LMFDB - Embedded newform 4896.2.a.bk.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4896,2,Mod(1,4896)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4896, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4896.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4896 = 2^{5} \cdot 3^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4896.a (trivial)

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:	yes
Analytic conductor:	$39.0947568296$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.102503232.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 15x^{4} - 16x^{3} + 27x^{2} + 42x + 12 $ x^6 - 15x^4 - 16x^3 + 27x^2 + 42x + 12
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.19689$ of defining polynomial
Character	$\chi$	$=$	4896.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.88202 q^{5} +2.71807 q^{7} +6.16328 q^{11} +3.30604 q^{13} -1.00000 q^{17} +1.04782 q^{19} +8.88136 q^{23} +10.0701 q^{25} -2.57598 q^{29} +2.71807 q^{31} -10.5516 q^{35} -8.95210 q^{37} -7.07008 q^{41} +9.50379 q^{43} +3.34051 q^{47} +0.387915 q^{49} -12.3761 q^{53} -23.9260 q^{55} -12.6472 q^{59} -1.18806 q^{61} -12.8341 q^{65} +10.5516 q^{67} -12.9490 q^{71} +6.00000 q^{73} +16.7523 q^{77} +15.0446 q^{79} +12.6472 q^{83} +3.88202 q^{85} +8.61208 q^{89} +8.98606 q^{91} -4.06765 q^{95} +16.3761 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 6 q^{5} + 6 q^{13} - 6 q^{17} + 12 q^{25} - 12 q^{29} + 12 q^{37} + 6 q^{41} + 30 q^{49} - 12 q^{53} + 24 q^{61} + 6 q^{65} + 36 q^{73} - 24 q^{77} + 6 q^{85} + 24 q^{89} + 36 q^{97}+O(q^{100}) $ 6 * q - 6 * q^5 + 6 * q^13 - 6 * q^17 + 12 * q^25 - 12 * q^29 + 12 * q^37 + 6 * q^41 + 30 * q^49 - 12 * q^53 + 24 * q^61 + 6 * q^65 + 36 * q^73 - 24 * q^77 + 6 * q^85 + 24 * q^89 + 36 * q^97

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	−3.88202	−1.73609	−0.868046	−	0.496483i	$-0.834624\pi$
$5$	−3.88202	−1.73609	−0.868046	+	0.496483i	$0.834624\pi$
$6$	0	0
$7$	2.71807	1.02733	0.513667	−	0.857989i	$-0.328287\pi$
$7$	2.71807	1.02733	0.513667	+	0.857989i	$0.328287\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	6.16328	1.85830	0.929150	−	0.369703i	$-0.120540\pi$
$11$	6.16328	1.85830	0.929150	+	0.369703i	$0.120540\pi$
$12$	0	0
$13$	3.30604	0.916931	0.458466	−	0.888712i	$-0.348399\pi$
$13$	3.30604	0.916931	0.458466	+	0.888712i	$0.348399\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0	0
$19$	1.04782	0.240386	0.120193	−	0.992751i	$-0.461649\pi$
$19$	1.04782	0.240386	0.120193	+	0.992751i	$0.461649\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	8.88136	1.85189	0.925945	−	0.377657i	$-0.123270\pi$
$23$	8.88136	1.85189	0.925945	+	0.377657i	$0.123270\pi$
$24$	0	0
$25$	10.0701	2.01402
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−2.57598	−0.478347	−0.239174	−	0.970977i	$-0.576876\pi$
$29$	−2.57598	−0.478347	−0.239174	+	0.970977i	$0.576876\pi$
$30$	0	0
$31$	2.71807	0.488180	0.244090	−	0.969753i	$-0.421511\pi$
$31$	2.71807	0.488180	0.244090	+	0.969753i	$0.421511\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−10.5516	−1.78355
$36$	0	0
$37$	−8.95210	−1.47172	−0.735858	−	0.677135i	$-0.763221\pi$
$37$	−8.95210	−1.47172	−0.735858	+	0.677135i	$0.763221\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−7.07008	−1.10416	−0.552081	−	0.833791i	$-0.686166\pi$
$41$	−7.07008	−1.10416	−0.552081	+	0.833791i	$0.686166\pi$
$42$	0	0
$43$	9.50379	1.44931	0.724657	−	0.689109i	$-0.241998\pi$
$43$	9.50379	1.44931	0.724657	+	0.689109i	$0.241998\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	3.34051	0.487263	0.243632	−	0.969868i	$-0.421661\pi$
$47$	3.34051	0.487263	0.243632	+	0.969868i	$0.421661\pi$
$48$	0	0
$49$	0.387915	0.0554165
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−12.3761	−1.69999	−0.849996	−	0.526789i	$-0.823396\pi$
$53$	−12.3761	−1.69999	−0.849996	+	0.526789i	$0.823396\pi$
$54$	0	0
$55$	−23.9260	−3.22618
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−12.6472	−1.64653	−0.823266	−	0.567656i	$-0.807850\pi$
$59$	−12.6472	−1.64653	−0.823266	+	0.567656i	$0.807850\pi$
$60$	0	0
$61$	−1.18806	−0.152116	−0.0760579	−	0.997103i	$-0.524233\pi$
$61$	−1.18806	−0.152116	−0.0760579	+	0.997103i	$0.524233\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−12.8341	−1.59188
$66$	0	0
$67$	10.5516	1.28908	0.644542	−	0.764569i	$-0.277048\pi$
$67$	10.5516	1.28908	0.644542	+	0.764569i	$0.277048\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−12.9490	−1.53676	−0.768382	−	0.639991i	$-0.778938\pi$
