LMFDB - Embedded newform 5175.2.a.bx.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [5175,2,Mod(1,5175)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage: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")

magma://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:traces = [4,2,0,4,0,0,3,9,0,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$41.3225830460$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.15317.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 4x^{2} + 5x + 2 $ x^4 - 2x^3 - 4x^2 + 5*x + 2
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			$2.69353$ 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)

$f(q)$	$=$	$q+2.69353 q^{2} +5.25508 q^{4} +2.74252 q^{7} +8.76763 q^{8} +3.38705 q^{11} +2.46356 q^{13} +7.38705 q^{14} +13.1057 q^{16} -2.64453 q^{17} +3.38705 q^{19} +9.12311 q^{22} -1.00000 q^{23} +6.63566 q^{26} +14.4122 q^{28} -9.20608 q^{29} -5.10809 q^{31} +17.7652 q^{32} -7.12311 q^{34} -5.50369 q^{37} +9.12311 q^{38} +1.20608 q^{41} -3.02511 q^{43} +17.7992 q^{44} -2.69353 q^{46} +8.21255 q^{47} +0.521423 q^{49} +12.9462 q^{52} -10.5417 q^{53} +24.0454 q^{56} -24.7968 q^{58} -8.28259 q^{59} +0.263945 q^{61} -13.7588 q^{62} +21.6397 q^{64} +7.66964 q^{67} -13.8972 q^{68} +0.0150161 q^{71} +5.53644 q^{73} -14.8243 q^{74} +17.7992 q^{76} +9.28906 q^{77} -8.67611 q^{79} +3.24861 q^{82} -3.52142 q^{83} -8.14822 q^{86} +29.6964 q^{88} +4.66318 q^{89} +6.75637 q^{91} -5.25508 q^{92} +22.1207 q^{94} +4.09799 q^{97} +1.40447 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{2} + 4 q^{4} + 3 q^{7} + 9 q^{8} - 4 q^{11} + 12 q^{14} + 8 q^{16} - q^{17} - 4 q^{19} + 20 q^{22} - 4 q^{23} + q^{26} + 22 q^{28} - 19 q^{29} - q^{31} + 20 q^{32} - 12 q^{34} + 3 q^{37} + 20 q^{38}+ \cdots + 16 q^{98}+O(q^{100}) $ 4 * q + 2 * q^2 + 4 * q^4 + 3 * q^7 + 9 * q^8 - 4 * q^11 + 12 * q^14 + 8 * q^16 - q^17 - 4 * q^19 + 20 * q^22 - 4 * q^23 + q^26 + 22 * q^28 - 19 * q^29 - q^31 + 20 * q^32 - 12 * q^34 + 3 * q^37 + 20 * q^38 - 13 * q^41 + 6 * q^43 + 18 * q^44 - 2 * q^46 + 6 * q^47 + 9 * q^49 + q^52 + 19 * q^53 + 10 * q^56 - 21 * q^58 - 23 * q^59 - 13 * q^62 + 27 * q^64 + 3 * q^67 - 4 * q^68 + 3 * q^71 + 32 * q^73 + 12 * q^74 + 18 * q^76 + 18 * q^77 + 2 * q^79 + 5 * q^82 - 21 * q^83 + 2 * q^86 + 14 * q^88 - 40 * q^91 - 4 * q^92 + 47 * q^94 + 18 * q^97 + 16 * 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 $

$2$	2.69353	1.90461	0.952305	−	0.305148i	$-0.0987059\pi$
$2$	2.69353	1.90461	0.952305	+	0.305148i	$0.0987059\pi$
$3$	0	0
$3$	0	0
$4$	5.25508	2.62754
$5$	0	0
$5$	0	0
$6$	0	0
$7$	2.74252	1.03658	0.518288	−	0.855206i	$-0.326570\pi$
$7$	2.74252	1.03658	0.518288	+	0.855206i	$0.326570\pi$
$8$	8.76763	3.09983
$9$	0	0
$10$	0	0
$11$	3.38705	1.02123	0.510617	−	0.859808i	$-0.329417\pi$
$11$	3.38705	1.02123	0.510617	+	0.859808i	$0.329417\pi$
$12$	0	0
$13$	2.46356	0.683269	0.341634	−	0.939833i	$-0.389020\pi$
$13$	2.46356	0.683269	0.341634	+	0.939833i	$0.389020\pi$
$14$	7.38705	1.97427
$15$	0	0
$16$	13.1057	3.27642
$17$	−2.64453	−0.641393	−0.320696	−	0.947182i	$-0.603917\pi$
$17$	−2.64453	−0.641393	−0.320696	+	0.947182i	$0.603917\pi$
$18$	0	0
$19$	3.38705	0.777043	0.388521	−	0.921440i	$-0.372986\pi$
$19$	3.38705	0.777043	0.388521	+	0.921440i	$0.372986\pi$
$20$	0	0
$21$	0	0
$22$	9.12311	1.94505
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	0	0
$26$	6.63566	1.30136
$27$	0	0
$28$	14.4122	2.72364
$29$	−9.20608	−1.70953	−0.854763	−	0.519018i	$-0.826298\pi$
$29$	−9.20608	−1.70953	−0.854763	+	0.519018i	$0.826298\pi$
$30$	0	0
$31$	−5.10809	−0.917440	−0.458720	−	0.888581i	$-0.651692\pi$
$31$	−5.10809	−0.917440	−0.458720	+	0.888581i	$0.651692\pi$
$32$	17.7652	3.14048
$33$	0	0
$34$	−7.12311	−1.22160
$35$	0	0
$36$	0	0
$37$	−5.50369	−0.904801	−0.452401	−	0.891815i	$-0.649432\pi$
$37$	−5.50369	−0.904801	−0.452401	+	0.891815i	$0.649432\pi$
$38$	9.12311	1.47996
$39$	0	0
$40$	0	0
$41$	1.20608	0.188358	0.0941792	−	0.995555i	$-0.469977\pi$
$41$	1.20608	0.188358	0.0941792	+	0.995555i	$0.469977\pi$
$42$	0	0
$43$	−3.02511	−0.461325	−0.230663	−	0.973034i	$-0.574089\pi$
$43$	−3.02511	−0.461325	−0.230663	+	0.973034i	$0.574089\pi$
$44$	17.7992	2.68333
$45$	0	0
$46$	−2.69353	−0.397139
$47$	8.21255	1.19792	0.598962	−	0.800778i	$-0.295580\pi$
$47$	8.21255	1.19792	0.598962	+	0.800778i	$0.295580\pi$
$48$	0	0
$49$	0.521423	0.0744891
$50$	0	0
$51$	0	0
$52$	12.9462	1.79532
$53$	−10.5417	−1.44802	−0.724009	−	0.689790i	$-0.757703\pi$
$53$	−10.5417	−1.44802	−0.724009	+	0.689790i	$0.757703\pi$
$54$	0	0
$55$	0	0
$56$	24.0454	3.21321
$57$	0	0
$58$	−24.7968	−3.25598
$59$	−8.28259	−1.07830	−0.539151	−	0.842209i	$-0.681255\pi$
$59$	−8.28259	−1.07830	−0.539151	+	0.842209i	$0.681255\pi$
$60$	0	0
$61$	0.263945	0.0337947	0.0168973	−	0.999857i	$-0.494621\pi$
$61$	0.263945	0.0337947	0.0168973	+	0.999857i	$0.494621\pi$
