LMFDB - Embedded newform 8004.2.a.j.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8004,2,Mod(1,8004)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8004, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("8004.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8004 = 2^{2} \cdot 3 \cdot 23 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8004.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:	$63.9122617778$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 3 x^{15} - 49 x^{14} + 130 x^{13} + 932 x^{12} - 2028 x^{11} - 8965 x^{10} + 14400 x^{9} + \cdots + 3888 $ x^16 - 3x^15 - 49x^14 + 130x^13 + 932x^12 - 2028x^11 - 8965x^10 + 14400x^9 + 46229x^8 - 47547x^7 - 122604x^6 + 65278x^5 + 151028x^4 - 17988x^3 - 67608x^2 - 8424*x + 3888
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-4.29726$ of defining polynomial
Character	$\chi$	$=$	8004.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -4.29726 q^{5} +3.54512 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -4.29726 q^{5} +3.54512 q^{7} +1.00000 q^{9} +0.161684 q^{11} +2.42045 q^{13} -4.29726 q^{15} +3.98980 q^{17} +7.43592 q^{19} +3.54512 q^{21} +1.00000 q^{23} +13.4665 q^{25} +1.00000 q^{27} -1.00000 q^{29} +2.05208 q^{31} +0.161684 q^{33} -15.2343 q^{35} -5.65285 q^{37} +2.42045 q^{39} -7.47256 q^{41} -2.80205 q^{43} -4.29726 q^{45} -5.80920 q^{47} +5.56789 q^{49} +3.98980 q^{51} +5.67820 q^{53} -0.694798 q^{55} +7.43592 q^{57} +14.1016 q^{59} +14.8564 q^{61} +3.54512 q^{63} -10.4013 q^{65} -9.34631 q^{67} +1.00000 q^{69} -11.6877 q^{71} +2.93865 q^{73} +13.4665 q^{75} +0.573189 q^{77} -0.741638 q^{79} +1.00000 q^{81} -7.89190 q^{83} -17.1452 q^{85} -1.00000 q^{87} +1.43329 q^{89} +8.58079 q^{91} +2.05208 q^{93} -31.9541 q^{95} -4.91986 q^{97} +0.161684 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 16 q^{3} + 3 q^{5} + 4 q^{7} + 16 q^{9}+O(q^{10}) $ 16 * q + 16 * q^3 + 3 * q^5 + 4 * q^7 + 16 * q^9	$ 16 q + 16 q^{3} + 3 q^{5} + 4 q^{7} + 16 q^{9} + 5 q^{11} + 6 q^{13} + 3 q^{15} + 3 q^{17} + 11 q^{19} + 4 q^{21} + 16 q^{23} + 27 q^{25} + 16 q^{27} - 16 q^{29} + 14 q^{31} + 5 q^{33} + 11 q^{35} + 4 q^{37} + 6 q^{39} + 11 q^{41} + 23 q^{43} + 3 q^{45} - 2 q^{47} + 34 q^{49} + 3 q^{51} + 19 q^{53} + 31 q^{55} + 11 q^{57} + 32 q^{59} + 19 q^{61} + 4 q^{63} + 6 q^{65} + 33 q^{67} + 16 q^{69} - 5 q^{71} + 23 q^{73} + 27 q^{75} + 42 q^{77} + 24 q^{79} + 16 q^{81} + 7 q^{83} - 16 q^{87} - 2 q^{89} + 25 q^{91} + 14 q^{93} + 7 q^{95} + 33 q^{97} + 5 q^{99}+O(q^{100}) $ 16 * q + 16 * q^3 + 3 * q^5 + 4 * q^7 + 16 * q^9 + 5 * q^11 + 6 * q^13 + 3 * q^15 + 3 * q^17 + 11 * q^19 + 4 * q^21 + 16 * q^23 + 27 * q^25 + 16 * q^27 - 16 * q^29 + 14 * q^31 + 5 * q^33 + 11 * q^35 + 4 * q^37 + 6 * q^39 + 11 * q^41 + 23 * q^43 + 3 * q^45 - 2 * q^47 + 34 * q^49 + 3 * q^51 + 19 * q^53 + 31 * q^55 + 11 * q^57 + 32 * q^59 + 19 * q^61 + 4 * q^63 + 6 * q^65 + 33 * q^67 + 16 * q^69 - 5 * q^71 + 23 * q^73 + 27 * q^75 + 42 * q^77 + 24 * q^79 + 16 * q^81 + 7 * q^83 - 16 * q^87 - 2 * q^89 + 25 * q^91 + 14 * q^93 + 7 * q^95 + 33 * 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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	−4.29726	−1.92179	−0.960897	−	0.276905i	$-0.910691\pi$
$5$	−4.29726	−1.92179	−0.960897	+	0.276905i	$0.910691\pi$
$6$	0	0
$7$	3.54512	1.33993	0.669965	−	0.742393i	$-0.266309\pi$
$7$	3.54512	1.33993	0.669965	+	0.742393i	$0.266309\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	0.161684	0.0487495	0.0243747	−	0.999703i	$-0.492241\pi$
$11$	0.161684	0.0487495	0.0243747	+	0.999703i	$0.492241\pi$
$12$	0	0
$13$	2.42045	0.671312	0.335656	−	0.941985i	$-0.391042\pi$
$13$	2.42045	0.671312	0.335656	+	0.941985i	$0.391042\pi$
$14$	0	0
$15$	−4.29726	−1.10955
$16$	0	0
$17$	3.98980	0.967669	0.483834	−	0.875160i	$-0.339244\pi$
$17$	3.98980	0.967669	0.483834	+	0.875160i	$0.339244\pi$
$18$	0	0
$19$	7.43592	1.70592	0.852959	−	0.521978i	$-0.174806\pi$
$19$	7.43592	1.70592	0.852959	+	0.521978i	$0.174806\pi$
$20$	0	0
$21$	3.54512	0.773609
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	13.4665	2.69329
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	2.05208	0.368565	0.184283	−	0.982873i	$-0.441004\pi$
$31$	2.05208	0.368565	0.184283	+	0.982873i	$0.441004\pi$
$32$	0	0
$33$	0.161684	0.0281455
$34$	0	0
$35$	−15.2343	−2.57507
$36$	0	0
$37$	−5.65285	−0.929322	−0.464661	−	0.885489i	$-0.653824\pi$
$37$	−5.65285	−0.929322	−0.464661	+	0.885489i	$0.653824\pi$
$38$	0	0
$39$	2.42045	0.387582
$40$	0	0
$41$	−7.47256	−1.16702	−0.583509	−	0.812107i	$-0.698321\pi$
$41$	−7.47256	−1.16702	−0.583509	+	0.812107i	$0.698321\pi$
$42$	0	0
$43$	−2.80205	−0.427309	−0.213655	−	0.976909i	$-0.568537\pi$
$43$	−2.80205	−0.427309	−0.213655	+	0.976909i	$0.568537\pi$
$44$	0	0
$45$	−4.29726	−0.640598
$46$	0	0
