LMFDB - Embedded newform 9300.2.a.u.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [9300,2,Mod(1,9300)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("9300.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9300, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 9300 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9300.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,0,-3,0,0,0,2,0,3,0,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$74.2608738798$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.404.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 5x - 1 $ x^3 - x^2 - 5*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1860)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.86620$ of defining polynomial
Character	$\chi$	$=$	9300.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -1.34889 q^{7} +1.00000 q^{9} +4.69779 q^{11} +2.38350 q^{13} +1.21509 q^{17} +1.03461 q^{19} +1.34889 q^{21} -6.24970 q^{23} -1.00000 q^{27} -5.59859 q^{29} +1.00000 q^{31} -4.69779 q^{33} -5.41811 q^{37} -2.38350 q^{39} +1.30221 q^{41} -5.73240 q^{43} -3.48270 q^{47} -5.18048 q^{49} -1.21509 q^{51} -2.51730 q^{53} -1.03461 q^{57} +7.86620 q^{59} -5.46479 q^{61} -1.34889 q^{63} +1.41811 q^{67} +6.24970 q^{69} +3.79698 q^{71} +1.41811 q^{73} -6.33682 q^{77} -1.81952 q^{79} +1.00000 q^{81} -13.2151 q^{83} +5.59859 q^{87} -2.20302 q^{89} -3.21509 q^{91} -1.00000 q^{93} -6.69779 q^{97} +4.69779 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{3} + 2 q^{7} + 3 q^{9} + 2 q^{11} - 2 q^{13} - 10 q^{17} - 2 q^{21} - 2 q^{23} - 3 q^{27} + 6 q^{29} + 3 q^{31} - 2 q^{33} - 4 q^{37} + 2 q^{39} + 16 q^{41} - 2 q^{43} - 12 q^{47} - 5 q^{49}+ \cdots + 2 q^{99}+O(q^{100}) $ 3 * q - 3 * q^3 + 2 * q^7 + 3 * q^9 + 2 * q^11 - 2 * q^13 - 10 * q^17 - 2 * q^21 - 2 * q^23 - 3 * q^27 + 6 * q^29 + 3 * q^31 - 2 * q^33 - 4 * q^37 + 2 * q^39 + 16 * q^41 - 2 * q^43 - 12 * q^47 - 5 * q^49 + 10 * q^51 - 6 * q^53 + 16 * q^59 + 14 * q^61 + 2 * q^63 - 8 * q^67 + 2 * q^69 + 10 * q^71 - 8 * q^73 - 28 * q^77 - 16 * q^79 + 3 * q^81 - 26 * q^83 - 6 * q^87 - 8 * q^89 + 4 * q^91 - 3 * q^93 - 8 * q^97 + 2 * 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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−1.34889	−0.509834	−0.254917	−	0.966963i	$-0.582048\pi$
$7$	−1.34889	−0.509834	−0.254917	+	0.966963i	$0.582048\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	4.69779	1.41644	0.708218	−	0.705994i	$-0.249499\pi$
$11$	4.69779	1.41644	0.708218	+	0.705994i	$0.249499\pi$
$12$	0	0
$13$	2.38350	0.661065	0.330532	−	0.943795i	$-0.392772\pi$
$13$	2.38350	0.661065	0.330532	+	0.943795i	$0.392772\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	1.21509	0.294703	0.147352	−	0.989084i	$-0.452925\pi$
$17$	1.21509	0.294703	0.147352	+	0.989084i	$0.452925\pi$
$18$	0	0
$19$	1.03461	0.237355	0.118678	−	0.992933i	$-0.462134\pi$
$19$	1.03461	0.237355	0.118678	+	0.992933i	$0.462134\pi$
$20$	0	0
$21$	1.34889	0.294353
$22$	0	0
$23$	−6.24970	−1.30315	−0.651576	−	0.758583i	$-0.725892\pi$
$23$	−6.24970	−1.30315	−0.651576	+	0.758583i	$0.725892\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−5.59859	−1.03963	−0.519816	−	0.854278i	$-0.674000\pi$
$29$	−5.59859	−1.03963	−0.519816	+	0.854278i	$0.674000\pi$
$30$	0	0
$31$	1.00000	0.179605
$31$	1.00000	0.179605
$32$	0	0
$33$	−4.69779	−0.817780
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−5.41811	−0.890732	−0.445366	−	0.895349i	$-0.646926\pi$
$37$	−5.41811	−0.890732	−0.445366	+	0.895349i	$0.646926\pi$
$38$	0	0
$39$	−2.38350	−0.381666
$40$	0	0
$41$	1.30221	0.203371	0.101686	−	0.994817i	$-0.467576\pi$
$41$	1.30221	0.203371	0.101686	+	0.994817i	$0.467576\pi$
$42$	0	0
$43$	−5.73240	−0.874182	−0.437091	−	0.899417i	$-0.643991\pi$
$43$	−5.73240	−0.874182	−0.437091	+	0.899417i	$0.643991\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−3.48270	−0.508003	−0.254002	−	0.967204i	$-0.581747\pi$
$47$	−3.48270	−0.508003	−0.254002	+	0.967204i	$0.581747\pi$
$48$	0	0
$49$	−5.18048	−0.740069
$50$	0	0
$51$	−1.21509	−0.170147
$52$	0	0
$53$	−2.51730	−0.345778	−0.172889	−	0.984941i	$-0.555310\pi$
$53$	−2.51730	−0.345778	−0.172889	+	0.984941i	$0.555310\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−1.03461	−0.137037
$58$	0	0
$59$	7.86620	1.02409	0.512046	−	0.858958i	$-0.328888\pi$
$59$	7.86620	1.02409	0.512046	+	0.858958i	$0.328888\pi$
$60$	0	0
$61$	−5.46479	−0.699695	−0.349848	−	0.936807i	$-0.613767\pi$
$61$	−5.46479	−0.699695	−0.349848	+	0.936807i	$0.613767\pi$
$62$	0	0
$63$	−1.34889	−0.169945
$64$	0	0
$65$	0	0
$66$	0	0
$67$	1.41811	0.173250	0.0866249	−	0.996241i	$-0.472392\pi$