$71$	−12.9490	−1.53676	−0.768382	+	0.639991i	$0.778938\pi$
$72$	0	0
$73$	6.00000	0.702247	0.351123	−	0.936329i	$-0.385800\pi$
$73$	6.00000	0.702247	0.351123	+	0.936329i	$0.385800\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	16.7523	1.90910
$78$	0	0
$79$	15.0446	1.69265	0.846327	−	0.532663i	$-0.178809\pi$
$79$	15.0446	1.69265	0.846327	+	0.532663i	$0.178809\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	12.6472	1.38822	0.694108	−	0.719871i	$-0.255799\pi$
$83$	12.6472	1.38822	0.694108	+	0.719871i	$0.255799\pi$
$84$	0	0
$85$	3.88202	0.421064
$86$	0	0
$87$	0	0
$88$	0	0
$89$	8.61208	0.912879	0.456440	−	0.889754i	$-0.349124\pi$
$89$	8.61208	0.912879	0.456440	+	0.889754i	$0.349124\pi$
$90$	0	0
$91$	8.98606	0.941995
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−4.06765	−0.417332
$96$	0	0
$97$	16.3761	1.66274	0.831372	−	0.555717i	$-0.187556\pi$
$97$	16.3761	1.66274	0.831372	+	0.555717i	$0.187556\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−9.76404	−0.971558	−0.485779	−	0.874082i	$-0.661464\pi$
$101$	−9.76404	−0.971558	−0.485779	+	0.874082i	$0.661464\pi$
$102$	0	0
$103$	−6.16328	−0.607287	−0.303643	−	0.952786i	$-0.598203\pi$
$103$	−6.16328	−0.607287	−0.303643	+	0.952786i	$0.598203\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−8.25892	−0.798420	−0.399210	−	0.916860i	$-0.630716\pi$
$107$	−8.25892	−0.798420	−0.399210	+	0.916860i	$0.630716\pi$
$108$	0	0
$109$	2.81194	0.269335	0.134667	−	0.990891i	$-0.457003\pi$
$109$	2.81194	0.269335	0.134667	+	0.990891i	$0.457003\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	12.2220	1.14975	0.574876	−	0.818240i	$-0.305050\pi$
$113$	12.2220	1.14975	0.574876	+	0.818240i	$0.305050\pi$
$114$	0	0
$115$	−34.4776	−3.21505
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−2.71807	−0.249165
$120$	0	0
$121$	26.9861	2.45328
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−19.6822	−1.76043
$126$	0	0
$127$	0.727141	0.0645233	0.0322617	−	0.999479i	$-0.489729\pi$
$127$	0.727141	0.0645233	0.0322617	+	0.999479i	$0.489729\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	4.06765	0.355392	0.177696	−	0.984085i	$-0.443136\pi$
$131$	4.06765	0.355392	0.177696	+	0.984085i	$0.443136\pi$
$132$	0	0
$133$	2.84804	0.246957
$134$	0	0
$135$	0	0
$136$	0	0
$137$	8.00000	0.683486	0.341743	−	0.939793i	$-0.388983\pi$
$137$	8.00000	0.683486	0.341743	+	0.939793i	$0.388983\pi$
$138$	0	0
$139$	−15.6671	−1.32886	−0.664432	−	0.747349i	$-0.731327\pi$
$139$	−15.6671	−1.32886	−0.664432	+	0.747349i	$0.731327\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	20.3761	1.70393
$144$	0	0
$145$	10.0000	0.830455
$146$	0	0
$147$	0	0
$148$	0	0
$149$	6.91600	0.566581	0.283290	−	0.959034i	$-0.408574\pi$
$149$	6.91600	0.566581	0.283290	+	0.959034i	$0.408574\pi$
$150$	0	0
$151$	−10.8723	−0.884774	−0.442387	−	0.896824i	$-0.645868\pi$
$151$	−10.8723	−0.884774	−0.442387	+	0.896824i	$0.645868\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−10.5516	−0.847526
$156$	0	0
$157$	−3.07008	−0.245019	−0.122510	−	0.992467i	$-0.539094\pi$
$157$	−3.07008	−0.245019	−0.122510	+	0.992467i	$0.539094\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	24.1402	1.90251
$162$	0	0
$163$	10.2309	0.801349	0.400674	−	0.916221i	$-0.368776\pi$
$163$	10.2309	0.801349	0.400674	+	0.916221i	$0.368776\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	3.44521	0.266599	0.133299	−	0.991076i	$-0.457443\pi$
$167$	3.44521	0.266599	0.133299	+	0.991076i	$0.457443\pi$
$168$	0	0
$169$	−2.07008	−0.159237
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−6.73006	−0.511677	−0.255839	−	0.966719i	$-0.582352\pi$
$173$	−6.73006	−0.511677	−0.255839	+	0.966719i	$0.582352\pi$
$174$	0	0
$175$	27.3712	2.06907
$176$	0	0
$177$	0	0
$178$	0	0
$179$	10.8723	0.812633	0.406316	−	0.913732i	$-0.366813\pi$
$179$	10.8723	0.812633	0.406316	+	0.913732i	$0.366813\pi$
$180$	0	0
$181$	12.0361	0.894637	0.447318	−	0.894375i	$-0.352379\pi$
$181$	12.0361	0.894637	0.447318	+	0.894375i	$0.352379\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	34.7523	2.55504
$186$	0	0
$187$	−6.16328	−0.450704
$188$	0	0
$189$	0	0
$190$	0	0
$191$	8.98606	0.650209	0.325104	−	0.945678i	$-0.394601\pi$
$191$	8.98606	0.650209	0.325104	+	0.945678i	$0.394601\pi$
$192$	0	0
$193$	−9.29212	−0.668862	−0.334431	−	0.942420i	$-0.608544\pi$
$193$	−9.29212	−0.668862	−0.334431	+	0.942420i	$0.608544\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−0.353938	−0.0252171	−0.0126085	−	0.999921i	$-0.504014\pi$