$47$	−5.80920	−0.847359	−0.423679	−	0.905812i	$-0.639262\pi$
$47$	−5.80920	−0.847359	−0.423679	+	0.905812i	$0.639262\pi$
$48$	0	0
$49$	5.56789	0.795412
$50$	0	0
$51$	3.98980	0.558684
$52$	0	0
$53$	5.67820	0.779960	0.389980	−	0.920823i	$-0.372482\pi$
$53$	5.67820	0.779960	0.389980	+	0.920823i	$0.372482\pi$
$54$	0	0
$55$	−0.694798	−0.0936865
$56$	0	0
$57$	7.43592	0.984912
$58$	0	0
$59$	14.1016	1.83587	0.917936	−	0.396729i	$-0.129855\pi$
$59$	14.1016	1.83587	0.917936	+	0.396729i	$0.129855\pi$
$60$	0	0
$61$	14.8564	1.90217	0.951083	−	0.308936i	$-0.0999730\pi$
$61$	14.8564	1.90217	0.951083	+	0.308936i	$0.0999730\pi$
$62$	0	0
$63$	3.54512	0.446643
$64$	0	0
$65$	−10.4013	−1.29012
$66$	0	0
$67$	−9.34631	−1.14183	−0.570917	−	0.821008i	$-0.693412\pi$
$67$	−9.34631	−1.14183	−0.570917	+	0.821008i	$0.693412\pi$
$68$	0	0
$69$	1.00000	0.120386
$70$	0	0
$71$	−11.6877	−1.38707	−0.693536	−	0.720422i	$-0.743948\pi$
$71$	−11.6877	−1.38707	−0.693536	+	0.720422i	$0.743948\pi$
$72$	0	0
$73$	2.93865	0.343943	0.171972	−	0.985102i	$-0.444986\pi$
$73$	2.93865	0.343943	0.171972	+	0.985102i	$0.444986\pi$
$74$	0	0
$75$	13.4665	1.55497
$76$	0	0
$77$	0.573189	0.0653209
$78$	0	0
$79$	−0.741638	−0.0834408	−0.0417204	−	0.999129i	$-0.513284\pi$
$79$	−0.741638	−0.0834408	−0.0417204	+	0.999129i	$0.513284\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−7.89190	−0.866249	−0.433124	−	0.901334i	$-0.642589\pi$
$83$	−7.89190	−0.866249	−0.433124	+	0.901334i	$0.642589\pi$
$84$	0	0
$85$	−17.1452	−1.85966
$86$	0	0
$87$	−1.00000	−0.107211
$88$	0	0
$89$	1.43329	0.151928	0.0759640	−	0.997111i	$-0.475797\pi$
$89$	1.43329	0.151928	0.0759640	+	0.997111i	$0.475797\pi$
$90$	0	0
$91$	8.58079	0.899511
$92$	0	0
$93$	2.05208	0.212791
$94$	0	0
$95$	−31.9541	−3.27842
$96$	0	0
$97$	−4.91986	−0.499537	−0.249768	−	0.968306i	$-0.580354\pi$
$97$	−4.91986	−0.499537	−0.249768	+	0.968306i	$0.580354\pi$
$98$	0	0
$99$	0.161684	0.0162498
$100$	0	0
$101$	−14.9474	−1.48732	−0.743659	−	0.668559i	$-0.766912\pi$
$101$	−14.9474	−1.48732	−0.743659	+	0.668559i	$0.766912\pi$
$102$	0	0
$103$	8.31164	0.818970	0.409485	−	0.912317i	$-0.365708\pi$
$103$	8.31164	0.818970	0.409485	+	0.912317i	$0.365708\pi$
$104$	0	0
$105$	−15.2343	−1.48672
$106$	0	0
$107$	13.6226	1.31695	0.658473	−	0.752604i	$-0.271203\pi$
$107$	13.6226	1.31695	0.658473	+	0.752604i	$0.271203\pi$
$108$	0	0
$109$	1.79623	0.172047	0.0860237	−	0.996293i	$-0.472584\pi$
$109$	1.79623	0.172047	0.0860237	+	0.996293i	$0.472584\pi$
$110$	0	0
$111$	−5.65285	−0.536544
$112$	0	0
$113$	7.84617	0.738106	0.369053	−	0.929408i	$-0.379682\pi$
$113$	7.84617	0.738106	0.369053	+	0.929408i	$0.379682\pi$
$114$	0	0
$115$	−4.29726	−0.400722
$116$	0	0
$117$	2.42045	0.223771
$118$	0	0
$119$	14.1443	1.29661
$120$	0	0
$121$	−10.9739	−0.997623
$122$	0	0
$123$	−7.47256	−0.673778
$124$	0	0
$125$	−36.3826	−3.25416
$126$	0	0
$127$	4.81437	0.427206	0.213603	−	0.976921i	$-0.431480\pi$
$127$	4.81437	0.427206	0.213603	+	0.976921i	$0.431480\pi$
$128$	0	0
$129$	−2.80205	−0.246707
$130$	0	0
$131$	3.42938	0.299626	0.149813	−	0.988714i	$-0.452133\pi$
$131$	3.42938	0.299626	0.149813	+	0.988714i	$0.452133\pi$
$132$	0	0
$133$	26.3613	2.28581
$134$	0	0
$135$	−4.29726	−0.369849
$136$	0	0
$137$	4.52628	0.386706	0.193353	−	0.981129i	$-0.438064\pi$
$137$	4.52628	0.386706	0.193353	+	0.981129i	$0.438064\pi$
$138$	0	0
$139$	12.9197	1.09583	0.547915	−	0.836534i	$-0.315422\pi$
$139$	12.9197	1.09583	0.547915	+	0.836534i	$0.315422\pi$
$140$	0	0
$141$	−5.80920	−0.489223
$142$	0	0
$143$	0.391348	0.0327261
$144$	0	0
$145$	4.29726	0.356868
$146$	0	0
$147$	5.56789	0.459232
$148$	0	0
$149$	6.26844	0.513531	0.256765	−	0.966474i	$-0.417343\pi$
$149$	6.26844	0.513531	0.256765	+	0.966474i	$0.417343\pi$
$150$	0	0
$151$	−22.1302	−1.80093	−0.900463	−	0.434932i	$-0.856772\pi$
$151$	−22.1302	−1.80093	−0.900463	+	0.434932i	$0.856772\pi$
$152$	0	0
$153$	3.98980	0.322556
$154$	0	0
$155$	−8.81835	−0.708307
$156$	0	0
$157$	−22.9965	−1.83532	−0.917662	−	0.397362i	$-0.869926\pi$
$157$	−22.9965	−1.83532	−0.917662	+	0.397362i	$0.869926\pi$
$158$	0	0
$159$	5.67820	0.450310
$160$	0	0
$161$	3.54512	0.279395
$162$	0	0
$163$	22.5673	1.76761	0.883805	−	0.467855i	$-0.154973\pi$
$163$	22.5673	1.76761	0.883805	+	0.467855i	$0.154973\pi$
$164$	0	0
$165$	−0.694798	−0.0540899
$166$	0	0
$167$	−15.3901	−1.19092	−0.595461	−	0.803385i	$-0.703030\pi$
$167$	−15.3901	−1.19092	−0.595461	+	0.803385i	$0.703030\pi$
$168$	0	0
$169$	−7.14142	−0.549340
$170$	0	0
$171$	7.43592	0.568639
$172$	0	0
$173$	21.5562	1.63889	0.819443	−	0.573161i	$-0.194283\pi$
$173$	21.5562	1.63889	0.819443	+	0.573161i	$0.194283\pi$
$174$	0	0
$175$	47.7403	3.60882
$176$	0	0
$177$	14.1016	1.05994
$178$	0	0
$179$	24.5968	1.83845	0.919227	−	0.393729i	$-0.128815\pi$
$179$	24.5968	1.83845	0.919227	+	0.393729i	$0.128815\pi$
$180$	0	0