$67$	1.41811	0.173250	0.0866249	+	0.996241i	$0.472392\pi$
$68$	0	0
$69$	6.24970	0.752376
$70$	0	0
$71$	3.79698	0.450619	0.225309	−	0.974287i	$-0.427661\pi$
$71$	3.79698	0.450619	0.225309	+	0.974287i	$0.427661\pi$
$72$	0	0
$73$	1.41811	0.165977	0.0829886	−	0.996550i	$-0.473553\pi$
$73$	1.41811	0.165977	0.0829886	+	0.996550i	$0.473553\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−6.33682	−0.722148
$78$	0	0
$79$	−1.81952	−0.204711	−0.102356	−	0.994748i	$-0.532638\pi$
$79$	−1.81952	−0.204711	−0.102356	+	0.994748i	$0.532638\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−13.2151	−1.45054	−0.725272	−	0.688462i	$-0.758286\pi$
$83$	−13.2151	−1.45054	−0.725272	+	0.688462i	$0.758286\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	5.59859	0.600232
$88$	0	0
$89$	−2.20302	−0.233519	−0.116760	−	0.993160i	$-0.537251\pi$
$89$	−2.20302	−0.233519	−0.116760	+	0.993160i	$0.537251\pi$
$90$	0	0
$91$	−3.21509	−0.337033
$92$	0	0
$93$	−1.00000	−0.103695
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−6.69779	−0.680057	−0.340029	−	0.940415i	$-0.610437\pi$
$97$	−6.69779	−0.680057	−0.340029	+	0.940415i	$0.610437\pi$
$98$	0	0
$99$	4.69779	0.472146
$100$	0	0
$101$	13.1280	1.30628	0.653141	−	0.757236i	$-0.273451\pi$
$101$	13.1280	1.30628	0.653141	+	0.757236i	$0.273451\pi$
$102$	0	0
$103$	16.4769	1.62351	0.811757	−	0.583995i	$-0.198511\pi$
$103$	16.4769	1.62351	0.811757	+	0.583995i	$0.198511\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−4.78491	−0.462574	−0.231287	−	0.972886i	$-0.574294\pi$
$107$	−4.78491	−0.462574	−0.231287	+	0.972886i	$0.574294\pi$
$108$	0	0
$109$	0.947489	0.0907530	0.0453765	−	0.998970i	$-0.485551\pi$
$109$	0.947489	0.0907530	0.0453765	+	0.998970i	$0.485551\pi$
$110$	0	0
$111$	5.41811	0.514264
$112$	0	0
$113$	−17.1972	−1.61778	−0.808888	−	0.587963i	$-0.799930\pi$
$113$	−17.1972	−1.61778	−0.808888	+	0.587963i	$0.799930\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	2.38350	0.220355
$118$	0	0
$119$	−1.63903	−0.150250
$120$	0	0
$121$	11.0692	1.00629
$122$	0	0
$123$	−1.30221	−0.117416
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−2.16258	−0.191898	−0.0959490	−	0.995386i	$-0.530589\pi$
$127$	−2.16258	−0.191898	−0.0959490	+	0.995386i	$0.530589\pi$
$128$	0	0
$129$	5.73240	0.504709
$130$	0	0
$131$	10.9700	0.958455	0.479228	−	0.877691i	$-0.340917\pi$
$131$	10.9700	0.958455	0.479228	+	0.877691i	$0.340917\pi$
$132$	0	0
$133$	−1.39558	−0.121012
$134$	0	0
$135$	0	0
$136$	0	0
$137$	8.67989	0.741573	0.370786	−	0.928718i	$-0.379088\pi$
$137$	8.67989	0.741573	0.370786	+	0.928718i	$0.379088\pi$
$138$	0	0
$139$	4.76700	0.404332	0.202166	−	0.979351i	$-0.435202\pi$
$139$	4.76700	0.404332	0.202166	+	0.979351i	$0.435202\pi$
$140$	0	0
$141$	3.48270	0.293296
$142$	0	0
$143$	11.1972	0.936356
$144$	0	0
$145$	0	0
$146$	0	0
$147$	5.18048	0.427279
$148$	0	0
$149$	−10.6978	−0.876397	−0.438198	−	0.898878i	$-0.644383\pi$
$149$	−10.6978	−0.876397	−0.438198	+	0.898878i	$0.644383\pi$
$150$	0	0
$151$	5.01671	0.408254	0.204127	−	0.978944i	$-0.434565\pi$
$151$	5.01671	0.408254	0.204127	+	0.978944i	$0.434565\pi$
$152$	0	0
$153$	1.21509	0.0982344
$154$	0	0
$155$	0	0
$156$	0	0
$157$	5.80161	0.463019	0.231510	−	0.972833i	$-0.425634\pi$
$157$	5.80161	0.463019	0.231510	+	0.972833i	$0.425634\pi$
$158$	0	0
$159$	2.51730	0.199635
$160$	0	0
$161$	8.43018	0.664392
$162$	0	0
$163$	10.3143	0.807877	0.403939	−	0.914786i	$-0.367641\pi$
$163$	10.3143	0.807877	0.403939	+	0.914786i	$0.367641\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	5.66318	0.438230	0.219115	−	0.975699i	$-0.429683\pi$
$167$	5.66318	0.438230	0.219115	+	0.975699i	$0.429683\pi$
$168$	0	0
$169$	−7.31892	−0.562994
$170$	0	0
$171$	1.03461	0.0791185
$172$	0	0
$173$	11.6632	0.886735	0.443368	−	0.896340i	$-0.353784\pi$
$173$	11.6632	0.886735	0.443368	+	0.896340i	$0.353784\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−7.86620	−0.591260
$178$	0	0
$179$	−11.8258	−0.883899	−0.441949	−	0.897040i	$-0.645713\pi$
$179$	−11.8258	−0.883899	−0.441949	+	0.897040i	$0.645713\pi$
$180$	0	0
$181$	−14.8604	−1.10456	−0.552281	−	0.833658i	$-0.686243\pi$
$181$	−14.8604	−1.10456	−0.552281	+	0.833658i	$0.686243\pi$
$182$	0	0
$183$	5.46479	0.403969
$184$	0	0
$185$	0	0
$186$	0	0
$187$	5.70825	0.417428
$188$	0	0
$189$	1.34889	0.0981176
$190$	0	0
$191$	−5.93541	−0.429472	−0.214736	−	0.976672i	$-0.568889\pi$
$191$	−5.93541	−0.429472	−0.214736	+	0.976672i	$0.568889\pi$
$192$	0	0
$193$	3.19719	0.230139	0.115069	−	0.993357i	$-0.463291\pi$
$193$	3.19719	0.230139	0.115069	+	0.993357i	$0.463291\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−18.7491	−1.33582	−0.667909	−	0.744243i	$-0.732811\pi$
