LMFDB - Embedded newform 6525.2.a.bw.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6525,2,Mod(1,6525)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6525, 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("6525.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6525 = 3^{2} \cdot 5^{2} \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6525.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:	$52.1023873189$
Analytic rank:	$0$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - x^{6} - 13x^{5} + 12x^{4} + 47x^{3} - 37x^{2} - 35x + 18 $ x^7 - x^6 - 13x^5 + 12x^4 + 47x^3 - 37x^2 - 35*x + 18
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 1305)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$2.43890$ of defining polynomial
Character	$\chi$	$=$	6525.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.43890 q^{2} +3.94822 q^{4} -3.59688 q^{7} +4.75150 q^{8} +6.58661 q^{11} -2.25623 q^{13} -8.77241 q^{14} +3.69199 q^{16} -0.256230 q^{17} +7.95099 q^{19} +16.0641 q^{22} -0.961264 q^{23} -5.50271 q^{26} -14.2013 q^{28} +1.00000 q^{29} -3.95099 q^{31} -0.498627 q^{32} -0.624919 q^{34} +3.76964 q^{37} +19.3917 q^{38} +5.00459 q^{41} +6.19835 q^{43} +26.0054 q^{44} -2.34442 q^{46} +7.02055 q^{47} +5.93752 q^{49} -8.90809 q^{52} -6.38493 q^{53} -17.0906 q^{56} +2.43890 q^{58} -3.19375 q^{59} +14.4072 q^{61} -9.63606 q^{62} -8.60008 q^{64} +9.35246 q^{67} -1.01165 q^{68} -11.9020 q^{71} +13.4965 q^{73} +9.19375 q^{74} +31.3923 q^{76} -23.6912 q^{77} -5.22222 q^{79} +12.2057 q^{82} -0.195206 q^{83} +15.1171 q^{86} +31.2963 q^{88} -3.72323 q^{89} +8.11538 q^{91} -3.79528 q^{92} +17.1224 q^{94} +2.97486 q^{97} +14.4810 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + q^{2} + 13 q^{4} - 10 q^{7} + 3 q^{11} - 6 q^{13} + 9 q^{14} + 21 q^{16} + 8 q^{17} + 10 q^{19} - 9 q^{22} + 11 q^{23} - 3 q^{26} - 25 q^{28} + 7 q^{29} + 18 q^{31} + q^{32} - q^{34} - 13 q^{37}+ \cdots - 20 q^{98}+O(q^{100}) $ 7 * q + q^2 + 13 * q^4 - 10 * q^7 + 3 * q^11 - 6 * q^13 + 9 * q^14 + 21 * q^16 + 8 * q^17 + 10 * q^19 - 9 * q^22 + 11 * q^23 - 3 * q^26 - 25 * q^28 + 7 * q^29 + 18 * q^31 + q^32 - q^34 - 13 * q^37 + 12 * q^38 + 13 * q^41 - 9 * q^43 + 37 * q^44 - 8 * q^46 + 2 * q^47 + 21 * q^49 + q^52 + 5 * q^53 + 30 * q^56 + q^58 + 8 * q^59 + 14 * q^61 - 8 * q^62 + 8 * q^64 - 14 * q^67 + 27 * q^68 + 8 * q^71 - 3 * q^73 + 34 * q^74 + 4 * q^76 - 28 * q^77 + 4 * q^79 + 20 * q^82 + 17 * q^83 - 4 * q^86 - 26 * q^88 + 20 * q^89 + 12 * q^91 + 60 * q^92 - 21 * q^94 - 13 * q^97 - 20 * 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 $

$2$	2.43890	1.72456	0.862280	−	0.506431i	$-0.169036\pi$
$2$	2.43890	1.72456	0.862280	+	0.506431i	$0.169036\pi$
$3$	0	0
$3$	0	0
$4$	3.94822	1.97411
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−3.59688	−1.35949	−0.679746	−	0.733448i	$-0.737910\pi$
$7$	−3.59688	−1.35949	−0.679746	+	0.733448i	$0.737910\pi$
$8$	4.75150	1.67991
$9$	0	0
$10$	0	0
$11$	6.58661	1.98594	0.992968	−	0.118382i	$-0.0377706\pi$
$11$	6.58661	1.98594	0.992968	+	0.118382i	$0.0377706\pi$
$12$	0	0
$13$	−2.25623	−0.625766	−0.312883	−	0.949792i	$-0.601295\pi$
$13$	−2.25623	−0.625766	−0.312883	+	0.949792i	$0.601295\pi$
$14$	−8.77241	−2.34453
$15$	0	0
$16$	3.69199	0.922997
$17$	−0.256230	−0.0621449	−0.0310725	−	0.999517i	$-0.509892\pi$
$17$	−0.256230	−0.0621449	−0.0310725	+	0.999517i	$0.509892\pi$
$18$	0	0
$19$	7.95099	1.82408	0.912041	−	0.410098i	$-0.134505\pi$
$19$	7.95099	1.82408	0.912041	+	0.410098i	$0.134505\pi$
$20$	0	0
$21$	0	0
$22$	16.0641	3.42487
$23$	−0.961264	−0.200437	−0.100219	−	0.994965i	$-0.531954\pi$
$23$	−0.961264	−0.200437	−0.100219	+	0.994965i	$0.531954\pi$
$24$	0	0
$25$	0	0
$26$	−5.50271	−1.07917
$27$	0	0
$28$	−14.2013	−2.68378
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	−3.95099	−0.709619	−0.354810	−	0.934939i	$-0.615454\pi$
$31$	−3.95099	−0.709619	−0.354810	+	0.934939i	$0.615454\pi$
$32$	−0.498627	−0.0881456
$33$	0	0
$34$	−0.624919	−0.107173
$35$	0	0
$36$	0	0
$37$	3.76964	0.619724	0.309862	−	0.950781i	$-0.399717\pi$
$37$	3.76964	0.619724	0.309862	+	0.950781i	$0.399717\pi$
$38$	19.3917	3.14574
$39$	0	0
$40$	0	0
$41$	5.00459	0.781586	0.390793	−	0.920479i	$-0.372201\pi$
$41$	5.00459	0.781586	0.390793	+	0.920479i	$0.372201\pi$
$42$	0	0
$43$	6.19835	0.945239	0.472620	−	0.881267i	$-0.343309\pi$
$43$	6.19835	0.945239	0.472620	+	0.881267i	$0.343309\pi$
$44$	26.0054	3.92045
$45$	0	0
$46$	−2.34442	−0.345667
$47$	7.02055	1.02405	0.512026	−	0.858970i	$-0.328895\pi$
$47$	7.02055	1.02405	0.512026	+	0.858970i	$0.328895\pi$
$48$	0	0
$49$	5.93752	0.848218
$50$	0	0
$51$	0	0
$52$	−8.90809	−1.23533
$53$	−6.38493	−0.877038	−0.438519	−	0.898722i	$-0.644497\pi$
$53$	−6.38493	−0.877038	−0.438519	+	0.898722i	$0.644497\pi$
$54$	0	0
$55$	0	0
