LMFDB - Embedded newform 28.8.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [28,8,Mod(1,28)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(28, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 8, names="a")

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

chi := DirichletCharacter("28.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 28 = 2^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	28.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:	$8.74678071356$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{1009}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 252 $ x^2 - x - 252
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$16.3824$ of defining polynomial
Character	$\chi$	$=$	28.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-24.7648 q^{3} +202.412 q^{5} -343.000 q^{7} -1573.71 q^{9} -7527.19 q^{11} -1700.28 q^{13} -5012.69 q^{15} -2481.86 q^{17} -39816.0 q^{19} +8494.31 q^{21} +25560.1 q^{23} -37154.2 q^{25} +93133.0 q^{27} +148830. q^{29} -33538.4 q^{31} +186409. q^{33} -69427.4 q^{35} +400557. q^{37} +42107.1 q^{39} -362750. q^{41} -324652. q^{43} -318538. q^{45} -708899. q^{47} +117649. q^{49} +61462.7 q^{51} -185924. q^{53} -1.52360e6 q^{55} +986034. q^{57} -1.19372e6 q^{59} +2.51204e6 q^{61} +539781. q^{63} -344158. q^{65} -2.85983e6 q^{67} -632989. q^{69} +3.22083e6 q^{71} +5.01287e6 q^{73} +920116. q^{75} +2.58182e6 q^{77} +5.94208e6 q^{79} +1.13528e6 q^{81} -1.02006e7 q^{83} -502359. q^{85} -3.68573e6 q^{87} +1.85373e6 q^{89} +583197. q^{91} +830571. q^{93} -8.05925e6 q^{95} -1.52142e7 q^{97} +1.18456e7 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 14 q^{3} - 294 q^{5} - 686 q^{7} - 2258 q^{9} - 3492 q^{11} - 16170 q^{13} - 24256 q^{15} - 29232 q^{17} - 3206 q^{19} - 4802 q^{21} - 9360 q^{23} + 131146 q^{25} - 18172 q^{27} + 184704 q^{29} + 165060 q^{31}+ \cdots + 9084332 q^{99}+O(q^{100}) $ 2 * q + 14 * q^3 - 294 * q^5 - 686 * q^7 - 2258 * q^9 - 3492 * q^11 - 16170 * q^13 - 24256 * q^15 - 29232 * q^17 - 3206 * q^19 - 4802 * q^21 - 9360 * q^23 + 131146 * q^25 - 18172 * q^27 + 184704 * q^29 + 165060 * q^31 + 342832 * q^33 + 100842 * q^35 + 286144 * q^37 - 518808 * q^39 - 116760 * q^41 - 294428 * q^43 + 21154 * q^45 - 1014132 * q^47 + 235298 * q^49 - 975500 * q^51 - 1396452 * q^53 - 3526712 * q^55 + 2405212 * q^57 - 2729286 * q^59 + 2466954 * q^61 + 774494 * q^63 + 6838788 * q^65 + 225176 * q^67 - 1986656 * q^69 + 1530312 * q^71 + 1143548 * q^73 + 7444234 * q^75 + 1197756 * q^77 + 7951176 * q^79 - 1682882 * q^81 - 18487854 * q^83 + 12776740 * q^85 - 2295076 * q^87 - 4652508 * q^89 + 5546310 * q^91 + 8529192 * q^93 - 26232912 * q^95 - 26702368 * q^97 + 9084332 * q^99

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
$3$	−24.7648	−0.529553	−0.264777	−	0.964310i	$-0.585298\pi$
$3$	−24.7648	−0.529553	−0.264777	+	0.964310i	$0.585298\pi$
$4$	0	0
$5$	202.412	0.724172	0.362086	−	0.932145i	$-0.382065\pi$
$5$	202.412	0.724172	0.362086	+	0.932145i	$0.382065\pi$
$6$	0	0
$7$	−343.000	−0.377964
$7$	−343.000	−0.377964
$8$	0	0
$9$	−1573.71	−0.719573
$10$	0	0
$11$	−7527.19	−1.70513	−0.852567	−	0.522619i	$-0.824955\pi$
$11$	−7527.19	−1.70513	−0.852567	+	0.522619i	$0.824955\pi$
$12$	0	0
$13$	−1700.28	−0.214644	−0.107322	−	0.994224i	$-0.534228\pi$
$13$	−1700.28	−0.214644	−0.107322	+	0.994224i	$0.534228\pi$
$14$	0	0
$15$	−5012.69	−0.383488
$16$	0	0
$17$	−2481.86	−0.122520	−0.0612599	−	0.998122i	$-0.519512\pi$
$17$	−2481.86	−0.122520	−0.0612599	+	0.998122i	$0.519512\pi$
$18$	0	0
$19$	−39816.0	−1.33174	−0.665871	−	0.746067i	$-0.731940\pi$
$19$	−39816.0	−1.33174	−0.665871	+	0.746067i	$0.731940\pi$
$20$	0	0
$21$	8494.31	0.200152
$22$	0	0
$23$	25560.1	0.438041	0.219020	−	0.975720i	$-0.429714\pi$
$23$	25560.1	0.438041	0.219020	+	0.975720i	$0.429714\pi$
$24$	0	0
$25$	−37154.2	−0.475574
$26$	0	0
$27$	93133.0	0.910606
$28$	0	0
$29$	148830.	1.13317	0.566587	−	0.824002i	$-0.308263\pi$
$29$	148830.	1.13317	0.566587	+	0.824002i	$0.308263\pi$
$30$	0	0
$31$	−33538.4	−0.202198	−0.101099	−	0.994876i	$-0.532236\pi$
$31$	−33538.4	−0.202198	−0.101099	+	0.994876i	$0.532236\pi$
$32$	0	0
$33$	186409.	0.902959
$34$	0	0
$35$	−69427.4	−0.273711
$36$	0	0
$37$	400557.	1.30005	0.650023	−	0.759915i	$-0.274759\pi$
$37$	400557.	1.30005	0.650023	+	0.759915i	$0.274759\pi$
$38$	0	0
$39$	42107.1	0.113666
$40$	0	0
$41$	−362750.	−0.821985	−0.410992	−	0.911639i	$-0.634818\pi$
$41$	−362750.	−0.821985	−0.410992	+	0.911639i	$0.634818\pi$
$42$	0	0
$43$	−324652.	−0.622699	−0.311350	−	0.950295i	$-0.600781\pi$
$43$	−324652.	−0.622699	−0.311350	+	0.950295i	$0.600781\pi$
$44$	0	0
$45$	−318538.	−0.521095
$46$	0	0
$47$	−708899.	−0.995960	−0.497980	−	0.867188i	$-0.665925\pi$
$47$	−708899.	−0.995960	−0.497980	+	0.867188i	$0.665925\pi$
$48$	0	0
$49$	117649.	0.142857
$50$	0	0
$51$	61462.7	0.0648808
$52$	0	0
$53$	−185924.	−0.171542	−0.0857709	−	0.996315i	$-0.527335\pi$
$53$	−185924.	−0.171542	−0.0857709	+	0.996315i	$0.527335\pi$
$54$	0	0
$55$	−1.52360e6	−1.23481
