LMFDB - Embedded newform 400.8.c.b.49.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [400,8,Mod(49,400)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(400, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 8, names="a")

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

chi := DirichletCharacter("400.49");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 400 = 2^{4} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	400.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:	$124.954010194$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 8)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			49.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	400.49
Dual form			400.8.c.b.49.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+84.0000i q^{3} -456.000i q^{7} -4869.00 q^{9} +O(q^{10})$	$q+84.0000i q^{3} -456.000i q^{7} -4869.00 q^{9} +2524.00 q^{11} -10778.0i q^{13} +11150.0i q^{17} +4124.00 q^{19} +38304.0 q^{21} -81704.0i q^{23} -225288. i q^{27} -99798.0 q^{29} +40480.0 q^{31} +212016. i q^{33} +419442. i q^{37} +905352. q^{39} +141402. q^{41} +690428. i q^{43} -682032. i q^{47} +615607. q^{49} -936600. q^{51} +1.81312e6i q^{53} +346416. i q^{57} -966028. q^{59} +1.88767e6 q^{61} +2.22026e6i q^{63} +2.96587e6i q^{67} +6.86314e6 q^{69} +2.54823e6 q^{71} -1.68033e6i q^{73} -1.15094e6i q^{77} +4.03806e6 q^{79} +8.27569e6 q^{81} +5.38576e6i q^{83} -8.38303e6i q^{87} +6.47305e6 q^{89} -4.91477e6 q^{91} +3.40032e6i q^{93} +6.06576e6i q^{97} -1.22894e7 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 9738 q^{9}+O(q^{10}) $ 2 * q - 9738 * q^9	$ 2 q - 9738 q^{9} + 5048 q^{11} + 8248 q^{19} + 76608 q^{21} - 199596 q^{29} + 80960 q^{31} + 1810704 q^{39} + 282804 q^{41} + 1231214 q^{49} - 1873200 q^{51} - 1932056 q^{59} + 3775340 q^{61} + 13726272 q^{69} + 5096464 q^{71} + 8076128 q^{79} + 16551378 q^{81} + 12946092 q^{89} - 9829536 q^{91} - 24578712 q^{99}+O(q^{100}) $ 2 * q - 9738 * q^9 + 5048 * q^11 + 8248 * q^19 + 76608 * q^21 - 199596 * q^29 + 80960 * q^31 + 1810704 * q^39 + 282804 * q^41 + 1231214 * q^49 - 1873200 * q^51 - 1932056 * q^59 + 3775340 * q^61 + 13726272 * q^69 + 5096464 * q^71 + 8076128 * q^79 + 16551378 * q^81 + 12946092 * q^89 - 9829536 * q^91 - 24578712 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/400\mathbb{Z}\right)^\times$.

$n$	$101$	$177$	$351$
$\chi(n)$	$1$	$-1$	$1$

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
$3$	84.0000i	1.79620i	0.439790	+	0.898100i	$0.355053\pi$
$3$	84.0000i	1.79620i	−0.439790	+	0.898100i	$0.644947\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 456.000i	− 0.502483i	−0.967924	−	0.251242i	$-0.919161\pi$
$7$	− 456.000i	− 0.502483i	0.967924	−	0.251242i	$-0.0808389\pi$
$8$	0	0
$9$	−4869.00	−2.22634
$10$	0	0
$11$	2524.00	0.571762	0.285881	−	0.958265i	$-0.407714\pi$
$11$	2524.00	0.571762	0.285881	+	0.958265i	$0.407714\pi$
$12$	0	0
$13$	− 10778.0i	− 1.36062i	−0.732925	−	0.680309i	$-0.761845\pi$
$13$	− 10778.0i	− 1.36062i	0.732925	−	0.680309i	$-0.238155\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	11150.0i	0.550432i	0.961382	+	0.275216i	$0.0887494\pi$
$17$	11150.0i	0.550432i	−0.961382	+	0.275216i	$0.911251\pi$
$18$	0	0
$19$	4124.00	0.137937	0.0689685	−	0.997619i	$-0.478029\pi$
$19$	4124.00	0.137937	0.0689685	+	0.997619i	$0.478029\pi$
$20$	0	0
$21$	38304.0	0.902561
$22$	0	0
$23$	− 81704.0i	− 1.40022i	−0.714036	−	0.700109i	$-0.753135\pi$
$23$	− 81704.0i	− 1.40022i	0.714036	−	0.700109i	$-0.246865\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 225288.i	− 2.20275i
$28$	0	0
$29$	−99798.0	−0.759852	−0.379926	−	0.925017i	$-0.624051\pi$
$29$	−99798.0	−0.759852	−0.379926	+	0.925017i	$0.624051\pi$
$30$	0	0
$31$	40480.0	0.244048	0.122024	−	0.992527i	$-0.461062\pi$
$31$	40480.0	0.244048	0.122024	+	0.992527i	$0.461062\pi$
$32$	0	0
$33$	212016.i	1.02700i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	419442.i	1.36134i	0.732591	+	0.680669i	$0.238311\pi$
$37$	419442.i	1.36134i	−0.732591	+	0.680669i	$0.761689\pi$
$38$	0	0
$39$	905352.	2.44394
$40$	0	0
$41$	141402.	0.320414	0.160207	−	0.987083i	$-0.448784\pi$
$41$	141402.	0.320414	0.160207	+	0.987083i	$0.448784\pi$
$42$	0	0
$43$	690428.i	1.32428i	0.749382	+	0.662138i	$0.230351\pi$
$43$	690428.i	1.32428i	−0.749382	+	0.662138i	$0.769649\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 682032.i	− 0.958213i	−0.877757	−	0.479107i	$-0.840961\pi$
$47$	− 682032.i	− 0.958213i	0.877757	−	0.479107i	$-0.159039\pi$
$48$	0	0
$49$	615607.	0.747510
$50$	0	0
$51$	−936600.	−0.988686
$52$	0	0
$53$	1.81312e6i	1.67286i	0.548071	+	0.836432i	$0.315362\pi$
$53$	1.81312e6i	1.67286i	−0.548071	+	0.836432i	$0.684638\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	346416.i	0.247763i
$58$	0	0
$59$	−966028.	−0.612361	−0.306181	−	0.951973i	$-0.599051\pi$
$59$	−966028.	−0.612361	−0.306181	+	0.951973i	$0.599051\pi$
$60$	0	0
$61$	1.88767e6	1.06481	0.532404	−	0.846490i	$-0.321289\pi$
$61$	1.88767e6	1.06481	0.532404	+	0.846490i	$0.321289\pi$
$62$	0	0
