LMFDB - Embedded newform 784.4.f.f.783.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [784,4,Mod(783,784)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(784, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("784.783");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 784 = 2^{4} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	784.f (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$46.2574974445$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{7})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 7x^{2} + 49 $ x^4 + 7*x^2 + 49
Coefficient ring:	$\Z[a_1, \ldots, a_{23}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 112)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			783.1
Root			$1.32288 + 2.29129i$ of defining polynomial
Character	$\chi$	$=$	784.783
Dual form			784.4.f.f.783.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.64575 q^{3} -8.66025i q^{5} -20.0000 q^{9} +O(q^{10})$	$q-2.64575 q^{3} -8.66025i q^{5} -20.0000 q^{9} -68.7386i q^{11} -51.9615i q^{13} +22.9129i q^{15} +8.66025i q^{17} -66.1438 q^{19} +50.4083i q^{23} +50.0000 q^{25} +124.350 q^{27} -168.000 q^{29} +224.889 q^{31} +181.865i q^{33} -245.000 q^{37} +137.477i q^{39} -45.0333i q^{41} -302.450i q^{43} +173.205i q^{45} +420.674 q^{47} -22.9129i q^{51} -345.000 q^{53} -595.294 q^{55} +175.000 q^{57} -436.549 q^{59} +396.640i q^{61} -450.000 q^{65} -444.510i q^{67} -133.368i q^{69} -45.8258i q^{71} +961.288i q^{73} -132.288 q^{75} -206.216i q^{79} +211.000 q^{81} +888.972 q^{83} +75.0000 q^{85} +444.486 q^{87} +1512.08i q^{89} -595.000 q^{93} +572.822i q^{95} -433.013i q^{97} +1374.77i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 80 q^{9}+O(q^{10}) $ 4 * q - 80 * q^9	$ 4 q - 80 q^{9} + 200 q^{25} - 672 q^{29} - 980 q^{37} - 1380 q^{53} + 700 q^{57} - 1800 q^{65} + 844 q^{81} + 300 q^{85} - 2380 q^{93}+O(q^{100}) $ 4 * q - 80 * q^9 + 200 * q^25 - 672 * q^29 - 980 * q^37 - 1380 * q^53 + 700 * q^57 - 1800 * q^65 + 844 * q^81 + 300 * q^85 - 2380 * q^93

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/784\mathbb{Z}\right)^\times$.

$n$	$197$	$687$	$689$
$\chi(n)$	$1$	$-1$	$-1$

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
$3$	−2.64575	−0.509175	−0.254588	−	0.967050i	$-0.581940\pi$
$3$	−2.64575	−0.509175	−0.254588	+	0.967050i	$0.581940\pi$
$4$	0	0
$5$	− 8.66025i	− 0.774597i	−0.921954	−	0.387298i	$-0.873408\pi$
$5$	− 8.66025i	− 0.774597i	0.921954	−	0.387298i	$-0.126592\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−20.0000	−0.740741
$10$	0	0
$11$	− 68.7386i	− 1.88413i	−0.335424	−	0.942067i	$-0.608880\pi$
$11$	− 68.7386i	− 1.88413i	0.335424	−	0.942067i	$-0.391120\pi$
$12$	0	0
$13$	− 51.9615i	− 1.10858i	−0.832324	−	0.554290i	$-0.812990\pi$
$13$	− 51.9615i	− 1.10858i	0.832324	−	0.554290i	$-0.187010\pi$
$14$	0	0
$15$	22.9129i	0.394405i
$16$	0	0
$17$	8.66025i	0.123554i	0.998090	+	0.0617771i	$0.0196768\pi$
$17$	8.66025i	0.123554i	−0.998090	+	0.0617771i	$0.980323\pi$
$18$	0	0
$19$	−66.1438	−0.798654	−0.399327	−	0.916809i	$-0.630756\pi$
$19$	−66.1438	−0.798654	−0.399327	+	0.916809i	$0.630756\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	50.4083i	0.456994i	0.973545	+	0.228497i	$0.0733811\pi$
$23$	50.4083i	0.456994i	−0.973545	+	0.228497i	$0.926619\pi$
$24$	0	0
$25$	50.0000	0.400000
$26$	0	0
$27$	124.350	0.886342
$28$	0	0
$29$	−168.000	−1.07575	−0.537876	−	0.843024i	$-0.680773\pi$
$29$	−168.000	−1.07575	−0.537876	+	0.843024i	$0.680773\pi$
$30$	0	0
$31$	224.889	1.30294	0.651471	−	0.758673i	$-0.274152\pi$
$31$	224.889	1.30294	0.651471	+	0.758673i	$0.274152\pi$
$32$	0	0
$33$	181.865i	0.959354i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−245.000	−1.08859	−0.544294	−	0.838895i	$-0.683202\pi$
$37$	−245.000	−1.08859	−0.544294	+	0.838895i	$0.683202\pi$
$38$	0	0
$39$	137.477i	0.564461i
$40$	0	0
$41$	− 45.0333i	− 0.171537i	−0.996315	−	0.0857686i	$-0.972665\pi$
$41$	− 45.0333i	− 0.171537i	0.996315	−	0.0857686i	$-0.0273346\pi$
$42$	0	0
$43$	− 302.450i	− 1.07263i	−0.844017	−	0.536316i	$-0.819815\pi$
$43$	− 302.450i	− 1.07263i	0.844017	−	0.536316i	$-0.180185\pi$
$44$	0	0
$45$	173.205i	0.573775i
$46$	0	0
$47$	420.674	1.30557	0.652784	−	0.757544i	$-0.273601\pi$
$47$	420.674	1.30557	0.652784	+	0.757544i	$0.273601\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	− 22.9129i	− 0.0629107i
$52$	0	0
$53$	−345.000	−0.894140	−0.447070	−	0.894499i	$-0.647532\pi$
$53$	−345.000	−0.894140	−0.447070	+	0.894499i	$0.647532\pi$
$54$	0	0
$55$	−595.294	−1.45944
$56$	0	0
$57$	175.000	0.406655
$58$	0	0
$59$	−436.549	−0.963285	−0.481643	−	0.876368i	$-0.659960\pi$
$59$	−436.549	−0.963285	−0.481643	+	0.876368i	$0.659960\pi$
$60$	0	0
$61$	396.640i	0.832533i	0.909243	+	0.416266i	$0.136662\pi$
