LMFDB - Embedded newform 6633.2.a.w.1.10

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			1.10
Root			$-0.0213052$ 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)

$f(q)$	$=$	$q+0.0213052 q^{2} -1.99955 q^{4} +1.68894 q^{5} -3.23910 q^{7} -0.0852109 q^{8} +O(q^{10})$	\(q+0.0213052 q^{2} -1.99955 q^{4} +1.68894 q^{5} -3.23910 q^{7} -0.0852109 q^{8} +0.0359832 q^{10} -1.00000 q^{11} -0.729835 q^{13} -0.0690095 q^{14} +3.99728 q^{16} -5.52036 q^{17} +4.57010 q^{19} -3.37712 q^{20} -0.0213052 q^{22} -2.78230 q^{23} -2.14748 q^{25} -0.0155493 q^{26} +6.47673 q^{28} -3.98163 q^{29} -3.98681 q^{31} +0.255584 q^{32} -0.117612 q^{34} -5.47065 q^{35} +1.37287 q^{37} +0.0973667 q^{38} -0.143916 q^{40} +6.10913 q^{41} -2.58984 q^{43} +1.99955 q^{44} -0.0592774 q^{46} +4.60719 q^{47} +3.49177 q^{49} -0.0457523 q^{50} +1.45934 q^{52} -5.81037 q^{53} -1.68894 q^{55} +0.276007 q^{56} -0.0848291 q^{58} -2.36004 q^{59} +13.1551 q^{61} -0.0849396 q^{62} -7.98911 q^{64} -1.23265 q^{65} -1.00000 q^{67} +11.0382 q^{68} -0.116553 q^{70} -9.41869 q^{71} +3.79882 q^{73} +0.0292491 q^{74} -9.13813 q^{76} +3.23910 q^{77} +1.84987 q^{79} +6.75117 q^{80} +0.130156 q^{82} -3.66734 q^{83} -9.32357 q^{85} -0.0551769 q^{86} +0.0852109 q^{88} +15.6158 q^{89} +2.36401 q^{91} +5.56334 q^{92} +0.0981568 q^{94} +7.71863 q^{95} -5.47596 q^{97} +0.0743928 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	0.0213052	0.0150650	0.00753251	−	0.999972i	$-0.497602\pi$
$2$	0.0213052	0.0150650	0.00753251	+	0.999972i	$0.497602\pi$
$3$	0	0
$3$	0	0
$4$	−1.99955	−0.999773
$5$	1.68894	0.755318	0.377659	−	0.925945i	$-0.376729\pi$
$5$	1.68894	0.755318	0.377659	+	0.925945i	$0.376729\pi$
$6$	0	0
$7$	−3.23910	−1.22427	−0.612133	−	0.790755i	$-0.709688\pi$
$7$	−3.23910	−1.22427	−0.612133	+	0.790755i	$0.709688\pi$
$8$	−0.0852109	−0.0301266
$9$	0	0
$10$	0.0359832	0.0113789
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−0.729835	−0.202420	−0.101210	−	0.994865i	$-0.532271\pi$
$13$	−0.729835	−0.202420	−0.101210	+	0.994865i	$0.532271\pi$
$14$	−0.0690095	−0.0184436
$15$	0	0
$16$	3.99728	0.999319
$17$	−5.52036	−1.33888	−0.669442	−	0.742864i	$-0.733467\pi$
$17$	−5.52036	−1.33888	−0.669442	+	0.742864i	$0.733467\pi$
$18$	0	0
$19$	4.57010	1.04845	0.524226	−	0.851579i	$-0.324355\pi$
$19$	4.57010	1.04845	0.524226	+	0.851579i	$0.324355\pi$
$20$	−3.37712	−0.755146
$21$	0	0
$22$	−0.0213052	−0.00454227
$23$	−2.78230	−0.580150	−0.290075	−	0.957004i	$-0.593680\pi$
$23$	−2.78230	−0.580150	−0.290075	+	0.957004i	$0.593680\pi$
$24$	0	0
$25$	−2.14748	−0.429495
$26$	−0.0155493	−0.00304946
$27$	0	0
$28$	6.47673	1.22399
$29$	−3.98163	−0.739369	−0.369685	−	0.929157i	$-0.620534\pi$
$29$	−3.98163	−0.739369	−0.369685	+	0.929157i	$0.620534\pi$
$30$	0	0
$31$	−3.98681	−0.716052	−0.358026	−	0.933712i	$-0.616550\pi$
$31$	−3.98681	−0.716052	−0.358026	+	0.933712i	$0.616550\pi$
$32$	0.255584	0.0451814
$33$	0	0
$34$	−0.117612	−0.0201703
$35$	−5.47065	−0.924709
$36$	0	0
$37$	1.37287	0.225698	0.112849	−	0.993612i	$-0.464002\pi$
$37$	1.37287	0.225698	0.112849	+	0.993612i	$0.464002\pi$
$38$	0.0973667	0.0157950
$39$	0	0
$40$	−0.143916	−0.0227552
$41$	6.10913	0.954085	0.477043	−	0.878880i	$-0.341709\pi$
$41$	6.10913	0.954085	0.477043	+	0.878880i	$0.341709\pi$
$42$	0	0
$43$	−2.58984	−0.394946	−0.197473	−	0.980308i	$-0.563274\pi$
$43$	−2.58984	−0.394946	−0.197473	+	0.980308i	$0.563274\pi$
$44$	1.99955	0.301443
$45$	0	0
$46$	−0.0592774	−0.00873998
$47$	4.60719	0.672027	0.336014	−	0.941857i	$-0.390921\pi$
$47$	4.60719	0.672027	0.336014	+	0.941857i	$0.390921\pi$
$48$	0	0
$49$	3.49177	0.498825
$50$	−0.0457523	−0.00647035
$51$	0	0
$52$	1.45934	0.202374
$53$	−5.81037	−0.798116	−0.399058	−	0.916926i	$-0.630663\pi$
$53$	−5.81037	−0.798116	−0.399058	+	0.916926i	$0.630663\pi$
$54$	0	0
$55$	−1.68894	−0.227737
$56$	0.276007	0.0368830
$57$	0	0
$58$	−0.0848291	−0.0111386
$59$	−2.36004	−0.307250	−0.153625	−	0.988129i	$-0.549095\pi$
$59$	−2.36004	−0.307250	−0.153625	+	0.988129i	$0.549095\pi$
$60$	0	0
$61$	13.1551	1.68434	0.842171	−	0.539210i	$-0.181277\pi$
$61$	13.1551	1.68434	0.842171	+	0.539210i	$0.181277\pi$
$62$	−0.0849396	−0.0107873
$63$	0	0
$64$	−7.98911	−0.998639
$65$	−1.23265	−0.152891
$66$	0	0
$67$	−1.00000	−0.122169
$67$	−1.00000	−0.122169
$68$	11.0382	1.33858
$69$	0	0
$70$	−0.116553	−0.0139308
$71$	−9.41869	−1.11779	−0.558897	−	0.829237i	$-0.688775\pi$
$71$	−9.41869	−1.11779	−0.558897	+	0.829237i	$0.688775\pi$
$72$	0	0
$73$	3.79882	0.444618	0.222309	−	0.974976i	$-0.428641\pi$
$73$	3.79882	0.444618	0.222309	+	0.974976i	$0.428641\pi$
$74$	0.0292491	0.00340014
$75$	0	0
$76$	−9.13813	−1.04821
$77$	3.23910	0.369130
$78$	0	0
$79$	1.84987	0.208126	0.104063	−	0.994571i	$-0.466816\pi$
$79$	1.84987	0.208126	0.104063	+	0.994571i	$0.466816\pi$
$80$	6.75117	0.754803
$81$	0	0
$82$	0.130156	0.0143733
$83$	−3.66734	−0.402542	−0.201271	−	0.979536i	$-0.564507\pi$
