LMFDB - Embedded newform 2175.2.a.w.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2175,2,Mod(1,2175)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2175, 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("2175.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2175 = 3 \cdot 5^{2} \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2175.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:	$17.3674624396$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	5.5.246832.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 5x^{3} + 6x^{2} + 7x - 2 $ x^5 - 2x^4 - 5x^3 + 6x^2 + 7x - 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 435)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.15351$ of defining polynomial
Character	$\chi$	$=$	2175.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-2.15351 q^{2} -1.00000 q^{3} +2.63760 q^{4} +2.15351 q^{6} -1.51591 q^{7} -1.37308 q^{8} +1.00000 q^{9} +1.88899 q^{11} -2.63760 q^{12} -0.484093 q^{13} +3.26452 q^{14} -2.31826 q^{16} -3.39257 q^{17} -2.15351 q^{18} +2.85121 q^{19} +1.51591 q^{21} -4.06795 q^{22} +1.28156 q^{23} +1.37308 q^{24} +1.04250 q^{26} -1.00000 q^{27} -3.99836 q^{28} +1.00000 q^{29} -5.47404 q^{31} +7.73856 q^{32} -1.88899 q^{33} +7.30594 q^{34} +2.63760 q^{36} -7.43995 q^{37} -6.14010 q^{38} +0.484093 q^{39} +8.28237 q^{41} -3.26452 q^{42} -1.43262 q^{43} +4.98240 q^{44} -2.75985 q^{46} +2.25328 q^{47} +2.31826 q^{48} -4.70203 q^{49} +3.39257 q^{51} -1.27684 q^{52} +5.62364 q^{53} +2.15351 q^{54} +2.08146 q^{56} -2.85121 q^{57} -2.15351 q^{58} -12.7907 q^{59} +7.40162 q^{61} +11.7884 q^{62} -1.51591 q^{63} -12.0285 q^{64} +4.06795 q^{66} +8.76646 q^{67} -8.94826 q^{68} -1.28156 q^{69} +14.5705 q^{71} -1.37308 q^{72} -2.80833 q^{73} +16.0220 q^{74} +7.52035 q^{76} -2.86353 q^{77} -1.04250 q^{78} -9.34617 q^{79} +1.00000 q^{81} -17.8362 q^{82} -1.79167 q^{83} +3.99836 q^{84} +3.08516 q^{86} -1.00000 q^{87} -2.59374 q^{88} +7.24703 q^{89} +0.733840 q^{91} +3.38025 q^{92} +5.47404 q^{93} -4.85246 q^{94} -7.73856 q^{96} -4.59510 q^{97} +10.1259 q^{98} +1.88899 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 3 q^{2} - 5 q^{3} + 5 q^{4} + 3 q^{6} - 8 q^{7} - 9 q^{8} + 5 q^{9} + 12 q^{11} - 5 q^{12} - 2 q^{13} + 6 q^{14} + q^{16} - 3 q^{18} - 2 q^{19} + 8 q^{21} - 14 q^{22} - 8 q^{23} + 9 q^{24} - 5 q^{27}+ \cdots + 12 q^{99}+O(q^{100}) $ 5 * q - 3 * q^2 - 5 * q^3 + 5 * q^4 + 3 * q^6 - 8 * q^7 - 9 * q^8 + 5 * q^9 + 12 * q^11 - 5 * q^12 - 2 * q^13 + 6 * q^14 + q^16 - 3 * q^18 - 2 * q^19 + 8 * q^21 - 14 * q^22 - 8 * q^23 + 9 * q^24 - 5 * q^27 + 6 * q^28 + 5 * q^29 + 2 * q^31 - q^32 - 12 * q^33 + 4 * q^34 + 5 * q^36 - 16 * q^37 + 14 * q^38 + 2 * q^39 - 14 * q^41 - 6 * q^42 + 20 * q^44 - 6 * q^46 - 2 * q^47 - q^48 - 7 * q^49 - 16 * q^52 - 26 * q^53 + 3 * q^54 - 2 * q^56 + 2 * q^57 - 3 * q^58 + 4 * q^59 - 12 * q^61 - 8 * q^63 - 9 * q^64 + 14 * q^66 - 12 * q^67 + 20 * q^68 + 8 * q^69 + 30 * q^71 - 9 * q^72 + 12 * q^73 + 2 * q^74 - 44 * q^76 - 18 * q^77 - 18 * q^79 + 5 * q^81 - 10 * q^82 - 2 * q^83 - 6 * q^84 + 30 * q^86 - 5 * q^87 - 42 * q^88 - 22 * q^89 - 12 * q^91 - 20 * q^92 - 2 * q^93 - 50 * q^94 + q^96 - 20 * q^97 + 9 * q^98 + 12 * q^99

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 $

$2$	−2.15351	−1.52276	−0.761380	−	0.648305i	$-0.775478\pi$
$2$	−2.15351	−1.52276	−0.761380	+	0.648305i	$0.775478\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	2.63760	1.31880
$5$	0	0
$5$	0	0
$6$	2.15351	0.879166
$7$	−1.51591	−0.572959	−0.286480	−	0.958086i	$-0.592485\pi$
$7$	−1.51591	−0.572959	−0.286480	+	0.958086i	$0.592485\pi$
$8$	−1.37308	−0.485458
$9$	1.00000	0.333333
$10$	0	0
$11$	1.88899	0.569552	0.284776	−	0.958594i	$-0.408081\pi$
$11$	1.88899	0.569552	0.284776	+	0.958594i	$0.408081\pi$
$12$	−2.63760	−0.761410
$13$	−0.484093	−0.134263	−0.0671316	−	0.997744i	$-0.521385\pi$
$13$	−0.484093	−0.134263	−0.0671316	+	0.997744i	$0.521385\pi$
$14$	3.26452	0.872480
$15$	0	0
$16$	−2.31826	−0.579565
$17$	−3.39257	−0.822820	−0.411410	−	0.911450i	$-0.634964\pi$
$17$	−3.39257	−0.822820	−0.411410	+	0.911450i	$0.634964\pi$
$18$	−2.15351	−0.507587
$19$	2.85121	0.654112	0.327056	−	0.945005i	$-0.393943\pi$
$19$	2.85121	0.654112	0.327056	+	0.945005i	$0.393943\pi$
$20$	0	0
$21$	1.51591	0.330798
$22$	−4.06795	−0.867291
$23$	1.28156	0.267224	0.133612	−	0.991034i	$-0.457342\pi$
$23$	1.28156	0.267224	0.133612	+	0.991034i	$0.457342\pi$
$24$	1.37308	0.280279
$25$	0	0
$26$	1.04250	0.204451
$27$	−1.00000	−0.192450
$28$	−3.99836	−0.755619
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	−5.47404	−0.983166	−0.491583	−	0.870831i	$-0.663582\pi$
$31$	−5.47404	−0.983166	−0.491583	+	0.870831i	$0.663582\pi$
$32$	7.73856	1.36800
$33$	−1.88899	−0.328831
$34$	7.30594	1.25296
$35$	0	0
$36$	2.63760	0.439600
$37$	−7.43995	−1.22312	−0.611561	−	0.791198i	$-0.709458\pi$
$37$	−7.43995	−1.22312	−0.611561	+	0.791198i	$0.709458\pi$
$38$	−6.14010	−0.996056
$39$	0.484093	0.0775169
$40$	0	0
$41$	8.28237	1.29349	0.646745	−	0.762707i	$-0.276130\pi$
$41$	8.28237	1.29349	0.646745	+	0.762707i	$0.276130\pi$
$42$	−3.26452	−0.503726
$43$	−1.43262	−0.218473	−0.109236	−	0.994016i	$-0.534841\pi$
$43$	−1.43262	−0.218473	−0.109236	+	0.994016i	$0.534841\pi$
$44$	4.98240	0.751125
$45$	0	0
$46$	−2.75985	−0.406918
$47$	2.25328	0.328674	0.164337	−	0.986404i	$-0.447451\pi$
$47$	2.25328	0.328674	0.164337	+	0.986404i	$0.447451\pi$
$48$	2.31826	0.334612
$49$	−4.70203	−0.671718
$50$	0	0
$51$	3.39257	0.475055
