LMFDB - Embedded newform 9200.2.a.cw.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9200,2,Mod(1,9200)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9200, 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("9200.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9200 = 2^{4} \cdot 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9200.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:	$73.4623698596$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	5.5.791953.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 7x^{3} + 7x^{2} + 9x + 2 $ x^5 - 2x^4 - 7x^3 + 7x^2 + 9x + 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 4600)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.72457$ of defining polynomial
Character	$\chi$	$=$	9200.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.724570 q^{3} -2.33840 q^{7} -2.47500 q^{9} +2.62783 q^{11} +4.29631 q^{13} -6.85404 q^{17} -2.87303 q^{19} +1.69433 q^{21} +1.00000 q^{23} +3.96702 q^{27} -5.03711 q^{29} +7.31504 q^{31} -1.90405 q^{33} +9.24142 q^{37} -3.11297 q^{39} -6.95078 q^{41} +7.01520 q^{43} +4.74690 q^{47} -1.53188 q^{49} +4.96623 q^{51} +12.3849 q^{53} +2.08171 q^{57} +2.14073 q^{59} -2.30567 q^{61} +5.78754 q^{63} -1.98376 q^{67} -0.724570 q^{69} +6.87990 q^{71} -13.2518 q^{73} -6.14492 q^{77} -16.6578 q^{79} +4.55062 q^{81} +9.09450 q^{83} +3.64974 q^{87} +0.676383 q^{89} -10.0465 q^{91} -5.30026 q^{93} -15.0759 q^{97} -6.50388 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 3 q^{3} - q^{7} + 4 q^{9} + 4 q^{11} + q^{13} - 5 q^{17} - 4 q^{19} - 6 q^{21} + 5 q^{23} + 6 q^{27} - 11 q^{29} - 4 q^{31} - 13 q^{33} - 6 q^{37} - 31 q^{39} - 8 q^{41} + 3 q^{43} + 2 q^{47} - 2 q^{49}+ \cdots - 37 q^{99}+O(q^{100}) $ 5 * q + 3 * q^3 - q^7 + 4 * q^9 + 4 * q^11 + q^13 - 5 * q^17 - 4 * q^19 - 6 * q^21 + 5 * q^23 + 6 * q^27 - 11 * q^29 - 4 * q^31 - 13 * q^33 - 6 * q^37 - 31 * q^39 - 8 * q^41 + 3 * q^43 + 2 * q^47 - 2 * q^49 + 5 * q^51 - 18 * q^53 - 27 * q^57 - 23 * q^59 - 26 * q^61 + 5 * q^63 + 3 * q^67 + 3 * q^69 + 2 * q^71 - 4 * q^73 - 15 * q^77 - 43 * q^79 - 3 * q^81 + 30 * q^83 + 27 * q^87 + 15 * q^89 + 19 * q^91 + 15 * q^93 - 8 * q^97 - 37 * q^99

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	−0.724570	−0.418331	−0.209165	−	0.977880i	$-0.567075\pi$
$3$	−0.724570	−0.418331	−0.209165	+	0.977880i	$0.567075\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−2.33840	−0.883833	−0.441916	−	0.897056i	$-0.645701\pi$
$7$	−2.33840	−0.883833	−0.441916	+	0.897056i	$0.645701\pi$
$8$	0	0
$9$	−2.47500	−0.825000
$10$	0	0
$11$	2.62783	0.792321	0.396160	−	0.918181i	$-0.370342\pi$
$11$	2.62783	0.792321	0.396160	+	0.918181i	$0.370342\pi$
$12$	0	0
$13$	4.29631	1.19158	0.595791	−	0.803140i	$-0.296839\pi$
$13$	4.29631	1.19158	0.595791	+	0.803140i	$0.296839\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−6.85404	−1.66235	−0.831175	−	0.556011i	$-0.812331\pi$
$17$	−6.85404	−1.66235	−0.831175	+	0.556011i	$0.812331\pi$
$18$	0	0
$19$	−2.87303	−0.659118	−0.329559	−	0.944135i	$-0.606900\pi$
$19$	−2.87303	−0.659118	−0.329559	+	0.944135i	$0.606900\pi$
$20$	0	0
$21$	1.69433	0.369734
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	0	0
$26$	0	0
$27$	3.96702	0.763453
$28$	0	0
$29$	−5.03711	−0.935368	−0.467684	−	0.883896i	$-0.654912\pi$
$29$	−5.03711	−0.935368	−0.467684	+	0.883896i	$0.654912\pi$
$30$	0	0
$31$	7.31504	1.31382	0.656910	−	0.753969i	$-0.271863\pi$
$31$	7.31504	1.31382	0.656910	+	0.753969i	$0.271863\pi$
$32$	0	0
$33$	−1.90405	−0.331452
$34$	0	0
$35$	0	0
$36$	0	0
$37$	9.24142	1.51928	0.759640	−	0.650344i	$-0.225375\pi$
$37$	9.24142	1.51928	0.759640	+	0.650344i	$0.225375\pi$
$38$	0	0
$39$	−3.11297	−0.498475
$40$	0	0
$41$	−6.95078	−1.08553	−0.542765	−	0.839885i	$-0.682623\pi$
$41$	−6.95078	−1.08553	−0.542765	+	0.839885i	$0.682623\pi$
$42$	0	0
$43$	7.01520	1.06981	0.534904	−	0.844913i	$-0.320348\pi$
$43$	7.01520	1.06981	0.534904	+	0.844913i	$0.320348\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	4.74690	0.692406	0.346203	−	0.938160i	$-0.387471\pi$
$47$	4.74690	0.692406	0.346203	+	0.938160i	$0.387471\pi$
$48$	0	0
$49$	−1.53188	−0.218840
$50$	0	0
$51$	4.96623	0.695412
$52$	0	0
$53$	12.3849	1.70120	0.850598	−	0.525817i	$-0.176240\pi$
$53$	12.3849	1.70120	0.850598	+	0.525817i	$0.176240\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	2.08171	0.275729
$58$	0	0
$59$	2.14073	0.278699	0.139349	−	0.990243i	$-0.455499\pi$
$59$	2.14073	0.278699	0.139349	+	0.990243i	$0.455499\pi$
$60$	0	0
$61$	−2.30567	−0.295210	−0.147605	−	0.989046i	$-0.547156\pi$
$61$	−2.30567	−0.295210	−0.147605	+	0.989046i	$0.547156\pi$
