LMFDB - Embedded newform 4725.2.a.bi.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4725,2,Mod(1,4725)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4725, 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("4725.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4725 = 3^{3} \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4725.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:	$37.7293149551$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.257.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 4x + 3 $ x^3 - x^2 - 4*x + 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$0.713538$ of defining polynomial
Character	$\chi$	$=$	4725.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.49086 q^{2} +4.20440 q^{4} -1.00000 q^{7} -5.49086 q^{8} -0.713538 q^{11} -0.777326 q^{13} +2.49086 q^{14} +5.26819 q^{16} -4.06379 q^{17} +2.35025 q^{19} +1.77733 q^{22} -3.20440 q^{23} +1.93621 q^{26} -4.20440 q^{28} +3.20440 q^{29} +5.75905 q^{31} -2.14061 q^{32} +10.1223 q^{34} +2.63148 q^{37} -5.85415 q^{38} -8.54942 q^{41} +7.71354 q^{43} -3.00000 q^{44} +7.98173 q^{46} +3.75905 q^{47} +1.00000 q^{49} -3.26819 q^{52} -11.6953 q^{53} +5.49086 q^{56} -7.98173 q^{58} +12.1041 q^{59} +1.34502 q^{61} -14.3450 q^{62} -5.20440 q^{64} +9.94518 q^{67} -17.0858 q^{68} -0.567690 q^{71} -6.83588 q^{73} -6.55465 q^{74} +9.88139 q^{76} +0.713538 q^{77} +6.83588 q^{79} +21.2954 q^{82} +6.24992 q^{83} -19.2134 q^{86} +3.91794 q^{88} -2.12758 q^{89} +0.777326 q^{91} -13.4726 q^{92} -9.36329 q^{94} -4.85415 q^{97} -2.49086 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 4 q^{4} - 3 q^{7} - 9 q^{8} - q^{11} + 4 q^{13} + 2 q^{16} - 7 q^{17} + 3 q^{19} - q^{22} - q^{23} + 11 q^{26} - 4 q^{28} + q^{29} - 4 q^{31} - 3 q^{32} + 12 q^{34} - 3 q^{37} - 13 q^{38} - 5 q^{41}+ \cdots - 10 q^{97}+O(q^{100}) $ 3 * q + 4 * q^4 - 3 * q^7 - 9 * q^8 - q^11 + 4 * q^13 + 2 * q^16 - 7 * q^17 + 3 * q^19 - q^22 - q^23 + 11 * q^26 - 4 * q^28 + q^29 - 4 * q^31 - 3 * q^32 + 12 * q^34 - 3 * q^37 - 13 * q^38 - 5 * q^41 + 22 * q^43 - 9 * q^44 + 9 * q^46 - 10 * q^47 + 3 * q^49 + 4 * q^52 - 19 * q^53 + 9 * q^56 - 9 * q^58 + 3 * q^59 - 8 * q^61 - 31 * q^62 - 7 * q^64 - 15 * q^67 - 3 * q^68 + 4 * q^71 - q^73 - 7 * q^74 - 10 * q^76 + q^77 + q^79 + 15 * q^82 - 10 * q^83 + q^86 + 2 * q^88 + 4 * q^89 - 4 * q^91 - 18 * q^92 - 31 * q^94 - 10 * q^97

Coefficient data

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

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

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

$2$	−2.49086	−1.76131	−0.880653	−	0.473761i	$-0.842896\pi$
$2$	−2.49086	−1.76131	−0.880653	+	0.473761i	$0.842896\pi$
$3$	0	0
$3$	0	0
$4$	4.20440	2.10220
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−5.49086	−1.94131
$9$	0	0
$10$	0	0
$11$	−0.713538	−0.215140	−0.107570	−	0.994198i	$-0.534307\pi$
$11$	−0.713538	−0.215140	−0.107570	+	0.994198i	$0.534307\pi$
$12$	0	0
$13$	−0.777326	−0.215591	−0.107796	−	0.994173i	$-0.534379\pi$
$13$	−0.777326	−0.215591	−0.107796	+	0.994173i	$0.534379\pi$
$14$	2.49086	0.665711
$15$	0	0
$16$	5.26819	1.31705
$17$	−4.06379	−0.985613	−0.492807	−	0.870139i	$-0.664029\pi$
$17$	−4.06379	−0.985613	−0.492807	+	0.870139i	$0.664029\pi$
$18$	0	0
$19$	2.35025	0.539184	0.269592	−	0.962975i	$-0.413111\pi$
$19$	2.35025	0.539184	0.269592	+	0.962975i	$0.413111\pi$
$20$	0	0
$21$	0	0
$22$	1.77733	0.378927
$23$	−3.20440	−0.668164	−0.334082	−	0.942544i	$-0.608426\pi$
$23$	−3.20440	−0.668164	−0.334082	+	0.942544i	$0.608426\pi$
$24$	0	0
$25$	0	0
$26$	1.93621	0.379722
$27$	0	0
$28$	−4.20440	−0.794557
$29$	3.20440	0.595042	0.297521	−	0.954715i	$-0.403840\pi$
$29$	3.20440	0.595042	0.297521	+	0.954715i	$0.403840\pi$
$30$	0	0
$31$	5.75905	1.03436	0.517178	−	0.855878i	$-0.326982\pi$
$31$	5.75905	1.03436	0.517178	+	0.855878i	$0.326982\pi$
$32$	−2.14061	−0.378411
$33$	0	0
$34$	10.1223	1.73597
$35$	0	0
$36$	0	0
$37$	2.63148	0.432612	0.216306	−	0.976326i	$-0.430599\pi$
$37$	2.63148	0.432612	0.216306	+	0.976326i	$0.430599\pi$
$38$	−5.85415	−0.949669
$39$	0	0
$40$	0	0
$41$	−8.54942	−1.33519	−0.667597	−	0.744523i	$-0.732677\pi$
$41$	−8.54942	−1.33519	−0.667597	+	0.744523i	$0.732677\pi$
$42$	0	0
$43$	7.71354	1.17630	0.588152	−	0.808751i	$-0.299856\pi$
$43$	7.71354	1.17630	0.588152	+	0.808751i	$0.299856\pi$
$44$	−3.00000	−0.452267
$45$	0	0
$46$	7.98173	1.17684
$47$	3.75905	0.548314	0.274157	−	0.961685i	$-0.411601\pi$
$47$	3.75905	0.548314	0.274157	+	0.961685i	$0.411601\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	−3.26819	−0.453216
$53$	−11.6953	−1.60647	−0.803234	−	0.595663i	$-0.796889\pi$
$53$	−11.6953	−1.60647	−0.803234	+	0.595663i	$0.796889\pi$
$54$	0	0
$55$	0	0
$56$	5.49086	0.733748
$57$	0	0
$58$	−7.98173	−1.04805
