LMFDB - Embedded newform 8960.2.a.bh.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8960,2,Mod(1,8960)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8960, 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("8960.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8960 = 2^{8} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8960.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:	$71.5459602111$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.940.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 7x - 4 $ x^3 - 7*x - 4
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2240)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.89511$ of defining polynomial
Character	$\chi$	$=$	8960.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.89511 q^{3} -1.00000 q^{5} -1.00000 q^{7} +5.38164 q^{9} +O(q^{10})$	$q-2.89511 q^{3} -1.00000 q^{5} -1.00000 q^{7} +5.38164 q^{9} +2.38164 q^{11} +3.40857 q^{13} +2.89511 q^{15} -1.92204 q^{17} -5.79021 q^{19} +2.89511 q^{21} +8.76328 q^{23} +1.00000 q^{25} -6.89511 q^{27} +5.86818 q^{29} -1.02693 q^{31} -6.89511 q^{33} +1.00000 q^{35} +2.81714 q^{37} -9.86818 q^{39} +11.7902 q^{41} -2.00000 q^{43} -5.38164 q^{45} +5.40857 q^{47} +1.00000 q^{49} +5.56450 q^{51} +6.97307 q^{53} -2.38164 q^{55} +16.7633 q^{57} -6.97307 q^{59} -14.7633 q^{61} -5.38164 q^{63} -3.40857 q^{65} +0.209787 q^{67} -25.3706 q^{69} +13.7902 q^{71} +14.7633 q^{73} -2.89511 q^{75} -2.38164 q^{77} -13.4486 q^{79} +3.81714 q^{81} -5.79021 q^{83} +1.92204 q^{85} -16.9890 q^{87} +1.23672 q^{89} -3.40857 q^{91} +2.97307 q^{93} +5.79021 q^{95} +5.92204 q^{97} +12.8171 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{5} - 3 q^{7} + 5 q^{9}+O(q^{10}) $ 3 * q - 3 * q^5 - 3 * q^7 + 5 * q^9	$ 3 q - 3 q^{5} - 3 q^{7} + 5 q^{9} - 4 q^{11} + 4 q^{13} - 2 q^{17} + 4 q^{23} + 3 q^{25} - 12 q^{27} + 4 q^{29} - 8 q^{31} - 12 q^{33} + 3 q^{35} - 4 q^{37} - 16 q^{39} + 18 q^{41} - 6 q^{43} - 5 q^{45} + 10 q^{47} + 3 q^{49} + 18 q^{51} + 16 q^{53} + 4 q^{55} + 28 q^{57} - 16 q^{59} - 22 q^{61} - 5 q^{63} - 4 q^{65} + 18 q^{67} - 24 q^{69} + 24 q^{71} + 22 q^{73} + 4 q^{77} + 8 q^{79} - q^{81} + 2 q^{85} - 10 q^{87} + 26 q^{89} - 4 q^{91} + 4 q^{93} + 14 q^{97} + 26 q^{99}+O(q^{100}) $ 3 * q - 3 * q^5 - 3 * q^7 + 5 * q^9 - 4 * q^11 + 4 * q^13 - 2 * q^17 + 4 * q^23 + 3 * q^25 - 12 * q^27 + 4 * q^29 - 8 * q^31 - 12 * q^33 + 3 * q^35 - 4 * q^37 - 16 * q^39 + 18 * q^41 - 6 * q^43 - 5 * q^45 + 10 * q^47 + 3 * q^49 + 18 * q^51 + 16 * q^53 + 4 * q^55 + 28 * q^57 - 16 * q^59 - 22 * q^61 - 5 * q^63 - 4 * q^65 + 18 * q^67 - 24 * q^69 + 24 * q^71 + 22 * q^73 + 4 * q^77 + 8 * q^79 - q^81 + 2 * q^85 - 10 * q^87 + 26 * q^89 - 4 * q^91 + 4 * q^93 + 14 * q^97 + 26 * 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
$3$	−2.89511	−1.67149	−0.835745	−	0.549117i	$-0.814964\pi$
$3$	−2.89511	−1.67149	−0.835745	+	0.549117i	$0.814964\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	5.38164	1.79388
$10$	0	0
$11$	2.38164	0.718092	0.359046	−	0.933320i	$-0.383102\pi$
$11$	2.38164	0.718092	0.359046	+	0.933320i	$0.383102\pi$
$12$	0	0
$13$	3.40857	0.945368	0.472684	−	0.881232i	$-0.343285\pi$
$13$	3.40857	0.945368	0.472684	+	0.881232i	$0.343285\pi$
$14$	0	0
$15$	2.89511	0.747513
$16$	0	0
$17$	−1.92204	−0.466162	−0.233081	−	0.972457i	$-0.574881\pi$
$17$	−1.92204	−0.466162	−0.233081	+	0.972457i	$0.574881\pi$
$18$	0	0
$19$	−5.79021	−1.32837	−0.664183	−	0.747570i	$-0.731220\pi$
$19$	−5.79021	−1.32837	−0.664183	+	0.747570i	$0.731220\pi$
$20$	0	0
$21$	2.89511	0.631764
$22$	0	0
$23$	8.76328	1.82727	0.913635	−	0.406534i	$-0.133263\pi$
$23$	8.76328	1.82727	0.913635	+	0.406534i	$0.133263\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−6.89511	−1.32696
$28$	0	0
$29$	5.86818	1.08969	0.544847	−	0.838536i	$-0.316588\pi$
$29$	5.86818	1.08969	0.544847	+	0.838536i	$0.316588\pi$
$30$	0	0
$31$	−1.02693	−0.184442	−0.0922210	−	0.995739i	$-0.529397\pi$
$31$	−1.02693	−0.184442	−0.0922210	+	0.995739i	$0.529397\pi$
$32$	0	0
$33$	−6.89511	−1.20028
$34$	0	0
$35$	1.00000	0.169031
$36$	0	0
$37$	2.81714	0.463135	0.231568	−	0.972819i	$-0.425615\pi$
$37$	2.81714	0.463135	0.231568	+	0.972819i	$0.425615\pi$
$38$	0	0
$39$	−9.86818	−1.58017
$40$	0	0
$41$	11.7902	1.84132	0.920661	−	0.390363i	$-0.127651\pi$
$41$	11.7902	1.84132	0.920661	+	0.390363i	$0.127651\pi$
$42$	0	0
$43$	−2.00000	−0.304997	−0.152499	−	0.988304i	$-0.548732\pi$
$43$	−2.00000	−0.304997	−0.152499	+	0.988304i	$0.548732\pi$
$44$	0	0
$45$	−5.38164	−0.802248
$46$	0	0
$47$	5.40857	0.788921	0.394461	−	0.918913i	$-0.370931\pi$
$47$	5.40857	0.788921	0.394461	+	0.918913i	$0.370931\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	5.56450	0.779186
$52$	0	0
$53$	6.97307	0.957825	0.478912	−	0.877863i	$-0.341031\pi$
$53$	6.97307	0.957825	0.478912	+	0.877863i	$0.341031\pi$
