LMFDB - Embedded newform 784.3.c.a.97.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [784,3,Mod(97,784)]

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([0, 0, 1]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("784.97");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 784 = 2^{4} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	784.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:	$21.3624527258$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 28)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			97.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	784.97
Dual form			784.3.c.a.97.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.73205i q^{3} -1.73205i q^{5} +6.00000 q^{9} +O(q^{10})$	$q-1.73205i q^{3} -1.73205i q^{5} +6.00000 q^{9} -15.0000 q^{11} +13.8564i q^{13} -3.00000 q^{15} +29.4449i q^{17} +15.5885i q^{19} +9.00000 q^{23} +22.0000 q^{25} -25.9808i q^{27} -6.00000 q^{29} +12.1244i q^{31} +25.9808i q^{33} +31.0000 q^{37} +24.0000 q^{39} -55.4256i q^{41} -10.0000 q^{43} -10.3923i q^{45} +43.3013i q^{47} +51.0000 q^{51} -57.0000 q^{53} +25.9808i q^{55} +27.0000 q^{57} +81.4064i q^{59} +81.4064i q^{61} +24.0000 q^{65} +49.0000 q^{67} -15.5885i q^{69} +126.000 q^{71} -25.9808i q^{73} -38.1051i q^{75} +73.0000 q^{79} +9.00000 q^{81} +13.8564i q^{83} +51.0000 q^{85} +10.3923i q^{87} -57.1577i q^{89} +21.0000 q^{93} +27.0000 q^{95} +27.7128i q^{97} -90.0000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 12 q^{9}+O(q^{10}) $ 2 * q + 12 * q^9	$ 2 q + 12 q^{9} - 30 q^{11} - 6 q^{15} + 18 q^{23} + 44 q^{25} - 12 q^{29} + 62 q^{37} + 48 q^{39} - 20 q^{43} + 102 q^{51} - 114 q^{53} + 54 q^{57} + 48 q^{65} + 98 q^{67} + 252 q^{71} + 146 q^{79} + 18 q^{81} + 102 q^{85} + 42 q^{93} + 54 q^{95} - 180 q^{99}+O(q^{100}) $ 2 * q + 12 * q^9 - 30 * q^11 - 6 * q^15 + 18 * q^23 + 44 * q^25 - 12 * q^29 + 62 * q^37 + 48 * q^39 - 20 * q^43 + 102 * q^51 - 114 * q^53 + 54 * q^57 + 48 * q^65 + 98 * q^67 + 252 * q^71 + 146 * q^79 + 18 * q^81 + 102 * q^85 + 42 * q^93 + 54 * q^95 - 180 * q^99

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
$3$	− 1.73205i	− 0.577350i	−0.957427	−	0.288675i	$-0.906785\pi$
$3$	− 1.73205i	− 0.577350i	0.957427	−	0.288675i	$-0.0932147\pi$
$4$	0	0
$5$	− 1.73205i	− 0.346410i	−0.984886	−	0.173205i	$-0.944588\pi$
$5$	− 1.73205i	− 0.346410i	0.984886	−	0.173205i	$-0.0554123\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	6.00000	0.666667
$10$	0	0
$11$	−15.0000	−1.36364	−0.681818	−	0.731522i	$-0.738810\pi$
$11$	−15.0000	−1.36364	−0.681818	+	0.731522i	$0.738810\pi$
$12$	0	0
$13$	13.8564i	1.06588i	0.846154	+	0.532939i	$0.178912\pi$
$13$	13.8564i	1.06588i	−0.846154	+	0.532939i	$0.821088\pi$
$14$	0	0
$15$	−3.00000	−0.200000
$16$	0	0
$17$	29.4449i	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$17$	29.4449i	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$18$	0	0
$19$	15.5885i	0.820445i	0.911985	+	0.410223i	$0.134549\pi$
$19$	15.5885i	0.820445i	−0.911985	+	0.410223i	$0.865451\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	9.00000	0.391304	0.195652	−	0.980673i	$-0.437318\pi$
$23$	9.00000	0.391304	0.195652	+	0.980673i	$0.437318\pi$
$24$	0	0
$25$	22.0000	0.880000
$26$	0	0
$27$	− 25.9808i	− 0.962250i
$28$	0	0
$29$	−6.00000	−0.206897	−0.103448	−	0.994635i	$-0.532988\pi$
$29$	−6.00000	−0.206897	−0.103448	+	0.994635i	$0.532988\pi$
$30$	0	0
$31$	12.1244i	0.391108i	0.980693	+	0.195554i	$0.0626505\pi$
$31$	12.1244i	0.391108i	−0.980693	+	0.195554i	$0.937349\pi$
$32$	0	0
$33$	25.9808i	0.787296i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	31.0000	0.837838	0.418919	−	0.908024i	$-0.362409\pi$
$37$	31.0000	0.837838	0.418919	+	0.908024i	$0.362409\pi$
$38$	0	0
$39$	24.0000	0.615385
$40$	0	0
$41$	− 55.4256i	− 1.35184i	−0.736973	−	0.675922i	$-0.763745\pi$
$41$	− 55.4256i	− 1.35184i	0.736973	−	0.675922i	$-0.236255\pi$
$42$	0	0
$43$	−10.0000	−0.232558	−0.116279	−	0.993217i	$-0.537097\pi$
$43$	−10.0000	−0.232558	−0.116279	+	0.993217i	$0.537097\pi$
$44$	0	0
$45$	− 10.3923i	− 0.230940i
$46$	0	0
$47$	43.3013i	0.921304i	0.887581	+	0.460652i	$0.152384\pi$
$47$	43.3013i	0.921304i	−0.887581	+	0.460652i	$0.847616\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	51.0000	1.00000
$52$	0	0
$53$	−57.0000	−1.07547	−0.537736	−	0.843113i	$-0.680720\pi$
$53$	−57.0000	−1.07547	−0.537736	+	0.843113i	$0.680720\pi$
$54$	0	0
$55$	25.9808i	0.472377i
$56$	0	0
$57$	27.0000	0.473684
$58$	0	0
$59$	81.4064i	1.37977i	0.723919	+	0.689885i	$0.242339\pi$
$59$	81.4064i	1.37977i	−0.723919	+	0.689885i	$0.757661\pi$
$60$	0	0
$61$	81.4064i	1.33453i	0.744820	+	0.667265i	$0.232535\pi$
$61$	81.4064i	1.33453i	−0.744820	+	0.667265i	$0.767465\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	24.0000	0.369231
$66$	0	0
$67$	49.0000	0.731343	0.365672	−	0.930744i	$-0.380839\pi$
$67$	49.0000	0.731343	0.365672	+	0.930744i	$0.380839\pi$
$68$	0	0
$69$	− 15.5885i	− 0.225920i
$70$	0	0