$63$	2.22026e6i	1.11870i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	2.96587e6i	1.20473i	0.798220	+	0.602365i	$0.205775\pi$
$67$	2.96587e6i	1.20473i	−0.798220	+	0.602365i	$0.794225\pi$
$68$	0	0
$69$	6.86314e6	2.51507
$70$	0	0
$71$	2.54823e6	0.844957	0.422479	−	0.906373i	$-0.361160\pi$
$71$	2.54823e6	0.844957	0.422479	+	0.906373i	$0.361160\pi$
$72$	0	0
$73$	− 1.68033e6i	− 0.505549i	−0.967525	−	0.252775i	$-0.918657\pi$
$73$	− 1.68033e6i	− 0.505549i	0.967525	−	0.252775i	$-0.0813431\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 1.15094e6i	− 0.287301i
$78$	0	0
$79$	4.03806e6	0.921464	0.460732	−	0.887539i	$-0.347587\pi$
$79$	4.03806e6	0.921464	0.460732	+	0.887539i	$0.347587\pi$
$80$	0	0
$81$	8.27569e6	1.73024
$82$	0	0
$83$	5.38576e6i	1.03389i	0.856019	+	0.516945i	$0.172931\pi$
$83$	5.38576e6i	1.03389i	−0.856019	+	0.516945i	$0.827069\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 8.38303e6i	− 1.36485i
$88$	0	0
$89$	6.47305e6	0.973293	0.486647	−	0.873599i	$-0.338220\pi$
$89$	6.47305e6	0.973293	0.486647	+	0.873599i	$0.338220\pi$
$90$	0	0
$91$	−4.91477e6	−0.683688
$92$	0	0
$93$	3.40032e6i	0.438359i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	6.06576e6i	0.674814i	0.941359	+	0.337407i	$0.109550\pi$
$97$	6.06576e6i	0.674814i	−0.941359	+	0.337407i	$0.890450\pi$
$98$	0	0
$99$	−1.22894e7	−1.27293
$100$	0	0
$101$	9.70069e6	0.936866	0.468433	−	0.883499i	$-0.344819\pi$
$101$	9.70069e6	0.936866	0.468433	+	0.883499i	$0.344819\pi$
$102$	0	0
$103$	− 4.10159e6i	− 0.369847i	−0.982753	−	0.184924i	$-0.940796\pi$
$103$	− 4.10159e6i	− 0.369847i	0.982753	−	0.184924i	$-0.0592037\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	72900.0i	0.00575287i	0.999996	+	0.00287643i	$0.000915598\pi$
$107$	72900.0i	0.00575287i	−0.999996	+	0.00287643i	$0.999084\pi$
$108$	0	0
$109$	−9.55841e6	−0.706957	−0.353478	−	0.935443i	$-0.615001\pi$
$109$	−9.55841e6	−0.706957	−0.353478	+	0.935443i	$0.615001\pi$
$110$	0	0
$111$	−3.52331e7	−2.44524
$112$	0	0
$113$	9.33890e6i	0.608865i	0.952534	+	0.304433i	$0.0984668\pi$
$113$	9.33890e6i	0.608865i	−0.952534	+	0.304433i	$0.901533\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	5.24781e7i	3.02920i
$118$	0	0
$119$	5.08440e6	0.276583
$120$	0	0
$121$	−1.31166e7	−0.673089
$122$	0	0
$123$	1.18778e7i	0.575529i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 3.59794e7i	− 1.55862i	−0.626637	−	0.779311i	$-0.715569\pi$
$127$	− 3.59794e7i	− 1.55862i	0.626637	−	0.779311i	$-0.284431\pi$
$128$	0	0
$129$	−5.79960e7	−2.37867
$130$	0	0
$131$	676052.	0.0262743	0.0131371	−	0.999914i	$-0.495818\pi$
$131$	676052.	0.0262743	0.0131371	+	0.999914i	$0.495818\pi$
$132$	0	0
$133$	− 1.88054e6i	− 0.0693111i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	2.95841e7i	0.982962i	0.870888	+	0.491481i	$0.163544\pi$
$137$	2.95841e7i	0.982962i	−0.870888	+	0.491481i	$0.836456\pi$
$138$	0	0
$139$	−3.19084e7	−1.00775	−0.503876	−	0.863776i	$-0.668093\pi$
$139$	−3.19084e7	−1.00775	−0.503876	+	0.863776i	$0.668093\pi$
$140$	0	0
$141$	5.72907e7	1.72114
$142$	0	0
$143$	− 2.72037e7i	− 0.777949i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	5.17110e7i	1.34268i
$148$	0	0
$149$	1.16603e7	0.288773	0.144386	−	0.989521i	$-0.453879\pi$
$149$	1.16603e7	0.288773	0.144386	+	0.989521i	$0.453879\pi$
$150$	0	0
$151$	1.76295e7	0.416698	0.208349	−	0.978055i	$-0.433191\pi$
$151$	1.76295e7	0.416698	0.208349	+	0.978055i	$0.433191\pi$
$152$	0	0
$153$	− 5.42894e7i	− 1.22545i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 6.34658e6i	− 0.130885i	−0.997856	−	0.0654427i	$-0.979154\pi$
$157$	− 6.34658e6i	− 0.130885i	0.997856	−	0.0654427i	$-0.0208460\pi$
$158$	0	0
$159$	−1.52302e8	−3.00480
$160$	0	0
$161$	−3.72570e7	−0.703587
$162$	0	0
$163$	− 8.04234e7i	− 1.45454i	−0.686351	−	0.727271i	$-0.740789\pi$
$163$	− 8.04234e7i	− 1.45454i	0.686351	−	0.727271i	$-0.259211\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	1.14767e8i	1.90682i	0.301673	+	0.953411i	$0.402455\pi$
$167$	1.14767e8i	1.90682i	−0.301673	+	0.953411i	$0.597545\pi$
$168$	0	0
$169$	−5.34168e7	−0.851283
$170$	0	0
$171$	−2.00798e7	−0.307095
$172$	0	0
$173$	− 6.33755e7i	− 0.930594i	−0.885155	−	0.465297i	$-0.845947\pi$
$173$	− 6.33755e7i	− 0.930594i	0.885155	−	0.465297i	$-0.154053\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 8.11464e7i	− 1.09992i
$178$	0	0
$179$	−1.13228e7	−0.147559	−0.0737796	−	0.997275i	$-0.523506\pi$
$179$	−1.13228e7	−0.147559	−0.0737796	+	0.997275i	$0.523506\pi$
$180$	0	0
$181$	−5.22650e6	−0.0655143	−0.0327571	−	0.999463i	$-0.510429\pi$
$181$	−5.22650e6	−0.0655143	−0.0327571	+	0.999463i	$0.510429\pi$
$182$	0	0
$183$	1.58564e8i	1.91261i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	2.81426e7i	0.314716i
$188$	0	0
$189$	−1.02731e8	−1.10684
$190$	0	0
$191$	8.50301e7	0.882990	0.441495	−	0.897264i	$-0.354448\pi$
$191$	8.50301e7	0.882990	0.441495	+	0.897264i	$0.354448\pi$
$192$	0	0
$193$	1.15092e8i	1.15237i	0.817319	+	0.576186i	$0.195460\pi$
$193$	1.15092e8i	1.15237i	−0.817319	+	0.576186i	$0.804540\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	1.38522e8i	1.29088i	0.763810	+	0.645441i	$0.223326\pi$
