LMFDB - Embedded newform 1336.2.a.d.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1336,2,Mod(1,1336)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1336, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1336.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1336 = 2^{3} \cdot 167 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1336.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:	$10.6680137100$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - x^{11} - 25 x^{10} + 20 x^{9} + 224 x^{8} - 135 x^{7} - 865 x^{6} + 330 x^{5} + 1341 x^{4} + \cdots + 64 $ x^12 - x^11 - 25x^10 + 20x^9 + 224x^8 - 135x^7 - 865x^6 + 330x^5 + 1341x^4 - 127x^3 - 652x^2 - 48x + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.68363$ of defining polynomial
Character	$\chi$	$=$	1336.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.68363 q^{3} +0.670957 q^{5} +4.03610 q^{7} +4.20190 q^{9} +O(q^{10})$	$q-2.68363 q^{3} +0.670957 q^{5} +4.03610 q^{7} +4.20190 q^{9} +3.61287 q^{11} +5.67581 q^{13} -1.80060 q^{15} +0.483968 q^{17} -4.68473 q^{19} -10.8314 q^{21} -5.75964 q^{23} -4.54982 q^{25} -3.22545 q^{27} +7.21351 q^{29} -7.44801 q^{31} -9.69563 q^{33} +2.70805 q^{35} +7.41358 q^{37} -15.2318 q^{39} +5.06486 q^{41} +4.17762 q^{43} +2.81929 q^{45} +4.81919 q^{47} +9.29007 q^{49} -1.29879 q^{51} -2.17230 q^{53} +2.42408 q^{55} +12.5721 q^{57} +7.78893 q^{59} -1.76338 q^{61} +16.9593 q^{63} +3.80822 q^{65} -3.20425 q^{67} +15.4568 q^{69} +5.72685 q^{71} +1.71677 q^{73} +12.2100 q^{75} +14.5819 q^{77} -15.3141 q^{79} -3.94976 q^{81} -6.41691 q^{83} +0.324722 q^{85} -19.3584 q^{87} +1.59782 q^{89} +22.9081 q^{91} +19.9877 q^{93} -3.14325 q^{95} -2.46140 q^{97} +15.1809 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - q^{3} + 8 q^{5} - 4 q^{7} + 15 q^{9}+O(q^{10}) $ 12 * q - q^3 + 8 * q^5 - 4 * q^7 + 15 * q^9	$ 12 q - q^{3} + 8 q^{5} - 4 q^{7} + 15 q^{9} + 6 q^{11} + 13 q^{13} + 4 q^{15} + 10 q^{17} - q^{19} + 13 q^{21} + 3 q^{23} + 18 q^{25} - 10 q^{27} + 29 q^{29} - 3 q^{31} + 3 q^{35} + 41 q^{37} + 10 q^{39} + 20 q^{41} - q^{43} + 42 q^{45} + 5 q^{47} + 20 q^{49} + 14 q^{51} + 39 q^{53} + 3 q^{55} + 3 q^{57} + 8 q^{59} + 30 q^{61} - 2 q^{63} + 21 q^{65} - 9 q^{67} + 33 q^{69} + 29 q^{71} + 12 q^{73} + q^{75} + 19 q^{77} - 2 q^{79} + 24 q^{81} - 5 q^{83} + 44 q^{85} - 2 q^{87} + 7 q^{89} - 4 q^{91} + 37 q^{93} + 18 q^{95} - 14 q^{97} + 5 q^{99}+O(q^{100}) $ 12 * q - q^3 + 8 * q^5 - 4 * q^7 + 15 * q^9 + 6 * q^11 + 13 * q^13 + 4 * q^15 + 10 * q^17 - q^19 + 13 * q^21 + 3 * q^23 + 18 * q^25 - 10 * q^27 + 29 * q^29 - 3 * q^31 + 3 * q^35 + 41 * q^37 + 10 * q^39 + 20 * q^41 - q^43 + 42 * q^45 + 5 * q^47 + 20 * q^49 + 14 * q^51 + 39 * q^53 + 3 * q^55 + 3 * q^57 + 8 * q^59 + 30 * q^61 - 2 * q^63 + 21 * q^65 - 9 * q^67 + 33 * q^69 + 29 * q^71 + 12 * q^73 + q^75 + 19 * q^77 - 2 * q^79 + 24 * q^81 - 5 * q^83 + 44 * q^85 - 2 * q^87 + 7 * q^89 - 4 * q^91 + 37 * q^93 + 18 * q^95 - 14 * q^97 + 5 * 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
$3$	−2.68363	−1.54940	−0.774699	−	0.632331i	$-0.782098\pi$
$3$	−2.68363	−1.54940	−0.774699	+	0.632331i	$0.782098\pi$
$4$	0	0
$5$	0.670957	0.300061	0.150031	−	0.988681i	$-0.452063\pi$
$5$	0.670957	0.300061	0.150031	+	0.988681i	$0.452063\pi$
$6$	0	0
$7$	4.03610	1.52550	0.762750	−	0.646693i	$-0.223848\pi$
$7$	4.03610	1.52550	0.762750	+	0.646693i	$0.223848\pi$
$8$	0	0
$9$	4.20190	1.40063
$10$	0	0
$11$	3.61287	1.08932	0.544661	−	0.838656i	$-0.316658\pi$
$11$	3.61287	1.08932	0.544661	+	0.838656i	$0.316658\pi$
$12$	0	0
$13$	5.67581	1.57419	0.787093	−	0.616834i	$-0.211585\pi$
$13$	5.67581	1.57419	0.787093	+	0.616834i	$0.211585\pi$
$14$	0	0
$15$	−1.80060	−0.464914
$16$	0	0
$17$	0.483968	0.117379	0.0586897	−	0.998276i	$-0.481308\pi$
$17$	0.483968	0.117379	0.0586897	+	0.998276i	$0.481308\pi$
$18$	0	0
$19$	−4.68473	−1.07475	−0.537375	−	0.843343i	$-0.680584\pi$
$19$	−4.68473	−1.07475	−0.537375	+	0.843343i	$0.680584\pi$
$20$	0	0
$21$	−10.8314	−2.36361
$22$	0	0
$23$	−5.75964	−1.20097	−0.600484	−	0.799637i	$-0.705025\pi$
$23$	−5.75964	−1.20097	−0.600484	+	0.799637i	$0.705025\pi$
$24$	0	0
$25$	−4.54982	−0.909963
$26$	0	0
$27$	−3.22545	−0.620738
$28$	0	0
$29$	7.21351	1.33951	0.669757	−	0.742580i	$-0.266398\pi$
$29$	7.21351	1.33951	0.669757	+	0.742580i	$0.266398\pi$
$30$	0	0
$31$	−7.44801	−1.33770	−0.668851	−	0.743396i	$-0.733214\pi$
$31$	−7.44801	−1.33770	−0.668851	+	0.743396i	$0.733214\pi$
$32$	0	0
$33$	−9.69563	−1.68779
$34$	0	0
$35$	2.70805	0.457743
$36$	0	0
$37$	7.41358	1.21879	0.609393	−	0.792868i	$-0.291413\pi$
$37$	7.41358	1.21879	0.609393	+	0.792868i	$0.291413\pi$
$38$	0	0
$39$	−15.2318	−2.43904
$40$	0	0
$41$	5.06486	0.790998	0.395499	−	0.918466i	$-0.370572\pi$
$41$	5.06486	0.790998	0.395499	+	0.918466i	$0.370572\pi$
$42$	0	0