$71$	126.000	1.77465	0.887324	−	0.461147i	$-0.152562\pi$
$71$	126.000	1.77465	0.887324	+	0.461147i	$0.152562\pi$
$72$	0	0
$73$	− 25.9808i	− 0.355901i	−0.984039	−	0.177950i	$-0.943053\pi$
$73$	− 25.9808i	− 0.355901i	0.984039	−	0.177950i	$-0.0569467\pi$
$74$	0	0
$75$	− 38.1051i	− 0.508068i
$76$	0	0
$77$	0	0
$78$	0	0
$79$	73.0000	0.924051	0.462025	−	0.886867i	$-0.347123\pi$
$79$	73.0000	0.924051	0.462025	+	0.886867i	$0.347123\pi$
$80$	0	0
$81$	9.00000	0.111111
$82$	0	0
$83$	13.8564i	0.166945i	0.996510	+	0.0834723i	$0.0266010\pi$
$83$	13.8564i	0.166945i	−0.996510	+	0.0834723i	$0.973399\pi$
$84$	0	0
$85$	51.0000	0.600000
$86$	0	0
$87$	10.3923i	0.119452i
$88$	0	0
$89$	− 57.1577i	− 0.642221i	−0.947042	−	0.321111i	$-0.895944\pi$
$89$	− 57.1577i	− 0.642221i	0.947042	−	0.321111i	$-0.104056\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	21.0000	0.225806
$94$	0	0
$95$	27.0000	0.284211
$96$	0	0
$97$	27.7128i	0.285699i	0.989744	+	0.142850i	$0.0456265\pi$
$97$	27.7128i	0.285699i	−0.989744	+	0.142850i	$0.954373\pi$
$98$	0	0
$99$	−90.0000	−0.909091
$100$	0	0
$101$	98.7269i	0.977494i	0.872426	+	0.488747i	$0.162546\pi$
$101$	98.7269i	0.977494i	−0.872426	+	0.488747i	$0.837454\pi$
$102$	0	0
$103$	71.0141i	0.689457i	0.938702	+	0.344729i	$0.112029\pi$
$103$	71.0141i	0.689457i	−0.938702	+	0.344729i	$0.887971\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−39.0000	−0.364486	−0.182243	−	0.983254i	$-0.558336\pi$
$107$	−39.0000	−0.364486	−0.182243	+	0.983254i	$0.558336\pi$
$108$	0	0
$109$	103.000	0.944954	0.472477	−	0.881343i	$-0.343360\pi$
$109$	103.000	0.944954	0.472477	+	0.881343i	$0.343360\pi$
$110$	0	0
$111$	− 53.6936i	− 0.483726i
$112$	0	0
$113$	−78.0000	−0.690265	−0.345133	−	0.938554i	$-0.612166\pi$
$113$	−78.0000	−0.690265	−0.345133	+	0.938554i	$0.612166\pi$
$114$	0	0
$115$	− 15.5885i	− 0.135552i
$116$	0	0
$117$	83.1384i	0.710585i
$118$	0	0
$119$	0	0
$120$	0	0
$121$	104.000	0.859504
$122$	0	0
$123$	−96.0000	−0.780488
$124$	0	0
$125$	− 81.4064i	− 0.651251i
$126$	0	0
$127$	−50.0000	−0.393701	−0.196850	−	0.980434i	$-0.563071\pi$
$127$	−50.0000	−0.393701	−0.196850	+	0.980434i	$0.563071\pi$
$128$	0	0
$129$	17.3205i	0.134268i
$130$	0	0
$131$	98.7269i	0.753640i	0.926286	+	0.376820i	$0.122983\pi$
$131$	98.7269i	0.753640i	−0.926286	+	0.376820i	$0.877017\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−45.0000	−0.333333
$136$	0	0
$137$	63.0000	0.459854	0.229927	−	0.973208i	$-0.426151\pi$
$137$	63.0000	0.459854	0.229927	+	0.973208i	$0.426151\pi$
$138$	0	0
$139$	− 235.559i	− 1.69467i	−0.531060	−	0.847334i	$-0.678206\pi$
$139$	− 235.559i	− 1.69467i	0.531060	−	0.847334i	$-0.321794\pi$
$140$	0	0
$141$	75.0000	0.531915
$142$	0	0
$143$	− 207.846i	− 1.45347i
$144$	0	0
$145$	10.3923i	0.0716711i
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−177.000	−1.18792	−0.593960	−	0.804495i	$-0.702436\pi$
$149$	−177.000	−1.18792	−0.593960	+	0.804495i	$0.702436\pi$
$150$	0	0
$151$	−55.0000	−0.364238	−0.182119	−	0.983276i	$-0.558296\pi$
$151$	−55.0000	−0.364238	−0.182119	+	0.983276i	$0.558296\pi$
$152$	0	0
$153$	176.669i	1.15470i
$154$	0	0
$155$	21.0000	0.135484
$156$	0	0
$157$	98.7269i	0.628834i	0.949285	+	0.314417i	$0.101809\pi$
$157$	98.7269i	0.628834i	−0.949285	+	0.314417i	$0.898191\pi$
$158$	0	0
$159$	98.7269i	0.620924i
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−23.0000	−0.141104	−0.0705521	−	0.997508i	$-0.522476\pi$
$163$	−23.0000	−0.141104	−0.0705521	+	0.997508i	$0.522476\pi$
$164$	0	0
$165$	45.0000	0.272727
$166$	0	0
$167$	277.128i	1.65945i	0.558172	+	0.829725i	$0.311503\pi$
$167$	277.128i	1.65945i	−0.558172	+	0.829725i	$0.688497\pi$
$168$	0	0
$169$	−23.0000	−0.136095
$170$	0	0
$171$	93.5307i	0.546963i
$172$	0	0
$173$	− 140.296i	− 0.810960i	−0.914104	−	0.405480i	$-0.867104\pi$
$173$	− 140.296i	− 0.810960i	0.914104	−	0.405480i	$-0.132896\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	141.000	0.796610
$178$	0	0
$179$	−63.0000	−0.351955	−0.175978	−	0.984394i	$-0.556309\pi$
$179$	−63.0000	−0.351955	−0.175978	+	0.984394i	$0.556309\pi$
$180$	0	0
$181$	124.708i	0.688993i	0.938788	+	0.344496i	$0.111950\pi$
$181$	124.708i	0.688993i	−0.938788	+	0.344496i	$0.888050\pi$
$182$	0	0
$183$	141.000	0.770492
$184$	0	0
$185$	− 53.6936i	− 0.290236i
$186$	0	0
$187$	− 441.673i	− 2.36189i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	57.0000	0.298429	0.149215	−	0.988805i	$-0.452325\pi$
$191$	57.0000	0.298429	0.149215	+	0.988805i	$0.452325\pi$
$192$	0	0
$193$	−209.000	−1.08290	−0.541451	−	0.840732i	$-0.682125\pi$
$193$	−209.000	−1.08290	−0.541451	+	0.840732i	$0.682125\pi$
$194$	0	0
$195$	− 41.5692i	− 0.213175i
$196$	0	0
$197$	−150.000	−0.761421	−0.380711	−	0.924694i	$-0.624321\pi$
$197$	−150.000	−0.761421	−0.380711	+	0.924694i	$0.624321\pi$
$198$	0	0
$199$	206.114i	1.03575i	0.855457	+	0.517874i	$0.173277\pi$
$199$	206.114i	1.03575i	−0.855457	+	0.517874i	$0.826723\pi$
$200$	0	0
$201$	− 84.8705i	− 0.422241i
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−96.0000	−0.468293
