LMFDB - Embedded newform 1344.4.c.h.673.8

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			673.8
Root			$-0.910182i$ of defining polynomial
Character	$\chi$	$=$	1344.673
Dual form			1344.4.c.h.673.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.00000i q^{3} -13.8002i q^{5} +7.00000 q^{7} -9.00000 q^{9} +O(q^{10})$	$q+3.00000i q^{3} -13.8002i q^{5} +7.00000 q^{7} -9.00000 q^{9} -9.03959i q^{11} +50.6976i q^{13} +41.4007 q^{15} +1.30157 q^{17} +14.9067i q^{19} +21.0000i q^{21} -62.4329 q^{23} -65.4469 q^{25} -27.0000i q^{27} +37.3470i q^{29} -32.6466 q^{31} +27.1188 q^{33} -96.6017i q^{35} +307.226i q^{37} -152.093 q^{39} +19.4537 q^{41} -201.834i q^{43} +124.202i q^{45} -273.997 q^{47} +49.0000 q^{49} +3.90470i q^{51} +29.5266i q^{53} -124.749 q^{55} -44.7200 q^{57} -625.595i q^{59} +758.719i q^{61} -63.0000 q^{63} +699.640 q^{65} +733.451i q^{67} -187.299i q^{69} -107.407 q^{71} +342.209 q^{73} -196.341i q^{75} -63.2771i q^{77} +800.686 q^{79} +81.0000 q^{81} +1097.68i q^{83} -17.9620i q^{85} -112.041 q^{87} +1295.70 q^{89} +354.883i q^{91} -97.9399i q^{93} +205.716 q^{95} -301.872 q^{97} +81.3563i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	− 13.8002i	− 1.23433i	−0.786833	−	0.617166i	$-0.788281\pi$
$5$	− 13.8002i	− 1.23433i	0.786833	−	0.617166i	$-0.211719\pi$
$6$	0	0
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	0	0
$9$	−9.00000	−0.333333
$10$	0	0
$11$	− 9.03959i	− 0.247776i	−0.992296	−	0.123888i	$-0.960464\pi$
$11$	− 9.03959i	− 0.247776i	0.992296	−	0.123888i	$-0.0395364\pi$
$12$	0	0
$13$	50.6976i	1.08161i	0.841146	+	0.540807i	$0.181881\pi$
$13$	50.6976i	1.08161i	−0.841146	+	0.540807i	$0.818119\pi$
$14$	0	0
$15$	41.4007	0.712642
$16$	0	0
$17$	1.30157	0.0185692	0.00928460	−	0.999957i	$-0.497045\pi$
$17$	1.30157	0.0185692	0.00928460	+	0.999957i	$0.497045\pi$
$18$	0	0
$19$	14.9067i	0.179991i	0.995942	+	0.0899953i	$0.0286852\pi$
$19$	14.9067i	0.179991i	−0.995942	+	0.0899953i	$0.971315\pi$
$20$	0	0
$21$	21.0000i	0.218218i
$22$	0	0
$23$	−62.4329	−0.566007	−0.283003	−	0.959119i	$-0.591331\pi$
$23$	−62.4329	−0.566007	−0.283003	+	0.959119i	$0.591331\pi$
$24$	0	0
$25$	−65.4469	−0.523575
$26$	0	0
$27$	− 27.0000i	− 0.192450i
$28$	0	0
$29$	37.3470i	0.239143i	0.992826	+	0.119572i	$0.0381521\pi$
$29$	37.3470i	0.239143i	−0.992826	+	0.119572i	$0.961848\pi$
$30$	0	0
$31$	−32.6466	−0.189145	−0.0945727	−	0.995518i	$-0.530148\pi$
$31$	−32.6466	−0.189145	−0.0945727	+	0.995518i	$0.530148\pi$
$32$	0	0
$33$	27.1188	0.143054
$34$	0	0
$35$	− 96.6017i	− 0.466534i
$36$	0	0
$37$	307.226i	1.36507i	0.730852	+	0.682536i	$0.239123\pi$
$37$	307.226i	1.36507i	−0.730852	+	0.682536i	$0.760877\pi$
$38$	0	0
$39$	−152.093	−0.624471
$40$	0	0
$41$	19.4537	0.0741014	0.0370507	−	0.999313i	$-0.488204\pi$
$41$	19.4537	0.0741014	0.0370507	+	0.999313i	$0.488204\pi$
$42$	0	0
$43$	− 201.834i	− 0.715799i	−0.933760	−	0.357899i	$-0.883493\pi$
$43$	− 201.834i	− 0.715799i	0.933760	−	0.357899i	$-0.116507\pi$
$44$	0	0
$45$	124.202i	0.411444i
$46$	0	0
$47$	−273.997	−0.850351	−0.425176	−	0.905111i	$-0.639788\pi$
$47$	−273.997	−0.850351	−0.425176	+	0.905111i	$0.639788\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	3.90470i	0.0107209i
$52$	0	0
$53$	29.5266i	0.0765245i	0.999268	+	0.0382622i	$0.0121822\pi$
$53$	29.5266i	0.0765245i	−0.999268	+	0.0382622i	$0.987818\pi$
$54$	0	0
$55$	−124.749	−0.305838
$56$	0	0
$57$	−44.7200	−0.103918
$58$	0	0
$59$	− 625.595i	− 1.38043i	−0.723603	−	0.690216i	$-0.757515\pi$
$59$	− 625.595i	− 1.38043i	0.723603	−	0.690216i	$-0.242485\pi$
$60$	0	0
$61$	758.719i	1.59253i	0.604951	+	0.796263i	$0.293193\pi$
$61$	758.719i	1.59253i	−0.604951	+	0.796263i	$0.706807\pi$
$62$	0	0
$63$	−63.0000	−0.125988
$64$	0	0
$65$	699.640	1.33507
$66$	0	0
$67$	733.451i	1.33739i	0.743536	+	0.668696i	$0.233147\pi$
$67$	733.451i	1.33739i	−0.743536	+	0.668696i	$0.766853\pi$
$68$	0	0
$69$	− 187.299i	− 0.326784i
$70$	0	0
$71$	−107.407	−0.179533	−0.0897665	−	0.995963i	$-0.528612\pi$
$71$	−107.407	−0.179533	−0.0897665	+	0.995963i	$0.528612\pi$
$72$	0	0
$73$	342.209	0.548665	0.274333	−	0.961635i	$-0.411543\pi$
$73$	342.209	0.548665	0.274333	+	0.961635i	$0.411543\pi$
$74$	0	0
$75$	− 196.341i	− 0.302286i
$76$	0	0
$77$	− 63.2771i	− 0.0936506i
$78$	0	0
$79$	800.686	1.14031	0.570153	−	0.821538i	$-0.306884\pi$
$79$	800.686	1.14031	0.570153	+	0.821538i	$0.306884\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	1097.68i	1.45164i	0.687884	+	0.725821i	$0.258540\pi$
$83$	1097.68i	1.45164i	−0.687884	+	0.725821i	$0.741460\pi$
$84$	0	0
$85$	− 17.9620i	− 0.0229206i
$86$	0	0
$87$	−112.041	−0.138069
$88$	0	0
