LMFDB - Embedded newform 2175.2.a.u.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:	$0$
Dimension:	$3$
Coefficient field:	3.3.469.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 5x + 4 $ x^3 - x^2 - 5*x + 4
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			$0.772866$ 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)

$f(q)$	$=$	\(q-0.772866 q^{2} +1.00000 q^{3} -1.40268 q^{4} -0.772866 q^{6} -2.17554 q^{7} +2.62981 q^{8} +1.00000 q^{9} +3.00000 q^{11} -1.40268 q^{12} +0.629813 q^{13} +1.68140 q^{14} +0.772866 q^{16} -4.17554 q^{17} -0.772866 q^{18} +4.80536 q^{19} -2.17554 q^{21} -2.31860 q^{22} -2.08408 q^{23} +2.62981 q^{24} -0.486761 q^{26} +1.00000 q^{27} +3.05159 q^{28} -1.00000 q^{29} -5.85695 q^{32} +3.00000 q^{33} +3.22713 q^{34} -1.40268 q^{36} -6.08408 q^{37} -3.71390 q^{38} +0.629813 q^{39} +0.824456 q^{41} +1.68140 q^{42} +8.72128 q^{43} -4.20804 q^{44} +1.61072 q^{46} +8.98090 q^{47} +0.772866 q^{48} -2.26701 q^{49} -4.17554 q^{51} -0.883426 q^{52} -6.88944 q^{53} -0.772866 q^{54} -5.72128 q^{56} +4.80536 q^{57} +0.772866 q^{58} +6.45427 q^{59} -2.80536 q^{61} -2.17554 q^{63} +2.98090 q^{64} -2.31860 q^{66} +11.0841 q^{67} +5.85695 q^{68} -2.08408 q^{69} +2.63719 q^{71} +2.62981 q^{72} +14.7863 q^{73} +4.70218 q^{74} -6.74037 q^{76} -6.52663 q^{77} -0.486761 q^{78} -12.0650 q^{79} +1.00000 q^{81} -0.637193 q^{82} +7.62981 q^{83} +3.05159 q^{84} -6.74037 q^{86} -1.00000 q^{87} +7.88944 q^{88} +17.8894 q^{89} -1.37019 q^{91} +2.92330 q^{92} -6.94103 q^{94} -5.85695 q^{96} -0.538351 q^{97} +1.75209 q^{98} +3.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - q^{2} + 3 q^{3} + 5 q^{4} - q^{6} + 4 q^{7} + 3 q^{9} + 9 q^{11} + 5 q^{12} - 6 q^{13} + 9 q^{14} + q^{16} - 2 q^{17} - q^{18} - 4 q^{19} + 4 q^{21} - 3 q^{22} - q^{23} + 13 q^{26} + 3 q^{27}+ \cdots + 9 q^{99}+O(q^{100}) $ 3 * q - q^2 + 3 * q^3 + 5 * q^4 - q^6 + 4 * q^7 + 3 * q^9 + 9 * q^11 + 5 * q^12 - 6 * q^13 + 9 * q^14 + q^16 - 2 * q^17 - q^18 - 4 * q^19 + 4 * q^21 - 3 * q^22 - q^23 + 13 * q^26 + 3 * q^27 + 21 * q^28 - 3 * q^29 - 11 * q^32 + 9 * q^33 + 11 * q^34 + 5 * q^36 - 13 * q^37 + 2 * q^38 - 6 * q^39 + 13 * q^41 + 9 * q^42 + 13 * q^43 + 15 * q^44 - 32 * q^46 - 2 * q^47 + q^48 + 9 * q^49 - 2 * q^51 - 25 * q^52 + 3 * q^53 - q^54 - 4 * q^56 - 4 * q^57 + q^58 + 22 * q^59 + 10 * q^61 + 4 * q^63 - 20 * q^64 - 3 * q^66 + 28 * q^67 + 11 * q^68 - q^69 - 3 * q^73 - 28 * q^74 - 36 * q^76 + 12 * q^77 + 13 * q^78 - 2 * q^79 + 3 * q^81 + 6 * q^82 + 15 * q^83 + 21 * q^84 - 36 * q^86 - 3 * q^87 + 30 * q^89 - 12 * q^91 + 14 * q^92 - 9 * q^94 - 11 * q^96 + q^97 + 50 * q^98 + 9 * 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 $

$2$	−0.772866	−0.546498	−0.273249	−	0.961943i	$-0.588098\pi$
$2$	−0.772866	−0.546498	−0.273249	+	0.961943i	$0.588098\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	−1.40268	−0.701339
$5$	0	0
$5$	0	0
$6$	−0.772866	−0.315521
$7$	−2.17554	−0.822278	−0.411139	−	0.911573i	$-0.634869\pi$
$7$	−2.17554	−0.822278	−0.411139	+	0.911573i	$0.634869\pi$
$8$	2.62981	0.929779
$9$	1.00000	0.333333
$10$	0	0
$11$	3.00000	0.904534	0.452267	−	0.891883i	$-0.350615\pi$
$11$	3.00000	0.904534	0.452267	+	0.891883i	$0.350615\pi$
$12$	−1.40268	−0.404918
$13$	0.629813	0.174679	0.0873394	−	0.996179i	$-0.472164\pi$
$13$	0.629813	0.174679	0.0873394	+	0.996179i	$0.472164\pi$
$14$	1.68140	0.449374
$15$	0	0
$16$	0.772866	0.193216
$17$	−4.17554	−1.01272	−0.506359	−	0.862323i	$-0.669009\pi$
$17$	−4.17554	−1.01272	−0.506359	+	0.862323i	$0.669009\pi$
$18$	−0.772866	−0.182166
$19$	4.80536	1.10242	0.551212	−	0.834365i	$-0.314165\pi$
$19$	4.80536	1.10242	0.551212	+	0.834365i	$0.314165\pi$
$20$	0	0
$21$	−2.17554	−0.474743
$22$	−2.31860	−0.494326
$23$	−2.08408	−0.434561	−0.217281	−	0.976109i	$-0.569719\pi$
$23$	−2.08408	−0.434561	−0.217281	+	0.976109i	$0.569719\pi$
$24$	2.62981	0.536808
$25$	0	0
$26$	−0.486761	−0.0954617
$27$	1.00000	0.192450
$28$	3.05159	0.576696
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	−5.85695	−1.03537
$33$	3.00000	0.522233
$34$	3.22713	0.553449
$35$	0	0
$36$	−1.40268	−0.233780
$37$	−6.08408	−1.00022	−0.500108	−	0.865963i	$-0.666707\pi$
$37$	−6.08408	−1.00022	−0.500108	+	0.865963i	$0.666707\pi$
$38$	−3.71390	−0.602473
$39$	0.629813	0.100851
$40$	0	0
$41$	0.824456	0.128758	0.0643792	−	0.997926i	$-0.479493\pi$
$41$	0.824456	0.128758	0.0643792	+	0.997926i	$0.479493\pi$
$42$	1.68140	0.259446
$43$	8.72128	1.32998	0.664991	−	0.746851i	$-0.268435\pi$
$43$	8.72128	1.32998	0.664991	+	0.746851i	$0.268435\pi$
$44$	−4.20804	−0.634385
$45$	0	0
$46$	1.61072	0.237487
$47$	8.98090	1.31000	0.655000	−	0.755629i	$-0.272669\pi$
$47$	8.98090	1.31000	0.655000	+	0.755629i	$0.272669\pi$
$48$	0.772866	0.111554
$49$	−2.26701	−0.323858
$50$	0	0
$51$	−4.17554	−0.584693
$52$	−0.883426	−0.122509
$53$	−6.88944	−0.946337	−0.473169	−	0.880972i	$-0.656890\pi$
$53$	−6.88944	−0.946337	−0.473169	+	0.880972i	$0.656890\pi$
$54$	−0.772866	−0.105174
$55$	0	0
$56$	−5.72128	−0.764538
$57$	4.80536	0.636485
$58$	0.772866	0.101482
$59$	6.45427	0.840274	0.420137	−	0.907461i	$-0.361982\pi$
$59$	6.45427	0.840274	0.420137	+	0.907461i	$0.361982\pi$
$60$	0	0
$61$	−2.80536	−0.359189	−0.179595	−	0.983741i	$-0.557479\pi$
$61$	−2.80536	−0.359189	−0.179595	+	0.983741i	$0.557479\pi$
$62$	0	0
$63$	−2.17554	−0.274093