$197$	−0.353938	−0.0252171	−0.0126085	+	0.999921i	$0.504014\pi$
$198$	0	0
$199$	15.0446	1.06649	0.533243	−	0.845962i	$-0.320973\pi$
$199$	15.0446	1.06649	0.533243	+	0.845962i	$0.320973\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−7.00169	−0.491423
$204$	0	0
$205$	27.4462	1.91693
$206$	0	0
$207$	0	0
$208$	0	0
$209$	6.45800	0.446709
$210$	0	0
$211$	−8.98606	−0.618626	−0.309313	−	0.950960i	$-0.600099\pi$
$211$	−8.98606	−0.618626	−0.309313	+	0.950960i	$0.600099\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−36.8939	−2.51614
$216$	0	0
$217$	7.38792	0.501524
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−3.30604	−0.222388
$222$	0	0
$223$	−8.25892	−0.553058	−0.276529	−	0.961006i	$-0.589184\pi$
$223$	−8.25892	−0.553058	−0.276529	+	0.961006i	$0.589184\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−10.9581	−0.727313	−0.363657	−	0.931533i	$-0.618472\pi$
$227$	−10.9581	−0.727313	−0.363657	+	0.931533i	$0.618472\pi$
$228$	0	0
$229$	−26.2803	−1.73665	−0.868327	−	0.495993i	$-0.834804\pi$
$229$	−26.2803	−1.73665	−0.868327	+	0.495993i	$0.834804\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	19.3060	1.26478	0.632390	−	0.774650i	$-0.282074\pi$
$233$	19.3060	1.26478	0.632390	+	0.774650i	$0.282074\pi$
$234$	0	0
$235$	−12.9679	−0.845934
$236$	0	0
$237$	0	0
$238$	0	0
$239$	6.89043	0.445705	0.222852	−	0.974852i	$-0.428463\pi$
$239$	6.89043	0.445705	0.222852	+	0.974852i	$0.428463\pi$
$240$	0	0
$241$	−0.539871	−0.0347762	−0.0173881	−	0.999849i	$-0.505535\pi$
$241$	−0.539871	−0.0347762	−0.0173881	+	0.999849i	$0.505535\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−1.50589	−0.0962081
$246$	0	0
$247$	3.46413	0.220417
$248$	0	0
$249$	0	0
$250$	0	0
$251$	12.3266	0.778046	0.389023	−	0.921228i	$-0.372813\pi$
$251$	12.3266	0.778046	0.389023	+	0.921228i	$0.372813\pi$
$252$	0	0
$253$	54.7383	3.44137
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−5.22417	−0.325875	−0.162937	−	0.986636i	$-0.552097\pi$
$257$	−5.22417	−0.325875	−0.162937	+	0.986636i	$0.552097\pi$
$258$	0	0
$259$	−24.3325	−1.51195
$260$	0	0
$261$	0	0
$262$	0	0
$263$	17.1214	1.05575	0.527874	−	0.849323i	$-0.322989\pi$
$263$	17.1214	1.05575	0.527874	+	0.849323i	$0.322989\pi$
$264$	0	0
$265$	48.0444	2.95134
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−9.26994	−0.565198	−0.282599	−	0.959238i	$-0.591197\pi$
$269$	−9.26994	−0.565198	−0.282599	+	0.959238i	$0.591197\pi$
$270$	0	0
$271$	−20.3761	−1.23776	−0.618879	−	0.785486i	$-0.712413\pi$
$271$	−20.3761	−1.23776	−0.618879	+	0.785486i	$0.712413\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	62.0648	3.74265
$276$	0	0
$277$	−23.3282	−1.40166	−0.700829	−	0.713330i	$-0.747186\pi$
$277$	−23.3282	−1.40166	−0.700829	+	0.713330i	$0.747186\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	12.6121	0.752374	0.376187	−	0.926544i	$-0.377235\pi$
$281$	12.6121	0.752374	0.376187	+	0.926544i	$0.377235\pi$
$282$	0	0
$283$	17.7627	1.05588	0.527942	−	0.849281i	$-0.322964\pi$
$283$	17.7627	1.05588	0.527942	+	0.849281i	$0.322964\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−19.2170	−1.13434
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	0	0
$292$	0	0
$293$	33.0562	1.93116	0.965581	−	0.260103i	$-0.0837565\pi$
$293$	33.0562	1.93116	0.965581	+	0.260103i	$0.0837565\pi$
$294$	0	0
$295$	49.0969	2.85853
$296$	0	0
$297$	0	0
$298$	0	0
$299$	29.3621	1.69806
$300$	0	0
$301$	25.8320	1.48893
$302$	0	0
$303$	0	0
$304$	0	0
$305$	4.61208	0.264087
$306$	0	0
$307$	0	0		−	1.00000i	$-0.5\pi$
$307$	0	0			1.00000i	$0.5\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	18.3852	1.04253	0.521263	−	0.853396i	$-0.325461\pi$
$311$	18.3852	1.04253	0.521263	+	0.853396i	$0.325461\pi$
$312$	0	0
$313$	−8.37613	−0.473447	−0.236723	−	0.971577i	$-0.576073\pi$
$313$	−8.37613	−0.473447	−0.236723	+	0.971577i	$0.576073\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−22.5760	−1.26799	−0.633997	−	0.773336i	$-0.718587\pi$
$317$	−22.5760	−1.26799	−0.633997	+	0.773336i	$0.718587\pi$
$318$	0	0
$319$	−15.8765	−0.888913
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−1.04782	−0.0583021
$324$	0	0
$325$	33.2921	1.84671
$326$	0	0
$327$	0	0
$328$	0	0
$329$	9.07974	0.500582
$330$	0	0
$331$	26.0216	1.43028	0.715139	−	0.698982i	$-0.246363\pi$
$331$	26.0216	1.43028	0.715139	+	0.698982i	$0.246363\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−40.9616	−2.23797
$336$	0	0
$337$	−16.1402	−0.879211	−0.439605	−	0.898191i	$-0.644882\pi$
