LMFDB - Embedded newform 8001.2.a.v.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8001,2,Mod(1,8001)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8001, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("8001.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8001 = 3^{2} \cdot 7 \cdot 127 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8001.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:	$63.8883066572$
Analytic rank:	$0$
Dimension:	$19$
Coefficient field:	$\mathbb{Q}[x]/(x^{19} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{19} - 4 x^{18} - 22 x^{17} + 101 x^{16} + 178 x^{15} - 1035 x^{14} - 583 x^{13} + 5572 x^{12} + \cdots + 210 $ x^19 - 4x^18 - 22x^17 + 101x^16 + 178x^15 - 1035x^14 - 583x^13 + 5572x^12 + 21x^11 - 17032x^10 + 4985x^9 + 29792x^8 - 13249x^7 - 28600x^6 + 14000x^5 + 13725x^4 - 5723x^3 - 2913x^2 + 608x + 210
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 2667)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$1.20823$ of defining polynomial
Character	$\chi$	$=$	8001.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.20823 q^{2} -0.540168 q^{4} -0.486834 q^{5} +1.00000 q^{7} +3.06912 q^{8} +O(q^{10})$	\(q-1.20823 q^{2} -0.540168 q^{4} -0.486834 q^{5} +1.00000 q^{7} +3.06912 q^{8} +0.588209 q^{10} -3.33645 q^{11} +2.96046 q^{13} -1.20823 q^{14} -2.62788 q^{16} -0.691924 q^{17} +0.556445 q^{19} +0.262972 q^{20} +4.03122 q^{22} -4.97719 q^{23} -4.76299 q^{25} -3.57693 q^{26} -0.540168 q^{28} -1.54090 q^{29} -1.24157 q^{31} -2.96314 q^{32} +0.836007 q^{34} -0.486834 q^{35} -7.43380 q^{37} -0.672316 q^{38} -1.49415 q^{40} +5.74868 q^{41} +9.66027 q^{43} +1.80225 q^{44} +6.01361 q^{46} -4.42787 q^{47} +1.00000 q^{49} +5.75481 q^{50} -1.59915 q^{52} +3.25257 q^{53} +1.62430 q^{55} +3.06912 q^{56} +1.86177 q^{58} -10.8233 q^{59} +9.14746 q^{61} +1.50011 q^{62} +8.83593 q^{64} -1.44125 q^{65} +10.3697 q^{67} +0.373756 q^{68} +0.588209 q^{70} +2.82644 q^{71} -4.24827 q^{73} +8.98178 q^{74} -0.300574 q^{76} -3.33645 q^{77} +13.7939 q^{79} +1.27934 q^{80} -6.94576 q^{82} -0.539810 q^{83} +0.336852 q^{85} -11.6719 q^{86} -10.2400 q^{88} +9.15032 q^{89} +2.96046 q^{91} +2.68852 q^{92} +5.34991 q^{94} -0.270896 q^{95} +11.8201 q^{97} -1.20823 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 19 q - 4 q^{2} + 22 q^{4} - 5 q^{5} + 19 q^{7} - 9 q^{8}+O(q^{10}) $ 19 * q - 4 * q^2 + 22 * q^4 - 5 * q^5 + 19 * q^7 - 9 * q^8	$ 19 q - 4 q^{2} + 22 q^{4} - 5 q^{5} + 19 q^{7} - 9 q^{8} + 9 q^{11} + 24 q^{13} - 4 q^{14} + 20 q^{16} - 17 q^{17} + 23 q^{19} - 5 q^{20} - 3 q^{22} + 17 q^{23} + 38 q^{25} - 28 q^{26} + 22 q^{28} - 2 q^{29} + 16 q^{31} - 17 q^{32} + 29 q^{34} - 5 q^{35} + 56 q^{37} - 2 q^{38} - 13 q^{40} + 7 q^{41} + 19 q^{43} + 29 q^{44} + 10 q^{46} - 25 q^{47} + 19 q^{49} + 9 q^{50} + 16 q^{52} - 18 q^{53} + 10 q^{55} - 9 q^{56} + 31 q^{58} - 11 q^{59} + 26 q^{61} - 26 q^{62} + 45 q^{64} - 27 q^{65} + 24 q^{67} - 14 q^{68} + 32 q^{71} + 51 q^{73} + 12 q^{76} + 9 q^{77} + 30 q^{79} + 30 q^{80} - 52 q^{82} - q^{83} + 44 q^{85} + 24 q^{86} - 30 q^{88} - 5 q^{89} + 24 q^{91} + 88 q^{92} + 7 q^{94} + 24 q^{95} + 5 q^{97} - 4 q^{98}+O(q^{100}) $ 19 * q - 4 * q^2 + 22 * q^4 - 5 * q^5 + 19 * q^7 - 9 * q^8 + 9 * q^11 + 24 * q^13 - 4 * q^14 + 20 * q^16 - 17 * q^17 + 23 * q^19 - 5 * q^20 - 3 * q^22 + 17 * q^23 + 38 * q^25 - 28 * q^26 + 22 * q^28 - 2 * q^29 + 16 * q^31 - 17 * q^32 + 29 * q^34 - 5 * q^35 + 56 * q^37 - 2 * q^38 - 13 * q^40 + 7 * q^41 + 19 * q^43 + 29 * q^44 + 10 * q^46 - 25 * q^47 + 19 * q^49 + 9 * q^50 + 16 * q^52 - 18 * q^53 + 10 * q^55 - 9 * q^56 + 31 * q^58 - 11 * q^59 + 26 * q^61 - 26 * q^62 + 45 * q^64 - 27 * q^65 + 24 * q^67 - 14 * q^68 + 32 * q^71 + 51 * q^73 + 12 * q^76 + 9 * q^77 + 30 * q^79 + 30 * q^80 - 52 * q^82 - q^83 + 44 * q^85 + 24 * q^86 - 30 * q^88 - 5 * q^89 + 24 * q^91 + 88 * q^92 + 7 * q^94 + 24 * q^95 + 5 * q^97 - 4 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−1.20823	−0.854351	−0.427176	−	0.904169i	$-0.640491\pi$
$2$	−1.20823	−0.854351	−0.427176	+	0.904169i	$0.640491\pi$
$3$	0	0
$3$	0	0
$4$	−0.540168	−0.270084
$5$	−0.486834	−0.217719	−0.108859	−	0.994057i	$-0.534720\pi$
$5$	−0.486834	−0.217719	−0.108859	+	0.994057i	$0.534720\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	3.06912	1.08510
$9$	0	0
$10$	0.588209	0.186008
$11$	−3.33645	−1.00598	−0.502989	−	0.864293i	$-0.667766\pi$
$11$	−3.33645	−1.00598	−0.502989	+	0.864293i	$0.667766\pi$
$12$	0	0
$13$	2.96046	0.821085	0.410542	−	0.911842i	$-0.365339\pi$
$13$	2.96046	0.821085	0.410542	+	0.911842i	$0.365339\pi$
$14$	−1.20823	−0.322914
$15$	0	0
$16$	−2.62788	−0.656970
$17$	−0.691924	−0.167816	−0.0839081	−	0.996473i	$-0.526740\pi$
$17$	−0.691924	−0.167816	−0.0839081	+	0.996473i	$0.526740\pi$
$18$	0	0
$19$	0.556445	0.127657	0.0638286	−	0.997961i	$-0.479669\pi$
$19$	0.556445	0.127657	0.0638286	+	0.997961i	$0.479669\pi$
$20$	0.262972	0.0588024
$21$	0	0
$22$	4.03122	0.859459
$23$	−4.97719	−1.03782	−0.518908	−	0.854830i	$-0.673661\pi$
$23$	−4.97719	−1.03782	−0.518908	+	0.854830i	$0.673661\pi$
$24$	0	0
$25$	−4.76299	−0.952599
$26$	−3.57693	−0.701495
$27$	0	0
$28$	−0.540168	−0.102082
$29$	−1.54090	−0.286138	−0.143069	−	0.989713i	$-0.545697\pi$
$29$	−1.54090	−0.286138	−0.143069	+	0.989713i	$0.545697\pi$
$30$	0	0
$31$	−1.24157	−0.222993	−0.111497	−	0.993765i	$-0.535564\pi$
