LMFDB - Embedded newform 8001.2.a.v.1.16

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			1.16
Root			$-1.91759$ 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)

$f(q)$	$=$	$q+1.91759 q^{2} +1.67713 q^{4} -1.63006 q^{5} +1.00000 q^{7} -0.619124 q^{8} +O(q^{10})$	\(q+1.91759 q^{2} +1.67713 q^{4} -1.63006 q^{5} +1.00000 q^{7} -0.619124 q^{8} -3.12578 q^{10} -4.53723 q^{11} +0.785019 q^{13} +1.91759 q^{14} -4.54149 q^{16} +4.58625 q^{17} +7.32031 q^{19} -2.73383 q^{20} -8.70052 q^{22} -1.77992 q^{23} -2.34290 q^{25} +1.50534 q^{26} +1.67713 q^{28} -4.77456 q^{29} +5.06545 q^{31} -7.47045 q^{32} +8.79452 q^{34} -1.63006 q^{35} -1.03335 q^{37} +14.0373 q^{38} +1.00921 q^{40} +10.6489 q^{41} +8.26855 q^{43} -7.60954 q^{44} -3.41315 q^{46} -7.59560 q^{47} +1.00000 q^{49} -4.49271 q^{50} +1.31658 q^{52} -0.151205 q^{53} +7.39596 q^{55} -0.619124 q^{56} -9.15563 q^{58} -4.69350 q^{59} -12.1937 q^{61} +9.71343 q^{62} -5.24224 q^{64} -1.27963 q^{65} +1.56249 q^{67} +7.69175 q^{68} -3.12578 q^{70} +13.2252 q^{71} +5.59318 q^{73} -1.98153 q^{74} +12.2771 q^{76} -4.53723 q^{77} -3.81705 q^{79} +7.40291 q^{80} +20.4201 q^{82} -0.734264 q^{83} -7.47586 q^{85} +15.8556 q^{86} +2.80911 q^{88} +5.16974 q^{89} +0.785019 q^{91} -2.98516 q^{92} -14.5652 q^{94} -11.9326 q^{95} -5.35014 q^{97} +1.91759 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	1.91759	1.35594	0.677969	−	0.735091i	$-0.262860\pi$
$2$	1.91759	1.35594	0.677969	+	0.735091i	$0.262860\pi$
$3$	0	0
$3$	0	0
$4$	1.67713	0.838567
$5$	−1.63006	−0.728986	−0.364493	−	0.931206i	$-0.618758\pi$
$5$	−1.63006	−0.728986	−0.364493	+	0.931206i	$0.618758\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−0.619124	−0.218893
$9$	0	0
$10$	−3.12578	−0.988459
$11$	−4.53723	−1.36803	−0.684013	−	0.729470i	$-0.739767\pi$
$11$	−4.53723	−1.36803	−0.684013	+	0.729470i	$0.739767\pi$
$12$	0	0
$13$	0.785019	0.217725	0.108863	−	0.994057i	$-0.465279\pi$
$13$	0.785019	0.217725	0.108863	+	0.994057i	$0.465279\pi$
$14$	1.91759	0.512496
$15$	0	0
$16$	−4.54149	−1.13537
$17$	4.58625	1.11233	0.556164	−	0.831073i	$-0.312273\pi$
$17$	4.58625	1.11233	0.556164	+	0.831073i	$0.312273\pi$
$18$	0	0
$19$	7.32031	1.67940	0.839698	−	0.543054i	$-0.182732\pi$
$19$	7.32031	1.67940	0.839698	+	0.543054i	$0.182732\pi$
$20$	−2.73383	−0.611303
$21$	0	0
$22$	−8.70052	−1.85496
$23$	−1.77992	−0.371139	−0.185569	−	0.982631i	$-0.559413\pi$
$23$	−1.77992	−0.371139	−0.185569	+	0.982631i	$0.559413\pi$
$24$	0	0
$25$	−2.34290	−0.468580
$26$	1.50534	0.295222
$27$	0	0
$28$	1.67713	0.316948
$29$	−4.77456	−0.886614	−0.443307	−	0.896370i	$-0.646195\pi$
$29$	−4.77456	−0.886614	−0.443307	+	0.896370i	$0.646195\pi$
$30$	0	0
$31$	5.06545	0.909781	0.454891	−	0.890547i	$-0.349678\pi$
$31$	5.06545	0.909781	0.454891	+	0.890547i	$0.349678\pi$
$32$	−7.47045	−1.32060
$33$	0	0
$34$	8.79452	1.50825
$35$	−1.63006	−0.275531
$36$	0	0
$37$	−1.03335	−0.169881	−0.0849407	−	0.996386i	$-0.527070\pi$
$37$	−1.03335	−0.169881	−0.0849407	+	0.996386i	$0.527070\pi$
$38$	14.0373	2.27716
$39$	0	0
$40$	1.00921	0.159570
$41$	10.6489	1.66307	0.831536	−	0.555471i	$-0.187462\pi$
$41$	10.6489	1.66307	0.831536	+	0.555471i	$0.187462\pi$
$42$	0	0
$43$	8.26855	1.26094	0.630471	−	0.776213i	$-0.282862\pi$
$43$	8.26855	1.26094	0.630471	+	0.776213i	$0.282862\pi$
$44$	−7.60954	−1.14718
$45$	0	0
$46$	−3.41315	−0.503241
$47$	−7.59560	−1.10793	−0.553966	−	0.832539i	$-0.686886\pi$
$47$	−7.59560	−1.10793	−0.553966	+	0.832539i	$0.686886\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−4.49271	−0.635365
$51$	0	0
$52$	1.31658	0.182577
$53$	−0.151205	−0.0207696	−0.0103848	−	0.999946i	$-0.503306\pi$
$53$	−0.151205	−0.0207696	−0.0103848	+	0.999946i	$0.503306\pi$
$54$	0	0
$55$	7.39596	0.997272
$56$	−0.619124	−0.0827339
$57$	0	0
$58$	−9.15563	−1.20219
$59$	−4.69350	−0.611042	−0.305521	−	0.952185i	$-0.598831\pi$
$59$	−4.69350	−0.611042	−0.305521	+	0.952185i	$0.598831\pi$
$60$	0	0
$61$	−12.1937	−1.56125	−0.780623	−	0.625002i	$-0.785098\pi$
$61$	−12.1937	−1.56125	−0.780623	+	0.625002i	$0.785098\pi$
$62$	9.71343	1.23361
$63$	0	0
$64$	−5.24224	−0.655280
$65$	−1.27963	−0.158718
$66$	0	0
$67$	1.56249	0.190889	0.0954444	−	0.995435i	$-0.469573\pi$
$67$	1.56249	0.190889	0.0954444	+	0.995435i	$0.469573\pi$
$68$	7.69175	0.932761
$69$	0	0
$70$	−3.12578	−0.373602
$71$	13.2252	1.56954	0.784768	−	0.619789i	$-0.212782\pi$
$71$	13.2252	1.56954	0.784768	+	0.619789i	$0.212782\pi$
$72$	0	0
$73$	5.59318	0.654632	0.327316	−	0.944915i	$-0.393856\pi$
$73$	5.59318	0.654632	0.327316	+	0.944915i	$0.393856\pi$
$74$	−1.98153	−0.230349
$75$	0	0
$76$	12.2771	1.40829
$77$	−4.53723	−0.517065
$78$	0	0
$79$	−3.81705	−0.429451	−0.214726	−	0.976674i	$-0.568886\pi$
$79$	−3.81705	−0.429451	−0.214726	+	0.976674i	$0.568886\pi$
$80$	7.40291	0.827670
$81$	0	0
