LMFDB - Embedded newform 1104.6.a.s.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1104,6,Mod(1,1104)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1104, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("1104.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 1104 = 2^{4} \cdot 3 \cdot 23 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	1104.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:	$177.063737074$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 3740x^{4} - 50049x^{3} + 3200252x^{2} + 86063268x + 576646848 $ x^6 - 2x^5 - 3740x^4 - 50049x^3 + 3200252x^2 + 86063268*x + 576646848
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	no (minimal twist has level 276)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$56.6655$ of defining polynomial
Character	$\chi$	$=$	1104.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-9.00000 q^{3} -55.3592 q^{5} +89.4396 q^{7} +81.0000 q^{9} +O(q^{10})$	$q-9.00000 q^{3} -55.3592 q^{5} +89.4396 q^{7} +81.0000 q^{9} -566.405 q^{11} -152.812 q^{13} +498.233 q^{15} -2061.71 q^{17} -611.113 q^{19} -804.957 q^{21} -529.000 q^{23} -60.3592 q^{25} -729.000 q^{27} +2670.41 q^{29} -7993.60 q^{31} +5097.64 q^{33} -4951.31 q^{35} +8765.25 q^{37} +1375.31 q^{39} +4206.28 q^{41} -7350.56 q^{43} -4484.10 q^{45} -8252.22 q^{47} -8807.55 q^{49} +18555.4 q^{51} +6531.12 q^{53} +31355.7 q^{55} +5500.02 q^{57} -52742.9 q^{59} +12100.7 q^{61} +7244.61 q^{63} +8459.57 q^{65} +2676.52 q^{67} +4761.00 q^{69} -76796.3 q^{71} -21549.0 q^{73} +543.233 q^{75} -50659.1 q^{77} +17113.0 q^{79} +6561.00 q^{81} -43897.1 q^{83} +114135. q^{85} -24033.7 q^{87} -69817.9 q^{89} -13667.5 q^{91} +71942.4 q^{93} +33830.8 q^{95} -82707.4 q^{97} -45878.8 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 54 q^{3} + 50 q^{5} + 154 q^{7} + 486 q^{9}+O(q^{10}) $ 6 * q - 54 * q^3 + 50 * q^5 + 154 * q^7 + 486 * q^9	$ 6 q - 54 q^{3} + 50 q^{5} + 154 q^{7} + 486 q^{9} - 68 q^{11} + 1076 q^{13} - 450 q^{15} + 1650 q^{17} + 226 q^{19} - 1386 q^{21} - 3174 q^{23} + 14994 q^{25} - 4374 q^{27} + 12760 q^{29} - 10980 q^{31} + 612 q^{33} - 8492 q^{35} + 18012 q^{37} - 9684 q^{39} + 15396 q^{41} + 7146 q^{43} + 4050 q^{45} - 23588 q^{47} + 56182 q^{49} - 14850 q^{51} + 9310 q^{53} - 52048 q^{55} - 2034 q^{57} - 5260 q^{59} + 16932 q^{61} + 12474 q^{63} + 30852 q^{65} - 24858 q^{67} + 28566 q^{69} - 101832 q^{71} + 33100 q^{73} - 134946 q^{75} + 182232 q^{77} - 100570 q^{79} + 39366 q^{81} - 222524 q^{83} + 23824 q^{85} - 114840 q^{87} + 28374 q^{89} - 18204 q^{91} + 98820 q^{93} - 481996 q^{95} + 47632 q^{97} - 5508 q^{99}+O(q^{100}) $ 6 * q - 54 * q^3 + 50 * q^5 + 154 * q^7 + 486 * q^9 - 68 * q^11 + 1076 * q^13 - 450 * q^15 + 1650 * q^17 + 226 * q^19 - 1386 * q^21 - 3174 * q^23 + 14994 * q^25 - 4374 * q^27 + 12760 * q^29 - 10980 * q^31 + 612 * q^33 - 8492 * q^35 + 18012 * q^37 - 9684 * q^39 + 15396 * q^41 + 7146 * q^43 + 4050 * q^45 - 23588 * q^47 + 56182 * q^49 - 14850 * q^51 + 9310 * q^53 - 52048 * q^55 - 2034 * q^57 - 5260 * q^59 + 16932 * q^61 + 12474 * q^63 + 30852 * q^65 - 24858 * q^67 + 28566 * q^69 - 101832 * q^71 + 33100 * q^73 - 134946 * q^75 + 182232 * q^77 - 100570 * q^79 + 39366 * q^81 - 222524 * q^83 + 23824 * q^85 - 114840 * q^87 + 28374 * q^89 - 18204 * q^91 + 98820 * q^93 - 481996 * q^95 + 47632 * q^97 - 5508 * 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
$3$	−9.00000	−0.577350
$3$	−9.00000	−0.577350
$4$	0	0
$5$	−55.3592	−0.990295	−0.495148	−	0.868809i	$-0.664886\pi$
$5$	−55.3592	−0.990295	−0.495148	+	0.868809i	$0.664886\pi$
$6$	0	0
$7$	89.4396	0.689898	0.344949	−	0.938621i	$-0.387896\pi$
$7$	89.4396	0.689898	0.344949	+	0.938621i	$0.387896\pi$
$8$	0	0
$9$	81.0000	0.333333
$10$	0	0
$11$	−566.405	−1.41138	−0.705692	−	0.708518i	$-0.749364\pi$
$11$	−566.405	−1.41138	−0.705692	+	0.708518i	$0.749364\pi$
$12$	0	0
$13$	−152.812	−0.250784	−0.125392	−	0.992107i	$-0.540019\pi$
$13$	−152.812	−0.250784	−0.125392	+	0.992107i	$0.540019\pi$
$14$	0	0
$15$	498.233	0.571747
$16$	0	0
$17$	−2061.71	−1.73024	−0.865119	−	0.501566i	$-0.832757\pi$
$17$	−2061.71	−1.73024	−0.865119	+	0.501566i	$0.832757\pi$
$18$	0	0
$19$	−611.113	−0.388363	−0.194181	−	0.980966i	$-0.562205\pi$
$19$	−611.113	−0.388363	−0.194181	+	0.980966i	$0.562205\pi$
$20$	0	0
$21$	−804.957	−0.398313
$22$	0	0
$23$	−529.000	−0.208514
$23$	−529.000	−0.208514
$24$	0	0
$25$	−60.3592	−0.0193149
$26$	0	0
$27$	−729.000	−0.192450
$28$	0	0
$29$	2670.41	0.589634	0.294817	−	0.955554i	$-0.404741\pi$
$29$	2670.41	0.589634	0.294817	+	0.955554i	$0.404741\pi$
$30$	0	0
$31$	−7993.60	−1.49396	−0.746979	−	0.664848i	$-0.768496\pi$
$31$	−7993.60	−1.49396	−0.746979	+	0.664848i	$0.768496\pi$
$32$	0	0
$33$	5097.64	0.814863
$34$	0	0
$35$	−4951.31	−0.683203
$36$	0	0
$37$	8765.25	1.05259	0.526296	−	0.850302i	$-0.323580\pi$
$37$	8765.25	1.05259	0.526296	+	0.850302i	$0.323580\pi$
$38$	0	0
$39$	1375.31	0.144790
$40$	0	0
$41$	4206.28	0.390786	0.195393	−	0.980725i	$-0.437402\pi$
$41$	4206.28	0.390786	0.195393	+	0.980725i	$0.437402\pi$
