LMFDB - Embedded newform 6633.2.a.w.1.11

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6633,2,Mod(1,6633)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6633, 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("6633.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6633 = 3^{2} \cdot 11 \cdot 67 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6633.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:	$52.9647716607$
Analytic rank:	$0$
Dimension:	$17$
Coefficient field:	$\mathbb{Q}[x]/(x^{17} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{17} - x^{16} - 26 x^{15} + 25 x^{14} + 272 x^{13} - 244 x^{12} - 1472 x^{11} + 1186 x^{10} + 4406 x^{9} + \cdots - 2 $ x^17 - x^16 - 26x^15 + 25x^14 + 272x^13 - 244x^12 - 1472x^11 + 1186x^10 + 4406x^9 - 3042x^8 - 7180x^7 + 4055x^6 + 5665x^5 - 2655x^4 - 1418x^3 + 810x^2 - 76*x - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 737)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.11
Root			$-0.880523$ of defining polynomial
Character	$\chi$	$=$	6633.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.880523 q^{2} -1.22468 q^{4} -1.19978 q^{5} -2.13292 q^{7} -2.83940 q^{8} +O(q^{10})$	\(q+0.880523 q^{2} -1.22468 q^{4} -1.19978 q^{5} -2.13292 q^{7} -2.83940 q^{8} -1.05643 q^{10} -1.00000 q^{11} -5.84305 q^{13} -1.87808 q^{14} -0.0508036 q^{16} +1.21751 q^{17} -6.59971 q^{19} +1.46935 q^{20} -0.880523 q^{22} -7.98884 q^{23} -3.56053 q^{25} -5.14494 q^{26} +2.61214 q^{28} +4.07798 q^{29} +3.41898 q^{31} +5.63408 q^{32} +1.07204 q^{34} +2.55903 q^{35} +1.14383 q^{37} -5.81120 q^{38} +3.40666 q^{40} -10.5148 q^{41} +7.66761 q^{43} +1.22468 q^{44} -7.03436 q^{46} -9.80313 q^{47} -2.45066 q^{49} -3.13513 q^{50} +7.15586 q^{52} -6.97907 q^{53} +1.19978 q^{55} +6.05622 q^{56} +3.59076 q^{58} +4.72951 q^{59} -1.58402 q^{61} +3.01049 q^{62} +5.06254 q^{64} +7.01037 q^{65} -1.00000 q^{67} -1.49105 q^{68} +2.25329 q^{70} +2.40587 q^{71} +6.89792 q^{73} +1.00717 q^{74} +8.08253 q^{76} +2.13292 q^{77} +2.80338 q^{79} +0.0609531 q^{80} -9.25853 q^{82} -17.5380 q^{83} -1.46074 q^{85} +6.75151 q^{86} +2.83940 q^{88} -16.6046 q^{89} +12.4627 q^{91} +9.78376 q^{92} -8.63188 q^{94} +7.91820 q^{95} +5.22255 q^{97} -2.15786 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 17 q - q^{2} + 19 q^{4} - 10 q^{5} + 20 q^{7}+O(q^{10}) $ 17 * q - q^2 + 19 * q^4 - 10 * q^5 + 20 * q^7	$ 17 q - q^{2} + 19 q^{4} - 10 q^{5} + 20 q^{7} + 10 q^{10} - 17 q^{11} + q^{13} + 11 q^{14} + 19 q^{16} - 2 q^{17} + 13 q^{19} - 3 q^{20} + q^{22} - 16 q^{23} + 33 q^{25} - 12 q^{26} + 44 q^{28} + 5 q^{29} + 16 q^{31} + 24 q^{32} + 4 q^{34} + 2 q^{35} + 29 q^{37} + 19 q^{38} + 31 q^{40} + 6 q^{41} + 19 q^{43} - 19 q^{44} - 33 q^{46} - 40 q^{47} + 23 q^{49} + 3 q^{50} - 28 q^{52} - 15 q^{53} + 10 q^{55} + 38 q^{56} - 12 q^{58} + 2 q^{59} - 6 q^{61} - 3 q^{62} - 4 q^{64} + 30 q^{65} - 17 q^{67} + 13 q^{68} + 71 q^{70} - 2 q^{71} + 41 q^{73} + 13 q^{74} + 21 q^{76} - 20 q^{77} + 41 q^{79} + 23 q^{80} - 8 q^{82} - 2 q^{83} - 36 q^{85} + 54 q^{86} - q^{89} + 16 q^{91} - 36 q^{92} + 12 q^{94} + 31 q^{95} + 3 q^{97} + 64 q^{98}+O(q^{100}) $ 17 * q - q^2 + 19 * q^4 - 10 * q^5 + 20 * q^7 + 10 * q^10 - 17 * q^11 + q^13 + 11 * q^14 + 19 * q^16 - 2 * q^17 + 13 * q^19 - 3 * q^20 + q^22 - 16 * q^23 + 33 * q^25 - 12 * q^26 + 44 * q^28 + 5 * q^29 + 16 * q^31 + 24 * q^32 + 4 * q^34 + 2 * q^35 + 29 * q^37 + 19 * q^38 + 31 * q^40 + 6 * q^41 + 19 * q^43 - 19 * q^44 - 33 * q^46 - 40 * q^47 + 23 * q^49 + 3 * q^50 - 28 * q^52 - 15 * q^53 + 10 * q^55 + 38 * q^56 - 12 * q^58 + 2 * q^59 - 6 * q^61 - 3 * q^62 - 4 * q^64 + 30 * q^65 - 17 * q^67 + 13 * q^68 + 71 * q^70 - 2 * q^71 + 41 * q^73 + 13 * q^74 + 21 * q^76 - 20 * q^77 + 41 * q^79 + 23 * q^80 - 8 * q^82 - 2 * q^83 - 36 * q^85 + 54 * q^86 - q^89 + 16 * q^91 - 36 * q^92 + 12 * q^94 + 31 * q^95 + 3 * q^97 + 64 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0.880523	0.622624	0.311312	−	0.950308i	$-0.399232\pi$
$2$	0.880523	0.622624	0.311312	+	0.950308i	$0.399232\pi$
$3$	0	0
$3$	0	0
$4$	−1.22468	−0.612339
$5$	−1.19978	−0.536558	−0.268279	−	0.963341i	$-0.586455\pi$
$5$	−1.19978	−0.536558	−0.268279	+	0.963341i	$0.586455\pi$
$6$	0	0
$7$	−2.13292	−0.806167	−0.403084	−	0.915163i	$-0.632062\pi$
$7$	−2.13292	−0.806167	−0.403084	+	0.915163i	$0.632062\pi$
$8$	−2.83940	−1.00388
$9$	0	0
$10$	−1.05643	−0.334074
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−5.84305	−1.62057	−0.810285	−	0.586036i	$-0.800688\pi$
$13$	−5.84305	−1.62057	−0.810285	+	0.586036i	$0.800688\pi$
$14$	−1.87808	−0.501939
$15$	0	0
$16$	−0.0508036	−0.0127009
$17$	1.21751	0.295289	0.147644	−	0.989041i	$-0.452831\pi$
$17$	1.21751	0.295289	0.147644	+	0.989041i	$0.452831\pi$
$18$	0	0
$19$	−6.59971	−1.51408	−0.757039	−	0.653370i	$-0.773355\pi$
$19$	−6.59971	−1.51408	−0.757039	+	0.653370i	$0.773355\pi$
$20$	1.46935	0.328556
$21$	0	0
$22$	−0.880523	−0.187728
$23$	−7.98884	−1.66579	−0.832894	−	0.553433i	$-0.813318\pi$
$23$	−7.98884	−1.66579	−0.832894	+	0.553433i	$0.813318\pi$
$24$	0	0
$25$	−3.56053	−0.712105
$26$	−5.14494	−1.00901
$27$	0	0
$28$	2.61214	0.493648
$29$	4.07798	0.757262	0.378631	−	0.925548i	$-0.376395\pi$
$29$	4.07798	0.757262	0.378631	+	0.925548i	$0.376395\pi$
$30$	0	0
$31$	3.41898	0.614068	0.307034	−	0.951699i	$-0.400664\pi$
$31$	3.41898	0.614068	0.307034	+	0.951699i	$0.400664\pi$
