LMFDB - Embedded newform 5175.2.a.bw.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5175,2,Mod(1,5175)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5175, 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("5175.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5175 = 3^{2} \cdot 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5175.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:	$41.3225830460$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.5744.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 5x^{2} - 2x + 1 $ x^4 - 5x^2 - 2x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 115)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-1.92022$ of defining polynomial
Character	$\chi$	$=$	5175.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.92022 q^{2} +1.68725 q^{4} +4.60747 q^{7} -0.600553 q^{8} +O(q^{10})$	$q+1.92022 q^{2} +1.68725 q^{4} +4.60747 q^{7} -0.600553 q^{8} -1.56592 q^{11} +2.60055 q^{13} +8.84736 q^{14} -4.52769 q^{16} -0.559006 q^{17} -1.16647 q^{19} -3.00692 q^{22} +1.00000 q^{23} +4.99364 q^{26} +7.77394 q^{28} +3.17339 q^{29} +10.0554 q^{31} -7.49306 q^{32} -1.07341 q^{34} +5.07341 q^{37} -2.23989 q^{38} +11.8127 q^{41} -2.76426 q^{43} -2.64210 q^{44} +1.92022 q^{46} -9.32298 q^{47} +14.2288 q^{49} +4.38778 q^{52} +5.54789 q^{53} -2.76703 q^{56} +6.09361 q^{58} -3.84044 q^{59} -4.29832 q^{61} +19.3086 q^{62} -5.33295 q^{64} +2.60747 q^{67} -0.943181 q^{68} -7.89582 q^{71} +9.90579 q^{73} +9.74208 q^{74} -1.96813 q^{76} -7.21494 q^{77} +12.0485 q^{79} +22.6830 q^{82} -13.3856 q^{83} -5.30800 q^{86} +0.940419 q^{88} +7.71551 q^{89} +11.9820 q^{91} +1.68725 q^{92} -17.9022 q^{94} -2.62130 q^{97} +27.3224 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{4} + 6 q^{7} - 6 q^{8}+O(q^{10}) $ 4 * q + 2 * q^4 + 6 * q^7 - 6 * q^8	$ 4 q + 2 q^{4} + 6 q^{7} - 6 q^{8} - 2 q^{11} + 14 q^{13} + 4 q^{14} + 2 q^{16} - 14 q^{17} - 4 q^{19} + 4 q^{22} + 4 q^{23} - 6 q^{26} + 18 q^{28} - 4 q^{29} - 2 q^{32} + 14 q^{34} + 2 q^{37} + 10 q^{38} + 8 q^{41} + 4 q^{43} - 6 q^{44} - 2 q^{47} + 18 q^{52} - 4 q^{53} - 14 q^{56} - 8 q^{61} + 28 q^{62} - 20 q^{64} - 2 q^{67} - 8 q^{68} + 24 q^{71} + 18 q^{73} + 36 q^{74} - 18 q^{76} - 4 q^{77} + 24 q^{79} + 32 q^{82} + 6 q^{83} - 14 q^{86} - 10 q^{88} + 8 q^{89} + 26 q^{91} + 2 q^{92} - 42 q^{94} + 34 q^{97} + 28 q^{98}+O(q^{100}) $ 4 * q + 2 * q^4 + 6 * q^7 - 6 * q^8 - 2 * q^11 + 14 * q^13 + 4 * q^14 + 2 * q^16 - 14 * q^17 - 4 * q^19 + 4 * q^22 + 4 * q^23 - 6 * q^26 + 18 * q^28 - 4 * q^29 - 2 * q^32 + 14 * q^34 + 2 * q^37 + 10 * q^38 + 8 * q^41 + 4 * q^43 - 6 * q^44 - 2 * q^47 + 18 * q^52 - 4 * q^53 - 14 * q^56 - 8 * q^61 + 28 * q^62 - 20 * q^64 - 2 * q^67 - 8 * q^68 + 24 * q^71 + 18 * q^73 + 36 * q^74 - 18 * q^76 - 4 * q^77 + 24 * q^79 + 32 * q^82 + 6 * q^83 - 14 * q^86 - 10 * q^88 + 8 * q^89 + 26 * q^91 + 2 * q^92 - 42 * q^94 + 34 * q^97 + 28 * q^98

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 $

$2$	1.92022	1.35780	0.678901	−	0.734230i	$-0.262457\pi$
$2$	1.92022	1.35780	0.678901	+	0.734230i	$0.262457\pi$
$3$	0	0
$3$	0	0
$4$	1.68725	0.843624
$5$	0	0
$5$	0	0
$6$	0	0
$7$	4.60747	1.74146	0.870730	−	0.491762i	$-0.163647\pi$
$7$	4.60747	1.74146	0.870730	+	0.491762i	$0.163647\pi$
$8$	−0.600553	−0.212327
$9$	0	0
$10$	0	0
$11$	−1.56592	−0.472143	−0.236072	−	0.971736i	$-0.575860\pi$
$11$	−1.56592	−0.472143	−0.236072	+	0.971736i	$0.575860\pi$
$12$	0	0
$13$	2.60055	0.721264	0.360632	−	0.932708i	$-0.382561\pi$
$13$	2.60055	0.721264	0.360632	+	0.932708i	$0.382561\pi$
$14$	8.84736	2.36456
$15$	0	0
$16$	−4.52769	−1.13192
$17$	−0.559006	−0.135579	−0.0677894	−	0.997700i	$-0.521595\pi$
$17$	−0.559006	−0.135579	−0.0677894	+	0.997700i	$0.521595\pi$
$18$	0	0
$19$	−1.16647	−0.267608	−0.133804	−	0.991008i	$-0.542719\pi$
$19$	−1.16647	−0.267608	−0.133804	+	0.991008i	$0.542719\pi$
$20$	0	0
$21$	0	0
$22$	−3.00692	−0.641077
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	0	0
$26$	4.99364	0.979332
$27$	0	0
$28$	7.77394	1.46914
$29$	3.17339	0.589284	0.294642	−	0.955608i	$-0.404800\pi$
$29$	3.17339	0.589284	0.294642	+	0.955608i	$0.404800\pi$
$30$	0	0
$31$	10.0554	1.80600	0.903000	−	0.429641i	$-0.141360\pi$
$31$	10.0554	1.80600	0.903000	+	0.429641i	$0.141360\pi$
$32$	−7.49306	−1.32460
$33$	0	0
$34$	−1.07341	−0.184089
$35$	0	0
$36$	0	0
$37$	5.07341	0.834064	0.417032	−	0.908892i	$-0.363070\pi$
$37$	5.07341	0.834064	0.417032	+	0.908892i	$0.363070\pi$
$38$	−2.23989	−0.363358
$39$	0	0
$40$	0	0
$41$	11.8127	1.84484	0.922419	−	0.386190i	$-0.126209\pi$
$41$	11.8127	1.84484	0.922419	+	0.386190i	$0.126209\pi$
$42$	0	0
$43$	−2.76426	−0.421546	−0.210773	−	0.977535i	$-0.567598\pi$
$43$	−2.76426	−0.421546	−0.210773	+	0.977535i	$0.567598\pi$
$44$	−2.64210	−0.398311
$45$	0	0
$46$	1.92022	0.283121
$47$	−9.32298	−1.35990	−0.679948	−	0.733260i	$-0.737998\pi$
$47$	−9.32298	−1.35990	−0.679948	+	0.733260i	$0.737998\pi$
$48$	0	0
$49$	14.2288	2.03268
$50$	0	0
