LMFDB - Embedded newform 4056.2.c.r.337.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4056,2,Mod(337,4056)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4056, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 1])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("4056.337"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 4056 = 2^{3} \cdot 3 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4056.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [12,0,12,0,0,0,0,0,12,0,0,0,0,0,0,0,-18,0,0,0,0,0,-24] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(23)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$32.3873230598$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 39x^{10} + 601x^{8} + 4599x^{6} + 17849x^{4} + 31203x^{2} + 16129 $ x^12 + 39x^10 + 601x^8 + 4599x^6 + 17849x^4 + 31203*x^2 + 16129
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			337.3
Root			$-2.72245i$ of defining polynomial
Character	$\chi$	$=$	4056.337
Dual form			4056.2.c.r.337.10

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -2.72245i q^{5} +3.31535i q^{7} +1.00000 q^{9} +0.656645i q^{11} -2.72245i q^{15} +1.16749 q^{17} -6.77137i q^{19} +3.31535i q^{21} -6.90073 q^{23} -2.41172 q^{25} +1.00000 q^{27} -5.82010 q^{29} +0.969574i q^{31} +0.656645i q^{33} +9.02586 q^{35} -9.93482i q^{37} -10.5937i q^{41} +5.07523 q^{43} -2.72245i q^{45} -8.24364i q^{47} -3.99153 q^{49} +1.16749 q^{51} +0.841166 q^{53} +1.78768 q^{55} -6.77137i q^{57} +0.128144i q^{59} +11.7223 q^{61} +3.31535i q^{63} +15.9422i q^{67} -6.90073 q^{69} -5.32056i q^{71} -3.75917i q^{73} -2.41172 q^{75} -2.17701 q^{77} +2.17759 q^{79} +1.00000 q^{81} +10.8237i q^{83} -3.17843i q^{85} -5.82010 q^{87} -4.09193i q^{89} +0.969574i q^{93} -18.4347 q^{95} -16.5008i q^{97} +0.656645i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 12 q^{3} + 12 q^{9} - 18 q^{17} - 24 q^{23} - 18 q^{25} + 12 q^{27} + 14 q^{29} + 12 q^{35} - 30 q^{43} - 26 q^{49} - 18 q^{51} + 44 q^{53} + 6 q^{55} + 50 q^{61} - 24 q^{69} - 18 q^{75} - 90 q^{77}+ \cdots - 94 q^{95}+O(q^{100}) $ 12 * q + 12 * q^3 + 12 * q^9 - 18 * q^17 - 24 * q^23 - 18 * q^25 + 12 * q^27 + 14 * q^29 + 12 * q^35 - 30 * q^43 - 26 * q^49 - 18 * q^51 + 44 * q^53 + 6 * q^55 + 50 * q^61 - 24 * q^69 - 18 * q^75 - 90 * q^77 + 28 * q^79 + 12 * q^81 + 14 * q^87 - 94 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4056\mathbb{Z}\right)^\times$.

$n$	$1015$	$2029$	$2705$	$3889$
$\chi(n)$	$1$	$1$	$1$	$-1$

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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	− 2.72245i	− 1.21752i	−0.793356	−	0.608758i	$-0.791668\pi$
$5$	− 2.72245i	− 1.21752i	0.793356	−	0.608758i	$-0.208332\pi$
$6$	0	0
$7$	3.31535i	1.25308i	0.779388	+	0.626542i	$0.215530\pi$
$7$	3.31535i	1.25308i	−0.779388	+	0.626542i	$0.784470\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	0.656645i	0.197986i	0.995088	+	0.0989930i	$0.0315621\pi$
$11$	0.656645i	0.197986i	−0.995088	+	0.0989930i	$0.968438\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	− 2.72245i	− 0.702933i
$16$	0	0
$17$	1.16749	0.283158	0.141579	−	0.989927i	$-0.454782\pi$
$17$	1.16749	0.283158	0.141579	+	0.989927i	$0.454782\pi$
$18$	0	0
$19$	− 6.77137i	− 1.55346i	−0.629835	−	0.776729i	$-0.716878\pi$
$19$	− 6.77137i	− 1.55346i	0.629835	−	0.776729i	$-0.283122\pi$
$20$	0	0
$21$	3.31535i	0.723468i
$22$	0	0
$23$	−6.90073	−1.43890	−0.719451	−	0.694543i	$-0.755607\pi$
$23$	−6.90073	−1.43890	−0.719451	+	0.694543i	$0.755607\pi$
$24$	0	0
$25$	−2.41172	−0.482344
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−5.82010	−1.08077	−0.540383	−	0.841419i	$-0.681721\pi$
$29$	−5.82010	−1.08077	−0.540383	+	0.841419i	$0.681721\pi$
$30$	0	0
$31$	0.969574i	0.174141i	0.996202	+	0.0870703i	$0.0277505\pi$
$31$	0.969574i	0.174141i	−0.996202	+	0.0870703i	$0.972250\pi$
$32$	0	0
$33$	0.656645i	0.114307i
$34$	0	0
$35$	9.02586	1.52565
$36$	0	0
$37$	− 9.93482i	− 1.63327i	−0.577151	−	0.816637i	$-0.695836\pi$
$37$	− 9.93482i	− 1.63327i	0.577151	−	0.816637i	$-0.304164\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 10.5937i	− 1.65445i	−0.561869	−	0.827227i	$-0.689917\pi$
$41$	− 10.5937i	− 1.65445i	0.561869	−	0.827227i	$-0.310083\pi$
$42$	0	0
$43$	5.07523	0.773965	0.386983	−	0.922087i	$-0.373517\pi$
$43$	5.07523	0.773965	0.386983	+	0.922087i	$0.373517\pi$
$44$	0	0
$45$	− 2.72245i	− 0.405839i
$46$	0	0
$47$	− 8.24364i	− 1.20246i	−0.799077	−	0.601229i	$-0.794678\pi$
$47$	− 8.24364i	− 1.20246i	0.799077	−	0.601229i	$-0.205322\pi$
$48$	0	0
$49$	−3.99153	−0.570218
$50$	0	0
$51$	1.16749	0.163481
$52$	0	0
$53$	0.841166	0.115543	0.0577715	−	0.998330i	$-0.481601\pi$
$53$	0.841166	0.115543	0.0577715	+	0.998330i	$0.481601\pi$
$54$	0	0
$55$	1.78768	0.241051
$56$	0	0
$57$	− 6.77137i	− 0.896889i
$58$	0	0
