LMFDB - Embedded newform 1344.4.c.b.673.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1344,4,Mod(673,1344)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1344, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 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("1344.673");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1344 = 2^{6} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1344.c (of order $2$, degree $1$, minimal)

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:	no
Analytic conductor:	$79.2985670477$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.14024243776.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 17x^{4} - 164x^{3} + 299x^{2} + 2466x + 13042 $ x^6 - 2x^5 - 17x^4 - 164x^3 + 299x^2 + 2466*x + 13042
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			673.6
Root			$5.82579 - 1.30833i$ of defining polynomial
Character	$\chi$	$=$	1344.673
Dual form			1344.4.c.b.673.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.00000i q^{3} +7.03493i q^{5} -7.00000 q^{7} -9.00000 q^{9} +O(q^{10})$	$q+3.00000i q^{3} +7.03493i q^{5} -7.00000 q^{7} -9.00000 q^{9} -46.7348i q^{11} +8.90643i q^{13} -21.1048 q^{15} +58.0811 q^{17} +135.122i q^{19} -21.0000i q^{21} +86.0112 q^{23} +75.5097 q^{25} -27.0000i q^{27} -300.014i q^{29} -183.712 q^{31} +140.205 q^{33} -49.2445i q^{35} +418.319i q^{37} -26.7193 q^{39} -106.081 q^{41} -162.206i q^{43} -63.3144i q^{45} +377.751 q^{47} +49.0000 q^{49} +174.243i q^{51} +709.548i q^{53} +328.777 q^{55} -405.367 q^{57} +753.348i q^{59} +667.954i q^{61} +63.0000 q^{63} -62.6562 q^{65} -447.506i q^{67} +258.034i q^{69} -384.548 q^{71} +1090.16 q^{73} +226.529i q^{75} +327.144i q^{77} -670.900 q^{79} +81.0000 q^{81} -447.801i q^{83} +408.597i q^{85} +900.043 q^{87} -70.5678 q^{89} -62.3450i q^{91} -551.137i q^{93} -950.576 q^{95} -1319.10 q^{97} +420.614i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 42 q^{7} - 54 q^{9}+O(q^{10}) $ 6 * q - 42 * q^7 - 54 * q^9	$ 6 q - 42 q^{7} - 54 q^{9} + 24 q^{15} - 72 q^{17} + 196 q^{23} + 246 q^{25} - 104 q^{31} + 408 q^{33} + 348 q^{39} - 216 q^{41} + 408 q^{47} + 294 q^{49} - 720 q^{55} + 336 q^{57} + 378 q^{63} - 2472 q^{65} + 1164 q^{71} - 276 q^{73} - 1352 q^{79} + 486 q^{81} - 720 q^{87} + 2520 q^{89} - 4648 q^{95} - 2340 q^{97}+O(q^{100}) $ 6 * q - 42 * q^7 - 54 * q^9 + 24 * q^15 - 72 * q^17 + 196 * q^23 + 246 * q^25 - 104 * q^31 + 408 * q^33 + 348 * q^39 - 216 * q^41 + 408 * q^47 + 294 * q^49 - 720 * q^55 + 336 * q^57 + 378 * q^63 - 2472 * q^65 + 1164 * q^71 - 276 * q^73 - 1352 * q^79 + 486 * q^81 - 720 * q^87 + 2520 * q^89 - 4648 * q^95 - 2340 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1344\mathbb{Z}\right)^\times$.

$n$	$127$	$449$	$577$	$1093$
$\chi(n)$	$1$	$1$	$1$	$-1$

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
$3$	3.00000i	0.577350i
$3$	3.00000i	0.577350i
$4$	0	0
$5$	7.03493i	0.629224i	0.949220	+	0.314612i	$0.101874\pi$
$5$	7.03493i	0.629224i	−0.949220	+	0.314612i	$0.898126\pi$
$6$	0	0
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	0	0
$9$	−9.00000	−0.333333
$10$	0	0
$11$	− 46.7348i	− 1.28101i	−0.767955	−	0.640504i	$-0.778726\pi$
$11$	− 46.7348i	− 1.28101i	0.767955	−	0.640504i	$-0.221274\pi$
$12$	0	0
$13$	8.90643i	0.190015i	0.995477	+	0.0950077i	$0.0302876\pi$
$13$	8.90643i	0.190015i	−0.995477	+	0.0950077i	$0.969712\pi$
$14$	0	0
$15$	−21.1048	−0.363282
$16$	0	0
$17$	58.0811	0.828632	0.414316	−	0.910133i	$-0.364021\pi$
$17$	58.0811	0.828632	0.414316	+	0.910133i	$0.364021\pi$
$18$	0	0
$19$	135.122i	1.63153i	0.578380	+	0.815767i	$0.303685\pi$
$19$	135.122i	1.63153i	−0.578380	+	0.815767i	$0.696315\pi$
$20$	0	0
$21$	− 21.0000i	− 0.218218i
$22$	0	0
$23$	86.0112	0.779764	0.389882	−	0.920865i	$-0.372516\pi$
$23$	86.0112	0.779764	0.389882	+	0.920865i	$0.372516\pi$
$24$	0	0
$25$	75.5097	0.604078
$26$	0	0
$27$	− 27.0000i	− 0.192450i
$28$	0	0
$29$	− 300.014i	− 1.92108i	−0.278147	−	0.960539i	$-0.589720\pi$
$29$	− 300.014i	− 1.92108i	0.278147	−	0.960539i	$-0.410280\pi$
$30$	0	0
$31$	−183.712	−1.06438	−0.532189	−	0.846626i	$-0.678630\pi$
$31$	−183.712	−1.06438	−0.532189	+	0.846626i	$0.678630\pi$
$32$	0	0
$33$	140.205	0.739590
$34$	0	0
$35$	− 49.2445i	− 0.237824i
$36$	0	0
$37$	418.319i	1.85868i	0.369225	+	0.929340i	$0.379623\pi$
$37$	418.319i	1.85868i	−0.369225	+	0.929340i	$0.620377\pi$
$38$	0	0
$39$	−26.7193	−0.109705
$40$	0	0
$41$	−106.081	−0.404075	−0.202038	−	0.979378i	$-0.564756\pi$
$41$	−106.081	−0.404075	−0.202038	+	0.979378i	$0.564756\pi$
$42$	0	0
$43$	− 162.206i	− 0.575259i	−0.957742	−	0.287629i	$-0.907133\pi$
$43$	− 162.206i	− 0.575259i	0.957742	−	0.287629i	$-0.0928671\pi$
$44$	0	0
$45$	− 63.3144i	− 0.209741i
$46$	0	0
$47$	377.751	1.17236	0.586178	−	0.810183i	$-0.300632\pi$
$47$	377.751	1.17236	0.586178	+	0.810183i	$0.300632\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	174.243i	0.478411i
$52$	0	0
