LMFDB - Embedded newform 6840.2.a.bf.1.1

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:	$0$
Dimension:	$3$
Coefficient field:	3.3.316.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 4x + 2 $ x^3 - x^2 - 4*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			1.1
Root			$0.470683$ 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)

$f(q)$	$=$	$q-1.00000 q^{5} -3.30777 q^{7} +O(q^{10})$	$q-1.00000 q^{5} -3.30777 q^{7} -2.24914 q^{11} -3.19051 q^{13} -3.55691 q^{17} -1.00000 q^{19} -1.55691 q^{23} +1.00000 q^{25} -4.24914 q^{29} -7.43965 q^{31} +3.30777 q^{35} -5.30777 q^{37} -2.36641 q^{41} +3.80605 q^{43} +9.55691 q^{47} +3.94137 q^{49} +3.55691 q^{53} +2.24914 q^{55} -0.498281 q^{59} -1.17590 q^{61} +3.19051 q^{65} -12.9966 q^{67} +10.3810 q^{71} -0.117266 q^{73} +7.43965 q^{77} +12.1725 q^{83} +3.55691 q^{85} -10.6302 q^{89} +10.5535 q^{91} +1.00000 q^{95} +7.68879 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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} + 2 q^{11} + 6 q^{17} - 3 q^{19} + 12 q^{23} + 3 q^{25} - 4 q^{29} - 4 q^{31} + 2 q^{35} - 8 q^{37} - 14 q^{43} + 12 q^{47} + 11 q^{49} - 6 q^{53} - 2 q^{55} + 16 q^{59} - 6 q^{61} - 4 q^{67} + 12 q^{71} - 2 q^{73} + 4 q^{77} + 4 q^{83} - 6 q^{85} - 4 q^{89} - 20 q^{91} + 3 q^{95} - 4 q^{97}+O(q^{100}) $ 3 * q - 3 * q^5 - 2 * q^7 + 2 * q^11 + 6 * q^17 - 3 * q^19 + 12 * q^23 + 3 * q^25 - 4 * q^29 - 4 * q^31 + 2 * q^35 - 8 * q^37 - 14 * q^43 + 12 * q^47 + 11 * q^49 - 6 * q^53 - 2 * q^55 + 16 * q^59 - 6 * q^61 - 4 * q^67 + 12 * q^71 - 2 * q^73 + 4 * q^77 + 4 * q^83 - 6 * q^85 - 4 * q^89 - 20 * q^91 + 3 * q^95 - 4 * 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
$7$	−3.30777	−1.25022	−0.625110	−	0.780536i	$-0.714946\pi$
$7$	−3.30777	−1.25022	−0.625110	+	0.780536i	$0.714946\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−2.24914	−0.678141	−0.339071	−	0.940761i	$-0.610113\pi$
$11$	−2.24914	−0.678141	−0.339071	+	0.940761i	$0.610113\pi$
$12$	0	0
$13$	−3.19051	−0.884888	−0.442444	−	0.896796i	$-0.645888\pi$
$13$	−3.19051	−0.884888	−0.442444	+	0.896796i	$0.645888\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−3.55691	−0.862678	−0.431339	−	0.902190i	$-0.641959\pi$
$17$	−3.55691	−0.862678	−0.431339	+	0.902190i	$0.641959\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−1.55691	−0.324639	−0.162320	−	0.986738i	$-0.551898\pi$
$23$	−1.55691	−0.324639	−0.162320	+	0.986738i	$0.551898\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−4.24914	−0.789046	−0.394523	−	0.918886i	$-0.629090\pi$
$29$	−4.24914	−0.789046	−0.394523	+	0.918886i	$0.629090\pi$
$30$	0	0
$31$	−7.43965	−1.33620	−0.668100	−	0.744071i	$-0.732892\pi$
$31$	−7.43965	−1.33620	−0.668100	+	0.744071i	$0.732892\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	3.30777	0.559116
$36$	0	0
$37$	−5.30777	−0.872593	−0.436296	−	0.899803i	$-0.643710\pi$
$37$	−5.30777	−0.872593	−0.436296	+	0.899803i	$0.643710\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−2.36641	−0.369571	−0.184785	−	0.982779i	$-0.559159\pi$
$41$	−2.36641	−0.369571	−0.184785	+	0.982779i	$0.559159\pi$
$42$	0	0
$43$	3.80605	0.580418	0.290209	−	0.956963i	$-0.406275\pi$
$43$	3.80605	0.580418	0.290209	+	0.956963i	$0.406275\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	9.55691	1.39402	0.697010	−	0.717062i	$-0.254513\pi$
$47$	9.55691	1.39402	0.697010	+	0.717062i	$0.254513\pi$
$48$	0	0
$49$	3.94137	0.563052
$50$	0	0
$51$	0	0
$52$	0	0
$53$	3.55691	0.488580	0.244290	−	0.969702i	$-0.421445\pi$
$53$	3.55691	0.488580	0.244290	+	0.969702i	$0.421445\pi$
$54$	0	0
$55$	2.24914	0.303274
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−0.498281	−0.0648707	−0.0324353	−	0.999474i	$-0.510326\pi$
$59$	−0.498281	−0.0648707	−0.0324353	+	0.999474i	$0.510326\pi$
$60$	0	0
$61$	−1.17590	−0.150559	−0.0752793	−	0.997162i	$-0.523985\pi$
$61$	−1.17590	−0.150559	−0.0752793	+	0.997162i	$0.523985\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	3.19051	0.395734
$66$	0	0
$67$	−12.9966	−1.58778	−0.793891	−	0.608060i	$-0.791948\pi$
$67$	−12.9966	−1.58778	−0.793891	+	0.608060i	$0.791948\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	10.3810	1.23200	0.616000	−	0.787746i	$-0.288752\pi$
$71$	10.3810	1.23200	0.616000	+	0.787746i	$0.288752\pi$
$72$	0	0
$73$	−0.117266	−0.0137250	−0.00686249	−	0.999976i	$-0.502184\pi$
$73$	−0.117266	−0.0137250	−0.00686249	+	0.999976i	$0.502184\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	7.43965	0.847827
