LMFDB - Embedded newform 6900.2.a.ba.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6900,2,Mod(1,6900)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6900, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("6900.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6900 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6900.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:	$55.0967773947$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.175557.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 8x^{2} + 3x + 2 $ x^4 - 2x^3 - 8x^2 + 3*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$3.84324$ of defining polynomial
Character	$\chi$	$=$	6900.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -1.32284 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -1.32284 q^{7} +1.00000 q^{9} -3.84324 q^{11} -4.36363 q^{13} +2.75924 q^{17} -5.92724 q^{19} +1.32284 q^{21} +1.00000 q^{23} -1.00000 q^{27} +0.761159 q^{29} -8.72969 q^{31} +3.84324 q^{33} +3.24076 q^{37} +4.36363 q^{39} +4.16608 q^{41} -9.72727 q^{43} -2.92724 q^{47} -5.25008 q^{49} -2.75924 q^{51} +11.2501 q^{53} +5.92724 q^{57} -3.08208 q^{59} -6.08400 q^{61} -1.32284 q^{63} +2.19755 q^{67} -1.00000 q^{69} -1.39560 q^{71} +9.20687 q^{73} +5.08400 q^{77} +14.3341 q^{79} +1.00000 q^{81} +3.03886 q^{83} -0.761159 q^{87} -18.3002 q^{89} +5.77240 q^{91} +8.72969 q^{93} +5.43832 q^{97} -3.84324 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{3} + 3 q^{7} + 4 q^{9}+O(q^{10}) $ 4 * q - 4 * q^3 + 3 * q^7 + 4 * q^9	$ 4 q - 4 q^{3} + 3 q^{7} + 4 q^{9} - 2 q^{11} + q^{13} + 10 q^{17} + 2 q^{19} - 3 q^{21} + 4 q^{23} - 4 q^{27} - q^{29} - 6 q^{31} + 2 q^{33} + 14 q^{37} - q^{39} - 5 q^{41} - 2 q^{43} + 14 q^{47} + 13 q^{49} - 10 q^{51} + 11 q^{53} - 2 q^{57} - 3 q^{59} - 12 q^{61} + 3 q^{63} + 12 q^{67} - 4 q^{69} - 23 q^{71} + 5 q^{73} + 8 q^{77} + 11 q^{79} + 4 q^{81} + 5 q^{83} + q^{87} + 6 q^{89} - 19 q^{91} + 6 q^{93} + 26 q^{97} - 2 q^{99}+O(q^{100}) $ 4 * q - 4 * q^3 + 3 * q^7 + 4 * q^9 - 2 * q^11 + q^13 + 10 * q^17 + 2 * q^19 - 3 * q^21 + 4 * q^23 - 4 * q^27 - q^29 - 6 * q^31 + 2 * q^33 + 14 * q^37 - q^39 - 5 * q^41 - 2 * q^43 + 14 * q^47 + 13 * q^49 - 10 * q^51 + 11 * q^53 - 2 * q^57 - 3 * q^59 - 12 * q^61 + 3 * q^63 + 12 * q^67 - 4 * q^69 - 23 * q^71 + 5 * q^73 + 8 * q^77 + 11 * q^79 + 4 * q^81 + 5 * q^83 + q^87 + 6 * q^89 - 19 * q^91 + 6 * q^93 + 26 * q^97 - 2 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−1.32284	−0.499988	−0.249994	−	0.968247i	$-0.580429\pi$
$7$	−1.32284	−0.499988	−0.249994	+	0.968247i	$0.580429\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−3.84324	−1.15878	−0.579390	−	0.815050i	$-0.696709\pi$
$11$	−3.84324	−1.15878	−0.579390	+	0.815050i	$0.696709\pi$
$12$	0	0
$13$	−4.36363	−1.21025	−0.605127	−	0.796129i	$-0.706878\pi$
$13$	−4.36363	−1.21025	−0.605127	+	0.796129i	$0.706878\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	2.75924	0.669213	0.334606	−	0.942358i	$-0.391397\pi$
$17$	2.75924	0.669213	0.334606	+	0.942358i	$0.391397\pi$
$18$	0	0
$19$	−5.92724	−1.35980	−0.679901	−	0.733304i	$-0.737977\pi$
$19$	−5.92724	−1.35980	−0.679901	+	0.733304i	$0.737977\pi$
$20$	0	0
$21$	1.32284	0.288668
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	0.761159	0.141344	0.0706719	−	0.997500i	$-0.477486\pi$
$29$	0.761159	0.141344	0.0706719	+	0.997500i	$0.477486\pi$
$30$	0	0
$31$	−8.72969	−1.56790	−0.783949	−	0.620825i	$-0.786798\pi$
$31$	−8.72969	−1.56790	−0.783949	+	0.620825i	$0.786798\pi$
$32$	0	0
$33$	3.84324	0.669022
$34$	0	0
$35$	0	0
$36$	0	0
$37$	3.24076	0.532778	0.266389	−	0.963866i	$-0.414169\pi$
$37$	3.24076	0.532778	0.266389	+	0.963866i	$0.414169\pi$
$38$	0	0
$39$	4.36363	0.698740
$40$	0	0
$41$	4.16608	0.650633	0.325316	−	0.945605i	$-0.394529\pi$
$41$	4.16608	0.650633	0.325316	+	0.945605i	$0.394529\pi$
$42$	0	0
$43$	−9.72727	−1.48339	−0.741697	−	0.670735i	$-0.765979\pi$
$43$	−9.72727	−1.48339	−0.741697	+	0.670735i	$0.765979\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−2.92724	−0.426982	−0.213491	−	0.976945i	$-0.568483\pi$
$47$	−2.92724	−0.426982	−0.213491	+	0.976945i	$0.568483\pi$
$48$	0	0
$49$	−5.25008	−0.750012
$50$	0	0
$51$	−2.75924	−0.386370
$52$	0	0
$53$	11.2501	1.54532	0.772659	−	0.634821i	$-0.218926\pi$
$53$	11.2501	1.54532	0.772659	+	0.634821i	$0.218926\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	5.92724	0.785082
$58$	0	0
$59$	−3.08208	−0.401252	−0.200626	−	0.979668i	$-0.564298\pi$
$59$	−3.08208	−0.401252	−0.200626	+	0.979668i	$0.564298\pi$
$60$	0	0
$61$	−6.08400	−0.778977	−0.389488	−	0.921031i	$-0.627348\pi$
$61$	−6.08400	−0.778977	−0.389488	+	0.921031i	$0.627348\pi$