$83$	−3.66734	−0.402542	−0.201271	+	0.979536i	$0.564507\pi$
$84$	0	0
$85$	−9.32357	−1.01128
$86$	−0.0551769	−0.00594987
$87$	0	0
$88$	0.0852109	0.00908352
$89$	15.6158	1.65528	0.827638	−	0.561262i	$-0.189684\pi$
$89$	15.6158	1.65528	0.827638	+	0.561262i	$0.189684\pi$
$90$	0	0
$91$	2.36401	0.247816
$92$	5.56334	0.580019
$93$	0	0
$94$	0.0981568	0.0101241
$95$	7.71863	0.791915
$96$	0	0
$97$	−5.47596	−0.556000	−0.278000	−	0.960581i	$-0.589671\pi$
$97$	−5.47596	−0.556000	−0.278000	+	0.960581i	$0.589671\pi$
$98$	0.0743928	0.00751481
$99$	0	0
$100$	4.29398	0.429398
$101$	−13.3662	−1.32999	−0.664995	−	0.746847i	$-0.731567\pi$
$101$	−13.3662	−1.32999	−0.664995	+	0.746847i	$0.731567\pi$
$102$	0	0
$103$	18.1594	1.78930	0.894651	−	0.446766i	$-0.147424\pi$
$103$	18.1594	1.78930	0.894651	+	0.446766i	$0.147424\pi$
$104$	0.0621899	0.00609823
$105$	0	0
$106$	−0.123791	−0.0120236
$107$	−6.52010	−0.630322	−0.315161	−	0.949038i	$-0.602058\pi$
$107$	−6.52010	−0.630322	−0.315161	+	0.949038i	$0.602058\pi$
$108$	0	0
$109$	−10.9067	−1.04467	−0.522334	−	0.852741i	$-0.674939\pi$
$109$	−10.9067	−1.04467	−0.522334	+	0.852741i	$0.674939\pi$
$110$	−0.0359832	−0.00343086
$111$	0	0
$112$	−12.9476	−1.22343
$113$	2.75543	0.259209	0.129605	−	0.991566i	$-0.458629\pi$
$113$	2.75543	0.259209	0.129605	+	0.991566i	$0.458629\pi$
$114$	0	0
$115$	−4.69915	−0.438198
$116$	7.96144	0.739201
$117$	0	0
$118$	−0.0502809	−0.00462873
$119$	17.8810	1.63915
$120$	0	0
$121$	1.00000	0.0909091
$122$	0.280272	0.0253746
$123$	0	0
$124$	7.97181	0.715890
$125$	−12.0717	−1.07972
$126$	0	0
$127$	19.4504	1.72594	0.862970	−	0.505254i	$-0.168601\pi$
$127$	19.4504	1.72594	0.862970	+	0.505254i	$0.168601\pi$
$128$	−0.681378	−0.0602259
$129$	0	0
$130$	−0.0262618	−0.00230331
$131$	−4.51463	−0.394445	−0.197223	−	0.980359i	$-0.563192\pi$
$131$	−4.51463	−0.394445	−0.197223	+	0.980359i	$0.563192\pi$
$132$	0	0
$133$	−14.8030	−1.28358
$134$	−0.0213052	−0.00184049
$135$	0	0
$136$	0.470395	0.0403361
$137$	0.0242608	0.00207274	0.00103637	−	0.999999i	$-0.499670\pi$
$137$	0.0242608	0.00207274	0.00103637	+	0.999999i	$0.499670\pi$
$138$	0	0
$139$	−10.6134	−0.900221	−0.450110	−	0.892973i	$-0.648615\pi$
$139$	−10.6134	−0.900221	−0.450110	+	0.892973i	$0.648615\pi$
$140$	10.9388	0.924499
$141$	0	0
$142$	−0.200667	−0.0168396
$143$	0.729835	0.0610319
$144$	0	0
$145$	−6.72473	−0.558459
$146$	0.0809345	0.00669818
$147$	0	0
$148$	−2.74511	−0.225646
$149$	4.52419	0.370636	0.185318	−	0.982679i	$-0.440668\pi$
$149$	4.52419	0.370636	0.185318	+	0.982679i	$0.440668\pi$
$150$	0	0
$151$	21.7361	1.76886	0.884428	−	0.466677i	$-0.154549\pi$
$151$	21.7361	1.76886	0.884428	+	0.466677i	$0.154549\pi$
$152$	−0.389423	−0.0315863
$153$	0	0
$154$	0.0690095	0.00556095
$155$	−6.73349	−0.540847
$156$	0	0
$157$	−3.24464	−0.258950	−0.129475	−	0.991583i	$-0.541329\pi$
$157$	−3.24464	−0.258950	−0.129475	+	0.991583i	$0.541329\pi$
$158$	0.0394117	0.00313542
$159$	0	0
$160$	0.431667	0.0341263
$161$	9.01216	0.710258
$162$	0	0
$163$	17.9557	1.40640	0.703198	−	0.710994i	$-0.251755\pi$
$163$	17.9557	1.40640	0.703198	+	0.710994i	$0.251755\pi$
$164$	−12.2155	−0.953869
$165$	0	0
$166$	−0.0781331	−0.00606431
$167$	19.2420	1.48899	0.744494	−	0.667629i	$-0.232691\pi$
$167$	19.2420	1.48899	0.744494	+	0.667629i	$0.232691\pi$
$168$	0	0
$169$	−12.4673	−0.959026
$170$	−0.198640	−0.0152350
$171$	0	0
$172$	5.17850	0.394857
$173$	−8.55134	−0.650147	−0.325073	−	0.945689i	$-0.605389\pi$
$173$	−8.55134	−0.650147	−0.325073	+	0.945689i	$0.605389\pi$
$174$	0	0
$175$	6.95589	0.525816
$176$	−3.99728	−0.301306
$177$	0	0
$178$	0.332698	0.0249368
$179$	23.6096	1.76467	0.882333	−	0.470626i	$-0.155972\pi$
$179$	23.6096	1.76467	0.882333	+	0.470626i	$0.155972\pi$
$180$	0	0
$181$	−6.81286	−0.506396	−0.253198	−	0.967414i	$-0.581482\pi$
$181$	−6.81286	−0.506396	−0.253198	+	0.967414i	$0.581482\pi$
$182$	0.0503656	0.00373335
$183$	0	0
$184$	0.237083	0.0174780
$185$	2.31869	0.170473
$186$	0	0
$187$	5.52036	0.403689
$188$	−9.21228	−0.671875
$189$	0	0
$190$	0.164447	0.0119302
$191$	−21.1010	−1.52682	−0.763408	−	0.645916i	$-0.776476\pi$
$191$	−21.1010	−1.52682	−0.763408	+	0.645916i	$0.776476\pi$
$192$	0	0
$193$	24.9449	1.79557	0.897785	−	0.440434i	$-0.145175\pi$
$193$	24.9449	1.79557	0.897785	+	0.440434i	$0.145175\pi$
$194$	−0.116666	−0.00837615
$195$	0	0
$196$	−6.98196	−0.498712
$197$	−7.39429	−0.526821	−0.263411	−	0.964684i	$-0.584847\pi$
$197$	−7.39429	−0.526821	−0.263411	+	0.964684i	$0.584847\pi$
$198$	0	0
$199$	19.3563	1.37213	0.686067	−	0.727538i	$-0.259336\pi$
$199$	19.3563	1.37213	0.686067	+	0.727538i	$0.259336\pi$
$200$	0.182988	0.0129392
$201$	0	0
$202$	−0.284770	−0.0200363
$203$	12.8969	0.905184
$204$	0	0
$205$	10.3180	0.720638
$206$	0.386890	0.0269559
$207$	0	0
$208$	−2.91735	−0.202282
$209$	−4.57010	−0.316120
$210$	0	0
$211$	−6.45727	−0.444537	−0.222268	−	0.974986i	$-0.571346\pi$
