LMFDB - Embedded newform 1344.4.c.e.673.7

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} + 386x^{10} + 54793x^{8} + 3447408x^{6} + 90154296x^{4} + 707138208x^{2} + 525876624 $ x^12 + 386x^10 + 54793x^8 + 3447408x^6 + 90154296x^4 + 707138208*x^2 + 525876624
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{22} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			673.7
Root			$6.47819i$ of defining polynomial
Character	$\chi$	$=$	1344.673
Dual form			1344.4.c.e.673.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.00000i q^{3} -16.7959i q^{5} -7.00000 q^{7} -9.00000 q^{9} +O(q^{10})$	$q+3.00000i q^{3} -16.7959i q^{5} -7.00000 q^{7} -9.00000 q^{9} +16.7471i q^{11} -86.0675i q^{13} +50.3878 q^{15} +87.5778 q^{17} +104.816i q^{19} -21.0000i q^{21} +171.028 q^{23} -157.104 q^{25} -27.0000i q^{27} -88.8053i q^{29} -53.1493 q^{31} -50.2414 q^{33} +117.572i q^{35} +215.676i q^{37} +258.202 q^{39} +396.005 q^{41} +132.965i q^{43} +151.164i q^{45} -325.107 q^{47} +49.0000 q^{49} +262.734i q^{51} -674.480i q^{53} +281.284 q^{55} -314.449 q^{57} -254.356i q^{59} -815.732i q^{61} +63.0000 q^{63} -1445.58 q^{65} -616.494i q^{67} +513.085i q^{69} -41.3946 q^{71} -7.03128 q^{73} -471.312i q^{75} -117.230i q^{77} -418.431 q^{79} +81.0000 q^{81} +93.4310i q^{83} -1470.95i q^{85} +266.416 q^{87} -88.6646 q^{89} +602.472i q^{91} -159.448i q^{93} +1760.49 q^{95} +379.317 q^{97} -150.724i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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} + 24 q^{17} + 80 q^{23} - 564 q^{25} - 640 q^{31} - 408 q^{33} + 120 q^{39} + 1416 q^{41} - 1536 q^{47} + 588 q^{49} + 1392 q^{55} - 336 q^{57} + 756 q^{63} - 2880 q^{65} + 1392 q^{71} + 2472 q^{73} - 544 q^{79} + 972 q^{81} + 720 q^{87} + 888 q^{89} + 2368 q^{95} - 2712 q^{97}+O(q^{100}) $ 12 * q - 84 * q^7 - 108 * q^9 - 24 * q^15 + 24 * q^17 + 80 * q^23 - 564 * q^25 - 640 * q^31 - 408 * q^33 + 120 * q^39 + 1416 * q^41 - 1536 * q^47 + 588 * q^49 + 1392 * q^55 - 336 * q^57 + 756 * q^63 - 2880 * q^65 + 1392 * q^71 + 2472 * q^73 - 544 * q^79 + 972 * q^81 + 720 * q^87 + 888 * q^89 + 2368 * q^95 - 2712 * 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
$5$	− 16.7959i	− 1.50228i	−0.660146	−	0.751138i	$-0.729505\pi$
$5$	− 16.7959i	− 1.50228i	0.660146	−	0.751138i	$-0.270495\pi$
$6$	0	0
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	0	0
$9$	−9.00000	−0.333333
$10$	0	0
$11$	16.7471i	0.459041i	0.973304	+	0.229521i	$0.0737158\pi$
$11$	16.7471i	0.459041i	−0.973304	+	0.229521i	$0.926284\pi$
$12$	0	0
$13$	− 86.0675i	− 1.83622i	−0.396330	−	0.918108i	$-0.629716\pi$
$13$	− 86.0675i	− 1.83622i	0.396330	−	0.918108i	$-0.370284\pi$
$14$	0	0
$15$	50.3878	0.867339
$16$	0	0
$17$	87.5778	1.24946	0.624728	−	0.780843i	$-0.285210\pi$
$17$	87.5778	1.24946	0.624728	+	0.780843i	$0.285210\pi$
$18$	0	0
$19$	104.816i	1.26561i	0.774313	+	0.632803i	$0.218096\pi$
$19$	104.816i	1.26561i	−0.774313	+	0.632803i	$0.781904\pi$
$20$	0	0
$21$	− 21.0000i	− 0.218218i
$22$	0	0
$23$	171.028	1.55052	0.775258	−	0.631644i	$-0.217620\pi$
$23$	171.028	1.55052	0.775258	+	0.631644i	$0.217620\pi$
$24$	0	0
$25$	−157.104	−1.25683
$26$	0	0
$27$	− 27.0000i	− 0.192450i
$28$	0	0
$29$	− 88.8053i	− 0.568646i	−0.958729	−	0.284323i	$-0.908231\pi$
$29$	− 88.8053i	− 0.568646i	0.958729	−	0.284323i	$-0.0917688\pi$
$30$	0	0
$31$	−53.1493	−0.307932	−0.153966	−	0.988076i	$-0.549205\pi$
$31$	−53.1493	−0.307932	−0.153966	+	0.988076i	$0.549205\pi$
$32$	0	0
$33$	−50.2414	−0.265028
$34$	0	0
$35$	117.572i	0.567807i
$36$	0	0
$37$	215.676i	0.958296i	0.877734	+	0.479148i	$0.159054\pi$
$37$	215.676i	0.958296i	−0.877734	+	0.479148i	$0.840946\pi$
$38$	0	0
$39$	258.202	1.06014
$40$	0	0
$41$	396.005	1.50843	0.754214	−	0.656629i	$-0.228018\pi$
$41$	396.005	1.50843	0.754214	+	0.656629i	$0.228018\pi$
$42$	0	0
$43$	132.965i	0.471556i	0.971807	+	0.235778i	$0.0757638\pi$
$43$	132.965i	0.471556i	−0.971807	+	0.235778i	$0.924236\pi$
$44$	0	0
$45$	151.164i	0.500758i
$46$	0	0
$47$	−325.107	−1.00897	−0.504487	−	0.863419i	$-0.668318\pi$
$47$	−325.107	−1.00897	−0.504487	+	0.863419i	$0.668318\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	262.734i	0.721374i
$52$	0	0
$53$	− 674.480i	− 1.74806i	−0.485875	−	0.874028i	$-0.661499\pi$
$53$	− 674.480i	− 1.74806i	0.485875	−	0.874028i	$-0.338501\pi$
$54$	0	0
$55$	281.284	0.689606
$56$	0	0
$57$	−314.449	−0.730698
$58$	0	0
$59$	− 254.356i	− 0.561260i	−0.959816	−	0.280630i	$-0.909457\pi$
$59$	− 254.356i	− 0.561260i	0.959816	−	0.280630i	$-0.0905435\pi$
$60$	0	0
$61$	− 815.732i	− 1.71219i	−0.516816	−	0.856096i	$-0.672883\pi$
$61$	− 815.732i	− 1.71219i	0.516816	−	0.856096i	$-0.327117\pi$
$62$	0	0
$63$	63.0000	0.125988
$64$	0	0
$65$	−1445.58	−2.75850
$66$	0	0