$78$	0	0
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	12.1725	1.33610	0.668051	−	0.744116i	$-0.267129\pi$
$83$	12.1725	1.33610	0.668051	+	0.744116i	$0.267129\pi$
$84$	0	0
$85$	3.55691	0.385802
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−10.6302	−1.12679	−0.563397	−	0.826186i	$-0.690506\pi$
$89$	−10.6302	−1.12679	−0.563397	+	0.826186i	$0.690506\pi$
$90$	0	0
$91$	10.5535	1.10630
$92$	0	0
$93$	0	0
$94$	0	0
$95$	1.00000	0.102598
$96$	0	0
$97$	7.68879	0.780678	0.390339	−	0.920671i	$-0.372358\pi$
$97$	7.68879	0.780678	0.390339	+	0.920671i	$0.372358\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−8.38101	−0.833942	−0.416971	−	0.908920i	$-0.636908\pi$
$101$	−8.38101	−0.833942	−0.416971	+	0.908920i	$0.636908\pi$
$102$	0	0
$103$	−3.76547	−0.371023	−0.185511	−	0.982642i	$-0.559394\pi$
$103$	−3.76547	−0.371023	−0.185511	+	0.982642i	$0.559394\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	6.11727	0.591378	0.295689	−	0.955284i	$-0.404451\pi$
$107$	6.11727	0.591378	0.295689	+	0.955284i	$0.404451\pi$
$108$	0	0
$109$	−15.4948	−1.48414	−0.742068	−	0.670324i	$-0.766155\pi$
$109$	−15.4948	−1.48414	−0.742068	+	0.670324i	$0.766155\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	8.94137	0.841133	0.420567	−	0.907262i	$-0.361831\pi$
$113$	8.94137	0.841133	0.420567	+	0.907262i	$0.361831\pi$
$114$	0	0
$115$	1.55691	0.145183
$116$	0	0
$117$	0	0
$118$	0	0
$119$	11.7655	1.07854
$120$	0	0
$121$	−5.94137	−0.540124
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−10.1173	−0.897762	−0.448881	−	0.893591i	$-0.648177\pi$
$127$	−10.1173	−0.897762	−0.448881	+	0.893591i	$0.648177\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	1.51633	0.132482	0.0662410	−	0.997804i	$-0.478899\pi$
$131$	1.51633	0.132482	0.0662410	+	0.997804i	$0.478899\pi$
$132$	0	0
$133$	3.30777	0.286820
$134$	0	0
$135$	0	0
$136$	0	0
$137$	11.8827	1.01521	0.507605	−	0.861590i	$-0.330531\pi$
$137$	11.8827	1.01521	0.507605	+	0.861590i	$0.330531\pi$
$138$	0	0
$139$	16.4983	1.39937	0.699683	−	0.714453i	$-0.253325\pi$
$139$	16.4983	1.39937	0.699683	+	0.714453i	$0.253325\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	7.17590	0.600079
$144$	0	0
$145$	4.24914	0.352872
$146$	0	0
$147$	0	0
$148$	0	0
$149$	7.23109	0.592394	0.296197	−	0.955127i	$-0.404281\pi$
$149$	7.23109	0.592394	0.296197	+	0.955127i	$0.404281\pi$
$150$	0	0
$151$	−3.67418	−0.299001	−0.149500	−	0.988762i	$-0.547766\pi$
$151$	−3.67418	−0.299001	−0.149500	+	0.988762i	$0.547766\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	7.43965	0.597567
$156$	0	0
$157$	0.381015	0.0304083	0.0152041	−	0.999884i	$-0.495160\pi$
$157$	0.381015	0.0304083	0.0152041	+	0.999884i	$0.495160\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	5.14992	0.405871
$162$	0	0
$163$	2.80949	0.220056	0.110028	−	0.993928i	$-0.464906\pi$
$163$	2.80949	0.220056	0.110028	+	0.993928i	$0.464906\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	7.17590	0.555288	0.277644	−	0.960684i	$-0.410446\pi$
$167$	7.17590	0.555288	0.277644	+	0.960684i	$0.410446\pi$
$168$	0	0
$169$	−2.82066	−0.216974
$170$	0	0
$171$	0	0
$172$	0	0
$173$	4.94137	0.375685	0.187843	−	0.982199i	$-0.439851\pi$
$173$	4.94137	0.375685	0.187843	+	0.982199i	$0.439851\pi$
$174$	0	0
$175$	−3.30777	−0.250044
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−2.61555	−0.195495	−0.0977476	−	0.995211i	$-0.531164\pi$
$179$	−2.61555	−0.195495	−0.0977476	+	0.995211i	$0.531164\pi$
$180$	0	0
$181$	−17.6121	−1.30910	−0.654549	−	0.756020i	$-0.727141\pi$
$181$	−17.6121	−1.30910	−0.654549	+	0.756020i	$0.727141\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	5.30777	0.390235
$186$	0	0
$187$	8.00000	0.585018
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−11.7440	−0.849765	−0.424882	−	0.905249i	$-0.639685\pi$
$191$	−11.7440	−0.849765	−0.424882	+	0.905249i	$0.639685\pi$
$192$	0	0
$193$	−0.574960	−0.0413865	−0.0206933	−	0.999786i	$-0.506587\pi$
$193$	−0.574960	−0.0413865	−0.0206933	+	0.999786i	$0.506587\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	11.5569	0.823396	0.411698	−	0.911320i	$-0.364936\pi$
$197$	11.5569	0.823396	0.411698	+	0.911320i	$0.364936\pi$
$198$	0	0
$199$	3.50172	0.248230	0.124115	−	0.992268i	$-0.460391\pi$
$199$	3.50172	0.248230	0.124115	+	0.992268i	$0.460391\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	14.0552	0.986481
$204$	0	0
$205$	2.36641	0.165277
