LMFDB - Embedded newform 1680.4.a.bo.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1680,4,Mod(1,1680)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1680, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("1680.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1680.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:	$99.1232088096$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 105)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ of defining polynomial
Character	$\chi$	$=$	1680.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.00000 q^{3} +5.00000 q^{5} +7.00000 q^{7} +9.00000 q^{9} +O(q^{10})$	$q+3.00000 q^{3} +5.00000 q^{5} +7.00000 q^{7} +9.00000 q^{9} +64.5685 q^{11} -32.3431 q^{13} +15.0000 q^{15} -56.3431 q^{17} +2.74517 q^{19} +21.0000 q^{21} -88.1665 q^{23} +25.0000 q^{25} +27.0000 q^{27} +246.735 q^{29} +110.912 q^{31} +193.706 q^{33} +35.0000 q^{35} +120.676 q^{37} -97.0294 q^{39} -176.274 q^{41} +443.362 q^{43} +45.0000 q^{45} +345.206 q^{47} +49.0000 q^{49} -169.029 q^{51} +260.981 q^{53} +322.843 q^{55} +8.23550 q^{57} -628.999 q^{59} -115.206 q^{61} +63.0000 q^{63} -161.716 q^{65} +951.480 q^{67} -264.500 q^{69} -356.264 q^{71} -656.754 q^{73} +75.0000 q^{75} +451.980 q^{77} -440.195 q^{79} +81.0000 q^{81} +54.4121 q^{83} -281.716 q^{85} +740.205 q^{87} -1018.78 q^{89} -226.402 q^{91} +332.735 q^{93} +13.7258 q^{95} -724.108 q^{97} +581.117 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 6 q^{3} + 10 q^{5} + 14 q^{7} + 18 q^{9}+O(q^{10}) $ 2 * q + 6 * q^3 + 10 * q^5 + 14 * q^7 + 18 * q^9	$ 2 q + 6 q^{3} + 10 q^{5} + 14 q^{7} + 18 q^{9} + 16 q^{11} - 76 q^{13} + 30 q^{15} - 124 q^{17} + 96 q^{19} + 42 q^{21} + 16 q^{23} + 50 q^{25} + 54 q^{27} + 188 q^{29} + 120 q^{31} + 48 q^{33} + 70 q^{35} - 132 q^{37} - 228 q^{39} + 100 q^{41} + 536 q^{43} + 90 q^{45} + 928 q^{47} + 98 q^{49} - 372 q^{51} + 884 q^{53} + 80 q^{55} + 288 q^{57} - 104 q^{59} - 468 q^{61} + 126 q^{63} - 380 q^{65} + 1688 q^{67} + 48 q^{69} + 136 q^{71} + 508 q^{73} + 150 q^{75} + 112 q^{77} + 432 q^{79} + 162 q^{81} + 584 q^{83} - 620 q^{85} + 564 q^{87} - 1404 q^{89} - 532 q^{91} + 360 q^{93} + 480 q^{95} - 1188 q^{97} + 144 q^{99}+O(q^{100}) $ 2 * q + 6 * q^3 + 10 * q^5 + 14 * q^7 + 18 * q^9 + 16 * q^11 - 76 * q^13 + 30 * q^15 - 124 * q^17 + 96 * q^19 + 42 * q^21 + 16 * q^23 + 50 * q^25 + 54 * q^27 + 188 * q^29 + 120 * q^31 + 48 * q^33 + 70 * q^35 - 132 * q^37 - 228 * q^39 + 100 * q^41 + 536 * q^43 + 90 * q^45 + 928 * q^47 + 98 * q^49 - 372 * q^51 + 884 * q^53 + 80 * q^55 + 288 * q^57 - 104 * q^59 - 468 * q^61 + 126 * q^63 - 380 * q^65 + 1688 * q^67 + 48 * q^69 + 136 * q^71 + 508 * q^73 + 150 * q^75 + 112 * q^77 + 432 * q^79 + 162 * q^81 + 584 * q^83 - 620 * q^85 + 564 * q^87 - 1404 * q^89 - 532 * q^91 + 360 * q^93 + 480 * q^95 - 1188 * q^97 + 144 * 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
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	0	0
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	0	0
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	64.5685	1.76983	0.884916	−	0.465751i	$-0.154216\pi$
$11$	64.5685	1.76983	0.884916	+	0.465751i	$0.154216\pi$
$12$	0	0
$13$	−32.3431	−0.690029	−0.345014	−	0.938597i	$-0.612126\pi$
$13$	−32.3431	−0.690029	−0.345014	+	0.938597i	$0.612126\pi$
$14$	0	0
$15$	15.0000	0.258199
$16$	0	0
$17$	−56.3431	−0.803836	−0.401918	−	0.915676i	$-0.631656\pi$
$17$	−56.3431	−0.803836	−0.401918	+	0.915676i	$0.631656\pi$
$18$	0	0
$19$	2.74517	0.0331465	0.0165733	−	0.999863i	$-0.494724\pi$
$19$	2.74517	0.0331465	0.0165733	+	0.999863i	$0.494724\pi$
$20$	0	0
$21$	21.0000	0.218218
$22$	0	0
$23$	−88.1665	−0.799304	−0.399652	−	0.916667i	$-0.630869\pi$
$23$	−88.1665	−0.799304	−0.399652	+	0.916667i	$0.630869\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	27.0000	0.192450
$28$	0	0
$29$	246.735	1.57992	0.789958	−	0.613161i	$-0.210102\pi$
$29$	246.735	1.57992	0.789958	+	0.613161i	$0.210102\pi$
$30$	0	0
$31$	110.912	0.642591	0.321296	−	0.946979i	$-0.395882\pi$
$31$	110.912	0.642591	0.321296	+	0.946979i	$0.395882\pi$
$32$	0	0
$33$	193.706	1.02181
$34$	0	0
$35$	35.0000	0.169031
$36$	0	0
$37$	120.676	0.536190	0.268095	−	0.963392i	$-0.413606\pi$
$37$	120.676	0.536190	0.268095	+	0.963392i	$0.413606\pi$
$38$	0	0
$39$	−97.0294	−0.398388
$40$	0	0
$41$	−176.274	−0.671449	−0.335724	−	0.941960i	$-0.608981\pi$
$41$	−176.274	−0.671449	−0.335724	+	0.941960i	$0.608981\pi$
$42$	0	0
$43$	443.362	1.57238	0.786188	−	0.617988i	$-0.212052\pi$
$43$	443.362	1.57238	0.786188	+	0.617988i	$0.212052\pi$
$44$	0	0
$45$	45.0000	0.149071
$46$	0	0
$47$	345.206	1.07135	0.535675	−	0.844424i	$-0.320057\pi$
$47$	345.206	1.07135	0.535675	+	0.844424i	$0.320057\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	−169.029	−0.464095
$52$	0	0
$53$	260.981	0.676386	0.338193	−	0.941077i	$-0.390184\pi$
