LMFDB - Embedded newform 3920.2.k.a.2351.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3920,2,Mod(2351,3920)]

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([1, 0, 0, 1]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("3920.2351");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3920 = 2^{4} \cdot 5 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3920.k (of order $2$, degree $1$, minimal)

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:	no
Analytic conductor:	$31.3013575923$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\Q(\zeta_{16})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 1 $ x^8 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2351.3
Root			$0.382683 + 0.923880i$ of defining polynomial
Character	$\chi$	$=$	3920.2351
Dual form			3920.2.k.a.2351.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.64885 q^{3} -1.00000i q^{5} -0.281305 q^{9} +O(q^{10})$	$q-1.64885 q^{3} -1.00000i q^{5} -0.281305 q^{9} +1.94495i q^{11} -1.64885i q^{13} +1.64885i q^{15} +0.378493i q^{17} -4.30864 q^{19} +5.92562i q^{23} -1.00000 q^{25} +5.41037 q^{27} +3.07786 q^{29} +10.4609 q^{31} -3.20692i q^{33} -2.81363 q^{37} +2.71870i q^{39} -7.75858i q^{41} +0.418749i q^{43} +0.281305i q^{45} -4.17279 q^{47} -0.624077i q^{51} -4.48822 q^{53} +1.94495 q^{55} +7.10429 q^{57} -1.80910 q^{59} -3.04373i q^{61} -1.64885 q^{65} -8.86831i q^{67} -9.77043i q^{69} -3.02228i q^{71} +2.92721i q^{73} +1.64885 q^{75} +9.35916i q^{79} -8.07695 q^{81} +6.41384 q^{83} +0.378493 q^{85} -5.07491 q^{87} -0.688730i q^{89} -17.2484 q^{93} +4.30864i q^{95} -14.4992i q^{97} -0.547123i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{3} + 16 q^{9}+O(q^{10}) $ 8 * q - 8 * q^3 + 16 * q^9	$ 8 q - 8 q^{3} + 16 q^{9} + 16 q^{19} - 8 q^{25} - 8 q^{27} + 8 q^{29} + 48 q^{31} + 24 q^{47} + 32 q^{53} - 8 q^{55} - 32 q^{57} + 16 q^{59} - 8 q^{65} + 8 q^{75} - 8 q^{81} - 48 q^{83} - 24 q^{85} - 24 q^{87} - 32 q^{93}+O(q^{100}) $ 8 * q - 8 * q^3 + 16 * q^9 + 16 * q^19 - 8 * q^25 - 8 * q^27 + 8 * q^29 + 48 * q^31 + 24 * q^47 + 32 * q^53 - 8 * q^55 - 32 * q^57 + 16 * q^59 - 8 * q^65 + 8 * q^75 - 8 * q^81 - 48 * q^83 - 24 * q^85 - 24 * q^87 - 32 * q^93

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3920\mathbb{Z}\right)^\times$.

$n$	$981$	$1471$	$3041$	$3137$
$\chi(n)$	$1$	$-1$	$-1$	$1$

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
$3$	−1.64885	−0.951962	−0.475981	−	0.879456i	$-0.657907\pi$
$3$	−1.64885	−0.951962	−0.475981	+	0.879456i	$0.657907\pi$
$4$	0	0
$5$	− 1.00000i	− 0.447214i
$5$	− 1.00000i	− 0.447214i
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−0.281305	−0.0937682
$10$	0	0
$11$	1.94495i	0.586424i	0.956047	+	0.293212i	$0.0947241\pi$
$11$	1.94495i	0.586424i	−0.956047	+	0.293212i	$0.905276\pi$
$12$	0	0
$13$	− 1.64885i	− 0.457308i	−0.973508	−	0.228654i	$-0.926568\pi$
$13$	− 1.64885i	− 0.457308i	0.973508	−	0.228654i	$-0.0734324\pi$
$14$	0	0
$15$	1.64885i	0.425730i
$16$	0	0
$17$	0.378493i	0.0917980i	0.998946	+	0.0458990i	$0.0146152\pi$
$17$	0.378493i	0.0917980i	−0.998946	+	0.0458990i	$0.985385\pi$
$18$	0	0
$19$	−4.30864	−0.988471	−0.494235	−	0.869328i	$-0.664552\pi$
$19$	−4.30864	−0.988471	−0.494235	+	0.869328i	$0.664552\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	5.92562i	1.23558i	0.786345	+	0.617788i	$0.211971\pi$
$23$	5.92562i	1.23558i	−0.786345	+	0.617788i	$0.788029\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	0	0
$27$	5.41037	1.04123
$28$	0	0
$29$	3.07786	0.571544	0.285772	−	0.958298i	$-0.407750\pi$
$29$	3.07786	0.571544	0.285772	+	0.958298i	$0.407750\pi$
$30$	0	0
$31$	10.4609	1.87883	0.939415	−	0.342781i	$-0.111369\pi$
$31$	10.4609	1.87883	0.939415	+	0.342781i	$0.111369\pi$
$32$	0	0
$33$	− 3.20692i	− 0.558253i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−2.81363	−0.462558	−0.231279	−	0.972887i	$-0.574291\pi$
$37$	−2.81363	−0.462558	−0.231279	+	0.972887i	$0.574291\pi$
$38$	0	0
$39$	2.71870i	0.435340i
$40$	0	0
$41$	− 7.75858i	− 1.21169i	−0.795584	−	0.605843i	$-0.792836\pi$
$41$	− 7.75858i	− 1.21169i	0.795584	−	0.605843i	$-0.207164\pi$
$42$	0	0
$43$	0.418749i	0.0638587i	0.999490	+	0.0319293i	$0.0101652\pi$
$43$	0.418749i	0.0638587i	−0.999490	+	0.0319293i	$0.989835\pi$
$44$	0	0
$45$	0.281305i	0.0419344i
$46$	0	0
$47$	−4.17279	−0.608664	−0.304332	−	0.952566i	$-0.598433\pi$
$47$	−4.17279	−0.608664	−0.304332	+	0.952566i	$0.598433\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	− 0.624077i	− 0.0873882i
$52$	0	0
$53$	−4.48822	−0.616505	−0.308253	−	0.951305i	$-0.599744\pi$
$53$	−4.48822	−0.616505	−0.308253	+	0.951305i	$0.599744\pi$
$54$	0	0
$55$	1.94495	0.262257
$56$	0	0
$57$	7.10429	0.940987
$58$	0	0
$59$	−1.80910	−0.235524	−0.117762	−	0.993042i	$-0.537572\pi$