$211$	−6.45727	−0.444537	−0.222268	+	0.974986i	$0.571346\pi$
$212$	11.6181	0.797935
$213$	0	0
$214$	−0.138912	−0.00949581
$215$	−4.37408	−0.298310
$216$	0	0
$217$	12.9137	0.876638
$218$	−0.232368	−0.0157380
$219$	0	0
$220$	3.37712	0.227685
$221$	4.02896	0.271017
$222$	0	0
$223$	9.58533	0.641881	0.320941	−	0.947099i	$-0.396001\pi$
$223$	9.58533	0.641881	0.320941	+	0.947099i	$0.396001\pi$
$224$	−0.827864	−0.0553140
$225$	0	0
$226$	0.0587049	0.00390499
$227$	16.0949	1.06826	0.534128	−	0.845404i	$-0.320640\pi$
$227$	16.0949	1.06826	0.534128	+	0.845404i	$0.320640\pi$
$228$	0	0
$229$	14.9046	0.984922	0.492461	−	0.870334i	$-0.336097\pi$
$229$	14.9046	0.984922	0.492461	+	0.870334i	$0.336097\pi$
$230$	−0.100116	−0.00660146
$231$	0	0
$232$	0.339278	0.0222747
$233$	27.0638	1.77301	0.886505	−	0.462719i	$-0.153126\pi$
$233$	27.0638	1.77301	0.886505	+	0.462719i	$0.153126\pi$
$234$	0	0
$235$	7.78127	0.507594
$236$	4.71900	0.307181
$237$	0	0
$238$	0.380958	0.0246938
$239$	14.2244	0.920101	0.460051	−	0.887893i	$-0.347831\pi$
$239$	14.2244	0.920101	0.460051	+	0.887893i	$0.347831\pi$
$240$	0	0
$241$	−14.6234	−0.941973	−0.470987	−	0.882140i	$-0.656102\pi$
$241$	−14.6234	−0.941973	−0.470987	+	0.882140i	$0.656102\pi$
$242$	0.0213052	0.00136955
$243$	0	0
$244$	−26.3043	−1.68396
$245$	5.89740	0.376771
$246$	0	0
$247$	−3.33542	−0.212228
$248$	0.339720	0.0215722
$249$	0	0
$250$	−0.257189	−0.0162660
$251$	−18.0005	−1.13618	−0.568090	−	0.822967i	$-0.692317\pi$
$251$	−18.0005	−1.13618	−0.568090	+	0.822967i	$0.692317\pi$
$252$	0	0
$253$	2.78230	0.174922
$254$	0.414393	0.0260013
$255$	0	0
$256$	15.9637	0.997731
$257$	−1.27828	−0.0797368	−0.0398684	−	0.999205i	$-0.512694\pi$
$257$	−1.27828	−0.0797368	−0.0398684	+	0.999205i	$0.512694\pi$
$258$	0	0
$259$	−4.44685	−0.276314
$260$	2.46474	0.152857
$261$	0	0
$262$	−0.0961849	−0.00594233
$263$	−2.15265	−0.132738	−0.0663691	−	0.997795i	$-0.521141\pi$
$263$	−2.15265	−0.132738	−0.0663691	+	0.997795i	$0.521141\pi$
$264$	0	0
$265$	−9.81338	−0.602831
$266$	−0.315381	−0.0193372
$267$	0	0
$268$	1.99955	0.122142
$269$	−10.8551	−0.661847	−0.330923	−	0.943658i	$-0.607360\pi$
$269$	−10.8551	−0.661847	−0.330923	+	0.943658i	$0.607360\pi$
$270$	0	0
$271$	−4.14708	−0.251917	−0.125959	−	0.992035i	$-0.540201\pi$
$271$	−4.14708	−0.251917	−0.125959	+	0.992035i	$0.540201\pi$
$272$	−22.0664	−1.33797
$273$	0	0
$274$	0.000516880 0	3.12258e−5 0
$275$	2.14748	0.129498
$276$	0	0
$277$	12.2158	0.733974	0.366987	−	0.930226i	$-0.380389\pi$
$277$	12.2158	0.733974	0.366987	+	0.930226i	$0.380389\pi$
$278$	−0.226121	−0.0135618
$279$	0	0
$280$	0.466159	0.0278584
$281$	4.12060	0.245814	0.122907	−	0.992418i	$-0.460778\pi$
$281$	4.12060	0.245814	0.122907	+	0.992418i	$0.460778\pi$
$282$	0	0
$283$	4.79827	0.285227	0.142614	−	0.989778i	$-0.454449\pi$
$283$	4.79827	0.285227	0.142614	+	0.989778i	$0.454449\pi$
$284$	18.8331	1.11754
$285$	0	0
$286$	0.0155493	0.000919446 0
$287$	−19.7881	−1.16805
$288$	0	0
$289$	13.4744	0.792613
$290$	−0.143271	−0.00841319
$291$	0	0
$292$	−7.59592	−0.444517
$293$	28.8052	1.68282	0.841409	−	0.540399i	$-0.181727\pi$
$293$	28.8052	1.68282	0.841409	+	0.540399i	$0.181727\pi$
$294$	0	0
$295$	−3.98596	−0.232072
$296$	−0.116983	−0.00679951
$297$	0	0
$298$	0.0963887	0.00558365
$299$	2.03062	0.117434
$300$	0	0
$301$	8.38874	0.483519
$302$	0.463090	0.0266478
$303$	0	0
$304$	18.2680	1.04774
$305$	22.2183	1.27221
$306$	0	0
$307$	10.0145	0.571558	0.285779	−	0.958296i	$-0.407748\pi$
$307$	10.0145	0.571558	0.285779	+	0.958296i	$0.407748\pi$
$308$	−6.47673	−0.369046
$309$	0	0
$310$	−0.143458	−0.00814787
$311$	−21.8223	−1.23743	−0.618713	−	0.785617i	$-0.712346\pi$
$311$	−21.8223	−1.23743	−0.618713	+	0.785617i	$0.712346\pi$
$312$	0	0
$313$	−12.1719	−0.687994	−0.343997	−	0.938971i	$-0.611781\pi$
$313$	−12.1719	−0.687994	−0.343997	+	0.938971i	$0.611781\pi$
$314$	−0.0691275	−0.00390109
$315$	0	0
$316$	−3.69889	−0.208079
$317$	16.9133	0.949945	0.474972	−	0.880001i	$-0.342458\pi$
$317$	16.9133	0.949945	0.474972	+	0.880001i	$0.342458\pi$
$318$	0	0
$319$	3.98163	0.222928
$320$	−13.4931	−0.754289
$321$	0	0
$322$	0.192006	0.0107000
$323$	−25.2286	−1.40376
$324$	0	0
$325$	1.56730	0.0869383
$326$	0.382548	0.0211874
$327$	0	0
$328$	−0.520565	−0.0287434
$329$	−14.9231	−0.822739
$330$	0	0
$331$	21.9687	1.20751	0.603753	−	0.797171i	$-0.293671\pi$
$331$	21.9687	1.20751	0.603753	+	0.797171i	$0.293671\pi$
$332$	7.33301	0.402451
$333$	0	0
$334$	0.409953	0.0224316
$335$	−1.68894	−0.0922767
$336$	0	0
$337$	5.26378	0.286736	0.143368	−	0.989669i	$-0.454207\pi$
$337$	5.26378	0.286736	0.143368	+	0.989669i	$0.454207\pi$
$338$	−0.265619	−0.0144477
$339$	0	0
$340$	18.6429	1.01105
$341$	3.98681	0.215898
$342$	0	0
$343$	11.3635	0.613571
$344$	0.220682	0.0118984
$345$	0	0
$346$	−0.182188	−0.00979447
$347$	14.5616	0.781707	0.390854	−	0.920453i	$-0.372180\pi$
$347$	14.5616	0.781707	0.390854	+	0.920453i	$0.372180\pi$