$56$	0	0
$57$	986034.	0.705228
$58$	0	0
$59$	−1.19372e6	−0.756692	−0.378346	−	0.925664i	$-0.623507\pi$
$59$	−1.19372e6	−0.756692	−0.378346	+	0.925664i	$0.623507\pi$
$60$	0	0
$61$	2.51204e6	1.41701	0.708504	−	0.705707i	$-0.249370\pi$
$61$	2.51204e6	1.41701	0.708504	+	0.705707i	$0.249370\pi$
$62$	0	0
$63$	539781.	0.271973
$64$	0	0
$65$	−344158.	−0.155440
$66$	0	0
$67$	−2.85983e6	−1.16166	−0.580829	−	0.814025i	$-0.697272\pi$
$67$	−2.85983e6	−1.16166	−0.580829	+	0.814025i	$0.697272\pi$
$68$	0	0
$69$	−632989.	−0.231966
$70$	0	0
$71$	3.22083e6	1.06798	0.533990	−	0.845491i	$-0.320692\pi$
$71$	3.22083e6	1.06798	0.533990	+	0.845491i	$0.320692\pi$
$72$	0	0
$73$	5.01287e6	1.50819	0.754095	−	0.656765i	$-0.228076\pi$
$73$	5.01287e6	1.50819	0.754095	+	0.656765i	$0.228076\pi$
$74$	0	0
$75$	920116.	0.251842
$76$	0	0
$77$	2.58182e6	0.644480
$78$	0	0
$79$	5.94208e6	1.35595	0.677975	−	0.735085i	$-0.262858\pi$
$79$	5.94208e6	1.35595	0.677975	+	0.735085i	$0.262858\pi$
$80$	0	0
$81$	1.13528e6	0.237359
$82$	0	0
$83$	−1.02006e7	−1.95819	−0.979094	−	0.203407i	$-0.934799\pi$
$83$	−1.02006e7	−1.95819	−0.979094	+	0.203407i	$0.934799\pi$
$84$	0	0
$85$	−502359.	−0.0887255
$86$	0	0
$87$	−3.68573e6	−0.600076
$88$	0	0
$89$	1.85373e6	0.278729	0.139365	−	0.990241i	$-0.455494\pi$
$89$	1.85373e6	0.278729	0.139365	+	0.990241i	$0.455494\pi$
$90$	0	0
$91$	583197.	0.0811280
$92$	0	0
$93$	830571.	0.107075
$94$	0	0
$95$	−8.05925e6	−0.964411
$96$	0	0
$97$	−1.52142e7	−1.69257	−0.846287	−	0.532727i	$-0.821167\pi$
$97$	−1.52142e7	−1.69257	−0.846287	+	0.532727i	$0.821167\pi$
$98$	0	0
$99$	1.18456e7	1.22697
$100$	0	0
$101$	−2.02298e7	−1.95374	−0.976869	−	0.213838i	$-0.931404\pi$
$101$	−2.02298e7	−1.95374	−0.976869	+	0.213838i	$0.931404\pi$
$102$	0	0
$103$	9.23180e6	0.832446	0.416223	−	0.909263i	$-0.363354\pi$
$103$	9.23180e6	0.832446	0.416223	+	0.909263i	$0.363354\pi$
$104$	0	0
$105$	1.71935e6	0.144945
$106$	0	0
$107$	9.59273e6	0.757005	0.378503	−	0.925600i	$-0.376439\pi$
$107$	9.59273e6	0.757005	0.378503	+	0.925600i	$0.376439\pi$
$108$	0	0
$109$	−1.01709e7	−0.752256	−0.376128	−	0.926568i	$-0.622745\pi$
$109$	−1.01709e7	−0.752256	−0.376128	+	0.926568i	$0.622745\pi$
$110$	0	0
$111$	−9.91970e6	−0.688443
$112$	0	0
$113$	1.09461e6	0.0713649	0.0356825	−	0.999363i	$-0.488640\pi$
$113$	1.09461e6	0.0713649	0.0356825	+	0.999363i	$0.488640\pi$
$114$	0	0
$115$	5.17367e6	0.317217
$116$	0	0
$117$	2.67575e6	0.154452
$118$	0	0
$119$	851279.	0.0463081
$120$	0	0
$121$	3.71714e7	1.90748
$122$	0	0
$123$	8.98342e6	0.435285
$124$	0	0
$125$	−2.33339e7	−1.06857
$126$	0	0
$127$	2.13370e7	0.924317	0.462158	−	0.886797i	$-0.347075\pi$
$127$	2.13370e7	0.924317	0.462158	+	0.886797i	$0.347075\pi$
$128$	0	0
$129$	8.03993e6	0.329753
$130$	0	0
$131$	−1.28333e7	−0.498758	−0.249379	−	0.968406i	$-0.580226\pi$
$131$	−1.28333e7	−0.498758	−0.249379	+	0.968406i	$0.580226\pi$
$132$	0	0
$133$	1.36569e7	0.503351
$134$	0	0
$135$	1.88513e7	0.659436
$136$	0	0
$137$	−4.08059e7	−1.35582	−0.677908	−	0.735146i	$-0.737113\pi$
$137$	−4.08059e7	−1.35582	−0.677908	+	0.735146i	$0.737113\pi$
$138$	0	0
$139$	1.52006e7	0.480074	0.240037	−	0.970764i	$-0.422840\pi$
$139$	1.52006e7	0.480074	0.240037	+	0.970764i	$0.422840\pi$
$140$	0	0
$141$	1.75557e7	0.527414
$142$	0	0
$143$	1.27983e7	0.365997
$144$	0	0
$145$	3.01250e7	0.820614
$146$	0	0
$147$	−2.91355e6	−0.0756505
$148$	0	0
$149$	−4.76985e7	−1.18128	−0.590640	−	0.806935i	$-0.701125\pi$
$149$	−4.76985e7	−1.18128	−0.590640	+	0.806935i	$0.701125\pi$
$150$	0	0
$151$	−7.67985e6	−0.181524	−0.0907619	−	0.995873i	$-0.528930\pi$
$151$	−7.67985e6	−0.181524	−0.0907619	+	0.995873i	$0.528930\pi$
$152$	0	0
$153$	3.90572e6	0.0881620
$154$	0	0
$155$	−6.78859e6	−0.146426
$156$	0	0
$157$	−7.37198e7	−1.52032	−0.760161	−	0.649735i	$-0.774880\pi$
$157$	−7.37198e7	−1.52032	−0.760161	+	0.649735i	$0.774880\pi$
$158$	0	0
$159$	4.60436e6	0.0908405
$160$	0	0
$161$	−8.76710e6	−0.165564
$162$	0	0
$163$	−5.42300e7	−0.980805	−0.490402	−	0.871496i	$-0.663150\pi$
$163$	−5.42300e7	−0.980805	−0.490402	+	0.871496i	$0.663150\pi$
$164$	0	0
$165$	3.77315e7	0.653898
$166$	0	0
$167$	−8.26795e7	−1.37369	−0.686847	−	0.726802i	$-0.741006\pi$
$167$	−8.26795e7	−1.37369	−0.686847	+	0.726802i	$0.741006\pi$
$168$	0	0
$169$	−5.98576e7	−0.953928
$170$	0	0
$171$	6.26587e7	0.958286
$172$	0	0
$173$	−6.21094e7	−0.912003	−0.456001	−	0.889979i	$-0.650719\pi$
$173$	−6.21094e7	−0.912003	−0.456001	+	0.889979i	$0.650719\pi$
$174$	0	0
$175$	1.27439e7	0.179750
$176$	0	0
$177$	2.95621e7	0.400709
$178$	0	0
$179$	4.74032e7	0.617763	0.308881	−	0.951101i	$-0.400045\pi$
$179$	4.74032e7	0.617763	0.308881	+	0.951101i	$0.400045\pi$
$180$	0	0
$181$	1.30477e8	1.63553	0.817767	−	0.575549i	$-0.195212\pi$
$181$	1.30477e8	1.63553	0.817767	+	0.575549i	$0.195212\pi$
$182$	0	0
$183$	−6.22101e7	−0.750381
$184$	0	0
$185$	8.10777e7	0.941457
$186$	0	0
$187$	1.86814e7	0.208913
$188$	0	0
$189$	−3.19446e7	−0.344177
$190$	0	0
$191$	−3.89381e7	−0.404351	−0.202175	−	0.979349i	$-0.564801\pi$
