LMFDB - Embedded newform 2300.2.a.l.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2300,2,Mod(1,2300)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2300, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("2300.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2300 = 2^{2} \cdot 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2300.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:	$18.3655924649$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.53121.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 7x^{2} - x + 2 $ x^4 - 7*x^2 - x + 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-2.50653$ of defining polynomial
Character	$\chi$	$=$	2300.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.50653 q^{3} +0.797915 q^{7} +3.28271 q^{9} +O(q^{10})$	$q+2.50653 q^{3} +0.797915 q^{7} +3.28271 q^{9} +4.21515 q^{11} -1.48479 q^{13} +3.01307 q^{17} -2.21515 q^{19} +2.00000 q^{21} +1.00000 q^{23} +0.708618 q^{27} -2.57409 q^{29} +8.78924 q^{31} +10.5654 q^{33} +9.57848 q^{37} -3.72168 q^{39} -1.26097 q^{41} -2.21515 q^{43} -2.86547 q^{47} -6.36333 q^{49} +7.55235 q^{51} +5.41724 q^{53} -5.55235 q^{57} -1.21515 q^{59} +2.58276 q^{61} +2.61932 q^{63} -8.56542 q^{67} +2.50653 q^{69} +6.73036 q^{71} +3.02174 q^{73} +3.36333 q^{77} +0.189019 q^{79} -8.07195 q^{81} -7.22822 q^{83} -6.45204 q^{87} +6.38682 q^{89} -1.18474 q^{91} +22.0305 q^{93} -13.3999 q^{97} +13.8371 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - q^{7} + 2 q^{9}+O(q^{10}) $ 4 * q - q^7 + 2 * q^9	$ 4 q - q^{7} + 2 q^{9} + q^{11} + q^{13} - 8 q^{17} + 7 q^{19} + 8 q^{21} + 4 q^{23} - 3 q^{27} - 5 q^{29} + 14 q^{31} + 20 q^{33} - 4 q^{37} + 11 q^{39} + 3 q^{41} + 7 q^{43} - 12 q^{47} + q^{49} + 28 q^{51} + 10 q^{53} - 20 q^{57} + 11 q^{59} + 22 q^{61} + 3 q^{63} - 12 q^{67} + 18 q^{71} + 9 q^{73} - 13 q^{77} + 25 q^{79} + 7 q^{83} + 9 q^{87} + 25 q^{91} + 11 q^{93} - 8 q^{97} - 9 q^{99}+O(q^{100}) $ 4 * q - q^7 + 2 * q^9 + q^11 + q^13 - 8 * q^17 + 7 * q^19 + 8 * q^21 + 4 * q^23 - 3 * q^27 - 5 * q^29 + 14 * q^31 + 20 * q^33 - 4 * q^37 + 11 * q^39 + 3 * q^41 + 7 * q^43 - 12 * q^47 + q^49 + 28 * q^51 + 10 * q^53 - 20 * q^57 + 11 * q^59 + 22 * q^61 + 3 * q^63 - 12 * q^67 + 18 * q^71 + 9 * q^73 - 13 * q^77 + 25 * q^79 + 7 * q^83 + 9 * q^87 + 25 * q^91 + 11 * q^93 - 8 * q^97 - 9 * 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
$3$	2.50653	1.44715	0.723574	−	0.690247i	$-0.242498\pi$
$3$	2.50653	1.44715	0.723574	+	0.690247i	$0.242498\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	0.797915	0.301583	0.150792	−	0.988566i	$-0.451818\pi$
$7$	0.797915	0.301583	0.150792	+	0.988566i	$0.451818\pi$
$8$	0	0
$9$	3.28271	1.09424
$10$	0	0
$11$	4.21515	1.27092	0.635458	−	0.772135i	$-0.280811\pi$
$11$	4.21515	1.27092	0.635458	+	0.772135i	$0.280811\pi$
$12$	0	0
$13$	−1.48479	−0.411808	−0.205904	−	0.978572i	$-0.566013\pi$
$13$	−1.48479	−0.411808	−0.205904	+	0.978572i	$0.566013\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	3.01307	0.730776	0.365388	−	0.930855i	$-0.380936\pi$
$17$	3.01307	0.730776	0.365388	+	0.930855i	$0.380936\pi$
$18$	0	0
$19$	−2.21515	−0.508191	−0.254095	−	0.967179i	$-0.581778\pi$
$19$	−2.21515	−0.508191	−0.254095	+	0.967179i	$0.581778\pi$
$20$	0	0
$21$	2.00000	0.436436
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	0	0
$26$	0	0
$27$	0.708618	0.136374
$28$	0	0
$29$	−2.57409	−0.477997	−0.238998	−	0.971020i	$-0.576819\pi$
$29$	−2.57409	−0.477997	−0.238998	+	0.971020i	$0.576819\pi$
$30$	0	0
$31$	8.78924	1.57859	0.789297	−	0.614011i	$-0.210445\pi$
$31$	8.78924	1.57859	0.789297	+	0.614011i	$0.210445\pi$
$32$	0	0
$33$	10.5654	1.83920
$34$	0	0
$35$	0	0
$36$	0	0
$37$	9.57848	1.57469	0.787346	−	0.616511i	$-0.211454\pi$
$37$	9.57848	1.57469	0.787346	+	0.616511i	$0.211454\pi$
$38$	0	0
$39$	−3.72168	−0.595946
$40$	0	0
$41$	−1.26097	−0.196930	−0.0984651	−	0.995141i	$-0.531393\pi$
$41$	−1.26097	−0.196930	−0.0984651	+	0.995141i	$0.531393\pi$
$42$	0	0
$43$	−2.21515	−0.337807	−0.168904	−	0.985633i	$-0.554023\pi$
$43$	−2.21515	−0.337807	−0.168904	+	0.985633i	$0.554023\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−2.86547	−0.417972	−0.208986	−	0.977919i	$-0.567016\pi$
$47$	−2.86547	−0.417972	−0.208986	+	0.977919i	$0.567016\pi$
$48$	0	0
$49$	−6.36333	−0.909047
$50$	0	0
$51$	7.55235	1.05754
$52$	0	0
$53$	5.41724	0.744115	0.372057	−	0.928210i	$-0.378652\pi$
$53$	5.41724	0.744115	0.372057	+	0.928210i	$0.378652\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−5.55235	−0.735427
$58$	0	0
$59$	−1.21515	−0.158199	−0.0790996	−	0.996867i	$-0.525205\pi$
$59$	−1.21515	−0.158199	−0.0790996	+	0.996867i	$0.525205\pi$
$60$	0	0