$348$	0	0
$349$	−13.3885	−0.716671	−0.358335	−	0.933593i	$-0.616656\pi$
$349$	−13.3885	−0.716671	−0.358335	+	0.933593i	$0.616656\pi$
$350$	0.148196	0.00792143
$351$	0	0
$352$	−0.255584	−0.0136227
$353$	−32.0925	−1.70811	−0.854056	−	0.520181i	$-0.825864\pi$
$353$	−32.0925	−1.70811	−0.854056	+	0.520181i	$0.825864\pi$
$354$	0	0
$355$	−15.9076	−0.844289
$356$	−31.2246	−1.65490
$357$	0	0
$358$	0.503007	0.0265847
$359$	−10.9632	−0.578617	−0.289309	−	0.957236i	$-0.593425\pi$
$359$	−10.9632	−0.578617	−0.289309	+	0.957236i	$0.593425\pi$
$360$	0	0
$361$	1.88581	0.0992533
$362$	−0.145149	−0.00762887
$363$	0	0
$364$	−4.72695	−0.247759
$365$	6.41599	0.335828
$366$	0	0
$367$	27.6434	1.44297	0.721487	−	0.692428i	$-0.243459\pi$
$367$	27.6434	1.44297	0.721487	+	0.692428i	$0.243459\pi$
$368$	−11.1216	−0.579755
$369$	0	0
$370$	0.0494000	0.00256819
$371$	18.8204	0.977105
$372$	0	0
$373$	7.30167	0.378066	0.189033	−	0.981971i	$-0.439465\pi$
$373$	7.30167	0.378066	0.189033	+	0.981971i	$0.439465\pi$
$374$	0.117612	0.00608158
$375$	0	0
$376$	−0.392583	−0.0202459
$377$	2.90593	0.149663
$378$	0	0
$379$	2.58896	0.132986	0.0664930	−	0.997787i	$-0.478819\pi$
$379$	2.58896	0.132986	0.0664930	+	0.997787i	$0.478819\pi$
$380$	−15.4338	−0.791735
$381$	0	0
$382$	−0.449561	−0.0230015
$383$	−5.51883	−0.281999	−0.141000	−	0.990010i	$-0.545032\pi$
$383$	−5.51883	−0.281999	−0.141000	+	0.990010i	$0.545032\pi$
$384$	0	0
$385$	5.47065	0.278810
$386$	0.531454	0.0270503
$387$	0	0
$388$	10.9494	0.555874
$389$	16.9637	0.860093	0.430047	−	0.902807i	$-0.358497\pi$
$389$	16.9637	0.860093	0.430047	+	0.902807i	$0.358497\pi$
$390$	0	0
$391$	15.3593	0.776755
$392$	−0.297537	−0.0150279
$393$	0	0
$394$	−0.157536	−0.00793657
$395$	3.12432	0.157201
$396$	0	0
$397$	30.4623	1.52886	0.764431	−	0.644706i	$-0.223020\pi$
$397$	30.4623	1.52886	0.764431	+	0.644706i	$0.223020\pi$
$398$	0.412390	0.0206712
$399$	0	0
$400$	−8.58406	−0.429203
$401$	12.0438	0.601438	0.300719	−	0.953713i	$-0.402773\pi$
$401$	12.0438	0.601438	0.300719	+	0.953713i	$0.402773\pi$
$402$	0	0
$403$	2.90971	0.144943
$404$	26.7264	1.32969
$405$	0	0
$406$	0.274770	0.0136366
$407$	−1.37287	−0.0680504
$408$	0	0
$409$	16.6409	0.822840	0.411420	−	0.911446i	$-0.365033\pi$
$409$	16.6409	0.822840	0.411420	+	0.911446i	$0.365033\pi$
$410$	0.219826	0.0108564
$411$	0	0
$412$	−36.3106	−1.78890
$413$	7.64439	0.376156
$414$	0	0
$415$	−6.19392	−0.304047
$416$	−0.186535	−0.00914561
$417$	0	0
$418$	−0.0973667	−0.00476236
$419$	−10.4342	−0.509745	−0.254872	−	0.966975i	$-0.582033\pi$
$419$	−10.4342	−0.509745	−0.254872	+	0.966975i	$0.582033\pi$
$420$	0	0
$421$	−26.2693	−1.28029	−0.640144	−	0.768255i	$-0.721125\pi$
$421$	−26.2693	−1.28029	−0.640144	+	0.768255i	$0.721125\pi$
$422$	−0.137573	−0.00669695
$423$	0	0
$424$	0.495107	0.0240445
$425$	11.8548	0.575045
$426$	0	0
$427$	−42.6108	−2.06208
$428$	13.0372	0.630179
$429$	0	0
$430$	−0.0931905	−0.00449404
$431$	4.91498	0.236746	0.118373	−	0.992969i	$-0.462232\pi$
$431$	4.91498	0.236746	0.118373	+	0.992969i	$0.462232\pi$
$432$	0	0
$433$	19.6738	0.945463	0.472732	−	0.881206i	$-0.343268\pi$
$433$	19.6738	0.945463	0.472732	+	0.881206i	$0.343268\pi$
$434$	0.275128	0.0132066
$435$	0	0
$436$	21.8084	1.04443
$437$	−12.7154	−0.608260
$438$	0	0
$439$	−9.73914	−0.464824	−0.232412	−	0.972617i	$-0.574662\pi$
$439$	−9.73914	−0.464824	−0.232412	+	0.972617i	$0.574662\pi$
$440$	0.143916	0.00686094
$441$	0	0
$442$	0.0858375	0.00408287
$443$	−28.0339	−1.33193	−0.665964	−	0.745983i	$-0.731980\pi$
$443$	−28.0339	−1.33193	−0.665964	+	0.745983i	$0.731980\pi$
$444$	0	0
$445$	26.3742	1.25026
$446$	0.204217	0.00966995
$447$	0	0
$448$	25.8775	1.22260
$449$	9.36598	0.442008	0.221004	−	0.975273i	$-0.429067\pi$
$449$	9.36598	0.442008	0.221004	+	0.975273i	$0.429067\pi$
$450$	0	0
$451$	−6.10913	−0.287668
$452$	−5.50962	−0.259151
$453$	0	0
$454$	0.342904	0.0160933
$455$	3.99267	0.187179
$456$	0	0
$457$	−2.21336	−0.103536	−0.0517682	−	0.998659i	$-0.516486\pi$
$457$	−2.21336	−0.103536	−0.0517682	+	0.998659i	$0.516486\pi$
$458$	0.317544	0.0148379
$459$	0	0
$460$	9.39616	0.438098
$461$	−0.291926	−0.0135964	−0.00679818	−	0.999977i	$-0.502164\pi$
$461$	−0.291926	−0.0135964	−0.00679818	+	0.999977i	$0.502164\pi$
$462$	0	0
$463$	20.8938	0.971019	0.485509	−	0.874231i	$-0.338634\pi$
$463$	20.8938	0.971019	0.485509	+	0.874231i	$0.338634\pi$
$464$	−15.9157	−0.738866
$465$	0	0
$466$	0.576599	0.0267104
$467$	−39.9976	−1.85087	−0.925435	−	0.378908i	$-0.876300\pi$
$467$	−39.9976	−1.85087	−0.925435	+	0.378908i	$0.876300\pi$
$468$	0	0
$469$	3.23910	0.149568
$470$	0.165781	0.00764691
$471$	0	0
$472$	0.201101	0.00925642
$473$	2.58984	0.119081
$474$	0	0
$475$	−9.81418	−0.450305
$476$	−35.7539	−1.63878
$477$	0	0
$478$	0.303053	0.0138613
$479$	−2.84358	−0.129926	−0.0649632	−	0.997888i	$-0.520693\pi$
$479$	−2.84358	−0.129926	−0.0649632	+	0.997888i	$0.520693\pi$
