LMFDB - Embedded newform 3920.2.a.cd.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3920,2,Mod(1,3920)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3920, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("3920.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3920 = 2^{4} \cdot 5 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3920.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:	$31.3013575923$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.16448.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 7x^{2} + 8x + 14 $ x^4 - 2x^3 - 7x^2 + 8*x + 14
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1960)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.87996$ of defining polynomial
Character	$\chi$	$=$	3920.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.87996 q^{3} +1.00000 q^{5} +5.29417 q^{9} +O(q^{10})$	$q-2.87996 q^{3} +1.00000 q^{5} +5.29417 q^{9} -1.46575 q^{11} +2.22129 q^{13} -2.87996 q^{15} +7.26580 q^{17} +5.48709 q^{19} -2.51547 q^{23} +1.00000 q^{25} -6.60713 q^{27} -7.12260 q^{29} +6.51547 q^{31} +4.22129 q^{33} +6.90131 q^{37} -6.39724 q^{39} -11.3199 q^{41} -3.31733 q^{43} +5.29417 q^{45} +8.36705 q^{47} -20.9252 q^{51} -7.00437 q^{53} -1.46575 q^{55} -15.8026 q^{57} -9.07107 q^{59} +11.2143 q^{61} +2.22129 q^{65} +6.41240 q^{67} +7.24445 q^{69} +10.5316 q^{71} +10.5217 q^{73} -2.87996 q^{75} -10.1911 q^{79} +3.14576 q^{81} -16.4186 q^{83} +7.26580 q^{85} +20.5128 q^{87} +9.83024 q^{89} -18.7643 q^{93} +5.48709 q^{95} -2.09423 q^{97} -7.75992 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{3} + 4 q^{5} + 6 q^{9}+O(q^{10}) $ 4 * q - 2 * q^3 + 4 * q^5 + 6 * q^9	$ 4 q - 2 q^{3} + 4 q^{5} + 6 q^{9} - 2 q^{11} + 10 q^{13} - 2 q^{15} + 6 q^{17} + 4 q^{23} + 4 q^{25} - 14 q^{27} - 2 q^{29} + 12 q^{31} + 18 q^{33} - 14 q^{39} + 12 q^{41} + 8 q^{43} + 6 q^{45} + 2 q^{47} - 2 q^{51} - 4 q^{53} - 2 q^{55} - 8 q^{57} - 8 q^{59} + 20 q^{61} + 10 q^{65} + 8 q^{67} + 24 q^{69} - 4 q^{71} + 16 q^{73} - 2 q^{75} - 22 q^{79} - 20 q^{81} - 36 q^{83} + 6 q^{85} + 18 q^{87} + 40 q^{89} - 32 q^{93} + 26 q^{97} - 12 q^{99}+O(q^{100}) $ 4 * q - 2 * q^3 + 4 * q^5 + 6 * q^9 - 2 * q^11 + 10 * q^13 - 2 * q^15 + 6 * q^17 + 4 * q^23 + 4 * q^25 - 14 * q^27 - 2 * q^29 + 12 * q^31 + 18 * q^33 - 14 * q^39 + 12 * q^41 + 8 * q^43 + 6 * q^45 + 2 * q^47 - 2 * q^51 - 4 * q^53 - 2 * q^55 - 8 * q^57 - 8 * q^59 + 20 * q^61 + 10 * q^65 + 8 * q^67 + 24 * q^69 - 4 * q^71 + 16 * q^73 - 2 * q^75 - 22 * q^79 - 20 * q^81 - 36 * q^83 + 6 * q^85 + 18 * q^87 + 40 * q^89 - 32 * q^93 + 26 * q^97 - 12 * q^99

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	0
$2$	0	0
$3$	−2.87996	−1.66275	−0.831373	−	0.555715i	$-0.812445\pi$
$3$	−2.87996	−1.66275	−0.831373	+	0.555715i	$0.812445\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	5.29417	1.76472
$10$	0	0
$11$	−1.46575	−0.441939	−0.220970	−	0.975281i	$-0.570922\pi$
$11$	−1.46575	−0.441939	−0.220970	+	0.975281i	$0.570922\pi$
$12$	0	0
$13$	2.22129	0.616076	0.308038	−	0.951374i	$-0.400328\pi$
$13$	2.22129	0.616076	0.308038	+	0.951374i	$0.400328\pi$
$14$	0	0
$15$	−2.87996	−0.743603
$16$	0	0
$17$	7.26580	1.76221	0.881107	−	0.472916i	$-0.156799\pi$
$17$	7.26580	1.76221	0.881107	+	0.472916i	$0.156799\pi$
$18$	0	0
$19$	5.48709	1.25883	0.629413	−	0.777071i	$-0.283295\pi$
$19$	5.48709	1.25883	0.629413	+	0.777071i	$0.283295\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−2.51547	−0.524512	−0.262256	−	0.964998i	$-0.584466\pi$
$23$	−2.51547	−0.524512	−0.262256	+	0.964998i	$0.584466\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−6.60713	−1.27154
$28$	0	0
$29$	−7.12260	−1.32263	−0.661317	−	0.750107i	$-0.730002\pi$
$29$	−7.12260	−1.32263	−0.661317	+	0.750107i	$0.730002\pi$
$30$	0	0
$31$	6.51547	1.17021	0.585106	−	0.810957i	$-0.301053\pi$
$31$	6.51547	1.17021	0.585106	+	0.810957i	$0.301053\pi$
$32$	0	0
$33$	4.22129	0.734833
$34$	0	0
$35$	0	0
$36$	0	0
$37$	6.90131	1.13457	0.567284	−	0.823522i	$-0.307994\pi$
$37$	6.90131	1.13457	0.567284	+	0.823522i	$0.307994\pi$
$38$	0	0
$39$	−6.39724	−1.02438
$40$	0	0
$41$	−11.3199	−1.76787	−0.883935	−	0.467609i	$-0.845115\pi$
$41$	−11.3199	−1.76787	−0.883935	+	0.467609i	$0.845115\pi$
$42$	0	0
$43$	−3.31733	−0.505888	−0.252944	−	0.967481i	$-0.581399\pi$
$43$	−3.31733	−0.505888	−0.252944	+	0.967481i	$0.581399\pi$
$44$	0	0
$45$	5.29417	0.789209
$46$	0	0
$47$	8.36705	1.22046	0.610230	−	0.792224i	$-0.291077\pi$
$47$	8.36705	1.22046	0.610230	+	0.792224i	$0.291077\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−20.9252	−2.93012
$52$	0	0
$53$	−7.00437	−0.962125	−0.481062	−	0.876686i	$-0.659749\pi$
$53$	−7.00437	−0.962125	−0.481062	+	0.876686i	$0.659749\pi$
$54$	0	0
$55$	−1.46575	−0.197641
$56$	0	0
$57$	−15.8026	−2.09311