$42$	0	0
$43$	−7350.56	−0.606247	−0.303123	−	0.952951i	$-0.598029\pi$
$43$	−7350.56	−0.606247	−0.303123	+	0.952951i	$0.598029\pi$
$44$	0	0
$45$	−4484.10	−0.330098
$46$	0	0
$47$	−8252.22	−0.544912	−0.272456	−	0.962168i	$-0.587836\pi$
$47$	−8252.22	−0.544912	−0.272456	+	0.962168i	$0.587836\pi$
$48$	0	0
$49$	−8807.55	−0.524041
$50$	0	0
$51$	18555.4	0.998954
$52$	0	0
$53$	6531.12	0.319373	0.159686	−	0.987168i	$-0.448952\pi$
$53$	6531.12	0.319373	0.159686	+	0.987168i	$0.448952\pi$
$54$	0	0
$55$	31355.7	1.39769
$56$	0	0
$57$	5500.02	0.224221
$58$	0	0
$59$	−52742.9	−1.97258	−0.986289	−	0.165026i	$-0.947229\pi$
$59$	−52742.9	−1.97258	−0.986289	+	0.165026i	$0.947229\pi$
$60$	0	0
$61$	12100.7	0.416375	0.208187	−	0.978089i	$-0.433244\pi$
$61$	12100.7	0.416375	0.208187	+	0.978089i	$0.433244\pi$
$62$	0	0
$63$	7244.61	0.229966
$64$	0	0
$65$	8459.57	0.248351
$66$	0	0
$67$	2676.52	0.0728424	0.0364212	−	0.999337i	$-0.488404\pi$
$67$	2676.52	0.0728424	0.0364212	+	0.999337i	$0.488404\pi$
$68$	0	0
$69$	4761.00	0.120386
$70$	0	0
$71$	−76796.3	−1.80798	−0.903991	−	0.427551i	$-0.859377\pi$
$71$	−76796.3	−1.80798	−0.903991	+	0.427551i	$0.859377\pi$
$72$	0	0
$73$	−21549.0	−0.473283	−0.236641	−	0.971597i	$-0.576047\pi$
$73$	−21549.0	−0.473283	−0.236641	+	0.971597i	$0.576047\pi$
$74$	0	0
$75$	543.233	0.0111515
$76$	0	0
$77$	−50659.1	−0.973712
$78$	0	0
$79$	17113.0	0.308503	0.154251	−	0.988032i	$-0.450703\pi$
$79$	17113.0	0.308503	0.154251	+	0.988032i	$0.450703\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	0	0
$83$	−43897.1	−0.699424	−0.349712	−	0.936857i	$-0.613721\pi$
$83$	−43897.1	−0.699424	−0.349712	+	0.936857i	$0.613721\pi$
$84$	0	0
$85$	114135.	1.71345
$86$	0	0
$87$	−24033.7	−0.340425
$88$	0	0
$89$	−69817.9	−0.934312	−0.467156	−	0.884175i	$-0.654721\pi$
$89$	−69817.9	−0.934312	−0.467156	+	0.884175i	$0.654721\pi$
$90$	0	0
$91$	−13667.5	−0.173016
$92$	0	0
$93$	71942.4	0.862537
$94$	0	0
$95$	33830.8	0.384594
$96$	0	0
$97$	−82707.4	−0.892514	−0.446257	−	0.894905i	$-0.647243\pi$
$97$	−82707.4	−0.892514	−0.446257	+	0.894905i	$0.647243\pi$
$98$	0	0
$99$	−45878.8	−0.470462
$100$	0	0
$101$	94362.6	0.920442	0.460221	−	0.887804i	$-0.347770\pi$
$101$	94362.6	0.920442	0.460221	+	0.887804i	$0.347770\pi$
$102$	0	0
$103$	−176719.	−1.64131	−0.820653	−	0.571427i	$-0.806390\pi$
$103$	−176719.	−1.64131	−0.820653	+	0.571427i	$0.806390\pi$
$104$	0	0
$105$	44561.8	0.394447
$106$	0	0
$107$	−45810.5	−0.386817	−0.193409	−	0.981118i	$-0.561954\pi$
$107$	−45810.5	−0.386817	−0.193409	+	0.981118i	$0.561954\pi$
$108$	0	0
$109$	68415.4	0.551554	0.275777	−	0.961222i	$-0.411065\pi$
$109$	68415.4	0.551554	0.275777	+	0.961222i	$0.411065\pi$
$110$	0	0
$111$	−78887.2	−0.607714
$112$	0	0
$113$	−214406.	−1.57957	−0.789787	−	0.613381i	$-0.789809\pi$
$113$	−214406.	−1.57957	−0.789787	+	0.613381i	$0.789809\pi$
$114$	0	0
$115$	29285.0	0.206491
$116$	0	0
$117$	−12377.8	−0.0835948
$118$	0	0
$119$	−184399.	−1.19369
$120$	0	0
$121$	159764.	0.992006
$122$	0	0
$123$	−37856.6	−0.225620
$124$	0	0
$125$	176339.	1.00942
$126$	0	0
$127$	243452.	1.33938	0.669689	−	0.742641i	$-0.266427\pi$
$127$	243452.	1.33938	0.669689	+	0.742641i	$0.266427\pi$
$128$	0	0
$129$	66155.0	0.350017
$130$	0	0
$131$	−266247.	−1.35552	−0.677762	−	0.735281i	$-0.737050\pi$
$131$	−266247.	−1.35552	−0.677762	+	0.735281i	$0.737050\pi$
$132$	0	0
$133$	−54657.8	−0.267931
$134$	0	0
$135$	40356.9	0.190582
$136$	0	0
$137$	−119521.	−0.544057	−0.272029	−	0.962289i	$-0.587695\pi$
$137$	−119521.	−0.544057	−0.272029	+	0.962289i	$0.587695\pi$
$138$	0	0
$139$	125586.	0.551321	0.275660	−	0.961255i	$-0.411103\pi$
$139$	125586.	0.551321	0.275660	+	0.961255i	$0.411103\pi$
$140$	0	0
$141$	74270.0	0.314605
$142$	0	0
$143$	86553.7	0.353953
$144$	0	0
$145$	−147832.	−0.583912
$146$	0	0
$147$	79267.9	0.302555
$148$	0	0
$149$	301611.	1.11296	0.556482	−	0.830860i	$-0.312151\pi$
$149$	301611.	1.11296	0.556482	+	0.830860i	$0.312151\pi$
$150$	0	0
$151$	−242172.	−0.864335	−0.432167	−	0.901793i	$-0.642251\pi$
$151$	−242172.	−0.864335	−0.432167	+	0.901793i	$0.642251\pi$
$152$	0	0
$153$	−166999.	−0.576746
$154$	0	0
$155$	442519.	1.47946
$156$	0	0
$157$	−7878.65	−0.0255095	−0.0127548	−	0.999919i	$-0.504060\pi$
$157$	−7878.65	−0.0255095	−0.0127548	+	0.999919i	$0.504060\pi$
$158$	0	0
$159$	−58780.1	−0.184390
$160$	0	0
$161$	−47313.6	−0.143854
$162$	0	0
$163$	514477.	1.51669	0.758346	−	0.651853i	$-0.226008\pi$
$163$	514477.	1.51669	0.758346	+	0.651853i	$0.226008\pi$
$164$	0	0
$165$	−282202.	−0.806955
$166$	0	0
$167$	85174.9	0.236331	0.118165	−	0.992994i	$-0.462299\pi$
$167$	85174.9	0.236331	0.118165	+	0.992994i	$0.462299\pi$
$168$	0	0
$169$	−347941.	−0.937107
$170$	0	0
$171$	−49500.2	−0.129454
$172$	0	0
$173$	−107137.	−0.272161	−0.136080	−	0.990698i	$-0.543451\pi$
$173$	−107137.	−0.272161	−0.136080	+	0.990698i	$0.543451\pi$
$174$	0	0
$175$	−5398.51	−0.0133253
$176$	0	0
$177$	474686.	1.13887
$178$	0	0
$179$	142427.	0.332245	0.166122	−	0.986105i	$-0.446875\pi$
$179$	142427.	0.332245	0.166122	+	0.986105i	$0.446875\pi$
