LMFDB - Embedded newform 6840.2.a.bk.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6840,2,Mod(1,6840)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6840, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("6840.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6840 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6840.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:	$54.6176749826$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.568.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 6x - 2 $ x^3 - x^2 - 6*x - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 2280)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.363328$ of defining polynomial
Character	$\chi$	$=$	6840.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{5} -2.77801 q^{7} +O(q^{10})$	$q+1.00000 q^{5} -2.77801 q^{7} +0.778008 q^{11} -5.50466 q^{13} +7.00933 q^{17} +1.00000 q^{19} +1.27334 q^{23} +1.00000 q^{25} -8.23132 q^{29} +5.00933 q^{31} -2.77801 q^{35} +8.51399 q^{37} -9.06068 q^{41} -10.5140 q^{43} +6.72666 q^{47} +0.717328 q^{49} -8.72666 q^{53} +0.778008 q^{55} +5.29200 q^{59} -7.73599 q^{61} -5.50466 q^{65} -1.45331 q^{67} +12.4626 q^{71} +14.5653 q^{73} -2.16131 q^{77} +2.28267 q^{79} -3.45331 q^{83} +7.00933 q^{85} +2.51399 q^{89} +15.2920 q^{91} +1.00000 q^{95} +8.51399 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{5} - 2 q^{7}+O(q^{10}) $ 3 * q + 3 * q^5 - 2 * q^7	$ 3 q + 3 q^{5} - 2 q^{7} - 4 q^{11} - 6 q^{13} + 3 q^{19} + 8 q^{23} + 3 q^{25} - 10 q^{29} - 6 q^{31} - 2 q^{35} - 6 q^{37} - 4 q^{41} + 16 q^{47} + 19 q^{49} - 22 q^{53} - 4 q^{55} - 22 q^{59} + 2 q^{61} - 6 q^{65} + 4 q^{67} + 8 q^{71} + 10 q^{73} - 36 q^{77} - 10 q^{79} - 2 q^{83} - 24 q^{89} + 8 q^{91} + 3 q^{95} - 6 q^{97}+O(q^{100}) $ 3 * q + 3 * q^5 - 2 * q^7 - 4 * q^11 - 6 * q^13 + 3 * q^19 + 8 * q^23 + 3 * q^25 - 10 * q^29 - 6 * q^31 - 2 * q^35 - 6 * q^37 - 4 * q^41 + 16 * q^47 + 19 * q^49 - 22 * q^53 - 4 * q^55 - 22 * q^59 + 2 * q^61 - 6 * q^65 + 4 * q^67 + 8 * q^71 + 10 * q^73 - 36 * q^77 - 10 * q^79 - 2 * q^83 - 24 * q^89 + 8 * q^91 + 3 * q^95 - 6 * q^97

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−2.77801	−1.04999	−0.524994	−	0.851106i	$-0.675932\pi$
$7$	−2.77801	−1.04999	−0.524994	+	0.851106i	$0.675932\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	0.778008	0.234578	0.117289	−	0.993098i	$-0.462580\pi$
$11$	0.778008	0.234578	0.117289	+	0.993098i	$0.462580\pi$
$12$	0	0
$13$	−5.50466	−1.52672	−0.763360	−	0.645974i	$-0.776452\pi$
$13$	−5.50466	−1.52672	−0.763360	+	0.645974i	$0.776452\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	7.00933	1.70001	0.850006	−	0.526773i	$-0.176598\pi$
$17$	7.00933	1.70001	0.850006	+	0.526773i	$0.176598\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	0	0
$22$	0	0
$23$	1.27334	0.265510	0.132755	−	0.991149i	$-0.457618\pi$
$23$	1.27334	0.265510	0.132755	+	0.991149i	$0.457618\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−8.23132	−1.52852	−0.764259	−	0.644909i	$-0.776895\pi$
$29$	−8.23132	−1.52852	−0.764259	+	0.644909i	$0.776895\pi$
$30$	0	0
$31$	5.00933	0.899702	0.449851	−	0.893104i	$-0.351477\pi$
$31$	5.00933	0.899702	0.449851	+	0.893104i	$0.351477\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−2.77801	−0.469569
$36$	0	0
$37$	8.51399	1.39969	0.699846	−	0.714294i	$-0.253252\pi$
$37$	8.51399	1.39969	0.699846	+	0.714294i	$0.253252\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−9.06068	−1.41504	−0.707520	−	0.706693i	$-0.750186\pi$
$41$	−9.06068	−1.41504	−0.707520	+	0.706693i	$0.750186\pi$
$42$	0	0
$43$	−10.5140	−1.60337	−0.801684	−	0.597747i	$-0.796063\pi$
$43$	−10.5140	−1.60337	−0.801684	+	0.597747i	$0.796063\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	6.72666	0.981184	0.490592	−	0.871389i	$-0.336781\pi$
$47$	6.72666	0.981184	0.490592	+	0.871389i	$0.336781\pi$
$48$	0	0
$49$	0.717328	0.102475
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−8.72666	−1.19870	−0.599349	−	0.800488i	$-0.704574\pi$
$53$	−8.72666	−1.19870	−0.599349	+	0.800488i	$0.704574\pi$
$54$	0	0
$55$	0.778008	0.104907
$56$	0	0
$57$	0	0
$58$	0	0
$59$	5.29200	0.688960	0.344480	−	0.938794i	$-0.388055\pi$
$59$	5.29200	0.688960	0.344480	+	0.938794i	$0.388055\pi$
$60$	0	0
$61$	−7.73599	−0.990491	−0.495246	−	0.868753i	$-0.664922\pi$
$61$	−7.73599	−0.990491	−0.495246	+	0.868753i	$0.664922\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−5.50466	−0.682770
$66$	0	0
$67$	−1.45331	−0.177550	−0.0887752	−	0.996052i	$-0.528295\pi$
