LMFDB - Embedded newform 2259.2.a.k.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2259,2,Mod(1,2259)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2259, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("2259.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2259 = 3^{2} \cdot 251 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2259.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:	$18.0382058166$
Analytic rank:	$0$
Dimension:	$17$
Coefficient field:	$\mathbb{Q}[x]/(x^{17} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{17} - 2 x^{16} - 28 x^{15} + 54 x^{14} + 317 x^{13} - 582 x^{12} - 1867 x^{11} + 3178 x^{10} + \cdots + 256 $ x^17 - 2x^16 - 28x^15 + 54x^14 + 317x^13 - 582x^12 - 1867x^11 + 3178x^10 + 6186x^9 - 9216x^8 - 11921x^7 + 13680x^6 + 13752x^5 - 9400x^4 - 8800x^3 + 1920x^2 + 2240x + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 251)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$0.779516$ of defining polynomial
Character	$\chi$	$=$	2259.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.779516 q^{2} -1.39236 q^{4} -1.66398 q^{5} +2.07256 q^{7} +2.64439 q^{8} +O(q^{10})$	\(q-0.779516 q^{2} -1.39236 q^{4} -1.66398 q^{5} +2.07256 q^{7} +2.64439 q^{8} +1.29710 q^{10} -4.31307 q^{11} +0.691540 q^{13} -1.61559 q^{14} +0.723363 q^{16} -4.59213 q^{17} +1.42603 q^{19} +2.31685 q^{20} +3.36210 q^{22} +7.21258 q^{23} -2.23118 q^{25} -0.539067 q^{26} -2.88574 q^{28} -6.96732 q^{29} +2.81029 q^{31} -5.85266 q^{32} +3.57964 q^{34} -3.44870 q^{35} +5.24691 q^{37} -1.11162 q^{38} -4.40021 q^{40} -4.77869 q^{41} -0.0696559 q^{43} +6.00532 q^{44} -5.62232 q^{46} +11.4139 q^{47} -2.70449 q^{49} +1.73924 q^{50} -0.962870 q^{52} -8.68993 q^{53} +7.17685 q^{55} +5.48067 q^{56} +5.43114 q^{58} +7.99534 q^{59} -7.84861 q^{61} -2.19066 q^{62} +3.11552 q^{64} -1.15071 q^{65} +14.4895 q^{67} +6.39387 q^{68} +2.68831 q^{70} -10.8406 q^{71} -0.113363 q^{73} -4.09005 q^{74} -1.98555 q^{76} -8.93909 q^{77} -6.17996 q^{79} -1.20366 q^{80} +3.72506 q^{82} +11.1760 q^{83} +7.64120 q^{85} +0.0542979 q^{86} -11.4054 q^{88} +2.35477 q^{89} +1.43326 q^{91} -10.0425 q^{92} -8.89734 q^{94} -2.37289 q^{95} -2.40260 q^{97} +2.10819 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 17 q - 2 q^{2} + 26 q^{4} - 3 q^{5} + 3 q^{7} - 6 q^{8}+O(q^{10}) $ 17 * q - 2 * q^2 + 26 * q^4 - 3 * q^5 + 3 * q^7 - 6 * q^8	$ 17 q - 2 q^{2} + 26 q^{4} - 3 q^{5} + 3 q^{7} - 6 q^{8} + 7 q^{10} + q^{11} + 22 q^{13} + 7 q^{14} + 40 q^{16} + q^{17} + 13 q^{19} + 14 q^{20} + 4 q^{22} + 2 q^{23} + 32 q^{25} + 9 q^{26} - 10 q^{28} - 28 q^{29} + 12 q^{31} - 4 q^{32} - 21 q^{34} + 15 q^{35} + 27 q^{37} + 37 q^{38} - 7 q^{40} + q^{41} + 9 q^{43} + 43 q^{44} + 4 q^{46} + 20 q^{47} + 32 q^{49} + 28 q^{50} - q^{52} - q^{53} - 11 q^{55} + 61 q^{56} - 46 q^{58} + 20 q^{59} + 59 q^{61} + 73 q^{62} + 54 q^{64} + 14 q^{65} + 15 q^{67} + 20 q^{68} - 11 q^{70} + 26 q^{71} + 8 q^{73} - 2 q^{74} + 38 q^{76} + 33 q^{79} + 29 q^{80} + 10 q^{82} + 67 q^{85} - 11 q^{86} + 27 q^{88} - 11 q^{89} - 2 q^{91} - 28 q^{92} + 29 q^{94} + 8 q^{95} - 10 q^{97} - 22 q^{98}+O(q^{100}) $ 17 * q - 2 * q^2 + 26 * q^4 - 3 * q^5 + 3 * q^7 - 6 * q^8 + 7 * q^10 + q^11 + 22 * q^13 + 7 * q^14 + 40 * q^16 + q^17 + 13 * q^19 + 14 * q^20 + 4 * q^22 + 2 * q^23 + 32 * q^25 + 9 * q^26 - 10 * q^28 - 28 * q^29 + 12 * q^31 - 4 * q^32 - 21 * q^34 + 15 * q^35 + 27 * q^37 + 37 * q^38 - 7 * q^40 + q^41 + 9 * q^43 + 43 * q^44 + 4 * q^46 + 20 * q^47 + 32 * q^49 + 28 * q^50 - q^52 - q^53 - 11 * q^55 + 61 * q^56 - 46 * q^58 + 20 * q^59 + 59 * q^61 + 73 * q^62 + 54 * q^64 + 14 * q^65 + 15 * q^67 + 20 * q^68 - 11 * q^70 + 26 * q^71 + 8 * q^73 - 2 * q^74 + 38 * q^76 + 33 * q^79 + 29 * q^80 + 10 * q^82 + 67 * q^85 - 11 * q^86 + 27 * q^88 - 11 * q^89 - 2 * q^91 - 28 * q^92 + 29 * q^94 + 8 * q^95 - 10 * q^97 - 22 * 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 $

$2$	−0.779516	−0.551201	−0.275600	−	0.961272i	$-0.588877\pi$
$2$	−0.779516	−0.551201	−0.275600	+	0.961272i	$0.588877\pi$
$3$	0	0
$3$	0	0
$4$	−1.39236	−0.696178
$5$	−1.66398	−0.744154	−0.372077	−	0.928202i	$-0.621354\pi$
$5$	−1.66398	−0.744154	−0.372077	+	0.928202i	$0.621354\pi$
$6$	0	0
$7$	2.07256	0.783355	0.391677	−	0.920103i	$-0.371895\pi$
$7$	2.07256	0.783355	0.391677	+	0.920103i	$0.371895\pi$
$8$	2.64439	0.934935
$9$	0	0
$10$	1.29710	0.410178
$11$	−4.31307	−1.30044	−0.650219	−	0.759747i	$-0.725323\pi$
$11$	−4.31307	−1.30044	−0.650219	+	0.759747i	$0.725323\pi$
$12$	0	0
$13$	0.691540	0.191799	0.0958994	−	0.995391i	$-0.469427\pi$
$13$	0.691540	0.191799	0.0958994	+	0.995391i	$0.469427\pi$
$14$	−1.61559	−0.431786
$15$	0	0
$16$	0.723363	0.180841
$17$	−4.59213	−1.11375	−0.556877	−	0.830595i	$-0.688001\pi$
$17$	−4.59213	−1.11375	−0.556877	+	0.830595i	$0.688001\pi$
$18$	0	0
$19$	1.42603	0.327155	0.163577	−	0.986531i	$-0.447697\pi$
$19$	1.42603	0.327155	0.163577	+	0.986531i	$0.447697\pi$
$20$	2.31685	0.518063
$21$	0	0
$22$	3.36210	0.716803
$23$	7.21258	1.50393	0.751964	−	0.659204i	$-0.229107\pi$
$23$	7.21258	1.50393	0.751964	+	0.659204i	$0.229107\pi$
$24$	0	0
$25$	−2.23118	−0.446235
$26$	−0.539067	−0.105720
$27$	0	0
$28$	−2.88574	−0.545354
$29$	−6.96732	−1.29380	−0.646900	−	0.762575i	$-0.723935\pi$
$29$	−6.96732	−1.29380	−0.646900	+	0.762575i	$0.723935\pi$
$30$	0	0
$31$	2.81029	0.504743	0.252371	−	0.967630i	$-0.418790\pi$
