LMFDB - Embedded newform 4200.2.a.bp.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4200,2,Mod(1,4200)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4200, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("4200.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4200 = 2^{3} \cdot 3 \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4200.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:	$33.5371688489$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.148.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 3x + 1 $ x^3 - x^2 - 3*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 840)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$0.311108$ of defining polynomial
Character	$\chi$	$=$	4200.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +1.00000 q^{7} +1.00000 q^{9} -5.05086 q^{11} -3.37778 q^{13} -7.18421 q^{17} +8.23506 q^{19} +1.00000 q^{21} +6.23506 q^{23} +1.00000 q^{27} -2.00000 q^{29} -4.62222 q^{31} -5.05086 q^{33} +4.85728 q^{37} -3.37778 q^{39} -3.37778 q^{41} -1.24443 q^{43} +1.00000 q^{49} -7.18421 q^{51} -4.62222 q^{53} +8.23506 q^{57} -11.6128 q^{59} +0.488863 q^{61} +1.00000 q^{63} -3.61285 q^{67} +6.23506 q^{69} -10.2953 q^{71} -16.2351 q^{73} -5.05086 q^{77} +1.24443 q^{79} +1.00000 q^{81} -11.6128 q^{83} -2.00000 q^{87} +6.99063 q^{89} -3.37778 q^{91} -4.62222 q^{93} +8.23506 q^{97} -5.05086 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{3} + 3 q^{7} + 3 q^{9}+O(q^{10}) $ 3 * q + 3 * q^3 + 3 * q^7 + 3 * q^9	$ 3 q + 3 q^{3} + 3 q^{7} + 3 q^{9} - 2 q^{11} - 10 q^{13} - 8 q^{17} - 2 q^{19} + 3 q^{21} - 8 q^{23} + 3 q^{27} - 6 q^{29} - 14 q^{31} - 2 q^{33} - 12 q^{37} - 10 q^{39} - 10 q^{41} - 4 q^{43} + 3 q^{49} - 8 q^{51} - 14 q^{53} - 2 q^{57} - 8 q^{59} + 2 q^{61} + 3 q^{63} + 16 q^{67} - 8 q^{69} - 18 q^{71} - 22 q^{73} - 2 q^{77} + 4 q^{79} + 3 q^{81} - 8 q^{83} - 6 q^{87} - 6 q^{89} - 10 q^{91} - 14 q^{93} - 2 q^{97} - 2 q^{99}+O(q^{100}) $ 3 * q + 3 * q^3 + 3 * q^7 + 3 * q^9 - 2 * q^11 - 10 * q^13 - 8 * q^17 - 2 * q^19 + 3 * q^21 - 8 * q^23 + 3 * q^27 - 6 * q^29 - 14 * q^31 - 2 * q^33 - 12 * q^37 - 10 * q^39 - 10 * q^41 - 4 * q^43 + 3 * q^49 - 8 * q^51 - 14 * q^53 - 2 * q^57 - 8 * q^59 + 2 * q^61 + 3 * q^63 + 16 * q^67 - 8 * q^69 - 18 * q^71 - 22 * q^73 - 2 * q^77 + 4 * q^79 + 3 * q^81 - 8 * q^83 - 6 * q^87 - 6 * q^89 - 10 * q^91 - 14 * q^93 - 2 * 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
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−5.05086	−1.52289	−0.761445	−	0.648229i	$-0.775510\pi$
$11$	−5.05086	−1.52289	−0.761445	+	0.648229i	$0.775510\pi$
$12$	0	0
$13$	−3.37778	−0.936829	−0.468414	−	0.883509i	$-0.655175\pi$
$13$	−3.37778	−0.936829	−0.468414	+	0.883509i	$0.655175\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−7.18421	−1.74243	−0.871213	−	0.490905i	$-0.836666\pi$
$17$	−7.18421	−1.74243	−0.871213	+	0.490905i	$0.836666\pi$
$18$	0	0
$19$	8.23506	1.88925	0.944627	−	0.328147i	$-0.106424\pi$
$19$	8.23506	1.88925	0.944627	+	0.328147i	$0.106424\pi$
$20$	0	0
$21$	1.00000	0.218218
$22$	0	0
$23$	6.23506	1.30010	0.650050	−	0.759891i	$-0.274748\pi$
$23$	6.23506	1.30010	0.650050	+	0.759891i	$0.274748\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−2.00000	−0.371391	−0.185695	−	0.982607i	$-0.559454\pi$
$29$	−2.00000	−0.371391	−0.185695	+	0.982607i	$0.559454\pi$
$30$	0	0
$31$	−4.62222	−0.830174	−0.415087	−	0.909782i	$-0.636249\pi$
$31$	−4.62222	−0.830174	−0.415087	+	0.909782i	$0.636249\pi$
$32$	0	0
$33$	−5.05086	−0.879241
$34$	0	0
$35$	0	0
$36$	0	0
$37$	4.85728	0.798532	0.399266	−	0.916835i	$-0.369265\pi$
$37$	4.85728	0.798532	0.399266	+	0.916835i	$0.369265\pi$
$38$	0	0
$39$	−3.37778	−0.540878
$40$	0	0
$41$	−3.37778	−0.527521	−0.263761	−	0.964588i	$-0.584963\pi$
$41$	−3.37778	−0.527521	−0.263761	+	0.964588i	$0.584963\pi$
$42$	0	0
$43$	−1.24443	−0.189774	−0.0948870	−	0.995488i	$-0.530249\pi$
$43$	−1.24443	−0.189774	−0.0948870	+	0.995488i	$0.530249\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−7.18421	−1.00599
$52$	0	0
$53$	−4.62222	−0.634910	−0.317455	−	0.948273i	$-0.602828\pi$
$53$	−4.62222	−0.634910	−0.317455	+	0.948273i	$0.602828\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	8.23506	1.09076
$58$	0	0
$59$	−11.6128	−1.51186	−0.755932	−	0.654650i	$-0.772816\pi$
$59$	−11.6128	−1.51186	−0.755932	+	0.654650i	$0.772816\pi$
$60$	0	0
$61$	0.488863	0.0625924	0.0312962	−	0.999510i	$-0.490036\pi$
$61$	0.488863	0.0625924	0.0312962	+	0.999510i	$0.490036\pi$
$62$	0	0
$63$	1.00000	0.125988
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−3.61285	−0.441380	−0.220690	−	0.975344i	$-0.570831\pi$
$67$	−3.61285	−0.441380	−0.220690	+	0.975344i	$0.570831\pi$