$337$	−16.1402	−0.879211	−0.439605	+	0.898191i	$0.644882\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	16.7523	0.907185
$342$	0	0
$343$	−17.9721	−0.970403
$344$	0	0
$345$	0	0
$346$	0	0
$347$	0.641353	0.0344296	0.0172148	−	0.999852i	$-0.494520\pi$
$347$	0.641353	0.0344296	0.0172148	+	0.999852i	$0.494520\pi$
$348$	0	0
$349$	12.2220	0.654231	0.327115	−	0.944984i	$-0.393923\pi$
$349$	12.2220	0.654231	0.327115	+	0.944984i	$0.393923\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−21.0562	−1.12071	−0.560353	−	0.828254i	$-0.689335\pi$
$353$	−21.0562	−1.12071	−0.560353	+	0.828254i	$0.689335\pi$
$354$	0	0
$355$	50.2683	2.66796
$356$	0	0
$357$	0	0
$358$	0	0
$359$	26.7488	1.41175	0.705873	−	0.708338i	$-0.250555\pi$
$359$	26.7488	1.41175	0.705873	+	0.708338i	$0.250555\pi$
$360$	0	0
$361$	−17.9021	−0.942215
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−23.2921	−1.21917
$366$	0	0
$367$	19.2359	1.00411	0.502053	−	0.864837i	$-0.332578\pi$
$367$	19.2359	1.00411	0.502053	+	0.864837i	$0.332578\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−33.6392	−1.74646
$372$	0	0
$373$	−14.6800	−0.760104	−0.380052	−	0.924965i	$-0.624094\pi$
$373$	−14.6800	−0.760104	−0.380052	+	0.924965i	$0.624094\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−8.51629	−0.438611
$378$	0	0
$379$	21.1032	1.08400	0.542000	−	0.840379i	$-0.317667\pi$
$379$	21.1032	1.08400	0.542000	+	0.840379i	$0.317667\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	7.53178	0.384856	0.192428	−	0.981311i	$-0.438364\pi$
$383$	7.53178	0.384856	0.192428	+	0.981311i	$0.438364\pi$
$384$	0	0
$385$	−65.0326	−3.31437
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−17.0562	−0.864782	−0.432391	−	0.901686i	$-0.642330\pi$
$389$	−17.0562	−0.864782	−0.432391	+	0.901686i	$0.642330\pi$
$390$	0	0
$391$	−8.88136	−0.449150
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−58.4036	−2.93860
$396$	0	0
$397$	24.4802	1.22863	0.614313	−	0.789063i	$-0.289433\pi$
$397$	24.4802	1.22863	0.614313	+	0.789063i	$0.289433\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−7.54200	−0.376630	−0.188315	−	0.982109i	$-0.560303\pi$
$401$	−7.54200	−0.376630	−0.188315	+	0.982109i	$0.560303\pi$
$402$	0	0
$403$	8.98606	0.447628
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−55.1744	−2.73489
$408$	0	0
$409$	1.23383	0.0610089	0.0305045	−	0.999535i	$-0.490289\pi$
$409$	1.23383	0.0610089	0.0305045	+	0.999535i	$0.490289\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−34.3761	−1.69154
$414$	0	0
$415$	−49.0969	−2.41007
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−21.7446	−1.06229	−0.531146	−	0.847280i	$-0.678238\pi$
$419$	−21.7446	−1.06229	−0.531146	+	0.847280i	$0.678238\pi$
$420$	0	0
$421$	10.8341	0.528023	0.264011	−	0.964520i	$-0.414954\pi$
$421$	10.8341	0.528023	0.264011	+	0.964520i	$0.414954\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−10.0701	−0.488471
$426$	0	0
$427$	−3.22924	−0.156274
$428$	0	0
$429$	0	0
$430$	0	0
$431$	28.0126	1.34932	0.674659	−	0.738130i	$-0.264291\pi$
$431$	28.0126	1.34932	0.674659	+	0.738130i	$0.264291\pi$
$432$	0	0
$433$	1.54200	0.0741039	0.0370519	−	0.999313i	$-0.488203\pi$
$433$	1.54200	0.0741039	0.0370519	+	0.999313i	$0.488203\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	9.30604	0.445168
$438$	0	0
$439$	6.90934	0.329765	0.164882	−	0.986313i	$-0.447276\pi$
$439$	6.90934	0.329765	0.164882	+	0.986313i	$0.447276\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	8.13530	0.386520	0.193260	−	0.981148i	$-0.438094\pi$
$443$	8.13530	0.386520	0.193260	+	0.981148i	$0.438094\pi$
$444$	0	0
$445$	−33.4323	−1.58484
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−0.775830	−0.0366137	−0.0183068	−	0.999832i	$-0.505828\pi$
$449$	−0.775830	−0.0366137	−0.0183068	+	0.999832i	$0.505828\pi$
$450$	0	0
$451$	−43.5749	−2.05186
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−34.8841	−1.63539
$456$	0	0
$457$	−9.54200	−0.446356	−0.223178	−	0.974778i	$-0.571643\pi$
$457$	−9.54200	−0.446356	−0.223178	+	0.974778i	$0.571643\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	7.90421	0.368136	0.184068	−	0.982914i	$-0.441073\pi$
$461$	7.90421	0.368136	0.184068	+	0.982914i	$0.441073\pi$
$462$	0	0
$463$	−8.98606	−0.417618	−0.208809	−	0.977956i	$-0.566959\pi$
$463$	−8.98606	−0.417618	−0.208809	+	0.977956i	$0.566959\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−20.4619	−0.946862	−0.473431	−	0.880831i	$-0.656985\pi$
$467$	−20.4619	−0.946862	−0.473431	+	0.880831i	$0.656985\pi$
$468$	0	0
$469$	28.6800	1.32432
$470$	0	0
$471$	0	0
$472$	0	0
