LMFDB - Embedded newform 6975.2.a.ca.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [6975,2,Mod(1,6975)] 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("6975.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(6975, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 6975 = 3^{2} \cdot 5^{2} \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6975.a (trivial)

Newform invariants

comment:select newform

sage:traces = [6,-1,0,7,0,0,2,3,0,0,-7,0,4,-10,0,17,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

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

Self dual:	yes
Analytic conductor:	$55.6956554098$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.75968016.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 9x^{4} + 9x^{3} + 14x^{2} - 6x - 6 $ x^6 - x^5 - 9x^4 + 9x^3 + 14x^2 - 6x - 6
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2325)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.864597$ of defining polynomial
Character	$\chi$	$=$	6975.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-0.864597 q^{2} -1.25247 q^{4} -4.28308 q^{7} +2.81208 q^{8} -0.353929 q^{11} +3.89854 q^{13} +3.70313 q^{14} +0.0736318 q^{16} +5.22766 q^{17} +8.23515 q^{19} +0.306006 q^{22} +1.28131 q^{23} -3.37067 q^{26} +5.36443 q^{28} -7.27223 q^{29} -1.00000 q^{31} -5.68782 q^{32} -4.51982 q^{34} +2.30601 q^{37} -7.12009 q^{38} +2.68303 q^{41} +6.32275 q^{43} +0.443286 q^{44} -1.10782 q^{46} -6.68146 q^{47} +11.3447 q^{49} -4.88282 q^{52} +4.33661 q^{53} -12.0443 q^{56} +6.28755 q^{58} -1.04616 q^{59} -0.983032 q^{61} +0.864597 q^{62} +4.77040 q^{64} +4.92592 q^{67} -6.54751 q^{68} -10.7048 q^{71} +4.56262 q^{73} -1.99377 q^{74} -10.3143 q^{76} +1.51590 q^{77} -8.87953 q^{79} -2.31974 q^{82} -2.37726 q^{83} -5.46663 q^{86} -0.995275 q^{88} -0.170731 q^{89} -16.6978 q^{91} -1.60481 q^{92} +5.77676 q^{94} -1.01227 q^{97} -9.80862 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - q^{2} + 7 q^{4} + 2 q^{7} + 3 q^{8} - 7 q^{11} + 4 q^{13} - 10 q^{14} + 17 q^{16} + 17 q^{19} + 2 q^{22} + q^{23} - 2 q^{26} + 22 q^{28} + 8 q^{29} - 6 q^{31} + 35 q^{32} - 13 q^{34} + 14 q^{37}+ \cdots - 37 q^{98}+O(q^{100}) $ 6 * q - q^2 + 7 * q^4 + 2 * q^7 + 3 * q^8 - 7 * q^11 + 4 * q^13 - 10 * q^14 + 17 * q^16 + 17 * q^19 + 2 * q^22 + q^23 - 2 * q^26 + 22 * q^28 + 8 * q^29 - 6 * q^31 + 35 * q^32 - 13 * q^34 + 14 * q^37 - 10 * q^38 - 18 * q^41 + 15 * q^43 - 4 * q^44 - 10 * q^46 + 13 * q^47 + 18 * q^49 - 38 * q^52 + 13 * q^53 - 32 * q^56 + 23 * q^58 - 32 * q^59 + q^62 + 29 * q^64 + 9 * q^67 - 9 * q^68 - q^71 + 14 * q^73 - 20 * q^74 + 42 * q^76 + 14 * q^77 + 13 * q^79 - 32 * q^82 - 9 * q^83 + 11 * q^86 + 36 * q^88 - 34 * q^89 - 2 * q^91 + 46 * q^92 + 19 * q^94 + 30 * q^97 - 37 * 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 $

$2$	−0.864597	−0.611362	−0.305681	−	0.952134i	$-0.598884\pi$
$2$	−0.864597	−0.611362	−0.305681	+	0.952134i	$0.598884\pi$
$3$	0	0
$3$	0	0
$4$	−1.25247	−0.626236
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−4.28308	−1.61885	−0.809425	−	0.587223i	$-0.800221\pi$
$7$	−4.28308	−1.61885	−0.809425	+	0.587223i	$0.800221\pi$
$8$	2.81208	0.994219
$9$	0	0
$10$	0	0
$11$	−0.353929	−0.106714	−0.0533568	−	0.998576i	$-0.516992\pi$
$11$	−0.353929	−0.106714	−0.0533568	+	0.998576i	$0.516992\pi$
$12$	0	0
$13$	3.89854	1.08126	0.540631	−	0.841260i	$-0.318186\pi$
$13$	3.89854	1.08126	0.540631	+	0.841260i	$0.318186\pi$
$14$	3.70313	0.989704
$15$	0	0
$16$	0.0736318	0.0184079
$17$	5.22766	1.26789	0.633947	−	0.773376i	$-0.281434\pi$
$17$	5.22766	1.26789	0.633947	+	0.773376i	$0.281434\pi$
$18$	0	0
$19$	8.23515	1.88927	0.944637	−	0.328118i	$-0.106414\pi$
$19$	8.23515	1.88927	0.944637	+	0.328118i	$0.106414\pi$
$20$	0	0
$21$	0	0
$22$	0.306006	0.0652407
$23$	1.28131	0.267172	0.133586	−	0.991037i	$-0.457351\pi$
$23$	1.28131	0.267172	0.133586	+	0.991037i	$0.457351\pi$
$24$	0	0
$25$	0	0
$26$	−3.37067	−0.661042
$27$	0	0
$28$	5.36443	1.01378
$29$	−7.27223	−1.35042	−0.675209	−	0.737626i	$-0.735947\pi$
$29$	−7.27223	−1.35042	−0.675209	+	0.737626i	$0.735947\pi$
$30$	0	0
$31$	−1.00000	−0.179605
$31$	−1.00000	−0.179605
$32$	−5.68782	−1.00547
$33$	0	0
$34$	−4.51982	−0.775143
$35$	0	0
$36$	0	0
$37$	2.30601	0.379105	0.189553	−	0.981871i	$-0.439296\pi$
$37$	2.30601	0.379105	0.189553	+	0.981871i	$0.439296\pi$
$38$	−7.12009	−1.15503
$39$	0	0
$40$	0	0
$41$	2.68303	0.419020	0.209510	−	0.977807i	$-0.432813\pi$
$41$	2.68303	0.419020	0.209510	+	0.977807i	$0.432813\pi$
$42$	0	0
$43$	6.32275	0.964210	0.482105	−	0.876114i	$-0.339872\pi$
$43$	6.32275	0.964210	0.482105	+	0.876114i	$0.339872\pi$
$44$	0.443286	0.0668279
$45$	0	0
$46$	−1.10782	−0.163339
$47$	−6.68146	−0.974590	−0.487295	−	0.873237i	$-0.662016\pi$
$47$	−6.68146	−0.974590	−0.487295	+	0.873237i	$0.662016\pi$
$48$	0	0
$49$	11.3447	1.62068
$50$	0	0
$51$	0	0
