LMFDB - Embedded newform 6008.2.a.e.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6008,2,Mod(1,6008)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6008, 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("6008.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6008 = 2^{3} \cdot 751 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6008.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:	$47.9741215344$
Analytic rank:	$0$
Dimension:	$50$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Character	$\chi$	$=$	6008.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.58697 q^{3} +0.0890087 q^{5} +4.21791 q^{7} +3.69244 q^{9} +O(q^{10})$	$q-2.58697 q^{3} +0.0890087 q^{5} +4.21791 q^{7} +3.69244 q^{9} -5.56038 q^{11} +5.65898 q^{13} -0.230263 q^{15} +4.66419 q^{17} +5.92080 q^{19} -10.9116 q^{21} -2.98182 q^{23} -4.99208 q^{25} -1.79132 q^{27} +6.14670 q^{29} -0.725876 q^{31} +14.3846 q^{33} +0.375431 q^{35} +11.8046 q^{37} -14.6396 q^{39} -3.43975 q^{41} +4.18359 q^{43} +0.328659 q^{45} -5.47405 q^{47} +10.7908 q^{49} -12.0662 q^{51} +2.96430 q^{53} -0.494922 q^{55} -15.3170 q^{57} +3.19404 q^{59} +2.66626 q^{61} +15.5744 q^{63} +0.503698 q^{65} -2.11614 q^{67} +7.71389 q^{69} +12.6906 q^{71} -5.71725 q^{73} +12.9144 q^{75} -23.4532 q^{77} +7.85668 q^{79} -6.44322 q^{81} -2.69956 q^{83} +0.415154 q^{85} -15.9013 q^{87} -12.3442 q^{89} +23.8691 q^{91} +1.87782 q^{93} +0.527003 q^{95} -4.06463 q^{97} -20.5314 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 50 q + 6 q^{3} + 23 q^{5} + 12 q^{7} + 56 q^{9}+O(q^{10}) $ 50 * q + 6 * q^3 + 23 * q^5 + 12 * q^7 + 56 * q^9	$ 50 q + 6 q^{3} + 23 q^{5} + 12 q^{7} + 56 q^{9} - 5 q^{11} + 36 q^{13} + 5 q^{15} + 14 q^{17} + 9 q^{19} + 30 q^{21} + 3 q^{23} + 71 q^{25} + 24 q^{27} + 61 q^{29} + 27 q^{31} + 24 q^{33} - 7 q^{35} + 56 q^{37} - 2 q^{39} + 10 q^{41} + 19 q^{43} + 76 q^{45} + 3 q^{47} + 82 q^{49} - q^{51} + 56 q^{53} + 7 q^{55} + 35 q^{57} - q^{59} + 67 q^{61} + 25 q^{63} + 27 q^{65} + 46 q^{67} + 68 q^{69} + 4 q^{71} + 62 q^{73} + 27 q^{75} + 71 q^{77} + 7 q^{79} + 74 q^{81} - q^{83} + 72 q^{85} + 25 q^{87} + 19 q^{89} + 45 q^{91} + 72 q^{93} - 24 q^{95} + 81 q^{97} + 16 q^{99}+O(q^{100}) $ 50 * q + 6 * q^3 + 23 * q^5 + 12 * q^7 + 56 * q^9 - 5 * q^11 + 36 * q^13 + 5 * q^15 + 14 * q^17 + 9 * q^19 + 30 * q^21 + 3 * q^23 + 71 * q^25 + 24 * q^27 + 61 * q^29 + 27 * q^31 + 24 * q^33 - 7 * q^35 + 56 * q^37 - 2 * q^39 + 10 * q^41 + 19 * q^43 + 76 * q^45 + 3 * q^47 + 82 * q^49 - q^51 + 56 * q^53 + 7 * q^55 + 35 * q^57 - q^59 + 67 * q^61 + 25 * q^63 + 27 * q^65 + 46 * q^67 + 68 * q^69 + 4 * q^71 + 62 * q^73 + 27 * q^75 + 71 * q^77 + 7 * q^79 + 74 * q^81 - q^83 + 72 * q^85 + 25 * q^87 + 19 * q^89 + 45 * q^91 + 72 * q^93 - 24 * q^95 + 81 * q^97 + 16 * 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
$3$	−2.58697	−1.49359	−0.746795	−	0.665054i	$-0.768408\pi$
$3$	−2.58697	−1.49359	−0.746795	+	0.665054i	$0.768408\pi$
$4$	0	0
$5$	0.0890087	0.0398059	0.0199029	−	0.999802i	$-0.493664\pi$
$5$	0.0890087	0.0398059	0.0199029	+	0.999802i	$0.493664\pi$
$6$	0	0
$7$	4.21791	1.59422	0.797111	−	0.603833i	$-0.206361\pi$
$7$	4.21791	1.59422	0.797111	+	0.603833i	$0.206361\pi$
$8$	0	0
$9$	3.69244	1.23081
$10$	0	0
$11$	−5.56038	−1.67652	−0.838260	−	0.545271i	$-0.816427\pi$
$11$	−5.56038	−1.67652	−0.838260	+	0.545271i	$0.816427\pi$
$12$	0	0
$13$	5.65898	1.56952	0.784759	−	0.619801i	$-0.212787\pi$
$13$	5.65898	1.56952	0.784759	+	0.619801i	$0.212787\pi$
$14$	0	0
$15$	−0.230263	−0.0594537
$16$	0	0
$17$	4.66419	1.13123	0.565617	−	0.824668i	$-0.308638\pi$
$17$	4.66419	1.13123	0.565617	+	0.824668i	$0.308638\pi$
$18$	0	0
$19$	5.92080	1.35833	0.679163	−	0.733988i	$-0.262343\pi$
$19$	5.92080	1.35833	0.679163	+	0.733988i	$0.262343\pi$
$20$	0	0
$21$	−10.9116	−2.38111
$22$	0	0
$23$	−2.98182	−0.621752	−0.310876	−	0.950450i	$-0.600622\pi$
$23$	−2.98182	−0.621752	−0.310876	+	0.950450i	$0.600622\pi$
$24$	0	0
$25$	−4.99208	−0.998415
$26$	0	0
$27$	−1.79132	−0.344739
$28$	0	0
$29$	6.14670	1.14141	0.570707	−	0.821154i	$-0.306669\pi$
$29$	6.14670	1.14141	0.570707	+	0.821154i	$0.306669\pi$
$30$	0	0
$31$	−0.725876	−0.130371	−0.0651856	−	0.997873i	$-0.520764\pi$
$31$	−0.725876	−0.130371	−0.0651856	+	0.997873i	$0.520764\pi$
$32$	0	0
$33$	14.3846	2.50403
$34$	0	0
$35$	0.375431	0.0634594
$36$	0	0
$37$	11.8046	1.94067	0.970334	−	0.241770i	$-0.0777279\pi$
$37$	11.8046	1.94067	0.970334	+	0.241770i	$0.0777279\pi$
$38$	0	0
$39$	−14.6396	−2.34422
$40$	0	0
$41$	−3.43975	−0.537199	−0.268600	−	0.963252i	$-0.586561\pi$
$41$	−3.43975	−0.537199	−0.268600	+	0.963252i	$0.586561\pi$
$42$	0	0
$43$	4.18359	0.637991	0.318995	−	0.947756i	$-0.396655\pi$
$43$	4.18359	0.637991	0.318995	+	0.947756i	$0.396655\pi$
$44$	0	0
$45$	0.328659	0.0489936
$46$	0	0
