LMFDB - Embedded newform 668.2.a.c.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [668,2,Mod(1,668)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(668, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("668.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 668 = 2^{2} \cdot 167 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	668.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:	$5.33400685502$
Analytic rank:	$1$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 11x^{5} - 7x^{4} + 21x^{3} + 17x^{2} - 4x - 4 $ x^7 - 11x^5 - 7x^4 + 21x^3 + 17x^2 - 4*x - 4
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$3.27771$ of defining polynomial
Character	$\chi$	$=$	668.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.685166 q^{3} +0.610181 q^{5} -0.184306 q^{7} -2.53055 q^{9} +O(q^{10})$	$q-0.685166 q^{3} +0.610181 q^{5} -0.184306 q^{7} -2.53055 q^{9} -5.07449 q^{11} +2.65594 q^{13} -0.418076 q^{15} -0.128237 q^{17} -7.41417 q^{19} +0.126280 q^{21} +3.99471 q^{23} -4.62768 q^{25} +3.78935 q^{27} -1.33375 q^{29} -8.19083 q^{31} +3.47687 q^{33} -0.112460 q^{35} +3.63984 q^{37} -1.81976 q^{39} -2.15173 q^{41} -2.76015 q^{43} -1.54409 q^{45} +4.94015 q^{47} -6.96603 q^{49} +0.0878636 q^{51} -0.756443 q^{53} -3.09636 q^{55} +5.07994 q^{57} -6.98791 q^{59} -0.248451 q^{61} +0.466395 q^{63} +1.62060 q^{65} -15.5507 q^{67} -2.73704 q^{69} +4.03018 q^{71} -1.57460 q^{73} +3.17073 q^{75} +0.935260 q^{77} +8.12950 q^{79} +4.99531 q^{81} -15.0509 q^{83} -0.0782477 q^{85} +0.913840 q^{87} +3.67917 q^{89} -0.489505 q^{91} +5.61208 q^{93} -4.52399 q^{95} +10.2997 q^{97} +12.8412 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q - 4 q^{3} - 2 q^{5} - 12 q^{7} + 7 q^{9}+O(q^{10}) $ 7 * q - 4 * q^3 - 2 * q^5 - 12 * q^7 + 7 * q^9	$ 7 q - 4 q^{3} - 2 q^{5} - 12 q^{7} + 7 q^{9} - 7 q^{11} - 9 q^{13} - 17 q^{15} - q^{17} - 11 q^{19} - 4 q^{21} - 19 q^{23} + 3 q^{25} - 16 q^{27} - 5 q^{29} - 13 q^{31} - 8 q^{33} - 7 q^{35} - 26 q^{37} - 17 q^{39} - 2 q^{41} - 24 q^{43} - 7 q^{45} - 11 q^{47} + 19 q^{49} + 8 q^{51} + 4 q^{53} - 4 q^{55} + 14 q^{57} - 4 q^{59} - 5 q^{61} - 21 q^{63} + 13 q^{65} - 42 q^{67} + 24 q^{69} + 9 q^{71} + 27 q^{73} + 25 q^{75} + 12 q^{77} - 8 q^{79} + 35 q^{81} + 16 q^{83} - 27 q^{85} + 3 q^{87} + 9 q^{89} - 2 q^{91} - 10 q^{93} + 10 q^{95} - 8 q^{97} - 5 q^{99}+O(q^{100}) $ 7 * q - 4 * q^3 - 2 * q^5 - 12 * q^7 + 7 * q^9 - 7 * q^11 - 9 * q^13 - 17 * q^15 - q^17 - 11 * q^19 - 4 * q^21 - 19 * q^23 + 3 * q^25 - 16 * q^27 - 5 * q^29 - 13 * q^31 - 8 * q^33 - 7 * q^35 - 26 * q^37 - 17 * q^39 - 2 * q^41 - 24 * q^43 - 7 * q^45 - 11 * q^47 + 19 * q^49 + 8 * q^51 + 4 * q^53 - 4 * q^55 + 14 * q^57 - 4 * q^59 - 5 * q^61 - 21 * q^63 + 13 * q^65 - 42 * q^67 + 24 * q^69 + 9 * q^71 + 27 * q^73 + 25 * q^75 + 12 * q^77 - 8 * q^79 + 35 * q^81 + 16 * q^83 - 27 * q^85 + 3 * q^87 + 9 * q^89 - 2 * q^91 - 10 * q^93 + 10 * q^95 - 8 * 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
$3$	−0.685166	−0.395581	−0.197791	−	0.980244i	$-0.563377\pi$
$3$	−0.685166	−0.395581	−0.197791	+	0.980244i	$0.563377\pi$
$4$	0	0
$5$	0.610181	0.272881	0.136441	−	0.990648i	$-0.456434\pi$
$5$	0.610181	0.272881	0.136441	+	0.990648i	$0.456434\pi$
$6$	0	0
$7$	−0.184306	−0.0696611	−0.0348306	−	0.999393i	$-0.511089\pi$
$7$	−0.184306	−0.0696611	−0.0348306	+	0.999393i	$0.511089\pi$
$8$	0	0
$9$	−2.53055	−0.843516
$10$	0	0
$11$	−5.07449	−1.53002	−0.765009	−	0.644020i	$-0.777265\pi$
$11$	−5.07449	−1.53002	−0.765009	+	0.644020i	$0.777265\pi$
$12$	0	0
$13$	2.65594	0.736624	0.368312	−	0.929702i	$-0.379936\pi$
$13$	2.65594	0.736624	0.368312	+	0.929702i	$0.379936\pi$
$14$	0	0
$15$	−0.418076	−0.107947
$16$	0	0
$17$	−0.128237	−0.0311020	−0.0155510	−	0.999879i	$-0.504950\pi$
$17$	−0.128237	−0.0311020	−0.0155510	+	0.999879i	$0.504950\pi$
$18$	0	0
$19$	−7.41417	−1.70093	−0.850464	−	0.526033i	$-0.823679\pi$
$19$	−7.41417	−1.70093	−0.850464	+	0.526033i	$0.823679\pi$
$20$	0	0
$21$	0.126280	0.0275566
$22$	0	0
$23$	3.99471	0.832955	0.416477	−	0.909146i	$-0.363264\pi$
$23$	3.99471	0.832955	0.416477	+	0.909146i	$0.363264\pi$
$24$	0	0
$25$	−4.62768	−0.925536
$26$	0	0
$27$	3.78935	0.729260
$28$	0	0
$29$	−1.33375	−0.247671	−0.123836	−	0.992303i	$-0.539520\pi$
$29$	−1.33375	−0.247671	−0.123836	+	0.992303i	$0.539520\pi$
$30$	0	0
$31$	−8.19083	−1.47112	−0.735559	−	0.677461i	$-0.763080\pi$
$31$	−8.19083	−1.47112	−0.735559	+	0.677461i	$0.763080\pi$
$32$	0	0
$33$	3.47687	0.605246
$34$	0	0
$35$	−0.112460	−0.0190092
$36$	0	0
$37$	3.63984	0.598385	0.299193	−	0.954193i	$-0.403283\pi$
$37$	3.63984	0.598385	0.299193	+	0.954193i	$0.403283\pi$
$38$	0	0
$39$	−1.81976	−0.291395
$40$	0	0
$41$	−2.15173	−0.336044	−0.168022	−	0.985783i	$-0.553738\pi$
$41$	−2.15173	−0.336044	−0.168022	+	0.985783i	$0.553738\pi$
$42$	0	0
$43$	−2.76015	−0.420919	−0.210460	−	0.977603i	$-0.567496\pi$
