LMFDB - Embedded newform 1672.2.a.i.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1672,2,Mod(1,1672)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1672, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1672.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1672 = 2^{3} \cdot 11 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1672.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:	$13.3509872180$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	6.6.106392688.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 9x^{4} + 12x^{3} + 25x^{2} - 10x - 16 $ x^6 - 2x^5 - 9x^4 + 12x^3 + 25x^2 - 10*x - 16
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.25613$ of defining polynomial
Character	$\chi$	$=$	1672.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.25613 q^{3} -4.20604 q^{5} +3.20604 q^{7} +2.09011 q^{9} +O(q^{10})$	$q-2.25613 q^{3} -4.20604 q^{5} +3.20604 q^{7} +2.09011 q^{9} -1.00000 q^{11} +0.723884 q^{13} +9.48935 q^{15} +0.655737 q^{17} +1.00000 q^{19} -7.23323 q^{21} -2.68543 q^{23} +12.6907 q^{25} +2.05282 q^{27} -2.64351 q^{29} +9.48935 q^{31} +2.25613 q^{33} -13.4847 q^{35} +4.20992 q^{37} -1.63317 q^{39} +5.04141 q^{41} -10.7480 q^{43} -8.79110 q^{45} -4.84428 q^{47} +3.27866 q^{49} -1.47943 q^{51} +2.80135 q^{53} +4.20604 q^{55} -2.25613 q^{57} -8.16817 q^{59} +1.66797 q^{61} +6.70098 q^{63} -3.04468 q^{65} -1.57845 q^{67} +6.05867 q^{69} -10.3445 q^{71} -3.35322 q^{73} -28.6319 q^{75} -3.20604 q^{77} -17.4366 q^{79} -10.9018 q^{81} +4.13518 q^{83} -2.75805 q^{85} +5.96409 q^{87} -15.8240 q^{89} +2.32080 q^{91} -21.4092 q^{93} -4.20604 q^{95} +5.57301 q^{97} -2.09011 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 4 q^{3} - 3 q^{5} - 3 q^{7} + 6 q^{9}+O(q^{10}) $ 6 * q - 4 * q^3 - 3 * q^5 - 3 * q^7 + 6 * q^9	$ 6 q - 4 q^{3} - 3 q^{5} - 3 q^{7} + 6 q^{9} - 6 q^{11} - 3 q^{13} - q^{15} - q^{17} + 6 q^{19} + 5 q^{21} - 4 q^{23} + 5 q^{25} - 16 q^{27} - 7 q^{29} - q^{31} + 4 q^{33} - 32 q^{35} + 5 q^{37} - 13 q^{39} - 6 q^{41} - 16 q^{43} - 4 q^{47} - 7 q^{49} - 35 q^{51} - 11 q^{53} + 3 q^{55} - 4 q^{57} - 18 q^{59} + 16 q^{61} - 6 q^{63} + 10 q^{65} - 18 q^{67} - 2 q^{69} - 7 q^{71} - 21 q^{73} - 23 q^{75} + 3 q^{77} - 22 q^{79} - 10 q^{81} - 16 q^{83} - 3 q^{87} - 13 q^{89} - 7 q^{91} - 9 q^{93} - 3 q^{95} - 15 q^{97} - 6 q^{99}+O(q^{100}) $ 6 * q - 4 * q^3 - 3 * q^5 - 3 * q^7 + 6 * q^9 - 6 * q^11 - 3 * q^13 - q^15 - q^17 + 6 * q^19 + 5 * q^21 - 4 * q^23 + 5 * q^25 - 16 * q^27 - 7 * q^29 - q^31 + 4 * q^33 - 32 * q^35 + 5 * q^37 - 13 * q^39 - 6 * q^41 - 16 * q^43 - 4 * q^47 - 7 * q^49 - 35 * q^51 - 11 * q^53 + 3 * q^55 - 4 * q^57 - 18 * q^59 + 16 * q^61 - 6 * q^63 + 10 * q^65 - 18 * q^67 - 2 * q^69 - 7 * q^71 - 21 * q^73 - 23 * q^75 + 3 * q^77 - 22 * q^79 - 10 * q^81 - 16 * q^83 - 3 * q^87 - 13 * q^89 - 7 * q^91 - 9 * q^93 - 3 * q^95 - 15 * q^97 - 6 * 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
$3$	−2.25613	−1.30258	−0.651288	−	0.758831i	$-0.725771\pi$
$3$	−2.25613	−1.30258	−0.651288	+	0.758831i	$0.725771\pi$
$4$	0	0
$5$	−4.20604	−1.88100	−0.940498	−	0.339799i	$-0.889641\pi$
$5$	−4.20604	−1.88100	−0.940498	+	0.339799i	$0.889641\pi$
$6$	0	0
$7$	3.20604	1.21177	0.605884	−	0.795553i	$-0.292820\pi$
$7$	3.20604	1.21177	0.605884	+	0.795553i	$0.292820\pi$
$8$	0	0
$9$	2.09011	0.696705
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	0.723884	0.200769	0.100385	−	0.994949i	$-0.467993\pi$
$13$	0.723884	0.200769	0.100385	+	0.994949i	$0.467993\pi$
$14$	0	0
$15$	9.48935	2.45014
$16$	0	0
$17$	0.655737	0.159040	0.0795198	−	0.996833i	$-0.474661\pi$
$17$	0.655737	0.159040	0.0795198	+	0.996833i	$0.474661\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	−7.23323	−1.57842
$22$	0	0
$23$	−2.68543	−0.559950	−0.279975	−	0.960007i	$-0.590326\pi$
$23$	−2.68543	−0.559950	−0.279975	+	0.960007i	$0.590326\pi$
$24$	0	0
$25$	12.6907	2.53815
$26$	0	0
$27$	2.05282	0.395065
$28$	0	0
$29$	−2.64351	−0.490887	−0.245443	−	0.969411i	$-0.578934\pi$
$29$	−2.64351	−0.490887	−0.245443	+	0.969411i	$0.578934\pi$
$30$	0	0
$31$	9.48935	1.70434	0.852169	−	0.523266i	$-0.175287\pi$
$31$	9.48935	1.70434	0.852169	+	0.523266i	$0.175287\pi$
$32$	0	0
$33$	2.25613	0.392742
$34$	0	0
$35$	−13.4847	−2.27933
$36$	0	0
$37$	4.20992	0.692107	0.346053	−	0.938215i	$-0.387522\pi$
$37$	4.20992	0.692107	0.346053	+	0.938215i	$0.387522\pi$
$38$	0	0
$39$	−1.63317	−0.261517
$40$	0	0
$41$	5.04141	0.787336	0.393668	−	0.919253i	$-0.371206\pi$
$41$	5.04141	0.787336	0.393668	+	0.919253i	$0.371206\pi$
$42$	0	0
$43$	−10.7480	−1.63905	−0.819526	−	0.573042i	$-0.805763\pi$
$43$	−10.7480	−1.63905	−0.819526	+	0.573042i	$0.805763\pi$
$44$	0	0
$45$	−8.79110	−1.31050
$46$	0	0
$47$	−4.84428	−0.706611	−0.353306	−	0.935508i	$-0.614942\pi$
$47$	−4.84428	−0.706611	−0.353306	+	0.935508i	$0.614942\pi$
