	−13.2773	−1.34121
$99$	6.38896	0.642114
$100$	−14.9918	−1.49918
$101$	7.47710	0.744000	0.372000	−	0.928233i	$-0.378672\pi$
$101$	7.47710	0.744000	0.372000	+	0.928233i	$0.378672\pi$
$102$	8.78603	0.869946
$103$	−4.32698	−0.426350	−0.213175	−	0.977014i	$-0.568381\pi$
$103$	−4.32698	−0.426350	−0.213175	+	0.977014i	$0.568381\pi$
$104$	45.7181	4.48303
$105$	12.0392	1.17491
$106$	−11.2189	−1.08968
$107$	14.6675	1.41796	0.708980	−	0.705229i	$-0.249156\pi$
$107$	14.6675	1.41796	0.708980	+	0.705229i	$0.249156\pi$
$108$	−6.48972	−0.624474
$109$	14.5655	1.39512	0.697559	−	0.716527i	$-0.254269\pi$
$109$	14.5655	1.39512	0.697559	+	0.716527i	$0.254269\pi$
$110$	−6.33852	−0.604354
$111$	−11.3582	−1.07807
$112$	−30.9982	−2.92905
$113$	−0.197391	−0.0185689	−0.00928447	−	0.999957i	$-0.502955\pi$
$113$	−0.197391	−0.0185689	−0.00928447	+	0.999957i	$0.502955\pi$
$114$	−31.8150	−2.97975
$115$	0.966825	0.0901569
$116$	24.4753	2.27248
$117$	−22.8422	−2.11177
$118$	−13.3427	−1.22830
$119$	−4.61172	−0.422756
$120$	24.4811	2.23481
$121$	−7.73678	−0.703343
$122$	−14.7467	−1.33510
$123$	29.9235	2.69811
$124$	37.9130	3.40468
$125$	−11.0517	−0.988496
$126$	31.9375	2.84522
$127$	−5.64632	−0.501030	−0.250515	−	0.968113i	$-0.580600\pi$
$127$	−5.64632	−0.501030	−0.250515	+	0.968113i	$0.580600\pi$
$128$	3.91600	0.346129
$129$	−6.94292	−0.611290
$130$	22.6619	1.98758
$131$	−4.02746	−0.351881	−0.175940	−	0.984401i	$-0.556297\pi$
$131$	−4.02746	−0.351881	−0.175940	+	0.984401i	$0.556297\pi$
$132$	−21.8405	−1.90098
$133$	16.6995	1.44803
$134$	−23.5479	−2.03423
$135$	−1.85635	−0.159769
$136$	−9.37768	−0.804130
$137$	−2.19168	−0.187248	−0.0936241	−	0.995608i	$-0.529845\pi$
$137$	−2.19168	−0.187248	−0.0936241	+	0.995608i	$0.529845\pi$
$138$	4.74032	0.403523
$139$	3.21434	0.272636	0.136318	−	0.990665i	$-0.456473\pi$
$139$	3.21434	0.272636	0.136318	+	0.990665i	$0.456473\pi$
$140$	−22.2677	−1.88196
$141$	−2.25365	−0.189791
$142$	2.58172	0.216653
$143$	−11.6669	−0.975635
$144$	31.4933	2.62445
$145$	7.00105	0.581406
$146$	35.8448	2.96653
$147$	−13.0864	−1.07935
$148$	21.0081	1.72685
$149$	20.1574	1.65136	0.825681	−	0.564138i	$-0.190791\pi$
$149$	20.1574	1.65136	0.825681	+	0.564138i	$0.190791\pi$
$150$	−21.0257	−1.71674
$151$	4.18949	0.340936	0.170468	−	0.985363i	$-0.445472\pi$
$151$	4.18949	0.340936	0.170468	+	0.985363i	$0.445472\pi$
$152$	33.9574	2.75431
$153$	4.68539	0.378792
$154$	16.3124	1.31449
$155$	10.8448	0.871077
$156$	78.0858	6.25187
$157$	−4.95612	−0.395541	−0.197771	−	0.980248i	$-0.563370\pi$
$157$	−4.95612	−0.395541	−0.197771	+	0.980248i	$0.563370\pi$
$158$	−5.32266	−0.423448
$159$	−11.0576	−0.876926
$160$	−12.0943	−0.956137
$161$	−2.48816	−0.196094
$162$	18.4216	1.44733
$163$	19.3866	1.51848	0.759238	−	0.650814i	$-0.225572\pi$
$163$	19.3866	1.51848	0.759238	+	0.650814i	$0.225572\pi$
$164$	−55.3463	−4.32182
$165$	−6.24738	−0.486358
$166$	4.31858	0.335187
$167$	9.30842	0.720307	0.360154	−	0.932893i	$-0.382724\pi$
$167$	9.30842	0.720307	0.360154	+	0.932893i	$0.382724\pi$
$168$	−63.0030	−4.86079
$169$	28.7123	2.20864
$170$	−4.64841	−0.356517
$171$	−16.9662	−1.29744
$172$	12.8416	0.979163
$173$	−22.8996	−1.74102	−0.870511	−	0.492149i	$-0.836212\pi$
$173$	−22.8996	−1.74102	−0.870511	+	0.492149i	$0.836212\pi$
$174$	34.3260	2.60225
$175$	11.0362	0.834260
$176$	16.0855	1.21249
$177$	−13.1509	−0.988481
$178$	17.6270	1.32120
$179$	13.1589	0.983539	0.491770	−	0.870725i	$-0.336350\pi$
$179$	13.1589	0.983539	0.491770	+	0.870725i	$0.336350\pi$
$180$	22.6234	1.68625
$181$	−1.68692	−0.125388	−0.0626939	−	0.998033i	$-0.519969\pi$
$181$	−1.68692	−0.125388	−0.0626939	+	0.998033i	$0.519969\pi$
$182$	−58.3213	−4.32306
$183$	−14.5347	−1.07443
$184$	−5.05954	−0.372994
$185$	6.00926	0.441810
$186$	53.1719	3.89875
$187$	2.39311	0.175002
$188$	4.16834	0.304007
$189$	4.77740	0.347504
$190$	16.8323	1.22114
$191$	23.7746	1.72027	0.860133	−	0.510069i	$-0.170380\pi$
$191$	23.7746	1.72027	0.860133	+	0.510069i	$0.170380\pi$
$192$	−13.7652	−0.993419
$193$	27.4443	1.97548	0.987742	−	0.156098i	$-0.0498915\pi$
$193$	27.4443	1.97548	0.987742	+	0.156098i	$0.0498915\pi$
$194$	8.36574	0.600625
$195$	22.3361	1.59952
$196$	24.2046	1.72890
$197$	−16.3197	−1.16273	−0.581365	−	0.813643i	$-0.697481\pi$
$197$	−16.3197	−1.16273	−0.581365	+	0.813643i	$0.697481\pi$
$198$	−16.5730	−1.17779
$199$	15.1751	1.07573	0.537867	−	0.843030i	$-0.319230\pi$
$199$	15.1751	1.07573	0.537867	+	0.843030i	$0.319230\pi$
$200$	22.4415	1.58686
$201$	−23.2093	−1.63706
$202$	−19.3957	−1.36467
$203$	−18.0175	−1.26458
$204$	−16.0169	−1.12141
$205$	−15.8316	−1.10572
$206$	11.2242	0.782030
$207$	2.52791	0.175702
$208$	−57.5101	−3.98761
$209$	−8.66567	−0.599417
$210$	−31.2298	−2.15506
$211$	−23.2325	−1.59939	−0.799694	−	0.600408i	$-0.795005\pi$
$211$	−23.2325	−1.59939	−0.799694	+	0.600408i	$0.795005\pi$
$212$	20.4521	1.40466
$213$	2.54460	0.174353
$214$	−38.0476	−2.60088
$215$	3.67328	0.250516
$216$	9.71457	0.660993
$217$	−27.9096	−1.89462
$218$	−37.7830	−2.55898
$219$	35.3294	2.38734