$51$	0	0
$52$	4.38778	0.608475
$53$	5.54789	0.762061	0.381030	−	0.924562i	$-0.375569\pi$
$53$	5.54789	0.762061	0.381030	+	0.924562i	$0.375569\pi$
$54$	0	0
$55$	0	0
$56$	−2.76703	−0.369760
$57$	0	0
$58$	6.09361	0.800130
$59$	−3.84044	−0.499983	−0.249991	−	0.968248i	$-0.580428\pi$
$59$	−3.84044	−0.499983	−0.249991	+	0.968248i	$0.580428\pi$
$60$	0	0
$61$	−4.29832	−0.550343	−0.275172	−	0.961395i	$-0.588735\pi$
$61$	−4.29832	−0.550343	−0.275172	+	0.961395i	$0.588735\pi$
$62$	19.3086	2.45219
$63$	0	0
$64$	−5.33295	−0.666619
$65$	0	0
$66$	0	0
$67$	2.60747	0.318553	0.159277	−	0.987234i	$-0.449084\pi$
$67$	2.60747	0.318553	0.159277	+	0.987234i	$0.449084\pi$
$68$	−0.943181	−0.114378
$69$	0	0
$70$	0	0
$71$	−7.89582	−0.937062	−0.468531	−	0.883447i	$-0.655217\pi$
$71$	−7.89582	−0.937062	−0.468531	+	0.883447i	$0.655217\pi$
$72$	0	0
$73$	9.90579	1.15938	0.579692	−	0.814835i	$-0.303173\pi$
$73$	9.90579	1.15938	0.579692	+	0.814835i	$0.303173\pi$
$74$	9.74208	1.13249
$75$	0	0
$76$	−1.96813	−0.225760
$77$	−7.21494	−0.822219
$78$	0	0
$79$	12.0485	1.35556	0.677779	−	0.735266i	$-0.262943\pi$
$79$	12.0485	1.35556	0.677779	+	0.735266i	$0.262943\pi$
$80$	0	0
$81$	0	0
$82$	22.6830	2.50492
$83$	−13.3856	−1.46926	−0.734628	−	0.678470i	$-0.762643\pi$
$83$	−13.3856	−1.46926	−0.734628	+	0.678470i	$0.762643\pi$
$84$	0	0
$85$	0	0
$86$	−5.30800	−0.572376
$87$	0	0
$88$	0.940419	0.100249
$89$	7.71551	0.817843	0.408921	−	0.912570i	$-0.365905\pi$
$89$	7.71551	0.817843	0.408921	+	0.912570i	$0.365905\pi$
$90$	0	0
$91$	11.9820	1.25605
$92$	1.68725	0.175908
$93$	0	0
$94$	−17.9022	−1.84647
$95$	0	0
$96$	0	0
$97$	−2.62130	−0.266153	−0.133076	−	0.991106i	$-0.542486\pi$
$97$	−2.62130	−0.266153	−0.133076	+	0.991106i	$0.542486\pi$
$98$	27.3224	2.75998
$99$	0	0
$100$	0	0
$101$	8.88199	0.883791	0.441895	−	0.897067i	$-0.354306\pi$
$101$	8.88199	0.883791	0.441895	+	0.897067i	$0.354306\pi$
$102$	0	0
$103$	−8.65593	−0.852894	−0.426447	−	0.904512i	$-0.640235\pi$
$103$	−8.65593	−0.852894	−0.426447	+	0.904512i	$0.640235\pi$
$104$	−1.56177	−0.153144
$105$	0	0
$106$	10.6532	1.03473
$107$	6.91662	0.668655	0.334327	−	0.942457i	$-0.391491\pi$
$107$	6.91662	0.668655	0.334327	+	0.942457i	$0.391491\pi$
$108$	0	0
$109$	−6.84736	−0.655858	−0.327929	−	0.944702i	$-0.606351\pi$
$109$	−6.84736	−0.655858	−0.327929	+	0.944702i	$0.606351\pi$
$110$	0	0
$111$	0	0
$112$	−20.8612	−1.97120
$113$	4.60747	0.433434	0.216717	−	0.976234i	$-0.430465\pi$
$113$	4.60747	0.433434	0.216717	+	0.976234i	$0.430465\pi$
$114$	0	0
$115$	0	0
$116$	5.35430	0.497134
$117$	0	0
$118$	−7.37450	−0.678877
$119$	−2.57560	−0.236105
$120$	0	0
$121$	−8.54789	−0.777081
$122$	−8.25372	−0.747257
$123$	0	0
$124$	16.9659	1.52358
$125$	0	0
$126$	0	0
$127$	11.5324	1.02334	0.511669	−	0.859182i	$-0.329027\pi$
$127$	11.5324	1.02334	0.511669	+	0.859182i	$0.329027\pi$
$128$	4.74568	0.419463
$129$	0	0
$130$	0	0
$131$	12.9639	1.13266	0.566332	−	0.824177i	$-0.308362\pi$
$131$	12.9639	1.13266	0.566332	+	0.824177i	$0.308362\pi$
$132$	0	0
$133$	−5.37450	−0.466028
$134$	5.00692	0.432532
$135$	0	0
$136$	0.335712	0.0287871
$137$	17.1457	1.46485	0.732427	−	0.680846i	$-0.238388\pi$
$137$	17.1457	1.46485	0.732427	+	0.680846i	$0.238388\pi$
$138$	0	0
$139$	−19.4798	−1.65225	−0.826127	−	0.563485i	$-0.809460\pi$
$139$	−19.4798	−1.65225	−0.826127	+	0.563485i	$0.809460\pi$
$140$	0	0
$141$	0	0
$142$	−15.1617	−1.27234
$143$	−4.07226	−0.340540
$144$	0	0
$145$	0	0
$146$	19.0213	1.57421
$147$	0	0
$148$	8.56011	0.703637
$149$	−13.2288	−1.08374	−0.541872	−	0.840461i	$-0.682284\pi$
$149$	−13.2288	−1.08374	−0.541872	+	0.840461i	$0.682284\pi$
$150$	0	0
$151$	−2.50749	−0.204057	−0.102028	−	0.994781i	$-0.532533\pi$
$151$	−2.50749	−0.204057	−0.102028	+	0.994781i	$0.532533\pi$
$152$	0.700529	0.0568204
$153$	0	0
$154$	−13.8543	−1.11641
$155$	0	0
$156$	0	0
$157$	−8.52824	−0.680628	−0.340314	−	0.940312i	$-0.610533\pi$
$157$	−8.52824	−0.680628	−0.340314	+	0.940312i	$0.610533\pi$
$158$	23.1357	1.84058
$159$	0	0
$160$	0	0
$161$	4.60747	0.363119
$162$	0	0
$163$	−7.73240	−0.605648	−0.302824	−	0.953046i	$-0.597929\pi$
$163$	−7.73240	−0.605648	−0.302824	+	0.953046i	$0.597929\pi$
$164$	19.9310	1.55635
$165$	0	0
$166$	−25.7032	−1.99496
$167$	14.3064	1.10706	0.553531	−	0.832829i	$-0.313280\pi$
$167$	14.3064	1.10706	0.553531	+	0.832829i	$0.313280\pi$
$168$	0	0
$169$	−6.23713	−0.479779
$170$	0	0
$171$	0	0
$172$	−4.66400	−0.355627
$173$	−15.4714	−1.17627	−0.588135	−	0.808763i	$-0.700138\pi$
$173$	−15.4714	−1.17627	−0.588135	+	0.808763i	$0.700138\pi$
$174$	0	0
$175$	0	0
$176$	7.09001	0.534430
$177$	0	0
$178$	14.8155	1.11047
$179$	−5.98612	−0.447424	−0.223712	−	0.974655i	$-0.571817\pi$
$179$	−5.98612	−0.447424	−0.223712	+	0.974655i	$0.571817\pi$
$180$	0	0
$181$	−5.69892	−0.423597	−0.211799	−	0.977313i	$-0.567932\pi$
$181$	−5.69892	−0.423597	−0.211799	+	0.977313i	$0.567932\pi$
$182$	23.0080	1.70547
$183$	0	0
$184$	−0.600553	−0.0442733
$185$	0	0
$186$	0	0
$187$	0.875359	0.0640126