$43$	4.17762	0.637081	0.318540	−	0.947909i	$-0.396807\pi$
$43$	4.17762	0.637081	0.318540	+	0.947909i	$0.396807\pi$
$44$	0	0
$45$	2.81929	0.420275
$46$	0	0
$47$	4.81919	0.702952	0.351476	−	0.936197i	$-0.385680\pi$
$47$	4.81919	0.702952	0.351476	+	0.936197i	$0.385680\pi$
$48$	0	0
$49$	9.29007	1.32715
$50$	0	0
$51$	−1.29879	−0.181867
$52$	0	0
$53$	−2.17230	−0.298388	−0.149194	−	0.988808i	$-0.547668\pi$
$53$	−2.17230	−0.298388	−0.149194	+	0.988808i	$0.547668\pi$
$54$	0	0
$55$	2.42408	0.326863
$56$	0	0
$57$	12.5721	1.66522
$58$	0	0
$59$	7.78893	1.01403	0.507016	−	0.861937i	$-0.330748\pi$
$59$	7.78893	1.01403	0.507016	+	0.861937i	$0.330748\pi$
$60$	0	0
$61$	−1.76338	−0.225778	−0.112889	−	0.993608i	$-0.536010\pi$
$61$	−1.76338	−0.225778	−0.112889	+	0.993608i	$0.536010\pi$
$62$	0	0
$63$	16.9593	2.13667
$64$	0	0
$65$	3.80822	0.472352
$66$	0	0
$67$	−3.20425	−0.391461	−0.195731	−	0.980658i	$-0.562708\pi$
$67$	−3.20425	−0.391461	−0.195731	+	0.980658i	$0.562708\pi$
$68$	0	0
$69$	15.4568	1.86078
$70$	0	0
$71$	5.72685	0.679652	0.339826	−	0.940488i	$-0.389632\pi$
$71$	5.72685	0.679652	0.339826	+	0.940488i	$0.389632\pi$
$72$	0	0
$73$	1.71677	0.200933	0.100466	−	0.994940i	$-0.467967\pi$
$73$	1.71677	0.200933	0.100466	+	0.994940i	$0.467967\pi$
$74$	0	0
$75$	12.2100	1.40989
$76$	0	0
$77$	14.5819	1.66176
$78$	0	0
$79$	−15.3141	−1.72298	−0.861488	−	0.507779i	$-0.830467\pi$
$79$	−15.3141	−1.72298	−0.861488	+	0.507779i	$0.830467\pi$
$80$	0	0
$81$	−3.94976	−0.438862
$82$	0	0
$83$	−6.41691	−0.704347	−0.352174	−	0.935935i	$-0.614557\pi$
$83$	−6.41691	−0.704347	−0.352174	+	0.935935i	$0.614557\pi$
$84$	0	0
$85$	0.324722	0.0352210
$86$	0	0
$87$	−19.3584	−2.07544
$88$	0	0
$89$	1.59782	0.169368	0.0846842	−	0.996408i	$-0.473012\pi$
$89$	1.59782	0.169368	0.0846842	+	0.996408i	$0.473012\pi$
$90$	0	0
$91$	22.9081	2.40142
$92$	0	0
$93$	19.9877	2.07263
$94$	0	0
$95$	−3.14325	−0.322491
$96$	0	0
$97$	−2.46140	−0.249917	−0.124958	−	0.992162i	$-0.539880\pi$
$97$	−2.46140	−0.249917	−0.124958	+	0.992162i	$0.539880\pi$
$98$	0	0
$99$	15.1809	1.52574
$100$	0	0
$101$	7.60074	0.756301	0.378151	−	0.925744i	$-0.376560\pi$
$101$	7.60074	0.756301	0.378151	+	0.925744i	$0.376560\pi$
$102$	0	0
$103$	−19.1243	−1.88437	−0.942187	−	0.335086i	$-0.891234\pi$
$103$	−19.1243	−1.88437	−0.942187	+	0.335086i	$0.891234\pi$
$104$	0	0
$105$	−7.26741	−0.709226
$106$	0	0
$107$	4.99014	0.482415	0.241208	−	0.970474i	$-0.422457\pi$
$107$	4.99014	0.482415	0.241208	+	0.970474i	$0.422457\pi$
$108$	0	0
$109$	1.33938	0.128289	0.0641445	−	0.997941i	$-0.479568\pi$
$109$	1.33938	0.128289	0.0641445	+	0.997941i	$0.479568\pi$
$110$	0	0
$111$	−19.8953	−1.88838
$112$	0	0
$113$	3.77947	0.355542	0.177771	−	0.984072i	$-0.443111\pi$
$113$	3.77947	0.355542	0.177771	+	0.984072i	$0.443111\pi$
$114$	0	0
$115$	−3.86447	−0.360364
$116$	0	0
$117$	23.8492	2.20486
$118$	0	0
$119$	1.95334	0.179062
$120$	0	0
$121$	2.05285	0.186623
$122$	0	0
$123$	−13.5922	−1.22557
$124$	0	0
$125$	−6.40751	−0.573106
$126$	0	0
$127$	11.7633	1.04383	0.521913	−	0.852999i	$-0.325219\pi$
$127$	11.7633	1.04383	0.521913	+	0.852999i	$0.325219\pi$
$128$	0	0
$129$	−11.2112	−0.987091
$130$	0	0
$131$	15.5855	1.36171	0.680854	−	0.732419i	$-0.261609\pi$
$131$	15.5855	1.36171	0.680854	+	0.732419i	$0.261609\pi$
$132$	0	0
$133$	−18.9080	−1.63953
$134$	0	0
$135$	−2.16414	−0.186259
$136$	0	0
$137$	19.9773	1.70678	0.853388	−	0.521277i	$-0.174544\pi$
$137$	19.9773	1.70678	0.853388	+	0.521277i	$0.174544\pi$
$138$	0	0
$139$	19.8177	1.68092	0.840459	−	0.541876i	$-0.182286\pi$
$139$	19.8177	1.68092	0.840459	+	0.541876i	$0.182286\pi$
$140$	0	0
$141$	−12.9330	−1.08915
$142$	0	0
$143$	20.5060	1.71480
$144$	0	0
$145$	4.83995	0.401936
$146$	0	0
$147$	−24.9312	−2.05629
$148$	0	0
$149$	14.6111	1.19699	0.598495	−	0.801127i	$-0.295766\pi$
$149$	14.6111	1.19699	0.598495	+	0.801127i	$0.295766\pi$
$150$	0	0
$151$	20.0063	1.62809	0.814043	−	0.580804i	$-0.197262\pi$
$151$	20.0063	1.62809	0.814043	+	0.580804i	$0.197262\pi$
$152$	0	0
$153$	2.03358	0.164405
$154$	0	0
$155$	−4.99729	−0.401392
$156$	0	0
$157$	−6.56240	−0.523737	−0.261868	−	0.965104i	$-0.584339\pi$
$157$	−6.56240	−0.523737	−0.261868	+	0.965104i	$0.584339\pi$
$158$	0	0
$159$	5.82966	0.462322
$160$	0	0
$161$	−23.2465	−1.83208
$162$	0	0
$163$	−13.2713	−1.03949	−0.519743	−	0.854323i	$-0.673972\pi$
$163$	−13.2713	−1.03949	−0.519743	+	0.854323i	$0.673972\pi$
$164$	0	0
$165$	−6.50535	−0.506441
$166$	0	0
$167$	−1.00000	−0.0773823
$167$	−1.00000	−0.0773823
$168$	0	0
$169$	19.2148	1.47806
$170$	0	0
$171$	−19.6847	−1.50533
$172$	0	0
$173$	−16.9869	−1.29149	−0.645744	−	0.763554i	$-0.723452\pi$
$173$	−16.9869	−1.29149	−0.645744	+	0.763554i	$0.723452\pi$
$174$	0	0
$175$	−18.3635	−1.38815
$176$	0	0
$177$	−20.9026	−1.57114
$178$	0	0
$179$	6.26246	0.468079	0.234039	−	0.972227i	$-0.424806\pi$
$179$	6.26246	0.468079	0.234039	+	0.972227i	$0.424806\pi$