$59$	12.1041	1.57582	0.787908	−	0.615793i	$-0.211164\pi$
$59$	12.1041	1.57582	0.787908	+	0.615793i	$0.211164\pi$
$60$	0	0
$61$	1.34502	0.172212	0.0861058	−	0.996286i	$-0.472558\pi$
$61$	1.34502	0.172212	0.0861058	+	0.996286i	$0.472558\pi$
$62$	−14.3450	−1.82182
$63$	0	0
$64$	−5.20440	−0.650550
$65$	0	0
$66$	0	0
$67$	9.94518	1.21500	0.607499	−	0.794321i	$-0.292173\pi$
$67$	9.94518	1.21500	0.607499	+	0.794321i	$0.292173\pi$
$68$	−17.0858	−2.07196
$69$	0	0
$70$	0	0
$71$	−0.567690	−0.0673724	−0.0336862	−	0.999432i	$-0.510725\pi$
$71$	−0.567690	−0.0673724	−0.0336862	+	0.999432i	$0.510725\pi$
$72$	0	0
$73$	−6.83588	−0.800079	−0.400040	−	0.916498i	$-0.631004\pi$
$73$	−6.83588	−0.800079	−0.400040	+	0.916498i	$0.631004\pi$
$74$	−6.55465	−0.761963
$75$	0	0
$76$	9.88139	1.13347
$77$	0.713538	0.0813152
$78$	0	0
$79$	6.83588	0.769096	0.384548	−	0.923105i	$-0.374357\pi$
$79$	6.83588	0.769096	0.384548	+	0.923105i	$0.374357\pi$
$80$	0	0
$81$	0	0
$82$	21.2954	2.35169
$83$	6.24992	0.686017	0.343009	−	0.939332i	$-0.388554\pi$
$83$	6.24992	0.686017	0.343009	+	0.939332i	$0.388554\pi$
$84$	0	0
$85$	0	0
$86$	−19.2134	−2.07183
$87$	0	0
$88$	3.91794	0.417654
$89$	−2.12758	−0.225523	−0.112761	−	0.993622i	$-0.535970\pi$
$89$	−2.12758	−0.225523	−0.112761	+	0.993622i	$0.535970\pi$
$90$	0	0
$91$	0.777326	0.0814859
$92$	−13.4726	−1.40461
$93$	0	0
$94$	−9.36329	−0.965749
$95$	0	0
$96$	0	0
$97$	−4.85415	−0.492864	−0.246432	−	0.969160i	$-0.579258\pi$
$97$	−4.85415	−0.492864	−0.246432	+	0.969160i	$0.579258\pi$
$98$	−2.49086	−0.251615
$99$	0	0
$100$	0	0
$101$	7.48563	0.744848	0.372424	−	0.928063i	$-0.378527\pi$
$101$	7.48563	0.744848	0.372424	+	0.928063i	$0.378527\pi$
$102$	0	0
$103$	−18.8814	−1.86044	−0.930220	−	0.367004i	$-0.880384\pi$
$103$	−18.8814	−1.86044	−0.930220	+	0.367004i	$0.880384\pi$
$104$	4.26819	0.418530
$105$	0	0
$106$	29.1313	2.82948
$107$	−16.4271	−1.58807	−0.794033	−	0.607875i	$-0.792022\pi$
$107$	−16.4271	−1.58807	−0.794033	+	0.607875i	$0.792022\pi$
$108$	0	0
$109$	−13.4088	−1.28433	−0.642165	−	0.766566i	$-0.721964\pi$
$109$	−13.4088	−1.28433	−0.642165	+	0.766566i	$0.721964\pi$
$110$	0	0
$111$	0	0
$112$	−5.26819	−0.497797
$113$	−7.61844	−0.716682	−0.358341	−	0.933591i	$-0.616658\pi$
$113$	−7.61844	−0.716682	−0.358341	+	0.933591i	$0.616658\pi$
$114$	0	0
$115$	0	0
$116$	13.4726	1.25090
$117$	0	0
$118$	−30.1496	−2.77549
$119$	4.06379	0.372527
$120$	0	0
$121$	−10.4909	−0.953715
$122$	−3.35025	−0.303317
$123$	0	0
$124$	24.2134	2.17442
$125$	0	0
$126$	0	0
$127$	19.7591	1.75333	0.876666	−	0.481099i	$-0.159762\pi$
$127$	19.7591	1.75333	0.876666	+	0.481099i	$0.159762\pi$
$128$	17.2447	1.52423
$129$	0	0
$130$	0	0
$131$	8.34502	0.729107	0.364554	−	0.931182i	$-0.381222\pi$
$131$	8.34502	0.729107	0.364554	+	0.931182i	$0.381222\pi$
$132$	0	0
$133$	−2.35025	−0.203793
$134$	−24.7721	−2.13998
$135$	0	0
$136$	22.3137	1.91338
$137$	6.30847	0.538969	0.269484	−	0.963005i	$-0.413147\pi$
$137$	6.30847	0.538969	0.269484	+	0.963005i	$0.413147\pi$
$138$	0	0
$139$	−13.8176	−1.17199	−0.585997	−	0.810313i	$-0.699297\pi$
$139$	−13.8176	−1.17199	−0.585997	+	0.810313i	$0.699297\pi$
$140$	0	0
$141$	0	0
$142$	1.41404	0.118663
$143$	0.554651	0.0463823
$144$	0	0
$145$	0	0
$146$	17.0272	1.40918
$147$	0	0
$148$	11.0638	0.909438
$149$	−2.64975	−0.217076	−0.108538	−	0.994092i	$-0.534617\pi$
$149$	−2.64975	−0.217076	−0.108538	+	0.994092i	$0.534617\pi$
$150$	0	0
$151$	−0.140614	−0.0114430	−0.00572149	−	0.999984i	$-0.501821\pi$
$151$	−0.140614	−0.0114430	−0.00572149	+	0.999984i	$0.501821\pi$
$152$	−12.9049	−1.04673
$153$	0	0
$154$	−1.77733	−0.143221
$155$	0	0
$156$	0	0
$157$	12.3905	0.988872	0.494436	−	0.869214i	$-0.335375\pi$
$157$	12.3905	0.988872	0.494436	+	0.869214i	$0.335375\pi$
$158$	−17.0272	−1.35461
$159$	0	0
$160$	0	0
$161$	3.20440	0.252542
$162$	0	0
$163$	−20.2369	−1.58507	−0.792537	−	0.609823i	$-0.791240\pi$
$163$	−20.2369	−1.58507	−0.792537	+	0.609823i	$0.791240\pi$
$164$	−35.9452	−2.80685
$165$	0	0
$166$	−15.5677	−1.20829
$167$	−10.0638	−0.778759	−0.389380	−	0.921077i	$-0.627311\pi$
$167$	−10.0638	−0.778759	−0.389380	+	0.921077i	$0.627311\pi$
$168$	0	0
$169$	−12.3958	−0.953520
$170$	0	0
$171$	0	0
$172$	32.4308	2.47283
$173$	−1.87766	−0.142756	−0.0713779	−	0.997449i	$-0.522740\pi$
$173$	−1.87766	−0.142756	−0.0713779	+	0.997449i	$0.522740\pi$
$174$	0	0
$175$	0	0
$176$	−3.75905	−0.283349
$177$	0	0
$178$	5.29950	0.397214
$179$	−15.0455	−1.12455	−0.562277	−	0.826949i	$-0.690075\pi$
$179$	−15.0455	−1.12455	−0.562277	+	0.826949i	$0.690075\pi$
$180$	0	0
$181$	−2.42708	−0.180403	−0.0902016	−	0.995924i	$-0.528751\pi$
$181$	−2.42708	−0.180403	−0.0902016	+	0.995924i	$0.528751\pi$
$182$	−1.93621	−0.143522
$183$	0	0
$184$	17.5949	1.29712
$185$	0	0
$186$	0	0
$187$	2.89967	0.212045
$188$	15.8046	1.15267
$189$	0	0
$190$	0	0
