LMFDB - Embedded newform 4056.2.c.p.337.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4056,2,Mod(337,4056)]

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

lf = mfeigenbasis(mf)

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")

//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: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$32.3873230598$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.649638144.4
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 14x^{6} + 75x^{4} - 170x^{2} + 169 $ x^8 - 14x^6 + 75x^4 - 170*x^2 + 169
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{8} $
Twist minimal:	no (minimal twist has level 312)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			337.2
Root			$-1.42055 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	4056.337
Dual form			4056.2.c.p.337.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -1.55452i q^{5} +2.96046i q^{7} +1.00000 q^{9} -2.24703i q^{11} -1.55452i q^{15} +1.01862 q^{17} +5.35607i q^{19} +2.96046i q^{21} -0.782926 q^{23} +2.58347 q^{25} +1.00000 q^{27} +2.69251 q^{29} -7.28657i q^{31} -2.24703i q^{33} +4.60209 q^{35} +7.80155i q^{37} -5.34474i q^{41} -3.12766 q^{43} -1.55452i q^{45} +7.13799i q^{47} -1.76430 q^{49} +1.01862 q^{51} +13.8388 q^{53} -3.49305 q^{55} +5.35607i q^{57} +4.35506i q^{59} +5.14528 q^{61} +2.96046i q^{63} +12.9977i q^{67} -0.782926 q^{69} -13.8501i q^{71} +8.30519i q^{73} +2.58347 q^{75} +6.65223 q^{77} +1.23570 q^{79} +1.00000 q^{81} -7.67389i q^{83} -1.58347i q^{85} +2.69251 q^{87} -0.494055i q^{89} -7.28657i q^{93} +8.32611 q^{95} +11.1057i q^{97} -2.24703i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{3} + 8 q^{9} - 24 q^{17} + 4 q^{23} - 4 q^{25} + 8 q^{27} - 12 q^{29} - 20 q^{35} + 16 q^{43} - 36 q^{49} - 24 q^{51} + 4 q^{53} + 20 q^{55} - 4 q^{61} + 4 q^{69} - 4 q^{75} + 56 q^{77} - 12 q^{79}+ \cdots + 68 q^{95}+O(q^{100}) $ 8 * q + 8 * q^3 + 8 * q^9 - 24 * q^17 + 4 * q^23 - 4 * q^25 + 8 * q^27 - 12 * q^29 - 20 * q^35 + 16 * q^43 - 36 * q^49 - 24 * q^51 + 4 * q^53 + 20 * q^55 - 4 * q^61 + 4 * q^69 - 4 * q^75 + 56 * q^77 - 12 * q^79 + 8 * q^81 - 12 * q^87 + 68 * 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
$5$	− 1.55452i	− 0.695202i	−0.937642	−	0.347601i	$-0.886996\pi$
$5$	− 1.55452i	− 0.695202i	0.937642	−	0.347601i	$-0.113004\pi$
$6$	0	0
$7$	2.96046i	1.11895i	0.828848	+	0.559474i	$0.188997\pi$
$7$	2.96046i	1.11895i	−0.828848	+	0.559474i	$0.811003\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	− 2.24703i	− 0.677504i	−0.940876	−	0.338752i	$-0.889995\pi$
$11$	− 2.24703i	− 0.677504i	0.940876	−	0.338752i	$-0.110005\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	− 1.55452i	− 0.401375i
$16$	0	0
$17$	1.01862	0.247052	0.123526	−	0.992341i	$-0.460580\pi$
$17$	1.01862	0.247052	0.123526	+	0.992341i	$0.460580\pi$
$18$	0	0
$19$	5.35607i	1.22877i	0.789008	+	0.614383i	$0.210595\pi$
$19$	5.35607i	1.22877i	−0.789008	+	0.614383i	$0.789405\pi$
$20$	0	0
$21$	2.96046i	0.646025i
$22$	0	0
$23$	−0.782926	−0.163251	−0.0816257	−	0.996663i	$-0.526011\pi$
$23$	−0.782926	−0.163251	−0.0816257	+	0.996663i	$0.526011\pi$
$24$	0	0
$25$	2.58347	0.516694
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	2.69251	0.499986	0.249993	−	0.968248i	$-0.419572\pi$
$29$	2.69251	0.499986	0.249993	+	0.968248i	$0.419572\pi$
$30$	0	0
$31$	− 7.28657i	− 1.30871i	−0.756189	−	0.654353i	$-0.772941\pi$
$31$	− 7.28657i	− 1.30871i	0.756189	−	0.654353i	$-0.227059\pi$
$32$	0	0
$33$	− 2.24703i	− 0.391157i
$34$	0	0
$35$	4.60209	0.777895
$36$	0	0
$37$	7.80155i	1.28257i	0.767304	+	0.641283i	$0.221598\pi$
$37$	7.80155i	1.28257i	−0.767304	+	0.641283i	$0.778402\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 5.34474i	− 0.834707i	−0.908744	−	0.417354i	$-0.862958\pi$
$41$	− 5.34474i	− 0.834707i	0.908744	−	0.417354i	$-0.137042\pi$
$42$	0	0
$43$	−3.12766	−0.476964	−0.238482	−	0.971147i	$-0.576650\pi$
$43$	−3.12766	−0.476964	−0.238482	+	0.971147i	$0.576650\pi$
$44$	0	0
$45$	− 1.55452i	− 0.231734i
$46$	0	0
$47$	7.13799i	1.04118i	0.853806	+	0.520591i	$0.174288\pi$
$47$	7.13799i	1.04118i	−0.853806	+	0.520591i	$0.825712\pi$
$48$	0	0
$49$	−1.76430	−0.252043
$50$	0	0
$51$	1.01862	0.142636
$52$	0	0
$53$	13.8388	1.90090	0.950452	−	0.310871i	$-0.100621\pi$
$53$	13.8388	1.90090	0.950452	+	0.310871i	$0.100621\pi$
$54$	0	0
$55$	−3.49305	−0.471003
$56$	0	0
$57$	5.35607i	0.709428i
$58$	0	0
$59$	4.35506i	0.566981i	0.958975	+	0.283490i	$0.0914924\pi$
$59$	4.35506i	0.566981i	−0.958975	+	0.283490i	$0.908508\pi$
$60$	0	0
$61$	5.14528	0.658785	0.329393	−	0.944193i	$-0.393156\pi$
$61$	5.14528	0.658785	0.329393	+	0.944193i	$0.393156\pi$
$62$	0	0
$63$	2.96046i	0.372982i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	12.9977i	1.58792i	0.607969	+	0.793961i	$0.291985\pi$
$67$	12.9977i	1.58792i	−0.607969	+	0.793961i	$0.708015\pi$