$31$	2.81029	0.504743	0.252371	+	0.967630i	$0.418790\pi$
$32$	−5.85266	−1.03461
$33$	0	0
$34$	3.57964	0.613903
$35$	−3.44870	−0.582936
$36$	0	0
$37$	5.24691	0.862587	0.431294	−	0.902212i	$-0.358057\pi$
$37$	5.24691	0.862587	0.431294	+	0.902212i	$0.358057\pi$
$38$	−1.11162	−0.180328
$39$	0	0
$40$	−4.40021	−0.695735
$41$	−4.77869	−0.746306	−0.373153	−	0.927770i	$-0.621723\pi$
$41$	−4.77869	−0.746306	−0.373153	+	0.927770i	$0.621723\pi$
$42$	0	0
$43$	−0.0696559	−0.0106224	−0.00531122	−	0.999986i	$-0.501691\pi$
$43$	−0.0696559	−0.0106224	−0.00531122	+	0.999986i	$0.501691\pi$
$44$	6.00532	0.905336
$45$	0	0
$46$	−5.62232	−0.828966
$47$	11.4139	1.66489	0.832446	−	0.554106i	$-0.186940\pi$
$47$	11.4139	1.66489	0.832446	+	0.554106i	$0.186940\pi$
$48$	0	0
$49$	−2.70449	−0.386356
$50$	1.73924	0.245965
$51$	0	0
$52$	−0.962870	−0.133526
$53$	−8.68993	−1.19365	−0.596827	−	0.802370i	$-0.703572\pi$
$53$	−8.68993	−1.19365	−0.596827	+	0.802370i	$0.703572\pi$
$54$	0	0
$55$	7.17685	0.967726
$56$	5.48067	0.732385
$57$	0	0
$58$	5.43114	0.713143
$59$	7.99534	1.04090	0.520452	−	0.853891i	$-0.325763\pi$
$59$	7.99534	1.04090	0.520452	+	0.853891i	$0.325763\pi$
$60$	0	0
$61$	−7.84861	−1.00491	−0.502456	−	0.864603i	$-0.667570\pi$
$61$	−7.84861	−1.00491	−0.502456	+	0.864603i	$0.667570\pi$
$62$	−2.19066	−0.278215
$63$	0	0
$64$	3.11552	0.389439
$65$	−1.15071	−0.142728
$66$	0	0
$67$	14.4895	1.77017	0.885087	−	0.465426i	$-0.154099\pi$
$67$	14.4895	1.77017	0.885087	+	0.465426i	$0.154099\pi$
$68$	6.39387	0.775371
$69$	0	0
$70$	2.68831	0.321315
$71$	−10.8406	−1.28654	−0.643268	−	0.765641i	$-0.722422\pi$
$71$	−10.8406	−1.28654	−0.643268	+	0.765641i	$0.722422\pi$
$72$	0	0
$73$	−0.113363	−0.0132682	−0.00663409	−	0.999978i	$-0.502112\pi$
$73$	−0.113363	−0.0132682	−0.00663409	+	0.999978i	$0.502112\pi$
$74$	−4.09005	−0.475459
$75$	0	0
$76$	−1.98555	−0.227758
$77$	−8.93909	−1.01870
$78$	0	0
$79$	−6.17996	−0.695300	−0.347650	−	0.937624i	$-0.613020\pi$
$79$	−6.17996	−0.695300	−0.347650	+	0.937624i	$0.613020\pi$
$80$	−1.20366	−0.134573
$81$	0	0
$82$	3.72506	0.411364
$83$	11.1760	1.22673	0.613363	−	0.789801i	$-0.289816\pi$
$83$	11.1760	1.22673	0.613363	+	0.789801i	$0.289816\pi$
$84$	0	0
$85$	7.64120	0.828805
$86$	0.0542979	0.00585509
$87$	0	0
$88$	−11.4054	−1.21582
$89$	2.35477	0.249605	0.124803	−	0.992182i	$-0.460170\pi$
$89$	2.35477	0.249605	0.124803	+	0.992182i	$0.460170\pi$
$90$	0	0
$91$	1.43326	0.150246
$92$	−10.0425	−1.04700
$93$	0	0
$94$	−8.89734	−0.917690
$95$	−2.37289	−0.243453
$96$	0	0
$97$	−2.40260	−0.243947	−0.121973	−	0.992533i	$-0.538922\pi$
$97$	−2.40260	−0.243947	−0.121973	+	0.992533i	$0.538922\pi$
$98$	2.10819	0.212960
$99$	0	0
$100$	3.10659	0.310659
$101$	0.196664	0.0195688	0.00978438	−	0.999952i	$-0.496885\pi$
$101$	0.196664	0.0195688	0.00978438	+	0.999952i	$0.496885\pi$
$102$	0	0
$103$	8.71179	0.858398	0.429199	−	0.903210i	$-0.358796\pi$
$103$	8.71179	0.858398	0.429199	+	0.903210i	$0.358796\pi$
$104$	1.82871	0.179319
$105$	0	0
$106$	6.77394	0.657943
$107$	4.84459	0.468344	0.234172	−	0.972195i	$-0.424762\pi$
$107$	4.84459	0.468344	0.234172	+	0.972195i	$0.424762\pi$
$108$	0	0
$109$	20.3490	1.94908	0.974541	−	0.224208i	$-0.0719796\pi$
$109$	20.3490	1.94908	0.974541	+	0.224208i	$0.0719796\pi$
$110$	−5.59446	−0.533411
$111$	0	0
$112$	1.49921	0.141663
$113$	16.1506	1.51932	0.759659	−	0.650322i	$-0.225366\pi$
$113$	16.1506	1.51932	0.759659	+	0.650322i	$0.225366\pi$
$114$	0	0
$115$	−12.0016	−1.11915
$116$	9.70099	0.900714
$117$	0	0
$118$	−6.23249	−0.573748
$119$	−9.51747	−0.872465
$120$	0	0
$121$	7.60253	0.691139
$122$	6.11811	0.553908
$123$	0	0
$124$	−3.91292	−0.351391
$125$	12.0325	1.07622
$126$	0	0
$127$	−21.5000	−1.90782	−0.953909	−	0.300097i	$-0.902981\pi$
$127$	−21.5000	−1.90782	−0.953909	+	0.300097i	$0.902981\pi$
$128$	9.27673	0.819955
$129$	0	0
$130$	0.896995	0.0786717
$131$	10.4234	0.910697	0.455349	−	0.890313i	$-0.349515\pi$
$131$	10.4234	0.910697	0.455349	+	0.890313i	$0.349515\pi$
$132$	0	0
$133$	2.95554	0.256278
$134$	−11.2948	−0.975721
$135$	0	0
$136$	−12.1434	−1.04129
$137$	17.0007	1.45246	0.726232	−	0.687449i	$-0.241270\pi$
$137$	17.0007	1.45246	0.726232	+	0.687449i	$0.241270\pi$
$138$	0	0
$139$	15.9865	1.35596	0.677980	−	0.735081i	$-0.262856\pi$
$139$	15.9865	1.35596	0.677980	+	0.735081i	$0.262856\pi$
$140$	4.80181	0.405827
$141$	0	0
$142$	8.45038	0.709140
$143$	−2.98266	−0.249422
$144$	0	0
$145$	11.5935	0.962785
$146$	0.0883686	0.00731343
$147$	0	0
$148$	−7.30557	−0.600514
$149$	14.2480	1.16725	0.583623	−	0.812025i	$-0.301635\pi$
$149$	14.2480	1.16725	0.583623	+	0.812025i	$0.301635\pi$
$150$	0	0
$151$	13.4756	1.09663	0.548316	−	0.836271i	$-0.315269\pi$
$151$	13.4756	1.09663	0.548316	+	0.836271i	$0.315269\pi$
$152$	3.77099	0.305868
$153$	0	0
$154$	6.96816	0.561511
$155$	−4.67626	−0.375606
$156$	0	0
$157$	10.2729	0.819870	0.409935	−	0.912115i	$-0.365551\pi$
$157$	10.2729	0.819870	0.409935	+	0.912115i	$0.365551\pi$
$158$	4.81738	0.383250
$159$	0	0
$160$	9.73870	0.769912
$161$	14.9485	1.17811
$162$	0	0
$163$	12.4576	0.975753	0.487877	−	0.872913i	$-0.337772\pi$
$163$	12.4576	0.975753	0.487877	+	0.872913i	$0.337772\pi$
$164$	6.65363	0.519561
$165$	0	0
$166$	−8.71188	−0.676173
$167$	17.4629	1.35132	0.675659	−	0.737214i	$-0.263859\pi$
$167$	17.4629	1.35132	0.675659	+	0.737214i	$0.263859\pi$
$168$	0	0
$169$	−12.5218	−0.963213