$52$	−4.88282	−0.677125
$53$	4.33661	0.595679	0.297840	−	0.954616i	$-0.403734\pi$
$53$	4.33661	0.595679	0.297840	+	0.954616i	$0.403734\pi$
$54$	0	0
$55$	0	0
$56$	−12.0443	−1.60949
$57$	0	0
$58$	6.28755	0.825595
$59$	−1.04616	−0.136198	−0.0680991	−	0.997679i	$-0.521693\pi$
$59$	−1.04616	−0.136198	−0.0680991	+	0.997679i	$0.521693\pi$
$60$	0	0
$61$	−0.983032	−0.125864	−0.0629321	−	0.998018i	$-0.520045\pi$
$61$	−0.983032	−0.125864	−0.0629321	+	0.998018i	$0.520045\pi$
$62$	0.864597	0.109804
$63$	0	0
$64$	4.77040	0.596301
$65$	0	0
$66$	0	0
$67$	4.92592	0.601796	0.300898	−	0.953656i	$-0.402714\pi$
$67$	4.92592	0.601796	0.300898	+	0.953656i	$0.402714\pi$
$68$	−6.54751	−0.794002
$69$	0	0
$70$	0	0
$71$	−10.7048	−1.27042	−0.635212	−	0.772338i	$-0.719087\pi$
$71$	−10.7048	−1.27042	−0.635212	+	0.772338i	$0.719087\pi$
$72$	0	0
$73$	4.56262	0.534015	0.267007	−	0.963695i	$-0.413965\pi$
$73$	4.56262	0.534015	0.267007	+	0.963695i	$0.413965\pi$
$74$	−1.99377	−0.231771
$75$	0	0
$76$	−10.3143	−1.18313
$77$	1.51590	0.172753
$78$	0	0
$79$	−8.87953	−0.999025	−0.499513	−	0.866307i	$-0.666488\pi$
$79$	−8.87953	−0.999025	−0.499513	+	0.866307i	$0.666488\pi$
$80$	0	0
$81$	0	0
$82$	−2.31974	−0.256173
$83$	−2.37726	−0.260938	−0.130469	−	0.991452i	$-0.541648\pi$
$83$	−2.37726	−0.260938	−0.130469	+	0.991452i	$0.541648\pi$
$84$	0	0
$85$	0	0
$86$	−5.46663	−0.589481
$87$	0	0
$88$	−0.995275	−0.106097
$89$	−0.170731	−0.0180974	−0.00904871	−	0.999959i	$-0.502880\pi$
$89$	−0.170731	−0.0180974	−0.00904871	+	0.999959i	$0.502880\pi$
$90$	0	0
$91$	−16.6978	−1.75040
$92$	−1.60481	−0.167313
$93$	0	0
$94$	5.77676	0.595828
$95$	0	0
$96$	0	0
$97$	−1.01227	−0.102780	−0.0513902	−	0.998679i	$-0.516365\pi$
$97$	−1.01227	−0.102780	−0.0513902	+	0.998679i	$0.516365\pi$
$98$	−9.80862	−0.990820
$99$	0	0
$100$	0	0
$101$	−17.9045	−1.78157	−0.890783	−	0.454428i	$-0.849844\pi$
$101$	−17.9045	−1.78157	−0.890783	+	0.454428i	$0.849844\pi$
$102$	0	0
$103$	8.49535	0.837072	0.418536	−	0.908200i	$-0.362543\pi$
$103$	8.49535	0.837072	0.418536	+	0.908200i	$0.362543\pi$
$104$	10.9630	1.07501
$105$	0	0
$106$	−3.74942	−0.364176
$107$	4.50652	0.435662	0.217831	−	0.975986i	$-0.430102\pi$
$107$	4.50652	0.435662	0.217831	+	0.975986i	$0.430102\pi$
$108$	0	0
$109$	14.3051	1.37018	0.685088	−	0.728460i	$-0.259764\pi$
$109$	14.3051	1.37018	0.685088	+	0.728460i	$0.259764\pi$
$110$	0	0
$111$	0	0
$112$	−0.315370	−0.0297997
$113$	−13.4229	−1.26272	−0.631360	−	0.775489i	$-0.717503\pi$
$113$	−13.4229	−1.26272	−0.631360	+	0.775489i	$0.717503\pi$
$114$	0	0
$115$	0	0
$116$	9.10827	0.845681
$117$	0	0
$118$	0.904506	0.0832665
$119$	−22.3905	−2.05253
$120$	0	0
$121$	−10.8747	−0.988612
$122$	0.849926	0.0769487
$123$	0	0
$124$	1.25247	0.112475
$125$	0	0
$126$	0	0
$127$	−13.7611	−1.22110	−0.610550	−	0.791977i	$-0.709052\pi$
$127$	−13.7611	−1.22110	−0.610550	+	0.791977i	$0.709052\pi$
$128$	7.25116	0.640918
$129$	0	0
$130$	0	0
$131$	−6.94358	−0.606663	−0.303332	−	0.952885i	$-0.598099\pi$
$131$	−6.94358	−0.606663	−0.303332	+	0.952885i	$0.598099\pi$
$132$	0	0
$133$	−35.2718	−3.05845
$134$	−4.25893	−0.367916
$135$	0	0
$136$	14.7006	1.26057
$137$	7.86949	0.672336	0.336168	−	0.941802i	$-0.390869\pi$
$137$	7.86949	0.672336	0.336168	+	0.941802i	$0.390869\pi$
$138$	0	0
$139$	7.69265	0.652482	0.326241	−	0.945287i	$-0.394218\pi$
$139$	7.69265	0.652482	0.326241	+	0.945287i	$0.394218\pi$
$140$	0	0
$141$	0	0
$142$	9.25532	0.776689
$143$	−1.37981	−0.115385
$144$	0	0
$145$	0	0
$146$	−3.94483	−0.326476
$147$	0	0
$148$	−2.88821	−0.237409
$149$	12.6799	1.03878	0.519388	−	0.854539i	$-0.326160\pi$
$149$	12.6799	1.03878	0.519388	+	0.854539i	$0.326160\pi$
$150$	0	0
$151$	−2.94960	−0.240035	−0.120017	−	0.992772i	$-0.538295\pi$
$151$	−2.94960	−0.240035	−0.120017	+	0.992772i	$0.538295\pi$
$152$	23.1579	1.87835
$153$	0	0
$154$	−1.31065	−0.105615
$155$	0	0
$156$	0	0
$157$	−3.49659	−0.279058	−0.139529	−	0.990218i	$-0.544559\pi$
$157$	−3.49659	−0.279058	−0.139529	+	0.990218i	$0.544559\pi$
$158$	7.67721	0.610766
$159$	0	0
$160$	0	0
$161$	−5.48795	−0.432511
$162$	0	0
$163$	−4.07999	−0.319570	−0.159785	−	0.987152i	$-0.551080\pi$
$163$	−4.07999	−0.319570	−0.159785	+	0.987152i	$0.551080\pi$
$164$	−3.36043	−0.262405
$165$	0	0
$166$	2.05537	0.159528
$167$	11.2784	0.872747	0.436373	−	0.899766i	$-0.356263\pi$
$167$	11.2784	0.872747	0.436373	+	0.899766i	$0.356263\pi$
$168$	0	0
$169$	2.19864	0.169126
$170$	0	0
$171$	0	0
$172$	−7.91906	−0.603823
$173$	22.4690	1.70829	0.854145	−	0.520036i	$-0.174081\pi$
$173$	22.4690	1.70829	0.854145	+	0.520036i	$0.174081\pi$
$174$	0	0
$175$	0	0
$176$	−0.0260604	−0.00196438
$177$	0	0
$178$	0.147613	0.0110641
$179$	−6.71387	−0.501818	−0.250909	−	0.968011i	$-0.580729\pi$
$179$	−6.71387	−0.501818	−0.250909	+	0.968011i	$0.580729\pi$
$180$	0	0
$181$	26.4167	1.96354	0.981769	−	0.190076i	$-0.0608733\pi$