$67$	− 616.494i	− 1.12413i	−0.827093	−	0.562065i	$-0.810007\pi$
$67$	− 616.494i	− 1.12413i	0.827093	−	0.562065i	$-0.189993\pi$
$68$	0	0
$69$	513.085i	0.895191i
$70$	0	0
$71$	−41.3946	−0.0691920	−0.0345960	−	0.999401i	$-0.511014\pi$
$71$	−41.3946	−0.0691920	−0.0345960	+	0.999401i	$0.511014\pi$
$72$	0	0
$73$	−7.03128	−0.0112733	−0.00563664	−	0.999984i	$-0.501794\pi$
$73$	−7.03128	−0.0112733	−0.00563664	+	0.999984i	$0.501794\pi$
$74$	0	0
$75$	− 471.312i	− 0.725632i
$76$	0	0
$77$	− 117.230i	− 0.173501i
$78$	0	0
$79$	−418.431	−0.595914	−0.297957	−	0.954579i	$-0.596305\pi$
$79$	−418.431	−0.595914	−0.297957	+	0.954579i	$0.596305\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	93.4310i	0.123559i	0.998090	+	0.0617794i	$0.0196775\pi$
$83$	93.4310i	0.123559i	−0.998090	+	0.0617794i	$0.980322\pi$
$84$	0	0
$85$	− 1470.95i	− 1.87703i
$86$	0	0
$87$	266.416	0.328308
$88$	0	0
$89$	−88.6646	−0.105600	−0.0528002	−	0.998605i	$-0.516815\pi$
$89$	−88.6646	−0.105600	−0.0528002	+	0.998605i	$0.516815\pi$
$90$	0	0
$91$	602.472i	0.694025i
$92$	0	0
$93$	− 159.448i	− 0.177785i
$94$	0	0
$95$	1760.49	1.90129
$96$	0	0
$97$	379.317	0.397050	0.198525	−	0.980096i	$-0.436385\pi$
$97$	379.317	0.397050	0.198525	+	0.980096i	$0.436385\pi$
$98$	0	0
$99$	− 150.724i	− 0.153014i
$100$	0	0
$101$	347.285i	0.342140i	0.985259	+	0.171070i	$0.0547225\pi$
$101$	347.285i	0.342140i	−0.985259	+	0.171070i	$0.945278\pi$
$102$	0	0
$103$	−823.404	−0.787693	−0.393847	−	0.919176i	$-0.628856\pi$
$103$	−823.404	−0.787693	−0.393847	+	0.919176i	$0.628856\pi$
$104$	0	0
$105$	−352.715	−0.327823
$106$	0	0
$107$	459.583i	0.415230i	0.978211	+	0.207615i	$0.0665701\pi$
$107$	459.583i	0.415230i	−0.978211	+	0.207615i	$0.933430\pi$
$108$	0	0
$109$	1237.74i	1.08765i	0.839197	+	0.543827i	$0.183025\pi$
$109$	1237.74i	1.08765i	−0.839197	+	0.543827i	$0.816975\pi$
$110$	0	0
$111$	−647.029	−0.553272
$112$	0	0
$113$	−1666.05	−1.38698	−0.693492	−	0.720465i	$-0.743929\pi$
$113$	−1666.05	−1.38698	−0.693492	+	0.720465i	$0.743929\pi$
$114$	0	0
$115$	− 2872.58i	− 2.32930i
$116$	0	0
$117$	774.607i	0.612072i
$118$	0	0
$119$	−613.045	−0.472250
$120$	0	0
$121$	1050.53	0.789281
$122$	0	0
$123$	1188.01i	0.870891i
$124$	0	0
$125$	539.215i	0.385831i
$126$	0	0
$127$	−2468.42	−1.72470	−0.862350	−	0.506312i	$-0.831008\pi$
$127$	−2468.42	−1.72470	−0.862350	+	0.506312i	$0.831008\pi$
$128$	0	0
$129$	−398.894	−0.272253
$130$	0	0
$131$	− 1031.12i	− 0.687705i	−0.939024	−	0.343853i	$-0.888268\pi$
$131$	− 1031.12i	− 0.687705i	0.939024	−	0.343853i	$-0.111732\pi$
$132$	0	0
$133$	− 733.714i	− 0.478354i
$134$	0	0
$135$	−453.491	−0.289113
$136$	0	0
$137$	−2896.25	−1.80615	−0.903077	−	0.429478i	$-0.858698\pi$
$137$	−2896.25	−1.80615	−0.903077	+	0.429478i	$0.858698\pi$
$138$	0	0
$139$	− 2834.18i	− 1.72944i	−0.502253	−	0.864721i	$-0.667495\pi$
$139$	− 2834.18i	− 1.72944i	0.502253	−	0.864721i	$-0.332505\pi$
$140$	0	0
$141$	− 975.322i	− 0.582531i
$142$	0	0
$143$	1441.38	0.842899
$144$	0	0
$145$	−1491.57	−0.854263
$146$	0	0
$147$	147.000i	0.0824786i
$148$	0	0
$149$	2323.88i	1.27771i	0.769325	+	0.638857i	$0.220593\pi$
$149$	2323.88i	1.27771i	−0.769325	+	0.638857i	$0.779407\pi$
$150$	0	0
$151$	−713.348	−0.384447	−0.192223	−	0.981351i	$-0.561570\pi$
$151$	−713.348	−0.384447	−0.192223	+	0.981351i	$0.561570\pi$
$152$	0	0
$153$	−788.201	−0.416485
$154$	0	0
$155$	892.692i	0.462598i
$156$	0	0
$157$	− 211.600i	− 0.107564i	−0.998553	−	0.0537818i	$-0.982872\pi$
$157$	− 211.600i	− 0.107564i	0.998553	−	0.0537818i	$-0.0171276\pi$
$158$	0	0
$159$	2023.44	1.00924
$160$	0	0
$161$	−1197.20	−0.586040
$162$	0	0
$163$	− 4058.62i	− 1.95028i	−0.221590	−	0.975140i	$-0.571125\pi$
$163$	− 4058.62i	− 1.95028i	0.221590	−	0.975140i	$-0.428875\pi$
$164$	0	0
$165$	843.852i	0.398144i
$166$	0	0
$167$	1372.81	0.636117	0.318058	−	0.948071i	$-0.396969\pi$
$167$	1372.81	0.636117	0.318058	+	0.948071i	$0.396969\pi$
$168$	0	0
$169$	−5210.61	−2.37169
$170$	0	0
$171$	− 943.347i	− 0.421869i
$172$	0	0
$173$	907.211i	0.398694i	0.979929	+	0.199347i	$0.0638820\pi$
$173$	907.211i	0.398694i	−0.979929	+	0.199347i	$0.936118\pi$
$174$	0	0
$175$	1099.73	0.475037
$176$	0	0
$177$	763.069	0.324044
$178$	0	0
$179$	− 1587.94i	− 0.663063i	−0.943444	−	0.331531i	$-0.892435\pi$
$179$	− 1587.94i	− 0.663063i	0.943444	−	0.331531i	$-0.107565\pi$
$180$	0	0
$181$	− 2306.28i	− 0.947098i	−0.880767	−	0.473549i	$-0.842973\pi$
$181$	− 2306.28i	− 0.947098i	0.880767	−	0.473549i	$-0.157027\pi$
$182$	0	0
$183$	2447.20	0.988535
$184$	0	0
$185$	3622.49	1.43962
$186$	0	0
$187$	1466.68i	0.573552i
$188$	0	0
$189$	189.000i	0.0727393i
$190$	0	0
$191$	1102.96	0.417841	0.208920	−	0.977933i	$-0.433005\pi$
$191$	1102.96	0.417841	0.208920	+	0.977933i	$0.433005\pi$
$192$	0	0
$193$	2785.42	1.03885	0.519427	−	0.854515i	$-0.326145\pi$