$188$	−15.7302	−1.14724
$189$	0	0
$190$	0	0
$191$	8.34402	0.603752	0.301876	−	0.953347i	$-0.402387\pi$
$191$	8.34402	0.603752	0.301876	+	0.953347i	$0.402387\pi$
$192$	0	0
$193$	16.0250	1.15350	0.576751	−	0.816920i	$-0.304320\pi$
$193$	16.0250	1.15350	0.576751	+	0.816920i	$0.304320\pi$
$194$	−5.03348	−0.361383
$195$	0	0
$196$	24.0075	1.71482
$197$	−5.78893	−0.412444	−0.206222	−	0.978505i	$-0.566117\pi$
$197$	−5.78893	−0.412444	−0.206222	+	0.978505i	$0.566117\pi$
$198$	0	0
$199$	−18.8085	−1.33330	−0.666651	−	0.745370i	$-0.732273\pi$
$199$	−18.8085	−1.33330	−0.666651	+	0.745370i	$0.732273\pi$
$200$	0	0
$201$	0	0
$202$	17.0554	1.20001
$203$	14.6213	1.02621
$204$	0	0
$205$	0	0
$206$	−16.6213	−1.15806
$207$	0	0
$208$	−11.7745	−0.816414
$209$	1.82661	0.126349
$210$	0	0
$211$	−4.88199	−0.336090	−0.168045	−	0.985779i	$-0.553745\pi$
$211$	−4.88199	−0.336090	−0.168045	+	0.985779i	$0.553745\pi$
$212$	9.36066	0.642893
$213$	0	0
$214$	13.2814	0.907900
$215$	0	0
$216$	0	0
$217$	46.3299	3.14508
$218$	−13.1484	−0.890525
$219$	0	0
$220$	0	0
$221$	−1.45372	−0.0977880
$222$	0	0
$223$	−13.4786	−0.902596	−0.451298	−	0.892373i	$-0.649039\pi$
$223$	−13.4786	−0.902596	−0.451298	+	0.892373i	$0.649039\pi$
$224$	−34.5240	−2.30673
$225$	0	0
$226$	8.84736	0.588518
$227$	−13.6894	−0.908598	−0.454299	−	0.890849i	$-0.650110\pi$
$227$	−13.6894	−0.908598	−0.454299	+	0.890849i	$0.650110\pi$
$228$	0	0
$229$	27.3964	1.81040	0.905202	−	0.424981i	$-0.139719\pi$
$229$	27.3964	1.81040	0.905202	+	0.424981i	$0.139719\pi$
$230$	0	0
$231$	0	0
$232$	−1.90579	−0.125121
$233$	0.0387841	0.00254083	0.00127041	−	0.999999i	$-0.499596\pi$
$233$	0.0387841	0.00254083	0.00127041	+	0.999999i	$0.499596\pi$
$234$	0	0
$235$	0	0
$236$	−6.47978	−0.421798
$237$	0	0
$238$	−4.94572	−0.320584
$239$	13.9030	0.899312	0.449656	−	0.893202i	$-0.351547\pi$
$239$	13.9030	0.899312	0.449656	+	0.893202i	$0.351547\pi$
$240$	0	0
$241$	−11.9972	−0.772810	−0.386405	−	0.922329i	$-0.626283\pi$
$241$	−11.9972	−0.772810	−0.386405	+	0.922329i	$0.626283\pi$
$242$	−16.4138	−1.05512
$243$	0	0
$244$	−7.25233	−0.464283
$245$	0	0
$246$	0	0
$247$	−3.03348	−0.193016
$248$	−6.03878	−0.383463
$249$	0	0
$250$	0	0
$251$	17.7806	1.12230	0.561150	−	0.827714i	$-0.310359\pi$
$251$	17.7806	1.12230	0.561150	+	0.827714i	$0.310359\pi$
$252$	0	0
$253$	−1.56592	−0.0984487
$254$	22.1448	1.38949
$255$	0	0
$256$	19.7786	1.23617
$257$	−10.2315	−0.638226	−0.319113	−	0.947717i	$-0.603385\pi$
$257$	−10.2315	−0.638226	−0.319113	+	0.947717i	$0.603385\pi$
$258$	0	0
$259$	23.3756	1.45249
$260$	0	0
$261$	0	0
$262$	24.8936	1.53793
$263$	3.99856	0.246562	0.123281	−	0.992372i	$-0.460658\pi$
$263$	3.99856	0.246562	0.123281	+	0.992372i	$0.460658\pi$
$264$	0	0
$265$	0	0
$266$	−10.3202	−0.632773
$267$	0	0
$268$	4.39945	0.268739
$269$	2.88084	0.175648	0.0878239	−	0.996136i	$-0.472009\pi$
$269$	2.88084	0.175648	0.0878239	+	0.996136i	$0.472009\pi$
$270$	0	0
$271$	14.1944	0.862250	0.431125	−	0.902292i	$-0.358117\pi$
$271$	14.1944	0.862250	0.431125	+	0.902292i	$0.358117\pi$
$272$	2.53100	0.153465
$273$	0	0
$274$	32.9235	1.98898
$275$	0	0
$276$	0	0
$277$	9.13576	0.548915	0.274457	−	0.961599i	$-0.411502\pi$
$277$	9.13576	0.548915	0.274457	+	0.961599i	$0.411502\pi$
$278$	−37.4055	−2.24343
$279$	0	0
$280$	0	0
$281$	−14.8640	−0.886709	−0.443355	−	0.896346i	$-0.646212\pi$
$281$	−14.8640	−0.886709	−0.443355	+	0.896346i	$0.646212\pi$
$282$	0	0
$283$	3.03878	0.180637	0.0903185	−	0.995913i	$-0.471212\pi$
$283$	3.03878	0.180637	0.0903185	+	0.995913i	$0.471212\pi$
$284$	−13.3222	−0.790528
$285$	0	0
$286$	−7.81964	−0.462385
$287$	54.4268	3.21271
$288$	0	0
$289$	−16.6875	−0.981618
$290$	0	0
$291$	0	0
$292$	16.7135	0.978085
$293$	19.2288	1.12336	0.561678	−	0.827356i	$-0.310156\pi$
$293$	19.2288	1.12336	0.561678	+	0.827356i	$0.310156\pi$
$294$	0	0
$295$	0	0
$296$	−3.04685	−0.177095
$297$	0	0
$298$	−25.4022	−1.47151
$299$	2.60055	0.150394
$300$	0	0
$301$	−12.7363	−0.734106
$302$	−4.81494	−0.277069
$303$	0	0
$304$	5.28144	0.302911
$305$	0	0
$306$	0	0
$307$	−20.5395	−1.17225	−0.586127	−	0.810220i	$-0.699348\pi$
$307$	−20.5395	−1.17225	−0.586127	+	0.810220i	$0.699348\pi$
$308$	−12.1734	−0.693643
$309$	0	0
$310$	0	0
$311$	9.83929	0.557935	0.278967	−	0.960301i	$-0.410008\pi$
$311$	9.83929	0.557935	0.278967	+	0.960301i	$0.410008\pi$
$312$	0	0
$313$	22.4659	1.26985	0.634925	−	0.772574i	$-0.281031\pi$
$313$	22.4659	1.26985	0.634925	+	0.772574i	$0.281031\pi$
$314$	−16.3761	−0.924157
$315$	0	0
$316$	20.3287	1.14358
$317$	17.8404	1.00202	0.501010	−	0.865442i	$-0.332962\pi$
$317$	17.8404	1.00202	0.501010	+	0.865442i	$0.332962\pi$
$318$	0	0
$319$	−4.96928	−0.278226
$320$	0	0
$321$	0	0
$322$	8.84736	0.493044
$323$	0.652066	0.0362819
$324$	0	0
$325$	0	0
$326$	−14.8479	−0.822350
$327$	0	0
$328$	−7.09416	−0.391710
$329$	−42.9554	−2.36821
$330$	0	0
$331$	13.3130	0.731750	0.365875	−	0.930664i	$-0.380770\pi$
$331$	13.3130	0.731750	0.365875	+	0.930664i	$0.380770\pi$