$82$	20.4201	2.25502
$83$	−0.734264	−0.0805960	−0.0402980	−	0.999188i	$-0.512831\pi$
$83$	−0.734264	−0.0805960	−0.0402980	+	0.999188i	$0.512831\pi$
$84$	0	0
$85$	−7.47586	−0.810871
$86$	15.8556	1.70976
$87$	0	0
$88$	2.80911	0.299452
$89$	5.16974	0.547991	0.273996	−	0.961731i	$-0.411655\pi$
$89$	5.16974	0.547991	0.273996	+	0.961731i	$0.411655\pi$
$90$	0	0
$91$	0.785019	0.0822923
$92$	−2.98516	−0.311225
$93$	0	0
$94$	−14.5652	−1.50229
$95$	−11.9326	−1.22426
$96$	0	0
$97$	−5.35014	−0.543224	−0.271612	−	0.962407i	$-0.587557\pi$
$97$	−5.35014	−0.543224	−0.271612	+	0.962407i	$0.587557\pi$
$98$	1.91759	0.193705
$99$	0	0
$100$	−3.92935	−0.392935
$101$	13.8634	1.37946	0.689731	−	0.724065i	$-0.257729\pi$
$101$	13.8634	1.37946	0.689731	+	0.724065i	$0.257729\pi$
$102$	0	0
$103$	17.4513	1.71953	0.859764	−	0.510691i	$-0.170610\pi$
$103$	17.4513	1.71953	0.859764	+	0.510691i	$0.170610\pi$
$104$	−0.486024	−0.0476586
$105$	0	0
$106$	−0.289949	−0.0281623
$107$	7.09496	0.685896	0.342948	−	0.939354i	$-0.388575\pi$
$107$	7.09496	0.685896	0.342948	+	0.939354i	$0.388575\pi$
$108$	0	0
$109$	17.4311	1.66959	0.834796	−	0.550559i	$-0.185586\pi$
$109$	17.4311	1.66959	0.834796	+	0.550559i	$0.185586\pi$
$110$	14.1824	1.35224
$111$	0	0
$112$	−4.54149	−0.429130
$113$	7.36358	0.692707	0.346353	−	0.938104i	$-0.387420\pi$
$113$	7.36358	0.692707	0.346353	+	0.938104i	$0.387420\pi$
$114$	0	0
$115$	2.90138	0.270555
$116$	−8.00758	−0.743485
$117$	0	0
$118$	−9.00019	−0.828534
$119$	4.58625	0.420420
$120$	0	0
$121$	9.58645	0.871495
$122$	−23.3825	−2.11695
$123$	0	0
$124$	8.49543	0.762912
$125$	11.9694	1.07057
$126$	0	0
$127$	1.00000	0.0887357
$127$	1.00000	0.0887357
$128$	4.88845	0.432082
$129$	0	0
$130$	−2.45380	−0.215212
$131$	−3.42397	−0.299153	−0.149577	−	0.988750i	$-0.547791\pi$
$131$	−3.42397	−0.299153	−0.149577	+	0.988750i	$0.547791\pi$
$132$	0	0
$133$	7.32031	0.634752
$134$	2.99621	0.258833
$135$	0	0
$136$	−2.83945	−0.243481
$137$	8.60770	0.735405	0.367703	−	0.929943i	$-0.380144\pi$
$137$	8.60770	0.735405	0.367703	+	0.929943i	$0.380144\pi$
$138$	0	0
$139$	11.1427	0.945108	0.472554	−	0.881302i	$-0.343332\pi$
$139$	11.1427	0.945108	0.472554	+	0.881302i	$0.343332\pi$
$140$	−2.73383	−0.231051
$141$	0	0
$142$	25.3604	2.12819
$143$	−3.56181	−0.297854
$144$	0	0
$145$	7.78283	0.646329
$146$	10.7254	0.887640
$147$	0	0
$148$	−1.73306	−0.142457
$149$	20.6563	1.69223	0.846117	−	0.532997i	$-0.178934\pi$
$149$	20.6563	1.69223	0.846117	+	0.532997i	$0.178934\pi$
$150$	0	0
$151$	1.83421	0.149266	0.0746330	−	0.997211i	$-0.476221\pi$
$151$	1.83421	0.149266	0.0746330	+	0.997211i	$0.476221\pi$
$152$	−4.53218	−0.367608
$153$	0	0
$154$	−8.70052	−0.701108
$155$	−8.25699	−0.663218
$156$	0	0
$157$	9.98994	0.797284	0.398642	−	0.917107i	$-0.369482\pi$
$157$	9.98994	0.797284	0.398642	+	0.917107i	$0.369482\pi$
$158$	−7.31952	−0.582309
$159$	0	0
$160$	12.1773	0.962699
$161$	−1.77992	−0.140277
$162$	0	0
$163$	5.49104	0.430092	0.215046	−	0.976604i	$-0.431010\pi$
$163$	5.49104	0.430092	0.215046	+	0.976604i	$0.431010\pi$
$164$	17.8596	1.39460
$165$	0	0
$166$	−1.40801	−0.109283
$167$	−1.35423	−0.104794	−0.0523969	−	0.998626i	$-0.516686\pi$
$167$	−1.35423	−0.104794	−0.0523969	+	0.998626i	$0.516686\pi$
$168$	0	0
$169$	−12.3837	−0.952596
$170$	−14.3356	−1.09949
$171$	0	0
$172$	13.8675	1.05738
$173$	5.51469	0.419274	0.209637	−	0.977779i	$-0.432772\pi$
$173$	5.51469	0.419274	0.209637	+	0.977779i	$0.432772\pi$
$174$	0	0
$175$	−2.34290	−0.177106
$176$	20.6058	1.55322
$177$	0	0
$178$	9.91342	0.743042
$179$	9.88587	0.738904	0.369452	−	0.929250i	$-0.379545\pi$
$179$	9.88587	0.738904	0.369452	+	0.929250i	$0.379545\pi$
$180$	0	0
$181$	3.10026	0.230441	0.115220	−	0.993340i	$-0.463243\pi$
$181$	3.10026	0.230441	0.115220	+	0.993340i	$0.463243\pi$
$182$	1.50534	0.111583
$183$	0	0
$184$	1.10199	0.0812398
$185$	1.68442	0.123841
$186$	0	0
$187$	−20.8088	−1.52169
$188$	−12.7388	−0.929075
$189$	0	0
$190$	−22.8817	−1.66001
$191$	−5.18368	−0.375078	−0.187539	−	0.982257i	$-0.560051\pi$
$191$	−5.18368	−0.375078	−0.187539	+	0.982257i	$0.560051\pi$
$192$	0	0
$193$	7.06423	0.508494	0.254247	−	0.967139i	$-0.418172\pi$
$193$	7.06423	0.508494	0.254247	+	0.967139i	$0.418172\pi$
$194$	−10.2594	−0.736578
$195$	0	0
$196$	1.67713	0.119795
$197$	4.38545	0.312450	0.156225	−	0.987721i	$-0.450067\pi$
$197$	4.38545	0.312450	0.156225	+	0.987721i	$0.450067\pi$
$198$	0	0
$199$	1.65748	0.117496	0.0587479	−	0.998273i	$-0.481289\pi$
$199$	1.65748	0.117496	0.0587479	+	0.998273i	$0.481289\pi$
$200$	1.45054	0.102569
$201$	0	0
$202$	26.5843	1.87047
$203$	−4.77456	−0.335109
$204$	0	0
$205$	−17.3583	−1.21236
$206$	33.4644	2.33157
$207$	0	0
$208$	−3.56516	−0.247199
$209$	−33.2139	−2.29746
$210$	0	0
$211$	−23.9651	−1.64983	−0.824913	−	0.565259i	$-0.808776\pi$
