LMFDB - Embedded newform 4020.2.a.i.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4020,2,Mod(1,4020)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4020, 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("4020.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4020 = 2^{2} \cdot 3 \cdot 5 \cdot 67 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4020.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:	$32.0998616126$
Analytic rank:	$0$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 21x^{5} - 12x^{4} + 93x^{3} + 18x^{2} - 120x + 48 $ x^7 - 21x^5 - 12x^4 + 93x^3 + 18x^2 - 120*x + 48
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$0.575750$ of defining polynomial
Character	$\chi$	$=$	4020.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +1.00000 q^{5} +1.48734 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +1.00000 q^{5} +1.48734 q^{7} +1.00000 q^{9} +1.67621 q^{11} +1.06167 q^{13} +1.00000 q^{15} -4.17087 q^{17} -1.78840 q^{19} +1.48734 q^{21} -0.687937 q^{23} +1.00000 q^{25} +1.00000 q^{27} +4.17528 q^{29} -0.0242963 q^{31} +1.67621 q^{33} +1.48734 q^{35} +3.83944 q^{37} +1.06167 q^{39} +10.0122 q^{41} +7.41026 q^{43} +1.00000 q^{45} +3.36990 q^{47} -4.78782 q^{49} -4.17087 q^{51} +9.22139 q^{53} +1.67621 q^{55} -1.78840 q^{57} +8.62084 q^{59} -7.04457 q^{61} +1.48734 q^{63} +1.06167 q^{65} -1.00000 q^{67} -0.687937 q^{69} +0.809704 q^{71} +5.89461 q^{73} +1.00000 q^{75} +2.49310 q^{77} -1.93005 q^{79} +1.00000 q^{81} +4.31505 q^{83} -4.17087 q^{85} +4.17528 q^{87} -4.77336 q^{89} +1.57906 q^{91} -0.0242963 q^{93} -1.78840 q^{95} +10.4092 q^{97} +1.67621 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + 7 q^{3} + 7 q^{5} + q^{7} + 7 q^{9}+O(q^{10}) $ 7 * q + 7 * q^3 + 7 * q^5 + q^7 + 7 * q^9	$ 7 q + 7 q^{3} + 7 q^{5} + q^{7} + 7 q^{9} + 5 q^{11} + 9 q^{13} + 7 q^{15} + 11 q^{17} + 2 q^{19} + q^{21} + 7 q^{23} + 7 q^{25} + 7 q^{27} + 8 q^{29} + 9 q^{31} + 5 q^{33} + q^{35} + 7 q^{37} + 9 q^{39} + 9 q^{41} + 5 q^{43} + 7 q^{45} + 2 q^{47} + 12 q^{49} + 11 q^{51} + 15 q^{53} + 5 q^{55} + 2 q^{57} + 11 q^{61} + q^{63} + 9 q^{65} - 7 q^{67} + 7 q^{69} + 18 q^{71} + 21 q^{73} + 7 q^{75} + 5 q^{77} + 5 q^{79} + 7 q^{81} + 6 q^{83} + 11 q^{85} + 8 q^{87} + 7 q^{89} - 9 q^{91} + 9 q^{93} + 2 q^{95} - 9 q^{97} + 5 q^{99}+O(q^{100}) $ 7 * q + 7 * q^3 + 7 * q^5 + q^7 + 7 * q^9 + 5 * q^11 + 9 * q^13 + 7 * q^15 + 11 * q^17 + 2 * q^19 + q^21 + 7 * q^23 + 7 * q^25 + 7 * q^27 + 8 * q^29 + 9 * q^31 + 5 * q^33 + q^35 + 7 * q^37 + 9 * q^39 + 9 * q^41 + 5 * q^43 + 7 * q^45 + 2 * q^47 + 12 * q^49 + 11 * q^51 + 15 * q^53 + 5 * q^55 + 2 * q^57 + 11 * q^61 + q^63 + 9 * q^65 - 7 * q^67 + 7 * q^69 + 18 * q^71 + 21 * q^73 + 7 * q^75 + 5 * q^77 + 5 * q^79 + 7 * q^81 + 6 * q^83 + 11 * q^85 + 8 * q^87 + 7 * q^89 - 9 * q^91 + 9 * q^93 + 2 * q^95 - 9 * q^97 + 5 * 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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	1.48734	0.562162	0.281081	−	0.959684i	$-0.409307\pi$
$7$	1.48734	0.562162	0.281081	+	0.959684i	$0.409307\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	1.67621	0.505397	0.252698	−	0.967545i	$-0.418682\pi$
$11$	1.67621	0.505397	0.252698	+	0.967545i	$0.418682\pi$
$12$	0	0
$13$	1.06167	0.294453	0.147226	−	0.989103i	$-0.452965\pi$
$13$	1.06167	0.294453	0.147226	+	0.989103i	$0.452965\pi$
$14$	0	0
$15$	1.00000	0.258199
$16$	0	0
$17$	−4.17087	−1.01158	−0.505792	−	0.862655i	$-0.668800\pi$
$17$	−4.17087	−1.01158	−0.505792	+	0.862655i	$0.668800\pi$
$18$	0	0
$19$	−1.78840	−0.410286	−0.205143	−	0.978732i	$-0.565766\pi$
$19$	−1.78840	−0.410286	−0.205143	+	0.978732i	$0.565766\pi$
$20$	0	0
$21$	1.48734	0.324564
$22$	0	0
$23$	−0.687937	−0.143445	−0.0717224	−	0.997425i	$-0.522850\pi$
$23$	−0.687937	−0.143445	−0.0717224	+	0.997425i	$0.522850\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	4.17528	0.775330	0.387665	−	0.921800i	$-0.373282\pi$
$29$	4.17528	0.775330	0.387665	+	0.921800i	$0.373282\pi$
$30$	0	0
$31$	−0.0242963	−0.00436374	−0.00218187	−	0.999998i	$-0.500695\pi$
$31$	−0.0242963	−0.00436374	−0.00218187	+	0.999998i	$0.500695\pi$
$32$	0	0
$33$	1.67621	0.291791
$34$	0	0
$35$	1.48734	0.251407
$36$	0	0
$37$	3.83944	0.631200	0.315600	−	0.948892i	$-0.397794\pi$
$37$	3.83944	0.631200	0.315600	+	0.948892i	$0.397794\pi$
$38$	0	0
$39$	1.06167	0.170002
$40$	0	0
$41$	10.0122	1.56364	0.781821	−	0.623503i	$-0.214291\pi$
$41$	10.0122	1.56364	0.781821	+	0.623503i	$0.214291\pi$
$42$	0	0
$43$	7.41026	1.13005	0.565027	−	0.825072i	$-0.308866\pi$
$43$	7.41026	1.13005	0.565027	+	0.825072i	$0.308866\pi$
$44$	0	0
$45$	1.00000	0.149071
$46$	0	0
$47$	3.36990	0.491551	0.245775	−	0.969327i	$-0.420957\pi$
$47$	3.36990	0.491551	0.245775	+	0.969327i	$0.420957\pi$
$48$	0	0
$49$	−4.78782	−0.683974
$50$	0	0
