LMFDB - Embedded newform 968.4.a.r.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [968,4,Mod(1,968)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(968, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("968.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 968 = 2^{3} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	968.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:	$57.1138488856$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 4 x^{9} - 193 x^{8} + 670 x^{7} + 10959 x^{6} - 33408 x^{5} - 177207 x^{4} + 365822 x^{3} + \cdots - 781744 $ x^10 - 4x^9 - 193x^8 + 670x^7 + 10959x^6 - 33408x^5 - 177207x^4 + 365822x^3 + 990744x^2 - 836120*x - 781744
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{8}\cdot 11^{2} $
Twist minimal:	no (minimal twist has level 88)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$-0.614830$ of defining polynomial
Character	$\chi$	$=$	968.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00320 q^{3} +6.07450 q^{5} -26.7097 q^{7} -25.9936 q^{9} +O(q^{10})$	$q+1.00320 q^{3} +6.07450 q^{5} -26.7097 q^{7} -25.9936 q^{9} +82.1765 q^{13} +6.09397 q^{15} -103.876 q^{17} -28.7485 q^{19} -26.7953 q^{21} +104.319 q^{23} -88.1005 q^{25} -53.1634 q^{27} +13.3542 q^{29} +333.592 q^{31} -162.248 q^{35} +71.0862 q^{37} +82.4398 q^{39} -319.439 q^{41} +103.144 q^{43} -157.898 q^{45} +402.885 q^{47} +370.407 q^{49} -104.209 q^{51} +14.0627 q^{53} -28.8407 q^{57} +246.416 q^{59} +832.874 q^{61} +694.280 q^{63} +499.181 q^{65} +63.6175 q^{67} +104.653 q^{69} +309.079 q^{71} -408.308 q^{73} -88.3828 q^{75} +855.680 q^{79} +648.493 q^{81} +524.312 q^{83} -630.998 q^{85} +13.3970 q^{87} -392.146 q^{89} -2194.91 q^{91} +334.661 q^{93} -174.633 q^{95} +1438.66 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 9 q^{3} + 13 q^{5} - 3 q^{7} + 141 q^{9}+O(q^{10}) $ 10 * q + 9 * q^3 + 13 * q^5 - 3 * q^7 + 141 * q^9	$ 10 q + 9 q^{3} + 13 q^{5} - 3 q^{7} + 141 q^{9} - 45 q^{13} + 120 q^{15} + 17 q^{17} + 147 q^{19} - 131 q^{21} + 164 q^{23} + 439 q^{25} + 420 q^{27} - 177 q^{29} + 275 q^{31} + 220 q^{35} + 745 q^{37} + 524 q^{39} - 967 q^{41} + 380 q^{43} - 44 q^{45} + 769 q^{47} + 503 q^{49} + 956 q^{51} + 701 q^{53} - 1293 q^{57} + 1291 q^{59} + 1359 q^{61} - 929 q^{63} - 173 q^{65} + 2260 q^{67} + 1988 q^{69} + 465 q^{71} - 111 q^{73} + 4584 q^{75} - 1827 q^{79} + 6874 q^{81} + 4947 q^{83} - 2609 q^{85} - 1303 q^{87} + 446 q^{89} + 2176 q^{91} + 4204 q^{93} - 108 q^{95} + 3511 q^{97}+O(q^{100}) $ 10 * q + 9 * q^3 + 13 * q^5 - 3 * q^7 + 141 * q^9 - 45 * q^13 + 120 * q^15 + 17 * q^17 + 147 * q^19 - 131 * q^21 + 164 * q^23 + 439 * q^25 + 420 * q^27 - 177 * q^29 + 275 * q^31 + 220 * q^35 + 745 * q^37 + 524 * q^39 - 967 * q^41 + 380 * q^43 - 44 * q^45 + 769 * q^47 + 503 * q^49 + 956 * q^51 + 701 * q^53 - 1293 * q^57 + 1291 * q^59 + 1359 * q^61 - 929 * q^63 - 173 * q^65 + 2260 * q^67 + 1988 * q^69 + 465 * q^71 - 111 * q^73 + 4584 * q^75 - 1827 * q^79 + 6874 * q^81 + 4947 * q^83 - 2609 * q^85 - 1303 * q^87 + 446 * q^89 + 2176 * q^91 + 4204 * q^93 - 108 * q^95 + 3511 * q^97

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
$3$	1.00320	0.193067	0.0965334	−	0.995330i	$-0.469225\pi$
$3$	1.00320	0.193067	0.0965334	+	0.995330i	$0.469225\pi$
$4$	0	0
$5$	6.07450	0.543320	0.271660	−	0.962393i	$-0.412427\pi$
$5$	6.07450	0.543320	0.271660	+	0.962393i	$0.412427\pi$
$6$	0	0
$7$	−26.7097	−1.44219	−0.721093	−	0.692838i	$-0.756360\pi$
$7$	−26.7097	−1.44219	−0.721093	+	0.692838i	$0.756360\pi$
$8$	0	0
$9$	−25.9936	−0.962725
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	82.1765	1.75320	0.876602	−	0.481216i	$-0.159805\pi$
$13$	82.1765	1.75320	0.876602	+	0.481216i	$0.159805\pi$
$14$	0	0
$15$	6.09397	0.104897
$16$	0	0
$17$	−103.876	−1.48198	−0.740992	−	0.671513i	$-0.765645\pi$
$17$	−103.876	−1.48198	−0.740992	+	0.671513i	$0.765645\pi$
$18$	0	0
$19$	−28.7485	−0.347125	−0.173562	−	0.984823i	$-0.555528\pi$
$19$	−28.7485	−0.347125	−0.173562	+	0.984823i	$0.555528\pi$
$20$	0	0
$21$	−26.7953	−0.278438
$22$	0	0
$23$	104.319	0.945739	0.472870	−	0.881132i	$-0.343218\pi$
$23$	104.319	0.945739	0.472870	+	0.881132i	$0.343218\pi$
$24$	0	0
$25$	−88.1005	−0.704804
$26$	0	0
$27$	−53.1634	−0.378937
$28$	0	0
$29$	13.3542	0.0855109	0.0427555	−	0.999086i	$-0.486386\pi$
$29$	13.3542	0.0855109	0.0427555	+	0.999086i	$0.486386\pi$
$30$	0	0
$31$	333.592	1.93274	0.966370	−	0.257154i	$-0.0827847\pi$
$31$	333.592	1.93274	0.966370	+	0.257154i	$0.0827847\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−162.248	−0.783569
$36$	0	0
$37$	71.0862	0.315851	0.157926	−	0.987451i	$-0.449519\pi$
$37$	71.0862	0.315851	0.157926	+	0.987451i	$0.449519\pi$
$38$	0	0
$39$	82.4398	0.338485
$40$	0	0
$41$	−319.439	−1.21678	−0.608390	−	0.793639i	$-0.708184\pi$
$41$	−319.439	−1.21678	−0.608390	+	0.793639i	$0.708184\pi$
$42$	0	0
