LMFDB - Embedded newform 6960.2.a.cl.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6960,2,Mod(1,6960)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6960, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("6960.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6960 = 2^{4} \cdot 3 \cdot 5 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6960.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:	$55.5758798068$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.469.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 5x + 4 $ x^3 - x^2 - 5*x + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 435)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$0.772866$ of defining polynomial
Character	$\chi$	$=$	6960.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +1.00000 q^{5} -2.17554 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +1.00000 q^{5} -2.17554 q^{7} +1.00000 q^{9} -3.00000 q^{11} -0.629813 q^{13} +1.00000 q^{15} +4.17554 q^{17} -4.80536 q^{19} -2.17554 q^{21} -2.08408 q^{23} +1.00000 q^{25} +1.00000 q^{27} -1.00000 q^{29} -3.00000 q^{33} -2.17554 q^{35} +6.08408 q^{37} -0.629813 q^{39} +0.824456 q^{41} +8.72128 q^{43} +1.00000 q^{45} +8.98090 q^{47} -2.26701 q^{49} +4.17554 q^{51} +6.88944 q^{53} -3.00000 q^{55} -4.80536 q^{57} -6.45427 q^{59} -2.80536 q^{61} -2.17554 q^{63} -0.629813 q^{65} +11.0841 q^{67} -2.08408 q^{69} -2.63719 q^{71} -14.7863 q^{73} +1.00000 q^{75} +6.52663 q^{77} +12.0650 q^{79} +1.00000 q^{81} +7.62981 q^{83} +4.17554 q^{85} -1.00000 q^{87} +17.8894 q^{89} +1.37019 q^{91} -4.80536 q^{95} +0.538351 q^{97} -3.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{3} + 3 q^{5} + 4 q^{7} + 3 q^{9}+O(q^{10}) $ 3 * q + 3 * q^3 + 3 * q^5 + 4 * q^7 + 3 * q^9	$ 3 q + 3 q^{3} + 3 q^{5} + 4 q^{7} + 3 q^{9} - 9 q^{11} + 6 q^{13} + 3 q^{15} + 2 q^{17} + 4 q^{19} + 4 q^{21} - q^{23} + 3 q^{25} + 3 q^{27} - 3 q^{29} - 9 q^{33} + 4 q^{35} + 13 q^{37} + 6 q^{39} + 13 q^{41} + 13 q^{43} + 3 q^{45} - 2 q^{47} + 9 q^{49} + 2 q^{51} - 3 q^{53} - 9 q^{55} + 4 q^{57} - 22 q^{59} + 10 q^{61} + 4 q^{63} + 6 q^{65} + 28 q^{67} - q^{69} + 3 q^{73} + 3 q^{75} - 12 q^{77} + 2 q^{79} + 3 q^{81} + 15 q^{83} + 2 q^{85} - 3 q^{87} + 30 q^{89} + 12 q^{91} + 4 q^{95} - q^{97} - 9 q^{99}+O(q^{100}) $ 3 * q + 3 * q^3 + 3 * q^5 + 4 * q^7 + 3 * q^9 - 9 * q^11 + 6 * q^13 + 3 * q^15 + 2 * q^17 + 4 * q^19 + 4 * q^21 - q^23 + 3 * q^25 + 3 * q^27 - 3 * q^29 - 9 * q^33 + 4 * q^35 + 13 * q^37 + 6 * q^39 + 13 * q^41 + 13 * q^43 + 3 * q^45 - 2 * q^47 + 9 * q^49 + 2 * q^51 - 3 * q^53 - 9 * q^55 + 4 * q^57 - 22 * q^59 + 10 * q^61 + 4 * q^63 + 6 * q^65 + 28 * q^67 - q^69 + 3 * q^73 + 3 * q^75 - 12 * q^77 + 2 * q^79 + 3 * q^81 + 15 * q^83 + 2 * q^85 - 3 * q^87 + 30 * q^89 + 12 * q^91 + 4 * q^95 - q^97 - 9 * 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
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−2.17554	−0.822278	−0.411139	−	0.911573i	$-0.634869\pi$
$7$	−2.17554	−0.822278	−0.411139	+	0.911573i	$0.634869\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−3.00000	−0.904534	−0.452267	−	0.891883i	$-0.649385\pi$
$11$	−3.00000	−0.904534	−0.452267	+	0.891883i	$0.649385\pi$
$12$	0	0
$13$	−0.629813	−0.174679	−0.0873394	−	0.996179i	$-0.527836\pi$
$13$	−0.629813	−0.174679	−0.0873394	+	0.996179i	$0.527836\pi$
$14$	0	0
$15$	1.00000	0.258199
$16$	0	0
$17$	4.17554	1.01272	0.506359	−	0.862323i	$-0.330991\pi$
$17$	4.17554	1.01272	0.506359	+	0.862323i	$0.330991\pi$
$18$	0	0
$19$	−4.80536	−1.10242	−0.551212	−	0.834365i	$-0.685835\pi$
$19$	−4.80536	−1.10242	−0.551212	+	0.834365i	$0.685835\pi$
$20$	0	0
$21$	−2.17554	−0.474743
$22$	0	0
$23$	−2.08408	−0.434561	−0.217281	−	0.976109i	$-0.569719\pi$
$23$	−2.08408	−0.434561	−0.217281	+	0.976109i	$0.569719\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	0	0
$33$	−3.00000	−0.522233
$34$	0	0
$35$	−2.17554	−0.367734
$36$	0	0
$37$	6.08408	1.00022	0.500108	−	0.865963i	$-0.333293\pi$
$37$	6.08408	1.00022	0.500108	+	0.865963i	$0.333293\pi$
$38$	0	0
$39$	−0.629813	−0.100851
$40$	0	0
$41$	0.824456	0.128758	0.0643792	−	0.997926i	$-0.479493\pi$
$41$	0.824456	0.128758	0.0643792	+	0.997926i	$0.479493\pi$
$42$	0	0
$43$	8.72128	1.32998	0.664991	−	0.746851i	$-0.268435\pi$
$43$	8.72128	1.32998	0.664991	+	0.746851i	$0.268435\pi$
$44$	0	0
$45$	1.00000	0.149071
$46$	0	0
$47$	8.98090	1.31000	0.655000	−	0.755629i	$-0.272669\pi$
$47$	8.98090	1.31000	0.655000	+	0.755629i	$0.272669\pi$
$48$	0	0
$49$	−2.26701	−0.323858
$50$	0	0
$51$	4.17554	0.584693
$52$	0	0
$53$	6.88944	0.946337	0.473169	−	0.880972i	$-0.343110\pi$
$53$	6.88944	0.946337	0.473169	+	0.880972i	$0.343110\pi$
$54$	0	0
$55$	−3.00000	−0.404520
$56$	0	0
$57$	−4.80536	−0.636485
$58$	0	0
$59$	−6.45427	−0.840274	−0.420137	−	0.907461i	$-0.638018\pi$