$53$	709.548i	1.83894i	0.393159	+	0.919470i	$0.371382\pi$
$53$	709.548i	1.83894i	−0.393159	+	0.919470i	$0.628618\pi$
$54$	0	0
$55$	328.777	0.806041
$56$	0	0
$57$	−405.367	−0.941967
$58$	0	0
$59$	753.348i	1.66233i	0.556024	+	0.831166i	$0.312326\pi$
$59$	753.348i	1.66233i	−0.556024	+	0.831166i	$0.687674\pi$
$60$	0	0
$61$	667.954i	1.40201i	0.713155	+	0.701007i	$0.247266\pi$
$61$	667.954i	1.40201i	−0.713155	+	0.701007i	$0.752734\pi$
$62$	0	0
$63$	63.0000	0.125988
$64$	0	0
$65$	−62.6562	−0.119562
$66$	0	0
$67$	− 447.506i	− 0.815993i	−0.912983	−	0.407997i	$-0.866227\pi$
$67$	− 447.506i	− 0.815993i	0.912983	−	0.407997i	$-0.133773\pi$
$68$	0	0
$69$	258.034i	0.450197i
$70$	0	0
$71$	−384.548	−0.642780	−0.321390	−	0.946947i	$-0.604150\pi$
$71$	−384.548	−0.642780	−0.321390	+	0.946947i	$0.604150\pi$
$72$	0	0
$73$	1090.16	1.74786	0.873930	−	0.486052i	$-0.161563\pi$
$73$	1090.16	1.74786	0.873930	+	0.486052i	$0.161563\pi$
$74$	0	0
$75$	226.529i	0.348764i
$76$	0	0
$77$	327.144i	0.484175i
$78$	0	0
$79$	−670.900	−0.955470	−0.477735	−	0.878504i	$-0.658542\pi$
$79$	−670.900	−0.955470	−0.477735	+	0.878504i	$0.658542\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	− 447.801i	− 0.592199i	−0.955157	−	0.296100i	$-0.904314\pi$
$83$	− 447.801i	− 0.592199i	0.955157	−	0.296100i	$-0.0956861\pi$
$84$	0	0
$85$	408.597i	0.521395i
$86$	0	0
$87$	900.043	1.10913
$88$	0	0
$89$	−70.5678	−0.0840468	−0.0420234	−	0.999117i	$-0.513380\pi$
$89$	−70.5678	−0.0840468	−0.0420234	+	0.999117i	$0.513380\pi$
$90$	0	0
$91$	− 62.3450i	− 0.0718191i
$92$	0	0
$93$	− 551.137i	− 0.614519i
$94$	0	0
$95$	−950.576	−1.02660
$96$	0	0
$97$	−1319.10	−1.38077	−0.690385	−	0.723442i	$-0.742559\pi$
$97$	−1319.10	−1.38077	−0.690385	+	0.723442i	$0.742559\pi$
$98$	0	0
$99$	420.614i	0.427003i
$100$	0	0
$101$	704.449i	0.694013i	0.937863	+	0.347006i	$0.112802\pi$
$101$	704.449i	0.694013i	−0.937863	+	0.347006i	$0.887198\pi$
$102$	0	0
$103$	−1553.21	−1.48585	−0.742925	−	0.669374i	$-0.766562\pi$
$103$	−1553.21	−1.48585	−0.742925	+	0.669374i	$0.766562\pi$
$104$	0	0
$105$	147.734	0.137308
$106$	0	0
$107$	− 1989.18i	− 1.79721i	−0.438758	−	0.898605i	$-0.644582\pi$
$107$	− 1989.18i	− 1.79721i	0.438758	−	0.898605i	$-0.355418\pi$
$108$	0	0
$109$	− 1307.97i	− 1.14937i	−0.818376	−	0.574683i	$-0.805125\pi$
$109$	− 1307.97i	− 1.14937i	0.818376	−	0.574683i	$-0.194875\pi$
$110$	0	0
$111$	−1254.96	−1.07311
$112$	0	0
$113$	−1342.12	−1.11731	−0.558653	−	0.829401i	$-0.688682\pi$
$113$	−1342.12	−1.11731	−0.558653	+	0.829401i	$0.688682\pi$
$114$	0	0
$115$	605.083i	0.490646i
$116$	0	0
$117$	− 80.1579i	− 0.0633385i
$118$	0	0
$119$	−406.568	−0.313193
$120$	0	0
$121$	−853.146	−0.640981
$122$	0	0
$123$	− 318.243i	− 0.233293i
$124$	0	0
$125$	1410.57i	1.00932i
$126$	0	0
$127$	499.179	0.348780	0.174390	−	0.984677i	$-0.444205\pi$
$127$	499.179	0.348780	0.174390	+	0.984677i	$0.444205\pi$
$128$	0	0
$129$	486.617	0.332126
$130$	0	0
$131$	2405.14i	1.60411i	0.597252	+	0.802053i	$0.296259\pi$
$131$	2405.14i	1.60411i	−0.597252	+	0.802053i	$0.703741\pi$
$132$	0	0
$133$	− 945.856i	− 0.616662i
$134$	0	0
$135$	189.943	0.121094
$136$	0	0
$137$	390.126	0.243290	0.121645	−	0.992574i	$-0.461183\pi$
$137$	390.126	0.243290	0.121645	+	0.992574i	$0.461183\pi$
$138$	0	0
$139$	224.663i	0.137091i	0.997648	+	0.0685456i	$0.0218359\pi$
$139$	224.663i	0.137091i	−0.997648	+	0.0685456i	$0.978164\pi$
$140$	0	0
$141$	1133.25i	0.676860i
$142$	0	0
$143$	416.241	0.243411
$144$	0	0
$145$	2110.58	1.20879
$146$	0	0
$147$	147.000i	0.0824786i
$148$	0	0
$149$	2478.96i	1.36298i	0.731826	+	0.681492i	$0.238668\pi$
$149$	2478.96i	1.36298i	−0.731826	+	0.681492i	$0.761332\pi$
$150$	0	0
$151$	−305.771	−0.164790	−0.0823951	−	0.996600i	$-0.526257\pi$
$151$	−305.771	−0.164790	−0.0823951	+	0.996600i	$0.526257\pi$
$152$	0	0
$153$	−522.730	−0.276211
$154$	0	0
$155$	− 1292.40i	− 0.669731i
$156$	0	0
$157$	3901.93i	1.98349i	0.128219	+	0.991746i	$0.459074\pi$
$157$	3901.93i	1.98349i	−0.128219	+	0.991746i	$0.540926\pi$
$158$	0	0
$159$	−2128.64	−1.06171
$160$	0	0
$161$	−602.079	−0.294723
$162$	0	0
$163$	− 3243.51i	− 1.55860i	−0.626651	−	0.779300i	$-0.715575\pi$
$163$	− 3243.51i	− 1.55860i	0.626651	−	0.779300i	$-0.284425\pi$
$164$	0	0
$165$	986.330i	0.465368i
$166$	0	0
$167$	−1468.42	−0.680416	−0.340208	−	0.940350i	$-0.610497\pi$
$167$	−1468.42	−0.680416	−0.340208	+	0.940350i	$0.610497\pi$
$168$	0	0
$169$	2117.68	0.963894
$170$	0	0
$171$	− 1216.10i	− 0.543845i
$172$	0	0
$173$	179.022i	0.0786752i	0.999226	+	0.0393376i	$0.0125248\pi$
$173$	179.022i	0.0786752i	−0.999226	+	0.0393376i	$0.987475\pi$
$174$	0	0
$175$	−528.568	−0.228320
$176$	0	0
$177$	−2260.04	−0.959748
$178$	0	0
$179$	936.199i	0.390921i	0.980712	+	0.195460i	$0.0626201\pi$
$179$	936.199i	0.390921i	−0.980712	+	0.195460i	$0.937380\pi$
$180$	0	0
$181$	− 583.791i	− 0.239740i	−0.992790	−	0.119870i	$-0.961752\pi$
$181$	− 583.791i	− 0.239740i	0.992790	−	0.119870i	$-0.0382477\pi$