$67$	−1.45331	−0.177550	−0.0887752	+	0.996052i	$0.528295\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	12.4626	1.47904	0.739522	−	0.673133i	$-0.235052\pi$
$71$	12.4626	1.47904	0.739522	+	0.673133i	$0.235052\pi$
$72$	0	0
$73$	14.5653	1.70474	0.852372	−	0.522935i	$-0.175163\pi$
$73$	14.5653	1.70474	0.852372	+	0.522935i	$0.175163\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−2.16131	−0.246304
$78$	0	0
$79$	2.28267	0.256821	0.128410	−	0.991721i	$-0.459013\pi$
$79$	2.28267	0.256821	0.128410	+	0.991721i	$0.459013\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−3.45331	−0.379050	−0.189525	−	0.981876i	$-0.560695\pi$
$83$	−3.45331	−0.379050	−0.189525	+	0.981876i	$0.560695\pi$
$84$	0	0
$85$	7.00933	0.760268
$86$	0	0
$87$	0	0
$88$	0	0
$89$	2.51399	0.266483	0.133241	−	0.991084i	$-0.457461\pi$
$89$	2.51399	0.266483	0.133241	+	0.991084i	$0.457461\pi$
$90$	0	0
$91$	15.2920	1.60304
$92$	0	0
$93$	0	0
$94$	0	0
$95$	1.00000	0.102598
$96$	0	0
$97$	8.51399	0.864465	0.432233	−	0.901762i	$-0.357726\pi$
$97$	8.51399	0.864465	0.432233	+	0.901762i	$0.357726\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−15.5560	−1.54788	−0.773941	−	0.633258i	$-0.781717\pi$
$101$	−15.5560	−1.54788	−0.773941	+	0.633258i	$0.781717\pi$
$102$	0	0
$103$	−1.45331	−0.143199	−0.0715996	−	0.997433i	$-0.522810\pi$
$103$	−1.45331	−0.143199	−0.0715996	+	0.997433i	$0.522810\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−14.0187	−1.35523	−0.677617	−	0.735415i	$-0.736987\pi$
$107$	−14.0187	−1.35523	−0.677617	+	0.735415i	$0.736987\pi$
$108$	0	0
$109$	−7.29200	−0.698447	−0.349224	−	0.937039i	$-0.613555\pi$
$109$	−7.29200	−0.698447	−0.349224	+	0.937039i	$0.613555\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−19.7360	−1.85661	−0.928303	−	0.371825i	$-0.878732\pi$
$113$	−19.7360	−1.85661	−0.928303	+	0.371825i	$0.878732\pi$
$114$	0	0
$115$	1.27334	0.118740
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−19.4720	−1.78499
$120$	0	0
$121$	−10.3947	−0.944973
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	1.45331	0.128961	0.0644803	−	0.997919i	$-0.479461\pi$
$127$	1.45331	0.128961	0.0644803	+	0.997919i	$0.479461\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−15.7873	−1.37935	−0.689673	−	0.724121i	$-0.742246\pi$
$131$	−15.7873	−1.37935	−0.689673	+	0.724121i	$0.742246\pi$
$132$	0	0
$133$	−2.77801	−0.240884
$134$	0	0
$135$	0	0
$136$	0	0
$137$	0.282672	0.0241503	0.0120752	−	0.999927i	$-0.496156\pi$
$137$	0.282672	0.0241503	0.0120752	+	0.999927i	$0.496156\pi$
$138$	0	0
$139$	16.4626	1.39634	0.698172	−	0.715931i	$-0.253997\pi$
$139$	16.4626	1.39634	0.698172	+	0.715931i	$0.253997\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−4.28267	−0.358135
$144$	0	0
$145$	−8.23132	−0.683574
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−14.0000	−1.14692	−0.573462	−	0.819232i	$-0.694400\pi$
$149$	−14.0000	−1.14692	−0.573462	+	0.819232i	$0.694400\pi$
$150$	0	0
$151$	−21.5747	−1.75572	−0.877861	−	0.478916i	$-0.841030\pi$
$151$	−21.5747	−1.75572	−0.877861	+	0.478916i	$0.841030\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	5.00933	0.402359
$156$	0	0
$157$	−13.8387	−1.10445	−0.552224	−	0.833696i	$-0.686221\pi$
$157$	−13.8387	−1.10445	−0.552224	+	0.833696i	$0.686221\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−3.53736	−0.278783
$162$	0	0
$163$	18.6167	1.45817	0.729086	−	0.684422i	$-0.239945\pi$
$163$	18.6167	1.45817	0.729086	+	0.684422i	$0.239945\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	9.73599	0.753393	0.376697	−	0.926337i	$-0.377060\pi$
$167$	9.73599	0.753393	0.376697	+	0.926337i	$0.377060\pi$
$168$	0	0
$169$	17.3013	1.33087
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−4.82936	−0.367169	−0.183585	−	0.983004i	$-0.558770\pi$
$173$	−4.82936	−0.367169	−0.183585	+	0.983004i	$0.558770\pi$
$174$	0	0
$175$	−2.77801	−0.209998
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−23.8387	−1.78179	−0.890894	−	0.454212i	$-0.849921\pi$
$179$	−23.8387	−1.78179	−0.890894	+	0.454212i	$0.849921\pi$
$180$	0	0
$181$	−14.7267	−1.09462	−0.547312	−	0.836929i	$-0.684349\pi$
$181$	−14.7267	−1.09462	−0.547312	+	0.836929i	$0.684349\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	8.51399	0.625961
$186$	0	0
$187$	5.45331	0.398786
$188$	0	0
$189$	0	0
$190$	0	0
$191$	3.22199	0.233135	0.116568	−	0.993183i	$-0.462811\pi$
$191$	3.22199	0.233135	0.116568	+	0.993183i	$0.462811\pi$
$192$	0	0
$193$	5.50466	0.396234	0.198117	−	0.980178i	$-0.436517\pi$
$193$	5.50466	0.396234	0.198117	+	0.980178i	$0.436517\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−6.17997	−0.440305	−0.220152	−	0.975466i	$-0.570655\pi$