$332$	−22.5848	−1.23950
$333$	0	0
$334$	27.4714	1.50317
$335$	0	0
$336$	0	0
$337$	3.08338	0.167962	0.0839812	−	0.996467i	$-0.473236\pi$
$337$	3.08338	0.167962	0.0839812	+	0.996467i	$0.473236\pi$
$338$	−11.9767	−0.651444
$339$	0	0
$340$	0	0
$341$	−15.7459	−0.852691
$342$	0	0
$343$	33.3063	1.79837
$344$	1.66009	0.0895058
$345$	0	0
$346$	−29.7085	−1.59714
$347$	−11.5617	−0.620666	−0.310333	−	0.950628i	$-0.600441\pi$
$347$	−11.5617	−0.620666	−0.310333	+	0.950628i	$0.600441\pi$
$348$	0	0
$349$	−16.4980	−0.883117	−0.441558	−	0.897232i	$-0.645574\pi$
$349$	−16.4980	−0.883117	−0.441558	+	0.897232i	$0.645574\pi$
$350$	0	0
$351$	0	0
$352$	11.7335	0.625400
$353$	−13.7252	−0.730518	−0.365259	−	0.930906i	$-0.619020\pi$
$353$	−13.7252	−0.730518	−0.365259	+	0.930906i	$0.619020\pi$
$354$	0	0
$355$	0	0
$356$	13.0180	0.689952
$357$	0	0
$358$	−11.4947	−0.607512
$359$	−15.1442	−0.799282	−0.399641	−	0.916672i	$-0.630865\pi$
$359$	−15.1442	−0.799282	−0.399641	+	0.916672i	$0.630865\pi$
$360$	0	0
$361$	−17.6393	−0.928386
$362$	−10.9432	−0.575161
$363$	0	0
$364$	20.2165	1.05964
$365$	0	0
$366$	0	0
$367$	11.2719	0.588390	0.294195	−	0.955745i	$-0.404949\pi$
$367$	11.2719	0.588390	0.294195	+	0.955745i	$0.404949\pi$
$368$	−4.52769	−0.236022
$369$	0	0
$370$	0	0
$371$	25.5617	1.32710
$372$	0	0
$373$	−35.3296	−1.82930	−0.914648	−	0.404252i	$-0.867532\pi$
$373$	−35.3296	−1.82930	−0.914648	+	0.404252i	$0.867532\pi$
$374$	1.68088	0.0869164
$375$	0	0
$376$	5.59894	0.288743
$377$	8.25257	0.425029
$378$	0	0
$379$	−36.8598	−1.89336	−0.946679	−	0.322178i	$-0.895585\pi$
$379$	−36.8598	−1.89336	−0.946679	+	0.322178i	$0.895585\pi$
$380$	0	0
$381$	0	0
$382$	16.0224	0.819775
$383$	4.27065	0.218220	0.109110	−	0.994030i	$-0.465200\pi$
$383$	4.27065	0.218220	0.109110	+	0.994030i	$0.465200\pi$
$384$	0	0
$385$	0	0
$386$	30.7714	1.56623
$387$	0	0
$388$	−4.42279	−0.224533
$389$	6.42988	0.326008	0.163004	−	0.986625i	$-0.447882\pi$
$389$	6.42988	0.326008	0.163004	+	0.986625i	$0.447882\pi$
$390$	0	0
$391$	−0.559006	−0.0282701
$392$	−8.54512	−0.431594
$393$	0	0
$394$	−11.1160	−0.560017
$395$	0	0
$396$	0	0
$397$	−33.9539	−1.70410	−0.852049	−	0.523462i	$-0.824640\pi$
$397$	−33.9539	−1.70410	−0.852049	+	0.523462i	$0.824640\pi$
$398$	−36.1165	−1.81036
$399$	0	0
$400$	0	0
$401$	−27.6421	−1.38038	−0.690189	−	0.723629i	$-0.742473\pi$
$401$	−27.6421	−1.38038	−0.690189	+	0.723629i	$0.742473\pi$
$402$	0	0
$403$	26.1495	1.30260
$404$	14.9861	0.745587
$405$	0	0
$406$	28.0761	1.39339
$407$	−7.94457	−0.393798
$408$	0	0
$409$	−13.2161	−0.653494	−0.326747	−	0.945112i	$-0.605952\pi$
$409$	−13.2161	−0.653494	−0.326747	+	0.945112i	$0.605952\pi$
$410$	0	0
$411$	0	0
$412$	−14.6047	−0.719522
$413$	−17.6947	−0.870700
$414$	0	0
$415$	0	0
$416$	−19.4861	−0.955384
$417$	0	0
$418$	3.50749	0.171557
$419$	−23.4772	−1.14694	−0.573468	−	0.819228i	$-0.694402\pi$
$419$	−23.4772	−1.14694	−0.573468	+	0.819228i	$0.694402\pi$
$420$	0	0
$421$	23.1138	1.12650	0.563249	−	0.826287i	$-0.309551\pi$
$421$	23.1138	1.12650	0.563249	+	0.826287i	$0.309551\pi$
$422$	−9.37450	−0.456343
$423$	0	0
$424$	−3.33180	−0.161806
$425$	0	0
$426$	0	0
$427$	−19.8044	−0.958401
$428$	11.6701	0.564093
$429$	0	0
$430$	0	0
$431$	9.52709	0.458904	0.229452	−	0.973320i	$-0.426307\pi$
$431$	9.52709	0.458904	0.229452	+	0.973320i	$0.426307\pi$
$432$	0	0
$433$	2.79445	0.134293	0.0671464	−	0.997743i	$-0.478611\pi$
$433$	2.79445	0.134293	0.0671464	+	0.997743i	$0.478611\pi$
$434$	88.9635	4.27039
$435$	0	0
$436$	−11.5532	−0.553298
$437$	−1.16647	−0.0558001
$438$	0	0
$439$	2.09578	0.100026	0.0500129	−	0.998749i	$-0.484074\pi$
$439$	2.09578	0.100026	0.0500129	+	0.998749i	$0.484074\pi$
$440$	0	0
$441$	0	0
$442$	−2.79147	−0.132777
$443$	27.0870	1.28694	0.643470	−	0.765471i	$-0.277494\pi$
$443$	27.0870	1.28694	0.643470	+	0.765471i	$0.277494\pi$
$444$	0	0
$445$	0	0
$446$	−25.8819	−1.22555
$447$	0	0
$448$	−24.5714	−1.16089
$449$	0.470272	0.0221935	0.0110967	−	0.999938i	$-0.496468\pi$
$449$	0.470272	0.0221935	0.0110967	+	0.999938i	$0.496468\pi$
$450$	0	0
$451$	−18.4978	−0.871028
$452$	7.77394	0.365656
$453$	0	0
$454$	−26.2867	−1.23370
$455$	0	0
$456$	0	0
$457$	5.83768	0.273075	0.136538	−	0.990635i	$-0.456403\pi$
$457$	5.83768	0.273075	0.136538	+	0.990635i	$0.456403\pi$
$458$	52.6071	2.45817
$459$	0	0
$460$	0	0
$461$	−10.8487	−0.505277	−0.252638	−	0.967561i	$-0.581298\pi$
$461$	−10.8487	−0.505277	−0.252638	+	0.967561i	$0.581298\pi$
$462$	0	0
$463$	1.06258	0.0493825	0.0246912	−	0.999695i	$-0.492140\pi$
$463$	1.06258	0.0493825	0.0246912	+	0.999695i	$0.492140\pi$
$464$	−14.3681	−0.667024
$465$	0	0
$466$	0.0744740	0.00344994
$467$	4.99885	0.231319	0.115660	−	0.993289i	$-0.463102\pi$
$467$	4.99885	0.231319	0.115660	+	0.993289i	$0.463102\pi$
$468$	0	0
$469$	12.0138	0.554747
$470$	0	0
$471$	0	0
$472$	2.30639	0.106160
$473$	4.32862	0.199030
$474$	0	0
$475$	0	0
$476$	−4.34568	−0.199184
$477$	0	0
$478$	26.6969	1.22109
$479$	−33.4283	−1.52738	−0.763688	−	0.645585i	$-0.776613\pi$