$62$	0	0
$63$	−1.32284	−0.166663
$64$	0	0
$65$	0	0
$66$	0	0
$67$	2.19755	0.268474	0.134237	−	0.990949i	$-0.457142\pi$
$67$	2.19755	0.268474	0.134237	+	0.990949i	$0.457142\pi$
$68$	0	0
$69$	−1.00000	−0.120386
$70$	0	0
$71$	−1.39560	−0.165628	−0.0828138	−	0.996565i	$-0.526391\pi$
$71$	−1.39560	−0.165628	−0.0828138	+	0.996565i	$0.526391\pi$
$72$	0	0
$73$	9.20687	1.07758	0.538791	−	0.842439i	$-0.318881\pi$
$73$	9.20687	1.07758	0.538791	+	0.842439i	$0.318881\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	5.08400	0.579376
$78$	0	0
$79$	14.3341	1.61271	0.806355	−	0.591431i	$-0.201437\pi$
$79$	14.3341	1.61271	0.806355	+	0.591431i	$0.201437\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	3.03886	0.333559	0.166779	−	0.985994i	$-0.446663\pi$
$83$	3.03886	0.333559	0.166779	+	0.985994i	$0.446663\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−0.761159	−0.0816049
$88$	0	0
$89$	−18.3002	−1.93982	−0.969908	−	0.243470i	$-0.921714\pi$
$89$	−18.3002	−1.93982	−0.969908	+	0.243470i	$0.921714\pi$
$90$	0	0
$91$	5.77240	0.605112
$92$	0	0
$93$	8.72969	0.905227
$94$	0	0
$95$	0	0
$96$	0	0
$97$	5.43832	0.552177	0.276089	−	0.961132i	$-0.410962\pi$
$97$	5.43832	0.552177	0.276089	+	0.961132i	$0.410962\pi$
$98$	0	0
$99$	−3.84324	−0.386260
$100$	0	0
$101$	−15.5316	−1.54546	−0.772728	−	0.634737i	$-0.781108\pi$
$101$	−15.5316	−1.54546	−0.772728	+	0.634737i	$0.781108\pi$
$102$	0	0
$103$	−9.09332	−0.895992	−0.447996	−	0.894036i	$-0.647862\pi$
$103$	−9.09332	−0.895992	−0.447996	+	0.894036i	$0.647862\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	2.92724	0.282987	0.141494	−	0.989939i	$-0.454810\pi$
$107$	2.92724	0.282987	0.141494	+	0.989939i	$0.454810\pi$
$108$	0	0
$109$	13.0525	1.25021	0.625103	−	0.780542i	$-0.285057\pi$
$109$	13.0525	1.25021	0.625103	+	0.780542i	$0.285057\pi$
$110$	0	0
$111$	−3.24076	−0.307600
$112$	0	0
$113$	17.0501	1.60394	0.801970	−	0.597365i	$-0.203786\pi$
$113$	17.0501	1.60394	0.801970	+	0.597365i	$0.203786\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−4.36363	−0.403418
$118$	0	0
$119$	−3.65004	−0.334598
$120$	0	0
$121$	3.77048	0.342771
$122$	0	0
$123$	−4.16608	−0.375643
$124$	0	0
$125$	0	0
$126$	0	0
$127$	0.959211	0.0851162	0.0425581	−	0.999094i	$-0.486449\pi$
$127$	0.959211	0.0851162	0.0425581	+	0.999094i	$0.486449\pi$
$128$	0	0
$129$	9.72727	0.856438
$130$	0	0
$131$	−3.07276	−0.268468	−0.134234	−	0.990950i	$-0.542857\pi$
$131$	−3.07276	−0.268468	−0.134234	+	0.990950i	$0.542857\pi$
$132$	0	0
$133$	7.84081	0.679885
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−2.55926	−0.218652	−0.109326	−	0.994006i	$-0.534869\pi$
$137$	−2.55926	−0.218652	−0.109326	+	0.994006i	$0.534869\pi$
$138$	0	0
$139$	22.9366	1.94545	0.972727	−	0.231954i	$-0.0745117\pi$
$139$	22.9366	1.94545	0.972727	+	0.231954i	$0.0745117\pi$
$140$	0	0
$141$	2.92724	0.246518
$142$	0	0
$143$	16.7705	1.40242
$144$	0	0
$145$	0	0
$146$	0	0
$147$	5.25008	0.433020
$148$	0	0
$149$	6.53214	0.535134	0.267567	−	0.963539i	$-0.413780\pi$
$149$	6.53214	0.535134	0.267567	+	0.963539i	$0.413780\pi$
$150$	0	0
$151$	2.32042	0.188833	0.0944165	−	0.995533i	$-0.469901\pi$
$151$	2.32042	0.188833	0.0944165	+	0.995533i	$0.469901\pi$
$152$	0	0
$153$	2.75924	0.223071
$154$	0	0
$155$	0	0
$156$	0	0
$157$	2.35239	0.187741	0.0938705	−	0.995584i	$-0.470076\pi$
$157$	2.35239	0.187741	0.0938705	+	0.995584i	$0.470076\pi$
$158$	0	0
$159$	−11.2501	−0.892190
$160$	0	0
$161$	−1.32284	−0.104255
$162$	0	0
$163$	15.6156	1.22311	0.611556	−	0.791201i	$-0.290544\pi$
$163$	15.6156	1.22311	0.611556	+	0.791201i	$0.290544\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−2.15869	−0.167044	−0.0835221	−	0.996506i	$-0.526617\pi$
$167$	−2.15869	−0.167044	−0.0835221	+	0.996506i	$0.526617\pi$
$168$	0	0
$169$	6.04129	0.464715
$170$	0	0
$171$	−5.92724	−0.453267
$172$	0	0
$173$	−22.4250	−1.70494	−0.852470	−	0.522776i	$-0.824896\pi$
$173$	−22.4250	−1.70494	−0.852470	+	0.522776i	$0.824896\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	3.08208	0.231663
$178$	0	0
$179$	9.25008	0.691384	0.345692	−	0.938348i	$-0.387644\pi$
$179$	9.25008	0.691384	0.345692	+	0.938348i	$0.387644\pi$
$180$	0	0
$181$	−15.3341	−1.13977	−0.569887	−	0.821723i	$-0.693013\pi$
$181$	−15.3341	−1.13977	−0.569887	+	0.821723i	$0.693013\pi$
$182$	0	0
$183$	6.08400	0.449742
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−10.6044	−0.775470
$188$	0	0
$189$	1.32284	0.0962227
$190$	0	0
$191$	−11.5204	−0.833586	−0.416793	−	0.909001i	$-0.636846\pi$
$191$	−11.5204	−0.833586	−0.416793	+	0.909001i	$0.636846\pi$
$192$	0	0
$193$	−2.40685	−0.173249	−0.0866243	−	0.996241i	$-0.527608\pi$