$59$	0.128144i	0.0166829i	0.999965	+	0.00834145i	$0.00265520\pi$
$59$	0.128144i	0.0166829i	−0.999965	+	0.00834145i	$0.997345\pi$
$60$	0	0
$61$	11.7223	1.50089	0.750443	−	0.660935i	$-0.229840\pi$
$61$	11.7223	1.50089	0.750443	+	0.660935i	$0.229840\pi$
$62$	0	0
$63$	3.31535i	0.417694i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	15.9422i	1.94765i	0.227303	+	0.973824i	$0.427009\pi$
$67$	15.9422i	1.94765i	−0.227303	+	0.973824i	$0.572991\pi$
$68$	0	0
$69$	−6.90073	−0.830750
$70$	0	0
$71$	− 5.32056i	− 0.631434i	−0.948853	−	0.315717i	$-0.897755\pi$
$71$	− 5.32056i	− 0.631434i	0.948853	−	0.315717i	$-0.102245\pi$
$72$	0	0
$73$	− 3.75917i	− 0.439977i	−0.975502	−	0.219989i	$-0.929398\pi$
$73$	− 3.75917i	− 0.439977i	0.975502	−	0.219989i	$-0.0706020\pi$
$74$	0	0
$75$	−2.41172	−0.278482
$76$	0	0
$77$	−2.17701	−0.248093
$78$	0	0
$79$	2.17759	0.244998	0.122499	−	0.992469i	$-0.460909\pi$
$79$	2.17759	0.244998	0.122499	+	0.992469i	$0.460909\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	10.8237i	1.18806i	0.804444	+	0.594029i	$0.202464\pi$
$83$	10.8237i	1.18806i	−0.804444	+	0.594029i	$0.797536\pi$
$84$	0	0
$85$	− 3.17843i	− 0.344749i
$86$	0	0
$87$	−5.82010	−0.623980
$88$	0	0
$89$	− 4.09193i	− 0.433743i	−0.976200	−	0.216872i	$-0.930415\pi$
$89$	− 4.09193i	− 0.433743i	0.976200	−	0.216872i	$-0.0695853\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0.969574i	0.100540i
$94$	0	0
$95$	−18.4347	−1.89136
$96$	0	0
$97$	− 16.5008i	− 1.67540i	−0.546127	−	0.837702i	$-0.683899\pi$
$97$	− 16.5008i	− 1.67540i	0.546127	−	0.837702i	$-0.316101\pi$
$98$	0	0
$99$	0.656645i	0.0659953i
$100$	0	0
$101$	−14.4134	−1.43419	−0.717096	−	0.696975i	$-0.754529\pi$
$101$	−14.4134	−1.43419	−0.717096	+	0.696975i	$0.754529\pi$
$102$	0	0
$103$	15.7309	1.55001	0.775007	−	0.631953i	$-0.217747\pi$
$103$	15.7309	1.55001	0.775007	+	0.631953i	$0.217747\pi$
$104$	0	0
$105$	9.02586	0.880834
$106$	0	0
$107$	9.27509	0.896656	0.448328	−	0.893869i	$-0.352020\pi$
$107$	9.27509	0.896656	0.448328	+	0.893869i	$0.352020\pi$
$108$	0	0
$109$	− 8.82889i	− 0.845655i	−0.906210	−	0.422827i	$-0.861038\pi$
$109$	− 8.82889i	− 0.845655i	0.906210	−	0.422827i	$-0.138962\pi$
$110$	0	0
$111$	− 9.93482i	− 0.942971i
$112$	0	0
$113$	6.45873	0.607586	0.303793	−	0.952738i	$-0.401747\pi$
$113$	6.45873	0.607586	0.303793	+	0.952738i	$0.401747\pi$
$114$	0	0
$115$	18.7869i	1.75189i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	3.87063i	0.354820i
$120$	0	0
$121$	10.5688	0.960802
$122$	0	0
$123$	− 10.5937i	− 0.955199i
$124$	0	0
$125$	− 7.04645i	− 0.630254i
$126$	0	0
$127$	4.90436	0.435191	0.217596	−	0.976039i	$-0.430179\pi$
$127$	4.90436	0.435191	0.217596	+	0.976039i	$0.430179\pi$
$128$	0	0
$129$	5.07523	0.446849
$130$	0	0
$131$	−13.9311	−1.21717	−0.608584	−	0.793489i	$-0.708262\pi$
$131$	−13.9311	−1.21717	−0.608584	+	0.793489i	$0.708262\pi$
$132$	0	0
$133$	22.4494	1.94661
$134$	0	0
$135$	− 2.72245i	− 0.234311i
$136$	0	0
$137$	− 17.4589i	− 1.49161i	−0.666163	−	0.745806i	$-0.732064\pi$
$137$	− 17.4589i	− 1.49161i	0.666163	−	0.745806i	$-0.267936\pi$
$138$	0	0
$139$	9.93881	0.842999	0.421499	−	0.906829i	$-0.361504\pi$
$139$	9.93881	0.842999	0.421499	+	0.906829i	$0.361504\pi$
$140$	0	0
$141$	− 8.24364i	− 0.694239i
$142$	0	0
$143$	0	0
$144$	0	0
$145$	15.8449i	1.31585i
$146$	0	0
$147$	−3.99153	−0.329216
$148$	0	0
$149$	14.7938i	1.21196i	0.795481	+	0.605979i	$0.207218\pi$
$149$	14.7938i	1.21196i	−0.795481	+	0.605979i	$0.792782\pi$
$150$	0	0
$151$	− 11.5920i	− 0.943340i	−0.881775	−	0.471670i	$-0.843651\pi$
$151$	− 11.5920i	− 0.943340i	0.881775	−	0.471670i	$-0.156349\pi$
$152$	0	0
$153$	1.16749	0.0943859
$154$	0	0
$155$	2.63961	0.212019
$156$	0	0
$157$	−16.3811	−1.30735	−0.653677	−	0.756773i	$-0.726775\pi$
$157$	−16.3811	−1.30735	−0.653677	+	0.756773i	$0.726775\pi$
$158$	0	0
$159$	0.841166	0.0667088
$160$	0	0
$161$	− 22.8783i	− 1.80306i
$162$	0	0
$163$	− 21.2719i	− 1.66615i	−0.553163	−	0.833073i	$-0.686579\pi$
$163$	− 21.2719i	− 1.66615i	0.553163	−	0.833073i	$-0.313421\pi$
$164$	0	0
$165$	1.78768	0.139171
$166$	0	0
$167$	− 10.5498i	− 0.816367i	−0.912900	−	0.408183i	$-0.866162\pi$
$167$	− 10.5498i	− 0.816367i	0.912900	−	0.408183i	$-0.133838\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	− 6.77137i	− 0.517819i
$172$	0	0
$173$	−20.4284	−1.55314	−0.776569	−	0.630032i	$-0.783042\pi$
$173$	−20.4284	−1.55314	−0.776569	+	0.630032i	$0.783042\pi$
$174$	0	0
$175$	− 7.99570i	− 0.604418i
$176$	0	0
$177$	0.128144i	0.00963188i
$178$	0	0
$179$	5.71037	0.426813	0.213406	−	0.976964i	$-0.431544\pi$
$179$	5.71037	0.426813	0.213406	+	0.976964i	$0.431544\pi$
$180$	0	0
$181$	−4.70324	−0.349589	−0.174795	−	0.984605i	$-0.555926\pi$
$181$	−4.70324	−0.349589	−0.174795	+	0.984605i	$0.555926\pi$
$182$	0	0
$183$	11.7223	0.866537
$184$	0	0
$185$	−27.0470	−1.98854
$186$	0	0
$187$	0.766626i	0.0560613i