$58$	0	0
$59$	−9.07107	−1.18095	−0.590476	−	0.807055i	$-0.701060\pi$
$59$	−9.07107	−1.18095	−0.590476	+	0.807055i	$0.701060\pi$
$60$	0	0
$61$	11.2143	1.43584	0.717920	−	0.696126i	$-0.245094\pi$
$61$	11.2143	1.43584	0.717920	+	0.696126i	$0.245094\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	2.22129	0.275518
$66$	0	0
$67$	6.41240	0.783400	0.391700	−	0.920093i	$-0.371887\pi$
$67$	6.41240	0.783400	0.391700	+	0.920093i	$0.371887\pi$
$68$	0	0
$69$	7.24445	0.872130
$70$	0	0
$71$	10.5316	1.24987	0.624935	−	0.780677i	$-0.285125\pi$
$71$	10.5316	1.24987	0.624935	+	0.780677i	$0.285125\pi$
$72$	0	0
$73$	10.5217	1.23147	0.615733	−	0.787955i	$-0.288860\pi$
$73$	10.5217	1.23147	0.615733	+	0.787955i	$0.288860\pi$
$74$	0	0
$75$	−2.87996	−0.332549
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−10.1911	−1.14659	−0.573295	−	0.819349i	$-0.694335\pi$
$79$	−10.1911	−1.14659	−0.573295	+	0.819349i	$0.694335\pi$
$80$	0	0
$81$	3.14576	0.349529
$82$	0	0
$83$	−16.4186	−1.80217	−0.901087	−	0.433638i	$-0.857230\pi$
$83$	−16.4186	−1.80217	−0.901087	+	0.433638i	$0.857230\pi$
$84$	0	0
$85$	7.26580	0.788086
$86$	0	0
$87$	20.5128	2.19920
$88$	0	0
$89$	9.83024	1.04200	0.521002	−	0.853556i	$-0.325559\pi$
$89$	9.83024	1.04200	0.521002	+	0.853556i	$0.325559\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−18.7643	−1.94577
$94$	0	0
$95$	5.48709	0.562964
$96$	0	0
$97$	−2.09423	−0.212636	−0.106318	−	0.994332i	$-0.533906\pi$
$97$	−2.09423	−0.212636	−0.106318	+	0.994332i	$0.533906\pi$
$98$	0	0
$99$	−7.75992	−0.779901
$100$	0	0
$101$	−13.6656	−1.35978	−0.679889	−	0.733315i	$-0.737972\pi$
$101$	−13.6656	−1.35978	−0.679889	+	0.733315i	$0.737972\pi$
$102$	0	0
$103$	−10.0239	−0.987685	−0.493843	−	0.869551i	$-0.664408\pi$
$103$	−10.0239	−0.987685	−0.493843	+	0.869551i	$0.664408\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	9.42040	0.910704	0.455352	−	0.890311i	$-0.349513\pi$
$107$	9.42040	0.910704	0.455352	+	0.890311i	$0.349513\pi$
$108$	0	0
$109$	−8.98559	−0.860663	−0.430332	−	0.902671i	$-0.641603\pi$
$109$	−8.98559	−0.860663	−0.430332	+	0.902671i	$0.641603\pi$
$110$	0	0
$111$	−19.8755	−1.88650
$112$	0	0
$113$	10.9476	1.02987	0.514933	−	0.857231i	$-0.327817\pi$
$113$	10.9476	1.02987	0.514933	+	0.857231i	$0.327817\pi$
$114$	0	0
$115$	−2.51547	−0.234569
$116$	0	0
$117$	11.7599	1.08721
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−8.85158	−0.804690
$122$	0	0
$123$	32.6009	2.93952
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	5.44696	0.483340	0.241670	−	0.970359i	$-0.422305\pi$
$127$	5.44696	0.483340	0.241670	+	0.970359i	$0.422305\pi$
$128$	0	0
$129$	9.55379	0.841164
$130$	0	0
$131$	−9.17414	−0.801548	−0.400774	−	0.916177i	$-0.631259\pi$
$131$	−9.17414	−0.801548	−0.400774	+	0.916177i	$0.631259\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−6.60713	−0.568651
$136$	0	0
$137$	11.3137	0.966595	0.483298	−	0.875456i	$-0.339439\pi$
$137$	11.3137	0.966595	0.483298	+	0.875456i	$0.339439\pi$
$138$	0	0
$139$	6.83099	0.579397	0.289698	−	0.957118i	$-0.406445\pi$
$139$	6.83099	0.579397	0.289698	+	0.957118i	$0.406445\pi$
$140$	0	0
$141$	−24.0968	−2.02932
$142$	0	0
$143$	−3.25586	−0.272268
$144$	0	0
$145$	−7.12260	−0.591500
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−1.96243	−0.160769	−0.0803844	−	0.996764i	$-0.525615\pi$
$149$	−1.96243	−0.160769	−0.0803844	+	0.996764i	$0.525615\pi$
$150$	0	0
$151$	7.32511	0.596109	0.298055	−	0.954549i	$-0.403662\pi$
$151$	7.32511	0.596109	0.298055	+	0.954549i	$0.403662\pi$
$152$	0	0
$153$	38.4664	3.10982
$154$	0	0
$155$	6.51547	0.523335
$156$	0	0
$157$	−7.46309	−0.595620	−0.297810	−	0.954625i	$-0.596256\pi$
$157$	−7.46309	−0.595620	−0.297810	+	0.954625i	$0.596256\pi$
$158$	0	0
$159$	20.1723	1.59977
$160$	0	0
$161$	0	0
$162$	0	0
$163$	12.2790	0.961767	0.480883	−	0.876785i	$-0.340316\pi$
$163$	12.2790	0.961767	0.480883	+	0.876785i	$0.340316\pi$
$164$	0	0
$165$	4.22129	0.328627
$166$	0	0
$167$	18.2300	1.41068	0.705342	−	0.708868i	$-0.250794\pi$
$167$	18.2300	1.41068	0.705342	+	0.708868i	$0.250794\pi$
$168$	0	0
$169$	−8.06585	−0.620450
$170$	0	0
$171$	29.0496	2.22148
$172$	0	0
$173$	17.9245	1.36277	0.681386	−	0.731924i	$-0.261378\pi$
$173$	17.9245	1.36277	0.681386	+	0.731924i	$0.261378\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	26.1243	1.96362
$178$	0	0
$179$	−0.0375672	−0.00280791	−0.00140395	−	0.999999i	$-0.500447\pi$
$179$	−0.0375672	−0.00280791	−0.00140395	+	0.999999i	$0.500447\pi$
$180$	0	0
$181$	−6.82105	−0.507004	−0.253502	−	0.967335i	$-0.581583\pi$
$181$	−6.82105	−0.507004	−0.253502	+	0.967335i	$0.581583\pi$
$182$	0	0
$183$	−32.2966	−2.38744
$184$	0	0
$185$	6.90131	0.507394
$186$	0	0
$187$	−10.6498	−0.778792
$188$	0	0
$189$	0	0
$190$	0	0