$51$	−4.17087	−0.584039
$52$	0	0
$53$	9.22139	1.26666	0.633328	−	0.773884i	$-0.281689\pi$
$53$	9.22139	1.26666	0.633328	+	0.773884i	$0.281689\pi$
$54$	0	0
$55$	1.67621	0.226020
$56$	0	0
$57$	−1.78840	−0.236879
$58$	0	0
$59$	8.62084	1.12234	0.561169	−	0.827701i	$-0.310352\pi$
$59$	8.62084	1.12234	0.561169	+	0.827701i	$0.310352\pi$
$60$	0	0
$61$	−7.04457	−0.901965	−0.450983	−	0.892533i	$-0.648926\pi$
$61$	−7.04457	−0.901965	−0.450983	+	0.892533i	$0.648926\pi$
$62$	0	0
$63$	1.48734	0.187387
$64$	0	0
$65$	1.06167	0.131683
$66$	0	0
$67$	−1.00000	−0.122169
$67$	−1.00000	−0.122169
$68$	0	0
$69$	−0.687937	−0.0828179
$70$	0	0
$71$	0.809704	0.0960942	0.0480471	−	0.998845i	$-0.484700\pi$
$71$	0.809704	0.0960942	0.0480471	+	0.998845i	$0.484700\pi$
$72$	0	0
$73$	5.89461	0.689912	0.344956	−	0.938619i	$-0.387894\pi$
$73$	5.89461	0.689912	0.344956	+	0.938619i	$0.387894\pi$
$74$	0	0
$75$	1.00000	0.115470
$76$	0	0
$77$	2.49310	0.284115
$78$	0	0
$79$	−1.93005	−0.217148	−0.108574	−	0.994088i	$-0.534628\pi$
$79$	−1.93005	−0.217148	−0.108574	+	0.994088i	$0.534628\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	4.31505	0.473639	0.236819	−	0.971554i	$-0.423895\pi$
$83$	4.31505	0.473639	0.236819	+	0.971554i	$0.423895\pi$
$84$	0	0
$85$	−4.17087	−0.452394
$86$	0	0
$87$	4.17528	0.447637
$88$	0	0
$89$	−4.77336	−0.505976	−0.252988	−	0.967469i	$-0.581413\pi$
$89$	−4.77336	−0.505976	−0.252988	+	0.967469i	$0.581413\pi$
$90$	0	0
$91$	1.57906	0.165530
$92$	0	0
$93$	−0.0242963	−0.00251940
$94$	0	0
$95$	−1.78840	−0.183486
$96$	0	0
$97$	10.4092	1.05689	0.528445	−	0.848967i	$-0.322775\pi$
$97$	10.4092	1.05689	0.528445	+	0.848967i	$0.322775\pi$
$98$	0	0
$99$	1.67621	0.168466
$100$	0	0
$101$	−15.5607	−1.54835	−0.774173	−	0.632974i	$-0.781834\pi$
$101$	−15.5607	−1.54835	−0.774173	+	0.632974i	$0.781834\pi$
$102$	0	0
$103$	−6.08194	−0.599272	−0.299636	−	0.954054i	$-0.596865\pi$
$103$	−6.08194	−0.599272	−0.299636	+	0.954054i	$0.596865\pi$
$104$	0	0
$105$	1.48734	0.145150
$106$	0	0
$107$	−14.9446	−1.44475	−0.722373	−	0.691503i	$-0.756949\pi$
$107$	−14.9446	−1.44475	−0.722373	+	0.691503i	$0.756949\pi$
$108$	0	0
$109$	4.57322	0.438035	0.219018	−	0.975721i	$-0.429715\pi$
$109$	4.57322	0.438035	0.219018	+	0.975721i	$0.429715\pi$
$110$	0	0
$111$	3.83944	0.364423
$112$	0	0
$113$	−13.7902	−1.29728	−0.648638	−	0.761097i	$-0.724661\pi$
$113$	−13.7902	−1.29728	−0.648638	+	0.761097i	$0.724661\pi$
$114$	0	0
$115$	−0.687937	−0.0641504
$116$	0	0
$117$	1.06167	0.0981510
$118$	0	0
$119$	−6.20351	−0.568675
$120$	0	0
$121$	−8.19032	−0.744574
$122$	0	0
$123$	10.0122	0.902769
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−6.36170	−0.564510	−0.282255	−	0.959339i	$-0.591082\pi$
$127$	−6.36170	−0.564510	−0.282255	+	0.959339i	$0.591082\pi$
$128$	0	0
$129$	7.41026	0.652437
$130$	0	0
$131$	21.6545	1.89197	0.945983	−	0.324216i	$-0.105100\pi$
$131$	21.6545	1.89197	0.945983	+	0.324216i	$0.105100\pi$
$132$	0	0
$133$	−2.65996	−0.230647
$134$	0	0
$135$	1.00000	0.0860663
$136$	0	0
$137$	10.3048	0.880401	0.440200	−	0.897900i	$-0.354907\pi$
$137$	10.3048	0.880401	0.440200	+	0.897900i	$0.354907\pi$
$138$	0	0
$139$	0.209219	0.0177457	0.00887285	−	0.999961i	$-0.497176\pi$
$139$	0.209219	0.0177457	0.00887285	+	0.999961i	$0.497176\pi$
$140$	0	0
$141$	3.36990	0.283797
$142$	0	0
$143$	1.77957	0.148815
$144$	0	0
$145$	4.17528	0.346738
$146$	0	0
$147$	−4.78782	−0.394892
$148$	0	0
$149$	9.70484	0.795052	0.397526	−	0.917591i	$-0.369869\pi$
$149$	9.70484	0.795052	0.397526	+	0.917591i	$0.369869\pi$
$150$	0	0
$151$	12.7873	1.04062	0.520309	−	0.853978i	$-0.325817\pi$
$151$	12.7873	1.04062	0.520309	+	0.853978i	$0.325817\pi$
$152$	0	0
$153$	−4.17087	−0.337195
$154$	0	0
$155$	−0.0242963	−0.00195152
$156$	0	0
$157$	4.60161	0.367248	0.183624	−	0.982997i	$-0.441217\pi$
$157$	4.60161	0.367248	0.183624	+	0.982997i	$0.441217\pi$
$158$	0	0
$159$	9.22139	0.731304
$160$	0	0
$161$	−1.02320	−0.0806392
$162$	0	0
$163$	7.04620	0.551901	0.275951	−	0.961172i	$-0.411007\pi$
$163$	7.04620	0.551901	0.275951	+	0.961172i	$0.411007\pi$
$164$	0	0
$165$	1.67621	0.130493
$166$	0	0
$167$	−23.5766	−1.82441	−0.912207	−	0.409730i	$-0.865623\pi$
$167$	−23.5766	−1.82441	−0.912207	+	0.409730i	$0.865623\pi$
$168$	0	0
$169$	−11.8729	−0.913297
$170$	0	0
$171$	−1.78840	−0.136762
$172$	0	0
$173$	9.69181	0.736855	0.368427	−	0.929657i	$-0.379896\pi$
$173$	9.69181	0.736855	0.368427	+	0.929657i	$0.379896\pi$
$174$	0	0
$175$	1.48734	0.112432
$176$	0	0
$177$	8.62084	0.647982
$178$	0	0
$179$	5.30677	0.396647	0.198323	−	0.980137i	$-0.436450\pi$
$179$	5.30677	0.396647	0.198323	+	0.980137i	$0.436450\pi$
$180$	0	0
$181$	−5.14996	−0.382794	−0.191397	−	0.981513i	$-0.561302\pi$
$181$	−5.14996	−0.382794	−0.191397	+	0.981513i	$0.561302\pi$
$182$	0	0
$183$	−7.04457	−0.520750
$184$	0	0
$185$	3.83944	0.282281