$56$	−17.0906	−2.28382
$57$	0	0
$58$	2.43890	0.320243
$59$	−3.19375	−0.415791	−0.207896	−	0.978151i	$-0.566661\pi$
$59$	−3.19375	−0.415791	−0.207896	+	0.978151i	$0.566661\pi$
$60$	0	0
$61$	14.4072	1.84465	0.922323	−	0.386419i	$-0.126288\pi$
$61$	14.4072	1.84465	0.922323	+	0.386419i	$0.126288\pi$
$62$	−9.63606	−1.22378
$63$	0	0
$64$	−8.60008	−1.07501
$65$	0	0
$66$	0	0
$67$	9.35246	1.14259	0.571293	−	0.820746i	$-0.306442\pi$
$67$	9.35246	1.14259	0.571293	+	0.820746i	$0.306442\pi$
$68$	−1.01165	−0.122681
$69$	0	0
$70$	0	0
$71$	−11.9020	−1.41251	−0.706253	−	0.707960i	$-0.749616\pi$
$71$	−11.9020	−1.41251	−0.706253	+	0.707960i	$0.749616\pi$
$72$	0	0
$73$	13.4965	1.57965	0.789824	−	0.613334i	$-0.210172\pi$
$73$	13.4965	1.57965	0.789824	+	0.613334i	$0.210172\pi$
$74$	9.19375	1.06875
$75$	0	0
$76$	31.3923	3.60094
$77$	−23.6912	−2.69986
$78$	0	0
$79$	−5.22222	−0.587545	−0.293773	−	0.955875i	$-0.594911\pi$
$79$	−5.22222	−0.587545	−0.293773	+	0.955875i	$0.594911\pi$
$80$	0	0
$81$	0	0
$82$	12.2057	1.34789
$83$	−0.195206	−0.0214266	−0.0107133	−	0.999943i	$-0.503410\pi$
$83$	−0.195206	−0.0214266	−0.0107133	+	0.999943i	$0.503410\pi$
$84$	0	0
$85$	0	0
$86$	15.1171	1.63012
$87$	0	0
$88$	31.2963	3.33619
$89$	−3.72323	−0.394661	−0.197331	−	0.980337i	$-0.563227\pi$
$89$	−3.72323	−0.394661	−0.197331	+	0.980337i	$0.563227\pi$
$90$	0	0
$91$	8.11538	0.850723
$92$	−3.79528	−0.395685
$93$	0	0
$94$	17.1224	1.76604
$95$	0	0
$96$	0	0
$97$	2.97486	0.302052	0.151026	−	0.988530i	$-0.451742\pi$
$97$	2.97486	0.302052	0.151026	+	0.988530i	$0.451742\pi$
$98$	14.4810	1.46280
$99$	0	0
$100$	0	0
$101$	14.4742	1.44024	0.720119	−	0.693850i	$-0.244087\pi$
$101$	14.4742	1.44024	0.720119	+	0.693850i	$0.244087\pi$
$102$	0	0
$103$	−4.77210	−0.470209	−0.235105	−	0.971970i	$-0.575543\pi$
$103$	−4.77210	−0.470209	−0.235105	+	0.971970i	$0.575543\pi$
$104$	−10.7205	−1.05123
$105$	0	0
$106$	−15.5722	−1.51251
$107$	11.9361	1.15391	0.576955	−	0.816776i	$-0.304241\pi$
$107$	11.9361	1.15391	0.576955	+	0.816776i	$0.304241\pi$
$108$	0	0
$109$	−4.30818	−0.412649	−0.206324	−	0.978484i	$-0.566150\pi$
$109$	−4.30818	−0.412649	−0.206324	+	0.978484i	$0.566150\pi$
$110$	0	0
$111$	0	0
$112$	−13.2796	−1.25481
$113$	9.98310	0.939131	0.469566	−	0.882898i	$-0.344410\pi$
$113$	9.98310	0.939131	0.469566	+	0.882898i	$0.344410\pi$
$114$	0	0
$115$	0	0
$116$	3.94822	0.366583
$117$	0	0
$118$	−7.78924	−0.717057
$119$	0.921628	0.0844855
$120$	0	0
$121$	32.3834	2.94394
$122$	35.1376	3.18121
$123$	0	0
$124$	−15.5994	−1.40087
$125$	0	0
$126$	0	0
$127$	−6.97486	−0.618919	−0.309460	−	0.950913i	$-0.600148\pi$
$127$	−6.97486	−0.618919	−0.309460	+	0.950913i	$0.600148\pi$
$128$	−19.9774	−1.76577
$129$	0	0
$130$	0	0
$131$	21.0735	1.84120	0.920602	−	0.390502i	$-0.127699\pi$
$131$	21.0735	1.84120	0.920602	+	0.390502i	$0.127699\pi$
$132$	0	0
$133$	−28.5987	−2.47983
$134$	22.8097	1.97046
$135$	0	0
$136$	−1.21748	−0.104398
$137$	−12.9493	−1.10634	−0.553168	−	0.833069i	$-0.686581\pi$
$137$	−12.9493	−1.10634	−0.553168	+	0.833069i	$0.686581\pi$
$138$	0	0
$139$	−8.75391	−0.742497	−0.371249	−	0.928533i	$-0.621070\pi$
$139$	−8.75391	−0.742497	−0.371249	+	0.928533i	$0.621070\pi$
$140$	0	0
$141$	0	0
$142$	−29.0277	−2.43595
$143$	−14.8609	−1.24273
$144$	0	0
$145$	0	0
$146$	32.9166	2.72420
$147$	0	0
$148$	14.8833	1.22340
$149$	−6.38751	−0.523285	−0.261643	−	0.965165i	$-0.584264\pi$
$149$	−6.38751	−0.523285	−0.261643	+	0.965165i	$0.584264\pi$
$150$	0	0
$151$	17.4892	1.42325	0.711626	−	0.702558i	$-0.247959\pi$
$151$	17.4892	1.42325	0.711626	+	0.702558i	$0.247959\pi$
$152$	37.7792	3.06429
$153$	0	0
$154$	−57.7804	−4.65608
$155$	0	0
$156$	0	0
$157$	6.42770	0.512986	0.256493	−	0.966546i	$-0.417433\pi$
$157$	6.42770	0.512986	0.256493	+	0.966546i	$0.417433\pi$
$158$	−12.7365	−1.01326
$159$	0	0
$160$	0	0
$161$	3.45755	0.272493
$162$	0	0
$163$	−17.2726	−1.35290	−0.676449	−	0.736490i	$-0.736482\pi$
$163$	−17.2726	−1.35290	−0.676449	+	0.736490i	$0.736482\pi$
$164$	19.7592	1.54294
$165$	0	0
$166$	−0.476087	−0.0369515
$167$	−21.4846	−1.66253	−0.831264	−	0.555878i	$-0.812382\pi$
$167$	−21.4846	−1.66253	−0.831264	+	0.555878i	$0.812382\pi$
$168$	0	0
$169$	−7.90943	−0.608417
$170$	0	0
$171$	0	0
$172$	24.4724	1.86600
$173$	−4.64188	−0.352916	−0.176458	−	0.984308i	$-0.556464\pi$
$173$	−4.64188	−0.352916	−0.176458	+	0.984308i	$0.556464\pi$
$174$	0	0
$175$	0	0
$176$	24.3177	1.83301
$177$	0	0
$178$	−9.08057	−0.680617
$179$	−12.3053	−0.919744	−0.459872	−	0.887985i	$-0.652105\pi$
$179$	−12.3053	−0.919744	−0.459872	+	0.887985i	$0.652105\pi$
$180$	0	0
$181$	−15.1061	−1.12283	−0.561415	−	0.827534i	$-0.689743\pi$
$181$	−15.1061	−1.12283	−0.561415	+	0.827534i	$0.689743\pi$
$182$	19.7926	1.46712
$183$	0	0
$184$	−4.56745	−0.336717
$185$	0	0
$186$	0	0
$187$	−1.68769	−0.123416