$191$	5.86202	0.424161	0.212080	−	0.977252i	$-0.431976\pi$
$191$	5.86202	0.424161	0.212080	+	0.977252i	$0.431976\pi$
$192$	0	0
$193$	−4.77168	−0.343473	−0.171736	−	0.985143i	$-0.554938\pi$
$193$	−4.77168	−0.343473	−0.171736	+	0.985143i	$0.554938\pi$
$194$	0	0
$195$	−6.39724	−0.458116
$196$	0	0
$197$	−9.28714	−0.661682	−0.330841	−	0.943687i	$-0.607332\pi$
$197$	−9.28714	−0.661682	−0.330841	+	0.943687i	$0.607332\pi$
$198$	0	0
$199$	3.20176	0.226967	0.113483	−	0.993540i	$-0.463799\pi$
$199$	3.20176	0.226967	0.113483	+	0.993540i	$0.463799\pi$
$200$	0	0
$201$	−18.4675	−1.30259
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−11.3199	−0.790616
$206$	0	0
$207$	−13.3173	−0.925619
$208$	0	0
$209$	−8.04269	−0.556325
$210$	0	0
$211$	26.1055	1.79718	0.898590	−	0.438790i	$-0.144593\pi$
$211$	26.1055	1.79718	0.898590	+	0.438790i	$0.144593\pi$
$212$	0	0
$213$	−30.3306	−2.07822
$214$	0	0
$215$	−3.31733	−0.226240
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−30.3020	−2.04762
$220$	0	0
$221$	16.1395	1.08566
$222$	0	0
$223$	23.8214	1.59520	0.797599	−	0.603188i	$-0.206103\pi$
$223$	23.8214	1.59520	0.797599	+	0.603188i	$0.206103\pi$
$224$	0	0
$225$	5.29417	0.352945
$226$	0	0
$227$	9.42199	0.625360	0.312680	−	0.949859i	$-0.398773\pi$
$227$	9.42199	0.625360	0.312680	+	0.949859i	$0.398773\pi$
$228$	0	0
$229$	9.93587	0.656581	0.328290	−	0.944577i	$-0.393528\pi$
$229$	9.93587	0.656581	0.328290	+	0.944577i	$0.393528\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	1.23707	0.0810434	0.0405217	−	0.999179i	$-0.487098\pi$
$233$	1.23707	0.0810434	0.0405217	+	0.999179i	$0.487098\pi$
$234$	0	0
$235$	8.36705	0.545806
$236$	0	0
$237$	29.3500	1.90649
$238$	0	0
$239$	−1.05710	−0.0683782	−0.0341891	−	0.999415i	$-0.510885\pi$
$239$	−1.05710	−0.0683782	−0.0341891	+	0.999415i	$0.510885\pi$
$240$	0	0
$241$	4.52903	0.291741	0.145870	−	0.989304i	$-0.453402\pi$
$241$	4.52903	0.291741	0.145870	+	0.989304i	$0.453402\pi$
$242$	0	0
$243$	10.7617	0.690366
$244$	0	0
$245$	0	0
$246$	0	0
$247$	12.1885	0.775533
$248$	0	0
$249$	47.2849	2.99656
$250$	0	0
$251$	−4.74198	−0.299311	−0.149656	−	0.988738i	$-0.547816\pi$
$251$	−4.74198	−0.299311	−0.149656	+	0.988738i	$0.547816\pi$
$252$	0	0
$253$	3.68704	0.231802
$254$	0	0
$255$	−20.9252	−1.31039
$256$	0	0
$257$	15.9297	0.993666	0.496833	−	0.867846i	$-0.334496\pi$
$257$	15.9297	0.993666	0.496833	+	0.867846i	$0.334496\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−37.7083	−2.33408
$262$	0	0
$263$	13.2746	0.818549	0.409275	−	0.912411i	$-0.365782\pi$
$263$	13.2746	0.818549	0.409275	+	0.912411i	$0.365782\pi$
$264$	0	0
$265$	−7.00437	−0.430275
$266$	0	0
$267$	−28.3107	−1.73259
$268$	0	0
$269$	24.8601	1.51575	0.757874	−	0.652401i	$-0.226238\pi$
$269$	24.8601	1.51575	0.757874	+	0.652401i	$0.226238\pi$
$270$	0	0
$271$	30.4285	1.84840	0.924201	−	0.381907i	$-0.124733\pi$
$271$	30.4285	1.84840	0.924201	+	0.381907i	$0.124733\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−1.46575	−0.0883879
$276$	0	0
$277$	−21.7297	−1.30561	−0.652807	−	0.757525i	$-0.726409\pi$
$277$	−21.7297	−1.30561	−0.652807	+	0.757525i	$0.726409\pi$
$278$	0	0
$279$	34.4940	2.06510
$280$	0	0
$281$	−0.285426	−0.0170271	−0.00851354	−	0.999964i	$-0.502710\pi$
$281$	−0.285426	−0.0170271	−0.00851354	+	0.999964i	$0.502710\pi$
$282$	0	0
$283$	−5.51463	−0.327810	−0.163905	−	0.986476i	$-0.552409\pi$
$283$	−5.51463	−0.327810	−0.163905	+	0.986476i	$0.552409\pi$
$284$	0	0
$285$	−15.8026	−0.936066
$286$	0	0
$287$	0	0
$288$	0	0
$289$	35.7918	2.10540
$290$	0	0
$291$	6.03129	0.353560
$292$	0	0
$293$	17.5777	1.02690	0.513450	−	0.858120i	$-0.328367\pi$
$293$	17.5777	1.02690	0.513450	+	0.858120i	$0.328367\pi$
$294$	0	0
$295$	−9.07107	−0.528138
$296$	0	0
$297$	9.68439	0.561945
$298$	0	0
$299$	−5.58760	−0.323139
$300$	0	0
$301$	0	0
$302$	0	0
$303$	39.3564	2.26097
$304$	0	0
$305$	11.2143	0.642127
$306$	0	0
$307$	25.4771	1.45405	0.727026	−	0.686610i	$-0.240902\pi$
$307$	25.4771	1.45405	0.727026	+	0.686610i	$0.240902\pi$
$308$	0	0
$309$	28.8685	1.64227
$310$	0	0
$311$	−21.5662	−1.22290	−0.611452	−	0.791281i	$-0.709414\pi$
$311$	−21.5662	−1.22290	−0.611452	+	0.791281i	$0.709414\pi$
$312$	0	0
$313$	2.97940	0.168406	0.0842030	−	0.996449i	$-0.473166\pi$
$313$	2.97940	0.168406	0.0842030	+	0.996449i	$0.473166\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−7.99638	−0.449121	−0.224561	−	0.974460i	$-0.572095\pi$
$317$	−7.99638	−0.449121	−0.224561	+	0.974460i	$0.572095\pi$
$318$	0	0
$319$	10.4399	0.584524
$320$	0	0
$321$	−27.1304	−1.51427
$322$	0	0
$323$	39.8681	2.21832
$324$	0	0
$325$	2.22129	0.123215
$326$	0	0
$327$	25.8781	1.43106
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−34.2966	−1.88511	−0.942557	−	0.334045i	$-0.891586\pi$
$331$	−34.2966	−1.88511	−0.942557	+	0.334045i	$0.891586\pi$