$62$	−13.7588	−1.74737
$63$	0	0
$64$	21.6397	2.70497
$65$	0	0
$66$	0	0
$67$	7.66964	0.936996	0.468498	−	0.883465i	$-0.344795\pi$
$67$	7.66964	0.936996	0.468498	+	0.883465i	$0.344795\pi$
$68$	−13.8972	−1.68528
$69$	0	0
$70$	0	0
$71$	0.0150161	0.00178208	0.000891041	−	1.00000i	$-0.499716\pi$
$71$	0.0150161	0.00178208	0.000891041		1.00000i	$0.499716\pi$
$72$	0	0
$73$	5.53644	0.647991	0.323996	−	0.946059i	$-0.394974\pi$
$73$	5.53644	0.647991	0.323996	+	0.946059i	$0.394974\pi$
$74$	−14.8243	−1.72329
$75$	0	0
$76$	17.7992	2.04171
$77$	9.28906	1.05859
$78$	0	0
$79$	−8.67611	−0.976138	−0.488069	−	0.872805i	$-0.662299\pi$
$79$	−8.67611	−0.976138	−0.488069	+	0.872805i	$0.662299\pi$
$80$	0	0
$81$	0	0
$82$	3.24861	0.358749
$83$	−3.52142	−0.386526	−0.193263	−	0.981147i	$-0.561907\pi$
$83$	−3.52142	−0.386526	−0.193263	+	0.981147i	$0.561907\pi$
$84$	0	0
$85$	0	0
$86$	−8.14822	−0.878645
$87$	0	0
$88$	29.6964	3.16565
$89$	4.66318	0.494296	0.247148	−	0.968978i	$-0.420507\pi$
$89$	4.66318	0.494296	0.247148	+	0.968978i	$0.420507\pi$
$90$	0	0
$91$	6.75637	0.708260
$92$	−5.25508	−0.547880
$93$	0	0
$94$	22.1207	2.28158
$95$	0	0
$96$	0	0
$97$	4.09799	0.416088	0.208044	−	0.978119i	$-0.433290\pi$
$97$	4.09799	0.416088	0.208044	+	0.978119i	$0.433290\pi$
$98$	1.40447	0.141873
$99$	0	0
$100$	0	0
$101$	−12.2955	−1.22345	−0.611725	−	0.791070i	$-0.709524\pi$
$101$	−12.2955	−1.22345	−0.611725	+	0.791070i	$0.709524\pi$
$102$	0	0
$103$	−6.05023	−0.596147	−0.298073	−	0.954543i	$-0.596344\pi$
$103$	−6.05023	−0.596147	−0.298073	+	0.954543i	$0.596344\pi$
$104$	21.5996	2.11801
$105$	0	0
$106$	−28.3944	−2.75791
$107$	13.1547	1.27171	0.635856	−	0.771808i	$-0.280647\pi$
$107$	13.1547	1.27171	0.635856	+	0.771808i	$0.280647\pi$
$108$	0	0
$109$	−17.1183	−1.63964	−0.819818	−	0.572624i	$-0.805925\pi$
$109$	−17.1183	−1.63964	−0.819818	+	0.572624i	$0.805925\pi$
$110$	0	0
$111$	0	0
$112$	35.9426	3.39626
$113$	4.38058	0.412091	0.206045	−	0.978542i	$-0.433941\pi$
$113$	4.38058	0.412091	0.206045	+	0.978542i	$0.433941\pi$
$114$	0	0
$115$	0	0
$116$	−48.3787	−4.49185
$117$	0	0
$118$	−22.3094	−2.05374
$119$	−7.25268	−0.664852
$120$	0	0
$121$	0.472110	0.0429191
$122$	0.710942	0.0643657
$123$	0	0
$124$	−26.8434	−2.41061
$125$	0	0
$126$	0	0
$127$	18.4588	1.63795	0.818975	−	0.573829i	$-0.194543\pi$
$127$	18.4588	1.63795	0.818975	+	0.573829i	$0.194543\pi$
$128$	22.7567	2.01143
$129$	0	0
$130$	0	0
$131$	3.86834	0.337979	0.168989	−	0.985618i	$-0.445950\pi$
$131$	3.86834	0.337979	0.168989	+	0.985618i	$0.445950\pi$
$132$	0	0
$133$	9.28906	0.805463
$134$	20.6584	1.78461
$135$	0	0
$136$	−23.1863	−1.98821
$137$	20.6713	1.76607	0.883034	−	0.469308i	$-0.155497\pi$
$137$	20.6713	1.76607	0.883034	+	0.469308i	$0.155497\pi$
$138$	0	0
$139$	12.5802	1.06704	0.533519	−	0.845788i	$-0.320869\pi$
$139$	12.5802	1.06704	0.533519	+	0.845788i	$0.320869\pi$
$140$	0	0
$141$	0	0
$142$	0.0404462	0.00339417
$143$	8.34420	0.697777
$144$	0	0
$145$	0	0
$146$	14.9125	1.23417
$147$	0	0
$148$	−28.9223	−2.37740
$149$	−11.4850	−0.940891	−0.470446	−	0.882429i	$-0.655907\pi$
$149$	−11.4850	−0.940891	−0.470446	+	0.882429i	$0.655907\pi$
$150$	0	0
$151$	−5.58667	−0.454636	−0.227318	−	0.973821i	$-0.572996\pi$
$151$	−5.58667	−0.454636	−0.227318	+	0.973821i	$0.572996\pi$
$152$	29.6964	2.40870
$153$	0	0
$154$	25.0203	2.01619
$155$	0	0
$156$	0	0
$157$	−5.86563	−0.468128	−0.234064	−	0.972221i	$-0.575203\pi$
$157$	−5.86563	−0.468128	−0.234064	+	0.972221i	$0.575203\pi$
$158$	−23.3693	−1.85916
$159$	0	0
$160$	0	0
$161$	−2.74252	−0.216141
$162$	0	0
$163$	−0.0465955	−0.00364964	−0.00182482	−	0.999998i	$-0.500581\pi$
$163$	−0.0465955	−0.00364964	−0.00182482	+	0.999998i	$0.500581\pi$
$164$	6.33805	0.494919
$165$	0	0
$166$	−9.48504	−0.736182
$167$	−2.24621	−0.173817	−0.0869085	−	0.996216i	$-0.527699\pi$
$167$	−2.24621	−0.173817	−0.0869085	+	0.996216i	$0.527699\pi$
$168$	0	0
$169$	−6.93087	−0.533144
$170$	0	0
$171$	0	0
$172$	−15.8972	−1.21215
$173$	8.01293	0.609212	0.304606	−	0.952478i	$-0.401475\pi$
$173$	8.01293	0.609212	0.304606	+	0.952478i	$0.401475\pi$
$174$	0	0
$175$	0	0
$176$	44.3896	3.34599
$177$	0	0
$178$	12.5604	0.941440
$179$	11.9615	0.894047	0.447024	−	0.894522i	$-0.352484\pi$
$179$	11.9615	0.894047	0.447024	+	0.894522i	$0.352484\pi$
$180$	0	0
$181$	−17.4802	−1.29930	−0.649648	−	0.760235i	$-0.725084\pi$
$181$	−17.4802	−1.29930	−0.649648	+	0.760235i	$0.725084\pi$
$182$	18.1984	1.34896
$183$	0	0
$184$	−8.76763	−0.646359
$185$	0	0
$186$	0	0
$187$	−8.95715	−0.655012
$188$	43.1576	3.14759
$189$	0	0
$190$	0	0
$191$	13.1231	0.949555	0.474777	−	0.880106i	$-0.342529\pi$
$191$	13.1231	0.949555	0.474777	+	0.880106i	$0.342529\pi$
$192$	0	0
$193$	12.5438	0.902924	0.451462	−	0.892290i	$-0.350903\pi$
$193$	12.5438	0.902924	0.451462	+	0.892290i	$0.350903\pi$
$194$	11.0380	0.792485
$195$	0	0