$220$	11.5551	0.779047
$221$	−8.55602	−0.575540
$222$	29.4633	1.97744
$223$	25.0591	1.67808	0.839042	−	0.544066i	$-0.183116\pi$
$223$	25.0591	1.67808	0.839042	+	0.544066i	$0.183116\pi$
$224$	31.1251	2.07963
$225$	−11.2125	−0.747501
$226$	0.512033	0.0340599
$227$	−20.5845	−1.36624	−0.683119	−	0.730307i	$-0.739377\pi$
$227$	−20.5845	−1.36624	−0.683119	+	0.730307i	$0.739377\pi$
$228$	57.9988	3.84106
$229$	−17.8907	−1.18225	−0.591125	−	0.806580i	$-0.701316\pi$
$229$	−17.8907	−1.18225	−0.591125	+	0.806580i	$0.701316\pi$
$230$	−2.50795	−0.165369
$231$	16.0779	1.05785
$232$	−36.6375	−2.40537
$233$	−7.39592	−0.484523	−0.242262	−	0.970211i	$-0.577889\pi$
$233$	−7.39592	−0.484523	−0.242262	+	0.970211i	$0.577889\pi$
$234$	59.2530	3.87349
$235$	1.19233	0.0777793
$236$	24.3238	1.58335
$237$	−5.24613	−0.340773
$238$	11.9628	0.775436
$239$	3.52167	0.227798	0.113899	−	0.993492i	$-0.463666\pi$
$239$	3.52167	0.227798	0.113899	+	0.993492i	$0.463666\pi$
$240$	−30.7955	−1.98784
$241$	23.8510	1.53638	0.768191	−	0.640221i	$-0.221157\pi$
$241$	23.8510	1.53638	0.768191	+	0.640221i	$0.221157\pi$
$242$	20.0693	1.29010
$243$	22.2738	1.42886
$244$	26.8833	1.72102
$245$	6.92360	0.442333
$246$	−77.6217	−4.94898
$247$	30.9821	1.97134
$248$	−56.7525	−3.60379
$249$	4.25648	0.269744
$250$	28.6683	1.81314
$251$	−23.6388	−1.49207	−0.746033	−	0.665909i	$-0.768044\pi$
$251$	−23.6388	−1.49207	−0.746033	+	0.665909i	$0.768044\pi$
$252$	−58.2222	−3.66765
$253$	1.29115	0.0811742
$254$	14.6466	0.919009
$255$	−4.58157	−0.286909
$256$	−20.9260	−1.30788
$257$	5.19007	0.323748	0.161874	−	0.986811i	$-0.448246\pi$
$257$	5.19007	0.323748	0.161874	+	0.986811i	$0.448246\pi$
$258$	18.0100	1.12125
$259$	−15.4651	−0.960952
$260$	−41.3127	−2.56211
$261$	18.3053	1.13307
$262$	10.4473	0.645434
$263$	−20.0118	−1.23398	−0.616990	−	0.786971i	$-0.711648\pi$
$263$	−20.0118	−1.23398	−0.616990	+	0.786971i	$0.711648\pi$
$264$	32.6935	2.01214
$265$	5.85023	0.359377
$266$	−43.3186	−2.65603
$267$	17.3736	1.06325
$268$	42.9279	2.62224
$269$	3.35935	0.204823	0.102412	−	0.994742i	$-0.467344\pi$
$269$	3.35935	0.204823	0.102412	+	0.994742i	$0.467344\pi$
$270$	4.81540	0.293056
$271$	−8.18476	−0.497189	−0.248594	−	0.968608i	$-0.579969\pi$
$271$	−8.18476	−0.497189	−0.248594	+	0.968608i	$0.579969\pi$
$272$	11.7965	0.715266
$273$	−57.4827	−3.47901
$274$	5.68525	0.343458
$275$	−5.72691	−0.345345
$276$	−8.64161	−0.520164
$277$	20.4935	1.23134	0.615668	−	0.788005i	$-0.288886\pi$
$277$	20.4935	1.23134	0.615668	+	0.788005i	$0.288886\pi$
$278$	−8.33802	−0.500081
$279$	28.3554	1.69759
$280$	33.3329	1.99202
$281$	−4.90167	−0.292409	−0.146205	−	0.989254i	$-0.546706\pi$
$281$	−4.90167	−0.292409	−0.146205	+	0.989254i	$0.546706\pi$
$282$	5.84599	0.348123
$283$	−5.39215	−0.320530	−0.160265	−	0.987074i	$-0.551235\pi$
$283$	−5.39215	−0.320530	−0.160265	+	0.987074i	$0.551235\pi$
$284$	−4.70648	−0.279278
$285$	16.5903	0.982723
$286$	30.2640	1.78955
$287$	40.7431	2.40499
$288$	−31.6223	−1.86336
$289$	−15.2450	−0.896764
$290$	−18.1608	−1.06644
$291$	8.24545	0.483357
$292$	−65.3451	−3.82403
$293$	−13.8832	−0.811067	−0.405533	−	0.914080i	$-0.632914\pi$
$293$	−13.8832	−0.811067	−0.405533	+	0.914080i	$0.632914\pi$
$294$	33.9462	1.97979
$295$	6.95772	0.405094
$296$	−31.4473	−1.82784
$297$	−2.47908	−0.143851
$298$	−52.2886	−3.02899
$299$	−4.61622	−0.266963
$300$	38.3298	2.21297
$301$	−9.45332	−0.544880
$302$	−10.8676	−0.625359
$303$	−19.1168	−1.09823
$304$	−42.7161	−2.44993
$305$	7.68983	0.440318
$306$	−12.1539	−0.694795
$307$	31.2437	1.78317	0.891587	−	0.452850i	$-0.149593\pi$
$307$	31.2437	1.78317	0.891587	+	0.452850i	$0.149593\pi$
$308$	−29.7376	−1.69446
$309$	11.0628	0.629343
$310$	−28.1316	−1.59777
$311$	−20.8264	−1.18096	−0.590478	−	0.807054i	$-0.701061\pi$
$311$	−20.8264	−1.18096	−0.590478	+	0.807054i	$0.701061\pi$
$312$	−116.888	−6.61748
$313$	−20.3774	−1.15180	−0.575899	−	0.817521i	$-0.695348\pi$
$313$	−20.3774	−1.15180	−0.575899	+	0.817521i	$0.695348\pi$
$314$	12.8562	0.725518
$315$	−16.6542	−0.938357
$316$	9.70322	0.545849
$317$	0.0877986	0.00493126	0.00246563	−	0.999997i	$-0.499215\pi$
$317$	0.0877986	0.00493126	0.00246563	+	0.999997i	$0.499215\pi$
$318$	28.6835	1.60849
$319$	9.34961	0.523478
$320$	7.28274	0.407118
$321$	−37.5005	−2.09307
$322$	6.45431	0.359685
$323$	−6.35504	−0.353604
$324$	−33.5825	−1.86570
$325$	20.4752	1.13576
$326$	−50.2890	−2.78525
$327$	−37.2397	−2.05936
$328$	82.8489	4.57456
$329$	−3.06852	−0.169173
$330$	16.2058	0.892098
$331$	−16.0236	−0.880738	−0.440369	−	0.897817i	$-0.645152\pi$
$331$	−16.0236	−0.880738	−0.440369	+	0.897817i	$0.645152\pi$
$332$	−7.87277	−0.432075
$333$	15.7121	0.861018
$334$	−24.1461	−1.32122
$335$	12.2793	0.670890
$336$	79.2533	4.32362
$337$	−4.12312	−0.224601	−0.112300	−	0.993674i	$-0.535822\pi$
$337$	−4.12312	−0.224601	−0.112300	+	0.993674i	$0.535822\pi$
$338$	−74.4800	−4.05118
$339$	0.504671	0.0274099
$340$	8.47405	0.459570
$341$	14.4828	0.784288
$342$	44.0106	2.37982
$343$	6.54996	0.353664
$344$	−19.2228	−1.03642