$473$	58.5746	2.69326
$474$	0	0
$475$	10.5516	0.484141
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−14.5269	−0.663751	−0.331876	−	0.943323i	$-0.607681\pi$
$479$	−14.5269	−0.663751	−0.331876	+	0.943323i	$0.607681\pi$
$480$	0	0
$481$	−29.5960	−1.34946
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−63.5725	−2.88668
$486$	0	0
$487$	−9.60850	−0.435403	−0.217701	−	0.976015i	$-0.569856\pi$
$487$	−9.60850	−0.435403	−0.217701	+	0.976015i	$0.569856\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	29.8799	1.34846	0.674230	−	0.738521i	$-0.264476\pi$
$491$	29.8799	1.34846	0.674230	+	0.738521i	$0.264476\pi$
$492$	0	0
$493$	2.57598	0.116016
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−35.1963	−1.57877
$498$	0	0
$499$	2.69916	0.120831	0.0604154	−	0.998173i	$-0.480757\pi$
$499$	2.69916	0.120831	0.0604154	+	0.998173i	$0.480757\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−2.20034	−0.0981083	−0.0490541	−	0.998796i	$-0.515621\pi$
$503$	−2.20034	−0.0981083	−0.0490541	+	0.998796i	$0.515621\pi$
$504$	0	0
$505$	37.9042	1.68672
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−15.3879	−0.682057	−0.341029	−	0.940053i	$-0.610775\pi$
$509$	−15.3879	−0.682057	−0.341029	+	0.940053i	$0.610775\pi$
$510$	0	0
$511$	16.3084	0.721443
$512$	0	0
$513$	0	0
$514$	0	0
$515$	23.9260	1.05431
$516$	0	0
$517$	20.5885	0.905481
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−35.3504	−1.54873	−0.774365	−	0.632739i	$-0.781931\pi$
$521$	−35.3504	−1.54873	−0.774365	+	0.632739i	$0.781931\pi$
$522$	0	0
$523$	35.2048	1.53940	0.769698	−	0.638408i	$-0.220407\pi$
$523$	35.2048	1.53940	0.769698	+	0.638408i	$0.220407\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−2.71807	−0.118401
$528$	0	0
$529$	55.8785	2.42950
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−23.3740	−1.01244
$534$	0	0
$535$	32.0613	1.38613
$536$	0	0
$537$	0	0
$538$	0	0
$539$	2.39083	0.102980
$540$	0	0
$541$	11.7279	0.504223	0.252112	−	0.967698i	$-0.418875\pi$
$541$	11.7279	0.504223	0.252112	+	0.967698i	$0.418875\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−10.9160	−0.467590
$546$	0	0
$547$	15.6671	0.669876	0.334938	−	0.942240i	$-0.391285\pi$
$547$	15.6671	0.669876	0.334938	+	0.942240i	$0.391285\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−2.69916	−0.114988
$552$	0	0
$553$	40.8924	1.73892
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−4.84804	−0.205418	−0.102709	−	0.994711i	$-0.532751\pi$
$557$	−4.84804	−0.205418	−0.102709	+	0.994711i	$0.532751\pi$
$558$	0	0
$559$	31.4199	1.32892
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−24.9738	−1.05252	−0.526260	−	0.850323i	$-0.676406\pi$
$563$	−24.9738	−1.05252	−0.526260	+	0.850323i	$0.676406\pi$
$564$	0	0
$565$	−47.4462	−1.99608
$566$	0	0
$567$	0	0
$568$	0	0
$569$	13.8598	0.581034	0.290517	−	0.956870i	$-0.406173\pi$
$569$	13.8598	0.581034	0.290517	+	0.956870i	$0.406173\pi$
$570$	0	0
$571$	−34.0711	−1.42583	−0.712917	−	0.701249i	$-0.752626\pi$
$571$	−34.0711	−1.42583	−0.712917	+	0.701249i	$0.752626\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	89.4360	3.72974
$576$	0	0
$577$	−47.0422	−1.95839	−0.979197	−	0.202911i	$-0.934960\pi$
$577$	−47.0422	−1.95839	−0.979197	+	0.202911i	$0.934960\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	34.3761	1.42616
$582$	0	0
$583$	−76.2776	−3.15910
$584$	0	0
$585$	0	0
$586$	0	0
$587$	21.7446	0.897495	0.448747	−	0.893659i	$-0.351870\pi$
$587$	21.7446	0.897495	0.448747	+	0.893659i	$0.351870\pi$
$588$	0	0
$589$	2.84804	0.117352
$590$	0	0
$591$	0	0
$592$	0	0
$593$	17.3879	0.714036	0.357018	−	0.934097i	$-0.383793\pi$
$593$	17.3879	0.714036	0.357018	+	0.934097i	$0.383793\pi$
$594$	0	0
$595$	10.5516	0.432574
$596$	0	0
$597$	0	0
$598$	0	0
$599$	31.3342	1.28028	0.640139	−	0.768259i	$-0.278877\pi$
$599$	31.3342	1.28028	0.640139	+	0.768259i	$0.278877\pi$
$600$	0	0
$601$	8.14017	0.332044	0.166022	−	0.986122i	$-0.446908\pi$
$601$	8.14017	0.332044	0.166022	+	0.986122i	$0.446908\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−104.761	−4.25912
$606$	0	0
$607$	−20.6902	−0.839789	−0.419895	−	0.907573i	$-0.637933\pi$
$607$	−20.6902	−0.839789	−0.419895	+	0.907573i	$0.637933\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	11.0439	0.446787
$612$	0	0
$613$	24.4580	0.987849	0.493925	−	0.869505i	$-0.335562\pi$
$613$	24.4580	0.987849	0.493925	+	0.869505i	$0.335562\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	39.4323	1.58748	0.793742	−	0.608254i	$-0.208130\pi$
