LMFDB - Embedded newform 2432.2.c.f.1217.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2432,2,Mod(1217,2432)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2432, 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("2432.1217");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2432 = 2^{7} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2432.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:	$19.4196177716$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.3182656.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{3} + 25x^{2} - 10x + 2 $ x^6 - 2x^3 + 25x^2 - 10*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1217.3
Root			$1.46962 + 1.46962i$ of defining polynomial
Character	$\chi$	$=$	2432.1217
Dual form			2432.2.c.f.1217.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.31955i q^{3} +0.319551i q^{5} +1.93923 q^{7} +1.25879 q^{9} +O(q^{10})$	$q-1.31955i q^{3} +0.319551i q^{5} +1.93923 q^{7} +1.25879 q^{9} -3.25879i q^{11} +4.25879i q^{13} +0.421664 q^{15} +3.00000 q^{17} +1.00000i q^{19} -2.55892i q^{21} -5.61968 q^{23} +4.89789 q^{25} -5.61968i q^{27} -6.55892i q^{29} +6.93923 q^{31} -4.30013 q^{33} +0.619684i q^{35} +7.57834i q^{37} +5.61968 q^{39} +4.30013 q^{41} +7.13726i q^{43} +0.402246i q^{45} +6.31955 q^{47} -3.23937 q^{49} -3.95865i q^{51} +5.31955i q^{53} +1.04135 q^{55} +1.31955 q^{57} -1.74121i q^{59} +2.19802i q^{61} +2.44108 q^{63} -1.36090 q^{65} -8.68045i q^{67} +7.41546i q^{69} +9.15667 q^{71} -15.9136 q^{73} -6.46301i q^{75} -6.31955i q^{77} +6.93923 q^{79} -3.63910 q^{81} -15.1178i q^{83} +0.958652i q^{85} -8.65483 q^{87} +5.69987 q^{89} +8.25879i q^{91} -9.15667i q^{93} -0.319551 q^{95} +17.1178 q^{97} -4.10211i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 6 q^{7} - 16 q^{9}+O(q^{10}) $ 6 * q - 6 * q^7 - 16 * q^9	$ 6 q - 6 q^{7} - 16 q^{9} + 32 q^{15} + 18 q^{17} - 22 q^{23} - 6 q^{25} + 24 q^{31} - 20 q^{33} + 22 q^{39} + 20 q^{41} + 32 q^{47} + 4 q^{49} + 24 q^{55} + 2 q^{57} + 44 q^{63} - 20 q^{65} - 4 q^{71} + 34 q^{73} + 24 q^{79} - 10 q^{81} + 54 q^{87} + 40 q^{89} + 4 q^{95} + 44 q^{97}+O(q^{100}) $ 6 * q - 6 * q^7 - 16 * q^9 + 32 * q^15 + 18 * q^17 - 22 * q^23 - 6 * q^25 + 24 * q^31 - 20 * q^33 + 22 * q^39 + 20 * q^41 + 32 * q^47 + 4 * q^49 + 24 * q^55 + 2 * q^57 + 44 * q^63 - 20 * q^65 - 4 * q^71 + 34 * q^73 + 24 * q^79 - 10 * q^81 + 54 * q^87 + 40 * q^89 + 4 * q^95 + 44 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2432\mathbb{Z}\right)^\times$.

$n$	$1407$	$1921$	$2053$
$\chi(n)$	$1$	$1$	$-1$

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
$3$	− 1.31955i	− 0.761843i	−0.924607	−	0.380922i	$-0.875607\pi$
$3$	− 1.31955i	− 0.761843i	0.924607	−	0.380922i	$-0.124393\pi$
$4$	0	0
$5$	0.319551i	0.142907i	0.997444	+	0.0714537i	$0.0227638\pi$
$5$	0.319551i	0.142907i	−0.997444	+	0.0714537i	$0.977236\pi$
$6$	0	0
$7$	1.93923	0.732962	0.366481	−	0.930426i	$-0.380562\pi$
$7$	1.93923	0.732962	0.366481	+	0.930426i	$0.380562\pi$
$8$	0	0
$9$	1.25879	0.419595
$10$	0	0
$11$	− 3.25879i	− 0.982561i	−0.871001	−	0.491280i	$-0.836529\pi$
$11$	− 3.25879i	− 0.982561i	0.871001	−	0.491280i	$-0.163471\pi$
$12$	0	0
$13$	4.25879i	1.18117i	0.806974	+	0.590587i	$0.201104\pi$
$13$	4.25879i	1.18117i	−0.806974	+	0.590587i	$0.798896\pi$
$14$	0	0
$15$	0.421664	0.108873
$16$	0	0
$17$	3.00000	0.727607	0.363803	−	0.931476i	$-0.381478\pi$
$17$	3.00000	0.727607	0.363803	+	0.931476i	$0.381478\pi$
$18$	0	0
$19$	1.00000i	0.229416i
$19$	1.00000i	0.229416i
$20$	0	0
$21$	− 2.55892i	− 0.558402i
$22$	0	0
$23$	−5.61968	−1.17179	−0.585893	−	0.810389i	$-0.699256\pi$
$23$	−5.61968	−1.17179	−0.585893	+	0.810389i	$0.699256\pi$
$24$	0	0
$25$	4.89789	0.979577
$26$	0	0
$27$	− 5.61968i	− 1.08151i
$28$	0	0
$29$	− 6.55892i	− 1.21796i	−0.793185	−	0.608980i	$-0.791579\pi$
$29$	− 6.55892i	− 1.21796i	0.793185	−	0.608980i	$-0.208421\pi$
$30$	0	0
$31$	6.93923	1.24632	0.623162	−	0.782093i	$-0.285848\pi$
$31$	6.93923	1.24632	0.623162	+	0.782093i	$0.285848\pi$
$32$	0	0
$33$	−4.30013	−0.748557
$34$	0	0
$35$	0.619684i	0.104746i
$36$	0	0
$37$	7.57834i	1.24587i	0.782273	+	0.622935i	$0.214060\pi$
$37$	7.57834i	1.24587i	−0.782273	+	0.622935i	$0.785940\pi$
$38$	0	0
$39$	5.61968	0.899870
$40$	0	0
$41$	4.30013	0.671568	0.335784	−	0.941939i	$-0.390999\pi$
$41$	4.30013	0.671568	0.335784	+	0.941939i	$0.390999\pi$
$42$	0	0
$43$	7.13726i	1.08842i	0.838949	+	0.544211i	$0.183171\pi$
$43$	7.13726i	1.08842i	−0.838949	+	0.544211i	$0.816829\pi$
$44$	0	0
$45$	0.402246i	0.0599633i
$46$	0	0
$47$	6.31955	0.921801	0.460901	−	0.887452i	$-0.347526\pi$
$47$	6.31955	0.921801	0.460901	+	0.887452i	$0.347526\pi$
$48$	0	0
$49$	−3.23937	−0.462767
$50$	0	0
$51$	− 3.95865i	− 0.554322i
$52$	0	0
$53$	5.31955i	0.730696i	0.930871	+	0.365348i	$0.119050\pi$
$53$	5.31955i	0.730696i	−0.930871	+	0.365348i	$0.880950\pi$
$54$	0	0
$55$	1.04135	0.140415
$56$	0	0
$57$	1.31955	0.174779
$58$	0	0
