LMFDB - Embedded newform 4275.2.a.u.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4275,2,Mod(1,4275)] 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("4275.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4275, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 4275 = 3^{2} \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4275.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,0,10,0,0,2,0,0,0,-6,0,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	yes
Analytic conductor:	$34.1360468641$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{7}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 7 $ x^2 - 7
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 285)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.64575$ of defining polynomial
Character	$\chi$	$=$	4275.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.64575 q^{2} +5.00000 q^{4} +3.64575 q^{7} +7.93725 q^{8} -5.64575 q^{11} +5.64575 q^{13} +9.64575 q^{14} +11.0000 q^{16} -4.00000 q^{17} -1.00000 q^{19} -14.9373 q^{22} -1.29150 q^{23} +14.9373 q^{26} +18.2288 q^{28} +6.93725 q^{29} +6.00000 q^{31} +13.2288 q^{32} -10.5830 q^{34} +1.64575 q^{37} -2.64575 q^{38} +4.35425 q^{41} -0.354249 q^{43} -28.2288 q^{44} -3.41699 q^{46} +9.29150 q^{47} +6.29150 q^{49} +28.2288 q^{52} +0.708497 q^{53} +28.9373 q^{56} +18.3542 q^{58} -0.708497 q^{59} -0.708497 q^{61} +15.8745 q^{62} +13.0000 q^{64} -14.5830 q^{67} -20.0000 q^{68} +3.29150 q^{71} -10.0000 q^{73} +4.35425 q^{74} -5.00000 q^{76} -20.5830 q^{77} -14.5830 q^{79} +11.5203 q^{82} +6.00000 q^{83} -0.937254 q^{86} -44.8118 q^{88} +1.06275 q^{89} +20.5830 q^{91} -6.45751 q^{92} +24.5830 q^{94} -12.9373 q^{97} +16.6458 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 10 q^{4} + 2 q^{7} - 6 q^{11} + 6 q^{13} + 14 q^{14} + 22 q^{16} - 8 q^{17} - 2 q^{19} - 14 q^{22} + 8 q^{23} + 14 q^{26} + 10 q^{28} - 2 q^{29} + 12 q^{31} - 2 q^{37} + 14 q^{41} - 6 q^{43} - 30 q^{44}+ \cdots + 28 q^{98}+O(q^{100}) $ 2 * q + 10 * q^4 + 2 * q^7 - 6 * q^11 + 6 * q^13 + 14 * q^14 + 22 * q^16 - 8 * q^17 - 2 * q^19 - 14 * q^22 + 8 * q^23 + 14 * q^26 + 10 * q^28 - 2 * q^29 + 12 * q^31 - 2 * q^37 + 14 * q^41 - 6 * q^43 - 30 * q^44 - 28 * q^46 + 8 * q^47 + 2 * q^49 + 30 * q^52 + 12 * q^53 + 42 * q^56 + 42 * q^58 - 12 * q^59 - 12 * q^61 + 26 * q^64 - 8 * q^67 - 40 * q^68 - 4 * q^71 - 20 * q^73 + 14 * q^74 - 10 * q^76 - 20 * q^77 - 8 * q^79 - 14 * q^82 + 12 * q^83 + 14 * q^86 - 42 * q^88 + 18 * q^89 + 20 * q^91 + 40 * q^92 + 28 * q^94 - 10 * q^97 + 28 * q^98

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 $

$2$	2.64575	1.87083	0.935414	−	0.353553i	$-0.115027\pi$
$2$	2.64575	1.87083	0.935414	+	0.353553i	$0.115027\pi$
$3$	0	0
$3$	0	0
$4$	5.00000	2.50000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	3.64575	1.37796	0.688982	−	0.724778i	$-0.258058\pi$
$7$	3.64575	1.37796	0.688982	+	0.724778i	$0.258058\pi$
$8$	7.93725	2.80624
$9$	0	0
$10$	0	0
$11$	−5.64575	−1.70226	−0.851129	−	0.524957i	$-0.824082\pi$
$11$	−5.64575	−1.70226	−0.851129	+	0.524957i	$0.824082\pi$
$12$	0	0
$13$	5.64575	1.56585	0.782925	−	0.622116i	$-0.213727\pi$
$13$	5.64575	1.56585	0.782925	+	0.622116i	$0.213727\pi$
$14$	9.64575	2.57794
$15$	0	0
$16$	11.0000	2.75000
$17$	−4.00000	−0.970143	−0.485071	−	0.874475i	$-0.661206\pi$
$17$	−4.00000	−0.970143	−0.485071	+	0.874475i	$0.661206\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	0	0
$22$	−14.9373	−3.18463
$23$	−1.29150	−0.269297	−0.134648	−	0.990893i	$-0.542991\pi$
$23$	−1.29150	−0.269297	−0.134648	+	0.990893i	$0.542991\pi$
$24$	0	0
$25$	0	0
$26$	14.9373	2.92944
$27$	0	0
$28$	18.2288	3.44491
$29$	6.93725	1.28822	0.644108	−	0.764935i	$-0.277229\pi$
$29$	6.93725	1.28822	0.644108	+	0.764935i	$0.277229\pi$
$30$	0	0
$31$	6.00000	1.07763	0.538816	−	0.842424i	$-0.318872\pi$
$31$	6.00000	1.07763	0.538816	+	0.842424i	$0.318872\pi$
$32$	13.2288	2.33854
$33$	0	0
$34$	−10.5830	−1.81497
$35$	0	0
$36$	0	0
$37$	1.64575	0.270560	0.135280	−	0.990807i	$-0.456807\pi$
$37$	1.64575	0.270560	0.135280	+	0.990807i	$0.456807\pi$
$38$	−2.64575	−0.429198
$39$	0	0
$40$	0	0
$41$	4.35425	0.680019	0.340010	−	0.940422i	$-0.389570\pi$
$41$	4.35425	0.680019	0.340010	+	0.940422i	$0.389570\pi$
$42$	0	0
$43$	−0.354249	−0.0540224	−0.0270112	−	0.999635i	$-0.508599\pi$
$43$	−0.354249	−0.0540224	−0.0270112	+	0.999635i	$0.508599\pi$
$44$	−28.2288	−4.25565
$45$	0	0
$46$	−3.41699	−0.503808
$47$	9.29150	1.35530	0.677652	−	0.735382i	$-0.262997\pi$
$47$	9.29150	1.35530	0.677652	+	0.735382i	$0.262997\pi$
$48$	0	0
$49$	6.29150	0.898786
$50$	0	0
$51$	0	0
$52$	28.2288	3.91462
$53$	0.708497	0.0973196	0.0486598	−	0.998815i	$-0.484505\pi$
$53$	0.708497	0.0973196	0.0486598	+	0.998815i	$0.484505\pi$
$54$	0	0
$55$	0	0
$56$	28.9373	3.86690