$43$	103.144	0.365797	0.182898	−	0.983132i	$-0.441452\pi$
$43$	103.144	0.365797	0.182898	+	0.983132i	$0.441452\pi$
$44$	0	0
$45$	−157.898	−0.523068
$46$	0	0
$47$	402.885	1.25036	0.625179	−	0.780481i	$-0.285026\pi$
$47$	402.885	1.25036	0.625179	+	0.780481i	$0.285026\pi$
$48$	0	0
$49$	370.407	1.07990
$50$	0	0
$51$	−104.209	−0.286122
$52$	0	0
$53$	14.0627	0.0364465	0.0182233	−	0.999834i	$-0.494199\pi$
$53$	14.0627	0.0364465	0.0182233	+	0.999834i	$0.494199\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−28.8407	−0.0670182
$58$	0	0
$59$	246.416	0.543739	0.271869	−	0.962334i	$-0.412358\pi$
$59$	246.416	0.543739	0.271869	+	0.962334i	$0.412358\pi$
$60$	0	0
$61$	832.874	1.74817	0.874087	−	0.485770i	$-0.161461\pi$
$61$	832.874	1.74817	0.874087	+	0.485770i	$0.161461\pi$
$62$	0	0
$63$	694.280	1.38843
$64$	0	0
$65$	499.181	0.952550
$66$	0	0
$67$	63.6175	0.116002	0.0580009	−	0.998317i	$-0.481527\pi$
$67$	63.6175	0.116002	0.0580009	+	0.998317i	$0.481527\pi$
$68$	0	0
$69$	104.653	0.182591
$70$	0	0
$71$	309.079	0.516633	0.258317	−	0.966060i	$-0.416832\pi$
$71$	309.079	0.516633	0.258317	+	0.966060i	$0.416832\pi$
$72$	0	0
$73$	−408.308	−0.654642	−0.327321	−	0.944913i	$-0.606146\pi$
$73$	−408.308	−0.654642	−0.327321	+	0.944913i	$0.606146\pi$
$74$	0	0
$75$	−88.3828	−0.136074
$76$	0	0
$77$	0	0
$78$	0	0
$79$	855.680	1.21863	0.609314	−	0.792929i	$-0.291445\pi$
$79$	855.680	1.21863	0.609314	+	0.792929i	$0.291445\pi$
$80$	0	0
$81$	648.493	0.889565
$82$	0	0
$83$	524.312	0.693382	0.346691	−	0.937979i	$-0.387305\pi$
$83$	524.312	0.693382	0.346691	+	0.937979i	$0.387305\pi$
$84$	0	0
$85$	−630.998	−0.805192
$86$	0	0
$87$	13.3970	0.0165093
$88$	0	0
$89$	−392.146	−0.467049	−0.233524	−	0.972351i	$-0.575026\pi$
$89$	−392.146	−0.467049	−0.233524	+	0.972351i	$0.575026\pi$
$90$	0	0
$91$	−2194.91	−2.52845
$92$	0	0
$93$	334.661	0.373148
$94$	0	0
$95$	−174.633	−0.188600
$96$	0	0
$97$	1438.66	1.50591	0.752957	−	0.658069i	$-0.228627\pi$
$97$	1438.66	1.50591	0.752957	+	0.658069i	$0.228627\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−730.622	−0.719798	−0.359899	−	0.932991i	$-0.617189\pi$
$101$	−730.622	−0.719798	−0.359899	+	0.932991i	$0.617189\pi$
$102$	0	0
$103$	272.665	0.260839	0.130420	−	0.991459i	$-0.458368\pi$
$103$	272.665	0.260839	0.130420	+	0.991459i	$0.458368\pi$
$104$	0	0
$105$	−162.768	−0.151281
$106$	0	0
$107$	−1845.38	−1.66729	−0.833645	−	0.552301i	$-0.813750\pi$
$107$	−1845.38	−1.66729	−0.833645	+	0.552301i	$0.813750\pi$
$108$	0	0
$109$	−1492.17	−1.31123	−0.655615	−	0.755096i	$-0.727590\pi$
$109$	−1492.17	−1.31123	−0.655615	+	0.755096i	$0.727590\pi$
$110$	0	0
$111$	71.3140	0.0609804
$112$	0	0
$113$	877.631	0.730624	0.365312	−	0.930885i	$-0.380962\pi$
$113$	877.631	0.730624	0.365312	+	0.930885i	$0.380962\pi$
$114$	0	0
$115$	633.685	0.513839
$116$	0	0
$117$	−2136.06	−1.68785
$118$	0	0
$119$	2774.51	2.13730
$120$	0	0
$121$	0	0
$122$	0	0
$123$	−320.462	−0.234920
$124$	0	0
$125$	−1294.48	−0.926254
$126$	0	0
$127$	197.150	0.137750	0.0688751	−	0.997625i	$-0.478059\pi$
$127$	197.150	0.137750	0.0688751	+	0.997625i	$0.478059\pi$
$128$	0	0
$129$	103.474	0.0706232
$130$	0	0
$131$	1597.71	1.06559	0.532796	−	0.846244i	$-0.321141\pi$
$131$	1597.71	1.06559	0.532796	+	0.846244i	$0.321141\pi$
$132$	0	0
$133$	767.864	0.500619
$134$	0	0
$135$	−322.941	−0.205884
$136$	0	0
$137$	1401.86	0.874225	0.437112	−	0.899407i	$-0.356001\pi$
$137$	1401.86	0.874225	0.437112	+	0.899407i	$0.356001\pi$
$138$	0	0
$139$	−79.6678	−0.0486139	−0.0243070	−	0.999705i	$-0.507738\pi$
$139$	−79.6678	−0.0486139	−0.0243070	+	0.999705i	$0.507738\pi$
$140$	0	0
$141$	404.176	0.241403
$142$	0	0
$143$	0	0
$144$	0	0
$145$	81.1202	0.0464598
$146$	0	0
$147$	371.594	0.208493
$148$	0	0
$149$	2509.33	1.37968	0.689841	−	0.723961i	$-0.257680\pi$
$149$	2509.33	1.37968	0.689841	+	0.723961i	$0.257680\pi$
$150$	0	0
$151$	−726.307	−0.391430	−0.195715	−	0.980661i	$-0.562703\pi$
$151$	−726.307	−0.391430	−0.195715	+	0.980661i	$0.562703\pi$
$152$	0	0
$153$	2700.12	1.42674
$154$	0	0
$155$	2026.41	1.05010
$156$	0	0
$157$	−596.004	−0.302970	−0.151485	−	0.988460i	$-0.548406\pi$
$157$	−596.004	−0.302970	−0.151485	+	0.988460i	$0.548406\pi$
$158$	0	0
$159$	14.1078	0.00703662
$160$	0	0
$161$	−2786.33	−1.36393
$162$	0	0
$163$	1800.34	0.865113	0.432557	−	0.901607i	$-0.357612\pi$
$163$	1800.34	0.865113	0.432557	+	0.901607i	$0.357612\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	1038.60	0.481251	0.240626	−	0.970618i	$-0.422647\pi$
$167$	1038.60	0.481251	0.240626	+	0.970618i	$0.422647\pi$
$168$	0	0
$169$	4555.97	2.07372
$170$	0	0
$171$	747.278	0.334186
$172$	0	0
$173$	877.749	0.385746	0.192873	−	0.981224i	$-0.438219\pi$
$173$	877.749	0.385746	0.192873	+	0.981224i	$0.438219\pi$
$174$	0	0
$175$	2353.13	1.01646
$176$	0	0
$177$	247.205	0.104978
$178$	0	0
$179$	927.009	0.387083	0.193542	−	0.981092i	$-0.438003\pi$
