LMFDB - Embedded newform 3584.2.b.i.1793.11

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3584,2,Mod(1793,3584)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3584, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("3584.1793");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3584 = 2^{9} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3584.b (of order $2$, degree $1$, not 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:	$28.6183840844$
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} - 4 x^{11} + 4 x^{10} + 8 x^{9} - 24 x^{8} + 24 x^{7} + 176 x^{6} + 160 x^{5} + 145 x^{4} + \cdots + 8 $ x^12 - 4x^11 + 4x^10 + 8x^9 - 24x^8 + 24x^7 + 176x^6 + 160x^5 + 145x^4 + 108x^3 + 68x^2 + 32*x + 8
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{14} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1793.11
Root			$-1.45042 + 0.600784i$ of defining polynomial
Character	$\chi$	$=$	3584.1793
Dual form			3584.2.b.i.1793.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.61578i q^{3} +3.96111i q^{5} -1.00000 q^{7} -3.84231 q^{9} +O(q^{10})$	$q+2.61578i q^{3} +3.96111i q^{5} -1.00000 q^{7} -3.84231 q^{9} +0.474480i q^{11} -1.64074i q^{13} -10.3614 q^{15} -1.49944 q^{17} +6.98607i q^{19} -2.61578i q^{21} +1.46340 q^{23} -10.6904 q^{25} -2.20331i q^{27} +6.47270i q^{29} -4.57012 q^{31} -1.24114 q^{33} -3.96111i q^{35} -8.73101i q^{37} +4.29183 q^{39} +7.51908 q^{41} -11.8730i q^{43} -15.2198i q^{45} -5.34862 q^{47} +1.00000 q^{49} -3.92222i q^{51} -11.8470i q^{53} -1.87947 q^{55} -18.2740 q^{57} +11.6162i q^{59} +7.76627i q^{61} +3.84231 q^{63} +6.49917 q^{65} +1.19299i q^{67} +3.82794i q^{69} -10.4122 q^{71} -13.9453 q^{73} -27.9637i q^{75} -0.474480i q^{77} +2.14017 q^{79} -5.76357 q^{81} +14.6551i q^{83} -5.93946i q^{85} -16.9312 q^{87} +7.82480 q^{89} +1.64074i q^{91} -11.9544i q^{93} -27.6726 q^{95} +9.03576 q^{97} -1.82310i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 12 q^{7} - 20 q^{9}+O(q^{10}) $ 12 * q - 12 * q^7 - 20 * q^9	$ 12 q - 12 q^{7} - 20 q^{9} - 16 q^{15} - 4 q^{25} - 16 q^{31} + 8 q^{33} + 8 q^{41} + 16 q^{47} + 12 q^{49} - 32 q^{55} + 8 q^{57} + 20 q^{63} - 16 q^{65} + 16 q^{71} - 32 q^{73} - 48 q^{79} + 20 q^{81} + 16 q^{87} - 32 q^{89} - 32 q^{95} + 32 q^{97}+O(q^{100}) $ 12 * q - 12 * q^7 - 20 * q^9 - 16 * q^15 - 4 * q^25 - 16 * q^31 + 8 * q^33 + 8 * q^41 + 16 * q^47 + 12 * q^49 - 32 * q^55 + 8 * q^57 + 20 * q^63 - 16 * q^65 + 16 * q^71 - 32 * q^73 - 48 * q^79 + 20 * q^81 + 16 * q^87 - 32 * q^89 - 32 * q^95 + 32 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3584\mathbb{Z}\right)^\times$.

$n$	$1023$	$1025$	$1541$
$\chi(n)$	$1$	$1$	$-1$

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
$3$	2.61578i	1.51022i	0.655597	+	0.755111i	$0.272417\pi$
$3$	2.61578i	1.51022i	−0.655597	+	0.755111i	$0.727583\pi$
$4$	0	0
$5$	3.96111i	1.77146i	0.464199	+	0.885731i	$0.346342\pi$
$5$	3.96111i	1.77146i	−0.464199	+	0.885731i	$0.653658\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	−3.84231	−1.28077
$10$	0	0
$11$	0.474480i	0.143061i	0.997438	+	0.0715306i	$0.0227883\pi$
$11$	0.474480i	0.143061i	−0.997438	+	0.0715306i	$0.977212\pi$
$12$	0	0
$13$	− 1.64074i	− 0.455061i	−0.973771	−	0.227530i	$-0.926935\pi$
$13$	− 1.64074i	− 0.455061i	0.973771	−	0.227530i	$-0.0730651\pi$
$14$	0	0
$15$	−10.3614	−2.67530
$16$	0	0
$17$	−1.49944	−0.363668	−0.181834	−	0.983329i	$-0.558203\pi$
$17$	−1.49944	−0.363668	−0.181834	+	0.983329i	$0.558203\pi$
$18$	0	0
$19$	6.98607i	1.60271i	0.598186	+	0.801357i	$0.295888\pi$
$19$	6.98607i	1.60271i	−0.598186	+	0.801357i	$0.704112\pi$
$20$	0	0
$21$	− 2.61578i	− 0.570810i
$22$	0	0
$23$	1.46340	0.305140	0.152570	−	0.988293i	$-0.451245\pi$
$23$	1.46340	0.305140	0.152570	+	0.988293i	$0.451245\pi$
$24$	0	0
$25$	−10.6904	−2.13808
$26$	0	0
$27$	− 2.20331i	− 0.424026i
$28$	0	0
$29$	6.47270i	1.20195i	0.799268	+	0.600975i	$0.205221\pi$
$29$	6.47270i	1.20195i	−0.799268	+	0.600975i	$0.794779\pi$
$30$	0	0
$31$	−4.57012	−0.820818	−0.410409	−	0.911902i	$-0.634614\pi$
$31$	−4.57012	−0.820818	−0.410409	+	0.911902i	$0.634614\pi$
$32$	0	0
$33$	−1.24114	−0.216054
$34$	0	0
$35$	− 3.96111i	− 0.669549i
$36$	0	0
$37$	− 8.73101i	− 1.43537i	−0.696369	−	0.717684i	$-0.745202\pi$
$37$	− 8.73101i	− 1.43537i	0.696369	−	0.717684i	$-0.254798\pi$
$38$	0	0
$39$	4.29183	0.687243
$40$	0	0
$41$	7.51908	1.17428	0.587142	−	0.809484i	$-0.300253\pi$
$41$	7.51908	1.17428	0.587142	+	0.809484i	$0.300253\pi$
$42$	0	0
$43$	− 11.8730i	− 1.81062i	−0.424752	−	0.905310i	$-0.639639\pi$
$43$	− 11.8730i	− 1.81062i	0.424752	−	0.905310i	$-0.360361\pi$
$44$	0	0
$45$	− 15.2198i	− 2.26884i
$46$	0	0
$47$	−5.34862	−0.780177	−0.390088	−	0.920777i	$-0.627556\pi$
$47$	−5.34862	−0.780177	−0.390088	+	0.920777i	$0.627556\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	− 3.92222i	− 0.549220i
$52$	0	0
$53$	− 11.8470i	− 1.62730i	−0.581352	−	0.813652i	$-0.697476\pi$