$61$	2.58276	0.330689	0.165344	−	0.986236i	$-0.447126\pi$
$61$	2.58276	0.330689	0.165344	+	0.986236i	$0.447126\pi$
$62$	0	0
$63$	2.61932	0.330004
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−8.56542	−1.04643	−0.523216	−	0.852200i	$-0.675268\pi$
$67$	−8.56542	−1.04643	−0.523216	+	0.852200i	$0.675268\pi$
$68$	0	0
$69$	2.50653	0.301751
$70$	0	0
$71$	6.73036	0.798747	0.399373	−	0.916788i	$-0.369228\pi$
$71$	6.73036	0.798747	0.399373	+	0.916788i	$0.369228\pi$
$72$	0	0
$73$	3.02174	0.353668	0.176834	−	0.984241i	$-0.443414\pi$
$73$	3.02174	0.353668	0.176834	+	0.984241i	$0.443414\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	3.36333	0.383287
$78$	0	0
$79$	0.189019	0.0212663	0.0106331	−	0.999943i	$-0.496615\pi$
$79$	0.189019	0.0212663	0.0106331	+	0.999943i	$0.496615\pi$
$80$	0	0
$81$	−8.07195	−0.896883
$82$	0	0
$83$	−7.22822	−0.793400	−0.396700	−	0.917948i	$-0.629845\pi$
$83$	−7.22822	−0.793400	−0.396700	+	0.917948i	$0.629845\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−6.45204	−0.691732
$88$	0	0
$89$	6.38682	0.677002	0.338501	−	0.940966i	$-0.390080\pi$
$89$	6.38682	0.677002	0.338501	+	0.940966i	$0.390080\pi$
$90$	0	0
$91$	−1.18474	−0.124194
$92$	0	0
$93$	22.0305	2.28446
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−13.3999	−1.36055	−0.680276	−	0.732956i	$-0.738140\pi$
$97$	−13.3999	−1.36055	−0.680276	+	0.732956i	$0.738140\pi$
$98$	0	0
$99$	13.8371	1.39068
$100$	0	0
$101$	17.1978	1.71125	0.855623	−	0.517600i	$-0.173174\pi$
$101$	17.1978	1.71125	0.855623	+	0.517600i	$0.173174\pi$
$102$	0	0
$103$	−0.393745	−0.0387968	−0.0193984	−	0.999812i	$-0.506175\pi$
$103$	−0.393745	−0.0387968	−0.0193984	+	0.999812i	$0.506175\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	0.969587	0.0937336	0.0468668	−	0.998901i	$-0.485076\pi$
$107$	0.969587	0.0937336	0.0468668	+	0.998901i	$0.485076\pi$
$108$	0	0
$109$	10.1612	0.973271	0.486635	−	0.873605i	$-0.338224\pi$
$109$	10.1612	0.973271	0.486635	+	0.873605i	$0.338224\pi$
$110$	0	0
$111$	24.0088	2.27881
$112$	0	0
$113$	−6.99572	−0.658102	−0.329051	−	0.944312i	$-0.606729\pi$
$113$	−6.99572	−0.658102	−0.329051	+	0.944312i	$0.606729\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−4.87414	−0.450615
$118$	0	0
$119$	2.40417	0.220390
$120$	0	0
$121$	6.76750	0.615227
$122$	0	0
$123$	−3.16066	−0.284987
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−4.78924	−0.424976	−0.212488	−	0.977164i	$-0.568157\pi$
$127$	−4.78924	−0.424976	−0.212488	+	0.977164i	$0.568157\pi$
$128$	0	0
$129$	−5.55235	−0.488857
$130$	0	0
$131$	−5.11543	−0.446937	−0.223469	−	0.974711i	$-0.571738\pi$
$131$	−5.11543	−0.446937	−0.223469	+	0.974711i	$0.571738\pi$
$132$	0	0
$133$	−1.76750	−0.153262
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−12.0261	−1.02746	−0.513731	−	0.857951i	$-0.671737\pi$
$137$	−12.0261	−1.02746	−0.513731	+	0.857951i	$0.671737\pi$
$138$	0	0
$139$	12.1041	1.02666	0.513329	−	0.858192i	$-0.328412\pi$
$139$	12.1041	1.02666	0.513329	+	0.858192i	$0.328412\pi$
$140$	0	0
$141$	−7.18240	−0.604867
$142$	0	0
$143$	−6.25863	−0.523373
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−15.9499	−1.31553
$148$	0	0
$149$	14.7701	1.21002	0.605009	−	0.796219i	$-0.293170\pi$
$149$	14.7701	1.21002	0.605009	+	0.796219i	$0.293170\pi$
$150$	0	0
$151$	−6.40048	−0.520863	−0.260432	−	0.965492i	$-0.583865\pi$
$151$	−6.40048	−0.520863	−0.260432	+	0.965492i	$0.583865\pi$
$152$	0	0
$153$	9.89102	0.799641
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−10.8780	−0.868155	−0.434078	−	0.900876i	$-0.642926\pi$
$157$	−10.8780	−0.868155	−0.434078	+	0.900876i	$0.642926\pi$
$158$	0	0
$159$	13.5785	1.07684
$160$	0	0
$161$	0.797915	0.0628845
$162$	0	0
$163$	−15.6237	−1.22374	−0.611872	−	0.790957i	$-0.709583\pi$
$163$	−15.6237	−1.22374	−0.611872	+	0.790957i	$0.709583\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	12.7502	0.986637	0.493318	−	0.869849i	$-0.335784\pi$
$167$	12.7502	0.986637	0.493318	+	0.869849i	$0.335784\pi$
$168$	0	0
$169$	−10.7954	−0.830414
$170$	0	0
$171$	−7.27170	−0.556081
$172$	0	0
$173$	0.569697	0.0433133	0.0216566	−	0.999765i	$-0.493106\pi$
$173$	0.569697	0.0433133	0.0216566	+	0.999765i	$0.493106\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−3.04582	−0.228938
$178$	0	0
$179$	3.99825	0.298843	0.149422	−	0.988774i	$-0.452259\pi$
$179$	3.99825	0.298843	0.149422	+	0.988774i	$0.452259\pi$
$180$	0	0
$181$	−11.2178	−0.833812	−0.416906	−	0.908950i	$-0.636886\pi$
$181$	−11.2178	−0.833812	−0.416906	+	0.908950i	$0.636886\pi$
$182$	0	0
$183$	6.47378	0.478556
$184$	0	0
$185$	0	0
$186$	0	0
$187$	12.7005	0.928755
$188$	0	0
$189$	0.565417	0.0411280
$190$	0	0
$191$	−0.0365585	−0.00264528	−0.00132264	−	0.999999i	$-0.500421\pi$