$59$	− 1.74121i	− 0.226687i	−0.993556	−	0.113343i	$-0.963844\pi$
$59$	− 1.74121i	− 0.226687i	0.993556	−	0.113343i	$-0.0361560\pi$
$60$	0	0
$61$	2.19802i	0.281428i	0.990050	+	0.140714i	$0.0449398\pi$
$61$	2.19802i	0.281428i	−0.990050	+	0.140714i	$0.955060\pi$
$62$	0	0
$63$	2.44108	0.307547
$64$	0	0
$65$	−1.36090	−0.168799
$66$	0	0
$67$	− 8.68045i	− 1.06049i	−0.847846	−	0.530243i	$-0.822101\pi$
$67$	− 8.68045i	− 1.06049i	0.847846	−	0.530243i	$-0.177899\pi$
$68$	0	0
$69$	7.41546i	0.892716i
$70$	0	0
$71$	9.15667	1.08670	0.543349	−	0.839507i	$-0.317156\pi$
$71$	9.15667	1.08670	0.543349	+	0.839507i	$0.317156\pi$
$72$	0	0
$73$	−15.9136	−1.86255	−0.931274	−	0.364320i	$-0.881301\pi$
$73$	−15.9136	−1.86255	−0.931274	+	0.364320i	$0.881301\pi$
$74$	0	0
$75$	− 6.46301i	− 0.746284i
$76$	0	0
$77$	− 6.31955i	− 0.720180i
$78$	0	0
$79$	6.93923	0.780725	0.390362	−	0.920661i	$-0.372350\pi$
$79$	6.93923	0.780725	0.390362	+	0.920661i	$0.372350\pi$
$80$	0	0
$81$	−3.63910	−0.404345
$82$	0	0
$83$	− 15.1178i	− 1.65940i	−0.558211	−	0.829699i	$-0.688512\pi$
$83$	− 15.1178i	− 1.65940i	0.558211	−	0.829699i	$-0.311488\pi$
$84$	0	0
$85$	0.958652i	0.103980i
$86$	0	0
$87$	−8.65483	−0.927895
$88$	0	0
$89$	5.69987	0.604185	0.302092	−	0.953279i	$-0.402315\pi$
$89$	5.69987	0.604185	0.302092	+	0.953279i	$0.402315\pi$
$90$	0	0
$91$	8.25879i	0.865756i
$92$	0	0
$93$	− 9.15667i	− 0.949503i
$94$	0	0
$95$	−0.319551	−0.0327852
$96$	0	0
$97$	17.1178	1.73805	0.869027	−	0.494765i	$-0.164746\pi$
$97$	17.1178	1.73805	0.869027	+	0.494765i	$0.164746\pi$
$98$	0	0
$99$	− 4.10211i	− 0.412278i
$100$	0	0
$101$	− 7.36090i	− 0.732437i	−0.930529	−	0.366218i	$-0.880652\pi$
$101$	− 7.36090i	− 0.732437i	0.930529	−	0.366218i	$-0.119348\pi$
$102$	0	0
$103$	−15.1178	−1.48960	−0.744802	−	0.667285i	$-0.767456\pi$
$103$	−15.1178	−1.48960	−0.744802	+	0.667285i	$0.767456\pi$
$104$	0	0
$105$	0.817705	0.0797998
$106$	0	0
$107$	− 12.1373i	− 1.17335i	−0.809821	−	0.586676i	$-0.800436\pi$
$107$	− 12.1373i	− 1.17335i	0.809821	−	0.586676i	$-0.199564\pi$
$108$	0	0
$109$	9.07649i	0.869370i	0.900583	+	0.434685i	$0.143140\pi$
$109$	9.07649i	0.869370i	−0.900583	+	0.434685i	$0.856860\pi$
$110$	0	0
$111$	10.0000	0.949158
$112$	0	0
$113$	−4.81770	−0.453212	−0.226606	−	0.973987i	$-0.572763\pi$
$113$	−4.81770	−0.453212	−0.226606	+	0.973987i	$0.572763\pi$
$114$	0	0
$115$	− 1.79577i	− 0.167457i
$116$	0	0
$117$	5.36090i	0.495615i
$118$	0	0
$119$	5.81770	0.533308
$120$	0	0
$121$	0.380316	0.0345742
$122$	0	0
$123$	− 5.67424i	− 0.511629i
$124$	0	0
$125$	3.16288i	0.282896i
$126$	0	0
$127$	18.7350	1.66246	0.831232	−	0.555926i	$-0.187636\pi$
$127$	18.7350	1.66246	0.831232	+	0.555926i	$0.187636\pi$
$128$	0	0
$129$	9.41797	0.829206
$130$	0	0
$131$	12.4155i	1.08474i	0.840139	+	0.542372i	$0.182474\pi$
$131$	12.4155i	1.08474i	−0.840139	+	0.542372i	$0.817526\pi$
$132$	0	0
$133$	1.93923i	0.168153i
$134$	0	0
$135$	1.79577	0.154556
$136$	0	0
$137$	−12.2782	−1.04900	−0.524499	−	0.851411i	$-0.675747\pi$
$137$	−12.2782	−1.04900	−0.524499	+	0.851411i	$0.675747\pi$
$138$	0	0
$139$	− 13.2588i	− 1.12460i	−0.826935	−	0.562298i	$-0.809917\pi$
$139$	− 13.2588i	− 1.12460i	0.826935	−	0.562298i	$-0.190083\pi$
$140$	0	0
$141$	− 8.33897i	− 0.702268i
$142$	0	0
$143$	13.8785	1.16058
$144$	0	0
$145$	2.09591	0.174056
$146$	0	0
$147$	4.27451i	0.352556i
$148$	0	0
$149$	7.55892i	0.619251i	0.950859	+	0.309625i	$0.100204\pi$
$149$	7.55892i	0.619251i	−0.950859	+	0.309625i	$0.899796\pi$
$150$	0	0
$151$	0.396041	0.0322294	0.0161147	−	0.999870i	$-0.494870\pi$
$151$	0.396041	0.0322294	0.0161147	+	0.999870i	$0.494870\pi$
$152$	0	0
$153$	3.77636	0.305300
$154$	0	0
$155$	2.21744i	0.178109i
$156$	0	0
$157$	− 11.0351i	− 0.880700i	−0.897826	−	0.440350i	$-0.854854\pi$
$157$	− 11.0351i	− 0.880700i	0.897826	−	0.440350i	$-0.145146\pi$
$158$	0	0
$159$	7.01942	0.556676
$160$	0	0
$161$	−10.8979	−0.858874
$162$	0	0
$163$	− 3.03514i	− 0.237731i	−0.992910	−	0.118865i	$-0.962074\pi$
$163$	− 3.03514i	− 0.237731i	0.992910	−	0.118865i	$-0.0379257\pi$
$164$	0	0
$165$	− 1.37411i	− 0.106974i
$166$	0	0
$167$	−9.97438	−0.771841	−0.385920	−	0.922532i	$-0.626116\pi$
$167$	−9.97438	−0.771841	−0.385920	+	0.922532i	$0.626116\pi$
$168$	0	0
$169$	−5.13726	−0.395173
$170$	0	0
$171$	1.25879i	0.0962617i
$172$	0	0
$173$	− 1.75694i	− 0.133578i	−0.997767	−	0.0667888i	$-0.978725\pi$
$173$	− 1.75694i	− 0.133578i	0.997767	−	0.0667888i	$-0.0212754\pi$
$174$	0	0
$175$	9.49815	0.717993
$176$	0	0
$177$	−2.29762	−0.172700
$178$	0	0
$179$	− 4.30013i	− 0.321407i	−0.987003	−	0.160704i	$-0.948624\pi$
$179$	− 4.30013i	− 0.321407i	0.987003	−	0.160704i	$-0.0513763\pi$
$180$	0	0
$181$	− 14.4349i	− 1.07294i	−0.843921	−	0.536468i	$-0.819758\pi$
$181$	− 14.4349i	− 1.07294i	0.843921	−	0.536468i	$-0.180242\pi$
$182$	0	0
$183$	2.90040	0.214404
$184$	0	0
$185$	−2.42166	−0.178044
$186$	0	0
$187$	− 9.77636i	− 0.714918i
$188$	0	0
$189$	− 10.8979i	− 0.792705i