$193$	−2.40685	−0.173249	−0.0866243	+	0.996241i	$0.527608\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−4.61564	−0.328851	−0.164425	−	0.986390i	$-0.552577\pi$
$197$	−4.61564	−0.328851	−0.164425	+	0.986390i	$0.552577\pi$
$198$	0	0
$199$	25.7779	1.82735	0.913673	−	0.406451i	$-0.133234\pi$
$199$	25.7779	1.82735	0.913673	+	0.406451i	$0.133234\pi$
$200$	0	0
$201$	−2.19755	−0.155003
$202$	0	0
$203$	−1.00689	−0.0706702
$204$	0	0
$205$	0	0
$206$	0	0
$207$	1.00000	0.0695048
$208$	0	0
$209$	22.7798	1.57571
$210$	0	0
$211$	−25.2663	−1.73940	−0.869702	−	0.493577i	$-0.835689\pi$
$211$	−25.2663	−1.73940	−0.869702	+	0.493577i	$0.835689\pi$
$212$	0	0
$213$	1.39560	0.0956251
$214$	0	0
$215$	0	0
$216$	0	0
$217$	11.5480	0.783930
$218$	0	0
$219$	−9.20687	−0.622143
$220$	0	0
$221$	−12.0403	−0.809917
$222$	0	0
$223$	−7.08897	−0.474713	−0.237356	−	0.971423i	$-0.576281\pi$
$223$	−7.08897	−0.474713	−0.237356	+	0.971423i	$0.576281\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	6.05446	0.401849	0.200924	−	0.979607i	$-0.435605\pi$
$227$	6.05446	0.401849	0.200924	+	0.979607i	$0.435605\pi$
$228$	0	0
$229$	−0.488425	−0.0322760	−0.0161380	−	0.999870i	$-0.505137\pi$
$229$	−0.488425	−0.0322760	−0.0161380	+	0.999870i	$0.505137\pi$
$230$	0	0
$231$	−5.08400	−0.334503
$232$	0	0
$233$	30.5114	1.99887	0.999435	−	0.0336230i	$-0.0107046\pi$
$233$	30.5114	1.99887	0.999435	+	0.0336230i	$0.0107046\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−14.3341	−0.931099
$238$	0	0
$239$	14.6457	0.947351	0.473675	−	0.880700i	$-0.342927\pi$
$239$	14.6457	0.947351	0.473675	+	0.880700i	$0.342927\pi$
$240$	0	0
$241$	17.1754	1.10636	0.553182	−	0.833060i	$-0.313413\pi$
$241$	17.1754	1.10636	0.553182	+	0.833060i	$0.313413\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	25.8643	1.64571
$248$	0	0
$249$	−3.03886	−0.192580
$250$	0	0
$251$	−22.5453	−1.42305	−0.711524	−	0.702662i	$-0.751995\pi$
$251$	−22.5453	−1.42305	−0.711524	+	0.702662i	$0.751995\pi$
$252$	0	0
$253$	−3.84324	−0.241622
$254$	0	0
$255$	0	0
$256$	0	0
$257$	16.5724	1.03376	0.516880	−	0.856058i	$-0.327093\pi$
$257$	16.5724	1.03376	0.516880	+	0.856058i	$0.327093\pi$
$258$	0	0
$259$	−4.28703	−0.266383
$260$	0	0
$261$	0.761159	0.0471146
$262$	0	0
$263$	10.7995	0.665927	0.332964	−	0.942940i	$-0.391951\pi$
$263$	10.7995	0.665927	0.332964	+	0.942940i	$0.391951\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	18.3002	1.11995
$268$	0	0
$269$	−7.32427	−0.446568	−0.223284	−	0.974753i	$-0.571678\pi$
$269$	−7.32427	−0.446568	−0.223284	+	0.974753i	$0.571678\pi$
$270$	0	0
$271$	12.0506	0.732022	0.366011	−	0.930610i	$-0.380723\pi$
$271$	12.0506	0.732022	0.366011	+	0.930610i	$0.380723\pi$
$272$	0	0
$273$	−5.77240	−0.349362
$274$	0	0
$275$	0	0
$276$	0	0
$277$	17.9995	1.08149	0.540743	−	0.841188i	$-0.318143\pi$
$277$	17.9995	1.08149	0.540743	+	0.841188i	$0.318143\pi$
$278$	0	0
$279$	−8.72969	−0.522633
$280$	0	0
$281$	−10.9234	−0.651635	−0.325817	−	0.945433i	$-0.605639\pi$
$281$	−10.9234	−0.651635	−0.325817	+	0.945433i	$0.605639\pi$
$282$	0	0
$283$	6.79810	0.404105	0.202053	−	0.979375i	$-0.435239\pi$
$283$	6.79810	0.404105	0.202053	+	0.979375i	$0.435239\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−5.51107	−0.325308
$288$	0	0
$289$	−9.38662	−0.552154
$290$	0	0
$291$	−5.43832	−0.318800
$292$	0	0
$293$	−22.6324	−1.32220	−0.661098	−	0.750299i	$-0.729909\pi$
$293$	−22.6324	−1.32220	−0.661098	+	0.750299i	$0.729909\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	3.84324	0.223007
$298$	0	0
$299$	−4.36363	−0.252355
$300$	0	0
$301$	12.8677	0.741679
$302$	0	0
$303$	15.5316	0.892269
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−25.0167	−1.42778	−0.713890	−	0.700258i	$-0.753068\pi$
$307$	−25.0167	−1.42778	−0.713890	+	0.700258i	$0.753068\pi$
$308$	0	0
$309$	9.09332	0.517301
$310$	0	0
$311$	27.1139	1.53749	0.768744	−	0.639557i	$-0.220882\pi$
$311$	27.1139	1.53749	0.768744	+	0.639557i	$0.220882\pi$
$312$	0	0
$313$	−1.27963	−0.0723289	−0.0361645	−	0.999346i	$-0.511514\pi$
$313$	−1.27963	−0.0723289	−0.0361645	+	0.999346i	$0.511514\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	27.4668	1.54269	0.771344	−	0.636419i	$-0.219585\pi$
$317$	27.4668	1.54269	0.771344	+	0.636419i	$0.219585\pi$
$318$	0	0
$319$	−2.92532	−0.163786
$320$	0	0
$321$	−2.92724	−0.163383
$322$	0	0
$323$	−16.3547	−0.909997
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−13.0525	−0.721807
$328$	0	0
$329$	3.87228	0.213486
$330$	0	0
$331$	−14.2551	−0.783529	−0.391764	−	0.920066i	$-0.628135\pi$
$331$	−14.2551	−0.783529	−0.391764	+	0.920066i	$0.628135\pi$
$332$	0	0
$333$	3.24076	0.177593
$334$	0	0
$335$	0	0
$336$	0	0