$188$	0	0
$189$	3.31535i	0.241156i
$190$	0	0
$191$	14.7128	1.06458	0.532292	−	0.846561i	$-0.321331\pi$
$191$	14.7128	1.06458	0.532292	+	0.846561i	$0.321331\pi$
$192$	0	0
$193$	1.50643i	0.108435i	0.998529	+	0.0542177i	$0.0172665\pi$
$193$	1.50643i	0.108435i	−0.998529	+	0.0542177i	$0.982734\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	17.4470i	1.24305i	0.783395	+	0.621524i	$0.213486\pi$
$197$	17.4470i	1.24305i	−0.783395	+	0.621524i	$0.786514\pi$
$198$	0	0
$199$	−17.4032	−1.23368	−0.616841	−	0.787088i	$-0.711588\pi$
$199$	−17.4032	−1.23368	−0.616841	+	0.787088i	$0.711588\pi$
$200$	0	0
$201$	15.9422i	1.12448i
$202$	0	0
$203$	− 19.2957i	− 1.35429i
$204$	0	0
$205$	−28.8407	−2.01432
$206$	0	0
$207$	−6.90073	−0.479634
$208$	0	0
$209$	4.44638	0.307563
$210$	0	0
$211$	−12.4002	−0.853662	−0.426831	−	0.904331i	$-0.640370\pi$
$211$	−12.4002	−0.853662	−0.426831	+	0.904331i	$0.640370\pi$
$212$	0	0
$213$	− 5.32056i	− 0.364559i
$214$	0	0
$215$	− 13.8170i	− 0.942315i
$216$	0	0
$217$	−3.21447	−0.218213
$218$	0	0
$219$	− 3.75917i	− 0.254021i
$220$	0	0
$221$	0	0
$222$	0	0
$223$	12.5356i	0.839448i	0.907652	+	0.419724i	$0.137873\pi$
$223$	12.5356i	0.839448i	−0.907652	+	0.419724i	$0.862127\pi$
$224$	0	0
$225$	−2.41172	−0.160781
$226$	0	0
$227$	− 1.16802i	− 0.0775243i	−0.999248	−	0.0387622i	$-0.987659\pi$
$227$	− 1.16802i	− 0.0775243i	0.999248	−	0.0387622i	$-0.0123415\pi$
$228$	0	0
$229$	21.9158i	1.44824i	0.689675	+	0.724119i	$0.257753\pi$
$229$	21.9158i	1.44824i	−0.689675	+	0.724119i	$0.742247\pi$
$230$	0	0
$231$	−2.17701	−0.143237
$232$	0	0
$233$	3.00659	0.196968	0.0984842	−	0.995139i	$-0.468601\pi$
$233$	3.00659	0.196968	0.0984842	+	0.995139i	$0.468601\pi$
$234$	0	0
$235$	−22.4429	−1.46401
$236$	0	0
$237$	2.17759	0.141450
$238$	0	0
$239$	− 12.7853i	− 0.827015i	−0.910501	−	0.413508i	$-0.864303\pi$
$239$	− 12.7853i	− 0.827015i	0.910501	−	0.413508i	$-0.135697\pi$
$240$	0	0
$241$	23.6123i	1.52100i	0.649339	+	0.760499i	$0.275046\pi$
$241$	23.6123i	1.52100i	−0.649339	+	0.760499i	$0.724954\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	10.8667i	0.694249i
$246$	0	0
$247$	0	0
$248$	0	0
$249$	10.8237i	0.685926i
$250$	0	0
$251$	−8.22001	−0.518843	−0.259421	−	0.965764i	$-0.583532\pi$
$251$	−8.22001	−0.518843	−0.259421	+	0.965764i	$0.583532\pi$
$252$	0	0
$253$	− 4.53133i	− 0.284882i
$254$	0	0
$255$	− 3.17843i	− 0.199041i
$256$	0	0
$257$	−27.7850	−1.73318	−0.866592	−	0.499018i	$-0.833694\pi$
$257$	−27.7850	−1.73318	−0.866592	+	0.499018i	$0.833694\pi$
$258$	0	0
$259$	32.9374	2.04663
$260$	0	0
$261$	−5.82010	−0.360255
$262$	0	0
$263$	10.5270	0.649120	0.324560	−	0.945865i	$-0.394784\pi$
$263$	10.5270	0.649120	0.324560	+	0.945865i	$0.394784\pi$
$264$	0	0
$265$	− 2.29003i	− 0.140676i
$266$	0	0
$267$	− 4.09193i	− 0.250422i
$268$	0	0
$269$	−2.51426	−0.153297	−0.0766486	−	0.997058i	$-0.524422\pi$
$269$	−2.51426	−0.153297	−0.0766486	+	0.997058i	$0.524422\pi$
$270$	0	0
$271$	− 4.83460i	− 0.293681i	−0.989160	−	0.146841i	$-0.953090\pi$
$271$	− 4.83460i	− 0.293681i	0.989160	−	0.146841i	$-0.0469104\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 1.58365i	− 0.0954974i
$276$	0	0
$277$	15.4274	0.926941	0.463471	−	0.886112i	$-0.346604\pi$
$277$	15.4274	0.926941	0.463471	+	0.886112i	$0.346604\pi$
$278$	0	0
$279$	0.969574i	0.0580469i
$280$	0	0
$281$	− 6.10796i	− 0.364370i	−0.983264	−	0.182185i	$-0.941683\pi$
$281$	− 6.10796i	− 0.364370i	0.983264	−	0.182185i	$-0.0583170\pi$
$282$	0	0
$283$	−11.3138	−0.672538	−0.336269	−	0.941766i	$-0.609165\pi$
$283$	−11.3138	−0.672538	−0.336269	+	0.941766i	$0.609165\pi$
$284$	0	0
$285$	−18.4347	−1.09198
$286$	0	0
$287$	35.1217	2.07317
$288$	0	0
$289$	−15.6370	−0.919822
$290$	0	0
$291$	− 16.5008i	− 0.967295i
$292$	0	0
$293$	28.0013i	1.63585i	0.575322	+	0.817927i	$0.304877\pi$
$293$	28.0013i	1.63585i	−0.575322	+	0.817927i	$0.695123\pi$
$294$	0	0
$295$	0.348865	0.0203117
$296$	0	0
$297$	0.656645i	0.0381024i
$298$	0	0
$299$	0	0
$300$	0	0
$301$	16.8261i	0.969843i
$302$	0	0
$303$	−14.4134	−0.828031
$304$	0	0
$305$	− 31.9134i	− 1.82735i
$306$	0	0
$307$	18.6077i	1.06199i	0.847373	+	0.530997i	$0.178183\pi$
$307$	18.6077i	1.06199i	−0.847373	+	0.530997i	$0.821817\pi$
$308$	0	0
$309$	15.7309	0.894900
$310$	0	0
$311$	−25.9076	−1.46909	−0.734543	−	0.678563i	$-0.762603\pi$
$311$	−25.9076	−1.46909	−0.734543	+	0.678563i	$0.762603\pi$
$312$	0	0
$313$	−0.799031	−0.0451639	−0.0225819	−	0.999745i	$-0.507189\pi$
$313$	−0.799031	−0.0451639	−0.0225819	+	0.999745i	$0.507189\pi$
$314$	0	0
$315$	9.02586	0.508550
$316$	0	0
$317$	− 32.7072i	− 1.83702i	−0.395400	−	0.918509i	$-0.629394\pi$
$317$	− 32.7072i	− 1.83702i	0.395400	−	0.918509i	$-0.370606\pi$
$318$	0	0
$319$	− 3.82174i	− 0.213976i
$320$	0	0
$321$	9.27509	0.517685
$322$	0	0
$323$	− 7.90550i	− 0.439874i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 8.82889i	− 0.488239i