$47$	−5.47405	−0.798472	−0.399236	−	0.916848i	$-0.630725\pi$
$47$	−5.47405	−0.798472	−0.399236	+	0.916848i	$0.630725\pi$
$48$	0	0
$49$	10.7908	1.54154
$50$	0	0
$51$	−12.0662	−1.68960
$52$	0	0
$53$	2.96430	0.407178	0.203589	−	0.979056i	$-0.434739\pi$
$53$	2.96430	0.407178	0.203589	+	0.979056i	$0.434739\pi$
$54$	0	0
$55$	−0.494922	−0.0667353
$56$	0	0
$57$	−15.3170	−2.02878
$58$	0	0
$59$	3.19404	0.415828	0.207914	−	0.978147i	$-0.433333\pi$
$59$	3.19404	0.415828	0.207914	+	0.978147i	$0.433333\pi$
$60$	0	0
$61$	2.66626	0.341379	0.170690	−	0.985325i	$-0.445400\pi$
$61$	2.66626	0.341379	0.170690	+	0.985325i	$0.445400\pi$
$62$	0	0
$63$	15.5744	1.96219
$64$	0	0
$65$	0.503698	0.0624760
$66$	0	0
$67$	−2.11614	−0.258528	−0.129264	−	0.991610i	$-0.541261\pi$
$67$	−2.11614	−0.258528	−0.129264	+	0.991610i	$0.541261\pi$
$68$	0	0
$69$	7.71389	0.928643
$70$	0	0
$71$	12.6906	1.50609	0.753046	−	0.657968i	$-0.228584\pi$
$71$	12.6906	1.50609	0.753046	+	0.657968i	$0.228584\pi$
$72$	0	0
$73$	−5.71725	−0.669153	−0.334577	−	0.942369i	$-0.608593\pi$
$73$	−5.71725	−0.669153	−0.334577	+	0.942369i	$0.608593\pi$
$74$	0	0
$75$	12.9144	1.49122
$76$	0	0
$77$	−23.4532	−2.67274
$78$	0	0
$79$	7.85668	0.883945	0.441973	−	0.897029i	$-0.354279\pi$
$79$	7.85668	0.883945	0.441973	+	0.897029i	$0.354279\pi$
$80$	0	0
$81$	−6.44322	−0.715913
$82$	0	0
$83$	−2.69956	−0.296315	−0.148158	−	0.988964i	$-0.547334\pi$
$83$	−2.69956	−0.296315	−0.148158	+	0.988964i	$0.547334\pi$
$84$	0	0
$85$	0.415154	0.0450297
$86$	0	0
$87$	−15.9013	−1.70480
$88$	0	0
$89$	−12.3442	−1.30848	−0.654241	−	0.756286i	$-0.727012\pi$
$89$	−12.3442	−1.30848	−0.654241	+	0.756286i	$0.727012\pi$
$90$	0	0
$91$	23.8691	2.50216
$92$	0	0
$93$	1.87782	0.194721
$94$	0	0
$95$	0.527003	0.0540693
$96$	0	0
$97$	−4.06463	−0.412701	−0.206350	−	0.978478i	$-0.566159\pi$
$97$	−4.06463	−0.412701	−0.206350	+	0.978478i	$0.566159\pi$
$98$	0	0
$99$	−20.5314	−2.06348
$100$	0	0
$101$	18.9006	1.88068	0.940340	−	0.340236i	$-0.110507\pi$
$101$	18.9006	1.88068	0.940340	+	0.340236i	$0.110507\pi$
$102$	0	0
$103$	−14.8998	−1.46812	−0.734062	−	0.679083i	$-0.762378\pi$
$103$	−14.8998	−1.46812	−0.734062	+	0.679083i	$0.762378\pi$
$104$	0	0
$105$	−0.971230	−0.0947823
$106$	0	0
$107$	−9.90443	−0.957498	−0.478749	−	0.877952i	$-0.658910\pi$
$107$	−9.90443	−0.957498	−0.478749	+	0.877952i	$0.658910\pi$
$108$	0	0
$109$	−16.0205	−1.53448	−0.767241	−	0.641359i	$-0.778371\pi$
$109$	−16.0205	−1.53448	−0.767241	+	0.641359i	$0.778371\pi$
$110$	0	0
$111$	−30.5382	−2.89856
$112$	0	0
$113$	8.17961	0.769473	0.384737	−	0.923026i	$-0.374292\pi$
$113$	8.17961	0.769473	0.384737	+	0.923026i	$0.374292\pi$
$114$	0	0
$115$	−0.265408	−0.0247494
$116$	0	0
$117$	20.8954	1.93178
$118$	0	0
$119$	19.6732	1.80344
$120$	0	0
$121$	19.9179	1.81072
$122$	0	0
$123$	8.89855	0.802355
$124$	0	0
$125$	−0.889382	−0.0795487
$126$	0	0
$127$	−9.32495	−0.827455	−0.413728	−	0.910401i	$-0.635773\pi$
$127$	−9.32495	−0.827455	−0.413728	+	0.910401i	$0.635773\pi$
$128$	0	0
$129$	−10.8228	−0.952897
$130$	0	0
$131$	5.34175	0.466711	0.233355	−	0.972392i	$-0.425029\pi$
$131$	5.34175	0.466711	0.233355	+	0.972392i	$0.425029\pi$
$132$	0	0
$133$	24.9734	2.16547
$134$	0	0
$135$	−0.159443	−0.0137226
$136$	0	0
$137$	−15.0180	−1.28307	−0.641537	−	0.767092i	$-0.721703\pi$
$137$	−15.0180	−1.28307	−0.641537	+	0.767092i	$0.721703\pi$
$138$	0	0
$139$	1.35192	0.114669	0.0573344	−	0.998355i	$-0.481740\pi$
$139$	1.35192	0.114669	0.0573344	+	0.998355i	$0.481740\pi$
$140$	0	0
$141$	14.1612	1.19259
$142$	0	0
$143$	−31.4661	−2.63133
$144$	0	0
$145$	0.547109	0.0454350
$146$	0	0
$147$	−27.9155	−2.30243
$148$	0	0
$149$	−10.8113	−0.885699	−0.442850	−	0.896596i	$-0.646032\pi$
$149$	−10.8113	−0.885699	−0.442850	+	0.896596i	$0.646032\pi$
$150$	0	0
$151$	−4.91695	−0.400136	−0.200068	−	0.979782i	$-0.564116\pi$
$151$	−4.91695	−0.400136	−0.200068	+	0.979782i	$0.564116\pi$
$152$	0	0
$153$	17.2222	1.39234
$154$	0	0
$155$	−0.0646092	−0.00518954
$156$	0	0
$157$	20.8790	1.66632	0.833162	−	0.553029i	$-0.186528\pi$
$157$	20.8790	1.66632	0.833162	+	0.553029i	$0.186528\pi$
$158$	0	0
$159$	−7.66858	−0.608158
$160$	0	0
$161$	−12.5770	−0.991210
$162$	0	0
$163$	−1.79156	−0.140326	−0.0701631	−	0.997536i	$-0.522352\pi$
$163$	−1.79156	−0.140326	−0.0701631	+	0.997536i	$0.522352\pi$
$164$	0	0
$165$	1.28035	0.0996752
$166$	0	0
$167$	−10.7548	−0.832231	−0.416116	−	0.909312i	$-0.636609\pi$
$167$	−10.7548	−0.832231	−0.416116	+	0.909312i	$0.636609\pi$
$168$	0	0
$169$	19.0240	1.46338
$170$	0	0
$171$	21.8622	1.67184
$172$	0	0
$173$	−19.0997	−1.45212	−0.726061	−	0.687630i	$-0.758651\pi$
$173$	−19.0997	−1.45212	−0.726061	+	0.687630i	$0.758651\pi$
$174$	0	0
$175$	−21.0561	−1.59170
$176$	0	0
$177$	−8.26290	−0.621077
$178$	0	0
$179$	−16.9659	−1.26809	−0.634045	−	0.773296i	$-0.718607\pi$
$179$	−16.9659	−1.26809	−0.634045	+	0.773296i	$0.718607\pi$