$206$	0	0
$207$	54.0000	0.260870
$208$	0	0
$209$	− 233.827i	− 1.11879i
$210$	0	0
$211$	−346.000	−1.63981	−0.819905	−	0.572499i	$-0.805974\pi$
$211$	−346.000	−1.63981	−0.819905	+	0.572499i	$0.805974\pi$
$212$	0	0
$213$	− 218.238i	− 1.02459i
$214$	0	0
$215$	17.3205i	0.0805605i
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−45.0000	−0.205479
$220$	0	0
$221$	−408.000	−1.84615
$222$	0	0
$223$	332.554i	1.49127i	0.666353	+	0.745636i	$0.267854\pi$
$223$	332.554i	1.49127i	−0.666353	+	0.745636i	$0.732146\pi$
$224$	0	0
$225$	132.000	0.586667
$226$	0	0
$227$	− 140.296i	− 0.618045i	−0.951055	−	0.309022i	$-0.899998\pi$
$227$	− 140.296i	− 0.618045i	0.951055	−	0.309022i	$-0.100002\pi$
$228$	0	0
$229$	− 168.009i	− 0.733663i	−0.930287	−	0.366832i	$-0.880442\pi$
$229$	− 168.009i	− 0.733663i	0.930287	−	0.366832i	$-0.119558\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−273.000	−1.17167	−0.585837	−	0.810429i	$-0.699234\pi$
$233$	−273.000	−1.17167	−0.585837	+	0.810429i	$0.699234\pi$
$234$	0	0
$235$	75.0000	0.319149
$236$	0	0
$237$	− 126.440i	− 0.533501i
$238$	0	0
$239$	222.000	0.928870	0.464435	−	0.885607i	$-0.346257\pi$
$239$	222.000	0.928870	0.464435	+	0.885607i	$0.346257\pi$
$240$	0	0
$241$	− 219.970i	− 0.912740i	−0.889790	−	0.456370i	$-0.849149\pi$
$241$	− 219.970i	− 0.912740i	0.889790	−	0.456370i	$-0.150851\pi$
$242$	0	0
$243$	− 249.415i	− 1.02640i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−216.000	−0.874494
$248$	0	0
$249$	24.0000	0.0963855
$250$	0	0
$251$	− 96.9948i	− 0.386434i	−0.981156	−	0.193217i	$-0.938108\pi$
$251$	− 96.9948i	− 0.386434i	0.981156	−	0.193217i	$-0.0618921\pi$
$252$	0	0
$253$	−135.000	−0.533597
$254$	0	0
$255$	− 88.3346i	− 0.346410i
$256$	0	0
$257$	192.258i	0.748084i	0.927412	+	0.374042i	$0.122028\pi$
$257$	192.258i	0.748084i	−0.927412	+	0.374042i	$0.877972\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−36.0000	−0.137931
$262$	0	0
$263$	153.000	0.581749	0.290875	−	0.956761i	$-0.406054\pi$
$263$	153.000	0.581749	0.290875	+	0.956761i	$0.406054\pi$
$264$	0	0
$265$	98.7269i	0.372554i
$266$	0	0
$267$	−99.0000	−0.370787
$268$	0	0
$269$	− 455.529i	− 1.69342i	−0.532057	−	0.846709i	$-0.678581\pi$
$269$	− 455.529i	− 1.69342i	0.532057	−	0.846709i	$-0.321419\pi$
$270$	0	0
$271$	71.0141i	0.262045i	0.991379	+	0.131022i	$0.0418259\pi$
$271$	71.0141i	0.262045i	−0.991379	+	0.131022i	$0.958174\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−330.000	−1.20000
$276$	0	0
$277$	359.000	1.29603	0.648014	−	0.761628i	$-0.275600\pi$
$277$	359.000	1.29603	0.648014	+	0.761628i	$0.275600\pi$
$278$	0	0
$279$	72.7461i	0.260739i
$280$	0	0
$281$	−222.000	−0.790036	−0.395018	−	0.918673i	$-0.629262\pi$
$281$	−222.000	−0.790036	−0.395018	+	0.918673i	$0.629262\pi$
$282$	0	0
$283$	− 1.73205i	− 0.00612032i	−0.999995	−	0.00306016i	$-0.999026\pi$
$283$	− 1.73205i	− 0.00612032i	0.999995	−	0.00306016i	$-0.000974081\pi$
$284$	0	0
$285$	− 46.7654i	− 0.164089i
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−578.000	−2.00000
$290$	0	0
$291$	48.0000	0.164948
$292$	0	0
$293$	235.559i	0.803955i	0.915649	+	0.401978i	$0.131677\pi$
$293$	235.559i	0.803955i	−0.915649	+	0.401978i	$0.868323\pi$
$294$	0	0
$295$	141.000	0.477966
$296$	0	0
$297$	389.711i	1.31216i
$298$	0	0
$299$	124.708i	0.417082i
$300$	0	0
$301$	0	0
$302$	0	0
$303$	171.000	0.564356
$304$	0	0
$305$	141.000	0.462295
$306$	0	0
$307$	− 96.9948i	− 0.315944i	−0.987444	−	0.157972i	$-0.949504\pi$
$307$	− 96.9948i	− 0.315944i	0.987444	−	0.157972i	$-0.0504956\pi$
$308$	0	0
$309$	123.000	0.398058
$310$	0	0
$311$	− 486.706i	− 1.56497i	−0.622668	−	0.782486i	$-0.713951\pi$
$311$	− 486.706i	− 1.56497i	0.622668	−	0.782486i	$-0.286049\pi$
$312$	0	0
$313$	− 223.435i	− 0.713848i	−0.934133	−	0.356924i	$-0.883825\pi$
$313$	− 223.435i	− 0.713848i	0.934133	−	0.356924i	$-0.116175\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	447.000	1.41009	0.705047	−	0.709160i	$-0.250926\pi$
$317$	447.000	1.41009	0.705047	+	0.709160i	$0.250926\pi$
$318$	0	0
$319$	90.0000	0.282132
$320$	0	0
$321$	67.5500i	0.210436i
$322$	0	0
$323$	−459.000	−1.42105
$324$	0	0
$325$	304.841i	0.937972i
$326$	0	0
$327$	− 178.401i	− 0.545570i
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−151.000	−0.456193	−0.228097	−	0.973638i	$-0.573250\pi$
$331$	−151.000	−0.456193	−0.228097	+	0.973638i	$0.573250\pi$
$332$	0	0
$333$	186.000	0.558559
$334$	0	0
$335$	− 84.8705i	− 0.253345i
$336$	0	0
$337$	274.000	0.813056	0.406528	−	0.913638i	$-0.366739\pi$
$337$	274.000	0.813056	0.406528	+	0.913638i	$0.366739\pi$
$338$	0	0
$339$	135.100i	0.398525i
$340$	0	0
$341$	− 181.865i	− 0.533329i
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−27.0000	−0.0782609
$346$	0	0
$347$	−543.000	−1.56484	−0.782421	−	0.622750i	$-0.786015\pi$
$347$	−543.000	−1.56484	−0.782421	+	0.622750i	$0.786015\pi$
$348$	0	0
$349$	180.133i	0.516141i	0.966126	+	0.258071i	$0.0830867\pi$
$349$	180.133i	0.516141i	−0.966126	+	0.258071i	$0.916913\pi$