$62$	0	0
$63$	5.78754	0.729162
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−1.98376	−0.242355	−0.121178	−	0.992631i	$-0.538667\pi$
$67$	−1.98376	−0.242355	−0.121178	+	0.992631i	$0.538667\pi$
$68$	0	0
$69$	−0.724570	−0.0872279
$70$	0	0
$71$	6.87990	0.816494	0.408247	−	0.912871i	$-0.366140\pi$
$71$	6.87990	0.816494	0.408247	+	0.912871i	$0.366140\pi$
$72$	0	0
$73$	−13.2518	−1.55101	−0.775505	−	0.631342i	$-0.782504\pi$
$73$	−13.2518	−1.55101	−0.775505	+	0.631342i	$0.782504\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−6.14492	−0.700279
$78$	0	0
$79$	−16.6578	−1.87415	−0.937076	−	0.349126i	$-0.886478\pi$
$79$	−16.6578	−1.87415	−0.937076	+	0.349126i	$0.886478\pi$
$80$	0	0
$81$	4.55062	0.505624
$82$	0	0
$83$	9.09450	0.998251	0.499126	−	0.866530i	$-0.333655\pi$
$83$	9.09450	0.998251	0.499126	+	0.866530i	$0.333655\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	3.64974	0.391293
$88$	0	0
$89$	0.676383	0.0716965	0.0358482	−	0.999357i	$-0.488587\pi$
$89$	0.676383	0.0716965	0.0358482	+	0.999357i	$0.488587\pi$
$90$	0	0
$91$	−10.0465	−1.05316
$92$	0	0
$93$	−5.30026	−0.549611
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−15.0759	−1.53073	−0.765365	−	0.643597i	$-0.777441\pi$
$97$	−15.0759	−1.53073	−0.765365	+	0.643597i	$0.777441\pi$
$98$	0	0
$99$	−6.50388	−0.653664
$100$	0	0
$101$	−10.7713	−1.07178	−0.535892	−	0.844286i	$-0.680025\pi$
$101$	−10.7713	−1.07178	−0.535892	+	0.844286i	$0.680025\pi$
$102$	0	0
$103$	−16.4501	−1.62088	−0.810438	−	0.585824i	$-0.800771\pi$
$103$	−16.4501	−1.62088	−0.810438	+	0.585824i	$0.800771\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	7.21143	0.697155	0.348578	−	0.937280i	$-0.386665\pi$
$107$	7.21143	0.697155	0.348578	+	0.937280i	$0.386665\pi$
$108$	0	0
$109$	18.6581	1.78712	0.893560	−	0.448943i	$-0.148199\pi$
$109$	18.6581	1.78712	0.893560	+	0.448943i	$0.148199\pi$
$110$	0	0
$111$	−6.69605	−0.635561
$112$	0	0
$113$	0.722608	0.0679772	0.0339886	−	0.999422i	$-0.489179\pi$
$113$	0.722608	0.0679772	0.0339886	+	0.999422i	$0.489179\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−10.6334	−0.983054
$118$	0	0
$119$	16.0275	1.46924
$120$	0	0
$121$	−4.09450	−0.372228
$122$	0	0
$123$	5.03633	0.454110
$124$	0	0
$125$	0	0
$126$	0	0
$127$	19.1029	1.69510	0.847552	−	0.530712i	$-0.178076\pi$
$127$	19.1029	1.69510	0.847552	+	0.530712i	$0.178076\pi$
$128$	0	0
$129$	−5.08300	−0.447534
$130$	0	0
$131$	−9.25213	−0.808362	−0.404181	−	0.914679i	$-0.632443\pi$
$131$	−9.25213	−0.808362	−0.404181	+	0.914679i	$0.632443\pi$
$132$	0	0
$133$	6.71829	0.582550
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−20.5484	−1.75557	−0.877783	−	0.479058i	$-0.840978\pi$
$137$	−20.5484	−1.75557	−0.877783	+	0.479058i	$0.840978\pi$
$138$	0	0
$139$	5.48875	0.465550	0.232775	−	0.972531i	$-0.425219\pi$
$139$	5.48875	0.465550	0.232775	+	0.972531i	$0.425219\pi$
$140$	0	0
$141$	−3.43946	−0.289655
$142$	0	0
$143$	11.2900	0.944115
$144$	0	0
$145$	0	0
$146$	0	0
$147$	1.10995	0.0915473
$148$	0	0
$149$	5.51461	0.451775	0.225887	−	0.974153i	$-0.427472\pi$
$149$	5.51461	0.451775	0.225887	+	0.974153i	$0.427472\pi$
$150$	0	0
$151$	−16.5363	−1.34570	−0.672851	−	0.739778i	$-0.734931\pi$
$151$	−16.5363	−1.34570	−0.672851	+	0.739778i	$0.734931\pi$
$152$	0	0
$153$	16.9638	1.37144
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−13.6888	−1.09249	−0.546244	−	0.837626i	$-0.683943\pi$
$157$	−13.6888	−1.09249	−0.546244	+	0.837626i	$0.683943\pi$
$158$	0	0
$159$	−8.97372	−0.711662
$160$	0	0
$161$	−2.33840	−0.184292
$162$	0	0
$163$	18.8266	1.47462	0.737308	−	0.675556i	$-0.236097\pi$
$163$	18.8266	1.47462	0.737308	+	0.675556i	$0.236097\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	3.68830	0.285409	0.142705	−	0.989765i	$-0.454420\pi$
$167$	3.68830	0.285409	0.142705	+	0.989765i	$0.454420\pi$
$168$	0	0
$169$	5.45825	0.419866
$170$	0	0
$171$	7.11074	0.543772
$172$	0	0
$173$	−7.94698	−0.604198	−0.302099	−	0.953277i	$-0.597687\pi$
$173$	−7.94698	−0.604198	−0.302099	+	0.953277i	$0.597687\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−1.55111	−0.116588
$178$	0	0
$179$	−5.72373	−0.427811	−0.213906	−	0.976854i	$-0.568619\pi$
$179$	−5.72373	−0.427811	−0.213906	+	0.976854i	$0.568619\pi$
$180$	0	0
$181$	−2.19348	−0.163040	−0.0815199	−	0.996672i	$-0.525977\pi$
$181$	−2.19348	−0.163040	−0.0815199	+	0.996672i	$0.525977\pi$
$182$	0	0
$183$	1.67062	0.123495
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−18.0113	−1.31711
$188$	0	0
$189$	−9.27648	−0.674765
$190$	0	0
$191$	14.4643	1.04660	0.523302	−	0.852148i	$-0.324700\pi$
$191$	14.4643	1.04660	0.523302	+	0.852148i	$0.324700\pi$
$192$	0	0
