LMFDB - Embedded newform 1344.4.c.f.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:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 2 x^{11} - x^{10} - 861 x^{8} - 2158 x^{7} + 8654 x^{6} + 118244 x^{5} + 707300 x^{4} + \cdots + 43264 $ x^12 - 2x^11 - x^10 - 861x^8 - 2158x^7 + 8654x^6 + 118244x^5 + 707300x^4 + 1646096x^3 + 1391904x^2 + 174720*x + 43264
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{22}\cdot 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			673.6
Root			$-0.0488747 - 0.182403i$ of defining polynomial
Character	$\chi$	$=$	1344.673
Dual form			1344.4.c.f.673.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000i q^{3} +17.1345i q^{5} -7.00000 q^{7} -9.00000 q^{9} +O(q^{10})$	$q-3.00000i q^{3} +17.1345i q^{5} -7.00000 q^{7} -9.00000 q^{9} +2.98040i q^{11} -39.8701i q^{13} +51.4036 q^{15} +132.530 q^{17} +83.2004i q^{19} +21.0000i q^{21} -80.0478 q^{23} -168.592 q^{25} +27.0000i q^{27} -139.743i q^{29} +204.541 q^{31} +8.94119 q^{33} -119.942i q^{35} -377.916i q^{37} -119.610 q^{39} +45.3733 q^{41} +540.789i q^{43} -154.211i q^{45} +362.473 q^{47} +49.0000 q^{49} -397.589i q^{51} -668.494i q^{53} -51.0677 q^{55} +249.601 q^{57} +757.392i q^{59} +690.035i q^{61} +63.0000 q^{63} +683.157 q^{65} +370.530i q^{67} +240.143i q^{69} -652.184 q^{71} +229.804 q^{73} +505.777i q^{75} -20.8628i q^{77} +43.3308 q^{79} +81.0000 q^{81} +802.404i q^{83} +2270.84i q^{85} -419.230 q^{87} -868.188 q^{89} +279.091i q^{91} -613.622i q^{93} -1425.60 q^{95} -1181.82 q^{97} -26.8236i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 84 q^{7} - 108 q^{9}+O(q^{10}) $ 12 * q - 84 * q^7 - 108 * q^9	$ 12 q - 84 q^{7} - 108 q^{9} + 24 q^{15} + 376 q^{17} - 336 q^{23} - 180 q^{25} + 192 q^{31} - 168 q^{33} - 504 q^{39} + 488 q^{41} - 448 q^{47} + 588 q^{49} - 3600 q^{55} - 432 q^{57} + 756 q^{63} + 1408 q^{65} - 5104 q^{71} - 1752 q^{73} - 1632 q^{79} + 972 q^{81} - 336 q^{87} - 3688 q^{89} - 2496 q^{95} - 1944 q^{97}+O(q^{100}) $ 12 * q - 84 * q^7 - 108 * q^9 + 24 * q^15 + 376 * q^17 - 336 * q^23 - 180 * q^25 + 192 * q^31 - 168 * q^33 - 504 * q^39 + 488 * q^41 - 448 * q^47 + 588 * q^49 - 3600 * q^55 - 432 * q^57 + 756 * q^63 + 1408 * q^65 - 5104 * q^71 - 1752 * q^73 - 1632 * q^79 + 972 * q^81 - 336 * q^87 - 3688 * q^89 - 2496 * q^95 - 1944 * 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
$5$	17.1345i	1.53256i	0.642507	+	0.766280i	$0.277894\pi$
$5$	17.1345i	1.53256i	−0.642507	+	0.766280i	$0.722106\pi$
$6$	0	0
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	0	0
$9$	−9.00000	−0.333333
$10$	0	0
$11$	2.98040i	0.0816931i	0.999165	+	0.0408465i	$0.0130055\pi$
$11$	2.98040i	0.0816931i	−0.999165	+	0.0408465i	$0.986995\pi$
$12$	0	0
$13$	− 39.8701i	− 0.850615i	−0.905049	−	0.425307i	$-0.860166\pi$
$13$	− 39.8701i	− 0.850615i	0.905049	−	0.425307i	$-0.139834\pi$
$14$	0	0
$15$	51.4036	0.884824
$16$	0	0
$17$	132.530	1.89077	0.945387	−	0.325949i	$-0.105684\pi$
$17$	132.530	1.89077	0.945387	+	0.325949i	$0.105684\pi$
$18$	0	0
$19$	83.2004i	1.00460i	0.864692	+	0.502302i	$0.167514\pi$
$19$	83.2004i	1.00460i	−0.864692	+	0.502302i	$0.832486\pi$
$20$	0	0
$21$	21.0000i	0.218218i
$22$	0	0
$23$	−80.0478	−0.725701	−0.362850	−	0.931847i	$-0.618196\pi$
$23$	−80.0478	−0.725701	−0.362850	+	0.931847i	$0.618196\pi$
$24$	0	0
$25$	−168.592	−1.34874
$26$	0	0
$27$	27.0000i	0.192450i
$28$	0	0
$29$	− 139.743i	− 0.894817i	−0.894330	−	0.447408i	$-0.852347\pi$
$29$	− 139.743i	− 0.894817i	0.894330	−	0.447408i	$-0.147653\pi$
$30$	0	0
$31$	204.541	1.18505	0.592525	−	0.805552i	$-0.298131\pi$
$31$	204.541	1.18505	0.592525	+	0.805552i	$0.298131\pi$
$32$	0	0
$33$	8.94119	0.0471655
$34$	0	0
$35$	− 119.942i	− 0.579253i
$36$	0	0
$37$	− 377.916i	− 1.67916i	−0.543234	−	0.839582i	$-0.682800\pi$
$37$	− 377.916i	− 1.67916i	0.543234	−	0.839582i	$-0.317200\pi$
$38$	0	0
$39$	−119.610	−0.491103
$40$	0	0
$41$	45.3733	0.172832	0.0864161	−	0.996259i	$-0.472459\pi$
$41$	45.3733	0.172832	0.0864161	+	0.996259i	$0.472459\pi$
$42$	0	0
$43$	540.789i	1.91790i	0.283582	+	0.958948i	$0.408477\pi$
$43$	540.789i	1.91790i	−0.283582	+	0.958948i	$0.591523\pi$
$44$	0	0
$45$	− 154.211i	− 0.510853i
$46$	0	0
$47$	362.473	1.12494	0.562470	−	0.826818i	$-0.309851\pi$
$47$	362.473	1.12494	0.562470	+	0.826818i	$0.309851\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	− 397.589i	− 1.09164i
$52$	0	0
$53$	− 668.494i	− 1.73254i	−0.499575	−	0.866270i	$-0.666511\pi$
$53$	− 668.494i	− 1.73254i	0.499575	−	0.866270i	$-0.333489\pi$
$54$	0	0
$55$	−51.0677	−0.125200
$56$	0	0
$57$	249.601	0.580009
$58$	0	0
$59$	757.392i	1.67125i	0.549297	+	0.835627i	$0.314895\pi$
$59$	757.392i	1.67125i	−0.549297	+	0.835627i	$0.685105\pi$
$60$	0	0
$61$	690.035i	1.44836i	0.689611	+	0.724180i	$0.257782\pi$
$61$	690.035i	1.44836i	−0.689611	+	0.724180i	$0.742218\pi$