$53$	− 11.8470i	− 1.62730i	0.581352	−	0.813652i	$-0.302524\pi$
$54$	0	0
$55$	−1.87947	−0.253427
$56$	0	0
$57$	−18.2740	−2.42046
$58$	0	0
$59$	11.6162i	1.51230i	0.654399	+	0.756149i	$0.272921\pi$
$59$	11.6162i	1.51230i	−0.654399	+	0.756149i	$0.727079\pi$
$60$	0	0
$61$	7.76627i	0.994368i	0.867645	+	0.497184i	$0.165633\pi$
$61$	7.76627i	0.994368i	−0.867645	+	0.497184i	$0.834367\pi$
$62$	0	0
$63$	3.84231	0.484086
$64$	0	0
$65$	6.49917	0.806122
$66$	0	0
$67$	1.19299i	0.145747i	0.997341	+	0.0728735i	$0.0232169\pi$
$67$	1.19299i	0.145747i	−0.997341	+	0.0728735i	$0.976783\pi$
$68$	0	0
$69$	3.82794i	0.460830i
$70$	0	0
$71$	−10.4122	−1.23570	−0.617848	−	0.786298i	$-0.711995\pi$
$71$	−10.4122	−1.23570	−0.617848	+	0.786298i	$0.711995\pi$
$72$	0	0
$73$	−13.9453	−1.63218	−0.816089	−	0.577927i	$-0.803862\pi$
$73$	−13.9453	−1.63218	−0.816089	+	0.577927i	$0.803862\pi$
$74$	0	0
$75$	− 27.9637i	− 3.22897i
$76$	0	0
$77$	− 0.474480i	− 0.0540720i
$78$	0	0
$79$	2.14017	0.240788	0.120394	−	0.992726i	$-0.461584\pi$
$79$	2.14017	0.240788	0.120394	+	0.992726i	$0.461584\pi$
$80$	0	0
$81$	−5.76357	−0.640397
$82$	0	0
$83$	14.6551i	1.60860i	0.594223	+	0.804301i	$0.297460\pi$
$83$	14.6551i	1.60860i	−0.594223	+	0.804301i	$0.702540\pi$
$84$	0	0
$85$	− 5.93946i	− 0.644225i
$86$	0	0
$87$	−16.9312	−1.81521
$88$	0	0
$89$	7.82480	0.829427	0.414713	−	0.909952i	$-0.363882\pi$
$89$	7.82480	0.829427	0.414713	+	0.909952i	$0.363882\pi$
$90$	0	0
$91$	1.64074i	0.171997i
$92$	0	0
$93$	− 11.9544i	− 1.23962i
$94$	0	0
$95$	−27.6726	−2.83915
$96$	0	0
$97$	9.03576	0.917443	0.458721	−	0.888580i	$-0.348308\pi$
$97$	9.03576	0.917443	0.458721	+	0.888580i	$0.348308\pi$
$98$	0	0
$99$	− 1.82310i	− 0.183229i
$100$	0	0
$101$	− 4.01611i	− 0.399618i	−0.979835	−	0.199809i	$-0.935968\pi$
$101$	− 4.01611i	− 0.399618i	0.979835	−	0.199809i	$-0.0640321\pi$
$102$	0	0
$103$	−1.00603	−0.0991269	−0.0495634	−	0.998771i	$-0.515783\pi$
$103$	−1.00603	−0.0991269	−0.0495634	+	0.998771i	$0.515783\pi$
$104$	0	0
$105$	10.3614	1.01117
$106$	0	0
$107$	7.56113i	0.730962i	0.930819	+	0.365481i	$0.119095\pi$
$107$	7.56113i	0.730962i	−0.930819	+	0.365481i	$0.880905\pi$
$108$	0	0
$109$	− 18.5927i	− 1.78086i	−0.455124	−	0.890428i	$-0.650405\pi$
$109$	− 18.5927i	− 1.78086i	0.455124	−	0.890428i	$-0.349595\pi$
$110$	0	0
$111$	22.8384	2.16773
$112$	0	0
$113$	19.0995	1.79673	0.898363	−	0.439254i	$-0.144757\pi$
$113$	19.0995	1.79673	0.898363	+	0.439254i	$0.144757\pi$
$114$	0	0
$115$	5.79669i	0.540545i
$116$	0	0
$117$	6.30425i	0.582828i
$118$	0	0
$119$	1.49944	0.137454
$120$	0	0
$121$	10.7749	0.979534
$122$	0	0
$123$	19.6683i	1.77343i
$124$	0	0
$125$	− 22.5402i	− 2.01606i
$126$	0	0
$127$	−0.391682	−0.0347562	−0.0173781	−	0.999849i	$-0.505532\pi$
$127$	−0.391682	−0.0347562	−0.0173781	+	0.999849i	$0.505532\pi$
$128$	0	0
$129$	31.0572	2.73444
$130$	0	0
$131$	0.251033i	0.0219329i	0.999940	+	0.0109664i	$0.00349079\pi$
$131$	0.251033i	0.0219329i	−0.999940	+	0.0109664i	$0.996509\pi$
$132$	0	0
$133$	− 6.98607i	− 0.605769i
$134$	0	0
$135$	8.72753	0.751146
$136$	0	0
$137$	2.73448	0.233623	0.116811	−	0.993154i	$-0.462733\pi$
$137$	2.73448	0.233623	0.116811	+	0.993154i	$0.462733\pi$
$138$	0	0
$139$	0.858361i	0.0728052i	0.999337	+	0.0364026i	$0.0115899\pi$
$139$	0.858361i	0.0728052i	−0.999337	+	0.0364026i	$0.988410\pi$
$140$	0	0
$141$	− 13.9908i	− 1.17824i
$142$	0	0
$143$	0.778501	0.0651015
$144$	0	0
$145$	−25.6391	−2.12921
$146$	0	0
$147$	2.61578i	0.215746i
$148$	0	0
$149$	− 14.6705i	− 1.20185i	−0.799305	−	0.600926i	$-0.794799\pi$
$149$	− 14.6705i	− 1.20185i	0.799305	−	0.600926i	$-0.205201\pi$
$150$	0	0
$151$	1.73895	0.141514	0.0707569	−	0.997494i	$-0.477459\pi$
$151$	1.73895	0.141514	0.0707569	+	0.997494i	$0.477459\pi$
$152$	0	0
$153$	5.76133	0.465776
$154$	0	0
$155$	− 18.1027i	− 1.45405i
$156$	0	0
$157$	13.6800i	1.09178i	0.837855	+	0.545892i	$0.183809\pi$
$157$	13.6800i	1.09178i	−0.837855	+	0.545892i	$0.816191\pi$
$158$	0	0
$159$	30.9890	2.45759
$160$	0	0
$161$	−1.46340	−0.115332
$162$	0	0
$163$	9.58899i	0.751067i	0.926809	+	0.375534i	$0.122541\pi$
$163$	9.58899i	0.751067i	−0.926809	+	0.375534i	$0.877459\pi$
$164$	0	0
$165$	− 4.91627i	− 0.382731i
$166$	0	0
$167$	−4.20838	−0.325654	−0.162827	−	0.986655i	$-0.552061\pi$
$167$	−4.20838	−0.325654	−0.162827	+	0.986655i	$0.552061\pi$
$168$	0	0
$169$	10.3080	0.792920
$170$	0	0
$171$	− 26.8427i	− 2.05271i
$172$	0	0
$173$	− 2.87587i	− 0.218648i	−0.994006	−	0.109324i	$-0.965131\pi$
$173$	− 2.87587i	− 0.218648i	0.994006	−	0.109324i	$-0.0348687\pi$
$174$	0	0
$175$	10.6904	0.808117
$176$	0	0
$177$	−30.3854	−2.28391
$178$	0	0
$179$	18.1207i	1.35441i	0.735796	+	0.677203i	$0.236808\pi$
$179$	18.1207i	1.35441i	−0.735796	+	0.677203i	$0.763192\pi$
$180$	0	0
$181$	18.1054i	1.34576i	0.739750	+	0.672881i	$0.234944\pi$
$181$	18.1054i	1.34576i	−0.739750	+	0.672881i	$0.765056\pi$
$182$	0	0
$183$	−20.3149	−1.50172
$184$	0	0
$185$	34.5845	2.54270