$193$	17.7400	1.27696	0.638478	−	0.769640i	$-0.279564\pi$
$193$	17.7400	1.27696	0.638478	+	0.769640i	$0.279564\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	24.0847	1.71596	0.857982	−	0.513679i	$-0.171718\pi$
$197$	24.0847	1.71596	0.857982	+	0.513679i	$0.171718\pi$
$198$	0	0
$199$	−7.78754	−0.552044	−0.276022	−	0.961151i	$-0.589016\pi$
$199$	−7.78754	−0.552044	−0.276022	+	0.961151i	$0.589016\pi$
$200$	0	0
$201$	1.43738	0.101385
$202$	0	0
$203$	11.7788	0.826709
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−2.47500	−0.172024
$208$	0	0
$209$	−7.54983	−0.522233
$210$	0	0
$211$	−16.5419	−1.13879	−0.569394	−	0.822065i	$-0.692822\pi$
$211$	−16.5419	−1.13879	−0.569394	+	0.822065i	$0.692822\pi$
$212$	0	0
$213$	−4.98497	−0.341565
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−17.1055	−1.16120
$218$	0	0
$219$	9.60187	0.648834
$220$	0	0
$221$	−29.4471	−1.98082
$222$	0	0
$223$	−21.4323	−1.43521	−0.717606	−	0.696449i	$-0.754762\pi$
$223$	−21.4323	−1.43521	−0.717606	+	0.696449i	$0.754762\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	18.6739	1.23943	0.619715	−	0.784827i	$-0.287248\pi$
$227$	18.6739	1.23943	0.619715	+	0.784827i	$0.287248\pi$
$228$	0	0
$229$	−1.85981	−0.122900	−0.0614499	−	0.998110i	$-0.519572\pi$
$229$	−1.85981	−0.122900	−0.0614499	+	0.998110i	$0.519572\pi$
$230$	0	0
$231$	4.45243	0.292948
$232$	0	0
$233$	5.39803	0.353637	0.176818	−	0.984244i	$-0.443420\pi$
$233$	5.39803	0.353637	0.176818	+	0.984244i	$0.443420\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	12.0698	0.784015
$238$	0	0
$239$	−21.1817	−1.37013	−0.685065	−	0.728482i	$-0.740226\pi$
$239$	−21.1817	−1.37013	−0.685065	+	0.728482i	$0.740226\pi$
$240$	0	0
$241$	−29.8152	−1.92057	−0.960283	−	0.279028i	$-0.909988\pi$
$241$	−29.8152	−1.92057	−0.960283	+	0.279028i	$0.909988\pi$
$242$	0	0
$243$	−15.1983	−0.974971
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−12.3434	−0.785392
$248$	0	0
$249$	−6.58960	−0.417599
$250$	0	0
$251$	14.0212	0.885013	0.442506	−	0.896765i	$-0.354089\pi$
$251$	14.0212	0.885013	0.442506	+	0.896765i	$0.354089\pi$
$252$	0	0
$253$	2.62783	0.165210
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−7.14991	−0.445999	−0.222999	−	0.974819i	$-0.571585\pi$
$257$	−7.14991	−0.445999	−0.222999	+	0.974819i	$0.571585\pi$
$258$	0	0
$259$	−21.6101	−1.34279
$260$	0	0
$261$	12.4668	0.771678
$262$	0	0
$263$	16.6723	1.02806	0.514030	−	0.857772i	$-0.328152\pi$
$263$	16.6723	1.02806	0.514030	+	0.857772i	$0.328152\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−0.490087	−0.0299928
$268$	0	0
$269$	−11.7884	−0.718752	−0.359376	−	0.933193i	$-0.617010\pi$
$269$	−11.7884	−0.718752	−0.359376	+	0.933193i	$0.617010\pi$
$270$	0	0
$271$	8.37717	0.508877	0.254438	−	0.967089i	$-0.418109\pi$
$271$	8.37717	0.508877	0.254438	+	0.967089i	$0.418109\pi$
$272$	0	0
$273$	7.27938	0.440568
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−3.99879	−0.240264	−0.120132	−	0.992758i	$-0.538332\pi$
$277$	−3.99879	−0.240264	−0.120132	+	0.992758i	$0.538332\pi$
$278$	0	0
$279$	−18.1047	−1.08390
$280$	0	0
$281$	21.9478	1.30930	0.654648	−	0.755934i	$-0.272817\pi$
$281$	21.9478	1.30930	0.654648	+	0.755934i	$0.272817\pi$
$282$	0	0
$283$	−10.6297	−0.631868	−0.315934	−	0.948781i	$-0.602318\pi$
$283$	−10.6297	−0.631868	−0.315934	+	0.948781i	$0.602318\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	16.2537	0.959427
$288$	0	0
$289$	29.9779	1.76341
$290$	0	0
$291$	10.9236	0.640351
$292$	0	0
$293$	−28.5108	−1.66562	−0.832810	−	0.553558i	$-0.813270\pi$
$293$	−28.5108	−1.66562	−0.832810	+	0.553558i	$0.813270\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	10.4247	0.604900
$298$	0	0
$299$	4.29631	0.248462
$300$	0	0
$301$	−16.4044	−0.945532
$302$	0	0
$303$	7.80456	0.448360
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−0.391652	−0.0223528	−0.0111764	−	0.999938i	$-0.503558\pi$
$307$	−0.391652	−0.0223528	−0.0111764	+	0.999938i	$0.503558\pi$
$308$	0	0
$309$	11.9192	0.678062
$310$	0	0
$311$	−22.5240	−1.27722	−0.638609	−	0.769531i	$-0.720490\pi$
$311$	−22.5240	−1.27722	−0.638609	+	0.769531i	$0.720490\pi$
$312$	0	0
$313$	−22.0468	−1.24616	−0.623078	−	0.782160i	$-0.714118\pi$
$313$	−22.0468	−1.24616	−0.623078	+	0.782160i	$0.714118\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−6.01192	−0.337663	−0.168831	−	0.985645i	$-0.553999\pi$
$317$	−6.01192	−0.337663	−0.168831	+	0.985645i	$0.553999\pi$
$318$	0	0
$319$	−13.2367	−0.741112
$320$	0	0
$321$	−5.22518	−0.291641
$322$	0	0
$323$	19.6919	1.09568
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−13.5191	−0.747607
$328$	0	0
$329$	−11.1002	−0.611971
$330$	0	0
