LMFDB - Embedded newform 1280.2.d.m.641.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1280,2,Mod(641,1280)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1280, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0]))

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

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

chi := DirichletCharacter("1280.641");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1280 = 2^{8} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1280.d (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:	$10.2208514587$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 3x^{2} + 1 $ x^4 + 3*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 640)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			641.3
Root			$0.618034i$ of defining polynomial
Character	$\chi$	$=$	1280.641
Dual form			1280.2.d.m.641.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.23607i q^{3} +1.00000i q^{5} +3.23607 q^{7} +1.47214 q^{9} +O(q^{10})$	$q+1.23607i q^{3} +1.00000i q^{5} +3.23607 q^{7} +1.47214 q^{9} +2.00000i q^{11} -4.47214i q^{13} -1.23607 q^{15} +4.47214 q^{17} -4.47214i q^{19} +4.00000i q^{21} +4.76393 q^{23} -1.00000 q^{25} +5.52786i q^{27} -2.00000i q^{29} +6.47214 q^{31} -2.47214 q^{33} +3.23607i q^{35} -6.94427i q^{37} +5.52786 q^{39} -12.4721 q^{41} +7.70820i q^{43} +1.47214i q^{45} -7.23607 q^{47} +3.47214 q^{49} +5.52786i q^{51} -0.472136i q^{53} -2.00000 q^{55} +5.52786 q^{57} +8.47214i q^{59} -6.00000i q^{61} +4.76393 q^{63} +4.47214 q^{65} +7.70820i q^{67} +5.88854i q^{69} +2.47214 q^{71} -4.47214 q^{73} -1.23607i q^{75} +6.47214i q^{77} +12.9443 q^{79} -2.41641 q^{81} -3.70820i q^{83} +4.47214i q^{85} +2.47214 q^{87} +14.9443 q^{89} -14.4721i q^{91} +8.00000i q^{93} +4.47214 q^{95} -16.4721 q^{97} +2.94427i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{7} - 12 q^{9}+O(q^{10}) $ 4 * q + 4 * q^7 - 12 * q^9	$ 4 q + 4 q^{7} - 12 q^{9} + 4 q^{15} + 28 q^{23} - 4 q^{25} + 8 q^{31} + 8 q^{33} + 40 q^{39} - 32 q^{41} - 20 q^{47} - 4 q^{49} - 8 q^{55} + 40 q^{57} + 28 q^{63} - 8 q^{71} + 16 q^{79} + 44 q^{81} - 8 q^{87} + 24 q^{89} - 48 q^{97}+O(q^{100}) $ 4 * q + 4 * q^7 - 12 * q^9 + 4 * q^15 + 28 * q^23 - 4 * q^25 + 8 * q^31 + 8 * q^33 + 40 * q^39 - 32 * q^41 - 20 * q^47 - 4 * q^49 - 8 * q^55 + 40 * q^57 + 28 * q^63 - 8 * q^71 + 16 * q^79 + 44 * q^81 - 8 * q^87 + 24 * q^89 - 48 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1280\mathbb{Z}\right)^\times$.

$n$	$257$	$261$	$511$
$\chi(n)$	$1$	$-1$	$1$

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
$3$	1.23607i	0.713644i	0.934172	+	0.356822i	$0.116140\pi$
$3$	1.23607i	0.713644i	−0.934172	+	0.356822i	$0.883860\pi$
$4$	0	0
$5$	1.00000i	0.447214i
$5$	1.00000i	0.447214i
$6$	0	0
$7$	3.23607	1.22312	0.611559	−	0.791199i	$-0.290543\pi$
$7$	3.23607	1.22312	0.611559	+	0.791199i	$0.290543\pi$
$8$	0	0
$9$	1.47214	0.490712
$10$	0	0
$11$	2.00000i	0.603023i	0.953463	+	0.301511i	$0.0974911\pi$
$11$	2.00000i	0.603023i	−0.953463	+	0.301511i	$0.902509\pi$
$12$	0	0
$13$	− 4.47214i	− 1.24035i	−0.784465	−	0.620174i	$-0.787062\pi$
$13$	− 4.47214i	− 1.24035i	0.784465	−	0.620174i	$-0.212938\pi$
$14$	0	0
$15$	−1.23607	−0.319151
$16$	0	0
$17$	4.47214	1.08465	0.542326	−	0.840168i	$-0.317544\pi$
$17$	4.47214	1.08465	0.542326	+	0.840168i	$0.317544\pi$
$18$	0	0
$19$	− 4.47214i	− 1.02598i	−0.858395	−	0.512989i	$-0.828538\pi$
$19$	− 4.47214i	− 1.02598i	0.858395	−	0.512989i	$-0.171462\pi$
$20$	0	0
$21$	4.00000i	0.872872i
$22$	0	0
$23$	4.76393	0.993348	0.496674	−	0.867937i	$-0.334554\pi$
$23$	4.76393	0.993348	0.496674	+	0.867937i	$0.334554\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	0	0
$27$	5.52786i	1.06384i
$28$	0	0
$29$	− 2.00000i	− 0.371391i	−0.982607	−	0.185695i	$-0.940546\pi$
$29$	− 2.00000i	− 0.371391i	0.982607	−	0.185695i	$-0.0594537\pi$
$30$	0	0
$31$	6.47214	1.16243	0.581215	−	0.813750i	$-0.302578\pi$
$31$	6.47214	1.16243	0.581215	+	0.813750i	$0.302578\pi$
$32$	0	0
$33$	−2.47214	−0.430344
$34$	0	0
$35$	3.23607i	0.546995i
$36$	0	0
$37$	− 6.94427i	− 1.14163i	−0.821078	−	0.570816i	$-0.806627\pi$
$37$	− 6.94427i	− 1.14163i	0.821078	−	0.570816i	$-0.193373\pi$
$38$	0	0
$39$	5.52786	0.885167
$40$	0	0
$41$	−12.4721	−1.94782	−0.973910	−	0.226934i	$-0.927130\pi$
$41$	−12.4721	−1.94782	−0.973910	+	0.226934i	$0.927130\pi$
$42$	0	0
$43$	7.70820i	1.17549i	0.809046	+	0.587745i	$0.199984\pi$
$43$	7.70820i	1.17549i	−0.809046	+	0.587745i	$0.800016\pi$
$44$	0	0
$45$	1.47214i	0.219453i
$46$	0	0
$47$	−7.23607	−1.05549	−0.527744	−	0.849403i	$-0.676962\pi$
$47$	−7.23607	−1.05549	−0.527744	+	0.849403i	$0.676962\pi$
$48$	0	0
$49$	3.47214	0.496019
$50$	0	0
$51$	5.52786i	0.774056i
$52$	0	0
$53$	− 0.472136i	− 0.0648529i	−0.999474	−	0.0324264i	$-0.989677\pi$
$53$	− 0.472136i	− 0.0648529i	0.999474	−	0.0324264i	$-0.0103235\pi$
$54$	0	0
$55$	−2.00000	−0.269680
$56$	0	0
$57$	5.52786	0.732183
$58$	0	0
$59$	8.47214i	1.10298i	0.834182	+	0.551489i	$0.185940\pi$
$59$	8.47214i	1.10298i	−0.834182	+	0.551489i	$0.814060\pi$