$64$	2.98090	0.372613
$65$	0	0
$66$	−2.31860	−0.285400
$67$	11.0841	1.35414	0.677068	−	0.735920i	$-0.263250\pi$
$67$	11.0841	1.35414	0.677068	+	0.735920i	$0.263250\pi$
$68$	5.85695	0.710259
$69$	−2.08408	−0.250894
$70$	0	0
$71$	2.63719	0.312977	0.156489	−	0.987680i	$-0.449983\pi$
$71$	2.63719	0.312977	0.156489	+	0.987680i	$0.449983\pi$
$72$	2.62981	0.309926
$73$	14.7863	1.73060	0.865300	−	0.501254i	$-0.167128\pi$
$73$	14.7863	1.73060	0.865300	+	0.501254i	$0.167128\pi$
$74$	4.70218	0.546617
$75$	0	0
$76$	−6.74037	−0.773174
$77$	−6.52663	−0.743779
$78$	−0.486761	−0.0551148
$79$	−12.0650	−1.35742	−0.678708	−	0.734408i	$-0.737460\pi$
$79$	−12.0650	−1.35742	−0.678708	+	0.734408i	$0.737460\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	−0.637193	−0.0703662
$83$	7.62981	0.837481	0.418740	−	0.908106i	$-0.362472\pi$
$83$	7.62981	0.837481	0.418740	+	0.908106i	$0.362472\pi$
$84$	3.05159	0.332956
$85$	0	0
$86$	−6.74037	−0.726833
$87$	−1.00000	−0.107211
$88$	7.88944	0.841017
$89$	17.8894	1.89628	0.948138	−	0.317858i	$-0.102963\pi$
$89$	17.8894	1.89628	0.948138	+	0.317858i	$0.102963\pi$
$90$	0	0
$91$	−1.37019	−0.143635
$92$	2.92330	0.304775
$93$	0	0
$94$	−6.94103	−0.715913
$95$	0	0
$96$	−5.85695	−0.597772
$97$	−0.538351	−0.0546613	−0.0273306	−	0.999626i	$-0.508701\pi$
$97$	−0.538351	−0.0546613	−0.0273306	+	0.999626i	$0.508701\pi$
$98$	1.75209	0.176988
$99$	3.00000	0.301511
$100$	0	0
$101$	13.8054	1.37368	0.686842	−	0.726807i	$-0.258996\pi$
$101$	13.8054	1.37368	0.686842	+	0.726807i	$0.258996\pi$
$102$	3.22713	0.319534
$103$	13.4426	1.32453	0.662267	−	0.749268i	$-0.269594\pi$
$103$	13.4426	1.32453	0.662267	+	0.749268i	$0.269594\pi$
$104$	1.65629	0.162413
$105$	0	0
$106$	5.32461	0.517172
$107$	9.25963	0.895162	0.447581	−	0.894243i	$-0.352286\pi$
$107$	9.25963	0.895162	0.447581	+	0.894243i	$0.352286\pi$
$108$	−1.40268	−0.134973
$109$	8.61072	0.824757	0.412378	−	0.911013i	$-0.364698\pi$
$109$	8.61072	0.824757	0.412378	+	0.911013i	$0.364698\pi$
$110$	0	0
$111$	−6.08408	−0.577476
$112$	−1.68140	−0.158878
$113$	−11.2670	−1.05991	−0.529955	−	0.848026i	$-0.677791\pi$
$113$	−11.2670	−1.05991	−0.529955	+	0.848026i	$0.677791\pi$
$114$	−3.71390	−0.347838
$115$	0	0
$116$	1.40268	0.130235
$117$	0.629813	0.0582263
$118$	−4.98828	−0.459209
$119$	9.08408	0.832736
$120$	0	0
$121$	−2.00000	−0.181818
$122$	2.16816	0.196296
$123$	0.824456	0.0743387
$124$	0	0
$125$	0	0
$126$	1.68140	0.149791
$127$	−16.7863	−1.48954	−0.744770	−	0.667321i	$-0.767441\pi$
$127$	−16.7863	−1.48954	−0.744770	+	0.667321i	$0.767441\pi$
$128$	9.41006	0.831740
$129$	8.72128	0.767865
$130$	0	0
$131$	11.2670	0.984403	0.492201	−	0.870481i	$-0.336192\pi$
$131$	11.2670	0.984403	0.492201	+	0.870481i	$0.336192\pi$
$132$	−4.20804	−0.366263
$133$	−10.4543	−0.906500
$134$	−8.56651	−0.740033
$135$	0	0
$136$	−10.9809	−0.941605
$137$	18.8703	1.61220	0.806101	−	0.591778i	$-0.201574\pi$
$137$	18.8703	1.61220	0.806101	+	0.591778i	$0.201574\pi$
$138$	1.61072	0.137113
$139$	3.80536	0.322766	0.161383	−	0.986892i	$-0.448405\pi$
$139$	3.80536	0.322766	0.161383	+	0.986892i	$0.448405\pi$
$140$	0	0
$141$	8.98090	0.756328
$142$	−2.03820	−0.171042
$143$	1.88944	0.158003
$144$	0.772866	0.0644055
$145$	0	0
$146$	−11.4278	−0.945771
$147$	−2.26701	−0.186980
$148$	8.53401	0.701492
$149$	4.45427	0.364908	0.182454	−	0.983214i	$-0.441596\pi$
$149$	4.45427	0.364908	0.182454	+	0.983214i	$0.441596\pi$
$150$	0	0
$151$	1.17554	0.0956644	0.0478322	−	0.998855i	$-0.484769\pi$
$151$	1.17554	0.0956644	0.0478322	+	0.998855i	$0.484769\pi$
$152$	12.6372	1.02501
$153$	−4.17554	−0.337573
$154$	5.04421	0.406474
$155$	0	0
$156$	−0.883426	−0.0707307
$157$	0.805358	0.0642745	0.0321373	−	0.999483i	$-0.489769\pi$
$157$	0.805358	0.0642745	0.0321373	+	0.999483i	$0.489769\pi$
$158$	9.32461	0.741826
$159$	−6.88944	−0.546368
$160$	0	0
$161$	4.53401	0.357330
$162$	−0.772866	−0.0607221
$163$	1.11056	0.0869858	0.0434929	−	0.999054i	$-0.486151\pi$
$163$	1.11056	0.0869858	0.0434929	+	0.999054i	$0.486151\pi$
$164$	−1.15645	−0.0903033
$165$	0	0
$166$	−5.89682	−0.457682
$167$	8.70218	0.673395	0.336697	−	0.941613i	$-0.390690\pi$
$167$	8.70218	0.673395	0.336697	+	0.941613i	$0.390690\pi$
$168$	−5.72128	−0.441406
$169$	−12.6033	−0.969487
$170$	0	0
$171$	4.80536	0.367475
$172$	−12.2331	−0.932769
$173$	7.69480	0.585025	0.292512	−	0.956262i	$-0.405509\pi$
$173$	7.69480	0.585025	0.292512	+	0.956262i	$0.405509\pi$
$174$	0.772866	0.0585908
$175$	0	0
$176$	2.31860	0.174771
$177$	6.45427	0.485133
$178$	−13.8261	−1.03631
$179$	−6.06498	−0.453318	−0.226659	−	0.973974i	$-0.572780\pi$
$179$	−6.06498	−0.453318	−0.226659	+	0.973974i	$0.572780\pi$
$180$	0	0
$181$	4.09146	0.304116	0.152058	−	0.988372i	$-0.451410\pi$
$181$	4.09146	0.304116	0.152058	+	0.988372i	$0.451410\pi$
$182$	1.05897	0.0784961
$183$	−2.80536	−0.207378
$184$	−5.48075	−0.404046
$185$	0	0
$186$	0	0
$187$	−12.5266	−0.916038
$188$	−12.5973	−0.918754
$189$	−2.17554	−0.158248
$190$	0	0
$191$	−0.267007	−0.0193199	−0.00965996	−	0.999953i	$-0.503075\pi$
$191$	−0.267007	−0.0193199	−0.00965996	+	0.999953i	$0.503075\pi$
$192$	2.98090	0.215128
$193$	−10.7022	−0.770360	−0.385180	−	0.922842i	$-0.625861\pi$
$193$	−10.7022	−0.770360	−0.385180	+	0.922842i	$0.625861\pi$
$194$	0.416073	0.0298723
$195$	0	0
$196$	3.17988	0.227134