$48$	0	0
$49$	3.27866	0.468380
$50$	0	0
$51$	−1.47943	−0.207161
$52$	0	0
$53$	2.80135	0.384795	0.192397	−	0.981317i	$-0.438374\pi$
$53$	2.80135	0.384795	0.192397	+	0.981317i	$0.438374\pi$
$54$	0	0
$55$	4.20604	0.567142
$56$	0	0
$57$	−2.25613	−0.298831
$58$	0	0
$59$	−8.16817	−1.06341	−0.531703	−	0.846931i	$-0.678448\pi$
$59$	−8.16817	−1.06341	−0.531703	+	0.846931i	$0.678448\pi$
$60$	0	0
$61$	1.66797	0.213562	0.106781	−	0.994283i	$-0.465946\pi$
$61$	1.66797	0.213562	0.106781	+	0.994283i	$0.465946\pi$
$62$	0	0
$63$	6.70098	0.844244
$64$	0	0
$65$	−3.04468	−0.377646
$66$	0	0
$67$	−1.57845	−0.192839	−0.0964193	−	0.995341i	$-0.530739\pi$
$67$	−1.57845	−0.192839	−0.0964193	+	0.995341i	$0.530739\pi$
$68$	0	0
$69$	6.05867	0.729378
$70$	0	0
$71$	−10.3445	−1.22766	−0.613832	−	0.789436i	$-0.710373\pi$
$71$	−10.3445	−1.22766	−0.613832	+	0.789436i	$0.710373\pi$
$72$	0	0
$73$	−3.35322	−0.392465	−0.196232	−	0.980557i	$-0.562871\pi$
$73$	−3.35322	−0.392465	−0.196232	+	0.980557i	$0.562871\pi$
$74$	0	0
$75$	−28.6319	−3.30613
$76$	0	0
$77$	−3.20604	−0.365362
$78$	0	0
$79$	−17.4366	−1.96177	−0.980884	−	0.194591i	$-0.937662\pi$
$79$	−17.4366	−1.96177	−0.980884	+	0.194591i	$0.937662\pi$
$80$	0	0
$81$	−10.9018	−1.21131
$82$	0	0
$83$	4.13518	0.453895	0.226948	−	0.973907i	$-0.427125\pi$
$83$	4.13518	0.453895	0.226948	+	0.973907i	$0.427125\pi$
$84$	0	0
$85$	−2.75805	−0.299153
$86$	0	0
$87$	5.96409	0.639417
$88$	0	0
$89$	−15.8240	−1.67734	−0.838668	−	0.544643i	$-0.816665\pi$
$89$	−15.8240	−1.67734	−0.838668	+	0.544643i	$0.816665\pi$
$90$	0	0
$91$	2.32080	0.243286
$92$	0	0
$93$	−21.4092	−2.22003
$94$	0	0
$95$	−4.20604	−0.431530
$96$	0	0
$97$	5.57301	0.565853	0.282926	−	0.959142i	$-0.408695\pi$
$97$	5.57301	0.565853	0.282926	+	0.959142i	$0.408695\pi$
$98$	0	0
$99$	−2.09011	−0.210064
$100$	0	0
$101$	7.98882	0.794917	0.397458	−	0.917620i	$-0.369892\pi$
$101$	7.98882	0.794917	0.397458	+	0.917620i	$0.369892\pi$
$102$	0	0
$103$	−14.7152	−1.44993	−0.724964	−	0.688787i	$-0.758144\pi$
$103$	−14.7152	−1.44993	−0.724964	+	0.688787i	$0.758144\pi$
$104$	0	0
$105$	30.4232	2.96900
$106$	0	0
$107$	−1.11242	−0.107542	−0.0537710	−	0.998553i	$-0.517124\pi$
$107$	−1.11242	−0.107542	−0.0537710	+	0.998553i	$0.517124\pi$
$108$	0	0
$109$	−8.03175	−0.769302	−0.384651	−	0.923062i	$-0.625678\pi$
$109$	−8.03175	−0.769302	−0.384651	+	0.923062i	$0.625678\pi$
$110$	0	0
$111$	−9.49812	−0.901522
$112$	0	0
$113$	10.1703	0.956742	0.478371	−	0.878158i	$-0.341227\pi$
$113$	10.1703	0.956742	0.478371	+	0.878158i	$0.341227\pi$
$114$	0	0
$115$	11.2950	1.05326
$116$	0	0
$117$	1.51300	0.139877
$118$	0	0
$119$	2.10232	0.192719
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−11.3741	−1.02556
$124$	0	0
$125$	−32.3475	−2.89325
$126$	0	0
$127$	−11.6874	−1.03709	−0.518544	−	0.855051i	$-0.673526\pi$
$127$	−11.6874	−1.03709	−0.518544	+	0.855051i	$0.673526\pi$
$128$	0	0
$129$	24.2488	2.13499
$130$	0	0
$131$	15.2777	1.33482	0.667409	−	0.744691i	$-0.267403\pi$
$131$	15.2777	1.33482	0.667409	+	0.744691i	$0.267403\pi$
$132$	0	0
$133$	3.20604	0.277999
$134$	0	0
$135$	−8.63422	−0.743115
$136$	0	0
$137$	−12.8730	−1.09982	−0.549909	−	0.835225i	$-0.685338\pi$
$137$	−12.8730	−1.09982	−0.549909	+	0.835225i	$0.685338\pi$
$138$	0	0
$139$	−13.1802	−1.11793	−0.558965	−	0.829191i	$-0.688801\pi$
$139$	−13.1802	−1.11793	−0.558965	+	0.829191i	$0.688801\pi$
$140$	0	0
$141$	10.9293	0.920415
$142$	0	0
$143$	−0.723884	−0.0605342
$144$	0	0
$145$	11.1187	0.923356
$146$	0	0
$147$	−7.39708	−0.610101
$148$	0	0
$149$	16.5422	1.35519	0.677594	−	0.735436i	$-0.263023\pi$
$149$	16.5422	1.35519	0.677594	+	0.735436i	$0.263023\pi$
$150$	0	0
$151$	16.0851	1.30899	0.654495	−	0.756067i	$-0.272881\pi$
$151$	16.0851	1.30899	0.654495	+	0.756067i	$0.272881\pi$
$152$	0	0
$153$	1.37057	0.110804
$154$	0	0
$155$	−39.9126	−3.20585
$156$	0	0
$157$	17.3684	1.38615	0.693074	−	0.720866i	$-0.256256\pi$
$157$	17.3684	1.38615	0.693074	+	0.720866i	$0.256256\pi$
$158$	0	0
$159$	−6.32020	−0.501224
$160$	0	0
$161$	−8.60958	−0.678530
$162$	0	0
$163$	−18.3725	−1.43904	−0.719521	−	0.694471i	$-0.755638\pi$
$163$	−18.3725	−1.43904	−0.719521	+	0.694471i	$0.755638\pi$
$164$	0	0
$165$	−9.48935	−0.738745
$166$	0	0
$167$	−2.62676	−0.203265	−0.101632	−	0.994822i	$-0.532407\pi$
$167$	−2.62676	−0.203265	−0.101632	+	0.994822i	$0.532407\pi$
$168$	0	0
$169$	−12.4760	−0.959692
$170$	0	0
$171$	2.09011	0.159835
$172$	0	0
$173$	−6.23475	−0.474020	−0.237010	−	0.971507i	$-0.576167\pi$
$173$	−6.23475	−0.474020	−0.237010	+	0.971507i	$0.576167\pi$
$174$	0	0
$175$	40.6869	3.07564
$176$	0	0
$177$	18.4284	1.38517
$178$	0	0
$179$	4.29357	0.320917	0.160458	−	0.987043i	$-0.448703\pi$
$179$	4.29357	0.320917	0.160458	+	0.987043i	$0.448703\pi$
$180$	0	0
$181$	4.52254	0.336158	0.168079	−	0.985774i	$-0.446244\pi$