$179$	927.009	0.387083	0.193542	+	0.981092i	$0.438003\pi$
$180$	0	0
$181$	1583.57	0.650310	0.325155	−	0.945661i	$-0.394584\pi$
$181$	1583.57	0.650310	0.325155	+	0.945661i	$0.394584\pi$
$182$	0	0
$183$	835.543	0.337514
$184$	0	0
$185$	431.813	0.171608
$186$	0	0
$187$	0	0
$188$	0	0
$189$	1419.98	0.546498
$190$	0	0
$191$	2504.87	0.948933	0.474466	−	0.880274i	$-0.342641\pi$
$191$	2504.87	0.948933	0.474466	+	0.880274i	$0.342641\pi$
$192$	0	0
$193$	3233.84	1.20610	0.603049	−	0.797704i	$-0.293952\pi$
$193$	3233.84	1.20610	0.603049	+	0.797704i	$0.293952\pi$
$194$	0	0
$195$	500.781	0.183906
$196$	0	0
$197$	−2262.30	−0.818185	−0.409092	−	0.912493i	$-0.634155\pi$
$197$	−2262.30	−0.818185	−0.409092	+	0.912493i	$0.634155\pi$
$198$	0	0
$199$	488.243	0.173923	0.0869614	−	0.996212i	$-0.472284\pi$
$199$	488.243	0.173923	0.0869614	+	0.996212i	$0.472284\pi$
$200$	0	0
$201$	63.8214	0.0223961
$202$	0	0
$203$	−356.687	−0.123323
$204$	0	0
$205$	−1940.43	−0.661100
$206$	0	0
$207$	−2711.62	−0.910487
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−3698.90	−1.20684	−0.603418	−	0.797425i	$-0.706195\pi$
$211$	−3698.90	−1.20684	−0.603418	+	0.797425i	$0.706195\pi$
$212$	0	0
$213$	310.070	0.0997447
$214$	0	0
$215$	626.547	0.198745
$216$	0	0
$217$	−8910.15	−2.78737
$218$	0	0
$219$	−409.617	−0.126390
$220$	0	0
$221$	−8536.20	−2.59822
$222$	0	0
$223$	−4494.02	−1.34951	−0.674757	−	0.738040i	$-0.735751\pi$
$223$	−4494.02	−1.34951	−0.674757	+	0.738040i	$0.735751\pi$
$224$	0	0
$225$	2290.05	0.678532
$226$	0	0
$227$	2887.89	0.844389	0.422194	−	0.906505i	$-0.361260\pi$
$227$	2887.89	0.844389	0.422194	+	0.906505i	$0.361260\pi$
$228$	0	0
$229$	2826.69	0.815690	0.407845	−	0.913051i	$-0.366280\pi$
$229$	2826.69	0.815690	0.407845	+	0.913051i	$0.366280\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	3248.03	0.913242	0.456621	−	0.889661i	$-0.349060\pi$
$233$	3248.03	0.913242	0.456621	+	0.889661i	$0.349060\pi$
$234$	0	0
$235$	2447.33	0.679344
$236$	0	0
$237$	858.422	0.235276
$238$	0	0
$239$	3726.85	1.00866	0.504330	−	0.863511i	$-0.331739\pi$
$239$	3726.85	1.00866	0.504330	+	0.863511i	$0.331739\pi$
$240$	0	0
$241$	−2906.13	−0.776766	−0.388383	−	0.921498i	$-0.626966\pi$
$241$	−2906.13	−0.776766	−0.388383	+	0.921498i	$0.626966\pi$
$242$	0	0
$243$	2085.98	0.550683
$244$	0	0
$245$	2250.04	0.586733
$246$	0	0
$247$	−2362.45	−0.608580
$248$	0	0
$249$	525.992	0.133869
$250$	0	0
$251$	5471.07	1.37582	0.687911	−	0.725795i	$-0.258528\pi$
$251$	5471.07	1.37582	0.687911	+	0.725795i	$0.258528\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	−633.020	−0.155456
$256$	0	0
$257$	−5352.44	−1.29913	−0.649565	−	0.760306i	$-0.725049\pi$
$257$	−5352.44	−1.29913	−0.649565	+	0.760306i	$0.725049\pi$
$258$	0	0
$259$	−1898.69	−0.455517
$260$	0	0
$261$	−347.124	−0.0823235
$262$	0	0
$263$	5152.87	1.20814	0.604068	−	0.796933i	$-0.293545\pi$
$263$	5152.87	1.20814	0.604068	+	0.796933i	$0.293545\pi$
$264$	0	0
$265$	85.4241	0.0198021
$266$	0	0
$267$	−393.402	−0.0901716
$268$	0	0
$269$	2450.74	0.555480	0.277740	−	0.960656i	$-0.410415\pi$
$269$	2450.74	0.555480	0.277740	+	0.960656i	$0.410415\pi$
$270$	0	0
$271$	−179.064	−0.0401378	−0.0200689	−	0.999799i	$-0.506389\pi$
$271$	−179.064	−0.0401378	−0.0200689	+	0.999799i	$0.506389\pi$
$272$	0	0
$273$	−2201.94	−0.488159
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−5106.41	−1.10763	−0.553817	−	0.832638i	$-0.686829\pi$
$277$	−5106.41	−1.10763	−0.553817	+	0.832638i	$0.686829\pi$
$278$	0	0
$279$	−8671.26	−1.86070
$280$	0	0
$281$	−5251.29	−1.11482	−0.557412	−	0.830236i	$-0.688206\pi$
$281$	−5251.29	−1.11482	−0.557412	+	0.830236i	$0.688206\pi$
$282$	0	0
$283$	6186.40	1.29945	0.649723	−	0.760171i	$-0.274885\pi$
$283$	6186.40	1.29945	0.649723	+	0.760171i	$0.274885\pi$
$284$	0	0
$285$	−175.193	−0.0364123
$286$	0	0
$287$	8532.11	1.75482
$288$	0	0
$289$	5877.32	1.19628
$290$	0	0
$291$	1443.27	0.290742
$292$	0	0
$293$	−3508.95	−0.699641	−0.349821	−	0.936817i	$-0.613757\pi$
$293$	−3508.95	−0.699641	−0.349821	+	0.936817i	$0.613757\pi$
$294$	0	0
$295$	1496.85	0.295424
$296$	0	0
$297$	0	0
$298$	0	0
$299$	8572.56	1.65807
$300$	0	0
$301$	−2754.94	−0.527548
$302$	0	0
$303$	−732.963	−0.138969
$304$	0	0
$305$	5059.29	0.949817
$306$	0	0
$307$	−1744.41	−0.324295	−0.162148	−	0.986767i	$-0.551842\pi$
$307$	−1744.41	−0.324295	−0.162148	+	0.986767i	$0.551842\pi$
$308$	0	0
$309$	273.539	0.0503594
$310$	0	0
$311$	−2926.97	−0.533676	−0.266838	−	0.963741i	$-0.585979\pi$
$311$	−2926.97	−0.533676	−0.266838	+	0.963741i	$0.585979\pi$
$312$	0	0
$313$	−6794.84	−1.22705	−0.613525	−	0.789675i	$-0.710249\pi$
$313$	−6794.84	−1.22705	−0.613525	+	0.789675i	$0.710249\pi$
$314$	0	0
$315$	4217.40	0.754361
$316$	0	0
$317$	−10367.2	−1.83684	−0.918420	−	0.395608i	$-0.870534\pi$
$317$	−10367.2	−1.83684	−0.918420	+	0.395608i	$0.870534\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	−1851.30	−0.321898