$53$	260.981	0.676386	0.338193	+	0.941077i	$0.390184\pi$
$54$	0	0
$55$	322.843	0.791493
$56$	0	0
$57$	8.23550	0.0191372
$58$	0	0
$59$	−628.999	−1.38794	−0.693972	−	0.720002i	$-0.744141\pi$
$59$	−628.999	−1.38794	−0.693972	+	0.720002i	$0.744141\pi$
$60$	0	0
$61$	−115.206	−0.241814	−0.120907	−	0.992664i	$-0.538580\pi$
$61$	−115.206	−0.241814	−0.120907	+	0.992664i	$0.538580\pi$
$62$	0	0
$63$	63.0000	0.125988
$64$	0	0
$65$	−161.716	−0.308590
$66$	0	0
$67$	951.480	1.73495	0.867476	−	0.497479i	$-0.165741\pi$
$67$	951.480	1.73495	0.867476	+	0.497479i	$0.165741\pi$
$68$	0	0
$69$	−264.500	−0.461478
$70$	0	0
$71$	−356.264	−0.595504	−0.297752	−	0.954643i	$-0.596237\pi$
$71$	−356.264	−0.595504	−0.297752	+	0.954643i	$0.596237\pi$
$72$	0	0
$73$	−656.754	−1.05298	−0.526488	−	0.850183i	$-0.676491\pi$
$73$	−656.754	−1.05298	−0.526488	+	0.850183i	$0.676491\pi$
$74$	0	0
$75$	75.0000	0.115470
$76$	0	0
$77$	451.980	0.668933
$78$	0	0
$79$	−440.195	−0.626909	−0.313455	−	0.949603i	$-0.601486\pi$
$79$	−440.195	−0.626909	−0.313455	+	0.949603i	$0.601486\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	54.4121	0.0719579	0.0359790	−	0.999353i	$-0.488545\pi$
$83$	54.4121	0.0719579	0.0359790	+	0.999353i	$0.488545\pi$
$84$	0	0
$85$	−281.716	−0.359487
$86$	0	0
$87$	740.205	0.912165
$88$	0	0
$89$	−1018.78	−1.21338	−0.606690	−	0.794938i	$-0.707503\pi$
$89$	−1018.78	−1.21338	−0.606690	+	0.794938i	$0.707503\pi$
$90$	0	0
$91$	−226.402	−0.260806
$92$	0	0
$93$	332.735	0.371000
$94$	0	0
$95$	13.7258	0.0148236
$96$	0	0
$97$	−724.108	−0.757959	−0.378979	−	0.925405i	$-0.623725\pi$
$97$	−724.108	−0.757959	−0.378979	+	0.925405i	$0.623725\pi$
$98$	0	0
$99$	581.117	0.589944
$100$	0	0
$101$	268.725	0.264744	0.132372	−	0.991200i	$-0.457741\pi$
$101$	268.725	0.264744	0.132372	+	0.991200i	$0.457741\pi$
$102$	0	0
$103$	1840.63	1.76080	0.880399	−	0.474233i	$-0.157275\pi$
$103$	1840.63	1.76080	0.880399	+	0.474233i	$0.157275\pi$
$104$	0	0
$105$	105.000	0.0975900
$106$	0	0
$107$	243.087	0.219627	0.109813	−	0.993952i	$-0.464975\pi$
$107$	243.087	0.219627	0.109813	+	0.993952i	$0.464975\pi$
$108$	0	0
$109$	−405.176	−0.356044	−0.178022	−	0.984027i	$-0.556970\pi$
$109$	−405.176	−0.356044	−0.178022	+	0.984027i	$0.556970\pi$
$110$	0	0
$111$	362.029	0.309570
$112$	0	0
$113$	−28.1766	−0.0234569	−0.0117285	−	0.999931i	$-0.503733\pi$
$113$	−28.1766	−0.0234569	−0.0117285	+	0.999931i	$0.503733\pi$
$114$	0	0
$115$	−440.833	−0.357460
$116$	0	0
$117$	−291.088	−0.230010
$118$	0	0
$119$	−394.402	−0.303822
$120$	0	0
$121$	2838.10	2.13230
$122$	0	0
$123$	−528.823	−0.387661
$124$	0	0
$125$	125.000	0.0894427
$126$	0	0
$127$	2740.90	1.91508	0.957541	−	0.288298i	$-0.0930892\pi$
$127$	2740.90	1.91508	0.957541	+	0.288298i	$0.0930892\pi$
$128$	0	0
$129$	1330.09	0.907811
$130$	0	0
$131$	1832.04	1.22188	0.610938	−	0.791678i	$-0.290792\pi$
$131$	1832.04	1.22188	0.610938	+	0.791678i	$0.290792\pi$
$132$	0	0
$133$	19.2162	0.0125282
$134$	0	0
$135$	135.000	0.0860663
$136$	0	0
$137$	382.747	0.238688	0.119344	−	0.992853i	$-0.461921\pi$
$137$	382.747	0.238688	0.119344	+	0.992853i	$0.461921\pi$
$138$	0	0
$139$	−3053.60	−1.86333	−0.931667	−	0.363314i	$-0.881645\pi$
$139$	−3053.60	−1.86333	−0.931667	+	0.363314i	$0.881645\pi$
$140$	0	0
$141$	1035.62	0.618545
$142$	0	0
$143$	−2088.35	−1.22123
$144$	0	0
$145$	1233.68	0.706560
$146$	0	0
$147$	147.000	0.0824786
$148$	0	0
$149$	3560.60	1.95769	0.978843	−	0.204611i	$-0.0655929\pi$
$149$	3560.60	1.95769	0.978843	+	0.204611i	$0.0655929\pi$
$150$	0	0
$151$	−3261.80	−1.75789	−0.878945	−	0.476923i	$-0.841752\pi$
$151$	−3261.80	−1.75789	−0.878945	+	0.476923i	$0.841752\pi$
$152$	0	0
$153$	−507.088	−0.267945
$154$	0	0
$155$	554.558	0.287376
$156$	0	0
$157$	2878.46	1.46322	0.731611	−	0.681723i	$-0.238769\pi$
$157$	2878.46	1.46322	0.731611	+	0.681723i	$0.238769\pi$
$158$	0	0
$159$	782.942	0.390512
$160$	0	0
$161$	−617.166	−0.302108
$162$	0	0
$163$	927.537	0.445708	0.222854	−	0.974852i	$-0.428463\pi$
$163$	927.537	0.445708	0.222854	+	0.974852i	$0.428463\pi$
$164$	0	0
$165$	968.528	0.456969
$166$	0	0
$167$	−1094.52	−0.507164	−0.253582	−	0.967314i	$-0.581609\pi$
$167$	−1094.52	−0.507164	−0.253582	+	0.967314i	$0.581609\pi$
$168$	0	0
$169$	−1150.92	−0.523860
$170$	0	0
$171$	24.7065	0.0110488
$172$	0	0
$173$	−1713.25	−0.752926	−0.376463	−	0.926432i	$-0.622860\pi$
$173$	−1713.25	−0.752926	−0.376463	+	0.926432i	$0.622860\pi$
$174$	0	0
$175$	175.000	0.0755929
$176$	0	0
$177$	−1887.00	−0.801330
$178$	0	0
$179$	4065.58	1.69763	0.848816	−	0.528689i	$-0.177316\pi$
$179$	4065.58	1.69763	0.848816	+	0.528689i	$0.177316\pi$
$180$	0	0
$181$	−2791.40	−1.14631	−0.573157	−	0.819445i	$-0.694282\pi$
$181$	−2791.40	−1.14631	−0.573157	+	0.819445i	$0.694282\pi$
$182$	0	0
$183$	−345.618	−0.139611
$184$	0	0
$185$	603.381	0.239792
$186$	0	0
$187$	−3637.99	−1.42266
$188$	0	0
$189$	189.000	0.0727393
$190$	0	0
$191$	634.185	0.240251	0.120126	−	0.992759i	$-0.461670\pi$