$193$	2785.42	1.03885	0.519427	+	0.854515i	$0.326145\pi$
$194$	0	0
$195$	− 4336.75i	− 1.59262i
$196$	0	0
$197$	− 785.571i	− 0.284110i	−0.989859	−	0.142055i	$-0.954629\pi$
$197$	− 785.571i	− 0.284110i	0.989859	−	0.142055i	$-0.0453710\pi$
$198$	0	0
$199$	5248.41	1.86960	0.934799	−	0.355176i	$-0.115579\pi$
$199$	5248.41	1.86960	0.934799	+	0.355176i	$0.115579\pi$
$200$	0	0
$201$	1849.48	0.649017
$202$	0	0
$203$	621.637i	0.214928i
$204$	0	0
$205$	− 6651.27i	− 2.26607i
$206$	0	0
$207$	−1539.26	−0.516839
$208$	0	0
$209$	−1755.37	−0.580965
$210$	0	0
$211$	− 3072.04i	− 1.00231i	−0.865357	−	0.501157i	$-0.832908\pi$
$211$	− 3072.04i	− 1.00231i	0.865357	−	0.501157i	$-0.167092\pi$
$212$	0	0
$213$	− 124.184i	− 0.0399480i
$214$	0	0
$215$	2233.26	0.708407
$216$	0	0
$217$	372.045	0.116387
$218$	0	0
$219$	− 21.0938i	− 0.00650863i
$220$	0	0
$221$	− 7537.60i	− 2.29427i
$222$	0	0
$223$	−643.250	−0.193162	−0.0965812	−	0.995325i	$-0.530791\pi$
$223$	−643.250	−0.193162	−0.0965812	+	0.995325i	$0.530791\pi$
$224$	0	0
$225$	1413.93	0.418944
$226$	0	0
$227$	− 3521.08i	− 1.02953i	−0.857332	−	0.514763i	$-0.827880\pi$
$227$	− 3521.08i	− 1.02953i	0.857332	−	0.514763i	$-0.172120\pi$
$228$	0	0
$229$	2164.66i	0.624651i	0.949975	+	0.312325i	$0.101108\pi$
$229$	2164.66i	0.624651i	−0.949975	+	0.312325i	$0.898892\pi$
$230$	0	0
$231$	351.690	0.100171
$232$	0	0
$233$	−554.223	−0.155830	−0.0779149	−	0.996960i	$-0.524826\pi$
$233$	−554.223	−0.155830	−0.0779149	+	0.996960i	$0.524826\pi$
$234$	0	0
$235$	5460.48i	1.51576i
$236$	0	0
$237$	− 1255.29i	− 0.344051i
$238$	0	0
$239$	−3005.14	−0.813331	−0.406665	−	0.913577i	$-0.633309\pi$
$239$	−3005.14	−0.813331	−0.406665	+	0.913577i	$0.633309\pi$
$240$	0	0
$241$	6394.32	1.70910	0.854552	−	0.519366i	$-0.173832\pi$
$241$	6394.32	1.70910	0.854552	+	0.519366i	$0.173832\pi$
$242$	0	0
$243$	243.000i	0.0641500i
$244$	0	0
$245$	− 823.001i	− 0.214611i
$246$	0	0
$247$	9021.27	2.32393
$248$	0	0
$249$	−280.293	−0.0713367
$250$	0	0
$251$	6516.96i	1.63883i	0.573199	+	0.819416i	$0.305702\pi$
$251$	6516.96i	1.63883i	−0.573199	+	0.819416i	$0.694298\pi$
$252$	0	0
$253$	2864.24i	0.711751i
$254$	0	0
$255$	4412.86	1.08370
$256$	0	0
$257$	−6886.65	−1.67151	−0.835754	−	0.549104i	$-0.814969\pi$
$257$	−6886.65	−1.67151	−0.835754	+	0.549104i	$0.814969\pi$
$258$	0	0
$259$	− 1509.73i	− 0.362202i
$260$	0	0
$261$	799.248i	0.189549i
$262$	0	0
$263$	−2179.75	−0.511060	−0.255530	−	0.966801i	$-0.582250\pi$
$263$	−2179.75	−0.511060	−0.255530	+	0.966801i	$0.582250\pi$
$264$	0	0
$265$	−11328.5	−2.62606
$266$	0	0
$267$	− 265.994i	− 0.0609684i
$268$	0	0
$269$	− 8357.77i	− 1.89436i	−0.320706	−	0.947179i	$-0.603920\pi$
$269$	− 8357.77i	− 1.89436i	0.320706	−	0.947179i	$-0.396080\pi$
$270$	0	0
$271$	6064.22	1.35932	0.679659	−	0.733528i	$-0.262128\pi$
$271$	6064.22	1.35932	0.679659	+	0.733528i	$0.262128\pi$
$272$	0	0
$273$	−1807.42	−0.400695
$274$	0	0
$275$	− 2631.04i	− 0.576937i
$276$	0	0
$277$	− 829.802i	− 0.179993i	−0.995942	−	0.0899963i	$-0.971314\pi$
$277$	− 829.802i	− 0.179993i	0.995942	−	0.0899963i	$-0.0286855\pi$
$278$	0	0
$279$	478.343	0.102644
$280$	0	0
$281$	1939.05	0.411650	0.205825	−	0.978589i	$-0.434012\pi$
$281$	1939.05	0.411650	0.205825	+	0.978589i	$0.434012\pi$
$282$	0	0
$283$	− 6575.04i	− 1.38108i	−0.723295	−	0.690540i	$-0.757373\pi$
$283$	− 6575.04i	− 1.38108i	0.723295	−	0.690540i	$-0.242627\pi$
$284$	0	0
$285$	5281.47i	1.09771i
$286$	0	0
$287$	−2772.03	−0.570132
$288$	0	0
$289$	2756.88	0.561140
$290$	0	0
$291$	1137.95i	0.229237i
$292$	0	0
$293$	− 2398.11i	− 0.478154i	−0.971001	−	0.239077i	$-0.923155\pi$
$293$	− 2398.11i	− 0.478154i	0.971001	−	0.239077i	$-0.0768448\pi$
$294$	0	0
$295$	−4272.15	−0.843168
$296$	0	0
$297$	452.173	0.0883425
$298$	0	0
$299$	− 14720.0i	− 2.84708i
$300$	0	0
$301$	− 930.752i	− 0.178231i
$302$	0	0
$303$	−1041.86	−0.197535
$304$	0	0
$305$	−13701.0	−2.57218
$306$	0	0
$307$	9061.67i	1.68461i	0.538997	+	0.842307i	$0.318803\pi$
$307$	9061.67i	1.68461i	−0.538997	+	0.842307i	$0.681197\pi$
$308$	0	0
$309$	− 2470.21i	− 0.454775i
$310$	0	0
$311$	−4507.86	−0.821920	−0.410960	−	0.911653i	$-0.634806\pi$
$311$	−4507.86	−0.821920	−0.410960	+	0.911653i	$0.634806\pi$
$312$	0	0
$313$	4793.37	0.865615	0.432807	−	0.901486i	$-0.357523\pi$
$313$	4793.37	0.865615	0.432807	+	0.901486i	$0.357523\pi$
$314$	0	0
$315$	− 1058.14i	− 0.189269i
$316$	0	0
$317$	1692.39i	0.299855i	0.988697	+	0.149927i	$0.0479040\pi$
$317$	1692.39i	0.299855i	−0.988697	+	0.149927i	$0.952096\pi$
$318$	0	0
$319$	1487.24	0.261032
$320$	0	0
$321$	−1378.75	−0.239733
$322$	0	0
$323$	9179.59i	1.58132i
$324$	0	0
$325$	13521.5i	2.30781i
$326$	0	0
$327$	−3713.23	−0.627958
$328$	0	0
$329$	2275.75	0.381356
$330$	0	0
$331$	− 8190.22i	− 1.36005i	−0.733191	−	0.680023i	$-0.761970\pi$
$331$	− 8190.22i	− 1.36005i	0.733191	−	0.680023i	$-0.238030\pi$