$328$	0	0
$329$	27.3305	1.50678
$330$	0	0
$331$	− 5.69662i	− 0.313115i	−0.987669	−	0.156557i	$-0.949960\pi$
$331$	− 5.69662i	− 0.313115i	0.987669	−	0.156557i	$-0.0500396\pi$
$332$	0	0
$333$	− 9.93482i	− 0.544425i
$334$	0	0
$335$	43.4018	2.37129
$336$	0	0
$337$	10.4046	0.566773	0.283387	−	0.959006i	$-0.408542\pi$
$337$	10.4046	0.566773	0.283387	+	0.959006i	$0.408542\pi$
$338$	0	0
$339$	6.45873	0.350790
$340$	0	0
$341$	−0.636666	−0.0344774
$342$	0	0
$343$	9.97413i	0.538553i
$344$	0	0
$345$	18.7869i	1.01145i
$346$	0	0
$347$	9.76863	0.524408	0.262204	−	0.965013i	$-0.415551\pi$
$347$	9.76863	0.524408	0.262204	+	0.965013i	$0.415551\pi$
$348$	0	0
$349$	− 6.11185i	− 0.327160i	−0.986530	−	0.163580i	$-0.947696\pi$
$349$	− 6.11185i	− 0.327160i	0.986530	−	0.163580i	$-0.0523041\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 21.1136i	− 1.12376i	−0.827217	−	0.561882i	$-0.810077\pi$
$353$	− 21.1136i	− 1.12376i	0.827217	−	0.561882i	$-0.189923\pi$
$354$	0	0
$355$	−14.4849	−0.768781
$356$	0	0
$357$	3.87063i	0.204856i
$358$	0	0
$359$	2.37346i	0.125267i	0.998037	+	0.0626333i	$0.0199499\pi$
$359$	2.37346i	0.125267i	−0.998037	+	0.0626333i	$0.980050\pi$
$360$	0	0
$361$	−26.8514	−1.41323
$362$	0	0
$363$	10.5688	0.554719
$364$	0	0
$365$	−10.2341	−0.535679
$366$	0	0
$367$	−34.1977	−1.78511	−0.892553	−	0.450943i	$-0.851088\pi$
$367$	−34.1977	−1.78511	−0.892553	+	0.450943i	$0.851088\pi$
$368$	0	0
$369$	− 10.5937i	− 0.551484i
$370$	0	0
$371$	2.78876i	0.144785i
$372$	0	0
$373$	4.01148	0.207707	0.103853	−	0.994593i	$-0.466883\pi$
$373$	4.01148	0.207707	0.103853	+	0.994593i	$0.466883\pi$
$374$	0	0
$375$	− 7.04645i	− 0.363877i
$376$	0	0
$377$	0	0
$378$	0	0
$379$	0.282802i	0.0145265i	0.999974	+	0.00726327i	$0.00231199\pi$
$379$	0.282802i	0.0145265i	−0.999974	+	0.00726327i	$0.997688\pi$
$380$	0	0
$381$	4.90436	0.251258
$382$	0	0
$383$	11.7130i	0.598508i	0.954173	+	0.299254i	$0.0967379\pi$
$383$	11.7130i	0.598508i	−0.954173	+	0.299254i	$0.903262\pi$
$384$	0	0
$385$	5.92679i	0.302057i
$386$	0	0
$387$	5.07523	0.257988
$388$	0	0
$389$	11.7485	0.595672	0.297836	−	0.954617i	$-0.403735\pi$
$389$	11.7485	0.595672	0.297836	+	0.954617i	$0.403735\pi$
$390$	0	0
$391$	−8.05653	−0.407436
$392$	0	0
$393$	−13.9311	−0.702733
$394$	0	0
$395$	− 5.92838i	− 0.298289i
$396$	0	0
$397$	− 1.55897i	− 0.0782427i	−0.999234	−	0.0391213i	$-0.987544\pi$
$397$	− 1.55897i	− 0.0782427i	0.999234	−	0.0391213i	$-0.0124559\pi$
$398$	0	0
$399$	22.4494	1.12388
$400$	0	0
$401$	− 5.75468i	− 0.287375i	−0.989623	−	0.143687i	$-0.954104\pi$
$401$	− 5.75468i	− 0.287375i	0.989623	−	0.143687i	$-0.0458960\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	− 2.72245i	− 0.135280i
$406$	0	0
$407$	6.52365	0.323365
$408$	0	0
$409$	− 22.5609i	− 1.11557i	−0.829987	−	0.557783i	$-0.811652\pi$
$409$	− 22.5609i	− 1.11557i	0.829987	−	0.557783i	$-0.188348\pi$
$410$	0	0
$411$	− 17.4589i	− 0.861183i
$412$	0	0
$413$	−0.424841	−0.0209051
$414$	0	0
$415$	29.4670	1.44648
$416$	0	0
$417$	9.93881	0.486706
$418$	0	0
$419$	14.5676	0.711675	0.355837	−	0.934548i	$-0.384196\pi$
$419$	14.5676	0.711675	0.355837	+	0.934548i	$0.384196\pi$
$420$	0	0
$421$	40.2281i	1.96060i	0.197525	+	0.980298i	$0.436710\pi$
$421$	40.2281i	1.96060i	−0.197525	+	0.980298i	$0.563290\pi$
$422$	0	0
$423$	− 8.24364i	− 0.400819i
$424$	0	0
$425$	−2.81566	−0.136580
$426$	0	0
$427$	38.8635i	1.88074i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	14.9559i	0.720402i	0.932875	+	0.360201i	$0.117292\pi$
$431$	14.9559i	0.720402i	−0.932875	+	0.360201i	$0.882708\pi$
$432$	0	0
$433$	30.1200	1.44748	0.723738	−	0.690074i	$-0.242422\pi$
$433$	30.1200	1.44748	0.723738	+	0.690074i	$0.242422\pi$
$434$	0	0
$435$	15.8449i	0.759706i
$436$	0	0
$437$	46.7274i	2.23527i
$438$	0	0
$439$	−10.7914	−0.515043	−0.257522	−	0.966273i	$-0.582906\pi$
$439$	−10.7914	−0.515043	−0.257522	+	0.966273i	$0.582906\pi$
$440$	0	0
$441$	−3.99153	−0.190073
$442$	0	0
$443$	30.6498	1.45622	0.728108	−	0.685463i	$-0.240400\pi$
$443$	30.6498	1.45622	0.728108	+	0.685463i	$0.240400\pi$
$444$	0	0
$445$	−11.1401	−0.528089
$446$	0	0
$447$	14.7938i	0.699724i
$448$	0	0
$449$	12.3953i	0.584970i	0.956270	+	0.292485i	$0.0944821\pi$
$449$	12.3953i	0.584970i	−0.956270	+	0.292485i	$0.905518\pi$
$450$	0	0
$451$	6.95628	0.327558
$452$	0	0
$453$	− 11.5920i	− 0.544638i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 21.5121i	− 1.00629i	−0.864201	−	0.503147i	$-0.832175\pi$
$457$	− 21.5121i	− 1.00629i	0.864201	−	0.503147i	$-0.167825\pi$
$458$	0	0
$459$	1.16749	0.0544938
$460$	0	0
$461$	20.2208i	0.941777i	0.882193	+	0.470889i	$0.156067\pi$
$461$	20.2208i	0.941777i	−0.882193	+	0.470889i	$0.843933\pi$
$462$	0	0
$463$	− 11.2424i	− 0.522477i	−0.965274	−	0.261239i	$-0.915869\pi$
$463$	− 11.2424i	− 0.522477i	0.965274	−	0.261239i	$-0.0841310\pi$