$60$	0	0
$61$	− 6.00000i	− 0.768221i	−0.923287	−	0.384111i	$-0.874508\pi$
$61$	− 6.00000i	− 0.768221i	0.923287	−	0.384111i	$-0.125492\pi$
$62$	0	0
$63$	4.76393	0.600199
$64$	0	0
$65$	4.47214	0.554700
$66$	0	0
$67$	7.70820i	0.941707i	0.882211	+	0.470853i	$0.156054\pi$
$67$	7.70820i	0.941707i	−0.882211	+	0.470853i	$0.843946\pi$
$68$	0	0
$69$	5.88854i	0.708897i
$70$	0	0
$71$	2.47214	0.293389	0.146694	−	0.989182i	$-0.453137\pi$
$71$	2.47214	0.293389	0.146694	+	0.989182i	$0.453137\pi$
$72$	0	0
$73$	−4.47214	−0.523424	−0.261712	−	0.965146i	$-0.584287\pi$
$73$	−4.47214	−0.523424	−0.261712	+	0.965146i	$0.584287\pi$
$74$	0	0
$75$	− 1.23607i	− 0.142729i
$76$	0	0
$77$	6.47214i	0.737568i
$78$	0	0
$79$	12.9443	1.45634	0.728172	−	0.685394i	$-0.240370\pi$
$79$	12.9443	1.45634	0.728172	+	0.685394i	$0.240370\pi$
$80$	0	0
$81$	−2.41641	−0.268490
$82$	0	0
$83$	− 3.70820i	− 0.407028i	−0.979072	−	0.203514i	$-0.934764\pi$
$83$	− 3.70820i	− 0.407028i	0.979072	−	0.203514i	$-0.0652363\pi$
$84$	0	0
$85$	4.47214i	0.485071i
$86$	0	0
$87$	2.47214	0.265041
$88$	0	0
$89$	14.9443	1.58409	0.792045	−	0.610463i	$-0.209017\pi$
$89$	14.9443	1.58409	0.792045	+	0.610463i	$0.209017\pi$
$90$	0	0
$91$	− 14.4721i	− 1.51709i
$92$	0	0
$93$	8.00000i	0.829561i
$94$	0	0
$95$	4.47214	0.458831
$96$	0	0
$97$	−16.4721	−1.67249	−0.836246	−	0.548354i	$-0.815254\pi$
$97$	−16.4721	−1.67249	−0.836246	+	0.548354i	$0.815254\pi$
$98$	0	0
$99$	2.94427i	0.295910i
$100$	0	0
$101$	10.0000i	0.995037i	0.867453	+	0.497519i	$0.165755\pi$
$101$	10.0000i	0.995037i	−0.867453	+	0.497519i	$0.834245\pi$
$102$	0	0
$103$	−3.23607	−0.318859	−0.159430	−	0.987209i	$-0.550966\pi$
$103$	−3.23607	−0.318859	−0.159430	+	0.987209i	$0.550966\pi$
$104$	0	0
$105$	−4.00000	−0.390360
$106$	0	0
$107$	17.2361i	1.66627i	0.553067	+	0.833137i	$0.313457\pi$
$107$	17.2361i	1.66627i	−0.553067	+	0.833137i	$0.686543\pi$
$108$	0	0
$109$	14.9443i	1.43140i	0.698407	+	0.715701i	$0.253893\pi$
$109$	14.9443i	1.43140i	−0.698407	+	0.715701i	$0.746107\pi$
$110$	0	0
$111$	8.58359	0.814719
$112$	0	0
$113$	−2.94427	−0.276974	−0.138487	−	0.990364i	$-0.544224\pi$
$113$	−2.94427	−0.276974	−0.138487	+	0.990364i	$0.544224\pi$
$114$	0	0
$115$	4.76393i	0.444239i
$116$	0	0
$117$	− 6.58359i	− 0.608653i
$118$	0	0
$119$	14.4721	1.32666
$120$	0	0
$121$	7.00000	0.636364
$122$	0	0
$123$	− 15.4164i	− 1.39005i
$124$	0	0
$125$	− 1.00000i	− 0.0894427i
$126$	0	0
$127$	−20.1803	−1.79072	−0.895358	−	0.445348i	$-0.853080\pi$
$127$	−20.1803	−1.79072	−0.895358	+	0.445348i	$0.853080\pi$
$128$	0	0
$129$	−9.52786	−0.838882
$130$	0	0
$131$	14.9443i	1.30569i	0.757493	+	0.652844i	$0.226424\pi$
$131$	14.9443i	1.30569i	−0.757493	+	0.652844i	$0.773576\pi$
$132$	0	0
$133$	− 14.4721i	− 1.25489i
$134$	0	0
$135$	−5.52786	−0.475763
$136$	0	0
$137$	−2.00000	−0.170872	−0.0854358	−	0.996344i	$-0.527228\pi$
$137$	−2.00000	−0.170872	−0.0854358	+	0.996344i	$0.527228\pi$
$138$	0	0
$139$	− 20.4721i	− 1.73642i	−0.496194	−	0.868212i	$-0.665269\pi$
$139$	− 20.4721i	− 1.73642i	0.496194	−	0.868212i	$-0.334731\pi$
$140$	0	0
$141$	− 8.94427i	− 0.753244i
$142$	0	0
$143$	8.94427	0.747958
$144$	0	0
$145$	2.00000	0.166091
$146$	0	0
$147$	4.29180i	0.353981i
$148$	0	0
$149$	− 6.94427i	− 0.568897i	−0.958691	−	0.284448i	$-0.908190\pi$
$149$	− 6.94427i	− 0.568897i	0.958691	−	0.284448i	$-0.0918105\pi$
$150$	0	0
$151$	−23.4164	−1.90560	−0.952800	−	0.303598i	$-0.901812\pi$
$151$	−23.4164	−1.90560	−0.952800	+	0.303598i	$0.901812\pi$
$152$	0	0
$153$	6.58359	0.532252
$154$	0	0
$155$	6.47214i	0.519854i
$156$	0	0
$157$	5.05573i	0.403491i	0.979438	+	0.201746i	$0.0646614\pi$
$157$	5.05573i	0.403491i	−0.979438	+	0.201746i	$0.935339\pi$
$158$	0	0
$159$	0.583592	0.0462819
$160$	0	0
$161$	15.4164	1.21498
$162$	0	0
$163$	− 11.7082i	− 0.917057i	−0.888680	−	0.458529i	$-0.848377\pi$
$163$	− 11.7082i	− 0.917057i	0.888680	−	0.458529i	$-0.151623\pi$
$164$	0	0
$165$	− 2.47214i	− 0.192456i
$166$	0	0
$167$	−14.6525	−1.13384	−0.566921	−	0.823772i	$-0.691866\pi$
$167$	−14.6525	−1.13384	−0.566921	+	0.823772i	$0.691866\pi$
$168$	0	0
$169$	−7.00000	−0.538462
$170$	0	0
$171$	− 6.58359i	− 0.503460i
$172$	0	0
$173$	− 10.9443i	− 0.832078i	−0.909347	−	0.416039i	$-0.863418\pi$
$173$	− 10.9443i	− 0.832078i	0.909347	−	0.416039i	$-0.136582\pi$
$174$	0	0
$175$	−3.23607	−0.244624
$176$	0	0
$177$	−10.4721	−0.787134
$178$	0	0
$179$	− 20.4721i	− 1.53016i	−0.643936	−	0.765080i	$-0.722700\pi$
$179$	− 20.4721i	− 1.53016i	0.643936	−	0.765080i	$-0.277300\pi$
$180$	0	0
$181$	− 10.9443i	− 0.813481i	−0.913544	−	0.406741i	$-0.866665\pi$
$181$	− 10.9443i	− 0.813481i	0.913544	−	0.406741i	$-0.133335\pi$
$182$	0	0
$183$	7.41641	0.548237
$184$	0	0
$185$	6.94427	0.510553
$186$	0	0
$187$	8.94427i	0.654070i
$188$	0	0
$189$	17.8885i	1.30120i
$190$	0	0
$191$	11.4164	0.826062	0.413031	−	0.910717i	$-0.364470\pi$
$191$	11.4164	0.826062	0.413031	+	0.910717i	$0.364470\pi$
$192$	0	0
$193$	−11.5279	−0.829794	−0.414897	−	0.909868i	$-0.636182\pi$
$193$	−11.5279	−0.829794	−0.414897	+	0.909868i	$0.636182\pi$