$332$	0	0
$333$	− 1941.09i	− 0.319432i
$334$	0	0
$335$	−10354.6	−1.68875
$336$	0	0
$337$	−749.966	−0.121226	−0.0606132	−	0.998161i	$-0.519306\pi$
$337$	−749.966	−0.121226	−0.0606132	+	0.998161i	$0.519306\pi$
$338$	0	0
$339$	− 4998.16i	− 0.800775i
$340$	0	0
$341$	− 890.098i	− 0.141353i
$342$	0	0
$343$	−343.000	−0.0539949
$344$	0	0
$345$	8617.75	1.34482
$346$	0	0
$347$	− 2496.80i	− 0.386269i	−0.981172	−	0.193134i	$-0.938135\pi$
$347$	− 2496.80i	− 0.386269i	0.981172	−	0.193134i	$-0.0618653\pi$
$348$	0	0
$349$	3217.47i	0.493488i	0.969081	+	0.246744i	$0.0793607\pi$
$349$	3217.47i	0.493488i	−0.969081	+	0.246744i	$0.920639\pi$
$350$	0	0
$351$	−2323.82	−0.353380
$352$	0	0
$353$	5979.79	0.901621	0.450811	−	0.892620i	$-0.351135\pi$
$353$	5979.79	0.901621	0.450811	+	0.892620i	$0.351135\pi$
$354$	0	0
$355$	695.261i	0.103945i
$356$	0	0
$357$	− 1839.13i	− 0.272654i
$358$	0	0
$359$	8368.91	1.23035	0.615173	−	0.788392i	$-0.289086\pi$
$359$	8368.91	1.23035	0.615173	+	0.788392i	$0.289086\pi$
$360$	0	0
$361$	−4127.46	−0.601759
$362$	0	0
$363$	3151.60i	0.455692i
$364$	0	0
$365$	118.097i	0.0169356i
$366$	0	0
$367$	−6101.03	−0.867769	−0.433885	−	0.900968i	$-0.642857\pi$
$367$	−6101.03	−0.867769	−0.433885	+	0.900968i	$0.642857\pi$
$368$	0	0
$369$	−3564.04	−0.502809
$370$	0	0
$371$	4721.36i	0.660703i
$372$	0	0
$373$	9137.00i	1.26835i	0.773188	+	0.634177i	$0.218661\pi$
$373$	9137.00i	1.26835i	−0.773188	+	0.634177i	$0.781339\pi$
$374$	0	0
$375$	−1617.64	−0.222759
$376$	0	0
$377$	−7643.25	−1.04416
$378$	0	0
$379$	− 3676.13i	− 0.498233i	−0.968474	−	0.249116i	$-0.919860\pi$
$379$	− 3676.13i	− 0.498233i	0.968474	−	0.249116i	$-0.0801402\pi$
$380$	0	0
$381$	− 7405.27i	− 0.995756i
$382$	0	0
$383$	−9039.82	−1.20604	−0.603020	−	0.797726i	$-0.706036\pi$
$383$	−9039.82	−1.20604	−0.603020	+	0.797726i	$0.706036\pi$
$384$	0	0
$385$	−1968.99	−0.260647
$386$	0	0
$387$	− 1196.68i	− 0.157185i
$388$	0	0
$389$	− 14698.1i	− 1.91574i	−0.287210	−	0.957868i	$-0.592728\pi$
$389$	− 14698.1i	− 1.91574i	0.287210	−	0.957868i	$-0.407272\pi$
$390$	0	0
$391$	14978.3	1.93730
$392$	0	0
$393$	3093.36	0.397047
$394$	0	0
$395$	7027.95i	0.895227i
$396$	0	0
$397$	9354.18i	1.18255i	0.806469	+	0.591276i	$0.201376\pi$
$397$	9354.18i	1.18255i	−0.806469	+	0.591276i	$0.798624\pi$
$398$	0	0
$399$	2201.14	0.276178
$400$	0	0
$401$	1874.63	0.233453	0.116726	−	0.993164i	$-0.462760\pi$
$401$	1874.63	0.233453	0.116726	+	0.993164i	$0.462760\pi$
$402$	0	0
$403$	4574.42i	0.565430i
$404$	0	0
$405$	− 1360.47i	− 0.166919i
$406$	0	0
$407$	−3611.96	−0.439897
$408$	0	0
$409$	−732.936	−0.0886097	−0.0443048	−	0.999018i	$-0.514107\pi$
$409$	−732.936	−0.0886097	−0.0443048	+	0.999018i	$0.514107\pi$
$410$	0	0
$411$	− 8688.74i	− 1.04278i
$412$	0	0
$413$	1780.49i	0.212136i
$414$	0	0
$415$	1569.26	0.185619
$416$	0	0
$417$	8502.55	0.998494
$418$	0	0
$419$	16146.8i	1.88263i	0.337528	+	0.941316i	$0.390409\pi$
$419$	16146.8i	1.88263i	−0.337528	+	0.941316i	$0.609591\pi$
$420$	0	0
$421$	− 6008.68i	− 0.695594i	−0.937570	−	0.347797i	$-0.886930\pi$
$421$	− 6008.68i	− 0.695594i	0.937570	−	0.347797i	$-0.113070\pi$
$422$	0	0
$423$	2925.97	0.336325
$424$	0	0
$425$	−13758.8	−1.57035
$426$	0	0
$427$	5710.12i	0.647148i
$428$	0	0
$429$	4324.15i	0.486648i
$430$	0	0
$431$	−878.122	−0.0981384	−0.0490692	−	0.998795i	$-0.515625\pi$
$431$	−878.122	−0.0981384	−0.0490692	+	0.998795i	$0.515625\pi$
$432$	0	0
$433$	−4183.78	−0.464341	−0.232171	−	0.972675i	$-0.574583\pi$
$433$	−4183.78	−0.464341	−0.232171	+	0.972675i	$0.574583\pi$
$434$	0	0
$435$	− 4474.71i	− 0.493209i
$436$	0	0
$437$	17926.6i	1.96234i
$438$	0	0
$439$	−7633.70	−0.829924	−0.414962	−	0.909839i	$-0.636205\pi$
$439$	−7633.70	−0.829924	−0.414962	+	0.909839i	$0.636205\pi$
$440$	0	0
$441$	−441.000	−0.0476190
$442$	0	0
$443$	11512.6i	1.23472i	0.786680	+	0.617361i	$0.211798\pi$
$443$	11512.6i	1.23472i	−0.786680	+	0.617361i	$0.788202\pi$
$444$	0	0
$445$	1489.21i	0.158641i
$446$	0	0
$447$	−6971.63	−0.737689
$448$	0	0
$449$	−7980.85	−0.838841	−0.419421	−	0.907792i	$-0.637767\pi$
$449$	−7980.85	−0.838841	−0.419421	+	0.907792i	$0.637767\pi$
$450$	0	0
$451$	6631.95i	0.692431i
$452$	0	0
$453$	− 2140.04i	− 0.221960i
$454$	0	0
$455$	10119.1	1.04262
$456$	0	0
$457$	−6783.55	−0.694357	−0.347178	−	0.937799i	$-0.612860\pi$
$457$	−6783.55	−0.694357	−0.347178	+	0.937799i	$0.612860\pi$
$458$	0	0
$459$	− 2364.60i	− 0.240458i
$460$	0	0
$461$	7512.04i	0.758938i	0.925204	+	0.379469i	$0.123893\pi$
$461$	7512.04i	0.758938i	−0.925204	+	0.379469i	$0.876107\pi$
$462$	0	0
$463$	−5806.86	−0.582867	−0.291434	−	0.956591i	$-0.594132\pi$
$463$	−5806.86	−0.582867	−0.291434	+	0.956591i	$0.594132\pi$
$464$	0	0
$465$	−2678.08	−0.267081
$466$	0	0
$467$	11426.1i	1.13220i	0.824338	+	0.566098i	$0.191548\pi$