$186$	0	0
$187$	− 0.711456i	− 0.0520268i
$188$	0	0
$189$	2.20331i	0.160267i
$190$	0	0
$191$	9.85508	0.713089	0.356544	−	0.934278i	$-0.383955\pi$
$191$	9.85508	0.713089	0.356544	+	0.934278i	$0.383955\pi$
$192$	0	0
$193$	−10.9837	−0.790622	−0.395311	−	0.918547i	$-0.629363\pi$
$193$	−10.9837	−0.790622	−0.395311	+	0.918547i	$0.629363\pi$
$194$	0	0
$195$	17.0004i	1.21742i
$196$	0	0
$197$	7.59087i	0.540827i	0.962744	+	0.270414i	$0.0871604\pi$
$197$	7.59087i	0.540827i	−0.962744	+	0.270414i	$0.912840\pi$
$198$	0	0
$199$	−9.18155	−0.650863	−0.325431	−	0.945566i	$-0.605510\pi$
$199$	−9.18155	−0.650863	−0.325431	+	0.945566i	$0.605510\pi$
$200$	0	0
$201$	−3.12060	−0.220110
$202$	0	0
$203$	− 6.47270i	− 0.454294i
$204$	0	0
$205$	29.7839i	2.08020i
$206$	0	0
$207$	−5.62285	−0.390815
$208$	0	0
$209$	−3.31475	−0.229286
$210$	0	0
$211$	13.7909i	0.949403i	0.880147	+	0.474702i	$0.157444\pi$
$211$	13.7909i	0.949403i	−0.880147	+	0.474702i	$0.842556\pi$
$212$	0	0
$213$	− 27.2359i	− 1.86617i
$214$	0	0
$215$	47.0303	3.20744
$216$	0	0
$217$	4.57012	0.310240
$218$	0	0
$219$	− 36.4779i	− 2.46495i
$220$	0	0
$221$	2.46020i	0.165491i
$222$	0	0
$223$	20.1870	1.35182	0.675912	−	0.736983i	$-0.263750\pi$
$223$	20.1870	1.35182	0.675912	+	0.736983i	$0.263750\pi$
$224$	0	0
$225$	41.0758	2.73838
$226$	0	0
$227$	− 11.7146i	− 0.777523i	−0.921339	−	0.388761i	$-0.872903\pi$
$227$	− 11.7146i	− 0.777523i	0.921339	−	0.388761i	$-0.127097\pi$
$228$	0	0
$229$	− 12.2430i	− 0.809040i	−0.914529	−	0.404520i	$-0.867439\pi$
$229$	− 12.2430i	− 0.809040i	0.914529	−	0.404520i	$-0.132561\pi$
$230$	0	0
$231$	1.24114	0.0816608
$232$	0	0
$233$	1.49453	0.0979098	0.0489549	−	0.998801i	$-0.484411\pi$
$233$	1.49453	0.0979098	0.0489549	+	0.998801i	$0.484411\pi$
$234$	0	0
$235$	− 21.1865i	− 1.38205i
$236$	0	0
$237$	5.59822i	0.363643i
$238$	0	0
$239$	−1.67314	−0.108226	−0.0541131	−	0.998535i	$-0.517233\pi$
$239$	−1.67314	−0.108226	−0.0541131	+	0.998535i	$0.517233\pi$
$240$	0	0
$241$	−10.3587	−0.667259	−0.333630	−	0.942704i	$-0.608273\pi$
$241$	−10.3587	−0.667259	−0.333630	+	0.942704i	$0.608273\pi$
$242$	0	0
$243$	− 21.6862i	− 1.39117i
$244$	0	0
$245$	3.96111i	0.253066i
$246$	0	0
$247$	11.4624	0.729332
$248$	0	0
$249$	−38.3344	−2.42935
$250$	0	0
$251$	− 0.142039i	− 0.00896543i	−0.999990	−	0.00448271i	$-0.998573\pi$
$251$	− 0.142039i	− 0.00896543i	0.999990	−	0.00448271i	$-0.00142690\pi$
$252$	0	0
$253$	0.694355i	0.0436537i
$254$	0	0
$255$	15.5363	0.972922
$256$	0	0
$257$	−9.18869	−0.573175	−0.286587	−	0.958054i	$-0.592521\pi$
$257$	−9.18869	−0.573175	−0.286587	+	0.958054i	$0.592521\pi$
$258$	0	0
$259$	8.73101i	0.542518i
$260$	0	0
$261$	− 24.8701i	− 1.53942i
$262$	0	0
$263$	16.6554	1.02702	0.513509	−	0.858084i	$-0.328345\pi$
$263$	16.6554	1.02702	0.513509	+	0.858084i	$0.328345\pi$
$264$	0	0
$265$	46.9271	2.88271
$266$	0	0
$267$	20.4680i	1.25262i
$268$	0	0
$269$	13.9696i	0.851740i	0.904784	+	0.425870i	$0.140032\pi$
$269$	13.9696i	0.851740i	−0.904784	+	0.425870i	$0.859968\pi$
$270$	0	0
$271$	−22.4746	−1.36524	−0.682618	−	0.730775i	$-0.739159\pi$
$271$	−22.4746	−1.36524	−0.682618	+	0.730775i	$0.739159\pi$
$272$	0	0
$273$	−4.29183	−0.259753
$274$	0	0
$275$	− 5.07237i	− 0.305875i
$276$	0	0
$277$	11.0507i	0.663970i	0.943285	+	0.331985i	$0.107718\pi$
$277$	11.0507i	0.663970i	−0.943285	+	0.331985i	$0.892282\pi$
$278$	0	0
$279$	17.5598	1.05128
$280$	0	0
$281$	−6.72334	−0.401081	−0.200540	−	0.979685i	$-0.564270\pi$
$281$	−6.72334	−0.401081	−0.200540	+	0.979685i	$0.564270\pi$
$282$	0	0
$283$	5.84512i	0.347456i	0.984794	+	0.173728i	$0.0555814\pi$
$283$	5.84512i	0.347456i	−0.984794	+	0.173728i	$0.944419\pi$
$284$	0	0
$285$	− 72.3854i	− 4.28774i
$286$	0	0
$287$	−7.51908	−0.443837
$288$	0	0
$289$	−14.7517	−0.867745
$290$	0	0
$291$	23.6356i	1.38554i
$292$	0	0
$293$	− 16.4187i	− 0.959190i	−0.877490	−	0.479595i	$-0.840784\pi$
$293$	− 16.4187i	− 0.959190i	0.877490	−	0.479595i	$-0.159216\pi$
$294$	0	0
$295$	−46.0129	−2.67898
$296$	0	0
$297$	1.04542	0.0606617
$298$	0	0
$299$	− 2.40107i	− 0.138857i
$300$	0	0
$301$	11.8730i	0.684350i
$302$	0	0
$303$	10.5053	0.603512
$304$	0	0
$305$	−30.7630	−1.76149
$306$	0	0
$307$	− 23.6812i	− 1.35156i	−0.737106	−	0.675778i	$-0.763808\pi$
$307$	− 23.6812i	− 1.35156i	0.737106	−	0.675778i	$-0.236192\pi$
$308$	0	0
$309$	− 2.63155i	− 0.149704i
$310$	0	0
$311$	−28.5347	−1.61805	−0.809026	−	0.587773i	$-0.800005\pi$
$311$	−28.5347	−1.61805	−0.809026	+	0.587773i	$0.800005\pi$
$312$	0	0
$313$	−7.93375	−0.448442	−0.224221	−	0.974538i	$-0.571984\pi$
$313$	−7.93375	−0.448442	−0.224221	+	0.974538i	$0.571984\pi$
$314$	0	0
$315$	15.2198i	0.857539i
$316$	0	0
$317$	3.29315i	0.184962i	0.995714	+	0.0924808i	$0.0294797\pi$
$317$	3.29315i	0.184962i	−0.995714	+	0.0924808i	$0.970520\pi$
$318$	0	0
$319$	−3.07117	−0.171952
$320$	0	0
$321$	−19.7783	−1.10391
$322$	0	0
$323$	− 10.4752i	− 0.582857i
$324$	0	0
$325$	17.5402i	0.972954i
$326$	0	0
$327$	48.6344	2.68949