$54$	0	0
$55$	−2.38164	−0.321141
$56$	0	0
$57$	16.7633	2.22035
$58$	0	0
$59$	−6.97307	−0.907816	−0.453908	−	0.891048i	$-0.649971\pi$
$59$	−6.97307	−0.907816	−0.453908	+	0.891048i	$0.649971\pi$
$60$	0	0
$61$	−14.7633	−1.89024	−0.945122	−	0.326717i	$-0.894058\pi$
$61$	−14.7633	−1.89024	−0.945122	+	0.326717i	$0.894058\pi$
$62$	0	0
$63$	−5.38164	−0.678023
$64$	0	0
$65$	−3.40857	−0.422781
$66$	0	0
$67$	0.209787	0.0256296	0.0128148	−	0.999918i	$-0.495921\pi$
$67$	0.209787	0.0256296	0.0128148	+	0.999918i	$0.495921\pi$
$68$	0	0
$69$	−25.3706	−3.05427
$70$	0	0
$71$	13.7902	1.63660	0.818299	−	0.574793i	$-0.194918\pi$
$71$	13.7902	1.63660	0.818299	+	0.574793i	$0.194918\pi$
$72$	0	0
$73$	14.7633	1.72791	0.863956	−	0.503568i	$-0.167980\pi$
$73$	14.7633	1.72791	0.863956	+	0.503568i	$0.167980\pi$
$74$	0	0
$75$	−2.89511	−0.334298
$76$	0	0
$77$	−2.38164	−0.271413
$78$	0	0
$79$	−13.4486	−1.51309	−0.756543	−	0.653944i	$-0.773113\pi$
$79$	−13.4486	−1.51309	−0.756543	+	0.653944i	$0.773113\pi$
$80$	0	0
$81$	3.81714	0.424127
$82$	0	0
$83$	−5.79021	−0.635558	−0.317779	−	0.948165i	$-0.602937\pi$
$83$	−5.79021	−0.635558	−0.317779	+	0.948165i	$0.602937\pi$
$84$	0	0
$85$	1.92204	0.208474
$86$	0	0
$87$	−16.9890	−1.82141
$88$	0	0
$89$	1.23672	0.131092	0.0655458	−	0.997850i	$-0.479121\pi$
$89$	1.23672	0.131092	0.0655458	+	0.997850i	$0.479121\pi$
$90$	0	0
$91$	−3.40857	−0.357315
$92$	0	0
$93$	2.97307	0.308293
$94$	0	0
$95$	5.79021	0.594063
$96$	0	0
$97$	5.92204	0.601292	0.300646	−	0.953736i	$-0.402798\pi$
$97$	5.92204	0.601292	0.300646	+	0.953736i	$0.402798\pi$
$98$	0	0
$99$	12.8171	1.28817
$100$	0	0
$101$	−9.58043	−0.953288	−0.476644	−	0.879096i	$-0.658147\pi$
$101$	−9.58043	−0.953288	−0.476644	+	0.879096i	$0.658147\pi$
$102$	0	0
$103$	−6.59143	−0.649473	−0.324736	−	0.945805i	$-0.605276\pi$
$103$	−6.59143	−0.649473	−0.324736	+	0.945805i	$0.605276\pi$
$104$	0	0
$105$	−2.89511	−0.282533
$106$	0	0
$107$	−0.0538591	−0.00520676	−0.00260338	−	0.999997i	$-0.500829\pi$
$107$	−0.0538591	−0.00520676	−0.00260338	+	0.999997i	$0.500829\pi$
$108$	0	0
$109$	9.44860	0.905012	0.452506	−	0.891761i	$-0.350530\pi$
$109$	9.44860	0.905012	0.452506	+	0.891761i	$0.350530\pi$
$110$	0	0
$111$	−8.15593	−0.774126
$112$	0	0
$113$	12.5535	1.18093	0.590467	−	0.807062i	$-0.298944\pi$
$113$	12.5535	1.18093	0.590467	+	0.807062i	$0.298944\pi$
$114$	0	0
$115$	−8.76328	−0.817180
$116$	0	0
$117$	18.3437	1.69588
$118$	0	0
$119$	1.92204	0.176193
$120$	0	0
$121$	−5.32778	−0.484344
$122$	0	0
$123$	−34.1339	−3.07775
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	2.97307	0.263817	0.131909	−	0.991262i	$-0.457889\pi$
$127$	2.97307	0.263817	0.131909	+	0.991262i	$0.457889\pi$
$128$	0	0
$129$	5.79021	0.509800
$130$	0	0
$131$	10.8171	0.945098	0.472549	−	0.881304i	$-0.343334\pi$
$131$	10.8171	0.945098	0.472549	+	0.881304i	$0.343334\pi$
$132$	0	0
$133$	5.79021	0.502075
$134$	0	0
$135$	6.89511	0.593436
$136$	0	0
$137$	−2.76328	−0.236083	−0.118042	−	0.993009i	$-0.537662\pi$
$137$	−2.76328	−0.236083	−0.118042	+	0.993009i	$0.537662\pi$
$138$	0	0
$139$	−2.81714	−0.238947	−0.119473	−	0.992837i	$-0.538121\pi$
$139$	−2.81714	−0.238947	−0.119473	+	0.992837i	$0.538121\pi$
$140$	0	0
$141$	−15.6584	−1.31867
$142$	0	0
$143$	8.11800	0.678861
$144$	0	0
$145$	−5.86818	−0.487326
$146$	0	0
$147$	−2.89511	−0.238784
$148$	0	0
$149$	−2.05386	−0.168259	−0.0841293	−	0.996455i	$-0.526811\pi$
$149$	−2.05386	−0.168259	−0.0841293	+	0.996455i	$0.526811\pi$
$150$	0	0
$151$	−9.71225	−0.790372	−0.395186	−	0.918601i	$-0.629320\pi$
$151$	−9.71225	−0.790372	−0.395186	+	0.918601i	$0.629320\pi$
$152$	0	0
$153$	−10.3437	−0.836239
$154$	0	0
$155$	1.02693	0.0824850
$156$	0	0
$157$	2.41957	0.193103	0.0965515	−	0.995328i	$-0.469219\pi$
$157$	2.41957	0.193103	0.0965515	+	0.995328i	$0.469219\pi$
$158$	0	0
$159$	−20.1878	−1.60100
$160$	0	0
$161$	−8.76328	−0.690643
$162$	0	0
$163$	−12.5535	−0.983266	−0.491633	−	0.870803i	$-0.663600\pi$
$163$	−12.5535	−0.983266	−0.491633	+	0.870803i	$0.663600\pi$
$164$	0	0
$165$	6.89511	0.536783
$166$	0	0
$167$	−24.5694	−1.90124	−0.950620	−	0.310359i	$-0.899551\pi$
$167$	−24.5694	−1.90124	−0.950620	+	0.310359i	$0.899551\pi$
$168$	0	0
$169$	−1.38164	−0.106280
$170$	0	0
$171$	−31.1609	−2.38293
$172$	0	0
$173$	−1.46243	−0.111187	−0.0555933	−	0.998453i	$-0.517705\pi$
$173$	−1.46243	−0.111187	−0.0555933	+	0.998453i	$0.517705\pi$
$174$	0	0
$175$	−1.00000	−0.0755929
$176$	0	0
$177$	20.1878	1.51741
$178$	0	0
$179$	−8.81714	−0.659024	−0.329512	−	0.944151i	$-0.606884\pi$
$179$	−8.81714	−0.659024	−0.329512	+	0.944151i	$0.606884\pi$
$180$	0	0
$181$	−17.3168	−1.28715	−0.643573	−	0.765385i	$-0.722549\pi$
$181$	−17.3168	−1.28715	−0.643573	+	0.765385i	$0.722549\pi$
$182$	0	0
$183$	42.7413	3.15953
$184$	0	0
$185$	−2.81714	−0.207120
$186$	0	0
$187$	−4.57760	−0.334747