$181$	26.4167	1.96354	0.981769	+	0.190076i	$0.0608733\pi$
$182$	14.4368	1.07013
$183$	0	0
$184$	3.60315	0.265627
$185$	0	0
$186$	0	0
$187$	−1.85022	−0.135302
$188$	8.36834	0.610324
$189$	0	0
$190$	0	0
$191$	19.9355	1.44248	0.721240	−	0.692685i	$-0.243573\pi$
$191$	19.9355	1.44248	0.721240	+	0.692685i	$0.243573\pi$
$192$	0	0
$193$	3.98672	0.286970	0.143485	−	0.989652i	$-0.454169\pi$
$193$	3.98672	0.286970	0.143485	+	0.989652i	$0.454169\pi$
$194$	0.875204	0.0628360
$195$	0	0
$196$	−14.2090	−1.01493
$197$	23.1920	1.65236	0.826179	−	0.563407i	$-0.190510\pi$
$197$	23.1920	1.65236	0.826179	+	0.563407i	$0.190510\pi$
$198$	0	0
$199$	−12.5109	−0.886876	−0.443438	−	0.896305i	$-0.646241\pi$
$199$	−12.5109	−0.886876	−0.443438	+	0.896305i	$0.646241\pi$
$200$	0	0
$201$	0	0
$202$	15.4802	1.08918
$203$	31.1475	2.18613
$204$	0	0
$205$	0	0
$206$	−7.34505	−0.511754
$207$	0	0
$208$	0.287057	0.0199038
$209$	−2.91466	−0.201611
$210$	0	0
$211$	23.1316	1.59245	0.796224	−	0.605002i	$-0.206828\pi$
$211$	23.1316	1.59245	0.796224	+	0.605002i	$0.206828\pi$
$212$	−5.43148	−0.373036
$213$	0	0
$214$	−3.89633	−0.266347
$215$	0	0
$216$	0	0
$217$	4.28308	0.290754
$218$	−12.3681	−0.837674
$219$	0	0
$220$	0	0
$221$	20.3803	1.37093
$222$	0	0
$223$	23.6450	1.58339	0.791693	−	0.610920i	$-0.209200\pi$
$223$	23.6450	1.58339	0.791693	+	0.610920i	$0.209200\pi$
$224$	24.3613	1.62771
$225$	0	0
$226$	11.6054	0.771980
$227$	26.4506	1.75559	0.877794	−	0.479039i	$-0.159015\pi$
$227$	26.4506	1.75559	0.877794	+	0.479039i	$0.159015\pi$
$228$	0	0
$229$	−7.61479	−0.503199	−0.251600	−	0.967831i	$-0.580957\pi$
$229$	−7.61479	−0.503199	−0.251600	+	0.967831i	$0.580957\pi$
$230$	0	0
$231$	0	0
$232$	−20.4501	−1.34261
$233$	−3.13938	−0.205668	−0.102834	−	0.994699i	$-0.532791\pi$
$233$	−3.13938	−0.205668	−0.102834	+	0.994699i	$0.532791\pi$
$234$	0	0
$235$	0	0
$236$	1.31028	0.0852923
$237$	0	0
$238$	19.3587	1.25484
$239$	−26.0939	−1.68787	−0.843936	−	0.536444i	$-0.819767\pi$
$239$	−26.0939	−1.68787	−0.843936	+	0.536444i	$0.819767\pi$
$240$	0	0
$241$	−5.22027	−0.336267	−0.168134	−	0.985764i	$-0.553774\pi$
$241$	−5.22027	−0.336267	−0.168134	+	0.985764i	$0.553774\pi$
$242$	9.40226	0.604400
$243$	0	0
$244$	1.23122	0.0788208
$245$	0	0
$246$	0	0
$247$	32.1051	2.04280
$248$	−2.81208	−0.178567
$249$	0	0
$250$	0	0
$251$	−29.8930	−1.88683	−0.943414	−	0.331617i	$-0.892406\pi$
$251$	−29.8930	−1.88683	−0.943414	+	0.331617i	$0.892406\pi$
$252$	0	0
$253$	−0.453493	−0.0285109
$254$	11.8978	0.746535
$255$	0	0
$256$	−15.8101	−0.988133
$257$	15.6602	0.976855	0.488427	−	0.872604i	$-0.337571\pi$
$257$	15.6602	0.976855	0.488427	+	0.872604i	$0.337571\pi$
$258$	0	0
$259$	−9.87680	−0.613714
$260$	0	0
$261$	0	0
$262$	6.00339	0.370891
$263$	−26.1422	−1.61200	−0.805998	−	0.591918i	$-0.798371\pi$
$263$	−26.1422	−1.61200	−0.805998	+	0.591918i	$0.798371\pi$
$264$	0	0
$265$	0	0
$266$	30.4959	1.86982
$267$	0	0
$268$	−6.16957	−0.376867
$269$	32.0456	1.95386	0.976928	−	0.213567i	$-0.0685083\pi$
$269$	32.0456	1.95386	0.976928	+	0.213567i	$0.0685083\pi$
$270$	0	0
$271$	−17.3240	−1.05236	−0.526179	−	0.850374i	$-0.676376\pi$
$271$	−17.3240	−1.05236	−0.526179	+	0.850374i	$0.676376\pi$
$272$	0.384922	0.0233393
$273$	0	0
$274$	−6.80394	−0.411041
$275$	0	0
$276$	0	0
$277$	16.6490	1.00034	0.500172	−	0.865926i	$-0.333270\pi$
$277$	16.6490	1.00034	0.500172	+	0.865926i	$0.333270\pi$
$278$	−6.65104	−0.398903
$279$	0	0
$280$	0	0
$281$	−13.1488	−0.784392	−0.392196	−	0.919882i	$-0.628284\pi$
$281$	−13.1488	−0.784392	−0.392196	+	0.919882i	$0.628284\pi$
$282$	0	0
$283$	25.9728	1.54392	0.771962	−	0.635669i	$-0.219276\pi$
$283$	25.9728	1.54392	0.771962	+	0.635669i	$0.219276\pi$
$284$	13.4074	0.795586
$285$	0	0
$286$	1.19298	0.0705422
$287$	−11.4916	−0.678330
$288$	0	0
$289$	10.3285	0.607558
$290$	0	0
$291$	0	0
$292$	−5.71456	−0.334419
$293$	−27.0981	−1.58309	−0.791543	−	0.611114i	$-0.790722\pi$
$293$	−27.0981	−1.58309	−0.791543	+	0.611114i	$0.790722\pi$
$294$	0	0
$295$	0	0
$296$	6.48467	0.376914
$297$	0	0
$298$	−10.9630	−0.635068
$299$	4.99525	0.288883
$300$	0	0
$301$	−27.0808	−1.56091
$302$	2.55021	0.146748
$303$	0	0
$304$	0.606369	0.0347776
$305$	0	0
$306$	0	0
$307$	−7.83898	−0.447394	−0.223697	−	0.974659i	$-0.571813\pi$
$307$	−7.83898	−0.447394	−0.223697	+	0.974659i	$0.571813\pi$
$308$	−1.89863	−0.108184
$309$	0	0
$310$	0	0
$311$	−0.0311685	−0.00176740	−0.000883702	−	1.00000i	$-0.500281\pi$
$311$	−0.0311685	−0.00176740	−0.000883702		1.00000i	$0.500281\pi$
$312$	0	0
$313$	−2.44289	−0.138080	−0.0690401	−	0.997614i	$-0.521994\pi$
$313$	−2.44289	−0.138080	−0.0690401	+	0.997614i	$0.521994\pi$
$314$	3.02314	0.170606
$315$	0	0
$316$	11.1214	0.625626
$317$	−4.03351	−0.226544	−0.113272	−	0.993564i	$-0.536133\pi$
$317$	−4.03351	−0.226544	−0.113272	+	0.993564i	$0.536133\pi$
$318$	0	0
$319$	2.57385	0.144108
$320$	0	0
$321$	0	0
$322$	4.74487	0.264421