$68$	0	0
$69$	6.23506	0.750613
$70$	0	0
$71$	−10.2953	−1.22183	−0.610913	−	0.791698i	$-0.709197\pi$
$71$	−10.2953	−1.22183	−0.610913	+	0.791698i	$0.709197\pi$
$72$	0	0
$73$	−16.2351	−1.90017	−0.950085	−	0.311991i	$-0.899004\pi$
$73$	−16.2351	−1.90017	−0.950085	+	0.311991i	$0.899004\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−5.05086	−0.575598
$78$	0	0
$79$	1.24443	0.140009	0.0700047	−	0.997547i	$-0.477699\pi$
$79$	1.24443	0.140009	0.0700047	+	0.997547i	$0.477699\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−11.6128	−1.27468	−0.637338	−	0.770585i	$-0.719964\pi$
$83$	−11.6128	−1.27468	−0.637338	+	0.770585i	$0.719964\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−2.00000	−0.214423
$88$	0	0
$89$	6.99063	0.741006	0.370503	−	0.928831i	$-0.379185\pi$
$89$	6.99063	0.741006	0.370503	+	0.928831i	$0.379185\pi$
$90$	0	0
$91$	−3.37778	−0.354088
$92$	0	0
$93$	−4.62222	−0.479301
$94$	0	0
$95$	0	0
$96$	0	0
$97$	8.23506	0.836144	0.418072	−	0.908414i	$-0.362706\pi$
$97$	8.23506	0.836144	0.418072	+	0.908414i	$0.362706\pi$
$98$	0	0
$99$	−5.05086	−0.507630
$100$	0	0
$101$	−9.47949	−0.943245	−0.471622	−	0.881801i	$-0.656331\pi$
$101$	−9.47949	−0.943245	−0.471622	+	0.881801i	$0.656331\pi$
$102$	0	0
$103$	−16.8573	−1.66100	−0.830499	−	0.557021i	$-0.811944\pi$
$103$	−16.8573	−1.66100	−0.830499	+	0.557021i	$0.811944\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−15.4795	−1.49646	−0.748230	−	0.663440i	$-0.769096\pi$
$107$	−15.4795	−1.49646	−0.748230	+	0.663440i	$0.769096\pi$
$108$	0	0
$109$	−1.61285	−0.154483	−0.0772414	−	0.997012i	$-0.524611\pi$
$109$	−1.61285	−0.154483	−0.0772414	+	0.997012i	$0.524611\pi$
$110$	0	0
$111$	4.85728	0.461033
$112$	0	0
$113$	−1.86665	−0.175599	−0.0877997	−	0.996138i	$-0.527984\pi$
$113$	−1.86665	−0.175599	−0.0877997	+	0.996138i	$0.527984\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−3.37778	−0.312276
$118$	0	0
$119$	−7.18421	−0.658575
$120$	0	0
$121$	14.5111	1.31919
$122$	0	0
$123$	−3.37778	−0.304565
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−12.8573	−1.14090	−0.570450	−	0.821333i	$-0.693231\pi$
$127$	−12.8573	−1.14090	−0.570450	+	0.821333i	$0.693231\pi$
$128$	0	0
$129$	−1.24443	−0.109566
$130$	0	0
$131$	21.7146	1.89721	0.948605	−	0.316463i	$-0.102495\pi$
$131$	21.7146	1.89721	0.948605	+	0.316463i	$0.102495\pi$
$132$	0	0
$133$	8.23506	0.714071
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−9.47949	−0.809888	−0.404944	−	0.914342i	$-0.632709\pi$
$137$	−9.47949	−0.809888	−0.404944	+	0.914342i	$0.632709\pi$
$138$	0	0
$139$	10.1334	0.859500	0.429750	−	0.902948i	$-0.358602\pi$
$139$	10.1334	0.859500	0.429750	+	0.902948i	$0.358602\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	17.0607	1.42669
$144$	0	0
$145$	0	0
$146$	0	0
$147$	1.00000	0.0824786
$148$	0	0
$149$	7.24443	0.593487	0.296743	−	0.954957i	$-0.404099\pi$
$149$	7.24443	0.593487	0.296743	+	0.954957i	$0.404099\pi$
$150$	0	0
$151$	−8.85728	−0.720795	−0.360398	−	0.932799i	$-0.617359\pi$
$151$	−8.85728	−0.720795	−0.360398	+	0.932799i	$0.617359\pi$
$152$	0	0
$153$	−7.18421	−0.580809
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−13.4795	−1.07578	−0.537890	−	0.843015i	$-0.680779\pi$
$157$	−13.4795	−1.07578	−0.537890	+	0.843015i	$0.680779\pi$
$158$	0	0
$159$	−4.62222	−0.366566
$160$	0	0
$161$	6.23506	0.491392
$162$	0	0
$163$	−4.00000	−0.313304	−0.156652	−	0.987654i	$-0.550070\pi$
$163$	−4.00000	−0.313304	−0.156652	+	0.987654i	$0.550070\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−2.10171	−0.162635	−0.0813176	−	0.996688i	$-0.525913\pi$
$167$	−2.10171	−0.162635	−0.0813176	+	0.996688i	$0.525913\pi$
$168$	0	0
$169$	−1.59057	−0.122352
$170$	0	0
$171$	8.23506	0.629751
$172$	0	0
$173$	−11.1842	−0.850320	−0.425160	−	0.905118i	$-0.639782\pi$
$173$	−11.1842	−0.850320	−0.425160	+	0.905118i	$0.639782\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−11.6128	−0.872875
$178$	0	0
$179$	20.6637	1.54448	0.772239	−	0.635332i	$-0.219137\pi$
$179$	20.6637	1.54448	0.772239	+	0.635332i	$0.219137\pi$
$180$	0	0
$181$	24.9590	1.85519	0.927594	−	0.373591i	$-0.121874\pi$
$181$	24.9590	1.85519	0.927594	+	0.373591i	$0.121874\pi$
$182$	0	0
$183$	0.488863	0.0361378
$184$	0	0
$185$	0	0
$186$	0	0
$187$	36.2864	2.65352
$188$	0	0
$189$	1.00000	0.0727393
$190$	0	0
$191$	9.52098	0.688914	0.344457	−	0.938802i	$-0.388063\pi$
$191$	9.52098	0.688914	0.344457	+	0.938802i	$0.388063\pi$
$192$	0	0
$193$	22.9590	1.65262	0.826312	−	0.563212i	$-0.190435\pi$
$193$	22.9590	1.65262	0.826312	+	0.563212i	$0.190435\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	19.8479	1.41411	0.707053	−	0.707161i	$-0.250024\pi$