$464$	0	0
$465$	2.63961	0.122409
$466$	0	0
$467$	27.1166	1.25481	0.627403	−	0.778695i	$-0.284118\pi$
$467$	27.1166	1.25481	0.627403	+	0.778695i	$0.284118\pi$
$468$	0	0
$469$	−52.8539	−2.44057
$470$	0	0
$471$	−16.3811	−0.754802
$472$	0	0
$473$	3.33262i	0.153234i
$474$	0	0
$475$	16.3307i	0.749302i
$476$	0	0
$477$	0.841166	0.0385144
$478$	0	0
$479$	− 13.8701i	− 0.633743i	−0.948468	−	0.316872i	$-0.897368\pi$
$479$	− 13.8701i	− 0.633743i	0.948468	−	0.316872i	$-0.102632\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	− 22.8783i	− 1.04100i
$484$	0	0
$485$	−44.9226	−2.03983
$486$	0	0
$487$	0.0497164i	0.00225287i	0.999999	+	0.00112643i	$0.000358555\pi$
$487$	0.0497164i	0.00225287i	−0.999999	+	0.00112643i	$0.999641\pi$
$488$	0	0
$489$	− 21.2719i	− 0.961950i
$490$	0	0
$491$	21.9646	0.991248	0.495624	−	0.868537i	$-0.334939\pi$
$491$	21.9646	0.991248	0.495624	+	0.868537i	$0.334939\pi$
$492$	0	0
$493$	−6.79491	−0.306027
$494$	0	0
$495$	1.78768	0.0803503
$496$	0	0
$497$	17.6395	0.791240
$498$	0	0
$499$	35.8347i	1.60418i	0.597202	+	0.802091i	$0.296279\pi$
$499$	35.8347i	1.60418i	−0.597202	+	0.802091i	$0.703721\pi$
$500$	0	0
$501$	− 10.5498i	− 0.471329i
$502$	0	0
$503$	−0.104369	−0.00465358	−0.00232679	−	0.999997i	$-0.500741\pi$
$503$	−0.104369	−0.00465358	−0.00232679	+	0.999997i	$0.500741\pi$
$504$	0	0
$505$	39.2399i	1.74615i
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 1.19452i	− 0.0529464i	−0.999650	−	0.0264732i	$-0.991572\pi$
$509$	− 1.19452i	− 0.0529464i	0.999650	−	0.0264732i	$-0.00842766\pi$
$510$	0	0
$511$	12.4629	0.551328
$512$	0	0
$513$	− 6.77137i	− 0.298963i
$514$	0	0
$515$	− 42.8266i	− 1.88717i
$516$	0	0
$517$	5.41314	0.238070
$518$	0	0
$519$	−20.4284	−0.896705
$520$	0	0
$521$	22.8446	1.00084	0.500420	−	0.865783i	$-0.333179\pi$
$521$	22.8446	1.00084	0.500420	+	0.865783i	$0.333179\pi$
$522$	0	0
$523$	38.3582	1.67728	0.838642	−	0.544682i	$-0.183350\pi$
$523$	38.3582	1.67728	0.838642	+	0.544682i	$0.183350\pi$
$524$	0	0
$525$	− 7.99570i	− 0.348961i
$526$	0	0
$527$	1.13197i	0.0493093i
$528$	0	0
$529$	24.6201	1.07044
$530$	0	0
$531$	0.128144i	0.00556097i
$532$	0	0
$533$	0	0
$534$	0	0
$535$	− 25.2509i	− 1.09169i
$536$	0	0
$537$	5.71037	0.246420
$538$	0	0
$539$	− 2.62102i	− 0.112895i
$540$	0	0
$541$	− 10.5126i	− 0.451970i	−0.974131	−	0.225985i	$-0.927440\pi$
$541$	− 10.5126i	− 0.451970i	0.974131	−	0.225985i	$-0.0725601\pi$
$542$	0	0
$543$	−4.70324	−0.201835
$544$	0	0
$545$	−24.0362	−1.02960
$546$	0	0
$547$	23.9358	1.02342	0.511711	−	0.859158i	$-0.329012\pi$
$547$	23.9358	1.02342	0.511711	+	0.859158i	$0.329012\pi$
$548$	0	0
$549$	11.7223	0.500296
$550$	0	0
$551$	39.4100i	1.67892i
$552$	0	0
$553$	7.21948i	0.307003i
$554$	0	0
$555$	−27.0470	−1.14808
$556$	0	0
$557$	0.843060i	0.0357216i	0.999840	+	0.0178608i	$0.00568557\pi$
$557$	0.843060i	0.0357216i	−0.999840	+	0.0178608i	$0.994314\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0.766626i	0.0323670i
$562$	0	0
$563$	18.4169	0.776179	0.388090	−	0.921622i	$-0.373135\pi$
$563$	18.4169	0.776179	0.388090	+	0.921622i	$0.373135\pi$
$564$	0	0
$565$	− 17.5836i	− 0.739746i
$566$	0	0
$567$	3.31535i	0.139231i
$568$	0	0
$569$	27.6396	1.15871	0.579356	−	0.815075i	$-0.303304\pi$
$569$	27.6396	1.15871	0.579356	+	0.815075i	$0.303304\pi$
$570$	0	0
$571$	−8.23866	−0.344777	−0.172389	−	0.985029i	$-0.555148\pi$
$571$	−8.23866	−0.344777	−0.172389	+	0.985029i	$0.555148\pi$
$572$	0	0
$573$	14.7128	0.614638
$574$	0	0
$575$	16.6426	0.694046
$576$	0	0
$577$	2.60340i	0.108381i	0.998531	+	0.0541906i	$0.0172578\pi$
$577$	2.60340i	0.108381i	−0.998531	+	0.0541906i	$0.982742\pi$
$578$	0	0
$579$	1.50643i	0.0626052i
$580$	0	0
$581$	−35.8844	−1.48874
$582$	0	0
$583$	0.552348i	0.0228759i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	8.62011i	0.355790i	0.984049	+	0.177895i	$0.0569287\pi$
$587$	8.62011i	0.355790i	−0.984049	+	0.177895i	$0.943071\pi$
$588$	0	0
$589$	6.56534	0.270520
$590$	0	0
$591$	17.4470i	0.717674i
$592$	0	0
$593$	39.3090i	1.61423i	0.590395	+	0.807114i	$0.298972\pi$
$593$	39.3090i	1.61423i	−0.590395	+	0.807114i	$0.701028\pi$
$594$	0	0
$595$	10.5376	0.431999
$596$	0	0
$597$	−17.4032	−0.712266
$598$	0	0
$599$	38.7451	1.58308	0.791542	−	0.611115i	$-0.209279\pi$
$599$	38.7451	1.58308	0.791542	+	0.611115i	$0.209279\pi$
$600$	0	0
$601$	−31.5673	−1.28766	−0.643828	−	0.765170i	$-0.722655\pi$
$601$	−31.5673	−1.28766	−0.643828	+	0.765170i	$0.722655\pi$
$602$	0	0
$603$	15.9422i	0.649216i
$604$	0	0
$605$	− 28.7731i	− 1.16979i
$606$	0	0
$607$	−39.5512	−1.60533	−0.802667	−	0.596427i	$-0.796586\pi$
$607$	−39.5512	−1.60533	−0.802667	+	0.596427i	$0.796586\pi$
$608$	0	0
$609$	− 19.2957i	− 0.781899i
$610$	0	0
$611$	0	0
$612$	0	0
$613$	42.8092i	1.72905i	0.502592	+	0.864524i	$0.332380\pi$
$613$	42.8092i	1.72905i	−0.502592	+	0.864524i	$0.667620\pi$