$323$	43.0506	2.39540
$324$	0	0
$325$	0	0
$326$	3.52755	0.195373
$327$	0	0
$328$	7.54490	0.416597
$329$	28.6172	1.57772
$330$	0	0
$331$	−11.0802	−0.609025	−0.304512	−	0.952508i	$-0.598493\pi$
$331$	−11.0802	−0.609025	−0.304512	+	0.952508i	$0.598493\pi$
$332$	2.97745	0.163409
$333$	0	0
$334$	−9.75125	−0.533565
$335$	0	0
$336$	0	0
$337$	10.4054	0.566817	0.283409	−	0.958999i	$-0.408535\pi$
$337$	10.4054	0.566817	0.283409	+	0.958999i	$0.408535\pi$
$338$	−1.90094	−0.103397
$339$	0	0
$340$	0	0
$341$	0.353929	0.0191663
$342$	0	0
$343$	−18.6088	−1.00478
$344$	17.7800	0.958636
$345$	0	0
$346$	−19.4267	−1.04438
$347$	−23.7995	−1.27762	−0.638812	−	0.769363i	$-0.720574\pi$
$347$	−23.7995	−1.27762	−0.638812	+	0.769363i	$0.720574\pi$
$348$	0	0
$349$	−26.1849	−1.40165	−0.700824	−	0.713335i	$-0.747184\pi$
$349$	−26.1849	−1.40165	−0.700824	+	0.713335i	$0.747184\pi$
$350$	0	0
$351$	0	0
$352$	2.01308	0.107298
$353$	14.3318	0.762807	0.381404	−	0.924409i	$-0.375441\pi$
$353$	14.3318	0.762807	0.381404	+	0.924409i	$0.375441\pi$
$354$	0	0
$355$	0	0
$356$	0.213836	0.0113333
$357$	0	0
$358$	5.80479	0.306793
$359$	−29.2468	−1.54359	−0.771793	−	0.635874i	$-0.780640\pi$
$359$	−29.2468	−1.54359	−0.771793	+	0.635874i	$0.780640\pi$
$360$	0	0
$361$	48.8177	2.56935
$362$	−22.8398	−1.20043
$363$	0	0
$364$	20.9135	1.09616
$365$	0	0
$366$	0	0
$367$	21.6111	1.12809	0.564046	−	0.825743i	$-0.309244\pi$
$367$	21.6111	1.12809	0.564046	+	0.825743i	$0.309244\pi$
$368$	0.0943452	0.00491808
$369$	0	0
$370$	0	0
$371$	−18.5740	−0.964315
$372$	0	0
$373$	−21.6784	−1.12247	−0.561233	−	0.827658i	$-0.689673\pi$
$373$	−21.6784	−1.12247	−0.561233	+	0.827658i	$0.689673\pi$
$374$	1.59970	0.0827183
$375$	0	0
$376$	−18.7888	−0.968957
$377$	−28.3511	−1.46016
$378$	0	0
$379$	35.3697	1.81682	0.908408	−	0.418084i	$-0.137298\pi$
$379$	35.3697	1.81682	0.908408	+	0.418084i	$0.137298\pi$
$380$	0	0
$381$	0	0
$382$	−17.2361	−0.881878
$383$	30.1843	1.54235	0.771173	−	0.636626i	$-0.219670\pi$
$383$	30.1843	1.54235	0.771173	+	0.636626i	$0.219670\pi$
$384$	0	0
$385$	0	0
$386$	−3.44690	−0.175443
$387$	0	0
$388$	1.26784	0.0643648
$389$	8.07568	0.409453	0.204727	−	0.978819i	$-0.434370\pi$
$389$	8.07568	0.409453	0.204727	+	0.978819i	$0.434370\pi$
$390$	0	0
$391$	6.69826	0.338746
$392$	31.9023	1.61131
$393$	0	0
$394$	−20.0517	−1.01019
$395$	0	0
$396$	0	0
$397$	−34.0991	−1.71139	−0.855693	−	0.517483i	$-0.826869\pi$
$397$	−34.0991	−1.71139	−0.855693	+	0.517483i	$0.826869\pi$
$398$	10.8169	0.542202
$399$	0	0
$400$	0	0
$401$	−35.4194	−1.76876	−0.884380	−	0.466768i	$-0.845418\pi$
$401$	−35.4194	−1.76876	−0.884380	+	0.466768i	$0.845418\pi$
$402$	0	0
$403$	−3.89854	−0.194200
$404$	22.4249	1.11568
$405$	0	0
$406$	−26.9300	−1.33652
$407$	−0.816162	−0.0404557
$408$	0	0
$409$	−4.67021	−0.230927	−0.115463	−	0.993312i	$-0.536835\pi$
$409$	−4.67021	−0.230927	−0.115463	+	0.993312i	$0.536835\pi$
$410$	0	0
$411$	0	0
$412$	−10.6402	−0.524205
$413$	4.48078	0.220485
$414$	0	0
$415$	0	0
$416$	−22.1742	−1.08718
$417$	0	0
$418$	2.52000	0.123257
$419$	−11.4046	−0.557150	−0.278575	−	0.960414i	$-0.589862\pi$
$419$	−11.4046	−0.557150	−0.278575	+	0.960414i	$0.589862\pi$
$420$	0	0
$421$	6.33585	0.308791	0.154395	−	0.988009i	$-0.450657\pi$
$421$	6.33585	0.308791	0.154395	+	0.988009i	$0.450657\pi$
$422$	−19.9995	−0.973563
$423$	0	0
$424$	12.1949	0.592236
$425$	0	0
$426$	0	0
$427$	4.21040	0.203755
$428$	−5.64430	−0.272827
$429$	0	0
$430$	0	0
$431$	2.01105	0.0968691	0.0484346	−	0.998826i	$-0.484577\pi$
$431$	2.01105	0.0968691	0.0484346	+	0.998826i	$0.484577\pi$
$432$	0	0
$433$	37.7917	1.81615	0.908077	−	0.418803i	$-0.137550\pi$
$433$	37.7917	1.81615	0.908077	+	0.418803i	$0.137550\pi$
$434$	−3.70313	−0.177756
$435$	0	0
$436$	−17.9167	−0.858054
$437$	10.5518	0.504761
$438$	0	0
$439$	14.3895	0.686771	0.343386	−	0.939194i	$-0.388426\pi$
$439$	14.3895	0.686771	0.343386	+	0.939194i	$0.388426\pi$
$440$	0	0
$441$	0	0
$442$	−17.6207	−0.838132
$443$	35.8785	1.70464	0.852320	−	0.523021i	$-0.175195\pi$
$443$	35.8785	1.70464	0.852320	+	0.523021i	$0.175195\pi$
$444$	0	0
$445$	0	0
$446$	−20.4434	−0.968022
$447$	0	0
$448$	−20.4320	−0.965321
$449$	35.4975	1.67523	0.837615	−	0.546261i	$-0.183949\pi$
$449$	35.4975	1.67523	0.837615	+	0.546261i	$0.183949\pi$
$450$	0	0
$451$	−0.949604	−0.0447151
$452$	16.8118	0.790762
$453$	0	0
$454$	−22.8691	−1.07330
$455$	0	0
$456$	0	0
$457$	33.8742	1.58457	0.792283	−	0.610153i	$-0.208892\pi$
$457$	33.8742	1.58457	0.792283	+	0.610153i	$0.208892\pi$
$458$	6.58372	0.307637
$459$	0	0
$460$	0	0
$461$	19.2748	0.897717	0.448859	−	0.893603i	$-0.351831\pi$
$461$	19.2748	0.897717	0.448859	+	0.893603i	$0.351831\pi$
$462$	0	0
$463$	1.98874	0.0924244	0.0462122	−	0.998932i	$-0.485285\pi$
$463$	1.98874	0.0924244	0.0462122	+	0.998932i	$0.485285\pi$
$464$	−0.535467	−0.0248584
$465$	0	0
$466$	2.71430	0.125737