$191$	634.185	0.240251	0.120126	+	0.992759i	$0.461670\pi$
$192$	0	0
$193$	−254.999	−0.0951049	−0.0475524	−	0.998869i	$-0.515142\pi$
$193$	−254.999	−0.0951049	−0.0475524	+	0.998869i	$0.515142\pi$
$194$	0	0
$195$	−485.147	−0.178165
$196$	0	0
$197$	4172.37	1.50898	0.754490	−	0.656311i	$-0.227884\pi$
$197$	4172.37	1.50898	0.754490	+	0.656311i	$0.227884\pi$
$198$	0	0
$199$	4626.48	1.64805	0.824026	−	0.566552i	$-0.191723\pi$
$199$	4626.48	1.64805	0.824026	+	0.566552i	$0.191723\pi$
$200$	0	0
$201$	2854.44	1.00168
$202$	0	0
$203$	1727.15	0.597152
$204$	0	0
$205$	−881.371	−0.300281
$206$	0	0
$207$	−793.499	−0.266435
$208$	0	0
$209$	177.251	0.0586638
$210$	0	0
$211$	1562.64	0.509843	0.254921	−	0.966962i	$-0.417950\pi$
$211$	1562.64	0.509843	0.254921	+	0.966962i	$0.417950\pi$
$212$	0	0
$213$	−1068.79	−0.343814
$214$	0	0
$215$	2216.81	0.703188
$216$	0	0
$217$	776.382	0.242877
$218$	0	0
$219$	−1970.26	−0.607935
$220$	0	0
$221$	1822.31	0.554670
$222$	0	0
$223$	1236.39	0.371278	0.185639	−	0.982618i	$-0.440564\pi$
$223$	1236.39	0.371278	0.185639	+	0.982618i	$0.440564\pi$
$224$	0	0
$225$	225.000	0.0666667
$226$	0	0
$227$	−4181.82	−1.22272	−0.611359	−	0.791353i	$-0.709377\pi$
$227$	−4181.82	−1.22272	−0.611359	+	0.791353i	$0.709377\pi$
$228$	0	0
$229$	−484.774	−0.139890	−0.0699449	−	0.997551i	$-0.522282\pi$
$229$	−484.774	−0.139890	−0.0699449	+	0.997551i	$0.522282\pi$
$230$	0	0
$231$	1355.94	0.386209
$232$	0	0
$233$	2080.54	0.584982	0.292491	−	0.956268i	$-0.405516\pi$
$233$	2080.54	0.584982	0.292491	+	0.956268i	$0.405516\pi$
$234$	0	0
$235$	1726.03	0.479123
$236$	0	0
$237$	−1320.59	−0.361946
$238$	0	0
$239$	−6814.10	−1.84422	−0.922108	−	0.386933i	$-0.873534\pi$
$239$	−6814.10	−1.84422	−0.922108	+	0.386933i	$0.873534\pi$
$240$	0	0
$241$	−3921.84	−1.04825	−0.524125	−	0.851642i	$-0.675607\pi$
$241$	−3921.84	−1.04825	−0.524125	+	0.851642i	$0.675607\pi$
$242$	0	0
$243$	243.000	0.0641500
$244$	0	0
$245$	245.000	0.0638877
$246$	0	0
$247$	−88.7873	−0.0228721
$248$	0	0
$249$	163.236	0.0415449
$250$	0	0
$251$	5219.10	1.31246	0.656228	−	0.754562i	$-0.272151\pi$
$251$	5219.10	1.31246	0.656228	+	0.754562i	$0.272151\pi$
$252$	0	0
$253$	−5692.78	−1.41463
$254$	0	0
$255$	−845.147	−0.207550
$256$	0	0
$257$	6975.71	1.69312	0.846562	−	0.532289i	$-0.178668\pi$
$257$	6975.71	1.69312	0.846562	+	0.532289i	$0.178668\pi$
$258$	0	0
$259$	844.733	0.202661
$260$	0	0
$261$	2220.62	0.526639
$262$	0	0
$263$	3607.36	0.845776	0.422888	−	0.906182i	$-0.361016\pi$
$263$	3607.36	0.845776	0.422888	+	0.906182i	$0.361016\pi$
$264$	0	0
$265$	1304.90	0.302489
$266$	0	0
$267$	−3056.35	−0.700546
$268$	0	0
$269$	5.88572	0.00133405	0.000667023	−	1.00000i	$-0.499788\pi$
$269$	5.88572	0.00133405	0.000667023		1.00000i	$0.499788\pi$
$270$	0	0
$271$	−6916.32	−1.55032	−0.775160	−	0.631765i	$-0.782331\pi$
$271$	−6916.32	−1.55032	−0.775160	+	0.631765i	$0.782331\pi$
$272$	0	0
$273$	−679.206	−0.150577
$274$	0	0
$275$	1614.21	0.353966
$276$	0	0
$277$	−2119.46	−0.459733	−0.229867	−	0.973222i	$-0.573829\pi$
$277$	−2119.46	−0.459733	−0.229867	+	0.973222i	$0.573829\pi$
$278$	0	0
$279$	998.205	0.214197
$280$	0	0
$281$	−239.917	−0.0509334	−0.0254667	−	0.999676i	$-0.508107\pi$
$281$	−239.917	−0.0509334	−0.0254667	+	0.999676i	$0.508107\pi$
$282$	0	0
$283$	4542.12	0.954067	0.477034	−	0.878885i	$-0.341712\pi$
$283$	4542.12	0.954067	0.477034	+	0.878885i	$0.341712\pi$
$284$	0	0
$285$	41.1775	0.00855840
$286$	0	0
$287$	−1233.92	−0.253784
$288$	0	0
$289$	−1738.45	−0.353847
$290$	0	0
$291$	−2172.32	−0.437608
$292$	0	0
$293$	−2171.70	−0.433010	−0.216505	−	0.976281i	$-0.569466\pi$
$293$	−2171.70	−0.433010	−0.216505	+	0.976281i	$0.569466\pi$
$294$	0	0
$295$	−3145.00	−0.620708
$296$	0	0
$297$	1743.35	0.340604
$298$	0	0
$299$	2851.58	0.551543
$300$	0	0
$301$	3103.54	0.594302
$302$	0	0
$303$	806.175	0.152850
$304$	0	0
$305$	−576.030	−0.108142
$306$	0	0
$307$	3508.64	0.652276	0.326138	−	0.945322i	$-0.394253\pi$
$307$	3508.64	0.652276	0.326138	+	0.945322i	$0.394253\pi$
$308$	0	0
$309$	5521.88	1.01660
$310$	0	0
$311$	3133.25	0.571287	0.285643	−	0.958336i	$-0.407793\pi$
$311$	3133.25	0.571287	0.285643	+	0.958336i	$0.407793\pi$
$312$	0	0
$313$	6389.59	1.15387	0.576935	−	0.816790i	$-0.304249\pi$
$313$	6389.59	1.15387	0.576935	+	0.816790i	$0.304249\pi$
$314$	0	0
$315$	315.000	0.0563436
$316$	0	0
$317$	1634.44	0.289587	0.144794	−	0.989462i	$-0.453748\pi$
$317$	1634.44	0.289587	0.144794	+	0.989462i	$0.453748\pi$
$318$	0	0
$319$	15931.3	2.79618
$320$	0	0
$321$	729.260	0.126802
$322$	0	0
$323$	−154.671	−0.0266444
$324$	0	0
$325$	−808.579	−0.138006
$326$	0	0
$327$	−1215.53	−0.205562
$328$	0	0
$329$	2416.44	0.404932
$330$	0	0
$331$	−4386.17	−0.728355	−0.364177	−	0.931330i	$-0.618650\pi$
$331$	−4386.17	−0.728355	−0.364177	+	0.931330i	$0.618650\pi$
$332$	0	0
$333$	1086.09	0.178730
$334$	0	0
$335$	4757.40	0.775894
$336$	0	0