$182$	0	0
$183$	−2003.86	−0.809453
$184$	0	0
$185$	−2942.84	−1.16953
$186$	0	0
$187$	− 2714.41i	− 1.06148i
$188$	0	0
$189$	189.000i	0.0727393i
$190$	0	0
$191$	−1045.38	−0.396026	−0.198013	−	0.980199i	$-0.563449\pi$
$191$	−1045.38	−0.396026	−0.198013	+	0.980199i	$0.563449\pi$
$192$	0	0
$193$	562.178	0.209671	0.104835	−	0.994490i	$-0.466568\pi$
$193$	562.178	0.209671	0.104835	+	0.994490i	$0.466568\pi$
$194$	0	0
$195$	− 187.969i	− 0.0690293i
$196$	0	0
$197$	1561.54i	0.564747i	0.959305	+	0.282374i	$0.0911218\pi$
$197$	1561.54i	0.564747i	−0.959305	+	0.282374i	$0.908878\pi$
$198$	0	0
$199$	−3770.70	−1.34321	−0.671603	−	0.740911i	$-0.734394\pi$
$199$	−3770.70	−1.34321	−0.671603	+	0.740911i	$0.734394\pi$
$200$	0	0
$201$	1342.52	0.471114
$202$	0	0
$203$	2100.10i	0.726099i
$204$	0	0
$205$	− 746.274i	− 0.254254i
$206$	0	0
$207$	−774.101	−0.259921
$208$	0	0
$209$	6314.92	2.09001
$210$	0	0
$211$	4716.75i	1.53893i	0.638687	+	0.769466i	$0.279478\pi$
$211$	4716.75i	1.53893i	−0.638687	+	0.769466i	$0.720522\pi$
$212$	0	0
$213$	− 1153.64i	− 0.371109i
$214$	0	0
$215$	1141.11	0.361966
$216$	0	0
$217$	1285.99	0.402297
$218$	0	0
$219$	3270.49i	1.00913i
$220$	0	0
$221$	517.296i	0.157453i
$222$	0	0
$223$	−4684.99	−1.40686	−0.703431	−	0.710764i	$-0.748350\pi$
$223$	−4684.99	−1.40686	−0.703431	+	0.710764i	$0.748350\pi$
$224$	0	0
$225$	−679.587	−0.201359
$226$	0	0
$227$	993.424i	0.290466i	0.989397	+	0.145233i	$0.0463932\pi$
$227$	993.424i	0.290466i	−0.989397	+	0.145233i	$0.953607\pi$
$228$	0	0
$229$	3998.74i	1.15390i	0.816778	+	0.576952i	$0.195758\pi$
$229$	3998.74i	1.15390i	−0.816778	+	0.576952i	$0.804242\pi$
$230$	0	0
$231$	−981.432	−0.279539
$232$	0	0
$233$	763.955	0.214800	0.107400	−	0.994216i	$-0.465747\pi$
$233$	763.955	0.214800	0.107400	+	0.994216i	$0.465747\pi$
$234$	0	0
$235$	2657.46i	0.737674i
$236$	0	0
$237$	− 2012.70i	− 0.551641i
$238$	0	0
$239$	−6244.06	−1.68994	−0.844968	−	0.534816i	$-0.820381\pi$
$239$	−6244.06	−1.68994	−0.844968	+	0.534816i	$0.820381\pi$
$240$	0	0
$241$	−4590.71	−1.22703	−0.613513	−	0.789684i	$-0.710244\pi$
$241$	−4590.71	−1.22703	−0.613513	+	0.789684i	$0.710244\pi$
$242$	0	0
$243$	243.000i	0.0641500i
$244$	0	0
$245$	344.712i	0.0898891i
$246$	0	0
$247$	−1203.46	−0.310017
$248$	0	0
$249$	1343.40	0.341906
$250$	0	0
$251$	− 894.864i	− 0.225033i	−0.993650	−	0.112517i	$-0.964109\pi$
$251$	− 894.864i	− 0.225033i	0.993650	−	0.112517i	$-0.0358911\pi$
$252$	0	0
$253$	− 4019.72i	− 0.998884i
$254$	0	0
$255$	−1225.79	−0.301027
$256$	0	0
$257$	1199.86	0.291226	0.145613	−	0.989342i	$-0.453485\pi$
$257$	1199.86	0.291226	0.145613	+	0.989342i	$0.453485\pi$
$258$	0	0
$259$	− 2928.23i	− 0.702515i
$260$	0	0
$261$	2700.13i	0.640359i
$262$	0	0
$263$	252.278	0.0591487	0.0295744	−	0.999563i	$-0.490585\pi$
$263$	252.278	0.0591487	0.0295744	+	0.999563i	$0.490585\pi$
$264$	0	0
$265$	−4991.62	−1.15711
$266$	0	0
$267$	− 211.703i	− 0.0485245i
$268$	0	0
$269$	4473.97i	1.01406i	0.861927	+	0.507032i	$0.169257\pi$
$269$	4473.97i	1.01406i	−0.861927	+	0.507032i	$0.830743\pi$
$270$	0	0
$271$	3665.60	0.821658	0.410829	−	0.911712i	$-0.365239\pi$
$271$	3665.60	0.821658	0.410829	+	0.911712i	$0.365239\pi$
$272$	0	0
$273$	187.035	0.0414648
$274$	0	0
$275$	− 3528.93i	− 0.773828i
$276$	0	0
$277$	3138.29i	0.680727i	0.940294	+	0.340363i	$0.110550\pi$
$277$	3138.29i	0.680727i	−0.940294	+	0.340363i	$0.889450\pi$
$278$	0	0
$279$	1653.41	0.354792
$280$	0	0
$281$	2545.87	0.540476	0.270238	−	0.962794i	$-0.412898\pi$
$281$	2545.87	0.540476	0.270238	+	0.962794i	$0.412898\pi$
$282$	0	0
$283$	− 1136.80i	− 0.238785i	−0.992847	−	0.119392i	$-0.961905\pi$
$283$	− 1136.80i	− 0.238785i	0.992847	−	0.119392i	$-0.0380946\pi$
$284$	0	0
$285$	− 2851.73i	− 0.592708i
$286$	0	0
$287$	742.568	0.152726
$288$	0	0
$289$	−1539.58	−0.313370
$290$	0	0
$291$	− 3957.31i	− 0.797188i
$292$	0	0
$293$	3554.88i	0.708800i	0.935094	+	0.354400i	$0.115315\pi$
$293$	3554.88i	0.708800i	−0.935094	+	0.354400i	$0.884685\pi$
$294$	0	0
$295$	−5299.76	−1.04598
$296$	0	0
$297$	−1261.84	−0.246530
$298$	0	0
$299$	766.053i	0.148167i
$300$	0	0
$301$	1135.44i	0.217427i
$302$	0	0
$303$	−2113.35	−0.400688
$304$	0	0
$305$	−4699.02	−0.882180
$306$	0	0
$307$	6131.92i	1.13996i	0.821659	+	0.569979i	$0.193049\pi$
$307$	6131.92i	1.13996i	−0.821659	+	0.569979i	$0.806951\pi$
$308$	0	0
$309$	− 4659.64i	− 0.857856i
$310$	0	0
$311$	6727.85	1.22669	0.613346	−	0.789814i	$-0.289823\pi$
$311$	6727.85	1.22669	0.613346	+	0.789814i	$0.289823\pi$
$312$	0	0
$313$	5502.24	0.993626	0.496813	−	0.867858i	$-0.334504\pi$
$313$	5502.24	0.993626	0.496813	+	0.867858i	$0.334504\pi$
$314$	0	0
$315$	443.201i	0.0792747i
$316$	0	0
$317$	6682.27i	1.18395i	0.805955	+	0.591977i	$0.201653\pi$
$317$	6682.27i	1.18395i	−0.805955	+	0.591977i	$0.798347\pi$
$318$	0	0
$319$	−14021.1	−2.46092
$320$	0	0
$321$	5967.55	1.03762
$322$	0	0
$323$	7848.05i	1.35194i
$324$	0	0
$325$	672.522i	0.114784i