$62$	0	0
$63$	63.0000	0.125988
$64$	0	0
$65$	683.157	1.30362
$66$	0	0
$67$	370.530i	0.675634i	0.941212	+	0.337817i	$0.109689\pi$
$67$	370.530i	0.675634i	−0.941212	+	0.337817i	$0.890311\pi$
$68$	0	0
$69$	240.143i	0.418984i
$70$	0	0
$71$	−652.184	−1.09014	−0.545070	−	0.838390i	$-0.683497\pi$
$71$	−652.184	−1.09014	−0.545070	+	0.838390i	$0.683497\pi$
$72$	0	0
$73$	229.804	0.368445	0.184223	−	0.982885i	$-0.441023\pi$
$73$	229.804	0.368445	0.184223	+	0.982885i	$0.441023\pi$
$74$	0	0
$75$	505.777i	0.778695i
$76$	0	0
$77$	− 20.8628i	− 0.0308771i
$78$	0	0
$79$	43.3308	0.0617101	0.0308550	−	0.999524i	$-0.490177\pi$
$79$	43.3308	0.0617101	0.0308550	+	0.999524i	$0.490177\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	802.404i	1.06115i	0.847639	+	0.530574i	$0.178024\pi$
$83$	802.404i	1.06115i	−0.847639	+	0.530574i	$0.821976\pi$
$84$	0	0
$85$	2270.84i	2.89773i
$86$	0	0
$87$	−419.230	−0.516623
$88$	0	0
$89$	−868.188	−1.03402	−0.517010	−	0.855980i	$-0.672955\pi$
$89$	−868.188	−1.03402	−0.517010	+	0.855980i	$0.672955\pi$
$90$	0	0
$91$	279.091i	0.321502i
$92$	0	0
$93$	− 613.622i	− 0.684189i
$94$	0	0
$95$	−1425.60	−1.53962
$96$	0	0
$97$	−1181.82	−1.23707	−0.618537	−	0.785756i	$-0.712274\pi$
$97$	−1181.82	−1.23707	−0.618537	+	0.785756i	$0.712274\pi$
$98$	0	0
$99$	− 26.8236i	− 0.0272310i
$100$	0	0
$101$	215.909i	0.212710i	0.994328	+	0.106355i	$0.0339180\pi$
$101$	215.909i	0.212710i	−0.994328	+	0.106355i	$0.966082\pi$
$102$	0	0
$103$	1204.93	1.15267	0.576334	−	0.817214i	$-0.304483\pi$
$103$	1204.93	1.15267	0.576334	+	0.817214i	$0.304483\pi$
$104$	0	0
$105$	−359.825	−0.334432
$106$	0	0
$107$	− 327.101i	− 0.295533i	−0.989022	−	0.147767i	$-0.952792\pi$
$107$	− 327.101i	− 0.295533i	0.989022	−	0.147767i	$-0.0472085\pi$
$108$	0	0
$109$	668.438i	0.587383i	0.955900	+	0.293692i	$0.0948839\pi$
$109$	668.438i	0.587383i	−0.955900	+	0.293692i	$0.905116\pi$
$110$	0	0
$111$	−1133.75	−0.969465
$112$	0	0
$113$	−725.333	−0.603837	−0.301919	−	0.953334i	$-0.597627\pi$
$113$	−725.333	−0.603837	−0.301919	+	0.953334i	$0.597627\pi$
$114$	0	0
$115$	− 1371.58i	− 1.11218i
$116$	0	0
$117$	358.831i	0.283538i
$118$	0	0
$119$	−927.708	−0.714646
$120$	0	0
$121$	1322.12	0.993326
$122$	0	0
$123$	− 136.120i	− 0.0997847i
$124$	0	0
$125$	− 746.937i	− 0.534465i
$126$	0	0
$127$	−672.231	−0.469692	−0.234846	−	0.972033i	$-0.575459\pi$
$127$	−672.231	−0.469692	−0.234846	+	0.972033i	$0.575459\pi$
$128$	0	0
$129$	1622.37	1.10730
$130$	0	0
$131$	− 2082.36i	− 1.38883i	−0.719575	−	0.694414i	$-0.755664\pi$
$131$	− 2082.36i	− 1.38883i	0.719575	−	0.694414i	$-0.244336\pi$
$132$	0	0
$133$	− 582.403i	− 0.379705i
$134$	0	0
$135$	−462.633	−0.294941
$136$	0	0
$137$	1221.50	0.761749	0.380875	−	0.924627i	$-0.375623\pi$
$137$	1221.50	0.761749	0.380875	+	0.924627i	$0.375623\pi$
$138$	0	0
$139$	2590.38i	1.58067i	0.612674	+	0.790336i	$0.290094\pi$
$139$	2590.38i	1.58067i	−0.612674	+	0.790336i	$0.709906\pi$
$140$	0	0
$141$	− 1087.42i	− 0.649484i
$142$	0	0
$143$	118.829	0.0694893
$144$	0	0
$145$	2394.44	1.37136
$146$	0	0
$147$	− 147.000i	− 0.0824786i
$148$	0	0
$149$	3406.10i	1.87274i	0.351015	+	0.936370i	$0.385837\pi$
$149$	3406.10i	1.87274i	−0.351015	+	0.936370i	$0.614163\pi$
$150$	0	0
$151$	−1795.43	−0.967619	−0.483809	−	0.875173i	$-0.660747\pi$
$151$	−1795.43	−0.967619	−0.483809	+	0.875173i	$0.660747\pi$
$152$	0	0
$153$	−1192.77	−0.630258
$154$	0	0
$155$	3504.71i	1.81616i
$156$	0	0
$157$	2968.25i	1.50887i	0.656377	+	0.754433i	$0.272088\pi$
$157$	2968.25i	1.50887i	−0.656377	+	0.754433i	$0.727912\pi$
$158$	0	0
$159$	−2005.48	−1.00028
$160$	0	0
$161$	560.335	0.274289
$162$	0	0
$163$	1529.60i	0.735016i	0.930020	+	0.367508i	$0.119789\pi$
$163$	1529.60i	0.735016i	−0.930020	+	0.367508i	$0.880211\pi$
$164$	0	0
$165$	153.203i	0.0722840i
$166$	0	0
$167$	−252.934	−0.117201	−0.0586007	−	0.998282i	$-0.518664\pi$
$167$	−252.934	−0.117201	−0.0586007	+	0.998282i	$0.518664\pi$
$168$	0	0
$169$	607.371	0.276455
$170$	0	0
$171$	− 748.804i	− 0.334868i
$172$	0	0
$173$	1196.52i	0.525838i	0.964818	+	0.262919i	$0.0846853\pi$
$173$	1196.52i	0.525838i	−0.964818	+	0.262919i	$0.915315\pi$
$174$	0	0
$175$	1180.15	0.509776
$176$	0	0
$177$	2272.17	0.964899
$178$	0	0
$179$	3863.73i	1.61335i	0.590998	+	0.806673i	$0.298734\pi$
$179$	3863.73i	1.61335i	−0.590998	+	0.806673i	$0.701266\pi$
$180$	0	0
$181$	1166.62i	0.479085i	0.970886	+	0.239542i	$0.0769974\pi$
$181$	1166.62i	0.479085i	−0.970886	+	0.239542i	$0.923003\pi$
$182$	0	0
$183$	2070.11	0.836211
$184$	0	0
$185$	6475.42	2.57342
$186$	0	0
$187$	394.991i	0.154463i
$188$	0	0
$189$	− 189.000i	− 0.0727393i
$190$	0	0
$191$	1734.96	0.657263	0.328631	−	0.944458i	$-0.393413\pi$
$191$	1734.96	0.657263	0.328631	+	0.944458i	$0.393413\pi$
$192$	0	0
$193$	−308.801	−0.115171	−0.0575854	−	0.998341i	$-0.518340\pi$
$193$	−308.801	−0.115171	−0.0575854	+	0.998341i	$0.518340\pi$
