LMFDB - Embedded newform 2151.2.a.i.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2151,2,Mod(1,2151)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2151, 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("2151.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2151 = 3^{2} \cdot 239 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2151.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:	$17.1758214748$
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} - 28 x^{15} - x^{14} + 319 x^{13} + 17 x^{12} - 1903 x^{11} - 91 x^{10} + 6377 x^{9} + 125 x^{8} + \cdots - 49 $ x^17 - 28x^15 - x^14 + 319x^13 + 17x^12 - 1903x^11 - 91x^10 + 6377x^9 + 125x^8 - 11967x^7 + 233x^6 + 11733x^5 - 503x^4 - 5015x^3 + 94x^2 + 609x - 49
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 239)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-2.33063$ of defining polynomial
Character	$\chi$	$=$	2151.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.33063 q^{2} +3.43183 q^{4} +3.31496 q^{5} +2.77554 q^{7} -3.33707 q^{8} +O(q^{10})$	\(q-2.33063 q^{2} +3.43183 q^{4} +3.31496 q^{5} +2.77554 q^{7} -3.33707 q^{8} -7.72595 q^{10} +4.79275 q^{11} -1.25221 q^{13} -6.46874 q^{14} +0.913810 q^{16} -1.06447 q^{17} +4.62057 q^{19} +11.3764 q^{20} -11.1701 q^{22} -8.57900 q^{23} +5.98898 q^{25} +2.91843 q^{26} +9.52517 q^{28} +8.15267 q^{29} +6.19940 q^{31} +4.54439 q^{32} +2.48088 q^{34} +9.20080 q^{35} -5.02385 q^{37} -10.7688 q^{38} -11.0623 q^{40} +1.20956 q^{41} +9.78544 q^{43} +16.4479 q^{44} +19.9945 q^{46} -8.13125 q^{47} +0.703595 q^{49} -13.9581 q^{50} -4.29736 q^{52} +0.913494 q^{53} +15.8878 q^{55} -9.26216 q^{56} -19.0009 q^{58} +3.60073 q^{59} -6.77631 q^{61} -14.4485 q^{62} -12.4189 q^{64} -4.15102 q^{65} +5.67699 q^{67} -3.65308 q^{68} -21.4436 q^{70} +9.29697 q^{71} +7.29879 q^{73} +11.7087 q^{74} +15.8570 q^{76} +13.3024 q^{77} -5.91731 q^{79} +3.02925 q^{80} -2.81903 q^{82} -6.67107 q^{83} -3.52868 q^{85} -22.8062 q^{86} -15.9937 q^{88} -2.70328 q^{89} -3.47554 q^{91} -29.4417 q^{92} +18.9509 q^{94} +15.3170 q^{95} +11.4312 q^{97} -1.63982 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 17 q + 22 q^{4} - 6 q^{5} + 5 q^{7} + 3 q^{8}+O(q^{10}) $ 17 * q + 22 * q^4 - 6 * q^5 + 5 * q^7 + 3 * q^8	$ 17 q + 22 q^{4} - 6 q^{5} + 5 q^{7} + 3 q^{8} + 5 q^{10} + q^{11} + 15 q^{13} + 3 q^{14} + 24 q^{16} - 4 q^{17} + 24 q^{19} - 4 q^{20} - 10 q^{22} + 9 q^{23} + 39 q^{25} + 12 q^{26} - 7 q^{28} + 2 q^{29} + 28 q^{31} + 31 q^{32} + 29 q^{34} + 24 q^{35} + 11 q^{37} + 19 q^{38} - 18 q^{40} - 20 q^{41} - 9 q^{43} + 43 q^{44} - 18 q^{46} + 18 q^{47} + 60 q^{49} + 61 q^{50} - q^{52} + 12 q^{53} - 10 q^{55} + 60 q^{56} - 38 q^{58} - q^{59} + 24 q^{61} + 33 q^{62} + 21 q^{64} - 2 q^{65} + 16 q^{67} + 10 q^{68} + 7 q^{70} - 12 q^{71} + 30 q^{73} + 21 q^{74} + 75 q^{76} + 15 q^{77} - 10 q^{79} - 32 q^{80} + 50 q^{82} + 16 q^{83} - 18 q^{85} + 3 q^{86} - 28 q^{88} - 65 q^{89} + 47 q^{91} - 24 q^{92} + 32 q^{94} + 37 q^{95} + 87 q^{97} + 9 q^{98}+O(q^{100}) $ 17 * q + 22 * q^4 - 6 * q^5 + 5 * q^7 + 3 * q^8 + 5 * q^10 + q^11 + 15 * q^13 + 3 * q^14 + 24 * q^16 - 4 * q^17 + 24 * q^19 - 4 * q^20 - 10 * q^22 + 9 * q^23 + 39 * q^25 + 12 * q^26 - 7 * q^28 + 2 * q^29 + 28 * q^31 + 31 * q^32 + 29 * q^34 + 24 * q^35 + 11 * q^37 + 19 * q^38 - 18 * q^40 - 20 * q^41 - 9 * q^43 + 43 * q^44 - 18 * q^46 + 18 * q^47 + 60 * q^49 + 61 * q^50 - q^52 + 12 * q^53 - 10 * q^55 + 60 * q^56 - 38 * q^58 - q^59 + 24 * q^61 + 33 * q^62 + 21 * q^64 - 2 * q^65 + 16 * q^67 + 10 * q^68 + 7 * q^70 - 12 * q^71 + 30 * q^73 + 21 * q^74 + 75 * q^76 + 15 * q^77 - 10 * q^79 - 32 * q^80 + 50 * q^82 + 16 * q^83 - 18 * q^85 + 3 * q^86 - 28 * q^88 - 65 * q^89 + 47 * q^91 - 24 * q^92 + 32 * q^94 + 37 * q^95 + 87 * q^97 + 9 * 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 $

$2$	−2.33063	−1.64800	−0.824002	−	0.566587i	$-0.808263\pi$
$2$	−2.33063	−1.64800	−0.824002	+	0.566587i	$0.808263\pi$
$3$	0	0
$3$	0	0
$4$	3.43183	1.71592
$5$	3.31496	1.48250	0.741248	−	0.671231i	$-0.234234\pi$
$5$	3.31496	1.48250	0.741248	+	0.671231i	$0.234234\pi$
$6$	0	0
$7$	2.77554	1.04905	0.524527	−	0.851394i	$-0.324242\pi$
$7$	2.77554	1.04905	0.524527	+	0.851394i	$0.324242\pi$
$8$	−3.33707	−1.17983
$9$	0	0
$10$	−7.72595	−2.44316
$11$	4.79275	1.44507	0.722534	−	0.691335i	$-0.242977\pi$
$11$	4.79275	1.44507	0.722534	+	0.691335i	$0.242977\pi$
$12$	0	0
$13$	−1.25221	−0.347299	−0.173650	−	0.984807i	$-0.555556\pi$
$13$	−1.25221	−0.347299	−0.173650	+	0.984807i	$0.555556\pi$
$14$	−6.46874	−1.72884
$15$	0	0
$16$	0.913810	0.228453
$17$	−1.06447	−0.258172	−0.129086	−	0.991633i	$-0.541204\pi$
$17$	−1.06447	−0.258172	−0.129086	+	0.991633i	$0.541204\pi$
$18$	0	0
$19$	4.62057	1.06003	0.530016	−	0.847988i	$-0.322186\pi$
$19$	4.62057	1.06003	0.530016	+	0.847988i	$0.322186\pi$
$20$	11.3764	2.54384
$21$	0	0
$22$	−11.1701	−2.38148
$23$	−8.57900	−1.78885	−0.894423	−	0.447223i	$-0.852413\pi$
$23$	−8.57900	−1.78885	−0.894423	+	0.447223i	$0.852413\pi$
$24$	0	0
$25$	5.98898	1.19780
$26$	2.91843	0.572351
$27$	0	0
$28$	9.52517	1.80009
$29$	8.15267	1.51391	0.756957	−	0.653465i	$-0.226685\pi$
$29$	8.15267	1.51391	0.756957	+	0.653465i	$0.226685\pi$
$30$	0	0
$31$	6.19940	1.11345	0.556723	−	0.830698i	$-0.312059\pi$
$31$	6.19940	1.11345	0.556723	+	0.830698i	$0.312059\pi$