$467$	−3.18274	−0.147279	−0.0736397	−	0.997285i	$-0.523461\pi$
$467$	−3.18274	−0.147279	−0.0736397	+	0.997285i	$0.523461\pi$
$468$	0	0
$469$	−21.0981	−0.974218
$470$	0	0
$471$	0	0
$472$	−2.94188	−0.135411
$473$	−2.23780	−0.102894
$474$	0	0
$475$	0	0
$476$	28.0435	1.28537
$477$	0	0
$478$	22.5607	1.03190
$479$	−9.28348	−0.424173	−0.212086	−	0.977251i	$-0.568026\pi$
$479$	−9.28348	−0.424173	−0.212086	+	0.977251i	$0.568026\pi$
$480$	0	0
$481$	8.99006	0.409912
$482$	4.51343	0.205581
$483$	0	0
$484$	13.6203	0.619105
$485$	0	0
$486$	0	0
$487$	−8.58292	−0.388929	−0.194465	−	0.980910i	$-0.562297\pi$
$487$	−8.58292	−0.388929	−0.194465	+	0.980910i	$0.562297\pi$
$488$	−2.76436	−0.125137
$489$	0	0
$490$	0	0
$491$	34.6474	1.56362	0.781808	−	0.623519i	$-0.214298\pi$
$491$	34.6474	1.56362	0.781808	+	0.623519i	$0.214298\pi$
$492$	0	0
$493$	−38.0168	−1.71219
$494$	−27.7580	−1.24889
$495$	0	0
$496$	−0.0736318	−0.00330616
$497$	45.8494	2.05663
$498$	0	0
$499$	−11.2295	−0.502703	−0.251352	−	0.967896i	$-0.580875\pi$
$499$	−11.2295	−0.502703	−0.251352	+	0.967896i	$0.580875\pi$
$500$	0	0
$501$	0	0
$502$	25.8454	1.15354
$503$	12.4460	0.554941	0.277470	−	0.960734i	$-0.410504\pi$
$503$	12.4460	0.554941	0.277470	+	0.960734i	$0.410504\pi$
$504$	0	0
$505$	0	0
$506$	0.392089	0.0174305
$507$	0	0
$508$	17.2354	0.764697
$509$	26.3376	1.16740	0.583698	−	0.811971i	$-0.301605\pi$
$509$	26.3376	1.16740	0.583698	+	0.811971i	$0.301605\pi$
$510$	0	0
$511$	−19.5421	−0.864490
$512$	−0.832921	−0.0368102
$513$	0	0
$514$	−13.5397	−0.597212
$515$	0	0
$516$	0	0
$517$	2.36476	0.104002
$518$	8.53945	0.375202
$519$	0	0
$520$	0	0
$521$	8.41110	0.368497	0.184248	−	0.982880i	$-0.441015\pi$
$521$	8.41110	0.368497	0.184248	+	0.982880i	$0.441015\pi$
$522$	0	0
$523$	8.89885	0.389120	0.194560	−	0.980891i	$-0.437672\pi$
$523$	8.89885	0.389120	0.194560	+	0.980891i	$0.437672\pi$
$524$	8.69664	0.379914
$525$	0	0
$526$	22.6025	0.985514
$527$	−5.22766	−0.227721
$528$	0	0
$529$	−21.3582	−0.928619
$530$	0	0
$531$	0	0
$532$	44.1769	1.91531
$533$	10.4599	0.453070
$534$	0	0
$535$	0	0
$536$	13.8521	0.598318
$537$	0	0
$538$	−27.7066	−1.19451
$539$	−4.01523	−0.172948
$540$	0	0
$541$	−27.0487	−1.16291	−0.581457	−	0.813577i	$-0.697517\pi$
$541$	−27.0487	−1.16291	−0.581457	+	0.813577i	$0.697517\pi$
$542$	14.9783	0.643372
$543$	0	0
$544$	−29.7340	−1.27483
$545$	0	0
$546$	0	0
$547$	20.4371	0.873829	0.436914	−	0.899503i	$-0.356071\pi$
$547$	20.4371	0.873829	0.436914	+	0.899503i	$0.356071\pi$
$548$	−9.85632	−0.421041
$549$	0	0
$550$	0	0
$551$	−59.8879	−2.55131
$552$	0	0
$553$	38.0317	1.61727
$554$	−14.3947	−0.611573
$555$	0	0
$556$	−9.63484	−0.408608
$557$	−33.5070	−1.41974	−0.709869	−	0.704333i	$-0.751246\pi$
$557$	−33.5070	−1.41974	−0.709869	+	0.704333i	$0.751246\pi$
$558$	0	0
$559$	24.6495	1.04256
$560$	0	0
$561$	0	0
$562$	11.3684	0.479547
$563$	1.03505	0.0436223	0.0218112	−	0.999762i	$-0.493057\pi$
$563$	1.03505	0.0436223	0.0218112	+	0.999762i	$0.493057\pi$
$564$	0	0
$565$	0	0
$566$	−22.4560	−0.943897
$567$	0	0
$568$	−30.1027	−1.26308
$569$	−2.63277	−0.110371	−0.0551857	−	0.998476i	$-0.517575\pi$
$569$	−2.63277	−0.110371	−0.0551857	+	0.998476i	$0.517575\pi$
$570$	0	0
$571$	26.6263	1.11428	0.557138	−	0.830420i	$-0.311899\pi$
$571$	26.6263	1.11428	0.557138	+	0.830420i	$0.311899\pi$
$572$	1.72817	0.0722584
$573$	0	0
$574$	9.93564	0.414705
$575$	0	0
$576$	0	0
$577$	20.4641	0.851933	0.425967	−	0.904739i	$-0.359934\pi$
$577$	20.4641	0.851933	0.425967	+	0.904739i	$0.359934\pi$
$578$	−8.92997	−0.371438
$579$	0	0
$580$	0	0
$581$	10.1820	0.422420
$582$	0	0
$583$	−1.53485	−0.0635670
$584$	12.8304	0.530928
$585$	0	0
$586$	23.4289	0.967839
$587$	−21.2696	−0.877888	−0.438944	−	0.898514i	$-0.644647\pi$
$587$	−21.2696	−0.877888	−0.438944	+	0.898514i	$0.644647\pi$
$588$	0	0
$589$	−8.23515	−0.339324
$590$	0	0
$591$	0	0
$592$	0.169795	0.00697854
$593$	39.3701	1.61674	0.808369	−	0.588676i	$-0.200351\pi$
$593$	39.3701	1.61674	0.808369	+	0.588676i	$0.200351\pi$
$594$	0	0
$595$	0	0
$596$	−15.8812	−0.650519
$597$	0	0
$598$	−4.31887	−0.176612
$599$	−28.4292	−1.16158	−0.580792	−	0.814052i	$-0.697257\pi$
$599$	−28.4292	−1.16158	−0.580792	+	0.814052i	$0.697257\pi$
$600$	0	0
$601$	20.6046	0.840480	0.420240	−	0.907413i	$-0.361946\pi$
$601$	20.6046	0.840480	0.420240	+	0.907413i	$0.361946\pi$
$602$	23.4140	0.954282
$603$	0	0
$604$	3.69429	0.150318
$605$	0	0
$606$	0	0
$607$	−18.8232	−0.764010	−0.382005	−	0.924160i	$-0.624766\pi$
$607$	−18.8232	−0.764010	−0.382005	+	0.924160i	$0.624766\pi$
$608$	−46.8400	−1.89961
$609$	0	0
$610$	0	0
$611$	−26.0479	−1.05379
$612$	0	0
$613$	−18.0049	−0.727213	−0.363606	−	0.931553i	$-0.618455\pi$
$613$	−18.0049	−0.727213	−0.363606	+	0.931553i	$0.618455\pi$
$614$	6.77755	0.273520
$615$	0	0
$616$	4.26284	0.171755
$617$	−26.2389	−1.05634	−0.528169	−	0.849139i	$-0.677121\pi$