$197$	19.8479	1.41411	0.707053	+	0.707161i	$0.250024\pi$
$198$	0	0
$199$	−8.23506	−0.583768	−0.291884	−	0.956454i	$-0.594282\pi$
$199$	−8.23506	−0.583768	−0.291884	+	0.956454i	$0.594282\pi$
$200$	0	0
$201$	−3.61285	−0.254831
$202$	0	0
$203$	−2.00000	−0.140372
$204$	0	0
$205$	0	0
$206$	0	0
$207$	6.23506	0.433367
$208$	0	0
$209$	−41.5941	−2.87712
$210$	0	0
$211$	−11.6128	−0.799461	−0.399731	−	0.916633i	$-0.630896\pi$
$211$	−11.6128	−0.799461	−0.399731	+	0.916633i	$0.630896\pi$
$212$	0	0
$213$	−10.2953	−0.705421
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−4.62222	−0.313776
$218$	0	0
$219$	−16.2351	−1.09706
$220$	0	0
$221$	24.2667	1.63236
$222$	0	0
$223$	−2.48886	−0.166667	−0.0833333	−	0.996522i	$-0.526557\pi$
$223$	−2.48886	−0.166667	−0.0833333	+	0.996522i	$0.526557\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	0.857279	0.0568996	0.0284498	−	0.999595i	$-0.490943\pi$
$227$	0.857279	0.0568996	0.0284498	+	0.999595i	$0.490943\pi$
$228$	0	0
$229$	−14.4701	−0.956213	−0.478106	−	0.878302i	$-0.658677\pi$
$229$	−14.4701	−0.956213	−0.478106	+	0.878302i	$0.658677\pi$
$230$	0	0
$231$	−5.05086	−0.332322
$232$	0	0
$233$	3.96836	0.259976	0.129988	−	0.991516i	$-0.458506\pi$
$233$	3.96836	0.259976	0.129988	+	0.991516i	$0.458506\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	1.24443	0.0808345
$238$	0	0
$239$	1.90813	0.123427	0.0617135	−	0.998094i	$-0.480344\pi$
$239$	1.90813	0.123427	0.0617135	+	0.998094i	$0.480344\pi$
$240$	0	0
$241$	0.755569	0.0486705	0.0243352	−	0.999704i	$-0.492253\pi$
$241$	0.755569	0.0486705	0.0243352	+	0.999704i	$0.492253\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−27.8163	−1.76991
$248$	0	0
$249$	−11.6128	−0.735934
$250$	0	0
$251$	−9.12399	−0.575901	−0.287950	−	0.957645i	$-0.592974\pi$
$251$	−9.12399	−0.575901	−0.287950	+	0.957645i	$0.592974\pi$
$252$	0	0
$253$	−31.4924	−1.97991
$254$	0	0
$255$	0	0
$256$	0	0
$257$	1.28592	0.0802134	0.0401067	−	0.999195i	$-0.487230\pi$
$257$	1.28592	0.0802134	0.0401067	+	0.999195i	$0.487230\pi$
$258$	0	0
$259$	4.85728	0.301817
$260$	0	0
$261$	−2.00000	−0.123797
$262$	0	0
$263$	−7.00937	−0.432216	−0.216108	−	0.976369i	$-0.569336\pi$
$263$	−7.00937	−0.432216	−0.216108	+	0.976369i	$0.569336\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	6.99063	0.427820
$268$	0	0
$269$	23.1941	1.41417	0.707083	−	0.707130i	$-0.250011\pi$
$269$	23.1941	1.41417	0.707083	+	0.707130i	$0.250011\pi$
$270$	0	0
$271$	7.11108	0.431967	0.215984	−	0.976397i	$-0.430704\pi$
$271$	7.11108	0.431967	0.215984	+	0.976397i	$0.430704\pi$
$272$	0	0
$273$	−3.37778	−0.204433
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−7.34614	−0.441387	−0.220693	−	0.975343i	$-0.570832\pi$
$277$	−7.34614	−0.441387	−0.220693	+	0.975343i	$0.570832\pi$
$278$	0	0
$279$	−4.62222	−0.276725
$280$	0	0
$281$	−19.9813	−1.19198	−0.595991	−	0.802991i	$-0.703241\pi$
$281$	−19.9813	−1.19198	−0.595991	+	0.802991i	$0.703241\pi$
$282$	0	0
$283$	−4.85728	−0.288735	−0.144368	−	0.989524i	$-0.546115\pi$
$283$	−4.85728	−0.288735	−0.144368	+	0.989524i	$0.546115\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−3.37778	−0.199384
$288$	0	0
$289$	34.6128	2.03605
$290$	0	0
$291$	8.23506	0.482748
$292$	0	0
$293$	11.1842	0.653388	0.326694	−	0.945130i	$-0.394065\pi$
$293$	11.1842	0.653388	0.326694	+	0.945130i	$0.394065\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−5.05086	−0.293080
$298$	0	0
$299$	−21.0607	−1.21797
$300$	0	0
$301$	−1.24443	−0.0717278
$302$	0	0
$303$	−9.47949	−0.544583
$304$	0	0
$305$	0	0
$306$	0	0
$307$	23.3461	1.33243	0.666217	−	0.745758i	$-0.267912\pi$
$307$	23.3461	1.33243	0.666217	+	0.745758i	$0.267912\pi$
$308$	0	0
$309$	−16.8573	−0.958977
$310$	0	0
$311$	−32.8573	−1.86317	−0.931583	−	0.363530i	$-0.881572\pi$
$311$	−32.8573	−1.86317	−0.931583	+	0.363530i	$0.881572\pi$
$312$	0	0
$313$	−28.8256	−1.62932	−0.814661	−	0.579938i	$-0.803077\pi$
$313$	−28.8256	−1.62932	−0.814661	+	0.579938i	$0.803077\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−25.3590	−1.42431	−0.712153	−	0.702024i	$-0.752280\pi$
$317$	−25.3590	−1.42431	−0.712153	+	0.702024i	$0.752280\pi$
$318$	0	0
$319$	10.1017	0.565587
$320$	0	0
$321$	−15.4795	−0.863981
$322$	0	0
$323$	−59.1624	−3.29188
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−1.61285	−0.0891907
$328$	0	0
$329$	0	0
$330$	0	0
$331$	1.89829	0.104339	0.0521697	−	0.998638i	$-0.483386\pi$
$331$	1.89829	0.104339	0.0521697	+	0.998638i	$0.483386\pi$
$332$	0	0
$333$	4.85728	0.266177
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−23.2257	−1.26518	−0.632592	−	0.774485i	$-0.718009\pi$
$337$	−23.2257	−1.26518	−0.632592	+	0.774485i	$0.718009\pi$
$338$	0	0
$339$	−1.86665	−0.101382
$340$	0	0
$341$	23.3461	1.26426