$350$	0	0
$351$	360.000	1.02564
$352$	0	0
$353$	− 303.109i	− 0.858665i	−0.903146	−	0.429333i	$-0.858749\pi$
$353$	− 303.109i	− 0.858665i	0.903146	−	0.429333i	$-0.141251\pi$
$354$	0	0
$355$	− 218.238i	− 0.614756i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−39.0000	−0.108635	−0.0543175	−	0.998524i	$-0.517298\pi$
$359$	−39.0000	−0.108635	−0.0543175	+	0.998524i	$0.517298\pi$
$360$	0	0
$361$	118.000	0.326870
$362$	0	0
$363$	− 180.133i	− 0.496235i
$364$	0	0
$365$	−45.0000	−0.123288
$366$	0	0
$367$	− 348.142i	− 0.948616i	−0.880359	−	0.474308i	$-0.842698\pi$
$367$	− 348.142i	− 0.948616i	0.880359	−	0.474308i	$-0.157302\pi$
$368$	0	0
$369$	− 332.554i	− 0.901230i
$370$	0	0
$371$	0	0
$372$	0	0
$373$	383.000	1.02681	0.513405	−	0.858147i	$-0.328384\pi$
$373$	383.000	1.02681	0.513405	+	0.858147i	$0.328384\pi$
$374$	0	0
$375$	−141.000	−0.376000
$376$	0	0
$377$	− 83.1384i	− 0.220526i
$378$	0	0
$379$	230.000	0.606860	0.303430	−	0.952854i	$-0.401868\pi$
$379$	230.000	0.606860	0.303430	+	0.952854i	$0.401868\pi$
$380$	0	0
$381$	86.6025i	0.227303i
$382$	0	0
$383$	320.429i	0.836630i	0.908302	+	0.418315i	$0.137379\pi$
$383$	320.429i	0.836630i	−0.908302	+	0.418315i	$0.862621\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−60.0000	−0.155039
$388$	0	0
$389$	−201.000	−0.516710	−0.258355	−	0.966050i	$-0.583180\pi$
$389$	−201.000	−0.516710	−0.258355	+	0.966050i	$0.583180\pi$
$390$	0	0
$391$	265.004i	0.677759i
$392$	0	0
$393$	171.000	0.435115
$394$	0	0
$395$	− 126.440i	− 0.320101i
$396$	0	0
$397$	53.6936i	0.135248i	0.997711	+	0.0676241i	$0.0215419\pi$
$397$	53.6936i	0.135248i	−0.997711	+	0.0676241i	$0.978458\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−177.000	−0.441397	−0.220698	−	0.975342i	$-0.570834\pi$
$401$	−177.000	−0.441397	−0.220698	+	0.975342i	$0.570834\pi$
$402$	0	0
$403$	−168.000	−0.416873
$404$	0	0
$405$	− 15.5885i	− 0.0384900i
$406$	0	0
$407$	−465.000	−1.14251
$408$	0	0
$409$	334.286i	0.817325i	0.912686	+	0.408662i	$0.134005\pi$
$409$	334.286i	0.817325i	−0.912686	+	0.408662i	$0.865995\pi$
$410$	0	0
$411$	− 109.119i	− 0.265497i
$412$	0	0
$413$	0	0
$414$	0	0
$415$	24.0000	0.0578313
$416$	0	0
$417$	−408.000	−0.978417
$418$	0	0
$419$	− 595.825i	− 1.42202i	−0.703183	−	0.711009i	$-0.748239\pi$
$419$	− 595.825i	− 1.42202i	0.703183	−	0.711009i	$-0.251761\pi$
$420$	0	0
$421$	−22.0000	−0.0522565	−0.0261283	−	0.999659i	$-0.508318\pi$
$421$	−22.0000	−0.0522565	−0.0261283	+	0.999659i	$0.508318\pi$
$422$	0	0
$423$	259.808i	0.614202i
$424$	0	0
$425$	647.787i	1.52420i
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−360.000	−0.839161
$430$	0	0
$431$	−327.000	−0.758701	−0.379350	−	0.925253i	$-0.623852\pi$
$431$	−327.000	−0.758701	−0.379350	+	0.925253i	$0.623852\pi$
$432$	0	0
$433$	27.7128i	0.0640019i	0.999488	+	0.0320009i	$0.0101880\pi$
$433$	27.7128i	0.0640019i	−0.999488	+	0.0320009i	$0.989812\pi$
$434$	0	0
$435$	18.0000	0.0413793
$436$	0	0
$437$	140.296i	0.321044i
$438$	0	0
$439$	597.558i	1.36118i	0.732665	+	0.680589i	$0.238276\pi$
$439$	597.558i	1.36118i	−0.732665	+	0.680589i	$0.761724\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	729.000	1.64560	0.822799	−	0.568332i	$-0.192411\pi$
$443$	729.000	1.64560	0.822799	+	0.568332i	$0.192411\pi$
$444$	0	0
$445$	−99.0000	−0.222472
$446$	0	0
$447$	306.573i	0.685846i
$448$	0	0
$449$	−270.000	−0.601336	−0.300668	−	0.953729i	$-0.597210\pi$
$449$	−270.000	−0.601336	−0.300668	+	0.953729i	$0.597210\pi$
$450$	0	0
$451$	831.384i	1.84342i
$452$	0	0
$453$	95.2628i	0.210293i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	799.000	1.74836	0.874179	−	0.485603i	$-0.161400\pi$
$457$	799.000	1.74836	0.874179	+	0.485603i	$0.161400\pi$
$458$	0	0
$459$	765.000	1.66667
$460$	0	0
$461$	− 429.549i	− 0.931776i	−0.884844	−	0.465888i	$-0.845735\pi$
$461$	− 429.549i	− 0.931776i	0.884844	−	0.465888i	$-0.154265\pi$
$462$	0	0
$463$	814.000	1.75810	0.879050	−	0.476730i	$-0.158178\pi$
$463$	814.000	1.75810	0.879050	+	0.476730i	$0.158178\pi$
$464$	0	0
$465$	− 36.3731i	− 0.0782216i
$466$	0	0
$467$	− 400.104i	− 0.856753i	−0.903600	−	0.428377i	$-0.859086\pi$
$467$	− 400.104i	− 0.856753i	0.903600	−	0.428377i	$-0.140914\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	171.000	0.363057
$472$	0	0
$473$	150.000	0.317125
$474$	0	0
$475$	342.946i	0.721992i
$476$	0	0
$477$	−342.000	−0.716981
$478$	0	0
$479$	538.668i	1.12457i	0.826944	+	0.562284i	$0.190077\pi$
$479$	538.668i	1.12457i	−0.826944	+	0.562284i	$0.809923\pi$
$480$	0	0
$481$	429.549i	0.893032i
$482$	0	0
$483$	0	0
$484$	0	0
$485$	48.0000	0.0989691
$486$	0	0
$487$	857.000	1.75975	0.879877	−	0.475202i	$-0.157625\pi$
$487$	857.000	1.75975	0.879877	+	0.475202i	$0.157625\pi$
$488$	0	0
$489$	39.8372i	0.0814666i
$490$	0	0
$491$	−570.000	−1.16090	−0.580448	−	0.814297i	$-0.697123\pi$
$491$	−570.000	−1.16090	−0.580448	+	0.814297i	$0.697123\pi$
$492$	0	0
$493$	− 176.669i	− 0.358355i
$494$	0	0
$495$	155.885i	0.314918i
$496$	0	0
$497$	0	0