$322$	0	0
$323$	2986.30	0.514434
$324$	0	0
$325$	−7239.78	−1.23566
$326$	0	0
$327$	−1496.95	−0.253155
$328$	0	0
$329$	−10760.9	−1.80325
$330$	0	0
$331$	−3982.48	−0.661320	−0.330660	−	0.943750i	$-0.607271\pi$
$331$	−3982.48	−0.661320	−0.330660	+	0.943750i	$0.607271\pi$
$332$	0	0
$333$	−1847.78	−0.304078
$334$	0	0
$335$	386.445	0.0630260
$336$	0	0
$337$	−7461.83	−1.20615	−0.603074	−	0.797685i	$-0.706058\pi$
$337$	−7461.83	−1.20615	−0.603074	+	0.797685i	$0.706058\pi$
$338$	0	0
$339$	880.443	0.141059
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−732.027	−0.115235
$344$	0	0
$345$	635.716	0.0992052
$346$	0	0
$347$	1992.95	0.308320	0.154160	−	0.988046i	$-0.450733\pi$
$347$	1992.95	0.308320	0.154160	+	0.988046i	$0.450733\pi$
$348$	0	0
$349$	−1412.29	−0.216613	−0.108307	−	0.994118i	$-0.534543\pi$
$349$	−1412.29	−0.216613	−0.108307	+	0.994118i	$0.534543\pi$
$350$	0	0
$351$	−4368.78	−0.664354
$352$	0	0
$353$	1224.00	0.184553	0.0922763	−	0.995733i	$-0.470586\pi$
$353$	1224.00	0.184553	0.0922763	+	0.995733i	$0.470586\pi$
$354$	0	0
$355$	1877.50	0.280697
$356$	0	0
$357$	2783.40	0.412642
$358$	0	0
$359$	5513.65	0.810583	0.405291	−	0.914188i	$-0.367170\pi$
$359$	5513.65	0.810583	0.405291	+	0.914188i	$0.367170\pi$
$360$	0	0
$361$	−6032.52	−0.879504
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−2480.27	−0.355680
$366$	0	0
$367$	2764.74	0.393238	0.196619	−	0.980480i	$-0.437004\pi$
$367$	2764.74	0.393238	0.196619	+	0.980480i	$0.437004\pi$
$368$	0	0
$369$	8303.36	1.17142
$370$	0	0
$371$	−375.611	−0.0525627
$372$	0	0
$373$	3977.72	0.552168	0.276084	−	0.961134i	$-0.410963\pi$
$373$	3977.72	0.552168	0.276084	+	0.961134i	$0.410963\pi$
$374$	0	0
$375$	−1298.63	−0.178829
$376$	0	0
$377$	1097.40	0.149918
$378$	0	0
$379$	−14533.5	−1.96976	−0.984878	−	0.173249i	$-0.944574\pi$
$379$	−14533.5	−1.96976	−0.984878	+	0.173249i	$0.944574\pi$
$380$	0	0
$381$	197.782	0.0265950
$382$	0	0
$383$	−5198.09	−0.693499	−0.346749	−	0.937958i	$-0.612715\pi$
$383$	−5198.09	−0.693499	−0.346749	+	0.937958i	$0.612715\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−2681.07	−0.352162
$388$	0	0
$389$	6181.85	0.805738	0.402869	−	0.915258i	$-0.368013\pi$
$389$	6181.85	0.805738	0.402869	+	0.915258i	$0.368013\pi$
$390$	0	0
$391$	−10836.3	−1.40157
$392$	0	0
$393$	1602.83	0.205730
$394$	0	0
$395$	5197.83	0.662104
$396$	0	0
$397$	−636.976	−0.0805263	−0.0402631	−	0.999189i	$-0.512820\pi$
$397$	−636.976	−0.0805263	−0.0402631	+	0.999189i	$0.512820\pi$
$398$	0	0
$399$	770.325	0.0966528
$400$	0	0
$401$	4495.73	0.559866	0.279933	−	0.960020i	$-0.409688\pi$
$401$	4495.73	0.559866	0.279933	+	0.960020i	$0.409688\pi$
$402$	0	0
$403$	27413.4	3.38849
$404$	0	0
$405$	3939.27	0.483318
$406$	0	0
$407$	0	0
$408$	0	0
$409$	3569.38	0.431527	0.215763	−	0.976446i	$-0.430776\pi$
$409$	3569.38	0.431527	0.215763	+	0.976446i	$0.430776\pi$
$410$	0	0
$411$	1406.35	0.168784
$412$	0	0
$413$	−6581.68	−0.784173
$414$	0	0
$415$	3184.93	0.376728
$416$	0	0
$417$	−79.9231	−0.00938574
$418$	0	0
$419$	−12382.2	−1.44370	−0.721850	−	0.692050i	$-0.756708\pi$
$419$	−12382.2	−1.44370	−0.721850	+	0.692050i	$0.756708\pi$
$420$	0	0
$421$	12996.9	1.50459	0.752294	−	0.658827i	$-0.228947\pi$
$421$	12996.9	1.50459	0.752294	+	0.658827i	$0.228947\pi$
$422$	0	0
$423$	−10472.4	−1.20375
$424$	0	0
$425$	9151.56	1.04451
$426$	0	0
$427$	−22245.8	−2.52119
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−3054.63	−0.341383	−0.170692	−	0.985324i	$-0.554600\pi$
$431$	−3054.63	−0.341383	−0.170692	+	0.985324i	$0.554600\pi$
$432$	0	0
$433$	−11674.8	−1.29574	−0.647870	−	0.761751i	$-0.724340\pi$
$433$	−11674.8	−1.29574	−0.647870	+	0.761751i	$0.724340\pi$
$434$	0	0
$435$	81.3802	0.00896984
$436$	0	0
$437$	−2999.02	−0.328289
$438$	0	0
$439$	9542.32	1.03743	0.518713	−	0.854948i	$-0.326411\pi$
$439$	9542.32	1.03743	0.518713	+	0.854948i	$0.326411\pi$
$440$	0	0
$441$	−9628.20	−1.03965
$442$	0	0
$443$	−9040.66	−0.969604	−0.484802	−	0.874624i	$-0.661108\pi$
$443$	−9040.66	−0.969604	−0.484802	+	0.874624i	$0.661108\pi$
$444$	0	0
$445$	−2382.09	−0.253757
$446$	0	0
$447$	2517.37	0.266371
$448$	0	0
$449$	−9642.72	−1.01351	−0.506757	−	0.862089i	$-0.669156\pi$
$449$	−9642.72	−1.01351	−0.506757	+	0.862089i	$0.669156\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	−728.634	−0.0755722
$454$	0	0
$455$	−13333.0	−1.37376
$456$	0	0
$457$	3403.31	0.348359	0.174180	−	0.984714i	$-0.444273\pi$
$457$	3403.31	0.348359	0.174180	+	0.984714i	$0.444273\pi$
$458$	0	0
$459$	5522.43	0.561579
$460$	0	0
$461$	2127.43	0.214933	0.107467	−	0.994209i	$-0.465726\pi$
$461$	2127.43	0.214933	0.107467	+	0.994209i	$0.465726\pi$
$462$	0	0
$463$	14223.3	1.42768	0.713839	−	0.700310i	$-0.246955\pi$
$463$	14223.3	1.42768	0.713839	+	0.700310i	$0.246955\pi$
$464$	0	0
$465$	2032.90	0.202739
$466$	0	0
$467$	−2991.19	−0.296393	−0.148197	−	0.988958i	$-0.547347\pi$