$31$	−1.24157	−0.222993	−0.111497	+	0.993765i	$0.535564\pi$
$32$	−2.96314	−0.523814
$33$	0	0
$34$	0.836007	0.143374
$35$	−0.486834	−0.0822899
$36$	0	0
$37$	−7.43380	−1.22211	−0.611055	−	0.791588i	$-0.709254\pi$
$37$	−7.43380	−1.22211	−0.611055	+	0.791588i	$0.709254\pi$
$38$	−0.672316	−0.109064
$39$	0	0
$40$	−1.49415	−0.236246
$41$	5.74868	0.897793	0.448897	−	0.893584i	$-0.351817\pi$
$41$	5.74868	0.897793	0.448897	+	0.893584i	$0.351817\pi$
$42$	0	0
$43$	9.66027	1.47318	0.736588	−	0.676341i	$-0.236436\pi$
$43$	9.66027	1.47318	0.736588	+	0.676341i	$0.236436\pi$
$44$	1.80225	0.271699
$45$	0	0
$46$	6.01361	0.886659
$47$	−4.42787	−0.645872	−0.322936	−	0.946421i	$-0.604670\pi$
$47$	−4.42787	−0.645872	−0.322936	+	0.946421i	$0.604670\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	5.75481	0.813854
$51$	0	0
$52$	−1.59915	−0.221762
$53$	3.25257	0.446775	0.223388	−	0.974730i	$-0.428288\pi$
$53$	3.25257	0.446775	0.223388	+	0.974730i	$0.428288\pi$
$54$	0	0
$55$	1.62430	0.219020
$56$	3.06912	0.410128
$57$	0	0
$58$	1.86177	0.244462
$59$	−10.8233	−1.40907	−0.704535	−	0.709669i	$-0.748844\pi$
$59$	−10.8233	−1.40907	−0.704535	+	0.709669i	$0.748844\pi$
$60$	0	0
$61$	9.14746	1.17121	0.585606	−	0.810596i	$-0.300856\pi$
$61$	9.14746	1.17121	0.585606	+	0.810596i	$0.300856\pi$
$62$	1.50011	0.190515
$63$	0	0
$64$	8.83593	1.10449
$65$	−1.44125	−0.178765
$66$	0	0
$67$	10.3697	1.26686	0.633430	−	0.773800i	$-0.281647\pi$
$67$	10.3697	1.26686	0.633430	+	0.773800i	$0.281647\pi$
$68$	0.373756	0.0453245
$69$	0	0
$70$	0.588209	0.0703045
$71$	2.82644	0.335437	0.167718	−	0.985835i	$-0.446360\pi$
$71$	2.82644	0.335437	0.167718	+	0.985835i	$0.446360\pi$
$72$	0	0
$73$	−4.24827	−0.497222	−0.248611	−	0.968603i	$-0.579974\pi$
$73$	−4.24827	−0.497222	−0.248611	+	0.968603i	$0.579974\pi$
$74$	8.98178	1.04411
$75$	0	0
$76$	−0.300574	−0.0344782
$77$	−3.33645	−0.380224
$78$	0	0
$79$	13.7939	1.55193	0.775965	−	0.630776i	$-0.217263\pi$
$79$	13.7939	1.55193	0.775965	+	0.630776i	$0.217263\pi$
$80$	1.27934	0.143035
$81$	0	0
$82$	−6.94576	−0.767031
$83$	−0.539810	−0.0592519	−0.0296259	−	0.999561i	$-0.509432\pi$
$83$	−0.539810	−0.0592519	−0.0296259	+	0.999561i	$0.509432\pi$
$84$	0	0
$85$	0.336852	0.0365367
$86$	−11.6719	−1.25861
$87$	0	0
$88$	−10.2400	−1.09158
$89$	9.15032	0.969932	0.484966	−	0.874533i	$-0.338832\pi$
$89$	9.15032	0.969932	0.484966	+	0.874533i	$0.338832\pi$
$90$	0	0
$91$	2.96046	0.310341
$92$	2.68852	0.280297
$93$	0	0
$94$	5.34991	0.551801
$95$	−0.270896	−0.0277933
$96$	0	0
$97$	11.8201	1.20015	0.600075	−	0.799944i	$-0.295137\pi$
$97$	11.8201	1.20015	0.600075	+	0.799944i	$0.295137\pi$
$98$	−1.20823	−0.122050
$99$	0	0
$100$	2.57282	0.257282
$101$	−11.7839	−1.17255	−0.586273	−	0.810113i	$-0.699405\pi$
$101$	−11.7839	−1.17255	−0.586273	+	0.810113i	$0.699405\pi$
$102$	0	0
$103$	−6.48469	−0.638955	−0.319478	−	0.947594i	$-0.603507\pi$
$103$	−6.48469	−0.638955	−0.319478	+	0.947594i	$0.603507\pi$
$104$	9.08602	0.890957
$105$	0	0
$106$	−3.92987	−0.381703
$107$	−7.40352	−0.715725	−0.357863	−	0.933774i	$-0.616494\pi$
$107$	−7.40352	−0.715725	−0.357863	+	0.933774i	$0.616494\pi$
$108$	0	0
$109$	−14.4941	−1.38828	−0.694141	−	0.719839i	$-0.744216\pi$
$109$	−14.4941	−1.38828	−0.694141	+	0.719839i	$0.744216\pi$
$110$	−1.96253	−0.187120
$111$	0	0
$112$	−2.62788	−0.248311
$113$	−12.1032	−1.13858	−0.569289	−	0.822138i	$-0.692781\pi$
$113$	−12.1032	−1.13858	−0.569289	+	0.822138i	$0.692781\pi$
$114$	0	0
$115$	2.42306	0.225952
$116$	0.832346	0.0772814
$117$	0	0
$118$	13.0771	1.20384
$119$	−0.691924	−0.0634286
$120$	0	0
$121$	0.131916	0.0119924
$122$	−11.0523	−1.00063
$123$	0	0
$124$	0.670660	0.0602270
$125$	4.75295	0.425117
$126$	0	0
$127$	1.00000	0.0887357
$127$	1.00000	0.0887357
$128$	−4.74960	−0.419809
$129$	0	0
$130$	1.74137	0.152728
$131$	11.3196	0.988996	0.494498	−	0.869179i	$-0.335352\pi$
$131$	11.3196	0.988996	0.494498	+	0.869179i	$0.335352\pi$
$132$	0	0
$133$	0.556445	0.0482499
$134$	−12.5290	−1.08234
$135$	0	0
$136$	−2.12360	−0.182097
$137$	−1.75217	−0.149698	−0.0748490	−	0.997195i	$-0.523847\pi$
$137$	−1.75217	−0.149698	−0.0748490	+	0.997195i	$0.523847\pi$
$138$	0	0
$139$	−12.3362	−1.04634	−0.523169	−	0.852229i	$-0.675251\pi$
$139$	−12.3362	−1.04634	−0.523169	+	0.852229i	$0.675251\pi$
$140$	0.262972	0.0222252
$141$	0	0
$142$	−3.41500	−0.286581
$143$	−9.87744	−0.825993
$144$	0	0
$145$	0.750163	0.0622976
$146$	5.13290	0.424802
$147$	0	0
$148$	4.01550	0.330072
$149$	−0.292352	−0.0239504	−0.0119752	−	0.999928i	$-0.503812\pi$
$149$	−0.292352	−0.0239504	−0.0119752	+	0.999928i	$0.503812\pi$
$150$	0	0
$151$	11.0695	0.900821	0.450411	−	0.892822i	$-0.351278\pi$
$151$	11.0695	0.900821	0.450411	+	0.892822i	$0.351278\pi$
$152$	1.70780	0.138521
$153$	0	0
$154$	4.03122	0.324845
$155$	0.604441	0.0485498
$156$	0	0
$157$	−9.72904	−0.776462	−0.388231	−	0.921562i	$-0.626914\pi$
$157$	−9.72904	−0.776462	−0.388231	+	0.921562i	$0.626914\pi$
$158$	−16.6662	−1.32589
$159$	0	0
$160$	1.44256	0.114044
$161$	−4.97719	−0.392257
$162$	0	0
$163$	−23.7172	−1.85768	−0.928839	−	0.370484i	$-0.879192\pi$
$163$	−23.7172	−1.85768	−0.928839	+	0.370484i	$0.879192\pi$
$164$	−3.10526	−0.242480
$165$	0	0
$166$	0.652217	0.0506219
$167$	12.0490	0.932380	0.466190	−	0.884685i	$-0.345626\pi$
$167$	12.0490	0.932380	0.466190	+	0.884685i	$0.345626\pi$
$168$	0	0
$169$	−4.23566	−0.325820