$345$	−2.47189	−0.133082
$346$	59.4017	3.19345
$347$	−32.1398	−1.72535	−0.862676	−	0.505757i	$-0.831213\pi$
$347$	−32.1398	−1.72535	−0.862676	+	0.505757i	$0.831213\pi$
$348$	−62.5763	−3.35444
$349$	2.28465	0.122295	0.0611474	−	0.998129i	$-0.480524\pi$
$349$	2.28465	0.122295	0.0611474	+	0.998129i	$0.480524\pi$
$350$	−28.6280	−1.53023
$351$	8.86339	0.473093
$352$	−16.1514	−0.860873
$353$	−25.2328	−1.34301	−0.671504	−	0.741001i	$-0.734352\pi$
$353$	−25.2328	−1.34301	−0.671504	+	0.741001i	$0.734352\pi$
$354$	34.1135	1.81311
$355$	−1.34627	−0.0714524
$356$	−32.1341	−1.70311
$357$	11.7908	0.624037
$358$	−34.1342	−1.80405
$359$	1.71510	0.0905197	0.0452598	−	0.998975i	$-0.485588\pi$
$359$	1.71510	0.0905197	0.0452598	+	0.998975i	$0.485588\pi$
$360$	−33.8654	−1.78486
$361$	4.01218	0.211168
$362$	4.37589	0.229991
$363$	19.7807	1.03822
$364$	106.320	5.57267
$365$	−18.6917	−0.978366
$366$	37.7030	1.97077
$367$	17.8761	0.933125	0.466562	−	0.884488i	$-0.345492\pi$
$367$	17.8761	0.933125	0.466562	+	0.884488i	$0.345492\pi$
$368$	6.36454	0.331775
$369$	−41.3939	−2.15488
$370$	−15.5881	−0.810385
$371$	−15.0558	−0.781657
$372$	−96.9324	−5.02571
$373$	−7.79765	−0.403747	−0.201873	−	0.979412i	$-0.564703\pi$
$373$	−7.79765	−0.403747	−0.201873	+	0.979412i	$0.564703\pi$
$374$	−6.20775	−0.320995
$375$	28.2560	1.45914
$376$	−6.23966	−0.321786
$377$	−33.4274	−1.72160
$378$	−12.3926	−0.637407
$379$	6.77977	0.348253	0.174127	−	0.984723i	$-0.444290\pi$
$379$	6.77977	0.348253	0.174127	+	0.984723i	$0.444290\pi$
$380$	−30.6853	−1.57412
$381$	14.4360	0.739579
$382$	−61.6714	−3.15538
$383$	32.0775	1.63908	0.819541	−	0.573021i	$-0.194229\pi$
$383$	32.0775	1.63908	0.819541	+	0.573021i	$0.194229\pi$
$384$	−10.0121	−0.510927
$385$	−8.50629	−0.433521
$386$	−71.1907	−3.62351
$387$	9.60434	0.488216
$388$	−15.2507	−0.774239
$389$	−12.2389	−0.620538	−0.310269	−	0.950649i	$-0.600419\pi$
$389$	−12.2389	−0.620538	−0.310269	+	0.950649i	$0.600419\pi$
$390$	−57.9400	−2.93390
$391$	0.946878	0.0478857
$392$	−36.2322	−1.83000
$393$	10.2970	0.519417
$394$	42.3334	2.13273
$395$	2.77556	0.139654
$396$	30.2126	1.51824
$397$	−3.43323	−0.172309	−0.0861545	−	0.996282i	$-0.527458\pi$
$397$	−3.43323	−0.172309	−0.0861545	+	0.996282i	$0.527458\pi$
$398$	−39.3643	−1.97316
$399$	−42.6957	−2.13746
$400$	−28.2299	−1.41149
$401$	−4.58578	−0.229003	−0.114501	−	0.993423i	$-0.536527\pi$
$401$	−4.58578	−0.229003	−0.114501	+	0.993423i	$0.536527\pi$
$402$	60.2052	3.00276
$403$	−51.7799	−2.57934
$404$	35.3583	1.75914
$405$	−9.60612	−0.477332
$406$	46.7374	2.31954
$407$	8.02511	0.397790
$408$	23.9760	1.18699
$409$	36.5184	1.80572	0.902860	−	0.429935i	$-0.141464\pi$
$409$	36.5184	1.80572	0.902860	+	0.429935i	$0.141464\pi$
$410$	41.0672	2.02816
$411$	5.60350	0.276400
$412$	−20.4618	−1.00808
$413$	−17.9059	−0.881094
$414$	−6.55741	−0.322279
$415$	−2.25197	−0.110545
$416$	57.7456	2.83121
$417$	−8.21813	−0.402443
$418$	22.4788	1.09948
$419$	−8.09665	−0.395547	−0.197773	−	0.980248i	$-0.563371\pi$
$419$	−8.09665	−0.395547	−0.197773	+	0.980248i	$0.563371\pi$
$420$	56.9320	2.77800
$421$	−1.13367	−0.0552517	−0.0276259	−	0.999618i	$-0.508795\pi$
$421$	−1.13367	−0.0552517	−0.0276259	+	0.999618i	$0.508795\pi$
$422$	60.2652	2.93366
$423$	3.11753	0.151580
$424$	−30.6151	−1.48680
$425$	−4.19987	−0.203724
$426$	−6.60071	−0.319806
$427$	−19.7901	−0.957708
$428$	69.3608	3.35268
$429$	29.8289	1.44015
$430$	−9.52852	−0.459506
$431$	−26.8927	−1.29537	−0.647687	−	0.761906i	$-0.724264\pi$
$431$	−26.8927	−1.29537	−0.647687	+	0.761906i	$0.724264\pi$
$432$	−12.2202	−0.587947
$433$	−31.2644	−1.50247	−0.751235	−	0.660034i	$-0.770542\pi$
$433$	−31.2644	−1.50247	−0.751235	+	0.660034i	$0.770542\pi$
$434$	72.3976	3.47520
$435$	−17.8997	−0.858223
$436$	68.8784	3.29868
$437$	−3.42873	−0.164018
$438$	−91.6447	−4.37895
$439$	−20.3674	−0.972084	−0.486042	−	0.873935i	$-0.661560\pi$
$439$	−20.3674	−0.972084	−0.486042	+	0.873935i	$0.661560\pi$
$440$	−17.2971	−0.824606
$441$	18.1028	0.862037
$442$	22.1944	1.05568
$443$	20.0036	0.950398	0.475199	−	0.879878i	$-0.342376\pi$
$443$	20.0036	0.950398	0.475199	+	0.879878i	$0.342376\pi$
$444$	−53.7116	−2.54904
$445$	−9.19182	−0.435734
$446$	−65.0037	−3.07801
$447$	−51.5367	−2.43760
$448$	−18.7424	−0.885495
$449$	24.6868	1.16504	0.582521	−	0.812816i	$-0.302066\pi$
$449$	24.6868	1.16504	0.582521	+	0.812816i	$0.302066\pi$
$450$	29.0854	1.37110
$451$	−21.1424	−0.995556
$452$	−0.933437	−0.0439052
$453$	−10.7113	−0.503262
$454$	53.3963	2.50601
$455$	30.4123	1.42575
$456$	−86.8194	−4.06569
$457$	−30.1555	−1.41062	−0.705308	−	0.708901i	$-0.749191\pi$
$457$	−30.1555	−1.41062	−0.705308	+	0.708901i	$0.749191\pi$
$458$	46.4086	2.16853
$459$	−1.81806	−0.0848595
$460$	4.57200	0.213171
$461$	−19.7920	−0.921804	−0.460902	−	0.887451i	$-0.652474\pi$
$461$	−19.7920	−0.921804	−0.460902	+	0.887451i	$0.652474\pi$
$462$	−41.7061	−1.94035
$463$	−11.5726	−0.537826	−0.268913	−	0.963164i	$-0.586664\pi$
$463$	−11.5726	−0.537826	−0.268913	+	0.963164i	$0.586664\pi$
$464$	46.0874	2.13955