$188$	27.7186	2.02159
$189$	0	0
$190$	0	0
$191$	13.3833	0.968380	0.484190	−	0.874963i	$-0.339114\pi$
$191$	13.3833	0.968380	0.484190	+	0.874963i	$0.339114\pi$
$192$	0	0
$193$	5.12776	0.369104	0.184552	−	0.982823i	$-0.440917\pi$
$193$	5.12776	0.369104	0.184552	+	0.982823i	$0.440917\pi$
$194$	7.25539	0.520907
$195$	0	0
$196$	23.4426	1.67447
$197$	−9.95230	−0.709072	−0.354536	−	0.935042i	$-0.615361\pi$
$197$	−9.95230	−0.709072	−0.354536	+	0.935042i	$0.615361\pi$
$198$	0	0
$199$	−21.3405	−1.51279	−0.756395	−	0.654115i	$-0.773041\pi$
$199$	−21.3405	−1.51279	−0.756395	+	0.654115i	$0.773041\pi$
$200$	0	0
$201$	0	0
$202$	35.3011	2.48378
$203$	−3.59688	−0.252451
$204$	0	0
$205$	0	0
$206$	−11.6387	−0.810904
$207$	0	0
$208$	−8.32997	−0.577580
$209$	52.3701	3.62251
$210$	0	0
$211$	17.9602	1.23643	0.618215	−	0.786009i	$-0.287856\pi$
$211$	17.9602	1.23643	0.618215	+	0.786009i	$0.287856\pi$
$212$	−25.2091	−1.73137
$213$	0	0
$214$	29.1110	1.98999
$215$	0	0
$216$	0	0
$217$	14.2112	0.964722
$218$	−10.5072	−0.711637
$219$	0	0
$220$	0	0
$221$	0.578114	0.0388882
$222$	0	0
$223$	−10.9175	−0.731088	−0.365544	−	0.930794i	$-0.619117\pi$
$223$	−10.9175	−0.731088	−0.365544	+	0.930794i	$0.619117\pi$
$224$	1.79350	0.119833
$225$	0	0
$226$	24.3478	1.61959
$227$	27.1891	1.80461	0.902303	−	0.431103i	$-0.141875\pi$
$227$	27.1891	1.80461	0.902303	+	0.431103i	$0.141875\pi$
$228$	0	0
$229$	3.93198	0.259832	0.129916	−	0.991525i	$-0.458529\pi$
$229$	3.93198	0.259832	0.129916	+	0.991525i	$0.458529\pi$
$230$	0	0
$231$	0	0
$232$	4.75150	0.311951
$233$	19.3343	1.26663	0.633315	−	0.773894i	$-0.281694\pi$
$233$	19.3343	1.26663	0.633315	+	0.773894i	$0.281694\pi$
$234$	0	0
$235$	0	0
$236$	−12.6096	−0.820817
$237$	0	0
$238$	2.24776	0.145700
$239$	10.8473	0.701656	0.350828	−	0.936440i	$-0.385900\pi$
$239$	10.8473	0.701656	0.350828	+	0.936440i	$0.385900\pi$
$240$	0	0
$241$	−22.4834	−1.44828	−0.724142	−	0.689651i	$-0.757764\pi$
$241$	−22.4834	−1.44828	−0.724142	+	0.689651i	$0.757764\pi$
$242$	78.9797	5.07701
$243$	0	0
$244$	56.8826	3.64153
$245$	0	0
$246$	0	0
$247$	−17.9393	−1.14145
$248$	−18.7732	−1.19210
$249$	0	0
$250$	0	0
$251$	−16.7271	−1.05581	−0.527903	−	0.849304i	$-0.677022\pi$
$251$	−16.7271	−1.05581	−0.527903	+	0.849304i	$0.677022\pi$
$252$	0	0
$253$	−6.33147	−0.398056
$254$	−17.0110	−1.06736
$255$	0	0
$256$	−31.5228	−1.97017
$257$	22.3892	1.39660	0.698299	−	0.715806i	$-0.253940\pi$
$257$	22.3892	1.39660	0.698299	+	0.715806i	$0.253940\pi$
$258$	0	0
$259$	−13.5589	−0.842510
$260$	0	0
$261$	0	0
$262$	51.3962	3.17527
$263$	−1.48565	−0.0916090	−0.0458045	−	0.998950i	$-0.514585\pi$
$263$	−1.48565	−0.0916090	−0.0458045	+	0.998950i	$0.514585\pi$
$264$	0	0
$265$	0	0
$266$	−69.7494	−4.27661
$267$	0	0
$268$	36.9256	2.25559
$269$	−2.08929	−0.127386	−0.0636931	−	0.997970i	$-0.520288\pi$
$269$	−2.08929	−0.127386	−0.0636931	+	0.997970i	$0.520288\pi$
$270$	0	0
$271$	14.2890	0.867992	0.433996	−	0.900915i	$-0.357103\pi$
$271$	14.2890	0.867992	0.433996	+	0.900915i	$0.357103\pi$
$272$	−0.945998	−0.0573596
$273$	0	0
$274$	−31.5821	−1.90794
$275$	0	0
$276$	0	0
$277$	13.5685	0.815255	0.407627	−	0.913148i	$-0.366356\pi$
$277$	13.5685	0.815255	0.407627	+	0.913148i	$0.366356\pi$
$278$	−21.3499	−1.28048
$279$	0	0
$280$	0	0
$281$	−16.5216	−0.985599	−0.492799	−	0.870143i	$-0.664026\pi$
$281$	−16.5216	−0.985599	−0.492799	+	0.870143i	$0.664026\pi$
$282$	0	0
$283$	20.7310	1.23233	0.616166	−	0.787617i	$-0.288685\pi$
$283$	20.7310	1.23233	0.616166	+	0.787617i	$0.288685\pi$
$284$	−46.9916	−2.78844
$285$	0	0
$286$	−36.2442	−2.14316
$287$	−18.0009	−1.06256
$288$	0	0
$289$	−16.9343	−0.996138
$290$	0	0
$291$	0	0
$292$	53.2872	3.11840
$293$	4.86682	0.284323	0.142161	−	0.989843i	$-0.454595\pi$
$293$	4.86682	0.284323	0.142161	+	0.989843i	$0.454595\pi$
$294$	0	0
$295$	0	0
$296$	17.9114	1.04108
$297$	0	0
$298$	−15.5785	−0.902437
$299$	2.16883	0.125427
$300$	0	0
$301$	−22.2947	−1.28504
$302$	42.6544	2.45448
$303$	0	0
$304$	29.3550	1.68362
$305$	0	0
$306$	0	0
$307$	−13.3624	−0.762631	−0.381315	−	0.924445i	$-0.624529\pi$
$307$	−13.3624	−0.762631	−0.381315	+	0.924445i	$0.624529\pi$
$308$	−93.5381	−5.32983
$309$	0	0
$310$	0	0
$311$	1.27926	0.0725400	0.0362700	−	0.999342i	$-0.488452\pi$
$311$	1.27926	0.0725400	0.0362700	+	0.999342i	$0.488452\pi$
$312$	0	0
$313$	−29.2467	−1.65312	−0.826560	−	0.562849i	$-0.809705\pi$
$313$	−29.2467	−1.65312	−0.826560	+	0.562849i	$0.809705\pi$
$314$	15.6765	0.884675
$315$	0	0
$316$	−20.6185	−1.15988
$317$	−25.3909	−1.42609	−0.713047	−	0.701116i	$-0.752685\pi$
$317$	−25.3909	−1.42609	−0.713047	+	0.701116i	$0.752685\pi$
$318$	0	0
$319$	6.58661	0.368779
$320$	0	0
$321$	0	0
$322$	8.43261	0.469931
$323$	−2.03728	−0.113357
$324$	0	0
$325$	0	0
$326$	−42.1262	−2.33315
$327$	0	0
$328$	23.7793	1.31299