$197$	−18.7491	−1.33582	−0.667909	+	0.744243i	$0.732811\pi$
$198$	0	0
$199$	−21.9129	−1.55336	−0.776681	−	0.629894i	$-0.783099\pi$
$199$	−21.9129	−1.55336	−0.776681	+	0.629894i	$0.783099\pi$
$200$	0	0
$201$	−1.41811	−0.100026
$202$	0	0
$203$	7.55191	0.530040
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−6.24970	−0.434384
$208$	0	0
$209$	4.86037	0.336199
$210$	0	0
$211$	1.66318	0.114498	0.0572490	−	0.998360i	$-0.481767\pi$
$211$	1.66318	0.114498	0.0572490	+	0.998360i	$0.481767\pi$
$212$	0	0
$213$	−3.79698	−0.260165
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−1.34889	−0.0915689
$218$	0	0
$219$	−1.41811	−0.0958270
$220$	0	0
$221$	2.89618	0.194818
$222$	0	0
$223$	−5.10382	−0.341777	−0.170889	−	0.985290i	$-0.554664\pi$
$223$	−5.10382	−0.341777	−0.170889	+	0.985290i	$0.554664\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−2.85412	−0.189435	−0.0947174	−	0.995504i	$-0.530195\pi$
$227$	−2.85412	−0.189435	−0.0947174	+	0.995504i	$0.530195\pi$
$228$	0	0
$229$	−18.0934	−1.19564	−0.597822	−	0.801629i	$-0.703967\pi$
$229$	−18.0934	−1.19564	−0.597822	+	0.801629i	$0.703967\pi$
$230$	0	0
$231$	6.33682	0.416932
$232$	0	0
$233$	9.01671	0.590704	0.295352	−	0.955389i	$-0.404563\pi$
$233$	9.01671	0.590704	0.295352	+	0.955389i	$0.404563\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	1.81952	0.118190
$238$	0	0
$239$	13.9642	0.903269	0.451634	−	0.892203i	$-0.350841\pi$
$239$	13.9642	0.903269	0.451634	+	0.892203i	$0.350841\pi$
$240$	0	0
$241$	13.1972	0.850106	0.425053	−	0.905169i	$-0.360256\pi$
$241$	13.1972	0.850106	0.425053	+	0.905169i	$0.360256\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	2.46599	0.156907
$248$	0	0
$249$	13.2151	0.837472
$250$	0	0
$251$	−4.36097	−0.275262	−0.137631	−	0.990484i	$-0.543949\pi$
$251$	−4.36097	−0.275262	−0.137631	+	0.990484i	$0.543949\pi$
$252$	0	0
$253$	−29.3598	−1.84583
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−11.6211	−0.724906	−0.362453	−	0.932002i	$-0.618061\pi$
$257$	−11.6211	−0.724906	−0.362453	+	0.932002i	$0.618061\pi$
$258$	0	0
$259$	7.30846	0.454125
$260$	0	0
$261$	−5.59859	−0.346544
$262$	0	0
$263$	6.16258	0.380001	0.190001	−	0.981784i	$-0.439151\pi$
$263$	6.16258	0.380001	0.190001	+	0.981784i	$0.439151\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	2.20302	0.134823
$268$	0	0
$269$	−22.4590	−1.36935	−0.684674	−	0.728850i	$-0.740055\pi$
$269$	−22.4590	−1.36935	−0.684674	+	0.728850i	$0.740055\pi$
$270$	0	0
$271$	−22.8362	−1.38720	−0.693601	−	0.720360i	$-0.743977\pi$
$271$	−22.8362	−1.38720	−0.693601	+	0.720360i	$0.743977\pi$
$272$	0	0
$273$	3.21509	0.194586
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−14.1851	−0.852301	−0.426150	−	0.904652i	$-0.640131\pi$
$277$	−14.1851	−0.852301	−0.426150	+	0.904652i	$0.640131\pi$
$278$	0	0
$279$	1.00000	0.0598684
$280$	0	0
$281$	12.8362	0.765745	0.382872	−	0.923801i	$-0.374935\pi$
$281$	12.8362	0.765745	0.382872	+	0.923801i	$0.374935\pi$
$282$	0	0
$283$	−5.70986	−0.339416	−0.169708	−	0.985494i	$-0.554282\pi$
$283$	−5.70986	−0.339416	−0.169708	+	0.985494i	$0.554282\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−1.75655	−0.103686
$288$	0	0
$289$	−15.5236	−0.913150
$290$	0	0
$291$	6.69779	0.392631
$292$	0	0
$293$	−8.01790	−0.468411	−0.234205	−	0.972187i	$-0.575249\pi$
$293$	−8.01790	−0.468411	−0.234205	+	0.972187i	$0.575249\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−4.69779	−0.272593
$298$	0	0
$299$	−14.8962	−0.861468
$300$	0	0
$301$	7.73240	0.445688
$302$	0	0
$303$	−13.1280	−0.754182
$304$	0	0
$305$	0	0
$306$	0	0
$307$	18.7445	1.06980	0.534902	−	0.844914i	$-0.320349\pi$
$307$	18.7445	1.06980	0.534902	+	0.844914i	$0.320349\pi$
$308$	0	0
$309$	−16.4769	−0.937336
$310$	0	0
$311$	−19.2260	−1.09020	−0.545102	−	0.838370i	$-0.683509\pi$
$311$	−19.2260	−1.09020	−0.545102	+	0.838370i	$0.683509\pi$
$312$	0	0
$313$	−14.5219	−0.820828	−0.410414	−	0.911899i	$-0.634616\pi$
$313$	−14.5219	−0.820828	−0.410414	+	0.911899i	$0.634616\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−0.378872	−0.0212796	−0.0106398	−	0.999943i	$-0.503387\pi$
$317$	−0.378872	−0.0212796	−0.0106398	+	0.999943i	$0.503387\pi$
$318$	0	0
$319$	−26.3010	−1.47257
$320$	0	0
$321$	4.78491	0.267067
$322$	0	0
$323$	1.25714	0.0699494
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−0.947489	−0.0523963
$328$	0	0
$329$	4.69779	0.258997
$330$	0	0
$331$	17.1459	0.942423	0.471211	−	0.882020i	$-0.343817\pi$
$331$	17.1459	0.942423	0.471211	+	0.882020i	$0.343817\pi$
$332$	0	0
$333$	−5.41811	−0.296911
$334$	0	0
