LMFDB - Embedded newform 3192.2.a.z.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3192,2,Mod(1,3192)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3192, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("3192.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3192 = 2^{3} \cdot 3 \cdot 7 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3192.a (trivial)

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:	yes
Analytic conductor:	$25.4882483252$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.2225.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 5x^{2} + 2x + 4 $ x^4 - x^3 - 5x^2 + 2x + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$1.13856$ of defining polynomial
Character	$\chi$	$=$	3192.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +3.23607 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +3.23607 q^{5} +1.00000 q^{7} +1.00000 q^{9} -3.68447 q^{11} -0.277129 q^{13} +3.23607 q^{15} +2.17127 q^{17} -1.00000 q^{19} +1.00000 q^{21} +2.27713 q^{23} +5.47214 q^{25} +1.00000 q^{27} -1.51320 q^{29} -1.40734 q^{31} -3.68447 q^{33} +3.23607 q^{35} +7.96160 q^{37} -0.277129 q^{39} +9.02639 q^{41} +4.55426 q^{43} +3.23607 q^{45} +2.44840 q^{47} +1.00000 q^{49} +2.17127 q^{51} +5.51320 q^{53} -11.9232 q^{55} -1.00000 q^{57} +11.3689 q^{59} +2.55426 q^{61} +1.00000 q^{63} -0.896807 q^{65} -3.68447 q^{67} +2.27713 q^{69} -12.4575 q^{71} +2.34255 q^{73} +5.47214 q^{75} -3.68447 q^{77} +12.1182 q^{79} +1.00000 q^{81} +1.38361 q^{83} +7.02639 q^{85} -1.51320 q^{87} +2.34255 q^{89} -0.277129 q^{91} -1.40734 q^{93} -3.23607 q^{95} +9.73194 q^{97} -3.68447 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{3} + 4 q^{5} + 4 q^{7} + 4 q^{9}+O(q^{10}) $ 4 * q + 4 * q^3 + 4 * q^5 + 4 * q^7 + 4 * q^9	$ 4 q + 4 q^{3} + 4 q^{5} + 4 q^{7} + 4 q^{9} + 4 q^{11} + 6 q^{13} + 4 q^{15} + 6 q^{17} - 4 q^{19} + 4 q^{21} + 2 q^{23} + 4 q^{25} + 4 q^{27} + 10 q^{29} + 6 q^{31} + 4 q^{33} + 4 q^{35} + 6 q^{37} + 6 q^{39} + 4 q^{41} + 4 q^{43} + 4 q^{45} + 4 q^{49} + 6 q^{51} + 6 q^{53} + 4 q^{55} - 4 q^{57} + 8 q^{59} - 4 q^{61} + 4 q^{63} + 16 q^{65} + 4 q^{67} + 2 q^{69} + 2 q^{71} + 4 q^{73} + 4 q^{75} + 4 q^{77} - 14 q^{79} + 4 q^{81} + 2 q^{83} - 4 q^{85} + 10 q^{87} + 4 q^{89} + 6 q^{91} + 6 q^{93} - 4 q^{95} + 4 q^{97} + 4 q^{99}+O(q^{100}) $ 4 * q + 4 * q^3 + 4 * q^5 + 4 * q^7 + 4 * q^9 + 4 * q^11 + 6 * q^13 + 4 * q^15 + 6 * q^17 - 4 * q^19 + 4 * q^21 + 2 * q^23 + 4 * q^25 + 4 * q^27 + 10 * q^29 + 6 * q^31 + 4 * q^33 + 4 * q^35 + 6 * q^37 + 6 * q^39 + 4 * q^41 + 4 * q^43 + 4 * q^45 + 4 * q^49 + 6 * q^51 + 6 * q^53 + 4 * q^55 - 4 * q^57 + 8 * q^59 - 4 * q^61 + 4 * q^63 + 16 * q^65 + 4 * q^67 + 2 * q^69 + 2 * q^71 + 4 * q^73 + 4 * q^75 + 4 * q^77 - 14 * q^79 + 4 * q^81 + 2 * q^83 - 4 * q^85 + 10 * q^87 + 4 * q^89 + 6 * q^91 + 6 * q^93 - 4 * q^95 + 4 * q^97 + 4 * q^99