$465$	−27.7271	−1.28581
$466$	19.1851	0.888732
$467$	−7.55882	−0.349781	−0.174890	−	0.984588i	$-0.555957\pi$
$467$	−7.55882	−0.349781	−0.174890	+	0.984588i	$0.555957\pi$
$468$	−108.018	−4.99314
$469$	−31.6013	−1.45921
$470$	−3.09293	−0.142666
$471$	12.6714	0.583866
$472$	−36.4108	−1.67594
$473$	4.90551	0.225556
$474$	13.6085	0.625060
$475$	15.2081	0.697796
$476$	−21.8083	−0.999581
$477$	15.2963	0.700369
$478$	−9.13523	−0.417836
$479$	−27.5038	−1.25668	−0.628340	−	0.777939i	$-0.716265\pi$
$479$	−27.5038	−1.25668	−0.628340	+	0.777939i	$0.716265\pi$
$480$	30.9216	1.41137
$481$	−28.6919	−1.30824
$482$	−61.8698	−2.81809
$483$	6.36150	0.289458
$484$	−36.5863	−1.66301
$485$	−4.36240	−0.198087
$486$	−57.7783	−2.62088
$487$	−12.3747	−0.560753	−0.280376	−	0.959890i	$-0.590459\pi$
$487$	−12.3747	−0.560753	−0.280376	+	0.959890i	$0.590459\pi$
$488$	−40.2420	−1.82167
$489$	−49.5659	−2.24145
$490$	−17.9599	−0.811345
$491$	−37.3263	−1.68451	−0.842257	−	0.539076i	$-0.818773\pi$
$491$	−37.3263	−1.68451	−0.842257	+	0.539076i	$0.818773\pi$
$492$	141.505	6.37952
$493$	6.85661	0.308806
$494$	−80.3679	−3.61592
$495$	8.64218	0.388437
$496$	71.3907	3.20554
$497$	3.46467	0.155412
$498$	−11.0414	−0.494775
$499$	−8.25297	−0.369454	−0.184727	−	0.982790i	$-0.559140\pi$
$499$	−8.25297	−0.369454	−0.184727	+	0.982790i	$0.559140\pi$
$500$	−52.2623	−2.33724
$501$	−23.7989	−1.06326
$502$	61.3192	2.73681
$503$	30.5399	1.36171	0.680854	−	0.732419i	$-0.261609\pi$
$503$	30.5399	1.36171	0.680854	+	0.732419i	$0.261609\pi$
$504$	87.1538	3.88214
$505$	10.1141	0.450071
$506$	−3.34927	−0.148893
$507$	−73.4091	−3.26021
$508$	−26.7008	−1.18465
$509$	−13.3944	−0.593698	−0.296849	−	0.954924i	$-0.595936\pi$
$509$	−13.3944	−0.593698	−0.296849	+	0.954924i	$0.595936\pi$
$510$	11.8846	0.526260
$511$	48.1037	2.12798
$512$	46.4504	2.05284
$513$	6.58334	0.290662
$514$	−13.4631	−0.593831
$515$	−5.85300	−0.257914
$516$	−32.8323	−1.44536
$517$	1.59231	0.0700298
$518$	40.1165	1.76262
$519$	58.5476	2.56995
$520$	61.8417	2.71194
$521$	13.5543	0.593827	0.296913	−	0.954904i	$-0.404043\pi$
$521$	13.5543	0.593827	0.296913	+	0.954904i	$0.404043\pi$
$522$	−47.4841	−2.07832
$523$	−2.45182	−0.107210	−0.0536052	−	0.998562i	$-0.517071\pi$
$523$	−2.45182	−0.107210	−0.0536052	+	0.998562i	$0.517071\pi$
$524$	−19.0454	−0.832001
$525$	−28.2164	−1.23146
$526$	51.9107	2.26342
$527$	10.6211	0.462662
$528$	−41.1261	−1.78978
$529$	−22.4891	−0.977788
$530$	−15.1756	−0.659184
$531$	18.1920	0.789465
$532$	78.9698	3.42377
$533$	75.5897	3.27415
$534$	−45.0673	−1.95025
$535$	19.8403	0.857772
$536$	−64.2594	−2.77558
$537$	−33.6434	−1.45182
$538$	−8.71420	−0.375696
$539$	9.24618	0.398261
$540$	−8.77848	−0.377766
$541$	8.64134	0.371520	0.185760	−	0.982595i	$-0.440525\pi$
$541$	8.64134	0.371520	0.185760	+	0.982595i	$0.440525\pi$
$542$	21.2313	0.911964
$543$	4.31297	0.185087
$544$	−11.8448	−0.507840
$545$	19.7023	0.843955
$546$	149.111	6.38135
$547$	−29.4698	−1.26004	−0.630019	−	0.776580i	$-0.716953\pi$
$547$	−29.4698	−1.26004	−0.630019	+	0.776580i	$0.716953\pi$
$548$	−10.3642	−0.442737
$549$	20.1062	0.858111
$550$	14.8556	0.633447
$551$	−24.8284	−1.05773
$552$	12.9358	0.550583
$553$	−7.14301	−0.303752
$554$	−53.1604	−2.25857
$555$	−15.3639	−0.652163
$556$	15.2002	0.644633
$557$	2.56885	0.108845	0.0544227	−	0.998518i	$-0.482668\pi$
$557$	2.56885	0.108845	0.0544227	+	0.998518i	$0.482668\pi$
$558$	−73.5541	−3.11380
$559$	−17.5385	−0.741800
$560$	−41.9304	−1.77188
$561$	−6.11849	−0.258323
$562$	12.7150	0.536349
$563$	8.36104	0.352376	0.176188	−	0.984357i	$-0.443623\pi$
$563$	8.36104	0.352376	0.176188	+	0.984357i	$0.443623\pi$
$564$	−10.6572	−0.448751
$565$	−0.267005	−0.0112330
$566$	13.9873	0.587929
$567$	24.7217	1.03821
$568$	7.04521	0.295610
$569$	−7.96825	−0.334046	−0.167023	−	0.985953i	$-0.553415\pi$
$569$	−7.96825	−0.334046	−0.167023	+	0.985953i	$0.553415\pi$
$570$	−43.0353	−1.80255
$571$	−3.22741	−0.135063	−0.0675315	−	0.997717i	$-0.521512\pi$
$571$	−3.22741	−0.135063	−0.0675315	+	0.997717i	$0.521512\pi$
$572$	−55.1714	−2.30683
$573$	−60.7847	−2.53932
$574$	−105.688	−4.41133
$575$	−2.26595	−0.0944968
$576$	19.0418	0.793408
$577$	20.3058	0.845342	0.422671	−	0.906283i	$-0.361093\pi$
$577$	20.3058	0.845342	0.422671	+	0.906283i	$0.361093\pi$
$578$	39.5456	1.64488
$579$	−70.1671	−2.91605
$580$	33.1071	1.37470
$581$	5.79553	0.240439
$582$	−21.3888	−0.886593
$583$	7.81274	0.323571
$584$	97.8162	4.04766
$585$	−30.8981	−1.27748
$586$	36.0132	1.48769
$587$	−5.24103	−0.216321	−0.108160	−	0.994133i	$-0.534496\pi$
$587$	−5.24103	−0.216321	−0.108160	+	0.994133i	$0.534496\pi$
$588$	−61.8841	−2.55206
$589$	−38.4599	−1.58471
$590$	−18.0484	−0.743040
$591$	41.7247	1.71633
$592$	39.5585	1.62585
$593$	2.58829	0.106288	0.0531442	−	0.998587i	$-0.483076\pi$
$593$	2.58829	0.106288	0.0531442	+	0.998587i	$0.483076\pi$
$594$	6.43076	0.263857
$595$	−6.23816	−0.255739
$596$	95.3221	3.90455
$597$	−38.7983	−1.58791
$598$	11.9745	0.489675
$599$	13.3577	0.545781	0.272890	−	0.962045i	$-0.412020\pi$