$57$	0	0
$58$	18.3542	2.41003
$59$	−0.708497	−0.0922385	−0.0461193	−	0.998936i	$-0.514685\pi$
$59$	−0.708497	−0.0922385	−0.0461193	+	0.998936i	$0.514685\pi$
$60$	0	0
$61$	−0.708497	−0.0907138	−0.0453569	−	0.998971i	$-0.514443\pi$
$61$	−0.708497	−0.0907138	−0.0453569	+	0.998971i	$0.514443\pi$
$62$	15.8745	2.01606
$63$	0	0
$64$	13.0000	1.62500
$65$	0	0
$66$	0	0
$67$	−14.5830	−1.78160	−0.890799	−	0.454398i	$-0.849854\pi$
$67$	−14.5830	−1.78160	−0.890799	+	0.454398i	$0.849854\pi$
$68$	−20.0000	−2.42536
$69$	0	0
$70$	0	0
$71$	3.29150	0.390629	0.195315	−	0.980741i	$-0.437427\pi$
$71$	3.29150	0.390629	0.195315	+	0.980741i	$0.437427\pi$
$72$	0	0
$73$	−10.0000	−1.17041	−0.585206	−	0.810885i	$-0.698986\pi$
$73$	−10.0000	−1.17041	−0.585206	+	0.810885i	$0.698986\pi$
$74$	4.35425	0.506171
$75$	0	0
$76$	−5.00000	−0.573539
$77$	−20.5830	−2.34565
$78$	0	0
$79$	−14.5830	−1.64072	−0.820358	−	0.571850i	$-0.806226\pi$
$79$	−14.5830	−1.64072	−0.820358	+	0.571850i	$0.806226\pi$
$80$	0	0
$81$	0	0
$82$	11.5203	1.27220
$83$	6.00000	0.658586	0.329293	−	0.944228i	$-0.393190\pi$
$83$	6.00000	0.658586	0.329293	+	0.944228i	$0.393190\pi$
$84$	0	0
$85$	0	0
$86$	−0.937254	−0.101067
$87$	0	0
$88$	−44.8118	−4.77695
$89$	1.06275	0.112651	0.0563254	−	0.998412i	$-0.482062\pi$
$89$	1.06275	0.112651	0.0563254	+	0.998412i	$0.482062\pi$
$90$	0	0
$91$	20.5830	2.15769
$92$	−6.45751	−0.673242
$93$	0	0
$94$	24.5830	2.53554
$95$	0	0
$96$	0	0
$97$	−12.9373	−1.31358	−0.656790	−	0.754074i	$-0.728086\pi$
$97$	−12.9373	−1.31358	−0.656790	+	0.754074i	$0.728086\pi$
$98$	16.6458	1.68147
$99$	0	0
$100$	0	0
$101$	1.29150	0.128509	0.0642547	−	0.997934i	$-0.479533\pi$
$101$	1.29150	0.128509	0.0642547	+	0.997934i	$0.479533\pi$
$102$	0	0
$103$	−10.5830	−1.04277	−0.521387	−	0.853320i	$-0.674585\pi$
$103$	−10.5830	−1.04277	−0.521387	+	0.853320i	$0.674585\pi$
$104$	44.8118	4.39415
$105$	0	0
$106$	1.87451	0.182068
$107$	0	0		−	1.00000i	$-0.5\pi$
$107$	0	0			1.00000i	$0.5\pi$
$108$	0	0
$109$	−5.29150	−0.506834	−0.253417	−	0.967357i	$-0.581554\pi$
$109$	−5.29150	−0.506834	−0.253417	+	0.967357i	$0.581554\pi$
$110$	0	0
$111$	0	0
$112$	40.1033	3.78940
$113$	−4.00000	−0.376288	−0.188144	−	0.982141i	$-0.560247\pi$
$113$	−4.00000	−0.376288	−0.188144	+	0.982141i	$0.560247\pi$
$114$	0	0
$115$	0	0
$116$	34.6863	3.22054
$117$	0	0
$118$	−1.87451	−0.172562
$119$	−14.5830	−1.33682
$120$	0	0
$121$	20.8745	1.89768
$122$	−1.87451	−0.169710
$123$	0	0
$124$	30.0000	2.69408
$125$	0	0
$126$	0	0
$127$	−4.00000	−0.354943	−0.177471	−	0.984126i	$-0.556792\pi$
$127$	−4.00000	−0.354943	−0.177471	+	0.984126i	$0.556792\pi$
$128$	7.93725	0.701561
$129$	0	0
$130$	0	0
$131$	−12.2288	−1.06843	−0.534216	−	0.845348i	$-0.679393\pi$
$131$	−12.2288	−1.06843	−0.534216	+	0.845348i	$0.679393\pi$
$132$	0	0
$133$	−3.64575	−0.316127
$134$	−38.5830	−3.33306
$135$	0	0
$136$	−31.7490	−2.72246
$137$	6.00000	0.512615	0.256307	−	0.966595i	$-0.417494\pi$
$137$	6.00000	0.512615	0.256307	+	0.966595i	$0.417494\pi$
$138$	0	0
$139$	−13.8745	−1.17682	−0.588410	−	0.808563i	$-0.700246\pi$
$139$	−13.8745	−1.17682	−0.588410	+	0.808563i	$0.700246\pi$
$140$	0	0
$141$	0	0
$142$	8.70850	0.730801
$143$	−31.8745	−2.66548
$144$	0	0
$145$	0	0
$146$	−26.4575	−2.18964
$147$	0	0
$148$	8.22876	0.676400
$149$	−0.583005	−0.0477617	−0.0238808	−	0.999715i	$-0.507602\pi$
$149$	−0.583005	−0.0477617	−0.0238808	+	0.999715i	$0.507602\pi$
$150$	0	0
$151$	12.5830	1.02399	0.511995	−	0.858988i	$-0.328907\pi$
$151$	12.5830	1.02399	0.511995	+	0.858988i	$0.328907\pi$
$152$	−7.93725	−0.643796
$153$	0	0
$154$	−54.4575	−4.38831
$155$	0	0
$156$	0	0
$157$	2.70850	0.216162	0.108081	−	0.994142i	$-0.465529\pi$
$157$	2.70850	0.216162	0.108081	+	0.994142i	$0.465529\pi$
$158$	−38.5830	−3.06950
$159$	0	0
$160$	0	0
$161$	−4.70850	−0.371082
$162$	0	0
$163$	7.64575	0.598861	0.299431	−	0.954118i	$-0.403203\pi$
$163$	7.64575	0.598861	0.299431	+	0.954118i	$0.403203\pi$
$164$	21.7712	1.70005
$165$	0	0
$166$	15.8745	1.23210
$167$	−10.7085	−0.828648	−0.414324	−	0.910129i	$-0.635982\pi$
$167$	−10.7085	−0.828648	−0.414324	+	0.910129i	$0.635982\pi$
$168$	0	0
$169$	18.8745	1.45189
$170$	0	0
$171$	0	0
$172$	−1.77124	−0.135056
$173$	0	0		−	1.00000i	$-0.5\pi$
$173$	0	0			1.00000i	$0.5\pi$
$174$	0	0
$175$	0	0
$176$	−62.1033	−4.68121
$177$	0	0
$178$	2.81176	0.210750
$179$	−3.29150	−0.246018	−0.123009	−	0.992406i	$-0.539254\pi$
$179$	−3.29150	−0.246018	−0.123009	+	0.992406i	$0.539254\pi$
$180$	0	0
$181$	6.70850	0.498639	0.249319	−	0.968421i	$-0.419793\pi$
$181$	6.70850	0.498639	0.249319	+	0.968421i	$0.419793\pi$
$182$	54.4575	4.03666
$183$	0	0
$184$	−10.2510	−0.755713
$185$	0	0
$186$	0	0
$187$	22.5830	1.65143
$188$	46.4575	3.38826
$189$	0	0
$190$	0	0
$191$	20.2288	1.46370	0.731851	−	0.681465i	$-0.238657\pi$