$59$	−6.45427	−0.840274	−0.420137	+	0.907461i	$0.638018\pi$
$60$	0	0
$61$	−2.80536	−0.359189	−0.179595	−	0.983741i	$-0.557479\pi$
$61$	−2.80536	−0.359189	−0.179595	+	0.983741i	$0.557479\pi$
$62$	0	0
$63$	−2.17554	−0.274093
$64$	0	0
$65$	−0.629813	−0.0781187
$66$	0	0
$67$	11.0841	1.35414	0.677068	−	0.735920i	$-0.263250\pi$
$67$	11.0841	1.35414	0.677068	+	0.735920i	$0.263250\pi$
$68$	0	0
$69$	−2.08408	−0.250894
$70$	0	0
$71$	−2.63719	−0.312977	−0.156489	−	0.987680i	$-0.550017\pi$
$71$	−2.63719	−0.312977	−0.156489	+	0.987680i	$0.550017\pi$
$72$	0	0
$73$	−14.7863	−1.73060	−0.865300	−	0.501254i	$-0.832872\pi$
$73$	−14.7863	−1.73060	−0.865300	+	0.501254i	$0.832872\pi$
$74$	0	0
$75$	1.00000	0.115470
$76$	0	0
$77$	6.52663	0.743779
$78$	0	0
$79$	12.0650	1.35742	0.678708	−	0.734408i	$-0.262540\pi$
$79$	12.0650	1.35742	0.678708	+	0.734408i	$0.262540\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	7.62981	0.837481	0.418740	−	0.908106i	$-0.362472\pi$
$83$	7.62981	0.837481	0.418740	+	0.908106i	$0.362472\pi$
$84$	0	0
$85$	4.17554	0.452901
$86$	0	0
$87$	−1.00000	−0.107211
$88$	0	0
$89$	17.8894	1.89628	0.948138	−	0.317858i	$-0.102963\pi$
$89$	17.8894	1.89628	0.948138	+	0.317858i	$0.102963\pi$
$90$	0	0
$91$	1.37019	0.143635
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−4.80536	−0.493019
$96$	0	0
$97$	0.538351	0.0546613	0.0273306	−	0.999626i	$-0.491299\pi$
$97$	0.538351	0.0546613	0.0273306	+	0.999626i	$0.491299\pi$
$98$	0	0
$99$	−3.00000	−0.301511
$100$	0	0
$101$	13.8054	1.37368	0.686842	−	0.726807i	$-0.258996\pi$
$101$	13.8054	1.37368	0.686842	+	0.726807i	$0.258996\pi$
$102$	0	0
$103$	13.4426	1.32453	0.662267	−	0.749268i	$-0.269594\pi$
$103$	13.4426	1.32453	0.662267	+	0.749268i	$0.269594\pi$
$104$	0	0
$105$	−2.17554	−0.212311
$106$	0	0
$107$	9.25963	0.895162	0.447581	−	0.894243i	$-0.352286\pi$
$107$	9.25963	0.895162	0.447581	+	0.894243i	$0.352286\pi$
$108$	0	0
$109$	8.61072	0.824757	0.412378	−	0.911013i	$-0.364698\pi$
$109$	8.61072	0.824757	0.412378	+	0.911013i	$0.364698\pi$
$110$	0	0
$111$	6.08408	0.577476
$112$	0	0
$113$	11.2670	1.05991	0.529955	−	0.848026i	$-0.322209\pi$
$113$	11.2670	1.05991	0.529955	+	0.848026i	$0.322209\pi$
$114$	0	0
$115$	−2.08408	−0.194342
$116$	0	0
$117$	−0.629813	−0.0582263
$118$	0	0
$119$	−9.08408	−0.832736
$120$	0	0
$121$	−2.00000	−0.181818
$122$	0	0
$123$	0.824456	0.0743387
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−16.7863	−1.48954	−0.744770	−	0.667321i	$-0.767441\pi$
$127$	−16.7863	−1.48954	−0.744770	+	0.667321i	$0.767441\pi$
$128$	0	0
$129$	8.72128	0.767865
$130$	0	0
$131$	−11.2670	−0.984403	−0.492201	−	0.870481i	$-0.663808\pi$
$131$	−11.2670	−0.984403	−0.492201	+	0.870481i	$0.663808\pi$
$132$	0	0
$133$	10.4543	0.906500
$134$	0	0
$135$	1.00000	0.0860663
$136$	0	0
$137$	−18.8703	−1.61220	−0.806101	−	0.591778i	$-0.798426\pi$
$137$	−18.8703	−1.61220	−0.806101	+	0.591778i	$0.798426\pi$
$138$	0	0
$139$	−3.80536	−0.322766	−0.161383	−	0.986892i	$-0.551595\pi$
$139$	−3.80536	−0.322766	−0.161383	+	0.986892i	$0.551595\pi$
$140$	0	0
$141$	8.98090	0.756328
$142$	0	0
$143$	1.88944	0.158003
$144$	0	0
$145$	−1.00000	−0.0830455
$146$	0	0
$147$	−2.26701	−0.186980
$148$	0	0
$149$	4.45427	0.364908	0.182454	−	0.983214i	$-0.441596\pi$
$149$	4.45427	0.364908	0.182454	+	0.983214i	$0.441596\pi$
$150$	0	0
$151$	−1.17554	−0.0956644	−0.0478322	−	0.998855i	$-0.515231\pi$
$151$	−1.17554	−0.0956644	−0.0478322	+	0.998855i	$0.515231\pi$
$152$	0	0
$153$	4.17554	0.337573
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−0.805358	−0.0642745	−0.0321373	−	0.999483i	$-0.510231\pi$
$157$	−0.805358	−0.0642745	−0.0321373	+	0.999483i	$0.510231\pi$
$158$	0	0
$159$	6.88944	0.546368
$160$	0	0
$161$	4.53401	0.357330
$162$	0	0
$163$	1.11056	0.0869858	0.0434929	−	0.999054i	$-0.486151\pi$
$163$	1.11056	0.0869858	0.0434929	+	0.999054i	$0.486151\pi$
$164$	0	0
$165$	−3.00000	−0.233550
$166$	0	0
$167$	8.70218	0.673395	0.336697	−	0.941613i	$-0.390690\pi$
$167$	8.70218	0.673395	0.336697	+	0.941613i	$0.390690\pi$
$168$	0	0
$169$	−12.6033	−0.969487
$170$	0	0
$171$	−4.80536	−0.367475
$172$	0	0
$173$	−7.69480	−0.585025	−0.292512	−	0.956262i	$-0.594491\pi$
$173$	−7.69480	−0.585025	−0.292512	+	0.956262i	$0.594491\pi$
$174$	0	0
$175$	−2.17554	−0.164456
$176$	0	0
$177$	−6.45427	−0.485133
$178$	0	0
$179$	6.06498	0.453318	0.226659	−	0.973974i	$-0.427220\pi$
$179$	6.06498	0.453318	0.226659	+	0.973974i	$0.427220\pi$
$180$	0	0
$181$	4.09146	0.304116	0.152058	−	0.988372i	$-0.451410\pi$
$181$	4.09146	0.304116	0.152058	+	0.988372i	$0.451410\pi$
$182$	0	0
$183$	−2.80536	−0.207378
$184$	0	0
$185$	6.08408	0.447311
$186$	0	0
$187$	−12.5266	−0.916038
$188$	0	0
$189$	−2.17554	−0.158248
$190$	0	0
$191$	0.267007	0.0193199	0.00965996	−	0.999953i	$-0.496925\pi$
$191$	0.267007	0.0193199	0.00965996	+	0.999953i	$0.496925\pi$
$192$	0	0