$180$	0	0
$181$	−69622.0	−0.157961	−0.0789805	−	0.996876i	$-0.525167\pi$
$181$	−69622.0	−0.157961	−0.0789805	+	0.996876i	$0.525167\pi$
$182$	0	0
$183$	−108906.	−0.240394
$184$	0	0
$185$	−485237.	−1.04238
$186$	0	0
$187$	1.16776e6	2.44203
$188$	0	0
$189$	−65201.5	−0.132771
$190$	0	0
$191$	−30005.5	−0.0595137	−0.0297569	−	0.999557i	$-0.509473\pi$
$191$	−30005.5	−0.0595137	−0.0297569	+	0.999557i	$0.509473\pi$
$192$	0	0
$193$	856517.	1.65517	0.827586	−	0.561339i	$-0.189714\pi$
$193$	856517.	1.65517	0.827586	+	0.561339i	$0.189714\pi$
$194$	0	0
$195$	−76136.2	−0.143385
$196$	0	0
$197$	608461.	1.11704	0.558518	−	0.829492i	$-0.311370\pi$
$197$	608461.	1.11704	0.558518	+	0.829492i	$0.311370\pi$
$198$	0	0
$199$	518926.	0.928907	0.464454	−	0.885597i	$-0.346251\pi$
$199$	518926.	0.928907	0.464454	+	0.885597i	$0.346251\pi$
$200$	0	0
$201$	−24088.7	−0.0420556
$202$	0	0
$203$	238840.	0.406787
$204$	0	0
$205$	−232856.	−0.386994
$206$	0	0
$207$	−42849.0	−0.0695048
$208$	0	0
$209$	346138.	0.548130
$210$	0	0
$211$	826298.	1.27770	0.638852	−	0.769330i	$-0.279410\pi$
$211$	826298.	1.27770	0.638852	+	0.769330i	$0.279410\pi$
$212$	0	0
$213$	691167.	1.04384
$214$	0	0
$215$	406921.	0.600363
$216$	0	0
$217$	−714945.	−1.03068
$218$	0	0
$219$	193941.	0.273250
$220$	0	0
$221$	315055.	0.433917
$222$	0	0
$223$	1.01705e6	1.36956	0.684779	−	0.728751i	$-0.259899\pi$
$223$	1.01705e6	1.36956	0.684779	+	0.728751i	$0.259899\pi$
$224$	0	0
$225$	−4889.09	−0.00643831
$226$	0	0
$227$	379096.	0.488298	0.244149	−	0.969738i	$-0.421491\pi$
$227$	379096.	0.488298	0.244149	+	0.969738i	$0.421491\pi$
$228$	0	0
$229$	−81468.6	−0.102660	−0.0513300	−	0.998682i	$-0.516346\pi$
$229$	−81468.6	−0.102660	−0.0513300	+	0.998682i	$0.516346\pi$
$230$	0	0
$231$	455932.	0.562173
$232$	0	0
$233$	977505.	1.17959	0.589793	−	0.807555i	$-0.299209\pi$
$233$	977505.	1.17959	0.589793	+	0.807555i	$0.299209\pi$
$234$	0	0
$235$	456836.	0.539623
$236$	0	0
$237$	−154017.	−0.178114
$238$	0	0
$239$	−1.02988e6	−1.16625	−0.583124	−	0.812383i	$-0.698170\pi$
$239$	−1.02988e6	−1.16625	−0.583124	+	0.812383i	$0.698170\pi$
$240$	0	0
$241$	−574594.	−0.637262	−0.318631	−	0.947879i	$-0.603223\pi$
$241$	−574594.	−0.637262	−0.318631	+	0.947879i	$0.603223\pi$
$242$	0	0
$243$	−59049.0	−0.0641500
$244$	0	0
$245$	487579.	0.518955
$246$	0	0
$247$	93385.7	0.0973953
$248$	0	0
$249$	395074.	0.403813
$250$	0	0
$251$	−888912.	−0.890583	−0.445292	−	0.895386i	$-0.646900\pi$
$251$	−888912.	−0.890583	−0.445292	+	0.895386i	$0.646900\pi$
$252$	0	0
$253$	299628.	0.294294
$254$	0	0
$255$	−1.02721e6	−0.989259
$256$	0	0
$257$	−1.43589e6	−1.35609	−0.678046	−	0.735020i	$-0.737173\pi$
$257$	−1.43589e6	−1.35609	−0.678046	+	0.735020i	$0.737173\pi$
$258$	0	0
$259$	783961.	0.726181
$260$	0	0
$261$	216303.	0.196545
$262$	0	0
$263$	−1.01109e6	−0.901367	−0.450683	−	0.892684i	$-0.648820\pi$
$263$	−1.01109e6	−0.901367	−0.450683	+	0.892684i	$0.648820\pi$
$264$	0	0
$265$	−361557.	−0.316273
$266$	0	0
$267$	628361.	0.539425
$268$	0	0
$269$	−201112.	−0.169456	−0.0847281	−	0.996404i	$-0.527002\pi$
$269$	−201112.	−0.169456	−0.0847281	+	0.996404i	$0.527002\pi$
$270$	0	0
$271$	61687.6	0.0510240	0.0255120	−	0.999675i	$-0.491878\pi$
$271$	61687.6	0.0510240	0.0255120	+	0.999675i	$0.491878\pi$
$272$	0	0
$273$	123007.	0.0998906
$274$	0	0
$275$	34187.8	0.0272608
$276$	0	0
$277$	−2.01468e6	−1.57763	−0.788817	−	0.614628i	$-0.789306\pi$
$277$	−2.01468e6	−1.57763	−0.788817	+	0.614628i	$0.789306\pi$
$278$	0	0
$279$	−647482.	−0.497986
$280$	0	0
$281$	764461.	0.577550	0.288775	−	0.957397i	$-0.406752\pi$
$281$	764461.	0.577550	0.288775	+	0.957397i	$0.406752\pi$
$282$	0	0
$283$	1.41975e6	1.05377	0.526886	−	0.849936i	$-0.323360\pi$
$283$	1.41975e6	1.05377	0.526886	+	0.849936i	$0.323360\pi$
$284$	0	0
$285$	−304477.	−0.222046
$286$	0	0
$287$	376209.	0.269602
$288$	0	0
$289$	2.83080e6	1.99372
$290$	0	0
$291$	744366.	0.515293
$292$	0	0
$293$	−1.17232e6	−0.797771	−0.398885	−	0.917001i	$-0.630603\pi$
$293$	−1.17232e6	−0.797771	−0.398885	+	0.917001i	$0.630603\pi$
$294$	0	0
$295$	2.91981e6	1.95344
$296$	0	0
$297$	412909.	0.271621
$298$	0	0
$299$	80837.8	0.0522921
$300$	0	0
$301$	−657431.	−0.418248
$302$	0	0
$303$	−849264.	−0.531418
$304$	0	0
$305$	−669883.	−0.412334
$306$	0	0
$307$	−2.87116e6	−1.73865	−0.869324	−	0.494243i	$-0.835445\pi$
$307$	−2.87116e6	−1.73865	−0.869324	+	0.494243i	$0.835445\pi$
$308$	0	0
$309$	1.59047e6	0.947608
$310$	0	0
$311$	−278189.	−0.163095	−0.0815473	−	0.996669i	$-0.525986\pi$
$311$	−278189.	−0.163095	−0.0815473	+	0.996669i	$0.525986\pi$
$312$	0	0
$313$	−378830.	−0.218566	−0.109283	−	0.994011i	$-0.534856\pi$
$313$	−378830.	−0.218566	−0.109283	+	0.994011i	$0.534856\pi$
$314$	0	0
$315$	−401056.	−0.227734
$316$	0	0
$317$	2.47100e6	1.38110	0.690550	−	0.723284i	$-0.257368\pi$
$317$	2.47100e6	1.38110	0.690550	+	0.723284i	$0.257368\pi$
$318$	0	0
$319$	−1.51253e6	−0.832200
$320$	0	0
$321$	412295.	0.223329
$322$	0	0
$323$	1.25994e6	0.671961
$324$	0	0
$325$	9223.64	0.00484388
$326$	0	0
$327$	−615739.	−0.318440