$190$	0	0
$191$	23.7350	1.71741	0.858703	−	0.512474i	$-0.171271\pi$
$191$	23.7350	1.71741	0.858703	+	0.512474i	$0.171271\pi$
$192$	0	0
$193$	−20.6962	−1.48974	−0.744872	−	0.667208i	$-0.767489\pi$
$193$	−20.6962	−1.48974	−0.744872	+	0.667208i	$0.767489\pi$
$194$	0	0
$195$	1.79577i	0.128598i
$196$	0	0
$197$	− 15.7569i	− 1.12264i	−0.827600	−	0.561318i	$-0.810295\pi$
$197$	− 15.7569i	− 1.12264i	0.827600	−	0.561318i	$-0.189705\pi$
$198$	0	0
$199$	−9.85654	−0.698712	−0.349356	−	0.936990i	$-0.613600\pi$
$199$	−9.85654	−0.698712	−0.349356	+	0.936990i	$0.613600\pi$
$200$	0	0
$201$	−11.4543	−0.807924
$202$	0	0
$203$	− 12.7193i	− 0.892719i
$204$	0	0
$205$	1.37411i	0.0959721i
$206$	0	0
$207$	−7.07398	−0.491675
$208$	0	0
$209$	3.25879	0.225415
$210$	0	0
$211$	− 12.1373i	− 0.835563i	−0.908548	−	0.417782i	$-0.862808\pi$
$211$	− 12.1373i	− 0.835563i	0.908548	−	0.417782i	$-0.137192\pi$
$212$	0	0
$213$	− 12.0827i	− 0.827893i
$214$	0	0
$215$	−2.28072	−0.155544
$216$	0	0
$217$	13.4568	0.913508
$218$	0	0
$219$	20.9988i	1.41897i
$220$	0	0
$221$	12.7764i	0.859431i
$222$	0	0
$223$	−18.5746	−1.24385	−0.621925	−	0.783077i	$-0.713649\pi$
$223$	−18.5746	−1.24385	−0.621925	+	0.783077i	$0.713649\pi$
$224$	0	0
$225$	6.16539	0.411026
$226$	0	0
$227$	21.2939i	1.41333i	0.707549	+	0.706664i	$0.249801\pi$
$227$	21.2939i	1.41333i	−0.707549	+	0.706664i	$0.750199\pi$
$228$	0	0
$229$	− 11.4762i	− 0.758370i	−0.925321	−	0.379185i	$-0.876204\pi$
$229$	− 11.4762i	− 0.758370i	0.925321	−	0.379185i	$-0.123796\pi$
$230$	0	0
$231$	−8.33897	−0.548664
$232$	0	0
$233$	17.6548	1.15661	0.578303	−	0.815822i	$-0.303715\pi$
$233$	17.6548	1.15661	0.578303	+	0.815822i	$0.303715\pi$
$234$	0	0
$235$	2.01942i	0.131732i
$236$	0	0
$237$	− 9.15667i	− 0.594790i
$238$	0	0
$239$	11.5432	0.746667	0.373334	−	0.927697i	$-0.378215\pi$
$239$	11.5432	0.746667	0.373334	+	0.927697i	$0.378215\pi$
$240$	0	0
$241$	9.57834	0.616995	0.308497	−	0.951225i	$-0.400174\pi$
$241$	9.57834	0.616995	0.308497	+	0.951225i	$0.400174\pi$
$242$	0	0
$243$	− 12.0571i	− 0.773462i
$244$	0	0
$245$	− 1.03514i	− 0.0661328i
$246$	0	0
$247$	−4.25879	−0.270980
$248$	0	0
$249$	−19.9488	−1.26420
$250$	0	0
$251$	0.223643i	0.0141162i	0.999975	+	0.00705811i	$0.00224668\pi$
$251$	0.223643i	0.0141162i	−0.999975	+	0.00705811i	$0.997753\pi$
$252$	0	0
$253$	18.3133i	1.15135i
$254$	0	0
$255$	1.26499	0.0792168
$256$	0	0
$257$	−0.0826952	−0.00515838	−0.00257919	−	0.999997i	$-0.500821\pi$
$257$	−0.0826952	−0.00515838	−0.00257919	+	0.999997i	$0.500821\pi$
$258$	0	0
$259$	14.6962i	0.913176i
$260$	0	0
$261$	− 8.25627i	− 0.511050i
$262$	0	0
$263$	−1.04135	−0.0642122	−0.0321061	−	0.999484i	$-0.510221\pi$
$263$	−1.04135	−0.0642122	−0.0321061	+	0.999484i	$0.510221\pi$
$264$	0	0
$265$	−1.69987	−0.104422
$266$	0	0
$267$	− 7.52126i	− 0.460294i
$268$	0	0
$269$	13.8528i	0.844623i	0.906451	+	0.422312i	$0.138781\pi$
$269$	13.8528i	0.844623i	−0.906451	+	0.422312i	$0.861219\pi$
$270$	0	0
$271$	−12.2976	−0.747027	−0.373514	−	0.927625i	$-0.621847\pi$
$271$	−12.2976	−0.747027	−0.373514	+	0.927625i	$0.621847\pi$
$272$	0	0
$273$	10.8979	0.659570
$274$	0	0
$275$	− 15.9612i	− 0.962494i
$276$	0	0
$277$	− 11.6804i	− 0.701810i	−0.936411	−	0.350905i	$-0.885874\pi$
$277$	− 11.6804i	− 0.701810i	0.936411	−	0.350905i	$-0.114126\pi$
$278$	0	0
$279$	8.73501	0.522951
$280$	0	0
$281$	−1.23937	−0.0739345	−0.0369673	−	0.999316i	$-0.511770\pi$
$281$	−1.23937	−0.0739345	−0.0369673	+	0.999316i	$0.511770\pi$
$282$	0	0
$283$	14.6585i	0.871359i	0.900102	+	0.435679i	$0.143492\pi$
$283$	14.6585i	0.871359i	−0.900102	+	0.435679i	$0.856508\pi$
$284$	0	0
$285$	0.421664i	0.0249772i
$286$	0	0
$287$	8.33897	0.492234
$288$	0	0
$289$	−8.00000	−0.470588
$290$	0	0
$291$	− 22.5879i	− 1.32412i
$292$	0	0
$293$	20.0157i	1.16933i	0.811274	+	0.584666i	$0.198774\pi$
$293$	20.0157i	1.16933i	−0.811274	+	0.584666i	$0.801226\pi$
$294$	0	0
$295$	0.556406	0.0323952
$296$	0	0
$297$	−18.3133	−1.06265
$298$	0	0
$299$	− 23.9330i	− 1.38408i
$300$	0	0
$301$	13.8408i	0.797771i
$302$	0	0
$303$	−9.71308	−0.558002
$304$	0	0
$305$	−0.702379	−0.0402181
$306$	0	0
$307$	19.4180i	1.10824i	0.832436	+	0.554121i	$0.186946\pi$
$307$	19.4180i	1.10824i	−0.832436	+	0.554121i	$0.813054\pi$
$308$	0	0
$309$	19.9488i	1.13485i
$310$	0	0
$311$	−29.8565	−1.69301	−0.846505	−	0.532382i	$-0.821297\pi$
$311$	−29.8565	−1.69301	−0.846505	+	0.532382i	$0.821297\pi$
$312$	0	0
$313$	17.4543	0.986575	0.493288	−	0.869866i	$-0.335795\pi$
$313$	17.4543	0.986575	0.493288	+	0.869866i	$0.335795\pi$
$314$	0	0
$315$	0.780049i	0.0439508i
$316$	0	0
$317$	11.5370i	0.647982i	0.946060	+	0.323991i	$0.105025\pi$
$317$	11.5370i	0.647982i	−0.946060	+	0.323991i	$0.894975\pi$
$318$	0	0
$319$	−21.3741	−1.19672
$320$	0	0
$321$	−16.0157	−0.893911
$322$	0	0
$323$	3.00000i	0.166924i
$324$	0	0
$325$	20.8591i	1.15705i
$326$	0	0
$327$	11.9769	0.662324
$328$	0	0
$329$	12.2551	0.675645
$330$	0	0