$193$	10.7022	0.770360	0.385180	−	0.922842i	$-0.374139\pi$
$193$	10.7022	0.770360	0.385180	+	0.922842i	$0.374139\pi$
$194$	0	0
$195$	−0.629813	−0.0451019
$196$	0	0
$197$	8.08408	0.575967	0.287984	−	0.957635i	$-0.407015\pi$
$197$	8.08408	0.575967	0.287984	+	0.957635i	$0.407015\pi$
$198$	0	0
$199$	15.8054	1.12041	0.560206	−	0.828353i	$-0.310722\pi$
$199$	15.8054	1.12041	0.560206	+	0.828353i	$0.310722\pi$
$200$	0	0
$201$	11.0841	0.781811
$202$	0	0
$203$	2.17554	0.152693
$204$	0	0
$205$	0.824456	0.0575825
$206$	0	0
$207$	−2.08408	−0.144854
$208$	0	0
$209$	14.4161	0.997181
$210$	0	0
$211$	25.2214	1.73631	0.868157	−	0.496289i	$-0.165304\pi$
$211$	25.2214	1.73631	0.868157	+	0.496289i	$0.165304\pi$
$212$	0	0
$213$	−2.63719	−0.180698
$214$	0	0
$215$	8.72128	0.594786
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−14.7863	−0.999163
$220$	0	0
$221$	−2.62981	−0.176900
$222$	0	0
$223$	12.5266	0.838845	0.419423	−	0.907791i	$-0.362233\pi$
$223$	12.5266	0.838845	0.419423	+	0.907791i	$0.362233\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	7.46165	0.495247	0.247624	−	0.968856i	$-0.420350\pi$
$227$	7.46165	0.495247	0.247624	+	0.968856i	$0.420350\pi$
$228$	0	0
$229$	21.7936	1.44016	0.720082	−	0.693889i	$-0.244104\pi$
$229$	21.7936	1.44016	0.720082	+	0.693889i	$0.244104\pi$
$230$	0	0
$231$	6.52663	0.429421
$232$	0	0
$233$	−9.69480	−0.635127	−0.317564	−	0.948237i	$-0.602865\pi$
$233$	−9.69480	−0.635127	−0.317564	+	0.948237i	$0.602865\pi$
$234$	0	0
$235$	8.98090	0.585849
$236$	0	0
$237$	12.0650	0.783705
$238$	0	0
$239$	10.7022	0.692266	0.346133	−	0.938185i	$-0.387495\pi$
$239$	10.7022	0.692266	0.346133	+	0.938185i	$0.387495\pi$
$240$	0	0
$241$	11.1682	0.719405	0.359702	−	0.933067i	$-0.382878\pi$
$241$	11.1682	0.719405	0.359702	+	0.933067i	$0.382878\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−2.26701	−0.144834
$246$	0	0
$247$	3.02648	0.192570
$248$	0	0
$249$	7.62981	0.483520
$250$	0	0
$251$	−14.3585	−0.906299	−0.453149	−	0.891435i	$-0.649700\pi$
$251$	−14.3585	−0.906299	−0.453149	+	0.891435i	$0.649700\pi$
$252$	0	0
$253$	6.25225	0.393075
$254$	0	0
$255$	4.17554	0.261483
$256$	0	0
$257$	−7.11056	−0.443545	−0.221772	−	0.975098i	$-0.571184\pi$
$257$	−7.11056	−0.443545	−0.221772	+	0.975098i	$0.571184\pi$
$258$	0	0
$259$	−13.2362	−0.822457
$260$	0	0
$261$	−1.00000	−0.0618984
$262$	0	0
$263$	−19.2214	−1.18524	−0.592622	−	0.805481i	$-0.701907\pi$
$263$	−19.2214	−1.18524	−0.592622	+	0.805481i	$0.701907\pi$
$264$	0	0
$265$	6.88944	0.423215
$266$	0	0
$267$	17.8894	1.09482
$268$	0	0
$269$	−9.88944	−0.602970	−0.301485	−	0.953471i	$-0.597482\pi$
$269$	−9.88944	−0.602970	−0.301485	+	0.953471i	$0.597482\pi$
$270$	0	0
$271$	13.7936	0.837904	0.418952	−	0.908008i	$-0.362398\pi$
$271$	13.7936	0.837904	0.418952	+	0.908008i	$0.362398\pi$
$272$	0	0
$273$	1.37019	0.0829275
$274$	0	0
$275$	−3.00000	−0.180907
$276$	0	0
$277$	−14.9661	−0.899228	−0.449614	−	0.893223i	$-0.648439\pi$
$277$	−14.9661	−0.899228	−0.449614	+	0.893223i	$0.648439\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	11.7287	0.699673	0.349836	−	0.936811i	$-0.386237\pi$
$281$	11.7287	0.699673	0.349836	+	0.936811i	$0.386237\pi$
$282$	0	0
$283$	3.96180	0.235505	0.117752	−	0.993043i	$-0.462431\pi$
$283$	3.96180	0.235505	0.117752	+	0.993043i	$0.462431\pi$
$284$	0	0
$285$	−4.80536	−0.284645
$286$	0	0
$287$	−1.79364	−0.105875
$288$	0	0
$289$	0.435171	0.0255983
$290$	0	0
$291$	0.538351	0.0315587
$292$	0	0
$293$	−3.26701	−0.190861	−0.0954303	−	0.995436i	$-0.530423\pi$
$293$	−3.26701	−0.190861	−0.0954303	+	0.995436i	$0.530423\pi$
$294$	0	0
$295$	−6.45427	−0.375782
$296$	0	0
$297$	−3.00000	−0.174078
$298$	0	0
$299$	1.31258	0.0759086
$300$	0	0
$301$	−18.9735	−1.09362
$302$	0	0
$303$	13.8054	0.793097
$304$	0	0
$305$	−2.80536	−0.160634
$306$	0	0
$307$	−20.4352	−1.16630	−0.583148	−	0.812366i	$-0.698179\pi$
$307$	−20.4352	−1.16630	−0.583148	+	0.812366i	$0.698179\pi$
$308$	0	0
$309$	13.4426	0.764720
$310$	0	0
$311$	−11.0000	−0.623753	−0.311876	−	0.950123i	$-0.600957\pi$
$311$	−11.0000	−0.623753	−0.311876	+	0.950123i	$0.600957\pi$
$312$	0	0
$313$	27.6683	1.56391	0.781953	−	0.623337i	$-0.214224\pi$
$313$	27.6683	1.56391	0.781953	+	0.623337i	$0.214224\pi$
$314$	0	0
$315$	−2.17554	−0.122578
$316$	0	0
$317$	−12.8777	−0.723285	−0.361642	−	0.932317i	$-0.617784\pi$
$317$	−12.8777	−0.723285	−0.361642	+	0.932317i	$0.617784\pi$
$318$	0	0
$319$	3.00000	0.167968
$320$	0	0
$321$	9.25963	0.516822
$322$	0	0
$323$	−20.0650	−1.11645
$324$	0	0
$325$	−0.629813	−0.0349358
$326$	0	0
$327$	8.61072	0.476174
$328$	0	0
$329$	−19.5384	−1.07718
$330$	0	0
$331$	−24.6874	−1.35694	−0.678472	−	0.734627i	$-0.737357\pi$
$331$	−24.6874	−1.35694	−0.678472	+	0.734627i	$0.737357\pi$
$332$	0	0
$333$	6.08408	0.333406
$334$	0	0
$335$	11.0841	0.605588
$336$	0	0