$467$	11426.1i	1.13220i	−0.824338	+	0.566098i	$0.808452\pi$
$468$	0	0
$469$	4315.46i	0.424881i
$470$	0	0
$471$	634.799	0.0621019
$472$	0	0
$473$	−2226.78	−0.216464
$474$	0	0
$475$	− 16467.0i	− 1.59065i
$476$	0	0
$477$	6070.32i	0.582686i
$478$	0	0
$479$	−8773.75	−0.836916	−0.418458	−	0.908236i	$-0.637429\pi$
$479$	−8773.75	−0.836916	−0.418458	+	0.908236i	$0.637429\pi$
$480$	0	0
$481$	18562.7	1.75964
$482$	0	0
$483$	− 3591.60i	− 0.338350i
$484$	0	0
$485$	− 6370.99i	− 0.596478i
$486$	0	0
$487$	−2676.42	−0.249035	−0.124518	−	0.992217i	$-0.539738\pi$
$487$	−2676.42	−0.249035	−0.124518	+	0.992217i	$0.539738\pi$
$488$	0	0
$489$	12175.9	1.12599
$490$	0	0
$491$	6353.86i	0.584003i	0.956418	+	0.292002i	$0.0943213\pi$
$491$	6353.86i	0.584003i	−0.956418	+	0.292002i	$0.905679\pi$
$492$	0	0
$493$	− 7777.38i	− 0.710498i
$494$	0	0
$495$	−2531.56	−0.229869
$496$	0	0
$497$	289.762	0.0261521
$498$	0	0
$499$	5120.91i	0.459406i	0.973261	+	0.229703i	$0.0737754\pi$
$499$	5120.91i	0.459406i	−0.973261	+	0.229703i	$0.926225\pi$
$500$	0	0
$501$	4118.44i	0.367262i
$502$	0	0
$503$	−13009.2	−1.15319	−0.576594	−	0.817031i	$-0.695619\pi$
$503$	−13009.2	−1.15319	−0.576594	+	0.817031i	$0.695619\pi$
$504$	0	0
$505$	5832.98	0.513989
$506$	0	0
$507$	− 15631.8i	− 1.36930i
$508$	0	0
$509$	6947.87i	0.605027i	0.953145	+	0.302513i	$0.0978257\pi$
$509$	6947.87i	0.605027i	−0.953145	+	0.302513i	$0.902174\pi$
$510$	0	0
$511$	49.2189	0.00426090
$512$	0	0
$513$	2830.04	0.243566
$514$	0	0
$515$	13829.8i	1.18333i
$516$	0	0
$517$	− 5444.62i	− 0.463161i
$518$	0	0
$519$	−2721.63	−0.230186
$520$	0	0
$521$	−3129.87	−0.263190	−0.131595	−	0.991304i	$-0.542010\pi$
$521$	−3129.87	−0.263190	−0.131595	+	0.991304i	$0.542010\pi$
$522$	0	0
$523$	1229.06i	0.102759i	0.998679	+	0.0513797i	$0.0163619\pi$
$523$	1229.06i	0.102759i	−0.998679	+	0.0513797i	$0.983638\pi$
$524$	0	0
$525$	3299.18i	0.274263i
$526$	0	0
$527$	−4654.70	−0.384747
$528$	0	0
$529$	17083.7	1.40410
$530$	0	0
$531$	2289.21i	0.187087i
$532$	0	0
$533$	− 34083.1i	− 2.76980i
$534$	0	0
$535$	7719.13	0.623789
$536$	0	0
$537$	4763.82	0.382819
$538$	0	0
$539$	820.610i	0.0655773i
$540$	0	0
$541$	− 16794.3i	− 1.33465i	−0.744768	−	0.667323i	$-0.767440\pi$
$541$	− 16794.3i	− 1.33465i	0.744768	−	0.667323i	$-0.232560\pi$
$542$	0	0
$543$	6918.85	0.546807
$544$	0	0
$545$	20789.1	1.63396
$546$	0	0
$547$	− 16833.2i	− 1.31578i	−0.753112	−	0.657892i	$-0.771448\pi$
$547$	− 16833.2i	− 1.31578i	0.753112	−	0.657892i	$-0.228552\pi$
$548$	0	0
$549$	7341.59i	0.570731i
$550$	0	0
$551$	9308.25	0.719682
$552$	0	0
$553$	2929.02	0.225234
$554$	0	0
$555$	10867.5i	0.831167i
$556$	0	0
$557$	5781.06i	0.439769i	0.975526	+	0.219884i	$0.0705680\pi$
$557$	5781.06i	0.439769i	−0.975526	+	0.219884i	$0.929432\pi$
$558$	0	0
$559$	11443.9	0.865879
$560$	0	0
$561$	−4400.04	−0.331140
$562$	0	0
$563$	− 6090.61i	− 0.455930i	−0.973669	−	0.227965i	$-0.926793\pi$
$563$	− 6090.61i	− 0.455930i	0.973669	−	0.227965i	$-0.0732072\pi$
$564$	0	0
$565$	27982.9i	2.08363i
$566$	0	0
$567$	−567.000	−0.0419961
$568$	0	0
$569$	8281.38	0.610147	0.305073	−	0.952329i	$-0.401319\pi$
$569$	8281.38	0.610147	0.305073	+	0.952329i	$0.401319\pi$
$570$	0	0
$571$	20754.1i	1.52107i	0.649296	+	0.760536i	$0.275064\pi$
$571$	20754.1i	1.52107i	−0.649296	+	0.760536i	$0.724936\pi$
$572$	0	0
$573$	3308.89i	0.241241i
$574$	0	0
$575$	−26869.2	−1.94874
$576$	0	0
$577$	24903.6	1.79679	0.898396	−	0.439186i	$-0.144733\pi$
$577$	24903.6	1.79679	0.898396	+	0.439186i	$0.144733\pi$
$578$	0	0
$579$	8356.25i	0.599782i
$580$	0	0
$581$	− 654.017i	− 0.0467009i
$582$	0	0
$583$	11295.6	0.802430
$584$	0	0
$585$	13010.3	0.919501
$586$	0	0
$587$	− 4369.08i	− 0.307208i	−0.988132	−	0.153604i	$-0.950912\pi$
$587$	− 4369.08i	− 0.307208i	0.988132	−	0.153604i	$-0.0490881\pi$
$588$	0	0
$589$	− 5570.91i	− 0.389720i
$590$	0	0
$591$	2356.71	0.164031
$592$	0	0
$593$	−2290.07	−0.158586	−0.0792932	−	0.996851i	$-0.525266\pi$
$593$	−2290.07	−0.158586	−0.0792932	+	0.996851i	$0.525266\pi$
$594$	0	0
$595$	10296.7i	0.709449i
$596$	0	0
$597$	15745.2i	1.07941i
$598$	0	0
$599$	13994.2	0.954570	0.477285	−	0.878749i	$-0.341621\pi$
$599$	13994.2	0.954570	0.477285	+	0.878749i	$0.341621\pi$
$600$	0	0
$601$	10702.8	0.726417	0.363209	−	0.931708i	$-0.381681\pi$
$601$	10702.8	0.726417	0.363209	+	0.931708i	$0.381681\pi$
$602$	0	0
$603$	5548.44i	0.374710i
$604$	0	0
$605$	− 17644.7i	− 1.18572i
$606$	0	0
$607$	−27802.6	−1.85910	−0.929551	−	0.368694i	$-0.879805\pi$
$607$	−27802.6	−1.85910	−0.929551	+	0.368694i	$0.879805\pi$
$608$	0	0
$609$	−1864.91	−0.124089
$610$	0	0
$611$	27981.2i	1.85269i
$612$	0	0
$613$	1984.09i	0.130728i	0.997861	+	0.0653641i	$0.0208209\pi$
$613$	1984.09i	0.130728i	−0.997861	+	0.0653641i	$0.979179\pi$
$614$	0	0
$615$	19953.8	1.30832
$616$	0	0