$52$	−1.27684	−0.177066
$53$	5.62364	0.772466	0.386233	−	0.922401i	$-0.373776\pi$
$53$	5.62364	0.772466	0.386233	+	0.922401i	$0.373776\pi$
$54$	2.15351	0.293055
$55$	0	0
$56$	2.08146	0.278147
$57$	−2.85121	−0.377652
$58$	−2.15351	−0.282770
$59$	−12.7907	−1.66520	−0.832601	−	0.553873i	$-0.813149\pi$
$59$	−12.7907	−1.66520	−0.832601	+	0.553873i	$0.813149\pi$
$60$	0	0
$61$	7.40162	0.947680	0.473840	−	0.880611i	$-0.342868\pi$
$61$	7.40162	0.947680	0.473840	+	0.880611i	$0.342868\pi$
$62$	11.7884	1.49713
$63$	−1.51591	−0.190986
$64$	−12.0285	−1.50357
$65$	0	0
$66$	4.06795	0.500731
$67$	8.76646	1.07099	0.535497	−	0.844537i	$-0.320124\pi$
$67$	8.76646	1.07099	0.535497	+	0.844537i	$0.320124\pi$
$68$	−8.94826	−1.08514
$69$	−1.28156	−0.154282
$70$	0	0
$71$	14.5705	1.72920	0.864598	−	0.502465i	$-0.167573\pi$
$71$	14.5705	1.72920	0.864598	+	0.502465i	$0.167573\pi$
$72$	−1.37308	−0.161819
$73$	−2.80833	−0.328691	−0.164345	−	0.986403i	$-0.552551\pi$
$73$	−2.80833	−0.328691	−0.164345	+	0.986403i	$0.552551\pi$
$74$	16.0220	1.86252
$75$	0	0
$76$	7.52035	0.862644
$77$	−2.86353	−0.326330
$78$	−1.04250	−0.118040
$79$	−9.34617	−1.05153	−0.525763	−	0.850631i	$-0.676220\pi$
$79$	−9.34617	−1.05153	−0.525763	+	0.850631i	$0.676220\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	−17.8362	−1.96967
$83$	−1.79167	−0.196661	−0.0983306	−	0.995154i	$-0.531350\pi$
$83$	−1.79167	−0.196661	−0.0983306	+	0.995154i	$0.531350\pi$
$84$	3.99836	0.436257
$85$	0	0
$86$	3.08516	0.332682
$87$	−1.00000	−0.107211
$88$	−2.59374	−0.276493
$89$	7.24703	0.768183	0.384092	−	0.923295i	$-0.374515\pi$
$89$	7.24703	0.768183	0.384092	+	0.923295i	$0.374515\pi$
$90$	0	0
$91$	0.733840	0.0769273
$92$	3.38025	0.352415
$93$	5.47404	0.567631
$94$	−4.85246	−0.500493
$95$	0	0
$96$	−7.73856	−0.789813
$97$	−4.59510	−0.466562	−0.233281	−	0.972409i	$-0.574946\pi$
$97$	−4.59510	−0.466562	−0.233281	+	0.972409i	$0.574946\pi$
$98$	10.1259	1.02287
$99$	1.88899	0.189851
$100$	0	0
$101$	−15.8611	−1.57823	−0.789117	−	0.614243i	$-0.789462\pi$
$101$	−15.8611	−1.57823	−0.789117	+	0.614243i	$0.789462\pi$
$102$	−7.30594	−0.723396
$103$	8.46995	0.834569	0.417284	−	0.908776i	$-0.362982\pi$
$103$	8.46995	0.834569	0.417284	+	0.908776i	$0.362982\pi$
$104$	0.664699	0.0651791
$105$	0	0
$106$	−12.1106	−1.17628
$107$	−0.445502	−0.0430684	−0.0215342	−	0.999768i	$-0.506855\pi$
$107$	−0.445502	−0.0430684	−0.0215342	+	0.999768i	$0.506855\pi$
$108$	−2.63760	−0.253803
$109$	−7.32651	−0.701752	−0.350876	−	0.936422i	$-0.614116\pi$
$109$	−7.32651	−0.701752	−0.350876	+	0.936422i	$0.614116\pi$
$110$	0	0
$111$	7.43995	0.706169
$112$	3.51427	0.332067
$113$	9.41884	0.886050	0.443025	−	0.896509i	$-0.353905\pi$
$113$	9.41884	0.886050	0.443025	+	0.896509i	$0.353905\pi$
$114$	6.14010	0.575073
$115$	0	0
$116$	2.63760	0.244895
$117$	−0.484093	−0.0447544
$118$	27.5448	2.53570
$119$	5.14283	0.471442
$120$	0	0
$121$	−7.43172	−0.675611
$122$	−15.9394	−1.44309
$123$	−8.28237	−0.746796
$124$	−14.4383	−1.29660
$125$	0	0
$126$	3.26452	0.290827
$127$	−2.22656	−0.197575	−0.0987875	−	0.995109i	$-0.531496\pi$
$127$	−2.22656	−0.197575	−0.0987875	+	0.995109i	$0.531496\pi$
$128$	10.4264	0.921576
$129$	1.43262	0.126135
$130$	0	0
$131$	15.3567	1.34172	0.670859	−	0.741585i	$-0.265925\pi$
$131$	15.3567	1.34172	0.670859	+	0.741585i	$0.265925\pi$
$132$	−4.98240	−0.433662
$133$	−4.32217	−0.374779
$134$	−18.8787	−1.63087
$135$	0	0
$136$	4.65828	0.399444
$137$	−1.72253	−0.147166	−0.0735828	−	0.997289i	$-0.523443\pi$
$137$	−1.72253	−0.147166	−0.0735828	+	0.997289i	$0.523443\pi$
$138$	2.75985	0.234934
$139$	−6.68293	−0.566838	−0.283419	−	0.958996i	$-0.591469\pi$
$139$	−6.68293	−0.566838	−0.283419	+	0.958996i	$0.591469\pi$
$140$	0	0
$141$	−2.25328	−0.189760
$142$	−31.3776	−2.63315
$143$	−0.914446	−0.0764698
$144$	−2.31826	−0.193188
$145$	0	0
$146$	6.04777	0.500517
$147$	4.70203	0.387816
$148$	−19.6236	−1.61305
$149$	0.474203	0.0388482	0.0194241	−	0.999811i	$-0.493817\pi$
$149$	0.474203	0.0388482	0.0194241	+	0.999811i	$0.493817\pi$
$150$	0	0
$151$	−12.5209	−1.01893	−0.509467	−	0.860490i	$-0.670157\pi$
$151$	−12.5209	−1.01893	−0.509467	+	0.860490i	$0.670157\pi$
$152$	−3.91494	−0.317544
$153$	−3.39257	−0.274273
$154$	6.16664	0.496922
$155$	0	0
$156$	1.27684	0.102229
$157$	23.8828	1.90606	0.953028	−	0.302881i	$-0.0979486\pi$
$157$	23.8828	1.90606	0.953028	+	0.302881i	$0.0979486\pi$
$158$	20.1271	1.60122
$159$	−5.62364	−0.445984
$160$	0	0
$161$	−1.94273	−0.153108
$162$	−2.15351	−0.169196
$163$	−14.7254	−1.15339	−0.576693	−	0.816961i	$-0.695657\pi$
$163$	−14.7254	−1.15339	−0.576693	+	0.816961i	$0.695657\pi$
$164$	21.8456	1.70585
$165$	0	0
$166$	3.85838	0.299468
$167$	−2.95547	−0.228701	−0.114351	−	0.993440i	$-0.536479\pi$
$167$	−2.95547	−0.228701	−0.114351	+	0.993440i	$0.536479\pi$
$168$	−2.08146	−0.160588
$169$	−12.7657	−0.981973
$170$	0	0
$171$	2.85121	0.218037
$172$	−3.77868	−0.288122
$173$	12.1974	0.927349	0.463675	−	0.886006i	$-0.346531\pi$
$173$	12.1974	0.927349	0.463675	+	0.886006i	$0.346531\pi$
$174$	2.15351	0.163257
$175$	0	0
$176$	−4.37917	−0.330092
$177$	12.7907	0.961405
$178$	−15.6065	−1.16976
$179$	−12.3747	−0.924929	−0.462464	−	0.886638i	$-0.653035\pi$
$179$	−12.3747	−0.924929	−0.462464	+	0.886638i	$0.653035\pi$
$180$	0	0
$181$	−13.1895	−0.980367	−0.490183	−	0.871619i	$-0.663070\pi$
$181$	−13.1895	−0.980367	−0.490183	+	0.871619i	$0.663070\pi$
$182$	−1.58033	−0.117142
$183$	−7.40162	−0.547143
$184$	−1.75969	−0.129726
$185$	0	0
$186$	−11.7884	−0.864367