$337$	−21.0915	−1.14893	−0.574463	−	0.818531i	$-0.694789\pi$
$337$	−21.0915	−1.14893	−0.574463	+	0.818531i	$0.694789\pi$
$338$	0	0
$339$	11.2670	0.611940
$340$	0	0
$341$	0	0
$342$	0	0
$343$	20.1608	1.08858
$344$	0	0
$345$	−2.08408	−0.112203
$346$	0	0
$347$	18.8245	1.01055	0.505275	−	0.862958i	$-0.331391\pi$
$347$	18.8245	1.01055	0.505275	+	0.862958i	$0.331391\pi$
$348$	0	0
$349$	17.4884	0.936135	0.468067	−	0.883693i	$-0.344950\pi$
$349$	17.4884	0.936135	0.468067	+	0.883693i	$0.344950\pi$
$350$	0	0
$351$	−0.629813	−0.0336169
$352$	0	0
$353$	−8.38928	−0.446517	−0.223258	−	0.974759i	$-0.571669\pi$
$353$	−8.38928	−0.446517	−0.223258	+	0.974759i	$0.571669\pi$
$354$	0	0
$355$	−2.63719	−0.139968
$356$	0	0
$357$	−9.08408	−0.480781
$358$	0	0
$359$	27.6948	1.46168	0.730838	−	0.682551i	$-0.239130\pi$
$359$	27.6948	1.46168	0.730838	+	0.682551i	$0.239130\pi$
$360$	0	0
$361$	4.09146	0.215340
$362$	0	0
$363$	−2.00000	−0.104973
$364$	0	0
$365$	−14.7863	−0.773948
$366$	0	0
$367$	−9.00738	−0.470181	−0.235091	−	0.971973i	$-0.575539\pi$
$367$	−9.00738	−0.470181	−0.235091	+	0.971973i	$0.575539\pi$
$368$	0	0
$369$	0.824456	0.0429194
$370$	0	0
$371$	−14.9883	−0.778153
$372$	0	0
$373$	−25.0533	−1.29721	−0.648604	−	0.761126i	$-0.724647\pi$
$373$	−25.0533	−1.29721	−0.648604	+	0.761126i	$0.724647\pi$
$374$	0	0
$375$	1.00000	0.0516398
$376$	0	0
$377$	0.629813	0.0324370
$378$	0	0
$379$	27.5725	1.41631	0.708153	−	0.706059i	$-0.249529\pi$
$379$	27.5725	1.41631	0.708153	+	0.706059i	$0.249529\pi$
$380$	0	0
$381$	−16.7863	−0.859986
$382$	0	0
$383$	13.0724	0.667967	0.333983	−	0.942579i	$-0.391607\pi$
$383$	13.0724	0.667967	0.333983	+	0.942579i	$0.391607\pi$
$384$	0	0
$385$	6.52663	0.332628
$386$	0	0
$387$	8.72128	0.443327
$388$	0	0
$389$	−23.0797	−1.17019	−0.585095	−	0.810965i	$-0.698943\pi$
$389$	−23.0797	−1.17019	−0.585095	+	0.810965i	$0.698943\pi$
$390$	0	0
$391$	−8.70218	−0.440088
$392$	0	0
$393$	−11.2670	−0.568345
$394$	0	0
$395$	12.0650	0.607055
$396$	0	0
$397$	−12.3129	−0.617966	−0.308983	−	0.951067i	$-0.599989\pi$
$397$	−12.3129	−0.617966	−0.308983	+	0.951067i	$0.599989\pi$
$398$	0	0
$399$	10.4543	0.523368
$400$	0	0
$401$	−36.4308	−1.81927	−0.909634	−	0.415410i	$-0.863638\pi$
$401$	−36.4308	−1.81927	−0.909634	+	0.415410i	$0.863638\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	1.00000	0.0496904
$406$	0	0
$407$	−18.2522	−0.904730
$408$	0	0
$409$	−22.5842	−1.11672	−0.558359	−	0.829599i	$-0.688569\pi$
$409$	−22.5842	−1.11672	−0.558359	+	0.829599i	$0.688569\pi$
$410$	0	0
$411$	−18.8703	−0.930805
$412$	0	0
$413$	14.0415	0.690939
$414$	0	0
$415$	7.62981	0.374533
$416$	0	0
$417$	−3.80536	−0.186349
$418$	0	0
$419$	2.90854	0.142091	0.0710457	−	0.997473i	$-0.477366\pi$
$419$	2.90854	0.142091	0.0710457	+	0.997473i	$0.477366\pi$
$420$	0	0
$421$	27.1183	1.32166	0.660831	−	0.750534i	$-0.270204\pi$
$421$	27.1183	1.32166	0.660831	+	0.750534i	$0.270204\pi$
$422$	0	0
$423$	8.98090	0.436666
$424$	0	0
$425$	4.17554	0.202544
$426$	0	0
$427$	6.10318	0.295354
$428$	0	0
$429$	1.88944	0.0912230
$430$	0	0
$431$	−19.5960	−0.943904	−0.471952	−	0.881624i	$-0.656450\pi$
$431$	−19.5960	−0.943904	−0.471952	+	0.881624i	$0.656450\pi$
$432$	0	0
$433$	24.1638	1.16124	0.580620	−	0.814175i	$-0.302810\pi$
$433$	24.1638	1.16124	0.580620	+	0.814175i	$0.302810\pi$
$434$	0	0
$435$	−1.00000	−0.0479463
$436$	0	0
$437$	10.0148	0.479071
$438$	0	0
$439$	30.4235	1.45203	0.726016	−	0.687678i	$-0.241370\pi$
$439$	30.4235	1.45203	0.726016	+	0.687678i	$0.241370\pi$
$440$	0	0
$441$	−2.26701	−0.107953
$442$	0	0
$443$	32.3853	1.53867	0.769335	−	0.638846i	$-0.220588\pi$
$443$	32.3853	1.53867	0.769335	+	0.638846i	$0.220588\pi$
$444$	0	0
$445$	17.8894	0.848041
$446$	0	0
$447$	4.45427	0.210680
$448$	0	0
$449$	−5.62243	−0.265339	−0.132670	−	0.991160i	$-0.542355\pi$
$449$	−5.62243	−0.265339	−0.132670	+	0.991160i	$0.542355\pi$
$450$	0	0
$451$	−2.47337	−0.116466
$452$	0	0
$453$	−1.17554	−0.0552319
$454$	0	0
$455$	1.37019	0.0642353
$456$	0	0
$457$	3.33199	0.155864	0.0779320	−	0.996959i	$-0.475168\pi$
$457$	3.33199	0.155864	0.0779320	+	0.996959i	$0.475168\pi$
$458$	0	0
$459$	4.17554	0.194898
$460$	0	0
$461$	23.3437	1.08722	0.543612	−	0.839336i	$-0.317056\pi$
$461$	23.3437	1.08722	0.543612	+	0.839336i	$0.317056\pi$
$462$	0	0
$463$	−4.13735	−0.192279	−0.0961394	−	0.995368i	$-0.530649\pi$
$463$	−4.13735	−0.192279	−0.0961394	+	0.995368i	$0.530649\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−35.7554	−1.65456	−0.827282	−	0.561786i	$-0.810114\pi$
$467$	−35.7554	−1.65456	−0.827282	+	0.561786i	$0.810114\pi$
$468$	0	0
$469$	−24.1139	−1.11348
$470$	0	0
$471$	−0.805358	−0.0371089
$472$	0	0
$473$	−26.1638	−1.20301
$474$	0	0
$475$	−4.80536	−0.220485
$476$	0	0
$477$	6.88944	0.315446
$478$	0	0
$479$	29.6107	1.35295	0.676474	−	0.736466i	$-0.263507\pi$