$181$	4.52254	0.336158	0.168079	+	0.985774i	$0.446244\pi$
$182$	0	0
$183$	−3.76316	−0.278181
$184$	0	0
$185$	−17.7071	−1.30185
$186$	0	0
$187$	−0.655737	−0.0479523
$188$	0	0
$189$	6.58140	0.478727
$190$	0	0
$191$	25.0370	1.81162	0.905808	−	0.423689i	$-0.139265\pi$
$191$	25.0370	1.81162	0.905808	+	0.423689i	$0.139265\pi$
$192$	0	0
$193$	4.60176	0.331242	0.165621	−	0.986189i	$-0.447037\pi$
$193$	4.60176	0.331242	0.165621	+	0.986189i	$0.447037\pi$
$194$	0	0
$195$	6.86919	0.491913
$196$	0	0
$197$	5.51765	0.393117	0.196558	−	0.980492i	$-0.437024\pi$
$197$	5.51765	0.393117	0.196558	+	0.980492i	$0.437024\pi$
$198$	0	0
$199$	−9.84534	−0.697917	−0.348959	−	0.937138i	$-0.613465\pi$
$199$	−9.84534	−0.697917	−0.348959	+	0.937138i	$0.613465\pi$
$200$	0	0
$201$	3.56119	0.251187
$202$	0	0
$203$	−8.47517	−0.594841
$204$	0	0
$205$	−21.2043	−1.48098
$206$	0	0
$207$	−5.61285	−0.390120
$208$	0	0
$209$	−1.00000	−0.0691714
$210$	0	0
$211$	−20.3536	−1.40120	−0.700601	−	0.713553i	$-0.747085\pi$
$211$	−20.3536	−1.40120	−0.700601	+	0.713553i	$0.747085\pi$
$212$	0	0
$213$	23.3385	1.59913
$214$	0	0
$215$	45.2064	3.08305
$216$	0	0
$217$	30.4232	2.06526
$218$	0	0
$219$	7.56530	0.511215
$220$	0	0
$221$	0.474677	0.0319303
$222$	0	0
$223$	5.69489	0.381358	0.190679	−	0.981652i	$-0.438931\pi$
$223$	5.69489	0.381358	0.190679	+	0.981652i	$0.438931\pi$
$224$	0	0
$225$	26.5251	1.76834
$226$	0	0
$227$	−16.0945	−1.06823	−0.534116	−	0.845411i	$-0.679356\pi$
$227$	−16.0945	−1.06823	−0.534116	+	0.845411i	$0.679356\pi$
$228$	0	0
$229$	10.1886	0.673280	0.336640	−	0.941633i	$-0.390709\pi$
$229$	10.1886	0.673280	0.336640	+	0.941633i	$0.390709\pi$
$230$	0	0
$231$	7.23323	0.475911
$232$	0	0
$233$	16.5112	1.08168	0.540841	−	0.841125i	$-0.318106\pi$
$233$	16.5112	1.08168	0.540841	+	0.841125i	$0.318106\pi$
$234$	0	0
$235$	20.3752	1.32913
$236$	0	0
$237$	39.3392	2.55535
$238$	0	0
$239$	−28.1261	−1.81933	−0.909664	−	0.415345i	$-0.863661\pi$
$239$	−28.1261	−1.81933	−0.909664	+	0.415345i	$0.863661\pi$
$240$	0	0
$241$	18.8092	1.21160	0.605802	−	0.795615i	$-0.292852\pi$
$241$	18.8092	1.21160	0.605802	+	0.795615i	$0.292852\pi$
$242$	0	0
$243$	18.4373	1.18276
$244$	0	0
$245$	−13.7902	−0.881021
$246$	0	0
$247$	0.723884	0.0460596
$248$	0	0
$249$	−9.32950	−0.591233
$250$	0	0
$251$	19.2974	1.21804	0.609020	−	0.793155i	$-0.291563\pi$
$251$	19.2974	1.21804	0.609020	+	0.793155i	$0.291563\pi$
$252$	0	0
$253$	2.68543	0.168831
$254$	0	0
$255$	6.22252	0.389669
$256$	0	0
$257$	−21.9480	−1.36908	−0.684540	−	0.728976i	$-0.739997\pi$
$257$	−21.9480	−1.36908	−0.684540	+	0.728976i	$0.739997\pi$
$258$	0	0
$259$	13.4972	0.838672
$260$	0	0
$261$	−5.52523	−0.342003
$262$	0	0
$263$	−9.31030	−0.574098	−0.287049	−	0.957916i	$-0.592674\pi$
$263$	−9.31030	−0.574098	−0.287049	+	0.957916i	$0.592674\pi$
$264$	0	0
$265$	−11.7826	−0.723797
$266$	0	0
$267$	35.7009	2.18486
$268$	0	0
$269$	−18.1406	−1.10605	−0.553026	−	0.833164i	$-0.686527\pi$
$269$	−18.1406	−1.10605	−0.553026	+	0.833164i	$0.686527\pi$
$270$	0	0
$271$	25.8372	1.56950	0.784750	−	0.619813i	$-0.212792\pi$
$271$	25.8372	1.56950	0.784750	+	0.619813i	$0.212792\pi$
$272$	0	0
$273$	−5.23602	−0.316898
$274$	0	0
$275$	−12.6907	−0.765280
$276$	0	0
$277$	−14.9438	−0.897886	−0.448943	−	0.893560i	$-0.648199\pi$
$277$	−14.9438	−0.897886	−0.448943	+	0.893560i	$0.648199\pi$
$278$	0	0
$279$	19.8338	1.18742
$280$	0	0
$281$	−24.4663	−1.45954	−0.729769	−	0.683694i	$-0.760372\pi$
$281$	−24.4663	−1.45954	−0.729769	+	0.683694i	$0.760372\pi$
$282$	0	0
$283$	8.68159	0.516067	0.258034	−	0.966136i	$-0.416925\pi$
$283$	8.68159	0.516067	0.258034	+	0.966136i	$0.416925\pi$
$284$	0	0
$285$	9.48935	0.562101
$286$	0	0
$287$	16.1629	0.954068
$288$	0	0
$289$	−16.5700	−0.974706
$290$	0	0
$291$	−12.5734	−0.737067
$292$	0	0
$293$	−18.2517	−1.06628	−0.533138	−	0.846028i	$-0.678987\pi$
$293$	−18.2517	−1.06628	−0.533138	+	0.846028i	$0.678987\pi$
$294$	0	0
$295$	34.3556	2.00026
$296$	0	0
$297$	−2.05282	−0.119117
$298$	0	0
$299$	−1.94394	−0.112421
$300$	0	0
$301$	−34.4584	−1.98615
$302$	0	0
$303$	−18.0238	−1.03544
$304$	0	0
$305$	−7.01555	−0.401709
$306$	0	0
$307$	5.22644	0.298289	0.149144	−	0.988815i	$-0.452348\pi$
$307$	5.22644	0.298289	0.149144	+	0.988815i	$0.452348\pi$
$308$	0	0
$309$	33.1993	1.88864
$310$	0	0
$311$	−5.53926	−0.314103	−0.157051	−	0.987590i	$-0.550199\pi$
$311$	−5.53926	−0.314103	−0.157051	+	0.987590i	$0.550199\pi$
$312$	0	0
$313$	−12.7911	−0.722998	−0.361499	−	0.932372i	$-0.617735\pi$
$313$	−12.7911	−0.722998	−0.361499	+	0.932372i	$0.617735\pi$
$314$	0	0
$315$	−28.1846	−1.58802
$316$	0	0
$317$	−29.2257	−1.64148	−0.820740	−	0.571301i	$-0.806439\pi$
$317$	−29.2257	−1.64148	−0.820740	+	0.571301i	$0.806439\pi$
$318$	0	0
$319$	2.64351	0.148008
$320$	0	0
$321$	2.50977	0.140082
$322$	0	0
$323$	0.655737	0.0364862