$480$	0	0
$481$	−1.00197	−0.0456857
$482$	−0.311553	−0.0141908
$483$	0	0
$484$	−1.99955	−0.0908885
$485$	−9.24858	−0.419956
$486$	0	0
$487$	−14.1280	−0.640202	−0.320101	−	0.947383i	$-0.603717\pi$
$487$	−14.1280	−0.640202	−0.320101	+	0.947383i	$0.603717\pi$
$488$	−1.12096	−0.0507435
$489$	0	0
$490$	0.125645	0.00567607
$491$	9.26772	0.418246	0.209123	−	0.977889i	$-0.432939\pi$
$491$	9.26772	0.418246	0.209123	+	0.977889i	$0.432939\pi$
$492$	0	0
$493$	21.9800	0.989930
$494$	−0.0710616	−0.00319721
$495$	0	0
$496$	−15.9364	−0.715565
$497$	30.5081	1.36848
$498$	0	0
$499$	22.3962	1.00259	0.501295	−	0.865277i	$-0.332857\pi$
$499$	22.3962	1.00259	0.501295	+	0.865277i	$0.332857\pi$
$500$	24.1379	1.07948
$501$	0	0
$502$	−0.383503	−0.0171166
$503$	37.9536	1.69227	0.846134	−	0.532971i	$-0.178924\pi$
$503$	37.9536	1.69227	0.846134	+	0.532971i	$0.178924\pi$
$504$	0	0
$505$	−22.5748	−1.00457
$506$	0.0592774	0.00263520
$507$	0	0
$508$	−38.8919	−1.72555
$509$	−2.17080	−0.0962190	−0.0481095	−	0.998842i	$-0.515320\pi$
$509$	−2.17080	−0.0962190	−0.0481095	+	0.998842i	$0.515320\pi$
$510$	0	0
$511$	−12.3048	−0.544331
$512$	1.70287	0.0752567
$513$	0	0
$514$	−0.0272339	−0.00120124
$515$	30.6702	1.35149
$516$	0	0
$517$	−4.60719	−0.202624
$518$	−0.0947408	−0.00416267
$519$	0	0
$520$	0.105035	0.00460610
$521$	−4.26585	−0.186890	−0.0934451	−	0.995624i	$-0.529788\pi$
$521$	−4.26585	−0.186890	−0.0934451	+	0.995624i	$0.529788\pi$
$522$	0	0
$523$	21.5339	0.941610	0.470805	−	0.882237i	$-0.343963\pi$
$523$	21.5339	0.941610	0.470805	+	0.882237i	$0.343963\pi$
$524$	9.02722	0.394356
$525$	0	0
$526$	−0.0458626	−0.00199970
$527$	22.0086	0.958711
$528$	0	0
$529$	−15.2588	−0.663425
$530$	−0.209076	−0.00908166
$531$	0	0
$532$	29.5993	1.28329
$533$	−4.45866	−0.193126
$534$	0	0
$535$	−11.0121	−0.476093
$536$	0.0852109	0.00368055
$537$	0	0
$538$	−0.231269	−0.00997073
$539$	−3.49177	−0.150401
$540$	0	0
$541$	5.66794	0.243684	0.121842	−	0.992550i	$-0.461120\pi$
$541$	5.66794	0.243684	0.121842	+	0.992550i	$0.461120\pi$
$542$	−0.0883542	−0.00379514
$543$	0	0
$544$	−1.41092	−0.0604927
$545$	−18.4207	−0.789057
$546$	0	0
$547$	2.42389	0.103638	0.0518190	−	0.998656i	$-0.483498\pi$
$547$	2.42389	0.103638	0.0518190	+	0.998656i	$0.483498\pi$
$548$	−0.0485106	−0.00207227
$549$	0	0
$550$	0.0457523	0.00195088
$551$	−18.1964	−0.775194
$552$	0	0
$553$	−5.99190	−0.254802
$554$	0.260259	0.0110573
$555$	0	0
$556$	21.2221	0.900017
$557$	−4.24123	−0.179707	−0.0898533	−	0.995955i	$-0.528640\pi$
$557$	−4.24123	−0.179707	−0.0898533	+	0.995955i	$0.528640\pi$
$558$	0	0
$559$	1.89015	0.0799450
$560$	−21.8677	−0.924080
$561$	0	0
$562$	0.0877900	0.00370320
$563$	19.3999	0.817609	0.408805	−	0.912622i	$-0.365946\pi$
$563$	19.3999	0.817609	0.408805	+	0.912622i	$0.365946\pi$
$564$	0	0
$565$	4.65377	0.195785
$566$	0.102228	0.00429695
$567$	0	0
$568$	0.802576	0.0336753
$569$	40.2978	1.68937	0.844686	−	0.535262i	$-0.179787\pi$
$569$	40.2978	1.68937	0.844686	+	0.535262i	$0.179787\pi$
$570$	0	0
$571$	−40.3296	−1.68774	−0.843870	−	0.536548i	$-0.819728\pi$
$571$	−40.3296	−1.68774	−0.843870	+	0.536548i	$0.819728\pi$
$572$	−1.45934	−0.0610180
$573$	0	0
$574$	−0.421588	−0.0175967
$575$	5.97493	0.249172
$576$	0	0
$577$	−4.68545	−0.195058	−0.0975289	−	0.995233i	$-0.531094\pi$
$577$	−4.68545	−0.195058	−0.0975289	+	0.995233i	$0.531094\pi$
$578$	0.287075	0.0119407
$579$	0	0
$580$	13.4464	0.558332
$581$	11.8789	0.492819
$582$	0	0
$583$	5.81037	0.240641
$584$	−0.323701	−0.0133948
$585$	0	0
$586$	0.613699	0.0253517
$587$	35.0050	1.44481	0.722405	−	0.691471i	$-0.243037\pi$
$587$	35.0050	1.44481	0.722405	+	0.691471i	$0.243037\pi$
$588$	0	0
$589$	−18.2201	−0.750747
$590$	−0.0849215	−0.00349616
$591$	0	0
$592$	5.48772	0.225544
$593$	−29.1322	−1.19632	−0.598159	−	0.801378i	$-0.704101\pi$
$593$	−29.1322	−1.19632	−0.598159	+	0.801378i	$0.704101\pi$
$594$	0	0
$595$	30.2000	1.23808
$596$	−9.04634	−0.370552
$597$	0	0
$598$	0.0432627	0.00176915
$599$	−18.4534	−0.753983	−0.376992	−	0.926217i	$-0.623041\pi$
$599$	−18.4534	−0.753983	−0.376992	+	0.926217i	$0.623041\pi$
$600$	0	0
$601$	−46.8174	−1.90972	−0.954860	−	0.297056i	$-0.903995\pi$
$601$	−46.8174	−1.90972	−0.954860	+	0.297056i	$0.903995\pi$
$602$	0.178723	0.00728422
$603$	0	0
$604$	−43.4623	−1.76845
$605$	1.68894	0.0686652
$606$	0	0
$607$	23.7040	0.962115	0.481057	−	0.876689i	$-0.340253\pi$
$607$	23.7040	0.962115	0.481057	+	0.876689i	$0.340253\pi$
$608$	1.16805	0.0473705
$609$	0	0
$610$	0.473363	0.0191659
$611$	−3.36249	−0.136032
$612$	0	0
$613$	2.93484	0.118537	0.0592685	−	0.998242i	$-0.481123\pi$
$613$	2.93484	0.118537	0.0592685	+	0.998242i	$0.481123\pi$
$614$	0.213361	0.00861053
$615$	0	0
$616$	−0.276007	−0.0111206
$617$	−13.8934	−0.559326	−0.279663	−	0.960098i	$-0.590223\pi$
$617$	−13.8934	−0.559326	−0.279663	+	0.960098i	$0.590223\pi$
$618$	0	0
$619$	37.0880	1.49069	0.745346	−	0.666678i	$-0.232284\pi$