$59$	−1.80910	−0.235524	−0.117762	+	0.993042i	$0.537572\pi$
$60$	0	0
$61$	− 3.04373i	− 0.389709i	−0.980832	−	0.194855i	$-0.937576\pi$
$61$	− 3.04373i	− 0.389709i	0.980832	−	0.194855i	$-0.0624235\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−1.64885	−0.204514
$66$	0	0
$67$	− 8.86831i	− 1.08344i	−0.840560	−	0.541718i	$-0.817774\pi$
$67$	− 8.86831i	− 1.08344i	0.840560	−	0.541718i	$-0.182226\pi$
$68$	0	0
$69$	− 9.77043i	− 1.17622i
$70$	0	0
$71$	− 3.02228i	− 0.358678i	−0.983787	−	0.179339i	$-0.942604\pi$
$71$	− 3.02228i	− 0.358678i	0.983787	−	0.179339i	$-0.0573959\pi$
$72$	0	0
$73$	2.92721i	0.342604i	0.985219	+	0.171302i	$0.0547973\pi$
$73$	2.92721i	0.342604i	−0.985219	+	0.171302i	$0.945203\pi$
$74$	0	0
$75$	1.64885	0.190392
$76$	0	0
$77$	0	0
$78$	0	0
$79$	9.35916i	1.05299i	0.850179	+	0.526494i	$0.176494\pi$
$79$	9.35916i	1.05299i	−0.850179	+	0.526494i	$0.823506\pi$
$80$	0	0
$81$	−8.07695	−0.897439
$82$	0	0
$83$	6.41384	0.704010	0.352005	−	0.935998i	$-0.385500\pi$
$83$	6.41384	0.704010	0.352005	+	0.935998i	$0.385500\pi$
$84$	0	0
$85$	0.378493	0.0410533
$86$	0	0
$87$	−5.07491	−0.544088
$88$	0	0
$89$	− 0.688730i	− 0.0730052i	−0.999334	−	0.0365026i	$-0.988378\pi$
$89$	− 0.688730i	− 0.0730052i	0.999334	−	0.0365026i	$-0.0116217\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−17.2484	−1.78858
$94$	0	0
$95$	4.30864i	0.442058i
$96$	0	0
$97$	− 14.4992i	− 1.47217i	−0.676890	−	0.736084i	$-0.736673\pi$
$97$	− 14.4992i	− 1.47217i	0.676890	−	0.736084i	$-0.263327\pi$
$98$	0	0
$99$	− 0.547123i	− 0.0549879i
$100$	0	0
$101$	− 8.81657i	− 0.877282i	−0.898662	−	0.438641i	$-0.855460\pi$
$101$	− 8.81657i	− 0.877282i	0.898662	−	0.438641i	$-0.144540\pi$
$102$	0	0
$103$	1.45672	0.143535	0.0717676	−	0.997421i	$-0.477136\pi$
$103$	1.45672	0.143535	0.0717676	+	0.997421i	$0.477136\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	13.6997i	1.32440i	0.749328	+	0.662199i	$0.230377\pi$
$107$	13.6997i	1.32440i	−0.749328	+	0.662199i	$0.769623\pi$
$108$	0	0
$109$	−10.5558	−1.01106	−0.505532	−	0.862808i	$-0.668704\pi$
$109$	−10.5558	−1.01106	−0.505532	+	0.862808i	$0.668704\pi$
$110$	0	0
$111$	4.63925	0.440338
$112$	0	0
$113$	10.8474	1.02044	0.510218	−	0.860045i	$-0.329565\pi$
$113$	10.8474	1.02044	0.510218	+	0.860045i	$0.329565\pi$
$114$	0	0
$115$	5.92562	0.552566
$116$	0	0
$117$	0.463828i	0.0428809i
$118$	0	0
$119$	0	0
$120$	0	0
$121$	7.21718	0.656107
$122$	0	0
$123$	12.7927i	1.15348i
$124$	0	0
$125$	1.00000i	0.0894427i
$126$	0	0
$127$	11.6985i	1.03807i	0.854753	+	0.519035i	$0.173709\pi$
$127$	11.6985i	1.03807i	−0.854753	+	0.519035i	$0.826291\pi$
$128$	0	0
$129$	− 0.690454i	− 0.0607911i
$130$	0	0
$131$	−7.75539	−0.677592	−0.338796	−	0.940860i	$-0.610020\pi$
$131$	−7.75539	−0.677592	−0.338796	+	0.940860i	$0.610020\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	− 5.41037i	− 0.465650i
$136$	0	0
$137$	−12.9008	−1.10219	−0.551097	−	0.834441i	$-0.685791\pi$
$137$	−12.9008	−1.10219	−0.551097	+	0.834441i	$0.685791\pi$
$138$	0	0
$139$	−0.651099	−0.0552255	−0.0276128	−	0.999619i	$-0.508791\pi$
$139$	−0.651099	−0.0552255	−0.0276128	+	0.999619i	$0.508791\pi$
$140$	0	0
$141$	6.88029	0.579425
$142$	0	0
$143$	3.20692	0.268176
$144$	0	0
$145$	− 3.07786i	− 0.255602i
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−13.8371	−1.13358	−0.566791	−	0.823862i	$-0.691815\pi$
$149$	−13.8371	−1.13358	−0.566791	+	0.823862i	$0.691815\pi$
$150$	0	0
$151$	5.07977i	0.413385i	0.978406	+	0.206693i	$0.0662700\pi$
$151$	5.07977i	0.413385i	−0.978406	+	0.206693i	$0.933730\pi$
$152$	0	0
$153$	− 0.106472i	− 0.00860773i
$154$	0	0
$155$	− 10.4609i	− 0.840239i
$156$	0	0
$157$	− 13.6569i	− 1.08994i	−0.838457	−	0.544968i	$-0.816542\pi$
$157$	− 13.6569i	− 1.08994i	0.838457	−	0.544968i	$-0.183458\pi$
$158$	0	0
$159$	7.40039	0.586889
$160$	0	0
$161$	0	0
$162$	0	0
$163$	− 14.3057i	− 1.12051i	−0.828321	−	0.560254i	$-0.810703\pi$
$163$	− 14.3057i	− 1.12051i	0.828321	−	0.560254i	$-0.189297\pi$
$164$	0	0
$165$	−3.20692	−0.249658
$166$	0	0
$167$	−22.3891	−1.73252	−0.866259	−	0.499595i	$-0.833482\pi$
$167$	−22.3891	−1.73252	−0.866259	+	0.499595i	$0.833482\pi$
$168$	0	0
$169$	10.2813	0.790870
$170$	0	0
$171$	1.21204	0.0926871
$172$	0	0
$173$	− 14.8583i	− 1.12966i	−0.825208	−	0.564829i	$-0.808942\pi$
$173$	− 14.8583i	− 1.12966i	0.825208	−	0.564829i	$-0.191058\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	2.98292	0.224210
$178$	0	0
$179$	− 20.1029i	− 1.50256i	−0.659981	−	0.751282i	$-0.729436\pi$
$179$	− 20.1029i	− 1.50256i	0.659981	−	0.751282i	$-0.270564\pi$
$180$	0	0
$181$	− 23.9690i	− 1.78160i	−0.454396	−	0.890800i	$-0.650145\pi$
$181$	− 23.9690i	− 1.78160i	0.454396	−	0.890800i	$-0.349855\pi$
$182$	0	0
$183$	5.01864i	0.370989i
$184$	0	0
$185$	2.81363i	0.206862i
$186$	0	0
$187$	−0.736149	−0.0538325