$180$	0	0
$181$	−17.3699	−1.29109	−0.645547	−	0.763720i	$-0.723371\pi$
$181$	−17.3699	−1.29109	−0.645547	+	0.763720i	$0.723371\pi$
$182$	0	0
$183$	4.73228	0.349820
$184$	0	0
$185$	4.97419	0.365710
$186$	0	0
$187$	1.74851	0.127864
$188$	0	0
$189$	−13.0182	−0.946936
$190$	0	0
$191$	1.85766	0.134416	0.0672078	−	0.997739i	$-0.478591\pi$
$191$	1.85766	0.134416	0.0672078	+	0.997739i	$0.478591\pi$
$192$	0	0
$193$	1.23794	0.0891088	0.0445544	−	0.999007i	$-0.485813\pi$
$193$	1.23794	0.0891088	0.0445544	+	0.999007i	$0.485813\pi$
$194$	0	0
$195$	−10.2199	−0.731861
$196$	0	0
$197$	−16.5015	−1.17569	−0.587843	−	0.808975i	$-0.700023\pi$
$197$	−16.5015	−1.17569	−0.587843	+	0.808975i	$0.700023\pi$
$198$	0	0
$199$	−9.15166	−0.648744	−0.324372	−	0.945930i	$-0.605153\pi$
$199$	−9.15166	−0.648744	−0.324372	+	0.945930i	$0.605153\pi$
$200$	0	0
$201$	8.59903	0.606529
$202$	0	0
$203$	29.1144	2.04343
$204$	0	0
$205$	3.39830	0.237348
$206$	0	0
$207$	−24.2014	−1.68211
$208$	0	0
$209$	−16.9253	−1.17075
$210$	0	0
$211$	27.3080	1.87996	0.939980	−	0.341229i	$-0.110843\pi$
$211$	27.3080	1.87996	0.939980	+	0.341229i	$0.110843\pi$
$212$	0	0
$213$	−15.3688	−1.05305
$214$	0	0
$215$	2.80300	0.191163
$216$	0	0
$217$	−30.0609	−2.04067
$218$	0	0
$219$	−4.60719	−0.311325
$220$	0	0
$221$	2.74691	0.184777
$222$	0	0
$223$	17.4971	1.17169	0.585845	−	0.810423i	$-0.300763\pi$
$223$	17.4971	1.17169	0.585845	+	0.810423i	$0.300763\pi$
$224$	0	0
$225$	−19.1179	−1.27452
$226$	0	0
$227$	−17.2958	−1.14797	−0.573983	−	0.818867i	$-0.694602\pi$
$227$	−17.2958	−1.14797	−0.573983	+	0.818867i	$0.694602\pi$
$228$	0	0
$229$	−3.20670	−0.211905	−0.105952	−	0.994371i	$-0.533789\pi$
$229$	−3.20670	−0.211905	−0.105952	+	0.994371i	$0.533789\pi$
$230$	0	0
$231$	−39.1325	−2.57473
$232$	0	0
$233$	−17.4105	−1.14060	−0.570299	−	0.821437i	$-0.693173\pi$
$233$	−17.4105	−1.14060	−0.570299	+	0.821437i	$0.693173\pi$
$234$	0	0
$235$	3.23347	0.210928
$236$	0	0
$237$	41.0976	2.66957
$238$	0	0
$239$	−1.26153	−0.0816018	−0.0408009	−	0.999167i	$-0.512991\pi$
$239$	−1.26153	−0.0816018	−0.0408009	+	0.999167i	$0.512991\pi$
$240$	0	0
$241$	21.5337	1.38711	0.693554	−	0.720405i	$-0.256044\pi$
$241$	21.5337	1.38711	0.693554	+	0.720405i	$0.256044\pi$
$242$	0	0
$243$	20.2761	1.30071
$244$	0	0
$245$	6.23324	0.398227
$246$	0	0
$247$	−26.5896	−1.69186
$248$	0	0
$249$	17.2206	1.09131
$250$	0	0
$251$	−16.8421	−1.06306	−0.531532	−	0.847038i	$-0.678383\pi$
$251$	−16.8421	−1.06306	−0.531532	+	0.847038i	$0.678383\pi$
$252$	0	0
$253$	−20.8089	−1.30824
$254$	0	0
$255$	−0.871434	−0.0545713
$256$	0	0
$257$	−10.3596	−0.646212	−0.323106	−	0.946363i	$-0.604727\pi$
$257$	−10.3596	−0.646212	−0.323106	+	0.946363i	$0.604727\pi$
$258$	0	0
$259$	29.9219	1.85926
$260$	0	0
$261$	30.3104	1.87617
$262$	0	0
$263$	22.5472	1.39032	0.695160	−	0.718855i	$-0.255334\pi$
$263$	22.5472	1.39032	0.695160	+	0.718855i	$0.255334\pi$
$264$	0	0
$265$	−1.45752	−0.0895347
$266$	0	0
$267$	−4.28796	−0.262419
$268$	0	0
$269$	−18.0316	−1.09941	−0.549703	−	0.835360i	$-0.685259\pi$
$269$	−18.0316	−1.09941	−0.549703	+	0.835360i	$0.685259\pi$
$270$	0	0
$271$	−31.9203	−1.93902	−0.969510	−	0.245050i	$-0.921196\pi$
$271$	−31.9203	−1.93902	−0.969510	+	0.245050i	$0.921196\pi$
$272$	0	0
$273$	−61.4770	−3.72076
$274$	0	0
$275$	−16.4379	−0.991243
$276$	0	0
$277$	12.2519	0.736148	0.368074	−	0.929796i	$-0.380017\pi$
$277$	12.2519	0.736148	0.368074	+	0.929796i	$0.380017\pi$
$278$	0	0
$279$	−31.2958	−1.87363
$280$	0	0
$281$	11.3103	0.674718	0.337359	−	0.941376i	$-0.390466\pi$
$281$	11.3103	0.674718	0.337359	+	0.941376i	$0.390466\pi$
$282$	0	0
$283$	13.0749	0.777221	0.388610	−	0.921402i	$-0.372955\pi$
$283$	13.0749	0.777221	0.388610	+	0.921402i	$0.372955\pi$
$284$	0	0
$285$	8.43534	0.499666
$286$	0	0
$287$	20.4422	1.20667
$288$	0	0
$289$	−16.7658	−0.986222
$290$	0	0
$291$	6.60549	0.387220
$292$	0	0
$293$	−26.2618	−1.53423	−0.767115	−	0.641510i	$-0.778308\pi$
$293$	−26.2618	−1.53423	−0.767115	+	0.641510i	$0.778308\pi$
$294$	0	0
$295$	5.22604	0.304272
$296$	0	0
$297$	−11.6531	−0.676184
$298$	0	0
$299$	−32.6906	−1.89055
$300$	0	0
$301$	16.8613	0.971868
$302$	0	0
$303$	−20.3976	−1.17181
$304$	0	0
$305$	−1.18315	−0.0677472
$306$	0	0
$307$	−15.8285	−0.903381	−0.451690	−	0.892175i	$-0.649179\pi$
$307$	−15.8285	−0.903381	−0.451690	+	0.892175i	$0.649179\pi$
$308$	0	0
$309$	51.3227	2.91965
$310$	0	0
$311$	9.77402	0.554234	0.277117	−	0.960836i	$-0.410621\pi$
$311$	9.77402	0.554234	0.277117	+	0.960836i	$0.410621\pi$
$312$	0	0
$313$	0.777166	0.0439280	0.0219640	−	0.999759i	$-0.493008\pi$
$313$	0.777166	0.0439280	0.0219640	+	0.999759i	$0.493008\pi$
$314$	0	0
$315$	11.3789	0.641130
$316$	0	0
$317$	28.9456	1.62575	0.812873	−	0.582442i	$-0.197902\pi$
$317$	28.9456	1.62575	0.812873	+	0.582442i	$0.197902\pi$
$318$	0	0
$319$	26.0615	1.45916