$89$	1295.70	1.54319	0.771594	−	0.636115i	$-0.219460\pi$
$89$	1295.70	1.54319	0.771594	+	0.636115i	$0.219460\pi$
$90$	0	0
$91$	354.883i	0.408812i
$92$	0	0
$93$	− 97.9399i	− 0.109203i
$94$	0	0
$95$	205.716	0.222168
$96$	0	0
$97$	−301.872	−0.315984	−0.157992	−	0.987440i	$-0.550502\pi$
$97$	−301.872	−0.315984	−0.157992	+	0.987440i	$0.550502\pi$
$98$	0	0
$99$	81.3563i	0.0825921i
$100$	0	0
$101$	− 404.892i	− 0.398893i	−0.979909	−	0.199447i	$-0.936086\pi$
$101$	− 404.892i	− 0.398893i	0.979909	−	0.199447i	$-0.0639145\pi$
$102$	0	0
$103$	444.944	0.425647	0.212823	−	0.977091i	$-0.431734\pi$
$103$	444.944	0.425647	0.212823	+	0.977091i	$0.431734\pi$
$104$	0	0
$105$	289.805	0.269353
$106$	0	0
$107$	− 264.030i	− 0.238549i	−0.992861	−	0.119275i	$-0.961943\pi$
$107$	− 264.030i	− 0.238549i	0.992861	−	0.119275i	$-0.0380569\pi$
$108$	0	0
$109$	93.4086i	0.0820818i	0.999157	+	0.0410409i	$0.0130674\pi$
$109$	93.4086i	0.0820818i	−0.999157	+	0.0410409i	$0.986933\pi$
$110$	0	0
$111$	−921.679	−0.788125
$112$	0	0
$113$	536.170	0.446359	0.223180	−	0.974777i	$-0.428356\pi$
$113$	536.170	0.446359	0.223180	+	0.974777i	$0.428356\pi$
$114$	0	0
$115$	861.589i	0.698640i
$116$	0	0
$117$	− 456.279i	− 0.360538i
$118$	0	0
$119$	9.11097	0.00701850
$120$	0	0
$121$	1249.29	0.938607
$122$	0	0
$123$	58.3611i	0.0427824i
$124$	0	0
$125$	− 821.848i	− 0.588067i
$126$	0	0
$127$	63.7997	0.0445772	0.0222886	−	0.999752i	$-0.492905\pi$
$127$	63.7997	0.0445772	0.0222886	+	0.999752i	$0.492905\pi$
$128$	0	0
$129$	605.501	0.413267
$130$	0	0
$131$	878.089i	0.585641i	0.956167	+	0.292821i	$0.0945939\pi$
$131$	878.089i	0.585641i	−0.956167	+	0.292821i	$0.905406\pi$
$132$	0	0
$133$	104.347i	0.0680301i
$134$	0	0
$135$	−372.607	−0.237547
$136$	0	0
$137$	582.172	0.363054	0.181527	−	0.983386i	$-0.441896\pi$
$137$	582.172	0.363054	0.181527	+	0.983386i	$0.441896\pi$
$138$	0	0
$139$	1968.29i	1.20106i	0.799601	+	0.600532i	$0.205044\pi$
$139$	1968.29i	1.20106i	−0.799601	+	0.600532i	$0.794956\pi$
$140$	0	0
$141$	− 821.990i	− 0.490950i
$142$	0	0
$143$	458.286	0.267998
$144$	0	0
$145$	515.397	0.295182
$146$	0	0
$147$	147.000i	0.0824786i
$148$	0	0
$149$	1085.29i	0.596712i	0.954455	+	0.298356i	$0.0964383\pi$
$149$	1085.29i	0.596712i	−0.954455	+	0.298356i	$0.903562\pi$
$150$	0	0
$151$	−784.336	−0.422705	−0.211352	−	0.977410i	$-0.567787\pi$
$151$	−784.336	−0.422705	−0.211352	+	0.977410i	$0.567787\pi$
$152$	0	0
$153$	−11.7141	−0.00618974
$154$	0	0
$155$	450.532i	0.233468i
$156$	0	0
$157$	2495.67i	1.26864i	0.773071	+	0.634320i	$0.218720\pi$
$157$	2495.67i	1.26864i	−0.773071	+	0.634320i	$0.781280\pi$
$158$	0	0
$159$	−88.5799	−0.0441814
$160$	0	0
$161$	−437.030	−0.213930
$162$	0	0
$163$	896.461i	0.430774i	0.976529	+	0.215387i	$0.0691014\pi$
$163$	896.461i	0.430774i	−0.976529	+	0.215387i	$0.930899\pi$
$164$	0	0
$165$	− 374.246i	− 0.176576i
$166$	0	0
$167$	−1159.58	−0.537312	−0.268656	−	0.963236i	$-0.586580\pi$
$167$	−1159.58	−0.537312	−0.268656	+	0.963236i	$0.586580\pi$
$168$	0	0
$169$	−373.249	−0.169890
$170$	0	0
$171$	− 134.160i	− 0.0599969i
$172$	0	0
$173$	3822.75i	1.67999i	0.542595	+	0.839995i	$0.317442\pi$
$173$	3822.75i	1.67999i	−0.542595	+	0.839995i	$0.682558\pi$
$174$	0	0
$175$	−458.128	−0.197893
$176$	0	0
$177$	1876.78	0.796993
$178$	0	0
$179$	3849.44i	1.60738i	0.595050	+	0.803689i	$0.297132\pi$
$179$	3849.44i	1.60738i	−0.595050	+	0.803689i	$0.702868\pi$
$180$	0	0
$181$	334.288i	0.137278i	0.997642	+	0.0686392i	$0.0218657\pi$
$181$	334.288i	0.137278i	−0.997642	+	0.0686392i	$0.978134\pi$
$182$	0	0
$183$	−2276.16	−0.919445
$184$	0	0
$185$	4239.80	1.68495
$186$	0	0
$187$	− 11.7656i	− 0.00460101i
$188$	0	0
$189$	− 189.000i	− 0.0727393i
$190$	0	0
$191$	1230.68	0.466223	0.233112	−	0.972450i	$-0.425109\pi$
$191$	1230.68	0.466223	0.233112	+	0.972450i	$0.425109\pi$
$192$	0	0
$193$	−3026.92	−1.12893	−0.564463	−	0.825459i	$-0.690917\pi$
$193$	−3026.92	−1.12893	−0.564463	+	0.825459i	$0.690917\pi$
$194$	0	0
$195$	2098.92i	0.770804i
$196$	0	0
$197$	202.135i	0.0731041i	0.999332	+	0.0365520i	$0.0116375\pi$
$197$	202.135i	0.0731041i	−0.999332	+	0.0365520i	$0.988363\pi$
$198$	0	0
$199$	−138.229	−0.0492402	−0.0246201	−	0.999697i	$-0.507838\pi$
$199$	−138.229	−0.0492402	−0.0246201	+	0.999697i	$0.507838\pi$
$200$	0	0
$201$	−2200.35	−0.772144
$202$	0	0
$203$	261.429i	0.0903877i
$204$	0	0
$205$	− 268.466i	− 0.0914657i
$206$	0	0
$207$	561.896	0.188669
$208$	0	0
$209$	134.750	0.0445974
$210$	0	0
$211$	4519.96i	1.47472i	0.675498	+	0.737362i	$0.263929\pi$
$211$	4519.96i	1.47472i	−0.675498	+	0.737362i	$0.736071\pi$
$212$	0	0
$213$	− 322.220i	− 0.103653i
$214$	0	0