$197$	1.38522e8i	1.29088i	−0.763810	+	0.645441i	$0.776674\pi$
$198$	0	0
$199$	−2.19614e7	−0.197548	−0.0987742	−	0.995110i	$-0.531492\pi$
$199$	−2.19614e7	−0.197548	−0.0987742	+	0.995110i	$0.531492\pi$
$200$	0	0
$201$	−2.49133e8	−2.16394
$202$	0	0
$203$	4.55079e7i	0.381813i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	3.97817e8i	3.11736i
$208$	0	0
$209$	1.04090e7	0.0788671
$210$	0	0
$211$	6.10208e7	0.447187	0.223594	−	0.974682i	$-0.428221\pi$
$211$	6.10208e7	0.447187	0.223594	+	0.974682i	$0.428221\pi$
$212$	0	0
$213$	2.14051e8i	1.51771i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 1.84589e7i	− 0.122630i
$218$	0	0
$219$	1.41147e8	0.908068
$220$	0	0
$221$	1.20175e8	0.748928
$222$	0	0
$223$	4.22448e7i	0.255098i	0.991832	+	0.127549i	$0.0407110\pi$
$223$	4.22448e7i	0.255098i	−0.991832	+	0.127549i	$0.959289\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 2.39102e8i	− 1.35673i	−0.734726	−	0.678364i	$-0.762689\pi$
$227$	− 2.39102e8i	− 1.35673i	0.734726	−	0.678364i	$-0.237311\pi$
$228$	0	0
$229$	4.67889e7	0.257465	0.128733	−	0.991679i	$-0.458909\pi$
$229$	4.67889e7	0.257465	0.128733	+	0.991679i	$0.458909\pi$
$230$	0	0
$231$	9.66793e7	0.516050
$232$	0	0
$233$	3.45225e8i	1.78795i	0.448113	+	0.893977i	$0.352096\pi$
$233$	3.45225e8i	1.78795i	−0.448113	+	0.893977i	$0.647904\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	3.39197e8i	1.65513i
$238$	0	0
$239$	2.34413e8	1.11068	0.555340	−	0.831624i	$-0.312588\pi$
$239$	2.34413e8	1.11068	0.555340	+	0.831624i	$0.312588\pi$
$240$	0	0
$241$	−1.09557e8	−0.504175	−0.252087	−	0.967705i	$-0.581117\pi$
$241$	−1.09557e8	−0.504175	−0.252087	+	0.967705i	$0.581117\pi$
$242$	0	0
$243$	2.02453e8i	0.905112i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 4.44485e7i	− 0.187680i
$248$	0	0
$249$	−4.52404e8	−1.85707
$250$	0	0
$251$	3.94031e8	1.57280	0.786398	−	0.617720i	$-0.211943\pi$
$251$	3.94031e8	1.57280	0.786398	+	0.617720i	$0.211943\pi$
$252$	0	0
$253$	− 2.06221e8i	− 0.800591i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 3.19064e8i	− 1.17250i	−0.810131	−	0.586248i	$-0.800604\pi$
$257$	− 3.19064e8i	− 1.17250i	0.810131	−	0.586248i	$-0.199396\pi$
$258$	0	0
$259$	1.91266e8	0.684050
$260$	0	0
$261$	4.85916e8	1.69169
$262$	0	0
$263$	− 2.19359e8i	− 0.743549i	−0.928323	−	0.371774i	$-0.878750\pi$
$263$	− 2.19359e8i	− 0.743549i	0.928323	−	0.371774i	$-0.121250\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	5.43736e8i	1.74823i
$268$	0	0
$269$	1.48033e8	0.463687	0.231844	−	0.972753i	$-0.425524\pi$
$269$	1.48033e8	0.463687	0.231844	+	0.972753i	$0.425524\pi$
$270$	0	0
$271$	3.69934e8	1.12910	0.564549	−	0.825399i	$-0.309050\pi$
$271$	3.69934e8	1.12910	0.564549	+	0.825399i	$0.309050\pi$
$272$	0	0
$273$	− 4.12841e8i	− 1.22804i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	3.95860e8i	1.11908i	0.828803	+	0.559541i	$0.189023\pi$
$277$	3.95860e8i	1.11908i	−0.828803	+	0.559541i	$0.810977\pi$
$278$	0	0
$279$	−1.97097e8	−0.543332
$280$	0	0
$281$	−5.97760e8	−1.60714	−0.803572	−	0.595208i	$-0.797070\pi$
$281$	−5.97760e8	−1.60714	−0.803572	+	0.595208i	$0.797070\pi$
$282$	0	0
$283$	− 8.05797e7i	− 0.211336i	−0.994401	−	0.105668i	$-0.966302\pi$
$283$	− 8.05797e7i	− 0.211336i	0.994401	−	0.105668i	$-0.0336981\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 6.44793e7i	− 0.161003i
$288$	0	0
$289$	2.86016e8	0.697025
$290$	0	0
$291$	−5.09524e8	−1.21210
$292$	0	0
$293$	7.54530e8i	1.75243i	0.481924	+	0.876213i	$0.339938\pi$
$293$	7.54530e8i	1.75243i	−0.481924	+	0.876213i	$0.660062\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 5.68627e8i	− 1.25945i
$298$	0	0
$299$	−8.80606e8	−1.90516
$300$	0	0
$301$	3.14835e8	0.665427
$302$	0	0
$303$	8.14858e8i	1.68280i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	8.20472e8i	1.61838i	0.587549	+	0.809188i	$0.300093\pi$
$307$	8.20472e8i	1.61838i	−0.587549	+	0.809188i	$0.699907\pi$
$308$	0	0
$309$	3.44534e8	0.664320
$310$	0	0
$311$	−6.53503e8	−1.23193	−0.615965	−	0.787773i	$-0.711234\pi$
$311$	−6.53503e8	−1.23193	−0.615965	+	0.787773i	$0.711234\pi$
$312$	0	0
$313$	6.63587e8i	1.22319i	0.791172	+	0.611594i	$0.209471\pi$
$313$	6.63587e8i	1.22319i	−0.791172	+	0.611594i	$0.790529\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	3.54718e8i	0.625426i	0.949848	+	0.312713i	$0.101238\pi$
$317$	3.54718e8i	0.625426i	−0.949848	+	0.312713i	$0.898762\pi$
$318$	0	0
$319$	−2.51890e8	−0.434454
$320$	0	0
$321$	−6.12360e6	−0.0103333
$322$	0	0
$323$	4.59826e7i	0.0759250i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 8.02906e8i	− 1.26984i
$328$	0	0
$329$	−3.11007e8	−0.481486
$330$	0	0
$331$	3.05543e8	0.463100	0.231550	−	0.972823i	$-0.425620\pi$
$331$	3.05543e8	0.463100	0.231550	+	0.972823i	$0.425620\pi$
$332$	0	0
$333$	− 2.04226e9i	− 3.03080i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 3.54965e7i	− 0.0505220i	−0.999681	−	0.0252610i	$-0.991958\pi$
$337$	− 3.54965e7i	− 0.0505220i	0.999681	−	0.0252610i	$-0.00804169\pi$
$338$	0	0
$339$	−7.84467e8	−1.09364
$340$	0	0
$341$	1.02172e8	0.139537
$342$	0	0