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 24.0247i	− 1.73837i	−0.494489	−	0.869184i	$-0.664645\pi$
$191$	− 24.0247i	− 1.73837i	0.494489	−	0.869184i	$-0.335355\pi$
$192$	0	0
$193$	−25.5332	−1.83792	−0.918959	−	0.394353i	$-0.870969\pi$
$193$	−25.5332	−1.83792	−0.918959	+	0.394353i	$0.870969\pi$
$194$	0	0
$195$	2.71870	0.194690
$196$	0	0
$197$	12.0872	0.861178	0.430589	−	0.902548i	$-0.358306\pi$
$197$	12.0872	0.861178	0.430589	+	0.902548i	$0.358306\pi$
$198$	0	0
$199$	−4.40204	−0.312053	−0.156026	−	0.987753i	$-0.549868\pi$
$199$	−4.40204	−0.312053	−0.156026	+	0.987753i	$0.549868\pi$
$200$	0	0
$201$	14.6225i	1.03139i
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−7.75858	−0.541883
$206$	0	0
$207$	− 1.66690i	− 0.115858i
$208$	0	0
$209$	− 8.38009i	− 0.579663i
$210$	0	0
$211$	− 19.7956i	− 1.36279i	−0.731916	−	0.681394i	$-0.761374\pi$
$211$	− 19.7956i	− 1.36279i	0.731916	−	0.681394i	$-0.238626\pi$
$212$	0	0
$213$	4.98327i	0.341448i
$214$	0	0
$215$	0.418749	0.0285585
$216$	0	0
$217$	0	0
$218$	0	0
$219$	− 4.82652i	− 0.326146i
$220$	0	0
$221$	0.624077	0.0419799
$222$	0	0
$223$	−10.8583	−0.727128	−0.363564	−	0.931569i	$-0.618440\pi$
$223$	−10.8583	−0.727128	−0.363564	+	0.931569i	$0.618440\pi$
$224$	0	0
$225$	0.281305	0.0187536
$226$	0	0
$227$	−22.3785	−1.48531	−0.742656	−	0.669673i	$-0.766434\pi$
$227$	−22.3785	−1.48531	−0.742656	+	0.669673i	$0.766434\pi$
$228$	0	0
$229$	− 4.30795i	− 0.284678i	−0.989818	−	0.142339i	$-0.954538\pi$
$229$	− 4.30795i	− 0.284678i	0.989818	−	0.142339i	$-0.0454622\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−20.1303	−1.31878	−0.659389	−	0.751802i	$-0.729185\pi$
$233$	−20.1303	−1.31878	−0.659389	+	0.751802i	$0.729185\pi$
$234$	0	0
$235$	4.17279i	0.272203i
$236$	0	0
$237$	− 15.4318i	− 1.00240i
$238$	0	0
$239$	19.5690i	1.26581i	0.774227	+	0.632907i	$0.218139\pi$
$239$	19.5690i	1.26581i	−0.774227	+	0.632907i	$0.781861\pi$
$240$	0	0
$241$	− 4.71804i	− 0.303915i	−0.988387	−	0.151958i	$-0.951442\pi$
$241$	− 4.71804i	− 0.303915i	0.988387	−	0.151958i	$-0.0485578\pi$
$242$	0	0
$243$	−2.91345	−0.186898
$244$	0	0
$245$	0	0
$246$	0	0
$247$	7.10429i	0.452035i
$248$	0	0
$249$	−10.5754	−0.670191
$250$	0	0
$251$	7.41994	0.468342	0.234171	−	0.972195i	$-0.424762\pi$
$251$	7.41994	0.468342	0.234171	+	0.972195i	$0.424762\pi$
$252$	0	0
$253$	−11.5250	−0.724571
$254$	0	0
$255$	−0.624077	−0.0390812
$256$	0	0
$257$	7.56940i	0.472166i	0.971733	+	0.236083i	$0.0758637\pi$
$257$	7.56940i	0.472166i	−0.971733	+	0.236083i	$0.924136\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−0.865815	−0.0535926
$262$	0	0
$263$	− 29.4753i	− 1.81753i	−0.417313	−	0.908763i	$-0.637028\pi$
$263$	− 29.4753i	− 1.81753i	0.417313	−	0.908763i	$-0.362972\pi$
$264$	0	0
$265$	4.48822i	0.275709i
$266$	0	0
$267$	1.13561i	0.0694982i
$268$	0	0
$269$	− 11.1455i	− 0.679550i	−0.940507	−	0.339775i	$-0.889649\pi$
$269$	− 11.1455i	− 0.679550i	0.940507	−	0.339775i	$-0.110351\pi$
$270$	0	0
$271$	20.7138	1.25827	0.629137	−	0.777294i	$-0.283408\pi$
$271$	20.7138	1.25827	0.629137	+	0.777294i	$0.283408\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 1.94495i	− 0.117285i
$276$	0	0
$277$	1.03760	0.0623433	0.0311717	−	0.999514i	$-0.490076\pi$
$277$	1.03760	0.0623433	0.0311717	+	0.999514i	$0.490076\pi$
$278$	0	0
$279$	−2.94269	−0.176175
$280$	0	0
$281$	25.7821	1.53803	0.769015	−	0.639231i	$-0.220747\pi$
$281$	25.7821	1.53803	0.769015	+	0.639231i	$0.220747\pi$
$282$	0	0
$283$	−15.3299	−0.911265	−0.455633	−	0.890168i	$-0.650587\pi$
$283$	−15.3299	−0.911265	−0.455633	+	0.890168i	$0.650587\pi$
$284$	0	0
$285$	− 7.10429i	− 0.420822i
$286$	0	0
$287$	0	0
$288$	0	0
$289$	16.8567	0.991573
$290$	0	0
$291$	23.9069i	1.40145i
$292$	0	0
$293$	11.9206i	0.696406i	0.937419	+	0.348203i	$0.113208\pi$
$293$	11.9206i	0.696406i	−0.937419	+	0.348203i	$0.886792\pi$
$294$	0	0
$295$	1.80910i	0.105330i
$296$	0	0
$297$	10.5229i	0.610599i
$298$	0	0
$299$	9.77043	0.565039
$300$	0	0
$301$	0	0
$302$	0	0
$303$	14.5372i	0.835139i
$304$	0	0
$305$	−3.04373	−0.174283
$306$	0	0
$307$	−7.82177	−0.446412	−0.223206	−	0.974771i	$-0.571652\pi$
$307$	−7.82177	−0.446412	−0.223206	+	0.974771i	$0.571652\pi$
$308$	0	0
$309$	−2.40191	−0.136640
$310$	0	0
$311$	32.0480	1.81728	0.908639	−	0.417583i	$-0.137123\pi$
$311$	32.0480	1.81728	0.908639	+	0.417583i	$0.137123\pi$
$312$	0	0
$313$	− 27.0468i	− 1.52878i	−0.644756	−	0.764389i	$-0.723041\pi$
$313$	− 27.0468i	− 1.52878i	0.644756	−	0.764389i	$-0.276959\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	1.36423	0.0766225	0.0383113	−	0.999266i	$-0.487802\pi$
$317$	1.36423	0.0766225	0.0383113	+	0.999266i	$0.487802\pi$
$318$	0	0
$319$	5.98627i	0.335167i
$320$	0	0
$321$	− 22.5887i	− 1.26078i
$322$	0	0
$323$	− 1.63079i	− 0.0907396i
$324$	0	0
$325$	1.64885i	0.0914616i
$326$	0	0
$327$	17.4049	0.962495