$188$	0	0
$189$	6.89511	0.501545
$190$	0	0
$191$	−15.5025	−1.12172	−0.560859	−	0.827911i	$-0.689529\pi$
$191$	−15.5025	−1.12172	−0.560859	+	0.827911i	$0.689529\pi$
$192$	0	0
$193$	−8.39757	−0.604470	−0.302235	−	0.953233i	$-0.597733\pi$
$193$	−8.39757	−0.604470	−0.302235	+	0.953233i	$0.597733\pi$
$194$	0	0
$195$	9.86818	0.706675
$196$	0	0
$197$	14.1339	1.00700	0.503500	−	0.863995i	$-0.332045\pi$
$197$	14.1339	1.00700	0.503500	+	0.863995i	$0.332045\pi$
$198$	0	0
$199$	20.4996	1.45318	0.726590	−	0.687071i	$-0.241104\pi$
$199$	20.4996	1.45318	0.726590	+	0.687071i	$0.241104\pi$
$200$	0	0
$201$	−0.607356	−0.0428396
$202$	0	0
$203$	−5.86818	−0.411865
$204$	0	0
$205$	−11.7902	−0.823464
$206$	0	0
$207$	47.1609	3.27791
$208$	0	0
$209$	−13.7902	−0.953889
$210$	0	0
$211$	−25.9621	−1.78730	−0.893651	−	0.448762i	$-0.851865\pi$
$211$	−25.9621	−1.78730	−0.893651	+	0.448762i	$0.851865\pi$
$212$	0	0
$213$	−39.9241	−2.73556
$214$	0	0
$215$	2.00000	0.136399
$216$	0	0
$217$	1.02693	0.0697125
$218$	0	0
$219$	−42.7413	−2.88819
$220$	0	0
$221$	−6.55140	−0.440695
$222$	0	0
$223$	24.5694	1.64529	0.822645	−	0.568555i	$-0.192497\pi$
$223$	24.5694	1.64529	0.822645	+	0.568555i	$0.192497\pi$
$224$	0	0
$225$	5.38164	0.358776
$226$	0	0
$227$	20.8412	1.38328	0.691641	−	0.722241i	$-0.256888\pi$
$227$	20.8412	1.38328	0.691641	+	0.722241i	$0.256888\pi$
$228$	0	0
$229$	−1.84407	−0.121860	−0.0609299	−	0.998142i	$-0.519407\pi$
$229$	−1.84407	−0.121860	−0.0609299	+	0.998142i	$0.519407\pi$
$230$	0	0
$231$	6.89511	0.453665
$232$	0	0
$233$	1.12900	0.0739631	0.0369816	−	0.999316i	$-0.488226\pi$
$233$	1.12900	0.0739631	0.0369816	+	0.999316i	$0.488226\pi$
$234$	0	0
$235$	−5.40857	−0.352816
$236$	0	0
$237$	38.9351	2.52911
$238$	0	0
$239$	6.13182	0.396635	0.198317	−	0.980138i	$-0.436452\pi$
$239$	6.13182	0.396635	0.198317	+	0.980138i	$0.436452\pi$
$240$	0	0
$241$	25.5804	1.64778	0.823890	−	0.566750i	$-0.191799\pi$
$241$	25.5804	1.64778	0.823890	+	0.566750i	$0.191799\pi$
$242$	0	0
$243$	9.63429	0.618040
$244$	0	0
$245$	−1.00000	−0.0638877
$246$	0	0
$247$	−19.7364	−1.25579
$248$	0	0
$249$	16.7633	1.06233
$250$	0	0
$251$	2.81714	0.177816	0.0889082	−	0.996040i	$-0.471662\pi$
$251$	2.81714	0.177816	0.0889082	+	0.996040i	$0.471662\pi$
$252$	0	0
$253$	20.8710	1.31215
$254$	0	0
$255$	−5.56450	−0.348462
$256$	0	0
$257$	−14.3437	−0.894736	−0.447368	−	0.894350i	$-0.647639\pi$
$257$	−14.3437	−0.894736	−0.447368	+	0.894350i	$0.647639\pi$
$258$	0	0
$259$	−2.81714	−0.175049
$260$	0	0
$261$	31.5804	1.95478
$262$	0	0
$263$	−10.2098	−0.629562	−0.314781	−	0.949164i	$-0.601931\pi$
$263$	−10.2098	−0.629562	−0.314781	+	0.949164i	$0.601931\pi$
$264$	0	0
$265$	−6.97307	−0.428352
$266$	0	0
$267$	−3.58043	−0.219119
$268$	0	0
$269$	−8.55350	−0.521516	−0.260758	−	0.965404i	$-0.583972\pi$
$269$	−8.55350	−0.521516	−0.260758	+	0.965404i	$0.583972\pi$
$270$	0	0
$271$	−19.5804	−1.18943	−0.594713	−	0.803938i	$-0.702734\pi$
$271$	−19.5804	−1.18943	−0.594713	+	0.803938i	$0.702734\pi$
$272$	0	0
$273$	9.86818	0.597249
$274$	0	0
$275$	2.38164	0.143618
$276$	0	0
$277$	12.7633	0.766871	0.383436	−	0.923568i	$-0.374741\pi$
$277$	12.7633	0.766871	0.383436	+	0.923568i	$0.374741\pi$
$278$	0	0
$279$	−5.52657	−0.330867
$280$	0	0
$281$	25.9621	1.54877	0.774384	−	0.632716i	$-0.218060\pi$
$281$	25.9621	1.54877	0.774384	+	0.632716i	$0.218060\pi$
$282$	0	0
$283$	28.5776	1.69876	0.849381	−	0.527780i	$-0.176975\pi$
$283$	28.5776	1.69876	0.849381	+	0.527780i	$0.176975\pi$
$284$	0	0
$285$	−16.7633	−0.992971
$286$	0	0
$287$	−11.7902	−0.695954
$288$	0	0
$289$	−13.3058	−0.782693
$290$	0	0
$291$	−17.1449	−1.00505
$292$	0	0
$293$	−12.1719	−0.711087	−0.355544	−	0.934660i	$-0.615704\pi$
$293$	−12.1719	−0.711087	−0.355544	+	0.934660i	$0.615704\pi$
$294$	0	0
$295$	6.97307	0.405988
$296$	0	0
$297$	−16.4217	−0.952882
$298$	0	0
$299$	29.8703	1.72744
$300$	0	0
$301$	2.00000	0.115278
$302$	0	0
$303$	27.7364	1.59341
$304$	0	0
$305$	14.7633	0.845343
$306$	0	0
$307$	−12.8412	−0.732889	−0.366444	−	0.930440i	$-0.619425\pi$
$307$	−12.8412	−0.732889	−0.366444	+	0.930440i	$0.619425\pi$
$308$	0	0
$309$	19.0829	1.08559
$310$	0	0
$311$	17.1829	0.974350	0.487175	−	0.873304i	$-0.338027\pi$
$311$	17.1829	0.974350	0.487175	+	0.873304i	$0.338027\pi$
$312$	0	0
$313$	15.2927	0.864393	0.432197	−	0.901779i	$-0.357739\pi$
$313$	15.2927	0.864393	0.432197	+	0.901779i	$0.357739\pi$
$314$	0	0
$315$	5.38164	0.303221
$316$	0	0
$317$	−30.8972	−1.73536	−0.867680	−	0.497123i	$-0.834390\pi$
$317$	−30.8972	−1.73536	−0.867680	+	0.497123i	$0.834390\pi$
$318$	0	0
$319$	13.9759	0.782500
$320$	0	0
$321$	0.155928	0.00870304
$322$	0	0
$323$	11.1290	0.619234
$324$	0	0
$325$	3.40857	0.189074
$326$	0	0
$327$	−27.3547	−1.51272
$328$	0	0
$329$	−5.40857	−0.298184
$330$	0	0
$331$	26.4514	1.45390	0.726951	−	0.686689i	$-0.240937\pi$