$331$	27.5619	1.51494	0.757469	−	0.652871i	$-0.226436\pi$
$331$	27.5619	1.51494	0.757469	+	0.652871i	$0.226436\pi$
$332$	0	0
$333$	−22.8725	−1.25341
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−11.8095	−0.643306	−0.321653	−	0.946858i	$-0.604239\pi$
$337$	−11.8095	−0.643306	−0.321653	+	0.946858i	$0.604239\pi$
$338$	0	0
$339$	−0.523580	−0.0284370
$340$	0	0
$341$	19.2227	1.04097
$342$	0	0
$343$	19.9510	1.07725
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−28.8288	−1.54761	−0.773806	−	0.633423i	$-0.781649\pi$
$347$	−28.8288	−1.54761	−0.773806	+	0.633423i	$0.781649\pi$
$348$	0	0
$349$	−26.1403	−1.39926	−0.699629	−	0.714506i	$-0.746651\pi$
$349$	−26.1403	−1.39926	−0.699629	+	0.714506i	$0.746651\pi$
$350$	0	0
$351$	17.0435	0.909716
$352$	0	0
$353$	23.8004	1.26677	0.633383	−	0.773839i	$-0.281666\pi$
$353$	23.8004	1.26677	0.633383	+	0.773839i	$0.281666\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−11.6130	−0.614628
$358$	0	0
$359$	−29.0102	−1.53110	−0.765551	−	0.643375i	$-0.777534\pi$
$359$	−29.0102	−1.53110	−0.765551	+	0.643375i	$0.777534\pi$
$360$	0	0
$361$	−10.7457	−0.565564
$362$	0	0
$363$	2.96675	0.155714
$364$	0	0
$365$	0	0
$366$	0	0
$367$	2.84461	0.148487	0.0742437	−	0.997240i	$-0.476346\pi$
$367$	2.84461	0.148487	0.0742437	+	0.997240i	$0.476346\pi$
$368$	0	0
$369$	17.2032	0.895562
$370$	0	0
$371$	−28.9609	−1.50357
$372$	0	0
$373$	−4.65308	−0.240928	−0.120464	−	0.992718i	$-0.538438\pi$
$373$	−4.65308	−0.240928	−0.120464	+	0.992718i	$0.538438\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−21.6410	−1.11457
$378$	0	0
$379$	−20.8607	−1.07154	−0.535772	−	0.844363i	$-0.679979\pi$
$379$	−20.8607	−1.07154	−0.535772	+	0.844363i	$0.679979\pi$
$380$	0	0
$381$	−13.8413	−0.709114
$382$	0	0
$383$	10.7116	0.547337	0.273669	−	0.961824i	$-0.411763\pi$
$383$	10.7116	0.547337	0.273669	+	0.961824i	$0.411763\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−17.3626	−0.882592
$388$	0	0
$389$	3.28523	0.166568	0.0832839	−	0.996526i	$-0.473459\pi$
$389$	3.28523	0.166568	0.0832839	+	0.996526i	$0.473459\pi$
$390$	0	0
$391$	−6.85404	−0.346624
$392$	0	0
$393$	6.70381	0.338163
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−6.00319	−0.301292	−0.150646	−	0.988588i	$-0.548135\pi$
$397$	−6.00319	−0.301292	−0.150646	+	0.988588i	$0.548135\pi$
$398$	0	0
$399$	−4.86787	−0.243698
$400$	0	0
$401$	−22.9420	−1.14567	−0.572835	−	0.819671i	$-0.694156\pi$
$401$	−22.9420	−1.14567	−0.572835	+	0.819671i	$0.694156\pi$
$402$	0	0
$403$	31.4277	1.56552
$404$	0	0
$405$	0	0
$406$	0	0
$407$	24.2849	1.20376
$408$	0	0
$409$	−9.04425	−0.447210	−0.223605	−	0.974680i	$-0.571782\pi$
$409$	−9.04425	−0.447210	−0.223605	+	0.974680i	$0.571782\pi$
$410$	0	0
$411$	14.8887	0.734407
$412$	0	0
$413$	−5.00588	−0.246323
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−3.97698	−0.194754
$418$	0	0
$419$	28.1921	1.37727	0.688637	−	0.725106i	$-0.258209\pi$
$419$	28.1921	1.37727	0.688637	+	0.725106i	$0.258209\pi$
$420$	0	0
$421$	−9.14897	−0.445893	−0.222947	−	0.974831i	$-0.571568\pi$
$421$	−9.14897	−0.445893	−0.222947	+	0.974831i	$0.571568\pi$
$422$	0	0
$423$	−11.7486	−0.571235
$424$	0	0
$425$	0	0
$426$	0	0
$427$	5.39157	0.260916
$428$	0	0
$429$	−8.18037	−0.394952
$430$	0	0
$431$	−32.5005	−1.56549	−0.782747	−	0.622340i	$-0.786182\pi$
$431$	−32.5005	−1.56549	−0.782747	+	0.622340i	$0.786182\pi$
$432$	0	0
$433$	−0.555560	−0.0266985	−0.0133492	−	0.999911i	$-0.504249\pi$
$433$	−0.555560	−0.0266985	−0.0133492	+	0.999911i	$0.504249\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−2.87303	−0.137436
$438$	0	0
$439$	25.8680	1.23461	0.617305	−	0.786724i	$-0.288224\pi$
$439$	25.8680	1.23461	0.617305	+	0.786724i	$0.288224\pi$
$440$	0	0
$441$	3.79140	0.180543
$442$	0	0
$443$	−30.2965	−1.43943	−0.719715	−	0.694270i	$-0.755728\pi$
$443$	−30.2965	−1.43943	−0.719715	+	0.694270i	$0.755728\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−3.99572	−0.188991
$448$	0	0
$449$	−31.6573	−1.49400	−0.747001	−	0.664823i	$-0.768507\pi$
$449$	−31.6573	−1.49400	−0.747001	+	0.664823i	$0.768507\pi$
$450$	0	0
$451$	−18.2655	−0.860088
$452$	0	0
$453$	11.9817	0.562949
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−7.34455	−0.343564	−0.171782	−	0.985135i	$-0.554952\pi$
$457$	−7.34455	−0.343564	−0.171782	+	0.985135i	$0.554952\pi$
$458$	0	0
$459$	−27.1901	−1.26913
$460$	0	0
$461$	15.6987	0.731163	0.365581	−	0.930779i	$-0.380870\pi$
$461$	15.6987	0.731163	0.365581	+	0.930779i	$0.380870\pi$
$462$	0	0
$463$	1.82421	0.0847782	0.0423891	−	0.999101i	$-0.486503\pi$
$463$	1.82421	0.0847782	0.0423891	+	0.999101i	$0.486503\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−1.43722	−0.0665067	−0.0332533	−	0.999447i	$-0.510587\pi$
