LMFDB - Embedded newform 6348.2.a.s.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6348,2,Mod(1,6348)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6348, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("6348.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6348 = 2^{2} \cdot 3 \cdot 23^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6348.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:	$50.6890352031$
Analytic rank:	$1$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 4x^{9} - 14x^{8} + 65x^{7} + 57x^{6} - 354x^{5} - 46x^{4} + 714x^{3} - 74x^{2} - 323x - 23 $ x^10 - 4x^9 - 14x^8 + 65x^7 + 57x^6 - 354x^5 - 46x^4 + 714x^3 - 74x^2 - 323*x - 23
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 276)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-0.633097$ of defining polynomial
Character	$\chi$	$=$	6348.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -1.52600 q^{5} -3.74770 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -1.52600 q^{5} -3.74770 q^{7} +1.00000 q^{9} -1.08855 q^{11} +3.02820 q^{13} -1.52600 q^{15} +3.10326 q^{17} -4.15119 q^{19} -3.74770 q^{21} -2.67134 q^{25} +1.00000 q^{27} +2.66395 q^{29} +6.80696 q^{31} -1.08855 q^{33} +5.71897 q^{35} +6.84075 q^{37} +3.02820 q^{39} +9.95527 q^{41} +4.00770 q^{43} -1.52600 q^{45} -1.56940 q^{47} +7.04524 q^{49} +3.10326 q^{51} -2.86932 q^{53} +1.66112 q^{55} -4.15119 q^{57} -14.7252 q^{59} -5.23520 q^{61} -3.74770 q^{63} -4.62102 q^{65} -4.63334 q^{67} -15.2454 q^{71} +3.21760 q^{73} -2.67134 q^{75} +4.07954 q^{77} -3.09813 q^{79} +1.00000 q^{81} +5.43710 q^{83} -4.73557 q^{85} +2.66395 q^{87} -15.0633 q^{89} -11.3488 q^{91} +6.80696 q^{93} +6.33470 q^{95} -3.83748 q^{97} -1.08855 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 10 q^{3} - 2 q^{5} - 11 q^{7} + 10 q^{9}+O(q^{10}) $ 10 * q + 10 * q^3 - 2 * q^5 - 11 * q^7 + 10 * q^9	$ 10 q + 10 q^{3} - 2 q^{5} - 11 q^{7} + 10 q^{9} - 11 q^{11} - 2 q^{15} - 13 q^{17} - 18 q^{19} - 11 q^{21} + 10 q^{25} + 10 q^{27} + 5 q^{29} + 15 q^{31} - 11 q^{33} - 13 q^{35} - 5 q^{37} + 24 q^{41} - 40 q^{43} - 2 q^{45} - 9 q^{47} + 5 q^{49} - 13 q^{51} - 6 q^{53} - 14 q^{55} - 18 q^{57} + 28 q^{59} - 39 q^{61} - 11 q^{63} - 14 q^{65} - 32 q^{67} - 33 q^{71} - 50 q^{73} + 10 q^{75} + 19 q^{77} - 33 q^{79} + 10 q^{81} - 29 q^{83} - 21 q^{85} + 5 q^{87} - 17 q^{89} - 36 q^{91} + 15 q^{93} - 42 q^{95} - 46 q^{97} - 11 q^{99}+O(q^{100}) $ 10 * q + 10 * q^3 - 2 * q^5 - 11 * q^7 + 10 * q^9 - 11 * q^11 - 2 * q^15 - 13 * q^17 - 18 * q^19 - 11 * q^21 + 10 * q^25 + 10 * q^27 + 5 * q^29 + 15 * q^31 - 11 * q^33 - 13 * q^35 - 5 * q^37 + 24 * q^41 - 40 * q^43 - 2 * q^45 - 9 * q^47 + 5 * q^49 - 13 * q^51 - 6 * q^53 - 14 * q^55 - 18 * q^57 + 28 * q^59 - 39 * q^61 - 11 * q^63 - 14 * q^65 - 32 * q^67 - 33 * q^71 - 50 * q^73 + 10 * q^75 + 19 * q^77 - 33 * q^79 + 10 * q^81 - 29 * q^83 - 21 * q^85 + 5 * q^87 - 17 * q^89 - 36 * q^91 + 15 * q^93 - 42 * q^95 - 46 * q^97 - 11 * q^99

Display 100 coefficients 10 coefficients

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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	−1.52600	−0.682446	−0.341223	−	0.939982i	$-0.610841\pi$
$5$	−1.52600	−0.682446	−0.341223	+	0.939982i	$0.610841\pi$
$6$	0	0
$7$	−3.74770	−1.41650	−0.708248	−	0.705963i	$-0.750514\pi$
$7$	−3.74770	−1.41650	−0.708248	+	0.705963i	$0.750514\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−1.08855	−0.328209	−0.164104	−	0.986443i	$-0.552473\pi$
$11$	−1.08855	−0.328209	−0.164104	+	0.986443i	$0.552473\pi$
$12$	0	0
$13$	3.02820	0.839872	0.419936	−	0.907554i	$-0.362053\pi$
$13$	3.02820	0.839872	0.419936	+	0.907554i	$0.362053\pi$
$14$	0	0
$15$	−1.52600	−0.394011
$16$	0	0
$17$	3.10326	0.752652	0.376326	−	0.926487i	$-0.377187\pi$
$17$	3.10326	0.752652	0.376326	+	0.926487i	$0.377187\pi$
$18$	0	0
$19$	−4.15119	−0.952348	−0.476174	−	0.879351i	$-0.657977\pi$
$19$	−4.15119	−0.952348	−0.476174	+	0.879351i	$0.657977\pi$
$20$	0	0
$21$	−3.74770	−0.817815
$22$	0	0
$23$	0	0
$23$	0	0
$24$	0	0
$25$	−2.67134	−0.534267
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	2.66395	0.494683	0.247342	−	0.968928i	$-0.420443\pi$
$29$	2.66395	0.494683	0.247342	+	0.968928i	$0.420443\pi$
$30$	0	0
$31$	6.80696	1.22257	0.611283	−	0.791412i	$-0.290654\pi$
$31$	6.80696	1.22257	0.611283	+	0.791412i	$0.290654\pi$
$32$	0	0
$33$	−1.08855	−0.189492
$34$	0	0
$35$	5.71897	0.966683
$36$	0	0
$37$	6.84075	1.12461	0.562306	−	0.826929i	$-0.309914\pi$
$37$	6.84075	1.12461	0.562306	+	0.826929i	$0.309914\pi$
$38$	0	0
$39$	3.02820	0.484900
$40$	0	0
$41$	9.95527	1.55475	0.777376	−	0.629036i	$-0.216550\pi$
$41$	9.95527	1.55475	0.777376	+	0.629036i	$0.216550\pi$
$42$	0	0
$43$	4.00770	0.611169	0.305585	−	0.952165i	$-0.401148\pi$
$43$	4.00770	0.611169	0.305585	+	0.952165i	$0.401148\pi$
$44$	0	0
$45$	−1.52600	−0.227482
$46$	0	0