$329$	−25.2520	−1.39219
$330$	0	0
$331$	−19.8405	−1.09053	−0.545266	−	0.838263i	$-0.683571\pi$
$331$	−19.8405	−1.09053	−0.545266	+	0.838263i	$0.683571\pi$
$332$	−0.770716	−0.0422985
$333$	0	0
$334$	−52.3987	−2.86713
$335$	0	0
$336$	0	0
$337$	−21.7192	−1.18312	−0.591560	−	0.806261i	$-0.701488\pi$
$337$	−21.7192	−1.18312	−0.591560	+	0.806261i	$0.701488\pi$
$338$	−19.2903	−1.04925
$339$	0	0
$340$	0	0
$341$	−26.0236	−1.40926
$342$	0	0
$343$	3.82160	0.206347
$344$	29.4515	1.58792
$345$	0	0
$346$	−11.3211	−0.608624
$347$	9.47114	0.508437	0.254219	−	0.967147i	$-0.418182\pi$
$347$	9.47114	0.508437	0.254219	+	0.967147i	$0.418182\pi$
$348$	0	0
$349$	25.8065	1.38139	0.690696	−	0.723145i	$-0.257304\pi$
$349$	25.8065	1.38139	0.690696	+	0.723145i	$0.257304\pi$
$350$	0	0
$351$	0	0
$352$	−3.28426	−0.175052
$353$	19.9743	1.06312	0.531561	−	0.847020i	$-0.321606\pi$
$353$	19.9743	1.06312	0.531561	+	0.847020i	$0.321606\pi$
$354$	0	0
$355$	0	0
$356$	−14.7001	−0.779104
$357$	0	0
$358$	−30.0114	−1.58615
$359$	−28.6956	−1.51450	−0.757248	−	0.653128i	$-0.773457\pi$
$359$	−28.6956	−1.51450	−0.757248	+	0.653128i	$0.773457\pi$
$360$	0	0
$361$	44.2183	2.32728
$362$	−36.8423	−1.93639
$363$	0	0
$364$	32.0413	1.67942
$365$	0	0
$366$	0	0
$367$	−19.0084	−0.992230	−0.496115	−	0.868257i	$-0.665241\pi$
$367$	−19.0084	−0.992230	−0.496115	+	0.868257i	$0.665241\pi$
$368$	−3.54898	−0.185003
$369$	0	0
$370$	0	0
$371$	22.9658	1.19233
$372$	0	0
$373$	−2.88462	−0.149360	−0.0746800	−	0.997208i	$-0.523794\pi$
$373$	−2.88462	−0.149360	−0.0746800	+	0.997208i	$0.523794\pi$
$374$	−4.11609	−0.212838
$375$	0	0
$376$	33.3581	1.72031
$377$	−2.25623	−0.116202
$378$	0	0
$379$	−21.1117	−1.08444	−0.542218	−	0.840238i	$-0.682415\pi$
$379$	−21.1117	−1.08444	−0.542218	+	0.840238i	$0.682415\pi$
$380$	0	0
$381$	0	0
$382$	32.6404	1.67003
$383$	−2.99855	−0.153219	−0.0766093	−	0.997061i	$-0.524409\pi$
$383$	−2.99855	−0.153219	−0.0766093	+	0.997061i	$0.524409\pi$
$384$	0	0
$385$	0	0
$386$	12.5061	0.636542
$387$	0	0
$388$	11.7454	0.596283
$389$	−10.9069	−0.552999	−0.276500	−	0.961014i	$-0.589174\pi$
$389$	−10.9069	−0.552999	−0.276500	+	0.961014i	$0.589174\pi$
$390$	0	0
$391$	0.246305	0.0124562
$392$	28.2122	1.42493
$393$	0	0
$394$	−24.2726	−1.22284
$395$	0	0
$396$	0	0
$397$	3.44546	0.172923	0.0864613	−	0.996255i	$-0.472444\pi$
$397$	3.44546	0.172923	0.0864613	+	0.996255i	$0.472444\pi$
$398$	−52.0474	−2.60890
$399$	0	0
$400$	0	0
$401$	−3.05491	−0.152555	−0.0762775	−	0.997087i	$-0.524303\pi$
$401$	−3.05491	−0.152555	−0.0762775	+	0.997087i	$0.524303\pi$
$402$	0	0
$403$	8.91435	0.444055
$404$	57.1474	2.84319
$405$	0	0
$406$	−8.77241	−0.435367
$407$	24.8291	1.23073
$408$	0	0
$409$	24.0609	1.18974	0.594868	−	0.803823i	$-0.297204\pi$
$409$	24.0609	1.18974	0.594868	+	0.803823i	$0.297204\pi$
$410$	0	0
$411$	0	0
$412$	−18.8413	−0.928244
$413$	11.4875	0.565265
$414$	0	0
$415$	0	0
$416$	1.12502	0.0551585
$417$	0	0
$418$	127.725	6.24724
$419$	−31.7868	−1.55289	−0.776443	−	0.630188i	$-0.782978\pi$
$419$	−31.7868	−1.55289	−0.776443	+	0.630188i	$0.782978\pi$
$420$	0	0
$421$	−21.4413	−1.04498	−0.522492	−	0.852644i	$-0.674997\pi$
$421$	−21.4413	−1.04498	−0.522492	+	0.852644i	$0.674997\pi$
$422$	43.8030	2.13230
$423$	0	0
$424$	−30.3380	−1.47334
$425$	0	0
$426$	0	0
$427$	−51.8208	−2.50778
$428$	47.1264	2.27794
$429$	0	0
$430$	0	0
$431$	−19.8462	−0.955959	−0.477980	−	0.878371i	$-0.658631\pi$
$431$	−19.8462	−0.955959	−0.477980	+	0.878371i	$0.658631\pi$
$432$	0	0
$433$	−2.40179	−0.115422	−0.0577112	−	0.998333i	$-0.518380\pi$
$433$	−2.40179	−0.115422	−0.0577112	+	0.998333i	$0.518380\pi$
$434$	34.6597	1.66372
$435$	0	0
$436$	−17.0096	−0.814613
$437$	−7.64301	−0.365615
$438$	0	0
$439$	3.51627	0.167822	0.0839111	−	0.996473i	$-0.473259\pi$
$439$	3.51627	0.167822	0.0839111	+	0.996473i	$0.473259\pi$
$440$	0	0
$441$	0	0
$442$	1.40996	0.0670650
$443$	−40.2174	−1.91079	−0.955393	−	0.295338i	$-0.904568\pi$
$443$	−40.2174	−1.91079	−0.955393	+	0.295338i	$0.904568\pi$
$444$	0	0
$445$	0	0
$446$	−26.6266	−1.26081
$447$	0	0
$448$	30.9334	1.46147
$449$	−12.8868	−0.608163	−0.304082	−	0.952646i	$-0.598350\pi$
$449$	−12.8868	−0.608163	−0.304082	+	0.952646i	$0.598350\pi$
$450$	0	0
$451$	32.9633	1.55218
$452$	39.4155	1.85395
$453$	0	0
$454$	66.3115	3.11215
$455$	0	0
$456$	0	0
$457$	−11.1142	−0.519901	−0.259950	−	0.965622i	$-0.583706\pi$
$457$	−11.1142	−0.519901	−0.259950	+	0.965622i	$0.583706\pi$
$458$	9.58969	0.448097
$459$	0	0
$460$	0	0
$461$	−15.3222	−0.713625	−0.356813	−	0.934176i	$-0.616137\pi$
$461$	−15.3222	−0.713625	−0.356813	+	0.934176i	$0.616137\pi$
$462$	0	0
$463$	−41.2367	−1.91643	−0.958216	−	0.286046i	$-0.907659\pi$
$463$	−41.2367	−1.91643	−0.958216	+	0.286046i	$0.907659\pi$
$464$	3.69199	0.171396
$465$	0	0
$466$	47.1543	2.18438