$206$	0	0
$207$	0	0
$208$	0	0
$209$	2.24914	0.155576
$210$	0	0
$211$	9.23109	0.635495	0.317747	−	0.948175i	$-0.397074\pi$
$211$	9.23109	0.635495	0.317747	+	0.948175i	$0.397074\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−3.80605	−0.259571
$216$	0	0
$217$	24.6087	1.67055
$218$	0	0
$219$	0	0
$220$	0	0
$221$	11.3484	0.763373
$222$	0	0
$223$	17.9931	1.20491	0.602454	−	0.798153i	$-0.294190\pi$
$223$	17.9931	1.20491	0.602454	+	0.798153i	$0.294190\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	5.94480	0.394571	0.197285	−	0.980346i	$-0.436787\pi$
$227$	5.94480	0.394571	0.197285	+	0.980346i	$0.436787\pi$
$228$	0	0
$229$	14.1725	0.936543	0.468271	−	0.883585i	$-0.344877\pi$
$229$	14.1725	0.936543	0.468271	+	0.883585i	$0.344877\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	13.3484	0.874480	0.437240	−	0.899345i	$-0.355956\pi$
$233$	13.3484	0.874480	0.437240	+	0.899345i	$0.355956\pi$
$234$	0	0
$235$	−9.55691	−0.623424
$236$	0	0
$237$	0	0
$238$	0	0
$239$	24.2423	1.56810	0.784051	−	0.620697i	$-0.213150\pi$
$239$	24.2423	1.56810	0.784051	+	0.620697i	$0.213150\pi$
$240$	0	0
$241$	−11.2311	−0.723458	−0.361729	−	0.932283i	$-0.617814\pi$
$241$	−11.2311	−0.723458	−0.361729	+	0.932283i	$0.617814\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−3.94137	−0.251805
$246$	0	0
$247$	3.19051	0.203007
$248$	0	0
$249$	0	0
$250$	0	0
$251$	7.48024	0.472148	0.236074	−	0.971735i	$-0.424139\pi$
$251$	7.48024	0.472148	0.236074	+	0.971735i	$0.424139\pi$
$252$	0	0
$253$	3.50172	0.220151
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−0.0551953	−0.00344299	−0.00172149	−	0.999999i	$-0.500548\pi$
$257$	−0.0551953	−0.00344299	−0.00172149	+	0.999999i	$0.500548\pi$
$258$	0	0
$259$	17.5569	1.09093
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−3.93793	−0.242823	−0.121412	−	0.992602i	$-0.538742\pi$
$263$	−3.93793	−0.242823	−0.121412	+	0.992602i	$0.538742\pi$
$264$	0	0
$265$	−3.55691	−0.218500
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−17.8681	−1.08944	−0.544719	−	0.838618i	$-0.683364\pi$
$269$	−17.8681	−1.08944	−0.544719	+	0.838618i	$0.683364\pi$
$270$	0	0
$271$	−27.8466	−1.69156	−0.845782	−	0.533529i	$-0.820865\pi$
$271$	−27.8466	−1.69156	−0.845782	+	0.533529i	$0.820865\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−2.24914	−0.135628
$276$	0	0
$277$	−11.9931	−0.720597	−0.360299	−	0.932837i	$-0.617325\pi$
$277$	−11.9931	−0.720597	−0.360299	+	0.932837i	$0.617325\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	10.8647	0.648133	0.324066	−	0.946034i	$-0.394950\pi$
$281$	10.8647	0.648133	0.324066	+	0.946034i	$0.394950\pi$
$282$	0	0
$283$	16.6922	0.992250	0.496125	−	0.868251i	$-0.334756\pi$
$283$	16.6922	0.992250	0.496125	+	0.868251i	$0.334756\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	7.82754	0.462045
$288$	0	0
$289$	−4.34836	−0.255786
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−31.1690	−1.82091	−0.910457	−	0.413604i	$-0.864270\pi$
$293$	−31.1690	−1.82091	−0.910457	+	0.413604i	$0.864270\pi$
$294$	0	0
$295$	0.498281	0.0290110
$296$	0	0
$297$	0	0
$298$	0	0
$299$	4.96735	0.287269
$300$	0	0
$301$	−12.5896	−0.725651
$302$	0	0
$303$	0	0
$304$	0	0
$305$	1.17590	0.0673318
$306$	0	0
$307$	−0.498281	−0.0284384	−0.0142192	−	0.999899i	$-0.504526\pi$
$307$	−0.498281	−0.0284384	−0.0142192	+	0.999899i	$0.504526\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−1.01805	−0.0577281	−0.0288640	−	0.999583i	$-0.509189\pi$
$311$	−1.01805	−0.0577281	−0.0288640	+	0.999583i	$0.509189\pi$
$312$	0	0
$313$	12.3810	0.699816	0.349908	−	0.936784i	$-0.386213\pi$
$313$	12.3810	0.699816	0.349908	+	0.936784i	$0.386213\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−4.78801	−0.268921	−0.134461	−	0.990919i	$-0.542930\pi$
$317$	−4.78801	−0.268921	−0.134461	+	0.990919i	$0.542930\pi$
$318$	0	0
$319$	9.55691	0.535084
$320$	0	0
$321$	0	0
$322$	0	0
$323$	3.55691	0.197912
$324$	0	0
$325$	−3.19051	−0.176978
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−31.6121	−1.74283
$330$	0	0
$331$	−12.4362	−0.683556	−0.341778	−	0.939781i	$-0.611029\pi$
$331$	−12.4362	−0.683556	−0.341778	+	0.939781i	$0.611029\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	12.9966	0.710078
$336$	0	0
$337$	−15.4543	−0.841847	−0.420923	−	0.907096i	$-0.638294\pi$
$337$	−15.4543	−0.841847	−0.420923	+	0.907096i	$0.638294\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	16.7328	0.906133
$342$	0	0
$343$	10.1173	0.546281