Display 100 coefficients 10 coefficients

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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	3.23607	1.44721	0.723607	−	0.690212i	$-0.242483\pi$
$5$	3.23607	1.44721	0.723607	+	0.690212i	$0.242483\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−3.68447	−1.11091	−0.555455	−	0.831547i	$-0.687456\pi$
$11$	−3.68447	−1.11091	−0.555455	+	0.831547i	$0.687456\pi$
$12$	0	0
$13$	−0.277129	−0.0768616	−0.0384308	−	0.999261i	$-0.512236\pi$
$13$	−0.277129	−0.0768616	−0.0384308	+	0.999261i	$0.512236\pi$
$14$	0	0
$15$	3.23607	0.835549
$16$	0	0
$17$	2.17127	0.526612	0.263306	−	0.964712i	$-0.415187\pi$
$17$	2.17127	0.526612	0.263306	+	0.964712i	$0.415187\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	1.00000	0.218218
$22$	0	0
$23$	2.27713	0.474814	0.237407	−	0.971410i	$-0.423702\pi$
$23$	2.27713	0.474814	0.237407	+	0.971410i	$0.423702\pi$
$24$	0	0
$25$	5.47214	1.09443
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−1.51320	−0.280994	−0.140497	−	0.990081i	$-0.544870\pi$
$29$	−1.51320	−0.280994	−0.140497	+	0.990081i	$0.544870\pi$
$30$	0	0
$31$	−1.40734	−0.252766	−0.126383	−	0.991982i	$-0.540337\pi$
$31$	−1.40734	−0.252766	−0.126383	+	0.991982i	$0.540337\pi$
$32$	0	0
$33$	−3.68447	−0.641384
$34$	0	0
$35$	3.23607	0.546995
$36$	0	0
$37$	7.96160	1.30888	0.654439	−	0.756114i	$-0.272905\pi$
$37$	7.96160	1.30888	0.654439	+	0.756114i	$0.272905\pi$
$38$	0	0
$39$	−0.277129	−0.0443761
$40$	0	0
$41$	9.02639	1.40969	0.704843	−	0.709363i	$-0.251017\pi$
$41$	9.02639	1.40969	0.704843	+	0.709363i	$0.251017\pi$
$42$	0	0
$43$	4.55426	0.694518	0.347259	−	0.937769i	$-0.387113\pi$
$43$	4.55426	0.694518	0.347259	+	0.937769i	$0.387113\pi$
$44$	0	0
$45$	3.23607	0.482405
$46$	0	0
$47$	2.44840	0.357136	0.178568	−	0.983928i	$-0.442853\pi$
$47$	2.44840	0.357136	0.178568	+	0.983928i	$0.442853\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	2.17127	0.304039
$52$	0	0
$53$	5.51320	0.757296	0.378648	−	0.925541i	$-0.376389\pi$
$53$	5.51320	0.757296	0.378648	+	0.925541i	$0.376389\pi$
$54$	0	0
$55$	−11.9232	−1.60772
$56$	0	0
$57$	−1.00000	−0.132453
$58$	0	0
$59$	11.3689	1.48011	0.740055	−	0.672546i	$-0.234799\pi$
$59$	11.3689	1.48011	0.740055	+	0.672546i	$0.234799\pi$
$60$	0	0
$61$	2.55426	0.327039	0.163520	−	0.986540i	$-0.447715\pi$
$61$	2.55426	0.327039	0.163520	+	0.986540i	$0.447715\pi$
$62$	0	0
$63$	1.00000	0.125988