$194$	0	0
$195$	5.52786i	0.395859i
$196$	0	0
$197$	− 0.472136i	− 0.0336383i	−0.999859	−	0.0168191i	$-0.994646\pi$
$197$	− 0.472136i	− 0.0336383i	0.999859	−	0.0168191i	$-0.00535395\pi$
$198$	0	0
$199$	0.944272	0.0669377	0.0334688	−	0.999440i	$-0.489345\pi$
$199$	0.944272	0.0669377	0.0334688	+	0.999440i	$0.489345\pi$
$200$	0	0
$201$	−9.52786	−0.672044
$202$	0	0
$203$	− 6.47214i	− 0.454255i
$204$	0	0
$205$	− 12.4721i	− 0.871092i
$206$	0	0
$207$	7.01316	0.487448
$208$	0	0
$209$	8.94427	0.618688
$210$	0	0
$211$	− 1.05573i	− 0.0726793i	−0.999339	−	0.0363397i	$-0.988430\pi$
$211$	− 1.05573i	− 0.0726793i	0.999339	−	0.0363397i	$-0.0115698\pi$
$212$	0	0
$213$	3.05573i	0.209375i
$214$	0	0
$215$	−7.70820	−0.525695
$216$	0	0
$217$	20.9443	1.42179
$218$	0	0
$219$	− 5.52786i	− 0.373538i
$220$	0	0
$221$	− 20.0000i	− 1.34535i
$222$	0	0
$223$	−8.76393	−0.586876	−0.293438	−	0.955978i	$-0.594799\pi$
$223$	−8.76393	−0.586876	−0.293438	+	0.955978i	$0.594799\pi$
$224$	0	0
$225$	−1.47214	−0.0981424
$226$	0	0
$227$	− 10.1803i	− 0.675693i	−0.941201	−	0.337846i	$-0.890302\pi$
$227$	− 10.1803i	− 0.675693i	0.941201	−	0.337846i	$-0.109698\pi$
$228$	0	0
$229$	− 2.94427i	− 0.194563i	−0.995257	−	0.0972815i	$-0.968985\pi$
$229$	− 2.94427i	− 0.194563i	0.995257	−	0.0972815i	$-0.0310147\pi$
$230$	0	0
$231$	−8.00000	−0.526361
$232$	0	0
$233$	−15.5279	−1.01726	−0.508632	−	0.860984i	$-0.669849\pi$
$233$	−15.5279	−1.01726	−0.508632	+	0.860984i	$0.669849\pi$
$234$	0	0
$235$	− 7.23607i	− 0.472029i
$236$	0	0
$237$	16.0000i	1.03931i
$238$	0	0
$239$	−12.9443	−0.837295	−0.418648	−	0.908149i	$-0.637496\pi$
$239$	−12.9443	−0.837295	−0.418648	+	0.908149i	$0.637496\pi$
$240$	0	0
$241$	26.3607	1.69804	0.849020	−	0.528360i	$-0.177193\pi$
$241$	26.3607	1.69804	0.849020	+	0.528360i	$0.177193\pi$
$242$	0	0
$243$	13.5967i	0.872232i
$244$	0	0
$245$	3.47214i	0.221827i
$246$	0	0
$247$	−20.0000	−1.27257
$248$	0	0
$249$	4.58359	0.290473
$250$	0	0
$251$	2.00000i	0.126239i	0.998006	+	0.0631194i	$0.0201049\pi$
$251$	2.00000i	0.126239i	−0.998006	+	0.0631194i	$0.979895\pi$
$252$	0	0
$253$	9.52786i	0.599012i
$254$	0	0
$255$	−5.52786	−0.346168
$256$	0	0
$257$	−2.94427	−0.183659	−0.0918293	−	0.995775i	$-0.529271\pi$
$257$	−2.94427	−0.183659	−0.0918293	+	0.995775i	$0.529271\pi$
$258$	0	0
$259$	− 22.4721i	− 1.39635i
$260$	0	0
$261$	− 2.94427i	− 0.182246i
$262$	0	0
$263$	17.7082	1.09193	0.545967	−	0.837806i	$-0.316162\pi$
$263$	17.7082	1.09193	0.545967	+	0.837806i	$0.316162\pi$
$264$	0	0
$265$	0.472136	0.0290031
$266$	0	0
$267$	18.4721i	1.13048i
$268$	0	0
$269$	− 23.8885i	− 1.45651i	−0.685306	−	0.728255i	$-0.740332\pi$
$269$	− 23.8885i	− 1.45651i	0.685306	−	0.728255i	$-0.259668\pi$
$270$	0	0
$271$	−24.3607	−1.47981	−0.739903	−	0.672714i	$-0.765129\pi$
$271$	−24.3607	−1.47981	−0.739903	+	0.672714i	$0.765129\pi$
$272$	0	0
$273$	17.8885	1.08266
$274$	0	0
$275$	− 2.00000i	− 0.120605i
$276$	0	0
$277$	10.9443i	0.657578i	0.944403	+	0.328789i	$0.106640\pi$
$277$	10.9443i	0.657578i	−0.944403	+	0.328789i	$0.893360\pi$
$278$	0	0
$279$	9.52786	0.570418
$280$	0	0
$281$	−3.52786	−0.210455	−0.105227	−	0.994448i	$-0.533557\pi$
$281$	−3.52786	−0.210455	−0.105227	+	0.994448i	$0.533557\pi$
$282$	0	0
$283$	− 8.29180i	− 0.492896i	−0.969156	−	0.246448i	$-0.920737\pi$
$283$	− 8.29180i	− 0.492896i	0.969156	−	0.246448i	$-0.0792635\pi$
$284$	0	0
$285$	5.52786i	0.327442i
$286$	0	0
$287$	−40.3607	−2.38242
$288$	0	0
$289$	3.00000	0.176471
$290$	0	0
$291$	− 20.3607i	− 1.19356i
$292$	0	0
$293$	23.8885i	1.39558i	0.716301	+	0.697792i	$0.245834\pi$
$293$	23.8885i	1.39558i	−0.716301	+	0.697792i	$0.754166\pi$
$294$	0	0
$295$	−8.47214	−0.493267
$296$	0	0
$297$	−11.0557	−0.641518
$298$	0	0
$299$	− 21.3050i	− 1.23210i
$300$	0	0
$301$	24.9443i	1.43776i
$302$	0	0
$303$	−12.3607	−0.710102
$304$	0	0
$305$	6.00000	0.343559
$306$	0	0
$307$	− 0.291796i	− 0.0166537i	−0.999965	−	0.00832684i	$-0.997349\pi$
$307$	− 0.291796i	− 0.0166537i	0.999965	−	0.00832684i	$-0.00265055\pi$
$308$	0	0
$309$	− 4.00000i	− 0.227552i
$310$	0	0
$311$	15.4164	0.874184	0.437092	−	0.899417i	$-0.356008\pi$
$311$	15.4164	0.874184	0.437092	+	0.899417i	$0.356008\pi$
$312$	0	0
$313$	28.8328	1.62973	0.814864	−	0.579653i	$-0.196812\pi$
$313$	28.8328	1.62973	0.814864	+	0.579653i	$0.196812\pi$
$314$	0	0
$315$	4.76393i	0.268417i
$316$	0	0
$317$	− 20.4721i	− 1.14983i	−0.818213	−	0.574915i	$-0.805035\pi$
$317$	− 20.4721i	− 1.14983i	0.818213	−	0.574915i	$-0.194965\pi$
$318$	0	0
$319$	4.00000	0.223957
$320$	0	0
$321$	−21.3050	−1.18913
$322$	0	0
$323$	− 20.0000i	− 1.11283i
$324$	0	0
$325$	4.47214i	0.248069i
$326$	0	0
$327$	−18.4721	−1.02151
$328$	0	0
$329$	−23.4164	−1.29099
$330$	0	0
$331$	− 15.8885i	− 0.873313i	−0.899628	−	0.436657i	$-0.856162\pi$
$331$	− 15.8885i	− 0.873313i	0.899628	−	0.436657i	$-0.143838\pi$
$332$	0	0
$333$	− 10.2229i	− 0.560212i
$334$	0	0
$335$	−7.70820	−0.421144
$336$	0	0
$337$	5.05573	0.275403	0.137702	−	0.990474i	$-0.456029\pi$
$337$	5.05573	0.275403	0.137702	+	0.990474i	$0.456029\pi$
$338$	0	0
$339$	− 3.63932i	− 0.197661i