$331$	11.3196i	0.622179i	0.950381	+	0.311089i	$0.100694\pi$
$331$	11.3196i	0.622179i	−0.950381	+	0.311089i	$0.899306\pi$
$332$	0	0
$333$	9.53950i	0.522761i
$334$	0	0
$335$	2.77384	0.151551
$336$	0	0
$337$	4.51757	0.246088	0.123044	−	0.992401i	$-0.460734\pi$
$337$	4.51757	0.246088	0.123044	+	0.992401i	$0.460734\pi$
$338$	0	0
$339$	6.35721i	0.345276i
$340$	0	0
$341$	− 22.6135i	− 1.22459i
$342$	0	0
$343$	−19.8565	−1.07215
$344$	0	0
$345$	−2.36962	−0.127576
$346$	0	0
$347$	17.6937i	0.949846i	0.880027	+	0.474923i	$0.157524\pi$
$347$	17.6937i	0.949846i	−0.880027	+	0.474923i	$0.842476\pi$
$348$	0	0
$349$	11.7193i	0.627319i	0.949536	+	0.313659i	$0.101555\pi$
$349$	11.7193i	0.627319i	−0.949536	+	0.313659i	$0.898445\pi$
$350$	0	0
$351$	23.9330	1.27745
$352$	0	0
$353$	−9.37662	−0.499067	−0.249534	−	0.968366i	$-0.580277\pi$
$353$	−9.37662	−0.499067	−0.249534	+	0.968366i	$0.580277\pi$
$354$	0	0
$355$	2.92602i	0.155297i
$356$	0	0
$357$	− 7.67676i	− 0.406297i
$358$	0	0
$359$	−28.4568	−1.50189	−0.750946	−	0.660363i	$-0.770402\pi$
$359$	−28.4568	−1.50189	−0.750946	+	0.660363i	$0.770402\pi$
$360$	0	0
$361$	−1.00000	−0.0526316
$362$	0	0
$363$	− 0.501846i	− 0.0263401i
$364$	0	0
$365$	− 5.08521i	− 0.266172i
$366$	0	0
$367$	−8.60027	−0.448930	−0.224465	−	0.974482i	$-0.572063\pi$
$367$	−8.60027	−0.448930	−0.224465	+	0.974482i	$0.572063\pi$
$368$	0	0
$369$	5.41295	0.281787
$370$	0	0
$371$	10.3159i	0.535573i
$372$	0	0
$373$	12.0157i	0.622151i	0.950385	+	0.311075i	$0.100689\pi$
$373$	12.0157i	0.622151i	−0.950385	+	0.311075i	$0.899311\pi$
$374$	0	0
$375$	4.17358	0.215523
$376$	0	0
$377$	27.9330	1.43862
$378$	0	0
$379$	23.5370i	1.20901i	0.796600	+	0.604507i	$0.206630\pi$
$379$	23.5370i	1.20901i	−0.796600	+	0.604507i	$0.793370\pi$
$380$	0	0
$381$	− 24.7218i	− 1.26654i
$382$	0	0
$383$	12.0571	0.616088	0.308044	−	0.951372i	$-0.400326\pi$
$383$	12.0571	0.616088	0.308044	+	0.951372i	$0.400326\pi$
$384$	0	0
$385$	2.01942	0.102919
$386$	0	0
$387$	8.98427i	0.456696i
$388$	0	0
$389$	17.3986i	0.882142i	0.897472	+	0.441071i	$0.145401\pi$
$389$	17.3986i	0.882142i	−0.897472	+	0.441071i	$0.854599\pi$
$390$	0	0
$391$	−16.8591	−0.852599
$392$	0	0
$393$	16.3828	0.826404
$394$	0	0
$395$	2.21744i	0.111571i
$396$	0	0
$397$	14.8371i	0.744654i	0.928102	+	0.372327i	$0.121440\pi$
$397$	14.8371i	0.744654i	−0.928102	+	0.372327i	$0.878560\pi$
$398$	0	0
$399$	2.55892	0.128106
$400$	0	0
$401$	−2.81770	−0.140709	−0.0703547	−	0.997522i	$-0.522413\pi$
$401$	−2.81770	−0.140709	−0.0703547	+	0.997522i	$0.522413\pi$
$402$	0	0
$403$	29.5527i	1.47213i
$404$	0	0
$405$	− 1.16288i	− 0.0577839i
$406$	0	0
$407$	24.6962	1.22414
$408$	0	0
$409$	−3.33528	−0.164919	−0.0824594	−	0.996594i	$-0.526277\pi$
$409$	−3.33528	−0.164919	−0.0824594	+	0.996594i	$0.526277\pi$
$410$	0	0
$411$	16.2017i	0.799172i
$412$	0	0
$413$	− 3.37662i	− 0.166153i
$414$	0	0
$415$	4.83092	0.237140
$416$	0	0
$417$	−17.4956	−0.856765
$418$	0	0
$419$	− 20.2357i	− 0.988577i	−0.869298	−	0.494289i	$-0.835429\pi$
$419$	− 20.2357i	− 0.988577i	0.869298	−	0.494289i	$-0.164571\pi$
$420$	0	0
$421$	− 5.83712i	− 0.284484i	−0.989832	−	0.142242i	$-0.954569\pi$
$421$	− 5.83712i	− 0.284484i	0.989832	−	0.142242i	$-0.0454311\pi$
$422$	0	0
$423$	7.95496	0.386783
$424$	0	0
$425$	14.6937	0.712747
$426$	0	0
$427$	4.26248i	0.206276i
$428$	0	0
$429$	− 18.3133i	− 0.884177i
$430$	0	0
$431$	13.1955	0.635605	0.317803	−	0.948157i	$-0.397055\pi$
$431$	13.1955	0.635605	0.317803	+	0.948157i	$0.397055\pi$
$432$	0	0
$433$	16.9781	0.815914	0.407957	−	0.913001i	$-0.366241\pi$
$433$	16.9781	0.815914	0.407957	+	0.913001i	$0.366241\pi$
$434$	0	0
$435$	− 2.76566i	− 0.132603i
$436$	0	0
$437$	− 5.61968i	− 0.268826i
$438$	0	0
$439$	6.38283	0.304636	0.152318	−	0.988332i	$-0.451326\pi$
$439$	6.38283	0.304636	0.152318	+	0.988332i	$0.451326\pi$
$440$	0	0
$441$	−4.07767	−0.194175
$442$	0	0
$443$	18.3766i	0.873100i	0.899680	+	0.436550i	$0.143800\pi$
$443$	18.3766i	0.873100i	−0.899680	+	0.436550i	$0.856200\pi$
$444$	0	0
$445$	1.82140i	0.0863425i
$446$	0	0
$447$	9.97438	0.471772
$448$	0	0
$449$	31.7569	1.49870	0.749351	−	0.662173i	$-0.230365\pi$
$449$	31.7569	1.49870	0.749351	+	0.662173i	$0.230365\pi$
$450$	0	0
$451$	− 14.0132i	− 0.659856i
$452$	0	0
$453$	− 0.522596i	− 0.0245537i
$454$	0	0
$455$	−2.63910	−0.123723
$456$	0	0
$457$	−30.9173	−1.44625	−0.723125	−	0.690717i	$-0.757295\pi$
$457$	−30.9173	−1.44625	−0.723125	+	0.690717i	$0.757295\pi$
$458$	0	0
$459$	− 16.8591i	− 0.786913i
$460$	0	0
$461$	24.2369i	1.12882i	0.825494	+	0.564411i	$0.190897\pi$
$461$	24.2369i	1.12882i	−0.825494	+	0.564411i	$0.809103\pi$
$462$	0	0
$463$	−20.5941	−0.957087	−0.478544	−	0.878064i	$-0.658835\pi$
$463$	−20.5941	−0.957087	−0.478544	+	0.878064i	$0.658835\pi$
$464$	0	0
$465$	2.92602	0.135691
$466$	0	0
$467$	1.85905i	0.0860267i	0.999074	+	0.0430133i	$0.0136958\pi$
$467$	1.85905i	0.0860267i	−0.999074	+	0.0430133i	$0.986304\pi$
$468$	0	0
$469$	− 16.8334i	− 0.777296i