$197$	−8.08408	−0.575967	−0.287984	−	0.957635i	$-0.592985\pi$
$197$	−8.08408	−0.575967	−0.287984	+	0.957635i	$0.592985\pi$
$198$	−2.31860	−0.164775
$199$	−15.8054	−1.12041	−0.560206	−	0.828353i	$-0.689278\pi$
$199$	−15.8054	−1.12041	−0.560206	+	0.828353i	$0.689278\pi$
$200$	0	0
$201$	11.0841	0.781811
$202$	−10.6697	−0.750716
$203$	2.17554	0.152693
$204$	5.85695	0.410068
$205$	0	0
$206$	−10.3893	−0.723856
$207$	−2.08408	−0.144854
$208$	0.486761	0.0337508
$209$	14.4161	0.997181
$210$	0	0
$211$	−25.2214	−1.73631	−0.868157	−	0.496289i	$-0.834696\pi$
$211$	−25.2214	−1.73631	−0.868157	+	0.496289i	$0.834696\pi$
$212$	9.66367	0.663704
$213$	2.63719	0.180698
$214$	−7.15645	−0.489205
$215$	0	0
$216$	2.62981	0.178936
$217$	0	0
$218$	−6.65493	−0.450728
$219$	14.7863	0.999163
$220$	0	0
$221$	−2.62981	−0.176900
$222$	4.70218	0.315589
$223$	12.5266	0.838845	0.419423	−	0.907791i	$-0.362233\pi$
$223$	12.5266	0.838845	0.419423	+	0.907791i	$0.362233\pi$
$224$	12.7420	0.851364
$225$	0	0
$226$	8.70788	0.579240
$227$	7.46165	0.495247	0.247624	−	0.968856i	$-0.420350\pi$
$227$	7.46165	0.495247	0.247624	+	0.968856i	$0.420350\pi$
$228$	−6.74037	−0.446392
$229$	21.7936	1.44016	0.720082	−	0.693889i	$-0.244104\pi$
$229$	21.7936	1.44016	0.720082	+	0.693889i	$0.244104\pi$
$230$	0	0
$231$	−6.52663	−0.429421
$232$	−2.62981	−0.172656
$233$	9.69480	0.635127	0.317564	−	0.948237i	$-0.397135\pi$
$233$	9.69480	0.635127	0.317564	+	0.948237i	$0.397135\pi$
$234$	−0.486761	−0.0318206
$235$	0	0
$236$	−9.05327	−0.589317
$237$	−12.0650	−0.783705
$238$	−7.02077	−0.455089
$239$	−10.7022	−0.692266	−0.346133	−	0.938185i	$-0.612505\pi$
$239$	−10.7022	−0.692266	−0.346133	+	0.938185i	$0.612505\pi$
$240$	0	0
$241$	11.1682	0.719405	0.359702	−	0.933067i	$-0.382878\pi$
$241$	11.1682	0.719405	0.359702	+	0.933067i	$0.382878\pi$
$242$	1.54573	0.0993634
$243$	1.00000	0.0641500
$244$	3.93502	0.251914
$245$	0	0
$246$	−0.637193	−0.0406260
$247$	3.02648	0.192570
$248$	0	0
$249$	7.62981	0.483520
$250$	0	0
$251$	14.3585	0.906299	0.453149	−	0.891435i	$-0.350300\pi$
$251$	14.3585	0.906299	0.453149	+	0.891435i	$0.350300\pi$
$252$	3.05159	0.192232
$253$	−6.25225	−0.393075
$254$	12.9735	0.814031
$255$	0	0
$256$	−13.2345	−0.827157
$257$	7.11056	0.443545	0.221772	−	0.975098i	$-0.428816\pi$
$257$	7.11056	0.443545	0.221772	+	0.975098i	$0.428816\pi$
$258$	−6.74037	−0.419637
$259$	13.2362	0.822457
$260$	0	0
$261$	−1.00000	−0.0618984
$262$	−8.70788	−0.537975
$263$	−19.2214	−1.18524	−0.592622	−	0.805481i	$-0.701907\pi$
$263$	−19.2214	−1.18524	−0.592622	+	0.805481i	$0.701907\pi$
$264$	7.88944	0.485561
$265$	0	0
$266$	8.07974	0.495401
$267$	17.8894	1.09482
$268$	−15.5474	−0.949709
$269$	−9.88944	−0.602970	−0.301485	−	0.953471i	$-0.597482\pi$
$269$	−9.88944	−0.602970	−0.301485	+	0.953471i	$0.597482\pi$
$270$	0	0
$271$	−13.7936	−0.837904	−0.418952	−	0.908008i	$-0.637602\pi$
$271$	−13.7936	−0.837904	−0.418952	+	0.908008i	$0.637602\pi$
$272$	−3.22713	−0.195674
$273$	−1.37019	−0.0829275
$274$	−14.5842	−0.881066
$275$	0	0
$276$	2.92330	0.175962
$277$	14.9661	0.899228	0.449614	−	0.893223i	$-0.351561\pi$
$277$	14.9661	0.899228	0.449614	+	0.893223i	$0.351561\pi$
$278$	−2.94103	−0.176391
$279$	0	0
$280$	0	0
$281$	11.7287	0.699673	0.349836	−	0.936811i	$-0.386237\pi$
$281$	11.7287	0.699673	0.349836	+	0.936811i	$0.386237\pi$
$282$	−6.94103	−0.413332
$283$	3.96180	0.235505	0.117752	−	0.993043i	$-0.462431\pi$
$283$	3.96180	0.235505	0.117752	+	0.993043i	$0.462431\pi$
$284$	−3.69914	−0.219503
$285$	0	0
$286$	−1.46028	−0.0863483
$287$	−1.79364	−0.105875
$288$	−5.85695	−0.345124
$289$	0.435171	0.0255983
$290$	0	0
$291$	−0.538351	−0.0315587
$292$	−20.7404	−1.21374
$293$	3.26701	0.190861	0.0954303	−	0.995436i	$-0.469577\pi$
$293$	3.26701	0.190861	0.0954303	+	0.995436i	$0.469577\pi$
$294$	1.75209	0.102184
$295$	0	0
$296$	−16.0000	−0.929981
$297$	3.00000	0.174078
$298$	−3.44255	−0.199422
$299$	−1.31258	−0.0759086
$300$	0	0
$301$	−18.9735	−1.09362
$302$	−0.908538	−0.0522805
$303$	13.8054	0.793097
$304$	3.71390	0.213007
$305$	0	0
$306$	3.22713	0.184483
$307$	−20.4352	−1.16630	−0.583148	−	0.812366i	$-0.698179\pi$
$307$	−20.4352	−1.16630	−0.583148	+	0.812366i	$0.698179\pi$
$308$	9.15477	0.521641
$309$	13.4426	0.764720
$310$	0	0
$311$	11.0000	0.623753	0.311876	−	0.950123i	$-0.399043\pi$
$311$	11.0000	0.623753	0.311876	+	0.950123i	$0.399043\pi$
$312$	1.65629	0.0937690
$313$	−27.6683	−1.56391	−0.781953	−	0.623337i	$-0.785776\pi$
$313$	−27.6683	−1.56391	−0.781953	+	0.623337i	$0.785776\pi$
$314$	−0.622433	−0.0351259
$315$	0	0
$316$	16.9233	0.952010
$317$	12.8777	0.723285	0.361642	−	0.932317i	$-0.382216\pi$
$317$	12.8777	0.723285	0.361642	+	0.932317i	$0.382216\pi$
$318$	5.32461	0.298589
$319$	−3.00000	−0.167968
$320$	0	0
$321$	9.25963	0.516822
$322$	−3.50418	−0.195280
$323$	−20.0650	−1.11645
$324$	−1.40268	−0.0779266
$325$	0	0
$326$	−0.858314	−0.0475376
$327$	8.61072	0.476174
$328$	2.16816	0.119717
$329$	−19.5384	−1.07718
$330$	0	0
$331$	24.6874	1.35694	0.678472	−	0.734627i	$-0.262643\pi$
$331$	24.6874	1.35694	0.678472	+	0.734627i	$0.262643\pi$
$332$	−10.7022	−0.587358
$333$	−6.08408	−0.333406
$334$	−6.72561	−0.368009
$335$	0	0
$336$	−1.68140	−0.0917281
$337$	21.0915	1.14893	0.574463	−	0.818531i	$-0.305211\pi$
$337$	21.0915	1.14893	0.574463	+	0.818531i	$0.305211\pi$
$338$	9.74068	0.529823
$339$	−11.2670	−0.611940