$32$	4.54439	0.803342
$33$	0	0
$34$	2.48088	0.425468
$35$	9.20080	1.55522
$36$	0	0
$37$	−5.02385	−0.825917	−0.412958	−	0.910750i	$-0.635504\pi$
$37$	−5.02385	−0.825917	−0.412958	+	0.910750i	$0.635504\pi$
$38$	−10.7688	−1.74694
$39$	0	0
$40$	−11.0623	−1.74910
$41$	1.20956	0.188901	0.0944506	−	0.995530i	$-0.469891\pi$
$41$	1.20956	0.188901	0.0944506	+	0.995530i	$0.469891\pi$
$42$	0	0
$43$	9.78544	1.49227	0.746133	−	0.665797i	$-0.231908\pi$
$43$	9.78544	1.49227	0.746133	+	0.665797i	$0.231908\pi$
$44$	16.4479	2.47962
$45$	0	0
$46$	19.9945	2.94802
$47$	−8.13125	−1.18606	−0.593032	−	0.805179i	$-0.702069\pi$
$47$	−8.13125	−1.18606	−0.593032	+	0.805179i	$0.702069\pi$
$48$	0	0
$49$	0.703595	0.100514
$50$	−13.9581	−1.97397
$51$	0	0
$52$	−4.29736	−0.595937
$53$	0.913494	0.125478	0.0627390	−	0.998030i	$-0.480016\pi$
$53$	0.913494	0.125478	0.0627390	+	0.998030i	$0.480016\pi$
$54$	0	0
$55$	15.8878	2.14231
$56$	−9.26216	−1.23771
$57$	0	0
$58$	−19.0009	−2.49493
$59$	3.60073	0.468775	0.234388	−	0.972143i	$-0.424692\pi$
$59$	3.60073	0.468775	0.234388	+	0.972143i	$0.424692\pi$
$60$	0	0
$61$	−6.77631	−0.867618	−0.433809	−	0.901005i	$-0.642831\pi$
$61$	−6.77631	−0.867618	−0.433809	+	0.901005i	$0.642831\pi$
$62$	−14.4485	−1.83496
$63$	0	0
$64$	−12.4189	−1.55236
$65$	−4.15102	−0.514870
$66$	0	0
$67$	5.67699	0.693555	0.346778	−	0.937947i	$-0.387276\pi$
$67$	5.67699	0.693555	0.346778	+	0.937947i	$0.387276\pi$
$68$	−3.65308	−0.443001
$69$	0	0
$70$	−21.4436	−2.56301
$71$	9.29697	1.10335	0.551674	−	0.834060i	$-0.313989\pi$
$71$	9.29697	1.10335	0.551674	+	0.834060i	$0.313989\pi$
$72$	0	0
$73$	7.29879	0.854259	0.427129	−	0.904190i	$-0.359525\pi$
$73$	7.29879	0.854259	0.427129	+	0.904190i	$0.359525\pi$
$74$	11.7087	1.36111
$75$	0	0
$76$	15.8570	1.81892
$77$	13.3024	1.51595
$78$	0	0
$79$	−5.91731	−0.665749	−0.332875	−	0.942971i	$-0.608019\pi$
$79$	−5.91731	−0.665749	−0.332875	+	0.942971i	$0.608019\pi$
$80$	3.02925	0.338680
$81$	0	0
$82$	−2.81903	−0.311310
$83$	−6.67107	−0.732245	−0.366122	−	0.930567i	$-0.619315\pi$
$83$	−6.67107	−0.732245	−0.366122	+	0.930567i	$0.619315\pi$
$84$	0	0
$85$	−3.52868	−0.382739
$86$	−22.8062	−2.45926
$87$	0	0
$88$	−15.9937	−1.70494
$89$	−2.70328	−0.286548	−0.143274	−	0.989683i	$-0.545763\pi$
$89$	−2.70328	−0.286548	−0.143274	+	0.989683i	$0.545763\pi$
$90$	0	0
$91$	−3.47554	−0.364336
$92$	−29.4417	−3.06951
$93$	0	0
$94$	18.9509	1.95464
$95$	15.3170	1.57149
$96$	0	0
$97$	11.4312	1.16067	0.580333	−	0.814380i	$-0.302923\pi$
$97$	11.4312	1.16067	0.580333	+	0.814380i	$0.302923\pi$
$98$	−1.63982	−0.165647
$99$	0	0
$100$	20.5532	2.05532
$101$	−12.1827	−1.21222	−0.606112	−	0.795379i	$-0.707272\pi$
$101$	−12.1827	−1.21222	−0.606112	+	0.795379i	$0.707272\pi$
$102$	0	0
$103$	−8.22914	−0.810841	−0.405421	−	0.914130i	$-0.632875\pi$
$103$	−8.22914	−0.810841	−0.405421	+	0.914130i	$0.632875\pi$
$104$	4.17870	0.409755
$105$	0	0
$106$	−2.12902	−0.206788
$107$	−15.1594	−1.46552	−0.732759	−	0.680488i	$-0.761768\pi$
$107$	−15.1594	−1.46552	−0.732759	+	0.680488i	$0.761768\pi$
$108$	0	0
$109$	−19.2403	−1.84289	−0.921445	−	0.388508i	$-0.872990\pi$
$109$	−19.2403	−1.84289	−0.921445	+	0.388508i	$0.872990\pi$
$110$	−37.0286	−3.53053
$111$	0	0
$112$	2.53631	0.239659
$113$	−1.41401	−0.133019	−0.0665095	−	0.997786i	$-0.521186\pi$
$113$	−1.41401	−0.133019	−0.0665095	+	0.997786i	$0.521186\pi$
$114$	0	0
$115$	−28.4391	−2.65196
$116$	27.9786	2.59775
$117$	0	0
$118$	−8.39197	−0.772543
$119$	−2.95447	−0.270836
$120$	0	0
$121$	11.9705	1.08822
$122$	15.7931	1.42984
$123$	0	0
$124$	21.2753	1.91058
$125$	3.27845	0.293233
$126$	0	0
$127$	−14.5825	−1.29399	−0.646993	−	0.762496i	$-0.723974\pi$
$127$	−14.5825	−1.29399	−0.646993	+	0.762496i	$0.723974\pi$
$128$	19.8551	1.75496
$129$	0	0
$130$	9.67448	0.848508
$131$	8.34256	0.728893	0.364446	−	0.931224i	$-0.381258\pi$
$131$	8.34256	0.728893	0.364446	+	0.931224i	$0.381258\pi$
$132$	0	0
$133$	12.8246	1.11203
$134$	−13.2310	−1.14298
$135$	0	0
$136$	3.55221	0.304600
$137$	−4.13372	−0.353168	−0.176584	−	0.984286i	$-0.556505\pi$
$137$	−4.13372	−0.353168	−0.176584	+	0.984286i	$0.556505\pi$
$138$	0	0
$139$	10.3311	0.876275	0.438138	−	0.898908i	$-0.355638\pi$
$139$	10.3311	0.876275	0.438138	+	0.898908i	$0.355638\pi$
$140$	31.5756	2.66863
$141$	0	0
$142$	−21.6678	−1.81832
$143$	−6.00151	−0.501871
$144$	0	0
$145$	27.0258	2.24437
$146$	−17.0108	−1.40782
$147$	0	0
$148$	−17.2410	−1.41720
$149$	8.90334	0.729390	0.364695	−	0.931127i	$-0.381173\pi$
$149$	8.90334	0.729390	0.364695	+	0.931127i	$0.381173\pi$
$150$	0	0
$151$	−12.2196	−0.994421	−0.497210	−	0.867630i	$-0.665642\pi$
$151$	−12.2196	−0.994421	−0.497210	+	0.867630i	$0.665642\pi$
$152$	−15.4192	−1.25066
$153$	0	0
$154$	−31.0031	−2.49830
$155$	20.5508	1.65068
$156$	0	0
$157$	−4.00384	−0.319541	−0.159771	−	0.987154i	$-0.551075\pi$
$157$	−4.00384	−0.319541	−0.159771	+	0.987154i	$0.551075\pi$
$158$	13.7911	1.09716
$159$	0	0
$160$	15.0645	1.19095
$161$	−23.8113	−1.87659
$162$	0	0
$163$	−2.63809	−0.206631	−0.103316	−	0.994649i	$-0.532945\pi$
$163$	−2.63809	−0.206631	−0.103316	+	0.994649i	$0.532945\pi$
$164$	4.15100	0.324139
$165$	0	0
$166$	15.5478	1.20674
$167$	−10.3363	−0.799850	−0.399925	−	0.916548i	$-0.630964\pi$
$167$	−10.3363	−0.799850	−0.399925	+	0.916548i	$0.630964\pi$
$168$	0	0
$169$	−11.4320	−0.879383