$337$	12.3243	0.671346	0.335673	−	0.941979i	$-0.391036\pi$
$337$	12.3243	0.671346	0.335673	+	0.941979i	$0.391036\pi$
$338$	0	0
$339$	−17.0501	−0.926035
$340$	0	0
$341$	33.5503	1.81685
$342$	0	0
$343$	16.2049	0.874985
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−30.6643	−1.64615	−0.823074	−	0.567935i	$-0.807743\pi$
$347$	−30.6643	−1.64615	−0.823074	+	0.567935i	$0.807743\pi$
$348$	0	0
$349$	14.0835	0.753873	0.376936	−	0.926239i	$-0.376978\pi$
$349$	14.0835	0.753873	0.376936	+	0.926239i	$0.376978\pi$
$350$	0	0
$351$	4.36363	0.232913
$352$	0	0
$353$	−26.9842	−1.43623	−0.718113	−	0.695926i	$-0.754994\pi$
$353$	−26.9842	−1.43623	−0.718113	+	0.695926i	$0.754994\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	3.65004	0.193180
$358$	0	0
$359$	−28.4343	−1.50071	−0.750353	−	0.661038i	$-0.770116\pi$
$359$	−28.4343	−1.50071	−0.750353	+	0.661038i	$0.770116\pi$
$360$	0	0
$361$	16.1322	0.849063
$362$	0	0
$363$	−3.77048	−0.197899
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−9.24574	−0.482623	−0.241312	−	0.970448i	$-0.577578\pi$
$367$	−9.24574	−0.482623	−0.241312	+	0.970448i	$0.577578\pi$
$368$	0	0
$369$	4.16608	0.216878
$370$	0	0
$371$	−14.8821	−0.772640
$372$	0	0
$373$	22.0107	1.13967	0.569837	−	0.821758i	$-0.307007\pi$
$373$	22.0107	1.13967	0.569837	+	0.821758i	$0.307007\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−3.32142	−0.171062
$378$	0	0
$379$	25.0983	1.28921	0.644606	−	0.764515i	$-0.277021\pi$
$379$	25.0983	1.28921	0.644606	+	0.764515i	$0.277021\pi$
$380$	0	0
$381$	−0.959211	−0.0491419
$382$	0	0
$383$	27.6113	1.41087	0.705436	−	0.708774i	$-0.250751\pi$
$383$	27.6113	1.41087	0.705436	+	0.708774i	$0.250751\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−9.72727	−0.494465
$388$	0	0
$389$	−0.581909	−0.0295040	−0.0147520	−	0.999891i	$-0.504696\pi$
$389$	−0.581909	−0.0295040	−0.0147520	+	0.999891i	$0.504696\pi$
$390$	0	0
$391$	2.75924	0.139541
$392$	0	0
$393$	3.07276	0.155000
$394$	0	0
$395$	0	0
$396$	0	0
$397$	19.1622	0.961725	0.480862	−	0.876796i	$-0.340324\pi$
$397$	19.1622	0.961725	0.480862	+	0.876796i	$0.340324\pi$
$398$	0	0
$399$	−7.84081	−0.392532
$400$	0	0
$401$	11.0859	0.553605	0.276802	−	0.960927i	$-0.410725\pi$
$401$	11.0859	0.553605	0.276802	+	0.960927i	$0.410725\pi$
$402$	0	0
$403$	38.0932	1.89756
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−12.4550	−0.617373
$408$	0	0
$409$	4.07661	0.201575	0.100788	−	0.994908i	$-0.467864\pi$
$409$	4.07661	0.201575	0.100788	+	0.994908i	$0.467864\pi$
$410$	0	0
$411$	2.55926	0.126239
$412$	0	0
$413$	4.07711	0.200621
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−22.9366	−1.12321
$418$	0	0
$419$	−13.1656	−0.643181	−0.321590	−	0.946879i	$-0.604217\pi$
$419$	−13.1656	−0.643181	−0.321590	+	0.946879i	$0.604217\pi$
$420$	0	0
$421$	−10.7404	−0.523457	−0.261728	−	0.965142i	$-0.584292\pi$
$421$	−10.7404	−0.523457	−0.261728	+	0.965142i	$0.584292\pi$
$422$	0	0
$423$	−2.92724	−0.142327
$424$	0	0
$425$	0	0
$426$	0	0
$427$	8.04818	0.389479
$428$	0	0
$429$	−16.7705	−0.809686
$430$	0	0
$431$	−27.3458	−1.31720	−0.658601	−	0.752492i	$-0.728852\pi$
$431$	−27.3458	−1.31720	−0.658601	+	0.752492i	$0.728852\pi$
$432$	0	0
$433$	22.4030	1.07662	0.538310	−	0.842747i	$-0.319063\pi$
$433$	22.4030	1.07662	0.538310	+	0.842747i	$0.319063\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−5.92724	−0.283538
$438$	0	0
$439$	12.6339	0.602985	0.301493	−	0.953469i	$-0.402515\pi$
$439$	12.6339	0.602985	0.301493	+	0.953469i	$0.402515\pi$
$440$	0	0
$441$	−5.25008	−0.250004
$442$	0	0
$443$	14.2506	0.677066	0.338533	−	0.940955i	$-0.390069\pi$
$443$	14.2506	0.677066	0.338533	+	0.940955i	$0.390069\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−6.53214	−0.308960
$448$	0	0
$449$	38.0589	1.79611	0.898056	−	0.439881i	$-0.144979\pi$
$449$	38.0589	1.79611	0.898056	+	0.439881i	$0.144979\pi$
$450$	0	0
$451$	−16.0112	−0.753940
$452$	0	0
$453$	−2.32042	−0.109023
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−8.18665	−0.382955	−0.191478	−	0.981497i	$-0.561328\pi$
$457$	−8.18665	−0.382955	−0.191478	+	0.981497i	$0.561328\pi$
$458$	0	0
$459$	−2.75924	−0.128790
$460$	0	0
$461$	28.6521	1.33446	0.667230	−	0.744852i	$-0.267480\pi$
$461$	28.6521	1.33446	0.667230	+	0.744852i	$0.267480\pi$
$462$	0	0
$463$	3.24624	0.150865	0.0754327	−	0.997151i	$-0.475966\pi$
$463$	3.24624	0.150865	0.0754327	+	0.997151i	$0.475966\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	35.7576	1.65467	0.827333	−	0.561711i	$-0.189857\pi$
$467$	35.7576	1.65467	0.827333	+	0.561711i	$0.189857\pi$
$468$	0	0
$469$	−2.90702	−0.134234
$470$	0	0
$471$	−2.35239	−0.108392
$472$	0	0
$473$	37.3842	1.71893
$474$	0	0
$475$	0	0