$340$	0	0
$341$	0	0
$342$	−3.71390	−0.200824
$343$	20.1608	1.08858
$344$	22.9353	1.23659
$345$	0	0
$346$	−5.94704	−0.319715
$347$	18.8245	1.01055	0.505275	−	0.862958i	$-0.331391\pi$
$347$	18.8245	1.01055	0.505275	+	0.862958i	$0.331391\pi$
$348$	1.40268	0.0751915
$349$	17.4884	0.936135	0.468067	−	0.883693i	$-0.344950\pi$
$349$	17.4884	0.936135	0.468067	+	0.883693i	$0.344950\pi$
$350$	0	0
$351$	0.629813	0.0336169
$352$	−17.5708	−0.936529
$353$	8.38928	0.446517	0.223258	−	0.974759i	$-0.428331\pi$
$353$	8.38928	0.446517	0.223258	+	0.974759i	$0.428331\pi$
$354$	−4.98828	−0.265124
$355$	0	0
$356$	−25.0931	−1.32993
$357$	9.08408	0.480781
$358$	4.68742	0.247738
$359$	−27.6948	−1.46168	−0.730838	−	0.682551i	$-0.760870\pi$
$359$	−27.6948	−1.46168	−0.730838	+	0.682551i	$0.760870\pi$
$360$	0	0
$361$	4.09146	0.215340
$362$	−3.16215	−0.166199
$363$	−2.00000	−0.104973
$364$	1.92193	0.100737
$365$	0	0
$366$	2.16816	0.113332
$367$	−9.00738	−0.470181	−0.235091	−	0.971973i	$-0.575539\pi$
$367$	−9.00738	−0.470181	−0.235091	+	0.971973i	$0.575539\pi$
$368$	−1.61072	−0.0839643
$369$	0.824456	0.0429194
$370$	0	0
$371$	14.9883	0.778153
$372$	0	0
$373$	25.0533	1.29721	0.648604	−	0.761126i	$-0.275353\pi$
$373$	25.0533	1.29721	0.648604	+	0.761126i	$0.275353\pi$
$374$	9.68140	0.500613
$375$	0	0
$376$	23.6181	1.21801
$377$	−0.629813	−0.0324370
$378$	1.68140	0.0864821
$379$	−27.5725	−1.41631	−0.708153	−	0.706059i	$-0.750471\pi$
$379$	−27.5725	−1.41631	−0.708153	+	0.706059i	$0.750471\pi$
$380$	0	0
$381$	−16.7863	−0.859986
$382$	0.206360	0.0105583
$383$	13.0724	0.667967	0.333983	−	0.942579i	$-0.391607\pi$
$383$	13.0724	0.667967	0.333983	+	0.942579i	$0.391607\pi$
$384$	9.41006	0.480205
$385$	0	0
$386$	8.27134	0.421000
$387$	8.72128	0.443327
$388$	0.755134	0.0383361
$389$	−23.0797	−1.17019	−0.585095	−	0.810965i	$-0.698943\pi$
$389$	−23.0797	−1.17019	−0.585095	+	0.810965i	$0.698943\pi$
$390$	0	0
$391$	8.70218	0.440088
$392$	−5.96180	−0.301117
$393$	11.2670	0.568345
$394$	6.24791	0.314765
$395$	0	0
$396$	−4.20804	−0.211462
$397$	12.3129	0.617966	0.308983	−	0.951067i	$-0.400011\pi$
$397$	12.3129	0.617966	0.308983	+	0.951067i	$0.400011\pi$
$398$	12.2154	0.612304
$399$	−10.4543	−0.523368
$400$	0	0
$401$	−36.4308	−1.81927	−0.909634	−	0.415410i	$-0.863638\pi$
$401$	−36.4308	−1.81927	−0.909634	+	0.415410i	$0.863638\pi$
$402$	−8.56651	−0.427258
$403$	0	0
$404$	−19.3645	−0.963419
$405$	0	0
$406$	−1.68140	−0.0834466
$407$	−18.2522	−0.904730
$408$	−10.9809	−0.543636
$409$	−22.5842	−1.11672	−0.558359	−	0.829599i	$-0.688569\pi$
$409$	−22.5842	−1.11672	−0.558359	+	0.829599i	$0.688569\pi$
$410$	0	0
$411$	18.8703	0.930805
$412$	−18.8556	−0.928948
$413$	−14.0415	−0.690939
$414$	1.61072	0.0791623
$415$	0	0
$416$	−3.68878	−0.180857
$417$	3.80536	0.186349
$418$	−11.1417	−0.544958
$419$	−2.90854	−0.142091	−0.0710457	−	0.997473i	$-0.522634\pi$
$419$	−2.90854	−0.142091	−0.0710457	+	0.997473i	$0.522634\pi$
$420$	0	0
$421$	27.1183	1.32166	0.660831	−	0.750534i	$-0.270204\pi$
$421$	27.1183	1.32166	0.660831	+	0.750534i	$0.270204\pi$
$422$	19.4928	0.948893
$423$	8.98090	0.436666
$424$	−18.1179	−0.879885
$425$	0	0
$426$	−2.03820	−0.0987509
$427$	6.10318	0.295354
$428$	−12.9883	−0.627812
$429$	1.88944	0.0912230
$430$	0	0
$431$	19.5960	0.943904	0.471952	−	0.881624i	$-0.343550\pi$
$431$	19.5960	0.943904	0.471952	+	0.881624i	$0.343550\pi$
$432$	0.772866	0.0371845
$433$	−24.1638	−1.16124	−0.580620	−	0.814175i	$-0.697190\pi$
$433$	−24.1638	−1.16124	−0.580620	+	0.814175i	$0.697190\pi$
$434$	0	0
$435$	0	0
$436$	−12.0781	−0.578435
$437$	−10.0148	−0.479071
$438$	−11.4278	−0.546041
$439$	−30.4235	−1.45203	−0.726016	−	0.687678i	$-0.758630\pi$
$439$	−30.4235	−1.45203	−0.726016	+	0.687678i	$0.758630\pi$
$440$	0	0
$441$	−2.26701	−0.107953
$442$	2.03249	0.0966758
$443$	32.3853	1.53867	0.769335	−	0.638846i	$-0.220588\pi$
$443$	32.3853	1.53867	0.769335	+	0.638846i	$0.220588\pi$
$444$	8.53401	0.405006
$445$	0	0
$446$	−9.68140	−0.458428
$447$	4.45427	0.210680
$448$	−6.48508	−0.306391
$449$	−5.62243	−0.265339	−0.132670	−	0.991160i	$-0.542355\pi$
$449$	−5.62243	−0.265339	−0.132670	+	0.991160i	$0.542355\pi$
$450$	0	0
$451$	2.47337	0.116466
$452$	15.8040	0.743357
$453$	1.17554	0.0552319
$454$	−5.76685	−0.270652
$455$	0	0
$456$	12.6372	0.591791
$457$	−3.33199	−0.155864	−0.0779320	−	0.996959i	$-0.524832\pi$
$457$	−3.33199	−0.155864	−0.0779320	+	0.996959i	$0.524832\pi$
$458$	−16.8436	−0.787048
$459$	−4.17554	−0.194898
$460$	0	0
$461$	23.3437	1.08722	0.543612	−	0.839336i	$-0.317056\pi$
$461$	23.3437	1.08722	0.543612	+	0.839336i	$0.317056\pi$
$462$	5.04421	0.234678
$463$	−4.13735	−0.192279	−0.0961394	−	0.995368i	$-0.530649\pi$
$463$	−4.13735	−0.192279	−0.0961394	+	0.995368i	$0.530649\pi$
$464$	−0.772866	−0.0358794
$465$	0	0
$466$	−7.49278	−0.347096
$467$	−35.7554	−1.65456	−0.827282	−	0.561786i	$-0.810114\pi$
$467$	−35.7554	−1.65456	−0.827282	+	0.561786i	$0.810114\pi$
$468$	−0.883426	−0.0408364
$469$	−24.1139	−1.11348
$470$	0	0
$471$	0.805358	0.0371089
$472$	16.9735	0.781270
$473$	26.1638	1.20301
$474$	9.32461	0.428294
$475$	0	0
$476$	−12.7420	−0.584031
$477$	−6.88944	−0.315446
$478$	8.27134	0.378322
$479$	−29.6107	−1.35295	−0.676474	−	0.736466i	$-0.736493\pi$
$479$	−29.6107	−1.35295	−0.676474	+	0.736466i	$0.736493\pi$
$480$	0	0
$481$	−3.83184	−0.174717