$211$	−23.9651	−1.64983	−0.824913	+	0.565259i	$0.808776\pi$
$212$	−0.253591	−0.0174167
$213$	0	0
$214$	13.6052	0.930032
$215$	−13.4782	−0.919209
$216$	0	0
$217$	5.06545	0.343865
$218$	33.4255	2.26386
$219$	0	0
$220$	12.4040	0.836279
$221$	3.60029	0.242182
$222$	0	0
$223$	−6.48293	−0.434129	−0.217065	−	0.976157i	$-0.569648\pi$
$223$	−6.48293	−0.434129	−0.217065	+	0.976157i	$0.569648\pi$
$224$	−7.47045	−0.499140
$225$	0	0
$226$	14.1203	0.939267
$227$	−10.4264	−0.692024	−0.346012	−	0.938230i	$-0.612464\pi$
$227$	−10.4264	−0.692024	−0.346012	+	0.938230i	$0.612464\pi$
$228$	0	0
$229$	−21.8797	−1.44585	−0.722925	−	0.690926i	$-0.757203\pi$
$229$	−21.8797	−1.44585	−0.722925	+	0.690926i	$0.757203\pi$
$230$	5.56364	0.366855
$231$	0	0
$232$	2.95605	0.194074
$233$	−19.7154	−1.29160	−0.645799	−	0.763508i	$-0.723476\pi$
$233$	−19.7154	−1.29160	−0.645799	+	0.763508i	$0.723476\pi$
$234$	0	0
$235$	12.3813	0.807667
$236$	−7.87163	−0.512399
$237$	0	0
$238$	8.79452	0.570064
$239$	18.5429	1.19944	0.599721	−	0.800209i	$-0.295278\pi$
$239$	18.5429	1.19944	0.599721	+	0.800209i	$0.295278\pi$
$240$	0	0
$241$	28.9917	1.86752	0.933761	−	0.357897i	$-0.116506\pi$
$241$	28.9917	1.86752	0.933761	+	0.357897i	$0.116506\pi$
$242$	18.3828	1.18169
$243$	0	0
$244$	−20.4505	−1.30921
$245$	−1.63006	−0.104141
$246$	0	0
$247$	5.74658	0.365646
$248$	−3.13614	−0.199145
$249$	0	0
$250$	22.9523	1.45163
$251$	−28.0776	−1.77225	−0.886123	−	0.463451i	$-0.846611\pi$
$251$	−28.0776	−1.77225	−0.886123	+	0.463451i	$0.846611\pi$
$252$	0	0
$253$	8.07590	0.507727
$254$	1.91759	0.120320
$255$	0	0
$256$	19.8585	1.24116
$257$	−3.12939	−0.195206	−0.0976029	−	0.995225i	$-0.531118\pi$
$257$	−3.12939	−0.195206	−0.0976029	+	0.995225i	$0.531118\pi$
$258$	0	0
$259$	−1.03335	−0.0642092
$260$	−2.14611	−0.133096
$261$	0	0
$262$	−6.56575	−0.405633
$263$	−23.7229	−1.46281	−0.731407	−	0.681941i	$-0.761136\pi$
$263$	−23.7229	−1.46281	−0.731407	+	0.681941i	$0.761136\pi$
$264$	0	0
$265$	0.246474	0.0151408
$266$	14.0373	0.860684
$267$	0	0
$268$	2.62051	0.160073
$269$	−4.27831	−0.260853	−0.130427	−	0.991458i	$-0.541635\pi$
$269$	−4.27831	−0.260853	−0.130427	+	0.991458i	$0.541635\pi$
$270$	0	0
$271$	12.9618	0.787375	0.393687	−	0.919244i	$-0.371199\pi$
$271$	12.9618	0.787375	0.393687	+	0.919244i	$0.371199\pi$
$272$	−20.8284	−1.26291
$273$	0	0
$274$	16.5060	0.997163
$275$	10.6303	0.641029
$276$	0	0
$277$	26.8719	1.61457	0.807287	−	0.590160i	$-0.200935\pi$
$277$	26.8719	1.61457	0.807287	+	0.590160i	$0.200935\pi$
$278$	21.3670	1.28151
$279$	0	0
$280$	1.00921	0.0603118
$281$	−7.21075	−0.430157	−0.215079	−	0.976597i	$-0.569001\pi$
$281$	−7.21075	−0.430157	−0.215079	+	0.976597i	$0.569001\pi$
$282$	0	0
$283$	7.06754	0.420121	0.210061	−	0.977688i	$-0.432634\pi$
$283$	7.06754	0.420121	0.210061	+	0.977688i	$0.432634\pi$
$284$	22.1803	1.31616
$285$	0	0
$286$	−6.83008	−0.403871
$287$	10.6489	0.628582
$288$	0	0
$289$	4.03365	0.237273
$290$	14.9242	0.876382
$291$	0	0
$292$	9.38050	0.548952
$293$	−6.10993	−0.356946	−0.178473	−	0.983945i	$-0.557116\pi$
$293$	−6.10993	−0.356946	−0.178473	+	0.983945i	$0.557116\pi$
$294$	0	0
$295$	7.65070	0.445441
$296$	0.639771	0.0371859
$297$	0	0
$298$	39.6103	2.29456
$299$	−1.39727	−0.0808062
$300$	0	0
$301$	8.26855	0.476591
$302$	3.51726	0.202395
$303$	0	0
$304$	−33.2451	−1.90674
$305$	19.8765	1.13813
$306$	0	0
$307$	−14.8475	−0.847389	−0.423695	−	0.905805i	$-0.639267\pi$
$307$	−14.8475	−0.847389	−0.423695	+	0.905805i	$0.639267\pi$
$308$	−7.60954	−0.433594
$309$	0	0
$310$	−15.8335	−0.899282
$311$	12.1323	0.687961	0.343981	−	0.938977i	$-0.388225\pi$
$311$	12.1323	0.687961	0.343981	+	0.938977i	$0.388225\pi$
$312$	0	0
$313$	0.476080	0.0269097	0.0134548	−	0.999909i	$-0.495717\pi$
$313$	0.476080	0.0269097	0.0134548	+	0.999909i	$0.495717\pi$
$314$	19.1566	1.08107
$315$	0	0
$316$	−6.40170	−0.360124
$317$	1.09825	0.0616840	0.0308420	−	0.999524i	$-0.490181\pi$
$317$	1.09825	0.0616840	0.0308420	+	0.999524i	$0.490181\pi$
$318$	0	0
$319$	21.6633	1.21291
$320$	8.54518	0.477690
$321$	0	0
$322$	−3.41315	−0.190207
$323$	33.5728	1.86804
$324$	0	0
$325$	−1.83922	−0.102022
$326$	10.5295	0.583177
$327$	0	0
$328$	−6.59296	−0.364035
$329$	−7.59560	−0.418759
$330$	0	0
$331$	21.4769	1.18048	0.590239	−	0.807228i	$-0.299033\pi$
$331$	21.4769	1.18048	0.590239	+	0.807228i	$0.299033\pi$
$332$	−1.23146	−0.0675851
$333$	0	0
$334$	−2.59686	−0.142094
$335$	−2.54696	−0.139155
$336$	0	0
$337$	−2.56002	−0.139453	−0.0697267	−	0.997566i	$-0.522213\pi$
$337$	−2.56002	−0.139453	−0.0697267	+	0.997566i	$0.522213\pi$
$338$	−23.7469	−1.29166
$339$	0	0
$340$	−12.5380	−0.679970
$341$	−22.9831	−1.24460
$342$	0	0
$343$	1.00000	0.0539949
$344$	−5.11926	−0.276012
$345$	0	0
$346$	10.5749	0.568509
$347$	−17.8141	−0.956312	−0.478156	−	0.878275i	$-0.658695\pi$
$347$	−17.8141	−0.956312	−0.478156	+	0.878275i	$0.658695\pi$