$194$	0	0
$195$	− 2049.47i	− 0.752644i
$196$	0	0
$197$	− 1297.96i	− 0.469420i	−0.972065	−	0.234710i	$-0.924586\pi$
$197$	− 1297.96i	− 0.469420i	0.972065	−	0.234710i	$-0.0754140\pi$
$198$	0	0
$199$	1750.53	0.623576	0.311788	−	0.950152i	$-0.399072\pi$
$199$	1750.53	0.623576	0.311788	+	0.950152i	$0.399072\pi$
$200$	0	0
$201$	1111.59	0.390078
$202$	0	0
$203$	978.203i	0.338209i
$204$	0	0
$205$	777.450i	0.264876i
$206$	0	0
$207$	720.430	0.241900
$208$	0	0
$209$	−247.970	−0.0820692
$210$	0	0
$211$	− 1022.70i	− 0.333676i	−0.985984	−	0.166838i	$-0.946644\pi$
$211$	− 1022.70i	− 0.333676i	0.985984	−	0.166838i	$-0.0533557\pi$
$212$	0	0
$213$	1956.55i	0.629393i
$214$	0	0
$215$	−9266.17	−2.93929
$216$	0	0
$217$	−1431.78	−0.447907
$218$	0	0
$219$	− 689.411i	− 0.212722i
$220$	0	0
$221$	− 5283.98i	− 1.60832i
$222$	0	0
$223$	255.416	0.0766992	0.0383496	−	0.999264i	$-0.487790\pi$
$223$	255.416	0.0766992	0.0383496	+	0.999264i	$0.487790\pi$
$224$	0	0
$225$	1517.33	0.449580
$226$	0	0
$227$	− 1041.15i	− 0.304422i	−0.988348	−	0.152211i	$-0.951361\pi$
$227$	− 1041.15i	− 0.304422i	0.988348	−	0.152211i	$-0.0486393\pi$
$228$	0	0
$229$	− 2592.03i	− 0.747975i	−0.927434	−	0.373988i	$-0.877990\pi$
$229$	− 2592.03i	− 0.747975i	0.927434	−	0.373988i	$-0.122010\pi$
$230$	0	0
$231$	−62.5884	−0.0178269
$232$	0	0
$233$	−1830.07	−0.514558	−0.257279	−	0.966337i	$-0.582826\pi$
$233$	−1830.07	−0.514558	−0.257279	+	0.966337i	$0.582826\pi$
$234$	0	0
$235$	6210.82i	1.72404i
$236$	0	0
$237$	− 129.992i	− 0.0356283i
$238$	0	0
$239$	135.227	0.0365989	0.0182994	−	0.999833i	$-0.494175\pi$
$239$	135.227	0.0365989	0.0182994	+	0.999833i	$0.494175\pi$
$240$	0	0
$241$	−312.681	−0.0835748	−0.0417874	−	0.999127i	$-0.513305\pi$
$241$	−312.681	−0.0835748	−0.0417874	+	0.999127i	$0.513305\pi$
$242$	0	0
$243$	− 243.000i	− 0.0641500i
$244$	0	0
$245$	839.592i	0.218937i
$246$	0	0
$247$	3317.21	0.854531
$248$	0	0
$249$	2407.21	0.612654
$250$	0	0
$251$	4385.72i	1.10289i	0.834213	+	0.551443i	$0.185922\pi$
$251$	4385.72i	1.10289i	−0.834213	+	0.551443i	$0.814078\pi$
$252$	0	0
$253$	− 238.574i	− 0.0592847i
$254$	0	0
$255$	6812.51	1.67300
$256$	0	0
$257$	6255.49	1.51831	0.759157	−	0.650908i	$-0.225612\pi$
$257$	6255.49	1.51831	0.759157	+	0.650908i	$0.225612\pi$
$258$	0	0
$259$	2645.41i	0.634664i
$260$	0	0
$261$	1257.69i	0.298272i
$262$	0	0
$263$	5952.39	1.39559	0.697795	−	0.716298i	$-0.254165\pi$
$263$	5952.39	1.39559	0.697795	+	0.716298i	$0.254165\pi$
$264$	0	0
$265$	11454.3	2.65522
$266$	0	0
$267$	2604.56i	0.596991i
$268$	0	0
$269$	2997.81i	0.679479i	0.940520	+	0.339739i	$0.110339\pi$
$269$	2997.81i	0.679479i	−0.940520	+	0.339739i	$0.889661\pi$
$270$	0	0
$271$	681.179	0.152689	0.0763444	−	0.997082i	$-0.475675\pi$
$271$	681.179	0.152689	0.0763444	+	0.997082i	$0.475675\pi$
$272$	0	0
$273$	837.273	0.185619
$274$	0	0
$275$	− 502.473i	− 0.110183i
$276$	0	0
$277$	1466.94i	0.318194i	0.987263	+	0.159097i	$0.0508583\pi$
$277$	1466.94i	0.318194i	−0.987263	+	0.159097i	$0.949142\pi$
$278$	0	0
$279$	−1840.86	−0.395017
$280$	0	0
$281$	−6558.87	−1.39242	−0.696209	−	0.717840i	$-0.745131\pi$
$281$	−6558.87	−1.39242	−0.696209	+	0.717840i	$0.745131\pi$
$282$	0	0
$283$	− 8554.70i	− 1.79690i	−0.439071	−	0.898452i	$-0.644692\pi$
$283$	− 8554.70i	− 1.79690i	0.439071	−	0.898452i	$-0.355308\pi$
$284$	0	0
$285$	4276.80i	0.888898i
$286$	0	0
$287$	−317.613	−0.0653244
$288$	0	0
$289$	12651.1	2.57503
$290$	0	0
$291$	3545.47i	0.714225i
$292$	0	0
$293$	1262.01i	0.251629i	0.992054	+	0.125815i	$0.0401544\pi$
$293$	1262.01i	0.251629i	−0.992054	+	0.125815i	$0.959846\pi$
$294$	0	0
$295$	−12977.6	−2.56130
$296$	0	0
$297$	−80.4707	−0.0157218
$298$	0	0
$299$	3191.52i	0.617292i
$300$	0	0
$301$	− 3785.52i	− 0.724896i
$302$	0	0
$303$	647.726	0.122808
$304$	0	0
$305$	−11823.4	−2.21970
$306$	0	0
$307$	234.049i	0.0435111i	0.999763	+	0.0217555i	$0.00692555\pi$
$307$	234.049i	0.0435111i	−0.999763	+	0.0217555i	$0.993074\pi$
$308$	0	0
$309$	− 3614.78i	− 0.665493i
$310$	0	0
$311$	−182.019	−0.0331876	−0.0165938	−	0.999862i	$-0.505282\pi$
$311$	−182.019	−0.0331876	−0.0165938	+	0.999862i	$0.505282\pi$
$312$	0	0
$313$	3515.22	0.634799	0.317399	−	0.948292i	$-0.397190\pi$
$313$	3515.22	0.634799	0.317399	+	0.948292i	$0.397190\pi$
$314$	0	0
$315$	1079.48i	0.193084i
$316$	0	0
$317$	− 3339.39i	− 0.591667i	−0.955239	−	0.295834i	$-0.904403\pi$
$317$	− 3339.39i	− 0.591667i	0.955239	−	0.295834i	$-0.0955974\pi$
$318$	0	0
$319$	416.491	0.0731003
$320$	0	0
$321$	−981.304	−0.170626
$322$	0	0
$323$	11026.5i	1.89948i
$324$	0	0
$325$	6721.81i	1.14726i
$326$	0	0
$327$	2005.32	0.339126
$328$	0	0
$329$	−2537.31	−0.425187
$330$	0	0
$331$	1594.18i	0.264725i	0.991201	+	0.132363i	$0.0422563\pi$
$331$	1594.18i	0.264725i	−0.991201	+	0.132363i	$0.957744\pi$
$332$	0	0
$333$	3401.25i	0.559721i
$334$	0	0