$326$	0	0
$327$	3923.91	0.663587
$328$	0	0
$329$	−2644.26	−0.443109
$330$	0	0
$331$	4768.64i	0.791868i	0.918279	+	0.395934i	$0.129579\pi$
$331$	4768.64i	0.791868i	−0.918279	+	0.395934i	$0.870421\pi$
$332$	0	0
$333$	− 3764.87i	− 0.619560i
$334$	0	0
$335$	3148.18	0.513442
$336$	0	0
$337$	4265.17	0.689432	0.344716	−	0.938707i	$-0.387975\pi$
$337$	4265.17	0.689432	0.344716	+	0.938707i	$0.387975\pi$
$338$	0	0
$339$	− 4026.35i	− 0.645077i
$340$	0	0
$341$	8585.76i	1.36348i
$342$	0	0
$343$	−343.000	−0.0539949
$344$	0	0
$345$	−1815.25	−0.283275
$346$	0	0
$347$	− 4490.33i	− 0.694678i	−0.937740	−	0.347339i	$-0.887085\pi$
$347$	− 4490.33i	− 0.694678i	0.937740	−	0.347339i	$-0.112915\pi$
$348$	0	0
$349$	− 295.697i	− 0.0453533i	−0.999743	−	0.0226767i	$-0.992781\pi$
$349$	− 295.697i	− 0.0453533i	0.999743	−	0.0226767i	$-0.00721883\pi$
$350$	0	0
$351$	240.474	0.0365685
$352$	0	0
$353$	−1255.22	−0.189260	−0.0946300	−	0.995513i	$-0.530167\pi$
$353$	−1255.22	−0.189260	−0.0946300	+	0.995513i	$0.530167\pi$
$354$	0	0
$355$	− 2705.27i	− 0.404453i
$356$	0	0
$357$	− 1219.70i	− 0.180822i
$358$	0	0
$359$	11982.9	1.76165	0.880827	−	0.473439i	$-0.156987\pi$
$359$	11982.9	1.76165	0.880827	+	0.473439i	$0.156987\pi$
$360$	0	0
$361$	−11399.0	−1.66191
$362$	0	0
$363$	− 2559.44i	− 0.370071i
$364$	0	0
$365$	7669.22i	1.09979i
$366$	0	0
$367$	5725.42	0.814345	0.407173	−	0.913351i	$-0.366515\pi$
$367$	5725.42	0.814345	0.407173	+	0.913351i	$0.366515\pi$
$368$	0	0
$369$	954.730	0.134692
$370$	0	0
$371$	− 4966.83i	− 0.695054i
$372$	0	0
$373$	− 2425.91i	− 0.336752i	−0.985723	−	0.168376i	$-0.946148\pi$
$373$	− 2425.91i	− 0.336752i	0.985723	−	0.168376i	$-0.0538523\pi$
$374$	0	0
$375$	−4231.72	−0.582733
$376$	0	0
$377$	2672.06	0.365034
$378$	0	0
$379$	− 3958.37i	− 0.536485i	−0.963351	−	0.268242i	$-0.913557\pi$
$379$	− 3958.37i	− 0.536485i	0.963351	−	0.268242i	$-0.0864428\pi$
$380$	0	0
$381$	1497.54i	0.201368i
$382$	0	0
$383$	−7610.32	−1.01532	−0.507662	−	0.861556i	$-0.669490\pi$
$383$	−7610.32	−1.01532	−0.507662	+	0.861556i	$0.669490\pi$
$384$	0	0
$385$	−2301.44	−0.304655
$386$	0	0
$387$	1459.85i	0.191753i
$388$	0	0
$389$	− 3731.93i	− 0.486417i	−0.969974	−	0.243209i	$-0.921800\pi$
$389$	− 3731.93i	− 0.486417i	0.969974	−	0.243209i	$-0.0781999\pi$
$390$	0	0
$391$	4995.63	0.646138
$392$	0	0
$393$	−7215.42	−0.926132
$394$	0	0
$395$	− 4719.74i	− 0.601204i
$396$	0	0
$397$	− 7134.28i	− 0.901913i	−0.892546	−	0.450956i	$-0.851083\pi$
$397$	− 7134.28i	− 0.901913i	0.892546	−	0.450956i	$-0.148917\pi$
$398$	0	0
$399$	2837.57	0.356030
$400$	0	0
$401$	10713.9	1.33423	0.667114	−	0.744956i	$-0.267529\pi$
$401$	10713.9	1.33423	0.667114	+	0.744956i	$0.267529\pi$
$402$	0	0
$403$	− 1636.22i	− 0.202248i
$404$	0	0
$405$	569.830i	0.0699137i
$406$	0	0
$407$	19550.1	2.38098
$408$	0	0
$409$	647.047	0.0782260	0.0391130	−	0.999235i	$-0.487547\pi$
$409$	647.047	0.0782260	0.0391130	+	0.999235i	$0.487547\pi$
$410$	0	0
$411$	1170.38i	0.140463i
$412$	0	0
$413$	− 5273.44i	− 0.628302i
$414$	0	0
$415$	3150.25	0.372626
$416$	0	0
$417$	−673.989	−0.0791496
$418$	0	0
$419$	− 9910.37i	− 1.15550i	−0.816215	−	0.577748i	$-0.803932\pi$
$419$	− 9910.37i	− 1.15550i	0.816215	−	0.577748i	$-0.196068\pi$
$420$	0	0
$421$	5284.82i	0.611797i	0.952064	+	0.305898i	$0.0989568\pi$
$421$	5284.82i	0.611797i	−0.952064	+	0.305898i	$0.901043\pi$
$422$	0	0
$423$	−3399.76	−0.390785
$424$	0	0
$425$	4385.69	0.500558
$426$	0	0
$427$	− 4675.68i	− 0.529911i
$428$	0	0
$429$	1248.72i	0.140534i
$430$	0	0
$431$	15887.7	1.77560	0.887802	−	0.460227i	$-0.152232\pi$
$431$	15887.7	1.77560	0.887802	+	0.460227i	$0.152232\pi$
$432$	0	0
$433$	−2612.44	−0.289944	−0.144972	−	0.989436i	$-0.546309\pi$
$433$	−2612.44	−0.289944	−0.144972	+	0.989436i	$0.546309\pi$
$434$	0	0
$435$	6331.74i	0.697894i
$436$	0	0
$437$	11622.0i	1.27221i
$438$	0	0
$439$	−8750.27	−0.951316	−0.475658	−	0.879630i	$-0.657790\pi$
$439$	−8750.27	−0.951316	−0.475658	+	0.879630i	$0.657790\pi$
$440$	0	0
$441$	−441.000	−0.0476190
$442$	0	0
$443$	10228.3i	1.09698i	0.836157	+	0.548490i	$0.184797\pi$
$443$	10228.3i	1.09698i	−0.836157	+	0.548490i	$0.815203\pi$
$444$	0	0
$445$	− 496.440i	− 0.0528842i
$446$	0	0
$447$	−7436.89	−0.786919
$448$	0	0
$449$	−16896.9	−1.77598	−0.887988	−	0.459866i	$-0.847897\pi$
$449$	−16896.9	−1.77598	−0.887988	+	0.459866i	$0.847897\pi$
$450$	0	0
$451$	4957.68i	0.517624i
$452$	0	0
$453$	− 917.314i	− 0.0951417i
$454$	0	0
$455$	438.593	0.0451903
$456$	0	0
$457$	15876.7	1.62512	0.812561	−	0.582876i	$-0.198073\pi$
$457$	15876.7	1.62512	0.812561	+	0.582876i	$0.198073\pi$
$458$	0	0
$459$	− 1568.19i	− 0.159470i
$460$	0	0
$461$	− 5034.64i	− 0.508648i	−0.967119	−	0.254324i	$-0.918147\pi$
$461$	− 5034.64i	− 0.508648i	0.967119	−	0.254324i	$-0.0818529\pi$
$462$	0	0
$463$	−5132.53	−0.515182	−0.257591	−	0.966254i	$-0.582929\pi$
$463$	−5132.53	−0.515182	−0.257591	+	0.966254i	$0.582929\pi$
$464$	0	0
$465$	3877.21	0.386670