$348$	0	0
$349$	−9.98746	−0.534616	−0.267308	−	0.963611i	$-0.586134\pi$
$349$	−9.98746	−0.534616	−0.267308	+	0.963611i	$0.586134\pi$
$350$	−4.49271	−0.240145
$351$	0	0
$352$	33.8951	1.80662
$353$	1.59328	0.0848018	0.0424009	−	0.999101i	$-0.486499\pi$
$353$	1.59328	0.0848018	0.0424009	+	0.999101i	$0.486499\pi$
$354$	0	0
$355$	−21.5578	−1.14417
$356$	8.67035	0.459527
$357$	0	0
$358$	18.9570	1.00191
$359$	15.8997	0.839153	0.419576	−	0.907720i	$-0.362179\pi$
$359$	15.8997	0.839153	0.419576	+	0.907720i	$0.362179\pi$
$360$	0	0
$361$	34.5870	1.82037
$362$	5.94501	0.312463
$363$	0	0
$364$	1.31658	0.0690076
$365$	−9.11722	−0.477217
$366$	0	0
$367$	30.3461	1.58405	0.792027	−	0.610486i	$-0.209026\pi$
$367$	30.3461	1.58405	0.792027	+	0.610486i	$0.209026\pi$
$368$	8.08348	0.421381
$369$	0	0
$370$	3.23002	0.167921
$371$	−0.151205	−0.00785017
$372$	0	0
$373$	−0.699026	−0.0361942	−0.0180971	−	0.999836i	$-0.505761\pi$
$373$	−0.699026	−0.0361942	−0.0180971	+	0.999836i	$0.505761\pi$
$374$	−39.9027	−2.06332
$375$	0	0
$376$	4.70262	0.242519
$377$	−3.74812	−0.193038
$378$	0	0
$379$	2.08955	0.107333	0.0536665	−	0.998559i	$-0.482909\pi$
$379$	2.08955	0.107333	0.0536665	+	0.998559i	$0.482909\pi$
$380$	−20.0125	−1.02662
$381$	0	0
$382$	−9.94015	−0.508583
$383$	−7.36514	−0.376341	−0.188170	−	0.982136i	$-0.560256\pi$
$383$	−7.36514	−0.376341	−0.188170	+	0.982136i	$0.560256\pi$
$384$	0	0
$385$	7.39596	0.376933
$386$	13.5463	0.689486
$387$	0	0
$388$	−8.97290	−0.455530
$389$	−25.3382	−1.28470	−0.642348	−	0.766413i	$-0.722040\pi$
$389$	−25.3382	−1.28470	−0.642348	+	0.766413i	$0.722040\pi$
$390$	0	0
$391$	−8.16314	−0.412828
$392$	−0.619124	−0.0312705
$393$	0	0
$394$	8.40948	0.423663
$395$	6.22202	0.313064
$396$	0	0
$397$	−22.4952	−1.12900	−0.564500	−	0.825433i	$-0.690931\pi$
$397$	−22.4952	−1.12900	−0.564500	+	0.825433i	$0.690931\pi$
$398$	3.17836	0.159317
$399$	0	0
$400$	10.6402	0.532012
$401$	−16.0902	−0.803508	−0.401754	−	0.915748i	$-0.631599\pi$
$401$	−16.0902	−0.803508	−0.401754	+	0.915748i	$0.631599\pi$
$402$	0	0
$403$	3.97647	0.198082
$404$	23.2508	1.15677
$405$	0	0
$406$	−9.15563	−0.454386
$407$	4.68854	0.232402
$408$	0	0
$409$	−26.5315	−1.31190	−0.655949	−	0.754805i	$-0.727731\pi$
$409$	−26.5315	−1.31190	−0.655949	+	0.754805i	$0.727731\pi$
$410$	−33.2860	−1.64388
$411$	0	0
$412$	29.2682	1.44194
$413$	−4.69350	−0.230952
$414$	0	0
$415$	1.19690	0.0587533
$416$	−5.86444	−0.287528
$417$	0	0
$418$	−63.6906	−3.11521
$419$	13.0559	0.637821	0.318911	−	0.947785i	$-0.396683\pi$
$419$	13.0559	0.637821	0.318911	+	0.947785i	$0.396683\pi$
$420$	0	0
$421$	−0.503178	−0.0245234	−0.0122617	−	0.999925i	$-0.503903\pi$
$421$	−0.503178	−0.0245234	−0.0122617	+	0.999925i	$0.503903\pi$
$422$	−45.9551	−2.23706
$423$	0	0
$424$	0.0936146	0.00454633
$425$	−10.7451	−0.521214
$426$	0	0
$427$	−12.1937	−0.590095
$428$	11.8992	0.575170
$429$	0	0
$430$	−25.8457	−1.24639
$431$	−19.3583	−0.932457	−0.466228	−	0.884664i	$-0.654388\pi$
$431$	−19.3583	−0.932457	−0.466228	+	0.884664i	$0.654388\pi$
$432$	0	0
$433$	−32.9626	−1.58408	−0.792041	−	0.610468i	$-0.790982\pi$
$433$	−32.9626	−1.58408	−0.792041	+	0.610468i	$0.790982\pi$
$434$	9.71343	0.466260
$435$	0	0
$436$	29.2342	1.40006
$437$	−13.0296	−0.623289
$438$	0	0
$439$	32.0599	1.53013	0.765067	−	0.643951i	$-0.222706\pi$
$439$	32.0599	1.53013	0.765067	+	0.643951i	$0.222706\pi$
$440$	−4.57902	−0.218296
$441$	0	0
$442$	6.90386	0.328383
$443$	19.0080	0.903096	0.451548	−	0.892247i	$-0.350872\pi$
$443$	19.0080	0.903096	0.451548	+	0.892247i	$0.350872\pi$
$444$	0	0
$445$	−8.42700	−0.399478
$446$	−12.4316	−0.588652
$447$	0	0
$448$	−5.24224	−0.247673
$449$	−16.2304	−0.765960	−0.382980	−	0.923757i	$-0.625102\pi$
$449$	−16.2304	−0.765960	−0.382980	+	0.923757i	$0.625102\pi$
$450$	0	0
$451$	−48.3163	−2.27513
$452$	12.3497	0.580881
$453$	0	0
$454$	−19.9935	−0.938341
$455$	−1.27963	−0.0599899
$456$	0	0
$457$	15.6324	0.731254	0.365627	−	0.930761i	$-0.380855\pi$
$457$	15.6324	0.731254	0.365627	+	0.930761i	$0.380855\pi$
$458$	−41.9562	−1.96048
$459$	0	0
$460$	4.86600	0.226878
$461$	21.3891	0.996191	0.498096	−	0.867122i	$-0.334033\pi$
$461$	21.3891	0.996191	0.498096	+	0.867122i	$0.334033\pi$
$462$	0	0
$463$	−18.8588	−0.876445	−0.438222	−	0.898867i	$-0.644392\pi$
$463$	−18.8588	−0.876445	−0.438222	+	0.898867i	$0.644392\pi$
$464$	21.6836	1.00664
$465$	0	0
$466$	−37.8059	−1.75133
$467$	−21.0632	−0.974689	−0.487344	−	0.873210i	$-0.662034\pi$
$467$	−21.0632	−0.974689	−0.487344	+	0.873210i	$0.662034\pi$
$468$	0	0
$469$	1.56249	0.0721492
$470$	23.7422	1.09515
$471$	0	0
$472$	2.90586	0.133753
$473$	−37.5163	−1.72500
$474$	0	0
$475$	−17.1508	−0.786930
$476$	7.69175	0.352551
$477$	0	0
$478$	35.5577	1.62637
$479$	−23.6039	−1.07849	−0.539245	−	0.842149i	$-0.681290\pi$