$479$	−33.4283	−1.52738	−0.763688	+	0.645585i	$0.776613\pi$
$480$	0	0
$481$	13.1937	0.601580
$482$	−23.0373	−1.04932
$483$	0	0
$484$	−14.4224	−0.655564
$485$	0	0
$486$	0	0
$487$	−10.7901	−0.488945	−0.244473	−	0.969656i	$-0.578615\pi$
$487$	−10.7901	−0.488945	−0.244473	+	0.969656i	$0.578615\pi$
$488$	2.58137	0.116853
$489$	0	0
$490$	0	0
$491$	−15.8127	−0.713618	−0.356809	−	0.934177i	$-0.616135\pi$
$491$	−15.8127	−0.713618	−0.356809	+	0.934177i	$0.616135\pi$
$492$	0	0
$493$	−1.77394	−0.0798944
$494$	−5.82495	−0.262077
$495$	0	0
$496$	−45.5276	−2.04425
$497$	−36.3798	−1.63185
$498$	0	0
$499$	12.9711	0.580668	0.290334	−	0.956925i	$-0.406234\pi$
$499$	12.9711	0.580668	0.290334	+	0.956925i	$0.406234\pi$
$500$	0	0
$501$	0	0
$502$	34.1426	1.52386
$503$	0.998849	0.0445365	0.0222682	−	0.999752i	$-0.492911\pi$
$503$	0.998849	0.0445365	0.0222682	+	0.999752i	$0.492911\pi$
$504$	0	0
$505$	0	0
$506$	−3.00692	−0.133674
$507$	0	0
$508$	19.4581	0.863313
$509$	20.6936	0.917228	0.458614	−	0.888636i	$-0.348346\pi$
$509$	20.6936	0.917228	0.458614	+	0.888636i	$0.348346\pi$
$510$	0	0
$511$	45.6406	2.01902
$512$	28.4880	1.25900
$513$	0	0
$514$	−19.6468	−0.866583
$515$	0	0
$516$	0	0
$517$	14.5991	0.642066
$518$	44.8863	1.97219
$519$	0	0
$520$	0	0
$521$	−39.2620	−1.72010	−0.860049	−	0.510212i	$-0.829567\pi$
$521$	−39.2620	−1.72010	−0.860049	+	0.510212i	$0.829567\pi$
$522$	0	0
$523$	−1.74162	−0.0761556	−0.0380778	−	0.999275i	$-0.512123\pi$
$523$	−1.74162	−0.0761556	−0.0380778	+	0.999275i	$0.512123\pi$
$524$	21.8734	0.955543
$525$	0	0
$526$	7.67812	0.334782
$527$	−5.62101	−0.244855
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	0	0
$532$	−9.06811	−0.393152
$533$	30.7196	1.33061
$534$	0	0
$535$	0	0
$536$	−1.56592	−0.0676375
$537$	0	0
$538$	5.53184	0.238495
$539$	−22.2811	−0.959717
$540$	0	0
$541$	−8.18117	−0.351736	−0.175868	−	0.984414i	$-0.556273\pi$
$541$	−8.18117	−0.351736	−0.175868	+	0.984414i	$0.556273\pi$
$542$	27.2564	1.17076
$543$	0	0
$544$	4.18866	0.179587
$545$	0	0
$546$	0	0
$547$	−24.2470	−1.03673	−0.518364	−	0.855160i	$-0.673459\pi$
$547$	−24.2470	−1.03673	−0.518364	+	0.855160i	$0.673459\pi$
$548$	28.9290	1.23579
$549$	0	0
$550$	0	0
$551$	−3.70168	−0.157697
$552$	0	0
$553$	55.5129	2.36065
$554$	17.5427	0.745317
$555$	0	0
$556$	−32.8672	−1.39388
$557$	35.0966	1.48709	0.743546	−	0.668685i	$-0.233142\pi$
$557$	35.0966	1.48709	0.743546	+	0.668685i	$0.233142\pi$
$558$	0	0
$559$	−7.18862	−0.304046
$560$	0	0
$561$	0	0
$562$	−28.5421	−1.20397
$563$	6.98179	0.294247	0.147124	−	0.989118i	$-0.452998\pi$
$563$	6.98179	0.294247	0.147124	+	0.989118i	$0.452998\pi$
$564$	0	0
$565$	0	0
$566$	5.83514	0.245269
$567$	0	0
$568$	4.74186	0.198964
$569$	−21.7280	−0.910886	−0.455443	−	0.890265i	$-0.650519\pi$
$569$	−21.7280	−0.910886	−0.455443	+	0.890265i	$0.650519\pi$
$570$	0	0
$571$	−4.71016	−0.197114	−0.0985570	−	0.995131i	$-0.531423\pi$
$571$	−4.71016	−0.197114	−0.0985570	+	0.995131i	$0.531423\pi$
$572$	−6.87092	−0.287288
$573$	0	0
$574$	104.511	4.36222
$575$	0	0
$576$	0	0
$577$	−9.49085	−0.395109	−0.197555	−	0.980292i	$-0.563300\pi$
$577$	−9.49085	−0.395109	−0.197555	+	0.980292i	$0.563300\pi$
$578$	−32.0437	−1.33284
$579$	0	0
$580$	0	0
$581$	−61.6736	−2.55865
$582$	0	0
$583$	−8.68756	−0.359802
$584$	−5.94895	−0.246169
$585$	0	0
$586$	36.9235	1.52530
$587$	−10.4952	−0.433184	−0.216592	−	0.976262i	$-0.569494\pi$
$587$	−10.4952	−0.433184	−0.216592	+	0.976262i	$0.569494\pi$
$588$	0	0
$589$	−11.7293	−0.483299
$590$	0	0
$591$	0	0
$592$	−22.9708	−0.944096
$593$	18.2138	0.747951	0.373975	−	0.927439i	$-0.377994\pi$
$593$	18.2138	0.747951	0.373975	+	0.927439i	$0.377994\pi$
$594$	0	0
$595$	0	0
$596$	−22.3202	−0.914272
$597$	0	0
$598$	4.99364	0.204205
$599$	20.8055	0.850091	0.425045	−	0.905172i	$-0.360258\pi$
$599$	20.8055	0.850091	0.425045	+	0.905172i	$0.360258\pi$
$600$	0	0
$601$	−47.4004	−1.93350	−0.966752	−	0.255716i	$-0.917689\pi$
$601$	−47.4004	−1.93350	−0.966752	+	0.255716i	$0.917689\pi$
$602$	−24.4564	−0.996770
$603$	0	0
$604$	−4.23076	−0.172147
$605$	0	0
$606$	0	0
$607$	−5.77118	−0.234245	−0.117123	−	0.993117i	$-0.537367\pi$
$607$	−5.77118	−0.234245	−0.117123	+	0.993117i	$0.537367\pi$
$608$	8.74046	0.354473
$609$	0	0
$610$	0	0
$611$	−24.2449	−0.980844
$612$	0	0
$613$	38.9329	1.57248	0.786242	−	0.617919i	$-0.212024\pi$
$613$	38.9329	1.57248	0.786242	+	0.617919i	$0.212024\pi$
$614$	−39.4404	−1.59169
$615$	0	0
$616$	4.33295	0.174580
$617$	−10.1653	−0.409241	−0.204620	−	0.978841i	$-0.565596\pi$
$617$	−10.1653	−0.409241	−0.204620	+	0.978841i	$0.565596\pi$
$618$	0	0
$619$	2.09923	0.0843752	0.0421876	−	0.999110i	$-0.486567\pi$
$619$	2.09923	0.0843752	0.0421876	+	0.999110i	$0.486567\pi$
$620$	0	0
$621$	0	0
$622$	18.8936	0.757565
$623$	35.5490	1.42424
$624$	0	0
$625$	0	0
$626$	43.1396	1.72420
$627$	0	0
$628$	−14.3893	−0.574194
$629$	−2.83607	−0.113081
$630$	0	0
$631$	−32.3066	−1.28611	−0.643053	−	0.765821i	$-0.722333\pi$