$191$	20.2288	1.46370	0.731851	+	0.681465i	$0.238657\pi$
$192$	0	0
$193$	6.35425	0.457389	0.228694	−	0.973498i	$-0.426554\pi$
$193$	6.35425	0.457389	0.228694	+	0.973498i	$0.426554\pi$
$194$	−34.2288	−2.45748
$195$	0	0
$196$	31.4575	2.24697
$197$	17.1660	1.22303	0.611514	−	0.791234i	$-0.290561\pi$
$197$	17.1660	1.22303	0.611514	+	0.791234i	$0.290561\pi$
$198$	0	0
$199$	−3.29150	−0.233328	−0.116664	−	0.993171i	$-0.537220\pi$
$199$	−3.29150	−0.233328	−0.116664	+	0.993171i	$0.537220\pi$
$200$	0	0
$201$	0	0
$202$	3.41699	0.240419
$203$	25.2915	1.77512
$204$	0	0
$205$	0	0
$206$	−28.0000	−1.95085
$207$	0	0
$208$	62.1033	4.30609
$209$	5.64575	0.390525
$210$	0	0
$211$	−18.5830	−1.27931	−0.639653	−	0.768663i	$-0.720922\pi$
$211$	−18.5830	−1.27931	−0.639653	+	0.768663i	$0.720922\pi$
$212$	3.54249	0.243299
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	21.8745	1.48494
$218$	−14.0000	−0.948200
$219$	0	0
$220$	0	0
$221$	−22.5830	−1.51910
$222$	0	0
$223$	18.5830	1.24441	0.622205	−	0.782854i	$-0.286237\pi$
$223$	18.5830	1.24441	0.622205	+	0.782854i	$0.286237\pi$
$224$	48.2288	3.22242
$225$	0	0
$226$	−10.5830	−0.703971
$227$	−21.2915	−1.41317	−0.706583	−	0.707630i	$-0.749764\pi$
$227$	−21.2915	−1.41317	−0.706583	+	0.707630i	$0.749764\pi$
$228$	0	0
$229$	19.2915	1.27482	0.637409	−	0.770525i	$-0.280006\pi$
$229$	19.2915	1.27482	0.637409	+	0.770525i	$0.280006\pi$
$230$	0	0
$231$	0	0
$232$	55.0627	3.61505
$233$	−18.0000	−1.17922	−0.589610	−	0.807688i	$-0.700718\pi$
$233$	−18.0000	−1.17922	−0.589610	+	0.807688i	$0.700718\pi$
$234$	0	0
$235$	0	0
$236$	−3.54249	−0.230596
$237$	0	0
$238$	−38.5830	−2.50096
$239$	−10.3542	−0.669761	−0.334880	−	0.942261i	$-0.608696\pi$
$239$	−10.3542	−0.669761	−0.334880	+	0.942261i	$0.608696\pi$
$240$	0	0
$241$	−4.58301	−0.295217	−0.147609	−	0.989046i	$-0.547158\pi$
$241$	−4.58301	−0.295217	−0.147609	+	0.989046i	$0.547158\pi$
$242$	55.2288	3.55024
$243$	0	0
$244$	−3.54249	−0.226784
$245$	0	0
$246$	0	0
$247$	−5.64575	−0.359231
$248$	47.6235	3.02410
$249$	0	0
$250$	0	0
$251$	21.6458	1.36627	0.683134	−	0.730293i	$-0.260617\pi$
$251$	21.6458	1.36627	0.683134	+	0.730293i	$0.260617\pi$
$252$	0	0
$253$	7.29150	0.458413
$254$	−10.5830	−0.664037
$255$	0	0
$256$	−5.00000	−0.312500
$257$	−26.5830	−1.65820	−0.829101	−	0.559099i	$-0.811147\pi$
$257$	−26.5830	−1.65820	−0.829101	+	0.559099i	$0.811147\pi$
$258$	0	0
$259$	6.00000	0.372822
$260$	0	0
$261$	0	0
$262$	−32.3542	−1.99885
$263$	−16.5830	−1.02255	−0.511276	−	0.859417i	$-0.670827\pi$
$263$	−16.5830	−1.02255	−0.511276	+	0.859417i	$0.670827\pi$
$264$	0	0
$265$	0	0
$266$	−9.64575	−0.591419
$267$	0	0
$268$	−72.9150	−4.45399
$269$	22.2288	1.35531	0.677656	−	0.735379i	$-0.262996\pi$
$269$	22.2288	1.35531	0.677656	+	0.735379i	$0.262996\pi$
$270$	0	0
$271$	−15.2915	−0.928893	−0.464446	−	0.885601i	$-0.653747\pi$
$271$	−15.2915	−0.928893	−0.464446	+	0.885601i	$0.653747\pi$
$272$	−44.0000	−2.66789
$273$	0	0
$274$	15.8745	0.959014
$275$	0	0
$276$	0	0
$277$	20.5830	1.23671	0.618356	−	0.785898i	$-0.287799\pi$
$277$	20.5830	1.23671	0.618356	+	0.785898i	$0.287799\pi$
$278$	−36.7085	−2.20163
$279$	0	0
$280$	0	0
$281$	5.77124	0.344284	0.172142	−	0.985072i	$-0.444931\pi$
$281$	5.77124	0.344284	0.172142	+	0.985072i	$0.444931\pi$
$282$	0	0
$283$	−25.5203	−1.51702	−0.758511	−	0.651660i	$-0.774073\pi$
$283$	−25.5203	−1.51702	−0.758511	+	0.651660i	$0.774073\pi$
$284$	16.4575	0.976574
$285$	0	0
$286$	−84.3320	−4.98666
$287$	15.8745	0.937043
$288$	0	0
$289$	−1.00000	−0.0588235
$290$	0	0
$291$	0	0
$292$	−50.0000	−2.92603
$293$	26.5830	1.55300	0.776498	−	0.630120i	$-0.216994\pi$
$293$	26.5830	1.55300	0.776498	+	0.630120i	$0.216994\pi$
$294$	0	0
$295$	0	0
$296$	13.0627	0.759257
$297$	0	0
$298$	−1.54249	−0.0893539
$299$	−7.29150	−0.421678
$300$	0	0
$301$	−1.29150	−0.0744410
$302$	33.2915	1.91571
$303$	0	0
$304$	−11.0000	−0.630893
$305$	0	0
$306$	0	0
$307$	−28.4575	−1.62416	−0.812078	−	0.583549i	$-0.801664\pi$
$307$	−28.4575	−1.62416	−0.812078	+	0.583549i	$0.801664\pi$
$308$	−102.915	−5.86413
$309$	0	0
$310$	0	0
$311$	−7.06275	−0.400492	−0.200246	−	0.979746i	$-0.564174\pi$
$311$	−7.06275	−0.400492	−0.200246	+	0.979746i	$0.564174\pi$
$312$	0	0
$313$	−6.70850	−0.379187	−0.189593	−	0.981863i	$-0.560717\pi$
$313$	−6.70850	−0.379187	−0.189593	+	0.981863i	$0.560717\pi$
$314$	7.16601	0.404401
$315$	0	0
$316$	−72.9150	−4.10179
$317$	−32.4575	−1.82300	−0.911498	−	0.411305i	$-0.865073\pi$
$317$	−32.4575	−1.82300	−0.911498	+	0.411305i	$0.865073\pi$
$318$	0	0
$319$	−39.1660	−2.19288
$320$	0	0
$321$	0	0
$322$	−12.4575	−0.694230
$323$	4.00000	0.222566
$324$	0	0
$325$	0	0
$326$	20.2288	1.12037
$327$	0	0
$328$	34.5608	1.90830
$329$	33.8745	1.86756
$330$	0	0
$331$	32.5830	1.79092	0.895462	−	0.445138i	$-0.146845\pi$