$479$	−23.6039	−1.07849	−0.539245	+	0.842149i	$0.681290\pi$
$480$	0	0
$481$	−0.811198	−0.0369875
$482$	55.5941	2.53224
$483$	0	0
$484$	16.0778	0.730807
$485$	8.72106	0.396003
$486$	0	0
$487$	−16.2006	−0.734119	−0.367060	−	0.930197i	$-0.619636\pi$
$487$	−16.2006	−0.734119	−0.367060	+	0.930197i	$0.619636\pi$
$488$	7.54942	0.341746
$489$	0	0
$490$	−3.12578	−0.141208
$491$	20.8074	0.939025	0.469512	−	0.882926i	$-0.344430\pi$
$491$	20.8074	0.939025	0.469512	+	0.882926i	$0.344430\pi$
$492$	0	0
$493$	−21.8973	−0.986206
$494$	11.0196	0.495794
$495$	0	0
$496$	−23.0047	−1.03294
$497$	13.2252	0.593229
$498$	0	0
$499$	−6.17234	−0.276312	−0.138156	−	0.990410i	$-0.544118\pi$
$499$	−6.17234	−0.276312	−0.138156	+	0.990410i	$0.544118\pi$
$500$	20.0742	0.897748
$501$	0	0
$502$	−53.8413	−2.40305
$503$	28.7432	1.28159	0.640797	−	0.767710i	$-0.278604\pi$
$503$	28.7432	1.28159	0.640797	+	0.767710i	$0.278604\pi$
$504$	0	0
$505$	−22.5982	−1.00561
$506$	15.4862	0.688447
$507$	0	0
$508$	1.67713	0.0744108
$509$	−13.2246	−0.586169	−0.293084	−	0.956087i	$-0.594682\pi$
$509$	−13.2246	−0.586169	−0.293084	+	0.956087i	$0.594682\pi$
$510$	0	0
$511$	5.59318	0.247428
$512$	28.3035	1.25085
$513$	0	0
$514$	−6.00087	−0.264687
$515$	−28.4467	−1.25351
$516$	0	0
$517$	34.4630	1.51568
$518$	−1.98153	−0.0870636
$519$	0	0
$520$	0.792249	0.0347424
$521$	32.9614	1.44407	0.722033	−	0.691858i	$-0.243208\pi$
$521$	32.9614	1.44407	0.722033	+	0.691858i	$0.243208\pi$
$522$	0	0
$523$	−3.22512	−0.141025	−0.0705123	−	0.997511i	$-0.522463\pi$
$523$	−3.22512	−0.141025	−0.0705123	+	0.997511i	$0.522463\pi$
$524$	−5.74245	−0.250860
$525$	0	0
$526$	−45.4906	−1.98349
$527$	23.2314	1.01198
$528$	0	0
$529$	−19.8319	−0.862256
$530$	0.472634	0.0205299
$531$	0	0
$532$	12.2771	0.532282
$533$	8.35955	0.362092
$534$	0	0
$535$	−11.5652	−0.500009
$536$	−0.967376	−0.0417843
$537$	0	0
$538$	−8.20403	−0.353701
$539$	−4.53723	−0.195432
$540$	0	0
$541$	−4.01886	−0.172784	−0.0863922	−	0.996261i	$-0.527534\pi$
$541$	−4.01886	−0.172784	−0.0863922	+	0.996261i	$0.527534\pi$
$542$	24.8554	1.06763
$543$	0	0
$544$	−34.2613	−1.46894
$545$	−28.4137	−1.21711
$546$	0	0
$547$	−3.36285	−0.143785	−0.0718926	−	0.997412i	$-0.522904\pi$
$547$	−3.36285	−0.143785	−0.0718926	+	0.997412i	$0.522904\pi$
$548$	14.4363	0.616686
$549$	0	0
$550$	20.3844	0.869196
$551$	−34.9513	−1.48898
$552$	0	0
$553$	−3.81705	−0.162317
$554$	51.5291	2.18926
$555$	0	0
$556$	18.6877	0.792536
$557$	36.3729	1.54117	0.770584	−	0.637339i	$-0.219965\pi$
$557$	36.3729	1.54117	0.770584	+	0.637339i	$0.219965\pi$
$558$	0	0
$559$	6.49097	0.274539
$560$	7.40291	0.312830
$561$	0	0
$562$	−13.8272	−0.583266
$563$	21.8814	0.922190	0.461095	−	0.887351i	$-0.347457\pi$
$563$	21.8814	0.922190	0.461095	+	0.887351i	$0.347457\pi$
$564$	0	0
$565$	−12.0031	−0.504974
$566$	13.5526	0.569658
$567$	0	0
$568$	−8.18801	−0.343561
$569$	1.52443	0.0639073	0.0319536	−	0.999489i	$-0.489827\pi$
$569$	1.52443	0.0639073	0.0319536	+	0.999489i	$0.489827\pi$
$570$	0	0
$571$	−0.407383	−0.0170485	−0.00852423	−	0.999964i	$-0.502713\pi$
$571$	−0.407383	−0.0170485	−0.00852423	+	0.999964i	$0.502713\pi$
$572$	−5.97363	−0.249770
$573$	0	0
$574$	20.4201	0.852318
$575$	4.17017	0.173908
$576$	0	0
$577$	−9.40086	−0.391363	−0.195681	−	0.980668i	$-0.562692\pi$
$577$	−9.40086	−0.391363	−0.195681	+	0.980668i	$0.562692\pi$
$578$	7.73486	0.321728
$579$	0	0
$580$	13.0529	0.541990
$581$	−0.734264	−0.0304624
$582$	0	0
$583$	0.686052	0.0284134
$584$	−3.46287	−0.143295
$585$	0	0
$586$	−11.7163	−0.483996
$587$	−7.47968	−0.308719	−0.154360	−	0.988015i	$-0.549331\pi$
$587$	−7.47968	−0.308719	−0.154360	+	0.988015i	$0.549331\pi$
$588$	0	0
$589$	37.0807	1.52788
$590$	14.6709	0.603990
$591$	0	0
$592$	4.69294	0.192879
$593$	−26.7023	−1.09653	−0.548266	−	0.836304i	$-0.684712\pi$
$593$	−26.7023	−1.09653	−0.548266	+	0.836304i	$0.684712\pi$
$594$	0	0
$595$	−7.47586	−0.306481
$596$	34.6434	1.41905
$597$	0	0
$598$	−2.67938	−0.109568
$599$	9.66950	0.395085	0.197543	−	0.980294i	$-0.436704\pi$
$599$	9.66950	0.395085	0.197543	+	0.980294i	$0.436704\pi$
$600$	0	0
$601$	21.1650	0.863340	0.431670	−	0.902032i	$-0.357925\pi$
$601$	21.1650	0.863340	0.431670	+	0.902032i	$0.357925\pi$
$602$	15.8556	0.646228
$603$	0	0
$604$	3.07622	0.125170
$605$	−15.6265	−0.635308
$606$	0	0
$607$	16.3837	0.664993	0.332496	−	0.943105i	$-0.392109\pi$
$607$	16.3837	0.664993	0.332496	+	0.943105i	$0.392109\pi$
$608$	−54.6860	−2.21781
$609$	0	0
$610$	38.1149	1.54323
$611$	−5.96269	−0.241225
$612$	0	0
$613$	−36.7506	−1.48434	−0.742171	−	0.670210i	$-0.766204\pi$
$613$	−36.7506	−1.48434	−0.742171	+	0.670210i	$0.766204\pi$
$614$	−28.4713	−1.14901
$615$	0	0
$616$	2.80911	0.113182
$617$	−27.8500	−1.12120	−0.560600	−	0.828087i	$-0.689429\pi$
$617$	−27.8500	−1.12120	−0.560600	+	0.828087i	$0.689429\pi$
$618$	0	0