$320$	0	0
$321$	−13.3917	−0.747453
$322$	0	0
$323$	−2.26726	−0.126154
$324$	0	0
$325$	−25.8239	−1.43245
$326$	0	0
$327$	−3.59440	−0.198771
$328$	0	0
$329$	19.4507	1.07235
$330$	0	0
$331$	−19.8449	−1.09078	−0.545388	−	0.838184i	$-0.683618\pi$
$331$	−19.8449	−1.09078	−0.545388	+	0.838184i	$0.683618\pi$
$332$	0	0
$333$	31.1511	1.70707
$334$	0	0
$335$	−2.14991	−0.117462
$336$	0	0
$337$	−31.8125	−1.73294	−0.866470	−	0.499230i	$-0.833616\pi$
$337$	−31.8125	−1.73294	−0.866470	+	0.499230i	$0.833616\pi$
$338$	0	0
$339$	−10.1427	−0.550876
$340$	0	0
$341$	−26.9087	−1.45719
$342$	0	0
$343$	9.24295	0.499072
$344$	0	0
$345$	10.3708	0.558347
$346$	0	0
$347$	−0.398828	−0.0214102	−0.0107051	−	0.999943i	$-0.503408\pi$
$347$	−0.398828	−0.0214102	−0.0107051	+	0.999943i	$0.503408\pi$
$348$	0	0
$349$	−5.60779	−0.300178	−0.150089	−	0.988672i	$-0.547956\pi$
$349$	−5.60779	−0.300178	−0.150089	+	0.988672i	$0.547956\pi$
$350$	0	0
$351$	−18.3070	−0.977157
$352$	0	0
$353$	−5.11919	−0.272467	−0.136234	−	0.990677i	$-0.543500\pi$
$353$	−5.11919	−0.272467	−0.136234	+	0.990677i	$0.543500\pi$
$354$	0	0
$355$	3.84247	0.203937
$356$	0	0
$357$	−5.24205	−0.277439
$358$	0	0
$359$	33.4509	1.76547	0.882736	−	0.469869i	$-0.155699\pi$
$359$	33.4509	1.76547	0.882736	+	0.469869i	$0.155699\pi$
$360$	0	0
$361$	2.94669	0.155089
$362$	0	0
$363$	−5.50911	−0.289153
$364$	0	0
$365$	1.15188	0.0602921
$366$	0	0
$367$	−28.0493	−1.46416	−0.732080	−	0.681218i	$-0.761450\pi$
$367$	−28.0493	−1.46416	−0.732080	+	0.681218i	$0.761450\pi$
$368$	0	0
$369$	21.2820	1.10790
$370$	0	0
$371$	−8.76761	−0.455192
$372$	0	0
$373$	−22.1220	−1.14543	−0.572716	−	0.819754i	$-0.694110\pi$
$373$	−22.1220	−1.14543	−0.572716	+	0.819754i	$0.694110\pi$
$374$	0	0
$375$	17.1954	0.887968
$376$	0	0
$377$	40.9425	2.10865
$378$	0	0
$379$	27.7730	1.42660	0.713301	−	0.700858i	$-0.247199\pi$
$379$	27.7730	1.42660	0.713301	+	0.700858i	$0.247199\pi$
$380$	0	0
$381$	−31.5685	−1.61730
$382$	0	0
$383$	26.3850	1.34821	0.674106	−	0.738635i	$-0.264529\pi$
$383$	26.3850	1.34821	0.674106	+	0.738635i	$0.264529\pi$
$384$	0	0
$385$	9.78383	0.498630
$386$	0	0
$387$	17.5539	0.892316
$388$	0	0
$389$	39.2318	1.98913	0.994567	−	0.104100i	$-0.0331963\pi$
$389$	39.2318	1.98913	0.994567	+	0.104100i	$0.0331963\pi$
$390$	0	0
$391$	−2.78748	−0.140969
$392$	0	0
$393$	−41.8257	−2.10983
$394$	0	0
$395$	−10.2751	−0.516998
$396$	0	0
$397$	32.6782	1.64007	0.820035	−	0.572313i	$-0.193954\pi$
$397$	32.6782	1.64007	0.820035	+	0.572313i	$0.193954\pi$
$398$	0	0
$399$	50.7422	2.54029
$400$	0	0
$401$	−7.01318	−0.350221	−0.175111	−	0.984549i	$-0.556028\pi$
$401$	−7.01318	−0.350221	−0.175111	+	0.984549i	$0.556028\pi$
$402$	0	0
$403$	−42.2735	−2.10579
$404$	0	0
$405$	−2.65012	−0.131685
$406$	0	0
$407$	26.7843	1.32765
$408$	0	0
$409$	8.36642	0.413693	0.206846	−	0.978373i	$-0.433680\pi$
$409$	8.36642	0.413693	0.206846	+	0.978373i	$0.433680\pi$
$410$	0	0
$411$	−53.6118	−2.64447
$412$	0	0
$413$	31.4369	1.54691
$414$	0	0
$415$	−4.30547	−0.211347
$416$	0	0
$417$	−53.1835	−2.60441
$418$	0	0
$419$	19.7339	0.964062	0.482031	−	0.876154i	$-0.339899\pi$
$419$	19.7339	0.964062	0.482031	+	0.876154i	$0.339899\pi$
$420$	0	0
$421$	9.21227	0.448978	0.224489	−	0.974477i	$-0.427929\pi$
$421$	9.21227	0.448978	0.224489	+	0.974477i	$0.427929\pi$
$422$	0	0
$423$	20.2497	0.984576
$424$	0	0
$425$	−2.20197	−0.106811
$426$	0	0
$427$	−7.11718	−0.344425
$428$	0	0
$429$	−55.0306	−2.65690
$430$	0	0
$431$	−28.7672	−1.38567	−0.692833	−	0.721098i	$-0.743638\pi$
$431$	−28.7672	−1.38567	−0.692833	+	0.721098i	$0.743638\pi$
$432$	0	0
$433$	28.3746	1.36360	0.681799	−	0.731539i	$-0.261198\pi$
$433$	28.3746	1.36360	0.681799	+	0.731539i	$0.261198\pi$
$434$	0	0
$435$	−12.9887	−0.622759
$436$	0	0
$437$	26.9824	1.29074
$438$	0	0
$439$	21.0106	1.00278	0.501390	−	0.865221i	$-0.332822\pi$
$439$	21.0106	1.00278	0.501390	+	0.865221i	$0.332822\pi$
$440$	0	0
$441$	39.0359	1.85885
$442$	0	0
$443$	−22.2382	−1.05657	−0.528285	−	0.849067i	$-0.677165\pi$
$443$	−22.2382	−1.05657	−0.528285	+	0.849067i	$0.677165\pi$
$444$	0	0
$445$	1.07207	0.0508209
$446$	0	0
$447$	−39.2109	−1.85461
$448$	0	0
$449$	11.2402	0.530456	0.265228	−	0.964186i	$-0.414553\pi$
$449$	11.2402	0.530456	0.265228	+	0.964186i	$0.414553\pi$
$450$	0	0
$451$	18.2987	0.861651
$452$	0	0
$453$	−53.6895	−2.52255
$454$	0	0
$455$	15.3704	0.720573
$456$	0	0
$457$	−21.7907	−1.01933	−0.509664	−	0.860373i	$-0.670230\pi$
$457$	−21.7907	−1.01933	−0.509664	+	0.860373i	$0.670230\pi$
$458$	0	0
$459$	−1.56101	−0.0728619
$460$	0	0
$461$	−10.8163	−0.503766	−0.251883	−	0.967758i	$-0.581050\pi$
$461$	−10.8163	−0.503766	−0.251883	+	0.967758i	$0.581050\pi$
$462$	0	0
$463$	−22.7847	−1.05889	−0.529447	−	0.848343i	$-0.677601\pi$
$463$	−22.7847	−1.05889	−0.529447	+	0.848343i	$0.677601\pi$
$464$	0	0
$465$	13.4109	0.621916