$331$	32.5830	1.79092	0.895462	+	0.445138i	$0.146845\pi$
$332$	30.0000	1.64646
$333$	0	0
$334$	−28.3320	−1.55026
$335$	0	0
$336$	0	0
$337$	0.937254	0.0510555	0.0255277	−	0.999674i	$-0.491873\pi$
$337$	0.937254	0.0510555	0.0255277	+	0.999674i	$0.491873\pi$
$338$	49.9373	2.71623
$339$	0	0
$340$	0	0
$341$	−33.8745	−1.83441
$342$	0	0
$343$	−2.58301	−0.139469
$344$	−2.81176	−0.151600
$345$	0	0
$346$	0	0
$347$	26.0000	1.39575	0.697877	−	0.716218i	$-0.254128\pi$
$347$	26.0000	1.39575	0.697877	+	0.716218i	$0.254128\pi$
$348$	0	0
$349$	−19.1660	−1.02593	−0.512967	−	0.858409i	$-0.671454\pi$
$349$	−19.1660	−1.02593	−0.512967	+	0.858409i	$0.671454\pi$
$350$	0	0
$351$	0	0
$352$	−74.6863	−3.98079
$353$	−20.5830	−1.09552	−0.547761	−	0.836635i	$-0.684520\pi$
$353$	−20.5830	−1.09552	−0.547761	+	0.836635i	$0.684520\pi$
$354$	0	0
$355$	0	0
$356$	5.31373	0.281627
$357$	0	0
$358$	−8.70850	−0.460258
$359$	−22.1033	−1.16657	−0.583283	−	0.812269i	$-0.698232\pi$
$359$	−22.1033	−1.16657	−0.583283	+	0.812269i	$0.698232\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	17.7490	0.932868
$363$	0	0
$364$	102.915	5.39421
$365$	0	0
$366$	0	0
$367$	3.64575	0.190307	0.0951533	−	0.995463i	$-0.469666\pi$
$367$	3.64575	0.190307	0.0951533	+	0.995463i	$0.469666\pi$
$368$	−14.2065	−0.740567
$369$	0	0
$370$	0	0
$371$	2.58301	0.134103
$372$	0	0
$373$	−20.9373	−1.08409	−0.542045	−	0.840349i	$-0.682350\pi$
$373$	−20.9373	−1.08409	−0.542045	+	0.840349i	$0.682350\pi$
$374$	59.7490	3.08955
$375$	0	0
$376$	73.7490	3.80332
$377$	39.1660	2.01715
$378$	0	0
$379$	10.0000	0.513665	0.256833	−	0.966456i	$-0.417321\pi$
$379$	10.0000	0.513665	0.256833	+	0.966456i	$0.417321\pi$
$380$	0	0
$381$	0	0
$382$	53.5203	2.73833
$383$	2.58301	0.131985	0.0659927	−	0.997820i	$-0.478979\pi$
$383$	2.58301	0.131985	0.0659927	+	0.997820i	$0.478979\pi$
$384$	0	0
$385$	0	0
$386$	16.8118	0.855696
$387$	0	0
$388$	−64.6863	−3.28395
$389$	6.00000	0.304212	0.152106	−	0.988364i	$-0.451394\pi$
$389$	6.00000	0.304212	0.152106	+	0.988364i	$0.451394\pi$
$390$	0	0
$391$	5.16601	0.261256
$392$	49.9373	2.52221
$393$	0	0
$394$	45.4170	2.28808
$395$	0	0
$396$	0	0
$397$	2.00000	0.100377	0.0501886	−	0.998740i	$-0.484018\pi$
$397$	2.00000	0.100377	0.0501886	+	0.998740i	$0.484018\pi$
$398$	−8.70850	−0.436518
$399$	0	0
$400$	0	0
$401$	−10.9373	−0.546180	−0.273090	−	0.961988i	$-0.588046\pi$
$401$	−10.9373	−0.546180	−0.273090	+	0.961988i	$0.588046\pi$
$402$	0	0
$403$	33.8745	1.68741
$404$	6.45751	0.321273
$405$	0	0
$406$	66.9150	3.32094
$407$	−9.29150	−0.460563
$408$	0	0
$409$	−17.2915	−0.855010	−0.427505	−	0.904013i	$-0.640607\pi$
$409$	−17.2915	−0.855010	−0.427505	+	0.904013i	$0.640607\pi$
$410$	0	0
$411$	0	0
$412$	−52.9150	−2.60694
$413$	−2.58301	−0.127101
$414$	0	0
$415$	0	0
$416$	74.6863	3.66180
$417$	0	0
$418$	14.9373	0.730605
$419$	−23.0627	−1.12669	−0.563344	−	0.826222i	$-0.690486\pi$
$419$	−23.0627	−1.12669	−0.563344	+	0.826222i	$0.690486\pi$
$420$	0	0
$421$	7.41699	0.361482	0.180741	−	0.983531i	$-0.442150\pi$
$421$	7.41699	0.361482	0.180741	+	0.983531i	$0.442150\pi$
$422$	−49.1660	−2.39336
$423$	0	0
$424$	5.62352	0.273102
$425$	0	0
$426$	0	0
$427$	−2.58301	−0.125000
$428$	0	0
$429$	0	0
$430$	0	0
$431$	7.29150	0.351219	0.175610	−	0.984460i	$-0.443810\pi$
$431$	7.29150	0.351219	0.175610	+	0.984460i	$0.443810\pi$
$432$	0	0
$433$	22.3542	1.07428	0.537138	−	0.843494i	$-0.319505\pi$
$433$	22.3542	1.07428	0.537138	+	0.843494i	$0.319505\pi$
$434$	57.8745	2.77807
$435$	0	0
$436$	−26.4575	−1.26709
$437$	1.29150	0.0617809
$438$	0	0
$439$	−22.5830	−1.07783	−0.538914	−	0.842361i	$-0.681165\pi$
$439$	−22.5830	−1.07783	−0.538914	+	0.842361i	$0.681165\pi$
$440$	0	0
$441$	0	0
$442$	−59.7490	−2.84197
$443$	37.2915	1.77177	0.885886	−	0.463902i	$-0.153551\pi$
$443$	37.2915	1.77177	0.885886	+	0.463902i	$0.153551\pi$
$444$	0	0
$445$	0	0
$446$	49.1660	2.32808
$447$	0	0
$448$	47.3948	2.23919
$449$	9.77124	0.461133	0.230567	−	0.973057i	$-0.425942\pi$
$449$	9.77124	0.461133	0.230567	+	0.973057i	$0.425942\pi$
$450$	0	0
$451$	−24.5830	−1.15757
$452$	−20.0000	−0.940721
$453$	0	0
$454$	−56.3320	−2.64379
$455$	0	0
$456$	0	0
$457$	−31.8745	−1.49103	−0.745513	−	0.666491i	$-0.767796\pi$
$457$	−31.8745	−1.49103	−0.745513	+	0.666491i	$0.767796\pi$
$458$	51.0405	2.38497
$459$	0	0
$460$	0	0
$461$	−25.7490	−1.19925	−0.599626	−	0.800281i	$-0.704684\pi$
$461$	−25.7490	−1.19925	−0.599626	+	0.800281i	$0.704684\pi$
$462$	0	0
$463$	−1.52026	−0.0706524	−0.0353262	−	0.999376i	$-0.511247\pi$
$463$	−1.52026	−0.0706524	−0.0353262	+	0.999376i	$0.511247\pi$
$464$	76.3098	3.54259
$465$	0	0
$466$	−47.6235	−2.20612
$467$	−11.8745	−0.549487	−0.274743	−	0.961518i	$-0.588593\pi$
$467$	−11.8745	−0.549487	−0.274743	+	0.961518i	$0.588593\pi$
$468$	0	0
$469$	−53.1660	−2.45498