$476$	0	0
$477$	11.2501	0.515106
$478$	0	0
$479$	17.6564	0.806743	0.403371	−	0.915036i	$-0.367838\pi$
$479$	17.6564	0.806743	0.403371	+	0.915036i	$0.367838\pi$
$480$	0	0
$481$	−14.1415	−0.644797
$482$	0	0
$483$	1.32284	0.0601915
$484$	0	0
$485$	0	0
$486$	0	0
$487$	13.1847	0.597457	0.298728	−	0.954338i	$-0.403438\pi$
$487$	13.1847	0.597457	0.298728	+	0.954338i	$0.403438\pi$
$488$	0	0
$489$	−15.6156	−0.706164
$490$	0	0
$491$	27.3503	1.23430	0.617151	−	0.786845i	$-0.288287\pi$
$491$	27.3503	1.23430	0.617151	+	0.786845i	$0.288287\pi$
$492$	0	0
$493$	2.10022	0.0945891
$494$	0	0
$495$	0	0
$496$	0	0
$497$	1.84616	0.0828118
$498$	0	0
$499$	17.1155	0.766194	0.383097	−	0.923708i	$-0.374858\pi$
$499$	17.1155	0.766194	0.383097	+	0.923708i	$0.374858\pi$
$500$	0	0
$501$	2.15869	0.0964430
$502$	0	0
$503$	28.1031	1.25306	0.626529	−	0.779398i	$-0.284475\pi$
$503$	28.1031	1.25306	0.626529	+	0.779398i	$0.284475\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−6.04129	−0.268303
$508$	0	0
$509$	−23.2462	−1.03037	−0.515186	−	0.857079i	$-0.672277\pi$
$509$	−23.2462	−1.03037	−0.515186	+	0.857079i	$0.672277\pi$
$510$	0	0
$511$	−12.1793	−0.538778
$512$	0	0
$513$	5.92724	0.261694
$514$	0	0
$515$	0	0
$516$	0	0
$517$	11.2501	0.494778
$518$	0	0
$519$	22.4250	0.984348
$520$	0	0
$521$	−44.5374	−1.95122	−0.975611	−	0.219509i	$-0.929555\pi$
$521$	−44.5374	−1.95122	−0.975611	+	0.219509i	$0.929555\pi$
$522$	0	0
$523$	1.47383	0.0644462	0.0322231	−	0.999481i	$-0.489741\pi$
$523$	1.47383	0.0644462	0.0322231	+	0.999481i	$0.489741\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−24.0873	−1.04926
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−3.08208	−0.133751
$532$	0	0
$533$	−18.1793	−0.787431
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−9.25008	−0.399171
$538$	0	0
$539$	20.1773	0.869099
$540$	0	0
$541$	3.84566	0.165338	0.0826690	−	0.996577i	$-0.473656\pi$
$541$	3.84566	0.165338	0.0826690	+	0.996577i	$0.473656\pi$
$542$	0	0
$543$	15.3341	0.658049
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−6.03097	−0.257866	−0.128933	−	0.991653i	$-0.541155\pi$
$547$	−6.03097	−0.257866	−0.128933	+	0.991653i	$0.541155\pi$
$548$	0	0
$549$	−6.08400	−0.259659
$550$	0	0
$551$	−4.51158	−0.192200
$552$	0	0
$553$	−18.9618	−0.806336
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−20.9139	−0.886151	−0.443075	−	0.896484i	$-0.646113\pi$
$557$	−20.9139	−0.886151	−0.443075	+	0.896484i	$0.646113\pi$
$558$	0	0
$559$	42.4462	1.79528
$560$	0	0
$561$	10.6044	0.447718
$562$	0	0
$563$	14.3744	0.605808	0.302904	−	0.953021i	$-0.402044\pi$
$563$	14.3744	0.605808	0.302904	+	0.953021i	$0.402044\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−1.32284	−0.0555542
$568$	0	0
$569$	16.1360	0.676458	0.338229	−	0.941064i	$-0.390172\pi$
$569$	16.1360	0.676458	0.338229	+	0.941064i	$0.390172\pi$
$570$	0	0
$571$	−16.3478	−0.684132	−0.342066	−	0.939676i	$-0.611127\pi$
$571$	−16.3478	−0.684132	−0.342066	+	0.939676i	$0.611127\pi$
$572$	0	0
$573$	11.5204	0.481271
$574$	0	0
$575$	0	0
$576$	0	0
$577$	33.9046	1.41147	0.705733	−	0.708478i	$-0.250618\pi$
$577$	33.9046	1.41147	0.705733	+	0.708478i	$0.250618\pi$
$578$	0	0
$579$	2.40685	0.100025
$580$	0	0
$581$	−4.01994	−0.166775
$582$	0	0
$583$	−43.2368	−1.79068
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−21.7045	−0.895840	−0.447920	−	0.894074i	$-0.647835\pi$
$587$	−21.7045	−0.895840	−0.447920	+	0.894074i	$0.647835\pi$
$588$	0	0
$589$	51.7430	2.13203
$590$	0	0
$591$	4.61564	0.189862
$592$	0	0
$593$	42.5169	1.74596	0.872980	−	0.487757i	$-0.162185\pi$
$593$	42.5169	1.74596	0.872980	+	0.487757i	$0.162185\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−25.7779	−1.05502
$598$	0	0
$599$	−4.94973	−0.202240	−0.101120	−	0.994874i	$-0.532243\pi$
$599$	−4.94973	−0.202240	−0.101120	+	0.994874i	$0.532243\pi$
$600$	0	0
$601$	29.3527	1.19732	0.598661	−	0.801002i	$-0.295700\pi$
$601$	29.3527	1.19732	0.598661	+	0.801002i	$0.295700\pi$
$602$	0	0
$603$	2.19755	0.0894912
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−17.2958	−0.702017	−0.351008	−	0.936372i	$-0.614161\pi$
$607$	−17.2958	−0.702017	−0.351008	+	0.936372i	$0.614161\pi$
$608$	0	0
$609$	1.00689	0.0408014
$610$	0	0
$611$	12.7734	0.516757
$612$	0	0
$613$	−18.3463	−0.741001	−0.370501	−	0.928832i	$-0.620814\pi$
$613$	−18.3463	−0.741001	−0.370501	+	0.928832i	$0.620814\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	25.0959	1.01032	0.505161	−	0.863025i	$-0.331433\pi$
$617$	25.0959	1.01032	0.505161	+	0.863025i	$0.331433\pi$
$618$	0	0
$619$	−40.9115	−1.64437	−0.822186	−	0.569219i	$-0.807246\pi$
$619$	−40.9115	−1.64437	−0.822186	+	0.569219i	$0.807246\pi$