$599$	13.3577	0.545781	0.272890	+	0.962045i	$0.412020\pi$
$600$	−57.3765	−2.34239
$601$	−5.18999	−0.211704	−0.105852	−	0.994382i	$-0.533757\pi$
$601$	−5.18999	−0.211704	−0.105852	+	0.994382i	$0.533757\pi$
$602$	24.5220	0.999442
$603$	32.1061	1.30746
$604$	19.8116	0.806123
$605$	−10.4653	−0.425476
$606$	49.5891	2.01442
$607$	11.6807	0.474106	0.237053	−	0.971497i	$-0.423818\pi$
$607$	11.6807	0.474106	0.237053	+	0.971497i	$0.423818\pi$
$608$	42.8910	1.73946
$609$	46.0654	1.86667
$610$	−19.9475	−0.807650
$611$	−5.69294	−0.230312
$612$	22.1567	0.895630
$613$	11.9965	0.484533	0.242267	−	0.970210i	$-0.422109\pi$
$613$	11.9965	0.484533	0.242267	+	0.970210i	$0.422109\pi$
$614$	−81.0465	−3.27077
$615$	40.4767	1.63218
$616$	44.5147	1.79355
$617$	−4.33665	−0.174587	−0.0872936	−	0.996183i	$-0.527822\pi$
$617$	−4.33665	−0.174587	−0.0872936	+	0.996183i	$0.527822\pi$
$618$	−28.6971	−1.15437
$619$	−28.4850	−1.14491	−0.572455	−	0.819936i	$-0.694009\pi$
$619$	−28.4850	−1.14491	−0.572455	+	0.819936i	$0.694009\pi$
$620$	51.2839	2.05961
$621$	−0.980894	−0.0393619
$622$	54.0238	2.16616
$623$	23.6555	0.947737
$624$	147.037	5.88618
$625$	0.901991	0.0360796
$626$	52.8591	2.11268
$627$	22.1556	0.884810
$628$	−23.4369	−0.935234
$629$	5.88528	0.234662
$630$	43.2011	1.72117
$631$	9.38154	0.373473	0.186737	−	0.982410i	$-0.440209\pi$
$631$	9.38154	0.373473	0.186737	+	0.982410i	$0.440209\pi$
$632$	−14.5249	−0.577770
$633$	59.3987	2.36088
$634$	−0.227750	−0.00904512
$635$	−7.63763	−0.303090
$636$	−52.2901	−2.07344
$637$	−33.0576	−1.30979
$638$	−24.2530	−0.960184
$639$	−3.52001	−0.139250
$640$	5.29707	0.209385
$641$	21.1841	0.836723	0.418361	−	0.908281i	$-0.362605\pi$
$641$	21.1841	0.836723	0.418361	+	0.908281i	$0.362605\pi$
$642$	97.2767	3.83920
$643$	38.6539	1.52436	0.762180	−	0.647365i	$-0.224129\pi$
$643$	38.6539	1.52436	0.762180	+	0.647365i	$0.224129\pi$
$644$	−11.7662	−0.463654
$645$	−9.39151	−0.369790
$646$	16.4850	0.648595
$647$	13.6422	0.536329	0.268164	−	0.963373i	$-0.413583\pi$
$647$	13.6422	0.536329	0.268164	+	0.963373i	$0.413583\pi$
$648$	50.2703	1.97480
$649$	9.29174	0.364733
$650$	−53.1129	−2.08326
$651$	71.3566	2.79669
$652$	91.6769	3.59034
$653$	−15.3012	−0.598783	−0.299392	−	0.954130i	$-0.596784\pi$
$653$	−15.3012	−0.598783	−0.299392	+	0.954130i	$0.596784\pi$
$654$	96.6001	3.77736
$655$	−5.44784	−0.212865
$656$	−104.218	−4.06903
$657$	−48.8721	−1.90668
$658$	7.95976	0.310304
$659$	−32.5309	−1.26722	−0.633612	−	0.773651i	$-0.718429\pi$
$659$	−32.5309	−1.26722	−0.633612	+	0.773651i	$0.718429\pi$
$660$	−29.5431	−1.14997
$661$	−19.1643	−0.745404	−0.372702	−	0.927951i	$-0.621569\pi$
$661$	−19.1643	−0.745404	−0.372702	+	0.927951i	$0.621569\pi$
$662$	41.5654	1.61549
$663$	21.8753	0.849564
$664$	11.7849	0.457342
$665$	22.5889	0.875961
$666$	−40.7573	−1.57931
$667$	3.69934	0.143239
$668$	44.0184	1.70312
$669$	−64.0690	−2.47705
$670$	−31.8526	−1.23057
$671$	10.2694	0.396447
$672$	−79.5778	−3.06978
$673$	−18.8208	−0.725489	−0.362745	−	0.931889i	$-0.618160\pi$
$673$	−18.8208	−0.725489	−0.362745	+	0.931889i	$0.618160\pi$
$674$	10.6954	0.411972
$675$	4.35075	0.167460
$676$	135.777	5.22220
$677$	−21.4497	−0.824379	−0.412189	−	0.911098i	$-0.635236\pi$
$677$	−21.4497	−0.824379	−0.412189	+	0.911098i	$0.635236\pi$
$678$	−1.30912	−0.0502765
$679$	11.2268	0.430845
$680$	−12.6850	−0.486446
$681$	52.6285	2.01673
$682$	−37.5685	−1.43857
$683$	28.6365	1.09575	0.547873	−	0.836562i	$-0.315438\pi$
$683$	28.6365	1.09575	0.547873	+	0.836562i	$0.315438\pi$
$684$	−80.2313	−3.06772
$685$	−2.96463	−0.113273
$686$	−16.9906	−0.648706
$687$	45.7413	1.74514
$688$	24.1809	0.921890
$689$	−27.9326	−1.06415
$690$	6.41211	0.244105
$691$	−9.20632	−0.350225	−0.175112	−	0.984548i	$-0.556029\pi$
$691$	−9.20632	−0.350225	−0.175112	+	0.984548i	$0.556029\pi$
$692$	−108.289	−4.11654
$693$	−22.2410	−0.844864
$694$	83.3708	3.16471
$695$	4.34795	0.164927
$696$	93.6715	3.55061
$697$	−15.5049	−0.587291
$698$	−5.92641	−0.224318
$699$	18.9092	0.715213
$700$	52.1890	1.97256
$701$	−16.4835	−0.622572	−0.311286	−	0.950316i	$-0.600760\pi$
$701$	−16.4835	−0.622572	−0.311286	+	0.950316i	$0.600760\pi$
$702$	−22.9917	−0.867766
$703$	−21.3111	−0.803765
$704$	9.72579	0.366555
$705$	−3.04845	−0.114811
$706$	65.4542	2.46340
$707$	−26.0290	−0.978920
$708$	−62.1890	−2.33721
$709$	45.2030	1.69763	0.848817	−	0.528686i	$-0.177315\pi$
$709$	45.2030	1.69763	0.848817	+	0.528686i	$0.177315\pi$
$710$	3.49223	0.131061
$711$	7.25712	0.272163
$712$	48.1021	1.80270
$713$	5.73039	0.214605
$714$	−30.5855	−1.14463
$715$	−15.7815	−0.590195
$716$	62.2267	2.32552
$717$	−9.00388	−0.336256
$718$	−4.44899	−0.166035
$719$	−13.2756	−0.495096	−0.247548	−	0.968876i	$-0.579625\pi$
$719$	−13.2756	−0.495096	−0.247548	+	0.968876i	$0.579625\pi$
$720$	42.6002	1.58762
$721$	15.0629	0.560972
$722$	−10.4076	−0.387332
$723$	−60.9802	−2.26788
$724$	−7.97725	−0.296472
$725$	−16.4084	−0.609393
$726$	−51.3113	−1.90434
$727$	35.3322	1.31040	0.655199	−	0.755457i	$-0.272585\pi$