$619$	−26.3590	−1.05946	−0.529730	−	0.848167i	$-0.677707\pi$
$619$	−26.3590	−1.05946	−0.529730	+	0.848167i	$0.677707\pi$
$620$	−13.8481	−0.556152
$621$	0	0
$622$	23.2648	0.932832
$623$	5.16974	0.207121
$624$	0	0
$625$	−7.79634	−0.311853
$626$	0.912925	0.0364878
$627$	0	0
$628$	16.7545	0.668576
$629$	−4.73919	−0.188964
$630$	0	0
$631$	24.1225	0.960303	0.480151	−	0.877186i	$-0.340582\pi$
$631$	24.1225	0.960303	0.480151	+	0.877186i	$0.340582\pi$
$632$	2.36323	0.0940041
$633$	0	0
$634$	2.10599	0.0836397
$635$	−1.63006	−0.0646870
$636$	0	0
$637$	0.785019	0.0311036
$638$	41.5412	1.64463
$639$	0	0
$640$	−7.96848	−0.314982
$641$	31.6483	1.25003	0.625017	−	0.780611i	$-0.285092\pi$
$641$	31.6483	1.25003	0.625017	+	0.780611i	$0.285092\pi$
$642$	0	0
$643$	−36.3625	−1.43400	−0.716998	−	0.697075i	$-0.754484\pi$
$643$	−36.3625	−1.43400	−0.716998	+	0.697075i	$0.754484\pi$
$644$	−2.98516	−0.117632
$645$	0	0
$646$	64.3786	2.53294
$647$	30.6486	1.20492	0.602460	−	0.798149i	$-0.294187\pi$
$647$	30.6486	1.20492	0.602460	+	0.798149i	$0.294187\pi$
$648$	0	0
$649$	21.2955	0.835921
$650$	−3.52686	−0.138335
$651$	0	0
$652$	9.20921	0.360660
$653$	5.09863	0.199525	0.0997625	−	0.995011i	$-0.468192\pi$
$653$	5.09863	0.199525	0.0997625	+	0.995011i	$0.468192\pi$
$654$	0	0
$655$	5.58128	0.218078
$656$	−48.3617	−1.88821
$657$	0	0
$658$	−14.5652	−0.567811
$659$	−25.9346	−1.01027	−0.505135	−	0.863041i	$-0.668557\pi$
$659$	−25.9346	−1.01027	−0.505135	+	0.863041i	$0.668557\pi$
$660$	0	0
$661$	−27.2692	−1.06065	−0.530324	−	0.847795i	$-0.677930\pi$
$661$	−27.2692	−1.06065	−0.530324	+	0.847795i	$0.677930\pi$
$662$	41.1838	1.60065
$663$	0	0
$664$	0.454601	0.0176419
$665$	−11.9326	−0.462725
$666$	0	0
$667$	8.49833	0.329057
$668$	−2.27123	−0.0878766
$669$	0	0
$670$	−4.88401	−0.188686
$671$	55.3257	2.13582
$672$	0	0
$673$	−28.5849	−1.10187	−0.550933	−	0.834549i	$-0.685728\pi$
$673$	−28.5849	−1.10187	−0.550933	+	0.834549i	$0.685728\pi$
$674$	−4.90906	−0.189090
$675$	0	0
$676$	−20.7692	−0.798815
$677$	−28.2536	−1.08588	−0.542938	−	0.839773i	$-0.682688\pi$
$677$	−28.2536	−1.08588	−0.542938	+	0.839773i	$0.682688\pi$
$678$	0	0
$679$	−5.35014	−0.205320
$680$	4.62849	0.177494
$681$	0	0
$682$	−44.0721	−1.68761
$683$	10.4079	0.398248	0.199124	−	0.979974i	$-0.436190\pi$
$683$	10.4079	0.398248	0.199124	+	0.979974i	$0.436190\pi$
$684$	0	0
$685$	−14.0311	−0.536100
$686$	1.91759	0.0732137
$687$	0	0
$688$	−37.5515	−1.43164
$689$	−0.118699	−0.00452206
$690$	0	0
$691$	−36.1483	−1.37515	−0.687574	−	0.726115i	$-0.741324\pi$
$691$	−36.1483	−1.37515	−0.687574	+	0.726115i	$0.741324\pi$
$692$	9.24887	0.351589
$693$	0	0
$694$	−34.1601	−1.29670
$695$	−18.1632	−0.688970
$696$	0	0
$697$	48.8383	1.84988
$698$	−19.1518	−0.724906
$699$	0	0
$700$	−3.92935	−0.148516
$701$	9.20252	0.347574	0.173787	−	0.984783i	$-0.444399\pi$
$701$	9.20252	0.347574	0.173787	+	0.984783i	$0.444399\pi$
$702$	0	0
$703$	−7.56444	−0.285298
$704$	23.7852	0.896440
$705$	0	0
$706$	3.05525	0.114986
$707$	13.8634	0.521388
$708$	0	0
$709$	7.72738	0.290208	0.145104	−	0.989416i	$-0.453648\pi$
$709$	7.72738	0.290208	0.145104	+	0.989416i	$0.453648\pi$
$710$	−41.3389	−1.55142
$711$	0	0
$712$	−3.20071	−0.119952
$713$	−9.01609	−0.337655
$714$	0	0
$715$	5.80597	0.217131
$716$	16.5799	0.619621
$717$	0	0
$718$	30.4890	1.13784
$719$	43.6917	1.62942	0.814712	−	0.579865i	$-0.196895\pi$
$719$	43.6917	1.62942	0.814712	+	0.579865i	$0.196895\pi$
$720$	0	0
$721$	17.4513	0.649921
$722$	66.3235	2.46831
$723$	0	0
$724$	5.19955	0.193240
$725$	11.1863	0.415449
$726$	0	0
$727$	12.4150	0.460447	0.230224	−	0.973138i	$-0.426054\pi$
$727$	12.4150	0.460447	0.230224	+	0.973138i	$0.426054\pi$
$728$	−0.486024	−0.0180132
$729$	0	0
$730$	−17.4831	−0.647077
$731$	37.9216	1.40258
$732$	0	0
$733$	−15.8586	−0.585752	−0.292876	−	0.956150i	$-0.594612\pi$
$733$	−15.8586	−0.585752	−0.292876	+	0.956150i	$0.594612\pi$
$734$	58.1913	2.14788
$735$	0	0
$736$	13.2968	0.490126
$737$	−7.08939	−0.261141
$738$	0	0
$739$	−50.8373	−1.87008	−0.935040	−	0.354542i	$-0.884637\pi$
$739$	−50.8373	−1.87008	−0.935040	+	0.354542i	$0.884637\pi$
$740$	2.82500	0.103849
$741$	0	0
$742$	−0.289949	−0.0106443
$743$	−19.4170	−0.712341	−0.356170	−	0.934421i	$-0.615918\pi$
$743$	−19.4170	−0.712341	−0.356170	+	0.934421i	$0.615918\pi$
$744$	0	0
$745$	−33.6711	−1.23361
$746$	−1.34044	−0.0490771
$747$	0	0
$748$	−34.8992	−1.27604
$749$	7.09496	0.259244
$750$	0	0
$751$	30.2814	1.10498	0.552491	−	0.833519i	$-0.313677\pi$
$751$	30.2814	1.10498	0.552491	+	0.833519i	$0.313677\pi$
$752$	34.4953	1.25792
$753$	0	0
$754$	−7.18734	−0.261748
$755$	−2.98988	−0.108813
$756$	0	0
$757$	−7.73773	−0.281233	−0.140616	−	0.990064i	$-0.544908\pi$
$757$	−7.73773	−0.281233	−0.140616	+	0.990064i	$0.544908\pi$
$758$	4.00689	0.145537
$759$	0	0
$760$	7.38774	0.267981