$620$	0	0
$621$	−1.00000	−0.0401286
$622$	0	0
$623$	24.2083	0.969885
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−22.7798	−0.909738
$628$	0	0
$629$	8.94203	0.356542
$630$	0	0
$631$	30.6677	1.22086	0.610430	−	0.792070i	$-0.290996\pi$
$631$	30.6677	1.22086	0.610430	+	0.792070i	$0.290996\pi$
$632$	0	0
$633$	25.2663	1.00425
$634$	0	0
$635$	0	0
$636$	0	0
$637$	22.9094	0.907705
$638$	0	0
$639$	−1.39560	−0.0552092
$640$	0	0
$641$	−0.685976	−0.0270944	−0.0135472	−	0.999908i	$-0.504312\pi$
$641$	−0.685976	−0.0270944	−0.0135472	+	0.999908i	$0.504312\pi$
$642$	0	0
$643$	−12.4879	−0.492476	−0.246238	−	0.969209i	$-0.579194\pi$
$643$	−12.4879	−0.492476	−0.246238	+	0.969209i	$0.579194\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−33.5361	−1.31844	−0.659220	−	0.751950i	$-0.729114\pi$
$647$	−33.5361	−1.31844	−0.659220	+	0.751950i	$0.729114\pi$
$648$	0	0
$649$	11.8452	0.464963
$650$	0	0
$651$	−11.5480	−0.452602
$652$	0	0
$653$	26.9749	1.05561	0.527805	−	0.849365i	$-0.323015\pi$
$653$	26.9749	1.05561	0.527805	+	0.849365i	$0.323015\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	9.20687	0.359194
$658$	0	0
$659$	8.05700	0.313856	0.156928	−	0.987610i	$-0.449841\pi$
$659$	8.05700	0.313856	0.156928	+	0.987610i	$0.449841\pi$
$660$	0	0
$661$	18.5376	0.721029	0.360515	−	0.932754i	$-0.382601\pi$
$661$	18.5376	0.721029	0.360515	+	0.932754i	$0.382601\pi$
$662$	0	0
$663$	12.0403	0.467606
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0.761159	0.0294722
$668$	0	0
$669$	7.08897	0.274076
$670$	0	0
$671$	23.3823	0.902663
$672$	0	0
$673$	−22.9391	−0.884238	−0.442119	−	0.896957i	$-0.645773\pi$
$673$	−22.9391	−0.884238	−0.442119	+	0.896957i	$0.645773\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−2.25991	−0.0868552	−0.0434276	−	0.999057i	$-0.513828\pi$
$677$	−2.25991	−0.0868552	−0.0434276	+	0.999057i	$0.513828\pi$
$678$	0	0
$679$	−7.19404	−0.276082
$680$	0	0
$681$	−6.05446	−0.232007
$682$	0	0
$683$	−34.5498	−1.32201	−0.661005	−	0.750381i	$-0.729870\pi$
$683$	−34.5498	−1.32201	−0.661005	+	0.750381i	$0.729870\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0.488425	0.0186346
$688$	0	0
$689$	−49.0912	−1.87023
$690$	0	0
$691$	25.1768	0.957772	0.478886	−	0.877877i	$-0.341041\pi$
$691$	25.1768	0.957772	0.478886	+	0.877877i	$0.341041\pi$
$692$	0	0
$693$	5.08400	0.193125
$694$	0	0
$695$	0	0
$696$	0	0
$697$	11.4952	0.435412
$698$	0	0
$699$	−30.5114	−1.15405
$700$	0	0
$701$	−10.0458	−0.379423	−0.189712	−	0.981840i	$-0.560755\pi$
$701$	−10.0458	−0.379423	−0.189712	+	0.981840i	$0.560755\pi$
$702$	0	0
$703$	−19.2088	−0.724473
$704$	0	0
$705$	0	0
$706$	0	0
$707$	20.5459	0.772709
$708$	0	0
$709$	8.43882	0.316926	0.158463	−	0.987365i	$-0.449346\pi$
$709$	8.43882	0.316926	0.158463	+	0.987365i	$0.449346\pi$
$710$	0	0
$711$	14.3341	0.537570
$712$	0	0
$713$	−8.72969	−0.326929
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−14.6457	−0.546953
$718$	0	0
$719$	−10.4271	−0.388864	−0.194432	−	0.980916i	$-0.562286\pi$
$719$	−10.4271	−0.388864	−0.194432	+	0.980916i	$0.562286\pi$
$720$	0	0
$721$	12.0290	0.447985
$722$	0	0
$723$	−17.1754	−0.638760
$724$	0	0
$725$	0	0
$726$	0	0
$727$	44.2442	1.64092	0.820462	−	0.571701i	$-0.193716\pi$
$727$	44.2442	1.64092	0.820462	+	0.571701i	$0.193716\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−26.8398	−0.992706
$732$	0	0
$733$	−41.4236	−1.53001	−0.765007	−	0.644022i	$-0.777265\pi$
$733$	−41.4236	−1.53001	−0.765007	+	0.644022i	$0.777265\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−8.44571	−0.311102
$738$	0	0
$739$	2.91792	0.107337	0.0536687	−	0.998559i	$-0.482908\pi$
$739$	2.91792	0.107337	0.0536687	+	0.998559i	$0.482908\pi$
$740$	0	0
$741$	−25.8643	−0.950149
$742$	0	0
$743$	−0.285101	−0.0104593	−0.00522967	−	0.999986i	$-0.501665\pi$
$743$	−0.285101	−0.0104593	−0.00522967	+	0.999986i	$0.501665\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	3.03886	0.111186
$748$	0	0
$749$	−3.87228	−0.141490
$750$	0	0
$751$	−38.8978	−1.41940	−0.709701	−	0.704503i	$-0.751170\pi$
$751$	−38.8978	−1.41940	−0.709701	+	0.704503i	$0.751170\pi$
$752$	0	0
$753$	22.5453	0.821597
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−35.8116	−1.30160	−0.650798	−	0.759251i	$-0.725565\pi$
$757$	−35.8116	−1.30160	−0.650798	+	0.759251i	$0.725565\pi$
$758$	0	0
$759$	3.84324	0.139501
$760$	0	0
$761$	−50.4529	−1.82892	−0.914459	−	0.404679i	$-0.867383\pi$
$761$	−50.4529	−1.82892	−0.914459	+	0.404679i	$0.867383\pi$
$762$	0	0
$763$	−17.2665	−0.625088
$764$	0	0
$765$	0	0
$766$	0	0
$767$	13.4491	0.485617
$768$	0	0
$769$	−0.197551	−0.00712387	−0.00356193	−	0.999994i	$-0.501134\pi$
$769$	−0.197551	−0.00712387	−0.00356193	+	0.999994i	$0.501134\pi$