$215$	−2785.35	−0.883533
$216$	0	0
$217$	−228.526	−0.0714903
$218$	0	0
$219$	1026.63i	0.316772i
$220$	0	0
$221$	65.9864i	0.0200847i
$222$	0	0
$223$	−3151.11	−0.946252	−0.473126	−	0.880995i	$-0.656874\pi$
$223$	−3151.11	−0.946252	−0.473126	+	0.880995i	$0.656874\pi$
$224$	0	0
$225$	589.022	0.174525
$226$	0	0
$227$	− 4211.08i	− 1.23127i	−0.788030	−	0.615637i	$-0.788899\pi$
$227$	− 4211.08i	− 1.23127i	0.788030	−	0.615637i	$-0.211101\pi$
$228$	0	0
$229$	1822.48i	0.525909i	0.964808	+	0.262954i	$0.0846969\pi$
$229$	1822.48i	0.525909i	−0.964808	+	0.262954i	$0.915303\pi$
$230$	0	0
$231$	189.831	0.0540692
$232$	0	0
$233$	3714.19	1.04431	0.522156	−	0.852850i	$-0.325128\pi$
$233$	3714.19	1.04431	0.522156	+	0.852850i	$0.325128\pi$
$234$	0	0
$235$	3781.22i	1.04962i
$236$	0	0
$237$	2402.06i	0.658356i
$238$	0	0
$239$	−5149.09	−1.39359	−0.696793	−	0.717272i	$-0.745390\pi$
$239$	−5149.09	−1.39359	−0.696793	+	0.717272i	$0.745390\pi$
$240$	0	0
$241$	7286.99	1.94770	0.973851	−	0.227188i	$-0.0729533\pi$
$241$	7286.99	1.94770	0.973851	+	0.227188i	$0.0729533\pi$
$242$	0	0
$243$	243.000i	0.0641500i
$244$	0	0
$245$	− 676.212i	− 0.176333i
$246$	0	0
$247$	−755.732	−0.194680
$248$	0	0
$249$	−3293.04	−0.838105
$250$	0	0
$251$	− 2202.95i	− 0.553981i	−0.960873	−	0.276991i	$-0.910663\pi$
$251$	− 2202.95i	− 0.553981i	0.960873	−	0.276991i	$-0.0893371\pi$
$252$	0	0
$253$	564.368i	0.140243i
$254$	0	0
$255$	53.8859	0.0132332
$256$	0	0
$257$	−382.218	−0.0927708	−0.0463854	−	0.998924i	$-0.514770\pi$
$257$	−382.218	−0.0927708	−0.0463854	+	0.998924i	$0.514770\pi$
$258$	0	0
$259$	2150.58i	0.515949i
$260$	0	0
$261$	− 336.123i	− 0.0797145i
$262$	0	0
$263$	−4374.34	−1.02560	−0.512801	−	0.858508i	$-0.671392\pi$
$263$	−4374.34	−1.02560	−0.512801	+	0.858508i	$0.671392\pi$
$264$	0	0
$265$	407.475	0.0944566
$266$	0	0
$267$	3887.09i	0.890960i
$268$	0	0
$269$	4182.99i	0.948110i	0.880495	+	0.474055i	$0.157210\pi$
$269$	4182.99i	0.948110i	−0.880495	+	0.474055i	$0.842790\pi$
$270$	0	0
$271$	−1227.13	−0.275065	−0.137533	−	0.990497i	$-0.543917\pi$
$271$	−1227.13	−0.275065	−0.137533	+	0.990497i	$0.543917\pi$
$272$	0	0
$273$	−1064.65	−0.236028
$274$	0	0
$275$	591.613i	0.129729i
$276$	0	0
$277$	− 2468.15i	− 0.535368i	−0.963507	−	0.267684i	$-0.913742\pi$
$277$	− 2468.15i	− 0.535368i	0.963507	−	0.267684i	$-0.0862584\pi$
$278$	0	0
$279$	293.820	0.0630485
$280$	0	0
$281$	−1588.55	−0.337241	−0.168621	−	0.985681i	$-0.553931\pi$
$281$	−1588.55	−0.337241	−0.168621	+	0.985681i	$0.553931\pi$
$282$	0	0
$283$	− 1186.01i	− 0.249121i	−0.992212	−	0.124560i	$-0.960248\pi$
$283$	− 1186.01i	− 0.249121i	0.992212	−	0.124560i	$-0.0397520\pi$
$284$	0	0
$285$	617.147i	0.128269i
$286$	0	0
$287$	136.176	0.0280077
$288$	0	0
$289$	−4911.31	−0.999655
$290$	0	0
$291$	− 905.616i	− 0.182433i
$292$	0	0
$293$	3491.33i	0.696128i	0.937471	+	0.348064i	$0.113161\pi$
$293$	3491.33i	0.696128i	−0.937471	+	0.348064i	$0.886839\pi$
$294$	0	0
$295$	−8633.36	−1.70391
$296$	0	0
$297$	−244.069	−0.0476846
$298$	0	0
$299$	− 3165.20i	− 0.612201i
$300$	0	0
$301$	− 1412.84i	− 0.270547i
$302$	0	0
$303$	1214.68	0.230301
$304$	0	0
$305$	10470.5	1.96570
$306$	0	0
$307$	− 1230.87i	− 0.228826i	−0.993433	−	0.114413i	$-0.963501\pi$
$307$	− 1230.87i	− 0.228826i	0.993433	−	0.114413i	$-0.0364987\pi$
$308$	0	0
$309$	1334.83i	0.245747i
$310$	0	0
$311$	5625.29	1.02566	0.512831	−	0.858490i	$-0.328597\pi$
$311$	5625.29	1.02566	0.512831	+	0.858490i	$0.328597\pi$
$312$	0	0
$313$	3114.06	0.562355	0.281178	−	0.959656i	$-0.409275\pi$
$313$	3114.06	0.562355	0.281178	+	0.959656i	$0.409275\pi$
$314$	0	0
$315$	869.416i	0.155511i
$316$	0	0
$317$	6729.84i	1.19238i	0.802842	+	0.596192i	$0.203320\pi$
$317$	6729.84i	1.19238i	−0.802842	+	0.596192i	$0.796680\pi$
$318$	0	0
$319$	337.601	0.0592541
$320$	0	0
$321$	792.091	0.137727
$322$	0	0
$323$	19.4020i	0.00334228i
$324$	0	0
$325$	− 3318.00i	− 0.566306i
$326$	0	0
$327$	−280.226	−0.0473900
$328$	0	0
$329$	−1917.98	−0.321402
$330$	0	0
$331$	− 2942.02i	− 0.488543i	−0.969707	−	0.244272i	$-0.921451\pi$
$331$	− 2942.02i	− 0.488543i	0.969707	−	0.244272i	$-0.0785488\pi$
$332$	0	0
$333$	− 2765.04i	− 0.455024i
$334$	0	0
$335$	10121.8	1.65079
$336$	0	0
$337$	−2629.36	−0.425016	−0.212508	−	0.977159i	$-0.568163\pi$
$337$	−2629.36	−0.425016	−0.212508	+	0.977159i	$0.568163\pi$
$338$	0	0
$339$	1608.51i	0.257706i
$340$	0	0
$341$	295.112i	0.0468658i
$342$	0	0
$343$	343.000	0.0539949
$344$	0	0
$345$	−2584.77	−0.403360
$346$	0	0
$347$	2984.39i	0.461701i	0.972989	+	0.230851i	$0.0741509\pi$
$347$	2984.39i	0.461701i	−0.972989	+	0.230851i	$0.925849\pi$
$348$	0	0