$482$	−8.63149	−0.393154
$483$	4.53401	0.206305
$484$	2.80536	0.127516
$485$	0	0
$486$	−0.772866	−0.0350579
$487$	4.33633	0.196498	0.0982489	−	0.995162i	$-0.468676\pi$
$487$	4.33633	0.196498	0.0982489	+	0.995162i	$0.468676\pi$
$488$	−7.37757	−0.333967
$489$	1.11056	0.0502213
$490$	0	0
$491$	22.5193	1.01628	0.508140	−	0.861275i	$-0.330333\pi$
$491$	22.5193	1.01628	0.508140	+	0.861275i	$0.330333\pi$
$492$	−1.15645	−0.0521366
$493$	4.17554	0.188057
$494$	−2.33906	−0.105239
$495$	0	0
$496$	0	0
$497$	−5.73733	−0.257354
$498$	−5.89682	−0.264243
$499$	−41.1638	−1.84275	−0.921373	−	0.388680i	$-0.872931\pi$
$499$	−41.1638	−1.84275	−0.921373	+	0.388680i	$0.872931\pi$
$500$	0	0
$501$	8.70218	0.388785
$502$	−11.0972	−0.495291
$503$	−31.5149	−1.40518	−0.702590	−	0.711595i	$-0.747973\pi$
$503$	−31.5149	−1.40518	−0.702590	+	0.711595i	$0.747973\pi$
$504$	−5.72128	−0.254846
$505$	0	0
$506$	4.83215	0.214815
$507$	−12.6033	−0.559734
$508$	23.5457	1.04467
$509$	22.6224	1.00272	0.501361	−	0.865238i	$-0.332833\pi$
$509$	22.6224	1.00272	0.501361	+	0.865238i	$0.332833\pi$
$510$	0	0
$511$	−32.1682	−1.42304
$512$	−8.59162	−0.379699
$513$	4.80536	0.212162
$514$	−5.49551	−0.242396
$515$	0	0
$516$	−12.2331	−0.538534
$517$	26.9427	1.18494
$518$	−10.2298	−0.449471
$519$	7.69480	0.337764
$520$	0	0
$521$	−2.06498	−0.0904686	−0.0452343	−	0.998976i	$-0.514403\pi$
$521$	−2.06498	−0.0904686	−0.0452343	+	0.998976i	$0.514403\pi$
$522$	0.772866	0.0338274
$523$	−24.5266	−1.07247	−0.536237	−	0.844067i	$-0.680155\pi$
$523$	−24.5266	−1.07247	−0.536237	+	0.844067i	$0.680155\pi$
$524$	−15.8040	−0.690401
$525$	0	0
$526$	14.8556	0.647734
$527$	0	0
$528$	2.31860	0.100904
$529$	−18.6566	−0.811157
$530$	0	0
$531$	6.45427	0.280091
$532$	14.6640	0.635764
$533$	0.519253	0.0224913
$534$	−13.8261	−0.598315
$535$	0	0
$536$	29.1491	1.25905
$537$	−6.06498	−0.261723
$538$	7.64321	0.329522
$539$	−6.80102	−0.292941
$540$	0	0
$541$	0.389285	0.0167367	0.00836833	−	0.999965i	$-0.497336\pi$
$541$	0.389285	0.0167367	0.00836833	+	0.999965i	$0.497336\pi$
$542$	10.6606	0.457913
$543$	4.09146	0.175581
$544$	24.4559	1.04854
$545$	0	0
$546$	1.05897	0.0453197
$547$	−21.4352	−0.916502	−0.458251	−	0.888823i	$-0.651524\pi$
$547$	−21.4352	−0.916502	−0.458251	+	0.888823i	$0.651524\pi$
$548$	−26.4690	−1.13070
$549$	−2.80536	−0.119730
$550$	0	0
$551$	−4.80536	−0.204715
$552$	−5.48075	−0.233276
$553$	26.2479	1.11617
$554$	−11.5668	−0.491427
$555$	0	0
$556$	−5.33769	−0.226369
$557$	19.7598	0.837249	0.418624	−	0.908159i	$-0.362512\pi$
$557$	19.7598	0.837249	0.418624	+	0.908159i	$0.362512\pi$
$558$	0	0
$559$	5.49278	0.232320
$560$	0	0
$561$	−12.5266	−0.528875
$562$	−9.06467	−0.382370
$563$	−6.46165	−0.272326	−0.136163	−	0.990686i	$-0.543477\pi$
$563$	−6.46165	−0.272326	−0.136163	+	0.990686i	$0.543477\pi$
$564$	−12.5973	−0.530443
$565$	0	0
$566$	−3.06194	−0.128703
$567$	−2.17554	−0.0913643
$568$	6.93533	0.291000
$569$	−14.9427	−0.626431	−0.313215	−	0.949682i	$-0.601406\pi$
$569$	−14.9427	−0.626431	−0.313215	+	0.949682i	$0.601406\pi$
$570$	0	0
$571$	35.1521	1.47107	0.735535	−	0.677487i	$-0.236931\pi$
$571$	35.1521	1.47107	0.735535	+	0.677487i	$0.236931\pi$
$572$	−2.65028	−0.110814
$573$	−0.267007	−0.0111544
$574$	1.38624	0.0578606
$575$	0	0
$576$	2.98090	0.124204
$577$	0.740373	0.0308222	0.0154111	−	0.999881i	$-0.495094\pi$
$577$	0.740373	0.0308222	0.0154111	+	0.999881i	$0.495094\pi$
$578$	−0.336329	−0.0139894
$579$	−10.7022	−0.444767
$580$	0	0
$581$	−16.5990	−0.688642
$582$	0.416073	0.0172468
$583$	−20.6683	−0.855994
$584$	38.8851	1.60908
$585$	0	0
$586$	−2.52496	−0.104305
$587$	18.5842	0.767054	0.383527	−	0.923530i	$-0.374709\pi$
$587$	18.5842	0.767054	0.383527	+	0.923530i	$0.374709\pi$
$588$	3.17988	0.131136
$589$	0	0
$590$	0	0
$591$	−8.08408	−0.332535
$592$	−4.70218	−0.193258
$593$	−24.0268	−0.986662	−0.493331	−	0.869842i	$-0.664221\pi$
$593$	−24.0268	−0.986662	−0.493331	+	0.869842i	$0.664221\pi$
$594$	−2.31860	−0.0951332
$595$	0	0
$596$	−6.24791	−0.255924
$597$	−15.8054	−0.646870
$598$	1.01445	0.0414839
$599$	22.3585	0.913542	0.456771	−	0.889584i	$-0.349006\pi$
$599$	22.3585	0.913542	0.456771	+	0.889584i	$0.349006\pi$
$600$	0	0
$601$	−20.3511	−0.830138	−0.415069	−	0.909790i	$-0.636243\pi$
$601$	−20.3511	−0.830138	−0.415069	+	0.909790i	$0.636243\pi$
$602$	14.6640	0.597659
$603$	11.0841	0.451379
$604$	−1.64891	−0.0670932
$605$	0	0
$606$	−10.6697	−0.433426
$607$	−19.4928	−0.791187	−0.395594	−	0.918426i	$-0.629461\pi$
$607$	−19.4928	−0.791187	−0.395594	+	0.918426i	$0.629461\pi$
$608$	−28.1447	−1.14142
$609$	2.17554	0.0881575
$610$	0	0
$611$	5.65629	0.228829
$612$	5.85695	0.236753
$613$	−39.1638	−1.58181	−0.790906	−	0.611938i	$-0.790390\pi$
$613$	−39.1638	−1.58181	−0.790906	+	0.611938i	$0.790390\pi$
$614$	15.7936	0.637379
$615$	0	0
$616$	−17.1638	−0.691550
$617$	−26.8321	−1.08022	−0.540111	−	0.841594i	$-0.681618\pi$
$617$	−26.8321	−1.08022	−0.540111	+	0.841594i	$0.681618\pi$
$618$	−10.3893	−0.417918
$619$	−33.4278	−1.34358	−0.671788	−	0.740743i	$-0.734473\pi$
$619$	−33.4278	−1.34358	−0.671788	+	0.740743i	$0.734473\pi$
$620$	0	0
$621$	−2.08408	−0.0836313
$622$	−8.50152	−0.340880
$623$	−38.9193	−1.55927
$624$	0.486761	0.0194860
$625$	0	0
$626$	21.3839	0.854672
$627$	14.4161	0.575722
$628$	−1.12966	−0.0450783
$629$	25.4044	1.01294
$630$	0	0