$170$	−5.95644	−0.456838
$171$	0	0
$172$	0.0969858	0.00739510
$173$	13.2820	1.00982	0.504908	−	0.863173i	$-0.331527\pi$
$173$	13.2820	1.00982	0.504908	+	0.863173i	$0.331527\pi$
$174$	0	0
$175$	−4.62425	−0.349560
$176$	−3.11991	−0.235172
$177$	0	0
$178$	−1.83558	−0.137583
$179$	−16.9053	−1.26357	−0.631783	−	0.775146i	$-0.717676\pi$
$179$	−16.9053	−1.26357	−0.631783	+	0.775146i	$0.717676\pi$
$180$	0	0
$181$	−8.95143	−0.665354	−0.332677	−	0.943041i	$-0.607952\pi$
$181$	−8.95143	−0.665354	−0.332677	+	0.943041i	$0.607952\pi$
$182$	−1.11725	−0.0828160
$183$	0	0
$184$	19.0729	1.40607
$185$	−8.73075	−0.641898
$186$	0	0
$187$	19.8062	1.44837
$188$	−15.8922	−1.15906
$189$	0	0
$190$	1.84970	0.134192
$191$	−6.87340	−0.497342	−0.248671	−	0.968588i	$-0.579994\pi$
$191$	−6.87340	−0.497342	−0.248671	+	0.968588i	$0.579994\pi$
$192$	0	0
$193$	10.7691	0.775174	0.387587	−	0.921833i	$-0.373309\pi$
$193$	10.7691	0.775174	0.387587	+	0.921833i	$0.373309\pi$
$194$	1.87286	0.134464
$195$	0	0
$196$	3.76561	0.268972
$197$	15.3371	1.09273	0.546363	−	0.837548i	$-0.316012\pi$
$197$	15.3371	1.09273	0.546363	+	0.837548i	$0.316012\pi$
$198$	0	0
$199$	−5.96752	−0.423026	−0.211513	−	0.977375i	$-0.567839\pi$
$199$	−5.96752	−0.423026	−0.211513	+	0.977375i	$0.567839\pi$
$200$	−5.90011	−0.417201
$201$	0	0
$202$	−0.153302	−0.0107863
$203$	−14.4402	−1.01350
$204$	0	0
$205$	7.95163	0.555366
$206$	−6.79098	−0.473150
$207$	0	0
$208$	0.500235	0.0346851
$209$	−6.15058	−0.425444
$210$	0	0
$211$	−0.312277	−0.0214981	−0.0107490	−	0.999942i	$-0.503422\pi$
$211$	−0.312277	−0.0214981	−0.0107490	+	0.999942i	$0.503422\pi$
$212$	12.0995	0.830995
$213$	0	0
$214$	−3.77644	−0.258152
$215$	0.115906	0.00790472
$216$	0	0
$217$	5.82449	0.395392
$218$	−15.8624	−1.07434
$219$	0	0
$220$	−9.99272	−0.673709
$221$	−3.17564	−0.213617
$222$	0	0
$223$	−9.92618	−0.664706	−0.332353	−	0.943155i	$-0.607843\pi$
$223$	−9.92618	−0.664706	−0.332353	+	0.943155i	$0.607843\pi$
$224$	−12.1300	−0.810470
$225$	0	0
$226$	−12.5896	−0.837449
$227$	21.8928	1.45307	0.726537	−	0.687128i	$-0.241129\pi$
$227$	21.8928	1.45307	0.726537	+	0.687128i	$0.241129\pi$
$228$	0	0
$229$	−3.74494	−0.247473	−0.123736	−	0.992315i	$-0.539488\pi$
$229$	−3.74494	−0.247473	−0.123736	+	0.992315i	$0.539488\pi$
$230$	9.35542	0.616878
$231$	0	0
$232$	−18.4243	−1.20962
$233$	−22.6997	−1.48710	−0.743552	−	0.668678i	$-0.766860\pi$
$233$	−22.6997	−1.48710	−0.743552	+	0.668678i	$0.766860\pi$
$234$	0	0
$235$	−18.9925	−1.23894
$236$	−11.1324	−0.724655
$237$	0	0
$238$	7.41902	0.480903
$239$	−4.32556	−0.279797	−0.139899	−	0.990166i	$-0.544678\pi$
$239$	−4.32556	−0.279797	−0.139899	+	0.990166i	$0.544678\pi$
$240$	0	0
$241$	−12.9477	−0.834034	−0.417017	−	0.908899i	$-0.636924\pi$
$241$	−12.9477	−0.834034	−0.417017	+	0.908899i	$0.636924\pi$
$242$	−5.92629	−0.380956
$243$	0	0
$244$	10.9280	0.699597
$245$	4.50021	0.287508
$246$	0	0
$247$	0.986160	0.0627478
$248$	7.43151	0.471901
$249$	0	0
$250$	−9.37954	−0.593214
$251$	−1.00000	−0.0631194
$251$	−1.00000	−0.0631194
$252$	0	0
$253$	−31.1083	−1.95576
$254$	16.7596	1.05159
$255$	0	0
$256$	−13.4624	−0.841399
$257$	0.394093	0.0245828	0.0122914	−	0.999924i	$-0.496087\pi$
$257$	0.394093	0.0245828	0.0122914	+	0.999924i	$0.496087\pi$
$258$	0	0
$259$	10.8746	0.675712
$260$	1.60219	0.0993639
$261$	0	0
$262$	−8.12521	−0.501977
$263$	−1.53116	−0.0944155	−0.0472077	−	0.998885i	$-0.515032\pi$
$263$	−1.53116	−0.0944155	−0.0472077	+	0.998885i	$0.515032\pi$
$264$	0	0
$265$	14.4599	0.888262
$266$	−2.30389	−0.141261
$267$	0	0
$268$	−20.1745	−1.23235
$269$	3.68392	0.224612	0.112306	−	0.993674i	$-0.464176\pi$
$269$	3.68392	0.224612	0.112306	+	0.993674i	$0.464176\pi$
$270$	0	0
$271$	10.8632	0.659892	0.329946	−	0.944000i	$-0.392969\pi$
$271$	10.8632	0.659892	0.329946	+	0.944000i	$0.392969\pi$
$272$	−3.32178	−0.201412
$273$	0	0
$274$	−13.2523	−0.800600
$275$	9.62321	0.580301
$276$	0	0
$277$	−3.92556	−0.235864	−0.117932	−	0.993022i	$-0.537627\pi$
$277$	−3.92556	−0.235864	−0.117932	+	0.993022i	$0.537627\pi$
$278$	−12.4618	−0.747406
$279$	0	0
$280$	−9.11971	−0.545007
$281$	1.81456	0.108248	0.0541239	−	0.998534i	$-0.482763\pi$
$281$	1.81456	0.108248	0.0541239	+	0.998534i	$0.482763\pi$
$282$	0	0
$283$	20.4093	1.21321	0.606604	−	0.795004i	$-0.292531\pi$
$283$	20.4093	1.21321	0.606604	+	0.795004i	$0.292531\pi$
$284$	15.0939	0.895658
$285$	0	0
$286$	2.32503	0.137482
$287$	−9.90412	−0.584622
$288$	0	0
$289$	4.08765	0.240450
$290$	−9.03729	−0.530688
$291$	0	0
$292$	0.157842	0.00923701
$293$	19.2694	1.12573	0.562865	−	0.826549i	$-0.309699\pi$
$293$	19.2694	1.12573	0.562865	+	0.826549i	$0.309699\pi$
$294$	0	0
$295$	−13.3041	−0.774593
$296$	13.8749	0.806463
$297$	0	0
$298$	−11.1066	−0.643386
$299$	4.98779	0.288452
$300$	0	0
$301$	−0.144366	−0.00832113
$302$	−10.5045	−0.604464
$303$	0	0
$304$	1.03154	0.0591629
$305$	13.0599	0.747808
$306$	0	0
$307$	19.9296	1.13744	0.568720	−	0.822531i	$-0.307439\pi$
$307$	19.9296	1.13744	0.568720	+	0.822531i	$0.307439\pi$
$308$	12.4464	0.709199
$309$	0	0
$310$	3.64522	0.207034
$311$	−29.0820	−1.64909	−0.824545	−	0.565797i	$-0.808569\pi$
$311$	−29.0820	−1.64909	−0.824545	+	0.565797i	$0.808569\pi$
$312$	0	0
$313$	−25.4848	−1.44049	−0.720243	−	0.693722i	$-0.755970\pi$
$313$	−25.4848	−1.44049	−0.720243	+	0.693722i	$0.755970\pi$
$314$	−8.00792	−0.451913
$315$	0	0
$316$	8.60470	0.484052