$181$	10.9273	0.812218	0.406109	−	0.913825i	$-0.366885\pi$
$181$	10.9273	0.812218	0.406109	+	0.913825i	$0.366885\pi$
$182$	0	0
$183$	14.8564	1.09822
$184$	0	0
$185$	24.2918	1.78597
$186$	0	0
$187$	0.645086	0.0471734
$188$	0	0
$189$	3.54512	0.257870
$190$	0	0
$191$	20.1377	1.45711	0.728557	−	0.684985i	$-0.240191\pi$
$191$	20.1377	1.45711	0.728557	+	0.684985i	$0.240191\pi$
$192$	0	0
$193$	−11.2775	−0.811771	−0.405885	−	0.913924i	$-0.633037\pi$
$193$	−11.2775	−0.811771	−0.405885	+	0.913924i	$0.633037\pi$
$194$	0	0
$195$	−10.4013	−0.744853
$196$	0	0
$197$	−6.42258	−0.457590	−0.228795	−	0.973475i	$-0.573478\pi$
$197$	−6.42258	−0.457590	−0.228795	+	0.973475i	$0.573478\pi$
$198$	0	0
$199$	0.180557	0.0127993	0.00639967	−	0.999980i	$-0.497963\pi$
$199$	0.180557	0.0127993	0.00639967	+	0.999980i	$0.497963\pi$
$200$	0	0
$201$	−9.34631	−0.659238
$202$	0	0
$203$	−3.54512	−0.248819
$204$	0	0
$205$	32.1115	2.24277
$206$	0	0
$207$	1.00000	0.0695048
$208$	0	0
$209$	1.20227	0.0831626
$210$	0	0
$211$	13.5250	0.931099	0.465549	−	0.885022i	$-0.345857\pi$
$211$	13.5250	0.931099	0.465549	+	0.885022i	$0.345857\pi$
$212$	0	0
$213$	−11.6877	−0.800827
$214$	0	0
$215$	12.0412	0.821200
$216$	0	0
$217$	7.27489	0.493852
$218$	0	0
$219$	2.93865	0.198576
$220$	0	0
$221$	9.65711	0.649608
$222$	0	0
$223$	−0.118035	−0.00790420	−0.00395210	−	0.999992i	$-0.501258\pi$
$223$	−0.118035	−0.00790420	−0.00395210	+	0.999992i	$0.501258\pi$
$224$	0	0
$225$	13.4665	0.897765
$226$	0	0
$227$	12.9065	0.856634	0.428317	−	0.903629i	$-0.359107\pi$
$227$	12.9065	0.856634	0.428317	+	0.903629i	$0.359107\pi$
$228$	0	0
$229$	−13.2700	−0.876909	−0.438455	−	0.898753i	$-0.644474\pi$
$229$	−13.2700	−0.876909	−0.438455	+	0.898753i	$0.644474\pi$
$230$	0	0
$231$	0.573189	0.0377130
$232$	0	0
$233$	1.94230	0.127244	0.0636222	−	0.997974i	$-0.479735\pi$
$233$	1.94230	0.127244	0.0636222	+	0.997974i	$0.479735\pi$
$234$	0	0
$235$	24.9637	1.62845
$236$	0	0
$237$	−0.741638	−0.0481746
$238$	0	0
$239$	17.3816	1.12432	0.562161	−	0.827028i	$-0.309970\pi$
$239$	17.3816	1.12432	0.562161	+	0.827028i	$0.309970\pi$
$240$	0	0
$241$	15.9745	1.02901	0.514504	−	0.857488i	$-0.327976\pi$
$241$	15.9745	1.02901	0.514504	+	0.857488i	$0.327976\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−23.9267	−1.52862
$246$	0	0
$247$	17.9983	1.14520
$248$	0	0
$249$	−7.89190	−0.500129
$250$	0	0
$251$	6.56916	0.414642	0.207321	−	0.978273i	$-0.433526\pi$
$251$	6.56916	0.414642	0.207321	+	0.978273i	$0.433526\pi$
$252$	0	0
$253$	0.161684	0.0101650
$254$	0	0
$255$	−17.1452	−1.07368
$256$	0	0
$257$	−22.8125	−1.42300	−0.711502	−	0.702684i	$-0.751985\pi$
$257$	−22.8125	−1.42300	−0.711502	+	0.702684i	$0.751985\pi$
$258$	0	0
$259$	−20.0400	−1.24523
$260$	0	0
$261$	−1.00000	−0.0618984
$262$	0	0
$263$	−26.5281	−1.63579	−0.817896	−	0.575366i	$-0.804860\pi$
$263$	−26.5281	−1.63579	−0.817896	+	0.575366i	$0.804860\pi$
$264$	0	0
$265$	−24.4007	−1.49892
$266$	0	0
$267$	1.43329	0.0877156
$268$	0	0
$269$	28.3257	1.72705	0.863525	−	0.504307i	$-0.168252\pi$
$269$	28.3257	1.72705	0.863525	+	0.504307i	$0.168252\pi$
$270$	0	0
$271$	32.1900	1.95540	0.977701	−	0.210002i	$-0.0673471\pi$
$271$	32.1900	1.95540	0.977701	+	0.210002i	$0.0673471\pi$
$272$	0	0
$273$	8.58079	0.519333
$274$	0	0
$275$	2.17731	0.131297
$276$	0	0
$277$	24.2130	1.45482	0.727409	−	0.686204i	$-0.240724\pi$
$277$	24.2130	1.45482	0.727409	+	0.686204i	$0.240724\pi$
$278$	0	0
$279$	2.05208	0.122855
$280$	0	0
$281$	−11.0584	−0.659687	−0.329844	−	0.944036i	$-0.606996\pi$
$281$	−11.0584	−0.659687	−0.329844	+	0.944036i	$0.606996\pi$
$282$	0	0
$283$	−18.1002	−1.07594	−0.537972	−	0.842963i	$-0.680809\pi$
$283$	−18.1002	−1.07594	−0.537972	+	0.842963i	$0.680809\pi$
$284$	0	0
$285$	−31.9541	−1.89280
$286$	0	0
$287$	−26.4911	−1.56372
$288$	0	0
$289$	−1.08150	−0.0636175
$290$	0	0
$291$	−4.91986	−0.288408
$292$	0	0
$293$	3.41385	0.199439	0.0997197	−	0.995016i	$-0.468205\pi$
$293$	3.41385	0.199439	0.0997197	+	0.995016i	$0.468205\pi$
$294$	0	0
$295$	−60.5983	−3.52817
$296$	0	0
$297$	0.161684	0.00938184
$298$	0	0
$299$	2.42045	0.139978
$300$	0	0
$301$	−9.93362	−0.572564
$302$	0	0
$303$	−14.9474	−0.858704
$304$	0	0
$305$	−63.8418	−3.65557
$306$	0	0
$307$	19.5836	1.11769	0.558846	−	0.829271i	$-0.311244\pi$
$307$	19.5836	1.11769	0.558846	+	0.829271i	$0.311244\pi$
$308$	0	0
$309$	8.31164	0.472833
$310$	0	0
$311$	18.5304	1.05076	0.525381	−	0.850867i	$-0.323923\pi$
$311$	18.5304	1.05076	0.525381	+	0.850867i	$0.323923\pi$
$312$	0	0
$313$	−0.248252	−0.0140320	−0.00701602	−	0.999975i	$-0.502233\pi$
$313$	−0.248252	−0.0140320	−0.00701602	+	0.999975i	$0.502233\pi$
$314$	0	0
$315$	−15.2343	−0.858357
$316$	0	0
$317$	4.94475	0.277725	0.138862	−	0.990312i	$-0.455655\pi$
$317$	4.94475	0.277725	0.138862	+	0.990312i	$0.455655\pi$
$318$	0	0
$319$	−0.161684	−0.00905255
$320$	0	0
$321$	13.6226	0.760339