$180$	0	0
$181$	8.82509	0.655964	0.327982	−	0.944684i	$-0.393631\pi$
$181$	8.82509	0.655964	0.327982	+	0.944684i	$0.393631\pi$
$182$	0	0
$183$	−6.89754	−0.509880
$184$	0	0
$185$	1.05071	0.0772500
$186$	0	0
$187$	−25.9347	−1.89653
$188$	0	0
$189$	−7.55562	−0.549591
$190$	0	0
$191$	14.0977	1.02007	0.510036	−	0.860153i	$-0.329632\pi$
$191$	14.0977	1.02007	0.510036	+	0.860153i	$0.329632\pi$
$192$	0	0
$193$	12.2396	0.881028	0.440514	−	0.897746i	$-0.354796\pi$
$193$	12.2396	0.881028	0.440514	+	0.897746i	$0.354796\pi$
$194$	0	0
$195$	−1.30305	−0.0933136
$196$	0	0
$197$	17.3032	1.23280	0.616401	−	0.787432i	$-0.288590\pi$
$197$	17.3032	1.23280	0.616401	+	0.787432i	$0.288590\pi$
$198$	0	0
$199$	7.97358	0.565232	0.282616	−	0.959233i	$-0.408798\pi$
$199$	7.97358	0.565232	0.282616	+	0.959233i	$0.408798\pi$
$200$	0	0
$201$	5.47440	0.386134
$202$	0	0
$203$	25.9262	1.81967
$204$	0	0
$205$	−0.306168	−0.0213837
$206$	0	0
$207$	−11.0102	−0.765260
$208$	0	0
$209$	−32.9219	−2.27726
$210$	0	0
$211$	1.21487	0.0836348	0.0418174	−	0.999125i	$-0.486685\pi$
$211$	1.21487	0.0836348	0.0418174	+	0.999125i	$0.486685\pi$
$212$	0	0
$213$	−32.8301	−2.24948
$214$	0	0
$215$	0.372375	0.0253958
$216$	0	0
$217$	−3.06168	−0.207840
$218$	0	0
$219$	14.7904	0.999441
$220$	0	0
$221$	26.3946	1.77549
$222$	0	0
$223$	−7.75290	−0.519172	−0.259586	−	0.965720i	$-0.583586\pi$
$223$	−7.75290	−0.519172	−0.259586	+	0.965720i	$0.583586\pi$
$224$	0	0
$225$	−18.4329	−1.22886
$226$	0	0
$227$	−17.9480	−1.19125	−0.595626	−	0.803262i	$-0.703096\pi$
$227$	−17.9480	−1.19125	−0.595626	+	0.803262i	$0.703096\pi$
$228$	0	0
$229$	11.2040	0.740383	0.370192	−	0.928955i	$-0.379292\pi$
$229$	11.2040	0.740383	0.370192	+	0.928955i	$0.379292\pi$
$230$	0	0
$231$	60.6729	3.99198
$232$	0	0
$233$	24.8767	1.62973	0.814864	−	0.579653i	$-0.196812\pi$
$233$	24.8767	1.62973	0.814864	+	0.579653i	$0.196812\pi$
$234$	0	0
$235$	−0.487238	−0.0317839
$236$	0	0
$237$	−20.3250	−1.32025
$238$	0	0
$239$	−2.12353	−0.137360	−0.0686798	−	0.997639i	$-0.521879\pi$
$239$	−2.12353	−0.137360	−0.0686798	+	0.997639i	$0.521879\pi$
$240$	0	0
$241$	19.9586	1.28565	0.642823	−	0.766015i	$-0.277763\pi$
$241$	19.9586	1.28565	0.642823	+	0.766015i	$0.277763\pi$
$242$	0	0
$243$	22.0424	1.41402
$244$	0	0
$245$	0.960474	0.0613624
$246$	0	0
$247$	33.5057	2.13191
$248$	0	0
$249$	6.98370	0.442574
$250$	0	0
$251$	19.5092	1.23141	0.615706	−	0.787976i	$-0.288871\pi$
$251$	19.5092	1.23141	0.615706	+	0.787976i	$0.288871\pi$
$252$	0	0
$253$	16.5801	1.04238
$254$	0	0
$255$	−1.07399	−0.0672560
$256$	0	0
$257$	8.20677	0.511924	0.255962	−	0.966687i	$-0.417608\pi$
$257$	8.20677	0.511924	0.255962	+	0.966687i	$0.417608\pi$
$258$	0	0
$259$	49.7909	3.09385
$260$	0	0
$261$	22.6963	1.40487
$262$	0	0
$263$	12.5489	0.773801	0.386901	−	0.922121i	$-0.373546\pi$
$263$	12.5489	0.773801	0.386901	+	0.922121i	$0.373546\pi$
$264$	0	0
$265$	0.263849	0.0162081
$266$	0	0
$267$	31.9341	1.95434
$268$	0	0
$269$	−21.4017	−1.30489	−0.652443	−	0.757837i	$-0.726256\pi$
$269$	−21.4017	−1.30489	−0.652443	+	0.757837i	$0.726256\pi$
$270$	0	0
$271$	−30.0583	−1.82591	−0.912955	−	0.408060i	$-0.866206\pi$
$271$	−30.0583	−1.82591	−0.912955	+	0.408060i	$0.866206\pi$
$272$	0	0
$273$	−61.7487	−3.73720
$274$	0	0
$275$	27.7579	1.67386
$276$	0	0
$277$	−2.25631	−0.135568	−0.0677842	−	0.997700i	$-0.521593\pi$
$277$	−2.25631	−0.135568	−0.0677842	+	0.997700i	$0.521593\pi$
$278$	0	0
$279$	−2.68025	−0.160462
$280$	0	0
$281$	23.4142	1.39677	0.698387	−	0.715720i	$-0.253901\pi$
$281$	23.4142	1.39677	0.698387	+	0.715720i	$0.253901\pi$
$282$	0	0
$283$	4.89680	0.291084	0.145542	−	0.989352i	$-0.453507\pi$
$283$	4.89680	0.291084	0.145542	+	0.989352i	$0.453507\pi$
$284$	0	0
$285$	−1.36334	−0.0807574
$286$	0	0
$287$	−14.5086	−0.856414
$288$	0	0
$289$	4.75471	0.279689
$290$	0	0
$291$	10.5151	0.616406
$292$	0	0
$293$	−30.9491	−1.80806	−0.904032	−	0.427465i	$-0.859407\pi$
$293$	−30.9491	−1.80806	−0.904032	+	0.427465i	$0.859407\pi$
$294$	0	0
$295$	0.284297	0.0165524
$296$	0	0
$297$	9.96041	0.577962
$298$	0	0
$299$	−16.8740	−0.975850
$300$	0	0
$301$	17.6460	1.01710
$302$	0	0
$303$	−48.8954	−2.80897
$304$	0	0
$305$	0.237320	0.0135889
$306$	0	0
$307$	23.2317	1.32590	0.662951	−	0.748663i	$-0.269304\pi$
$307$	23.2317	1.32590	0.662951	+	0.748663i	$0.269304\pi$
$308$	0	0
$309$	38.5455	2.19278
$310$	0	0
$311$	−21.8622	−1.23969	−0.619847	−	0.784723i	$-0.712805\pi$
$311$	−21.8622	−1.23969	−0.619847	+	0.784723i	$0.712805\pi$
$312$	0	0
$313$	−13.2754	−0.750369	−0.375184	−	0.926950i	$-0.622421\pi$
$313$	−13.2754	−0.750369	−0.375184	+	0.926950i	$0.622421\pi$
$314$	0	0
$315$	1.38625	0.0781066
$316$	0	0
$317$	25.0392	1.40634	0.703172	−	0.711020i	$-0.251766\pi$
$317$	25.0392	1.40634	0.703172	+	0.711020i	$0.251766\pi$
$318$	0	0
$319$	−34.1780	−1.91360
$320$	0	0