$328$	0	0
$329$	0	0
$330$	0	0
$331$	− 31.5159i	− 1.73227i	−0.499811	−	0.866135i	$-0.666597\pi$
$331$	− 31.5159i	− 1.73227i	0.499811	−	0.866135i	$-0.333403\pi$
$332$	0	0
$333$	0.791487	0.0433732
$334$	0	0
$335$	−8.86831	−0.484528
$336$	0	0
$337$	−10.7045	−0.583111	−0.291556	−	0.956554i	$-0.594173\pi$
$337$	−10.7045	−0.583111	−0.291556	+	0.956554i	$0.594173\pi$
$338$	0	0
$339$	−17.8857	−0.971417
$340$	0	0
$341$	20.3459i	1.10179i
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−9.77043	−0.526022
$346$	0	0
$347$	− 5.47236i	− 0.293772i	−0.989153	−	0.146886i	$-0.953075\pi$
$347$	− 5.47236i	− 0.293772i	0.989153	−	0.146886i	$-0.0469250\pi$
$348$	0	0
$349$	− 17.8822i	− 0.957212i	−0.878030	−	0.478606i	$-0.841142\pi$
$349$	− 17.8822i	− 0.957212i	0.878030	−	0.478606i	$-0.158858\pi$
$350$	0	0
$351$	− 8.92087i	− 0.476161i
$352$	0	0
$353$	27.2899i	1.45249i	0.687434	+	0.726247i	$0.258737\pi$
$353$	27.2899i	1.45249i	−0.687434	+	0.726247i	$0.741263\pi$
$354$	0	0
$355$	−3.02228	−0.160406
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 13.5113i	− 0.713098i	−0.934277	−	0.356549i	$-0.883953\pi$
$359$	− 13.5113i	− 0.713098i	0.934277	−	0.356549i	$-0.116047\pi$
$360$	0	0
$361$	−0.435586	−0.0229256
$362$	0	0
$363$	−11.9000	−0.624589
$364$	0	0
$365$	2.92721	0.153217
$366$	0	0
$367$	−13.3080	−0.694670	−0.347335	−	0.937741i	$-0.612913\pi$
$367$	−13.3080	−0.694670	−0.347335	+	0.937741i	$0.612913\pi$
$368$	0	0
$369$	2.18252i	0.113618i
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−33.1008	−1.71390	−0.856948	−	0.515402i	$-0.827643\pi$
$373$	−33.1008	−1.71390	−0.856948	+	0.515402i	$0.827643\pi$
$374$	0	0
$375$	− 1.64885i	− 0.0851461i
$376$	0	0
$377$	− 5.07491i	− 0.261371i
$378$	0	0
$379$	− 21.0794i	− 1.08278i	−0.840773	−	0.541388i	$-0.817899\pi$
$379$	− 21.0794i	− 1.08278i	0.840773	−	0.541388i	$-0.182101\pi$
$380$	0	0
$381$	− 19.2890i	− 0.988204i
$382$	0	0
$383$	13.8731	0.708884	0.354442	−	0.935078i	$-0.384671\pi$
$383$	13.8731	0.708884	0.354442	+	0.935078i	$0.384671\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 0.117796i	− 0.00598791i
$388$	0	0
$389$	20.8145	1.05534	0.527668	−	0.849451i	$-0.323067\pi$
$389$	20.8145	1.05534	0.527668	+	0.849451i	$0.323067\pi$
$390$	0	0
$391$	−2.24280	−0.113423
$392$	0	0
$393$	12.7875	0.645042
$394$	0	0
$395$	9.35916	0.470910
$396$	0	0
$397$	− 19.2024i	− 0.963742i	−0.876242	−	0.481871i	$-0.839957\pi$
$397$	− 19.2024i	− 0.963742i	0.876242	−	0.481871i	$-0.160043\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	33.7981	1.68780	0.843898	−	0.536504i	$-0.180255\pi$
$401$	33.7981	1.68780	0.843898	+	0.536504i	$0.180255\pi$
$402$	0	0
$403$	− 17.2484i	− 0.859204i
$404$	0	0
$405$	8.07695i	0.401347i
$406$	0	0
$407$	− 5.47236i	− 0.271255i
$408$	0	0
$409$	5.12106i	0.253220i	0.991953	+	0.126610i	$0.0404097\pi$
$409$	5.12106i	0.253220i	−0.991953	+	0.126610i	$0.959590\pi$
$410$	0	0
$411$	21.2715	1.04925
$412$	0	0
$413$	0	0
$414$	0	0
$415$	− 6.41384i	− 0.314843i
$416$	0	0
$417$	1.07356	0.0525726
$418$	0	0
$419$	−12.0985	−0.591052	−0.295526	−	0.955335i	$-0.595495\pi$
$419$	−12.0985	−0.591052	−0.295526	+	0.955335i	$0.595495\pi$
$420$	0	0
$421$	−14.2455	−0.694281	−0.347140	−	0.937813i	$-0.612847\pi$
$421$	−14.2455	−0.694281	−0.347140	+	0.937813i	$0.612847\pi$
$422$	0	0
$423$	1.17383	0.0570733
$424$	0	0
$425$	− 0.378493i	− 0.0183596i
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−5.28772	−0.255293
$430$	0	0
$431$	20.2287i	0.974382i	0.873296	+	0.487191i	$0.161978\pi$
$431$	20.2287i	0.974382i	−0.873296	+	0.487191i	$0.838022\pi$
$432$	0	0
$433$	31.6972i	1.52327i	0.648006	+	0.761635i	$0.275603\pi$
$433$	31.6972i	1.52327i	−0.648006	+	0.761635i	$0.724397\pi$
$434$	0	0
$435$	5.07491i	0.243323i
$436$	0	0
$437$	− 25.5314i	− 1.22133i
$438$	0	0
$439$	−41.4147	−1.97662	−0.988308	−	0.152474i	$-0.951276\pi$
$439$	−41.4147	−1.97662	−0.988308	+	0.152474i	$0.951276\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 1.24815i	− 0.0593015i	−0.999560	−	0.0296508i	$-0.990560\pi$
$443$	− 1.24815i	− 0.0593015i	0.999560	−	0.0296508i	$-0.00943952\pi$
$444$	0	0
$445$	−0.688730	−0.0326489
$446$	0	0
$447$	22.8153	1.07913
$448$	0	0
$449$	14.1508	0.667819	0.333910	−	0.942605i	$-0.391632\pi$
$449$	14.1508	0.667819	0.333910	+	0.942605i	$0.391632\pi$
$450$	0	0
$451$	15.0900	0.710562
$452$	0	0
$453$	− 8.37576i	− 0.393527i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	30.8606	1.44360	0.721799	−	0.692102i	$-0.243315\pi$
$457$	30.8606	1.44360	0.721799	+	0.692102i	$0.243315\pi$
$458$	0	0
$459$	2.04779i	0.0955824i
$460$	0	0
$461$	− 14.9259i	− 0.695170i	−0.937648	−	0.347585i	$-0.887002\pi$
$461$	− 14.9259i	− 0.695170i	0.937648	−	0.347585i	$-0.112998\pi$
$462$	0	0
$463$	− 11.5945i	− 0.538841i	−0.963023	−	0.269421i	$-0.913168\pi$
$463$	− 11.5945i	− 0.538841i	0.963023	−	0.269421i	$-0.0868322\pi$
