LMFDB - Embedded newform 2175.2.a.bb.1.3

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:	$0$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 2x^{6} - 10x^{5} + 19x^{4} + 24x^{3} - 44x^{2} - 3x + 14 $ x^7 - 2x^6 - 10x^5 + 19x^4 + 24x^3 - 44x^2 - 3x + 14
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-0.560139$ 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)

$f(q)$	$=$	\(q-0.560139 q^{2} +1.00000 q^{3} -1.68624 q^{4} -0.560139 q^{6} +4.36313 q^{7} +2.06481 q^{8} +1.00000 q^{9} -3.72689 q^{11} -1.68624 q^{12} -2.53237 q^{13} -2.44396 q^{14} +2.21591 q^{16} +5.11270 q^{17} -0.560139 q^{18} +2.09916 q^{19} +4.36313 q^{21} +2.08758 q^{22} +6.47404 q^{23} +2.06481 q^{24} +1.41848 q^{26} +1.00000 q^{27} -7.35730 q^{28} -1.00000 q^{29} -9.94570 q^{31} -5.37083 q^{32} -3.72689 q^{33} -2.86382 q^{34} -1.68624 q^{36} +5.37249 q^{37} -1.17582 q^{38} -2.53237 q^{39} +7.96557 q^{41} -2.44396 q^{42} -10.7771 q^{43} +6.28445 q^{44} -3.62636 q^{46} +9.38424 q^{47} +2.21591 q^{48} +12.0369 q^{49} +5.11270 q^{51} +4.27019 q^{52} -5.64265 q^{53} -0.560139 q^{54} +9.00902 q^{56} +2.09916 q^{57} +0.560139 q^{58} +4.24056 q^{59} +9.81848 q^{61} +5.57097 q^{62} +4.36313 q^{63} -1.42341 q^{64} +2.08758 q^{66} +0.520614 q^{67} -8.62126 q^{68} +6.47404 q^{69} +3.22415 q^{71} +2.06481 q^{72} -10.1718 q^{73} -3.00934 q^{74} -3.53970 q^{76} -16.2609 q^{77} +1.41848 q^{78} +1.81484 q^{79} +1.00000 q^{81} -4.46182 q^{82} -1.35377 q^{83} -7.35730 q^{84} +6.03667 q^{86} -1.00000 q^{87} -7.69532 q^{88} +14.2314 q^{89} -11.0490 q^{91} -10.9168 q^{92} -9.94570 q^{93} -5.25648 q^{94} -5.37083 q^{96} +14.0094 q^{97} -6.74233 q^{98} -3.72689 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + 2 q^{2} + 7 q^{3} + 10 q^{4} + 2 q^{6} - q^{7} + 3 q^{8} + 7 q^{9} + 4 q^{11} + 10 q^{12} - q^{13} + 15 q^{14} + 12 q^{16} + 8 q^{17} + 2 q^{18} + 15 q^{19} - q^{21} + 3 q^{22} + 14 q^{23} + 3 q^{24}+ \cdots + 4 q^{99}+O(q^{100}) $ 7 * q + 2 * q^2 + 7 * q^3 + 10 * q^4 + 2 * q^6 - q^7 + 3 * q^8 + 7 * q^9 + 4 * q^11 + 10 * q^12 - q^13 + 15 * q^14 + 12 * q^16 + 8 * q^17 + 2 * q^18 + 15 * q^19 - q^21 + 3 * q^22 + 14 * q^23 + 3 * q^24 + 6 * q^26 + 7 * q^27 - 24 * q^28 - 7 * q^29 + 5 * q^31 + 18 * q^32 + 4 * q^33 + 7 * q^34 + 10 * q^36 - 6 * q^37 - 18 * q^38 - q^39 + 22 * q^41 + 15 * q^42 - 19 * q^43 + 15 * q^44 - 4 * q^46 + 22 * q^47 + 12 * q^48 + 12 * q^49 + 8 * q^51 + 11 * q^52 + 10 * q^53 + 2 * q^54 + 14 * q^56 + 15 * q^57 - 2 * q^58 + 6 * q^59 + 23 * q^61 + 40 * q^62 - q^63 + 5 * q^64 + 3 * q^66 - 13 * q^67 - 7 * q^68 + 14 * q^69 + 26 * q^71 + 3 * q^72 - 24 * q^73 - 10 * q^74 + 46 * q^76 + 4 * q^77 + 6 * q^78 + 14 * q^79 + 7 * q^81 + 16 * q^82 + 10 * q^83 - 24 * q^84 + 44 * q^86 - 7 * q^87 - 66 * q^88 + 14 * q^89 + 13 * q^91 + 58 * q^92 + 5 * q^93 - 3 * q^94 + 18 * q^96 - 31 * q^97 - 59 * q^98 + 4 * 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 $

$2$	−0.560139	−0.396078	−0.198039	−	0.980194i	$-0.563457\pi$
$2$	−0.560139	−0.396078	−0.198039	+	0.980194i	$0.563457\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	−1.68624	−0.843122
$5$	0	0
$5$	0	0
$6$	−0.560139	−0.228676
$7$	4.36313	1.64911	0.824554	−	0.565784i	$-0.191426\pi$
$7$	4.36313	1.64911	0.824554	+	0.565784i	$0.191426\pi$
$8$	2.06481	0.730020
$9$	1.00000	0.333333
$10$	0	0
$11$	−3.72689	−1.12370	−0.561850	−	0.827239i	$-0.689910\pi$
$11$	−3.72689	−1.12370	−0.561850	+	0.827239i	$0.689910\pi$
$12$	−1.68624	−0.486777
$13$	−2.53237	−0.702352	−0.351176	−	0.936309i	$-0.614218\pi$
$13$	−2.53237	−0.702352	−0.351176	+	0.936309i	$0.614218\pi$
$14$	−2.44396	−0.653175
$15$	0	0
$16$	2.21591	0.553977
$17$	5.11270	1.24001	0.620006	−	0.784597i	$-0.287130\pi$
$17$	5.11270	1.24001	0.620006	+	0.784597i	$0.287130\pi$
$18$	−0.560139	−0.132026
$19$	2.09916	0.481581	0.240791	−	0.970577i	$-0.422593\pi$
$19$	2.09916	0.481581	0.240791	+	0.970577i	$0.422593\pi$
$20$	0	0
$21$	4.36313	0.952113
$22$	2.08758	0.445073
$23$	6.47404	1.34993	0.674966	−	0.737849i	$-0.264158\pi$
$23$	6.47404	1.34993	0.674966	+	0.737849i	$0.264158\pi$
$24$	2.06481	0.421477
$25$	0	0
$26$	1.41848	0.278186
$27$	1.00000	0.192450
$28$	−7.35730	−1.39040
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	−9.94570	−1.78630	−0.893150	−	0.449759i	$-0.851510\pi$
$31$	−9.94570	−1.78630	−0.893150	+	0.449759i	$0.851510\pi$
$32$	−5.37083	−0.949438
$33$	−3.72689	−0.648768
$34$	−2.86382	−0.491141
$35$	0	0
$36$	−1.68624	−0.281041
$37$	5.37249	0.883232	0.441616	−	0.897204i	$-0.354405\pi$
$37$	5.37249	0.883232	0.441616	+	0.897204i	$0.354405\pi$
$38$	−1.17582	−0.190744
$39$	−2.53237	−0.405503
$40$	0	0
$41$	7.96557	1.24401	0.622006	−	0.783012i	$-0.286318\pi$
$41$	7.96557	1.24401	0.622006	+	0.783012i	$0.286318\pi$
$42$	−2.44396	−0.377111
$43$	−10.7771	−1.64349	−0.821746	−	0.569854i	$-0.807000\pi$
$43$	−10.7771	−1.64349	−0.821746	+	0.569854i	$0.807000\pi$
$44$	6.28445	0.947416
$45$	0	0
$46$	−3.62636	−0.534678
$47$	9.38424	1.36883	0.684416	−	0.729092i	$-0.260057\pi$
$47$	9.38424	1.36883	0.684416	+	0.729092i	$0.260057\pi$
$48$	2.21591	0.319839
$49$	12.0369	1.71955
$50$	0	0
$51$	5.11270	0.715921
$52$	4.27019	0.592169
$53$	−5.64265	−0.775077	−0.387539	−	0.921853i	$-0.626675\pi$
$53$	−5.64265	−0.775077	−0.387539	+	0.921853i	$0.626675\pi$
$54$	−0.560139	−0.0762252
$55$	0	0
$56$	9.00902	1.20388
$57$	2.09916	0.278041
$58$	0.560139	0.0735498
$59$	4.24056	0.552073	0.276037	−	0.961147i	$-0.410979\pi$