$344$	0	0
$345$	0	0
$346$	0	0
$347$	12.8241	0.688434	0.344217	−	0.938890i	$-0.388144\pi$
$347$	12.8241	0.688434	0.344217	+	0.938890i	$0.388144\pi$
$348$	0	0
$349$	28.8793	1.54587	0.772937	−	0.634483i	$-0.218787\pi$
$349$	28.8793	1.54587	0.772937	+	0.634483i	$0.218787\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−2.00000	−0.106449	−0.0532246	−	0.998583i	$-0.516950\pi$
$353$	−2.00000	−0.106449	−0.0532246	+	0.998583i	$0.516950\pi$
$354$	0	0
$355$	−10.3810	−0.550967
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−19.7440	−1.04205	−0.521024	−	0.853542i	$-0.674450\pi$
$359$	−19.7440	−1.04205	−0.521024	+	0.853542i	$0.674450\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0.117266	0.00613800
$366$	0	0
$367$	13.6888	0.714549	0.357274	−	0.933999i	$-0.383706\pi$
$367$	13.6888	0.714549	0.357274	+	0.933999i	$0.383706\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−11.7655	−0.610833
$372$	0	0
$373$	30.1510	1.56116	0.780579	−	0.625057i	$-0.214924\pi$
$373$	30.1510	1.56116	0.780579	+	0.625057i	$0.214924\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	13.5569	0.698217
$378$	0	0
$379$	−25.6673	−1.31844	−0.659220	−	0.751950i	$-0.729114\pi$
$379$	−25.6673	−1.31844	−0.659220	+	0.751950i	$0.729114\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	23.8759	1.22000	0.610000	−	0.792402i	$-0.291170\pi$
$383$	23.8759	1.22000	0.610000	+	0.792402i	$0.291170\pi$
$384$	0	0
$385$	−7.43965	−0.379160
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−29.3484	−1.48802	−0.744010	−	0.668168i	$-0.767079\pi$
$389$	−29.3484	−1.48802	−0.744010	+	0.668168i	$0.767079\pi$
$390$	0	0
$391$	5.53781	0.280059
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−12.1173	−0.608148	−0.304074	−	0.952648i	$-0.598347\pi$
$397$	−12.1173	−0.608148	−0.304074	+	0.952648i	$0.598347\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	27.4734	1.37195	0.685977	−	0.727623i	$-0.259375\pi$
$401$	27.4734	1.37195	0.685977	+	0.727623i	$0.259375\pi$
$402$	0	0
$403$	23.7363	1.18239
$404$	0	0
$405$	0	0
$406$	0	0
$407$	11.9379	0.591741
$408$	0	0
$409$	32.9605	1.62979	0.814895	−	0.579608i	$-0.196794\pi$
$409$	32.9605	1.62979	0.814895	+	0.579608i	$0.196794\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	1.64820	0.0811027
$414$	0	0
$415$	−12.1725	−0.597523
$416$	0	0
$417$	0	0
$418$	0	0
$419$	14.7474	0.720459	0.360229	−	0.932864i	$-0.382698\pi$
$419$	14.7474	0.720459	0.360229	+	0.932864i	$0.382698\pi$
$420$	0	0
$421$	24.9897	1.21792	0.608961	−	0.793200i	$-0.291586\pi$
$421$	24.9897	1.21792	0.608961	+	0.793200i	$0.291586\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−3.55691	−0.172536
$426$	0	0
$427$	3.88961	0.188231
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−10.8501	−0.522630	−0.261315	−	0.965254i	$-0.584156\pi$
$431$	−10.8501	−0.522630	−0.261315	+	0.965254i	$0.584156\pi$
$432$	0	0
$433$	−11.0371	−0.530412	−0.265206	−	0.964192i	$-0.585440\pi$
$433$	−11.0371	−0.530412	−0.265206	+	0.964192i	$0.585440\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	1.55691	0.0744773
$438$	0	0
$439$	30.2277	1.44269	0.721344	−	0.692577i	$-0.243525\pi$
$439$	30.2277	1.44269	0.721344	+	0.692577i	$0.243525\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−21.9018	−1.04059	−0.520294	−	0.853987i	$-0.674178\pi$
$443$	−21.9018	−1.04059	−0.520294	+	0.853987i	$0.674178\pi$
$444$	0	0
$445$	10.6302	0.503918
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−3.01805	−0.142430	−0.0712152	−	0.997461i	$-0.522688\pi$
$449$	−3.01805	−0.142430	−0.0712152	+	0.997461i	$0.522688\pi$
$450$	0	0
$451$	5.32238	0.250621
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−10.5535	−0.494755
$456$	0	0
$457$	13.5017	0.631584	0.315792	−	0.948828i	$-0.397730\pi$
$457$	13.5017	0.631584	0.315792	+	0.948828i	$0.397730\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	3.64820	0.169914	0.0849568	−	0.996385i	$-0.472925\pi$
$461$	3.64820	0.169914	0.0849568	+	0.996385i	$0.472925\pi$
$462$	0	0
$463$	−8.53887	−0.396835	−0.198417	−	0.980118i	$-0.563580\pi$
$463$	−8.53887	−0.396835	−0.198417	+	0.980118i	$0.563580\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−22.5535	−1.04365	−0.521825	−	0.853052i	$-0.674749\pi$
$467$	−22.5535	−1.04365	−0.521825	+	0.853052i	$0.674749\pi$
$468$	0	0
$469$	42.9897	1.98508
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−8.56035	−0.393605
$474$	0	0
$475$	−1.00000	−0.0458831
$476$	0	0
$477$	0	0
$478$	0	0