$32$	5.63408	0.995973
$33$	0	0
$34$	1.07204	0.183854
$35$	2.55903	0.432555
$36$	0	0
$37$	1.14383	0.188044	0.0940221	−	0.995570i	$-0.470028\pi$
$37$	1.14383	0.188044	0.0940221	+	0.995570i	$0.470028\pi$
$38$	−5.81120	−0.942701
$39$	0	0
$40$	3.40666	0.538640
$41$	−10.5148	−1.64214	−0.821069	−	0.570829i	$-0.806622\pi$
$41$	−10.5148	−1.64214	−0.821069	+	0.570829i	$0.806622\pi$
$42$	0	0
$43$	7.66761	1.16930	0.584650	−	0.811286i	$-0.301232\pi$
$43$	7.66761	1.16930	0.584650	+	0.811286i	$0.301232\pi$
$44$	1.22468	0.184627
$45$	0	0
$46$	−7.03436	−1.03716
$47$	−9.80313	−1.42993	−0.714966	−	0.699159i	$-0.753558\pi$
$47$	−9.80313	−1.42993	−0.714966	+	0.699159i	$0.753558\pi$
$48$	0	0
$49$	−2.45066	−0.350095
$50$	−3.13513	−0.443374
$51$	0	0
$52$	7.15586	0.992339
$53$	−6.97907	−0.958650	−0.479325	−	0.877638i	$-0.659118\pi$
$53$	−6.97907	−0.958650	−0.479325	+	0.877638i	$0.659118\pi$
$54$	0	0
$55$	1.19978	0.161778
$56$	6.05622	0.809296
$57$	0	0
$58$	3.59076	0.471490
$59$	4.72951	0.615730	0.307865	−	0.951430i	$-0.400385\pi$
$59$	4.72951	0.615730	0.307865	+	0.951430i	$0.400385\pi$
$60$	0	0
$61$	−1.58402	−0.202813	−0.101407	−	0.994845i	$-0.532334\pi$
$61$	−1.58402	−0.202813	−0.101407	+	0.994845i	$0.532334\pi$
$62$	3.01049	0.382333
$63$	0	0
$64$	5.06254	0.632818
$65$	7.01037	0.869530
$66$	0	0
$67$	−1.00000	−0.122169
$67$	−1.00000	−0.122169
$68$	−1.49105	−0.180817
$69$	0	0
$70$	2.25329	0.269319
$71$	2.40587	0.285524	0.142762	−	0.989757i	$-0.454402\pi$
$71$	2.40587	0.285524	0.142762	+	0.989757i	$0.454402\pi$
$72$	0	0
$73$	6.89792	0.807341	0.403670	−	0.914904i	$-0.367734\pi$
$73$	6.89792	0.807341	0.403670	+	0.914904i	$0.367734\pi$
$74$	1.00717	0.117081
$75$	0	0
$76$	8.08253	0.927129
$77$	2.13292	0.243069
$78$	0	0
$79$	2.80338	0.315405	0.157703	−	0.987487i	$-0.449591\pi$
$79$	2.80338	0.315405	0.157703	+	0.987487i	$0.449591\pi$
$80$	0.0609531	0.00681477
$81$	0	0
$82$	−9.25853	−1.02243
$83$	−17.5380	−1.92505	−0.962524	−	0.271196i	$-0.912581\pi$
$83$	−17.5380	−1.92505	−0.962524	+	0.271196i	$0.912581\pi$
$84$	0	0
$85$	−1.46074	−0.158439
$86$	6.75151	0.728034
$87$	0	0
$88$	2.83940	0.302682
$89$	−16.6046	−1.76009	−0.880043	−	0.474894i	$-0.842486\pi$
$89$	−16.6046	−1.76009	−0.880043	+	0.474894i	$0.842486\pi$
$90$	0	0
$91$	12.4627	1.30645
$92$	9.78376	1.02003
$93$	0	0
$94$	−8.63188	−0.890310
$95$	7.91820	0.812390
$96$	0	0
$97$	5.22255	0.530270	0.265135	−	0.964211i	$-0.414583\pi$
$97$	5.22255	0.530270	0.265135	+	0.964211i	$0.414583\pi$
$98$	−2.15786	−0.217977
$99$	0	0
$100$	4.36050	0.436050
$101$	13.7327	1.36646	0.683229	−	0.730204i	$-0.260575\pi$
$101$	13.7327	1.36646	0.683229	+	0.730204i	$0.260575\pi$
$102$	0	0
$103$	14.5717	1.43579	0.717895	−	0.696151i	$-0.245106\pi$
$103$	14.5717	1.43579	0.717895	+	0.696151i	$0.245106\pi$
$104$	16.5908	1.62686
$105$	0	0
$106$	−6.14524	−0.596878
$107$	2.31604	0.223900	0.111950	−	0.993714i	$-0.464290\pi$
$107$	2.31604	0.223900	0.111950	+	0.993714i	$0.464290\pi$
$108$	0	0
$109$	−13.7225	−1.31438	−0.657189	−	0.753726i	$-0.728255\pi$
$109$	−13.7225	−1.31438	−0.657189	+	0.753726i	$0.728255\pi$
$110$	1.05643	0.100727
$111$	0	0
$112$	0.108360	0.0102390
$113$	16.7717	1.57775	0.788874	−	0.614555i	$-0.210665\pi$
$113$	16.7717	1.57775	0.788874	+	0.614555i	$0.210665\pi$
$114$	0	0
$115$	9.58485	0.893792
$116$	−4.99422	−0.463701
$117$	0	0
$118$	4.16445	0.383368
$119$	−2.59684	−0.238052
$120$	0	0
$121$	1.00000	0.0909091
$122$	−1.39477	−0.126277
$123$	0	0
$124$	−4.18716	−0.376018
$125$	10.2708	0.918644
$126$	0	0
$127$	−10.4543	−0.927665	−0.463833	−	0.885923i	$-0.653526\pi$
$127$	−10.4543	−0.927665	−0.463833	+	0.885923i	$0.653526\pi$
$128$	−6.81047	−0.601966
$129$	0	0
$130$	6.17280	0.541390
$131$	−1.34121	−0.117182	−0.0585912	−	0.998282i	$-0.518661\pi$
$131$	−1.34121	−0.117182	−0.0585912	+	0.998282i	$0.518661\pi$
$132$	0	0
$133$	14.0766	1.22060
$134$	−0.880523	−0.0760656
$135$	0	0
$136$	−3.45699	−0.296435
$137$	8.96031	0.765531	0.382765	−	0.923846i	$-0.374972\pi$
$137$	8.96031	0.765531	0.382765	+	0.923846i	$0.374972\pi$
$138$	0	0
$139$	19.0046	1.61195	0.805973	−	0.591952i	$-0.201643\pi$
$139$	19.0046	1.61195	0.805973	+	0.591952i	$0.201643\pi$
$140$	−3.13399	−0.264871
$141$	0	0
$142$	2.11842	0.177774
$143$	5.84305	0.488620
$144$	0	0
$145$	−4.89268	−0.406315
$146$	6.07378	0.502670
$147$	0	0
$148$	−1.40082	−0.115147
$149$	−10.7526	−0.880888	−0.440444	−	0.897780i	$-0.645179\pi$
$149$	−10.7526	−0.880888	−0.440444	+	0.897780i	$0.645179\pi$
$150$	0	0
$151$	0.702373	0.0571583	0.0285791	−	0.999592i	$-0.490902\pi$
$151$	0.702373	0.0571583	0.0285791	+	0.999592i	$0.490902\pi$
$152$	18.7392	1.51995
$153$	0	0
$154$	1.87808	0.151340
$155$	−4.10203	−0.329483
$156$	0	0
$157$	15.9764	1.27505	0.637526	−	0.770429i	$-0.279958\pi$
$157$	15.9764	1.27505	0.637526	+	0.770429i	$0.279958\pi$
$158$	2.46844	0.196379
$159$	0	0
$160$	−6.75965	−0.534397
$161$	17.0395	1.34290
$162$	0	0
$163$	−16.7640	−1.31306	−0.656530	−	0.754300i	$-0.727976\pi$
$163$	−16.7640	−1.31306	−0.656530	+	0.754300i	$0.727976\pi$
$164$	12.8773	1.00555
$165$	0	0
$166$	−15.4426	−1.19858
$167$	2.55176	0.197461	0.0987304	−	0.995114i	$-0.468522\pi$
$167$	2.55176	0.197461	0.0987304	+	0.995114i	$0.468522\pi$
$168$	0	0
$169$	21.1412	1.62625
$170$	−1.28621	−0.0986482
$171$	0	0
$172$	−9.39036	−0.716008
$173$	−4.11560	−0.312903	−0.156452	−	0.987686i	$-0.550006\pi$