$64$	0	0
$65$	−0.896807	−0.111235
$66$	0	0
$67$	−3.68447	−0.450130	−0.225065	−	0.974344i	$-0.572259\pi$
$67$	−3.68447	−0.450130	−0.225065	+	0.974344i	$0.572259\pi$
$68$	0	0
$69$	2.27713	0.274134
$70$	0	0
$71$	−12.4575	−1.47843	−0.739215	−	0.673470i	$-0.764803\pi$
$71$	−12.4575	−1.47843	−0.739215	+	0.673470i	$0.764803\pi$
$72$	0	0
$73$	2.34255	0.274175	0.137087	−	0.990559i	$-0.456226\pi$
$73$	2.34255	0.274175	0.137087	+	0.990559i	$0.456226\pi$
$74$	0	0
$75$	5.47214	0.631868
$76$	0	0
$77$	−3.68447	−0.419884
$78$	0	0
$79$	12.1182	1.36340	0.681702	−	0.731630i	$-0.261240\pi$
$79$	12.1182	1.36340	0.681702	+	0.731630i	$0.261240\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	1.38361	0.151871	0.0759355	−	0.997113i	$-0.475806\pi$
$83$	1.38361	0.151871	0.0759355	+	0.997113i	$0.475806\pi$
$84$	0	0
$85$	7.02639	0.762119
$86$	0	0
$87$	−1.51320	−0.162232
$88$	0	0
$89$	2.34255	0.248310	0.124155	−	0.992263i	$-0.460378\pi$
$89$	2.34255	0.248310	0.124155	+	0.992263i	$0.460378\pi$
$90$	0	0
$91$	−0.277129	−0.0290510
$92$	0	0
$93$	−1.40734	−0.145935
$94$	0	0
$95$	−3.23607	−0.332014
$96$	0	0
$97$	9.73194	0.988128	0.494064	−	0.869425i	$-0.335511\pi$
$97$	9.73194	0.988128	0.494064	+	0.869425i	$0.335511\pi$
$98$	0	0
$99$	−3.68447	−0.370303
$100$	0	0
$101$	−1.70820	−0.169973	−0.0849863	−	0.996382i	$-0.527085\pi$
$101$	−1.70820	−0.169973	−0.0849863	+	0.996382i	$0.527085\pi$
$102$	0	0
$103$	−9.96160	−0.981546	−0.490773	−	0.871288i	$-0.663285\pi$
$103$	−9.96160	−0.981546	−0.490773	+	0.871288i	$0.663285\pi$
$104$	0	0
$105$	3.23607	0.315808
$106$	0	0
$107$	1.04106	0.100643	0.0503216	−	0.998733i	$-0.483975\pi$
$107$	1.04106	0.100643	0.0503216	+	0.998733i	$0.483975\pi$
$108$	0	0
$109$	3.74989	0.359175	0.179587	−	0.983742i	$-0.442524\pi$
$109$	3.74989	0.359175	0.179587	+	0.983742i	$0.442524\pi$
$110$	0	0
$111$	7.96160	0.755682
$112$	0	0
$113$	−10.4575	−0.983756	−0.491878	−	0.870664i	$-0.663689\pi$
$113$	−10.4575	−0.983756	−0.491878	+	0.870664i	$0.663689\pi$
$114$	0	0
$115$	7.36894	0.687157
$116$	0	0
$117$	−0.277129	−0.0256205
$118$	0	0
$119$	2.17127	0.199040
$120$	0	0
$121$	2.57533	0.234121
$122$	0	0
$123$	9.02639	0.813882
$124$	0	0
$125$	1.52786	0.136656
$126$	0	0
$127$	−18.0361	−1.60044	−0.800222	−	0.599704i	$-0.795285\pi$
$127$	−18.0361	−1.60044	−0.800222	+	0.599704i	$0.795285\pi$
$128$	0	0
$129$	4.55426	0.400980
$130$	0	0
$131$	5.98533	0.522941	0.261470	−	0.965211i	$-0.415793\pi$
$131$	5.98533	0.522941	0.261470	+	0.965211i	$0.415793\pi$
$132$	0	0
$133$	−1.00000	−0.0867110
$134$	0	0
$135$	3.23607	0.278516
$136$	0	0
$137$	−16.1836	−1.38266	−0.691330	−	0.722539i	$-0.742975\pi$