$322$	0	0
$323$	29.6678	1.65076
$324$	0	0
$325$	32.5949	1.80804
$326$	0	0
$327$	1.79623	0.0993316
$328$	0	0
$329$	−20.5943	−1.13540
$330$	0	0
$331$	31.6537	1.73984	0.869921	−	0.493191i	$-0.164170\pi$
$331$	31.6537	1.73984	0.869921	+	0.493191i	$0.164170\pi$
$332$	0	0
$333$	−5.65285	−0.309774
$334$	0	0
$335$	40.1635	2.19437
$336$	0	0
$337$	0.331821	0.0180755	0.00903773	−	0.999959i	$-0.497123\pi$
$337$	0.331821	0.0180755	0.00903773	+	0.999959i	$0.497123\pi$
$338$	0	0
$339$	7.84617	0.426146
$340$	0	0
$341$	0.331789	0.0179674
$342$	0	0
$343$	−5.07701	−0.274133
$344$	0	0
$345$	−4.29726	−0.231357
$346$	0	0
$347$	−12.9088	−0.692978	−0.346489	−	0.938054i	$-0.612626\pi$
$347$	−12.9088	−0.692978	−0.346489	+	0.938054i	$0.612626\pi$
$348$	0	0
$349$	5.15461	0.275920	0.137960	−	0.990438i	$-0.455945\pi$
$349$	5.15461	0.275920	0.137960	+	0.990438i	$0.455945\pi$
$350$	0	0
$351$	2.42045	0.129194
$352$	0	0
$353$	−25.7647	−1.37132	−0.685659	−	0.727923i	$-0.740486\pi$
$353$	−25.7647	−1.37132	−0.685659	+	0.727923i	$0.740486\pi$
$354$	0	0
$355$	50.2251	2.66567
$356$	0	0
$357$	14.1443	0.748597
$358$	0	0
$359$	−37.3146	−1.96939	−0.984695	−	0.174286i	$-0.944238\pi$
$359$	−37.3146	−1.96939	−0.984695	+	0.174286i	$0.944238\pi$
$360$	0	0
$361$	36.2930	1.91016
$362$	0	0
$363$	−10.9739	−0.575978
$364$	0	0
$365$	−12.6282	−0.660988
$366$	0	0
$367$	14.9939	0.782673	0.391337	−	0.920248i	$-0.372013\pi$
$367$	14.9939	0.782673	0.391337	+	0.920248i	$0.372013\pi$
$368$	0	0
$369$	−7.47256	−0.389006
$370$	0	0
$371$	20.1299	1.04509
$372$	0	0
$373$	−10.0591	−0.520842	−0.260421	−	0.965495i	$-0.583861\pi$
$373$	−10.0591	−0.520842	−0.260421	+	0.965495i	$0.583861\pi$
$374$	0	0
$375$	−36.3826	−1.87879
$376$	0	0
$377$	−2.42045	−0.124660
$378$	0	0
$379$	−36.5534	−1.87762	−0.938810	−	0.344435i	$-0.888070\pi$
$379$	−36.5534	−1.87762	−0.938810	+	0.344435i	$0.888070\pi$
$380$	0	0
$381$	4.81437	0.246647
$382$	0	0
$383$	−3.76778	−0.192525	−0.0962623	−	0.995356i	$-0.530689\pi$
$383$	−3.76778	−0.192525	−0.0962623	+	0.995356i	$0.530689\pi$
$384$	0	0
$385$	−2.46314	−0.125533
$386$	0	0
$387$	−2.80205	−0.142436
$388$	0	0
$389$	−17.8158	−0.903296	−0.451648	−	0.892196i	$-0.649164\pi$
$389$	−17.8158	−0.903296	−0.451648	+	0.892196i	$0.649164\pi$
$390$	0	0
$391$	3.98980	0.201773
$392$	0	0
$393$	3.42938	0.172989
$394$	0	0
$395$	3.18701	0.160356
$396$	0	0
$397$	−36.0963	−1.81162	−0.905812	−	0.423681i	$-0.860738\pi$
$397$	−36.0963	−1.81162	−0.905812	+	0.423681i	$0.860738\pi$
$398$	0	0
$399$	26.3613	1.31971
$400$	0	0
$401$	−25.7298	−1.28488	−0.642442	−	0.766334i	$-0.722079\pi$
$401$	−25.7298	−1.28488	−0.642442	+	0.766334i	$0.722079\pi$
$402$	0	0
$403$	4.96697	0.247422
$404$	0	0
$405$	−4.29726	−0.213533
$406$	0	0
$407$	−0.913974	−0.0453040
$408$	0	0
$409$	−2.87957	−0.142386	−0.0711928	−	0.997463i	$-0.522681\pi$
$409$	−2.87957	−0.142386	−0.0711928	+	0.997463i	$0.522681\pi$
$410$	0	0
$411$	4.52628	0.223265
$412$	0	0
$413$	49.9919	2.45994
$414$	0	0
$415$	33.9136	1.66475
$416$	0	0
$417$	12.9197	0.632678
$418$	0	0
$419$	37.4858	1.83130	0.915651	−	0.401973i	$-0.131676\pi$
$419$	37.4858	1.83130	0.915651	+	0.401973i	$0.131676\pi$
$420$	0	0
$421$	23.1785	1.12965	0.564826	−	0.825210i	$-0.308943\pi$
$421$	23.1785	1.12965	0.564826	+	0.825210i	$0.308943\pi$
$422$	0	0
$423$	−5.80920	−0.282453
$424$	0	0
$425$	53.7285	2.60622
$426$	0	0
$427$	52.6677	2.54877
$428$	0	0
$429$	0.391348	0.0188944
$430$	0	0
$431$	11.1831	0.538671	0.269335	−	0.963046i	$-0.413196\pi$
$431$	11.1831	0.538671	0.269335	+	0.963046i	$0.413196\pi$
$432$	0	0
$433$	14.3062	0.687514	0.343757	−	0.939059i	$-0.388300\pi$
$433$	14.3062	0.687514	0.343757	+	0.939059i	$0.388300\pi$
$434$	0	0
$435$	4.29726	0.206038
$436$	0	0
$437$	7.43592	0.355708
$438$	0	0
$439$	6.97421	0.332861	0.166431	−	0.986053i	$-0.446776\pi$
$439$	6.97421	0.332861	0.166431	+	0.986053i	$0.446776\pi$
$440$	0	0
$441$	5.56789	0.265137
$442$	0	0
$443$	15.1626	0.720396	0.360198	−	0.932876i	$-0.382709\pi$
$443$	15.1626	0.720396	0.360198	+	0.932876i	$0.382709\pi$
$444$	0	0
$445$	−6.15920	−0.291974
$446$	0	0
$447$	6.26844	0.296487
$448$	0	0
$449$	16.0503	0.757460	0.378730	−	0.925507i	$-0.376361\pi$
$449$	16.0503	0.757460	0.378730	+	0.925507i	$0.376361\pi$
$450$	0	0
$451$	−1.20819	−0.0568915
$452$	0	0
$453$	−22.1302	−1.03977
$454$	0	0
$455$	−36.8739	−1.72868
$456$	0	0
$457$	−5.93598	−0.277673	−0.138837	−	0.990315i	$-0.544336\pi$
$457$	−5.93598	−0.277673	−0.138837	+	0.990315i	$0.544336\pi$
$458$	0	0
$459$	3.98980	0.186228
$460$	0	0
$461$	15.8202	0.736819	0.368410	−	0.929664i	$-0.379902\pi$
$461$	15.8202	0.736819	0.368410	+	0.929664i	$0.379902\pi$
$462$	0	0
$463$	21.4169	0.995330	0.497665	−	0.867369i	$-0.334191\pi$
$463$	21.4169	0.995330	0.497665	+	0.867369i	$0.334191\pi$
$464$	0	0
$465$	−8.81835	−0.408941
$466$	0	0