$321$	25.6225	1.43011
$322$	0	0
$323$	27.6158	1.53658
$324$	0	0
$325$	−28.2500	−1.56703
$326$	0	0
$327$	41.4445	2.29189
$328$	0	0
$329$	−23.0891	−1.27294
$330$	0	0
$331$	17.9055	0.984174	0.492087	−	0.870546i	$-0.336234\pi$
$331$	17.9055	0.984174	0.492087	+	0.870546i	$0.336234\pi$
$332$	0	0
$333$	43.5878	2.38860
$334$	0	0
$335$	−0.188355	−0.0102909
$336$	0	0
$337$	7.72574	0.420848	0.210424	−	0.977610i	$-0.432516\pi$
$337$	7.72574	0.420848	0.210424	+	0.977610i	$0.432516\pi$
$338$	0	0
$339$	−21.1604	−1.14928
$340$	0	0
$341$	4.03615	0.218570
$342$	0	0
$343$	15.9892	0.863337
$344$	0	0
$345$	0.686603	0.0369654
$346$	0	0
$347$	−4.90582	−0.263358	−0.131679	−	0.991292i	$-0.542037\pi$
$347$	−4.90582	−0.263358	−0.131679	+	0.991292i	$0.542037\pi$
$348$	0	0
$349$	−0.611795	−0.0327487	−0.0163743	−	0.999866i	$-0.505212\pi$
$349$	−0.611795	−0.0327487	−0.0163743	+	0.999866i	$0.505212\pi$
$350$	0	0
$351$	−10.1370	−0.541074
$352$	0	0
$353$	10.5794	0.563083	0.281542	−	0.959549i	$-0.409154\pi$
$353$	10.5794	0.563083	0.281542	+	0.959549i	$0.409154\pi$
$354$	0	0
$355$	1.12957	0.0599513
$356$	0	0
$357$	−50.8940	−2.69359
$358$	0	0
$359$	−1.20906	−0.0638119	−0.0319060	−	0.999491i	$-0.510158\pi$
$359$	−1.20906	−0.0638119	−0.0319060	+	0.999491i	$0.510158\pi$
$360$	0	0
$361$	16.0559	0.845047
$362$	0	0
$363$	−51.5270	−2.70447
$364$	0	0
$365$	−0.508884	−0.0266362
$366$	0	0
$367$	−26.1402	−1.36451	−0.682254	−	0.731116i	$-0.739000\pi$
$367$	−26.1402	−1.36451	−0.682254	+	0.731116i	$0.739000\pi$
$368$	0	0
$369$	−12.7011	−0.661191
$370$	0	0
$371$	12.5032	0.649132
$372$	0	0
$373$	27.0290	1.39951	0.699755	−	0.714383i	$-0.253292\pi$
$373$	27.0290	1.39951	0.699755	+	0.714383i	$0.253292\pi$
$374$	0	0
$375$	2.30081	0.118813
$376$	0	0
$377$	34.7840	1.79147
$378$	0	0
$379$	16.1069	0.827357	0.413679	−	0.910423i	$-0.364244\pi$
$379$	16.1069	0.827357	0.413679	+	0.910423i	$0.364244\pi$
$380$	0	0
$381$	24.1234	1.23588
$382$	0	0
$383$	0.795719	0.0406594	0.0203297	−	0.999793i	$-0.493528\pi$
$383$	0.795719	0.0406594	0.0203297	+	0.999793i	$0.493528\pi$
$384$	0	0
$385$	−2.08754	−0.106391
$386$	0	0
$387$	15.4476	0.785247
$388$	0	0
$389$	−28.1735	−1.42845	−0.714226	−	0.699915i	$-0.753221\pi$
$389$	−28.1735	−1.42845	−0.714226	+	0.699915i	$0.753221\pi$
$390$	0	0
$391$	−13.9078	−0.703346
$392$	0	0
$393$	−13.8190	−0.697075
$394$	0	0
$395$	0.699312	0.0351862
$396$	0	0
$397$	12.0999	0.607276	0.303638	−	0.952787i	$-0.401799\pi$
$397$	12.0999	0.607276	0.303638	+	0.952787i	$0.401799\pi$
$398$	0	0
$399$	−64.6056	−3.23433
$400$	0	0
$401$	36.2367	1.80958	0.904788	−	0.425863i	$-0.140029\pi$
$401$	36.2367	1.80958	0.904788	+	0.425863i	$0.140029\pi$
$402$	0	0
$403$	−4.10771	−0.204620
$404$	0	0
$405$	−0.573502	−0.0284976
$406$	0	0
$407$	−65.6382	−3.25357
$408$	0	0
$409$	10.6678	0.527490	0.263745	−	0.964592i	$-0.415042\pi$
$409$	10.6678	0.527490	0.263745	+	0.964592i	$0.415042\pi$
$410$	0	0
$411$	38.8512	1.91639
$412$	0	0
$413$	13.4722	0.662922
$414$	0	0
$415$	−0.240284	−0.0117951
$416$	0	0
$417$	−3.49739	−0.171268
$418$	0	0
$419$	23.2757	1.13709	0.568546	−	0.822651i	$-0.307506\pi$
$419$	23.2757	1.13709	0.568546	+	0.822651i	$0.307506\pi$
$420$	0	0
$421$	11.9908	0.584395	0.292197	−	0.956358i	$-0.405614\pi$
$421$	11.9908	0.584395	0.292197	+	0.956358i	$0.405614\pi$
$422$	0	0
$423$	−20.2126	−0.982769
$424$	0	0
$425$	−23.2840	−1.12944
$426$	0	0
$427$	11.2460	0.544234
$428$	0	0
$429$	81.4019	3.93012
$430$	0	0
$431$	3.76114	0.181168	0.0905840	−	0.995889i	$-0.471127\pi$
$431$	3.76114	0.181168	0.0905840	+	0.995889i	$0.471127\pi$
$432$	0	0
$433$	4.30609	0.206937	0.103469	−	0.994633i	$-0.467006\pi$
$433$	4.30609	0.206937	0.103469	+	0.994633i	$0.467006\pi$
$434$	0	0
$435$	−1.41536	−0.0678612
$436$	0	0
$437$	−17.6547	−0.844541
$438$	0	0
$439$	−3.06838	−0.146446	−0.0732230	−	0.997316i	$-0.523328\pi$
$439$	−3.06838	−0.146446	−0.0732230	+	0.997316i	$0.523328\pi$
$440$	0	0
$441$	39.8443	1.89735
$442$	0	0
$443$	−3.96278	−0.188277	−0.0941387	−	0.995559i	$-0.530010\pi$
$443$	−3.96278	−0.188277	−0.0941387	+	0.995559i	$0.530010\pi$
$444$	0	0
$445$	−1.09874	−0.0520853
$446$	0	0
$447$	27.9687	1.32287
$448$	0	0
$449$	−28.6478	−1.35197	−0.675986	−	0.736914i	$-0.736282\pi$
$449$	−28.6478	−1.35197	−0.675986	+	0.736914i	$0.736282\pi$
$450$	0	0
$451$	19.1263	0.900624
$452$	0	0
$453$	12.7200	0.597639
$454$	0	0
$455$	2.12455	0.0996006
$456$	0	0
$457$	8.32907	0.389618	0.194809	−	0.980841i	$-0.437591\pi$
$457$	8.32907	0.389618	0.194809	+	0.980841i	$0.437591\pi$
$458$	0	0
$459$	−8.35505	−0.389980
$460$	0	0
$461$	−37.6682	−1.75438	−0.877192	−	0.480139i	$-0.840586\pi$
$461$	−37.6682	−1.75438	−0.877192	+	0.480139i	$0.840586\pi$
$462$	0	0
$463$	32.1495	1.49411	0.747057	−	0.664760i	$-0.231466\pi$
$463$	32.1495	1.49411	0.747057	+	0.664760i	$0.231466\pi$
$464$	0	0
$465$	0.167142	0.00775104