$173$	−4.11560	−0.312903	−0.156452	+	0.987686i	$0.550006\pi$
$174$	0	0
$175$	7.59431	0.574076
$176$	0.0508036	0.00382946
$177$	0	0
$178$	−14.6208	−1.09587
$179$	−15.3646	−1.14840	−0.574202	−	0.818714i	$-0.694688\pi$
$179$	−15.3646	−1.14840	−0.574202	+	0.818714i	$0.694688\pi$
$180$	0	0
$181$	3.59132	0.266940	0.133470	−	0.991053i	$-0.457388\pi$
$181$	3.59132	0.266940	0.133470	+	0.991053i	$0.457388\pi$
$182$	10.9737	0.813427
$183$	0	0
$184$	22.6835	1.67225
$185$	−1.37234	−0.100897
$186$	0	0
$187$	−1.21751	−0.0890328
$188$	12.0057	0.875604
$189$	0	0
$190$	6.97216	0.505814
$191$	−12.3713	−0.895158	−0.447579	−	0.894244i	$-0.647714\pi$
$191$	−12.3713	−0.895158	−0.447579	+	0.894244i	$0.647714\pi$
$192$	0	0
$193$	−7.52220	−0.541460	−0.270730	−	0.962655i	$-0.587265\pi$
$193$	−7.52220	−0.541460	−0.270730	+	0.962655i	$0.587265\pi$
$194$	4.59858	0.330159
$195$	0	0
$196$	3.00127	0.214377
$197$	−5.52541	−0.393669	−0.196835	−	0.980437i	$-0.563066\pi$
$197$	−5.52541	−0.393669	−0.196835	+	0.980437i	$0.563066\pi$
$198$	0	0
$199$	−3.42561	−0.242835	−0.121417	−	0.992602i	$-0.538744\pi$
$199$	−3.42561	−0.242835	−0.121417	+	0.992602i	$0.538744\pi$
$200$	10.1098	0.714869
$201$	0	0
$202$	12.0920	0.850790
$203$	−8.69800	−0.610480
$204$	0	0
$205$	12.6155	0.881102
$206$	12.8307	0.893957
$207$	0	0
$208$	0.296848	0.0205827
$209$	6.59971	0.456511
$210$	0	0
$211$	−23.7799	−1.63708	−0.818538	−	0.574453i	$-0.805215\pi$
$211$	−23.7799	−1.63708	−0.818538	+	0.574453i	$0.805215\pi$
$212$	8.54713	0.587019
$213$	0	0
$214$	2.03933	0.139406
$215$	−9.19944	−0.627397
$216$	0	0
$217$	−7.29241	−0.495041
$218$	−12.0830	−0.818363
$219$	0	0
$220$	−1.46935	−0.0990633
$221$	−7.11395	−0.478536
$222$	0	0
$223$	5.06995	0.339509	0.169755	−	0.985486i	$-0.445703\pi$
$223$	5.06995	0.339509	0.169755	+	0.985486i	$0.445703\pi$
$224$	−12.0170	−0.802921
$225$	0	0
$226$	14.7679	0.982343
$227$	−10.8376	−0.719316	−0.359658	−	0.933084i	$-0.617107\pi$
$227$	−10.8376	−0.719316	−0.359658	+	0.933084i	$0.617107\pi$
$228$	0	0
$229$	−25.3009	−1.67193	−0.835966	−	0.548782i	$-0.815092\pi$
$229$	−25.3009	−1.67193	−0.835966	+	0.548782i	$0.815092\pi$
$230$	8.43968	0.556496
$231$	0	0
$232$	−11.5790	−0.760201
$233$	−10.8962	−0.713831	−0.356915	−	0.934137i	$-0.616172\pi$
$233$	−10.8962	−0.713831	−0.356915	+	0.934137i	$0.616172\pi$
$234$	0	0
$235$	11.7616	0.767242
$236$	−5.79214	−0.377036
$237$	0	0
$238$	−2.28658	−0.148217
$239$	26.8946	1.73966	0.869832	−	0.493347i	$-0.164227\pi$
$239$	26.8946	1.73966	0.869832	+	0.493347i	$0.164227\pi$
$240$	0	0
$241$	14.3252	0.922769	0.461385	−	0.887200i	$-0.347353\pi$
$241$	14.3252	0.922769	0.461385	+	0.887200i	$0.347353\pi$
$242$	0.880523	0.0566022
$243$	0	0
$244$	1.93992	0.124191
$245$	2.94026	0.187846
$246$	0	0
$247$	38.5624	2.45367
$248$	−9.70788	−0.616451
$249$	0	0
$250$	9.04363	0.571970
$251$	10.9658	0.692154	0.346077	−	0.938206i	$-0.387514\pi$
$251$	10.9658	0.692154	0.346077	+	0.938206i	$0.387514\pi$
$252$	0	0
$253$	7.98884	0.502254
$254$	−9.20522	−0.577587
$255$	0	0
$256$	−16.1219	−1.00762
$257$	−16.2384	−1.01292	−0.506461	−	0.862263i	$-0.669047\pi$
$257$	−16.2384	−1.01292	−0.506461	+	0.862263i	$0.669047\pi$
$258$	0	0
$259$	−2.43969	−0.151595
$260$	−8.58546	−0.532448
$261$	0	0
$262$	−1.18097	−0.0729606
$263$	−3.84928	−0.237357	−0.118678	−	0.992933i	$-0.537866\pi$
$263$	−3.84928	−0.237357	−0.118678	+	0.992933i	$0.537866\pi$
$264$	0	0
$265$	8.37336	0.514371
$266$	12.3948	0.759974
$267$	0	0
$268$	1.22468	0.0748092
$269$	−19.4751	−1.18742	−0.593709	−	0.804680i	$-0.702337\pi$
$269$	−19.4751	−1.18742	−0.593709	+	0.804680i	$0.702337\pi$
$270$	0	0
$271$	21.2733	1.29226	0.646132	−	0.763226i	$-0.276386\pi$
$271$	21.2733	1.29226	0.646132	+	0.763226i	$0.276386\pi$
$272$	−0.0618537	−0.00375043
$273$	0	0
$274$	7.88976	0.476638
$275$	3.56053	0.214708
$276$	0	0
$277$	24.7157	1.48502	0.742510	−	0.669835i	$-0.233635\pi$
$277$	24.7157	1.48502	0.742510	+	0.669835i	$0.233635\pi$
$278$	16.7340	1.00364
$279$	0	0
$280$	−7.26613	−0.434234
$281$	16.0378	0.956734	0.478367	−	0.878160i	$-0.341229\pi$
$281$	16.0378	0.956734	0.478367	+	0.878160i	$0.341229\pi$
$282$	0	0
$283$	22.9817	1.36612	0.683060	−	0.730362i	$-0.260649\pi$
$283$	22.9817	1.36612	0.683060	+	0.730362i	$0.260649\pi$
$284$	−2.94641	−0.174838
$285$	0	0
$286$	5.14494	0.304227
$287$	22.4272	1.32384
$288$	0	0
$289$	−15.5177	−0.912805
$290$	−4.30812	−0.252981
$291$	0	0
$292$	−8.44774	−0.494367
$293$	26.2167	1.53159	0.765797	−	0.643082i	$-0.222345\pi$
$293$	26.2167	1.53159	0.765797	+	0.643082i	$0.222345\pi$
$294$	0	0
$295$	−5.67438	−0.330375
$296$	−3.24779	−0.188774
$297$	0	0
$298$	−9.46792	−0.548462
$299$	46.6792	2.69953
$300$	0	0
$301$	−16.3544	−0.942651
$302$	0.618456	0.0355881
$303$	0	0
$304$	0.335289	0.0192301
$305$	1.90048	0.108821
$306$	0	0
$307$	15.7065	0.896415	0.448207	−	0.893930i	$-0.352063\pi$
$307$	15.7065	0.896415	0.448207	+	0.893930i	$0.352063\pi$
$308$	−2.61214	−0.148840
$309$	0	0
$310$	−3.61193	−0.205144
$311$	5.79581	0.328650	0.164325	−	0.986406i	$-0.447455\pi$
$311$	5.79581	0.328650	0.164325	+	0.986406i	$0.447455\pi$
$312$	0	0
$313$	−12.8207	−0.724667	−0.362333	−	0.932049i	$-0.618020\pi$
$313$	−12.8207	−0.724667	−0.362333	+	0.932049i	$0.618020\pi$
$314$	14.0676	0.793878
$315$	0	0
$316$	−3.43325	−0.193135
$317$	10.5364	0.591781	0.295890	−	0.955222i	$-0.404384\pi$