$328$	0	0
$329$	5.34862	0.294879
$330$	0	0
$331$	27.6962i	1.52232i	0.648563	+	0.761161i	$0.275370\pi$
$331$	27.6962i	1.52232i	−0.648563	+	0.761161i	$0.724630\pi$
$332$	0	0
$333$	33.5473i	1.83838i
$334$	0	0
$335$	−4.72557	−0.258185
$336$	0	0
$337$	−4.40892	−0.240169	−0.120085	−	0.992764i	$-0.538317\pi$
$337$	−4.40892	−0.240169	−0.120085	+	0.992764i	$0.538317\pi$
$338$	0	0
$339$	49.9600i	2.71345i
$340$	0	0
$341$	− 2.16843i	− 0.117427i
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−15.1629	−0.816342
$346$	0	0
$347$	10.0694i	0.540551i	0.962783	+	0.270276i	$0.0871148\pi$
$347$	10.0694i	0.540551i	−0.962783	+	0.270276i	$0.912885\pi$
$348$	0	0
$349$	− 25.1406i	− 1.34575i	−0.739758	−	0.672873i	$-0.765060\pi$
$349$	− 25.1406i	− 1.34575i	0.739758	−	0.672873i	$-0.234940\pi$
$350$	0	0
$351$	−3.61506	−0.192958
$352$	0	0
$353$	15.5341	0.826796	0.413398	−	0.910550i	$-0.364342\pi$
$353$	15.5341	0.826796	0.413398	+	0.910550i	$0.364342\pi$
$354$	0	0
$355$	− 41.2437i	− 2.18899i
$356$	0	0
$357$	3.92222i	0.207586i
$358$	0	0
$359$	35.7682	1.88777	0.943886	−	0.330271i	$-0.107140\pi$
$359$	35.7682	1.88777	0.943886	+	0.330271i	$0.107140\pi$
$360$	0	0
$361$	−29.8052	−1.56869
$362$	0	0
$363$	28.1847i	1.47931i
$364$	0	0
$365$	− 55.2390i	− 2.89134i
$366$	0	0
$367$	17.0682	0.890951	0.445475	−	0.895294i	$-0.353035\pi$
$367$	17.0682	0.890951	0.445475	+	0.895294i	$0.353035\pi$
$368$	0	0
$369$	−28.8907	−1.50399
$370$	0	0
$371$	11.8470i	0.615063i
$372$	0	0
$373$	30.4199i	1.57508i	0.616262	+	0.787542i	$0.288646\pi$
$373$	30.4199i	1.57508i	−0.616262	+	0.787542i	$0.711354\pi$
$374$	0	0
$375$	58.9602	3.04469
$376$	0	0
$377$	10.6200	0.546960
$378$	0	0
$379$	− 6.18496i	− 0.317700i	−0.987303	−	0.158850i	$-0.949221\pi$
$379$	− 6.18496i	− 0.317700i	0.987303	−	0.158850i	$-0.0507787\pi$
$380$	0	0
$381$	− 1.02456i	− 0.0524896i
$382$	0	0
$383$	−27.5219	−1.40630	−0.703152	−	0.711039i	$-0.748225\pi$
$383$	−27.5219	−1.40630	−0.703152	+	0.711039i	$0.748225\pi$
$384$	0	0
$385$	1.87947	0.0957865
$386$	0	0
$387$	45.6199i	2.31899i
$388$	0	0
$389$	3.79558i	0.192444i	0.995360	+	0.0962219i	$0.0306758\pi$
$389$	3.79558i	0.192444i	−0.995360	+	0.0962219i	$0.969324\pi$
$390$	0	0
$391$	−2.19429	−0.110970
$392$	0	0
$393$	−0.656648	−0.0331235
$394$	0	0
$395$	8.47744i	0.426547i
$396$	0	0
$397$	1.69172i	0.0849049i	0.999098	+	0.0424524i	$0.0135171\pi$
$397$	1.69172i	0.0849049i	−0.999098	+	0.0424524i	$0.986483\pi$
$398$	0	0
$399$	18.2740	0.914846
$400$	0	0
$401$	−12.4593	−0.622190	−0.311095	−	0.950379i	$-0.600696\pi$
$401$	−12.4593	−0.622190	−0.311095	+	0.950379i	$0.600696\pi$
$402$	0	0
$403$	7.49840i	0.373522i
$404$	0	0
$405$	− 22.8301i	− 1.13444i
$406$	0	0
$407$	4.14269	0.205345
$408$	0	0
$409$	−32.3618	−1.60019	−0.800095	−	0.599873i	$-0.795218\pi$
$409$	−32.3618	−1.60019	−0.800095	+	0.599873i	$0.795218\pi$
$410$	0	0
$411$	7.15281i	0.352822i
$412$	0	0
$413$	− 11.6162i	− 0.571595i
$414$	0	0
$415$	−58.0503	−2.84958
$416$	0	0
$417$	−2.24528	−0.109952
$418$	0	0
$419$	− 19.7882i	− 0.966719i	−0.875422	−	0.483359i	$-0.839416\pi$
$419$	− 19.7882i	− 0.966719i	0.875422	−	0.483359i	$-0.160584\pi$
$420$	0	0
$421$	− 12.2485i	− 0.596957i	−0.954416	−	0.298478i	$-0.903521\pi$
$421$	− 12.2485i	− 0.596957i	0.954416	−	0.298478i	$-0.0964790\pi$
$422$	0	0
$423$	20.5511	0.999227
$424$	0	0
$425$	16.0296	0.777550
$426$	0	0
$427$	− 7.76627i	− 0.375836i
$428$	0	0
$429$	2.03639i	0.0983177i
$430$	0	0
$431$	−28.5603	−1.37570	−0.687850	−	0.725853i	$-0.741445\pi$
$431$	−28.5603	−1.37570	−0.687850	+	0.725853i	$0.741445\pi$
$432$	0	0
$433$	−23.6727	−1.13764	−0.568820	−	0.822462i	$-0.692600\pi$
$433$	−23.6727	−1.13764	−0.568820	+	0.822462i	$0.692600\pi$
$434$	0	0
$435$	− 67.0662i	− 3.21558i
$436$	0	0
$437$	10.2234i	0.489053i
$438$	0	0
$439$	−31.7217	−1.51399	−0.756996	−	0.653419i	$-0.773334\pi$
$439$	−31.7217	−1.51399	−0.756996	+	0.653419i	$0.773334\pi$
$440$	0	0
$441$	−3.84231	−0.182967
$442$	0	0
$443$	− 22.3547i	− 1.06210i	−0.847340	−	0.531051i	$-0.821797\pi$
$443$	− 22.3547i	− 1.06210i	0.847340	−	0.531051i	$-0.178203\pi$
$444$	0	0
$445$	30.9949i	1.46930i
$446$	0	0
$447$	38.3747	1.81506
$448$	0	0
$449$	16.8475	0.795083	0.397542	−	0.917584i	$-0.369863\pi$
$449$	16.8475	0.795083	0.397542	+	0.917584i	$0.369863\pi$
$450$	0	0
$451$	3.56765i	0.167994i
$452$	0	0
$453$	4.54871i	0.213717i
$454$	0	0
$455$	−6.49917	−0.304686
$456$	0	0
$457$	8.90256	0.416444	0.208222	−	0.978082i	$-0.433232\pi$
$457$	8.90256	0.416444	0.208222	+	0.978082i	$0.433232\pi$
$458$	0	0
$459$	3.30373i	0.154205i
$460$	0	0
$461$	28.8632i	1.34429i	0.740418	+	0.672147i	$0.234628\pi$
$461$	28.8632i	1.34429i	−0.740418	+	0.672147i	$0.765372\pi$
$462$	0	0
$463$	2.27185	0.105582	0.0527908	−	0.998606i	$-0.483188\pi$
$463$	2.27185	0.105582	0.0527908	+	0.998606i	$0.483188\pi$
$464$	0	0
$465$	47.3528	2.19593
$466$	0	0
$467$	− 24.6253i	− 1.13952i	−0.821811	−	0.569760i	$-0.807036\pi$