$498$	0	0
$499$	425.000	0.851703	0.425852	−	0.904793i	$-0.359975\pi$
$499$	425.000	0.851703	0.425852	+	0.904793i	$0.359975\pi$
$500$	0	0
$501$	480.000	0.958084
$502$	0	0
$503$	− 193.990i	− 0.385665i	−0.981232	−	0.192833i	$-0.938233\pi$
$503$	− 193.990i	− 0.385665i	0.981232	−	0.192833i	$-0.0617675\pi$
$504$	0	0
$505$	171.000	0.338614
$506$	0	0
$507$	39.8372i	0.0785743i
$508$	0	0
$509$	386.247i	0.758836i	0.925225	+	0.379418i	$0.123876\pi$
$509$	386.247i	0.758836i	−0.925225	+	0.379418i	$0.876124\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	405.000	0.789474
$514$	0	0
$515$	123.000	0.238835
$516$	0	0
$517$	− 649.519i	− 1.25632i
$518$	0	0
$519$	−243.000	−0.468208
$520$	0	0
$521$	− 247.683i	− 0.475400i	−0.971339	−	0.237700i	$-0.923607\pi$
$521$	− 247.683i	− 0.475400i	0.971339	−	0.237700i	$-0.0763935\pi$
$522$	0	0
$523$	791.547i	1.51347i	0.653719	+	0.756737i	$0.273208\pi$
$523$	791.547i	1.51347i	−0.653719	+	0.756737i	$0.726792\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−357.000	−0.677419
$528$	0	0
$529$	−448.000	−0.846881
$530$	0	0
$531$	488.438i	0.919846i
$532$	0	0
$533$	768.000	1.44090
$534$	0	0
$535$	67.5500i	0.126262i
$536$	0	0
$537$	109.119i	0.203201i
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−721.000	−1.33272	−0.666359	−	0.745631i	$-0.732148\pi$
$541$	−721.000	−1.33272	−0.666359	+	0.745631i	$0.732148\pi$
$542$	0	0
$543$	216.000	0.397790
$544$	0	0
$545$	− 178.401i	− 0.327342i
$546$	0	0
$547$	118.000	0.215722	0.107861	−	0.994166i	$-0.465600\pi$
$547$	118.000	0.215722	0.107861	+	0.994166i	$0.465600\pi$
$548$	0	0
$549$	488.438i	0.889687i
$550$	0	0
$551$	− 93.5307i	− 0.169747i
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−93.0000	−0.167568
$556$	0	0
$557$	−633.000	−1.13645	−0.568223	−	0.822875i	$-0.692369\pi$
$557$	−633.000	−1.13645	−0.568223	+	0.822875i	$0.692369\pi$
$558$	0	0
$559$	− 138.564i	− 0.247878i
$560$	0	0
$561$	−765.000	−1.36364
$562$	0	0
$563$	275.396i	0.489158i	0.969629	+	0.244579i	$0.0786498\pi$
$563$	275.396i	0.489158i	−0.969629	+	0.244579i	$0.921350\pi$
$564$	0	0
$565$	135.100i	0.239115i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	447.000	0.785589	0.392794	−	0.919626i	$-0.371508\pi$
$569$	447.000	0.785589	0.392794	+	0.919626i	$0.371508\pi$
$570$	0	0
$571$	−527.000	−0.922942	−0.461471	−	0.887155i	$-0.652678\pi$
$571$	−527.000	−0.922942	−0.461471	+	0.887155i	$0.652678\pi$
$572$	0	0
$573$	− 98.7269i	− 0.172298i
$574$	0	0
$575$	198.000	0.344348
$576$	0	0
$577$	168.009i	0.291177i	0.989345	+	0.145588i	$0.0465075\pi$
$577$	168.009i	0.291177i	−0.989345	+	0.145588i	$0.953493\pi$
$578$	0	0
$579$	361.999i	0.625214i
$580$	0	0
$581$	0	0
$582$	0	0
$583$	855.000	1.46655
$584$	0	0
$585$	144.000	0.246154
$586$	0	0
$587$	− 956.092i	− 1.62878i	−0.580320	−	0.814388i	$-0.697073\pi$
$587$	− 956.092i	− 1.62878i	0.580320	−	0.814388i	$-0.302927\pi$
$588$	0	0
$589$	−189.000	−0.320883
$590$	0	0
$591$	259.808i	0.439607i
$592$	0	0
$593$	303.109i	0.511145i	0.966790	+	0.255572i	$0.0822639\pi$
$593$	303.109i	0.511145i	−0.966790	+	0.255572i	$0.917736\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	357.000	0.597990
$598$	0	0
$599$	−807.000	−1.34725	−0.673623	−	0.739075i	$-0.735263\pi$
$599$	−807.000	−1.34725	−0.673623	+	0.739075i	$0.735263\pi$
$600$	0	0
$601$	387.979i	0.645556i	0.946475	+	0.322778i	$0.104617\pi$
$601$	387.979i	0.645556i	−0.946475	+	0.322778i	$0.895383\pi$
$602$	0	0
$603$	294.000	0.487562
$604$	0	0
$605$	− 180.133i	− 0.297741i
$606$	0	0
$607$	− 732.657i	− 1.20701i	−0.797358	−	0.603507i	$-0.793770\pi$
$607$	− 732.657i	− 1.20701i	0.797358	−	0.603507i	$-0.206230\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−600.000	−0.981997
$612$	0	0
$613$	503.000	0.820555	0.410277	−	0.911961i	$-0.365432\pi$
$613$	503.000	0.820555	0.410277	+	0.911961i	$0.365432\pi$
$614$	0	0
$615$	166.277i	0.270369i
$616$	0	0
$617$	930.000	1.50729	0.753647	−	0.657280i	$-0.228293\pi$
$617$	930.000	1.50729	0.753647	+	0.657280i	$0.228293\pi$
$618$	0	0
$619$	− 528.275i	− 0.853434i	−0.904385	−	0.426717i	$-0.859670\pi$
$619$	− 528.275i	− 0.853434i	0.904385	−	0.426717i	$-0.140330\pi$
$620$	0	0
$621$	− 233.827i	− 0.376533i
$622$	0	0
$623$	0	0
$624$	0	0
$625$	409.000	0.654400
$626$	0	0
$627$	−405.000	−0.645933
$628$	0	0
$629$	912.791i	1.45118i
$630$	0	0
$631$	−194.000	−0.307448	−0.153724	−	0.988114i	$-0.549127\pi$
$631$	−194.000	−0.307448	−0.153724	+	0.988114i	$0.549127\pi$
$632$	0	0
$633$	599.290i	0.946745i
$634$	0	0
$635$	86.6025i	0.136382i
$636$	0	0
$637$	0	0
$638$	0	0
$639$	756.000	1.18310
$640$	0	0
$641$	−33.0000	−0.0514821	−0.0257410	−	0.999669i	$-0.508195\pi$
$641$	−33.0000	−0.0514821	−0.0257410	+	0.999669i	$0.508195\pi$
$642$	0	0
$643$	346.410i	0.538741i	0.963037	+	0.269370i	$0.0868155\pi$
$643$	346.410i	0.538741i	−0.963037	+	0.269370i	$0.913184\pi$
$644$	0	0
$645$	30.0000	0.0465116
$646$	0	0
$647$	− 763.834i	− 1.18058i	−0.807192	−	0.590289i	$-0.799014\pi$
$647$	− 763.834i	− 1.18058i	0.807192	−	0.590289i	$-0.200986\pi$