$631$	−32.3066	−1.28611	−0.643053	+	0.765821i	$0.722333\pi$
$632$	−7.23574	−0.287822
$633$	0	0
$634$	34.2576	1.36054
$635$	0	0
$636$	0	0
$637$	37.0027	1.46610
$638$	−9.54212	−0.377776
$639$	0	0
$640$	0	0
$641$	−14.5758	−0.575711	−0.287856	−	0.957674i	$-0.592942\pi$
$641$	−14.5758	−0.575711	−0.287856	+	0.957674i	$0.592942\pi$
$642$	0	0
$643$	−42.5090	−1.67639	−0.838196	−	0.545369i	$-0.816389\pi$
$643$	−42.5090	−1.67639	−0.838196	+	0.545369i	$0.816389\pi$
$644$	7.77394	0.306336
$645$	0	0
$646$	1.25211	0.0492636
$647$	3.04979	0.119900	0.0599498	−	0.998201i	$-0.480906\pi$
$647$	3.04979	0.119900	0.0599498	+	0.998201i	$0.480906\pi$
$648$	0	0
$649$	6.01383	0.236064
$650$	0	0
$651$	0	0
$652$	−13.0465	−0.510939
$653$	−17.9600	−0.702830	−0.351415	−	0.936220i	$-0.614299\pi$
$653$	−17.9600	−0.702830	−0.351415	+	0.936220i	$0.614299\pi$
$654$	0	0
$655$	0	0
$656$	−53.4844	−2.08821
$657$	0	0
$658$	−82.4838	−3.21555
$659$	−46.3411	−1.80519	−0.902597	−	0.430486i	$-0.858342\pi$
$659$	−46.3411	−1.80519	−0.902597	+	0.430486i	$0.858342\pi$
$660$	0	0
$661$	1.23165	0.0479056	0.0239528	−	0.999713i	$-0.492375\pi$
$661$	1.23165	0.0479056	0.0239528	+	0.999713i	$0.492375\pi$
$662$	25.5639	0.993570
$663$	0	0
$664$	8.03874	0.311963
$665$	0	0
$666$	0	0
$667$	3.17339	0.122874
$668$	24.1384	0.933944
$669$	0	0
$670$	0	0
$671$	6.73083	0.259841
$672$	0	0
$673$	42.4664	1.63696	0.818479	−	0.574537i	$-0.194818\pi$
$673$	42.4664	1.63696	0.818479	+	0.574537i	$0.194818\pi$
$674$	5.92077	0.228060
$675$	0	0
$676$	−10.5236	−0.404753
$677$	−1.41777	−0.0544893	−0.0272447	−	0.999629i	$-0.508673\pi$
$677$	−1.41777	−0.0544893	−0.0272447	+	0.999629i	$0.508673\pi$
$678$	0	0
$679$	−12.0776	−0.463495
$680$	0	0
$681$	0	0
$682$	−30.2357	−1.15778
$683$	−4.60886	−0.176353	−0.0881766	−	0.996105i	$-0.528104\pi$
$683$	−4.60886	−0.176353	−0.0881766	+	0.996105i	$0.528104\pi$
$684$	0	0
$685$	0	0
$686$	63.9555	2.44183
$687$	0	0
$688$	12.5157	0.477158
$689$	14.4276	0.549647
$690$	0	0
$691$	−16.8903	−0.642537	−0.321269	−	0.946988i	$-0.604109\pi$
$691$	−16.8903	−0.642537	−0.321269	+	0.946988i	$0.604109\pi$
$692$	−26.1041	−0.992330
$693$	0	0
$694$	−22.2011	−0.842741
$695$	0	0
$696$	0	0
$697$	−6.60338	−0.250121
$698$	−31.6798	−1.19910
$699$	0	0
$700$	0	0
$701$	11.9543	0.451508	0.225754	−	0.974184i	$-0.427516\pi$
$701$	11.9543	0.451508	0.225754	+	0.974184i	$0.427516\pi$
$702$	0	0
$703$	−5.91801	−0.223202
$704$	8.35098	0.314740
$705$	0	0
$706$	−26.3554	−0.991899
$707$	40.9235	1.53909
$708$	0	0
$709$	−12.6755	−0.476040	−0.238020	−	0.971260i	$-0.576498\pi$
$709$	−12.6755	−0.476040	−0.238020	+	0.971260i	$0.576498\pi$
$710$	0	0
$711$	0	0
$712$	−4.63357	−0.173650
$713$	10.0554	0.376577
$714$	0	0
$715$	0	0
$716$	−10.1001	−0.377457
$717$	0	0
$718$	−29.0803	−1.08527
$719$	34.7351	1.29540	0.647700	−	0.761896i	$-0.275731\pi$
$719$	34.7351	1.29540	0.647700	+	0.761896i	$0.275731\pi$
$720$	0	0
$721$	−39.8819	−1.48528
$722$	−33.8714	−1.26056
$723$	0	0
$724$	−9.61549	−0.357357
$725$	0	0
$726$	0	0
$727$	27.6631	1.02597	0.512984	−	0.858398i	$-0.328540\pi$
$727$	27.6631	1.02597	0.512984	+	0.858398i	$0.328540\pi$
$728$	−7.19580	−0.266694
$729$	0	0
$730$	0	0
$731$	1.54524	0.0571528
$732$	0	0
$733$	−2.14239	−0.0791309	−0.0395654	−	0.999217i	$-0.512597\pi$
$733$	−2.14239	−0.0791309	−0.0395654	+	0.999217i	$0.512597\pi$
$734$	21.6446	0.798917
$735$	0	0
$736$	−7.49306	−0.276198
$737$	−4.08309	−0.150403
$738$	0	0
$739$	−42.6249	−1.56798	−0.783991	−	0.620772i	$-0.786819\pi$
$739$	−42.6249	−1.56798	−0.783991	+	0.620772i	$0.786819\pi$
$740$	0	0
$741$	0	0
$742$	49.0841	1.80194
$743$	−23.4687	−0.860982	−0.430491	−	0.902595i	$-0.641660\pi$
$743$	−23.4687	−0.860982	−0.430491	+	0.902595i	$0.641660\pi$
$744$	0	0
$745$	0	0
$746$	−67.8406	−2.48382
$747$	0	0
$748$	1.47695	0.0540026
$749$	31.8681	1.16444
$750$	0	0
$751$	−27.2579	−0.994654	−0.497327	−	0.867563i	$-0.665685\pi$
$751$	−27.2579	−0.994654	−0.497327	+	0.867563i	$0.665685\pi$
$752$	42.2116	1.53930
$753$	0	0
$754$	15.8468	0.577105
$755$	0	0
$756$	0	0
$757$	−12.7382	−0.462976	−0.231488	−	0.972838i	$-0.574359\pi$
$757$	−12.7382	−0.462976	−0.231488	+	0.972838i	$0.574359\pi$
$758$	−70.7789	−2.57080
$759$	0	0
$760$	0	0
$761$	−6.61715	−0.239871	−0.119936	−	0.992782i	$-0.538269\pi$
$761$	−6.61715	−0.239871	−0.119936	+	0.992782i	$0.538269\pi$
$762$	0	0
$763$	−31.5490	−1.14215
$764$	14.0784	0.509340
$765$	0	0
$766$	8.20060	0.296300
$767$	−9.98727	−0.360619
$768$	0	0
$769$	−27.2116	−0.981275	−0.490638	−	0.871364i	$-0.663236\pi$
$769$	−27.2116	−0.981275	−0.490638	+	0.871364i	$0.663236\pi$
$770$	0	0
$771$	0	0
$772$	27.0381	0.973121
$773$	−22.9249	−0.824552	−0.412276	−	0.911059i	$-0.635266\pi$
$773$	−22.9249	−0.824552	−0.412276	+	0.911059i	$0.635266\pi$
$774$	0	0
$775$	0	0
$776$	1.57423	0.0565116
$777$	0	0
$778$	12.3468	0.442654
$779$	−13.7792	−0.493693
$780$	0	0
$781$	12.3642	0.442427
$782$	−1.07341	−0.0383852
$783$	0	0
$784$	−64.4235	−2.30084
$785$	0	0
$786$	0	0