$619$	37.0880	1.49069	0.745346	+	0.666678i	$0.232284\pi$
$620$	13.4639	0.540724
$621$	0	0
$622$	−0.464926	−0.0186419
$623$	−50.5813	−2.02650
$624$	0	0
$625$	−9.65097	−0.386039
$626$	−0.259323	−0.0103646
$627$	0	0
$628$	6.48780	0.258891
$629$	−7.57872	−0.302183
$630$	0	0
$631$	−38.6365	−1.53810	−0.769048	−	0.639191i	$-0.779269\pi$
$631$	−38.6365	−1.53810	−0.769048	+	0.639191i	$0.779269\pi$
$632$	−0.157629	−0.00627014
$633$	0	0
$634$	0.360340	0.0143109
$635$	32.8505	1.30363
$636$	0	0
$637$	−2.54842	−0.100972
$638$	0.0848291	0.00335842
$639$	0	0
$640$	−1.15081	−0.0454897
$641$	18.5153	0.731312	0.365656	−	0.930750i	$-0.380845\pi$
$641$	18.5153	0.731312	0.365656	+	0.930750i	$0.380845\pi$
$642$	0	0
$643$	23.9238	0.943464	0.471732	−	0.881742i	$-0.343629\pi$
$643$	23.9238	0.943464	0.471732	+	0.881742i	$0.343629\pi$
$644$	−18.0202	−0.710097
$645$	0	0
$646$	−0.537500	−0.0211476
$647$	26.5747	1.04476	0.522379	−	0.852713i	$-0.325044\pi$
$647$	26.5747	1.04476	0.522379	+	0.852713i	$0.325044\pi$
$648$	0	0
$649$	2.36004	0.0926395
$650$	0.0333916	0.00130973
$651$	0	0
$652$	−35.9032	−1.40608
$653$	13.5585	0.530585	0.265293	−	0.964168i	$-0.414531\pi$
$653$	13.5585	0.530585	0.265293	+	0.964168i	$0.414531\pi$
$654$	0	0
$655$	−7.62495	−0.297932
$656$	24.4199	0.953436
$657$	0	0
$658$	−0.317940	−0.0123946
$659$	11.9234	0.464471	0.232235	−	0.972660i	$-0.425396\pi$
$659$	11.9234	0.464471	0.232235	+	0.972660i	$0.425396\pi$
$660$	0	0
$661$	−37.4282	−1.45579	−0.727893	−	0.685690i	$-0.759500\pi$
$661$	−37.4282	−1.45579	−0.727893	+	0.685690i	$0.759500\pi$
$662$	0.468046	0.0181911
$663$	0	0
$664$	0.312497	0.0121272
$665$	−25.0014	−0.969514
$666$	0	0
$667$	11.0781	0.428945
$668$	−38.4752	−1.48865
$669$	0	0
$670$	−0.0359832	−0.00139015
$671$	−13.1551	−0.507848
$672$	0	0
$673$	37.7695	1.45591	0.727953	−	0.685627i	$-0.240472\pi$
$673$	37.7695	1.45591	0.727953	+	0.685627i	$0.240472\pi$
$674$	0.112146	0.00431969
$675$	0	0
$676$	24.9290	0.958809
$677$	−15.3913	−0.591534	−0.295767	−	0.955260i	$-0.595575\pi$
$677$	−15.3913	−0.591534	−0.295767	+	0.955260i	$0.595575\pi$
$678$	0	0
$679$	17.7372	0.680691
$680$	0.794470	0.0304666
$681$	0	0
$682$	0.0849396	0.00325251
$683$	10.2969	0.393998	0.196999	−	0.980404i	$-0.436880\pi$
$683$	10.2969	0.393998	0.196999	+	0.980404i	$0.436880\pi$
$684$	0	0
$685$	0.0409751	0.00156558
$686$	0.242101	0.00924346
$687$	0	0
$688$	−10.3523	−0.394677
$689$	4.24061	0.161555
$690$	0	0
$691$	20.3625	0.774626	0.387313	−	0.921948i	$-0.373403\pi$
$691$	20.3625	0.774626	0.387313	+	0.921948i	$0.373403\pi$
$692$	17.0988	0.649999
$693$	0	0
$694$	0.310237	0.0117764
$695$	−17.9255	−0.679953
$696$	0	0
$697$	−33.7246	−1.27741
$698$	−0.285244	−0.0107967
$699$	0	0
$700$	−13.9086	−0.525697
$701$	27.5233	1.03954	0.519770	−	0.854306i	$-0.326018\pi$
$701$	27.5233	1.03954	0.519770	+	0.854306i	$0.326018\pi$
$702$	0	0
$703$	6.27413	0.236633
$704$	7.98911	0.301101
$705$	0	0
$706$	−0.683736	−0.0257327
$707$	43.2946	1.62826
$708$	0	0
$709$	−18.8640	−0.708452	−0.354226	−	0.935160i	$-0.615256\pi$
$709$	−18.8640	−0.708452	−0.354226	+	0.935160i	$0.615256\pi$
$710$	−0.338914	−0.0127192
$711$	0	0
$712$	−1.33064	−0.0498679
$713$	11.0925	0.415418
$714$	0	0
$715$	1.23265	0.0460985
$716$	−47.2085	−1.76426
$717$	0	0
$718$	−0.233573	−0.00871688
$719$	−26.0117	−0.970072	−0.485036	−	0.874494i	$-0.661193\pi$
$719$	−26.0117	−0.970072	−0.485036	+	0.874494i	$0.661193\pi$
$720$	0	0
$721$	−58.8202	−2.19058
$722$	0.0401775	0.00149525
$723$	0	0
$724$	13.6226	0.506281
$725$	8.55044	0.317556
$726$	0	0
$727$	6.63210	0.245971	0.122985	−	0.992408i	$-0.460753\pi$
$727$	6.63210	0.245971	0.122985	+	0.992408i	$0.460753\pi$
$728$	−0.201440	−0.00746584
$729$	0	0
$730$	0.136694	0.00505926
$731$	14.2968	0.528788
$732$	0	0
$733$	19.9131	0.735509	0.367754	−	0.929923i	$-0.380127\pi$
$733$	19.9131	0.735509	0.367754	+	0.929923i	$0.380127\pi$
$734$	0.588947	0.0217384
$735$	0	0
$736$	−0.711114	−0.0262120
$737$	1.00000	0.0368355
$738$	0	0
$739$	−10.8311	−0.398427	−0.199213	−	0.979956i	$-0.563839\pi$
$739$	−10.8311	−0.398427	−0.199213	+	0.979956i	$0.563839\pi$
$740$	−4.63633	−0.170435
$741$	0	0
$742$	0.400971	0.0147201
$743$	40.7728	1.49581	0.747904	−	0.663806i	$-0.231060\pi$
$743$	40.7728	1.49581	0.747904	+	0.663806i	$0.231060\pi$
$744$	0	0
$745$	7.64110	0.279948
$746$	0.155563	0.00569557
$747$	0	0
$748$	−11.0382	−0.403597
$749$	21.1193	0.771681
$750$	0	0
$751$	35.2767	1.28726	0.643632	−	0.765335i	$-0.277427\pi$
$751$	35.2767	1.28726	0.643632	+	0.765335i	$0.277427\pi$
$752$	18.4162	0.671570
$753$	0	0
$754$	0.0619113	0.00225468
$755$	36.7109	1.33605
$756$	0	0
$757$	11.3604	0.412900	0.206450	−	0.978457i	$-0.433809\pi$
$757$	11.3604	0.412900	0.206450	+	0.978457i	$0.433809\pi$
$758$	0.0551583	0.00200344
$759$	0	0
$760$	−0.657712	−0.0238577
$761$	14.0722	0.510115	0.255058	−	0.966926i	$-0.417906\pi$