$464$	0	0
$465$	17.2484i	0.799875i
$466$	0	0
$467$	−29.3687	−1.35902	−0.679511	−	0.733666i	$-0.737808\pi$
$467$	−29.3687	−1.35902	−0.679511	+	0.733666i	$0.737808\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	22.5181i	1.03758i
$472$	0	0
$473$	−0.814446	−0.0374483
$474$	0	0
$475$	4.30864	0.197694
$476$	0	0
$477$	1.26256	0.0578086
$478$	0	0
$479$	−8.25744	−0.377292	−0.188646	−	0.982045i	$-0.560410\pi$
$479$	−8.25744	−0.377292	−0.188646	+	0.982045i	$0.560410\pi$
$480$	0	0
$481$	4.63925i	0.211531i
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−14.4992	−0.658374
$486$	0	0
$487$	− 31.0659i	− 1.40773i	−0.710334	−	0.703865i	$-0.751456\pi$
$487$	− 31.0659i	− 1.40773i	0.710334	−	0.703865i	$-0.248544\pi$
$488$	0	0
$489$	23.5879i	1.06668i
$490$	0	0
$491$	22.3871i	1.01031i	0.863028	+	0.505157i	$0.168565\pi$
$491$	22.3871i	1.01031i	−0.863028	+	0.505157i	$0.831435\pi$
$492$	0	0
$493$	1.16495i	0.0524665i
$494$	0	0
$495$	−0.547123	−0.0245913
$496$	0	0
$497$	0	0
$498$	0	0
$499$	33.7176i	1.50941i	0.656067	+	0.754703i	$0.272219\pi$
$499$	33.7176i	1.50941i	−0.656067	+	0.754703i	$0.727781\pi$
$500$	0	0
$501$	36.9161	1.64929
$502$	0	0
$503$	10.1760	0.453724	0.226862	−	0.973927i	$-0.427153\pi$
$503$	10.1760	0.453724	0.226862	+	0.973927i	$0.427153\pi$
$504$	0	0
$505$	−8.81657	−0.392332
$506$	0	0
$507$	−16.9523	−0.752878
$508$	0	0
$509$	4.48106i	0.198620i	0.995057	+	0.0993098i	$0.0316635\pi$
$509$	4.48106i	0.198620i	−0.995057	+	0.0993098i	$0.968337\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−23.3114	−1.02922
$514$	0	0
$515$	− 1.45672i	− 0.0641909i
$516$	0	0
$517$	− 8.11586i	− 0.356935i
$518$	0	0
$519$	24.4991i	1.07539i
$520$	0	0
$521$	− 0.338641i	− 0.0148361i	−0.999972	−	0.00741806i	$-0.997639\pi$
$521$	− 0.338641i	− 0.0148361i	0.999972	−	0.00741806i	$-0.00236126\pi$
$522$	0	0
$523$	−30.1801	−1.31968	−0.659841	−	0.751405i	$-0.729376\pi$
$523$	−30.1801	−1.31968	−0.659841	+	0.751405i	$0.729376\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	3.95937i	0.172473i
$528$	0	0
$529$	−12.1129	−0.526649
$530$	0	0
$531$	0.508907	0.0220847
$532$	0	0
$533$	−12.7927	−0.554114
$534$	0	0
$535$	13.6997	0.592289
$536$	0	0
$537$	33.1467i	1.43038i
$538$	0	0
$539$	0	0
$540$	0	0
$541$	40.3676	1.73554	0.867770	−	0.496966i	$-0.165553\pi$
$541$	40.3676	1.73554	0.867770	+	0.496966i	$0.165553\pi$
$542$	0	0
$543$	39.5212i	1.69602i
$544$	0	0
$545$	10.5558i	0.452162i
$546$	0	0
$547$	25.0225i	1.06988i	0.844889	+	0.534942i	$0.179666\pi$
$547$	25.0225i	1.06988i	−0.844889	+	0.534942i	$0.820334\pi$
$548$	0	0
$549$	0.856215i	0.0365424i
$550$	0	0
$551$	−13.2614	−0.564954
$552$	0	0
$553$	0	0
$554$	0	0
$555$	− 4.63925i	− 0.196925i
$556$	0	0
$557$	22.5131	0.953909	0.476955	−	0.878928i	$-0.341741\pi$
$557$	22.5131	0.953909	0.476955	+	0.878928i	$0.341741\pi$
$558$	0	0
$559$	0.690454	0.0292031
$560$	0	0
$561$	1.21380	0.0512465
$562$	0	0
$563$	28.0447	1.18194	0.590972	−	0.806692i	$-0.298744\pi$
$563$	28.0447	1.18194	0.590972	+	0.806692i	$0.298744\pi$
$564$	0	0
$565$	− 10.8474i	− 0.456353i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	14.8084	0.620802	0.310401	−	0.950606i	$-0.399537\pi$
$569$	14.8084	0.620802	0.310401	+	0.950606i	$0.399537\pi$
$570$	0	0
$571$	− 11.8699i	− 0.496740i	−0.968665	−	0.248370i	$-0.920105\pi$
$571$	− 11.8699i	− 0.496740i	0.968665	−	0.248370i	$-0.0798949\pi$
$572$	0	0
$573$	39.6131i	1.65486i
$574$	0	0
$575$	− 5.92562i	− 0.247115i
$576$	0	0
$577$	19.7011i	0.820167i	0.912048	+	0.410084i	$0.134500\pi$
$577$	19.7011i	0.820167i	−0.912048	+	0.410084i	$0.865500\pi$
$578$	0	0
$579$	42.1003	1.74963
$580$	0	0
$581$	0	0
$582$	0	0
$583$	− 8.72936i	− 0.361533i
$584$	0	0
$585$	0.463828	0.0191769
$586$	0	0
$587$	16.9141	0.698119	0.349059	−	0.937101i	$-0.386501\pi$
$587$	16.9141	0.698119	0.349059	+	0.937101i	$0.386501\pi$
$588$	0	0
$589$	−45.0722	−1.85717
$590$	0	0
$591$	−19.9300	−0.819809
$592$	0	0
$593$	− 21.5902i	− 0.886605i	−0.896372	−	0.443302i	$-0.853807\pi$
$593$	− 21.5902i	− 0.886605i	0.896372	−	0.443302i	$-0.146193\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	7.25830	0.297062
$598$	0	0
$599$	11.1056i	0.453764i	0.973922	+	0.226882i	$0.0728532\pi$
$599$	11.1056i	0.453764i	−0.973922	+	0.226882i	$0.927147\pi$
$600$	0	0
$601$	4.86537i	0.198462i	0.995064	+	0.0992312i	$0.0316383\pi$
$601$	4.86537i	0.198462i	−0.995064	+	0.0992312i	$0.968362\pi$
$602$	0	0
$603$	2.49470i	0.101592i
$604$	0	0
$605$	− 7.21718i	− 0.293420i
$606$	0	0
$607$	11.1045	0.450718	0.225359	−	0.974276i	$-0.427645\pi$
$607$	11.1045	0.450718	0.225359	+	0.974276i	$0.427645\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	6.88029i	0.278347i
$612$	0	0
$613$	−0.742432	−0.0299865	−0.0149933	−	0.999888i	$-0.504773\pi$
$613$	−0.742432	−0.0299865	−0.0149933	+	0.999888i	$0.504773\pi$