$467$	−10.0144	−0.463412	−0.231706	−	0.972786i	$-0.574431\pi$
$467$	−10.0144	−0.463412	−0.231706	+	0.972786i	$0.574431\pi$
$468$	0	0
$469$	−33.1338	−1.52998
$470$	0	0
$471$	−22.9965	−1.05962
$472$	0	0
$473$	−0.453047	−0.0208311
$474$	0	0
$475$	100.136	4.59454
$476$	0	0
$477$	5.67820	0.259987
$478$	0	0
$479$	2.38039	0.108763	0.0543814	−	0.998520i	$-0.482681\pi$
$479$	2.38039	0.108763	0.0543814	+	0.998520i	$0.482681\pi$
$480$	0	0
$481$	−13.6824	−0.623865
$482$	0	0
$483$	3.54512	0.161309
$484$	0	0
$485$	21.1420	0.960007
$486$	0	0
$487$	21.1654	0.959098	0.479549	−	0.877515i	$-0.340800\pi$
$487$	21.1654	0.959098	0.479549	+	0.877515i	$0.340800\pi$
$488$	0	0
$489$	22.5673	1.02053
$490$	0	0
$491$	26.1269	1.17909	0.589546	−	0.807735i	$-0.299307\pi$
$491$	26.1269	1.17909	0.589546	+	0.807735i	$0.299307\pi$
$492$	0	0
$493$	−3.98980	−0.179692
$494$	0	0
$495$	−0.694798	−0.0312288
$496$	0	0
$497$	−41.4343	−1.85858
$498$	0	0
$499$	21.9600	0.983065	0.491533	−	0.870859i	$-0.336437\pi$
$499$	21.9600	0.983065	0.491533	+	0.870859i	$0.336437\pi$
$500$	0	0
$501$	−15.3901	−0.687579
$502$	0	0
$503$	−29.8753	−1.33207	−0.666037	−	0.745918i	$-0.732011\pi$
$503$	−29.8753	−1.33207	−0.666037	+	0.745918i	$0.732011\pi$
$504$	0	0
$505$	64.2328	2.85832
$506$	0	0
$507$	−7.14142	−0.317162
$508$	0	0
$509$	1.76813	0.0783708	0.0391854	−	0.999232i	$-0.487524\pi$
$509$	1.76813	0.0783708	0.0391854	+	0.999232i	$0.487524\pi$
$510$	0	0
$511$	10.4179	0.460860
$512$	0	0
$513$	7.43592	0.328304
$514$	0	0
$515$	−35.7173	−1.57389
$516$	0	0
$517$	−0.939253	−0.0413083
$518$	0	0
$519$	21.5562	0.946211
$520$	0	0
$521$	18.1591	0.795567	0.397783	−	0.917479i	$-0.369780\pi$
$521$	18.1591	0.795567	0.397783	+	0.917479i	$0.369780\pi$
$522$	0	0
$523$	18.5795	0.812425	0.406213	−	0.913779i	$-0.366849\pi$
$523$	18.5795	0.812425	0.406213	+	0.913779i	$0.366849\pi$
$524$	0	0
$525$	47.7403	2.08356
$526$	0	0
$527$	8.18741	0.356649
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	14.1016	0.611957
$532$	0	0
$533$	−18.0870	−0.783433
$534$	0	0
$535$	−58.5399	−2.53090
$536$	0	0
$537$	24.5968	1.06143
$538$	0	0
$539$	0.900237	0.0387760
$540$	0	0
$541$	−33.9990	−1.46173	−0.730865	−	0.682522i	$-0.760883\pi$
$541$	−33.9990	−1.46173	−0.730865	+	0.682522i	$0.760883\pi$
$542$	0	0
$543$	10.9273	0.468934
$544$	0	0
$545$	−7.71886	−0.330640
$546$	0	0
$547$	−42.1184	−1.80085	−0.900427	−	0.435007i	$-0.856746\pi$
$547$	−42.1184	−1.80085	−0.900427	+	0.435007i	$0.856746\pi$
$548$	0	0
$549$	14.8564	0.634055
$550$	0	0
$551$	−7.43592	−0.316781
$552$	0	0
$553$	−2.62920	−0.111805
$554$	0	0
$555$	24.2918	1.03113
$556$	0	0
$557$	−18.9780	−0.804122	−0.402061	−	0.915613i	$-0.631706\pi$
$557$	−18.9780	−0.804122	−0.402061	+	0.915613i	$0.631706\pi$
$558$	0	0
$559$	−6.78223	−0.286858
$560$	0	0
$561$	0.645086	0.0272356
$562$	0	0
$563$	−30.7441	−1.29571	−0.647854	−	0.761765i	$-0.724333\pi$
$563$	−30.7441	−1.29571	−0.647854	+	0.761765i	$0.724333\pi$
$564$	0	0
$565$	−33.7171	−1.41849
$566$	0	0
$567$	3.54512	0.148881
$568$	0	0
$569$	−20.8939	−0.875916	−0.437958	−	0.898995i	$-0.644298\pi$
$569$	−20.8939	−0.875916	−0.437958	+	0.898995i	$0.644298\pi$
$570$	0	0
$571$	−37.3266	−1.56207	−0.781034	−	0.624488i	$-0.785308\pi$
$571$	−37.3266	−1.56207	−0.781034	+	0.624488i	$0.785308\pi$
$572$	0	0
$573$	20.1377	0.841265
$574$	0	0
$575$	13.4665	0.561591
$576$	0	0
$577$	9.26532	0.385720	0.192860	−	0.981226i	$-0.438224\pi$
$577$	9.26532	0.385720	0.192860	+	0.981226i	$0.438224\pi$
$578$	0	0
$579$	−11.2775	−0.468676
$580$	0	0
$581$	−27.9778	−1.16071
$582$	0	0
$583$	0.918072	0.0380227
$584$	0	0
$585$	−10.4013	−0.430041
$586$	0	0
$587$	36.6545	1.51289	0.756447	−	0.654056i	$-0.226934\pi$
$587$	36.6545	1.51289	0.756447	+	0.654056i	$0.226934\pi$
$588$	0	0
$589$	15.2591	0.628742
$590$	0	0
$591$	−6.42258	−0.264189
$592$	0	0
$593$	46.2980	1.90123	0.950616	−	0.310368i	$-0.100452\pi$
$593$	46.2980	1.90123	0.950616	+	0.310368i	$0.100452\pi$
$594$	0	0
$595$	−60.7819	−2.49181
$596$	0	0
$597$	0.180557	0.00738970
$598$	0	0
$599$	22.9770	0.938813	0.469407	−	0.882982i	$-0.344468\pi$
$599$	22.9770	0.938813	0.469407	+	0.882982i	$0.344468\pi$
$600$	0	0
$601$	−44.8715	−1.83035	−0.915173	−	0.403061i	$-0.867946\pi$
$601$	−44.8715	−1.83035	−0.915173	+	0.403061i	$0.867946\pi$
$602$	0	0
$603$	−9.34631	−0.380611
$604$	0	0
$605$	47.1576	1.91723
$606$	0	0
$607$	−17.5239	−0.711271	−0.355636	−	0.934625i	$-0.615736\pi$
$607$	−17.5239	−0.711271	−0.355636	+	0.934625i	$0.615736\pi$
$608$	0	0
$609$	−3.54512	−0.143656
$610$	0	0
$611$	−14.0609	−0.568842
$612$	0	0
$613$	13.7233	0.554279	0.277140	−	0.960830i	$-0.410614\pi$
$613$	13.7233	0.554279	0.277140	+	0.960830i	$0.410614\pi$
$614$	0	0
$615$	32.1115	1.29486
$616$	0	0
$617$	−6.01447	−0.242133	−0.121067	−	0.992644i	$-0.538632\pi$
$617$	−6.01447	−0.242133	−0.121067	+	0.992644i	$0.538632\pi$
$618$	0	0