$761$	42.4083	1.53730	0.768650	−	0.639669i	$-0.220929\pi$
$761$	42.4083	1.53730	0.768650	+	0.639669i	$0.220929\pi$
$762$	0	0
$763$	17.4311	0.631047
$764$	−8.69373	−0.314528
$765$	0	0
$766$	−14.1233	−0.510295
$767$	−3.68449	−0.133039
$768$	0	0
$769$	40.9692	1.47739	0.738694	−	0.674041i	$-0.235443\pi$
$769$	40.9692	1.47739	0.738694	+	0.674041i	$0.235443\pi$
$770$	14.1824	0.511098
$771$	0	0
$772$	11.8477	0.426406
$773$	−24.7232	−0.889232	−0.444616	−	0.895721i	$-0.646660\pi$
$773$	−24.7232	−0.889232	−0.444616	+	0.895721i	$0.646660\pi$
$774$	0	0
$775$	−11.8678	−0.426305
$776$	3.31240	0.118908
$777$	0	0
$778$	−48.5881	−1.74197
$779$	77.9530	2.79295
$780$	0	0
$781$	−60.0055	−2.14717
$782$	−15.6535	−0.559769
$783$	0	0
$784$	−4.54149	−0.162196
$785$	−16.2842	−0.581209
$786$	0	0
$787$	38.2059	1.36189	0.680946	−	0.732334i	$-0.261569\pi$
$787$	38.2059	1.36189	0.680946	+	0.732334i	$0.261569\pi$
$788$	7.35499	0.262011
$789$	0	0
$790$	11.9313	0.424495
$791$	7.36358	0.261819
$792$	0	0
$793$	−9.57230	−0.339922
$794$	−43.1364	−1.53085
$795$	0	0
$796$	2.77982	0.0985280
$797$	39.8094	1.41012	0.705061	−	0.709147i	$-0.250920\pi$
$797$	39.8094	1.41012	0.705061	+	0.709147i	$0.250920\pi$
$798$	0	0
$799$	−34.8353	−1.23238
$800$	17.5025	0.618807
$801$	0	0
$802$	−30.8544	−1.08951
$803$	−25.3775	−0.895553
$804$	0	0
$805$	2.90138	0.102260
$806$	7.62523	0.268587
$807$	0	0
$808$	−8.58318	−0.301955
$809$	19.8034	0.696250	0.348125	−	0.937448i	$-0.386818\pi$
$809$	19.8034	0.696250	0.348125	+	0.937448i	$0.386818\pi$
$810$	0	0
$811$	−48.6032	−1.70669	−0.853344	−	0.521348i	$-0.825430\pi$
$811$	−48.6032	−1.70669	−0.853344	+	0.521348i	$0.825430\pi$
$812$	−8.00758	−0.281011
$813$	0	0
$814$	8.99068	0.315123
$815$	−8.95074	−0.313531
$816$	0	0
$817$	60.5284	2.11762
$818$	−50.8764	−1.77885
$819$	0	0
$820$	−29.1122	−1.01664
$821$	27.5660	0.962060	0.481030	−	0.876704i	$-0.340263\pi$
$821$	27.5660	0.962060	0.481030	+	0.876704i	$0.340263\pi$
$822$	0	0
$823$	50.5300	1.76136	0.880682	−	0.473708i	$-0.157085\pi$
$823$	50.5300	1.76136	0.880682	+	0.473708i	$0.157085\pi$
$824$	−10.8045	−0.376393
$825$	0	0
$826$	−9.00019	−0.313157
$827$	34.5514	1.20147	0.600735	−	0.799448i	$-0.294875\pi$
$827$	34.5514	1.20147	0.600735	+	0.799448i	$0.294875\pi$
$828$	0	0
$829$	45.2920	1.57305	0.786527	−	0.617555i	$-0.211877\pi$
$829$	45.2920	1.57305	0.786527	+	0.617555i	$0.211877\pi$
$830$	2.29515	0.0796658
$831$	0	0
$832$	−4.11526	−0.142671
$833$	4.58625	0.158904
$834$	0	0
$835$	2.20749	0.0763932
$836$	−55.7042	−1.92657
$837$	0	0
$838$	25.0358	0.864846
$839$	45.4572	1.56936	0.784679	−	0.619902i	$-0.212828\pi$
$839$	45.4572	1.56936	0.784679	+	0.619902i	$0.212828\pi$
$840$	0	0
$841$	−6.20354	−0.213915
$842$	−0.964886	−0.0332522
$843$	0	0
$844$	−40.1927	−1.38349
$845$	20.1863	0.694429
$846$	0	0
$847$	9.58645	0.329394
$848$	0.686696	0.0235812
$849$	0	0
$850$	−20.6047	−0.706734
$851$	1.83928	0.0630496
$852$	0	0
$853$	7.10046	0.243115	0.121558	−	0.992584i	$-0.461211\pi$
$853$	7.10046	0.243115	0.121558	+	0.992584i	$0.461211\pi$
$854$	−23.3825	−0.800132
$855$	0	0
$856$	−4.39266	−0.150138
$857$	−38.1705	−1.30388	−0.651940	−	0.758270i	$-0.726045\pi$
$857$	−38.1705	−1.30388	−0.651940	+	0.758270i	$0.726045\pi$
$858$	0	0
$859$	−0.939629	−0.0320597	−0.0160299	−	0.999872i	$-0.505103\pi$
$859$	−0.939629	−0.0320597	−0.0160299	+	0.999872i	$0.505103\pi$
$860$	−22.6048	−0.770818
$861$	0	0
$862$	−37.1212	−1.26435
$863$	−37.0524	−1.26128	−0.630639	−	0.776077i	$-0.717207\pi$
$863$	−37.0524	−1.26128	−0.630639	+	0.776077i	$0.717207\pi$
$864$	0	0
$865$	−8.98928	−0.305645
$866$	−63.2086	−2.14792
$867$	0	0
$868$	8.49543	0.288354
$869$	17.3188	0.587501
$870$	0	0
$871$	1.22659	0.0415613
$872$	−10.7920	−0.365463
$873$	0	0
$874$	−24.9853	−0.845140
$875$	11.9694	0.404639
$876$	0	0
$877$	8.41666	0.284210	0.142105	−	0.989852i	$-0.454613\pi$
$877$	8.41666	0.284210	0.142105	+	0.989852i	$0.454613\pi$
$878$	61.4775	2.07477
$879$	0	0
$880$	−33.5887	−1.13227
$881$	16.1308	0.543462	0.271731	−	0.962373i	$-0.412404\pi$
$881$	16.1308	0.543462	0.271731	+	0.962373i	$0.412404\pi$
$882$	0	0
$883$	−54.9590	−1.84952	−0.924759	−	0.380553i	$-0.875734\pi$
$883$	−54.9590	−1.84952	−0.924759	+	0.380553i	$0.875734\pi$
$884$	6.03817	0.203086
$885$	0	0
$886$	36.4494	1.22454
$887$	6.72375	0.225761	0.112881	−	0.993609i	$-0.463992\pi$
$887$	6.72375	0.225761	0.112881	+	0.993609i	$0.463992\pi$
$888$	0	0
$889$	1.00000	0.0335389
$890$	−16.1595	−0.541667
$891$	0	0
$892$	−10.8727	−0.364046
$893$	−55.6022	−1.86066
$894$	0	0
$895$	−16.1146	−0.538651
$896$	4.88845	0.163312
$897$	0	0
$898$	−31.1232	−1.03859
$899$	−24.1853	−0.806625
$900$	0	0
$901$	−0.693463	−0.0231026
$902$	−92.6506	−3.08493
$903$	0	0
$904$	−4.55897	−0.151629
$905$	−5.05362	−0.167988
$906$	0	0
$907$	−44.3476	−1.47254	−0.736269	−	0.676689i	$-0.763414\pi$