$317$	10.5364	0.591781	0.295890	+	0.955222i	$0.404384\pi$
$318$	0	0
$319$	−4.07798	−0.228323
$320$	−6.07394	−0.339543
$321$	0	0
$322$	15.0037	0.836124
$323$	−8.03519	−0.447090
$324$	0	0
$325$	20.8043	1.15402
$326$	−14.7611	−0.817542
$327$	0	0
$328$	29.8558	1.64851
$329$	20.9093	1.15276
$330$	0	0
$331$	33.5990	1.84677	0.923385	−	0.383876i	$-0.125411\pi$
$331$	33.5990	1.84677	0.923385	+	0.383876i	$0.125411\pi$
$332$	21.4784	1.17878
$333$	0	0
$334$	2.24688	0.122944
$335$	1.19978	0.0655510
$336$	0	0
$337$	−28.9201	−1.57538	−0.787689	−	0.616073i	$-0.788723\pi$
$337$	−28.9201	−1.57538	−0.787689	+	0.616073i	$0.788723\pi$
$338$	18.6153	1.01254
$339$	0	0
$340$	1.78894	0.0970187
$341$	−3.41898	−0.185148
$342$	0	0
$343$	20.1575	1.08840
$344$	−21.7714	−1.17384
$345$	0	0
$346$	−3.62388	−0.194821
$347$	7.79291	0.418345	0.209173	−	0.977879i	$-0.432923\pi$
$347$	7.79291	0.418345	0.209173	+	0.977879i	$0.432923\pi$
$348$	0	0
$349$	3.17947	0.170193	0.0850965	−	0.996373i	$-0.472880\pi$
$349$	3.17947	0.170193	0.0850965	+	0.996373i	$0.472880\pi$
$350$	6.68697	0.357433
$351$	0	0
$352$	−5.63408	−0.300297
$353$	−30.7310	−1.63565	−0.817824	−	0.575469i	$-0.804820\pi$
$353$	−30.7310	−1.63565	−0.817824	+	0.575469i	$0.804820\pi$
$354$	0	0
$355$	−2.88651	−0.153200
$356$	20.3353	1.07777
$357$	0	0
$358$	−13.5289	−0.715024
$359$	−12.9718	−0.684625	−0.342312	−	0.939586i	$-0.611210\pi$
$359$	−12.9718	−0.684625	−0.342312	+	0.939586i	$0.611210\pi$
$360$	0	0
$361$	24.5562	1.29243
$362$	3.16224	0.166203
$363$	0	0
$364$	−15.2629	−0.799991
$365$	−8.27599	−0.433185
$366$	0	0
$367$	29.4691	1.53827	0.769137	−	0.639084i	$-0.220686\pi$
$367$	29.4691	1.53827	0.769137	+	0.639084i	$0.220686\pi$
$368$	0.405862	0.0211570
$369$	0	0
$370$	−1.20838	−0.0628206
$371$	14.8858	0.772832
$372$	0	0
$373$	−20.0414	−1.03771	−0.518853	−	0.854864i	$-0.673641\pi$
$373$	−20.0414	−1.03771	−0.518853	+	0.854864i	$0.673641\pi$
$374$	−1.07204	−0.0554340
$375$	0	0
$376$	27.8350	1.43548
$377$	−23.8278	−1.22720
$378$	0	0
$379$	−32.0060	−1.64404	−0.822019	−	0.569460i	$-0.807152\pi$
$379$	−32.0060	−1.64404	−0.822019	+	0.569460i	$0.807152\pi$
$380$	−9.69725	−0.497459
$381$	0	0
$382$	−10.8932	−0.557347
$383$	−7.66141	−0.391480	−0.195740	−	0.980656i	$-0.562711\pi$
$383$	−7.66141	−0.391480	−0.195740	+	0.980656i	$0.562711\pi$
$384$	0	0
$385$	−2.55903	−0.130420
$386$	−6.62347	−0.337126
$387$	0	0
$388$	−6.39595	−0.324705
$389$	−22.9145	−1.16181	−0.580906	−	0.813971i	$-0.697302\pi$
$389$	−22.9145	−1.16181	−0.580906	+	0.813971i	$0.697302\pi$
$390$	0	0
$391$	−9.72646	−0.491888
$392$	6.95842	0.351453
$393$	0	0
$394$	−4.86525	−0.245108
$395$	−3.36345	−0.169233
$396$	0	0
$397$	9.57928	0.480770	0.240385	−	0.970678i	$-0.422726\pi$
$397$	9.57928	0.480770	0.240385	+	0.970678i	$0.422726\pi$
$398$	−3.01632	−0.151195
$399$	0	0
$400$	0.180888	0.00904438
$401$	−26.3388	−1.31530	−0.657649	−	0.753324i	$-0.728449\pi$
$401$	−26.3388	−1.31530	−0.657649	+	0.753324i	$0.728449\pi$
$402$	0	0
$403$	−19.9773	−0.995140
$404$	−16.8182	−0.836737
$405$	0	0
$406$	−7.65879	−0.380099
$407$	−1.14383	−0.0566974
$408$	0	0
$409$	−30.5525	−1.51072	−0.755362	−	0.655308i	$-0.772539\pi$
$409$	−30.5525	−1.51072	−0.755362	+	0.655308i	$0.772539\pi$
$410$	11.1082	0.548595
$411$	0	0
$412$	−17.8456	−0.879191
$413$	−10.0877	−0.496382
$414$	0	0
$415$	21.0418	1.03290
$416$	−32.9202	−1.61404
$417$	0	0
$418$	5.81120	0.284235
$419$	7.01285	0.342600	0.171300	−	0.985219i	$-0.445203\pi$
$419$	7.01285	0.342600	0.171300	+	0.985219i	$0.445203\pi$
$420$	0	0
$421$	17.6448	0.859953	0.429977	−	0.902840i	$-0.358522\pi$
$421$	17.6448	0.859953	0.429977	+	0.902840i	$0.358522\pi$
$422$	−20.9388	−1.01928
$423$	0	0
$424$	19.8164	0.962370
$425$	−4.33496	−0.210277
$426$	0	0
$427$	3.37859	0.163502
$428$	−2.83641	−0.137103
$429$	0	0
$430$	−8.10032	−0.390632
$431$	−23.0512	−1.11033	−0.555167	−	0.831739i	$-0.687346\pi$
$431$	−23.0512	−1.11033	−0.555167	+	0.831739i	$0.687346\pi$
$432$	0	0
$433$	−0.780429	−0.0375050	−0.0187525	−	0.999824i	$-0.505969\pi$
$433$	−0.780429	−0.0375050	−0.0187525	+	0.999824i	$0.505969\pi$
$434$	−6.42114	−0.308224
$435$	0	0
$436$	16.8057	0.804845
$437$	52.7240	2.52213
$438$	0	0
$439$	−29.1867	−1.39301	−0.696504	−	0.717553i	$-0.745262\pi$
$439$	−29.1867	−1.39301	−0.696504	+	0.717553i	$0.745262\pi$
$440$	−3.40666	−0.162406
$441$	0	0
$442$	−6.26399	−0.297948
$443$	−8.29707	−0.394206	−0.197103	−	0.980383i	$-0.563153\pi$
$443$	−8.29707	−0.394206	−0.197103	+	0.980383i	$0.563153\pi$
$444$	0	0
$445$	19.9219	0.944388
$446$	4.46421	0.211386
$447$	0	0
$448$	−10.7980	−0.510157
$449$	−15.3448	−0.724166	−0.362083	−	0.932146i	$-0.617934\pi$
$449$	−15.3448	−0.724166	−0.362083	+	0.932146i	$0.617934\pi$
$450$	0	0
$451$	10.5148	0.495123
$452$	−20.5399	−0.966117
$453$	0	0
$454$	−9.54275	−0.447864
$455$	−14.9526	−0.700986
$456$	0	0
$457$	33.5036	1.56723	0.783615	−	0.621246i	$-0.213373\pi$
$457$	33.5036	1.56723	0.783615	+	0.621246i	$0.213373\pi$
$458$	−22.2780	−1.04098
$459$	0	0
$460$	−11.7384	−0.547304
$461$	−14.7470	−0.686835	−0.343417	−	0.939183i	$-0.611585\pi$
$461$	−14.7470	−0.686835	−0.343417	+	0.939183i	$0.611585\pi$
$462$	0	0
$463$	8.55163	0.397428	0.198714	−	0.980058i	$-0.436324\pi$
$463$	8.55163	0.397428	0.198714	+	0.980058i	$0.436324\pi$
$464$	−0.207176	−0.00961791
$465$	0	0
$466$	−9.59432	−0.444448
$467$	31.2896	1.44791	0.723955	−	0.689847i	$-0.242322\pi$
