LMFDB - Embedded newform 6900.2.a.l.1.1

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:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{13}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 3 $ x^2 - x - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.30278$ 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)

$f(q)$	$=$	$q-1.00000 q^{3} -4.30278 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -4.30278 q^{7} +1.00000 q^{9} +4.30278 q^{11} -4.30278 q^{13} +1.60555 q^{17} -2.60555 q^{19} +4.30278 q^{21} -1.00000 q^{23} -1.00000 q^{27} -2.60555 q^{29} +8.90833 q^{31} -4.30278 q^{33} +9.60555 q^{37} +4.30278 q^{39} +3.00000 q^{41} +6.21110 q^{43} -8.81665 q^{47} +11.5139 q^{49} -1.60555 q^{51} -10.3028 q^{53} +2.60555 q^{57} +14.1194 q^{59} -6.90833 q^{61} -4.30278 q^{63} -0.0916731 q^{67} +1.00000 q^{69} -2.90833 q^{71} +1.78890 q^{73} -18.5139 q^{77} -5.00000 q^{79} +1.00000 q^{81} +1.78890 q^{83} +2.60555 q^{87} -9.00000 q^{89} +18.5139 q^{91} -8.90833 q^{93} +9.51388 q^{97} +4.30278 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} - 5 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^3 - 5 * q^7 + 2 * q^9	$ 2 q - 2 q^{3} - 5 q^{7} + 2 q^{9} + 5 q^{11} - 5 q^{13} - 4 q^{17} + 2 q^{19} + 5 q^{21} - 2 q^{23} - 2 q^{27} + 2 q^{29} + 7 q^{31} - 5 q^{33} + 12 q^{37} + 5 q^{39} + 6 q^{41} - 2 q^{43} + 4 q^{47} + 5 q^{49} + 4 q^{51} - 17 q^{53} - 2 q^{57} + 3 q^{59} - 3 q^{61} - 5 q^{63} - 11 q^{67} + 2 q^{69} + 5 q^{71} + 18 q^{73} - 19 q^{77} - 10 q^{79} + 2 q^{81} + 18 q^{83} - 2 q^{87} - 18 q^{89} + 19 q^{91} - 7 q^{93} + q^{97} + 5 q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 - 5 * q^7 + 2 * q^9 + 5 * q^11 - 5 * q^13 - 4 * q^17 + 2 * q^19 + 5 * q^21 - 2 * q^23 - 2 * q^27 + 2 * q^29 + 7 * q^31 - 5 * q^33 + 12 * q^37 + 5 * q^39 + 6 * q^41 - 2 * q^43 + 4 * q^47 + 5 * q^49 + 4 * q^51 - 17 * q^53 - 2 * q^57 + 3 * q^59 - 3 * q^61 - 5 * q^63 - 11 * q^67 + 2 * q^69 + 5 * q^71 + 18 * q^73 - 19 * q^77 - 10 * q^79 + 2 * q^81 + 18 * q^83 - 2 * q^87 - 18 * q^89 + 19 * q^91 - 7 * q^93 + q^97 + 5 * 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
$7$	−4.30278	−1.62630	−0.813148	−	0.582057i	$-0.802248\pi$
$7$	−4.30278	−1.62630	−0.813148	+	0.582057i	$0.802248\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	4.30278	1.29734	0.648668	−	0.761072i	$-0.275326\pi$
$11$	4.30278	1.29734	0.648668	+	0.761072i	$0.275326\pi$
$12$	0	0
$13$	−4.30278	−1.19338	−0.596688	−	0.802474i	$-0.703517\pi$
$13$	−4.30278	−1.19338	−0.596688	+	0.802474i	$0.703517\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	1.60555	0.389403	0.194702	−	0.980863i	$-0.437626\pi$
$17$	1.60555	0.389403	0.194702	+	0.980863i	$0.437626\pi$
$18$	0	0
$19$	−2.60555	−0.597754	−0.298877	−	0.954292i	$-0.596612\pi$
$19$	−2.60555	−0.597754	−0.298877	+	0.954292i	$0.596612\pi$
$20$	0	0
$21$	4.30278	0.938943
$22$	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
$29$	−2.60555	−0.483839	−0.241919	−	0.970296i	$-0.577777\pi$
$29$	−2.60555	−0.483839	−0.241919	+	0.970296i	$0.577777\pi$
$30$	0	0
$31$	8.90833	1.59998	0.799991	−	0.600012i	$-0.204837\pi$
$31$	8.90833	1.59998	0.799991	+	0.600012i	$0.204837\pi$
$32$	0	0
$33$	−4.30278	−0.749017
$34$	0	0
$35$	0	0
$36$	0	0
$37$	9.60555	1.57914	0.789571	−	0.613659i	$-0.210303\pi$
$37$	9.60555	1.57914	0.789571	+	0.613659i	$0.210303\pi$
$38$	0	0
$39$	4.30278	0.688996
$40$	0	0
$41$	3.00000	0.468521	0.234261	−	0.972174i	$-0.424733\pi$
$41$	3.00000	0.468521	0.234261	+	0.972174i	$0.424733\pi$
$42$	0	0
$43$	6.21110	0.947184	0.473592	−	0.880744i	$-0.342957\pi$
$43$	6.21110	0.947184	0.473592	+	0.880744i	$0.342957\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−8.81665	−1.28604	−0.643021	−	0.765849i	$-0.722319\pi$
$47$	−8.81665	−1.28604	−0.643021	+	0.765849i	$0.722319\pi$
$48$	0	0
$49$	11.5139	1.64484
$50$	0	0
$51$	−1.60555	−0.224822
$52$	0	0
$53$	−10.3028	−1.41520	−0.707598	−	0.706616i	$-0.750221\pi$
$53$	−10.3028	−1.41520	−0.707598	+	0.706616i	$0.750221\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	2.60555	0.345114
$58$	0	0
$59$	14.1194	1.83819	0.919097	−	0.394032i	$-0.128920\pi$
$59$	14.1194	1.83819	0.919097	+	0.394032i	$0.128920\pi$
$60$	0	0
$61$	−6.90833	−0.884521	−0.442260	−	0.896887i	$-0.645823\pi$
$61$	−6.90833	−0.884521	−0.442260	+	0.896887i	$0.645823\pi$
$62$	0	0
$63$	−4.30278	−0.542099
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−0.0916731	−0.0111997	−0.00559983	−	0.999984i	$-0.501782\pi$
$67$	−0.0916731	−0.0111997	−0.00559983	+	0.999984i	$0.501782\pi$
$68$	0	0
$69$	1.00000	0.120386
$70$	0	0