$337$	1713.98	0.277051	0.138526	−	0.990359i	$-0.455764\pi$
$337$	1713.98	0.277051	0.138526	+	0.990359i	$0.455764\pi$
$338$	0	0
$339$	−84.5299	−0.0135429
$340$	0	0
$341$	7161.41	1.13728
$342$	0	0
$343$	343.000	0.0539949
$344$	0	0
$345$	−1322.50	−0.206379
$346$	0	0
$347$	1744.83	0.269935	0.134967	−	0.990850i	$-0.456907\pi$
$347$	1744.83	0.269935	0.134967	+	0.990850i	$0.456907\pi$
$348$	0	0
$349$	7046.78	1.08082	0.540409	−	0.841403i	$-0.318270\pi$
$349$	7046.78	1.08082	0.540409	+	0.841403i	$0.318270\pi$
$350$	0	0
$351$	−873.265	−0.132796
$352$	0	0
$353$	−12668.5	−1.91013	−0.955064	−	0.296400i	$-0.904214\pi$
$353$	−12668.5	−1.91013	−0.955064	+	0.296400i	$0.904214\pi$
$354$	0	0
$355$	−1781.32	−0.266317
$356$	0	0
$357$	−1183.21	−0.175411
$358$	0	0
$359$	−37.7844	−0.00555483	−0.00277742	−	0.999996i	$-0.500884\pi$
$359$	−37.7844	−0.00555483	−0.00277742	+	0.999996i	$0.500884\pi$
$360$	0	0
$361$	−6851.46	−0.998901
$362$	0	0
$363$	8514.29	1.23109
$364$	0	0
$365$	−3283.77	−0.470905
$366$	0	0
$367$	759.829	0.108073	0.0540364	−	0.998539i	$-0.482791\pi$
$367$	759.829	0.108073	0.0540364	+	0.998539i	$0.482791\pi$
$368$	0	0
$369$	−1586.47	−0.223816
$370$	0	0
$371$	1826.86	0.255650
$372$	0	0
$373$	719.320	0.0998525	0.0499263	−	0.998753i	$-0.484101\pi$
$373$	719.320	0.0998525	0.0499263	+	0.998753i	$0.484101\pi$
$374$	0	0
$375$	375.000	0.0516398
$376$	0	0
$377$	−7980.19	−1.09019
$378$	0	0
$379$	−572.559	−0.0775999	−0.0388000	−	0.999247i	$-0.512354\pi$
$379$	−572.559	−0.0775999	−0.0388000	+	0.999247i	$0.512354\pi$
$380$	0	0
$381$	8222.69	1.10567
$382$	0	0
$383$	−4513.18	−0.602122	−0.301061	−	0.953605i	$-0.597341\pi$
$383$	−4513.18	−0.602122	−0.301061	+	0.953605i	$0.597341\pi$
$384$	0	0
$385$	2259.90	0.299156
$386$	0	0
$387$	3990.26	0.524125
$388$	0	0
$389$	−6902.13	−0.899619	−0.449810	−	0.893124i	$-0.648508\pi$
$389$	−6902.13	−0.899619	−0.449810	+	0.893124i	$0.648508\pi$
$390$	0	0
$391$	4967.58	0.642510
$392$	0	0
$393$	5496.11	0.705451
$394$	0	0
$395$	−2200.98	−0.280362
$396$	0	0
$397$	4124.58	0.521427	0.260714	−	0.965416i	$-0.416042\pi$
$397$	4124.58	0.521427	0.260714	+	0.965416i	$0.416042\pi$
$398$	0	0
$399$	57.6485	0.00723317
$400$	0	0
$401$	−1002.50	−0.124844	−0.0624219	−	0.998050i	$-0.519882\pi$
$401$	−1002.50	−0.124844	−0.0624219	+	0.998050i	$0.519882\pi$
$402$	0	0
$403$	−3587.23	−0.443406
$404$	0	0
$405$	405.000	0.0496904
$406$	0	0
$407$	7791.89	0.948967
$408$	0	0
$409$	10335.0	1.24947	0.624736	−	0.780836i	$-0.285206\pi$
$409$	10335.0	1.24947	0.624736	+	0.780836i	$0.285206\pi$
$410$	0	0
$411$	1148.24	0.137807
$412$	0	0
$413$	−4402.99	−0.524594
$414$	0	0
$415$	272.061	0.0321806
$416$	0	0
$417$	−9160.81	−1.07580
$418$	0	0
$419$	−3183.21	−0.371145	−0.185573	−	0.982631i	$-0.559414\pi$
$419$	−3183.21	−0.371145	−0.185573	+	0.982631i	$0.559414\pi$
$420$	0	0
$421$	−6944.34	−0.803911	−0.401956	−	0.915659i	$-0.631669\pi$
$421$	−6944.34	−0.803911	−0.401956	+	0.915659i	$0.631669\pi$
$422$	0	0
$423$	3106.85	0.357117
$424$	0	0
$425$	−1408.58	−0.160767
$426$	0	0
$427$	−806.442	−0.0913969
$428$	0	0
$429$	−6265.05	−0.705080
$430$	0	0
$431$	−3868.41	−0.432331	−0.216166	−	0.976357i	$-0.569355\pi$
$431$	−3868.41	−0.432331	−0.216166	+	0.976357i	$0.569355\pi$
$432$	0	0
$433$	−6132.96	−0.680673	−0.340336	−	0.940304i	$-0.610541\pi$
$433$	−6132.96	−0.680673	−0.340336	+	0.940304i	$0.610541\pi$
$434$	0	0
$435$	3701.03	0.407932
$436$	0	0
$437$	−242.032	−0.0264942
$438$	0	0
$439$	4090.14	0.444673	0.222337	−	0.974970i	$-0.428632\pi$
$439$	4090.14	0.444673	0.222337	+	0.974970i	$0.428632\pi$
$440$	0	0
$441$	441.000	0.0476190
$442$	0	0
$443$	−12434.5	−1.33359	−0.666795	−	0.745241i	$-0.732334\pi$
$443$	−12434.5	−1.33359	−0.666795	+	0.745241i	$0.732334\pi$
$444$	0	0
$445$	−5093.92	−0.542640
$446$	0	0
$447$	10681.8	1.13027
$448$	0	0
$449$	883.046	0.0928141	0.0464071	−	0.998923i	$-0.485223\pi$
$449$	883.046	0.0928141	0.0464071	+	0.998923i	$0.485223\pi$
$450$	0	0
$451$	−11381.8	−1.18835
$452$	0	0
$453$	−9785.40	−1.01492
$454$	0	0
$455$	−1132.01	−0.116636
$456$	0	0
$457$	−9068.44	−0.928235	−0.464118	−	0.885774i	$-0.653629\pi$
$457$	−9068.44	−0.928235	−0.464118	+	0.885774i	$0.653629\pi$
$458$	0	0
$459$	−1521.26	−0.154698
$460$	0	0
$461$	12508.9	1.26377	0.631885	−	0.775063i	$-0.282282\pi$
$461$	12508.9	1.26377	0.631885	+	0.775063i	$0.282282\pi$
$462$	0	0
$463$	−12688.7	−1.27363	−0.636817	−	0.771015i	$-0.719749\pi$
$463$	−12688.7	−1.27363	−0.636817	+	0.771015i	$0.719749\pi$
$464$	0	0
$465$	1663.68	0.165916
$466$	0	0
$467$	10136.5	1.00442	0.502208	−	0.864747i	$-0.332521\pi$
$467$	10136.5	1.00442	0.502208	+	0.864747i	$0.332521\pi$
$468$	0	0
$469$	6660.36	0.655750
$470$	0	0
$471$	8635.37	0.844791
$472$	0	0
$473$	28627.3	2.78284
$474$	0	0
$475$	68.6292	0.00662931
$476$	0	0
$477$	2348.83	0.225462
$478$	0	0
$479$	11361.1	1.08372	0.541861	−	0.840468i	$-0.317720\pi$
$479$	11361.1	1.08372	0.541861	+	0.840468i	$0.317720\pi$