$191$	−22.2394	−1.60919	−0.804595	−	0.593824i	$-0.797618\pi$
$191$	−22.2394	−1.60919	−0.804595	+	0.593824i	$0.797618\pi$
$192$	0	0
$193$	14.2682	1.02705	0.513523	−	0.858076i	$-0.328340\pi$
$193$	14.2682	1.02705	0.513523	+	0.858076i	$0.328340\pi$
$194$	12.0910	0.868085
$195$	0	0
$196$	4.20440	0.300314
$197$	2.08206	0.148341	0.0741703	−	0.997246i	$-0.476369\pi$
$197$	2.08206	0.148341	0.0741703	+	0.997246i	$0.476369\pi$
$198$	0	0
$199$	−21.2772	−1.50830	−0.754149	−	0.656703i	$-0.771950\pi$
$199$	−21.2772	−1.50830	−0.754149	+	0.656703i	$0.771950\pi$
$200$	0	0
$201$	0	0
$202$	−18.6457	−1.31191
$203$	−3.20440	−0.224905
$204$	0	0
$205$	0	0
$206$	47.0310	3.27680
$207$	0	0
$208$	−4.09510	−0.283944
$209$	−1.67699	−0.116000
$210$	0	0
$211$	7.18613	0.494714	0.247357	−	0.968924i	$-0.420438\pi$
$211$	7.18613	0.494714	0.247357	+	0.968924i	$0.420438\pi$
$212$	−49.1716	−3.37712
$213$	0	0
$214$	40.9176	2.79707
$215$	0	0
$216$	0	0
$217$	−5.75905	−0.390950
$218$	33.3995	2.26210
$219$	0	0
$220$	0	0
$221$	3.15889	0.212490
$222$	0	0
$223$	−26.2081	−1.75503	−0.877513	−	0.479552i	$-0.840799\pi$
$223$	−26.2081	−1.75503	−0.877513	+	0.479552i	$0.840799\pi$
$224$	2.14061	0.143026
$225$	0	0
$226$	18.9765	1.26230
$227$	23.8448	1.58264	0.791319	−	0.611403i	$-0.209395\pi$
$227$	23.8448	1.58264	0.791319	+	0.611403i	$0.209395\pi$
$228$	0	0
$229$	3.58596	0.236967	0.118484	−	0.992956i	$-0.462197\pi$
$229$	3.58596	0.236967	0.118484	+	0.992956i	$0.462197\pi$
$230$	0	0
$231$	0	0
$232$	−17.5949	−1.15516
$233$	27.1626	1.77948	0.889741	−	0.456465i	$-0.150885\pi$
$233$	27.1626	1.77948	0.889741	+	0.456465i	$0.150885\pi$
$234$	0	0
$235$	0	0
$236$	50.8904	3.31268
$237$	0	0
$238$	−10.1223	−0.656134
$239$	−5.54568	−0.358720	−0.179360	−	0.983783i	$-0.557403\pi$
$239$	−5.54568	−0.358720	−0.179360	+	0.983783i	$0.557403\pi$
$240$	0	0
$241$	12.7355	0.820369	0.410184	−	0.912003i	$-0.365464\pi$
$241$	12.7355	0.820369	0.410184	+	0.912003i	$0.365464\pi$
$242$	26.1313	1.67978
$243$	0	0
$244$	5.65498	0.362023
$245$	0	0
$246$	0	0
$247$	−1.82691	−0.116243
$248$	−31.6222	−2.00801
$249$	0	0
$250$	0	0
$251$	−8.88663	−0.560919	−0.280460	−	0.959866i	$-0.590487\pi$
$251$	−8.88663	−0.560919	−0.280460	+	0.959866i	$0.590487\pi$
$252$	0	0
$253$	2.28646	0.143749
$254$	−49.2171	−3.08816
$255$	0	0
$256$	−32.5453	−2.03408
$257$	6.51287	0.406262	0.203131	−	0.979152i	$-0.434888\pi$
$257$	6.51287	0.406262	0.203131	+	0.979152i	$0.434888\pi$
$258$	0	0
$259$	−2.63148	−0.163512
$260$	0	0
$261$	0	0
$262$	−20.7863	−1.28418
$263$	22.0545	1.35994	0.679969	−	0.733241i	$-0.261993\pi$
$263$	22.0545	1.35994	0.679969	+	0.733241i	$0.261993\pi$
$264$	0	0
$265$	0	0
$266$	5.85415	0.358941
$267$	0	0
$268$	41.8135	2.55417
$269$	27.2954	1.66423	0.832116	−	0.554602i	$-0.187129\pi$
$269$	27.2954	1.66423	0.832116	+	0.554602i	$0.187129\pi$
$270$	0	0
$271$	−19.2537	−1.16958	−0.584788	−	0.811186i	$-0.698822\pi$
$271$	−19.2537	−1.16958	−0.584788	+	0.811186i	$0.698822\pi$
$272$	−21.4088	−1.29810
$273$	0	0
$274$	−15.7135	−0.949290
$275$	0	0
$276$	0	0
$277$	16.1951	0.973069	0.486535	−	0.873661i	$-0.338261\pi$
$277$	16.1951	0.973069	0.486535	+	0.873661i	$0.338261\pi$
$278$	34.4178	2.06424
$279$	0	0
$280$	0	0
$281$	−2.02351	−0.120712	−0.0603562	−	0.998177i	$-0.519224\pi$
$281$	−2.02351	−0.120712	−0.0603562	+	0.998177i	$0.519224\pi$
$282$	0	0
$283$	26.6770	1.58578	0.792891	−	0.609363i	$-0.208575\pi$
$283$	26.6770	1.58578	0.792891	+	0.609363i	$0.208575\pi$
$284$	−2.38680	−0.141630
$285$	0	0
$286$	−1.38156	−0.0816934
$287$	8.54942	0.504656
$288$	0	0
$289$	−0.485629	−0.0285664
$290$	0	0
$291$	0	0
$292$	−28.7408	−1.68193
$293$	−9.95042	−0.581310	−0.290655	−	0.956828i	$-0.593873\pi$
$293$	−9.95042	−0.581310	−0.290655	+	0.956828i	$0.593873\pi$
$294$	0	0
$295$	0	0
$296$	−14.4491	−0.839836
$297$	0	0
$298$	6.60017	0.382337
$299$	2.49086	0.144050
$300$	0	0
$301$	−7.71354	−0.444601
$302$	0.350250	0.0201546
$303$	0	0
$304$	12.3816	0.710131
$305$	0	0
$306$	0	0
$307$	−3.37226	−0.192465	−0.0962325	−	0.995359i	$-0.530679\pi$
$307$	−3.37226	−0.192465	−0.0962325	+	0.995359i	$0.530679\pi$
$308$	3.00000	0.170941
$309$	0	0
$310$	0	0
$311$	8.95449	0.507762	0.253881	−	0.967235i	$-0.418293\pi$
$311$	8.95449	0.507762	0.253881	+	0.967235i	$0.418293\pi$
$312$	0	0
$313$	5.89443	0.333173	0.166587	−	0.986027i	$-0.446725\pi$
$313$	5.89443	0.333173	0.166587	+	0.986027i	$0.446725\pi$
$314$	−30.8631	−1.74171
$315$	0	0
$316$	28.7408	1.61680
$317$	1.61844	0.0909006	0.0454503	−	0.998967i	$-0.485528\pi$
$317$	1.61844	0.0909006	0.0454503	+	0.998967i	$0.485528\pi$
$318$	0	0
$319$	−2.28646	−0.128017
$320$	0	0
$321$	0	0
$322$	−7.98173	−0.444804
$323$	−9.55092	−0.531427
$324$	0	0
$325$	0	0
$326$	50.4073	2.79180
$327$	0	0
$328$	46.9437	2.59203
$329$	−3.75905	−0.207243
$330$	0	0
$331$	−17.7225	−0.974117	−0.487059	−	0.873369i	$-0.661930\pi$