$191$	−0.0365585	−0.00264528	−0.00132264	+	0.999999i	$0.500421\pi$
$192$	0	0
$193$	10.2201	0.735661	0.367831	−	0.929893i	$-0.380101\pi$
$193$	10.2201	0.735661	0.367831	+	0.929893i	$0.380101\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−17.4657	−1.24438	−0.622190	−	0.782867i	$-0.713757\pi$
$197$	−17.4657	−1.24438	−0.622190	+	0.782867i	$0.713757\pi$
$198$	0	0
$199$	−4.37640	−0.310235	−0.155117	−	0.987896i	$-0.549576\pi$
$199$	−4.37640	−0.310235	−0.155117	+	0.987896i	$0.549576\pi$
$200$	0	0
$201$	−21.4695	−1.51434
$202$	0	0
$203$	−2.05390	−0.144156
$204$	0	0
$205$	0	0
$206$	0	0
$207$	3.28271	0.228164
$208$	0	0
$209$	−9.33720	−0.645868
$210$	0	0
$211$	11.6455	0.801706	0.400853	−	0.916142i	$-0.368714\pi$
$211$	11.6455	0.801706	0.400853	+	0.916142i	$0.368714\pi$
$212$	0	0
$213$	16.8699	1.15590
$214$	0	0
$215$	0	0
$216$	0	0
$217$	7.01307	0.476078
$218$	0	0
$219$	7.57409	0.511810
$220$	0	0
$221$	−4.47378	−0.300939
$222$	0	0
$223$	−18.4068	−1.23261	−0.616306	−	0.787507i	$-0.711371\pi$
$223$	−18.4068	−1.23261	−0.616306	+	0.787507i	$0.711371\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−27.2747	−1.81029	−0.905144	−	0.425105i	$-0.860237\pi$
$227$	−27.2747	−1.81029	−0.905144	+	0.425105i	$0.860237\pi$
$228$	0	0
$229$	12.8606	0.849853	0.424926	−	0.905228i	$-0.360300\pi$
$229$	12.8606	0.849853	0.424926	+	0.905228i	$0.360300\pi$
$230$	0	0
$231$	8.43030	0.554673
$232$	0	0
$233$	24.8847	1.63025	0.815125	−	0.579285i	$-0.196668\pi$
$233$	24.8847	1.63025	0.815125	+	0.579285i	$0.196668\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0.473782	0.0307754
$238$	0	0
$239$	−17.8160	−1.15242	−0.576209	−	0.817302i	$-0.695469\pi$
$239$	−17.8160	−1.15242	−0.576209	+	0.817302i	$0.695469\pi$
$240$	0	0
$241$	13.2004	0.850315	0.425158	−	0.905119i	$-0.360219\pi$
$241$	13.2004	0.850315	0.425158	+	0.905119i	$0.360219\pi$
$242$	0	0
$243$	−22.3585	−1.43430
$244$	0	0
$245$	0	0
$246$	0	0
$247$	3.28904	0.209277
$248$	0	0
$249$	−18.1178	−1.14817
$250$	0	0
$251$	−28.1266	−1.77533	−0.887666	−	0.460488i	$-0.847675\pi$
$251$	−28.1266	−1.77533	−0.887666	+	0.460488i	$0.847675\pi$
$252$	0	0
$253$	4.21515	0.265004
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−17.1416	−1.06926	−0.534631	−	0.845086i	$-0.679549\pi$
$257$	−17.1416	−1.06926	−0.534631	+	0.845086i	$0.679549\pi$
$258$	0	0
$259$	7.64281	0.474901
$260$	0	0
$261$	−8.44999	−0.523041
$262$	0	0
$263$	−22.7093	−1.40032	−0.700158	−	0.713988i	$-0.746887\pi$
$263$	−22.7093	−1.40032	−0.700158	+	0.713988i	$0.746887\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	16.0088	0.979722
$268$	0	0
$269$	−19.3758	−1.18136	−0.590682	−	0.806904i	$-0.701141\pi$
$269$	−19.3758	−1.18136	−0.590682	+	0.806904i	$0.701141\pi$
$270$	0	0
$271$	32.0549	1.94720	0.973598	−	0.228268i	$-0.0733061\pi$
$271$	32.0549	1.94720	0.973598	+	0.228268i	$0.0733061\pi$
$272$	0	0
$273$	−2.96959	−0.179728
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−6.36099	−0.382195	−0.191098	−	0.981571i	$-0.561205\pi$
$277$	−6.36099	−0.382195	−0.191098	+	0.981571i	$0.561205\pi$
$278$	0	0
$279$	28.8525	1.72736
$280$	0	0
$281$	−14.7440	−0.879554	−0.439777	−	0.898107i	$-0.644943\pi$
$281$	−14.7440	−0.879554	−0.439777	+	0.898107i	$0.644943\pi$
$282$	0	0
$283$	−8.76136	−0.520809	−0.260404	−	0.965500i	$-0.583856\pi$
$283$	−8.76136	−0.520809	−0.260404	+	0.965500i	$0.583856\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−1.00615	−0.0593909
$288$	0	0
$289$	−7.92143	−0.465967
$290$	0	0
$291$	−33.5873	−1.96892
$292$	0	0
$293$	−10.2047	−0.596166	−0.298083	−	0.954540i	$-0.596347\pi$
$293$	−10.2047	−0.596166	−0.298083	+	0.954540i	$0.596347\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	2.98693	0.173319
$298$	0	0
$299$	−1.48479	−0.0858678
$300$	0	0
$301$	−1.76750	−0.101877
$302$	0	0
$303$	43.1069	2.47642
$304$	0	0
$305$	0	0
$306$	0	0
$307$	26.7267	1.52537	0.762686	−	0.646769i	$-0.223880\pi$
$307$	26.7267	1.52537	0.762686	+	0.646769i	$0.223880\pi$
$308$	0	0
$309$	−0.986934	−0.0561447
$310$	0	0
$311$	26.7807	1.51859	0.759297	−	0.650745i	$-0.225543\pi$
$311$	26.7807	1.51859	0.759297	+	0.650745i	$0.225543\pi$
$312$	0	0
$313$	−12.9214	−0.730362	−0.365181	−	0.930936i	$-0.618993\pi$
$313$	−12.9214	−0.730362	−0.365181	+	0.930936i	$0.618993\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−14.1717	−0.795960	−0.397980	−	0.917394i	$-0.630289\pi$
$317$	−14.1717	−0.795960	−0.397980	+	0.917394i	$0.630289\pi$
$318$	0	0
$319$	−10.8502	−0.607493
$320$	0	0
$321$	2.43030	0.135646
$322$	0	0
$323$	−6.67440	−0.371373
$324$	0	0
$325$	0	0
$326$	0	0
$327$	25.4695	1.40847
$328$	0	0
$329$	−2.28640	−0.126053
$330$	0	0
$331$	30.6766	1.68614	0.843068	−	0.537807i	$-0.180747\pi$