$137$	−16.1836	−1.38266	−0.691330	+	0.722539i	$0.742975\pi$
$138$	0	0
$139$	−8.60172	−0.729589	−0.364794	−	0.931088i	$-0.618861\pi$
$139$	−8.60172	−0.729589	−0.364794	+	0.931088i	$0.618861\pi$
$140$	0	0
$141$	2.44840	0.206193
$142$	0	0
$143$	1.02107	0.0853863
$144$	0	0
$145$	−4.89681	−0.406658
$146$	0	0
$147$	1.00000	0.0824786
$148$	0	0
$149$	1.56626	0.128313	0.0641567	−	0.997940i	$-0.479564\pi$
$149$	1.56626	0.128313	0.0641567	+	0.997940i	$0.479564\pi$
$150$	0	0
$151$	−14.5903	−1.18735	−0.593673	−	0.804707i	$-0.702323\pi$
$151$	−14.5903	−1.18735	−0.593673	+	0.804707i	$0.702323\pi$
$152$	0	0
$153$	2.17127	0.175537
$154$	0	0
$155$	−4.55426	−0.365807
$156$	0	0
$157$	−18.8675	−1.50579	−0.752894	−	0.658142i	$-0.771343\pi$
$157$	−18.8675	−1.50579	−0.752894	+	0.658142i	$0.771343\pi$
$158$	0	0
$159$	5.51320	0.437225
$160$	0	0
$161$	2.27713	0.179463
$162$	0	0
$163$	−8.00000	−0.626608	−0.313304	−	0.949653i	$-0.601436\pi$
$163$	−8.00000	−0.626608	−0.313304	+	0.949653i	$0.601436\pi$
$164$	0	0
$165$	−11.9232	−0.928220
$166$	0	0
$167$	−7.71149	−0.596733	−0.298367	−	0.954451i	$-0.596442\pi$
$167$	−7.71149	−0.596733	−0.298367	+	0.954451i	$0.596442\pi$
$168$	0	0
$169$	−12.9232	−0.994092
$170$	0	0
$171$	−1.00000	−0.0764719
$172$	0	0
$173$	−1.02639	−0.0780352	−0.0390176	−	0.999239i	$-0.512423\pi$
$173$	−1.02639	−0.0780352	−0.0390176	+	0.999239i	$0.512423\pi$
$174$	0	0
$175$	5.47214	0.413655
$176$	0	0
$177$	11.3689	0.854542
$178$	0	0
$179$	4.62171	0.345443	0.172721	−	0.984971i	$-0.444744\pi$
$179$	4.62171	0.345443	0.172721	+	0.984971i	$0.444744\pi$
$180$	0	0
$181$	22.6725	1.68523	0.842615	−	0.538516i	$-0.181015\pi$
$181$	22.6725	1.68523	0.842615	+	0.538516i	$0.181015\pi$
$182$	0	0
$183$	2.55426	0.188816
$184$	0	0
$185$	25.7643	1.89423
$186$	0	0
$187$	−8.00000	−0.585018
$188$	0	0
$189$	1.00000	0.0727393
$190$	0	0
$191$	2.27713	0.164767	0.0823836	−	0.996601i	$-0.473747\pi$
$191$	2.27713	0.164767	0.0823836	+	0.996601i	$0.473747\pi$
$192$	0	0
$193$	−8.81469	−0.634495	−0.317247	−	0.948343i	$-0.602759\pi$
$193$	−8.81469	−0.634495	−0.317247	+	0.948343i	$0.602759\pi$
$194$	0	0
$195$	−0.896807	−0.0642217
$196$	0	0
$197$	−12.3516	−0.880016	−0.440008	−	0.897994i	$-0.645024\pi$
$197$	−12.3516	−0.880016	−0.440008	+	0.897994i	$0.645024\pi$
$198$	0	0
$199$	−4.21171	−0.298560	−0.149280	−	0.988795i	$-0.547696\pi$
$199$	−4.21171	−0.298560	−0.149280	+	0.988795i	$0.547696\pi$
$200$	0	0
$201$	−3.68447	−0.259883
$202$	0	0
$203$	−1.51320	−0.106206
$204$	0	0
$205$	29.2100	2.04012
$206$	0	0
$207$	2.27713	0.158271
$208$	0	0
$209$	3.68447	0.254860
$210$	0	0
$211$	−7.89618	−0.543595	−0.271798	−	0.962354i	$-0.587618\pi$