$727$	35.3322	1.31040	0.655199	+	0.755457i	$0.272585\pi$
$728$	−159.152	−5.89856
$729$	−35.6428	−1.32010
$730$	48.4863	1.79456
$731$	3.59750	0.133058
$732$	−68.7327	−2.54043
$733$	22.6399	0.836223	0.418111	−	0.908396i	$-0.362692\pi$
$733$	22.6399	0.836223	0.418111	+	0.908396i	$0.362692\pi$
$734$	−46.3708	−1.71158
$735$	−17.7016	−0.652935
$736$	−6.39060	−0.235561
$737$	16.3985	0.604047
$738$	107.376	3.95257
$739$	−4.34868	−0.159969	−0.0799843	−	0.996796i	$-0.525487\pi$
$739$	−4.34868	−0.159969	−0.0799843	+	0.996796i	$0.525487\pi$
$740$	28.4171	1.04463
$741$	−79.2123	−2.90994
$742$	39.0548	1.43375
$743$	31.3521	1.15020	0.575099	−	0.818084i	$-0.304964\pi$
$743$	31.3521	1.15020	0.575099	+	0.818084i	$0.304964\pi$
$744$	145.100	5.31962
$745$	27.2664	0.998965
$746$	20.2272	0.740569
$747$	−5.88811	−0.215435
$748$	11.3167	0.413781
$749$	−51.0598	−1.86569
$750$	−73.2964	−2.67641
$751$	−0.113322	−0.00413519	−0.00206759	−	0.999998i	$-0.500658\pi$
$751$	−0.113322	−0.00413519	−0.00206759	+	0.999998i	$0.500658\pi$
$752$	7.84905	0.286225
$753$	60.4375	2.20246
$754$	86.7109	3.15782
$755$	5.66702	0.206244
$756$	22.5917	0.821653
$757$	−13.7245	−0.498827	−0.249413	−	0.968397i	$-0.580238\pi$
$757$	−13.7245	−0.498827	−0.249413	+	0.968397i	$0.580238\pi$
$758$	−17.5868	−0.638780
$759$	−3.30111	−0.119823
$760$	45.9334	1.66618
$761$	−20.3247	−0.736769	−0.368385	−	0.929674i	$-0.620089\pi$
$761$	−20.3247	−0.736769	−0.368385	+	0.929674i	$0.620089\pi$
$762$	−37.4471	−1.35657
$763$	−50.7047	−1.83563
$764$	112.427	4.06747
$765$	6.33781	0.229144
$766$	−83.2092	−3.00647
$767$	−33.2205	−1.19952
$768$	53.5018	1.93058
$769$	−36.3709	−1.31157	−0.655784	−	0.754948i	$-0.727662\pi$
$769$	−36.3709	−1.31157	−0.655784	+	0.754948i	$0.727662\pi$
$770$	22.0654	0.795182
$771$	−13.2695	−0.477890
$772$	129.781	4.67091
$773$	−27.7345	−0.997539	−0.498770	−	0.866735i	$-0.666215\pi$
$773$	−27.7345	−0.997539	−0.498770	+	0.866735i	$0.666215\pi$
$774$	−24.9137	−0.895506
$775$	−25.4171	−0.913009
$776$	22.8291	0.819517
$777$	39.5397	1.41848
$778$	31.7479	1.13822
$779$	56.1448	2.01159
$780$	105.625	3.78197
$781$	−1.79788	−0.0643333
$782$	−2.45621	−0.0878339
$783$	−7.10293	−0.253838
$784$	45.5776	1.62777
$785$	−6.70402	−0.239277
$786$	−26.7106	−0.952736
$787$	−6.78578	−0.241887	−0.120943	−	0.992659i	$-0.538592\pi$
$787$	−6.78578	−0.241887	−0.120943	+	0.992659i	$0.538592\pi$
$788$	−77.1739	−2.74921
$789$	51.1643	1.82150
$790$	−7.19983	−0.256158
$791$	0.687148	0.0244322
$792$	−45.2258	−1.60703
$793$	−36.7160	−1.30382
$794$	8.90583	0.316056
$795$	−14.9573	−0.530482
$796$	71.7612	2.54351
$797$	18.8237	0.666771	0.333385	−	0.942791i	$-0.391809\pi$
$797$	18.8237	0.666771	0.333385	+	0.942791i	$0.391809\pi$
$798$	110.753	3.92061
$799$	1.16773	0.0413115
$800$	28.3455	1.00216
$801$	−24.0334	−0.849177
$802$	11.8956	0.420047
$803$	−24.9619	−0.880887
$804$	−109.754	−3.87073
$805$	−3.36567	−0.118624
$806$	134.318	4.73113
$807$	−8.58890	−0.302343
$808$	−52.9285	−1.86202
$809$	47.7833	1.67997	0.839986	−	0.542608i	$-0.182563\pi$
$809$	47.7833	1.67997	0.839986	+	0.542608i	$0.182563\pi$
$810$	24.9184	0.875542
$811$	1.01471	0.0356312	0.0178156	−	0.999841i	$-0.494329\pi$
$811$	1.01471	0.0356312	0.0178156	+	0.999841i	$0.494329\pi$
$812$	−85.2024	−2.99002
$813$	20.9261	0.733909
$814$	−20.8172	−0.729643
$815$	26.2237	0.918578
$816$	−30.1601	−1.05582
$817$	−13.0269	−0.455752
$818$	−94.7291	−3.31213
$819$	79.5174	2.77856
$820$	−74.8655	−2.61442
$821$	33.8184	1.18027	0.590136	−	0.807304i	$-0.299074\pi$
$821$	33.8184	1.18027	0.590136	+	0.807304i	$0.299074\pi$
$822$	−14.5355	−0.506985
$823$	−26.9371	−0.938968	−0.469484	−	0.882941i	$-0.655560\pi$
$823$	−26.9371	−0.938968	−0.469484	+	0.882941i	$0.655560\pi$
$824$	30.6296	1.06703
$825$	14.6420	0.509770
$826$	46.4482	1.61614
$827$	−44.3101	−1.54081	−0.770406	−	0.637553i	$-0.779947\pi$
$827$	−44.3101	−1.54081	−0.770406	+	0.637553i	$0.779947\pi$
$828$	11.9542	0.415436
$829$	−47.8095	−1.66049	−0.830246	−	0.557397i	$-0.811800\pi$
$829$	−47.8095	−1.66049	−0.830246	+	0.557397i	$0.811800\pi$
$830$	5.84163	0.202766
$831$	−52.3960	−1.81760
$832$	−34.7723	−1.20551
$833$	6.78076	0.234939
$834$	21.3179	0.738178
$835$	12.5913	0.435739
$836$	−40.9789	−1.41729
$837$	−11.0026	−0.380307
$838$	21.0028	0.725529
$839$	−4.08428	−0.141005	−0.0705025	−	0.997512i	$-0.522460\pi$
$839$	−4.08428	−0.141005	−0.0705025	+	0.997512i	$0.522460\pi$
$840$	−85.2225	−2.94046
$841$	−2.21202	−0.0762767
$842$	2.94075	0.101345
$843$	12.5322	0.431630
$844$	−109.864	−3.78166
$845$	38.8384	1.33608
$846$	−8.08691	−0.278034
$847$	26.9329	0.925426
$848$	38.5117	1.32250
$849$	13.7862	0.473140
$850$	10.8945	0.373678
$851$	3.17528	0.108847
$852$	12.0331	0.412247
$853$	16.2710	0.557107	0.278553	−	0.960421i	$-0.410145\pi$
$853$	16.2710	0.557107	0.278553	+	0.960421i	$0.410145\pi$
$854$	51.3356	1.75667
$855$	−22.9498	−0.784866
$856$	−103.827	−3.54875
$857$	23.9996	0.819810	0.409905	−	0.912128i	$-0.365562\pi$
$857$	23.9996	0.819810	0.409905	+	0.912128i	$0.365562\pi$
$858$	−77.3764	−2.64159