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−7.47949	−0.401520	−0.200760	−	0.979640i	$-0.564341\pi$
$347$	−7.47949	−0.401520	−0.200760	+	0.979640i	$0.564341\pi$
$348$	0	0
$349$	29.2257	1.56442	0.782208	−	0.623018i	$-0.214094\pi$
$349$	29.2257	1.56442	0.782208	+	0.623018i	$0.214094\pi$
$350$	0	0
$351$	−3.37778	−0.180293
$352$	0	0
$353$	30.4099	1.61856	0.809278	−	0.587426i	$-0.199859\pi$
$353$	30.4099	1.61856	0.809278	+	0.587426i	$0.199859\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−7.18421	−0.380229
$358$	0	0
$359$	−16.1936	−0.854664	−0.427332	−	0.904095i	$-0.640546\pi$
$359$	−16.1936	−0.854664	−0.427332	+	0.904095i	$0.640546\pi$
$360$	0	0
$361$	48.8163	2.56928
$362$	0	0
$363$	14.5111	0.761637
$364$	0	0
$365$	0	0
$366$	0	0
$367$	5.51114	0.287679	0.143840	−	0.989601i	$-0.454055\pi$
$367$	5.51114	0.287679	0.143840	+	0.989601i	$0.454055\pi$
$368$	0	0
$369$	−3.37778	−0.175840
$370$	0	0
$371$	−4.62222	−0.239973
$372$	0	0
$373$	−1.63158	−0.0844802	−0.0422401	−	0.999107i	$-0.513449\pi$
$373$	−1.63158	−0.0844802	−0.0422401	+	0.999107i	$0.513449\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	6.75557	0.347929
$378$	0	0
$379$	20.8573	1.07137	0.535683	−	0.844419i	$-0.320054\pi$
$379$	20.8573	1.07137	0.535683	+	0.844419i	$0.320054\pi$
$380$	0	0
$381$	−12.8573	−0.658698
$382$	0	0
$383$	−20.0830	−1.02619	−0.513096	−	0.858331i	$-0.671502\pi$
$383$	−20.0830	−1.02619	−0.513096	+	0.858331i	$0.671502\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−1.24443	−0.0632580
$388$	0	0
$389$	33.2257	1.68461	0.842305	−	0.539001i	$-0.181198\pi$
$389$	33.2257	1.68461	0.842305	+	0.539001i	$0.181198\pi$
$390$	0	0
$391$	−44.7940	−2.26533
$392$	0	0
$393$	21.7146	1.09535
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−22.7239	−1.14048	−0.570241	−	0.821478i	$-0.693150\pi$
$397$	−22.7239	−1.14048	−0.570241	+	0.821478i	$0.693150\pi$
$398$	0	0
$399$	8.23506	0.412269
$400$	0	0
$401$	12.7556	0.636983	0.318491	−	0.947926i	$-0.396824\pi$
$401$	12.7556	0.636983	0.318491	+	0.947926i	$0.396824\pi$
$402$	0	0
$403$	15.6128	0.777731
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−24.5334	−1.21608
$408$	0	0
$409$	27.4479	1.35721	0.678604	−	0.734504i	$-0.262585\pi$
$409$	27.4479	1.35721	0.678604	+	0.734504i	$0.262585\pi$
$410$	0	0
$411$	−9.47949	−0.467589
$412$	0	0
$413$	−11.6128	−0.571431
$414$	0	0
$415$	0	0
$416$	0	0
$417$	10.1334	0.496232
$418$	0	0
$419$	2.36842	0.115705	0.0578524	−	0.998325i	$-0.481575\pi$
$419$	2.36842	0.115705	0.0578524	+	0.998325i	$0.481575\pi$
$420$	0	0
$421$	39.3274	1.91670	0.958350	−	0.285596i	$-0.0921914\pi$
$421$	39.3274	1.91670	0.958350	+	0.285596i	$0.0921914\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	0.488863	0.0236577
$428$	0	0
$429$	17.0607	0.823698
$430$	0	0
$431$	8.19358	0.394671	0.197335	−	0.980336i	$-0.436771\pi$
$431$	8.19358	0.394671	0.197335	+	0.980336i	$0.436771\pi$
$432$	0	0
$433$	14.6035	0.701798	0.350899	−	0.936413i	$-0.385876\pi$
$433$	14.6035	0.701798	0.350899	+	0.936413i	$0.385876\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	51.3461	2.45622
$438$	0	0
$439$	5.27607	0.251813	0.125907	−	0.992042i	$-0.459816\pi$
$439$	5.27607	0.251813	0.125907	+	0.992042i	$0.459816\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	−18.5018	−0.879046	−0.439523	−	0.898231i	$-0.644852\pi$
$443$	−18.5018	−0.879046	−0.439523	+	0.898231i	$0.644852\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	7.24443	0.342650
$448$	0	0
$449$	11.7146	0.552844	0.276422	−	0.961036i	$-0.410851\pi$
$449$	11.7146	0.552844	0.276422	+	0.961036i	$0.410851\pi$
$450$	0	0
$451$	17.0607	0.803357
$452$	0	0
$453$	−8.85728	−0.416151
$454$	0	0
$455$	0	0
$456$	0	0
$457$	18.9590	0.886864	0.443432	−	0.896308i	$-0.353761\pi$
$457$	18.9590	0.886864	0.443432	+	0.896308i	$0.353761\pi$
$458$	0	0
$459$	−7.18421	−0.335330
$460$	0	0
$461$	−15.1111	−0.703793	−0.351897	−	0.936039i	$-0.614463\pi$
$461$	−15.1111	−0.703793	−0.351897	+	0.936039i	$0.614463\pi$
$462$	0	0
$463$	22.5718	1.04900	0.524501	−	0.851410i	$-0.324252\pi$
$463$	22.5718	1.04900	0.524501	+	0.851410i	$0.324252\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	22.3684	1.03509	0.517543	−	0.855657i	$-0.326847\pi$
$467$	22.3684	1.03509	0.517543	+	0.855657i	$0.326847\pi$
$468$	0	0
$469$	−3.61285	−0.166826
$470$	0	0
$471$	−13.4795	−0.621102
$472$	0	0
$473$	6.28544	0.289005
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−4.62222	−0.211637
$478$	0	0
$479$	−32.8573	−1.50129	−0.750644	−	0.660707i	$-0.770256\pi$
$479$	−32.8573	−1.50129	−0.750644	+	0.660707i	$0.770256\pi$
$480$	0	0
$481$	−16.4068	−0.748088
$482$	0	0
$483$	6.23506	0.283705