$43$	−2.76015	−0.420919	−0.210460	+	0.977603i	$0.567496\pi$
$44$	0	0
$45$	−1.54409	−0.230180
$46$	0	0
$47$	4.94015	0.720595	0.360298	−	0.932837i	$-0.382675\pi$
$47$	4.94015	0.720595	0.360298	+	0.932837i	$0.382675\pi$
$48$	0	0
$49$	−6.96603	−0.995147
$50$	0	0
$51$	0.0878636	0.0123034
$52$	0	0
$53$	−0.756443	−0.103905	−0.0519527	−	0.998650i	$-0.516545\pi$
$53$	−0.756443	−0.103905	−0.0519527	+	0.998650i	$0.516545\pi$
$54$	0	0
$55$	−3.09636	−0.417513
$56$	0	0
$57$	5.07994	0.672855
$58$	0	0
$59$	−6.98791	−0.909748	−0.454874	−	0.890556i	$-0.650316\pi$
$59$	−6.98791	−0.909748	−0.454874	+	0.890556i	$0.650316\pi$
$60$	0	0
$61$	−0.248451	−0.0318109	−0.0159055	−	0.999874i	$-0.505063\pi$
$61$	−0.248451	−0.0318109	−0.0159055	+	0.999874i	$0.505063\pi$
$62$	0	0
$63$	0.466395	0.0587603
$64$	0	0
$65$	1.62060	0.201011
$66$	0	0
$67$	−15.5507	−1.89982	−0.949908	−	0.312529i	$-0.898824\pi$
$67$	−15.5507	−1.89982	−0.949908	+	0.312529i	$0.898824\pi$
$68$	0	0
$69$	−2.73704	−0.329501
$70$	0	0
$71$	4.03018	0.478295	0.239147	−	0.970983i	$-0.423132\pi$
$71$	4.03018	0.478295	0.239147	+	0.970983i	$0.423132\pi$
$72$	0	0
$73$	−1.57460	−0.184292	−0.0921462	−	0.995745i	$-0.529373\pi$
$73$	−1.57460	−0.184292	−0.0921462	+	0.995745i	$0.529373\pi$
$74$	0	0
$75$	3.17073	0.366124
$76$	0	0
$77$	0.935260	0.106583
$78$	0	0
$79$	8.12950	0.914640	0.457320	−	0.889302i	$-0.348809\pi$
$79$	8.12950	0.914640	0.457320	+	0.889302i	$0.348809\pi$
$80$	0	0
$81$	4.99531	0.555034
$82$	0	0
$83$	−15.0509	−1.65205	−0.826027	−	0.563631i	$-0.809404\pi$
$83$	−15.0509	−1.65205	−0.826027	+	0.563631i	$0.809404\pi$
$84$	0	0
$85$	−0.0782477	−0.00848715
$86$	0	0
$87$	0.913840	0.0979740
$88$	0	0
$89$	3.67917	0.389991	0.194996	−	0.980804i	$-0.437531\pi$
$89$	3.67917	0.389991	0.194996	+	0.980804i	$0.437531\pi$
$90$	0	0
$91$	−0.489505	−0.0513141
$92$	0	0
$93$	5.61208	0.581946
$94$	0	0
$95$	−4.52399	−0.464152
$96$	0	0
$97$	10.2997	1.04577	0.522886	−	0.852402i	$-0.324855\pi$
$97$	10.2997	1.04577	0.522886	+	0.852402i	$0.324855\pi$
$98$	0	0
$99$	12.8412	1.29059
$100$	0	0
$101$	16.1953	1.61149	0.805745	−	0.592263i	$-0.201765\pi$
$101$	16.1953	1.61149	0.805745	+	0.592263i	$0.201765\pi$
$102$	0	0
$103$	2.33475	0.230049	0.115025	−	0.993363i	$-0.463305\pi$
$103$	2.33475	0.230049	0.115025	+	0.993363i	$0.463305\pi$
$104$	0	0
$105$	0.0770539	0.00751969
$106$	0	0
$107$	5.75331	0.556194	0.278097	−	0.960553i	$-0.410296\pi$
$107$	5.75331	0.556194	0.278097	+	0.960553i	$0.410296\pi$
$108$	0	0
$109$	12.2756	1.17578	0.587892	−	0.808939i	$-0.299958\pi$
$109$	12.2756	1.17578	0.587892	+	0.808939i	$0.299958\pi$
$110$	0	0
$111$	−2.49389	−0.236710
$112$	0	0
$113$	14.0646	1.32308	0.661542	−	0.749908i	$-0.269902\pi$
$113$	14.0646	1.32308	0.661542	+	0.749908i	$0.269902\pi$
$114$	0	0
$115$	2.43750	0.227298
$116$	0	0
$117$	−6.72097	−0.621354
$118$	0	0
$119$	0.0236348	0.00216660
$120$	0	0
$121$	14.7505	1.34095
$122$	0	0
$123$	1.47429	0.132933
$124$	0	0
$125$	−5.87463	−0.525443
$126$	0	0
$127$	−22.3104	−1.97973	−0.989866	−	0.142004i	$-0.954646\pi$
$127$	−22.3104	−1.97973	−0.989866	+	0.142004i	$0.954646\pi$
$128$	0	0
$129$	1.89116	0.166508
$130$	0	0
$131$	3.05168	0.266627	0.133313	−	0.991074i	$-0.457438\pi$
$131$	3.05168	0.266627	0.133313	+	0.991074i	$0.457438\pi$
$132$	0	0
$133$	1.36648	0.118489
$134$	0	0
$135$	2.31219	0.199001
$136$	0	0
$137$	1.65628	0.141505	0.0707526	−	0.997494i	$-0.477460\pi$
$137$	1.65628	0.141505	0.0707526	+	0.997494i	$0.477460\pi$
$138$	0	0
$139$	9.85669	0.836034	0.418017	−	0.908439i	$-0.362725\pi$
$139$	9.85669	0.836034	0.418017	+	0.908439i	$0.362725\pi$
$140$	0	0
$141$	−3.38483	−0.285054
$142$	0	0
$143$	−13.4775	−1.12705
$144$	0	0
$145$	−0.813829	−0.0675848
$146$	0	0
$147$	4.77289	0.393661
$148$	0	0
$149$	−15.9076	−1.30320	−0.651599	−	0.758564i	$-0.725901\pi$
$149$	−15.9076	−1.30320	−0.651599	+	0.758564i	$0.725901\pi$
$150$	0	0
$151$	14.6612	1.19311	0.596557	−	0.802571i	$-0.296535\pi$
$151$	14.6612	1.19311	0.596557	+	0.802571i	$0.296535\pi$
$152$	0	0
$153$	0.324509	0.0262350
$154$	0	0
$155$	−4.99789	−0.401440
$156$	0	0
$157$	−4.30181	−0.343322	−0.171661	−	0.985156i	$-0.554913\pi$
$157$	−4.30181	−0.343322	−0.171661	+	0.985156i	$0.554913\pi$
$158$	0	0
$159$	0.518289	0.0411030
$160$	0	0
$161$	−0.736250	−0.0580246
$162$	0	0
$163$	18.6533	1.46104	0.730520	−	0.682891i	$-0.239278\pi$
$163$	18.6533	1.46104	0.730520	+	0.682891i	$0.239278\pi$
$164$	0	0
$165$	2.12152	0.165160
$166$	0	0
$167$	1.00000	0.0773823
$167$	1.00000	0.0773823
$168$	0	0
$169$	−5.94600	−0.457385
$170$	0	0
$171$	18.7619	1.43476
$172$	0	0
$173$	22.4896	1.70985	0.854927	−	0.518748i	$-0.173602\pi$
$173$	22.4896	1.70985	0.854927	+	0.518748i	$0.173602\pi$
$174$	0	0
$175$	0.852909	0.0644739
$176$	0	0
$177$	4.78788	0.359879
$178$	0	0
$179$	6.85112	0.512077	0.256038	−	0.966667i	$-0.417583\pi$
$179$	6.85112	0.512077	0.256038	+	0.966667i	$0.417583\pi$
$180$	0	0