$191$	−3.89381e7	−0.404351	−0.202175	+	0.979349i	$0.564801\pi$
$192$	0	0
$193$	1.31913e8	1.32080	0.660401	−	0.750913i	$-0.270386\pi$
$193$	1.31913e8	1.32080	0.660401	+	0.750913i	$0.270386\pi$
$194$	0	0
$195$	8.52300e6	0.0823135
$196$	0	0
$197$	2.26076e7	0.210680	0.105340	−	0.994436i	$-0.466407\pi$
$197$	2.26076e7	0.210680	0.105340	+	0.994436i	$0.466407\pi$
$198$	0	0
$199$	9.24230e6	0.0831370	0.0415685	−	0.999136i	$-0.486765\pi$
$199$	9.24230e6	0.0831370	0.0415685	+	0.999136i	$0.486765\pi$
$200$	0	0
$201$	7.08230e7	0.615160
$202$	0	0
$203$	−5.10486e7	−0.428300
$204$	0	0
$205$	−7.34251e7	−0.595259
$206$	0	0
$207$	−4.02240e7	−0.315202
$208$	0	0
$209$	2.99703e8	2.27080
$210$	0	0
$211$	−5.91116e7	−0.433196	−0.216598	−	0.976261i	$-0.569496\pi$
$211$	−5.91116e7	−0.433196	−0.216598	+	0.976261i	$0.569496\pi$
$212$	0	0
$213$	−7.97630e7	−0.565552
$214$	0	0
$215$	−6.57136e7	−0.450942
$216$	0	0
$217$	1.15037e7	0.0764237
$218$	0	0
$219$	−1.24142e8	−0.798668
$220$	0	0
$221$	4.21987e6	0.0262982
$222$	0	0
$223$	2.50841e8	1.51472	0.757358	−	0.652999i	$-0.226490\pi$
$223$	2.50841e8	1.51472	0.757358	+	0.652999i	$0.226490\pi$
$224$	0	0
$225$	5.84699e7	0.342210
$226$	0	0
$227$	8.10085e7	0.459664	0.229832	−	0.973230i	$-0.426182\pi$
$227$	8.10085e7	0.459664	0.229832	+	0.973230i	$0.426182\pi$
$228$	0	0
$229$	−3.13422e8	−1.72467	−0.862333	−	0.506342i	$-0.830998\pi$
$229$	−3.13422e8	−1.72467	−0.862333	+	0.506342i	$0.830998\pi$
$230$	0	0
$231$	−6.39383e7	−0.341286
$232$	0	0
$233$	−2.36331e7	−0.122398	−0.0611992	−	0.998126i	$-0.519492\pi$
$233$	−2.36331e7	−0.122398	−0.0611992	+	0.998126i	$0.519492\pi$
$234$	0	0
$235$	−1.43490e8	−0.721247
$236$	0	0
$237$	−1.47154e8	−0.718048
$238$	0	0
$239$	−2.65137e8	−1.25626	−0.628128	−	0.778110i	$-0.716178\pi$
$239$	−2.65137e8	−1.25626	−0.628128	+	0.778110i	$0.716178\pi$
$240$	0	0
$241$	4.76463e6	0.0219265	0.0109632	−	0.999940i	$-0.496510\pi$
$241$	4.76463e6	0.0219265	0.0109632	+	0.999940i	$0.496510\pi$
$242$	0	0
$243$	−2.31797e8	−1.03630
$244$	0	0
$245$	2.38136e7	0.103453
$246$	0	0
$247$	6.76985e7	0.285851
$248$	0	0
$249$	2.52617e8	1.03697
$250$	0	0
$251$	2.80773e8	1.12072	0.560360	−	0.828249i	$-0.310663\pi$
$251$	2.80773e8	1.12072	0.560360	+	0.828249i	$0.310663\pi$
$252$	0	0
$253$	−1.92395e8	−0.746917
$254$	0	0
$255$	1.24408e7	0.0469849
$256$	0	0
$257$	5.85988e7	0.215339	0.107669	−	0.994187i	$-0.465661\pi$
$257$	5.85988e7	0.215339	0.107669	+	0.994187i	$0.465661\pi$
$258$	0	0
$259$	−1.37391e8	−0.491371
$260$	0	0
$261$	−2.34214e8	−0.815402
$262$	0	0
$263$	1.23092e8	0.417238	0.208619	−	0.977997i	$-0.433103\pi$
$263$	1.23092e8	0.417238	0.208619	+	0.977997i	$0.433103\pi$
$264$	0	0
$265$	−3.76333e7	−0.124226
$266$	0	0
$267$	−4.59073e7	−0.147602
$268$	0	0
$269$	−4.82404e7	−0.151105	−0.0755524	−	0.997142i	$-0.524072\pi$
$269$	−4.82404e7	−0.151105	−0.0755524	+	0.997142i	$0.524072\pi$
$270$	0	0
$271$	5.43236e8	1.65804	0.829021	−	0.559217i	$-0.188898\pi$
$271$	5.43236e8	1.65804	0.829021	+	0.559217i	$0.188898\pi$
$272$	0	0
$273$	−1.44427e7	−0.0429616
$274$	0	0
$275$	2.79667e8	0.810917
$276$	0	0
$277$	−2.02874e8	−0.573519	−0.286760	−	0.958003i	$-0.592578\pi$
$277$	−2.02874e8	−0.573519	−0.286760	+	0.958003i	$0.592578\pi$
$278$	0	0
$279$	5.27797e7	0.145496
$280$	0	0
$281$	6.91877e8	1.86019	0.930094	−	0.367321i	$-0.119725\pi$
$281$	6.91877e8	1.86019	0.930094	+	0.367321i	$0.119725\pi$
$282$	0	0
$283$	3.80825e8	0.998788	0.499394	−	0.866375i	$-0.333556\pi$
$283$	3.80825e8	0.998788	0.499394	+	0.866375i	$0.333556\pi$
$284$	0	0
$285$	1.99585e8	0.510707
$286$	0	0
$287$	1.24423e8	0.310681
$288$	0	0
$289$	−4.04179e8	−0.984989
$290$	0	0
$291$	3.76776e8	0.896308
$292$	0	0
$293$	−9.72201e7	−0.225798	−0.112899	−	0.993606i	$-0.536014\pi$
$293$	−9.72201e7	−0.225798	−0.112899	+	0.993606i	$0.536014\pi$
$294$	0	0
$295$	−2.41623e8	−0.547976
$296$	0	0
$297$	−7.01029e8	−1.55270
$298$	0	0
$299$	−4.34593e7	−0.0940229
$300$	0	0
$301$	1.11356e8	0.235358
$302$	0	0
$303$	5.00986e8	1.03461
$304$	0	0
$305$	5.08468e8	1.02616
$306$	0	0
$307$	−8.60584e8	−1.69750	−0.848749	−	0.528797i	$-0.822643\pi$
$307$	−8.60584e8	−1.69750	−0.848749	+	0.528797i	$0.822643\pi$
$308$	0	0
$309$	−2.28623e8	−0.440825
$310$	0	0
$311$	4.68830e8	0.883800	0.441900	−	0.897064i	$-0.354305\pi$
$311$	4.68830e8	0.883800	0.441900	+	0.897064i	$0.354305\pi$
$312$	0	0
$313$	−5.34360e8	−0.984984	−0.492492	−	0.870317i	$-0.663914\pi$
$313$	−5.34360e8	−0.984984	−0.492492	+	0.870317i	$0.663914\pi$
$314$	0	0
$315$	1.09258e8	0.196955
$316$	0	0
$317$	−3.36916e8	−0.594038	−0.297019	−	0.954872i	$-0.595993\pi$
$317$	−3.36916e8	−0.594038	−0.297019	+	0.954872i	$0.595993\pi$
$318$	0	0
$319$	−1.12027e9	−1.93221
$320$	0	0
$321$	−2.37562e8	−0.400875
$322$	0	0
$323$	9.88178e7	0.163165
$324$	0	0
$325$	6.31727e7	0.102079
$326$	0	0
$327$	2.51879e8	0.398359
$328$	0	0
$329$	2.43152e8	0.376438
$330$	0	0
$331$	5.55287e8	0.841626	0.420813	−	0.907147i	$-0.361745\pi$
$331$	5.55287e8	0.841626	0.420813	+	0.907147i	$0.361745\pi$
$332$	0	0
$333$	−6.30359e8	−0.935478