$614$	0	0
$615$	−28.8407	−1.16297
$616$	0	0
$617$	− 19.9368i	− 0.802626i	−0.915941	−	0.401313i	$-0.868554\pi$
$617$	− 19.9368i	− 0.802626i	0.915941	−	0.401313i	$-0.131446\pi$
$618$	0	0
$619$	38.1555i	1.53360i	0.641887	+	0.766799i	$0.278152\pi$
$619$	38.1555i	1.53360i	−0.641887	+	0.766799i	$0.721848\pi$
$620$	0	0
$621$	−6.90073	−0.276917
$622$	0	0
$623$	13.5662	0.543516
$624$	0	0
$625$	−31.2422	−1.24969
$626$	0	0
$627$	4.44638	0.177571
$628$	0	0
$629$	− 11.5988i	− 0.462474i
$630$	0	0
$631$	− 30.6027i	− 1.21827i	−0.793065	−	0.609137i	$-0.791516\pi$
$631$	− 30.6027i	− 1.21827i	0.793065	−	0.609137i	$-0.208484\pi$
$632$	0	0
$633$	−12.4002	−0.492862
$634$	0	0
$635$	− 13.3519i	− 0.529852i
$636$	0	0
$637$	0	0
$638$	0	0
$639$	− 5.32056i	− 0.210478i
$640$	0	0
$641$	−25.7211	−1.01592	−0.507962	−	0.861379i	$-0.669601\pi$
$641$	−25.7211	−1.01592	−0.507962	+	0.861379i	$0.669601\pi$
$642$	0	0
$643$	30.1373i	1.18850i	0.804281	+	0.594249i	$0.202551\pi$
$643$	30.1373i	1.18850i	−0.804281	+	0.594249i	$0.797449\pi$
$644$	0	0
$645$	− 13.8170i	− 0.544046i
$646$	0	0
$647$	−6.75127	−0.265420	−0.132710	−	0.991155i	$-0.542368\pi$
$647$	−6.75127	−0.265420	−0.132710	+	0.991155i	$0.542368\pi$
$648$	0	0
$649$	−0.0841450	−0.00330298
$650$	0	0
$651$	−3.21447	−0.125985
$652$	0	0
$653$	−1.20999	−0.0473507	−0.0236754	−	0.999720i	$-0.507537\pi$
$653$	−1.20999	−0.0473507	−0.0236754	+	0.999720i	$0.507537\pi$
$654$	0	0
$655$	37.9268i	1.48192i
$656$	0	0
$657$	− 3.75917i	− 0.146659i
$658$	0	0
$659$	−13.6149	−0.530361	−0.265181	−	0.964199i	$-0.585432\pi$
$659$	−13.6149	−0.530361	−0.265181	+	0.964199i	$0.585432\pi$
$660$	0	0
$661$	− 10.7711i	− 0.418949i	−0.977814	−	0.209474i	$-0.932825\pi$
$661$	− 10.7711i	− 0.418949i	0.977814	−	0.209474i	$-0.0671753\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 61.1174i	− 2.37003i
$666$	0	0
$667$	40.1629	1.55512
$668$	0	0
$669$	12.5356i	0.484655i
$670$	0	0
$671$	7.69739i	0.297155i
$672$	0	0
$673$	46.0338	1.77447	0.887236	−	0.461316i	$-0.152623\pi$
$673$	46.0338	1.77447	0.887236	+	0.461316i	$0.152623\pi$
$674$	0	0
$675$	−2.41172	−0.0928272
$676$	0	0
$677$	1.84261	0.0708172	0.0354086	−	0.999373i	$-0.488727\pi$
$677$	1.84261	0.0708172	0.0354086	+	0.999373i	$0.488727\pi$
$678$	0	0
$679$	54.7059	2.09942
$680$	0	0
$681$	− 1.16802i	− 0.0447587i
$682$	0	0
$683$	− 23.7758i	− 0.909757i	−0.890554	−	0.454878i	$-0.849683\pi$
$683$	− 23.7758i	− 0.909757i	0.890554	−	0.454878i	$-0.150317\pi$
$684$	0	0
$685$	−47.5309	−1.81606
$686$	0	0
$687$	21.9158i	0.836140i
$688$	0	0
$689$	0	0
$690$	0	0
$691$	− 1.67973i	− 0.0639001i	−0.999489	−	0.0319501i	$-0.989828\pi$
$691$	− 1.67973i	− 0.0639001i	0.999489	−	0.0319501i	$-0.0101718\pi$
$692$	0	0
$693$	−2.17701	−0.0826976
$694$	0	0
$695$	− 27.0579i	− 1.02636i
$696$	0	0
$697$	− 12.3680i	− 0.468471i
$698$	0	0
$699$	3.00659	0.113720
$700$	0	0
$701$	−17.8738	−0.675086	−0.337543	−	0.941310i	$-0.609596\pi$
$701$	−17.8738	−0.675086	−0.337543	+	0.941310i	$0.609596\pi$
$702$	0	0
$703$	−67.2723	−2.53722
$704$	0	0
$705$	−22.4429	−0.845247
$706$	0	0
$707$	− 47.7856i	− 1.79716i
$708$	0	0
$709$	4.27227i	0.160448i	0.996777	+	0.0802242i	$0.0255636\pi$
$709$	4.27227i	0.160448i	−0.996777	+	0.0802242i	$0.974436\pi$
$710$	0	0
$711$	2.17759	0.0816661
$712$	0	0
$713$	− 6.69077i	− 0.250571i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 12.7853i	− 0.477478i
$718$	0	0
$719$	27.5341	1.02685	0.513424	−	0.858135i	$-0.328377\pi$
$719$	27.5341	1.02685	0.513424	+	0.858135i	$0.328377\pi$
$720$	0	0
$721$	52.1534i	1.94230i
$722$	0	0
$723$	23.6123i	0.878149i
$724$	0	0
$725$	14.0365	0.521301
$726$	0	0
$727$	28.2674	1.04838	0.524189	−	0.851602i	$-0.324369\pi$
$727$	28.2674	1.04838	0.524189	+	0.851602i	$0.324369\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	5.92528	0.219154
$732$	0	0
$733$	− 25.2904i	− 0.934123i	−0.884225	−	0.467062i	$-0.845313\pi$
$733$	− 25.2904i	− 0.934123i	0.884225	−	0.467062i	$-0.154687\pi$
$734$	0	0
$735$	10.8667i	0.400825i
$736$	0	0
$737$	−10.4684	−0.385607
$738$	0	0
$739$	8.06509i	0.296679i	0.988936	+	0.148340i	$0.0473929\pi$
$739$	8.06509i	0.296679i	−0.988936	+	0.148340i	$0.952607\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	14.2716i	0.523575i	0.965126	+	0.261787i	$0.0843119\pi$
$743$	14.2716i	0.523575i	−0.965126	+	0.261787i	$0.915688\pi$
$744$	0	0
$745$	40.2754	1.47558
$746$	0	0
$747$	10.8237i	0.396019i
$748$	0	0
$749$	30.7501i	1.12359i
$750$	0	0
$751$	−33.0043	−1.20434	−0.602171	−	0.798367i	$-0.705698\pi$
$751$	−33.0043	−1.20434	−0.602171	+	0.798367i	$0.705698\pi$
$752$	0	0
$753$	−8.22001	−0.299554
$754$	0	0
$755$	−31.5585	−1.14853
$756$	0	0
$757$	35.2365	1.28069	0.640347	−	0.768086i	$-0.278791\pi$
$757$	35.2365	1.28069	0.640347	+	0.768086i	$0.278791\pi$
$758$	0	0
$759$	− 4.53133i	− 0.164477i
$760$	0	0