$331$	−17.7225	−0.974117	−0.487059	+	0.873369i	$0.661930\pi$
$332$	26.2772	1.44215
$333$	0	0
$334$	25.0675	1.37163
$335$	0	0
$336$	0	0
$337$	−25.4230	−1.38488	−0.692440	−	0.721476i	$-0.743464\pi$
$337$	−25.4230	−1.38488	−0.692440	+	0.721476i	$0.743464\pi$
$338$	30.8762	1.67944
$339$	0	0
$340$	0	0
$341$	−4.10930	−0.222531
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	−42.3540	−2.28357
$345$	0	0
$346$	4.67699	0.251437
$347$	−23.0597	−1.23791	−0.618955	−	0.785426i	$-0.712444\pi$
$347$	−23.0597	−1.23791	−0.618955	+	0.785426i	$0.712444\pi$
$348$	0	0
$349$	2.37633	0.127202	0.0636009	−	0.997975i	$-0.479742\pi$
$349$	2.37633	0.127202	0.0636009	+	0.997975i	$0.479742\pi$
$350$	0	0
$351$	0	0
$352$	1.52741	0.0814112
$353$	−16.6733	−0.887428	−0.443714	−	0.896168i	$-0.646339\pi$
$353$	−16.6733	−0.887428	−0.443714	+	0.896168i	$0.646339\pi$
$354$	0	0
$355$	0	0
$356$	−8.94518	−0.474094
$357$	0	0
$358$	37.4763	1.98069
$359$	7.00523	0.369722	0.184861	−	0.982765i	$-0.440816\pi$
$359$	7.00523	0.369722	0.184861	+	0.982765i	$0.440816\pi$
$360$	0	0
$361$	−13.4763	−0.709280
$362$	6.04551	0.317745
$363$	0	0
$364$	3.26819	0.171300
$365$	0	0
$366$	0	0
$367$	−18.4763	−0.964456	−0.482228	−	0.876046i	$-0.660172\pi$
$367$	−18.4763	−0.964456	−0.482228	+	0.876046i	$0.660172\pi$
$368$	−16.8814	−0.880003
$369$	0	0
$370$	0	0
$371$	11.6953	0.607188
$372$	0	0
$373$	−26.9672	−1.39631	−0.698154	−	0.715948i	$-0.745995\pi$
$373$	−26.9672	−1.39631	−0.698154	+	0.715948i	$0.745995\pi$
$374$	−7.22267	−0.373476
$375$	0	0
$376$	−20.6404	−1.06445
$377$	−2.49086	−0.128286
$378$	0	0
$379$	−15.1443	−0.777913	−0.388956	−	0.921256i	$-0.627164\pi$
$379$	−15.1443	−0.777913	−0.388956	+	0.921256i	$0.627164\pi$
$380$	0	0
$381$	0	0
$382$	55.3954	2.83428
$383$	−32.2354	−1.64715	−0.823575	−	0.567207i	$-0.808024\pi$
$383$	−32.2354	−1.64715	−0.823575	+	0.567207i	$0.808024\pi$
$384$	0	0
$385$	0	0
$386$	−35.5401	−1.80894
$387$	0	0
$388$	−20.4088	−1.03610
$389$	27.0910	1.37357	0.686785	−	0.726861i	$-0.259021\pi$
$389$	27.0910	1.37357	0.686785	+	0.726861i	$0.259021\pi$
$390$	0	0
$391$	13.0220	0.658551
$392$	−5.49086	−0.277330
$393$	0	0
$394$	−5.18613	−0.261273
$395$	0	0
$396$	0	0
$397$	21.2902	1.06852	0.534262	−	0.845319i	$-0.320590\pi$
$397$	21.2902	1.06852	0.534262	+	0.845319i	$0.320590\pi$
$398$	52.9985	2.65657
$399$	0	0
$400$	0	0
$401$	−27.7758	−1.38706	−0.693529	−	0.720428i	$-0.743945\pi$
$401$	−27.7758	−1.38706	−0.693529	+	0.720428i	$0.743945\pi$
$402$	0	0
$403$	−4.47666	−0.222998
$404$	31.4726	1.56582
$405$	0	0
$406$	7.98173	0.396126
$407$	−1.87766	−0.0930721
$408$	0	0
$409$	−18.6404	−0.921711	−0.460855	−	0.887475i	$-0.652457\pi$
$409$	−18.6404	−0.921711	−0.460855	+	0.887475i	$0.652457\pi$
$410$	0	0
$411$	0	0
$412$	−79.3850	−3.91102
$413$	−12.1041	−0.595602
$414$	0	0
$415$	0	0
$416$	1.66395	0.0815821
$417$	0	0
$418$	4.17716	0.204312
$419$	−3.35025	−0.163670	−0.0818352	−	0.996646i	$-0.526078\pi$
$419$	−3.35025	−0.163670	−0.0818352	+	0.996646i	$0.526078\pi$
$420$	0	0
$421$	−6.74078	−0.328526	−0.164263	−	0.986417i	$-0.552524\pi$
$421$	−6.74078	−0.328526	−0.164263	+	0.986417i	$0.552524\pi$
$422$	−17.8997	−0.871342
$423$	0	0
$424$	64.2171	3.11866
$425$	0	0
$426$	0	0
$427$	−1.34502	−0.0650899
$428$	−69.0660	−3.33843
$429$	0	0
$430$	0	0
$431$	11.7445	0.565713	0.282857	−	0.959162i	$-0.408718\pi$
$431$	11.7445	0.565713	0.282857	+	0.959162i	$0.408718\pi$
$432$	0	0
$433$	5.83588	0.280454	0.140227	−	0.990119i	$-0.455217\pi$
$433$	5.83588	0.280454	0.140227	+	0.990119i	$0.455217\pi$
$434$	14.3450	0.688583
$435$	0	0
$436$	−56.3760	−2.69992
$437$	−7.53114	−0.360263
$438$	0	0
$439$	−37.4125	−1.78560	−0.892802	−	0.450450i	$-0.851263\pi$
$439$	−37.4125	−1.78560	−0.892802	+	0.450450i	$0.851263\pi$
$440$	0	0
$441$	0	0
$442$	−7.86836	−0.374260
$443$	−3.25922	−0.154850	−0.0774251	−	0.996998i	$-0.524670\pi$
$443$	−3.25922	−0.154850	−0.0774251	+	0.996998i	$0.524670\pi$
$444$	0	0
$445$	0	0
$446$	65.2809	3.09114
$447$	0	0
$448$	5.20440	0.245885
$449$	−17.5103	−0.826362	−0.413181	−	0.910649i	$-0.635582\pi$
$449$	−17.5103	−0.826362	−0.413181	+	0.910649i	$0.635582\pi$
$450$	0	0
$451$	6.10033	0.287253
$452$	−32.0310	−1.50661
$453$	0	0
$454$	−59.3943	−2.78751
$455$	0	0
$456$	0	0
$457$	31.6718	1.48154	0.740771	−	0.671757i	$-0.234460\pi$
$457$	31.6718	1.48154	0.740771	+	0.671757i	$0.234460\pi$
$458$	−8.93214	−0.417372
$459$	0	0
$460$	0	0
$461$	−21.4256	−0.997889	−0.498944	−	0.866634i	$-0.666279\pi$
$461$	−21.4256	−0.997889	−0.498944	+	0.866634i	$0.666279\pi$
$462$	0	0
$463$	−36.1261	−1.67892	−0.839461	−	0.543421i	$-0.817129\pi$
$463$	−36.1261	−1.67892	−0.839461	+	0.543421i	$0.817129\pi$
$464$	16.8814	0.783699
$465$	0	0
$466$	−67.6584	−3.13421
$467$	−21.2954	−0.985435	−0.492718	−	0.870189i	$-0.663996\pi$