$467$	−2991.19	−0.296393	−0.148197	+	0.988958i	$0.547347\pi$
$468$	0	0
$469$	−1699.20	−0.167296
$470$	0	0
$471$	−597.914	−0.0584935
$472$	0	0
$473$	0	0
$474$	0	0
$475$	2532.76	0.244655
$476$	0	0
$477$	−365.541	−0.0350880
$478$	0	0
$479$	−17489.0	−1.66826	−0.834128	−	0.551570i	$-0.814029\pi$
$479$	−17489.0	−1.66826	−0.834128	+	0.551570i	$0.814029\pi$
$480$	0	0
$481$	5841.61	0.553752
$482$	0	0
$483$	−2795.25	−0.263330
$484$	0	0
$485$	8739.14	0.818193
$486$	0	0
$487$	10994.1	1.02298	0.511489	−	0.859290i	$-0.329094\pi$
$487$	10994.1	1.02298	0.511489	+	0.859290i	$0.329094\pi$
$488$	0	0
$489$	1806.11	0.167025
$490$	0	0
$491$	−21268.6	−1.95487	−0.977433	−	0.211244i	$-0.932249\pi$
$491$	−21268.6	−1.95487	−0.977433	+	0.211244i	$0.932249\pi$
$492$	0	0
$493$	−1387.19	−0.126726
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−8255.41	−0.745082
$498$	0	0
$499$	3306.52	0.296633	0.148317	−	0.988940i	$-0.452615\pi$
$499$	3306.52	0.296633	0.148317	+	0.988940i	$0.452615\pi$
$500$	0	0
$501$	1041.92	0.0929136
$502$	0	0
$503$	5078.26	0.450156	0.225078	−	0.974341i	$-0.427736\pi$
$503$	5078.26	0.450156	0.225078	+	0.974341i	$0.427736\pi$
$504$	0	0
$505$	−4438.16	−0.391080
$506$	0	0
$507$	4570.57	0.400367
$508$	0	0
$509$	2057.27	0.179149	0.0895744	−	0.995980i	$-0.471449\pi$
$509$	2057.27	0.179149	0.0895744	+	0.995980i	$0.471449\pi$
$510$	0	0
$511$	10905.8	0.944117
$512$	0	0
$513$	1528.37	0.131538
$514$	0	0
$515$	1656.30	0.141719
$516$	0	0
$517$	0	0
$518$	0	0
$519$	880.562	0.0744747
$520$	0	0
$521$	−13584.7	−1.14234	−0.571168	−	0.820833i	$-0.693509\pi$
$521$	−13584.7	−1.14234	−0.571168	+	0.820833i	$0.693509\pi$
$522$	0	0
$523$	4068.26	0.340139	0.170069	−	0.985432i	$-0.445601\pi$
$523$	4068.26	0.340139	0.170069	+	0.985432i	$0.445601\pi$
$524$	0	0
$525$	2360.68	0.196244
$526$	0	0
$527$	−34652.4	−2.86429
$528$	0	0
$529$	−1284.56	−0.105577
$530$	0	0
$531$	−6405.23	−0.523471
$532$	0	0
$533$	−26250.3	−2.13326
$534$	0	0
$535$	−11209.8	−0.905872
$536$	0	0
$537$	929.980	0.0747329
$538$	0	0
$539$	0	0
$540$	0	0
$541$	19884.1	1.58020	0.790098	−	0.612981i	$-0.210030\pi$
$541$	19884.1	1.58020	0.790098	+	0.612981i	$0.210030\pi$
$542$	0	0
$543$	1588.65	0.125553
$544$	0	0
$545$	−9064.19	−0.712417
$546$	0	0
$547$	3811.32	0.297916	0.148958	−	0.988844i	$-0.452408\pi$
$547$	3811.32	0.297916	0.148958	+	0.988844i	$0.452408\pi$
$548$	0	0
$549$	−21649.4	−1.68301
$550$	0	0
$551$	−383.914	−0.0296830
$552$	0	0
$553$	−22854.9	−1.75749
$554$	0	0
$555$	433.197	0.0331319
$556$	0	0
$557$	15397.2	1.17128	0.585638	−	0.810573i	$-0.300844\pi$
$557$	15397.2	1.17128	0.585638	+	0.810573i	$0.300844\pi$
$558$	0	0
$559$	8475.99	0.641317
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−18752.4	−1.40376	−0.701881	−	0.712294i	$-0.747656\pi$
$563$	−18752.4	−1.40376	−0.701881	+	0.712294i	$0.747656\pi$
$564$	0	0
$565$	5331.17	0.396963
$566$	0	0
$567$	−17321.0	−1.28292
$568$	0	0
$569$	3445.50	0.253854	0.126927	−	0.991912i	$-0.459489\pi$
$569$	3445.50	0.253854	0.126927	+	0.991912i	$0.459489\pi$
$570$	0	0
$571$	20827.4	1.52645	0.763223	−	0.646135i	$-0.223616\pi$
$571$	20827.4	1.52645	0.763223	+	0.646135i	$0.223616\pi$
$572$	0	0
$573$	2512.90	0.183207
$574$	0	0
$575$	−9190.55	−0.666561
$576$	0	0
$577$	3937.49	0.284090	0.142045	−	0.989860i	$-0.454632\pi$
$577$	3937.49	0.284090	0.142045	+	0.989860i	$0.454632\pi$
$578$	0	0
$579$	3244.20	0.232857
$580$	0	0
$581$	−14004.2	−0.999986
$582$	0	0
$583$	0	0
$584$	0	0
$585$	−12975.5	−0.917044
$586$	0	0
$587$	−9657.58	−0.679064	−0.339532	−	0.940594i	$-0.610269\pi$
$587$	−9657.58	−0.679064	−0.339532	+	0.940594i	$0.610269\pi$
$588$	0	0
$589$	−9590.30	−0.670902
$590$	0	0
$591$	−2269.55	−0.157964
$592$	0	0
$593$	4048.54	0.280360	0.140180	−	0.990126i	$-0.455232\pi$
$593$	4048.54	0.280360	0.140180	+	0.990126i	$0.455232\pi$
$594$	0	0
$595$	16853.7	1.16124
$596$	0	0
$597$	489.808	0.0335787
$598$	0	0
$599$	880.647	0.0600706	0.0300353	−	0.999549i	$-0.490438\pi$
$599$	880.647	0.0600706	0.0300353	+	0.999549i	$0.490438\pi$
$600$	0	0
$601$	2500.43	0.169708	0.0848541	−	0.996393i	$-0.472958\pi$
$601$	2500.43	0.169708	0.0848541	+	0.996393i	$0.472958\pi$
$602$	0	0
$603$	−1653.65	−0.111678
$604$	0	0
$605$	0	0
$606$	0	0
$607$	20857.7	1.39471	0.697353	−	0.716728i	$-0.254361\pi$
$607$	20857.7	1.39471	0.697353	+	0.716728i	$0.254361\pi$
$608$	0	0
$609$	−357.830	−0.0238095
$610$	0	0
$611$	33107.7	2.19213
$612$	0	0
$613$	−2145.90	−0.141390	−0.0706949	−	0.997498i	$-0.522522\pi$
$613$	−2145.90	−0.141390	−0.0706949	+	0.997498i	$0.522522\pi$
$614$	0	0
$615$	−1946.65	−0.127637
$616$	0	0
$617$	−23302.7	−1.52047	−0.760235	−	0.649649i	$-0.774916\pi$
$617$	−23302.7	−1.52047	−0.760235	+	0.649649i	$0.774916\pi$
$618$	0	0
$619$	29666.2	1.92631	0.963155	−	0.268945i	$-0.0866751\pi$
$619$	29666.2	1.92631	0.963155	+	0.268945i	$0.0866751\pi$
$620$	0	0
$621$	−5545.95	−0.358376
$622$	0	0