$61$	396.640i	0.832533i	−0.909243	+	0.416266i	$0.863338\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−450.000	−0.858702
$66$	0	0
$67$	− 444.510i	− 0.810530i	−0.914199	−	0.405265i	$-0.867179\pi$
$67$	− 444.510i	− 0.810530i	0.914199	−	0.405265i	$-0.132821\pi$
$68$	0	0
$69$	− 133.368i	− 0.232690i
$70$	0	0
$71$	− 45.8258i	− 0.0765988i	−0.999266	−	0.0382994i	$-0.987806\pi$
$71$	− 45.8258i	− 0.0765988i	0.999266	−	0.0382994i	$-0.0121941\pi$
$72$	0	0
$73$	961.288i	1.54124i	0.637297	+	0.770618i	$0.280052\pi$
$73$	961.288i	1.54124i	−0.637297	+	0.770618i	$0.719948\pi$
$74$	0	0
$75$	−132.288	−0.203670
$76$	0	0
$77$	0	0
$78$	0	0
$79$	− 206.216i	− 0.293685i	−0.989160	−	0.146842i	$-0.953089\pi$
$79$	− 206.216i	− 0.293685i	0.989160	−	0.146842i	$-0.0469110\pi$
$80$	0	0
$81$	211.000	0.289438
$82$	0	0
$83$	888.972	1.17563	0.587816	−	0.808995i	$-0.299988\pi$
$83$	888.972	1.17563	0.587816	+	0.808995i	$0.299988\pi$
$84$	0	0
$85$	75.0000	0.0957046
$86$	0	0
$87$	444.486	0.547746
$88$	0	0
$89$	1512.08i	1.80090i	0.434958	+	0.900451i	$0.356763\pi$
$89$	1512.08i	1.80090i	−0.434958	+	0.900451i	$0.643237\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−595.000	−0.663426
$94$	0	0
$95$	572.822i	0.618635i
$96$	0	0
$97$	− 433.013i	− 0.453255i	−0.973981	−	0.226628i	$-0.927230\pi$
$97$	− 433.013i	− 0.453255i	0.973981	−	0.226628i	$-0.0727701\pi$
$98$	0	0
$99$	1374.77i	1.39566i
$100$	0	0
$101$	833.116i	0.820774i	0.911911	+	0.410387i	$0.134606\pi$
$101$	833.116i	0.820774i	−0.911911	+	0.410387i	$0.865394\pi$
$102$	0	0
$103$	−1934.04	−1.85017	−0.925083	−	0.379766i	$-0.876005\pi$
$103$	−1934.04	−1.85017	−0.925083	+	0.379766i	$0.876005\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	792.786i	0.716275i	0.933669	+	0.358138i	$0.116588\pi$
$107$	792.786i	0.716275i	−0.933669	+	0.358138i	$0.883412\pi$
$108$	0	0
$109$	−763.000	−0.670478	−0.335239	−	0.942133i	$-0.608817\pi$
$109$	−763.000	−0.670478	−0.335239	+	0.942133i	$0.608817\pi$
$110$	0	0
$111$	648.209	0.554282
$112$	0	0
$113$	−1500.00	−1.24874	−0.624372	−	0.781127i	$-0.714645\pi$
$113$	−1500.00	−1.24874	−0.624372	+	0.781127i	$0.714645\pi$
$114$	0	0
$115$	436.549	0.353986
$116$	0	0
$117$	1039.23i	0.821170i
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−3394.00	−2.54996
$122$	0	0
$123$	119.147i	0.0873424i
$124$	0	0
$125$	− 1515.54i	− 1.08444i
$126$	0	0
$127$	45.8258i	0.0320187i	0.999872	+	0.0160094i	$0.00509616\pi$
$127$	45.8258i	0.0320187i	−0.999872	+	0.0160094i	$0.994904\pi$
$128$	0	0
$129$	800.207i	0.546158i
$130$	0	0
$131$	2024.00	1.34991	0.674953	−	0.737861i	$-0.264164\pi$
$131$	2024.00	1.34991	0.674953	+	0.737861i	$0.264164\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	− 1076.91i	− 0.686557i
$136$	0	0
$137$	−315.000	−0.196440	−0.0982199	−	0.995165i	$-0.531315\pi$
$137$	−315.000	−0.196440	−0.0982199	+	0.995165i	$0.531315\pi$
$138$	0	0
$139$	−2275.35	−1.38843	−0.694217	−	0.719766i	$-0.744249\pi$
$139$	−2275.35	−1.38843	−0.694217	+	0.719766i	$0.744249\pi$
$140$	0	0
$141$	−1113.00	−0.664762
$142$	0	0
$143$	−3571.76	−2.08871
$144$	0	0
$145$	1454.92i	0.833274i
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−1563.00	−0.859369	−0.429684	−	0.902979i	$-0.641375\pi$
$149$	−1563.00	−0.859369	−0.429684	+	0.902979i	$0.641375\pi$
$150$	0	0
$151$	297.867i	0.160531i	0.996774	+	0.0802653i	$0.0255767\pi$
$151$	297.867i	0.160531i	−0.996774	+	0.0802653i	$0.974423\pi$
$152$	0	0
$153$	− 173.205i	− 0.0915216i
$154$	0	0
$155$	− 1947.59i	− 1.00926i
$156$	0	0
$157$	718.801i	0.365392i	0.983169	+	0.182696i	$0.0584824\pi$
$157$	718.801i	0.365392i	−0.983169	+	0.182696i	$0.941518\pi$
$158$	0	0
$159$	912.784	0.455274
$160$	0	0
$161$	0	0
$162$	0	0
$163$	− 2130.90i	− 1.02396i	−0.858999	−	0.511978i	$-0.828913\pi$
$163$	− 2130.90i	− 1.02396i	0.858999	−	0.511978i	$-0.171087\pi$
$164$	0	0
$165$	1575.00	0.743113
$166$	0	0
$167$	−746.102	−0.345719	−0.172860	−	0.984946i	$-0.555301\pi$
$167$	−746.102	−0.345719	−0.172860	+	0.984946i	$0.555301\pi$
$168$	0	0
$169$	−503.000	−0.228949
$170$	0	0
$171$	1322.88	0.591595
$172$	0	0
$173$	1082.53i	0.475742i	0.971297	+	0.237871i	$0.0764495\pi$
$173$	1082.53i	0.475742i	−0.971297	+	0.237871i	$0.923550\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	1155.00	0.490481
$178$	0	0
$179$	1489.34i	0.621890i	0.950428	+	0.310945i	$0.100645\pi$
$179$	1489.34i	0.621890i	−0.950428	+	0.310945i	$0.899355\pi$
$180$	0	0
$181$	− 2833.64i	− 1.16366i	−0.813310	−	0.581830i	$-0.802337\pi$
$181$	− 2833.64i	− 1.16366i	0.813310	−	0.581830i	$-0.197663\pi$
$182$	0	0
$183$	− 1049.41i	− 0.423905i
$184$	0	0
$185$	2121.76i	0.843217i
$186$	0	0
$187$	595.294	0.232793
$188$	0	0
$189$	0	0
$190$	0	0
$191$	4880.44i	1.84888i	0.381325	+	0.924441i	$0.375468\pi$
$191$	4880.44i	1.84888i	−0.381325	+	0.924441i	$0.624532\pi$
$192$	0	0
$193$	−1955.00	−0.729140	−0.364570	−	0.931176i	$-0.618784\pi$