$467$	− 24.6253i	− 1.13952i	0.821811	−	0.569760i	$-0.192964\pi$
$468$	0	0
$469$	− 1.19299i	− 0.0550872i
$470$	0	0
$471$	−35.7839	−1.64884
$472$	0	0
$473$	5.63352	0.259029
$474$	0	0
$475$	− 74.6837i	− 3.42672i
$476$	0	0
$477$	45.5197i	2.08420i
$478$	0	0
$479$	−10.4872	−0.479172	−0.239586	−	0.970875i	$-0.577012\pi$
$479$	−10.4872	−0.479172	−0.239586	+	0.970875i	$0.577012\pi$
$480$	0	0
$481$	−14.3254	−0.653180
$482$	0	0
$483$	− 3.82794i	− 0.174177i
$484$	0	0
$485$	35.7916i	1.62521i
$486$	0	0
$487$	−7.85086	−0.355756	−0.177878	−	0.984053i	$-0.556923\pi$
$487$	−7.85086	−0.355756	−0.177878	+	0.984053i	$0.556923\pi$
$488$	0	0
$489$	−25.0827	−1.13428
$490$	0	0
$491$	1.78393i	0.0805077i	0.999189	+	0.0402538i	$0.0128167\pi$
$491$	1.78393i	0.0805077i	−0.999189	+	0.0402538i	$0.987183\pi$
$492$	0	0
$493$	− 9.70545i	− 0.437111i
$494$	0	0
$495$	7.22150	0.324582
$496$	0	0
$497$	10.4122	0.467049
$498$	0	0
$499$	25.4194i	1.13793i	0.822362	+	0.568965i	$0.192656\pi$
$499$	25.4194i	1.13793i	−0.822362	+	0.568965i	$0.807344\pi$
$500$	0	0
$501$	− 11.0082i	− 0.491810i
$502$	0	0
$503$	−9.35354	−0.417054	−0.208527	−	0.978017i	$-0.566867\pi$
$503$	−9.35354	−0.417054	−0.208527	+	0.978017i	$0.566867\pi$
$504$	0	0
$505$	15.9082	0.707908
$506$	0	0
$507$	26.9634i	1.19748i
$508$	0	0
$509$	− 29.8128i	− 1.32143i	−0.750637	−	0.660715i	$-0.770253\pi$
$509$	− 29.8128i	− 1.32143i	0.750637	−	0.660715i	$-0.229747\pi$
$510$	0	0
$511$	13.9453	0.616905
$512$	0	0
$513$	15.3924	0.679593
$514$	0	0
$515$	− 3.98499i	− 0.175599i
$516$	0	0
$517$	− 2.53781i	− 0.111613i
$518$	0	0
$519$	7.52265	0.330208
$520$	0	0
$521$	39.1964	1.71723	0.858614	−	0.512623i	$-0.171326\pi$
$521$	39.1964	1.71723	0.858614	+	0.512623i	$0.171326\pi$
$522$	0	0
$523$	− 8.74926i	− 0.382579i	−0.981534	−	0.191289i	$-0.938733\pi$
$523$	− 8.74926i	− 0.382579i	0.981534	−	0.191289i	$-0.0612669\pi$
$524$	0	0
$525$	27.9637i	1.22044i
$526$	0	0
$527$	6.85264	0.298506
$528$	0	0
$529$	−20.8585	−0.906889
$530$	0	0
$531$	− 44.6330i	− 1.93691i
$532$	0	0
$533$	− 12.3369i	− 0.534370i
$534$	0	0
$535$	−29.9504	−1.29487
$536$	0	0
$537$	−47.3998	−2.04545
$538$	0	0
$539$	0.474480i	0.0204373i
$540$	0	0
$541$	15.6773i	0.674019i	0.941501	+	0.337009i	$0.109415\pi$
$541$	15.6773i	0.674019i	−0.941501	+	0.337009i	$0.890585\pi$
$542$	0	0
$543$	−47.3597	−2.03240
$544$	0	0
$545$	73.6476	3.15472
$546$	0	0
$547$	28.5429i	1.22041i	0.792245	+	0.610203i	$0.208912\pi$
$547$	28.5429i	1.22041i	−0.792245	+	0.610203i	$0.791088\pi$
$548$	0	0
$549$	− 29.8404i	− 1.27356i
$550$	0	0
$551$	−45.2187	−1.92638
$552$	0	0
$553$	−2.14017	−0.0910093
$554$	0	0
$555$	90.4654i	3.84004i
$556$	0	0
$557$	9.96613i	0.422279i	0.977456	+	0.211139i	$0.0677174\pi$
$557$	9.96613i	0.422279i	−0.977456	+	0.211139i	$0.932283\pi$
$558$	0	0
$559$	−19.4806	−0.823942
$560$	0	0
$561$	1.86101	0.0785720
$562$	0	0
$563$	− 35.1144i	− 1.47990i	−0.672664	−	0.739948i	$-0.734850\pi$
$563$	− 35.1144i	− 1.47990i	0.672664	−	0.739948i	$-0.265150\pi$
$564$	0	0
$565$	75.6550i	3.18283i
$566$	0	0
$567$	5.76357	0.242047
$568$	0	0
$569$	−17.8594	−0.748703	−0.374352	−	0.927287i	$-0.622135\pi$
$569$	−17.8594	−0.748703	−0.374352	+	0.927287i	$0.622135\pi$
$570$	0	0
$571$	− 42.4376i	− 1.77596i	−0.459885	−	0.887978i	$-0.652110\pi$
$571$	− 42.4376i	− 1.77596i	0.459885	−	0.887978i	$-0.347890\pi$
$572$	0	0
$573$	25.7787i	1.07692i
$574$	0	0
$575$	−15.6443	−0.652413
$576$	0	0
$577$	4.88058	0.203181	0.101591	−	0.994826i	$-0.467607\pi$
$577$	4.88058	0.203181	0.101591	+	0.994826i	$0.467607\pi$
$578$	0	0
$579$	− 28.7309i	− 1.19401i
$580$	0	0
$581$	− 14.6551i	− 0.607994i
$582$	0	0
$583$	5.62114	0.232804
$584$	0	0
$585$	−24.9718	−1.03246
$586$	0	0
$587$	5.70462i	0.235455i	0.993046	+	0.117727i	$0.0375609\pi$
$587$	5.70462i	0.235455i	−0.993046	+	0.117727i	$0.962439\pi$
$588$	0	0
$589$	− 31.9272i	− 1.31554i
$590$	0	0
$591$	−19.8561	−0.816769
$592$	0	0
$593$	−2.29469	−0.0942318	−0.0471159	−	0.998889i	$-0.515003\pi$
$593$	−2.29469	−0.0942318	−0.0471159	+	0.998889i	$0.515003\pi$
$594$	0	0
$595$	5.93946i	0.243494i
$596$	0	0
$597$	− 24.0169i	− 0.982947i
$598$	0	0
$599$	−22.1319	−0.904283	−0.452142	−	0.891946i	$-0.649340\pi$
$599$	−22.1319	−0.904283	−0.452142	+	0.891946i	$0.649340\pi$
$600$	0	0
$601$	17.2799	0.704860	0.352430	−	0.935838i	$-0.385355\pi$
$601$	17.2799	0.704860	0.352430	+	0.935838i	$0.385355\pi$
$602$	0	0
$603$	− 4.58384i	− 0.186669i
$604$	0	0
$605$	42.6804i	1.73521i
$606$	0	0
$607$	19.1842	0.778664	0.389332	−	0.921097i	$-0.372706\pi$
$607$	19.1842	0.778664	0.389332	+	0.921097i	$0.372706\pi$
$608$	0	0
$609$	16.9312	0.686085
$610$	0	0
$611$	8.77572i	0.355028i
$612$	0	0
$613$	− 36.0658i	− 1.45668i	−0.685214	−	0.728341i	$-0.740291\pi$
$613$	− 36.0658i	− 1.45668i	0.685214	−	0.728341i	$-0.259709\pi$
$614$	0	0
$615$	−77.9081	−3.14156
$616$	0	0
$617$	−32.4364	−1.30584	−0.652919	−	0.757427i	$-0.726456\pi$
$617$	−32.4364	−1.30584	−0.652919	+	0.757427i	$0.726456\pi$
$618$	0	0