$181$	−15.3603	−1.14172	−0.570862	−	0.821046i	$-0.693391\pi$
$181$	−15.3603	−1.14172	−0.570862	+	0.821046i	$0.693391\pi$
$182$	0	0
$183$	0.170230	0.0125838
$184$	0	0
$185$	2.22096	0.163288
$186$	0	0
$187$	0.650737	0.0475866
$188$	0	0
$189$	−0.698399	−0.0508011
$190$	0	0
$191$	14.9438	1.08130	0.540648	−	0.841249i	$-0.318179\pi$
$191$	14.9438	1.08130	0.540648	+	0.841249i	$0.318179\pi$
$192$	0	0
$193$	−21.3141	−1.53422	−0.767111	−	0.641514i	$-0.778307\pi$
$193$	−21.3141	−1.53422	−0.767111	+	0.641514i	$0.778307\pi$
$194$	0	0
$195$	−1.11038	−0.0795161
$196$	0	0
$197$	−6.98752	−0.497840	−0.248920	−	0.968524i	$-0.580076\pi$
$197$	−6.98752	−0.497840	−0.248920	+	0.968524i	$0.580076\pi$
$198$	0	0
$199$	−8.93797	−0.633596	−0.316798	−	0.948493i	$-0.602608\pi$
$199$	−8.93797	−0.633596	−0.316798	+	0.948493i	$0.602608\pi$
$200$	0	0
$201$	10.6548	0.751531
$202$	0	0
$203$	0.245818	0.0172531
$204$	0	0
$205$	−1.31295	−0.0917002
$206$	0	0
$207$	−10.1088	−0.702610
$208$	0	0
$209$	37.6232	2.60245
$210$	0	0
$211$	−26.6715	−1.83614	−0.918070	−	0.396419i	$-0.870253\pi$
$211$	−26.6715	−1.83614	−0.918070	+	0.396419i	$0.870253\pi$
$212$	0	0
$213$	−2.76135	−0.189204
$214$	0	0
$215$	−1.68419	−0.114861
$216$	0	0
$217$	1.50962	0.102480
$218$	0	0
$219$	1.07886	0.0729026
$220$	0	0
$221$	−0.340589	−0.0229105
$222$	0	0
$223$	−3.46480	−0.232020	−0.116010	−	0.993248i	$-0.537011\pi$
$223$	−3.46480	−0.232020	−0.116010	+	0.993248i	$0.537011\pi$
$224$	0	0
$225$	11.7106	0.780704
$226$	0	0
$227$	−20.8856	−1.38622	−0.693112	−	0.720830i	$-0.743761\pi$
$227$	−20.8856	−1.38622	−0.693112	+	0.720830i	$0.743761\pi$
$228$	0	0
$229$	−10.4052	−0.687593	−0.343796	−	0.939044i	$-0.611713\pi$
$229$	−10.4052	−0.687593	−0.343796	+	0.939044i	$0.611713\pi$
$230$	0	0
$231$	−0.640809	−0.0421621
$232$	0	0
$233$	5.91067	0.387221	0.193611	−	0.981078i	$-0.437980\pi$
$233$	5.91067	0.387221	0.193611	+	0.981078i	$0.437980\pi$
$234$	0	0
$235$	3.01439	0.196637
$236$	0	0
$237$	−5.57006	−0.361814
$238$	0	0
$239$	10.2644	0.663946	0.331973	−	0.943289i	$-0.392286\pi$
$239$	10.2644	0.663946	0.331973	+	0.943289i	$0.392286\pi$
$240$	0	0
$241$	−0.544851	−0.0350970	−0.0175485	−	0.999846i	$-0.505586\pi$
$241$	−0.544851	−0.0350970	−0.0175485	+	0.999846i	$0.505586\pi$
$242$	0	0
$243$	−14.7907	−0.948821
$244$	0	0
$245$	−4.25054	−0.271557
$246$	0	0
$247$	−19.6916	−1.25294
$248$	0	0
$249$	10.3124	0.653521
$250$	0	0
$251$	1.46231	0.0923003	0.0461501	−	0.998935i	$-0.485305\pi$
$251$	1.46231	0.0923003	0.0461501	+	0.998935i	$0.485305\pi$
$252$	0	0
$253$	−20.2711	−1.27444
$254$	0	0
$255$	0.0536127	0.00335736
$256$	0	0
$257$	−5.88199	−0.366909	−0.183454	−	0.983028i	$-0.558728\pi$
$257$	−5.88199	−0.366909	−0.183454	+	0.983028i	$0.558728\pi$
$258$	0	0
$259$	−0.670844	−0.0416842
$260$	0	0
$261$	3.37512	0.208914
$262$	0	0
$263$	25.6134	1.57939	0.789694	−	0.613501i	$-0.210240\pi$
$263$	25.6134	1.57939	0.789694	+	0.613501i	$0.210240\pi$
$264$	0	0
$265$	−0.461567	−0.0283538
$266$	0	0
$267$	−2.52084	−0.154273
$268$	0	0
$269$	−32.1214	−1.95847	−0.979237	−	0.202717i	$-0.935023\pi$
$269$	−32.1214	−1.95847	−0.979237	+	0.202717i	$0.935023\pi$
$270$	0	0
$271$	−19.4060	−1.17883	−0.589415	−	0.807831i	$-0.700642\pi$
$271$	−19.4060	−1.17883	−0.589415	+	0.807831i	$0.700642\pi$
$272$	0	0
$273$	0.335392	0.0202989
$274$	0	0
$275$	23.4831	1.41609
$276$	0	0
$277$	14.5086	0.871739	0.435869	−	0.900010i	$-0.356441\pi$
$277$	14.5086	0.871739	0.435869	+	0.900010i	$0.356441\pi$
$278$	0	0
$279$	20.7273	1.24091
$280$	0	0
$281$	14.7215	0.878212	0.439106	−	0.898435i	$-0.355295\pi$
$281$	14.7215	0.878212	0.439106	+	0.898435i	$0.355295\pi$
$282$	0	0
$283$	−20.5585	−1.22207	−0.611037	−	0.791602i	$-0.709247\pi$
$283$	−20.5585	−1.22207	−0.611037	+	0.791602i	$0.709247\pi$
$284$	0	0
$285$	3.09969	0.183610
$286$	0	0
$287$	0.396577	0.0234092
$288$	0	0
$289$	−16.9836	−0.999033
$290$	0	0
$291$	−7.05699	−0.413688
$292$	0	0
$293$	14.3988	0.841188	0.420594	−	0.907249i	$-0.361822\pi$
$293$	14.3988	0.841188	0.420594	+	0.907249i	$0.361822\pi$
$294$	0	0
$295$	−4.26389	−0.248253
$296$	0	0
$297$	−19.2290	−1.11578
$298$	0	0
$299$	10.6097	0.613575
$300$	0	0
$301$	0.508713	0.0293217
$302$	0	0
$303$	−11.0965	−0.637475
$304$	0	0
$305$	−0.151600	−0.00868060
$306$	0	0
$307$	1.89578	0.108198	0.0540991	−	0.998536i	$-0.482771\pi$
$307$	1.89578	0.108198	0.0540991	+	0.998536i	$0.482771\pi$
$308$	0	0
$309$	−1.59969	−0.0910032
$310$	0	0
$311$	15.4944	0.878609	0.439305	−	0.898338i	$-0.355225\pi$
$311$	15.4944	0.878609	0.439305	+	0.898338i	$0.355225\pi$
$312$	0	0
$313$	20.2090	1.14228	0.571139	−	0.820854i	$-0.306502\pi$
$313$	20.2090	1.14228	0.571139	+	0.820854i	$0.306502\pi$
$314$	0	0
$315$	0.284586	0.0160346
$316$	0	0
$317$	−23.5767	−1.32420	−0.662099	−	0.749417i	$-0.730334\pi$
$317$	−23.5767	−1.32420	−0.662099	+	0.749417i	$0.730334\pi$
$318$	0	0
$319$	6.76810	0.378941
$320$	0	0
$321$	−3.94198	−0.220020