$187$	−6.40853	−0.468638
$188$	5.94325	0.433456
$189$	1.51591	0.110266
$190$	0	0
$191$	−6.10636	−0.441841	−0.220920	−	0.975292i	$-0.570906\pi$
$191$	−6.10636	−0.441841	−0.220920	+	0.975292i	$0.570906\pi$
$192$	12.0285	0.868085
$193$	−19.6580	−1.41501	−0.707507	−	0.706707i	$-0.750180\pi$
$193$	−19.6580	−1.41501	−0.707507	+	0.706707i	$0.750180\pi$
$194$	9.89560	0.710463
$195$	0	0
$196$	−12.4021	−0.885862
$197$	−24.2075	−1.72471	−0.862357	−	0.506301i	$-0.831013\pi$
$197$	−24.2075	−1.72471	−0.862357	+	0.506301i	$0.831013\pi$
$198$	−4.06795	−0.289097
$199$	0.542480	0.0384554	0.0192277	−	0.999815i	$-0.493879\pi$
$199$	0.542480	0.0384554	0.0192277	+	0.999815i	$0.493879\pi$
$200$	0	0
$201$	−8.76646	−0.618339
$202$	34.1569	2.40327
$203$	−1.51591	−0.106396
$204$	8.94826	0.626503
$205$	0	0
$206$	−18.2401	−1.27085
$207$	1.28156	0.0890747
$208$	1.12225	0.0778142
$209$	5.38590	0.372551
$210$	0	0
$211$	−1.20999	−0.0832989	−0.0416495	−	0.999132i	$-0.513261\pi$
$211$	−1.20999	−0.0832989	−0.0416495	+	0.999132i	$0.513261\pi$
$212$	14.8329	1.01873
$213$	−14.5705	−0.998351
$214$	0.959394	0.0655828
$215$	0	0
$216$	1.37308	0.0934264
$217$	8.29813	0.563314
$218$	15.7777	1.06860
$219$	2.80833	0.189770
$220$	0	0
$221$	1.64232	0.110474
$222$	−16.0220	−1.07533
$223$	−15.8267	−1.05983	−0.529917	−	0.848049i	$-0.677777\pi$
$223$	−15.8267	−1.05983	−0.529917	+	0.848049i	$0.677777\pi$
$224$	−11.7309	−0.783806
$225$	0	0
$226$	−20.2836	−1.34924
$227$	8.03996	0.533631	0.266815	−	0.963748i	$-0.414029\pi$
$227$	8.03996	0.533631	0.266815	+	0.963748i	$0.414029\pi$
$228$	−7.52035	−0.498048
$229$	−0.0946339	−0.00625359	−0.00312679	−	0.999995i	$-0.500995\pi$
$229$	−0.0946339	−0.00625359	−0.00312679	+	0.999995i	$0.500995\pi$
$230$	0	0
$231$	2.86353	0.188407
$232$	−1.37308	−0.0901472
$233$	−18.6579	−1.22232	−0.611159	−	0.791508i	$-0.709297\pi$
$233$	−18.6579	−1.22232	−0.611159	+	0.791508i	$0.709297\pi$
$234$	1.04250	0.0681502
$235$	0	0
$236$	−33.7367	−2.19607
$237$	9.34617	0.607099
$238$	−11.0751	−0.717894
$239$	−25.3243	−1.63809	−0.819047	−	0.573726i	$-0.805497\pi$
$239$	−25.3243	−1.63809	−0.819047	+	0.573726i	$0.805497\pi$
$240$	0	0
$241$	−28.5508	−1.83912	−0.919559	−	0.392951i	$-0.871454\pi$
$241$	−28.5508	−1.83912	−0.919559	+	0.392951i	$0.871454\pi$
$242$	16.0043	1.02879
$243$	−1.00000	−0.0641500
$244$	19.5225	1.24980
$245$	0	0
$246$	17.8362	1.13719
$247$	−1.38025	−0.0878232
$248$	7.51630	0.477285
$249$	1.79167	0.113542
$250$	0	0
$251$	−16.4952	−1.04117	−0.520584	−	0.853810i	$-0.674286\pi$
$251$	−16.4952	−1.04117	−0.520584	+	0.853810i	$0.674286\pi$
$252$	−3.99836	−0.251873
$253$	2.42086	0.152198
$254$	4.79491	0.300860
$255$	0	0
$256$	1.60362	0.100226
$257$	−13.7659	−0.858694	−0.429347	−	0.903140i	$-0.641256\pi$
$257$	−13.7659	−0.858694	−0.429347	+	0.903140i	$0.641256\pi$
$258$	−3.08516	−0.192074
$259$	11.2783	0.700798
$260$	0	0
$261$	1.00000	0.0618984
$262$	−33.0707	−2.04312
$263$	16.7039	1.03001	0.515004	−	0.857188i	$-0.327790\pi$
$263$	16.7039	1.03001	0.515004	+	0.857188i	$0.327790\pi$
$264$	2.59374	0.159633
$265$	0	0
$266$	9.30783	0.570700
$267$	−7.24703	−0.443511
$268$	23.1224	1.41243
$269$	−17.3518	−1.05796	−0.528979	−	0.848635i	$-0.677425\pi$
$269$	−17.3518	−1.05796	−0.528979	+	0.848635i	$0.677425\pi$
$270$	0	0
$271$	8.27747	0.502821	0.251410	−	0.967881i	$-0.419106\pi$
$271$	8.27747	0.502821	0.251410	+	0.967881i	$0.419106\pi$
$272$	7.86487	0.476878
$273$	−0.733840	−0.0444140
$274$	3.70948	0.224098
$275$	0	0
$276$	−3.38025	−0.203467
$277$	−18.0560	−1.08488	−0.542439	−	0.840095i	$-0.682499\pi$
$277$	−18.0560	−1.08488	−0.542439	+	0.840095i	$0.682499\pi$
$278$	14.3917	0.863159
$279$	−5.47404	−0.327722
$280$	0	0
$281$	−18.6241	−1.11102	−0.555510	−	0.831510i	$-0.687477\pi$
$281$	−18.6241	−1.11102	−0.555510	+	0.831510i	$0.687477\pi$
$282$	4.85246	0.288960
$283$	−16.6920	−0.992235	−0.496117	−	0.868256i	$-0.665241\pi$
$283$	−16.6920	−0.992235	−0.496117	+	0.868256i	$0.665241\pi$
$284$	38.4311	2.28046
$285$	0	0
$286$	1.96927	0.116445
$287$	−12.5553	−0.741116
$288$	7.73856	0.455999
$289$	−5.49045	−0.322968
$290$	0	0
$291$	4.59510	0.269370
$292$	−7.40727	−0.433478
$293$	−32.9866	−1.92710	−0.963549	−	0.267532i	$-0.913792\pi$
$293$	−32.9866	−1.92710	−0.963549	+	0.267532i	$0.913792\pi$
$294$	−10.1259	−0.590552
$295$	0	0
$296$	10.2157	0.593773
$297$	−1.88899	−0.109610
$298$	−1.02120	−0.0591566
$299$	−0.620395	−0.0358783
$300$	0	0
$301$	2.17172	0.125176
$302$	26.9638	1.55159
$303$	15.8611	0.911194
$304$	−6.60984	−0.379100
$305$	0	0
$306$	7.30594	0.417653
$307$	5.20522	0.297077	0.148539	−	0.988907i	$-0.452543\pi$
$307$	5.20522	0.297077	0.148539	+	0.988907i	$0.452543\pi$
$308$	−7.55286	−0.430364
$309$	−8.46995	−0.481838
$310$	0	0
$311$	15.1777	0.860651	0.430325	−	0.902674i	$-0.358399\pi$
$311$	15.1777	0.860651	0.430325	+	0.902674i	$0.358399\pi$
$312$	−0.664699	−0.0376312
$313$	6.91509	0.390864	0.195432	−	0.980717i	$-0.437389\pi$
$313$	6.91509	0.390864	0.195432	+	0.980717i	$0.437389\pi$
$314$	−51.4319	−2.90247
$315$	0	0
$316$	−24.6515	−1.38675
$317$	−24.6694	−1.38557	−0.692786	−	0.721144i	$-0.743617\pi$
$317$	−24.6694	−1.38557	−0.692786	+	0.721144i	$0.743617\pi$
$318$	12.1106	0.679126
$319$	1.88899	0.105763
$320$	0	0
$321$	0.445502	0.0248655
$322$	4.18368	0.233148
$323$	−9.67293	−0.538216
$324$	2.63760	0.146533
$325$	0	0
$326$	31.7114	1.75633
$327$	7.32651	0.405157
$328$	−11.3724	−0.627934
$329$	−3.41576	−0.188317
$330$	0	0
$331$	−4.24031	−0.233069	−0.116534	−	0.993187i	$-0.537178\pi$