$71$	−2.90833	−0.345155	−0.172577	−	0.984996i	$-0.555210\pi$
$71$	−2.90833	−0.345155	−0.172577	+	0.984996i	$0.555210\pi$
$72$	0	0
$73$	1.78890	0.209375	0.104687	−	0.994505i	$-0.466616\pi$
$73$	1.78890	0.209375	0.104687	+	0.994505i	$0.466616\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−18.5139	−2.10985
$78$	0	0
$79$	−5.00000	−0.562544	−0.281272	−	0.959628i	$-0.590756\pi$
$79$	−5.00000	−0.562544	−0.281272	+	0.959628i	$0.590756\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	1.78890	0.196357	0.0981785	−	0.995169i	$-0.468698\pi$
$83$	1.78890	0.196357	0.0981785	+	0.995169i	$0.468698\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	2.60555	0.279344
$88$	0	0
$89$	−9.00000	−0.953998	−0.476999	−	0.878904i	$-0.658275\pi$
$89$	−9.00000	−0.953998	−0.476999	+	0.878904i	$0.658275\pi$
$90$	0	0
$91$	18.5139	1.94078
$92$	0	0
$93$	−8.90833	−0.923750
$94$	0	0
$95$	0	0
$96$	0	0
$97$	9.51388	0.965988	0.482994	−	0.875624i	$-0.339549\pi$
$97$	9.51388	0.965988	0.482994	+	0.875624i	$0.339549\pi$
$98$	0	0
$99$	4.30278	0.432445
$100$	0	0
$101$	−9.11943	−0.907417	−0.453709	−	0.891150i	$-0.649899\pi$
$101$	−9.11943	−0.907417	−0.453709	+	0.891150i	$0.649899\pi$
$102$	0	0
$103$	5.81665	0.573132	0.286566	−	0.958061i	$-0.407486\pi$
$103$	5.81665	0.573132	0.286566	+	0.958061i	$0.407486\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	2.21110	0.213755	0.106878	−	0.994272i	$-0.465915\pi$
$107$	2.21110	0.213755	0.106878	+	0.994272i	$0.465915\pi$
$108$	0	0
$109$	−14.2111	−1.36118	−0.680588	−	0.732666i	$-0.738276\pi$
$109$	−14.2111	−1.36118	−0.680588	+	0.732666i	$0.738276\pi$
$110$	0	0
$111$	−9.60555	−0.911719
$112$	0	0
$113$	−10.6972	−1.00631	−0.503155	−	0.864196i	$-0.667828\pi$
$113$	−10.6972	−1.00631	−0.503155	+	0.864196i	$0.667828\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−4.30278	−0.397792
$118$	0	0
$119$	−6.90833	−0.633285
$120$	0	0
$121$	7.51388	0.683080
$122$	0	0
$123$	−3.00000	−0.270501
$124$	0	0
$125$	0	0
$126$	0	0
$127$	4.78890	0.424946	0.212473	−	0.977167i	$-0.431848\pi$
$127$	4.78890	0.424946	0.212473	+	0.977167i	$0.431848\pi$
$128$	0	0
$129$	−6.21110	−0.546857
$130$	0	0
$131$	14.8167	1.29454	0.647269	−	0.762262i	$-0.275911\pi$
$131$	14.8167	1.29454	0.647269	+	0.762262i	$0.275911\pi$
$132$	0	0
$133$	11.2111	0.972126
$134$	0	0
$135$	0	0
$136$	0	0
$137$	6.00000	0.512615	0.256307	−	0.966595i	$-0.417494\pi$
$137$	6.00000	0.512615	0.256307	+	0.966595i	$0.417494\pi$
$138$	0	0
$139$	−11.6972	−0.992146	−0.496073	−	0.868281i	$-0.665225\pi$
$139$	−11.6972	−0.992146	−0.496073	+	0.868281i	$0.665225\pi$
$140$	0	0
$141$	8.81665	0.742496
$142$	0	0
$143$	−18.5139	−1.54821
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−11.5139	−0.949649
$148$	0	0
$149$	−5.60555	−0.459225	−0.229612	−	0.973282i	$-0.573746\pi$
$149$	−5.60555	−0.459225	−0.229612	+	0.973282i	$0.573746\pi$
$150$	0	0
$151$	−2.81665	−0.229216	−0.114608	−	0.993411i	$-0.536561\pi$
$151$	−2.81665	−0.229216	−0.114608	+	0.993411i	$0.536561\pi$
$152$	0	0
$153$	1.60555	0.129801
$154$	0	0
$155$	0	0
$156$	0	0
$157$	10.4222	0.831783	0.415891	−	0.909414i	$-0.363470\pi$
$157$	10.4222	0.831783	0.415891	+	0.909414i	$0.363470\pi$
$158$	0	0
$159$	10.3028	0.817063
$160$	0	0
$161$	4.30278	0.339106
$162$	0	0
$163$	5.60555	0.439061	0.219530	−	0.975606i	$-0.429548\pi$
$163$	5.60555	0.439061	0.219530	+	0.975606i	$0.429548\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−8.69722	−0.673011	−0.336506	−	0.941681i	$-0.609245\pi$
$167$	−8.69722	−0.673011	−0.336506	+	0.941681i	$0.609245\pi$
$168$	0	0
$169$	5.51388	0.424144
$170$	0	0
$171$	−2.60555	−0.199251
$172$	0	0
$173$	23.3305	1.77379	0.886894	−	0.461973i	$-0.152858\pi$
$173$	23.3305	1.77379	0.886894	+	0.461973i	$0.152858\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−14.1194	−1.06128
$178$	0	0
$179$	−12.7250	−0.951110	−0.475555	−	0.879686i	$-0.657753\pi$
$179$	−12.7250	−0.951110	−0.475555	+	0.879686i	$0.657753\pi$
$180$	0	0
$181$	5.39445	0.400966	0.200483	−	0.979697i	$-0.435749\pi$
$181$	5.39445	0.400966	0.200483	+	0.979697i	$0.435749\pi$
$182$	0	0
$183$	6.90833	0.510678
$184$	0	0
$185$	0	0
$186$	0	0
$187$	6.90833	0.505187
$188$	0	0
$189$	4.30278	0.312981
$190$	0	0
$191$	−13.4222	−0.971197	−0.485598	−	0.874182i	$-0.661398\pi$
$191$	−13.4222	−0.971197	−0.485598	+	0.874182i	$0.661398\pi$
$192$	0	0
$193$	−19.4222	−1.39804	−0.699020	−	0.715102i	$-0.746380\pi$
$193$	−19.4222	−1.39804	−0.699020	+	0.715102i	$0.746380\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−5.39445	−0.384339	−0.192169	−	0.981362i	$-0.561552\pi$
$197$	−5.39445	−0.384339	−0.192169	+	0.981362i	$0.561552\pi$
$198$	0	0