$466$	0	0
$467$	35.6401	1.64923	0.824614	−	0.565695i	$-0.191392\pi$
$467$	35.6401	1.64923	0.824614	+	0.565695i	$0.191392\pi$
$468$	0	0
$469$	−8.92569	−0.412150
$470$	0	0
$471$	−54.0134	−2.48881
$472$	0	0
$473$	−23.2623	−1.06960
$474$	0	0
$475$	−29.5571	−1.35617
$476$	0	0
$477$	10.9455	0.501160
$478$	0	0
$479$	27.3687	1.25051	0.625254	−	0.780421i	$-0.284995\pi$
$479$	27.3687	1.25051	0.625254	+	0.780421i	$0.284995\pi$
$480$	0	0
$481$	66.8020	3.04591
$482$	0	0
$483$	32.5365	1.48046
$484$	0	0
$485$	−0.361787	−0.0164279
$486$	0	0
$487$	−11.6603	−0.528378	−0.264189	−	0.964471i	$-0.585104\pi$
$487$	−11.6603	−0.528378	−0.264189	+	0.964471i	$0.585104\pi$
$488$	0	0
$489$	4.63473	0.209590
$490$	0	0
$491$	−17.2858	−0.780097	−0.390048	−	0.920794i	$-0.627542\pi$
$491$	−17.2858	−0.780097	−0.390048	+	0.920794i	$0.627542\pi$
$492$	0	0
$493$	28.6694	1.29120
$494$	0	0
$495$	−1.82747	−0.0821387
$496$	0	0
$497$	53.5277	2.40104
$498$	0	0
$499$	38.9749	1.74476	0.872378	−	0.488832i	$-0.162577\pi$
$499$	38.9749	1.74476	0.872378	+	0.488832i	$0.162577\pi$
$500$	0	0
$501$	27.8224	1.24301
$502$	0	0
$503$	32.0605	1.42951	0.714753	−	0.699377i	$-0.246539\pi$
$503$	32.0605	1.42951	0.714753	+	0.699377i	$0.246539\pi$
$504$	0	0
$505$	1.68232	0.0748621
$506$	0	0
$507$	−49.2146	−2.18570
$508$	0	0
$509$	18.1814	0.805877	0.402939	−	0.915227i	$-0.367989\pi$
$509$	18.1814	0.805877	0.402939	+	0.915227i	$0.367989\pi$
$510$	0	0
$511$	−24.1148	−1.06678
$512$	0	0
$513$	−10.6060	−0.468268
$514$	0	0
$515$	−1.32621	−0.0584400
$516$	0	0
$517$	30.4378	1.33865
$518$	0	0
$519$	49.4104	2.16888
$520$	0	0
$521$	35.9460	1.57482	0.787412	−	0.616427i	$-0.211420\pi$
$521$	35.9460	1.57482	0.787412	+	0.616427i	$0.211420\pi$
$522$	0	0
$523$	28.4882	1.24570	0.622850	−	0.782341i	$-0.285975\pi$
$523$	28.4882	1.24570	0.622850	+	0.782341i	$0.285975\pi$
$524$	0	0
$525$	54.4717	2.37734
$526$	0	0
$527$	−3.38562	−0.147480
$528$	0	0
$529$	−14.1088	−0.613425
$530$	0	0
$531$	11.7938	0.511807
$532$	0	0
$533$	−19.4655	−0.843143
$534$	0	0
$535$	−0.881580	−0.0381140
$536$	0	0
$537$	43.8904	1.89401
$538$	0	0
$539$	−60.0010	−2.58442
$540$	0	0
$541$	30.4441	1.30889	0.654447	−	0.756108i	$-0.272902\pi$
$541$	30.4441	1.30889	0.654447	+	0.756108i	$0.272902\pi$
$542$	0	0
$543$	−22.8303	−0.979741
$544$	0	0
$545$	−1.42596	−0.0610814
$546$	0	0
$547$	−38.5946	−1.65018	−0.825092	−	0.564998i	$-0.808877\pi$
$547$	−38.5946	−1.65018	−0.825092	+	0.564998i	$0.808877\pi$
$548$	0	0
$549$	9.84498	0.420174
$550$	0	0
$551$	36.3934	1.55041
$552$	0	0
$553$	33.1388	1.40920
$554$	0	0
$555$	−2.71817	−0.115380
$556$	0	0
$557$	−36.2175	−1.53459	−0.767293	−	0.641297i	$-0.778397\pi$
$557$	−36.2175	−1.53459	−0.767293	+	0.641297i	$0.778397\pi$
$558$	0	0
$559$	23.6748	1.00134
$560$	0	0
$561$	67.0924	2.83265
$562$	0	0
$563$	−21.6524	−0.912540	−0.456270	−	0.889841i	$-0.650815\pi$
$563$	−21.6524	−0.912540	−0.456270	+	0.889841i	$0.650815\pi$
$564$	0	0
$565$	0.728056	0.0306296
$566$	0	0
$567$	−27.1769	−1.14132
$568$	0	0
$569$	5.26799	0.220846	0.110423	−	0.993885i	$-0.464779\pi$
$569$	5.26799	0.220846	0.110423	+	0.993885i	$0.464779\pi$
$570$	0	0
$571$	−18.9409	−0.792654	−0.396327	−	0.918109i	$-0.629715\pi$
$571$	−18.9409	−0.792654	−0.396327	+	0.918109i	$0.629715\pi$
$572$	0	0
$573$	−36.4703	−1.52357
$574$	0	0
$575$	14.8855	0.620767
$576$	0	0
$577$	31.8242	1.32486	0.662429	−	0.749125i	$-0.269526\pi$
$577$	31.8242	1.32486	0.662429	+	0.749125i	$0.269526\pi$
$578$	0	0
$579$	−31.6636	−1.31590
$580$	0	0
$581$	−11.3865	−0.472392
$582$	0	0
$583$	−16.4827	−0.682642
$584$	0	0
$585$	1.85987	0.0768963
$586$	0	0
$587$	27.8655	1.15013	0.575066	−	0.818107i	$-0.304977\pi$
$587$	27.8655	1.15013	0.575066	+	0.818107i	$0.304977\pi$
$588$	0	0
$589$	−4.29777	−0.177086
$590$	0	0
$591$	−44.7629	−1.84130
$592$	0	0
$593$	7.37337	0.302788	0.151394	−	0.988473i	$-0.451624\pi$
$593$	7.37337	0.302788	0.151394	+	0.988473i	$0.451624\pi$
$594$	0	0
$595$	1.75108	0.0717874
$596$	0	0
$597$	−20.6274	−0.844225
$598$	0	0
$599$	14.1996	0.580181	0.290090	−	0.956999i	$-0.406315\pi$
$599$	14.1996	0.580181	0.290090	+	0.956999i	$0.406315\pi$
$600$	0	0
$601$	−18.3842	−0.749906	−0.374953	−	0.927044i	$-0.622341\pi$
$601$	−18.3842	−0.749906	−0.374953	+	0.927044i	$0.622341\pi$
$602$	0	0
$603$	−7.81371	−0.318199
$604$	0	0
$605$	1.77286	0.0720772
$606$	0	0
$607$	−37.9474	−1.54024	−0.770119	−	0.637900i	$-0.779803\pi$
$607$	−37.9474	−1.54024	−0.770119	+	0.637900i	$0.779803\pi$
$608$	0	0
$609$	−67.0705	−2.71783
$610$	0	0
$611$	−30.9775	−1.25322
$612$	0	0
$613$	31.9514	1.29050	0.645252	−	0.763970i	$-0.276752\pi$
$613$	31.9514	1.29050	0.645252	+	0.763970i	$0.276752\pi$
$614$	0	0
$615$	0.792048	0.0319385
$616$	0	0
$617$	−17.2855	−0.695887	−0.347943	−	0.937516i	$-0.613120\pi$
$617$	−17.2855	−0.695887	−0.347943	+	0.937516i	$0.613120\pi$
$618$	0	0