$467$	−1.43722	−0.0665067	−0.0332533	+	0.999447i	$0.510587\pi$
$468$	0	0
$469$	4.63884	0.214202
$470$	0	0
$471$	9.91852	0.457021
$472$	0	0
$473$	18.4348	0.847632
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−30.6526	−1.40349
$478$	0	0
$479$	−26.5858	−1.21474	−0.607368	−	0.794420i	$-0.707775\pi$
$479$	−26.5858	−1.21474	−0.607368	+	0.794420i	$0.707775\pi$
$480$	0	0
$481$	39.7040	1.81035
$482$	0	0
$483$	1.69433	0.0770949
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−33.8112	−1.53213	−0.766066	−	0.642762i	$-0.777788\pi$
$487$	−33.8112	−1.53213	−0.766066	+	0.642762i	$0.777788\pi$
$488$	0	0
$489$	−13.6412	−0.616877
$490$	0	0
$491$	−22.4820	−1.01460	−0.507298	−	0.861770i	$-0.669356\pi$
$491$	−22.4820	−1.01460	−0.507298	+	0.861770i	$0.669356\pi$
$492$	0	0
$493$	34.5246	1.55491
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−16.0880	−0.721644
$498$	0	0
$499$	−22.6152	−1.01240	−0.506198	−	0.862417i	$-0.668949\pi$
$499$	−22.6152	−1.01240	−0.506198	+	0.862417i	$0.668949\pi$
$500$	0	0
$501$	−2.67243	−0.119395
$502$	0	0
$503$	−16.3142	−0.727414	−0.363707	−	0.931513i	$-0.618489\pi$
$503$	−16.3142	−0.727414	−0.363707	+	0.931513i	$0.618489\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−3.95488	−0.175643
$508$	0	0
$509$	−8.01935	−0.355451	−0.177726	−	0.984080i	$-0.556874\pi$
$509$	−8.01935	−0.355451	−0.177726	+	0.984080i	$0.556874\pi$
$510$	0	0
$511$	30.9881	1.37083
$512$	0	0
$513$	−11.3973	−0.503205
$514$	0	0
$515$	0	0
$516$	0	0
$517$	12.4740	0.548608
$518$	0	0
$519$	5.75814	0.252754
$520$	0	0
$521$	18.0130	0.789162	0.394581	−	0.918861i	$-0.370890\pi$
$521$	18.0130	0.789162	0.394581	+	0.918861i	$0.370890\pi$
$522$	0	0
$523$	1.91940	0.0839294	0.0419647	−	0.999119i	$-0.486638\pi$
$523$	1.91940	0.0839294	0.0419647	+	0.999119i	$0.486638\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−50.1376	−2.18403
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−5.29830	−0.229927
$532$	0	0
$533$	−29.8627	−1.29350
$534$	0	0
$535$	0	0
$536$	0	0
$537$	4.14724	0.178967
$538$	0	0
$539$	−4.02552	−0.173391
$540$	0	0
$541$	−42.0297	−1.80700	−0.903500	−	0.428589i	$-0.859011\pi$
$541$	−42.0297	−1.80700	−0.903500	+	0.428589i	$0.859011\pi$
$542$	0	0
$543$	1.58933	0.0682045
$544$	0	0
$545$	0	0
$546$	0	0
$547$	36.0750	1.54246	0.771228	−	0.636559i	$-0.219643\pi$
$547$	36.0750	1.54246	0.771228	+	0.636559i	$0.219643\pi$
$548$	0	0
$549$	5.70652	0.243548
$550$	0	0
$551$	14.4718	0.616518
$552$	0	0
$553$	38.9527	1.65644
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−31.9500	−1.35377	−0.676883	−	0.736091i	$-0.736670\pi$
$557$	−31.9500	−1.35377	−0.676883	+	0.736091i	$0.736670\pi$
$558$	0	0
$559$	30.1395	1.27476
$560$	0	0
$561$	13.0504	0.550989
$562$	0	0
$563$	6.26575	0.264070	0.132035	−	0.991245i	$-0.457849\pi$
$563$	6.26575	0.264070	0.132035	+	0.991245i	$0.457849\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−10.6412	−0.446887
$568$	0	0
$569$	−12.7997	−0.536593	−0.268296	−	0.963336i	$-0.586461\pi$
$569$	−12.7997	−0.536593	−0.268296	+	0.963336i	$0.586461\pi$
$570$	0	0
$571$	−13.7690	−0.576217	−0.288108	−	0.957598i	$-0.593026\pi$
$571$	−13.7690	−0.576217	−0.288108	+	0.957598i	$0.593026\pi$
$572$	0	0
$573$	−10.4804	−0.437826
$574$	0	0
$575$	0	0
$576$	0	0
$577$	13.5830	0.565470	0.282735	−	0.959198i	$-0.408758\pi$
$577$	13.5830	0.565470	0.282735	+	0.959198i	$0.408758\pi$
$578$	0	0
$579$	−12.8539	−0.534190
$580$	0	0
$581$	−21.2666	−0.882287
$582$	0	0
$583$	32.5454	1.34789
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−19.3233	−0.797557	−0.398778	−	0.917047i	$-0.630566\pi$
$587$	−19.3233	−0.797557	−0.398778	+	0.917047i	$0.630566\pi$
$588$	0	0
$589$	−21.0163	−0.865962
$590$	0	0
$591$	−17.4511	−0.717840
$592$	0	0
$593$	−25.1344	−1.03215	−0.516073	−	0.856545i	$-0.672607\pi$
$593$	−25.1344	−1.03215	−0.516073	+	0.856545i	$0.672607\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	5.64262	0.230937
$598$	0	0
$599$	−1.88678	−0.0770918	−0.0385459	−	0.999257i	$-0.512273\pi$
$599$	−1.88678	−0.0770918	−0.0385459	+	0.999257i	$0.512273\pi$
$600$	0	0
$601$	13.2105	0.538866	0.269433	−	0.963019i	$-0.413164\pi$
$601$	13.2105	0.538866	0.269433	+	0.963019i	$0.413164\pi$
$602$	0	0
$603$	4.90982	0.199943
$604$	0	0
$605$	0	0
$606$	0	0
$607$	3.50465	0.142249	0.0711246	−	0.997467i	$-0.477341\pi$
$607$	3.50465	0.142249	0.0711246	+	0.997467i	$0.477341\pi$
$608$	0	0
$609$	−8.53455	−0.345838
$610$	0	0
$611$	20.3941	0.825058
$612$	0	0
$613$	7.59433	0.306732	0.153366	−	0.988169i	$-0.450989\pi$