$197$	−6.17997	−0.440305	−0.220152	+	0.975466i	$0.570655\pi$
$198$	0	0
$199$	−17.0280	−1.20708	−0.603541	−	0.797332i	$-0.706244\pi$
$199$	−17.0280	−1.20708	−0.603541	+	0.797332i	$0.706244\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	22.8667	1.60493
$204$	0	0
$205$	−9.06068	−0.632825
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0.778008	0.0538159
$210$	0	0
$211$	8.56534	0.589663	0.294831	−	0.955549i	$-0.404737\pi$
$211$	8.56534	0.589663	0.294831	+	0.955549i	$0.404737\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−10.5140	−0.717048
$216$	0	0
$217$	−13.9160	−0.944677
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−38.5840	−2.59544
$222$	0	0
$223$	−1.98134	−0.132681	−0.0663403	−	0.997797i	$-0.521132\pi$
$223$	−1.98134	−0.132681	−0.0663403	+	0.997797i	$0.521132\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−0.282672	−0.0187616	−0.00938081	−	0.999956i	$-0.502986\pi$
$227$	−0.282672	−0.0187616	−0.00938081	+	0.999956i	$0.502986\pi$
$228$	0	0
$229$	4.26401	0.281774	0.140887	−	0.990026i	$-0.455005\pi$
$229$	4.26401	0.281774	0.140887	+	0.990026i	$0.455005\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	6.62395	0.433950	0.216975	−	0.976177i	$-0.430381\pi$
$233$	6.62395	0.433950	0.216975	+	0.976177i	$0.430381\pi$
$234$	0	0
$235$	6.72666	0.438799
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−4.21266	−0.272495	−0.136247	−	0.990675i	$-0.543504\pi$
$239$	−4.21266	−0.272495	−0.136247	+	0.990675i	$0.543504\pi$
$240$	0	0
$241$	−25.1120	−1.61761	−0.808804	−	0.588078i	$-0.799885\pi$
$241$	−25.1120	−1.61761	−0.808804	+	0.588078i	$0.799885\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0.717328	0.0458284
$246$	0	0
$247$	−5.50466	−0.350253
$248$	0	0
$249$	0	0
$250$	0	0
$251$	22.3340	1.40971	0.704856	−	0.709351i	$-0.251012\pi$
$251$	22.3340	1.40971	0.704856	+	0.709351i	$0.251012\pi$
$252$	0	0
$253$	0.990671	0.0622830
$254$	0	0
$255$	0	0
$256$	0	0
$257$	0.829359	0.0517340	0.0258670	−	0.999665i	$-0.491765\pi$
$257$	0.829359	0.0517340	0.0258670	+	0.999665i	$0.491765\pi$
$258$	0	0
$259$	−23.6519	−1.46966
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−5.73599	−0.353696	−0.176848	−	0.984238i	$-0.556590\pi$
$263$	−5.73599	−0.353696	−0.176848	+	0.984238i	$0.556590\pi$
$264$	0	0
$265$	−8.72666	−0.536074
$266$	0	0
$267$	0	0
$268$	0	0
$269$	15.1379	0.922977	0.461488	−	0.887146i	$-0.347316\pi$
$269$	15.1379	0.922977	0.461488	+	0.887146i	$0.347316\pi$
$270$	0	0
$271$	−24.6680	−1.49848	−0.749239	−	0.662300i	$-0.769580\pi$
$271$	−24.6680	−1.49848	−0.749239	+	0.662300i	$0.769580\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0.778008	0.0469156
$276$	0	0
$277$	−4.28267	−0.257321	−0.128660	−	0.991689i	$-0.541068\pi$
$277$	−4.28267	−0.257321	−0.128660	+	0.991689i	$0.541068\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−30.5140	−1.82031	−0.910156	−	0.414265i	$-0.864039\pi$
$281$	−30.5140	−1.82031	−0.910156	+	0.414265i	$0.864039\pi$
$282$	0	0
$283$	17.5233	1.04165	0.520827	−	0.853662i	$-0.325624\pi$
$283$	17.5233	1.04165	0.520827	+	0.853662i	$0.325624\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	25.1706	1.48578
$288$	0	0
$289$	32.1307	1.89004
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−5.71733	−0.334010	−0.167005	−	0.985956i	$-0.553410\pi$
$293$	−5.71733	−0.334010	−0.167005	+	0.985956i	$0.553410\pi$
$294$	0	0
$295$	5.29200	0.308112
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−7.00933	−0.405360
$300$	0	0
$301$	29.2080	1.68352
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−7.73599	−0.442961
$306$	0	0
$307$	−8.46264	−0.482988	−0.241494	−	0.970402i	$-0.577637\pi$
$307$	−8.46264	−0.482988	−0.241494	+	0.970402i	$0.577637\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	10.2313	0.580165	0.290082	−	0.957002i	$-0.406317\pi$
$311$	10.2313	0.580165	0.290082	+	0.957002i	$0.406317\pi$
$312$	0	0
$313$	30.1400	1.70361	0.851807	−	0.523855i	$-0.175507\pi$
$313$	30.1400	1.70361	0.851807	+	0.523855i	$0.175507\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−7.83869	−0.440265	−0.220132	−	0.975470i	$-0.570649\pi$
$317$	−7.83869	−0.440265	−0.220132	+	0.975470i	$0.570649\pi$
$318$	0	0
$319$	−6.40403	−0.358557
$320$	0	0
$321$	0	0
$322$	0	0
$323$	7.00933	0.390009
$324$	0	0
$325$	−5.50466	−0.305344
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−18.6867	−1.03023
$330$	0	0
$331$	−7.29200	−0.400805	−0.200402	−	0.979714i	$-0.564225\pi$