$322$	0	0
$323$	0.950770	0.0529023
$324$	0	0
$325$	−12.2908	−0.681772
$326$	0	0
$327$	−8.41080	−0.465118
$328$	0	0
$329$	−0.910500	−0.0501975
$330$	0	0
$331$	−18.2487	−1.00304	−0.501521	−	0.865146i	$-0.667226\pi$
$331$	−18.2487	−1.00304	−0.501521	+	0.865146i	$0.667226\pi$
$332$	0	0
$333$	−9.21078	−0.504747
$334$	0	0
$335$	−9.48873	−0.518424
$336$	0	0
$337$	−13.1625	−0.717009	−0.358504	−	0.933528i	$-0.616713\pi$
$337$	−13.1625	−0.717009	−0.358504	+	0.933528i	$0.616713\pi$
$338$	0	0
$339$	−9.63658	−0.523387
$340$	0	0
$341$	41.5643	2.25083
$342$	0	0
$343$	2.57402	0.138984
$344$	0	0
$345$	−1.67009	−0.0899147
$346$	0	0
$347$	−20.2699	−1.08814	−0.544072	−	0.839038i	$-0.683118\pi$
$347$	−20.2699	−1.08814	−0.544072	+	0.839038i	$0.683118\pi$
$348$	0	0
$349$	−15.5167	−0.830591	−0.415296	−	0.909686i	$-0.636322\pi$
$349$	−15.5167	−0.830591	−0.415296	+	0.909686i	$0.636322\pi$
$350$	0	0
$351$	10.0643	0.537190
$352$	0	0
$353$	−10.3568	−0.551237	−0.275619	−	0.961267i	$-0.588883\pi$
$353$	−10.3568	−0.551237	−0.275619	+	0.961267i	$0.588883\pi$
$354$	0	0
$355$	2.45914	0.130518
$356$	0	0
$357$	−0.0161938	−0.000857066 0
$358$	0	0
$359$	1.12973	0.0596249	0.0298125	−	0.999556i	$-0.490509\pi$
$359$	1.12973	0.0596249	0.0298125	+	0.999556i	$0.490509\pi$
$360$	0	0
$361$	35.9700	1.89316
$362$	0	0
$363$	−10.1065	−0.530455
$364$	0	0
$365$	−0.960789	−0.0502900
$366$	0	0
$367$	8.33352	0.435006	0.217503	−	0.976060i	$-0.430209\pi$
$367$	8.33352	0.435006	0.217503	+	0.976060i	$0.430209\pi$
$368$	0	0
$369$	5.44506	0.283458
$370$	0	0
$371$	0.139417	0.00723817
$372$	0	0
$373$	−33.1243	−1.71511	−0.857556	−	0.514391i	$-0.828018\pi$
$373$	−33.1243	−1.71511	−0.857556	+	0.514391i	$0.828018\pi$
$374$	0	0
$375$	4.02510	0.207855
$376$	0	0
$377$	−3.54235	−0.182440
$378$	0	0
$379$	1.19041	0.0611472	0.0305736	−	0.999533i	$-0.490267\pi$
$379$	1.19041	0.0611472	0.0305736	+	0.999533i	$0.490267\pi$
$380$	0	0
$381$	15.2864	0.783145
$382$	0	0
$383$	−5.60495	−0.286399	−0.143200	−	0.989694i	$-0.545739\pi$
$383$	−5.60495	−0.286399	−0.143200	+	0.989694i	$0.545739\pi$
$384$	0	0
$385$	0.570678	0.0290844
$386$	0	0
$387$	6.98469	0.355052
$388$	0	0
$389$	−29.7226	−1.50700	−0.753498	−	0.657450i	$-0.771635\pi$
$389$	−29.7226	−1.50700	−0.753498	+	0.657450i	$0.771635\pi$
$390$	0	0
$391$	−0.512269	−0.0259066
$392$	0	0
$393$	−2.09091	−0.105473
$394$	0	0
$395$	4.96047	0.249588
$396$	0	0
$397$	1.62739	0.0816765	0.0408382	−	0.999166i	$-0.486997\pi$
$397$	1.62739	0.0816765	0.0408382	+	0.999166i	$0.486997\pi$
$398$	0	0
$399$	−0.936264	−0.0468718
$400$	0	0
$401$	−15.9783	−0.797918	−0.398959	−	0.916969i	$-0.630628\pi$
$401$	−15.9783	−0.797918	−0.398959	+	0.916969i	$0.630628\pi$
$402$	0	0
$403$	−21.7543	−1.08366
$404$	0	0
$405$	3.04804	0.151458
$406$	0	0
$407$	−18.4703	−0.915540
$408$	0	0
$409$	6.78228	0.335362	0.167681	−	0.985841i	$-0.446372\pi$
$409$	6.78228	0.335362	0.167681	+	0.985841i	$0.446372\pi$
$410$	0	0
$411$	−1.13482	−0.0559768
$412$	0	0
$413$	1.28791	0.0633741
$414$	0	0
$415$	−9.18379	−0.450815
$416$	0	0
$417$	−6.75348	−0.330719
$418$	0	0
$419$	−23.3434	−1.14040	−0.570200	−	0.821506i	$-0.693134\pi$
$419$	−23.3434	−1.14040	−0.570200	+	0.821506i	$0.693134\pi$
$420$	0	0
$421$	−14.2208	−0.693078	−0.346539	−	0.938036i	$-0.612643\pi$
$421$	−14.2208	−0.693078	−0.346539	+	0.938036i	$0.612643\pi$
$422$	0	0
$423$	−12.5013	−0.607833
$424$	0	0
$425$	0.593439	0.0287860
$426$	0	0
$427$	0.0457911	0.00221598
$428$	0	0
$429$	9.23435	0.445839
$430$	0	0
$431$	25.9317	1.24909	0.624543	−	0.780990i	$-0.285285\pi$
$431$	25.9317	1.24909	0.624543	+	0.780990i	$0.285285\pi$
$432$	0	0
$433$	28.9884	1.39309	0.696546	−	0.717512i	$-0.254719\pi$
$433$	28.9884	1.39309	0.696546	+	0.717512i	$0.254719\pi$
$434$	0	0
$435$	0.557608	0.0267353
$436$	0	0
$437$	−29.6175	−1.41680
$438$	0	0
$439$	−10.8487	−0.517778	−0.258889	−	0.965907i	$-0.583356\pi$
$439$	−10.8487	−0.517778	−0.258889	+	0.965907i	$0.583356\pi$
$440$	0	0
$441$	17.6279	0.839422
$442$	0	0
$443$	8.31119	0.394876	0.197438	−	0.980315i	$-0.436738\pi$
$443$	8.31119	0.394876	0.197438	+	0.980315i	$0.436738\pi$
$444$	0	0
$445$	2.24496	0.106421
$446$	0	0
$447$	10.8993	0.515520
$448$	0	0
$449$	−25.5362	−1.20513	−0.602563	−	0.798071i	$-0.705854\pi$
$449$	−25.5362	−1.20513	−0.602563	+	0.798071i	$0.705854\pi$
$450$	0	0
$451$	10.9189	0.514153
$452$	0	0
$453$	−10.0454	−0.471973
$454$	0	0
$455$	−0.298687	−0.0140027
$456$	0	0
$457$	−15.4963	−0.724888	−0.362444	−	0.932006i	$-0.618058\pi$
$457$	−15.4963	−0.724888	−0.362444	+	0.932006i	$0.618058\pi$
$458$	0	0
$459$	−0.485934	−0.0226814
$460$	0	0
$461$	−2.39851	−0.111710	−0.0558549	−	0.998439i	$-0.517788\pi$
$461$	−2.39851	−0.111710	−0.0558549	+	0.998439i	$0.517788\pi$
$462$	0	0
$463$	26.6280	1.23751	0.618753	−	0.785585i	$-0.287638\pi$
$463$	26.6280	1.23751	0.618753	+	0.785585i	$0.287638\pi$
$464$	0	0
$465$	3.42439	0.158802
$466$	0	0