$328$	0	0
$329$	−738075.	−0.375933
$330$	0	0
$331$	−3.53121e6	−1.77155	−0.885776	−	0.464112i	$-0.846373\pi$
$331$	−3.53121e6	−1.77155	−0.885776	+	0.464112i	$0.846373\pi$
$332$	0	0
$333$	709985.	0.350864
$334$	0	0
$335$	−148170.	−0.0721355
$336$	0	0
$337$	607874.	0.291567	0.145784	−	0.989316i	$-0.453430\pi$
$337$	607874.	0.291567	0.145784	+	0.989316i	$0.453430\pi$
$338$	0	0
$339$	1.92965e6	0.911967
$340$	0	0
$341$	4.52761e6	2.10855
$342$	0	0
$343$	−2.29096e6	−1.05143
$344$	0	0
$345$	−263565.	−0.119218
$346$	0	0
$347$	−1.81627e6	−0.809759	−0.404880	−	0.914370i	$-0.632687\pi$
$347$	−1.81627e6	−0.809759	−0.404880	+	0.914370i	$0.632687\pi$
$348$	0	0
$349$	807021.	0.354667	0.177334	−	0.984151i	$-0.443253\pi$
$349$	807021.	0.354667	0.177334	+	0.984151i	$0.443253\pi$
$350$	0	0
$351$	111400.	0.0482635
$352$	0	0
$353$	1.91288e6	0.817055	0.408527	−	0.912746i	$-0.366042\pi$
$353$	1.91288e6	0.817055	0.408527	+	0.912746i	$0.366042\pi$
$354$	0	0
$355$	4.25138e6	1.79044
$356$	0	0
$357$	1.65959e6	0.689176
$358$	0	0
$359$	4.09405e6	1.67655	0.838275	−	0.545247i	$-0.183564\pi$
$359$	4.09405e6	1.67655	0.838275	+	0.545247i	$0.183564\pi$
$360$	0	0
$361$	−2.10264e6	−0.849174
$362$	0	0
$363$	−1.43787e6	−0.572735
$364$	0	0
$365$	1.19294e6	0.468690
$366$	0	0
$367$	2.96338e6	1.14848	0.574239	−	0.818688i	$-0.305298\pi$
$367$	2.96338e6	1.14848	0.574239	+	0.818688i	$0.305298\pi$
$368$	0	0
$369$	340709.	0.130262
$370$	0	0
$371$	584141.	0.220335
$372$	0	0
$373$	4.56228e6	1.69789	0.848945	−	0.528481i	$-0.177238\pi$
$373$	4.56228e6	1.69789	0.848945	+	0.528481i	$0.177238\pi$
$374$	0	0
$375$	−1.58705e6	−0.582791
$376$	0	0
$377$	−408071.	−0.147871
$378$	0	0
$379$	2.32376e6	0.830984	0.415492	−	0.909597i	$-0.363610\pi$
$379$	2.32376e6	0.830984	0.415492	+	0.909597i	$0.363610\pi$
$380$	0	0
$381$	−2.19106e6	−0.773291
$382$	0	0
$383$	5.00143e6	1.74220	0.871098	−	0.491109i	$-0.163408\pi$
$383$	5.00143e6	1.74220	0.871098	+	0.491109i	$0.163408\pi$
$384$	0	0
$385$	2.80445e6	0.964262
$386$	0	0
$387$	−595395.	−0.202082
$388$	0	0
$389$	−1.39556e6	−0.467601	−0.233800	−	0.972285i	$-0.575116\pi$
$389$	−1.39556e6	−0.467601	−0.233800	+	0.972285i	$0.575116\pi$
$390$	0	0
$391$	1.09065e6	0.360780
$392$	0	0
$393$	2.39623e6	0.782612
$394$	0	0
$395$	−947364.	−0.305509
$396$	0	0
$397$	4.13119e6	1.31553	0.657763	−	0.753225i	$-0.271503\pi$
$397$	4.13119e6	1.31553	0.657763	+	0.753225i	$0.271503\pi$
$398$	0	0
$399$	491920.	0.154690
$400$	0	0
$401$	5.28892e6	1.64250	0.821251	−	0.570567i	$-0.193276\pi$
$401$	5.28892e6	1.64250	0.821251	+	0.570567i	$0.193276\pi$
$402$	0	0
$403$	1.22152e6	0.374661
$404$	0	0
$405$	−363212.	−0.110033
$406$	0	0
$407$	−4.96468e6	−1.48561
$408$	0	0
$409$	−1.19086e6	−0.352009	−0.176005	−	0.984389i	$-0.556317\pi$
$409$	−1.19086e6	−0.352009	−0.176005	+	0.984389i	$0.556317\pi$
$410$	0	0
$411$	1.07569e6	0.314112
$412$	0	0
$413$	−4.71731e6	−1.36088
$414$	0	0
$415$	2.43011e6	0.692636
$416$	0	0
$417$	−1.13027e6	−0.318305
$418$	0	0
$419$	1.94768e6	0.541979	0.270990	−	0.962582i	$-0.412649\pi$
$419$	1.94768e6	0.541979	0.270990	+	0.962582i	$0.412649\pi$
$420$	0	0
$421$	−2.56649e6	−0.705723	−0.352861	−	0.935676i	$-0.614791\pi$
$421$	−2.56649e6	−0.705723	−0.352861	+	0.935676i	$0.614791\pi$
$422$	0	0
$423$	−668430.	−0.181637
$424$	0	0
$425$	124443.	0.0334195
$426$	0	0
$427$	1.08228e6	0.287256
$428$	0	0
$429$	−778983.	−0.204355
$430$	0	0
$431$	−4.09690e6	−1.06234	−0.531169	−	0.847266i	$-0.678247\pi$
$431$	−4.09690e6	−1.06234	−0.531169	+	0.847266i	$0.678247\pi$
$432$	0	0
$433$	3.30578e6	0.847334	0.423667	−	0.905818i	$-0.360743\pi$
$433$	3.30578e6	0.847334	0.423667	+	0.905818i	$0.360743\pi$
$434$	0	0
$435$	1.33048e6	0.337122
$436$	0	0
$437$	323279.	0.0809793
$438$	0	0
$439$	−612748.	−0.151747	−0.0758735	−	0.997117i	$-0.524175\pi$
$439$	−612748.	−0.151747	−0.0758735	+	0.997117i	$0.524175\pi$
$440$	0	0
$441$	−713412.	−0.174680
$442$	0	0
$443$	−2.63430e6	−0.637759	−0.318879	−	0.947795i	$-0.603307\pi$
$443$	−2.63430e6	−0.637759	−0.318879	+	0.947795i	$0.603307\pi$
$444$	0	0
$445$	3.86506e6	0.925245
$446$	0	0
$447$	−2.71450e6	−0.642570
$448$	0	0
$449$	7.88042e6	1.84473	0.922366	−	0.386316i	$-0.126253\pi$
$449$	7.88042e6	1.84473	0.922366	+	0.386316i	$0.126253\pi$
$450$	0	0
$451$	−2.38246e6	−0.551549
$452$	0	0
$453$	2.17955e6	0.499024
$454$	0	0
$455$	756621.	0.171337
$456$	0	0
$457$	−3.19498e6	−0.715613	−0.357806	−	0.933796i	$-0.616475\pi$
$457$	−3.19498e6	−0.715613	−0.357806	+	0.933796i	$0.616475\pi$
$458$	0	0
$459$	1.50299e6	0.332985
$460$	0	0
$461$	−411819.	−0.0902514	−0.0451257	−	0.998981i	$-0.514369\pi$
$461$	−411819.	−0.0902514	−0.0451257	+	0.998981i	$0.514369\pi$
$462$	0	0
$463$	2.78917e6	0.604676	0.302338	−	0.953201i	$-0.402233\pi$
$463$	2.78917e6	0.604676	0.302338	+	0.953201i	$0.402233\pi$
$464$	0	0
$465$	−3.98267e6	−0.854166
$466$	0	0
$467$	791413.	0.167923	0.0839617	−	0.996469i	$-0.473243\pi$
$467$	791413.	0.167923	0.0839617	+	0.996469i	$0.473243\pi$
$468$	0	0
$469$	239387.	0.0502538
$470$	0	0
$471$	70907.9	0.0147279
$472$	0	0