$331$	26.4514	1.45390	0.726951	+	0.686689i	$0.240937\pi$
$332$	0	0
$333$	15.1609	0.830810
$334$	0	0
$335$	−0.209787	−0.0114619
$336$	0	0
$337$	−0.553497	−0.0301509	−0.0150754	−	0.999886i	$-0.504799\pi$
$337$	−0.553497	−0.0301509	−0.0150754	+	0.999886i	$0.504799\pi$
$338$	0	0
$339$	−36.3437	−1.97392
$340$	0	0
$341$	−2.44578	−0.132446
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	25.3706	1.36591
$346$	0	0
$347$	30.0801	1.61478	0.807391	−	0.590016i	$-0.200879\pi$
$347$	30.0801	1.61478	0.807391	+	0.590016i	$0.200879\pi$
$348$	0	0
$349$	23.2147	1.24266	0.621328	−	0.783551i	$-0.286594\pi$
$349$	23.2147	1.24266	0.621328	+	0.783551i	$0.286594\pi$
$350$	0	0
$351$	−23.5025	−1.25447
$352$	0	0
$353$	−30.7654	−1.63748	−0.818738	−	0.574167i	$-0.805326\pi$
$353$	−30.7654	−1.63748	−0.818738	+	0.574167i	$0.805326\pi$
$354$	0	0
$355$	−13.7902	−0.731909
$356$	0	0
$357$	−5.56450	−0.294505
$358$	0	0
$359$	11.4245	0.602962	0.301481	−	0.953472i	$-0.402519\pi$
$359$	11.4245	0.602962	0.301481	+	0.953472i	$0.402519\pi$
$360$	0	0
$361$	14.5266	0.764556
$362$	0	0
$363$	15.4245	0.809576
$364$	0	0
$365$	−14.7633	−0.772746
$366$	0	0
$367$	−20.9890	−1.09562	−0.547808	−	0.836604i	$-0.684538\pi$
$367$	−20.9890	−1.09562	−0.547808	+	0.836604i	$0.684538\pi$
$368$	0	0
$369$	63.4507	3.30311
$370$	0	0
$371$	−6.97307	−0.362024
$372$	0	0
$373$	23.5804	1.22095	0.610474	−	0.792036i	$-0.290979\pi$
$373$	23.5804	1.22095	0.610474	+	0.792036i	$0.290979\pi$
$374$	0	0
$375$	2.89511	0.149503
$376$	0	0
$377$	20.0021	1.03016
$378$	0	0
$379$	24.3976	1.25322	0.626609	−	0.779333i	$-0.284442\pi$
$379$	24.3976	1.25322	0.626609	+	0.779333i	$0.284442\pi$
$380$	0	0
$381$	−8.60736	−0.440968
$382$	0	0
$383$	−0.311856	−0.0159351	−0.00796754	−	0.999968i	$-0.502536\pi$
$383$	−0.311856	−0.0159351	−0.00796754	+	0.999968i	$0.502536\pi$
$384$	0	0
$385$	2.38164	0.121380
$386$	0	0
$387$	−10.7633	−0.547128
$388$	0	0
$389$	−0.0779639	−0.00395293	−0.00197646	−	0.999998i	$-0.500629\pi$
$389$	−0.0779639	−0.00395293	−0.00197646	+	0.999998i	$0.500629\pi$
$390$	0	0
$391$	−16.8433	−0.851805
$392$	0	0
$393$	−31.3168	−1.57972
$394$	0	0
$395$	13.4486	0.676673
$396$	0	0
$397$	26.8813	1.34913	0.674566	−	0.738214i	$-0.264331\pi$
$397$	26.8813	1.34913	0.674566	+	0.738214i	$0.264331\pi$
$398$	0	0
$399$	−16.7633	−0.839214
$400$	0	0
$401$	23.1449	1.15580	0.577901	−	0.816107i	$-0.303872\pi$
$401$	23.1449	1.15580	0.577901	+	0.816107i	$0.303872\pi$
$402$	0	0
$403$	−3.50036	−0.174365
$404$	0	0
$405$	−3.81714	−0.189675
$406$	0	0
$407$	6.70942	0.332574
$408$	0	0
$409$	−16.5535	−0.818518	−0.409259	−	0.912418i	$-0.634213\pi$
$409$	−16.5535	−0.818518	−0.409259	+	0.912418i	$0.634213\pi$
$410$	0	0
$411$	8.00000	0.394611
$412$	0	0
$413$	6.97307	0.343122
$414$	0	0
$415$	5.79021	0.284230
$416$	0	0
$417$	8.15593	0.399398
$418$	0	0
$419$	−5.44650	−0.266079	−0.133040	−	0.991111i	$-0.542474\pi$
$419$	−5.44650	−0.266079	−0.133040	+	0.991111i	$0.542474\pi$
$420$	0	0
$421$	−3.92204	−0.191148	−0.0955742	−	0.995422i	$-0.530469\pi$
$421$	−3.92204	−0.191148	−0.0955742	+	0.995422i	$0.530469\pi$
$422$	0	0
$423$	29.1070	1.41523
$424$	0	0
$425$	−1.92204	−0.0932324
$426$	0	0
$427$	14.7633	0.714445
$428$	0	0
$429$	−23.5025	−1.13471
$430$	0	0
$431$	−29.1367	−1.40347	−0.701734	−	0.712439i	$-0.747590\pi$
$431$	−29.1367	−1.40347	−0.701734	+	0.712439i	$0.747590\pi$
$432$	0	0
$433$	40.3976	1.94138	0.970692	−	0.240328i	$-0.0772551\pi$
$433$	40.3976	1.94138	0.970692	+	0.240328i	$0.0772551\pi$
$434$	0	0
$435$	16.9890	0.814560
$436$	0	0
$437$	−50.7413	−2.42728
$438$	0	0
$439$	−25.7902	−1.23090	−0.615450	−	0.788176i	$-0.711026\pi$
$439$	−25.7902	−1.23090	−0.615450	+	0.788176i	$0.711026\pi$
$440$	0	0
$441$	5.38164	0.256269
$442$	0	0
$443$	36.7413	1.74563	0.872815	−	0.488050i	$-0.162292\pi$
$443$	36.7413	1.74563	0.872815	+	0.488050i	$0.162292\pi$
$444$	0	0
$445$	−1.23672	−0.0586260
$446$	0	0
$447$	5.94614	0.281243
$448$	0	0
$449$	−21.5425	−1.01665	−0.508327	−	0.861164i	$-0.669736\pi$
$449$	−21.5425	−1.01665	−0.508327	+	0.861164i	$0.669736\pi$
$450$	0	0
$451$	28.0801	1.32224
$452$	0	0
$453$	28.1180	1.32110
$454$	0	0
$455$	3.40857	0.159796
$456$	0	0
$457$	17.8441	0.834710	0.417355	−	0.908743i	$-0.362957\pi$
$457$	17.8441	0.834710	0.417355	+	0.908743i	$0.362957\pi$
$458$	0	0
$459$	13.2526	0.618580
$460$	0	0
$461$	−3.07514	−0.143224	−0.0716118	−	0.997433i	$-0.522814\pi$
$461$	−3.07514	−0.143224	−0.0716118	+	0.997433i	$0.522814\pi$
$462$	0	0
$463$	21.1070	0.980925	0.490463	−	0.871462i	$-0.336828\pi$
$463$	21.1070	0.980925	0.490463	+	0.871462i	$0.336828\pi$
$464$	0	0
$465$	−2.97307	−0.137873
$466$	0	0
$467$	−3.31468	−0.153385	−0.0766926	−	0.997055i	$-0.524436\pi$
$467$	−3.31468	−0.153385	−0.0766926	+	0.997055i	$0.524436\pi$
$468$	0	0
$469$	−0.209787	−0.00968706
$470$	0	0
$471$	−7.00492	−0.322770