$470$	0	0
$471$	−14.5614	−0.670955
$472$	0	0
$473$	23.2588	1.06944
$474$	0	0
$475$	4.89789i	0.224730i
$476$	0	0
$477$	6.69617i	0.306597i
$478$	0	0
$479$	−6.91361	−0.315891	−0.157946	−	0.987448i	$-0.550487\pi$
$479$	−6.91361	−0.315891	−0.157946	+	0.987448i	$0.550487\pi$
$480$	0	0
$481$	−32.2745	−1.47159
$482$	0	0
$483$	14.3803i	0.654327i
$484$	0	0
$485$	5.47002i	0.248381i
$486$	0	0
$487$	−39.5527	−1.79230	−0.896152	−	0.443747i	$-0.853649\pi$
$487$	−39.5527	−1.79230	−0.896152	+	0.443747i	$0.853649\pi$
$488$	0	0
$489$	−4.00502	−0.181113
$490$	0	0
$491$	− 11.7958i	− 0.532336i	−0.963927	−	0.266168i	$-0.914242\pi$
$491$	− 11.7958i	− 0.532336i	0.963927	−	0.266168i	$-0.0857576\pi$
$492$	0	0
$493$	− 19.6768i	− 0.886197i
$494$	0	0
$495$	1.31083	0.0589176
$496$	0	0
$497$	17.7569	0.796508
$498$	0	0
$499$	− 1.01573i	− 0.0454701i	−0.999742	−	0.0227351i	$-0.992763\pi$
$499$	− 1.01573i	− 0.0454701i	0.999742	−	0.0227351i	$-0.00723742\pi$
$500$	0	0
$501$	13.1617i	0.588021i
$502$	0	0
$503$	−43.0897	−1.92127	−0.960637	−	0.277805i	$-0.910393\pi$
$503$	−43.0897	−1.92127	−0.960637	+	0.277805i	$0.910393\pi$
$504$	0	0
$505$	2.35218	0.104671
$506$	0	0
$507$	6.77887i	0.301060i
$508$	0	0
$509$	35.1881i	1.55969i	0.625975	+	0.779843i	$0.284701\pi$
$509$	35.1881i	1.55969i	−0.625975	+	0.779843i	$0.715299\pi$
$510$	0	0
$511$	−30.8602	−1.36518
$512$	0	0
$513$	5.61968	0.248115
$514$	0	0
$515$	− 4.83092i	− 0.212876i
$516$	0	0
$517$	− 20.5941i	− 0.905726i
$518$	0	0
$519$	−2.31837	−0.101765
$520$	0	0
$521$	−14.2174	−0.622877	−0.311439	−	0.950266i	$-0.600811\pi$
$521$	−14.2174	−0.622877	−0.311439	+	0.950266i	$0.600811\pi$
$522$	0	0
$523$	− 4.38032i	− 0.191538i	−0.995404	−	0.0957689i	$-0.969469\pi$
$523$	− 4.38032i	− 0.191538i	0.995404	−	0.0957689i	$-0.0305310\pi$
$524$	0	0
$525$	− 12.5333i	− 0.546998i
$526$	0	0
$527$	20.8177	0.906833
$528$	0	0
$529$	8.58085	0.373080
$530$	0	0
$531$	− 2.19182i	− 0.0951167i
$532$	0	0
$533$	18.3133i	0.793239i
$534$	0	0
$535$	3.87847	0.167681
$536$	0	0
$537$	−5.67424	−0.244862
$538$	0	0
$539$	10.5564i	0.454697i
$540$	0	0
$541$	14.9198i	0.641453i	0.947172	+	0.320727i	$0.103927\pi$
$541$	14.9198i	0.641453i	−0.947172	+	0.320727i	$0.896073\pi$
$542$	0	0
$543$	−19.0476	−0.817409
$544$	0	0
$545$	−2.90040	−0.124239
$546$	0	0
$547$	46.0703i	1.96982i	0.173058	+	0.984912i	$0.444635\pi$
$547$	46.0703i	1.96982i	−0.173058	+	0.984912i	$0.555365\pi$
$548$	0	0
$549$	2.76684i	0.118086i
$550$	0	0
$551$	6.55892	0.279419
$552$	0	0
$553$	13.4568	0.572242
$554$	0	0
$555$	3.19551i	0.135642i
$556$	0	0
$557$	− 44.2295i	− 1.87406i	−0.349245	−	0.937031i	$-0.613562\pi$
$557$	− 44.2295i	− 1.87406i	0.349245	−	0.937031i	$-0.386438\pi$
$558$	0	0
$559$	−30.3960	−1.28562
$560$	0	0
$561$	−12.9004	−0.544655
$562$	0	0
$563$	2.61348i	0.110145i	0.998482	+	0.0550725i	$0.0175390\pi$
$563$	2.61348i	0.110145i	−0.998482	+	0.0550725i	$0.982461\pi$
$564$	0	0
$565$	− 1.53950i	− 0.0647673i
$566$	0	0
$567$	−7.05707	−0.296369
$568$	0	0
$569$	−24.0000	−1.00613	−0.503066	−	0.864248i	$-0.667795\pi$
$569$	−24.0000	−1.00613	−0.503066	+	0.864248i	$0.667795\pi$
$570$	0	0
$571$	33.4312i	1.39905i	0.714607	+	0.699526i	$0.246605\pi$
$571$	33.4312i	1.39905i	−0.714607	+	0.699526i	$0.753395\pi$
$572$	0	0
$573$	− 31.3196i	− 1.30839i
$574$	0	0
$575$	−27.5246	−1.14785
$576$	0	0
$577$	12.6742	0.527636	0.263818	−	0.964573i	$-0.415018\pi$
$577$	12.6742	0.527636	0.263818	+	0.964573i	$0.415018\pi$
$578$	0	0
$579$	27.3097i	1.13495i
$580$	0	0
$581$	− 29.3170i	− 1.21628i
$582$	0	0
$583$	17.3353	0.717954
$584$	0	0
$585$	−1.71308	−0.0708271
$586$	0	0
$587$	21.5721i	0.890377i	0.895437	+	0.445189i	$0.146863\pi$
$587$	21.5721i	0.890377i	−0.895437	+	0.445189i	$0.853137\pi$
$588$	0	0
$589$	6.93923i	0.285926i
$590$	0	0
$591$	−20.7921	−0.855272
$592$	0	0
$593$	−4.76063	−0.195496	−0.0977479	−	0.995211i	$-0.531164\pi$
$593$	−4.76063	−0.195496	−0.0977479	+	0.995211i	$0.531164\pi$
$594$	0	0
$595$	1.85905i	0.0762137i
$596$	0	0
$597$	13.0062i	0.532309i
$598$	0	0
$599$	40.9268	1.67222	0.836112	−	0.548558i	$-0.184823\pi$
$599$	40.9268	1.67222	0.836112	+	0.548558i	$0.184823\pi$
$600$	0	0
$601$	25.5139	1.04073	0.520366	−	0.853943i	$-0.325795\pi$
$601$	25.5139	1.04073	0.520366	+	0.853943i	$0.325795\pi$
$602$	0	0
$603$	− 10.9268i	− 0.444975i
$604$	0	0
$605$	0.121530i	0.00494091i
$606$	0	0
$607$	−33.1955	−1.34736	−0.673682	−	0.739021i	$-0.735288\pi$
$607$	−33.1955	−1.34736	−0.673682	+	0.739021i	$0.735288\pi$
$608$	0	0
$609$	−16.7837	−0.680112
$610$	0	0
$611$	26.9136i	1.08881i
$612$	0	0
$613$	− 39.3159i	− 1.58795i	−0.607949	−	0.793976i	$-0.708007\pi$
$613$	− 39.3159i	− 1.58795i	0.607949	−	0.793976i	$-0.291993\pi$
$614$	0	0
$615$	1.81321	0.0731157
$616$	0	0
$617$	−2.45932	−0.0990084	−0.0495042	−	0.998774i	$-0.515764\pi$
$617$	−2.45932	−0.0990084	−0.0495042	+	0.998774i	$0.515764\pi$
$618$	0	0
$619$	− 45.5139i	− 1.82936i	−0.404182	−	0.914679i	$-0.632444\pi$