$466$	0	0
$467$	6.93996	0.321143	0.160571	−	0.987024i	$-0.448666\pi$
$467$	6.93996	0.321143	0.160571	+	0.987024i	$0.448666\pi$
$468$	0	0
$469$	−12.9327	−0.597175
$470$	0	0
$471$	17.6111	0.811476
$472$	0	0
$473$	15.0932	0.693986
$474$	0	0
$475$	21.3147	0.977984
$476$	0	0
$477$	−9.12778	−0.417932
$478$	0	0
$479$	−23.1576	−1.05810	−0.529049	−	0.848591i	$-0.677451\pi$
$479$	−23.1576	−1.05810	−0.529049	+	0.848591i	$0.677451\pi$
$480$	0	0
$481$	42.0781	1.91860
$482$	0	0
$483$	62.3850	2.83862
$484$	0	0
$485$	−1.65149	−0.0749903
$486$	0	0
$487$	15.3917	0.697463	0.348731	−	0.937223i	$-0.386613\pi$
$487$	15.3917	0.697463	0.348731	+	0.937223i	$0.386613\pi$
$488$	0	0
$489$	35.6152	1.61058
$490$	0	0
$491$	−24.7877	−1.11865	−0.559326	−	0.828948i	$-0.688940\pi$
$491$	−24.7877	−1.11865	−0.559326	+	0.828948i	$0.688940\pi$
$492$	0	0
$493$	3.49111	0.157232
$494$	0	0
$495$	10.1857	0.457815
$496$	0	0
$497$	23.1141	1.03681
$498$	0	0
$499$	−20.1074	−0.900133	−0.450066	−	0.892995i	$-0.648600\pi$
$499$	−20.1074	−0.900133	−0.450066	+	0.892995i	$0.648600\pi$
$500$	0	0
$501$	2.68363	0.119896
$502$	0	0
$503$	−10.6161	−0.473351	−0.236675	−	0.971589i	$-0.576058\pi$
$503$	−10.6161	−0.473351	−0.236675	+	0.971589i	$0.576058\pi$
$504$	0	0
$505$	5.09977	0.226937
$506$	0	0
$507$	−51.5655	−2.29011
$508$	0	0
$509$	33.9087	1.50298	0.751489	−	0.659746i	$-0.229336\pi$
$509$	33.9087	1.50298	0.751489	+	0.659746i	$0.229336\pi$
$510$	0	0
$511$	6.92905	0.306523
$512$	0	0
$513$	15.1104	0.667139
$514$	0	0
$515$	−12.8316	−0.565427
$516$	0	0
$517$	17.4111	0.765741
$518$	0	0
$519$	45.5865	2.00103
$520$	0	0
$521$	−41.5293	−1.81943	−0.909717	−	0.415229i	$-0.863701\pi$
$521$	−41.5293	−1.81943	−0.909717	+	0.415229i	$0.863701\pi$
$522$	0	0
$523$	−28.4938	−1.24595	−0.622974	−	0.782242i	$-0.714076\pi$
$523$	−28.4938	−1.24595	−0.622974	+	0.782242i	$0.714076\pi$
$524$	0	0
$525$	49.2809	2.15080
$526$	0	0
$527$	−3.60460	−0.157019
$528$	0	0
$529$	10.1735	0.442325
$530$	0	0
$531$	32.7283	1.42029
$532$	0	0
$533$	28.7472	1.24518
$534$	0	0
$535$	3.34817	0.144754
$536$	0	0
$537$	−16.8062	−0.725240
$538$	0	0
$539$	33.5638	1.44570
$540$	0	0
$541$	−19.9542	−0.857897	−0.428949	−	0.903329i	$-0.641116\pi$
$541$	−19.9542	−0.857897	−0.428949	+	0.903329i	$0.641116\pi$
$542$	0	0
$543$	46.6145	2.00042
$544$	0	0
$545$	0.898664	0.0384945
$546$	0	0
$547$	0.518020	0.0221490	0.0110745	−	0.999939i	$-0.496475\pi$
$547$	0.518020	0.0221490	0.0110745	+	0.999939i	$0.496475\pi$
$548$	0	0
$549$	−7.40955	−0.316232
$550$	0	0
$551$	−33.7933	−1.43964
$552$	0	0
$553$	−61.8093	−2.62840
$554$	0	0
$555$	−13.3489	−0.566630
$556$	0	0
$557$	−41.3143	−1.75054	−0.875271	−	0.483633i	$-0.839317\pi$
$557$	−41.3143	−1.75054	−0.875271	+	0.483633i	$0.839317\pi$
$558$	0	0
$559$	23.7114	1.00288
$560$	0	0
$561$	−4.69237	−0.198112
$562$	0	0
$563$	14.2788	0.601778	0.300889	−	0.953659i	$-0.402717\pi$
$563$	14.2788	0.601778	0.300889	+	0.953659i	$0.402717\pi$
$564$	0	0
$565$	2.53586	0.106684
$566$	0	0
$567$	−15.9416	−0.669485
$568$	0	0
$569$	13.6991	0.574298	0.287149	−	0.957886i	$-0.407293\pi$
$569$	13.6991	0.574298	0.287149	+	0.957886i	$0.407293\pi$
$570$	0	0
$571$	26.4910	1.10861	0.554306	−	0.832313i	$-0.312984\pi$
$571$	26.4910	1.10861	0.554306	+	0.832313i	$0.312984\pi$
$572$	0	0
$573$	−4.98528	−0.208263
$574$	0	0
$575$	26.2053	1.09284
$576$	0	0
$577$	−28.5908	−1.19025	−0.595125	−	0.803633i	$-0.702897\pi$
$577$	−28.5908	−1.19025	−0.595125	+	0.803633i	$0.702897\pi$
$578$	0	0
$579$	−3.32218	−0.138065
$580$	0	0
$581$	−25.8993	−1.07448
$582$	0	0
$583$	−7.84824	−0.325041
$584$	0	0
$585$	16.0018	0.661591
$586$	0	0
$587$	−13.2266	−0.545920	−0.272960	−	0.962025i	$-0.588003\pi$
$587$	−13.2266	−0.545920	−0.272960	+	0.962025i	$0.588003\pi$
$588$	0	0
$589$	34.8919	1.43770
$590$	0	0
$591$	44.2841	1.82160
$592$	0	0
$593$	−14.9125	−0.612383	−0.306191	−	0.951970i	$-0.599055\pi$
$593$	−14.9125	−0.612383	−0.306191	+	0.951970i	$0.599055\pi$
$594$	0	0
$595$	1.31061	0.0537297
$596$	0	0
$597$	24.5597	1.00516
$598$	0	0
$599$	23.7546	0.970587	0.485293	−	0.874351i	$-0.338713\pi$
$599$	23.7546	0.970587	0.485293	+	0.874351i	$0.338713\pi$
$600$	0	0
$601$	−33.1958	−1.35409	−0.677043	−	0.735943i	$-0.736739\pi$
$601$	−33.1958	−1.35409	−0.677043	+	0.735943i	$0.736739\pi$
$602$	0	0
$603$	−13.4639	−0.548293
$604$	0	0
$605$	1.37738	0.0559983
$606$	0	0
$607$	−24.2160	−0.982899	−0.491449	−	0.870906i	$-0.663533\pi$
$607$	−24.2160	−0.982899	−0.491449	+	0.870906i	$0.663533\pi$
$608$	0	0
$609$	−78.1325	−3.16609
$610$	0	0
$611$	27.3528	1.10658
$612$	0	0
$613$	−9.23272	−0.372906	−0.186453	−	0.982464i	$-0.559699\pi$
$613$	−9.23272	−0.372906	−0.186453	+	0.982464i	$0.559699\pi$
$614$	0	0
$615$	−9.11980	−0.367746
$616$	0	0
$617$	−32.3847	−1.30376	−0.651880	−	0.758322i	$-0.726020\pi$
$617$	−32.3847	−1.30376	−0.651880	+	0.758322i	$0.726020\pi$