$196$	2.74012	0.195723
$197$	3.88544	0.276826	0.138413	−	0.990375i	$-0.455800\pi$
$197$	3.88544	0.276826	0.138413	+	0.990375i	$0.455800\pi$
$198$	0	0
$199$	22.1612	1.57096	0.785481	−	0.618886i	$-0.212416\pi$
$199$	22.1612	1.57096	0.785481	+	0.618886i	$0.212416\pi$
$200$	0	0
$201$	0	0
$202$	−33.1183	−2.33020
$203$	−25.2479	−1.77205
$204$	0	0
$205$	0	0
$206$	−16.2964	−1.13543
$207$	0	0
$208$	32.2867	2.23868
$209$	11.4721	0.793542
$210$	0	0
$211$	4.81048	0.331167	0.165584	−	0.986196i	$-0.447049\pi$
$211$	4.81048	0.331167	0.165584	+	0.986196i	$0.447049\pi$
$212$	−55.3976	−3.80473
$213$	0	0
$214$	35.4325	2.42211
$215$	0	0
$216$	0	0
$217$	−14.0090	−0.950996
$218$	−46.1086	−3.12287
$219$	0	0
$220$	0	0
$221$	−6.51496	−0.438243
$222$	0	0
$223$	−13.7815	−0.922876	−0.461438	−	0.887172i	$-0.652666\pi$
$223$	−13.7815	−0.922876	−0.461438	+	0.887172i	$0.652666\pi$
$224$	48.7215	3.25534
$225$	0	0
$226$	11.7992	0.784872
$227$	29.1012	1.93151	0.965757	−	0.259447i	$-0.0835402\pi$
$227$	29.1012	1.93151	0.965757	+	0.259447i	$0.0835402\pi$
$228$	0	0
$229$	−9.38705	−0.620314	−0.310157	−	0.950685i	$-0.600382\pi$
$229$	−9.38705	−0.620314	−0.310157	+	0.950685i	$0.600382\pi$
$230$	0	0
$231$	0	0
$232$	−80.7156	−5.29924
$233$	−12.6725	−0.830202	−0.415101	−	0.909775i	$-0.636254\pi$
$233$	−12.6725	−0.830202	−0.415101	+	0.909775i	$0.636254\pi$
$234$	0	0
$235$	0	0
$236$	−43.5257	−2.83328
$237$	0	0
$238$	−19.5353	−1.26628
$239$	−19.1883	−1.24119	−0.620596	−	0.784131i	$-0.713109\pi$
$239$	−19.1883	−1.24119	−0.620596	+	0.784131i	$0.713109\pi$
$240$	0	0
$241$	16.7741	1.08051	0.540257	−	0.841500i	$-0.318327\pi$
$241$	16.7741	1.08051	0.540257	+	0.841500i	$0.318327\pi$
$242$	1.27164	0.0817442
$243$	0	0
$244$	1.38705	0.0887968
$245$	0	0
$246$	0	0
$247$	8.34420	0.530929
$248$	−44.7859	−2.84391
$249$	0	0
$250$	0	0
$251$	22.8114	1.43984	0.719921	−	0.694056i	$-0.244178\pi$
$251$	22.8114	1.43984	0.719921	+	0.694056i	$0.244178\pi$
$252$	0	0
$253$	−3.38705	−0.212942
$254$	49.7191	3.11966
$255$	0	0
$256$	18.0162	1.12602
$257$	−23.7526	−1.48165	−0.740824	−	0.671699i	$-0.765565\pi$
$257$	−23.7526	−1.48165	−0.740824	+	0.671699i	$0.765565\pi$
$258$	0	0
$259$	−15.0940	−0.937895
$260$	0	0
$261$	0	0
$262$	10.4195	0.643718
$263$	16.5895	1.02295	0.511476	−	0.859297i	$-0.329099\pi$
$263$	16.5895	1.02295	0.511476	+	0.859297i	$0.329099\pi$
$264$	0	0
$265$	0	0
$266$	25.0203	1.53409
$267$	0	0
$268$	40.3046	2.46199
$269$	8.47495	0.516727	0.258363	−	0.966048i	$-0.416817\pi$
$269$	8.47495	0.516727	0.258363	+	0.966048i	$0.416817\pi$
$270$	0	0
$271$	−21.3029	−1.29406	−0.647030	−	0.762465i	$-0.723989\pi$
$271$	−21.3029	−1.29406	−0.647030	+	0.762465i	$0.723989\pi$
$272$	−34.6584	−2.10147
$273$	0	0
$274$	55.6787	3.36367
$275$	0	0
$276$	0	0
$277$	−21.9615	−1.31954	−0.659770	−	0.751467i	$-0.729346\pi$
$277$	−21.9615	−1.31954	−0.659770	+	0.751467i	$0.729346\pi$
$278$	33.8851	2.03229
$279$	0	0
$280$	0	0
$281$	−19.9020	−1.18725	−0.593627	−	0.804740i	$-0.702305\pi$
$281$	−19.9020	−1.18725	−0.593627	+	0.804740i	$0.702305\pi$
$282$	0	0
$283$	8.87781	0.527731	0.263865	−	0.964559i	$-0.415003\pi$
$283$	8.87781	0.527731	0.263865	+	0.964559i	$0.415003\pi$
$284$	0.0789107	0.00468249
$285$	0	0
$286$	22.4753	1.32899
$287$	3.30771	0.195248
$288$	0	0
$289$	−10.0065	−0.588616
$290$	0	0
$291$	0	0
$292$	29.0944	1.70262
$293$	23.9709	1.40039	0.700197	−	0.713950i	$-0.253096\pi$
$293$	23.9709	1.40039	0.700197	+	0.713950i	$0.253096\pi$
$294$	0	0
$295$	0	0
$296$	−48.2543	−2.80473
$297$	0	0
$298$	−30.9353	−1.79203
$299$	−2.46356	−0.142471
$300$	0	0
$301$	−8.29644	−0.478199
$302$	−15.0478	−0.865905
$303$	0	0
$304$	44.3896	2.54592
$305$	0	0
$306$	0	0
$307$	−12.0632	−0.688481	−0.344240	−	0.938882i	$-0.611864\pi$
$307$	−12.0632	−0.688481	−0.344240	+	0.938882i	$0.611864\pi$
$308$	48.8147	2.78148
$309$	0	0
$310$	0	0
$311$	6.81257	0.386305	0.193153	−	0.981169i	$-0.438129\pi$
$311$	6.81257	0.386305	0.193153	+	0.981169i	$0.438129\pi$
$312$	0	0
$313$	6.38539	0.360923	0.180462	−	0.983582i	$-0.442241\pi$
$313$	6.38539	0.360923	0.180462	+	0.983582i	$0.442241\pi$
$314$	−15.7992	−0.891601
$315$	0	0
$316$	−45.5936	−2.56484
$317$	16.8114	0.944222	0.472111	−	0.881539i	$-0.343492\pi$
$317$	16.8114	0.944222	0.472111	+	0.881539i	$0.343492\pi$
$318$	0	0
$319$	−31.1815	−1.74583
$320$	0	0
$321$	0	0
$322$	−7.38705	−0.411664
$323$	−8.95715	−0.498389
$324$	0	0
$325$	0	0
$326$	−0.125506	−0.00695115
$327$	0	0
$328$	10.5745	0.583878
$329$	22.5231	1.24174
$330$	0	0
$331$	3.29114	0.180898	0.0904488	−	0.995901i	$-0.471170\pi$
$331$	3.29114	0.180898	0.0904488	+	0.995901i	$0.471170\pi$
$332$	−18.5054	−1.01561
$333$	0	0
$334$	−6.05023	−0.331054
$335$	0	0
$336$	0	0
$337$	7.90201	0.430450	0.215225	−	0.976565i	$-0.430952\pi$
$337$	7.90201	0.430450	0.215225	+	0.976565i	$0.430952\pi$
$338$	−18.6685	−1.01543
$339$	0	0