$324$	0	0
$325$	9.18661	0.509582
$326$	0	0
$327$	18.1207	1.00207
$328$	0	0
$329$	−15.5309	−0.856249
$330$	0	0
$331$	−15.8740	−0.872513	−0.436256	−	0.899822i	$-0.643696\pi$
$331$	−15.8740	−0.872513	−0.436256	+	0.899822i	$0.643696\pi$
$332$	0	0
$333$	8.79922	0.482194
$334$	0	0
$335$	6.63903	0.362729
$336$	0	0
$337$	2.93726	0.160003	0.0800014	−	0.996795i	$-0.474508\pi$
$337$	2.93726	0.160003	0.0800014	+	0.996795i	$0.474508\pi$
$338$	0	0
$339$	−22.9455	−1.24623
$340$	0	0
$341$	−9.48935	−0.513877
$342$	0	0
$343$	−11.9307	−0.644200
$344$	0	0
$345$	−25.4830	−1.37196
$346$	0	0
$347$	−8.74992	−0.469721	−0.234860	−	0.972029i	$-0.575463\pi$
$347$	−8.74992	−0.469721	−0.234860	+	0.972029i	$0.575463\pi$
$348$	0	0
$349$	−18.4164	−0.985807	−0.492904	−	0.870084i	$-0.664064\pi$
$349$	−18.4164	−0.985807	−0.492904	+	0.870084i	$0.664064\pi$
$350$	0	0
$351$	1.48600	0.0793169
$352$	0	0
$353$	−19.9919	−1.06406	−0.532031	−	0.846725i	$-0.678571\pi$
$353$	−19.9919	−1.06406	−0.532031	+	0.846725i	$0.678571\pi$
$354$	0	0
$355$	43.5093	2.30923
$356$	0	0
$357$	−4.74310	−0.251031
$358$	0	0
$359$	3.29606	0.173960	0.0869798	−	0.996210i	$-0.472278\pi$
$359$	3.29606	0.173960	0.0869798	+	0.996210i	$0.472278\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−2.25613	−0.118416
$364$	0	0
$365$	14.1038	0.738225
$366$	0	0
$367$	−25.2683	−1.31900	−0.659498	−	0.751707i	$-0.729231\pi$
$367$	−25.2683	−1.31900	−0.659498	+	0.751707i	$0.729231\pi$
$368$	0	0
$369$	10.5371	0.548541
$370$	0	0
$371$	8.98122	0.466282
$372$	0	0
$373$	−5.25095	−0.271884	−0.135942	−	0.990717i	$-0.543406\pi$
$373$	−5.25095	−0.271884	−0.135942	+	0.990717i	$0.543406\pi$
$374$	0	0
$375$	72.9801	3.76868
$376$	0	0
$377$	−1.91359	−0.0985550
$378$	0	0
$379$	−24.3964	−1.25316	−0.626580	−	0.779357i	$-0.715546\pi$
$379$	−24.3964	−1.25316	−0.626580	+	0.779357i	$0.715546\pi$
$380$	0	0
$381$	26.3682	1.35089
$382$	0	0
$383$	29.5585	1.51037	0.755184	−	0.655512i	$-0.227547\pi$
$383$	29.5585	1.51037	0.755184	+	0.655512i	$0.227547\pi$
$384$	0	0
$385$	13.4847	0.687244
$386$	0	0
$387$	−22.4645	−1.14194
$388$	0	0
$389$	33.0958	1.67803	0.839013	−	0.544112i	$-0.183133\pi$
$389$	33.0958	1.67803	0.839013	+	0.544112i	$0.183133\pi$
$390$	0	0
$391$	−1.76093	−0.0890543
$392$	0	0
$393$	−34.4684	−1.73870
$394$	0	0
$395$	73.3389	3.69008
$396$	0	0
$397$	−1.55966	−0.0782769	−0.0391385	−	0.999234i	$-0.512461\pi$
$397$	−1.55966	−0.0782769	−0.0391385	+	0.999234i	$0.512461\pi$
$398$	0	0
$399$	−7.23323	−0.362114
$400$	0	0
$401$	17.9881	0.898282	0.449141	−	0.893461i	$-0.351730\pi$
$401$	17.9881	0.898282	0.449141	+	0.893461i	$0.351730\pi$
$402$	0	0
$403$	6.86919	0.342179
$404$	0	0
$405$	45.8532	2.27846
$406$	0	0
$407$	−4.20992	−0.208678
$408$	0	0
$409$	31.8247	1.57363	0.786816	−	0.617188i	$-0.211728\pi$
$409$	31.8247	1.57363	0.786816	+	0.617188i	$0.211728\pi$
$410$	0	0
$411$	29.0432	1.43260
$412$	0	0
$413$	−26.1874	−1.28860
$414$	0	0
$415$	−17.3927	−0.853775
$416$	0	0
$417$	29.7362	1.45619
$418$	0	0
$419$	−22.0175	−1.07563	−0.537814	−	0.843064i	$-0.680750\pi$
$419$	−22.0175	−1.07563	−0.537814	+	0.843064i	$0.680750\pi$
$420$	0	0
$421$	−17.5466	−0.855171	−0.427585	−	0.903975i	$-0.640636\pi$
$421$	−17.5466	−0.855171	−0.427585	+	0.903975i	$0.640636\pi$
$422$	0	0
$423$	−10.1251	−0.492300
$424$	0	0
$425$	8.32178	0.403666
$426$	0	0
$427$	5.34758	0.258788
$428$	0	0
$429$	1.63317	0.0788504
$430$	0	0
$431$	0.854799	0.0411742	0.0205871	−	0.999788i	$-0.493446\pi$
$431$	0.854799	0.0411742	0.0205871	+	0.999788i	$0.493446\pi$
$432$	0	0
$433$	−17.0259	−0.818214	−0.409107	−	0.912486i	$-0.634160\pi$
$433$	−17.0259	−0.818214	−0.409107	+	0.912486i	$0.634160\pi$
$434$	0	0
$435$	−25.0852	−1.20274
$436$	0	0
$437$	−2.68543	−0.128461
$438$	0	0
$439$	2.81936	0.134561	0.0672804	−	0.997734i	$-0.478568\pi$
$439$	2.81936	0.134561	0.0672804	+	0.997734i	$0.478568\pi$
$440$	0	0
$441$	6.85278	0.326323
$442$	0	0
$443$	−19.4356	−0.923414	−0.461707	−	0.887033i	$-0.652763\pi$
$443$	−19.4356	−0.923414	−0.461707	+	0.887033i	$0.652763\pi$
$444$	0	0
$445$	66.5561	3.15506
$446$	0	0
$447$	−37.3213	−1.76524
$448$	0	0
$449$	13.7032	0.646693	0.323347	−	0.946281i	$-0.395192\pi$
$449$	13.7032	0.646693	0.323347	+	0.946281i	$0.395192\pi$
$450$	0	0
$451$	−5.04141	−0.237391
$452$	0	0
$453$	−36.2901	−1.70506
$454$	0	0
$455$	−9.76135	−0.457619
$456$	0	0
$457$	−11.2586	−0.526653	−0.263326	−	0.964707i	$-0.584820\pi$
$457$	−11.2586	−0.526653	−0.263326	+	0.964707i	$0.584820\pi$
$458$	0	0
$459$	1.34611	0.0628310
$460$	0	0
$461$	17.5115	0.815594	0.407797	−	0.913073i	$-0.366297\pi$
$461$	17.5115	0.815594	0.407797	+	0.913073i	$0.366297\pi$
$462$	0	0
$463$	−20.4636	−0.951022	−0.475511	−	0.879710i	$-0.657737\pi$
$463$	−20.4636	−0.951022	−0.475511	+	0.879710i	$0.657737\pi$
$464$	0	0
$465$	90.0479	4.17587
$466$	0	0