$467$	−21.2954	−0.985435	−0.492718	+	0.870189i	$0.663996\pi$
$468$	0	0
$469$	−9.94518	−0.459226
$470$	0	0
$471$	0	0
$472$	−66.4618	−3.05915
$473$	−5.50390	−0.253070
$474$	0	0
$475$	0	0
$476$	17.0858	0.783126
$477$	0	0
$478$	13.8135	0.631816
$479$	−40.1548	−1.83472	−0.917360	−	0.398058i	$-0.869684\pi$
$479$	−40.1548	−1.83472	−0.917360	+	0.398058i	$0.869684\pi$
$480$	0	0
$481$	−2.04551	−0.0932675
$482$	−31.7225	−1.44492
$483$	0	0
$484$	−44.1078	−2.00490
$485$	0	0
$486$	0	0
$487$	9.46479	0.428890	0.214445	−	0.976736i	$-0.431206\pi$
$487$	9.46479	0.428890	0.214445	+	0.976736i	$0.431206\pi$
$488$	−7.38530	−0.334317
$489$	0	0
$490$	0	0
$491$	−19.4271	−0.876732	−0.438366	−	0.898797i	$-0.644443\pi$
$491$	−19.4271	−0.876732	−0.438366	+	0.898797i	$0.644443\pi$
$492$	0	0
$493$	−13.0220	−0.586482
$494$	4.55058	0.204740
$495$	0	0
$496$	30.3398	1.36230
$497$	0.567690	0.0254644
$498$	0	0
$499$	12.1679	0.544708	0.272354	−	0.962197i	$-0.412198\pi$
$499$	12.1679	0.544708	0.272354	+	0.962197i	$0.412198\pi$
$500$	0	0
$501$	0	0
$502$	22.1354	0.987950
$503$	−14.7408	−0.657259	−0.328629	−	0.944459i	$-0.606587\pi$
$503$	−14.7408	−0.657259	−0.328629	+	0.944459i	$0.606587\pi$
$504$	0	0
$505$	0	0
$506$	−5.69527	−0.253185
$507$	0	0
$508$	83.0750	3.68586
$509$	35.4036	1.56924	0.784618	−	0.619980i	$-0.212859\pi$
$509$	35.4036	1.56924	0.784618	+	0.619980i	$0.212859\pi$
$510$	0	0
$511$	6.83588	0.302401
$512$	46.5767	2.05842
$513$	0	0
$514$	−16.2227	−0.715551
$515$	0	0
$516$	0	0
$517$	−2.68223	−0.117964
$518$	6.55465	0.287995
$519$	0	0
$520$	0	0
$521$	20.9907	0.919619	0.459810	−	0.888018i	$-0.347918\pi$
$521$	20.9907	0.919619	0.459810	+	0.888018i	$0.347918\pi$
$522$	0	0
$523$	9.64045	0.421547	0.210774	−	0.977535i	$-0.432402\pi$
$523$	9.64045	0.421547	0.210774	+	0.977535i	$0.432402\pi$
$524$	35.0858	1.53273
$525$	0	0
$526$	−54.9347	−2.39527
$527$	−23.4036	−1.01948
$528$	0	0
$529$	−12.7318	−0.553557
$530$	0	0
$531$	0	0
$532$	−9.88139	−0.428413
$533$	6.64568	0.287856
$534$	0	0
$535$	0	0
$536$	−54.6076	−2.35869
$537$	0	0
$538$	−67.9892	−2.93122
$539$	−0.713538	−0.0307343
$540$	0	0
$541$	−30.4983	−1.31123	−0.655613	−	0.755097i	$-0.727590\pi$
$541$	−30.4983	−1.31123	−0.655613	+	0.755097i	$0.727590\pi$
$542$	47.9582	2.05998
$543$	0	0
$544$	8.69900	0.372967
$545$	0	0
$546$	0	0
$547$	4.11711	0.176035	0.0880174	−	0.996119i	$-0.471947\pi$
$547$	4.11711	0.176035	0.0880174	+	0.996119i	$0.471947\pi$
$548$	26.5233	1.13302
$549$	0	0
$550$	0	0
$551$	7.53114	0.320838
$552$	0	0
$553$	−6.83588	−0.290691
$554$	−40.3398	−1.71387
$555$	0	0
$556$	−58.0948	−2.46377
$557$	9.49610	0.402363	0.201181	−	0.979554i	$-0.435522\pi$
$557$	9.49610	0.402363	0.201181	+	0.979554i	$0.435522\pi$
$558$	0	0
$559$	−5.99593	−0.253601
$560$	0	0
$561$	0	0
$562$	5.04028	0.212611
$563$	−29.0403	−1.22390	−0.611951	−	0.790896i	$-0.709615\pi$
$563$	−29.0403	−1.22390	−0.611951	+	0.790896i	$0.709615\pi$
$564$	0	0
$565$	0	0
$566$	−66.4487	−2.79305
$567$	0	0
$568$	3.11711	0.130791
$569$	7.10930	0.298037	0.149019	−	0.988834i	$-0.452389\pi$
$569$	7.10930	0.298037	0.149019	+	0.988834i	$0.452389\pi$
$570$	0	0
$571$	−17.8553	−0.747222	−0.373611	−	0.927586i	$-0.621880\pi$
$571$	−17.8553	−0.747222	−0.373611	+	0.927586i	$0.621880\pi$
$572$	2.33198	0.0975049
$573$	0	0
$574$	−21.2954	−0.888854
$575$	0	0
$576$	0	0
$577$	−0.180894	−0.00753073	−0.00376536	−	0.999993i	$-0.501199\pi$
$577$	−0.180894	−0.00753073	−0.00376536	+	0.999993i	$0.501199\pi$
$578$	1.20964	0.0503142
$579$	0	0
$580$	0	0
$581$	−6.24992	−0.259290
$582$	0	0
$583$	8.34502	0.345615
$584$	37.5349	1.55320
$585$	0	0
$586$	24.7851	1.02386
$587$	−6.46362	−0.266782	−0.133391	−	0.991063i	$-0.542587\pi$
$587$	−6.46362	−0.266782	−0.133391	+	0.991063i	$0.542587\pi$
$588$	0	0
$589$	13.5352	0.557709
$590$	0	0
$591$	0	0
$592$	13.8631	0.569771
$593$	−12.5807	−0.516629	−0.258314	−	0.966061i	$-0.583167\pi$
$593$	−12.5807	−0.516629	−0.258314	+	0.966061i	$0.583167\pi$
$594$	0	0
$595$	0	0
$596$	−11.1406	−0.456337
$597$	0	0
$598$	−6.20440	−0.253717
$599$	29.3357	1.19862	0.599312	−	0.800515i	$-0.295441\pi$
$599$	29.3357	1.19862	0.599312	+	0.800515i	$0.295441\pi$
$600$	0	0
$601$	−34.8344	−1.42092	−0.710462	−	0.703736i	$-0.751514\pi$
$601$	−34.8344	−1.42092	−0.710462	+	0.703736i	$0.751514\pi$
$602$	19.2134	0.783079
$603$	0	0
$604$	−0.591197	−0.0240555
$605$	0	0
$606$	0	0
$607$	16.9542	0.688148	0.344074	−	0.938943i	$-0.388193\pi$
$607$	16.9542	0.688148	0.344074	+	0.938943i	$0.388193\pi$
$608$	−5.03098	−0.204033
$609$	0	0
$610$	0	0
$611$	−2.92201	−0.118212
$612$	0	0
$613$	−4.03655	−0.163035	−0.0815173	−	0.996672i	$-0.525977\pi$
$613$	−4.03655	−0.163035	−0.0815173	+	0.996672i	$0.525977\pi$
$614$	8.39983	0.338990
$615$	0	0
$616$	−3.91794	−0.157858
$617$	0.142113	0.00572127	0.00286063	−	0.999996i	$-0.499089\pi$