$467$	28.0043	1.29588	0.647941	−	0.761690i	$-0.275630\pi$
$467$	28.0043	1.29588	0.647941	+	0.761690i	$0.275630\pi$
$468$	0	0
$469$	−33.6397	−1.55334
$470$	0	0
$471$	0	0
$472$	−15.1751	−0.698492
$473$	40.8261	1.87718
$474$	0	0
$475$	0	0
$476$	3.63879	0.166784
$477$	0	0
$478$	26.4555	1.21005
$479$	−6.36308	−0.290737	−0.145368	−	0.989378i	$-0.546437\pi$
$479$	−6.36308	−0.290737	−0.145368	+	0.989378i	$0.546437\pi$
$480$	0	0
$481$	−8.50517	−0.387802
$482$	−54.8347	−2.49765
$483$	0	0
$484$	127.857	5.81166
$485$	0	0
$486$	0	0
$487$	−3.88748	−0.176159	−0.0880793	−	0.996113i	$-0.528073\pi$
$487$	−3.88748	−0.176159	−0.0880793	+	0.996113i	$0.528073\pi$
$488$	68.4556	3.09884
$489$	0	0
$490$	0	0
$491$	9.96901	0.449895	0.224948	−	0.974371i	$-0.427779\pi$
$491$	9.96901	0.449895	0.224948	+	0.974371i	$0.427779\pi$
$492$	0	0
$493$	−0.256230	−0.0115400
$494$	−43.7520	−1.96850
$495$	0	0
$496$	−14.5870	−0.654976
$497$	42.8100	1.92029
$498$	0	0
$499$	6.60090	0.295497	0.147749	−	0.989025i	$-0.452797\pi$
$499$	6.60090	0.295497	0.147749	+	0.989025i	$0.452797\pi$
$500$	0	0
$501$	0	0
$502$	−40.7957	−1.82080
$503$	32.5121	1.44964	0.724821	−	0.688937i	$-0.241922\pi$
$503$	32.5121	1.44964	0.724821	+	0.688937i	$0.241922\pi$
$504$	0	0
$505$	0	0
$506$	−15.4418	−0.686472
$507$	0	0
$508$	−27.5383	−1.22181
$509$	4.46788	0.198035	0.0990177	−	0.995086i	$-0.468430\pi$
$509$	4.46788	0.198035	0.0990177	+	0.995086i	$0.468430\pi$
$510$	0	0
$511$	−48.5453	−2.14752
$512$	−36.9259	−1.63191
$513$	0	0
$514$	54.6049	2.40852
$515$	0	0
$516$	0	0
$517$	46.2416	2.03370
$518$	−33.0688	−1.45296
$519$	0	0
$520$	0	0
$521$	11.1982	0.490604	0.245302	−	0.969447i	$-0.421113\pi$
$521$	11.1982	0.490604	0.245302	+	0.969447i	$0.421113\pi$
$522$	0	0
$523$	−21.3797	−0.934867	−0.467434	−	0.884028i	$-0.654821\pi$
$523$	−21.3797	−0.934867	−0.467434	+	0.884028i	$0.654821\pi$
$524$	83.2029	3.63474
$525$	0	0
$526$	−3.62334	−0.157985
$527$	1.01236	0.0440992
$528$	0	0
$529$	−22.0760	−0.959825
$530$	0	0
$531$	0	0
$532$	−112.914	−4.89545
$533$	−11.2915	−0.489090
$534$	0	0
$535$	0	0
$536$	44.4383	1.91944
$537$	0	0
$538$	−5.09556	−0.219685
$539$	39.1081	1.68451
$540$	0	0
$541$	6.28388	0.270165	0.135083	−	0.990834i	$-0.456870\pi$
$541$	6.28388	0.270165	0.135083	+	0.990834i	$0.456870\pi$
$542$	34.8493	1.49691
$543$	0	0
$544$	0.127763	0.00547780
$545$	0	0
$546$	0	0
$547$	−27.3897	−1.17110	−0.585551	−	0.810636i	$-0.699122\pi$
$547$	−27.3897	−1.17110	−0.585551	+	0.810636i	$0.699122\pi$
$548$	−51.1268	−2.18403
$549$	0	0
$550$	0	0
$551$	7.95099	0.338724
$552$	0	0
$553$	18.7837	0.798763
$554$	33.0923	1.40596
$555$	0	0
$556$	−34.5624	−1.46577
$557$	−26.3046	−1.11456	−0.557280	−	0.830325i	$-0.688155\pi$
$557$	−26.3046	−1.11456	−0.557280	+	0.830325i	$0.688155\pi$
$558$	0	0
$559$	−13.9849	−0.591498
$560$	0	0
$561$	0	0
$562$	−40.2946	−1.69972
$563$	−21.8053	−0.918984	−0.459492	−	0.888182i	$-0.651969\pi$
$563$	−21.8053	−0.918984	−0.459492	+	0.888182i	$0.651969\pi$
$564$	0	0
$565$	0	0
$566$	50.5608	2.12523
$567$	0	0
$568$	−56.5523	−2.37288
$569$	7.77161	0.325803	0.162901	−	0.986642i	$-0.447915\pi$
$569$	7.77161	0.325803	0.162901	+	0.986642i	$0.447915\pi$
$570$	0	0
$571$	−16.9439	−0.709079	−0.354539	−	0.935041i	$-0.615362\pi$
$571$	−16.9439	−0.709079	−0.354539	+	0.935041i	$0.615362\pi$
$572$	−58.6741	−2.45329
$573$	0	0
$574$	−43.9023	−1.83245
$575$	0	0
$576$	0	0
$577$	7.67750	0.319619	0.159809	−	0.987148i	$-0.448912\pi$
$577$	7.67750	0.319619	0.159809	+	0.987148i	$0.448912\pi$
$578$	−41.3011	−1.71790
$579$	0	0
$580$	0	0
$581$	0.702132	0.0291293
$582$	0	0
$583$	−42.0550	−1.74174
$584$	64.1287	2.65366
$585$	0	0
$586$	11.8697	0.490332
$587$	7.58590	0.313104	0.156552	−	0.987670i	$-0.449962\pi$
$587$	7.58590	0.313104	0.156552	+	0.987670i	$0.449962\pi$
$588$	0	0
$589$	−31.4143	−1.29440
$590$	0	0
$591$	0	0
$592$	13.9175	0.572004
$593$	3.02067	0.124044	0.0620220	−	0.998075i	$-0.480245\pi$
$593$	3.02067	0.124044	0.0620220	+	0.998075i	$0.480245\pi$
$594$	0	0
$595$	0	0
$596$	−25.2193	−1.03302
$597$	0	0
$598$	5.28956	0.216306
$599$	28.2147	1.15282	0.576410	−	0.817161i	$-0.304453\pi$
$599$	28.2147	1.15282	0.576410	+	0.817161i	$0.304453\pi$
$600$	0	0
$601$	7.40917	0.302226	0.151113	−	0.988516i	$-0.451714\pi$
$601$	7.40917	0.302226	0.151113	+	0.988516i	$0.451714\pi$
$602$	−54.3744	−2.21614
$603$	0	0
$604$	69.0512	2.80965
$605$	0	0
$606$	0	0
$607$	31.4998	1.27854	0.639269	−	0.768983i	$-0.279237\pi$
$607$	31.4998	1.27854	0.639269	+	0.768983i	$0.279237\pi$
$608$	−3.96458	−0.160785
$609$	0	0
$610$	0	0
$611$	−15.8400	−0.640816
$612$	0	0
$613$	−26.6641	−1.07695	−0.538477	−	0.842640i	$-0.681000\pi$
$613$	−26.6641	−1.07695	−0.538477	+	0.842640i	$0.681000\pi$
$614$	−32.5894	−1.31520
$615$	0	0
$616$	−112.569	−4.53553
$617$	−42.6460	−1.71686	−0.858431	−	0.512929i	$-0.828560\pi$