$331$	−7.29200	−0.400805	−0.200402	+	0.979714i	$0.564225\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−1.45331	−0.0794030
$336$	0	0
$337$	−10.4953	−0.571717	−0.285859	−	0.958272i	$-0.592279\pi$
$337$	−10.4953	−0.571717	−0.285859	+	0.958272i	$0.592279\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	3.89730	0.211050
$342$	0	0
$343$	17.4533	0.942390
$344$	0	0
$345$	0	0
$346$	0	0
$347$	5.43466	0.291748	0.145874	−	0.989303i	$-0.453401\pi$
$347$	5.43466	0.291748	0.145874	+	0.989303i	$0.453401\pi$
$348$	0	0
$349$	−1.43466	−0.0767954	−0.0383977	−	0.999263i	$-0.512225\pi$
$349$	−1.43466	−0.0767954	−0.0383977	+	0.999263i	$0.512225\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	5.73599	0.305296	0.152648	−	0.988281i	$-0.451220\pi$
$353$	5.73599	0.305296	0.152648	+	0.988281i	$0.451220\pi$
$354$	0	0
$355$	12.4626	0.661448
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−28.7780	−1.51885	−0.759423	−	0.650598i	$-0.774518\pi$
$359$	−28.7780	−1.51885	−0.759423	+	0.650598i	$0.774518\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	0	0
$364$	0	0
$365$	14.5653	0.762385
$366$	0	0
$367$	−31.9087	−1.66562	−0.832810	−	0.553559i	$-0.813269\pi$
$367$	−31.9087	−1.66562	−0.832810	+	0.553559i	$0.813269\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	24.2427	1.25862
$372$	0	0
$373$	−23.5233	−1.21799	−0.608996	−	0.793174i	$-0.708427\pi$
$373$	−23.5233	−1.21799	−0.608996	+	0.793174i	$0.708427\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	45.3107	2.33362
$378$	0	0
$379$	27.2920	1.40190	0.700948	−	0.713212i	$-0.252761\pi$
$379$	27.2920	1.40190	0.700948	+	0.713212i	$0.252761\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−5.65872	−0.289147	−0.144574	−	0.989494i	$-0.546181\pi$
$383$	−5.65872	−0.289147	−0.144574	+	0.989494i	$0.546181\pi$
$384$	0	0
$385$	−2.16131	−0.110151
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−30.9253	−1.56797	−0.783987	−	0.620777i	$-0.786817\pi$
$389$	−30.9253	−1.56797	−0.783987	+	0.620777i	$0.786817\pi$
$390$	0	0
$391$	8.92528	0.451371
$392$	0	0
$393$	0	0
$394$	0	0
$395$	2.28267	0.114854
$396$	0	0
$397$	−11.1893	−0.561575	−0.280787	−	0.959770i	$-0.590596\pi$
$397$	−11.1893	−0.561575	−0.280787	+	0.959770i	$0.590596\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	7.60737	0.379894	0.189947	−	0.981794i	$-0.439168\pi$
$401$	7.60737	0.379894	0.189947	+	0.981794i	$0.439168\pi$
$402$	0	0
$403$	−27.5747	−1.37359
$404$	0	0
$405$	0	0
$406$	0	0
$407$	6.62395	0.328337
$408$	0	0
$409$	−18.6680	−0.923076	−0.461538	−	0.887121i	$-0.652702\pi$
$409$	−18.6680	−0.923076	−0.461538	+	0.887121i	$0.652702\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−14.7012	−0.723400
$414$	0	0
$415$	−3.45331	−0.169516
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−30.8994	−1.50953	−0.754766	−	0.655994i	$-0.772250\pi$
$419$	−30.8994	−1.50953	−0.754766	+	0.655994i	$0.772250\pi$
$420$	0	0
$421$	−21.7360	−1.05935	−0.529674	−	0.848202i	$-0.677686\pi$
$421$	−21.7360	−1.05935	−0.529674	+	0.848202i	$0.677686\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	7.00933	0.340002
$426$	0	0
$427$	21.4906	1.04000
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−24.1400	−1.16278	−0.581392	−	0.813624i	$-0.697492\pi$
$431$	−24.1400	−1.16278	−0.581392	+	0.813624i	$0.697492\pi$
$432$	0	0
$433$	−6.53265	−0.313939	−0.156970	−	0.987603i	$-0.550172\pi$
$433$	−6.53265	−0.313939	−0.156970	+	0.987603i	$0.550172\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	1.27334	0.0609123
$438$	0	0
$439$	27.2080	1.29856	0.649282	−	0.760547i	$-0.275069\pi$
$439$	27.2080	1.29856	0.649282	+	0.760547i	$0.275069\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	36.1587	1.71795	0.858975	−	0.512017i	$-0.171102\pi$
$443$	36.1587	1.71795	0.858975	+	0.512017i	$0.171102\pi$
$444$	0	0
$445$	2.51399	0.119175
$446$	0	0
$447$	0	0
$448$	0	0
$449$	38.6540	1.82420	0.912098	−	0.409973i	$-0.134462\pi$
$449$	38.6540	1.82420	0.912098	+	0.409973i	$0.134462\pi$
$450$	0	0
$451$	−7.04928	−0.331938
$452$	0	0
$453$	0	0
$454$	0	0
$455$	15.2920	0.716900
$456$	0	0
$457$	40.1587	1.87854	0.939272	−	0.343174i	$-0.111502\pi$
$457$	40.1587	1.87854	0.939272	+	0.343174i	$0.111502\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	22.9253	1.06774	0.533868	−	0.845568i	$-0.320738\pi$
$461$	22.9253	1.06774	0.533868	+	0.845568i	$0.320738\pi$
$462$	0	0
$463$	−5.11929	−0.237914	−0.118957	−	0.992899i	$-0.537955\pi$
$463$	−5.11929	−0.237914	−0.118957	+	0.992899i	$0.537955\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−35.0280	−1.62090	−0.810451	−	0.585807i	$-0.800778\pi$