$317$	−22.9102	−1.28677	−0.643383	−	0.765544i	$-0.722470\pi$
$317$	−22.9102	−1.28677	−0.643383	+	0.765544i	$0.722470\pi$
$318$	0	0
$319$	30.0505	1.68251
$320$	−5.18415	−0.289803
$321$	0	0
$322$	−11.6526	−0.649375
$323$	−6.54853	−0.364370
$324$	0	0
$325$	−1.54295	−0.0855874
$326$	−9.71088	−0.537836
$327$	0	0
$328$	−12.6367	−0.697747
$329$	23.6561	1.30420
$330$	0	0
$331$	24.7108	1.35823	0.679113	−	0.734034i	$-0.262365\pi$
$331$	24.7108	1.35823	0.679113	+	0.734034i	$0.262365\pi$
$332$	−15.5610	−0.854020
$333$	0	0
$334$	−13.6126	−0.744848
$335$	−24.1102	−1.31728
$336$	0	0
$337$	17.8271	0.971102	0.485551	−	0.874208i	$-0.338619\pi$
$337$	17.8271	0.971102	0.485551	+	0.874208i	$0.338619\pi$
$338$	9.76092	0.530924
$339$	0	0
$340$	−10.6393	−0.576995
$341$	−12.1210	−0.656387
$342$	0	0
$343$	−20.1131	−1.08601
$344$	−0.184198	−0.00993128
$345$	0	0
$346$	−10.3536	−0.556611
$347$	26.3885	1.41661	0.708305	−	0.705907i	$-0.249460\pi$
$347$	26.3885	1.41661	0.708305	+	0.705907i	$0.249460\pi$
$348$	0	0
$349$	−16.0414	−0.858677	−0.429339	−	0.903144i	$-0.641253\pi$
$349$	−16.0414	−0.858677	−0.429339	+	0.903144i	$0.641253\pi$
$350$	3.60468	0.192678
$351$	0	0
$352$	25.2429	1.34545
$353$	−9.77533	−0.520289	−0.260144	−	0.965570i	$-0.583770\pi$
$353$	−9.77533	−0.520289	−0.260144	+	0.965570i	$0.583770\pi$
$354$	0	0
$355$	18.0384	0.957381
$356$	−3.27868	−0.173769
$357$	0	0
$358$	13.1780	0.696478
$359$	−9.93170	−0.524175	−0.262088	−	0.965044i	$-0.584411\pi$
$359$	−9.93170	−0.524175	−0.262088	+	0.965044i	$0.584411\pi$
$360$	0	0
$361$	−16.9664	−0.892970
$362$	6.97778	0.366744
$363$	0	0
$364$	−1.99561	−0.104598
$365$	0.188634	0.00987357
$366$	0	0
$367$	7.88573	0.411632	0.205816	−	0.978591i	$-0.434015\pi$
$367$	7.88573	0.411632	0.205816	+	0.978591i	$0.434015\pi$
$368$	5.21732	0.271972
$369$	0	0
$370$	6.80576	0.353814
$371$	−18.0104	−0.935054
$372$	0	0
$373$	−9.68592	−0.501518	−0.250759	−	0.968050i	$-0.580680\pi$
$373$	−9.68592	−0.501518	−0.250759	+	0.968050i	$0.580680\pi$
$374$	−15.4392	−0.798342
$375$	0	0
$376$	30.1829	1.55657
$377$	−4.81818	−0.248149
$378$	0	0
$379$	6.40384	0.328943	0.164471	−	0.986382i	$-0.447408\pi$
$379$	6.40384	0.328943	0.164471	+	0.986382i	$0.447408\pi$
$380$	3.30390	0.169487
$381$	0	0
$382$	5.35793	0.274135
$383$	3.01090	0.153850	0.0769248	−	0.997037i	$-0.475490\pi$
$383$	3.01090	0.153850	0.0769248	+	0.997037i	$0.475490\pi$
$384$	0	0
$385$	14.8745	0.758072
$386$	−8.39465	−0.427277
$387$	0	0
$388$	3.34527	0.169830
$389$	3.05939	0.155117	0.0775586	−	0.996988i	$-0.475287\pi$
$389$	3.05939	0.155117	0.0775586	+	0.996988i	$0.475287\pi$
$390$	0	0
$391$	−33.1211	−1.67501
$392$	−7.15174	−0.361217
$393$	0	0
$394$	−11.9555	−0.602312
$395$	10.2833	0.517410
$396$	0	0
$397$	11.8686	0.595668	0.297834	−	0.954618i	$-0.403736\pi$
$397$	11.8686	0.595668	0.297834	+	0.954618i	$0.403736\pi$
$398$	4.65177	0.233172
$399$	0	0
$400$	−1.61395	−0.0806976
$401$	3.21244	0.160422	0.0802109	−	0.996778i	$-0.474441\pi$
$401$	3.21244	0.160422	0.0802109	+	0.996778i	$0.474441\pi$
$402$	0	0
$403$	1.94343	0.0968090
$404$	−0.273826	−0.0136233
$405$	0	0
$406$	11.2564	0.558644
$407$	−22.6303	−1.12174
$408$	0	0
$409$	−39.8967	−1.97277	−0.986383	−	0.164462i	$-0.947411\pi$
$409$	−39.8967	−1.97277	−0.986383	+	0.164462i	$0.947411\pi$
$410$	−6.19842	−0.306118
$411$	0	0
$412$	−12.1299	−0.597598
$413$	16.5708	0.815397
$414$	0	0
$415$	−18.5966	−0.912873
$416$	−4.04735	−0.198438
$417$	0	0
$418$	4.79447	0.234505
$419$	−21.7394	−1.06204	−0.531020	−	0.847359i	$-0.678191\pi$
$419$	−21.7394	−1.06204	−0.531020	+	0.847359i	$0.678191\pi$
$420$	0	0
$421$	4.07730	0.198715	0.0993577	−	0.995052i	$-0.468321\pi$
$421$	4.07730	0.198715	0.0993577	+	0.995052i	$0.468321\pi$
$422$	0.243425	0.0118498
$423$	0	0
$424$	−22.9796	−1.11599
$425$	10.2459	0.496997
$426$	0	0
$427$	−16.2667	−0.787202
$428$	−6.74539	−0.326051
$429$	0	0
$430$	−0.0903505	−0.00435709
$431$	−12.9221	−0.622436	−0.311218	−	0.950339i	$-0.600737\pi$
$431$	−12.9221	−0.622436	−0.311218	+	0.950339i	$0.600737\pi$
$432$	0	0
$433$	24.6228	1.18330	0.591649	−	0.806196i	$-0.298477\pi$
$433$	24.6228	1.18330	0.591649	+	0.806196i	$0.298477\pi$
$434$	−4.54029	−0.217941
$435$	0	0
$436$	−28.3331	−1.35691
$437$	10.2854	0.492017
$438$	0	0
$439$	−21.7282	−1.03703	−0.518515	−	0.855068i	$-0.673515\pi$
$439$	−21.7282	−1.03703	−0.518515	+	0.855068i	$0.673515\pi$
$440$	18.9784	0.904760
$441$	0	0
$442$	2.47546	0.117746
$443$	35.5501	1.68904	0.844518	−	0.535527i	$-0.179887\pi$
$443$	35.5501	1.68904	0.844518	+	0.535527i	$0.179887\pi$
$444$	0	0
$445$	−3.91829	−0.185745
$446$	7.73761	0.366387
$447$	0	0
$448$	6.45710	0.305069
$449$	30.4969	1.43924	0.719619	−	0.694370i	$-0.244317\pi$
$449$	30.4969	1.43924	0.719619	+	0.694370i	$0.244317\pi$
$450$	0	0
$451$	20.6108	0.970524
$452$	−22.4873	−1.05772
$453$	0	0
$454$	−17.0658	−0.800935
$455$	−2.38491	−0.111806
$456$	0	0
$457$	23.1694	1.08382	0.541908	−	0.840437i	$-0.317702\pi$
$457$	23.1694	1.08382	0.541908	+	0.840437i	$0.317702\pi$
$458$	2.91924	0.136407
$459$	0	0
$460$	16.7105	0.779130
$461$	−24.0832	−1.12167	−0.560833	−	0.827929i	$-0.689519\pi$
$461$	−24.0832	−1.12167	−0.560833	+	0.827929i	$0.689519\pi$
$462$	0	0
$463$	5.04661	0.234536	0.117268	−	0.993100i	$-0.462586\pi$
$463$	5.04661	0.234536	0.117268	+	0.993100i	$0.462586\pi$
$464$	−5.03991	−0.233972
$465$	0	0
$466$	17.6947	0.819693