$859$	−36.3440	−1.24004	−0.620020	−	0.784586i	$-0.712876\pi$
$859$	−36.3440	−1.24004	−0.620020	+	0.784586i	$0.712876\pi$
$860$	17.3705	0.592329
$861$	−104.168	−3.55005
$862$	69.7598	2.37603
$863$	3.32836	0.113299	0.0566494	−	0.998394i	$-0.481958\pi$
$863$	3.32836	0.113299	0.0566494	+	0.998394i	$0.481958\pi$
$864$	12.2703	0.417443
$865$	−30.9757	−1.05320
$866$	81.1001	2.75589
$867$	38.9770	1.32373
$868$	−131.981	−4.47973
$869$	3.70665	0.125739
$870$	46.4319	1.57419
$871$	−58.6290	−1.98657
$872$	−103.105	−3.49158
$873$	−11.4062	−0.386040
$874$	8.89416	0.300849
$875$	38.4728	1.30062
$876$	167.068	5.64472
$877$	−41.1013	−1.38789	−0.693946	−	0.720028i	$-0.744129\pi$
$877$	−41.1013	−1.38789	−0.693946	+	0.720028i	$0.744129\pi$
$878$	52.8333	1.78304
$879$	35.4954	1.19723
$880$	21.7585	0.733479
$881$	−1.47662	−0.0497485	−0.0248743	−	0.999691i	$-0.507919\pi$
$881$	−1.47662	−0.0497485	−0.0248743	+	0.999691i	$0.507919\pi$
$882$	−46.9588	−1.58118
$883$	46.5478	1.56646	0.783229	−	0.621734i	$-0.213571\pi$
$883$	46.5478	1.56646	0.783229	+	0.621734i	$0.213571\pi$
$884$	−40.4604	−1.36083
$885$	−17.7889	−0.597966
$886$	−51.8894	−1.74326
$887$	−52.6043	−1.76628	−0.883140	−	0.469109i	$-0.844575\pi$
$887$	−52.6043	−1.76628	−0.883140	+	0.469109i	$0.844575\pi$
$888$	80.4017	2.69811
$889$	19.6557	0.659232
$890$	23.8437	0.799241
$891$	−12.8286	−0.429773
$892$	118.502	3.96773
$893$	−4.22847	−0.141500
$894$	133.687	4.47115
$895$	17.7997	0.594977
$896$	−13.6322	−0.455420
$897$	11.8023	0.394069
$898$	−64.0377	−2.13697
$899$	41.4953	1.38395
$900$	−53.0227	−1.76742
$901$	5.72953	0.190878
$902$	54.8435	1.82609
$903$	24.1694	0.804307
$904$	1.39728	0.0464728
$905$	−2.28185	−0.0758514
$906$	27.7853	0.923103
$907$	−50.3528	−1.67194	−0.835968	−	0.548778i	$-0.815093\pi$
$907$	−50.3528	−1.67194	−0.835968	+	0.548778i	$0.815093\pi$
$908$	−97.3415	−3.23039
$909$	26.4448	0.877118
$910$	−78.8897	−2.61517
$911$	−8.04471	−0.266533	−0.133267	−	0.991080i	$-0.542547\pi$
$911$	−8.04471	−0.266533	−0.133267	+	0.991080i	$0.542547\pi$
$912$	109.213	3.61639
$913$	−3.00741	−0.0995308
$914$	78.2237	2.58741
$915$	−19.6607	−0.649961
$916$	−84.6030	−2.79536
$917$	14.0202	0.462988
$918$	4.71605	0.155653
$919$	−31.8562	−1.05084	−0.525420	−	0.850843i	$-0.676092\pi$
$919$	−31.8562	−1.05084	−0.525420	+	0.850843i	$0.676092\pi$
$920$	−6.84390	−0.225637
$921$	−79.8811	−2.63217
$922$	51.3406	1.69081
$923$	6.42791	0.211577
$924$	76.0304	2.50122
$925$	−14.0839	−0.463077
$926$	30.0195	0.986503
$927$	−15.3035	−0.502634
$928$	−46.2761	−1.51909
$929$	0.419424	0.0137609	0.00688043	−	0.999976i	$-0.497810\pi$
$929$	0.419424	0.0137609	0.00688043	+	0.999976i	$0.497810\pi$
$930$	71.9243	2.35849
$931$	−24.5537	−0.804716
$932$	−34.9744	−1.14563
$933$	53.2471	1.74323
$934$	19.6076	0.641582
$935$	3.23710	0.105864
$936$	161.694	5.28514
$937$	38.7218	1.26499	0.632493	−	0.774566i	$-0.282032\pi$
$937$	38.7218	1.26499	0.632493	+	0.774566i	$0.282032\pi$
$938$	81.9739	2.67654
$939$	52.0991	1.70019
$940$	5.63841	0.183905
$941$	16.3083	0.531637	0.265818	−	0.964023i	$-0.414358\pi$
$941$	16.3083	0.531637	0.265818	+	0.964023i	$0.414358\pi$
$942$	−32.8696	−1.07095
$943$	−8.36537	−0.272414
$944$	45.8022	1.49073
$945$	6.46226	0.210217
$946$	−12.7249	−0.413723
$947$	−21.3367	−0.693350	−0.346675	−	0.937985i	$-0.612689\pi$
$947$	−21.3367	−0.693350	−0.346675	+	0.937985i	$0.612689\pi$
$948$	−24.8083	−0.805737
$949$	89.2455	2.89703
$950$	−39.4500	−1.27993
$951$	−0.224476	−0.00727912
$952$	32.6452	1.05804
$953$	13.1162	0.424877	0.212438	−	0.977174i	$-0.431860\pi$
$953$	13.1162	0.424877	0.212438	+	0.977174i	$0.431860\pi$
$954$	−39.6787	−1.28465
$955$	32.1592	1.04065
$956$	16.6535	0.538614
$957$	−23.9042	−0.772714
$958$	71.3450	2.30505
$959$	7.62960	0.246373
$960$	−18.6199	−0.600953
$961$	33.2774	1.07347
$962$	74.4271	2.39963
$963$	51.8755	1.67166
$964$	112.789	3.63268
$965$	37.1232	1.19504
$966$	−16.5018	−0.530937
$967$	3.12461	0.100481	0.0502404	−	0.998737i	$-0.484001\pi$
$967$	3.12461	0.100481	0.0502404	+	0.998737i	$0.484001\pi$
$968$	54.7666	1.76027
$969$	16.2480	0.521961
$970$	11.3161	0.363339
$971$	54.9392	1.76308	0.881541	−	0.472108i	$-0.156507\pi$
$971$	54.9392	1.76308	0.881541	+	0.472108i	$0.156507\pi$
$972$	105.330	3.37846
$973$	−11.1896	−0.358722
$974$	32.1002	1.02856
$975$	−52.3492	−1.67652
$976$	50.6216	1.62036
$977$	−9.52654	−0.304781	−0.152391	−	0.988320i	$-0.548697\pi$
$977$	−9.52654	−0.304781	−0.152391	+	0.988320i	$0.548697\pi$
$978$	128.574	4.11136
$979$	−12.2753	−0.392320
$980$	32.7409	1.04587
$981$	51.5147	1.64474
$982$	96.8248	3.08980
$983$	30.5804	0.975362	0.487681	−	0.873022i	$-0.337843\pi$
$983$	30.5804	0.975362	0.487681	+	0.873022i	$0.337843\pi$
$984$	−211.821	−6.75259
$985$	−22.0752	−0.703375
$986$	−17.7861	−0.566425
$987$	7.84531	0.249719
$988$	146.511	4.66113
$989$	1.94096	0.0617188
$990$	−22.4179	−0.712487
$991$	−18.1473	−0.576468	−0.288234	−	0.957560i	$-0.593068\pi$
$991$	−18.1473	−0.576468	−0.288234	+	0.957560i	$0.593068\pi$
$992$	−71.6830	−2.27594