$617$	−42.6460	−1.71686	−0.858431	+	0.512929i	$0.828560\pi$
$618$	0	0
$619$	−23.6243	−0.949541	−0.474771	−	0.880110i	$-0.657469\pi$
$619$	−23.6243	−0.949541	−0.474771	+	0.880110i	$0.657469\pi$
$620$	0	0
$621$	0	0
$622$	3.11997	0.125100
$623$	13.3920	0.536539
$624$	0	0
$625$	0	0
$626$	−71.3296	−2.85090
$627$	0	0
$628$	25.3779	1.01269
$629$	−0.965894	−0.0385127
$630$	0	0
$631$	22.5629	0.898213	0.449107	−	0.893478i	$-0.351742\pi$
$631$	22.5629	0.898213	0.449107	+	0.893478i	$0.351742\pi$
$632$	−24.8134	−0.987023
$633$	0	0
$634$	−61.9257	−2.45939
$635$	0	0
$636$	0	0
$637$	−13.3964	−0.530785
$638$	16.0641	0.635982
$639$	0	0
$640$	0	0
$641$	6.65061	0.262683	0.131342	−	0.991337i	$-0.458072\pi$
$641$	6.65061	0.262683	0.131342	+	0.991337i	$0.458072\pi$
$642$	0	0
$643$	−13.5539	−0.534514	−0.267257	−	0.963625i	$-0.586117\pi$
$643$	−13.5539	−0.534514	−0.267257	+	0.963625i	$0.586117\pi$
$644$	13.6512	0.537931
$645$	0	0
$646$	−4.96872	−0.195492
$647$	6.91189	0.271734	0.135867	−	0.990727i	$-0.456618\pi$
$647$	6.91189	0.271734	0.135867	+	0.990727i	$0.456618\pi$
$648$	0	0
$649$	−21.0360	−0.825735
$650$	0	0
$651$	0	0
$652$	−68.1961	−2.67077
$653$	−22.0261	−0.861947	−0.430974	−	0.902364i	$-0.641830\pi$
$653$	−22.0261	−0.861947	−0.430974	+	0.902364i	$0.641830\pi$
$654$	0	0
$655$	0	0
$656$	18.4769	0.721402
$657$	0	0
$658$	−61.5871	−2.40092
$659$	−24.9205	−0.970766	−0.485383	−	0.874302i	$-0.661320\pi$
$659$	−24.9205	−0.970766	−0.485383	+	0.874302i	$0.661320\pi$
$660$	0	0
$661$	−24.5688	−0.955617	−0.477809	−	0.878464i	$-0.658569\pi$
$661$	−24.5688	−0.955617	−0.477809	+	0.878464i	$0.658569\pi$
$662$	−48.3889	−1.88069
$663$	0	0
$664$	−0.927522	−0.0359948
$665$	0	0
$666$	0	0
$667$	−0.961264	−0.0372203
$668$	−84.8258	−3.28201
$669$	0	0
$670$	0	0
$671$	94.8942	3.66335
$672$	0	0
$673$	−20.8663	−0.804337	−0.402169	−	0.915566i	$-0.631743\pi$
$673$	−20.8663	−0.804337	−0.402169	+	0.915566i	$0.631743\pi$
$674$	−52.9709	−2.04036
$675$	0	0
$676$	−31.2281	−1.20108
$677$	−12.9901	−0.499250	−0.249625	−	0.968343i	$-0.580307\pi$
$677$	−12.9901	−0.499250	−0.249625	+	0.968343i	$0.580307\pi$
$678$	0	0
$679$	−10.7002	−0.410637
$680$	0	0
$681$	0	0
$682$	−63.4690	−2.43035
$683$	25.4388	0.973388	0.486694	−	0.873573i	$-0.338203\pi$
$683$	25.4388	0.973388	0.486694	+	0.873573i	$0.338203\pi$
$684$	0	0
$685$	0	0
$686$	9.32048	0.355858
$687$	0	0
$688$	22.8842	0.872453
$689$	14.4059	0.548820
$690$	0	0
$691$	7.63166	0.290322	0.145161	−	0.989408i	$-0.453630\pi$
$691$	7.63166	0.290322	0.145161	+	0.989408i	$0.453630\pi$
$692$	−18.3271	−0.696694
$693$	0	0
$694$	23.0991	0.876831
$695$	0	0
$696$	0	0
$697$	−1.28233	−0.0485716
$698$	62.9395	2.38230
$699$	0	0
$700$	0	0
$701$	20.1233	0.760045	0.380023	−	0.924977i	$-0.375916\pi$
$701$	20.1233	0.760045	0.380023	+	0.924977i	$0.375916\pi$
$702$	0	0
$703$	29.9724	1.13043
$704$	−56.6453	−2.13490
$705$	0	0
$706$	48.7152	1.83342
$707$	−52.0620	−1.95799
$708$	0	0
$709$	−0.0528446	−0.00198462	−0.000992311	−	1.00000i	$-0.500316\pi$
$709$	−0.0528446	−0.00198462	−0.000992311		1.00000i	$0.500316\pi$
$710$	0	0
$711$	0	0
$712$	−17.6909	−0.662995
$713$	3.79795	0.142234
$714$	0	0
$715$	0	0
$716$	−48.5841	−1.81567
$717$	0	0
$718$	−69.9856	−2.61184
$719$	3.64149	0.135805	0.0679024	−	0.997692i	$-0.478369\pi$
$719$	3.64149	0.135805	0.0679024	+	0.997692i	$0.478369\pi$
$720$	0	0
$721$	17.1647	0.639246
$722$	107.844	4.01353
$723$	0	0
$724$	−59.6423	−2.21659
$725$	0	0
$726$	0	0
$727$	7.68378	0.284976	0.142488	−	0.989797i	$-0.454490\pi$
$727$	7.68378	0.284976	0.142488	+	0.989797i	$0.454490\pi$
$728$	38.5603	1.42914
$729$	0	0
$730$	0	0
$731$	−1.58820	−0.0587418
$732$	0	0
$733$	43.4669	1.60549	0.802743	−	0.596325i	$-0.203373\pi$
$733$	43.4669	1.60549	0.802743	+	0.596325i	$0.203373\pi$
$734$	−46.3595	−1.71116
$735$	0	0
$736$	0.479312	0.0176677
$737$	61.6010	2.26910
$738$	0	0
$739$	38.2405	1.40670	0.703350	−	0.710844i	$-0.251687\pi$
$739$	38.2405	1.40670	0.703350	+	0.710844i	$0.251687\pi$
$740$	0	0
$741$	0	0
$742$	56.0113	2.05624
$743$	42.7317	1.56767	0.783836	−	0.620968i	$-0.213260\pi$
$743$	42.7317	1.56767	0.783836	+	0.620968i	$0.213260\pi$
$744$	0	0
$745$	0	0
$746$	−7.03529	−0.257580
$747$	0	0
$748$	−6.66335	−0.243636
$749$	−42.9328	−1.56873
$750$	0	0
$751$	30.7352	1.12154	0.560772	−	0.827971i	$-0.310505\pi$
$751$	30.7352	1.12154	0.560772	+	0.827971i	$0.310505\pi$
$752$	25.9198	0.945197
$753$	0	0
$754$	−5.50271	−0.200397
$755$	0	0
$756$	0	0
$757$	47.7626	1.73596	0.867980	−	0.496599i	$-0.165418\pi$
$757$	47.7626	1.73596	0.867980	+	0.496599i	$0.165418\pi$
$758$	−51.4893	−1.87018
$759$	0	0
$760$	0	0
$761$	7.02492	0.254653	0.127327	−	0.991861i	$-0.459360\pi$
$761$	7.02492	0.254653	0.127327	+	0.991861i	$0.459360\pi$
$762$	0	0
$763$	15.4960	0.560992
$764$	52.8401	1.91169
$765$	0	0
$766$	−7.31315	−0.264235