$619$	11.9239i	0.479261i	0.970864	+	0.239630i	$0.0770263\pi$
$619$	11.9239i	0.479261i	−0.970864	+	0.239630i	$0.922974\pi$
$620$	0	0
$621$	− 3.22432i	− 0.129388i
$622$	0	0
$623$	−7.82480	−0.313494
$624$	0	0
$625$	35.8323	1.43329
$626$	0	0
$627$	− 8.67067i	− 0.346273i
$628$	0	0
$629$	13.0916i	0.521998i
$630$	0	0
$631$	4.85753	0.193375	0.0966877	−	0.995315i	$-0.469175\pi$
$631$	4.85753	0.193375	0.0966877	+	0.995315i	$0.469175\pi$
$632$	0	0
$633$	−36.0739	−1.43381
$634$	0	0
$635$	− 1.55150i	− 0.0615692i
$636$	0	0
$637$	− 1.64074i	− 0.0650087i
$638$	0	0
$639$	40.0068	1.58264
$640$	0	0
$641$	0.0152206	0.000601179 0	0.000300590	−	1.00000i	$-0.499904\pi$
$641$	0.0152206	0.000601179 0	0.000300590		1.00000i	$0.499904\pi$
$642$	0	0
$643$	1.94065i	0.0765316i	0.999268	+	0.0382658i	$0.0121834\pi$
$643$	1.94065i	0.0765316i	−0.999268	+	0.0382658i	$0.987817\pi$
$644$	0	0
$645$	123.021i	4.84395i
$646$	0	0
$647$	18.0180	0.708362	0.354181	−	0.935177i	$-0.384760\pi$
$647$	18.0180	0.708362	0.354181	+	0.935177i	$0.384760\pi$
$648$	0	0
$649$	−5.51165	−0.216351
$650$	0	0
$651$	11.9544i	0.468531i
$652$	0	0
$653$	− 13.8033i	− 0.540166i	−0.962837	−	0.270083i	$-0.912949\pi$
$653$	− 13.8033i	− 0.540166i	0.962837	−	0.270083i	$-0.0870512\pi$
$654$	0	0
$655$	−0.994369	−0.0388532
$656$	0	0
$657$	53.5823	2.09044
$658$	0	0
$659$	7.62840i	0.297160i	0.988900	+	0.148580i	$0.0474703\pi$
$659$	7.62840i	0.297160i	−0.988900	+	0.148580i	$0.952530\pi$
$660$	0	0
$661$	43.7350i	1.70109i	0.525899	+	0.850547i	$0.323729\pi$
$661$	43.7350i	1.70109i	−0.525899	+	0.850547i	$0.676271\pi$
$662$	0	0
$663$	−6.43535	−0.249928
$664$	0	0
$665$	27.6726	1.07310
$666$	0	0
$667$	9.47216i	0.366764i
$668$	0	0
$669$	52.8048i	2.04155i
$670$	0	0
$671$	−3.68494	−0.142255
$672$	0	0
$673$	−26.3076	−1.01408	−0.507041	−	0.861922i	$-0.669261\pi$
$673$	−26.3076	−1.01408	−0.507041	+	0.861922i	$0.669261\pi$
$674$	0	0
$675$	23.5542i	0.906600i
$676$	0	0
$677$	44.7023i	1.71805i	0.511933	+	0.859025i	$0.328930\pi$
$677$	44.7023i	1.71805i	−0.511933	+	0.859025i	$0.671070\pi$
$678$	0	0
$679$	−9.03576	−0.346761
$680$	0	0
$681$	30.6427	1.17423
$682$	0	0
$683$	38.9563i	1.49062i	0.666718	+	0.745310i	$0.267699\pi$
$683$	38.9563i	1.49062i	−0.666718	+	0.745310i	$0.732301\pi$
$684$	0	0
$685$	10.8316i	0.413853i
$686$	0	0
$687$	32.0250	1.22183
$688$	0	0
$689$	−19.4378	−0.740522
$690$	0	0
$691$	49.5932i	1.88661i	0.331922	+	0.943307i	$0.392303\pi$
$691$	49.5932i	1.88661i	−0.331922	+	0.943307i	$0.607697\pi$
$692$	0	0
$693$	1.82310i	0.0692539i
$694$	0	0
$695$	−3.40006	−0.128972
$696$	0	0
$697$	−11.2744	−0.427050
$698$	0	0
$699$	3.90936i	0.147866i
$700$	0	0
$701$	15.8117i	0.597200i	0.954378	+	0.298600i	$0.0965197\pi$
$701$	15.8117i	0.597200i	−0.954378	+	0.298600i	$0.903480\pi$
$702$	0	0
$703$	60.9954	2.30049
$704$	0	0
$705$	55.4192	2.08721
$706$	0	0
$707$	4.01611i	0.151041i
$708$	0	0
$709$	− 3.79972i	− 0.142701i	−0.997451	−	0.0713507i	$-0.977269\pi$
$709$	− 3.79972i	− 0.142701i	0.997451	−	0.0713507i	$-0.0227309\pi$
$710$	0	0
$711$	−8.22320	−0.308394
$712$	0	0
$713$	−6.68792	−0.250465
$714$	0	0
$715$	3.08373i	0.115325i
$716$	0	0
$717$	− 4.37656i	− 0.163446i
$718$	0	0
$719$	−14.8712	−0.554603	−0.277301	−	0.960783i	$-0.589440\pi$
$719$	−14.8712	−0.554603	−0.277301	+	0.960783i	$0.589440\pi$
$720$	0	0
$721$	1.00603	0.0374664
$722$	0	0
$723$	− 27.0960i	− 1.00771i
$724$	0	0
$725$	− 69.1956i	− 2.56986i
$726$	0	0
$727$	−21.7844	−0.807937	−0.403969	−	0.914773i	$-0.632369\pi$
$727$	−21.7844	−0.807937	−0.403969	+	0.914773i	$0.632369\pi$
$728$	0	0
$729$	39.4355	1.46058
$730$	0	0
$731$	17.8029i	0.658465i
$732$	0	0
$733$	− 13.6416i	− 0.503863i	−0.967745	−	0.251931i	$-0.918934\pi$
$733$	− 13.6416i	− 0.503863i	0.967745	−	0.251931i	$-0.0810657\pi$
$734$	0	0
$735$	−10.3614	−0.382186
$736$	0	0
$737$	−0.566050	−0.0208507
$738$	0	0
$739$	− 17.5833i	− 0.646811i	−0.946260	−	0.323405i	$-0.895172\pi$
$739$	− 17.5833i	− 0.646811i	0.946260	−	0.323405i	$-0.104828\pi$
$740$	0	0
$741$	29.9830i	1.10145i
$742$	0	0
$743$	−28.6102	−1.04961	−0.524803	−	0.851223i	$-0.675861\pi$
$743$	−28.6102	−1.04961	−0.524803	+	0.851223i	$0.675861\pi$
$744$	0	0
$745$	58.1113	2.12903
$746$	0	0
$747$	− 56.3093i	− 2.06025i
$748$	0	0
$749$	− 7.56113i	− 0.276278i
$750$	0	0
$751$	−44.0524	−1.60749	−0.803747	−	0.594972i	$-0.797163\pi$
$751$	−44.0524	−1.60749	−0.803747	+	0.594972i	$0.797163\pi$
$752$	0	0
$753$	0.371543	0.0135398
$754$	0	0
$755$	6.88817i	0.250686i
$756$	0	0
$757$	− 6.18214i	− 0.224694i	−0.993669	−	0.112347i	$-0.964163\pi$
$757$	− 6.18214i	− 0.224694i	0.993669	−	0.112347i	$-0.0358368\pi$
$758$	0	0
$759$	−1.81628	−0.0659268
$760$	0	0
$761$	37.6107	1.36339	0.681694	−	0.731637i	$-0.261244\pi$
$761$	37.6107	1.36339	0.681694	+	0.731637i	$0.261244\pi$
$762$	0	0
$763$	18.5927i	0.673100i
$764$	0	0
$765$	22.8212i	0.825104i
$766$	0	0
$767$	19.0592	0.688187
$768$	0	0
$769$	23.1628	0.835271	0.417635	−	0.908615i	$-0.362859\pi$