$467$	−11.1897	−0.517796	−0.258898	−	0.965905i	$-0.583359\pi$
$467$	−11.1897	−0.517796	−0.258898	+	0.965905i	$0.583359\pi$
$468$	0	0
$469$	−5.06057	−0.233676
$470$	0	0
$471$	−39.1853	−1.80556
$472$	0	0
$473$	10.7480	0.494193
$474$	0	0
$475$	12.6907	0.582291
$476$	0	0
$477$	5.85514	0.268088
$478$	0	0
$479$	38.8664	1.77585	0.887926	−	0.459986i	$-0.152145\pi$
$479$	38.8664	1.77585	0.887926	+	0.459986i	$0.152145\pi$
$480$	0	0
$481$	3.04749	0.138954
$482$	0	0
$483$	19.4243	0.883837
$484$	0	0
$485$	−23.4403	−1.06437
$486$	0	0
$487$	−23.6592	−1.07210	−0.536050	−	0.844186i	$-0.680084\pi$
$487$	−23.6592	−1.07210	−0.536050	+	0.844186i	$0.680084\pi$
$488$	0	0
$489$	41.4506	1.87446
$490$	0	0
$491$	5.74101	0.259088	0.129544	−	0.991574i	$-0.458649\pi$
$491$	5.74101	0.259088	0.129544	+	0.991574i	$0.458649\pi$
$492$	0	0
$493$	−1.73345	−0.0780704
$494$	0	0
$495$	8.79110	0.395130
$496$	0	0
$497$	−33.1648	−1.48764
$498$	0	0
$499$	−4.85986	−0.217557	−0.108779	−	0.994066i	$-0.534694\pi$
$499$	−4.85986	−0.217557	−0.108779	+	0.994066i	$0.534694\pi$
$500$	0	0
$501$	5.92630	0.264768
$502$	0	0
$503$	5.08324	0.226650	0.113325	−	0.993558i	$-0.463850\pi$
$503$	5.08324	0.226650	0.113325	+	0.993558i	$0.463850\pi$
$504$	0	0
$505$	−33.6012	−1.49524
$506$	0	0
$507$	28.1474	1.25007
$508$	0	0
$509$	35.2272	1.56142	0.780708	−	0.624896i	$-0.214859\pi$
$509$	35.2272	1.56142	0.780708	+	0.624896i	$0.214859\pi$
$510$	0	0
$511$	−10.7505	−0.475576
$512$	0	0
$513$	2.05282	0.0906341
$514$	0	0
$515$	61.8925	2.72731
$516$	0	0
$517$	4.84428	0.213051
$518$	0	0
$519$	14.0664	0.617447
$520$	0	0
$521$	−13.2922	−0.582344	−0.291172	−	0.956671i	$-0.594045\pi$
$521$	−13.2922	−0.582344	−0.291172	+	0.956671i	$0.594045\pi$
$522$	0	0
$523$	−29.0749	−1.27136	−0.635679	−	0.771954i	$-0.719280\pi$
$523$	−29.0749	−1.27136	−0.635679	+	0.771954i	$0.719280\pi$
$524$	0	0
$525$	−91.7949	−4.00626
$526$	0	0
$527$	6.22252	0.271057
$528$	0	0
$529$	−15.7885	−0.686456
$530$	0	0
$531$	−17.0724	−0.740880
$532$	0	0
$533$	3.64939	0.158073
$534$	0	0
$535$	4.67889	0.202286
$536$	0	0
$537$	−9.68685	−0.418018
$538$	0	0
$539$	−3.27866	−0.141222
$540$	0	0
$541$	25.8688	1.11219	0.556093	−	0.831120i	$-0.312300\pi$
$541$	25.8688	1.11219	0.556093	+	0.831120i	$0.312300\pi$
$542$	0	0
$543$	−10.2034	−0.437871
$544$	0	0
$545$	33.7818	1.44705
$546$	0	0
$547$	10.5966	0.453076	0.226538	−	0.974002i	$-0.427259\pi$
$547$	10.5966	0.453076	0.226538	+	0.974002i	$0.427259\pi$
$548$	0	0
$549$	3.48626	0.148790
$550$	0	0
$551$	−2.64351	−0.112617
$552$	0	0
$553$	−55.9023	−2.37721
$554$	0	0
$555$	39.9494	1.69576
$556$	0	0
$557$	46.0565	1.95148	0.975738	−	0.218939i	$-0.0702597\pi$
$557$	46.0565	1.95148	0.975738	+	0.218939i	$0.0702597\pi$
$558$	0	0
$559$	−7.78029	−0.329071
$560$	0	0
$561$	1.47943	0.0624615
$562$	0	0
$563$	−37.4009	−1.57626	−0.788129	−	0.615510i	$-0.788950\pi$
$563$	−37.4009	−1.57626	−0.788129	+	0.615510i	$0.788950\pi$
$564$	0	0
$565$	−42.7767	−1.79963
$566$	0	0
$567$	−34.9514	−1.46782
$568$	0	0
$569$	6.67373	0.279777	0.139889	−	0.990167i	$-0.455326\pi$
$569$	6.67373	0.279777	0.139889	+	0.990167i	$0.455326\pi$
$570$	0	0
$571$	−28.1302	−1.17721	−0.588606	−	0.808420i	$-0.700323\pi$
$571$	−28.1302	−1.17721	−0.588606	+	0.808420i	$0.700323\pi$
$572$	0	0
$573$	−56.4867	−2.35977
$574$	0	0
$575$	−34.0800	−1.42124
$576$	0	0
$577$	21.1279	0.879567	0.439784	−	0.898104i	$-0.355055\pi$
$577$	21.1279	0.879567	0.439784	+	0.898104i	$0.355055\pi$
$578$	0	0
$579$	−10.3822	−0.431468
$580$	0	0
$581$	13.2575	0.550015
$582$	0	0
$583$	−2.80135	−0.116020
$584$	0	0
$585$	−6.36373	−0.263108
$586$	0	0
$587$	20.6634	0.852869	0.426435	−	0.904518i	$-0.359769\pi$
$587$	20.6634	0.852869	0.426435	+	0.904518i	$0.359769\pi$
$588$	0	0
$589$	9.48935	0.391002
$590$	0	0
$591$	−12.4485	−0.512064
$592$	0	0
$593$	−7.59734	−0.311985	−0.155993	−	0.987758i	$-0.549858\pi$
$593$	−7.59734	−0.311985	−0.155993	+	0.987758i	$0.549858\pi$
$594$	0	0
$595$	−8.84242	−0.362504
$596$	0	0
$597$	22.2123	0.909091
$598$	0	0
$599$	33.9894	1.38877	0.694385	−	0.719604i	$-0.255677\pi$
$599$	33.9894	1.38877	0.694385	+	0.719604i	$0.255677\pi$
$600$	0	0
$601$	22.0050	0.897602	0.448801	−	0.893632i	$-0.351851\pi$
$601$	22.0050	0.897602	0.448801	+	0.893632i	$0.351851\pi$
$602$	0	0
$603$	−3.29915	−0.134352
$604$	0	0
$605$	−4.20604	−0.171000
$606$	0	0
$607$	−11.9767	−0.486120	−0.243060	−	0.970011i	$-0.578151\pi$
$607$	−11.9767	−0.486120	−0.243060	+	0.970011i	$0.578151\pi$
$608$	0	0
$609$	19.1211	0.774825
$610$	0	0
$611$	−3.50670	−0.141866
$612$	0	0
$613$	19.7909	0.799345	0.399673	−	0.916658i	$-0.369124\pi$
$613$	19.7909	0.799345	0.399673	+	0.916658i	$0.369124\pi$
$614$	0	0
$615$	47.8397	1.92908
$616$	0	0
$617$	−31.1026	−1.25214	−0.626072	−	0.779765i	$-0.715338\pi$
$617$	−31.1026	−1.25214	−0.626072	+	0.779765i	$0.715338\pi$
$618$	0	0