$47$	−1.56940	−0.228921	−0.114461	−	0.993428i	$-0.536514\pi$
$47$	−1.56940	−0.228921	−0.114461	+	0.993428i	$0.536514\pi$
$48$	0	0
$49$	7.04524	1.00646
$50$	0	0
$51$	3.10326	0.434544
$52$	0	0
$53$	−2.86932	−0.394132	−0.197066	−	0.980390i	$-0.563141\pi$
$53$	−2.86932	−0.394132	−0.197066	+	0.980390i	$0.563141\pi$
$54$	0	0
$55$	1.66112	0.223985
$56$	0	0
$57$	−4.15119	−0.549838
$58$	0	0
$59$	−14.7252	−1.91706	−0.958529	−	0.284996i	$-0.908008\pi$
$59$	−14.7252	−1.91706	−0.958529	+	0.284996i	$0.908008\pi$
$60$	0	0
$61$	−5.23520	−0.670299	−0.335149	−	0.942165i	$-0.608787\pi$
$61$	−5.23520	−0.670299	−0.335149	+	0.942165i	$0.608787\pi$
$62$	0	0
$63$	−3.74770	−0.472166
$64$	0	0
$65$	−4.62102	−0.573167
$66$	0	0
$67$	−4.63334	−0.566052	−0.283026	−	0.959112i	$-0.591338\pi$
$67$	−4.63334	−0.566052	−0.283026	+	0.959112i	$0.591338\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−15.2454	−1.80929	−0.904646	−	0.426164i	$-0.859865\pi$
$71$	−15.2454	−1.80929	−0.904646	+	0.426164i	$0.859865\pi$
$72$	0	0
$73$	3.21760	0.376592	0.188296	−	0.982112i	$-0.439704\pi$
$73$	3.21760	0.376592	0.188296	+	0.982112i	$0.439704\pi$
$74$	0	0
$75$	−2.67134	−0.308459
$76$	0	0
$77$	4.07954	0.464907
$78$	0	0
$79$	−3.09813	−0.348567	−0.174284	−	0.984696i	$-0.555761\pi$
$79$	−3.09813	−0.348567	−0.174284	+	0.984696i	$0.555761\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	5.43710	0.596799	0.298399	−	0.954441i	$-0.403547\pi$
$83$	5.43710	0.596799	0.298399	+	0.954441i	$0.403547\pi$
$84$	0	0
$85$	−4.73557	−0.513645
$86$	0	0
$87$	2.66395	0.285606
$88$	0	0
$89$	−15.0633	−1.59670	−0.798351	−	0.602193i	$-0.794294\pi$
$89$	−15.0633	−1.59670	−0.798351	+	0.602193i	$0.794294\pi$
$90$	0	0
$91$	−11.3488	−1.18968
$92$	0	0
$93$	6.80696	0.705848
$94$	0	0
$95$	6.33470	0.649926
$96$	0	0
$97$	−3.83748	−0.389637	−0.194818	−	0.980839i	$-0.562412\pi$
$97$	−3.83748	−0.389637	−0.194818	+	0.980839i	$0.562412\pi$
$98$	0	0
$99$	−1.08855	−0.109403
$100$	0	0
$101$	−7.41434	−0.737754	−0.368877	−	0.929478i	$-0.620258\pi$
$101$	−7.41434	−0.737754	−0.368877	+	0.929478i	$0.620258\pi$
$102$	0	0
$103$	−13.2633	−1.30687	−0.653436	−	0.756982i	$-0.726673\pi$
$103$	−13.2633	−1.30687	−0.653436	+	0.756982i	$0.726673\pi$
$104$	0	0
$105$	5.71897	0.558115
$106$	0	0
$107$	−0.233329	−0.0225568	−0.0112784	−	0.999936i	$-0.503590\pi$
$107$	−0.233329	−0.0225568	−0.0112784	+	0.999936i	$0.503590\pi$
$108$	0	0
$109$	−11.8531	−1.13532	−0.567662	−	0.823262i	$-0.692152\pi$
$109$	−11.8531	−1.13532	−0.567662	+	0.823262i	$0.692152\pi$
$110$	0	0
$111$	6.84075	0.649295
$112$	0	0
$113$	−7.97715	−0.750427	−0.375214	−	0.926938i	$-0.622431\pi$
$113$	−7.97715	−0.750427	−0.375214	+	0.926938i	$0.622431\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	3.02820	0.279957
$118$	0	0
$119$	−11.6301	−1.06613
$120$	0	0
$121$	−9.81507	−0.892279
$122$	0	0
$123$	9.95527	0.897637
$124$	0	0
$125$	11.7064	1.04705
$126$	0	0
$127$	−10.2688	−0.911206	−0.455603	−	0.890183i	$-0.650576\pi$
$127$	−10.2688	−0.911206	−0.455603	+	0.890183i	$0.650576\pi$
$128$	0	0
$129$	4.00770	0.352859
$130$	0	0
$131$	−20.3938	−1.78182	−0.890908	−	0.454184i	$-0.849931\pi$
$131$	−20.3938	−1.78182	−0.890908	+	0.454184i	$0.849931\pi$
$132$	0	0
$133$	15.5574	1.34900
$134$	0	0
$135$	−1.52600	−0.131337
$136$	0	0
$137$	9.77788	0.835380	0.417690	−	0.908590i	$-0.362840\pi$
$137$	9.77788	0.835380	0.417690	+	0.908590i	$0.362840\pi$
$138$	0	0
$139$	−3.43414	−0.291280	−0.145640	−	0.989338i	$-0.546524\pi$
$139$	−3.43414	−0.291280	−0.145640	+	0.989338i	$0.546524\pi$
$140$	0	0
$141$	−1.56940	−0.132168
$142$	0	0
$143$	−3.29633	−0.275653
$144$	0	0
$145$	−4.06518	−0.337595
$146$	0	0
$147$	7.04524	0.581081
$148$	0	0
$149$	20.3791	1.66952	0.834761	−	0.550613i	$-0.185606\pi$
$149$	20.3791	1.66952	0.834761	+	0.550613i	$0.185606\pi$
$150$	0	0
$151$	14.0012	1.13940	0.569702	−	0.821851i	$-0.307059\pi$
$151$	14.0012	1.13940	0.569702	+	0.821851i	$0.307059\pi$
$152$	0	0
$153$	3.10326	0.250884
$154$	0	0
$155$	−10.3874	−0.834335
$156$	0	0
$157$	18.0066	1.43708	0.718542	−	0.695483i	$-0.244810\pi$
$157$	18.0066	1.43708	0.718542	+	0.695483i	$0.244810\pi$
$158$	0	0
$159$	−2.86932	−0.227552
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−11.2628	−0.882174	−0.441087	−	0.897464i	$-0.645407\pi$
$163$	−11.2628	−0.882174	−0.441087	+	0.897464i	$0.645407\pi$
$164$	0	0
$165$	1.66112	0.129318
$166$	0	0
$167$	18.6172	1.44064	0.720321	−	0.693641i	$-0.243995\pi$
$167$	18.6172	1.44064	0.720321	+	0.693641i	$0.243995\pi$
$168$	0	0
$169$	−3.83000	−0.294616
$170$	0	0
$171$	−4.15119	−0.317449
$172$	0	0
$173$	11.6030	0.882161	0.441080	−	0.897468i	$-0.354595\pi$
$173$	11.6030	0.882161	0.441080	+	0.897468i	$0.354595\pi$
$174$	0	0
$175$	10.0114	0.756788
$176$	0	0
$177$	−14.7252	−1.10681
$178$	0	0
$179$	11.6981	0.874360	0.437180	−	0.899374i	$-0.355977\pi$
$179$	11.6981	0.874360	0.437180	+	0.899374i	$0.355977\pi$