$199$	−12.6056	−0.893584	−0.446792	−	0.894638i	$-0.647434\pi$
$199$	−12.6056	−0.893584	−0.446792	+	0.894638i	$0.647434\pi$
$200$	0	0
$201$	0.0916731	0.00646612
$202$	0	0
$203$	11.2111	0.786865
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−1.00000	−0.0695048
$208$	0	0
$209$	−11.2111	−0.775488
$210$	0	0
$211$	2.18335	0.150308	0.0751539	−	0.997172i	$-0.476055\pi$
$211$	2.18335	0.150308	0.0751539	+	0.997172i	$0.476055\pi$
$212$	0	0
$213$	2.90833	0.199275
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−38.3305	−2.60205
$218$	0	0
$219$	−1.78890	−0.120882
$220$	0	0
$221$	−6.90833	−0.464704
$222$	0	0
$223$	−18.5139	−1.23978	−0.619890	−	0.784688i	$-0.712823\pi$
$223$	−18.5139	−1.23978	−0.619890	+	0.784688i	$0.712823\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−11.6056	−0.770287	−0.385144	−	0.922857i	$-0.625848\pi$
$227$	−11.6056	−0.770287	−0.385144	+	0.922857i	$0.625848\pi$
$228$	0	0
$229$	−23.6056	−1.55990	−0.779949	−	0.625843i	$-0.784755\pi$
$229$	−23.6056	−1.55990	−0.779949	+	0.625843i	$0.784755\pi$
$230$	0	0
$231$	18.5139	1.21812
$232$	0	0
$233$	−14.3028	−0.937006	−0.468503	−	0.883462i	$-0.655206\pi$
$233$	−14.3028	−0.937006	−0.468503	+	0.883462i	$0.655206\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	5.00000	0.324785
$238$	0	0
$239$	21.8167	1.41120	0.705601	−	0.708609i	$-0.250677\pi$
$239$	21.8167	1.41120	0.705601	+	0.708609i	$0.250677\pi$
$240$	0	0
$241$	−2.69722	−0.173743	−0.0868717	−	0.996220i	$-0.527687\pi$
$241$	−2.69722	−0.173743	−0.0868717	+	0.996220i	$0.527687\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	11.2111	0.713345
$248$	0	0
$249$	−1.78890	−0.113367
$250$	0	0
$251$	21.9361	1.38459	0.692297	−	0.721613i	$-0.256599\pi$
$251$	21.9361	1.38459	0.692297	+	0.721613i	$0.256599\pi$
$252$	0	0
$253$	−4.30278	−0.270513
$254$	0	0
$255$	0	0
$256$	0	0
$257$	15.9083	0.992334	0.496167	−	0.868227i	$-0.334740\pi$
$257$	15.9083	0.992334	0.496167	+	0.868227i	$0.334740\pi$
$258$	0	0
$259$	−41.3305	−2.56815
$260$	0	0
$261$	−2.60555	−0.161280
$262$	0	0
$263$	9.72498	0.599668	0.299834	−	0.953991i	$-0.403069\pi$
$263$	9.72498	0.599668	0.299834	+	0.953991i	$0.403069\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	9.00000	0.550791
$268$	0	0
$269$	−22.6056	−1.37828	−0.689142	−	0.724626i	$-0.742013\pi$
$269$	−22.6056	−1.37828	−0.689142	+	0.724626i	$0.742013\pi$
$270$	0	0
$271$	22.6333	1.37488	0.687438	−	0.726243i	$-0.258735\pi$
$271$	22.6333	1.37488	0.687438	+	0.726243i	$0.258735\pi$
$272$	0	0
$273$	−18.5139	−1.12051
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−29.3028	−1.76063	−0.880317	−	0.474387i	$-0.842670\pi$
$277$	−29.3028	−1.76063	−0.880317	+	0.474387i	$0.842670\pi$
$278$	0	0
$279$	8.90833	0.533328
$280$	0	0
$281$	−7.00000	−0.417585	−0.208792	−	0.977960i	$-0.566953\pi$
$281$	−7.00000	−0.417585	−0.208792	+	0.977960i	$0.566953\pi$
$282$	0	0
$283$	−7.42221	−0.441204	−0.220602	−	0.975364i	$-0.570802\pi$
$283$	−7.42221	−0.441204	−0.220602	+	0.975364i	$0.570802\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−12.9083	−0.761954
$288$	0	0
$289$	−14.4222	−0.848365
$290$	0	0
$291$	−9.51388	−0.557713
$292$	0	0
$293$	−25.4222	−1.48518	−0.742591	−	0.669746i	$-0.766403\pi$
$293$	−25.4222	−1.48518	−0.742591	+	0.669746i	$0.766403\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−4.30278	−0.249672
$298$	0	0
$299$	4.30278	0.248836
$300$	0	0
$301$	−26.7250	−1.54040
$302$	0	0
$303$	9.11943	0.523898
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−20.4222	−1.16556	−0.582778	−	0.812631i	$-0.698034\pi$
$307$	−20.4222	−1.16556	−0.582778	+	0.812631i	$0.698034\pi$
$308$	0	0
$309$	−5.81665	−0.330898
$310$	0	0
$311$	29.6056	1.67878	0.839388	−	0.543532i	$-0.182913\pi$
$311$	29.6056	1.67878	0.839388	+	0.543532i	$0.182913\pi$
$312$	0	0
$313$	23.8444	1.34777	0.673883	−	0.738838i	$-0.264625\pi$
$313$	23.8444	1.34777	0.673883	+	0.738838i	$0.264625\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	16.1194	0.905357	0.452679	−	0.891674i	$-0.350468\pi$
$317$	16.1194	0.905357	0.452679	+	0.891674i	$0.350468\pi$
$318$	0	0
$319$	−11.2111	−0.627701
$320$	0	0
$321$	−2.21110	−0.123412
$322$	0	0
$323$	−4.18335	−0.232768
$324$	0	0
$325$	0	0
$326$	0	0
$327$	14.2111	0.785876
$328$	0	0
$329$	37.9361	2.09148
$330$	0	0
$331$	−8.51388	−0.467965	−0.233983	−	0.972241i	$-0.575176\pi$
$331$	−8.51388	−0.467965	−0.233983	+	0.972241i	$0.575176\pi$
$332$	0	0
$333$	9.60555	0.526381
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−29.4222	−1.60273	−0.801365	−	0.598176i	$-0.795892\pi$
$337$	−29.4222	−1.60273	−0.801365	+	0.598176i	$0.795892\pi$
$338$	0	0
$339$	10.6972	0.580993