$614$	0	0
$615$	12.7927	0.515852
$616$	0	0
$617$	−2.97801	−0.119890	−0.0599451	−	0.998202i	$-0.519093\pi$
$617$	−2.97801	−0.119890	−0.0599451	+	0.998202i	$0.519093\pi$
$618$	0	0
$619$	11.4982	0.462153	0.231077	−	0.972936i	$-0.425775\pi$
$619$	11.4982	0.462153	0.231077	+	0.972936i	$0.425775\pi$
$620$	0	0
$621$	32.0598i	1.28651i
$622$	0	0
$623$	0	0
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	13.8175i	0.551817i
$628$	0	0
$629$	− 1.06494i	− 0.0424619i
$630$	0	0
$631$	− 2.70002i	− 0.107486i	−0.998555	−	0.0537431i	$-0.982885\pi$
$631$	− 2.70002i	− 0.107486i	0.998555	−	0.0537431i	$-0.0171152\pi$
$632$	0	0
$633$	32.6400i	1.29732i
$634$	0	0
$635$	11.6985	0.464239
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0.850180i	0.0336326i
$640$	0	0
$641$	11.4334	0.451593	0.225796	−	0.974175i	$-0.427502\pi$
$641$	11.4334	0.451593	0.225796	+	0.974175i	$0.427502\pi$
$642$	0	0
$643$	3.46367	0.136594	0.0682968	−	0.997665i	$-0.478244\pi$
$643$	3.46367	0.136594	0.0682968	+	0.997665i	$0.478244\pi$
$644$	0	0
$645$	−0.690454	−0.0271866
$646$	0	0
$647$	8.29679	0.326181	0.163090	−	0.986611i	$-0.447854\pi$
$647$	8.29679	0.326181	0.163090	+	0.986611i	$0.447854\pi$
$648$	0	0
$649$	− 3.51859i	− 0.138117i
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−32.7681	−1.28231	−0.641157	−	0.767410i	$-0.721545\pi$
$653$	−32.7681	−1.28231	−0.641157	+	0.767410i	$0.721545\pi$
$654$	0	0
$655$	7.75539i	0.303028i
$656$	0	0
$657$	− 0.823437i	− 0.0321253i
$658$	0	0
$659$	− 31.0121i	− 1.20806i	−0.796962	−	0.604030i	$-0.793561\pi$
$659$	− 31.0121i	− 1.20806i	0.796962	−	0.604030i	$-0.206439\pi$
$660$	0	0
$661$	− 32.5296i	− 1.26525i	−0.774457	−	0.632627i	$-0.781977\pi$
$661$	− 32.5296i	− 1.26525i	0.774457	−	0.632627i	$-0.218023\pi$
$662$	0	0
$663$	−1.02901	−0.0399633
$664$	0	0
$665$	0	0
$666$	0	0
$667$	18.2382i	0.706186i
$668$	0	0
$669$	17.9037	0.692198
$670$	0	0
$671$	5.91989	0.228535
$672$	0	0
$673$	−38.4444	−1.48192	−0.740962	−	0.671547i	$-0.765630\pi$
$673$	−38.4444	−1.48192	−0.740962	+	0.671547i	$0.765630\pi$
$674$	0	0
$675$	−5.41037	−0.208245
$676$	0	0
$677$	44.3110i	1.70301i	0.524347	+	0.851505i	$0.324310\pi$
$677$	44.3110i	1.70301i	−0.524347	+	0.851505i	$0.675690\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	36.8987	1.41396
$682$	0	0
$683$	12.4929i	0.478027i	0.971016	+	0.239014i	$0.0768241\pi$
$683$	12.4929i	0.478027i	−0.971016	+	0.239014i	$0.923176\pi$
$684$	0	0
$685$	12.9008i	0.492916i
$686$	0	0
$687$	7.10315i	0.271002i
$688$	0	0
$689$	7.40039i	0.281933i
$690$	0	0
$691$	−40.3103	−1.53348	−0.766739	−	0.641959i	$-0.778122\pi$
$691$	−40.3103	−1.53348	−0.766739	+	0.641959i	$0.778122\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0.651099i	0.0246976i
$696$	0	0
$697$	2.93657	0.111230
$698$	0	0
$699$	33.1918	1.25543
$700$	0	0
$701$	−16.0954	−0.607915	−0.303958	−	0.952686i	$-0.598308\pi$
$701$	−16.0954	−0.607915	−0.303958	+	0.952686i	$0.598308\pi$
$702$	0	0
$703$	12.1229	0.457225
$704$	0	0
$705$	− 6.88029i	− 0.259127i
$706$	0	0
$707$	0	0
$708$	0	0
$709$	6.71100	0.252037	0.126019	−	0.992028i	$-0.459780\pi$
$709$	6.71100	0.252037	0.126019	+	0.992028i	$0.459780\pi$
$710$	0	0
$711$	− 2.63277i	− 0.0987368i
$712$	0	0
$713$	61.9872i	2.32144i
$714$	0	0
$715$	− 3.20692i	− 0.119932i
$716$	0	0
$717$	− 32.2663i	− 1.20501i
$718$	0	0
$719$	−32.2397	−1.20234	−0.601169	−	0.799122i	$-0.705298\pi$
$719$	−32.2397	−1.20234	−0.601169	+	0.799122i	$0.705298\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	7.77932i	0.289316i
$724$	0	0
$725$	−3.07786	−0.114309
$726$	0	0
$727$	−10.4259	−0.386676	−0.193338	−	0.981132i	$-0.561931\pi$
$727$	−10.4259	−0.386676	−0.193338	+	0.981132i	$0.561931\pi$
$728$	0	0
$729$	29.0347	1.07536
$730$	0	0
$731$	−0.158494	−0.00586210
$732$	0	0
$733$	− 43.2358i	− 1.59695i	−0.602028	−	0.798475i	$-0.705641\pi$
$733$	− 43.2358i	− 1.59695i	0.602028	−	0.798475i	$-0.294359\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	17.2484	0.635353
$738$	0	0
$739$	− 13.0914i	− 0.481576i	−0.970578	−	0.240788i	$-0.922594\pi$
$739$	− 13.0914i	− 0.481576i	0.970578	−	0.240788i	$-0.0774059\pi$
$740$	0	0
$741$	− 11.7139i	− 0.430321i
$742$	0	0
$743$	− 5.95202i	− 0.218359i	−0.994022	−	0.109179i	$-0.965178\pi$
$743$	− 5.95202i	− 0.218359i	0.994022	−	0.109179i	$-0.0348223\pi$
$744$	0	0
$745$	13.8371i	0.506953i
$746$	0	0
$747$	−1.80424	−0.0660138
$748$	0	0
$749$	0	0
$750$	0	0
$751$	17.9764i	0.655970i	0.944683	+	0.327985i	$0.106370\pi$
$751$	17.9764i	0.655970i	−0.944683	+	0.327985i	$0.893630\pi$
$752$	0	0
$753$	−12.2343	−0.445844
$754$	0	0
$755$	5.07977	0.184872
$756$	0	0
$757$	−7.39520	−0.268783	−0.134392	−	0.990928i	$-0.542908\pi$
$757$	−7.39520	−0.268783	−0.134392	+	0.990928i	$0.542908\pi$
$758$	0	0
$759$	19.0030	0.689764
$760$	0	0