$334$	0	0
$335$	−5.78865e8	−0.841241
$336$	0	0
$337$	8.29504e8	1.18063	0.590315	−	0.807173i	$-0.299003\pi$
$337$	8.29504e8	1.18063	0.590315	+	0.807173i	$0.299003\pi$
$338$	0	0
$339$	−2.71077e7	−0.0377915
$340$	0	0
$341$	2.52450e8	0.344774
$342$	0	0
$343$	−4.03536e7	−0.0539949
$344$	0	0
$345$	−1.28125e8	−0.167983
$346$	0	0
$347$	−1.40234e9	−1.80177	−0.900887	−	0.434054i	$-0.857083\pi$
$347$	−1.40234e9	−1.80177	−0.900887	+	0.434054i	$0.857083\pi$
$348$	0	0
$349$	−2.03183e8	−0.255857	−0.127929	−	0.991783i	$-0.540833\pi$
$349$	−2.03183e8	−0.255857	−0.127929	+	0.991783i	$0.540833\pi$
$350$	0	0
$351$	−1.58352e8	−0.195456
$352$	0	0
$353$	−8.77208e8	−1.06143	−0.530714	−	0.847551i	$-0.678076\pi$
$353$	−8.77208e8	−1.06143	−0.530714	+	0.847551i	$0.678076\pi$
$354$	0	0
$355$	6.51935e8	0.773402
$356$	0	0
$357$	−2.10817e7	−0.0245226
$358$	0	0
$359$	9.36785e8	1.06859	0.534293	−	0.845300i	$-0.320578\pi$
$359$	9.36785e8	1.06859	0.534293	+	0.845300i	$0.320578\pi$
$360$	0	0
$361$	6.91443e8	0.773537
$362$	0	0
$363$	−9.20540e8	−1.01011
$364$	0	0
$365$	1.01467e9	1.09219
$366$	0	0
$367$	−1.14703e8	−0.121127	−0.0605637	−	0.998164i	$-0.519290\pi$
$367$	−1.14703e8	−0.121127	−0.0605637	+	0.998164i	$0.519290\pi$
$368$	0	0
$369$	5.70862e8	0.591478
$370$	0	0
$371$	6.37719e7	0.0648367
$372$	0	0
$373$	1.13809e9	1.13552	0.567761	−	0.823193i	$-0.307810\pi$
$373$	1.13809e9	1.13552	0.567761	+	0.823193i	$0.307810\pi$
$374$	0	0
$375$	5.77859e8	0.565865
$376$	0	0
$377$	−2.53053e8	−0.243230
$378$	0	0
$379$	1.02843e9	0.970372	0.485186	−	0.874411i	$-0.338752\pi$
$379$	1.02843e9	0.970372	0.485186	+	0.874411i	$0.338752\pi$
$380$	0	0
$381$	−5.28406e8	−0.489475
$382$	0	0
$383$	−3.92102e8	−0.356618	−0.178309	−	0.983975i	$-0.557063\pi$
$383$	−3.92102e8	−0.356618	−0.178309	+	0.983975i	$0.557063\pi$
$384$	0	0
$385$	5.22593e8	0.466714
$386$	0	0
$387$	5.10907e8	0.448078
$388$	0	0
$389$	4.73450e8	0.407803	0.203901	−	0.978991i	$-0.434638\pi$
$389$	4.73450e8	0.407803	0.203901	+	0.978991i	$0.434638\pi$
$390$	0	0
$391$	−6.34365e7	−0.0536686
$392$	0	0
$393$	3.17814e8	0.264119
$394$	0	0
$395$	1.20275e9	0.981942
$396$	0	0
$397$	−1.02400e9	−0.821358	−0.410679	−	0.911780i	$-0.634708\pi$
$397$	−1.02400e9	−0.821358	−0.410679	+	0.911780i	$0.634708\pi$
$398$	0	0
$399$	−3.38210e8	−0.266551
$400$	0	0
$401$	1.85861e9	1.43941	0.719703	−	0.694283i	$-0.244278\pi$
$401$	1.85861e9	1.43941	0.719703	+	0.694283i	$0.244278\pi$
$402$	0	0
$403$	5.70248e7	0.0434007
$404$	0	0
$405$	2.29795e8	0.171889
$406$	0	0
$407$	−3.01507e9	−2.21675
$408$	0	0
$409$	−2.18677e9	−1.58042	−0.790209	−	0.612838i	$-0.790028\pi$
$409$	−2.18677e9	−1.58042	−0.790209	+	0.612838i	$0.790028\pi$
$410$	0	0
$411$	1.01055e9	0.717977
$412$	0	0
$413$	4.09445e8	0.286003
$414$	0	0
$415$	−2.06474e9	−1.41807
$416$	0	0
$417$	−3.76439e8	−0.254225
$418$	0	0
$419$	−2.05243e9	−1.36307	−0.681535	−	0.731785i	$-0.738687\pi$
$419$	−2.05243e9	−1.36307	−0.681535	+	0.731785i	$0.738687\pi$
$420$	0	0
$421$	2.64104e9	1.72499	0.862497	−	0.506062i	$-0.168899\pi$
$421$	2.64104e9	1.72499	0.862497	+	0.506062i	$0.168899\pi$
$422$	0	0
$423$	1.11560e9	0.716666
$424$	0	0
$425$	9.22117e7	0.0582673
$426$	0	0
$427$	−8.61630e8	−0.535579
$428$	0	0
$429$	−3.16948e8	−0.193815
$430$	0	0
$431$	−1.96735e9	−1.18362	−0.591810	−	0.806078i	$-0.701586\pi$
$431$	−1.96735e9	−1.18362	−0.591810	+	0.806078i	$0.701586\pi$
$432$	0	0
$433$	2.06071e7	0.0121986	0.00609930	−	0.999981i	$-0.498059\pi$
$433$	2.06071e7	0.0121986	0.00609930	+	0.999981i	$0.498059\pi$
$434$	0	0
$435$	−7.46038e8	−0.434559
$436$	0	0
$437$	−1.01770e9	−0.583357
$438$	0	0
$439$	8.42596e8	0.475328	0.237664	−	0.971347i	$-0.423618\pi$
$439$	8.42596e8	0.475328	0.237664	+	0.971347i	$0.423618\pi$
$440$	0	0
$441$	−1.85145e8	−0.102796
$442$	0	0
$443$	−1.61074e9	−0.880261	−0.440130	−	0.897934i	$-0.645068\pi$
$443$	−1.61074e9	−0.880261	−0.440130	+	0.897934i	$0.645068\pi$
$444$	0	0
$445$	3.75219e8	0.201848
$446$	0	0
$447$	1.18124e9	0.625551
$448$	0	0
$449$	−2.53247e9	−1.32033	−0.660164	−	0.751121i	$-0.729513\pi$
$449$	−2.53247e9	−1.32033	−0.660164	+	0.751121i	$0.729513\pi$
$450$	0	0
$451$	2.73049e9	1.40159
$452$	0	0
$453$	1.90190e8	0.0961265
$454$	0	0
$455$	1.18046e8	0.0587506
$456$	0	0
$457$	−2.99719e9	−1.46895	−0.734476	−	0.678634i	$-0.762572\pi$
$457$	−2.99719e9	−1.46895	−0.734476	+	0.678634i	$0.762572\pi$
$458$	0	0
$459$	−2.31143e8	−0.111567
$460$	0	0
$461$	3.49092e9	1.65954	0.829768	−	0.558108i	$-0.188472\pi$
$461$	3.49092e9	1.65954	0.829768	+	0.558108i	$0.188472\pi$
$462$	0	0
$463$	1.06382e9	0.498122	0.249061	−	0.968488i	$-0.419878\pi$
$463$	1.06382e9	0.498122	0.249061	+	0.968488i	$0.419878\pi$
$464$	0	0
$465$	1.68118e8	0.0775405
$466$	0	0
$467$	−4.06159e9	−1.84539	−0.922694	−	0.385534i	$-0.874017\pi$
$467$	−4.06159e9	−1.84539	−0.922694	+	0.385534i	$0.874017\pi$
$468$	0	0
$469$	9.80922e8	0.439066
$470$	0	0
$471$	1.82565e9	0.805091
$472$	0	0
$473$	2.44372e9	1.06179
$474$	0	0
$475$	1.47933e9	0.633342
$476$	0	0
$477$	2.92590e8	0.123437
$478$	0	0