$332$	0	0
$333$	36.5367	2.00220
$334$	0	0
$335$	6.41240	0.350347
$336$	0	0
$337$	23.8719	1.30038	0.650192	−	0.759770i	$-0.274689\pi$
$337$	23.8719	1.30038	0.650192	+	0.759770i	$0.274689\pi$
$338$	0	0
$339$	−31.5287	−1.71241
$340$	0	0
$341$	−9.55003	−0.517163
$342$	0	0
$343$	0	0
$344$	0	0
$345$	7.24445	0.390028
$346$	0	0
$347$	−6.35565	−0.341189	−0.170595	−	0.985341i	$-0.554569\pi$
$347$	−6.35565	−0.341189	−0.170595	+	0.985341i	$0.554569\pi$
$348$	0	0
$349$	−9.24821	−0.495045	−0.247523	−	0.968882i	$-0.579616\pi$
$349$	−9.24821	−0.495045	−0.247523	+	0.968882i	$0.579616\pi$
$350$	0	0
$351$	−14.6764	−0.783368
$352$	0	0
$353$	17.9608	0.955959	0.477979	−	0.878371i	$-0.341369\pi$
$353$	17.9608	0.955959	0.477979	+	0.878371i	$0.341369\pi$
$354$	0	0
$355$	10.5316	0.558959
$356$	0	0
$357$	0	0
$358$	0	0
$359$	20.2843	1.07056	0.535281	−	0.844674i	$-0.320206\pi$
$359$	20.2843	1.07056	0.535281	+	0.844674i	$0.320206\pi$
$360$	0	0
$361$	11.1082	0.584642
$362$	0	0
$363$	25.4922	1.33799
$364$	0	0
$365$	10.5217	0.550729
$366$	0	0
$367$	−25.1439	−1.31250	−0.656249	−	0.754544i	$-0.727858\pi$
$367$	−25.1439	−1.31250	−0.656249	+	0.754544i	$0.727858\pi$
$368$	0	0
$369$	−59.9295	−3.11980
$370$	0	0
$371$	0	0
$372$	0	0
$373$	0.737732	0.0381983	0.0190992	−	0.999818i	$-0.493920\pi$
$373$	0.737732	0.0381983	0.0190992	+	0.999818i	$0.493920\pi$
$374$	0	0
$375$	−2.87996	−0.148721
$376$	0	0
$377$	−15.8214	−0.814843
$378$	0	0
$379$	7.11120	0.365278	0.182639	−	0.983180i	$-0.441536\pi$
$379$	7.11120	0.365278	0.182639	+	0.983180i	$0.441536\pi$
$380$	0	0
$381$	−15.6870	−0.803672
$382$	0	0
$383$	−28.6842	−1.46569	−0.732846	−	0.680394i	$-0.761809\pi$
$383$	−28.6842	−1.46569	−0.732846	+	0.680394i	$0.761809\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−17.5625	−0.892754
$388$	0	0
$389$	13.5189	0.685434	0.342717	−	0.939439i	$-0.388653\pi$
$389$	13.5189	0.685434	0.342717	+	0.939439i	$0.388653\pi$
$390$	0	0
$391$	−18.2769	−0.924302
$392$	0	0
$393$	26.4212	1.33277
$394$	0	0
$395$	−10.1911	−0.512770
$396$	0	0
$397$	4.02391	0.201954	0.100977	−	0.994889i	$-0.467803\pi$
$397$	4.02391	0.201954	0.100977	+	0.994889i	$0.467803\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−24.3796	−1.21746	−0.608729	−	0.793379i	$-0.708320\pi$
$401$	−24.3796	−1.21746	−0.608729	+	0.793379i	$0.708320\pi$
$402$	0	0
$403$	14.4728	0.720940
$404$	0	0
$405$	3.14576	0.156314
$406$	0	0
$407$	−10.1156	−0.501410
$408$	0	0
$409$	8.49522	0.420062	0.210031	−	0.977695i	$-0.432644\pi$
$409$	8.49522	0.420062	0.210031	+	0.977695i	$0.432644\pi$
$410$	0	0
$411$	−32.5830	−1.60720
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−16.4186	−0.805957
$416$	0	0
$417$	−19.6730	−0.963390
$418$	0	0
$419$	−3.07469	−0.150209	−0.0751043	−	0.997176i	$-0.523929\pi$
$419$	−3.07469	−0.150209	−0.0751043	+	0.997176i	$0.523929\pi$
$420$	0	0
$421$	22.4790	1.09556	0.547780	−	0.836623i	$-0.315473\pi$
$421$	22.4790	1.09556	0.547780	+	0.836623i	$0.315473\pi$
$422$	0	0
$423$	44.2966	2.15378
$424$	0	0
$425$	7.26580	0.352443
$426$	0	0
$427$	0	0
$428$	0	0
$429$	9.37674	0.452713
$430$	0	0
$431$	5.17254	0.249153	0.124576	−	0.992210i	$-0.460243\pi$
$431$	5.17254	0.249153	0.124576	+	0.992210i	$0.460243\pi$
$432$	0	0
$433$	1.78030	0.0855557	0.0427779	−	0.999085i	$-0.486379\pi$
$433$	1.78030	0.0855557	0.0427779	+	0.999085i	$0.486379\pi$
$434$	0	0
$435$	20.5128	0.983514
$436$	0	0
$437$	−13.8026	−0.660269
$438$	0	0
$439$	25.7865	1.23072	0.615361	−	0.788245i	$-0.289010\pi$
$439$	25.7865	1.23072	0.615361	+	0.788245i	$0.289010\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	12.4426	0.591165	0.295583	−	0.955317i	$-0.404486\pi$
$443$	12.4426	0.591165	0.295583	+	0.955317i	$0.404486\pi$
$444$	0	0
$445$	9.83024	0.465998
$446$	0	0
$447$	5.65173	0.267318
$448$	0	0
$449$	−25.3574	−1.19669	−0.598344	−	0.801239i	$-0.704174\pi$
$449$	−25.3574	−1.19669	−0.598344	+	0.801239i	$0.704174\pi$
$450$	0	0
$451$	16.5921	0.781292
$452$	0	0
$453$	−21.0960	−0.991178
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−34.7039	−1.62338	−0.811690	−	0.584088i	$-0.801452\pi$
$457$	−34.7039	−1.62338	−0.811690	+	0.584088i	$0.801452\pi$
$458$	0	0
$459$	−48.0061	−2.24073
$460$	0	0
$461$	−40.8762	−1.90380	−0.951898	−	0.306414i	$-0.900871\pi$
$461$	−40.8762	−1.90380	−0.951898	+	0.306414i	$0.900871\pi$
$462$	0	0
$463$	25.1236	1.16759	0.583796	−	0.811901i	$-0.301567\pi$
$463$	25.1236	1.16759	0.583796	+	0.811901i	$0.301567\pi$
$464$	0	0
$465$	−18.7643	−0.870173
$466$	0	0
$467$	27.1766	1.25758	0.628792	−	0.777574i	$-0.283550\pi$
$467$	27.1766	1.25758	0.628792	+	0.777574i	$0.283550\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	21.4934	0.990364
$472$	0	0
$473$	4.86237	0.223572
$474$	0	0
$475$	5.48709	0.251765
$476$	0	0