$180$	0	0
$181$	−14.9103	−1.10828	−0.554139	−	0.832424i	$-0.686952\pi$
$181$	−14.9103	−1.10828	−0.554139	+	0.832424i	$0.686952\pi$
$182$	0	0
$183$	−5.23520	−0.386997
$184$	0	0
$185$	−10.4390	−0.767487
$186$	0	0
$187$	−3.37805	−0.247027
$188$	0	0
$189$	−3.74770	−0.272605
$190$	0	0
$191$	8.97108	0.649125	0.324562	−	0.945864i	$-0.394783\pi$
$191$	8.97108	0.649125	0.324562	+	0.945864i	$0.394783\pi$
$192$	0	0
$193$	−22.3444	−1.60838	−0.804191	−	0.594371i	$-0.797401\pi$
$193$	−22.3444	−1.60838	−0.804191	+	0.594371i	$0.797401\pi$
$194$	0	0
$195$	−4.62102	−0.330918
$196$	0	0
$197$	5.12354	0.365037	0.182519	−	0.983202i	$-0.441575\pi$
$197$	5.12354	0.365037	0.182519	+	0.983202i	$0.441575\pi$
$198$	0	0
$199$	−26.5858	−1.88462	−0.942308	−	0.334746i	$-0.891350\pi$
$199$	−26.5858	−1.88462	−0.942308	+	0.334746i	$0.891350\pi$
$200$	0	0
$201$	−4.63334	−0.326810
$202$	0	0
$203$	−9.98369	−0.700717
$204$	0	0
$205$	−15.1917	−1.06104
$206$	0	0
$207$	0	0
$208$	0	0
$209$	4.51876	0.312569
$210$	0	0
$211$	−12.3102	−0.847469	−0.423735	−	0.905786i	$-0.639281\pi$
$211$	−12.3102	−0.847469	−0.423735	+	0.905786i	$0.639281\pi$
$212$	0	0
$213$	−15.2454	−1.04460
$214$	0	0
$215$	−6.11574	−0.417090
$216$	0	0
$217$	−25.5104	−1.73176
$218$	0	0
$219$	3.21760	0.217425
$220$	0	0
$221$	9.39731	0.632131
$222$	0	0
$223$	−9.64540	−0.645904	−0.322952	−	0.946415i	$-0.604675\pi$
$223$	−9.64540	−0.645904	−0.322952	+	0.946415i	$0.604675\pi$
$224$	0	0
$225$	−2.67134	−0.178089
$226$	0	0
$227$	−18.6180	−1.23572	−0.617859	−	0.786289i	$-0.712000\pi$
$227$	−18.6180	−1.23572	−0.617859	+	0.786289i	$0.712000\pi$
$228$	0	0
$229$	−6.22102	−0.411097	−0.205548	−	0.978647i	$-0.565898\pi$
$229$	−6.22102	−0.411097	−0.205548	+	0.978647i	$0.565898\pi$
$230$	0	0
$231$	4.07954	0.268414
$232$	0	0
$233$	−3.70185	−0.242516	−0.121258	−	0.992621i	$-0.538693\pi$
$233$	−3.70185	−0.242516	−0.121258	+	0.992621i	$0.538693\pi$
$234$	0	0
$235$	2.39491	0.156226
$236$	0	0
$237$	−3.09813	−0.201245
$238$	0	0
$239$	−16.0825	−1.04029	−0.520146	−	0.854077i	$-0.674123\pi$
$239$	−16.0825	−1.04029	−0.520146	+	0.854077i	$0.674123\pi$
$240$	0	0
$241$	28.8474	1.85822	0.929111	−	0.369800i	$-0.120574\pi$
$241$	28.8474	1.85822	0.929111	+	0.369800i	$0.120574\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−10.7510	−0.686857
$246$	0	0
$247$	−12.5706	−0.799850
$248$	0	0
$249$	5.43710	0.344562
$250$	0	0
$251$	0.182845	0.0115411	0.00577053	−	0.999983i	$-0.498163\pi$
$251$	0.182845	0.0115411	0.00577053	+	0.999983i	$0.498163\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	−4.73557	−0.296553
$256$	0	0
$257$	−24.5862	−1.53365	−0.766823	−	0.641858i	$-0.778164\pi$
$257$	−24.5862	−1.53365	−0.766823	+	0.641858i	$0.778164\pi$
$258$	0	0
$259$	−25.6371	−1.59301
$260$	0	0
$261$	2.66395	0.164894
$262$	0	0
$263$	−11.7692	−0.725718	−0.362859	−	0.931844i	$-0.618199\pi$
$263$	−11.7692	−0.725718	−0.362859	+	0.931844i	$0.618199\pi$
$264$	0	0
$265$	4.37858	0.268974
$266$	0	0
$267$	−15.0633	−0.921856
$268$	0	0
$269$	4.58740	0.279699	0.139849	−	0.990173i	$-0.455338\pi$
$269$	4.58740	0.279699	0.139849	+	0.990173i	$0.455338\pi$
$270$	0	0
$271$	−7.98352	−0.484964	−0.242482	−	0.970156i	$-0.577962\pi$
$271$	−7.98352	−0.484964	−0.242482	+	0.970156i	$0.577962\pi$
$272$	0	0
$273$	−11.3488	−0.686859
$274$	0	0
$275$	2.90787	0.175351
$276$	0	0
$277$	5.12957	0.308206	0.154103	−	0.988055i	$-0.450751\pi$
$277$	5.12957	0.308206	0.154103	+	0.988055i	$0.450751\pi$
$278$	0	0
$279$	6.80696	0.407522
$280$	0	0
$281$	26.6343	1.58887	0.794434	−	0.607350i	$-0.207767\pi$
$281$	26.6343	1.58887	0.794434	+	0.607350i	$0.207767\pi$
$282$	0	0
$283$	3.54397	0.210667	0.105334	−	0.994437i	$-0.466409\pi$
$283$	3.54397	0.210667	0.105334	+	0.994437i	$0.466409\pi$
$284$	0	0
$285$	6.33470	0.375235
$286$	0	0
$287$	−37.3094	−2.20230
$288$	0	0
$289$	−7.36975	−0.433515
$290$	0	0
$291$	−3.83748	−0.224957
$292$	0	0
$293$	−28.6676	−1.67478	−0.837390	−	0.546606i	$-0.815919\pi$
$293$	−28.6676	−1.67478	−0.837390	+	0.546606i	$0.815919\pi$
$294$	0	0
$295$	22.4706	1.30829
$296$	0	0
$297$	−1.08855	−0.0631638
$298$	0	0
$299$	0	0
$300$	0	0
$301$	−15.0197	−0.865719
$302$	0	0
$303$	−7.41434	−0.425943
$304$	0	0
$305$	7.98890	0.457443
$306$	0	0
$307$	21.7367	1.24058	0.620289	−	0.784373i	$-0.287015\pi$
$307$	21.7367	1.24058	0.620289	+	0.784373i	$0.287015\pi$
$308$	0	0
$309$	−13.2633	−0.754522
$310$	0	0
$311$	23.1949	1.31526	0.657630	−	0.753341i	$-0.271559\pi$
$311$	23.1949	1.31526	0.657630	+	0.753341i	$0.271559\pi$
$312$	0	0
$313$	10.2058	0.576865	0.288432	−	0.957500i	$-0.406866\pi$
$313$	10.2058	0.576865	0.288432	+	0.957500i	$0.406866\pi$
$314$	0	0
$315$	5.71897	0.322228
$316$	0	0
$317$	15.1189	0.849161	0.424581	−	0.905390i	$-0.360422\pi$
$317$	15.1189	0.849161	0.424581	+	0.905390i	$0.360422\pi$
$318$	0	0
$319$	−2.89983	−0.162360
$320$	0	0
$321$	−0.233329	−0.0130232