$473$	4.16339e6	0.855647
$474$	0	0
$475$	36886.3	0.00750121
$476$	0	0
$477$	529020.	0.106458
$478$	0	0
$479$	495749.	0.0987240	0.0493620	−	0.998781i	$-0.484281\pi$
$479$	495749.	0.0987240	0.0493620	+	0.998781i	$0.484281\pi$
$480$	0	0
$481$	−1.33944e6	−0.263973
$482$	0	0
$483$	425822.	0.0830540
$484$	0	0
$485$	4.57861e6	0.883852
$486$	0	0
$487$	−9.55276e6	−1.82518	−0.912591	−	0.408873i	$-0.865922\pi$
$487$	−9.55276e6	−1.82518	−0.912591	+	0.408873i	$0.865922\pi$
$488$	0	0
$489$	−4.63030e6	−0.875662
$490$	0	0
$491$	−6.71995e6	−1.25795	−0.628974	−	0.777426i	$-0.716525\pi$
$491$	−6.71995e6	−1.25795	−0.628974	+	0.777426i	$0.716525\pi$
$492$	0	0
$493$	−5.50561e6	−1.02021
$494$	0	0
$495$	2.53981e6	0.465896
$496$	0	0
$497$	−6.86863e6	−1.24732
$498$	0	0
$499$	7.76019e6	1.39515	0.697576	−	0.716511i	$-0.254262\pi$
$499$	7.76019e6	1.39515	0.697576	+	0.716511i	$0.254262\pi$
$500$	0	0
$501$	−766574.	−0.136446
$502$	0	0
$503$	−3.84599e6	−0.677779	−0.338889	−	0.940826i	$-0.610051\pi$
$503$	−3.84599e6	−0.677779	−0.338889	+	0.940826i	$0.610051\pi$
$504$	0	0
$505$	−5.22384e6	−0.911510
$506$	0	0
$507$	3.13147e6	0.541039
$508$	0	0
$509$	−7.39636e6	−1.26539	−0.632693	−	0.774402i	$-0.718051\pi$
$509$	−7.39636e6	−1.26539	−0.632693	+	0.774402i	$0.718051\pi$
$510$	0	0
$511$	−1.92734e6	−0.326517
$512$	0	0
$513$	445502.	0.0747405
$514$	0	0
$515$	9.78301e6	1.62538
$516$	0	0
$517$	4.67410e6	0.769080
$518$	0	0
$519$	964235.	0.157132
$520$	0	0
$521$	−5.65355e6	−0.912487	−0.456244	−	0.889855i	$-0.650805\pi$
$521$	−5.65355e6	−0.912487	−0.456244	+	0.889855i	$0.650805\pi$
$522$	0	0
$523$	1.47849e6	0.236354	0.118177	−	0.992993i	$-0.462295\pi$
$523$	1.47849e6	0.236354	0.118177	+	0.992993i	$0.462295\pi$
$524$	0	0
$525$	48586.5	0.00769339
$526$	0	0
$527$	1.64805e7	2.58490
$528$	0	0
$529$	279841.	0.0434783
$530$	0	0
$531$	−4.27218e6	−0.657526
$532$	0	0
$533$	−642772.	−0.0980030
$534$	0	0
$535$	2.53603e6	0.383063
$536$	0	0
$537$	−1.28184e6	−0.191822
$538$	0	0
$539$	4.98864e6	0.739623
$540$	0	0
$541$	6.56939e6	0.965010	0.482505	−	0.875893i	$-0.339727\pi$
$541$	6.56939e6	0.965010	0.482505	+	0.875893i	$0.339727\pi$
$542$	0	0
$543$	626598.	0.0911989
$544$	0	0
$545$	−3.78742e6	−0.546201
$546$	0	0
$547$	−7.51534e6	−1.07394	−0.536971	−	0.843601i	$-0.680431\pi$
$547$	−7.51534e6	−1.07394	−0.536971	+	0.843601i	$0.680431\pi$
$548$	0	0
$549$	980153.	0.138792
$550$	0	0
$551$	−1.63192e6	−0.228992
$552$	0	0
$553$	1.53058e6	0.212836
$554$	0	0
$555$	4.36713e6	0.601816
$556$	0	0
$557$	−868099.	−0.118558	−0.0592790	−	0.998241i	$-0.518880\pi$
$557$	−868099.	−0.118558	−0.0592790	+	0.998241i	$0.518880\pi$
$558$	0	0
$559$	1.12326e6	0.152037
$560$	0	0
$561$	−1.05099e7	−1.40991
$562$	0	0
$563$	8.46829e6	1.12596	0.562982	−	0.826469i	$-0.309654\pi$
$563$	8.46829e6	1.12596	0.562982	+	0.826469i	$0.309654\pi$
$564$	0	0
$565$	1.18693e7	1.56424
$566$	0	0
$567$	586814.	0.0766553
$568$	0	0
$569$	−6.08419e6	−0.787811	−0.393906	−	0.919151i	$-0.628876\pi$
$569$	−6.08419e6	−0.787811	−0.393906	+	0.919151i	$0.628876\pi$
$570$	0	0
$571$	5.57999e6	0.716215	0.358107	−	0.933680i	$-0.383422\pi$
$571$	5.57999e6	0.716215	0.358107	+	0.933680i	$0.383422\pi$
$572$	0	0
$573$	270049.	0.0343603
$574$	0	0
$575$	31930.0	0.00402744
$576$	0	0
$577$	−7.21155e6	−0.901756	−0.450878	−	0.892586i	$-0.648889\pi$
$577$	−7.21155e6	−0.901756	−0.450878	+	0.892586i	$0.648889\pi$
$578$	0	0
$579$	−7.70866e6	−0.955614
$580$	0	0
$581$	−3.92614e6	−0.482531
$582$	0	0
$583$	−3.69926e6	−0.450758
$584$	0	0
$585$	685225.	0.0827835
$586$	0	0
$587$	5.58416e6	0.668902	0.334451	−	0.942413i	$-0.391449\pi$
$587$	5.58416e6	0.668902	0.334451	+	0.942413i	$0.391449\pi$
$588$	0	0
$589$	4.88500e6	0.580198
$590$	0	0
$591$	−5.47615e6	−0.644921
$592$	0	0
$593$	−3.76868e6	−0.440102	−0.220051	−	0.975488i	$-0.570622\pi$
$593$	−3.76868e6	−0.440102	−0.220051	+	0.975488i	$0.570622\pi$
$594$	0	0
$595$	1.02082e7	1.18210
$596$	0	0
$597$	−4.67033e6	−0.536305
$598$	0	0
$599$	8.97583e6	1.02213	0.511067	−	0.859541i	$-0.329251\pi$
$599$	8.97583e6	1.02213	0.511067	+	0.859541i	$0.329251\pi$
$600$	0	0
$601$	3.28500e6	0.370979	0.185489	−	0.982646i	$-0.440613\pi$
$601$	3.28500e6	0.370979	0.185489	+	0.982646i	$0.440613\pi$
$602$	0	0
$603$	216798.	0.0242808
$604$	0	0
$605$	−8.84439e6	−0.982379
$606$	0	0
$607$	−1.30113e7	−1.43334	−0.716669	−	0.697414i	$-0.754334\pi$
$607$	−1.30113e7	−1.43334	−0.716669	+	0.697414i	$0.754334\pi$
$608$	0	0
$609$	−2.14956e6	−0.234859
$610$	0	0
$611$	1.26104e6	0.136655
$612$	0	0
$613$	−5.19705e6	−0.558606	−0.279303	−	0.960203i	$-0.590103\pi$
$613$	−5.19705e6	−0.558606	−0.279303	+	0.960203i	$0.590103\pi$
$614$	0	0
$615$	2.09571e6	0.223431
$616$	0	0
$617$	8.74006e6	0.924276	0.462138	−	0.886808i	$-0.347083\pi$
$617$	8.74006e6	0.924276	0.462138	+	0.886808i	$0.347083\pi$
$618$	0	0
$619$	2.84962e6	0.298924	0.149462	−	0.988768i	$-0.452246\pi$
$619$	2.84962e6	0.298924	0.149462	+	0.988768i	$0.452246\pi$
$620$	0	0
$621$	385641.	0.0401286
$622$	0	0
$623$	−6.24449e6	−0.644580
$624$	0	0
$625$	−9.57336e6	−0.980312
$626$	0	0