$349$	− 2450.58i	− 0.375864i	−0.982182	−	0.187932i	$-0.939822\pi$
$349$	− 2450.58i	− 0.375864i	0.982182	−	0.187932i	$-0.0601784\pi$
$350$	0	0
$351$	1368.84	0.208157
$352$	0	0
$353$	3980.70	0.600202	0.300101	−	0.953907i	$-0.402980\pi$
$353$	3980.70	0.600202	0.300101	+	0.953907i	$0.402980\pi$
$354$	0	0
$355$	1482.24i	0.221603i
$356$	0	0
$357$	27.3329i	0.00405213i
$358$	0	0
$359$	1807.77	0.265767	0.132883	−	0.991132i	$-0.457576\pi$
$359$	1807.77	0.265767	0.132883	+	0.991132i	$0.457576\pi$
$360$	0	0
$361$	6636.79	0.967603
$362$	0	0
$363$	3747.86i	0.541905i
$364$	0	0
$365$	− 4722.57i	− 0.677235i
$366$	0	0
$367$	−1718.88	−0.244481	−0.122241	−	0.992500i	$-0.539008\pi$
$367$	−1718.88	−0.244481	−0.122241	+	0.992500i	$0.539008\pi$
$368$	0	0
$369$	−175.083	−0.0247005
$370$	0	0
$371$	206.687i	0.0289235i
$372$	0	0
$373$	− 8236.73i	− 1.14338i	−0.820469	−	0.571691i	$-0.806288\pi$
$373$	− 8236.73i	− 1.14338i	0.820469	−	0.571691i	$-0.193712\pi$
$374$	0	0
$375$	2465.54	0.339520
$376$	0	0
$377$	−1893.40	−0.258661
$378$	0	0
$379$	702.905i	0.0952659i	0.998865	+	0.0476330i	$0.0151678\pi$
$379$	702.905i	0.0952659i	−0.998865	+	0.0476330i	$0.984832\pi$
$380$	0	0
$381$	191.399i	0.0257367i
$382$	0	0
$383$	8552.42	1.14101	0.570507	−	0.821293i	$-0.306747\pi$
$383$	8552.42	1.14101	0.570507	+	0.821293i	$0.306747\pi$
$384$	0	0
$385$	−873.240	−0.115596
$386$	0	0
$387$	1816.50i	0.238600i
$388$	0	0
$389$	5627.96i	0.733545i	0.930311	+	0.366773i	$0.119537\pi$
$389$	5627.96i	0.733545i	−0.930311	+	0.366773i	$0.880463\pi$
$390$	0	0
$391$	−81.2606	−0.0105103
$392$	0	0
$393$	−2634.27	−0.338120
$394$	0	0
$395$	− 11049.7i	− 1.40752i
$396$	0	0
$397$	9872.80i	1.24811i	0.781379	+	0.624057i	$0.214517\pi$
$397$	9872.80i	1.24811i	−0.781379	+	0.624057i	$0.785483\pi$
$398$	0	0
$399$	−313.040	−0.0392772
$400$	0	0
$401$	15672.1	1.95168	0.975842	−	0.218479i	$-0.0701095\pi$
$401$	15672.1	1.95168	0.975842	+	0.218479i	$0.0701095\pi$
$402$	0	0
$403$	− 1655.11i	− 0.204582i
$404$	0	0
$405$	− 1117.82i	− 0.137148i
$406$	0	0
$407$	2777.20	0.338233
$408$	0	0
$409$	12951.5	1.56580	0.782898	−	0.622150i	$-0.213740\pi$
$409$	12951.5	1.56580	0.782898	+	0.622150i	$0.213740\pi$
$410$	0	0
$411$	1746.52i	0.209609i
$412$	0	0
$413$	− 4379.16i	− 0.521754i
$414$	0	0
$415$	15148.3	1.79181
$416$	0	0
$417$	−5904.86	−0.693434
$418$	0	0
$419$	− 4474.53i	− 0.521707i	−0.965378	−	0.260853i	$-0.915996\pi$
$419$	− 4474.53i	− 0.521707i	0.965378	−	0.260853i	$-0.0840039\pi$
$420$	0	0
$421$	− 11461.4i	− 1.32683i	−0.748250	−	0.663416i	$-0.769106\pi$
$421$	− 11461.4i	− 1.32683i	0.748250	−	0.663416i	$-0.230894\pi$
$422$	0	0
$423$	2465.97	0.283450
$424$	0	0
$425$	−85.1835	−0.00972237
$426$	0	0
$427$	5311.04i	0.601918i
$428$	0	0
$429$	1374.86i	0.154729i
$430$	0	0
$431$	11478.6	1.28284	0.641418	−	0.767191i	$-0.278346\pi$
$431$	11478.6	1.28284	0.641418	+	0.767191i	$0.278346\pi$
$432$	0	0
$433$	−6977.91	−0.774450	−0.387225	−	0.921985i	$-0.626566\pi$
$433$	−6977.91	−0.774450	−0.387225	+	0.921985i	$0.626566\pi$
$434$	0	0
$435$	1546.19i	0.170424i
$436$	0	0
$437$	− 930.666i	− 0.101876i
$438$	0	0
$439$	−12332.6	−1.34078	−0.670391	−	0.742008i	$-0.733874\pi$
$439$	−12332.6	−1.34078	−0.670391	+	0.742008i	$0.733874\pi$
$440$	0	0
$441$	−441.000	−0.0476190
$442$	0	0
$443$	6404.12i	0.686838i	0.939182	+	0.343419i	$0.111585\pi$
$443$	6404.12i	0.686838i	−0.939182	+	0.343419i	$0.888415\pi$
$444$	0	0
$445$	− 17881.0i	− 1.90481i
$446$	0	0
$447$	−3255.86	−0.344512
$448$	0	0
$449$	1011.06	0.106269	0.0531346	−	0.998587i	$-0.483079\pi$
$449$	1011.06	0.106269	0.0531346	+	0.998587i	$0.483079\pi$
$450$	0	0
$451$	− 175.853i	− 0.0183606i
$452$	0	0
$453$	− 2353.01i	− 0.244049i
$454$	0	0
$455$	4897.48	0.504610
$456$	0	0
$457$	−8076.06	−0.826657	−0.413328	−	0.910582i	$-0.635634\pi$
$457$	−8076.06	−0.826657	−0.413328	+	0.910582i	$0.635634\pi$
$458$	0	0
$459$	− 35.1423i	− 0.00357365i
$460$	0	0
$461$	10882.8i	1.09948i	0.835335	+	0.549742i	$0.185274\pi$
$461$	10882.8i	1.09948i	−0.835335	+	0.549742i	$0.814726\pi$
$462$	0	0
$463$	−7423.39	−0.745128	−0.372564	−	0.928007i	$-0.621521\pi$
$463$	−7423.39	−0.745128	−0.372564	+	0.928007i	$0.621521\pi$
$464$	0	0
$465$	−1351.60	−0.134793
$466$	0	0
$467$	− 15704.8i	− 1.55617i	−0.628160	−	0.778084i	$-0.716192\pi$
$467$	− 15704.8i	− 1.55617i	0.628160	−	0.778084i	$-0.283808\pi$
$468$	0	0
$469$	5134.16i	0.505487i
$470$	0	0
$471$	−7487.02	−0.732450
$472$	0	0
$473$	−1824.49	−0.177358
$474$	0	0
$475$	− 975.594i	− 0.0942386i
$476$	0	0
$477$	− 265.740i	− 0.0255082i
$478$	0	0
$479$	12984.5	1.23857	0.619286	−	0.785166i	$-0.287422\pi$
$479$	12984.5	1.23857	0.619286	+	0.785166i	$0.287422\pi$
$480$	0	0