$331$	30.6766	1.68614	0.843068	+	0.537807i	$0.180747\pi$
$332$	0	0
$333$	31.4434	1.72309
$334$	0	0
$335$	0	0
$336$	0	0
$337$	29.4260	1.60294	0.801469	−	0.598037i	$-0.204052\pi$
$337$	29.4260	1.60294	0.801469	+	0.598037i	$0.204052\pi$
$338$	0	0
$339$	−17.5350	−0.952371
$340$	0	0
$341$	37.0480	2.00626
$342$	0	0
$343$	−10.6628	−0.575737
$344$	0	0
$345$	0	0
$346$	0	0
$347$	13.9157	0.747033	0.373516	−	0.927624i	$-0.378152\pi$
$347$	13.9157	0.747033	0.373516	+	0.927624i	$0.378152\pi$
$348$	0	0
$349$	10.2931	0.550979	0.275489	−	0.961304i	$-0.411160\pi$
$349$	10.2931	0.550979	0.275489	+	0.961304i	$0.411160\pi$
$350$	0	0
$351$	−1.05215	−0.0561597
$352$	0	0
$353$	−13.8960	−0.739609	−0.369805	−	0.929110i	$-0.620575\pi$
$353$	−13.8960	−0.739609	−0.369805	+	0.929110i	$0.620575\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	6.02613	0.318937
$358$	0	0
$359$	−32.8328	−1.73285	−0.866425	−	0.499307i	$-0.833588\pi$
$359$	−32.8328	−1.73285	−0.866425	+	0.499307i	$0.833588\pi$
$360$	0	0
$361$	−14.0931	−0.741742
$362$	0	0
$363$	16.9630	0.890325
$364$	0	0
$365$	0	0
$366$	0	0
$367$	3.47114	0.181192	0.0905961	−	0.995888i	$-0.471123\pi$
$367$	3.47114	0.181192	0.0905961	+	0.995888i	$0.471123\pi$
$368$	0	0
$369$	−4.13939	−0.215488
$370$	0	0
$371$	4.32249	0.224413
$372$	0	0
$373$	−24.7440	−1.28120	−0.640598	−	0.767876i	$-0.721314\pi$
$373$	−24.7440	−1.28120	−0.640598	+	0.767876i	$0.721314\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	3.82199	0.196843
$378$	0	0
$379$	37.7832	1.94079	0.970397	−	0.241517i	$-0.0776450\pi$
$379$	37.7832	1.94079	0.970397	+	0.241517i	$0.0776450\pi$
$380$	0	0
$381$	−12.0044	−0.615004
$382$	0	0
$383$	8.51151	0.434918	0.217459	−	0.976069i	$-0.430223\pi$
$383$	8.51151	0.434918	0.217459	+	0.976069i	$0.430223\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−7.27170	−0.369641
$388$	0	0
$389$	−28.2309	−1.43136	−0.715681	−	0.698428i	$-0.753883\pi$
$389$	−28.2309	−1.43136	−0.715681	+	0.698428i	$0.753883\pi$
$390$	0	0
$391$	3.01307	0.152377
$392$	0	0
$393$	−12.8220	−0.646784
$394$	0	0
$395$	0	0
$396$	0	0
$397$	12.0714	0.605844	0.302922	−	0.953015i	$-0.402038\pi$
$397$	12.0714	0.605844	0.302922	+	0.953015i	$0.402038\pi$
$398$	0	0
$399$	−4.43030	−0.221793
$400$	0	0
$401$	−32.1612	−1.60606	−0.803028	−	0.595941i	$-0.796779\pi$
$401$	−32.1612	−1.60606	−0.803028	+	0.595941i	$0.796779\pi$
$402$	0	0
$403$	−13.0502	−0.650077
$404$	0	0
$405$	0	0
$406$	0	0
$407$	40.3748	2.00130
$408$	0	0
$409$	17.6704	0.873746	0.436873	−	0.899523i	$-0.356086\pi$
$409$	17.6704	0.873746	0.436873	+	0.899523i	$0.356086\pi$
$410$	0	0
$411$	−30.1439	−1.48689
$412$	0	0
$413$	−0.969587	−0.0477103
$414$	0	0
$415$	0	0
$416$	0	0
$417$	30.3394	1.48573
$418$	0	0
$419$	−16.8710	−0.824204	−0.412102	−	0.911138i	$-0.635205\pi$
$419$	−16.8710	−0.824204	−0.412102	+	0.911138i	$0.635205\pi$
$420$	0	0
$421$	8.29519	0.404283	0.202141	−	0.979356i	$-0.435210\pi$
$421$	8.29519	0.404283	0.202141	+	0.979356i	$0.435210\pi$
$422$	0	0
$423$	−9.40651	−0.457360
$424$	0	0
$425$	0	0
$426$	0	0
$427$	2.06083	0.0997303
$428$	0	0
$429$	−15.6875	−0.757398
$430$	0	0
$431$	−18.3213	−0.882507	−0.441254	−	0.897382i	$-0.645466\pi$
$431$	−18.3213	−0.882507	−0.441254	+	0.897382i	$0.645466\pi$
$432$	0	0
$433$	−2.39538	−0.115115	−0.0575574	−	0.998342i	$-0.518331\pi$
$433$	−2.39538	−0.115115	−0.0575574	+	0.998342i	$0.518331\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−2.21515	−0.105965
$438$	0	0
$439$	3.00234	0.143294	0.0716469	−	0.997430i	$-0.477175\pi$
$439$	3.00234	0.143294	0.0716469	+	0.997430i	$0.477175\pi$
$440$	0	0
$441$	−20.8890	−0.994713
$442$	0	0
$443$	17.0790	0.811447	0.405724	−	0.913996i	$-0.367020\pi$
$443$	17.0790	0.811447	0.405724	+	0.913996i	$0.367020\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	37.0219	1.75107
$448$	0	0
$449$	−8.01999	−0.378487	−0.189243	−	0.981930i	$-0.560603\pi$
$449$	−8.01999	−0.378487	−0.189243	+	0.981930i	$0.560603\pi$
$450$	0	0
$451$	−5.31518	−0.250282
$452$	0	0
$453$	−16.0430	−0.753766
$454$	0	0
$455$	0	0
$456$	0	0
$457$	26.7081	1.24935	0.624677	−	0.780883i	$-0.285231\pi$
$457$	26.7081	1.24935	0.624677	+	0.780883i	$0.285231\pi$
$458$	0	0
$459$	2.13511	0.0996586
$460$	0	0
$461$	14.7895	0.688818	0.344409	−	0.938820i	$-0.388079\pi$
$461$	14.7895	0.688818	0.344409	+	0.938820i	$0.388079\pi$
$462$	0	0
$463$	−1.64545	−0.0764708	−0.0382354	−	0.999269i	$-0.512174\pi$
$463$	−1.64545	−0.0764708	−0.0382354	+	0.999269i	$0.512174\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−22.3591	−1.03465	−0.517327	−	0.855788i	$-0.673073\pi$
$467$	−22.3591	−1.03465	−0.517327	+	0.855788i	$0.673073\pi$
$468$	0	0
$469$	−6.83447	−0.315587
$470$	0	0
$471$	−27.2659	−1.25635