$322$	0	0
$323$	−12.8822	−0.716787
$324$	0	0
$325$	−8.08934	−0.448716
$326$	0	0
$327$	−11.8531	−0.655479
$328$	0	0
$329$	5.88165	0.324266
$330$	0	0
$331$	29.6873	1.63176	0.815882	−	0.578219i	$-0.196252\pi$
$331$	29.6873	1.63176	0.815882	+	0.578219i	$0.196252\pi$
$332$	0	0
$333$	6.84075	0.374871
$334$	0	0
$335$	7.07045	0.386300
$336$	0	0
$337$	−7.70975	−0.419977	−0.209989	−	0.977704i	$-0.567343\pi$
$337$	−7.70975	−0.419977	−0.209989	+	0.977704i	$0.567343\pi$
$338$	0	0
$339$	−7.97715	−0.433259
$340$	0	0
$341$	−7.40968	−0.401257
$342$	0	0
$343$	−0.169536	−0.00915409
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−8.06394	−0.432895	−0.216447	−	0.976294i	$-0.569447\pi$
$347$	−8.06394	−0.432895	−0.216447	+	0.976294i	$0.569447\pi$
$348$	0	0
$349$	−15.5375	−0.831701	−0.415850	−	0.909433i	$-0.636516\pi$
$349$	−15.5375	−0.831701	−0.415850	+	0.909433i	$0.636516\pi$
$350$	0	0
$351$	3.02820	0.161633
$352$	0	0
$353$	8.58323	0.456840	0.228420	−	0.973563i	$-0.426644\pi$
$353$	8.58323	0.456840	0.228420	+	0.973563i	$0.426644\pi$
$354$	0	0
$355$	23.2644	1.23474
$356$	0	0
$357$	−11.6301	−0.615530
$358$	0	0
$359$	33.7964	1.78370	0.891852	−	0.452328i	$-0.149406\pi$
$359$	33.7964	1.78370	0.891852	+	0.452328i	$0.149406\pi$
$360$	0	0
$361$	−1.76765	−0.0930341
$362$	0	0
$363$	−9.81507	−0.515157
$364$	0	0
$365$	−4.91005	−0.257004
$366$	0	0
$367$	13.4110	0.700047	0.350023	−	0.936741i	$-0.386174\pi$
$367$	13.4110	0.700047	0.350023	+	0.936741i	$0.386174\pi$
$368$	0	0
$369$	9.95527	0.518251
$370$	0	0
$371$	10.7534	0.558287
$372$	0	0
$373$	−30.0491	−1.55589	−0.777943	−	0.628335i	$-0.783737\pi$
$373$	−30.0491	−1.55589	−0.777943	+	0.628335i	$0.783737\pi$
$374$	0	0
$375$	11.7064	0.604517
$376$	0	0
$377$	8.06698	0.415471
$378$	0	0
$379$	−18.9122	−0.971453	−0.485727	−	0.874111i	$-0.661445\pi$
$379$	−18.9122	−0.971453	−0.485727	+	0.874111i	$0.661445\pi$
$380$	0	0
$381$	−10.2688	−0.526085
$382$	0	0
$383$	−7.83223	−0.400208	−0.200104	−	0.979775i	$-0.564128\pi$
$383$	−7.83223	−0.400208	−0.200104	+	0.979775i	$0.564128\pi$
$384$	0	0
$385$	−6.22536	−0.317274
$386$	0	0
$387$	4.00770	0.203723
$388$	0	0
$389$	−7.40652	−0.375525	−0.187763	−	0.982214i	$-0.560124\pi$
$389$	−7.40652	−0.375525	−0.187763	+	0.982214i	$0.560124\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	−20.3938	−1.02873
$394$	0	0
$395$	4.72774	0.237878
$396$	0	0
$397$	−1.86646	−0.0936748	−0.0468374	−	0.998903i	$-0.514914\pi$
$397$	−1.86646	−0.0936748	−0.0468374	+	0.998903i	$0.514914\pi$
$398$	0	0
$399$	15.5574	0.778844
$400$	0	0
$401$	−10.8245	−0.540550	−0.270275	−	0.962783i	$-0.587115\pi$
$401$	−10.8245	−0.540550	−0.270275	+	0.962783i	$0.587115\pi$
$402$	0	0
$403$	20.6128	1.02680
$404$	0	0
$405$	−1.52600	−0.0758274
$406$	0	0
$407$	−7.44647	−0.369108
$408$	0	0
$409$	−6.84470	−0.338449	−0.169224	−	0.985578i	$-0.554126\pi$
$409$	−6.84470	−0.338449	−0.169224	+	0.985578i	$0.554126\pi$
$410$	0	0
$411$	9.77788	0.482307
$412$	0	0
$413$	55.1856	2.71551
$414$	0	0
$415$	−8.29699	−0.407283
$416$	0	0
$417$	−3.43414	−0.168170
$418$	0	0
$419$	28.7335	1.40372	0.701862	−	0.712313i	$-0.252352\pi$
$419$	28.7335	1.40372	0.701862	+	0.712313i	$0.252352\pi$
$420$	0	0
$421$	−27.6553	−1.34784	−0.673918	−	0.738806i	$-0.735390\pi$
$421$	−27.6553	−1.34784	−0.673918	+	0.738806i	$0.735390\pi$
$422$	0	0
$423$	−1.56940	−0.0763071
$424$	0	0
$425$	−8.28986	−0.402117
$426$	0	0
$427$	19.6200	0.949476
$428$	0	0
$429$	−3.29633	−0.159149
$430$	0	0
$431$	28.6098	1.37809	0.689044	−	0.724720i	$-0.258031\pi$
$431$	28.6098	1.37809	0.689044	+	0.724720i	$0.258031\pi$
$432$	0	0
$433$	−27.2900	−1.31147	−0.655736	−	0.754990i	$-0.727641\pi$
$433$	−27.2900	−1.31147	−0.655736	+	0.754990i	$0.727641\pi$
$434$	0	0
$435$	−4.06518	−0.194910
$436$	0	0
$437$	0	0
$438$	0	0
$439$	−0.947265	−0.0452105	−0.0226052	−	0.999744i	$-0.507196\pi$
$439$	−0.947265	−0.0452105	−0.0226052	+	0.999744i	$0.507196\pi$
$440$	0	0
$441$	7.04524	0.335487
$442$	0	0
$443$	−21.8546	−1.03834	−0.519172	−	0.854670i	$-0.673759\pi$
$443$	−21.8546	−1.03834	−0.519172	+	0.854670i	$0.673759\pi$
$444$	0	0
$445$	22.9865	1.08966
$446$	0	0
$447$	20.3791	0.963899
$448$	0	0
$449$	5.34044	0.252031	0.126016	−	0.992028i	$-0.459781\pi$
$449$	5.34044	0.252031	0.126016	+	0.992028i	$0.459781\pi$
$450$	0	0
$451$	−10.8368	−0.510284
$452$	0	0
$453$	14.0012	0.657835
$454$	0	0
$455$	17.3182	0.811889
$456$	0	0
$457$	−29.9206	−1.39963	−0.699814	−	0.714325i	$-0.746734\pi$
$457$	−29.9206	−1.39963	−0.699814	+	0.714325i	$0.746734\pi$
$458$	0	0
$459$	3.10326	0.144848
$460$	0	0
$461$	−2.50571	−0.116702	−0.0583512	−	0.998296i	$-0.518584\pi$
$461$	−2.50571	−0.116702	−0.0583512	+	0.998296i	$0.518584\pi$
$462$	0	0
$463$	42.4289	1.97184	0.985918	−	0.167227i	$-0.0534814\pi$
$463$	42.4289	1.97184	0.985918	+	0.167227i	$0.0534814\pi$
$464$	0	0
$465$	−10.3874	−0.481704
$466$	0	0