$170$	8.22404	0.630755
$171$	0	0
$172$	33.5820	2.56060
$173$	7.63985	0.580847	0.290424	−	0.956898i	$-0.406204\pi$
$173$	7.63985	0.580847	0.290424	+	0.956898i	$0.406204\pi$
$174$	0	0
$175$	16.6226	1.25655
$176$	4.37966	0.330130
$177$	0	0
$178$	6.30035	0.472231
$179$	25.7412	1.92398	0.961992	−	0.273077i	$-0.0880413\pi$
$179$	25.7412	1.92398	0.961992	+	0.273077i	$0.0880413\pi$
$180$	0	0
$181$	19.6261	1.45880	0.729398	−	0.684089i	$-0.239800\pi$
$181$	19.6261	1.45880	0.729398	+	0.684089i	$0.239800\pi$
$182$	8.10020	0.600427
$183$	0	0
$184$	28.6287	2.11054
$185$	−16.6539	−1.22442
$186$	0	0
$187$	−5.10174	−0.373076
$188$	−27.9051	−2.03519
$189$	0	0
$190$	−35.6983	−2.58983
$191$	−12.1193	−0.876922	−0.438461	−	0.898750i	$-0.644476\pi$
$191$	−12.1193	−0.876922	−0.438461	+	0.898750i	$0.644476\pi$
$192$	0	0
$193$	13.6963	0.985883	0.492942	−	0.870062i	$-0.335922\pi$
$193$	13.6963	0.985883	0.492942	+	0.870062i	$0.335922\pi$
$194$	−26.6419	−1.91278
$195$	0	0
$196$	2.41462	0.172473
$197$	10.0011	0.712548	0.356274	−	0.934382i	$-0.384047\pi$
$197$	10.0011	0.712548	0.356274	+	0.934382i	$0.384047\pi$
$198$	0	0
$199$	−27.9937	−1.98442	−0.992211	−	0.124568i	$-0.960246\pi$
$199$	−27.9937	−1.98442	−0.992211	+	0.124568i	$0.960246\pi$
$200$	−19.9857	−1.41320
$201$	0	0
$202$	28.3934	1.99775
$203$	22.6280	1.58818
$204$	0	0
$205$	4.00964	0.280046
$206$	19.1791	1.33627
$207$	0	0
$208$	−1.14428	−0.0793415
$209$	22.1452	1.53182
$210$	0	0
$211$	26.2512	1.80720	0.903602	−	0.428373i	$-0.140913\pi$
$211$	26.2512	1.80720	0.903602	+	0.428373i	$0.140913\pi$
$212$	3.13496	0.215310
$213$	0	0
$214$	35.3310	2.41518
$215$	32.4384	2.21228
$216$	0	0
$217$	17.2067	1.16806
$218$	44.8421	3.03709
$219$	0	0
$220$	54.5242	3.67602
$221$	1.33294	0.0896629
$222$	0	0
$223$	17.2448	1.15480	0.577398	−	0.816463i	$-0.304068\pi$
$223$	17.2448	1.15480	0.577398	+	0.816463i	$0.304068\pi$
$224$	12.6131	0.842749
$225$	0	0
$226$	3.29554	0.219216
$227$	−9.00070	−0.597397	−0.298699	−	0.954347i	$-0.596553\pi$
$227$	−9.00070	−0.597397	−0.298699	+	0.954347i	$0.596553\pi$
$228$	0	0
$229$	8.95516	0.591774	0.295887	−	0.955223i	$-0.404385\pi$
$229$	8.95516	0.591774	0.295887	+	0.955223i	$0.404385\pi$
$230$	66.2809	4.37044
$231$	0	0
$232$	−27.2061	−1.78616
$233$	17.6582	1.15683	0.578415	−	0.815743i	$-0.303672\pi$
$233$	17.6582	1.15683	0.578415	+	0.815743i	$0.303672\pi$
$234$	0	0
$235$	−26.9548	−1.75834
$236$	12.3571	0.804379
$237$	0	0
$238$	6.88578	0.446339
$239$	−1.00000	−0.0646846
$239$	−1.00000	−0.0646846
$240$	0	0
$241$	−19.2677	−1.24114	−0.620570	−	0.784151i	$-0.713099\pi$
$241$	−19.2677	−1.24114	−0.620570	+	0.784151i	$0.713099\pi$
$242$	−27.8987	−1.79340
$243$	0	0
$244$	−23.2552	−1.48876
$245$	2.33239	0.149011
$246$	0	0
$247$	−5.78590	−0.368148
$248$	−20.6878	−1.31368
$249$	0	0
$250$	−7.64085	−0.483250
$251$	10.0266	0.632874	0.316437	−	0.948613i	$-0.397513\pi$
$251$	10.0266	0.632874	0.316437	+	0.948613i	$0.397513\pi$
$252$	0	0
$253$	−41.1170	−2.58500
$254$	33.9864	2.13249
$255$	0	0
$256$	−21.4370	−1.33981
$257$	5.79006	0.361174	0.180587	−	0.983559i	$-0.442200\pi$
$257$	5.79006	0.361174	0.180587	+	0.983559i	$0.442200\pi$
$258$	0	0
$259$	−13.9439	−0.866431
$260$	−14.2456	−0.883474
$261$	0	0
$262$	−19.4434	−1.20122
$263$	−17.7764	−1.09614	−0.548069	−	0.836433i	$-0.684637\pi$
$263$	−17.7764	−1.09614	−0.548069	+	0.836433i	$0.684637\pi$
$264$	0	0
$265$	3.02820	0.186021
$266$	−29.8893	−1.83263
$267$	0	0
$268$	19.4825	1.19008
$269$	23.2253	1.41607	0.708036	−	0.706176i	$-0.249581\pi$
$269$	23.2253	1.41607	0.708036	+	0.706176i	$0.249581\pi$
$270$	0	0
$271$	−30.5130	−1.85354	−0.926768	−	0.375635i	$-0.877425\pi$
$271$	−30.5130	−1.85354	−0.926768	+	0.375635i	$0.877425\pi$
$272$	−0.972724	−0.0589800
$273$	0	0
$274$	9.63417	0.582021
$275$	28.7037	1.73090
$276$	0	0
$277$	4.68969	0.281776	0.140888	−	0.990026i	$-0.455004\pi$
$277$	4.68969	0.281776	0.140888	+	0.990026i	$0.455004\pi$
$278$	−24.0780	−1.44410
$279$	0	0
$280$	−30.7037	−1.83490
$281$	−8.56986	−0.511235	−0.255618	−	0.966778i	$-0.582279\pi$
$281$	−8.56986	−0.511235	−0.255618	+	0.966778i	$0.582279\pi$
$282$	0	0
$283$	14.8587	0.883260	0.441630	−	0.897197i	$-0.354400\pi$
$283$	14.8587	0.883260	0.441630	+	0.897197i	$0.354400\pi$
$284$	31.9057	1.89325
$285$	0	0
$286$	13.9873	0.827086
$287$	3.35717	0.198168
$288$	0	0
$289$	−15.8669	−0.933347
$290$	−62.9872	−3.69873
$291$	0	0
$292$	25.0482	1.46584
$293$	−23.4843	−1.37197	−0.685985	−	0.727616i	$-0.740628\pi$
$293$	−23.4843	−1.37197	−0.685985	+	0.727616i	$0.740628\pi$
$294$	0	0
$295$	11.9363	0.694958
$296$	16.7650	0.974444
$297$	0	0
$298$	−20.7504	−1.20204
$299$	10.7427	0.621265
$300$	0	0
$301$	27.1598	1.56547
$302$	28.4795	1.63881
$303$	0	0
$304$	4.22232	0.242167
$305$	−22.4632	−1.28624
$306$	0	0
$307$	12.8402	0.732826	0.366413	−	0.930452i	$-0.380586\pi$
$307$	12.8402	0.732826	0.366413	+	0.930452i	$0.380586\pi$
$308$	45.6518	2.60125
$309$	0	0
$310$	−47.8963	−2.72033
$311$	18.7234	1.06170	0.530852	−	0.847465i	$-0.321872\pi$
$311$	18.7234	1.06170	0.530852	+	0.847465i	$0.321872\pi$
$312$	0	0
$313$	−12.1678	−0.687763	−0.343882	−	0.939013i	$-0.611742\pi$
$313$	−12.1678	−0.687763	−0.343882	+	0.939013i	$0.611742\pi$
$314$	9.33147	0.526605
$315$	0	0
$316$	−20.3072	−1.14237
$317$	17.1089	0.960930	0.480465	−	0.877014i	$-0.340468\pi$