$613$	7.59433	0.306732	0.153366	+	0.988169i	$0.450989\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	13.4277	0.540579	0.270290	−	0.962779i	$-0.412881\pi$
$617$	13.4277	0.540579	0.270290	+	0.962779i	$0.412881\pi$
$618$	0	0
$619$	45.0584	1.81105	0.905525	−	0.424292i	$-0.139477\pi$
$619$	45.0584	1.81105	0.905525	+	0.424292i	$0.139477\pi$
$620$	0	0
$621$	3.96702	0.159191
$622$	0	0
$623$	−1.58166	−0.0633677
$624$	0	0
$625$	0	0
$626$	0	0
$627$	5.47038	0.218466
$628$	0	0
$629$	−63.3411	−2.52557
$630$	0	0
$631$	14.9030	0.593280	0.296640	−	0.954989i	$-0.404134\pi$
$631$	14.9030	0.593280	0.296640	+	0.954989i	$0.404134\pi$
$632$	0	0
$633$	11.9857	0.476390
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−6.58142	−0.260765
$638$	0	0
$639$	−17.0278	−0.673608
$640$	0	0
$641$	37.7852	1.49243	0.746213	−	0.665707i	$-0.231870\pi$
$641$	37.7852	1.49243	0.746213	+	0.665707i	$0.231870\pi$
$642$	0	0
$643$	−18.8777	−0.744465	−0.372232	−	0.928140i	$-0.621408\pi$
$643$	−18.8777	−0.744465	−0.372232	+	0.928140i	$0.621408\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	20.5272	0.807009	0.403505	−	0.914978i	$-0.367792\pi$
$647$	20.5272	0.807009	0.403505	+	0.914978i	$0.367792\pi$
$648$	0	0
$649$	5.62547	0.220819
$650$	0	0
$651$	12.3941	0.485764
$652$	0	0
$653$	3.07302	0.120256	0.0601282	−	0.998191i	$-0.480849\pi$
$653$	3.07302	0.120256	0.0601282	+	0.998191i	$0.480849\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	32.7983	1.27958
$658$	0	0
$659$	20.4014	0.794725	0.397363	−	0.917662i	$-0.369925\pi$
$659$	20.4014	0.794725	0.397363	+	0.917662i	$0.369925\pi$
$660$	0	0
$661$	−4.17682	−0.162460	−0.0812298	−	0.996695i	$-0.525885\pi$
$661$	−4.17682	−0.162460	−0.0812298	+	0.996695i	$0.525885\pi$
$662$	0	0
$663$	21.3365	0.828639
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−5.03711	−0.195038
$668$	0	0
$669$	15.5292	0.600393
$670$	0	0
$671$	−6.05890	−0.233901
$672$	0	0
$673$	45.0529	1.73666	0.868331	−	0.495984i	$-0.165193\pi$
$673$	45.0529	1.73666	0.868331	+	0.495984i	$0.165193\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	16.5395	0.635665	0.317833	−	0.948147i	$-0.397045\pi$
$677$	16.5395	0.635665	0.317833	+	0.948147i	$0.397045\pi$
$678$	0	0
$679$	35.2536	1.35291
$680$	0	0
$681$	−13.5305	−0.518492
$682$	0	0
$683$	−26.9328	−1.03055	−0.515277	−	0.857024i	$-0.672311\pi$
$683$	−26.9328	−1.03055	−0.515277	+	0.857024i	$0.672311\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	1.34756	0.0514128
$688$	0	0
$689$	53.2093	2.02711
$690$	0	0
$691$	−44.1583	−1.67986	−0.839931	−	0.542693i	$-0.817405\pi$
$691$	−44.1583	−1.67986	−0.839931	+	0.542693i	$0.817405\pi$
$692$	0	0
$693$	15.2087	0.577730
$694$	0	0
$695$	0	0
$696$	0	0
$697$	47.6410	1.80453
$698$	0	0
$699$	−3.91125	−0.147937
$700$	0	0
$701$	−15.6476	−0.591002	−0.295501	−	0.955342i	$-0.595487\pi$
$701$	−15.6476	−0.591002	−0.295501	+	0.955342i	$0.595487\pi$
$702$	0	0
$703$	−26.5508	−1.00138
$704$	0	0
$705$	0	0
$706$	0	0
$707$	25.1876	0.947279
$708$	0	0
$709$	8.79068	0.330141	0.165070	−	0.986282i	$-0.447215\pi$
$709$	8.79068	0.330141	0.165070	+	0.986282i	$0.447215\pi$
$710$	0	0
$711$	41.2281	1.54617
$712$	0	0
$713$	7.31504	0.273951
$714$	0	0
$715$	0	0
$716$	0	0
$717$	15.3476	0.573167
$718$	0	0
$719$	−39.3081	−1.46594	−0.732972	−	0.680259i	$-0.761867\pi$
$719$	−39.3081	−1.46594	−0.732972	+	0.680259i	$0.761867\pi$
$720$	0	0
$721$	38.4669	1.43258
$722$	0	0
$723$	21.6032	0.803431
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−16.1156	−0.597695	−0.298848	−	0.954301i	$-0.596602\pi$
$727$	−16.1156	−0.597695	−0.298848	+	0.954301i	$0.596602\pi$
$728$	0	0
$729$	−2.63962	−0.0977638
$730$	0	0
$731$	−48.0825	−1.77840
$732$	0	0
$733$	−9.66306	−0.356913	−0.178457	−	0.983948i	$-0.557110\pi$
$733$	−9.66306	−0.356913	−0.178457	+	0.983948i	$0.557110\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−5.21300	−0.192023
$738$	0	0
$739$	−24.3518	−0.895796	−0.447898	−	0.894085i	$-0.647827\pi$
$739$	−24.3518	−0.895796	−0.447898	+	0.894085i	$0.647827\pi$
$740$	0	0
$741$	8.94366	0.328553
$742$	0	0
$743$	32.0702	1.17654	0.588271	−	0.808664i	$-0.299809\pi$
$743$	32.0702	1.17654	0.588271	+	0.808664i	$0.299809\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−22.5089	−0.823557
$748$	0	0
$749$	−16.8632	−0.616168
$750$	0	0
$751$	−1.88705	−0.0688596	−0.0344298	−	0.999407i	$-0.510962\pi$
$751$	−1.88705	−0.0688596	−0.0344298	+	0.999407i	$0.510962\pi$
$752$	0	0
$753$	−10.1594	−0.370228
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−5.13762	−0.186730	−0.0933651	−	0.995632i	$-0.529762\pi$
$757$	−5.13762	−0.186730	−0.0933651	+	0.995632i	$0.529762\pi$
$758$	0	0
$759$	−1.90405	−0.0691125
$760$	0	0