$648$	0	0
$649$	− 1221.10i	− 1.88150i
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−33.0000	−0.0505360	−0.0252680	−	0.999681i	$-0.508044\pi$
$653$	−33.0000	−0.0505360	−0.0252680	+	0.999681i	$0.508044\pi$
$654$	0	0
$655$	171.000	0.261069
$656$	0	0
$657$	− 155.885i	− 0.237267i
$658$	0	0
$659$	870.000	1.32018	0.660091	−	0.751186i	$-0.270518\pi$
$659$	870.000	1.32018	0.660091	+	0.751186i	$0.270518\pi$
$660$	0	0
$661$	1013.25i	1.53290i	0.642301	+	0.766452i	$0.277980\pi$
$661$	1013.25i	1.53290i	−0.642301	+	0.766452i	$0.722020\pi$
$662$	0	0
$663$	706.677i	1.06588i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−54.0000	−0.0809595
$668$	0	0
$669$	576.000	0.860987
$670$	0	0
$671$	− 1221.10i	− 1.81981i
$672$	0	0
$673$	−14.0000	−0.0208024	−0.0104012	−	0.999946i	$-0.503311\pi$
$673$	−14.0000	−0.0208024	−0.0104012	+	0.999946i	$0.503311\pi$
$674$	0	0
$675$	− 571.577i	− 0.846780i
$676$	0	0
$677$	275.396i	0.406789i	0.979097	+	0.203394i	$0.0651974\pi$
$677$	275.396i	0.406789i	−0.979097	+	0.203394i	$0.934803\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−243.000	−0.356828
$682$	0	0
$683$	273.000	0.399707	0.199854	−	0.979826i	$-0.435953\pi$
$683$	273.000	0.399707	0.199854	+	0.979826i	$0.435953\pi$
$684$	0	0
$685$	− 109.119i	− 0.159298i
$686$	0	0
$687$	−291.000	−0.423581
$688$	0	0
$689$	− 789.815i	− 1.14632i
$690$	0	0
$691$	154.153i	0.223086i	0.993760	+	0.111543i	$0.0355793\pi$
$691$	154.153i	0.223086i	−0.993760	+	0.111543i	$0.964421\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−408.000	−0.587050
$696$	0	0
$697$	1632.00	2.34146
$698$	0	0
$699$	472.850i	0.676466i
$700$	0	0
$701$	906.000	1.29244	0.646220	−	0.763151i	$-0.276349\pi$
$701$	906.000	1.29244	0.646220	+	0.763151i	$0.276349\pi$
$702$	0	0
$703$	483.242i	0.687400i
$704$	0	0
$705$	− 129.904i	− 0.184261i
$706$	0	0
$707$	0	0
$708$	0	0
$709$	703.000	0.991537	0.495769	−	0.868455i	$-0.334886\pi$
$709$	703.000	0.991537	0.495769	+	0.868455i	$0.334886\pi$
$710$	0	0
$711$	438.000	0.616034
$712$	0	0
$713$	109.119i	0.153042i
$714$	0	0
$715$	−360.000	−0.503497
$716$	0	0
$717$	− 384.515i	− 0.536284i
$718$	0	0
$719$	− 594.093i	− 0.826277i	−0.910668	−	0.413139i	$-0.864432\pi$
$719$	− 594.093i	− 0.826277i	0.910668	−	0.413139i	$-0.135568\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−381.000	−0.526971
$724$	0	0
$725$	−132.000	−0.182069
$726$	0	0
$727$	− 27.7128i	− 0.0381194i	−0.999818	−	0.0190597i	$-0.993933\pi$
$727$	− 27.7128i	− 0.0381194i	0.999818	−	0.0190597i	$-0.00606726\pi$
$728$	0	0
$729$	−351.000	−0.481481
$730$	0	0
$731$	− 294.449i	− 0.402803i
$732$	0	0
$733$	968.216i	1.32090i	0.750872	+	0.660448i	$0.229633\pi$
$733$	968.216i	1.32090i	−0.750872	+	0.660448i	$0.770367\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−735.000	−0.997286
$738$	0	0
$739$	577.000	0.780785	0.390392	−	0.920649i	$-0.372339\pi$
$739$	577.000	0.780785	0.390392	+	0.920649i	$0.372339\pi$
$740$	0	0
$741$	374.123i	0.504889i
$742$	0	0
$743$	−162.000	−0.218035	−0.109017	−	0.994040i	$-0.534770\pi$
$743$	−162.000	−0.218035	−0.109017	+	0.994040i	$0.534770\pi$
$744$	0	0
$745$	306.573i	0.411507i
$746$	0	0
$747$	83.1384i	0.111296i
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−775.000	−1.03196	−0.515979	−	0.856601i	$-0.672572\pi$
$751$	−775.000	−1.03196	−0.515979	+	0.856601i	$0.672572\pi$
$752$	0	0
$753$	−168.000	−0.223108
$754$	0	0
$755$	95.2628i	0.126176i
$756$	0	0
$757$	−950.000	−1.25495	−0.627477	−	0.778635i	$-0.715912\pi$
$757$	−950.000	−1.25495	−0.627477	+	0.778635i	$0.715912\pi$
$758$	0	0
$759$	233.827i	0.308072i
$760$	0	0
$761$	− 195.722i	− 0.257190i	−0.991697	−	0.128595i	$-0.958953\pi$
$761$	− 195.722i	− 0.257190i	0.991697	−	0.128595i	$-0.0410468\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	306.000	0.400000
$766$	0	0
$767$	−1128.00	−1.47066
$768$	0	0
$769$	− 914.523i	− 1.18924i	−0.804008	−	0.594618i	$-0.797303\pi$
$769$	− 914.523i	− 1.18924i	0.804008	−	0.594618i	$-0.202697\pi$
$770$	0	0
$771$	333.000	0.431907
$772$	0	0
$773$	− 178.401i	− 0.230791i	−0.993320	−	0.115395i	$-0.963187\pi$
$773$	− 178.401i	− 0.230791i	0.993320	−	0.115395i	$-0.0368135\pi$
$774$	0	0
$775$	266.736i	0.344175i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	864.000	1.10911
$780$	0	0
$781$	−1890.00	−2.41997
$782$	0	0
$783$	155.885i	0.199086i
$784$	0	0
$785$	171.000	0.217834
$786$	0	0
$787$	− 334.286i	− 0.424760i	−0.977187	−	0.212380i	$-0.931879\pi$
$787$	− 334.286i	− 0.424760i	0.977187	−	0.212380i	$-0.0681214\pi$
$788$	0	0
$789$	− 265.004i	− 0.335873i
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−1128.00	−1.42245
$794$	0	0
$795$	171.000	0.215094
$796$	0	0
$797$	− 872.954i	− 1.09530i	−0.836708	−	0.547650i	$-0.815523\pi$
$797$	− 872.954i	− 1.09530i	0.836708	−	0.547650i	$-0.184477\pi$
$798$	0	0
$799$	−1275.00	−1.59574
$800$	0	0
$801$	− 342.946i	− 0.428147i
$802$	0	0
$803$	389.711i	0.485319i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−789.000	−0.977695
$808$	0	0