$787$	13.0620	0.465610	0.232805	−	0.972523i	$-0.425210\pi$
$787$	13.0620	0.465610	0.232805	+	0.972523i	$0.425210\pi$
$788$	−9.76736	−0.347948
$789$	0	0
$790$	0	0
$791$	21.2288	0.754808
$792$	0	0
$793$	−11.1780	−0.396943
$794$	−65.1990	−2.31383
$795$	0	0
$796$	−31.7347	−1.12480
$797$	−20.9305	−0.741395	−0.370697	−	0.928754i	$-0.620881\pi$
$797$	−20.9305	−0.741395	−0.370697	+	0.928754i	$0.620881\pi$
$798$	0	0
$799$	5.21160	0.184373
$800$	0	0
$801$	0	0
$802$	−53.0788	−1.87428
$803$	−15.5117	−0.547396
$804$	0	0
$805$	0	0
$806$	50.2129	1.76867
$807$	0	0
$808$	−5.33410	−0.187653
$809$	−18.0664	−0.635180	−0.317590	−	0.948228i	$-0.602874\pi$
$809$	−18.0664	−0.635180	−0.317590	+	0.948228i	$0.602874\pi$
$810$	0	0
$811$	34.8658	1.22430	0.612152	−	0.790740i	$-0.290304\pi$
$811$	34.8658	1.22430	0.612152	+	0.790740i	$0.290304\pi$
$812$	24.6698	0.865739
$813$	0	0
$814$	−15.2553	−0.534699
$815$	0	0
$816$	0	0
$817$	3.22444	0.112809
$818$	−25.3778	−0.887314
$819$	0	0
$820$	0	0
$821$	−14.5989	−0.509507	−0.254753	−	0.967006i	$-0.581994\pi$
$821$	−14.5989	−0.509507	−0.254753	+	0.967006i	$0.581994\pi$
$822$	0	0
$823$	25.8126	0.899771	0.449886	−	0.893086i	$-0.351465\pi$
$823$	25.8126	0.899771	0.449886	+	0.893086i	$0.351465\pi$
$824$	5.19834	0.181093
$825$	0	0
$826$	−33.9778	−1.18224
$827$	−54.9097	−1.90940	−0.954698	−	0.297577i	$-0.903822\pi$
$827$	−54.9097	−1.90940	−0.954698	+	0.297577i	$0.903822\pi$
$828$	0	0
$829$	−27.4809	−0.954452	−0.477226	−	0.878781i	$-0.658358\pi$
$829$	−27.4809	−0.954452	−0.477226	+	0.878781i	$0.658358\pi$
$830$	0	0
$831$	0	0
$832$	−13.8686	−0.480808
$833$	−7.95396	−0.275589
$834$	0	0
$835$	0	0
$836$	3.08194	0.106591
$837$	0	0
$838$	−45.0814	−1.55731
$839$	1.46577	0.0506041	0.0253020	−	0.999680i	$-0.491945\pi$
$839$	1.46577	0.0506041	0.0253020	+	0.999680i	$0.491945\pi$
$840$	0	0
$841$	−18.9296	−0.652744
$842$	44.3836	1.52956
$843$	0	0
$844$	−8.23713	−0.283534
$845$	0	0
$846$	0	0
$847$	−39.3841	−1.35325
$848$	−25.1191	−0.862594
$849$	0	0
$850$	0	0
$851$	5.07341	0.173914
$852$	0	0
$853$	35.8044	1.22592	0.612959	−	0.790115i	$-0.289979\pi$
$853$	35.8044	1.22592	0.612959	+	0.790115i	$0.289979\pi$
$854$	−38.0288	−1.30132
$855$	0	0
$856$	−4.15379	−0.141974
$857$	−28.1663	−0.962141	−0.481070	−	0.876682i	$-0.659752\pi$
$857$	−28.1663	−0.962141	−0.481070	+	0.876682i	$0.659752\pi$
$858$	0	0
$859$	20.3185	0.693258	0.346629	−	0.938002i	$-0.387326\pi$
$859$	20.3185	0.693258	0.346629	+	0.938002i	$0.387326\pi$
$860$	0	0
$861$	0	0
$862$	18.2941	0.623100
$863$	15.1357	0.515224	0.257612	−	0.966248i	$-0.417064\pi$
$863$	15.1357	0.515224	0.257612	+	0.966248i	$0.417064\pi$
$864$	0	0
$865$	0	0
$866$	5.36597	0.182343
$867$	0	0
$868$	78.1700	2.65326
$869$	−18.8670	−0.640018
$870$	0	0
$871$	6.78086	0.229761
$872$	4.11220	0.139257
$873$	0	0
$874$	−2.23989	−0.0757654
$875$	0	0
$876$	0	0
$877$	12.9668	0.437856	0.218928	−	0.975741i	$-0.429744\pi$
$877$	12.9668	0.437856	0.218928	+	0.975741i	$0.429744\pi$
$878$	4.02435	0.135815
$879$	0	0
$880$	0	0
$881$	34.3573	1.15753	0.578763	−	0.815496i	$-0.303536\pi$
$881$	34.3573	1.15753	0.578763	+	0.815496i	$0.303536\pi$
$882$	0	0
$883$	−14.7201	−0.495372	−0.247686	−	0.968840i	$-0.579670\pi$
$883$	−14.7201	−0.495372	−0.247686	+	0.968840i	$0.579670\pi$
$884$	−2.45279	−0.0824963
$885$	0	0
$886$	52.0129	1.74741
$887$	−20.9401	−0.703101	−0.351550	−	0.936169i	$-0.614345\pi$
$887$	−20.9401	−0.703101	−0.351550	+	0.936169i	$0.614345\pi$
$888$	0	0
$889$	53.1354	1.78210
$890$	0	0
$891$	0	0
$892$	−22.7418	−0.761451
$893$	10.8750	0.363919
$894$	0	0
$895$	0	0
$896$	21.8656	0.730477
$897$	0	0
$898$	0.903025	0.0301343
$899$	31.9097	1.06425
$900$	0	0
$901$	−3.10130	−0.103319
$902$	−35.5199	−1.18268
$903$	0	0
$904$	−2.76703	−0.0920300
$905$	0	0
$906$	0	0
$907$	−40.2008	−1.33485	−0.667423	−	0.744679i	$-0.732603\pi$
$907$	−40.2008	−1.33485	−0.667423	+	0.744679i	$0.732603\pi$
$908$	−23.0974	−0.766515
$909$	0	0
$910$	0	0
$911$	−54.6058	−1.80917	−0.904586	−	0.426292i	$-0.859820\pi$
$911$	−54.6058	−1.80917	−0.904586	+	0.426292i	$0.859820\pi$
$912$	0	0
$913$	20.9608	0.693700
$914$	11.2096	0.370782
$915$	0	0
$916$	46.2245	1.52730
$917$	59.7309	1.97249
$918$	0	0
$919$	13.8019	0.455284	0.227642	−	0.973745i	$-0.426898\pi$
$919$	13.8019	0.455284	0.227642	+	0.973745i	$0.426898\pi$
$920$	0	0
$921$	0	0
$922$	−20.8320	−0.686065
$923$	−20.5335	−0.675868
$924$	0	0
$925$	0	0
$926$	2.04040	0.0670516
$927$	0	0
$928$	−23.7784	−0.780565
$929$	−19.2775	−0.632475	−0.316237	−	0.948680i	$-0.602420\pi$
$929$	−19.2775	−0.632475	−0.316237	+	0.948680i	$0.602420\pi$
$930$	0	0
$931$	−16.5975	−0.543961
$932$	0.0654384	0.00214350
$933$	0	0
$934$	9.59889	0.314085
$935$	0	0
$936$	0	0
$937$	48.0855	1.57089	0.785443	−	0.618934i	$-0.212435\pi$
$937$	48.0855	1.57089	0.785443	+	0.618934i	$0.212435\pi$
$938$	23.0692	0.753237
$939$	0	0
$940$	0	0
$941$	30.6853	1.00031	0.500156	−	0.865935i	$-0.333276\pi$
$941$	30.6853	1.00031	0.500156	+	0.865935i	$0.333276\pi$
$942$	0	0