$477$	−37.0824	−1.69789
$478$	0	0
$479$	2.07288	0.0947123	0.0473561	−	0.998878i	$-0.484920\pi$
$479$	2.07288	0.0947123	0.0473561	+	0.998878i	$0.484920\pi$
$480$	0	0
$481$	15.3298	0.698980
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−2.09423	−0.0950939
$486$	0	0
$487$	−33.3490	−1.51119	−0.755594	−	0.655040i	$-0.772652\pi$
$487$	−33.3490	−1.51119	−0.755594	+	0.655040i	$0.772652\pi$
$488$	0	0
$489$	−35.3631	−1.59917
$490$	0	0
$491$	−36.4487	−1.64491	−0.822453	−	0.568833i	$-0.807395\pi$
$491$	−36.4487	−1.64491	−0.822453	+	0.568833i	$0.807395\pi$
$492$	0	0
$493$	−51.7514	−2.33077
$494$	0	0
$495$	−7.75992	−0.348783
$496$	0	0
$497$	0	0
$498$	0	0
$499$	7.49368	0.335463	0.167732	−	0.985833i	$-0.446356\pi$
$499$	7.49368	0.335463	0.167732	+	0.985833i	$0.446356\pi$
$500$	0	0
$501$	−52.5018	−2.34561
$502$	0	0
$503$	15.6381	0.697267	0.348634	−	0.937259i	$-0.386646\pi$
$503$	15.6381	0.697267	0.348634	+	0.937259i	$0.386646\pi$
$504$	0	0
$505$	−13.6656	−0.608111
$506$	0	0
$507$	23.2293	1.03165
$508$	0	0
$509$	7.21051	0.319600	0.159800	−	0.987149i	$-0.448915\pi$
$509$	7.21051	0.319600	0.159800	+	0.987149i	$0.448915\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−36.2540	−1.60065
$514$	0	0
$515$	−10.0239	−0.441706
$516$	0	0
$517$	−12.2640	−0.539370
$518$	0	0
$519$	−51.6218	−2.26594
$520$	0	0
$521$	14.8236	0.649434	0.324717	−	0.945811i	$-0.394731\pi$
$521$	14.8236	0.649434	0.324717	+	0.945811i	$0.394731\pi$
$522$	0	0
$523$	4.27283	0.186838	0.0934189	−	0.995627i	$-0.470220\pi$
$523$	4.27283	0.186838	0.0934189	+	0.995627i	$0.470220\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	47.3401	2.06217
$528$	0	0
$529$	−16.6724	−0.724888
$530$	0	0
$531$	−48.0238	−2.08406
$532$	0	0
$533$	−25.1448	−1.08914
$534$	0	0
$535$	9.42040	0.407279
$536$	0	0
$537$	0.108192	0.00466883
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−0.942899	−0.0405384	−0.0202692	−	0.999795i	$-0.506452\pi$
$541$	−0.942899	−0.0405384	−0.0202692	+	0.999795i	$0.506452\pi$
$542$	0	0
$543$	19.6444	0.843020
$544$	0	0
$545$	−8.98559	−0.384900
$546$	0	0
$547$	11.1871	0.478327	0.239164	−	0.970979i	$-0.423127\pi$
$547$	11.1871	0.478327	0.239164	+	0.970979i	$0.423127\pi$
$548$	0	0
$549$	59.3703	2.53386
$550$	0	0
$551$	−39.0824	−1.66497
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−19.8755	−0.843668
$556$	0	0
$557$	28.7305	1.21735	0.608675	−	0.793420i	$-0.291701\pi$
$557$	28.7305	1.21735	0.608675	+	0.793420i	$0.291701\pi$
$558$	0	0
$559$	−7.36877	−0.311666
$560$	0	0
$561$	30.6711	1.29493
$562$	0	0
$563$	−2.44741	−0.103146	−0.0515729	−	0.998669i	$-0.516423\pi$
$563$	−2.44741	−0.103146	−0.0515729	+	0.998669i	$0.516423\pi$
$564$	0	0
$565$	10.9476	0.460570
$566$	0	0
$567$	0	0
$568$	0	0
$569$	35.5058	1.48848	0.744240	−	0.667912i	$-0.232812\pi$
$569$	35.5058	1.48848	0.744240	+	0.667912i	$0.232812\pi$
$570$	0	0
$571$	−5.78874	−0.242251	−0.121126	−	0.992637i	$-0.538650\pi$
$571$	−5.78874	−0.242251	−0.121126	+	0.992637i	$0.538650\pi$
$572$	0	0
$573$	−16.8824	−0.705272
$574$	0	0
$575$	−2.51547	−0.104902
$576$	0	0
$577$	2.20773	0.0919090	0.0459545	−	0.998944i	$-0.485367\pi$
$577$	2.20773	0.0919090	0.0459545	+	0.998944i	$0.485367\pi$
$578$	0	0
$579$	13.7422	0.571108
$580$	0	0
$581$	0	0
$582$	0	0
$583$	10.2666	0.425201
$584$	0	0
$585$	11.7599	0.486213
$586$	0	0
$587$	−39.6719	−1.63744	−0.818718	−	0.574196i	$-0.805315\pi$
$587$	−39.6719	−1.63744	−0.818718	+	0.574196i	$0.805315\pi$
$588$	0	0
$589$	35.7510	1.47309
$590$	0	0
$591$	26.7466	1.10021
$592$	0	0
$593$	−8.93309	−0.366838	−0.183419	−	0.983035i	$-0.558717\pi$
$593$	−8.93309	−0.366838	−0.183419	+	0.983035i	$0.558717\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−9.22095	−0.377388
$598$	0	0
$599$	−18.3900	−0.751395	−0.375697	−	0.926742i	$-0.622597\pi$
$599$	−18.3900	−0.751395	−0.375697	+	0.926742i	$0.622597\pi$
$600$	0	0
$601$	17.5063	0.714096	0.357048	−	0.934086i	$-0.383783\pi$
$601$	17.5063	0.714096	0.357048	+	0.934086i	$0.383783\pi$
$602$	0	0
$603$	33.9484	1.38248
$604$	0	0
$605$	−8.85158	−0.359868
$606$	0	0
$607$	45.1439	1.83233	0.916166	−	0.400798i	$-0.131267\pi$
$607$	45.1439	1.83233	0.916166	+	0.400798i	$0.131267\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	18.5857	0.751897
$612$	0	0
$613$	−11.8557	−0.478849	−0.239424	−	0.970915i	$-0.576959\pi$
$613$	−11.8557	−0.478849	−0.239424	+	0.970915i	$0.576959\pi$
$614$	0	0
$615$	32.6009	1.31459
$616$	0	0
$617$	−20.8174	−0.838078	−0.419039	−	0.907968i	$-0.637633\pi$
$617$	−20.8174	−0.838078	−0.419039	+	0.907968i	$0.637633\pi$
$618$	0	0
$619$	8.78149	0.352958	0.176479	−	0.984304i	$-0.443529\pi$
$619$	8.78149	0.352958	0.176479	+	0.984304i	$0.443529\pi$
$620$	0	0
$621$	16.6200	0.666939
$622$	0	0
$623$	0	0
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	23.1626	0.925027