$186$	0	0
$187$	−6.99126	−0.511251
$188$	0	0
$189$	1.48734	0.108188
$190$	0	0
$191$	24.1432	1.74694	0.873472	−	0.486875i	$-0.161863\pi$
$191$	24.1432	1.74694	0.873472	+	0.486875i	$0.161863\pi$
$192$	0	0
$193$	7.71657	0.555451	0.277725	−	0.960661i	$-0.410419\pi$
$193$	7.71657	0.555451	0.277725	+	0.960661i	$0.410419\pi$
$194$	0	0
$195$	1.06167	0.0760274
$196$	0	0
$197$	1.37321	0.0978371	0.0489186	−	0.998803i	$-0.484423\pi$
$197$	1.37321	0.0978371	0.0489186	+	0.998803i	$0.484423\pi$
$198$	0	0
$199$	−14.4107	−1.02155	−0.510775	−	0.859715i	$-0.670641\pi$
$199$	−14.4107	−1.02155	−0.510775	+	0.859715i	$0.670641\pi$
$200$	0	0
$201$	−1.00000	−0.0705346
$202$	0	0
$203$	6.21006	0.435861
$204$	0	0
$205$	10.0122	0.699282
$206$	0	0
$207$	−0.687937	−0.0478149
$208$	0	0
$209$	−2.99773	−0.207357
$210$	0	0
$211$	−19.0617	−1.31226	−0.656132	−	0.754646i	$-0.727809\pi$
$211$	−19.0617	−1.31226	−0.656132	+	0.754646i	$0.727809\pi$
$212$	0	0
$213$	0.809704	0.0554800
$214$	0	0
$215$	7.41026	0.505376
$216$	0	0
$217$	−0.0361368	−0.00245313
$218$	0	0
$219$	5.89461	0.398321
$220$	0	0
$221$	−4.42807	−0.297864
$222$	0	0
$223$	−11.3441	−0.759660	−0.379830	−	0.925056i	$-0.624017\pi$
$223$	−11.3441	−0.759660	−0.379830	+	0.925056i	$0.624017\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	22.3381	1.48263	0.741315	−	0.671157i	$-0.234202\pi$
$227$	22.3381	1.48263	0.741315	+	0.671157i	$0.234202\pi$
$228$	0	0
$229$	3.92047	0.259072	0.129536	−	0.991575i	$-0.458651\pi$
$229$	3.92047	0.259072	0.129536	+	0.991575i	$0.458651\pi$
$230$	0	0
$231$	2.49310	0.164034
$232$	0	0
$233$	24.7839	1.62365	0.811825	−	0.583901i	$-0.198474\pi$
$233$	24.7839	1.62365	0.811825	+	0.583901i	$0.198474\pi$
$234$	0	0
$235$	3.36990	0.219828
$236$	0	0
$237$	−1.93005	−0.125370
$238$	0	0
$239$	2.54658	0.164725	0.0823624	−	0.996602i	$-0.473753\pi$
$239$	2.54658	0.164725	0.0823624	+	0.996602i	$0.473753\pi$
$240$	0	0
$241$	23.5277	1.51555	0.757777	−	0.652514i	$-0.226286\pi$
$241$	23.5277	1.51555	0.757777	+	0.652514i	$0.226286\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−4.78782	−0.305882
$246$	0	0
$247$	−1.89868	−0.120810
$248$	0	0
$249$	4.31505	0.273455
$250$	0	0
$251$	−24.6505	−1.55593	−0.777964	−	0.628309i	$-0.783747\pi$
$251$	−24.6505	−1.55593	−0.777964	+	0.628309i	$0.783747\pi$
$252$	0	0
$253$	−1.15313	−0.0724965
$254$	0	0
$255$	−4.17087	−0.261190
$256$	0	0
$257$	−23.9468	−1.49376	−0.746880	−	0.664959i	$-0.768449\pi$
$257$	−23.9468	−1.49376	−0.746880	+	0.664959i	$0.768449\pi$
$258$	0	0
$259$	5.71055	0.354836
$260$	0	0
$261$	4.17528	0.258443
$262$	0	0
$263$	−13.7064	−0.845170	−0.422585	−	0.906323i	$-0.638877\pi$
$263$	−13.7064	−0.845170	−0.422585	+	0.906323i	$0.638877\pi$
$264$	0	0
$265$	9.22139	0.566466
$266$	0	0
$267$	−4.77336	−0.292125
$268$	0	0
$269$	14.8587	0.905953	0.452977	−	0.891522i	$-0.350362\pi$
$269$	14.8587	0.905953	0.452977	+	0.891522i	$0.350362\pi$
$270$	0	0
$271$	−13.8834	−0.843357	−0.421679	−	0.906745i	$-0.638559\pi$
$271$	−13.8834	−0.843357	−0.421679	+	0.906745i	$0.638559\pi$
$272$	0	0
$273$	1.57906	0.0955690
$274$	0	0
$275$	1.67621	0.101079
$276$	0	0
$277$	−0.199049	−0.0119597	−0.00597984	−	0.999982i	$-0.501903\pi$
$277$	−0.199049	−0.0119597	−0.00597984	+	0.999982i	$0.501903\pi$
$278$	0	0
$279$	−0.0242963	−0.00145458
$280$	0	0
$281$	−6.81087	−0.406302	−0.203151	−	0.979147i	$-0.565118\pi$
$281$	−6.81087	−0.406302	−0.203151	+	0.979147i	$0.565118\pi$
$282$	0	0
$283$	−6.10862	−0.363120	−0.181560	−	0.983380i	$-0.558115\pi$
$283$	−6.10862	−0.363120	−0.181560	+	0.983380i	$0.558115\pi$
$284$	0	0
$285$	−1.78840	−0.105936
$286$	0	0
$287$	14.8915	0.879020
$288$	0	0
$289$	0.396159	0.0233035
$290$	0	0
$291$	10.4092	0.610196
$292$	0	0
$293$	13.1785	0.769896	0.384948	−	0.922938i	$-0.374219\pi$
$293$	13.1785	0.769896	0.384948	+	0.922938i	$0.374219\pi$
$294$	0	0
$295$	8.62084	0.501925
$296$	0	0
$297$	1.67621	0.0972636
$298$	0	0
$299$	−0.730359	−0.0422377
$300$	0	0
$301$	11.0216	0.635274
$302$	0	0
$303$	−15.5607	−0.893938
$304$	0	0
$305$	−7.04457	−0.403371
$306$	0	0
$307$	−4.27714	−0.244109	−0.122055	−	0.992523i	$-0.538948\pi$
$307$	−4.27714	−0.244109	−0.122055	+	0.992523i	$0.538948\pi$
$308$	0	0
$309$	−6.08194	−0.345990
$310$	0	0
$311$	8.83160	0.500794	0.250397	−	0.968143i	$-0.419439\pi$
$311$	8.83160	0.500794	0.250397	+	0.968143i	$0.419439\pi$
$312$	0	0
$313$	19.4705	1.10054	0.550268	−	0.834988i	$-0.314526\pi$
$313$	19.4705	1.10054	0.550268	+	0.834988i	$0.314526\pi$
$314$	0	0
$315$	1.48734	0.0838022
$316$	0	0
$317$	3.20488	0.180004	0.0900021	−	0.995942i	$-0.471313\pi$
$317$	3.20488	0.180004	0.0900021	+	0.995942i	$0.471313\pi$
$318$	0	0
$319$	6.99865	0.391849
$320$	0	0
$321$	−14.9446	−0.834125
$322$	0	0
$323$	7.45917	0.415039
$324$	0	0
$325$	1.06167	0.0588906
$326$	0	0
$327$	4.57322	0.252900
$328$	0	0
$329$	5.01220	0.276331