$617$	−26.2389	−1.05634	−0.528169	+	0.849139i	$0.677121\pi$
$618$	0	0
$619$	30.1624	1.21233	0.606166	−	0.795339i	$-0.292707\pi$
$619$	30.1624	1.21233	0.606166	+	0.795339i	$0.292707\pi$
$620$	0	0
$621$	0	0
$622$	0.0269482	0.00108052
$623$	0.731253	0.0292970
$624$	0	0
$625$	0	0
$626$	2.11211	0.0844170
$627$	0	0
$628$	4.37938	0.174756
$629$	12.0550	0.480665
$630$	0	0
$631$	30.7023	1.22224	0.611121	−	0.791538i	$-0.290719\pi$
$631$	30.7023	1.22224	0.611121	+	0.791538i	$0.290719\pi$
$632$	−24.9699	−0.993250
$633$	0	0
$634$	3.48736	0.138501
$635$	0	0
$636$	0	0
$637$	44.2279	1.75237
$638$	−2.22534	−0.0881022
$639$	0	0
$640$	0	0
$641$	−4.11497	−0.162531	−0.0812657	−	0.996692i	$-0.525896\pi$
$641$	−4.11497	−0.162531	−0.0812657	+	0.996692i	$0.525896\pi$
$642$	0	0
$643$	−24.2235	−0.955281	−0.477641	−	0.878555i	$-0.658508\pi$
$643$	−24.2235	−0.955281	−0.477641	+	0.878555i	$0.658508\pi$
$644$	6.87351	0.270854
$645$	0	0
$646$	−37.2214	−1.46446
$647$	−32.4635	−1.27627	−0.638136	−	0.769924i	$-0.720294\pi$
$647$	−32.4635	−1.27627	−0.638136	+	0.769924i	$0.720294\pi$
$648$	0	0
$649$	0.370266	0.0145342
$650$	0	0
$651$	0	0
$652$	5.11008	0.200126
$653$	17.3165	0.677648	0.338824	−	0.940850i	$-0.389971\pi$
$653$	17.3165	0.677648	0.338824	+	0.940850i	$0.389971\pi$
$654$	0	0
$655$	0	0
$656$	0.197557	0.00771329
$657$	0	0
$658$	−24.7423	−0.964556
$659$	−10.3947	−0.404922	−0.202461	−	0.979290i	$-0.564894\pi$
$659$	−10.3947	−0.404922	−0.202461	+	0.979290i	$0.564894\pi$
$660$	0	0
$661$	35.1133	1.36575	0.682874	−	0.730536i	$-0.260730\pi$
$661$	35.1133	1.36575	0.682874	+	0.730536i	$0.260730\pi$
$662$	9.57994	0.372335
$663$	0	0
$664$	−6.68503	−0.259430
$665$	0	0
$666$	0	0
$667$	−9.31799	−0.360794
$668$	−14.1259	−0.546546
$669$	0	0
$670$	0	0
$671$	0.347923	0.0134314
$672$	0	0
$673$	16.3883	0.631724	0.315862	−	0.948805i	$-0.397706\pi$
$673$	16.3883	0.631724	0.315862	+	0.948805i	$0.397706\pi$
$674$	−8.99646	−0.346531
$675$	0	0
$676$	−2.75374	−0.105913
$677$	11.3973	0.438036	0.219018	−	0.975721i	$-0.429715\pi$
$677$	11.3973	0.438036	0.219018	+	0.975721i	$0.429715\pi$
$678$	0	0
$679$	4.33562	0.166386
$680$	0	0
$681$	0	0
$682$	−0.306006	−0.0117176
$683$	33.7148	1.29006	0.645031	−	0.764157i	$-0.276845\pi$
$683$	33.7148	1.29006	0.645031	+	0.764157i	$0.276845\pi$
$684$	0	0
$685$	0	0
$686$	16.0891	0.614286
$687$	0	0
$688$	0.465555	0.0177491
$689$	16.9065	0.644085
$690$	0	0
$691$	−26.4611	−1.00663	−0.503313	−	0.864104i	$-0.667886\pi$
$691$	−26.4611	−1.00663	−0.503313	+	0.864104i	$0.667886\pi$
$692$	−28.1418	−1.06979
$693$	0	0
$694$	20.5770	0.781091
$695$	0	0
$696$	0	0
$697$	14.0260	0.531273
$698$	22.6394	0.856914
$699$	0	0
$700$	0	0
$701$	48.7391	1.84085	0.920425	−	0.390919i	$-0.127843\pi$
$701$	48.7391	1.84085	0.920425	+	0.390919i	$0.127843\pi$
$702$	0	0
$703$	18.9903	0.716233
$704$	−1.68838	−0.0636334
$705$	0	0
$706$	−12.3913	−0.466352
$707$	76.6864	2.88409
$708$	0	0
$709$	13.1850	0.495172	0.247586	−	0.968866i	$-0.420363\pi$
$709$	13.1850	0.495172	0.247586	+	0.968866i	$0.420363\pi$
$710$	0	0
$711$	0	0
$712$	−0.480108	−0.0179928
$713$	−1.28131	−0.0479855
$714$	0	0
$715$	0	0
$716$	8.40893	0.314257
$717$	0	0
$718$	25.2867	0.943690
$719$	12.3742	0.461481	0.230740	−	0.973015i	$-0.425885\pi$
$719$	12.3742	0.461481	0.230740	+	0.973015i	$0.425885\pi$
$720$	0	0
$721$	−36.3862	−1.35509
$722$	−42.2077	−1.57081
$723$	0	0
$724$	−33.0862	−1.22964
$725$	0	0
$726$	0	0
$727$	−18.0273	−0.668597	−0.334299	−	0.942467i	$-0.608499\pi$
$727$	−18.0273	−0.668597	−0.334299	+	0.942467i	$0.608499\pi$
$728$	−46.9554	−1.74028
$729$	0	0
$730$	0	0
$731$	33.0532	1.22252
$732$	0	0
$733$	16.6357	0.614453	0.307226	−	0.951636i	$-0.400599\pi$
$733$	16.6357	0.614453	0.307226	+	0.951636i	$0.400599\pi$
$734$	−18.6849	−0.689673
$735$	0	0
$736$	−7.28786	−0.268634
$737$	−1.74342	−0.0642198
$738$	0	0
$739$	−3.71551	−0.136677	−0.0683385	−	0.997662i	$-0.521770\pi$
$739$	−3.71551	−0.136677	−0.0683385	+	0.997662i	$0.521770\pi$
$740$	0	0
$741$	0	0
$742$	16.0590	0.589546
$743$	−18.8360	−0.691028	−0.345514	−	0.938414i	$-0.612295\pi$
$743$	−18.8360	−0.691028	−0.345514	+	0.938414i	$0.612295\pi$
$744$	0	0
$745$	0	0
$746$	18.7431	0.686233
$747$	0	0
$748$	2.31735	0.0847308
$749$	−19.3018	−0.705272
$750$	0	0
$751$	−12.6253	−0.460703	−0.230351	−	0.973108i	$-0.573988\pi$
$751$	−12.6253	−0.460703	−0.230351	+	0.973108i	$0.573988\pi$
$752$	−0.491967	−0.0179402
$753$	0	0
$754$	24.5123	0.892684
$755$	0	0
$756$	0	0
$757$	−52.2392	−1.89866	−0.949332	−	0.314274i	$-0.898239\pi$
$757$	−52.2392	−1.89866	−0.949332	+	0.314274i	$0.898239\pi$
$758$	−30.5805	−1.11073
$759$	0	0
$760$	0	0
$761$	−1.78070	−0.0645502	−0.0322751	−	0.999479i	$-0.510275\pi$
$761$	−1.78070	−0.0645502	−0.0322751	+	0.999479i	$0.510275\pi$
$762$	0	0
$763$	−61.2697	−2.21811
$764$	−24.9686	−0.903333
$765$	0	0
$766$	−26.0972	−0.942932
$767$	−4.07850	−0.147266