$68$	0	0
$69$	−0.782926	−0.0942532
$70$	0	0
$71$	− 13.8501i	− 1.64371i	−0.569699	−	0.821854i	$-0.692940\pi$
$71$	− 13.8501i	− 1.64371i	0.569699	−	0.821854i	$-0.307060\pi$
$72$	0	0
$73$	8.30519i	0.972049i	0.873945	+	0.486025i	$0.161553\pi$
$73$	8.30519i	0.972049i	−0.873945	+	0.486025i	$0.838447\pi$
$74$	0	0
$75$	2.58347	0.298313
$76$	0	0
$77$	6.65223	0.758092
$78$	0	0
$79$	1.23570	0.139027	0.0695133	−	0.997581i	$-0.477855\pi$
$79$	1.23570	0.139027	0.0695133	+	0.997581i	$0.477855\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	− 7.67389i	− 0.842318i	−0.906987	−	0.421159i	$-0.861623\pi$
$83$	− 7.67389i	− 0.842318i	0.906987	−	0.421159i	$-0.138377\pi$
$84$	0	0
$85$	− 1.58347i	− 0.171751i
$86$	0	0
$87$	2.69251	0.288667
$88$	0	0
$89$	− 0.494055i	− 0.0523697i	−0.999657	−	0.0261849i	$-0.991664\pi$
$89$	− 0.494055i	− 0.0523697i	0.999657	−	0.0261849i	$-0.00833586\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	− 7.28657i	− 0.755582i
$94$	0	0
$95$	8.32611	0.854241
$96$	0	0
$97$	11.1057i	1.12762i	0.825906	+	0.563808i	$0.190664\pi$
$97$	11.1057i	1.12762i	−0.825906	+	0.563808i	$0.809336\pi$
$98$	0	0
$99$	− 2.24703i	− 0.225835i
$100$	0	0
$101$	15.1866	1.51112	0.755560	−	0.655080i	$-0.227365\pi$
$101$	15.1866	1.51112	0.755560	+	0.655080i	$0.227365\pi$
$102$	0	0
$103$	1.12766	0.111112	0.0555559	−	0.998456i	$-0.482307\pi$
$103$	1.12766	0.111112	0.0555559	+	0.998456i	$0.482307\pi$
$104$	0	0
$105$	4.60209	0.449118
$106$	0	0
$107$	−17.0382	−1.64715	−0.823575	−	0.567208i	$-0.808024\pi$
$107$	−17.0382	−1.64715	−0.823575	+	0.567208i	$0.808024\pi$
$108$	0	0
$109$	− 15.2866i	− 1.46419i	−0.681204	−	0.732094i	$-0.738543\pi$
$109$	− 15.2866i	− 1.46419i	0.681204	−	0.732094i	$-0.261457\pi$
$110$	0	0
$111$	7.80155i	0.740490i
$112$	0	0
$113$	7.01862	0.660256	0.330128	−	0.943936i	$-0.392908\pi$
$113$	7.01862	0.660256	0.330128	+	0.943936i	$0.392908\pi$
$114$	0	0
$115$	1.21707i	0.113493i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	3.01559i	0.276438i
$120$	0	0
$121$	5.95087	0.540988
$122$	0	0
$123$	− 5.34474i	− 0.481919i
$124$	0	0
$125$	− 11.7887i	− 1.05441i
$126$	0	0
$127$	14.8015	1.31342	0.656712	−	0.754141i	$-0.271947\pi$
$127$	14.8015	1.31342	0.656712	+	0.754141i	$0.271947\pi$
$128$	0	0
$129$	−3.12766	−0.275375
$130$	0	0
$131$	5.81916	0.508423	0.254211	−	0.967149i	$-0.418184\pi$
$131$	5.81916	0.508423	0.254211	+	0.967149i	$0.418184\pi$
$132$	0	0
$133$	−15.8564	−1.37492
$134$	0	0
$135$	− 1.55452i	− 0.133792i
$136$	0	0
$137$	17.3674i	1.48380i	0.670512	+	0.741899i	$0.266074\pi$
$137$	17.3674i	1.48380i	−0.670512	+	0.741899i	$0.733926\pi$
$138$	0	0
$139$	5.88792	0.499407	0.249704	−	0.968322i	$-0.419667\pi$
$139$	5.88792	0.499407	0.249704	+	0.968322i	$0.419667\pi$
$140$	0	0
$141$	7.13799i	0.601127i
$142$	0	0
$143$	0	0
$144$	0	0
$145$	− 4.18556i	− 0.347592i
$146$	0	0
$147$	−1.76430	−0.145517
$148$	0	0
$149$	10.5845i	0.867114i	0.901126	+	0.433557i	$0.142742\pi$
$149$	10.5845i	0.867114i	−0.901126	+	0.433557i	$0.857258\pi$
$150$	0	0
$151$	2.24703i	0.182861i	0.995811	+	0.0914303i	$0.0291439\pi$
$151$	2.24703i	0.182861i	−0.995811	+	0.0914303i	$0.970856\pi$
$152$	0	0
$153$	1.01862	0.0823507
$154$	0	0
$155$	−11.3271	−0.909816
$156$	0	0
$157$	−8.45782	−0.675007	−0.337504	−	0.941324i	$-0.609583\pi$
$157$	−8.45782	−0.675007	−0.337504	+	0.941324i	$0.609583\pi$
$158$	0	0
$159$	13.8388	1.09749
$160$	0	0
$161$	− 2.31782i	− 0.182670i
$162$	0	0
$163$	− 4.20019i	− 0.328985i	−0.986378	−	0.164492i	$-0.947401\pi$
$163$	− 4.20019i	− 0.328985i	0.986378	−	0.164492i	$-0.0525986\pi$
$164$	0	0
$165$	−3.49305	−0.271934
$166$	0	0
$167$	18.7121i	1.44799i	0.689806	+	0.723994i	$0.257696\pi$
$167$	18.7121i	1.44799i	−0.689806	+	0.723994i	$0.742304\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	5.35607i	0.409589i
$172$	0	0
$173$	−2.89096	−0.219796	−0.109898	−	0.993943i	$-0.535052\pi$
$173$	−2.89096	−0.219796	−0.109898	+	0.993943i	$0.535052\pi$
$174$	0	0
$175$	7.64824i	0.578153i
$176$	0	0
$177$	4.35506i	0.327346i
$178$	0	0
$179$	22.8574	1.70844	0.854222	−	0.519909i	$-0.174034\pi$
$179$	22.8574	1.70844	0.854222	+	0.519909i	$0.174034\pi$
$180$	0	0
$181$	8.09042	0.601356	0.300678	−	0.953726i	$-0.402787\pi$
$181$	8.09042	0.601356	0.300678	+	0.953726i	$0.402787\pi$
$182$	0	0
$183$	5.14528	0.380350
$184$	0	0
$185$	12.1277	0.891643
$186$	0	0
$187$	− 2.28887i	− 0.167379i
$188$	0	0
$189$	2.96046i	0.215342i
$190$	0	0
$191$	−15.3850	−1.11322	−0.556610	−	0.830774i	$-0.687898\pi$
$191$	−15.3850	−1.11322	−0.556610	+	0.830774i	$0.687898\pi$
$192$	0	0
$193$	− 0.347036i	− 0.0249802i	−0.999922	−	0.0124901i	$-0.996024\pi$
$193$	− 0.347036i	− 0.0249802i	0.999922	−	0.0124901i	$-0.00397582\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 25.8564i	− 1.84219i	−0.389335	−	0.921096i	$-0.627295\pi$
$197$	− 25.8564i	− 1.84219i	0.389335	−	0.921096i	$-0.372705\pi$