$770$	0	0
$771$	−16.5724	−0.596841
$772$	0	0
$773$	14.8914	0.535607	0.267804	−	0.963474i	$-0.413702\pi$
$773$	14.8914	0.535607	0.267804	+	0.963474i	$0.413702\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	4.28703	0.153796
$778$	0	0
$779$	−24.6934	−0.884732
$780$	0	0
$781$	5.36363	0.191926
$782$	0	0
$783$	−0.761159	−0.0272016
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−30.0643	−1.07168	−0.535838	−	0.844321i	$-0.680004\pi$
$787$	−30.0643	−1.07168	−0.535838	+	0.844321i	$0.680004\pi$
$788$	0	0
$789$	−10.7995	−0.384473
$790$	0	0
$791$	−22.5546	−0.801950
$792$	0	0
$793$	26.5484	0.942760
$794$	0	0
$795$	0	0
$796$	0	0
$797$	20.6275	0.730664	0.365332	−	0.930877i	$-0.380956\pi$
$797$	20.6275	0.730664	0.365332	+	0.930877i	$0.380956\pi$
$798$	0	0
$799$	−8.07695	−0.285742
$800$	0	0
$801$	−18.3002	−0.646606
$802$	0	0
$803$	−35.3842	−1.24868
$804$	0	0
$805$	0	0
$806$	0	0
$807$	7.32427	0.257826
$808$	0	0
$809$	−6.46097	−0.227155	−0.113578	−	0.993529i	$-0.536231\pi$
$809$	−6.46097	−0.227155	−0.113578	+	0.993529i	$0.536231\pi$
$810$	0	0
$811$	43.4072	1.52423	0.762116	−	0.647440i	$-0.224160\pi$
$811$	43.4072	1.52423	0.762116	+	0.647440i	$0.224160\pi$
$812$	0	0
$813$	−12.0506	−0.422633
$814$	0	0
$815$	0	0
$816$	0	0
$817$	57.6558	2.01712
$818$	0	0
$819$	5.77240	0.201704
$820$	0	0
$821$	1.93414	0.0675018	0.0337509	−	0.999430i	$-0.489255\pi$
$821$	1.93414	0.0675018	0.0337509	+	0.999430i	$0.489255\pi$
$822$	0	0
$823$	22.5890	0.787404	0.393702	−	0.919238i	$-0.371194\pi$
$823$	22.5890	0.787404	0.393702	+	0.919238i	$0.371194\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−43.2069	−1.50245	−0.751225	−	0.660046i	$-0.770537\pi$
$827$	−43.2069	−1.50245	−0.751225	+	0.660046i	$0.770537\pi$
$828$	0	0
$829$	39.5983	1.37531	0.687654	−	0.726039i	$-0.258641\pi$
$829$	39.5983	1.37531	0.687654	+	0.726039i	$0.258641\pi$
$830$	0	0
$831$	−17.9995	−0.624396
$832$	0	0
$833$	−14.4862	−0.501918
$834$	0	0
$835$	0	0
$836$	0	0
$837$	8.72969	0.301742
$838$	0	0
$839$	−4.33425	−0.149635	−0.0748175	−	0.997197i	$-0.523837\pi$
$839$	−4.33425	−0.149635	−0.0748175	+	0.997197i	$0.523837\pi$
$840$	0	0
$841$	−28.4206	−0.980022
$842$	0	0
$843$	10.9234	0.376222
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−4.98775	−0.171381
$848$	0	0
$849$	−6.79810	−0.233310
$850$	0	0
$851$	3.24076	0.111092
$852$	0	0
$853$	−30.5090	−1.04461	−0.522304	−	0.852759i	$-0.674927\pi$
$853$	−30.5090	−1.04461	−0.522304	+	0.852759i	$0.674927\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−0.909223	−0.0310585	−0.0155292	−	0.999879i	$-0.504943\pi$
$857$	−0.909223	−0.0310585	−0.0155292	+	0.999879i	$0.504943\pi$
$858$	0	0
$859$	−13.8442	−0.472357	−0.236178	−	0.971710i	$-0.575895\pi$
$859$	−13.8442	−0.472357	−0.236178	+	0.971710i	$0.575895\pi$
$860$	0	0
$861$	5.51107	0.187817
$862$	0	0
$863$	−16.6767	−0.567680	−0.283840	−	0.958872i	$-0.591608\pi$
$863$	−16.6767	−0.567680	−0.283840	+	0.958872i	$0.591608\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	9.38662	0.318786
$868$	0	0
$869$	−55.0893	−1.86878
$870$	0	0
$871$	−9.58930	−0.324921
$872$	0	0
$873$	5.43832	0.184059
$874$	0	0
$875$	0	0
$876$	0	0
$877$	53.2485	1.79807	0.899037	−	0.437873i	$-0.144268\pi$
$877$	53.2485	1.79807	0.899037	+	0.437873i	$0.144268\pi$
$878$	0	0
$879$	22.6324	0.763370
$880$	0	0
$881$	43.7812	1.47503	0.737513	−	0.675332i	$-0.236000\pi$
$881$	43.7812	1.47503	0.737513	+	0.675332i	$0.236000\pi$
$882$	0	0
$883$	−17.9524	−0.604148	−0.302074	−	0.953285i	$-0.597679\pi$
$883$	−17.9524	−0.604148	−0.302074	+	0.953285i	$0.597679\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	53.4631	1.79511	0.897557	−	0.440898i	$-0.145340\pi$
$887$	53.4631	1.79511	0.897557	+	0.440898i	$0.145340\pi$
$888$	0	0
$889$	−1.26889	−0.0425571
$890$	0	0
$891$	−3.84324	−0.128753
$892$	0	0
$893$	17.3505	0.580611
$894$	0	0
$895$	0	0
$896$	0	0
$897$	4.36363	0.145697
$898$	0	0
$899$	−6.64469	−0.221613
$900$	0	0
$901$	31.0416	1.03415
$902$	0	0
$903$	−12.8677	−0.428209
$904$	0	0
$905$	0	0
$906$	0	0
$907$	23.9651	0.795748	0.397874	−	0.917440i	$-0.369748\pi$
$907$	23.9651	0.795748	0.397874	+	0.917440i	$0.369748\pi$
$908$	0	0
$909$	−15.5316	−0.515152
$910$	0	0
$911$	8.52073	0.282304	0.141152	−	0.989988i	$-0.454919\pi$
$911$	8.52073	0.282304	0.141152	+	0.989988i	$0.454919\pi$
$912$	0	0
$913$	−11.6791	−0.386521
$914$	0	0
$915$	0	0
$916$	0	0
$917$	4.06478	0.134231
$918$	0	0
$919$	−6.59812	−0.217652	−0.108826	−	0.994061i	$-0.534709\pi$
$919$	−6.59812	−0.217652	−0.108826	+	0.994061i	$0.534709\pi$
$920$	0	0
$921$	25.0167	0.824329
$922$	0	0
$923$	6.08990	0.200451
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−9.09332	−0.298664