$467$	−35.0280	−1.62090	−0.810451	+	0.585807i	$0.800778\pi$
$468$	0	0
$469$	4.03731	0.186426
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−8.17997	−0.376115
$474$	0	0
$475$	1.00000	0.0458831
$476$	0	0
$477$	0	0
$478$	0	0
$479$	40.8153	1.86490	0.932450	−	0.361299i	$-0.117667\pi$
$479$	40.8153	1.86490	0.932450	+	0.361299i	$0.117667\pi$
$480$	0	0
$481$	−46.8667	−2.13694
$482$	0	0
$483$	0	0
$484$	0	0
$485$	8.51399	0.386601
$486$	0	0
$487$	−21.9160	−0.993107	−0.496553	−	0.868006i	$-0.665401\pi$
$487$	−21.9160	−0.993107	−0.496553	+	0.868006i	$0.665401\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	15.9927	0.721742	0.360871	−	0.932616i	$-0.382479\pi$
$491$	15.9927	0.721742	0.360871	+	0.932616i	$0.382479\pi$
$492$	0	0
$493$	−57.6960	−2.59850
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−34.6213	−1.55298
$498$	0	0
$499$	−0.668047	−0.0299059	−0.0149530	−	0.999888i	$-0.504760\pi$
$499$	−0.668047	−0.0299059	−0.0149530	+	0.999888i	$0.504760\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−12.4881	−0.556816	−0.278408	−	0.960463i	$-0.589807\pi$
$503$	−12.4881	−0.556816	−0.278408	+	0.960463i	$0.589807\pi$
$504$	0	0
$505$	−15.5560	−0.692234
$506$	0	0
$507$	0	0
$508$	0	0
$509$	1.32469	0.0587161	0.0293580	−	0.999569i	$-0.490654\pi$
$509$	1.32469	0.0587161	0.0293580	+	0.999569i	$0.490654\pi$
$510$	0	0
$511$	−40.4626	−1.78996
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−1.45331	−0.0640406
$516$	0	0
$517$	5.23339	0.230164
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−32.3272	−1.41628	−0.708141	−	0.706071i	$-0.750466\pi$
$521$	−32.3272	−1.41628	−0.708141	+	0.706071i	$0.750466\pi$
$522$	0	0
$523$	33.2334	1.45319	0.726597	−	0.687063i	$-0.241101\pi$
$523$	33.2334	1.45319	0.726597	+	0.687063i	$0.241101\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	35.1120	1.52950
$528$	0	0
$529$	−21.3786	−0.929504
$530$	0	0
$531$	0	0
$532$	0	0
$533$	49.8760	2.16037
$534$	0	0
$535$	−14.0187	−0.606079
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0.558087	0.0240385
$540$	0	0
$541$	11.0934	0.476941	0.238471	−	0.971150i	$-0.423354\pi$
$541$	11.0934	0.476941	0.238471	+	0.971150i	$0.423354\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−7.29200	−0.312355
$546$	0	0
$547$	−5.55602	−0.237558	−0.118779	−	0.992921i	$-0.537898\pi$
$547$	−5.55602	−0.237558	−0.118779	+	0.992921i	$0.537898\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−8.23132	−0.350666
$552$	0	0
$553$	−6.34128	−0.269659
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−30.2427	−1.28143	−0.640713	−	0.767781i	$-0.721361\pi$
$557$	−30.2427	−1.28143	−0.640713	+	0.767781i	$0.721361\pi$
$558$	0	0
$559$	57.8760	2.44789
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−32.8853	−1.38595	−0.692976	−	0.720961i	$-0.743701\pi$
$563$	−32.8853	−1.38595	−0.692976	+	0.720961i	$0.743701\pi$
$564$	0	0
$565$	−19.7360	−0.830299
$566$	0	0
$567$	0	0
$568$	0	0
$569$	9.48601	0.397674	0.198837	−	0.980033i	$-0.436284\pi$
$569$	9.48601	0.397674	0.198837	+	0.980033i	$0.436284\pi$
$570$	0	0
$571$	−45.0653	−1.88592	−0.942962	−	0.332900i	$-0.891973\pi$
$571$	−45.0653	−1.88592	−0.942962	+	0.332900i	$0.891973\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	1.27334	0.0531021
$576$	0	0
$577$	−34.6680	−1.44325	−0.721625	−	0.692284i	$-0.756604\pi$
$577$	−34.6680	−1.44325	−0.721625	+	0.692284i	$0.756604\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	9.59333	0.397998
$582$	0	0
$583$	−6.78941	−0.281189
$584$	0	0
$585$	0	0
$586$	0	0
$587$	35.3879	1.46062	0.730308	−	0.683118i	$-0.239377\pi$
$587$	35.3879	1.46062	0.730308	+	0.683118i	$0.239377\pi$
$588$	0	0
$589$	5.00933	0.206406
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−25.2080	−1.03517	−0.517583	−	0.855633i	$-0.673168\pi$
$593$	−25.2080	−1.03517	−0.517583	+	0.855633i	$0.673168\pi$
$594$	0	0
$595$	−19.4720	−0.798273
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−32.6027	−1.33211	−0.666054	−	0.745903i	$-0.732018\pi$
$599$	−32.6027	−1.33211	−0.666054	+	0.745903i	$0.732018\pi$
$600$	0	0
$601$	5.89730	0.240556	0.120278	−	0.992740i	$-0.461621\pi$
$601$	5.89730	0.240556	0.120278	+	0.992740i	$0.461621\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−10.3947	−0.422605
$606$	0	0
$607$	8.46264	0.343488	0.171744	−	0.985142i	$-0.445060\pi$
$607$	8.46264	0.343488	0.171744	+	0.985142i	$0.445060\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−37.0280	−1.49799
$612$	0	0
$613$	46.4413	1.87575	0.937874	−	0.346976i	$-0.112791\pi$