$198$	0	0
$199$	7.34574	0.520726	0.260363	−	0.965511i	$-0.416158\pi$
$199$	7.34574	0.520726	0.260363	+	0.965511i	$0.416158\pi$
$200$	0	0
$201$	12.9977i	0.916787i
$202$	0	0
$203$	7.97105i	0.559458i
$204$	0	0
$205$	−8.30850	−0.580291
$206$	0	0
$207$	−0.782926	−0.0544171
$208$	0	0
$209$	12.0352	0.832494
$210$	0	0
$211$	−28.6952	−1.97546	−0.987729	−	0.156175i	$-0.950084\pi$
$211$	−28.6952	−1.97546	−0.987729	+	0.156175i	$0.950084\pi$
$212$	0	0
$213$	− 13.8501i	− 0.948995i
$214$	0	0
$215$	4.86201i	0.331586i
$216$	0	0
$217$	21.5716	1.46437
$218$	0	0
$219$	8.30519i	0.561213i
$220$	0	0
$221$	0	0
$222$	0	0
$223$	2.13698i	0.143103i	0.997437	+	0.0715514i	$0.0227950\pi$
$223$	2.13698i	0.143103i	−0.997437	+	0.0715514i	$0.977205\pi$
$224$	0	0
$225$	2.58347	0.172231
$226$	0	0
$227$	18.3860i	1.22032i	0.792277	+	0.610162i	$0.208895\pi$
$227$	18.3860i	1.22032i	−0.792277	+	0.610162i	$0.791105\pi$
$228$	0	0
$229$	1.74669i	0.115424i	0.998333	+	0.0577122i	$0.0183806\pi$
$229$	1.74669i	0.115424i	−0.998333	+	0.0577122i	$0.981619\pi$
$230$	0	0
$231$	6.65223	0.437684
$232$	0	0
$233$	26.3525	1.72641	0.863204	−	0.504855i	$-0.168454\pi$
$233$	26.3525	1.72641	0.863204	+	0.504855i	$0.168454\pi$
$234$	0	0
$235$	11.0961	0.723833
$236$	0	0
$237$	1.23570	0.0802671
$238$	0	0
$239$	− 6.36336i	− 0.411611i	−0.978593	−	0.205806i	$-0.934019\pi$
$239$	− 6.36336i	− 0.411611i	0.978593	−	0.205806i	$-0.0659815\pi$
$240$	0	0
$241$	− 9.01862i	− 0.580940i	−0.956884	−	0.290470i	$-0.906188\pi$
$241$	− 9.01862i	− 0.580940i	0.956884	−	0.290470i	$-0.0938117\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	2.74265i	0.175221i
$246$	0	0
$247$	0	0
$248$	0	0
$249$	− 7.67389i	− 0.486313i
$250$	0	0
$251$	18.2760	1.15357	0.576785	−	0.816896i	$-0.304307\pi$
$251$	18.2760	1.15357	0.576785	+	0.816896i	$0.304307\pi$
$252$	0	0
$253$	1.75926i	0.110603i
$254$	0	0
$255$	− 1.58347i	− 0.0991606i
$256$	0	0
$257$	−26.4263	−1.64843	−0.824214	−	0.566279i	$-0.808383\pi$
$257$	−26.4263	−1.64843	−0.824214	+	0.566279i	$0.808383\pi$
$258$	0	0
$259$	−23.0961	−1.43512
$260$	0	0
$261$	2.69251	0.166662
$262$	0	0
$263$	4.82017	0.297224	0.148612	−	0.988896i	$-0.452519\pi$
$263$	4.82017	0.297224	0.148612	+	0.988896i	$0.452519\pi$
$264$	0	0
$265$	− 21.5127i	− 1.32151i
$266$	0	0
$267$	− 0.494055i	− 0.0302357i
$268$	0	0
$269$	−1.64034	−0.100013	−0.0500066	−	0.998749i	$-0.515924\pi$
$269$	−1.64034	−0.100013	−0.0500066	+	0.998749i	$0.515924\pi$
$270$	0	0
$271$	− 12.7300i	− 0.773294i	−0.922228	−	0.386647i	$-0.873633\pi$
$271$	− 12.7300i	− 0.773294i	0.922228	−	0.386647i	$-0.126367\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 5.80512i	− 0.350062i
$276$	0	0
$277$	−24.5008	−1.47211	−0.736055	−	0.676922i	$-0.763313\pi$
$277$	−24.5008	−1.47211	−0.736055	+	0.676922i	$0.763313\pi$
$278$	0	0
$279$	− 7.28657i	− 0.436236i
$280$	0	0
$281$	1.93853i	0.115643i	0.998327	+	0.0578215i	$0.0184154\pi$
$281$	1.93853i	0.115643i	−0.998327	+	0.0578215i	$0.981585\pi$
$282$	0	0
$283$	−8.07853	−0.480219	−0.240109	−	0.970746i	$-0.577183\pi$
$283$	−8.07853	−0.480219	−0.240109	+	0.970746i	$0.577183\pi$
$284$	0	0
$285$	8.32611	0.493196
$286$	0	0
$287$	15.8229	0.933994
$288$	0	0
$289$	−15.9624	−0.938965
$290$	0	0
$291$	11.1057i	0.651030i
$292$	0	0
$293$	− 28.3411i	− 1.65571i	−0.560944	−	0.827854i	$-0.689562\pi$
$293$	− 28.3411i	− 1.65571i	0.560944	−	0.827854i	$-0.310438\pi$
$294$	0	0
$295$	6.77003	0.394166
$296$	0	0
$297$	− 2.24703i	− 0.130386i
$298$	0	0
$299$	0	0
$300$	0	0
$301$	− 9.25931i	− 0.533698i
$302$	0	0
$303$	15.1866	0.872445
$304$	0	0
$305$	− 7.99844i	− 0.457989i
$306$	0	0
$307$	− 30.7025i	− 1.75229i	−0.482051	−	0.876143i	$-0.660108\pi$
$307$	− 30.7025i	− 1.75229i	0.482051	−	0.876143i	$-0.339892\pi$
$308$	0	0
$309$	1.12766	0.0641504
$310$	0	0
$311$	13.7338	0.778772	0.389386	−	0.921075i	$-0.372687\pi$
$311$	13.7338	0.778772	0.389386	+	0.921075i	$0.372687\pi$
$312$	0	0
$313$	32.1886	1.81941	0.909703	−	0.415260i	$-0.136309\pi$
$313$	32.1886	1.81941	0.909703	+	0.415260i	$0.136309\pi$
$314$	0	0
$315$	4.60209	0.259298
$316$	0	0
$317$	4.54723i	0.255398i	0.991813	+	0.127699i	$0.0407591\pi$
$317$	4.54723i	0.255398i	−0.991813	+	0.127699i	$0.959241\pi$
$318$	0	0
$319$	− 6.05014i	− 0.338743i
$320$	0	0
$321$	−17.0382	−0.950982
$322$	0	0
$323$	5.45581i	0.303569i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 15.2866i	− 0.845349i
$328$	0	0
$329$	−21.1317	−1.16503
$330$	0	0
$331$	− 14.7899i	− 0.812928i	−0.913667	−	0.406464i	$-0.866762\pi$
$331$	− 14.7899i	− 0.812928i	0.913667	−	0.406464i	$-0.133238\pi$
$332$	0	0
$333$	7.80155i	0.427522i
$334$	0	0
$335$	20.2052	1.10393
$336$	0	0
$337$	−3.21808	−0.175300	−0.0876500	−	0.996151i	$-0.527936\pi$
$337$	−3.21808	−0.175300	−0.0876500	+	0.996151i	$0.527936\pi$
$338$	0	0
$339$	7.01862	0.381199
$340$	0	0
$341$	−16.3731	−0.886654
$342$	0	0
$343$	15.5001i	0.836924i
$344$	0	0