$623$	10474.1	0.673572
$624$	0	0
$625$	3149.25	0.201552
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−7384.18	−0.468087
$630$	0	0
$631$	−12224.9	−0.771261	−0.385630	−	0.922653i	$-0.626016\pi$
$631$	−12224.9	−0.771261	−0.385630	+	0.922653i	$0.626016\pi$
$632$	0	0
$633$	−3710.75	−0.233000
$634$	0	0
$635$	1197.59	0.0748424
$636$	0	0
$637$	30438.7	1.89329
$638$	0	0
$639$	−8034.08	−0.497376
$640$	0	0
$641$	8068.91	0.497196	0.248598	−	0.968607i	$-0.420030\pi$
$641$	8068.91	0.497196	0.248598	+	0.968607i	$0.420030\pi$
$642$	0	0
$643$	22154.7	1.35878	0.679389	−	0.733778i	$-0.262245\pi$
$643$	22154.7	1.35878	0.679389	+	0.733778i	$0.262245\pi$
$644$	0	0
$645$	628.554	0.0383710
$646$	0	0
$647$	18371.7	1.11633	0.558164	−	0.829730i	$-0.311506\pi$
$647$	18371.7	1.11633	0.558164	+	0.829730i	$0.311506\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	−8938.70	−0.538149
$652$	0	0
$653$	13966.3	0.836973	0.418486	−	0.908223i	$-0.362561\pi$
$653$	13966.3	0.836973	0.418486	+	0.908223i	$0.362561\pi$
$654$	0	0
$655$	9705.29	0.578957
$656$	0	0
$657$	10613.4	0.630241
$658$	0	0
$659$	11074.6	0.654634	0.327317	−	0.944915i	$-0.393855\pi$
$659$	11074.6	0.654634	0.327317	+	0.944915i	$0.393855\pi$
$660$	0	0
$661$	−9471.84	−0.557355	−0.278678	−	0.960385i	$-0.589896\pi$
$661$	−9471.84	−0.557355	−0.278678	+	0.960385i	$0.589896\pi$
$662$	0	0
$663$	−8563.55	−0.501630
$664$	0	0
$665$	4664.39	0.271996
$666$	0	0
$667$	1393.10	0.0808710
$668$	0	0
$669$	−4508.42	−0.260546
$670$	0	0
$671$	0	0
$672$	0	0
$673$	21421.9	1.22698	0.613488	−	0.789704i	$-0.289766\pi$
$673$	21421.9	1.22698	0.613488	+	0.789704i	$0.289766\pi$
$674$	0	0
$675$	4683.72	0.267076
$676$	0	0
$677$	22221.8	1.26152	0.630762	−	0.775976i	$-0.282742\pi$
$677$	22221.8	1.26152	0.630762	+	0.775976i	$0.282742\pi$
$678$	0	0
$679$	−38426.1	−2.17181
$680$	0	0
$681$	2897.15	0.163023
$682$	0	0
$683$	−30048.5	−1.68342	−0.841709	−	0.539931i	$-0.818450\pi$
$683$	−30048.5	−1.68342	−0.841709	+	0.539931i	$0.818450\pi$
$684$	0	0
$685$	8515.59	0.474984
$686$	0	0
$687$	2835.75	0.157483
$688$	0	0
$689$	1155.63	0.0638982
$690$	0	0
$691$	−27578.9	−1.51831	−0.759155	−	0.650910i	$-0.774387\pi$
$691$	−27578.9	−1.51831	−0.759155	+	0.650910i	$0.774387\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−483.942	−0.0264129
$696$	0	0
$697$	33182.2	1.80325
$698$	0	0
$699$	3258.43	0.176317
$700$	0	0
$701$	−15938.5	−0.858756	−0.429378	−	0.903125i	$-0.641267\pi$
$701$	−15938.5	−0.858756	−0.429378	+	0.903125i	$0.641267\pi$
$702$	0	0
$703$	−2043.62	−0.109640
$704$	0	0
$705$	2455.17	0.131159
$706$	0	0
$707$	19514.7	1.03808
$708$	0	0
$709$	20279.0	1.07418	0.537089	−	0.843526i	$-0.319524\pi$
$709$	20279.0	1.07418	0.537089	+	0.843526i	$0.319524\pi$
$710$	0	0
$711$	−22242.2	−1.17320
$712$	0	0
$713$	34800.0	1.82787
$714$	0	0
$715$	0	0
$716$	0	0
$717$	3738.79	0.194739
$718$	0	0
$719$	1841.51	0.0955170	0.0477585	−	0.998859i	$-0.484792\pi$
$719$	1841.51	0.0955170	0.0477585	+	0.998859i	$0.484792\pi$
$720$	0	0
$721$	−7282.79	−0.376179
$722$	0	0
$723$	−2915.45	−0.149968
$724$	0	0
$725$	−1176.51	−0.0602684
$726$	0	0
$727$	−5678.11	−0.289669	−0.144834	−	0.989456i	$-0.546265\pi$
$727$	−5678.11	−0.289669	−0.144834	+	0.989456i	$0.546265\pi$
$728$	0	0
$729$	−15416.6	−0.783247
$730$	0	0
$731$	−10714.2	−0.542106
$732$	0	0
$733$	−200.997	−0.0101282	−0.00506412	−	0.999987i	$-0.501612\pi$
$733$	−200.997	−0.0101282	−0.00506412	+	0.999987i	$0.501612\pi$
$734$	0	0
$735$	2257.25	0.113279
$736$	0	0
$737$	0	0
$738$	0	0
$739$	−2831.33	−0.140937	−0.0704684	−	0.997514i	$-0.522449\pi$
$739$	−2831.33	−0.140937	−0.0704684	+	0.997514i	$0.522449\pi$
$740$	0	0
$741$	−2370.02	−0.117497
$742$	0	0
$743$	2213.01	0.109270	0.0546349	−	0.998506i	$-0.482601\pi$
$743$	2213.01	0.109270	0.0546349	+	0.998506i	$0.482601\pi$
$744$	0	0
$745$	15242.9	0.749609
$746$	0	0
$747$	−13628.7	−0.667536
$748$	0	0
$749$	49289.6	2.40454
$750$	0	0
$751$	−3565.52	−0.173246	−0.0866229	−	0.996241i	$-0.527608\pi$
$751$	−3565.52	−0.173246	−0.0866229	+	0.996241i	$0.527608\pi$
$752$	0	0
$753$	5488.61	0.265625
$754$	0	0
$755$	−4411.95	−0.212672
$756$	0	0
$757$	29497.0	1.41623	0.708116	−	0.706096i	$-0.249545\pi$
$757$	29497.0	1.41623	0.708116	+	0.706096i	$0.249545\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−4012.34	−0.191127	−0.0955633	−	0.995423i	$-0.530465\pi$
$761$	−4012.34	−0.191127	−0.0955633	+	0.995423i	$0.530465\pi$
$762$	0	0
$763$	39855.4	1.89104
$764$	0	0
$765$	16401.9	0.775178
$766$	0	0
$767$	20249.6	0.953285
$768$	0	0
$769$	21327.7	1.00013	0.500064	−	0.865989i	$-0.333310\pi$
$769$	21327.7	1.00013	0.500064	+	0.865989i	$0.333310\pi$
$770$	0	0
$771$	−5369.60	−0.250819
$772$	0	0
$773$	−35031.9	−1.63002	−0.815012	−	0.579444i	$-0.803270\pi$
$773$	−35031.9	−1.63002	−0.815012	+	0.579444i	$0.803270\pi$
$774$	0	0
$775$	−29389.6	−1.36220
$776$	0	0
$777$	−1904.77	−0.0879451
$778$	0	0