$470$	0	0
$471$	0	0
$472$	−5.62352	−0.258844
$473$	2.00000	0.0919601
$474$	0	0
$475$	0	0
$476$	−72.9150	−3.34205
$477$	0	0
$478$	−27.3948	−1.25301
$479$	−9.64575	−0.440726	−0.220363	−	0.975418i	$-0.570724\pi$
$479$	−9.64575	−0.440726	−0.220363	+	0.975418i	$0.570724\pi$
$480$	0	0
$481$	9.29150	0.423656
$482$	−12.1255	−0.552301
$483$	0	0
$484$	104.373	4.74421
$485$	0	0
$486$	0	0
$487$	13.8745	0.628714	0.314357	−	0.949305i	$-0.398211\pi$
$487$	13.8745	0.628714	0.314357	+	0.949305i	$0.398211\pi$
$488$	−5.62352	−0.254565
$489$	0	0
$490$	0	0
$491$	32.2288	1.45446	0.727232	−	0.686392i	$-0.240807\pi$
$491$	32.2288	1.45446	0.727232	+	0.686392i	$0.240807\pi$
$492$	0	0
$493$	−27.7490	−1.24975
$494$	−14.9373	−0.672059
$495$	0	0
$496$	66.0000	2.96349
$497$	12.0000	0.538274
$498$	0	0
$499$	43.0405	1.92676	0.963379	−	0.268143i	$-0.0864100\pi$
$499$	43.0405	1.92676	0.963379	+	0.268143i	$0.0864100\pi$
$500$	0	0
$501$	0	0
$502$	57.2693	2.55605
$503$	−2.00000	−0.0891756	−0.0445878	−	0.999005i	$-0.514197\pi$
$503$	−2.00000	−0.0891756	−0.0445878	+	0.999005i	$0.514197\pi$
$504$	0	0
$505$	0	0
$506$	19.2915	0.857612
$507$	0	0
$508$	−20.0000	−0.887357
$509$	−24.1033	−1.06836	−0.534179	−	0.845371i	$-0.679379\pi$
$509$	−24.1033	−1.06836	−0.534179	+	0.845371i	$0.679379\pi$
$510$	0	0
$511$	−36.4575	−1.61279
$512$	−29.1033	−1.28619
$513$	0	0
$514$	−70.3320	−3.10221
$515$	0	0
$516$	0	0
$517$	−52.4575	−2.30708
$518$	15.8745	0.697486
$519$	0	0
$520$	0	0
$521$	36.1033	1.58171	0.790856	−	0.612002i	$-0.209635\pi$
$521$	36.1033	1.58171	0.790856	+	0.612002i	$0.209635\pi$
$522$	0	0
$523$	12.7085	0.555704	0.277852	−	0.960624i	$-0.410378\pi$
$523$	12.7085	0.555704	0.277852	+	0.960624i	$0.410378\pi$
$524$	−61.1438	−2.67108
$525$	0	0
$526$	−43.8745	−1.91302
$527$	−24.0000	−1.04546
$528$	0	0
$529$	−21.3320	−0.927479
$530$	0	0
$531$	0	0
$532$	−18.2288	−0.790317
$533$	24.5830	1.06481
$534$	0	0
$535$	0	0
$536$	−115.749	−4.99960
$537$	0	0
$538$	58.8118	2.53556
$539$	−35.5203	−1.52997
$540$	0	0
$541$	30.0000	1.28980	0.644900	−	0.764267i	$-0.276899\pi$
$541$	30.0000	1.28980	0.644900	+	0.764267i	$0.276899\pi$
$542$	−40.4575	−1.73780
$543$	0	0
$544$	−52.9150	−2.26871
$545$	0	0
$546$	0	0
$547$	29.8745	1.27734	0.638671	−	0.769480i	$-0.279485\pi$
$547$	29.8745	1.27734	0.638671	+	0.769480i	$0.279485\pi$
$548$	30.0000	1.28154
$549$	0	0
$550$	0	0
$551$	−6.93725	−0.295537
$552$	0	0
$553$	−53.1660	−2.26085
$554$	54.4575	2.31368
$555$	0	0
$556$	−69.3725	−2.94205
$557$	−32.5830	−1.38059	−0.690293	−	0.723530i	$-0.742518\pi$
$557$	−32.5830	−1.38059	−0.690293	+	0.723530i	$0.742518\pi$
$558$	0	0
$559$	−2.00000	−0.0845910
$560$	0	0
$561$	0	0
$562$	15.2693	0.644095
$563$	39.8745	1.68051	0.840255	−	0.542191i	$-0.182405\pi$
$563$	39.8745	1.68051	0.840255	+	0.542191i	$0.182405\pi$
$564$	0	0
$565$	0	0
$566$	−67.5203	−2.83809
$567$	0	0
$568$	26.1255	1.09620
$569$	22.9373	0.961580	0.480790	−	0.876836i	$-0.340350\pi$
$569$	22.9373	0.961580	0.480790	+	0.876836i	$0.340350\pi$
$570$	0	0
$571$	27.2915	1.14211	0.571057	−	0.820910i	$-0.306534\pi$
$571$	27.2915	1.14211	0.571057	+	0.820910i	$0.306534\pi$
$572$	−159.373	−6.66370
$573$	0	0
$574$	42.0000	1.75305
$575$	0	0
$576$	0	0
$577$	2.70850	0.112756	0.0563781	−	0.998409i	$-0.482045\pi$
$577$	2.70850	0.112756	0.0563781	+	0.998409i	$0.482045\pi$
$578$	−2.64575	−0.110049
$579$	0	0
$580$	0	0
$581$	21.8745	0.907508
$582$	0	0
$583$	−4.00000	−0.165663
$584$	−79.3725	−3.28446
$585$	0	0
$586$	70.3320	2.90539
$587$	−22.7085	−0.937280	−0.468640	−	0.883389i	$-0.655256\pi$
$587$	−22.7085	−0.937280	−0.468640	+	0.883389i	$0.655256\pi$
$588$	0	0
$589$	−6.00000	−0.247226
$590$	0	0
$591$	0	0
$592$	18.1033	0.744040
$593$	24.5830	1.00950	0.504752	−	0.863265i	$-0.331584\pi$
$593$	24.5830	1.00950	0.504752	+	0.863265i	$0.331584\pi$
$594$	0	0
$595$	0	0
$596$	−2.91503	−0.119404
$597$	0	0
$598$	−19.2915	−0.788888
$599$	−30.5830	−1.24959	−0.624794	−	0.780790i	$-0.714817\pi$
$599$	−30.5830	−1.24959	−0.624794	+	0.780790i	$0.714817\pi$
$600$	0	0
$601$	33.2915	1.35799	0.678994	−	0.734143i	$-0.262416\pi$
$601$	33.2915	1.35799	0.678994	+	0.734143i	$0.262416\pi$
$602$	−3.41699	−0.139266
$603$	0	0
$604$	62.9150	2.55998
$605$	0	0
$606$	0	0
$607$	−31.0405	−1.25990	−0.629948	−	0.776637i	$-0.716924\pi$
$607$	−31.0405	−1.25990	−0.629948	+	0.776637i	$0.716924\pi$
$608$	−13.2288	−0.536497
$609$	0	0
$610$	0	0
$611$	52.4575	2.12220
$612$	0	0
$613$	−22.4575	−0.907050	−0.453525	−	0.891243i	$-0.649834\pi$
$613$	−22.4575	−0.907050	−0.453525	+	0.891243i	$0.649834\pi$
$614$	−75.2915	−3.03852
$615$	0	0
$616$	−163.373	−6.58247
$617$	12.0000	0.483102	0.241551	−	0.970388i	$-0.422344\pi$
$617$	12.0000	0.483102	0.241551	+	0.970388i	$0.422344\pi$
$618$	0	0
$619$	12.7085	0.510798	0.255399	−	0.966836i	$-0.417793\pi$