$631$	−32.3705	−1.28865	−0.644325	−	0.764752i	$-0.722861\pi$
$631$	−32.3705	−1.28865	−0.644325	+	0.764752i	$0.722861\pi$
$632$	−31.7287	−1.26210
$633$	−25.2214	−1.00246
$634$	−9.95275	−0.395274
$635$	0	0
$636$	9.66367	0.383189
$637$	−1.42779	−0.0565711
$638$	2.31860	0.0917941
$639$	2.63719	0.104326
$640$	0	0
$641$	33.0415	1.30506	0.652531	−	0.757762i	$-0.273707\pi$
$641$	33.0415	1.30506	0.652531	+	0.757762i	$0.273707\pi$
$642$	−7.15645	−0.282442
$643$	−27.4734	−1.08344	−0.541722	−	0.840558i	$-0.682227\pi$
$643$	−27.4734	−1.08344	−0.541722	+	0.840558i	$0.682227\pi$
$644$	−6.35976	−0.250610
$645$	0	0
$646$	15.5075	0.610136
$647$	23.2023	0.912178	0.456089	−	0.889934i	$-0.349250\pi$
$647$	23.2023	0.912178	0.456089	+	0.889934i	$0.349250\pi$
$648$	2.62981	0.103309
$649$	19.3628	0.760057
$650$	0	0
$651$	0	0
$652$	−1.55776	−0.0610066
$653$	28.9159	1.13157	0.565784	−	0.824554i	$-0.308574\pi$
$653$	28.9159	1.13157	0.565784	+	0.824554i	$0.308574\pi$
$654$	−6.65493	−0.260228
$655$	0	0
$656$	0.637193	0.0248782
$657$	14.7863	0.576867
$658$	15.1005	0.588680
$659$	36.1832	1.40950	0.704749	−	0.709456i	$-0.251059\pi$
$659$	36.1832	1.40950	0.704749	+	0.709456i	$0.251059\pi$
$660$	0	0
$661$	45.0918	1.75387	0.876933	−	0.480612i	$-0.159585\pi$
$661$	45.0918	1.75387	0.876933	+	0.480612i	$0.159585\pi$
$662$	−19.0801	−0.741567
$663$	−2.62981	−0.102133
$664$	20.0650	0.778672
$665$	0	0
$666$	4.70218	0.182206
$667$	2.08408	0.0806960
$668$	−12.2064	−0.472278
$669$	12.5266	0.484308
$670$	0	0
$671$	−8.41607	−0.324899
$672$	12.7420	0.491535
$673$	23.1491	0.892331	0.446165	−	0.894950i	$-0.352789\pi$
$673$	23.1491	0.892331	0.446165	+	0.894950i	$0.352789\pi$
$674$	−16.3009	−0.627886
$675$	0	0
$676$	17.6784	0.679940
$677$	−38.1521	−1.46630	−0.733152	−	0.680064i	$-0.761952\pi$
$677$	−38.1521	−1.46630	−0.733152	+	0.680064i	$0.761952\pi$
$678$	8.70788	0.334424
$679$	1.17121	0.0449468
$680$	0	0
$681$	7.46165	0.285931
$682$	0	0
$683$	0.228811	0.00875520	0.00437760	−	0.999990i	$-0.498607\pi$
$683$	0.228811	0.00875520	0.00437760	+	0.999990i	$0.498607\pi$
$684$	−6.74037	−0.257725
$685$	0	0
$686$	−15.5816	−0.594907
$687$	21.7936	0.831479
$688$	6.74037	0.256974
$689$	−4.33906	−0.165305
$690$	0	0
$691$	16.2258	0.617257	0.308629	−	0.951183i	$-0.400130\pi$
$691$	16.2258	0.617257	0.308629	+	0.951183i	$0.400130\pi$
$692$	−10.7933	−0.410301
$693$	−6.52663	−0.247926
$694$	−14.5488	−0.552264
$695$	0	0
$696$	−2.62981	−0.0996828
$697$	−3.44255	−0.130396
$698$	−13.5162	−0.511596
$699$	9.69480	0.366691
$700$	0	0
$701$	9.69914	0.366331	0.183166	−	0.983082i	$-0.441366\pi$
$701$	9.69914	0.366331	0.183166	+	0.983082i	$0.441366\pi$
$702$	−0.486761	−0.0183716
$703$	−29.2362	−1.10266
$704$	8.94271	0.337041
$705$	0	0
$706$	−6.48379	−0.244021
$707$	−30.0342	−1.12955
$708$	−9.05327	−0.340243
$709$	49.4884	1.85858	0.929289	−	0.369354i	$-0.120421\pi$
$709$	49.4884	1.85858	0.929289	+	0.369354i	$0.120421\pi$
$710$	0	0
$711$	−12.0650	−0.452472
$712$	47.0459	1.76312
$713$	0	0
$714$	−7.02077	−0.262746
$715$	0	0
$716$	8.50722	0.317930
$717$	−10.7022	−0.399680
$718$	21.4044	0.798803
$719$	15.7139	0.586029	0.293015	−	0.956108i	$-0.405342\pi$
$719$	15.7139	0.586029	0.293015	+	0.956108i	$0.405342\pi$
$720$	0	0
$721$	−29.2449	−1.08914
$722$	−3.16215	−0.117683
$723$	11.1682	0.415348
$724$	−5.73901	−0.213289
$725$	0	0
$726$	1.54573	0.0573675
$727$	−6.51925	−0.241786	−0.120893	−	0.992666i	$-0.538576\pi$
$727$	−6.51925	−0.241786	−0.120893	+	0.992666i	$0.538576\pi$
$728$	−3.60334	−0.133548
$729$	1.00000	0.0370370
$730$	0	0
$731$	−36.4161	−1.34690
$732$	3.93502	0.145442
$733$	6.76716	0.249951	0.124975	−	0.992160i	$-0.460115\pi$
$733$	6.76716	0.249951	0.124975	+	0.992160i	$0.460115\pi$
$734$	6.96149	0.256953
$735$	0	0
$736$	12.2064	0.449932
$737$	33.2522	1.22486
$738$	−0.637193	−0.0234554
$739$	−41.1183	−1.51256	−0.756280	−	0.654248i	$-0.772985\pi$
$739$	−41.1183	−1.51256	−0.756280	+	0.654248i	$0.772985\pi$
$740$	0	0
$741$	3.02648	0.111180
$742$	−11.5839	−0.425259
$743$	−36.5534	−1.34101	−0.670507	−	0.741903i	$-0.733924\pi$
$743$	−36.5534	−1.34101	−0.670507	+	0.741903i	$0.733924\pi$
$744$	0	0
$745$	0	0
$746$	−19.3628	−0.708923
$747$	7.62981	0.279160
$748$	17.5708	0.642454
$749$	−20.1447	−0.736072
$750$	0	0
$751$	−45.1980	−1.64930	−0.824649	−	0.565645i	$-0.808627\pi$
$751$	−45.1980	−1.64930	−0.824649	+	0.565645i	$0.808627\pi$
$752$	6.94103	0.253113
$753$	14.3585	0.523252
$754$	0.486761	0.0177268
$755$	0	0
$756$	3.05159	0.110985
$757$	−53.4737	−1.94353	−0.971767	−	0.235943i	$-0.924182\pi$
$757$	−53.4737	−1.94353	−0.971767	+	0.235943i	$0.924182\pi$
$758$	21.3099	0.774009
$759$	−6.25225	−0.226942
$760$	0	0
$761$	−49.4693	−1.79326	−0.896631	−	0.442778i	$-0.853993\pi$
$761$	−49.4693	−1.79326	−0.896631	+	0.442778i	$0.853993\pi$
$762$	12.9735	0.469981
$763$	−18.7330	−0.678180
$764$	0.374525	0.0135498
$765$	0	0
$766$	−10.1032	−0.365043
$767$	4.06498	0.146778
$768$	−13.2345	−0.477559
$769$	−14.2331	−0.513260	−0.256630	−	0.966510i	$-0.582612\pi$
$769$	−14.2331	−0.513260	−0.256630	+	0.966510i	$0.582612\pi$
$770$	0	0
$771$	7.11056	0.256081
$772$	15.0117	0.540284
$773$	−21.7936	−0.783863	−0.391931	−	0.919994i	$-0.628193\pi$
$773$	−21.7936	−0.783863	−0.391931	+	0.919994i	$0.628193\pi$
$774$	−6.74037	−0.242278
$775$	0	0
$776$	−1.41576	−0.0508229
$777$	13.2362	0.474846