$479$	33.6267	1.53644	0.768222	−	0.640184i	$-0.221142\pi$
$479$	33.6267	1.53644	0.768222	+	0.640184i	$0.221142\pi$
$480$	0	0
$481$	16.9345	0.772146
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−7.68879	−0.349130
$486$	0	0
$487$	11.8466	0.536823	0.268411	−	0.963304i	$-0.413501\pi$
$487$	11.8466	0.536823	0.268411	+	0.963304i	$0.413501\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−35.3269	−1.59428	−0.797140	−	0.603795i	$-0.793655\pi$
$491$	−35.3269	−1.59428	−0.797140	+	0.603795i	$0.793655\pi$
$492$	0	0
$493$	15.1138	0.680693
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−34.3380	−1.54027
$498$	0	0
$499$	30.7259	1.37548	0.687741	−	0.725956i	$-0.258602\pi$
$499$	30.7259	1.37548	0.687741	+	0.725956i	$0.258602\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−42.1656	−1.88007	−0.940035	−	0.341077i	$-0.889208\pi$
$503$	−42.1656	−1.88007	−0.940035	+	0.341077i	$0.889208\pi$
$504$	0	0
$505$	8.38101	0.372950
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−41.7440	−1.85027	−0.925135	−	0.379639i	$-0.876048\pi$
$509$	−41.7440	−1.85027	−0.925135	+	0.379639i	$0.876048\pi$
$510$	0	0
$511$	0.387890	0.0171593
$512$	0	0
$513$	0	0
$514$	0	0
$515$	3.76547	0.165926
$516$	0	0
$517$	−21.4948	−0.945342
$518$	0	0
$519$	0	0
$520$	0	0
$521$	18.7405	0.821038	0.410519	−	0.911852i	$-0.365348\pi$
$521$	18.7405	0.821038	0.410519	+	0.911852i	$0.365348\pi$
$522$	0	0
$523$	−14.7259	−0.643920	−0.321960	−	0.946753i	$-0.604342\pi$
$523$	−14.7259	−0.643920	−0.321960	+	0.946753i	$0.604342\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	26.4622	1.15271
$528$	0	0
$529$	−20.5760	−0.894609
$530$	0	0
$531$	0	0
$532$	0	0
$533$	7.55004	0.327028
$534$	0	0
$535$	−6.11727	−0.264472
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−8.86469	−0.381829
$540$	0	0
$541$	−40.2277	−1.72952	−0.864761	−	0.502184i	$-0.832530\pi$
$541$	−40.2277	−1.72952	−0.864761	+	0.502184i	$0.832530\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	15.4948	0.663726
$546$	0	0
$547$	37.7294	1.61319	0.806596	−	0.591103i	$-0.201307\pi$
$547$	37.7294	1.61319	0.806596	+	0.591103i	$0.201307\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	4.24914	0.181019
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	19.9931	0.847136	0.423568	−	0.905864i	$-0.360777\pi$
$557$	19.9931	0.847136	0.423568	+	0.905864i	$0.360777\pi$
$558$	0	0
$559$	−12.1432	−0.513605
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−44.0483	−1.85642	−0.928208	−	0.372063i	$-0.878651\pi$
$563$	−44.0483	−1.85642	−0.928208	+	0.372063i	$0.878651\pi$
$564$	0	0
$565$	−8.94137	−0.376166
$566$	0	0
$567$	0	0
$568$	0	0
$569$	16.7474	0.702088	0.351044	−	0.936359i	$-0.385827\pi$
$569$	16.7474	0.702088	0.351044	+	0.936359i	$0.385827\pi$
$570$	0	0
$571$	36.1396	1.51240	0.756198	−	0.654343i	$-0.227055\pi$
$571$	36.1396	1.51240	0.756198	+	0.654343i	$0.227055\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−1.55691	−0.0649278
$576$	0	0
$577$	−16.3810	−0.681951	−0.340975	−	0.940072i	$-0.610757\pi$
$577$	−16.3810	−0.681951	−0.340975	+	0.940072i	$0.610757\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−40.2637	−1.67042
$582$	0	0
$583$	−8.00000	−0.331326
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−12.5604	−0.518421	−0.259211	−	0.965821i	$-0.583462\pi$
$587$	−12.5604	−0.518421	−0.259211	+	0.965821i	$0.583462\pi$
$588$	0	0
$589$	7.43965	0.306545
$590$	0	0
$591$	0	0
$592$	0	0
$593$	38.9966	1.60140	0.800698	−	0.599068i	$-0.204462\pi$
$593$	38.9966	1.60140	0.800698	+	0.599068i	$0.204462\pi$
$594$	0	0
$595$	−11.7655	−0.482337
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−17.3415	−0.708554	−0.354277	−	0.935141i	$-0.615273\pi$
$599$	−17.3415	−0.708554	−0.354277	+	0.935141i	$0.615273\pi$
$600$	0	0
$601$	−15.9119	−0.649062	−0.324531	−	0.945875i	$-0.605206\pi$
$601$	−15.9119	−0.649062	−0.324531	+	0.945875i	$0.605206\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	5.94137	0.241551
$606$	0	0
$607$	−24.2637	−0.984835	−0.492418	−	0.870359i	$-0.663887\pi$
$607$	−24.2637	−0.984835	−0.492418	+	0.870359i	$0.663887\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−30.4914	−1.23355
$612$	0	0
$613$	−44.7259	−1.80646	−0.903232	−	0.429153i	$-0.858812\pi$
$613$	−44.7259	−1.80646	−0.903232	+	0.429153i	$0.858812\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	10.6707	0.429588	0.214794	−	0.976659i	$-0.431092\pi$
$617$	10.6707	0.429588	0.214794	+	0.976659i	$0.431092\pi$
$618$	0	0