$59$	4.24056	0.552073	0.276037	+	0.961147i	$0.410979\pi$
$60$	0	0
$61$	9.81848	1.25713	0.628564	−	0.777758i	$-0.283643\pi$
$61$	9.81848	1.25713	0.628564	+	0.777758i	$0.283643\pi$
$62$	5.57097	0.707514
$63$	4.36313	0.549702
$64$	−1.42341	−0.177926
$65$	0	0
$66$	2.08758	0.256963
$67$	0.520614	0.0636031	0.0318015	−	0.999494i	$-0.489876\pi$
$67$	0.520614	0.0636031	0.0318015	+	0.999494i	$0.489876\pi$
$68$	−8.62126	−1.04548
$69$	6.47404	0.779383
$70$	0	0
$71$	3.22415	0.382636	0.191318	−	0.981528i	$-0.438724\pi$
$71$	3.22415	0.382636	0.191318	+	0.981528i	$0.438724\pi$
$72$	2.06481	0.243340
$73$	−10.1718	−1.19052	−0.595262	−	0.803532i	$-0.702952\pi$
$73$	−10.1718	−1.19052	−0.595262	+	0.803532i	$0.702952\pi$
$74$	−3.00934	−0.349829
$75$	0	0
$76$	−3.53970	−0.406032
$77$	−16.2609	−1.85310
$78$	1.41848	0.160611
$79$	1.81484	0.204185	0.102093	−	0.994775i	$-0.467446\pi$
$79$	1.81484	0.204185	0.102093	+	0.994775i	$0.467446\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	−4.46182	−0.492726
$83$	−1.35377	−0.148595	−0.0742976	−	0.997236i	$-0.523671\pi$
$83$	−1.35377	−0.148595	−0.0742976	+	0.997236i	$0.523671\pi$
$84$	−7.35730	−0.802747
$85$	0	0
$86$	6.03667	0.650951
$87$	−1.00000	−0.107211
$88$	−7.69532	−0.820323
$89$	14.2314	1.50853	0.754263	−	0.656572i	$-0.227994\pi$
$89$	14.2314	1.50853	0.754263	+	0.656572i	$0.227994\pi$
$90$	0	0
$91$	−11.0490	−1.15825
$92$	−10.9168	−1.13816
$93$	−9.94570	−1.03132
$94$	−5.25648	−0.542164
$95$	0	0
$96$	−5.37083	−0.548159
$97$	14.0094	1.42244	0.711221	−	0.702968i	$-0.248142\pi$
$97$	14.0094	1.42244	0.711221	+	0.702968i	$0.248142\pi$
$98$	−6.74233	−0.681078
$99$	−3.72689	−0.374567
$100$	0	0
$101$	−1.82364	−0.181459	−0.0907294	−	0.995876i	$-0.528920\pi$
$101$	−1.82364	−0.181459	−0.0907294	+	0.995876i	$0.528920\pi$
$102$	−2.86382	−0.283561
$103$	10.6248	1.04689	0.523445	−	0.852059i	$-0.324647\pi$
$103$	10.6248	1.04689	0.523445	+	0.852059i	$0.324647\pi$
$104$	−5.22885	−0.512731
$105$	0	0
$106$	3.16067	0.306991
$107$	−12.2121	−1.18059	−0.590293	−	0.807189i	$-0.700988\pi$
$107$	−12.2121	−1.18059	−0.590293	+	0.807189i	$0.700988\pi$
$108$	−1.68624	−0.162259
$109$	−7.10102	−0.680155	−0.340077	−	0.940397i	$-0.610453\pi$
$109$	−7.10102	−0.680155	−0.340077	+	0.940397i	$0.610453\pi$
$110$	0	0
$111$	5.37249	0.509934
$112$	9.66829	0.913568
$113$	2.21008	0.207907	0.103953	−	0.994582i	$-0.466851\pi$
$113$	2.21008	0.207907	0.103953	+	0.994582i	$0.466851\pi$
$114$	−1.17582	−0.110126
$115$	0	0
$116$	1.68624	0.156564
$117$	−2.53237	−0.234117
$118$	−2.37530	−0.218664
$119$	22.3074	2.04491
$120$	0	0
$121$	2.88971	0.262701
$122$	−5.49971	−0.497920
$123$	7.96557	0.718231
$124$	16.7709	1.50607
$125$	0	0
$126$	−2.44396	−0.217725
$127$	−12.1699	−1.07990	−0.539950	−	0.841697i	$-0.681557\pi$
$127$	−12.1699	−1.07990	−0.539950	+	0.841697i	$0.681557\pi$
$128$	11.5390	1.01991
$129$	−10.7771	−0.948870
$130$	0	0
$131$	0.754636	0.0659329	0.0329664	−	0.999456i	$-0.489505\pi$
$131$	0.754636	0.0659329	0.0329664	+	0.999456i	$0.489505\pi$
$132$	6.28445	0.546991
$133$	9.15892	0.794179
$134$	−0.291616	−0.0251918
$135$	0	0
$136$	10.5567	0.905234
$137$	19.2792	1.64713	0.823567	−	0.567219i	$-0.191981\pi$
$137$	19.2792	1.64713	0.823567	+	0.567219i	$0.191981\pi$
$138$	−3.62636	−0.308697
$139$	21.1062	1.79020	0.895102	−	0.445862i	$-0.147103\pi$
$139$	21.1062	1.79020	0.895102	+	0.445862i	$0.147103\pi$
$140$	0	0
$141$	9.38424	0.790296
$142$	−1.80597	−0.151554
$143$	9.43785	0.789233
$144$	2.21591	0.184659
$145$	0	0
$146$	5.69765	0.471540
$147$	12.0369	0.992786
$148$	−9.05933	−0.744672
$149$	−7.12028	−0.583316	−0.291658	−	0.956523i	$-0.594207\pi$
$149$	−7.12028	−0.583316	−0.291658	+	0.956523i	$0.594207\pi$
$150$	0	0
$151$	5.46682	0.444884	0.222442	−	0.974946i	$-0.428597\pi$
$151$	5.46682	0.444884	0.222442	+	0.974946i	$0.428597\pi$
$152$	4.33437	0.351564
$153$	5.11270	0.413337
$154$	9.10836	0.733973
$155$	0	0
$156$	4.27019	0.341889
$157$	−14.6519	−1.16935	−0.584675	−	0.811268i	$-0.698778\pi$
$157$	−14.6519	−1.16935	−0.584675	+	0.811268i	$0.698778\pi$
$158$	−1.01656	−0.0808732
$159$	−5.64265	−0.447491
$160$	0	0
$161$	28.2471	2.22618
$162$	−0.560139	−0.0440087
$163$	13.9352	1.09149	0.545743	−	0.837952i	$-0.316247\pi$
$163$	13.9352	1.09149	0.545743	+	0.837952i	$0.316247\pi$
$164$	−13.4319	−1.04885
$165$	0	0
$166$	0.758297	0.0588553
$167$	25.1850	1.94887	0.974436	−	0.224663i	$-0.0721282\pi$
$167$	25.1850	1.94887	0.974436	+	0.224663i	$0.0721282\pi$
$168$	9.00902	0.695061
$169$	−6.58712	−0.506702
$170$	0	0
$171$	2.09916	0.160527
$172$	18.1728	1.38566
$173$	12.7333	0.968097	0.484048	−	0.875041i	$-0.339166\pi$
$173$	12.7333	0.968097	0.484048	+	0.875041i	$0.339166\pi$
$174$	0.560139	0.0424640
$175$	0	0
$176$	−8.25845	−0.622504
$177$	4.24056	0.318740
$178$	−7.97157	−0.597494
$179$	−7.19567	−0.537830	−0.268915	−	0.963164i	$-0.586665\pi$
$179$	−7.19567	−0.537830	−0.268915	+	0.963164i	$0.586665\pi$
$180$	0	0
$181$	6.47508	0.481289	0.240644	−	0.970613i	$-0.422641\pi$
$181$	6.47508	0.481289	0.240644	+	0.970613i	$0.422641\pi$
$182$	6.18899	0.458759
$183$	9.81848	0.725803
$184$	13.3677	0.985477
$185$	0	0
$186$	5.57097	0.408483
$187$	−19.0545	−1.39340
$188$	−15.8241	−1.15409
$189$	4.36313	0.317371
$190$	0	0
$191$	18.0212	1.30397	0.651983	−	0.758234i	$-0.273937\pi$
$191$	18.0212	1.30397	0.651983	+	0.758234i	$0.273937\pi$
$192$	−1.42341	−0.102725
$193$	−10.6003	−0.763025	−0.381512	−	0.924364i	$-0.624597\pi$
$193$	−10.6003	−0.763025	−0.381512	+	0.924364i	$0.624597\pi$