$345$	1.21707i	0.0655251i
$346$	0	0
$347$	−4.30345	−0.231021	−0.115511	−	0.993306i	$-0.536850\pi$
$347$	−4.30345	−0.231021	−0.115511	+	0.993306i	$0.536850\pi$
$348$	0	0
$349$	− 11.1284i	− 0.595689i	−0.954614	−	0.297845i	$-0.903732\pi$
$349$	− 11.1284i	− 0.595689i	0.954614	−	0.297845i	$-0.0962678\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 10.8295i	− 0.576396i	−0.957571	−	0.288198i	$-0.906944\pi$
$353$	− 10.8295i	− 0.576396i	0.957571	−	0.288198i	$-0.0930561\pi$
$354$	0	0
$355$	−21.5303	−1.14271
$356$	0	0
$357$	3.01559i	0.159602i
$358$	0	0
$359$	32.0616i	1.69215i	0.533067	+	0.846073i	$0.321039\pi$
$359$	32.0616i	1.69215i	−0.533067	+	0.846073i	$0.678961\pi$
$360$	0	0
$361$	−9.68746	−0.509866
$362$	0	0
$363$	5.95087	0.312340
$364$	0	0
$365$	12.9106	0.675771
$366$	0	0
$367$	0.222122	0.0115947	0.00579734	−	0.999983i	$-0.498155\pi$
$367$	0.222122	0.0115947	0.00579734	+	0.999983i	$0.498155\pi$
$368$	0	0
$369$	− 5.34474i	− 0.278236i
$370$	0	0
$371$	40.9691i	2.12701i
$372$	0	0
$373$	24.0961	1.24765	0.623826	−	0.781564i	$-0.285578\pi$
$373$	24.0961	1.24765	0.623826	+	0.781564i	$0.285578\pi$
$374$	0	0
$375$	− 11.7887i	− 0.608763i
$376$	0	0
$377$	0	0
$378$	0	0
$379$	16.3518i	0.839933i	0.907540	+	0.419967i	$0.137958\pi$
$379$	16.3518i	0.839933i	−0.907540	+	0.419967i	$0.862042\pi$
$380$	0	0
$381$	14.8015	0.758306
$382$	0	0
$383$	19.4967i	0.996237i	0.867109	+	0.498119i	$0.165976\pi$
$383$	19.4967i	0.996237i	−0.867109	+	0.498119i	$0.834024\pi$
$384$	0	0
$385$	− 10.3410i	− 0.527027i
$386$	0	0
$387$	−3.12766	−0.158988
$388$	0	0
$389$	−29.6972	−1.50571	−0.752854	−	0.658187i	$-0.771323\pi$
$389$	−29.6972	−1.50571	−0.752854	+	0.658187i	$0.771323\pi$
$390$	0	0
$391$	−0.797505	−0.0403316
$392$	0	0
$393$	5.81916	0.293538
$394$	0	0
$395$	− 1.92091i	− 0.0966517i
$396$	0	0
$397$	− 15.8743i	− 0.796708i	−0.917232	−	0.398354i	$-0.869582\pi$
$397$	− 15.8743i	− 0.796708i	0.917232	−	0.398354i	$-0.130418\pi$
$398$	0	0
$399$	−15.8564	−0.793813
$400$	0	0
$401$	− 11.5402i	− 0.576288i	−0.957587	−	0.288144i	$-0.906962\pi$
$401$	− 11.5402i	− 0.576288i	0.957587	−	0.288144i	$-0.0930381\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	− 1.55452i	− 0.0772447i
$406$	0	0
$407$	17.5303	0.868944
$408$	0	0
$409$	35.0979i	1.73548i	0.497019	+	0.867739i	$0.334428\pi$
$409$	35.0979i	1.73548i	−0.497019	+	0.867739i	$0.665572\pi$
$410$	0	0
$411$	17.3674i	0.856671i
$412$	0	0
$413$	−12.8930	−0.634422
$414$	0	0
$415$	−11.9292	−0.585582
$416$	0	0
$417$	5.88792	0.288333
$418$	0	0
$419$	22.6284	1.10547	0.552736	−	0.833356i	$-0.313584\pi$
$419$	22.6284	1.10547	0.552736	+	0.833356i	$0.313584\pi$
$420$	0	0
$421$	− 13.3833i	− 0.652261i	−0.945325	−	0.326130i	$-0.894255\pi$
$421$	− 13.3833i	− 0.652261i	0.945325	−	0.326130i	$-0.105745\pi$
$422$	0	0
$423$	7.13799i	0.347061i
$424$	0	0
$425$	2.63158	0.127650
$426$	0	0
$427$	15.2324i	0.737146i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	20.0616i	0.966333i	0.875529	+	0.483166i	$0.160513\pi$
$431$	20.0616i	0.966333i	−0.875529	+	0.483166i	$0.839487\pi$
$432$	0	0
$433$	−37.8445	−1.81869	−0.909346	−	0.416041i	$-0.863417\pi$
$433$	−37.8445	−1.81869	−0.909346	+	0.416041i	$0.863417\pi$
$434$	0	0
$435$	− 4.18556i	− 0.200682i
$436$	0	0
$437$	− 4.19340i	− 0.200598i
$438$	0	0
$439$	25.6796	1.22562	0.612811	−	0.790230i	$-0.290039\pi$
$439$	25.6796	1.22562	0.612811	+	0.790230i	$0.290039\pi$
$440$	0	0
$441$	−1.76430	−0.0840145
$442$	0	0
$443$	−37.4822	−1.78083	−0.890416	−	0.455148i	$-0.849586\pi$
$443$	−37.4822	−1.78083	−0.890416	+	0.455148i	$0.849586\pi$
$444$	0	0
$445$	−0.768019	−0.0364076
$446$	0	0
$447$	10.5845i	0.500628i
$448$	0	0
$449$	− 5.92091i	− 0.279425i	−0.990192	−	0.139713i	$-0.955382\pi$
$449$	− 5.92091i	− 0.279425i	0.990192	−	0.139713i	$-0.0446179\pi$
$450$	0	0
$451$	−12.0098	−0.565518
$452$	0	0
$453$	2.24703i	0.105575i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	40.6184i	1.90005i	0.312175	+	0.950025i	$0.398942\pi$
$457$	40.6184i	1.90005i	−0.312175	+	0.950025i	$0.601058\pi$
$458$	0	0
$459$	1.01862	0.0475452
$460$	0	0
$461$	− 7.86627i	− 0.366369i	−0.983079	−	0.183184i	$-0.941360\pi$
$461$	− 7.86627i	− 0.366369i	0.983079	−	0.183184i	$-0.0586405\pi$
$462$	0	0
$463$	− 3.96775i	− 0.184397i	−0.995741	−	0.0921984i	$-0.970611\pi$
$463$	− 3.96775i	− 0.184397i	0.995741	−	0.0921984i	$-0.0293894\pi$
$464$	0	0
$465$	−11.3271	−0.525283
$466$	0	0
$467$	1.41249	0.0653622	0.0326811	−	0.999466i	$-0.489595\pi$
$467$	1.41249	0.0653622	0.0326811	+	0.999466i	$0.489595\pi$
$468$	0	0
$469$	−38.4791	−1.77680
$470$	0	0
$471$	−8.45782	−0.389716
$472$	0	0
$473$	7.02794i	0.323145i
$474$	0	0
$475$	13.8372i	0.634896i
$476$	0	0
$477$	13.8388	0.633635
$478$	0	0
$479$	21.4368i	0.979474i	0.871870	+	0.489737i	$0.162907\pi$
$479$	21.4368i	0.979474i	−0.871870	+	0.489737i	$0.837093\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	− 2.31782i	− 0.105464i