$467$	31.2896	1.44791	0.723955	+	0.689847i	$0.242322\pi$
$468$	0	0
$469$	2.13292	0.0984890
$470$	10.3564	0.477703
$471$	0	0
$472$	−13.4290	−0.618120
$473$	−7.66761	−0.352557
$474$	0	0
$475$	23.4984	1.07818
$476$	3.18030	0.145769
$477$	0	0
$478$	23.6813	1.08316
$479$	37.3694	1.70745	0.853725	−	0.520724i	$-0.174338\pi$
$479$	37.3694	1.70745	0.853725	+	0.520724i	$0.174338\pi$
$480$	0	0
$481$	−6.68344	−0.304739
$482$	12.6137	0.574538
$483$	0	0
$484$	−1.22468	−0.0556672
$485$	−6.26592	−0.284521
$486$	0	0
$487$	−27.5046	−1.24635	−0.623176	−	0.782082i	$-0.714158\pi$
$487$	−27.5046	−1.24635	−0.623176	+	0.782082i	$0.714158\pi$
$488$	4.49768	0.203601
$489$	0	0
$490$	2.58896	0.116957
$491$	15.8241	0.714129	0.357065	−	0.934080i	$-0.383778\pi$
$491$	15.8241	0.714129	0.357065	+	0.934080i	$0.383778\pi$
$492$	0	0
$493$	4.96497	0.223611
$494$	33.9551	1.52771
$495$	0	0
$496$	−0.173697	−0.00779921
$497$	−5.13152	−0.230180
$498$	0	0
$499$	16.4213	0.735116	0.367558	−	0.930001i	$-0.380194\pi$
$499$	16.4213	0.735116	0.367558	+	0.930001i	$0.380194\pi$
$500$	−12.5784	−0.562522
$501$	0	0
$502$	9.65563	0.430952
$503$	−20.8827	−0.931113	−0.465556	−	0.885018i	$-0.654146\pi$
$503$	−20.8827	−0.931113	−0.465556	+	0.885018i	$0.654146\pi$
$504$	0	0
$505$	−16.4763	−0.733184
$506$	7.03436	0.312715
$507$	0	0
$508$	12.8031	0.568046
$509$	0.675315	0.0299328	0.0149664	−	0.999888i	$-0.495236\pi$
$509$	0.675315	0.0299328	0.0149664	+	0.999888i	$0.495236\pi$
$510$	0	0
$511$	−14.7127	−0.650852
$512$	−0.574735	−0.0253999
$513$	0	0
$514$	−14.2983	−0.630670
$515$	−17.4828	−0.770385
$516$	0	0
$517$	9.80313	0.431141
$518$	−2.14820	−0.0943867
$519$	0	0
$520$	−19.9053	−0.872905
$521$	13.4339	0.588549	0.294274	−	0.955721i	$-0.404922\pi$
$521$	13.4339	0.588549	0.294274	+	0.955721i	$0.404922\pi$
$522$	0	0
$523$	−12.1551	−0.531504	−0.265752	−	0.964041i	$-0.585620\pi$
$523$	−12.1551	−0.531504	−0.265752	+	0.964041i	$0.585620\pi$
$524$	1.64256	0.0717554
$525$	0	0
$526$	−3.38938	−0.147784
$527$	4.16263	0.181327
$528$	0	0
$529$	40.8215	1.77485
$530$	7.37293	0.320260
$531$	0	0
$532$	−17.2394	−0.747421
$533$	61.4386	2.66120
$534$	0	0
$535$	−2.77874	−0.120136
$536$	2.83940	0.122644
$537$	0	0
$538$	−17.1483	−0.739314
$539$	2.45066	0.105557
$540$	0	0
$541$	−8.91691	−0.383368	−0.191684	−	0.981457i	$-0.561395\pi$
$541$	−8.91691	−0.383368	−0.191684	+	0.981457i	$0.561395\pi$
$542$	18.7317	0.804594
$543$	0	0
$544$	6.85952	0.294099
$545$	16.4640	0.705240
$546$	0	0
$547$	−26.3894	−1.12833	−0.564164	−	0.825663i	$-0.690801\pi$
$547$	−26.3894	−1.12833	−0.564164	+	0.825663i	$0.690801\pi$
$548$	−10.9735	−0.468765
$549$	0	0
$550$	3.13513	0.133682
$551$	−26.9135	−1.14655
$552$	0	0
$553$	−5.97939	−0.254269
$554$	21.7627	0.924609
$555$	0	0
$556$	−23.2745	−0.987058
$557$	−4.35999	−0.184739	−0.0923694	−	0.995725i	$-0.529444\pi$
$557$	−4.35999	−0.184739	−0.0923694	+	0.995725i	$0.529444\pi$
$558$	0	0
$559$	−44.8022	−1.89493
$560$	−0.130008	−0.00549384
$561$	0	0
$562$	14.1216	0.595685
$563$	−8.56152	−0.360825	−0.180412	−	0.983591i	$-0.557743\pi$
$563$	−8.56152	−0.360825	−0.180412	+	0.983591i	$0.557743\pi$
$564$	0	0
$565$	−20.1223	−0.846553
$566$	20.2359	0.850579
$567$	0	0
$568$	−6.83123	−0.286632
$569$	−16.8448	−0.706169	−0.353085	−	0.935591i	$-0.614867\pi$
$569$	−16.8448	−0.706169	−0.353085	+	0.935591i	$0.614867\pi$
$570$	0	0
$571$	−9.93057	−0.415581	−0.207791	−	0.978173i	$-0.566627\pi$
$571$	−9.93057	−0.415581	−0.207791	+	0.978173i	$0.566627\pi$
$572$	−7.15586	−0.299202
$573$	0	0
$574$	19.7477	0.824253
$575$	28.4445	1.18622
$576$	0	0
$577$	2.80699	0.116856	0.0584282	−	0.998292i	$-0.481391\pi$
$577$	2.80699	0.116856	0.0584282	+	0.998292i	$0.481391\pi$
$578$	−13.6637	−0.568334
$579$	0	0
$580$	5.99196	0.248803
$581$	37.4072	1.55191
$582$	0	0
$583$	6.97907	0.289044
$584$	−19.5860	−0.810474
$585$	0	0
$586$	23.0844	0.953607
$587$	4.91349	0.202801	0.101401	−	0.994846i	$-0.467668\pi$
$587$	4.91349	0.202801	0.101401	+	0.994846i	$0.467668\pi$
$588$	0	0
$589$	−22.5643	−0.929746
$590$	−4.99642	−0.205699
$591$	0	0
$592$	−0.0581106	−0.00238833
$593$	−7.29789	−0.299688	−0.149844	−	0.988710i	$-0.547877\pi$
$593$	−7.29789	−0.299688	−0.149844	+	0.988710i	$0.547877\pi$
$594$	0	0
$595$	3.11564	0.127729
$596$	13.1685	0.539402
$597$	0	0
$598$	41.1021	1.68079
$599$	−21.8695	−0.893565	−0.446782	−	0.894643i	$-0.647430\pi$
$599$	−21.8695	−0.893565	−0.446782	+	0.894643i	$0.647430\pi$
$600$	0	0
$601$	10.7048	0.436657	0.218328	−	0.975875i	$-0.429940\pi$
$601$	10.7048	0.436657	0.218328	+	0.975875i	$0.429940\pi$
$602$	−14.4004	−0.586917
$603$	0	0
$604$	−0.860181	−0.0350003
$605$	−1.19978	−0.0487780
$606$	0	0
$607$	42.4767	1.72408	0.862039	−	0.506842i	$-0.169187\pi$
$607$	42.4767	1.72408	0.862039	+	0.506842i	$0.169187\pi$
$608$	−37.1833	−1.50798
$609$	0	0
$610$	1.67342	0.0677547
$611$	57.2801	2.31731
$612$	0	0
$613$	−24.5244	−0.990533	−0.495266	−	0.868741i	$-0.664930\pi$
$613$	−24.5244	−0.990533	−0.495266	+	0.868741i	$0.664930\pi$
$614$	13.8299	0.558129
$615$	0	0
$616$	−6.05622	−0.244012
$617$	−10.2040	−0.410798	−0.205399	−	0.978678i	$-0.565849\pi$
$617$	−10.2040	−0.410798	−0.205399	+	0.978678i	$0.565849\pi$
$618$	0	0
$619$	−35.0080	−1.40709	−0.703545	−	0.710651i	$-0.748401\pi$
$619$	−35.0080	−1.40709	−0.703545	+	0.710651i	$0.748401\pi$
$620$	5.02367	0.201755
$621$	0	0
$622$	5.10334	0.204625