$617$	39.4323	1.58748	0.793742	+	0.608254i	$0.208130\pi$
$618$	0	0
$619$	−22.5575	−0.906663	−0.453331	−	0.891342i	$-0.649765\pi$
$619$	−22.5575	−0.906663	−0.453331	+	0.891342i	$0.649765\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	23.4083	0.937832
$624$	0	0
$625$	26.0562	1.04225
$626$	0	0
$627$	0	0
$628$	0	0
$629$	8.95210	0.356944
$630$	0	0
$631$	−25.8122	−1.02757	−0.513784	−	0.857919i	$-0.671757\pi$
$631$	−25.8122	−1.02757	−0.513784	+	0.857919i	$0.671757\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−2.82278	−0.112018
$636$	0	0
$637$	1.28246	0.0508131
$638$	0	0
$639$	0	0
$640$	0	0
$641$	38.5982	1.52454	0.762268	−	0.647262i	$-0.224086\pi$
$641$	38.5982	1.52454	0.762268	+	0.647262i	$0.224086\pi$
$642$	0	0
$643$	5.64555	0.222639	0.111319	−	0.993785i	$-0.464492\pi$
$643$	5.64555	0.222639	0.111319	+	0.993785i	$0.464492\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−26.5394	−1.04337	−0.521685	−	0.853138i	$-0.674696\pi$
$647$	−26.5394	−1.04337	−0.521685	+	0.853138i	$0.674696\pi$
$648$	0	0
$649$	−77.9486	−3.05975
$650$	0	0
$651$	0	0
$652$	0	0
$653$	8.32610	0.325825	0.162913	−	0.986640i	$-0.447911\pi$
$653$	8.32610	0.325825	0.162913	+	0.986640i	$0.447911\pi$
$654$	0	0
$655$	−15.7907	−0.616994
$656$	0	0
$657$	0	0
$658$	0	0
$659$	24.9738	0.972842	0.486421	−	0.873725i	$-0.338302\pi$
$659$	24.9738	0.972842	0.486421	+	0.873725i	$0.338302\pi$
$660$	0	0
$661$	−1.60995	−0.0626200	−0.0313100	−	0.999510i	$-0.509968\pi$
$661$	−1.60995	−0.0626200	−0.0313100	+	0.999510i	$0.509968\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−11.0562	−0.428740
$666$	0	0
$667$	−22.8782	−0.885847
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−7.32237	−0.282677
$672$	0	0
$673$	−45.1284	−1.73957	−0.869786	−	0.493430i	$-0.835743\pi$
$673$	−45.1284	−1.73957	−0.869786	+	0.493430i	$0.835743\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	5.03398	0.193471	0.0967357	−	0.995310i	$-0.469160\pi$
$677$	5.03398	0.193471	0.0967357	+	0.995310i	$0.469160\pi$
$678$	0	0
$679$	44.5115	1.70819
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−18.2804	−0.699482	−0.349741	−	0.936846i	$-0.613730\pi$
$683$	−18.2804	−0.699482	−0.349741	+	0.936846i	$0.613730\pi$
$684$	0	0
$685$	−31.0562	−1.18660
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−40.9160	−1.55878
$690$	0	0
$691$	−41.6029	−1.58265	−0.791325	−	0.611396i	$-0.790608\pi$
$691$	−41.6029	−1.58265	−0.791325	+	0.611396i	$0.790608\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	60.8199	2.30703
$696$	0	0
$697$	7.07008	0.267799
$698$	0	0
$699$	0	0
$700$	0	0
$701$	21.2921	0.804192	0.402096	−	0.915597i	$-0.368282\pi$
$701$	21.2921	0.804192	0.402096	+	0.915597i	$0.368282\pi$
$702$	0	0
$703$	−9.38017	−0.353780
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−26.5394	−0.998116
$708$	0	0
$709$	−11.1201	−0.417624	−0.208812	−	0.977956i	$-0.566960\pi$
$709$	−11.1201	−0.417624	−0.208812	+	0.977956i	$0.566960\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	24.1402	0.904056
$714$	0	0
$715$	−79.1004	−2.95819
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−23.9071	−0.891584	−0.445792	−	0.895137i	$-0.647078\pi$
$719$	−23.9071	−0.891584	−0.445792	+	0.895137i	$0.647078\pi$
$720$	0	0
$721$	−16.7523	−0.623886
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−25.9403	−0.963399
$726$	0	0
$727$	25.2945	0.938121	0.469060	−	0.883166i	$-0.344593\pi$
$727$	25.2945	0.938121	0.469060	+	0.883166i	$0.344593\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−9.50379	−0.351511
$732$	0	0
$733$	37.7362	1.39382	0.696909	−	0.717159i	$-0.254558\pi$
$733$	37.7362	1.39382	0.696909	+	0.717159i	$0.254558\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	65.0326	2.39551
$738$	0	0
$739$	22.1510	0.814839	0.407420	−	0.913241i	$-0.366429\pi$
$739$	22.1510	0.814839	0.407420	+	0.913241i	$0.366429\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−0.831845	−0.0305174	−0.0152587	−	0.999884i	$-0.504857\pi$
$743$	−0.831845	−0.0305174	−0.0152587	+	0.999884i	$0.504857\pi$
$744$	0	0
$745$	−26.8480	−0.983636
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−22.4483	−0.820244
$750$	0	0
$751$	20.6902	0.754996	0.377498	−	0.926010i	$-0.376785\pi$
$751$	20.6902	0.754996	0.377498	+	0.926010i	$0.376785\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	42.2064	1.53605
$756$	0	0
$757$	0.149826	0.00544553	0.00272277	−	0.999996i	$-0.499133\pi$
$757$	0.149826	0.00544553	0.00272277	+	0.999996i	$0.499133\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	2.77583	0.100624	0.0503119	−	0.998734i	$-0.483978\pi$