$484$	0	0
$485$	17.2641	0.783922
$486$	0	0
$487$	− 11.1898i	− 0.507059i	−0.967328	−	0.253529i	$-0.918409\pi$
$487$	− 11.1898i	− 0.507059i	0.967328	−	0.253529i	$-0.0815914\pi$
$488$	0	0
$489$	− 4.20019i	− 0.189939i
$490$	0	0
$491$	−19.9665	−0.901073	−0.450537	−	0.892758i	$-0.648767\pi$
$491$	−19.9665	−0.901073	−0.450537	+	0.892758i	$0.648767\pi$
$492$	0	0
$493$	2.74265	0.123523
$494$	0	0
$495$	−3.49305	−0.157001
$496$	0	0
$497$	41.0027	1.83922
$498$	0	0
$499$	4.15156i	0.185849i	0.995673	+	0.0929247i	$0.0296216\pi$
$499$	4.15156i	0.185849i	−0.995673	+	0.0929247i	$0.970378\pi$
$500$	0	0
$501$	18.7121i	0.835997i
$502$	0	0
$503$	−22.3488	−0.996483	−0.498241	−	0.867038i	$-0.666021\pi$
$503$	−22.3488	−0.996483	−0.498241	+	0.867038i	$0.666021\pi$
$504$	0	0
$505$	− 23.6078i	− 1.05053i
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 34.6523i	− 1.53594i	−0.640487	−	0.767969i	$-0.721267\pi$
$509$	− 34.6523i	− 1.53594i	0.640487	−	0.767969i	$-0.278733\pi$
$510$	0	0
$511$	−24.5872	−1.08767
$512$	0	0
$513$	5.35607i	0.236476i
$514$	0	0
$515$	− 1.75297i	− 0.0772452i
$516$	0	0
$517$	16.0393	0.705406
$518$	0	0
$519$	−2.89096	−0.126899
$520$	0	0
$521$	25.0186	1.09609	0.548043	−	0.836450i	$-0.315373\pi$
$521$	25.0186	1.09609	0.548043	+	0.836450i	$0.315373\pi$
$522$	0	0
$523$	−27.7484	−1.21335	−0.606676	−	0.794949i	$-0.707497\pi$
$523$	−27.7484	−1.21335	−0.606676	+	0.794949i	$0.707497\pi$
$524$	0	0
$525$	7.64824i	0.333797i
$526$	0	0
$527$	− 7.42226i	− 0.323319i
$528$	0	0
$529$	−22.3870	−0.973349
$530$	0	0
$531$	4.35506i	0.188994i
$532$	0	0
$533$	0	0
$534$	0	0
$535$	26.4863i	1.14510i
$536$	0	0
$537$	22.8574	0.986370
$538$	0	0
$539$	3.96444i	0.170761i
$540$	0	0
$541$	− 31.4592i	− 1.35254i	−0.736655	−	0.676269i	$-0.763596\pi$
$541$	− 31.4592i	− 1.35254i	0.736655	−	0.676269i	$-0.236404\pi$
$542$	0	0
$543$	8.09042	0.347193
$544$	0	0
$545$	−23.7633	−1.01791
$546$	0	0
$547$	−18.1510	−0.776081	−0.388040	−	0.921642i	$-0.626848\pi$
$547$	−18.1510	−0.776081	−0.388040	+	0.921642i	$0.626848\pi$
$548$	0	0
$549$	5.14528	0.219595
$550$	0	0
$551$	14.4213i	0.614366i
$552$	0	0
$553$	3.65822i	0.155563i
$554$	0	0
$555$	12.1277	0.514791
$556$	0	0
$557$	14.8977i	0.631235i	0.948886	+	0.315618i	$0.102212\pi$
$557$	14.8977i	0.631235i	−0.948886	+	0.315618i	$0.897788\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	− 2.28887i	− 0.0966362i
$562$	0	0
$563$	6.23873	0.262931	0.131466	−	0.991321i	$-0.458032\pi$
$563$	6.23873	0.262931	0.131466	+	0.991321i	$0.458032\pi$
$564$	0	0
$565$	− 10.9106i	− 0.459012i
$566$	0	0
$567$	2.96046i	0.124327i
$568$	0	0
$569$	15.2268	0.638342	0.319171	−	0.947697i	$-0.396595\pi$
$569$	15.2268	0.638342	0.319171	+	0.947697i	$0.396595\pi$
$570$	0	0
$571$	14.8801	0.622712	0.311356	−	0.950293i	$-0.399217\pi$
$571$	14.8801	0.622712	0.311356	+	0.950293i	$0.399217\pi$
$572$	0	0
$573$	−15.3850	−0.642718
$574$	0	0
$575$	−2.02266	−0.0843509
$576$	0	0
$577$	− 15.6078i	− 0.649762i	−0.945755	−	0.324881i	$-0.894676\pi$
$577$	− 15.6078i	− 0.649762i	0.945755	−	0.324881i	$-0.105324\pi$
$578$	0	0
$579$	− 0.347036i	− 0.0144223i
$580$	0	0
$581$	22.7182	0.942510
$582$	0	0
$583$	− 31.0961i	− 1.28787i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	40.2301i	1.66047i	0.557411	+	0.830236i	$0.311795\pi$
$587$	40.2301i	1.66047i	−0.557411	+	0.830236i	$0.688205\pi$
$588$	0	0
$589$	39.0274	1.60809
$590$	0	0
$591$	− 25.8564i	− 1.06359i
$592$	0	0
$593$	37.7225i	1.54908i	0.632527	+	0.774538i	$0.282018\pi$
$593$	37.7225i	1.54908i	−0.632527	+	0.774538i	$0.717982\pi$
$594$	0	0
$595$	4.68779	0.192181
$596$	0	0
$597$	7.34574	0.300641
$598$	0	0
$599$	−30.0518	−1.22788	−0.613942	−	0.789351i	$-0.710417\pi$
$599$	−30.0518	−1.22788	−0.613942	+	0.789351i	$0.710417\pi$
$600$	0	0
$601$	−3.58347	−0.146173	−0.0730863	−	0.997326i	$-0.523285\pi$
$601$	−3.58347	−0.146173	−0.0730863	+	0.997326i	$0.523285\pi$
$602$	0	0
$603$	12.9977i	0.529307i
$604$	0	0
$605$	− 9.25074i	− 0.376096i
$606$	0	0
$607$	−23.5679	−0.956590	−0.478295	−	0.878199i	$-0.658745\pi$
$607$	−23.5679	−0.956590	−0.478295	+	0.878199i	$0.658745\pi$
$608$	0	0
$609$	7.97105i	0.323003i
$610$	0	0
$611$	0	0
$612$	0	0
$613$	10.2390i	0.413549i	0.978389	+	0.206775i	$0.0662967\pi$
$613$	10.2390i	0.413549i	−0.978389	+	0.206775i	$0.933703\pi$
$614$	0	0
$615$	−8.30850	−0.335031
$616$	0	0
$617$	− 37.7804i	− 1.52098i	−0.649350	−	0.760490i	$-0.724959\pi$
$617$	− 37.7804i	− 1.52098i	0.649350	−	0.760490i	$-0.275041\pi$
$618$	0	0
$619$	20.1165i	0.808551i	0.914637	+	0.404275i	$0.132476\pi$
$619$	20.1165i	0.808551i	−0.914637	+	0.404275i	$0.867524\pi$
$620$	0	0
$621$	−0.782926	−0.0314177
$622$	0	0
$623$	1.46263	0.0585990
$624$	0	0
$625$	−5.40836	−0.216334
$626$	0	0
$627$	12.0352	0.480641
$628$	0	0
$629$	7.94682i	0.316861i
$630$	0	0
$631$	− 23.6225i	− 0.940395i	−0.882561	−	0.470198i	$-0.844183\pi$