$619$	− 45.5139i	− 1.82936i	0.404182	−	0.914679i	$-0.367556\pi$
$620$	0	0
$621$	31.5808i	1.26730i
$622$	0	0
$623$	11.0534	0.442844
$624$	0	0
$625$	23.4787	0.939149
$626$	0	0
$627$	− 4.30013i	− 0.171731i
$628$	0	0
$629$	22.7350i	0.906504i
$630$	0	0
$631$	−24.6329	−0.980620	−0.490310	−	0.871548i	$-0.663116\pi$
$631$	−24.6329	−0.980620	−0.490310	+	0.871548i	$0.663116\pi$
$632$	0	0
$633$	−16.0157	−0.636568
$634$	0	0
$635$	5.98679i	0.237578i
$636$	0	0
$637$	− 13.7958i	− 0.546608i
$638$	0	0
$639$	11.5263	0.455973
$640$	0	0
$641$	30.2357	1.19424	0.597119	−	0.802153i	$-0.296312\pi$
$641$	30.2357	1.19424	0.597119	+	0.802153i	$0.296312\pi$
$642$	0	0
$643$	23.4945i	0.926531i	0.886220	+	0.463266i	$0.153322\pi$
$643$	23.4945i	0.926531i	−0.886220	+	0.463266i	$0.846678\pi$
$644$	0	0
$645$	3.00952i	0.118500i
$646$	0	0
$647$	−31.0959	−1.22251	−0.611253	−	0.791435i	$-0.709334\pi$
$647$	−31.0959	−1.22251	−0.611253	+	0.791435i	$0.709334\pi$
$648$	0	0
$649$	−5.67424	−0.222734
$650$	0	0
$651$	− 17.7569i	− 0.695949i
$652$	0	0
$653$	21.9550i	0.859164i	0.903028	+	0.429582i	$0.141339\pi$
$653$	21.9550i	0.859164i	−0.903028	+	0.429582i	$0.858661\pi$
$654$	0	0
$655$	−3.96737	−0.155018
$656$	0	0
$657$	−20.0318	−0.781516
$658$	0	0
$659$	− 32.2976i	− 1.25814i	−0.777350	−	0.629068i	$-0.783437\pi$
$659$	− 32.2976i	− 1.25814i	0.777350	−	0.629068i	$-0.216563\pi$
$660$	0	0
$661$	19.5808i	0.761607i	0.924656	+	0.380803i	$0.124353\pi$
$661$	19.5808i	0.761607i	−0.924656	+	0.380803i	$0.875647\pi$
$662$	0	0
$663$	16.8591	0.654751
$664$	0	0
$665$	−0.619684	−0.0240303
$666$	0	0
$667$	36.8591i	1.42719i
$668$	0	0
$669$	24.5102i	0.947619i
$670$	0	0
$671$	7.16288	0.276520
$672$	0	0
$673$	40.8309	1.57392	0.786958	−	0.617006i	$-0.211655\pi$
$673$	40.8309	1.57392	0.786958	+	0.617006i	$0.211655\pi$
$674$	0	0
$675$	− 27.5246i	− 1.05942i
$676$	0	0
$677$	− 4.43739i	− 0.170543i	−0.996358	−	0.0852714i	$-0.972824\pi$
$677$	− 4.43739i	− 0.170543i	0.996358	−	0.0852714i	$-0.0271757\pi$
$678$	0	0
$679$	33.1955	1.27393
$680$	0	0
$681$	28.0984	1.07673
$682$	0	0
$683$	13.8785i	0.531045i	0.964105	+	0.265522i	$0.0855444\pi$
$683$	13.8785i	0.531045i	−0.964105	+	0.265522i	$0.914456\pi$
$684$	0	0
$685$	− 3.92351i	− 0.149910i
$686$	0	0
$687$	−15.1435	−0.577759
$688$	0	0
$689$	−22.6548	−0.863080
$690$	0	0
$691$	36.4543i	1.38679i	0.720559	+	0.693393i	$0.243885\pi$
$691$	36.4543i	1.38679i	−0.720559	+	0.693393i	$0.756115\pi$
$692$	0	0
$693$	− 7.95496i	− 0.302184i
$694$	0	0
$695$	4.23686	0.160713
$696$	0	0
$697$	12.9004	0.488637
$698$	0	0
$699$	− 23.2964i	− 0.881152i
$700$	0	0
$701$	3.11784i	0.117759i	0.998265	+	0.0588796i	$0.0187528\pi$
$701$	3.11784i	0.117759i	−0.998265	+	0.0588796i	$0.981247\pi$
$702$	0	0
$703$	−7.57834	−0.285822
$704$	0	0
$705$	2.66472	0.100359
$706$	0	0
$707$	− 14.2745i	− 0.536848i
$708$	0	0
$709$	24.3133i	0.913107i	0.889696	+	0.456553i	$0.150916\pi$
$709$	24.3133i	0.913107i	−0.889696	+	0.456553i	$0.849084\pi$
$710$	0	0
$711$	8.73501	0.327588
$712$	0	0
$713$	−38.9963	−1.46042
$714$	0	0
$715$	4.43488i	0.165855i
$716$	0	0
$717$	− 15.2318i	− 0.568843i
$718$	0	0
$719$	18.0608	0.673553	0.336776	−	0.941585i	$-0.390663\pi$
$719$	18.0608	0.673553	0.336776	+	0.941585i	$0.390663\pi$
$720$	0	0
$721$	−29.3170	−1.09182
$722$	0	0
$723$	− 12.6391i	− 0.470053i
$724$	0	0
$725$	− 32.1248i	− 1.19309i
$726$	0	0
$727$	−27.4531	−1.01818	−0.509090	−	0.860713i	$-0.670018\pi$
$727$	−27.4531	−1.01818	−0.509090	+	0.860713i	$0.670018\pi$
$728$	0	0
$729$	−26.8272	−0.993601
$730$	0	0
$731$	21.4118i	0.791943i
$732$	0	0
$733$	− 22.6003i	− 0.834760i	−0.908732	−	0.417380i	$-0.862948\pi$
$733$	− 22.6003i	− 0.834760i	0.908732	−	0.417380i	$-0.137052\pi$
$734$	0	0
$735$	−1.36592	−0.0503828
$736$	0	0
$737$	−28.2877	−1.04199
$738$	0	0
$739$	− 6.13356i	− 0.225627i	−0.993616	−	0.112813i	$-0.964014\pi$
$739$	− 6.13356i	− 0.225627i	0.993616	−	0.112813i	$-0.0359862\pi$
$740$	0	0
$741$	5.61968i	0.206444i
$742$	0	0
$743$	0.817705	0.0299987	0.0149993	−	0.999888i	$-0.495225\pi$
$743$	0.817705	0.0299987	0.0149993	+	0.999888i	$0.495225\pi$
$744$	0	0
$745$	−2.41546	−0.0884956
$746$	0	0
$747$	− 19.0301i	− 0.696276i
$748$	0	0
$749$	− 23.5370i	− 0.860023i
$750$	0	0
$751$	−28.6003	−1.04364	−0.521819	−	0.853056i	$-0.674746\pi$
$751$	−28.6003	−1.04364	−0.521819	+	0.853056i	$0.674746\pi$
$752$	0	0
$753$	0.295108	0.0107543
$754$	0	0
$755$	0.126555i	0.00460582i
$756$	0	0
$757$	22.9901i	0.835589i	0.908541	+	0.417795i	$0.137197\pi$
$757$	22.9901i	0.835589i	−0.908541	+	0.417795i	$0.862803\pi$
$758$	0	0
$759$	24.1654	0.877148
$760$	0	0
$761$	−46.6669	−1.69167	−0.845836	−	0.533443i	$-0.820898\pi$
$761$	−46.6669	−1.69167	−0.845836	+	0.533443i	$0.820898\pi$
$762$	0	0
$763$	17.6014i	0.637215i
$764$	0	0
$765$	1.20674i	0.0436297i
$766$	0	0
$767$	7.41546	0.267757
$768$	0	0
$769$	−0.674244	−0.0243139	−0.0121569	−	0.999926i	$-0.503870\pi$
$769$	−0.674244	−0.0243139	−0.0121569	+	0.999926i	$0.503870\pi$