$335$	0	0
$336$	0	0
$337$	2.04668	0.111490	0.0557450	−	0.998445i	$-0.482247\pi$
$337$	2.04668	0.111490	0.0557450	+	0.998445i	$0.482247\pi$
$338$	0	0
$339$	17.1972	0.934023
$340$	0	0
$341$	4.69779	0.254400
$342$	0	0
$343$	16.4302	0.887147
$344$	0	0
$345$	0	0
$346$	0	0
$347$	9.26641	0.497447	0.248723	−	0.968575i	$-0.419989\pi$
$347$	9.26641	0.497447	0.248723	+	0.968575i	$0.419989\pi$
$348$	0	0
$349$	−3.41348	−0.182719	−0.0913597	−	0.995818i	$-0.529121\pi$
$349$	−3.41348	−0.182719	−0.0913597	+	0.995818i	$0.529121\pi$
$350$	0	0
$351$	−2.38350	−0.127222
$352$	0	0
$353$	20.6799	1.10068	0.550340	−	0.834941i	$-0.314498\pi$
$353$	20.6799	1.10068	0.550340	+	0.834941i	$0.314498\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	1.63903	0.0867467
$358$	0	0
$359$	−33.6227	−1.77454	−0.887270	−	0.461250i	$-0.847401\pi$
$359$	−33.6227	−1.77454	−0.887270	+	0.461250i	$0.847401\pi$
$360$	0	0
$361$	−17.9296	−0.943662
$362$	0	0
$363$	−11.0692	−0.580983
$364$	0	0
$365$	0	0
$366$	0	0
$367$	16.0000	0.835193	0.417597	−	0.908633i	$-0.362873\pi$
$367$	16.0000	0.835193	0.417597	+	0.908633i	$0.362873\pi$
$368$	0	0
$369$	1.30221	0.0677904
$370$	0	0
$371$	3.39558	0.176290
$372$	0	0
$373$	−35.2664	−1.82603	−0.913013	−	0.407931i	$-0.866250\pi$
$373$	−35.2664	−1.82603	−0.913013	+	0.407931i	$0.866250\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−13.3443	−0.687265
$378$	0	0
$379$	−19.1038	−0.981298	−0.490649	−	0.871357i	$-0.663240\pi$
$379$	−19.1038	−0.981298	−0.490649	+	0.871357i	$0.663240\pi$
$380$	0	0
$381$	2.16258	0.110792
$382$	0	0
$383$	−20.0062	−1.02227	−0.511136	−	0.859500i	$-0.670775\pi$
$383$	−20.0062	−1.02227	−0.511136	+	0.859500i	$0.670775\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−5.73240	−0.291394
$388$	0	0
$389$	−13.7036	−0.694801	−0.347400	−	0.937717i	$-0.612936\pi$
$389$	−13.7036	−0.694801	−0.347400	+	0.937717i	$0.612936\pi$
$390$	0	0
$391$	−7.59396	−0.384043
$392$	0	0
$393$	−10.9700	−0.553364
$394$	0	0
$395$	0	0
$396$	0	0
$397$	27.3598	1.37315	0.686574	−	0.727060i	$-0.259114\pi$
$397$	27.3598	1.37315	0.686574	+	0.727060i	$0.259114\pi$
$398$	0	0
$399$	1.39558	0.0698662
$400$	0	0
$401$	−9.79698	−0.489238	−0.244619	−	0.969619i	$-0.578663\pi$
$401$	−9.79698	−0.489238	−0.244619	+	0.969619i	$0.578663\pi$
$402$	0	0
$403$	2.38350	0.118731
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−25.4531	−1.26167
$408$	0	0
$409$	16.2559	0.803805	0.401902	−	0.915683i	$-0.368349\pi$
$409$	16.2559	0.803805	0.401902	+	0.915683i	$0.368349\pi$
$410$	0	0
$411$	−8.67989	−0.428147
$412$	0	0
$413$	−10.6107	−0.522117
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−4.76700	−0.233441
$418$	0	0
$419$	−25.6920	−1.25513	−0.627567	−	0.778562i	$-0.715949\pi$
$419$	−25.6920	−1.25513	−0.627567	+	0.778562i	$0.715949\pi$
$420$	0	0
$421$	−12.8541	−0.626472	−0.313236	−	0.949675i	$-0.601413\pi$
$421$	−12.8541	−0.626472	−0.313236	+	0.949675i	$0.601413\pi$
$422$	0	0
$423$	−3.48270	−0.169334
$424$	0	0
$425$	0	0
$426$	0	0
$427$	7.37143	0.356728
$428$	0	0
$429$	−11.1972	−0.540605
$430$	0	0
$431$	36.8891	1.77689	0.888444	−	0.458985i	$-0.151787\pi$
$431$	36.8891	1.77689	0.888444	+	0.458985i	$0.151787\pi$
$432$	0	0
$433$	−0.512673	−0.0246375	−0.0123188	−	0.999924i	$-0.503921\pi$
$433$	−0.512673	−0.0246375	−0.0123188	+	0.999924i	$0.503921\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−6.46599	−0.309310
$438$	0	0
$439$	−24.7219	−1.17991	−0.589957	−	0.807435i	$-0.700855\pi$
$439$	−24.7219	−1.17991	−0.589957	+	0.807435i	$0.700855\pi$
$440$	0	0
$441$	−5.18048	−0.246690
$442$	0	0
$443$	−35.4018	−1.68199	−0.840996	−	0.541042i	$-0.818030\pi$
$443$	−35.4018	−1.68199	−0.840996	+	0.541042i	$0.818030\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	10.6978	0.505988
$448$	0	0
$449$	−8.53984	−0.403020	−0.201510	−	0.979486i	$-0.564585\pi$
$449$	−8.53984	−0.403020	−0.201510	+	0.979486i	$0.564585\pi$
$450$	0	0
$451$	6.11751	0.288063
$452$	0	0
$453$	−5.01671	−0.235705
$454$	0	0
$455$	0	0
$456$	0	0
$457$	23.3131	1.09054	0.545270	−	0.838260i	$-0.316427\pi$
$457$	23.3131	1.09054	0.545270	+	0.838260i	$0.316427\pi$
$458$	0	0
$459$	−1.21509	−0.0567157
$460$	0	0
$461$	−28.7958	−1.34115	−0.670577	−	0.741840i	$-0.733953\pi$
$461$	−28.7958	−1.34115	−0.670577	+	0.741840i	$0.733953\pi$
$462$	0	0
$463$	18.7670	0.872177	0.436088	−	0.899904i	$-0.356364\pi$
$463$	18.7670	0.872177	0.436088	+	0.899904i	$0.356364\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	3.48270	0.161160	0.0805800	−	0.996748i	$-0.474323\pi$
$467$	3.48270	0.161160	0.0805800	+	0.996748i	$0.474323\pi$
$468$	0	0
$469$	−1.91288	−0.0883286
$470$	0	0
$471$	−5.80161	−0.267324