$768$	0	0
$769$	32.5936	1.17536	0.587678	−	0.809095i	$-0.300042\pi$
$769$	32.5936	1.17536	0.587678	+	0.809095i	$0.300042\pi$
$770$	0	0
$771$	0	0
$772$	−4.99325	−0.179711
$773$	23.1130	0.831317	0.415659	−	0.909521i	$-0.363551\pi$
$773$	23.1130	0.831317	0.415659	+	0.909521i	$0.363551\pi$
$774$	0	0
$775$	0	0
$776$	−2.84658	−0.102186
$777$	0	0
$778$	−6.98220	−0.250324
$779$	22.0952	0.791643
$780$	0	0
$781$	3.78873	0.135572
$782$	−5.79130	−0.207096
$783$	0	0
$784$	0.835333	0.0298333
$785$	0	0
$786$	0	0
$787$	−29.9984	−1.06933	−0.534664	−	0.845065i	$-0.679562\pi$
$787$	−29.9984	−1.06933	−0.534664	+	0.845065i	$0.679562\pi$
$788$	−29.0473	−1.03477
$789$	0	0
$790$	0	0
$791$	57.4913	2.04416
$792$	0	0
$793$	−3.83239	−0.136092
$794$	29.4820	1.04628
$795$	0	0
$796$	15.6696	0.555394
$797$	−18.4489	−0.653493	−0.326747	−	0.945112i	$-0.605952\pi$
$797$	−18.4489	−0.653493	−0.326747	+	0.945112i	$0.605952\pi$
$798$	0	0
$799$	−34.9284	−1.23568
$800$	0	0
$801$	0	0
$802$	30.6235	1.08135
$803$	−1.61484	−0.0569866
$804$	0	0
$805$	0	0
$806$	3.37067	0.118727
$807$	0	0
$808$	−50.3489	−1.77127
$809$	38.1306	1.34060	0.670301	−	0.742089i	$-0.266165\pi$
$809$	38.1306	1.34060	0.670301	+	0.742089i	$0.266165\pi$
$810$	0	0
$811$	−3.36292	−0.118088	−0.0590440	−	0.998255i	$-0.518805\pi$
$811$	−3.36292	−0.118088	−0.0590440	+	0.998255i	$0.518805\pi$
$812$	−39.0114	−1.36903
$813$	0	0
$814$	0.705651	0.0247331
$815$	0	0
$816$	0	0
$817$	52.0688	1.82166
$818$	4.03785	0.141180
$819$	0	0
$820$	0	0
$821$	6.17661	0.215565	0.107783	−	0.994174i	$-0.465625\pi$
$821$	6.17661	0.215565	0.107783	+	0.994174i	$0.465625\pi$
$822$	0	0
$823$	−7.45159	−0.259746	−0.129873	−	0.991531i	$-0.541457\pi$
$823$	−7.45159	−0.259746	−0.129873	+	0.991531i	$0.541457\pi$
$824$	23.8896	0.832233
$825$	0	0
$826$	−3.87407	−0.134796
$827$	0.0604717	0.00210281	0.00105140	−	0.999999i	$-0.499665\pi$
$827$	0.0604717	0.00210281	0.00105140	+	0.999999i	$0.499665\pi$
$828$	0	0
$829$	13.5457	0.470463	0.235231	−	0.971939i	$-0.424415\pi$
$829$	13.5457	0.470463	0.235231	+	0.971939i	$0.424415\pi$
$830$	0	0
$831$	0	0
$832$	18.5976	0.644757
$833$	59.3065	2.05485
$834$	0	0
$835$	0	0
$836$	3.65053	0.126256
$837$	0	0
$838$	9.86036	0.340621
$839$	16.1362	0.557085	0.278542	−	0.960424i	$-0.410149\pi$
$839$	16.1362	0.557085	0.278542	+	0.960424i	$0.410149\pi$
$840$	0	0
$841$	23.8853	0.823631
$842$	−5.47796	−0.188783
$843$	0	0
$844$	−28.9718	−0.997249
$845$	0	0
$846$	0	0
$847$	46.5773	1.60042
$848$	0.319312	0.0109652
$849$	0	0
$850$	0	0
$851$	2.95471	0.101286
$852$	0	0
$853$	3.88888	0.133153	0.0665763	−	0.997781i	$-0.478792\pi$
$853$	3.88888	0.133153	0.0665763	+	0.997781i	$0.478792\pi$
$854$	−3.64030	−0.124568
$855$	0	0
$856$	12.6727	0.433144
$857$	30.7605	1.05076	0.525379	−	0.850869i	$-0.323924\pi$
$857$	30.7605	1.05076	0.525379	+	0.850869i	$0.323924\pi$
$858$	0	0
$859$	18.8972	0.644765	0.322382	−	0.946610i	$-0.395516\pi$
$859$	18.8972	0.644765	0.322382	+	0.946610i	$0.395516\pi$
$860$	0	0
$861$	0	0
$862$	−1.73875	−0.0592221
$863$	−46.3032	−1.57618	−0.788089	−	0.615561i	$-0.788929\pi$
$863$	−46.3032	−1.57618	−0.788089	+	0.615561i	$0.788929\pi$
$864$	0	0
$865$	0	0
$866$	−32.6746	−1.11033
$867$	0	0
$868$	−5.36443	−0.182081
$869$	3.14272	0.106610
$870$	0	0
$871$	19.2039	0.650699
$872$	40.2269	1.36226
$873$	0	0
$874$	−9.12304	−0.308592
$875$	0	0
$876$	0	0
$877$	14.7762	0.498957	0.249479	−	0.968380i	$-0.419741\pi$
$877$	14.7762	0.498957	0.249479	+	0.968380i	$0.419741\pi$
$878$	−12.4411	−0.419866
$879$	0	0
$880$	0	0
$881$	2.84895	0.0959835	0.0479917	−	0.998848i	$-0.484718\pi$
$881$	2.84895	0.0959835	0.0479917	+	0.998848i	$0.484718\pi$
$882$	0	0
$883$	15.8536	0.533516	0.266758	−	0.963764i	$-0.414048\pi$
$883$	15.8536	0.533516	0.266758	+	0.963764i	$0.414048\pi$
$884$	−25.5257	−0.858523
$885$	0	0
$886$	−31.0205	−1.04215
$887$	−5.21365	−0.175057	−0.0875286	−	0.996162i	$-0.527897\pi$
$887$	−5.21365	−0.175057	−0.0875286	+	0.996162i	$0.527897\pi$
$888$	0	0
$889$	58.9399	1.97678
$890$	0	0
$891$	0	0
$892$	−29.6147	−0.991573
$893$	−55.0228	−1.84127
$894$	0	0
$895$	0	0
$896$	−31.0572	−1.03755
$897$	0	0
$898$	−30.6910	−1.02417
$899$	7.27223	0.242542
$900$	0	0
$901$	22.6703	0.755258
$902$	0.821024	0.0273371
$903$	0	0
$904$	−37.7463	−1.25542
$905$	0	0
$906$	0	0
$907$	−13.8569	−0.460111	−0.230055	−	0.973178i	$-0.573891\pi$
$907$	−13.8569	−0.460111	−0.230055	+	0.973178i	$0.573891\pi$
$908$	−33.1286	−1.09941
$909$	0	0
$910$	0	0
$911$	5.44928	0.180543	0.0902713	−	0.995917i	$-0.471227\pi$
$911$	5.44928	0.180543	0.0902713	+	0.995917i	$0.471227\pi$
$912$	0	0
$913$	0.841380	0.0278456
$914$	−29.2875	−0.968744
$915$	0	0
$916$	9.53731	0.315122
$917$	29.7399	0.982097
$918$	0	0
$919$	−36.4516	−1.20243	−0.601214	−	0.799088i	$-0.705316\pi$
$919$	−36.4516	−1.20243	−0.601214	+	0.799088i	$0.705316\pi$
$920$	0	0
$921$	0	0
$922$	−16.6649	−0.548830