$619$	31.9095	1.28255	0.641276	−	0.767311i	$-0.278406\pi$
$619$	31.9095	1.28255	0.641276	+	0.767311i	$0.278406\pi$
$620$	0	0
$621$	5.34138	0.214342
$622$	0	0
$623$	−52.0667	−2.08601
$624$	0	0
$625$	24.8812	0.995249
$626$	0	0
$627$	85.1682	3.40129
$628$	0	0
$629$	55.0590	2.19535
$630$	0	0
$631$	30.6529	1.22027	0.610136	−	0.792296i	$-0.291115\pi$
$631$	30.6529	1.22027	0.610136	+	0.792296i	$0.291115\pi$
$632$	0	0
$633$	−3.14283	−0.124916
$634$	0	0
$635$	−0.830001	−0.0329376
$636$	0	0
$637$	61.0648	2.41948
$638$	0	0
$639$	46.8591	1.85372
$640$	0	0
$641$	−38.1552	−1.50704	−0.753521	−	0.657424i	$-0.771646\pi$
$641$	−38.1552	−1.50704	−0.753521	+	0.657424i	$0.771646\pi$
$642$	0	0
$643$	12.8788	0.507890	0.253945	−	0.967219i	$-0.418272\pi$
$643$	12.8788	0.507890	0.253945	+	0.967219i	$0.418272\pi$
$644$	0	0
$645$	−0.963326	−0.0379309
$646$	0	0
$647$	35.2593	1.38619	0.693093	−	0.720849i	$-0.256248\pi$
$647$	35.2593	1.38619	0.693093	+	0.720849i	$0.256248\pi$
$648$	0	0
$649$	−17.7601	−0.697144
$650$	0	0
$651$	7.92049	0.310428
$652$	0	0
$653$	−31.0639	−1.21563	−0.607813	−	0.794080i	$-0.707953\pi$
$653$	−31.0639	−1.21563	−0.607813	+	0.794080i	$0.707953\pi$
$654$	0	0
$655$	0.475462	0.0185778
$656$	0	0
$657$	−21.1106	−0.823602
$658$	0	0
$659$	−5.31684	−0.207115	−0.103557	−	0.994623i	$-0.533023\pi$
$659$	−5.31684	−0.207115	−0.103557	+	0.994623i	$0.533023\pi$
$660$	0	0
$661$	7.89945	0.307253	0.153627	−	0.988129i	$-0.450905\pi$
$661$	7.89945	0.307253	0.153627	+	0.988129i	$0.450905\pi$
$662$	0	0
$663$	−68.2821	−2.65186
$664$	0	0
$665$	2.22285	0.0861985
$666$	0	0
$667$	−18.3283	−0.709676
$668$	0	0
$669$	20.0565	0.775431
$670$	0	0
$671$	−14.8254	−0.572328
$672$	0	0
$673$	−26.0511	−1.00420	−0.502099	−	0.864810i	$-0.667439\pi$
$673$	−26.0511	−1.00420	−0.502099	+	0.864810i	$0.667439\pi$
$674$	0	0
$675$	8.94240	0.344193
$676$	0	0
$677$	36.7302	1.41166	0.705829	−	0.708383i	$-0.250575\pi$
$677$	36.7302	1.41166	0.705829	+	0.708383i	$0.250575\pi$
$678$	0	0
$679$	−17.1443	−0.657936
$680$	0	0
$681$	46.4311	1.77924
$682$	0	0
$683$	15.9340	0.609696	0.304848	−	0.952401i	$-0.401394\pi$
$683$	15.9340	0.609696	0.304848	+	0.952401i	$0.401394\pi$
$684$	0	0
$685$	−1.33673	−0.0510739
$686$	0	0
$687$	−28.9845	−1.10583
$688$	0	0
$689$	16.7749	0.639073
$690$	0	0
$691$	29.3833	1.11779	0.558896	−	0.829237i	$-0.311225\pi$
$691$	29.3833	1.11779	0.558896	+	0.829237i	$0.311225\pi$
$692$	0	0
$693$	−86.5995	−3.28964
$694$	0	0
$695$	0.120333	0.00456449
$696$	0	0
$697$	−16.0437	−0.607697
$698$	0	0
$699$	−64.3554	−2.43415
$700$	0	0
$701$	28.0036	1.05768	0.528841	−	0.848721i	$-0.322627\pi$
$701$	28.0036	1.05768	0.528841	+	0.848721i	$0.322627\pi$
$702$	0	0
$703$	69.8928	2.63606
$704$	0	0
$705$	1.26047	0.0474721
$706$	0	0
$707$	79.7211	2.99822
$708$	0	0
$709$	−22.3354	−0.838824	−0.419412	−	0.907796i	$-0.637764\pi$
$709$	−22.3354	−0.838824	−0.419412	+	0.907796i	$0.637764\pi$
$710$	0	0
$711$	29.0103	1.08797
$712$	0	0
$713$	2.16443	0.0810585
$714$	0	0
$715$	−2.80075	−0.104742
$716$	0	0
$717$	5.49351	0.205159
$718$	0	0
$719$	−27.0098	−1.00729	−0.503647	−	0.863910i	$-0.668009\pi$
$719$	−27.0098	−1.00729	−0.503647	+	0.863910i	$0.668009\pi$
$720$	0	0
$721$	−62.8462	−2.34051
$722$	0	0
$723$	−51.6323	−1.92023
$724$	0	0
$725$	−30.6848	−1.13960
$726$	0	0
$727$	−14.5021	−0.537853	−0.268926	−	0.963161i	$-0.586669\pi$
$727$	−14.5021	−0.537853	−0.268926	+	0.963161i	$0.586669\pi$
$728$	0	0
$729$	−37.6935	−1.39605
$730$	0	0
$731$	19.5131	0.721716
$732$	0	0
$733$	−53.3408	−1.97019	−0.985094	−	0.172019i	$-0.944971\pi$
$733$	−53.3408	−1.97019	−0.985094	+	0.172019i	$0.944971\pi$
$734$	0	0
$735$	−2.48472	−0.0916503
$736$	0	0
$737$	11.7665	0.433426
$738$	0	0
$739$	−38.1112	−1.40194	−0.700971	−	0.713190i	$-0.747250\pi$
$739$	−38.1112	−1.40194	−0.700971	+	0.713190i	$0.747250\pi$
$740$	0	0
$741$	−86.6783	−3.18421
$742$	0	0
$743$	−6.84011	−0.250939	−0.125470	−	0.992097i	$-0.540044\pi$
$743$	−6.84011	−0.250939	−0.125470	+	0.992097i	$0.540044\pi$
$744$	0	0
$745$	−0.962303	−0.0352560
$746$	0	0
$747$	−9.96796	−0.364709
$748$	0	0
$749$	−41.7760	−1.52646
$750$	0	0
$751$	−1.00000	−0.0364905
$751$	−1.00000	−0.0364905
$752$	0	0
$753$	−50.4699	−1.83923
$754$	0	0
$755$	−0.437651	−0.0159278
$756$	0	0
$757$	−0.408020	−0.0148297	−0.00741486	−	0.999973i	$-0.502360\pi$
$757$	−0.408020	−0.0148297	−0.00741486	+	0.999973i	$0.502360\pi$
$758$	0	0
$759$	−42.8922	−1.55689
$760$	0	0
$761$	4.96730	0.180065	0.0900323	−	0.995939i	$-0.471303\pi$
$761$	4.96730	0.180065	0.0900323	+	0.995939i	$0.471303\pi$
$762$	0	0
$763$	−67.5729	−2.44630
$764$	0	0
$765$	1.53293	0.0554232
$766$	0	0
$767$	18.0750	0.652650
$768$	0	0
$769$	−49.6772	−1.79141	−0.895703	−	0.444654i	$-0.853327\pi$
$769$	−49.6772	−1.79141	−0.895703	+	0.444654i	$0.853327\pi$
$770$	0	0
$771$	−21.2307	−0.764605
$772$	0	0