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−3.40943	−0.154496	−0.0772479	−	0.997012i	$-0.524613\pi$
$487$	−3.40943	−0.154496	−0.0772479	+	0.997012i	$0.524613\pi$
$488$	0	0
$489$	−4.00000	−0.180886
$490$	0	0
$491$	−33.7877	−1.52482	−0.762409	−	0.647096i	$-0.775983\pi$
$491$	−33.7877	−1.52482	−0.762409	+	0.647096i	$0.775983\pi$
$492$	0	0
$493$	14.3684	0.647121
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−10.2953	−0.461807
$498$	0	0
$499$	13.6316	0.610233	0.305117	−	0.952315i	$-0.401305\pi$
$499$	13.6316	0.610233	0.305117	+	0.952315i	$0.401305\pi$
$500$	0	0
$501$	−2.10171	−0.0938975
$502$	0	0
$503$	−7.34614	−0.327548	−0.163774	−	0.986498i	$-0.552367\pi$
$503$	−7.34614	−0.327548	−0.163774	+	0.986498i	$0.552367\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−1.59057	−0.0706398
$508$	0	0
$509$	−15.9684	−0.707785	−0.353892	−	0.935286i	$-0.615142\pi$
$509$	−15.9684	−0.707785	−0.353892	+	0.935286i	$0.615142\pi$
$510$	0	0
$511$	−16.2351	−0.718197
$512$	0	0
$513$	8.23506	0.363587
$514$	0	0
$515$	0	0
$516$	0	0
$517$	0	0
$518$	0	0
$519$	−11.1842	−0.490932
$520$	0	0
$521$	−1.09234	−0.0478564	−0.0239282	−	0.999714i	$-0.507617\pi$
$521$	−1.09234	−0.0478564	−0.0239282	+	0.999714i	$0.507617\pi$
$522$	0	0
$523$	−17.5111	−0.765709	−0.382854	−	0.923809i	$-0.625059\pi$
$523$	−17.5111	−0.765709	−0.382854	+	0.923809i	$0.625059\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	33.2070	1.44652
$528$	0	0
$529$	15.8760	0.690262
$530$	0	0
$531$	−11.6128	−0.503955
$532$	0	0
$533$	11.4094	0.494197
$534$	0	0
$535$	0	0
$536$	0	0
$537$	20.6637	0.891705
$538$	0	0
$539$	−5.05086	−0.217556
$540$	0	0
$541$	−15.3274	−0.658977	−0.329488	−	0.944160i	$-0.606876\pi$
$541$	−15.3274	−0.658977	−0.329488	+	0.944160i	$0.606876\pi$
$542$	0	0
$543$	24.9590	1.07109
$544$	0	0
$545$	0	0
$546$	0	0
$547$	5.32741	0.227783	0.113892	−	0.993493i	$-0.463668\pi$
$547$	5.32741	0.227783	0.113892	+	0.993493i	$0.463668\pi$
$548$	0	0
$549$	0.488863	0.0208641
$550$	0	0
$551$	−16.4701	−0.701651
$552$	0	0
$553$	1.24443	0.0529186
$554$	0	0
$555$	0	0
$556$	0	0
$557$	0.355509	0.0150634	0.00753171	−	0.999972i	$-0.497603\pi$
$557$	0.355509	0.0150634	0.00753171	+	0.999972i	$0.497603\pi$
$558$	0	0
$559$	4.20342	0.177786
$560$	0	0
$561$	36.2864	1.53201
$562$	0	0
$563$	−16.4701	−0.694133	−0.347067	−	0.937841i	$-0.612822\pi$
$563$	−16.4701	−0.694133	−0.347067	+	0.937841i	$0.612822\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	1.00000	0.0419961
$568$	0	0
$569$	−20.9590	−0.878647	−0.439323	−	0.898329i	$-0.644782\pi$
$569$	−20.9590	−0.878647	−0.439323	+	0.898329i	$0.644782\pi$
$570$	0	0
$571$	17.5111	0.732818	0.366409	−	0.930454i	$-0.380587\pi$
$571$	17.5111	0.732818	0.366409	+	0.930454i	$0.380587\pi$
$572$	0	0
$573$	9.52098	0.397745
$574$	0	0
$575$	0	0
$576$	0	0
$577$	0.152089	0.00633155	0.00316577	−	0.999995i	$-0.498992\pi$
$577$	0.152089	0.00633155	0.00316577	+	0.999995i	$0.498992\pi$
$578$	0	0
$579$	22.9590	0.954143
$580$	0	0
$581$	−11.6128	−0.481782
$582$	0	0
$583$	23.3461	0.966898
$584$	0	0
$585$	0	0
$586$	0	0
$587$	34.1847	1.41095	0.705476	−	0.708733i	$-0.250733\pi$
$587$	34.1847	1.41095	0.705476	+	0.708733i	$0.250733\pi$
$588$	0	0
$589$	−38.0642	−1.56841
$590$	0	0
$591$	19.8479	0.816434
$592$	0	0
$593$	19.3047	0.792747	0.396374	−	0.918089i	$-0.370269\pi$
$593$	19.3047	0.792747	0.396374	+	0.918089i	$0.370269\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−8.23506	−0.337039
$598$	0	0
$599$	−25.9081	−1.05858	−0.529289	−	0.848442i	$-0.677541\pi$
$599$	−25.9081	−1.05858	−0.529289	+	0.848442i	$0.677541\pi$
$600$	0	0
$601$	−22.7368	−0.927455	−0.463727	−	0.885978i	$-0.653488\pi$
$601$	−22.7368	−0.927455	−0.463727	+	0.885978i	$0.653488\pi$
$602$	0	0
$603$	−3.61285	−0.147127
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−32.9403	−1.33700	−0.668502	−	0.743711i	$-0.733064\pi$
$607$	−32.9403	−1.33700	−0.668502	+	0.743711i	$0.733064\pi$
$608$	0	0
$609$	−2.00000	−0.0810441
$610$	0	0
$611$	0	0
$612$	0	0
$613$	−13.1240	−0.530073	−0.265036	−	0.964238i	$-0.585384\pi$
$613$	−13.1240	−0.530073	−0.265036	+	0.964238i	$0.585384\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	18.2538	0.734870	0.367435	−	0.930049i	$-0.380236\pi$
$617$	18.2538	0.734870	0.367435	+	0.930049i	$0.380236\pi$
$618$	0	0
$619$	7.64449	0.307258	0.153629	−	0.988129i	$-0.450904\pi$
$619$	7.64449	0.307258	0.153629	+	0.988129i	$0.450904\pi$
$620$	0	0
$621$	6.23506	0.250204
$622$	0	0
$623$	6.99063	0.280074
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−41.5941	−1.66111
$628$	0	0
$629$	−34.8957	−1.39138
$630$	0	0
$631$	28.6735	1.14148	0.570738	−	0.821132i	$-0.306657\pi$
$631$	28.6735	1.14148	0.570738	+	0.821132i	$0.306657\pi$