$613$	46.4413	1.87575	0.937874	+	0.346976i	$0.112791\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−4.42533	−0.178157	−0.0890785	−	0.996025i	$-0.528392\pi$
$617$	−4.42533	−0.178157	−0.0890785	+	0.996025i	$0.528392\pi$
$618$	0	0
$619$	28.9907	1.16523	0.582617	−	0.812747i	$-0.302029\pi$
$619$	28.9907	1.16523	0.582617	+	0.812747i	$0.302029\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−6.98389	−0.279804
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	59.6774	2.37949
$630$	0	0
$631$	−1.98134	−0.0788760	−0.0394380	−	0.999222i	$-0.512557\pi$
$631$	−1.98134	−0.0788760	−0.0394380	+	0.999222i	$0.512557\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	1.45331	0.0576730
$636$	0	0
$637$	−3.94865	−0.156451
$638$	0	0
$639$	0	0
$640$	0	0
$641$	4.17271	0.164812	0.0824061	−	0.996599i	$-0.473740\pi$
$641$	4.17271	0.164812	0.0824061	+	0.996599i	$0.473740\pi$
$642$	0	0
$643$	−14.6167	−0.576426	−0.288213	−	0.957566i	$-0.593061\pi$
$643$	−14.6167	−0.576426	−0.288213	+	0.957566i	$0.593061\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−1.17064	−0.0460226	−0.0230113	−	0.999735i	$-0.507325\pi$
$647$	−1.17064	−0.0460226	−0.0230113	+	0.999735i	$0.507325\pi$
$648$	0	0
$649$	4.11722	0.161615
$650$	0	0
$651$	0	0
$652$	0	0
$653$	17.6519	0.690774	0.345387	−	0.938460i	$-0.387748\pi$
$653$	17.6519	0.690774	0.345387	+	0.938460i	$0.387748\pi$
$654$	0	0
$655$	−15.7873	−0.616862
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−28.5067	−1.11046	−0.555232	−	0.831695i	$-0.687371\pi$
$659$	−28.5067	−1.11046	−0.555232	+	0.831695i	$0.687371\pi$
$660$	0	0
$661$	29.5161	1.14804	0.574021	−	0.818841i	$-0.305383\pi$
$661$	29.5161	1.14804	0.574021	+	0.818841i	$0.305383\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−2.77801	−0.107727
$666$	0	0
$667$	−10.4813	−0.405838
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−6.01866	−0.232348
$672$	0	0
$673$	1.50466	0.0580005	0.0290003	−	0.999579i	$-0.490768\pi$
$673$	1.50466	0.0580005	0.0290003	+	0.999579i	$0.490768\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	9.85735	0.378849	0.189424	−	0.981895i	$-0.439338\pi$
$677$	9.85735	0.378849	0.189424	+	0.981895i	$0.439338\pi$
$678$	0	0
$679$	−23.6519	−0.907678
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−1.65872	−0.0634691	−0.0317346	−	0.999496i	$-0.510103\pi$
$683$	−1.65872	−0.0634691	−0.0317346	+	0.999496i	$0.510103\pi$
$684$	0	0
$685$	0.282672	0.0108004
$686$	0	0
$687$	0	0
$688$	0	0
$689$	48.0373	1.83008
$690$	0	0
$691$	19.2147	0.730963	0.365481	−	0.930819i	$-0.380904\pi$
$691$	19.2147	0.730963	0.365481	+	0.930819i	$0.380904\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	16.4626	0.624464
$696$	0	0
$697$	−63.5093	−2.40559
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−34.5653	−1.30552	−0.652758	−	0.757567i	$-0.726388\pi$
$701$	−34.5653	−1.30552	−0.652758	+	0.757567i	$0.726388\pi$
$702$	0	0
$703$	8.51399	0.321111
$704$	0	0
$705$	0	0
$706$	0	0
$707$	43.2147	1.62526
$708$	0	0
$709$	2.84802	0.106960	0.0534798	−	0.998569i	$-0.482969\pi$
$709$	2.84802	0.106960	0.0534798	+	0.998569i	$0.482969\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	6.37860	0.238880
$714$	0	0
$715$	−4.28267	−0.160163
$716$	0	0
$717$	0	0
$718$	0	0
$719$	6.12862	0.228559	0.114279	−	0.993449i	$-0.463544\pi$
$719$	6.12862	0.228559	0.114279	+	0.993449i	$0.463544\pi$
$720$	0	0
$721$	4.03731	0.150357
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−8.23132	−0.305704
$726$	0	0
$727$	−10.6753	−0.395925	−0.197963	−	0.980210i	$-0.563432\pi$
$727$	−10.6753	−0.395925	−0.197963	+	0.980210i	$0.563432\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−73.6960	−2.72575
$732$	0	0
$733$	40.6426	1.50117	0.750585	−	0.660774i	$-0.229772\pi$
$733$	40.6426	1.50117	0.750585	+	0.660774i	$0.229772\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−1.13069	−0.0416495
$738$	0	0
$739$	26.2241	0.964668	0.482334	−	0.875987i	$-0.339789\pi$
$739$	26.2241	0.964668	0.482334	+	0.875987i	$0.339789\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	8.56534	0.314232	0.157116	−	0.987580i	$-0.449780\pi$
$743$	8.56534	0.314232	0.157116	+	0.987580i	$0.449780\pi$
$744$	0	0
$745$	−14.0000	−0.512920
$746$	0	0
$747$	0	0
$748$	0	0
$749$	38.9439	1.42298
$750$	0	0
$751$	−41.3693	−1.50959	−0.754793	−	0.655963i	$-0.772263\pi$
$751$	−41.3693	−1.50959	−0.754793	+	0.655963i	$0.772263\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−21.5747	−0.785183
$756$	0	0
$757$	34.1986	1.24297	0.621485	−	0.783426i	$-0.286530\pi$