$467$	15.4324	0.714128	0.357064	−	0.934080i	$-0.383778\pi$
$467$	15.4324	0.714128	0.357064	+	0.934080i	$0.383778\pi$
$468$	0	0
$469$	30.0304	1.38667
$470$	14.8050	0.682903
$471$	0	0
$472$	21.1428	0.973178
$473$	0.300431	0.0138138
$474$	0	0
$475$	−3.18173	−0.145988
$476$	13.2517	0.607391
$477$	0	0
$478$	3.37184	0.154224
$479$	−3.51010	−0.160381	−0.0801903	−	0.996780i	$-0.525553\pi$
$479$	−3.51010	−0.160381	−0.0801903	+	0.996780i	$0.525553\pi$
$480$	0	0
$481$	3.62845	0.165443
$482$	10.0929	0.459720
$483$	0	0
$484$	−10.5854	−0.481156
$485$	3.99787	0.181534
$486$	0	0
$487$	25.8708	1.17232	0.586158	−	0.810196i	$-0.300640\pi$
$487$	25.8708	1.17232	0.586158	+	0.810196i	$0.300640\pi$
$488$	−20.7548	−0.939526
$489$	0	0
$490$	−3.50799	−0.158475
$491$	20.4715	0.923868	0.461934	−	0.886914i	$-0.347156\pi$
$491$	20.4715	0.923868	0.461934	+	0.886914i	$0.347156\pi$
$492$	0	0
$493$	31.9948	1.44098
$494$	−0.768727	−0.0345867
$495$	0	0
$496$	2.03286	0.0912781
$497$	−22.4677	−1.00781
$498$	0	0
$499$	−31.6343	−1.41615	−0.708073	−	0.706139i	$-0.750435\pi$
$499$	−31.6343	−1.41615	−0.708073	+	0.706139i	$0.750435\pi$
$500$	−16.7535	−0.749241
$501$	0	0
$502$	0.779516	0.0347915
$503$	7.48882	0.333910	0.166955	−	0.985965i	$-0.446607\pi$
$503$	7.48882	0.333910	0.166955	+	0.985965i	$0.446607\pi$
$504$	0	0
$505$	−0.327244	−0.0145622
$506$	24.2494	1.07802
$507$	0	0
$508$	29.9357	1.32818
$509$	−3.71394	−0.164618	−0.0823088	−	0.996607i	$-0.526229\pi$
$509$	−3.71394	−0.164618	−0.0823088	+	0.996607i	$0.526229\pi$
$510$	0	0
$511$	−0.234953	−0.0103937
$512$	−8.05932	−0.356175
$513$	0	0
$514$	−0.307202	−0.0135501
$515$	−14.4962	−0.638780
$516$	0	0
$517$	−49.2290	−2.16509
$518$	−8.47688	−0.372453
$519$	0	0
$520$	−3.04293	−0.133441
$521$	17.6405	0.772846	0.386423	−	0.922322i	$-0.373711\pi$
$521$	17.6405	0.772846	0.386423	+	0.922322i	$0.373711\pi$
$522$	0	0
$523$	−11.7757	−0.514915	−0.257458	−	0.966290i	$-0.582885\pi$
$523$	−11.7757	−0.514915	−0.257458	+	0.966290i	$0.582885\pi$
$524$	−14.5131	−0.634007
$525$	0	0
$526$	1.19356	0.0520419
$527$	−12.9052	−0.562160
$528$	0	0
$529$	29.0214	1.26180
$530$	−11.2717	−0.489611
$531$	0	0
$532$	−4.11516	−0.178415
$533$	−3.30466	−0.143141
$534$	0	0
$535$	−8.06130	−0.348520
$536$	38.3159	1.65500
$537$	0	0
$538$	−2.87167	−0.123807
$539$	11.6646	0.502432
$540$	0	0
$541$	17.0425	0.732715	0.366357	−	0.930474i	$-0.380605\pi$
$541$	17.0425	0.732715	0.366357	+	0.930474i	$0.380605\pi$
$542$	−8.46804	−0.363733
$543$	0	0
$544$	26.8762	1.15231
$545$	−33.8603	−1.45042
$546$	0	0
$547$	−20.9090	−0.894004	−0.447002	−	0.894533i	$-0.647508\pi$
$547$	−20.9090	−0.894004	−0.447002	+	0.894533i	$0.647508\pi$
$548$	−23.6710	−1.01117
$549$	0	0
$550$	−7.50144	−0.319863
$551$	−9.93563	−0.423272
$552$	0	0
$553$	−12.8084	−0.544666
$554$	3.06004	0.130008
$555$	0	0
$556$	−22.2589	−0.943989
$557$	14.3266	0.607037	0.303518	−	0.952826i	$-0.401839\pi$
$557$	14.3266	0.607037	0.303518	+	0.952826i	$0.401839\pi$
$558$	0	0
$559$	−0.0481699	−0.00203737
$560$	−2.49466	−0.105419
$561$	0	0
$562$	−1.41448	−0.0596663
$563$	2.48884	0.104892	0.0524460	−	0.998624i	$-0.483298\pi$
$563$	2.48884	0.104892	0.0524460	+	0.998624i	$0.483298\pi$
$564$	0	0
$565$	−26.8742	−1.13061
$566$	−15.9094	−0.668722
$567$	0	0
$568$	−28.6667	−1.20283
$569$	5.07011	0.212550	0.106275	−	0.994337i	$-0.466108\pi$
$569$	5.07011	0.212550	0.106275	+	0.994337i	$0.466108\pi$
$570$	0	0
$571$	−35.7515	−1.49615	−0.748077	−	0.663612i	$-0.769023\pi$
$571$	−35.7515	−1.49615	−0.748077	+	0.663612i	$0.769023\pi$
$572$	4.15292	0.173642
$573$	0	0
$574$	7.72042	0.322244
$575$	−16.0925	−0.671106
$576$	0	0
$577$	12.8559	0.535197	0.267599	−	0.963530i	$-0.413770\pi$
$577$	12.8559	0.535197	0.267599	+	0.963530i	$0.413770\pi$
$578$	−3.18639	−0.132536
$579$	0	0
$580$	−16.1422	−0.670270
$581$	23.1630	0.960962
$582$	0	0
$583$	37.4803	1.55227
$584$	−0.299778	−0.0124049
$585$	0	0
$586$	−15.0208	−0.620504
$587$	−30.4149	−1.25536	−0.627679	−	0.778472i	$-0.715995\pi$
$587$	−30.4149	−1.25536	−0.627679	+	0.778472i	$0.715995\pi$
$588$	0	0
$589$	4.00757	0.165129
$590$	10.3707	0.426956
$591$	0	0
$592$	3.79543	0.155991
$593$	−28.6314	−1.17575	−0.587876	−	0.808951i	$-0.700036\pi$
$593$	−28.6314	−1.17575	−0.587876	+	0.808951i	$0.700036\pi$
$594$	0	0
$595$	15.8369	0.649248
$596$	−19.8383	−0.812610
$597$	0	0
$598$	−3.88806	−0.158995
$599$	11.0037	0.449600	0.224800	−	0.974405i	$-0.427827\pi$
$599$	11.0037	0.449600	0.224800	+	0.974405i	$0.427827\pi$
$600$	0	0
$601$	0.423259	0.0172651	0.00863255	−	0.999963i	$-0.497252\pi$
$601$	0.423259	0.0172651	0.00863255	+	0.999963i	$0.497252\pi$
$602$	0.112536	0.00458661
$603$	0	0
$604$	−18.7629	−0.763451
$605$	−12.6504	−0.514314
$606$	0	0
$607$	−40.7175	−1.65267	−0.826336	−	0.563177i	$-0.809579\pi$
$607$	−40.7175	−1.65267	−0.826336	+	0.563177i	$0.809579\pi$
$608$	−8.34609	−0.338479
$609$	0	0
$610$	−10.1804	−0.412193
$611$	7.89319	0.319324
$612$	0	0
$613$	4.87460	0.196883	0.0984417	−	0.995143i	$-0.468614\pi$
$613$	4.87460	0.196883	0.0984417	+	0.995143i	$0.468614\pi$
$614$	−15.5354	−0.626958
$615$	0	0
$616$	−23.6385	−0.952422
$617$	7.06448	0.284405	0.142203	−	0.989838i	$-0.454582\pi$
$617$	7.06448	0.284405	0.142203	+	0.989838i	$0.454582\pi$
$618$	0	0
$619$	−10.4543	−0.420195	−0.210098	−	0.977680i	$-0.567378\pi$
$619$	−10.4543	−0.420195	−0.210098	+	0.977680i	$0.567378\pi$
$620$	6.51101	0.261489
$621$	0	0