$170$	−0.406996	−0.0312152
$171$	0	0
$172$	−5.21817	−0.397882
$173$	13.0592	0.992872	0.496436	−	0.868073i	$-0.334642\pi$
$173$	13.0592	0.992872	0.496436	+	0.868073i	$0.334642\pi$
$174$	0	0
$175$	−4.76299	−0.360048
$176$	8.76780	0.660898
$177$	0	0
$178$	−11.0557	−0.828662
$179$	11.7072	0.875040	0.437520	−	0.899209i	$-0.355857\pi$
$179$	11.7072	0.875040	0.437520	+	0.899209i	$0.355857\pi$
$180$	0	0
$181$	23.3643	1.73666	0.868329	−	0.495988i	$-0.165194\pi$
$181$	23.3643	1.73666	0.868329	+	0.495988i	$0.165194\pi$
$182$	−3.57693	−0.265140
$183$	0	0
$184$	−15.2756	−1.12613
$185$	3.61903	0.266076
$186$	0	0
$187$	2.30857	0.168820
$188$	2.39180	0.174440
$189$	0	0
$190$	0.327306	0.0237453
$191$	−7.47087	−0.540573	−0.270287	−	0.962780i	$-0.587118\pi$
$191$	−7.47087	−0.540573	−0.270287	+	0.962780i	$0.587118\pi$
$192$	0	0
$193$	2.23193	0.160658	0.0803291	−	0.996768i	$-0.474403\pi$
$193$	2.23193	0.160658	0.0803291	+	0.996768i	$0.474403\pi$
$194$	−14.2815	−1.02535
$195$	0	0
$196$	−0.540168	−0.0385835
$197$	24.6516	1.75636	0.878179	−	0.478333i	$-0.158759\pi$
$197$	24.6516	1.75636	0.878179	+	0.478333i	$0.158759\pi$
$198$	0	0
$199$	5.25622	0.372604	0.186302	−	0.982493i	$-0.440350\pi$
$199$	5.25622	0.372604	0.186302	+	0.982493i	$0.440350\pi$
$200$	−14.6182	−1.03366
$201$	0	0
$202$	14.2378	1.00177
$203$	−1.54090	−0.108150
$204$	0	0
$205$	−2.79865	−0.195466
$206$	7.83502	0.545892
$207$	0	0
$208$	−7.77974	−0.539428
$209$	−1.85655	−0.128420
$210$	0	0
$211$	−25.9511	−1.78655	−0.893274	−	0.449513i	$-0.851598\pi$
$211$	−25.9511	−1.78655	−0.893274	+	0.449513i	$0.851598\pi$
$212$	−1.75694	−0.120667
$213$	0	0
$214$	8.94519	0.611481
$215$	−4.70294	−0.320738
$216$	0	0
$217$	−1.24157	−0.0842836
$218$	17.5123	1.18608
$219$	0	0
$220$	−0.877394	−0.0591539
$221$	−2.04842	−0.137791
$222$	0	0
$223$	−5.08231	−0.340337	−0.170168	−	0.985415i	$-0.554431\pi$
$223$	−5.08231	−0.340337	−0.170168	+	0.985415i	$0.554431\pi$
$224$	−2.96314	−0.197983
$225$	0	0
$226$	14.6236	0.972745
$227$	27.3641	1.81622	0.908111	−	0.418729i	$-0.137524\pi$
$227$	27.3641	1.81622	0.908111	+	0.418729i	$0.137524\pi$
$228$	0	0
$229$	−19.5981	−1.29508	−0.647540	−	0.762031i	$-0.724202\pi$
$229$	−19.5981	−1.29508	−0.647540	+	0.762031i	$0.724202\pi$
$230$	−2.92763	−0.193042
$231$	0	0
$232$	−4.72921	−0.310488
$233$	1.12255	0.0735409	0.0367705	−	0.999324i	$-0.488293\pi$
$233$	1.12255	0.0735409	0.0367705	+	0.999324i	$0.488293\pi$
$234$	0	0
$235$	2.15564	0.140618
$236$	5.84639	0.380568
$237$	0	0
$238$	0.836007	0.0541903
$239$	−20.6348	−1.33476	−0.667378	−	0.744719i	$-0.732584\pi$
$239$	−20.6348	−1.33476	−0.667378	+	0.744719i	$0.732584\pi$
$240$	0	0
$241$	16.6369	1.07168	0.535838	−	0.844321i	$-0.319996\pi$
$241$	16.6369	1.07168	0.535838	+	0.844321i	$0.319996\pi$
$242$	−0.159386	−0.0102457
$243$	0	0
$244$	−4.94117	−0.316326
$245$	−0.486834	−0.0311027
$246$	0	0
$247$	1.64733	0.104817
$248$	−3.81054	−0.241970
$249$	0	0
$250$	−5.74268	−0.363199
$251$	9.39146	0.592784	0.296392	−	0.955066i	$-0.404217\pi$
$251$	9.39146	0.592784	0.296392	+	0.955066i	$0.404217\pi$
$252$	0	0
$253$	16.6061	1.04402
$254$	−1.20823	−0.0758114
$255$	0	0
$256$	−11.9332	−0.745827
$257$	16.5814	1.03432	0.517159	−	0.855889i	$-0.326990\pi$
$257$	16.5814	1.03432	0.517159	+	0.855889i	$0.326990\pi$
$258$	0	0
$259$	−7.43380	−0.461914
$260$	0.778519	0.0482817
$261$	0	0
$262$	−13.6767	−0.844950
$263$	29.0182	1.78934	0.894668	−	0.446731i	$-0.147412\pi$
$263$	29.0182	1.78934	0.894668	+	0.446731i	$0.147412\pi$
$264$	0	0
$265$	−1.58346	−0.0972714
$266$	−0.672316	−0.0412223
$267$	0	0
$268$	−5.60138	−0.342159
$269$	1.32881	0.0810187	0.0405093	−	0.999179i	$-0.487102\pi$
$269$	1.32881	0.0810187	0.0405093	+	0.999179i	$0.487102\pi$
$270$	0	0
$271$	17.7182	1.07630	0.538152	−	0.842848i	$-0.319122\pi$
$271$	17.7182	1.07630	0.538152	+	0.842848i	$0.319122\pi$
$272$	1.81829	0.110250
$273$	0	0
$274$	2.11703	0.127895
$275$	15.8915	0.958294
$276$	0	0
$277$	−24.1077	−1.44849	−0.724247	−	0.689540i	$-0.757813\pi$
$277$	−24.1077	−1.44849	−0.724247	+	0.689540i	$0.757813\pi$
$278$	14.9050	0.893941
$279$	0	0
$280$	−1.49415	−0.0892926
$281$	−13.8431	−0.825810	−0.412905	−	0.910774i	$-0.635486\pi$
$281$	−13.8431	−0.825810	−0.412905	+	0.910774i	$0.635486\pi$
$282$	0	0
$283$	15.6326	0.929260	0.464630	−	0.885505i	$-0.346187\pi$
$283$	15.6326	0.929260	0.464630	+	0.885505i	$0.346187\pi$
$284$	−1.52675	−0.0905961
$285$	0	0
$286$	11.9343	0.705688
$287$	5.74868	0.339334
$288$	0	0
$289$	−16.5212	−0.971838
$290$	−0.906373	−0.0532240
$291$	0	0
$292$	2.29478	0.134292
$293$	−10.3552	−0.604955	−0.302477	−	0.953157i	$-0.597814\pi$
$293$	−10.3552	−0.604955	−0.302477	+	0.953157i	$0.597814\pi$
$294$	0	0
$295$	5.26913	0.306781
$296$	−22.8152	−1.32611
$297$	0	0
$298$	0.353230	0.0204620
$299$	−14.7348	−0.852134
$300$	0	0
$301$	9.66027	0.556808
$302$	−13.3745	−0.769618
$303$	0	0
$304$	−1.46227	−0.0838670
$305$	−4.45329	−0.254995
$306$	0	0
$307$	7.23631	0.412998	0.206499	−	0.978447i	$-0.433793\pi$
$307$	7.23631	0.412998	0.206499	+	0.978447i	$0.433793\pi$
$308$	1.80225	0.102693
$309$	0	0
$310$	−0.730306	−0.0414786
$311$	23.7205	1.34507	0.672533	−	0.740068i	$-0.265207\pi$
$311$	23.7205	1.34507	0.672533	+	0.740068i	$0.265207\pi$
$312$	0	0
$313$	30.9911	1.75172	0.875860	−	0.482566i	$-0.160295\pi$
$313$	30.9911	1.75172	0.875860	+	0.482566i	$0.160295\pi$
$314$	11.7550	0.663371
$315$	0	0