$340$	0	0
$341$	−17.3014	−0.936921
$342$	0	0
$343$	−17.7676	−0.959362
$344$	−26.5231	−1.43003
$345$	0	0
$346$	21.5830	1.16031
$347$	−19.4024	−1.04158	−0.520789	−	0.853686i	$-0.674362\pi$
$347$	−19.4024	−1.04158	−0.520789	+	0.853686i	$0.674362\pi$
$348$	0	0
$349$	−33.1664	−1.77536	−0.887680	−	0.460462i	$-0.847684\pi$
$349$	−33.1664	−1.77536	−0.887680	+	0.460462i	$0.847684\pi$
$350$	0	0
$351$	0	0
$352$	60.1717	3.20716
$353$	−29.8960	−1.59121	−0.795603	−	0.605819i	$-0.792846\pi$
$353$	−29.8960	−1.59121	−0.795603	+	0.605819i	$0.792846\pi$
$354$	0	0
$355$	0	0
$356$	24.5054	1.29878
$357$	0	0
$358$	32.2187	1.70281
$359$	24.9725	1.31800	0.659000	−	0.752143i	$-0.270980\pi$
$359$	24.9725	1.31800	0.659000	+	0.752143i	$0.270980\pi$
$360$	0	0
$361$	−7.52789	−0.396205
$362$	−47.0835	−2.47465
$363$	0	0
$364$	35.5052	1.86098
$365$	0	0
$366$	0	0
$367$	−15.8179	−0.825686	−0.412843	−	0.910802i	$-0.635464\pi$
$367$	−15.8179	−0.825686	−0.412843	+	0.910802i	$0.635464\pi$
$368$	−13.1057	−0.683181
$369$	0	0
$370$	0	0
$371$	−28.9109	−1.50098
$372$	0	0
$373$	−22.6283	−1.17165	−0.585826	−	0.810437i	$-0.699230\pi$
$373$	−22.6283	−1.17165	−0.585826	+	0.810437i	$0.699230\pi$
$374$	−24.1263	−1.24754
$375$	0	0
$376$	72.0046	3.71335
$377$	−22.6797	−1.16807
$378$	0	0
$379$	10.2721	0.527641	0.263821	−	0.964572i	$-0.415017\pi$
$379$	10.2721	0.527641	0.263821	+	0.964572i	$0.415017\pi$
$380$	0	0
$381$	0	0
$382$	35.3474	1.80853
$383$	6.21463	0.317553	0.158776	−	0.987315i	$-0.449245\pi$
$383$	6.21463	0.317553	0.158776	+	0.987315i	$0.449245\pi$
$384$	0	0
$385$	0	0
$386$	33.7871	1.71972
$387$	0	0
$388$	21.5353	1.09329
$389$	−27.6414	−1.40147	−0.700737	−	0.713420i	$-0.747145\pi$
$389$	−27.6414	−1.40147	−0.700737	+	0.713420i	$0.747145\pi$
$390$	0	0
$391$	2.64453	0.133740
$392$	4.57165	0.230903
$393$	0	0
$394$	10.4655	0.527246
$395$	0	0
$396$	0	0
$397$	−1.07171	−0.0537875	−0.0268938	−	0.999638i	$-0.508562\pi$
$397$	−1.07171	−0.0537875	−0.0268938	+	0.999638i	$0.508562\pi$
$398$	59.6916	2.99207
$399$	0	0
$400$	0	0
$401$	−1.49798	−0.0748053	−0.0374027	−	0.999300i	$-0.511908\pi$
$401$	−1.49798	−0.0748053	−0.0374027	+	0.999300i	$0.511908\pi$
$402$	0	0
$403$	−12.5841	−0.626858
$404$	−64.6139	−3.21466
$405$	0	0
$406$	−68.0058	−3.37507
$407$	−18.6413	−0.924014
$408$	0	0
$409$	20.9373	1.03528	0.517642	−	0.855597i	$-0.326810\pi$
$409$	20.9373	1.03528	0.517642	+	0.855597i	$0.326810\pi$
$410$	0	0
$411$	0	0
$412$	−31.7944	−1.56640
$413$	−22.7152	−1.11774
$414$	0	0
$415$	0	0
$416$	43.7657	2.14579
$417$	0	0
$418$	30.9004	1.51139
$419$	18.7564	0.916309	0.458154	−	0.888873i	$-0.348511\pi$
$419$	18.7564	0.916309	0.458154	+	0.888873i	$0.348511\pi$
$420$	0	0
$421$	20.6163	1.00478	0.502388	−	0.864642i	$-0.332455\pi$
$421$	20.6163	1.00478	0.502388	+	0.864642i	$0.332455\pi$
$422$	12.9572	0.630744
$423$	0	0
$424$	−92.4261	−4.48861
$425$	0	0
$426$	0	0
$427$	0.723874	0.0350307
$428$	69.1289	3.34147
$429$	0	0
$430$	0	0
$431$	−27.0155	−1.30129	−0.650646	−	0.759381i	$-0.725502\pi$
$431$	−27.0155	−1.30129	−0.650646	+	0.759381i	$0.725502\pi$
$432$	0	0
$433$	25.2900	1.21536	0.607679	−	0.794183i	$-0.292101\pi$
$433$	25.2900	1.21536	0.607679	+	0.794183i	$0.292101\pi$
$434$	−37.7337	−1.81128
$435$	0	0
$436$	−89.9580	−4.30821
$437$	−3.38705	−0.162025
$438$	0	0
$439$	34.4110	1.64235	0.821174	−	0.570679i	$-0.193320\pi$
$439$	34.4110	1.64235	0.821174	+	0.570679i	$0.193320\pi$
$440$	0	0
$441$	0	0
$442$	−17.5482	−0.834683
$443$	−29.8281	−1.41717	−0.708587	−	0.705623i	$-0.750667\pi$
$443$	−29.8281	−1.41717	−0.708587	+	0.705623i	$0.750667\pi$
$444$	0	0
$445$	0	0
$446$	−37.1208	−1.75772
$447$	0	0
$448$	59.3474	2.80390
$449$	2.67456	0.126220	0.0631102	−	0.998007i	$-0.479898\pi$
$449$	2.67456	0.126220	0.0631102	+	0.998007i	$0.479898\pi$
$450$	0	0
$451$	4.08506	0.192358
$452$	23.0203	1.08278
$453$	0	0
$454$	78.3848	3.67878
$455$	0	0
$456$	0	0
$457$	30.9037	1.44561	0.722806	−	0.691051i	$-0.242852\pi$
$457$	30.9037	1.44561	0.722806	+	0.691051i	$0.242852\pi$
$458$	−25.2843	−1.18146
$459$	0	0
$460$	0	0
$461$	26.6022	1.23899	0.619493	−	0.785002i	$-0.287338\pi$
$461$	26.6022	1.23899	0.619493	+	0.785002i	$0.287338\pi$
$462$	0	0
$463$	−26.7013	−1.24092	−0.620458	−	0.784239i	$-0.713053\pi$
$463$	−26.7013	−1.24092	−0.620458	+	0.784239i	$0.713053\pi$
$464$	−120.652	−5.60113
$465$	0	0
$466$	−34.1336	−1.58121
$467$	35.2503	1.63119	0.815596	−	0.578622i	$-0.196410\pi$
$467$	35.2503	1.63119	0.815596	+	0.578622i	$0.196410\pi$
$468$	0	0
$469$	21.0342	0.971267
$470$	0	0
$471$	0	0
$472$	−72.6187	−3.34255
$473$	−10.2462	−0.471121
$474$	0	0
$475$	0	0
$476$	−38.1134	−1.74692
$477$	0	0
$478$	−51.6843	−2.36398
$479$	39.0705	1.78518	0.892589	−	0.450871i	$-0.148886\pi$
$479$	39.0705	1.78518	0.892589	+	0.450871i	$0.148886\pi$
$480$	0	0
$481$	−13.5587	−0.618222
$482$	45.1815	2.05796
$483$	0	0
$484$	2.48098	0.112772