$761$	2.77583	0.100624	0.0503119	+	0.998734i	$0.483978\pi$
$762$	0	0
$763$	7.64305	0.276697
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−41.8123	−1.50976
$768$	0	0
$769$	−8.45800	−0.305003	−0.152502	−	0.988303i	$-0.548733\pi$
$769$	−8.45800	−0.305003	−0.152502	+	0.988303i	$0.548733\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	18.0000	0.647415	0.323708	−	0.946157i	$-0.395071\pi$
$773$	18.0000	0.647415	0.323708	+	0.946157i	$0.395071\pi$
$774$	0	0
$775$	27.3712	0.983203
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−7.40816	−0.265425
$780$	0	0
$781$	−79.8084	−2.85577
$782$	0	0
$783$	0	0
$784$	0	0
$785$	11.9181	0.425376
$786$	0	0
$787$	53.9295	1.92238	0.961189	−	0.275889i	$-0.0889723\pi$
$787$	53.9295	1.92238	0.961189	+	0.275889i	$0.0889723\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	33.2204	1.18118
$792$	0	0
$793$	−3.92779	−0.139480
$794$	0	0
$795$	0	0
$796$	0	0
$797$	26.9882	0.955971	0.477986	−	0.878368i	$-0.341367\pi$
$797$	26.9882	0.955971	0.477986	+	0.878368i	$0.341367\pi$
$798$	0	0
$799$	−3.34051	−0.118179
$800$	0	0
$801$	0	0
$802$	0	0
$803$	36.9797	1.30499
$804$	0	0
$805$	−93.7126	−3.30294
$806$	0	0
$807$	0	0
$808$	0	0
$809$	24.7383	0.869753	0.434877	−	0.900490i	$-0.356792\pi$
$809$	24.7383	0.869753	0.434877	+	0.900490i	$0.356792\pi$
$810$	0	0
$811$	1.45428	0.0510667	0.0255334	−	0.999674i	$-0.491872\pi$
$811$	1.45428	0.0510667	0.0255334	+	0.999674i	$0.491872\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−39.7167	−1.39122
$816$	0	0
$817$	9.95824	0.348395
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−26.1859	−0.913895	−0.456948	−	0.889494i	$-0.651057\pi$
$821$	−26.1859	−0.913895	−0.456948	+	0.889494i	$0.651057\pi$
$822$	0	0
$823$	−0.0189166	−0.000659392 0	−0.000329696	−	1.00000i	$-0.500105\pi$
$823$	−0.0189166	−0.000659392 0	−0.000329696		1.00000i	$0.500105\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	14.2986	0.497210	0.248605	−	0.968605i	$-0.420028\pi$
$827$	14.2986	0.497210	0.248605	+	0.968605i	$0.420028\pi$
$828$	0	0
$829$	−2.00000	−0.0694629	−0.0347314	−	0.999397i	$-0.511058\pi$
$829$	−2.00000	−0.0694629	−0.0347314	+	0.999397i	$0.511058\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−0.387915	−0.0134405
$834$	0	0
$835$	−13.3744	−0.462840
$836$	0	0
$837$	0	0
$838$	0	0
$839$	33.3251	1.15051	0.575255	−	0.817974i	$-0.304903\pi$
$839$	33.3251	1.15051	0.575255	+	0.817974i	$0.304903\pi$
$840$	0	0
$841$	−22.3643	−0.771184
$842$	0	0
$843$	0	0
$844$	0	0
$845$	8.03611	0.276450
$846$	0	0
$847$	73.3501	2.52034
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−79.5068	−2.72546
$852$	0	0
$853$	−41.6321	−1.42546	−0.712729	−	0.701440i	$-0.752541\pi$
$853$	−41.6321	−1.42546	−0.712729	+	0.701440i	$0.752541\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−13.5281	−0.462110	−0.231055	−	0.972941i	$-0.574218\pi$
$857$	−13.5281	−0.462110	−0.231055	+	0.972941i	$0.574218\pi$
$858$	0	0
$859$	−17.9721	−0.613201	−0.306600	−	0.951838i	$-0.599192\pi$
$859$	−17.9721	−0.613201	−0.306600	+	0.951838i	$0.599192\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−6.68102	−0.227424	−0.113712	−	0.993514i	$-0.536274\pi$
$863$	−6.68102	−0.227424	−0.113712	+	0.993514i	$0.536274\pi$
$864$	0	0
$865$	26.1262	0.888319
$866$	0	0
$867$	0	0
$868$	0	0
$869$	92.7244	3.14546
$870$	0	0
$871$	34.8841	1.18200
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−53.4975	−1.80855
$876$	0	0
$877$	30.6204	1.03398	0.516988	−	0.855993i	$-0.327053\pi$
$877$	30.6204	1.03398	0.516988	+	0.855993i	$0.327053\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−2.54413	−0.0857140	−0.0428570	−	0.999081i	$-0.513646\pi$
$881$	−2.54413	−0.0857140	−0.0428570	+	0.999081i	$0.513646\pi$
$882$	0	0
$883$	−33.8363	−1.13868	−0.569340	−	0.822102i	$-0.692801\pi$
$883$	−33.8363	−1.13868	−0.569340	+	0.822102i	$0.692801\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−17.2261	−0.578395	−0.289197	−	0.957270i	$-0.593388\pi$
$887$	−17.2261	−0.578395	−0.289197	+	0.957270i	$0.593388\pi$
$888$	0	0
$889$	1.97642	0.0662871
$890$	0	0
$891$	0	0
$892$	0	0
$893$	3.50024	0.117131
$894$	0	0
$895$	−42.2064	−1.41081
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−7.00169	−0.233520
$900$	0	0
$901$	12.3761	0.412309
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−46.7244	−1.55317
$906$	0	0
$907$	−1.24487	−0.0413353	−0.0206677	−	0.999786i	$-0.506579\pi$