$340$	0	0
$341$	38.3305	2.07571
$342$	0	0
$343$	−19.4222	−1.04870
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−21.6333	−1.16134	−0.580668	−	0.814140i	$-0.697209\pi$
$347$	−21.6333	−1.16134	−0.580668	+	0.814140i	$0.697209\pi$
$348$	0	0
$349$	−31.6056	−1.69181	−0.845903	−	0.533336i	$-0.820938\pi$
$349$	−31.6056	−1.69181	−0.845903	+	0.533336i	$0.820938\pi$
$350$	0	0
$351$	4.30278	0.229665
$352$	0	0
$353$	−33.5139	−1.78376	−0.891882	−	0.452268i	$-0.850615\pi$
$353$	−33.5139	−1.78376	−0.891882	+	0.452268i	$0.850615\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	6.90833	0.365627
$358$	0	0
$359$	−8.39445	−0.443042	−0.221521	−	0.975156i	$-0.571102\pi$
$359$	−8.39445	−0.443042	−0.221521	+	0.975156i	$0.571102\pi$
$360$	0	0
$361$	−12.2111	−0.642690
$362$	0	0
$363$	−7.51388	−0.394376
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−30.6333	−1.59905	−0.799523	−	0.600636i	$-0.794914\pi$
$367$	−30.6333	−1.59905	−0.799523	+	0.600636i	$0.794914\pi$
$368$	0	0
$369$	3.00000	0.156174
$370$	0	0
$371$	44.3305	2.30153
$372$	0	0
$373$	7.60555	0.393801	0.196900	−	0.980424i	$-0.436912\pi$
$373$	7.60555	0.393801	0.196900	+	0.980424i	$0.436912\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	11.2111	0.577401
$378$	0	0
$379$	−18.7889	−0.965121	−0.482560	−	0.875863i	$-0.660293\pi$
$379$	−18.7889	−0.965121	−0.482560	+	0.875863i	$0.660293\pi$
$380$	0	0
$381$	−4.78890	−0.245343
$382$	0	0
$383$	2.30278	0.117666	0.0588332	−	0.998268i	$-0.481262\pi$
$383$	2.30278	0.117666	0.0588332	+	0.998268i	$0.481262\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	6.21110	0.315728
$388$	0	0
$389$	6.30278	0.319563	0.159782	−	0.987152i	$-0.448921\pi$
$389$	6.30278	0.319563	0.159782	+	0.987152i	$0.448921\pi$
$390$	0	0
$391$	−1.60555	−0.0811962
$392$	0	0
$393$	−14.8167	−0.747401
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−18.7889	−0.942988	−0.471494	−	0.881869i	$-0.656285\pi$
$397$	−18.7889	−0.942988	−0.471494	+	0.881869i	$0.656285\pi$
$398$	0	0
$399$	−11.2111	−0.561257
$400$	0	0
$401$	−28.9083	−1.44361	−0.721806	−	0.692095i	$-0.756688\pi$
$401$	−28.9083	−1.44361	−0.721806	+	0.692095i	$0.756688\pi$
$402$	0	0
$403$	−38.3305	−1.90938
$404$	0	0
$405$	0	0
$406$	0	0
$407$	41.3305	2.04868
$408$	0	0
$409$	22.2389	1.09964	0.549820	−	0.835283i	$-0.314696\pi$
$409$	22.2389	1.09964	0.549820	+	0.835283i	$0.314696\pi$
$410$	0	0
$411$	−6.00000	−0.295958
$412$	0	0
$413$	−60.7527	−2.98945
$414$	0	0
$415$	0	0
$416$	0	0
$417$	11.6972	0.572816
$418$	0	0
$419$	25.1194	1.22716	0.613582	−	0.789631i	$-0.289728\pi$
$419$	25.1194	1.22716	0.613582	+	0.789631i	$0.289728\pi$
$420$	0	0
$421$	36.1472	1.76171	0.880853	−	0.473390i	$-0.156970\pi$
$421$	36.1472	1.76171	0.880853	+	0.473390i	$0.156970\pi$
$422$	0	0
$423$	−8.81665	−0.428680
$424$	0	0
$425$	0	0
$426$	0	0
$427$	29.7250	1.43849
$428$	0	0
$429$	18.5139	0.893858
$430$	0	0
$431$	−12.0278	−0.579357	−0.289678	−	0.957124i	$-0.593548\pi$
$431$	−12.0278	−0.579357	−0.289678	+	0.957124i	$0.593548\pi$
$432$	0	0
$433$	−2.39445	−0.115070	−0.0575349	−	0.998343i	$-0.518324\pi$
$433$	−2.39445	−0.115070	−0.0575349	+	0.998343i	$0.518324\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	2.60555	0.124640
$438$	0	0
$439$	25.8444	1.23349	0.616743	−	0.787164i	$-0.288452\pi$
$439$	25.8444	1.23349	0.616743	+	0.787164i	$0.288452\pi$
$440$	0	0
$441$	11.5139	0.548280
$442$	0	0
$443$	−22.0000	−1.04525	−0.522626	−	0.852562i	$-0.675047\pi$
$443$	−22.0000	−1.04525	−0.522626	+	0.852562i	$0.675047\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	5.60555	0.265133
$448$	0	0
$449$	8.30278	0.391832	0.195916	−	0.980621i	$-0.437232\pi$
$449$	8.30278	0.391832	0.195916	+	0.980621i	$0.437232\pi$
$450$	0	0
$451$	12.9083	0.607829
$452$	0	0
$453$	2.81665	0.132338
$454$	0	0
$455$	0	0
$456$	0	0
$457$	25.6333	1.19908	0.599538	−	0.800346i	$-0.295351\pi$
$457$	25.6333	1.19908	0.599538	+	0.800346i	$0.295351\pi$
$458$	0	0
$459$	−1.60555	−0.0749407
$460$	0	0
$461$	−24.9083	−1.16010	−0.580048	−	0.814582i	$-0.696966\pi$
$461$	−24.9083	−1.16010	−0.580048	+	0.814582i	$0.696966\pi$
$462$	0	0
$463$	−9.09167	−0.422526	−0.211263	−	0.977429i	$-0.567758\pi$
$463$	−9.09167	−0.422526	−0.211263	+	0.977429i	$0.567758\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	34.6056	1.60135	0.800677	−	0.599096i	$-0.204473\pi$
$467$	34.6056	1.60135	0.800677	+	0.599096i	$0.204473\pi$
$468$	0	0
$469$	0.394449	0.0182139
$470$	0	0
$471$	−10.4222	−0.480230
$472$	0	0
$473$	26.7250	1.22882
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−10.3028	−0.471732
$478$	0	0
$479$	4.18335	0.191142	0.0955710	−	0.995423i	$-0.469532\pi$