$779$	9183.40	0.422374
$780$	0	0
$781$	0	0
$782$	0	0
$783$	−709.956	−0.0324033
$784$	0	0
$785$	−3620.43	−0.164610
$786$	0	0
$787$	−6605.92	−0.299207	−0.149603	−	0.988746i	$-0.547800\pi$
$787$	−6605.92	−0.299207	−0.149603	+	0.988746i	$0.547800\pi$
$788$	0	0
$789$	5169.39	0.233251
$790$	0	0
$791$	−23441.2	−1.05370
$792$	0	0
$793$	68442.6	3.06490
$794$	0	0
$795$	85.6979	0.00382313
$796$	0	0
$797$	8918.42	0.396370	0.198185	−	0.980165i	$-0.436495\pi$
$797$	8918.42	0.396370	0.198185	+	0.980165i	$0.436495\pi$
$798$	0	0
$799$	−41850.3	−1.85301
$800$	0	0
$801$	10193.3	0.449640
$802$	0	0
$803$	0	0
$804$	0	0
$805$	−16925.5	−0.741052
$806$	0	0
$807$	2458.59	0.107245
$808$	0	0
$809$	14598.4	0.634429	0.317214	−	0.948354i	$-0.397253\pi$
$809$	14598.4	0.634429	0.317214	+	0.948354i	$0.397253\pi$
$810$	0	0
$811$	−38930.1	−1.68560	−0.842799	−	0.538229i	$-0.819094\pi$
$811$	−38930.1	−1.68560	−0.842799	+	0.538229i	$0.819094\pi$
$812$	0	0
$813$	−179.637	−0.00774927
$814$	0	0
$815$	10936.2	0.470033
$816$	0	0
$817$	−2965.23	−0.126977
$818$	0	0
$819$	57053.5	2.43420
$820$	0	0
$821$	−16851.6	−0.716352	−0.358176	−	0.933654i	$-0.616601\pi$
$821$	−16851.6	−0.716352	−0.358176	+	0.933654i	$0.616601\pi$
$822$	0	0
$823$	32099.8	1.35957	0.679787	−	0.733409i	$-0.262072\pi$
$823$	32099.8	1.35957	0.679787	+	0.733409i	$0.262072\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	31223.7	1.31288	0.656441	−	0.754378i	$-0.272061\pi$
$827$	31223.7	1.31288	0.656441	+	0.754378i	$0.272061\pi$
$828$	0	0
$829$	−4263.24	−0.178611	−0.0893055	−	0.996004i	$-0.528465\pi$
$829$	−4263.24	−0.178611	−0.0893055	+	0.996004i	$0.528465\pi$
$830$	0	0
$831$	−5122.78	−0.213847
$832$	0	0
$833$	−38476.5	−1.60040
$834$	0	0
$835$	6308.95	0.261473
$836$	0	0
$837$	−17734.9	−0.732387
$838$	0	0
$839$	−27031.7	−1.11232	−0.556162	−	0.831074i	$-0.687727\pi$
$839$	−27031.7	−1.11232	−0.556162	+	0.831074i	$0.687727\pi$
$840$	0	0
$841$	−24210.7	−0.992688
$842$	0	0
$843$	−5268.12	−0.215235
$844$	0	0
$845$	27675.2	1.12670
$846$	0	0
$847$	0	0
$848$	0	0
$849$	6206.22	0.250880
$850$	0	0
$851$	7415.64	0.298713
$852$	0	0
$853$	−23561.8	−0.945767	−0.472884	−	0.881125i	$-0.656787\pi$
$853$	−23561.8	−0.945767	−0.472884	+	0.881125i	$0.656787\pi$
$854$	0	0
$855$	4539.34	0.181570
$856$	0	0
$857$	25690.2	1.02399	0.511996	−	0.858988i	$-0.328906\pi$
$857$	25690.2	1.02399	0.511996	+	0.858988i	$0.328906\pi$
$858$	0	0
$859$	−38906.7	−1.54538	−0.772689	−	0.634785i	$-0.781089\pi$
$859$	−38906.7	−1.54538	−0.772689	+	0.634785i	$0.781089\pi$
$860$	0	0
$861$	8559.45	0.338798
$862$	0	0
$863$	29165.7	1.15042	0.575209	−	0.818007i	$-0.304921\pi$
$863$	29165.7	1.15042	0.575209	+	0.818007i	$0.304921\pi$
$864$	0	0
$865$	5331.89	0.209583
$866$	0	0
$867$	5896.15	0.230962
$868$	0	0
$869$	0	0
$870$	0	0
$871$	5227.86	0.203375
$872$	0	0
$873$	−37395.9	−1.44978
$874$	0	0
$875$	34575.1	1.33583
$876$	0	0
$877$	22564.5	0.868812	0.434406	−	0.900717i	$-0.356958\pi$
$877$	22564.5	0.868812	0.434406	+	0.900717i	$0.356958\pi$
$878$	0	0
$879$	−3520.19	−0.135077
$880$	0	0
$881$	1485.31	0.0568008	0.0284004	−	0.999597i	$-0.490959\pi$
$881$	1485.31	0.0568008	0.0284004	+	0.999597i	$0.490959\pi$
$882$	0	0
$883$	33378.6	1.27212	0.636058	−	0.771641i	$-0.280564\pi$
$883$	33378.6	1.27212	0.636058	+	0.771641i	$0.280564\pi$
$884$	0	0
$885$	1501.65	0.0570366
$886$	0	0
$887$	−51592.0	−1.95297	−0.976487	−	0.215575i	$-0.930838\pi$
$887$	−51592.0	−1.95297	−0.976487	+	0.215575i	$0.930838\pi$
$888$	0	0
$889$	−5265.83	−0.198662
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−11582.4	−0.434030
$894$	0	0
$895$	5631.12	0.210310
$896$	0	0
$897$	8600.03	0.320119
$898$	0	0
$899$	4454.87	0.165270
$900$	0	0
$901$	−1460.79	−0.0540132
$902$	0	0
$903$	−2763.76	−0.101852
$904$	0	0
$905$	9619.42	0.353326
$906$	0	0
$907$	14723.5	0.539013	0.269506	−	0.962999i	$-0.413139\pi$
$907$	14723.5	0.539013	0.269506	+	0.962999i	$0.413139\pi$
$908$	0	0
$909$	18991.5	0.692967
$910$	0	0
$911$	1602.51	0.0582806	0.0291403	−	0.999575i	$-0.490723\pi$
$911$	1602.51	0.0582806	0.0291403	+	0.999575i	$0.490723\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	5075.50	0.183378
$916$	0	0
$917$	−42674.3	−1.53678
$918$	0	0
$919$	2334.19	0.0837844	0.0418922	−	0.999122i	$-0.486661\pi$
$919$	2334.19	0.0837844	0.0418922	+	0.999122i	$0.486661\pi$
$920$	0	0
$921$	−1750.00	−0.0626106
$922$	0	0
$923$	25399.0	0.905763
$924$	0	0
$925$	−6262.73	−0.222613
$926$	0	0
$927$	−7087.54	−0.251117
$928$	0	0
$929$	9082.59	0.320765	0.160382	−	0.987055i	$-0.448727\pi$
$929$	9082.59	0.320765	0.160382	+	0.987055i	$0.448727\pi$
$930$	0	0
$931$	−10648.7	−0.374861
$932$	0	0
$933$	−2936.35	−0.103035
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−16688.9	−0.581861	−0.290931	−	0.956744i	$-0.593965\pi$
$937$	−16688.9	−0.581861	−0.290931	+	0.956744i	$0.593965\pi$
$938$	0	0
$939$	−6816.61	−0.236903
$940$	0	0