$907$	−1.24487	−0.0413353	−0.0206677	+	0.999786i	$0.506579\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−21.2079	−0.702650	−0.351325	−	0.936254i	$-0.614269\pi$
$911$	−21.2079	−0.702650	−0.351325	+	0.936254i	$0.614269\pi$
$912$	0	0
$913$	77.9486	2.57972
$914$	0	0
$915$	0	0
$916$	0	0
$917$	11.0562	0.365107
$918$	0	0
$919$	18.2804	0.603016	0.301508	−	0.953464i	$-0.402510\pi$
$919$	18.2804	0.603016	0.301508	+	0.953464i	$0.402510\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−42.8100	−1.40911
$924$	0	0
$925$	−90.1484	−2.96406
$926$	0	0
$927$	0	0
$928$	0	0
$929$	38.8785	1.27556	0.637781	−	0.770218i	$-0.279852\pi$
$929$	38.8785	1.27556	0.637781	+	0.770218i	$0.279852\pi$
$930$	0	0
$931$	0.406464	0.0133213
$932$	0	0
$933$	0	0
$934$	0	0
$935$	23.9260	0.782464
$936$	0	0
$937$	−15.8363	−0.517348	−0.258674	−	0.965965i	$-0.583286\pi$
$937$	−15.8363	−0.517348	−0.258674	+	0.965965i	$0.583286\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−26.8563	−0.875491	−0.437745	−	0.899099i	$-0.644223\pi$
$941$	−26.8563	−0.875491	−0.437745	+	0.899099i	$0.644223\pi$
$942$	0	0
$943$	−62.7919	−2.04479
$944$	0	0
$945$	0	0
$946$	0	0
$947$	19.8583	0.645310	0.322655	−	0.946517i	$-0.395425\pi$
$947$	19.8583	0.645310	0.322655	+	0.946517i	$0.395425\pi$
$948$	0	0
$949$	19.8363	0.643912
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−5.36434	−0.173768	−0.0868840	−	0.996218i	$-0.527691\pi$
$953$	−5.36434	−0.173768	−0.0868840	+	0.996218i	$0.527691\pi$
$954$	0	0
$955$	−34.8841	−1.12882
$956$	0	0
$957$	0	0
$958$	0	0
$959$	21.7446	0.702169
$960$	0	0
$961$	−23.6121	−0.761680
$962$	0	0
$963$	0	0
$964$	0	0
$965$	36.0722	1.16121
$966$	0	0
$967$	−38.1388	−1.22646	−0.613230	−	0.789904i	$-0.710130\pi$
$967$	−38.1388	−1.22646	−0.613230	+	0.789904i	$0.710130\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	21.4239	0.687526	0.343763	−	0.939057i	$-0.388298\pi$
$971$	21.4239	0.687526	0.343763	+	0.939057i	$0.388298\pi$
$972$	0	0
$973$	−42.5842	−1.36519
$974$	0	0
$975$	0	0
$976$	0	0
$977$	30.7480	0.983715	0.491858	−	0.870676i	$-0.336318\pi$
$977$	30.7480	0.983715	0.491858	+	0.870676i	$0.336318\pi$
$978$	0	0
$979$	53.0787	1.69640
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−46.8965	−1.49577	−0.747884	−	0.663830i	$-0.768930\pi$
$983$	−46.8965	−1.49577	−0.747884	+	0.663830i	$0.768930\pi$
$984$	0	0
$985$	1.37400	0.0437792
$986$	0	0
$987$	0	0
$988$	0	0
$989$	84.4066	2.68397
$990$	0	0
$991$	−8.36362	−0.265679	−0.132840	−	0.991138i	$-0.542410\pi$
$991$	−8.36362	−0.265679	−0.132840	+	0.991138i	$0.542410\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−58.4036	−1.85152
$996$	0	0
$997$	−46.1041	−1.46013	−0.730065	−	0.683378i	$-0.760510\pi$
$997$	−46.1041	−1.46013	−0.730065	+	0.683378i	$0.760510\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4896.2.a.bk.1.2	yes	6
3.2	odd	2		4896.2.a.bl.1.6	yes	6
4.3	odd	2	inner	4896.2.a.bk.1.1	✓	6
8.3	odd	2		9792.2.a.dr.1.5		6
8.5	even	2		9792.2.a.dr.1.6		6
12.11	even	2		4896.2.a.bl.1.5	yes	6
24.5	odd	2		9792.2.a.dq.1.2		6
24.11	even	2		9792.2.a.dq.1.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4896.2.a.bk.1.1	✓	6	4.3	odd	2	inner
4896.2.a.bk.1.2	yes	6	1.1	even	1	trivial
4896.2.a.bl.1.5	yes	6	12.11	even	2
4896.2.a.bl.1.6	yes	6	3.2	odd	2
9792.2.a.dq.1.1		6	24.11	even	2
9792.2.a.dq.1.2		6	24.5	odd	2
9792.2.a.dr.1.5		6	8.3	odd	2
9792.2.a.dr.1.6		6	8.5	even	2

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:	\( N \)	\(=\)	\( 4896 = 2^{5} \cdot 3^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4896.a (trivial)

\(f(q)\)	\(=\)	\(q-3.88202 q^{5} +2.71807 q^{7} +6.16328 q^{11} +3.30604 q^{13} -1.00000 q^{17} +1.04782 q^{19} +8.88136 q^{23} +10.0701 q^{25} -2.57598 q^{29} +2.71807 q^{31} -10.5516 q^{35} -8.95210 q^{37} -7.07008 q^{41} +9.50379 q^{43} +3.34051 q^{47} +0.387915 q^{49} -12.3761 q^{53} -23.9260 q^{55} -12.6472 q^{59} -1.18806 q^{61} -12.8341 q^{65} +10.5516 q^{67} -12.9490 q^{71} +6.00000 q^{73} +16.7523 q^{77} +15.0446 q^{79} +12.6472 q^{83} +3.88202 q^{85} +8.61208 q^{89} +8.98606 q^{91} -4.06765 q^{95} +16.3761 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 6 q^{5} + 6 q^{13} - 6 q^{17} + 12 q^{25} - 12 q^{29} + 12 q^{37} + 6 q^{41} + 30 q^{49} - 12 q^{53} + 24 q^{61} + 6 q^{65} + 36 q^{73} - 24 q^{77} + 6 q^{85} + 24 q^{89} + 36 q^{97}+O(q^{100}) \) 6 * q - 6 * q^5 + 6 * q^13 - 6 * q^17 + 12 * q^25 - 12 * q^29 + 12 * q^37 + 6 * q^41 + 30 * q^49 - 12 * q^53 + 24 * q^61 + 6 * q^65 + 36 * q^73 - 24 * q^77 + 6 * q^85 + 24 * q^89 + 36 * q^97

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