$335$	−6348.87	−1.03545
$336$	0	0
$337$	−12224.7	−1.97603	−0.988017	−	0.154342i	$-0.950674\pi$
$337$	−12224.7	−1.97603	−0.988017	+	0.154342i	$0.950674\pi$
$338$	0	0
$339$	2176.00i	0.348626i
$340$	0	0
$341$	609.612i	0.0968104i
$342$	0	0
$343$	−343.000	−0.0539949
$344$	0	0
$345$	−4114.75	−0.642117
$346$	0	0
$347$	− 3116.35i	− 0.482117i	−0.970511	−	0.241058i	$-0.922505\pi$
$347$	− 3116.35i	− 0.482117i	0.970511	−	0.241058i	$-0.0774945\pi$
$348$	0	0
$349$	− 7826.76i	− 1.20045i	−0.799831	−	0.600225i	$-0.795078\pi$
$349$	− 7826.76i	− 1.20045i	0.799831	−	0.600225i	$-0.204922\pi$
$350$	0	0
$351$	1076.49	0.163701
$352$	0	0
$353$	4326.14	0.652287	0.326143	−	0.945320i	$-0.394251\pi$
$353$	4326.14	0.652287	0.326143	+	0.945320i	$0.394251\pi$
$354$	0	0
$355$	− 11174.9i	− 1.67071i
$356$	0	0
$357$	2783.12i	0.412601i
$358$	0	0
$359$	−3871.77	−0.569204	−0.284602	−	0.958646i	$-0.591861\pi$
$359$	−3871.77	−0.569204	−0.284602	+	0.958646i	$0.591861\pi$
$360$	0	0
$361$	−63.3136	−0.00923073
$362$	0	0
$363$	− 3966.35i	− 0.573497i
$364$	0	0
$365$	3937.58i	0.564664i
$366$	0	0
$367$	8206.26	1.16720	0.583601	−	0.812041i	$-0.301643\pi$
$367$	8206.26	1.16720	0.583601	+	0.812041i	$0.301643\pi$
$368$	0	0
$369$	−408.360	−0.0576107
$370$	0	0
$371$	4679.46i	0.654839i
$372$	0	0
$373$	− 2149.21i	− 0.298343i	−0.988811	−	0.149172i	$-0.952339\pi$
$373$	− 2149.21i	− 0.298343i	0.988811	−	0.149172i	$-0.0476607\pi$
$374$	0	0
$375$	−2240.81	−0.308573
$376$	0	0
$377$	−5571.59	−0.761144
$378$	0	0
$379$	− 658.658i	− 0.0892690i	−0.999003	−	0.0446345i	$-0.985788\pi$
$379$	− 658.658i	− 0.0892690i	0.999003	−	0.0446345i	$-0.0142123\pi$
$380$	0	0
$381$	2016.69i	0.271177i
$382$	0	0
$383$	−3084.93	−0.411573	−0.205786	−	0.978597i	$-0.565975\pi$
$383$	−3084.93	−0.411573	−0.205786	+	0.978597i	$0.565975\pi$
$384$	0	0
$385$	357.474	0.0473210
$386$	0	0
$387$	− 4867.10i	− 0.639299i
$388$	0	0
$389$	4413.01i	0.575189i	0.957752	+	0.287595i	$0.0928556\pi$
$389$	4413.01i	0.575189i	−0.957752	+	0.287595i	$0.907144\pi$
$390$	0	0
$391$	−10608.7	−1.37214
$392$	0	0
$393$	−6247.08	−0.801840
$394$	0	0
$395$	742.454i	0.0945744i
$396$	0	0
$397$	3900.51i	0.493101i	0.969130	+	0.246551i	$0.0792972\pi$
$397$	3900.51i	0.493101i	−0.969130	+	0.246551i	$0.920703\pi$
$398$	0	0
$399$	−1747.21	−0.219223
$400$	0	0
$401$	−8593.07	−1.07012	−0.535059	−	0.844814i	$-0.679711\pi$
$401$	−8593.07	−1.07012	−0.535059	+	0.844814i	$0.679711\pi$
$402$	0	0
$403$	− 8155.06i	− 1.00802i
$404$	0	0
$405$	1387.90i	0.170284i
$406$	0	0
$407$	1126.34	0.137176
$408$	0	0
$409$	11212.2	1.35552	0.677758	−	0.735285i	$-0.262952\pi$
$409$	11212.2	1.35552	0.677758	+	0.735285i	$0.262952\pi$
$410$	0	0
$411$	− 3664.49i	− 0.439796i
$412$	0	0
$413$	− 5301.74i	− 0.631675i
$414$	0	0
$415$	−13748.8	−1.62627
$416$	0	0
$417$	7771.15	0.912601
$418$	0	0
$419$	7948.88i	0.926798i	0.886150	+	0.463399i	$0.153370\pi$
$419$	7948.88i	0.926798i	−0.886150	+	0.463399i	$0.846630\pi$
$420$	0	0
$421$	11113.8i	1.28658i	0.765622	+	0.643291i	$0.222432\pi$
$421$	11113.8i	1.28658i	−0.765622	+	0.643291i	$0.777568\pi$
$422$	0	0
$423$	−3262.26	−0.374980
$424$	0	0
$425$	−22343.5	−2.55016
$426$	0	0
$427$	− 4830.25i	− 0.547429i
$428$	0	0
$429$	− 356.487i	− 0.0401197i
$430$	0	0
$431$	−3571.39	−0.399137	−0.199568	−	0.979884i	$-0.563954\pi$
$431$	−3571.39	−0.399137	−0.199568	+	0.979884i	$0.563954\pi$
$432$	0	0
$433$	−13245.8	−1.47010	−0.735050	−	0.678013i	$-0.762841\pi$
$433$	−13245.8	−1.47010	−0.735050	+	0.678013i	$0.762841\pi$
$434$	0	0
$435$	− 7183.31i	− 0.791755i
$436$	0	0
$437$	− 6660.01i	− 0.729043i
$438$	0	0
$439$	−7028.36	−0.764112	−0.382056	−	0.924139i	$-0.624784\pi$
$439$	−7028.36	−0.764112	−0.382056	+	0.924139i	$0.624784\pi$
$440$	0	0
$441$	−441.000	−0.0476190
$442$	0	0
$443$	− 133.981i	− 0.0143694i	−0.999974	−	0.00718469i	$-0.997713\pi$
$443$	− 133.981i	− 0.0143694i	0.999974	−	0.00718469i	$-0.00228698\pi$
$444$	0	0
$445$	− 14876.0i	− 1.58470i
$446$	0	0
$447$	10218.3	1.08123
$448$	0	0
$449$	2783.94	0.292611	0.146306	−	0.989239i	$-0.453262\pi$
$449$	2783.94	0.292611	0.146306	+	0.989239i	$0.453262\pi$
$450$	0	0
$451$	135.230i	0.0141192i
$452$	0	0
$453$	5386.30i	0.558655i
$454$	0	0
$455$	−4782.10	−0.492721
$456$	0	0
$457$	19117.6	1.95686	0.978428	−	0.206590i	$-0.0662367\pi$
$457$	19117.6	1.95686	0.978428	+	0.206590i	$0.0662367\pi$
$458$	0	0
$459$	3578.30i	0.363880i
$460$	0	0
$461$	6684.09i	0.675291i	0.941273	+	0.337645i	$0.109630\pi$
$461$	6684.09i	0.675291i	−0.941273	+	0.337645i	$0.890370\pi$
$462$	0	0
$463$	12556.5	1.26037	0.630185	−	0.776445i	$-0.282979\pi$
$463$	12556.5	1.26037	0.630185	+	0.776445i	$0.282979\pi$
$464$	0	0
$465$	10514.1	1.04856
$466$	0	0
$467$	5380.24i	0.533122i	0.963818	+	0.266561i	$0.0858873\pi$
$467$	5380.24i	0.533122i	−0.963818	+	0.266561i	$0.914113\pi$
$468$	0	0
$469$	− 2593.71i	− 0.255366i
$470$	0	0
$471$	8904.75	0.871145