$479$	4.18335	0.191142	0.0955710	+	0.995423i	$0.469532\pi$
$480$	0	0
$481$	−41.3305	−1.88451
$482$	0	0
$483$	−4.30278	−0.195783
$484$	0	0
$485$	0	0
$486$	0	0
$487$	10.2389	0.463967	0.231983	−	0.972720i	$-0.425478\pi$
$487$	10.2389	0.463967	0.231983	+	0.972720i	$0.425478\pi$
$488$	0	0
$489$	−5.60555	−0.253492
$490$	0	0
$491$	−28.5416	−1.28807	−0.644033	−	0.764998i	$-0.722740\pi$
$491$	−28.5416	−1.28807	−0.644033	+	0.764998i	$0.722740\pi$
$492$	0	0
$493$	−4.18335	−0.188408
$494$	0	0
$495$	0	0
$496$	0	0
$497$	12.5139	0.561324
$498$	0	0
$499$	−28.2111	−1.26290	−0.631451	−	0.775416i	$-0.717540\pi$
$499$	−28.2111	−1.26290	−0.631451	+	0.775416i	$0.717540\pi$
$500$	0	0
$501$	8.69722	0.388563
$502$	0	0
$503$	−21.4222	−0.955169	−0.477584	−	0.878586i	$-0.658488\pi$
$503$	−21.4222	−0.955169	−0.477584	+	0.878586i	$0.658488\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−5.51388	−0.244880
$508$	0	0
$509$	−9.69722	−0.429822	−0.214911	−	0.976634i	$-0.568946\pi$
$509$	−9.69722	−0.429822	−0.214911	+	0.976634i	$0.568946\pi$
$510$	0	0
$511$	−7.69722	−0.340505
$512$	0	0
$513$	2.60555	0.115038
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−37.9361	−1.66843
$518$	0	0
$519$	−23.3305	−1.02410
$520$	0	0
$521$	−10.6056	−0.464638	−0.232319	−	0.972640i	$-0.574631\pi$
$521$	−10.6056	−0.464638	−0.232319	+	0.972640i	$0.574631\pi$
$522$	0	0
$523$	18.0278	0.788299	0.394149	−	0.919046i	$-0.371039\pi$
$523$	18.0278	0.788299	0.394149	+	0.919046i	$0.371039\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	14.3028	0.623039
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	14.1194	0.612731
$532$	0	0
$533$	−12.9083	−0.559122
$534$	0	0
$535$	0	0
$536$	0	0
$537$	12.7250	0.549123
$538$	0	0
$539$	49.5416	2.13391
$540$	0	0
$541$	9.02776	0.388134	0.194067	−	0.980988i	$-0.437832\pi$
$541$	9.02776	0.388134	0.194067	+	0.980988i	$0.437832\pi$
$542$	0	0
$543$	−5.39445	−0.231498
$544$	0	0
$545$	0	0
$546$	0	0
$547$	19.0000	0.812381	0.406191	−	0.913788i	$-0.366857\pi$
$547$	19.0000	0.812381	0.406191	+	0.913788i	$0.366857\pi$
$548$	0	0
$549$	−6.90833	−0.294840
$550$	0	0
$551$	6.78890	0.289217
$552$	0	0
$553$	21.5139	0.914863
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−43.4500	−1.84103	−0.920517	−	0.390703i	$-0.872232\pi$
$557$	−43.4500	−1.84103	−0.920517	+	0.390703i	$0.872232\pi$
$558$	0	0
$559$	−26.7250	−1.13035
$560$	0	0
$561$	−6.90833	−0.291670
$562$	0	0
$563$	−14.9361	−0.629481	−0.314740	−	0.949178i	$-0.601917\pi$
$563$	−14.9361	−0.629481	−0.314740	+	0.949178i	$0.601917\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−4.30278	−0.180700
$568$	0	0
$569$	−28.4500	−1.19268	−0.596342	−	0.802730i	$-0.703380\pi$
$569$	−28.4500	−1.19268	−0.596342	+	0.802730i	$0.703380\pi$
$570$	0	0
$571$	−24.2111	−1.01320	−0.506602	−	0.862180i	$-0.669098\pi$
$571$	−24.2111	−1.01320	−0.506602	+	0.862180i	$0.669098\pi$
$572$	0	0
$573$	13.4222	0.560721
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−24.7250	−1.02931	−0.514657	−	0.857396i	$-0.672081\pi$
$577$	−24.7250	−1.02931	−0.514657	+	0.857396i	$0.672081\pi$
$578$	0	0
$579$	19.4222	0.807159
$580$	0	0
$581$	−7.69722	−0.319335
$582$	0	0
$583$	−44.3305	−1.83598
$584$	0	0
$585$	0	0
$586$	0	0
$587$	41.7250	1.72217	0.861087	−	0.508457i	$-0.169784\pi$
$587$	41.7250	1.72217	0.861087	+	0.508457i	$0.169784\pi$
$588$	0	0
$589$	−23.2111	−0.956397
$590$	0	0
$591$	5.39445	0.221898
$592$	0	0
$593$	−1.18335	−0.0485942	−0.0242971	−	0.999705i	$-0.507735\pi$
$593$	−1.18335	−0.0485942	−0.0242971	+	0.999705i	$0.507735\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	12.6056	0.515911
$598$	0	0
$599$	−35.0555	−1.43233	−0.716165	−	0.697931i	$-0.754104\pi$
$599$	−35.0555	−1.43233	−0.716165	+	0.697931i	$0.754104\pi$
$600$	0	0
$601$	−17.6333	−0.719278	−0.359639	−	0.933092i	$-0.617100\pi$
$601$	−17.6333	−0.719278	−0.359639	+	0.933092i	$0.617100\pi$
$602$	0	0
$603$	−0.0916731	−0.00373322
$604$	0	0
$605$	0	0
$606$	0	0
$607$	6.69722	0.271832	0.135916	−	0.990720i	$-0.456602\pi$
$607$	6.69722	0.271832	0.135916	+	0.990720i	$0.456602\pi$
$608$	0	0
$609$	−11.2111	−0.454297
$610$	0	0
$611$	37.9361	1.53473
$612$	0	0
$613$	39.3028	1.58742	0.793712	−	0.608294i	$-0.208146\pi$
$613$	39.3028	1.58742	0.793712	+	0.608294i	$0.208146\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−28.1194	−1.13205	−0.566023	−	0.824390i	$-0.691519\pi$
$617$	−28.1194	−1.13205	−0.566023	+	0.824390i	$0.691519\pi$
$618$	0	0
$619$	39.9361	1.60517	0.802583	−	0.596540i	$-0.203458\pi$
$619$	39.9361	1.60517	0.802583	+	0.596540i	$0.203458\pi$
$620$	0	0
$621$	1.00000	0.0401286
$622$	0	0
$623$	38.7250	1.55148