$340$	0	0
$341$	12.9443i	0.700972i
$342$	0	0
$343$	−11.4164	−0.616428
$344$	0	0
$345$	−5.88854	−0.317029
$346$	0	0
$347$	17.2361i	0.925281i	0.886546	+	0.462640i	$0.153098\pi$
$347$	17.2361i	0.925281i	−0.886546	+	0.462640i	$0.846902\pi$
$348$	0	0
$349$	− 14.9443i	− 0.799949i	−0.916526	−	0.399974i	$-0.869019\pi$
$349$	− 14.9443i	− 0.799949i	0.916526	−	0.399974i	$-0.130981\pi$
$350$	0	0
$351$	24.7214	1.31953
$352$	0	0
$353$	5.05573	0.269089	0.134545	−	0.990908i	$-0.457043\pi$
$353$	5.05573	0.269089	0.134545	+	0.990908i	$0.457043\pi$
$354$	0	0
$355$	2.47214i	0.131207i
$356$	0	0
$357$	17.8885i	0.946762i
$358$	0	0
$359$	15.0557	0.794611	0.397305	−	0.917686i	$-0.369945\pi$
$359$	15.0557	0.794611	0.397305	+	0.917686i	$0.369945\pi$
$360$	0	0
$361$	−1.00000	−0.0526316
$362$	0	0
$363$	8.65248i	0.454137i
$364$	0	0
$365$	− 4.47214i	− 0.234082i
$366$	0	0
$367$	−18.2918	−0.954824	−0.477412	−	0.878680i	$-0.658425\pi$
$367$	−18.2918	−0.954824	−0.477412	+	0.878680i	$0.658425\pi$
$368$	0	0
$369$	−18.3607	−0.955819
$370$	0	0
$371$	− 1.52786i	− 0.0793227i
$372$	0	0
$373$	− 5.05573i	− 0.261776i	−0.991397	−	0.130888i	$-0.958217\pi$
$373$	− 5.05573i	− 0.261776i	0.991397	−	0.130888i	$-0.0417828\pi$
$374$	0	0
$375$	1.23607	0.0638303
$376$	0	0
$377$	−8.94427	−0.460653
$378$	0	0
$379$	− 15.5279i	− 0.797613i	−0.917035	−	0.398806i	$-0.869425\pi$
$379$	− 15.5279i	− 0.797613i	0.917035	−	0.398806i	$-0.130575\pi$
$380$	0	0
$381$	− 24.9443i	− 1.27793i
$382$	0	0
$383$	10.2918	0.525886	0.262943	−	0.964811i	$-0.415307\pi$
$383$	10.2918	0.525886	0.262943	+	0.964811i	$0.415307\pi$
$384$	0	0
$385$	−6.47214	−0.329851
$386$	0	0
$387$	11.3475i	0.576827i
$388$	0	0
$389$	− 11.8885i	− 0.602773i	−0.953502	−	0.301387i	$-0.902551\pi$
$389$	− 11.8885i	− 0.602773i	0.953502	−	0.301387i	$-0.0974495\pi$
$390$	0	0
$391$	21.3050	1.07744
$392$	0	0
$393$	−18.4721	−0.931796
$394$	0	0
$395$	12.9443i	0.651297i
$396$	0	0
$397$	− 12.4721i	− 0.625959i	−0.949760	−	0.312979i	$-0.898673\pi$
$397$	− 12.4721i	− 0.625959i	0.949760	−	0.312979i	$-0.101327\pi$
$398$	0	0
$399$	17.8885	0.895547
$400$	0	0
$401$	−2.00000	−0.0998752	−0.0499376	−	0.998752i	$-0.515902\pi$
$401$	−2.00000	−0.0998752	−0.0499376	+	0.998752i	$0.515902\pi$
$402$	0	0
$403$	− 28.9443i	− 1.44182i
$404$	0	0
$405$	− 2.41641i	− 0.120072i
$406$	0	0
$407$	13.8885	0.688430
$408$	0	0
$409$	−17.4164	−0.861186	−0.430593	−	0.902546i	$-0.641696\pi$
$409$	−17.4164	−0.861186	−0.430593	+	0.902546i	$0.641696\pi$
$410$	0	0
$411$	− 2.47214i	− 0.121941i
$412$	0	0
$413$	27.4164i	1.34907i
$414$	0	0
$415$	3.70820	0.182029
$416$	0	0
$417$	25.3050	1.23919
$418$	0	0
$419$	− 4.47214i	− 0.218478i	−0.994016	−	0.109239i	$-0.965159\pi$
$419$	− 4.47214i	− 0.218478i	0.994016	−	0.109239i	$-0.0348414\pi$
$420$	0	0
$421$	28.8328i	1.40523i	0.711572	+	0.702613i	$0.247983\pi$
$421$	28.8328i	1.40523i	−0.711572	+	0.702613i	$0.752017\pi$
$422$	0	0
$423$	−10.6525	−0.517941
$424$	0	0
$425$	−4.47214	−0.216930
$426$	0	0
$427$	− 19.4164i	− 0.939626i
$428$	0	0
$429$	11.0557i	0.533776i
$430$	0	0
$431$	−24.3607	−1.17341	−0.586706	−	0.809800i	$-0.699576\pi$
$431$	−24.3607	−1.17341	−0.586706	+	0.809800i	$0.699576\pi$
$432$	0	0
$433$	−0.472136	−0.0226894	−0.0113447	−	0.999936i	$-0.503611\pi$
$433$	−0.472136	−0.0226894	−0.0113447	+	0.999936i	$0.503611\pi$
$434$	0	0
$435$	2.47214i	0.118530i
$436$	0	0
$437$	− 21.3050i	− 1.01915i
$438$	0	0
$439$	−15.0557	−0.718571	−0.359285	−	0.933228i	$-0.616980\pi$
$439$	−15.0557	−0.718571	−0.359285	+	0.933228i	$0.616980\pi$
$440$	0	0
$441$	5.11146	0.243403
$442$	0	0
$443$	4.65248i	0.221046i	0.993874	+	0.110523i	$0.0352526\pi$
$443$	4.65248i	0.221046i	−0.993874	+	0.110523i	$0.964747\pi$
$444$	0	0
$445$	14.9443i	0.708426i
$446$	0	0
$447$	8.58359	0.405990
$448$	0	0
$449$	−1.41641	−0.0668444	−0.0334222	−	0.999441i	$-0.510641\pi$
$449$	−1.41641	−0.0668444	−0.0334222	+	0.999441i	$0.510641\pi$
$450$	0	0
$451$	− 24.9443i	− 1.17458i
$452$	0	0
$453$	− 28.9443i	− 1.35992i
$454$	0	0
$455$	14.4721	0.678464
$456$	0	0
$457$	−3.88854	−0.181898	−0.0909492	−	0.995856i	$-0.528990\pi$
$457$	−3.88854	−0.181898	−0.0909492	+	0.995856i	$0.528990\pi$
$458$	0	0
$459$	24.7214i	1.15389i
$460$	0	0
$461$	41.7771i	1.94575i	0.231325	+	0.972876i	$0.425694\pi$
$461$	41.7771i	1.94575i	−0.231325	+	0.972876i	$0.574306\pi$
$462$	0	0
$463$	1.12461	0.0522651	0.0261326	−	0.999658i	$-0.491681\pi$
$463$	1.12461	0.0522651	0.0261326	+	0.999658i	$0.491681\pi$
$464$	0	0
$465$	−8.00000	−0.370991
$466$	0	0
$467$	41.5967i	1.92487i	0.271516	+	0.962434i	$0.412475\pi$
$467$	41.5967i	1.92487i	−0.271516	+	0.962434i	$0.587525\pi$
$468$	0	0
$469$	24.9443i	1.15182i
$470$	0	0
$471$	−6.24922	−0.287949
$472$	0	0
$473$	−15.4164	−0.708847
$474$	0	0
$475$	4.47214i	0.205196i
$476$	0	0
$477$	− 0.695048i	− 0.0318241i
$478$	0	0
$479$	12.9443	0.591439	0.295719	−	0.955275i	$-0.404441\pi$
$479$	12.9443	0.591439	0.295719	+	0.955275i	$0.404441\pi$
$480$	0	0
$481$	−31.0557	−1.41602
$482$	0	0
$483$	19.0557i	0.867066i
$484$	0	0
$485$	− 16.4721i	− 0.747961i
$486$	0	0