$472$	0	0
$473$	−1611.77	−0.156679
$474$	0	0
$475$	− 14027.0i	− 1.35495i
$476$	0	0
$477$	6016.44i	0.577514i
$478$	0	0
$479$	9401.83	0.896828	0.448414	−	0.893826i	$-0.351989\pi$
$479$	9401.83	0.896828	0.448414	+	0.893826i	$0.351989\pi$
$480$	0	0
$481$	−15067.6	−1.42832
$482$	0	0
$483$	− 1681.00i	− 0.158361i
$484$	0	0
$485$	− 20250.0i	− 1.89589i
$486$	0	0
$487$	6836.64	0.636135	0.318067	−	0.948068i	$-0.396966\pi$
$487$	6836.64	0.636135	0.318067	+	0.948068i	$0.396966\pi$
$488$	0	0
$489$	4588.80	0.424362
$490$	0	0
$491$	44.2581i	0.00406790i	0.999998	+	0.00203395i	$0.000647427\pi$
$491$	44.2581i	0.00406790i	−0.999998	+	0.00203395i	$0.999353\pi$
$492$	0	0
$493$	− 18520.1i	− 1.69190i
$494$	0	0
$495$	459.610	0.0417332
$496$	0	0
$497$	4565.29	0.412034
$498$	0	0
$499$	7884.29i	0.707313i	0.935375	+	0.353657i	$0.115062\pi$
$499$	7884.29i	0.707313i	−0.935375	+	0.353657i	$0.884938\pi$
$500$	0	0
$501$	758.803i	0.0676663i
$502$	0	0
$503$	−14818.8	−1.31360	−0.656798	−	0.754066i	$-0.728090\pi$
$503$	−14818.8	−1.31360	−0.656798	+	0.754066i	$0.728090\pi$
$504$	0	0
$505$	−3699.50	−0.325991
$506$	0	0
$507$	− 1822.11i	− 0.159611i
$508$	0	0
$509$	5035.72i	0.438515i	0.975667	+	0.219258i	$0.0703635\pi$
$509$	5035.72i	0.438515i	−0.975667	+	0.219258i	$0.929637\pi$
$510$	0	0
$511$	−1608.63	−0.139259
$512$	0	0
$513$	−2246.41	−0.193336
$514$	0	0
$515$	20645.8i	1.76653i
$516$	0	0
$517$	1080.31i	0.0918998i
$518$	0	0
$519$	3589.57	0.303593
$520$	0	0
$521$	−13373.1	−1.12454	−0.562269	−	0.826954i	$-0.690071\pi$
$521$	−13373.1	−1.12454	−0.562269	+	0.826954i	$0.690071\pi$
$522$	0	0
$523$	− 7349.16i	− 0.614448i	−0.951637	−	0.307224i	$-0.900600\pi$
$523$	− 7349.16i	− 0.614448i	0.951637	−	0.307224i	$-0.0994000\pi$
$524$	0	0
$525$	− 3540.44i	− 0.294319i
$526$	0	0
$527$	27107.7	2.24066
$528$	0	0
$529$	−5759.35	−0.473358
$530$	0	0
$531$	− 6816.52i	− 0.557085i
$532$	0	0
$533$	− 1809.04i	− 0.147014i
$534$	0	0
$535$	5604.73	0.452923
$536$	0	0
$537$	11591.2	0.931465
$538$	0	0
$539$	146.039i	0.0116704i
$540$	0	0
$541$	8183.14i	0.650315i	0.945660	+	0.325158i	$0.105417\pi$
$541$	8183.14i	0.650315i	−0.945660	+	0.325158i	$0.894583\pi$
$542$	0	0
$543$	3499.87	0.276600
$544$	0	0
$545$	−11453.4	−0.900200
$546$	0	0
$547$	11819.9i	0.923916i	0.886902	+	0.461958i	$0.152853\pi$
$547$	11819.9i	0.923916i	−0.886902	+	0.461958i	$0.847147\pi$
$548$	0	0
$549$	− 6210.32i	− 0.482787i
$550$	0	0
$551$	11626.7	0.898937
$552$	0	0
$553$	−303.316	−0.0233242
$554$	0	0
$555$	− 19426.3i	− 1.48576i
$556$	0	0
$557$	− 22731.0i	− 1.72916i	−0.502491	−	0.864582i	$-0.667583\pi$
$557$	− 22731.0i	− 1.72916i	0.502491	−	0.864582i	$-0.332417\pi$
$558$	0	0
$559$	21561.3	1.63139
$560$	0	0
$561$	1184.97	0.0891794
$562$	0	0
$563$	7860.21i	0.588399i	0.955744	+	0.294199i	$0.0950530\pi$
$563$	7860.21i	0.588399i	−0.955744	+	0.294199i	$0.904947\pi$
$564$	0	0
$565$	− 12428.3i	− 0.925417i
$566$	0	0
$567$	−567.000	−0.0419961
$568$	0	0
$569$	−11633.8	−0.857146	−0.428573	−	0.903507i	$-0.640983\pi$
$569$	−11633.8	−0.857146	−0.428573	+	0.903507i	$0.640983\pi$
$570$	0	0
$571$	− 8319.09i	− 0.609707i	−0.952399	−	0.304854i	$-0.901392\pi$
$571$	− 8319.09i	− 0.609707i	0.952399	−	0.304854i	$-0.0986076\pi$
$572$	0	0
$573$	− 5204.87i	− 0.379471i
$574$	0	0
$575$	13495.5	0.978782
$576$	0	0
$577$	−12339.9	−0.890324	−0.445162	−	0.895450i	$-0.646854\pi$
$577$	−12339.9	−0.890324	−0.445162	+	0.895450i	$0.646854\pi$
$578$	0	0
$579$	926.402i	0.0664939i
$580$	0	0
$581$	− 5616.83i	− 0.401076i
$582$	0	0
$583$	1992.38	0.141537
$584$	0	0
$585$	−6148.41	−0.434539
$586$	0	0
$587$	− 22785.2i	− 1.60212i	−0.598584	−	0.801060i	$-0.704270\pi$
$587$	− 22785.2i	− 1.60212i	0.598584	−	0.801060i	$-0.295730\pi$
$588$	0	0
$589$	17017.9i	1.19051i
$590$	0	0
$591$	−3893.87	−0.271020
$592$	0	0
$593$	6786.01	0.469929	0.234965	−	0.972004i	$-0.424503\pi$
$593$	6786.01	0.469929	0.234965	+	0.972004i	$0.424503\pi$
$594$	0	0
$595$	− 15895.8i	− 1.09524i
$596$	0	0
$597$	− 5251.59i	− 0.360022i
$598$	0	0
$599$	−10498.4	−0.716114	−0.358057	−	0.933700i	$-0.616561\pi$
$599$	−10498.4	−0.716114	−0.358057	+	0.933700i	$0.616561\pi$
$600$	0	0
$601$	11360.3	0.771045	0.385522	−	0.922698i	$-0.374021\pi$
$601$	11360.3	0.771045	0.385522	+	0.922698i	$0.374021\pi$
$602$	0	0
$603$	− 3334.77i	− 0.225211i
$604$	0	0
$605$	22653.9i	1.52233i
$606$	0	0
$607$	5460.38	0.365124	0.182562	−	0.983194i	$-0.441561\pi$
$607$	5460.38	0.365124	0.182562	+	0.983194i	$0.441561\pi$
$608$	0	0
$609$	2934.61	0.195265
$610$	0	0
$611$	− 14451.9i	− 0.956890i
$612$	0	0
$613$	17628.3i	1.16150i	0.814082	+	0.580750i	$0.197241\pi$
$613$	17628.3i	1.16150i	−0.814082	+	0.580750i	$0.802759\pi$
$614$	0	0
$615$	2332.35	0.152926
$616$	0	0
$617$	−17543.6	−1.14470	−0.572351	−	0.820009i	$-0.693968\pi$
$617$	−17543.6	−1.14470	−0.572351	+	0.820009i	$0.693968\pi$
$618$	0	0