$467$	−13.8625	−0.641478	−0.320739	−	0.947168i	$-0.603931\pi$
$467$	−13.8625	−0.641478	−0.320739	+	0.947168i	$0.603931\pi$
$468$	0	0
$469$	17.3643	0.801811
$470$	0	0
$471$	18.0066	0.829701
$472$	0	0
$473$	−4.36257	−0.200591
$474$	0	0
$475$	11.0892	0.508808
$476$	0	0
$477$	−2.86932	−0.131377
$478$	0	0
$479$	−12.6335	−0.577240	−0.288620	−	0.957444i	$-0.593196\pi$
$479$	−12.6335	−0.577240	−0.288620	+	0.957444i	$0.593196\pi$
$480$	0	0
$481$	20.7152	0.944530
$482$	0	0
$483$	0	0
$484$	0	0
$485$	5.85597	0.265906
$486$	0	0
$487$	−15.6003	−0.706917	−0.353459	−	0.935450i	$-0.614994\pi$
$487$	−15.6003	−0.706917	−0.353459	+	0.935450i	$0.614994\pi$
$488$	0	0
$489$	−11.2628	−0.509323
$490$	0	0
$491$	−43.5993	−1.96761	−0.983804	−	0.179246i	$-0.942634\pi$
$491$	−43.5993	−1.96761	−0.983804	+	0.179246i	$0.942634\pi$
$492$	0	0
$493$	8.26695	0.372325
$494$	0	0
$495$	1.66112	0.0746617
$496$	0	0
$497$	57.1350	2.56286
$498$	0	0
$499$	10.0683	0.450718	0.225359	−	0.974276i	$-0.427644\pi$
$499$	10.0683	0.450718	0.225359	+	0.974276i	$0.427644\pi$
$500$	0	0
$501$	18.6172	0.831755
$502$	0	0
$503$	−6.89346	−0.307364	−0.153682	−	0.988120i	$-0.549113\pi$
$503$	−6.89346	−0.307364	−0.153682	+	0.988120i	$0.549113\pi$
$504$	0	0
$505$	11.3143	0.503478
$506$	0	0
$507$	−3.83000	−0.170096
$508$	0	0
$509$	19.4733	0.863140	0.431570	−	0.902079i	$-0.357960\pi$
$509$	19.4733	0.863140	0.431570	+	0.902079i	$0.357960\pi$
$510$	0	0
$511$	−12.0586	−0.533441
$512$	0	0
$513$	−4.15119	−0.183279
$514$	0	0
$515$	20.2397	0.891869
$516$	0	0
$517$	1.70837	0.0751340
$518$	0	0
$519$	11.6030	0.509316
$520$	0	0
$521$	−33.4732	−1.46649	−0.733243	−	0.679967i	$-0.761994\pi$
$521$	−33.4732	−1.46649	−0.733243	+	0.679967i	$0.761994\pi$
$522$	0	0
$523$	−31.9255	−1.39601	−0.698003	−	0.716095i	$-0.745928\pi$
$523$	−31.9255	−1.39601	−0.698003	+	0.716095i	$0.745928\pi$
$524$	0	0
$525$	10.0114	0.436931
$526$	0	0
$527$	21.1238	0.920167
$528$	0	0
$529$	0	0
$530$	0	0
$531$	−14.7252	−0.639019
$532$	0	0
$533$	30.1466	1.30579
$534$	0	0
$535$	0.356059	0.0153938
$536$	0	0
$537$	11.6981	0.504812
$538$	0	0
$539$	−7.66906	−0.330330
$540$	0	0
$541$	−31.8214	−1.36811	−0.684054	−	0.729431i	$-0.739785\pi$
$541$	−31.8214	−1.36811	−0.684054	+	0.729431i	$0.739785\pi$
$542$	0	0
$543$	−14.9103	−0.639864
$544$	0	0
$545$	18.0878	0.774797
$546$	0	0
$547$	36.8614	1.57608	0.788040	−	0.615624i	$-0.211096\pi$
$547$	36.8614	1.57608	0.788040	+	0.615624i	$0.211096\pi$
$548$	0	0
$549$	−5.23520	−0.223433
$550$	0	0
$551$	−11.0586	−0.471111
$552$	0	0
$553$	11.6109	0.493744
$554$	0	0
$555$	−10.4390	−0.443109
$556$	0	0
$557$	33.9776	1.43968	0.719840	−	0.694140i	$-0.244215\pi$
$557$	33.9776	1.43968	0.719840	+	0.694140i	$0.244215\pi$
$558$	0	0
$559$	12.1361	0.513304
$560$	0	0
$561$	−3.37805	−0.142621
$562$	0	0
$563$	−18.7216	−0.789020	−0.394510	−	0.918892i	$-0.629086\pi$
$563$	−18.7216	−0.789020	−0.394510	+	0.918892i	$0.629086\pi$
$564$	0	0
$565$	12.1731	0.512126
$566$	0	0
$567$	−3.74770	−0.157389
$568$	0	0
$569$	31.3575	1.31457	0.657286	−	0.753641i	$-0.271704\pi$
$569$	31.3575	1.31457	0.657286	+	0.753641i	$0.271704\pi$
$570$	0	0
$571$	7.48760	0.313346	0.156673	−	0.987651i	$-0.449923\pi$
$571$	7.48760	0.313346	0.156673	+	0.987651i	$0.449923\pi$
$572$	0	0
$573$	8.97108	0.374772
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−41.2244	−1.71619	−0.858097	−	0.513488i	$-0.828353\pi$
$577$	−41.2244	−1.71619	−0.858097	+	0.513488i	$0.828353\pi$
$578$	0	0
$579$	−22.3444	−0.928600
$580$	0	0
$581$	−20.3766	−0.845363
$582$	0	0
$583$	3.12339	0.129358
$584$	0	0
$585$	−4.62102	−0.191056
$586$	0	0
$587$	13.8617	0.572135	0.286067	−	0.958209i	$-0.407652\pi$
$587$	13.8617	0.572135	0.286067	+	0.958209i	$0.407652\pi$
$588$	0	0
$589$	−28.2569	−1.16431
$590$	0	0
$591$	5.12354	0.210754
$592$	0	0
$593$	−14.6461	−0.601443	−0.300722	−	0.953712i	$-0.597227\pi$
$593$	−14.6461	−0.601443	−0.300722	+	0.953712i	$0.597227\pi$
$594$	0	0
$595$	17.7475	0.727576
$596$	0	0
$597$	−26.5858	−1.08808
$598$	0	0
$599$	−25.6306	−1.04724	−0.523618	−	0.851953i	$-0.675418\pi$
$599$	−25.6306	−1.04724	−0.523618	+	0.851953i	$0.675418\pi$
$600$	0	0
$601$	−16.7648	−0.683849	−0.341924	−	0.939727i	$-0.611079\pi$
$601$	−16.7648	−0.683849	−0.341924	+	0.939727i	$0.611079\pi$
$602$	0	0
$603$	−4.63334	−0.188684
$604$	0	0
$605$	14.9778	0.608932
$606$	0	0
$607$	10.3917	0.421785	0.210893	−	0.977509i	$-0.432363\pi$
$607$	10.3917	0.421785	0.210893	+	0.977509i	$0.432363\pi$
$608$	0	0
$609$	−9.98369	−0.404559
$610$	0	0
$611$	−4.75247	−0.192264
$612$	0	0
$613$	22.3003	0.900700	0.450350	−	0.892852i	$-0.351299\pi$
$613$	22.3003	0.900700	0.450350	+	0.892852i	$0.351299\pi$
$614$	0	0
$615$	−15.1917	−0.612589
$616$	0	0
$617$	−11.8432	−0.476791	−0.238395	−	0.971168i	$-0.576621\pi$
$617$	−11.8432	−0.476791	−0.238395	+	0.971168i	$0.576621\pi$
$618$	0	0
$619$	−37.9545	−1.52552	−0.762759	−	0.646682i	$-0.776156\pi$