$631$	− 23.6225i	− 0.940395i	0.882561	−	0.470198i	$-0.155817\pi$
$632$	0	0
$633$	−28.6952	−1.14053
$634$	0	0
$635$	− 23.0093i	− 0.913096i
$636$	0	0
$637$	0	0
$638$	0	0
$639$	− 13.8501i	− 0.547902i
$640$	0	0
$641$	−3.57370	−0.141153	−0.0705763	−	0.997506i	$-0.522484\pi$
$641$	−3.57370	−0.141153	−0.0705763	+	0.997506i	$0.522484\pi$
$642$	0	0
$643$	− 32.6876i	− 1.28907i	−0.764573	−	0.644537i	$-0.777050\pi$
$643$	− 32.6876i	− 1.28907i	0.764573	−	0.644537i	$-0.222950\pi$
$644$	0	0
$645$	4.86201i	0.191442i
$646$	0	0
$647$	−37.0998	−1.45855	−0.729273	−	0.684223i	$-0.760141\pi$
$647$	−37.0998	−1.45855	−0.729273	+	0.684223i	$0.760141\pi$
$648$	0	0
$649$	9.78594	0.384132
$650$	0	0
$651$	21.5716	0.845457
$652$	0	0
$653$	25.9143	1.01410	0.507052	−	0.861915i	$-0.330735\pi$
$653$	25.9143	1.01410	0.507052	+	0.861915i	$0.330735\pi$
$654$	0	0
$655$	− 9.04601i	− 0.353457i
$656$	0	0
$657$	8.30519i	0.324016i
$658$	0	0
$659$	−14.9861	−0.583776	−0.291888	−	0.956453i	$-0.594283\pi$
$659$	−14.9861	−0.583776	−0.291888	+	0.956453i	$0.594283\pi$
$660$	0	0
$661$	− 12.1765i	− 0.473611i	−0.971557	−	0.236806i	$-0.923900\pi$
$661$	− 12.1765i	− 0.473611i	0.971557	−	0.236806i	$-0.0761004\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	24.6491i	0.955851i
$666$	0	0
$667$	−2.10803	−0.0816234
$668$	0	0
$669$	2.13698i	0.0826205i
$670$	0	0
$671$	− 11.5616i	− 0.446330i
$672$	0	0
$673$	−38.8564	−1.49780	−0.748902	−	0.662681i	$-0.769419\pi$
$673$	−38.8564	−1.49780	−0.748902	+	0.662681i	$0.769419\pi$
$674$	0	0
$675$	2.58347	0.0994377
$676$	0	0
$677$	−25.3624	−0.974754	−0.487377	−	0.873192i	$-0.662046\pi$
$677$	−25.3624	−0.974754	−0.487377	+	0.873192i	$0.662046\pi$
$678$	0	0
$679$	−32.8780	−1.26174
$680$	0	0
$681$	18.3860i	0.704554i
$682$	0	0
$683$	15.9189i	0.609120i	0.952493	+	0.304560i	$0.0985094\pi$
$683$	15.9189i	0.609120i	−0.952493	+	0.304560i	$0.901491\pi$
$684$	0	0
$685$	26.9980	1.03154
$686$	0	0
$687$	1.74669i	0.0666403i
$688$	0	0
$689$	0	0
$690$	0	0
$691$	− 3.57084i	− 0.135841i	−0.997691	−	0.0679206i	$-0.978364\pi$
$691$	− 3.57084i	− 0.135841i	0.997691	−	0.0679206i	$-0.0216365\pi$
$692$	0	0
$693$	6.65223	0.252697
$694$	0	0
$695$	− 9.15289i	− 0.347189i
$696$	0	0
$697$	− 5.44426i	− 0.206216i
$698$	0	0
$699$	26.3525	0.996742
$700$	0	0
$701$	1.56585	0.0591414	0.0295707	−	0.999563i	$-0.490586\pi$
$701$	1.56585	0.0591414	0.0295707	+	0.999563i	$0.490586\pi$
$702$	0	0
$703$	−41.7856	−1.57597
$704$	0	0
$705$	11.0961	0.417905
$706$	0	0
$707$	44.9592i	1.69086i
$708$	0	0
$709$	− 40.7930i	− 1.53201i	−0.642833	−	0.766006i	$-0.722241\pi$
$709$	− 40.7930i	− 1.53201i	0.642833	−	0.766006i	$-0.277759\pi$
$710$	0	0
$711$	1.23570	0.0463422
$712$	0	0
$713$	5.70485i	0.213648i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 6.36336i	− 0.237644i
$718$	0	0
$719$	0.831053	0.0309930	0.0154965	−	0.999880i	$-0.495067\pi$
$719$	0.831053	0.0309930	0.0154965	+	0.999880i	$0.495067\pi$
$720$	0	0
$721$	3.33839i	0.124328i
$722$	0	0
$723$	− 9.01862i	− 0.335406i
$724$	0	0
$725$	6.95601	0.258340
$726$	0	0
$727$	9.85844	0.365629	0.182815	−	0.983147i	$-0.441479\pi$
$727$	9.85844	0.365629	0.182815	+	0.983147i	$0.441479\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−3.18590	−0.117835
$732$	0	0
$733$	11.6467i	0.430180i	0.976594	+	0.215090i	$0.0690045\pi$
$733$	11.6467i	0.430180i	−0.976594	+	0.215090i	$0.930996\pi$
$734$	0	0
$735$	2.74265i	0.101164i
$736$	0	0
$737$	29.2062	1.07582
$738$	0	0
$739$	28.1132i	1.03416i	0.855937	+	0.517080i	$0.172981\pi$
$739$	28.1132i	1.03416i	−0.855937	+	0.517080i	$0.827019\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 16.8078i	− 0.616619i	−0.951286	−	0.308309i	$-0.900237\pi$
$743$	− 16.8078i	− 0.616619i	0.951286	−	0.308309i	$-0.0997632\pi$
$744$	0	0
$745$	16.4538	0.602820
$746$	0	0
$747$	− 7.67389i	− 0.280773i
$748$	0	0
$749$	− 50.4410i	− 1.84307i
$750$	0	0
$751$	−18.4378	−0.672806	−0.336403	−	0.941718i	$-0.609211\pi$
$751$	−18.4378	−0.672806	−0.336403	+	0.941718i	$0.609211\pi$
$752$	0	0
$753$	18.2760	0.666014
$754$	0	0
$755$	3.49305	0.127125
$756$	0	0
$757$	−1.44805	−0.0526303	−0.0263151	−	0.999654i	$-0.508377\pi$
$757$	−1.44805	−0.0526303	−0.0263151	+	0.999654i	$0.508377\pi$
$758$	0	0
$759$	1.75926i	0.0638570i
$760$	0	0
$761$	− 14.7121i	− 0.533314i	−0.963791	−	0.266657i	$-0.914081\pi$
$761$	− 14.7121i	− 0.533314i	0.963791	−	0.266657i	$-0.0859192\pi$
$762$	0	0
$763$	45.2552	1.63835
$764$	0	0
$765$	− 1.58347i	− 0.0572504i
$766$	0	0
$767$	0	0
$768$	0	0
$769$	9.10703i	0.328408i	0.986426	+	0.164204i	$0.0525055\pi$
$769$	9.10703i	0.328408i	−0.986426	+	0.164204i	$0.947494\pi$
$770$	0	0
$771$	−26.4263	−0.951720
$772$	0	0
$773$	− 23.2027i	− 0.834543i	−0.908782	−	0.417272i	$-0.862986\pi$
$773$	− 23.2027i	− 0.834543i	0.908782	−	0.417272i	$-0.137014\pi$
$774$	0	0
$775$	− 18.8246i	− 0.676200i
$776$	0	0
$777$	−23.0961	−0.828570