$619$	12.7085	0.510798	0.255399	+	0.966836i	$0.417793\pi$
$620$	0	0
$621$	0	0
$622$	−18.6863	−0.749251
$623$	3.87451	0.155229
$624$	0	0
$625$	0	0
$626$	−17.7490	−0.709393
$627$	0	0
$628$	13.5425	0.540404
$629$	−6.58301	−0.262482
$630$	0	0
$631$	6.58301	0.262065	0.131033	−	0.991378i	$-0.458171\pi$
$631$	6.58301	0.262065	0.131033	+	0.991378i	$0.458171\pi$
$632$	−115.749	−4.60425
$633$	0	0
$634$	−85.8745	−3.41051
$635$	0	0
$636$	0	0
$637$	35.5203	1.40736
$638$	−103.624	−4.10249
$639$	0	0
$640$	0	0
$641$	1.06275	0.0419759	0.0209880	−	0.999780i	$-0.493319\pi$
$641$	1.06275	0.0419759	0.0209880	+	0.999780i	$0.493319\pi$
$642$	0	0
$643$	8.35425	0.329459	0.164730	−	0.986339i	$-0.447325\pi$
$643$	8.35425	0.329459	0.164730	+	0.986339i	$0.447325\pi$
$644$	−23.5425	−0.927704
$645$	0	0
$646$	10.5830	0.416383
$647$	−24.5830	−0.966458	−0.483229	−	0.875494i	$-0.660536\pi$
$647$	−24.5830	−0.966458	−0.483229	+	0.875494i	$0.660536\pi$
$648$	0	0
$649$	4.00000	0.157014
$650$	0	0
$651$	0	0
$652$	38.2288	1.49715
$653$	30.5830	1.19681	0.598403	−	0.801195i	$-0.295802\pi$
$653$	30.5830	1.19681	0.598403	+	0.801195i	$0.295802\pi$
$654$	0	0
$655$	0	0
$656$	47.8967	1.87005
$657$	0	0
$658$	89.6235	3.49389
$659$	−46.5830	−1.81462	−0.907308	−	0.420466i	$-0.861866\pi$
$659$	−46.5830	−1.81462	−0.907308	+	0.420466i	$0.861866\pi$
$660$	0	0
$661$	−9.29150	−0.361398	−0.180699	−	0.983538i	$-0.557836\pi$
$661$	−9.29150	−0.361398	−0.180699	+	0.983538i	$0.557836\pi$
$662$	86.2065	3.35051
$663$	0	0
$664$	47.6235	1.84815
$665$	0	0
$666$	0	0
$667$	−8.95948	−0.346913
$668$	−53.5425	−2.07162
$669$	0	0
$670$	0	0
$671$	4.00000	0.154418
$672$	0	0
$673$	28.9373	1.11545	0.557725	−	0.830026i	$-0.311675\pi$
$673$	28.9373	1.11545	0.557725	+	0.830026i	$0.311675\pi$
$674$	2.47974	0.0955160
$675$	0	0
$676$	94.3725	3.62971
$677$	−12.4575	−0.478781	−0.239391	−	0.970923i	$-0.576948\pi$
$677$	−12.4575	−0.478781	−0.239391	+	0.970923i	$0.576948\pi$
$678$	0	0
$679$	−47.1660	−1.81007
$680$	0	0
$681$	0	0
$682$	−89.6235	−3.43186
$683$	−43.7490	−1.67401	−0.837005	−	0.547196i	$-0.815695\pi$
$683$	−43.7490	−1.67401	−0.837005	+	0.547196i	$0.815695\pi$
$684$	0	0
$685$	0	0
$686$	−6.83399	−0.260923
$687$	0	0
$688$	−3.89674	−0.148562
$689$	4.00000	0.152388
$690$	0	0
$691$	31.0405	1.18084	0.590418	−	0.807097i	$-0.298963\pi$
$691$	31.0405	1.18084	0.590418	+	0.807097i	$0.298963\pi$
$692$	0	0
$693$	0	0
$694$	68.7895	2.61122
$695$	0	0
$696$	0	0
$697$	−17.4170	−0.659716
$698$	−50.7085	−1.91934
$699$	0	0
$700$	0	0
$701$	4.58301	0.173098	0.0865489	−	0.996248i	$-0.472416\pi$
$701$	4.58301	0.173098	0.0865489	+	0.996248i	$0.472416\pi$
$702$	0	0
$703$	−1.64575	−0.0620707
$704$	−73.3948	−2.76617
$705$	0	0
$706$	−54.4575	−2.04954
$707$	4.70850	0.177081
$708$	0	0
$709$	−15.2915	−0.574284	−0.287142	−	0.957888i	$-0.592705\pi$
$709$	−15.2915	−0.574284	−0.287142	+	0.957888i	$0.592705\pi$
$710$	0	0
$711$	0	0
$712$	8.43529	0.316126
$713$	−7.74902	−0.290203
$714$	0	0
$715$	0	0
$716$	−16.4575	−0.615046
$717$	0	0
$718$	−58.4797	−2.18244
$719$	14.8118	0.552386	0.276193	−	0.961102i	$-0.410927\pi$
$719$	14.8118	0.552386	0.276193	+	0.961102i	$0.410927\pi$
$720$	0	0
$721$	−38.5830	−1.43691
$722$	2.64575	0.0984647
$723$	0	0
$724$	33.5425	1.24660
$725$	0	0
$726$	0	0
$727$	44.1033	1.63570	0.817850	−	0.575432i	$-0.195166\pi$
$727$	44.1033	1.63570	0.817850	+	0.575432i	$0.195166\pi$
$728$	163.373	6.05499
$729$	0	0
$730$	0	0
$731$	1.41699	0.0524094
$732$	0	0
$733$	−28.5830	−1.05574	−0.527869	−	0.849326i	$-0.677009\pi$
$733$	−28.5830	−1.05574	−0.527869	+	0.849326i	$0.677009\pi$
$734$	9.64575	0.356031
$735$	0	0
$736$	−17.0850	−0.629760
$737$	82.3320	3.03274
$738$	0	0
$739$	−20.0000	−0.735712	−0.367856	−	0.929883i	$-0.619908\pi$
$739$	−20.0000	−0.735712	−0.367856	+	0.929883i	$0.619908\pi$
$740$	0	0
$741$	0	0
$742$	6.83399	0.250884
$743$	10.5830	0.388253	0.194126	−	0.980977i	$-0.437813\pi$
$743$	10.5830	0.388253	0.194126	+	0.980977i	$0.437813\pi$
$744$	0	0
$745$	0	0
$746$	−55.3948	−2.02815
$747$	0	0
$748$	112.915	4.12858
$749$	0	0
$750$	0	0
$751$	−23.4170	−0.854498	−0.427249	−	0.904134i	$-0.640517\pi$
$751$	−23.4170	−0.854498	−0.427249	+	0.904134i	$0.640517\pi$
$752$	102.207	3.72709
$753$	0	0
$754$	103.624	3.77375
$755$	0	0
$756$	0	0
$757$	−7.87451	−0.286204	−0.143102	−	0.989708i	$-0.545708\pi$
$757$	−7.87451	−0.286204	−0.143102	+	0.989708i	$0.545708\pi$
$758$	26.4575	0.960980
$759$	0	0
$760$	0	0
$761$	37.7490	1.36840	0.684200	−	0.729294i	$-0.260151\pi$
$761$	37.7490	1.36840	0.684200	+	0.729294i	$0.260151\pi$
$762$	0	0
$763$	−19.2915	−0.698399
$764$	101.144	3.65925
$765$	0	0
$766$	6.83399	0.246922
$767$	−4.00000	−0.144432
$768$	0	0
$769$	45.7490	1.64975	0.824876	−	0.565314i	$-0.191245\pi$
$769$	45.7490	1.64975	0.824876	+	0.565314i	$0.191245\pi$