$617$	0.142113	0.00572127	0.00286063	+	0.999996i	$0.499089\pi$
$618$	0	0
$619$	−35.1716	−1.41367	−0.706833	−	0.707381i	$-0.749877\pi$
$619$	−35.1716	−1.41367	−0.706833	+	0.707381i	$0.749877\pi$
$620$	0	0
$621$	0	0
$622$	−22.3044	−0.894325
$623$	2.12758	0.0852395
$624$	0	0
$625$	0	0
$626$	−14.6822	−0.586820
$627$	0	0
$628$	52.0948	2.07881
$629$	−10.6938	−0.426388
$630$	0	0
$631$	−4.17832	−0.166336	−0.0831682	−	0.996536i	$-0.526504\pi$
$631$	−4.17832	−0.166336	−0.0831682	+	0.996536i	$0.526504\pi$
$632$	−37.5349	−1.49306
$633$	0	0
$634$	−4.03131	−0.160104
$635$	0	0
$636$	0	0
$637$	−0.777326	−0.0307988
$638$	5.69527	0.225478
$639$	0	0
$640$	0	0
$641$	−30.6262	−1.20966	−0.604832	−	0.796353i	$-0.706760\pi$
$641$	−30.6262	−1.20966	−0.604832	+	0.796353i	$0.706760\pi$
$642$	0	0
$643$	37.7135	1.48728	0.743638	−	0.668582i	$-0.233098\pi$
$643$	37.7135	1.48728	0.743638	+	0.668582i	$0.233098\pi$
$644$	13.4726	0.530894
$645$	0	0
$646$	23.7900	0.936006
$647$	−26.1861	−1.02948	−0.514741	−	0.857345i	$-0.672112\pi$
$647$	−26.1861	−1.02948	−0.514741	+	0.857345i	$0.672112\pi$
$648$	0	0
$649$	−8.63671	−0.339021
$650$	0	0
$651$	0	0
$652$	−85.0840	−3.33215
$653$	−2.14061	−0.0837687	−0.0418843	−	0.999122i	$-0.513336\pi$
$653$	−2.14061	−0.0837687	−0.0418843	+	0.999122i	$0.513336\pi$
$654$	0	0
$655$	0	0
$656$	−45.0399	−1.75851
$657$	0	0
$658$	9.36329	0.365019
$659$	48.2119	1.87807	0.939034	−	0.343824i	$-0.111723\pi$
$659$	48.2119	1.87807	0.939034	+	0.343824i	$0.111723\pi$
$660$	0	0
$661$	−4.10033	−0.159484	−0.0797422	−	0.996816i	$-0.525410\pi$
$661$	−4.10033	−0.159484	−0.0797422	+	0.996816i	$0.525410\pi$
$662$	44.1443	1.71572
$663$	0	0
$664$	−34.3174	−1.33177
$665$	0	0
$666$	0	0
$667$	−10.2682	−0.397586
$668$	−42.3122	−1.63711
$669$	0	0
$670$	0	0
$671$	−0.959719	−0.0370496
$672$	0	0
$673$	−45.4047	−1.75022	−0.875112	−	0.483921i	$-0.839212\pi$
$673$	−45.4047	−1.75022	−0.875112	+	0.483921i	$0.839212\pi$
$674$	63.3252	2.43920
$675$	0	0
$676$	−52.1168	−2.00449
$677$	37.0127	1.42251	0.711257	−	0.702932i	$-0.248126\pi$
$677$	37.0127	1.42251	0.711257	+	0.702932i	$0.248126\pi$
$678$	0	0
$679$	4.85415	0.186285
$680$	0	0
$681$	0	0
$682$	10.2357	0.391946
$683$	−40.4633	−1.54828	−0.774142	−	0.633012i	$-0.781819\pi$
$683$	−40.4633	−1.54828	−0.774142	+	0.633012i	$0.781819\pi$
$684$	0	0
$685$	0	0
$686$	2.49086	0.0951016
$687$	0	0
$688$	40.6364	1.54925
$689$	9.09103	0.346341
$690$	0	0
$691$	−7.72658	−0.293933	−0.146966	−	0.989141i	$-0.546951\pi$
$691$	−7.72658	−0.293933	−0.146966	+	0.989141i	$0.546951\pi$
$692$	−7.89443	−0.300101
$693$	0	0
$694$	57.4386	2.18034
$695$	0	0
$696$	0	0
$697$	34.7430	1.31599
$698$	−5.91911	−0.224041
$699$	0	0
$700$	0	0
$701$	−19.0638	−0.720029	−0.360015	−	0.932947i	$-0.617228\pi$
$701$	−19.0638	−0.720029	−0.360015	+	0.932947i	$0.617228\pi$
$702$	0	0
$703$	6.18463	0.233258
$704$	3.71354	0.139959
$705$	0	0
$706$	41.5308	1.56303
$707$	−7.48563	−0.281526
$708$	0	0
$709$	−26.9660	−1.01273	−0.506365	−	0.862319i	$-0.669011\pi$
$709$	−26.9660	−1.01273	−0.506365	+	0.862319i	$0.669011\pi$
$710$	0	0
$711$	0	0
$712$	11.6822	0.437810
$713$	−18.4543	−0.691120
$714$	0	0
$715$	0	0
$716$	−63.2574	−2.36404
$717$	0	0
$718$	−17.4491	−0.651194
$719$	−25.3760	−0.946365	−0.473182	−	0.880964i	$-0.656895\pi$
$719$	−25.3760	−0.946365	−0.473182	+	0.880964i	$0.656895\pi$
$720$	0	0
$721$	18.8814	0.703180
$722$	33.5677	1.24926
$723$	0	0
$724$	−10.2044	−0.379244
$725$	0	0
$726$	0	0
$727$	4.06786	0.150868	0.0754342	−	0.997151i	$-0.475966\pi$
$727$	4.06786	0.150868	0.0754342	+	0.997151i	$0.475966\pi$
$728$	−4.26819	−0.158190
$729$	0	0
$730$	0	0
$731$	−31.3462	−1.15938
$732$	0	0
$733$	44.1455	1.63055	0.815276	−	0.579073i	$-0.196585\pi$
$733$	44.1455	1.63055	0.815276	+	0.579073i	$0.196585\pi$
$734$	46.0220	1.69870
$735$	0	0
$736$	6.85939	0.252840
$737$	−7.09626	−0.261394
$738$	0	0
$739$	50.2667	1.84909	0.924545	−	0.381073i	$-0.124445\pi$
$739$	50.2667	1.84909	0.924545	+	0.381073i	$0.124445\pi$
$740$	0	0
$741$	0	0
$742$	−29.1313	−1.06944
$743$	4.04808	0.148510	0.0742549	−	0.997239i	$-0.476342\pi$
$743$	4.04808	0.148510	0.0742549	+	0.997239i	$0.476342\pi$
$744$	0	0
$745$	0	0
$746$	67.1716	2.45933
$747$	0	0
$748$	12.1914	0.445760
$749$	16.4271	0.600232
$750$	0	0
$751$	−15.8318	−0.577711	−0.288855	−	0.957373i	$-0.593275\pi$
$751$	−15.8318	−0.577711	−0.288855	+	0.957373i	$0.593275\pi$
$752$	19.8034	0.722156
$753$	0	0
$754$	6.20440	0.225951
$755$	0	0
$756$	0	0
$757$	−28.1026	−1.02141	−0.510703	−	0.859757i	$-0.670615\pi$
$757$	−28.1026	−1.02141	−0.510703	+	0.859757i	$0.670615\pi$
$758$	37.7225	1.37014
$759$	0	0
$760$	0	0
$761$	−4.94552	−0.179275	−0.0896374	−	0.995974i	$-0.528571\pi$
$761$	−4.94552	−0.179275	−0.0896374	+	0.995974i	$0.528571\pi$
$762$	0	0
$763$	13.4088	0.485431
$764$	−93.5036	−3.38284
$765$	0	0