$481$	−15575.6	−1.47648
$482$	0	0
$483$	− 1311.09i	− 0.123513i
$484$	0	0
$485$	4165.91i	0.390029i
$486$	0	0
$487$	−10080.4	−0.937960	−0.468980	−	0.883209i	$-0.655378\pi$
$487$	−10080.4	−0.937960	−0.468980	+	0.883209i	$0.655378\pi$
$488$	0	0
$489$	−2689.38	−0.248708
$490$	0	0
$491$	− 15508.0i	− 1.42539i	−0.701474	−	0.712695i	$-0.747474\pi$
$491$	− 15508.0i	− 1.42539i	0.701474	−	0.712695i	$-0.252526\pi$
$492$	0	0
$493$	48.6096i	0.00444070i
$494$	0	0
$495$	1122.74	0.101946
$496$	0	0
$497$	−751.848	−0.0678571
$498$	0	0
$499$	− 15965.8i	− 1.43232i	−0.697936	−	0.716161i	$-0.745898\pi$
$499$	− 15965.8i	− 1.43232i	0.697936	−	0.716161i	$-0.254102\pi$
$500$	0	0
$501$	− 3478.75i	− 0.310217i
$502$	0	0
$503$	−4854.84	−0.430351	−0.215175	−	0.976575i	$-0.569032\pi$
$503$	−4854.84	−0.430351	−0.215175	+	0.976575i	$0.569032\pi$
$504$	0	0
$505$	−5587.61	−0.492367
$506$	0	0
$507$	− 1119.75i	− 0.0980862i
$508$	0	0
$509$	17962.8i	1.56422i	0.623142	+	0.782108i	$0.285856\pi$
$509$	17962.8i	1.56422i	−0.623142	+	0.782108i	$0.714144\pi$
$510$	0	0
$511$	2395.46	0.207376
$512$	0	0
$513$	402.480	0.0346392
$514$	0	0
$515$	− 6140.34i	− 0.525390i
$516$	0	0
$517$	2476.82i	0.210697i
$518$	0	0
$519$	−11468.2	−0.969942
$520$	0	0
$521$	−6089.86	−0.512095	−0.256048	−	0.966664i	$-0.582420\pi$
$521$	−6089.86	−0.512095	−0.256048	+	0.966664i	$0.582420\pi$
$522$	0	0
$523$	10064.6i	0.841481i	0.907181	+	0.420740i	$0.138230\pi$
$523$	10064.6i	0.841481i	−0.907181	+	0.420740i	$0.861770\pi$
$524$	0	0
$525$	− 1374.38i	− 0.114253i
$526$	0	0
$527$	−42.4918	−0.00351228
$528$	0	0
$529$	−8269.14	−0.679636
$530$	0	0
$531$	5630.35i	0.460144i
$532$	0	0
$533$	986.256i	0.0801491i
$534$	0	0
$535$	−3643.68	−0.294449
$536$	0	0
$537$	−11548.3	−0.928020
$538$	0	0
$539$	− 442.940i	− 0.0353966i
$540$	0	0
$541$	− 1714.23i	− 0.136230i	−0.997677	−	0.0681149i	$-0.978302\pi$
$541$	− 1714.23i	− 0.136230i	0.997677	−	0.0681149i	$-0.0216984\pi$
$542$	0	0
$543$	−1002.86	−0.0792578
$544$	0	0
$545$	1289.06	0.101316
$546$	0	0
$547$	− 21550.4i	− 1.68451i	−0.539079	−	0.842255i	$-0.681227\pi$
$547$	− 21550.4i	− 1.68451i	0.539079	−	0.842255i	$-0.318773\pi$
$548$	0	0
$549$	− 6828.47i	− 0.530842i
$550$	0	0
$551$	−556.718	−0.0430436
$552$	0	0
$553$	5604.80	0.430995
$554$	0	0
$555$	12719.4i	0.972808i
$556$	0	0
$557$	− 19313.9i	− 1.46922i	−0.678489	−	0.734610i	$-0.737365\pi$
$557$	− 19313.9i	− 1.46922i	0.678489	−	0.734610i	$-0.262635\pi$
$558$	0	0
$559$	10232.5	0.774218
$560$	0	0
$561$	35.2969	0.00265639
$562$	0	0
$563$	− 16117.4i	− 1.20651i	−0.797547	−	0.603257i	$-0.793869\pi$
$563$	− 16117.4i	− 1.20651i	0.797547	−	0.603257i	$-0.206131\pi$
$564$	0	0
$565$	− 7399.28i	− 0.550956i
$566$	0	0
$567$	567.000	0.0419961
$568$	0	0
$569$	−6162.20	−0.454012	−0.227006	−	0.973893i	$-0.572894\pi$
$569$	−6162.20	−0.454012	−0.227006	+	0.973893i	$0.572894\pi$
$570$	0	0
$571$	305.052i	0.0223573i	0.999938	+	0.0111787i	$0.00355835\pi$
$571$	305.052i	0.0223573i	−0.999938	+	0.0111787i	$0.996442\pi$
$572$	0	0
$573$	3692.03i	0.269174i
$574$	0	0
$575$	4086.04	0.296347
$576$	0	0
$577$	−23824.2	−1.71892	−0.859458	−	0.511206i	$-0.829199\pi$
$577$	−23824.2	−1.71892	−0.859458	+	0.511206i	$0.829199\pi$
$578$	0	0
$579$	− 9080.77i	− 0.651785i
$580$	0	0
$581$	7683.77i	0.548669i
$582$	0	0
$583$	266.909	0.0189610
$584$	0	0
$585$	−6296.76	−0.445024
$586$	0	0
$587$	− 4028.40i	− 0.283254i	−0.989920	−	0.141627i	$-0.954767\pi$
$587$	− 4028.40i	− 0.283254i	0.989920	−	0.141627i	$-0.0452333\pi$
$588$	0	0
$589$	− 486.652i	− 0.0340444i
$590$	0	0
$591$	−606.404	−0.0422067
$592$	0	0
$593$	5755.42	0.398562	0.199281	−	0.979942i	$-0.436139\pi$
$593$	5755.42	0.398562	0.199281	+	0.979942i	$0.436139\pi$
$594$	0	0
$595$	− 125.734i	− 0.00866316i
$596$	0	0
$597$	− 414.687i	− 0.0284288i
$598$	0	0
$599$	−4561.05	−0.311117	−0.155559	−	0.987827i	$-0.549718\pi$
$599$	−4561.05	−0.311117	−0.155559	+	0.987827i	$0.549718\pi$
$600$	0	0
$601$	−20192.3	−1.37049	−0.685243	−	0.728314i	$-0.740304\pi$
$601$	−20192.3	−1.37049	−0.685243	+	0.728314i	$0.740304\pi$
$602$	0	0
$603$	− 6601.06i	− 0.445797i
$604$	0	0
$605$	− 17240.5i	− 1.15855i
$606$	0	0
$607$	11163.2	0.746462	0.373231	−	0.927738i	$-0.378250\pi$
$607$	11163.2	0.746462	0.373231	+	0.927738i	$0.378250\pi$
$608$	0	0
$609$	−784.286	−0.0521854
$610$	0	0
$611$	− 13891.0i	− 0.919752i
$612$	0	0
$613$	15405.0i	1.01501i	0.861648	+	0.507506i	$0.169432\pi$
$613$	15405.0i	1.01501i	−0.861648	+	0.507506i	$0.830568\pi$
$614$	0	0
$615$	805.397	0.0528077
$616$	0	0
$617$	20356.8	1.32825	0.664127	−	0.747620i	$-0.268803\pi$
$617$	20356.8	1.32825	0.664127	+	0.747620i	$0.268803\pi$
$618$	0	0