$480$	0	0
$481$	−3903.05	−0.369987
$482$	0	0
$483$	−1851.50	−0.174422
$484$	0	0
$485$	−3620.54	−0.338969
$486$	0	0
$487$	−7929.53	−0.737826	−0.368913	−	0.929464i	$-0.620270\pi$
$487$	−7929.53	−0.737826	−0.368913	+	0.929464i	$0.620270\pi$
$488$	0	0
$489$	2782.61	0.257329
$490$	0	0
$491$	−8111.51	−0.745555	−0.372777	−	0.927921i	$-0.621595\pi$
$491$	−8111.51	−0.745555	−0.372777	+	0.927921i	$0.621595\pi$
$492$	0	0
$493$	−13901.8	−1.26999
$494$	0	0
$495$	2905.58	0.263831
$496$	0	0
$497$	−2493.85	−0.225079
$498$	0	0
$499$	−16816.6	−1.50865	−0.754324	−	0.656502i	$-0.772035\pi$
$499$	−16816.6	−1.50865	−0.754324	+	0.656502i	$0.772035\pi$
$500$	0	0
$501$	−3283.55	−0.292811
$502$	0	0
$503$	17764.6	1.57472	0.787362	−	0.616491i	$-0.211446\pi$
$503$	17764.6	1.57472	0.787362	+	0.616491i	$0.211446\pi$
$504$	0	0
$505$	1343.62	0.118397
$506$	0	0
$507$	−3452.76	−0.302451
$508$	0	0
$509$	13908.8	1.21120	0.605598	−	0.795771i	$-0.292934\pi$
$509$	13908.8	1.21120	0.605598	+	0.795771i	$0.292934\pi$
$510$	0	0
$511$	−4597.27	−0.397987
$512$	0	0
$513$	74.1195	0.00637905
$514$	0	0
$515$	9203.13	0.787453
$516$	0	0
$517$	22289.5	1.89611
$518$	0	0
$519$	−5139.76	−0.434702
$520$	0	0
$521$	−8639.68	−0.726510	−0.363255	−	0.931690i	$-0.618335\pi$
$521$	−8639.68	−0.726510	−0.363255	+	0.931690i	$0.618335\pi$
$522$	0	0
$523$	−23242.2	−1.94323	−0.971617	−	0.236561i	$-0.923980\pi$
$523$	−23242.2	−1.94323	−0.971617	+	0.236561i	$0.923980\pi$
$524$	0	0
$525$	525.000	0.0436436
$526$	0	0
$527$	−6249.11	−0.516538
$528$	0	0
$529$	−4393.66	−0.361113
$530$	0	0
$531$	−5660.99	−0.462648
$532$	0	0
$533$	5701.26	0.463319
$534$	0	0
$535$	1215.43	0.0982201
$536$	0	0
$537$	12196.8	0.980128
$538$	0	0
$539$	3163.86	0.252833
$540$	0	0
$541$	11395.2	0.905577	0.452789	−	0.891618i	$-0.350429\pi$
$541$	11395.2	0.905577	0.452789	+	0.891618i	$0.350429\pi$
$542$	0	0
$543$	−8374.19	−0.661825
$544$	0	0
$545$	−2025.88	−0.159228
$546$	0	0
$547$	7870.21	0.615184	0.307592	−	0.951518i	$-0.400477\pi$
$547$	7870.21	0.615184	0.307592	+	0.951518i	$0.400477\pi$
$548$	0	0
$549$	−1036.85	−0.0806045
$550$	0	0
$551$	677.329	0.0523687
$552$	0	0
$553$	−3081.37	−0.236949
$554$	0	0
$555$	1810.14	0.138444
$556$	0	0
$557$	17769.8	1.35176	0.675880	−	0.737012i	$-0.263764\pi$
$557$	17769.8	1.35176	0.675880	+	0.737012i	$0.263764\pi$
$558$	0	0
$559$	−14339.7	−1.08498
$560$	0	0
$561$	−10914.0	−0.821370
$562$	0	0
$563$	−15192.8	−1.13730	−0.568651	−	0.822579i	$-0.692534\pi$
$563$	−15192.8	−1.13730	−0.568651	+	0.822579i	$0.692534\pi$
$564$	0	0
$565$	−140.883	−0.0104903
$566$	0	0
$567$	567.000	0.0419961
$568$	0	0
$569$	−23300.0	−1.71667	−0.858335	−	0.513090i	$-0.828501\pi$
$569$	−23300.0	−1.71667	−0.858335	+	0.513090i	$0.828501\pi$
$570$	0	0
$571$	−10638.2	−0.779673	−0.389837	−	0.920884i	$-0.627469\pi$
$571$	−10638.2	−0.779673	−0.389837	+	0.920884i	$0.627469\pi$
$572$	0	0
$573$	1902.55	0.138709
$574$	0	0
$575$	−2204.16	−0.159861
$576$	0	0
$577$	−897.258	−0.0647372	−0.0323686	−	0.999476i	$-0.510305\pi$
$577$	−897.258	−0.0647372	−0.0323686	+	0.999476i	$0.510305\pi$
$578$	0	0
$579$	−764.997	−0.0549088
$580$	0	0
$581$	380.885	0.0271975
$582$	0	0
$583$	16851.1	1.19709
$584$	0	0
$585$	−1455.44	−0.102863
$586$	0	0
$587$	14712.9	1.03452	0.517261	−	0.855828i	$-0.326952\pi$
$587$	14712.9	1.03452	0.517261	+	0.855828i	$0.326952\pi$
$588$	0	0
$589$	304.471	0.0212997
$590$	0	0
$591$	12517.1	0.871210
$592$	0	0
$593$	−7216.29	−0.499726	−0.249863	−	0.968281i	$-0.580386\pi$
$593$	−7216.29	−0.499726	−0.249863	+	0.968281i	$0.580386\pi$
$594$	0	0
$595$	−1972.01	−0.135873
$596$	0	0
$597$	13879.4	0.951503
$598$	0	0
$599$	20885.5	1.42464	0.712320	−	0.701855i	$-0.247645\pi$
$599$	20885.5	1.42464	0.712320	+	0.701855i	$0.247645\pi$
$600$	0	0
$601$	−11047.7	−0.749823	−0.374911	−	0.927061i	$-0.622327\pi$
$601$	−11047.7	−0.749823	−0.374911	+	0.927061i	$0.622327\pi$
$602$	0	0
$603$	8563.32	0.578317
$604$	0	0
$605$	14190.5	0.953595
$606$	0	0
$607$	9434.94	0.630894	0.315447	−	0.948943i	$-0.397846\pi$
$607$	9434.94	0.630894	0.315447	+	0.948943i	$0.397846\pi$
$608$	0	0
$609$	5181.44	0.344766
$610$	0	0
$611$	−11165.0	−0.739263
$612$	0	0
$613$	−17662.6	−1.16376	−0.581881	−	0.813274i	$-0.697683\pi$
$613$	−17662.6	−1.16376	−0.581881	+	0.813274i	$0.697683\pi$
$614$	0	0
$615$	−2644.11	−0.173367
$616$	0	0
$617$	−10817.4	−0.705820	−0.352910	−	0.935657i	$-0.614808\pi$
$617$	−10817.4	−0.705820	−0.352910	+	0.935657i	$0.614808\pi$
$618$	0	0
$619$	−29073.9	−1.88785	−0.943926	−	0.330158i	$-0.892898\pi$
$619$	−29073.9	−1.88785	−0.943926	+	0.330158i	$0.892898\pi$
$620$	0	0
$621$	−2380.50	−0.153826
$622$	0	0
$623$	−7131.49	−0.458615
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	531.754	0.0338696
$628$	0	0
$629$	−6799.28	−0.431009
$630$	0	0
$631$	2203.17	0.138996	0.0694981	−	0.997582i	$-0.477860\pi$
$631$	2203.17	0.138996	0.0694981	+	0.997582i	$0.477860\pi$
$632$	0	0