$317$	17.1089	0.960930	0.480465	+	0.877014i	$0.340468\pi$
$318$	0	0
$319$	39.0737	2.18771
$320$	−41.1682	−2.30137
$321$	0	0
$322$	55.4954	3.09264
$323$	−4.91846	−0.273670
$324$	0	0
$325$	−7.49944	−0.415994
$326$	6.14842	0.340529
$327$	0	0
$328$	−4.03638	−0.222872
$329$	−22.5686	−1.24424
$330$	0	0
$331$	9.74464	0.535614	0.267807	−	0.963473i	$-0.413701\pi$
$331$	9.74464	0.535614	0.267807	+	0.963473i	$0.413701\pi$
$332$	−22.8940	−1.25647
$333$	0	0
$334$	24.0902	1.31816
$335$	18.8190	1.02819
$336$	0	0
$337$	−6.41342	−0.349361	−0.174681	−	0.984625i	$-0.555889\pi$
$337$	−6.41342	−0.349361	−0.174681	+	0.984625i	$0.555889\pi$
$338$	26.6437	1.44923
$339$	0	0
$340$	−12.1098	−0.656748
$341$	29.7122	1.60901
$342$	0	0
$343$	−17.4759	−0.943610
$344$	−32.6547	−1.76062
$345$	0	0
$346$	−17.8057	−0.957239
$347$	7.23505	0.388398	0.194199	−	0.980962i	$-0.437789\pi$
$347$	7.23505	0.388398	0.194199	+	0.980962i	$0.437789\pi$
$348$	0	0
$349$	−10.9723	−0.587331	−0.293666	−	0.955908i	$-0.594875\pi$
$349$	−10.9723	−0.587331	−0.293666	+	0.955908i	$0.594875\pi$
$350$	−38.7412	−2.07080
$351$	0	0
$352$	21.7801	1.16088
$353$	7.11947	0.378931	0.189465	−	0.981887i	$-0.439324\pi$
$353$	7.11947	0.378931	0.189465	+	0.981887i	$0.439324\pi$
$354$	0	0
$355$	30.8191	1.63571
$356$	−9.27722	−0.491692
$357$	0	0
$358$	−59.9931	−3.17073
$359$	−28.0566	−1.48077	−0.740385	−	0.672183i	$-0.765357\pi$
$359$	−28.0566	−1.48077	−0.740385	+	0.672183i	$0.765357\pi$
$360$	0	0
$361$	2.34966	0.123666
$362$	−45.7412	−2.40410
$363$	0	0
$364$	−11.9275	−0.625170
$365$	24.1952	1.26644
$366$	0	0
$367$	−22.0653	−1.15180	−0.575899	−	0.817521i	$-0.695348\pi$
$367$	−22.0653	−1.15180	−0.575899	+	0.817521i	$0.695348\pi$
$368$	−7.83958	−0.408666
$369$	0	0
$370$	38.8141	2.01785
$371$	2.53543	0.131633
$372$	0	0
$373$	5.74963	0.297705	0.148852	−	0.988859i	$-0.452442\pi$
$373$	5.74963	0.297705	0.148852	+	0.988859i	$0.452442\pi$
$374$	11.8903	0.614831
$375$	0	0
$376$	27.1345	1.39936
$377$	−10.2088	−0.525781
$378$	0	0
$379$	17.6977	0.909071	0.454536	−	0.890729i	$-0.349805\pi$
$379$	17.6977	0.909071	0.454536	+	0.890729i	$0.349805\pi$
$380$	52.5654	2.69655
$381$	0	0
$382$	28.2456	1.44517
$383$	−33.3941	−1.70636	−0.853180	−	0.521616i	$-0.825329\pi$
$383$	−33.3941	−1.70636	−0.853180	+	0.521616i	$0.825329\pi$
$384$	0	0
$385$	44.0971	2.24740
$386$	−31.9211	−1.62474
$387$	0	0
$388$	39.2301	1.99160
$389$	−16.7137	−0.847418	−0.423709	−	0.905798i	$-0.639272\pi$
$389$	−16.7137	−0.847418	−0.423709	+	0.905798i	$0.639272\pi$
$390$	0	0
$391$	9.13209	0.461829
$392$	−2.34795	−0.118589
$393$	0	0
$394$	−23.3088	−1.17428
$395$	−19.6157	−0.986971
$396$	0	0
$397$	12.0709	0.605820	0.302910	−	0.953019i	$-0.402042\pi$
$397$	12.0709	0.605820	0.302910	+	0.953019i	$0.402042\pi$
$398$	65.2430	3.27034
$399$	0	0
$400$	5.47280	0.273640
$401$	−6.52014	−0.325600	−0.162800	−	0.986659i	$-0.552053\pi$
$401$	−6.52014	−0.325600	−0.162800	+	0.986659i	$0.552053\pi$
$402$	0	0
$403$	−7.76293	−0.386699
$404$	−41.8090	−2.08008
$405$	0	0
$406$	−52.7375	−2.61732
$407$	−24.0781	−1.19351
$408$	0	0
$409$	−1.40694	−0.0695689	−0.0347845	−	0.999395i	$-0.511074\pi$
$409$	−1.40694	−0.0695689	−0.0347845	+	0.999395i	$0.511074\pi$
$410$	−9.34499	−0.461516
$411$	0	0
$412$	−28.2410	−1.39134
$413$	9.99395	0.491770
$414$	0	0
$415$	−22.1143	−1.08555
$416$	−5.69051	−0.279000
$417$	0	0
$418$	−51.6123	−2.52444
$419$	21.5449	1.05254	0.526269	−	0.850318i	$-0.323590\pi$
$419$	21.5449	1.05254	0.526269	+	0.850318i	$0.323590\pi$
$420$	0	0
$421$	33.2781	1.62188	0.810939	−	0.585131i	$-0.198957\pi$
$421$	33.2781	1.62188	0.810939	+	0.585131i	$0.198957\pi$
$422$	−61.1817	−2.97828
$423$	0	0
$424$	−3.04839	−0.148043
$425$	−6.37509	−0.309237
$426$	0	0
$427$	−18.8079	−0.910178
$428$	−52.0247	−2.51471
$429$	0	0
$430$	−75.6018	−3.64584
$431$	6.34119	0.305444	0.152722	−	0.988269i	$-0.451196\pi$
$431$	6.34119	0.305444	0.152722	+	0.988269i	$0.451196\pi$
$432$	0	0
$433$	33.4176	1.60595	0.802975	−	0.596013i	$-0.203249\pi$
$433$	33.4176	1.60595	0.802975	+	0.596013i	$0.203249\pi$
$434$	−40.1023	−1.92497
$435$	0	0
$436$	−66.0296	−3.16225
$437$	−39.6399	−1.89623
$438$	0	0
$439$	−0.828712	−0.0395522	−0.0197761	−	0.999804i	$-0.506295\pi$
$439$	−0.828712	−0.0395522	−0.0197761	+	0.999804i	$0.506295\pi$
$440$	−53.0187	−2.52757
$441$	0	0
$442$	−3.10658	−0.147765
$443$	−14.2324	−0.676202	−0.338101	−	0.941110i	$-0.609784\pi$
$443$	−14.2324	−0.676202	−0.338101	+	0.941110i	$0.609784\pi$
$444$	0	0
$445$	−8.96129	−0.424806
$446$	−40.1912	−1.90311
$447$	0	0
$448$	−34.4691	−1.62851
$449$	−8.75016	−0.412946	−0.206473	−	0.978452i	$-0.566198\pi$
$449$	−8.75016	−0.412946	−0.206473	+	0.978452i	$0.566198\pi$
$450$	0	0
$451$	5.79711	0.272975
$452$	−4.85265	−0.228250
$453$	0	0
$454$	20.9773	0.984513
$455$	−11.5213	−0.540127
$456$	0	0
$457$	26.8783	1.25732	0.628658	−	0.777682i	$-0.283605\pi$
$457$	26.8783	1.25732	0.628658	+	0.777682i	$0.283605\pi$
$458$	−20.8712	−0.975245
$459$	0	0
$460$	−97.5982	−4.55054
$461$	−32.2497	−1.50202	−0.751009	−	0.660293i	$-0.770432\pi$
$461$	−32.2497	−1.50202	−0.751009	+	0.660293i	$0.770432\pi$
$462$	0	0
$463$	−12.9558	−0.602107	−0.301053	−	0.953607i	$-0.597338\pi$
$463$	−12.9558	−0.602107	−0.301053	+	0.953607i	$0.597338\pi$
$464$	7.45000	0.345857
$465$	0	0
$466$	−41.1548	−1.90646
$467$	15.9917	0.740007	0.370004	−	0.929030i	$-0.379356\pi$