$773$	−17.0274	−0.612434	−0.306217	−	0.951962i	$-0.599063\pi$
$773$	−17.0274	−0.612434	−0.306217	+	0.951962i	$0.599063\pi$
$774$	0	0
$775$	3.62363	0.130165
$776$	0	0
$777$	−128.808	−4.62095
$778$	0	0
$779$	−20.3661	−0.729691
$780$	0	0
$781$	−70.5644	−2.52499
$782$	0	0
$783$	−11.0107	−0.393490
$784$	0	0
$785$	1.85841	0.0663295
$786$	0	0
$787$	−47.6401	−1.69819	−0.849094	−	0.528242i	$-0.822851\pi$
$787$	−47.6401	−1.69819	−0.849094	+	0.528242i	$0.822851\pi$
$788$	0	0
$789$	−32.4638	−1.15574
$790$	0	0
$791$	34.5009	1.22671
$792$	0	0
$793$	15.0883	0.535800
$794$	0	0
$795$	−0.682570	−0.0242083
$796$	0	0
$797$	12.1881	0.431724	0.215862	−	0.976424i	$-0.430744\pi$
$797$	12.1881	0.431724	0.215862	+	0.976424i	$0.430744\pi$
$798$	0	0
$799$	−25.5320	−0.903258
$800$	0	0
$801$	−45.5802	−1.61050
$802$	0	0
$803$	31.7901	1.12185
$804$	0	0
$805$	−1.11947	−0.0394560
$806$	0	0
$807$	55.3657	1.94897
$808$	0	0
$809$	−11.2380	−0.395105	−0.197553	−	0.980292i	$-0.563299\pi$
$809$	−11.2380	−0.395105	−0.197553	+	0.980292i	$0.563299\pi$
$810$	0	0
$811$	−21.0646	−0.739679	−0.369840	−	0.929096i	$-0.620587\pi$
$811$	−21.0646	−0.739679	−0.369840	+	0.929096i	$0.620587\pi$
$812$	0	0
$813$	77.7600	2.72716
$814$	0	0
$815$	−0.159465	−0.00558581
$816$	0	0
$817$	24.7702	0.866599
$818$	0	0
$819$	88.1350	3.07969
$820$	0	0
$821$	−34.1236	−1.19092	−0.595460	−	0.803385i	$-0.703030\pi$
$821$	−34.1236	−1.19092	−0.595460	+	0.803385i	$0.703030\pi$
$822$	0	0
$823$	40.9583	1.42772	0.713858	−	0.700290i	$-0.246946\pi$
$823$	40.9583	1.42772	0.713858	+	0.700290i	$0.246946\pi$
$824$	0	0
$825$	−71.8089	−2.50007
$826$	0	0
$827$	−39.1072	−1.35989	−0.679945	−	0.733263i	$-0.737997\pi$
$827$	−39.1072	−1.35989	−0.679945	+	0.733263i	$0.737997\pi$
$828$	0	0
$829$	2.04428	0.0710008	0.0355004	−	0.999370i	$-0.488698\pi$
$829$	2.04428	0.0710008	0.0355004	+	0.999370i	$0.488698\pi$
$830$	0	0
$831$	5.83701	0.202484
$832$	0	0
$833$	50.3303	1.74384
$834$	0	0
$835$	−0.957270	−0.0331277
$836$	0	0
$837$	1.30027	0.0449440
$838$	0	0
$839$	−19.4503	−0.671500	−0.335750	−	0.941951i	$-0.608990\pi$
$839$	−19.4503	−0.671500	−0.335750	+	0.941951i	$0.608990\pi$
$840$	0	0
$841$	8.78189	0.302824
$842$	0	0
$843$	−60.5720	−2.08621
$844$	0	0
$845$	1.69330	0.0582513
$846$	0	0
$847$	84.0119	2.88668
$848$	0	0
$849$	−12.6679	−0.434761
$850$	0	0
$851$	−35.1992	−1.20661
$852$	0	0
$853$	−6.54966	−0.224256	−0.112128	−	0.993694i	$-0.535767\pi$
$853$	−6.54966	−0.224256	−0.112128	+	0.993694i	$0.535767\pi$
$854$	0	0
$855$	1.94592	0.0665492
$856$	0	0
$857$	1.39320	0.0475908	0.0237954	−	0.999717i	$-0.492425\pi$
$857$	1.39320	0.0475908	0.0237954	+	0.999717i	$0.492425\pi$
$858$	0	0
$859$	−33.8037	−1.15337	−0.576683	−	0.816968i	$-0.695653\pi$
$859$	−33.8037	−1.15337	−0.576683	+	0.816968i	$0.695653\pi$
$860$	0	0
$861$	37.5333	1.27913
$862$	0	0
$863$	0.551415	0.0187704	0.00938520	−	0.999956i	$-0.497013\pi$
$863$	0.551415	0.0187704	0.00938520	+	0.999956i	$0.497013\pi$
$864$	0	0
$865$	−1.70004	−0.0578030
$866$	0	0
$867$	−12.3003	−0.417740
$868$	0	0
$869$	−43.6861	−1.48195
$870$	0	0
$871$	−11.9752	−0.405764
$872$	0	0
$873$	−15.0084	−0.507957
$874$	0	0
$875$	−3.75133	−0.126818
$876$	0	0
$877$	−18.0494	−0.609486	−0.304743	−	0.952435i	$-0.598571\pi$
$877$	−18.0494	−0.609486	−0.304743	+	0.952435i	$0.598571\pi$
$878$	0	0
$879$	80.0645	2.70051
$880$	0	0
$881$	7.82956	0.263785	0.131892	−	0.991264i	$-0.457895\pi$
$881$	7.82956	0.263785	0.131892	+	0.991264i	$0.457895\pi$
$882$	0	0
$883$	−1.21190	−0.0407836	−0.0203918	−	0.999792i	$-0.506491\pi$
$883$	−1.21190	−0.0407836	−0.0203918	+	0.999792i	$0.506491\pi$
$884$	0	0
$885$	−0.735469	−0.0247225
$886$	0	0
$887$	−6.09078	−0.204508	−0.102254	−	0.994758i	$-0.532605\pi$
$887$	−6.09078	−0.204508	−0.102254	+	0.994758i	$0.532605\pi$
$888$	0	0
$889$	−39.3318	−1.31915
$890$	0	0
$891$	35.8268	1.20024
$892$	0	0
$893$	−32.4108	−1.08458
$894$	0	0
$895$	−1.51011	−0.0504775
$896$	0	0
$897$	43.6527	1.45752
$898$	0	0
$899$	−4.46174	−0.148807
$900$	0	0
$901$	13.8261	0.460614
$902$	0	0
$903$	−45.6498	−1.51913
$904$	0	0
$905$	0.785510	0.0261112
$906$	0	0
$907$	−17.1667	−0.570012	−0.285006	−	0.958526i	$-0.591996\pi$
$907$	−17.1667	−0.570012	−0.285006	+	0.958526i	$0.591996\pi$
$908$	0	0
$909$	69.7893	2.31476
$910$	0	0
$911$	14.3889	0.476725	0.238363	−	0.971176i	$-0.423389\pi$
$911$	14.3889	0.476725	0.238363	+	0.971176i	$0.423389\pi$
$912$	0	0
$913$	15.0106	0.496778
$914$	0	0
$915$	−0.613940	−0.0202962
$916$	0	0
$917$	22.5310	0.744040
$918$	0	0
$919$	43.1639	1.42384	0.711922	−	0.702258i	$-0.247825\pi$
$919$	43.1639	1.42384	0.711922	+	0.702258i	$0.247825\pi$
$920$	0	0
$921$	−60.0997	−1.98035
$922$	0	0
$923$	71.8155	2.36384
$924$	0	0
$925$	−58.9296	−1.93759
$926$	0	0
$927$	−55.0167	−1.80698
$928$	0	0
$929$	−7.22440	−0.237025	−0.118512	−	0.992953i	$-0.537813\pi$