$487$	−12.7639	−0.578389	−0.289194	−	0.957270i	$-0.593387\pi$
$487$	−12.7639	−0.578389	−0.289194	+	0.957270i	$0.593387\pi$
$488$	0	0
$489$	14.4721	0.654453
$490$	0	0
$491$	21.0557i	0.950232i	0.879923	+	0.475116i	$0.157594\pi$
$491$	21.0557i	0.950232i	−0.879923	+	0.475116i	$0.842406\pi$
$492$	0	0
$493$	− 8.94427i	− 0.402830i
$494$	0	0
$495$	−2.94427	−0.132335
$496$	0	0
$497$	8.00000	0.358849
$498$	0	0
$499$	14.5836i	0.652851i	0.945223	+	0.326426i	$0.105844\pi$
$499$	14.5836i	0.652851i	−0.945223	+	0.326426i	$0.894156\pi$
$500$	0	0
$501$	− 18.1115i	− 0.809160i
$502$	0	0
$503$	−5.12461	−0.228495	−0.114248	−	0.993452i	$-0.536446\pi$
$503$	−5.12461	−0.228495	−0.114248	+	0.993452i	$0.536446\pi$
$504$	0	0
$505$	−10.0000	−0.444994
$506$	0	0
$507$	− 8.65248i	− 0.384270i
$508$	0	0
$509$	− 16.8328i	− 0.746101i	−0.927811	−	0.373051i	$-0.878312\pi$
$509$	− 16.8328i	− 0.746101i	0.927811	−	0.373051i	$-0.121688\pi$
$510$	0	0
$511$	−14.4721	−0.640210
$512$	0	0
$513$	24.7214	1.09147
$514$	0	0
$515$	− 3.23607i	− 0.142598i
$516$	0	0
$517$	− 14.4721i	− 0.636484i
$518$	0	0
$519$	13.5279	0.593807
$520$	0	0
$521$	15.8885	0.696090	0.348045	−	0.937478i	$-0.386846\pi$
$521$	15.8885	0.696090	0.348045	+	0.937478i	$0.386846\pi$
$522$	0	0
$523$	− 20.0689i	− 0.877551i	−0.898597	−	0.438776i	$-0.855412\pi$
$523$	− 20.0689i	− 0.877551i	0.898597	−	0.438776i	$-0.144588\pi$
$524$	0	0
$525$	− 4.00000i	− 0.174574i
$526$	0	0
$527$	28.9443	1.26083
$528$	0	0
$529$	−0.304952	−0.0132588
$530$	0	0
$531$	12.4721i	0.541245i
$532$	0	0
$533$	55.7771i	2.41597i
$534$	0	0
$535$	−17.2361	−0.745180
$536$	0	0
$537$	25.3050	1.09199
$538$	0	0
$539$	6.94427i	0.299111i
$540$	0	0
$541$	23.8885i	1.02705i	0.858075	+	0.513524i	$0.171660\pi$
$541$	23.8885i	1.02705i	−0.858075	+	0.513524i	$0.828340\pi$
$542$	0	0
$543$	13.5279	0.580536
$544$	0	0
$545$	−14.9443	−0.640142
$546$	0	0
$547$	20.6525i	0.883036i	0.897252	+	0.441518i	$0.145560\pi$
$547$	20.6525i	0.883036i	−0.897252	+	0.441518i	$0.854440\pi$
$548$	0	0
$549$	− 8.83282i	− 0.376975i
$550$	0	0
$551$	−8.94427	−0.381039
$552$	0	0
$553$	41.8885	1.78128
$554$	0	0
$555$	8.58359i	0.364353i
$556$	0	0
$557$	13.0557i	0.553189i	0.960987	+	0.276594i	$0.0892059\pi$
$557$	13.0557i	0.553189i	−0.960987	+	0.276594i	$0.910794\pi$
$558$	0	0
$559$	34.4721	1.45802
$560$	0	0
$561$	−11.0557	−0.466773
$562$	0	0
$563$	4.29180i	0.180878i	0.995902	+	0.0904388i	$0.0288270\pi$
$563$	4.29180i	0.180878i	−0.995902	+	0.0904388i	$0.971173\pi$
$564$	0	0
$565$	− 2.94427i	− 0.123866i
$566$	0	0
$567$	−7.81966	−0.328395
$568$	0	0
$569$	34.3607	1.44047	0.720237	−	0.693728i	$-0.244033\pi$
$569$	34.3607	1.44047	0.720237	+	0.693728i	$0.244033\pi$
$570$	0	0
$571$	16.8328i	0.704431i	0.935919	+	0.352216i	$0.114572\pi$
$571$	16.8328i	0.704431i	−0.935919	+	0.352216i	$0.885428\pi$
$572$	0	0
$573$	14.1115i	0.589515i
$574$	0	0
$575$	−4.76393	−0.198670
$576$	0	0
$577$	19.8885	0.827971	0.413985	−	0.910283i	$-0.364136\pi$
$577$	19.8885	0.827971	0.413985	+	0.910283i	$0.364136\pi$
$578$	0	0
$579$	− 14.2492i	− 0.592178i
$580$	0	0
$581$	− 12.0000i	− 0.497844i
$582$	0	0
$583$	0.944272	0.0391077
$584$	0	0
$585$	6.58359	0.272198
$586$	0	0
$587$	− 8.65248i	− 0.357126i	−0.983928	−	0.178563i	$-0.942855\pi$
$587$	− 8.65248i	− 0.357126i	0.983928	−	0.178563i	$-0.0571448\pi$
$588$	0	0
$589$	− 28.9443i	− 1.19263i
$590$	0	0
$591$	0.583592	0.0240058
$592$	0	0
$593$	−14.0000	−0.574911	−0.287456	−	0.957794i	$-0.592809\pi$
$593$	−14.0000	−0.574911	−0.287456	+	0.957794i	$0.592809\pi$
$594$	0	0
$595$	14.4721i	0.593300i
$596$	0	0
$597$	1.16718i	0.0477697i
$598$	0	0
$599$	21.8885	0.894342	0.447171	−	0.894449i	$-0.352432\pi$
$599$	21.8885	0.894342	0.447171	+	0.894449i	$0.352432\pi$
$600$	0	0
$601$	22.3607	0.912111	0.456056	−	0.889951i	$-0.349262\pi$
$601$	22.3607	0.912111	0.456056	+	0.889951i	$0.349262\pi$
$602$	0	0
$603$	11.3475i	0.462107i
$604$	0	0
$605$	7.00000i	0.284590i
$606$	0	0
$607$	6.87539	0.279063	0.139532	−	0.990218i	$-0.455440\pi$
$607$	6.87539	0.279063	0.139532	+	0.990218i	$0.455440\pi$
$608$	0	0
$609$	8.00000	0.324176
$610$	0	0
$611$	32.3607i	1.30917i
$612$	0	0
$613$	− 19.5279i	− 0.788723i	−0.918955	−	0.394362i	$-0.870966\pi$
$613$	− 19.5279i	− 0.788723i	0.918955	−	0.394362i	$-0.129034\pi$
$614$	0	0
$615$	15.4164	0.621650
$616$	0	0
$617$	−25.4164	−1.02323	−0.511613	−	0.859216i	$-0.670952\pi$
$617$	−25.4164	−1.02323	−0.511613	+	0.859216i	$0.670952\pi$
$618$	0	0
$619$	− 20.4721i	− 0.822845i	−0.911445	−	0.411422i	$-0.865032\pi$
$619$	− 20.4721i	− 0.822845i	0.911445	−	0.411422i	$-0.134968\pi$
$620$	0	0
$621$	26.3344i	1.05676i
$622$	0	0
$623$	48.3607	1.93753
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	11.0557i	0.441523i
$628$	0	0
$629$	− 31.0557i	− 1.23827i
$630$	0	0
$631$	34.4721	1.37231	0.686157	−	0.727453i	$-0.259296\pi$
$631$	34.4721	1.37231	0.686157	+	0.727453i	$0.259296\pi$
$632$	0	0
$633$	1.30495	0.0518672
$634$	0	0
$635$	− 20.1803i	− 0.800832i
$636$	0	0
$637$	− 15.5279i	− 0.615236i
$638$	0	0
$639$	3.63932	0.143969
$640$	0	0