$479$	1.65959e9	0.689964	0.344982	−	0.938609i	$-0.387885\pi$
$479$	1.65959e9	0.689964	0.344982	+	0.938609i	$0.387885\pi$
$480$	0	0
$481$	−6.81061e8	−0.279047
$482$	0	0
$483$	2.17115e8	0.0876748
$484$	0	0
$485$	−3.07954e9	−1.22572
$486$	0	0
$487$	2.58745e9	1.01513	0.507563	−	0.861615i	$-0.330547\pi$
$487$	2.58745e9	1.01513	0.507563	+	0.861615i	$0.330547\pi$
$488$	0	0
$489$	1.34299e9	0.519389
$490$	0	0
$491$	−8.22361e8	−0.313529	−0.156764	−	0.987636i	$-0.550106\pi$
$491$	−8.22361e8	−0.313529	−0.156764	+	0.987636i	$0.550106\pi$
$492$	0	0
$493$	−3.69375e8	−0.138836
$494$	0	0
$495$	2.39769e9	0.888537
$496$	0	0
$497$	−1.10474e9	−0.403658
$498$	0	0
$499$	2.97850e9	1.07311	0.536557	−	0.843864i	$-0.319725\pi$
$499$	2.97850e9	1.07311	0.536557	+	0.843864i	$0.319725\pi$
$500$	0	0
$501$	2.04754e9	0.727444
$502$	0	0
$503$	−3.64779e7	−0.0127803	−0.00639016	−	0.999980i	$-0.502034\pi$
$503$	−3.64779e7	−0.0127803	−0.00639016	+	0.999980i	$0.502034\pi$
$504$	0	0
$505$	−4.09476e9	−1.41484
$506$	0	0
$507$	1.48236e9	0.505156
$508$	0	0
$509$	−1.14995e9	−0.386516	−0.193258	−	0.981148i	$-0.561905\pi$
$509$	−1.14995e9	−0.386516	−0.193258	+	0.981148i	$0.561905\pi$
$510$	0	0
$511$	−1.71941e9	−0.570043
$512$	0	0
$513$	−3.70818e9	−1.21269
$514$	0	0
$515$	1.86863e9	0.602834
$516$	0	0
$517$	5.33602e9	1.69824
$518$	0	0
$519$	1.53812e9	0.482954
$520$	0	0
$521$	−1.90283e9	−0.589478	−0.294739	−	0.955578i	$-0.595233\pi$
$521$	−1.90283e9	−0.589478	−0.294739	+	0.955578i	$0.595233\pi$
$522$	0	0
$523$	2.53692e9	0.775445	0.387723	−	0.921776i	$-0.373262\pi$
$523$	2.53692e9	0.775445	0.387723	+	0.921776i	$0.373262\pi$
$524$	0	0
$525$	−3.15600e8	−0.0951873
$526$	0	0
$527$	8.32378e7	0.0247733
$528$	0	0
$529$	−2.75151e9	−0.808120
$530$	0	0
$531$	1.87856e9	0.544495
$532$	0	0
$533$	6.16778e8	0.176434
$534$	0	0
$535$	1.94169e9	0.548202
$536$	0	0
$537$	−1.17393e9	−0.327138
$538$	0	0
$539$	−8.85566e8	−0.243590
$540$	0	0
$541$	−2.06525e9	−0.560766	−0.280383	−	0.959888i	$-0.590462\pi$
$541$	−2.06525e9	−0.560766	−0.280383	+	0.959888i	$0.590462\pi$
$542$	0	0
$543$	−3.23124e9	−0.866103
$544$	0	0
$545$	−2.05871e9	−0.544763
$546$	0	0
$547$	−5.74689e9	−1.50133	−0.750667	−	0.660680i	$-0.770268\pi$
$547$	−5.74689e9	−1.50133	−0.750667	+	0.660680i	$0.770268\pi$
$548$	0	0
$549$	−3.95321e9	−1.01964
$550$	0	0
$551$	−5.92581e9	−1.50910
$552$	0	0
$553$	−2.03813e9	−0.512501
$554$	0	0
$555$	−2.00787e9	−0.498552
$556$	0	0
$557$	4.98174e9	1.22148	0.610742	−	0.791829i	$-0.290871\pi$
$557$	4.98174e9	1.22148	0.610742	+	0.791829i	$0.290871\pi$
$558$	0	0
$559$	5.52000e8	0.133659
$560$	0	0
$561$	−4.62641e8	−0.110630
$562$	0	0
$563$	−6.24012e8	−0.147372	−0.0736858	−	0.997282i	$-0.523476\pi$
$563$	−6.24012e8	−0.147372	−0.0736858	+	0.997282i	$0.523476\pi$
$564$	0	0
$565$	2.21563e8	0.0516805
$566$	0	0
$567$	−3.89401e8	−0.0897132
$568$	0	0
$569$	4.25825e9	0.969032	0.484516	−	0.874782i	$-0.338996\pi$
$569$	4.25825e9	0.969032	0.484516	+	0.874782i	$0.338996\pi$
$570$	0	0
$571$	4.51572e8	0.101508	0.0507541	−	0.998711i	$-0.483838\pi$
$571$	4.51572e8	0.101508	0.0507541	+	0.998711i	$0.483838\pi$
$572$	0	0
$573$	9.64294e8	0.214125
$574$	0	0
$575$	−9.49664e8	−0.208321
$576$	0	0
$577$	−1.38624e8	−0.0300415	−0.0150208	−	0.999887i	$-0.504781\pi$
$577$	−1.38624e8	−0.0300415	−0.0150208	+	0.999887i	$0.504781\pi$
$578$	0	0
$579$	−3.26680e9	−0.699435
$580$	0	0
$581$	3.49882e9	0.740126
$582$	0	0
$583$	1.39948e9	0.292501
$584$	0	0
$585$	5.41604e8	0.111850
$586$	0	0
$587$	4.56072e9	0.930680	0.465340	−	0.885132i	$-0.345932\pi$
$587$	4.56072e9	0.930680	0.465340	+	0.885132i	$0.345932\pi$
$588$	0	0
$589$	1.33537e9	0.269276
$590$	0	0
$591$	−5.59873e8	−0.111566
$592$	0	0
$593$	−2.47185e9	−0.486777	−0.243389	−	0.969929i	$-0.578259\pi$
$593$	−2.47185e9	−0.486777	−0.243389	+	0.969929i	$0.578259\pi$
$594$	0	0
$595$	1.72309e8	0.0335351
$596$	0	0
$597$	−2.28883e8	−0.0440255
$598$	0	0
$599$	1.49970e9	0.285108	0.142554	−	0.989787i	$-0.454468\pi$
$599$	1.49970e9	0.285108	0.142554	+	0.989787i	$0.454468\pi$
$600$	0	0
$601$	−3.00231e9	−0.564151	−0.282076	−	0.959392i	$-0.591023\pi$
$601$	−3.00231e9	−0.564151	−0.282076	+	0.959392i	$0.591023\pi$
$602$	0	0
$603$	4.50054e9	0.835899
$604$	0	0
$605$	7.52394e9	1.38134
$606$	0	0
$607$	6.29912e9	1.14319	0.571597	−	0.820535i	$-0.306324\pi$
$607$	6.29912e9	1.14319	0.571597	+	0.820535i	$0.306324\pi$
$608$	0	0
$609$	1.26421e9	0.226808
$610$	0	0
$611$	1.20533e9	0.213777
$612$	0	0
$613$	−7.17649e9	−1.25835	−0.629174	−	0.777265i	$-0.716607\pi$
$613$	−7.17649e9	−1.25835	−0.629174	+	0.777265i	$0.716607\pi$
$614$	0	0
$615$	1.81835e9	0.315221
$616$	0	0
$617$	7.21636e9	1.23686	0.618430	−	0.785840i	$-0.287769\pi$
$617$	7.21636e9	1.23686	0.618430	+	0.785840i	$0.287769\pi$
$618$	0	0
$619$	−2.99602e9	−0.507724	−0.253862	−	0.967240i	$-0.581701\pi$
$619$	−2.99602e9	−0.507724	−0.253862	+	0.967240i	$0.581701\pi$
$620$	0	0
$621$	2.38048e9	0.398882
$622$	0	0
$623$	−6.35831e8	−0.105350
$624$	0	0
$625$	−1.82040e9	−0.298255
$626$	0	0
$627$	−7.42206e9	−1.20251
$628$	0	0