$627$	−3.11524e6	−0.316463
$628$	0	0
$629$	−1.80714e7	−1.82123
$630$	0	0
$631$	−2.16608e6	−0.216571	−0.108286	−	0.994120i	$-0.534536\pi$
$631$	−2.16608e6	−0.216571	−0.108286	+	0.994120i	$0.534536\pi$
$632$	0	0
$633$	−7.43668e6	−0.737683
$634$	0	0
$635$	−1.34773e7	−1.32638
$636$	0	0
$637$	1.34590e6	0.131421
$638$	0	0
$639$	−6.22050e6	−0.602661
$640$	0	0
$641$	3.91985e6	0.376811	0.188406	−	0.982091i	$-0.439668\pi$
$641$	3.91985e6	0.376811	0.188406	+	0.982091i	$0.439668\pi$
$642$	0	0
$643$	−1.89197e7	−1.80462	−0.902311	−	0.431086i	$-0.858130\pi$
$643$	−1.89197e7	−1.80462	−0.902311	+	0.431086i	$0.858130\pi$
$644$	0	0
$645$	−3.66229e6	−0.346620
$646$	0	0
$647$	4.12346e6	0.387259	0.193629	−	0.981075i	$-0.437974\pi$
$647$	4.12346e6	0.387259	0.193629	+	0.981075i	$0.437974\pi$
$648$	0	0
$649$	2.98739e7	2.78407
$650$	0	0
$651$	6.43450e6	0.595062
$652$	0	0
$653$	1.29569e6	0.118910	0.0594548	−	0.998231i	$-0.481064\pi$
$653$	1.29569e6	0.118910	0.0594548	+	0.998231i	$0.481064\pi$
$654$	0	0
$655$	1.47392e7	1.34237
$656$	0	0
$657$	−1.74547e6	−0.157761
$658$	0	0
$659$	−2.09327e7	−1.87764	−0.938818	−	0.344412i	$-0.888078\pi$
$659$	−2.09327e7	−1.87764	−0.938818	+	0.344412i	$0.888078\pi$
$660$	0	0
$661$	9.37904e6	0.834939	0.417469	−	0.908691i	$-0.362917\pi$
$661$	9.37904e6	0.834939	0.417469	+	0.908691i	$0.362917\pi$
$662$	0	0
$663$	−2.83550e6	−0.250522
$664$	0	0
$665$	3.02581e6	0.265331
$666$	0	0
$667$	−1.41265e6	−0.122947
$668$	0	0
$669$	−9.15346e6	−0.790715
$670$	0	0
$671$	−6.85387e6	−0.587665
$672$	0	0
$673$	2.22005e7	1.88940	0.944701	−	0.327933i	$-0.106352\pi$
$673$	2.22005e7	1.88940	0.944701	+	0.327933i	$0.106352\pi$
$674$	0	0
$675$	44001.9	0.00371716
$676$	0	0
$677$	−7.30083e6	−0.612210	−0.306105	−	0.951998i	$-0.599026\pi$
$677$	−7.30083e6	−0.612210	−0.306105	+	0.951998i	$0.599026\pi$
$678$	0	0
$679$	−7.39732e6	−0.615744
$680$	0	0
$681$	−3.41187e6	−0.281919
$682$	0	0
$683$	−1.33754e7	−1.09712	−0.548559	−	0.836112i	$-0.684823\pi$
$683$	−1.33754e7	−1.09712	−0.548559	+	0.836112i	$0.684823\pi$
$684$	0	0
$685$	6.61661e6	0.538777
$686$	0	0
$687$	733218.	0.0592708
$688$	0	0
$689$	−998036.	−0.0800937
$690$	0	0
$691$	1.26880e6	0.101087	0.0505436	−	0.998722i	$-0.483905\pi$
$691$	1.26880e6	0.101087	0.0505436	+	0.998722i	$0.483905\pi$
$692$	0	0
$693$	−4.10338e6	−0.324571
$694$	0	0
$695$	−6.95234e6	−0.545971
$696$	0	0
$697$	−8.67215e6	−0.676153
$698$	0	0
$699$	−8.79755e6	−0.681034
$700$	0	0
$701$	1.46823e7	1.12850	0.564248	−	0.825606i	$-0.309166\pi$
$701$	1.46823e7	1.12850	0.564248	+	0.825606i	$0.309166\pi$
$702$	0	0
$703$	−5.35656e6	−0.408788
$704$	0	0
$705$	−4.11153e6	−0.311552
$706$	0	0
$707$	8.43976e6	0.635011
$708$	0	0
$709$	−1.72881e7	−1.29161	−0.645804	−	0.763503i	$-0.723478\pi$
$709$	−1.72881e7	−1.29161	−0.645804	+	0.763503i	$0.723478\pi$
$710$	0	0
$711$	1.38616e6	0.102834
$712$	0	0
$713$	4.22861e6	0.311512
$714$	0	0
$715$	−4.79154e6	−0.350518
$716$	0	0
$717$	9.26890e6	0.673333
$718$	0	0
$719$	7.78727e6	0.561776	0.280888	−	0.959741i	$-0.409371\pi$
$719$	7.78727e6	0.561776	0.280888	+	0.959741i	$0.409371\pi$
$720$	0	0
$721$	−1.58057e7	−1.13233
$722$	0	0
$723$	5.17134e6	0.367924
$724$	0	0
$725$	−161184.	−0.0113887
$726$	0	0
$727$	1.08956e7	0.764566	0.382283	−	0.924045i	$-0.375138\pi$
$727$	1.08956e7	0.764566	0.382283	+	0.924045i	$0.375138\pi$
$728$	0	0
$729$	531441.	0.0370370
$730$	0	0
$731$	1.51547e7	1.04895
$732$	0	0
$733$	−136697.	−0.00939721	−0.00469861	−	0.999989i	$-0.501496\pi$
$733$	−136697.	−0.00939721	−0.00469861	+	0.999989i	$0.501496\pi$
$734$	0	0
$735$	−4.38821e6	−0.299619
$736$	0	0
$737$	−1.51600e6	−0.102809
$738$	0	0
$739$	−2.22208e7	−1.49675	−0.748374	−	0.663277i	$-0.769165\pi$
$739$	−2.22208e7	−1.49675	−0.748374	+	0.663277i	$0.769165\pi$
$740$	0	0
$741$	−840472.	−0.0562312
$742$	0	0
$743$	−2.35607e7	−1.56573	−0.782863	−	0.622194i	$-0.786242\pi$
$743$	−2.35607e7	−1.56573	−0.782863	+	0.622194i	$0.786242\pi$
$744$	0	0
$745$	−1.66969e7	−1.10216
$746$	0	0
$747$	−3.55566e6	−0.233141
$748$	0	0
$749$	−4.09728e6	−0.266864
$750$	0	0
$751$	1.88055e7	1.21671	0.608353	−	0.793666i	$-0.291830\pi$
$751$	1.88055e7	1.21671	0.608353	+	0.793666i	$0.291830\pi$
$752$	0	0
$753$	8.00021e6	0.514178
$754$	0	0
$755$	1.34065e7	0.855947
$756$	0	0
$757$	1.54515e7	0.980011	0.490005	−	0.871719i	$-0.336995\pi$
$757$	1.54515e7	0.980011	0.490005	+	0.871719i	$0.336995\pi$
$758$	0	0
$759$	−2.69665e6	−0.169911
$760$	0	0
$761$	−1.37995e7	−0.863776	−0.431888	−	0.901927i	$-0.642152\pi$
$761$	−1.37995e7	−0.863776	−0.431888	+	0.901927i	$0.642152\pi$
$762$	0	0
$763$	6.11905e6	0.380516
$764$	0	0
$765$	9.24492e6	0.571149
$766$	0	0
$767$	8.05978e6	0.494692
$768$	0	0
$769$	2.73527e7	1.66795	0.833977	−	0.551799i	$-0.186059\pi$
$769$	2.73527e7	1.66795	0.833977	+	0.551799i	$0.186059\pi$
$770$	0	0
$771$	1.29230e7	0.782940
$772$	0	0
$773$	−5.88968e6	−0.354522	−0.177261	−	0.984164i	$-0.556724\pi$
$773$	−5.88968e6	−0.354522	−0.177261	+	0.984164i	$0.556724\pi$
$774$	0	0
$775$	482487.	0.0288557
$776$	0	0
$777$	−7.05565e6	−0.419261
$778$	0	0
$779$	−2.57052e6	−0.151767