$618$	0	0
$619$	−22.4560	−0.902584	−0.451292	−	0.892376i	$-0.649037\pi$
$619$	−22.4560	−0.902584	−0.451292	+	0.892376i	$0.649037\pi$
$620$	0	0
$621$	18.5774	0.745487
$622$	0	0
$623$	6.44895	0.258372
$624$	0	0
$625$	18.4499	0.737997
$626$	0	0
$627$	45.4214	1.81396
$628$	0	0
$629$	3.58794	0.143060
$630$	0	0
$631$	−13.3537	−0.531602	−0.265801	−	0.964028i	$-0.585636\pi$
$631$	−13.3537	−0.531602	−0.265801	+	0.964028i	$0.585636\pi$
$632$	0	0
$633$	−73.2847	−2.91281
$634$	0	0
$635$	7.89268	0.313212
$636$	0	0
$637$	52.7287	2.08919
$638$	0	0
$639$	24.0636	0.951942
$640$	0	0
$641$	−16.9142	−0.668071	−0.334035	−	0.942561i	$-0.608411\pi$
$641$	−16.9142	−0.668071	−0.334035	+	0.942561i	$0.608411\pi$
$642$	0	0
$643$	−35.9322	−1.41703	−0.708514	−	0.705697i	$-0.750634\pi$
$643$	−35.9322	−1.41703	−0.708514	+	0.705697i	$0.750634\pi$
$644$	0	0
$645$	−7.52223	−0.296188
$646$	0	0
$647$	31.0731	1.22161	0.610806	−	0.791781i	$-0.290846\pi$
$647$	31.0731	1.22161	0.610806	+	0.791781i	$0.290846\pi$
$648$	0	0
$649$	28.1404	1.10461
$650$	0	0
$651$	80.6724	3.16180
$652$	0	0
$653$	−29.9689	−1.17278	−0.586388	−	0.810031i	$-0.699450\pi$
$653$	−29.9689	−1.17278	−0.586388	+	0.810031i	$0.699450\pi$
$654$	0	0
$655$	10.4572	0.408595
$656$	0	0
$657$	7.21369	0.281433
$658$	0	0
$659$	−7.44116	−0.289866	−0.144933	−	0.989441i	$-0.546297\pi$
$659$	−7.44116	−0.289866	−0.144933	+	0.989441i	$0.546297\pi$
$660$	0	0
$661$	2.29179	0.0891404	0.0445702	−	0.999006i	$-0.485808\pi$
$661$	2.29179	0.0891404	0.0445702	+	0.999006i	$0.485808\pi$
$662$	0	0
$663$	−7.37170	−0.286293
$664$	0	0
$665$	−12.6865	−0.491960
$666$	0	0
$667$	−41.5472	−1.60871
$668$	0	0
$669$	−46.9557	−1.81541
$670$	0	0
$671$	−6.37088	−0.245945
$672$	0	0
$673$	−6.61324	−0.254922	−0.127461	−	0.991844i	$-0.540683\pi$
$673$	−6.61324	−0.254922	−0.127461	+	0.991844i	$0.540683\pi$
$674$	0	0
$675$	14.6752	0.564849
$676$	0	0
$677$	3.99613	0.153584	0.0767919	−	0.997047i	$-0.475532\pi$
$677$	3.99613	0.153584	0.0767919	+	0.997047i	$0.475532\pi$
$678$	0	0
$679$	−9.93443	−0.381248
$680$	0	0
$681$	46.4157	1.77865
$682$	0	0
$683$	−16.1482	−0.617893	−0.308946	−	0.951079i	$-0.599976\pi$
$683$	−16.1482	−0.617893	−0.308946	+	0.951079i	$0.599976\pi$
$684$	0	0
$685$	13.4039	0.512137
$686$	0	0
$687$	8.60561	0.328325
$688$	0	0
$689$	−12.3296	−0.469719
$690$	0	0
$691$	−36.5258	−1.38951	−0.694754	−	0.719247i	$-0.744487\pi$
$691$	−36.5258	−1.38951	−0.694754	+	0.719247i	$0.744487\pi$
$692$	0	0
$693$	61.2716	2.32752
$694$	0	0
$695$	13.2968	0.504378
$696$	0	0
$697$	2.45123	0.0928469
$698$	0	0
$699$	46.7233	1.76724
$700$	0	0
$701$	−16.8595	−0.636774	−0.318387	−	0.947961i	$-0.603141\pi$
$701$	−16.8595	−0.636774	−0.318387	+	0.947961i	$0.603141\pi$
$702$	0	0
$703$	−34.7306	−1.30989
$704$	0	0
$705$	−8.67745	−0.326812
$706$	0	0
$707$	30.6773	1.15374
$708$	0	0
$709$	−11.8900	−0.446537	−0.223269	−	0.974757i	$-0.571673\pi$
$709$	−11.8900	−0.446537	−0.223269	+	0.974757i	$0.571673\pi$
$710$	0	0
$711$	−64.3484	−2.41325
$712$	0	0
$713$	42.8979	1.60654
$714$	0	0
$715$	13.7586	0.514543
$716$	0	0
$717$	3.38549	0.126434
$718$	0	0
$719$	4.76973	0.177881	0.0889404	−	0.996037i	$-0.471652\pi$
$719$	4.76973	0.177881	0.0889404	+	0.996037i	$0.471652\pi$
$720$	0	0
$721$	−77.1876	−2.87462
$722$	0	0
$723$	−57.7886	−2.14918
$724$	0	0
$725$	−32.8201	−1.21891
$726$	0	0
$727$	−52.1932	−1.93574	−0.967869	−	0.251456i	$-0.919091\pi$
$727$	−52.1932	−1.93574	−0.967869	+	0.251456i	$0.919091\pi$
$728$	0	0
$729$	−42.5643	−1.57645
$730$	0	0
$731$	2.02183	0.0747802
$732$	0	0
$733$	2.84094	0.104932	0.0524662	−	0.998623i	$-0.483292\pi$
$733$	2.84094	0.104932	0.0524662	+	0.998623i	$0.483292\pi$
$734$	0	0
$735$	−16.7277	−0.617012
$736$	0	0
$737$	−11.5765	−0.426428
$738$	0	0
$739$	23.5042	0.864614	0.432307	−	0.901726i	$-0.357700\pi$
$739$	23.5042	0.864614	0.432307	+	0.901726i	$0.357700\pi$
$740$	0	0
$741$	71.3569	2.62136
$742$	0	0
$743$	22.5458	0.827127	0.413563	−	0.910475i	$-0.364284\pi$
$743$	22.5458	0.827127	0.413563	+	0.910475i	$0.364284\pi$
$744$	0	0
$745$	9.80343	0.359170
$746$	0	0
$747$	−26.9632	−0.986531
$748$	0	0
$749$	20.1407	0.735925
$750$	0	0
$751$	26.9679	0.984071	0.492035	−	0.870575i	$-0.336253\pi$
$751$	26.9679	0.984071	0.492035	+	0.870575i	$0.336253\pi$
$752$	0	0
$753$	45.1980	1.64711
$754$	0	0
$755$	13.4233	0.488525
$756$	0	0
$757$	−7.13714	−0.259404	−0.129702	−	0.991553i	$-0.541402\pi$
$757$	−7.13714	−0.259404	−0.129702	+	0.991553i	$0.541402\pi$
$758$	0	0
$759$	55.8434	2.02699
$760$	0	0
$761$	−6.25433	−0.226719	−0.113360	−	0.993554i	$-0.536161\pi$
$761$	−6.25433	−0.226719	−0.113360	+	0.993554i	$0.536161\pi$
$762$	0	0
$763$	5.40585	0.195705
$764$	0	0
$765$	1.36445	0.0493316
$766$	0	0
$767$	44.2085	1.59628
$768$	0	0
$769$	−17.8178	−0.642527	−0.321264	−	0.946990i	$-0.604108\pi$
$769$	−17.8178	−0.642527	−0.321264	+	0.946990i	$0.604108\pi$