$466$	0	0
$467$	4039.62i	0.400281i	0.979767	+	0.200140i	$0.0641398\pi$
$467$	4039.62i	0.400281i	−0.979767	+	0.200140i	$0.935860\pi$
$468$	0	0
$469$	3132.54i	0.308417i
$470$	0	0
$471$	−11705.8	−1.14517
$472$	0	0
$473$	−7580.66	−0.736911
$474$	0	0
$475$	10203.0i	0.985574i
$476$	0	0
$477$	− 6385.93i	− 0.612980i
$478$	0	0
$479$	4253.93	0.405777	0.202889	−	0.979202i	$-0.434967\pi$
$479$	4253.93	0.405777	0.202889	+	0.979202i	$0.434967\pi$
$480$	0	0
$481$	−3725.73	−0.353178
$482$	0	0
$483$	− 1806.24i	− 0.170159i
$484$	0	0
$485$	− 9279.81i	− 0.868813i
$486$	0	0
$487$	15258.2	1.41974	0.709872	−	0.704331i	$-0.248753\pi$
$487$	15258.2	1.41974	0.709872	+	0.704331i	$0.248753\pi$
$488$	0	0
$489$	9730.54	0.899858
$490$	0	0
$491$	5040.05i	0.463247i	0.972805	+	0.231624i	$0.0744038\pi$
$491$	5040.05i	0.463247i	−0.972805	+	0.231624i	$0.925596\pi$
$492$	0	0
$493$	− 17425.2i	− 1.59187i
$494$	0	0
$495$	−2958.99	−0.268680
$496$	0	0
$497$	2691.83	0.242948
$498$	0	0
$499$	− 19346.0i	− 1.73556i	−0.496949	−	0.867780i	$-0.665546\pi$
$499$	− 19346.0i	− 1.73556i	0.496949	−	0.867780i	$-0.334454\pi$
$500$	0	0
$501$	− 4405.25i	− 0.392838i
$502$	0	0
$503$	−3875.01	−0.343495	−0.171748	−	0.985141i	$-0.554941\pi$
$503$	−3875.01	−0.343495	−0.171748	+	0.985141i	$0.554941\pi$
$504$	0	0
$505$	−4955.75	−0.436689
$506$	0	0
$507$	6353.03i	0.556505i
$508$	0	0
$509$	3312.25i	0.288434i	0.989546	+	0.144217i	$0.0460663\pi$
$509$	3312.25i	0.288434i	−0.989546	+	0.144217i	$0.953934\pi$
$510$	0	0
$511$	−7631.13	−0.660629
$512$	0	0
$513$	3648.30	0.313989
$514$	0	0
$515$	− 10926.7i	− 0.934932i
$516$	0	0
$517$	− 17654.2i	− 1.50180i
$518$	0	0
$519$	−537.067	−0.0454231
$520$	0	0
$521$	2600.64	0.218687	0.109344	−	0.994004i	$-0.465125\pi$
$521$	2600.64	0.218687	0.109344	+	0.994004i	$0.465125\pi$
$522$	0	0
$523$	12888.8i	1.07760i	0.842433	+	0.538801i	$0.181123\pi$
$523$	12888.8i	1.07760i	−0.842433	+	0.538801i	$0.818877\pi$
$524$	0	0
$525$	− 1585.70i	− 0.131821i
$526$	0	0
$527$	−10670.2	−0.881977
$528$	0	0
$529$	−4769.07	−0.391967
$530$	0	0
$531$	− 6780.13i	− 0.554111i
$532$	0	0
$533$	− 944.804i	− 0.0767805i
$534$	0	0
$535$	13993.8	1.13085
$536$	0	0
$537$	−2808.60	−0.225698
$538$	0	0
$539$	− 2290.01i	− 0.183001i
$540$	0	0
$541$	− 10524.6i	− 0.836396i	−0.908356	−	0.418198i	$-0.862662\pi$
$541$	− 10524.6i	− 0.836396i	0.908356	−	0.418198i	$-0.137338\pi$
$542$	0	0
$543$	1751.37	0.138414
$544$	0	0
$545$	9201.49	0.723208
$546$	0	0
$547$	15317.7i	1.19733i	0.801000	+	0.598664i	$0.204302\pi$
$547$	15317.7i	1.19733i	−0.801000	+	0.598664i	$0.795698\pi$
$548$	0	0
$549$	− 6011.59i	− 0.467338i
$550$	0	0
$551$	40538.6	3.13430
$552$	0	0
$553$	4696.30	0.361134
$554$	0	0
$555$	− 8828.53i	− 0.675226i
$556$	0	0
$557$	15585.0i	1.18556i	0.805365	+	0.592780i	$0.201969\pi$
$557$	15585.0i	1.18556i	−0.805365	+	0.592780i	$0.798031\pi$
$558$	0	0
$559$	1444.67	0.109308
$560$	0	0
$561$	8143.24	0.612848
$562$	0	0
$563$	7087.53i	0.530557i	0.964172	+	0.265279i	$0.0854639\pi$
$563$	7087.53i	0.530557i	−0.964172	+	0.265279i	$0.914536\pi$
$564$	0	0
$565$	− 9441.70i	− 0.703036i
$566$	0	0
$567$	−567.000	−0.0419961
$568$	0	0
$569$	−23842.6	−1.75665	−0.878326	−	0.478062i	$-0.841339\pi$
$569$	−23842.6	−1.75665	−0.878326	+	0.478062i	$0.841339\pi$
$570$	0	0
$571$	5520.47i	0.404596i	0.979324	+	0.202298i	$0.0648410\pi$
$571$	5520.47i	0.404596i	−0.979324	+	0.202298i	$0.935159\pi$
$572$	0	0
$573$	− 3136.14i	− 0.228646i
$574$	0	0
$575$	6494.68	0.471038
$576$	0	0
$577$	−16052.7	−1.15820	−0.579102	−	0.815255i	$-0.696597\pi$
$577$	−16052.7	−1.15820	−0.579102	+	0.815255i	$0.696597\pi$
$578$	0	0
$579$	1686.53i	0.121053i
$580$	0	0
$581$	3134.61i	0.223830i
$582$	0	0
$583$	33160.6	2.35570
$584$	0	0
$585$	563.906	0.0398541
$586$	0	0
$587$	− 5290.81i	− 0.372019i	−0.982548	−	0.186009i	$-0.940445\pi$
$587$	− 5290.81i	− 0.372019i	0.982548	−	0.186009i	$-0.0595555\pi$
$588$	0	0
$589$	− 24823.6i	− 1.73657i
$590$	0	0
$591$	−4684.62	−0.326057
$592$	0	0
$593$	18659.3	1.29215	0.646075	−	0.763274i	$-0.276409\pi$
$593$	18659.3	1.29215	0.646075	+	0.763274i	$0.276409\pi$
$594$	0	0
$595$	− 2860.18i	− 0.197069i
$596$	0	0
$597$	− 11312.1i	− 0.775500i
$598$	0	0
$599$	7546.39	0.514753	0.257377	−	0.966311i	$-0.417142\pi$
$599$	7546.39	0.514753	0.257377	+	0.966311i	$0.417142\pi$
$600$	0	0
$601$	15764.7	1.06998	0.534988	−	0.844860i	$-0.320316\pi$
$601$	15764.7	1.06998	0.534988	+	0.844860i	$0.320316\pi$
$602$	0	0
$603$	4027.55i	0.271998i
$604$	0	0
$605$	− 6001.83i	− 0.403321i
$606$	0	0
$607$	−9668.75	−0.646528	−0.323264	−	0.946309i	$-0.604780\pi$
$607$	−9668.75	−0.646528	−0.323264	+	0.946309i	$0.604780\pi$
$608$	0	0
$609$	−6300.30	−0.419213
$610$	0	0
$611$	3364.42i	0.222766i
$612$	0	0
$613$	− 16371.0i	− 1.07866i	−0.842096	−	0.539328i	$-0.818678\pi$
$613$	− 16371.0i	− 1.07866i	0.842096	−	0.539328i	$-0.181322\pi$
$614$	0	0
$615$	2238.82	0.146793
$616$	0	0
$617$	4262.29	0.278109	0.139055	−	0.990285i	$-0.455594\pi$