$767$	7.20584	0.260188
$768$	0	0
$769$	−25.1269	−0.906100	−0.453050	−	0.891485i	$-0.649664\pi$
$769$	−25.1269	−0.906100	−0.453050	+	0.891485i	$0.649664\pi$
$770$	0	0
$771$	0	0
$772$	20.2455	0.728652
$773$	−10.8598	−0.390599	−0.195300	−	0.980744i	$-0.562568\pi$
$773$	−10.8598	−0.390599	−0.195300	+	0.980744i	$0.562568\pi$
$774$	0	0
$775$	0	0
$776$	14.1351	0.507420
$777$	0	0
$778$	−26.6007	−0.953681
$779$	39.7915	1.42568
$780$	0	0
$781$	−78.3937	−2.80515
$782$	0.600712	0.0214814
$783$	0	0
$784$	21.9213	0.782902
$785$	0	0
$786$	0	0
$787$	−16.3114	−0.581438	−0.290719	−	0.956809i	$-0.593894\pi$
$787$	−16.3114	−0.581438	−0.290719	+	0.956809i	$0.593894\pi$
$788$	−39.2939	−1.39979
$789$	0	0
$790$	0	0
$791$	−35.9080	−1.27674
$792$	0	0
$793$	−32.5059	−1.15432
$794$	8.40312	0.298216
$795$	0	0
$796$	−84.2571	−2.98641
$797$	4.55126	0.161214	0.0806070	−	0.996746i	$-0.474314\pi$
$797$	4.55126	0.161214	0.0806070	+	0.996746i	$0.474314\pi$
$798$	0	0
$799$	−1.79887	−0.0636396
$800$	0	0
$801$	0	0
$802$	−7.45061	−0.263090
$803$	88.8962	3.13708
$804$	0	0
$805$	0	0
$806$	21.7412	0.765800
$807$	0	0
$808$	68.7743	2.41947
$809$	2.72709	0.0958793	0.0479396	−	0.998850i	$-0.484734\pi$
$809$	2.72709	0.0958793	0.0479396	+	0.998850i	$0.484734\pi$
$810$	0	0
$811$	−39.1620	−1.37516	−0.687581	−	0.726107i	$-0.741328\pi$
$811$	−39.1620	−1.37516	−0.687581	+	0.726107i	$0.741328\pi$
$812$	−14.2013	−0.498366
$813$	0	0
$814$	60.5556	2.12247
$815$	0	0
$816$	0	0
$817$	49.2830	1.72419
$818$	58.6821	2.05177
$819$	0	0
$820$	0	0
$821$	6.22315	0.217190	0.108595	−	0.994086i	$-0.465365\pi$
$821$	6.22315	0.217190	0.108595	+	0.994086i	$0.465365\pi$
$822$	0	0
$823$	−43.5896	−1.51944	−0.759718	−	0.650253i	$-0.774663\pi$
$823$	−43.5896	−1.51944	−0.759718	+	0.650253i	$0.774663\pi$
$824$	−22.6747	−0.789909
$825$	0	0
$826$	28.0169	0.974833
$827$	−0.982635	−0.0341696	−0.0170848	−	0.999854i	$-0.505439\pi$
$827$	−0.982635	−0.0341696	−0.0170848	+	0.999854i	$0.505439\pi$
$828$	0	0
$829$	−11.5045	−0.399568	−0.199784	−	0.979840i	$-0.564024\pi$
$829$	−11.5045	−0.399568	−0.199784	+	0.979840i	$0.564024\pi$
$830$	0	0
$831$	0	0
$832$	19.4037	0.672704
$833$	−1.52137	−0.0527124
$834$	0	0
$835$	0	0
$836$	206.768	7.15123
$837$	0	0
$838$	−77.5247	−2.67805
$839$	23.6588	0.816791	0.408396	−	0.912805i	$-0.366088\pi$
$839$	23.6588	0.816791	0.408396	+	0.912805i	$0.366088\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	−52.2930	−1.80214
$843$	0	0
$844$	70.9107	2.44085
$845$	0	0
$846$	0	0
$847$	−116.479	−4.00227
$848$	−23.5731	−0.809503
$849$	0	0
$850$	0	0
$851$	−3.62362	−0.124216
$852$	0	0
$853$	7.87029	0.269474	0.134737	−	0.990881i	$-0.456981\pi$
$853$	7.87029	0.269474	0.134737	+	0.990881i	$0.456981\pi$
$854$	−126.385	−4.32482
$855$	0	0
$856$	56.7145	1.93846
$857$	18.9403	0.646988	0.323494	−	0.946230i	$-0.395142\pi$
$857$	18.9403	0.646988	0.323494	+	0.946230i	$0.395142\pi$
$858$	0	0
$859$	−16.6320	−0.567477	−0.283739	−	0.958902i	$-0.591575\pi$
$859$	−16.6320	−0.567477	−0.283739	+	0.958902i	$0.591575\pi$
$860$	0	0
$861$	0	0
$862$	−48.4029	−1.64861
$863$	8.81066	0.299918	0.149959	−	0.988692i	$-0.452086\pi$
$863$	8.81066	0.299918	0.149959	+	0.988692i	$0.452086\pi$
$864$	0	0
$865$	0	0
$866$	−5.85771	−0.199053
$867$	0	0
$868$	56.1091	1.90447
$869$	−34.3967	−1.16683
$870$	0	0
$871$	−21.1013	−0.714991
$872$	−20.4703	−0.693212
$873$	0	0
$874$	−18.6405	−0.630524
$875$	0	0
$876$	0	0
$877$	16.2007	0.547060	0.273530	−	0.961863i	$-0.411809\pi$
$877$	16.2007	0.547060	0.273530	+	0.961863i	$0.411809\pi$
$878$	8.57581	0.289419
$879$	0	0
$880$	0	0
$881$	−54.7427	−1.84433	−0.922164	−	0.386799i	$-0.873581\pi$
$881$	−54.7427	−1.84433	−0.922164	+	0.386799i	$0.873581\pi$
$882$	0	0
$883$	−42.5560	−1.43212	−0.716062	−	0.698036i	$-0.754057\pi$
$883$	−42.5560	−1.43212	−0.716062	+	0.698036i	$0.754057\pi$
$884$	2.28252	0.0767695
$885$	0	0
$886$	−98.0861	−3.29527
$887$	52.9851	1.77907	0.889533	−	0.456871i	$-0.151030\pi$
$887$	52.9851	1.77907	0.889533	+	0.456871i	$0.151030\pi$
$888$	0	0
$889$	25.0877	0.841415
$890$	0	0
$891$	0	0
$892$	−43.1046	−1.44325
$893$	55.8203	1.86796
$894$	0	0
$895$	0	0
$896$	71.8564	2.40055
$897$	0	0
$898$	−31.4295	−1.04881
$899$	−3.95099	−0.131773
$900$	0	0
$901$	1.63601	0.0545035
$902$	80.3940	2.67683
$903$	0	0
$904$	47.4347	1.57766
$905$	0	0
$906$	0	0
$907$	54.3483	1.80461	0.902303	−	0.431103i	$-0.141876\pi$
$907$	54.3483	1.80461	0.902303	+	0.431103i	$0.141876\pi$
$908$	107.349	3.56249
$909$	0	0
$910$	0	0
$911$	46.7376	1.54849	0.774243	−	0.632888i	$-0.218131\pi$
$911$	46.7376	1.54849	0.774243	+	0.632888i	$0.218131\pi$
$912$	0	0
$913$	−1.28574	−0.0425519
$914$	−27.1064	−0.896600
$915$	0	0
$916$	15.5243	0.512937
$917$	−75.7989	−2.50310
$918$	0	0
$919$	10.9218	0.360277	0.180139	−	0.983641i	$-0.442345\pi$