$766$	80.2939	2.90114
$767$	−9.40880	−0.339732
$768$	0	0
$769$	−24.0571	−0.867520	−0.433760	−	0.901029i	$-0.642813\pi$
$769$	−24.0571	−0.867520	−0.433760	+	0.901029i	$0.642813\pi$
$770$	0	0
$771$	0	0
$772$	59.9892	2.15906
$773$	−16.6027	−0.597159	−0.298579	−	0.954385i	$-0.596513\pi$
$773$	−16.6027	−0.597159	−0.298579	+	0.954385i	$0.596513\pi$
$774$	0	0
$775$	0	0
$776$	26.6535	0.956804
$777$	0	0
$778$	−67.4801	−2.41928
$779$	−20.0933	−0.719916
$780$	0	0
$781$	0.405068	0.0144945
$782$	−32.4360	−1.15991
$783$	0	0
$784$	5.26819	0.188150
$785$	0	0
$786$	0	0
$787$	−22.1470	−0.789456	−0.394728	−	0.918798i	$-0.629161\pi$
$787$	−22.1470	−0.789456	−0.394728	+	0.918798i	$0.629161\pi$
$788$	8.75382	0.311842
$789$	0	0
$790$	0	0
$791$	7.61844	0.270881
$792$	0	0
$793$	−1.04551	−0.0371273
$794$	−53.0310	−1.88200
$795$	0	0
$796$	−89.4577	−3.17074
$797$	26.3775	0.934339	0.467169	−	0.884168i	$-0.345274\pi$
$797$	26.3775	0.934339	0.467169	+	0.884168i	$0.345274\pi$
$798$	0	0
$799$	−15.2760	−0.540426
$800$	0	0
$801$	0	0
$802$	69.1858	2.44304
$803$	4.87766	0.172129
$804$	0	0
$805$	0	0
$806$	11.1507	0.392768
$807$	0	0
$808$	−41.1026	−1.44598
$809$	−8.26039	−0.290420	−0.145210	−	0.989401i	$-0.546386\pi$
$809$	−8.26039	−0.290420	−0.145210	+	0.989401i	$0.546386\pi$
$810$	0	0
$811$	9.03771	0.317357	0.158678	−	0.987330i	$-0.449277\pi$
$811$	9.03771	0.317357	0.158678	+	0.987330i	$0.449277\pi$
$812$	−13.4726	−0.472795
$813$	0	0
$814$	4.67699	0.163928
$815$	0	0
$816$	0	0
$817$	18.1287	0.634244
$818$	46.4308	1.62342
$819$	0	0
$820$	0	0
$821$	−29.8606	−1.04214	−0.521070	−	0.853514i	$-0.674467\pi$
$821$	−29.8606	−1.04214	−0.521070	+	0.853514i	$0.674467\pi$
$822$	0	0
$823$	36.1704	1.26082	0.630411	−	0.776262i	$-0.282886\pi$
$823$	36.1704	1.26082	0.630411	+	0.776262i	$0.282886\pi$
$824$	103.675	3.61170
$825$	0	0
$826$	30.1496	1.04904
$827$	−51.9161	−1.80530	−0.902650	−	0.430376i	$-0.858381\pi$
$827$	−51.9161	−1.80530	−0.902650	+	0.430376i	$0.858381\pi$
$828$	0	0
$829$	19.4909	0.676946	0.338473	−	0.940976i	$-0.390090\pi$
$829$	19.4909	0.676946	0.338473	+	0.940976i	$0.390090\pi$
$830$	0	0
$831$	0	0
$832$	4.04551	0.140253
$833$	−4.06379	−0.140802
$834$	0	0
$835$	0	0
$836$	−7.05075	−0.243855
$837$	0	0
$838$	8.34502	0.288274
$839$	28.2552	0.975476	0.487738	−	0.872990i	$-0.337822\pi$
$839$	28.2552	0.975476	0.487738	+	0.872990i	$0.337822\pi$
$840$	0	0
$841$	−18.7318	−0.645925
$842$	16.7904	0.578634
$843$	0	0
$844$	30.2134	1.03999
$845$	0	0
$846$	0	0
$847$	10.4909	0.360470
$848$	−61.6129	−2.11579
$849$	0	0
$850$	0	0
$851$	−8.43231	−0.289056
$852$	0	0
$853$	−41.8799	−1.43394	−0.716970	−	0.697104i	$-0.754472\pi$
$853$	−41.8799	−1.43394	−0.716970	+	0.697104i	$0.754472\pi$
$854$	3.35025	0.114643
$855$	0	0
$856$	90.1988	3.08293
$857$	46.2682	1.58049	0.790246	−	0.612790i	$-0.209953\pi$
$857$	46.2682	1.58049	0.790246	+	0.612790i	$0.209953\pi$
$858$	0	0
$859$	26.6587	0.909584	0.454792	−	0.890598i	$-0.349714\pi$
$859$	26.6587	0.909584	0.454792	+	0.890598i	$0.349714\pi$
$860$	0	0
$861$	0	0
$862$	−29.2540	−0.996395
$863$	−38.7770	−1.31998	−0.659992	−	0.751273i	$-0.729440\pi$
$863$	−38.7770	−1.31998	−0.659992	+	0.751273i	$0.729440\pi$
$864$	0	0
$865$	0	0
$866$	−14.5364	−0.493966
$867$	0	0
$868$	−24.2134	−0.821855
$869$	−4.87766	−0.165463
$870$	0	0
$871$	−7.73065	−0.261943
$872$	73.6259	2.49329
$873$	0	0
$874$	18.7591	0.634534
$875$	0	0
$876$	0	0
$877$	−57.3890	−1.93789	−0.968945	−	0.247277i	$-0.920464\pi$
$877$	−57.3890	−1.93789	−0.968945	+	0.247277i	$0.920464\pi$
$878$	93.1895	3.14499
$879$	0	0
$880$	0	0
$881$	57.3316	1.93155	0.965776	−	0.259377i	$-0.0835173\pi$
$881$	57.3316	1.93155	0.965776	+	0.259377i	$0.0835173\pi$
$882$	0	0
$883$	−5.36329	−0.180489	−0.0902445	−	0.995920i	$-0.528765\pi$
$883$	−5.36329	−0.180489	−0.0902445	+	0.995920i	$0.528765\pi$
$884$	13.2812	0.446696
$885$	0	0
$886$	8.11827	0.272739
$887$	20.6729	0.694129	0.347064	−	0.937841i	$-0.387179\pi$
$887$	20.6729	0.694129	0.347064	+	0.937841i	$0.387179\pi$
$888$	0	0
$889$	−19.7591	−0.662697
$890$	0	0
$891$	0	0
$892$	−110.190	−3.68942
$893$	8.83471	0.295642
$894$	0	0
$895$	0	0
$896$	−17.2447	−0.576104
$897$	0	0
$898$	43.6158	1.45548
$899$	18.4543	0.615486
$900$	0	0
$901$	47.5271	1.58336
$902$	−15.1951	−0.505941
$903$	0	0
$904$	41.8318	1.39131
$905$	0	0
$906$	0	0
$907$	28.8120	0.956688	0.478344	−	0.878173i	$-0.341237\pi$
$907$	28.8120	0.956688	0.478344	+	0.878173i	$0.341237\pi$
$908$	100.253	3.32702
$909$	0	0
$910$	0	0
$911$	24.3827	0.807836	0.403918	−	0.914795i	$-0.367648\pi$
$911$	24.3827	0.807836	0.403918	+	0.914795i	$0.367648\pi$
$912$	0	0
$913$	−4.45955	−0.147590
$914$	−78.8900	−2.60945
$915$	0	0
$916$	15.0768	0.498152
$917$	−8.34502	−0.275577
$918$	0	0
$919$	13.2029	0.435524	0.217762	−	0.976002i	$-0.430124\pi$