$330$	0	0
$331$	−28.1447	−1.54697	−0.773487	−	0.633813i	$-0.781489\pi$
$331$	−28.1447	−1.54697	−0.773487	+	0.633813i	$0.781489\pi$
$332$	0	0
$333$	3.83944	0.210400
$334$	0	0
$335$	−1.00000	−0.0546358
$336$	0	0
$337$	−27.4643	−1.49608	−0.748038	−	0.663656i	$-0.769004\pi$
$337$	−27.4643	−1.49608	−0.748038	+	0.663656i	$0.769004\pi$
$338$	0	0
$339$	−13.7902	−0.748983
$340$	0	0
$341$	−0.0407256	−0.00220542
$342$	0	0
$343$	−17.5325	−0.946666
$344$	0	0
$345$	−0.687937	−0.0370373
$346$	0	0
$347$	−7.07216	−0.379653	−0.189827	−	0.981818i	$-0.560793\pi$
$347$	−7.07216	−0.379653	−0.189827	+	0.981818i	$0.560793\pi$
$348$	0	0
$349$	12.0715	0.646173	0.323087	−	0.946369i	$-0.395279\pi$
$349$	12.0715	0.646173	0.323087	+	0.946369i	$0.395279\pi$
$350$	0	0
$351$	1.06167	0.0566675
$352$	0	0
$353$	24.4356	1.30058	0.650289	−	0.759687i	$-0.274648\pi$
$353$	24.4356	1.30058	0.650289	+	0.759687i	$0.274648\pi$
$354$	0	0
$355$	0.809704	0.0429747
$356$	0	0
$357$	−6.20351	−0.328324
$358$	0	0
$359$	4.48487	0.236703	0.118351	−	0.992972i	$-0.462239\pi$
$359$	4.48487	0.236703	0.118351	+	0.992972i	$0.462239\pi$
$360$	0	0
$361$	−15.8016	−0.831665
$362$	0	0
$363$	−8.19032	−0.429880
$364$	0	0
$365$	5.89461	0.308538
$366$	0	0
$367$	−3.46057	−0.180641	−0.0903203	−	0.995913i	$-0.528789\pi$
$367$	−3.46057	−0.180641	−0.0903203	+	0.995913i	$0.528789\pi$
$368$	0	0
$369$	10.0122	0.521214
$370$	0	0
$371$	13.7154	0.712066
$372$	0	0
$373$	−31.9822	−1.65598	−0.827988	−	0.560746i	$-0.810514\pi$
$373$	−31.9822	−1.65598	−0.827988	+	0.560746i	$0.810514\pi$
$374$	0	0
$375$	1.00000	0.0516398
$376$	0	0
$377$	4.43275	0.228298
$378$	0	0
$379$	−11.5280	−0.592152	−0.296076	−	0.955164i	$-0.595678\pi$
$379$	−11.5280	−0.592152	−0.296076	+	0.955164i	$0.595678\pi$
$380$	0	0
$381$	−6.36170	−0.325920
$382$	0	0
$383$	14.2657	0.728944	0.364472	−	0.931214i	$-0.381249\pi$
$383$	14.2657	0.728944	0.364472	+	0.931214i	$0.381249\pi$
$384$	0	0
$385$	2.49310	0.127060
$386$	0	0
$387$	7.41026	0.376685
$388$	0	0
$389$	−17.4064	−0.882538	−0.441269	−	0.897375i	$-0.645472\pi$
$389$	−17.4064	−0.882538	−0.441269	+	0.897375i	$0.645472\pi$
$390$	0	0
$391$	2.86930	0.145107
$392$	0	0
$393$	21.6545	1.09233
$394$	0	0
$395$	−1.93005	−0.0971115
$396$	0	0
$397$	4.89389	0.245617	0.122809	−	0.992430i	$-0.460810\pi$
$397$	4.89389	0.245617	0.122809	+	0.992430i	$0.460810\pi$
$398$	0	0
$399$	−2.65996	−0.133164
$400$	0	0
$401$	−7.24642	−0.361869	−0.180935	−	0.983495i	$-0.557912\pi$
$401$	−7.24642	−0.361869	−0.180935	+	0.983495i	$0.557912\pi$
$402$	0	0
$403$	−0.0257945	−0.00128491
$404$	0	0
$405$	1.00000	0.0496904
$406$	0	0
$407$	6.43571	0.319006
$408$	0	0
$409$	−20.5403	−1.01565	−0.507825	−	0.861460i	$-0.669550\pi$
$409$	−20.5403	−1.01565	−0.507825	+	0.861460i	$0.669550\pi$
$410$	0	0
$411$	10.3048	0.508300
$412$	0	0
$413$	12.8221	0.630936
$414$	0	0
$415$	4.31505	0.211818
$416$	0	0
$417$	0.209219	0.0102455
$418$	0	0
$419$	−21.7992	−1.06496	−0.532480	−	0.846443i	$-0.678740\pi$
$419$	−21.7992	−1.06496	−0.532480	+	0.846443i	$0.678740\pi$
$420$	0	0
$421$	−5.90750	−0.287914	−0.143957	−	0.989584i	$-0.545983\pi$
$421$	−5.90750	−0.287914	−0.143957	+	0.989584i	$0.545983\pi$
$422$	0	0
$423$	3.36990	0.163850
$424$	0	0
$425$	−4.17087	−0.202317
$426$	0	0
$427$	−10.4777	−0.507051
$428$	0	0
$429$	1.77957	0.0859187
$430$	0	0
$431$	−31.9226	−1.53766	−0.768830	−	0.639453i	$-0.779161\pi$
$431$	−31.9226	−1.53766	−0.768830	+	0.639453i	$0.779161\pi$
$432$	0	0
$433$	0.892546	0.0428930	0.0214465	−	0.999770i	$-0.493173\pi$
$433$	0.892546	0.0428930	0.0214465	+	0.999770i	$0.493173\pi$
$434$	0	0
$435$	4.17528	0.200189
$436$	0	0
$437$	1.23030	0.0588534
$438$	0	0
$439$	22.1087	1.05519	0.527594	−	0.849497i	$-0.323094\pi$
$439$	22.1087	1.05519	0.527594	+	0.849497i	$0.323094\pi$
$440$	0	0
$441$	−4.78782	−0.227991
$442$	0	0
$443$	−4.79585	−0.227858	−0.113929	−	0.993489i	$-0.536344\pi$
$443$	−4.79585	−0.227858	−0.113929	+	0.993489i	$0.536344\pi$
$444$	0	0
$445$	−4.77336	−0.226279
$446$	0	0
$447$	9.70484	0.459023
$448$	0	0
$449$	6.42594	0.303259	0.151629	−	0.988437i	$-0.451548\pi$
$449$	6.42594	0.303259	0.151629	+	0.988437i	$0.451548\pi$
$450$	0	0
$451$	16.7825	0.790259
$452$	0	0
$453$	12.7873	0.600801
$454$	0	0
$455$	1.57906	0.0740274
$456$	0	0
$457$	7.02742	0.328729	0.164364	−	0.986400i	$-0.447443\pi$
$457$	7.02742	0.328729	0.164364	+	0.986400i	$0.447443\pi$
$458$	0	0
$459$	−4.17087	−0.194680
$460$	0	0
$461$	26.8303	1.24961	0.624806	−	0.780780i	$-0.285178\pi$
$461$	26.8303	1.24961	0.624806	+	0.780780i	$0.285178\pi$
$462$	0	0
$463$	20.8477	0.968875	0.484438	−	0.874826i	$-0.339024\pi$
$463$	20.8477	0.968875	0.484438	+	0.874826i	$0.339024\pi$
$464$	0	0
$465$	−0.0242963	−0.00112671
$466$	0	0
$467$	−16.3535	−0.756749	−0.378375	−	0.925653i	$-0.623517\pi$
$467$	−16.3535	−0.756749	−0.378375	+	0.925653i	$0.623517\pi$