$619$	−14.4997	−0.582792	−0.291396	−	0.956602i	$-0.594120\pi$
$619$	−14.4997	−0.582792	−0.291396	+	0.956602i	$0.594120\pi$
$620$	0	0
$621$	1.00000	0.0401286
$622$	0	0
$623$	5.08117	0.203573
$624$	0	0
$625$	89.0134	3.56054
$626$	0	0
$627$	1.20227	0.0480140
$628$	0	0
$629$	−22.5537	−0.899276
$630$	0	0
$631$	−8.13468	−0.323837	−0.161918	−	0.986804i	$-0.551768\pi$
$631$	−8.13468	−0.323837	−0.161918	+	0.986804i	$0.551768\pi$
$632$	0	0
$633$	13.5250	0.537570
$634$	0	0
$635$	−20.6886	−0.821002
$636$	0	0
$637$	13.4768	0.533970
$638$	0	0
$639$	−11.6877	−0.462358
$640$	0	0
$641$	34.3493	1.35672	0.678358	−	0.734732i	$-0.262692\pi$
$641$	34.3493	1.35672	0.678358	+	0.734732i	$0.262692\pi$
$642$	0	0
$643$	33.0356	1.30280	0.651399	−	0.758735i	$-0.274182\pi$
$643$	33.0356	1.30280	0.651399	+	0.758735i	$0.274182\pi$
$644$	0	0
$645$	12.0412	0.474120
$646$	0	0
$647$	11.7367	0.461417	0.230709	−	0.973023i	$-0.425896\pi$
$647$	11.7367	0.461417	0.230709	+	0.973023i	$0.425896\pi$
$648$	0	0
$649$	2.28000	0.0894978
$650$	0	0
$651$	7.27489	0.285125
$652$	0	0
$653$	15.8894	0.621801	0.310900	−	0.950443i	$-0.399369\pi$
$653$	15.8894	0.621801	0.310900	+	0.950443i	$0.399369\pi$
$654$	0	0
$655$	−14.7369	−0.575819
$656$	0	0
$657$	2.93865	0.114648
$658$	0	0
$659$	−7.06429	−0.275186	−0.137593	−	0.990489i	$-0.543937\pi$
$659$	−7.06429	−0.275186	−0.137593	+	0.990489i	$0.543937\pi$
$660$	0	0
$661$	−48.1367	−1.87230	−0.936151	−	0.351599i	$-0.885638\pi$
$661$	−48.1367	−1.87230	−0.936151	+	0.351599i	$0.885638\pi$
$662$	0	0
$663$	9.65711	0.375051
$664$	0	0
$665$	−113.281	−4.39286
$666$	0	0
$667$	−1.00000	−0.0387202
$668$	0	0
$669$	−0.118035	−0.00456349
$670$	0	0
$671$	2.40204	0.0927296
$672$	0	0
$673$	−11.0585	−0.426274	−0.213137	−	0.977022i	$-0.568368\pi$
$673$	−11.0585	−0.426274	−0.213137	+	0.977022i	$0.568368\pi$
$674$	0	0
$675$	13.4665	0.518325
$676$	0	0
$677$	14.2806	0.548849	0.274424	−	0.961609i	$-0.411513\pi$
$677$	14.2806	0.548849	0.274424	+	0.961609i	$0.411513\pi$
$678$	0	0
$679$	−17.4415	−0.669344
$680$	0	0
$681$	12.9065	0.494578
$682$	0	0
$683$	15.7111	0.601169	0.300585	−	0.953755i	$-0.402818\pi$
$683$	15.7111	0.601169	0.300585	+	0.953755i	$0.402818\pi$
$684$	0	0
$685$	−19.4506	−0.743169
$686$	0	0
$687$	−13.2700	−0.506284
$688$	0	0
$689$	13.7438	0.523597
$690$	0	0
$691$	−5.86530	−0.223126	−0.111563	−	0.993757i	$-0.535586\pi$
$691$	−5.86530	−0.223126	−0.111563	+	0.993757i	$0.535586\pi$
$692$	0	0
$693$	0.573189	0.0217736
$694$	0	0
$695$	−55.5192	−2.10596
$696$	0	0
$697$	−29.8140	−1.12929
$698$	0	0
$699$	1.94230	0.0734646
$700$	0	0
$701$	−14.6443	−0.553109	−0.276555	−	0.960998i	$-0.589193\pi$
$701$	−14.6443	−0.553109	−0.276555	+	0.960998i	$0.589193\pi$
$702$	0	0
$703$	−42.0341	−1.58535
$704$	0	0
$705$	24.9637	0.940186
$706$	0	0
$707$	−52.9902	−1.99290
$708$	0	0
$709$	−44.3384	−1.66516	−0.832582	−	0.553901i	$-0.813139\pi$
$709$	−44.3384	−1.66516	−0.832582	+	0.553901i	$0.813139\pi$
$710$	0	0
$711$	−0.741638	−0.0278136
$712$	0	0
$713$	2.05208	0.0768512
$714$	0	0
$715$	−1.68172	−0.0628929
$716$	0	0
$717$	17.3816	0.649128
$718$	0	0
$719$	1.77384	0.0661532	0.0330766	−	0.999453i	$-0.489469\pi$
$719$	1.77384	0.0661532	0.0330766	+	0.999453i	$0.489469\pi$
$720$	0	0
$721$	29.4658	1.09736
$722$	0	0
$723$	15.9745	0.594099
$724$	0	0
$725$	−13.4665	−0.500132
$726$	0	0
$727$	33.7989	1.25353	0.626766	−	0.779207i	$-0.284378\pi$
$727$	33.7989	1.25353	0.626766	+	0.779207i	$0.284378\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−11.1796	−0.413494
$732$	0	0
$733$	−4.74695	−0.175333	−0.0876664	−	0.996150i	$-0.527941\pi$
$733$	−4.74695	−0.175333	−0.0876664	+	0.996150i	$0.527941\pi$
$734$	0	0
$735$	−23.9267	−0.882549
$736$	0	0
$737$	−1.51115	−0.0556638
$738$	0	0
$739$	−17.7880	−0.654342	−0.327171	−	0.944965i	$-0.606095\pi$
$739$	−17.7880	−0.654342	−0.327171	+	0.944965i	$0.606095\pi$
$740$	0	0
$741$	17.9983	0.661183
$742$	0	0
$743$	−36.6763	−1.34552	−0.672762	−	0.739859i	$-0.734892\pi$
$743$	−36.6763	−1.34552	−0.672762	+	0.739859i	$0.734892\pi$
$744$	0	0
$745$	−26.9371	−0.986901
$746$	0	0
$747$	−7.89190	−0.288750
$748$	0	0
$749$	48.2938	1.76462
$750$	0	0
$751$	−41.5848	−1.51745	−0.758726	−	0.651410i	$-0.774178\pi$
$751$	−41.5848	−1.51745	−0.758726	+	0.651410i	$0.774178\pi$
$752$	0	0
$753$	6.56916	0.239394
$754$	0	0
$755$	95.0991	3.46101
$756$	0	0
$757$	29.0852	1.05712	0.528559	−	0.848896i	$-0.322732\pi$
$757$	29.0852	1.05712	0.528559	+	0.848896i	$0.322732\pi$
$758$	0	0
$759$	0.161684	0.00586875
$760$	0	0
$761$	2.70655	0.0981123	0.0490562	−	0.998796i	$-0.484379\pi$
$761$	2.70655	0.0981123	0.0490562	+	0.998796i	$0.484379\pi$
$762$	0	0
$763$	6.36784	0.230531
$764$	0	0
$765$	−17.1452	−0.619887
$766$	0	0
$767$	34.1322	1.23244
$768$	0	0
$769$	−3.47909	−0.125459	−0.0627295	−	0.998031i	$-0.519981\pi$
$769$	−3.47909	−0.125459	−0.0627295	+	0.998031i	$0.519981\pi$