$211$	−7.89618	−0.543595	−0.271798	+	0.962354i	$0.587618\pi$
$212$	0	0
$213$	−12.4575	−0.853572
$214$	0	0
$215$	14.7379	1.00512
$216$	0	0
$217$	−1.40734	−0.0955367
$218$	0	0
$219$	2.34255	0.158295
$220$	0	0
$221$	−0.601722	−0.0404762
$222$	0	0
$223$	20.0144	1.34026	0.670131	−	0.742243i	$-0.266238\pi$
$223$	20.0144	1.34026	0.670131	+	0.742243i	$0.266238\pi$
$224$	0	0
$225$	5.47214	0.364809
$226$	0	0
$227$	1.23278	0.0818224	0.0409112	−	0.999163i	$-0.486974\pi$
$227$	1.23278	0.0818224	0.0409112	+	0.999163i	$0.486974\pi$
$228$	0	0
$229$	−15.2868	−1.01018	−0.505091	−	0.863066i	$-0.668541\pi$
$229$	−15.2868	−1.01018	−0.505091	+	0.863066i	$0.668541\pi$
$230$	0	0
$231$	−3.68447	−0.242420
$232$	0	0
$233$	14.6017	0.956591	0.478295	−	0.878199i	$-0.341255\pi$
$233$	14.6017	0.956591	0.478295	+	0.878199i	$0.341255\pi$
$234$	0	0
$235$	7.92320	0.516853
$236$	0	0
$237$	12.1182	0.787162
$238$	0	0
$239$	−10.6197	−0.686930	−0.343465	−	0.939165i	$-0.611601\pi$
$239$	−10.6197	−0.686930	−0.343465	+	0.939165i	$0.611601\pi$
$240$	0	0
$241$	8.32085	0.535993	0.267997	−	0.963420i	$-0.413638\pi$
$241$	8.32085	0.535993	0.267997	+	0.963420i	$0.413638\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	3.23607	0.206745
$246$	0	0
$247$	0.277129	0.0176333
$248$	0	0
$249$	1.38361	0.0876827
$250$	0	0
$251$	−3.80828	−0.240377	−0.120188	−	0.992751i	$-0.538350\pi$
$251$	−3.80828	−0.240377	−0.120188	+	0.992751i	$0.538350\pi$
$252$	0	0
$253$	−8.39001	−0.527476
$254$	0	0
$255$	7.02639	0.440010
$256$	0	0
$257$	5.83576	0.364025	0.182012	−	0.983296i	$-0.441739\pi$
$257$	5.83576	0.364025	0.182012	+	0.983296i	$0.441739\pi$
$258$	0	0
$259$	7.96160	0.494710
$260$	0	0
$261$	−1.51320	−0.0936645
$262$	0	0
$263$	0.490092	0.0302204	0.0151102	−	0.999886i	$-0.495190\pi$
$263$	0.490092	0.0302204	0.0151102	+	0.999886i	$0.495190\pi$
$264$	0	0
$265$	17.8411	1.09597
$266$	0	0
$267$	2.34255	0.143362
$268$	0	0
$269$	17.2381	1.05103	0.525513	−	0.850786i	$-0.323874\pi$
$269$	17.2381	1.05103	0.525513	+	0.850786i	$0.323874\pi$
$270$	0	0
$271$	9.88854	0.600686	0.300343	−	0.953831i	$-0.402899\pi$
$271$	9.88854	0.600686	0.300343	+	0.953831i	$0.402899\pi$
$272$	0	0
$273$	−0.277129	−0.0167726
$274$	0	0
$275$	−20.1619	−1.21581
$276$	0	0
$277$	−6.76596	−0.406527	−0.203264	−	0.979124i	$-0.565155\pi$
$277$	−6.76596	−0.406527	−0.203264	+	0.979124i	$0.565155\pi$
$278$	0	0
$279$	−1.40734	−0.0842554
$280$	0	0
$281$	8.11617	0.484170	0.242085	−	0.970255i	$-0.422169\pi$
$281$	8.11617	0.484170	0.242085	+	0.970255i	$0.422169\pi$
$282$	0	0
$283$	−17.2868	−1.02759	−0.513797	−	0.857912i	$-0.671762\pi$
$283$	−17.2868	−1.02759	−0.513797	+	0.857912i	$0.671762\pi$