$472$	0	0
$473$	−4.76328	−0.219016
$474$	0	0
$475$	−5.79021	−0.265673
$476$	0	0
$477$	37.5266	1.71822
$478$	0	0
$479$	19.3168	0.882606	0.441303	−	0.897358i	$-0.354516\pi$
$479$	19.3168	0.882606	0.441303	+	0.897358i	$0.354516\pi$
$480$	0	0
$481$	9.60243	0.437833
$482$	0	0
$483$	25.3706	1.15440
$484$	0	0
$485$	−5.92204	−0.268906
$486$	0	0
$487$	−3.65629	−0.165682	−0.0828412	−	0.996563i	$-0.526399\pi$
$487$	−3.65629	−0.165682	−0.0828412	+	0.996563i	$0.526399\pi$
$488$	0	0
$489$	36.3437	1.64352
$490$	0	0
$491$	4.32778	0.195310	0.0976550	−	0.995220i	$-0.468866\pi$
$491$	4.32778	0.195310	0.0976550	+	0.995220i	$0.468866\pi$
$492$	0	0
$493$	−11.2788	−0.507974
$494$	0	0
$495$	−12.8171	−0.576088
$496$	0	0
$497$	−13.7902	−0.618576
$498$	0	0
$499$	29.1229	1.30372	0.651860	−	0.758339i	$-0.273989\pi$
$499$	29.1229	1.30372	0.651860	+	0.758339i	$0.273989\pi$
$500$	0	0
$501$	71.1311	3.17790
$502$	0	0
$503$	30.1719	1.34530	0.672648	−	0.739962i	$-0.265157\pi$
$503$	30.1719	1.34530	0.672648	+	0.739962i	$0.265157\pi$
$504$	0	0
$505$	9.58043	0.426323
$506$	0	0
$507$	4.00000	0.177646
$508$	0	0
$509$	12.5535	0.556424	0.278212	−	0.960520i	$-0.410258\pi$
$509$	12.5535	0.556424	0.278212	+	0.960520i	$0.410258\pi$
$510$	0	0
$511$	−14.7633	−0.653089
$512$	0	0
$513$	39.9241	1.76269
$514$	0	0
$515$	6.59143	0.290453
$516$	0	0
$517$	12.8813	0.566518
$518$	0	0
$519$	4.23389	0.185847
$520$	0	0
$521$	8.66121	0.379455	0.189727	−	0.981837i	$-0.439240\pi$
$521$	8.66121	0.379455	0.189727	+	0.981837i	$0.439240\pi$
$522$	0	0
$523$	9.79021	0.428096	0.214048	−	0.976823i	$-0.431335\pi$
$523$	9.79021	0.428096	0.214048	+	0.976823i	$0.431335\pi$
$524$	0	0
$525$	2.89511	0.126353
$526$	0	0
$527$	1.97380	0.0859799
$528$	0	0
$529$	53.7951	2.33892
$530$	0	0
$531$	−37.5266	−1.62851
$532$	0	0
$533$	40.1878	1.74073
$534$	0	0
$535$	0.0538591	0.00232853
$536$	0	0
$537$	25.5266	1.10155
$538$	0	0
$539$	2.38164	0.102585
$540$	0	0
$541$	−31.5025	−1.35440	−0.677198	−	0.735801i	$-0.736806\pi$
$541$	−31.5025	−1.35440	−0.677198	+	0.735801i	$0.736806\pi$
$542$	0	0
$543$	50.1339	2.15145
$544$	0	0
$545$	−9.44860	−0.404734
$546$	0	0
$547$	−30.7633	−1.31534	−0.657672	−	0.753305i	$-0.728458\pi$
$547$	−30.7633	−1.31534	−0.657672	+	0.753305i	$0.728458\pi$
$548$	0	0
$549$	−79.4507	−3.39087
$550$	0	0
$551$	−33.9780	−1.44751
$552$	0	0
$553$	13.4486	0.571893
$554$	0	0
$555$	8.15593	0.346200
$556$	0	0
$557$	17.1070	0.724847	0.362423	−	0.932014i	$-0.381949\pi$
$557$	17.1070	0.724847	0.362423	+	0.932014i	$0.381949\pi$
$558$	0	0
$559$	−6.81714	−0.288334
$560$	0	0
$561$	13.2526	0.559527
$562$	0	0
$563$	−2.62936	−0.110814	−0.0554072	−	0.998464i	$-0.517646\pi$
$563$	−2.62936	−0.110814	−0.0554072	+	0.998464i	$0.517646\pi$
$564$	0	0
$565$	−12.5535	−0.528130
$566$	0	0
$567$	−3.81714	−0.160305
$568$	0	0
$569$	28.7413	1.20490	0.602449	−	0.798158i	$-0.294192\pi$
$569$	28.7413	1.20490	0.602449	+	0.798158i	$0.294192\pi$
$570$	0	0
$571$	7.18286	0.300593	0.150297	−	0.988641i	$-0.451977\pi$
$571$	7.18286	0.300593	0.150297	+	0.988641i	$0.451977\pi$
$572$	0	0
$573$	44.8813	1.87494
$574$	0	0
$575$	8.76328	0.365454
$576$	0	0
$577$	20.2878	0.844590	0.422295	−	0.906458i	$-0.361225\pi$
$577$	20.2878	0.844590	0.422295	+	0.906458i	$0.361225\pi$
$578$	0	0
$579$	24.3119	1.01037
$580$	0	0
$581$	5.79021	0.240219
$582$	0	0
$583$	16.6074	0.687806
$584$	0	0
$585$	−18.3437	−0.758419
$586$	0	0
$587$	−43.7364	−1.80519	−0.902596	−	0.430488i	$-0.858341\pi$
$587$	−43.7364	−1.80519	−0.902596	+	0.430488i	$0.858341\pi$
$588$	0	0
$589$	5.94614	0.245006
$590$	0	0
$591$	−40.9192	−1.68319
$592$	0	0
$593$	−3.86818	−0.158847	−0.0794235	−	0.996841i	$-0.525308\pi$
$593$	−3.86818	−0.158847	−0.0794235	+	0.996841i	$0.525308\pi$
$594$	0	0
$595$	−1.92204	−0.0787958
$596$	0	0
$597$	−59.3486	−2.42898
$598$	0	0
$599$	−41.8682	−1.71069	−0.855344	−	0.518061i	$-0.826654\pi$
$599$	−41.8682	−1.71069	−0.855344	+	0.518061i	$0.826654\pi$
$600$	0	0
$601$	−22.2955	−0.909452	−0.454726	−	0.890631i	$-0.650263\pi$
$601$	−22.2955	−0.909452	−0.454726	+	0.890631i	$0.650263\pi$
$602$	0	0
$603$	1.12900	0.0459764
$604$	0	0
$605$	5.32778	0.216605
$606$	0	0
$607$	−11.6984	−0.474824	−0.237412	−	0.971409i	$-0.576299\pi$
$607$	−11.6984	−0.474824	−0.237412	+	0.971409i	$0.576299\pi$
$608$	0	0
$609$	16.9890	0.688429
$610$	0	0
$611$	18.4355	0.745821
$612$	0	0
$613$	29.4784	1.19062	0.595310	−	0.803496i	$-0.297029\pi$
$613$	29.4784	1.19062	0.595310	+	0.803496i	$0.297029\pi$
$614$	0	0
$615$	34.1339	1.37641
$616$	0	0
$617$	−29.7682	−1.19842	−0.599211	−	0.800591i	$-0.704519\pi$
$617$	−29.7682	−1.19842	−0.599211	+	0.800591i	$0.704519\pi$
$618$	0	0
$619$	34.0857	1.37002	0.685010	−	0.728533i	$-0.259798\pi$
$619$	34.0857	1.37002	0.685010	+	0.728533i	$0.259798\pi$
$620$	0	0
$621$	−60.4238	−2.42472
$622$	0	0
$623$	−1.23672	−0.0495480