$770$	0	0
$771$	−22.8125	−0.821571
$772$	0	0
$773$	−45.4744	−1.63560	−0.817800	−	0.575503i	$-0.804806\pi$
$773$	−45.4744	−1.63560	−0.817800	+	0.575503i	$0.804806\pi$
$774$	0	0
$775$	27.6343	0.992655
$776$	0	0
$777$	−20.0400	−0.718932
$778$	0	0
$779$	−55.5654	−1.99084
$780$	0	0
$781$	−1.88971	−0.0676191
$782$	0	0
$783$	−1.00000	−0.0357371
$784$	0	0
$785$	98.8222	3.52711
$786$	0	0
$787$	30.4430	1.08517	0.542587	−	0.840000i	$-0.317445\pi$
$787$	30.4430	1.08517	0.542587	+	0.840000i	$0.317445\pi$
$788$	0	0
$789$	−26.5281	−0.944425
$790$	0	0
$791$	27.8156	0.989010
$792$	0	0
$793$	35.9592	1.27695
$794$	0	0
$795$	−24.4007	−0.865404
$796$	0	0
$797$	8.83629	0.312997	0.156499	−	0.987678i	$-0.449979\pi$
$797$	8.83629	0.312997	0.156499	+	0.987678i	$0.449979\pi$
$798$	0	0
$799$	−23.1775	−0.819963
$800$	0	0
$801$	1.43329	0.0506426
$802$	0	0
$803$	0.475132	0.0167671
$804$	0	0
$805$	−15.2343	−0.536939
$806$	0	0
$807$	28.3257	0.997112
$808$	0	0
$809$	40.4858	1.42340	0.711702	−	0.702481i	$-0.247924\pi$
$809$	40.4858	1.42340	0.711702	+	0.702481i	$0.247924\pi$
$810$	0	0
$811$	−14.4687	−0.508066	−0.254033	−	0.967196i	$-0.581757\pi$
$811$	−14.4687	−0.508066	−0.254033	+	0.967196i	$0.581757\pi$
$812$	0	0
$813$	32.1900	1.12895
$814$	0	0
$815$	−96.9778	−3.39698
$816$	0	0
$817$	−20.8359	−0.728954
$818$	0	0
$819$	8.58079	0.299837
$820$	0	0
$821$	−7.96813	−0.278090	−0.139045	−	0.990286i	$-0.544403\pi$
$821$	−7.96813	−0.278090	−0.139045	+	0.990286i	$0.544403\pi$
$822$	0	0
$823$	−21.2549	−0.740901	−0.370450	−	0.928852i	$-0.620797\pi$
$823$	−21.2549	−0.740901	−0.370450	+	0.928852i	$0.620797\pi$
$824$	0	0
$825$	2.17731	0.0758042
$826$	0	0
$827$	−19.0576	−0.662697	−0.331349	−	0.943508i	$-0.607504\pi$
$827$	−19.0576	−0.662697	−0.331349	+	0.943508i	$0.607504\pi$
$828$	0	0
$829$	−35.2113	−1.22294	−0.611469	−	0.791268i	$-0.709421\pi$
$829$	−35.2113	−1.22294	−0.611469	+	0.791268i	$0.709421\pi$
$830$	0	0
$831$	24.2130	0.839940
$832$	0	0
$833$	22.2148	0.769696
$834$	0	0
$835$	66.1353	2.28871
$836$	0	0
$837$	2.05208	0.0709304
$838$	0	0
$839$	2.68690	0.0927622	0.0463811	−	0.998924i	$-0.485231\pi$
$839$	2.68690	0.0927622	0.0463811	+	0.998924i	$0.485231\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	−11.0584	−0.380871
$844$	0	0
$845$	30.6886	1.05572
$846$	0	0
$847$	−38.9037	−1.33675
$848$	0	0
$849$	−18.1002	−0.621197
$850$	0	0
$851$	−5.65285	−0.193777
$852$	0	0
$853$	3.18917	0.109195	0.0545976	−	0.998508i	$-0.482612\pi$
$853$	3.18917	0.109195	0.0545976	+	0.998508i	$0.482612\pi$
$854$	0	0
$855$	−31.9541	−1.09281
$856$	0	0
$857$	7.13026	0.243565	0.121782	−	0.992557i	$-0.461139\pi$
$857$	7.13026	0.243565	0.121782	+	0.992557i	$0.461139\pi$
$858$	0	0
$859$	−36.2899	−1.23819	−0.619097	−	0.785314i	$-0.712501\pi$
$859$	−36.2899	−1.23819	−0.619097	+	0.785314i	$0.712501\pi$
$860$	0	0
$861$	−26.4911	−0.902815
$862$	0	0
$863$	21.1719	0.720701	0.360350	−	0.932817i	$-0.382657\pi$
$863$	21.1719	0.720701	0.360350	+	0.932817i	$0.382657\pi$
$864$	0	0
$865$	−92.6325	−3.14960
$866$	0	0
$867$	−1.08150	−0.0367296
$868$	0	0
$869$	−0.119911	−0.00406770
$870$	0	0
$871$	−22.6223	−0.766527
$872$	0	0
$873$	−4.91986	−0.166512
$874$	0	0
$875$	−128.981	−4.36035
$876$	0	0
$877$	26.5380	0.896125	0.448063	−	0.894002i	$-0.352114\pi$
$877$	26.5380	0.896125	0.448063	+	0.894002i	$0.352114\pi$
$878$	0	0
$879$	3.41385	0.115146
$880$	0	0
$881$	2.83899	0.0956481	0.0478241	−	0.998856i	$-0.484771\pi$
$881$	2.83899	0.0956481	0.0478241	+	0.998856i	$0.484771\pi$
$882$	0	0
$883$	34.7316	1.16881	0.584406	−	0.811461i	$-0.301327\pi$
$883$	34.7316	1.16881	0.584406	+	0.811461i	$0.301327\pi$
$884$	0	0
$885$	−60.5983	−2.03699
$886$	0	0
$887$	−45.9535	−1.54297	−0.771484	−	0.636249i	$-0.780485\pi$
$887$	−45.9535	−1.54297	−0.771484	+	0.636249i	$0.780485\pi$
$888$	0	0
$889$	17.0675	0.572426
$890$	0	0
$891$	0.161684	0.00541661
$892$	0	0
$893$	−43.1968	−1.44552
$894$	0	0
$895$	−105.699	−3.53313
$896$	0	0
$897$	2.42045	0.0808165
$898$	0	0
$899$	−2.05208	−0.0684409
$900$	0	0
$901$	22.6549	0.754743
$902$	0	0
$903$	−9.93362	−0.330570
$904$	0	0
$905$	−46.9574	−1.56092
$906$	0	0
$907$	36.4184	1.20925	0.604626	−	0.796510i	$-0.293323\pi$
$907$	36.4184	1.20925	0.604626	+	0.796510i	$0.293323\pi$
$908$	0	0
$909$	−14.9474	−0.495773
$910$	0	0
$911$	13.3952	0.443802	0.221901	−	0.975069i	$-0.428774\pi$
$911$	13.3952	0.443802	0.221901	+	0.975069i	$0.428774\pi$
$912$	0	0
$913$	−1.27599	−0.0422292
$914$	0	0
$915$	−63.8418	−2.11054
$916$	0	0
$917$	12.1576	0.401478
$918$	0	0
$919$	13.7689	0.454196	0.227098	−	0.973872i	$-0.427076\pi$
$919$	13.7689	0.454196	0.227098	+	0.973872i	$0.427076\pi$
$920$	0	0
$921$	19.5836	0.645300
$922$	0	0
$923$	−28.2895	−0.931159
$924$	0	0
$925$	−76.1239	−2.50294
$926$	0	0
$927$	8.31164	0.272990
$928$	0	0
$929$	−9.33349	−0.306222	−0.153111	−	0.988209i	$-0.548929\pi$