$316$	−7.45100	−0.419152
$317$	−5.13371	−0.288338	−0.144169	−	0.989553i	$-0.546051\pi$
$317$	−5.13371	−0.288338	−0.144169	+	0.989553i	$0.546051\pi$
$318$	0	0
$319$	5.14114	0.287849
$320$	−4.30163	−0.240468
$321$	0	0
$322$	6.01361	0.335125
$323$	−0.385018	−0.0214229
$324$	0	0
$325$	−14.1007	−0.782164
$326$	28.6560	1.58711
$327$	0	0
$328$	17.6434	0.974193
$329$	−4.42787	−0.244117
$330$	0	0
$331$	25.3612	1.39398	0.696989	−	0.717082i	$-0.254523\pi$
$331$	25.3612	1.39398	0.696989	+	0.717082i	$0.254523\pi$
$332$	0.291588	0.0160030
$333$	0	0
$334$	−14.5580	−0.796580
$335$	−5.04831	−0.275819
$336$	0	0
$337$	−30.9030	−1.68339	−0.841696	−	0.539952i	$-0.818442\pi$
$337$	−30.9030	−1.68339	−0.841696	+	0.539952i	$0.818442\pi$
$338$	5.11767	0.278365
$339$	0	0
$340$	−0.181957	−0.00986799
$341$	4.14246	0.224327
$342$	0	0
$343$	1.00000	0.0539949
$344$	29.6485	1.59854
$345$	0	0
$346$	−15.7786	−0.848261
$347$	12.4171	0.666587	0.333293	−	0.942823i	$-0.391840\pi$
$347$	12.4171	0.666587	0.333293	+	0.942823i	$0.391840\pi$
$348$	0	0
$349$	17.2201	0.921768	0.460884	−	0.887460i	$-0.347532\pi$
$349$	17.2201	0.921768	0.460884	+	0.887460i	$0.347532\pi$
$350$	5.75481	0.307608
$351$	0	0
$352$	9.88638	0.526946
$353$	−12.0591	−0.641841	−0.320921	−	0.947106i	$-0.603992\pi$
$353$	−12.0591	−0.641841	−0.320921	+	0.947106i	$0.603992\pi$
$354$	0	0
$355$	−1.37601	−0.0730308
$356$	−4.94271	−0.261963
$357$	0	0
$358$	−14.1451	−0.747592
$359$	3.14917	0.166207	0.0831035	−	0.996541i	$-0.473517\pi$
$359$	3.14917	0.166207	0.0831035	+	0.996541i	$0.473517\pi$
$360$	0	0
$361$	−18.6904	−0.983704
$362$	−28.2296	−1.48372
$363$	0	0
$364$	−1.59915	−0.0838181
$365$	2.06820	0.108254
$366$	0	0
$367$	−18.4980	−0.965589	−0.482794	−	0.875734i	$-0.660378\pi$
$367$	−18.4980	−0.965589	−0.482794	+	0.875734i	$0.660378\pi$
$368$	13.0795	0.681814
$369$	0	0
$370$	−4.37263	−0.227322
$371$	3.25257	0.168865
$372$	0	0
$373$	3.34770	0.173337	0.0866686	−	0.996237i	$-0.472378\pi$
$373$	3.34770	0.173337	0.0866686	+	0.996237i	$0.472378\pi$
$374$	−2.78930	−0.144231
$375$	0	0
$376$	−13.5897	−0.700834
$377$	−4.56178	−0.234944
$378$	0	0
$379$	6.87229	0.353006	0.176503	−	0.984300i	$-0.443522\pi$
$379$	6.87229	0.353006	0.176503	+	0.984300i	$0.443522\pi$
$380$	0.146329	0.00750654
$381$	0	0
$382$	9.02657	0.461839
$383$	−3.87185	−0.197842	−0.0989210	−	0.995095i	$-0.531539\pi$
$383$	−3.87185	−0.197842	−0.0989210	+	0.995095i	$0.531539\pi$
$384$	0	0
$385$	1.62430	0.0827819
$386$	−2.69670	−0.137258
$387$	0	0
$388$	−6.38485	−0.324142
$389$	23.9772	1.21569	0.607847	−	0.794054i	$-0.292034\pi$
$389$	23.9772	1.21569	0.607847	+	0.794054i	$0.292034\pi$
$390$	0	0
$391$	3.44384	0.174162
$392$	3.06912	0.155014
$393$	0	0
$394$	−29.7850	−1.50055
$395$	−6.71531	−0.337884
$396$	0	0
$397$	9.65076	0.484358	0.242179	−	0.970232i	$-0.422138\pi$
$397$	9.65076	0.484358	0.242179	+	0.970232i	$0.422138\pi$
$398$	−6.35075	−0.318334
$399$	0	0
$400$	12.5166	0.625829
$401$	2.46700	0.123196	0.0615981	−	0.998101i	$-0.480380\pi$
$401$	2.46700	0.123196	0.0615981	+	0.998101i	$0.480380\pi$
$402$	0	0
$403$	−3.67564	−0.183096
$404$	6.36531	0.316686
$405$	0	0
$406$	1.86177	0.0923981
$407$	24.8025	1.22942
$408$	0	0
$409$	−21.4104	−1.05867	−0.529337	−	0.848412i	$-0.677559\pi$
$409$	−21.4104	−1.05867	−0.529337	+	0.848412i	$0.677559\pi$
$410$	3.38143	0.166997
$411$	0	0
$412$	3.50282	0.172572
$413$	−10.8233	−0.532578
$414$	0	0
$415$	0.262798	0.0129002
$416$	−8.77227	−0.430096
$417$	0	0
$418$	2.24315	0.109716
$419$	−31.1401	−1.52129	−0.760646	−	0.649167i	$-0.775118\pi$
$419$	−31.1401	−1.52129	−0.760646	+	0.649167i	$0.775118\pi$
$420$	0	0
$421$	−4.42223	−0.215526	−0.107763	−	0.994177i	$-0.534369\pi$
$421$	−4.42223	−0.215526	−0.107763	+	0.994177i	$0.534369\pi$
$422$	31.3550	1.52634
$423$	0	0
$424$	9.98254	0.484795
$425$	3.29563	0.159862
$426$	0	0
$427$	9.14746	0.442677
$428$	3.99915	0.193306
$429$	0	0
$430$	5.68226	0.274023
$431$	17.2618	0.831473	0.415737	−	0.909485i	$-0.363524\pi$
$431$	17.2618	0.831473	0.415737	+	0.909485i	$0.363524\pi$
$432$	0	0
$433$	36.4416	1.75127	0.875635	−	0.482973i	$-0.160443\pi$
$433$	36.4416	1.75127	0.875635	+	0.482973i	$0.160443\pi$
$434$	1.50011	0.0720078
$435$	0	0
$436$	7.82925	0.374953
$437$	−2.76953	−0.132485
$438$	0	0
$439$	−13.2699	−0.633337	−0.316668	−	0.948536i	$-0.602564\pi$
$439$	−13.2699	−0.633337	−0.316668	+	0.948536i	$0.602564\pi$
$440$	4.98516	0.237658
$441$	0	0
$442$	2.47497	0.117722
$443$	−13.3154	−0.632635	−0.316318	−	0.948653i	$-0.602447\pi$
$443$	−13.3154	−0.632635	−0.316318	+	0.948653i	$0.602447\pi$
$444$	0	0
$445$	−4.45468	−0.211172
$446$	6.14063	0.290767
$447$	0	0
$448$	8.83593	0.417459
$449$	30.8559	1.45618	0.728089	−	0.685483i	$-0.240409\pi$
$449$	30.8559	1.45618	0.728089	+	0.685483i	$0.240409\pi$
$450$	0	0
$451$	−19.1802	−0.903161
$452$	6.53779	0.307512
$453$	0	0
$454$	−33.0623	−1.55169
$455$	−1.44125	−0.0675670
$456$	0	0
$457$	14.5223	0.679323	0.339662	−	0.940548i	$-0.389687\pi$
$457$	14.5223	0.679323	0.339662	+	0.940548i	$0.389687\pi$
$458$	23.6791	1.10645
$459$	0	0
$460$	−1.30886	−0.0610260
$461$	−19.4555	−0.906135	−0.453068	−	0.891476i	$-0.649670\pi$
$461$	−19.4555	−0.906135	−0.453068	+	0.891476i	$0.649670\pi$
$462$	0	0
$463$	40.8834	1.90001	0.950007	−	0.312229i	$-0.101076\pi$
$463$	40.8834	1.90001	0.950007	+	0.312229i	$0.101076\pi$
$464$	4.04931	0.187984
$465$	0	0
$466$	−1.35631	−0.0628298