$622$	22.6699	0.908980
$623$	4.88041	0.195529
$624$	0	0
$625$	−8.86597	−0.354639
$626$	19.8658	0.793997
$627$	0	0
$628$	−14.3036	−0.570775
$629$	−24.0945	−0.960711
$630$	0	0
$631$	−8.50745	−0.338676	−0.169338	−	0.985558i	$-0.554163\pi$
$631$	−8.50745	−0.338676	−0.169338	+	0.985558i	$0.554163\pi$
$632$	−16.3423	−0.650060
$633$	0	0
$634$	17.8589	0.709267
$635$	35.7755	1.41971
$636$	0	0
$637$	−1.87026	−0.0741026
$638$	−23.4248	−0.927398
$639$	0	0
$640$	−15.4363	−0.610172
$641$	−17.1989	−0.679315	−0.339657	−	0.940549i	$-0.610311\pi$
$641$	−17.1989	−0.679315	−0.339657	+	0.940549i	$0.610311\pi$
$642$	0	0
$643$	−30.4905	−1.20243	−0.601215	−	0.799088i	$-0.705316\pi$
$643$	−30.4905	−1.20243	−0.601215	+	0.799088i	$0.705316\pi$
$644$	−20.8137	−0.820173
$645$	0	0
$646$	5.10468	0.200841
$647$	23.5857	0.927250	0.463625	−	0.886031i	$-0.346548\pi$
$647$	23.5857	0.927250	0.463625	+	0.886031i	$0.346548\pi$
$648$	0	0
$649$	−34.4844	−1.35363
$650$	1.20275	0.0471758
$651$	0	0
$652$	−17.3454	−0.679297
$653$	−11.0925	−0.434081	−0.217041	−	0.976163i	$-0.569640\pi$
$653$	−11.0925	−0.434081	−0.217041	+	0.976163i	$0.569640\pi$
$654$	0	0
$655$	−17.3443	−0.677699
$656$	−3.45673	−0.134963
$657$	0	0
$658$	−18.4403	−0.718877
$659$	−31.0542	−1.20970	−0.604850	−	0.796339i	$-0.706767\pi$
$659$	−31.0542	−1.20970	−0.604850	+	0.796339i	$0.706767\pi$
$660$	0	0
$661$	7.18698	0.279541	0.139771	−	0.990184i	$-0.455364\pi$
$661$	7.18698	0.279541	0.139771	+	0.990184i	$0.455364\pi$
$662$	−19.2624	−0.748655
$663$	0	0
$664$	29.5538	1.14691
$665$	−4.91796	−0.190710
$666$	0	0
$667$	−50.2524	−1.94578
$668$	−24.3145	−0.940757
$669$	0	0
$670$	18.7943	0.726086
$671$	33.8515	1.30682
$672$	0	0
$673$	−29.4721	−1.13607	−0.568034	−	0.823005i	$-0.692296\pi$
$673$	−29.4721	−1.13607	−0.568034	+	0.823005i	$0.692296\pi$
$674$	−13.8965	−0.535272
$675$	0	0
$676$	17.4348	0.670567
$677$	46.4622	1.78569	0.892843	−	0.450368i	$-0.148707\pi$
$677$	46.4622	1.78569	0.892843	+	0.450368i	$0.148707\pi$
$678$	0	0
$679$	−4.97953	−0.191097
$680$	20.2064	0.774878
$681$	0	0
$682$	9.44848	0.361801
$683$	9.60289	0.367444	0.183722	−	0.982978i	$-0.441185\pi$
$683$	9.60289	0.367444	0.183722	+	0.982978i	$0.441185\pi$
$684$	0	0
$685$	−28.2887	−1.08086
$686$	15.6785	0.598609
$687$	0	0
$688$	−0.0503866	−0.00192097
$689$	−6.00944	−0.228941
$690$	0	0
$691$	8.92238	0.339423	0.169712	−	0.985494i	$-0.445716\pi$
$691$	8.92238	0.339423	0.169712	+	0.985494i	$0.445716\pi$
$692$	−18.4933	−0.703011
$693$	0	0
$694$	−20.5703	−0.780836
$695$	−26.6012	−1.00904
$696$	0	0
$697$	21.9444	0.831202
$698$	12.5045	0.473304
$699$	0	0
$700$	6.43860	0.243356
$701$	3.79254	0.143242	0.0716212	−	0.997432i	$-0.477183\pi$
$701$	3.79254	0.143242	0.0716212	+	0.997432i	$0.477183\pi$
$702$	0	0
$703$	7.48228	0.282199
$704$	−13.4374	−0.506442
$705$	0	0
$706$	7.62003	0.286783
$707$	0.407597	0.0153293
$708$	0	0
$709$	−13.0451	−0.489920	−0.244960	−	0.969533i	$-0.578775\pi$
$709$	−13.0451	−0.489920	−0.244960	+	0.969533i	$0.578775\pi$
$710$	−14.0612	−0.527709
$711$	0	0
$712$	6.22694	0.233364
$713$	20.2694	0.759097
$714$	0	0
$715$	4.96308	0.185609
$716$	23.5383	0.879666
$717$	0	0
$718$	7.74192	0.288926
$719$	49.9742	1.86372	0.931861	−	0.362814i	$-0.118184\pi$
$719$	49.9742	1.86372	0.931861	+	0.362814i	$0.118184\pi$
$720$	0	0
$721$	18.0557	0.672430
$722$	13.2256	0.492206
$723$	0	0
$724$	12.4636	0.463205
$725$	15.5453	0.577339
$726$	0	0
$727$	44.0322	1.63306	0.816532	−	0.577301i	$-0.195894\pi$
$727$	44.0322	1.63306	0.816532	+	0.577301i	$0.195894\pi$
$728$	3.79010	0.140471
$729$	0	0
$730$	−0.147043	−0.00544232
$731$	0.319869	0.0118308
$732$	0	0
$733$	26.0442	0.961963	0.480981	−	0.876731i	$-0.340280\pi$
$733$	26.0442	0.961963	0.480981	+	0.876731i	$0.340280\pi$
$734$	−6.14705	−0.226892
$735$	0	0
$736$	−42.2128	−1.55599
$737$	−62.4941	−2.30200
$738$	0	0
$739$	48.8930	1.79856	0.899278	−	0.437377i	$-0.144092\pi$
$739$	48.8930	1.79856	0.899278	+	0.437377i	$0.144092\pi$
$740$	12.1563	0.446875
$741$	0	0
$742$	14.0394	0.515403
$743$	−12.1636	−0.446241	−0.223120	−	0.974791i	$-0.571624\pi$
$743$	−12.1636	−0.446241	−0.223120	+	0.974791i	$0.571624\pi$
$744$	0	0
$745$	−23.7084	−0.868610
$746$	7.55033	0.276437
$747$	0	0
$748$	−27.5772	−1.00832
$749$	10.0407	0.366880
$750$	0	0
$751$	−24.7050	−0.901500	−0.450750	−	0.892650i	$-0.648843\pi$
$751$	−24.7050	−0.901500	−0.450750	+	0.892650i	$0.648843\pi$
$752$	8.25642	0.301081
$753$	0	0
$754$	3.75585	0.136780
$755$	−22.4232	−0.816063
$756$	0	0
$757$	−11.9146	−0.433042	−0.216521	−	0.976278i	$-0.569471\pi$
$757$	−11.9146	−0.433042	−0.216521	+	0.976278i	$0.569471\pi$
$758$	−4.99189	−0.181314
$759$	0	0
$760$	−6.27485	−0.227613
$761$	7.04012	0.255204	0.127602	−	0.991825i	$-0.459272\pi$
$761$	7.04012	0.255204	0.127602	+	0.991825i	$0.459272\pi$
$762$	0	0
$763$	42.1746	1.52682
$764$	9.57022	0.346238
$765$	0	0
$766$	−2.34704	−0.0848021
$767$	5.52910	0.199644
$768$	0	0
$769$	12.7885	0.461164	0.230582	−	0.973053i	$-0.425937\pi$
$769$	12.7885	0.461164	0.230582	+	0.973053i	$0.425937\pi$
$770$	−11.5949	−0.417850
$771$	0	0
$772$	−14.9944	−0.539659
$773$	−43.1745	−1.55288	−0.776439	−	0.630192i	$-0.782976\pi$
$773$	−43.1745	−1.55288	−0.776439	+	0.630192i	$0.782976\pi$
$774$	0	0
$775$	−6.27025	−0.225234
$776$	−6.35342	−0.228074
$777$	0	0
$778$	−2.38484	−0.0855008
$779$	−6.81457	−0.244157
$780$	0	0
$781$	46.7560	1.67306