$943$	11.8127	0.384675
$944$	17.3883	0.565942
$945$	0	0
$946$	8.31191	0.270244
$947$	12.9280	0.420103	0.210051	−	0.977690i	$-0.432637\pi$
$947$	12.9280	0.420103	0.210051	+	0.977690i	$0.432637\pi$
$948$	0	0
$949$	25.7605	0.836222
$950$	0	0
$951$	0	0
$952$	1.54678	0.0501316
$953$	6.38285	0.206761	0.103380	−	0.994642i	$-0.467034\pi$
$953$	6.38285	0.206761	0.103380	+	0.994642i	$0.467034\pi$
$954$	0	0
$955$	0	0
$956$	23.4579	0.758681
$957$	0	0
$958$	−64.1897	−2.07387
$959$	78.9982	2.55098
$960$	0	0
$961$	70.1107	2.26163
$962$	25.3348	0.816826
$963$	0	0
$964$	−20.2423	−0.651961
$965$	0	0
$966$	0	0
$967$	−38.6028	−1.24138	−0.620691	−	0.784056i	$-0.713148\pi$
$967$	−38.6028	−1.24138	−0.620691	+	0.784056i	$0.713148\pi$
$968$	5.13346	0.164996
$969$	0	0
$970$	0	0
$971$	24.0216	0.770889	0.385444	−	0.922731i	$-0.374048\pi$
$971$	24.0216	0.770889	0.385444	+	0.922731i	$0.374048\pi$
$972$	0	0
$973$	−89.7525	−2.87733
$974$	−20.7193	−0.663890
$975$	0	0
$976$	19.4615	0.622946
$977$	−45.5558	−1.45746	−0.728730	−	0.684801i	$-0.759889\pi$
$977$	−45.5558	−1.45746	−0.728730	+	0.684801i	$0.759889\pi$
$978$	0	0
$979$	−12.0819	−0.386139
$980$	0	0
$981$	0	0
$982$	−30.3639	−0.968952
$983$	29.1396	0.929408	0.464704	−	0.885466i	$-0.346161\pi$
$983$	29.1396	0.929408	0.464704	+	0.885466i	$0.346161\pi$
$984$	0	0
$985$	0	0
$986$	−3.40636	−0.108481
$987$	0	0
$988$	−5.11823	−0.162833
$989$	−2.76426	−0.0878985
$990$	0	0
$991$	−20.0415	−0.636639	−0.318320	−	0.947983i	$-0.603119\pi$
$991$	−20.0415	−0.636639	−0.318320	+	0.947983i	$0.603119\pi$
$992$	−75.3456	−2.39222
$993$	0	0
$994$	−69.8572	−2.21573
$995$	0	0
$996$	0	0
$997$	0.340153	0.0107728	0.00538638	−	0.999985i	$-0.498285\pi$
$997$	0.340153	0.0107728	0.00538638	+	0.999985i	$0.498285\pi$
$998$	24.9074	0.788431
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5175.2.a.bw.1.4		4
3.2	odd	2		575.2.a.j.1.1		4
5.2	odd	4		1035.2.b.e.829.7		8
5.3	odd	4		1035.2.b.e.829.2		8
5.4	even	2		5175.2.a.bv.1.1		4
12.11	even	2		9200.2.a.ck.1.2		4
15.2	even	4		115.2.b.b.24.2	✓	8
15.8	even	4		115.2.b.b.24.7	yes	8
15.14	odd	2		575.2.a.i.1.4		4
60.23	odd	4		1840.2.e.d.369.3		8
60.47	odd	4		1840.2.e.d.369.6		8
60.59	even	2		9200.2.a.cq.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
115.2.b.b.24.2	✓	8	15.2	even	4
115.2.b.b.24.7	yes	8	15.8	even	4
575.2.a.i.1.4		4	15.14	odd	2
575.2.a.j.1.1		4	3.2	odd	2
1035.2.b.e.829.2		8	5.3	odd	4
1035.2.b.e.829.7		8	5.2	odd	4
1840.2.e.d.369.3		8	60.23	odd	4
1840.2.e.d.369.6		8	60.47	odd	4
5175.2.a.bv.1.1		4	5.4	even	2
5175.2.a.bw.1.4		4	1.1	even	1	trivial
9200.2.a.ck.1.2		4	12.11	even	2
9200.2.a.cq.1.3		4	60.59	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 5175 = 3^{2} \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5175.a (trivial)

\(f(q)\)	\(=\)	\(q+1.92022 q^{2} +1.68725 q^{4} +4.60747 q^{7} -0.600553 q^{8} +O(q^{10})\)	\(q+1.92022 q^{2} +1.68725 q^{4} +4.60747 q^{7} -0.600553 q^{8} -1.56592 q^{11} +2.60055 q^{13} +8.84736 q^{14} -4.52769 q^{16} -0.559006 q^{17} -1.16647 q^{19} -3.00692 q^{22} +1.00000 q^{23} +4.99364 q^{26} +7.77394 q^{28} +3.17339 q^{29} +10.0554 q^{31} -7.49306 q^{32} -1.07341 q^{34} +5.07341 q^{37} -2.23989 q^{38} +11.8127 q^{41} -2.76426 q^{43} -2.64210 q^{44} +1.92022 q^{46} -9.32298 q^{47} +14.2288 q^{49} +4.38778 q^{52} +5.54789 q^{53} -2.76703 q^{56} +6.09361 q^{58} -3.84044 q^{59} -4.29832 q^{61} +19.3086 q^{62} -5.33295 q^{64} +2.60747 q^{67} -0.943181 q^{68} -7.89582 q^{71} +9.90579 q^{73} +9.74208 q^{74} -1.96813 q^{76} -7.21494 q^{77} +12.0485 q^{79} +22.6830 q^{82} -13.3856 q^{83} -5.30800 q^{86} +0.940419 q^{88} +7.71551 q^{89} +11.9820 q^{91} +1.68725 q^{92} -17.9022 q^{94} -2.62130 q^{97} +27.3224 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{4} + 6 q^{7} - 6 q^{8}+O(q^{10}) \) 4 * q + 2 * q^4 + 6 * q^7 - 6 * q^8	\( 4 q + 2 q^{4} + 6 q^{7} - 6 q^{8} - 2 q^{11} + 14 q^{13} + 4 q^{14} + 2 q^{16} - 14 q^{17} - 4 q^{19} + 4 q^{22} + 4 q^{23} - 6 q^{26} + 18 q^{28} - 4 q^{29} - 2 q^{32} + 14 q^{34} + 2 q^{37} + 10 q^{38} + 8 q^{41} + 4 q^{43} - 6 q^{44} - 2 q^{47} + 18 q^{52} - 4 q^{53} - 14 q^{56} - 8 q^{61} + 28 q^{62} - 20 q^{64} - 2 q^{67} - 8 q^{68} + 24 q^{71} + 18 q^{73} + 36 q^{74} - 18 q^{76} - 4 q^{77} + 24 q^{79} + 32 q^{82} + 6 q^{83} - 14 q^{86} - 10 q^{88} + 8 q^{89} + 26 q^{91} + 2 q^{92} - 42 q^{94} + 34 q^{97} + 28 q^{98}+O(q^{100}) \) 4 * q + 2 * q^4 + 6 * q^7 - 6 * q^8 - 2 * q^11 + 14 * q^13 + 4 * q^14 + 2 * q^16 - 14 * q^17 - 4 * q^19 + 4 * q^22 + 4 * q^23 - 6 * q^26 + 18 * q^28 - 4 * q^29 - 2 * q^32 + 14 * q^34 + 2 * q^37 + 10 * q^38 + 8 * q^41 + 4 * q^43 - 6 * q^44 - 2 * q^47 + 18 * q^52 - 4 * q^53 - 14 * q^56 - 8 * q^61 + 28 * q^62 - 20 * q^64 - 2 * q^67 - 8 * q^68 + 24 * q^71 + 18 * q^73 + 36 * q^74 - 18 * q^76 - 4 * q^77 + 24 * q^79 + 32 * q^82 + 6 * q^83 - 14 * q^86 - 10 * q^88 + 8 * q^89 + 26 * q^91 + 2 * q^92 - 42 * q^94 + 34 * q^97 + 28 * q^98

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