$467$	22.6995	1.05041	0.525204	−	0.850976i	$-0.323989\pi$
$467$	22.6995	1.05041	0.525204	+	0.850976i	$0.323989\pi$
$468$	0	0
$469$	2.86608	0.132343
$470$	0	0
$471$	2.94745	0.135812
$472$	0	0
$473$	14.0064	0.644014
$474$	0	0
$475$	34.3104	1.57427
$476$	0	0
$477$	1.91421	0.0876458
$478$	0	0
$479$	−25.2174	−1.15221	−0.576107	−	0.817374i	$-0.695429\pi$
$479$	−25.2174	−1.15221	−0.576107	+	0.817374i	$0.695429\pi$
$480$	0	0
$481$	9.66717	0.440785
$482$	0	0
$483$	0.504454	0.0229534
$484$	0	0
$485$	6.28466	0.285372
$486$	0	0
$487$	−10.2524	−0.464580	−0.232290	−	0.972647i	$-0.574622\pi$
$487$	−10.2524	−0.464580	−0.232290	+	0.972647i	$0.574622\pi$
$488$	0	0
$489$	−12.7806	−0.577960
$490$	0	0
$491$	19.6373	0.886221	0.443110	−	0.896467i	$-0.353875\pi$
$491$	19.6373	0.886221	0.443110	+	0.896467i	$0.353875\pi$
$492$	0	0
$493$	0.171036	0.00770306
$494$	0	0
$495$	7.83548	0.352179
$496$	0	0
$497$	−0.742787	−0.0333185
$498$	0	0
$499$	14.3837	0.643902	0.321951	−	0.946756i	$-0.395661\pi$
$499$	14.3837	0.643902	0.321951	+	0.946756i	$0.395661\pi$
$500$	0	0
$501$	−0.685166	−0.0306110
$502$	0	0
$503$	37.4799	1.67115	0.835573	−	0.549380i	$-0.185136\pi$
$503$	37.4799	1.67115	0.835573	+	0.549380i	$0.185136\pi$
$504$	0	0
$505$	9.88205	0.439745
$506$	0	0
$507$	4.07400	0.180933
$508$	0	0
$509$	−21.0508	−0.933060	−0.466530	−	0.884505i	$-0.654496\pi$
$509$	−21.0508	−0.933060	−0.466530	+	0.884505i	$0.654496\pi$
$510$	0	0
$511$	0.290208	0.0128380
$512$	0	0
$513$	−28.0949	−1.24042
$514$	0	0
$515$	1.42462	0.0627762
$516$	0	0
$517$	−25.0688	−1.10252
$518$	0	0
$519$	−15.4091	−0.676386
$520$	0	0
$521$	−13.2544	−0.580685	−0.290342	−	0.956923i	$-0.593769\pi$
$521$	−13.2544	−0.580685	−0.290342	+	0.956923i	$0.593769\pi$
$522$	0	0
$523$	−4.16816	−0.182261	−0.0911304	−	0.995839i	$-0.529048\pi$
$523$	−4.16816	−0.182261	−0.0911304	+	0.995839i	$0.529048\pi$
$524$	0	0
$525$	−0.584385	−0.0255046
$526$	0	0
$527$	1.05037	0.0457547
$528$	0	0
$529$	−7.04228	−0.306186
$530$	0	0
$531$	17.6832	0.767387
$532$	0	0
$533$	−5.71486	−0.247538
$534$	0	0
$535$	3.51056	0.151775
$536$	0	0
$537$	−4.69416	−0.202568
$538$	0	0
$539$	35.3491	1.52259
$540$	0	0
$541$	−20.0822	−0.863402	−0.431701	−	0.902017i	$-0.642086\pi$
$541$	−20.0822	−0.863402	−0.431701	+	0.902017i	$0.642086\pi$
$542$	0	0
$543$	10.5244	0.451645
$544$	0	0
$545$	7.49031	0.320850
$546$	0	0
$547$	−20.6504	−0.882949	−0.441474	−	0.897274i	$-0.645544\pi$
$547$	−20.6504	−0.882949	−0.441474	+	0.897274i	$0.645544\pi$
$548$	0	0
$549$	0.628717	0.0268330
$550$	0	0
$551$	9.88865	0.421271
$552$	0	0
$553$	−1.49832	−0.0637149
$554$	0	0
$555$	−1.52173	−0.0645937
$556$	0	0
$557$	−31.4316	−1.33180	−0.665900	−	0.746041i	$-0.731952\pi$
$557$	−31.4316	−1.33180	−0.665900	+	0.746041i	$0.731952\pi$
$558$	0	0
$559$	−7.33079	−0.310059
$560$	0	0
$561$	−0.445863	−0.0188244
$562$	0	0
$563$	−5.67831	−0.239312	−0.119656	−	0.992815i	$-0.538179\pi$
$563$	−5.67831	−0.239312	−0.119656	+	0.992815i	$0.538179\pi$
$564$	0	0
$565$	8.58194	0.361045
$566$	0	0
$567$	−0.920666	−0.0386643
$568$	0	0
$569$	−5.58704	−0.234221	−0.117110	−	0.993119i	$-0.537363\pi$
$569$	−5.58704	−0.234221	−0.117110	+	0.993119i	$0.537363\pi$
$570$	0	0
$571$	−5.21501	−0.218241	−0.109121	−	0.994029i	$-0.534803\pi$
$571$	−5.21501	−0.218241	−0.109121	+	0.994029i	$0.534803\pi$
$572$	0	0
$573$	−10.2390	−0.427740
$574$	0	0
$575$	−18.4862	−0.770930
$576$	0	0
$577$	20.4310	0.850554	0.425277	−	0.905063i	$-0.360177\pi$
$577$	20.4310	0.850554	0.425277	+	0.905063i	$0.360177\pi$
$578$	0	0
$579$	14.6037	0.606909
$580$	0	0
$581$	2.77398	0.115084
$582$	0	0
$583$	3.83856	0.158977
$584$	0	0
$585$	−4.10101	−0.169556
$586$	0	0
$587$	41.9436	1.73120	0.865599	−	0.500739i	$-0.166938\pi$
$587$	41.9436	1.73120	0.865599	+	0.500739i	$0.166938\pi$
$588$	0	0
$589$	60.7283	2.50226
$590$	0	0
$591$	4.78761	0.196936
$592$	0	0
$593$	23.5669	0.967775	0.483887	−	0.875130i	$-0.339224\pi$
$593$	23.5669	0.967775	0.483887	+	0.875130i	$0.339224\pi$
$594$	0	0
$595$	0.0144215	0.000591225 0
$596$	0	0
$597$	6.12399	0.250638
$598$	0	0
$599$	24.8325	1.01463	0.507314	−	0.861761i	$-0.330638\pi$
$599$	24.8325	1.01463	0.507314	+	0.861761i	$0.330638\pi$
$600$	0	0
$601$	33.1836	1.35359	0.676795	−	0.736172i	$-0.263369\pi$
$601$	33.1836	1.35359	0.676795	+	0.736172i	$0.263369\pi$
$602$	0	0
$603$	39.3517	1.60252
$604$	0	0
$605$	9.00046	0.365921
$606$	0	0
$607$	14.5976	0.592500	0.296250	−	0.955110i	$-0.404264\pi$
$607$	14.5976	0.592500	0.296250	+	0.955110i	$0.404264\pi$
$608$	0	0
$609$	−0.168426	−0.00682498
$610$	0	0
$611$	13.1207	0.530808
$612$	0	0
$613$	6.58591	0.266002	0.133001	−	0.991116i	$-0.457539\pi$
$613$	6.58591	0.266002	0.133001	+	0.991116i	$0.457539\pi$
$614$	0	0
$615$	0.899587	0.0362749
$616$	0	0
$617$	−35.0859	−1.41250	−0.706252	−	0.707960i	$-0.749616\pi$
$617$	−35.0859	−1.41250	−0.706252	+	0.707960i	$0.749616\pi$
$618$	0	0
$619$	18.3113	0.735994	0.367997	−	0.929827i	$-0.380044\pi$