$194$	−7.84723	−0.563398
$195$	0	0
$196$	−20.2971	−1.44979
$197$	−2.28624	−0.162888	−0.0814441	−	0.996678i	$-0.525953\pi$
$197$	−2.28624	−0.162888	−0.0814441	+	0.996678i	$0.525953\pi$
$198$	2.08758	0.148358
$199$	−22.7385	−1.61189	−0.805946	−	0.591989i	$-0.798343\pi$
$199$	−22.7385	−1.61189	−0.805946	+	0.591989i	$0.798343\pi$
$200$	0	0
$201$	0.520614	0.0367213
$202$	1.02149	0.0718718
$203$	−4.36313	−0.306232
$204$	−8.62126	−0.603609
$205$	0	0
$206$	−5.95135	−0.414650
$207$	6.47404	0.449977
$208$	−5.61149	−0.389087
$209$	−7.82336	−0.541153
$210$	0	0
$211$	16.9446	1.16651	0.583256	−	0.812289i	$-0.301779\pi$
$211$	16.9446	1.16651	0.583256	+	0.812289i	$0.301779\pi$
$212$	9.51488	0.653485
$213$	3.22415	0.220915
$214$	6.84046	0.467604
$215$	0	0
$216$	2.06481	0.140492
$217$	−43.3944	−2.94580
$218$	3.97756	0.269394
$219$	−10.1718	−0.687350
$220$	0	0
$221$	−12.9472	−0.870925
$222$	−3.00934	−0.201974
$223$	7.13150	0.477560	0.238780	−	0.971074i	$-0.423252\pi$
$223$	7.13150	0.477560	0.238780	+	0.971074i	$0.423252\pi$
$224$	−23.4336	−1.56573
$225$	0	0
$226$	−1.23795	−0.0823474
$227$	2.42109	0.160693	0.0803467	−	0.996767i	$-0.474397\pi$
$227$	2.42109	0.160693	0.0803467	+	0.996767i	$0.474397\pi$
$228$	−3.53970	−0.234423
$229$	14.4538	0.955132	0.477566	−	0.878596i	$-0.341519\pi$
$229$	14.4538	0.955132	0.477566	+	0.878596i	$0.341519\pi$
$230$	0	0
$231$	−16.2609	−1.06989
$232$	−2.06481	−0.135561
$233$	10.0591	0.658994	0.329497	−	0.944157i	$-0.393121\pi$
$233$	10.0591	0.658994	0.329497	+	0.944157i	$0.393121\pi$
$234$	1.41848	0.0927287
$235$	0	0
$236$	−7.15061	−0.465465
$237$	1.81484	0.117886
$238$	−12.4952	−0.809945
$239$	−14.3621	−0.929006	−0.464503	−	0.885572i	$-0.653767\pi$
$239$	−14.3621	−0.929006	−0.464503	+	0.885572i	$0.653767\pi$
$240$	0	0
$241$	8.58457	0.552981	0.276490	−	0.961017i	$-0.410829\pi$
$241$	8.58457	0.552981	0.276490	+	0.961017i	$0.410829\pi$
$242$	−1.61864	−0.104050
$243$	1.00000	0.0641500
$244$	−16.5564	−1.05991
$245$	0	0
$246$	−4.46182	−0.284476
$247$	−5.31585	−0.338240
$248$	−20.5360	−1.30404
$249$	−1.35377	−0.0857915
$250$	0	0
$251$	20.6767	1.30510	0.652552	−	0.757744i	$-0.273699\pi$
$251$	20.6767	1.30510	0.652552	+	0.757744i	$0.273699\pi$
$252$	−7.35730	−0.463466
$253$	−24.1281	−1.51692
$254$	6.81681	0.427725
$255$	0	0
$256$	−3.61662	−0.226039
$257$	−22.6024	−1.40990	−0.704949	−	0.709258i	$-0.749030\pi$
$257$	−22.6024	−1.40990	−0.704949	+	0.709258i	$0.749030\pi$
$258$	6.03667	0.375827
$259$	23.4409	1.45654
$260$	0	0
$261$	−1.00000	−0.0618984
$262$	−0.422701	−0.0261146
$263$	18.7013	1.15317	0.576585	−	0.817037i	$-0.304385\pi$
$263$	18.7013	1.15317	0.576585	+	0.817037i	$0.304385\pi$
$264$	−7.69532	−0.473614
$265$	0	0
$266$	−5.13027	−0.314557
$267$	14.2314	0.870948
$268$	−0.877882	−0.0536252
$269$	28.1751	1.71787	0.858933	−	0.512088i	$-0.171128\pi$
$269$	28.1751	1.71787	0.858933	+	0.512088i	$0.171128\pi$
$270$	0	0
$271$	−19.1366	−1.16246	−0.581232	−	0.813738i	$-0.697429\pi$
$271$	−19.1366	−1.16246	−0.581232	+	0.813738i	$0.697429\pi$
$272$	11.3293	0.686938
$273$	−11.0490	−0.668718
$274$	−10.7990	−0.652393
$275$	0	0
$276$	−10.9168	−0.657115
$277$	18.7312	1.12545	0.562725	−	0.826644i	$-0.309753\pi$
$277$	18.7312	1.12545	0.562725	+	0.826644i	$0.309753\pi$
$278$	−11.8224	−0.709060
$279$	−9.94570	−0.595433
$280$	0	0
$281$	−9.02236	−0.538229	−0.269114	−	0.963108i	$-0.586731\pi$
$281$	−9.02236	−0.538229	−0.269114	+	0.963108i	$0.586731\pi$
$282$	−5.25648	−0.313019
$283$	−23.2193	−1.38025	−0.690123	−	0.723692i	$-0.742444\pi$
$283$	−23.2193	−1.38025	−0.690123	+	0.723692i	$0.742444\pi$
$284$	−5.43670	−0.322609
$285$	0	0
$286$	−5.28651	−0.312598
$287$	34.7548	2.05151
$288$	−5.37083	−0.316479
$289$	9.13969	0.537629
$290$	0	0
$291$	14.0094	0.821248
$292$	17.1522	1.00376
$293$	4.66226	0.272372	0.136186	−	0.990683i	$-0.456516\pi$
$293$	4.66226	0.272372	0.136186	+	0.990683i	$0.456516\pi$
$294$	−6.74233	−0.393221
$295$	0	0
$296$	11.0932	0.644777
$297$	−3.72689	−0.216256
$298$	3.98834	0.231039
$299$	−16.3946	−0.948127
$300$	0	0
$301$	−47.0218	−2.71029
$302$	−3.06218	−0.176209
$303$	−1.82364	−0.104765
$304$	4.65156	0.266785
$305$	0	0
$306$	−2.86382	−0.163714
$307$	−27.9383	−1.59452	−0.797262	−	0.603633i	$-0.793719\pi$
$307$	−27.9383	−1.59452	−0.797262	+	0.603633i	$0.793719\pi$
$308$	27.4199	1.56239
$309$	10.6248	0.604422
$310$	0	0
$311$	−34.7023	−1.96779	−0.983895	−	0.178748i	$-0.942795\pi$
$311$	−34.7023	−1.96779	−0.983895	+	0.178748i	$0.942795\pi$
$312$	−5.22885	−0.296025
$313$	−19.1085	−1.08008	−0.540038	−	0.841641i	$-0.681590\pi$
$313$	−19.1085	−1.08008	−0.540038	+	0.841641i	$0.681590\pi$
$314$	8.20711	0.463154
$315$	0	0
$316$	−3.06026	−0.172153
$317$	−16.7578	−0.941214	−0.470607	−	0.882343i	$-0.655965\pi$
$317$	−16.7578	−0.941214	−0.470607	+	0.882343i	$0.655965\pi$
$318$	3.16067	0.177241
$319$	3.72689	0.208666
$320$	0	0
$321$	−12.2121	−0.681612
$322$	−15.8223	−0.881742
$323$	10.7324	0.597167
$324$	−1.68624	−0.0936802
$325$	0	0
$326$	−7.80563	−0.432314
$327$	−7.10102	−0.392687
$328$	16.4474	0.908154
$329$	40.9446	2.25735
$330$	0	0
$331$	2.42129	0.133086	0.0665430	−	0.997784i	$-0.478803\pi$
$331$	2.42129	0.133086	0.0665430	+	0.997784i	$0.478803\pi$
$332$	2.28278	0.125284
$333$	5.37249	0.294411
$334$	−14.1071	−0.771906
$335$	0	0
$336$	9.66829	0.527449
$337$	−5.94477	−0.323832	−0.161916	−	0.986805i	$-0.551767\pi$
$337$	−5.94477	−0.323832	−0.161916	+	0.986805i	$0.551767\pi$
$338$	3.68970	0.200693