$923$	−41.7331	−1.37366
$924$	0	0
$925$	0	0
$926$	−1.71945	−0.0565048
$927$	0	0
$928$	41.3631	1.35781
$929$	7.33460	0.240641	0.120320	−	0.992735i	$-0.461608\pi$
$929$	7.33460	0.240641	0.120320	+	0.992735i	$0.461608\pi$
$930$	0	0
$931$	93.4256	3.06190
$932$	3.93199	0.128797
$933$	0	0
$934$	2.75178	0.0900411
$935$	0	0
$936$	0	0
$937$	−49.1036	−1.60415	−0.802073	−	0.597227i	$-0.796269\pi$
$937$	−49.1036	−1.60415	−0.802073	+	0.597227i	$0.796269\pi$
$938$	18.2413	0.595600
$939$	0	0
$940$	0	0
$941$	20.2169	0.659053	0.329527	−	0.944146i	$-0.393111\pi$
$941$	20.2169	0.659053	0.329527	+	0.944146i	$0.393111\pi$
$942$	0	0
$943$	3.43780	0.111950
$944$	−0.0770305	−0.00250713
$945$	0	0
$946$	1.93480	0.0629057
$947$	30.1596	0.980055	0.490028	−	0.871707i	$-0.336987\pi$
$947$	30.1596	0.980055	0.490028	+	0.871707i	$0.336987\pi$
$948$	0	0
$949$	17.7876	0.577409
$950$	0	0
$951$	0	0
$952$	−62.9638	−2.04067
$953$	−14.6953	−0.476026	−0.238013	−	0.971262i	$-0.576496\pi$
$953$	−14.6953	−0.476026	−0.238013	+	0.971262i	$0.576496\pi$
$954$	0	0
$955$	0	0
$956$	32.6818	1.05701
$957$	0	0
$958$	8.02646	0.259323
$959$	−33.7056	−1.08841
$960$	0	0
$961$	1.00000	0.0322581
$962$	−7.77278	−0.250604
$963$	0	0
$964$	6.53825	0.210583
$965$	0	0
$966$	0	0
$967$	−50.8330	−1.63468	−0.817340	−	0.576155i	$-0.804552\pi$
$967$	−50.8330	−1.63468	−0.817340	+	0.576155i	$0.804552\pi$
$968$	−30.5806	−0.982897
$969$	0	0
$970$	0	0
$971$	−36.0990	−1.15847	−0.579236	−	0.815160i	$-0.696649\pi$
$971$	−36.0990	−1.15847	−0.579236	+	0.815160i	$0.696649\pi$
$972$	0	0
$973$	−32.9482	−1.05627
$974$	7.42077	0.237777
$975$	0	0
$976$	−0.0723824	−0.00231690
$977$	8.14233	0.260496	0.130248	−	0.991481i	$-0.458423\pi$
$977$	8.14233	0.260496	0.130248	+	0.991481i	$0.458423\pi$
$978$	0	0
$979$	0.0604265	0.00193124
$980$	0	0
$981$	0	0
$982$	−29.9561	−0.955936
$983$	31.3377	0.999517	0.499759	−	0.866165i	$-0.333422\pi$
$983$	31.3377	0.999517	0.499759	+	0.866165i	$0.333422\pi$
$984$	0	0
$985$	0	0
$986$	32.8692	1.04677
$987$	0	0
$988$	−40.2107	−1.27927
$989$	8.10140	0.257610
$990$	0	0
$991$	−56.4797	−1.79414	−0.897068	−	0.441893i	$-0.854307\pi$
$991$	−56.4797	−1.79414	−0.897068	+	0.441893i	$0.854307\pi$
$992$	5.68782	0.180588
$993$	0	0
$994$	−39.6412	−1.25734
$995$	0	0
$996$	0	0
$997$	47.4373	1.50236	0.751178	−	0.660100i	$-0.229486\pi$
$997$	47.4373	1.50236	0.751178	+	0.660100i	$0.229486\pi$
$998$	9.70903	0.307334
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6975.2.a.ca.1.3		6
3.2	odd	2		2325.2.a.bb.1.4	yes	6
5.4	even	2		6975.2.a.cc.1.4		6
15.2	even	4		2325.2.c.r.1024.9		12
15.8	even	4		2325.2.c.r.1024.4		12
15.14	odd	2		2325.2.a.y.1.3	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2325.2.a.y.1.3	✓	6	15.14	odd	2
2325.2.a.bb.1.4	yes	6	3.2	odd	2
2325.2.c.r.1024.4		12	15.8	even	4
2325.2.c.r.1024.9		12	15.2	even	4
6975.2.a.ca.1.3		6	1.1	even	1	trivial
6975.2.a.cc.1.4		6	5.4	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:	\( N \)	\(=\)	\( 6975 = 3^{2} \cdot 5^{2} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6975.a (trivial)

\(f(q)\)	\(=\)	\(q-0.864597 q^{2} -1.25247 q^{4} -4.28308 q^{7} +2.81208 q^{8} -0.353929 q^{11} +3.89854 q^{13} +3.70313 q^{14} +0.0736318 q^{16} +5.22766 q^{17} +8.23515 q^{19} +0.306006 q^{22} +1.28131 q^{23} -3.37067 q^{26} +5.36443 q^{28} -7.27223 q^{29} -1.00000 q^{31} -5.68782 q^{32} -4.51982 q^{34} +2.30601 q^{37} -7.12009 q^{38} +2.68303 q^{41} +6.32275 q^{43} +0.443286 q^{44} -1.10782 q^{46} -6.68146 q^{47} +11.3447 q^{49} -4.88282 q^{52} +4.33661 q^{53} -12.0443 q^{56} +6.28755 q^{58} -1.04616 q^{59} -0.983032 q^{61} +0.864597 q^{62} +4.77040 q^{64} +4.92592 q^{67} -6.54751 q^{68} -10.7048 q^{71} +4.56262 q^{73} -1.99377 q^{74} -10.3143 q^{76} +1.51590 q^{77} -8.87953 q^{79} -2.31974 q^{82} -2.37726 q^{83} -5.46663 q^{86} -0.995275 q^{88} -0.170731 q^{89} -16.6978 q^{91} -1.60481 q^{92} +5.77676 q^{94} -1.01227 q^{97} -9.80862 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - q^{2} + 7 q^{4} + 2 q^{7} + 3 q^{8} - 7 q^{11} + 4 q^{13} - 10 q^{14} + 17 q^{16} + 17 q^{19} + 2 q^{22} + q^{23} - 2 q^{26} + 22 q^{28} + 8 q^{29} - 6 q^{31} + 35 q^{32} - 13 q^{34} + 14 q^{37}+ \cdots - 37 q^{98}+O(q^{100}) \) 6 * q - q^2 + 7 * q^4 + 2 * q^7 + 3 * q^8 - 7 * q^11 + 4 * q^13 - 10 * q^14 + 17 * q^16 + 17 * q^19 + 2 * q^22 + q^23 - 2 * q^26 + 22 * q^28 + 8 * q^29 - 6 * q^31 + 35 * q^32 - 13 * q^34 + 14 * q^37 - 10 * q^38 - 18 * q^41 + 15 * q^43 - 4 * q^44 - 10 * q^46 + 13 * q^47 + 18 * q^49 - 38 * q^52 + 13 * q^53 - 32 * q^56 + 23 * q^58 - 32 * q^59 + q^62 + 29 * q^64 + 9 * q^67 - 9 * q^68 - q^71 + 14 * q^73 - 20 * q^74 + 42 * q^76 + 14 * q^77 + 13 * q^79 - 32 * q^82 - 9 * q^83 + 11 * q^86 + 36 * q^88 - 34 * q^89 - 2 * q^91 + 46 * q^92 + 19 * q^94 + 30 * q^97 - 37 * q^98

Introduction

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