$761$	14.0722	0.510115	0.255058	+	0.966926i	$0.417906\pi$
$762$	0	0
$763$	35.3278	1.27895
$764$	42.1925	1.52647
$765$	0	0
$766$	−0.117580	−0.00424832
$767$	1.72244	0.0621936
$768$	0	0
$769$	−39.3322	−1.41835	−0.709177	−	0.705031i	$-0.750933\pi$
$769$	−39.3322	−1.41835	−0.709177	+	0.705031i	$0.750933\pi$
$770$	0.116553	0.00420028
$771$	0	0
$772$	−49.8784	−1.79516
$773$	30.3126	1.09027	0.545134	−	0.838349i	$-0.316479\pi$
$773$	30.3126	1.09027	0.545134	+	0.838349i	$0.316479\pi$
$774$	0	0
$775$	8.56158	0.307541
$776$	0.466612	0.0167504
$777$	0	0
$778$	0.361414	0.0129573
$779$	27.9193	1.00031
$780$	0	0
$781$	9.41869	0.337027
$782$	0.327233	0.0117018
$783$	0	0
$784$	13.9576	0.498485
$785$	−5.48000	−0.195590
$786$	0	0
$787$	18.4590	0.657991	0.328995	−	0.944332i	$-0.393290\pi$
$787$	18.4590	0.657991	0.328995	+	0.944332i	$0.393290\pi$
$788$	14.7852	0.526701
$789$	0	0
$790$	0.0665640	0.00236824
$791$	−8.92513	−0.317341
$792$	0	0
$793$	−9.60108	−0.340944
$794$	0.649005	0.0230323
$795$	0	0
$796$	−38.7039	−1.37182
$797$	7.52953	0.266709	0.133355	−	0.991068i	$-0.457425\pi$
$797$	7.52953	0.266709	0.133355	+	0.991068i	$0.457425\pi$
$798$	0	0
$799$	−25.4333	−0.899767
$800$	−0.548862	−0.0194052
$801$	0	0
$802$	0.256595	0.00906067
$803$	−3.79882	−0.134057
$804$	0	0
$805$	15.2210	0.536470
$806$	0.0619919	0.00218357
$807$	0	0
$808$	1.13895	0.0400681
$809$	−30.3665	−1.06763	−0.533815	−	0.845601i	$-0.679242\pi$
$809$	−30.3665	−1.06763	−0.533815	+	0.845601i	$0.679242\pi$
$810$	0	0
$811$	−25.7742	−0.905056	−0.452528	−	0.891750i	$-0.649478\pi$
$811$	−25.7742	−0.905056	−0.452528	+	0.891750i	$0.649478\pi$
$812$	−25.7879	−0.904979
$813$	0	0
$814$	−0.0292491	−0.00102518
$815$	30.3261	1.06228
$816$	0	0
$817$	−11.8358	−0.414082
$818$	0.354537	0.0123961
$819$	0	0
$820$	−20.6312	−0.720474
$821$	−14.9872	−0.523058	−0.261529	−	0.965196i	$-0.584227\pi$
$821$	−14.9872	−0.523058	−0.261529	+	0.965196i	$0.584227\pi$
$822$	0	0
$823$	−30.0200	−1.04643	−0.523216	−	0.852200i	$-0.675268\pi$
$823$	−30.0200	−1.04643	−0.523216	+	0.852200i	$0.675268\pi$
$824$	−1.54738	−0.0539056
$825$	0	0
$826$	0.162865	0.00566680
$827$	6.82458	0.237314	0.118657	−	0.992935i	$-0.462141\pi$
$827$	6.82458	0.237314	0.118657	+	0.992935i	$0.462141\pi$
$828$	0	0
$829$	−21.3151	−0.740303	−0.370151	−	0.928971i	$-0.620694\pi$
$829$	−21.3151	−0.740303	−0.370151	+	0.928971i	$0.620694\pi$
$830$	−0.131962	−0.00458048
$831$	0	0
$832$	5.83073	0.202144
$833$	−19.2759	−0.667869
$834$	0	0
$835$	32.4986	1.12466
$836$	9.13813	0.316049
$837$	0	0
$838$	−0.222303	−0.00767931
$839$	−12.1586	−0.419761	−0.209881	−	0.977727i	$-0.567308\pi$
$839$	−12.1586	−0.419761	−0.209881	+	0.977727i	$0.567308\pi$
$840$	0	0
$841$	−13.1467	−0.453333
$842$	−0.559671	−0.0192876
$843$	0	0
$844$	12.9116	0.444436
$845$	−21.0566	−0.724369
$846$	0	0
$847$	−3.23910	−0.111297
$848$	−23.2257	−0.797573
$849$	0	0
$850$	0.252569	0.00866306
$851$	−3.81973	−0.130939
$852$	0	0
$853$	−2.79550	−0.0957161	−0.0478580	−	0.998854i	$-0.515240\pi$
$853$	−2.79550	−0.0957161	−0.0478580	+	0.998854i	$0.515240\pi$
$854$	−0.907830	−0.0310653
$855$	0	0
$856$	0.555584	0.0189895
$857$	−56.8900	−1.94333	−0.971663	−	0.236372i	$-0.924042\pi$
$857$	−56.8900	−1.94333	−0.971663	+	0.236372i	$0.924042\pi$
$858$	0	0
$859$	−23.0825	−0.787565	−0.393783	−	0.919204i	$-0.628834\pi$
$859$	−23.0825	−0.787565	−0.393783	+	0.919204i	$0.628834\pi$
$860$	8.74618	0.298242
$861$	0	0
$862$	0.104714	0.00356659
$863$	−14.5240	−0.494402	−0.247201	−	0.968964i	$-0.579511\pi$
$863$	−14.5240	−0.494402	−0.247201	+	0.968964i	$0.579511\pi$
$864$	0	0
$865$	−14.4427	−0.491067
$866$	0.419154	0.0142434
$867$	0	0
$868$	−25.8215	−0.876439
$869$	−1.84987	−0.0627524
$870$	0	0
$871$	0.729835	0.0247295
$872$	0.929367	0.0314723
$873$	0	0
$874$	−0.270904	−0.00916345
$875$	39.1014	1.32187
$876$	0	0
$877$	46.3287	1.56441	0.782205	−	0.623022i	$-0.214095\pi$
$877$	46.3287	1.56441	0.782205	+	0.623022i	$0.214095\pi$
$878$	−0.207494	−0.00700258
$879$	0	0
$880$	−6.75117	−0.227582
$881$	−8.62388	−0.290546	−0.145273	−	0.989392i	$-0.546406\pi$
$881$	−8.62388	−0.290546	−0.145273	+	0.989392i	$0.546406\pi$
$882$	0	0
$883$	9.35621	0.314861	0.157431	−	0.987530i	$-0.449679\pi$
$883$	9.35621	0.314861	0.157431	+	0.987530i	$0.449679\pi$
$884$	−8.05608	−0.270955
$885$	0	0
$886$	−0.597266	−0.0200655
$887$	5.62070	0.188725	0.0943624	−	0.995538i	$-0.469919\pi$
$887$	5.62070	0.188725	0.0943624	+	0.995538i	$0.469919\pi$
$888$	0	0
$889$	−63.0017	−2.11301
$890$	0.561907	0.0188352
$891$	0	0
$892$	−19.1663	−0.641735
$893$	21.0553	0.704589
$894$	0	0
$895$	39.8753	1.33288
$896$	2.20705	0.0737325
$897$	0	0
$898$	0.199544	0.00665885
$899$	15.8740	0.529427
$900$	0	0
$901$	32.0754	1.06859
$902$	−0.130156	−0.00433372
$903$	0	0
$904$	−0.234793	−0.00780910
$905$	−11.5065	−0.382490
$906$	0	0
$907$	19.8426	0.658864	0.329432	−	0.944179i	$-0.393143\pi$
$907$	19.8426	0.658864	0.329432	+	0.944179i	$0.393143\pi$