$778$	0	0
$779$	28.6268	1.02566
$780$	0	0
$781$	−31.1216	−1.11362
$782$	0	0
$783$	2.69251	0.0962224
$784$	0	0
$785$	13.1478i	0.469267i
$786$	0	0
$787$	− 12.7879i	− 0.455840i	−0.973680	−	0.227920i	$-0.926808\pi$
$787$	− 12.7879i	− 0.455840i	0.973680	−	0.227920i	$-0.0731925\pi$
$788$	0	0
$789$	4.82017	0.171603
$790$	0	0
$791$	20.7783i	0.738792i
$792$	0	0
$793$	0	0
$794$	0	0
$795$	− 21.5127i	− 0.762976i
$796$	0	0
$797$	−40.4330	−1.43221	−0.716106	−	0.697992i	$-0.754077\pi$
$797$	−40.4330	−1.43221	−0.716106	+	0.697992i	$0.754077\pi$
$798$	0	0
$799$	7.27091i	0.257226i
$800$	0	0
$801$	− 0.494055i	− 0.0174566i
$802$	0	0
$803$	18.6620	0.658568
$804$	0	0
$805$	−3.60310	−0.126992
$806$	0	0
$807$	−1.64034	−0.0577426
$808$	0	0
$809$	44.5607	1.56667	0.783335	−	0.621599i	$-0.213517\pi$
$809$	44.5607	1.56667	0.783335	+	0.621599i	$0.213517\pi$
$810$	0	0
$811$	1.90102i	0.0667537i	0.999443	+	0.0333769i	$0.0106262\pi$
$811$	1.90102i	0.0667537i	−0.999443	+	0.0333769i	$0.989374\pi$
$812$	0	0
$813$	− 12.7300i	− 0.446461i
$814$	0	0
$815$	−6.52929	−0.228711
$816$	0	0
$817$	− 16.7520i	− 0.586077i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 43.0062i	− 1.50093i	−0.660913	−	0.750463i	$-0.729831\pi$
$821$	− 43.0062i	− 1.50093i	0.660913	−	0.750463i	$-0.270169\pi$
$822$	0	0
$823$	−32.1490	−1.12064	−0.560321	−	0.828275i	$-0.689322\pi$
$823$	−32.1490	−1.12064	−0.560321	+	0.828275i	$0.689322\pi$
$824$	0	0
$825$	− 5.80512i	− 0.202108i
$826$	0	0
$827$	− 2.20149i	− 0.0765533i	−0.999267	−	0.0382766i	$-0.987813\pi$
$827$	− 2.20149i	− 0.0765533i	0.999267	−	0.0382766i	$-0.0121868\pi$
$828$	0	0
$829$	−37.7013	−1.30942	−0.654709	−	0.755881i	$-0.727209\pi$
$829$	−37.7013	−1.30942	−0.654709	+	0.755881i	$0.727209\pi$
$830$	0	0
$831$	−24.5008	−0.849923
$832$	0	0
$833$	−1.79716	−0.0622679
$834$	0	0
$835$	29.0884	1.00665
$836$	0	0
$837$	− 7.28657i	− 0.251861i
$838$	0	0
$839$	− 27.8776i	− 0.962442i	−0.876599	−	0.481221i	$-0.840194\pi$
$839$	− 27.8776i	− 0.962442i	0.876599	−	0.481221i	$-0.159806\pi$
$840$	0	0
$841$	−21.7504	−0.750014
$842$	0	0
$843$	1.93853i	0.0667665i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	17.6173i	0.605337i
$848$	0	0
$849$	−8.07853	−0.277254
$850$	0	0
$851$	− 6.10803i	− 0.209381i
$852$	0	0
$853$	3.38475i	0.115891i	0.998320	+	0.0579457i	$0.0184550\pi$
$853$	3.38475i	0.115891i	−0.998320	+	0.0579457i	$0.981545\pi$
$854$	0	0
$855$	8.32611	0.284747
$856$	0	0
$857$	−0.567882	−0.0193985	−0.00969925	−	0.999953i	$-0.503087\pi$
$857$	−0.567882	−0.0193985	−0.00969925	+	0.999953i	$0.503087\pi$
$858$	0	0
$859$	−43.1391	−1.47189	−0.735944	−	0.677043i	$-0.763261\pi$
$859$	−43.1391	−1.47189	−0.735944	+	0.677043i	$0.763261\pi$
$860$	0	0
$861$	15.8229	0.539242
$862$	0	0
$863$	− 21.6466i	− 0.736860i	−0.929656	−	0.368430i	$-0.879895\pi$
$863$	− 21.6466i	− 0.736860i	0.929656	−	0.368430i	$-0.120105\pi$
$864$	0	0
$865$	4.49406i	0.152802i
$866$	0	0
$867$	−15.9624	−0.542112
$868$	0	0
$869$	− 2.77664i	− 0.0941911i
$870$	0	0
$871$	0	0
$872$	0	0
$873$	11.1057i	0.375872i
$874$	0	0
$875$	34.8998	1.17983
$876$	0	0
$877$	42.3494i	1.43004i	0.699105	+	0.715019i	$0.253582\pi$
$877$	42.3494i	1.43004i	−0.699105	+	0.715019i	$0.746418\pi$
$878$	0	0
$879$	− 28.3411i	− 0.955923i
$880$	0	0
$881$	29.1903	0.983445	0.491722	−	0.870752i	$-0.336367\pi$
$881$	29.1903	0.983445	0.491722	+	0.870752i	$0.336367\pi$
$882$	0	0
$883$	−41.6461	−1.40150	−0.700751	−	0.713406i	$-0.747152\pi$
$883$	−41.6461	−1.40150	−0.700751	+	0.713406i	$0.747152\pi$
$884$	0	0
$885$	6.77003	0.227572
$886$	0	0
$887$	40.6471	1.36480	0.682398	−	0.730981i	$-0.260937\pi$
$887$	40.6471	1.36480	0.682398	+	0.730981i	$0.260937\pi$
$888$	0	0
$889$	43.8193i	1.46965i
$890$	0	0
$891$	− 2.24703i	− 0.0752783i
$892$	0	0
$893$	−38.2315	−1.27937
$894$	0	0
$895$	− 35.5323i	− 1.18771i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 19.6191i	− 0.654335i
$900$	0	0
$901$	14.0965	0.469622
$902$	0	0
$903$	− 9.25931i	− 0.308130i
$904$	0	0
$905$	− 12.5767i	− 0.418064i
$906$	0	0
$907$	−37.7561	−1.25367	−0.626836	−	0.779151i	$-0.715650\pi$
$907$	−37.7561	−1.25367	−0.626836	+	0.779151i	$0.715650\pi$
$908$	0	0
$909$	15.1866	0.503706
$910$	0	0
$911$	−42.4249	−1.40560	−0.702801	−	0.711387i	$-0.748067\pi$
$911$	−42.4249	−1.40560	−0.702801	+	0.711387i	$0.748067\pi$
$912$	0	0
$913$	−17.2434	−0.570674
$914$	0	0
$915$	− 7.99844i	− 0.264420i
$916$	0	0
$917$	17.2274i	0.568898i
$918$	0	0
$919$	49.8346	1.64389	0.821947	−	0.569565i	$-0.192888\pi$
$919$	49.8346	1.64389	0.821947	+	0.569565i	$0.192888\pi$
$920$	0	0
$921$	− 30.7025i	− 1.01168i
$922$	0	0
$923$	0	0
$924$	0	0
$925$	20.1550i	0.662694i
$926$	0	0
$927$	1.12766	0.0370373
$928$	0	0
$929$	− 44.1893i	− 1.44980i	−0.688853	−	0.724901i	$-0.741885\pi$
$929$	− 44.1893i	− 1.44980i	0.688853	−	0.724901i	$-0.258115\pi$
$930$	0	0
$931$	− 9.44973i	− 0.309703i
$932$	0	0
$933$	13.7338	0.449624