$479$	29.6107	1.35295	0.676474	+	0.736466i	$0.263507\pi$
$480$	0	0
$481$	−3.83184	−0.174717
$482$	0	0
$483$	4.53401	0.206305
$484$	0	0
$485$	0.538351	0.0244453
$486$	0	0
$487$	4.33633	0.196498	0.0982489	−	0.995162i	$-0.468676\pi$
$487$	4.33633	0.196498	0.0982489	+	0.995162i	$0.468676\pi$
$488$	0	0
$489$	1.11056	0.0502213
$490$	0	0
$491$	−22.5193	−1.01628	−0.508140	−	0.861275i	$-0.669667\pi$
$491$	−22.5193	−1.01628	−0.508140	+	0.861275i	$0.669667\pi$
$492$	0	0
$493$	−4.17554	−0.188057
$494$	0	0
$495$	−3.00000	−0.134840
$496$	0	0
$497$	5.73733	0.257354
$498$	0	0
$499$	41.1638	1.84275	0.921373	−	0.388680i	$-0.127069\pi$
$499$	41.1638	1.84275	0.921373	+	0.388680i	$0.127069\pi$
$500$	0	0
$501$	8.70218	0.388785
$502$	0	0
$503$	−31.5149	−1.40518	−0.702590	−	0.711595i	$-0.747973\pi$
$503$	−31.5149	−1.40518	−0.702590	+	0.711595i	$0.747973\pi$
$504$	0	0
$505$	13.8054	0.614330
$506$	0	0
$507$	−12.6033	−0.559734
$508$	0	0
$509$	22.6224	1.00272	0.501361	−	0.865238i	$-0.332833\pi$
$509$	22.6224	1.00272	0.501361	+	0.865238i	$0.332833\pi$
$510$	0	0
$511$	32.1682	1.42304
$512$	0	0
$513$	−4.80536	−0.212162
$514$	0	0
$515$	13.4426	0.592350
$516$	0	0
$517$	−26.9427	−1.18494
$518$	0	0
$519$	−7.69480	−0.337764
$520$	0	0
$521$	−2.06498	−0.0904686	−0.0452343	−	0.998976i	$-0.514403\pi$
$521$	−2.06498	−0.0904686	−0.0452343	+	0.998976i	$0.514403\pi$
$522$	0	0
$523$	−24.5266	−1.07247	−0.536237	−	0.844067i	$-0.680155\pi$
$523$	−24.5266	−1.07247	−0.536237	+	0.844067i	$0.680155\pi$
$524$	0	0
$525$	−2.17554	−0.0949485
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−18.6566	−0.811157
$530$	0	0
$531$	−6.45427	−0.280091
$532$	0	0
$533$	−0.519253	−0.0224913
$534$	0	0
$535$	9.25963	0.400329
$536$	0	0
$537$	6.06498	0.261723
$538$	0	0
$539$	6.80102	0.292941
$540$	0	0
$541$	0.389285	0.0167367	0.00836833	−	0.999965i	$-0.497336\pi$
$541$	0.389285	0.0167367	0.00836833	+	0.999965i	$0.497336\pi$
$542$	0	0
$543$	4.09146	0.175581
$544$	0	0
$545$	8.61072	0.368843
$546$	0	0
$547$	−21.4352	−0.916502	−0.458251	−	0.888823i	$-0.651524\pi$
$547$	−21.4352	−0.916502	−0.458251	+	0.888823i	$0.651524\pi$
$548$	0	0
$549$	−2.80536	−0.119730
$550$	0	0
$551$	4.80536	0.204715
$552$	0	0
$553$	−26.2479	−1.11617
$554$	0	0
$555$	6.08408	0.258255
$556$	0	0
$557$	−19.7598	−0.837249	−0.418624	−	0.908159i	$-0.637488\pi$
$557$	−19.7598	−0.837249	−0.418624	+	0.908159i	$0.637488\pi$
$558$	0	0
$559$	−5.49278	−0.232320
$560$	0	0
$561$	−12.5266	−0.528875
$562$	0	0
$563$	−6.46165	−0.272326	−0.136163	−	0.990686i	$-0.543477\pi$
$563$	−6.46165	−0.272326	−0.136163	+	0.990686i	$0.543477\pi$
$564$	0	0
$565$	11.2670	0.474007
$566$	0	0
$567$	−2.17554	−0.0913643
$568$	0	0
$569$	−14.9427	−0.626431	−0.313215	−	0.949682i	$-0.601406\pi$
$569$	−14.9427	−0.626431	−0.313215	+	0.949682i	$0.601406\pi$
$570$	0	0
$571$	−35.1521	−1.47107	−0.735535	−	0.677487i	$-0.763069\pi$
$571$	−35.1521	−1.47107	−0.735535	+	0.677487i	$0.763069\pi$
$572$	0	0
$573$	0.267007	0.0111544
$574$	0	0
$575$	−2.08408	−0.0869122
$576$	0	0
$577$	−0.740373	−0.0308222	−0.0154111	−	0.999881i	$-0.504906\pi$
$577$	−0.740373	−0.0308222	−0.0154111	+	0.999881i	$0.504906\pi$
$578$	0	0
$579$	10.7022	0.444767
$580$	0	0
$581$	−16.5990	−0.688642
$582$	0	0
$583$	−20.6683	−0.855994
$584$	0	0
$585$	−0.629813	−0.0260396
$586$	0	0
$587$	18.5842	0.767054	0.383527	−	0.923530i	$-0.374709\pi$
$587$	18.5842	0.767054	0.383527	+	0.923530i	$0.374709\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	8.08408	0.332535
$592$	0	0
$593$	24.0268	0.986662	0.493331	−	0.869842i	$-0.335779\pi$
$593$	24.0268	0.986662	0.493331	+	0.869842i	$0.335779\pi$
$594$	0	0
$595$	−9.08408	−0.372411
$596$	0	0
$597$	15.8054	0.646870
$598$	0	0
$599$	−22.3585	−0.913542	−0.456771	−	0.889584i	$-0.650994\pi$
$599$	−22.3585	−0.913542	−0.456771	+	0.889584i	$0.650994\pi$
$600$	0	0
$601$	−20.3511	−0.830138	−0.415069	−	0.909790i	$-0.636243\pi$
$601$	−20.3511	−0.830138	−0.415069	+	0.909790i	$0.636243\pi$
$602$	0	0
$603$	11.0841	0.451379
$604$	0	0
$605$	−2.00000	−0.0813116
$606$	0	0
$607$	−19.4928	−0.791187	−0.395594	−	0.918426i	$-0.629461\pi$
$607$	−19.4928	−0.791187	−0.395594	+	0.918426i	$0.629461\pi$
$608$	0	0
$609$	2.17554	0.0881575
$610$	0	0
$611$	−5.65629	−0.228829
$612$	0	0
$613$	39.1638	1.58181	0.790906	−	0.611938i	$-0.209610\pi$
$613$	39.1638	1.58181	0.790906	+	0.611938i	$0.209610\pi$
$614$	0	0
$615$	0.824456	0.0332453
$616$	0	0
$617$	26.8321	1.08022	0.540111	−	0.841594i	$-0.318382\pi$
$617$	26.8321	1.08022	0.540111	+	0.841594i	$0.318382\pi$
$618$	0	0
$619$	33.4278	1.34358	0.671788	−	0.740743i	$-0.265527\pi$
$619$	33.4278	1.34358	0.671788	+	0.740743i	$0.265527\pi$
$620$	0	0
$621$	−2.08408	−0.0836313
$622$	0	0
$623$	−38.9193	−1.55927
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	14.4161	0.575722
$628$	0	0
$629$	25.4044	1.01294
$630$	0	0