$617$	4262.29	0.278109	0.139055	+	0.990285i	$0.455594\pi$
$618$	0	0
$619$	832.354i	0.0540471i	0.999635	+	0.0270235i	$0.00860291\pi$
$619$	832.354i	0.0540471i	−0.999635	+	0.0270235i	$0.991397\pi$
$620$	0	0
$621$	− 2322.30i	− 0.150066i
$622$	0	0
$623$	493.974	0.0317667
$624$	0	0
$625$	−484.574	−0.0310128
$626$	0	0
$627$	18944.7i	1.20667i
$628$	0	0
$629$	24296.4i	1.54016i
$630$	0	0
$631$	5055.10	0.318923	0.159462	−	0.987204i	$-0.449024\pi$
$631$	5055.10	0.318923	0.159462	+	0.987204i	$0.449024\pi$
$632$	0	0
$633$	−14150.3	−0.888503
$634$	0	0
$635$	3511.69i	0.219460i
$636$	0	0
$637$	436.415i	0.0271451i
$638$	0	0
$639$	3460.93	0.214260
$640$	0	0
$641$	6257.92	0.385605	0.192803	−	0.981238i	$-0.438242\pi$
$641$	6257.92	0.385605	0.192803	+	0.981238i	$0.438242\pi$
$642$	0	0
$643$	− 9642.14i	− 0.591367i	−0.955286	−	0.295683i	$-0.904453\pi$
$643$	− 9642.14i	− 0.591367i	0.955286	−	0.295683i	$-0.0955473\pi$
$644$	0	0
$645$	3423.32i	0.208981i
$646$	0	0
$647$	7537.26	0.457991	0.228996	−	0.973427i	$-0.426456\pi$
$647$	7537.26	0.457991	0.228996	+	0.973427i	$0.426456\pi$
$648$	0	0
$649$	35207.6	2.12946
$650$	0	0
$651$	3857.96i	0.232266i
$652$	0	0
$653$	30534.9i	1.82990i	0.403573	+	0.914948i	$0.367768\pi$
$653$	30534.9i	1.82990i	−0.403573	+	0.914948i	$0.632232\pi$
$654$	0	0
$655$	−16920.0	−1.00934
$656$	0	0
$657$	−9811.46	−0.582620
$658$	0	0
$659$	23623.0i	1.39639i	0.715906	+	0.698196i	$0.246014\pi$
$659$	23623.0i	1.39639i	−0.715906	+	0.698196i	$0.753986\pi$
$660$	0	0
$661$	− 2829.16i	− 0.166477i	−0.996530	−	0.0832387i	$-0.973474\pi$
$661$	− 2829.16i	− 0.166477i	0.996530	−	0.0832387i	$-0.0265264\pi$
$662$	0	0
$663$	−1551.89	−0.0909054
$664$	0	0
$665$	6654.03	0.388018
$666$	0	0
$667$	− 25804.6i	− 1.49799i
$668$	0	0
$669$	− 14055.0i	− 0.812252i
$670$	0	0
$671$	31216.8	1.79599
$672$	0	0
$673$	−6504.77	−0.372571	−0.186286	−	0.982496i	$-0.559645\pi$
$673$	−6504.77	−0.372571	−0.186286	+	0.982496i	$0.559645\pi$
$674$	0	0
$675$	− 2038.76i	− 0.116255i
$676$	0	0
$677$	− 26593.2i	− 1.50969i	−0.655905	−	0.754844i	$-0.727713\pi$
$677$	− 26593.2i	− 1.50969i	0.655905	−	0.754844i	$-0.272287\pi$
$678$	0	0
$679$	9233.73	0.521882
$680$	0	0
$681$	−2980.27	−0.167701
$682$	0	0
$683$	− 167.019i	− 0.00935696i	−0.999989	−	0.00467848i	$-0.998511\pi$
$683$	− 167.019i	− 0.00935696i	0.999989	−	0.00467848i	$-0.00148921\pi$
$684$	0	0
$685$	2744.51i	0.153084i
$686$	0	0
$687$	−11996.2	−0.666207
$688$	0	0
$689$	−6319.54	−0.349427
$690$	0	0
$691$	− 22975.8i	− 1.26489i	−0.774604	−	0.632447i	$-0.782051\pi$
$691$	− 22975.8i	− 1.26489i	0.774604	−	0.632447i	$-0.217949\pi$
$692$	0	0
$693$	− 2944.30i	− 0.161392i
$694$	0	0
$695$	−1580.49	−0.0862610
$696$	0	0
$697$	−6161.31	−0.334830
$698$	0	0
$699$	2291.87i	0.124015i
$700$	0	0
$701$	− 8303.14i	− 0.447368i	−0.974662	−	0.223684i	$-0.928192\pi$
$701$	− 8303.14i	− 0.447368i	0.974662	−	0.223684i	$-0.0718084\pi$
$702$	0	0
$703$	−56524.1	−3.03250
$704$	0	0
$705$	−7972.37	−0.425896
$706$	0	0
$707$	− 4931.14i	− 0.262312i
$708$	0	0
$709$	− 28886.7i	− 1.53013i	−0.643954	−	0.765064i	$-0.722707\pi$
$709$	− 28886.7i	− 1.53013i	0.643954	−	0.765064i	$-0.277293\pi$
$710$	0	0
$711$	6038.10	0.318490
$712$	0	0
$713$	−15801.3	−0.829964
$714$	0	0
$715$	2928.23i	0.153160i
$716$	0	0
$717$	− 18732.2i	− 0.975685i
$718$	0	0
$719$	10415.5	0.540242	0.270121	−	0.962826i	$-0.412936\pi$
$719$	10415.5	0.540242	0.270121	+	0.962826i	$0.412936\pi$
$720$	0	0
$721$	10872.5	0.561599
$722$	0	0
$723$	− 13772.1i	− 0.708424i
$724$	0	0
$725$	− 22654.0i	− 1.16048i
$726$	0	0
$727$	1858.15	0.0947938	0.0473969	−	0.998876i	$-0.484907\pi$
$727$	1858.15	0.0947938	0.0473969	+	0.998876i	$0.484907\pi$
$728$	0	0
$729$	−729.000	−0.0370370
$730$	0	0
$731$	− 9421.08i	− 0.476678i
$732$	0	0
$733$	− 1707.92i	− 0.0860622i	−0.999074	−	0.0430311i	$-0.986299\pi$
$733$	− 1707.92i	− 0.0860622i	0.999074	−	0.0430311i	$-0.0137015\pi$
$734$	0	0
$735$	−1034.14	−0.0518975
$736$	0	0
$737$	−20914.1	−1.04529
$738$	0	0
$739$	17529.6i	0.872579i	0.899806	+	0.436290i	$0.143708\pi$
$739$	17529.6i	0.872579i	−0.899806	+	0.436290i	$0.856292\pi$
$740$	0	0
$741$	− 3610.37i	− 0.178988i
$742$	0	0
$743$	5683.78	0.280643	0.140321	−	0.990106i	$-0.455186\pi$
$743$	5683.78	0.280643	0.140321	+	0.990106i	$0.455186\pi$
$744$	0	0
$745$	−17439.3	−0.857621
$746$	0	0
$747$	4030.21i	0.197400i
$748$	0	0
$749$	13924.3i	0.679282i
$750$	0	0
$751$	−26693.5	−1.29702	−0.648508	−	0.761208i	$-0.724607\pi$
$751$	−26693.5	−1.29702	−0.648508	+	0.761208i	$0.724607\pi$
$752$	0	0
$753$	2684.59	0.129923
$754$	0	0
$755$	− 2151.08i	− 0.103690i
$756$	0	0
$757$	− 8730.07i	− 0.419154i	−0.977792	−	0.209577i	$-0.932791\pi$
$757$	− 8730.07i	− 0.419154i	0.977792	−	0.209577i	$-0.0672088\pi$
$758$	0	0
$759$	12059.2	0.576706
$760$	0	0
$761$	15999.6	0.762134	0.381067	−	0.924547i	$-0.375557\pi$
$761$	15999.6	0.762134	0.381067	+	0.924547i	$0.375557\pi$
$762$	0	0
$763$	9155.79i	0.434419i