$809$	−465.000	−0.574784	−0.287392	−	0.957813i	$-0.592788\pi$
$809$	−465.000	−0.574784	−0.287392	+	0.957813i	$0.592788\pi$
$810$	0	0
$811$	− 124.708i	− 0.153770i	−0.997040	−	0.0768851i	$-0.975503\pi$
$811$	− 124.708i	− 0.153770i	0.997040	−	0.0768851i	$-0.0244975\pi$
$812$	0	0
$813$	123.000	0.151292
$814$	0	0
$815$	39.8372i	0.0488800i
$816$	0	0
$817$	− 155.885i	− 0.190801i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−753.000	−0.917174	−0.458587	−	0.888649i	$-0.651644\pi$
$821$	−753.000	−0.917174	−0.458587	+	0.888649i	$0.651644\pi$
$822$	0	0
$823$	25.0000	0.0303767	0.0151883	−	0.999885i	$-0.495165\pi$
$823$	25.0000	0.0303767	0.0151883	+	0.999885i	$0.495165\pi$
$824$	0	0
$825$	571.577i	0.692820i
$826$	0	0
$827$	246.000	0.297461	0.148730	−	0.988878i	$-0.452481\pi$
$827$	246.000	0.297461	0.148730	+	0.988878i	$0.452481\pi$
$828$	0	0
$829$	− 1480.90i	− 1.78637i	−0.449686	−	0.893187i	$-0.648464\pi$
$829$	− 1480.90i	− 1.78637i	0.449686	−	0.893187i	$-0.351536\pi$
$830$	0	0
$831$	− 621.806i	− 0.748263i
$832$	0	0
$833$	0	0
$834$	0	0
$835$	480.000	0.574850
$836$	0	0
$837$	315.000	0.376344
$838$	0	0
$839$	360.267i	0.429400i	0.976680	+	0.214700i	$0.0688774\pi$
$839$	360.267i	0.429400i	−0.976680	+	0.214700i	$0.931123\pi$
$840$	0	0
$841$	−805.000	−0.957194
$842$	0	0
$843$	384.515i	0.456127i
$844$	0	0
$845$	39.8372i	0.0471446i
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−3.00000	−0.00353357
$850$	0	0
$851$	279.000	0.327850
$852$	0	0
$853$	1205.51i	1.41326i	0.707585	+	0.706628i	$0.249785\pi$
$853$	1205.51i	1.41326i	−0.707585	+	0.706628i	$0.750215\pi$
$854$	0	0
$855$	162.000	0.189474
$856$	0	0
$857$	195.722i	0.228380i	0.993459	+	0.114190i	$0.0364273\pi$
$857$	195.722i	0.228380i	−0.993459	+	0.114190i	$0.963573\pi$
$858$	0	0
$859$	− 1425.48i	− 1.65946i	−0.558164	−	0.829731i	$-0.688494\pi$
$859$	− 1425.48i	− 1.65946i	0.558164	−	0.829731i	$-0.311506\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−855.000	−0.990730	−0.495365	−	0.868685i	$-0.664966\pi$
$863$	−855.000	−0.990730	−0.495365	+	0.868685i	$0.664966\pi$
$864$	0	0
$865$	−243.000	−0.280925
$866$	0	0
$867$	1001.13i	1.15470i
$868$	0	0
$869$	−1095.00	−1.26007
$870$	0	0
$871$	678.964i	0.779522i
$872$	0	0
$873$	166.277i	0.190466i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−241.000	−0.274800	−0.137400	−	0.990516i	$-0.543875\pi$
$877$	−241.000	−0.274800	−0.137400	+	0.990516i	$0.543875\pi$
$878$	0	0
$879$	408.000	0.464164
$880$	0	0
$881$	− 1330.22i	− 1.50989i	−0.655787	−	0.754946i	$-0.727663\pi$
$881$	− 1330.22i	− 1.50989i	0.655787	−	0.754946i	$-0.272337\pi$
$882$	0	0
$883$	1286.00	1.45640	0.728199	−	0.685365i	$-0.240358\pi$
$883$	1286.00	1.45640	0.728199	+	0.685365i	$0.240358\pi$
$884$	0	0
$885$	− 244.219i	− 0.275954i
$886$	0	0
$887$	− 1286.91i	− 1.45086i	−0.688295	−	0.725431i	$-0.741641\pi$
$887$	− 1286.91i	− 1.45086i	0.688295	−	0.725431i	$-0.258359\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−135.000	−0.151515
$892$	0	0
$893$	−675.000	−0.755879
$894$	0	0
$895$	109.119i	0.121921i
$896$	0	0
$897$	216.000	0.240803
$898$	0	0
$899$	− 72.7461i	− 0.0809189i
$900$	0	0
$901$	− 1678.36i	− 1.86277i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	216.000	0.238674
$906$	0	0
$907$	737.000	0.812569	0.406284	−	0.913747i	$-0.366824\pi$
$907$	737.000	0.812569	0.406284	+	0.913747i	$0.366824\pi$
$908$	0	0
$909$	592.361i	0.651663i
$910$	0	0
$911$	−1266.00	−1.38968	−0.694841	−	0.719164i	$-0.744525\pi$
$911$	−1266.00	−1.38968	−0.694841	+	0.719164i	$0.744525\pi$
$912$	0	0
$913$	− 207.846i	− 0.227652i
$914$	0	0
$915$	− 244.219i	− 0.266906i
$916$	0	0
$917$	0	0
$918$	0	0
$919$	473.000	0.514690	0.257345	−	0.966320i	$-0.417152\pi$
$919$	473.000	0.514690	0.257345	+	0.966320i	$0.417152\pi$
$920$	0	0
$921$	−168.000	−0.182410
$922$	0	0
$923$	1745.91i	1.89156i
$924$	0	0
$925$	682.000	0.737297
$926$	0	0
$927$	426.084i	0.459638i
$928$	0	0
$929$	− 334.286i	− 0.359834i	−0.983682	−	0.179917i	$-0.942417\pi$
$929$	− 334.286i	− 0.359834i	0.983682	−	0.179917i	$-0.0575829\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−843.000	−0.903537
$934$	0	0
$935$	−765.000	−0.818182
$936$	0	0
$937$	859.097i	0.916859i	0.888731	+	0.458430i	$0.151588\pi$
$937$	859.097i	0.916859i	−0.888731	+	0.458430i	$0.848412\pi$
$938$	0	0
$939$	−387.000	−0.412141
$940$	0	0
$941$	− 122.976i	− 0.130686i	−0.997863	−	0.0653430i	$-0.979186\pi$
$941$	− 122.976i	− 0.130686i	0.997863	−	0.0653430i	$-0.0208142\pi$
$942$	0	0
$943$	− 498.831i	− 0.528983i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	201.000	0.212249	0.106125	−	0.994353i	$-0.466156\pi$
$947$	201.000	0.212249	0.106125	+	0.994353i	$0.466156\pi$
$948$	0	0
$949$	360.000	0.379347
$950$	0	0
$951$	− 774.227i	− 0.814119i
$952$	0	0
$953$	66.0000	0.0692550	0.0346275	−	0.999400i	$-0.488976\pi$
$953$	66.0000	0.0692550	0.0346275	+	0.999400i	$0.488976\pi$
$954$	0	0
$955$	− 98.7269i	− 0.103379i
$956$	0	0
$957$	− 155.885i	− 0.162889i
$958$	0	0
$959$	0	0
$960$	0	0