$941$	22357.1	0.774518	0.387259	−	0.921971i	$-0.373422\pi$
$941$	22357.1	0.774518	0.387259	+	0.921971i	$0.373422\pi$
$942$	0	0
$943$	−33323.5	−1.15076
$944$	0	0
$945$	8625.65	0.296923
$946$	0	0
$947$	−39734.1	−1.36345	−0.681724	−	0.731610i	$-0.738769\pi$
$947$	−39734.1	−1.36345	−0.681724	+	0.731610i	$0.738769\pi$
$948$	0	0
$949$	−33553.3	−1.14772
$950$	0	0
$951$	−10400.4	−0.354633
$952$	0	0
$953$	16514.8	0.561349	0.280674	−	0.959803i	$-0.409442\pi$
$953$	16514.8	0.561349	0.280674	+	0.959803i	$0.409442\pi$
$954$	0	0
$955$	15215.8	0.515574
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−37443.2	−1.26080
$960$	0	0
$961$	81492.9	2.73549
$962$	0	0
$963$	47968.1	1.60514
$964$	0	0
$965$	19644.0	0.655297
$966$	0	0
$967$	−8015.15	−0.266546	−0.133273	−	0.991079i	$-0.542549\pi$
$967$	−8015.15	−0.266546	−0.133273	+	0.991079i	$0.542549\pi$
$968$	0	0
$969$	2995.87	0.0993200
$970$	0	0
$971$	−2803.91	−0.0926691	−0.0463345	−	0.998926i	$-0.514754\pi$
$971$	−2803.91	−0.0926691	−0.0463345	+	0.998926i	$0.514754\pi$
$972$	0	0
$973$	2127.90	0.0701104
$974$	0	0
$975$	−7262.98	−0.238566
$976$	0	0
$977$	−13053.3	−0.427443	−0.213722	−	0.976895i	$-0.568559\pi$
$977$	−13053.3	−0.427443	−0.213722	+	0.976895i	$0.568559\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	38786.9	1.26235
$982$	0	0
$983$	−18628.5	−0.604431	−0.302216	−	0.953240i	$-0.597726\pi$
$983$	−18628.5	−0.604431	−0.302216	+	0.953240i	$0.597726\pi$
$984$	0	0
$985$	−13742.4	−0.444536
$986$	0	0
$987$	−10795.4	−0.348148
$988$	0	0
$989$	10759.8	0.345949
$990$	0	0
$991$	35006.9	1.12213	0.561065	−	0.827772i	$-0.310392\pi$
$991$	35006.9	1.12213	0.561065	+	0.827772i	$0.310392\pi$
$992$	0	0
$993$	−3995.24	−0.127679
$994$	0	0
$995$	2965.83	0.0944957
$996$	0	0
$997$	39681.8	1.26052	0.630258	−	0.776385i	$-0.282949\pi$
$997$	39681.8	1.26052	0.630258	+	0.776385i	$0.282949\pi$
$998$	0	0
$999$	−3779.18	−0.119688

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	968.4.a.r.1.6		10
4.3	odd	2		1936.4.a.by.1.5		10
11.2	odd	10		88.4.i.b.81.3	yes	20
11.6	odd	10		88.4.i.b.25.3	✓	20
11.10	odd	2		968.4.a.s.1.6		10
44.35	even	10		176.4.m.f.81.3		20
44.39	even	10		176.4.m.f.113.3		20
44.43	even	2		1936.4.a.bx.1.5		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
88.4.i.b.25.3	✓	20	11.6	odd	10
88.4.i.b.81.3	yes	20	11.2	odd	10
176.4.m.f.81.3		20	44.35	even	10
176.4.m.f.113.3		20	44.39	even	10
968.4.a.r.1.6		10	1.1	even	1	trivial
968.4.a.s.1.6		10	11.10	odd	2
1936.4.a.bx.1.5		10	44.43	even	2
1936.4.a.by.1.5		10	4.3	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:	\( N \)	\(=\)	\( 968 = 2^{3} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	968.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00320 q^{3} +6.07450 q^{5} -26.7097 q^{7} -25.9936 q^{9} +O(q^{10})\)	\(q+1.00320 q^{3} +6.07450 q^{5} -26.7097 q^{7} -25.9936 q^{9} +82.1765 q^{13} +6.09397 q^{15} -103.876 q^{17} -28.7485 q^{19} -26.7953 q^{21} +104.319 q^{23} -88.1005 q^{25} -53.1634 q^{27} +13.3542 q^{29} +333.592 q^{31} -162.248 q^{35} +71.0862 q^{37} +82.4398 q^{39} -319.439 q^{41} +103.144 q^{43} -157.898 q^{45} +402.885 q^{47} +370.407 q^{49} -104.209 q^{51} +14.0627 q^{53} -28.8407 q^{57} +246.416 q^{59} +832.874 q^{61} +694.280 q^{63} +499.181 q^{65} +63.6175 q^{67} +104.653 q^{69} +309.079 q^{71} -408.308 q^{73} -88.3828 q^{75} +855.680 q^{79} +648.493 q^{81} +524.312 q^{83} -630.998 q^{85} +13.3970 q^{87} -392.146 q^{89} -2194.91 q^{91} +334.661 q^{93} -174.633 q^{95} +1438.66 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 9 q^{3} + 13 q^{5} - 3 q^{7} + 141 q^{9}+O(q^{10}) \) 10 * q + 9 * q^3 + 13 * q^5 - 3 * q^7 + 141 * q^9	\( 10 q + 9 q^{3} + 13 q^{5} - 3 q^{7} + 141 q^{9} - 45 q^{13} + 120 q^{15} + 17 q^{17} + 147 q^{19} - 131 q^{21} + 164 q^{23} + 439 q^{25} + 420 q^{27} - 177 q^{29} + 275 q^{31} + 220 q^{35} + 745 q^{37} + 524 q^{39} - 967 q^{41} + 380 q^{43} - 44 q^{45} + 769 q^{47} + 503 q^{49} + 956 q^{51} + 701 q^{53} - 1293 q^{57} + 1291 q^{59} + 1359 q^{61} - 929 q^{63} - 173 q^{65} + 2260 q^{67} + 1988 q^{69} + 465 q^{71} - 111 q^{73} + 4584 q^{75} - 1827 q^{79} + 6874 q^{81} + 4947 q^{83} - 2609 q^{85} - 1303 q^{87} + 446 q^{89} + 2176 q^{91} + 4204 q^{93} - 108 q^{95} + 3511 q^{97}+O(q^{100}) \) 10 * q + 9 * q^3 + 13 * q^5 - 3 * q^7 + 141 * q^9 - 45 * q^13 + 120 * q^15 + 17 * q^17 + 147 * q^19 - 131 * q^21 + 164 * q^23 + 439 * q^25 + 420 * q^27 - 177 * q^29 + 275 * q^31 + 220 * q^35 + 745 * q^37 + 524 * q^39 - 967 * q^41 + 380 * q^43 - 44 * q^45 + 769 * q^47 + 503 * q^49 + 956 * q^51 + 701 * q^53 - 1293 * q^57 + 1291 * q^59 + 1359 * q^61 - 929 * q^63 - 173 * q^65 + 2260 * q^67 + 1988 * q^69 + 465 * q^71 - 111 * q^73 + 4584 * q^75 - 1827 * q^79 + 6874 * q^81 + 4947 * q^83 - 2609 * q^85 - 1303 * q^87 + 446 * q^89 + 2176 * q^91 + 4204 * q^93 - 108 * q^95 + 3511 * q^97

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