$472$	0	0
$473$	−26.9296	−1.23822
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−2.51730	−0.115259
$478$	0	0
$479$	−7.02998	−0.321208	−0.160604	−	0.987019i	$-0.551344\pi$
$479$	−7.02998	−0.321208	−0.160604	+	0.987019i	$0.551344\pi$
$480$	0	0
$481$	−12.9141	−0.588831
$482$	0	0
$483$	−8.43018	−0.383587
$484$	0	0
$485$	0	0
$486$	0	0
$487$	16.4543	0.745617	0.372809	−	0.927908i	$-0.378395\pi$
$487$	16.4543	0.745617	0.372809	+	0.927908i	$0.378395\pi$
$488$	0	0
$489$	−10.3143	−0.466428
$490$	0	0
$491$	−11.1972	−0.505322	−0.252661	−	0.967555i	$-0.581306\pi$
$491$	−11.1972	−0.505322	−0.252661	+	0.967555i	$0.581306\pi$
$492$	0	0
$493$	−6.80281	−0.306383
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−5.12173	−0.229741
$498$	0	0
$499$	−2.74286	−0.122787	−0.0613935	−	0.998114i	$-0.519554\pi$
$499$	−2.74286	−0.122787	−0.0613935	+	0.998114i	$0.519554\pi$
$500$	0	0
$501$	−5.66318	−0.253012
$502$	0	0
$503$	15.9821	0.712606	0.356303	−	0.934370i	$-0.384037\pi$
$503$	15.9821	0.712606	0.356303	+	0.934370i	$0.384037\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	7.31892	0.325045
$508$	0	0
$509$	9.49357	0.420795	0.210398	−	0.977616i	$-0.432524\pi$
$509$	9.49357	0.420795	0.210398	+	0.977616i	$0.432524\pi$
$510$	0	0
$511$	−1.91288	−0.0846209
$512$	0	0
$513$	−1.03461	−0.0456791
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−16.3610	−0.719555
$518$	0	0
$519$	−11.6632	−0.511957
$520$	0	0
$521$	6.83622	0.299500	0.149750	−	0.988724i	$-0.452153\pi$
$521$	6.83622	0.299500	0.149750	+	0.988724i	$0.452153\pi$
$522$	0	0
$523$	27.1972	1.18925	0.594625	−	0.804003i	$-0.297301\pi$
$523$	27.1972	1.18925	0.594625	+	0.804003i	$0.297301\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	1.21509	0.0529303
$528$	0	0
$529$	16.0588	0.698207
$530$	0	0
$531$	7.86620	0.341364
$532$	0	0
$533$	3.10382	0.134442
$534$	0	0
$535$	0	0
$536$	0	0
$537$	11.8258	0.510319
$538$	0	0
$539$	−24.3368	−1.04826
$540$	0	0
$541$	41.6787	1.79191	0.895953	−	0.444148i	$-0.146494\pi$
$541$	41.6787	1.79191	0.895953	+	0.444148i	$0.146494\pi$
$542$	0	0
$543$	14.8604	0.637720
$544$	0	0
$545$	0	0
$546$	0	0
$547$	19.0362	0.813930	0.406965	−	0.913444i	$-0.366587\pi$
$547$	19.0362	0.813930	0.406965	+	0.913444i	$0.366587\pi$
$548$	0	0
$549$	−5.46479	−0.233232
$550$	0	0
$551$	−5.79235	−0.246762
$552$	0	0
$553$	2.45433	0.104369
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−36.2831	−1.53736	−0.768682	−	0.639631i	$-0.779087\pi$
$557$	−36.2831	−1.53736	−0.768682	+	0.639631i	$0.779087\pi$
$558$	0	0
$559$	−13.6632	−0.577891
$560$	0	0
$561$	−5.70825	−0.241002
$562$	0	0
$563$	−24.0755	−1.01466	−0.507330	−	0.861752i	$-0.669367\pi$
$563$	−24.0755	−1.01466	−0.507330	+	0.861752i	$0.669367\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−1.34889	−0.0566482
$568$	0	0
$569$	−11.4360	−0.479423	−0.239711	−	0.970844i	$-0.577053\pi$
$569$	−11.4360	−0.479423	−0.239711	+	0.970844i	$0.577053\pi$
$570$	0	0
$571$	24.2318	1.01407	0.507035	−	0.861926i	$-0.330742\pi$
$571$	24.2318	1.01407	0.507035	+	0.861926i	$0.330742\pi$
$572$	0	0
$573$	5.93541	0.247955
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−1.53401	−0.0638616	−0.0319308	−	0.999490i	$-0.510166\pi$
$577$	−1.53401	−0.0638616	−0.0319308	+	0.999490i	$0.510166\pi$
$578$	0	0
$579$	−3.19719	−0.132871
$580$	0	0
$581$	17.8258	0.739537
$582$	0	0
$583$	−11.8258	−0.489773
$584$	0	0
$585$	0	0
$586$	0	0
$587$	18.1626	0.749650	0.374825	−	0.927096i	$-0.377703\pi$
$587$	18.1626	0.749650	0.374825	+	0.927096i	$0.377703\pi$
$588$	0	0
$589$	1.03461	0.0426303
$590$	0	0
$591$	18.7491	0.771235
$592$	0	0
$593$	19.6274	0.806000	0.403000	−	0.915200i	$-0.367968\pi$
$593$	19.6274	0.806000	0.403000	+	0.915200i	$0.367968\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	21.9129	0.896835
$598$	0	0
$599$	−0.609054	−0.0248853	−0.0124426	−	0.999923i	$-0.503961\pi$
$599$	−0.609054	−0.0248853	−0.0124426	+	0.999923i	$0.503961\pi$
$600$	0	0
$601$	−23.6274	−0.963781	−0.481890	−	0.876232i	$-0.660050\pi$
$601$	−23.6274	−0.963781	−0.481890	+	0.876232i	$0.660050\pi$
$602$	0	0
$603$	1.41811	0.0577499
$604$	0	0
$605$	0	0
$606$	0	0
$607$	41.1505	1.67025	0.835124	−	0.550062i	$-0.185396\pi$
$607$	41.1505	1.67025	0.835124	+	0.550062i	$0.185396\pi$
$608$	0	0
$609$	−7.55191	−0.306019
$610$	0	0
$611$	−8.30101	−0.335823
$612$	0	0
$613$	−45.8966	−1.85375	−0.926873	−	0.375375i	$-0.877514\pi$
$613$	−45.8966	−1.85375	−0.926873	+	0.375375i	$0.877514\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−46.8604	−1.88653	−0.943264	−	0.332044i	$-0.892262\pi$
$617$	−46.8604	−1.88653	−0.943264	+	0.332044i	$0.892262\pi$