$339$	2.21008	0.120035
$340$	0	0
$341$	37.0665	2.00727
$342$	−1.17582	−0.0635813
$343$	21.9766	1.18662
$344$	−22.2526	−1.19978
$345$	0	0
$346$	−7.13243	−0.383442
$347$	−24.4339	−1.31168	−0.655841	−	0.754899i	$-0.727686\pi$
$347$	−24.4339	−1.31168	−0.655841	+	0.754899i	$0.727686\pi$
$348$	1.68624	0.0903922
$349$	−15.3090	−0.819472	−0.409736	−	0.912204i	$-0.634379\pi$
$349$	−15.3090	−0.819472	−0.409736	+	0.912204i	$0.634379\pi$
$350$	0	0
$351$	−2.53237	−0.135168
$352$	20.0165	1.06688
$353$	17.8418	0.949621	0.474810	−	0.880088i	$-0.342517\pi$
$353$	17.8418	0.949621	0.474810	+	0.880088i	$0.342517\pi$
$354$	−2.37530	−0.126246
$355$	0	0
$356$	−23.9976	−1.27187
$357$	22.3074	1.18063
$358$	4.03058	0.213023
$359$	−13.7962	−0.728136	−0.364068	−	0.931372i	$-0.618612\pi$
$359$	−13.7962	−0.728136	−0.364068	+	0.931372i	$0.618612\pi$
$360$	0	0
$361$	−14.5935	−0.768079
$362$	−3.62694	−0.190628
$363$	2.88971	0.151671
$364$	18.6314	0.976549
$365$	0	0
$366$	−5.49971	−0.287474
$367$	−0.0965282	−0.00503873	−0.00251936	−	0.999997i	$-0.500802\pi$
$367$	−0.0965282	−0.00503873	−0.00251936	+	0.999997i	$0.500802\pi$
$368$	14.3459	0.747831
$369$	7.96557	0.414671
$370$	0	0
$371$	−24.6196	−1.27819
$372$	16.7709	0.869530
$373$	−36.5838	−1.89424	−0.947119	−	0.320882i	$-0.896021\pi$
$373$	−36.5838	−1.89424	−0.947119	+	0.320882i	$0.896021\pi$
$374$	10.6731	0.551895
$375$	0	0
$376$	19.3767	0.999275
$377$	2.53237	0.130423
$378$	−2.44396	−0.125704
$379$	−0.562286	−0.0288827	−0.0144413	−	0.999896i	$-0.504597\pi$
$379$	−0.562286	−0.0288827	−0.0144413	+	0.999896i	$0.504597\pi$
$380$	0	0
$381$	−12.1699	−0.623480
$382$	−10.0944	−0.516472
$383$	−3.31052	−0.169159	−0.0845797	−	0.996417i	$-0.526955\pi$
$383$	−3.31052	−0.169159	−0.0845797	+	0.996417i	$0.526955\pi$
$384$	11.5390	0.588846
$385$	0	0
$386$	5.93763	0.302217
$387$	−10.7771	−0.547831
$388$	−23.6233	−1.19929
$389$	−8.36922	−0.424336	−0.212168	−	0.977233i	$-0.568052\pi$
$389$	−8.36922	−0.424336	−0.212168	+	0.977233i	$0.568052\pi$
$390$	0	0
$391$	33.0998	1.67393
$392$	24.8539	1.25531
$393$	0.754636	0.0380664
$394$	1.28061	0.0645164
$395$	0	0
$396$	6.28445	0.315805
$397$	−1.72859	−0.0867555	−0.0433778	−	0.999059i	$-0.513812\pi$
$397$	−1.72859	−0.0867555	−0.0433778	+	0.999059i	$0.513812\pi$
$398$	12.7367	0.638435
$399$	9.15892	0.458520
$400$	0	0
$401$	−20.6867	−1.03304	−0.516522	−	0.856274i	$-0.672774\pi$
$401$	−20.6867	−1.03304	−0.516522	+	0.856274i	$0.672774\pi$
$402$	−0.291616	−0.0145445
$403$	25.1861	1.25461
$404$	3.07510	0.152992
$405$	0	0
$406$	2.44396	0.121292
$407$	−20.0227	−0.992487
$408$	10.5567	0.522637
$409$	−36.4935	−1.80449	−0.902243	−	0.431228i	$-0.858081\pi$
$409$	−36.4935	−1.80449	−0.902243	+	0.431228i	$0.858081\pi$
$410$	0	0
$411$	19.2792	0.950973
$412$	−17.9160	−0.882656
$413$	18.5021	0.910428
$414$	−3.62636	−0.178226
$415$	0	0
$416$	13.6009	0.666840
$417$	21.1062	1.03357
$418$	4.38217	0.214339
$419$	−28.1697	−1.37618	−0.688090	−	0.725625i	$-0.741551\pi$
$419$	−28.1697	−1.37618	−0.688090	+	0.725625i	$0.741551\pi$
$420$	0	0
$421$	−11.1213	−0.542017	−0.271008	−	0.962577i	$-0.587357\pi$
$421$	−11.1213	−0.542017	−0.271008	+	0.962577i	$0.587357\pi$
$422$	−9.49130	−0.462029
$423$	9.38424	0.456277
$424$	−11.6510	−0.565822
$425$	0	0
$426$	−1.80597	−0.0874996
$427$	42.8393	2.07314
$428$	20.5925	0.995378
$429$	9.43785	0.455664
$430$	0	0
$431$	6.56816	0.316377	0.158189	−	0.987409i	$-0.449435\pi$
$431$	6.56816	0.316377	0.158189	+	0.987409i	$0.449435\pi$
$432$	2.21591	0.106613
$433$	1.10855	0.0532734	0.0266367	−	0.999645i	$-0.491520\pi$
$433$	1.10855	0.0532734	0.0266367	+	0.999645i	$0.491520\pi$
$434$	24.3069	1.16677
$435$	0	0
$436$	11.9741	0.573453
$437$	13.5901	0.650102
$438$	5.69765	0.272244
$439$	26.4625	1.26299	0.631494	−	0.775381i	$-0.282442\pi$
$439$	26.4625	1.26299	0.631494	+	0.775381i	$0.282442\pi$
$440$	0	0
$441$	12.0369	0.573185
$442$	7.25224	0.344954
$443$	−10.7820	−0.512267	−0.256133	−	0.966641i	$-0.582449\pi$
$443$	−10.7820	−0.512267	−0.256133	+	0.966641i	$0.582449\pi$
$444$	−9.05933	−0.429937
$445$	0	0
$446$	−3.99463	−0.189151
$447$	−7.12028	−0.336778
$448$	−6.21050	−0.293418
$449$	−3.56594	−0.168287	−0.0841436	−	0.996454i	$-0.526815\pi$
$449$	−3.56594	−0.168287	−0.0841436	+	0.996454i	$0.526815\pi$
$450$	0	0
$451$	−29.6868	−1.39790
$452$	−3.72674	−0.175291
$453$	5.46682	0.256854
$454$	−1.35615	−0.0636471
$455$	0	0
$456$	4.33437	0.202976
$457$	0.384118	0.0179683	0.00898415	−	0.999960i	$-0.497140\pi$
$457$	0.384118	0.0179683	0.00898415	+	0.999960i	$0.497140\pi$
$458$	−8.09612	−0.378307
$459$	5.11270	0.238640
$460$	0	0
$461$	0.168528	0.00784912	0.00392456	−	0.999992i	$-0.498751\pi$
$461$	0.168528	0.00784912	0.00392456	+	0.999992i	$0.498751\pi$
$462$	9.10836	0.423759
$463$	−11.6694	−0.542325	−0.271163	−	0.962534i	$-0.587408\pi$
$463$	−11.6694	−0.542325	−0.271163	+	0.962534i	$0.587408\pi$
$464$	−2.21591	−0.102871
$465$	0	0
$466$	−5.63450	−0.261013
$467$	14.6874	0.679651	0.339826	−	0.940488i	$-0.389632\pi$
$467$	14.6874	0.679651	0.339826	+	0.940488i	$0.389632\pi$
$468$	4.27019	0.197390
$469$	2.27150	0.104888
$470$	0	0
$471$	−14.6519	−0.675124
$472$	8.75594	0.403025
$473$	40.1651	1.84679
$474$	−1.01656	−0.0466922
$475$	0	0
$476$	−37.6157	−1.72411
$477$	−5.64265	−0.258359
$478$	8.04476	0.367959
$479$	12.8124	0.585412	0.292706	−	0.956202i	$-0.405444\pi$
$479$	12.8124	0.585412	0.292706	+	0.956202i	$0.405444\pi$
$480$	0	0
$481$	−13.6051	−0.620340