$623$	35.4163	1.41892
$624$	0	0
$625$	5.47999	0.219200
$626$	−11.2889	−0.451195
$627$	0	0
$628$	−19.5659	−0.780765
$629$	1.39262	0.0555273
$630$	0	0
$631$	12.2855	0.489077	0.244538	−	0.969640i	$-0.421364\pi$
$631$	12.2855	0.489077	0.244538	+	0.969640i	$0.421364\pi$
$632$	−7.95994	−0.316630
$633$	0	0
$634$	9.27750	0.368457
$635$	12.5428	0.497746
$636$	0	0
$637$	14.3193	0.567353
$638$	−3.59076	−0.142159
$639$	0	0
$640$	8.17106	0.322990
$641$	−10.9112	−0.430966	−0.215483	−	0.976508i	$-0.569133\pi$
$641$	−10.9112	−0.430966	−0.215483	+	0.976508i	$0.569133\pi$
$642$	0	0
$643$	4.35547	0.171763	0.0858815	−	0.996305i	$-0.472629\pi$
$643$	4.35547	0.171763	0.0858815	+	0.996305i	$0.472629\pi$
$644$	−20.8680	−0.822313
$645$	0	0
$646$	−7.07517	−0.278369
$647$	25.6129	1.00695	0.503474	−	0.864010i	$-0.332055\pi$
$647$	25.6129	1.00695	0.503474	+	0.864010i	$0.332055\pi$
$648$	0	0
$649$	−4.72951	−0.185650
$650$	18.3187	0.718519
$651$	0	0
$652$	20.5305	0.804038
$653$	−11.8234	−0.462684	−0.231342	−	0.972872i	$-0.574312\pi$
$653$	−11.8234	−0.462684	−0.231342	+	0.972872i	$0.574312\pi$
$654$	0	0
$655$	1.60916	0.0628751
$656$	0.534190	0.0208566
$657$	0	0
$658$	18.4111	0.717739
$659$	−25.3846	−0.988843	−0.494421	−	0.869222i	$-0.664620\pi$
$659$	−25.3846	−0.988843	−0.494421	+	0.869222i	$0.664620\pi$
$660$	0	0
$661$	−32.9454	−1.28143	−0.640713	−	0.767780i	$-0.721361\pi$
$661$	−32.9454	−1.28143	−0.640713	+	0.767780i	$0.721361\pi$
$662$	29.5847	1.14984
$663$	0	0
$664$	49.7975	1.93252
$665$	−16.8889	−0.654922
$666$	0	0
$667$	−32.5783	−1.26144
$668$	−3.12508	−0.120913
$669$	0	0
$670$	1.05643	0.0408136
$671$	1.58402	0.0611506
$672$	0	0
$673$	20.5056	0.790434	0.395217	−	0.918588i	$-0.370669\pi$
$673$	20.5056	0.790434	0.395217	+	0.918588i	$0.370669\pi$
$674$	−25.4648	−0.980868
$675$	0	0
$676$	−25.8912	−0.995816
$677$	−1.88304	−0.0723711	−0.0361856	−	0.999345i	$-0.511521\pi$
$677$	−1.88304	−0.0723711	−0.0361856	+	0.999345i	$0.511521\pi$
$678$	0	0
$679$	−11.1393	−0.427486
$680$	4.14763	0.159054
$681$	0	0
$682$	−3.01049	−0.115278
$683$	−14.4838	−0.554206	−0.277103	−	0.960840i	$-0.589374\pi$
$683$	−14.4838	−0.554206	−0.277103	+	0.960840i	$0.589374\pi$
$684$	0	0
$685$	−10.7504	−0.410752
$686$	17.7491	0.677665
$687$	0	0
$688$	−0.389542	−0.0148512
$689$	40.7791	1.55356
$690$	0	0
$691$	−41.4774	−1.57788	−0.788938	−	0.614473i	$-0.789369\pi$
$691$	−41.4774	−1.57788	−0.788938	+	0.614473i	$0.789369\pi$
$692$	5.04029	0.191603
$693$	0	0
$694$	6.86183	0.260472
$695$	−22.8013	−0.864903
$696$	0	0
$697$	−12.8018	−0.484904
$698$	2.79960	0.105966
$699$	0	0
$700$	−9.30059	−0.351529
$701$	22.3543	0.844309	0.422154	−	0.906524i	$-0.361274\pi$
$701$	22.3543	0.844309	0.422154	+	0.906524i	$0.361274\pi$
$702$	0	0
$703$	−7.54893	−0.284713
$704$	−5.06254	−0.190802
$705$	0	0
$706$	−27.0594	−1.01839
$707$	−29.2908	−1.10159
$708$	0	0
$709$	14.1646	0.531964	0.265982	−	0.963978i	$-0.414304\pi$
$709$	14.1646	0.531964	0.265982	+	0.963978i	$0.414304\pi$
$710$	−2.54164	−0.0953860
$711$	0	0
$712$	47.1472	1.76692
$713$	−27.3137	−1.02291
$714$	0	0
$715$	−7.01037	−0.262173
$716$	18.8167	0.703213
$717$	0	0
$718$	−11.4220	−0.426264
$719$	29.2832	1.09208	0.546040	−	0.837759i	$-0.316134\pi$
$719$	29.2832	1.09208	0.546040	+	0.837759i	$0.316134\pi$
$720$	0	0
$721$	−31.0802	−1.15749
$722$	21.6223	0.804698
$723$	0	0
$724$	−4.39821	−0.163458
$725$	−14.5198	−0.539251
$726$	0	0
$727$	13.8539	0.513812	0.256906	−	0.966436i	$-0.417297\pi$
$727$	13.8539	0.513812	0.256906	+	0.966436i	$0.417297\pi$
$728$	−35.3868	−1.31152
$729$	0	0
$730$	−7.28720	−0.269711
$731$	9.33536	0.345281
$732$	0	0
$733$	10.8446	0.400555	0.200277	−	0.979739i	$-0.435816\pi$
$733$	10.8446	0.400555	0.200277	+	0.979739i	$0.435816\pi$
$734$	25.9482	0.957766
$735$	0	0
$736$	−45.0097	−1.65908
$737$	1.00000	0.0368355
$738$	0	0
$739$	40.7749	1.49993	0.749965	−	0.661477i	$-0.230070\pi$
$739$	40.7749	1.49993	0.749965	+	0.661477i	$0.230070\pi$
$740$	1.68068	0.0617830
$741$	0	0
$742$	13.1073	0.481184
$743$	−28.4796	−1.04482	−0.522408	−	0.852695i	$-0.674966\pi$
$743$	−28.4796	−1.04482	−0.522408	+	0.852695i	$0.674966\pi$
$744$	0	0
$745$	12.9008	0.472647
$746$	−17.6469	−0.646100
$747$	0	0
$748$	1.49105	0.0545183
$749$	−4.93993	−0.180501
$750$	0	0
$751$	−1.74875	−0.0638127	−0.0319063	−	0.999491i	$-0.510158\pi$
$751$	−1.74875	−0.0638127	−0.0319063	+	0.999491i	$0.510158\pi$
$752$	0.498034	0.0181614
$753$	0	0
$754$	−20.9810	−0.764082
$755$	−0.842693	−0.0306687
$756$	0	0
$757$	−1.67179	−0.0607623	−0.0303811	−	0.999538i	$-0.509672\pi$
$757$	−1.67179	−0.0607623	−0.0303811	+	0.999538i	$0.509672\pi$
$758$	−28.1820	−1.02362
$759$	0	0
$760$	−22.4830	−0.815543
$761$	−36.5968	−1.32663	−0.663317	−	0.748339i	$-0.730852\pi$
$761$	−36.5968	−1.32663	−0.663317	+	0.748339i	$0.730852\pi$
$762$	0	0
$763$	29.2690	1.05961
$764$	15.1509	0.548141
$765$	0	0
$766$	−6.74605	−0.243745
$767$	−27.6348	−0.997834
$768$	0	0
$769$	−24.4752	−0.882599	−0.441300	−	0.897360i	$-0.645482\pi$
$769$	−24.4752	−0.882599	−0.441300	+	0.897360i	$0.645482\pi$
$770$	−2.25329	−0.0812028
$771$	0	0
$772$	9.21228	0.331557
$773$	5.82796	0.209617	0.104809	−	0.994492i	$-0.466577\pi$
$773$	5.82796	0.209617	0.104809	+	0.994492i	$0.466577\pi$
$774$	0	0
$775$	−12.1734	−0.437281
$776$	−14.8289	−0.532328
$777$	0	0
$778$	−20.1768	−0.723372
$779$	69.3947	2.48632
$780$	0	0