$619$	47.3776	1.90427	0.952133	−	0.305685i	$-0.0988853\pi$
$619$	47.3776	1.90427	0.952133	+	0.305685i	$0.0988853\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	35.1621	1.40874
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	18.8793	0.752767
$630$	0	0
$631$	10.4622	0.416493	0.208247	−	0.978076i	$-0.433224\pi$
$631$	10.4622	0.416493	0.208247	+	0.978076i	$0.433224\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	10.1173	0.401491
$636$	0	0
$637$	−12.5750	−0.498238
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−20.6233	−0.814571	−0.407285	−	0.913301i	$-0.633525\pi$
$641$	−20.6233	−0.814571	−0.407285	+	0.913301i	$0.633525\pi$
$642$	0	0
$643$	34.4216	1.35746	0.678728	−	0.734390i	$-0.262532\pi$
$643$	34.4216	1.35746	0.678728	+	0.734390i	$0.262532\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	25.8207	1.01511	0.507557	−	0.861618i	$-0.330548\pi$
$647$	25.8207	1.01511	0.507557	+	0.861618i	$0.330548\pi$
$648$	0	0
$649$	1.12070	0.0439915
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−45.3155	−1.77333	−0.886666	−	0.462410i	$-0.846985\pi$
$653$	−45.3155	−1.77333	−0.886666	+	0.462410i	$0.846985\pi$
$654$	0	0
$655$	−1.51633	−0.0592478
$656$	0	0
$657$	0	0
$658$	0	0
$659$	45.3415	1.76625	0.883127	−	0.469134i	$-0.155434\pi$
$659$	45.3415	1.76625	0.883127	+	0.469134i	$0.155434\pi$
$660$	0	0
$661$	−17.6121	−0.685032	−0.342516	−	0.939512i	$-0.611279\pi$
$661$	−17.6121	−0.685032	−0.342516	+	0.939512i	$0.611279\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−3.30777	−0.128270
$666$	0	0
$667$	6.61555	0.256155
$668$	0	0
$669$	0	0
$670$	0	0
$671$	2.64476	0.102100
$672$	0	0
$673$	−13.8061	−0.532184	−0.266092	−	0.963948i	$-0.585733\pi$
$673$	−13.8061	−0.532184	−0.266092	+	0.963948i	$0.585733\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	23.7914	0.914380	0.457190	−	0.889369i	$-0.348856\pi$
$677$	23.7914	0.914380	0.457190	+	0.889369i	$0.348856\pi$
$678$	0	0
$679$	−25.4328	−0.976020
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−7.11383	−0.272203	−0.136102	−	0.990695i	$-0.543457\pi$
$683$	−7.11383	−0.272203	−0.136102	+	0.990695i	$0.543457\pi$
$684$	0	0
$685$	−11.8827	−0.454016
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−11.3484	−0.432338
$690$	0	0
$691$	−18.1465	−0.690325	−0.345162	−	0.938543i	$-0.612176\pi$
$691$	−18.1465	−0.690325	−0.345162	+	0.938543i	$0.612176\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−16.4983	−0.625815
$696$	0	0
$697$	8.41711	0.318821
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−24.9897	−0.943847	−0.471924	−	0.881639i	$-0.656440\pi$
$701$	−24.9897	−0.943847	−0.471924	+	0.881639i	$0.656440\pi$
$702$	0	0
$703$	5.30777	0.200186
$704$	0	0
$705$	0	0
$706$	0	0
$707$	27.7225	1.04261
$708$	0	0
$709$	−31.9311	−1.19920	−0.599598	−	0.800301i	$-0.704673\pi$
$709$	−31.9311	−1.19920	−0.599598	+	0.800301i	$0.704673\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	11.5829	0.433783
$714$	0	0
$715$	−7.17590	−0.268363
$716$	0	0
$717$	0	0
$718$	0	0
$719$	38.5941	1.43932	0.719658	−	0.694329i	$-0.244299\pi$
$719$	38.5941	1.43932	0.719658	+	0.694329i	$0.244299\pi$
$720$	0	0
$721$	12.4553	0.463860
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−4.24914	−0.157809
$726$	0	0
$727$	−38.0337	−1.41059	−0.705296	−	0.708913i	$-0.749186\pi$
$727$	−38.0337	−1.41059	−0.705296	+	0.708913i	$0.749186\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−13.5378	−0.500714
$732$	0	0
$733$	14.5275	0.536585	0.268293	−	0.963337i	$-0.413541\pi$
$733$	14.5275	0.536585	0.268293	+	0.963337i	$0.413541\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	29.2311	1.07674
$738$	0	0
$739$	−5.46563	−0.201056	−0.100528	−	0.994934i	$-0.532053\pi$
$739$	−5.46563	−0.201056	−0.100528	+	0.994934i	$0.532053\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	44.5795	1.63546	0.817731	−	0.575601i	$-0.195232\pi$
$743$	44.5795	1.63546	0.817731	+	0.575601i	$0.195232\pi$
$744$	0	0
$745$	−7.23109	−0.264927
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−20.2345	−0.739354
$750$	0	0
$751$	7.43965	0.271477	0.135738	−	0.990745i	$-0.456659\pi$
$751$	7.43965	0.271477	0.135738	+	0.990745i	$0.456659\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	3.67418	0.133717
$756$	0	0
$757$	−23.6190	−0.858447	−0.429223	−	0.903198i	$-0.641213\pi$
$757$	−23.6190	−0.858447	−0.429223	+	0.903198i	$0.641213\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−18.9966	−0.688625	−0.344312	−	0.938855i	$-0.611888\pi$