$472$	0	0
$473$	−9.33720	−0.429325
$474$	0	0
$475$	0	0
$476$	0	0
$477$	17.7832	0.814237
$478$	0	0
$479$	14.9245	0.681916	0.340958	−	0.940078i	$-0.389249\pi$
$479$	14.9245	0.681916	0.340958	+	0.940078i	$0.389249\pi$
$480$	0	0
$481$	−14.2221	−0.648471
$482$	0	0
$483$	2.00000	0.0910032
$484$	0	0
$485$	0	0
$486$	0	0
$487$	20.9994	0.951574	0.475787	−	0.879560i	$-0.342163\pi$
$487$	20.9994	0.951574	0.475787	+	0.879560i	$0.342163\pi$
$488$	0	0
$489$	−39.1614	−1.77094
$490$	0	0
$491$	−37.9743	−1.71376	−0.856878	−	0.515520i	$-0.827599\pi$
$491$	−37.9743	−1.71376	−0.856878	+	0.515520i	$0.827599\pi$
$492$	0	0
$493$	−7.75590	−0.349308
$494$	0	0
$495$	0	0
$496$	0	0
$497$	5.37025	0.240889
$498$	0	0
$499$	20.9630	0.938431	0.469216	−	0.883084i	$-0.344537\pi$
$499$	20.9630	0.938431	0.469216	+	0.883084i	$0.344537\pi$
$500$	0	0
$501$	31.9587	1.42781
$502$	0	0
$503$	10.9157	0.486706	0.243353	−	0.969938i	$-0.421753\pi$
$503$	10.9157	0.486706	0.243353	+	0.969938i	$0.421753\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−27.0590	−1.20173
$508$	0	0
$509$	−36.0068	−1.59598	−0.797988	−	0.602674i	$-0.794102\pi$
$509$	−36.0068	−1.59598	−0.797988	+	0.602674i	$0.794102\pi$
$510$	0	0
$511$	2.41109	0.106660
$512$	0	0
$513$	−1.56970	−0.0693038
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−12.0784	−0.531207
$518$	0	0
$519$	1.42796	0.0626807
$520$	0	0
$521$	9.73973	0.426705	0.213353	−	0.976975i	$-0.431562\pi$
$521$	9.73973	0.426705	0.213353	+	0.976975i	$0.431562\pi$
$522$	0	0
$523$	−22.8414	−0.998784	−0.499392	−	0.866376i	$-0.666443\pi$
$523$	−22.8414	−0.998784	−0.499392	+	0.866376i	$0.666443\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	26.4826	1.15360
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−3.98899	−0.173107
$532$	0	0
$533$	1.87228	0.0810974
$534$	0	0
$535$	0	0
$536$	0	0
$537$	10.0217	0.432470
$538$	0	0
$539$	−26.8224	−1.15532
$540$	0	0
$541$	−43.1869	−1.85675	−0.928375	−	0.371645i	$-0.878794\pi$
$541$	−43.1869	−1.85675	−0.928375	+	0.371645i	$0.878794\pi$
$542$	0	0
$543$	−28.1178	−1.20665
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−28.3438	−1.21189	−0.605946	−	0.795506i	$-0.707205\pi$
$547$	−28.3438	−1.21189	−0.605946	+	0.795506i	$0.707205\pi$
$548$	0	0
$549$	8.47846	0.361852
$550$	0	0
$551$	5.70200	0.242913
$552$	0	0
$553$	0.150821	0.00641356
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−19.8214	−0.839860	−0.419930	−	0.907557i	$-0.637945\pi$
$557$	−19.8214	−0.839860	−0.419930	+	0.907557i	$0.637945\pi$
$558$	0	0
$559$	3.28904	0.139112
$560$	0	0
$561$	31.8343	1.34405
$562$	0	0
$563$	35.2109	1.48396	0.741981	−	0.670421i	$-0.233887\pi$
$563$	35.2109	1.48396	0.741981	+	0.670421i	$0.233887\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−6.44073	−0.270485
$568$	0	0
$569$	−9.83875	−0.412462	−0.206231	−	0.978503i	$-0.566120\pi$
$569$	−9.83875	−0.412462	−0.206231	+	0.978503i	$0.566120\pi$
$570$	0	0
$571$	32.5434	1.36190	0.680949	−	0.732331i	$-0.261567\pi$
$571$	32.5434	1.36190	0.680949	+	0.732331i	$0.261567\pi$
$572$	0	0
$573$	−0.0916351	−0.00382811
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−16.7305	−0.696499	−0.348249	−	0.937402i	$-0.613224\pi$
$577$	−16.7305	−0.696499	−0.348249	+	0.937402i	$0.613224\pi$
$578$	0	0
$579$	25.6171	1.06461
$580$	0	0
$581$	−5.76750	−0.239276
$582$	0	0
$583$	22.8345	0.945707
$584$	0	0
$585$	0	0
$586$	0	0
$587$	5.51257	0.227528	0.113764	−	0.993508i	$-0.463709\pi$
$587$	5.51257	0.227528	0.113764	+	0.993508i	$0.463709\pi$
$588$	0	0
$589$	−19.4695	−0.802227
$590$	0	0
$591$	−43.7783	−1.80080
$592$	0	0
$593$	−2.61582	−0.107419	−0.0537094	−	0.998557i	$-0.517104\pi$
$593$	−2.61582	−0.107419	−0.0537094	+	0.998557i	$0.517104\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−10.9696	−0.448955
$598$	0	0
$599$	−6.72402	−0.274736	−0.137368	−	0.990520i	$-0.543864\pi$
$599$	−6.72402	−0.274736	−0.137368	+	0.990520i	$0.543864\pi$
$600$	0	0
$601$	−38.7249	−1.57962	−0.789811	−	0.613350i	$-0.789821\pi$
$601$	−38.7249	−1.57962	−0.789811	+	0.613350i	$0.789821\pi$
$602$	0	0
$603$	−28.1178	−1.14504
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−24.4095	−0.990749	−0.495375	−	0.868679i	$-0.664969\pi$
$607$	−24.4095	−0.990749	−0.495375	+	0.868679i	$0.664969\pi$
$608$	0	0
$609$	−5.14818	−0.208615
$610$	0	0
$611$	4.25463	0.172124
$612$	0	0
$613$	−42.8224	−1.72958	−0.864790	−	0.502133i	$-0.832549\pi$
$613$	−42.8224	−1.72958	−0.864790	+	0.502133i	$0.832549\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−40.6177	−1.63521	−0.817603	−	0.575782i	$-0.804698\pi$
$617$	−40.6177	−1.63521	−0.817603	+	0.575782i	$0.804698\pi$
$618$	0	0
$619$	13.6132	0.547160	0.273580	−	0.961849i	$-0.411792\pi$