$907$	−44.3476	−1.47254	−0.736269	+	0.676689i	$0.763414\pi$
$908$	−17.4865	−0.580308
$909$	0	0
$910$	−2.45380	−0.0813426
$911$	26.5885	0.880916	0.440458	−	0.897773i	$-0.354816\pi$
$911$	26.5885	0.880916	0.440458	+	0.897773i	$0.354816\pi$
$912$	0	0
$913$	3.33153	0.110257
$914$	29.9765	0.991535
$915$	0	0
$916$	−36.6952	−1.21244
$917$	−3.42397	−0.113069
$918$	0	0
$919$	−6.91175	−0.227998	−0.113999	−	0.993481i	$-0.536366\pi$
$919$	−6.91175	−0.227998	−0.113999	+	0.993481i	$0.536366\pi$
$920$	−1.79631	−0.0592227
$921$	0	0
$922$	41.0155	1.35077
$923$	10.3820	0.341727
$924$	0	0
$925$	2.42103	0.0796030
$926$	−36.1634	−1.18840
$927$	0	0
$928$	35.6681	1.17086
$929$	−50.6624	−1.66218	−0.831090	−	0.556138i	$-0.812283\pi$
$929$	−50.6624	−1.66218	−0.831090	+	0.556138i	$0.812283\pi$
$930$	0	0
$931$	7.32031	0.239914
$932$	−33.0653	−1.08309
$933$	0	0
$934$	−40.3905	−1.32162
$935$	33.9197	1.10929
$936$	0	0
$937$	27.0802	0.884671	0.442335	−	0.896850i	$-0.354150\pi$
$937$	27.0802	0.884671	0.442335	+	0.896850i	$0.354150\pi$
$938$	2.99621	0.0978298
$939$	0	0
$940$	20.7651	0.677283
$941$	53.5291	1.74500	0.872500	−	0.488614i	$-0.162497\pi$
$941$	53.5291	1.74500	0.872500	+	0.488614i	$0.162497\pi$
$942$	0	0
$943$	−18.9541	−0.617230
$944$	21.3155	0.693760
$945$	0	0
$946$	−71.9407	−2.33899
$947$	−45.4863	−1.47811	−0.739053	−	0.673647i	$-0.764727\pi$
$947$	−45.4863	−1.47811	−0.739053	+	0.673647i	$0.764727\pi$
$948$	0	0
$949$	4.39075	0.142530
$950$	−32.8880	−1.06703
$951$	0	0
$952$	−2.83945	−0.0920272
$953$	32.2357	1.04422	0.522108	−	0.852880i	$-0.325146\pi$
$953$	32.2357	1.04422	0.522108	+	0.852880i	$0.325146\pi$
$954$	0	0
$955$	8.44972	0.273427
$956$	31.0990	1.00581
$957$	0	0
$958$	−45.2625	−1.46237
$959$	8.60770	0.277957
$960$	0	0
$961$	−5.34123	−0.172298
$962$	−1.55554	−0.0501527
$963$	0	0
$964$	48.6230	1.56604
$965$	−11.5151	−0.370685
$966$	0	0
$967$	25.3030	0.813690	0.406845	−	0.913497i	$-0.366629\pi$
$967$	25.3030	0.813690	0.406845	+	0.913497i	$0.366629\pi$
$968$	−5.93520	−0.190765
$969$	0	0
$970$	16.7234	0.536955
$971$	−20.9204	−0.671367	−0.335683	−	0.941975i	$-0.608967\pi$
$971$	−20.9204	−0.671367	−0.335683	+	0.941975i	$0.608967\pi$
$972$	0	0
$973$	11.1427	0.357217
$974$	−31.0660	−0.995420
$975$	0	0
$976$	55.3776	1.77260
$977$	31.9865	1.02334	0.511669	−	0.859183i	$-0.329027\pi$
$977$	31.9865	1.02334	0.511669	+	0.859183i	$0.329027\pi$
$978$	0	0
$979$	−23.4563	−0.749667
$980$	−2.73383	−0.0873290
$981$	0	0
$982$	39.9000	1.27326
$983$	21.4363	0.683712	0.341856	−	0.939752i	$-0.388944\pi$
$983$	21.4363	0.683712	0.341856	+	0.939752i	$0.388944\pi$
$984$	0	0
$985$	−7.14856	−0.227772
$986$	−41.9900	−1.33723
$987$	0	0
$988$	9.63779	0.306619
$989$	−14.7173	−0.467984
$990$	0	0
$991$	−47.1601	−1.49809	−0.749044	−	0.662520i	$-0.769487\pi$
$991$	−47.1601	−1.49809	−0.749044	+	0.662520i	$0.769487\pi$
$992$	−37.8412	−1.20146
$993$	0	0
$994$	25.3604	0.804382
$995$	−2.70180	−0.0856527
$996$	0	0
$997$	50.2481	1.59137	0.795687	−	0.605708i	$-0.207110\pi$
$997$	50.2481	1.59137	0.795687	+	0.605708i	$0.207110\pi$
$998$	−11.8360	−0.374662
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8001.2.a.v.1.16		19
3.2	odd	2		2667.2.a.q.1.4	✓	19

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2667.2.a.q.1.4	✓	19	3.2	odd	2
8001.2.a.v.1.16		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)

\(f(q)\)	\(=\)	\(q+1.91759 q^{2} +1.67713 q^{4} -1.63006 q^{5} +1.00000 q^{7} -0.619124 q^{8} +O(q^{10})\)	\(q+1.91759 q^{2} +1.67713 q^{4} -1.63006 q^{5} +1.00000 q^{7} -0.619124 q^{8} -3.12578 q^{10} -4.53723 q^{11} +0.785019 q^{13} +1.91759 q^{14} -4.54149 q^{16} +4.58625 q^{17} +7.32031 q^{19} -2.73383 q^{20} -8.70052 q^{22} -1.77992 q^{23} -2.34290 q^{25} +1.50534 q^{26} +1.67713 q^{28} -4.77456 q^{29} +5.06545 q^{31} -7.47045 q^{32} +8.79452 q^{34} -1.63006 q^{35} -1.03335 q^{37} +14.0373 q^{38} +1.00921 q^{40} +10.6489 q^{41} +8.26855 q^{43} -7.60954 q^{44} -3.41315 q^{46} -7.59560 q^{47} +1.00000 q^{49} -4.49271 q^{50} +1.31658 q^{52} -0.151205 q^{53} +7.39596 q^{55} -0.619124 q^{56} -9.15563 q^{58} -4.69350 q^{59} -12.1937 q^{61} +9.71343 q^{62} -5.24224 q^{64} -1.27963 q^{65} +1.56249 q^{67} +7.69175 q^{68} -3.12578 q^{70} +13.2252 q^{71} +5.59318 q^{73} -1.98153 q^{74} +12.2771 q^{76} -4.53723 q^{77} -3.81705 q^{79} +7.40291 q^{80} +20.4201 q^{82} -0.734264 q^{83} -7.47586 q^{85} +15.8556 q^{86} +2.80911 q^{88} +5.16974 q^{89} +0.785019 q^{91} -2.98516 q^{92} -14.5652 q^{94} -11.9326 q^{95} -5.35014 q^{97} +1.91759 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 19 q - 4 q^{2} + 22 q^{4} - 5 q^{5} + 19 q^{7} - 9 q^{8}+O(q^{10}) \) 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

Representations

Groups

Database

Properties

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