$619$	19443.1i	1.26250i	0.775581	+	0.631249i	$0.217457\pi$
$619$	19443.1i	1.26250i	−0.775581	+	0.631249i	$0.782543\pi$
$620$	0	0
$621$	1685.69i	0.108928i
$622$	0	0
$623$	9069.89	0.583270
$624$	0	0
$625$	−19522.6	−1.24944
$626$	0	0
$627$	404.250i	0.0257483i
$628$	0	0
$629$	399.876i	0.0253483i
$630$	0	0
$631$	1972.21	0.124425	0.0622126	−	0.998063i	$-0.480184\pi$
$631$	1972.21	0.124425	0.0622126	+	0.998063i	$0.480184\pi$
$632$	0	0
$633$	−13559.9	−0.851432
$634$	0	0
$635$	− 880.452i	− 0.0550231i
$636$	0	0
$637$	2484.18i	0.154516i
$638$	0	0
$639$	966.661	0.0598443
$640$	0	0
$641$	−2538.56	−0.156423	−0.0782115	−	0.996937i	$-0.524921\pi$
$641$	−2538.56	−0.156423	−0.0782115	+	0.996937i	$0.524921\pi$
$642$	0	0
$643$	8973.74i	0.550373i	0.961391	+	0.275186i	$0.0887396\pi$
$643$	8973.74i	0.550373i	−0.961391	+	0.275186i	$0.911260\pi$
$644$	0	0
$645$	− 8356.06i	− 0.510108i
$646$	0	0
$647$	−31664.1	−1.92403	−0.962013	−	0.273003i	$-0.911983\pi$
$647$	−31664.1	−1.92403	−0.962013	+	0.273003i	$0.911983\pi$
$648$	0	0
$649$	−5655.12	−0.342038
$650$	0	0
$651$	− 685.579i	− 0.0412749i
$652$	0	0
$653$	− 19639.4i	− 1.17695i	−0.808514	−	0.588477i	$-0.799728\pi$
$653$	− 19639.4i	− 1.17695i	0.808514	−	0.588477i	$-0.200272\pi$
$654$	0	0
$655$	12117.9	0.722876
$656$	0	0
$657$	−3079.88	−0.182888
$658$	0	0
$659$	− 20127.3i	− 1.18975i	−0.803817	−	0.594876i	$-0.797201\pi$
$659$	− 20127.3i	− 1.18975i	0.803817	−	0.594876i	$-0.202799\pi$
$660$	0	0
$661$	− 10898.2i	− 0.641290i	−0.947199	−	0.320645i	$-0.896100\pi$
$661$	− 10898.2i	− 0.641290i	0.947199	−	0.320645i	$-0.103900\pi$
$662$	0	0
$663$	−197.959	−0.0115959
$664$	0	0
$665$	1440.01	0.0839717
$666$	0	0
$667$	− 2331.68i	− 0.135357i
$668$	0	0
$669$	− 9453.34i	− 0.546319i
$670$	0	0
$671$	6858.51	0.394590
$672$	0	0
$673$	−14208.3	−0.813802	−0.406901	−	0.913472i	$-0.633391\pi$
$673$	−14208.3	−0.813802	−0.406901	+	0.913472i	$0.633391\pi$
$674$	0	0
$675$	1767.07i	0.100762i
$676$	0	0
$677$	8372.78i	0.475321i	0.971348	+	0.237660i	$0.0763805\pi$
$677$	8372.78i	0.475321i	−0.971348	+	0.237660i	$0.923619\pi$
$678$	0	0
$679$	−2113.10	−0.119431
$680$	0	0
$681$	12633.2	0.710876
$682$	0	0
$683$	− 34724.7i	− 1.94539i	−0.232081	−	0.972696i	$-0.574554\pi$
$683$	− 34724.7i	− 1.94539i	0.232081	−	0.972696i	$-0.425446\pi$
$684$	0	0
$685$	− 8034.12i	− 0.448129i
$686$	0	0
$687$	−5467.45	−0.303634
$688$	0	0
$689$	−1496.93	−0.0827700
$690$	0	0
$691$	10289.8i	0.566489i	0.959048	+	0.283245i	$0.0914108\pi$
$691$	10289.8i	0.566489i	−0.959048	+	0.283245i	$0.908589\pi$
$692$	0	0
$693$	569.494i	0.0312169i
$694$	0	0
$695$	27162.8	1.48251
$696$	0	0
$697$	25.3203	0.00137600
$698$	0	0
$699$	11142.6i	0.602933i
$700$	0	0
$701$	− 14676.9i	− 0.790782i	−0.918513	−	0.395391i	$-0.870609\pi$
$701$	− 14676.9i	− 0.790782i	0.918513	−	0.395391i	$-0.129391\pi$
$702$	0	0
$703$	−4579.72	−0.245700
$704$	0	0
$705$	−11343.7	−0.605996
$706$	0	0
$707$	− 2834.24i	− 0.150768i
$708$	0	0
$709$	896.286i	0.0474764i	0.999718	+	0.0237382i	$0.00755681\pi$
$709$	896.286i	0.0474764i	−0.999718	+	0.0237382i	$0.992443\pi$
$710$	0	0
$711$	−7206.18	−0.380102
$712$	0	0
$713$	2038.22	0.107058
$714$	0	0
$715$	− 6324.46i	− 0.330799i
$716$	0	0
$717$	− 15447.3i	− 0.804587i
$718$	0	0
$719$	8717.37	0.452160	0.226080	−	0.974109i	$-0.427409\pi$
$719$	8717.37	0.452160	0.226080	+	0.974109i	$0.427409\pi$
$720$	0	0
$721$	3114.61	0.160879
$722$	0	0
$723$	21861.0i	1.12451i
$724$	0	0
$725$	− 2444.24i	− 0.125209i
$726$	0	0
$727$	2221.29	0.113319	0.0566595	−	0.998394i	$-0.481955\pi$
$727$	2221.29	0.113319	0.0566595	+	0.998394i	$0.481955\pi$
$728$	0	0
$729$	−729.000	−0.0370370
$730$	0	0
$731$	− 262.700i	− 0.0132918i
$732$	0	0
$733$	32736.2i	1.64958i	0.565442	+	0.824788i	$0.308705\pi$
$733$	32736.2i	1.64958i	−0.565442	+	0.824788i	$0.691295\pi$
$734$	0	0
$735$	2028.64	0.101806
$736$	0	0
$737$	6630.09	0.331374
$738$	0	0
$739$	17029.1i	0.847667i	0.905740	+	0.423834i	$0.139316\pi$
$739$	17029.1i	0.847667i	−0.905740	+	0.423834i	$0.860684\pi$
$740$	0	0
$741$	− 2267.20i	− 0.112399i
$742$	0	0
$743$	−30283.4	−1.49528	−0.747639	−	0.664105i	$-0.768813\pi$
$743$	−30283.4	−1.49528	−0.747639	+	0.664105i	$0.768813\pi$
$744$	0	0
$745$	14977.2	0.736541
$746$	0	0
$747$	− 9879.13i	− 0.483880i
$748$	0	0
$749$	− 1848.21i	− 0.0901632i
$750$	0	0
$751$	39087.9	1.89925	0.949625	−	0.313387i	$-0.101464\pi$
$751$	39087.9	1.89925	0.949625	+	0.313387i	$0.101464\pi$
$752$	0	0
$753$	6608.86	0.319841
$754$	0	0
$755$	10824.0i	0.521758i
$756$	0	0
$757$	− 16887.7i	− 0.810824i	−0.914134	−	0.405412i	$-0.867128\pi$
$757$	− 16887.7i	− 0.810824i	0.914134	−	0.405412i	$-0.132872\pi$
$758$	0	0