$619$	−37.9545	−1.52552	−0.762759	+	0.646682i	$0.776156\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	56.4525	2.26172
$624$	0	0
$625$	−4.50729	−0.180292
$626$	0	0
$627$	4.51876	0.180462
$628$	0	0
$629$	21.2287	0.846442
$630$	0	0
$631$	−7.26754	−0.289316	−0.144658	−	0.989482i	$-0.546208\pi$
$631$	−7.26754	−0.289316	−0.144658	+	0.989482i	$0.546208\pi$
$632$	0	0
$633$	−12.3102	−0.489287
$634$	0	0
$635$	15.6701	0.621849
$636$	0	0
$637$	21.3344	0.845299
$638$	0	0
$639$	−15.2454	−0.603097
$640$	0	0
$641$	26.1585	1.03320	0.516600	−	0.856227i	$-0.327198\pi$
$641$	26.1585	1.03320	0.516600	+	0.856227i	$0.327198\pi$
$642$	0	0
$643$	−33.0605	−1.30378	−0.651889	−	0.758314i	$-0.726023\pi$
$643$	−33.0605	−1.30378	−0.651889	+	0.758314i	$0.726023\pi$
$644$	0	0
$645$	−6.11574	−0.240807
$646$	0	0
$647$	23.8719	0.938500	0.469250	−	0.883065i	$-0.344524\pi$
$647$	23.8719	0.938500	0.469250	+	0.883065i	$0.344524\pi$
$648$	0	0
$649$	16.0291	0.629195
$650$	0	0
$651$	−25.5104	−0.999832
$652$	0	0
$653$	−40.9679	−1.60320	−0.801598	−	0.597863i	$-0.796017\pi$
$653$	−40.9679	−1.60320	−0.801598	+	0.597863i	$0.796017\pi$
$654$	0	0
$655$	31.1209	1.21599
$656$	0	0
$657$	3.21760	0.125531
$658$	0	0
$659$	−5.49630	−0.214105	−0.107053	−	0.994253i	$-0.534141\pi$
$659$	−5.49630	−0.214105	−0.107053	+	0.994253i	$0.534141\pi$
$660$	0	0
$661$	20.7817	0.808314	0.404157	−	0.914690i	$-0.367565\pi$
$661$	20.7817	0.808314	0.404157	+	0.914690i	$0.367565\pi$
$662$	0	0
$663$	9.39731	0.364961
$664$	0	0
$665$	−23.7405	−0.920618
$666$	0	0
$667$	0	0
$668$	0	0
$669$	−9.64540	−0.372913
$670$	0	0
$671$	5.69876	0.219998
$672$	0	0
$673$	−11.6037	−0.447291	−0.223646	−	0.974671i	$-0.571796\pi$
$673$	−11.6037	−0.447291	−0.223646	+	0.974671i	$0.571796\pi$
$674$	0	0
$675$	−2.67134	−0.102820
$676$	0	0
$677$	8.79702	0.338097	0.169048	−	0.985608i	$-0.445931\pi$
$677$	8.79702	0.338097	0.169048	+	0.985608i	$0.445931\pi$
$678$	0	0
$679$	14.3817	0.551919
$680$	0	0
$681$	−18.6180	−0.713442
$682$	0	0
$683$	−28.8344	−1.10332	−0.551658	−	0.834070i	$-0.686005\pi$
$683$	−28.8344	−1.10332	−0.551658	+	0.834070i	$0.686005\pi$
$684$	0	0
$685$	−14.9210	−0.570102
$686$	0	0
$687$	−6.22102	−0.237347
$688$	0	0
$689$	−8.68889	−0.331020
$690$	0	0
$691$	9.54501	0.363109	0.181555	−	0.983381i	$-0.441887\pi$
$691$	9.54501	0.363109	0.181555	+	0.983381i	$0.441887\pi$
$692$	0	0
$693$	4.07954	0.154969
$694$	0	0
$695$	5.24048	0.198783
$696$	0	0
$697$	30.8939	1.17019
$698$	0	0
$699$	−3.70185	−0.140017
$700$	0	0
$701$	10.7859	0.407379	0.203690	−	0.979036i	$-0.434707\pi$
$701$	10.7859	0.407379	0.203690	+	0.979036i	$0.434707\pi$
$702$	0	0
$703$	−28.3972	−1.07102
$704$	0	0
$705$	2.39491	0.0901974
$706$	0	0
$707$	27.7867	1.04503
$708$	0	0
$709$	6.79613	0.255234	0.127617	−	0.991824i	$-0.459267\pi$
$709$	6.79613	0.255234	0.127617	+	0.991824i	$0.459267\pi$
$710$	0	0
$711$	−3.09813	−0.116189
$712$	0	0
$713$	0	0
$714$	0	0
$715$	5.03019	0.188119
$716$	0	0
$717$	−16.0825	−0.600613
$718$	0	0
$719$	37.2522	1.38927	0.694636	−	0.719362i	$-0.255566\pi$
$719$	37.2522	1.38927	0.694636	+	0.719362i	$0.255566\pi$
$720$	0	0
$721$	49.7068	1.85118
$722$	0	0
$723$	28.8474	1.07285
$724$	0	0
$725$	−7.11631	−0.264293
$726$	0	0
$727$	−46.1330	−1.71098	−0.855489	−	0.517820i	$-0.826744\pi$
$727$	−46.1330	−1.71098	−0.855489	+	0.517820i	$0.826744\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	12.4370	0.459998
$732$	0	0
$733$	−34.8197	−1.28610	−0.643048	−	0.765826i	$-0.722331\pi$
$733$	−34.8197	−1.28610	−0.643048	+	0.765826i	$0.722331\pi$
$734$	0	0
$735$	−10.7510	−0.396557
$736$	0	0
$737$	5.04360	0.185783
$738$	0	0
$739$	−21.1003	−0.776188	−0.388094	−	0.921620i	$-0.626866\pi$
$739$	−21.1003	−0.776188	−0.388094	+	0.921620i	$0.626866\pi$
$740$	0	0
$741$	−12.5706	−0.461793
$742$	0	0
$743$	28.2810	1.03753	0.518764	−	0.854917i	$-0.326392\pi$
$743$	28.2810	1.03753	0.518764	+	0.854917i	$0.326392\pi$
$744$	0	0
$745$	−31.0984	−1.13936
$746$	0	0
$747$	5.43710	0.198933
$748$	0	0
$749$	0.874447	0.0319516
$750$	0	0
$751$	−4.26497	−0.155631	−0.0778154	−	0.996968i	$-0.524794\pi$
$751$	−4.26497	−0.155631	−0.0778154	+	0.996968i	$0.524794\pi$
$752$	0	0
$753$	0.182845	0.00666323
$754$	0	0
$755$	−21.3658	−0.777582
$756$	0	0
$757$	37.4552	1.36133	0.680665	−	0.732595i	$-0.261691\pi$
$757$	37.4552	1.36133	0.680665	+	0.732595i	$0.261691\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	42.6179	1.54490	0.772449	−	0.635077i	$-0.219032\pi$
$761$	42.6179	1.54490	0.772449	+	0.635077i	$0.219032\pi$
$762$	0	0
$763$	44.4219	1.60818
$764$	0	0
$765$	−4.73557	−0.171215
$766$	0	0
$767$	−44.5908	−1.61008
$768$	0	0
$769$	−18.9716	−0.684133	−0.342067	−	0.939676i	$-0.611127\pi$
$769$	−18.9716	−0.684133	−0.342067	+	0.939676i	$0.611127\pi$
$770$	0	0
$771$	−24.5862	−0.885452
$772$	0	0
$773$	52.8570	1.90113	0.950567	−	0.310521i	$-0.100503\pi$