$631$	32.3705	1.28865	0.644325	−	0.764752i	$-0.277139\pi$
$631$	32.3705	1.28865	0.644325	+	0.764752i	$0.277139\pi$
$632$	0	0
$633$	25.2214	1.00246
$634$	0	0
$635$	−16.7863	−0.666142
$636$	0	0
$637$	1.42779	0.0565711
$638$	0	0
$639$	−2.63719	−0.104326
$640$	0	0
$641$	33.0415	1.30506	0.652531	−	0.757762i	$-0.273707\pi$
$641$	33.0415	1.30506	0.652531	+	0.757762i	$0.273707\pi$
$642$	0	0
$643$	−27.4734	−1.08344	−0.541722	−	0.840558i	$-0.682227\pi$
$643$	−27.4734	−1.08344	−0.541722	+	0.840558i	$0.682227\pi$
$644$	0	0
$645$	8.72128	0.343400
$646$	0	0
$647$	23.2023	0.912178	0.456089	−	0.889934i	$-0.349250\pi$
$647$	23.2023	0.912178	0.456089	+	0.889934i	$0.349250\pi$
$648$	0	0
$649$	19.3628	0.760057
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−28.9159	−1.13157	−0.565784	−	0.824554i	$-0.691426\pi$
$653$	−28.9159	−1.13157	−0.565784	+	0.824554i	$0.691426\pi$
$654$	0	0
$655$	−11.2670	−0.440238
$656$	0	0
$657$	−14.7863	−0.576867
$658$	0	0
$659$	−36.1832	−1.40950	−0.704749	−	0.709456i	$-0.748941\pi$
$659$	−36.1832	−1.40950	−0.704749	+	0.709456i	$0.748941\pi$
$660$	0	0
$661$	45.0918	1.75387	0.876933	−	0.480612i	$-0.159585\pi$
$661$	45.0918	1.75387	0.876933	+	0.480612i	$0.159585\pi$
$662$	0	0
$663$	−2.62981	−0.102133
$664$	0	0
$665$	10.4543	0.405399
$666$	0	0
$667$	2.08408	0.0806960
$668$	0	0
$669$	12.5266	0.484308
$670$	0	0
$671$	8.41607	0.324899
$672$	0	0
$673$	−23.1491	−0.892331	−0.446165	−	0.894950i	$-0.647211\pi$
$673$	−23.1491	−0.892331	−0.446165	+	0.894950i	$0.647211\pi$
$674$	0	0
$675$	1.00000	0.0384900
$676$	0	0
$677$	38.1521	1.46630	0.733152	−	0.680064i	$-0.238048\pi$
$677$	38.1521	1.46630	0.733152	+	0.680064i	$0.238048\pi$
$678$	0	0
$679$	−1.17121	−0.0449468
$680$	0	0
$681$	7.46165	0.285931
$682$	0	0
$683$	0.228811	0.00875520	0.00437760	−	0.999990i	$-0.498607\pi$
$683$	0.228811	0.00875520	0.00437760	+	0.999990i	$0.498607\pi$
$684$	0	0
$685$	−18.8703	−0.720999
$686$	0	0
$687$	21.7936	0.831479
$688$	0	0
$689$	−4.33906	−0.165305
$690$	0	0
$691$	−16.2258	−0.617257	−0.308629	−	0.951183i	$-0.599870\pi$
$691$	−16.2258	−0.617257	−0.308629	+	0.951183i	$0.599870\pi$
$692$	0	0
$693$	6.52663	0.247926
$694$	0	0
$695$	−3.80536	−0.144345
$696$	0	0
$697$	3.44255	0.130396
$698$	0	0
$699$	−9.69480	−0.366691
$700$	0	0
$701$	9.69914	0.366331	0.183166	−	0.983082i	$-0.441366\pi$
$701$	9.69914	0.366331	0.183166	+	0.983082i	$0.441366\pi$
$702$	0	0
$703$	−29.2362	−1.10266
$704$	0	0
$705$	8.98090	0.338240
$706$	0	0
$707$	−30.0342	−1.12955
$708$	0	0
$709$	49.4884	1.85858	0.929289	−	0.369354i	$-0.120421\pi$
$709$	49.4884	1.85858	0.929289	+	0.369354i	$0.120421\pi$
$710$	0	0
$711$	12.0650	0.452472
$712$	0	0
$713$	0	0
$714$	0	0
$715$	1.88944	0.0706610
$716$	0	0
$717$	10.7022	0.399680
$718$	0	0
$719$	−15.7139	−0.586029	−0.293015	−	0.956108i	$-0.594658\pi$
$719$	−15.7139	−0.586029	−0.293015	+	0.956108i	$0.594658\pi$
$720$	0	0
$721$	−29.2449	−1.08914
$722$	0	0
$723$	11.1682	0.415348
$724$	0	0
$725$	−1.00000	−0.0371391
$726$	0	0
$727$	−6.51925	−0.241786	−0.120893	−	0.992666i	$-0.538576\pi$
$727$	−6.51925	−0.241786	−0.120893	+	0.992666i	$0.538576\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	36.4161	1.34690
$732$	0	0
$733$	−6.76716	−0.249951	−0.124975	−	0.992160i	$-0.539885\pi$
$733$	−6.76716	−0.249951	−0.124975	+	0.992160i	$0.539885\pi$
$734$	0	0
$735$	−2.26701	−0.0836198
$736$	0	0
$737$	−33.2522	−1.22486
$738$	0	0
$739$	41.1183	1.51256	0.756280	−	0.654248i	$-0.227015\pi$
$739$	41.1183	1.51256	0.756280	+	0.654248i	$0.227015\pi$
$740$	0	0
$741$	3.02648	0.111180
$742$	0	0
$743$	−36.5534	−1.34101	−0.670507	−	0.741903i	$-0.733924\pi$
$743$	−36.5534	−1.34101	−0.670507	+	0.741903i	$0.733924\pi$
$744$	0	0
$745$	4.45427	0.163192
$746$	0	0
$747$	7.62981	0.279160
$748$	0	0
$749$	−20.1447	−0.736072
$750$	0	0
$751$	45.1980	1.64930	0.824649	−	0.565645i	$-0.191373\pi$
$751$	45.1980	1.64930	0.824649	+	0.565645i	$0.191373\pi$
$752$	0	0
$753$	−14.3585	−0.523252
$754$	0	0
$755$	−1.17554	−0.0427824
$756$	0	0
$757$	53.4737	1.94353	0.971767	−	0.235943i	$-0.0758178\pi$
$757$	53.4737	1.94353	0.971767	+	0.235943i	$0.0758178\pi$
$758$	0	0
$759$	6.25225	0.226942
$760$	0	0
$761$	−49.4693	−1.79326	−0.896631	−	0.442778i	$-0.853993\pi$
$761$	−49.4693	−1.79326	−0.896631	+	0.442778i	$0.853993\pi$
$762$	0	0
$763$	−18.7330	−0.678180
$764$	0	0
$765$	4.17554	0.150967
$766$	0	0
$767$	4.06498	0.146778
$768$	0	0
$769$	−14.2331	−0.513260	−0.256630	−	0.966510i	$-0.582612\pi$
$769$	−14.2331	−0.513260	−0.256630	+	0.966510i	$0.582612\pi$
$770$	0	0
$771$	−7.11056	−0.256081
$772$	0	0
$773$	21.7936	0.783863	0.391931	−	0.919994i	$-0.371807\pi$
$773$	21.7936	0.783863	0.391931	+	0.919994i	$0.371807\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−13.2362	−0.474846
$778$	0	0
$779$	−3.96180	−0.141946
$780$	0	0
$781$	7.91158	0.283099
$782$	0	0