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	39.9241	1.59442
$628$	0	0
$629$	−5.41465	−0.215896
$630$	0	0
$631$	30.1318	1.19953	0.599764	−	0.800177i	$-0.295261\pi$
$631$	30.1318	1.19953	0.599764	+	0.800177i	$0.295261\pi$
$632$	0	0
$633$	75.1630	2.98746
$634$	0	0
$635$	−2.97307	−0.117983
$636$	0	0
$637$	3.40857	0.135053
$638$	0	0
$639$	74.2140	2.93586
$640$	0	0
$641$	−41.1609	−1.62576	−0.812878	−	0.582434i	$-0.802100\pi$
$641$	−41.1609	−1.62576	−0.812878	+	0.582434i	$0.802100\pi$
$642$	0	0
$643$	20.1098	0.793054	0.396527	−	0.918023i	$-0.370215\pi$
$643$	20.1098	0.793054	0.396527	+	0.918023i	$0.370215\pi$
$644$	0	0
$645$	−5.79021	−0.227989
$646$	0	0
$647$	−4.31186	−0.169517	−0.0847583	−	0.996402i	$-0.527012\pi$
$647$	−4.31186	−0.169517	−0.0847583	+	0.996402i	$0.527012\pi$
$648$	0	0
$649$	−16.6074	−0.651896
$650$	0	0
$651$	−2.97307	−0.116524
$652$	0	0
$653$	−26.8972	−1.05257	−0.526285	−	0.850309i	$-0.676415\pi$
$653$	−26.8972	−1.05257	−0.526285	+	0.850309i	$0.676415\pi$
$654$	0	0
$655$	−10.8171	−0.422661
$656$	0	0
$657$	79.4507	3.09967
$658$	0	0
$659$	28.4674	1.10893	0.554465	−	0.832207i	$-0.312923\pi$
$659$	28.4674	1.10893	0.554465	+	0.832207i	$0.312923\pi$
$660$	0	0
$661$	5.12900	0.199495	0.0997475	−	0.995013i	$-0.468197\pi$
$661$	5.12900	0.199495	0.0997475	+	0.995013i	$0.468197\pi$
$662$	0	0
$663$	18.9670	0.736617
$664$	0	0
$665$	−5.79021	−0.224535
$666$	0	0
$667$	51.4245	1.99116
$668$	0	0
$669$	−71.1311	−2.75009
$670$	0	0
$671$	−35.1609	−1.35737
$672$	0	0
$673$	46.3437	1.78642	0.893209	−	0.449641i	$-0.148448\pi$
$673$	46.3437	1.78642	0.893209	+	0.449641i	$0.148448\pi$
$674$	0	0
$675$	−6.89511	−0.265393
$676$	0	0
$677$	−13.8061	−0.530613	−0.265307	−	0.964164i	$-0.585473\pi$
$677$	−13.8061	−0.530613	−0.265307	+	0.964164i	$0.585473\pi$
$678$	0	0
$679$	−5.92204	−0.227267
$680$	0	0
$681$	−60.3376	−2.31214
$682$	0	0
$683$	−0.817143	−0.0312671	−0.0156335	−	0.999878i	$-0.504977\pi$
$683$	−0.817143	−0.0312671	−0.0156335	+	0.999878i	$0.504977\pi$
$684$	0	0
$685$	2.76328	0.105580
$686$	0	0
$687$	5.33879	0.203687
$688$	0	0
$689$	23.7682	0.905497
$690$	0	0
$691$	21.1070	0.802948	0.401474	−	0.915870i	$-0.368498\pi$
$691$	21.1070	0.802948	0.401474	+	0.915870i	$0.368498\pi$
$692$	0	0
$693$	−12.8171	−0.486883
$694$	0	0
$695$	2.81714	0.106860
$696$	0	0
$697$	−22.6612	−0.858355
$698$	0	0
$699$	−3.26857	−0.123629
$700$	0	0
$701$	38.5556	1.45622	0.728112	−	0.685458i	$-0.240398\pi$
$701$	38.5556	1.45622	0.728112	+	0.685458i	$0.240398\pi$
$702$	0	0
$703$	−16.3119	−0.615213
$704$	0	0
$705$	15.6584	0.589729
$706$	0	0
$707$	9.58043	0.360309
$708$	0	0
$709$	−40.9213	−1.53683	−0.768416	−	0.639951i	$-0.778955\pi$
$709$	−40.9213	−1.53683	−0.768416	+	0.639951i	$0.778955\pi$
$710$	0	0
$711$	−72.3756	−2.71430
$712$	0	0
$713$	−8.99927	−0.337025
$714$	0	0
$715$	−8.11800	−0.303596
$716$	0	0
$717$	−17.7523	−0.662971
$718$	0	0
$719$	40.9511	1.52722	0.763609	−	0.645680i	$-0.223426\pi$
$719$	40.9511	1.52722	0.763609	+	0.645680i	$0.223426\pi$
$720$	0	0
$721$	6.59143	0.245478
$722$	0	0
$723$	−74.0581	−2.75425
$724$	0	0
$725$	5.86818	0.217939
$726$	0	0
$727$	42.6336	1.58119	0.790596	−	0.612339i	$-0.209771\pi$
$727$	42.6336	1.58119	0.790596	+	0.612339i	$0.209771\pi$
$728$	0	0
$729$	−39.3437	−1.45717
$730$	0	0
$731$	3.84407	0.142178
$732$	0	0
$733$	−26.8813	−0.992883	−0.496441	−	0.868070i	$-0.665360\pi$
$733$	−26.8813	−0.992883	−0.496441	+	0.868070i	$0.665360\pi$
$734$	0	0
$735$	2.89511	0.106788
$736$	0	0
$737$	0.499637	0.0184044
$738$	0	0
$739$	−28.2201	−1.03809	−0.519046	−	0.854746i	$-0.673713\pi$
$739$	−28.2201	−1.03809	−0.519046	+	0.854746i	$0.673713\pi$
$740$	0	0
$741$	57.1388	2.09905
$742$	0	0
$743$	−35.6605	−1.30826	−0.654128	−	0.756384i	$-0.726964\pi$
$743$	−35.6605	−1.30826	−0.654128	+	0.756384i	$0.726964\pi$
$744$	0	0
$745$	2.05386	0.0752476
$746$	0	0
$747$	−31.1609	−1.14012
$748$	0	0
$749$	0.0538591	0.00196797
$750$	0	0
$751$	−28.7654	−1.04966	−0.524832	−	0.851206i	$-0.675872\pi$
$751$	−28.7654	−1.04966	−0.524832	+	0.851206i	$0.675872\pi$
$752$	0	0
$753$	−8.15593	−0.297219
$754$	0	0
$755$	9.71225	0.353465
$756$	0	0
$757$	−50.7413	−1.84422	−0.922112	−	0.386924i	$-0.873538\pi$
$757$	−50.7413	−1.84422	−0.922112	+	0.386924i	$0.873538\pi$
$758$	0	0
$759$	−60.4238	−2.19324
$760$	0	0
$761$	−21.8923	−0.793595	−0.396797	−	0.917906i	$-0.629878\pi$
$761$	−21.8923	−0.793595	−0.396797	+	0.917906i	$0.629878\pi$
$762$	0	0
$763$	−9.44860	−0.342062
$764$	0	0
$765$	10.3437	0.373978
$766$	0	0
$767$	−23.7682	−0.858220
$768$	0	0
$769$	−43.8221	−1.58026	−0.790132	−	0.612937i	$-0.789988\pi$
$769$	−43.8221	−1.58026	−0.790132	+	0.612937i	$0.789988\pi$
$770$	0	0
$771$	41.5266	1.49554
$772$	0	0
$773$	40.9351	1.47233	0.736167	−	0.676800i	$-0.236634\pi$
$773$	40.9351	1.47233	0.736167	+	0.676800i	$0.236634\pi$
$774$	0	0
$775$	−1.02693	−0.0368884