$467$	15.9917	0.740007	0.370004	+	0.929030i	$0.379356\pi$
$468$	0	0
$469$	15.7567	0.727576
$470$	62.8216	2.89775
$471$	0	0
$472$	−12.0159	−0.553076
$473$	46.8992	2.15643
$474$	0	0
$475$	27.6725	1.26970
$476$	−10.1393	−0.464732
$477$	0	0
$478$	2.33063	0.106601
$479$	−20.0604	−0.916583	−0.458292	−	0.888802i	$-0.651538\pi$
$479$	−20.0604	−0.916583	−0.458292	+	0.888802i	$0.651538\pi$
$480$	0	0
$481$	6.29090	0.286840
$482$	44.9058	2.04540
$483$	0	0
$484$	41.0806	1.86730
$485$	37.8941	1.72068
$486$	0	0
$487$	−38.3663	−1.73854	−0.869272	−	0.494334i	$-0.835412\pi$
$487$	−38.3663	−1.73854	−0.869272	+	0.494334i	$0.835412\pi$
$488$	22.6130	1.02364
$489$	0	0
$490$	−5.43594	−0.245571
$491$	−39.8010	−1.79619	−0.898097	−	0.439798i	$-0.855050\pi$
$491$	−39.8010	−1.79619	−0.898097	+	0.439798i	$0.855050\pi$
$492$	0	0
$493$	−8.67827	−0.390850
$494$	13.4848	0.606710
$495$	0	0
$496$	5.66508	0.254370
$497$	25.8041	1.15747
$498$	0	0
$499$	−8.15613	−0.365118	−0.182559	−	0.983195i	$-0.558438\pi$
$499$	−8.15613	−0.365118	−0.182559	+	0.983195i	$0.558438\pi$
$500$	11.2511	0.503164
$501$	0	0
$502$	−23.3683	−1.04298
$503$	28.3046	1.26204	0.631020	−	0.775767i	$-0.282637\pi$
$503$	28.3046	1.26204	0.631020	+	0.775767i	$0.282637\pi$
$504$	0	0
$505$	−40.3852	−1.79712
$506$	95.8285	4.26010
$507$	0	0
$508$	−50.0447	−2.22037
$509$	18.8564	0.835797	0.417898	−	0.908494i	$-0.362767\pi$
$509$	18.8564	0.835797	0.417898	+	0.908494i	$0.362767\pi$
$510$	0	0
$511$	20.2581	0.896163
$512$	10.2516	0.453061
$513$	0	0
$514$	−13.4945	−0.595217
$515$	−27.2793	−1.20207
$516$	0	0
$517$	−38.9710	−1.71394
$518$	32.4980	1.42788
$519$	0	0
$520$	13.8522	0.607461
$521$	−26.9203	−1.17940	−0.589701	−	0.807622i	$-0.700754\pi$
$521$	−26.9203	−1.17940	−0.589701	+	0.807622i	$0.700754\pi$
$522$	0	0
$523$	−0.524393	−0.0229301	−0.0114651	−	0.999934i	$-0.503650\pi$
$523$	−0.524393	−0.0229301	−0.0114651	+	0.999934i	$0.503650\pi$
$524$	28.6303	1.25072
$525$	0	0
$526$	41.4301	1.80644
$527$	−6.59908	−0.287460
$528$	0	0
$529$	50.5993	2.19997
$530$	−7.05761	−0.306563
$531$	0	0
$532$	44.0117	1.90815
$533$	−1.51462	−0.0656053
$534$	0	0
$535$	−50.2530	−2.17263
$536$	−18.9445	−0.818279
$537$	0	0
$538$	−54.1296	−2.33369
$539$	3.37215	0.145249
$540$	0	0
$541$	28.2336	1.21386	0.606929	−	0.794756i	$-0.292401\pi$
$541$	28.2336	1.21386	0.606929	+	0.794756i	$0.292401\pi$
$542$	71.1146	3.05463
$543$	0	0
$544$	−4.83736	−0.207400
$545$	−63.7810	−2.73208
$546$	0	0
$547$	18.6927	0.799240	0.399620	−	0.916681i	$-0.369142\pi$
$547$	18.6927	0.799240	0.399620	+	0.916681i	$0.369142\pi$
$548$	−14.1862	−0.606006
$549$	0	0
$550$	−66.8977	−2.85253
$551$	37.6700	1.60480
$552$	0	0
$553$	−16.4237	−0.698407
$554$	−10.9299	−0.464368
$555$	0	0
$556$	35.4547	1.50362
$557$	8.50044	0.360175	0.180088	−	0.983651i	$-0.442362\pi$
$557$	8.50044	0.360175	0.180088	+	0.983651i	$0.442362\pi$
$558$	0	0
$559$	−12.2534	−0.518263
$560$	8.40778	0.355294
$561$	0	0
$562$	19.9732	0.842517
$563$	−8.35082	−0.351945	−0.175973	−	0.984395i	$-0.556307\pi$
$563$	−8.35082	−0.351945	−0.175973	+	0.984395i	$0.556307\pi$
$564$	0	0
$565$	−4.68740	−0.197200
$566$	−34.6302	−1.45562
$567$	0	0
$568$	−31.0247	−1.30177
$569$	−20.7713	−0.870779	−0.435389	−	0.900242i	$-0.643389\pi$
$569$	−20.7713	−0.870779	−0.435389	+	0.900242i	$0.643389\pi$
$570$	0	0
$571$	−0.912685	−0.0381947	−0.0190973	−	0.999818i	$-0.506079\pi$
$571$	−0.912685	−0.0381947	−0.0190973	+	0.999818i	$0.506079\pi$
$572$	−20.5962	−0.861170
$573$	0	0
$574$	−7.82432	−0.326581
$575$	−51.3795	−2.14267
$576$	0	0
$577$	−14.4920	−0.603308	−0.301654	−	0.953417i	$-0.597539\pi$
$577$	−14.4920	−0.603308	−0.301654	+	0.953417i	$0.597539\pi$
$578$	36.9799	1.53816
$579$	0	0
$580$	92.7481	3.85115
$581$	−18.5158	−0.768164
$582$	0	0
$583$	4.37815	0.181324
$584$	−24.3566	−1.00788
$585$	0	0
$586$	54.7333	2.26101
$587$	−22.6296	−0.934023	−0.467011	−	0.884251i	$-0.654669\pi$
$587$	−22.6296	−0.934023	−0.467011	+	0.884251i	$0.654669\pi$
$588$	0	0
$589$	28.6448	1.18029
$590$	−27.8191	−1.14529
$591$	0	0
$592$	−4.59085	−0.188683
$593$	0.675768	0.0277505	0.0138752	−	0.999904i	$-0.495583\pi$
$593$	0.675768	0.0277505	0.0138752	+	0.999904i	$0.495583\pi$
$594$	0	0
$595$	−9.79397	−0.401514
$596$	30.5548	1.25157
$597$	0	0
$598$	−25.0372	−1.02385
$599$	18.6406	0.761636	0.380818	−	0.924650i	$-0.375642\pi$
$599$	18.6406	0.761636	0.380818	+	0.924650i	$0.375642\pi$
$600$	0	0
$601$	23.4366	0.956001	0.478000	−	0.878360i	$-0.341362\pi$
$601$	23.4366	0.956001	0.478000	+	0.878360i	$0.341362\pi$
$602$	−63.2995	−2.57990
$603$	0	0
$604$	−41.9358	−1.70634
$605$	39.6816	1.61329
$606$	0	0
$607$	15.1933	0.616677	0.308338	−	0.951277i	$-0.400227\pi$
$607$	15.1933	0.616677	0.308338	+	0.951277i	$0.400227\pi$
$608$	20.9977	0.851568
$609$	0	0
$610$	52.3535	2.11973
$611$	10.1820	0.411919
$612$	0	0
$613$	6.09812	0.246301	0.123150	−	0.992388i	$-0.460700\pi$
$613$	6.09812	0.246301	0.123150	+	0.992388i	$0.460700\pi$
$614$	−29.9256	−1.20770
$615$	0	0
$616$	−44.3912	−1.78857
$617$	16.2072	0.652477	0.326239	−	0.945287i	$-0.394219\pi$
$617$	16.2072	0.652477	0.326239	+	0.945287i	$0.394219\pi$
$618$	0	0
$619$	−3.82801	−0.153861	−0.0769304	−	0.997036i	$-0.524512\pi$
$619$	−3.82801	−0.153861	−0.0769304	+	0.997036i	$0.524512\pi$
$620$	70.5269	2.83243
$621$	0	0
$622$	−43.6372	−1.74969
$623$	−7.50306	−0.300604