$919$	13.2029	0.435524	0.217762	+	0.976002i	$0.430124\pi$
$920$	0	0
$921$	0	0
$922$	53.3682	1.75759
$923$	0.441280	0.0145249
$924$	0	0
$925$	0	0
$926$	89.9851	2.95709
$927$	0	0
$928$	−6.85939	−0.225170
$929$	4.91944	0.161402	0.0807008	−	0.996738i	$-0.474284\pi$
$929$	4.91944	0.161402	0.0807008	+	0.996738i	$0.474284\pi$
$930$	0	0
$931$	2.35025	0.0770263
$932$	114.203	3.74083
$933$	0	0
$934$	53.0440	1.73565
$935$	0	0
$936$	0	0
$937$	−15.1261	−0.494147	−0.247074	−	0.968997i	$-0.579469\pi$
$937$	−15.1261	−0.494147	−0.247074	+	0.968997i	$0.579469\pi$
$938$	24.7721	0.808837
$939$	0	0
$940$	0	0
$941$	−15.2462	−0.497011	−0.248506	−	0.968630i	$-0.579939\pi$
$941$	−15.2462	−0.497011	−0.248506	+	0.968630i	$0.579939\pi$
$942$	0	0
$943$	27.3958	0.892129
$944$	63.7665	2.07542
$945$	0	0
$946$	13.7095	0.445733
$947$	25.7132	0.835567	0.417783	−	0.908547i	$-0.362807\pi$
$947$	25.7132	0.835567	0.417783	+	0.908547i	$0.362807\pi$
$948$	0	0
$949$	5.31370	0.172490
$950$	0	0
$951$	0	0
$952$	−22.3137	−0.723191
$953$	−9.92724	−0.321575	−0.160787	−	0.986989i	$-0.551403\pi$
$953$	−9.92724	−0.321575	−0.160787	+	0.986989i	$0.551403\pi$
$954$	0	0
$955$	0	0
$956$	−23.3163	−0.754102
$957$	0	0
$958$	100.020	3.23151
$959$	−6.30847	−0.203711
$960$	0	0
$961$	2.16669	0.0698932
$962$	5.09510	0.164273
$963$	0	0
$964$	53.5453	1.72458
$965$	0	0
$966$	0	0
$967$	−3.17716	−0.102171	−0.0510853	−	0.998694i	$-0.516268\pi$
$967$	−3.17716	−0.102171	−0.0510853	+	0.998694i	$0.516268\pi$
$968$	57.6039	1.85146
$969$	0	0
$970$	0	0
$971$	−33.1440	−1.06364	−0.531821	−	0.846857i	$-0.678492\pi$
$971$	−33.1440	−1.06364	−0.531821	+	0.846857i	$0.678492\pi$
$972$	0	0
$973$	13.8176	0.442972
$974$	−23.5755	−0.755408
$975$	0	0
$976$	7.08580	0.226811
$977$	−4.63521	−0.148294	−0.0741468	−	0.997247i	$-0.523623\pi$
$977$	−4.63521	−0.148294	−0.0741468	+	0.997247i	$0.523623\pi$
$978$	0	0
$979$	1.51811	0.0485189
$980$	0	0
$981$	0	0
$982$	48.3902	1.54419
$983$	−22.2552	−0.709829	−0.354915	−	0.934899i	$-0.615490\pi$
$983$	−22.2552	−0.709829	−0.354915	+	0.934899i	$0.615490\pi$
$984$	0	0
$985$	0	0
$986$	32.4360	1.03297
$987$	0	0
$988$	−7.68106	−0.244367
$989$	−24.7173	−0.785964
$990$	0	0
$991$	−0.610635	−0.0193975	−0.00969873	−	0.999953i	$-0.503087\pi$
$991$	−0.610635	−0.0193975	−0.00969873	+	0.999953i	$0.503087\pi$
$992$	−12.3279	−0.391411
$993$	0	0
$994$	−1.41404	−0.0448505
$995$	0	0
$996$	0	0
$997$	35.9101	1.13729	0.568643	−	0.822585i	$-0.307469\pi$
$997$	35.9101	1.13729	0.568643	+	0.822585i	$0.307469\pi$
$998$	−30.3085	−0.959398
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4725.2.a.bi.1.1	✓	3
3.2	odd	2		4725.2.a.bj.1.3	yes	3
5.4	even	2		4725.2.a.bl.1.3	yes	3
15.14	odd	2		4725.2.a.bk.1.1	yes	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4725.2.a.bi.1.1	✓	3	1.1	even	1	trivial
4725.2.a.bj.1.3	yes	3	3.2	odd	2
4725.2.a.bk.1.1	yes	3	15.14	odd	2
4725.2.a.bl.1.3	yes	3	5.4	even	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 4725 = 3^{3} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4725.a (trivial)

\(f(q)\)	\(=\)	\(q-2.49086 q^{2} +4.20440 q^{4} -1.00000 q^{7} -5.49086 q^{8} -0.713538 q^{11} -0.777326 q^{13} +2.49086 q^{14} +5.26819 q^{16} -4.06379 q^{17} +2.35025 q^{19} +1.77733 q^{22} -3.20440 q^{23} +1.93621 q^{26} -4.20440 q^{28} +3.20440 q^{29} +5.75905 q^{31} -2.14061 q^{32} +10.1223 q^{34} +2.63148 q^{37} -5.85415 q^{38} -8.54942 q^{41} +7.71354 q^{43} -3.00000 q^{44} +7.98173 q^{46} +3.75905 q^{47} +1.00000 q^{49} -3.26819 q^{52} -11.6953 q^{53} +5.49086 q^{56} -7.98173 q^{58} +12.1041 q^{59} +1.34502 q^{61} -14.3450 q^{62} -5.20440 q^{64} +9.94518 q^{67} -17.0858 q^{68} -0.567690 q^{71} -6.83588 q^{73} -6.55465 q^{74} +9.88139 q^{76} +0.713538 q^{77} +6.83588 q^{79} +21.2954 q^{82} +6.24992 q^{83} -19.2134 q^{86} +3.91794 q^{88} -2.12758 q^{89} +0.777326 q^{91} -13.4726 q^{92} -9.36329 q^{94} -4.85415 q^{97} -2.49086 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 4 q^{4} - 3 q^{7} - 9 q^{8} - q^{11} + 4 q^{13} + 2 q^{16} - 7 q^{17} + 3 q^{19} - q^{22} - q^{23} + 11 q^{26} - 4 q^{28} + q^{29} - 4 q^{31} - 3 q^{32} + 12 q^{34} - 3 q^{37} - 13 q^{38} - 5 q^{41}+ \cdots - 10 q^{97}+O(q^{100}) \) 3 * q + 4 * q^4 - 3 * q^7 - 9 * q^8 - q^11 + 4 * q^13 + 2 * q^16 - 7 * q^17 + 3 * q^19 - q^22 - q^23 + 11 * q^26 - 4 * q^28 + q^29 - 4 * q^31 - 3 * q^32 + 12 * q^34 - 3 * q^37 - 13 * q^38 - 5 * q^41 + 22 * q^43 - 9 * q^44 + 9 * q^46 - 10 * q^47 + 3 * q^49 + 4 * q^52 - 19 * q^53 + 9 * q^56 - 9 * q^58 + 3 * q^59 - 8 * q^61 - 31 * q^62 - 7 * q^64 - 15 * q^67 - 3 * q^68 + 4 * q^71 - q^73 - 7 * q^74 - 10 * q^76 + q^77 + q^79 + 15 * q^82 - 10 * q^83 + q^86 + 2 * q^88 + 4 * q^89 - 4 * q^91 - 18 * q^92 - 31 * q^94 - 10 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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