$485$	0	0
$486$	0	0
$487$	−24.0839	−1.09135	−0.545673	−	0.837998i	$-0.683726\pi$
$487$	−24.0839	−1.09135	−0.545673	+	0.837998i	$0.683726\pi$
$488$	2.31417	0.104758
$489$	0	0
$490$	0	0
$491$	−5.60051	−0.252748	−0.126374	−	0.991983i	$-0.540334\pi$
$491$	−5.60051	−0.252748	−0.126374	+	0.991983i	$0.540334\pi$
$492$	0	0
$493$	24.3458	1.09648
$494$	22.4753	1.01121
$495$	0	0
$496$	−66.9450	−3.00592
$497$	0.0411819	0.00184726
$498$	0	0
$499$	9.53319	0.426764	0.213382	−	0.976969i	$-0.431552\pi$
$499$	9.53319	0.426764	0.213382	+	0.976969i	$0.431552\pi$
$500$	0	0
$501$	0	0
$502$	61.4431	2.74234
$503$	14.0568	0.626762	0.313381	−	0.949627i	$-0.398538\pi$
$503$	14.0568	0.626762	0.313381	+	0.949627i	$0.398538\pi$
$504$	0	0
$505$	0	0
$506$	−9.12311	−0.405572
$507$	0	0
$508$	97.0022	4.30378
$509$	−21.7105	−0.962302	−0.481151	−	0.876638i	$-0.659781\pi$
$509$	−21.7105	−0.962302	−0.481151	+	0.876638i	$0.659781\pi$
$510$	0	0
$511$	15.1838	0.671692
$512$	3.01385	0.133194
$513$	0	0
$514$	−63.9783	−2.82196
$515$	0	0
$516$	0	0
$517$	27.8163	1.22336
$518$	−40.6560	−1.78632
$519$	0	0
$520$	0	0
$521$	8.87689	0.388904	0.194452	−	0.980912i	$-0.437707\pi$
$521$	8.87689	0.388904	0.194452	+	0.980912i	$0.437707\pi$
$522$	0	0
$523$	0.0550279	0.00240620	0.00120310	−	0.999999i	$-0.499617\pi$
$523$	0.0550279	0.00240620	0.00120310	+	0.999999i	$0.499617\pi$
$524$	20.3285	0.888053
$525$	0	0
$526$	44.6842	1.94833
$527$	13.5085	0.588439
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	0	0
$532$	48.8147	2.11639
$533$	2.97126	0.128699
$534$	0	0
$535$	0	0
$536$	67.2446	2.90453
$537$	0	0
$538$	22.8275	0.984163
$539$	1.76609	0.0760708
$540$	0	0
$541$	5.99070	0.257560	0.128780	−	0.991673i	$-0.458894\pi$
$541$	5.99070	0.257560	0.128780	+	0.991673i	$0.458894\pi$
$542$	−57.3799	−2.46468
$543$	0	0
$544$	−46.9807	−2.01428
$545$	0	0
$546$	0	0
$547$	0.408533	0.0174676	0.00873380	−	0.999962i	$-0.497220\pi$
$547$	0.408533	0.0174676	0.00873380	+	0.999962i	$0.497220\pi$
$548$	108.629	4.64042
$549$	0	0
$550$	0	0
$551$	−31.1815	−1.32837
$552$	0	0
$553$	−23.7944	−1.01184
$554$	−59.1539	−2.51321
$555$	0	0
$556$	66.1099	2.80369
$557$	−5.46406	−0.231519	−0.115760	−	0.993277i	$-0.536930\pi$
$557$	−5.46406	−0.231519	−0.115760	+	0.993277i	$0.536930\pi$
$558$	0	0
$559$	−7.45255	−0.315209
$560$	0	0
$561$	0	0
$562$	−53.6066	−2.26126
$563$	−41.8212	−1.76255	−0.881277	−	0.472601i	$-0.843315\pi$
$563$	−41.8212	−1.76255	−0.881277	+	0.472601i	$0.843315\pi$
$564$	0	0
$565$	0	0
$566$	23.9126	1.00512
$567$	0	0
$568$	0.131656	0.00552415
$569$	−39.5127	−1.65646	−0.828230	−	0.560388i	$-0.810652\pi$
$569$	−39.5127	−1.65646	−0.828230	+	0.560388i	$0.810652\pi$
$570$	0	0
$571$	−24.4924	−1.02498	−0.512488	−	0.858694i	$-0.671276\pi$
$571$	−24.4924	−1.02498	−0.512488	+	0.858694i	$0.671276\pi$
$572$	43.8494	1.83344
$573$	0	0
$574$	8.90939	0.371871
$575$	0	0
$576$	0	0
$577$	36.3382	1.51278	0.756390	−	0.654121i	$-0.226961\pi$
$577$	36.3382	1.51278	0.756390	+	0.654121i	$0.226961\pi$
$578$	−26.9527	−1.12108
$579$	0	0
$580$	0	0
$581$	−9.65758	−0.400664
$582$	0	0
$583$	−35.7054	−1.47877
$584$	48.5415	2.00866
$585$	0	0
$586$	64.5662	2.66720
$587$	5.89358	0.243254	0.121627	−	0.992576i	$-0.461189\pi$
$587$	5.89358	0.243254	0.121627	+	0.992576i	$0.461189\pi$
$588$	0	0
$589$	−17.3014	−0.712890
$590$	0	0
$591$	0	0
$592$	−72.1296	−2.96451
$593$	−11.0931	−0.455538	−0.227769	−	0.973715i	$-0.573143\pi$
$593$	−11.0931	−0.455538	−0.227769	+	0.973715i	$0.573143\pi$
$594$	0	0
$595$	0	0
$596$	−60.3548	−2.47223
$597$	0	0
$598$	−6.63566	−0.271352
$599$	−40.1864	−1.64197	−0.820986	−	0.570949i	$-0.806575\pi$
$599$	−40.1864	−1.64197	−0.820986	+	0.570949i	$0.806575\pi$
$600$	0	0
$601$	41.1965	1.68044	0.840220	−	0.542246i	$-0.182426\pi$
$601$	41.1965	1.68044	0.840220	+	0.542246i	$0.182426\pi$
$602$	−22.3467	−0.910782
$603$	0	0
$604$	−29.3584	−1.19457
$605$	0	0
$606$	0	0
$607$	16.6283	0.674924	0.337462	−	0.941339i	$-0.390432\pi$
$607$	16.6283	0.674924	0.337462	+	0.941339i	$0.390432\pi$
$608$	60.1717	2.44029
$609$	0	0
$610$	0	0
$611$	20.2321	0.818504
$612$	0	0
$613$	−25.5052	−1.03015	−0.515073	−	0.857146i	$-0.672235\pi$
$613$	−25.5052	−1.03015	−0.515073	+	0.857146i	$0.672235\pi$
$614$	−32.4924	−1.31129
$615$	0	0
$616$	81.4431	3.28143
$617$	−3.14742	−0.126710	−0.0633552	−	0.997991i	$-0.520180\pi$
$617$	−3.14742	−0.126710	−0.0633552	+	0.997991i	$0.520180\pi$
$618$	0	0
$619$	−39.2665	−1.57825	−0.789127	−	0.614230i	$-0.789467\pi$
$619$	−39.2665	−1.57825	−0.789127	+	0.614230i	$0.789467\pi$
$620$	0	0
$621$	0	0
$622$	18.3498	0.735761
$623$	12.7889	0.512375
$624$	0	0
$625$	0	0
$626$	17.1992	0.687418
$627$	0	0
$628$	−30.8243	−1.23002
$629$	14.5547	0.580333
$630$	0	0
$631$	18.2916	0.728179	0.364089	−	0.931364i	$-0.381380\pi$
$631$	18.2916	0.728179	0.364089	+	0.931364i	$0.381380\pi$
$632$	−76.0689	−3.02586