$770$	0	0
$771$	27.8013	1.00124
$772$	0	0
$773$	33.2292	1.19517	0.597586	−	0.801805i	$-0.296127\pi$
$773$	33.2292	1.19517	0.597586	+	0.801805i	$0.296127\pi$
$774$	0	0
$775$	33.8871	1.21726
$776$	0	0
$777$	−80.2995	−2.88073
$778$	0	0
$779$	−23.7275	−0.850125
$780$	0	0
$781$	20.6904	0.740360
$782$	0	0
$783$	−23.2668	−0.831488
$784$	0	0
$785$	−4.40309	−0.157153
$786$	0	0
$787$	−0.166829	−0.00594681	−0.00297341	−	0.999996i	$-0.500946\pi$
$787$	−0.166829	−0.00594681	−0.00297341	+	0.999996i	$0.500946\pi$
$788$	0	0
$789$	−60.5084	−2.15416
$790$	0	0
$791$	15.2543	0.542380
$792$	0	0
$793$	−10.0086	−0.355417
$794$	0	0
$795$	3.91145	0.138725
$796$	0	0
$797$	−13.9316	−0.493484	−0.246742	−	0.969081i	$-0.579360\pi$
$797$	−13.9316	−0.493484	−0.246742	+	0.969081i	$0.579360\pi$
$798$	0	0
$799$	2.33233	0.0825121
$800$	0	0
$801$	6.71387	0.237223
$802$	0	0
$803$	6.20248	0.218881
$804$	0	0
$805$	−15.5974	−0.549735
$806$	0	0
$807$	48.3902	1.70342
$808$	0	0
$809$	−29.7581	−1.04624	−0.523120	−	0.852259i	$-0.675232\pi$
$809$	−29.7581	−1.04624	−0.523120	+	0.852259i	$0.675232\pi$
$810$	0	0
$811$	−8.85111	−0.310805	−0.155402	−	0.987851i	$-0.549667\pi$
$811$	−8.85111	−0.310805	−0.155402	+	0.987851i	$0.549667\pi$
$812$	0	0
$813$	85.6625	3.00431
$814$	0	0
$815$	−8.90445	−0.311909
$816$	0	0
$817$	−19.5710	−0.684703
$818$	0	0
$819$	96.2575	3.36351
$820$	0	0
$821$	12.9205	0.450928	0.225464	−	0.974251i	$-0.427610\pi$
$821$	12.9205	0.450928	0.225464	+	0.974251i	$0.427610\pi$
$822$	0	0
$823$	−21.0346	−0.733222	−0.366611	−	0.930374i	$-0.619482\pi$
$823$	−21.0346	−0.733222	−0.366611	+	0.930374i	$0.619482\pi$
$824$	0	0
$825$	44.1134	1.53583
$826$	0	0
$827$	53.8843	1.87374	0.936870	−	0.349678i	$-0.113709\pi$
$827$	53.8843	1.87374	0.936870	+	0.349678i	$0.113709\pi$
$828$	0	0
$829$	56.3809	1.95819	0.979096	−	0.203400i	$-0.0651993\pi$
$829$	56.3809	1.95819	0.979096	+	0.203400i	$0.0651993\pi$
$830$	0	0
$831$	−32.8797	−1.14059
$832$	0	0
$833$	4.49610	0.155780
$834$	0	0
$835$	−0.670957	−0.0232194
$836$	0	0
$837$	24.0232	0.830363
$838$	0	0
$839$	20.7827	0.717499	0.358750	−	0.933434i	$-0.383203\pi$
$839$	20.7827	0.717499	0.358750	+	0.933434i	$0.383203\pi$
$840$	0	0
$841$	23.0347	0.794300
$842$	0	0
$843$	−30.3528	−1.04541
$844$	0	0
$845$	12.8923	0.443509
$846$	0	0
$847$	8.28551	0.284694
$848$	0	0
$849$	−35.0882	−1.20422
$850$	0	0
$851$	−42.6996	−1.46372
$852$	0	0
$853$	1.02450	0.0350781	0.0175390	−	0.999846i	$-0.494417\pi$
$853$	1.02450	0.0350781	0.0175390	+	0.999846i	$0.494417\pi$
$854$	0	0
$855$	−13.2076	−0.451691
$856$	0	0
$857$	−12.0625	−0.412047	−0.206023	−	0.978547i	$-0.566052\pi$
$857$	−12.0625	−0.412047	−0.206023	+	0.978547i	$0.566052\pi$
$858$	0	0
$859$	−0.915426	−0.0312339	−0.0156170	−	0.999878i	$-0.504971\pi$
$859$	−0.915426	−0.0312339	−0.0156170	+	0.999878i	$0.504971\pi$
$860$	0	0
$861$	−54.8595	−1.86961
$862$	0	0
$863$	40.3018	1.37189	0.685944	−	0.727654i	$-0.259390\pi$
$863$	40.3018	1.37189	0.685944	+	0.727654i	$0.259390\pi$
$864$	0	0
$865$	−11.3975	−0.387525
$866$	0	0
$867$	44.9932	1.52805
$868$	0	0
$869$	−55.3280	−1.87688
$870$	0	0
$871$	−18.1867	−0.616233
$872$	0	0
$873$	−10.3425	−0.350042
$874$	0	0
$875$	−25.8613	−0.874273
$876$	0	0
$877$	33.4557	1.12972	0.564860	−	0.825187i	$-0.308930\pi$
$877$	33.4557	1.12972	0.564860	+	0.825187i	$0.308930\pi$
$878$	0	0
$879$	70.4770	2.37713
$880$	0	0
$881$	30.1042	1.01424	0.507119	−	0.861876i	$-0.330711\pi$
$881$	30.1042	1.01424	0.507119	+	0.861876i	$0.330711\pi$
$882$	0	0
$883$	10.3155	0.347143	0.173571	−	0.984821i	$-0.444469\pi$
$883$	10.3155	0.347143	0.173571	+	0.984821i	$0.444469\pi$
$884$	0	0
$885$	−14.0248	−0.471438
$886$	0	0
$887$	−37.6562	−1.26437	−0.632186	−	0.774816i	$-0.717842\pi$
$887$	−37.6562	−1.26437	−0.632186	+	0.774816i	$0.717842\pi$
$888$	0	0
$889$	47.4779	1.59236
$890$	0	0
$891$	−14.2700	−0.478062
$892$	0	0
$893$	−22.5766	−0.755498
$894$	0	0
$895$	4.20184	0.140452
$896$	0	0
$897$	87.7297	2.92921
$898$	0	0
$899$	−53.7263	−1.79187
$900$	0	0
$901$	−1.05132	−0.0350247
$902$	0	0
$903$	−45.2495	−1.50581
$904$	0	0
$905$	−11.6545	−0.387407
$906$	0	0
$907$	−28.2552	−0.938199	−0.469100	−	0.883145i	$-0.655421\pi$
$907$	−28.2552	−0.938199	−0.469100	+	0.883145i	$0.655421\pi$
$908$	0	0
$909$	31.9375	1.05930
$910$	0	0
$911$	56.8338	1.88299	0.941493	−	0.337032i	$-0.109423\pi$
$911$	56.8338	1.88299	0.941493	+	0.337032i	$0.109423\pi$
$912$	0	0
$913$	−23.1835	−0.767261
$914$	0	0
$915$	3.17515	0.104967
$916$	0	0
$917$	62.9044	2.07729
$918$	0	0
$919$	−55.7206	−1.83805	−0.919027	−	0.394196i	$-0.871023\pi$
$919$	−55.7206	−1.83805	−0.919027	+	0.394196i	$0.871023\pi$
$920$	0	0
$921$	42.4779	1.39970
$922$	0	0
$923$	32.5045	1.06990
$924$	0	0
$925$	−33.7304	−1.10905
$926$	0	0
$927$	−80.3584	−2.63932
$928$	0	0
$929$	−39.3897	−1.29233	−0.646166	−	0.763197i	$-0.723629\pi$