$624$	0	0
$625$	0	0
$626$	0	0
$627$	11.2111	0.447728
$628$	0	0
$629$	15.4222	0.614924
$630$	0	0
$631$	0.936083	0.0372649	0.0186324	−	0.999826i	$-0.494069\pi$
$631$	0.936083	0.0372649	0.0186324	+	0.999826i	$0.494069\pi$
$632$	0	0
$633$	−2.18335	−0.0867802
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−49.5416	−1.96291
$638$	0	0
$639$	−2.90833	−0.115052
$640$	0	0
$641$	15.6972	0.620003	0.310002	−	0.950736i	$-0.399670\pi$
$641$	15.6972	0.620003	0.310002	+	0.950736i	$0.399670\pi$
$642$	0	0
$643$	0.302776	0.0119403	0.00597015	−	0.999982i	$-0.498100\pi$
$643$	0.302776	0.0119403	0.00597015	+	0.999982i	$0.498100\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	47.2111	1.85606	0.928030	−	0.372506i	$-0.121501\pi$
$647$	47.2111	1.85606	0.928030	+	0.372506i	$0.121501\pi$
$648$	0	0
$649$	60.7527	2.38475
$650$	0	0
$651$	38.3305	1.50229
$652$	0	0
$653$	20.4500	0.800269	0.400134	−	0.916456i	$-0.368963\pi$
$653$	20.4500	0.800269	0.400134	+	0.916456i	$0.368963\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	1.78890	0.0697915
$658$	0	0
$659$	23.7250	0.924194	0.462097	−	0.886829i	$-0.347097\pi$
$659$	23.7250	0.924194	0.462097	+	0.886829i	$0.347097\pi$
$660$	0	0
$661$	18.5416	0.721186	0.360593	−	0.932723i	$-0.382574\pi$
$661$	18.5416	0.721186	0.360593	+	0.932723i	$0.382574\pi$
$662$	0	0
$663$	6.90833	0.268297
$664$	0	0
$665$	0	0
$666$	0	0
$667$	2.60555	0.100887
$668$	0	0
$669$	18.5139	0.715788
$670$	0	0
$671$	−29.7250	−1.14752
$672$	0	0
$673$	13.5778	0.523386	0.261693	−	0.965151i	$-0.415719\pi$
$673$	13.5778	0.523386	0.261693	+	0.965151i	$0.415719\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	8.33053	0.320168	0.160084	−	0.987103i	$-0.448823\pi$
$677$	8.33053	0.320168	0.160084	+	0.987103i	$0.448823\pi$
$678$	0	0
$679$	−40.9361	−1.57098
$680$	0	0
$681$	11.6056	0.444726
$682$	0	0
$683$	−6.21110	−0.237661	−0.118831	−	0.992915i	$-0.537915\pi$
$683$	−6.21110	−0.237661	−0.118831	+	0.992915i	$0.537915\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	23.6056	0.900608
$688$	0	0
$689$	44.3305	1.68886
$690$	0	0
$691$	−19.4222	−0.738856	−0.369428	−	0.929259i	$-0.620446\pi$
$691$	−19.4222	−0.738856	−0.369428	+	0.929259i	$0.620446\pi$
$692$	0	0
$693$	−18.5139	−0.703284
$694$	0	0
$695$	0	0
$696$	0	0
$697$	4.81665	0.182444
$698$	0	0
$699$	14.3028	0.540981
$700$	0	0
$701$	21.0000	0.793159	0.396580	−	0.918000i	$-0.370197\pi$
$701$	21.0000	0.793159	0.396580	+	0.918000i	$0.370197\pi$
$702$	0	0
$703$	−25.0278	−0.943940
$704$	0	0
$705$	0	0
$706$	0	0
$707$	39.2389	1.47573
$708$	0	0
$709$	−12.6056	−0.473411	−0.236706	−	0.971581i	$-0.576068\pi$
$709$	−12.6056	−0.473411	−0.236706	+	0.971581i	$0.576068\pi$
$710$	0	0
$711$	−5.00000	−0.187515
$712$	0	0
$713$	−8.90833	−0.333619
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−21.8167	−0.814758
$718$	0	0
$719$	−41.0000	−1.52904	−0.764521	−	0.644599i	$-0.777024\pi$
$719$	−41.0000	−1.52904	−0.764521	+	0.644599i	$0.777024\pi$
$720$	0	0
$721$	−25.0278	−0.932082
$722$	0	0
$723$	2.69722	0.100311
$724$	0	0
$725$	0	0
$726$	0	0
$727$	41.6333	1.54409	0.772047	−	0.635565i	$-0.219233\pi$
$727$	41.6333	1.54409	0.772047	+	0.635565i	$0.219233\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	9.97224	0.368837
$732$	0	0
$733$	−36.4500	−1.34631	−0.673155	−	0.739501i	$-0.735061\pi$
$733$	−36.4500	−1.34631	−0.673155	+	0.739501i	$0.735061\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−0.394449	−0.0145297
$738$	0	0
$739$	−21.0917	−0.775870	−0.387935	−	0.921687i	$-0.626811\pi$
$739$	−21.0917	−0.775870	−0.387935	+	0.921687i	$0.626811\pi$
$740$	0	0
$741$	−11.2111	−0.411850
$742$	0	0
$743$	20.1472	0.739129	0.369564	−	0.929205i	$-0.379507\pi$
$743$	20.1472	0.739129	0.369564	+	0.929205i	$0.379507\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	1.78890	0.0654523
$748$	0	0
$749$	−9.51388	−0.347630
$750$	0	0
$751$	−31.6972	−1.15665	−0.578324	−	0.815807i	$-0.696293\pi$
$751$	−31.6972	−1.15665	−0.578324	+	0.815807i	$0.696293\pi$
$752$	0	0
$753$	−21.9361	−0.799395
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−37.0917	−1.34812	−0.674060	−	0.738677i	$-0.735451\pi$
$757$	−37.0917	−1.34812	−0.674060	+	0.738677i	$0.735451\pi$
$758$	0	0
$759$	4.30278	0.156181
$760$	0	0
$761$	14.4500	0.523811	0.261905	−	0.965094i	$-0.415649\pi$
$761$	14.4500	0.523811	0.261905	+	0.965094i	$0.415649\pi$
$762$	0	0
$763$	61.1472	2.21368
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−60.7527	−2.19365
$768$	0	0
$769$	43.3028	1.56154	0.780769	−	0.624820i	$-0.214828\pi$
$769$	43.3028	1.56154	0.780769	+	0.624820i	$0.214828\pi$
$770$	0	0
$771$	−15.9083	−0.572924