$193$	−1955.00	−0.729140	−0.364570	+	0.931176i	$0.618784\pi$
$194$	0	0
$195$	1190.59	0.437230
$196$	0	0
$197$	3180.00	1.15008	0.575040	−	0.818126i	$-0.304987\pi$
$197$	3180.00	1.15008	0.575040	+	0.818126i	$0.304987\pi$
$198$	0	0
$199$	3346.88	1.19223	0.596115	−	0.802899i	$-0.296710\pi$
$199$	3346.88	1.19223	0.596115	+	0.802899i	$0.296710\pi$
$200$	0	0
$201$	1176.06i	0.412702i
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−390.000	−0.132872
$206$	0	0
$207$	− 1008.17i	− 0.338514i
$208$	0	0
$209$	4546.63i	1.50477i
$210$	0	0
$211$	3803.54i	1.24098i	0.784215	+	0.620489i	$0.213066\pi$
$211$	3803.54i	1.24098i	−0.784215	+	0.620489i	$0.786934\pi$
$212$	0	0
$213$	121.244i	0.0390022i
$214$	0	0
$215$	−2619.29	−0.830857
$216$	0	0
$217$	0	0
$218$	0	0
$219$	− 2543.33i	− 0.784759i
$220$	0	0
$221$	450.000	0.136970
$222$	0	0
$223$	4942.26	1.48412	0.742059	−	0.670334i	$-0.233849\pi$
$223$	4942.26	1.48412	0.742059	+	0.670334i	$0.233849\pi$
$224$	0	0
$225$	−1000.00	−0.296296
$226$	0	0
$227$	2246.24	0.656777	0.328388	−	0.944543i	$-0.393494\pi$
$227$	2246.24	0.656777	0.328388	+	0.944543i	$0.393494\pi$
$228$	0	0
$229$	− 4858.40i	− 1.40198i	−0.713174	−	0.700988i	$-0.752743\pi$
$229$	− 4858.40i	− 1.40198i	0.713174	−	0.700988i	$-0.247257\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	1365.00	0.383795	0.191897	−	0.981415i	$-0.438536\pi$
$233$	1365.00	0.383795	0.191897	+	0.981415i	$0.438536\pi$
$234$	0	0
$235$	− 3643.15i	− 1.01129i
$236$	0	0
$237$	545.596i	0.149537i
$238$	0	0
$239$	− 1283.12i	− 0.347273i	−0.984810	−	0.173636i	$-0.944448\pi$
$239$	− 1283.12i	− 0.347273i	0.984810	−	0.173636i	$-0.0555518\pi$
$240$	0	0
$241$	− 5689.79i	− 1.52079i	−0.649458	−	0.760397i	$-0.725004\pi$
$241$	− 5689.79i	− 1.52079i	0.649458	−	0.760397i	$-0.274996\pi$
$242$	0	0
$243$	−3915.71	−1.03372
$244$	0	0
$245$	0	0
$246$	0	0
$247$	3436.93i	0.885371i
$248$	0	0
$249$	−2352.00	−0.598602
$250$	0	0
$251$	−1190.59	−0.299399	−0.149700	−	0.988732i	$-0.547831\pi$
$251$	−1190.59	−0.299399	−0.149700	+	0.988732i	$0.547831\pi$
$252$	0	0
$253$	3465.00	0.861038
$254$	0	0
$255$	−198.431	−0.0487304
$256$	0	0
$257$	1203.78i	0.292177i	0.989272	+	0.146088i	$0.0466684\pi$
$257$	1203.78i	0.292177i	−0.989272	+	0.146088i	$0.953332\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	3360.00	0.796854
$262$	0	0
$263$	− 5961.93i	− 1.39783i	−0.715206	−	0.698913i	$-0.753667\pi$
$263$	− 5961.93i	− 1.39783i	0.715206	−	0.698913i	$-0.246333\pi$
$264$	0	0
$265$	2987.79i	0.692597i
$266$	0	0
$267$	− 4000.59i	− 0.916974i
$268$	0	0
$269$	3815.71i	0.864862i	0.901667	+	0.432431i	$0.142344\pi$
$269$	3815.71i	0.864862i	−0.901667	+	0.432431i	$0.857656\pi$
$270$	0	0
$271$	4828.50	1.08233	0.541163	−	0.840918i	$-0.317984\pi$
$271$	4828.50	1.08233	0.541163	+	0.840918i	$0.317984\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 3436.93i	− 0.753654i
$276$	0	0
$277$	2515.00	0.545530	0.272765	−	0.962081i	$-0.412062\pi$
$277$	2515.00	0.545530	0.272765	+	0.962081i	$0.412062\pi$
$278$	0	0
$279$	−4497.78	−0.965143
$280$	0	0
$281$	−5208.00	−1.10563	−0.552817	−	0.833303i	$-0.686447\pi$
$281$	−5208.00	−1.10563	−0.552817	+	0.833303i	$0.686447\pi$
$282$	0	0
$283$	−7098.55	−1.49104	−0.745521	−	0.666482i	$-0.767799\pi$
$283$	−7098.55	−1.49104	−0.745521	+	0.666482i	$0.767799\pi$
$284$	0	0
$285$	− 1515.54i	− 0.314993i
$286$	0	0
$287$	0	0
$288$	0	0
$289$	4838.00	0.984734
$290$	0	0
$291$	1145.64i	0.230786i
$292$	0	0
$293$	311.769i	0.0621630i	0.999517	+	0.0310815i	$0.00989514\pi$
$293$	311.769i	0.0621630i	−0.999517	+	0.0310815i	$0.990105\pi$
$294$	0	0
$295$	3780.62i	0.746158i
$296$	0	0
$297$	− 8547.67i	− 1.66999i
$298$	0	0
$299$	2619.29	0.506614
$300$	0	0
$301$	0	0
$302$	0	0
$303$	− 2204.22i	− 0.417918i
$304$	0	0
$305$	3435.00	0.644877
$306$	0	0
$307$	−391.571	−0.0727953	−0.0363976	−	0.999337i	$-0.511588\pi$
$307$	−391.571	−0.0727953	−0.0363976	+	0.999337i	$0.511588\pi$
$308$	0	0
$309$	5117.00	0.942058
$310$	0	0
$311$	2420.86	0.441397	0.220699	−	0.975342i	$-0.429166\pi$
$311$	2420.86	0.441397	0.220699	+	0.975342i	$0.429166\pi$
$312$	0	0
$313$	− 6192.08i	− 1.11820i	−0.829100	−	0.559101i	$-0.811146\pi$
$313$	− 6192.08i	− 1.11820i	0.829100	−	0.559101i	$-0.188854\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	5595.00	0.991314	0.495657	−	0.868518i	$-0.334927\pi$
$317$	5595.00	0.991314	0.495657	+	0.868518i	$0.334927\pi$
$318$	0	0
$319$	11548.1i	2.02686i
$320$	0	0
$321$	− 2097.51i	− 0.364710i
$322$	0	0
$323$	− 572.822i	− 0.0986770i
$324$	0	0
$325$	− 2598.08i	− 0.443432i
$326$	0	0
$327$	2018.71	0.341391
$328$	0	0
$329$	0	0
$330$	0	0
$331$	7675.81i	1.27463i	0.770605	+	0.637313i	$0.219954\pi$
$331$	7675.81i	1.27463i	−0.770605	+	0.637313i	$0.780046\pi$
$332$	0	0
$333$	4900.00	0.806361
$334$	0	0
$335$	−3849.57	−0.627834