$284$	0	0
$285$	−3.23607	−0.191688
$286$	0	0
$287$	9.02639	0.532811
$288$	0	0
$289$	−12.2856	−0.722680
$290$	0	0
$291$	9.73194	0.570496
$292$	0	0
$293$	−17.8939	−1.04537	−0.522685	−	0.852526i	$-0.675070\pi$
$293$	−17.8939	−1.04537	−0.522685	+	0.852526i	$0.675070\pi$
$294$	0	0
$295$	36.7907	2.14204
$296$	0	0
$297$	−3.68447	−0.213795
$298$	0	0
$299$	−0.631057	−0.0364950
$300$	0	0
$301$	4.55426	0.262503
$302$	0	0
$303$	−1.70820	−0.0981338
$304$	0	0
$305$	8.26575	0.473295
$306$	0	0
$307$	−19.2381	−1.09798	−0.548988	−	0.835830i	$-0.684987\pi$
$307$	−19.2381	−1.09798	−0.548988	+	0.835830i	$0.684987\pi$
$308$	0	0
$309$	−9.96160	−0.566696
$310$	0	0
$311$	−1.08478	−0.0615123	−0.0307562	−	0.999527i	$-0.509792\pi$
$311$	−1.08478	−0.0615123	−0.0307562	+	0.999527i	$0.509792\pi$
$312$	0	0
$313$	6.31321	0.356844	0.178422	−	0.983954i	$-0.442901\pi$
$313$	6.31321	0.356844	0.178422	+	0.983954i	$0.442901\pi$
$314$	0	0
$315$	3.23607	0.182332
$316$	0	0
$317$	26.1149	1.46676	0.733380	−	0.679819i	$-0.237942\pi$
$317$	26.1149	1.46676	0.733380	+	0.679819i	$0.237942\pi$
$318$	0	0
$319$	5.57533	0.312159
$320$	0	0
$321$	1.04106	0.0581063
$322$	0	0
$323$	−2.17127	−0.120813
$324$	0	0
$325$	−1.51648	−0.0841195
$326$	0	0
$327$	3.74989	0.207370
$328$	0	0
$329$	2.44840	0.134985
$330$	0	0
$331$	−33.1062	−1.81968	−0.909841	−	0.414958i	$-0.863796\pi$
$331$	−33.1062	−1.81968	−0.909841	+	0.414958i	$0.863796\pi$
$332$	0	0
$333$	7.96160	0.436293
$334$	0	0
$335$	−11.9232	−0.651434
$336$	0	0
$337$	−14.3425	−0.781288	−0.390644	−	0.920542i	$-0.627748\pi$
$337$	−14.3425	−0.781288	−0.390644	+	0.920542i	$0.627748\pi$
$338$	0	0
$339$	−10.4575	−0.567972
$340$	0	0
$341$	5.18531	0.280801
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	7.36894	0.396731
$346$	0	0
$347$	23.6551	1.26987	0.634937	−	0.772564i	$-0.281026\pi$
$347$	23.6551	1.26987	0.634937	+	0.772564i	$0.281026\pi$
$348$	0	0
$349$	3.96534	0.212260	0.106130	−	0.994352i	$-0.466154\pi$
$349$	3.96534	0.212260	0.106130	+	0.994352i	$0.466154\pi$
$350$	0	0
$351$	−0.277129	−0.0147920
$352$	0	0
$353$	12.3009	0.654709	0.327354	−	0.944902i	$-0.393843\pi$
$353$	12.3009	0.654709	0.327354	+	0.944902i	$0.393843\pi$
$354$	0	0
$355$	−40.3132	−2.13960
$356$	0	0
$357$	2.17127	0.114916
$358$	0	0
$359$	−4.40671	−0.232578	−0.116289	−	0.993215i	$-0.537100\pi$
$359$	−4.40671	−0.232578	−0.116289	+	0.993215i	$0.537100\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	2.57533	0.135170
$364$	0	0
$365$	7.58065	0.396789
$366$	0	0
$367$	1.39702	0.0729239	0.0364620	−	0.999335i	$-0.488391\pi$
$367$	1.39702	0.0729239	0.0364620	+	0.999335i	$0.488391\pi$