$482$	−4.80855	−0.219023
$483$	28.2471	1.28529
$484$	−4.87276	−0.221489
$485$	0	0
$486$	−0.560139	−0.0254084
$487$	−12.9556	−0.587076	−0.293538	−	0.955947i	$-0.594833\pi$
$487$	−12.9556	−0.587076	−0.293538	+	0.955947i	$0.594833\pi$
$488$	20.2733	0.917728
$489$	13.9352	0.630170
$490$	0	0
$491$	−18.6710	−0.842609	−0.421304	−	0.906919i	$-0.638428\pi$
$491$	−18.6710	−0.842609	−0.421304	+	0.906919i	$0.638428\pi$
$492$	−13.4319	−0.605557
$493$	−5.11270	−0.230264
$494$	2.97762	0.133969
$495$	0	0
$496$	−22.0388	−0.989570
$497$	14.0674	0.631008
$498$	0.758297	0.0339801
$499$	−31.9283	−1.42931	−0.714653	−	0.699479i	$-0.753415\pi$
$499$	−31.9283	−1.42931	−0.714653	+	0.699479i	$0.753415\pi$
$500$	0	0
$501$	25.1850	1.12518
$502$	−11.5818	−0.516923
$503$	−23.7384	−1.05844	−0.529221	−	0.848484i	$-0.677516\pi$
$503$	−23.7384	−1.05844	−0.529221	+	0.848484i	$0.677516\pi$
$504$	9.00902	0.401294
$505$	0	0
$506$	13.5151	0.600818
$507$	−6.58712	−0.292544
$508$	20.5213	0.910487
$509$	−9.32935	−0.413516	−0.206758	−	0.978392i	$-0.566291\pi$
$509$	−9.32935	−0.413516	−0.206758	+	0.978392i	$0.566291\pi$
$510$	0	0
$511$	−44.3811	−1.96330
$512$	−21.0521	−0.930382
$513$	2.09916	0.0926804
$514$	12.6605	0.558430
$515$	0	0
$516$	18.1728	0.800014
$517$	−34.9740	−1.53816
$518$	−13.1301	−0.576905
$519$	12.7333	0.558931
$520$	0	0
$521$	7.78334	0.340994	0.170497	−	0.985358i	$-0.445463\pi$
$521$	7.78334	0.340994	0.170497	+	0.985358i	$0.445463\pi$
$522$	0.560139	0.0245166
$523$	−21.8373	−0.954879	−0.477439	−	0.878665i	$-0.658435\pi$
$523$	−21.8373	−0.954879	−0.477439	+	0.878665i	$0.658435\pi$
$524$	−1.27250	−0.0555895
$525$	0	0
$526$	−10.4753	−0.456746
$527$	−50.8494	−2.21503
$528$	−8.25845	−0.359403
$529$	18.9133	0.822315
$530$	0	0
$531$	4.24056	0.184024
$532$	−15.4442	−0.669590
$533$	−20.1717	−0.873735
$534$	−7.97157	−0.344963
$535$	0	0
$536$	1.07497	0.0464315
$537$	−7.19567	−0.310516
$538$	−15.7820	−0.680409
$539$	−44.8602	−1.93226
$540$	0	0
$541$	8.09048	0.347837	0.173918	−	0.984760i	$-0.444357\pi$
$541$	8.09048	0.347837	0.173918	+	0.984760i	$0.444357\pi$
$542$	10.7191	0.460426
$543$	6.47508	0.277872
$544$	−27.4595	−1.17731
$545$	0	0
$546$	6.18899	0.264865
$547$	−26.3530	−1.12677	−0.563386	−	0.826194i	$-0.690502\pi$
$547$	−26.3530	−1.12677	−0.563386	+	0.826194i	$0.690502\pi$
$548$	−32.5094	−1.38873
$549$	9.81848	0.419042
$550$	0	0
$551$	−2.09916	−0.0894274
$552$	13.3677	0.568966
$553$	7.91836	0.336723
$554$	−10.4921	−0.445766
$555$	0	0
$556$	−35.5902	−1.50936
$557$	−45.0121	−1.90722	−0.953612	−	0.301038i	$-0.902667\pi$
$557$	−45.0121	−1.90722	−0.953612	+	0.301038i	$0.902667\pi$
$558$	5.57097	0.235838
$559$	27.2915	1.15431
$560$	0	0
$561$	−19.0545	−0.804480
$562$	5.05377	0.213181
$563$	11.0447	0.465480	0.232740	−	0.972539i	$-0.425231\pi$
$563$	11.0447	0.465480	0.232740	+	0.972539i	$0.425231\pi$
$564$	−15.8241	−0.666316
$565$	0	0
$566$	13.0061	0.546685
$567$	4.36313	0.183234
$568$	6.65725	0.279332
$569$	44.8201	1.87896	0.939479	−	0.342607i	$-0.111310\pi$
$569$	44.8201	1.87896	0.939479	+	0.342607i	$0.111310\pi$
$570$	0	0
$571$	17.4506	0.730287	0.365143	−	0.930951i	$-0.381020\pi$
$571$	17.4506	0.730287	0.365143	+	0.930951i	$0.381020\pi$
$572$	−15.9145	−0.665420
$573$	18.0212	0.752845
$574$	−19.4675	−0.812558
$575$	0	0
$576$	−1.42341	−0.0593085
$577$	−7.31868	−0.304681	−0.152340	−	0.988328i	$-0.548681\pi$
$577$	−7.31868	−0.304681	−0.152340	+	0.988328i	$0.548681\pi$
$578$	−5.11950	−0.212943
$579$	−10.6003	−0.440533
$580$	0	0
$581$	−5.90666	−0.245049
$582$	−7.84723	−0.325278
$583$	21.0295	0.870954
$584$	−21.0029	−0.869107
$585$	0	0
$586$	−2.61151	−0.107881
$587$	−8.44561	−0.348587	−0.174294	−	0.984694i	$-0.555764\pi$
$587$	−8.44561	−0.348587	−0.174294	+	0.984694i	$0.555764\pi$
$588$	−20.2971	−0.837040
$589$	−20.8777	−0.860249
$590$	0	0
$591$	−2.28624	−0.0940435
$592$	11.9049	0.489290
$593$	−21.1951	−0.870377	−0.435189	−	0.900339i	$-0.643318\pi$
$593$	−21.1951	−0.870377	−0.435189	+	0.900339i	$0.643318\pi$
$594$	2.08758	0.0856543
$595$	0	0
$596$	12.0065	0.491807
$597$	−22.7385	−0.930627
$598$	9.18328	0.375532
$599$	37.1252	1.51689	0.758447	−	0.651735i	$-0.225959\pi$
$599$	37.1252	1.51689	0.758447	+	0.651735i	$0.225959\pi$
$600$	0	0
$601$	40.9314	1.66963	0.834813	−	0.550534i	$-0.185576\pi$
$601$	40.9314	1.66963	0.834813	+	0.550534i	$0.185576\pi$
$602$	26.3388	1.07349
$603$	0.520614	0.0212010
$604$	−9.21840	−0.375091
$605$	0	0
$606$	1.02149	0.0414952
$607$	−17.6432	−0.716114	−0.358057	−	0.933700i	$-0.616561\pi$
$607$	−17.6432	−0.716114	−0.358057	+	0.933700i	$0.616561\pi$
$608$	−11.2743	−0.457232
$609$	−4.36313	−0.176803
$610$	0	0
$611$	−23.7643	−0.961402
$612$	−8.62126	−0.348494
$613$	5.40476	0.218296	0.109148	−	0.994026i	$-0.465188\pi$
$613$	5.40476	0.218296	0.109148	+	0.994026i	$0.465188\pi$
$614$	15.6493	0.631556
$615$	0	0
$616$	−33.5756	−1.35280
$617$	5.47583	0.220449	0.110224	−	0.993907i	$-0.464843\pi$
$617$	5.47583	0.220449	0.110224	+	0.993907i	$0.464843\pi$
$618$	−5.95135	−0.239398
$619$	7.53825	0.302988	0.151494	−	0.988458i	$-0.451592\pi$
$619$	7.53825	0.302988	0.151494	+	0.988458i	$0.451592\pi$
$620$	0	0
$621$	6.47404	0.259794
$622$	19.4381	0.779398
$623$	62.0935	2.48772
$624$	−5.61149	−0.224639
$625$	0	0
$626$	10.7034	0.427794
$627$	−7.82336	−0.312435
$628$	24.7067	0.985905
$629$	27.4679	1.09522
$630$	0	0
$631$	7.52265	0.299472	0.149736	−	0.988726i	$-0.452158\pi$
$631$	7.52265	0.299472	0.149736	+	0.988726i	$0.452158\pi$