$619$	18.3113	0.735994	0.367997	+	0.929827i	$0.380044\pi$
$620$	0	0
$621$	15.1373	0.607441
$622$	0	0
$623$	−0.678093	−0.0271672
$624$	0	0
$625$	19.5538	0.782152
$626$	0	0
$627$	−25.7781	−1.02948
$628$	0	0
$629$	−0.466761	−0.0186110
$630$	0	0
$631$	−24.0740	−0.958373	−0.479186	−	0.877713i	$-0.659068\pi$
$631$	−24.0740	−0.958373	−0.479186	+	0.877713i	$0.659068\pi$
$632$	0	0
$633$	18.2744	0.726342
$634$	0	0
$635$	−13.6134	−0.540232
$636$	0	0
$637$	−18.5013	−0.733049
$638$	0	0
$639$	−10.1986	−0.403449
$640$	0	0
$641$	4.77393	0.188559	0.0942795	−	0.995546i	$-0.469945\pi$
$641$	4.77393	0.188559	0.0942795	+	0.995546i	$0.469945\pi$
$642$	0	0
$643$	0.669908	0.0264186	0.0132093	−	0.999913i	$-0.495795\pi$
$643$	0.669908	0.0264186	0.0132093	+	0.999913i	$0.495795\pi$
$644$	0	0
$645$	1.15395	0.0454368
$646$	0	0
$647$	24.8779	0.978049	0.489025	−	0.872270i	$-0.337353\pi$
$647$	24.8779	0.978049	0.489025	+	0.872270i	$0.337353\pi$
$648$	0	0
$649$	35.4601	1.39193
$650$	0	0
$651$	−1.03434	−0.0405390
$652$	0	0
$653$	−30.7916	−1.20497	−0.602484	−	0.798131i	$-0.705822\pi$
$653$	−30.7916	−1.20497	−0.602484	+	0.798131i	$0.705822\pi$
$654$	0	0
$655$	1.86208	0.0727575
$656$	0	0
$657$	3.98459	0.155454
$658$	0	0
$659$	−33.8998	−1.32055	−0.660274	−	0.751025i	$-0.729560\pi$
$659$	−33.8998	−1.32055	−0.660274	+	0.751025i	$0.729560\pi$
$660$	0	0
$661$	33.4469	1.30094	0.650468	−	0.759534i	$-0.274573\pi$
$661$	33.4469	1.30094	0.650468	+	0.759534i	$0.274573\pi$
$662$	0	0
$663$	0.233360	0.00906295
$664$	0	0
$665$	0.833799	0.0323333
$666$	0	0
$667$	−5.32794	−0.206299
$668$	0	0
$669$	2.37397	0.0917829
$670$	0	0
$671$	1.26076	0.0486712
$672$	0	0
$673$	−20.2228	−0.779532	−0.389766	−	0.920914i	$-0.627444\pi$
$673$	−20.2228	−0.779532	−0.389766	+	0.920914i	$0.627444\pi$
$674$	0	0
$675$	−17.5359	−0.674956
$676$	0	0
$677$	31.5078	1.21094	0.605471	−	0.795868i	$-0.292985\pi$
$677$	31.5078	1.21094	0.605471	+	0.795868i	$0.292985\pi$
$678$	0	0
$679$	−1.89829	−0.0728497
$680$	0	0
$681$	14.3101	0.548364
$682$	0	0
$683$	−14.9258	−0.571119	−0.285560	−	0.958361i	$-0.592179\pi$
$683$	−14.9258	−0.571119	−0.285560	+	0.958361i	$0.592179\pi$
$684$	0	0
$685$	1.01063	0.0386141
$686$	0	0
$687$	7.12927	0.271999
$688$	0	0
$689$	−2.00906	−0.0765392
$690$	0	0
$691$	−7.34883	−0.279563	−0.139781	−	0.990182i	$-0.544640\pi$
$691$	−7.34883	−0.279563	−0.139781	+	0.990182i	$0.544640\pi$
$692$	0	0
$693$	−2.36672	−0.0899042
$694$	0	0
$695$	6.01437	0.228138
$696$	0	0
$697$	0.275931	0.0104516
$698$	0	0
$699$	−4.04980	−0.153177
$700$	0	0
$701$	37.5538	1.41839	0.709193	−	0.705014i	$-0.249059\pi$
$701$	37.5538	1.41839	0.709193	+	0.705014i	$0.249059\pi$
$702$	0	0
$703$	−26.9864	−1.01781
$704$	0	0
$705$	−2.06536	−0.0777859
$706$	0	0
$707$	−2.98489	−0.112258
$708$	0	0
$709$	−43.3729	−1.62890	−0.814451	−	0.580232i	$-0.802962\pi$
$709$	−43.3729	−1.62890	−0.814451	+	0.580232i	$0.802962\pi$
$710$	0	0
$711$	−20.5721	−0.771513
$712$	0	0
$713$	−32.7200	−1.22537
$714$	0	0
$715$	−8.22373	−0.307550
$716$	0	0
$717$	−7.03279	−0.262644
$718$	0	0
$719$	36.6696	1.36755	0.683774	−	0.729694i	$-0.260338\pi$
$719$	36.6696	1.36755	0.683774	+	0.729694i	$0.260338\pi$
$720$	0	0
$721$	−0.430308	−0.0160255
$722$	0	0
$723$	0.373314	0.0138837
$724$	0	0
$725$	6.17216	0.229228
$726$	0	0
$727$	40.9797	1.51985	0.759926	−	0.650009i	$-0.225235\pi$
$727$	40.9797	1.51985	0.759926	+	0.650009i	$0.225235\pi$
$728$	0	0
$729$	−4.85186	−0.179699
$730$	0	0
$731$	0.353953	0.0130914
$732$	0	0
$733$	−21.8632	−0.807535	−0.403768	−	0.914862i	$-0.632300\pi$
$733$	−21.8632	−0.807535	−0.403768	+	0.914862i	$0.632300\pi$
$734$	0	0
$735$	2.91233	0.107423
$736$	0	0
$737$	78.9117	2.90675
$738$	0	0
$739$	−23.2901	−0.856741	−0.428370	−	0.903603i	$-0.640912\pi$
$739$	−23.2901	−0.856741	−0.428370	+	0.903603i	$0.640912\pi$
$740$	0	0
$741$	13.4920	0.495641
$742$	0	0
$743$	9.64619	0.353884	0.176942	−	0.984221i	$-0.443379\pi$
$743$	9.64619	0.353884	0.176942	+	0.984221i	$0.443379\pi$
$744$	0	0
$745$	−9.70649	−0.355618
$746$	0	0
$747$	38.0871	1.39353
$748$	0	0
$749$	−1.06037	−0.0387451
$750$	0	0
$751$	−12.2736	−0.447870	−0.223935	−	0.974604i	$-0.571890\pi$
$751$	−12.2736	−0.447870	−0.223935	+	0.974604i	$0.571890\pi$
$752$	0	0
$753$	−1.00193	−0.0365122
$754$	0	0
$755$	8.94600	0.325578
$756$	0	0
$757$	−25.4007	−0.923206	−0.461603	−	0.887087i	$-0.652725\pi$
$757$	−25.4007	−0.923206	−0.461603	+	0.887087i	$0.652725\pi$
$758$	0	0
$759$	13.8891	0.504142
$760$	0	0
$761$	−46.1364	−1.67244	−0.836221	−	0.548392i	$-0.815240\pi$
$761$	−46.1364	−1.67244	−0.836221	+	0.548392i	$0.815240\pi$
$762$	0	0
$763$	−2.26246	−0.0819065
$764$	0	0
$765$	0.198009	0.00715905
$766$	0	0
$767$	−18.5594	−0.670143
$768$	0	0
$769$	2.04597	0.0737797	0.0368899	−	0.999319i	$-0.488255\pi$
$769$	2.04597	0.0737797	0.0368899	+	0.999319i	$0.488255\pi$
$770$	0	0
$771$	4.03015	0.145142
$772$	0	0