$761$	40.9531i	1.48455i	0.670096	+	0.742275i	$0.266253\pi$
$761$	40.9531i	1.48455i	−0.670096	+	0.742275i	$0.733747\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−0.106472	−0.00384949
$766$	0	0
$767$	2.98292i	0.107707i
$768$	0	0
$769$	39.6509i	1.42985i	0.699203	+	0.714923i	$0.253538\pi$
$769$	39.6509i	1.42985i	−0.699203	+	0.714923i	$0.746462\pi$
$770$	0	0
$771$	− 12.4808i	− 0.449484i
$772$	0	0
$773$	24.4728i	0.880224i	0.897943	+	0.440112i	$0.145061\pi$
$773$	24.4728i	0.880224i	−0.897943	+	0.440112i	$0.854939\pi$
$774$	0	0
$775$	−10.4609	−0.375766
$776$	0	0
$777$	0	0
$778$	0	0
$779$	33.4290i	1.19772i
$780$	0	0
$781$	5.87817	0.210337
$782$	0	0
$783$	16.6523	0.595106
$784$	0	0
$785$	−13.6569	−0.487434
$786$	0	0
$787$	50.7437	1.80882	0.904409	−	0.426666i	$-0.140312\pi$
$787$	50.7437	1.80882	0.904409	+	0.426666i	$0.140312\pi$
$788$	0	0
$789$	48.6003i	1.73022i
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−5.01864	−0.178217
$794$	0	0
$795$	− 7.40039i	− 0.262465i
$796$	0	0
$797$	32.2616i	1.14277i	0.820684	+	0.571383i	$0.193593\pi$
$797$	32.2616i	1.14277i	−0.820684	+	0.571383i	$0.806407\pi$
$798$	0	0
$799$	− 1.57937i	− 0.0558742i
$800$	0	0
$801$	0.193743i	0.00684557i
$802$	0	0
$803$	−5.69327	−0.200911
$804$	0	0
$805$	0	0
$806$	0	0
$807$	18.3771i	0.646906i
$808$	0	0
$809$	36.2860	1.27575	0.637874	−	0.770140i	$-0.279814\pi$
$809$	36.2860	1.27575	0.637874	+	0.770140i	$0.279814\pi$
$810$	0	0
$811$	21.7349	0.763215	0.381607	−	0.924325i	$-0.375371\pi$
$811$	21.7349	0.763215	0.381607	+	0.924325i	$0.375371\pi$
$812$	0	0
$813$	−34.1539	−1.19783
$814$	0	0
$815$	−14.3057	−0.501107
$816$	0	0
$817$	− 1.80424i	− 0.0631225i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	15.6160	0.545001	0.272501	−	0.962156i	$-0.412149\pi$
$821$	15.6160	0.545001	0.272501	+	0.962156i	$0.412149\pi$
$822$	0	0
$823$	− 11.0982i	− 0.386858i	−0.981114	−	0.193429i	$-0.938039\pi$
$823$	− 11.0982i	− 0.386858i	0.981114	−	0.193429i	$-0.0619609\pi$
$824$	0	0
$825$	3.20692i	0.111651i
$826$	0	0
$827$	− 3.73713i	− 0.129953i	−0.997887	−	0.0649763i	$-0.979303\pi$
$827$	− 3.73713i	− 0.129953i	0.997887	−	0.0649763i	$-0.0206972\pi$
$828$	0	0
$829$	− 20.4507i	− 0.710283i	−0.934813	−	0.355141i	$-0.884433\pi$
$829$	− 20.4507i	− 0.710283i	0.934813	−	0.355141i	$-0.115567\pi$
$830$	0	0
$831$	−1.71084	−0.0593485
$832$	0	0
$833$	0	0
$834$	0	0
$835$	22.3891i	0.774806i
$836$	0	0
$837$	56.5972	1.95629
$838$	0	0
$839$	−26.9839	−0.931586	−0.465793	−	0.884894i	$-0.654231\pi$
$839$	−26.9839	−0.931586	−0.465793	+	0.884894i	$0.654231\pi$
$840$	0	0
$841$	−19.5268	−0.673338
$842$	0	0
$843$	−42.5107	−1.46415
$844$	0	0
$845$	− 10.2813i	− 0.353688i
$846$	0	0
$847$	0	0
$848$	0	0
$849$	25.2766	0.867490
$850$	0	0
$851$	− 16.6725i	− 0.571526i
$852$	0	0
$853$	30.5741i	1.04684i	0.852075	+	0.523419i	$0.175344\pi$
$853$	30.5741i	1.04684i	−0.852075	+	0.523419i	$0.824656\pi$
$854$	0	0
$855$	− 1.21204i	− 0.0414509i
$856$	0	0
$857$	18.2902i	0.624780i	0.949954	+	0.312390i	$0.101130\pi$
$857$	18.2902i	0.624780i	−0.949954	+	0.312390i	$0.898870\pi$
$858$	0	0
$859$	24.8198	0.846839	0.423420	−	0.905934i	$-0.360830\pi$
$859$	24.8198	0.846839	0.423420	+	0.905934i	$0.360830\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	21.4880i	0.731461i	0.930721	+	0.365731i	$0.119181\pi$
$863$	21.4880i	0.731461i	−0.930721	+	0.365731i	$0.880819\pi$
$864$	0	0
$865$	−14.8583	−0.505199
$866$	0	0
$867$	−27.7942	−0.943940
$868$	0	0
$869$	−18.2031	−0.617497
$870$	0	0
$871$	−14.6225	−0.495464
$872$	0	0
$873$	4.07868i	0.138043i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	43.5055	1.46908	0.734539	−	0.678567i	$-0.237399\pi$
$877$	43.5055	1.46908	0.734539	+	0.678567i	$0.237399\pi$
$878$	0	0
$879$	− 19.6552i	− 0.662952i
$880$	0	0
$881$	− 1.68783i	− 0.0568644i	−0.999596	−	0.0284322i	$-0.990949\pi$
$881$	− 1.68783i	− 0.0568644i	0.999596	−	0.0284322i	$-0.00905146\pi$
$882$	0	0
$883$	57.2325i	1.92603i	0.269453	+	0.963014i	$0.413157\pi$
$883$	57.2325i	1.92603i	−0.269453	+	0.963014i	$0.586843\pi$
$884$	0	0
$885$	− 2.98292i	− 0.100270i
$886$	0	0
$887$	−5.39918	−0.181287	−0.0906433	−	0.995883i	$-0.528892\pi$
$887$	−5.39918	−0.181287	−0.0906433	+	0.995883i	$0.528892\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	− 15.7092i	− 0.526280i
$892$	0	0
$893$	17.9791	0.601647
$894$	0	0
$895$	−20.1029	−0.671967
$896$	0	0
$897$	−16.1099	−0.537895
$898$	0	0
$899$	32.1971	1.07383
$900$	0	0
$901$	− 1.69876i	− 0.0565939i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−23.9690	−0.796756
$906$	0	0
$907$	37.9392i	1.25975i	0.776697	+	0.629875i	$0.216894\pi$
$907$	37.9392i	1.25975i	−0.776697	+	0.629875i	$0.783106\pi$
$908$	0	0
$909$	2.48014i	0.0822611i
$910$	0	0
$911$	− 30.3352i	− 1.00505i	−0.864563	−	0.502525i	$-0.832405\pi$
$911$	− 30.3352i	− 1.00505i	0.864563	−	0.502525i	$-0.167595\pi$