$619$	− 27868.2i	− 1.80956i	−0.425880	−	0.904779i	$-0.640036\pi$
$619$	− 27868.2i	− 1.80956i	0.425880	−	0.904779i	$-0.359964\pi$
$620$	0	0
$621$	− 2161.29i	− 0.139661i
$622$	0	0
$623$	6077.31	0.390823
$624$	0	0
$625$	−8275.64	−0.529641
$626$	0	0
$627$	743.911i	0.0473827i
$628$	0	0
$629$	− 50085.1i	− 3.17492i
$630$	0	0
$631$	21948.0	1.38469	0.692343	−	0.721569i	$-0.256579\pi$
$631$	21948.0	1.38469	0.692343	+	0.721569i	$0.256579\pi$
$632$	0	0
$633$	−3068.11	−0.192648
$634$	0	0
$635$	− 11518.4i	− 0.719830i
$636$	0	0
$637$	− 1953.64i	− 0.121516i
$638$	0	0
$639$	5869.65	0.363380
$640$	0	0
$641$	5186.49	0.319585	0.159793	−	0.987151i	$-0.448917\pi$
$641$	5186.49	0.319585	0.159793	+	0.987151i	$0.448917\pi$
$642$	0	0
$643$	21931.1i	1.34507i	0.740066	+	0.672534i	$0.234794\pi$
$643$	21931.1i	1.34507i	−0.740066	+	0.672534i	$0.765206\pi$
$644$	0	0
$645$	27798.5i	1.69700i
$646$	0	0
$647$	−19475.1	−1.18338	−0.591690	−	0.806166i	$-0.701539\pi$
$647$	−19475.1	−1.18338	−0.591690	+	0.806166i	$0.701539\pi$
$648$	0	0
$649$	−2257.33	−0.136530
$650$	0	0
$651$	4295.35i	0.258599i
$652$	0	0
$653$	− 29761.6i	− 1.78355i	−0.452477	−	0.891776i	$-0.649460\pi$
$653$	− 29761.6i	− 1.78355i	0.452477	−	0.891776i	$-0.350540\pi$
$654$	0	0
$655$	35680.3	2.12846
$656$	0	0
$657$	−2068.23	−0.122815
$658$	0	0
$659$	− 12257.3i	− 0.724547i	−0.932072	−	0.362274i	$-0.882001\pi$
$659$	− 12257.3i	− 0.724547i	0.932072	−	0.362274i	$-0.117999\pi$
$660$	0	0
$661$	− 9737.88i	− 0.573010i	−0.958079	−	0.286505i	$-0.907506\pi$
$661$	− 9737.88i	− 0.573010i	0.958079	−	0.286505i	$-0.0924935\pi$
$662$	0	0
$663$	−15851.9	−0.928564
$664$	0	0
$665$	9979.21	0.581921
$666$	0	0
$667$	11186.1i	0.649369i
$668$	0	0
$669$	− 766.248i	− 0.0442823i
$670$	0	0
$671$	−2056.58	−0.118321
$672$	0	0
$673$	33809.6	1.93650	0.968250	−	0.249984i	$-0.0804252\pi$
$673$	33809.6	1.93650	0.968250	+	0.249984i	$0.0804252\pi$
$674$	0	0
$675$	− 4552.00i	− 0.259565i
$676$	0	0
$677$	− 29402.3i	− 1.66916i	−0.550887	−	0.834580i	$-0.685711\pi$
$677$	− 29402.3i	− 1.66916i	0.550887	−	0.834580i	$-0.314289\pi$
$678$	0	0
$679$	8272.77	0.467570
$680$	0	0
$681$	−3123.46	−0.175758
$682$	0	0
$683$	− 5540.58i	− 0.310402i	−0.987883	−	0.155201i	$-0.950398\pi$
$683$	− 5540.58i	− 0.310402i	0.987883	−	0.155201i	$-0.0496025\pi$
$684$	0	0
$685$	20929.8i	1.16743i
$686$	0	0
$687$	−7776.10	−0.431844
$688$	0	0
$689$	−26652.9	−1.47372
$690$	0	0
$691$	− 33967.2i	− 1.87000i	−0.354642	−	0.935002i	$-0.615397\pi$
$691$	− 33967.2i	− 1.87000i	0.354642	−	0.935002i	$-0.384603\pi$
$692$	0	0
$693$	187.765i	0.0102924i
$694$	0	0
$695$	−44385.0	−2.42247
$696$	0	0
$697$	6013.31	0.326787
$698$	0	0
$699$	5490.22i	0.297080i
$700$	0	0
$701$	11444.0i	0.616597i	0.951290	+	0.308299i	$0.0997596\pi$
$701$	11444.0i	0.616597i	−0.951290	+	0.308299i	$0.900240\pi$
$702$	0	0
$703$	31442.8	1.68690
$704$	0	0
$705$	18632.4	0.995374
$706$	0	0
$707$	− 1511.36i	− 0.0803968i
$708$	0	0
$709$	30373.3i	1.60888i	0.594036	+	0.804438i	$0.297534\pi$
$709$	30373.3i	1.60888i	−0.594036	+	0.804438i	$0.702466\pi$
$710$	0	0
$711$	−389.977	−0.0205700
$712$	0	0
$713$	−16373.0	−0.859992
$714$	0	0
$715$	2036.08i	0.106497i
$716$	0	0
$717$	− 405.682i	− 0.0211304i
$718$	0	0
$719$	−36362.7	−1.88609	−0.943046	−	0.332663i	$-0.892053\pi$
$719$	−36362.7	−1.88609	−0.943046	+	0.332663i	$0.892053\pi$
$720$	0	0
$721$	−8434.48	−0.435668
$722$	0	0
$723$	938.042i	0.0482519i
$724$	0	0
$725$	23559.7i	1.20688i
$726$	0	0
$727$	−34240.1	−1.74676	−0.873379	−	0.487041i	$-0.838076\pi$
$727$	−34240.1	−1.74676	−0.873379	+	0.487041i	$0.838076\pi$
$728$	0	0
$729$	−729.000	−0.0370370
$730$	0	0
$731$	71670.6i	3.62631i
$732$	0	0
$733$	30023.1i	1.51286i	0.654074	+	0.756431i	$0.273059\pi$
$733$	30023.1i	1.51286i	−0.654074	+	0.756431i	$0.726941\pi$
$734$	0	0
$735$	2518.78	0.126403
$736$	0	0
$737$	−1104.33	−0.0551946
$738$	0	0
$739$	− 32000.0i	− 1.59288i	−0.604717	−	0.796440i	$-0.706714\pi$
$739$	− 32000.0i	− 1.59288i	0.604717	−	0.796440i	$-0.293286\pi$
$740$	0	0
$741$	− 9951.64i	− 0.493364i
$742$	0	0
$743$	−8412.86	−0.415394	−0.207697	−	0.978193i	$-0.566597\pi$
$743$	−8412.86	−0.415394	−0.207697	+	0.978193i	$0.566597\pi$
$744$	0	0
$745$	−58361.9	−2.87009
$746$	0	0
$747$	− 7221.64i	− 0.353716i
$748$	0	0
$749$	2289.71i	0.111701i
$750$	0	0
$751$	3889.62	0.188994	0.0944969	−	0.995525i	$-0.469876\pi$
$751$	3889.62	0.188994	0.0944969	+	0.995525i	$0.469876\pi$
$752$	0	0
$753$	13157.2	0.636751
$754$	0	0
$755$	− 30763.9i	− 1.48293i
$756$	0	0
$757$	− 17814.2i	− 0.855309i	−0.903942	−	0.427654i	$-0.859340\pi$
$757$	− 17814.2i	− 0.855309i	0.903942	−	0.427654i	$-0.140660\pi$
$758$	0	0
$759$	−715.723	−0.0342281
$760$	0	0
$761$	9356.95	0.445715	0.222857	−	0.974851i	$-0.428462\pi$
$761$	9356.95	0.445715	0.222857	+	0.974851i	$0.428462\pi$
$762$	0	0
$763$	− 4679.07i	− 0.222010i
$764$	0	0
$765$	− 20437.5i	− 0.965909i