$619$	13.6132	0.547160	0.273580	+	0.961849i	$0.411792\pi$
$620$	0	0
$621$	0.708618	0.0284359
$622$	0	0
$623$	5.09614	0.204173
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−23.4040	−0.934666
$628$	0	0
$629$	28.8606	1.15075
$630$	0	0
$631$	−37.7108	−1.50124	−0.750621	−	0.660733i	$-0.770246\pi$
$631$	−37.7108	−1.50124	−0.750621	+	0.660733i	$0.770246\pi$
$632$	0	0
$633$	29.1897	1.16019
$634$	0	0
$635$	0	0
$636$	0	0
$637$	9.44824	0.374353
$638$	0	0
$639$	22.0938	0.874017
$640$	0	0
$641$	−30.0261	−1.18596	−0.592980	−	0.805217i	$-0.702049\pi$
$641$	−30.0261	−1.18596	−0.592980	+	0.805217i	$0.702049\pi$
$642$	0	0
$643$	24.5030	0.966302	0.483151	−	0.875537i	$-0.339492\pi$
$643$	24.5030	0.966302	0.483151	+	0.875537i	$0.339492\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	19.3535	0.760864	0.380432	−	0.924809i	$-0.375775\pi$
$647$	19.3535	0.760864	0.380432	+	0.924809i	$0.375775\pi$
$648$	0	0
$649$	−5.12205	−0.201058
$650$	0	0
$651$	17.5785	0.688955
$652$	0	0
$653$	36.2953	1.42034	0.710172	−	0.704028i	$-0.248617\pi$
$653$	36.2953	1.42034	0.710172	+	0.704028i	$0.248617\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	9.91949	0.386996
$658$	0	0
$659$	30.0553	1.17079	0.585394	−	0.810749i	$-0.300940\pi$
$659$	30.0553	1.17079	0.585394	+	0.810749i	$0.300940\pi$
$660$	0	0
$661$	1.29947	0.0505435	0.0252717	−	0.999681i	$-0.491955\pi$
$661$	1.29947	0.0505435	0.0252717	+	0.999681i	$0.491955\pi$
$662$	0	0
$663$	−11.2137	−0.435503
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−2.57409	−0.0996692
$668$	0	0
$669$	−46.1373	−1.78377
$670$	0	0
$671$	10.8867	0.420278
$672$	0	0
$673$	−50.4441	−1.94448	−0.972238	−	0.233994i	$-0.924820\pi$
$673$	−50.4441	−1.94448	−0.972238	+	0.233994i	$0.924820\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	1.84754	0.0710067	0.0355034	−	0.999370i	$-0.488697\pi$
$677$	1.84754	0.0710067	0.0355034	+	0.999370i	$0.488697\pi$
$678$	0	0
$679$	−10.6920	−0.410320
$680$	0	0
$681$	−68.3650	−2.61975
$682$	0	0
$683$	−32.0332	−1.22572	−0.612858	−	0.790193i	$-0.709980\pi$
$683$	−32.0332	−1.22572	−0.612858	+	0.790193i	$0.709980\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	32.2355	1.22986
$688$	0	0
$689$	−8.04348	−0.306432
$690$	0	0
$691$	25.5585	0.972291	0.486146	−	0.873878i	$-0.338402\pi$
$691$	25.5585	0.972291	0.486146	+	0.873878i	$0.338402\pi$
$692$	0	0
$693$	11.0408	0.419407
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−3.79938	−0.143912
$698$	0	0
$699$	62.3743	2.35921
$700$	0	0
$701$	22.5707	0.852484	0.426242	−	0.904609i	$-0.359837\pi$
$701$	22.5707	0.852484	0.426242	+	0.904609i	$0.359837\pi$
$702$	0	0
$703$	−21.2178	−0.800244
$704$	0	0
$705$	0	0
$706$	0	0
$707$	13.7224	0.516083
$708$	0	0
$709$	13.6916	0.514198	0.257099	−	0.966385i	$-0.417233\pi$
$709$	13.6916	0.514198	0.257099	+	0.966385i	$0.417233\pi$
$710$	0	0
$711$	0.620494	0.0232703
$712$	0	0
$713$	8.78924	0.329160
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−44.6563	−1.66772
$718$	0	0
$719$	19.5003	0.727239	0.363619	−	0.931548i	$-0.381541\pi$
$719$	19.5003	0.727239	0.363619	+	0.931548i	$0.381541\pi$
$720$	0	0
$721$	−0.314175	−0.0117005
$722$	0	0
$723$	33.0874	1.23053
$724$	0	0
$725$	0	0
$726$	0	0
$727$	50.2432	1.86342	0.931708	−	0.363209i	$-0.118319\pi$
$727$	50.2432	1.86342	0.931708	+	0.363209i	$0.118319\pi$
$728$	0	0
$729$	−31.8264	−1.17876
$730$	0	0
$731$	−6.67440	−0.246862
$732$	0	0
$733$	−20.7355	−0.765881	−0.382941	−	0.923773i	$-0.625089\pi$
$733$	−20.7355	−0.765881	−0.382941	+	0.923773i	$0.625089\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−36.1045	−1.32993
$738$	0	0
$739$	33.6672	1.23847	0.619234	−	0.785207i	$-0.287443\pi$
$739$	33.6672	1.23847	0.619234	+	0.785207i	$0.287443\pi$
$740$	0	0
$741$	8.24410	0.302854
$742$	0	0
$743$	−36.5412	−1.34056	−0.670282	−	0.742106i	$-0.733827\pi$
$743$	−36.5412	−1.34056	−0.670282	+	0.742106i	$0.733827\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−23.7281	−0.868167
$748$	0	0
$749$	0.773648	0.0282685
$750$	0	0
$751$	25.3495	0.925016	0.462508	−	0.886615i	$-0.346950\pi$
$751$	25.3495	0.925016	0.462508	+	0.886615i	$0.346950\pi$
$752$	0	0
$753$	−70.5001	−2.56917
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−9.14818	−0.332496	−0.166248	−	0.986084i	$-0.553165\pi$
$757$	−9.14818	−0.332496	−0.166248	+	0.986084i	$0.553165\pi$
$758$	0	0
$759$	10.5654	0.383500
$760$	0	0
$761$	20.9609	0.759833	0.379916	−	0.925021i	$-0.375953\pi$
$761$	20.9609	0.759833	0.379916	+	0.925021i	$0.375953\pi$
$762$	0	0
$763$	8.10781	0.293522
$764$	0	0
$765$	0	0
$766$	0	0
$767$	1.80425	0.0651477
$768$	0	0
$769$	36.5915	1.31953	0.659763	−	0.751474i	$-0.270657\pi$
$769$	36.5915	1.31953	0.659763	+	0.751474i	$0.270657\pi$