$624$	0	0
$625$	−19.0770	−0.763079
$626$	28.3586	1.13344
$627$	0	0
$628$	−13.7405	−0.548306
$629$	5.34774	0.213228
$630$	0	0
$631$	18.0354	0.717976	0.358988	−	0.933342i	$-0.383122\pi$
$631$	18.0354	0.717976	0.358988	+	0.933342i	$0.383122\pi$
$632$	19.7465	0.785473
$633$	0	0
$634$	−39.8744	−1.58362
$635$	−48.3404	−1.91833
$636$	0	0
$637$	−0.881046	−0.0349083
$638$	−91.0664	−3.60535
$639$	0	0
$640$	65.8189	2.60172
$641$	42.0933	1.66258	0.831292	−	0.555836i	$-0.187602\pi$
$641$	42.0933	1.66258	0.831292	+	0.555836i	$0.187602\pi$
$642$	0	0
$643$	−6.48199	−0.255625	−0.127812	−	0.991798i	$-0.540796\pi$
$643$	−6.48199	−0.255625	−0.127812	+	0.991798i	$0.540796\pi$
$644$	−81.7165	−3.22008
$645$	0	0
$646$	11.4631	0.451010
$647$	−26.2949	−1.03376	−0.516879	−	0.856059i	$-0.672906\pi$
$647$	−26.2949	−1.03376	−0.516879	+	0.856059i	$0.672906\pi$
$648$	0	0
$649$	17.2574	0.677412
$650$	17.4784	0.685560
$651$	0	0
$652$	−9.05350	−0.354562
$653$	−5.35698	−0.209635	−0.104817	−	0.994491i	$-0.533426\pi$
$653$	−5.35698	−0.209635	−0.104817	+	0.994491i	$0.533426\pi$
$654$	0	0
$655$	27.6553	1.08058
$656$	1.10531	0.0431550
$657$	0	0
$658$	52.5989	2.05052
$659$	−10.1530	−0.395504	−0.197752	−	0.980252i	$-0.563364\pi$
$659$	−10.1530	−0.395504	−0.197752	+	0.980252i	$0.563364\pi$
$660$	0	0
$661$	37.2867	1.45028	0.725142	−	0.688599i	$-0.241774\pi$
$661$	37.2867	1.45028	0.725142	+	0.688599i	$0.241774\pi$
$662$	−22.7111	−0.882694
$663$	0	0
$664$	22.2618	0.863926
$665$	42.5129	1.64858
$666$	0	0
$667$	−69.9418	−2.70816
$668$	−35.4726	−1.37248
$669$	0	0
$670$	−43.8602	−1.69447
$671$	−32.4772	−1.25377
$672$	0	0
$673$	0.0639909	0.00246667	0.00123333	−	0.999999i	$-0.499607\pi$
$673$	0.0639909	0.00246667	0.00123333	+	0.999999i	$0.499607\pi$
$674$	14.9473	0.575749
$675$	0	0
$676$	−39.2326	−1.50895
$677$	−15.5711	−0.598445	−0.299223	−	0.954183i	$-0.596727\pi$
$677$	−15.5711	−0.598445	−0.299223	+	0.954183i	$0.596727\pi$
$678$	0	0
$679$	31.7278	1.21760
$680$	11.7755	0.451568
$681$	0	0
$682$	−69.2481	−2.65165
$683$	0.291245	0.0111442	0.00557208	−	0.999984i	$-0.498226\pi$
$683$	0.291245	0.0111442	0.00557208	+	0.999984i	$0.498226\pi$
$684$	0	0
$685$	−13.7031	−0.523570
$686$	40.7298	1.55507
$687$	0	0
$688$	8.94204	0.340912
$689$	−1.14388	−0.0435785
$690$	0	0
$691$	−2.77266	−0.105477	−0.0527385	−	0.998608i	$-0.516795\pi$
$691$	−2.77266	−0.105477	−0.0527385	+	0.998608i	$0.516795\pi$
$692$	26.2187	0.996685
$693$	0	0
$694$	−16.8622	−0.640081
$695$	34.2473	1.29908
$696$	0	0
$697$	−1.28754	−0.0487690
$698$	25.5722	0.967924
$699$	0	0
$700$	57.0461	2.15614
$701$	25.5664	0.965629	0.482815	−	0.875723i	$-0.339614\pi$
$701$	25.5664	0.965629	0.482815	+	0.875723i	$0.339614\pi$
$702$	0	0
$703$	−23.2131	−0.875497
$704$	−59.5207	−2.24327
$705$	0	0
$706$	−16.5928	−0.624480
$707$	−33.8135	−1.27169
$708$	0	0
$709$	−13.2045	−0.495904	−0.247952	−	0.968772i	$-0.579758\pi$
$709$	−13.2045	−0.495904	−0.247952	+	0.968772i	$0.579758\pi$
$710$	−71.8280	−2.69566
$711$	0	0
$712$	9.02105	0.338078
$713$	−53.1847	−1.99178
$714$	0	0
$715$	−19.8948	−0.744023
$716$	88.3393	3.30140
$717$	0	0
$718$	65.3895	2.44031
$719$	22.0503	0.822339	0.411169	−	0.911559i	$-0.365121\pi$
$719$	22.0503	0.822339	0.411169	+	0.911559i	$0.365121\pi$
$720$	0	0
$721$	−22.8403	−0.850616
$722$	−5.47618	−0.203802
$723$	0	0
$724$	67.3535	2.50317
$725$	48.8262	1.81336
$726$	0	0
$727$	−18.9999	−0.704666	−0.352333	−	0.935875i	$-0.614612\pi$
$727$	−18.9999	−0.704666	−0.352333	+	0.935875i	$0.614612\pi$
$728$	11.5981	0.429855
$729$	0	0
$730$	−56.3901	−2.08709
$731$	−10.4163	−0.385261
$732$	0	0
$733$	3.83925	0.141806	0.0709029	−	0.997483i	$-0.477412\pi$
$733$	3.83925	0.141806	0.0709029	+	0.997483i	$0.477412\pi$
$734$	51.4260	1.89817
$735$	0	0
$736$	−38.9863	−1.43705
$737$	27.2084	1.00223
$738$	0	0
$739$	−37.8585	−1.39265	−0.696323	−	0.717728i	$-0.745182\pi$
$739$	−37.8585	−1.39265	−0.696323	+	0.717728i	$0.745182\pi$
$740$	−57.1534	−2.10100
$741$	0	0
$742$	−5.90916	−0.216932
$743$	−37.7800	−1.38601	−0.693006	−	0.720931i	$-0.743714\pi$
$743$	−37.7800	−1.38601	−0.693006	+	0.720931i	$0.743714\pi$
$744$	0	0
$745$	29.5143	1.08132
$746$	−13.4003	−0.490619
$747$	0	0
$748$	−17.5083	−0.640167
$749$	−42.0756	−1.53741
$750$	0	0
$751$	22.5351	0.822316	0.411158	−	0.911564i	$-0.365125\pi$
$751$	22.5351	0.822316	0.411158	+	0.911564i	$0.365125\pi$
$752$	−7.43042	−0.270959
$753$	0	0
$754$	23.7930	0.866490
$755$	−40.5077	−1.47423
$756$	0	0
$757$	5.00131	0.181776	0.0908878	−	0.995861i	$-0.471030\pi$
$757$	5.00131	0.181776	0.0908878	+	0.995861i	$0.471030\pi$
$758$	−41.2468	−1.49815
$759$	0	0
$760$	−51.1140	−1.85410
$761$	12.1788	0.441480	0.220740	−	0.975333i	$-0.429153\pi$
$761$	12.1788	0.441480	0.220740	+	0.975333i	$0.429153\pi$
$762$	0	0
$763$	−53.4022	−1.93329
$764$	−41.5914	−1.50472
$765$	0	0
$766$	77.8293	2.81209
$767$	−4.50886	−0.162805
$768$	0	0
$769$	11.4968	0.414586	0.207293	−	0.978279i	$-0.433535\pi$
$769$	11.4968	0.414586	0.207293	+	0.978279i	$0.433535\pi$
$770$	−102.774	−3.70372
$771$	0	0
$772$	47.0035	1.69169
$773$	27.6934	0.996063	0.498032	−	0.867159i	$-0.334056\pi$
$773$	27.6934	0.996063	0.498032	+	0.867159i	$0.334056\pi$
$774$	0	0
$775$	37.1281	1.33368
$776$	−38.1468	−1.36939
$777$	0	0
$778$	38.9534	1.39655
$779$	5.58885	0.200241
$780$	0	0
$781$	44.5581	1.59441
$782$	−21.2835	−0.761097