$761$	−18.9966	−0.688625	−0.344312	+	0.938855i	$0.611888\pi$
$762$	0	0
$763$	51.2534	1.85550
$764$	0	0
$765$	0	0
$766$	0	0
$767$	1.58977	0.0574032
$768$	0	0
$769$	−11.2311	−0.405004	−0.202502	−	0.979282i	$-0.564907\pi$
$769$	−11.2311	−0.405004	−0.202502	+	0.979282i	$0.564907\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−26.0191	−0.935842	−0.467921	−	0.883770i	$-0.654997\pi$
$773$	−26.0191	−0.935842	−0.467921	+	0.883770i	$0.654997\pi$
$774$	0	0
$775$	−7.43965	−0.267240
$776$	0	0
$777$	0	0
$778$	0	0
$779$	2.36641	0.0847853
$780$	0	0
$781$	−23.3484	−0.835470
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−0.381015	−0.0135990
$786$	0	0
$787$	14.1984	0.506120	0.253060	−	0.967451i	$-0.418563\pi$
$787$	14.1984	0.506120	0.253060	+	0.967451i	$0.418563\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−29.5760	−1.05160
$792$	0	0
$793$	3.75172	0.133227
$794$	0	0
$795$	0	0
$796$	0	0
$797$	34.1725	1.21045	0.605225	−	0.796054i	$-0.293083\pi$
$797$	34.1725	1.21045	0.605225	+	0.796054i	$0.293083\pi$
$798$	0	0
$799$	−33.9931	−1.20259
$800$	0	0
$801$	0	0
$802$	0	0
$803$	0.263748	0.00930748
$804$	0	0
$805$	−5.14992	−0.181511
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−20.1173	−0.707285	−0.353643	−	0.935381i	$-0.615057\pi$
$809$	−20.1173	−0.707285	−0.353643	+	0.935381i	$0.615057\pi$
$810$	0	0
$811$	−0.762030	−0.0267585	−0.0133792	−	0.999910i	$-0.504259\pi$
$811$	−0.762030	−0.0267585	−0.0133792	+	0.999910i	$0.504259\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−2.80949	−0.0984122
$816$	0	0
$817$	−3.80605	−0.133157
$818$	0	0
$819$	0	0
$820$	0	0
$821$	4.50516	0.157231	0.0786155	−	0.996905i	$-0.474950\pi$
$821$	4.50516	0.157231	0.0786155	+	0.996905i	$0.474950\pi$
$822$	0	0
$823$	−11.1836	−0.389837	−0.194918	−	0.980819i	$-0.562444\pi$
$823$	−11.1836	−0.389837	−0.194918	+	0.980819i	$0.562444\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	16.2829	0.566210	0.283105	−	0.959089i	$-0.408635\pi$
$827$	16.2829	0.566210	0.283105	+	0.959089i	$0.408635\pi$
$828$	0	0
$829$	15.3845	0.534324	0.267162	−	0.963652i	$-0.413914\pi$
$829$	15.3845	0.534324	0.267162	+	0.963652i	$0.413914\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−14.0191	−0.485733
$834$	0	0
$835$	−7.17590	−0.248332
$836$	0	0
$837$	0	0
$838$	0	0
$839$	16.0812	0.555184	0.277592	−	0.960699i	$-0.410464\pi$
$839$	16.0812	0.555184	0.277592	+	0.960699i	$0.410464\pi$
$840$	0	0
$841$	−10.9448	−0.377407
$842$	0	0
$843$	0	0
$844$	0	0
$845$	2.82066	0.0970337
$846$	0	0
$847$	19.6527	0.675275
$848$	0	0
$849$	0	0
$850$	0	0
$851$	8.26375	0.283278
$852$	0	0
$853$	−19.9119	−0.681772	−0.340886	−	0.940105i	$-0.610727\pi$
$853$	−19.9119	−0.681772	−0.340886	+	0.940105i	$0.610727\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	28.1364	0.961120	0.480560	−	0.876962i	$-0.340433\pi$
$857$	28.1364	0.961120	0.480560	+	0.876962i	$0.340433\pi$
$858$	0	0
$859$	33.5760	1.14560	0.572799	−	0.819696i	$-0.305857\pi$
$859$	33.5760	1.14560	0.572799	+	0.819696i	$0.305857\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−32.8724	−1.11899	−0.559495	−	0.828834i	$-0.689005\pi$
$863$	−32.8724	−1.11899	−0.559495	+	0.828834i	$0.689005\pi$
$864$	0	0
$865$	−4.94137	−0.168012
$866$	0	0
$867$	0	0
$868$	0	0
$869$	0	0
$870$	0	0
$871$	41.4656	1.40501
$872$	0	0
$873$	0	0
$874$	0	0
$875$	3.30777	0.111823
$876$	0	0
$877$	−24.1579	−0.815753	−0.407876	−	0.913037i	$-0.633731\pi$
$877$	−24.1579	−0.815753	−0.407876	+	0.913037i	$0.633731\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−32.4914	−1.09466	−0.547332	−	0.836916i	$-0.684356\pi$
$881$	−32.4914	−1.09466	−0.547332	+	0.836916i	$0.684356\pi$
$882$	0	0
$883$	26.3404	0.886426	0.443213	−	0.896416i	$-0.353839\pi$
$883$	26.3404	0.886426	0.443213	+	0.896416i	$0.353839\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−0.996562	−0.0334613	−0.0167306	−	0.999860i	$-0.505326\pi$
$887$	−0.996562	−0.0334613	−0.0167306	+	0.999860i	$0.505326\pi$
$888$	0	0
$889$	33.4656	1.12240
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−9.55691	−0.319810
$894$	0	0
$895$	2.61555	0.0874281
$896$	0	0
$897$	0	0
$898$	0	0
$899$	31.6121	1.05432
$900$	0	0
$901$	−12.6516	−0.421487
$902$	0	0
$903$	0	0
$904$	0	0
$905$	17.6121	0.585446
$906$	0	0
$907$	43.1430	1.43254	0.716271	−	0.697823i	$-0.245848\pi$
$907$	43.1430	1.43254	0.716271	+	0.697823i	$0.245848\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−19.6413	−0.650746	−0.325373	−	0.945586i	$-0.605490\pi$