$993$	40.9678	1.30007
$994$	−8.98738	−0.285062
$995$	20.5270	0.650748
$996$	20.1284	0.637793
$997$	−25.0452	−0.793188	−0.396594	−	0.917994i	$-0.629808\pi$
$997$	−25.0452	−0.793188	−0.396594	+	0.917994i	$0.629808\pi$
$998$	21.4083	0.677667
$999$	−6.09671	−0.192891

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4027.2.a.b.1.11	✓	159

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4027.2.a.b.1.11	✓	159	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:	\( N \)	\(=\)	\( 4027 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4027.a (trivial)

\(f(q)\)	\(=\)	\(q-2.59401 q^{2} -2.55671 q^{3} +4.72888 q^{4} +1.35267 q^{5} +6.63213 q^{6} -3.48116 q^{7} -7.07874 q^{8} +3.53677 q^{9} +O(q^{10})\)	\(q-2.59401 q^{2} -2.55671 q^{3} +4.72888 q^{4} +1.35267 q^{5} +6.63213 q^{6} -3.48116 q^{7} -7.07874 q^{8} +3.53677 q^{9} -3.50885 q^{10} +1.80644 q^{11} -12.0904 q^{12} -6.45851 q^{13} +9.03015 q^{14} -3.45840 q^{15} +8.90456 q^{16} +1.32477 q^{17} -9.17441 q^{18} -4.79710 q^{19} +6.39664 q^{20} +8.90031 q^{21} -4.68592 q^{22} +0.714751 q^{23} +18.0983 q^{24} -3.17027 q^{25} +16.7534 q^{26} -1.37236 q^{27} -16.4620 q^{28} +5.17571 q^{29} +8.97111 q^{30} +8.01732 q^{31} -8.94102 q^{32} -4.61854 q^{33} -3.43646 q^{34} -4.70887 q^{35} +16.7250 q^{36} +4.44250 q^{37} +12.4437 q^{38} +16.5125 q^{39} -9.57523 q^{40} -11.7039 q^{41} -23.0875 q^{42} +2.71557 q^{43} +8.54244 q^{44} +4.78409 q^{45} -1.85407 q^{46} +0.881464 q^{47} -22.7664 q^{48} +5.11845 q^{49} +8.22371 q^{50} -3.38705 q^{51} -30.5415 q^{52} +4.32494 q^{53} +3.55991 q^{54} +2.44352 q^{55} +24.6422 q^{56} +12.2648 q^{57} -13.4258 q^{58} +5.14368 q^{59} -16.3543 q^{60} +5.68491 q^{61} -20.7970 q^{62} -12.3120 q^{63} +5.38396 q^{64} -8.73626 q^{65} +11.9805 q^{66} +9.07780 q^{67} +6.26467 q^{68} -1.82741 q^{69} +12.2149 q^{70} -0.995263 q^{71} -25.0359 q^{72} -13.8183 q^{73} -11.5239 q^{74} +8.10547 q^{75} -22.6849 q^{76} -6.28850 q^{77} -42.8336 q^{78} +2.05191 q^{79} +12.0450 q^{80} -7.10158 q^{81} +30.3600 q^{82} -1.66483 q^{83} +42.0885 q^{84} +1.79198 q^{85} -7.04421 q^{86} -13.2328 q^{87} -12.7873 q^{88} -6.79529 q^{89} -12.4100 q^{90} +22.4831 q^{91} +3.37997 q^{92} -20.4980 q^{93} -2.28653 q^{94} -6.48892 q^{95} +22.8596 q^{96} -3.22502 q^{97} -13.2773 q^{98} +6.38896 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 159 q - 22 q^{2} - 19 q^{3} + 148 q^{4} - 70 q^{5} - 23 q^{6} - 19 q^{7} - 66 q^{8} + 126 q^{9}+O(q^{10}) \) 159 * q - 22 * q^2 - 19 * q^3 + 148 * q^4 - 70 * q^5 - 23 * q^6 - 19 * q^7 - 66 * q^8 + 126 * q^9	\( 159 q - 22 q^{2} - 19 q^{3} + 148 q^{4} - 70 q^{5} - 23 q^{6} - 19 q^{7} - 66 q^{8} + 126 q^{9} - 23 q^{10} - 33 q^{11} - 57 q^{12} - 90 q^{13} - 28 q^{14} - 22 q^{15} + 130 q^{16} - 145 q^{17} - 50 q^{18} - 28 q^{19} - 121 q^{20} - 69 q^{21} - 26 q^{22} - 79 q^{23} - 62 q^{24} + 123 q^{25} - 40 q^{26} - 70 q^{27} - 43 q^{28} - 109 q^{29} - 43 q^{30} - 21 q^{31} - 139 q^{32} - 83 q^{33} - 93 q^{35} + 75 q^{36} - 65 q^{37} - 122 q^{38} - 18 q^{39} - 43 q^{40} - 71 q^{41} - 88 q^{42} - 72 q^{43} - 79 q^{44} - 181 q^{45} - 11 q^{46} - 114 q^{47} - 118 q^{48} + 118 q^{49} - 77 q^{50} - 29 q^{51} - 169 q^{52} - 220 q^{53} - 80 q^{54} - 37 q^{55} - 72 q^{56} - 90 q^{57} - 8 q^{58} - 60 q^{59} - 42 q^{60} - 108 q^{61} - 152 q^{62} - 65 q^{63} + 114 q^{64} - 81 q^{65} - 40 q^{66} - 50 q^{67} - 319 q^{68} - 103 q^{69} + 4 q^{70} - 7 q^{71} - 129 q^{72} - 94 q^{73} - 79 q^{74} - 59 q^{75} - 46 q^{76} - 329 q^{77} + 8 q^{78} - 18 q^{79} - 190 q^{80} + 59 q^{81} - 56 q^{82} - 201 q^{83} - 71 q^{84} - 26 q^{85} - 52 q^{86} - 126 q^{87} - 66 q^{88} - 114 q^{89} - 33 q^{90} - 30 q^{91} - 204 q^{92} - 125 q^{93} + 9 q^{94} - 84 q^{95} - 88 q^{96} - 56 q^{97} - 110 q^{98} - 46 q^{99}+O(q^{100}) \) 159 * q - 22 * q^2 - 19 * q^3 + 148 * q^4 - 70 * q^5 - 23 * q^6 - 19 * q^7 - 66 * q^8 + 126 * q^9 - 23 * q^10 - 33 * q^11 - 57 * q^12 - 90 * q^13 - 28 * q^14 - 22 * q^15 + 130 * q^16 - 145 * q^17 - 50 * q^18 - 28 * q^19 - 121 * q^20 - 69 * q^21 - 26 * q^22 - 79 * q^23 - 62 * q^24 + 123 * q^25 - 40 * q^26 - 70 * q^27 - 43 * q^28 - 109 * q^29 - 43 * q^30 - 21 * q^31 - 139 * q^32 - 83 * q^33 - 93 * q^35 + 75 * q^36 - 65 * q^37 - 122 * q^38 - 18 * q^39 - 43 * q^40 - 71 * q^41 - 88 * q^42 - 72 * q^43 - 79 * q^44 - 181 * q^45 - 11 * q^46 - 114 * q^47 - 118 * q^48 + 118 * q^49 - 77 * q^50 - 29 * q^51 - 169 * q^52 - 220 * q^53 - 80 * q^54 - 37 * q^55 - 72 * q^56 - 90 * q^57 - 8 * q^58 - 60 * q^59 - 42 * q^60 - 108 * q^61 - 152 * q^62 - 65 * q^63 + 114 * q^64 - 81 * q^65 - 40 * q^66 - 50 * q^67 - 319 * q^68 - 103 * q^69 + 4 * q^70 - 7 * q^71 - 129 * q^72 - 94 * q^73 - 79 * q^74 - 59 * q^75 - 46 * q^76 - 329 * q^77 + 8 * q^78 - 18 * q^79 - 190 * q^80 + 59 * q^81 - 56 * q^82 - 201 * q^83 - 71 * q^84 - 26 * q^85 - 52 * q^86 - 126 * q^87 - 66 * q^88 - 114 * q^89 - 33 * q^90 - 30 * q^91 - 204 * q^92 - 125 * q^93 + 9 * q^94 - 84 * q^95 - 88 * q^96 - 56 * q^97 - 110 * q^98 - 46 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more