$343$	− 6.56252e8i	− 0.878095i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 1.90594e8i	− 0.244882i	−0.992476	−	0.122441i	$-0.960928\pi$
$347$	− 1.90594e8i	− 0.244882i	0.992476	−	0.122441i	$-0.0390723\pi$
$348$	0	0
$349$	−8.60864e8	−1.08404	−0.542020	−	0.840366i	$-0.682340\pi$
$349$	−8.60864e8	−1.08404	−0.542020	+	0.840366i	$0.682340\pi$
$350$	0	0
$351$	−2.42815e9	−2.99710
$352$	0	0
$353$	− 1.04544e9i	− 1.26500i	−0.774562	−	0.632498i	$-0.782030\pi$
$353$	− 1.04544e9i	− 1.26500i	0.774562	−	0.632498i	$-0.217970\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	4.27090e8i	0.496798i
$358$	0	0
$359$	7.63303e8	0.870696	0.435348	−	0.900262i	$-0.356625\pi$
$359$	7.63303e8	0.870696	0.435348	+	0.900262i	$0.356625\pi$
$360$	0	0
$361$	−8.76864e8	−0.980973
$362$	0	0
$363$	− 1.10179e9i	− 1.20900i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 1.38692e9i	− 1.46460i	−0.680980	−	0.732302i	$-0.738446\pi$
$367$	− 1.38692e9i	− 1.46460i	0.680980	−	0.732302i	$-0.261554\pi$
$368$	0	0
$369$	−6.88486e8	−0.713351
$370$	0	0
$371$	8.26782e8	0.840586
$372$	0	0
$373$	4.77105e8i	0.476029i	0.971262	+	0.238015i	$0.0764966\pi$
$373$	4.77105e8i	0.476029i	−0.971262	+	0.238015i	$0.923503\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	1.07562e9i	1.03387i
$378$	0	0
$379$	−3.92468e8	−0.370311	−0.185156	−	0.982709i	$-0.559279\pi$
$379$	−3.92468e8	−0.370311	−0.185156	+	0.982709i	$0.559279\pi$
$380$	0	0
$381$	3.02227e9	2.79960
$382$	0	0
$383$	− 2.10409e9i	− 1.91368i	−0.290617	−	0.956839i	$-0.593861\pi$
$383$	− 2.10409e9i	− 1.91368i	0.290617	−	0.956839i	$-0.406139\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 3.36169e9i	− 2.94829i
$388$	0	0
$389$	1.26019e9	1.08546	0.542730	−	0.839907i	$-0.317391\pi$
$389$	1.26019e9	1.08546	0.542730	+	0.839907i	$0.317391\pi$
$390$	0	0
$391$	9.11000e8	0.770725
$392$	0	0
$393$	5.67884e7i	0.0471939i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	9.81298e8i	0.787107i	0.919302	+	0.393554i	$0.128754\pi$
$397$	9.81298e8i	0.787107i	−0.919302	+	0.393554i	$0.871246\pi$
$398$	0	0
$399$	1.57966e8	0.124497
$400$	0	0
$401$	9.09981e8	0.704737	0.352369	−	0.935861i	$-0.385376\pi$
$401$	9.09981e8	0.704737	0.352369	+	0.935861i	$0.385376\pi$
$402$	0	0
$403$	− 4.36293e8i	− 0.332056i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	1.05867e9i	0.778361i
$408$	0	0
$409$	3.55609e7	0.0257004	0.0128502	−	0.999917i	$-0.495910\pi$
$409$	3.55609e7	0.0257004	0.0128502	+	0.999917i	$0.495910\pi$
$410$	0	0
$411$	−2.48507e9	−1.76560
$412$	0	0
$413$	4.40509e8i	0.307701i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 2.68031e9i	− 1.81013i
$418$	0	0
$419$	2.65360e9	1.76233	0.881163	−	0.472813i	$-0.156761\pi$
$419$	2.65360e9	1.76233	0.881163	+	0.472813i	$0.156761\pi$
$420$	0	0
$421$	−1.12113e9	−0.732264	−0.366132	−	0.930563i	$-0.619318\pi$
$421$	−1.12113e9	−0.732264	−0.366132	+	0.930563i	$0.619318\pi$
$422$	0	0
$423$	3.32081e9i	2.13331i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 8.60778e8i	− 0.535049i
$428$	0	0
$429$	2.28511e9	1.39735
$430$	0	0
$431$	1.06344e9	0.639799	0.319900	−	0.947451i	$-0.396351\pi$
$431$	1.06344e9	0.639799	0.319900	+	0.947451i	$0.396351\pi$
$432$	0	0
$433$	− 7.05962e8i	− 0.417901i	−0.977926	−	0.208951i	$-0.932995\pi$
$433$	− 7.05962e8i	− 0.417901i	0.977926	−	0.208951i	$-0.0670048\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 3.36947e8i	− 0.193142i
$438$	0	0
$439$	1.48506e9	0.837760	0.418880	−	0.908042i	$-0.362423\pi$
$439$	1.48506e9	0.837760	0.418880	+	0.908042i	$0.362423\pi$
$440$	0	0
$441$	−2.99739e9	−1.66421
$442$	0	0
$443$	7.22153e8i	0.394654i	0.980338	+	0.197327i	$0.0632260\pi$
$443$	7.22153e8i	0.394654i	−0.980338	+	0.197327i	$0.936774\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	9.79462e8i	0.518694i
$448$	0	0
$449$	1.22968e9	0.641109	0.320554	−	0.947230i	$-0.396131\pi$
$449$	1.22968e9	0.641109	0.320554	+	0.947230i	$0.396131\pi$
$450$	0	0
$451$	3.56899e8	0.183201
$452$	0	0
$453$	1.48088e9i	0.748473i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	8.85551e7i	0.0434017i	0.999765	+	0.0217009i	$0.00690814\pi$
$457$	8.85551e7i	0.0434017i	−0.999765	+	0.0217009i	$0.993092\pi$
$458$	0	0
$459$	2.51196e9	1.21246
$460$	0	0
$461$	2.10937e8	0.100277	0.0501384	−	0.998742i	$-0.484034\pi$
$461$	2.10937e8	0.100277	0.0501384	+	0.998742i	$0.484034\pi$
$462$	0	0
$463$	3.29775e9i	1.54413i	0.635543	+	0.772066i	$0.280776\pi$
$463$	3.29775e9i	1.54413i	−0.635543	+	0.772066i	$0.719224\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	8.82873e7i	0.0401134i	0.999799	+	0.0200567i	$0.00638467\pi$
$467$	8.82873e7i	0.0401134i	−0.999799	+	0.0200567i	$0.993615\pi$
$468$	0	0
$469$	1.35244e9	0.605357
$470$	0	0
$471$	5.33113e8	0.235096
$472$	0	0
$473$	1.74264e9i	0.757171i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 8.82807e9i	− 3.72436i
$478$	0	0
$479$	−4.51507e9	−1.87711	−0.938557	−	0.345125i	$-0.887836\pi$
$479$	−4.51507e9	−1.87711	−0.938557	+	0.345125i	$0.887836\pi$
$480$	0	0
$481$	4.52075e9	1.85226
$482$	0	0
$483$	− 3.12959e9i	− 1.26378i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	3.31338e9i	1.29993i	0.759965	+	0.649964i	$0.225216\pi$