$773$	33.1862	1.19362	0.596812	−	0.802381i	$-0.296434\pi$
$773$	33.1862	1.19362	0.596812	+	0.802381i	$0.296434\pi$
$774$	0	0
$775$	37.9045	1.36157
$776$	0	0
$777$	0.459640	0.0164895
$778$	0	0
$779$	15.9533	0.571587
$780$	0	0
$781$	−20.4511	−0.731799
$782$	0	0
$783$	−5.05404	−0.180617
$784$	0	0
$785$	−2.62488	−0.0936861
$786$	0	0
$787$	−12.0389	−0.429140	−0.214570	−	0.976709i	$-0.568835\pi$
$787$	−12.0389	−0.429140	−0.214570	+	0.976709i	$0.568835\pi$
$788$	0	0
$789$	−17.5494	−0.624776
$790$	0	0
$791$	−2.59219	−0.0921676
$792$	0	0
$793$	−0.659870	−0.0234327
$794$	0	0
$795$	0.316250	0.0112162
$796$	0	0
$797$	−26.5238	−0.939520	−0.469760	−	0.882794i	$-0.655659\pi$
$797$	−26.5238	−0.939520	−0.469760	+	0.882794i	$0.655659\pi$
$798$	0	0
$799$	−0.633509	−0.0224119
$800$	0	0
$801$	−9.31031	−0.328964
$802$	0	0
$803$	7.99027	0.281971
$804$	0	0
$805$	−0.449246	−0.0158338
$806$	0	0
$807$	22.0085	0.774736
$808$	0	0
$809$	−38.4463	−1.35170	−0.675850	−	0.737039i	$-0.736223\pi$
$809$	−38.4463	−1.35170	−0.675850	+	0.737039i	$0.736223\pi$
$810$	0	0
$811$	21.5375	0.756283	0.378142	−	0.925748i	$-0.376563\pi$
$811$	21.5375	0.756283	0.378142	+	0.925748i	$0.376563\pi$
$812$	0	0
$813$	13.2963	0.466323
$814$	0	0
$815$	11.3819	0.398691
$816$	0	0
$817$	20.4642	0.715953
$818$	0	0
$819$	1.23872	0.0432842
$820$	0	0
$821$	17.2545	0.602185	0.301093	−	0.953595i	$-0.402649\pi$
$821$	17.2545	0.602185	0.301093	+	0.953595i	$0.402649\pi$
$822$	0	0
$823$	5.77925	0.201452	0.100726	−	0.994914i	$-0.467883\pi$
$823$	5.77925	0.201452	0.100726	+	0.994914i	$0.467883\pi$
$824$	0	0
$825$	−16.0898	−0.560177
$826$	0	0
$827$	6.80055	0.236478	0.118239	−	0.992985i	$-0.462275\pi$
$827$	6.80055	0.236478	0.118239	+	0.992985i	$0.462275\pi$
$828$	0	0
$829$	17.1422	0.595374	0.297687	−	0.954664i	$-0.403785\pi$
$829$	17.1422	0.595374	0.297687	+	0.954664i	$0.403785\pi$
$830$	0	0
$831$	−9.94082	−0.344843
$832$	0	0
$833$	0.893302	0.0309511
$834$	0	0
$835$	0.610181	0.0211162
$836$	0	0
$837$	−31.0379	−1.07283
$838$	0	0
$839$	−16.8702	−0.582425	−0.291212	−	0.956658i	$-0.594059\pi$
$839$	−16.8702	−0.582425	−0.291212	+	0.956658i	$0.594059\pi$
$840$	0	0
$841$	−27.2211	−0.938659
$842$	0	0
$843$	−10.0867	−0.347404
$844$	0	0
$845$	−3.62814	−0.124812
$846$	0	0
$847$	−2.71860	−0.0934123
$848$	0	0
$849$	14.0860	0.483430
$850$	0	0
$851$	14.5401	0.498428
$852$	0	0
$853$	26.8145	0.918112	0.459056	−	0.888407i	$-0.348188\pi$
$853$	26.8145	0.918112	0.459056	+	0.888407i	$0.348188\pi$
$854$	0	0
$855$	11.4482	0.391519
$856$	0	0
$857$	17.3952	0.594209	0.297105	−	0.954845i	$-0.403979\pi$
$857$	17.3952	0.594209	0.297105	+	0.954845i	$0.403979\pi$
$858$	0	0
$859$	38.7412	1.32183	0.660916	−	0.750460i	$-0.270168\pi$
$859$	38.7412	1.32183	0.660916	+	0.750460i	$0.270168\pi$
$860$	0	0
$861$	−0.271721	−0.00926024
$862$	0	0
$863$	32.1233	1.09349	0.546745	−	0.837299i	$-0.315867\pi$
$863$	32.1233	1.09349	0.546745	+	0.837299i	$0.315867\pi$
$864$	0	0
$865$	13.7227	0.466587
$866$	0	0
$867$	11.6366	0.395198
$868$	0	0
$869$	−41.2531	−1.39941
$870$	0	0
$871$	−41.3016	−1.39945
$872$	0	0
$873$	−26.0638	−0.882126
$874$	0	0
$875$	1.08273	0.0366029
$876$	0	0
$877$	−27.3296	−0.922854	−0.461427	−	0.887178i	$-0.652662\pi$
$877$	−27.3296	−0.922854	−0.461427	+	0.887178i	$0.652662\pi$
$878$	0	0
$879$	−9.86558	−0.332758
$880$	0	0
$881$	4.90030	0.165095	0.0825476	−	0.996587i	$-0.473694\pi$
$881$	4.90030	0.165095	0.0825476	+	0.996587i	$0.473694\pi$
$882$	0	0
$883$	−53.6493	−1.80544	−0.902722	−	0.430225i	$-0.858434\pi$
$883$	−53.6493	−1.80544	−0.902722	+	0.430225i	$0.858434\pi$
$884$	0	0
$885$	2.92148	0.0982043
$886$	0	0
$887$	−52.1938	−1.75250	−0.876248	−	0.481861i	$-0.839961\pi$
$887$	−52.1938	−1.75250	−0.876248	+	0.481861i	$0.839961\pi$
$888$	0	0
$889$	4.11195	0.137910
$890$	0	0
$891$	−25.3487	−0.849212
$892$	0	0
$893$	−36.6271	−1.22568
$894$	0	0
$895$	4.18043	0.139736
$896$	0	0
$897$	−7.26941	−0.242719
$898$	0	0
$899$	10.9245	0.364353
$900$	0	0
$901$	0.0970038	0.00323167
$902$	0	0
$903$	−0.348553	−0.0115991
$904$	0	0
$905$	−9.37259	−0.311555
$906$	0	0
$907$	−8.77307	−0.291305	−0.145653	−	0.989336i	$-0.546528\pi$
$907$	−8.77307	−0.291305	−0.145653	+	0.989336i	$0.546528\pi$
$908$	0	0
$909$	−40.9829	−1.35932
$910$	0	0
$911$	−54.0083	−1.78937	−0.894687	−	0.446693i	$-0.852602\pi$
$911$	−54.0083	−1.78937	−0.894687	+	0.446693i	$0.852602\pi$
$912$	0	0
$913$	76.3758	2.52767
$914$	0	0
$915$	0.103871	0.00343388
$916$	0	0
$917$	−0.562444	−0.0185735
$918$	0	0
$919$	4.59108	0.151446	0.0757228	−	0.997129i	$-0.475874\pi$
$919$	4.59108	0.151446	0.0757228	+	0.997129i	$0.475874\pi$
$920$	0	0
$921$	−1.29893	−0.0428011
$922$	0	0
$923$	10.7039	0.352323
$924$	0	0
$925$	−16.8440	−0.553827
$926$	0	0
$927$	−5.90819	−0.194050
$928$	0	0
$929$	−20.6593	−0.677808	−0.338904	−	0.940821i	$-0.610056\pi$
$929$	−20.6593	−0.677808	−0.338904	+	0.940821i	$0.610056\pi$
$930$	0	0