$619$	−14.4469	−0.580671	−0.290335	−	0.956925i	$-0.593767\pi$
$619$	−14.4469	−0.580671	−0.290335	+	0.956925i	$0.593767\pi$
$620$	0	0
$621$	−5.51269	−0.221217
$622$	0	0
$623$	−50.7322	−2.03254
$624$	0	0
$625$	72.6010	2.90404
$626$	0	0
$627$	2.25613	0.0901011
$628$	0	0
$629$	2.76060	0.110072
$630$	0	0
$631$	−4.05580	−0.161459	−0.0807295	−	0.996736i	$-0.525725\pi$
$631$	−4.05580	−0.161459	−0.0807295	+	0.996736i	$0.525725\pi$
$632$	0	0
$633$	45.9204	1.82517
$634$	0	0
$635$	49.1575	1.95076
$636$	0	0
$637$	2.37337	0.0940363
$638$	0	0
$639$	−21.6212	−0.855320
$640$	0	0
$641$	−32.8261	−1.29655	−0.648277	−	0.761405i	$-0.724510\pi$
$641$	−32.8261	−1.29655	−0.648277	+	0.761405i	$0.724510\pi$
$642$	0	0
$643$	22.0138	0.868139	0.434070	−	0.900879i	$-0.357077\pi$
$643$	22.0138	0.868139	0.434070	+	0.900879i	$0.357077\pi$
$644$	0	0
$645$	−101.991	−4.01591
$646$	0	0
$647$	42.5626	1.67331	0.836654	−	0.547731i	$-0.184508\pi$
$647$	42.5626	1.67331	0.836654	+	0.547731i	$0.184508\pi$
$648$	0	0
$649$	8.16817	0.320629
$650$	0	0
$651$	−68.6387	−2.69016
$652$	0	0
$653$	−37.0860	−1.45129	−0.725645	−	0.688070i	$-0.758458\pi$
$653$	−37.0860	−1.45129	−0.725645	+	0.688070i	$0.758458\pi$
$654$	0	0
$655$	−64.2585	−2.51079
$656$	0	0
$657$	−7.00862	−0.273432
$658$	0	0
$659$	−33.0772	−1.28850	−0.644252	−	0.764814i	$-0.722831\pi$
$659$	−33.0772	−1.28850	−0.644252	+	0.764814i	$0.722831\pi$
$660$	0	0
$661$	37.6789	1.46554	0.732770	−	0.680477i	$-0.238227\pi$
$661$	37.6789	1.46554	0.732770	+	0.680477i	$0.238227\pi$
$662$	0	0
$663$	−1.07093	−0.0415916
$664$	0	0
$665$	−13.4847	−0.522914
$666$	0	0
$667$	7.09894	0.274872
$668$	0	0
$669$	−12.8484	−0.496748
$670$	0	0
$671$	−1.66797	−0.0643914
$672$	0	0
$673$	−37.0065	−1.42650	−0.713248	−	0.700912i	$-0.752777\pi$
$673$	−37.0065	−1.42650	−0.713248	+	0.700912i	$0.752777\pi$
$674$	0	0
$675$	26.0518	1.00273
$676$	0	0
$677$	−23.1551	−0.889923	−0.444961	−	0.895550i	$-0.646783\pi$
$677$	−23.1551	−0.889923	−0.444961	+	0.895550i	$0.646783\pi$
$678$	0	0
$679$	17.8672	0.685682
$680$	0	0
$681$	36.3114	1.39145
$682$	0	0
$683$	−7.10245	−0.271768	−0.135884	−	0.990725i	$-0.543387\pi$
$683$	−7.10245	−0.271768	−0.135884	+	0.990725i	$0.543387\pi$
$684$	0	0
$685$	54.1445	2.06875
$686$	0	0
$687$	−22.9867	−0.876999
$688$	0	0
$689$	2.02785	0.0772549
$690$	0	0
$691$	15.3153	0.582622	0.291311	−	0.956628i	$-0.405909\pi$
$691$	15.3153	0.582622	0.291311	+	0.956628i	$0.405909\pi$
$692$	0	0
$693$	−6.70098	−0.254549
$694$	0	0
$695$	55.4364	2.10282
$696$	0	0
$697$	3.30584	0.125218
$698$	0	0
$699$	−37.2513	−1.40897
$700$	0	0
$701$	32.0180	1.20930	0.604651	−	0.796490i	$-0.293312\pi$
$701$	32.0180	1.20930	0.604651	+	0.796490i	$0.293312\pi$
$702$	0	0
$703$	4.20992	0.158780
$704$	0	0
$705$	−45.9691	−1.73130
$706$	0	0
$707$	25.6124	0.963254
$708$	0	0
$709$	−9.61140	−0.360964	−0.180482	−	0.983578i	$-0.557766\pi$
$709$	−9.61140	−0.360964	−0.180482	+	0.983578i	$0.557766\pi$
$710$	0	0
$711$	−36.4445	−1.36677
$712$	0	0
$713$	−25.4830	−0.954345
$714$	0	0
$715$	3.04468	0.113865
$716$	0	0
$717$	63.4562	2.36981
$718$	0	0
$719$	−18.3795	−0.685440	−0.342720	−	0.939438i	$-0.611348\pi$
$719$	−18.3795	−0.685440	−0.342720	+	0.939438i	$0.611348\pi$
$720$	0	0
$721$	−47.1773	−1.75698
$722$	0	0
$723$	−42.4359	−1.57821
$724$	0	0
$725$	−33.5480	−1.24594
$726$	0	0
$727$	−11.6194	−0.430941	−0.215471	−	0.976510i	$-0.569129\pi$
$727$	−11.6194	−0.430941	−0.215471	+	0.976510i	$0.569129\pi$
$728$	0	0
$729$	−8.89168	−0.329322
$730$	0	0
$731$	−7.04785	−0.260674
$732$	0	0
$733$	−23.4420	−0.865850	−0.432925	−	0.901430i	$-0.642519\pi$
$733$	−23.4420	−0.865850	−0.432925	+	0.901430i	$0.642519\pi$
$734$	0	0
$735$	31.1124	1.14760
$736$	0	0
$737$	1.57845	0.0581430
$738$	0	0
$739$	14.0970	0.518567	0.259284	−	0.965801i	$-0.416514\pi$
$739$	14.0970	0.518567	0.259284	+	0.965801i	$0.416514\pi$
$740$	0	0
$741$	−1.63317	−0.0599962
$742$	0	0
$743$	−39.6727	−1.45545	−0.727725	−	0.685869i	$-0.759422\pi$
$743$	−39.6727	−1.45545	−0.727725	+	0.685869i	$0.759422\pi$
$744$	0	0
$745$	−69.5770	−2.54910
$746$	0	0
$747$	8.64300	0.316231
$748$	0	0
$749$	−3.56647	−0.130316
$750$	0	0
$751$	−5.16229	−0.188375	−0.0941874	−	0.995554i	$-0.530025\pi$
$751$	−5.16229	−0.188375	−0.0941874	+	0.995554i	$0.530025\pi$
$752$	0	0
$753$	−43.5374	−1.58659
$754$	0	0
$755$	−67.6546	−2.46220
$756$	0	0
$757$	28.3285	1.02962	0.514809	−	0.857305i	$-0.327863\pi$
$757$	28.3285	1.02962	0.514809	+	0.857305i	$0.327863\pi$
$758$	0	0
$759$	−6.05867	−0.219916
$760$	0	0
$761$	−31.0318	−1.12490	−0.562451	−	0.826831i	$-0.690141\pi$
$761$	−31.0318	−1.12490	−0.562451	+	0.826831i	$0.690141\pi$
$762$	0	0
$763$	−25.7501	−0.932215
$764$	0	0
$765$	−5.76465	−0.208421
$766$	0	0
$767$	−5.91281	−0.213499
$768$	0	0
$769$	40.6414	1.46557	0.732783	−	0.680462i	$-0.238221\pi$
$769$	40.6414	1.46557	0.732783	+	0.680462i	$0.238221\pi$