$487$	3.31338e9i	1.29993i	−0.759965	+	0.649964i	$0.774784\pi$
$488$	0	0
$489$	6.75557e9	2.61265
$490$	0	0
$491$	4.01694e9	1.53147	0.765737	−	0.643154i	$-0.222374\pi$
$491$	4.01694e9	1.53147	0.765737	+	0.643154i	$0.222374\pi$
$492$	0	0
$493$	− 1.11275e9i	− 0.418247i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 1.16199e9i	− 0.424577i
$498$	0	0
$499$	2.70976e9	0.976290	0.488145	−	0.872763i	$-0.337674\pi$
$499$	2.70976e9	0.976290	0.488145	+	0.872763i	$0.337674\pi$
$500$	0	0
$501$	−9.64045e9	−3.42504
$502$	0	0
$503$	− 3.04579e8i	− 0.106712i	−0.998576	−	0.0533558i	$-0.983008\pi$
$503$	− 3.04579e8i	− 0.106712i	0.998576	−	0.0533558i	$-0.0169918\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 4.48701e9i	− 1.52908i
$508$	0	0
$509$	1.88202e8	0.0632575	0.0316287	−	0.999500i	$-0.489931\pi$
$509$	1.88202e8	0.0632575	0.0316287	+	0.999500i	$0.489931\pi$
$510$	0	0
$511$	−7.66229e8	−0.254030
$512$	0	0
$513$	− 9.29088e8i	− 0.303841i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 1.72145e9i	− 0.547870i
$518$	0	0
$519$	5.32355e9	1.67153
$520$	0	0
$521$	4.14963e9	1.28552	0.642758	−	0.766069i	$-0.277790\pi$
$521$	4.14963e9	1.28552	0.642758	+	0.766069i	$0.277790\pi$
$522$	0	0
$523$	− 2.51360e9i	− 0.768318i	−0.923267	−	0.384159i	$-0.874491\pi$
$523$	− 2.51360e9i	− 0.768318i	0.923267	−	0.384159i	$-0.125509\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	4.51352e8i	0.134332i
$528$	0	0
$529$	−3.27072e9	−0.960613
$530$	0	0
$531$	4.70359e9	1.36332
$532$	0	0
$533$	− 1.52403e9i	− 0.435962i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 9.51112e8i	− 0.265046i
$538$	0	0
$539$	1.55379e9	0.427398
$540$	0	0
$541$	−1.32416e9	−0.359543	−0.179772	−	0.983708i	$-0.557536\pi$
$541$	−1.32416e9	−0.359543	−0.179772	+	0.983708i	$0.557536\pi$
$542$	0	0
$543$	− 4.39026e8i	− 0.117677i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	5.58047e8i	0.145786i	0.997340	+	0.0728929i	$0.0232231\pi$
$547$	5.58047e8i	0.145786i	−0.997340	+	0.0728929i	$0.976777\pi$
$548$	0	0
$549$	−9.19107e9	−2.37062
$550$	0	0
$551$	−4.11567e8	−0.104812
$552$	0	0
$553$	− 1.84136e9i	− 0.463020i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	3.30331e9i	0.809946i	0.914329	+	0.404973i	$0.132719\pi$
$557$	3.30331e9i	0.809946i	−0.914329	+	0.404973i	$0.867281\pi$
$558$	0	0
$559$	7.44143e9	1.80184
$560$	0	0
$561$	−2.36398e9	−0.565293
$562$	0	0
$563$	1.22011e8i	0.0288152i	0.999896	+	0.0144076i	$0.00458623\pi$
$563$	1.22011e8i	0.0288152i	−0.999896	+	0.0144076i	$0.995414\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 3.77371e9i	− 0.869417i
$568$	0	0
$569$	−5.00925e8	−0.113993	−0.0569967	−	0.998374i	$-0.518152\pi$
$569$	−5.00925e8	−0.113993	−0.0569967	+	0.998374i	$0.518152\pi$
$570$	0	0
$571$	6.98702e9	1.57060	0.785300	−	0.619116i	$-0.212509\pi$
$571$	6.98702e9	1.57060	0.785300	+	0.619116i	$0.212509\pi$
$572$	0	0
$573$	7.14253e9i	1.58603i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	8.16573e9i	1.76962i	0.465954	+	0.884809i	$0.345711\pi$
$577$	8.16573e9i	1.76962i	−0.465954	+	0.884809i	$0.654289\pi$
$578$	0	0
$579$	−9.66769e9	−2.06989
$580$	0	0
$581$	2.45591e9	0.519512
$582$	0	0
$583$	4.57631e9i	0.956479i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	8.53182e9i	1.74104i	0.492135	+	0.870519i	$0.336217\pi$
$587$	8.53182e9i	1.74104i	−0.492135	+	0.870519i	$0.663783\pi$
$588$	0	0
$589$	1.66940e8	0.0336632
$590$	0	0
$591$	−1.16358e10	−2.31868
$592$	0	0
$593$	− 1.71175e9i	− 0.337092i	−0.985694	−	0.168546i	$-0.946093\pi$
$593$	− 1.71175e9i	− 0.337092i	0.985694	−	0.168546i	$-0.0539072\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 1.84475e9i	− 0.354836i
$598$	0	0
$599$	−4.77362e9	−0.907516	−0.453758	−	0.891125i	$-0.649917\pi$
$599$	−4.77362e9	−0.907516	−0.453758	+	0.891125i	$0.649917\pi$
$600$	0	0
$601$	7.89998e8	0.148445	0.0742224	−	0.997242i	$-0.476353\pi$
$601$	7.89998e8	0.148445	0.0742224	+	0.997242i	$0.476353\pi$
$602$	0	0
$603$	− 1.44408e10i	− 2.68214i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 1.82652e9i	− 0.331485i	−0.986169	−	0.165743i	$-0.946998\pi$
$607$	− 1.82652e9i	− 0.331485i	0.986169	−	0.165743i	$-0.0530021\pi$
$608$	0	0
$609$	−3.82266e9	−0.685813
$610$	0	0
$611$	−7.35094e9	−1.30376
$612$	0	0
$613$	− 6.90339e9i	− 1.21046i	−0.796050	−	0.605231i	$-0.793081\pi$
$613$	− 6.90339e9i	− 1.21046i	0.796050	−	0.605231i	$-0.206919\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	5.69235e9i	0.975649i	0.872942	+	0.487825i	$0.162209\pi$
$617$	5.69235e9i	0.975649i	−0.872942	+	0.487825i	$0.837791\pi$
$618$	0	0
$619$	−4.28594e9	−0.726321	−0.363161	−	0.931727i	$-0.618302\pi$
$619$	−4.28594e9	−0.726321	−0.363161	+	0.931727i	$0.618302\pi$
$620$	0	0
$621$	−1.84069e10	−3.08433
$622$	0	0
$623$	− 2.95171e9i	− 0.489064i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	8.74354e8i	0.141661i
$628$	0	0
$629$	−4.67678e9	−0.749324
$630$	0	0
$631$	5.61602e8	0.0889869	0.0444935	−	0.999010i	$-0.485833\pi$
$631$	5.61602e8	0.0889869	0.0444935	+	0.999010i	$0.485833\pi$
$632$	0	0
$633$	5.12575e9i	0.803238i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 6.63501e9i	− 1.01708i