$467$	22.7282	1.05174	0.525869	−	0.850566i	$-0.323740\pi$
$467$	22.7282	1.05174	0.525869	+	0.850566i	$0.323740\pi$
$468$	0	0
$469$	10.3697	0.478828
$470$	−2.60452	−0.120137
$471$	0	0
$472$	−33.2179	−1.52898
$473$	−32.2310	−1.48198
$474$	0	0
$475$	−2.65034	−0.121606
$476$	0.373756	0.0171311
$477$	0	0
$478$	24.9317	1.14035
$479$	20.8734	0.953729	0.476865	−	0.878977i	$-0.341773\pi$
$479$	20.8734	0.953729	0.476865	+	0.878977i	$0.341773\pi$
$480$	0	0
$481$	−22.0075	−1.00346
$482$	−20.1012	−0.915587
$483$	0	0
$484$	−0.0712570	−0.00323896
$485$	−5.75443	−0.261295
$486$	0	0
$487$	−9.56100	−0.433250	−0.216625	−	0.976255i	$-0.569505\pi$
$487$	−9.56100	−0.433250	−0.216625	+	0.976255i	$0.569505\pi$
$488$	28.0747	1.27088
$489$	0	0
$490$	0.588209	0.0265726
$491$	33.1329	1.49527	0.747634	−	0.664111i	$-0.231190\pi$
$491$	33.1329	1.49527	0.747634	+	0.664111i	$0.231190\pi$
$492$	0	0
$493$	1.06619	0.0480186
$494$	−1.99037	−0.0895508
$495$	0	0
$496$	3.26271	0.146500
$497$	2.82644	0.126783
$498$	0	0
$499$	−3.65216	−0.163493	−0.0817467	−	0.996653i	$-0.526050\pi$
$499$	−3.65216	−0.163493	−0.0817467	+	0.996653i	$0.526050\pi$
$500$	−2.56740	−0.114817
$501$	0	0
$502$	−11.3471	−0.506445
$503$	21.5098	0.959075	0.479537	−	0.877521i	$-0.340805\pi$
$503$	21.5098	0.959075	0.479537	+	0.877521i	$0.340805\pi$
$504$	0	0
$505$	5.73682	0.255285
$506$	−20.0641	−0.891959
$507$	0	0
$508$	−0.540168	−0.0239661
$509$	−0.500742	−0.0221950	−0.0110975	−	0.999938i	$-0.503533\pi$
$509$	−0.500742	−0.0221950	−0.0110975	+	0.999938i	$0.503533\pi$
$510$	0	0
$511$	−4.24827	−0.187932
$512$	23.9174	1.05701
$513$	0	0
$514$	−20.0342	−0.883671
$515$	3.15696	0.139112
$516$	0	0
$517$	14.7734	0.649733
$518$	8.98178	0.394637
$519$	0	0
$520$	−4.42338	−0.193978
$521$	38.1592	1.67179	0.835893	−	0.548893i	$-0.184950\pi$
$521$	38.1592	1.67179	0.835893	+	0.548893i	$0.184950\pi$
$522$	0	0
$523$	34.7513	1.51957	0.759783	−	0.650176i	$-0.225305\pi$
$523$	34.7513	1.51957	0.759783	+	0.650176i	$0.225305\pi$
$524$	−6.11448	−0.267112
$525$	0	0
$526$	−35.0608	−1.52872
$527$	0.859076	0.0374219
$528$	0	0
$529$	1.77239	0.0770606
$530$	1.91320	0.0831039
$531$	0	0
$532$	−0.300574	−0.0130315
$533$	17.0188	0.737164
$534$	0	0
$535$	3.60428	0.155827
$536$	31.8258	1.37467
$537$	0	0
$538$	−1.60551	−0.0692184
$539$	−3.33645	−0.143711
$540$	0	0
$541$	−24.4412	−1.05081	−0.525405	−	0.850852i	$-0.676086\pi$
$541$	−24.4412	−1.05081	−0.525405	+	0.850852i	$0.676086\pi$
$542$	−21.4078	−0.919542
$543$	0	0
$544$	2.05027	0.0879046
$545$	7.05621	0.302255
$546$	0	0
$547$	0.0873940	0.00373670	0.00186835	−	0.999998i	$-0.499405\pi$
$547$	0.0873940	0.00373670	0.00186835	+	0.999998i	$0.499405\pi$
$548$	0.946467	0.0404311
$549$	0	0
$550$	−19.2007	−0.818719
$551$	−0.857426	−0.0365276
$552$	0	0
$553$	13.7939	0.586574
$554$	29.1278	1.23752
$555$	0	0
$556$	6.66360	0.282600
$557$	33.8597	1.43468	0.717342	−	0.696721i	$-0.245359\pi$
$557$	33.8597	1.43468	0.717342	+	0.696721i	$0.245359\pi$
$558$	0	0
$559$	28.5989	1.20960
$560$	1.27934	0.0540620
$561$	0	0
$562$	16.7257	0.705532
$563$	−13.7648	−0.580116	−0.290058	−	0.957009i	$-0.593675\pi$
$563$	−13.7648	−0.580116	−0.290058	+	0.957009i	$0.593675\pi$
$564$	0	0
$565$	5.89227	0.247890
$566$	−18.8878	−0.793914
$567$	0	0
$568$	8.67468	0.363982
$569$	−22.2522	−0.932861	−0.466431	−	0.884558i	$-0.654460\pi$
$569$	−22.2522	−0.932861	−0.466431	+	0.884558i	$0.654460\pi$
$570$	0	0
$571$	15.4552	0.646779	0.323389	−	0.946266i	$-0.395178\pi$
$571$	15.4552	0.646779	0.323389	+	0.946266i	$0.395178\pi$
$572$	5.33548	0.223088
$573$	0	0
$574$	−6.94576	−0.289910
$575$	23.7063	0.988621
$576$	0	0
$577$	38.8006	1.61529	0.807645	−	0.589670i	$-0.200742\pi$
$577$	38.8006	1.61529	0.807645	+	0.589670i	$0.200742\pi$
$578$	19.9615	0.830291
$579$	0	0
$580$	−0.405214	−0.0168256
$581$	−0.539810	−0.0223951
$582$	0	0
$583$	−10.8521	−0.449446
$584$	−13.0384	−0.539534
$585$	0	0
$586$	12.5115	0.516844
$587$	−0.382995	−0.0158079	−0.00790394	−	0.999969i	$-0.502516\pi$
$587$	−0.382995	−0.0158079	−0.00790394	+	0.999969i	$0.502516\pi$
$588$	0	0
$589$	−0.690868	−0.0284667
$590$	−6.36635	−0.262099
$591$	0	0
$592$	19.5351	0.802890
$593$	−16.2475	−0.667205	−0.333602	−	0.942714i	$-0.608264\pi$
$593$	−16.2475	−0.667205	−0.333602	+	0.942714i	$0.608264\pi$
$594$	0	0
$595$	0.336852	0.0138096
$596$	0.157919	0.00646862
$597$	0	0
$598$	17.8031	0.728022
$599$	−18.6566	−0.762289	−0.381144	−	0.924516i	$-0.624470\pi$
$599$	−18.6566	−0.762289	−0.381144	+	0.924516i	$0.624470\pi$
$600$	0	0
$601$	9.10957	0.371587	0.185794	−	0.982589i	$-0.440514\pi$
$601$	9.10957	0.371587	0.185794	+	0.982589i	$0.440514\pi$
$602$	−11.6719	−0.475710
$603$	0	0
$604$	−5.97938	−0.243298
$605$	−0.0642213	−0.00261097
$606$	0	0
$607$	26.7513	1.08580	0.542902	−	0.839796i	$-0.317326\pi$
$607$	26.7513	1.08580	0.542902	+	0.839796i	$0.317326\pi$
$608$	−1.64882	−0.0668687
$609$	0	0
$610$	5.38063	0.217855
$611$	−13.1086	−0.530315
$612$	0	0
$613$	35.0822	1.41696	0.708478	−	0.705733i	$-0.249382\pi$
$613$	35.0822	1.41696	0.708478	+	0.705733i	$0.249382\pi$
$614$	−8.74317	−0.352845
$615$	0	0
$616$	−10.2400	−0.412580
$617$	2.43811	0.0981545	0.0490773	−	0.998795i	$-0.484372\pi$
$617$	2.43811	0.0981545	0.0490773	+	0.998795i	$0.484372\pi$
$618$	0	0
$619$	15.1723	0.609828	0.304914	−	0.952380i	$-0.401372\pi$
$619$	15.1723	0.609828	0.304914	+	0.952380i	$0.401372\pi$