$919$	10.9218	0.360277	0.180139	+	0.983641i	$0.442345\pi$
$920$	0	0
$921$	0	0
$922$	−37.3692	−1.23069
$923$	26.8536	0.883898
$924$	0	0
$925$	0	0
$926$	−100.572	−3.30500
$927$	0	0
$928$	−0.498627	−0.0163682
$929$	52.9110	1.73595	0.867976	−	0.496606i	$-0.165421\pi$
$929$	52.9110	1.73595	0.867976	+	0.496606i	$0.165421\pi$
$930$	0	0
$931$	47.2092	1.54722
$932$	76.3359	2.50047
$933$	0	0
$934$	68.2995	2.23483
$935$	0	0
$936$	0	0
$937$	−27.0005	−0.882067	−0.441033	−	0.897491i	$-0.645388\pi$
$937$	−27.0005	−0.882067	−0.441033	+	0.897491i	$0.645388\pi$
$938$	−82.0437	−2.67882
$939$	0	0
$940$	0	0
$941$	9.49673	0.309584	0.154792	−	0.987947i	$-0.450529\pi$
$941$	9.49673	0.309584	0.154792	+	0.987947i	$0.450529\pi$
$942$	0	0
$943$	−4.81074	−0.156659
$944$	−11.7913	−0.383774
$945$	0	0
$946$	99.5706	3.23732
$947$	25.3434	0.823549	0.411775	−	0.911286i	$-0.364909\pi$
$947$	25.3434	0.823549	0.411775	+	0.911286i	$0.364909\pi$
$948$	0	0
$949$	−30.4512	−0.988489
$950$	0	0
$951$	0	0
$952$	4.37912	0.141928
$953$	0.515249	0.0166906	0.00834528	−	0.999965i	$-0.497344\pi$
$953$	0.515249	0.0166906	0.00834528	+	0.999965i	$0.497344\pi$
$954$	0	0
$955$	0	0
$956$	42.8277	1.38515
$957$	0	0
$958$	−15.5189	−0.501393
$959$	46.5772	1.50406
$960$	0	0
$961$	−15.3897	−0.496440
$962$	−20.7432	−0.668788
$963$	0	0
$964$	−88.7694	−2.85907
$965$	0	0
$966$	0	0
$967$	41.0700	1.32072	0.660360	−	0.750949i	$-0.270403\pi$
$967$	41.0700	1.32072	0.660360	+	0.750949i	$0.270403\pi$
$968$	153.870	4.94556
$969$	0	0
$970$	0	0
$971$	35.9638	1.15413	0.577066	−	0.816697i	$-0.304198\pi$
$971$	35.9638	1.15413	0.577066	+	0.816697i	$0.304198\pi$
$972$	0	0
$973$	31.4868	1.00942
$974$	−9.48117	−0.303796
$975$	0	0
$976$	53.1910	1.70260
$977$	−56.4987	−1.80755	−0.903777	−	0.428004i	$-0.859217\pi$
$977$	−56.4987	−1.80755	−0.903777	+	0.428004i	$0.859217\pi$
$978$	0	0
$979$	−24.5234	−0.783772
$980$	0	0
$981$	0	0
$982$	24.3134	0.775871
$983$	3.49908	0.111603	0.0558017	−	0.998442i	$-0.482229\pi$
$983$	3.49908	0.111603	0.0558017	+	0.998442i	$0.482229\pi$
$984$	0	0
$985$	0	0
$986$	−0.624919	−0.0199015
$987$	0	0
$988$	−70.8281	−2.25334
$989$	−5.95825	−0.189461
$990$	0	0
$991$	2.42787	0.0771237	0.0385618	−	0.999256i	$-0.487722\pi$
$991$	2.42787	0.0771237	0.0385618	+	0.999256i	$0.487722\pi$
$992$	1.97007	0.0625498
$993$	0	0
$994$	104.409	3.31166
$995$	0	0
$996$	0	0
$997$	28.5463	0.904070	0.452035	−	0.892000i	$-0.350698\pi$
$997$	28.5463	0.904070	0.452035	+	0.892000i	$0.350698\pi$
$998$	16.0989	0.509602
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6525.2.a.bw.1.6		7
3.2	odd	2		6525.2.a.bv.1.2		7
5.4	even	2		1305.2.a.s.1.2	✓	7
15.14	odd	2		1305.2.a.t.1.6	yes	7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1305.2.a.s.1.2	✓	7	5.4	even	2
1305.2.a.t.1.6	yes	7	15.14	odd	2
6525.2.a.bv.1.2		7	3.2	odd	2
6525.2.a.bw.1.6		7	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 6525 = 3^{2} \cdot 5^{2} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6525.a (trivial)

\(f(q)\)	\(=\)	\(q+2.43890 q^{2} +3.94822 q^{4} -3.59688 q^{7} +4.75150 q^{8} +6.58661 q^{11} -2.25623 q^{13} -8.77241 q^{14} +3.69199 q^{16} -0.256230 q^{17} +7.95099 q^{19} +16.0641 q^{22} -0.961264 q^{23} -5.50271 q^{26} -14.2013 q^{28} +1.00000 q^{29} -3.95099 q^{31} -0.498627 q^{32} -0.624919 q^{34} +3.76964 q^{37} +19.3917 q^{38} +5.00459 q^{41} +6.19835 q^{43} +26.0054 q^{44} -2.34442 q^{46} +7.02055 q^{47} +5.93752 q^{49} -8.90809 q^{52} -6.38493 q^{53} -17.0906 q^{56} +2.43890 q^{58} -3.19375 q^{59} +14.4072 q^{61} -9.63606 q^{62} -8.60008 q^{64} +9.35246 q^{67} -1.01165 q^{68} -11.9020 q^{71} +13.4965 q^{73} +9.19375 q^{74} +31.3923 q^{76} -23.6912 q^{77} -5.22222 q^{79} +12.2057 q^{82} -0.195206 q^{83} +15.1171 q^{86} +31.2963 q^{88} -3.72323 q^{89} +8.11538 q^{91} -3.79528 q^{92} +17.1224 q^{94} +2.97486 q^{97} +14.4810 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + q^{2} + 13 q^{4} - 10 q^{7} + 3 q^{11} - 6 q^{13} + 9 q^{14} + 21 q^{16} + 8 q^{17} + 10 q^{19} - 9 q^{22} + 11 q^{23} - 3 q^{26} - 25 q^{28} + 7 q^{29} + 18 q^{31} + q^{32} - q^{34} - 13 q^{37}+ \cdots - 20 q^{98}+O(q^{100}) \) 7 * q + q^2 + 13 * q^4 - 10 * q^7 + 3 * q^11 - 6 * q^13 + 9 * q^14 + 21 * q^16 + 8 * q^17 + 10 * q^19 - 9 * q^22 + 11 * q^23 - 3 * q^26 - 25 * q^28 + 7 * q^29 + 18 * q^31 + q^32 - q^34 - 13 * q^37 + 12 * q^38 + 13 * q^41 - 9 * q^43 + 37 * q^44 - 8 * q^46 + 2 * q^47 + 21 * q^49 + q^52 + 5 * q^53 + 30 * q^56 + q^58 + 8 * q^59 + 14 * q^61 - 8 * q^62 + 8 * q^64 - 14 * q^67 + 27 * q^68 + 8 * q^71 - 3 * q^73 + 34 * q^74 + 4 * q^76 - 28 * q^77 + 4 * q^79 + 20 * q^82 + 17 * q^83 - 4 * q^86 - 26 * q^88 + 20 * q^89 + 12 * q^91 + 60 * q^92 - 21 * q^94 - 13 * q^97 - 20 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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