$628$	0	0
$629$	50.1435	1.99935
$630$	0	0
$631$	−22.1004	−0.879803	−0.439902	−	0.898046i	$-0.644987\pi$
$631$	−22.1004	−0.879803	−0.439902	+	0.898046i	$0.644987\pi$
$632$	0	0
$633$	−75.1829	−2.98825
$634$	0	0
$635$	5.44696	0.216156
$636$	0	0
$637$	0	0
$638$	0	0
$639$	55.7561	2.20568
$640$	0	0
$641$	−43.8577	−1.73227	−0.866137	−	0.499807i	$-0.833404\pi$
$641$	−43.8577	−1.73227	−0.866137	+	0.499807i	$0.833404\pi$
$642$	0	0
$643$	−37.9895	−1.49816	−0.749079	−	0.662480i	$-0.769504\pi$
$643$	−37.9895	−1.49816	−0.749079	+	0.662480i	$0.769504\pi$
$644$	0	0
$645$	9.55379	0.376180
$646$	0	0
$647$	−33.3705	−1.31193	−0.655964	−	0.754792i	$-0.727738\pi$
$647$	−33.3705	−1.31193	−0.655964	+	0.754792i	$0.727738\pi$
$648$	0	0
$649$	13.2959	0.521909
$650$	0	0
$651$	0	0
$652$	0	0
$653$	15.0632	0.589468	0.294734	−	0.955579i	$-0.404769\pi$
$653$	15.0632	0.589468	0.294734	+	0.955579i	$0.404769\pi$
$654$	0	0
$655$	−9.17414	−0.358463
$656$	0	0
$657$	55.7035	2.17320
$658$	0	0
$659$	0.153540	0.00598106	0.00299053	−	0.999996i	$-0.499048\pi$
$659$	0.153540	0.00598106	0.00299053	+	0.999996i	$0.499048\pi$
$660$	0	0
$661$	12.3056	0.478632	0.239316	−	0.970942i	$-0.423077\pi$
$661$	12.3056	0.478632	0.239316	+	0.970942i	$0.423077\pi$
$662$	0	0
$663$	−46.4811	−1.80518
$664$	0	0
$665$	0	0
$666$	0	0
$667$	17.9167	0.693737
$668$	0	0
$669$	−68.6047	−2.65241
$670$	0	0
$671$	−16.4373	−0.634554
$672$	0	0
$673$	4.82480	0.185983	0.0929913	−	0.995667i	$-0.470357\pi$
$673$	4.82480	0.185983	0.0929913	+	0.995667i	$0.470357\pi$
$674$	0	0
$675$	−6.60713	−0.254309
$676$	0	0
$677$	16.5041	0.634303	0.317151	−	0.948375i	$-0.397274\pi$
$677$	16.5041	0.634303	0.317151	+	0.948375i	$0.397274\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−27.1350	−1.03981
$682$	0	0
$683$	−32.3621	−1.23830	−0.619152	−	0.785272i	$-0.712523\pi$
$683$	−32.3621	−1.23830	−0.619152	+	0.785272i	$0.712523\pi$
$684$	0	0
$685$	11.3137	0.432275
$686$	0	0
$687$	−28.6149	−1.09173
$688$	0	0
$689$	−15.5588	−0.592742
$690$	0	0
$691$	−2.32446	−0.0884265	−0.0442132	−	0.999022i	$-0.514078\pi$
$691$	−2.32446	−0.0884265	−0.0442132	+	0.999022i	$0.514078\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	6.83099	0.259114
$696$	0	0
$697$	−82.2481	−3.11537
$698$	0	0
$699$	−3.56272	−0.134755
$700$	0	0
$701$	−6.77071	−0.255726	−0.127863	−	0.991792i	$-0.540812\pi$
$701$	−6.77071	−0.255726	−0.127863	+	0.991792i	$0.540812\pi$
$702$	0	0
$703$	37.8681	1.42822
$704$	0	0
$705$	−24.0968	−0.907538
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−6.23742	−0.234251	−0.117126	−	0.993117i	$-0.537368\pi$
$709$	−6.23742	−0.234251	−0.117126	+	0.993117i	$0.537368\pi$
$710$	0	0
$711$	−53.9535	−2.02341
$712$	0	0
$713$	−16.3895	−0.613790
$714$	0	0
$715$	−3.25586	−0.121762
$716$	0	0
$717$	3.04441	0.113696
$718$	0	0
$719$	−4.94931	−0.184578	−0.0922891	−	0.995732i	$-0.529418\pi$
$719$	−4.94931	−0.184578	−0.0922891	+	0.995732i	$0.529418\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−13.0434	−0.485091
$724$	0	0
$725$	−7.12260	−0.264527
$726$	0	0
$727$	18.5847	0.689269	0.344635	−	0.938737i	$-0.388003\pi$
$727$	18.5847	0.689269	0.344635	+	0.938737i	$0.388003\pi$
$728$	0	0
$729$	−40.4306	−1.49743
$730$	0	0
$731$	−24.1031	−0.891484
$732$	0	0
$733$	−24.5696	−0.907498	−0.453749	−	0.891130i	$-0.649914\pi$
$733$	−24.5696	−0.907498	−0.453749	+	0.891130i	$0.649914\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−9.39896	−0.346215
$738$	0	0
$739$	8.42950	0.310084	0.155042	−	0.987908i	$-0.450449\pi$
$739$	8.42950	0.310084	0.155042	+	0.987908i	$0.450449\pi$
$740$	0	0
$741$	−35.1023	−1.28951
$742$	0	0
$743$	11.4705	0.420811	0.210405	−	0.977614i	$-0.432522\pi$
$743$	11.4705	0.420811	0.210405	+	0.977614i	$0.432522\pi$
$744$	0	0
$745$	−1.96243	−0.0718980
$746$	0	0
$747$	−86.9229	−3.18034
$748$	0	0
$749$	0	0
$750$	0	0
$751$	16.3029	0.594902	0.297451	−	0.954737i	$-0.403863\pi$
$751$	16.3029	0.594902	0.297451	+	0.954737i	$0.403863\pi$
$752$	0	0
$753$	13.6567	0.497679
$754$	0	0
$755$	7.32511	0.266588
$756$	0	0
$757$	25.0875	0.911821	0.455910	−	0.890026i	$-0.349314\pi$
$757$	25.0875	0.911821	0.455910	+	0.890026i	$0.349314\pi$
$758$	0	0
$759$	−10.6185	−0.385428
$760$	0	0
$761$	49.0849	1.77933	0.889664	−	0.456616i	$-0.150939\pi$
$761$	49.0849	1.77933	0.889664	+	0.456616i	$0.150939\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	38.4664	1.39076
$766$	0	0
$767$	−20.1495	−0.727557
$768$	0	0
$769$	−27.0548	−0.975619	−0.487810	−	0.872950i	$-0.662204\pi$
$769$	−27.0548	−0.975619	−0.487810	+	0.872950i	$0.662204\pi$
$770$	0	0
$771$	−45.8769	−1.65221
$772$	0	0
$773$	−41.0828	−1.47764	−0.738822	−	0.673900i	$-0.764618\pi$
$773$	−41.0828	−1.47764	−0.738822	+	0.673900i	$0.764618\pi$
$774$	0	0
$775$	6.51547	0.234043