$620$	−0.326500	−0.0131125
$621$	0	0
$622$	−28.6599	−1.14916
$623$	9.15032	0.366600
$624$	0	0
$625$	21.5011	0.860043
$626$	−37.4445	−1.49658
$627$	0	0
$628$	5.25532	0.209710
$629$	5.14363	0.205090
$630$	0	0
$631$	−10.2068	−0.406325	−0.203162	−	0.979145i	$-0.565122\pi$
$631$	−10.2068	−0.406325	−0.203162	+	0.979145i	$0.565122\pi$
$632$	42.3350	1.68400
$633$	0	0
$634$	6.20273	0.246342
$635$	−0.486834	−0.0193194
$636$	0	0
$637$	2.96046	0.117298
$638$	−6.21171	−0.245924
$639$	0	0
$640$	2.31227	0.0914003
$641$	26.5871	1.05013	0.525063	−	0.851063i	$-0.324042\pi$
$641$	26.5871	1.05013	0.525063	+	0.851063i	$0.324042\pi$
$642$	0	0
$643$	18.5503	0.731552	0.365776	−	0.930703i	$-0.380804\pi$
$643$	18.5503	0.731552	0.365776	+	0.930703i	$0.380804\pi$
$644$	2.68852	0.105942
$645$	0	0
$646$	0.465192	0.0183027
$647$	−29.1301	−1.14522	−0.572612	−	0.819826i	$-0.694070\pi$
$647$	−29.1301	−1.14522	−0.572612	+	0.819826i	$0.694070\pi$
$648$	0	0
$649$	36.1113	1.41749
$650$	17.0369	0.668243
$651$	0	0
$652$	12.8113	0.501729
$653$	−11.0653	−0.433020	−0.216510	−	0.976280i	$-0.569467\pi$
$653$	−11.0653	−0.433020	−0.216510	+	0.976280i	$0.569467\pi$
$654$	0	0
$655$	−5.51075	−0.215323
$656$	−15.1069	−0.589824
$657$	0	0
$658$	5.34991	0.208561
$659$	23.4739	0.914414	0.457207	−	0.889360i	$-0.348850\pi$
$659$	23.4739	0.914414	0.457207	+	0.889360i	$0.348850\pi$
$660$	0	0
$661$	23.1407	0.900069	0.450035	−	0.893011i	$-0.351412\pi$
$661$	23.1407	0.900069	0.450035	+	0.893011i	$0.351412\pi$
$662$	−30.6423	−1.19095
$663$	0	0
$664$	−1.65674	−0.0642941
$665$	−0.270896	−0.0105049
$666$	0	0
$667$	7.66935	0.296959
$668$	−6.50849	−0.251821
$669$	0	0
$670$	6.09955	0.235646
$671$	−30.5201	−1.17821
$672$	0	0
$673$	−49.6462	−1.91372	−0.956860	−	0.290551i	$-0.906162\pi$
$673$	−49.6462	−1.91372	−0.956860	+	0.290551i	$0.906162\pi$
$674$	37.3380	1.43821
$675$	0	0
$676$	2.28797	0.0879989
$677$	28.5588	1.09760	0.548802	−	0.835952i	$-0.315084\pi$
$677$	28.5588	1.09760	0.548802	+	0.835952i	$0.315084\pi$
$678$	0	0
$679$	11.8201	0.453614
$680$	1.03384	0.0396459
$681$	0	0
$682$	−5.00506	−0.191654
$683$	29.1869	1.11680	0.558402	−	0.829570i	$-0.311415\pi$
$683$	29.1869	1.11680	0.558402	+	0.829570i	$0.311415\pi$
$684$	0	0
$685$	0.853016	0.0325921
$686$	−1.20823	−0.0461306
$687$	0	0
$688$	−25.3860	−0.967833
$689$	9.62913	0.366840
$690$	0	0
$691$	−32.1744	−1.22397	−0.611986	−	0.790868i	$-0.709629\pi$
$691$	−32.1744	−1.22397	−0.611986	+	0.790868i	$0.709629\pi$
$692$	−7.05416	−0.268159
$693$	0	0
$694$	−15.0028	−0.569499
$695$	6.00566	0.227807
$696$	0	0
$697$	−3.97765	−0.150664
$698$	−20.8059	−0.787514
$699$	0	0
$700$	2.57282	0.0972434
$701$	−18.0672	−0.682388	−0.341194	−	0.939993i	$-0.610831\pi$
$701$	−18.0672	−0.682388	−0.341194	+	0.939993i	$0.610831\pi$
$702$	0	0
$703$	−4.13650	−0.156011
$704$	−29.4807	−1.11109
$705$	0	0
$706$	14.5702	0.548358
$707$	−11.7839	−0.443181
$708$	0	0
$709$	15.1868	0.570353	0.285176	−	0.958475i	$-0.407948\pi$
$709$	15.1868	0.570353	0.285176	+	0.958475i	$0.407948\pi$
$710$	1.66254	0.0623940
$711$	0	0
$712$	28.0834	1.05247
$713$	6.17955	0.231426
$714$	0	0
$715$	4.80867	0.179834
$716$	−6.32388	−0.236335
$717$	0	0
$718$	−3.80494	−0.141999
$719$	−19.6906	−0.734335	−0.367168	−	0.930155i	$-0.619672\pi$
$719$	−19.6906	−0.734335	−0.367168	+	0.930155i	$0.619672\pi$
$720$	0	0
$721$	−6.48469	−0.241502
$722$	22.5824	0.840428
$723$	0	0
$724$	−12.6207	−0.469044
$725$	7.33930	0.272575
$726$	0	0
$727$	−35.5316	−1.31779	−0.658897	−	0.752233i	$-0.728977\pi$
$727$	−35.5316	−1.31779	−0.658897	+	0.752233i	$0.728977\pi$
$728$	9.08602	0.336750
$729$	0	0
$730$	−2.49887	−0.0924873
$731$	−6.68417	−0.247223
$732$	0	0
$733$	−13.6230	−0.503178	−0.251589	−	0.967834i	$-0.580953\pi$
$733$	−13.6230	−0.503178	−0.251589	+	0.967834i	$0.580953\pi$
$734$	22.3499	0.824952
$735$	0	0
$736$	14.7481	0.543623
$737$	−34.5980	−1.27443
$738$	0	0
$739$	−46.3582	−1.70531	−0.852656	−	0.522472i	$-0.825010\pi$
$739$	−46.3582	−1.70531	−0.852656	+	0.522472i	$0.825010\pi$
$740$	−1.95488	−0.0718629
$741$	0	0
$742$	−3.92987	−0.144270
$743$	11.2265	0.411859	0.205930	−	0.978567i	$-0.433978\pi$
$743$	11.2265	0.411859	0.205930	+	0.978567i	$0.433978\pi$
$744$	0	0
$745$	0.142327	0.00521445
$746$	−4.04481	−0.148091
$747$	0	0
$748$	−1.24702	−0.0455955
$749$	−7.40352	−0.270519
$750$	0	0
$751$	44.8285	1.63582	0.817908	−	0.575349i	$-0.195134\pi$
$751$	44.8285	1.63582	0.817908	+	0.575349i	$0.195134\pi$
$752$	11.6359	0.424318
$753$	0	0
$754$	5.51170	0.200724
$755$	−5.38899	−0.196126
$756$	0	0
$757$	30.5896	1.11180	0.555898	−	0.831250i	$-0.312374\pi$
$757$	30.5896	1.11180	0.555898	+	0.831250i	$0.312374\pi$
$758$	−8.30334	−0.301591
$759$	0	0
$760$	−0.831412	−0.0301585
$761$	−45.8271	−1.66123	−0.830616	−	0.556846i	$-0.812011\pi$
$761$	−45.8271	−1.66123	−0.830616	+	0.556846i	$0.812011\pi$
$762$	0	0
$763$	−14.4941	−0.524721
$764$	4.03553	0.146000
$765$	0	0
$766$	4.67810	0.169027
$767$	−32.0419	−1.15697
$768$	0	0
$769$	18.1925	0.656037	0.328019	−	0.944671i	$-0.393619\pi$
$769$	18.1925	0.656037	0.328019	+	0.944671i	$0.393619\pi$
$770$	−1.96253	−0.0707248
$771$	0	0
$772$	−1.20562	−0.0433912
$773$	−27.5311	−0.990227	−0.495113	−	0.868828i	$-0.664873\pi$
$773$	−27.5311	−0.990227	−0.495113	+	0.868828i	$0.664873\pi$
$774$	0	0
$775$	5.91361	0.212423
$776$	36.2773	1.30228
$777$	0	0
$778$	−28.9701	−1.03863
$779$	3.19882	0.114610