$633$	4687.93	0.294358
$634$	0	0
$635$	13704.5	0.856451
$636$	0	0
$637$	−1584.81	−0.0985755
$638$	0	0
$639$	−3206.38	−0.198501
$640$	0	0
$641$	22466.5	1.38436	0.692180	−	0.721725i	$-0.256651\pi$
$641$	22466.5	1.38436	0.692180	+	0.721725i	$0.256651\pi$
$642$	0	0
$643$	12347.0	0.757257	0.378629	−	0.925549i	$-0.376396\pi$
$643$	12347.0	0.757257	0.378629	+	0.925549i	$0.376396\pi$
$644$	0	0
$645$	6650.44	0.405986
$646$	0	0
$647$	24114.0	1.46525	0.732626	−	0.680631i	$-0.238294\pi$
$647$	24114.0	1.46525	0.732626	+	0.680631i	$0.238294\pi$
$648$	0	0
$649$	−40613.6	−2.45643
$650$	0	0
$651$	2329.15	0.140225
$652$	0	0
$653$	7843.33	0.470035	0.235018	−	0.971991i	$-0.424485\pi$
$653$	7843.33	0.470035	0.235018	+	0.971991i	$0.424485\pi$
$654$	0	0
$655$	9160.18	0.546440
$656$	0	0
$657$	−5910.78	−0.350992
$658$	0	0
$659$	−21242.8	−1.25569	−0.627846	−	0.778338i	$-0.716063\pi$
$659$	−21242.8	−1.25569	−0.627846	+	0.778338i	$0.716063\pi$
$660$	0	0
$661$	−22221.7	−1.30760	−0.653801	−	0.756667i	$-0.726827\pi$
$661$	−22221.7	−1.30760	−0.653801	+	0.756667i	$0.726827\pi$
$662$	0	0
$663$	5466.94	0.320239
$664$	0	0
$665$	96.0808	0.00560279
$666$	0	0
$667$	−21753.8	−1.26283
$668$	0	0
$669$	3709.18	0.214358
$670$	0	0
$671$	−7438.69	−0.427969
$672$	0	0
$673$	−3787.85	−0.216955	−0.108478	−	0.994099i	$-0.534598\pi$
$673$	−3787.85	−0.216955	−0.108478	+	0.994099i	$0.534598\pi$
$674$	0	0
$675$	675.000	0.0384900
$676$	0	0
$677$	−11296.8	−0.641314	−0.320657	−	0.947195i	$-0.603904\pi$
$677$	−11296.8	−0.641314	−0.320657	+	0.947195i	$0.603904\pi$
$678$	0	0
$679$	−5068.75	−0.286481
$680$	0	0
$681$	−12545.5	−0.705937
$682$	0	0
$683$	−4807.14	−0.269312	−0.134656	−	0.990892i	$-0.542993\pi$
$683$	−4807.14	−0.269312	−0.134656	+	0.990892i	$0.542993\pi$
$684$	0	0
$685$	1913.73	0.106745
$686$	0	0
$687$	−1454.32	−0.0807654
$688$	0	0
$689$	−8440.94	−0.466726
$690$	0	0
$691$	−5393.47	−0.296928	−0.148464	−	0.988918i	$-0.547433\pi$
$691$	−5393.47	−0.296928	−0.148464	+	0.988918i	$0.547433\pi$
$692$	0	0
$693$	4067.82	0.222978
$694$	0	0
$695$	−15268.0	−0.833308
$696$	0	0
$697$	9931.84	0.539735
$698$	0	0
$699$	6241.63	0.337740
$700$	0	0
$701$	2404.77	0.129568	0.0647838	−	0.997899i	$-0.479364\pi$
$701$	2404.77	0.129568	0.0647838	+	0.997899i	$0.479364\pi$
$702$	0	0
$703$	331.276	0.0177729
$704$	0	0
$705$	5178.09	0.276622
$706$	0	0
$707$	1881.07	0.100064
$708$	0	0
$709$	−21617.3	−1.14507	−0.572535	−	0.819881i	$-0.694040\pi$
$709$	−21617.3	−1.14507	−0.572535	+	0.819881i	$0.694040\pi$
$710$	0	0
$711$	−3961.76	−0.208970
$712$	0	0
$713$	−9778.70	−0.513626
$714$	0	0
$715$	−10441.7	−0.546153
$716$	0	0
$717$	−20442.3	−1.06476
$718$	0	0
$719$	18228.6	0.945498	0.472749	−	0.881197i	$-0.343262\pi$
$719$	18228.6	0.945498	0.472749	+	0.881197i	$0.343262\pi$
$720$	0	0
$721$	12884.4	0.665519
$722$	0	0
$723$	−11765.5	−0.605207
$724$	0	0
$725$	6168.38	0.315983
$726$	0	0
$727$	−20196.5	−1.03033	−0.515164	−	0.857092i	$-0.672269\pi$
$727$	−20196.5	−1.03033	−0.515164	+	0.857092i	$0.672269\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	−24980.4	−1.26393
$732$	0	0
$733$	−15264.9	−0.769196	−0.384598	−	0.923084i	$-0.625660\pi$
$733$	−15264.9	−0.769196	−0.384598	+	0.923084i	$0.625660\pi$
$734$	0	0
$735$	735.000	0.0368856
$736$	0	0
$737$	61435.7	3.07057
$738$	0	0
$739$	−13906.0	−0.692207	−0.346103	−	0.938196i	$-0.612495\pi$
$739$	−13906.0	−0.692207	−0.346103	+	0.938196i	$0.612495\pi$
$740$	0	0
$741$	−266.362	−0.0132052
$742$	0	0
$743$	−4592.87	−0.226778	−0.113389	−	0.993551i	$-0.536171\pi$
$743$	−4592.87	−0.226778	−0.113389	+	0.993551i	$0.536171\pi$
$744$	0	0
$745$	17803.0	0.875504
$746$	0	0
$747$	489.709	0.0239860
$748$	0	0
$749$	1701.61	0.0830111
$750$	0	0
$751$	8390.80	0.407702	0.203851	−	0.979002i	$-0.434654\pi$
$751$	8390.80	0.407702	0.203851	+	0.979002i	$0.434654\pi$
$752$	0	0
$753$	15657.3	0.757747
$754$	0	0
$755$	−16309.0	−0.786153
$756$	0	0
$757$	−1368.89	−0.0657240	−0.0328620	−	0.999460i	$-0.510462\pi$
$757$	−1368.89	−0.0657240	−0.0328620	+	0.999460i	$0.510462\pi$
$758$	0	0
$759$	−17078.4	−0.816739
$760$	0	0
$761$	−1623.77	−0.0773478	−0.0386739	−	0.999252i	$-0.512313\pi$
$761$	−1623.77	−0.0773478	−0.0386739	+	0.999252i	$0.512313\pi$
$762$	0	0
$763$	−2836.23	−0.134572
$764$	0	0
$765$	−2535.44	−0.119829
$766$	0	0
$767$	20343.8	0.957722
$768$	0	0
$769$	−26842.5	−1.25873	−0.629366	−	0.777109i	$-0.716685\pi$
$769$	−26842.5	−1.25873	−0.629366	+	0.777109i	$0.716685\pi$
$770$	0	0
$771$	20927.1	0.977526
$772$	0	0
$773$	−20961.4	−0.975330	−0.487665	−	0.873031i	$-0.662151\pi$
$773$	−20961.4	−0.975330	−0.487665	+	0.873031i	$0.662151\pi$
$774$	0	0
$775$	2772.79	0.128518
$776$	0	0
$777$	2534.20	0.117006
$778$	0	0
$779$	−483.902	−0.0222562
$780$	0	0
$781$	−23003.5	−1.05394
$782$	0	0
$783$	6661.85	0.304055
$784$	0	0
$785$	14392.3	0.654373
$786$	0	0
$787$	−35333.2	−1.60037	−0.800187	−	0.599751i	$-0.795266\pi$