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−62.1133	−2.22544
$780$	0	0
$781$	−15.4367	−0.552367
$782$	0	0
$783$	47.0600	1.68179
$784$	0	0
$785$	−7.46309	−0.266369
$786$	0	0
$787$	19.7475	0.703921	0.351960	−	0.936015i	$-0.385515\pi$
$787$	19.7475	0.703921	0.351960	+	0.936015i	$0.385515\pi$
$788$	0	0
$789$	−38.2304	−1.36104
$790$	0	0
$791$	0	0
$792$	0	0
$793$	24.9102	0.884587
$794$	0	0
$795$	20.1723	0.715439
$796$	0	0
$797$	−35.9488	−1.27337	−0.636685	−	0.771124i	$-0.719695\pi$
$797$	−35.9488	−1.27337	−0.636685	+	0.771124i	$0.719695\pi$
$798$	0	0
$799$	60.7933	2.15071
$800$	0	0
$801$	52.0430	1.83885
$802$	0	0
$803$	−15.4221	−0.544234
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−71.5962	−2.52030
$808$	0	0
$809$	39.1137	1.37516	0.687582	−	0.726107i	$-0.258672\pi$
$809$	39.1137	1.37516	0.687582	+	0.726107i	$0.258672\pi$
$810$	0	0
$811$	−43.4017	−1.52404	−0.762019	−	0.647555i	$-0.775792\pi$
$811$	−43.4017	−1.52404	−0.762019	+	0.647555i	$0.775792\pi$
$812$	0	0
$813$	−87.6330	−3.07342
$814$	0	0
$815$	12.2790	0.430115
$816$	0	0
$817$	−18.2025	−0.636825
$818$	0	0
$819$	0	0
$820$	0	0
$821$	13.0135	0.454175	0.227088	−	0.973874i	$-0.427080\pi$
$821$	13.0135	0.454175	0.227088	+	0.973874i	$0.427080\pi$
$822$	0	0
$823$	0.0590066	0.00205684	0.00102842	−	0.999999i	$-0.499673\pi$
$823$	0.0590066	0.00205684	0.00102842	+	0.999999i	$0.499673\pi$
$824$	0	0
$825$	4.22129	0.146967
$826$	0	0
$827$	−0.519093	−0.0180506	−0.00902531	−	0.999959i	$-0.502873\pi$
$827$	−0.519093	−0.0180506	−0.00902531	+	0.999959i	$0.502873\pi$
$828$	0	0
$829$	50.1574	1.74204	0.871019	−	0.491250i	$-0.163460\pi$
$829$	50.1574	1.74204	0.871019	+	0.491250i	$0.163460\pi$
$830$	0	0
$831$	62.5808	2.17090
$832$	0	0
$833$	0	0
$834$	0	0
$835$	18.2300	0.630877
$836$	0	0
$837$	−43.0486	−1.48798
$838$	0	0
$839$	−9.44696	−0.326145	−0.163073	−	0.986614i	$-0.552141\pi$
$839$	−9.44696	−0.326145	−0.163073	+	0.986614i	$0.552141\pi$
$840$	0	0
$841$	21.7315	0.749360
$842$	0	0
$843$	0.822016	0.0283117
$844$	0	0
$845$	−8.06585	−0.277474
$846$	0	0
$847$	0	0
$848$	0	0
$849$	15.8819	0.545066
$850$	0	0
$851$	−17.3600	−0.595094
$852$	0	0
$853$	21.2482	0.727525	0.363762	−	0.931492i	$-0.381492\pi$
$853$	21.2482	0.727525	0.363762	+	0.931492i	$0.381492\pi$
$854$	0	0
$855$	29.0496	0.993476
$856$	0	0
$857$	6.56435	0.224234	0.112117	−	0.993695i	$-0.464237\pi$
$857$	6.56435	0.224234	0.112117	+	0.993695i	$0.464237\pi$
$858$	0	0
$859$	−21.2506	−0.725062	−0.362531	−	0.931972i	$-0.618087\pi$
$859$	−21.2506	−0.725062	−0.362531	+	0.931972i	$0.618087\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	1.43459	0.0488341	0.0244170	−	0.999702i	$-0.492227\pi$
$863$	1.43459	0.0488341	0.0244170	+	0.999702i	$0.492227\pi$
$864$	0	0
$865$	17.9245	0.609450
$866$	0	0
$867$	−103.079	−3.50075
$868$	0	0
$869$	14.9376	0.506723
$870$	0	0
$871$	14.2438	0.482634
$872$	0	0
$873$	−11.0872	−0.375245
$874$	0	0
$875$	0	0
$876$	0	0
$877$	22.6039	0.763280	0.381640	−	0.924311i	$-0.375359\pi$
$877$	22.6039	0.763280	0.381640	+	0.924311i	$0.375359\pi$
$878$	0	0
$879$	−50.6231	−1.70747
$880$	0	0
$881$	25.2167	0.849572	0.424786	−	0.905294i	$-0.360349\pi$
$881$	25.2167	0.849572	0.424786	+	0.905294i	$0.360349\pi$
$882$	0	0
$883$	42.0869	1.41634	0.708168	−	0.706044i	$-0.249522\pi$
$883$	42.0869	1.41634	0.708168	+	0.706044i	$0.249522\pi$
$884$	0	0
$885$	26.1243	0.878159
$886$	0	0
$887$	−23.0721	−0.774686	−0.387343	−	0.921936i	$-0.626607\pi$
$887$	−23.0721	−0.774686	−0.387343	+	0.921936i	$0.626607\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−4.61089	−0.154471
$892$	0	0
$893$	45.9108	1.53635
$894$	0	0
$895$	−0.0375672	−0.00125573
$896$	0	0
$897$	16.0921	0.537298
$898$	0	0
$899$	−46.4071	−1.54776
$900$	0	0
$901$	−50.8924	−1.69547
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−6.82105	−0.226739
$906$	0	0
$907$	−22.3923	−0.743525	−0.371763	−	0.928328i	$-0.621246\pi$
$907$	−22.3923	−0.743525	−0.371763	+	0.928328i	$0.621246\pi$
$908$	0	0
$909$	−72.3481	−2.39963
$910$	0	0
$911$	54.7584	1.81423	0.907114	−	0.420886i	$-0.138281\pi$
$911$	54.7584	1.81423	0.907114	+	0.420886i	$0.138281\pi$
$912$	0	0
$913$	24.0655	0.796452
$914$	0	0
$915$	−32.2966	−1.06769
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−56.0590	−1.84922	−0.924608	−	0.380919i	$-0.875608\pi$
$919$	−56.0590	−1.84922	−0.924608	+	0.380919i	$0.875608\pi$
$920$	0	0
$921$	−73.3729	−2.41772
$922$	0	0
$923$	23.3938	0.770016
$924$	0	0
$925$	6.90131	0.226914
$926$	0	0
$927$	−53.0683	−1.74299
$928$	0	0
$929$	37.5879	1.23322	0.616610	−	0.787269i	$-0.288506\pi$
$929$	37.5879	1.23322	0.616610	+	0.787269i	$0.288506\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	62.1097	2.03338
$934$	0	0
$935$	−10.6498	−0.348286
$936$	0	0