$783$	0	0
$784$	0.642952	0.0229626
$785$	−13.2726	−0.473719
$786$	0	0
$787$	−20.9162	−0.745582	−0.372791	−	0.927915i	$-0.621599\pi$
$787$	−20.9162	−0.745582	−0.372791	+	0.927915i	$0.621599\pi$
$788$	34.3220	1.22267
$789$	0	0
$790$	45.7168	1.62653
$791$	−3.92464	−0.139544
$792$	0	0
$793$	8.48534	0.301323
$794$	−28.1327	−0.998393
$795$	0	0
$796$	−96.0698	−3.40510
$797$	33.0323	1.17007	0.585033	−	0.811010i	$-0.301082\pi$
$797$	33.0323	1.17007	0.585033	+	0.811010i	$0.301082\pi$
$798$	0	0
$799$	8.65547	0.306208
$800$	27.2163	0.962241
$801$	0	0
$802$	15.1960	0.536590
$803$	34.9813	1.23446
$804$	0	0
$805$	−78.9337	−2.78205
$806$	18.0925	0.637282
$807$	0	0
$808$	40.6546	1.43022
$809$	16.2259	0.570473	0.285236	−	0.958457i	$-0.407928\pi$
$809$	16.2259	0.570473	0.285236	+	0.958457i	$0.407928\pi$
$810$	0	0
$811$	19.6495	0.689986	0.344993	−	0.938605i	$-0.387881\pi$
$811$	19.6495	0.689986	0.344993	+	0.938605i	$0.387881\pi$
$812$	77.6556	2.72518
$813$	0	0
$814$	56.1171	1.96690
$815$	−8.74519	−0.306330
$816$	0	0
$817$	45.2143	1.58185
$818$	3.27907	0.114650
$819$	0	0
$820$	13.7604	0.480535
$821$	48.4963	1.69253	0.846267	−	0.532760i	$-0.178845\pi$
$821$	48.4963	1.69253	0.846267	+	0.532760i	$0.178845\pi$
$822$	0	0
$823$	41.8599	1.45915	0.729573	−	0.683903i	$-0.239719\pi$
$823$	41.8599	1.45915	0.729573	+	0.683903i	$0.239719\pi$
$824$	27.4612	0.956657
$825$	0	0
$826$	−23.2922	−0.810439
$827$	0.00604556	0.000210225 0	0.000105112	−	1.00000i	$-0.499967\pi$
$827$	0.00604556	0.000210225 0	0.000105112		1.00000i	$0.499967\pi$
$828$	0	0
$829$	15.6513	0.543593	0.271797	−	0.962355i	$-0.412382\pi$
$829$	15.6513	0.543593	0.271797	+	0.962355i	$0.412382\pi$
$830$	51.5403	1.78899
$831$	0	0
$832$	15.5510	0.539135
$833$	−0.748955	−0.0259498
$834$	0	0
$835$	−34.2646	−1.18578
$836$	75.9987	2.62847
$837$	0	0
$838$	−50.2133	−1.73459
$839$	−17.3524	−0.599070	−0.299535	−	0.954085i	$-0.596832\pi$
$839$	−17.3524	−0.599070	−0.299535	+	0.954085i	$0.596832\pi$
$840$	0	0
$841$	37.4661	1.29193
$842$	−77.5590	−2.67286
$843$	0	0
$844$	90.0896	3.10101
$845$	−37.8966	−1.30368
$846$	0	0
$847$	33.2244	1.14160
$848$	0.834760	0.0286658
$849$	0	0
$850$	14.8580	0.509624
$851$	43.0997	1.47744
$852$	0	0
$853$	−23.9343	−0.819496	−0.409748	−	0.912199i	$-0.634383\pi$
$853$	−23.9343	−0.819496	−0.409748	+	0.912199i	$0.634383\pi$
$854$	43.8342	1.49998
$855$	0	0
$856$	50.5881	1.72907
$857$	30.0565	1.02671	0.513354	−	0.858177i	$-0.328403\pi$
$857$	30.0565	1.02671	0.513354	+	0.858177i	$0.328403\pi$
$858$	0	0
$859$	7.27182	0.248111	0.124056	−	0.992275i	$-0.460410\pi$
$859$	7.27182	0.248111	0.124056	+	0.992275i	$0.460410\pi$
$860$	111.323	3.79609
$861$	0	0
$862$	−14.7790	−0.503374
$863$	−55.7614	−1.89814	−0.949070	−	0.315064i	$-0.897974\pi$
$863$	−55.7614	−1.89814	−0.949070	+	0.315064i	$0.897974\pi$
$864$	0	0
$865$	25.3258	0.861104
$866$	−77.8841	−2.64661
$867$	0	0
$868$	59.0504	2.00430
$869$	−28.3602	−0.962053
$870$	0	0
$871$	−7.10876	−0.240871
$872$	64.2064	2.17430
$873$	0	0
$874$	92.3858	3.12500
$875$	9.09945	0.307618
$876$	0	0
$877$	−23.3675	−0.789063	−0.394531	−	0.918882i	$-0.629093\pi$
$877$	−23.3675	−0.789063	−0.394531	+	0.918882i	$0.629093\pi$
$878$	1.93142	0.0651822
$879$	0	0
$880$	14.5184	0.489416
$881$	14.9846	0.504844	0.252422	−	0.967617i	$-0.418773\pi$
$881$	14.9846	0.504844	0.252422	+	0.967617i	$0.418773\pi$
$882$	0	0
$883$	−31.3043	−1.05348	−0.526738	−	0.850028i	$-0.676585\pi$
$883$	−31.3043	−1.05348	−0.526738	+	0.850028i	$0.676585\pi$
$884$	4.57441	0.153854
$885$	0	0
$886$	33.1704	1.11438
$887$	−8.79029	−0.295149	−0.147575	−	0.989051i	$-0.547147\pi$
$887$	−8.79029	−0.295149	−0.147575	+	0.989051i	$0.547147\pi$
$888$	0	0
$889$	−40.4742	−1.35746
$890$	20.8854	0.700082
$891$	0	0
$892$	59.1812	1.98153
$893$	−37.5710	−1.25727
$894$	0	0
$895$	85.3310	2.85230
$896$	55.1085	1.84105
$897$	0	0
$898$	20.3934	0.680536
$899$	50.5417	1.68566
$900$	0	0
$901$	−0.972387	−0.0323949
$902$	−13.5109	−0.449864
$903$	0	0
$904$	4.71866	0.156940
$905$	65.0598	2.16266
$906$	0	0
$907$	−42.5846	−1.41400	−0.707000	−	0.707213i	$-0.749952\pi$
$907$	−42.5846	−1.41400	−0.707000	+	0.707213i	$0.749952\pi$
$908$	−30.8889	−1.02508
$909$	0	0
$910$	26.8519	0.890131
$911$	21.4909	0.712024	0.356012	−	0.934481i	$-0.384136\pi$
$911$	21.4909	0.712024	0.356012	+	0.934481i	$0.384136\pi$
$912$	0	0
$913$	−31.9728	−1.05814
$914$	−62.6434	−2.07206
$915$	0	0
$916$	30.7326	1.01543
$917$	23.1551	0.764647
$918$	0	0
$919$	−32.5847	−1.07487	−0.537435	−	0.843305i	$-0.680607\pi$
$919$	−32.5847	−1.07487	−0.537435	+	0.843305i	$0.680607\pi$
$920$	94.9032	3.12887
$921$	0	0
$922$	75.1620	2.47533
$923$	−11.6417	−0.383192
$924$	0	0
$925$	−30.0878	−0.989280
$926$	30.1951	0.992274
$927$	0	0
$928$	37.0489	1.21619
$929$	−31.8009	−1.04335	−0.521677	−	0.853143i	$-0.674693\pi$
$929$	−31.8009	−1.04335	−0.521677	+	0.853143i	$0.674693\pi$
$930$	0	0
$931$	3.25101	0.106547
$932$	60.6001	1.98502
$933$	0	0
$934$	−37.2707	−1.21953
$935$	−16.9121	−0.553084
$936$	0	0
$937$	−33.1871	−1.08418	−0.542088	−	0.840322i	$-0.682366\pi$
$937$	−33.1871	−1.08418	−0.542088	+	0.840322i	$0.682366\pi$
$938$	−36.7230	−1.19905
$939$	0	0
$940$	−92.5043	−3.01716
$941$	40.8928	1.33307	0.666534	−	0.745474i	$-0.267777\pi$
$941$	40.8928	1.33307	0.666534	+	0.745474i	$0.267777\pi$
$942$	0	0
$943$	−10.3768	−0.337915