$787$	−35333.2	−1.60037	−0.800187	+	0.599751i	$0.795266\pi$
$788$	0	0
$789$	10822.1	0.488309
$790$	0	0
$791$	−197.236	−0.00886589
$792$	0	0
$793$	3726.13	0.166858
$794$	0	0
$795$	3914.71	0.174642
$796$	0	0
$797$	−8137.04	−0.361642	−0.180821	−	0.983516i	$-0.557875\pi$
$797$	−8137.04	−0.361642	−0.180821	+	0.983516i	$0.557875\pi$
$798$	0	0
$799$	−19450.0	−0.861191
$800$	0	0
$801$	−9169.05	−0.404460
$802$	0	0
$803$	−42405.6	−1.86359
$804$	0	0
$805$	−3085.83	−0.135107
$806$	0	0
$807$	17.6572	0.000770212 0
$808$	0	0
$809$	−36281.0	−1.57673	−0.788364	−	0.615209i	$-0.789072\pi$
$809$	−36281.0	−1.57673	−0.788364	+	0.615209i	$0.789072\pi$
$810$	0	0
$811$	34237.2	1.48240	0.741202	−	0.671282i	$-0.234256\pi$
$811$	34237.2	1.48240	0.741202	+	0.671282i	$0.234256\pi$
$812$	0	0
$813$	−20749.0	−0.895077
$814$	0	0
$815$	4637.69	0.199326
$816$	0	0
$817$	1217.10	0.0521188
$818$	0	0
$819$	−2037.62	−0.0869355
$820$	0	0
$821$	−23247.8	−0.988250	−0.494125	−	0.869391i	$-0.664511\pi$
$821$	−23247.8	−0.988250	−0.494125	+	0.869391i	$0.664511\pi$
$822$	0	0
$823$	−42934.0	−1.81845	−0.909225	−	0.416306i	$-0.863325\pi$
$823$	−42934.0	−1.81845	−0.909225	+	0.416306i	$0.863325\pi$
$824$	0	0
$825$	4842.64	0.204363
$826$	0	0
$827$	781.391	0.0328557	0.0164278	−	0.999865i	$-0.494771\pi$
$827$	781.391	0.0328557	0.0164278	+	0.999865i	$0.494771\pi$
$828$	0	0
$829$	−33493.3	−1.40322	−0.701611	−	0.712561i	$-0.747535\pi$
$829$	−33493.3	−1.40322	−0.701611	+	0.712561i	$0.747535\pi$
$830$	0	0
$831$	−6358.39	−0.265427
$832$	0	0
$833$	−2760.81	−0.114834
$834$	0	0
$835$	−5472.59	−0.226811
$836$	0	0
$837$	2994.62	0.123667
$838$	0	0
$839$	−15155.7	−0.623639	−0.311819	−	0.950141i	$-0.600938\pi$
$839$	−15155.7	−0.623639	−0.311819	+	0.950141i	$0.600938\pi$
$840$	0	0
$841$	36489.2	1.49613
$842$	0	0
$843$	−719.752	−0.0294064
$844$	0	0
$845$	−5754.60	−0.234277
$846$	0	0
$847$	19866.7	0.805935
$848$	0	0
$849$	13626.4	0.550831
$850$	0	0
$851$	−10639.6	−0.428579
$852$	0	0
$853$	−2917.48	−0.117107	−0.0585537	−	0.998284i	$-0.518649\pi$
$853$	−2917.48	−0.117107	−0.0585537	+	0.998284i	$0.518649\pi$
$854$	0	0
$855$	123.532	0.00494119
$856$	0	0
$857$	−31560.7	−1.25799	−0.628993	−	0.777411i	$-0.716532\pi$
$857$	−31560.7	−1.25799	−0.628993	+	0.777411i	$0.716532\pi$
$858$	0	0
$859$	1404.81	0.0557991	0.0278995	−	0.999611i	$-0.491118\pi$
$859$	1404.81	0.0557991	0.0278995	+	0.999611i	$0.491118\pi$
$860$	0	0
$861$	−3701.76	−0.146522
$862$	0	0
$863$	9808.24	0.386879	0.193439	−	0.981112i	$-0.438036\pi$
$863$	9808.24	0.386879	0.193439	+	0.981112i	$0.438036\pi$
$864$	0	0
$865$	−8566.27	−0.336719
$866$	0	0
$867$	−5215.35	−0.204294
$868$	0	0
$869$	−28422.8	−1.10952
$870$	0	0
$871$	−30773.9	−1.19717
$872$	0	0
$873$	−6516.97	−0.252653
$874$	0	0
$875$	875.000	0.0338062
$876$	0	0
$877$	−7196.05	−0.277073	−0.138537	−	0.990357i	$-0.544240\pi$
$877$	−7196.05	−0.277073	−0.138537	+	0.990357i	$0.544240\pi$
$878$	0	0
$879$	−6515.10	−0.249999
$880$	0	0
$881$	−3183.27	−0.121733	−0.0608667	−	0.998146i	$-0.519386\pi$
$881$	−3183.27	−0.121733	−0.0608667	+	0.998146i	$0.519386\pi$
$882$	0	0
$883$	−25392.5	−0.967751	−0.483876	−	0.875137i	$-0.660771\pi$
$883$	−25392.5	−0.967751	−0.483876	+	0.875137i	$0.660771\pi$
$884$	0	0
$885$	−9434.99	−0.358366
$886$	0	0
$887$	30634.2	1.15964	0.579818	−	0.814746i	$-0.303124\pi$
$887$	30634.2	1.15964	0.579818	+	0.814746i	$0.303124\pi$
$888$	0	0
$889$	19186.3	0.723833
$890$	0	0
$891$	5230.05	0.196648
$892$	0	0
$893$	947.648	0.0355116
$894$	0	0
$895$	20327.9	0.759204
$896$	0	0
$897$	8554.75	0.318433
$898$	0	0
$899$	27365.8	1.01524
$900$	0	0
$901$	−14704.5	−0.543704
$902$	0	0
$903$	9310.61	0.343120
$904$	0	0
$905$	−13957.0	−0.512648
$906$	0	0
$907$	28089.9	1.02834	0.514172	−	0.857687i	$-0.328099\pi$
$907$	28089.9	1.02834	0.514172	+	0.857687i	$0.328099\pi$
$908$	0	0
$909$	2418.52	0.0882480
$910$	0	0
$911$	36102.7	1.31299	0.656495	−	0.754330i	$-0.272038\pi$
$911$	36102.7	1.31299	0.656495	+	0.754330i	$0.272038\pi$
$912$	0	0
$913$	3513.31	0.127353
$914$	0	0
$915$	−1728.09	−0.0624360
$916$	0	0
$917$	12824.3	0.461826
$918$	0	0
$919$	14533.8	0.521682	0.260841	−	0.965382i	$-0.416000\pi$
$919$	14533.8	0.521682	0.260841	+	0.965382i	$0.416000\pi$
$920$	0	0
$921$	10525.9	0.376592
$922$	0	0
$923$	11522.7	0.410915
$924$	0	0
$925$	3016.90	0.107238
$926$	0	0
$927$	16565.6	0.586933
$928$	0	0
$929$	16539.6	0.584118	0.292059	−	0.956400i	$-0.405660\pi$
$929$	16539.6	0.584118	0.292059	+	0.956400i	$0.405660\pi$
$930$	0	0
$931$	134.513	0.00473522
$932$	0	0
$933$	9399.74	0.329833
$934$	0	0
$935$	−18190.0	−0.636231
$936$	0	0
$937$	30212.3	1.05335	0.526677	−	0.850065i	$-0.323438\pi$
$937$	30212.3	1.05335	0.526677	+	0.850065i	$0.323438\pi$
$938$	0	0
$939$	19168.8	0.666187
$940$	0	0
$941$	−26414.4	−0.915074	−0.457537	−	0.889191i	$-0.651268\pi$
$941$	−26414.4	−0.915074	−0.457537	+	0.889191i	$0.651268\pi$
$942$	0	0
$943$	15541.5	0.536692
$944$	0	0