$908$	−32.1825	−1.06801
$909$	0	0
$910$	0.0850645	0.00281986
$911$	−51.6342	−1.71072	−0.855358	−	0.518037i	$-0.826663\pi$
$911$	−51.6342	−1.71072	−0.855358	+	0.518037i	$0.826663\pi$
$912$	0	0
$913$	3.66734	0.121371
$914$	−0.0471559	−0.00155978
$915$	0	0
$916$	−29.8024	−0.984699
$917$	14.6234	0.482906
$918$	0	0
$919$	34.4013	1.13479	0.567396	−	0.823445i	$-0.307951\pi$
$919$	34.4013	1.13479	0.567396	+	0.823445i	$0.307951\pi$
$920$	0.400419	0.0132014
$921$	0	0
$922$	−0.00621953	−0.000204829 0
$923$	6.87409	0.226264
$924$	0	0
$925$	−2.94820	−0.0969361
$926$	0.445146	0.0146284
$927$	0	0
$928$	−1.01764	−0.0334057
$929$	−12.2043	−0.400410	−0.200205	−	0.979754i	$-0.564161\pi$
$929$	−12.2043	−0.400410	−0.200205	+	0.979754i	$0.564161\pi$
$930$	0	0
$931$	15.9578	0.522994
$932$	−54.1154	−1.77261
$933$	0	0
$934$	−0.852155	−0.0278834
$935$	9.32357	0.304913
$936$	0	0
$937$	−11.6150	−0.379446	−0.189723	−	0.981838i	$-0.560759\pi$
$937$	−11.6150	−0.379446	−0.189723	+	0.981838i	$0.560759\pi$
$938$	0.0690095	0.00225324
$939$	0	0
$940$	−15.5590	−0.507479
$941$	14.8692	0.484722	0.242361	−	0.970186i	$-0.422078\pi$
$941$	14.8692	0.484722	0.242361	+	0.970186i	$0.422078\pi$
$942$	0	0
$943$	−16.9974	−0.553513
$944$	−9.43371	−0.307041
$945$	0	0
$946$	0.0551769	0.00179395
$947$	−54.2849	−1.76402	−0.882011	−	0.471228i	$-0.843811\pi$
$947$	−54.2849	−1.76402	−0.882011	+	0.471228i	$0.843811\pi$
$948$	0	0
$949$	−2.77251	−0.0899996
$950$	−0.209093	−0.00678386
$951$	0	0
$952$	−1.52366	−0.0493820
$953$	−1.76356	−0.0571274	−0.0285637	−	0.999592i	$-0.509093\pi$
$953$	−1.76356	−0.0571274	−0.0285637	+	0.999592i	$0.509093\pi$
$954$	0	0
$955$	−35.6384	−1.15323
$956$	−28.4424	−0.919892
$957$	0	0
$958$	−0.0605829	−0.00195734
$959$	−0.0785831	−0.00253758
$960$	0	0
$961$	−15.1053	−0.487269
$962$	−0.0213470	−0.000688256 0
$963$	0	0
$964$	29.2401	0.941760
$965$	42.1304	1.35623
$966$	0	0
$967$	18.9404	0.609081	0.304540	−	0.952499i	$-0.401497\pi$
$967$	18.9404	0.609081	0.304540	+	0.952499i	$0.401497\pi$
$968$	−0.0852109	−0.00273878
$969$	0	0
$970$	−0.197042	−0.00632665
$971$	−25.6428	−0.822917	−0.411459	−	0.911428i	$-0.634981\pi$
$971$	−25.6428	−0.822917	−0.411459	+	0.911428i	$0.634981\pi$
$972$	0	0
$973$	34.3780	1.10211
$974$	−0.301000	−0.00964466
$975$	0	0
$976$	52.5847	1.68320
$977$	−28.9751	−0.926996	−0.463498	−	0.886098i	$-0.653406\pi$
$977$	−28.9751	−0.926996	−0.463498	+	0.886098i	$0.653406\pi$
$978$	0	0
$979$	−15.6158	−0.499085
$980$	−11.7921	−0.376686
$981$	0	0
$982$	0.197450	0.00630089
$983$	−36.9212	−1.17760	−0.588802	−	0.808277i	$-0.700400\pi$
$983$	−36.9212	−1.17760	−0.588802	+	0.808277i	$0.700400\pi$
$984$	0	0
$985$	−12.4885	−0.397917
$986$	0.468288	0.0149133
$987$	0	0
$988$	6.66933	0.212179
$989$	7.20571	0.229128
$990$	0	0
$991$	−7.59922	−0.241397	−0.120699	−	0.992689i	$-0.538513\pi$
$991$	−7.59922	−0.241397	−0.120699	+	0.992689i	$0.538513\pi$
$992$	−1.01897	−0.0323522
$993$	0	0
$994$	0.649980	0.0206161
$995$	32.6917	1.03640
$996$	0	0
$997$	15.2536	0.483088	0.241544	−	0.970390i	$-0.422346\pi$
$997$	15.2536	0.483088	0.241544	+	0.970390i	$0.422346\pi$
$998$	0.477154	0.0151040
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6633.2.a.w.1.10		17
3.2	odd	2		737.2.a.f.1.8	✓	17
33.32	even	2		8107.2.a.o.1.10		17

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
737.2.a.f.1.8	✓	17	3.2	odd	2
6633.2.a.w.1.10		17	1.1	even	1	trivial
8107.2.a.o.1.10		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)

\(f(q)\)	\(=\)	\(q+0.0213052 q^{2} -1.99955 q^{4} +1.68894 q^{5} -3.23910 q^{7} -0.0852109 q^{8} +O(q^{10})\)	\(q+0.0213052 q^{2} -1.99955 q^{4} +1.68894 q^{5} -3.23910 q^{7} -0.0852109 q^{8} +0.0359832 q^{10} -1.00000 q^{11} -0.729835 q^{13} -0.0690095 q^{14} +3.99728 q^{16} -5.52036 q^{17} +4.57010 q^{19} -3.37712 q^{20} -0.0213052 q^{22} -2.78230 q^{23} -2.14748 q^{25} -0.0155493 q^{26} +6.47673 q^{28} -3.98163 q^{29} -3.98681 q^{31} +0.255584 q^{32} -0.117612 q^{34} -5.47065 q^{35} +1.37287 q^{37} +0.0973667 q^{38} -0.143916 q^{40} +6.10913 q^{41} -2.58984 q^{43} +1.99955 q^{44} -0.0592774 q^{46} +4.60719 q^{47} +3.49177 q^{49} -0.0457523 q^{50} +1.45934 q^{52} -5.81037 q^{53} -1.68894 q^{55} +0.276007 q^{56} -0.0848291 q^{58} -2.36004 q^{59} +13.1551 q^{61} -0.0849396 q^{62} -7.98911 q^{64} -1.23265 q^{65} -1.00000 q^{67} +11.0382 q^{68} -0.116553 q^{70} -9.41869 q^{71} +3.79882 q^{73} +0.0292491 q^{74} -9.13813 q^{76} +3.23910 q^{77} +1.84987 q^{79} +6.75117 q^{80} +0.130156 q^{82} -3.66734 q^{83} -9.32357 q^{85} -0.0551769 q^{86} +0.0852109 q^{88} +15.6158 q^{89} +2.36401 q^{91} +5.56334 q^{92} +0.0981568 q^{94} +7.71863 q^{95} -5.47596 q^{97} +0.0743928 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 17 q - q^{2} + 19 q^{4} - 10 q^{5} + 20 q^{7}+O(q^{10}) \) 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

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