$770$	0	0
$771$	0.109120i	0.00392988i
$772$	0	0
$773$	51.5684i	1.85479i	0.374086	+	0.927394i	$0.377956\pi$
$773$	51.5684i	1.85479i	−0.374086	+	0.927394i	$0.622044\pi$
$774$	0	0
$775$	33.9876	1.22087
$776$	0	0
$777$	19.3923	0.695697
$778$	0	0
$779$	4.30013i	0.154068i
$780$	0	0
$781$	− 29.8396i	− 1.06775i
$782$	0	0
$783$	−36.8591	−1.31724
$784$	0	0
$785$	3.52629	0.125859
$786$	0	0
$787$	− 40.2902i	− 1.43619i	−0.695944	−	0.718096i	$-0.745014\pi$
$787$	− 40.2902i	− 1.43619i	0.695944	−	0.718096i	$-0.254986\pi$
$788$	0	0
$789$	1.37411i	0.0489196i
$790$	0	0
$791$	−9.34266	−0.332187
$792$	0	0
$793$	−9.36090	−0.332415
$794$	0	0
$795$	2.24306i	0.0795532i
$796$	0	0
$797$	16.0728i	0.569328i	0.958627	+	0.284664i	$0.0918820\pi$
$797$	16.0728i	0.569328i	−0.958627	+	0.284664i	$0.908118\pi$
$798$	0	0
$799$	18.9587	0.670709
$800$	0	0
$801$	7.17491	0.253513
$802$	0	0
$803$	51.8591i	1.83007i
$804$	0	0
$805$	− 3.48243i	− 0.122739i
$806$	0	0
$807$	18.2795	0.643470
$808$	0	0
$809$	−43.7094	−1.53674	−0.768370	−	0.640006i	$-0.778932\pi$
$809$	−43.7094	−1.53674	−0.768370	+	0.640006i	$0.778932\pi$
$810$	0	0
$811$	6.30265i	0.221316i	0.993859	+	0.110658i	$0.0352958\pi$
$811$	6.30265i	0.221316i	−0.993859	+	0.110658i	$0.964704\pi$
$812$	0	0
$813$	16.2273i	0.569117i
$814$	0	0
$815$	0.969882	0.0339735
$816$	0	0
$817$	−7.13726	−0.249701
$818$	0	0
$819$	10.3960i	0.363267i
$820$	0	0
$821$	− 49.1505i	− 1.71536i	−0.514181	−	0.857682i	$-0.671904\pi$
$821$	− 49.1505i	− 1.71536i	0.514181	−	0.857682i	$-0.328096\pi$
$822$	0	0
$823$	44.9231	1.56592	0.782961	−	0.622071i	$-0.213708\pi$
$823$	44.9231	1.56592	0.782961	+	0.622071i	$0.213708\pi$
$824$	0	0
$825$	−21.0616	−0.733270
$826$	0	0
$827$	− 44.3547i	− 1.54236i	−0.636615	−	0.771182i	$-0.719666\pi$
$827$	− 44.3547i	− 1.54236i	0.636615	−	0.771182i	$-0.280334\pi$
$828$	0	0
$829$	− 33.4857i	− 1.16301i	−0.813544	−	0.581504i	$-0.802464\pi$
$829$	− 33.4857i	− 1.16301i	0.813544	−	0.581504i	$-0.197536\pi$
$830$	0	0
$831$	−15.4129	−0.534669
$832$	0	0
$833$	−9.71810	−0.336712
$834$	0	0
$835$	− 3.18732i	− 0.110302i
$836$	0	0
$837$	− 38.9963i	− 1.34791i
$838$	0	0
$839$	−16.9136	−0.583923	−0.291961	−	0.956430i	$-0.594308\pi$
$839$	−16.9136	−0.583923	−0.291961	+	0.956430i	$0.594308\pi$
$840$	0	0
$841$	−14.0194	−0.483428
$842$	0	0
$843$	1.63541i	0.0563265i
$844$	0	0
$845$	− 1.64161i	− 0.0564732i
$846$	0	0
$847$	0.737522	0.0253416
$848$	0	0
$849$	19.3427	0.663838
$850$	0	0
$851$	− 42.5879i	− 1.45989i
$852$	0	0
$853$	44.1530i	1.51177i	0.654705	+	0.755885i	$0.272793\pi$
$853$	44.1530i	1.51177i	−0.654705	+	0.755885i	$0.727207\pi$
$854$	0	0
$855$	−0.402246	−0.0137565
$856$	0	0
$857$	46.3887	1.58461	0.792303	−	0.610128i	$-0.208882\pi$
$857$	46.3887	1.58461	0.792303	+	0.610128i	$0.208882\pi$
$858$	0	0
$859$	34.3378i	1.17159i	0.810459	+	0.585795i	$0.199218\pi$
$859$	34.3378i	1.17159i	−0.810459	+	0.585795i	$0.800782\pi$
$860$	0	0
$861$	− 11.0037i	− 0.375005i
$862$	0	0
$863$	−24.0132	−0.817419	−0.408710	−	0.912664i	$-0.634021\pi$
$863$	−24.0132	−0.817419	−0.408710	+	0.912664i	$0.634021\pi$
$864$	0	0
$865$	0.561431	0.0190892
$866$	0	0
$867$	10.5564i	0.358514i
$868$	0	0
$869$	− 22.6135i	− 0.767110i
$870$	0	0
$871$	36.9682	1.25262
$872$	0	0
$873$	21.5477	0.729279
$874$	0	0
$875$	6.13356i	0.207352i
$876$	0	0
$877$	− 44.7325i	− 1.51051i	−0.655432	−	0.755255i	$-0.727513\pi$
$877$	− 44.7325i	− 1.51051i	0.655432	−	0.755255i	$-0.272487\pi$
$878$	0	0
$879$	26.4118	0.890847
$880$	0	0
$881$	−24.6073	−0.829040	−0.414520	−	0.910040i	$-0.636051\pi$
$881$	−24.6073	−0.829040	−0.414520	+	0.910040i	$0.636051\pi$
$882$	0	0
$883$	27.9806i	0.941622i	0.882234	+	0.470811i	$0.156039\pi$
$883$	27.9806i	0.941622i	−0.882234	+	0.470811i	$0.843961\pi$
$884$	0	0
$885$	− 0.734207i	− 0.0246801i
$886$	0	0
$887$	−46.3060	−1.55480	−0.777401	−	0.629005i	$-0.783462\pi$
$887$	−46.3060	−1.55480	−0.777401	+	0.629005i	$0.783462\pi$
$888$	0	0
$889$	36.3316	1.21852
$890$	0	0
$891$	11.8591i	0.397293i
$892$	0	0
$893$	6.31955i	0.211476i
$894$	0	0
$895$	1.37411	0.0459315
$896$	0	0
$897$	−31.5808	−1.05445
$898$	0	0
$899$	− 45.5139i	− 1.51797i
$900$	0	0
$901$	15.9587i	0.531660i
$902$	0	0
$903$	18.2637	0.607776
$904$	0	0
$905$	4.61268	0.153331
$906$	0	0
$907$	− 6.99380i	− 0.232225i	−0.993236	−	0.116113i	$-0.962957\pi$
$907$	− 6.99380i	− 0.232225i	0.993236	−	0.116113i	$-0.0370433\pi$
$908$	0	0
$909$	− 9.26579i	− 0.307327i
$910$	0	0
$911$	−2.19182	−0.0726181	−0.0363090	−	0.999341i	$-0.511560\pi$
$911$	−2.19182	−0.0726181	−0.0363090	+	0.999341i	$0.511560\pi$
$912$	0	0
$913$	−49.2658	−1.63046
$914$	0	0
$915$	0.926825i	0.0306399i
$916$	0	0
$917$	24.0765i	0.795076i
$918$	0	0
$919$	−29.3716	−0.968880	−0.484440	−	0.874825i	$-0.660977\pi$
$919$	−29.3716	−0.968880	−0.484440	+	0.874825i	$0.660977\pi$
$920$	0	0
$921$	25.6230	0.844307
$922$	0	0
$923$	38.9963i	1.28358i
$924$	0	0
$925$	37.1178i	1.22043i
$926$	0	0
$927$	−19.0301	−0.625031