$368$	0	0
$369$	9.02639	0.469895
$370$	0	0
$371$	5.51320	0.286231
$372$	0	0
$373$	−9.20096	−0.476407	−0.238204	−	0.971215i	$-0.576559\pi$
$373$	−9.20096	−0.476407	−0.238204	+	0.971215i	$0.576559\pi$
$374$	0	0
$375$	1.52786	0.0788986
$376$	0	0
$377$	0.419350	0.0215976
$378$	0	0
$379$	18.4224	0.946293	0.473146	−	0.880984i	$-0.343118\pi$
$379$	18.4224	0.946293	0.473146	+	0.880984i	$0.343118\pi$
$380$	0	0
$381$	−18.0361	−0.924017
$382$	0	0
$383$	16.9149	0.864313	0.432156	−	0.901799i	$-0.357753\pi$
$383$	16.9149	0.864313	0.432156	+	0.901799i	$0.357753\pi$
$384$	0	0
$385$	−11.9232	−0.607663
$386$	0	0
$387$	4.55426	0.231506
$388$	0	0
$389$	30.7470	1.55893	0.779466	−	0.626444i	$-0.215490\pi$
$389$	30.7470	1.55893	0.779466	+	0.626444i	$0.215490\pi$
$390$	0	0
$391$	4.94427	0.250043
$392$	0	0
$393$	5.98533	0.301920
$394$	0	0
$395$	39.2153	1.97314
$396$	0	0
$397$	−0.206386	−0.0103582	−0.00517912	−	0.999987i	$-0.501649\pi$
$397$	−0.206386	−0.0103582	−0.00517912	+	0.999987i	$0.501649\pi$
$398$	0	0
$399$	−1.00000	−0.0500626
$400$	0	0
$401$	28.7179	1.43410	0.717052	−	0.697020i	$-0.245491\pi$
$401$	28.7179	1.43410	0.717052	+	0.697020i	$0.245491\pi$
$402$	0	0
$403$	0.390015	0.0194280
$404$	0	0
$405$	3.23607	0.160802
$406$	0	0
$407$	−29.3343	−1.45405
$408$	0	0
$409$	−13.6077	−0.672856	−0.336428	−	0.941709i	$-0.609219\pi$
$409$	−13.6077	−0.672856	−0.336428	+	0.941709i	$0.609219\pi$
$410$	0	0
$411$	−16.1836	−0.798280
$412$	0	0
$413$	11.3689	0.559429
$414$	0	0
$415$	4.47746	0.219790
$416$	0	0
$417$	−8.60172	−0.421228
$418$	0	0
$419$	−2.06213	−0.100742	−0.0503709	−	0.998731i	$-0.516040\pi$
$419$	−2.06213	−0.100742	−0.0503709	+	0.998731i	$0.516040\pi$
$420$	0	0
$421$	−23.5422	−1.14738	−0.573690	−	0.819073i	$-0.694488\pi$
$421$	−23.5422	−1.14738	−0.573690	+	0.819073i	$0.694488\pi$
$422$	0	0
$423$	2.44840	0.119045
$424$	0	0
$425$	11.8815	0.576338
$426$	0	0
$427$	2.55426	0.123609
$428$	0	0
$429$	1.02107	0.0492978
$430$	0	0
$431$	−30.6745	−1.47754	−0.738769	−	0.673958i	$-0.764593\pi$
$431$	−30.6745	−1.47754	−0.738769	+	0.673958i	$0.764593\pi$
$432$	0	0
$433$	−31.5796	−1.51762	−0.758809	−	0.651313i	$-0.774218\pi$
$433$	−31.5796	−1.51762	−0.758809	+	0.651313i	$0.774218\pi$
$434$	0	0
$435$	−4.89681	−0.234784
$436$	0	0
$437$	−2.27713	−0.108930
$438$	0	0
$439$	−12.4631	−0.594830	−0.297415	−	0.954748i	$-0.596124\pi$
$439$	−12.4631	−0.594830	−0.297415	+	0.954748i	$0.596124\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	−20.5045	−0.974197	−0.487099	−	0.873347i	$-0.661945\pi$
$443$	−20.5045	−0.974197	−0.487099	+	0.873347i	$0.661945\pi$
$444$	0	0
$445$	7.58065	0.359357
$446$	0	0
$447$	1.56626	0.0740818