$780$	0	0
$781$	4.34978e7	2.55176
$782$	0	0
$783$	−1.94673e6	−0.113475
$784$	0	0
$785$	436156.	0.0252620
$786$	0	0
$787$	−2.79492e7	−1.60855	−0.804273	−	0.594261i	$-0.797445\pi$
$787$	−2.79492e7	−1.60855	−0.804273	+	0.594261i	$0.797445\pi$
$788$	0	0
$789$	9.09983e6	0.520404
$790$	0	0
$791$	−1.91764e7	−1.08974
$792$	0	0
$793$	−1.84913e6	−0.104420
$794$	0	0
$795$	3.25402e6	0.182600
$796$	0	0
$797$	−7.35189e6	−0.409971	−0.204986	−	0.978765i	$-0.565715\pi$
$797$	−7.35189e6	−0.409971	−0.204986	+	0.978765i	$0.565715\pi$
$798$	0	0
$799$	1.70137e7	0.942827
$800$	0	0
$801$	−5.65525e6	−0.311437
$802$	0	0
$803$	1.22055e7	0.667984
$804$	0	0
$805$	2.61924e6	0.142458
$806$	0	0
$807$	1.81001e6	0.0978356
$808$	0	0
$809$	1.42386e7	0.764885	0.382442	−	0.923979i	$-0.375083\pi$
$809$	1.42386e7	0.764885	0.382442	+	0.923979i	$0.375083\pi$
$810$	0	0
$811$	−8.88470e6	−0.474341	−0.237171	−	0.971468i	$-0.576220\pi$
$811$	−8.88470e6	−0.474341	−0.237171	+	0.971468i	$0.576220\pi$
$812$	0	0
$813$	−555188.	−0.0294587
$814$	0	0
$815$	−2.84811e7	−1.50197
$816$	0	0
$817$	4.49203e6	0.235444
$818$	0	0
$819$	−1.10707e6	−0.0576719
$820$	0	0
$821$	−2.65514e7	−1.37477	−0.687383	−	0.726295i	$-0.741241\pi$
$821$	−2.65514e7	−1.37477	−0.687383	+	0.726295i	$0.741241\pi$
$822$	0	0
$823$	4.07451e6	0.209689	0.104844	−	0.994489i	$-0.466566\pi$
$823$	4.07451e6	0.209689	0.104844	+	0.994489i	$0.466566\pi$
$824$	0	0
$825$	−307690.	−0.0157390
$826$	0	0
$827$	−4.00280e6	−0.203517	−0.101758	−	0.994809i	$-0.532447\pi$
$827$	−4.00280e6	−0.203517	−0.101758	+	0.994809i	$0.532447\pi$
$828$	0	0
$829$	−5.39078e6	−0.272436	−0.136218	−	0.990679i	$-0.543495\pi$
$829$	−5.39078e6	−0.272436	−0.136218	+	0.990679i	$0.543495\pi$
$830$	0	0
$831$	1.81321e7	0.910847
$832$	0	0
$833$	1.81586e7	0.906715
$834$	0	0
$835$	−4.71522e6	−0.234037
$836$	0	0
$837$	5.82733e6	0.287512
$838$	0	0
$839$	−3.51735e7	−1.72509	−0.862544	−	0.505982i	$-0.831130\pi$
$839$	−3.51735e7	−1.72509	−0.862544	+	0.505982i	$0.831130\pi$
$840$	0	0
$841$	−1.33801e7	−0.652332
$842$	0	0
$843$	−6.88015e6	−0.333449
$844$	0	0
$845$	1.92618e7	0.928013
$846$	0	0
$847$	1.42892e7	0.684383
$848$	0	0
$849$	−1.27778e7	−0.608395
$850$	0	0
$851$	−4.63682e6	−0.219481
$852$	0	0
$853$	−8.99577e6	−0.423317	−0.211658	−	0.977344i	$-0.567886\pi$
$853$	−8.99577e6	−0.423317	−0.211658	+	0.977344i	$0.567886\pi$
$854$	0	0
$855$	2.74029e6	0.128198
$856$	0	0
$857$	9.43061e6	0.438619	0.219310	−	0.975655i	$-0.429620\pi$
$857$	9.43061e6	0.438619	0.219310	+	0.975655i	$0.429620\pi$
$858$	0	0
$859$	2.38413e7	1.10242	0.551211	−	0.834366i	$-0.314166\pi$
$859$	2.38413e7	1.10242	0.551211	+	0.834366i	$0.314166\pi$
$860$	0	0
$861$	−3.38588e6	−0.155655
$862$	0	0
$863$	−1.73159e7	−0.791441	−0.395720	−	0.918371i	$-0.629505\pi$
$863$	−1.73159e7	−0.791441	−0.395720	+	0.918371i	$0.629505\pi$
$864$	0	0
$865$	5.93103e6	0.269519
$866$	0	0
$867$	−2.54772e7	−1.15108
$868$	0	0
$869$	−9.69291e6	−0.435416
$870$	0	0
$871$	−409006.	−0.0182677
$872$	0	0
$873$	−6.69930e6	−0.297505
$874$	0	0
$875$	1.57717e7	0.696399
$876$	0	0
$877$	2.80717e6	0.123245	0.0616226	−	0.998100i	$-0.480372\pi$
$877$	2.80717e6	0.123245	0.0616226	+	0.998100i	$0.480372\pi$
$878$	0	0
$879$	1.05509e7	0.460593
$880$	0	0
$881$	−1.47396e7	−0.639801	−0.319900	−	0.947451i	$-0.603649\pi$
$881$	−1.47396e7	−0.639801	−0.319900	+	0.947451i	$0.603649\pi$
$882$	0	0
$883$	9.09719e6	0.392650	0.196325	−	0.980539i	$-0.437099\pi$
$883$	9.09719e6	0.392650	0.196325	+	0.980539i	$0.437099\pi$
$884$	0	0
$885$	−2.62783e7	−1.12782
$886$	0	0
$887$	−1.51626e7	−0.647090	−0.323545	−	0.946213i	$-0.604875\pi$
$887$	−1.51626e7	−0.647090	−0.323545	+	0.946213i	$0.604875\pi$
$888$	0	0
$889$	2.17742e7	0.924035
$890$	0	0
$891$	−3.71618e6	−0.156821
$892$	0	0
$893$	5.04304e6	0.211623
$894$	0	0
$895$	−7.88462e6	−0.329021
$896$	0	0
$897$	−727540.	−0.0301909
$898$	0	0
$899$	−2.13462e7	−0.880888
$900$	0	0
$901$	−1.34653e7	−0.552591
$902$	0	0
$903$	5.91688e6	0.241476
$904$	0	0
$905$	3.85422e6	0.156428
$906$	0	0
$907$	1.46577e7	0.591626	0.295813	−	0.955246i	$-0.404409\pi$
$907$	1.46577e7	0.591626	0.295813	+	0.955246i	$0.404409\pi$
$908$	0	0
$909$	7.64337e6	0.306814
$910$	0	0
$911$	−7.26826e6	−0.290158	−0.145079	−	0.989420i	$-0.546344\pi$
$911$	−7.26826e6	−0.290158	−0.145079	+	0.989420i	$0.546344\pi$
$912$	0	0
$913$	2.48635e7	0.987156
$914$	0	0
$915$	6.02894e6	0.238061
$916$	0	0
$917$	−2.38131e7	−0.935173
$918$	0	0
$919$	1.32408e7	0.517159	0.258580	−	0.965990i	$-0.416746\pi$
$919$	1.32408e7	0.517159	0.258580	+	0.965990i	$0.416746\pi$
$920$	0	0
$921$	2.58404e7	1.00381
$922$	0	0
$923$	1.17354e7	0.453414
$924$	0	0
$925$	−529063.	−0.0203307
$926$	0	0
$927$	−1.43142e7	−0.547102
$928$	0	0
$929$	−3.21714e7	−1.22301	−0.611507	−	0.791239i	$-0.709436\pi$
$929$	−3.21714e7	−1.22301	−0.611507	+	0.791239i	$0.709436\pi$
$930$	0	0
$931$	5.38241e6	0.203518
$932$	0	0
$933$	2.50370e6	0.0941627
$934$	0	0
$935$	−6.46465e7	−2.41833
$936$	0	0
$937$	−4.34923e7	−1.61832	−0.809158	−	0.587592i	$-0.800076\pi$
$937$	−4.34923e7	−1.61832	−0.809158	+	0.587592i	$0.800076\pi$