$638$	0	0
$639$	−1.24073e10	−1.88116
$640$	0	0
$641$	5.17445e9	0.775998	0.387999	−	0.921660i	$-0.373166\pi$
$641$	5.17445e9	0.775998	0.387999	+	0.921660i	$0.373166\pi$
$642$	0	0
$643$	1.04374e10i	1.54830i	0.633004	+	0.774148i	$0.281822\pi$
$643$	1.04374e10i	1.54830i	−0.633004	+	0.774148i	$0.718178\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 9.71623e8i	− 0.141037i	−0.997510	−	0.0705185i	$-0.977535\pi$
$647$	− 9.71623e8i	− 0.141037i	0.997510	−	0.0705185i	$-0.0224654\pi$
$648$	0	0
$649$	−2.43825e9	−0.350125
$650$	0	0
$651$	1.55055e9	0.220268
$652$	0	0
$653$	− 7.25223e9i	− 1.01924i	−0.860400	−	0.509619i	$-0.829786\pi$
$653$	− 7.25223e9i	− 1.01924i	0.860400	−	0.509619i	$-0.170214\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	8.18151e9i	1.12552i
$658$	0	0
$659$	3.81924e9	0.519851	0.259925	−	0.965629i	$-0.416302\pi$
$659$	3.81924e9	0.519851	0.259925	+	0.965629i	$0.416302\pi$
$660$	0	0
$661$	1.07881e10	1.45292	0.726459	−	0.687210i	$-0.241165\pi$
$661$	1.07881e10	1.45292	0.726459	+	0.687210i	$0.241165\pi$
$662$	0	0
$663$	1.00947e10i	1.34523i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	8.15390e9i	1.06396i
$668$	0	0
$669$	−3.54857e9	−0.458207
$670$	0	0
$671$	4.76448e9	0.608817
$672$	0	0
$673$	− 6.34833e9i	− 0.802798i	−0.915903	−	0.401399i	$-0.868524\pi$
$673$	− 6.34833e9i	− 0.802798i	0.915903	−	0.401399i	$-0.131476\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 8.82566e9i	− 1.09317i	−0.837404	−	0.546584i	$-0.815928\pi$
$677$	− 8.82566e9i	− 1.09317i	0.837404	−	0.546584i	$-0.184072\pi$
$678$	0	0
$679$	2.76599e9	0.339083
$680$	0	0
$681$	2.00846e10	2.43696
$682$	0	0
$683$	− 4.92331e9i	− 0.591268i	−0.955301	−	0.295634i	$-0.904469\pi$
$683$	− 4.92331e9i	− 0.591268i	0.955301	−	0.295634i	$-0.0955309\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	3.93027e9i	0.462459i
$688$	0	0
$689$	1.95418e10	2.27613
$690$	0	0
$691$	5.68449e9	0.655418	0.327709	−	0.944779i	$-0.393723\pi$
$691$	5.68449e9	0.655418	0.327709	+	0.944779i	$0.393723\pi$
$692$	0	0
$693$	5.60395e9i	0.639628i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	1.57663e9i	0.176366i
$698$	0	0
$699$	−2.89989e10	−3.21152
$700$	0	0
$701$	−1.70567e9	−0.187017	−0.0935085	−	0.995618i	$-0.529808\pi$
$701$	−1.70567e9	−0.187017	−0.0935085	+	0.995618i	$0.529808\pi$
$702$	0	0
$703$	1.72978e9i	0.187779i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 4.42351e9i	− 0.470760i
$708$	0	0
$709$	−4.52189e9	−0.476495	−0.238248	−	0.971204i	$-0.576573\pi$
$709$	−4.52189e9	−0.476495	−0.238248	+	0.971204i	$0.576573\pi$
$710$	0	0
$711$	−1.96613e10	−2.05149
$712$	0	0
$713$	− 3.30738e9i	− 0.341720i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	1.96907e10i	1.99500i
$718$	0	0
$719$	−3.09206e9	−0.310239	−0.155120	−	0.987896i	$-0.549576\pi$
$719$	−3.09206e9	−0.310239	−0.155120	+	0.987896i	$0.549576\pi$
$720$	0	0
$721$	−1.87033e9	−0.185842
$722$	0	0
$723$	− 9.20280e9i	− 0.905599i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	1.44622e10i	1.39593i	0.716132	+	0.697965i	$0.245911\pi$
$727$	1.44622e10i	1.39593i	−0.716132	+	0.697965i	$0.754089\pi$
$728$	0	0
$729$	1.09288e9	0.104478
$730$	0	0
$731$	−7.69827e9	−0.728924
$732$	0	0
$733$	3.15415e9i	0.295814i	0.989001	+	0.147907i	$0.0472536\pi$
$733$	3.15415e9i	0.295814i	−0.989001	+	0.147907i	$0.952746\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	7.48585e9i	0.688819i
$738$	0	0
$739$	1.54236e10	1.40582	0.702912	−	0.711277i	$-0.251883\pi$
$739$	1.54236e10	1.40582	0.702912	+	0.711277i	$0.251883\pi$
$740$	0	0
$741$	3.73367e9	0.337111
$742$	0	0
$743$	1.59520e10i	1.42677i	0.700772	+	0.713385i	$0.252839\pi$
$743$	1.59520e10i	1.42677i	−0.700772	+	0.713385i	$0.747161\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 2.62233e10i	− 2.30179i
$748$	0	0
$749$	3.32424e7	0.00289072
$750$	0	0
$751$	−6.13964e9	−0.528936	−0.264468	−	0.964395i	$-0.585196\pi$
$751$	−6.13964e9	−0.528936	−0.264468	+	0.964395i	$0.585196\pi$
$752$	0	0
$753$	3.30986e10i	2.82506i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	1.42818e10i	1.19660i	0.801273	+	0.598299i	$0.204157\pi$
$757$	1.42818e10i	1.19660i	−0.801273	+	0.598299i	$0.795843\pi$
$758$	0	0
$759$	1.73226e10	1.43802
$760$	0	0
$761$	1.47536e10	1.21353	0.606767	−	0.794880i	$-0.292466\pi$
$761$	1.47536e10	1.21353	0.606767	+	0.794880i	$0.292466\pi$
$762$	0	0
$763$	4.35863e9i	0.355234i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	1.04118e10i	0.833190i
$768$	0	0
$769$	−1.97592e10	−1.56685	−0.783424	−	0.621487i	$-0.786529\pi$
$769$	−1.97592e10	−1.56685	−0.783424	+	0.621487i	$0.786529\pi$
$770$	0	0
$771$	2.68014e10	2.10604
$772$	0	0
$773$	1.01370e10i	0.789374i	0.918816	+	0.394687i	$0.129147\pi$
$773$	1.01370e10i	0.789374i	−0.918816	+	0.394687i	$0.870853\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	1.60663e10i	1.22869i
$778$	0	0
$779$	5.83142e8	0.0441970
$780$	0	0
$781$	6.43174e9	0.483114
$782$	0	0
$783$	2.24833e10i	1.67376i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 1.27882e10i	− 0.935188i	−0.883943	−	0.467594i	$-0.845121\pi$