$770$	0	0
$771$	49.5175	1.78333
$772$	0	0
$773$	−0.817958	−0.0294199	−0.0147099	−	0.999892i	$-0.504682\pi$
$773$	−0.817958	−0.0294199	−0.0147099	+	0.999892i	$0.504682\pi$
$774$	0	0
$775$	120.427	4.32586
$776$	0	0
$777$	−30.4513	−1.09243
$778$	0	0
$779$	5.04141	0.180627
$780$	0	0
$781$	10.3445	0.370155
$782$	0	0
$783$	−5.42663	−0.193932
$784$	0	0
$785$	−73.0521	−2.60734
$786$	0	0
$787$	−11.2343	−0.400459	−0.200229	−	0.979749i	$-0.564169\pi$
$787$	−11.2343	−0.400459	−0.200229	+	0.979749i	$0.564169\pi$
$788$	0	0
$789$	21.0052	0.747806
$790$	0	0
$791$	32.6063	1.15935
$792$	0	0
$793$	1.20742	0.0428767
$794$	0	0
$795$	26.5830	0.942801
$796$	0	0
$797$	4.03810	0.143037	0.0715184	−	0.997439i	$-0.477216\pi$
$797$	4.03810	0.143037	0.0715184	+	0.997439i	$0.477216\pi$
$798$	0	0
$799$	−3.17658	−0.112379
$800$	0	0
$801$	−33.0739	−1.16861
$802$	0	0
$803$	3.35322	0.118333
$804$	0	0
$805$	36.2122	1.27631
$806$	0	0
$807$	40.9275	1.44072
$808$	0	0
$809$	47.1270	1.65690	0.828448	−	0.560066i	$-0.189224\pi$
$809$	47.1270	1.65690	0.828448	+	0.560066i	$0.189224\pi$
$810$	0	0
$811$	−37.6251	−1.32119	−0.660597	−	0.750740i	$-0.729697\pi$
$811$	−37.6251	−1.32119	−0.660597	+	0.750740i	$0.729697\pi$
$812$	0	0
$813$	−58.2921	−2.04439
$814$	0	0
$815$	77.2752	2.70683
$816$	0	0
$817$	−10.7480	−0.376024
$818$	0	0
$819$	4.85073	0.169498
$820$	0	0
$821$	−54.5948	−1.90537	−0.952685	−	0.303959i	$-0.901691\pi$
$821$	−54.5948	−1.90537	−0.952685	+	0.303959i	$0.901691\pi$
$822$	0	0
$823$	−5.10600	−0.177984	−0.0889920	−	0.996032i	$-0.528365\pi$
$823$	−5.10600	−0.177984	−0.0889920	+	0.996032i	$0.528365\pi$
$824$	0	0
$825$	28.6319	0.996835
$826$	0	0
$827$	−21.9650	−0.763798	−0.381899	−	0.924204i	$-0.624730\pi$
$827$	−21.9650	−0.763798	−0.381899	+	0.924204i	$0.624730\pi$
$828$	0	0
$829$	30.3940	1.05563	0.527813	−	0.849360i	$-0.323012\pi$
$829$	30.3940	1.05563	0.527813	+	0.849360i	$0.323012\pi$
$830$	0	0
$831$	33.7151	1.16957
$832$	0	0
$833$	2.14994	0.0744910
$834$	0	0
$835$	11.0482	0.382340
$836$	0	0
$837$	19.4799	0.673324
$838$	0	0
$839$	−5.38580	−0.185938	−0.0929692	−	0.995669i	$-0.529636\pi$
$839$	−5.38580	−0.185938	−0.0929692	+	0.995669i	$0.529636\pi$
$840$	0	0
$841$	−22.0119	−0.759030
$842$	0	0
$843$	55.1991	1.90116
$844$	0	0
$845$	52.4745	1.80518
$846$	0	0
$847$	3.20604	0.110161
$848$	0	0
$849$	−19.5868	−0.672217
$850$	0	0
$851$	−11.3054	−0.387545
$852$	0	0
$853$	36.1015	1.23609	0.618046	−	0.786142i	$-0.287925\pi$
$853$	36.1015	1.23609	0.618046	+	0.786142i	$0.287925\pi$
$854$	0	0
$855$	−8.79110	−0.300649
$856$	0	0
$857$	−36.5886	−1.24984	−0.624922	−	0.780687i	$-0.714869\pi$
$857$	−36.5886	−1.24984	−0.624922	+	0.780687i	$0.714869\pi$
$858$	0	0
$859$	−21.7813	−0.743168	−0.371584	−	0.928399i	$-0.621185\pi$
$859$	−21.7813	−0.743168	−0.371584	+	0.928399i	$0.621185\pi$
$860$	0	0
$861$	−36.4656	−1.24275
$862$	0	0
$863$	46.9178	1.59710	0.798550	−	0.601929i	$-0.205601\pi$
$863$	46.9178	1.59710	0.798550	+	0.601929i	$0.205601\pi$
$864$	0	0
$865$	26.2236	0.891629
$866$	0	0
$867$	37.3841	1.26963
$868$	0	0
$869$	17.4366	0.591496
$870$	0	0
$871$	−1.14262	−0.0387161
$872$	0	0
$873$	11.6482	0.394233
$874$	0	0
$875$	−103.707	−3.50594
$876$	0	0
$877$	−41.9565	−1.41677	−0.708385	−	0.705827i	$-0.750576\pi$
$877$	−41.9565	−1.41677	−0.708385	+	0.705827i	$0.750576\pi$
$878$	0	0
$879$	41.1782	1.38891
$880$	0	0
$881$	−22.6747	−0.763931	−0.381965	−	0.924177i	$-0.624753\pi$
$881$	−22.6747	−0.763931	−0.381965	+	0.924177i	$0.624753\pi$
$882$	0	0
$883$	52.7680	1.77578	0.887892	−	0.460052i	$-0.152169\pi$
$883$	52.7680	1.77578	0.887892	+	0.460052i	$0.152169\pi$
$884$	0	0
$885$	−77.5107	−2.60549
$886$	0	0
$887$	−46.4618	−1.56004	−0.780018	−	0.625758i	$-0.784790\pi$
$887$	−46.4618	−1.56004	−0.780018	+	0.625758i	$0.784790\pi$
$888$	0	0
$889$	−37.4701	−1.25671
$890$	0	0
$891$	10.9018	0.365223
$892$	0	0
$893$	−4.84428	−0.162108
$894$	0	0
$895$	−18.0589	−0.603643
$896$	0	0
$897$	4.38577	0.146437
$898$	0	0
$899$	−25.0852	−0.836637
$900$	0	0
$901$	1.83695	0.0611976
$902$	0	0
$903$	77.7426	2.58711
$904$	0	0
$905$	−19.0220	−0.632312
$906$	0	0
$907$	−40.1762	−1.33403	−0.667014	−	0.745045i	$-0.732428\pi$
$907$	−40.1762	−1.33403	−0.667014	+	0.745045i	$0.732428\pi$
$908$	0	0
$909$	16.6975	0.553823
$910$	0	0
$911$	−34.8397	−1.15429	−0.577145	−	0.816642i	$-0.695833\pi$
$911$	−34.8397	−1.15429	−0.577145	+	0.816642i	$0.695833\pi$
$912$	0	0
$913$	−4.13518	−0.136855
$914$	0	0
$915$	15.8280	0.523257
$916$	0	0
$917$	48.9808	1.61749
$918$	0	0
$919$	38.3172	1.26397	0.631983	−	0.774982i	$-0.282241\pi$
$919$	38.3172	1.26397	0.631983	+	0.774982i	$0.282241\pi$
$920$	0	0
$921$	−11.7915	−0.388544
$922$	0	0
$923$	−7.48821	−0.246477
$924$	0	0
$925$	53.4270	1.75667
$926$	0	0
$927$	−30.7564	−1.01017
$928$	0	0
$929$	3.20133	0.105032	0.0525161	−	0.998620i	$-0.483276\pi$