$783$	−1.00000	−0.0357371
$784$	0	0
$785$	−0.805358	−0.0287444
$786$	0	0
$787$	14.1829	0.505567	0.252783	−	0.967523i	$-0.418654\pi$
$787$	14.1829	0.505567	0.252783	+	0.967523i	$0.418654\pi$
$788$	0	0
$789$	−19.2214	−0.684301
$790$	0	0
$791$	−24.5119	−0.871542
$792$	0	0
$793$	1.76685	0.0627427
$794$	0	0
$795$	6.88944	0.244343
$796$	0	0
$797$	−2.00000	−0.0708436	−0.0354218	−	0.999372i	$-0.511277\pi$
$797$	−2.00000	−0.0708436	−0.0354218	+	0.999372i	$0.511277\pi$
$798$	0	0
$799$	37.5002	1.32666
$800$	0	0
$801$	17.8894	0.632092
$802$	0	0
$803$	44.3588	1.56539
$804$	0	0
$805$	4.53401	0.159803
$806$	0	0
$807$	−9.88944	−0.348125
$808$	0	0
$809$	30.6757	1.07850	0.539250	−	0.842146i	$-0.318708\pi$
$809$	30.6757	1.07850	0.539250	+	0.842146i	$0.318708\pi$
$810$	0	0
$811$	−36.6609	−1.28734	−0.643670	−	0.765303i	$-0.722589\pi$
$811$	−36.6609	−1.28734	−0.643670	+	0.765303i	$0.722589\pi$
$812$	0	0
$813$	13.7936	0.483764
$814$	0	0
$815$	1.11056	0.0389012
$816$	0	0
$817$	−41.9088	−1.46620
$818$	0	0
$819$	1.37019	0.0478782
$820$	0	0
$821$	−36.1447	−1.26146	−0.630730	−	0.776002i	$-0.717244\pi$
$821$	−36.1447	−1.26146	−0.630730	+	0.776002i	$0.717244\pi$
$822$	0	0
$823$	10.4395	0.363898	0.181949	−	0.983308i	$-0.441759\pi$
$823$	10.4395	0.363898	0.181949	+	0.983308i	$0.441759\pi$
$824$	0	0
$825$	−3.00000	−0.104447
$826$	0	0
$827$	34.1447	1.18733	0.593664	−	0.804713i	$-0.297681\pi$
$827$	34.1447	1.18733	0.593664	+	0.804713i	$0.297681\pi$
$828$	0	0
$829$	42.6874	1.48260	0.741298	−	0.671176i	$-0.234211\pi$
$829$	42.6874	1.48260	0.741298	+	0.671176i	$0.234211\pi$
$830$	0	0
$831$	−14.9661	−0.519170
$832$	0	0
$833$	−9.46599	−0.327977
$834$	0	0
$835$	8.70218	0.301151
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−13.9926	−0.483079	−0.241539	−	0.970391i	$-0.577652\pi$
$839$	−13.9926	−0.483079	−0.241539	+	0.970391i	$0.577652\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	11.7287	0.403956
$844$	0	0
$845$	−12.6033	−0.433568
$846$	0	0
$847$	4.35109	0.149505
$848$	0	0
$849$	3.96180	0.135969
$850$	0	0
$851$	−12.6797	−0.434655
$852$	0	0
$853$	34.6978	1.18803	0.594016	−	0.804453i	$-0.297542\pi$
$853$	34.6978	1.18803	0.594016	+	0.804453i	$0.297542\pi$
$854$	0	0
$855$	−4.80536	−0.164340
$856$	0	0
$857$	48.0077	1.63991	0.819956	−	0.572427i	$-0.193998\pi$
$857$	48.0077	1.63991	0.819956	+	0.572427i	$0.193998\pi$
$858$	0	0
$859$	−16.2627	−0.554875	−0.277438	−	0.960744i	$-0.589485\pi$
$859$	−16.2627	−0.554875	−0.277438	+	0.960744i	$0.589485\pi$
$860$	0	0
$861$	−1.79364	−0.0611271
$862$	0	0
$863$	−13.5605	−0.461604	−0.230802	−	0.973001i	$-0.574135\pi$
$863$	−13.5605	−0.461604	−0.230802	+	0.973001i	$0.574135\pi$
$864$	0	0
$865$	−7.69480	−0.261631
$866$	0	0
$867$	0.435171	0.0147792
$868$	0	0
$869$	−36.1950	−1.22783
$870$	0	0
$871$	−6.98090	−0.236539
$872$	0	0
$873$	0.538351	0.0182204
$874$	0	0
$875$	−2.17554	−0.0735468
$876$	0	0
$877$	−8.01476	−0.270639	−0.135320	−	0.990802i	$-0.543206\pi$
$877$	−8.01476	−0.270639	−0.135320	+	0.990802i	$0.543206\pi$
$878$	0	0
$879$	−3.26701	−0.110193
$880$	0	0
$881$	−44.1564	−1.48767	−0.743834	−	0.668364i	$-0.766995\pi$
$881$	−44.1564	−1.48767	−0.743834	+	0.668364i	$0.766995\pi$
$882$	0	0
$883$	20.7022	0.696684	0.348342	−	0.937368i	$-0.386745\pi$
$883$	20.7022	0.696684	0.348342	+	0.937368i	$0.386745\pi$
$884$	0	0
$885$	−6.45427	−0.216958
$886$	0	0
$887$	−48.5387	−1.62977	−0.814884	−	0.579624i	$-0.803200\pi$
$887$	−48.5387	−1.62977	−0.814884	+	0.579624i	$0.803200\pi$
$888$	0	0
$889$	36.5193	1.22482
$890$	0	0
$891$	−3.00000	−0.100504
$892$	0	0
$893$	−43.1564	−1.44418
$894$	0	0
$895$	6.06498	0.202730
$896$	0	0
$897$	1.31258	0.0438259
$898$	0	0
$899$	0	0
$900$	0	0
$901$	28.7672	0.958373
$902$	0	0
$903$	−18.9735	−0.631399
$904$	0	0
$905$	4.09146	0.136005
$906$	0	0
$907$	26.5149	0.880413	0.440207	−	0.897896i	$-0.354905\pi$
$907$	26.5149	0.880413	0.440207	+	0.897896i	$0.354905\pi$
$908$	0	0
$909$	13.8054	0.457895
$910$	0	0
$911$	−53.8469	−1.78403	−0.892014	−	0.452008i	$-0.850708\pi$
$911$	−53.8469	−1.78403	−0.892014	+	0.452008i	$0.850708\pi$
$912$	0	0
$913$	−22.8894	−0.757530
$914$	0	0
$915$	−2.80536	−0.0927423
$916$	0	0
$917$	24.5119	0.809453
$918$	0	0
$919$	11.4084	0.376328	0.188164	−	0.982138i	$-0.439746\pi$
$919$	11.4084	0.376328	0.188164	+	0.982138i	$0.439746\pi$
$920$	0	0
$921$	−20.4352	−0.673362
$922$	0	0
$923$	1.66094	0.0546705
$924$	0	0
$925$	6.08408	0.200043
$926$	0	0
$927$	13.4426	0.441511
$928$	0	0
$929$	13.7554	0.451301	0.225651	−	0.974208i	$-0.427549\pi$
$929$	13.7554	0.451301	0.225651	+	0.974208i	$0.427549\pi$
$930$	0	0
$931$	10.8938	0.357029
$932$	0	0
$933$	−11.0000	−0.360124
$934$	0	0
$935$	−12.5266	−0.409665
$936$	0	0
$937$	1.29379	0.0422664	0.0211332	−	0.999777i	$-0.493273\pi$
$937$	1.29379	0.0422664	0.0211332	+	0.999777i	$0.493273\pi$
$938$	0	0
$939$	27.6683	0.902921