$770$	0	0
$771$	−42.9659	−1.54738
$772$	0	0
$773$	−38.1219	−1.37115	−0.685574	−	0.728003i	$-0.740449\pi$
$773$	−38.1219	−1.37115	−0.685574	+	0.728003i	$0.740449\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	19.1570	0.687252
$778$	0	0
$779$	2.79324	0.100078
$780$	0	0
$781$	28.3695	1.01514
$782$	0	0
$783$	−1.82405	−0.0651861
$784$	0	0
$785$	0	0
$786$	0	0
$787$	20.4495	0.728946	0.364473	−	0.931214i	$-0.381249\pi$
$787$	20.4495	0.728946	0.364473	+	0.931214i	$0.381249\pi$
$788$	0	0
$789$	−56.9217	−2.02646
$790$	0	0
$791$	−5.58199	−0.198473
$792$	0	0
$793$	−3.83487	−0.136180
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−27.5177	−0.974725	−0.487363	−	0.873200i	$-0.662041\pi$
$797$	−27.5177	−0.974725	−0.487363	+	0.873200i	$0.662041\pi$
$798$	0	0
$799$	−8.63386	−0.305444
$800$	0	0
$801$	20.9661	0.740800
$802$	0	0
$803$	12.7371	0.449482
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−48.5661	−1.70961
$808$	0	0
$809$	16.1917	0.569268	0.284634	−	0.958636i	$-0.408128\pi$
$809$	16.1917	0.569268	0.284634	+	0.958636i	$0.408128\pi$
$810$	0	0
$811$	−17.2872	−0.607036	−0.303518	−	0.952826i	$-0.598161\pi$
$811$	−17.2872	−0.607036	−0.303518	+	0.952826i	$0.598161\pi$
$812$	0	0
$813$	80.3467	2.81788
$814$	0	0
$815$	0	0
$816$	0	0
$817$	4.90690	0.171671
$818$	0	0
$819$	−3.88915	−0.135898
$820$	0	0
$821$	−32.6395	−1.13913	−0.569564	−	0.821947i	$-0.692888\pi$
$821$	−32.6395	−1.13913	−0.569564	+	0.821947i	$0.692888\pi$
$822$	0	0
$823$	39.7419	1.38532	0.692658	−	0.721266i	$-0.256440\pi$
$823$	39.7419	1.38532	0.692658	+	0.721266i	$0.256440\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−9.46764	−0.329222	−0.164611	−	0.986359i	$-0.552637\pi$
$827$	−9.46764	−0.329222	−0.164611	+	0.986359i	$0.552637\pi$
$828$	0	0
$829$	43.6920	1.51748	0.758742	−	0.651391i	$-0.225814\pi$
$829$	43.6920	1.51748	0.758742	+	0.651391i	$0.225814\pi$
$830$	0	0
$831$	−15.9440	−0.553093
$832$	0	0
$833$	−19.1731	−0.664310
$834$	0	0
$835$	0	0
$836$	0	0
$837$	6.22822	0.215279
$838$	0	0
$839$	−1.37212	−0.0473708	−0.0236854	−	0.999719i	$-0.507540\pi$
$839$	−1.37212	−0.0473708	−0.0236854	+	0.999719i	$0.507540\pi$
$840$	0	0
$841$	−22.3741	−0.771519
$842$	0	0
$843$	−36.9564	−1.27284
$844$	0	0
$845$	0	0
$846$	0	0
$847$	5.39989	0.185542
$848$	0	0
$849$	−21.9606	−0.753687
$850$	0	0
$851$	9.57848	0.328346
$852$	0	0
$853$	−18.6367	−0.638107	−0.319054	−	0.947737i	$-0.603365\pi$
$853$	−18.6367	−0.638107	−0.319054	+	0.947737i	$0.603365\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−21.9201	−0.748776	−0.374388	−	0.927272i	$-0.622147\pi$
$857$	−21.9201	−0.748776	−0.374388	+	0.927272i	$0.622147\pi$
$858$	0	0
$859$	43.1587	1.47256	0.736278	−	0.676679i	$-0.236581\pi$
$859$	43.1587	1.47256	0.736278	+	0.676679i	$0.236581\pi$
$860$	0	0
$861$	−2.52194	−0.0859474
$862$	0	0
$863$	38.0107	1.29390	0.646950	−	0.762532i	$-0.276044\pi$
$863$	38.0107	1.29390	0.646950	+	0.762532i	$0.276044\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−19.8553	−0.674322
$868$	0	0
$869$	0.796743	0.0270277
$870$	0	0
$871$	12.7179	0.430929
$872$	0	0
$873$	−43.9879	−1.48877
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−27.9976	−0.945411	−0.472706	−	0.881220i	$-0.656723\pi$
$877$	−27.9976	−0.945411	−0.472706	+	0.881220i	$0.656723\pi$
$878$	0	0
$879$	−25.5785	−0.862741
$880$	0	0
$881$	−8.84326	−0.297937	−0.148968	−	0.988842i	$-0.547595\pi$
$881$	−8.84326	−0.297937	−0.148968	+	0.988842i	$0.547595\pi$
$882$	0	0
$883$	19.3282	0.650447	0.325224	−	0.945637i	$-0.394560\pi$
$883$	19.3282	0.650447	0.325224	+	0.945637i	$0.394560\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	22.4857	0.754995	0.377498	−	0.926011i	$-0.376785\pi$
$887$	22.4857	0.754995	0.377498	+	0.926011i	$0.376785\pi$
$888$	0	0
$889$	−3.82141	−0.128166
$890$	0	0
$891$	−34.0245	−1.13986
$892$	0	0
$893$	6.34745	0.212409
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−3.72168	−0.124263
$898$	0	0
$899$	−22.6243	−0.754563
$900$	0	0
$901$	16.3225	0.543781
$902$	0	0
$903$	−4.43030	−0.147431
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−28.5934	−0.949429	−0.474714	−	0.880140i	$-0.657449\pi$
$907$	−28.5934	−0.949429	−0.474714	+	0.880140i	$0.657449\pi$
$908$	0	0
$909$	56.4554	1.87251
$910$	0	0
$911$	−42.2774	−1.40071	−0.700356	−	0.713794i	$-0.746975\pi$
$911$	−42.2774	−1.40071	−0.700356	+	0.713794i	$0.746975\pi$
$912$	0	0
$913$	−30.4680	−1.00834
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−4.08168	−0.134789
$918$	0	0
$919$	−36.3783	−1.20001	−0.600004	−	0.799997i	$-0.704834\pi$
$919$	−36.3783	−1.20001	−0.600004	+	0.799997i	$0.704834\pi$
$920$	0	0
$921$	66.9913	2.20744
$922$	0	0
$923$	−9.99319	−0.328930