$928$	0	0
$929$	21.7288	0.712899	0.356449	−	0.934315i	$-0.383987\pi$
$929$	21.7288	0.712899	0.356449	+	0.934315i	$0.383987\pi$
$930$	0	0
$931$	− 3.23937i	− 0.106166i
$932$	0	0
$933$	39.3972i	1.28981i
$934$	0	0
$935$	3.12404	0.102167
$936$	0	0
$937$	0.917305	0.0299670	0.0149835	−	0.999888i	$-0.495230\pi$
$937$	0.917305	0.0299670	0.0149835	+	0.999888i	$0.495230\pi$
$938$	0	0
$939$	− 23.0318i	− 0.751615i
$940$	0	0
$941$	− 19.3716i	− 0.631496i	−0.948843	−	0.315748i	$-0.897745\pi$
$941$	− 19.3716i	− 0.631496i	0.948843	−	0.315748i	$-0.102255\pi$
$942$	0	0
$943$	−24.1654	−0.786933
$944$	0	0
$945$	3.48243	0.113283
$946$	0	0
$947$	− 46.7532i	− 1.51928i	−0.650346	−	0.759638i	$-0.725376\pi$
$947$	− 46.7532i	− 1.51928i	0.650346	−	0.759638i	$-0.274624\pi$
$948$	0	0
$949$	− 67.7727i	− 2.19999i
$950$	0	0
$951$	15.2236	0.493660
$952$	0	0
$953$	13.7313	0.444801	0.222400	−	0.974955i	$-0.428611\pi$
$953$	13.7313	0.444801	0.222400	+	0.974955i	$0.428611\pi$
$954$	0	0
$955$	7.58454i	0.245430i
$956$	0	0
$957$	28.2042i	0.911713i
$958$	0	0
$959$	−23.8103	−0.768875
$960$	0	0
$961$	17.1530	0.553322
$962$	0	0
$963$	− 15.2782i	− 0.492333i
$964$	0	0
$965$	− 6.61348i	− 0.212895i
$966$	0	0
$967$	8.47371	0.272496	0.136248	−	0.990675i	$-0.456496\pi$
$967$	8.47371	0.272496	0.136248	+	0.990675i	$0.456496\pi$
$968$	0	0
$969$	3.95865	0.127170
$970$	0	0
$971$	− 59.1054i	− 1.89678i	−0.317101	−	0.948392i	$-0.602709\pi$
$971$	− 59.1054i	− 1.89678i	0.317101	−	0.948392i	$-0.397291\pi$
$972$	0	0
$973$	− 25.7119i	− 0.824286i
$974$	0	0
$975$	27.5246	0.881492
$976$	0	0
$977$	−38.9268	−1.24538	−0.622690	−	0.782469i	$-0.713960\pi$
$977$	−38.9268	−1.24538	−0.622690	+	0.782469i	$0.713960\pi$
$978$	0	0
$979$	− 18.5746i	− 0.593648i
$980$	0	0
$981$	11.4254i	0.364784i
$982$	0	0
$983$	51.5089	1.64288	0.821439	−	0.570297i	$-0.193172\pi$
$983$	51.5089	1.64288	0.821439	+	0.570297i	$0.193172\pi$
$984$	0	0
$985$	5.03514	0.160433
$986$	0	0
$987$	− 16.1712i	− 0.514736i
$988$	0	0
$989$	− 40.1091i	− 1.27540i
$990$	0	0
$991$	−28.9707	−0.920284	−0.460142	−	0.887845i	$-0.652202\pi$
$991$	−28.9707	−0.920284	−0.460142	+	0.887845i	$0.652202\pi$
$992$	0	0
$993$	14.9367	0.474003
$994$	0	0
$995$	− 3.14967i	− 0.0998511i
$996$	0	0
$997$	27.9111i	0.883953i	0.897027	+	0.441977i	$0.145723\pi$
$997$	27.9111i	0.883953i	−0.897027	+	0.441977i	$0.854277\pi$
$998$	0	0
$999$	42.5879	1.34742

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2432.2.c.f.1217.3	✓	6
4.3	odd	2		2432.2.c.g.1217.4	yes	6
8.3	odd	2		2432.2.c.g.1217.3	yes	6
8.5	even	2	inner	2432.2.c.f.1217.4	yes	6
16.3	odd	4		4864.2.a.bc.1.2		3
16.5	even	4		4864.2.a.bd.1.2		3
16.11	odd	4		4864.2.a.bf.1.2		3
16.13	even	4		4864.2.a.be.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2432.2.c.f.1217.3	✓	6	1.1	even	1	trivial
2432.2.c.f.1217.4	yes	6	8.5	even	2	inner
2432.2.c.g.1217.3	yes	6	8.3	odd	2
2432.2.c.g.1217.4	yes	6	4.3	odd	2
4864.2.a.bc.1.2		3	16.3	odd	4
4864.2.a.bd.1.2		3	16.5	even	4
4864.2.a.be.1.2		3	16.13	even	4
4864.2.a.bf.1.2		3	16.11	odd	4

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:	\( N \)	\(=\)	\( 2432 = 2^{7} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2432.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.31955i q^{3} +0.319551i q^{5} +1.93923 q^{7} +1.25879 q^{9} +O(q^{10})\)	\(q-1.31955i q^{3} +0.319551i q^{5} +1.93923 q^{7} +1.25879 q^{9} -3.25879i q^{11} +4.25879i q^{13} +0.421664 q^{15} +3.00000 q^{17} +1.00000i q^{19} -2.55892i q^{21} -5.61968 q^{23} +4.89789 q^{25} -5.61968i q^{27} -6.55892i q^{29} +6.93923 q^{31} -4.30013 q^{33} +0.619684i q^{35} +7.57834i q^{37} +5.61968 q^{39} +4.30013 q^{41} +7.13726i q^{43} +0.402246i q^{45} +6.31955 q^{47} -3.23937 q^{49} -3.95865i q^{51} +5.31955i q^{53} +1.04135 q^{55} +1.31955 q^{57} -1.74121i q^{59} +2.19802i q^{61} +2.44108 q^{63} -1.36090 q^{65} -8.68045i q^{67} +7.41546i q^{69} +9.15667 q^{71} -15.9136 q^{73} -6.46301i q^{75} -6.31955i q^{77} +6.93923 q^{79} -3.63910 q^{81} -15.1178i q^{83} +0.958652i q^{85} -8.65483 q^{87} +5.69987 q^{89} +8.25879i q^{91} -9.15667i q^{93} -0.319551 q^{95} +17.1178 q^{97} -4.10211i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 6 q^{7} - 16 q^{9}+O(q^{10}) \) 6 * q - 6 * q^7 - 16 * q^9	\( 6 q - 6 q^{7} - 16 q^{9} + 32 q^{15} + 18 q^{17} - 22 q^{23} - 6 q^{25} + 24 q^{31} - 20 q^{33} + 22 q^{39} + 20 q^{41} + 32 q^{47} + 4 q^{49} + 24 q^{55} + 2 q^{57} + 44 q^{63} - 20 q^{65} - 4 q^{71} + 34 q^{73} + 24 q^{79} - 10 q^{81} + 54 q^{87} + 40 q^{89} + 4 q^{95} + 44 q^{97}+O(q^{100}) \) 6 * q - 6 * q^7 - 16 * q^9 + 32 * q^15 + 18 * q^17 - 22 * q^23 - 6 * q^25 + 24 * q^31 - 20 * q^33 + 22 * q^39 + 20 * q^41 + 32 * q^47 + 4 * q^49 + 24 * q^55 + 2 * q^57 + 44 * q^63 - 20 * q^65 - 4 * q^71 + 34 * q^73 + 24 * q^79 - 10 * q^81 + 54 * q^87 + 40 * q^89 + 4 * q^95 + 44 * q^97

Introduction

L-functions

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