$940$	0	0
$941$	−49.8586	−1.62534	−0.812672	−	0.582721i	$-0.801988\pi$
$941$	−49.8586	−1.62534	−0.812672	+	0.582721i	$0.801988\pi$
$942$	0	0
$943$	−1.71823	−0.0559534
$944$	0	0
$945$	−2.17554	−0.0707705
$946$	0	0
$947$	21.1109	0.686011	0.343006	−	0.939333i	$-0.388555\pi$
$947$	21.1109	0.686011	0.343006	+	0.939333i	$0.388555\pi$
$948$	0	0
$949$	9.31258	0.302299
$950$	0	0
$951$	−12.8777	−0.417589
$952$	0	0
$953$	−42.7672	−1.38536	−0.692682	−	0.721243i	$-0.743571\pi$
$953$	−42.7672	−1.38536	−0.692682	+	0.721243i	$0.743571\pi$
$954$	0	0
$955$	0.267007	0.00864013
$956$	0	0
$957$	3.00000	0.0969762
$958$	0	0
$959$	41.0533	1.32568
$960$	0	0
$961$	−31.0000	−1.00000
$962$	0	0
$963$	9.25963	0.298387
$964$	0	0
$965$	10.7022	0.344515
$966$	0	0
$967$	33.0342	1.06231	0.531154	−	0.847276i	$-0.321759\pi$
$967$	33.0342	1.06231	0.531154	+	0.847276i	$0.321759\pi$
$968$	0	0
$969$	−20.0650	−0.644580
$970$	0	0
$971$	26.0841	0.837078	0.418539	−	0.908199i	$-0.362542\pi$
$971$	26.0841	0.837078	0.418539	+	0.908199i	$0.362542\pi$
$972$	0	0
$973$	8.27872	0.265404
$974$	0	0
$975$	−0.629813	−0.0201702
$976$	0	0
$977$	−43.6948	−1.39792	−0.698960	−	0.715161i	$-0.746354\pi$
$977$	−43.6948	−1.39792	−0.698960	+	0.715161i	$0.746354\pi$
$978$	0	0
$979$	−53.6683	−1.71525
$980$	0	0
$981$	8.61072	0.274919
$982$	0	0
$983$	−2.55745	−0.0815700	−0.0407850	−	0.999168i	$-0.512986\pi$
$983$	−2.55745	−0.0815700	−0.0407850	+	0.999168i	$0.512986\pi$
$984$	0	0
$985$	8.08408	0.257580
$986$	0	0
$987$	−19.5384	−0.621913
$988$	0	0
$989$	−18.1759	−0.577959
$990$	0	0
$991$	−14.7287	−0.467871	−0.233936	−	0.972252i	$-0.575160\pi$
$991$	−14.7287	−0.467871	−0.233936	+	0.972252i	$0.575160\pi$
$992$	0	0
$993$	−24.6874	−0.783432
$994$	0	0
$995$	15.8054	0.501064
$996$	0	0
$997$	33.0992	1.04826	0.524130	−	0.851638i	$-0.324390\pi$
$997$	33.0992	1.04826	0.524130	+	0.851638i	$0.324390\pi$
$998$	0	0
$999$	6.08408	0.192492

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6960.2.a.cl.1.1		3
4.3	odd	2		435.2.a.i.1.2	✓	3
12.11	even	2		1305.2.a.q.1.2		3
20.3	even	4		2175.2.c.m.349.3		6
20.7	even	4		2175.2.c.m.349.4		6
20.19	odd	2		2175.2.a.u.1.2		3
60.59	even	2		6525.2.a.bf.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
435.2.a.i.1.2	✓	3	4.3	odd	2
1305.2.a.q.1.2		3	12.11	even	2
2175.2.a.u.1.2		3	20.19	odd	2
2175.2.c.m.349.3		6	20.3	even	4
2175.2.c.m.349.4		6	20.7	even	4
6525.2.a.bf.1.2		3	60.59	even	2
6960.2.a.cl.1.1		3	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:	\( N \)	\(=\)	\( 6960 = 2^{4} \cdot 3 \cdot 5 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6960.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +1.00000 q^{5} -2.17554 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +1.00000 q^{5} -2.17554 q^{7} +1.00000 q^{9} -3.00000 q^{11} -0.629813 q^{13} +1.00000 q^{15} +4.17554 q^{17} -4.80536 q^{19} -2.17554 q^{21} -2.08408 q^{23} +1.00000 q^{25} +1.00000 q^{27} -1.00000 q^{29} -3.00000 q^{33} -2.17554 q^{35} +6.08408 q^{37} -0.629813 q^{39} +0.824456 q^{41} +8.72128 q^{43} +1.00000 q^{45} +8.98090 q^{47} -2.26701 q^{49} +4.17554 q^{51} +6.88944 q^{53} -3.00000 q^{55} -4.80536 q^{57} -6.45427 q^{59} -2.80536 q^{61} -2.17554 q^{63} -0.629813 q^{65} +11.0841 q^{67} -2.08408 q^{69} -2.63719 q^{71} -14.7863 q^{73} +1.00000 q^{75} +6.52663 q^{77} +12.0650 q^{79} +1.00000 q^{81} +7.62981 q^{83} +4.17554 q^{85} -1.00000 q^{87} +17.8894 q^{89} +1.37019 q^{91} -4.80536 q^{95} +0.538351 q^{97} -3.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{3} + 3 q^{5} + 4 q^{7} + 3 q^{9}+O(q^{10}) \) 3 * q + 3 * q^3 + 3 * q^5 + 4 * q^7 + 3 * q^9	\( 3 q + 3 q^{3} + 3 q^{5} + 4 q^{7} + 3 q^{9} - 9 q^{11} + 6 q^{13} + 3 q^{15} + 2 q^{17} + 4 q^{19} + 4 q^{21} - q^{23} + 3 q^{25} + 3 q^{27} - 3 q^{29} - 9 q^{33} + 4 q^{35} + 13 q^{37} + 6 q^{39} + 13 q^{41} + 13 q^{43} + 3 q^{45} - 2 q^{47} + 9 q^{49} + 2 q^{51} - 3 q^{53} - 9 q^{55} + 4 q^{57} - 22 q^{59} + 10 q^{61} + 4 q^{63} + 6 q^{65} + 28 q^{67} - q^{69} + 3 q^{73} + 3 q^{75} - 12 q^{77} + 2 q^{79} + 3 q^{81} + 15 q^{83} + 2 q^{85} - 3 q^{87} + 30 q^{89} + 12 q^{91} + 4 q^{95} - q^{97} - 9 q^{99}+O(q^{100}) \) 3 * q + 3 * q^3 + 3 * q^5 + 4 * q^7 + 3 * q^9 - 9 * q^11 + 6 * q^13 + 3 * q^15 + 2 * q^17 + 4 * q^19 + 4 * q^21 - q^23 + 3 * q^25 + 3 * q^27 - 3 * q^29 - 9 * q^33 + 4 * q^35 + 13 * q^37 + 6 * q^39 + 13 * q^41 + 13 * q^43 + 3 * q^45 - 2 * q^47 + 9 * q^49 + 2 * q^51 - 3 * q^53 - 9 * q^55 + 4 * q^57 - 22 * q^59 + 10 * q^61 + 4 * q^63 + 6 * q^65 + 28 * q^67 - q^69 + 3 * q^73 + 3 * q^75 - 12 * q^77 + 2 * q^79 + 3 * q^81 + 15 * q^83 + 2 * q^85 - 3 * q^87 + 30 * q^89 + 12 * q^91 + 4 * q^95 - q^97 - 9 * q^99

Introduction

L-functions

Modular forms

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Database

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