$617$	18463.6	1.20473	0.602365	−	0.798221i	$-0.294225\pi$
$617$	18463.6	1.20473	0.602365	+	0.798221i	$0.294225\pi$
$618$	0	0
$619$	23153.9i	1.50345i	0.659479	+	0.751723i	$0.270777\pi$
$619$	23153.9i	1.50345i	−0.659479	+	0.751723i	$0.729223\pi$
$620$	0	0
$621$	− 4617.77i	− 0.298397i
$622$	0	0
$623$	620.652	0.0399132
$624$	0	0
$625$	−10581.4	−0.677207
$626$	0	0
$627$	− 5266.12i	− 0.335421i
$628$	0	0
$629$	18888.5i	1.19735i
$630$	0	0
$631$	8575.68	0.541034	0.270517	−	0.962715i	$-0.412805\pi$
$631$	8575.68	0.541034	0.270517	+	0.962715i	$0.412805\pi$
$632$	0	0
$633$	9216.13	0.578686
$634$	0	0
$635$	41459.5i	2.59098i
$636$	0	0
$637$	− 4217.31i	− 0.262317i
$638$	0	0
$639$	372.551	0.0230640
$640$	0	0
$641$	16725.9	1.03063	0.515313	−	0.857002i	$-0.327676\pi$
$641$	16725.9	1.03063	0.515313	+	0.857002i	$0.327676\pi$
$642$	0	0
$643$	− 16700.9i	− 1.02429i	−0.858898	−	0.512147i	$-0.828850\pi$
$643$	− 16700.9i	− 1.02429i	0.858898	−	0.512147i	$-0.171150\pi$
$644$	0	0
$645$	6699.79i	0.408999i
$646$	0	0
$647$	26113.8	1.58677	0.793384	−	0.608721i	$-0.208317\pi$
$647$	26113.8	1.58677	0.793384	+	0.608721i	$0.208317\pi$
$648$	0	0
$649$	4259.74	0.257642
$650$	0	0
$651$	1116.13i	0.0671962i
$652$	0	0
$653$	− 11577.7i	− 0.693832i	−0.937896	−	0.346916i	$-0.887229\pi$
$653$	− 11577.7i	− 0.693832i	0.937896	−	0.346916i	$-0.112771\pi$
$654$	0	0
$655$	−17318.6	−1.03312
$656$	0	0
$657$	63.2815	0.00375776
$658$	0	0
$659$	− 2887.60i	− 0.170690i	−0.996351	−	0.0853450i	$-0.972801\pi$
$659$	− 2887.60i	− 0.170690i	0.996351	−	0.0853450i	$-0.0271993\pi$
$660$	0	0
$661$	− 23731.4i	− 1.39644i	−0.715884	−	0.698219i	$-0.753976\pi$
$661$	− 23731.4i	− 1.39644i	0.715884	−	0.698219i	$-0.246024\pi$
$662$	0	0
$663$	22612.8	1.32460
$664$	0	0
$665$	−12323.4	−0.718620
$666$	0	0
$667$	− 15188.2i	− 0.881695i
$668$	0	0
$669$	− 1929.75i	− 0.111522i
$670$	0	0
$671$	13661.2	0.785967
$672$	0	0
$673$	20270.1	1.16100	0.580501	−	0.814260i	$-0.302857\pi$
$673$	20270.1	1.16100	0.580501	+	0.814260i	$0.302857\pi$
$674$	0	0
$675$	4241.80i	0.241877i
$676$	0	0
$677$	− 20382.3i	− 1.15710i	−0.815648	−	0.578549i	$-0.803619\pi$
$677$	− 20382.3i	− 1.15710i	0.815648	−	0.578549i	$-0.196381\pi$
$678$	0	0
$679$	−2655.22	−0.150071
$680$	0	0
$681$	10563.2	0.594397
$682$	0	0
$683$	− 28372.0i	− 1.58949i	−0.606940	−	0.794747i	$-0.707603\pi$
$683$	− 28372.0i	− 1.58949i	0.606940	−	0.794747i	$-0.292397\pi$
$684$	0	0
$685$	48645.2i	2.71334i
$686$	0	0
$687$	−6493.99	−0.360642
$688$	0	0
$689$	−58050.8	−3.20981
$690$	0	0
$691$	− 7836.02i	− 0.431398i	−0.976460	−	0.215699i	$-0.930797\pi$
$691$	− 7836.02i	− 0.431398i	0.976460	−	0.215699i	$-0.0692030\pi$
$692$	0	0
$693$	1055.07i	0.0578338i
$694$	0	0
$695$	−47602.8	−2.59810
$696$	0	0
$697$	34681.2	1.88471
$698$	0	0
$699$	− 1662.67i	− 0.0899684i
$700$	0	0
$701$	29663.9i	1.59827i	0.601150	+	0.799136i	$0.294709\pi$
$701$	29663.9i	1.59827i	−0.601150	+	0.799136i	$0.705291\pi$
$702$	0	0
$703$	−22606.4	−1.21283
$704$	0	0
$705$	−16381.5	−0.875122
$706$	0	0
$707$	− 2431.00i	− 0.129317i
$708$	0	0
$709$	7147.30i	0.378593i	0.981920	+	0.189296i	$0.0606207\pi$
$709$	7147.30i	0.378593i	−0.981920	+	0.189296i	$0.939379\pi$
$710$	0	0
$711$	3765.88	0.198638
$712$	0	0
$713$	−9090.03	−0.477453
$714$	0	0
$715$	− 24209.4i	− 1.26627i
$716$	0	0
$717$	− 9015.41i	− 0.469577i
$718$	0	0
$719$	−5522.54	−0.286448	−0.143224	−	0.989690i	$-0.545747\pi$
$719$	−5522.54	−0.286448	−0.143224	+	0.989690i	$0.545747\pi$
$720$	0	0
$721$	5763.83	0.297720
$722$	0	0
$723$	19182.9i	0.986752i
$724$	0	0
$725$	13951.7i	0.714692i
$726$	0	0
$727$	5800.09	0.295892	0.147946	−	0.988995i	$-0.452734\pi$
$727$	5800.09	0.295892	0.147946	+	0.988995i	$0.452734\pi$
$728$	0	0
$729$	−729.000	−0.0370370
$730$	0	0
$731$	11644.7i	0.589188i
$732$	0	0
$733$	30899.2i	1.55701i	0.627639	+	0.778505i	$0.284021\pi$
$733$	30899.2i	1.55701i	−0.627639	+	0.778505i	$0.715979\pi$
$734$	0	0
$735$	2469.00	0.123906
$736$	0	0
$737$	10324.5	0.516022
$738$	0	0
$739$	− 38603.4i	− 1.92158i	−0.277271	−	0.960792i	$-0.589430\pi$
$739$	− 38603.4i	− 1.92158i	0.277271	−	0.960792i	$-0.410570\pi$
$740$	0	0
$741$	27063.8i	1.34172i
$742$	0	0
$743$	−23492.9	−1.15999	−0.579994	−	0.814621i	$-0.696945\pi$
$743$	−23492.9	−1.15999	−0.579994	+	0.814621i	$0.696945\pi$
$744$	0	0
$745$	39031.7	1.91948
$746$	0	0
$747$	− 840.879i	− 0.0411863i
$748$	0	0
$749$	− 3217.08i	− 0.156942i
$750$	0	0
$751$	3638.34	0.176784	0.0883921	−	0.996086i	$-0.471827\pi$
$751$	3638.34	0.176784	0.0883921	+	0.996086i	$0.471827\pi$
$752$	0	0
$753$	−19550.9	−0.946180
$754$	0	0
$755$	11981.4i	0.577545i
$756$	0	0
$757$	− 17271.1i	− 0.829232i	−0.909997	−	0.414616i	$-0.863916\pi$
$757$	− 17271.1i	− 0.829232i	0.909997	−	0.414616i	$-0.136084\pi$
$758$	0	0
$759$	−8592.71	−0.410930
$760$	0	0