$336$	0	0
$337$	−3220.00	−0.520488	−0.260244	−	0.965543i	$-0.583803\pi$
$337$	−3220.00	−0.520488	−0.260244	+	0.965543i	$0.583803\pi$
$338$	0	0
$339$	3968.63	0.635830
$340$	0	0
$341$	− 15458.6i	− 2.45492i
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−1155.00	−0.180241
$346$	0	0
$347$	6997.59i	1.08257i	0.840840	+	0.541283i	$0.182061\pi$
$347$	6997.59i	1.08257i	−0.840840	+	0.541283i	$0.817939\pi$
$348$	0	0
$349$	− 4021.82i	− 0.616857i	−0.951247	−	0.308429i	$-0.900197\pi$
$349$	− 4021.82i	− 0.616857i	0.951247	−	0.308429i	$-0.0998031\pi$
$350$	0	0
$351$	− 6461.43i	− 0.982580i
$352$	0	0
$353$	− 6434.57i	− 0.970191i	−0.874461	−	0.485096i	$-0.838785\pi$
$353$	− 6434.57i	− 0.970191i	0.874461	−	0.485096i	$-0.161215\pi$
$354$	0	0
$355$	−396.863	−0.0593332
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 10883.6i	− 1.60004i	−0.599971	−	0.800021i	$-0.704821\pi$
$359$	− 10883.6i	− 1.60004i	0.599971	−	0.800021i	$-0.295179\pi$
$360$	0	0
$361$	−2484.00	−0.362152
$362$	0	0
$363$	8979.68	1.29838
$364$	0	0
$365$	8325.00	1.19384
$366$	0	0
$367$	−3368.04	−0.479047	−0.239524	−	0.970891i	$-0.576991\pi$
$367$	−3368.04	−0.479047	−0.239524	+	0.970891i	$0.576991\pi$
$368$	0	0
$369$	900.666i	0.127065i
$370$	0	0
$371$	0	0
$372$	0	0
$373$	4835.00	0.671171	0.335586	−	0.942010i	$-0.391066\pi$
$373$	4835.00	0.671171	0.335586	+	0.942010i	$0.391066\pi$
$374$	0	0
$375$	4009.75i	0.552167i
$376$	0	0
$377$	8729.54i	1.19256i
$378$	0	0
$379$	3941.02i	0.534133i	0.963678	+	0.267066i	$0.0860543\pi$
$379$	3941.02i	0.534133i	−0.963678	+	0.267066i	$0.913946\pi$
$380$	0	0
$381$	− 121.244i	− 0.0163031i
$382$	0	0
$383$	−8469.05	−1.12989	−0.564945	−	0.825128i	$-0.691103\pi$
$383$	−8469.05	−1.12989	−0.564945	+	0.825128i	$0.691103\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	6049.00i	0.794542i
$388$	0	0
$389$	−6951.00	−0.905989	−0.452994	−	0.891513i	$-0.649644\pi$
$389$	−6951.00	−0.905989	−0.452994	+	0.891513i	$0.649644\pi$
$390$	0	0
$391$	−436.549	−0.0564635
$392$	0	0
$393$	−5355.00	−0.687338
$394$	0	0
$395$	−1785.88	−0.227487
$396$	0	0
$397$	− 4477.35i	− 0.566025i	−0.959116	−	0.283012i	$-0.908666\pi$
$397$	− 4477.35i	− 0.566025i	0.959116	−	0.283012i	$-0.0913337\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	6909.00	0.860396	0.430198	−	0.902734i	$-0.358444\pi$
$401$	6909.00	0.860396	0.430198	+	0.902734i	$0.358444\pi$
$402$	0	0
$403$	− 11685.6i	− 1.44442i
$404$	0	0
$405$	− 1827.31i	− 0.224197i
$406$	0	0
$407$	16841.0i	2.05105i
$408$	0	0
$409$	− 6725.55i	− 0.813098i	−0.913629	−	0.406549i	$-0.866732\pi$
$409$	− 6725.55i	− 0.813098i	0.913629	−	0.406549i	$-0.133268\pi$
$410$	0	0
$411$	833.412	0.100022
$412$	0	0
$413$	0	0
$414$	0	0
$415$	− 7698.73i	− 0.910640i
$416$	0	0
$417$	6020.00	0.706956
$418$	0	0
$419$	−6508.55	−0.758863	−0.379431	−	0.925220i	$-0.623880\pi$
$419$	−6508.55	−0.758863	−0.379431	+	0.925220i	$0.623880\pi$
$420$	0	0
$421$	7576.00	0.877035	0.438517	−	0.898723i	$-0.355504\pi$
$421$	7576.00	0.877035	0.438517	+	0.898723i	$0.355504\pi$
$422$	0	0
$423$	−8413.49	−0.967087
$424$	0	0
$425$	433.013i	0.0494217i
$426$	0	0
$427$	0	0
$428$	0	0
$429$	9450.00	1.06352
$430$	0	0
$431$	− 16382.7i	− 1.83092i	−0.402407	−	0.915461i	$-0.631826\pi$
$431$	− 16382.7i	− 1.83092i	0.402407	−	0.915461i	$-0.368174\pi$
$432$	0	0
$433$	15952.2i	1.77047i	0.465145	+	0.885235i	$0.346002\pi$
$433$	15952.2i	1.77047i	−0.465145	+	0.885235i	$0.653998\pi$
$434$	0	0
$435$	− 3849.36i	− 0.424282i
$436$	0	0
$437$	− 3334.20i	− 0.364980i
$438$	0	0
$439$	−383.634	−0.0417081	−0.0208540	−	0.999783i	$-0.506639\pi$
$439$	−383.634	−0.0417081	−0.0208540	+	0.999783i	$0.506639\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	10856.1i	1.16431i	0.813077	+	0.582156i	$0.197791\pi$
$443$	10856.1i	1.16431i	−0.813077	+	0.582156i	$0.802209\pi$
$444$	0	0
$445$	13095.0	1.39497
$446$	0	0
$447$	4135.31	0.437569
$448$	0	0
$449$	−5868.00	−0.616766	−0.308383	−	0.951262i	$-0.599788\pi$
$449$	−5868.00	−0.616766	−0.308383	+	0.951262i	$0.599788\pi$
$450$	0	0
$451$	−3095.53	−0.323199
$452$	0	0
$453$	− 788.083i	− 0.0817381i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	6935.00	0.709859	0.354930	−	0.934893i	$-0.384505\pi$
$457$	6935.00	0.709859	0.354930	+	0.934893i	$0.384505\pi$
$458$	0	0
$459$	1076.91i	0.109511i
$460$	0	0
$461$	2961.81i	0.299230i	0.988744	+	0.149615i	$0.0478034\pi$
$461$	2961.81i	0.299230i	−0.988744	+	0.149615i	$0.952197\pi$
$462$	0	0
$463$	− 11154.0i	− 1.11959i	−0.828631	−	0.559795i	$-0.810880\pi$
$463$	− 11154.0i	− 1.11959i	0.828631	−	0.559795i	$-0.189120\pi$
$464$	0	0
$465$	5152.85i	0.513888i
$466$	0	0
$467$	−8643.67	−0.856491	−0.428246	−	0.903662i	$-0.640868\pi$
$467$	−8643.67	−0.856491	−0.428246	+	0.903662i	$0.640868\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	− 1901.77i	− 0.186049i
$472$	0	0