$787$	− 1.27882e10i	− 0.935188i	0.883943	−	0.467594i	$-0.154879\pi$
$788$	0	0
$789$	1.84261e10	1.33556
$790$	0	0
$791$	4.25854e9	0.305945
$792$	0	0
$793$	− 2.03453e10i	− 1.44880i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	7.38617e9i	0.516791i	0.966039	+	0.258396i	$0.0831938\pi$
$797$	7.38617e9i	0.516791i	−0.966039	+	0.258396i	$0.916806\pi$
$798$	0	0
$799$	7.60466e9	0.527431
$800$	0	0
$801$	−3.15173e10	−2.16688
$802$	0	0
$803$	− 4.24114e9i	− 0.289054i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	1.24348e10i	0.832875i
$808$	0	0
$809$	−1.53742e10	−1.02087	−0.510437	−	0.859915i	$-0.670516\pi$
$809$	−1.53742e10	−1.02087	−0.510437	+	0.859915i	$0.670516\pi$
$810$	0	0
$811$	−9.77882e9	−0.643744	−0.321872	−	0.946783i	$-0.604312\pi$
$811$	−9.77882e9	−0.643744	−0.321872	+	0.946783i	$0.604312\pi$
$812$	0	0
$813$	3.10745e10i	2.02809i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	2.84733e9i	0.182667i
$818$	0	0
$819$	2.39300e10	1.52212
$820$	0	0
$821$	1.83470e10	1.15708	0.578540	−	0.815654i	$-0.303623\pi$
$821$	1.83470e10	1.15708	0.578540	+	0.815654i	$0.303623\pi$
$822$	0	0
$823$	3.16960e10i	1.98201i	0.133829	+	0.991004i	$0.457273\pi$
$823$	3.16960e10i	1.98201i	−0.133829	+	0.991004i	$0.542727\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 6.12845e9i	− 0.376774i	−0.982095	−	0.188387i	$-0.939674\pi$
$827$	− 6.12845e9i	− 0.376774i	0.982095	−	0.188387i	$-0.0603260\pi$
$828$	0	0
$829$	1.24652e10	0.759904	0.379952	−	0.925006i	$-0.375940\pi$
$829$	1.24652e10	0.759904	0.379952	+	0.925006i	$0.375940\pi$
$830$	0	0
$831$	−3.32522e10	−2.01010
$832$	0	0
$833$	6.86402e9i	0.411454i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 9.11966e9i	− 0.537575i
$838$	0	0
$839$	−1.82237e10	−1.06530	−0.532648	−	0.846337i	$-0.678803\pi$
$839$	−1.82237e10	−1.06530	−0.532648	+	0.846337i	$0.678803\pi$
$840$	0	0
$841$	−7.29024e9	−0.422625
$842$	0	0
$843$	− 5.02118e10i	− 2.88675i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	5.98117e9i	0.338216i
$848$	0	0
$849$	6.76870e9	0.379602
$850$	0	0
$851$	3.42701e10	1.90617
$852$	0	0
$853$	− 2.48619e10i	− 1.37155i	−0.727812	−	0.685777i	$-0.759463\pi$
$853$	− 2.48619e10i	− 1.37155i	0.727812	−	0.685777i	$-0.240537\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 2.96761e10i	− 1.61055i	−0.592902	−	0.805275i	$-0.702018\pi$
$857$	− 2.96761e10i	− 1.61055i	0.592902	−	0.805275i	$-0.297982\pi$
$858$	0	0
$859$	1.14772e10	0.617819	0.308910	−	0.951091i	$-0.400036\pi$
$859$	1.14772e10	0.617819	0.308910	+	0.951091i	$0.400036\pi$
$860$	0	0
$861$	5.41626e9	0.289194
$862$	0	0
$863$	− 2.13485e10i	− 1.13066i	−0.824866	−	0.565328i	$-0.808750\pi$
$863$	− 2.13485e10i	− 1.13066i	0.824866	−	0.565328i	$-0.191250\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	2.40254e10i	1.25200i
$868$	0	0
$869$	1.01921e10	0.526858
$870$	0	0
$871$	3.19661e10	1.63918
$872$	0	0
$873$	− 2.95342e10i	− 1.50236i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	7.92753e9i	0.396862i	0.980115	+	0.198431i	$0.0635846\pi$
$877$	7.92753e9i	0.396862i	−0.980115	+	0.198431i	$0.936415\pi$
$878$	0	0
$879$	−6.33805e10	−3.14771
$880$	0	0
$881$	7.32045e9	0.360680	0.180340	−	0.983604i	$-0.442280\pi$
$881$	7.32045e9	0.360680	0.180340	+	0.983604i	$0.442280\pi$
$882$	0	0
$883$	3.54988e9i	0.173521i	0.996229	+	0.0867604i	$0.0276514\pi$
$883$	3.54988e9i	0.173521i	−0.996229	+	0.0867604i	$0.972349\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	5.80634e9i	0.279364i	0.990196	+	0.139682i	$0.0446080\pi$
$887$	5.80634e9i	0.279364i	−0.990196	+	0.139682i	$0.955392\pi$
$888$	0	0
$889$	−1.64066e10	−0.783182
$890$	0	0
$891$	2.08878e10	0.989285
$892$	0	0
$893$	− 2.81270e9i	− 0.132173i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 7.39709e10i	− 3.42206i
$898$	0	0
$899$	−4.03982e9	−0.185440
$900$	0	0
$901$	−2.02163e10	−0.920798
$902$	0	0
$903$	2.64462e10i	1.19524i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	1.78240e10i	0.793196i	0.917992	+	0.396598i	$0.129809\pi$
$907$	1.78240e10i	0.793196i	−0.917992	+	0.396598i	$0.870191\pi$
$908$	0	0
$909$	−4.72326e10	−2.08578
$910$	0	0
$911$	−1.87703e10	−0.822538	−0.411269	−	0.911514i	$-0.634914\pi$
$911$	−1.87703e10	−0.822538	−0.411269	+	0.911514i	$0.634914\pi$
$912$	0	0
$913$	1.35937e10i	0.591138i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 3.08280e8i	− 0.0132024i
$918$	0	0
$919$	3.75844e10	1.59736	0.798681	−	0.601754i	$-0.205531\pi$
$919$	3.75844e10	1.59736	0.798681	+	0.601754i	$0.205531\pi$
$920$	0	0
$921$	−6.89197e10	−2.90693
$922$	0	0
$923$	− 2.74648e10i	− 1.14966i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	1.99707e10i	0.823404i
$928$	0	0
$929$	1.92372e10	0.787205	0.393602	−	0.919281i	$-0.371229\pi$
$929$	1.92372e10	0.787205	0.393602	+	0.919281i	$0.371229\pi$
$930$	0	0
$931$	2.53876e9	0.103109
$932$	0	0
$933$	− 5.48943e10i	− 2.21279i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 1.04732e9i	− 0.0415900i	−0.999784	−	0.0207950i	$-0.993380\pi$
$937$	− 1.04732e9i	− 0.0415900i	0.999784	−	0.0207950i	$-0.00661973\pi$
$938$	0	0
$939$	−5.57413e10	−2.19709
$940$	0	0