$761$	30.7212i	1.11364i	0.830632	+	0.556822i	$0.187979\pi$
$761$	30.7212i	1.11364i	−0.830632	+	0.556822i	$0.812021\pi$
$762$	0	0
$763$	29.2709	1.05968
$764$	0	0
$765$	− 3.17843i	− 0.114916i
$766$	0	0
$767$	0	0
$768$	0	0
$769$	6.95972i	0.250974i	0.992095	+	0.125487i	$0.0400494\pi$
$769$	6.95972i	0.250974i	−0.992095	+	0.125487i	$0.959951\pi$
$770$	0	0
$771$	−27.7850	−1.00065
$772$	0	0
$773$	40.5717i	1.45926i	0.683840	+	0.729632i	$0.260308\pi$
$773$	40.5717i	1.45926i	−0.683840	+	0.729632i	$0.739692\pi$
$774$	0	0
$775$	− 2.33834i	− 0.0839957i
$776$	0	0
$777$	32.9374	1.18162
$778$	0	0
$779$	−71.7336	−2.57012
$780$	0	0
$781$	3.49372	0.125015
$782$	0	0
$783$	−5.82010	−0.207993
$784$	0	0
$785$	44.5967i	1.59172i
$786$	0	0
$787$	− 4.88731i	− 0.174214i	−0.996199	−	0.0871068i	$-0.972238\pi$
$787$	− 4.88731i	− 0.174214i	0.996199	−	0.0871068i	$-0.0277622\pi$
$788$	0	0
$789$	10.5270	0.374769
$790$	0	0
$791$	21.4129i	0.761356i
$792$	0	0
$793$	0	0
$794$	0	0
$795$	− 2.29003i	− 0.0812191i
$796$	0	0
$797$	24.5982	0.871314	0.435657	−	0.900113i	$-0.356516\pi$
$797$	24.5982	0.871314	0.435657	+	0.900113i	$0.356516\pi$
$798$	0	0
$799$	− 9.62436i	− 0.340485i
$800$	0	0
$801$	− 4.09193i	− 0.144581i
$802$	0	0
$803$	2.46844	0.0871093
$804$	0	0
$805$	−62.2850	−2.19526
$806$	0	0
$807$	−2.51426	−0.0885061
$808$	0	0
$809$	13.6525	0.479996	0.239998	−	0.970773i	$-0.422853\pi$
$809$	13.6525	0.479996	0.239998	+	0.970773i	$0.422853\pi$
$810$	0	0
$811$	44.2951i	1.55541i	0.628628	+	0.777706i	$0.283617\pi$
$811$	44.2951i	1.55541i	−0.628628	+	0.777706i	$0.716383\pi$
$812$	0	0
$813$	− 4.83460i	− 0.169557i
$814$	0	0
$815$	−57.9117	−2.02856
$816$	0	0
$817$	− 34.3662i	− 1.20232i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 32.2133i	− 1.12425i	−0.827051	−	0.562127i	$-0.809983\pi$
$821$	− 32.2133i	− 1.12425i	0.827051	−	0.562127i	$-0.190017\pi$
$822$	0	0
$823$	28.6321	0.998052	0.499026	−	0.866587i	$-0.333691\pi$
$823$	28.6321	0.998052	0.499026	+	0.866587i	$0.333691\pi$
$824$	0	0
$825$	− 1.58365i	− 0.0551355i
$826$	0	0
$827$	− 40.8863i	− 1.42176i	−0.703315	−	0.710879i	$-0.748298\pi$
$827$	− 40.8863i	− 1.42176i	0.703315	−	0.710879i	$-0.251702\pi$
$828$	0	0
$829$	−10.7016	−0.371681	−0.185841	−	0.982580i	$-0.559501\pi$
$829$	−10.7016	−0.371681	−0.185841	+	0.982580i	$0.559501\pi$
$830$	0	0
$831$	15.4274	0.535170
$832$	0	0
$833$	−4.66007	−0.161462
$834$	0	0
$835$	−28.7212	−0.993939
$836$	0	0
$837$	0.969574i	0.0335134i
$838$	0	0
$839$	− 16.1362i	− 0.557085i	−0.960424	−	0.278543i	$-0.910149\pi$
$839$	− 16.1362i	− 0.557085i	0.960424	−	0.278543i	$-0.0898513\pi$
$840$	0	0
$841$	4.87357	0.168054
$842$	0	0
$843$	− 6.10796i	− 0.210369i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	35.0393i	1.20396i
$848$	0	0
$849$	−11.3138	−0.388290
$850$	0	0
$851$	68.5575i	2.35012i
$852$	0	0
$853$	− 15.7776i	− 0.540215i	−0.962830	−	0.270108i	$-0.912941\pi$
$853$	− 15.7776i	− 0.540215i	0.962830	−	0.270108i	$-0.0870593\pi$
$854$	0	0
$855$	−18.4347	−0.630453
$856$	0	0
$857$	−21.5463	−0.736006	−0.368003	−	0.929825i	$-0.619958\pi$
$857$	−21.5463	−0.736006	−0.368003	+	0.929825i	$0.619958\pi$
$858$	0	0
$859$	−4.07749	−0.139122	−0.0695611	−	0.997578i	$-0.522160\pi$
$859$	−4.07749	−0.139122	−0.0695611	+	0.997578i	$0.522160\pi$
$860$	0	0
$861$	35.1217	1.19694
$862$	0	0
$863$	− 17.5883i	− 0.598712i	−0.954141	−	0.299356i	$-0.903228\pi$
$863$	− 17.5883i	− 0.598712i	0.954141	−	0.299356i	$-0.0967718\pi$
$864$	0	0
$865$	55.6151i	1.89097i
$866$	0	0
$867$	−15.6370	−0.531059
$868$	0	0
$869$	1.42991i	0.0485062i
$870$	0	0
$871$	0	0
$872$	0	0
$873$	− 16.5008i	− 0.558468i
$874$	0	0
$875$	23.3614	0.789761
$876$	0	0
$877$	5.07451i	0.171354i	0.996323	+	0.0856770i	$0.0273053\pi$
$877$	5.07451i	0.171354i	−0.996323	+	0.0856770i	$0.972695\pi$
$878$	0	0
$879$	28.0013i	0.944461i
$880$	0	0
$881$	−49.8471	−1.67939	−0.839696	−	0.543057i	$-0.817267\pi$
$881$	−49.8471	−1.67939	−0.839696	+	0.543057i	$0.817267\pi$
$882$	0	0
$883$	−46.7712	−1.57398	−0.786989	−	0.616967i	$-0.788361\pi$
$883$	−46.7712	−1.57398	−0.786989	+	0.616967i	$0.788361\pi$
$884$	0	0
$885$	0.348865	0.0117270
$886$	0	0
$887$	−19.5889	−0.657730	−0.328865	−	0.944377i	$-0.606666\pi$
$887$	−19.5889	−0.657730	−0.328865	+	0.944377i	$0.606666\pi$
$888$	0	0
$889$	16.2596i	0.545331i
$890$	0	0
$891$	0.656645i	0.0219984i
$892$	0	0
$893$	−55.8207	−1.86797
$894$	0	0
$895$	− 15.5462i	− 0.519651i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 5.64302i	− 0.188205i
$900$	0	0
$901$	0.982053	0.0327169
$902$	0	0
$903$	16.8261i	0.559939i
$904$	0	0
$905$	12.8043i	0.425630i
$906$	0	0
$907$	17.5223	0.581818	0.290909	−	0.956751i	$-0.406042\pi$
$907$	17.5223	0.581818	0.290909	+	0.956751i	$0.406042\pi$
$908$	0	0
$909$	−14.4134	−0.478064
$910$	0	0
$911$	−11.6916	−0.387360	−0.193680	−	0.981065i	$-0.562042\pi$
$911$	−11.6916	−0.387360	−0.193680	+	0.981065i	$0.562042\pi$