$780$	0	0
$781$	−9.43028	−0.337442
$782$	−4.16096	−0.148796
$783$	0	0
$784$	−2.62788	−0.0938529
$785$	4.73642	0.169050
$786$	0	0
$787$	−30.5551	−1.08917	−0.544586	−	0.838705i	$-0.683313\pi$
$787$	−30.5551	−1.08917	−0.544586	+	0.838705i	$0.683313\pi$
$788$	−13.3160	−0.474364
$789$	0	0
$790$	8.11367	0.288672
$791$	−12.1032	−0.430342
$792$	0	0
$793$	27.0807	0.961665
$794$	−11.6604	−0.413812
$795$	0	0
$796$	−2.83924	−0.100634
$797$	37.6677	1.33426	0.667129	−	0.744943i	$-0.267523\pi$
$797$	37.6677	1.33426	0.667129	+	0.744943i	$0.267523\pi$
$798$	0	0
$799$	3.06375	0.108388
$800$	14.1134	0.498985
$801$	0	0
$802$	−2.98072	−0.105253
$803$	14.1741	0.500194
$804$	0	0
$805$	2.42306	0.0854017
$806$	4.44103	0.156429
$807$	0	0
$808$	−36.1663	−1.27233
$809$	12.1123	0.425845	0.212923	−	0.977069i	$-0.431702\pi$
$809$	12.1123	0.425845	0.212923	+	0.977069i	$0.431702\pi$
$810$	0	0
$811$	43.5325	1.52863	0.764317	−	0.644841i	$-0.223076\pi$
$811$	43.5325	1.52863	0.764317	+	0.644841i	$0.223076\pi$
$812$	0.832346	0.0292096
$813$	0	0
$814$	−29.9673	−1.05035
$815$	11.5464	0.404451
$816$	0	0
$817$	5.37540	0.188062
$818$	25.8687	0.904479
$819$	0	0
$820$	1.51174	0.0527924
$821$	23.3163	0.813745	0.406873	−	0.913485i	$-0.366619\pi$
$821$	23.3163	0.813745	0.406873	+	0.913485i	$0.366619\pi$
$822$	0	0
$823$	−2.59914	−0.0906004	−0.0453002	−	0.998973i	$-0.514424\pi$
$823$	−2.59914	−0.0906004	−0.0453002	+	0.998973i	$0.514424\pi$
$824$	−19.9023	−0.693329
$825$	0	0
$826$	13.0771	0.455009
$827$	12.6692	0.440550	0.220275	−	0.975438i	$-0.429305\pi$
$827$	12.6692	0.440550	0.220275	+	0.975438i	$0.429305\pi$
$828$	0	0
$829$	−6.91219	−0.240070	−0.120035	−	0.992770i	$-0.538301\pi$
$829$	−6.91219	−0.240070	−0.120035	+	0.992770i	$0.538301\pi$
$830$	−0.317521	−0.0110213
$831$	0	0
$832$	26.1585	0.906881
$833$	−0.691924	−0.0239738
$834$	0	0
$835$	−5.86586	−0.202997
$836$	1.00285	0.0346843
$837$	0	0
$838$	37.6245	1.29972
$839$	−0.427700	−0.0147658	−0.00738292	−	0.999973i	$-0.502350\pi$
$839$	−0.427700	−0.0147658	−0.00738292	+	0.999973i	$0.502350\pi$
$840$	0	0
$841$	−26.6256	−0.918125
$842$	5.34310	0.184135
$843$	0	0
$844$	14.0180	0.482518
$845$	2.06206	0.0709371
$846$	0	0
$847$	0.131916	0.00453270
$848$	−8.54738	−0.293518
$849$	0	0
$850$	−3.98190	−0.136578
$851$	36.9994	1.26832
$852$	0	0
$853$	−16.3166	−0.558668	−0.279334	−	0.960194i	$-0.590114\pi$
$853$	−16.3166	−0.558668	−0.279334	+	0.960194i	$0.590114\pi$
$854$	−11.0523	−0.378201
$855$	0	0
$856$	−22.7223	−0.776632
$857$	−5.34493	−0.182579	−0.0912897	−	0.995824i	$-0.529099\pi$
$857$	−5.34493	−0.182579	−0.0912897	+	0.995824i	$0.529099\pi$
$858$	0	0
$859$	32.6604	1.11436	0.557179	−	0.830392i	$-0.311884\pi$
$859$	32.6604	1.11436	0.557179	+	0.830392i	$0.311884\pi$
$860$	2.54038	0.0866263
$861$	0	0
$862$	−20.8563	−0.710370
$863$	23.4800	0.799268	0.399634	−	0.916675i	$-0.369137\pi$
$863$	23.4800	0.799268	0.399634	+	0.916675i	$0.369137\pi$
$864$	0	0
$865$	−6.35765	−0.216167
$866$	−44.0300	−1.49620
$867$	0	0
$868$	0.670660	0.0227637
$869$	−46.0225	−1.56121
$870$	0	0
$871$	30.6991	1.04020
$872$	−44.4841	−1.50642
$873$	0	0
$874$	3.34624	0.113188
$875$	4.75295	0.160679
$876$	0	0
$877$	−23.4223	−0.790916	−0.395458	−	0.918484i	$-0.629414\pi$
$877$	−23.4223	−0.790916	−0.395458	+	0.918484i	$0.629414\pi$
$878$	16.0331	0.541092
$879$	0	0
$880$	−4.26846	−0.143890
$881$	−3.19683	−0.107704	−0.0538520	−	0.998549i	$-0.517150\pi$
$881$	−3.19683	−0.107704	−0.0538520	+	0.998549i	$0.517150\pi$
$882$	0	0
$883$	−2.91600	−0.0981312	−0.0490656	−	0.998796i	$-0.515624\pi$
$883$	−2.91600	−0.0981312	−0.0490656	+	0.998796i	$0.515624\pi$
$884$	1.10649	0.0372153
$885$	0	0
$886$	16.0882	0.540492
$887$	30.6363	1.02867	0.514334	−	0.857590i	$-0.328039\pi$
$887$	30.6363	1.02867	0.514334	+	0.857590i	$0.328039\pi$
$888$	0	0
$889$	1.00000	0.0335389
$890$	5.38230	0.180415
$891$	0	0
$892$	2.74530	0.0919196
$893$	−2.46387	−0.0824501
$894$	0	0
$895$	−5.69948	−0.190513
$896$	−4.74960	−0.158673
$897$	0	0
$898$	−37.2811	−1.24409
$899$	1.91314	0.0638069
$900$	0	0
$901$	−2.25053	−0.0749762
$902$	23.1742	0.771616
$903$	0	0
$904$	−37.1463	−1.23547
$905$	−11.3746	−0.378103
$906$	0	0
$907$	−28.0303	−0.930732	−0.465366	−	0.885118i	$-0.654077\pi$
$907$	−28.0303	−0.930732	−0.465366	+	0.885118i	$0.654077\pi$
$908$	−14.7812	−0.490533
$909$	0	0
$910$	1.74137	0.0577259
$911$	49.5167	1.64056	0.820280	−	0.571962i	$-0.193817\pi$
$911$	49.5167	1.64056	0.820280	+	0.571962i	$0.193817\pi$
$912$	0	0
$913$	1.80105	0.0596061
$914$	−17.5463	−0.580381
$915$	0	0
$916$	10.5863	0.349781
$917$	11.3196	0.373805
$918$	0	0
$919$	34.0996	1.12484	0.562420	−	0.826851i	$-0.309870\pi$
$919$	34.0996	1.12484	0.562420	+	0.826851i	$0.309870\pi$
$920$	7.43667	0.245180
$921$	0	0
$922$	23.5069	0.774158
$923$	8.36757	0.275422
$924$	0	0
$925$	35.4071	1.16418
$926$	−49.3968	−1.62328
$927$	0	0
$928$	4.56591	0.149883
$929$	36.7891	1.20701	0.603506	−	0.797359i	$-0.293770\pi$
$929$	36.7891	1.20701	0.603506	+	0.797359i	$0.293770\pi$
$930$	0	0
$931$	0.556445	0.0182367
$932$	−0.606368	−0.0198622
$933$	0	0
$934$	−27.4611	−0.898553
$935$	−1.12389	−0.0367552
$936$	0	0
$937$	−8.71731	−0.284782	−0.142391	−	0.989810i	$-0.545479\pi$
$937$	−8.71731	−0.284782	−0.142391	+	0.989810i	$0.545479\pi$
$938$	−12.5290	−0.409087
$939$	0	0
$940$	−1.16441	−0.0379788
$941$	−14.2029	−0.463001	−0.231500	−	0.972835i	$-0.574363\pi$