$618$	0	0
$619$	11.8529	0.476409	0.238205	−	0.971215i	$-0.423441\pi$
$619$	11.8529	0.476409	0.238205	+	0.971215i	$0.423441\pi$
$620$	0	0
$621$	6.24970	0.250792
$622$	0	0
$623$	2.97164	0.119056
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−4.86037	−0.194104
$628$	0	0
$629$	−6.58351	−0.262502
$630$	0	0
$631$	−17.2571	−0.686996	−0.343498	−	0.939153i	$-0.611612\pi$
$631$	−17.2571	−0.686996	−0.343498	+	0.939153i	$0.611612\pi$
$632$	0	0
$633$	−1.66318	−0.0661055
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−12.3477	−0.489234
$638$	0	0
$639$	3.79698	0.150206
$640$	0	0
$641$	−8.15795	−0.322220	−0.161110	−	0.986936i	$-0.551507\pi$
$641$	−8.15795	−0.322220	−0.161110	+	0.986936i	$0.551507\pi$
$642$	0	0
$643$	−7.19719	−0.283829	−0.141915	−	0.989879i	$-0.545326\pi$
$643$	−7.19719	−0.283829	−0.141915	+	0.989879i	$0.545326\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	15.6873	0.616733	0.308366	−	0.951268i	$-0.400218\pi$
$647$	15.6873	0.616733	0.308366	+	0.951268i	$0.400218\pi$
$648$	0	0
$649$	36.9537	1.45056
$650$	0	0
$651$	1.34889	0.0528673
$652$	0	0
$653$	37.8771	1.48224	0.741122	−	0.671370i	$-0.234294\pi$
$653$	37.8771	1.48224	0.741122	+	0.671370i	$0.234294\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	1.41811	0.0553258
$658$	0	0
$659$	−46.3298	−1.80475	−0.902376	−	0.430949i	$-0.858179\pi$
$659$	−46.3298	−1.80475	−0.902376	+	0.430949i	$0.858179\pi$
$660$	0	0
$661$	−9.19719	−0.357729	−0.178865	−	0.983874i	$-0.557242\pi$
$661$	−9.19719	−0.357729	−0.178865	+	0.983874i	$0.557242\pi$
$662$	0	0
$663$	−2.89618	−0.112478
$664$	0	0
$665$	0	0
$666$	0	0
$667$	34.9895	1.35480
$668$	0	0
$669$	5.10382	0.197325
$670$	0	0
$671$	−25.6724	−0.991074
$672$	0	0
$673$	20.4169	0.787014	0.393507	−	0.919322i	$-0.371262\pi$
$673$	20.4169	0.787014	0.393507	+	0.919322i	$0.371262\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−15.7744	−0.606261	−0.303131	−	0.952949i	$-0.598032\pi$
$677$	−15.7744	−0.606261	−0.303131	+	0.952949i	$0.598032\pi$
$678$	0	0
$679$	9.03461	0.346716
$680$	0	0
$681$	2.85412	0.109370
$682$	0	0
$683$	−11.4920	−0.439728	−0.219864	−	0.975531i	$-0.570561\pi$
$683$	−11.4920	−0.439728	−0.219864	+	0.975531i	$0.570561\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	18.0934	0.690305
$688$	0	0
$689$	−6.00000	−0.228582
$690$	0	0
$691$	13.1638	0.500774	0.250387	−	0.968146i	$-0.419442\pi$
$691$	13.1638	0.500774	0.250387	+	0.968146i	$0.419442\pi$
$692$	0	0
$693$	−6.33682	−0.240716
$694$	0	0
$695$	0	0
$696$	0	0
$697$	1.58231	0.0599342
$698$	0	0
$699$	−9.01671	−0.341043
$700$	0	0
$701$	−33.1372	−1.25158	−0.625788	−	0.779993i	$-0.715223\pi$
$701$	−33.1372	−1.25158	−0.625788	+	0.779993i	$0.715223\pi$
$702$	0	0
$703$	−5.60562	−0.211420
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−17.7082	−0.665987
$708$	0	0
$709$	6.86037	0.257647	0.128823	−	0.991668i	$-0.458880\pi$
$709$	6.86037	0.257647	0.128823	+	0.991668i	$0.458880\pi$
$710$	0	0
$711$	−1.81952	−0.0682372
$712$	0	0
$713$	−6.24970	−0.234053
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−13.9642	−0.521502
$718$	0	0
$719$	−9.90663	−0.369455	−0.184728	−	0.982790i	$-0.559140\pi$
$719$	−9.90663	−0.369455	−0.184728	+	0.982790i	$0.559140\pi$
$720$	0	0
$721$	−22.2256	−0.827723
$722$	0	0
$723$	−13.1972	−0.490809
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−15.4423	−0.572722	−0.286361	−	0.958122i	$-0.592446\pi$
$727$	−15.4423	−0.572722	−0.286361	+	0.958122i	$0.592446\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−6.96539	−0.257624
$732$	0	0
$733$	−40.9296	−1.51177	−0.755884	−	0.654705i	$-0.772793\pi$
$733$	−40.9296	−1.51177	−0.755884	+	0.654705i	$0.772793\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	6.66198	0.245397
$738$	0	0
$739$	50.4123	1.85445	0.927223	−	0.374510i	$-0.122189\pi$
$739$	50.4123	1.85445	0.927223	+	0.374510i	$0.122189\pi$
$740$	0	0
$741$	−2.46599	−0.0905904
$742$	0	0
$743$	−21.3505	−0.783274	−0.391637	−	0.920120i	$-0.628091\pi$
$743$	−21.3505	−0.783274	−0.391637	+	0.920120i	$0.628091\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−13.2151	−0.483515
$748$	0	0
$749$	6.45433	0.235836
$750$	0	0
$751$	32.1960	1.17485	0.587424	−	0.809279i	$-0.300142\pi$
$751$	32.1960	1.17485	0.587424	+	0.809279i	$0.300142\pi$
$752$	0	0
$753$	4.36097	0.158923
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−28.2334	−1.02616	−0.513080	−	0.858341i	$-0.671496\pi$
$757$	−28.2334	−1.02616	−0.513080	+	0.858341i	$0.671496\pi$
$758$	0	0
$759$	29.3598	1.06569
$760$	0	0
$761$	−33.9145	−1.22940	−0.614700	−	0.788761i	$-0.710723\pi$
$761$	−33.9145	−1.22940	−0.614700	+	0.788761i	$0.710723\pi$