$773$	52.8570	1.90113	0.950567	+	0.310521i	$0.100503\pi$
$774$	0	0
$775$	−18.1837	−0.653177
$776$	0	0
$777$	−25.6371	−0.919725
$778$	0	0
$779$	−41.3262	−1.48066
$780$	0	0
$781$	16.5953	0.593826
$782$	0	0
$783$	2.66395	0.0952019
$784$	0	0
$785$	−27.4780	−0.980733
$786$	0	0
$787$	27.4620	0.978916	0.489458	−	0.872027i	$-0.337195\pi$
$787$	27.4620	0.978916	0.489458	+	0.872027i	$0.337195\pi$
$788$	0	0
$789$	−11.7692	−0.418993
$790$	0	0
$791$	29.8960	1.06298
$792$	0	0
$793$	−15.8532	−0.562965
$794$	0	0
$795$	4.37858	0.155292
$796$	0	0
$797$	38.3985	1.36015	0.680073	−	0.733144i	$-0.261948\pi$
$797$	38.3985	1.36015	0.680073	+	0.733144i	$0.261948\pi$
$798$	0	0
$799$	−4.87028	−0.172298
$800$	0	0
$801$	−15.0633	−0.532234
$802$	0	0
$803$	−3.50251	−0.123601
$804$	0	0
$805$	0	0
$806$	0	0
$807$	4.58740	0.161484
$808$	0	0
$809$	12.4879	0.439051	0.219526	−	0.975607i	$-0.429549\pi$
$809$	12.4879	0.439051	0.219526	+	0.975607i	$0.429549\pi$
$810$	0	0
$811$	17.6655	0.620318	0.310159	−	0.950685i	$-0.399618\pi$
$811$	17.6655	0.620318	0.310159	+	0.950685i	$0.399618\pi$
$812$	0	0
$813$	−7.98352	−0.279994
$814$	0	0
$815$	17.1871	0.602036
$816$	0	0
$817$	−16.6367	−0.582045
$818$	0	0
$819$	−11.3488	−0.396558
$820$	0	0
$821$	−13.2365	−0.461958	−0.230979	−	0.972959i	$-0.574193\pi$
$821$	−13.2365	−0.461958	−0.230979	+	0.972959i	$0.574193\pi$
$822$	0	0
$823$	−39.9101	−1.39118	−0.695590	−	0.718439i	$-0.744857\pi$
$823$	−39.9101	−1.39118	−0.695590	+	0.718439i	$0.744857\pi$
$824$	0	0
$825$	2.90787	0.101239
$826$	0	0
$827$	−0.108053	−0.00375736	−0.00187868	−	0.999998i	$-0.500598\pi$
$827$	−0.108053	−0.00375736	−0.00187868	+	0.999998i	$0.500598\pi$
$828$	0	0
$829$	50.3641	1.74922	0.874608	−	0.484830i	$-0.161119\pi$
$829$	50.3641	1.74922	0.874608	+	0.484830i	$0.161119\pi$
$830$	0	0
$831$	5.12957	0.177943
$832$	0	0
$833$	21.8632	0.757516
$834$	0	0
$835$	−28.4098	−0.983161
$836$	0	0
$837$	6.80696	0.235283
$838$	0	0
$839$	3.85365	0.133043	0.0665213	−	0.997785i	$-0.478810\pi$
$839$	3.85365	0.133043	0.0665213	+	0.997785i	$0.478810\pi$
$840$	0	0
$841$	−21.9034	−0.755288
$842$	0	0
$843$	26.6343	0.917334
$844$	0	0
$845$	5.84457	0.201059
$846$	0	0
$847$	36.7839	1.26391
$848$	0	0
$849$	3.54397	0.121629
$850$	0	0
$851$	0	0
$852$	0	0
$853$	−0.513597	−0.0175852	−0.00879262	−	0.999961i	$-0.502799\pi$
$853$	−0.513597	−0.0175852	−0.00879262	+	0.999961i	$0.502799\pi$
$854$	0	0
$855$	6.33470	0.216642
$856$	0	0
$857$	38.9015	1.32885	0.664425	−	0.747355i	$-0.268676\pi$
$857$	38.9015	1.32885	0.664425	+	0.747355i	$0.268676\pi$
$858$	0	0
$859$	−47.7947	−1.63073	−0.815367	−	0.578944i	$-0.803465\pi$
$859$	−47.7947	−1.63073	−0.815367	+	0.578944i	$0.803465\pi$
$860$	0	0
$861$	−37.3094	−1.27150
$862$	0	0
$863$	7.63659	0.259953	0.129976	−	0.991517i	$-0.458510\pi$
$863$	7.63659	0.259953	0.129976	+	0.991517i	$0.458510\pi$
$864$	0	0
$865$	−17.7062	−0.602027
$866$	0	0
$867$	−7.36975	−0.250290
$868$	0	0
$869$	3.37246	0.114403
$870$	0	0
$871$	−14.0307	−0.475411
$872$	0	0
$873$	−3.83748	−0.129879
$874$	0	0
$875$	−43.8722	−1.48315
$876$	0	0
$877$	−2.45849	−0.0830174	−0.0415087	−	0.999138i	$-0.513216\pi$
$877$	−2.45849	−0.0830174	−0.0415087	+	0.999138i	$0.513216\pi$
$878$	0	0
$879$	−28.6676	−0.966934
$880$	0	0
$881$	6.53688	0.220233	0.110117	−	0.993919i	$-0.464878\pi$
$881$	6.53688	0.220233	0.110117	+	0.993919i	$0.464878\pi$
$882$	0	0
$883$	−32.5031	−1.09382	−0.546908	−	0.837193i	$-0.684195\pi$
$883$	−32.5031	−1.09382	−0.546908	+	0.837193i	$0.684195\pi$
$884$	0	0
$885$	22.4706	0.755341
$886$	0	0
$887$	14.2567	0.478693	0.239347	−	0.970934i	$-0.423067\pi$
$887$	14.2567	0.478693	0.239347	+	0.970934i	$0.423067\pi$
$888$	0	0
$889$	38.4842	1.29072
$890$	0	0
$891$	−1.08855	−0.0364677
$892$	0	0
$893$	6.51489	0.218013
$894$	0	0
$895$	−17.8513	−0.596704
$896$	0	0
$897$	0	0
$898$	0	0
$899$	18.1334	0.604783
$900$	0	0
$901$	−8.90427	−0.296644
$902$	0	0
$903$	−15.0197	−0.499823
$904$	0	0
$905$	22.7531	0.756340
$906$	0	0
$907$	27.7414	0.921138	0.460569	−	0.887624i	$-0.347645\pi$
$907$	27.7414	0.921138	0.460569	+	0.887624i	$0.347645\pi$
$908$	0	0
$909$	−7.41434	−0.245918
$910$	0	0
$911$	−17.0585	−0.565173	−0.282587	−	0.959242i	$-0.591192\pi$
$911$	−17.0585	−0.565173	−0.282587	+	0.959242i	$0.591192\pi$
$912$	0	0
$913$	−5.91853	−0.195875
$914$	0	0
$915$	7.98890	0.264105
$916$	0	0
$917$	76.4299	2.52394
$918$	0	0
$919$	−32.9304	−1.08627	−0.543137	−	0.839644i	$-0.682764\pi$
$919$	−32.9304	−1.08627	−0.543137	+	0.839644i	$0.682764\pi$
$920$	0	0
$921$	21.7367	0.716248
$922$	0	0
$923$	−46.1660	−1.51957
$924$	0	0
$925$	−18.2739	−0.600843
$926$	0	0
$927$	−13.2633	−0.435624
$928$	0	0
$929$	−56.0413	−1.83865	−0.919327	−	0.393494i	$-0.871266\pi$
$929$	−56.0413	−1.83865	−0.919327	+	0.393494i	$0.871266\pi$
$930$	0	0
$931$	−29.2461	−0.958502
$932$	0	0
$933$	23.1949	0.759366
$934$	0	0
$935$	5.15489	0.168583
$936$	0	0