$912$	0	0
$913$	−7.10735	−0.235219
$914$	0	0
$915$	− 31.9134i	− 1.05502i
$916$	0	0
$917$	− 46.1865i	− 1.52521i
$918$	0	0
$919$	−42.7980	−1.41177	−0.705887	−	0.708324i	$-0.749452\pi$
$919$	−42.7980	−1.41177	−0.705887	+	0.708324i	$0.749452\pi$
$920$	0	0
$921$	18.6077i	0.613143i
$922$	0	0
$923$	0	0
$924$	0	0
$925$	23.9600i	0.787801i
$926$	0	0
$927$	15.7309	0.516671
$928$	0	0
$929$	27.8135i	0.912532i	0.889843	+	0.456266i	$0.150813\pi$
$929$	27.8135i	0.912532i	−0.889843	+	0.456266i	$0.849187\pi$
$930$	0	0
$931$	27.0281i	0.885810i
$932$	0	0
$933$	−25.9076	−0.848177
$934$	0	0
$935$	2.08710	0.0682555
$936$	0	0
$937$	48.2544	1.57640	0.788201	−	0.615417i	$-0.211013\pi$
$937$	48.2544	1.57640	0.788201	+	0.615417i	$0.211013\pi$
$938$	0	0
$939$	−0.799031	−0.0260754
$940$	0	0
$941$	− 13.8449i	− 0.451332i	−0.974205	−	0.225666i	$-0.927544\pi$
$941$	− 13.8449i	− 0.451332i	0.974205	−	0.225666i	$-0.0724557\pi$
$942$	0	0
$943$	73.1041i	2.38060i
$944$	0	0
$945$	9.02586	0.293611
$946$	0	0
$947$	18.6924i	0.607422i	0.952764	+	0.303711i	$0.0982258\pi$
$947$	18.6924i	0.607422i	−0.952764	+	0.303711i	$0.901774\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	− 32.7072i	− 1.06060i
$952$	0	0
$953$	−44.2242	−1.43256	−0.716281	−	0.697812i	$-0.754157\pi$
$953$	−44.2242	−1.43256	−0.716281	+	0.697812i	$0.754157\pi$
$954$	0	0
$955$	− 40.0550i	− 1.29615i
$956$	0	0
$957$	− 3.82174i	− 0.123539i
$958$	0	0
$959$	57.8822	1.86911
$960$	0	0
$961$	30.0599	0.969675
$962$	0	0
$963$	9.27509	0.298885
$964$	0	0
$965$	4.10118	0.132022
$966$	0	0
$967$	36.5221i	1.17447i	0.809416	+	0.587235i	$0.199784\pi$
$967$	36.5221i	1.17447i	−0.809416	+	0.587235i	$0.800216\pi$
$968$	0	0
$969$	− 7.90550i	− 0.253961i
$970$	0	0
$971$	34.0922	1.09407	0.547035	−	0.837110i	$-0.315756\pi$
$971$	34.0922	1.09407	0.547035	+	0.837110i	$0.315756\pi$
$972$	0	0
$973$	32.9506i	1.05635i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 59.7205i	− 1.91063i	−0.295591	−	0.955315i	$-0.595517\pi$
$977$	− 59.7205i	− 1.91063i	0.295591	−	0.955315i	$-0.404483\pi$
$978$	0	0
$979$	2.68694	0.0858751
$980$	0	0
$981$	− 8.82889i	− 0.281885i
$982$	0	0
$983$	49.8000i	1.58837i	0.607674	+	0.794186i	$0.292103\pi$
$983$	49.8000i	1.58837i	−0.607674	+	0.794186i	$0.707897\pi$
$984$	0	0
$985$	47.4986	1.51343
$986$	0	0
$987$	27.3305	0.869940
$988$	0	0
$989$	−35.0228	−1.11366
$990$	0	0
$991$	−17.1731	−0.545522	−0.272761	−	0.962082i	$-0.587937\pi$
$991$	−17.1731	−0.545522	−0.272761	+	0.962082i	$0.587937\pi$
$992$	0	0
$993$	− 5.69662i	− 0.180777i
$994$	0	0
$995$	47.3794i	1.50203i
$996$	0	0
$997$	−34.9494	−1.10686	−0.553430	−	0.832896i	$-0.686681\pi$
$997$	−34.9494	−1.10686	−0.553430	+	0.832896i	$0.686681\pi$
$998$	0	0
$999$	− 9.93482i	− 0.314324i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4056.2.c.r.337.3		12
13.5	odd	4		4056.2.a.bh.1.2	✓	6
13.8	odd	4		4056.2.a.bi.1.5	yes	6
13.12	even	2	inner	4056.2.c.r.337.10		12
52.31	even	4		8112.2.a.ct.1.2		6
52.47	even	4		8112.2.a.cu.1.5		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4056.2.a.bh.1.2	✓	6	13.5	odd	4
4056.2.a.bi.1.5	yes	6	13.8	odd	4
4056.2.c.r.337.3		12	1.1	even	1	trivial
4056.2.c.r.337.10		12	13.12	even	2	inner
8112.2.a.ct.1.2		6	52.31	even	4
8112.2.a.cu.1.5		6	52.47	even	4

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 4056 = 2^{3} \cdot 3 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4056.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -2.72245i q^{5} +3.31535i q^{7} +1.00000 q^{9} +0.656645i q^{11} -2.72245i q^{15} +1.16749 q^{17} -6.77137i q^{19} +3.31535i q^{21} -6.90073 q^{23} -2.41172 q^{25} +1.00000 q^{27} -5.82010 q^{29} +0.969574i q^{31} +0.656645i q^{33} +9.02586 q^{35} -9.93482i q^{37} -10.5937i q^{41} +5.07523 q^{43} -2.72245i q^{45} -8.24364i q^{47} -3.99153 q^{49} +1.16749 q^{51} +0.841166 q^{53} +1.78768 q^{55} -6.77137i q^{57} +0.128144i q^{59} +11.7223 q^{61} +3.31535i q^{63} +15.9422i q^{67} -6.90073 q^{69} -5.32056i q^{71} -3.75917i q^{73} -2.41172 q^{75} -2.17701 q^{77} +2.17759 q^{79} +1.00000 q^{81} +10.8237i q^{83} -3.17843i q^{85} -5.82010 q^{87} -4.09193i q^{89} +0.969574i q^{93} -18.4347 q^{95} -16.5008i q^{97} +0.656645i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 12 q^{3} + 12 q^{9} - 18 q^{17} - 24 q^{23} - 18 q^{25} + 12 q^{27} + 14 q^{29} + 12 q^{35} - 30 q^{43} - 26 q^{49} - 18 q^{51} + 44 q^{53} + 6 q^{55} + 50 q^{61} - 24 q^{69} - 18 q^{75} - 90 q^{77}+ \cdots - 94 q^{95}+O(q^{100}) \) 12 * q + 12 * q^3 + 12 * q^9 - 18 * q^17 - 24 * q^23 - 18 * q^25 + 12 * q^27 + 14 * q^29 + 12 * q^35 - 30 * q^43 - 26 * q^49 - 18 * q^51 + 44 * q^53 + 6 * q^55 + 50 * q^61 - 24 * q^69 - 18 * q^75 - 90 * q^77 + 28 * q^79 + 12 * q^81 + 14 * q^87 - 94 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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