$766$	0	0
$767$	30197.3	1.42159
$768$	0	0
$769$	32968.7	1.54601	0.773005	−	0.634400i	$-0.218753\pi$
$769$	32968.7	1.54601	0.773005	+	0.634400i	$0.218753\pi$
$770$	0	0
$771$	− 18766.5i	− 0.876598i
$772$	0	0
$773$	11914.1i	0.554359i	0.960818	+	0.277180i	$0.0893997\pi$
$773$	11914.1i	0.554359i	−0.960818	+	0.277180i	$0.910600\pi$
$774$	0	0
$775$	−34484.0	−1.59832
$776$	0	0
$777$	7936.24	0.366423
$778$	0	0
$779$	3775.08i	0.173628i
$780$	0	0
$781$	− 1943.77i	− 0.0890569i
$782$	0	0
$783$	3773.07	0.172208
$784$	0	0
$785$	−50859.6	−2.31243
$786$	0	0
$787$	29231.8i	1.32402i	0.749496	+	0.662009i	$0.230296\pi$
$787$	29231.8i	1.32402i	−0.749496	+	0.662009i	$0.769704\pi$
$788$	0	0
$789$	− 17857.2i	− 0.805744i
$790$	0	0
$791$	5077.33	0.228229
$792$	0	0
$793$	27511.8	1.23200
$794$	0	0
$795$	− 34363.0i	− 1.53299i
$796$	0	0
$797$	− 13241.6i	− 0.588507i	−0.955727	−	0.294254i	$-0.904929\pi$
$797$	− 13241.6i	− 0.588507i	0.955727	−	0.294254i	$-0.0950710\pi$
$798$	0	0
$799$	48038.5	2.12701
$800$	0	0
$801$	7813.69	0.344673
$802$	0	0
$803$	684.906i	0.0300994i
$804$	0	0
$805$	9601.08i	0.420365i
$806$	0	0
$807$	8993.44	0.392297
$808$	0	0
$809$	−10389.0	−0.451491	−0.225746	−	0.974186i	$-0.572482\pi$
$809$	−10389.0	−0.451491	−0.225746	+	0.974186i	$0.572482\pi$
$810$	0	0
$811$	− 10637.7i	− 0.460593i	−0.973120	−	0.230297i	$-0.926030\pi$
$811$	− 10637.7i	− 0.460593i	0.973120	−	0.230297i	$-0.0739697\pi$
$812$	0	0
$813$	− 2043.54i	− 0.0881549i
$814$	0	0
$815$	−26209.0	−1.12646
$816$	0	0
$817$	−44993.9	−1.92673
$818$	0	0
$819$	− 2511.82i	− 0.107167i
$820$	0	0
$821$	4915.12i	0.208939i	0.994528	+	0.104470i	$0.0333145\pi$
$821$	4915.12i	0.208939i	−0.994528	+	0.104470i	$0.966686\pi$
$822$	0	0
$823$	22017.8	0.932555	0.466278	−	0.884638i	$-0.345595\pi$
$823$	22017.8	0.932555	0.466278	+	0.884638i	$0.345595\pi$
$824$	0	0
$825$	−1507.42	−0.0636140
$826$	0	0
$827$	− 4114.43i	− 0.173002i	−0.996252	−	0.0865010i	$-0.972431\pi$
$827$	− 4114.43i	− 0.173002i	0.996252	−	0.0865010i	$-0.0275686\pi$
$828$	0	0
$829$	− 3132.86i	− 0.131253i	−0.997844	−	0.0656264i	$-0.979095\pi$
$829$	− 3132.86i	− 0.131253i	0.997844	−	0.0656264i	$-0.0209046\pi$
$830$	0	0
$831$	4400.82	0.183710
$832$	0	0
$833$	6493.96	0.270111
$834$	0	0
$835$	− 4333.91i	− 0.179618i
$836$	0	0
$837$	5522.59i	0.228063i
$838$	0	0
$839$	11892.0	0.489341	0.244670	−	0.969606i	$-0.421320\pi$
$839$	11892.0	0.489341	0.244670	+	0.969606i	$0.421320\pi$
$840$	0	0
$841$	4860.80	0.199303
$842$	0	0
$843$	19676.6i	0.803912i
$844$	0	0
$845$	10407.0i	0.423684i
$846$	0	0
$847$	−9254.82	−0.375442
$848$	0	0
$849$	−25664.1	−1.03744
$850$	0	0
$851$	30251.4i	1.21857i
$852$	0	0
$853$	28626.0i	1.14905i	0.818488	+	0.574524i	$0.194813\pi$
$853$	28626.0i	1.14905i	−0.818488	+	0.574524i	$0.805187\pi$
$854$	0	0
$855$	12830.4	0.513206
$856$	0	0
$857$	−41513.8	−1.65471	−0.827353	−	0.561683i	$-0.810154\pi$
$857$	−41513.8	−1.65471	−0.827353	+	0.561683i	$0.810154\pi$
$858$	0	0
$859$	− 23085.1i	− 0.916944i	−0.888709	−	0.458472i	$-0.848397\pi$
$859$	− 23085.1i	− 0.916944i	0.888709	−	0.458472i	$-0.151603\pi$
$860$	0	0
$861$	952.839i	0.0377151i
$862$	0	0
$863$	−12260.9	−0.483622	−0.241811	−	0.970323i	$-0.577741\pi$
$863$	−12260.9	−0.483622	−0.241811	+	0.970323i	$0.577741\pi$
$864$	0	0
$865$	−20501.9	−0.805879
$866$	0	0
$867$	− 37953.4i	− 1.48669i
$868$	0	0
$869$	129.143i	0.00504129i
$870$	0	0
$871$	14773.1	0.574704
$872$	0	0
$873$	10636.4	0.412358
$874$	0	0
$875$	5228.56i	0.202009i
$876$	0	0
$877$	− 15969.9i	− 0.614896i	−0.951565	−	0.307448i	$-0.900525\pi$
$877$	− 15969.9i	− 0.614896i	0.951565	−	0.307448i	$-0.0994750\pi$
$878$	0	0
$879$	3786.03	0.145278
$880$	0	0
$881$	42590.6	1.62873	0.814367	−	0.580350i	$-0.197084\pi$
$881$	42590.6	1.62873	0.814367	+	0.580350i	$0.197084\pi$
$882$	0	0
$883$	− 3087.03i	− 0.117652i	−0.998268	−	0.0588261i	$-0.981264\pi$
$883$	− 3087.03i	− 0.117652i	0.998268	−	0.0588261i	$-0.0187357\pi$
$884$	0	0
$885$	38932.7i	1.47877i
$886$	0	0
$887$	−43586.1	−1.64992	−0.824960	−	0.565191i	$-0.808803\pi$
$887$	−43586.1	−1.64992	−0.824960	+	0.565191i	$0.808803\pi$
$888$	0	0
$889$	4705.61	0.177527
$890$	0	0
$891$	241.412i	0.00907701i
$892$	0	0
$893$	30158.0i	1.13012i
$894$	0	0
$895$	−66203.3	−2.47255
$896$	0	0
$897$	9574.55	0.356394
$898$	0	0
$899$	− 28583.2i	− 1.06040i
$900$	0	0
$901$	− 88595.3i	− 3.27584i
$902$	0	0
$903$	−11356.6	−0.418519
$904$	0	0
$905$	−19989.5	−0.734226
$906$	0	0
$907$	38325.2i	1.40305i	0.712645	+	0.701525i	$0.247497\pi$
$907$	38325.2i	1.40305i	−0.712645	+	0.701525i	$0.752503\pi$
$908$	0	0
$909$	− 1943.18i	− 0.0709033i
$910$	0	0
$911$	40999.9	1.49109	0.745547	−	0.666454i	$-0.232189\pi$
$911$	40999.9	1.49109	0.745547	+	0.666454i	$0.232189\pi$
$912$	0	0
$913$	−2391.48	−0.0866884
$914$	0	0
$915$	35470.3i	1.28154i
$916$	0	0
$917$	14576.5i	0.524928i
$918$	0	0