$944$	3.29038	0.107093
$945$	0	0
$946$	−109.305	−3.55380
$947$	−24.2247	−0.787198	−0.393599	−	0.919282i	$-0.628770\pi$
$947$	−24.2247	−0.787198	−0.393599	+	0.919282i	$0.628770\pi$
$948$	0	0
$949$	−9.13959	−0.296684
$950$	−64.4944	−2.09247
$951$	0	0
$952$	9.85929	0.319541
$953$	23.5109	0.761594	0.380797	−	0.924659i	$-0.375650\pi$
$953$	23.5109	0.761594	0.380797	+	0.924659i	$0.375650\pi$
$954$	0	0
$955$	−40.1751	−1.30003
$956$	−3.43183	−0.110993
$957$	0	0
$958$	46.7534	1.51053
$959$	−11.4733	−0.370492
$960$	0	0
$961$	7.43260	0.239761
$962$	−14.6618	−0.472714
$963$	0	0
$964$	−66.1234	−2.12969
$965$	45.4028	1.46157
$966$	0	0
$967$	−7.77610	−0.250062	−0.125031	−	0.992153i	$-0.539903\pi$
$967$	−7.77610	−0.250062	−0.125031	+	0.992153i	$0.539903\pi$
$968$	−39.9463	−1.28392
$969$	0	0
$970$	−88.3171	−2.83569
$971$	−38.6297	−1.23969	−0.619843	−	0.784726i	$-0.712804\pi$
$971$	−38.6297	−1.23969	−0.619843	+	0.784726i	$0.712804\pi$
$972$	0	0
$973$	28.6744	0.919260
$974$	89.4177	2.86513
$975$	0	0
$976$	−6.19227	−0.198210
$977$	−3.46063	−0.110715	−0.0553577	−	0.998467i	$-0.517630\pi$
$977$	−3.46063	−0.110715	−0.0553577	+	0.998467i	$0.517630\pi$
$978$	0	0
$979$	−12.9562	−0.414081
$980$	8.00438	0.255690
$981$	0	0
$982$	92.7614	2.96013
$983$	−34.0614	−1.08639	−0.543195	−	0.839607i	$-0.682785\pi$
$983$	−34.0614	−1.08639	−0.543195	+	0.839607i	$0.682785\pi$
$984$	0	0
$985$	33.1532	1.05635
$986$	20.2258	0.644122
$987$	0	0
$988$	−19.8563	−0.631712
$989$	−83.9493	−2.66943
$990$	0	0
$991$	−11.1868	−0.355360	−0.177680	−	0.984088i	$-0.556859\pi$
$991$	−11.1868	−0.355360	−0.177680	+	0.984088i	$0.556859\pi$
$992$	28.1725	0.894478
$993$	0	0
$994$	−60.1397	−1.90752
$995$	−92.7982	−2.94190
$996$	0	0
$997$	−43.9284	−1.39123	−0.695613	−	0.718417i	$-0.744867\pi$
$997$	−43.9284	−1.39123	−0.695613	+	0.718417i	$0.744867\pi$
$998$	19.0089	0.601716
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2151.2.a.i.1.2		17
3.2	odd	2		239.2.a.b.1.16	✓	17
12.11	even	2		3824.2.a.p.1.6		17
15.14	odd	2		5975.2.a.g.1.2		17

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
239.2.a.b.1.16	✓	17	3.2	odd	2
2151.2.a.i.1.2		17	1.1	even	1	trivial
3824.2.a.p.1.6		17	12.11	even	2
5975.2.a.g.1.2		17	15.14	odd	2

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:	\( N \)	\(=\)	\( 2151 = 3^{2} \cdot 239 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2151.a (trivial)

\(f(q)\)	\(=\)	\(q-2.33063 q^{2} +3.43183 q^{4} +3.31496 q^{5} +2.77554 q^{7} -3.33707 q^{8} +O(q^{10})\)	\(q-2.33063 q^{2} +3.43183 q^{4} +3.31496 q^{5} +2.77554 q^{7} -3.33707 q^{8} -7.72595 q^{10} +4.79275 q^{11} -1.25221 q^{13} -6.46874 q^{14} +0.913810 q^{16} -1.06447 q^{17} +4.62057 q^{19} +11.3764 q^{20} -11.1701 q^{22} -8.57900 q^{23} +5.98898 q^{25} +2.91843 q^{26} +9.52517 q^{28} +8.15267 q^{29} +6.19940 q^{31} +4.54439 q^{32} +2.48088 q^{34} +9.20080 q^{35} -5.02385 q^{37} -10.7688 q^{38} -11.0623 q^{40} +1.20956 q^{41} +9.78544 q^{43} +16.4479 q^{44} +19.9945 q^{46} -8.13125 q^{47} +0.703595 q^{49} -13.9581 q^{50} -4.29736 q^{52} +0.913494 q^{53} +15.8878 q^{55} -9.26216 q^{56} -19.0009 q^{58} +3.60073 q^{59} -6.77631 q^{61} -14.4485 q^{62} -12.4189 q^{64} -4.15102 q^{65} +5.67699 q^{67} -3.65308 q^{68} -21.4436 q^{70} +9.29697 q^{71} +7.29879 q^{73} +11.7087 q^{74} +15.8570 q^{76} +13.3024 q^{77} -5.91731 q^{79} +3.02925 q^{80} -2.81903 q^{82} -6.67107 q^{83} -3.52868 q^{85} -22.8062 q^{86} -15.9937 q^{88} -2.70328 q^{89} -3.47554 q^{91} -29.4417 q^{92} +18.9509 q^{94} +15.3170 q^{95} +11.4312 q^{97} -1.63982 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 17 q + 22 q^{4} - 6 q^{5} + 5 q^{7} + 3 q^{8}+O(q^{10}) \) 17 * q + 22 * q^4 - 6 * q^5 + 5 * q^7 + 3 * q^8	\( 17 q + 22 q^{4} - 6 q^{5} + 5 q^{7} + 3 q^{8} + 5 q^{10} + q^{11} + 15 q^{13} + 3 q^{14} + 24 q^{16} - 4 q^{17} + 24 q^{19} - 4 q^{20} - 10 q^{22} + 9 q^{23} + 39 q^{25} + 12 q^{26} - 7 q^{28} + 2 q^{29} + 28 q^{31} + 31 q^{32} + 29 q^{34} + 24 q^{35} + 11 q^{37} + 19 q^{38} - 18 q^{40} - 20 q^{41} - 9 q^{43} + 43 q^{44} - 18 q^{46} + 18 q^{47} + 60 q^{49} + 61 q^{50} - q^{52} + 12 q^{53} - 10 q^{55} + 60 q^{56} - 38 q^{58} - q^{59} + 24 q^{61} + 33 q^{62} + 21 q^{64} - 2 q^{65} + 16 q^{67} + 10 q^{68} + 7 q^{70} - 12 q^{71} + 30 q^{73} + 21 q^{74} + 75 q^{76} + 15 q^{77} - 10 q^{79} - 32 q^{80} + 50 q^{82} + 16 q^{83} - 18 q^{85} + 3 q^{86} - 28 q^{88} - 65 q^{89} + 47 q^{91} - 24 q^{92} + 32 q^{94} + 37 q^{95} + 87 q^{97} + 9 q^{98}+O(q^{100}) \) 17 * q + 22 * q^4 - 6 * q^5 + 5 * q^7 + 3 * q^8 + 5 * q^10 + q^11 + 15 * q^13 + 3 * q^14 + 24 * q^16 - 4 * q^17 + 24 * q^19 - 4 * q^20 - 10 * q^22 + 9 * q^23 + 39 * q^25 + 12 * q^26 - 7 * q^28 + 2 * q^29 + 28 * q^31 + 31 * q^32 + 29 * q^34 + 24 * q^35 + 11 * q^37 + 19 * q^38 - 18 * q^40 - 20 * q^41 - 9 * q^43 + 43 * q^44 - 18 * q^46 + 18 * q^47 + 60 * q^49 + 61 * q^50 - q^52 + 12 * q^53 - 10 * q^55 + 60 * q^56 - 38 * q^58 - q^59 + 24 * q^61 + 33 * q^62 + 21 * q^64 - 2 * q^65 + 16 * q^67 + 10 * q^68 + 7 * q^70 - 12 * q^71 + 30 * q^73 + 21 * q^74 + 75 * q^76 + 15 * q^77 - 10 * q^79 - 32 * q^80 + 50 * q^82 + 16 * q^83 - 18 * q^85 + 3 * q^86 - 28 * q^88 - 65 * q^89 + 47 * q^91 - 24 * q^92 + 32 * q^94 + 37 * q^95 + 87 * q^97 + 9 * q^98

Introduction

L-functions

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