$772$	0	0
$773$	−9.21110	−0.331300	−0.165650	−	0.986185i	$-0.552972\pi$
$773$	−9.21110	−0.331300	−0.165650	+	0.986185i	$0.552972\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	41.3305	1.48272
$778$	0	0
$779$	−7.81665	−0.280061
$780$	0	0
$781$	−12.5139	−0.447782
$782$	0	0
$783$	2.60555	0.0931148
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−51.6611	−1.84152	−0.920759	−	0.390132i	$-0.872429\pi$
$787$	−51.6611	−1.84152	−0.920759	+	0.390132i	$0.872429\pi$
$788$	0	0
$789$	−9.72498	−0.346218
$790$	0	0
$791$	46.0278	1.63656
$792$	0	0
$793$	29.7250	1.05557
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−32.4500	−1.14944	−0.574718	−	0.818351i	$-0.694888\pi$
$797$	−32.4500	−1.14944	−0.574718	+	0.818351i	$0.694888\pi$
$798$	0	0
$799$	−14.1556	−0.500789
$800$	0	0
$801$	−9.00000	−0.317999
$802$	0	0
$803$	7.69722	0.271629
$804$	0	0
$805$	0	0
$806$	0	0
$807$	22.6056	0.795753
$808$	0	0
$809$	−30.6333	−1.07701	−0.538505	−	0.842622i	$-0.681011\pi$
$809$	−30.6333	−1.07701	−0.538505	+	0.842622i	$0.681011\pi$
$810$	0	0
$811$	2.57779	0.0905186	0.0452593	−	0.998975i	$-0.485589\pi$
$811$	2.57779	0.0905186	0.0452593	+	0.998975i	$0.485589\pi$
$812$	0	0
$813$	−22.6333	−0.793785
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−16.1833	−0.566184
$818$	0	0
$819$	18.5139	0.646927
$820$	0	0
$821$	7.60555	0.265436	0.132718	−	0.991154i	$-0.457630\pi$
$821$	7.60555	0.265436	0.132718	+	0.991154i	$0.457630\pi$
$822$	0	0
$823$	13.3305	0.464673	0.232337	−	0.972635i	$-0.425363\pi$
$823$	13.3305	0.464673	0.232337	+	0.972635i	$0.425363\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	22.8167	0.793413	0.396706	−	0.917946i	$-0.370153\pi$
$827$	22.8167	0.793413	0.396706	+	0.917946i	$0.370153\pi$
$828$	0	0
$829$	35.3944	1.22930	0.614650	−	0.788800i	$-0.289297\pi$
$829$	35.3944	1.22930	0.614650	+	0.788800i	$0.289297\pi$
$830$	0	0
$831$	29.3028	1.01650
$832$	0	0
$833$	18.4861	0.640506
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−8.90833	−0.307917
$838$	0	0
$839$	−26.0917	−0.900785	−0.450392	−	0.892831i	$-0.648716\pi$
$839$	−26.0917	−0.900785	−0.450392	+	0.892831i	$0.648716\pi$
$840$	0	0
$841$	−22.2111	−0.765900
$842$	0	0
$843$	7.00000	0.241093
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−32.3305	−1.11089
$848$	0	0
$849$	7.42221	0.254729
$850$	0	0
$851$	−9.60555	−0.329274
$852$	0	0
$853$	16.7889	0.574841	0.287420	−	0.957805i	$-0.407202\pi$
$853$	16.7889	0.574841	0.287420	+	0.957805i	$0.407202\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	40.1194	1.37045	0.685227	−	0.728330i	$-0.259703\pi$
$857$	40.1194	1.37045	0.685227	+	0.728330i	$0.259703\pi$
$858$	0	0
$859$	−47.5416	−1.62210	−0.811050	−	0.584977i	$-0.801103\pi$
$859$	−47.5416	−1.62210	−0.811050	+	0.584977i	$0.801103\pi$
$860$	0	0
$861$	12.9083	0.439915
$862$	0	0
$863$	−22.8167	−0.776688	−0.388344	−	0.921514i	$-0.626953\pi$
$863$	−22.8167	−0.776688	−0.388344	+	0.921514i	$0.626953\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	14.4222	0.489804
$868$	0	0
$869$	−21.5139	−0.729808
$870$	0	0
$871$	0.394449	0.0133654
$872$	0	0
$873$	9.51388	0.321996
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−28.9083	−0.976165	−0.488082	−	0.872798i	$-0.662303\pi$
$877$	−28.9083	−0.976165	−0.488082	+	0.872798i	$0.662303\pi$
$878$	0	0
$879$	25.4222	0.857470
$880$	0	0
$881$	−26.9361	−0.907500	−0.453750	−	0.891129i	$-0.649914\pi$
$881$	−26.9361	−0.907500	−0.453750	+	0.891129i	$0.649914\pi$
$882$	0	0
$883$	35.2111	1.18495	0.592474	−	0.805590i	$-0.298151\pi$
$883$	35.2111	1.18495	0.592474	+	0.805590i	$0.298151\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	14.3028	0.480240	0.240120	−	0.970743i	$-0.422813\pi$
$887$	14.3028	0.480240	0.240120	+	0.970743i	$0.422813\pi$
$888$	0	0
$889$	−20.6056	−0.691088
$890$	0	0
$891$	4.30278	0.144148
$892$	0	0
$893$	22.9722	0.768737
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−4.30278	−0.143665
$898$	0	0
$899$	−23.2111	−0.774134
$900$	0	0
$901$	−16.5416	−0.551082
$902$	0	0
$903$	26.7250	0.889352
$904$	0	0
$905$	0	0
$906$	0	0
$907$	27.0278	0.897442	0.448721	−	0.893672i	$-0.351880\pi$
$907$	27.0278	0.897442	0.448721	+	0.893672i	$0.351880\pi$
$908$	0	0
$909$	−9.11943	−0.302472
$910$	0	0
$911$	15.8444	0.524949	0.262474	−	0.964939i	$-0.415461\pi$
$911$	15.8444	0.524949	0.262474	+	0.964939i	$0.415461\pi$
$912$	0	0
$913$	7.69722	0.254741
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−63.7527	−2.10530
$918$	0	0
$919$	37.4500	1.23536	0.617680	−	0.786429i	$-0.288073\pi$
$919$	37.4500	1.23536	0.617680	+	0.786429i	$0.288073\pi$
$920$	0	0
$921$	20.4222	0.672935
$922$	0	0