$929$	−9.33349	−0.306222	−0.153111	+	0.988209i	$0.548929\pi$
$930$	0	0
$931$	41.4024	1.35691
$932$	0	0
$933$	18.5304	0.606658
$934$	0	0
$935$	−2.77210	−0.0906575
$936$	0	0
$937$	−4.55343	−0.148754	−0.0743770	−	0.997230i	$-0.523697\pi$
$937$	−4.55343	−0.148754	−0.0743770	+	0.997230i	$0.523697\pi$
$938$	0	0
$939$	−0.248252	−0.00810140
$940$	0	0
$941$	−6.31675	−0.205920	−0.102960	−	0.994685i	$-0.532831\pi$
$941$	−6.31675	−0.205920	−0.102960	+	0.994685i	$0.532831\pi$
$942$	0	0
$943$	−7.47256	−0.243340
$944$	0	0
$945$	−15.2343	−0.495572
$946$	0	0
$947$	36.0044	1.16999	0.584994	−	0.811038i	$-0.301097\pi$
$947$	36.0044	1.16999	0.584994	+	0.811038i	$0.301097\pi$
$948$	0	0
$949$	7.11286	0.230893
$950$	0	0
$951$	4.94475	0.160344
$952$	0	0
$953$	8.08380	0.261860	0.130930	−	0.991392i	$-0.458204\pi$
$953$	8.08380	0.261860	0.130930	+	0.991392i	$0.458204\pi$
$954$	0	0
$955$	−86.5371	−2.80027
$956$	0	0
$957$	−0.161684	−0.00522649
$958$	0	0
$959$	16.0462	0.518159
$960$	0	0
$961$	−26.7889	−0.864160
$962$	0	0
$963$	13.6226	0.438982
$964$	0	0
$965$	48.4623	1.56006
$966$	0	0
$967$	44.6911	1.43717	0.718585	−	0.695439i	$-0.244790\pi$
$967$	44.6911	1.43717	0.718585	+	0.695439i	$0.244790\pi$
$968$	0	0
$969$	29.6678	0.953069
$970$	0	0
$971$	33.3864	1.07142	0.535710	−	0.844402i	$-0.320044\pi$
$971$	33.3864	1.07142	0.535710	+	0.844402i	$0.320044\pi$
$972$	0	0
$973$	45.8018	1.46834
$974$	0	0
$975$	32.5949	1.04387
$976$	0	0
$977$	54.3669	1.73935	0.869675	−	0.493624i	$-0.164328\pi$
$977$	54.3669	1.73935	0.869675	+	0.493624i	$0.164328\pi$
$978$	0	0
$979$	0.231739	0.00740641
$980$	0	0
$981$	1.79623	0.0573491
$982$	0	0
$983$	−3.86378	−0.123236	−0.0616178	−	0.998100i	$-0.519626\pi$
$983$	−3.86378	−0.123236	−0.0616178	+	0.998100i	$0.519626\pi$
$984$	0	0
$985$	27.5995	0.879393
$986$	0	0
$987$	−20.5943	−0.655524
$988$	0	0
$989$	−2.80205	−0.0891001
$990$	0	0
$991$	−16.0230	−0.508988	−0.254494	−	0.967074i	$-0.581909\pi$
$991$	−16.0230	−0.508988	−0.254494	+	0.967074i	$0.581909\pi$
$992$	0	0
$993$	31.6537	1.00450
$994$	0	0
$995$	−0.775900	−0.0245977
$996$	0	0
$997$	7.95503	0.251938	0.125969	−	0.992034i	$-0.459796\pi$
$997$	7.95503	0.251938	0.125969	+	0.992034i	$0.459796\pi$
$998$	0	0
$999$	−5.65285	−0.178848

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8004.2.a.j.1.1	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8004.2.a.j.1.1	✓	16	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:	\( N \)	\(=\)	\( 8004 = 2^{2} \cdot 3 \cdot 23 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8004.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -4.29726 q^{5} +3.54512 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -4.29726 q^{5} +3.54512 q^{7} +1.00000 q^{9} +0.161684 q^{11} +2.42045 q^{13} -4.29726 q^{15} +3.98980 q^{17} +7.43592 q^{19} +3.54512 q^{21} +1.00000 q^{23} +13.4665 q^{25} +1.00000 q^{27} -1.00000 q^{29} +2.05208 q^{31} +0.161684 q^{33} -15.2343 q^{35} -5.65285 q^{37} +2.42045 q^{39} -7.47256 q^{41} -2.80205 q^{43} -4.29726 q^{45} -5.80920 q^{47} +5.56789 q^{49} +3.98980 q^{51} +5.67820 q^{53} -0.694798 q^{55} +7.43592 q^{57} +14.1016 q^{59} +14.8564 q^{61} +3.54512 q^{63} -10.4013 q^{65} -9.34631 q^{67} +1.00000 q^{69} -11.6877 q^{71} +2.93865 q^{73} +13.4665 q^{75} +0.573189 q^{77} -0.741638 q^{79} +1.00000 q^{81} -7.89190 q^{83} -17.1452 q^{85} -1.00000 q^{87} +1.43329 q^{89} +8.58079 q^{91} +2.05208 q^{93} -31.9541 q^{95} -4.91986 q^{97} +0.161684 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 16 q^{3} + 3 q^{5} + 4 q^{7} + 16 q^{9}+O(q^{10}) \) 16 * q + 16 * q^3 + 3 * q^5 + 4 * q^7 + 16 * q^9	\( 16 q + 16 q^{3} + 3 q^{5} + 4 q^{7} + 16 q^{9} + 5 q^{11} + 6 q^{13} + 3 q^{15} + 3 q^{17} + 11 q^{19} + 4 q^{21} + 16 q^{23} + 27 q^{25} + 16 q^{27} - 16 q^{29} + 14 q^{31} + 5 q^{33} + 11 q^{35} + 4 q^{37} + 6 q^{39} + 11 q^{41} + 23 q^{43} + 3 q^{45} - 2 q^{47} + 34 q^{49} + 3 q^{51} + 19 q^{53} + 31 q^{55} + 11 q^{57} + 32 q^{59} + 19 q^{61} + 4 q^{63} + 6 q^{65} + 33 q^{67} + 16 q^{69} - 5 q^{71} + 23 q^{73} + 27 q^{75} + 42 q^{77} + 24 q^{79} + 16 q^{81} + 7 q^{83} - 16 q^{87} - 2 q^{89} + 25 q^{91} + 14 q^{93} + 7 q^{95} + 33 q^{97} + 5 q^{99}+O(q^{100}) \) 16 * q + 16 * q^3 + 3 * q^5 + 4 * q^7 + 16 * q^9 + 5 * q^11 + 6 * q^13 + 3 * q^15 + 3 * q^17 + 11 * q^19 + 4 * q^21 + 16 * q^23 + 27 * q^25 + 16 * q^27 - 16 * q^29 + 14 * q^31 + 5 * q^33 + 11 * q^35 + 4 * q^37 + 6 * q^39 + 11 * q^41 + 23 * q^43 + 3 * q^45 - 2 * q^47 + 34 * q^49 + 3 * q^51 + 19 * q^53 + 31 * q^55 + 11 * q^57 + 32 * q^59 + 19 * q^61 + 4 * q^63 + 6 * q^65 + 33 * q^67 + 16 * q^69 - 5 * q^71 + 23 * q^73 + 27 * q^75 + 42 * q^77 + 24 * q^79 + 16 * q^81 + 7 * q^83 - 16 * q^87 - 2 * q^89 + 25 * q^91 + 14 * q^93 + 7 * q^95 + 33 * q^97 + 5 * q^99

Introduction

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