$762$	0	0
$763$	−1.27806	−0.0462690
$764$	0	0
$765$	0	0
$766$	0	0
$767$	18.7491	0.676991
$768$	0	0
$769$	36.5990	1.31979	0.659897	−	0.751356i	$-0.270600\pi$
$769$	36.5990	1.31979	0.659897	+	0.751356i	$0.270600\pi$
$770$	0	0
$771$	11.6211	0.418525
$772$	0	0
$773$	33.2634	1.19640	0.598201	−	0.801346i	$-0.295883\pi$
$773$	33.2634	1.19640	0.598201	+	0.801346i	$0.295883\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−7.30846	−0.262189
$778$	0	0
$779$	1.34728	0.0482713
$780$	0	0
$781$	17.8374	0.638273
$782$	0	0
$783$	5.59859	0.200077
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−16.2559	−0.579462	−0.289731	−	0.957108i	$-0.593566\pi$
$787$	−16.2559	−0.579462	−0.289731	+	0.957108i	$0.593566\pi$
$788$	0	0
$789$	−6.16258	−0.219394
$790$	0	0
$791$	23.1972	0.824797
$792$	0	0
$793$	−13.0253	−0.462544
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−24.7157	−0.875475	−0.437737	−	0.899103i	$-0.644220\pi$
$797$	−24.7157	−0.875475	−0.437737	+	0.899103i	$0.644220\pi$
$798$	0	0
$799$	−4.23180	−0.149710
$800$	0	0
$801$	−2.20302	−0.0778398
$802$	0	0
$803$	6.66198	0.235096
$804$	0	0
$805$	0	0
$806$	0	0
$807$	22.4590	0.790593
$808$	0	0
$809$	19.8211	0.696874	0.348437	−	0.937332i	$-0.386713\pi$
$809$	19.8211	0.696874	0.348437	+	0.937332i	$0.386713\pi$
$810$	0	0
$811$	11.2330	0.394444	0.197222	−	0.980359i	$-0.436808\pi$
$811$	11.2330	0.394444	0.197222	+	0.980359i	$0.436808\pi$
$812$	0	0
$813$	22.8362	0.800901
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−5.93078	−0.207492
$818$	0	0
$819$	−3.21509	−0.112344
$820$	0	0
$821$	22.9700	0.801659	0.400830	−	0.916153i	$-0.368722\pi$
$821$	22.9700	0.801659	0.400830	+	0.916153i	$0.368722\pi$
$822$	0	0
$823$	−0.697788	−0.0243234	−0.0121617	−	0.999926i	$-0.503871\pi$
$823$	−0.697788	−0.0243234	−0.0121617	+	0.999926i	$0.503871\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−30.0755	−1.04583	−0.522913	−	0.852386i	$-0.675155\pi$
$827$	−30.0755	−1.04583	−0.522913	+	0.852386i	$0.675155\pi$
$828$	0	0
$829$	−16.6620	−0.578695	−0.289347	−	0.957224i	$-0.593438\pi$
$829$	−16.6620	−0.578695	−0.289347	+	0.957224i	$0.593438\pi$
$830$	0	0
$831$	14.1851	0.492076
$832$	0	0
$833$	−6.29477	−0.218101
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−1.00000	−0.0345651
$838$	0	0
$839$	16.9584	0.585468	0.292734	−	0.956194i	$-0.405435\pi$
$839$	16.9584	0.585468	0.292734	+	0.956194i	$0.405435\pi$
$840$	0	0
$841$	2.34426	0.0808367
$842$	0	0
$843$	−12.8362	−0.442103
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−14.9312	−0.513042
$848$	0	0
$849$	5.70986	0.195962
$850$	0	0
$851$	33.8616	1.16076
$852$	0	0
$853$	5.43138	0.185967	0.0929835	−	0.995668i	$-0.470360\pi$
$853$	5.43138	0.185967	0.0929835	+	0.995668i	$0.470360\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	11.6632	0.398407	0.199203	−	0.979958i	$-0.436165\pi$
$857$	11.6632	0.398407	0.199203	+	0.979958i	$0.436165\pi$
$858$	0	0
$859$	54.9779	1.87582	0.937911	−	0.346877i	$-0.112758\pi$
$859$	54.9779	1.87582	0.937911	+	0.346877i	$0.112758\pi$
$860$	0	0
$861$	1.75655	0.0598629
$862$	0	0
$863$	−20.4636	−0.696589	−0.348294	−	0.937385i	$-0.613239\pi$
$863$	−20.4636	−0.696589	−0.348294	+	0.937385i	$0.613239\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	15.5236	0.527207
$868$	0	0
$869$	−8.54770	−0.289961
$870$	0	0
$871$	3.38007	0.114529
$872$	0	0
$873$	−6.69779	−0.226686
$874$	0	0
$875$	0	0
$876$	0	0
$877$	33.1855	1.12060	0.560298	−	0.828291i	$-0.310687\pi$
$877$	33.1855	1.12060	0.560298	+	0.828291i	$0.310687\pi$
$878$	0	0
$879$	8.01790	0.270437
$880$	0	0
$881$	−5.30684	−0.178792	−0.0893960	−	0.995996i	$-0.528494\pi$
$881$	−5.30684	−0.178792	−0.0893960	+	0.995996i	$0.528494\pi$
$882$	0	0
$883$	−29.6032	−0.996228	−0.498114	−	0.867112i	$-0.665974\pi$
$883$	−29.6032	−0.996228	−0.498114	+	0.867112i	$0.665974\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−47.2727	−1.58726	−0.793630	−	0.608401i	$-0.791811\pi$
$887$	−47.2727	−1.58726	−0.793630	+	0.608401i	$0.791811\pi$
$888$	0	0
$889$	2.91709	0.0978362
$890$	0	0
$891$	4.69779	0.157382
$892$	0	0
$893$	−3.60323	−0.120577
$894$	0	0
$895$	0	0
$896$	0	0
$897$	14.8962	0.497369
$898$	0	0
$899$	−5.59859	−0.186724
$900$	0	0
$901$	−3.05876	−0.101902
$902$	0	0
$903$	−7.73240	−0.257318
$904$	0	0
$905$	0	0
$906$	0	0
$907$	20.5010	0.680725	0.340363	−	0.940294i	$-0.389450\pi$
$907$	20.5010	0.680725	0.340363	+	0.940294i	$0.389450\pi$
$908$	0	0
$909$	13.1280	0.435427
$910$	0	0
$911$	7.62737	0.252706	0.126353	−	0.991985i	$-0.459673\pi$
$911$	7.62737	0.252706	0.126353	+	0.991985i	$0.459673\pi$
$912$	0	0
$913$	−62.0817	−2.05460