$629$	−9.94127e8	−0.159281
$630$	0	0
$631$	3.74531e7	0.00593452	0.00296726	−	0.999996i	$-0.499055\pi$
$631$	3.74531e7	0.00593452	0.00296726	+	0.999996i	$0.499055\pi$
$632$	0	0
$633$	1.46389e9	0.229400
$634$	0	0
$635$	4.31888e9	0.669365
$636$	0	0
$637$	−2.00037e8	−0.0306635
$638$	0	0
$639$	−5.06864e9	−0.768490
$640$	0	0
$641$	8.58701e9	1.28777	0.643885	−	0.765122i	$-0.277321\pi$
$641$	8.58701e9	1.28777	0.643885	+	0.765122i	$0.277321\pi$
$642$	0	0
$643$	−5.15511e9	−0.764715	−0.382358	−	0.924014i	$-0.624888\pi$
$643$	−5.15511e9	−0.764715	−0.382358	+	0.924014i	$0.624888\pi$
$644$	0	0
$645$	1.62738e9	0.238798
$646$	0	0
$647$	2.12153e9	0.307954	0.153977	−	0.988074i	$-0.450792\pi$
$647$	2.12153e9	0.307954	0.153977	+	0.988074i	$0.450792\pi$
$648$	0	0
$649$	8.98533e9	1.29026
$650$	0	0
$651$	−2.84886e8	−0.0404704
$652$	0	0
$653$	9.90394e9	1.39191	0.695956	−	0.718085i	$-0.254981\pi$
$653$	9.90394e9	1.39191	0.695956	+	0.718085i	$0.254981\pi$
$654$	0	0
$655$	−2.59763e9	−0.361187
$656$	0	0
$657$	−7.88878e9	−1.08525
$658$	0	0
$659$	−9.80160e9	−1.33413	−0.667065	−	0.745000i	$-0.732450\pi$
$659$	−9.80160e9	−1.33413	−0.667065	+	0.745000i	$0.732450\pi$
$660$	0	0
$661$	−3.31625e9	−0.446624	−0.223312	−	0.974747i	$-0.571687\pi$
$661$	−3.31625e9	−0.446624	−0.223312	+	0.974747i	$0.571687\pi$
$662$	0	0
$663$	−1.04504e8	−0.0139263
$664$	0	0
$665$	2.76432e9	0.364513
$666$	0	0
$667$	3.80410e9	0.496376
$668$	0	0
$669$	−6.21202e9	−0.802123
$670$	0	0
$671$	−1.89086e10	−2.41619
$672$	0	0
$673$	3.23323e9	0.408869	0.204434	−	0.978880i	$-0.434465\pi$
$673$	3.23323e9	0.408869	0.204434	+	0.978880i	$0.434465\pi$
$674$	0	0
$675$	−3.46029e9	−0.433061
$676$	0	0
$677$	−7.81606e9	−0.968116	−0.484058	−	0.875036i	$-0.660838\pi$
$677$	−7.81606e9	−0.968116	−0.484058	+	0.875036i	$0.660838\pi$
$678$	0	0
$679$	5.21847e9	0.639733
$680$	0	0
$681$	−2.00616e9	−0.243416
$682$	0	0
$683$	7.39597e9	0.888225	0.444112	−	0.895971i	$-0.353519\pi$
$683$	7.39597e9	0.888225	0.444112	+	0.895971i	$0.353519\pi$
$684$	0	0
$685$	−8.25962e9	−0.981845
$686$	0	0
$687$	7.76182e9	0.913303
$688$	0	0
$689$	3.16123e8	0.0368205
$690$	0	0
$691$	−3.91291e9	−0.451156	−0.225578	−	0.974225i	$-0.572427\pi$
$691$	−3.91291e9	−0.451156	−0.225578	+	0.974225i	$0.572427\pi$
$692$	0	0
$693$	−4.06304e9	−0.463750
$694$	0	0
$695$	3.07679e9	0.347657
$696$	0	0
$697$	9.00295e8	0.100709
$698$	0	0
$699$	5.85269e8	0.0648165
$700$	0	0
$701$	−3.78702e9	−0.415226	−0.207613	−	0.978211i	$-0.566570\pi$
$701$	−3.78702e9	−0.415226	−0.207613	+	0.978211i	$0.566570\pi$
$702$	0	0
$703$	−1.59486e10	−1.73132
$704$	0	0
$705$	3.55349e9	0.381939
$706$	0	0
$707$	6.93882e9	0.738444
$708$	0	0
$709$	−6.02193e8	−0.0634561	−0.0317281	−	0.999497i	$-0.510101\pi$
$709$	−6.02193e8	−0.0634561	−0.0317281	+	0.999497i	$0.510101\pi$
$710$	0	0
$711$	−9.35109e9	−0.975705
$712$	0	0
$713$	−8.57244e8	−0.0885709
$714$	0	0
$715$	2.59054e9	0.265045
$716$	0	0
$717$	6.56606e9	0.665254
$718$	0	0
$719$	4.09602e9	0.410971	0.205485	−	0.978660i	$-0.434123\pi$
$719$	4.09602e9	0.410971	0.205485	+	0.978660i	$0.434123\pi$
$720$	0	0
$721$	−3.16651e9	−0.314635
$722$	0	0
$723$	−1.17995e8	−0.0116113
$724$	0	0
$725$	−5.52966e9	−0.538909
$726$	0	0
$727$	−5.00935e9	−0.483516	−0.241758	−	0.970337i	$-0.577724\pi$
$727$	−5.00935e9	−0.483516	−0.241758	+	0.970337i	$0.577724\pi$
$728$	0	0
$729$	3.25753e9	0.311417
$730$	0	0
$731$	8.05741e8	0.0762930
$732$	0	0
$733$	8.02679e9	0.752797	0.376399	−	0.926458i	$-0.377162\pi$
$733$	8.02679e9	0.752797	0.376399	+	0.926458i	$0.377162\pi$
$734$	0	0
$735$	−5.89738e8	−0.0547840
$736$	0	0
$737$	2.15265e10	1.98078
$738$	0	0
$739$	5.79998e9	0.528654	0.264327	−	0.964433i	$-0.414850\pi$
$739$	5.79998e9	0.528654	0.264327	+	0.964433i	$0.414850\pi$
$740$	0	0
$741$	−1.67654e9	−0.151373
$742$	0	0
$743$	1.67943e10	1.50211	0.751056	−	0.660239i	$-0.229545\pi$
$743$	1.67943e10	1.50211	0.751056	+	0.660239i	$0.229545\pi$
$744$	0	0
$745$	−9.65477e9	−0.855451
$746$	0	0
$747$	1.60528e10	1.40906
$748$	0	0
$749$	−3.29031e9	−0.286121
$750$	0	0
$751$	−4.86529e9	−0.419150	−0.209575	−	0.977793i	$-0.567208\pi$
$751$	−4.86529e9	−0.419150	−0.209575	+	0.977793i	$0.567208\pi$
$752$	0	0
$753$	−6.95327e9	−0.593481
$754$	0	0
$755$	−1.55450e9	−0.131455
$756$	0	0
$757$	9.00246e9	0.754268	0.377134	−	0.926159i	$-0.376910\pi$
$757$	9.00246e9	0.754268	0.377134	+	0.926159i	$0.376910\pi$
$758$	0	0
$759$	4.76462e9	0.395533
$760$	0	0
$761$	−1.37707e10	−1.13268	−0.566342	−	0.824170i	$-0.691642\pi$
$761$	−1.37707e10	−1.13268	−0.566342	+	0.824170i	$0.691642\pi$
$762$	0	0
$763$	3.48861e9	0.284326
$764$	0	0
$765$	7.90566e8	0.0638445
$766$	0	0
$767$	2.02966e9	0.162420
$768$	0	0
$769$	1.80342e10	1.43006	0.715030	−	0.699094i	$-0.246413\pi$
$769$	1.80342e10	1.43006	0.715030	+	0.699094i	$0.246413\pi$
$770$	0	0
$771$	−1.45118e9	−0.114033
$772$	0	0
$773$	1.91942e10	1.49466	0.747328	−	0.664455i	$-0.231336\pi$
$773$	1.91942e10	1.49466	0.747328	+	0.664455i	$0.231336\pi$
$774$	0	0
$775$	1.24609e9	0.0961602
$776$	0	0
$777$	3.40246e9	0.260207