$757$	34.1986	1.24297	0.621485	+	0.783426i	$0.286530\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−6.00000	−0.217500	−0.108750	−	0.994069i	$-0.534685\pi$
$761$	−6.00000	−0.217500	−0.108750	+	0.994069i	$0.534685\pi$
$762$	0	0
$763$	20.2572	0.733361
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−29.1307	−1.05185
$768$	0	0
$769$	−24.5840	−0.886522	−0.443261	−	0.896393i	$-0.646178\pi$
$769$	−24.5840	−0.886522	−0.443261	+	0.896393i	$0.646178\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−42.2173	−1.51845	−0.759225	−	0.650828i	$-0.774422\pi$
$773$	−42.2173	−1.51845	−0.759225	+	0.650828i	$0.774422\pi$
$774$	0	0
$775$	5.00933	0.179940
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−9.06068	−0.324633
$780$	0	0
$781$	9.69603	0.346951
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−13.8387	−0.493924
$786$	0	0
$787$	24.8226	0.884829	0.442415	−	0.896811i	$-0.354122\pi$
$787$	24.8226	0.884829	0.442415	+	0.896811i	$0.354122\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	54.8267	1.94941
$792$	0	0
$793$	42.5840	1.51220
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−11.1706	−0.395684	−0.197842	−	0.980234i	$-0.563393\pi$
$797$	−11.1706	−0.395684	−0.197842	+	0.980234i	$0.563393\pi$
$798$	0	0
$799$	47.1493	1.66802
$800$	0	0
$801$	0	0
$802$	0	0
$803$	11.3320	0.399896
$804$	0	0
$805$	−3.53736	−0.124675
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−24.9066	−0.875670	−0.437835	−	0.899055i	$-0.644255\pi$
$809$	−24.9066	−0.875670	−0.437835	+	0.899055i	$0.644255\pi$
$810$	0	0
$811$	43.6774	1.53372	0.766860	−	0.641814i	$-0.221818\pi$
$811$	43.6774	1.53372	0.766860	+	0.641814i	$0.221818\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	18.6167	0.652114
$816$	0	0
$817$	−10.5140	−0.367838
$818$	0	0
$819$	0	0
$820$	0	0
$821$	16.8039	0.586461	0.293230	−	0.956042i	$-0.405270\pi$
$821$	16.8039	0.586461	0.293230	+	0.956042i	$0.405270\pi$
$822$	0	0
$823$	8.33402	0.290506	0.145253	−	0.989395i	$-0.453600\pi$
$823$	8.33402	0.290506	0.145253	+	0.989395i	$0.453600\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	11.7173	0.407451	0.203726	−	0.979028i	$-0.434695\pi$
$827$	11.7173	0.407451	0.203726	+	0.979028i	$0.434695\pi$
$828$	0	0
$829$	16.2173	0.563250	0.281625	−	0.959525i	$-0.409127\pi$
$829$	16.2173	0.563250	0.281625	+	0.959525i	$0.409127\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	5.02799	0.174209
$834$	0	0
$835$	9.73599	0.336928
$836$	0	0
$837$	0	0
$838$	0	0
$839$	35.6120	1.22946	0.614731	−	0.788737i	$-0.289264\pi$
$839$	35.6120	1.22946	0.614731	+	0.788737i	$0.289264\pi$
$840$	0	0
$841$	38.7546	1.33637
$842$	0	0
$843$	0	0
$844$	0	0
$845$	17.3013	0.595184
$846$	0	0
$847$	28.8766	0.992211
$848$	0	0
$849$	0	0
$850$	0	0
$851$	10.8412	0.371633
$852$	0	0
$853$	−24.3854	−0.834939	−0.417470	−	0.908691i	$-0.637083\pi$
$853$	−24.3854	−0.834939	−0.417470	+	0.908691i	$0.637083\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	39.2733	1.34155	0.670776	−	0.741660i	$-0.265961\pi$
$857$	39.2733	1.34155	0.670776	+	0.741660i	$0.265961\pi$
$858$	0	0
$859$	15.4720	0.527897	0.263948	−	0.964537i	$-0.414975\pi$
$859$	15.4720	0.527897	0.263948	+	0.964537i	$0.414975\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−5.65872	−0.192625	−0.0963125	−	0.995351i	$-0.530705\pi$
$863$	−5.65872	−0.192625	−0.0963125	+	0.995351i	$0.530705\pi$
$864$	0	0
$865$	−4.82936	−0.164203
$866$	0	0
$867$	0	0
$868$	0	0
$869$	1.77594	0.0602445
$870$	0	0
$871$	8.00000	0.271070
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−2.77801	−0.0939138
$876$	0	0
$877$	−3.06068	−0.103352	−0.0516759	−	0.998664i	$-0.516456\pi$
$877$	−3.06068	−0.103352	−0.0516759	+	0.998664i	$0.516456\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−12.6867	−0.427426	−0.213713	−	0.976896i	$-0.568556\pi$
$881$	−12.6867	−0.427426	−0.213713	+	0.976896i	$0.568556\pi$
$882$	0	0
$883$	−27.0793	−0.911292	−0.455646	−	0.890161i	$-0.650592\pi$
$883$	−27.0793	−0.911292	−0.455646	+	0.890161i	$0.650592\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	12.8880	0.432736	0.216368	−	0.976312i	$-0.430579\pi$
$887$	12.8880	0.432736	0.216368	+	0.976312i	$0.430579\pi$
$888$	0	0
$889$	−4.03731	−0.135407
$890$	0	0
$891$	0	0
$892$	0	0
$893$	6.72666	0.225099
$894$	0	0
$895$	−23.8387	−0.796839
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−41.2334	−1.37521
$900$	0	0
$901$	−61.1680	−2.03780
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−14.7267	−0.489531
$906$	0	0
$907$	−13.5560	−0.450120	−0.225060	−	0.974345i	$-0.572258\pi$