$641$	0.472136	0.0186482	0.00932412	−	0.999957i	$-0.497032\pi$
$641$	0.472136	0.0186482	0.00932412	+	0.999957i	$0.497032\pi$
$642$	0	0
$643$	22.1803i	0.874707i	0.899290	+	0.437354i	$0.144084\pi$
$643$	22.1803i	0.874707i	−0.899290	+	0.437354i	$0.855916\pi$
$644$	0	0
$645$	− 9.52786i	− 0.375159i
$646$	0	0
$647$	−12.7639	−0.501802	−0.250901	−	0.968013i	$-0.580727\pi$
$647$	−12.7639	−0.501802	−0.250901	+	0.968013i	$0.580727\pi$
$648$	0	0
$649$	−16.9443	−0.665121
$650$	0	0
$651$	25.8885i	1.01465i
$652$	0	0
$653$	− 49.4164i	− 1.93381i	−0.255130	−	0.966907i	$-0.582118\pi$
$653$	− 49.4164i	− 1.93381i	0.255130	−	0.966907i	$-0.417882\pi$
$654$	0	0
$655$	−14.9443	−0.583921
$656$	0	0
$657$	−6.58359	−0.256850
$658$	0	0
$659$	21.4164i	0.834265i	0.908846	+	0.417132i	$0.136965\pi$
$659$	21.4164i	0.834265i	−0.908846	+	0.417132i	$0.863035\pi$
$660$	0	0
$661$	− 35.8885i	− 1.39590i	−0.716145	−	0.697951i	$-0.754095\pi$
$661$	− 35.8885i	− 1.39590i	0.716145	−	0.697951i	$-0.245905\pi$
$662$	0	0
$663$	24.7214	0.960098
$664$	0	0
$665$	14.4721	0.561205
$666$	0	0
$667$	− 9.52786i	− 0.368920i
$668$	0	0
$669$	− 10.8328i	− 0.418821i
$670$	0	0
$671$	12.0000	0.463255
$672$	0	0
$673$	−23.3050	−0.898340	−0.449170	−	0.893446i	$-0.648280\pi$
$673$	−23.3050	−0.898340	−0.449170	+	0.893446i	$0.648280\pi$
$674$	0	0
$675$	− 5.52786i	− 0.212768i
$676$	0	0
$677$	30.3607i	1.16686i	0.812165	+	0.583428i	$0.198289\pi$
$677$	30.3607i	1.16686i	−0.812165	+	0.583428i	$0.801711\pi$
$678$	0	0
$679$	−53.3050	−2.04566
$680$	0	0
$681$	12.5836	0.482204
$682$	0	0
$683$	− 24.2918i	− 0.929500i	−0.885442	−	0.464750i	$-0.846144\pi$
$683$	− 24.2918i	− 0.929500i	0.885442	−	0.464750i	$-0.153856\pi$
$684$	0	0
$685$	− 2.00000i	− 0.0764161i
$686$	0	0
$687$	3.63932	0.138849
$688$	0	0
$689$	−2.11146	−0.0804401
$690$	0	0
$691$	− 30.0000i	− 1.14125i	−0.821209	−	0.570627i	$-0.806700\pi$
$691$	− 30.0000i	− 1.14125i	0.821209	−	0.570627i	$-0.193300\pi$
$692$	0	0
$693$	9.52786i	0.361934i
$694$	0	0
$695$	20.4721	0.776552
$696$	0	0
$697$	−55.7771	−2.11271
$698$	0	0
$699$	− 19.1935i	− 0.725965i
$700$	0	0
$701$	− 9.05573i	− 0.342030i	−0.985268	−	0.171015i	$-0.945295\pi$
$701$	− 9.05573i	− 0.342030i	0.985268	−	0.171015i	$-0.0547047\pi$
$702$	0	0
$703$	−31.0557	−1.17129
$704$	0	0
$705$	8.94427	0.336861
$706$	0	0
$707$	32.3607i	1.21705i
$708$	0	0
$709$	18.0000i	0.676004i	0.941145	+	0.338002i	$0.109751\pi$
$709$	18.0000i	0.676004i	−0.941145	+	0.338002i	$0.890249\pi$
$710$	0	0
$711$	19.0557	0.714646
$712$	0	0
$713$	30.8328	1.15470
$714$	0	0
$715$	8.94427i	0.334497i
$716$	0	0
$717$	− 16.0000i	− 0.597531i
$718$	0	0
$719$	22.8328	0.851520	0.425760	−	0.904836i	$-0.360007\pi$
$719$	22.8328	0.851520	0.425760	+	0.904836i	$0.360007\pi$
$720$	0	0
$721$	−10.4721	−0.390003
$722$	0	0
$723$	32.5836i	1.21180i
$724$	0	0
$725$	2.00000i	0.0742781i
$726$	0	0
$727$	−1.70820	−0.0633538	−0.0316769	−	0.999498i	$-0.510085\pi$
$727$	−1.70820	−0.0633538	−0.0316769	+	0.999498i	$0.510085\pi$
$728$	0	0
$729$	−24.0557	−0.890953
$730$	0	0
$731$	34.4721i	1.27500i
$732$	0	0
$733$	51.8885i	1.91655i	0.285853	+	0.958274i	$0.407723\pi$
$733$	51.8885i	1.91655i	−0.285853	+	0.958274i	$0.592277\pi$
$734$	0	0
$735$	−4.29180	−0.158305
$736$	0	0
$737$	−15.4164	−0.567871
$738$	0	0
$739$	49.1935i	1.80961i	0.425824	+	0.904806i	$0.359984\pi$
$739$	49.1935i	1.80961i	−0.425824	+	0.904806i	$0.640016\pi$
$740$	0	0
$741$	− 24.7214i	− 0.908162i
$742$	0	0
$743$	30.6525	1.12453	0.562265	−	0.826957i	$-0.309930\pi$
$743$	30.6525	1.12453	0.562265	+	0.826957i	$0.309930\pi$
$744$	0	0
$745$	6.94427	0.254418
$746$	0	0
$747$	− 5.45898i	− 0.199734i
$748$	0	0
$749$	55.7771i	2.03805i
$750$	0	0
$751$	16.3607	0.597010	0.298505	−	0.954408i	$-0.403512\pi$
$751$	16.3607	0.597010	0.298505	+	0.954408i	$0.403512\pi$
$752$	0	0
$753$	−2.47214	−0.0900896
$754$	0	0
$755$	− 23.4164i	− 0.852210i
$756$	0	0
$757$	− 19.8885i	− 0.722861i	−0.932399	−	0.361431i	$-0.882288\pi$
$757$	− 19.8885i	− 0.722861i	0.932399	−	0.361431i	$-0.117712\pi$
$758$	0	0
$759$	−11.7771	−0.427481
$760$	0	0
$761$	−3.88854	−0.140960	−0.0704798	−	0.997513i	$-0.522453\pi$
$761$	−3.88854	−0.140960	−0.0704798	+	0.997513i	$0.522453\pi$
$762$	0	0
$763$	48.3607i	1.75077i
$764$	0	0
$765$	6.58359i	0.238030i
$766$	0	0
$767$	37.8885	1.36808
$768$	0	0
$769$	−14.9443	−0.538904	−0.269452	−	0.963014i	$-0.586843\pi$
$769$	−14.9443	−0.538904	−0.269452	+	0.963014i	$0.586843\pi$
$770$	0	0
$771$	− 3.63932i	− 0.131067i
$772$	0	0
$773$	− 18.3607i	− 0.660388i	−0.943913	−	0.330194i	$-0.892886\pi$
$773$	− 18.3607i	− 0.660388i	0.943913	−	0.330194i	$-0.107114\pi$
$774$	0	0
$775$	−6.47214	−0.232486
$776$	0	0
$777$	27.7771	0.996497
$778$	0	0
$779$	55.7771i	1.99842i
$780$	0	0
$781$	4.94427i	0.176920i
$782$	0	0
$783$	11.0557	0.395099
$784$	0	0
$785$	−5.05573	−0.180447
$786$	0	0
$787$	− 44.0689i	− 1.57089i	−0.618934	−	0.785443i	$-0.712435\pi$
$787$	− 44.0689i	− 1.57089i	0.618934	−	0.785443i	$-0.287565\pi$
$788$	0	0
$789$	21.8885i	0.779253i
$790$	0	0
$791$	−9.52786	−0.338772
$792$	0	0
$793$	−26.8328	−0.952861
$794$	0	0