$761$	−31691.1	−1.50960	−0.754798	−	0.655957i	$-0.772265\pi$
$761$	−31691.1	−1.50960	−0.754798	+	0.655957i	$0.772265\pi$
$762$	0	0
$763$	− 8664.21i	− 0.411095i
$764$	0	0
$765$	13238.6i	0.625675i
$766$	0	0
$767$	−21891.8	−1.03060
$768$	0	0
$769$	12281.7	0.575929	0.287964	−	0.957641i	$-0.407022\pi$
$769$	12281.7	0.575929	0.287964	+	0.957641i	$0.407022\pi$
$770$	0	0
$771$	− 20660.0i	− 0.965046i
$772$	0	0
$773$	− 25116.4i	− 1.16866i	−0.811516	−	0.584330i	$-0.801357\pi$
$773$	− 25116.4i	− 1.16866i	0.811516	−	0.584330i	$-0.198643\pi$
$774$	0	0
$775$	8349.95	0.387018
$776$	0	0
$777$	4529.20	0.209117
$778$	0	0
$779$	41507.8i	1.90908i
$780$	0	0
$781$	− 693.241i	− 0.0317620i
$782$	0	0
$783$	−2397.74	−0.109436
$784$	0	0
$785$	−3554.02	−0.161590
$786$	0	0
$787$	37337.5i	1.69115i	0.533854	+	0.845577i	$0.320743\pi$
$787$	37337.5i	1.69115i	−0.533854	+	0.845577i	$0.679257\pi$
$788$	0	0
$789$	− 6539.24i	− 0.295061i
$790$	0	0
$791$	11662.4	0.524230
$792$	0	0
$793$	−70208.0	−3.14396
$794$	0	0
$795$	− 33985.6i	− 1.51616i
$796$	0	0
$797$	− 2081.90i	− 0.0925278i	−0.998929	−	0.0462639i	$-0.985268\pi$
$797$	− 2081.90i	− 0.0925278i	0.998929	−	0.0462639i	$-0.0147315\pi$
$798$	0	0
$799$	−28472.2	−1.26067
$800$	0	0
$801$	797.981	0.0352001
$802$	0	0
$803$	− 117.754i	− 0.00517490i
$804$	0	0
$805$	20108.1i	0.880394i
$806$	0	0
$807$	25073.3	1.09371
$808$	0	0
$809$	8865.83	0.385298	0.192649	−	0.981268i	$-0.438292\pi$
$809$	8865.83	0.385298	0.192649	+	0.981268i	$0.438292\pi$
$810$	0	0
$811$	822.660i	0.0356196i	0.999841	+	0.0178098i	$0.00566933\pi$
$811$	822.660i	0.0356196i	−0.999841	+	0.0178098i	$0.994331\pi$
$812$	0	0
$813$	18192.7i	0.784803i
$814$	0	0
$815$	−68168.4	−2.92986
$816$	0	0
$817$	−13936.9	−0.596804
$818$	0	0
$819$	− 5422.25i	− 0.231342i
$820$	0	0
$821$	30286.4i	1.28746i	0.765254	+	0.643729i	$0.222613\pi$
$821$	30286.4i	1.28746i	−0.765254	+	0.643729i	$0.777387\pi$
$822$	0	0
$823$	12399.7	0.525186	0.262593	−	0.964907i	$-0.415422\pi$
$823$	12399.7	0.525186	0.262593	+	0.964907i	$0.415422\pi$
$824$	0	0
$825$	7893.12	0.333095
$826$	0	0
$827$	− 2303.98i	− 0.0968767i	−0.998826	−	0.0484384i	$-0.984576\pi$
$827$	− 2303.98i	− 0.0968767i	0.998826	−	0.0484384i	$-0.0154244\pi$
$828$	0	0
$829$	− 22788.3i	− 0.954728i	−0.878706	−	0.477364i	$-0.841592\pi$
$829$	− 22788.3i	− 0.954728i	0.878706	−	0.477364i	$-0.158408\pi$
$830$	0	0
$831$	2489.41	0.103919
$832$	0	0
$833$	4291.31	0.178494
$834$	0	0
$835$	− 23057.7i	− 0.955622i
$836$	0	0
$837$	1435.03i	0.0592615i
$838$	0	0
$839$	7391.54	0.304153	0.152077	−	0.988369i	$-0.451404\pi$
$839$	7391.54	0.304153	0.152077	+	0.988369i	$0.451404\pi$
$840$	0	0
$841$	16502.6	0.676642
$842$	0	0
$843$	5817.14i	0.237666i
$844$	0	0
$845$	87517.1i	3.56293i
$846$	0	0
$847$	−7353.73	−0.298320
$848$	0	0
$849$	19725.1	0.797366
$850$	0	0
$851$	36886.7i	1.48585i
$852$	0	0
$853$	− 9728.81i	− 0.390514i	−0.980752	−	0.195257i	$-0.937446\pi$
$853$	− 9728.81i	− 0.390514i	0.980752	−	0.195257i	$-0.0625541\pi$
$854$	0	0
$855$	−15844.4	−0.633763
$856$	0	0
$857$	15450.3	0.615835	0.307917	−	0.951413i	$-0.400368\pi$
$857$	15450.3	0.615835	0.307917	+	0.951413i	$0.400368\pi$
$858$	0	0
$859$	34565.3i	1.37294i	0.727159	+	0.686469i	$0.240840\pi$
$859$	34565.3i	1.37294i	−0.727159	+	0.686469i	$0.759160\pi$
$860$	0	0
$861$	− 8316.10i	− 0.329166i
$862$	0	0
$863$	25414.1	1.00244	0.501219	−	0.865320i	$-0.332885\pi$
$863$	25414.1	1.00244	0.501219	+	0.865320i	$0.332885\pi$
$864$	0	0
$865$	15237.5	0.598948
$866$	0	0
$867$	8270.64i	0.323974i
$868$	0	0
$869$	− 7007.53i	− 0.273549i
$870$	0	0
$871$	−53060.0	−2.06415
$872$	0	0
$873$	−3413.86	−0.132350
$874$	0	0
$875$	− 3774.50i	− 0.145830i
$876$	0	0
$877$	− 34606.2i	− 1.33246i	−0.745746	−	0.666230i	$-0.767907\pi$
$877$	− 34606.2i	− 1.33246i	0.745746	−	0.666230i	$-0.232093\pi$
$878$	0	0
$879$	7194.33	0.276062
$880$	0	0
$881$	−35078.6	−1.34146	−0.670732	−	0.741700i	$-0.734020\pi$
$881$	−35078.6	−1.34146	−0.670732	+	0.741700i	$0.734020\pi$
$882$	0	0
$883$	11466.0i	0.436988i	0.975838	+	0.218494i	$0.0701145\pi$
$883$	11466.0i	0.436988i	−0.975838	+	0.218494i	$0.929886\pi$
$884$	0	0
$885$	− 12816.5i	− 0.486803i
$886$	0	0
$887$	47449.3	1.79616	0.898079	−	0.439835i	$-0.144963\pi$
$887$	47449.3	1.79616	0.898079	+	0.439835i	$0.144963\pi$
$888$	0	0
$889$	17279.0	0.651876
$890$	0	0
$891$	1356.52i	0.0510046i
$892$	0	0
$893$	− 34076.5i	− 1.27696i
$894$	0	0
$895$	−26671.0	−0.996103
$896$	0	0
$897$	44159.9	1.64376
$898$	0	0
$899$	4719.94i	0.175104i
$900$	0	0
$901$	− 59069.5i	− 2.18412i
$902$	0	0
$903$	2792.25	0.102902
$904$	0	0
$905$	−38736.2	−1.42280
$906$	0	0
$907$	− 4677.16i	− 0.171227i	−0.996328	−	0.0856134i	$-0.972715\pi$
$907$	− 4677.16i	− 0.171227i	0.996328	−	0.0856134i	$-0.0272850\pi$
$908$	0	0
$909$	− 3125.57i	− 0.114047i
$910$	0	0