$473$	−20790.0	−2.02098
$474$	0	0
$475$	−3307.19	−0.319462
$476$	0	0
$477$	6900.00	0.662326
$478$	0	0
$479$	16231.7	1.54832	0.774159	−	0.632991i	$-0.218173\pi$
$479$	16231.7	1.54832	0.774159	+	0.632991i	$0.218173\pi$
$480$	0	0
$481$	12730.6i	1.20679i
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−3750.00	−0.351090
$486$	0	0
$487$	3826.45i	0.356043i	0.984027	+	0.178022i	$0.0569697\pi$
$487$	3826.45i	0.356043i	−0.984027	+	0.178022i	$0.943030\pi$
$488$	0	0
$489$	5637.83i	0.521373i
$490$	0	0
$491$	8890.20i	0.817126i	0.912730	+	0.408563i	$0.133970\pi$
$491$	8890.20i	0.817126i	−0.912730	+	0.408563i	$0.866030\pi$
$492$	0	0
$493$	− 1454.92i	− 0.132914i
$494$	0	0
$495$	11905.9	1.08107
$496$	0	0
$497$	0	0
$498$	0	0
$499$	− 3047.41i	− 0.273389i	−0.990613	−	0.136694i	$-0.956352\pi$
$499$	− 3047.41i	− 0.273389i	0.990613	−	0.136694i	$-0.0436478\pi$
$500$	0	0
$501$	1974.00	0.176032
$502$	0	0
$503$	7715.01	0.683887	0.341944	−	0.939720i	$-0.388915\pi$
$503$	7715.01	0.683887	0.341944	+	0.939720i	$0.388915\pi$
$504$	0	0
$505$	7215.00	0.635769
$506$	0	0
$507$	1330.81	0.116575
$508$	0	0
$509$	− 16068.2i	− 1.39924i	−0.714516	−	0.699619i	$-0.753353\pi$
$509$	− 16068.2i	− 1.39924i	0.714516	−	0.699619i	$-0.246647\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−8225.00	−0.707880
$514$	0	0
$515$	16749.3i	1.43313i
$516$	0	0
$517$	− 28916.6i	− 2.45986i
$518$	0	0
$519$	− 2864.11i	− 0.242236i
$520$	0	0
$521$	13328.1i	1.12076i	0.828236	+	0.560380i	$0.189345\pi$
$521$	13328.1i	1.12076i	−0.828236	+	0.560380i	$0.810655\pi$
$522$	0	0
$523$	−775.205	−0.0648133	−0.0324066	−	0.999475i	$-0.510317\pi$
$523$	−775.205	−0.0648133	−0.0324066	+	0.999475i	$0.510317\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	1947.59i	0.160984i
$528$	0	0
$529$	9626.00	0.791156
$530$	0	0
$531$	8730.98	0.713545
$532$	0	0
$533$	−2340.00	−0.190163
$534$	0	0
$535$	6865.72	0.554825
$536$	0	0
$537$	− 3940.42i	− 0.316651i
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−359.000	−0.0285298	−0.0142649	−	0.999898i	$-0.504541\pi$
$541$	−359.000	−0.0285298	−0.0142649	+	0.999898i	$0.504541\pi$
$542$	0	0
$543$	7497.09i	0.592507i
$544$	0	0
$545$	6607.77i	0.519350i
$546$	0	0
$547$	− 24984.2i	− 1.95292i	−0.215698	−	0.976460i	$-0.569203\pi$
$547$	− 24984.2i	− 1.95292i	0.215698	−	0.976460i	$-0.430797\pi$
$548$	0	0
$549$	− 7932.79i	− 0.616691i
$550$	0	0
$551$	11112.2	0.859154
$552$	0	0
$553$	0	0
$554$	0	0
$555$	− 5613.66i	− 0.429345i
$556$	0	0
$557$	−14775.0	−1.12394	−0.561972	−	0.827156i	$-0.689957\pi$
$557$	−14775.0	−1.12394	−0.561972	+	0.827156i	$0.689957\pi$
$558$	0	0
$559$	−15715.8	−1.18910
$560$	0	0
$561$	−1575.00	−0.118532
$562$	0	0
$563$	2674.85	0.200234	0.100117	−	0.994976i	$-0.468078\pi$
$563$	2674.85	0.200234	0.100117	+	0.994976i	$0.468078\pi$
$564$	0	0
$565$	12990.4i	0.967273i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−24381.0	−1.79632	−0.898159	−	0.439671i	$-0.855095\pi$
$569$	−24381.0	−1.79632	−0.898159	+	0.439671i	$0.855095\pi$
$570$	0	0
$571$	− 4926.27i	− 0.361047i	−0.983571	−	0.180524i	$-0.942221\pi$
$571$	− 4926.27i	− 0.361047i	0.983571	−	0.180524i	$-0.0577792\pi$
$572$	0	0
$573$	− 12912.4i	− 0.941405i
$574$	0	0
$575$	2520.42i	0.182798i
$576$	0	0
$577$	1099.85i	0.0793543i	0.999213	+	0.0396772i	$0.0126330\pi$
$577$	1099.85i	0.0793543i	−0.999213	+	0.0396772i	$0.987367\pi$
$578$	0	0
$579$	5172.44	0.371260
$580$	0	0
$581$	0	0
$582$	0	0
$583$	23714.8i	1.68468i
$584$	0	0
$585$	9000.00	0.636076
$586$	0	0
$587$	−10064.4	−0.707673	−0.353836	−	0.935307i	$-0.615123\pi$
$587$	−10064.4	−0.707673	−0.353836	+	0.935307i	$0.615123\pi$
$588$	0	0
$589$	−14875.0	−1.04060
$590$	0	0
$591$	−8413.49	−0.585592
$592$	0	0
$593$	20498.8i	1.41954i	0.704435	+	0.709769i	$0.251201\pi$
$593$	20498.8i	1.41954i	−0.704435	+	0.709769i	$0.748799\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−8855.00	−0.607054
$598$	0	0
$599$	− 12120.9i	− 0.826790i	−0.910552	−	0.413395i	$-0.864343\pi$
$599$	− 12120.9i	− 0.826790i	0.910552	−	0.413395i	$-0.135657\pi$
$600$	0	0
$601$	2476.83i	0.168107i	0.996461	+	0.0840533i	$0.0267866\pi$
$601$	2476.83i	0.168107i	−0.996461	+	0.0840533i	$0.973213\pi$
$602$	0	0
$603$	8890.20i	0.600393i
$604$	0	0
$605$	29392.9i	1.97519i
$606$	0	0
$607$	−5177.74	−0.346224	−0.173112	−	0.984902i	$-0.555382\pi$
$607$	−5177.74	−0.346224	−0.173112	+	0.984902i	$0.555382\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 21858.9i	− 1.44733i
$612$	0	0
$613$	12845.0	0.846337	0.423169	−	0.906051i	$-0.360918\pi$
$613$	12845.0	0.846337	0.423169	+	0.906051i	$0.360918\pi$
$614$	0	0
$615$	1031.84	0.0676552
$616$	0	0
$617$	−24360.0	−1.58946	−0.794730	−	0.606963i	$-0.792387\pi$
$617$	−24360.0	−1.58946	−0.794730	+	0.606963i	$0.792387\pi$
$618$	0	0
$619$	463.006	0.0300643	0.0150321	−	0.999887i	$-0.495215\pi$