$468$	0	0
$469$	−1.48734	−0.0686790
$470$	0	0
$471$	4.60161	0.212031
$472$	0	0
$473$	12.4212	0.571125
$474$	0	0
$475$	−1.78840	−0.0820573
$476$	0	0
$477$	9.22139	0.422219
$478$	0	0
$479$	4.91185	0.224428	0.112214	−	0.993684i	$-0.464206\pi$
$479$	4.91185	0.224428	0.112214	+	0.993684i	$0.464206\pi$
$480$	0	0
$481$	4.07620	0.185859
$482$	0	0
$483$	−1.02320	−0.0465571
$484$	0	0
$485$	10.4092	0.472656
$486$	0	0
$487$	−7.70973	−0.349361	−0.174681	−	0.984625i	$-0.555889\pi$
$487$	−7.70973	−0.349361	−0.174681	+	0.984625i	$0.555889\pi$
$488$	0	0
$489$	7.04620	0.318640
$490$	0	0
$491$	4.26466	0.192462	0.0962308	−	0.995359i	$-0.469321\pi$
$491$	4.26466	0.192462	0.0962308	+	0.995359i	$0.469321\pi$
$492$	0	0
$493$	−17.4145	−0.784312
$494$	0	0
$495$	1.67621	0.0753401
$496$	0	0
$497$	1.20431	0.0540205
$498$	0	0
$499$	−29.9173	−1.33928	−0.669642	−	0.742684i	$-0.733552\pi$
$499$	−29.9173	−1.33928	−0.669642	+	0.742684i	$0.733552\pi$
$500$	0	0
$501$	−23.5766	−1.05333
$502$	0	0
$503$	1.83184	0.0816777	0.0408389	−	0.999166i	$-0.486997\pi$
$503$	1.83184	0.0816777	0.0408389	+	0.999166i	$0.486997\pi$
$504$	0	0
$505$	−15.5607	−0.692441
$506$	0	0
$507$	−11.8729	−0.527293
$508$	0	0
$509$	11.2017	0.496507	0.248254	−	0.968695i	$-0.420143\pi$
$509$	11.2017	0.496507	0.248254	+	0.968695i	$0.420143\pi$
$510$	0	0
$511$	8.76730	0.387843
$512$	0	0
$513$	−1.78840	−0.0789597
$514$	0	0
$515$	−6.08194	−0.268002
$516$	0	0
$517$	5.64867	0.248428
$518$	0	0
$519$	9.69181	0.425423
$520$	0	0
$521$	28.3989	1.24418	0.622089	−	0.782947i	$-0.286284\pi$
$521$	28.3989	1.24418	0.622089	+	0.782947i	$0.286284\pi$
$522$	0	0
$523$	33.0621	1.44571	0.722853	−	0.691002i	$-0.242830\pi$
$523$	33.0621	1.44571	0.722853	+	0.691002i	$0.242830\pi$
$524$	0	0
$525$	1.48734	0.0649129
$526$	0	0
$527$	0.101337	0.00441429
$528$	0	0
$529$	−22.5267	−0.979424
$530$	0	0
$531$	8.62084	0.374113
$532$	0	0
$533$	10.6296	0.460419
$534$	0	0
$535$	−14.9446	−0.646110
$536$	0	0
$537$	5.30677	0.229004
$538$	0	0
$539$	−8.02539	−0.345678
$540$	0	0
$541$	−23.6867	−1.01837	−0.509186	−	0.860656i	$-0.670054\pi$
$541$	−23.6867	−1.01837	−0.509186	+	0.860656i	$0.670054\pi$
$542$	0	0
$543$	−5.14996	−0.221006
$544$	0	0
$545$	4.57322	0.195895
$546$	0	0
$547$	−31.6133	−1.35169	−0.675844	−	0.737045i	$-0.736221\pi$
$547$	−31.6133	−1.35169	−0.675844	+	0.737045i	$0.736221\pi$
$548$	0	0
$549$	−7.04457	−0.300655
$550$	0	0
$551$	−7.46706	−0.318107
$552$	0	0
$553$	−2.87065	−0.122072
$554$	0	0
$555$	3.83944	0.162975
$556$	0	0
$557$	16.0147	0.678563	0.339282	−	0.940685i	$-0.389816\pi$
$557$	16.0147	0.678563	0.339282	+	0.940685i	$0.389816\pi$
$558$	0	0
$559$	7.86722	0.332748
$560$	0	0
$561$	−6.99126	−0.295171
$562$	0	0
$563$	20.4321	0.861110	0.430555	−	0.902564i	$-0.358318\pi$
$563$	20.4321	0.861110	0.430555	+	0.902564i	$0.358318\pi$
$564$	0	0
$565$	−13.7902	−0.580159
$566$	0	0
$567$	1.48734	0.0624625
$568$	0	0
$569$	−12.3592	−0.518124	−0.259062	−	0.965861i	$-0.583413\pi$
$569$	−12.3592	−0.518124	−0.259062	+	0.965861i	$0.583413\pi$
$570$	0	0
$571$	−2.30778	−0.0965777	−0.0482889	−	0.998833i	$-0.515377\pi$
$571$	−2.30778	−0.0965777	−0.0482889	+	0.998833i	$0.515377\pi$
$572$	0	0
$573$	24.1432	1.00860
$574$	0	0
$575$	−0.687937	−0.0286890
$576$	0	0
$577$	14.1824	0.590421	0.295210	−	0.955432i	$-0.404610\pi$
$577$	14.1824	0.590421	0.295210	+	0.955432i	$0.404610\pi$
$578$	0	0
$579$	7.71657	0.320690
$580$	0	0
$581$	6.41795	0.266262
$582$	0	0
$583$	15.4570	0.640163
$584$	0	0
$585$	1.06167	0.0438945
$586$	0	0
$587$	11.6091	0.479159	0.239580	−	0.970877i	$-0.422990\pi$
$587$	11.6091	0.479159	0.239580	+	0.970877i	$0.422990\pi$
$588$	0	0
$589$	0.0434514	0.00179038
$590$	0	0
$591$	1.37321	0.0564863
$592$	0	0
$593$	−20.6914	−0.849694	−0.424847	−	0.905265i	$-0.639672\pi$
$593$	−20.6914	−0.849694	−0.424847	+	0.905265i	$0.639672\pi$
$594$	0	0
$595$	−6.20351	−0.254319
$596$	0	0
$597$	−14.4107	−0.589792
$598$	0	0
$599$	−33.2857	−1.36002	−0.680009	−	0.733204i	$-0.738024\pi$
$599$	−33.2857	−1.36002	−0.680009	+	0.733204i	$0.738024\pi$
$600$	0	0
$601$	2.72700	0.111237	0.0556184	−	0.998452i	$-0.482287\pi$
$601$	2.72700	0.111237	0.0556184	+	0.998452i	$0.482287\pi$
$602$	0	0
$603$	−1.00000	−0.0407231
$604$	0	0
$605$	−8.19032	−0.332984
$606$	0	0
$607$	−28.0090	−1.13685	−0.568424	−	0.822735i	$-0.692447\pi$
$607$	−28.0090	−1.13685	−0.568424	+	0.822735i	$0.692447\pi$
$608$	0	0
$609$	6.21006	0.251644
$610$	0	0
$611$	3.57771	0.144739
$612$	0	0
$613$	−4.51765	−0.182466	−0.0912331	−	0.995830i	$-0.529081\pi$
$613$	−4.51765	−0.182466	−0.0912331	+	0.995830i	$0.529081\pi$
$614$	0	0
$615$	10.0122	0.403730
$616$	0	0
$617$	−17.2953	−0.696284	−0.348142	−	0.937442i	$-0.613187\pi$
$617$	−17.2953	−0.696284	−0.348142	+	0.937442i	$0.613187\pi$
$618$	0	0
$619$	−41.5408	−1.66967	−0.834833	−	0.550503i	$-0.814436\pi$