$769$	23.1628	0.835271	0.417635	+	0.908615i	$0.362859\pi$
$770$	0	0
$771$	− 24.0356i	− 0.865621i
$772$	0	0
$773$	8.25369i	0.296865i	0.988923	+	0.148432i	$0.0474227\pi$
$773$	8.25369i	0.296865i	−0.988923	+	0.148432i	$0.952577\pi$
$774$	0	0
$775$	48.8563	1.75497
$776$	0	0
$777$	−22.8384	−0.819323
$778$	0	0
$779$	52.5288i	1.88204i
$780$	0	0
$781$	− 4.94036i	− 0.176780i
$782$	0	0
$783$	14.2613	0.509658
$784$	0	0
$785$	−54.1880	−1.93405
$786$	0	0
$787$	− 19.7287i	− 0.703251i	−0.936141	−	0.351626i	$-0.885629\pi$
$787$	− 19.7287i	− 0.703251i	0.936141	−	0.351626i	$-0.114371\pi$
$788$	0	0
$789$	43.5670i	1.55103i
$790$	0	0
$791$	−19.0995	−0.679098
$792$	0	0
$793$	12.7425	0.452498
$794$	0	0
$795$	122.751i	4.35353i
$796$	0	0
$797$	− 14.5934i	− 0.516924i	−0.966021	−	0.258462i	$-0.916784\pi$
$797$	− 14.5934i	− 0.516924i	0.966021	−	0.258462i	$-0.0832156\pi$
$798$	0	0
$799$	8.01995	0.283726
$800$	0	0
$801$	−30.0653	−1.06231
$802$	0	0
$803$	− 6.61678i	− 0.233501i
$804$	0	0
$805$	− 5.79669i	− 0.204307i
$806$	0	0
$807$	−36.5413	−1.28632
$808$	0	0
$809$	−27.2053	−0.956488	−0.478244	−	0.878227i	$-0.658727\pi$
$809$	−27.2053	−0.956488	−0.478244	+	0.878227i	$0.658727\pi$
$810$	0	0
$811$	− 21.3643i	− 0.750201i	−0.926984	−	0.375101i	$-0.877608\pi$
$811$	− 21.3643i	− 0.750201i	0.926984	−	0.375101i	$-0.122392\pi$
$812$	0	0
$813$	− 58.7887i	− 2.06181i
$814$	0	0
$815$	−37.9830	−1.33049
$816$	0	0
$817$	82.9458	2.90191
$818$	0	0
$819$	− 6.30425i	− 0.220288i
$820$	0	0
$821$	− 13.5442i	− 0.472695i	−0.971669	−	0.236348i	$-0.924050\pi$
$821$	− 13.5442i	− 0.472695i	0.971669	−	0.236348i	$-0.0759504\pi$
$822$	0	0
$823$	30.7225	1.07092	0.535460	−	0.844561i	$-0.320138\pi$
$823$	30.7225	1.07092	0.535460	+	0.844561i	$0.320138\pi$
$824$	0	0
$825$	13.2682	0.461940
$826$	0	0
$827$	− 18.5321i	− 0.644425i	−0.946667	−	0.322212i	$-0.895573\pi$
$827$	− 18.5321i	− 0.644425i	0.946667	−	0.322212i	$-0.104427\pi$
$828$	0	0
$829$	40.5916i	1.40980i	0.709305	+	0.704902i	$0.249009\pi$
$829$	40.5916i	1.40980i	−0.709305	+	0.704902i	$0.750991\pi$
$830$	0	0
$831$	−28.9061	−1.00274
$832$	0	0
$833$	−1.49944	−0.0519526
$834$	0	0
$835$	− 16.6699i	− 0.576884i
$836$	0	0
$837$	10.0694i	0.348048i
$838$	0	0
$839$	19.9144	0.687521	0.343760	−	0.939057i	$-0.388299\pi$
$839$	19.9144	0.687521	0.343760	+	0.939057i	$0.388299\pi$
$840$	0	0
$841$	−12.8958	−0.444684
$842$	0	0
$843$	− 17.5868i	− 0.605721i
$844$	0	0
$845$	40.8309i	1.40463i
$846$	0	0
$847$	−10.7749	−0.370229
$848$	0	0
$849$	−15.2896	−0.524736
$850$	0	0
$851$	− 12.7770i	− 0.437989i
$852$	0	0
$853$	18.0652i	0.618542i	0.950974	+	0.309271i	$0.100085\pi$
$853$	18.0652i	0.618542i	−0.950974	+	0.309271i	$0.899915\pi$
$854$	0	0
$855$	106.327	3.63630
$856$	0	0
$857$	−2.40661	−0.0822083	−0.0411041	−	0.999155i	$-0.513088\pi$
$857$	−2.40661	−0.0822083	−0.0411041	+	0.999155i	$0.513088\pi$
$858$	0	0
$859$	− 16.5152i	− 0.563491i	−0.959489	−	0.281745i	$-0.909087\pi$
$859$	− 16.5152i	− 0.563491i	0.959489	−	0.281745i	$-0.0909133\pi$
$860$	0	0
$861$	− 19.6683i	− 0.670293i
$862$	0	0
$863$	31.5905	1.07535	0.537676	−	0.843151i	$-0.319302\pi$
$863$	31.5905	1.07535	0.537676	+	0.843151i	$0.319302\pi$
$864$	0	0
$865$	11.3916	0.387327
$866$	0	0
$867$	− 38.5871i	− 1.31049i
$868$	0	0
$869$	1.01547i	0.0344474i
$870$	0	0
$871$	1.95739	0.0663237
$872$	0	0
$873$	−34.7182	−1.17503
$874$	0	0
$875$	22.5402i	0.761998i
$876$	0	0
$877$	23.3676i	0.789068i	0.918881	+	0.394534i	$0.129094\pi$
$877$	23.3676i	0.789068i	−0.918881	+	0.394534i	$0.870906\pi$
$878$	0	0
$879$	42.9477	1.44859
$880$	0	0
$881$	−21.6927	−0.730846	−0.365423	−	0.930842i	$-0.619076\pi$
$881$	−21.6927	−0.730846	−0.365423	+	0.930842i	$0.619076\pi$
$882$	0	0
$883$	31.3641i	1.05549i	0.849404	+	0.527743i	$0.176962\pi$
$883$	31.3641i	1.05549i	−0.849404	+	0.527743i	$0.823038\pi$
$884$	0	0
$885$	− 120.360i	− 4.04585i
$886$	0	0
$887$	−38.0341	−1.27706	−0.638531	−	0.769596i	$-0.720458\pi$
$887$	−38.0341	−1.27706	−0.638531	+	0.769596i	$0.720458\pi$
$888$	0	0
$889$	0.391682	0.0131366
$890$	0	0
$891$	− 2.73470i	− 0.0916159i
$892$	0	0
$893$	− 37.3658i	− 1.25040i
$894$	0	0
$895$	−71.7781	−2.39928
$896$	0	0
$897$	6.28067	0.209706
$898$	0	0
$899$	− 29.5810i	− 0.986582i
$900$	0	0
$901$	17.7638i	0.591799i
$902$	0	0
$903$	−31.0572	−1.03352
$904$	0	0
$905$	−71.7174	−2.38397
$906$	0	0
$907$	37.6510i	1.25018i	0.780552	+	0.625090i	$0.214938\pi$
$907$	37.6510i	1.25018i	−0.780552	+	0.625090i	$0.785062\pi$
$908$	0	0
$909$	15.4311i	0.511819i
$910$	0	0
$911$	32.1821	1.06624	0.533120	−	0.846040i	$-0.321019\pi$
$911$	32.1821	1.06624	0.533120	+	0.846040i	$0.321019\pi$
$912$	0	0
$913$	−6.95353	−0.230128
$914$	0	0
$915$	− 80.4693i	− 2.66023i
$916$	0	0
$917$	− 0.251033i	− 0.00828984i
$918$	0	0
$919$	−18.9243	−0.624256	−0.312128	−	0.950040i	$-0.601042\pi$
$919$	−18.9243	−0.624256	−0.312128	+	0.950040i	$0.601042\pi$
$920$	0	0
$921$	61.9447	2.04115
$922$	0	0
$923$	17.0837i	0.562316i
$924$	0	0