$633$	0	0
$634$	45.2819	1.79837
$635$	0	0
$636$	0	0
$637$	1.28456	0.0508960
$638$	−83.9881	−3.32512
$639$	0	0
$640$	0	0
$641$	−28.5304	−1.12688	−0.563441	−	0.826157i	$-0.690523\pi$
$641$	−28.5304	−1.12688	−0.563441	+	0.826157i	$0.690523\pi$
$642$	0	0
$643$	12.2430	0.482815	0.241408	−	0.970424i	$-0.422391\pi$
$643$	12.2430	0.482815	0.241408	+	0.970424i	$0.422391\pi$
$644$	−14.4122	−0.567919
$645$	0	0
$646$	−24.1263	−0.949237
$647$	3.11947	0.122639	0.0613196	−	0.998118i	$-0.480469\pi$
$647$	3.11947	0.122639	0.0613196	+	0.998118i	$0.480469\pi$
$648$	0	0
$649$	−28.0536	−1.10120
$650$	0	0
$651$	0	0
$652$	−0.244863	−0.00958958
$653$	−21.7301	−0.850364	−0.425182	−	0.905108i	$-0.639790\pi$
$653$	−21.7301	−0.850364	−0.425182	+	0.905108i	$0.639790\pi$
$654$	0	0
$655$	0	0
$656$	15.8065	0.617141
$657$	0	0
$658$	60.6665	2.36503
$659$	12.2333	0.476541	0.238270	−	0.971199i	$-0.423420\pi$
$659$	12.2333	0.476541	0.238270	+	0.971199i	$0.423420\pi$
$660$	0	0
$661$	22.0000	0.855701	0.427850	−	0.903850i	$-0.359271\pi$
$661$	22.0000	0.855701	0.427850	+	0.903850i	$0.359271\pi$
$662$	8.86477	0.344539
$663$	0	0
$664$	−30.8746	−1.19817
$665$	0	0
$666$	0	0
$667$	9.20608	0.356461
$668$	−11.8040	−0.456711
$669$	0	0
$670$	0	0
$671$	0.893994	0.0345123
$672$	0	0
$673$	−10.7900	−0.415925	−0.207963	−	0.978137i	$-0.566683\pi$
$673$	−10.7900	−0.415925	−0.207963	+	0.978137i	$0.566683\pi$
$674$	21.2843	0.819839
$675$	0	0
$676$	−36.4223	−1.40086
$677$	23.6721	0.909793	0.454896	−	0.890544i	$-0.349676\pi$
$677$	23.6721	0.909793	0.454896	+	0.890544i	$0.349676\pi$
$678$	0	0
$679$	11.2388	0.431307
$680$	0	0
$681$	0	0
$682$	−46.6016	−1.78447
$683$	−10.3583	−0.396350	−0.198175	−	0.980167i	$-0.563501\pi$
$683$	−10.3583	−0.396350	−0.198175	+	0.980167i	$0.563501\pi$
$684$	0	0
$685$	0	0
$686$	−47.8576	−1.82721
$687$	0	0
$688$	−39.6462	−1.51150
$689$	−25.9702	−0.989386
$690$	0	0
$691$	13.5578	0.515763	0.257882	−	0.966177i	$-0.416976\pi$
$691$	13.5578	0.515763	0.257882	+	0.966177i	$0.416976\pi$
$692$	42.1086	1.60073
$693$	0	0
$694$	−52.2610	−1.98380
$695$	0	0
$696$	0	0
$697$	−3.18952	−0.120812
$698$	−89.3347	−3.38137
$699$	0	0
$700$	0	0
$701$	21.3774	0.807415	0.403708	−	0.914888i	$-0.367721\pi$
$701$	21.3774	0.807415	0.403708	+	0.914888i	$0.367721\pi$
$702$	0	0
$703$	−18.6413	−0.703069
$704$	73.2948	2.76240
$705$	0	0
$706$	−80.5257	−3.03063
$707$	−33.7207	−1.26820
$708$	0	0
$709$	30.3719	1.14064	0.570320	−	0.821422i	$-0.306819\pi$
$709$	30.3719	1.14064	0.570320	+	0.821422i	$0.306819\pi$
$710$	0	0
$711$	0	0
$712$	40.8850	1.53223
$713$	5.10809	0.191299
$714$	0	0
$715$	0	0
$716$	62.8588	2.34914
$717$	0	0
$718$	67.2642	2.51028
$719$	−8.41851	−0.313958	−0.156979	−	0.987602i	$-0.550175\pi$
$719$	−8.41851	−0.313958	−0.156979	+	0.987602i	$0.550175\pi$
$720$	0	0
$721$	−16.5929	−0.617951
$722$	−20.2766	−0.754615
$723$	0	0
$724$	−91.8600	−3.41395
$725$	0	0
$726$	0	0
$727$	−19.7505	−0.732507	−0.366253	−	0.930515i	$-0.619360\pi$
$727$	−19.7505	−0.732507	−0.366253	+	0.930515i	$0.619360\pi$
$728$	59.2374	2.19548
$729$	0	0
$730$	0	0
$731$	8.00000	0.295891
$732$	0	0
$733$	−19.7280	−0.728670	−0.364335	−	0.931268i	$-0.618704\pi$
$733$	−19.7280	−0.728670	−0.364335	+	0.931268i	$0.618704\pi$
$734$	−42.6058	−1.57261
$735$	0	0
$736$	−17.7652	−0.654835
$737$	25.9775	0.956892
$738$	0	0
$739$	−0.180969	−0.00665704	−0.00332852	−	0.999994i	$-0.501060\pi$
$739$	−0.180969	−0.00665704	−0.00332852	+	0.999994i	$0.501060\pi$
$740$	0	0
$741$	0	0
$742$	−77.8723	−2.85878
$743$	6.05023	0.221961	0.110981	−	0.993823i	$-0.464601\pi$
$743$	6.05023	0.221961	0.110981	+	0.993823i	$0.464601\pi$
$744$	0	0
$745$	0	0
$746$	−60.9500	−2.23154
$747$	0	0
$748$	−47.0705	−1.72107
$749$	36.0770	1.31823
$750$	0	0
$751$	−21.5659	−0.786952	−0.393476	−	0.919335i	$-0.628728\pi$
$751$	−21.5659	−0.786952	−0.393476	+	0.919335i	$0.628728\pi$
$752$	107.631	3.92490
$753$	0	0
$754$	−61.0885	−2.22471
$755$	0	0
$756$	0	0
$757$	24.7798	0.900638	0.450319	−	0.892868i	$-0.351310\pi$
$757$	24.7798	0.900638	0.450319	+	0.892868i	$0.351310\pi$
$758$	27.6681	1.00495
$759$	0	0
$760$	0	0
$761$	−2.29916	−0.0833443	−0.0416722	−	0.999131i	$-0.513269\pi$
$761$	−2.29916	−0.0833443	−0.0416722	+	0.999131i	$0.513269\pi$
$762$	0	0
$763$	−46.9473	−1.69961
$764$	68.9629	2.49499
$765$	0	0
$766$	16.7393	0.604814
$767$	−20.4047	−0.736770
$768$	0	0
$769$	−10.3692	−0.373923	−0.186961	−	0.982367i	$-0.559864\pi$
$769$	−10.3692	−0.373923	−0.186961	+	0.982367i	$0.559864\pi$
$770$	0	0
$771$	0	0
$772$	65.9187	2.37247
$773$	12.7864	0.459895	0.229947	−	0.973203i	$-0.426145\pi$
$773$	12.7864	0.459895	0.229947	+	0.973203i	$0.426145\pi$
$774$	0	0
$775$	0	0
$776$	35.9297	1.28980
$777$	0	0
$778$	−74.4528	−2.66926
$779$	4.08506	0.146362
$780$	0	0
$781$	0.0508602	0.00181992
$782$	7.12311	0.254722
$783$	0	0
$784$	6.83361	0.244058
$785$	0	0
$786$	0	0