$776$	0	0
$777$	8.15593	0.292592
$778$	0	0
$779$	−68.2678	−2.44595
$780$	0	0
$781$	32.8433	1.17523
$782$	0	0
$783$	−40.4617	−1.44598
$784$	0	0
$785$	−2.41957	−0.0863583
$786$	0	0
$787$	10.3196	0.367854	0.183927	−	0.982940i	$-0.441119\pi$
$787$	10.3196	0.367854	0.183927	+	0.982940i	$0.441119\pi$
$788$	0	0
$789$	29.5584	1.05231
$790$	0	0
$791$	−12.5535	−0.446351
$792$	0	0
$793$	−50.3217	−1.78698
$794$	0	0
$795$	20.1878	0.715987
$796$	0	0
$797$	38.9890	1.38106	0.690531	−	0.723303i	$-0.257377\pi$
$797$	38.9890	1.38106	0.690531	+	0.723303i	$0.257377\pi$
$798$	0	0
$799$	−10.3955	−0.367765
$800$	0	0
$801$	6.65557	0.235163
$802$	0	0
$803$	35.1609	1.24080
$804$	0	0
$805$	8.76328	0.308865
$806$	0	0
$807$	24.7633	0.871709
$808$	0	0
$809$	19.6722	0.691638	0.345819	−	0.938301i	$-0.387601\pi$
$809$	19.6722	0.691638	0.345819	+	0.938301i	$0.387601\pi$
$810$	0	0
$811$	33.8703	1.18935	0.594673	−	0.803968i	$-0.297281\pi$
$811$	33.8703	1.18935	0.594673	+	0.803968i	$0.297281\pi$
$812$	0	0
$813$	56.6874	1.98811
$814$	0	0
$815$	12.5535	0.439730
$816$	0	0
$817$	11.5804	0.405148
$818$	0	0
$819$	−18.3437	−0.640981
$820$	0	0
$821$	−24.8731	−0.868077	−0.434039	−	0.900894i	$-0.642912\pi$
$821$	−24.8731	−0.868077	−0.434039	+	0.900894i	$0.642912\pi$
$822$	0	0
$823$	−1.44650	−0.0504219	−0.0252110	−	0.999682i	$-0.508026\pi$
$823$	−1.44650	−0.0504219	−0.0252110	+	0.999682i	$0.508026\pi$
$824$	0	0
$825$	−6.89511	−0.240057
$826$	0	0
$827$	24.2416	0.842964	0.421482	−	0.906837i	$-0.361510\pi$
$827$	24.2416	0.842964	0.421482	+	0.906837i	$0.361510\pi$
$828$	0	0
$829$	−40.5854	−1.40959	−0.704794	−	0.709412i	$-0.748960\pi$
$829$	−40.5854	−1.40959	−0.704794	+	0.709412i	$0.748960\pi$
$830$	0	0
$831$	−36.9511	−1.28182
$832$	0	0
$833$	−1.92204	−0.0665946
$834$	0	0
$835$	24.5694	0.850260
$836$	0	0
$837$	7.08079	0.244748
$838$	0	0
$839$	12.0482	0.415950	0.207975	−	0.978134i	$-0.433313\pi$
$839$	12.0482	0.415950	0.207975	+	0.978134i	$0.433313\pi$
$840$	0	0
$841$	5.43550	0.187431
$842$	0	0
$843$	−75.1630	−2.58875
$844$	0	0
$845$	1.38164	0.0475299
$846$	0	0
$847$	5.32778	0.183065
$848$	0	0
$849$	−82.7352	−2.83946
$850$	0	0
$851$	24.6874	0.846274
$852$	0	0
$853$	36.3217	1.24363	0.621816	−	0.783164i	$-0.286395\pi$
$853$	36.3217	1.24363	0.621816	+	0.783164i	$0.286395\pi$
$854$	0	0
$855$	31.1609	1.06568
$856$	0	0
$857$	−39.9780	−1.36562	−0.682811	−	0.730595i	$-0.739243\pi$
$857$	−39.9780	−1.36562	−0.682811	+	0.730595i	$0.739243\pi$
$858$	0	0
$859$	46.9454	1.60176	0.800878	−	0.598827i	$-0.204366\pi$
$859$	46.9454	1.60176	0.800878	+	0.598827i	$0.204366\pi$
$860$	0	0
$861$	34.1339	1.16328
$862$	0	0
$863$	16.2955	0.554705	0.277353	−	0.960768i	$-0.410543\pi$
$863$	16.2955	0.554705	0.277353	+	0.960768i	$0.410543\pi$
$864$	0	0
$865$	1.46243	0.0497241
$866$	0	0
$867$	38.5216	1.30826
$868$	0	0
$869$	−32.0298	−1.08653
$870$	0	0
$871$	0.715074	0.0242294
$872$	0	0
$873$	31.8703	1.07865
$874$	0	0
$875$	1.00000	0.0338062
$876$	0	0
$877$	36.9511	1.24775	0.623874	−	0.781525i	$-0.285558\pi$
$877$	36.9511	1.24775	0.623874	+	0.781525i	$0.285558\pi$
$878$	0	0
$879$	35.2388	1.18858
$880$	0	0
$881$	11.9780	0.403549	0.201774	−	0.979432i	$-0.435329\pi$
$881$	11.9780	0.403549	0.201774	+	0.979432i	$0.435329\pi$
$882$	0	0
$883$	26.6556	0.897031	0.448516	−	0.893775i	$-0.351953\pi$
$883$	26.6556	0.897031	0.448516	+	0.893775i	$0.351953\pi$
$884$	0	0
$885$	−20.1878	−0.678605
$886$	0	0
$887$	−4.31186	−0.144778	−0.0723890	−	0.997376i	$-0.523062\pi$
$887$	−4.31186	−0.144778	−0.0723890	+	0.997376i	$0.523062\pi$
$888$	0	0
$889$	−2.97307	−0.0997136
$890$	0	0
$891$	9.09107	0.304562
$892$	0	0
$893$	−31.3168	−1.04798
$894$	0	0
$895$	8.81714	0.294725
$896$	0	0
$897$	−86.4776	−2.88740
$898$	0	0
$899$	−6.02620	−0.200985
$900$	0	0
$901$	−13.4025	−0.446502
$902$	0	0
$903$	−5.79021	−0.192686
$904$	0	0
$905$	17.3168	0.575629
$906$	0	0
$907$	−11.0269	−0.366143	−0.183072	−	0.983100i	$-0.558604\pi$
$907$	−11.0269	−0.366143	−0.183072	+	0.983100i	$0.558604\pi$
$908$	0	0
$909$	−51.5584	−1.71008
$910$	0	0
$911$	23.0049	0.762187	0.381094	−	0.924536i	$-0.375548\pi$
$911$	23.0049	0.762187	0.381094	+	0.924536i	$0.375548\pi$
$912$	0	0
$913$	−13.7902	−0.456389
$914$	0	0
$915$	−42.7413	−1.41298
$916$	0	0
$917$	−10.8171	−0.357214
$918$	0	0
$919$	−5.13675	−0.169446	−0.0847228	−	0.996405i	$-0.527000\pi$
$919$	−5.13675	−0.169446	−0.0847228	+	0.996405i	$0.527000\pi$
$920$	0	0
$921$	37.1768	1.22502
$922$	0	0
$923$	47.0049	1.54719
$924$	0	0
$925$	2.81714	0.0926271
$926$	0	0
$927$	−35.4727	−1.16508
$928$	0	0
$929$	15.0588	0.494063	0.247031	−	0.969007i	$-0.420545\pi$
$929$	15.0588	0.494063	0.247031	+	0.969007i	$0.420545\pi$
$930$	0	0
$931$	−5.79021	−0.189767
$932$	0	0
$933$	−49.7462	−1.62862
$934$	0	0
$935$	4.57760	0.149704
$936$	0	0
$937$	−6.76118	−0.220878	−0.110439	−	0.993883i	$-0.535226\pi$