$781$	−2.40587	−0.0860887
$782$	−8.56437	−0.306261
$783$	0	0
$784$	0.124502	0.00444652
$785$	−19.1681	−0.684139
$786$	0	0
$787$	−4.03916	−0.143980	−0.0719902	−	0.997405i	$-0.522935\pi$
$787$	−4.03916	−0.143980	−0.0719902	+	0.997405i	$0.522935\pi$
$788$	6.76685	0.241059
$789$	0	0
$790$	−2.96159	−0.105369
$791$	−35.7726	−1.27193
$792$	0	0
$793$	9.25553	0.328673
$794$	8.43477	0.299339
$795$	0	0
$796$	4.19527	0.148697
$797$	19.3523	0.685495	0.342747	−	0.939428i	$-0.388643\pi$
$797$	19.3523	0.685495	0.342747	+	0.939428i	$0.388643\pi$
$798$	0	0
$799$	−11.9354	−0.422243
$800$	−20.0603	−0.709238
$801$	0	0
$802$	−23.1920	−0.818937
$803$	−6.89792	−0.243422
$804$	0	0
$805$	−20.4437	−0.720546
$806$	−17.5905	−0.619598
$807$	0	0
$808$	−38.9928	−1.37176
$809$	54.3466	1.91072	0.955362	−	0.295438i	$-0.0954655\pi$
$809$	54.3466	1.91072	0.955362	+	0.295438i	$0.0954655\pi$
$810$	0	0
$811$	−7.27875	−0.255591	−0.127796	−	0.991801i	$-0.540790\pi$
$811$	−7.27875	−0.255591	−0.127796	+	0.991801i	$0.540790\pi$
$812$	10.6523	0.373821
$813$	0	0
$814$	−1.00717	−0.0353012
$815$	20.1131	0.704533
$816$	0	0
$817$	−50.6040	−1.77041
$818$	−26.9022	−0.940613
$819$	0	0
$820$	−15.4499	−0.539534
$821$	−19.0382	−0.664437	−0.332218	−	0.943203i	$-0.607797\pi$
$821$	−19.0382	−0.664437	−0.332218	+	0.943203i	$0.607797\pi$
$822$	0	0
$823$	−26.8713	−0.936676	−0.468338	−	0.883549i	$-0.655147\pi$
$823$	−26.8713	−0.936676	−0.468338	+	0.883549i	$0.655147\pi$
$824$	−41.3749	−1.44136
$825$	0	0
$826$	−8.88242	−0.309059
$827$	−28.1622	−0.979297	−0.489649	−	0.871920i	$-0.662875\pi$
$827$	−28.1622	−0.979297	−0.489649	+	0.871920i	$0.662875\pi$
$828$	0	0
$829$	36.8920	1.28131	0.640656	−	0.767828i	$-0.278663\pi$
$829$	36.8920	1.28131	0.640656	+	0.767828i	$0.278663\pi$
$830$	18.5278	0.643108
$831$	0	0
$832$	−29.5807	−1.02553
$833$	−2.98370	−0.103379
$834$	0	0
$835$	−3.06155	−0.105949
$836$	−8.08253	−0.279540
$837$	0	0
$838$	6.17498	0.213311
$839$	−21.6831	−0.748583	−0.374291	−	0.927311i	$-0.622114\pi$
$839$	−21.6831	−0.748583	−0.374291	+	0.927311i	$0.622114\pi$
$840$	0	0
$841$	−12.3701	−0.426554
$842$	15.5366	0.535428
$843$	0	0
$844$	29.1227	1.00245
$845$	−25.3648	−0.872576
$846$	0	0
$847$	−2.13292	−0.0732879
$848$	0.354562	0.0121757
$849$	0	0
$850$	−3.81704	−0.130923
$851$	−9.13785	−0.313242
$852$	0	0
$853$	8.85756	0.303277	0.151639	−	0.988436i	$-0.451545\pi$
$853$	8.85756	0.303277	0.151639	+	0.988436i	$0.451545\pi$
$854$	2.97493	0.101800
$855$	0	0
$856$	−6.57619	−0.224769
$857$	−13.2864	−0.453856	−0.226928	−	0.973912i	$-0.572868\pi$
$857$	−13.2864	−0.453856	−0.226928	+	0.973912i	$0.572868\pi$
$858$	0	0
$859$	34.7269	1.18487	0.592433	−	0.805619i	$-0.298167\pi$
$859$	34.7269	1.18487	0.592433	+	0.805619i	$0.298167\pi$
$860$	11.2664	0.384180
$861$	0	0
$862$	−20.2971	−0.691321
$863$	−45.2505	−1.54035	−0.770173	−	0.637835i	$-0.779830\pi$
$863$	−45.2505	−1.54035	−0.770173	+	0.637835i	$0.779830\pi$
$864$	0	0
$865$	4.93781	0.167891
$866$	−0.687186	−0.0233515
$867$	0	0
$868$	8.93086	0.303133
$869$	−2.80338	−0.0950983
$870$	0	0
$871$	5.84305	0.197984
$872$	38.9638	1.31948
$873$	0	0
$874$	46.4247	1.57034
$875$	−21.9067	−0.740581
$876$	0	0
$877$	−25.6084	−0.864736	−0.432368	−	0.901697i	$-0.642322\pi$
$877$	−25.6084	−0.864736	−0.432368	+	0.901697i	$0.642322\pi$
$878$	−25.6996	−0.867319
$879$	0	0
$880$	−0.0609531	−0.00205473
$881$	−57.0760	−1.92294	−0.961469	−	0.274914i	$-0.911351\pi$
$881$	−57.0760	−1.92294	−0.961469	+	0.274914i	$0.911351\pi$
$882$	0	0
$883$	−26.8318	−0.902963	−0.451482	−	0.892280i	$-0.649104\pi$
$883$	−26.8318	−0.902963	−0.451482	+	0.892280i	$0.649104\pi$
$884$	8.71230	0.293026
$885$	0	0
$886$	−7.30576	−0.245442
$887$	4.18328	0.140461	0.0702304	−	0.997531i	$-0.477627\pi$
$887$	4.18328	0.140461	0.0702304	+	0.997531i	$0.477627\pi$
$888$	0	0
$889$	22.2981	0.747853
$890$	17.5417	0.587999
$891$	0	0
$892$	−6.20906	−0.207895
$893$	64.6978	2.16503
$894$	0	0
$895$	18.4341	0.616185
$896$	14.5262	0.485285
$897$	0	0
$898$	−13.5115	−0.450883
$899$	13.9425	0.465010
$900$	0	0
$901$	−8.49706	−0.283078
$902$	9.25853	0.308275
$903$	0	0
$904$	−47.6216	−1.58387
$905$	−4.30879	−0.143229
$906$	0	0
$907$	25.6397	0.851353	0.425677	−	0.904875i	$-0.360036\pi$
$907$	25.6397	0.851353	0.425677	+	0.904875i	$0.360036\pi$
$908$	13.2726	0.440466
$909$	0	0
$910$	−13.1661	−0.436451
$911$	18.7528	0.621307	0.310653	−	0.950523i	$-0.399452\pi$
$911$	18.7528	0.621307	0.310653	+	0.950523i	$0.399452\pi$
$912$	0	0
$913$	17.5380	0.580424
$914$	29.5007	0.975795
$915$	0	0
$916$	30.9855	1.02379
$917$	2.86070	0.0944686
$918$	0	0
$919$	22.7419	0.750187	0.375093	−	0.926987i	$-0.377611\pi$
$919$	22.7419	0.750187	0.375093	+	0.926987i	$0.377611\pi$
$920$	−27.2153	−0.897261
$921$	0	0
$922$	−12.9850	−0.427640
$923$	−14.0576	−0.462711
$924$	0	0
$925$	−4.07263	−0.133907
$926$	7.52990	0.247448
$927$	0	0
$928$	22.9757	0.754213
$929$	−49.1933	−1.61398	−0.806990	−	0.590566i	$-0.798905\pi$
$929$	−49.1933	−1.61398	−0.806990	+	0.590566i	$0.798905\pi$
$930$	0	0
$931$	16.1737	0.530070
$932$	13.3443	0.437107
$933$	0	0
$934$	27.5512	0.901503
$935$	1.46074	0.0477713
$936$	0	0
$937$	52.3956	1.71169	0.855845	−	0.517233i	$-0.173038\pi$
$937$	52.3956	1.71169	0.855845	+	0.517233i	$0.173038\pi$
$938$	1.87808	0.0613216
$939$	0	0
$940$	−14.4042	−0.469813
$941$	−25.9025	−0.844398	−0.422199	−	0.906503i	$-0.638742\pi$