$787$	34.9239	1.24490	0.622451	−	0.782659i	$-0.286137\pi$
$787$	34.9239	1.24490	0.622451	+	0.782659i	$0.286137\pi$
$788$	20.4183	0.727372
$789$	0	0
$790$	0	0
$791$	12.0138	0.427163
$792$	0	0
$793$	0.650244	0.0230908
$794$	−2.88667	−0.102444
$795$	0	0
$796$	116.459	4.12776
$797$	4.87844	0.172803	0.0864016	−	0.996260i	$-0.472463\pi$
$797$	4.87844	0.172803	0.0864016	+	0.996260i	$0.472463\pi$
$798$	0	0
$799$	−21.7183	−0.768339
$800$	0	0
$801$	0	0
$802$	−4.03483	−0.142475
$803$	18.7522	0.661751
$804$	0	0
$805$	0	0
$806$	−33.8956	−1.19392
$807$	0	0
$808$	−107.803	−3.79248
$809$	−18.1498	−0.638112	−0.319056	−	0.947736i	$-0.603366\pi$
$809$	−18.1498	−0.638112	−0.319056	+	0.947736i	$0.603366\pi$
$810$	0	0
$811$	17.8019	0.625110	0.312555	−	0.949900i	$-0.398815\pi$
$811$	17.8019	0.625110	0.312555	+	0.949900i	$0.398815\pi$
$812$	−132.680	−4.65614
$813$	0	0
$814$	−50.2107	−1.75989
$815$	0	0
$816$	0	0
$817$	−10.2462	−0.358470
$818$	56.3952	1.97181
$819$	0	0
$820$	0	0
$821$	−32.9701	−1.15066	−0.575332	−	0.817920i	$-0.695127\pi$
$821$	−32.9701	−1.15066	−0.575332	+	0.817920i	$0.695127\pi$
$822$	0	0
$823$	39.5819	1.37974	0.689869	−	0.723935i	$-0.257668\pi$
$823$	39.5819	1.37974	0.689869	+	0.723935i	$0.257668\pi$
$824$	−53.0462	−1.84795
$825$	0	0
$826$	−61.1839	−2.12886
$827$	−1.50849	−0.0524554	−0.0262277	−	0.999656i	$-0.508349\pi$
$827$	−1.50849	−0.0524554	−0.0262277	+	0.999656i	$0.508349\pi$
$828$	0	0
$829$	−7.66718	−0.266292	−0.133146	−	0.991096i	$-0.542508\pi$
$829$	−7.66718	−0.266292	−0.133146	+	0.991096i	$0.542508\pi$
$830$	0	0
$831$	0	0
$832$	53.3108	1.84822
$833$	−1.37892	−0.0477767
$834$	0	0
$835$	0	0
$836$	60.2868	2.08506
$837$	0	0
$838$	50.5207	1.74521
$839$	8.53602	0.294696	0.147348	−	0.989085i	$-0.452926\pi$
$839$	8.53602	0.294696	0.147348	+	0.989085i	$0.452926\pi$
$840$	0	0
$841$	55.7519	1.92248
$842$	55.5305	1.91371
$843$	0	0
$844$	25.2795	0.870155
$845$	0	0
$846$	0	0
$847$	1.29477	0.0444889
$848$	−138.157	−4.74432
$849$	0	0
$850$	0	0
$851$	5.50369	0.188664
$852$	0	0
$853$	48.0665	1.64577	0.822883	−	0.568211i	$-0.192364\pi$
$853$	48.0665	1.64577	0.822883	+	0.568211i	$0.192364\pi$
$854$	1.94977	0.0667199
$855$	0	0
$856$	115.335	3.94209
$857$	29.4313	1.00535	0.502677	−	0.864474i	$-0.332348\pi$
$857$	29.4313	1.00535	0.502677	+	0.864474i	$0.332348\pi$
$858$	0	0
$859$	−33.5308	−1.14406	−0.572029	−	0.820234i	$-0.693843\pi$
$859$	−33.5308	−1.14406	−0.572029	+	0.820234i	$0.693843\pi$
$860$	0	0
$861$	0	0
$862$	−72.7670	−2.47845
$863$	6.79483	0.231299	0.115649	−	0.993290i	$-0.463105\pi$
$863$	6.79483	0.231299	0.115649	+	0.993290i	$0.463105\pi$
$864$	0	0
$865$	0	0
$866$	68.1192	2.31478
$867$	0	0
$868$	−73.6186	−2.49878
$869$	−29.3864	−0.996866
$870$	0	0
$871$	18.8946	0.640220
$872$	−150.087	−5.08259
$873$	0	0
$874$	−9.12311	−0.308594
$875$	0	0
$876$	0	0
$877$	21.9904	0.742563	0.371281	−	0.928520i	$-0.378918\pi$
$877$	21.9904	0.742563	0.371281	+	0.928520i	$0.378918\pi$
$878$	92.6869	3.12803
$879$	0	0
$880$	0	0
$881$	−3.66242	−0.123390	−0.0616951	−	0.998095i	$-0.519651\pi$
$881$	−3.66242	−0.123390	−0.0616951	+	0.998095i	$0.519651\pi$
$882$	0	0
$883$	−10.6640	−0.358874	−0.179437	−	0.983769i	$-0.557428\pi$
$883$	−10.6640	−0.358874	−0.179437	+	0.983769i	$0.557428\pi$
$884$	−34.2366	−1.15150
$885$	0	0
$886$	−80.3427	−2.69916
$887$	0.481294	0.0161603	0.00808014	−	0.999967i	$-0.497428\pi$
$887$	0.481294	0.0161603	0.00808014	+	0.999967i	$0.497428\pi$
$888$	0	0
$889$	50.6235	1.69786
$890$	0	0
$891$	0	0
$892$	−72.4228	−2.42489
$893$	27.8163	0.930837
$894$	0	0
$895$	0	0
$896$	62.4107	2.08499
$897$	0	0
$898$	7.20400	0.240401
$899$	47.0255	1.56839
$900$	0	0
$901$	27.8779	0.928748
$902$	11.0032	0.366367
$903$	0	0
$904$	38.4074	1.27741
$905$	0	0
$906$	0	0
$907$	50.3078	1.67044	0.835222	−	0.549913i	$-0.185339\pi$
$907$	50.3078	1.67044	0.835222	+	0.549913i	$0.185339\pi$
$908$	152.929	5.07513
$909$	0	0
$910$	0	0
$911$	39.5103	1.30903	0.654517	−	0.756047i	$-0.272872\pi$
$911$	39.5103	1.30903	0.654517	+	0.756047i	$0.272872\pi$
$912$	0	0
$913$	−11.9272	−0.394734
$914$	83.2398	2.75333
$915$	0	0
$916$	−49.3297	−1.62990
$917$	10.6090	0.350341
$918$	0	0
$919$	−10.0307	−0.330881	−0.165441	−	0.986220i	$-0.552905\pi$
$919$	−10.0307	−0.330881	−0.165441	+	0.986220i	$0.552905\pi$
$920$	0	0
$921$	0	0
$922$	71.6536	2.35979
$923$	0.0369930	0.00121764
$924$	0	0
$925$	0	0
$926$	−71.9207	−2.36346
$927$	0	0
$928$	−163.548	−5.36873
$929$	−5.83676	−0.191498	−0.0957490	−	0.995406i	$-0.530525\pi$
$929$	−5.83676	−0.191498	−0.0957490	+	0.995406i	$0.530525\pi$
$930$	0	0
$931$	1.76609	0.0578812
$932$	−66.5949	−2.18139
$933$	0	0
$934$	94.9477	3.10678
$935$	0	0
$936$	0	0
$937$	−37.5175	−1.22564	−0.612822	−	0.790221i	$-0.709966\pi$
$937$	−37.5175	−1.22564	−0.612822	+	0.790221i	$0.709966\pi$
$938$	56.6560	1.84989
$939$	0	0
$940$	0	0