$937$	−55.4322	−1.81089	−0.905446	−	0.424461i	$-0.860464\pi$
$937$	−55.4322	−1.81089	−0.905446	+	0.424461i	$0.860464\pi$
$938$	0	0
$939$	−8.58057	−0.280016
$940$	0	0
$941$	15.1539	0.494003	0.247001	−	0.969015i	$-0.420555\pi$
$941$	15.1539	0.494003	0.247001	+	0.969015i	$0.420555\pi$
$942$	0	0
$943$	28.4748	0.927269
$944$	0	0
$945$	0	0
$946$	0	0
$947$	19.6907	0.639861	0.319930	−	0.947441i	$-0.396340\pi$
$947$	19.6907	0.639861	0.319930	+	0.947441i	$0.396340\pi$
$948$	0	0
$949$	23.3717	0.758677
$950$	0	0
$951$	23.0293	0.746775
$952$	0	0
$953$	44.8194	1.45184	0.725921	−	0.687778i	$-0.241414\pi$
$953$	44.8194	1.45184	0.725921	+	0.687778i	$0.241414\pi$
$954$	0	0
$955$	5.86202	0.189691
$956$	0	0
$957$	−30.0666	−0.971915
$958$	0	0
$959$	0	0
$960$	0	0
$961$	11.4513	0.369398
$962$	0	0
$963$	49.8732	1.60714
$964$	0	0
$965$	−4.77168	−0.153606
$966$	0	0
$967$	−32.3195	−1.03932	−0.519662	−	0.854372i	$-0.673942\pi$
$967$	−32.3195	−1.03932	−0.519662	+	0.854372i	$0.673942\pi$
$968$	0	0
$969$	−114.819	−3.68851
$970$	0	0
$971$	−12.5807	−0.403733	−0.201866	−	0.979413i	$-0.564701\pi$
$971$	−12.5807	−0.403733	−0.201866	+	0.979413i	$0.564701\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−6.39724	−0.204876
$976$	0	0
$977$	42.1169	1.34744	0.673720	−	0.738987i	$-0.264695\pi$
$977$	42.1169	1.34744	0.673720	+	0.738987i	$0.264695\pi$
$978$	0	0
$979$	−14.4086	−0.460502
$980$	0	0
$981$	−47.5713	−1.51883
$982$	0	0
$983$	−26.7102	−0.851923	−0.425962	−	0.904741i	$-0.640064\pi$
$983$	−26.7102	−0.851923	−0.425962	+	0.904741i	$0.640064\pi$
$984$	0	0
$985$	−9.28714	−0.295913
$986$	0	0
$987$	0	0
$988$	0	0
$989$	8.34465	0.265344
$990$	0	0
$991$	7.44903	0.236626	0.118313	−	0.992976i	$-0.462251\pi$
$991$	7.44903	0.236626	0.118313	+	0.992976i	$0.462251\pi$
$992$	0	0
$993$	98.7730	3.13447
$994$	0	0
$995$	3.20176	0.101503
$996$	0	0
$997$	53.9068	1.70724	0.853622	−	0.520892i	$-0.174401\pi$
$997$	53.9068	1.70724	0.853622	+	0.520892i	$0.174401\pi$
$998$	0	0
$999$	−45.5978	−1.44265

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3920.2.a.cd.1.1		4
4.3	odd	2		1960.2.a.y.1.4	yes	4
7.6	odd	2		3920.2.a.ce.1.4		4
20.19	odd	2		9800.2.a.cl.1.1		4
28.3	even	6		1960.2.q.y.961.4		8
28.11	odd	6		1960.2.q.x.961.1		8
28.19	even	6		1960.2.q.y.361.4		8
28.23	odd	6		1960.2.q.x.361.1		8
28.27	even	2		1960.2.a.x.1.1	✓	4
140.139	even	2		9800.2.a.cs.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1960.2.a.x.1.1	✓	4	28.27	even	2
1960.2.a.y.1.4	yes	4	4.3	odd	2
1960.2.q.x.361.1		8	28.23	odd	6
1960.2.q.x.961.1		8	28.11	odd	6
1960.2.q.y.361.4		8	28.19	even	6
1960.2.q.y.961.4		8	28.3	even	6
3920.2.a.cd.1.1		4	1.1	even	1	trivial
3920.2.a.ce.1.4		4	7.6	odd	2
9800.2.a.cl.1.1		4	20.19	odd	2
9800.2.a.cs.1.4		4	140.139	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:	\( N \)	\(=\)	\( 3920 = 2^{4} \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3920.a (trivial)

\(f(q)\)	\(=\)	\(q-2.87996 q^{3} +1.00000 q^{5} +5.29417 q^{9} +O(q^{10})\)	\(q-2.87996 q^{3} +1.00000 q^{5} +5.29417 q^{9} -1.46575 q^{11} +2.22129 q^{13} -2.87996 q^{15} +7.26580 q^{17} +5.48709 q^{19} -2.51547 q^{23} +1.00000 q^{25} -6.60713 q^{27} -7.12260 q^{29} +6.51547 q^{31} +4.22129 q^{33} +6.90131 q^{37} -6.39724 q^{39} -11.3199 q^{41} -3.31733 q^{43} +5.29417 q^{45} +8.36705 q^{47} -20.9252 q^{51} -7.00437 q^{53} -1.46575 q^{55} -15.8026 q^{57} -9.07107 q^{59} +11.2143 q^{61} +2.22129 q^{65} +6.41240 q^{67} +7.24445 q^{69} +10.5316 q^{71} +10.5217 q^{73} -2.87996 q^{75} -10.1911 q^{79} +3.14576 q^{81} -16.4186 q^{83} +7.26580 q^{85} +20.5128 q^{87} +9.83024 q^{89} -18.7643 q^{93} +5.48709 q^{95} -2.09423 q^{97} -7.75992 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{3} + 4 q^{5} + 6 q^{9}+O(q^{10}) \) 4 * q - 2 * q^3 + 4 * q^5 + 6 * q^9	\( 4 q - 2 q^{3} + 4 q^{5} + 6 q^{9} - 2 q^{11} + 10 q^{13} - 2 q^{15} + 6 q^{17} + 4 q^{23} + 4 q^{25} - 14 q^{27} - 2 q^{29} + 12 q^{31} + 18 q^{33} - 14 q^{39} + 12 q^{41} + 8 q^{43} + 6 q^{45} + 2 q^{47} - 2 q^{51} - 4 q^{53} - 2 q^{55} - 8 q^{57} - 8 q^{59} + 20 q^{61} + 10 q^{65} + 8 q^{67} + 24 q^{69} - 4 q^{71} + 16 q^{73} - 2 q^{75} - 22 q^{79} - 20 q^{81} - 36 q^{83} + 6 q^{85} + 18 q^{87} + 40 q^{89} - 32 q^{93} + 26 q^{97} - 12 q^{99}+O(q^{100}) \) 4 * q - 2 * q^3 + 4 * q^5 + 6 * q^9 - 2 * q^11 + 10 * q^13 - 2 * q^15 + 6 * q^17 + 4 * q^23 + 4 * q^25 - 14 * q^27 - 2 * q^29 + 12 * q^31 + 18 * q^33 - 14 * q^39 + 12 * q^41 + 8 * q^43 + 6 * q^45 + 2 * q^47 - 2 * q^51 - 4 * q^53 - 2 * q^55 - 8 * q^57 - 8 * q^59 + 20 * q^61 + 10 * q^65 + 8 * q^67 + 24 * q^69 - 4 * q^71 + 16 * q^73 - 2 * q^75 - 22 * q^79 - 20 * q^81 - 36 * q^83 + 6 * q^85 + 18 * q^87 + 40 * q^89 - 32 * q^93 + 26 * q^97 - 12 * q^99

Introduction

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