$912$	0	0
$913$	12.4746i	0.412848i
$914$	0	0
$915$	5.01864	0.165911
$916$	0	0
$917$	0	0
$918$	0	0
$919$	− 45.0441i	− 1.48587i	−0.669365	−	0.742933i	$-0.733434\pi$
$919$	− 45.0441i	− 1.48587i	0.669365	−	0.742933i	$-0.266566\pi$
$920$	0	0
$921$	12.8969	0.424967
$922$	0	0
$923$	−4.98327	−0.164026
$924$	0	0
$925$	2.81363	0.0925116
$926$	0	0
$927$	−0.409783	−0.0134590
$928$	0	0
$929$	− 8.99315i	− 0.295056i	−0.989058	−	0.147528i	$-0.952868\pi$
$929$	− 8.99315i	− 0.295056i	0.989058	−	0.147528i	$-0.0471316\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−52.8423	−1.72998
$934$	0	0
$935$	0.736149i	0.0240746i
$936$	0	0
$937$	− 2.80485i	− 0.0916305i	−0.998950	−	0.0458153i	$-0.985411\pi$
$937$	− 2.80485i	− 0.0916305i	0.998950	−	0.0458153i	$-0.0145886\pi$
$938$	0	0
$939$	44.5961i	1.45534i
$940$	0	0
$941$	− 19.8241i	− 0.646246i	−0.946357	−	0.323123i	$-0.895267\pi$
$941$	− 19.8241i	− 0.646246i	0.946357	−	0.323123i	$-0.104733\pi$
$942$	0	0
$943$	45.9744	1.49713
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 49.7939i	− 1.61808i	−0.587751	−	0.809042i	$-0.699986\pi$
$947$	− 49.7939i	− 1.61808i	0.587751	−	0.809042i	$-0.300014\pi$
$948$	0	0
$949$	4.82652	0.156675
$950$	0	0
$951$	−2.24940	−0.0729417
$952$	0	0
$953$	29.7448	0.963528	0.481764	−	0.876301i	$-0.339996\pi$
$953$	29.7448	0.963528	0.481764	+	0.876301i	$0.339996\pi$
$954$	0	0
$955$	−24.0247	−0.777421
$956$	0	0
$957$	− 9.87044i	− 0.319066i
$958$	0	0
$959$	0	0
$960$	0	0
$961$	78.4301	2.53000
$962$	0	0
$963$	− 3.85378i	− 0.124186i
$964$	0	0
$965$	25.5332i	0.821942i
$966$	0	0
$967$	− 35.8335i	− 1.15233i	−0.817334	−	0.576164i	$-0.804549\pi$
$967$	− 35.8335i	− 1.15233i	0.817334	−	0.576164i	$-0.195451\pi$
$968$	0	0
$969$	2.68892i	0.0863807i
$970$	0	0
$971$	3.11816	0.100066	0.0500332	−	0.998748i	$-0.484067\pi$
$971$	3.11816	0.100066	0.0500332	+	0.998748i	$0.484067\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	− 2.71870i	− 0.0870679i
$976$	0	0
$977$	−7.80319	−0.249646	−0.124823	−	0.992179i	$-0.539836\pi$
$977$	−7.80319	−0.249646	−0.124823	+	0.992179i	$0.539836\pi$
$978$	0	0
$979$	1.33954	0.0428120
$980$	0	0
$981$	2.96940	0.0948057
$982$	0	0
$983$	−19.7262	−0.629167	−0.314584	−	0.949230i	$-0.601865\pi$
$983$	−19.7262	−0.629167	−0.314584	+	0.949230i	$0.601865\pi$
$984$	0	0
$985$	− 12.0872i	− 0.385131i
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−2.48135	−0.0789023
$990$	0	0
$991$	− 31.2442i	− 0.992504i	−0.868179	−	0.496252i	$-0.834709\pi$
$991$	− 31.2442i	− 0.992504i	0.868179	−	0.496252i	$-0.165291\pi$
$992$	0	0
$993$	51.9649i	1.64905i
$994$	0	0
$995$	4.40204i	0.139554i
$996$	0	0
$997$	− 20.7304i	− 0.656540i	−0.944584	−	0.328270i	$-0.893534\pi$
$997$	− 20.7304i	− 0.656540i	0.944584	−	0.328270i	$-0.106466\pi$
$998$	0	0
$999$	−15.2228	−0.481627

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3920.2.k.a.2351.3	✓	8
4.3	odd	2		3920.2.k.c.2351.5	yes	8
7.6	odd	2		3920.2.k.c.2351.6	yes	8
28.27	even	2	inner	3920.2.k.a.2351.4	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3920.2.k.a.2351.3	✓	8	1.1	even	1	trivial
3920.2.k.a.2351.4	yes	8	28.27	even	2	inner
3920.2.k.c.2351.5	yes	8	4.3	odd	2
3920.2.k.c.2351.6	yes	8	7.6	odd	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.k (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.64885 q^{3} -1.00000i q^{5} -0.281305 q^{9} +O(q^{10})\)	\(q-1.64885 q^{3} -1.00000i q^{5} -0.281305 q^{9} +1.94495i q^{11} -1.64885i q^{13} +1.64885i q^{15} +0.378493i q^{17} -4.30864 q^{19} +5.92562i q^{23} -1.00000 q^{25} +5.41037 q^{27} +3.07786 q^{29} +10.4609 q^{31} -3.20692i q^{33} -2.81363 q^{37} +2.71870i q^{39} -7.75858i q^{41} +0.418749i q^{43} +0.281305i q^{45} -4.17279 q^{47} -0.624077i q^{51} -4.48822 q^{53} +1.94495 q^{55} +7.10429 q^{57} -1.80910 q^{59} -3.04373i q^{61} -1.64885 q^{65} -8.86831i q^{67} -9.77043i q^{69} -3.02228i q^{71} +2.92721i q^{73} +1.64885 q^{75} +9.35916i q^{79} -8.07695 q^{81} +6.41384 q^{83} +0.378493 q^{85} -5.07491 q^{87} -0.688730i q^{89} -17.2484 q^{93} +4.30864i q^{95} -14.4992i q^{97} -0.547123i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{3} + 16 q^{9}+O(q^{10}) \) 8 * q - 8 * q^3 + 16 * q^9	\( 8 q - 8 q^{3} + 16 q^{9} + 16 q^{19} - 8 q^{25} - 8 q^{27} + 8 q^{29} + 48 q^{31} + 24 q^{47} + 32 q^{53} - 8 q^{55} - 32 q^{57} + 16 q^{59} - 8 q^{65} + 8 q^{75} - 8 q^{81} - 48 q^{83} - 24 q^{85} - 24 q^{87} - 32 q^{93}+O(q^{100}) \) 8 * q - 8 * q^3 + 16 * q^9 + 16 * q^19 - 8 * q^25 - 8 * q^27 + 8 * q^29 + 48 * q^31 + 24 * q^47 + 32 * q^53 - 8 * q^55 - 32 * q^57 + 16 * q^59 - 8 * q^65 + 8 * q^75 - 8 * q^81 - 48 * q^83 - 24 * q^85 - 24 * q^87 - 32 * q^93

Introduction

L-functions

Modular forms

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Database

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