$945$	945.000	0.0325300
$946$	0	0
$947$	−10187.3	−0.349570	−0.174785	−	0.984607i	$-0.555923\pi$
$947$	−10187.3	−0.349570	−0.174785	+	0.984607i	$0.555923\pi$
$948$	0	0
$949$	21241.5	0.726583
$950$	0	0
$951$	4903.31	0.167193
$952$	0	0
$953$	2211.39	0.0751669	0.0375834	−	0.999293i	$-0.488034\pi$
$953$	2211.39	0.0751669	0.0375834	+	0.999293i	$0.488034\pi$
$954$	0	0
$955$	3170.92	0.107444
$956$	0	0
$957$	47794.0	1.61438
$958$	0	0
$959$	2679.23	0.0902156
$960$	0	0
$961$	−17489.6	−0.587077
$962$	0	0
$963$	2187.78	0.0732089
$964$	0	0
$965$	−1275.00	−0.0425322
$966$	0	0
$967$	−7955.89	−0.264575	−0.132287	−	0.991211i	$-0.542232\pi$
$967$	−7955.89	−0.264575	−0.132287	+	0.991211i	$0.542232\pi$
$968$	0	0
$969$	−464.014	−0.0153832
$970$	0	0
$971$	−53071.2	−1.75400	−0.877001	−	0.480488i	$-0.840459\pi$
$971$	−53071.2	−1.75400	−0.877001	+	0.480488i	$0.840459\pi$
$972$	0	0
$973$	−21375.2	−0.704274
$974$	0	0
$975$	−2425.74	−0.0796777
$976$	0	0
$977$	−22448.2	−0.735089	−0.367545	−	0.930006i	$-0.619801\pi$
$977$	−22448.2	−0.735089	−0.367545	+	0.930006i	$0.619801\pi$
$978$	0	0
$979$	−65781.4	−2.14748
$980$	0	0
$981$	−3646.58	−0.118681
$982$	0	0
$983$	−21712.8	−0.704509	−0.352254	−	0.935904i	$-0.614585\pi$
$983$	−21712.8	−0.704509	−0.352254	+	0.935904i	$0.614585\pi$
$984$	0	0
$985$	20861.9	0.674837
$986$	0	0
$987$	7249.33	0.233788
$988$	0	0
$989$	−39089.7	−1.25681
$990$	0	0
$991$	37849.2	1.21324	0.606620	−	0.794992i	$-0.292525\pi$
$991$	37849.2	1.21324	0.606620	+	0.794992i	$0.292525\pi$
$992$	0	0
$993$	−13158.5	−0.420516
$994$	0	0
$995$	23132.4	0.737031
$996$	0	0
$997$	−39573.1	−1.25707	−0.628533	−	0.777783i	$-0.716344\pi$
$997$	−39573.1	−1.25707	−0.628533	+	0.777783i	$0.716344\pi$
$998$	0	0
$999$	3258.26	0.103190

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1680.4.a.bo.1.2		2
4.3	odd	2		105.4.a.e.1.2	✓	2
12.11	even	2		315.4.a.k.1.1		2
20.3	even	4		525.4.d.l.274.2		4
20.7	even	4		525.4.d.l.274.3		4
20.19	odd	2		525.4.a.l.1.1		2
28.27	even	2		735.4.a.o.1.2		2
60.59	even	2		1575.4.a.q.1.2		2
84.83	odd	2		2205.4.a.bb.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.4.a.e.1.2	✓	2	4.3	odd	2
315.4.a.k.1.1		2	12.11	even	2
525.4.a.l.1.1		2	20.19	odd	2
525.4.d.l.274.2		4	20.3	even	4
525.4.d.l.274.3		4	20.7	even	4
735.4.a.o.1.2		2	28.27	even	2
1575.4.a.q.1.2		2	60.59	even	2
1680.4.a.bo.1.2		2	1.1	even	1	trivial
2205.4.a.bb.1.1		2	84.83	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:	\( N \)	\(=\)	\( 1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1680.a (trivial)

\(f(q)\)	\(=\)	\(q+3.00000 q^{3} +5.00000 q^{5} +7.00000 q^{7} +9.00000 q^{9} +O(q^{10})\)	\(q+3.00000 q^{3} +5.00000 q^{5} +7.00000 q^{7} +9.00000 q^{9} +64.5685 q^{11} -32.3431 q^{13} +15.0000 q^{15} -56.3431 q^{17} +2.74517 q^{19} +21.0000 q^{21} -88.1665 q^{23} +25.0000 q^{25} +27.0000 q^{27} +246.735 q^{29} +110.912 q^{31} +193.706 q^{33} +35.0000 q^{35} +120.676 q^{37} -97.0294 q^{39} -176.274 q^{41} +443.362 q^{43} +45.0000 q^{45} +345.206 q^{47} +49.0000 q^{49} -169.029 q^{51} +260.981 q^{53} +322.843 q^{55} +8.23550 q^{57} -628.999 q^{59} -115.206 q^{61} +63.0000 q^{63} -161.716 q^{65} +951.480 q^{67} -264.500 q^{69} -356.264 q^{71} -656.754 q^{73} +75.0000 q^{75} +451.980 q^{77} -440.195 q^{79} +81.0000 q^{81} +54.4121 q^{83} -281.716 q^{85} +740.205 q^{87} -1018.78 q^{89} -226.402 q^{91} +332.735 q^{93} +13.7258 q^{95} -724.108 q^{97} +581.117 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 6 q^{3} + 10 q^{5} + 14 q^{7} + 18 q^{9}+O(q^{10}) \) 2 * q + 6 * q^3 + 10 * q^5 + 14 * q^7 + 18 * q^9	\( 2 q + 6 q^{3} + 10 q^{5} + 14 q^{7} + 18 q^{9} + 16 q^{11} - 76 q^{13} + 30 q^{15} - 124 q^{17} + 96 q^{19} + 42 q^{21} + 16 q^{23} + 50 q^{25} + 54 q^{27} + 188 q^{29} + 120 q^{31} + 48 q^{33} + 70 q^{35} - 132 q^{37} - 228 q^{39} + 100 q^{41} + 536 q^{43} + 90 q^{45} + 928 q^{47} + 98 q^{49} - 372 q^{51} + 884 q^{53} + 80 q^{55} + 288 q^{57} - 104 q^{59} - 468 q^{61} + 126 q^{63} - 380 q^{65} + 1688 q^{67} + 48 q^{69} + 136 q^{71} + 508 q^{73} + 150 q^{75} + 112 q^{77} + 432 q^{79} + 162 q^{81} + 584 q^{83} - 620 q^{85} + 564 q^{87} - 1404 q^{89} - 532 q^{91} + 360 q^{93} + 480 q^{95} - 1188 q^{97} + 144 q^{99}+O(q^{100}) \) 2 * q + 6 * q^3 + 10 * q^5 + 14 * q^7 + 18 * q^9 + 16 * q^11 - 76 * q^13 + 30 * q^15 - 124 * q^17 + 96 * q^19 + 42 * q^21 + 16 * q^23 + 50 * q^25 + 54 * q^27 + 188 * q^29 + 120 * q^31 + 48 * q^33 + 70 * q^35 - 132 * q^37 - 228 * q^39 + 100 * q^41 + 536 * q^43 + 90 * q^45 + 928 * q^47 + 98 * q^49 - 372 * q^51 + 884 * q^53 + 80 * q^55 + 288 * q^57 - 104 * q^59 - 468 * q^61 + 126 * q^63 - 380 * q^65 + 1688 * q^67 + 48 * q^69 + 136 * q^71 + 508 * q^73 + 150 * q^75 + 112 * q^77 + 432 * q^79 + 162 * q^81 + 584 * q^83 - 620 * q^85 + 564 * q^87 - 1404 * q^89 - 532 * q^91 + 360 * q^93 + 480 * q^95 - 1188 * q^97 + 144 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more