$919$	−51974.3	−1.86559	−0.932793	−	0.360414i	$-0.882636\pi$
$919$	−51974.3	−1.86559	−0.932793	+	0.360414i	$0.882636\pi$
$920$	0	0
$921$	702.148	0.0251211
$922$	0	0
$923$	26002.7i	0.927289i
$924$	0	0
$925$	63713.8i	2.26475i
$926$	0	0
$927$	−10844.3	−0.384223
$928$	0	0
$929$	1073.91	0.0379266	0.0189633	−	0.999820i	$-0.493963\pi$
$929$	1073.91	0.0379266	0.0189633	+	0.999820i	$0.493963\pi$
$930$	0	0
$931$	4076.82i	0.143515i
$932$	0	0
$933$	546.057i	0.0191609i
$934$	0	0
$935$	−6767.99	−0.236724
$936$	0	0
$937$	5315.51	0.185326	0.0926629	−	0.995698i	$-0.470462\pi$
$937$	5315.51	0.185326	0.0926629	+	0.995698i	$0.470462\pi$
$938$	0	0
$939$	− 10545.7i	− 0.366501i
$940$	0	0
$941$	− 24770.6i	− 0.858127i	−0.903274	−	0.429064i	$-0.858844\pi$
$941$	− 24770.6i	− 0.858127i	0.903274	−	0.429064i	$-0.141156\pi$
$942$	0	0
$943$	−3632.03	−0.125424
$944$	0	0
$945$	3238.43	0.111477
$946$	0	0
$947$	− 1529.99i	− 0.0525005i	−0.999655	−	0.0262502i	$-0.991643\pi$
$947$	− 1529.99i	− 0.0525005i	0.999655	−	0.0262502i	$-0.00835667\pi$
$948$	0	0
$949$	− 9162.31i	− 0.313405i
$950$	0	0
$951$	−10018.2	−0.341599
$952$	0	0
$953$	29634.5	1.00730	0.503649	−	0.863909i	$-0.331991\pi$
$953$	29634.5	1.00730	0.503649	+	0.863909i	$0.331991\pi$
$954$	0	0
$955$	29727.7i	1.00729i
$956$	0	0
$957$	− 1249.47i	− 0.0422045i
$958$	0	0
$959$	−8550.49	−0.287914
$960$	0	0
$961$	12045.8	0.404344
$962$	0	0
$963$	2943.91i	0.0985112i
$964$	0	0
$965$	− 5291.16i	− 0.176506i
$966$	0	0
$967$	25369.8	0.843680	0.421840	−	0.906670i	$-0.361384\pi$
$967$	25369.8	0.843680	0.421840	+	0.906670i	$0.361384\pi$
$968$	0	0
$969$	33079.6	1.09667
$970$	0	0
$971$	42309.9i	1.39834i	0.714954	+	0.699171i	$0.246447\pi$
$971$	42309.9i	1.39834i	−0.714954	+	0.699171i	$0.753553\pi$
$972$	0	0
$973$	− 18132.7i	− 0.597438i
$974$	0	0
$975$	20165.4	0.662370
$976$	0	0
$977$	37869.9	1.24009	0.620044	−	0.784567i	$-0.287115\pi$
$977$	37869.9	1.24009	0.620044	+	0.784567i	$0.287115\pi$
$978$	0	0
$979$	− 2587.54i	− 0.0844722i
$980$	0	0
$981$	− 6015.95i	− 0.195794i
$982$	0	0
$983$	20941.4	0.679477	0.339739	−	0.940520i	$-0.389661\pi$
$983$	20941.4	0.679477	0.339739	+	0.940520i	$0.389661\pi$
$984$	0	0
$985$	22239.9	0.719414
$986$	0	0
$987$	7611.94i	0.245482i
$988$	0	0
$989$	− 43289.0i	− 1.39182i
$990$	0	0
$991$	−21465.4	−0.688065	−0.344033	−	0.938958i	$-0.611793\pi$
$991$	−21465.4	−0.688065	−0.344033	+	0.938958i	$0.611793\pi$
$992$	0	0
$993$	4782.54	0.152839
$994$	0	0
$995$	29994.5i	0.955668i
$996$	0	0
$997$	− 8105.46i	− 0.257475i	−0.991679	−	0.128737i	$-0.958908\pi$
$997$	− 8105.46i	− 0.257475i	0.991679	−	0.128737i	$-0.0410925\pi$
$998$	0	0
$999$	10203.7	0.323155

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1344.4.c.f.673.6	✓	12
4.3	odd	2		1344.4.c.g.673.12	yes	12
8.3	odd	2		1344.4.c.g.673.1	yes	12
8.5	even	2	inner	1344.4.c.f.673.7	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1344.4.c.f.673.6	✓	12	1.1	even	1	trivial
1344.4.c.f.673.7	yes	12	8.5	even	2	inner
1344.4.c.g.673.1	yes	12	8.3	odd	2
1344.4.c.g.673.12	yes	12	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-3.00000i q^{3} +17.1345i q^{5} -7.00000 q^{7} -9.00000 q^{9} +O(q^{10})\)	\(q-3.00000i q^{3} +17.1345i q^{5} -7.00000 q^{7} -9.00000 q^{9} +2.98040i q^{11} -39.8701i q^{13} +51.4036 q^{15} +132.530 q^{17} +83.2004i q^{19} +21.0000i q^{21} -80.0478 q^{23} -168.592 q^{25} +27.0000i q^{27} -139.743i q^{29} +204.541 q^{31} +8.94119 q^{33} -119.942i q^{35} -377.916i q^{37} -119.610 q^{39} +45.3733 q^{41} +540.789i q^{43} -154.211i q^{45} +362.473 q^{47} +49.0000 q^{49} -397.589i q^{51} -668.494i q^{53} -51.0677 q^{55} +249.601 q^{57} +757.392i q^{59} +690.035i q^{61} +63.0000 q^{63} +683.157 q^{65} +370.530i q^{67} +240.143i q^{69} -652.184 q^{71} +229.804 q^{73} +505.777i q^{75} -20.8628i q^{77} +43.3308 q^{79} +81.0000 q^{81} +802.404i q^{83} +2270.84i q^{85} -419.230 q^{87} -868.188 q^{89} +279.091i q^{91} -613.622i q^{93} -1425.60 q^{95} -1181.82 q^{97} -26.8236i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 84 q^{7} - 108 q^{9}+O(q^{10}) \) 12 * q - 84 * q^7 - 108 * q^9	\( 12 q - 84 q^{7} - 108 q^{9} + 24 q^{15} + 376 q^{17} - 336 q^{23} - 180 q^{25} + 192 q^{31} - 168 q^{33} - 504 q^{39} + 488 q^{41} - 448 q^{47} + 588 q^{49} - 3600 q^{55} - 432 q^{57} + 756 q^{63} + 1408 q^{65} - 5104 q^{71} - 1752 q^{73} - 1632 q^{79} + 972 q^{81} - 336 q^{87} - 3688 q^{89} - 2496 q^{95} - 1944 q^{97}+O(q^{100}) \) 12 * q - 84 * q^7 - 108 * q^9 + 24 * q^15 + 376 * q^17 - 336 * q^23 - 180 * q^25 + 192 * q^31 - 168 * q^33 - 504 * q^39 + 488 * q^41 - 448 * q^47 + 588 * q^49 - 3600 * q^55 - 432 * q^57 + 756 * q^63 + 1408 * q^65 - 5104 * q^71 - 1752 * q^73 - 1632 * q^79 + 972 * q^81 - 336 * q^87 - 3688 * q^89 - 2496 * q^95 - 1944 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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