$331$	−4.24031	−0.233069	−0.116534	+	0.993187i	$0.537178\pi$
$332$	−4.72571	−0.259357
$333$	−7.43995	−0.407707
$334$	6.36463	0.348257
$335$	0	0
$336$	−3.51427	−0.191719
$337$	26.2776	1.43143	0.715717	−	0.698390i	$-0.246100\pi$
$337$	26.2776	1.43143	0.715717	+	0.698390i	$0.246100\pi$
$338$	27.4910	1.49531
$339$	−9.41884	−0.511561
$340$	0	0
$341$	−10.3404	−0.559964
$342$	−6.14010	−0.332019
$343$	17.7392	0.957826
$344$	1.96711	0.106059
$345$	0	0
$346$	−26.2672	−1.41213
$347$	−21.7745	−1.16892	−0.584458	−	0.811424i	$-0.698693\pi$
$347$	−21.7745	−1.16892	−0.584458	+	0.811424i	$0.698693\pi$
$348$	−2.63760	−0.141390
$349$	8.94807	0.478979	0.239490	−	0.970899i	$-0.423020\pi$
$349$	8.94807	0.478979	0.239490	+	0.970899i	$0.423020\pi$
$350$	0	0
$351$	0.484093	0.0258390
$352$	14.6180	0.779145
$353$	−7.14944	−0.380526	−0.190263	−	0.981733i	$-0.560934\pi$
$353$	−7.14944	−0.380526	−0.190263	+	0.981733i	$0.560934\pi$
$354$	−27.5448	−1.46399
$355$	0	0
$356$	19.1148	1.01308
$357$	−5.14283	−0.272187
$358$	26.6490	1.40845
$359$	31.4641	1.66061	0.830305	−	0.557309i	$-0.188166\pi$
$359$	31.4641	1.66061	0.830305	+	0.557309i	$0.188166\pi$
$360$	0	0
$361$	−10.8706	−0.572137
$362$	28.4037	1.49286
$363$	7.43172	0.390064
$364$	1.93558	0.101452
$365$	0	0
$366$	15.9394	0.833168
$367$	27.7722	1.44970	0.724849	−	0.688908i	$-0.241910\pi$
$367$	27.7722	1.44970	0.724849	+	0.688908i	$0.241910\pi$
$368$	−2.97099	−0.154874
$369$	8.28237	0.431163
$370$	0	0
$371$	−8.52491	−0.442592
$372$	14.4383	0.748593
$373$	15.9293	0.824790	0.412395	−	0.911005i	$-0.364692\pi$
$373$	15.9293	0.824790	0.412395	+	0.911005i	$0.364692\pi$
$374$	13.8008	0.713624
$375$	0	0
$376$	−3.09394	−0.159558
$377$	−0.484093	−0.0249320
$378$	−3.26452	−0.167909
$379$	18.4181	0.946073	0.473037	−	0.881043i	$-0.343158\pi$
$379$	18.4181	0.946073	0.473037	+	0.881043i	$0.343158\pi$
$380$	0	0
$381$	2.22656	0.114070
$382$	13.1501	0.672818
$383$	14.1973	0.725447	0.362724	−	0.931897i	$-0.381847\pi$
$383$	14.1973	0.725447	0.362724	+	0.931897i	$0.381847\pi$
$384$	−10.4264	−0.532072
$385$	0	0
$386$	42.3337	2.15473
$387$	−1.43262	−0.0728242
$388$	−12.1201	−0.615303
$389$	−25.0087	−1.26799	−0.633995	−	0.773337i	$-0.718586\pi$
$389$	−25.0087	−1.26799	−0.633995	+	0.773337i	$0.718586\pi$
$390$	0	0
$391$	−4.34779	−0.219877
$392$	6.45626	0.326091
$393$	−15.3567	−0.774641
$394$	52.1311	2.62633
$395$	0	0
$396$	4.98240	0.250375
$397$	3.67651	0.184519	0.0922595	−	0.995735i	$-0.470591\pi$
$397$	3.67651	0.184519	0.0922595	+	0.995735i	$0.470591\pi$
$398$	−1.16824	−0.0585584
$399$	4.32217	0.216379
$400$	0	0
$401$	−4.68500	−0.233958	−0.116979	−	0.993134i	$-0.537321\pi$
$401$	−4.68500	−0.233958	−0.116979	+	0.993134i	$0.537321\pi$
$402$	18.8787	0.941582
$403$	2.64994	0.132003
$404$	−41.8352	−2.08138
$405$	0	0
$406$	3.26452	0.162015
$407$	−14.0540	−0.696630
$408$	−4.65828	−0.230619
$409$	36.8798	1.82359	0.911794	−	0.410647i	$-0.134697\pi$
$409$	36.8798	1.82359	0.911794	+	0.410647i	$0.134697\pi$
$410$	0	0
$411$	1.72253	0.0849661
$412$	22.3403	1.10063
$413$	19.3895	0.954092
$414$	−2.75985	−0.135639
$415$	0	0
$416$	−3.74618	−0.183672
$417$	6.68293	0.327264
$418$	−11.5986	−0.567305
$419$	19.1400	0.935050	0.467525	−	0.883980i	$-0.345146\pi$
$419$	19.1400	0.935050	0.467525	+	0.883980i	$0.345146\pi$
$420$	0	0
$421$	29.5114	1.43830	0.719149	−	0.694856i	$-0.244532\pi$
$421$	29.5114	1.43830	0.719149	+	0.694856i	$0.244532\pi$
$422$	2.60572	0.126844
$423$	2.25328	0.109558
$424$	−7.72171	−0.375000
$425$	0	0
$426$	31.3776	1.52025
$427$	−11.2202	−0.542982
$428$	−1.17506	−0.0567986
$429$	0.914446	0.0441499
$430$	0	0
$431$	13.6750	0.658703	0.329352	−	0.944207i	$-0.393170\pi$
$431$	13.6750	0.658703	0.329352	+	0.944207i	$0.393170\pi$
$432$	2.31826	0.111537
$433$	−34.7905	−1.67192	−0.835961	−	0.548788i	$-0.815089\pi$
$433$	−34.7905	−1.67192	−0.835961	+	0.548788i	$0.815089\pi$
$434$	−17.8701	−0.857793
$435$	0	0
$436$	−19.3244	−0.925472
$437$	3.65400	0.174794
$438$	−6.04777	−0.288974
$439$	−17.0869	−0.815513	−0.407757	−	0.913091i	$-0.633689\pi$
$439$	−17.0869	−0.815513	−0.407757	+	0.913091i	$0.633689\pi$
$440$	0	0
$441$	−4.70203	−0.223906
$442$	−3.53675	−0.168226
$443$	−10.2997	−0.489351	−0.244676	−	0.969605i	$-0.578682\pi$
$443$	−10.2997	−0.489351	−0.244676	+	0.969605i	$0.578682\pi$
$444$	19.6236	0.931297
$445$	0	0
$446$	34.0830	1.61388
$447$	−0.474203	−0.0224290
$448$	18.2341	0.861482
$449$	15.3331	0.723615	0.361808	−	0.932253i	$-0.382160\pi$
$449$	15.3331	0.723615	0.361808	+	0.932253i	$0.382160\pi$
$450$	0	0
$451$	15.6453	0.736709
$452$	24.8431	1.16852
$453$	12.5209	0.588281
$454$	−17.3141	−0.812592
$455$	0	0
$456$	3.91494	0.183334
$457$	−10.1841	−0.476394	−0.238197	−	0.971217i	$-0.576556\pi$
$457$	−10.1841	−0.476394	−0.238197	+	0.971217i	$0.576556\pi$
$458$	0.203795	0.00952272
$459$	3.39257	0.158352
$460$	0	0
$461$	−37.2567	−1.73522	−0.867609	−	0.497247i	$-0.834344\pi$
$461$	−37.2567	−1.73522	−0.867609	+	0.497247i	$0.834344\pi$
$462$	−6.16664	−0.286898
$463$	24.2457	1.12680	0.563398	−	0.826186i	$-0.309494\pi$
$463$	24.2457	1.12680	0.563398	+	0.826186i	$0.309494\pi$
$464$	−2.31826	−0.107623
$465$	0	0
$466$	40.1799	1.86130
$467$	28.5544	1.32134	0.660670	−	0.750676i	$-0.270272\pi$
$467$	28.5544	1.32134	0.660670	+	0.750676i	$0.270272\pi$
$468$	−1.27684	−0.0590221
$469$	−13.2891	−0.613636
$470$	0	0
$471$	−23.8828	−1.10046
$472$	17.5626	0.808385
$473$	−2.70620	−0.124431
$474$	−20.1271	−0.924466
$475$	0	0
$476$	13.5647	0.621738
$477$	5.62364	0.257489