$911$	−19.6413	−0.650746	−0.325373	+	0.945586i	$0.605490\pi$
$912$	0	0
$913$	−27.3776	−0.906066
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−5.01567	−0.165632
$918$	0	0
$919$	−37.8827	−1.24964	−0.624818	−	0.780770i	$-0.714827\pi$
$919$	−37.8827	−1.24964	−0.624818	+	0.780770i	$0.714827\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−33.1207	−1.09018
$924$	0	0
$925$	−5.30777	−0.174519
$926$	0	0
$927$	0	0
$928$	0	0
$929$	5.39133	0.176884	0.0884419	−	0.996081i	$-0.471811\pi$
$929$	5.39133	0.176884	0.0884419	+	0.996081i	$0.471811\pi$
$930$	0	0
$931$	−3.94137	−0.129173
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−8.00000	−0.261628
$936$	0	0
$937$	26.7328	0.873323	0.436661	−	0.899626i	$-0.356161\pi$
$937$	26.7328	0.873323	0.436661	+	0.899626i	$0.356161\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	4.35953	0.142117	0.0710583	−	0.997472i	$-0.477362\pi$
$941$	4.35953	0.142117	0.0710583	+	0.997472i	$0.477362\pi$
$942$	0	0
$943$	3.68429	0.119977
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−31.9379	−1.03784	−0.518922	−	0.854822i	$-0.673666\pi$
$947$	−31.9379	−1.03784	−0.518922	+	0.854822i	$0.673666\pi$
$948$	0	0
$949$	0.374139	0.0121451
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−1.82754	−0.0591998	−0.0295999	−	0.999562i	$-0.509423\pi$
$953$	−1.82754	−0.0591998	−0.0295999	+	0.999562i	$0.509423\pi$
$954$	0	0
$955$	11.7440	0.380026
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−39.3054	−1.26924
$960$	0	0
$961$	24.3484	0.785431
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0.574960	0.0185086
$966$	0	0
$967$	−0.926759	−0.0298026	−0.0149013	−	0.999889i	$-0.504743\pi$
$967$	−0.926759	−0.0298026	−0.0149013	+	0.999889i	$0.504743\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	58.1035	1.86463	0.932315	−	0.361647i	$-0.117785\pi$
$971$	58.1035	1.86463	0.932315	+	0.361647i	$0.117785\pi$
$972$	0	0
$973$	−54.5726	−1.74952
$974$	0	0
$975$	0	0
$976$	0	0
$977$	8.02598	0.256774	0.128387	−	0.991724i	$-0.459020\pi$
$977$	8.02598	0.256774	0.128387	+	0.991724i	$0.459020\pi$
$978$	0	0
$979$	23.9087	0.764126
$980$	0	0
$981$	0	0
$982$	0	0
$983$	44.8103	1.42923	0.714614	−	0.699519i	$-0.246602\pi$
$983$	44.8103	1.42923	0.714614	+	0.699519i	$0.246602\pi$
$984$	0	0
$985$	−11.5569	−0.368234
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−5.92570	−0.188426
$990$	0	0
$991$	−36.7620	−1.16778	−0.583892	−	0.811831i	$-0.698471\pi$
$991$	−36.7620	−1.16778	−0.583892	+	0.811831i	$0.698471\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−3.50172	−0.111012
$996$	0	0
$997$	26.3741	0.835277	0.417639	−	0.908613i	$-0.362858\pi$
$997$	26.3741	0.835277	0.417639	+	0.908613i	$0.362858\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6840.2.a.bf.1.1		3
3.2	odd	2		2280.2.a.s.1.1	✓	3
12.11	even	2		4560.2.a.bv.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2280.2.a.s.1.1	✓	3	3.2	odd	2
4560.2.a.bv.1.3		3	12.11	even	2
6840.2.a.bf.1.1		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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{5} -3.30777 q^{7} +O(q^{10})\)	\(q-1.00000 q^{5} -3.30777 q^{7} -2.24914 q^{11} -3.19051 q^{13} -3.55691 q^{17} -1.00000 q^{19} -1.55691 q^{23} +1.00000 q^{25} -4.24914 q^{29} -7.43965 q^{31} +3.30777 q^{35} -5.30777 q^{37} -2.36641 q^{41} +3.80605 q^{43} +9.55691 q^{47} +3.94137 q^{49} +3.55691 q^{53} +2.24914 q^{55} -0.498281 q^{59} -1.17590 q^{61} +3.19051 q^{65} -12.9966 q^{67} +10.3810 q^{71} -0.117266 q^{73} +7.43965 q^{77} +12.1725 q^{83} +3.55691 q^{85} -10.6302 q^{89} +10.5535 q^{91} +1.00000 q^{95} +7.68879 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{5} - 2 q^{7}+O(q^{10}) \) 3 * q - 3 * q^5 - 2 * q^7	\( 3 q - 3 q^{5} - 2 q^{7} + 2 q^{11} + 6 q^{17} - 3 q^{19} + 12 q^{23} + 3 q^{25} - 4 q^{29} - 4 q^{31} + 2 q^{35} - 8 q^{37} - 14 q^{43} + 12 q^{47} + 11 q^{49} - 6 q^{53} - 2 q^{55} + 16 q^{59} - 6 q^{61} - 4 q^{67} + 12 q^{71} - 2 q^{73} + 4 q^{77} + 4 q^{83} - 6 q^{85} - 4 q^{89} - 20 q^{91} + 3 q^{95} - 4 q^{97}+O(q^{100}) \) 3 * q - 3 * q^5 - 2 * q^7 + 2 * q^11 + 6 * q^17 - 3 * q^19 + 12 * q^23 + 3 * q^25 - 4 * q^29 - 4 * q^31 + 2 * q^35 - 8 * q^37 - 14 * q^43 + 12 * q^47 + 11 * q^49 - 6 * q^53 - 2 * q^55 + 16 * q^59 - 6 * q^61 - 4 * q^67 + 12 * q^71 - 2 * q^73 + 4 * q^77 + 4 * q^83 - 6 * q^85 - 4 * q^89 - 20 * q^91 + 3 * q^95 - 4 * q^97

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