$619$	−41.5408	−1.66967	−0.834833	+	0.550503i	$0.814436\pi$
$620$	0	0
$621$	−0.687937	−0.0276060
$622$	0	0
$623$	−7.09962	−0.284440
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−2.99773	−0.119718
$628$	0	0
$629$	−16.0138	−0.638512
$630$	0	0
$631$	−33.2815	−1.32492	−0.662458	−	0.749099i	$-0.730487\pi$
$631$	−33.2815	−1.32492	−0.662458	+	0.749099i	$0.730487\pi$
$632$	0	0
$633$	−19.0617	−0.757636
$634$	0	0
$635$	−6.36170	−0.252456
$636$	0	0
$637$	−5.08306	−0.201398
$638$	0	0
$639$	0.809704	0.0320314
$640$	0	0
$641$	−0.0485597	−0.00191799	−0.000958996	−	1.00000i	$-0.500305\pi$
$641$	−0.0485597	−0.00191799	−0.000958996		1.00000i	$0.500305\pi$
$642$	0	0
$643$	6.74902	0.266155	0.133078	−	0.991106i	$-0.457514\pi$
$643$	6.74902	0.266155	0.133078	+	0.991106i	$0.457514\pi$
$644$	0	0
$645$	7.41026	0.291779
$646$	0	0
$647$	−35.7123	−1.40400	−0.701998	−	0.712179i	$-0.747708\pi$
$647$	−35.7123	−1.40400	−0.701998	+	0.712179i	$0.747708\pi$
$648$	0	0
$649$	14.4503	0.567226
$650$	0	0
$651$	−0.0361368	−0.00141631
$652$	0	0
$653$	23.8780	0.934418	0.467209	−	0.884147i	$-0.345260\pi$
$653$	23.8780	0.934418	0.467209	+	0.884147i	$0.345260\pi$
$654$	0	0
$655$	21.6545	0.846113
$656$	0	0
$657$	5.89461	0.229971
$658$	0	0
$659$	−2.36635	−0.0921800	−0.0460900	−	0.998937i	$-0.514676\pi$
$659$	−2.36635	−0.0921800	−0.0460900	+	0.998937i	$0.514676\pi$
$660$	0	0
$661$	9.62740	0.374462	0.187231	−	0.982316i	$-0.440049\pi$
$661$	9.62740	0.374462	0.187231	+	0.982316i	$0.440049\pi$
$662$	0	0
$663$	−4.42807	−0.171972
$664$	0	0
$665$	−2.65996	−0.103149
$666$	0	0
$667$	−2.87233	−0.111217
$668$	0	0
$669$	−11.3441	−0.438590
$670$	0	0
$671$	−11.8082	−0.455850
$672$	0	0
$673$	18.0699	0.696544	0.348272	−	0.937393i	$-0.386768\pi$
$673$	18.0699	0.696544	0.348272	+	0.937393i	$0.386768\pi$
$674$	0	0
$675$	1.00000	0.0384900
$676$	0	0
$677$	−22.8015	−0.876331	−0.438166	−	0.898894i	$-0.644372\pi$
$677$	−22.8015	−0.876331	−0.438166	+	0.898894i	$0.644372\pi$
$678$	0	0
$679$	15.4820	0.594144
$680$	0	0
$681$	22.3381	0.855997
$682$	0	0
$683$	13.1529	0.503283	0.251642	−	0.967820i	$-0.419030\pi$
$683$	13.1529	0.503283	0.251642	+	0.967820i	$0.419030\pi$
$684$	0	0
$685$	10.3048	0.393727
$686$	0	0
$687$	3.92047	0.149575
$688$	0	0
$689$	9.79003	0.372970
$690$	0	0
$691$	−39.9482	−1.51970	−0.759851	−	0.650098i	$-0.774728\pi$
$691$	−39.9482	−1.51970	−0.759851	+	0.650098i	$0.774728\pi$
$692$	0	0
$693$	2.49310	0.0947049
$694$	0	0
$695$	0.209219	0.00793612
$696$	0	0
$697$	−41.7595	−1.58176
$698$	0	0
$699$	24.7839	0.937415
$700$	0	0
$701$	−27.5085	−1.03898	−0.519491	−	0.854476i	$-0.673878\pi$
$701$	−27.5085	−1.03898	−0.519491	+	0.854476i	$0.673878\pi$
$702$	0	0
$703$	−6.86644	−0.258973
$704$	0	0
$705$	3.36990	0.126918
$706$	0	0
$707$	−23.1440	−0.870421
$708$	0	0
$709$	39.4588	1.48191	0.740954	−	0.671556i	$-0.234374\pi$
$709$	39.4588	1.48191	0.740954	+	0.671556i	$0.234374\pi$
$710$	0	0
$711$	−1.93005	−0.0723827
$712$	0	0
$713$	0.0167143	0.000625955 0
$714$	0	0
$715$	1.77957	0.0665523
$716$	0	0
$717$	2.54658	0.0951039
$718$	0	0
$719$	10.0647	0.375351	0.187675	−	0.982231i	$-0.439905\pi$
$719$	10.0647	0.375351	0.187675	+	0.982231i	$0.439905\pi$
$720$	0	0
$721$	−9.04592	−0.336888
$722$	0	0
$723$	23.5277	0.875005
$724$	0	0
$725$	4.17528	0.155066
$726$	0	0
$727$	−38.5271	−1.42889	−0.714445	−	0.699692i	$-0.753321\pi$
$727$	−38.5271	−1.42889	−0.714445	+	0.699692i	$0.753321\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−30.9072	−1.14315
$732$	0	0
$733$	32.8432	1.21309	0.606546	−	0.795049i	$-0.292555\pi$
$733$	32.8432	1.21309	0.606546	+	0.795049i	$0.292555\pi$
$734$	0	0
$735$	−4.78782	−0.176601
$736$	0	0
$737$	−1.67621	−0.0617440
$738$	0	0
$739$	−3.53869	−0.130173	−0.0650863	−	0.997880i	$-0.520732\pi$
$739$	−3.53869	−0.130173	−0.0650863	+	0.997880i	$0.520732\pi$
$740$	0	0
$741$	−1.89868	−0.0697497
$742$	0	0
$743$	39.4392	1.44688	0.723442	−	0.690386i	$-0.242559\pi$
$743$	39.4392	1.44688	0.723442	+	0.690386i	$0.242559\pi$
$744$	0	0
$745$	9.70484	0.355558
$746$	0	0
$747$	4.31505	0.157880
$748$	0	0
$749$	−22.2277	−0.812182
$750$	0	0
$751$	22.9325	0.836819	0.418409	−	0.908259i	$-0.362588\pi$
$751$	22.9325	0.836819	0.418409	+	0.908259i	$0.362588\pi$
$752$	0	0
$753$	−24.6505	−0.898316
$754$	0	0
$755$	12.7873	0.465378
$756$	0	0
$757$	3.26931	0.118825	0.0594126	−	0.998234i	$-0.481077\pi$
$757$	3.26931	0.118825	0.0594126	+	0.998234i	$0.481077\pi$
$758$	0	0
$759$	−1.15313	−0.0418559
$760$	0	0
$761$	−12.1133	−0.439107	−0.219553	−	0.975601i	$-0.570460\pi$
$761$	−12.1133	−0.439107	−0.219553	+	0.975601i	$0.570460\pi$
$762$	0	0
$763$	6.80194	0.246247
$764$	0	0
$765$	−4.17087	−0.150798
$766$	0	0
$767$	9.15245	0.330476
$768$	0	0
$769$	−20.2424	−0.729960	−0.364980	−	0.931015i	$-0.618924\pi$
$769$	−20.2424	−0.729960	−0.364980	+	0.931015i	$0.618924\pi$