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−1.29255	−0.0424529
$928$	0	0
$929$	11.0567	0.362757	0.181379	−	0.983413i	$-0.441944\pi$
$929$	11.0567	0.362757	0.181379	+	0.983413i	$0.441944\pi$
$930$	0	0
$931$	14.0957	0.461969
$932$	0	0
$933$	67.1267	2.19763
$934$	0	0
$935$	0	0
$936$	0	0
$937$	44.7081	1.46055	0.730276	−	0.683153i	$-0.239392\pi$
$937$	44.7081	1.46055	0.730276	+	0.683153i	$0.239392\pi$
$938$	0	0
$939$	−32.3880	−1.05694
$940$	0	0
$941$	25.1343	0.819356	0.409678	−	0.912230i	$-0.365641\pi$
$941$	25.1343	0.819356	0.409678	+	0.912230i	$0.365641\pi$
$942$	0	0
$943$	−1.26097	−0.0410628
$944$	0	0
$945$	0	0
$946$	0	0
$947$	50.2426	1.63266	0.816332	−	0.577583i	$-0.196004\pi$
$947$	50.2426	1.63266	0.816332	+	0.577583i	$0.196004\pi$
$948$	0	0
$949$	−4.48666	−0.145643
$950$	0	0
$951$	−35.5218	−1.15187
$952$	0	0
$953$	37.2648	1.20712	0.603562	−	0.797316i	$-0.293747\pi$
$953$	37.2648	1.20712	0.603562	+	0.797316i	$0.293747\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−27.1963	−0.879133
$958$	0	0
$959$	−9.59583	−0.309866
$960$	0	0
$961$	46.2508	1.49196
$962$	0	0
$963$	3.18287	0.102567
$964$	0	0
$965$	0	0
$966$	0	0
$967$	6.29267	0.202359	0.101179	−	0.994868i	$-0.467738\pi$
$967$	6.29267	0.202359	0.101179	+	0.994868i	$0.467738\pi$
$968$	0	0
$969$	−16.7296	−0.537432
$970$	0	0
$971$	11.1451	0.357665	0.178832	−	0.983880i	$-0.442768\pi$
$971$	11.1451	0.357665	0.178832	+	0.983880i	$0.442768\pi$
$972$	0	0
$973$	9.65805	0.309623
$974$	0	0
$975$	0	0
$976$	0	0
$977$	31.1605	0.996912	0.498456	−	0.866915i	$-0.333901\pi$
$977$	31.1605	0.996912	0.498456	+	0.866915i	$0.333901\pi$
$978$	0	0
$979$	26.9214	0.860413
$980$	0	0
$981$	33.3564	1.06499
$982$	0	0
$983$	−34.4322	−1.09822	−0.549108	−	0.835752i	$-0.685032\pi$
$983$	−34.4322	−1.09822	−0.549108	+	0.835752i	$0.685032\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−5.73094	−0.182418
$988$	0	0
$989$	−2.21515	−0.0704377
$990$	0	0
$991$	5.54092	0.176013	0.0880066	−	0.996120i	$-0.471950\pi$
$991$	5.54092	0.176013	0.0880066	+	0.996120i	$0.471950\pi$
$992$	0	0
$993$	76.8918	2.44009
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−12.0230	−0.380773	−0.190387	−	0.981709i	$-0.560974\pi$
$997$	−12.0230	−0.380773	−0.190387	+	0.981709i	$0.560974\pi$
$998$	0	0
$999$	6.78749	0.214747

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2300.2.a.l.1.4	✓	4
4.3	odd	2		9200.2.a.co.1.1		4
5.2	odd	4		2300.2.c.j.1749.2		8
5.3	odd	4		2300.2.c.j.1749.7		8
5.4	even	2		2300.2.a.m.1.1	yes	4
20.19	odd	2		9200.2.a.cm.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2300.2.a.l.1.4	✓	4	1.1	even	1	trivial
2300.2.a.m.1.1	yes	4	5.4	even	2
2300.2.c.j.1749.2		8	5.2	odd	4
2300.2.c.j.1749.7		8	5.3	odd	4
9200.2.a.cm.1.4		4	20.19	odd	2
9200.2.a.co.1.1		4	4.3	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:	\( N \)	\(=\)	\( 2300 = 2^{2} \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2300.a (trivial)

\(f(q)\)	\(=\)	\(q+2.50653 q^{3} +0.797915 q^{7} +3.28271 q^{9} +O(q^{10})\)	\(q+2.50653 q^{3} +0.797915 q^{7} +3.28271 q^{9} +4.21515 q^{11} -1.48479 q^{13} +3.01307 q^{17} -2.21515 q^{19} +2.00000 q^{21} +1.00000 q^{23} +0.708618 q^{27} -2.57409 q^{29} +8.78924 q^{31} +10.5654 q^{33} +9.57848 q^{37} -3.72168 q^{39} -1.26097 q^{41} -2.21515 q^{43} -2.86547 q^{47} -6.36333 q^{49} +7.55235 q^{51} +5.41724 q^{53} -5.55235 q^{57} -1.21515 q^{59} +2.58276 q^{61} +2.61932 q^{63} -8.56542 q^{67} +2.50653 q^{69} +6.73036 q^{71} +3.02174 q^{73} +3.36333 q^{77} +0.189019 q^{79} -8.07195 q^{81} -7.22822 q^{83} -6.45204 q^{87} +6.38682 q^{89} -1.18474 q^{91} +22.0305 q^{93} -13.3999 q^{97} +13.8371 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - q^{7} + 2 q^{9}+O(q^{10}) \) 4 * q - q^7 + 2 * q^9	\( 4 q - q^{7} + 2 q^{9} + q^{11} + q^{13} - 8 q^{17} + 7 q^{19} + 8 q^{21} + 4 q^{23} - 3 q^{27} - 5 q^{29} + 14 q^{31} + 20 q^{33} - 4 q^{37} + 11 q^{39} + 3 q^{41} + 7 q^{43} - 12 q^{47} + q^{49} + 28 q^{51} + 10 q^{53} - 20 q^{57} + 11 q^{59} + 22 q^{61} + 3 q^{63} - 12 q^{67} + 18 q^{71} + 9 q^{73} - 13 q^{77} + 25 q^{79} + 7 q^{83} + 9 q^{87} + 25 q^{91} + 11 q^{93} - 8 q^{97} - 9 q^{99}+O(q^{100}) \) 4 * q - q^7 + 2 * q^9 + q^11 + q^13 - 8 * q^17 + 7 * q^19 + 8 * q^21 + 4 * q^23 - 3 * q^27 - 5 * q^29 + 14 * q^31 + 20 * q^33 - 4 * q^37 + 11 * q^39 + 3 * q^41 + 7 * q^43 - 12 * q^47 + q^49 + 28 * q^51 + 10 * q^53 - 20 * q^57 + 11 * q^59 + 22 * q^61 + 3 * q^63 - 12 * q^67 + 18 * q^71 + 9 * q^73 - 13 * q^77 + 25 * q^79 + 7 * q^83 + 9 * q^87 + 25 * q^91 + 11 * q^93 - 8 * q^97 - 9 * q^99

Introduction

L-functions

Modular forms

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Database

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