$761$	39.4855	1.43135	0.715674	−	0.698435i	$-0.246120\pi$
$761$	39.4855	1.43135	0.715674	+	0.698435i	$0.246120\pi$
$762$	0	0
$763$	−43.6301	−1.57952
$764$	0	0
$765$	0	0
$766$	0	0
$767$	9.19722	0.332092
$768$	0	0
$769$	−1.32121	−0.0476439	−0.0238220	−	0.999716i	$-0.507583\pi$
$769$	−1.32121	−0.0476439	−0.0238220	+	0.999716i	$0.507583\pi$
$770$	0	0
$771$	5.18061	0.186575
$772$	0	0
$773$	33.7473	1.21381	0.606903	−	0.794776i	$-0.292412\pi$
$773$	33.7473	1.21381	0.606903	+	0.794776i	$0.292412\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	15.6581	0.561730
$778$	0	0
$779$	19.9698	0.715492
$780$	0	0
$781$	18.0792	0.646926
$782$	0	0
$783$	−19.9823	−0.714110
$784$	0	0
$785$	0	0
$786$	0	0
$787$	13.1800	0.469816	0.234908	−	0.972018i	$-0.424521\pi$
$787$	13.1800	0.469816	0.234908	+	0.972018i	$0.424521\pi$
$788$	0	0
$789$	−12.0803	−0.430069
$790$	0	0
$791$	−1.68975	−0.0600805
$792$	0	0
$793$	−9.90584	−0.351767
$794$	0	0
$795$	0	0
$796$	0	0
$797$	21.0017	0.743919	0.371959	−	0.928249i	$-0.378686\pi$
$797$	21.0017	0.743919	0.371959	+	0.928249i	$0.378686\pi$
$798$	0	0
$799$	−32.5354	−1.15102
$800$	0	0
$801$	−1.67405	−0.0591496
$802$	0	0
$803$	−34.8236	−1.22890
$804$	0	0
$805$	0	0
$806$	0	0
$807$	8.54153	0.300676
$808$	0	0
$809$	4.45831	0.156746	0.0783729	−	0.996924i	$-0.475028\pi$
$809$	4.45831	0.156746	0.0783729	+	0.996924i	$0.475028\pi$
$810$	0	0
$811$	−27.9804	−0.982525	−0.491263	−	0.871012i	$-0.663464\pi$
$811$	−27.9804	−0.982525	−0.491263	+	0.871012i	$0.663464\pi$
$812$	0	0
$813$	−6.06984	−0.212879
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−20.1549	−0.705130
$818$	0	0
$819$	24.8651	0.868855
$820$	0	0
$821$	29.0587	1.01416	0.507078	−	0.861900i	$-0.330726\pi$
$821$	29.0587	1.01416	0.507078	+	0.861900i	$0.330726\pi$
$822$	0	0
$823$	6.83168	0.238138	0.119069	−	0.992886i	$-0.462009\pi$
$823$	6.83168	0.238138	0.119069	+	0.992886i	$0.462009\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−31.5165	−1.09594	−0.547968	−	0.836499i	$-0.684598\pi$
$827$	−31.5165	−1.09594	−0.547968	+	0.836499i	$0.684598\pi$
$828$	0	0
$829$	−22.9202	−0.796052	−0.398026	−	0.917374i	$-0.630305\pi$
$829$	−22.9202	−0.796052	−0.398026	+	0.917374i	$0.630305\pi$
$830$	0	0
$831$	2.89741	0.100510
$832$	0	0
$833$	10.4996	0.363788
$834$	0	0
$835$	0	0
$836$	0	0
$837$	29.0189	1.00304
$838$	0	0
$839$	−19.6003	−0.676676	−0.338338	−	0.941025i	$-0.609865\pi$
$839$	−19.6003	−0.676676	−0.338338	+	0.941025i	$0.609865\pi$
$840$	0	0
$841$	−3.62750	−0.125086
$842$	0	0
$843$	−15.9027	−0.547718
$844$	0	0
$845$	0	0
$846$	0	0
$847$	9.57459	0.328987
$848$	0	0
$849$	7.70193	0.264330
$850$	0	0
$851$	9.24142	0.316792
$852$	0	0
$853$	−9.76951	−0.334502	−0.167251	−	0.985914i	$-0.553489\pi$
$853$	−9.76951	−0.334502	−0.167251	+	0.985914i	$0.553489\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−26.5308	−0.906273	−0.453137	−	0.891441i	$-0.649695\pi$
$857$	−26.5308	−0.906273	−0.453137	+	0.891441i	$0.649695\pi$
$858$	0	0
$859$	−37.1341	−1.26700	−0.633499	−	0.773744i	$-0.718382\pi$
$859$	−37.1341	−1.26700	−0.633499	+	0.773744i	$0.718382\pi$
$860$	0	0
$861$	−11.7770	−0.401358
$862$	0	0
$863$	−6.15457	−0.209504	−0.104752	−	0.994498i	$-0.533405\pi$
$863$	−6.15457	−0.209504	−0.104752	+	0.994498i	$0.533405\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−21.7211	−0.737687
$868$	0	0
$869$	−43.7739	−1.48493
$870$	0	0
$871$	−8.52286	−0.288786
$872$	0	0
$873$	37.3129	1.26285
$874$	0	0
$875$	0	0
$876$	0	0
$877$	22.0440	0.744373	0.372186	−	0.928158i	$-0.378608\pi$
$877$	22.0440	0.744373	0.372186	+	0.928158i	$0.378608\pi$
$878$	0	0
$879$	20.6581	0.696780
$880$	0	0
$881$	11.0196	0.371260	0.185630	−	0.982620i	$-0.440567\pi$
$881$	11.0196	0.371260	0.185630	+	0.982620i	$0.440567\pi$
$882$	0	0
$883$	33.8860	1.14035	0.570177	−	0.821522i	$-0.306875\pi$
$883$	33.8860	1.14035	0.570177	+	0.821522i	$0.306875\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−3.89387	−0.130743	−0.0653717	−	0.997861i	$-0.520823\pi$
$887$	−3.89387	−0.130743	−0.0653717	+	0.997861i	$0.520823\pi$
$888$	0	0
$889$	−44.6701	−1.49819
$890$	0	0
$891$	11.9582	0.400616
$892$	0	0
$893$	−13.6380	−0.456377
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−3.11297	−0.103939
$898$	0	0
$899$	−36.8467	−1.22891
$900$	0	0
$901$	−84.8866	−2.82798
$902$	0	0
$903$	11.8861	0.395545
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−30.5048	−1.01289	−0.506447	−	0.862271i	$-0.669041\pi$
$907$	−30.5048	−1.01289	−0.506447	+	0.862271i	$0.669041\pi$
$908$	0	0
$909$	26.6590	0.884222
$910$	0	0
$911$	15.8042	0.523616	0.261808	−	0.965120i	$-0.415681\pi$
$911$	15.8042	0.523616	0.261808	+	0.965120i	$0.415681\pi$
$912$	0	0