$770$	0	0
$771$	−23.9468	−0.862422
$772$	0	0
$773$	44.3716	1.59594	0.797968	−	0.602700i	$-0.205908\pi$
$773$	44.3716	1.59594	0.797968	+	0.602700i	$0.205908\pi$
$774$	0	0
$775$	−0.0242963	−0.000872747 0
$776$	0	0
$777$	5.71055	0.204865
$778$	0	0
$779$	−17.9058	−0.641541
$780$	0	0
$781$	1.35724	0.0485657
$782$	0	0
$783$	4.17528	0.149212
$784$	0	0
$785$	4.60161	0.164238
$786$	0	0
$787$	−28.5317	−1.01705	−0.508523	−	0.861048i	$-0.669808\pi$
$787$	−28.5317	−1.01705	−0.508523	+	0.861048i	$0.669808\pi$
$788$	0	0
$789$	−13.7064	−0.487959
$790$	0	0
$791$	−20.5108	−0.729279
$792$	0	0
$793$	−7.47898	−0.265586
$794$	0	0
$795$	9.22139	0.327049
$796$	0	0
$797$	40.4584	1.43311	0.716555	−	0.697530i	$-0.245718\pi$
$797$	40.4584	1.43311	0.716555	+	0.697530i	$0.245718\pi$
$798$	0	0
$799$	−14.0554	−0.497245
$800$	0	0
$801$	−4.77336	−0.168659
$802$	0	0
$803$	9.88061	0.348679
$804$	0	0
$805$	−1.02320	−0.0360629
$806$	0	0
$807$	14.8587	0.523052
$808$	0	0
$809$	−7.26876	−0.255556	−0.127778	−	0.991803i	$-0.540785\pi$
$809$	−7.26876	−0.255556	−0.127778	+	0.991803i	$0.540785\pi$
$810$	0	0
$811$	−35.2858	−1.23905	−0.619525	−	0.784977i	$-0.712675\pi$
$811$	−35.2858	−1.23905	−0.619525	+	0.784977i	$0.712675\pi$
$812$	0	0
$813$	−13.8834	−0.486913
$814$	0	0
$815$	7.04620	0.246818
$816$	0	0
$817$	−13.2525	−0.463646
$818$	0	0
$819$	1.57906	0.0551768
$820$	0	0
$821$	28.6033	0.998263	0.499132	−	0.866526i	$-0.333652\pi$
$821$	28.6033	0.998263	0.499132	+	0.866526i	$0.333652\pi$
$822$	0	0
$823$	−9.98246	−0.347967	−0.173983	−	0.984749i	$-0.555664\pi$
$823$	−9.98246	−0.347967	−0.173983	+	0.984749i	$0.555664\pi$
$824$	0	0
$825$	1.67621	0.0583582
$826$	0	0
$827$	−24.2743	−0.844101	−0.422051	−	0.906572i	$-0.638690\pi$
$827$	−24.2743	−0.844101	−0.422051	+	0.906572i	$0.638690\pi$
$828$	0	0
$829$	−35.8762	−1.24603	−0.623017	−	0.782209i	$-0.714093\pi$
$829$	−35.8762	−1.24603	−0.623017	+	0.782209i	$0.714093\pi$
$830$	0	0
$831$	−0.199049	−0.00690493
$832$	0	0
$833$	19.9694	0.691897
$834$	0	0
$835$	−23.5766	−0.815903
$836$	0	0
$837$	−0.0242963	−0.000839801 0
$838$	0	0
$839$	13.8796	0.479178	0.239589	−	0.970874i	$-0.422987\pi$
$839$	13.8796	0.479178	0.239589	+	0.970874i	$0.422987\pi$
$840$	0	0
$841$	−11.5671	−0.398864
$842$	0	0
$843$	−6.81087	−0.234579
$844$	0	0
$845$	−11.8729	−0.408439
$846$	0	0
$847$	−12.1818	−0.418571
$848$	0	0
$849$	−6.10862	−0.209647
$850$	0	0
$851$	−2.64129	−0.0905423
$852$	0	0
$853$	3.08657	0.105682	0.0528411	−	0.998603i	$-0.483172\pi$
$853$	3.08657	0.105682	0.0528411	+	0.998603i	$0.483172\pi$
$854$	0	0
$855$	−1.78840	−0.0611619
$856$	0	0
$857$	−9.25234	−0.316054	−0.158027	−	0.987435i	$-0.550513\pi$
$857$	−9.25234	−0.316054	−0.158027	+	0.987435i	$0.550513\pi$
$858$	0	0
$859$	−21.3173	−0.727337	−0.363669	−	0.931528i	$-0.618476\pi$
$859$	−21.3173	−0.727337	−0.363669	+	0.931528i	$0.618476\pi$
$860$	0	0
$861$	14.8915	0.507502
$862$	0	0
$863$	4.89919	0.166770	0.0833852	−	0.996517i	$-0.473427\pi$
$863$	4.89919	0.166770	0.0833852	+	0.996517i	$0.473427\pi$
$864$	0	0
$865$	9.69181	0.329531
$866$	0	0
$867$	0.396159	0.0134543
$868$	0	0
$869$	−3.23518	−0.109746
$870$	0	0
$871$	−1.06167	−0.0359732
$872$	0	0
$873$	10.4092	0.352297
$874$	0	0
$875$	1.48734	0.0502813
$876$	0	0
$877$	−12.3424	−0.416774	−0.208387	−	0.978046i	$-0.566821\pi$
$877$	−12.3424	−0.416774	−0.208387	+	0.978046i	$0.566821\pi$
$878$	0	0
$879$	13.1785	0.444500
$880$	0	0
$881$	19.2882	0.649836	0.324918	−	0.945742i	$-0.394663\pi$
$881$	19.2882	0.649836	0.324918	+	0.945742i	$0.394663\pi$
$882$	0	0
$883$	−28.9980	−0.975861	−0.487931	−	0.872882i	$-0.662248\pi$
$883$	−28.9980	−0.975861	−0.487931	+	0.872882i	$0.662248\pi$
$884$	0	0
$885$	8.62084	0.289786
$886$	0	0
$887$	16.3722	0.549726	0.274863	−	0.961483i	$-0.411367\pi$
$887$	16.3722	0.549726	0.274863	+	0.961483i	$0.411367\pi$
$888$	0	0
$889$	−9.46202	−0.317346
$890$	0	0
$891$	1.67621	0.0561552
$892$	0	0
$893$	−6.02673	−0.201677
$894$	0	0
$895$	5.30677	0.177386
$896$	0	0
$897$	−0.730359	−0.0243860
$898$	0	0
$899$	−0.101444	−0.00338333
$900$	0	0
$901$	−38.4612	−1.28133
$902$	0	0
$903$	11.0216	0.366775
$904$	0	0
$905$	−5.14996	−0.171190
$906$	0	0
$907$	−8.66532	−0.287727	−0.143864	−	0.989598i	$-0.545953\pi$
$907$	−8.66532	−0.287727	−0.143864	+	0.989598i	$0.545953\pi$
$908$	0	0
$909$	−15.5607	−0.516115
$910$	0	0
$911$	−56.3567	−1.86718	−0.933590	−	0.358344i	$-0.883341\pi$
$911$	−56.3567	−1.86718	−0.933590	+	0.358344i	$0.883341\pi$
$912$	0	0
$913$	7.23294	0.239375
$914$	0	0
$915$	−7.04457	−0.232886
$916$	0	0
$917$	32.2077	1.06359
$918$	0	0
$919$	6.62058	0.218393	0.109196	−	0.994020i	$-0.465172\pi$
$919$	6.62058	0.218393	0.109196	+	0.994020i	$0.465172\pi$
$920$	0	0
$921$	−4.27714	−0.140937
$922$	0	0
$923$	0.859635	0.0282952
$924$	0	0
$925$	3.83944	0.126240
$926$	0	0
$927$	−6.08194	−0.199757