$941$	−25.9025	−0.844398	−0.422199	+	0.906503i	$0.638742\pi$
$942$	0	0
$943$	84.0011	2.73545
$944$	−0.240276	−0.00782033
$945$	0	0
$946$	−6.75151	−0.219510
$947$	−50.4641	−1.63986	−0.819932	−	0.572461i	$-0.805989\pi$
$947$	−50.4641	−1.63986	−0.819932	+	0.572461i	$0.805989\pi$
$948$	0	0
$949$	−40.3049	−1.30835
$950$	20.6909	0.671302
$951$	0	0
$952$	7.37348	0.238976
$953$	33.9411	1.09946	0.549731	−	0.835342i	$-0.314730\pi$
$953$	33.9411	1.09946	0.549731	+	0.835342i	$0.314730\pi$
$954$	0	0
$955$	14.8429	0.480304
$956$	−32.9372	−1.06527
$957$	0	0
$958$	32.9046	1.06310
$959$	−19.1116	−0.617146
$960$	0	0
$961$	−19.3106	−0.622921
$962$	−5.88493	−0.189738
$963$	0	0
$964$	−17.5438	−0.565048
$965$	9.02498	0.290525
$966$	0	0
$967$	50.3928	1.62052	0.810261	−	0.586069i	$-0.199325\pi$
$967$	50.3928	1.62052	0.810261	+	0.586069i	$0.199325\pi$
$968$	−2.83940	−0.0912619
$969$	0	0
$970$	−5.51729	−0.177149
$971$	−48.3094	−1.55032	−0.775161	−	0.631764i	$-0.782331\pi$
$971$	−48.3094	−1.55032	−0.775161	+	0.631764i	$0.782331\pi$
$972$	0	0
$973$	−40.5352	−1.29950
$974$	−24.2184	−0.776008
$975$	0	0
$976$	0.0804741	0.00257591
$977$	18.5555	0.593642	0.296821	−	0.954933i	$-0.404073\pi$
$977$	18.5555	0.593642	0.296821	+	0.954933i	$0.404073\pi$
$978$	0	0
$979$	16.6046	0.530686
$980$	−3.60087	−0.115026
$981$	0	0
$982$	13.9334	0.444634
$983$	−38.7712	−1.23661	−0.618305	−	0.785938i	$-0.712180\pi$
$983$	−38.7712	−1.23661	−0.618305	+	0.785938i	$0.712180\pi$
$984$	0	0
$985$	6.62928	0.211226
$986$	4.37177	0.139225
$987$	0	0
$988$	−47.2266	−1.50248
$989$	−61.2553	−1.94780
$990$	0	0
$991$	−42.7509	−1.35803	−0.679013	−	0.734126i	$-0.737592\pi$
$991$	−42.7509	−1.35803	−0.679013	+	0.734126i	$0.737592\pi$
$992$	19.2628	0.611595
$993$	0	0
$994$	−4.51842	−0.143316
$995$	4.10997	0.130295
$996$	0	0
$997$	−2.36238	−0.0748174	−0.0374087	−	0.999300i	$-0.511910\pi$
$997$	−2.36238	−0.0748174	−0.0374087	+	0.999300i	$0.511910\pi$
$998$	14.4593	0.457701
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6633.2.a.w.1.11		17
3.2	odd	2		737.2.a.f.1.7	✓	17
33.32	even	2		8107.2.a.o.1.11		17

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
737.2.a.f.1.7	✓	17	3.2	odd	2
6633.2.a.w.1.11		17	1.1	even	1	trivial
8107.2.a.o.1.11		17	33.32	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 6633 = 3^{2} \cdot 11 \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6633.a (trivial)

\(f(q)\)	\(=\)	\(q+0.880523 q^{2} -1.22468 q^{4} -1.19978 q^{5} -2.13292 q^{7} -2.83940 q^{8} +O(q^{10})\)	\(q+0.880523 q^{2} -1.22468 q^{4} -1.19978 q^{5} -2.13292 q^{7} -2.83940 q^{8} -1.05643 q^{10} -1.00000 q^{11} -5.84305 q^{13} -1.87808 q^{14} -0.0508036 q^{16} +1.21751 q^{17} -6.59971 q^{19} +1.46935 q^{20} -0.880523 q^{22} -7.98884 q^{23} -3.56053 q^{25} -5.14494 q^{26} +2.61214 q^{28} +4.07798 q^{29} +3.41898 q^{31} +5.63408 q^{32} +1.07204 q^{34} +2.55903 q^{35} +1.14383 q^{37} -5.81120 q^{38} +3.40666 q^{40} -10.5148 q^{41} +7.66761 q^{43} +1.22468 q^{44} -7.03436 q^{46} -9.80313 q^{47} -2.45066 q^{49} -3.13513 q^{50} +7.15586 q^{52} -6.97907 q^{53} +1.19978 q^{55} +6.05622 q^{56} +3.59076 q^{58} +4.72951 q^{59} -1.58402 q^{61} +3.01049 q^{62} +5.06254 q^{64} +7.01037 q^{65} -1.00000 q^{67} -1.49105 q^{68} +2.25329 q^{70} +2.40587 q^{71} +6.89792 q^{73} +1.00717 q^{74} +8.08253 q^{76} +2.13292 q^{77} +2.80338 q^{79} +0.0609531 q^{80} -9.25853 q^{82} -17.5380 q^{83} -1.46074 q^{85} +6.75151 q^{86} +2.83940 q^{88} -16.6046 q^{89} +12.4627 q^{91} +9.78376 q^{92} -8.63188 q^{94} +7.91820 q^{95} +5.22255 q^{97} -2.15786 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 17 q - q^{2} + 19 q^{4} - 10 q^{5} + 20 q^{7}+O(q^{10}) \) 17 * q - q^2 + 19 * q^4 - 10 * q^5 + 20 * q^7	\( 17 q - q^{2} + 19 q^{4} - 10 q^{5} + 20 q^{7} + 10 q^{10} - 17 q^{11} + q^{13} + 11 q^{14} + 19 q^{16} - 2 q^{17} + 13 q^{19} - 3 q^{20} + q^{22} - 16 q^{23} + 33 q^{25} - 12 q^{26} + 44 q^{28} + 5 q^{29} + 16 q^{31} + 24 q^{32} + 4 q^{34} + 2 q^{35} + 29 q^{37} + 19 q^{38} + 31 q^{40} + 6 q^{41} + 19 q^{43} - 19 q^{44} - 33 q^{46} - 40 q^{47} + 23 q^{49} + 3 q^{50} - 28 q^{52} - 15 q^{53} + 10 q^{55} + 38 q^{56} - 12 q^{58} + 2 q^{59} - 6 q^{61} - 3 q^{62} - 4 q^{64} + 30 q^{65} - 17 q^{67} + 13 q^{68} + 71 q^{70} - 2 q^{71} + 41 q^{73} + 13 q^{74} + 21 q^{76} - 20 q^{77} + 41 q^{79} + 23 q^{80} - 8 q^{82} - 2 q^{83} - 36 q^{85} + 54 q^{86} - q^{89} + 16 q^{91} - 36 q^{92} + 12 q^{94} + 31 q^{95} + 3 q^{97} + 64 q^{98}+O(q^{100}) \) 17 * q - q^2 + 19 * q^4 - 10 * q^5 + 20 * q^7 + 10 * q^10 - 17 * q^11 + q^13 + 11 * q^14 + 19 * q^16 - 2 * q^17 + 13 * q^19 - 3 * q^20 + q^22 - 16 * q^23 + 33 * q^25 - 12 * q^26 + 44 * q^28 + 5 * q^29 + 16 * q^31 + 24 * q^32 + 4 * q^34 + 2 * q^35 + 29 * q^37 + 19 * q^38 + 31 * q^40 + 6 * q^41 + 19 * q^43 - 19 * q^44 - 33 * q^46 - 40 * q^47 + 23 * q^49 + 3 * q^50 - 28 * q^52 - 15 * q^53 + 10 * q^55 + 38 * q^56 - 12 * q^58 + 2 * q^59 - 6 * q^61 - 3 * q^62 - 4 * q^64 + 30 * q^65 - 17 * q^67 + 13 * q^68 + 71 * q^70 - 2 * q^71 + 41 * q^73 + 13 * q^74 + 21 * q^76 - 20 * q^77 + 41 * q^79 + 23 * q^80 - 8 * q^82 - 2 * q^83 - 36 * q^85 + 54 * q^86 - q^89 + 16 * q^91 - 36 * q^92 + 12 * q^94 + 31 * q^95 + 3 * q^97 + 64 * q^98

Introduction

L-functions

Modular forms

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Database

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