$619$	463.006	0.0300643	0.0150321	+	0.999887i	$0.495215\pi$
$620$	0	0
$621$	6268.29i	0.405053i
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−6875.00	−0.440000
$626$	0	0
$627$	− 12029.3i	− 0.766192i
$628$	0	0
$629$	− 2121.76i	− 0.134500i
$630$	0	0
$631$	− 2337.11i	− 0.147447i	−0.997279	−	0.0737235i	$-0.976512\pi$
$631$	− 2337.11i	− 0.147447i	0.997279	−	0.0737235i	$-0.0234882\pi$
$632$	0	0
$633$	− 10063.2i	− 0.631875i
$634$	0	0
$635$	396.863	0.0248016
$636$	0	0
$637$	0	0
$638$	0	0
$639$	916.515i	0.0567399i
$640$	0	0
$641$	−11109.0	−0.684523	−0.342261	−	0.939605i	$-0.611193\pi$
$641$	−11109.0	−0.684523	−0.342261	+	0.939605i	$0.611193\pi$
$642$	0	0
$643$	−2772.75	−0.170057	−0.0850284	−	0.996379i	$-0.527098\pi$
$643$	−2772.75	−0.170057	−0.0850284	+	0.996379i	$0.527098\pi$
$644$	0	0
$645$	6930.00	0.423052
$646$	0	0
$647$	−16184.1	−0.983402	−0.491701	−	0.870764i	$-0.663625\pi$
$647$	−16184.1	−0.983402	−0.491701	+	0.870764i	$0.663625\pi$
$648$	0	0
$649$	30007.8i	1.81496i
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−15225.0	−0.912405	−0.456202	−	0.889876i	$-0.650791\pi$
$653$	−15225.0	−0.912405	−0.456202	+	0.889876i	$0.650791\pi$
$654$	0	0
$655$	− 17528.4i	− 1.04563i
$656$	0	0
$657$	− 19225.8i	− 1.14166i
$658$	0	0
$659$	− 1328.95i	− 0.0785560i	−0.999228	−	0.0392780i	$-0.987494\pi$
$659$	− 1328.95i	− 0.0785560i	0.999228	−	0.0392780i	$-0.0125058\pi$
$660$	0	0
$661$	6635.49i	0.390455i	0.980758	+	0.195227i	$0.0625445\pi$
$661$	6635.49i	0.390455i	−0.980758	+	0.195227i	$0.937456\pi$
$662$	0	0
$663$	−1190.59	−0.0697415
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 8468.60i	− 0.491612i
$668$	0	0
$669$	−13076.0	−0.755676
$670$	0	0
$671$	27264.5	1.56860
$672$	0	0
$673$	−21980.0	−1.25894	−0.629470	−	0.777025i	$-0.716728\pi$
$673$	−21980.0	−1.25894	−0.629470	+	0.777025i	$0.716728\pi$
$674$	0	0
$675$	6217.52	0.354537
$676$	0	0
$677$	961.288i	0.0545721i	0.999628	+	0.0272860i	$0.00868650\pi$
$677$	961.288i	0.0545721i	−0.999628	+	0.0272860i	$0.991314\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−5943.00	−0.334414
$682$	0	0
$683$	3524.00i	0.197426i	0.995116	+	0.0987131i	$0.0314726\pi$
$683$	3524.00i	0.197426i	−0.995116	+	0.0987131i	$0.968527\pi$
$684$	0	0
$685$	2727.98i	0.152162i
$686$	0	0
$687$	12854.1i	0.713851i
$688$	0	0
$689$	17926.7i	0.991225i
$690$	0	0
$691$	−15490.9	−0.852823	−0.426411	−	0.904529i	$-0.640222\pi$
$691$	−15490.9	−0.852823	−0.426411	+	0.904529i	$0.640222\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	19705.1i	1.07548i
$696$	0	0
$697$	390.000	0.0211941
$698$	0	0
$699$	−3611.45	−0.195419
$700$	0	0
$701$	22974.0	1.23783	0.618913	−	0.785460i	$-0.287573\pi$
$701$	22974.0	1.23783	0.618913	+	0.785460i	$0.287573\pi$
$702$	0	0
$703$	16205.2	0.869405
$704$	0	0
$705$	9638.86i	0.514923i
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−10297.0	−0.545433	−0.272716	−	0.962094i	$-0.587922\pi$
$709$	−10297.0	−0.545433	−0.272716	+	0.962094i	$0.587922\pi$
$710$	0	0
$711$	4124.32i	0.217544i
$712$	0	0
$713$	11336.3i	0.595437i
$714$	0	0
$715$	30932.4i	1.61791i
$716$	0	0
$717$	3394.82i	0.176823i
$718$	0	0
$719$	−21311.5	−1.10540	−0.552702	−	0.833379i	$-0.686403\pi$
$719$	−21311.5	−1.10540	−0.552702	+	0.833379i	$0.686403\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	15053.8i	0.774350i
$724$	0	0
$725$	−8400.00	−0.430301
$726$	0	0
$727$	10773.5	0.549611	0.274805	−	0.961500i	$-0.411387\pi$
$727$	10773.5	0.549611	0.274805	+	0.961500i	$0.411387\pi$
$728$	0	0
$729$	4663.00	0.236905
$730$	0	0
$731$	2619.29	0.132528
$732$	0	0
$733$	5932.27i	0.298927i	0.988767	+	0.149464i	$0.0477547\pi$
$733$	5932.27i	0.298927i	−0.988767	+	0.149464i	$0.952245\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−30555.0	−1.52715
$738$	0	0
$739$	31734.3i	1.57966i	0.613328	+	0.789828i	$0.289830\pi$
$739$	31734.3i	1.57966i	−0.613328	+	0.789828i	$0.710170\pi$
$740$	0	0
$741$	− 9093.27i	− 0.450809i
$742$	0	0
$743$	− 11575.6i	− 0.571557i	−0.958296	−	0.285779i	$-0.907748\pi$
$743$	− 11575.6i	− 0.571557i	0.958296	−	0.285779i	$-0.0922522\pi$
$744$	0	0
$745$	13536.0i	0.665664i
$746$	0	0
$747$	−17779.4	−0.870838
$748$	0	0
$749$	0	0
$750$	0	0
$751$	− 30130.4i	− 1.46401i	−0.681297	−	0.732007i	$-0.738584\pi$
$751$	− 30130.4i	− 1.46401i	0.681297	−	0.732007i	$-0.261416\pi$
$752$	0	0
$753$	3150.00	0.152447
$754$	0	0
$755$	2579.61	0.124346
$756$	0	0
$757$	11900.0	0.571351	0.285676	−	0.958326i	$-0.407782\pi$
$757$	11900.0	0.571351	0.285676	+	0.958326i	$0.407782\pi$
$758$	0	0
$759$	−9167.53	−0.438419
$760$	0	0
$761$	− 11752.0i	− 0.559801i	−0.960029	−	0.279900i	$-0.909699\pi$
$761$	− 11752.0i	− 0.559801i	0.960029	−	0.279900i	$-0.0903014\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−1500.00	−0.0708923
$766$	0	0
$767$	22683.7i	1.06788i
$768$	0	0