$931$	51.6474	1.69267
$932$	0	0
$933$	−10.6163	−0.347561
$934$	0	0
$935$	0.397067	0.0129855
$936$	0	0
$937$	−6.25885	−0.204468	−0.102234	−	0.994760i	$-0.532599\pi$
$937$	−6.25885	−0.204468	−0.102234	+	0.994760i	$0.532599\pi$
$938$	0	0
$939$	−13.8465	−0.451863
$940$	0	0
$941$	−56.1332	−1.82989	−0.914945	−	0.403579i	$-0.867766\pi$
$941$	−56.1332	−1.82989	−0.914945	+	0.403579i	$0.867766\pi$
$942$	0	0
$943$	−8.59555	−0.279910
$944$	0	0
$945$	−0.426150	−0.0138627
$946$	0	0
$947$	−40.8312	−1.32684	−0.663418	−	0.748249i	$-0.730895\pi$
$947$	−40.8312	−1.32684	−0.663418	+	0.748249i	$0.730895\pi$
$948$	0	0
$949$	−4.18203	−0.135754
$950$	0	0
$951$	16.1539	0.523827
$952$	0	0
$953$	−48.9462	−1.58552	−0.792762	−	0.609531i	$-0.791358\pi$
$953$	−48.9462	−1.58552	−0.792762	+	0.609531i	$0.791358\pi$
$954$	0	0
$955$	9.11843	0.295066
$956$	0	0
$957$	−4.63728	−0.149902
$958$	0	0
$959$	−0.305262	−0.00985741
$960$	0	0
$961$	36.0898	1.16419
$962$	0	0
$963$	−14.5590	−0.469158
$964$	0	0
$965$	−13.0055	−0.418661
$966$	0	0
$967$	−21.9422	−0.705614	−0.352807	−	0.935696i	$-0.614773\pi$
$967$	−21.9422	−0.705614	−0.352807	+	0.935696i	$0.614773\pi$
$968$	0	0
$969$	−0.651436	−0.0209271
$970$	0	0
$971$	14.8431	0.476338	0.238169	−	0.971224i	$-0.423453\pi$
$971$	14.8431	0.476338	0.238169	+	0.971224i	$0.423453\pi$
$972$	0	0
$973$	−1.81665	−0.0582391
$974$	0	0
$975$	8.42126	0.269696
$976$	0	0
$977$	10.7071	0.342551	0.171275	−	0.985223i	$-0.445211\pi$
$977$	10.7071	0.342551	0.171275	+	0.985223i	$0.445211\pi$
$978$	0	0
$979$	−18.6699	−0.596693
$980$	0	0
$981$	−31.0639	−0.991793
$982$	0	0
$983$	−4.05032	−0.129185	−0.0645925	−	0.997912i	$-0.520575\pi$
$983$	−4.05032	−0.129185	−0.0645925	+	0.997912i	$0.520575\pi$
$984$	0	0
$985$	−4.26365	−0.135851
$986$	0	0
$987$	0.623844	0.0198572
$988$	0	0
$989$	−11.0260	−0.350607
$990$	0	0
$991$	−23.0339	−0.731696	−0.365848	−	0.930675i	$-0.619221\pi$
$991$	−23.0339	−0.731696	−0.365848	+	0.930675i	$0.619221\pi$
$992$	0	0
$993$	12.5034	0.396784
$994$	0	0
$995$	−5.45378	−0.172896
$996$	0	0
$997$	33.8866	1.07320	0.536600	−	0.843837i	$-0.319709\pi$
$997$	33.8866	1.07320	0.536600	+	0.843837i	$0.319709\pi$
$998$	0	0
$999$	13.7926	0.436378

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	668.2.a.c.1.4	✓	7
3.2	odd	2		6012.2.a.g.1.3		7
4.3	odd	2		2672.2.a.k.1.4		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
668.2.a.c.1.4	✓	7	1.1	even	1	trivial
2672.2.a.k.1.4		7	4.3	odd	2
6012.2.a.g.1.3		7	3.2	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:	\( N \)	\(=\)	\( 668 = 2^{2} \cdot 167 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	668.a (trivial)

\(f(q)\)	\(=\)	\(q-0.685166 q^{3} +0.610181 q^{5} -0.184306 q^{7} -2.53055 q^{9} +O(q^{10})\)	\(q-0.685166 q^{3} +0.610181 q^{5} -0.184306 q^{7} -2.53055 q^{9} -5.07449 q^{11} +2.65594 q^{13} -0.418076 q^{15} -0.128237 q^{17} -7.41417 q^{19} +0.126280 q^{21} +3.99471 q^{23} -4.62768 q^{25} +3.78935 q^{27} -1.33375 q^{29} -8.19083 q^{31} +3.47687 q^{33} -0.112460 q^{35} +3.63984 q^{37} -1.81976 q^{39} -2.15173 q^{41} -2.76015 q^{43} -1.54409 q^{45} +4.94015 q^{47} -6.96603 q^{49} +0.0878636 q^{51} -0.756443 q^{53} -3.09636 q^{55} +5.07994 q^{57} -6.98791 q^{59} -0.248451 q^{61} +0.466395 q^{63} +1.62060 q^{65} -15.5507 q^{67} -2.73704 q^{69} +4.03018 q^{71} -1.57460 q^{73} +3.17073 q^{75} +0.935260 q^{77} +8.12950 q^{79} +4.99531 q^{81} -15.0509 q^{83} -0.0782477 q^{85} +0.913840 q^{87} +3.67917 q^{89} -0.489505 q^{91} +5.61208 q^{93} -4.52399 q^{95} +10.2997 q^{97} +12.8412 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - 4 q^{3} - 2 q^{5} - 12 q^{7} + 7 q^{9}+O(q^{10}) \) 7 * q - 4 * q^3 - 2 * q^5 - 12 * q^7 + 7 * q^9	\( 7 q - 4 q^{3} - 2 q^{5} - 12 q^{7} + 7 q^{9} - 7 q^{11} - 9 q^{13} - 17 q^{15} - q^{17} - 11 q^{19} - 4 q^{21} - 19 q^{23} + 3 q^{25} - 16 q^{27} - 5 q^{29} - 13 q^{31} - 8 q^{33} - 7 q^{35} - 26 q^{37} - 17 q^{39} - 2 q^{41} - 24 q^{43} - 7 q^{45} - 11 q^{47} + 19 q^{49} + 8 q^{51} + 4 q^{53} - 4 q^{55} + 14 q^{57} - 4 q^{59} - 5 q^{61} - 21 q^{63} + 13 q^{65} - 42 q^{67} + 24 q^{69} + 9 q^{71} + 27 q^{73} + 25 q^{75} + 12 q^{77} - 8 q^{79} + 35 q^{81} + 16 q^{83} - 27 q^{85} + 3 q^{87} + 9 q^{89} - 2 q^{91} - 10 q^{93} + 10 q^{95} - 8 q^{97} - 5 q^{99}+O(q^{100}) \) 7 * q - 4 * q^3 - 2 * q^5 - 12 * q^7 + 7 * q^9 - 7 * q^11 - 9 * q^13 - 17 * q^15 - q^17 - 11 * q^19 - 4 * q^21 - 19 * q^23 + 3 * q^25 - 16 * q^27 - 5 * q^29 - 13 * q^31 - 8 * q^33 - 7 * q^35 - 26 * q^37 - 17 * q^39 - 2 * q^41 - 24 * q^43 - 7 * q^45 - 11 * q^47 + 19 * q^49 + 8 * q^51 + 4 * q^53 - 4 * q^55 + 14 * q^57 - 4 * q^59 - 5 * q^61 - 21 * q^63 + 13 * q^65 - 42 * q^67 + 24 * q^69 + 9 * q^71 + 27 * q^73 + 25 * q^75 + 12 * q^77 - 8 * q^79 + 35 * q^81 + 16 * q^83 - 27 * q^85 + 3 * q^87 + 9 * q^89 - 2 * q^91 - 10 * q^93 + 10 * q^95 - 8 * q^97 - 5 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more