$937$	−17.2239	−0.562680	−0.281340	−	0.959608i	$-0.590779\pi$
$937$	−17.2239	−0.562680	−0.281340	+	0.959608i	$0.590779\pi$
$938$	0	0
$939$	10.2058	0.333053
$940$	0	0
$941$	17.6290	0.574689	0.287344	−	0.957827i	$-0.407228\pi$
$941$	17.6290	0.574689	0.287344	+	0.957827i	$0.407228\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	5.71897	0.186038
$946$	0	0
$947$	20.7322	0.673706	0.336853	−	0.941557i	$-0.390637\pi$
$947$	20.7322	0.673706	0.336853	+	0.941557i	$0.390637\pi$
$948$	0	0
$949$	9.74354	0.316289
$950$	0	0
$951$	15.1189	0.490263
$952$	0	0
$953$	−13.9658	−0.452398	−0.226199	−	0.974081i	$-0.572630\pi$
$953$	−13.9658	−0.452398	−0.226199	+	0.974081i	$0.572630\pi$
$954$	0	0
$955$	−13.6898	−0.442993
$956$	0	0
$957$	−2.89983	−0.0937383
$958$	0	0
$959$	−36.6445	−1.18331
$960$	0	0
$961$	15.3347	0.494666
$962$	0	0
$963$	−0.233329	−0.00751893
$964$	0	0
$965$	34.0974	1.09763
$966$	0	0
$967$	18.2603	0.587212	0.293606	−	0.955926i	$-0.405145\pi$
$967$	18.2603	0.587212	0.293606	+	0.955926i	$0.405145\pi$
$968$	0	0
$969$	−12.8822	−0.413837
$970$	0	0
$971$	−10.4937	−0.336760	−0.168380	−	0.985722i	$-0.553854\pi$
$971$	−10.4937	−0.336760	−0.168380	+	0.985722i	$0.553854\pi$
$972$	0	0
$973$	12.8701	0.412597
$974$	0	0
$975$	−8.08934	−0.259066
$976$	0	0
$977$	29.4306	0.941569	0.470785	−	0.882248i	$-0.343971\pi$
$977$	29.4306	0.941569	0.470785	+	0.882248i	$0.343971\pi$
$978$	0	0
$979$	16.3970	0.524052
$980$	0	0
$981$	−11.8531	−0.378441
$982$	0	0
$983$	−19.2499	−0.613975	−0.306987	−	0.951714i	$-0.599321\pi$
$983$	−19.2499	−0.613975	−0.306987	+	0.951714i	$0.599321\pi$
$984$	0	0
$985$	−7.81851	−0.249118
$986$	0	0
$987$	5.88165	0.187215
$988$	0	0
$989$	0	0
$990$	0	0
$991$	50.0956	1.59134	0.795670	−	0.605730i	$-0.207119\pi$
$991$	50.0956	1.59134	0.795670	+	0.605730i	$0.207119\pi$
$992$	0	0
$993$	29.6873	0.942099
$994$	0	0
$995$	40.5698	1.28615
$996$	0	0
$997$	−36.4817	−1.15539	−0.577694	−	0.816254i	$-0.696047\pi$
$997$	−36.4817	−1.15539	−0.577694	+	0.816254i	$0.696047\pi$
$998$	0	0
$999$	6.84075	0.216432

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6348.2.a.s.1.4		10
23.3	even	11		276.2.i.a.193.2	yes	20
23.8	even	11		276.2.i.a.133.2	✓	20
23.22	odd	2		6348.2.a.t.1.7		10
69.8	odd	22		828.2.q.c.685.1		20
69.26	odd	22		828.2.q.c.469.1		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
276.2.i.a.133.2	✓	20	23.8	even	11
276.2.i.a.193.2	yes	20	23.3	even	11
828.2.q.c.469.1		20	69.26	odd	22
828.2.q.c.685.1		20	69.8	odd	22
6348.2.a.s.1.4		10	1.1	even	1	trivial
6348.2.a.t.1.7		10	23.22	odd	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:	\( N \)	\(=\)	\( 6348 = 2^{2} \cdot 3 \cdot 23^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6348.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -1.52600 q^{5} -3.74770 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -1.52600 q^{5} -3.74770 q^{7} +1.00000 q^{9} -1.08855 q^{11} +3.02820 q^{13} -1.52600 q^{15} +3.10326 q^{17} -4.15119 q^{19} -3.74770 q^{21} -2.67134 q^{25} +1.00000 q^{27} +2.66395 q^{29} +6.80696 q^{31} -1.08855 q^{33} +5.71897 q^{35} +6.84075 q^{37} +3.02820 q^{39} +9.95527 q^{41} +4.00770 q^{43} -1.52600 q^{45} -1.56940 q^{47} +7.04524 q^{49} +3.10326 q^{51} -2.86932 q^{53} +1.66112 q^{55} -4.15119 q^{57} -14.7252 q^{59} -5.23520 q^{61} -3.74770 q^{63} -4.62102 q^{65} -4.63334 q^{67} -15.2454 q^{71} +3.21760 q^{73} -2.67134 q^{75} +4.07954 q^{77} -3.09813 q^{79} +1.00000 q^{81} +5.43710 q^{83} -4.73557 q^{85} +2.66395 q^{87} -15.0633 q^{89} -11.3488 q^{91} +6.80696 q^{93} +6.33470 q^{95} -3.83748 q^{97} -1.08855 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 10 q^{3} - 2 q^{5} - 11 q^{7} + 10 q^{9}+O(q^{10}) \) 10 * q + 10 * q^3 - 2 * q^5 - 11 * q^7 + 10 * q^9	\( 10 q + 10 q^{3} - 2 q^{5} - 11 q^{7} + 10 q^{9} - 11 q^{11} - 2 q^{15} - 13 q^{17} - 18 q^{19} - 11 q^{21} + 10 q^{25} + 10 q^{27} + 5 q^{29} + 15 q^{31} - 11 q^{33} - 13 q^{35} - 5 q^{37} + 24 q^{41} - 40 q^{43} - 2 q^{45} - 9 q^{47} + 5 q^{49} - 13 q^{51} - 6 q^{53} - 14 q^{55} - 18 q^{57} + 28 q^{59} - 39 q^{61} - 11 q^{63} - 14 q^{65} - 32 q^{67} - 33 q^{71} - 50 q^{73} + 10 q^{75} + 19 q^{77} - 33 q^{79} + 10 q^{81} - 29 q^{83} - 21 q^{85} + 5 q^{87} - 17 q^{89} - 36 q^{91} + 15 q^{93} - 42 q^{95} - 46 q^{97} - 11 q^{99}+O(q^{100}) \) 10 * q + 10 * q^3 - 2 * q^5 - 11 * q^7 + 10 * q^9 - 11 * q^11 - 2 * q^15 - 13 * q^17 - 18 * q^19 - 11 * q^21 + 10 * q^25 + 10 * q^27 + 5 * q^29 + 15 * q^31 - 11 * q^33 - 13 * q^35 - 5 * q^37 + 24 * q^41 - 40 * q^43 - 2 * q^45 - 9 * q^47 + 5 * q^49 - 13 * q^51 - 6 * q^53 - 14 * q^55 - 18 * q^57 + 28 * q^59 - 39 * q^61 - 11 * q^63 - 14 * q^65 - 32 * q^67 - 33 * q^71 - 50 * q^73 + 10 * q^75 + 19 * q^77 - 33 * q^79 + 10 * q^81 - 29 * q^83 - 21 * q^85 + 5 * q^87 - 17 * q^89 - 36 * q^91 + 15 * q^93 - 42 * q^95 - 46 * q^97 - 11 * q^99

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