$795$	0.583592i	0.0206979i
$796$	0	0
$797$	− 25.4164i	− 0.900295i	−0.892954	−	0.450148i	$-0.851371\pi$
$797$	− 25.4164i	− 0.900295i	0.892954	−	0.450148i	$-0.148629\pi$
$798$	0	0
$799$	−32.3607	−1.14484
$800$	0	0
$801$	22.0000	0.777332
$802$	0	0
$803$	− 8.94427i	− 0.315637i
$804$	0	0
$805$	15.4164i	0.543357i
$806$	0	0
$807$	29.5279	1.03943
$808$	0	0
$809$	12.1115	0.425816	0.212908	−	0.977072i	$-0.431707\pi$
$809$	12.1115	0.425816	0.212908	+	0.977072i	$0.431707\pi$
$810$	0	0
$811$	2.00000i	0.0702295i	0.999383	+	0.0351147i	$0.0111797\pi$
$811$	2.00000i	0.0702295i	−0.999383	+	0.0351147i	$0.988820\pi$
$812$	0	0
$813$	− 30.1115i	− 1.05605i
$814$	0	0
$815$	11.7082	0.410120
$816$	0	0
$817$	34.4721	1.20603
$818$	0	0
$819$	− 21.3050i	− 0.744455i
$820$	0	0
$821$	− 21.0557i	− 0.734850i	−0.930053	−	0.367425i	$-0.880239\pi$
$821$	− 21.0557i	− 0.734850i	0.930053	−	0.367425i	$-0.119761\pi$
$822$	0	0
$823$	1.70820	0.0595442	0.0297721	−	0.999557i	$-0.490522\pi$
$823$	1.70820	0.0595442	0.0297721	+	0.999557i	$0.490522\pi$
$824$	0	0
$825$	2.47214	0.0860687
$826$	0	0
$827$	6.18034i	0.214911i	0.994210	+	0.107456i	$0.0342704\pi$
$827$	6.18034i	0.214911i	−0.994210	+	0.107456i	$0.965730\pi$
$828$	0	0
$829$	− 38.0000i	− 1.31979i	−0.751356	−	0.659897i	$-0.770600\pi$
$829$	− 38.0000i	− 1.31979i	0.751356	−	0.659897i	$-0.229400\pi$
$830$	0	0
$831$	−13.5279	−0.469276
$832$	0	0
$833$	15.5279	0.538009
$834$	0	0
$835$	− 14.6525i	− 0.507070i
$836$	0	0
$837$	35.7771i	1.23664i
$838$	0	0
$839$	−4.00000	−0.138095	−0.0690477	−	0.997613i	$-0.521996\pi$
$839$	−4.00000	−0.138095	−0.0690477	+	0.997613i	$0.521996\pi$
$840$	0	0
$841$	25.0000	0.862069
$842$	0	0
$843$	− 4.36068i	− 0.150190i
$844$	0	0
$845$	− 7.00000i	− 0.240807i
$846$	0	0
$847$	22.6525	0.778348
$848$	0	0
$849$	10.2492	0.351752
$850$	0	0
$851$	− 33.0820i	− 1.13404i
$852$	0	0
$853$	7.52786i	0.257749i	0.991661	+	0.128875i	$0.0411365\pi$
$853$	7.52786i	0.257749i	−0.991661	+	0.128875i	$0.958864\pi$
$854$	0	0
$855$	6.58359	0.225154
$856$	0	0
$857$	−24.8328	−0.848273	−0.424136	−	0.905598i	$-0.639422\pi$
$857$	−24.8328	−0.848273	−0.424136	+	0.905598i	$0.639422\pi$
$858$	0	0
$859$	− 49.4164i	− 1.68607i	−0.537862	−	0.843033i	$-0.680768\pi$
$859$	− 49.4164i	− 1.68607i	0.537862	−	0.843033i	$-0.319232\pi$
$860$	0	0
$861$	− 49.8885i	− 1.70020i
$862$	0	0
$863$	18.2918	0.622660	0.311330	−	0.950302i	$-0.399226\pi$
$863$	18.2918	0.622660	0.311330	+	0.950302i	$0.399226\pi$
$864$	0	0
$865$	10.9443	0.372116
$866$	0	0
$867$	3.70820i	0.125937i
$868$	0	0
$869$	25.8885i	0.878209i
$870$	0	0
$871$	34.4721	1.16804
$872$	0	0
$873$	−24.2492	−0.820712
$874$	0	0
$875$	− 3.23607i	− 0.109399i
$876$	0	0
$877$	51.8885i	1.75215i	0.482173	+	0.876076i	$0.339848\pi$
$877$	51.8885i	1.75215i	−0.482173	+	0.876076i	$0.660152\pi$
$878$	0	0
$879$	−29.5279	−0.995950
$880$	0	0
$881$	−24.4721	−0.824487	−0.412244	−	0.911074i	$-0.635255\pi$
$881$	−24.4721	−0.824487	−0.412244	+	0.911074i	$0.635255\pi$
$882$	0	0
$883$	− 27.7082i	− 0.932455i	−0.884665	−	0.466228i	$-0.845613\pi$
$883$	− 27.7082i	− 0.932455i	0.884665	−	0.466228i	$-0.154387\pi$
$884$	0	0
$885$	− 10.4721i	− 0.352017i
$886$	0	0
$887$	−11.5967	−0.389381	−0.194690	−	0.980865i	$-0.562370\pi$
$887$	−11.5967	−0.389381	−0.194690	+	0.980865i	$0.562370\pi$
$888$	0	0
$889$	−65.3050	−2.19026
$890$	0	0
$891$	− 4.83282i	− 0.161905i
$892$	0	0
$893$	32.3607i	1.08291i
$894$	0	0
$895$	20.4721	0.684308
$896$	0	0
$897$	26.3344	0.879279
$898$	0	0
$899$	− 12.9443i	− 0.431716i
$900$	0	0
$901$	− 2.11146i	− 0.0703428i
$902$	0	0
$903$	−30.8328	−1.02605
$904$	0	0
$905$	10.9443	0.363800
$906$	0	0
$907$	9.23607i	0.306679i	0.988174	+	0.153339i	$0.0490027\pi$
$907$	9.23607i	0.306679i	−0.988174	+	0.153339i	$0.950997\pi$
$908$	0	0
$909$	14.7214i	0.488277i
$910$	0	0
$911$	−53.3050	−1.76607	−0.883036	−	0.469305i	$-0.844504\pi$
$911$	−53.3050	−1.76607	−0.883036	+	0.469305i	$0.844504\pi$
$912$	0	0
$913$	7.41641	0.245447
$914$	0	0
$915$	7.41641i	0.245179i
$916$	0	0
$917$	48.3607i	1.59701i
$918$	0	0
$919$	5.88854	0.194245	0.0971226	−	0.995272i	$-0.469036\pi$
$919$	5.88854	0.194245	0.0971226	+	0.995272i	$0.469036\pi$
$920$	0	0
$921$	0.360680	0.0118848
$922$	0	0
$923$	− 11.0557i	− 0.363904i
$924$	0	0
$925$	6.94427i	0.228326i
$926$	0	0
$927$	−4.76393	−0.156468
$928$	0	0
$929$	24.4721	0.802905	0.401452	−	0.915880i	$-0.368506\pi$
$929$	24.4721	0.802905	0.401452	+	0.915880i	$0.368506\pi$
$930$	0	0
$931$	− 15.5279i	− 0.508905i
$932$	0	0
$933$	19.0557i	0.623857i
$934$	0	0
$935$	−8.94427	−0.292509
$936$	0	0
$937$	3.52786	0.115250	0.0576251	−	0.998338i	$-0.481647\pi$
$937$	3.52786	0.115250	0.0576251	+	0.998338i	$0.481647\pi$
$938$	0	0
$939$	35.6393i	1.16305i
$940$	0	0
$941$	44.8328i	1.46151i	0.682641	+	0.730754i	$0.260831\pi$
$941$	44.8328i	1.46151i	−0.682641	+	0.730754i	$0.739169\pi$
$942$	0	0
$943$	−59.4164	−1.93486
$944$	0	0
$945$	−17.8885	−0.581914
$946$	0	0
$947$	21.8197i	0.709044i	0.935048	+	0.354522i	$0.115356\pi$
$947$	21.8197i	0.709044i	−0.935048	+	0.354522i	$0.884644\pi$
$948$	0	0