$770$	0	0
$771$	0	0
$772$	31.7712	1.14347
$773$	−19.2915	−0.693867	−0.346934	−	0.937890i	$-0.612777\pi$
$773$	−19.2915	−0.693867	−0.346934	+	0.937890i	$0.612777\pi$
$774$	0	0
$775$	0	0
$776$	−102.686	−3.68622
$777$	0	0
$778$	15.8745	0.569129
$779$	−4.35425	−0.156007
$780$	0	0
$781$	−18.5830	−0.664952
$782$	13.6680	0.488766
$783$	0	0
$784$	69.2065	2.47166
$785$	0	0
$786$	0	0
$787$	−6.12549	−0.218350	−0.109175	−	0.994023i	$-0.534821\pi$
$787$	−6.12549	−0.218350	−0.109175	+	0.994023i	$0.534821\pi$
$788$	85.8301	3.05757
$789$	0	0
$790$	0	0
$791$	−14.5830	−0.518512
$792$	0	0
$793$	−4.00000	−0.142044
$794$	5.29150	0.187788
$795$	0	0
$796$	−16.4575	−0.583321
$797$	40.0000	1.41687	0.708436	−	0.705775i	$-0.249401\pi$
$797$	40.0000	1.41687	0.708436	+	0.705775i	$0.249401\pi$
$798$	0	0
$799$	−37.1660	−1.31484
$800$	0	0
$801$	0	0
$802$	−28.9373	−1.02181
$803$	56.4575	1.99234
$804$	0	0
$805$	0	0
$806$	89.6235	3.15685
$807$	0	0
$808$	10.2510	0.360628
$809$	40.5830	1.42682	0.713411	−	0.700746i	$-0.247149\pi$
$809$	40.5830	1.42682	0.713411	+	0.700746i	$0.247149\pi$
$810$	0	0
$811$	−46.3320	−1.62694	−0.813469	−	0.581609i	$-0.802423\pi$
$811$	−46.3320	−1.62694	−0.813469	+	0.581609i	$0.802423\pi$
$812$	126.458	4.43779
$813$	0	0
$814$	−24.5830	−0.861634
$815$	0	0
$816$	0	0
$817$	0.354249	0.0123936
$818$	−45.7490	−1.59958
$819$	0	0
$820$	0	0
$821$	47.6235	1.66207	0.831036	−	0.556218i	$-0.187748\pi$
$821$	47.6235	1.66207	0.831036	+	0.556218i	$0.187748\pi$
$822$	0	0
$823$	22.2288	0.774846	0.387423	−	0.921902i	$-0.373365\pi$
$823$	22.2288	0.774846	0.387423	+	0.921902i	$0.373365\pi$
$824$	−84.0000	−2.92628
$825$	0	0
$826$	−6.83399	−0.237785
$827$	−22.4575	−0.780924	−0.390462	−	0.920619i	$-0.627685\pi$
$827$	−22.4575	−0.780924	−0.390462	+	0.920619i	$0.627685\pi$
$828$	0	0
$829$	13.2915	0.461633	0.230816	−	0.972997i	$-0.425860\pi$
$829$	13.2915	0.461633	0.230816	+	0.972997i	$0.425860\pi$
$830$	0	0
$831$	0	0
$832$	73.3948	2.54451
$833$	−25.1660	−0.871951
$834$	0	0
$835$	0	0
$836$	28.2288	0.976312
$837$	0	0
$838$	−61.0183	−2.10784
$839$	−40.4575	−1.39675	−0.698374	−	0.715733i	$-0.746093\pi$
$839$	−40.4575	−1.39675	−0.698374	+	0.715733i	$0.746093\pi$
$840$	0	0
$841$	19.1255	0.659500
$842$	19.6235	0.676271
$843$	0	0
$844$	−92.9150	−3.19827
$845$	0	0
$846$	0	0
$847$	76.1033	2.61494
$848$	7.79347	0.267629
$849$	0	0
$850$	0	0
$851$	−2.12549	−0.0728609
$852$	0	0
$853$	−25.2915	−0.865965	−0.432982	−	0.901402i	$-0.642539\pi$
$853$	−25.2915	−0.865965	−0.432982	+	0.901402i	$0.642539\pi$
$854$	−6.83399	−0.233854
$855$	0	0
$856$	0	0
$857$	−43.0405	−1.47024	−0.735118	−	0.677939i	$-0.762873\pi$
$857$	−43.0405	−1.47024	−0.735118	+	0.677939i	$0.762873\pi$
$858$	0	0
$859$	50.3320	1.71731	0.858653	−	0.512557i	$-0.171302\pi$
$859$	50.3320	1.71731	0.858653	+	0.512557i	$0.171302\pi$
$860$	0	0
$861$	0	0
$862$	19.2915	0.657071
$863$	−20.0000	−0.680808	−0.340404	−	0.940279i	$-0.610564\pi$
$863$	−20.0000	−0.680808	−0.340404	+	0.940279i	$0.610564\pi$
$864$	0	0
$865$	0	0
$866$	59.1438	2.00979
$867$	0	0
$868$	109.373	3.71235
$869$	82.3320	2.79292
$870$	0	0
$871$	−82.3320	−2.78971
$872$	−42.0000	−1.42230
$873$	0	0
$874$	3.41699	0.115582
$875$	0	0
$876$	0	0
$877$	40.9373	1.38235	0.691176	−	0.722686i	$-0.257093\pi$
$877$	40.9373	1.38235	0.691176	+	0.722686i	$0.257093\pi$
$878$	−59.7490	−2.01643
$879$	0	0
$880$	0	0
$881$	31.8745	1.07388	0.536940	−	0.843621i	$-0.319580\pi$
$881$	31.8745	1.07388	0.536940	+	0.843621i	$0.319580\pi$
$882$	0	0
$883$	36.8118	1.23881	0.619407	−	0.785070i	$-0.287373\pi$
$883$	36.8118	1.23881	0.619407	+	0.785070i	$0.287373\pi$
$884$	−112.915	−3.79774
$885$	0	0
$886$	98.6640	3.31468
$887$	−24.0000	−0.805841	−0.402921	−	0.915235i	$-0.632005\pi$
$887$	−24.0000	−0.805841	−0.402921	+	0.915235i	$0.632005\pi$
$888$	0	0
$889$	−14.5830	−0.489098
$890$	0	0
$891$	0	0
$892$	92.9150	3.11103
$893$	−9.29150	−0.310928
$894$	0	0
$895$	0	0
$896$	28.9373	0.966726
$897$	0	0
$898$	25.8523	0.862702
$899$	41.6235	1.38822
$900$	0	0
$901$	−2.83399	−0.0944139
$902$	−65.0405	−2.16561
$903$	0	0
$904$	−31.7490	−1.05596
$905$	0	0
$906$	0	0
$907$	−27.0405	−0.897866	−0.448933	−	0.893566i	$-0.648196\pi$
$907$	−27.0405	−0.897866	−0.448933	+	0.893566i	$0.648196\pi$
$908$	−106.458	−3.53292
$909$	0	0
$910$	0	0
$911$	−5.41699	−0.179473	−0.0897365	−	0.995966i	$-0.528602\pi$
$911$	−5.41699	−0.179473	−0.0897365	+	0.995966i	$0.528602\pi$
$912$	0	0
$913$	−33.8745	−1.12108
$914$	−84.3320	−2.78946
$915$	0	0
$916$	96.4575	3.18705
$917$	−44.5830	−1.47226
$918$	0	0
$919$	57.1660	1.88573	0.942866	−	0.333171i	$-0.108119\pi$
$919$	57.1660	1.88573	0.942866	+	0.333171i	$0.108119\pi$
$920$	0	0
$921$	0	0
$922$	−68.1255	−2.24359
$923$	18.5830	0.611667
$924$	0	0
$925$	0	0
$926$	−4.02223	−0.132179