$928$	0	0
$929$	−36.6853	−1.20361	−0.601803	−	0.798645i	$-0.705551\pi$
$929$	−36.6853	−1.20361	−0.601803	+	0.798645i	$0.705551\pi$
$930$	0	0
$931$	8.56252	0.280625
$932$	0	0
$933$	8.83160	0.289134
$934$	0	0
$935$	−6.99126	−0.228639
$936$	0	0
$937$	53.9678	1.76305	0.881525	−	0.472137i	$-0.156517\pi$
$937$	53.9678	1.76305	0.881525	+	0.472137i	$0.156517\pi$
$938$	0	0
$939$	19.4705	0.635395
$940$	0	0
$941$	−14.5190	−0.473307	−0.236654	−	0.971594i	$-0.576051\pi$
$941$	−14.5190	−0.473307	−0.236654	+	0.971594i	$0.576051\pi$
$942$	0	0
$943$	−6.88775	−0.224296
$944$	0	0
$945$	1.48734	0.0483832
$946$	0	0
$947$	−19.3563	−0.628994	−0.314497	−	0.949258i	$-0.601836\pi$
$947$	−19.3563	−0.628994	−0.314497	+	0.949258i	$0.601836\pi$
$948$	0	0
$949$	6.25811	0.203147
$950$	0	0
$951$	3.20488	0.103925
$952$	0	0
$953$	−18.7874	−0.608583	−0.304292	−	0.952579i	$-0.598420\pi$
$953$	−18.7874	−0.608583	−0.304292	+	0.952579i	$0.598420\pi$
$954$	0	0
$955$	24.1432	0.781257
$956$	0	0
$957$	6.99865	0.226234
$958$	0	0
$959$	15.3268	0.494928
$960$	0	0
$961$	−30.9994	−0.999981
$962$	0	0
$963$	−14.9446	−0.481582
$964$	0	0
$965$	7.71657	0.248405
$966$	0	0
$967$	29.6498	0.953474	0.476737	−	0.879046i	$-0.341819\pi$
$967$	29.6498	0.953474	0.476737	+	0.879046i	$0.341819\pi$
$968$	0	0
$969$	7.45917	0.239623
$970$	0	0
$971$	22.4867	0.721633	0.360816	−	0.932637i	$-0.382498\pi$
$971$	22.4867	0.721633	0.360816	+	0.932637i	$0.382498\pi$
$972$	0	0
$973$	0.311180	0.00997596
$974$	0	0
$975$	1.06167	0.0340005
$976$	0	0
$977$	41.8110	1.33765	0.668827	−	0.743419i	$-0.266797\pi$
$977$	41.8110	1.33765	0.668827	+	0.743419i	$0.266797\pi$
$978$	0	0
$979$	−8.00116	−0.255718
$980$	0	0
$981$	4.57322	0.146012
$982$	0	0
$983$	6.45090	0.205752	0.102876	−	0.994694i	$-0.467196\pi$
$983$	6.45090	0.205752	0.102876	+	0.994694i	$0.467196\pi$
$984$	0	0
$985$	1.37321	0.0437541
$986$	0	0
$987$	5.01220	0.159540
$988$	0	0
$989$	−5.09779	−0.162100
$990$	0	0
$991$	24.5228	0.778991	0.389496	−	0.921028i	$-0.372649\pi$
$991$	24.5228	0.778991	0.389496	+	0.921028i	$0.372649\pi$
$992$	0	0
$993$	−28.1447	−0.893145
$994$	0	0
$995$	−14.4107	−0.456851
$996$	0	0
$997$	36.1014	1.14334	0.571672	−	0.820483i	$-0.306295\pi$
$997$	36.1014	1.14334	0.571672	+	0.820483i	$0.306295\pi$
$998$	0	0
$999$	3.83944	0.121474

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4020.2.a.i.1.5	✓	7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4020.2.a.i.1.5	✓	7	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 4020 = 2^{2} \cdot 3 \cdot 5 \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4020.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +1.00000 q^{5} +1.48734 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +1.00000 q^{5} +1.48734 q^{7} +1.00000 q^{9} +1.67621 q^{11} +1.06167 q^{13} +1.00000 q^{15} -4.17087 q^{17} -1.78840 q^{19} +1.48734 q^{21} -0.687937 q^{23} +1.00000 q^{25} +1.00000 q^{27} +4.17528 q^{29} -0.0242963 q^{31} +1.67621 q^{33} +1.48734 q^{35} +3.83944 q^{37} +1.06167 q^{39} +10.0122 q^{41} +7.41026 q^{43} +1.00000 q^{45} +3.36990 q^{47} -4.78782 q^{49} -4.17087 q^{51} +9.22139 q^{53} +1.67621 q^{55} -1.78840 q^{57} +8.62084 q^{59} -7.04457 q^{61} +1.48734 q^{63} +1.06167 q^{65} -1.00000 q^{67} -0.687937 q^{69} +0.809704 q^{71} +5.89461 q^{73} +1.00000 q^{75} +2.49310 q^{77} -1.93005 q^{79} +1.00000 q^{81} +4.31505 q^{83} -4.17087 q^{85} +4.17528 q^{87} -4.77336 q^{89} +1.57906 q^{91} -0.0242963 q^{93} -1.78840 q^{95} +10.4092 q^{97} +1.67621 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 7 q^{3} + 7 q^{5} + q^{7} + 7 q^{9}+O(q^{10}) \) 7 * q + 7 * q^3 + 7 * q^5 + q^7 + 7 * q^9	\( 7 q + 7 q^{3} + 7 q^{5} + q^{7} + 7 q^{9} + 5 q^{11} + 9 q^{13} + 7 q^{15} + 11 q^{17} + 2 q^{19} + q^{21} + 7 q^{23} + 7 q^{25} + 7 q^{27} + 8 q^{29} + 9 q^{31} + 5 q^{33} + q^{35} + 7 q^{37} + 9 q^{39} + 9 q^{41} + 5 q^{43} + 7 q^{45} + 2 q^{47} + 12 q^{49} + 11 q^{51} + 15 q^{53} + 5 q^{55} + 2 q^{57} + 11 q^{61} + q^{63} + 9 q^{65} - 7 q^{67} + 7 q^{69} + 18 q^{71} + 21 q^{73} + 7 q^{75} + 5 q^{77} + 5 q^{79} + 7 q^{81} + 6 q^{83} + 11 q^{85} + 8 q^{87} + 7 q^{89} - 9 q^{91} + 9 q^{93} + 2 q^{95} - 9 q^{97} + 5 q^{99}+O(q^{100}) \) 7 * q + 7 * q^3 + 7 * q^5 + q^7 + 7 * q^9 + 5 * q^11 + 9 * q^13 + 7 * q^15 + 11 * q^17 + 2 * q^19 + q^21 + 7 * q^23 + 7 * q^25 + 7 * q^27 + 8 * q^29 + 9 * q^31 + 5 * q^33 + q^35 + 7 * q^37 + 9 * q^39 + 9 * q^41 + 5 * q^43 + 7 * q^45 + 2 * q^47 + 12 * q^49 + 11 * q^51 + 15 * q^53 + 5 * q^55 + 2 * q^57 + 11 * q^61 + q^63 + 9 * q^65 - 7 * q^67 + 7 * q^69 + 18 * q^71 + 21 * q^73 + 7 * q^75 + 5 * q^77 + 5 * q^79 + 7 * q^81 + 6 * q^83 + 11 * q^85 + 8 * q^87 + 7 * q^89 - 9 * q^91 + 9 * q^93 + 2 * q^95 - 9 * q^97 + 5 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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