$929$	−39.3897	−1.29233	−0.646166	+	0.763197i	$0.723629\pi$
$930$	0	0
$931$	−43.5215	−1.42636
$932$	0	0
$933$	−26.2299	−0.858729
$934$	0	0
$935$	1.17318	0.0383670
$936$	0	0
$937$	−20.3586	−0.665085	−0.332543	−	0.943088i	$-0.607907\pi$
$937$	−20.3586	−0.665085	−0.332543	+	0.943088i	$0.607907\pi$
$938$	0	0
$939$	−2.08563	−0.0680619
$940$	0	0
$941$	20.9591	0.683246	0.341623	−	0.939837i	$-0.389023\pi$
$941$	20.9591	0.683246	0.341623	+	0.939837i	$0.389023\pi$
$942$	0	0
$943$	−29.1718	−0.949963
$944$	0	0
$945$	−8.73466	−0.284139
$946$	0	0
$947$	26.6423	0.865758	0.432879	−	0.901452i	$-0.357498\pi$
$947$	26.6423	0.865758	0.432879	+	0.901452i	$0.357498\pi$
$948$	0	0
$949$	9.74406	0.316306
$950$	0	0
$951$	−77.6793	−2.51893
$952$	0	0
$953$	−8.85198	−0.286744	−0.143372	−	0.989669i	$-0.545794\pi$
$953$	−8.85198	−0.286744	−0.143372	+	0.989669i	$0.545794\pi$
$954$	0	0
$955$	1.24641	0.0403329
$956$	0	0
$957$	−69.9395	−2.26082
$958$	0	0
$959$	80.6303	2.60369
$960$	0	0
$961$	24.4729	0.789448
$962$	0	0
$963$	20.9681	0.675686
$964$	0	0
$965$	0.830604	0.0267381
$966$	0	0
$967$	21.7517	0.699488	0.349744	−	0.936845i	$-0.386269\pi$
$967$	21.7517	0.699488	0.349744	+	0.936845i	$0.386269\pi$
$968$	0	0
$969$	6.08449	0.195462
$970$	0	0
$971$	21.4217	0.687454	0.343727	−	0.939070i	$-0.388311\pi$
$971$	21.4217	0.687454	0.343727	+	0.939070i	$0.388311\pi$
$972$	0	0
$973$	79.9862	2.56424
$974$	0	0
$975$	69.3019	2.21944
$976$	0	0
$977$	−9.67678	−0.309588	−0.154794	−	0.987947i	$-0.549471\pi$
$977$	−9.67678	−0.309588	−0.154794	+	0.987947i	$0.549471\pi$
$978$	0	0
$979$	5.77271	0.184497
$980$	0	0
$981$	5.62792	0.179686
$982$	0	0
$983$	−26.9146	−0.858443	−0.429221	−	0.903199i	$-0.641212\pi$
$983$	−26.9146	−0.858443	−0.429221	+	0.903199i	$0.641212\pi$
$984$	0	0
$985$	−11.0718	−0.352777
$986$	0	0
$987$	−52.1986	−1.66150
$988$	0	0
$989$	−24.0616	−0.765114
$990$	0	0
$991$	−34.4948	−1.09576	−0.547882	−	0.836556i	$-0.684566\pi$
$991$	−34.4948	−1.09576	−0.547882	+	0.836556i	$0.684566\pi$
$992$	0	0
$993$	53.2565	1.69005
$994$	0	0
$995$	−6.14037	−0.194663
$996$	0	0
$997$	27.5325	0.871963	0.435981	−	0.899956i	$-0.356401\pi$
$997$	27.5325	0.871963	0.435981	+	0.899956i	$0.356401\pi$
$998$	0	0
$999$	−23.9121	−0.756546

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1336.2.a.d.1.2	✓	12
4.3	odd	2		2672.2.a.p.1.11		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1336.2.a.d.1.2	✓	12	1.1	even	1	trivial
2672.2.a.p.1.11		12	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:	\( N \)	\(=\)	\( 1336 = 2^{3} \cdot 167 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1336.a (trivial)

\(f(q)\)	\(=\)	\(q-2.68363 q^{3} +0.670957 q^{5} +4.03610 q^{7} +4.20190 q^{9} +O(q^{10})\)	\(q-2.68363 q^{3} +0.670957 q^{5} +4.03610 q^{7} +4.20190 q^{9} +3.61287 q^{11} +5.67581 q^{13} -1.80060 q^{15} +0.483968 q^{17} -4.68473 q^{19} -10.8314 q^{21} -5.75964 q^{23} -4.54982 q^{25} -3.22545 q^{27} +7.21351 q^{29} -7.44801 q^{31} -9.69563 q^{33} +2.70805 q^{35} +7.41358 q^{37} -15.2318 q^{39} +5.06486 q^{41} +4.17762 q^{43} +2.81929 q^{45} +4.81919 q^{47} +9.29007 q^{49} -1.29879 q^{51} -2.17230 q^{53} +2.42408 q^{55} +12.5721 q^{57} +7.78893 q^{59} -1.76338 q^{61} +16.9593 q^{63} +3.80822 q^{65} -3.20425 q^{67} +15.4568 q^{69} +5.72685 q^{71} +1.71677 q^{73} +12.2100 q^{75} +14.5819 q^{77} -15.3141 q^{79} -3.94976 q^{81} -6.41691 q^{83} +0.324722 q^{85} -19.3584 q^{87} +1.59782 q^{89} +22.9081 q^{91} +19.9877 q^{93} -3.14325 q^{95} -2.46140 q^{97} +15.1809 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - q^{3} + 8 q^{5} - 4 q^{7} + 15 q^{9}+O(q^{10}) \) 12 * q - q^3 + 8 * q^5 - 4 * q^7 + 15 * q^9	\( 12 q - q^{3} + 8 q^{5} - 4 q^{7} + 15 q^{9} + 6 q^{11} + 13 q^{13} + 4 q^{15} + 10 q^{17} - q^{19} + 13 q^{21} + 3 q^{23} + 18 q^{25} - 10 q^{27} + 29 q^{29} - 3 q^{31} + 3 q^{35} + 41 q^{37} + 10 q^{39} + 20 q^{41} - q^{43} + 42 q^{45} + 5 q^{47} + 20 q^{49} + 14 q^{51} + 39 q^{53} + 3 q^{55} + 3 q^{57} + 8 q^{59} + 30 q^{61} - 2 q^{63} + 21 q^{65} - 9 q^{67} + 33 q^{69} + 29 q^{71} + 12 q^{73} + q^{75} + 19 q^{77} - 2 q^{79} + 24 q^{81} - 5 q^{83} + 44 q^{85} - 2 q^{87} + 7 q^{89} - 4 q^{91} + 37 q^{93} + 18 q^{95} - 14 q^{97} + 5 q^{99}+O(q^{100}) \) 12 * q - q^3 + 8 * q^5 - 4 * q^7 + 15 * q^9 + 6 * q^11 + 13 * q^13 + 4 * q^15 + 10 * q^17 - q^19 + 13 * q^21 + 3 * q^23 + 18 * q^25 - 10 * q^27 + 29 * q^29 - 3 * q^31 + 3 * q^35 + 41 * q^37 + 10 * q^39 + 20 * q^41 - q^43 + 42 * q^45 + 5 * q^47 + 20 * q^49 + 14 * q^51 + 39 * q^53 + 3 * q^55 + 3 * q^57 + 8 * q^59 + 30 * q^61 - 2 * q^63 + 21 * q^65 - 9 * q^67 + 33 * q^69 + 29 * q^71 + 12 * q^73 + q^75 + 19 * q^77 - 2 * q^79 + 24 * q^81 - 5 * q^83 + 44 * q^85 - 2 * q^87 + 7 * q^89 - 4 * q^91 + 37 * q^93 + 18 * q^95 - 14 * q^97 + 5 * q^99

Introduction

L-functions

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