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−14.7974	−0.488653
$918$	0	0
$919$	−6.92959	−0.228586	−0.114293	−	0.993447i	$-0.536460\pi$
$919$	−6.92959	−0.228586	−0.114293	+	0.993447i	$0.536460\pi$
$920$	0	0
$921$	−18.7445	−0.617651
$922$	0	0
$923$	9.05012	0.297888
$924$	0	0
$925$	0	0
$926$	0	0
$927$	16.4769	0.541171
$928$	0	0
$929$	−39.6678	−1.30146	−0.650729	−	0.759310i	$-0.725537\pi$
$929$	−39.6678	−1.30146	−0.650729	+	0.759310i	$0.725537\pi$
$930$	0	0
$931$	−5.35977	−0.175659
$932$	0	0
$933$	19.2260	0.629430
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−31.6966	−1.03548	−0.517741	−	0.855537i	$-0.673227\pi$
$937$	−31.6966	−1.03548	−0.517741	+	0.855537i	$0.673227\pi$
$938$	0	0
$939$	14.5219	0.473905
$940$	0	0
$941$	−36.8557	−1.20146	−0.600731	−	0.799451i	$-0.705124\pi$
$941$	−36.8557	−1.20146	−0.600731	+	0.799451i	$0.705124\pi$
$942$	0	0
$943$	−8.13843	−0.265024
$944$	0	0
$945$	0	0
$946$	0	0
$947$	34.6107	1.12470	0.562348	−	0.826901i	$-0.309898\pi$
$947$	34.6107	1.12470	0.562348	+	0.826901i	$0.309898\pi$
$948$	0	0
$949$	3.38007	0.109722
$950$	0	0
$951$	0.378872	0.0122858
$952$	0	0
$953$	−4.14468	−0.134259	−0.0671297	−	0.997744i	$-0.521384\pi$
$953$	−4.14468	−0.134259	−0.0671297	+	0.997744i	$0.521384\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	26.3010	0.850191
$958$	0	0
$959$	−11.7082	−0.378079
$960$	0	0
$961$	1.00000	0.0322581
$962$	0	0
$963$	−4.78491	−0.154191
$964$	0	0
$965$	0	0
$966$	0	0
$967$	24.7553	0.796078	0.398039	−	0.917368i	$-0.369691\pi$
$967$	24.7553	0.796078	0.398039	+	0.917368i	$0.369691\pi$
$968$	0	0
$969$	−1.25714	−0.0403853
$970$	0	0
$971$	39.9930	1.28344	0.641718	−	0.766941i	$-0.278222\pi$
$971$	39.9930	1.28344	0.641718	+	0.766941i	$0.278222\pi$
$972$	0	0
$973$	−6.43018	−0.206142
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−10.8604	−0.347454	−0.173727	−	0.984794i	$-0.555581\pi$
$977$	−10.8604	−0.347454	−0.173727	+	0.984794i	$0.555581\pi$
$978$	0	0
$979$	−10.3493	−0.330765
$980$	0	0
$981$	0.947489	0.0302510
$982$	0	0
$983$	−5.03461	−0.160579	−0.0802895	−	0.996772i	$-0.525584\pi$
$983$	−5.03461	−0.160579	−0.0802895	+	0.996772i	$0.525584\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−4.69779	−0.149532
$988$	0	0
$989$	35.8258	1.13919
$990$	0	0
$991$	55.0230	1.74786	0.873931	−	0.486050i	$-0.161563\pi$
$991$	55.0230	1.74786	0.873931	+	0.486050i	$0.161563\pi$
$992$	0	0
$993$	−17.1459	−0.544108
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−5.40723	−0.171249	−0.0856244	−	0.996327i	$-0.527289\pi$
$997$	−5.40723	−0.171249	−0.0856244	+	0.996327i	$0.527289\pi$
$998$	0	0
$999$	5.41811	0.171421

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9300.2.a.u.1.1		3
5.2	odd	4		9300.2.g.q.3349.4		6
5.3	odd	4		9300.2.g.q.3349.3		6
5.4	even	2		1860.2.a.g.1.3	✓	3
15.14	odd	2		5580.2.a.j.1.3		3
20.19	odd	2		7440.2.a.bn.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1860.2.a.g.1.3	✓	3	5.4	even	2
5580.2.a.j.1.3		3	15.14	odd	2
7440.2.a.bn.1.1		3	20.19	odd	2
9300.2.a.u.1.1		3	1.1	even	1	trivial
9300.2.g.q.3349.3		6	5.3	odd	4
9300.2.g.q.3349.4		6	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 9300 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9300.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -1.34889 q^{7} +1.00000 q^{9} +4.69779 q^{11} +2.38350 q^{13} +1.21509 q^{17} +1.03461 q^{19} +1.34889 q^{21} -6.24970 q^{23} -1.00000 q^{27} -5.59859 q^{29} +1.00000 q^{31} -4.69779 q^{33} -5.41811 q^{37} -2.38350 q^{39} +1.30221 q^{41} -5.73240 q^{43} -3.48270 q^{47} -5.18048 q^{49} -1.21509 q^{51} -2.51730 q^{53} -1.03461 q^{57} +7.86620 q^{59} -5.46479 q^{61} -1.34889 q^{63} +1.41811 q^{67} +6.24970 q^{69} +3.79698 q^{71} +1.41811 q^{73} -6.33682 q^{77} -1.81952 q^{79} +1.00000 q^{81} -13.2151 q^{83} +5.59859 q^{87} -2.20302 q^{89} -3.21509 q^{91} -1.00000 q^{93} -6.69779 q^{97} +4.69779 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{3} + 2 q^{7} + 3 q^{9} + 2 q^{11} - 2 q^{13} - 10 q^{17} - 2 q^{21} - 2 q^{23} - 3 q^{27} + 6 q^{29} + 3 q^{31} - 2 q^{33} - 4 q^{37} + 2 q^{39} + 16 q^{41} - 2 q^{43} - 12 q^{47} - 5 q^{49}+ \cdots + 2 q^{99}+O(q^{100}) \) 3 * q - 3 * q^3 + 2 * q^7 + 3 * q^9 + 2 * q^11 - 2 * q^13 - 10 * q^17 - 2 * q^21 - 2 * q^23 - 3 * q^27 + 6 * q^29 + 3 * q^31 - 2 * q^33 - 4 * q^37 + 2 * q^39 + 16 * q^41 - 2 * q^43 - 12 * q^47 - 5 * q^49 + 10 * q^51 - 6 * q^53 + 16 * q^59 + 14 * q^61 + 2 * q^63 - 8 * q^67 + 2 * q^69 + 10 * q^71 - 8 * q^73 - 28 * q^77 - 16 * q^79 + 3 * q^81 - 26 * q^83 - 6 * q^87 - 8 * q^89 + 4 * q^91 - 3 * q^93 - 8 * q^97 + 2 * q^99

Introduction

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