$778$	17.8375	0.639507
$779$	3.96180	0.141946
$780$	0	0
$781$	7.91158	0.283099
$782$	−6.72561	−0.240507
$783$	−1.00000	−0.0357371
$784$	−1.75209	−0.0625747
$785$	0	0
$786$	−8.70788	−0.310600
$787$	14.1829	0.505567	0.252783	−	0.967523i	$-0.418654\pi$
$787$	14.1829	0.505567	0.252783	+	0.967523i	$0.418654\pi$
$788$	11.3394	0.403948
$789$	−19.2214	−0.684301
$790$	0	0
$791$	24.5119	0.871542
$792$	7.88944	0.280339
$793$	−1.76685	−0.0627427
$794$	−9.51621	−0.337718
$795$	0	0
$796$	22.1698	0.785789
$797$	2.00000	0.0708436	0.0354218	−	0.999372i	$-0.488723\pi$
$797$	2.00000	0.0708436	0.0354218	+	0.999372i	$0.488723\pi$
$798$	8.07974	0.286020
$799$	−37.5002	−1.32666
$800$	0	0
$801$	17.8894	0.632092
$802$	28.1561	0.994228
$803$	44.3588	1.56539
$804$	−15.5474	−0.548315
$805$	0	0
$806$	0	0
$807$	−9.88944	−0.348125
$808$	36.3055	1.27722
$809$	30.6757	1.07850	0.539250	−	0.842146i	$-0.318708\pi$
$809$	30.6757	1.07850	0.539250	+	0.842146i	$0.318708\pi$
$810$	0	0
$811$	36.6609	1.28734	0.643670	−	0.765303i	$-0.277411\pi$
$811$	36.6609	1.28734	0.643670	+	0.765303i	$0.277411\pi$
$812$	−3.05159	−0.107090
$813$	−13.7936	−0.483764
$814$	14.1065	0.494434
$815$	0	0
$816$	−3.22713	−0.112972
$817$	41.9088	1.46620
$818$	17.4546	0.610285
$819$	−1.37019	−0.0478782
$820$	0	0
$821$	−36.1447	−1.26146	−0.630730	−	0.776002i	$-0.717244\pi$
$821$	−36.1447	−1.26146	−0.630730	+	0.776002i	$0.717244\pi$
$822$	−14.5842	−0.508684
$823$	10.4395	0.363898	0.181949	−	0.983308i	$-0.441759\pi$
$823$	10.4395	0.363898	0.181949	+	0.983308i	$0.441759\pi$
$824$	35.3514	1.23152
$825$	0	0
$826$	10.8522	0.377597
$827$	34.1447	1.18733	0.593664	−	0.804713i	$-0.297681\pi$
$827$	34.1447	1.18733	0.593664	+	0.804713i	$0.297681\pi$
$828$	2.92330	0.101592
$829$	42.6874	1.48260	0.741298	−	0.671176i	$-0.234211\pi$
$829$	42.6874	1.48260	0.741298	+	0.671176i	$0.234211\pi$
$830$	0	0
$831$	14.9661	0.519170
$832$	1.87741	0.0650875
$833$	9.46599	0.327977
$834$	−2.94103	−0.101840
$835$	0	0
$836$	−20.2211	−0.699362
$837$	0	0
$838$	2.24791	0.0776527
$839$	13.9926	0.483079	0.241539	−	0.970391i	$-0.422348\pi$
$839$	13.9926	0.483079	0.241539	+	0.970391i	$0.422348\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	−20.9588	−0.722287
$843$	11.7287	0.403956
$844$	35.3776	1.21775
$845$	0	0
$846$	−6.94103	−0.238638
$847$	4.35109	0.149505
$848$	−5.32461	−0.182848
$849$	3.96180	0.135969
$850$	0	0
$851$	12.6797	0.434655
$852$	−3.69914	−0.126730
$853$	−34.6978	−1.18803	−0.594016	−	0.804453i	$-0.702458\pi$
$853$	−34.6978	−1.18803	−0.594016	+	0.804453i	$0.702458\pi$
$854$	−4.71694	−0.161410
$855$	0	0
$856$	24.3511	0.832303
$857$	−48.0077	−1.63991	−0.819956	−	0.572427i	$-0.806002\pi$
$857$	−48.0077	−1.63991	−0.819956	+	0.572427i	$0.806002\pi$
$858$	−1.46028	−0.0498532
$859$	16.2627	0.554875	0.277438	−	0.960744i	$-0.410515\pi$
$859$	16.2627	0.554875	0.277438	+	0.960744i	$0.410515\pi$
$860$	0	0
$861$	−1.79364	−0.0611271
$862$	−15.1450	−0.515842
$863$	−13.5605	−0.461604	−0.230802	−	0.973001i	$-0.574135\pi$
$863$	−13.5605	−0.461604	−0.230802	+	0.973001i	$0.574135\pi$
$864$	−5.85695	−0.199257
$865$	0	0
$866$	18.6754	0.634616
$867$	0.435171	0.0147792
$868$	0	0
$869$	−36.1950	−1.22783
$870$	0	0
$871$	6.98090	0.236539
$872$	22.6446	0.766842
$873$	−0.538351	−0.0182204
$874$	7.74006	0.261812
$875$	0	0
$876$	−20.7404	−0.700752
$877$	8.01476	0.270639	0.135320	−	0.990802i	$-0.456794\pi$
$877$	8.01476	0.270639	0.135320	+	0.990802i	$0.456794\pi$
$878$	23.5132	0.793533
$879$	3.26701	0.110193
$880$	0	0
$881$	−44.1564	−1.48767	−0.743834	−	0.668364i	$-0.766995\pi$
$881$	−44.1564	−1.48767	−0.743834	+	0.668364i	$0.766995\pi$
$882$	1.75209	0.0589960
$883$	20.7022	0.696684	0.348342	−	0.937368i	$-0.386745\pi$
$883$	20.7022	0.696684	0.348342	+	0.937368i	$0.386745\pi$
$884$	3.68878	0.124067
$885$	0	0
$886$	−25.0294	−0.840881
$887$	−48.5387	−1.62977	−0.814884	−	0.579624i	$-0.803200\pi$
$887$	−48.5387	−1.62977	−0.814884	+	0.579624i	$0.803200\pi$
$888$	−16.0000	−0.536925
$889$	36.5193	1.22482
$890$	0	0
$891$	3.00000	0.100504
$892$	−17.5708	−0.588315
$893$	43.1564	1.44418
$894$	−3.44255	−0.115136
$895$	0	0
$896$	−20.4720	−0.683922
$897$	−1.31258	−0.0438259
$898$	4.34538	0.145007
$899$	0	0
$900$	0	0
$901$	28.7672	0.958373
$902$	−1.91158	−0.0636487
$903$	−18.9735	−0.631399
$904$	−29.6301	−0.985483
$905$	0	0
$906$	−0.908538	−0.0301841
$907$	26.5149	0.880413	0.440207	−	0.897896i	$-0.354905\pi$
$907$	26.5149	0.880413	0.440207	+	0.897896i	$0.354905\pi$
$908$	−10.4663	−0.347336
$909$	13.8054	0.457895
$910$	0	0
$911$	53.8469	1.78403	0.892014	−	0.452008i	$-0.149292\pi$
$911$	53.8469	1.78403	0.892014	+	0.452008i	$0.149292\pi$
$912$	3.71390	0.122979
$913$	22.8894	0.757530
$914$	2.57518	0.0851794
$915$	0	0
$916$	−30.5695	−1.01004
$917$	−24.5119	−0.809453
$918$	3.22713	0.106511
$919$	−11.4084	−0.376328	−0.188164	−	0.982138i	$-0.560254\pi$
$919$	−11.4084	−0.376328	−0.188164	+	0.982138i	$0.560254\pi$
$920$	0	0
$921$	−20.4352	−0.673362
$922$	−18.0415	−0.594167
$923$	1.66094	0.0546705
$924$	9.15477	0.301170
$925$	0	0
$926$	3.19761	0.105080
$927$	13.4426	0.441511
$928$	5.85695	0.192264
$929$	13.7554	0.451301	0.225651	−	0.974208i	$-0.427549\pi$
$929$	13.7554	0.451301	0.225651	+	0.974208i	$0.427549\pi$
$930$	0	0
$931$	−10.8938	−0.357029
$932$	−13.5987	−0.445440
$933$	11.0000	0.360124
$934$	27.6342	0.904217