$949$	20.0000i	0.649227i
$950$	0	0
$951$	25.3050	0.820569
$952$	0	0
$953$	−45.7771	−1.48287	−0.741433	−	0.671027i	$-0.765853\pi$
$953$	−45.7771	−1.48287	−0.741433	+	0.671027i	$0.765853\pi$
$954$	0	0
$955$	11.4164i	0.369426i
$956$	0	0
$957$	4.94427i	0.159826i
$958$	0	0
$959$	−6.47214	−0.208996
$960$	0	0
$961$	10.8885	0.351243
$962$	0	0
$963$	25.3738i	0.817660i
$964$	0	0
$965$	− 11.5279i	− 0.371095i
$966$	0	0
$967$	1.34752	0.0433335	0.0216667	−	0.999765i	$-0.493103\pi$
$967$	1.34752	0.0433335	0.0216667	+	0.999765i	$0.493103\pi$
$968$	0	0
$969$	24.7214	0.794164
$970$	0	0
$971$	− 23.8885i	− 0.766620i	−0.923620	−	0.383310i	$-0.874784\pi$
$971$	− 23.8885i	− 0.766620i	0.923620	−	0.383310i	$-0.125216\pi$
$972$	0	0
$973$	− 66.2492i	− 2.12385i
$974$	0	0
$975$	−5.52786	−0.177033
$976$	0	0
$977$	−39.3050	−1.25748	−0.628738	−	0.777617i	$-0.716428\pi$
$977$	−39.3050	−1.25748	−0.628738	+	0.777617i	$0.716428\pi$
$978$	0	0
$979$	29.8885i	0.955242i
$980$	0	0
$981$	22.0000i	0.702406i
$982$	0	0
$983$	−39.0132	−1.24433	−0.622163	−	0.782888i	$-0.713746\pi$
$983$	−39.0132	−1.24433	−0.622163	+	0.782888i	$0.713746\pi$
$984$	0	0
$985$	0.472136	0.0150435
$986$	0	0
$987$	− 28.9443i	− 0.921306i
$988$	0	0
$989$	36.7214i	1.16767i
$990$	0	0
$991$	−17.5279	−0.556791	−0.278395	−	0.960467i	$-0.589803\pi$
$991$	−17.5279	−0.556791	−0.278395	+	0.960467i	$0.589803\pi$
$992$	0	0
$993$	19.6393	0.623235
$994$	0	0
$995$	0.944272i	0.0299354i
$996$	0	0
$997$	34.5836i	1.09527i	0.836716	+	0.547637i	$0.184472\pi$
$997$	34.5836i	1.09527i	−0.836716	+	0.547637i	$0.815528\pi$
$998$	0	0
$999$	38.3870	1.21451

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1280.2.d.m.641.3		4
4.3	odd	2		1280.2.d.k.641.2		4
8.3	odd	2		1280.2.d.k.641.3		4
8.5	even	2	inner	1280.2.d.m.641.2		4
16.3	odd	4		640.2.a.j.1.2	yes	2
16.5	even	4		640.2.a.i.1.2	✓	2
16.11	odd	4		640.2.a.k.1.1	yes	2
16.13	even	4		640.2.a.l.1.1	yes	2
48.5	odd	4		5760.2.a.ch.1.1		2
48.11	even	4		5760.2.a.ci.1.2		2
48.29	odd	4		5760.2.a.bw.1.1		2
48.35	even	4		5760.2.a.cd.1.2		2
80.3	even	4		3200.2.c.v.2049.3		4
80.13	odd	4		3200.2.c.x.2049.2		4
80.19	odd	4		3200.2.a.bk.1.1		2
80.27	even	4		3200.2.c.w.2049.3		4
80.29	even	4		3200.2.a.bf.1.2		2
80.37	odd	4		3200.2.c.u.2049.2		4
80.43	even	4		3200.2.c.w.2049.2		4
80.53	odd	4		3200.2.c.u.2049.3		4
80.59	odd	4		3200.2.a.be.1.2		2
80.67	even	4		3200.2.c.v.2049.2		4
80.69	even	4		3200.2.a.bl.1.1		2
80.77	odd	4		3200.2.c.x.2049.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
640.2.a.i.1.2	✓	2	16.5	even	4
640.2.a.j.1.2	yes	2	16.3	odd	4
640.2.a.k.1.1	yes	2	16.11	odd	4
640.2.a.l.1.1	yes	2	16.13	even	4
1280.2.d.k.641.2		4	4.3	odd	2
1280.2.d.k.641.3		4	8.3	odd	2
1280.2.d.m.641.2		4	8.5	even	2	inner
1280.2.d.m.641.3		4	1.1	even	1	trivial
3200.2.a.be.1.2		2	80.59	odd	4
3200.2.a.bf.1.2		2	80.29	even	4
3200.2.a.bk.1.1		2	80.19	odd	4
3200.2.a.bl.1.1		2	80.69	even	4
3200.2.c.u.2049.2		4	80.37	odd	4
3200.2.c.u.2049.3		4	80.53	odd	4
3200.2.c.v.2049.2		4	80.67	even	4
3200.2.c.v.2049.3		4	80.3	even	4
3200.2.c.w.2049.2		4	80.43	even	4
3200.2.c.w.2049.3		4	80.27	even	4
3200.2.c.x.2049.2		4	80.13	odd	4
3200.2.c.x.2049.3		4	80.77	odd	4
5760.2.a.bw.1.1		2	48.29	odd	4
5760.2.a.cd.1.2		2	48.35	even	4
5760.2.a.ch.1.1		2	48.5	odd	4
5760.2.a.ci.1.2		2	48.11	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:	\( N \)	\(=\)	\( 1280 = 2^{8} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1280.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.23607i q^{3} +1.00000i q^{5} +3.23607 q^{7} +1.47214 q^{9} +O(q^{10})\)	\(q+1.23607i q^{3} +1.00000i q^{5} +3.23607 q^{7} +1.47214 q^{9} +2.00000i q^{11} -4.47214i q^{13} -1.23607 q^{15} +4.47214 q^{17} -4.47214i q^{19} +4.00000i q^{21} +4.76393 q^{23} -1.00000 q^{25} +5.52786i q^{27} -2.00000i q^{29} +6.47214 q^{31} -2.47214 q^{33} +3.23607i q^{35} -6.94427i q^{37} +5.52786 q^{39} -12.4721 q^{41} +7.70820i q^{43} +1.47214i q^{45} -7.23607 q^{47} +3.47214 q^{49} +5.52786i q^{51} -0.472136i q^{53} -2.00000 q^{55} +5.52786 q^{57} +8.47214i q^{59} -6.00000i q^{61} +4.76393 q^{63} +4.47214 q^{65} +7.70820i q^{67} +5.88854i q^{69} +2.47214 q^{71} -4.47214 q^{73} -1.23607i q^{75} +6.47214i q^{77} +12.9443 q^{79} -2.41641 q^{81} -3.70820i q^{83} +4.47214i q^{85} +2.47214 q^{87} +14.9443 q^{89} -14.4721i q^{91} +8.00000i q^{93} +4.47214 q^{95} -16.4721 q^{97} +2.94427i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{7} - 12 q^{9}+O(q^{10}) \) 4 * q + 4 * q^7 - 12 * q^9	\( 4 q + 4 q^{7} - 12 q^{9} + 4 q^{15} + 28 q^{23} - 4 q^{25} + 8 q^{31} + 8 q^{33} + 40 q^{39} - 32 q^{41} - 20 q^{47} - 4 q^{49} - 8 q^{55} + 40 q^{57} + 28 q^{63} - 8 q^{71} + 16 q^{79} + 44 q^{81} - 8 q^{87} + 24 q^{89} - 48 q^{97}+O(q^{100}) \) 4 * q + 4 * q^7 - 12 * q^9 + 4 * q^15 + 28 * q^23 - 4 * q^25 + 8 * q^31 + 8 * q^33 + 40 * q^39 - 32 * q^41 - 20 * q^47 - 4 * q^49 - 8 * q^55 + 40 * q^57 + 28 * q^63 - 8 * q^71 + 16 * q^79 + 44 * q^81 - 8 * q^87 + 24 * q^89 - 48 * q^97

Introduction

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