$925$	93.3377i	3.06893i
$926$	0	0
$927$	3.86547	0.126959
$928$	0	0
$929$	−21.3296	−0.699801	−0.349901	−	0.936787i	$-0.613785\pi$
$929$	−21.3296	−0.699801	−0.349901	+	0.936787i	$0.613785\pi$
$930$	0	0
$931$	6.98607i	0.228959i
$932$	0	0
$933$	− 74.6404i	− 2.44362i
$934$	0	0
$935$	2.81815	0.0921635
$936$	0	0
$937$	−32.3621	−1.05722	−0.528611	−	0.848864i	$-0.677287\pi$
$937$	−32.3621	−1.05722	−0.528611	+	0.848864i	$0.677287\pi$
$938$	0	0
$939$	− 20.7530i	− 0.677248i
$940$	0	0
$941$	− 9.86763i	− 0.321676i	−0.986981	−	0.160838i	$-0.948580\pi$
$941$	− 9.86763i	− 0.321676i	0.986981	−	0.160838i	$-0.0514196\pi$
$942$	0	0
$943$	11.0034	0.358321
$944$	0	0
$945$	−8.72753	−0.283907
$946$	0	0
$947$	− 36.4716i	− 1.18517i	−0.805509	−	0.592584i	$-0.798108\pi$
$947$	− 36.4716i	− 1.18517i	0.805509	−	0.592584i	$-0.201892\pi$
$948$	0	0
$949$	22.8807i	0.742740i
$950$	0	0
$951$	−8.61416	−0.279333
$952$	0	0
$953$	27.3528	0.886045	0.443023	−	0.896510i	$-0.353906\pi$
$953$	27.3528	0.886045	0.443023	+	0.896510i	$0.353906\pi$
$954$	0	0
$955$	39.0371i	1.26321i
$956$	0	0
$957$	− 8.03350i	− 0.259686i
$958$	0	0
$959$	−2.73448	−0.0883010
$960$	0	0
$961$	−10.1140	−0.326258
$962$	0	0
$963$	− 29.0522i	− 0.936195i
$964$	0	0
$965$	− 43.5075i	− 1.40056i
$966$	0	0
$967$	49.4333	1.58967	0.794834	−	0.606827i	$-0.207558\pi$
$967$	49.4333	1.58967	0.794834	+	0.606827i	$0.207558\pi$
$968$	0	0
$969$	27.4009	0.880243
$970$	0	0
$971$	− 52.9442i	− 1.69906i	−0.527539	−	0.849531i	$-0.676885\pi$
$971$	− 52.9442i	− 1.69906i	0.527539	−	0.849531i	$-0.323115\pi$
$972$	0	0
$973$	− 0.858361i	− 0.0275178i
$974$	0	0
$975$	−45.8813	−1.46938
$976$	0	0
$977$	−10.5254	−0.336738	−0.168369	−	0.985724i	$-0.553850\pi$
$977$	−10.5254	−0.336738	−0.168369	+	0.985724i	$0.553850\pi$
$978$	0	0
$979$	3.71271i	0.118659i
$980$	0	0
$981$	71.4389i	2.28087i
$982$	0	0
$983$	−41.2224	−1.31479	−0.657395	−	0.753546i	$-0.728342\pi$
$983$	−41.2224	−1.31479	−0.657395	+	0.753546i	$0.728342\pi$
$984$	0	0
$985$	−30.0683	−0.958054
$986$	0	0
$987$	13.9908i	0.445333i
$988$	0	0
$989$	− 17.3750i	− 0.552493i
$990$	0	0
$991$	−15.4822	−0.491808	−0.245904	−	0.969294i	$-0.579085\pi$
$991$	−15.4822	−0.491808	−0.245904	+	0.969294i	$0.579085\pi$
$992$	0	0
$993$	−72.4473	−2.29904
$994$	0	0
$995$	− 36.3691i	− 1.15298i
$996$	0	0
$997$	16.9510i	0.536845i	0.963301	+	0.268422i	$0.0865023\pi$
$997$	16.9510i	0.536845i	−0.963301	+	0.268422i	$0.913498\pi$
$998$	0	0
$999$	−19.2371	−0.608634

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3584.2.b.i.1793.11		12
4.3	odd	2		3584.2.b.k.1793.2		12
8.3	odd	2		3584.2.b.k.1793.11		12
8.5	even	2	inner	3584.2.b.i.1793.2		12
16.3	odd	4		3584.2.a.k.1.5	yes	6
16.5	even	4		3584.2.a.l.1.5	yes	6
16.11	odd	4		3584.2.a.e.1.2	✓	6
16.13	even	4		3584.2.a.f.1.2	yes	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3584.2.a.e.1.2	✓	6	16.11	odd	4
3584.2.a.f.1.2	yes	6	16.13	even	4
3584.2.a.k.1.5	yes	6	16.3	odd	4
3584.2.a.l.1.5	yes	6	16.5	even	4
3584.2.b.i.1793.2		12	8.5	even	2	inner
3584.2.b.i.1793.11		12	1.1	even	1	trivial
3584.2.b.k.1793.2		12	4.3	odd	2
3584.2.b.k.1793.11		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:	\( N \)	\(=\)	\( 3584 = 2^{9} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3584.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.61578i q^{3} +3.96111i q^{5} -1.00000 q^{7} -3.84231 q^{9} +O(q^{10})\)	\(q+2.61578i q^{3} +3.96111i q^{5} -1.00000 q^{7} -3.84231 q^{9} +0.474480i q^{11} -1.64074i q^{13} -10.3614 q^{15} -1.49944 q^{17} +6.98607i q^{19} -2.61578i q^{21} +1.46340 q^{23} -10.6904 q^{25} -2.20331i q^{27} +6.47270i q^{29} -4.57012 q^{31} -1.24114 q^{33} -3.96111i q^{35} -8.73101i q^{37} +4.29183 q^{39} +7.51908 q^{41} -11.8730i q^{43} -15.2198i q^{45} -5.34862 q^{47} +1.00000 q^{49} -3.92222i q^{51} -11.8470i q^{53} -1.87947 q^{55} -18.2740 q^{57} +11.6162i q^{59} +7.76627i q^{61} +3.84231 q^{63} +6.49917 q^{65} +1.19299i q^{67} +3.82794i q^{69} -10.4122 q^{71} -13.9453 q^{73} -27.9637i q^{75} -0.474480i q^{77} +2.14017 q^{79} -5.76357 q^{81} +14.6551i q^{83} -5.93946i q^{85} -16.9312 q^{87} +7.82480 q^{89} +1.64074i q^{91} -11.9544i q^{93} -27.6726 q^{95} +9.03576 q^{97} -1.82310i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 12 q^{7} - 20 q^{9}+O(q^{10}) \) 12 * q - 12 * q^7 - 20 * q^9	\( 12 q - 12 q^{7} - 20 q^{9} - 16 q^{15} - 4 q^{25} - 16 q^{31} + 8 q^{33} + 8 q^{41} + 16 q^{47} + 12 q^{49} - 32 q^{55} + 8 q^{57} + 20 q^{63} - 16 q^{65} + 16 q^{71} - 32 q^{73} - 48 q^{79} + 20 q^{81} + 16 q^{87} - 32 q^{89} - 32 q^{95} + 32 q^{97}+O(q^{100}) \) 12 * q - 12 * q^7 - 20 * q^9 - 16 * q^15 - 4 * q^25 - 16 * q^31 + 8 * q^33 + 8 * q^41 + 16 * q^47 + 12 * q^49 - 32 * q^55 + 8 * q^57 + 20 * q^63 - 16 * q^65 + 16 * q^71 - 32 * q^73 - 48 * q^79 + 20 * q^81 + 16 * q^87 - 32 * q^89 - 32 * q^95 + 32 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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