$632$	3.74729	0.149059
$633$	16.9446	0.673485
$634$	9.38672	0.372794
$635$	0	0
$636$	9.51488	0.377290
$637$	−30.4818	−1.20773
$638$	−2.08758	−0.0826479
$639$	3.22415	0.127545
$640$	0	0
$641$	−27.1885	−1.07388	−0.536941	−	0.843620i	$-0.680420\pi$
$641$	−27.1885	−1.07388	−0.536941	+	0.843620i	$0.680420\pi$
$642$	6.84046	0.269971
$643$	−9.59932	−0.378560	−0.189280	−	0.981923i	$-0.560615\pi$
$643$	−9.59932	−0.378560	−0.189280	+	0.981923i	$0.560615\pi$
$644$	−47.6315	−1.87694
$645$	0	0
$646$	−6.01163	−0.236525
$647$	36.3223	1.42798	0.713989	−	0.700157i	$-0.246887\pi$
$647$	36.3223	1.42798	0.713989	+	0.700157i	$0.246887\pi$
$648$	2.06481	0.0811134
$649$	−15.8041	−0.620365
$650$	0	0
$651$	−43.3944	−1.70076
$652$	−23.4981	−0.920257
$653$	−12.3135	−0.481866	−0.240933	−	0.970542i	$-0.577453\pi$
$653$	−12.3135	−0.481866	−0.240933	+	0.970542i	$0.577453\pi$
$654$	3.97756	0.155535
$655$	0	0
$656$	17.6510	0.689155
$657$	−10.1718	−0.396841
$658$	−22.9347	−0.894087
$659$	−12.8085	−0.498947	−0.249473	−	0.968382i	$-0.580258\pi$
$659$	−12.8085	−0.498947	−0.249473	+	0.968382i	$0.580258\pi$
$660$	0	0
$661$	−14.1700	−0.551148	−0.275574	−	0.961280i	$-0.588868\pi$
$661$	−14.1700	−0.551148	−0.275574	+	0.961280i	$0.588868\pi$
$662$	−1.35626	−0.0527125
$663$	−12.9472	−0.502829
$664$	−2.79527	−0.108478
$665$	0	0
$666$	−3.00934	−0.116610
$667$	−6.47404	−0.250676
$668$	−42.4680	−1.64314
$669$	7.13150	0.275720
$670$	0	0
$671$	−36.5924	−1.41263
$672$	−23.4336	−0.903972
$673$	4.26452	0.164385	0.0821927	−	0.996616i	$-0.473808\pi$
$673$	4.26452	0.164385	0.0821927	+	0.996616i	$0.473808\pi$
$674$	3.32990	0.128263
$675$	0	0
$676$	11.1075	0.427212
$677$	30.5652	1.17472	0.587358	−	0.809327i	$-0.300168\pi$
$677$	30.5652	1.17472	0.587358	+	0.809327i	$0.300168\pi$
$678$	−1.23795	−0.0475433
$679$	61.1250	2.34576
$680$	0	0
$681$	2.42109	0.0927764
$682$	−20.7624	−0.795034
$683$	34.6493	1.32582	0.662909	−	0.748700i	$-0.269321\pi$
$683$	34.6493	1.32582	0.662909	+	0.748700i	$0.269321\pi$
$684$	−3.53970	−0.135344
$685$	0	0
$686$	−12.3099	−0.469995
$687$	14.4538	0.551446
$688$	−23.8811	−0.910457
$689$	14.2892	0.544377
$690$	0	0
$691$	36.5754	1.39140	0.695698	−	0.718335i	$-0.255095\pi$
$691$	36.5754	1.39140	0.695698	+	0.718335i	$0.255095\pi$
$692$	−21.4715	−0.816224
$693$	−16.2609	−0.617701
$694$	13.6864	0.519528
$695$	0	0
$696$	−2.06481	−0.0782664
$697$	40.7255	1.54259
$698$	8.57517	0.324575
$699$	10.0591	0.380470
$700$	0	0
$701$	30.4230	1.14906	0.574530	−	0.818483i	$-0.305185\pi$
$701$	30.4230	1.14906	0.574530	+	0.818483i	$0.305185\pi$
$702$	1.41848	0.0535369
$703$	11.2777	0.425348
$704$	5.30487	0.199935
$705$	0	0
$706$	−9.99386	−0.376124
$707$	−7.95677	−0.299245
$708$	−7.15061	−0.268736
$709$	−3.88470	−0.145893	−0.0729464	−	0.997336i	$-0.523240\pi$
$709$	−3.88470	−0.145893	−0.0729464	+	0.997336i	$0.523240\pi$
$710$	0	0
$711$	1.81484	0.0680617
$712$	29.3851	1.10125
$713$	−64.3889	−2.41138
$714$	−12.4952	−0.467622
$715$	0	0
$716$	12.1337	0.453456
$717$	−14.3621	−0.536362
$718$	7.72779	0.288399
$719$	−8.31279	−0.310015	−0.155007	−	0.987913i	$-0.549540\pi$
$719$	−8.31279	−0.310015	−0.155007	+	0.987913i	$0.549540\pi$
$720$	0	0
$721$	46.3573	1.72643
$722$	8.17439	0.304219
$723$	8.58457	0.319263
$724$	−10.9186	−0.405785
$725$	0	0
$726$	−1.61864	−0.0600734
$727$	−10.5865	−0.392633	−0.196316	−	0.980541i	$-0.562898\pi$
$727$	−10.5865	−0.392633	−0.196316	+	0.980541i	$0.562898\pi$
$728$	−22.8141	−0.845549
$729$	1.00000	0.0370370
$730$	0	0
$731$	−55.1000	−2.03795
$732$	−16.5564	−0.611940
$733$	−22.7841	−0.841549	−0.420775	−	0.907165i	$-0.638242\pi$
$733$	−22.7841	−0.841549	−0.420775	+	0.907165i	$0.638242\pi$
$734$	0.0540692	0.00199573
$735$	0	0
$736$	−34.7710	−1.28168
$737$	−1.94027	−0.0714708
$738$	−4.46182	−0.164242
$739$	28.1480	1.03544	0.517720	−	0.855550i	$-0.326781\pi$
$739$	28.1480	1.03544	0.517720	+	0.855550i	$0.326781\pi$
$740$	0	0
$741$	−5.31585	−0.195283
$742$	13.7904	0.506261
$743$	−47.6786	−1.74916	−0.874579	−	0.484883i	$-0.838862\pi$
$743$	−47.6786	−1.74916	−0.874579	+	0.484883i	$0.838862\pi$
$744$	−20.5360	−0.752885
$745$	0	0
$746$	20.4920	0.750266
$747$	−1.35377	−0.0495317
$748$	32.1305	1.17481
$749$	−53.2829	−1.94691
$750$	0	0
$751$	10.8278	0.395111	0.197556	−	0.980292i	$-0.436700\pi$
$751$	10.8278	0.395111	0.197556	+	0.980292i	$0.436700\pi$
$752$	20.7946	0.758302
$753$	20.6767	0.753502
$754$	−1.41848	−0.0516579
$755$	0	0
$756$	−7.35730	−0.267582
$757$	45.1777	1.64201	0.821007	−	0.570919i	$-0.193413\pi$
$757$	45.1777	1.64201	0.821007	+	0.570919i	$0.193413\pi$
$758$	0.314958	0.0114398
$759$	−24.1281	−0.875793
$760$	0	0
$761$	−8.52024	−0.308858	−0.154429	−	0.988004i	$-0.549354\pi$
$761$	−8.52024	−0.308858	−0.154429	+	0.988004i	$0.549354\pi$
$762$	6.81681	0.246947
$763$	−30.9827	−1.12165
$764$	−30.3881	−1.09940
$765$	0	0
$766$	1.85435	0.0670003
$767$	−10.7386	−0.387750
$768$	−3.61662	−0.130503
$769$	17.7232	0.639114	0.319557	−	0.947567i	$-0.396466\pi$
$769$	17.7232	0.639114	0.319557	+	0.947567i	$0.396466\pi$
$770$	0	0
$771$	−22.6024	−0.814005
$772$	17.8747	0.643323
$773$	−29.8111	−1.07223	−0.536116	−	0.844144i	$-0.680109\pi$
$773$	−29.8111	−1.07223	−0.536116	+	0.844144i	$0.680109\pi$
$774$	6.03667	0.216984
$775$	0	0
$776$	28.9268	1.03841
$777$	23.4409	0.840936
$778$	4.68793	0.168070
$779$	16.7210	0.599093
$780$	0	0
$781$	−12.0161	−0.429968
$782$	−18.5405	−0.663007
$783$	−1.00000	−0.0357371
$784$	26.6726	0.952594
$785$	0	0