$935$	0	0
$936$	1.65629	0.0541376
$937$	−1.29379	−0.0422664	−0.0211332	−	0.999777i	$-0.506727\pi$
$937$	−1.29379	−0.0422664	−0.0211332	+	0.999777i	$0.506727\pi$
$938$	18.6368	0.608514
$939$	−27.6683	−0.902921
$940$	0	0
$941$	−49.8586	−1.62534	−0.812672	−	0.582721i	$-0.801988\pi$
$941$	−49.8586	−1.62534	−0.812672	+	0.582721i	$0.801988\pi$
$942$	−0.622433	−0.0202800
$943$	−1.71823	−0.0559534
$944$	4.98828	0.162355
$945$	0	0
$946$	−20.2211	−0.657445
$947$	21.1109	0.686011	0.343006	−	0.939333i	$-0.388555\pi$
$947$	21.1109	0.686011	0.343006	+	0.939333i	$0.388555\pi$
$948$	16.9233	0.549643
$949$	9.31258	0.302299
$950$	0	0
$951$	12.8777	0.417589
$952$	23.8894	0.774261
$953$	42.7672	1.38536	0.692682	−	0.721243i	$-0.256429\pi$
$953$	42.7672	1.38536	0.692682	+	0.721243i	$0.256429\pi$
$954$	5.32461	0.172391
$955$	0	0
$956$	15.0117	0.485514
$957$	−3.00000	−0.0969762
$958$	22.8851	0.739384
$959$	−41.0533	−1.32568
$960$	0	0
$961$	−31.0000	−1.00000
$962$	2.96149	0.0954824
$963$	9.25963	0.298387
$964$	−15.6653	−0.504547
$965$	0	0
$966$	−3.50418	−0.112745
$967$	33.0342	1.06231	0.531154	−	0.847276i	$-0.321759\pi$
$967$	33.0342	1.06231	0.531154	+	0.847276i	$0.321759\pi$
$968$	−5.25963	−0.169051
$969$	−20.0650	−0.644580
$970$	0	0
$971$	−26.0841	−0.837078	−0.418539	−	0.908199i	$-0.637458\pi$
$971$	−26.0841	−0.837078	−0.418539	+	0.908199i	$0.637458\pi$
$972$	−1.40268	−0.0449909
$973$	−8.27872	−0.265404
$974$	−3.35140	−0.107386
$975$	0	0
$976$	−2.16816	−0.0694012
$977$	43.6948	1.39792	0.698960	−	0.715161i	$-0.253646\pi$
$977$	43.6948	1.39792	0.698960	+	0.715161i	$0.253646\pi$
$978$	−0.858314	−0.0274458
$979$	53.6683	1.71525
$980$	0	0
$981$	8.61072	0.274919
$982$	−17.4044	−0.555395
$983$	−2.55745	−0.0815700	−0.0407850	−	0.999168i	$-0.512986\pi$
$983$	−2.55745	−0.0815700	−0.0407850	+	0.999168i	$0.512986\pi$
$984$	2.16816	0.0691186
$985$	0	0
$986$	−3.22713	−0.102773
$987$	−19.5384	−0.621913
$988$	−4.24518	−0.135057
$989$	−18.1759	−0.577959
$990$	0	0
$991$	14.7287	0.467871	0.233936	−	0.972252i	$-0.424840\pi$
$991$	14.7287	0.467871	0.233936	+	0.972252i	$0.424840\pi$
$992$	0	0
$993$	24.6874	0.783432
$994$	4.43419	0.140644
$995$	0	0
$996$	−10.7022	−0.339111
$997$	−33.0992	−1.04826	−0.524130	−	0.851638i	$-0.675610\pi$
$997$	−33.0992	−1.04826	−0.524130	+	0.851638i	$0.675610\pi$
$998$	31.8141	1.00706
$999$	−6.08408	−0.192492

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2175.2.a.u.1.2		3
3.2	odd	2		6525.2.a.bf.1.2		3
5.2	odd	4		2175.2.c.m.349.3		6
5.3	odd	4		2175.2.c.m.349.4		6
5.4	even	2		435.2.a.i.1.2	✓	3
15.14	odd	2		1305.2.a.q.1.2		3
20.19	odd	2		6960.2.a.cl.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
435.2.a.i.1.2	✓	3	5.4	even	2
1305.2.a.q.1.2		3	15.14	odd	2
2175.2.a.u.1.2		3	1.1	even	1	trivial
2175.2.c.m.349.3		6	5.2	odd	4
2175.2.c.m.349.4		6	5.3	odd	4
6525.2.a.bf.1.2		3	3.2	odd	2
6960.2.a.cl.1.1		3	20.19	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)

\(f(q)\)	\(=\)	\(q-0.772866 q^{2} +1.00000 q^{3} -1.40268 q^{4} -0.772866 q^{6} -2.17554 q^{7} +2.62981 q^{8} +1.00000 q^{9} +3.00000 q^{11} -1.40268 q^{12} +0.629813 q^{13} +1.68140 q^{14} +0.772866 q^{16} -4.17554 q^{17} -0.772866 q^{18} +4.80536 q^{19} -2.17554 q^{21} -2.31860 q^{22} -2.08408 q^{23} +2.62981 q^{24} -0.486761 q^{26} +1.00000 q^{27} +3.05159 q^{28} -1.00000 q^{29} -5.85695 q^{32} +3.00000 q^{33} +3.22713 q^{34} -1.40268 q^{36} -6.08408 q^{37} -3.71390 q^{38} +0.629813 q^{39} +0.824456 q^{41} +1.68140 q^{42} +8.72128 q^{43} -4.20804 q^{44} +1.61072 q^{46} +8.98090 q^{47} +0.772866 q^{48} -2.26701 q^{49} -4.17554 q^{51} -0.883426 q^{52} -6.88944 q^{53} -0.772866 q^{54} -5.72128 q^{56} +4.80536 q^{57} +0.772866 q^{58} +6.45427 q^{59} -2.80536 q^{61} -2.17554 q^{63} +2.98090 q^{64} -2.31860 q^{66} +11.0841 q^{67} +5.85695 q^{68} -2.08408 q^{69} +2.63719 q^{71} +2.62981 q^{72} +14.7863 q^{73} +4.70218 q^{74} -6.74037 q^{76} -6.52663 q^{77} -0.486761 q^{78} -12.0650 q^{79} +1.00000 q^{81} -0.637193 q^{82} +7.62981 q^{83} +3.05159 q^{84} -6.74037 q^{86} -1.00000 q^{87} +7.88944 q^{88} +17.8894 q^{89} -1.37019 q^{91} +2.92330 q^{92} -6.94103 q^{94} -5.85695 q^{96} -0.538351 q^{97} +1.75209 q^{98} +3.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - q^{2} + 3 q^{3} + 5 q^{4} - q^{6} + 4 q^{7} + 3 q^{9} + 9 q^{11} + 5 q^{12} - 6 q^{13} + 9 q^{14} + q^{16} - 2 q^{17} - q^{18} - 4 q^{19} + 4 q^{21} - 3 q^{22} - q^{23} + 13 q^{26} + 3 q^{27}+ \cdots + 9 q^{99}+O(q^{100}) \) 3 * q - q^2 + 3 * q^3 + 5 * q^4 - q^6 + 4 * q^7 + 3 * q^9 + 9 * q^11 + 5 * q^12 - 6 * q^13 + 9 * q^14 + q^16 - 2 * q^17 - q^18 - 4 * q^19 + 4 * q^21 - 3 * q^22 - q^23 + 13 * q^26 + 3 * q^27 + 21 * q^28 - 3 * q^29 - 11 * q^32 + 9 * q^33 + 11 * q^34 + 5 * q^36 - 13 * q^37 + 2 * q^38 - 6 * q^39 + 13 * q^41 + 9 * q^42 + 13 * q^43 + 15 * q^44 - 32 * q^46 - 2 * q^47 + q^48 + 9 * q^49 - 2 * q^51 - 25 * q^52 + 3 * q^53 - q^54 - 4 * q^56 - 4 * q^57 + q^58 + 22 * q^59 + 10 * q^61 + 4 * q^63 - 20 * q^64 - 3 * q^66 + 28 * q^67 + 11 * q^68 - q^69 - 3 * q^73 - 28 * q^74 - 36 * q^76 + 12 * q^77 + 13 * q^78 - 2 * q^79 + 3 * q^81 + 6 * q^82 + 15 * q^83 + 21 * q^84 - 36 * q^86 - 3 * q^87 + 30 * q^89 - 12 * q^91 + 14 * q^92 - 9 * q^94 - 11 * q^96 + q^97 + 50 * q^98 + 9 * q^99

Introduction

L-functions

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