$764$	0	0
$765$	− 3677.37i	− 0.173798i
$766$	0	0
$767$	−6709.65	−0.315869
$768$	0	0
$769$	25437.2	1.19283	0.596416	−	0.802675i	$-0.296591\pi$
$769$	25437.2	1.19283	0.596416	+	0.802675i	$0.296591\pi$
$770$	0	0
$771$	3599.57i	0.168139i
$772$	0	0
$773$	− 33698.6i	− 1.56799i	−0.620767	−	0.783995i	$-0.713179\pi$
$773$	− 33698.6i	− 1.56799i	0.620767	−	0.783995i	$-0.286821\pi$
$774$	0	0
$775$	−13872.1	−0.642966
$776$	0	0
$777$	8784.69	0.405597
$778$	0	0
$779$	− 14333.9i	− 0.659263i
$780$	0	0
$781$	17971.8i	0.823406i
$782$	0	0
$783$	−8100.38	−0.369711
$784$	0	0
$785$	−27449.8	−1.24806
$786$	0	0
$787$	8264.06i	0.374310i	0.982330	+	0.187155i	$0.0599267\pi$
$787$	8264.06i	0.374310i	−0.982330	+	0.187155i	$0.940073\pi$
$788$	0	0
$789$	756.834i	0.0341495i
$790$	0	0
$791$	9394.81	0.422302
$792$	0	0
$793$	−5949.09	−0.266404
$794$	0	0
$795$	− 14974.9i	− 0.668055i
$796$	0	0
$797$	− 3961.93i	− 0.176084i	−0.996117	−	0.0880418i	$-0.971939\pi$
$797$	− 3961.93i	− 0.176084i	0.996117	−	0.0880418i	$-0.0280609\pi$
$798$	0	0
$799$	21940.2	0.971451
$800$	0	0
$801$	635.110	0.0280156
$802$	0	0
$803$	− 50948.5i	− 2.23902i
$804$	0	0
$805$	− 4235.58i	− 0.185447i
$806$	0	0
$807$	−13421.9	−0.585470
$808$	0	0
$809$	11017.9	0.478823	0.239412	−	0.970918i	$-0.423045\pi$
$809$	11017.9	0.478823	0.239412	+	0.970918i	$0.423045\pi$
$810$	0	0
$811$	− 37833.0i	− 1.63810i	−0.573726	−	0.819048i	$-0.694502\pi$
$811$	− 37833.0i	− 1.63810i	0.573726	−	0.819048i	$-0.305498\pi$
$812$	0	0
$813$	10996.8i	0.474385i
$814$	0	0
$815$	22817.9	0.980708
$816$	0	0
$817$	21917.6	0.938555
$818$	0	0
$819$	561.105i	0.0239397i
$820$	0	0
$821$	6700.96i	0.284854i	0.989805	+	0.142427i	$0.0454906\pi$
$821$	6700.96i	0.284854i	−0.989805	+	0.142427i	$0.954509\pi$
$822$	0	0
$823$	24716.5	1.04686	0.523428	−	0.852070i	$-0.324653\pi$
$823$	24716.5	1.04686	0.523428	+	0.852070i	$0.324653\pi$
$824$	0	0
$825$	10586.8	0.446770
$826$	0	0
$827$	37063.9i	1.55845i	0.626744	+	0.779225i	$0.284387\pi$
$827$	37063.9i	1.55845i	−0.626744	+	0.779225i	$0.715613\pi$
$828$	0	0
$829$	46151.4i	1.93354i	0.255650	+	0.966769i	$0.417711\pi$
$829$	46151.4i	1.93354i	−0.255650	+	0.966769i	$0.582289\pi$
$830$	0	0
$831$	−9414.86	−0.393018
$832$	0	0
$833$	2845.97	0.118376
$834$	0	0
$835$	− 10330.2i	− 0.428134i
$836$	0	0
$837$	4960.23i	0.204840i
$838$	0	0
$839$	40084.7	1.64944	0.824719	−	0.565542i	$-0.191333\pi$
$839$	40084.7	1.64944	0.824719	+	0.565542i	$0.191333\pi$
$840$	0	0
$841$	−65619.5	−2.69054
$842$	0	0
$843$	7637.60i	0.312044i
$844$	0	0
$845$	14897.7i	0.606505i
$846$	0	0
$847$	5972.02	0.242268
$848$	0	0
$849$	3410.41	0.137862
$850$	0	0
$851$	35980.1i	1.44933i
$852$	0	0
$853$	− 7132.84i	− 0.286312i	−0.989700	−	0.143156i	$-0.954275\pi$
$853$	− 7132.84i	− 0.286312i	0.989700	−	0.143156i	$-0.0457250\pi$
$854$	0	0
$855$	8555.18	0.342200
$856$	0	0
$857$	28261.8	1.12649	0.563246	−	0.826289i	$-0.309552\pi$
$857$	28261.8	1.12649	0.563246	+	0.826289i	$0.309552\pi$
$858$	0	0
$859$	10098.8i	0.401126i	0.979681	+	0.200563i	$0.0642771\pi$
$859$	10098.8i	0.401126i	−0.979681	+	0.200563i	$0.935723\pi$
$860$	0	0
$861$	2227.70i	0.0881765i
$862$	0	0
$863$	28249.9	1.11430	0.557148	−	0.830413i	$-0.311896\pi$
$863$	28249.9	1.11430	0.557148	+	0.830413i	$0.311896\pi$
$864$	0	0
$865$	−1259.41	−0.0495043
$866$	0	0
$867$	− 4618.75i	− 0.180924i
$868$	0	0
$869$	31354.4i	1.22396i
$870$	0	0
$871$	3985.68	0.155051
$872$	0	0
$873$	11871.9	0.460257
$874$	0	0
$875$	− 9874.01i	− 0.381488i
$876$	0	0
$877$	− 34787.4i	− 1.33944i	−0.742615	−	0.669718i	$-0.766415\pi$
$877$	− 34787.4i	− 1.33944i	0.742615	−	0.669718i	$-0.233585\pi$
$878$	0	0
$879$	−10664.6	−0.409226
$880$	0	0
$881$	−13237.3	−0.506217	−0.253108	−	0.967438i	$-0.581453\pi$
$881$	−13237.3	−0.506217	−0.253108	+	0.967438i	$0.581453\pi$
$882$	0	0
$883$	16543.2i	0.630491i	0.949010	+	0.315246i	$0.102087\pi$
$883$	16543.2i	0.630491i	−0.949010	+	0.315246i	$0.897913\pi$
$884$	0	0
$885$	− 15899.3i	− 0.603896i
$886$	0	0
$887$	−5551.40	−0.210144	−0.105072	−	0.994465i	$-0.533507\pi$
$887$	−5551.40	−0.210144	−0.105072	+	0.994465i	$0.533507\pi$
$888$	0	0
$889$	−3494.26	−0.131826
$890$	0	0
$891$	− 3785.52i	− 0.142334i
$892$	0	0
$893$	51042.6i	1.91274i
$894$	0	0
$895$	−6586.10	−0.245977
$896$	0	0
$897$	−2298.16	−0.0855444
$898$	0	0
$899$	55116.3i	2.04475i
$900$	0	0
$901$	41211.3i	1.52380i
$902$	0	0
$903$	−3406.32	−0.125532
$904$	0	0
$905$	4106.93	0.150850
$906$	0	0
$907$	− 24188.3i	− 0.885511i	−0.896642	−	0.442756i	$-0.854001\pi$
$907$	− 24188.3i	− 0.885511i	0.896642	−	0.442756i	$-0.145999\pi$
$908$	0	0
$909$	− 6340.04i	− 0.231338i
$910$	0	0
$911$	−8539.95	−0.310583	−0.155292	−	0.987869i	$-0.549632\pi$
$911$	−8539.95	−0.310583	−0.155292	+	0.987869i	$0.549632\pi$
$912$	0	0
$913$	−20927.9	−0.758612
$914$	0	0
$915$	− 14097.0i	− 0.509327i
$916$	0	0
$917$	− 16836.0i	− 0.606295i
$918$	0	0
$919$	−8958.34	−0.321554	−0.160777	−	0.986991i	$-0.551400\pi$