$786$	−0.422701	−0.0150772
$787$	4.46894	0.159301	0.0796503	−	0.996823i	$-0.474620\pi$
$787$	4.46894	0.159301	0.0796503	+	0.996823i	$0.474620\pi$
$788$	3.85517	0.137335
$789$	18.7013	0.665783
$790$	0	0
$791$	9.64287	0.342861
$792$	−7.69532	−0.273441
$793$	−24.8640	−0.882946
$794$	0.968252	0.0343620
$795$	0	0
$796$	38.3427	1.35902
$797$	−13.4589	−0.476737	−0.238368	−	0.971175i	$-0.576613\pi$
$797$	−13.4589	−0.476737	−0.238368	+	0.971175i	$0.576613\pi$
$798$	−5.13027	−0.181610
$799$	47.9788	1.69737
$800$	0	0
$801$	14.2314	0.502842
$802$	11.5874	0.409166
$803$	37.9093	1.33779
$804$	−0.877882	−0.0309605
$805$	0	0
$806$	−14.1077	−0.496924
$807$	28.1751	0.991810
$808$	−3.76546	−0.132469
$809$	−47.1877	−1.65903	−0.829516	−	0.558484i	$-0.811383\pi$
$809$	−47.1877	−1.65903	−0.829516	+	0.558484i	$0.811383\pi$
$810$	0	0
$811$	32.6796	1.14754	0.573769	−	0.819017i	$-0.305481\pi$
$811$	32.6796	1.14754	0.573769	+	0.819017i	$0.305481\pi$
$812$	7.35730	0.258191
$813$	−19.1366	−0.671149
$814$	11.2155	0.393102
$815$	0	0
$816$	11.3293	0.396604
$817$	−22.6229	−0.791475
$818$	20.4414	0.714717
$819$	−11.0490	−0.386085
$820$	0	0
$821$	20.0420	0.699470	0.349735	−	0.936849i	$-0.386272\pi$
$821$	20.0420	0.699470	0.349735	+	0.936849i	$0.386272\pi$
$822$	−10.7990	−0.376659
$823$	1.64700	0.0574108	0.0287054	−	0.999588i	$-0.490862\pi$
$823$	1.64700	0.0574108	0.0287054	+	0.999588i	$0.490862\pi$
$824$	21.9381	0.764251
$825$	0	0
$826$	−10.3637	−0.360601
$827$	−1.65948	−0.0577057	−0.0288528	−	0.999584i	$-0.509185\pi$
$827$	−1.65948	−0.0577057	−0.0288528	+	0.999584i	$0.509185\pi$
$828$	−10.9168	−0.379386
$829$	−30.2298	−1.04992	−0.524962	−	0.851126i	$-0.675920\pi$
$829$	−30.2298	−1.04992	−0.524962	+	0.851126i	$0.675920\pi$
$830$	0	0
$831$	18.7312	0.649779
$832$	3.60458	0.124966
$833$	61.5410	2.13227
$834$	−11.8224	−0.409376
$835$	0	0
$836$	13.1921	0.456258
$837$	−9.94570	−0.343774
$838$	15.7789	0.545075
$839$	−1.43638	−0.0495895	−0.0247947	−	0.999693i	$-0.507893\pi$
$839$	−1.43638	−0.0495895	−0.0247947	+	0.999693i	$0.507893\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	6.22944	0.214681
$843$	−9.02236	−0.310746
$844$	−28.5727	−0.983511
$845$	0	0
$846$	−5.25648	−0.180721
$847$	12.6082	0.433222
$848$	−12.5036	−0.429375
$849$	−23.2193	−0.796886
$850$	0	0
$851$	34.7817	1.19230
$852$	−5.43670	−0.186258
$853$	0.913705	0.0312847	0.0156423	−	0.999878i	$-0.495021\pi$
$853$	0.913705	0.0312847	0.0156423	+	0.999878i	$0.495021\pi$
$854$	−23.9959	−0.821124
$855$	0	0
$856$	−25.2156	−0.861852
$857$	53.1479	1.81550	0.907748	−	0.419516i	$-0.137800\pi$
$857$	53.1479	1.81550	0.907748	+	0.419516i	$0.137800\pi$
$858$	−5.28651	−0.180478
$859$	−10.2236	−0.348826	−0.174413	−	0.984673i	$-0.555803\pi$
$859$	−10.2236	−0.348826	−0.174413	+	0.984673i	$0.555803\pi$
$860$	0	0
$861$	34.7548	1.18444
$862$	−3.67908	−0.125310
$863$	−45.2465	−1.54021	−0.770104	−	0.637918i	$-0.779796\pi$
$863$	−45.2465	−1.54021	−0.770104	+	0.637918i	$0.779796\pi$
$864$	−5.37083	−0.182720
$865$	0	0
$866$	−0.620940	−0.0211004
$867$	9.13969	0.310400
$868$	73.1735	2.48367
$869$	−6.76370	−0.229443
$870$	0	0
$871$	−1.31838	−0.0446718
$872$	−14.6623	−0.496527
$873$	14.0094	0.474148
$874$	−7.61234	−0.257491
$875$	0	0
$876$	17.1522	0.579520
$877$	19.3183	0.652332	0.326166	−	0.945312i	$-0.394243\pi$
$877$	19.3183	0.652332	0.326166	+	0.945312i	$0.394243\pi$
$878$	−14.8227	−0.500242
$879$	4.66226	0.157254
$880$	0	0
$881$	5.54756	0.186902	0.0934510	−	0.995624i	$-0.470210\pi$
$881$	5.54756	0.186902	0.0934510	+	0.995624i	$0.470210\pi$
$882$	−6.74233	−0.227026
$883$	−32.1271	−1.08116	−0.540581	−	0.841292i	$-0.681795\pi$
$883$	−32.1271	−1.08116	−0.540581	+	0.841292i	$0.681795\pi$
$884$	21.8322	0.734296
$885$	0	0
$886$	6.03940	0.202898
$887$	−18.8095	−0.631560	−0.315780	−	0.948832i	$-0.602266\pi$
$887$	−18.8095	−0.631560	−0.315780	+	0.948832i	$0.602266\pi$
$888$	11.0932	0.372262
$889$	−53.0986	−1.78087
$890$	0	0
$891$	−3.72689	−0.124856
$892$	−12.0254	−0.402642
$893$	19.6991	0.659204
$894$	3.98834	0.133390
$895$	0	0
$896$	50.3460	1.68194
$897$	−16.3946	−0.547401
$898$	1.99742	0.0666549
$899$	9.94570	0.331708
$900$	0	0
$901$	−28.8492	−0.961105
$902$	16.6287	0.553676
$903$	−47.0218	−1.56479
$904$	4.56340	0.151776
$905$	0	0
$906$	−3.06218	−0.101734
$907$	3.04313	0.101045	0.0505227	−	0.998723i	$-0.483911\pi$
$907$	3.04313	0.101045	0.0505227	+	0.998723i	$0.483911\pi$
$908$	−4.08255	−0.135484
$909$	−1.82364	−0.0604863
$910$	0	0
$911$	−52.8072	−1.74958	−0.874790	−	0.484501i	$-0.839001\pi$
$911$	−52.8072	−1.74958	−0.874790	+	0.484501i	$0.839001\pi$
$912$	4.65156	0.154028
$913$	5.04534	0.166976
$914$	−0.215160	−0.00711685
$915$	0	0
$916$	−24.3726	−0.805293
$917$	3.29257	0.108730
$918$	−2.86382	−0.0945202
$919$	33.4001	1.10177	0.550884	−	0.834582i	$-0.314291\pi$
$919$	33.4001	1.10177	0.550884	+	0.834582i	$0.314291\pi$
$920$	0	0
$921$	−27.9383	−0.920599
$922$	−0.0943990	−0.00310886
$923$	−8.16473	−0.268745
$924$	27.4199	0.902047
$925$	0	0
$926$	6.53651	0.214803
$927$	10.6248	0.348963
$928$	5.37083	0.176306
$929$	−4.79788	−0.157413	−0.0787067	−	0.996898i	$-0.525079\pi$
$929$	−4.79788	−0.157413	−0.0787067	+	0.996898i	$0.525079\pi$
$930$	0	0
$931$	25.2674	0.828106
$932$	−16.9621	−0.555613
$933$	−34.7023	−1.13610
$934$	−8.22698	−0.269195
$935$	0	0
$936$	−5.22885	−0.170910
$937$	−27.5261	−0.899238	−0.449619	−	0.893220i	$-0.648440\pi$
$937$	−27.5261	−0.899238	−0.449619	+	0.893220i	$0.648440\pi$
$938$	−1.27236	−0.0415440