$632$	0	0
$633$	−11.6128	−0.461569
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−3.37778	−0.133833
$638$	0	0
$639$	−10.2953	−0.407275
$640$	0	0
$641$	15.4479	0.610153	0.305077	−	0.952328i	$-0.401318\pi$
$641$	15.4479	0.610153	0.305077	+	0.952328i	$0.401318\pi$
$642$	0	0
$643$	25.8350	1.01883	0.509417	−	0.860520i	$-0.329861\pi$
$643$	25.8350	1.01883	0.509417	+	0.860520i	$0.329861\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−23.6128	−0.928317	−0.464158	−	0.885752i	$-0.653643\pi$
$647$	−23.6128	−0.928317	−0.464158	+	0.885752i	$0.653643\pi$
$648$	0	0
$649$	58.6548	2.30240
$650$	0	0
$651$	−4.62222	−0.181159
$652$	0	0
$653$	36.7052	1.43639	0.718193	−	0.695844i	$-0.244970\pi$
$653$	36.7052	1.43639	0.718193	+	0.695844i	$0.244970\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−16.2351	−0.633390
$658$	0	0
$659$	37.9911	1.47992	0.739962	−	0.672649i	$-0.234844\pi$
$659$	37.9911	1.47992	0.739962	+	0.672649i	$0.234844\pi$
$660$	0	0
$661$	−49.2257	−1.91466	−0.957329	−	0.289001i	$-0.906677\pi$
$661$	−49.2257	−1.91466	−0.957329	+	0.289001i	$0.906677\pi$
$662$	0	0
$663$	24.2667	0.942441
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−12.4701	−0.482845
$668$	0	0
$669$	−2.48886	−0.0962250
$670$	0	0
$671$	−2.46917	−0.0953214
$672$	0	0
$673$	−11.2257	−0.432719	−0.216359	−	0.976314i	$-0.569418\pi$
$673$	−11.2257	−0.432719	−0.216359	+	0.976314i	$0.569418\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	9.55262	0.367137	0.183569	−	0.983007i	$-0.441235\pi$
$677$	9.55262	0.367137	0.183569	+	0.983007i	$0.441235\pi$
$678$	0	0
$679$	8.23506	0.316033
$680$	0	0
$681$	0.857279	0.0328510
$682$	0	0
$683$	−17.9684	−0.687540	−0.343770	−	0.939054i	$-0.611704\pi$
$683$	−17.9684	−0.687540	−0.343770	+	0.939054i	$0.611704\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−14.4701	−0.552070
$688$	0	0
$689$	15.6128	0.594802
$690$	0	0
$691$	0.355509	0.0135242	0.00676211	−	0.999977i	$-0.497848\pi$
$691$	0.355509	0.0135242	0.00676211	+	0.999977i	$0.497848\pi$
$692$	0	0
$693$	−5.05086	−0.191866
$694$	0	0
$695$	0	0
$696$	0	0
$697$	24.2667	0.919167
$698$	0	0
$699$	3.96836	0.150097
$700$	0	0
$701$	13.2257	0.499528	0.249764	−	0.968307i	$-0.419647\pi$
$701$	13.2257	0.499528	0.249764	+	0.968307i	$0.419647\pi$
$702$	0	0
$703$	40.0000	1.50863
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−9.47949	−0.356513
$708$	0	0
$709$	−10.9777	−0.412277	−0.206139	−	0.978523i	$-0.566090\pi$
$709$	−10.9777	−0.412277	−0.206139	+	0.978523i	$0.566090\pi$
$710$	0	0
$711$	1.24443	0.0466698
$712$	0	0
$713$	−28.8198	−1.07931
$714$	0	0
$715$	0	0
$716$	0	0
$717$	1.90813	0.0712606
$718$	0	0
$719$	−21.9813	−0.819763	−0.409881	−	0.912139i	$-0.634430\pi$
$719$	−21.9813	−0.819763	−0.409881	+	0.912139i	$0.634430\pi$
$720$	0	0
$721$	−16.8573	−0.627798
$722$	0	0
$723$	0.755569	0.0280999
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−13.0607	−0.484395	−0.242197	−	0.970227i	$-0.577868\pi$
$727$	−13.0607	−0.484395	−0.242197	+	0.970227i	$0.577868\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	8.94025	0.330667
$732$	0	0
$733$	13.5625	0.500941	0.250471	−	0.968124i	$-0.419415\pi$
$733$	13.5625	0.500941	0.250471	+	0.968124i	$0.419415\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	18.2480	0.672173
$738$	0	0
$739$	10.2854	0.378356	0.189178	−	0.981943i	$-0.439418\pi$
$739$	10.2854	0.378356	0.189178	+	0.981943i	$0.439418\pi$
$740$	0	0
$741$	−27.8163	−1.02186
$742$	0	0
$743$	21.4608	0.787319	0.393659	−	0.919256i	$-0.371209\pi$
$743$	21.4608	0.787319	0.393659	+	0.919256i	$0.371209\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−11.6128	−0.424892
$748$	0	0
$749$	−15.4795	−0.565608
$750$	0	0
$751$	46.0642	1.68091	0.840454	−	0.541883i	$-0.182288\pi$
$751$	46.0642	1.68091	0.840454	+	0.541883i	$0.182288\pi$
$752$	0	0
$753$	−9.12399	−0.332497
$754$	0	0
$755$	0	0
$756$	0	0
$757$	27.1052	0.985157	0.492579	−	0.870268i	$-0.336054\pi$
$757$	27.1052	0.985157	0.492579	+	0.870268i	$0.336054\pi$
$758$	0	0
$759$	−31.4924	−1.14310
$760$	0	0
$761$	39.4608	1.43045	0.715226	−	0.698894i	$-0.246324\pi$
$761$	39.4608	1.43045	0.715226	+	0.698894i	$0.246324\pi$
$762$	0	0
$763$	−1.61285	−0.0583890
$764$	0	0
$765$	0	0
$766$	0	0
$767$	39.2257	1.41636
$768$	0	0
$769$	−7.51114	−0.270859	−0.135429	−	0.990787i	$-0.543241\pi$
$769$	−7.51114	−0.270859	−0.135429	+	0.990787i	$0.543241\pi$
$770$	0	0
$771$	1.28592	0.0463112
$772$	0	0
$773$	1.46965	0.0528596	0.0264298	−	0.999651i	$-0.491586\pi$
$773$	1.46965	0.0528596	0.0264298	+	0.999651i	$0.491586\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	4.85728	0.174254
$778$	0	0
$779$	−27.8163	−0.996621
$780$	0	0
$781$	52.0000	1.86071
$782$	0	0