$769$	− 23573.2i	− 1.10542i	−0.833372	−	0.552712i	$-0.813593\pi$
$769$	− 23573.2i	− 1.10542i	0.833372	−	0.552712i	$-0.186407\pi$
$770$	0	0
$771$	− 3184.89i	− 0.148769i
$772$	0	0
$773$	38668.0i	1.79921i	0.436700	+	0.899607i	$0.356147\pi$
$773$	38668.0i	1.79921i	−0.436700	+	0.899607i	$0.643853\pi$
$774$	0	0
$775$	11244.4	0.521177
$776$	0	0
$777$	0	0
$778$	0	0
$779$	2978.67i	0.136999i
$780$	0	0
$781$	−3150.00	−0.144322
$782$	0	0
$783$	−20890.9	−0.953484
$784$	0	0
$785$	6225.00	0.283032
$786$	0	0
$787$	−10680.9	−0.483778	−0.241889	−	0.970304i	$-0.577767\pi$
$787$	−10680.9	−0.483778	−0.241889	+	0.970304i	$0.577767\pi$
$788$	0	0
$789$	15773.8i	0.711739i
$790$	0	0
$791$	0	0
$792$	0	0
$793$	20610.0	0.922929
$794$	0	0
$795$	− 7904.94i	− 0.352653i
$796$	0	0
$797$	24196.7i	1.07540i	0.843137	+	0.537699i	$0.180706\pi$
$797$	24196.7i	1.07540i	−0.843137	+	0.537699i	$0.819294\pi$
$798$	0	0
$799$	3643.15i	0.161308i
$800$	0	0
$801$	− 30241.6i	− 1.33400i
$802$	0	0
$803$	66077.6	2.90390
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 10095.4i	− 0.440366i
$808$	0	0
$809$	−18597.0	−0.808202	−0.404101	−	0.914714i	$-0.632416\pi$
$809$	−18597.0	−0.808202	−0.404101	+	0.914714i	$0.632416\pi$
$810$	0	0
$811$	−4339.03	−0.187872	−0.0939359	−	0.995578i	$-0.529945\pi$
$811$	−4339.03	−0.187872	−0.0939359	+	0.995578i	$0.529945\pi$
$812$	0	0
$813$	−12775.0	−0.551093
$814$	0	0
$815$	−18454.1	−0.793153
$816$	0	0
$817$	20005.2i	0.856662i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	4713.00	0.200347	0.100173	−	0.994970i	$-0.468060\pi$
$821$	4713.00	0.200347	0.100173	+	0.994970i	$0.468060\pi$
$822$	0	0
$823$	14018.1i	0.593731i	0.954919	+	0.296865i	$0.0959412\pi$
$823$	14018.1i	0.593731i	−0.954919	+	0.296865i	$0.904059\pi$
$824$	0	0
$825$	9093.27i	0.383742i
$826$	0	0
$827$	− 35111.7i	− 1.47636i	−0.674601	−	0.738182i	$-0.735684\pi$
$827$	− 35111.7i	− 1.47636i	0.674601	−	0.738182i	$-0.264316\pi$
$828$	0	0
$829$	− 9933.31i	− 0.416162i	−0.978112	−	0.208081i	$-0.933278\pi$
$829$	− 9933.31i	− 0.416162i	0.978112	−	0.208081i	$-0.0667217\pi$
$830$	0	0
$831$	−6654.06	−0.277770
$832$	0	0
$833$	0	0
$834$	0	0
$835$	6461.43i	0.267793i
$836$	0	0
$837$	27965.0	1.15485
$838$	0	0
$839$	−24605.5	−1.01249	−0.506243	−	0.862391i	$-0.668966\pi$
$839$	−24605.5	−1.01249	−0.506243	+	0.862391i	$0.668966\pi$
$840$	0	0
$841$	3835.00	0.157243
$842$	0	0
$843$	13779.1	0.562961
$844$	0	0
$845$	4356.11i	0.177343i
$846$	0	0
$847$	0	0
$848$	0	0
$849$	18781.0	0.759202
$850$	0	0
$851$	− 12350.0i	− 0.497478i
$852$	0	0
$853$	− 13769.8i	− 0.552719i	−0.961054	−	0.276359i	$-0.910872\pi$
$853$	− 13769.8i	− 0.552719i	0.961054	−	0.276359i	$-0.0891280\pi$
$854$	0	0
$855$	− 11456.4i	− 0.458248i
$856$	0	0
$857$	39170.3i	1.56130i	0.624969	+	0.780649i	$0.285111\pi$
$857$	39170.3i	1.56130i	−0.624969	+	0.780649i	$0.714889\pi$
$858$	0	0
$859$	11046.0	0.438749	0.219374	−	0.975641i	$-0.429598\pi$
$859$	11046.0	0.438749	0.219374	+	0.975641i	$0.429598\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 5540.33i	− 0.218534i	−0.994012	−	0.109267i	$-0.965150\pi$
$863$	− 5540.33i	− 0.218534i	0.994012	−	0.109267i	$-0.0348504\pi$
$864$	0	0
$865$	9375.00	0.368508
$866$	0	0
$867$	−12800.1	−0.501402
$868$	0	0
$869$	−14175.0	−0.553342
$870$	0	0
$871$	−23097.4	−0.898537
$872$	0	0
$873$	8660.25i	0.335745i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−16555.0	−0.637426	−0.318713	−	0.947851i	$-0.603251\pi$
$877$	−16555.0	−0.637426	−0.318713	+	0.947851i	$0.603251\pi$
$878$	0	0
$879$	− 824.864i	− 0.0316518i
$880$	0	0
$881$	8701.82i	0.332772i	0.986061	+	0.166386i	$0.0532097\pi$
$881$	8701.82i	0.332772i	−0.986061	+	0.166386i	$0.946790\pi$
$882$	0	0
$883$	24241.8i	0.923899i	0.886906	+	0.461949i	$0.152850\pi$
$883$	24241.8i	0.923899i	−0.886906	+	0.461949i	$0.847150\pi$
$884$	0	0
$885$	− 10002.6i	− 0.379925i
$886$	0	0
$887$	−16819.0	−0.636672	−0.318336	−	0.947978i	$-0.603124\pi$
$887$	−16819.0	−0.636672	−0.318336	+	0.947978i	$0.603124\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	− 14503.9i	− 0.545339i
$892$	0	0
$893$	−27825.0	−1.04270
$894$	0	0
$895$	12898.0	0.481714
$896$	0	0
$897$	−6930.00	−0.257955
$898$	0	0
$899$	−37781.3	−1.40164
$900$	0	0
$901$	− 2987.79i	− 0.110475i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−24540.0	−0.901367
$906$	0	0
$907$	− 23522.4i	− 0.861133i	−0.902559	−	0.430566i	$-0.858314\pi$
$907$	− 23522.4i	− 0.861133i	0.902559	−	0.430566i	$-0.141686\pi$
$908$	0	0
$909$	− 16662.3i	− 0.607981i
$910$	0	0
$911$	− 7836.20i	− 0.284989i	−0.989796	−	0.142495i	$-0.954488\pi$
$911$	− 7836.20i	− 0.284989i	0.989796	−	0.142495i	$-0.0455123\pi$
$912$	0	0
$913$	− 61106.8i	− 2.21505i
$914$	0	0
$915$	−9088.16	−0.328355
$916$	0	0
$917$	0	0
$918$	0	0
$919$	3597.32i	0.129124i	0.997914	+	0.0645619i	$0.0205650\pi$