$937$	−6.76118	−0.220878	−0.110439	+	0.993883i	$0.535226\pi$
$938$	0	0
$939$	−44.2739	−1.44482
$940$	0	0
$941$	−43.5584	−1.41996	−0.709982	−	0.704220i	$-0.751297\pi$
$941$	−43.5584	−1.41996	−0.709982	+	0.704220i	$0.751297\pi$
$942$	0	0
$943$	103.321	3.36459
$944$	0	0
$945$	−6.89511	−0.224298
$946$	0	0
$947$	29.2091	0.949167	0.474583	−	0.880210i	$-0.342599\pi$
$947$	29.2091	0.949167	0.474583	+	0.880210i	$0.342599\pi$
$948$	0	0
$949$	50.3217	1.63351
$950$	0	0
$951$	89.4507	2.90064
$952$	0	0
$953$	37.3168	1.20881	0.604405	−	0.796678i	$-0.293411\pi$
$953$	37.3168	1.20881	0.604405	+	0.796678i	$0.293411\pi$
$954$	0	0
$955$	15.5025	0.501648
$956$	0	0
$957$	−40.4617	−1.30794
$958$	0	0
$959$	2.76328	0.0892311
$960$	0	0
$961$	−29.9454	−0.965981
$962$	0	0
$963$	−0.289850	−0.00934030
$964$	0	0
$965$	8.39757	0.270327
$966$	0	0
$967$	6.62936	0.213186	0.106593	−	0.994303i	$-0.466006\pi$
$967$	6.62936	0.213186	0.106593	+	0.994303i	$0.466006\pi$
$968$	0	0
$969$	−32.2196	−1.03504
$970$	0	0
$971$	−40.4996	−1.29970	−0.649848	−	0.760065i	$-0.725167\pi$
$971$	−40.4996	−1.29970	−0.649848	+	0.760065i	$0.725167\pi$
$972$	0	0
$973$	2.81714	0.0903134
$974$	0	0
$975$	−9.86818	−0.316035
$976$	0	0
$977$	−33.0531	−1.05746	−0.528732	−	0.848789i	$-0.677332\pi$
$977$	−33.0531	−1.05746	−0.528732	+	0.848789i	$0.677332\pi$
$978$	0	0
$979$	2.94542	0.0941359
$980$	0	0
$981$	50.8490	1.62348
$982$	0	0
$983$	9.40857	0.300087	0.150043	−	0.988679i	$-0.452059\pi$
$983$	9.40857	0.300087	0.150043	+	0.988679i	$0.452059\pi$
$984$	0	0
$985$	−14.1339	−0.450344
$986$	0	0
$987$	15.6584	0.498412
$988$	0	0
$989$	−17.5266	−0.557312
$990$	0	0
$991$	−15.0049	−0.476647	−0.238324	−	0.971186i	$-0.576598\pi$
$991$	−15.0049	−0.476647	−0.238324	+	0.971186i	$0.576598\pi$
$992$	0	0
$993$	−76.5797	−2.43018
$994$	0	0
$995$	−20.4996	−0.649882
$996$	0	0
$997$	−36.1719	−1.14557	−0.572787	−	0.819704i	$-0.694138\pi$
$997$	−36.1719	−1.14557	−0.572787	+	0.819704i	$0.694138\pi$
$998$	0	0
$999$	−19.4245	−0.614564

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8960.2.a.bh.1.1		3
4.3	odd	2		8960.2.a.bk.1.3		3
8.3	odd	2		8960.2.a.bq.1.1		3
8.5	even	2		8960.2.a.bn.1.3		3
16.3	odd	4		2240.2.b.e.1121.6	yes	6
16.5	even	4		2240.2.b.f.1121.6	yes	6
16.11	odd	4		2240.2.b.e.1121.1	✓	6
16.13	even	4		2240.2.b.f.1121.1	yes	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2240.2.b.e.1121.1	✓	6	16.11	odd	4
2240.2.b.e.1121.6	yes	6	16.3	odd	4
2240.2.b.f.1121.1	yes	6	16.13	even	4
2240.2.b.f.1121.6	yes	6	16.5	even	4
8960.2.a.bh.1.1		3	1.1	even	1	trivial
8960.2.a.bk.1.3		3	4.3	odd	2
8960.2.a.bn.1.3		3	8.5	even	2
8960.2.a.bq.1.1		3	8.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 8960 = 2^{8} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8960.a (trivial)

\(f(q)\)	\(=\)	\(q-2.89511 q^{3} -1.00000 q^{5} -1.00000 q^{7} +5.38164 q^{9} +O(q^{10})\)	\(q-2.89511 q^{3} -1.00000 q^{5} -1.00000 q^{7} +5.38164 q^{9} +2.38164 q^{11} +3.40857 q^{13} +2.89511 q^{15} -1.92204 q^{17} -5.79021 q^{19} +2.89511 q^{21} +8.76328 q^{23} +1.00000 q^{25} -6.89511 q^{27} +5.86818 q^{29} -1.02693 q^{31} -6.89511 q^{33} +1.00000 q^{35} +2.81714 q^{37} -9.86818 q^{39} +11.7902 q^{41} -2.00000 q^{43} -5.38164 q^{45} +5.40857 q^{47} +1.00000 q^{49} +5.56450 q^{51} +6.97307 q^{53} -2.38164 q^{55} +16.7633 q^{57} -6.97307 q^{59} -14.7633 q^{61} -5.38164 q^{63} -3.40857 q^{65} +0.209787 q^{67} -25.3706 q^{69} +13.7902 q^{71} +14.7633 q^{73} -2.89511 q^{75} -2.38164 q^{77} -13.4486 q^{79} +3.81714 q^{81} -5.79021 q^{83} +1.92204 q^{85} -16.9890 q^{87} +1.23672 q^{89} -3.40857 q^{91} +2.97307 q^{93} +5.79021 q^{95} +5.92204 q^{97} +12.8171 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{5} - 3 q^{7} + 5 q^{9}+O(q^{10}) \) 3 * q - 3 * q^5 - 3 * q^7 + 5 * q^9	\( 3 q - 3 q^{5} - 3 q^{7} + 5 q^{9} - 4 q^{11} + 4 q^{13} - 2 q^{17} + 4 q^{23} + 3 q^{25} - 12 q^{27} + 4 q^{29} - 8 q^{31} - 12 q^{33} + 3 q^{35} - 4 q^{37} - 16 q^{39} + 18 q^{41} - 6 q^{43} - 5 q^{45} + 10 q^{47} + 3 q^{49} + 18 q^{51} + 16 q^{53} + 4 q^{55} + 28 q^{57} - 16 q^{59} - 22 q^{61} - 5 q^{63} - 4 q^{65} + 18 q^{67} - 24 q^{69} + 24 q^{71} + 22 q^{73} + 4 q^{77} + 8 q^{79} - q^{81} + 2 q^{85} - 10 q^{87} + 26 q^{89} - 4 q^{91} + 4 q^{93} + 14 q^{97} + 26 q^{99}+O(q^{100}) \) 3 * q - 3 * q^5 - 3 * q^7 + 5 * q^9 - 4 * q^11 + 4 * q^13 - 2 * q^17 + 4 * q^23 + 3 * q^25 - 12 * q^27 + 4 * q^29 - 8 * q^31 - 12 * q^33 + 3 * q^35 - 4 * q^37 - 16 * q^39 + 18 * q^41 - 6 * q^43 - 5 * q^45 + 10 * q^47 + 3 * q^49 + 18 * q^51 + 16 * q^53 + 4 * q^55 + 28 * q^57 - 16 * q^59 - 22 * q^61 - 5 * q^63 - 4 * q^65 + 18 * q^67 - 24 * q^69 + 24 * q^71 + 22 * q^73 + 4 * q^77 + 8 * q^79 - q^81 + 2 * q^85 - 10 * q^87 + 26 * q^89 - 4 * q^91 + 4 * q^93 + 14 * q^97 + 26 * q^99

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