$783$	−2.00000	−0.0714742
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−16.2034	−0.577590	−0.288795	−	0.957391i	$-0.593255\pi$
$787$	−16.2034	−0.577590	−0.288795	+	0.957391i	$0.593255\pi$
$788$	0	0
$789$	−7.00937	−0.249540
$790$	0	0
$791$	−1.86665	−0.0663703
$792$	0	0
$793$	−1.65127	−0.0586384
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−0.161933	−0.00573597	−0.00286799	−	0.999996i	$-0.500913\pi$
$797$	−0.161933	−0.00573597	−0.00286799	+	0.999996i	$0.500913\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	6.99063	0.247002
$802$	0	0
$803$	82.0010	2.89375
$804$	0	0
$805$	0	0
$806$	0	0
$807$	23.1941	0.816469
$808$	0	0
$809$	−42.9403	−1.50970	−0.754849	−	0.655898i	$-0.772290\pi$
$809$	−42.9403	−1.50970	−0.754849	+	0.655898i	$0.772290\pi$
$810$	0	0
$811$	−41.2958	−1.45009	−0.725045	−	0.688701i	$-0.758181\pi$
$811$	−41.2958	−1.45009	−0.725045	+	0.688701i	$0.758181\pi$
$812$	0	0
$813$	7.11108	0.249396
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−10.2480	−0.358531
$818$	0	0
$819$	−3.37778	−0.118029
$820$	0	0
$821$	−43.2070	−1.50793	−0.753967	−	0.656913i	$-0.771862\pi$
$821$	−43.2070	−1.50793	−0.753967	+	0.656913i	$0.771862\pi$
$822$	0	0
$823$	11.1427	0.388411	0.194205	−	0.980961i	$-0.437787\pi$
$823$	11.1427	0.388411	0.194205	+	0.980961i	$0.437787\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−36.9906	−1.28629	−0.643145	−	0.765744i	$-0.722371\pi$
$827$	−36.9906	−1.28629	−0.643145	+	0.765744i	$0.722371\pi$
$828$	0	0
$829$	−22.6735	−0.787485	−0.393742	−	0.919221i	$-0.628820\pi$
$829$	−22.6735	−0.787485	−0.393742	+	0.919221i	$0.628820\pi$
$830$	0	0
$831$	−7.34614	−0.254835
$832$	0	0
$833$	−7.18421	−0.248918
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−4.62222	−0.159767
$838$	0	0
$839$	−21.8983	−0.756013	−0.378006	−	0.925803i	$-0.623390\pi$
$839$	−21.8983	−0.756013	−0.378006	+	0.925803i	$0.623390\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	0	0
$843$	−19.9813	−0.688191
$844$	0	0
$845$	0	0
$846$	0	0
$847$	14.5111	0.498609
$848$	0	0
$849$	−4.85728	−0.166701
$850$	0	0
$851$	30.2854	1.03817
$852$	0	0
$853$	−52.4010	−1.79418	−0.897088	−	0.441851i	$-0.854322\pi$
$853$	−52.4010	−1.79418	−0.897088	+	0.441851i	$0.854322\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	24.5116	0.837301	0.418650	−	0.908147i	$-0.362503\pi$
$857$	24.5116	0.837301	0.418650	+	0.908147i	$0.362503\pi$
$858$	0	0
$859$	57.4795	1.96118	0.980588	−	0.196082i	$-0.0628218\pi$
$859$	57.4795	1.96118	0.980588	+	0.196082i	$0.0628218\pi$
$860$	0	0
$861$	−3.37778	−0.115115
$862$	0	0
$863$	−21.1941	−0.721454	−0.360727	−	0.932671i	$-0.617471\pi$
$863$	−21.1941	−0.721454	−0.360727	+	0.932671i	$0.617471\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	34.6128	1.17551
$868$	0	0
$869$	−6.28544	−0.213219
$870$	0	0
$871$	12.2034	0.413497
$872$	0	0
$873$	8.23506	0.278715
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−8.38715	−0.283214	−0.141607	−	0.989923i	$-0.545227\pi$
$877$	−8.38715	−0.283214	−0.141607	+	0.989923i	$0.545227\pi$
$878$	0	0
$879$	11.1842	0.377234
$880$	0	0
$881$	29.1753	0.982941	0.491471	−	0.870894i	$-0.336459\pi$
$881$	29.1753	0.982941	0.491471	+	0.870894i	$0.336459\pi$
$882$	0	0
$883$	19.8796	0.669000	0.334500	−	0.942396i	$-0.391433\pi$
$883$	19.8796	0.669000	0.334500	+	0.942396i	$0.391433\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−14.6923	−0.493319	−0.246659	−	0.969102i	$-0.579333\pi$
$887$	−14.6923	−0.493319	−0.246659	+	0.969102i	$0.579333\pi$
$888$	0	0
$889$	−12.8573	−0.431219
$890$	0	0
$891$	−5.05086	−0.169210
$892$	0	0
$893$	0	0
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−21.0607	−0.703196
$898$	0	0
$899$	9.24443	0.308319
$900$	0	0
$901$	33.2070	1.10628
$902$	0	0
$903$	−1.24443	−0.0414121
$904$	0	0
$905$	0	0
$906$	0	0
$907$	19.8796	0.660090	0.330045	−	0.943965i	$-0.392936\pi$
$907$	19.8796	0.660090	0.330045	+	0.943965i	$0.392936\pi$
$908$	0	0
$909$	−9.47949	−0.314415
$910$	0	0
$911$	−53.3372	−1.76714	−0.883571	−	0.468297i	$-0.844868\pi$
$911$	−53.3372	−1.76714	−0.883571	+	0.468297i	$0.844868\pi$
$912$	0	0
$913$	58.6548	1.94119
$914$	0	0
$915$	0	0
$916$	0	0
$917$	21.7146	0.717078
$918$	0	0
$919$	−21.1240	−0.696816	−0.348408	−	0.937343i	$-0.613278\pi$
$919$	−21.1240	−0.696816	−0.348408	+	0.937343i	$0.613278\pi$
$920$	0	0
$921$	23.3461	0.769282
$922$	0	0
$923$	34.7753	1.14464
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−16.8573	−0.553666
$928$	0	0
$929$	2.72393	0.0893691	0.0446846	−	0.999001i	$-0.485772\pi$
$929$	2.72393	0.0893691	0.0446846	+	0.999001i	$0.485772\pi$
$930$	0	0
$931$	8.23506	0.269893
$932$	0	0
$933$	−32.8573	−1.07570
$934$	0	0
$935$	0	0
$936$	0	0
$937$	15.3145	0.500303	0.250151	−	0.968207i	$-0.419520\pi$