$941$	−45.6189	−1.48713	−0.743566	−	0.668662i	$-0.766867\pi$
$941$	−45.6189	−1.48713	−0.743566	+	0.668662i	$0.766867\pi$
$942$	0	0
$943$	−1.20608	−0.0392754
$944$	−108.549	−3.53297
$945$	0	0
$946$	−27.5984	−0.897302
$947$	−17.3357	−0.563333	−0.281667	−	0.959512i	$-0.590887\pi$
$947$	−17.3357	−0.563333	−0.281667	+	0.959512i	$0.590887\pi$
$948$	0	0
$949$	13.6394	0.442752
$950$	0	0
$951$	0	0
$952$	−63.5888	−2.06093
$953$	14.2706	0.462269	0.231135	−	0.972922i	$-0.425756\pi$
$953$	14.2706	0.462269	0.231135	+	0.972922i	$0.425756\pi$
$954$	0	0
$955$	0	0
$956$	−100.836	−3.26128
$957$	0	0
$958$	105.237	3.40007
$959$	56.6915	1.83066
$960$	0	0
$961$	−4.90742	−0.158304
$962$	−36.5206	−1.17747
$963$	0	0
$964$	88.1492	2.83909
$965$	0	0
$966$	0	0
$967$	4.07663	0.131096	0.0655478	−	0.997849i	$-0.479121\pi$
$967$	4.07663	0.131096	0.0655478	+	0.997849i	$0.479121\pi$
$968$	4.13929	0.133042
$969$	0	0
$970$	0	0
$971$	20.2988	0.651419	0.325709	−	0.945470i	$-0.394397\pi$
$971$	20.2988	0.651419	0.325709	+	0.945470i	$0.394397\pi$
$972$	0	0
$973$	34.5015	1.10607
$974$	−64.8706	−2.07859
$975$	0	0
$976$	3.45918	0.110726
$977$	−40.3782	−1.29181	−0.645907	−	0.763416i	$-0.723521\pi$
$977$	−40.3782	−1.29181	−0.645907	+	0.763416i	$0.723521\pi$
$978$	0	0
$979$	15.7944	0.504792
$980$	0	0
$981$	0	0
$982$	−15.0851	−0.481386
$983$	−53.1628	−1.69563	−0.847815	−	0.530292i	$-0.822082\pi$
$983$	−53.1628	−1.69563	−0.847815	+	0.530292i	$0.822082\pi$
$984$	0	0
$985$	0	0
$986$	65.5759	2.08836
$987$	0	0
$988$	43.8494	1.39504
$989$	3.02511	0.0961930
$990$	0	0
$991$	36.6746	1.16501	0.582503	−	0.812829i	$-0.302073\pi$
$991$	36.6746	1.16501	0.582503	+	0.812829i	$0.302073\pi$
$992$	−90.7464	−2.88120
$993$	0	0
$994$	0.110925	0.00351831
$995$	0	0
$996$	0	0
$997$	11.1189	0.352140	0.176070	−	0.984378i	$-0.443661\pi$
$997$	11.1189	0.352140	0.176070	+	0.984378i	$0.443661\pi$
$998$	25.6779	0.812819
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5175.2.a.bx.1.4		4
3.2	odd	2		575.2.a.h.1.1		4
5.4	even	2		1035.2.a.o.1.1		4
12.11	even	2		9200.2.a.cl.1.1		4
15.2	even	4		575.2.b.e.24.1		8
15.8	even	4		575.2.b.e.24.8		8
15.14	odd	2		115.2.a.c.1.4	✓	4
60.59	even	2		1840.2.a.u.1.4		4
105.104	even	2		5635.2.a.v.1.4		4
120.29	odd	2		7360.2.a.cj.1.3		4
120.59	even	2		7360.2.a.cg.1.2		4
345.344	even	2		2645.2.a.m.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
115.2.a.c.1.4	✓	4	15.14	odd	2
575.2.a.h.1.1		4	3.2	odd	2
575.2.b.e.24.1		8	15.2	even	4
575.2.b.e.24.8		8	15.8	even	4
1035.2.a.o.1.1		4	5.4	even	2
1840.2.a.u.1.4		4	60.59	even	2
2645.2.a.m.1.4		4	345.344	even	2
5175.2.a.bx.1.4		4	1.1	even	1	trivial
5635.2.a.v.1.4		4	105.104	even	2
7360.2.a.cg.1.2		4	120.59	even	2
7360.2.a.cj.1.3		4	120.29	odd	2
9200.2.a.cl.1.1		4	12.11	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:	\( N \)	\(=\)	\( 5175 = 3^{2} \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5175.a (trivial)

\(f(q)\)	\(=\)	\(q+2.69353 q^{2} +5.25508 q^{4} +2.74252 q^{7} +8.76763 q^{8} +3.38705 q^{11} +2.46356 q^{13} +7.38705 q^{14} +13.1057 q^{16} -2.64453 q^{17} +3.38705 q^{19} +9.12311 q^{22} -1.00000 q^{23} +6.63566 q^{26} +14.4122 q^{28} -9.20608 q^{29} -5.10809 q^{31} +17.7652 q^{32} -7.12311 q^{34} -5.50369 q^{37} +9.12311 q^{38} +1.20608 q^{41} -3.02511 q^{43} +17.7992 q^{44} -2.69353 q^{46} +8.21255 q^{47} +0.521423 q^{49} +12.9462 q^{52} -10.5417 q^{53} +24.0454 q^{56} -24.7968 q^{58} -8.28259 q^{59} +0.263945 q^{61} -13.7588 q^{62} +21.6397 q^{64} +7.66964 q^{67} -13.8972 q^{68} +0.0150161 q^{71} +5.53644 q^{73} -14.8243 q^{74} +17.7992 q^{76} +9.28906 q^{77} -8.67611 q^{79} +3.24861 q^{82} -3.52142 q^{83} -8.14822 q^{86} +29.6964 q^{88} +4.66318 q^{89} +6.75637 q^{91} -5.25508 q^{92} +22.1207 q^{94} +4.09799 q^{97} +1.40447 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{2} + 4 q^{4} + 3 q^{7} + 9 q^{8} - 4 q^{11} + 12 q^{14} + 8 q^{16} - q^{17} - 4 q^{19} + 20 q^{22} - 4 q^{23} + q^{26} + 22 q^{28} - 19 q^{29} - q^{31} + 20 q^{32} - 12 q^{34} + 3 q^{37} + 20 q^{38}+ \cdots + 16 q^{98}+O(q^{100}) \) 4 * q + 2 * q^2 + 4 * q^4 + 3 * q^7 + 9 * q^8 - 4 * q^11 + 12 * q^14 + 8 * q^16 - q^17 - 4 * q^19 + 20 * q^22 - 4 * q^23 + q^26 + 22 * q^28 - 19 * q^29 - q^31 + 20 * q^32 - 12 * q^34 + 3 * q^37 + 20 * q^38 - 13 * q^41 + 6 * q^43 + 18 * q^44 - 2 * q^46 + 6 * q^47 + 9 * q^49 + q^52 + 19 * q^53 + 10 * q^56 - 21 * q^58 - 23 * q^59 - 13 * q^62 + 27 * q^64 + 3 * q^67 - 4 * q^68 + 3 * q^71 + 32 * q^73 + 12 * q^74 + 18 * q^76 + 18 * q^77 + 2 * q^79 + 5 * q^82 - 21 * q^83 + 2 * q^86 + 14 * q^88 - 40 * q^91 - 4 * q^92 + 47 * q^94 + 18 * q^97 + 16 * q^98

Introduction

L-functions

Modular forms

Varieties

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Representations

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Database

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