$919$	−8958.34	−0.321554	−0.160777	+	0.986991i	$0.551400\pi$
$920$	0	0
$921$	−18395.8	−0.658155
$922$	0	0
$923$	− 3424.95i	− 0.122138i
$924$	0	0
$925$	31587.1i	1.12279i
$926$	0	0
$927$	13978.9	0.495283
$928$	0	0
$929$	52535.5	1.85537	0.927683	−	0.373369i	$-0.121797\pi$
$929$	52535.5	1.85537	0.927683	+	0.373369i	$0.121797\pi$
$930$	0	0
$931$	6620.99i	0.233076i
$932$	0	0
$933$	20183.6i	0.708231i
$934$	0	0
$935$	19095.7	0.667911
$936$	0	0
$937$	24569.9	0.856633	0.428316	−	0.903629i	$-0.359107\pi$
$937$	24569.9	0.856633	0.428316	+	0.903629i	$0.359107\pi$
$938$	0	0
$939$	16506.7i	0.573670i
$940$	0	0
$941$	16009.7i	0.554625i	0.960780	+	0.277312i	$0.0894437\pi$
$941$	16009.7i	0.554625i	−0.960780	+	0.277312i	$0.910556\pi$
$942$	0	0
$943$	−9124.17	−0.315084
$944$	0	0
$945$	−1329.60	−0.0457693
$946$	0	0
$947$	− 9512.18i	− 0.326404i	−0.986593	−	0.163202i	$-0.947818\pi$
$947$	− 9512.18i	− 0.326404i	0.986593	−	0.163202i	$-0.0521822\pi$
$948$	0	0
$949$	9709.45i	0.332120i
$950$	0	0
$951$	−20046.8	−0.683557
$952$	0	0
$953$	−25507.1	−0.867004	−0.433502	−	0.901153i	$-0.642722\pi$
$953$	−25507.1	−0.867004	−0.433502	+	0.901153i	$0.642722\pi$
$954$	0	0
$955$	− 7354.17i	− 0.249189i
$956$	0	0
$957$	− 42063.4i	− 1.42081i
$958$	0	0
$959$	−2730.88	−0.0919549
$960$	0	0
$961$	3959.20	0.132899
$962$	0	0
$963$	17902.6i	0.599070i
$964$	0	0
$965$	3954.88i	0.131930i
$966$	0	0
$967$	17237.5	0.573238	0.286619	−	0.958045i	$-0.407469\pi$
$967$	17237.5	0.573238	0.286619	+	0.958045i	$0.407469\pi$
$968$	0	0
$969$	−23544.1	−0.780544
$970$	0	0
$971$	2441.24i	0.0806829i	0.999186	+	0.0403414i	$0.0128446\pi$
$971$	2441.24i	0.0806829i	−0.999186	+	0.0403414i	$0.987155\pi$
$972$	0	0
$973$	− 1572.64i	− 0.0518156i
$974$	0	0
$975$	−2017.57	−0.0662706
$976$	0	0
$977$	−44441.0	−1.45527	−0.727633	−	0.685967i	$-0.759379\pi$
$977$	−44441.0	−1.45527	−0.727633	+	0.685967i	$0.759379\pi$
$978$	0	0
$979$	3297.97i	0.107665i
$980$	0	0
$981$	11771.7i	0.383122i
$982$	0	0
$983$	−15793.5	−0.512447	−0.256223	−	0.966618i	$-0.582478\pi$
$983$	−15793.5	−0.512447	−0.256223	+	0.966618i	$0.582478\pi$
$984$	0	0
$985$	−10985.3	−0.355352
$986$	0	0
$987$	− 7932.78i	− 0.255829i
$988$	0	0
$989$	− 13951.5i	− 0.448566i
$990$	0	0
$991$	9580.56	0.307100	0.153550	−	0.988141i	$-0.450929\pi$
$991$	9580.56	0.307100	0.153550	+	0.988141i	$0.450929\pi$
$992$	0	0
$993$	−14305.9	−0.457185
$994$	0	0
$995$	− 26526.6i	− 0.845177i
$996$	0	0
$997$	− 22405.3i	− 0.711719i	−0.934539	−	0.355860i	$-0.884188\pi$
$997$	− 22405.3i	− 0.711719i	0.934539	−	0.355860i	$-0.115812\pi$
$998$	0	0
$999$	11294.6	0.357703

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1344.4.c.b.673.6	yes	6
4.3	odd	2		1344.4.c.c.673.3	yes	6
8.3	odd	2		1344.4.c.c.673.4	yes	6
8.5	even	2	inner	1344.4.c.b.673.1	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1344.4.c.b.673.1	✓	6	8.5	even	2	inner
1344.4.c.b.673.6	yes	6	1.1	even	1	trivial
1344.4.c.c.673.3	yes	6	4.3	odd	2
1344.4.c.c.673.4	yes	6	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:	\( N \)	\(=\)	\( 1344 = 2^{6} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1344.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+3.00000i q^{3} +7.03493i q^{5} -7.00000 q^{7} -9.00000 q^{9} +O(q^{10})\)	\(q+3.00000i q^{3} +7.03493i q^{5} -7.00000 q^{7} -9.00000 q^{9} -46.7348i q^{11} +8.90643i q^{13} -21.1048 q^{15} +58.0811 q^{17} +135.122i q^{19} -21.0000i q^{21} +86.0112 q^{23} +75.5097 q^{25} -27.0000i q^{27} -300.014i q^{29} -183.712 q^{31} +140.205 q^{33} -49.2445i q^{35} +418.319i q^{37} -26.7193 q^{39} -106.081 q^{41} -162.206i q^{43} -63.3144i q^{45} +377.751 q^{47} +49.0000 q^{49} +174.243i q^{51} +709.548i q^{53} +328.777 q^{55} -405.367 q^{57} +753.348i q^{59} +667.954i q^{61} +63.0000 q^{63} -62.6562 q^{65} -447.506i q^{67} +258.034i q^{69} -384.548 q^{71} +1090.16 q^{73} +226.529i q^{75} +327.144i q^{77} -670.900 q^{79} +81.0000 q^{81} -447.801i q^{83} +408.597i q^{85} +900.043 q^{87} -70.5678 q^{89} -62.3450i q^{91} -551.137i q^{93} -950.576 q^{95} -1319.10 q^{97} +420.614i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 42 q^{7} - 54 q^{9}+O(q^{10}) \) 6 * q - 42 * q^7 - 54 * q^9	\( 6 q - 42 q^{7} - 54 q^{9} + 24 q^{15} - 72 q^{17} + 196 q^{23} + 246 q^{25} - 104 q^{31} + 408 q^{33} + 348 q^{39} - 216 q^{41} + 408 q^{47} + 294 q^{49} - 720 q^{55} + 336 q^{57} + 378 q^{63} - 2472 q^{65} + 1164 q^{71} - 276 q^{73} - 1352 q^{79} + 486 q^{81} - 720 q^{87} + 2520 q^{89} - 4648 q^{95} - 2340 q^{97}+O(q^{100}) \) 6 * q - 42 * q^7 - 54 * q^9 + 24 * q^15 - 72 * q^17 + 196 * q^23 + 246 * q^25 - 104 * q^31 + 408 * q^33 + 348 * q^39 - 216 * q^41 + 408 * q^47 + 294 * q^49 - 720 * q^55 + 336 * q^57 + 378 * q^63 - 2472 * q^65 + 1164 * q^71 - 276 * q^73 - 1352 * q^79 + 486 * q^81 - 720 * q^87 + 2520 * q^89 - 4648 * q^95 - 2340 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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