$913$	23.8988	0.790935
$914$	0	0
$915$	0	0
$916$	0	0
$917$	21.6352	0.714457
$918$	0	0
$919$	41.2196	1.35971	0.679855	−	0.733347i	$-0.262043\pi$
$919$	41.2196	1.35971	0.679855	+	0.733347i	$0.262043\pi$
$920$	0	0
$921$	0.283779	0.00935085
$922$	0	0
$923$	29.5582	0.972919
$924$	0	0
$925$	0	0
$926$	0	0
$927$	40.7140	1.33722
$928$	0	0
$929$	13.3600	0.438326	0.219163	−	0.975688i	$-0.429667\pi$
$929$	13.3600	0.438326	0.219163	+	0.975688i	$0.429667\pi$
$930$	0	0
$931$	4.40113	0.144241
$932$	0	0
$933$	16.3202	0.534299
$934$	0	0
$935$	0	0
$936$	0	0
$937$	16.6980	0.545501	0.272751	−	0.962085i	$-0.412067\pi$
$937$	16.6980	0.545501	0.272751	+	0.962085i	$0.412067\pi$
$938$	0	0
$939$	15.9744	0.521305
$940$	0	0
$941$	16.3650	0.533482	0.266741	−	0.963768i	$-0.414053\pi$
$941$	16.3650	0.533482	0.266741	+	0.963768i	$0.414053\pi$
$942$	0	0
$943$	−6.95078	−0.226349
$944$	0	0
$945$	0	0
$946$	0	0
$947$	7.88894	0.256356	0.128178	−	0.991751i	$-0.459087\pi$
$947$	7.88894	0.256356	0.128178	+	0.991751i	$0.459087\pi$
$948$	0	0
$949$	−56.9339	−1.84815
$950$	0	0
$951$	4.35605	0.141255
$952$	0	0
$953$	−2.26143	−0.0732549	−0.0366275	−	0.999329i	$-0.511661\pi$
$953$	−2.26143	−0.0732549	−0.0366275	+	0.999329i	$0.511661\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	9.59090	0.310030
$958$	0	0
$959$	48.0504	1.55163
$960$	0	0
$961$	22.5099	0.726125
$962$	0	0
$963$	−17.8483	−0.575153
$964$	0	0
$965$	0	0
$966$	0	0
$967$	23.1416	0.744182	0.372091	−	0.928196i	$-0.378641\pi$
$967$	23.1416	0.744182	0.372091	+	0.928196i	$0.378641\pi$
$968$	0	0
$969$	−14.2681	−0.458358
$970$	0	0
$971$	27.9414	0.896683	0.448341	−	0.893862i	$-0.352015\pi$
$971$	27.9414	0.896683	0.448341	+	0.893862i	$0.352015\pi$
$972$	0	0
$973$	−12.8349	−0.411468
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−9.62964	−0.308079	−0.154040	−	0.988065i	$-0.549228\pi$
$977$	−9.62964	−0.308079	−0.154040	+	0.988065i	$0.549228\pi$
$978$	0	0
$979$	1.77742	0.0568066
$980$	0	0
$981$	−46.1787	−1.47437
$982$	0	0
$983$	23.3104	0.743487	0.371743	−	0.928336i	$-0.378760\pi$
$983$	23.3104	0.743487	0.371743	+	0.928336i	$0.378760\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	8.04283	0.256006
$988$	0	0
$989$	7.01520	0.223071
$990$	0	0
$991$	2.62769	0.0834712	0.0417356	−	0.999129i	$-0.486711\pi$
$991$	2.62769	0.0834712	0.0417356	+	0.999129i	$0.486711\pi$
$992$	0	0
$993$	−19.9705	−0.633745
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−23.4056	−0.741264	−0.370632	−	0.928780i	$-0.620859\pi$
$997$	−23.4056	−0.741264	−0.370632	+	0.928780i	$0.620859\pi$
$998$	0	0
$999$	36.6609	1.15990

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9200.2.a.cw.1.2		5
4.3	odd	2		4600.2.a.bc.1.4	✓	5
5.4	even	2		9200.2.a.cs.1.4		5
20.3	even	4		4600.2.e.v.4049.6		10
20.7	even	4		4600.2.e.v.4049.5		10
20.19	odd	2		4600.2.a.bg.1.2	yes	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4600.2.a.bc.1.4	✓	5	4.3	odd	2
4600.2.a.bg.1.2	yes	5	20.19	odd	2
4600.2.e.v.4049.5		10	20.7	even	4
4600.2.e.v.4049.6		10	20.3	even	4
9200.2.a.cs.1.4		5	5.4	even	2
9200.2.a.cw.1.2		5	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 9200 = 2^{4} \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9200.a (trivial)

\(f(q)\)	\(=\)	\(q-0.724570 q^{3} -2.33840 q^{7} -2.47500 q^{9} +2.62783 q^{11} +4.29631 q^{13} -6.85404 q^{17} -2.87303 q^{19} +1.69433 q^{21} +1.00000 q^{23} +3.96702 q^{27} -5.03711 q^{29} +7.31504 q^{31} -1.90405 q^{33} +9.24142 q^{37} -3.11297 q^{39} -6.95078 q^{41} +7.01520 q^{43} +4.74690 q^{47} -1.53188 q^{49} +4.96623 q^{51} +12.3849 q^{53} +2.08171 q^{57} +2.14073 q^{59} -2.30567 q^{61} +5.78754 q^{63} -1.98376 q^{67} -0.724570 q^{69} +6.87990 q^{71} -13.2518 q^{73} -6.14492 q^{77} -16.6578 q^{79} +4.55062 q^{81} +9.09450 q^{83} +3.64974 q^{87} +0.676383 q^{89} -10.0465 q^{91} -5.30026 q^{93} -15.0759 q^{97} -6.50388 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 3 q^{3} - q^{7} + 4 q^{9} + 4 q^{11} + q^{13} - 5 q^{17} - 4 q^{19} - 6 q^{21} + 5 q^{23} + 6 q^{27} - 11 q^{29} - 4 q^{31} - 13 q^{33} - 6 q^{37} - 31 q^{39} - 8 q^{41} + 3 q^{43} + 2 q^{47} - 2 q^{49}+ \cdots - 37 q^{99}+O(q^{100}) \) 5 * q + 3 * q^3 - q^7 + 4 * q^9 + 4 * q^11 + q^13 - 5 * q^17 - 4 * q^19 - 6 * q^21 + 5 * q^23 + 6 * q^27 - 11 * q^29 - 4 * q^31 - 13 * q^33 - 6 * q^37 - 31 * q^39 - 8 * q^41 + 3 * q^43 + 2 * q^47 - 2 * q^49 + 5 * q^51 - 18 * q^53 - 27 * q^57 - 23 * q^59 - 26 * q^61 + 5 * q^63 + 3 * q^67 + 3 * q^69 + 2 * q^71 - 4 * q^73 - 15 * q^77 - 43 * q^79 - 3 * q^81 + 30 * q^83 + 27 * q^87 + 15 * q^89 + 19 * q^91 + 15 * q^93 - 8 * q^97 - 37 * q^99

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