$782$	25.8184	0.923265
$783$	0	0
$784$	−1.95633	−0.0698689
$785$	−17.0939	−0.610109
$786$	0	0
$787$	22.3365	0.796212	0.398106	−	0.917339i	$-0.369668\pi$
$787$	22.3365	0.796212	0.398106	+	0.917339i	$0.369668\pi$
$788$	−21.3547	−0.760731
$789$	0	0
$790$	−8.01601	−0.285197
$791$	33.4731	1.19016
$792$	0	0
$793$	−5.42763	−0.192741
$794$	−9.25176	−0.328333
$795$	0	0
$796$	8.30890	0.294501
$797$	−39.2565	−1.39054	−0.695269	−	0.718750i	$-0.744715\pi$
$797$	−39.2565	−1.39054	−0.695269	+	0.718750i	$0.744715\pi$
$798$	0	0
$799$	−52.4142	−1.85428
$800$	13.0583	0.461681
$801$	0	0
$802$	−2.50415	−0.0884246
$803$	0.488944	0.0172545
$804$	0	0
$805$	−24.8740	−0.876694
$806$	−1.51493	−0.0533612
$807$	0	0
$808$	0.520056	0.0182955
$809$	4.53751	0.159530	0.0797651	−	0.996814i	$-0.474583\pi$
$809$	4.53751	0.159530	0.0797651	+	0.996814i	$0.474583\pi$
$810$	0	0
$811$	12.1635	0.427118	0.213559	−	0.976930i	$-0.431494\pi$
$811$	12.1635	0.427118	0.213559	+	0.976930i	$0.431494\pi$
$812$	20.1059	0.705578
$813$	0	0
$814$	17.6407	0.618305
$815$	−20.7291	−0.726110
$816$	0	0
$817$	−0.0993317	−0.00347518
$818$	31.1001	1.08739
$819$	0	0
$820$	−11.0715	−0.386633
$821$	12.4541	0.434650	0.217325	−	0.976099i	$-0.430267\pi$
$821$	12.4541	0.434650	0.217325	+	0.976099i	$0.430267\pi$
$822$	0	0
$823$	−21.5162	−0.750009	−0.375005	−	0.927023i	$-0.622359\pi$
$823$	−21.5162	−0.750009	−0.375005	+	0.927023i	$0.622359\pi$
$824$	23.0374	0.802546
$825$	0	0
$826$	−12.9172	−0.449448
$827$	−21.6544	−0.752997	−0.376499	−	0.926417i	$-0.622872\pi$
$827$	−21.6544	−0.752997	−0.376499	+	0.926417i	$0.622872\pi$
$828$	0	0
$829$	23.6207	0.820382	0.410191	−	0.912000i	$-0.365462\pi$
$829$	23.6207	0.820382	0.410191	+	0.912000i	$0.365462\pi$
$830$	14.4964	0.503177
$831$	0	0
$832$	2.15450	0.0746940
$833$	12.4194	0.430306
$834$	0	0
$835$	−29.0578	−1.00559
$836$	8.56379	0.296185
$837$	0	0
$838$	16.9462	0.585397
$839$	−7.82348	−0.270096	−0.135048	−	0.990839i	$-0.543119\pi$
$839$	−7.82348	−0.270096	−0.135048	+	0.990839i	$0.543119\pi$
$840$	0	0
$841$	19.5436	0.673916
$842$	−3.17832	−0.109532
$843$	0	0
$844$	0.434801	0.0149665
$845$	20.8360	0.716779
$846$	0	0
$847$	15.7567	0.541407
$848$	−6.28598	−0.215861
$849$	0	0
$850$	−7.98680	−0.273945
$851$	37.8438	1.29727
$852$	0	0
$853$	−20.1784	−0.690896	−0.345448	−	0.938438i	$-0.612273\pi$
$853$	−20.1784	−0.690896	−0.345448	+	0.938438i	$0.612273\pi$
$854$	12.6802	0.433906
$855$	0	0
$856$	12.8110	0.437871
$857$	7.59263	0.259359	0.129680	−	0.991556i	$-0.458605\pi$
$857$	7.59263	0.259359	0.129680	+	0.991556i	$0.458605\pi$
$858$	0	0
$859$	−26.8210	−0.915122	−0.457561	−	0.889178i	$-0.651277\pi$
$859$	−26.8210	−0.915122	−0.457561	+	0.889178i	$0.651277\pi$
$860$	−0.161382	−0.00550309
$861$	0	0
$862$	10.0730	0.343087
$863$	2.72517	0.0927660	0.0463830	−	0.998924i	$-0.485231\pi$
$863$	2.72517	0.0927660	0.0463830	+	0.998924i	$0.485231\pi$
$864$	0	0
$865$	−22.1010	−0.751458
$866$	−19.1939	−0.652235
$867$	0	0
$868$	−8.10977	−0.275263
$869$	26.6546	0.904195
$870$	0	0
$871$	10.0201	0.339517
$872$	53.8108	1.82226
$873$	0	0
$874$	−8.01762	−0.271200
$875$	24.9381	0.843063
$876$	0	0
$877$	44.9386	1.51747	0.758735	−	0.651400i	$-0.225818\pi$
$877$	44.9386	1.51747	0.758735	+	0.651400i	$0.225818\pi$
$878$	16.9375	0.571612
$879$	0	0
$880$	5.19147	0.175004
$881$	56.3147	1.89729	0.948645	−	0.316344i	$-0.102455\pi$
$881$	56.3147	1.89729	0.948645	+	0.316344i	$0.102455\pi$
$882$	0	0
$883$	−46.7714	−1.57398	−0.786992	−	0.616963i	$-0.788363\pi$
$883$	−46.7714	−1.57398	−0.786992	+	0.616963i	$0.788363\pi$
$884$	4.42162	0.148715
$885$	0	0
$886$	−27.7119	−0.930998
$887$	43.3251	1.45472	0.727358	−	0.686259i	$-0.240748\pi$
$887$	43.3251	1.45472	0.727358	+	0.686259i	$0.240748\pi$
$888$	0	0
$889$	−44.5601	−1.49450
$890$	3.05437	0.102383
$891$	0	0
$892$	13.8208	0.462754
$893$	16.2766	0.544677
$894$	0	0
$895$	28.1301	0.940287
$896$	19.2266	0.642315
$897$	0	0
$898$	−23.7728	−0.793309
$899$	−19.5802	−0.653036
$900$	0	0
$901$	39.9053	1.32944
$902$	−16.0664	−0.534954
$903$	0	0
$904$	42.7085	1.42046
$905$	14.8950	0.495126
$906$	0	0
$907$	34.0973	1.13218	0.566091	−	0.824343i	$-0.308455\pi$
$907$	34.0973	1.13218	0.566091	+	0.824343i	$0.308455\pi$
$908$	−30.4825	−1.01160
$909$	0	0
$910$	1.85908	0.0616278
$911$	9.81857	0.325304	0.162652	−	0.986684i	$-0.447995\pi$
$911$	9.81857	0.325304	0.162652	+	0.986684i	$0.447995\pi$
$912$	0	0
$913$	−48.2029	−1.59528
$914$	−18.0609	−0.597401
$915$	0	0
$916$	5.21429	0.172285
$917$	21.6032	0.713399
$918$	0	0
$919$	38.2375	1.26134	0.630670	−	0.776051i	$-0.282780\pi$
$919$	38.2375	1.26134	0.630670	+	0.776051i	$0.282780\pi$
$920$	−31.7369	−1.04634
$921$	0	0
$922$	18.7732	0.618263
$923$	−7.49668	−0.246756
$924$	0	0
$925$	−11.7068	−0.384917
$926$	−3.93391	−0.129276
$927$	0	0
$928$	40.7774	1.33858
$929$	8.67986	0.284777	0.142388	−	0.989811i	$-0.454522\pi$
$929$	8.67986	0.284777	0.142388	+	0.989811i	$0.454522\pi$
$930$	0	0
$931$	−3.85669	−0.126398
$932$	31.6060	1.03529
$933$	0	0
$934$	−12.0298	−0.393628
$935$	−32.9570	−1.07781
$936$	0	0
$937$	25.5923	0.836063	0.418032	−	0.908432i	$-0.362720\pi$
$937$	25.5923	0.836063	0.418032	+	0.908432i	$0.362720\pi$
$938$	−23.4091	−0.764335
$939$	0	0
$940$	26.4444	0.862520
$941$	12.9227	0.421267	0.210633	−	0.977565i	$-0.432447\pi$
$941$	12.9227	0.421267	0.210633	+	0.977565i	$0.432447\pi$
$942$	0	0
$943$	−34.4667	−1.12239