$928$	0	0
$929$	1.28398	0.0421260	0.0210630	−	0.999778i	$-0.493295\pi$
$929$	1.28398	0.0421260	0.0210630	+	0.999778i	$0.493295\pi$
$930$	0	0
$931$	31.1185	1.01987
$932$	0	0
$933$	−27.1139	−0.887669
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−44.0122	−1.43782	−0.718908	−	0.695105i	$-0.755358\pi$
$937$	−44.0122	−1.43782	−0.718908	+	0.695105i	$0.755358\pi$
$938$	0	0
$939$	1.27963	0.0417591
$940$	0	0
$941$	17.1317	0.558477	0.279239	−	0.960222i	$-0.409918\pi$
$941$	17.1317	0.558477	0.279239	+	0.960222i	$0.409918\pi$
$942$	0	0
$943$	4.16608	0.135666
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−2.30320	−0.0748440	−0.0374220	−	0.999300i	$-0.511915\pi$
$947$	−2.30320	−0.0748440	−0.0374220	+	0.999300i	$0.511915\pi$
$948$	0	0
$949$	−40.1754	−1.30415
$950$	0	0
$951$	−27.4668	−0.890671
$952$	0	0
$953$	−14.8222	−0.480137	−0.240069	−	0.970756i	$-0.577170\pi$
$953$	−14.8222	−0.480137	−0.240069	+	0.970756i	$0.577170\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	2.92532	0.0945621
$958$	0	0
$959$	3.38550	0.109323
$960$	0	0
$961$	45.2075	1.45831
$962$	0	0
$963$	2.92724	0.0943290
$964$	0	0
$965$	0	0
$966$	0	0
$967$	6.98248	0.224541	0.112271	−	0.993678i	$-0.464188\pi$
$967$	6.98248	0.224541	0.112271	+	0.993678i	$0.464188\pi$
$968$	0	0
$969$	16.3547	0.525387
$970$	0	0
$971$	14.3885	0.461750	0.230875	−	0.972983i	$-0.425841\pi$
$971$	14.3885	0.461750	0.230875	+	0.972983i	$0.425841\pi$
$972$	0	0
$973$	−30.3415	−0.972703
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−55.0868	−1.76238	−0.881191	−	0.472761i	$-0.843258\pi$
$977$	−55.0868	−1.76238	−0.881191	+	0.472761i	$0.843258\pi$
$978$	0	0
$979$	70.3320	2.24782
$980$	0	0
$981$	13.0525	0.416735
$982$	0	0
$983$	1.13846	0.0363113	0.0181556	−	0.999835i	$-0.494221\pi$
$983$	1.13846	0.0363113	0.0181556	+	0.999835i	$0.494221\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−3.87228	−0.123256
$988$	0	0
$989$	−9.72727	−0.309309
$990$	0	0
$991$	−7.33794	−0.233097	−0.116549	−	0.993185i	$-0.537183\pi$
$991$	−7.33794	−0.233097	−0.116549	+	0.993185i	$0.537183\pi$
$992$	0	0
$993$	14.2551	0.452371
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−19.0077	−0.601981	−0.300990	−	0.953627i	$-0.597317\pi$
$997$	−19.0077	−0.601981	−0.300990	+	0.953627i	$0.597317\pi$
$998$	0	0
$999$	−3.24076	−0.102533

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6900.2.a.ba.1.2	✓	4
5.2	odd	4		6900.2.f.s.6349.6		8
5.3	odd	4		6900.2.f.s.6349.3		8
5.4	even	2		6900.2.a.bb.1.3	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6900.2.a.ba.1.2	✓	4	1.1	even	1	trivial
6900.2.a.bb.1.3	yes	4	5.4	even	2
6900.2.f.s.6349.3		8	5.3	odd	4
6900.2.f.s.6349.6		8	5.2	odd	4

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:	\( N \)	\(=\)	\( 6900 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6900.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -1.32284 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -1.32284 q^{7} +1.00000 q^{9} -3.84324 q^{11} -4.36363 q^{13} +2.75924 q^{17} -5.92724 q^{19} +1.32284 q^{21} +1.00000 q^{23} -1.00000 q^{27} +0.761159 q^{29} -8.72969 q^{31} +3.84324 q^{33} +3.24076 q^{37} +4.36363 q^{39} +4.16608 q^{41} -9.72727 q^{43} -2.92724 q^{47} -5.25008 q^{49} -2.75924 q^{51} +11.2501 q^{53} +5.92724 q^{57} -3.08208 q^{59} -6.08400 q^{61} -1.32284 q^{63} +2.19755 q^{67} -1.00000 q^{69} -1.39560 q^{71} +9.20687 q^{73} +5.08400 q^{77} +14.3341 q^{79} +1.00000 q^{81} +3.03886 q^{83} -0.761159 q^{87} -18.3002 q^{89} +5.77240 q^{91} +8.72969 q^{93} +5.43832 q^{97} -3.84324 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{3} + 3 q^{7} + 4 q^{9}+O(q^{10}) \) 4 * q - 4 * q^3 + 3 * q^7 + 4 * q^9	\( 4 q - 4 q^{3} + 3 q^{7} + 4 q^{9} - 2 q^{11} + q^{13} + 10 q^{17} + 2 q^{19} - 3 q^{21} + 4 q^{23} - 4 q^{27} - q^{29} - 6 q^{31} + 2 q^{33} + 14 q^{37} - q^{39} - 5 q^{41} - 2 q^{43} + 14 q^{47} + 13 q^{49} - 10 q^{51} + 11 q^{53} - 2 q^{57} - 3 q^{59} - 12 q^{61} + 3 q^{63} + 12 q^{67} - 4 q^{69} - 23 q^{71} + 5 q^{73} + 8 q^{77} + 11 q^{79} + 4 q^{81} + 5 q^{83} + q^{87} + 6 q^{89} - 19 q^{91} + 6 q^{93} + 26 q^{97} - 2 q^{99}+O(q^{100}) \) 4 * q - 4 * q^3 + 3 * q^7 + 4 * q^9 - 2 * q^11 + q^13 + 10 * q^17 + 2 * q^19 - 3 * q^21 + 4 * q^23 - 4 * q^27 - q^29 - 6 * q^31 + 2 * q^33 + 14 * q^37 - q^39 - 5 * q^41 - 2 * q^43 + 14 * q^47 + 13 * q^49 - 10 * q^51 + 11 * q^53 - 2 * q^57 - 3 * q^59 - 12 * q^61 + 3 * q^63 + 12 * q^67 - 4 * q^69 - 23 * q^71 + 5 * q^73 + 8 * q^77 + 11 * q^79 + 4 * q^81 + 5 * q^83 + q^87 + 6 * q^89 - 19 * q^91 + 6 * q^93 + 26 * q^97 - 2 * q^99

Introduction

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