$759$	−1693.10	−0.0809694
$760$	0	0
$761$	−20357.1	−0.969705	−0.484852	−	0.874596i	$-0.661127\pi$
$761$	−20357.1	−0.969705	−0.484852	+	0.874596i	$0.661127\pi$
$762$	0	0
$763$	653.860i	0.0310240i
$764$	0	0
$765$	161.658i	0.00764019i
$766$	0	0
$767$	31716.2	1.49310
$768$	0	0
$769$	−31570.3	−1.48044	−0.740218	−	0.672367i	$-0.765278\pi$
$769$	−31570.3	−1.48044	−0.740218	+	0.672367i	$0.765278\pi$
$770$	0	0
$771$	− 1146.65i	− 0.0535613i
$772$	0	0
$773$	− 3695.86i	− 0.171968i	−0.996297	−	0.0859838i	$-0.972597\pi$
$773$	− 3695.86i	− 0.171968i	0.996297	−	0.0859838i	$-0.0274033\pi$
$774$	0	0
$775$	2136.62	0.0990318
$776$	0	0
$777$	−6451.75	−0.297883
$778$	0	0
$779$	289.989i	0.0133376i
$780$	0	0
$781$	970.914i	0.0444840i
$782$	0	0
$783$	1008.37	0.0460232
$784$	0	0
$785$	34440.9	1.56592
$786$	0	0
$787$	36218.2i	1.64046i	0.572036	+	0.820229i	$0.306154\pi$
$787$	36218.2i	1.64046i	−0.572036	+	0.820229i	$0.693846\pi$
$788$	0	0
$789$	− 13123.0i	− 0.592131i
$790$	0	0
$791$	3753.19	0.168708
$792$	0	0
$793$	−38465.3	−1.72250
$794$	0	0
$795$	1222.43i	0.0545345i
$796$	0	0
$797$	12815.9i	0.569587i	0.958589	+	0.284794i	$0.0919251\pi$
$797$	12815.9i	0.569587i	−0.958589	+	0.284794i	$0.908075\pi$
$798$	0	0
$799$	−356.625	−0.0157903
$800$	0	0
$801$	−11661.3	−0.514396
$802$	0	0
$803$	− 3093.43i	− 0.135946i
$804$	0	0
$805$	6031.12i	0.264061i
$806$	0	0
$807$	−12549.0	−0.547392
$808$	0	0
$809$	16477.2	0.716080	0.358040	−	0.933706i	$-0.383445\pi$
$809$	16477.2	0.716080	0.358040	+	0.933706i	$0.383445\pi$
$810$	0	0
$811$	− 28389.1i	− 1.22919i	−0.788842	−	0.614596i	$-0.789319\pi$
$811$	− 28389.1i	− 1.22919i	0.788842	−	0.614596i	$-0.210681\pi$
$812$	0	0
$813$	− 3681.38i	− 0.158809i
$814$	0	0
$815$	12371.4	0.531718
$816$	0	0
$817$	3008.67	0.128837
$818$	0	0
$819$	− 3193.95i	− 0.136271i
$820$	0	0
$821$	45205.5i	1.92166i	0.277138	+	0.960830i	$0.410614\pi$
$821$	45205.5i	1.92166i	−0.277138	+	0.960830i	$0.589386\pi$
$822$	0	0
$823$	−4872.70	−0.206381	−0.103191	−	0.994662i	$-0.532905\pi$
$823$	−4872.70	−0.206381	−0.103191	+	0.994662i	$0.532905\pi$
$824$	0	0
$825$	−1774.84	−0.0748993
$826$	0	0
$827$	− 20947.1i	− 0.880778i	−0.897807	−	0.440389i	$-0.854840\pi$
$827$	− 20947.1i	− 0.880778i	0.897807	−	0.440389i	$-0.145160\pi$
$828$	0	0
$829$	− 25995.0i	− 1.08907i	−0.838737	−	0.544537i	$-0.816705\pi$
$829$	− 25995.0i	− 1.08907i	0.838737	−	0.544537i	$-0.183295\pi$
$830$	0	0
$831$	7404.46	0.309095
$832$	0	0
$833$	63.7768	0.00265274
$834$	0	0
$835$	16002.5i	0.663222i
$836$	0	0
$837$	881.459i	0.0364011i
$838$	0	0
$839$	31337.3	1.28949	0.644746	−	0.764397i	$-0.276963\pi$
$839$	31337.3	1.28949	0.644746	+	0.764397i	$0.276963\pi$
$840$	0	0
$841$	22994.2	0.942810
$842$	0	0
$843$	− 4765.64i	− 0.194706i
$844$	0	0
$845$	5150.93i	0.209701i
$846$	0	0
$847$	8745.00	0.354760
$848$	0	0
$849$	3558.04	0.143830
$850$	0	0
$851$	− 19181.0i	− 0.772640i
$852$	0	0
$853$	7641.96i	0.306748i	0.988168	+	0.153374i	$0.0490138\pi$
$853$	7641.96i	0.306748i	−0.988168	+	0.153374i	$0.950986\pi$
$854$	0	0
$855$	−1851.44	−0.0740560
$856$	0	0
$857$	−18271.5	−0.728290	−0.364145	−	0.931342i	$-0.618639\pi$
$857$	−18271.5	−0.728290	−0.364145	+	0.931342i	$0.618639\pi$
$858$	0	0
$859$	18836.6i	0.748190i	0.927390	+	0.374095i	$0.122047\pi$
$859$	18836.6i	0.748190i	−0.927390	+	0.374095i	$0.877953\pi$
$860$	0	0
$861$	408.528i	0.0161702i
$862$	0	0
$863$	−26302.5	−1.03748	−0.518742	−	0.854931i	$-0.673599\pi$
$863$	−26302.5	−1.03748	−0.518742	+	0.854931i	$0.673599\pi$
$864$	0	0
$865$	52754.9	2.07366
$866$	0	0
$867$	− 14733.9i	− 0.577151i
$868$	0	0
$869$	− 7237.88i	− 0.282541i
$870$	0	0
$871$	−37184.2	−1.44654
$872$	0	0
$873$	2716.85	0.105328
$874$	0	0
$875$	− 5752.94i	− 0.222268i
$876$	0	0
$877$	− 39775.9i	− 1.53151i	−0.643130	−	0.765757i	$-0.722365\pi$
$877$	− 39775.9i	− 1.53151i	0.643130	−	0.765757i	$-0.277635\pi$
$878$	0	0
$879$	−10474.0	−0.401910
$880$	0	0
$881$	35303.3	1.35005	0.675027	−	0.737793i	$-0.264132\pi$
$881$	35303.3	1.35005	0.675027	+	0.737793i	$0.264132\pi$
$882$	0	0
$883$	− 21710.3i	− 0.827416i	−0.910410	−	0.413708i	$-0.864233\pi$
$883$	− 21710.3i	− 0.827416i	0.910410	−	0.413708i	$-0.135767\pi$
$884$	0	0
$885$	− 25900.1i	− 0.983754i
$886$	0	0
$887$	15410.7	0.583359	0.291680	−	0.956516i	$-0.405786\pi$
$887$	15410.7	0.583359	0.291680	+	0.956516i	$0.405786\pi$
$888$	0	0
$889$	446.598	0.0168486
$890$	0	0
$891$	− 732.207i	− 0.0275307i
$892$	0	0
$893$	− 4084.37i	− 0.153055i
$894$	0	0
$895$	53123.2	1.98404
$896$	0	0
$897$	9495.60	0.353455
$898$	0	0
$899$	− 1219.25i	− 0.0452329i
$900$	0	0
$901$	38.4309i	0.00142100i
$902$	0	0
$903$	4238.51	0.156200
$904$	0	0