$927$	0	0
$928$	91.7712	3.01254
$929$	−19.8745	−0.652061	−0.326031	−	0.945359i	$-0.605711\pi$
$929$	−19.8745	−0.652061	−0.326031	+	0.945359i	$0.605711\pi$
$930$	0	0
$931$	−6.29150	−0.206196
$932$	−90.0000	−2.94805
$933$	0	0
$934$	−31.4170	−1.02800
$935$	0	0
$936$	0	0
$937$	−51.8745	−1.69467	−0.847333	−	0.531062i	$-0.821793\pi$
$937$	−51.8745	−1.69467	−0.847333	+	0.531062i	$0.821793\pi$
$938$	−140.664	−4.59284
$939$	0	0
$940$	0	0
$941$	38.2288	1.24622	0.623111	−	0.782133i	$-0.285869\pi$
$941$	38.2288	1.24622	0.623111	+	0.782133i	$0.285869\pi$
$942$	0	0
$943$	−5.62352	−0.183127
$944$	−7.79347	−0.253656
$945$	0	0
$946$	5.29150	0.172042
$947$	32.5830	1.05881	0.529403	−	0.848371i	$-0.322416\pi$
$947$	32.5830	1.05881	0.529403	+	0.848371i	$0.322416\pi$
$948$	0	0
$949$	−56.4575	−1.83269
$950$	0	0
$951$	0	0
$952$	−115.749	−3.75145
$953$	10.5830	0.342817	0.171409	−	0.985200i	$-0.445168\pi$
$953$	10.5830	0.342817	0.171409	+	0.985200i	$0.445168\pi$
$954$	0	0
$955$	0	0
$956$	−51.7712	−1.67440
$957$	0	0
$958$	−25.5203	−0.824522
$959$	21.8745	0.706365
$960$	0	0
$961$	5.00000	0.161290
$962$	24.5830	0.792588
$963$	0	0
$964$	−22.9150	−0.738043
$965$	0	0
$966$	0	0
$967$	36.3542	1.16907	0.584537	−	0.811367i	$-0.301276\pi$
$967$	36.3542	1.16907	0.584537	+	0.811367i	$0.301276\pi$
$968$	165.686	5.32536
$969$	0	0
$970$	0	0
$971$	0	0		−	1.00000i	$-0.5\pi$
$971$	0	0			1.00000i	$0.5\pi$
$972$	0	0
$973$	−50.5830	−1.62162
$974$	36.7085	1.17622
$975$	0	0
$976$	−7.79347	−0.249463
$977$	23.0405	0.737131	0.368566	−	0.929602i	$-0.379849\pi$
$977$	23.0405	0.737131	0.368566	+	0.929602i	$0.379849\pi$
$978$	0	0
$979$	−6.00000	−0.191761
$980$	0	0
$981$	0	0
$982$	85.2693	2.72105
$983$	−18.4575	−0.588703	−0.294352	−	0.955697i	$-0.595104\pi$
$983$	−18.4575	−0.588703	−0.294352	+	0.955697i	$0.595104\pi$
$984$	0	0
$985$	0	0
$986$	−73.4170	−2.33807
$987$	0	0
$988$	−28.2288	−0.898076
$989$	0.457513	0.0145481
$990$	0	0
$991$	27.7490	0.881477	0.440738	−	0.897636i	$-0.354717\pi$
$991$	27.7490	0.881477	0.440738	+	0.897636i	$0.354717\pi$
$992$	79.3725	2.52008
$993$	0	0
$994$	31.7490	1.00702
$995$	0	0
$996$	0	0
$997$	−37.2915	−1.18103	−0.590517	−	0.807025i	$-0.701076\pi$
$997$	−37.2915	−1.18103	−0.590517	+	0.807025i	$0.701076\pi$
$998$	113.875	3.60463
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4275.2.a.u.1.2		2
3.2	odd	2		1425.2.a.p.1.1		2
5.4	even	2		855.2.a.g.1.1		2
15.2	even	4		1425.2.c.i.799.1		4
15.8	even	4		1425.2.c.i.799.4		4
15.14	odd	2		285.2.a.d.1.2	✓	2
60.59	even	2		4560.2.a.bo.1.2		2
285.284	even	2		5415.2.a.s.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
285.2.a.d.1.2	✓	2	15.14	odd	2
855.2.a.g.1.1		2	5.4	even	2
1425.2.a.p.1.1		2	3.2	odd	2
1425.2.c.i.799.1		4	15.2	even	4
1425.2.c.i.799.4		4	15.8	even	4
4275.2.a.u.1.2		2	1.1	even	1	trivial
4560.2.a.bo.1.2		2	60.59	even	2
5415.2.a.s.1.1		2	285.284	even	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 4275 = 3^{2} \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4275.a (trivial)

\(f(q)\)	\(=\)	\(q+2.64575 q^{2} +5.00000 q^{4} +3.64575 q^{7} +7.93725 q^{8} -5.64575 q^{11} +5.64575 q^{13} +9.64575 q^{14} +11.0000 q^{16} -4.00000 q^{17} -1.00000 q^{19} -14.9373 q^{22} -1.29150 q^{23} +14.9373 q^{26} +18.2288 q^{28} +6.93725 q^{29} +6.00000 q^{31} +13.2288 q^{32} -10.5830 q^{34} +1.64575 q^{37} -2.64575 q^{38} +4.35425 q^{41} -0.354249 q^{43} -28.2288 q^{44} -3.41699 q^{46} +9.29150 q^{47} +6.29150 q^{49} +28.2288 q^{52} +0.708497 q^{53} +28.9373 q^{56} +18.3542 q^{58} -0.708497 q^{59} -0.708497 q^{61} +15.8745 q^{62} +13.0000 q^{64} -14.5830 q^{67} -20.0000 q^{68} +3.29150 q^{71} -10.0000 q^{73} +4.35425 q^{74} -5.00000 q^{76} -20.5830 q^{77} -14.5830 q^{79} +11.5203 q^{82} +6.00000 q^{83} -0.937254 q^{86} -44.8118 q^{88} +1.06275 q^{89} +20.5830 q^{91} -6.45751 q^{92} +24.5830 q^{94} -12.9373 q^{97} +16.6458 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 10 q^{4} + 2 q^{7} - 6 q^{11} + 6 q^{13} + 14 q^{14} + 22 q^{16} - 8 q^{17} - 2 q^{19} - 14 q^{22} + 8 q^{23} + 14 q^{26} + 10 q^{28} - 2 q^{29} + 12 q^{31} - 2 q^{37} + 14 q^{41} - 6 q^{43} - 30 q^{44}+ \cdots + 28 q^{98}+O(q^{100}) \) 2 * q + 10 * q^4 + 2 * q^7 - 6 * q^11 + 6 * q^13 + 14 * q^14 + 22 * q^16 - 8 * q^17 - 2 * q^19 - 14 * q^22 + 8 * q^23 + 14 * q^26 + 10 * q^28 - 2 * q^29 + 12 * q^31 - 2 * q^37 + 14 * q^41 - 6 * q^43 - 30 * q^44 - 28 * q^46 + 8 * q^47 + 2 * q^49 + 30 * q^52 + 12 * q^53 + 42 * q^56 + 42 * q^58 - 12 * q^59 - 12 * q^61 + 26 * q^64 - 8 * q^67 - 40 * q^68 - 4 * q^71 - 20 * q^73 + 14 * q^74 - 10 * q^76 - 20 * q^77 - 8 * q^79 - 14 * q^82 + 12 * q^83 + 14 * q^86 - 42 * q^88 + 18 * q^89 + 20 * q^91 + 40 * q^92 + 28 * q^94 - 10 * q^97 + 28 * q^98

Introduction

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