$929$	3.20133	0.105032	0.0525161	+	0.998620i	$0.483276\pi$
$930$	0	0
$931$	3.27866	0.107454
$932$	0	0
$933$	12.4973	0.409143
$934$	0	0
$935$	2.75805	0.0901980
$936$	0	0
$937$	−38.8123	−1.26794	−0.633972	−	0.773356i	$-0.718576\pi$
$937$	−38.8123	−1.26794	−0.633972	+	0.773356i	$0.718576\pi$
$938$	0	0
$939$	28.8585	0.941761
$940$	0	0
$941$	46.2516	1.50776	0.753879	−	0.657013i	$-0.228180\pi$
$941$	46.2516	1.50776	0.753879	+	0.657013i	$0.228180\pi$
$942$	0	0
$943$	−13.5383	−0.440869
$944$	0	0
$945$	−27.6816	−0.900483
$946$	0	0
$947$	−7.01420	−0.227931	−0.113965	−	0.993485i	$-0.536355\pi$
$947$	−7.01420	−0.227931	−0.113965	+	0.993485i	$0.536355\pi$
$948$	0	0
$949$	−2.42734	−0.0787949
$950$	0	0
$951$	65.9370	2.13815
$952$	0	0
$953$	13.8853	0.449788	0.224894	−	0.974383i	$-0.427796\pi$
$953$	13.8853	0.449788	0.224894	+	0.974383i	$0.427796\pi$
$954$	0	0
$955$	−105.307	−3.40764
$956$	0	0
$957$	−5.96409	−0.192792
$958$	0	0
$959$	−41.2714	−1.33272
$960$	0	0
$961$	59.0479	1.90477
$962$	0	0
$963$	−2.32509	−0.0749250
$964$	0	0
$965$	−19.3552	−0.623065
$966$	0	0
$967$	−34.4880	−1.10906	−0.554529	−	0.832164i	$-0.687102\pi$
$967$	−34.4880	−1.10906	−0.554529	+	0.832164i	$0.687102\pi$
$968$	0	0
$969$	−1.47943	−0.0475260
$970$	0	0
$971$	8.98878	0.288464	0.144232	−	0.989544i	$-0.453929\pi$
$971$	8.98878	0.288464	0.144232	+	0.989544i	$0.453929\pi$
$972$	0	0
$973$	−42.2562	−1.35467
$974$	0	0
$975$	−20.7262	−0.663769
$976$	0	0
$977$	−49.8473	−1.59476	−0.797378	−	0.603480i	$-0.793780\pi$
$977$	−49.8473	−1.59476	−0.797378	+	0.603480i	$0.793780\pi$
$978$	0	0
$979$	15.8240	0.505736
$980$	0	0
$981$	−16.7873	−0.535977
$982$	0	0
$983$	−38.4740	−1.22713	−0.613565	−	0.789644i	$-0.710265\pi$
$983$	−38.4740	−1.22713	−0.613565	+	0.789644i	$0.710265\pi$
$984$	0	0
$985$	−23.2074	−0.739451
$986$	0	0
$987$	35.0398	1.11533
$988$	0	0
$989$	28.8629	0.917788
$990$	0	0
$991$	26.0741	0.828271	0.414135	−	0.910215i	$-0.364084\pi$
$991$	26.0741	0.828271	0.414135	+	0.910215i	$0.364084\pi$
$992$	0	0
$993$	35.8137	1.13651
$994$	0	0
$995$	41.4098	1.31278
$996$	0	0
$997$	51.2926	1.62445	0.812227	−	0.583341i	$-0.198255\pi$
$997$	51.2926	1.62445	0.812227	+	0.583341i	$0.198255\pi$
$998$	0	0
$999$	8.64220	0.273427

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1672.2.a.i.1.2	✓	6
4.3	odd	2		3344.2.a.z.1.5		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1672.2.a.i.1.2	✓	6	1.1	even	1	trivial
3344.2.a.z.1.5		6	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1672 = 2^{3} \cdot 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1672.a (trivial)

\(f(q)\)	\(=\)	\(q-2.25613 q^{3} -4.20604 q^{5} +3.20604 q^{7} +2.09011 q^{9} +O(q^{10})\)	\(q-2.25613 q^{3} -4.20604 q^{5} +3.20604 q^{7} +2.09011 q^{9} -1.00000 q^{11} +0.723884 q^{13} +9.48935 q^{15} +0.655737 q^{17} +1.00000 q^{19} -7.23323 q^{21} -2.68543 q^{23} +12.6907 q^{25} +2.05282 q^{27} -2.64351 q^{29} +9.48935 q^{31} +2.25613 q^{33} -13.4847 q^{35} +4.20992 q^{37} -1.63317 q^{39} +5.04141 q^{41} -10.7480 q^{43} -8.79110 q^{45} -4.84428 q^{47} +3.27866 q^{49} -1.47943 q^{51} +2.80135 q^{53} +4.20604 q^{55} -2.25613 q^{57} -8.16817 q^{59} +1.66797 q^{61} +6.70098 q^{63} -3.04468 q^{65} -1.57845 q^{67} +6.05867 q^{69} -10.3445 q^{71} -3.35322 q^{73} -28.6319 q^{75} -3.20604 q^{77} -17.4366 q^{79} -10.9018 q^{81} +4.13518 q^{83} -2.75805 q^{85} +5.96409 q^{87} -15.8240 q^{89} +2.32080 q^{91} -21.4092 q^{93} -4.20604 q^{95} +5.57301 q^{97} -2.09011 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 4 q^{3} - 3 q^{5} - 3 q^{7} + 6 q^{9}+O(q^{10}) \) 6 * q - 4 * q^3 - 3 * q^5 - 3 * q^7 + 6 * q^9	\( 6 q - 4 q^{3} - 3 q^{5} - 3 q^{7} + 6 q^{9} - 6 q^{11} - 3 q^{13} - q^{15} - q^{17} + 6 q^{19} + 5 q^{21} - 4 q^{23} + 5 q^{25} - 16 q^{27} - 7 q^{29} - q^{31} + 4 q^{33} - 32 q^{35} + 5 q^{37} - 13 q^{39} - 6 q^{41} - 16 q^{43} - 4 q^{47} - 7 q^{49} - 35 q^{51} - 11 q^{53} + 3 q^{55} - 4 q^{57} - 18 q^{59} + 16 q^{61} - 6 q^{63} + 10 q^{65} - 18 q^{67} - 2 q^{69} - 7 q^{71} - 21 q^{73} - 23 q^{75} + 3 q^{77} - 22 q^{79} - 10 q^{81} - 16 q^{83} - 3 q^{87} - 13 q^{89} - 7 q^{91} - 9 q^{93} - 3 q^{95} - 15 q^{97} - 6 q^{99}+O(q^{100}) \) 6 * q - 4 * q^3 - 3 * q^5 - 3 * q^7 + 6 * q^9 - 6 * q^11 - 3 * q^13 - q^15 - q^17 + 6 * q^19 + 5 * q^21 - 4 * q^23 + 5 * q^25 - 16 * q^27 - 7 * q^29 - q^31 + 4 * q^33 - 32 * q^35 + 5 * q^37 - 13 * q^39 - 6 * q^41 - 16 * q^43 - 4 * q^47 - 7 * q^49 - 35 * q^51 - 11 * q^53 + 3 * q^55 - 4 * q^57 - 18 * q^59 + 16 * q^61 - 6 * q^63 + 10 * q^65 - 18 * q^67 - 2 * q^69 - 7 * q^71 - 21 * q^73 - 23 * q^75 + 3 * q^77 - 22 * q^79 - 10 * q^81 - 16 * q^83 - 3 * q^87 - 13 * q^89 - 7 * q^91 - 9 * q^93 - 3 * q^95 - 15 * q^97 - 6 * q^99

Introduction

L-functions

Modular forms

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