$919$	3597.32i	0.129124i	−0.997914	+	0.0645619i	$0.979435\pi$
$920$	0	0
$921$	1036.00	0.0370655
$922$	0	0
$923$	−2381.18	−0.0849159
$924$	0	0
$925$	−12250.0	−0.435435
$926$	0	0
$927$	38680.9	1.37049
$928$	0	0
$929$	− 43833.0i	− 1.54802i	−0.633171	−	0.774012i	$-0.718247\pi$
$929$	− 43833.0i	− 1.54802i	0.633171	−	0.774012i	$-0.281753\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−6405.00	−0.224748
$934$	0	0
$935$	− 5155.40i	− 0.180320i
$936$	0	0
$937$	13995.0i	0.487936i	0.969783	+	0.243968i	$0.0784491\pi$
$937$	13995.0i	0.487936i	−0.969783	+	0.243968i	$0.921551\pi$
$938$	0	0
$939$	16382.7i	0.569360i
$940$	0	0
$941$	11041.8i	0.382522i	0.981539	+	0.191261i	$0.0612577\pi$
$941$	11041.8i	0.382522i	−0.981539	+	0.191261i	$0.938742\pi$
$942$	0	0
$943$	2270.05	0.0783915
$944$	0	0
$945$	0	0
$946$	0	0
$947$	36876.0i	1.26537i	0.774408	+	0.632687i	$0.218048\pi$
$947$	36876.0i	1.26537i	−0.774408	+	0.632687i	$0.781952\pi$
$948$	0	0
$949$	49950.0	1.70858
$950$	0	0
$951$	−14803.0	−0.504752
$952$	0	0
$953$	−16980.0	−0.577163	−0.288581	−	0.957455i	$-0.593184\pi$
$953$	−16980.0	−0.577163	−0.288581	+	0.957455i	$0.593184\pi$
$954$	0	0
$955$	42265.9	1.43214
$956$	0	0
$957$	− 30553.4i	− 1.03203i
$958$	0	0
$959$	0	0
$960$	0	0
$961$	20784.0	0.697660
$962$	0	0
$963$	− 15855.7i	− 0.530574i
$964$	0	0
$965$	16930.8i	0.564789i
$966$	0	0
$967$	7982.85i	0.265472i	0.991151	+	0.132736i	$0.0423762\pi$
$967$	7982.85i	0.265472i	−0.991151	+	0.132736i	$0.957624\pi$
$968$	0	0
$969$	1515.54i	0.0502439i
$970$	0	0
$971$	−29725.0	−0.982411	−0.491206	−	0.871044i	$-0.663444\pi$
$971$	−29725.0	−0.982411	−0.491206	+	0.871044i	$0.663444\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	6873.86i	0.225784i
$976$	0	0
$977$	−43155.0	−1.41315	−0.706577	−	0.707637i	$-0.749761\pi$
$977$	−43155.0	−1.41315	−0.706577	+	0.707637i	$0.749761\pi$
$978$	0	0
$979$	103938.	3.39314
$980$	0	0
$981$	15260.0	0.496651
$982$	0	0
$983$	49552.3	1.60780	0.803902	−	0.594761i	$-0.202753\pi$
$983$	49552.3	1.60780	0.803902	+	0.594761i	$0.202753\pi$
$984$	0	0
$985$	− 27539.6i	− 0.890848i
$986$	0	0
$987$	0	0
$988$	0	0
$989$	15246.0	0.490187
$990$	0	0
$991$	4559.66i	0.146158i	0.997326	+	0.0730789i	$0.0232825\pi$
$991$	4559.66i	0.146158i	−0.997326	+	0.0730789i	$0.976717\pi$
$992$	0	0
$993$	− 20308.3i	− 0.649007i
$994$	0	0
$995$	− 28984.8i	− 0.923497i
$996$	0	0
$997$	− 41716.4i	− 1.32515i	−0.748996	−	0.662574i	$-0.769464\pi$
$997$	− 41716.4i	− 1.32515i	0.748996	−	0.662574i	$-0.230536\pi$
$998$	0	0
$999$	−30465.8	−0.964861

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	784.4.f.f.783.1		4
4.3	odd	2	inner	784.4.f.f.783.3		4
7.2	even	3		112.4.p.e.31.2	yes	4
7.3	odd	6		112.4.p.e.47.1	yes	4
7.6	odd	2	inner	784.4.f.f.783.4		4
28.3	even	6		112.4.p.e.47.2	yes	4
28.23	odd	6		112.4.p.e.31.1	✓	4
28.27	even	2	inner	784.4.f.f.783.2		4
56.3	even	6		448.4.p.e.383.1		4
56.37	even	6		448.4.p.e.255.1		4
56.45	odd	6		448.4.p.e.383.2		4
56.51	odd	6		448.4.p.e.255.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
112.4.p.e.31.1	✓	4	28.23	odd	6
112.4.p.e.31.2	yes	4	7.2	even	3
112.4.p.e.47.1	yes	4	7.3	odd	6
112.4.p.e.47.2	yes	4	28.3	even	6
448.4.p.e.255.1		4	56.37	even	6
448.4.p.e.255.2		4	56.51	odd	6
448.4.p.e.383.1		4	56.3	even	6
448.4.p.e.383.2		4	56.45	odd	6
784.4.f.f.783.1		4	1.1	even	1	trivial
784.4.f.f.783.2		4	28.27	even	2	inner
784.4.f.f.783.3		4	4.3	odd	2	inner
784.4.f.f.783.4		4	7.6	odd	2	inner

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:	\( N \)	\(=\)	\( 784 = 2^{4} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	784.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.64575 q^{3} -8.66025i q^{5} -20.0000 q^{9} +O(q^{10})\)	\(q-2.64575 q^{3} -8.66025i q^{5} -20.0000 q^{9} -68.7386i q^{11} -51.9615i q^{13} +22.9129i q^{15} +8.66025i q^{17} -66.1438 q^{19} +50.4083i q^{23} +50.0000 q^{25} +124.350 q^{27} -168.000 q^{29} +224.889 q^{31} +181.865i q^{33} -245.000 q^{37} +137.477i q^{39} -45.0333i q^{41} -302.450i q^{43} +173.205i q^{45} +420.674 q^{47} -22.9129i q^{51} -345.000 q^{53} -595.294 q^{55} +175.000 q^{57} -436.549 q^{59} +396.640i q^{61} -450.000 q^{65} -444.510i q^{67} -133.368i q^{69} -45.8258i q^{71} +961.288i q^{73} -132.288 q^{75} -206.216i q^{79} +211.000 q^{81} +888.972 q^{83} +75.0000 q^{85} +444.486 q^{87} +1512.08i q^{89} -595.000 q^{93} +572.822i q^{95} -433.013i q^{97} +1374.77i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 80 q^{9}+O(q^{10}) \) 4 * q - 80 * q^9	\( 4 q - 80 q^{9} + 200 q^{25} - 672 q^{29} - 980 q^{37} - 1380 q^{53} + 700 q^{57} - 1800 q^{65} + 844 q^{81} + 300 q^{85} - 2380 q^{93}+O(q^{100}) \) 4 * q - 80 * q^9 + 200 * q^25 - 672 * q^29 - 980 * q^37 - 1380 * q^53 + 700 * q^57 - 1800 * q^65 + 844 * q^81 + 300 * q^85 - 2380 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more