$478$	54.5362	2.49443
$479$	20.8455	0.952455	0.476227	−	0.879322i	$-0.342004\pi$
$479$	20.8455	0.952455	0.476227	+	0.879322i	$0.342004\pi$
$480$	0	0
$481$	3.60163	0.164220
$482$	61.4844	2.80054
$483$	1.94273	0.0883972
$484$	−19.6019	−0.890996
$485$	0	0
$486$	2.15351	0.0976852
$487$	5.31743	0.240956	0.120478	−	0.992716i	$-0.461557\pi$
$487$	5.31743	0.240956	0.120478	+	0.992716i	$0.461557\pi$
$488$	−10.1630	−0.460058
$489$	14.7254	0.665907
$490$	0	0
$491$	34.6711	1.56469	0.782343	−	0.622848i	$-0.214025\pi$
$491$	34.6711	1.56469	0.782343	+	0.622848i	$0.214025\pi$
$492$	−21.8456	−0.984876
$493$	−3.39257	−0.152794
$494$	2.97238	0.133734
$495$	0	0
$496$	12.6902	0.569809
$497$	−22.0875	−0.990758
$498$	−3.85838	−0.172898
$499$	−37.2788	−1.66883	−0.834414	−	0.551138i	$-0.814194\pi$
$499$	−37.2788	−1.66883	−0.834414	+	0.551138i	$0.814194\pi$
$500$	0	0
$501$	2.95547	0.132041
$502$	35.5226	1.58545
$503$	−10.7589	−0.479717	−0.239858	−	0.970808i	$-0.577101\pi$
$503$	−10.7589	−0.479717	−0.239858	+	0.970808i	$0.577101\pi$
$504$	2.08146	0.0927158
$505$	0	0
$506$	−5.21333	−0.231761
$507$	12.7657	0.566943
$508$	−5.87277	−0.260562
$509$	−0.424406	−0.0188115	−0.00940573	−	0.999956i	$-0.502994\pi$
$509$	−0.424406	−0.0188115	−0.00940573	+	0.999956i	$0.502994\pi$
$510$	0	0
$511$	4.25717	0.188326
$512$	−24.3063	−1.07420
$513$	−2.85121	−0.125884
$514$	29.6450	1.30759
$515$	0	0
$516$	3.77868	0.166347
$517$	4.25642	0.187197
$518$	−24.2879	−1.06715
$519$	−12.1974	−0.535405
$520$	0	0
$521$	6.39104	0.279996	0.139998	−	0.990152i	$-0.455290\pi$
$521$	6.39104	0.279996	0.139998	+	0.990152i	$0.455290\pi$
$522$	−2.15351	−0.0942565
$523$	−24.8564	−1.08689	−0.543446	−	0.839444i	$-0.682881\pi$
$523$	−24.8564	−1.08689	−0.543446	+	0.839444i	$0.682881\pi$
$524$	40.5048	1.76946
$525$	0	0
$526$	−35.9721	−1.56846
$527$	18.5711	0.808968
$528$	4.37917	0.190579
$529$	−21.3576	−0.928591
$530$	0	0
$531$	−12.7907	−0.555067
$532$	−11.4002	−0.494260
$533$	−4.00944	−0.173668
$534$	15.6065	0.675361
$535$	0	0
$536$	−12.0371	−0.519922
$537$	12.3747	0.534008
$538$	37.3673	1.61102
$539$	−8.88207	−0.382578
$540$	0	0
$541$	13.2290	0.568759	0.284379	−	0.958712i	$-0.408212\pi$
$541$	13.2290	0.568759	0.284379	+	0.958712i	$0.408212\pi$
$542$	−17.8256	−0.765676
$543$	13.1895	0.566015
$544$	−26.2536	−1.12561
$545$	0	0
$546$	1.58033	0.0676319
$547$	−43.7132	−1.86904	−0.934521	−	0.355908i	$-0.884172\pi$
$547$	−43.7132	−1.86904	−0.934521	+	0.355908i	$0.884172\pi$
$548$	−4.54334	−0.194082
$549$	7.40162	0.315893
$550$	0	0
$551$	2.85121	0.121466
$552$	1.75969	0.0748973
$553$	14.1679	0.602481
$554$	38.8837	1.65201
$555$	0	0
$556$	−17.6269	−0.747547
$557$	−18.7407	−0.794067	−0.397033	−	0.917804i	$-0.629960\pi$
$557$	−18.7407	−0.794067	−0.397033	+	0.917804i	$0.629960\pi$
$558$	11.7884	0.499042
$559$	0.693521	0.0293328
$560$	0	0
$561$	6.40853	0.270568
$562$	40.1071	1.69182
$563$	−6.52697	−0.275079	−0.137539	−	0.990496i	$-0.543919\pi$
$563$	−6.52697	−0.275079	−0.137539	+	0.990496i	$0.543919\pi$
$564$	−5.94325	−0.250256
$565$	0	0
$566$	35.9463	1.51094
$567$	−1.51591	−0.0636621
$568$	−20.0064	−0.839451
$569$	24.4702	1.02585	0.512923	−	0.858435i	$-0.328563\pi$
$569$	24.4702	1.02585	0.512923	+	0.858435i	$0.328563\pi$
$570$	0	0
$571$	45.1164	1.88806	0.944031	−	0.329856i	$-0.107000\pi$
$571$	45.1164	1.88806	0.944031	+	0.329856i	$0.107000\pi$
$572$	−2.41194	−0.100848
$573$	6.10636	0.255097
$574$	27.0380	1.12854
$575$	0	0
$576$	−12.0285	−0.501189
$577$	−25.4834	−1.06089	−0.530444	−	0.847720i	$-0.677975\pi$
$577$	−25.4834	−1.06089	−0.530444	+	0.847720i	$0.677975\pi$
$578$	11.8237	0.491803
$579$	19.6580	0.816958
$580$	0	0
$581$	2.71600	0.112679
$582$	−9.89560	−0.410186
$583$	10.6230	0.439959
$584$	3.85607	0.159565
$585$	0	0
$586$	71.0370	2.93451
$587$	42.3930	1.74975	0.874873	−	0.484352i	$-0.160945\pi$
$587$	42.3930	1.74975	0.874873	+	0.484352i	$0.160945\pi$
$588$	12.4021	0.511453
$589$	−15.6076	−0.643101
$590$	0	0
$591$	24.2075	0.995764
$592$	17.2477	0.708878
$593$	−5.71303	−0.234606	−0.117303	−	0.993096i	$-0.537425\pi$
$593$	−5.71303	−0.234606	−0.117303	+	0.993096i	$0.537425\pi$
$594$	4.06795	0.166910
$595$	0	0
$596$	1.25076	0.0512331
$597$	−0.542480	−0.0222022
$598$	1.33603	0.0546341
$599$	−14.0026	−0.572132	−0.286066	−	0.958210i	$-0.592348\pi$
$599$	−14.0026	−0.572132	−0.286066	+	0.958210i	$0.592348\pi$
$600$	0	0
$601$	−22.6834	−0.925277	−0.462639	−	0.886547i	$-0.653097\pi$
$601$	−22.6834	−0.925277	−0.462639	+	0.886547i	$0.653097\pi$
$602$	−4.67682	−0.190613
$603$	8.76646	0.356998
$604$	−33.0250	−1.34377
$605$	0	0
$606$	−34.1569	−1.38753
$607$	27.7581	1.12667	0.563333	−	0.826230i	$-0.309519\pi$
$607$	27.7581	1.12667	0.563333	+	0.826230i	$0.309519\pi$
$608$	22.0642	0.894823
$609$	1.51591	0.0614277
$610$	0	0
$611$	−1.09080	−0.0441289
$612$	−8.94826	−0.361712
$613$	12.0619	0.487176	0.243588	−	0.969879i	$-0.421675\pi$
$613$	12.0619	0.487176	0.243588	+	0.969879i	$0.421675\pi$
$614$	−11.2095	−0.452378
$615$	0	0
$616$	3.93186	0.158419
$617$	−45.0988	−1.81561	−0.907804	−	0.419394i	$-0.862243\pi$
$617$	−45.0988	−1.81561	−0.907804	+	0.419394i	$0.862243\pi$
$618$	18.2401	0.733725
$619$	−35.5206	−1.42769	−0.713846	−	0.700302i	$-0.753049\pi$
$619$	−35.5206	−1.42769	−0.713846	+	0.700302i	$0.753049\pi$
$620$	0	0
$621$	−1.28156	−0.0514273
$622$	−32.6854	−1.31057
$623$	−10.9858	−0.440138
$624$	−1.12225	−0.0449261
$625$	0	0
$626$	−14.8917	−0.595192
$627$	−5.38590	−0.215092
$628$	62.9934	2.51371
$629$	25.2406	1.00641