$937$	15.3145	0.500303	0.250151	+	0.968207i	$0.419520\pi$
$938$	0	0
$939$	−28.8256	−0.940689
$940$	0	0
$941$	−30.6035	−0.997645	−0.498822	−	0.866704i	$-0.666234\pi$
$941$	−30.6035	−0.997645	−0.498822	+	0.866704i	$0.666234\pi$
$942$	0	0
$943$	−21.0607	−0.685831
$944$	0	0
$945$	0	0
$946$	0	0
$947$	2.74266	0.0891245	0.0445623	−	0.999007i	$-0.485811\pi$
$947$	2.74266	0.0891245	0.0445623	+	0.999007i	$0.485811\pi$
$948$	0	0
$949$	54.8385	1.78013
$950$	0	0
$951$	−25.3590	−0.822323
$952$	0	0
$953$	47.8479	1.54995	0.774973	−	0.631994i	$-0.217763\pi$
$953$	47.8479	1.54995	0.774973	+	0.631994i	$0.217763\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	10.1017	0.326542
$958$	0	0
$959$	−9.47949	−0.306109
$960$	0	0
$961$	−9.63512	−0.310810
$962$	0	0
$963$	−15.4795	−0.498820
$964$	0	0
$965$	0	0
$966$	0	0
$967$	38.7181	1.24509	0.622545	−	0.782584i	$-0.286099\pi$
$967$	38.7181	1.24509	0.622545	+	0.782584i	$0.286099\pi$
$968$	0	0
$969$	−59.1624	−1.90057
$970$	0	0
$971$	13.7778	0.442152	0.221076	−	0.975257i	$-0.429043\pi$
$971$	13.7778	0.442152	0.221076	+	0.975257i	$0.429043\pi$
$972$	0	0
$973$	10.1334	0.324860
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−4.03164	−0.128984	−0.0644918	−	0.997918i	$-0.520543\pi$
$977$	−4.03164	−0.128984	−0.0644918	+	0.997918i	$0.520543\pi$
$978$	0	0
$979$	−35.3087	−1.12847
$980$	0	0
$981$	−1.61285	−0.0514943
$982$	0	0
$983$	−47.4924	−1.51477	−0.757386	−	0.652967i	$-0.773524\pi$
$983$	−47.4924	−1.51477	−0.757386	+	0.652967i	$0.773524\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−7.75911	−0.246725
$990$	0	0
$991$	1.16146	0.0368949	0.0184474	−	0.999830i	$-0.494128\pi$
$991$	1.16146	0.0368949	0.0184474	+	0.999830i	$0.494128\pi$
$992$	0	0
$993$	1.89829	0.0602404
$994$	0	0
$995$	0	0
$996$	0	0
$997$	18.4572	0.584546	0.292273	−	0.956335i	$-0.405588\pi$
$997$	18.4572	0.584546	0.292273	+	0.956335i	$0.405588\pi$
$998$	0	0
$999$	4.85728	0.153678

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4200.2.a.bp.1.1		3
4.3	odd	2		8400.2.a.di.1.3		3
5.2	odd	4		840.2.t.d.169.2	✓	6
5.3	odd	4		840.2.t.d.169.5	yes	6
5.4	even	2		4200.2.a.bn.1.1		3
15.2	even	4		2520.2.t.k.1009.3		6
15.8	even	4		2520.2.t.k.1009.4		6
20.3	even	4		1680.2.t.j.1009.2		6
20.7	even	4		1680.2.t.j.1009.5		6
20.19	odd	2		8400.2.a.dl.1.3		3
60.23	odd	4		5040.2.t.z.1009.4		6
60.47	odd	4		5040.2.t.z.1009.3		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
840.2.t.d.169.2	✓	6	5.2	odd	4
840.2.t.d.169.5	yes	6	5.3	odd	4
1680.2.t.j.1009.2		6	20.3	even	4
1680.2.t.j.1009.5		6	20.7	even	4
2520.2.t.k.1009.3		6	15.2	even	4
2520.2.t.k.1009.4		6	15.8	even	4
4200.2.a.bn.1.1		3	5.4	even	2
4200.2.a.bp.1.1		3	1.1	even	1	trivial
5040.2.t.z.1009.3		6	60.47	odd	4
5040.2.t.z.1009.4		6	60.23	odd	4
8400.2.a.di.1.3		3	4.3	odd	2
8400.2.a.dl.1.3		3	20.19	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 4200 = 2^{3} \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4200.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +1.00000 q^{7} +1.00000 q^{9} -5.05086 q^{11} -3.37778 q^{13} -7.18421 q^{17} +8.23506 q^{19} +1.00000 q^{21} +6.23506 q^{23} +1.00000 q^{27} -2.00000 q^{29} -4.62222 q^{31} -5.05086 q^{33} +4.85728 q^{37} -3.37778 q^{39} -3.37778 q^{41} -1.24443 q^{43} +1.00000 q^{49} -7.18421 q^{51} -4.62222 q^{53} +8.23506 q^{57} -11.6128 q^{59} +0.488863 q^{61} +1.00000 q^{63} -3.61285 q^{67} +6.23506 q^{69} -10.2953 q^{71} -16.2351 q^{73} -5.05086 q^{77} +1.24443 q^{79} +1.00000 q^{81} -11.6128 q^{83} -2.00000 q^{87} +6.99063 q^{89} -3.37778 q^{91} -4.62222 q^{93} +8.23506 q^{97} -5.05086 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{3} + 3 q^{7} + 3 q^{9}+O(q^{10}) \) 3 * q + 3 * q^3 + 3 * q^7 + 3 * q^9	\( 3 q + 3 q^{3} + 3 q^{7} + 3 q^{9} - 2 q^{11} - 10 q^{13} - 8 q^{17} - 2 q^{19} + 3 q^{21} - 8 q^{23} + 3 q^{27} - 6 q^{29} - 14 q^{31} - 2 q^{33} - 12 q^{37} - 10 q^{39} - 10 q^{41} - 4 q^{43} + 3 q^{49} - 8 q^{51} - 14 q^{53} - 2 q^{57} - 8 q^{59} + 2 q^{61} + 3 q^{63} + 16 q^{67} - 8 q^{69} - 18 q^{71} - 22 q^{73} - 2 q^{77} + 4 q^{79} + 3 q^{81} - 8 q^{83} - 6 q^{87} - 6 q^{89} - 10 q^{91} - 14 q^{93} - 2 q^{97} - 2 q^{99}+O(q^{100}) \) 3 * q + 3 * q^3 + 3 * q^7 + 3 * q^9 - 2 * q^11 - 10 * q^13 - 8 * q^17 - 2 * q^19 + 3 * q^21 - 8 * q^23 + 3 * q^27 - 6 * q^29 - 14 * q^31 - 2 * q^33 - 12 * q^37 - 10 * q^39 - 10 * q^41 - 4 * q^43 + 3 * q^49 - 8 * q^51 - 14 * q^53 - 2 * q^57 - 8 * q^59 + 2 * q^61 + 3 * q^63 + 16 * q^67 - 8 * q^69 - 18 * q^71 - 22 * q^73 - 2 * q^77 + 4 * q^79 + 3 * q^81 - 8 * q^83 - 6 * q^87 - 6 * q^89 - 10 * q^91 - 14 * q^93 - 2 * q^97 - 2 * q^99

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