$938$	0	0
$939$	3.40947e6	0.126189
$940$	0	0
$941$	−5.07902e7	−1.86985	−0.934924	−	0.354848i	$-0.884533\pi$
$941$	−5.07902e7	−1.86985	−0.934924	+	0.354848i	$0.884533\pi$
$942$	0	0
$943$	−2.22512e6	−0.0814845
$944$	0	0
$945$	3.60950e6	0.131482
$946$	0	0
$947$	1.46907e7	0.532315	0.266157	−	0.963930i	$-0.414246\pi$
$947$	1.46907e7	0.532315	0.266157	+	0.963930i	$0.414246\pi$
$948$	0	0
$949$	3.29296e6	0.118692
$950$	0	0
$951$	−2.22390e7	−0.797379
$952$	0	0
$953$	−1.76454e7	−0.629359	−0.314679	−	0.949198i	$-0.601897\pi$
$953$	−1.76454e7	−0.629359	−0.314679	+	0.949198i	$0.601897\pi$
$954$	0	0
$955$	1.66108e6	0.0589362
$956$	0	0
$957$	1.36128e7	0.480471
$958$	0	0
$959$	−1.06900e7	−0.375344
$960$	0	0
$961$	3.52685e7	1.23191
$962$	0	0
$963$	−3.71065e6	−0.128939
$964$	0	0
$965$	−4.74161e7	−1.63911
$966$	0	0
$967$	−2.58751e6	−0.0889849	−0.0444925	−	0.999010i	$-0.514167\pi$
$967$	−2.58751e6	−0.0889849	−0.0444925	+	0.999010i	$0.514167\pi$
$968$	0	0
$969$	−1.13395e7	−0.387957
$970$	0	0
$971$	1.90358e7	0.647922	0.323961	−	0.946070i	$-0.394985\pi$
$971$	1.90358e7	0.647922	0.323961	+	0.946070i	$0.394985\pi$
$972$	0	0
$973$	1.12324e7	0.380355
$974$	0	0
$975$	−83012.7	−0.00279662
$976$	0	0
$977$	−5.02122e7	−1.68296	−0.841478	−	0.540291i	$-0.818314\pi$
$977$	−5.02122e7	−1.68296	−0.841478	+	0.540291i	$0.818314\pi$
$978$	0	0
$979$	3.95452e7	1.31867
$980$	0	0
$981$	5.54165e6	0.183851
$982$	0	0
$983$	−1.84354e7	−0.608512	−0.304256	−	0.952590i	$-0.598408\pi$
$983$	−1.84354e7	−0.608512	−0.304256	+	0.952590i	$0.598408\pi$
$984$	0	0
$985$	−3.36839e7	−1.10620
$986$	0	0
$987$	6.64268e6	0.217045
$988$	0	0
$989$	3.88845e6	0.126411
$990$	0	0
$991$	−4.91898e7	−1.59108	−0.795539	−	0.605903i	$-0.792812\pi$
$991$	−4.91898e7	−1.59108	−0.795539	+	0.605903i	$0.792812\pi$
$992$	0	0
$993$	3.17809e7	1.02281
$994$	0	0
$995$	−2.87273e7	−0.919893
$996$	0	0
$997$	−3.68176e7	−1.17305	−0.586527	−	0.809930i	$-0.699505\pi$
$997$	−3.68176e7	−1.17305	−0.586527	+	0.809930i	$0.699505\pi$
$998$	0	0
$999$	−6.38987e6	−0.202571

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1104.6.a.s.1.2		6
4.3	odd	2		276.6.a.d.1.2	✓	6
12.11	even	2		828.6.a.e.1.5		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
276.6.a.d.1.2	✓	6	4.3	odd	2
828.6.a.e.1.5		6	12.11	even	2
1104.6.a.s.1.2		6	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:	\( N \)	\(=\)	\( 1104 = 2^{4} \cdot 3 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	1104.a (trivial)

\(f(q)\)	\(=\)	\(q-9.00000 q^{3} -55.3592 q^{5} +89.4396 q^{7} +81.0000 q^{9} +O(q^{10})\)	\(q-9.00000 q^{3} -55.3592 q^{5} +89.4396 q^{7} +81.0000 q^{9} -566.405 q^{11} -152.812 q^{13} +498.233 q^{15} -2061.71 q^{17} -611.113 q^{19} -804.957 q^{21} -529.000 q^{23} -60.3592 q^{25} -729.000 q^{27} +2670.41 q^{29} -7993.60 q^{31} +5097.64 q^{33} -4951.31 q^{35} +8765.25 q^{37} +1375.31 q^{39} +4206.28 q^{41} -7350.56 q^{43} -4484.10 q^{45} -8252.22 q^{47} -8807.55 q^{49} +18555.4 q^{51} +6531.12 q^{53} +31355.7 q^{55} +5500.02 q^{57} -52742.9 q^{59} +12100.7 q^{61} +7244.61 q^{63} +8459.57 q^{65} +2676.52 q^{67} +4761.00 q^{69} -76796.3 q^{71} -21549.0 q^{73} +543.233 q^{75} -50659.1 q^{77} +17113.0 q^{79} +6561.00 q^{81} -43897.1 q^{83} +114135. q^{85} -24033.7 q^{87} -69817.9 q^{89} -13667.5 q^{91} +71942.4 q^{93} +33830.8 q^{95} -82707.4 q^{97} -45878.8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 54 q^{3} + 50 q^{5} + 154 q^{7} + 486 q^{9}+O(q^{10}) \) 6 * q - 54 * q^3 + 50 * q^5 + 154 * q^7 + 486 * q^9	\( 6 q - 54 q^{3} + 50 q^{5} + 154 q^{7} + 486 q^{9} - 68 q^{11} + 1076 q^{13} - 450 q^{15} + 1650 q^{17} + 226 q^{19} - 1386 q^{21} - 3174 q^{23} + 14994 q^{25} - 4374 q^{27} + 12760 q^{29} - 10980 q^{31} + 612 q^{33} - 8492 q^{35} + 18012 q^{37} - 9684 q^{39} + 15396 q^{41} + 7146 q^{43} + 4050 q^{45} - 23588 q^{47} + 56182 q^{49} - 14850 q^{51} + 9310 q^{53} - 52048 q^{55} - 2034 q^{57} - 5260 q^{59} + 16932 q^{61} + 12474 q^{63} + 30852 q^{65} - 24858 q^{67} + 28566 q^{69} - 101832 q^{71} + 33100 q^{73} - 134946 q^{75} + 182232 q^{77} - 100570 q^{79} + 39366 q^{81} - 222524 q^{83} + 23824 q^{85} - 114840 q^{87} + 28374 q^{89} - 18204 q^{91} + 98820 q^{93} - 481996 q^{95} + 47632 q^{97} - 5508 q^{99}+O(q^{100}) \) 6 * q - 54 * q^3 + 50 * q^5 + 154 * q^7 + 486 * q^9 - 68 * q^11 + 1076 * q^13 - 450 * q^15 + 1650 * q^17 + 226 * q^19 - 1386 * q^21 - 3174 * q^23 + 14994 * q^25 - 4374 * q^27 + 12760 * q^29 - 10980 * q^31 + 612 * q^33 - 8492 * q^35 + 18012 * q^37 - 9684 * q^39 + 15396 * q^41 + 7146 * q^43 + 4050 * q^45 - 23588 * q^47 + 56182 * q^49 - 14850 * q^51 + 9310 * q^53 - 52048 * q^55 - 2034 * q^57 - 5260 * q^59 + 16932 * q^61 + 12474 * q^63 + 30852 * q^65 - 24858 * q^67 + 28566 * q^69 - 101832 * q^71 + 33100 * q^73 - 134946 * q^75 + 182232 * q^77 - 100570 * q^79 + 39366 * q^81 - 222524 * q^83 + 23824 * q^85 - 114840 * q^87 + 28374 * q^89 - 18204 * q^91 + 98820 * q^93 - 481996 * q^95 + 47632 * q^97 - 5508 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more