$630$	0	0
$631$	−32.0402	−1.27550	−0.637750	−	0.770243i	$-0.720135\pi$
$631$	−32.0402	−1.27550	−0.637750	+	0.770243i	$0.720135\pi$
$632$	12.8330	0.510471
$633$	1.20999	0.0480927
$634$	53.1258	2.10989
$635$	0	0
$636$	−14.8329	−0.588164
$637$	2.27622	0.0901870
$638$	−4.06795	−0.161052
$639$	14.5705	0.576398
$640$	0	0
$641$	−28.4963	−1.12554	−0.562769	−	0.826614i	$-0.690264\pi$
$641$	−28.4963	−1.12554	−0.562769	+	0.826614i	$0.690264\pi$
$642$	−0.959394	−0.0378642
$643$	43.1625	1.70216	0.851081	−	0.525035i	$-0.175948\pi$
$643$	43.1625	1.70216	0.851081	+	0.525035i	$0.175948\pi$
$644$	−5.12414	−0.201920
$645$	0	0
$646$	20.8307	0.819575
$647$	−32.5875	−1.28115	−0.640573	−	0.767898i	$-0.721303\pi$
$647$	−32.5875	−1.28115	−0.640573	+	0.767898i	$0.721303\pi$
$648$	−1.37308	−0.0539397
$649$	−24.1614	−0.948418
$650$	0	0
$651$	−8.29813	−0.325229
$652$	−38.8398	−1.52109
$653$	−28.1951	−1.10336	−0.551679	−	0.834057i	$-0.686013\pi$
$653$	−28.1951	−1.10336	−0.551679	+	0.834057i	$0.686013\pi$
$654$	−15.7777	−0.616957
$655$	0	0
$656$	−19.2007	−0.749661
$657$	−2.80833	−0.109564
$658$	7.35587	0.286762
$659$	42.7564	1.66555	0.832776	−	0.553611i	$-0.186750\pi$
$659$	42.7564	1.66555	0.832776	+	0.553611i	$0.186750\pi$
$660$	0	0
$661$	17.4726	0.679606	0.339803	−	0.940497i	$-0.389640\pi$
$661$	17.4726	0.679606	0.339803	+	0.940497i	$0.389640\pi$
$662$	9.13155	0.354908
$663$	−1.64232	−0.0637824
$664$	2.46011	0.0954707
$665$	0	0
$666$	16.0220	0.620840
$667$	1.28156	0.0496223
$668$	−7.79535	−0.301611
$669$	15.8267	0.611896
$670$	0	0
$671$	13.9816	0.539753
$672$	11.7309	0.452531
$673$	−12.5821	−0.485006	−0.242503	−	0.970151i	$-0.577968\pi$
$673$	−12.5821	−0.485006	−0.242503	+	0.970151i	$0.577968\pi$
$674$	−56.5891	−2.17973
$675$	0	0
$676$	−33.6707	−1.29503
$677$	−27.7829	−1.06778	−0.533892	−	0.845553i	$-0.679271\pi$
$677$	−27.7829	−1.06778	−0.533892	+	0.845553i	$0.679271\pi$
$678$	20.2836	0.778985
$679$	6.96575	0.267321
$680$	0	0
$681$	−8.03996	−0.308092
$682$	22.2681	0.852691
$683$	−41.1587	−1.57489	−0.787446	−	0.616383i	$-0.788597\pi$
$683$	−41.1587	−1.57489	−0.787446	+	0.616383i	$0.788597\pi$
$684$	7.52035	0.287548
$685$	0	0
$686$	−38.2015	−1.45854
$687$	0.0946339	0.00361051
$688$	3.32119	0.126619
$689$	−2.72236	−0.103714
$690$	0	0
$691$	−6.51859	−0.247979	−0.123989	−	0.992284i	$-0.539569\pi$
$691$	−6.51859	−0.247979	−0.123989	+	0.992284i	$0.539569\pi$
$692$	32.1718	1.22299
$693$	−2.86353	−0.108777
$694$	46.8916	1.77998
$695$	0	0
$696$	1.37308	0.0520465
$697$	−28.0985	−1.06431
$698$	−19.2698	−0.729371
$699$	18.6579	0.705706
$700$	0	0
$701$	3.82210	0.144359	0.0721793	−	0.997392i	$-0.477005\pi$
$701$	3.82210	0.144359	0.0721793	+	0.997392i	$0.477005\pi$
$702$	−1.04250	−0.0393466
$703$	−21.2129	−0.800058
$704$	−22.7218	−0.856359
$705$	0	0
$706$	15.3964	0.579450
$707$	24.0439	0.904264
$708$	33.7367	1.26790
$709$	4.67767	0.175673	0.0878367	−	0.996135i	$-0.472005\pi$
$709$	4.67767	0.175673	0.0878367	+	0.996135i	$0.472005\pi$
$710$	0	0
$711$	−9.34617	−0.350509
$712$	−9.95076	−0.372920
$713$	−7.01532	−0.262726
$714$	11.0751	0.414476
$715$	0	0
$716$	−32.6395	−1.21980
$717$	25.3243	0.945754
$718$	−67.7582	−2.52871
$719$	26.0976	0.973276	0.486638	−	0.873604i	$-0.338223\pi$
$719$	26.0976	0.973276	0.486638	+	0.873604i	$0.338223\pi$
$720$	0	0
$721$	−12.8397	−0.478174
$722$	23.4100	0.871228
$723$	28.5508	1.06182
$724$	−34.7886	−1.29291
$725$	0	0
$726$	−16.0043	−0.593975
$727$	42.0811	1.56070	0.780352	−	0.625341i	$-0.215040\pi$
$727$	42.0811	1.56070	0.780352	+	0.625341i	$0.215040\pi$
$728$	−1.00762	−0.0373449
$729$	1.00000	0.0370370
$730$	0	0
$731$	4.86027	0.179764
$732$	−19.5225	−0.721573
$733$	−5.06122	−0.186940	−0.0934702	−	0.995622i	$-0.529796\pi$
$733$	−5.06122	−0.186940	−0.0934702	+	0.995622i	$0.529796\pi$
$734$	−59.8077	−2.20754
$735$	0	0
$736$	9.91744	0.365562
$737$	16.5598	0.609986
$738$	−17.8362	−0.656558
$739$	−26.3474	−0.969203	−0.484601	−	0.874735i	$-0.661035\pi$
$739$	−26.3474	−0.969203	−0.484601	+	0.874735i	$0.661035\pi$
$740$	0	0
$741$	1.38025	0.0507047
$742$	18.3585	0.673961
$743$	−1.17798	−0.0432158	−0.0216079	−	0.999767i	$-0.506879\pi$
$743$	−1.17798	−0.0432158	−0.0216079	+	0.999767i	$0.506879\pi$
$744$	−7.51630	−0.275561
$745$	0	0
$746$	−34.3040	−1.25596
$747$	−1.79167	−0.0655537
$748$	−16.9032	−0.618041
$749$	0.675340	0.0246764
$750$	0	0
$751$	−18.1100	−0.660844	−0.330422	−	0.943833i	$-0.607191\pi$
$751$	−18.1100	−0.660844	−0.330422	+	0.943833i	$0.607191\pi$
$752$	−5.22369	−0.190488
$753$	16.4952	0.601119
$754$	1.04250	0.0379655
$755$	0	0
$756$	3.99836	0.145419
$757$	−4.02431	−0.146266	−0.0731331	−	0.997322i	$-0.523300\pi$
$757$	−4.02431	−0.146266	−0.0731331	+	0.997322i	$0.523300\pi$
$758$	−39.6635	−1.44064
$759$	−2.42086	−0.0878715
$760$	0	0
$761$	35.2780	1.27883	0.639414	−	0.768863i	$-0.279177\pi$
$761$	35.2780	1.27883	0.639414	+	0.768863i	$0.279177\pi$
$762$	−4.79491	−0.173701
$763$	11.1063	0.402075
$764$	−16.1062	−0.582700
$765$	0	0
$766$	−30.5740	−1.10468
$767$	6.19186	0.223575
$768$	−1.60362	−0.0578658
$769$	37.6829	1.35888	0.679440	−	0.733731i	$-0.262223\pi$
$769$	37.6829	1.35888	0.679440	+	0.733731i	$0.262223\pi$
$770$	0	0
$771$	13.7659	0.495767
$772$	−51.8500	−1.86612
$773$	−25.4551	−0.915558	−0.457779	−	0.889066i	$-0.651355\pi$
$773$	−25.4551	−0.915558	−0.457779	+	0.889066i	$0.651355\pi$
$774$	3.08516	0.110894
$775$	0	0
$776$	6.30945	0.226496
$777$	−11.2783	−0.404606
$778$	53.8564	1.93085
$779$	23.6148	0.846087
$780$	0	0
$781$	27.5234	0.984866