$939$	−19.1085	−0.623582
$940$	0	0
$941$	16.6083	0.541415	0.270707	−	0.962662i	$-0.412742\pi$
$941$	16.6083	0.541415	0.270707	+	0.962662i	$0.412742\pi$
$942$	8.20711	0.267402
$943$	51.5694	1.67933
$944$	9.39669	0.305836
$945$	0	0
$946$	−22.4980	−0.731473
$947$	37.8994	1.23156	0.615782	−	0.787916i	$-0.288840\pi$
$947$	37.8994	1.23156	0.615782	+	0.787916i	$0.288840\pi$
$948$	−3.06026	−0.0993926
$949$	25.7588	0.836167
$950$	0	0
$951$	−16.7578	−0.543410
$952$	46.0604	1.49283
$953$	39.3478	1.27460	0.637301	−	0.770615i	$-0.280051\pi$
$953$	39.3478	1.27460	0.637301	+	0.770615i	$0.280051\pi$
$954$	3.16067	0.102330
$955$	0	0
$956$	24.2180	0.783265
$957$	3.72689	0.120473
$958$	−7.17671	−0.231869
$959$	84.1176	2.71630
$960$	0	0
$961$	67.9169	2.19087
$962$	7.62075	0.245703
$963$	−12.2121	−0.393529
$964$	−14.4757	−0.466230
$965$	0	0
$966$	−15.8223	−0.509074
$967$	−34.8122	−1.11949	−0.559743	−	0.828666i	$-0.689100\pi$
$967$	−34.8122	−1.11949	−0.559743	+	0.828666i	$0.689100\pi$
$968$	5.96671	0.191777
$969$	10.7324	0.344774
$970$	0	0
$971$	−16.7398	−0.537207	−0.268603	−	0.963251i	$-0.586562\pi$
$971$	−16.7398	−0.537207	−0.268603	+	0.963251i	$0.586562\pi$
$972$	−1.68624	−0.0540863
$973$	92.0890	2.95224
$974$	7.25696	0.232528
$975$	0	0
$976$	21.7569	0.696420
$977$	−12.8886	−0.412344	−0.206172	−	0.978516i	$-0.566101\pi$
$977$	−12.8886	−0.412344	−0.206172	+	0.978516i	$0.566101\pi$
$978$	−7.80563	−0.249597
$979$	−53.0389	−1.69513
$980$	0	0
$981$	−7.10102	−0.226718
$982$	10.4583	0.333739
$983$	20.3218	0.648166	0.324083	−	0.946029i	$-0.394944\pi$
$983$	20.3218	0.648166	0.324083	+	0.946029i	$0.394944\pi$
$984$	16.4474	0.524323
$985$	0	0
$986$	2.86382	0.0912027
$987$	40.9446	1.30328
$988$	8.96383	0.285177
$989$	−69.7714	−2.21860
$990$	0	0
$991$	−13.1802	−0.418683	−0.209341	−	0.977843i	$-0.567132\pi$
$991$	−13.1802	−0.418683	−0.209341	+	0.977843i	$0.567132\pi$
$992$	53.4167	1.69598
$993$	2.42129	0.0768373
$994$	−7.87968	−0.249928
$995$	0	0
$996$	2.28278	0.0723327
$997$	57.5645	1.82309	0.911543	−	0.411206i	$-0.134892\pi$
$997$	57.5645	1.82309	0.911543	+	0.411206i	$0.134892\pi$
$998$	17.8843	0.566116
$999$	5.37249	0.169978

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2175.2.a.bb.1.3	yes	7
3.2	odd	2		6525.2.a.bu.1.5		7
5.2	odd	4		2175.2.c.o.349.7		14
5.3	odd	4		2175.2.c.o.349.8		14
5.4	even	2		2175.2.a.ba.1.5	✓	7
15.14	odd	2		6525.2.a.bx.1.3		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2175.2.a.ba.1.5	✓	7	5.4	even	2
2175.2.a.bb.1.3	yes	7	1.1	even	1	trivial
2175.2.c.o.349.7		14	5.2	odd	4
2175.2.c.o.349.8		14	5.3	odd	4
6525.2.a.bu.1.5		7	3.2	odd	2
6525.2.a.bx.1.3		7	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:	\( N \)	\(=\)	\( 2175 = 3 \cdot 5^{2} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2175.a (trivial)

\(f(q)\)	\(=\)	\(q-0.560139 q^{2} +1.00000 q^{3} -1.68624 q^{4} -0.560139 q^{6} +4.36313 q^{7} +2.06481 q^{8} +1.00000 q^{9} -3.72689 q^{11} -1.68624 q^{12} -2.53237 q^{13} -2.44396 q^{14} +2.21591 q^{16} +5.11270 q^{17} -0.560139 q^{18} +2.09916 q^{19} +4.36313 q^{21} +2.08758 q^{22} +6.47404 q^{23} +2.06481 q^{24} +1.41848 q^{26} +1.00000 q^{27} -7.35730 q^{28} -1.00000 q^{29} -9.94570 q^{31} -5.37083 q^{32} -3.72689 q^{33} -2.86382 q^{34} -1.68624 q^{36} +5.37249 q^{37} -1.17582 q^{38} -2.53237 q^{39} +7.96557 q^{41} -2.44396 q^{42} -10.7771 q^{43} +6.28445 q^{44} -3.62636 q^{46} +9.38424 q^{47} +2.21591 q^{48} +12.0369 q^{49} +5.11270 q^{51} +4.27019 q^{52} -5.64265 q^{53} -0.560139 q^{54} +9.00902 q^{56} +2.09916 q^{57} +0.560139 q^{58} +4.24056 q^{59} +9.81848 q^{61} +5.57097 q^{62} +4.36313 q^{63} -1.42341 q^{64} +2.08758 q^{66} +0.520614 q^{67} -8.62126 q^{68} +6.47404 q^{69} +3.22415 q^{71} +2.06481 q^{72} -10.1718 q^{73} -3.00934 q^{74} -3.53970 q^{76} -16.2609 q^{77} +1.41848 q^{78} +1.81484 q^{79} +1.00000 q^{81} -4.46182 q^{82} -1.35377 q^{83} -7.35730 q^{84} +6.03667 q^{86} -1.00000 q^{87} -7.69532 q^{88} +14.2314 q^{89} -11.0490 q^{91} -10.9168 q^{92} -9.94570 q^{93} -5.25648 q^{94} -5.37083 q^{96} +14.0094 q^{97} -6.74233 q^{98} -3.72689 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 2 q^{2} + 7 q^{3} + 10 q^{4} + 2 q^{6} - q^{7} + 3 q^{8} + 7 q^{9} + 4 q^{11} + 10 q^{12} - q^{13} + 15 q^{14} + 12 q^{16} + 8 q^{17} + 2 q^{18} + 15 q^{19} - q^{21} + 3 q^{22} + 14 q^{23} + 3 q^{24}+ \cdots + 4 q^{99}+O(q^{100}) \) 7 * q + 2 * q^2 + 7 * q^3 + 10 * q^4 + 2 * q^6 - q^7 + 3 * q^8 + 7 * q^9 + 4 * q^11 + 10 * q^12 - q^13 + 15 * q^14 + 12 * q^16 + 8 * q^17 + 2 * q^18 + 15 * q^19 - q^21 + 3 * q^22 + 14 * q^23 + 3 * q^24 + 6 * q^26 + 7 * q^27 - 24 * q^28 - 7 * q^29 + 5 * q^31 + 18 * q^32 + 4 * q^33 + 7 * q^34 + 10 * q^36 - 6 * q^37 - 18 * q^38 - q^39 + 22 * q^41 + 15 * q^42 - 19 * q^43 + 15 * q^44 - 4 * q^46 + 22 * q^47 + 12 * q^48 + 12 * q^49 + 8 * q^51 + 11 * q^52 + 10 * q^53 + 2 * q^54 + 14 * q^56 + 15 * q^57 - 2 * q^58 + 6 * q^59 + 23 * q^61 + 40 * q^62 - q^63 + 5 * q^64 + 3 * q^66 - 13 * q^67 - 7 * q^68 + 14 * q^69 + 26 * q^71 + 3 * q^72 - 24 * q^73 - 10 * q^74 + 46 * q^76 + 4 * q^77 + 6 * q^78 + 14 * q^79 + 7 * q^81 + 16 * q^82 + 10 * q^83 - 24 * q^84 + 44 * q^86 - 7 * q^87 - 66 * q^88 + 14 * q^89 + 13 * q^91 + 58 * q^92 + 5 * q^93 - 3 * q^94 + 18 * q^96 - 31 * q^97 - 59 * q^98 + 4 * q^99

Introduction

L-functions

Modular forms

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Database

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