$907$	−13.5560	−0.450120	−0.225060	+	0.974345i	$0.572258\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	17.0934	0.566329	0.283164	−	0.959071i	$-0.408616\pi$
$911$	17.0934	0.566329	0.283164	+	0.959071i	$0.408616\pi$
$912$	0	0
$913$	−2.68670	−0.0889169
$914$	0	0
$915$	0	0
$916$	0	0
$917$	43.8573	1.44830
$918$	0	0
$919$	24.7708	0.817112	0.408556	−	0.912733i	$-0.366033\pi$
$919$	24.7708	0.817112	0.408556	+	0.912733i	$0.366033\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−68.6027	−2.25808
$924$	0	0
$925$	8.51399	0.279938
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−1.57467	−0.0516634	−0.0258317	−	0.999666i	$-0.508223\pi$
$929$	−1.57467	−0.0516634	−0.0258317	+	0.999666i	$0.508223\pi$
$930$	0	0
$931$	0.717328	0.0235095
$932$	0	0
$933$	0	0
$934$	0	0
$935$	5.45331	0.178342
$936$	0	0
$937$	−27.5933	−0.901435	−0.450717	−	0.892667i	$-0.648832\pi$
$937$	−27.5933	−0.901435	−0.450717	+	0.892667i	$0.648832\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−52.3713	−1.70726	−0.853628	−	0.520882i	$-0.825603\pi$
$941$	−52.3713	−1.70726	−0.853628	+	0.520882i	$0.825603\pi$
$942$	0	0
$943$	−11.5374	−0.375708
$944$	0	0
$945$	0	0
$946$	0	0
$947$	46.2427	1.50269	0.751343	−	0.659912i	$-0.229406\pi$
$947$	46.2427	1.50269	0.751343	+	0.659912i	$0.229406\pi$
$948$	0	0
$949$	−80.1773	−2.60267
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−37.1893	−1.20468	−0.602340	−	0.798240i	$-0.705765\pi$
$953$	−37.1893	−1.20468	−0.602340	+	0.798240i	$0.705765\pi$
$954$	0	0
$955$	3.22199	0.104261
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−0.785266	−0.0253576
$960$	0	0
$961$	−5.90663	−0.190536
$962$	0	0
$963$	0	0
$964$	0	0
$965$	5.50466	0.177201
$966$	0	0
$967$	42.2500	1.35867	0.679334	−	0.733829i	$-0.262269\pi$
$967$	42.2500	1.35867	0.679334	+	0.733829i	$0.262269\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	7.73599	0.248260	0.124130	−	0.992266i	$-0.460386\pi$
$971$	7.73599	0.248260	0.124130	+	0.992266i	$0.460386\pi$
$972$	0	0
$973$	−45.7333	−1.46614
$974$	0	0
$975$	0	0
$976$	0	0
$977$	38.9880	1.24734	0.623669	−	0.781689i	$-0.285641\pi$
$977$	38.9880	1.24734	0.623669	+	0.781689i	$0.285641\pi$
$978$	0	0
$979$	1.95591	0.0625110
$980$	0	0
$981$	0	0
$982$	0	0
$983$	7.39470	0.235854	0.117927	−	0.993022i	$-0.462375\pi$
$983$	7.39470	0.235854	0.117927	+	0.993022i	$0.462375\pi$
$984$	0	0
$985$	−6.17997	−0.196910
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−13.3879	−0.425711
$990$	0	0
$991$	44.6613	1.41871	0.709356	−	0.704850i	$-0.248986\pi$
$991$	44.6613	1.41871	0.709356	+	0.704850i	$0.248986\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−17.0280	−0.539823
$996$	0	0
$997$	−8.74531	−0.276967	−0.138483	−	0.990365i	$-0.544223\pi$
$997$	−8.74531	−0.276967	−0.138483	+	0.990365i	$0.544223\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6840.2.a.bk.1.2		3
3.2	odd	2		2280.2.a.r.1.2	✓	3
12.11	even	2		4560.2.a.bt.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2280.2.a.r.1.2	✓	3	3.2	odd	2
4560.2.a.bt.1.2		3	12.11	even	2
6840.2.a.bk.1.2		3	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 6840 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6840.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{5} -2.77801 q^{7} +O(q^{10})\)	\(q+1.00000 q^{5} -2.77801 q^{7} +0.778008 q^{11} -5.50466 q^{13} +7.00933 q^{17} +1.00000 q^{19} +1.27334 q^{23} +1.00000 q^{25} -8.23132 q^{29} +5.00933 q^{31} -2.77801 q^{35} +8.51399 q^{37} -9.06068 q^{41} -10.5140 q^{43} +6.72666 q^{47} +0.717328 q^{49} -8.72666 q^{53} +0.778008 q^{55} +5.29200 q^{59} -7.73599 q^{61} -5.50466 q^{65} -1.45331 q^{67} +12.4626 q^{71} +14.5653 q^{73} -2.16131 q^{77} +2.28267 q^{79} -3.45331 q^{83} +7.00933 q^{85} +2.51399 q^{89} +15.2920 q^{91} +1.00000 q^{95} +8.51399 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{5} - 2 q^{7}+O(q^{10}) \) 3 * q + 3 * q^5 - 2 * q^7	\( 3 q + 3 q^{5} - 2 q^{7} - 4 q^{11} - 6 q^{13} + 3 q^{19} + 8 q^{23} + 3 q^{25} - 10 q^{29} - 6 q^{31} - 2 q^{35} - 6 q^{37} - 4 q^{41} + 16 q^{47} + 19 q^{49} - 22 q^{53} - 4 q^{55} - 22 q^{59} + 2 q^{61} - 6 q^{65} + 4 q^{67} + 8 q^{71} + 10 q^{73} - 36 q^{77} - 10 q^{79} - 2 q^{83} - 24 q^{89} + 8 q^{91} + 3 q^{95} - 6 q^{97}+O(q^{100}) \) 3 * q + 3 * q^5 - 2 * q^7 - 4 * q^11 - 6 * q^13 + 3 * q^19 + 8 * q^23 + 3 * q^25 - 10 * q^29 - 6 * q^31 - 2 * q^35 - 6 * q^37 - 4 * q^41 + 16 * q^47 + 19 * q^49 - 22 * q^53 - 4 * q^55 - 22 * q^59 + 2 * q^61 - 6 * q^65 + 4 * q^67 + 8 * q^71 + 10 * q^73 - 36 * q^77 - 10 * q^79 - 2 * q^83 - 24 * q^89 + 8 * q^91 + 3 * q^95 - 6 * q^97

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