$778$	0	0
$779$	1.44433e10	1.09467
$780$	0	0
$781$	−2.42438e10	−1.82105
$782$	0	0
$783$	1.38610e10	1.03188
$784$	0	0
$785$	−1.49218e10	−1.10098
$786$	0	0
$787$	−4.00491e8	−0.0292874	−0.0146437	−	0.999893i	$-0.504661\pi$
$787$	−4.00491e8	−0.0292874	−0.0146437	+	0.999893i	$0.504661\pi$
$788$	0	0
$789$	−3.04834e9	−0.220950
$790$	0	0
$791$	−3.75451e8	−0.0269734
$792$	0	0
$793$	−4.27118e9	−0.304153
$794$	0	0
$795$	9.31980e8	0.0657842
$796$	0	0
$797$	−2.57300e10	−1.80026	−0.900129	−	0.435623i	$-0.856528\pi$
$797$	−2.57300e10	−1.80026	−0.900129	+	0.435623i	$0.856528\pi$
$798$	0	0
$799$	1.75939e9	0.122025
$800$	0	0
$801$	−2.91723e9	−0.200566
$802$	0	0
$803$	−3.77328e10	−2.57167
$804$	0	0
$805$	−1.77457e9	−0.119897
$806$	0	0
$807$	1.19466e9	0.0800180
$808$	0	0
$809$	1.42308e10	0.944954	0.472477	−	0.881343i	$-0.343360\pi$
$809$	1.42308e10	0.944954	0.472477	+	0.881343i	$0.343360\pi$
$810$	0	0
$811$	7.90561e9	0.520430	0.260215	−	0.965551i	$-0.416207\pi$
$811$	7.90561e9	0.520430	0.260215	+	0.965551i	$0.416207\pi$
$812$	0	0
$813$	−1.34531e10	−0.878022
$814$	0	0
$815$	−1.09768e10	−0.710272
$816$	0	0
$817$	1.29263e10	0.829275
$818$	0	0
$819$	−9.17781e8	−0.0583775
$820$	0	0
$821$	−1.10081e10	−0.694245	−0.347122	−	0.937820i	$-0.612841\pi$
$821$	−1.10081e10	−0.694245	−0.347122	+	0.937820i	$0.612841\pi$
$822$	0	0
$823$	2.34709e10	1.46767	0.733837	−	0.679325i	$-0.237727\pi$
$823$	2.34709e10	1.46767	0.733837	+	0.679325i	$0.237727\pi$
$824$	0	0
$825$	−6.92588e9	−0.429424
$826$	0	0
$827$	−8.50710e9	−0.523013	−0.261506	−	0.965202i	$-0.584219\pi$
$827$	−8.50710e9	−0.523013	−0.261506	+	0.965202i	$0.584219\pi$
$828$	0	0
$829$	−1.03685e9	−0.0632083	−0.0316042	−	0.999500i	$-0.510062\pi$
$829$	−1.03685e9	−0.0632083	−0.0316042	+	0.999500i	$0.510062\pi$
$830$	0	0
$831$	5.02414e9	0.303709
$832$	0	0
$833$	−2.91989e8	−0.0175028
$834$	0	0
$835$	−1.67353e10	−0.994791
$836$	0	0
$837$	−3.12353e9	−0.184123
$838$	0	0
$839$	3.03376e10	1.77343	0.886716	−	0.462315i	$-0.152981\pi$
$839$	3.03376e10	1.77343	0.886716	+	0.462315i	$0.152981\pi$
$840$	0	0
$841$	4.90042e9	0.284084
$842$	0	0
$843$	−1.71342e10	−0.985069
$844$	0	0
$845$	−1.21159e10	−0.690808
$846$	0	0
$847$	−1.27498e10	−0.720959
$848$	0	0
$849$	−9.43105e9	−0.528912
$850$	0	0
$851$	1.02383e10	0.569472
$852$	0	0
$853$	−2.76476e9	−0.152523	−0.0762617	−	0.997088i	$-0.524298\pi$
$853$	−2.76476e9	−0.152523	−0.0762617	+	0.997088i	$0.524298\pi$
$854$	0	0
$855$	1.26829e10	0.693964
$856$	0	0
$857$	1.38837e10	0.753483	0.376742	−	0.926318i	$-0.377044\pi$
$857$	1.38837e10	0.753483	0.376742	+	0.926318i	$0.377044\pi$
$858$	0	0
$859$	1.45970e10	0.785758	0.392879	−	0.919590i	$-0.371479\pi$
$859$	1.45970e10	0.785758	0.392879	+	0.919590i	$0.371479\pi$
$860$	0	0
$861$	−3.08131e9	−0.164522
$862$	0	0
$863$	−3.56600e10	−1.88861	−0.944307	−	0.329065i	$-0.893266\pi$
$863$	−3.56600e10	−1.88861	−0.944307	+	0.329065i	$0.893266\pi$
$864$	0	0
$865$	−1.25717e10	−0.660447
$866$	0	0
$867$	1.00094e10	0.521604
$868$	0	0
$869$	−4.47271e10	−2.31208
$870$	0	0
$871$	4.86252e9	0.249344
$872$	0	0
$873$	2.39427e10	1.21793
$874$	0	0
$875$	8.00354e9	0.403882
$876$	0	0
$877$	5.28586e9	0.264616	0.132308	−	0.991209i	$-0.457761\pi$
$877$	5.28586e9	0.264616	0.132308	+	0.991209i	$0.457761\pi$
$878$	0	0
$879$	2.40763e9	0.119572
$880$	0	0
$881$	5.21604e9	0.256996	0.128498	−	0.991710i	$-0.458984\pi$
$881$	5.21604e9	0.256996	0.128498	+	0.991710i	$0.458984\pi$
$882$	0	0
$883$	1.21534e10	0.594065	0.297032	−	0.954867i	$-0.404003\pi$
$883$	1.21534e10	0.594065	0.297032	+	0.954867i	$0.404003\pi$
$884$	0	0
$885$	5.98374e9	0.290182
$886$	0	0
$887$	−1.17188e10	−0.563833	−0.281916	−	0.959439i	$-0.590970\pi$
$887$	−1.17188e10	−0.563833	−0.281916	+	0.959439i	$0.590970\pi$
$888$	0	0
$889$	−7.31860e9	−0.349359
$890$	0	0
$891$	−8.54546e9	−0.404728
$892$	0	0
$893$	2.82255e10	1.32636
$894$	0	0
$895$	9.59499e9	0.447367
$896$	0	0
$897$	1.07626e9	0.0497902
$898$	0	0
$899$	−4.99152e9	−0.229126
$900$	0	0
$901$	4.61437e8	0.0210173
$902$	0	0
$903$	−2.75770e9	−0.124635
$904$	0	0
$905$	2.64102e10	1.18441
$906$	0	0
$907$	1.59433e10	0.709503	0.354751	−	0.934961i	$-0.384566\pi$
$907$	1.59433e10	0.709503	0.354751	+	0.934961i	$0.384566\pi$
$908$	0	0
$909$	3.18357e10	1.40586
$910$	0	0
$911$	−1.93529e10	−0.848071	−0.424035	−	0.905646i	$-0.639387\pi$
$911$	−1.93529e10	−0.848071	−0.424035	+	0.905646i	$0.639387\pi$
$912$	0	0
$913$	7.67822e10	3.33897
$914$	0	0
$915$	−1.25921e10	−0.543405
$916$	0	0
$917$	4.40183e9	0.188513
$918$	0	0
$919$	−1.32527e10	−0.563248	−0.281624	−	0.959525i	$-0.590873\pi$
$919$	−1.32527e10	−0.563248	−0.281624	+	0.959525i	$0.590873\pi$
$920$	0	0
$921$	2.13122e10	0.898915
$922$	0	0
$923$	−5.47632e9	−0.229236
$924$	0	0
$925$	−1.48824e10	−0.618268
$926$	0	0
$927$	−1.45281e10	−0.599006
$928$	0	0
$929$	−2.41624e10	−0.988745	−0.494373	−	0.869250i	$-0.664602\pi$
$929$	−2.41624e10	−0.988745	−0.494373	+	0.869250i	$0.664602\pi$
$930$	0	0
$931$	−4.68431e9	−0.190249
$932$	0	0
$933$	−1.16105e10	−0.468019
$934$	0	0