$929$	−7.22440	−0.237025	−0.118512	+	0.992953i	$0.537813\pi$
$930$	0	0
$931$	63.8901	2.09391
$932$	0	0
$933$	56.5570	1.85159
$934$	0	0
$935$	−2.30841	−0.0754932
$936$	0	0
$937$	−40.6497	−1.32797	−0.663984	−	0.747747i	$-0.731136\pi$
$937$	−40.6497	−1.32797	−0.663984	+	0.747747i	$0.731136\pi$
$938$	0	0
$939$	34.3431	1.12074
$940$	0	0
$941$	−14.8638	−0.484547	−0.242274	−	0.970208i	$-0.577893\pi$
$941$	−14.8638	−0.484547	−0.242274	+	0.970208i	$0.577893\pi$
$942$	0	0
$943$	10.2567	0.334005
$944$	0	0
$945$	−0.672516	−0.0218769
$946$	0	0
$947$	−32.4110	−1.05322	−0.526608	−	0.850109i	$-0.676536\pi$
$947$	−32.4110	−1.05322	−0.526608	+	0.850109i	$0.676536\pi$
$948$	0	0
$949$	−32.3538	−1.05025
$950$	0	0
$951$	−64.7759	−2.10050
$952$	0	0
$953$	40.2154	1.30270	0.651352	−	0.758776i	$-0.274202\pi$
$953$	40.2154	1.30270	0.651352	+	0.758776i	$0.274202\pi$
$954$	0	0
$955$	1.25482	0.0406049
$956$	0	0
$957$	88.4176	2.85814
$958$	0	0
$959$	−63.3446	−2.04551
$960$	0	0
$961$	−30.4731	−0.983003
$962$	0	0
$963$	−36.5715	−1.17850
$964$	0	0
$965$	1.08943	0.0350701
$966$	0	0
$967$	48.2295	1.55096	0.775478	−	0.631374i	$-0.217509\pi$
$967$	48.2295	1.55096	0.775478	+	0.631374i	$0.217509\pi$
$968$	0	0
$969$	−71.4413	−2.29502
$970$	0	0
$971$	−11.2576	−0.361272	−0.180636	−	0.983550i	$-0.557816\pi$
$971$	−11.2576	−0.361272	−0.180636	+	0.983550i	$0.557816\pi$
$972$	0	0
$973$	5.70230	0.182807
$974$	0	0
$975$	73.0821	2.34050
$976$	0	0
$977$	27.2631	0.872223	0.436111	−	0.899893i	$-0.356355\pi$
$977$	27.2631	0.872223	0.436111	+	0.899893i	$0.356355\pi$
$978$	0	0
$979$	68.6385	2.19369
$980$	0	0
$981$	−59.1546	−1.88866
$982$	0	0
$983$	13.0263	0.415475	0.207737	−	0.978185i	$-0.433390\pi$
$983$	13.0263	0.415475	0.207737	+	0.978185i	$0.433390\pi$
$984$	0	0
$985$	1.54013	0.0490728
$986$	0	0
$987$	59.7308	1.90125
$988$	0	0
$989$	−12.4747	−0.396672
$990$	0	0
$991$	51.8539	1.64719	0.823597	−	0.567175i	$-0.191964\pi$
$991$	51.8539	1.64719	0.823597	+	0.567175i	$0.191964\pi$
$992$	0	0
$993$	−46.3210	−1.46995
$994$	0	0
$995$	0.709718	0.0224996
$996$	0	0
$997$	−22.0678	−0.698895	−0.349448	−	0.936956i	$-0.613631\pi$
$997$	−22.0678	−0.698895	−0.349448	+	0.936956i	$0.613631\pi$
$998$	0	0
$999$	−21.1458	−0.669024

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6008.2.a.e.1.6	✓	50

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6008.2.a.e.1.6	✓	50	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 6008 = 2^{3} \cdot 751 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6008.a (trivial)

\(f(q)\)	\(=\)	\(q-2.58697 q^{3} +0.0890087 q^{5} +4.21791 q^{7} +3.69244 q^{9} +O(q^{10})\)	\(q-2.58697 q^{3} +0.0890087 q^{5} +4.21791 q^{7} +3.69244 q^{9} -5.56038 q^{11} +5.65898 q^{13} -0.230263 q^{15} +4.66419 q^{17} +5.92080 q^{19} -10.9116 q^{21} -2.98182 q^{23} -4.99208 q^{25} -1.79132 q^{27} +6.14670 q^{29} -0.725876 q^{31} +14.3846 q^{33} +0.375431 q^{35} +11.8046 q^{37} -14.6396 q^{39} -3.43975 q^{41} +4.18359 q^{43} +0.328659 q^{45} -5.47405 q^{47} +10.7908 q^{49} -12.0662 q^{51} +2.96430 q^{53} -0.494922 q^{55} -15.3170 q^{57} +3.19404 q^{59} +2.66626 q^{61} +15.5744 q^{63} +0.503698 q^{65} -2.11614 q^{67} +7.71389 q^{69} +12.6906 q^{71} -5.71725 q^{73} +12.9144 q^{75} -23.4532 q^{77} +7.85668 q^{79} -6.44322 q^{81} -2.69956 q^{83} +0.415154 q^{85} -15.9013 q^{87} -12.3442 q^{89} +23.8691 q^{91} +1.87782 q^{93} +0.527003 q^{95} -4.06463 q^{97} -20.5314 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 50 q + 6 q^{3} + 23 q^{5} + 12 q^{7} + 56 q^{9}+O(q^{10}) \) 50 * q + 6 * q^3 + 23 * q^5 + 12 * q^7 + 56 * q^9	\( 50 q + 6 q^{3} + 23 q^{5} + 12 q^{7} + 56 q^{9} - 5 q^{11} + 36 q^{13} + 5 q^{15} + 14 q^{17} + 9 q^{19} + 30 q^{21} + 3 q^{23} + 71 q^{25} + 24 q^{27} + 61 q^{29} + 27 q^{31} + 24 q^{33} - 7 q^{35} + 56 q^{37} - 2 q^{39} + 10 q^{41} + 19 q^{43} + 76 q^{45} + 3 q^{47} + 82 q^{49} - q^{51} + 56 q^{53} + 7 q^{55} + 35 q^{57} - q^{59} + 67 q^{61} + 25 q^{63} + 27 q^{65} + 46 q^{67} + 68 q^{69} + 4 q^{71} + 62 q^{73} + 27 q^{75} + 71 q^{77} + 7 q^{79} + 74 q^{81} - q^{83} + 72 q^{85} + 25 q^{87} + 19 q^{89} + 45 q^{91} + 72 q^{93} - 24 q^{95} + 81 q^{97} + 16 q^{99}+O(q^{100}) \) 50 * q + 6 * q^3 + 23 * q^5 + 12 * q^7 + 56 * q^9 - 5 * q^11 + 36 * q^13 + 5 * q^15 + 14 * q^17 + 9 * q^19 + 30 * q^21 + 3 * q^23 + 71 * q^25 + 24 * q^27 + 61 * q^29 + 27 * q^31 + 24 * q^33 - 7 * q^35 + 56 * q^37 - 2 * q^39 + 10 * q^41 + 19 * q^43 + 76 * q^45 + 3 * q^47 + 82 * q^49 - q^51 + 56 * q^53 + 7 * q^55 + 35 * q^57 - q^59 + 67 * q^61 + 25 * q^63 + 27 * q^65 + 46 * q^67 + 68 * q^69 + 4 * q^71 + 62 * q^73 + 27 * q^75 + 71 * q^77 + 7 * q^79 + 74 * q^81 - q^83 + 72 * q^85 + 25 * q^87 + 19 * q^89 + 45 * q^91 + 72 * q^93 - 24 * q^95 + 81 * q^97 + 16 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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