$941$	−14.2029	−0.463001	−0.231500	+	0.972835i	$0.574363\pi$
$942$	0	0
$943$	−28.6123	−0.931744
$944$	28.4423	0.925717
$945$	0	0
$946$	38.9426	1.26613
$947$	47.6944	1.54986	0.774930	−	0.632047i	$-0.217785\pi$
$947$	47.6944	1.54986	0.774930	+	0.632047i	$0.217785\pi$
$948$	0	0
$949$	−12.5768	−0.408261
$950$	3.20224	0.103894
$951$	0	0
$952$	−2.12360	−0.0688262
$953$	−27.4401	−0.888873	−0.444436	−	0.895810i	$-0.646596\pi$
$953$	−27.4401	−0.888873	−0.444436	+	0.895810i	$0.646596\pi$
$954$	0	0
$955$	3.63707	0.117693
$956$	11.1463	0.360497
$957$	0	0
$958$	−25.2199	−0.814820
$959$	−1.75217	−0.0565805
$960$	0	0
$961$	−29.4585	−0.950274
$962$	26.5902	0.857303
$963$	0	0
$964$	−8.98671	−0.289442
$965$	−1.08658	−0.0349783
$966$	0	0
$967$	11.3899	0.366275	0.183138	−	0.983087i	$-0.441375\pi$
$967$	11.3899	0.366275	0.183138	+	0.983087i	$0.441375\pi$
$968$	0.404867	0.0130129
$969$	0	0
$970$	6.95270	0.223238
$971$	56.1074	1.80057	0.900286	−	0.435299i	$-0.143358\pi$
$971$	56.1074	1.80057	0.900286	+	0.435299i	$0.143358\pi$
$972$	0	0
$973$	−12.3362	−0.395479
$974$	11.5519	0.370148
$975$	0	0
$976$	−24.0385	−0.769452
$977$	29.0159	0.928300	0.464150	−	0.885757i	$-0.346360\pi$
$977$	29.0159	0.928300	0.464150	+	0.885757i	$0.346360\pi$
$978$	0	0
$979$	−30.5296	−0.975730
$980$	0.262972	0.00840034
$981$	0	0
$982$	−40.0324	−1.27748
$983$	−32.9269	−1.05020	−0.525102	−	0.851039i	$-0.675973\pi$
$983$	−32.9269	−1.05020	−0.525102	+	0.851039i	$0.675973\pi$
$984$	0	0
$985$	−12.0013	−0.382392
$986$	−1.28820	−0.0410248
$987$	0	0
$988$	−0.889838	−0.0283095
$989$	−48.0810	−1.52889
$990$	0	0
$991$	−13.4717	−0.427944	−0.213972	−	0.976840i	$-0.568640\pi$
$991$	−13.4717	−0.427944	−0.213972	+	0.976840i	$0.568640\pi$
$992$	3.67896	0.116807
$993$	0	0
$994$	−3.41500	−0.108317
$995$	−2.55890	−0.0811227
$996$	0	0
$997$	−5.54506	−0.175614	−0.0878069	−	0.996138i	$-0.527986\pi$
$997$	−5.54506	−0.175614	−0.0878069	+	0.996138i	$0.527986\pi$
$998$	4.41267	0.139681
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8001.2.a.v.1.8		19
3.2	odd	2		2667.2.a.q.1.12	✓	19

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2667.2.a.q.1.12	✓	19	3.2	odd	2
8001.2.a.v.1.8		19	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:	\( N \)	\(=\)	\( 8001 = 3^{2} \cdot 7 \cdot 127 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8001.a (trivial)

\(f(q)\)	\(=\)	\(q-1.20823 q^{2} -0.540168 q^{4} -0.486834 q^{5} +1.00000 q^{7} +3.06912 q^{8} +O(q^{10})\)	\(q-1.20823 q^{2} -0.540168 q^{4} -0.486834 q^{5} +1.00000 q^{7} +3.06912 q^{8} +0.588209 q^{10} -3.33645 q^{11} +2.96046 q^{13} -1.20823 q^{14} -2.62788 q^{16} -0.691924 q^{17} +0.556445 q^{19} +0.262972 q^{20} +4.03122 q^{22} -4.97719 q^{23} -4.76299 q^{25} -3.57693 q^{26} -0.540168 q^{28} -1.54090 q^{29} -1.24157 q^{31} -2.96314 q^{32} +0.836007 q^{34} -0.486834 q^{35} -7.43380 q^{37} -0.672316 q^{38} -1.49415 q^{40} +5.74868 q^{41} +9.66027 q^{43} +1.80225 q^{44} +6.01361 q^{46} -4.42787 q^{47} +1.00000 q^{49} +5.75481 q^{50} -1.59915 q^{52} +3.25257 q^{53} +1.62430 q^{55} +3.06912 q^{56} +1.86177 q^{58} -10.8233 q^{59} +9.14746 q^{61} +1.50011 q^{62} +8.83593 q^{64} -1.44125 q^{65} +10.3697 q^{67} +0.373756 q^{68} +0.588209 q^{70} +2.82644 q^{71} -4.24827 q^{73} +8.98178 q^{74} -0.300574 q^{76} -3.33645 q^{77} +13.7939 q^{79} +1.27934 q^{80} -6.94576 q^{82} -0.539810 q^{83} +0.336852 q^{85} -11.6719 q^{86} -10.2400 q^{88} +9.15032 q^{89} +2.96046 q^{91} +2.68852 q^{92} +5.34991 q^{94} -0.270896 q^{95} +11.8201 q^{97} -1.20823 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 19 q - 4 q^{2} + 22 q^{4} - 5 q^{5} + 19 q^{7} - 9 q^{8}+O(q^{10}) \) 19 * q - 4 * q^2 + 22 * q^4 - 5 * q^5 + 19 * q^7 - 9 * q^8	\( 19 q - 4 q^{2} + 22 q^{4} - 5 q^{5} + 19 q^{7} - 9 q^{8} + 9 q^{11} + 24 q^{13} - 4 q^{14} + 20 q^{16} - 17 q^{17} + 23 q^{19} - 5 q^{20} - 3 q^{22} + 17 q^{23} + 38 q^{25} - 28 q^{26} + 22 q^{28} - 2 q^{29} + 16 q^{31} - 17 q^{32} + 29 q^{34} - 5 q^{35} + 56 q^{37} - 2 q^{38} - 13 q^{40} + 7 q^{41} + 19 q^{43} + 29 q^{44} + 10 q^{46} - 25 q^{47} + 19 q^{49} + 9 q^{50} + 16 q^{52} - 18 q^{53} + 10 q^{55} - 9 q^{56} + 31 q^{58} - 11 q^{59} + 26 q^{61} - 26 q^{62} + 45 q^{64} - 27 q^{65} + 24 q^{67} - 14 q^{68} + 32 q^{71} + 51 q^{73} + 12 q^{76} + 9 q^{77} + 30 q^{79} + 30 q^{80} - 52 q^{82} - q^{83} + 44 q^{85} + 24 q^{86} - 30 q^{88} - 5 q^{89} + 24 q^{91} + 88 q^{92} + 7 q^{94} + 24 q^{95} + 5 q^{97} - 4 q^{98}+O(q^{100}) \) 19 * q - 4 * q^2 + 22 * q^4 - 5 * q^5 + 19 * q^7 - 9 * q^8 + 9 * q^11 + 24 * q^13 - 4 * q^14 + 20 * q^16 - 17 * q^17 + 23 * q^19 - 5 * q^20 - 3 * q^22 + 17 * q^23 + 38 * q^25 - 28 * q^26 + 22 * q^28 - 2 * q^29 + 16 * q^31 - 17 * q^32 + 29 * q^34 - 5 * q^35 + 56 * q^37 - 2 * q^38 - 13 * q^40 + 7 * q^41 + 19 * q^43 + 29 * q^44 + 10 * q^46 - 25 * q^47 + 19 * q^49 + 9 * q^50 + 16 * q^52 - 18 * q^53 + 10 * q^55 - 9 * q^56 + 31 * q^58 - 11 * q^59 + 26 * q^61 - 26 * q^62 + 45 * q^64 - 27 * q^65 + 24 * q^67 - 14 * q^68 + 32 * q^71 + 51 * q^73 + 12 * q^76 + 9 * q^77 + 30 * q^79 + 30 * q^80 - 52 * q^82 - q^83 + 44 * q^85 + 24 * q^86 - 30 * q^88 - 5 * q^89 + 24 * q^91 + 88 * q^92 + 7 * q^94 + 24 * q^95 + 5 * q^97 - 4 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

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Groups

Database

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