$935$	3.78135e9	0.151289
$936$	0	0
$937$	2.73784e10	1.08723	0.543613	−	0.839336i	$-0.317056\pi$
$937$	2.73784e10	1.08723	0.543613	+	0.839336i	$0.317056\pi$
$938$	0	0
$939$	1.32333e10	0.521601
$940$	0	0
$941$	−1.02226e10	−0.399944	−0.199972	−	0.979802i	$-0.564085\pi$
$941$	−1.02226e10	−0.399944	−0.199972	+	0.979802i	$0.564085\pi$
$942$	0	0
$943$	−9.27191e9	−0.360063
$944$	0	0
$945$	−6.46599e9	−0.249243
$946$	0	0
$947$	1.35222e9	0.0517397	0.0258698	−	0.999665i	$-0.491764\pi$
$947$	1.35222e9	0.0517397	0.0258698	+	0.999665i	$0.491764\pi$
$948$	0	0
$949$	−8.52330e9	−0.323725
$950$	0	0
$951$	8.34365e9	0.314575
$952$	0	0
$953$	−2.33119e10	−0.872476	−0.436238	−	0.899831i	$-0.643689\pi$
$953$	−2.33119e10	−0.872476	−0.436238	+	0.899831i	$0.643689\pi$
$954$	0	0
$955$	−7.88156e9	−0.292820
$956$	0	0
$957$	2.77432e10	1.02321
$958$	0	0
$959$	1.39964e10	0.512451
$960$	0	0
$961$	−2.63878e10	−0.959116
$962$	0	0
$963$	−1.50961e10	−0.544721
$964$	0	0
$965$	2.67009e10	0.956488
$966$	0	0
$967$	2.07860e10	0.739229	0.369615	−	0.929185i	$-0.379490\pi$
$967$	2.07860e10	0.739229	0.369615	+	0.929185i	$0.379490\pi$
$968$	0	0
$969$	−2.44720e9	−0.0864045
$970$	0	0
$971$	−5.56486e10	−1.95068	−0.975341	−	0.220702i	$-0.929165\pi$
$971$	−5.56486e10	−1.95068	−0.975341	+	0.220702i	$0.929165\pi$
$972$	0	0
$973$	−5.21380e9	−0.181451
$974$	0	0
$975$	−1.56446e9	−0.0540565
$976$	0	0
$977$	−3.74779e10	−1.28571	−0.642857	−	0.765986i	$-0.722251\pi$
$977$	−3.74779e10	−1.28571	−0.642857	+	0.765986i	$0.722251\pi$
$978$	0	0
$979$	−1.39534e10	−0.475270
$980$	0	0
$981$	1.60060e10	0.541303
$982$	0	0
$983$	−2.07940e10	−0.698234	−0.349117	−	0.937079i	$-0.613518\pi$
$983$	−2.07940e10	−0.698234	−0.349117	+	0.937079i	$0.613518\pi$
$984$	0	0
$985$	4.57607e9	0.152569
$986$	0	0
$987$	−6.02161e9	−0.199344
$988$	0	0
$989$	−8.29812e9	−0.272768
$990$	0	0
$991$	−4.17472e10	−1.36260	−0.681301	−	0.732003i	$-0.738586\pi$
$991$	−4.17472e10	−1.36260	−0.681301	+	0.732003i	$0.738586\pi$
$992$	0	0
$993$	−1.37515e10	−0.445686
$994$	0	0
$995$	1.87076e9	0.0602055
$996$	0	0
$997$	−1.18879e10	−0.379902	−0.189951	−	0.981794i	$-0.560833\pi$
$997$	−1.18879e10	−0.379902	−0.189951	+	0.981794i	$0.560833\pi$
$998$	0	0
$999$	3.73051e10	1.18383

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	28.8.a.b.1.1	✓	2
3.2	odd	2		252.8.a.f.1.1		2
4.3	odd	2		112.8.a.h.1.2		2
7.2	even	3		196.8.e.b.165.2		4
7.3	odd	6		196.8.e.c.177.1		4
7.4	even	3		196.8.e.b.177.2		4
7.5	odd	6		196.8.e.c.165.1		4
7.6	odd	2		196.8.a.a.1.2		2
8.3	odd	2		448.8.a.q.1.1		2
8.5	even	2		448.8.a.o.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
28.8.a.b.1.1	✓	2	1.1	even	1	trivial
112.8.a.h.1.2		2	4.3	odd	2
196.8.a.a.1.2		2	7.6	odd	2
196.8.e.b.165.2		4	7.2	even	3
196.8.e.b.177.2		4	7.4	even	3
196.8.e.c.165.1		4	7.5	odd	6
196.8.e.c.177.1		4	7.3	odd	6
252.8.a.f.1.1		2	3.2	odd	2
448.8.a.o.1.2		2	8.5	even	2
448.8.a.q.1.1		2	8.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 28 = 2^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	28.a (trivial)

\(f(q)\)	\(=\)	\(q-24.7648 q^{3} +202.412 q^{5} -343.000 q^{7} -1573.71 q^{9} -7527.19 q^{11} -1700.28 q^{13} -5012.69 q^{15} -2481.86 q^{17} -39816.0 q^{19} +8494.31 q^{21} +25560.1 q^{23} -37154.2 q^{25} +93133.0 q^{27} +148830. q^{29} -33538.4 q^{31} +186409. q^{33} -69427.4 q^{35} +400557. q^{37} +42107.1 q^{39} -362750. q^{41} -324652. q^{43} -318538. q^{45} -708899. q^{47} +117649. q^{49} +61462.7 q^{51} -185924. q^{53} -1.52360e6 q^{55} +986034. q^{57} -1.19372e6 q^{59} +2.51204e6 q^{61} +539781. q^{63} -344158. q^{65} -2.85983e6 q^{67} -632989. q^{69} +3.22083e6 q^{71} +5.01287e6 q^{73} +920116. q^{75} +2.58182e6 q^{77} +5.94208e6 q^{79} +1.13528e6 q^{81} -1.02006e7 q^{83} -502359. q^{85} -3.68573e6 q^{87} +1.85373e6 q^{89} +583197. q^{91} +830571. q^{93} -8.05925e6 q^{95} -1.52142e7 q^{97} +1.18456e7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 14 q^{3} - 294 q^{5} - 686 q^{7} - 2258 q^{9} - 3492 q^{11} - 16170 q^{13} - 24256 q^{15} - 29232 q^{17} - 3206 q^{19} - 4802 q^{21} - 9360 q^{23} + 131146 q^{25} - 18172 q^{27} + 184704 q^{29} + 165060 q^{31}+ \cdots + 9084332 q^{99}+O(q^{100}) \) 2 * q + 14 * q^3 - 294 * q^5 - 686 * q^7 - 2258 * q^9 - 3492 * q^11 - 16170 * q^13 - 24256 * q^15 - 29232 * q^17 - 3206 * q^19 - 4802 * q^21 - 9360 * q^23 + 131146 * q^25 - 18172 * q^27 + 184704 * q^29 + 165060 * q^31 + 342832 * q^33 + 100842 * q^35 + 286144 * q^37 - 518808 * q^39 - 116760 * q^41 - 294428 * q^43 + 21154 * q^45 - 1014132 * q^47 + 235298 * q^49 - 975500 * q^51 - 1396452 * q^53 - 3526712 * q^55 + 2405212 * q^57 - 2729286 * q^59 + 2466954 * q^61 + 774494 * q^63 + 6838788 * q^65 + 225176 * q^67 - 1986656 * q^69 + 1530312 * q^71 + 1143548 * q^73 + 7444234 * q^75 + 1197756 * q^77 + 7951176 * q^79 - 1682882 * q^81 - 18487854 * q^83 + 12776740 * q^85 - 2295076 * q^87 - 4652508 * q^89 + 5546310 * q^91 + 8529192 * q^93 - 26232912 * q^95 - 26702368 * q^97 + 9084332 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more