$944$	5.78354	0.188238
$945$	0	0
$946$	−0.234190	−0.00761419
$947$	−51.5358	−1.67469	−0.837345	−	0.546675i	$-0.815893\pi$
$947$	−51.5358	−1.67469	−0.837345	+	0.546675i	$0.815893\pi$
$948$	0	0
$949$	−0.0783954	−0.00254482
$950$	2.48021	0.0804687
$951$	0	0
$952$	−25.1679	−0.815698
$953$	20.2468	0.655859	0.327929	−	0.944702i	$-0.393649\pi$
$953$	20.2468	0.655859	0.327929	+	0.944702i	$0.393649\pi$
$954$	0	0
$955$	11.4372	0.370099
$956$	6.02271	0.194788
$957$	0	0
$958$	2.73618	0.0884019
$959$	35.2349	1.13779
$960$	0	0
$961$	−23.1023	−0.745235
$962$	−2.82844	−0.0911925
$963$	0	0
$964$	18.0278	0.580636
$965$	−17.9195	−0.576848
$966$	0	0
$967$	16.6869	0.536616	0.268308	−	0.963333i	$-0.413535\pi$
$967$	16.6869	0.536616	0.268308	+	0.963333i	$0.413535\pi$
$968$	20.1041	0.646170
$969$	0	0
$970$	−3.11640	−0.100062
$971$	−37.5064	−1.20364	−0.601819	−	0.798633i	$-0.705557\pi$
$971$	−37.5064	−1.20364	−0.601819	+	0.798633i	$0.705557\pi$
$972$	0	0
$973$	33.1331	1.06220
$974$	−20.1667	−0.646182
$975$	0	0
$976$	−5.67739	−0.181729
$977$	8.01221	0.256333	0.128167	−	0.991753i	$-0.459091\pi$
$977$	8.01221	0.256333	0.128167	+	0.991753i	$0.459091\pi$
$978$	0	0
$979$	−10.1563	−0.324596
$980$	−6.26589	−0.200157
$981$	0	0
$982$	−15.9579	−0.509237
$983$	−27.5705	−0.879362	−0.439681	−	0.898154i	$-0.644909\pi$
$983$	−27.5705	−0.879362	−0.439681	+	0.898154i	$0.644909\pi$
$984$	0	0
$985$	−25.5207	−0.813156
$986$	−24.9405	−0.794267
$987$	0	0
$988$	−1.37308	−0.0436836
$989$	−0.502399	−0.0159754
$990$	0	0
$991$	11.0011	0.349463	0.174731	−	0.984616i	$-0.444094\pi$
$991$	11.0011	0.349463	0.174731	+	0.984616i	$0.444094\pi$
$992$	−16.4477	−0.522214
$993$	0	0
$994$	17.5139	0.555508
$995$	9.92982	0.314796
$996$	0	0
$997$	−29.1077	−0.921849	−0.460925	−	0.887439i	$-0.652482\pi$
$997$	−29.1077	−0.921849	−0.460925	+	0.887439i	$0.652482\pi$
$998$	24.6595	0.780581
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2259.2.a.k.1.8		17
3.2	odd	2		251.2.a.b.1.10	✓	17
12.11	even	2		4016.2.a.k.1.17		17
15.14	odd	2		6275.2.a.e.1.8		17

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
251.2.a.b.1.10	✓	17	3.2	odd	2
2259.2.a.k.1.8		17	1.1	even	1	trivial
4016.2.a.k.1.17		17	12.11	even	2
6275.2.a.e.1.8		17	15.14	odd	2

Overview	Random
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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:	\( N \)	\(=\)	\( 2259 = 3^{2} \cdot 251 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2259.a (trivial)

\(f(q)\)	\(=\)	\(q-0.779516 q^{2} -1.39236 q^{4} -1.66398 q^{5} +2.07256 q^{7} +2.64439 q^{8} +O(q^{10})\)	\(q-0.779516 q^{2} -1.39236 q^{4} -1.66398 q^{5} +2.07256 q^{7} +2.64439 q^{8} +1.29710 q^{10} -4.31307 q^{11} +0.691540 q^{13} -1.61559 q^{14} +0.723363 q^{16} -4.59213 q^{17} +1.42603 q^{19} +2.31685 q^{20} +3.36210 q^{22} +7.21258 q^{23} -2.23118 q^{25} -0.539067 q^{26} -2.88574 q^{28} -6.96732 q^{29} +2.81029 q^{31} -5.85266 q^{32} +3.57964 q^{34} -3.44870 q^{35} +5.24691 q^{37} -1.11162 q^{38} -4.40021 q^{40} -4.77869 q^{41} -0.0696559 q^{43} +6.00532 q^{44} -5.62232 q^{46} +11.4139 q^{47} -2.70449 q^{49} +1.73924 q^{50} -0.962870 q^{52} -8.68993 q^{53} +7.17685 q^{55} +5.48067 q^{56} +5.43114 q^{58} +7.99534 q^{59} -7.84861 q^{61} -2.19066 q^{62} +3.11552 q^{64} -1.15071 q^{65} +14.4895 q^{67} +6.39387 q^{68} +2.68831 q^{70} -10.8406 q^{71} -0.113363 q^{73} -4.09005 q^{74} -1.98555 q^{76} -8.93909 q^{77} -6.17996 q^{79} -1.20366 q^{80} +3.72506 q^{82} +11.1760 q^{83} +7.64120 q^{85} +0.0542979 q^{86} -11.4054 q^{88} +2.35477 q^{89} +1.43326 q^{91} -10.0425 q^{92} -8.89734 q^{94} -2.37289 q^{95} -2.40260 q^{97} +2.10819 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 17 q - 2 q^{2} + 26 q^{4} - 3 q^{5} + 3 q^{7} - 6 q^{8}+O(q^{10}) \) 17 * q - 2 * q^2 + 26 * q^4 - 3 * q^5 + 3 * q^7 - 6 * q^8	\( 17 q - 2 q^{2} + 26 q^{4} - 3 q^{5} + 3 q^{7} - 6 q^{8} + 7 q^{10} + q^{11} + 22 q^{13} + 7 q^{14} + 40 q^{16} + q^{17} + 13 q^{19} + 14 q^{20} + 4 q^{22} + 2 q^{23} + 32 q^{25} + 9 q^{26} - 10 q^{28} - 28 q^{29} + 12 q^{31} - 4 q^{32} - 21 q^{34} + 15 q^{35} + 27 q^{37} + 37 q^{38} - 7 q^{40} + q^{41} + 9 q^{43} + 43 q^{44} + 4 q^{46} + 20 q^{47} + 32 q^{49} + 28 q^{50} - q^{52} - q^{53} - 11 q^{55} + 61 q^{56} - 46 q^{58} + 20 q^{59} + 59 q^{61} + 73 q^{62} + 54 q^{64} + 14 q^{65} + 15 q^{67} + 20 q^{68} - 11 q^{70} + 26 q^{71} + 8 q^{73} - 2 q^{74} + 38 q^{76} + 33 q^{79} + 29 q^{80} + 10 q^{82} + 67 q^{85} - 11 q^{86} + 27 q^{88} - 11 q^{89} - 2 q^{91} - 28 q^{92} + 29 q^{94} + 8 q^{95} - 10 q^{97} - 22 q^{98}+O(q^{100}) \) 17 * q - 2 * q^2 + 26 * q^4 - 3 * q^5 + 3 * q^7 - 6 * q^8 + 7 * q^10 + q^11 + 22 * q^13 + 7 * q^14 + 40 * q^16 + q^17 + 13 * q^19 + 14 * q^20 + 4 * q^22 + 2 * q^23 + 32 * q^25 + 9 * q^26 - 10 * q^28 - 28 * q^29 + 12 * q^31 - 4 * q^32 - 21 * q^34 + 15 * q^35 + 27 * q^37 + 37 * q^38 - 7 * q^40 + q^41 + 9 * q^43 + 43 * q^44 + 4 * q^46 + 20 * q^47 + 32 * q^49 + 28 * q^50 - q^52 - q^53 - 11 * q^55 + 61 * q^56 - 46 * q^58 + 20 * q^59 + 59 * q^61 + 73 * q^62 + 54 * q^64 + 14 * q^65 + 15 * q^67 + 20 * q^68 - 11 * q^70 + 26 * q^71 + 8 * q^73 - 2 * q^74 + 38 * q^76 + 33 * q^79 + 29 * q^80 + 10 * q^82 + 67 * q^85 - 11 * q^86 + 27 * q^88 - 11 * q^89 - 2 * q^91 - 28 * q^92 + 29 * q^94 + 8 * q^95 - 10 * q^97 - 22 * q^98

Introduction

L-functions

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