$934$	0	0
$935$	−3.55810	−0.116362
$936$	0	0
$937$	−22.9644	−0.750215	−0.375107	−	0.926981i	$-0.622394\pi$
$937$	−22.9644	−0.750215	−0.375107	+	0.926981i	$0.622394\pi$
$938$	0	0
$939$	32.1886	1.05043
$940$	0	0
$941$	− 6.09255i	− 0.198611i	−0.995057	−	0.0993057i	$-0.968338\pi$
$941$	− 6.09255i	− 0.198611i	0.995057	−	0.0993057i	$-0.0316622\pi$
$942$	0	0
$943$	4.18453i	0.136267i
$944$	0	0
$945$	4.60209	0.149706
$946$	0	0
$947$	− 41.6982i	− 1.35501i	−0.735518	−	0.677505i	$-0.763061\pi$
$947$	− 41.6982i	− 1.35501i	0.735518	−	0.677505i	$-0.236939\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	4.54723i	0.147454i
$952$	0	0
$953$	−35.3850	−1.14623	−0.573117	−	0.819474i	$-0.694266\pi$
$953$	−35.3850	−1.14623	−0.573117	+	0.819474i	$0.694266\pi$
$954$	0	0
$955$	23.9163i	0.773914i
$956$	0	0
$957$	− 6.05014i	− 0.195573i
$958$	0	0
$959$	−51.4154	−1.66029
$960$	0	0
$961$	−22.0941	−0.712713
$962$	0	0
$963$	−17.0382	−0.549050
$964$	0	0
$965$	−0.539474	−0.0173663
$966$	0	0
$967$	4.08684i	0.131424i	0.997839	+	0.0657120i	$0.0209319\pi$
$967$	4.08684i	0.131424i	−0.997839	+	0.0657120i	$0.979068\pi$
$968$	0	0
$969$	5.45581i	0.175266i
$970$	0	0
$971$	−5.72402	−0.183693	−0.0918463	−	0.995773i	$-0.529277\pi$
$971$	−5.72402	−0.183693	−0.0918463	+	0.995773i	$0.529277\pi$
$972$	0	0
$973$	17.4309i	0.558810i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 9.04498i	− 0.289375i	−0.989477	−	0.144687i	$-0.953782\pi$
$977$	− 9.04498i	− 0.289375i	0.989477	−	0.144687i	$-0.0462176\pi$
$978$	0	0
$979$	−1.11016	−0.0354807
$980$	0	0
$981$	− 15.2866i	− 0.488063i
$982$	0	0
$983$	− 7.05061i	− 0.224879i	−0.993659	−	0.112440i	$-0.964133\pi$
$983$	− 7.05061i	− 0.224879i	0.993659	−	0.112440i	$-0.0358665\pi$
$984$	0	0
$985$	−40.1943	−1.28070
$986$	0	0
$987$	−21.1317	−0.672630
$988$	0	0
$989$	2.44873	0.0778650
$990$	0	0
$991$	−19.9292	−0.633072	−0.316536	−	0.948580i	$-0.602520\pi$
$991$	−19.9292	−0.633072	−0.316536	+	0.948580i	$0.602520\pi$
$992$	0	0
$993$	− 14.7899i	− 0.469344i
$994$	0	0
$995$	− 11.4191i	− 0.362010i
$996$	0	0
$997$	−22.1395	−0.701164	−0.350582	−	0.936532i	$-0.614016\pi$
$997$	−22.1395	−0.701164	−0.350582	+	0.936532i	$0.614016\pi$
$998$	0	0
$999$	7.80155i	0.246830i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4056.2.c.p.337.2		8
13.5	odd	4		4056.2.a.bd.1.2		4
13.8	odd	4		4056.2.a.be.1.3		4
13.9	even	3		312.2.bf.b.49.1	✓	8
13.10	even	6		312.2.bf.b.121.4	yes	8
13.12	even	2	inner	4056.2.c.p.337.7		8
39.23	odd	6		936.2.bi.c.433.1		8
39.35	odd	6		936.2.bi.c.361.4		8
52.23	odd	6		624.2.bv.g.433.4		8
52.31	even	4		8112.2.a.cq.1.2		4
52.35	odd	6		624.2.bv.g.49.1		8
52.47	even	4		8112.2.a.cs.1.3		4
156.23	even	6		1872.2.by.m.433.1		8
156.35	even	6		1872.2.by.m.1297.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
312.2.bf.b.49.1	✓	8	13.9	even	3
312.2.bf.b.121.4	yes	8	13.10	even	6
624.2.bv.g.49.1		8	52.35	odd	6
624.2.bv.g.433.4		8	52.23	odd	6
936.2.bi.c.361.4		8	39.35	odd	6
936.2.bi.c.433.1		8	39.23	odd	6
1872.2.by.m.433.1		8	156.23	even	6
1872.2.by.m.1297.4		8	156.35	even	6
4056.2.a.bd.1.2		4	13.5	odd	4
4056.2.a.be.1.3		4	13.8	odd	4
4056.2.c.p.337.2		8	1.1	even	1	trivial
4056.2.c.p.337.7		8	13.12	even	2	inner
8112.2.a.cq.1.2		4	52.31	even	4
8112.2.a.cs.1.3		4	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 $\F_{q}$

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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -1.55452i q^{5} +2.96046i q^{7} +1.00000 q^{9} -2.24703i q^{11} -1.55452i q^{15} +1.01862 q^{17} +5.35607i q^{19} +2.96046i q^{21} -0.782926 q^{23} +2.58347 q^{25} +1.00000 q^{27} +2.69251 q^{29} -7.28657i q^{31} -2.24703i q^{33} +4.60209 q^{35} +7.80155i q^{37} -5.34474i q^{41} -3.12766 q^{43} -1.55452i q^{45} +7.13799i q^{47} -1.76430 q^{49} +1.01862 q^{51} +13.8388 q^{53} -3.49305 q^{55} +5.35607i q^{57} +4.35506i q^{59} +5.14528 q^{61} +2.96046i q^{63} +12.9977i q^{67} -0.782926 q^{69} -13.8501i q^{71} +8.30519i q^{73} +2.58347 q^{75} +6.65223 q^{77} +1.23570 q^{79} +1.00000 q^{81} -7.67389i q^{83} -1.58347i q^{85} +2.69251 q^{87} -0.494055i q^{89} -7.28657i q^{93} +8.32611 q^{95} +11.1057i q^{97} -2.24703i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{3} + 8 q^{9} - 24 q^{17} + 4 q^{23} - 4 q^{25} + 8 q^{27} - 12 q^{29} - 20 q^{35} + 16 q^{43} - 36 q^{49} - 24 q^{51} + 4 q^{53} + 20 q^{55} - 4 q^{61} + 4 q^{69} - 4 q^{75} + 56 q^{77} - 12 q^{79}+ \cdots + 68 q^{95}+O(q^{100}) \) 8 * q + 8 * q^3 + 8 * q^9 - 24 * q^17 + 4 * q^23 - 4 * q^25 + 8 * q^27 - 12 * q^29 - 20 * q^35 + 16 * q^43 - 36 * q^49 - 24 * q^51 + 4 * q^53 + 20 * q^55 - 4 * q^61 + 4 * q^69 - 4 * q^75 + 56 * q^77 - 12 * q^79 + 8 * q^81 - 12 * q^87 + 68 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

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Database

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