$923$	12.5139	0.411899
$924$	0	0
$925$	0	0
$926$	0	0
$927$	5.81665	0.191044
$928$	0	0
$929$	34.1194	1.11942	0.559711	−	0.828688i	$-0.310912\pi$
$929$	34.1194	1.11942	0.559711	+	0.828688i	$0.310912\pi$
$930$	0	0
$931$	−30.0000	−0.983210
$932$	0	0
$933$	−29.6056	−0.969242
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−11.0278	−0.360261	−0.180131	−	0.983643i	$-0.557652\pi$
$937$	−11.0278	−0.360261	−0.180131	+	0.983643i	$0.557652\pi$
$938$	0	0
$939$	−23.8444	−0.778133
$940$	0	0
$941$	−59.9638	−1.95477	−0.977383	−	0.211478i	$-0.932172\pi$
$941$	−59.9638	−1.95477	−0.977383	+	0.211478i	$0.932172\pi$
$942$	0	0
$943$	−3.00000	−0.0976934
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−59.7805	−1.94261	−0.971303	−	0.237848i	$-0.923558\pi$
$947$	−59.7805	−1.94261	−0.971303	+	0.237848i	$0.923558\pi$
$948$	0	0
$949$	−7.69722	−0.249862
$950$	0	0
$951$	−16.1194	−0.522708
$952$	0	0
$953$	−55.4222	−1.79530	−0.897651	−	0.440708i	$-0.854728\pi$
$953$	−55.4222	−1.79530	−0.897651	+	0.440708i	$0.854728\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	11.2111	0.362403
$958$	0	0
$959$	−25.8167	−0.833663
$960$	0	0
$961$	48.3583	1.55994
$962$	0	0
$963$	2.21110	0.0712518
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−56.6333	−1.82120	−0.910602	−	0.413284i	$-0.864382\pi$
$967$	−56.6333	−1.82120	−0.910602	+	0.413284i	$0.864382\pi$
$968$	0	0
$969$	4.18335	0.134388
$970$	0	0
$971$	−22.6056	−0.725447	−0.362723	−	0.931897i	$-0.618153\pi$
$971$	−22.6056	−0.725447	−0.362723	+	0.931897i	$0.618153\pi$
$972$	0	0
$973$	50.3305	1.61352
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−8.97224	−0.287048	−0.143524	−	0.989647i	$-0.545843\pi$
$977$	−8.97224	−0.287048	−0.143524	+	0.989647i	$0.545843\pi$
$978$	0	0
$979$	−38.7250	−1.23766
$980$	0	0
$981$	−14.2111	−0.453726
$982$	0	0
$983$	34.5416	1.10171	0.550854	−	0.834602i	$-0.314302\pi$
$983$	34.5416	1.10171	0.550854	+	0.834602i	$0.314302\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−37.9361	−1.20752
$988$	0	0
$989$	−6.21110	−0.197502
$990$	0	0
$991$	−21.8167	−0.693029	−0.346514	−	0.938045i	$-0.612635\pi$
$991$	−21.8167	−0.693029	−0.346514	+	0.938045i	$0.612635\pi$
$992$	0	0
$993$	8.51388	0.270180
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−3.88057	−0.122899	−0.0614495	−	0.998110i	$-0.519572\pi$
$997$	−3.88057	−0.122899	−0.0614495	+	0.998110i	$0.519572\pi$
$998$	0	0
$999$	−9.60555	−0.303906

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6900.2.a.l.1.1	✓	2
5.2	odd	4		6900.2.f.q.6349.3		4
5.3	odd	4		6900.2.f.q.6349.2		4
5.4	even	2		6900.2.a.w.1.2	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6900.2.a.l.1.1	✓	2	1.1	even	1	trivial
6900.2.a.w.1.2	yes	2	5.4	even	2
6900.2.f.q.6349.2		4	5.3	odd	4
6900.2.f.q.6349.3		4	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -4.30278 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -4.30278 q^{7} +1.00000 q^{9} +4.30278 q^{11} -4.30278 q^{13} +1.60555 q^{17} -2.60555 q^{19} +4.30278 q^{21} -1.00000 q^{23} -1.00000 q^{27} -2.60555 q^{29} +8.90833 q^{31} -4.30278 q^{33} +9.60555 q^{37} +4.30278 q^{39} +3.00000 q^{41} +6.21110 q^{43} -8.81665 q^{47} +11.5139 q^{49} -1.60555 q^{51} -10.3028 q^{53} +2.60555 q^{57} +14.1194 q^{59} -6.90833 q^{61} -4.30278 q^{63} -0.0916731 q^{67} +1.00000 q^{69} -2.90833 q^{71} +1.78890 q^{73} -18.5139 q^{77} -5.00000 q^{79} +1.00000 q^{81} +1.78890 q^{83} +2.60555 q^{87} -9.00000 q^{89} +18.5139 q^{91} -8.90833 q^{93} +9.51388 q^{97} +4.30278 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} - 5 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^3 - 5 * q^7 + 2 * q^9	\( 2 q - 2 q^{3} - 5 q^{7} + 2 q^{9} + 5 q^{11} - 5 q^{13} - 4 q^{17} + 2 q^{19} + 5 q^{21} - 2 q^{23} - 2 q^{27} + 2 q^{29} + 7 q^{31} - 5 q^{33} + 12 q^{37} + 5 q^{39} + 6 q^{41} - 2 q^{43} + 4 q^{47} + 5 q^{49} + 4 q^{51} - 17 q^{53} - 2 q^{57} + 3 q^{59} - 3 q^{61} - 5 q^{63} - 11 q^{67} + 2 q^{69} + 5 q^{71} + 18 q^{73} - 19 q^{77} - 10 q^{79} + 2 q^{81} + 18 q^{83} - 2 q^{87} - 18 q^{89} + 19 q^{91} - 7 q^{93} + q^{97} + 5 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 - 5 * q^7 + 2 * q^9 + 5 * q^11 - 5 * q^13 - 4 * q^17 + 2 * q^19 + 5 * q^21 - 2 * q^23 - 2 * q^27 + 2 * q^29 + 7 * q^31 - 5 * q^33 + 12 * q^37 + 5 * q^39 + 6 * q^41 - 2 * q^43 + 4 * q^47 + 5 * q^49 + 4 * q^51 - 17 * q^53 - 2 * q^57 + 3 * q^59 - 3 * q^61 - 5 * q^63 - 11 * q^67 + 2 * q^69 + 5 * q^71 + 18 * q^73 - 19 * q^77 - 10 * q^79 + 2 * q^81 + 18 * q^83 - 2 * q^87 - 18 * q^89 + 19 * q^91 - 7 * q^93 + q^97 + 5 * q^99

Introduction

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