$782$	9.36301	0.334820
$783$	−1.00000	−0.0357371
$784$	10.9005	0.389304
$785$	0	0
$786$	33.0707	1.17959
$787$	27.0582	0.964521	0.482261	−	0.876028i	$-0.339816\pi$
$787$	27.0582	0.964521	0.482261	+	0.876028i	$0.339816\pi$
$788$	−63.8498	−2.27455
$789$	−16.7039	−0.594676
$790$	0	0
$791$	−14.2781	−0.507670
$792$	−2.59374	−0.0921644
$793$	−3.58307	−0.127238
$794$	−7.91741	−0.280978
$795$	0	0
$796$	1.43085	0.0507150
$797$	43.5501	1.54262	0.771312	−	0.636457i	$-0.219601\pi$
$797$	43.5501	1.54262	0.771312	+	0.636457i	$0.219601\pi$
$798$	−9.30783	−0.329494
$799$	−7.64441	−0.270440
$800$	0	0
$801$	7.24703	0.256061
$802$	10.0892	0.356262
$803$	−5.30491	−0.187206
$804$	−23.1224	−0.815466
$805$	0	0
$806$	−5.70667	−0.201009
$807$	17.3518	0.610812
$808$	21.7785	0.766166
$809$	32.0173	1.12567	0.562834	−	0.826570i	$-0.309711\pi$
$809$	32.0173	1.12567	0.562834	+	0.826570i	$0.309711\pi$
$810$	0	0
$811$	4.19698	0.147376	0.0736880	−	0.997281i	$-0.476523\pi$
$811$	4.19698	0.147376	0.0736880	+	0.997281i	$0.476523\pi$
$812$	−3.99836	−0.140315
$813$	−8.27747	−0.290304
$814$	30.2654	1.06080
$815$	0	0
$816$	−7.86487	−0.275325
$817$	−4.08470	−0.142906
$818$	−79.4210	−2.77689
$819$	0.733840	0.0256424
$820$	0	0
$821$	47.5461	1.65937	0.829686	−	0.558230i	$-0.188519\pi$
$821$	47.5461	1.65937	0.829686	+	0.558230i	$0.188519\pi$
$822$	−3.70948	−0.129383
$823$	−13.3520	−0.465422	−0.232711	−	0.972546i	$-0.574760\pi$
$823$	−13.3520	−0.465422	−0.232711	+	0.972546i	$0.574760\pi$
$824$	−11.6299	−0.405148
$825$	0	0
$826$	−41.7554	−1.45285
$827$	1.07408	0.0373493	0.0186747	−	0.999826i	$-0.494055\pi$
$827$	1.07408	0.0373493	0.0186747	+	0.999826i	$0.494055\pi$
$828$	3.38025	0.117472
$829$	−53.3629	−1.85337	−0.926684	−	0.375840i	$-0.877354\pi$
$829$	−53.3629	−1.85337	−0.926684	+	0.375840i	$0.877354\pi$
$830$	0	0
$831$	18.0560	0.626355
$832$	5.82293	0.201874
$833$	15.9520	0.552703
$834$	−14.3917	−0.498345
$835$	0	0
$836$	14.2059	0.491320
$837$	5.47404	0.189210
$838$	−41.2182	−1.42386
$839$	1.82410	0.0629751	0.0314875	−	0.999504i	$-0.489976\pi$
$839$	1.82410	0.0629751	0.0314875	+	0.999504i	$0.489976\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	−63.5531	−2.19019
$843$	18.6241	0.641448
$844$	−3.19146	−0.109855
$845$	0	0
$846$	−4.85246	−0.166831
$847$	11.2658	0.387097
$848$	−13.0371	−0.447694
$849$	16.6920	0.572867
$850$	0	0
$851$	−9.53476	−0.326847
$852$	−38.4311	−1.31663
$853$	26.3499	0.902203	0.451102	−	0.892473i	$-0.351031\pi$
$853$	26.3499	0.902203	0.451102	+	0.892473i	$0.351031\pi$
$854$	24.1627	0.826831
$855$	0	0
$856$	0.611711	0.0209079
$857$	−39.7239	−1.35694	−0.678472	−	0.734627i	$-0.737357\pi$
$857$	−39.7239	−1.35694	−0.678472	+	0.734627i	$0.737357\pi$
$858$	−1.96927	−0.0672297
$859$	−39.1847	−1.33696	−0.668482	−	0.743729i	$-0.733055\pi$
$859$	−39.1847	−1.33696	−0.668482	+	0.743729i	$0.733055\pi$
$860$	0	0
$861$	12.5553	0.427884
$862$	−29.4493	−1.00305
$863$	−23.6601	−0.805399	−0.402699	−	0.915332i	$-0.631928\pi$
$863$	−23.6601	−0.805399	−0.402699	+	0.915332i	$0.631928\pi$
$864$	−7.73856	−0.263271
$865$	0	0
$866$	74.9216	2.54594
$867$	5.49045	0.186465
$868$	21.8872	0.742899
$869$	−17.6548	−0.598898
$870$	0	0
$871$	−4.24378	−0.143795
$872$	10.0599	0.340671
$873$	−4.59510	−0.155521
$874$	−7.86892	−0.266170
$875$	0	0
$876$	7.40727	0.250268
$877$	52.7791	1.78222	0.891112	−	0.453783i	$-0.149926\pi$
$877$	52.7791	1.78222	0.891112	+	0.453783i	$0.149926\pi$
$878$	36.7968	1.24183
$879$	32.9866	1.11261
$880$	0	0
$881$	−15.0520	−0.507115	−0.253558	−	0.967320i	$-0.581601\pi$
$881$	−15.0520	−0.507115	−0.253558	+	0.967320i	$0.581601\pi$
$882$	10.1259	0.340955
$883$	−37.7972	−1.27198	−0.635989	−	0.771698i	$-0.719408\pi$
$883$	−37.7972	−1.27198	−0.635989	+	0.771698i	$0.719408\pi$
$884$	4.33179	0.145694
$885$	0	0
$886$	22.1804	0.745165
$887$	4.81353	0.161622	0.0808112	−	0.996729i	$-0.474249\pi$
$887$	4.81353	0.161622	0.0808112	+	0.996729i	$0.474249\pi$
$888$	−10.2157	−0.342815
$889$	3.37526	0.113202
$890$	0	0
$891$	1.88899	0.0632835
$892$	−41.7446	−1.39771
$893$	6.42457	0.214990
$894$	1.02120	0.0341541
$895$	0	0
$896$	−15.8055	−0.528026
$897$	0.620395	0.0207144
$898$	−33.0201	−1.10189
$899$	−5.47404	−0.182569
$900$	0	0
$901$	−19.0786	−0.635600
$902$	−33.6923	−1.12183
$903$	−2.17172	−0.0722703
$904$	−12.9328	−0.430140
$905$	0	0
$906$	−26.9638	−0.895812
$907$	−43.5796	−1.44704	−0.723518	−	0.690305i	$-0.757476\pi$
$907$	−43.5796	−1.44704	−0.723518	+	0.690305i	$0.757476\pi$
$908$	21.2062	0.703753
$909$	−15.8611	−0.526078
$910$	0	0
$911$	−15.9296	−0.527770	−0.263885	−	0.964554i	$-0.585004\pi$
$911$	−15.9296	−0.527770	−0.263885	+	0.964554i	$0.585004\pi$
$912$	6.60984	0.218874
$913$	−3.38444	−0.112009
$914$	21.9316	0.725434
$915$	0	0
$916$	−0.249607	−0.00824724
$917$	−23.2793	−0.768750
$918$	−7.30594	−0.241132
$919$	44.8752	1.48030	0.740148	−	0.672444i	$-0.234755\pi$
$919$	44.8752	1.48030	0.740148	+	0.672444i	$0.234755\pi$
$920$	0	0
$921$	−5.20522	−0.171518
$922$	80.2327	2.64232
$923$	−7.05345	−0.232167
$924$	7.55286	0.248471
$925$	0	0
$926$	−52.2134	−1.71584
$927$	8.46995	0.278190
$928$	7.73856	0.254031
$929$	−34.3242	−1.12614	−0.563071	−	0.826409i	$-0.690380\pi$
$929$	−34.3242	−1.12614	−0.563071	+	0.826409i	$0.690380\pi$
$930$	0	0
$931$	−13.4065	−0.439379
$932$	−49.2121	−1.61200
$933$	−15.1777	−0.496897
$934$	−61.4922	−2.01209
$935$	0	0
$936$	0.664699	0.0217264
$937$	−21.0494	−0.687653	−0.343827	−	0.939033i	$-0.611723\pi$
$937$	−21.0494	−0.687653	−0.343827	+	0.939033i	$0.611723\pi$