$905$	4613.25	0.169447
$906$	0	0
$907$	− 7462.18i	− 0.273184i	−0.990627	−	0.136592i	$-0.956385\pi$
$907$	− 7462.18i	− 0.273184i	0.990627	−	0.136592i	$-0.0436149\pi$
$908$	0	0
$909$	3644.03i	0.132964i
$910$	0	0
$911$	4681.98	0.170276	0.0851378	−	0.996369i	$-0.472867\pi$
$911$	4681.98	0.170276	0.0851378	+	0.996369i	$0.472867\pi$
$912$	0	0
$913$	9922.59	0.359682
$914$	0	0
$915$	31411.5i	1.13490i
$916$	0	0
$917$	6146.63i	0.221352i
$918$	0	0
$919$	−2846.62	−0.102178	−0.0510888	−	0.998694i	$-0.516269\pi$
$919$	−2846.62	−0.102178	−0.0510888	+	0.998694i	$0.516269\pi$
$920$	0	0
$921$	3692.62	0.132113
$922$	0	0
$923$	− 5445.27i	− 0.194186i
$924$	0	0
$925$	− 20107.0i	− 0.714718i
$926$	0	0
$927$	−4004.50	−0.141882
$928$	0	0
$929$	−35272.0	−1.24568	−0.622839	−	0.782350i	$-0.714021\pi$
$929$	−35272.0	−1.24568	−0.622839	+	0.782350i	$0.714021\pi$
$930$	0	0
$931$	730.426i	0.0257129i
$932$	0	0
$933$	16875.9i	0.592166i
$934$	0	0
$935$	−162.369	−0.00567917
$936$	0	0
$937$	6983.20	0.243470	0.121735	−	0.992563i	$-0.461154\pi$
$937$	6983.20	0.243470	0.121735	+	0.992563i	$0.461154\pi$
$938$	0	0
$939$	9342.19i	0.324676i
$940$	0	0
$941$	− 4614.35i	− 0.159855i	−0.996801	−	0.0799274i	$-0.974531\pi$
$941$	− 4614.35i	− 0.159855i	0.996801	−	0.0799274i	$-0.0254688\pi$
$942$	0	0
$943$	−1214.55	−0.0419419
$944$	0	0
$945$	−2608.25	−0.0897844
$946$	0	0
$947$	36604.3i	1.25605i	0.778194	+	0.628025i	$0.216136\pi$
$947$	36604.3i	1.25605i	−0.778194	+	0.628025i	$0.783864\pi$
$948$	0	0
$949$	17349.2i	0.593444i
$950$	0	0
$951$	−20189.5	−0.688423
$952$	0	0
$953$	52783.0	1.79413	0.897067	−	0.441895i	$-0.145694\pi$
$953$	52783.0	1.79413	0.897067	+	0.441895i	$0.145694\pi$
$954$	0	0
$955$	− 16983.6i	− 0.575474i
$956$	0	0
$957$	1012.80i	0.0342104i
$958$	0	0
$959$	4075.21	0.137221
$960$	0	0
$961$	−28725.2	−0.964224
$962$	0	0
$963$	2376.27i	0.0795165i
$964$	0	0
$965$	41772.3i	1.39347i
$966$	0	0
$967$	−49366.0	−1.64168	−0.820840	−	0.571158i	$-0.806494\pi$
$967$	−49366.0	−1.64168	−0.820840	+	0.571158i	$0.806494\pi$
$968$	0	0
$969$	−58.2061	−0.00192967
$970$	0	0
$971$	− 12250.5i	− 0.404878i	−0.979295	−	0.202439i	$-0.935113\pi$
$971$	− 12250.5i	− 0.404878i	0.979295	−	0.202439i	$-0.0648868\pi$
$972$	0	0
$973$	13778.0i	0.453959i
$974$	0	0
$975$	9954.00	0.326957
$976$	0	0
$977$	−9536.66	−0.312287	−0.156144	−	0.987734i	$-0.549906\pi$
$977$	−9536.66	−0.312287	−0.156144	+	0.987734i	$0.549906\pi$
$978$	0	0
$979$	− 11712.6i	− 0.382365i
$980$	0	0
$981$	− 840.677i	− 0.0273606i
$982$	0	0
$983$	−47967.2	−1.55638	−0.778188	−	0.628032i	$-0.783861\pi$
$983$	−47967.2	−1.55638	−0.778188	+	0.628032i	$0.783861\pi$
$984$	0	0
$985$	2789.51	0.0902347
$986$	0	0
$987$	− 5753.93i	− 0.185562i
$988$	0	0
$989$	12601.1i	0.405147i
$990$	0	0
$991$	−43201.5	−1.38480	−0.692401	−	0.721513i	$-0.743447\pi$
$991$	−43201.5	−1.38480	−0.692401	+	0.721513i	$0.743447\pi$
$992$	0	0
$993$	8826.05	0.282061
$994$	0	0
$995$	1907.59i	0.0607787i
$996$	0	0
$997$	13736.8i	0.436359i	0.975909	+	0.218180i	$0.0700119\pi$
$997$	13736.8i	0.436359i	−0.975909	+	0.218180i	$0.929988\pi$
$998$	0	0
$999$	8295.11	0.262708

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1344.4.c.h.673.8	yes	12
4.3	odd	2		1344.4.c.e.673.2	✓	12
8.3	odd	2		1344.4.c.e.673.11	yes	12
8.5	even	2	inner	1344.4.c.h.673.5	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1344.4.c.e.673.2	✓	12	4.3	odd	2
1344.4.c.e.673.11	yes	12	8.3	odd	2
1344.4.c.h.673.5	yes	12	8.5	even	2	inner
1344.4.c.h.673.8	yes	12	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q+3.00000i q^{3} -13.8002i q^{5} +7.00000 q^{7} -9.00000 q^{9} +O(q^{10})\)	\(q+3.00000i q^{3} -13.8002i q^{5} +7.00000 q^{7} -9.00000 q^{9} -9.03959i q^{11} +50.6976i q^{13} +41.4007 q^{15} +1.30157 q^{17} +14.9067i q^{19} +21.0000i q^{21} -62.4329 q^{23} -65.4469 q^{25} -27.0000i q^{27} +37.3470i q^{29} -32.6466 q^{31} +27.1188 q^{33} -96.6017i q^{35} +307.226i q^{37} -152.093 q^{39} +19.4537 q^{41} -201.834i q^{43} +124.202i q^{45} -273.997 q^{47} +49.0000 q^{49} +3.90470i q^{51} +29.5266i q^{53} -124.749 q^{55} -44.7200 q^{57} -625.595i q^{59} +758.719i q^{61} -63.0000 q^{63} +699.640 q^{65} +733.451i q^{67} -187.299i q^{69} -107.407 q^{71} +342.209 q^{73} -196.341i q^{75} -63.2771i q^{77} +800.686 q^{79} +81.0000 q^{81} +1097.68i q^{83} -17.9620i q^{85} -112.041 q^{87} +1295.70 q^{89} +354.883i q^{91} -97.9399i q^{93} +205.716 q^{95} -301.872 q^{97} +81.3563i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 84 q^{7} - 108 q^{9}+O(q^{10}) \) 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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