$941$	−7.97861e9	−0.312150	−0.156075	−	0.987745i	$-0.549884\pi$
$941$	−7.97861e9	−0.312150	−0.156075	+	0.987745i	$0.549884\pi$
$942$	0	0
$943$	− 1.15531e10i	− 0.448650i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 4.26943e9i	− 0.163360i	−0.996659	−	0.0816799i	$-0.973971\pi$
$947$	− 4.26943e9i	− 0.163360i	0.996659	−	0.0816799i	$-0.0260285\pi$
$948$	0	0
$949$	−1.81106e10	−0.687860
$950$	0	0
$951$	−2.97963e10	−1.12339
$952$	0	0
$953$	− 1.06048e10i	− 0.396897i	−0.980111	−	0.198449i	$-0.936410\pi$
$953$	− 1.06048e10i	− 0.396897i	0.980111	−	0.198449i	$-0.0635903\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 2.11588e10i	− 0.780367i
$958$	0	0
$959$	1.34904e10	0.493922
$960$	0	0
$961$	−2.58740e10	−0.940441
$962$	0	0
$963$	− 3.54950e8i	− 0.0128078i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 1.65090e10i	− 0.587123i	−0.955940	−	0.293562i	$-0.905159\pi$
$967$	− 1.65090e10i	− 0.587123i	0.955940	−	0.293562i	$-0.0948406\pi$
$968$	0	0
$969$	−3.86254e9	−0.136377
$970$	0	0
$971$	2.46094e10	0.862649	0.431324	−	0.902197i	$-0.358046\pi$
$971$	2.46094e10	0.862649	0.431324	+	0.902197i	$0.358046\pi$
$972$	0	0
$973$	1.45503e10i	0.506379i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	2.53886e10i	0.870979i	0.900194	+	0.435489i	$0.143425\pi$
$977$	2.53886e10i	0.870979i	−0.900194	+	0.435489i	$0.856575\pi$
$978$	0	0
$979$	1.63380e10	0.556492
$980$	0	0
$981$	4.65399e10	1.57392
$982$	0	0
$983$	− 1.87585e10i	− 0.629884i	−0.949111	−	0.314942i	$-0.898015\pi$
$983$	− 1.87585e10i	− 0.629884i	0.949111	−	0.314942i	$-0.101985\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 2.61246e10i	− 0.864846i
$988$	0	0
$989$	5.64107e10	1.85428
$990$	0	0
$991$	3.59792e9	0.117434	0.0587170	−	0.998275i	$-0.481299\pi$
$991$	3.59792e9	0.117434	0.0587170	+	0.998275i	$0.481299\pi$
$992$	0	0
$993$	2.56656e10i	0.831821i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	1.34287e10i	0.429143i	0.976708	+	0.214571i	$0.0688354\pi$
$997$	1.34287e10i	0.429143i	−0.976708	+	0.214571i	$0.931165\pi$
$998$	0	0
$999$	9.44952e10	2.99868

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	400.8.c.b.49.2		2
4.3	odd	2		200.8.c.a.49.1		2
5.2	odd	4		16.8.a.c.1.1		1
5.3	odd	4		400.8.a.b.1.1		1
5.4	even	2	inner	400.8.c.b.49.1		2
15.2	even	4		144.8.a.g.1.1		1
20.3	even	4		200.8.a.i.1.1		1
20.7	even	4		8.8.a.a.1.1	✓	1
20.19	odd	2		200.8.c.a.49.2		2
40.27	even	4		64.8.a.g.1.1		1
40.37	odd	4		64.8.a.a.1.1		1
60.47	odd	4		72.8.a.d.1.1		1
80.27	even	4		256.8.b.e.129.2		2
80.37	odd	4		256.8.b.c.129.1		2
80.67	even	4		256.8.b.e.129.1		2
80.77	odd	4		256.8.b.c.129.2		2
120.77	even	4		576.8.a.k.1.1		1
120.107	odd	4		576.8.a.j.1.1		1
140.27	odd	4		392.8.a.d.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8.8.a.a.1.1	✓	1	20.7	even	4
16.8.a.c.1.1		1	5.2	odd	4
64.8.a.a.1.1		1	40.37	odd	4
64.8.a.g.1.1		1	40.27	even	4
72.8.a.d.1.1		1	60.47	odd	4
144.8.a.g.1.1		1	15.2	even	4
200.8.a.i.1.1		1	20.3	even	4
200.8.c.a.49.1		2	4.3	odd	2
200.8.c.a.49.2		2	20.19	odd	2
256.8.b.c.129.1		2	80.37	odd	4
256.8.b.c.129.2		2	80.77	odd	4
256.8.b.e.129.1		2	80.67	even	4
256.8.b.e.129.2		2	80.27	even	4
392.8.a.d.1.1		1	140.27	odd	4
400.8.a.b.1.1		1	5.3	odd	4
400.8.c.b.49.1		2	5.4	even	2	inner
400.8.c.b.49.2		2	1.1	even	1	trivial
576.8.a.j.1.1		1	120.107	odd	4
576.8.a.k.1.1		1	120.77	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:	\( N \)	\(=\)	\( 400 = 2^{4} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	400.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+84.0000i q^{3} -456.000i q^{7} -4869.00 q^{9} +O(q^{10})\)	\(q+84.0000i q^{3} -456.000i q^{7} -4869.00 q^{9} +2524.00 q^{11} -10778.0i q^{13} +11150.0i q^{17} +4124.00 q^{19} +38304.0 q^{21} -81704.0i q^{23} -225288. i q^{27} -99798.0 q^{29} +40480.0 q^{31} +212016. i q^{33} +419442. i q^{37} +905352. q^{39} +141402. q^{41} +690428. i q^{43} -682032. i q^{47} +615607. q^{49} -936600. q^{51} +1.81312e6i q^{53} +346416. i q^{57} -966028. q^{59} +1.88767e6 q^{61} +2.22026e6i q^{63} +2.96587e6i q^{67} +6.86314e6 q^{69} +2.54823e6 q^{71} -1.68033e6i q^{73} -1.15094e6i q^{77} +4.03806e6 q^{79} +8.27569e6 q^{81} +5.38576e6i q^{83} -8.38303e6i q^{87} +6.47305e6 q^{89} -4.91477e6 q^{91} +3.40032e6i q^{93} +6.06576e6i q^{97} -1.22894e7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 9738 q^{9}+O(q^{10}) \) 2 * q - 9738 * q^9	\( 2 q - 9738 q^{9} + 5048 q^{11} + 8248 q^{19} + 76608 q^{21} - 199596 q^{29} + 80960 q^{31} + 1810704 q^{39} + 282804 q^{41} + 1231214 q^{49} - 1873200 q^{51} - 1932056 q^{59} + 3775340 q^{61} + 13726272 q^{69} + 5096464 q^{71} + 8076128 q^{79} + 16551378 q^{81} + 12946092 q^{89} - 9829536 q^{91} - 24578712 q^{99}+O(q^{100}) \) 2 * q - 9738 * q^9 + 5048 * q^11 + 8248 * q^19 + 76608 * q^21 - 199596 * q^29 + 80960 * q^31 + 1810704 * q^39 + 282804 * q^41 + 1231214 * q^49 - 1873200 * q^51 - 1932056 * q^59 + 3775340 * q^61 + 13726272 * q^69 + 5096464 * q^71 + 8076128 * q^79 + 16551378 * q^81 + 12946092 * q^89 - 9829536 * q^91 - 24578712 * q^99

Introduction

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