$938$	28.6183	0.934421
$939$	−6.91509	−0.225665
$940$	0	0
$941$	46.2494	1.50769	0.753844	−	0.657053i	$-0.228197\pi$
$941$	46.2494	1.50769	0.753844	+	0.657053i	$0.228197\pi$
$942$	51.4319	1.67574
$943$	10.6144	0.345651
$944$	29.6521	0.965093
$945$	0	0
$946$	5.82784	0.189479
$947$	21.7174	0.705720	0.352860	−	0.935676i	$-0.385209\pi$
$947$	21.7174	0.705720	0.352860	+	0.935676i	$0.385209\pi$
$948$	24.6515	0.800642
$949$	1.35949	0.0441310
$950$	0	0
$951$	24.6694	0.799960
$952$	−7.06152	−0.228865
$953$	16.6122	0.538120	0.269060	−	0.963123i	$-0.413287\pi$
$953$	16.6122	0.538120	0.269060	+	0.963123i	$0.413287\pi$
$954$	−12.1106	−0.392094
$955$	0	0
$956$	−66.7955	−2.16032
$957$	−1.88899	−0.0610623
$958$	−44.8910	−1.45036
$959$	2.61119	0.0843198
$960$	0	0
$961$	−1.03492	−0.0333845
$962$	−7.75614	−0.250068
$963$	−0.445502	−0.0143561
$964$	−75.3056	−2.42543
$965$	0	0
$966$	−4.18368	−0.134608
$967$	−12.1886	−0.391960	−0.195980	−	0.980608i	$-0.562789\pi$
$967$	−12.1886	−0.391960	−0.195980	+	0.980608i	$0.562789\pi$
$968$	10.2044	0.327981
$969$	9.67293	0.310739
$970$	0	0
$971$	−10.0067	−0.321131	−0.160566	−	0.987025i	$-0.551332\pi$
$971$	−10.0067	−0.321131	−0.160566	+	0.987025i	$0.551332\pi$
$972$	−2.63760	−0.0846011
$973$	10.1307	0.324775
$974$	−11.4511	−0.366918
$975$	0	0
$976$	−17.1589	−0.549242
$977$	−23.0188	−0.736435	−0.368218	−	0.929740i	$-0.620032\pi$
$977$	−23.0188	−0.736435	−0.368218	+	0.929740i	$0.620032\pi$
$978$	−31.7114	−1.01402
$979$	13.6895	0.437520
$980$	0	0
$981$	−7.32651	−0.233917
$982$	−74.6645	−2.38264
$983$	−54.9480	−1.75257	−0.876285	−	0.481793i	$-0.839986\pi$
$983$	−54.9480	−1.75257	−0.876285	+	0.481793i	$0.839986\pi$
$984$	11.3724	0.362538
$985$	0	0
$986$	7.30594	0.232668
$987$	3.41576	0.108725
$988$	−3.64055	−0.115821
$989$	−1.83599	−0.0583811
$990$	0	0
$991$	−8.69215	−0.276115	−0.138058	−	0.990424i	$-0.544086\pi$
$991$	−8.69215	−0.276115	−0.138058	+	0.990424i	$0.544086\pi$
$992$	−42.3612	−1.34497
$993$	4.24031	0.134562
$994$	47.5656	1.50869
$995$	0	0
$996$	4.72571	0.149740
$997$	18.8275	0.596274	0.298137	−	0.954523i	$-0.403635\pi$
$997$	18.8275	0.596274	0.298137	+	0.954523i	$0.403635\pi$
$998$	80.2802	2.54123
$999$	7.43995	0.235390

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2175.2.a.w.1.2		5
3.2	odd	2		6525.2.a.bs.1.4		5
5.2	odd	4		435.2.c.e.349.2	✓	10
5.3	odd	4		435.2.c.e.349.9	yes	10
5.4	even	2		2175.2.a.z.1.4		5
15.2	even	4		1305.2.c.j.784.9		10
15.8	even	4		1305.2.c.j.784.2		10
15.14	odd	2		6525.2.a.bl.1.2		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
435.2.c.e.349.2	✓	10	5.2	odd	4
435.2.c.e.349.9	yes	10	5.3	odd	4
1305.2.c.j.784.2		10	15.8	even	4
1305.2.c.j.784.9		10	15.2	even	4
2175.2.a.w.1.2		5	1.1	even	1	trivial
2175.2.a.z.1.4		5	5.4	even	2
6525.2.a.bl.1.2		5	15.14	odd	2
6525.2.a.bs.1.4		5	3.2	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:	\( N \)	\(=\)	\( 2175 = 3 \cdot 5^{2} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2175.a (trivial)

\(f(q)\)	\(=\)	\(q-2.15351 q^{2} -1.00000 q^{3} +2.63760 q^{4} +2.15351 q^{6} -1.51591 q^{7} -1.37308 q^{8} +1.00000 q^{9} +1.88899 q^{11} -2.63760 q^{12} -0.484093 q^{13} +3.26452 q^{14} -2.31826 q^{16} -3.39257 q^{17} -2.15351 q^{18} +2.85121 q^{19} +1.51591 q^{21} -4.06795 q^{22} +1.28156 q^{23} +1.37308 q^{24} +1.04250 q^{26} -1.00000 q^{27} -3.99836 q^{28} +1.00000 q^{29} -5.47404 q^{31} +7.73856 q^{32} -1.88899 q^{33} +7.30594 q^{34} +2.63760 q^{36} -7.43995 q^{37} -6.14010 q^{38} +0.484093 q^{39} +8.28237 q^{41} -3.26452 q^{42} -1.43262 q^{43} +4.98240 q^{44} -2.75985 q^{46} +2.25328 q^{47} +2.31826 q^{48} -4.70203 q^{49} +3.39257 q^{51} -1.27684 q^{52} +5.62364 q^{53} +2.15351 q^{54} +2.08146 q^{56} -2.85121 q^{57} -2.15351 q^{58} -12.7907 q^{59} +7.40162 q^{61} +11.7884 q^{62} -1.51591 q^{63} -12.0285 q^{64} +4.06795 q^{66} +8.76646 q^{67} -8.94826 q^{68} -1.28156 q^{69} +14.5705 q^{71} -1.37308 q^{72} -2.80833 q^{73} +16.0220 q^{74} +7.52035 q^{76} -2.86353 q^{77} -1.04250 q^{78} -9.34617 q^{79} +1.00000 q^{81} -17.8362 q^{82} -1.79167 q^{83} +3.99836 q^{84} +3.08516 q^{86} -1.00000 q^{87} -2.59374 q^{88} +7.24703 q^{89} +0.733840 q^{91} +3.38025 q^{92} +5.47404 q^{93} -4.85246 q^{94} -7.73856 q^{96} -4.59510 q^{97} +10.1259 q^{98} +1.88899 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 3 q^{2} - 5 q^{3} + 5 q^{4} + 3 q^{6} - 8 q^{7} - 9 q^{8} + 5 q^{9} + 12 q^{11} - 5 q^{12} - 2 q^{13} + 6 q^{14} + q^{16} - 3 q^{18} - 2 q^{19} + 8 q^{21} - 14 q^{22} - 8 q^{23} + 9 q^{24} - 5 q^{27}+ \cdots + 12 q^{99}+O(q^{100}) \) 5 * q - 3 * q^2 - 5 * q^3 + 5 * q^4 + 3 * q^6 - 8 * q^7 - 9 * q^8 + 5 * q^9 + 12 * q^11 - 5 * q^12 - 2 * q^13 + 6 * q^14 + q^16 - 3 * q^18 - 2 * q^19 + 8 * q^21 - 14 * q^22 - 8 * q^23 + 9 * q^24 - 5 * q^27 + 6 * q^28 + 5 * q^29 + 2 * q^31 - q^32 - 12 * q^33 + 4 * q^34 + 5 * q^36 - 16 * q^37 + 14 * q^38 + 2 * q^39 - 14 * q^41 - 6 * q^42 + 20 * q^44 - 6 * q^46 - 2 * q^47 - q^48 - 7 * q^49 - 16 * q^52 - 26 * q^53 + 3 * q^54 - 2 * q^56 + 2 * q^57 - 3 * q^58 + 4 * q^59 - 12 * q^61 - 8 * q^63 - 9 * q^64 + 14 * q^66 - 12 * q^67 + 20 * q^68 + 8 * q^69 + 30 * q^71 - 9 * q^72 + 12 * q^73 + 2 * q^74 - 44 * q^76 - 18 * q^77 - 18 * q^79 + 5 * q^81 - 10 * q^82 - 2 * q^83 - 6 * q^84 + 30 * q^86 - 5 * q^87 - 42 * q^88 - 22 * q^89 - 12 * q^91 - 20 * q^92 - 2 * q^93 - 50 * q^94 + q^96 - 20 * q^97 + 9 * q^98 + 12 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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