$911$	−3778.20	−0.137406	−0.0687032	−	0.997637i	$-0.521886\pi$
$911$	−3778.20	−0.137406	−0.0687032	+	0.997637i	$0.521886\pi$
$912$	0	0
$913$	−1564.70	−0.0567186
$914$	0	0
$915$	− 41103.0i	− 1.48505i
$916$	0	0
$917$	7217.84i	0.259928i
$918$	0	0
$919$	35997.1	1.29209	0.646047	−	0.763298i	$-0.276421\pi$
$919$	35997.1	1.29209	0.646047	+	0.763298i	$0.276421\pi$
$920$	0	0
$921$	−27185.0	−0.972613
$922$	0	0
$923$	3562.73i	0.127052i
$924$	0	0
$925$	− 33883.6i	− 1.20442i
$926$	0	0
$927$	7410.63	0.262564
$928$	0	0
$929$	−17396.5	−0.614383	−0.307191	−	0.951648i	$-0.599389\pi$
$929$	−17396.5	−0.614383	−0.307191	+	0.951648i	$0.599389\pi$
$930$	0	0
$931$	5136.00i	0.180801i
$932$	0	0
$933$	− 13523.6i	− 0.474536i
$934$	0	0
$935$	24634.3	0.861633
$936$	0	0
$937$	−23754.5	−0.828203	−0.414101	−	0.910231i	$-0.635904\pi$
$937$	−23754.5	−0.828203	−0.414101	+	0.910231i	$0.635904\pi$
$938$	0	0
$939$	14380.1i	0.499763i
$940$	0	0
$941$	− 11411.2i	− 0.395318i	−0.980271	−	0.197659i	$-0.936666\pi$
$941$	− 11411.2i	− 0.395318i	0.980271	−	0.197659i	$-0.0633338\pi$
$942$	0	0
$943$	67728.0	2.33884
$944$	0	0
$945$	3174.43	0.109274
$946$	0	0
$947$	− 21209.8i	− 0.727799i	−0.931438	−	0.363899i	$-0.881445\pi$
$947$	− 21209.8i	− 0.727799i	0.931438	−	0.363899i	$-0.118555\pi$
$948$	0	0
$949$	605.164i	0.0207002i
$950$	0	0
$951$	−5077.16	−0.173121
$952$	0	0
$953$	30132.6	1.02423	0.512115	−	0.858917i	$-0.328862\pi$
$953$	30132.6	1.02423	0.512115	+	0.858917i	$0.328862\pi$
$954$	0	0
$955$	− 18525.3i	− 0.627712i
$956$	0	0
$957$	4461.71i	0.150707i
$958$	0	0
$959$	20273.7	0.682662
$960$	0	0
$961$	−26966.2	−0.905178
$962$	0	0
$963$	− 4136.25i	− 0.138410i
$964$	0	0
$965$	− 46783.7i	− 1.56064i
$966$	0	0
$967$	24799.9	0.824727	0.412363	−	0.911019i	$-0.364703\pi$
$967$	24799.9	0.824727	0.412363	+	0.911019i	$0.364703\pi$
$968$	0	0
$969$	−27538.8	−0.912975
$970$	0	0
$971$	11909.6i	0.393611i	0.980443	+	0.196805i	$0.0630567\pi$
$971$	11909.6i	0.393611i	−0.980443	+	0.196805i	$0.936943\pi$
$972$	0	0
$973$	19839.3i	0.653668i
$974$	0	0
$975$	−40564.6	−1.33242
$976$	0	0
$977$	2267.66	0.0742566	0.0371283	−	0.999311i	$-0.488179\pi$
$977$	2267.66	0.0742566	0.0371283	+	0.999311i	$0.488179\pi$
$978$	0	0
$979$	− 1484.88i	− 0.0484749i
$980$	0	0
$981$	− 11139.7i	− 0.362552i
$982$	0	0
$983$	−2731.79	−0.0886374	−0.0443187	−	0.999017i	$-0.514112\pi$
$983$	−2731.79	−0.0886374	−0.0443187	+	0.999017i	$0.514112\pi$
$984$	0	0
$985$	−13194.4	−0.426811
$986$	0	0
$987$	6827.25i	0.220176i
$988$	0	0
$989$	22740.7i	0.731155i
$990$	0	0
$991$	−18409.5	−0.590108	−0.295054	−	0.955481i	$-0.595338\pi$
$991$	−18409.5	−0.590108	−0.295054	+	0.955481i	$0.595338\pi$
$992$	0	0
$993$	24570.7	0.785223
$994$	0	0
$995$	− 88152.1i	− 2.80865i
$996$	0	0
$997$	23664.1i	0.751706i	0.926679	+	0.375853i	$0.122650\pi$
$997$	23664.1i	0.751706i	−0.926679	+	0.375853i	$0.877350\pi$
$998$	0	0
$999$	5823.26	0.184424

Twists

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

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

\(f(q)\)	\(=\)	\(q+3.00000i q^{3} -16.7959i q^{5} -7.00000 q^{7} -9.00000 q^{9} +O(q^{10})\)	\(q+3.00000i q^{3} -16.7959i q^{5} -7.00000 q^{7} -9.00000 q^{9} +16.7471i q^{11} -86.0675i q^{13} +50.3878 q^{15} +87.5778 q^{17} +104.816i q^{19} -21.0000i q^{21} +171.028 q^{23} -157.104 q^{25} -27.0000i q^{27} -88.8053i q^{29} -53.1493 q^{31} -50.2414 q^{33} +117.572i q^{35} +215.676i q^{37} +258.202 q^{39} +396.005 q^{41} +132.965i q^{43} +151.164i q^{45} -325.107 q^{47} +49.0000 q^{49} +262.734i q^{51} -674.480i q^{53} +281.284 q^{55} -314.449 q^{57} -254.356i q^{59} -815.732i q^{61} +63.0000 q^{63} -1445.58 q^{65} -616.494i q^{67} +513.085i q^{69} -41.3946 q^{71} -7.03128 q^{73} -471.312i q^{75} -117.230i q^{77} -418.431 q^{79} +81.0000 q^{81} +93.4310i q^{83} -1470.95i q^{85} +266.416 q^{87} -88.6646 q^{89} +602.472i q^{91} -159.448i q^{93} +1760.49 q^{95} +379.317 q^{97} -150.724i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 84 q^{7} - 108 q^{9}+O(q^{10}) \) 12 * q - 84 * q^7 - 108 * q^9	\( 12 q - 84 q^{7} - 108 q^{9} - 24 q^{15} + 24 q^{17} + 80 q^{23} - 564 q^{25} - 640 q^{31} - 408 q^{33} + 120 q^{39} + 1416 q^{41} - 1536 q^{47} + 588 q^{49} + 1392 q^{55} - 336 q^{57} + 756 q^{63} - 2880 q^{65} + 1392 q^{71} + 2472 q^{73} - 544 q^{79} + 972 q^{81} + 720 q^{87} + 888 q^{89} + 2368 q^{95} - 2712 q^{97}+O(q^{100}) \) 12 * q - 84 * q^7 - 108 * q^9 - 24 * q^15 + 24 * q^17 + 80 * q^23 - 564 * q^25 - 640 * q^31 - 408 * q^33 + 120 * q^39 + 1416 * q^41 - 1536 * q^47 + 588 * q^49 + 1392 * q^55 - 336 * q^57 + 756 * q^63 - 2880 * q^65 + 1392 * q^71 + 2472 * q^73 - 544 * q^79 + 972 * q^81 + 720 * q^87 + 888 * q^89 + 2368 * q^95 - 2712 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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