$961$	814.000	0.847034
$962$	0	0
$963$	−234.000	−0.242991
$964$	0	0
$965$	361.999i	0.375128i
$966$	0	0
$967$	−194.000	−0.200620	−0.100310	−	0.994956i	$-0.531984\pi$
$967$	−194.000	−0.200620	−0.100310	+	0.994956i	$0.531984\pi$
$968$	0	0
$969$	795.011i	0.820445i
$970$	0	0
$971$	1484.37i	1.52870i	0.644802	+	0.764350i	$0.276940\pi$
$971$	1484.37i	1.52870i	−0.644802	+	0.764350i	$0.723060\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	528.000	0.541538
$976$	0	0
$977$	−1521.00	−1.55681	−0.778403	−	0.627765i	$-0.783970\pi$
$977$	−1521.00	−1.55681	−0.778403	+	0.627765i	$0.783970\pi$
$978$	0	0
$979$	857.365i	0.875756i
$980$	0	0
$981$	618.000	0.629969
$982$	0	0
$983$	− 1124.10i	− 1.14354i	−0.820414	−	0.571771i	$-0.806257\pi$
$983$	− 1124.10i	− 1.14354i	0.820414	−	0.571771i	$-0.193743\pi$
$984$	0	0
$985$	259.808i	0.263764i
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−90.0000	−0.0910010
$990$	0	0
$991$	−823.000	−0.830474	−0.415237	−	0.909713i	$-0.636301\pi$
$991$	−823.000	−0.830474	−0.415237	+	0.909713i	$0.636301\pi$
$992$	0	0
$993$	261.540i	0.263383i
$994$	0	0
$995$	357.000	0.358794
$996$	0	0
$997$	126.440i	0.126820i	0.997988	+	0.0634101i	$0.0201976\pi$
$997$	126.440i	0.126820i	−0.997988	+	0.0634101i	$0.979802\pi$
$998$	0	0
$999$	− 805.404i	− 0.806210i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	784.3.c.a.97.1		2
4.3	odd	2		196.3.b.a.97.2		2
7.2	even	3		784.3.s.b.129.1		2
7.3	odd	6		784.3.s.b.705.1		2
7.4	even	3		112.3.s.a.33.1		2
7.5	odd	6		112.3.s.a.17.1		2
7.6	odd	2	inner	784.3.c.a.97.2		2
12.11	even	2		1764.3.d.a.685.2		2
21.5	even	6		1008.3.cg.c.577.1		2
21.11	odd	6		1008.3.cg.c.145.1		2
28.3	even	6		196.3.h.a.117.1		2
28.11	odd	6		28.3.h.a.5.1	✓	2
28.19	even	6		28.3.h.a.17.1	yes	2
28.23	odd	6		196.3.h.a.129.1		2
28.27	even	2		196.3.b.a.97.1		2
56.5	odd	6		448.3.s.b.129.1		2
56.11	odd	6		448.3.s.a.257.1		2
56.19	even	6		448.3.s.a.129.1		2
56.53	even	6		448.3.s.b.257.1		2
84.11	even	6		252.3.z.a.145.1		2
84.23	even	6		1764.3.z.f.325.1		2
84.47	odd	6		252.3.z.a.73.1		2
84.59	odd	6		1764.3.z.f.901.1		2
84.83	odd	2		1764.3.d.a.685.1		2
140.19	even	6		700.3.s.a.101.1		2
140.39	odd	6		700.3.s.a.201.1		2
140.47	odd	12		700.3.o.a.549.2		4
140.67	even	12		700.3.o.a.649.1		4
140.103	odd	12		700.3.o.a.549.1		4
140.123	even	12		700.3.o.a.649.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
28.3.h.a.5.1	✓	2	28.11	odd	6
28.3.h.a.17.1	yes	2	28.19	even	6
112.3.s.a.17.1		2	7.5	odd	6
112.3.s.a.33.1		2	7.4	even	3
196.3.b.a.97.1		2	28.27	even	2
196.3.b.a.97.2		2	4.3	odd	2
196.3.h.a.117.1		2	28.3	even	6
196.3.h.a.129.1		2	28.23	odd	6
252.3.z.a.73.1		2	84.47	odd	6
252.3.z.a.145.1		2	84.11	even	6
448.3.s.a.129.1		2	56.19	even	6
448.3.s.a.257.1		2	56.11	odd	6
448.3.s.b.129.1		2	56.5	odd	6
448.3.s.b.257.1		2	56.53	even	6
700.3.o.a.549.1		4	140.103	odd	12
700.3.o.a.549.2		4	140.47	odd	12
700.3.o.a.649.1		4	140.67	even	12
700.3.o.a.649.2		4	140.123	even	12
700.3.s.a.101.1		2	140.19	even	6
700.3.s.a.201.1		2	140.39	odd	6
784.3.c.a.97.1		2	1.1	even	1	trivial
784.3.c.a.97.2		2	7.6	odd	2	inner
784.3.s.b.129.1		2	7.2	even	3
784.3.s.b.705.1		2	7.3	odd	6
1008.3.cg.c.145.1		2	21.11	odd	6
1008.3.cg.c.577.1		2	21.5	even	6
1764.3.d.a.685.1		2	84.83	odd	2
1764.3.d.a.685.2		2	12.11	even	2
1764.3.z.f.325.1		2	84.23	even	6
1764.3.z.f.901.1		2	84.59	odd	6

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:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	784.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.73205i q^{3} -1.73205i q^{5} +6.00000 q^{9} +O(q^{10})\)	\(q-1.73205i q^{3} -1.73205i q^{5} +6.00000 q^{9} -15.0000 q^{11} +13.8564i q^{13} -3.00000 q^{15} +29.4449i q^{17} +15.5885i q^{19} +9.00000 q^{23} +22.0000 q^{25} -25.9808i q^{27} -6.00000 q^{29} +12.1244i q^{31} +25.9808i q^{33} +31.0000 q^{37} +24.0000 q^{39} -55.4256i q^{41} -10.0000 q^{43} -10.3923i q^{45} +43.3013i q^{47} +51.0000 q^{51} -57.0000 q^{53} +25.9808i q^{55} +27.0000 q^{57} +81.4064i q^{59} +81.4064i q^{61} +24.0000 q^{65} +49.0000 q^{67} -15.5885i q^{69} +126.000 q^{71} -25.9808i q^{73} -38.1051i q^{75} +73.0000 q^{79} +9.00000 q^{81} +13.8564i q^{83} +51.0000 q^{85} +10.3923i q^{87} -57.1577i q^{89} +21.0000 q^{93} +27.0000 q^{95} +27.7128i q^{97} -90.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 12 q^{9}+O(q^{10}) \) 2 * q + 12 * q^9	\( 2 q + 12 q^{9} - 30 q^{11} - 6 q^{15} + 18 q^{23} + 44 q^{25} - 12 q^{29} + 62 q^{37} + 48 q^{39} - 20 q^{43} + 102 q^{51} - 114 q^{53} + 54 q^{57} + 48 q^{65} + 98 q^{67} + 252 q^{71} + 146 q^{79} + 18 q^{81} + 102 q^{85} + 42 q^{93} + 54 q^{95} - 180 q^{99}+O(q^{100}) \) 2 * q + 12 * q^9 - 30 * q^11 - 6 * q^15 + 18 * q^23 + 44 * q^25 - 12 * q^29 + 62 * q^37 + 48 * q^39 - 20 * q^43 + 102 * q^51 - 114 * q^53 + 54 * q^57 + 48 * q^65 + 98 * q^67 + 252 * q^71 + 146 * q^79 + 18 * q^81 + 102 * q^85 + 42 * q^93 + 54 * q^95 - 180 * q^99

Introduction

L-functions

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