$448$	0	0
$449$	5.38235	0.254009	0.127004	−	0.991902i	$-0.459464\pi$
$449$	5.38235	0.254009	0.127004	+	0.991902i	$0.459464\pi$
$450$	0	0
$451$	−33.2575	−1.56603
$452$	0	0
$453$	−14.5903	−0.685514
$454$	0	0
$455$	−0.896807	−0.0420429
$456$	0	0
$457$	−15.9179	−0.744607	−0.372303	−	0.928111i	$-0.621432\pi$
$457$	−15.9179	−0.744607	−0.372303	+	0.928111i	$0.621432\pi$
$458$	0	0
$459$	2.17127	0.101346
$460$	0	0
$461$	36.7886	1.71342	0.856709	−	0.515800i	$-0.172505\pi$
$461$	36.7886	1.71342	0.856709	+	0.515800i	$0.172505\pi$
$462$	0	0
$463$	−1.10851	−0.0515170	−0.0257585	−	0.999668i	$-0.508200\pi$
$463$	−1.10851	−0.0515170	−0.0257585	+	0.999668i	$0.508200\pi$
$464$	0	0
$465$	−4.55426	−0.211199
$466$	0	0
$467$	−30.3279	−1.40341	−0.701704	−	0.712469i	$-0.747577\pi$
$467$	−30.3279	−1.40341	−0.701704	+	0.712469i	$0.747577\pi$
$468$	0	0
$469$	−3.68447	−0.170133
$470$	0	0
$471$	−18.8675	−0.869367
$472$	0	0
$473$	−16.7800	−0.771547
$474$	0	0
$475$	−5.47214	−0.251079
$476$	0	0
$477$	5.51320	0.252432
$478$	0	0
$479$	−17.1322	−0.782792	−0.391396	−	0.920222i	$-0.628008\pi$
$479$	−17.1322	−0.782792	−0.391396	+	0.920222i	$0.628008\pi$
$480$	0	0
$481$	−2.20639	−0.100603
$482$	0	0
$483$	2.27713	0.103613
$484$	0	0
$485$	31.4932	1.43003
$486$	0	0
$487$	22.5429	1.02152	0.510758	−	0.859725i	$-0.329365\pi$
$487$	22.5429	1.02152	0.510758	+	0.859725i	$0.329365\pi$
$488$	0	0
$489$	−8.00000	−0.361773
$490$	0	0
$491$	8.58128	0.387268	0.193634	−	0.981074i	$-0.437973\pi$
$491$	8.58128	0.387268	0.193634	+	0.981074i	$0.437973\pi$
$492$	0	0
$493$	−3.28557	−0.147974
$494$	0	0
$495$	−11.9232	−0.535908
$496$	0	0
$497$	−12.4575	−0.558794
$498$	0	0
$499$	−18.6504	−0.834909	−0.417454	−	0.908698i	$-0.637078\pi$
$499$	−18.6504	−0.834909	−0.417454	+	0.908698i	$0.637078\pi$
$500$	0	0
$501$	−7.71149	−0.344524
$502$	0	0
$503$	1.81735	0.0810315	0.0405157	−	0.999179i	$-0.487100\pi$
$503$	1.81735	0.0810315	0.0405157	+	0.999179i	$0.487100\pi$
$504$	0	0
$505$	−5.52786	−0.245987
$506$	0	0
$507$	−12.9232	−0.573939
$508$	0	0
$509$	17.9232	0.794432	0.397216	−	0.917725i	$-0.369976\pi$
$509$	17.9232	0.794432	0.397216	+	0.917725i	$0.369976\pi$
$510$	0	0
$511$	2.34255	0.103628
$512$	0	0
$513$	−1.00000	−0.0441511
$514$	0	0
$515$	−32.2364	−1.42051
$516$	0	0
$517$	−9.02107	−0.396746
$518$	0	0
$519$	−1.02639	−0.0450537
$520$	0	0
$521$	−16.3660	−0.717008	−0.358504	−	0.933528i	$-0.616713\pi$
$521$	−16.3660	−0.717008	−0.358504	+	0.933528i	$0.616713\pi$
$522$	0	0
$523$	−39.7590	−1.73854	−0.869269	−	0.494340i	$-0.835410\pi$
$523$	−39.7590	−1.73854	−0.869269	+	0.494340i	$0.835410\pi$
$524$	0	0
$525$	5.47214	0.238824
$526$	0	0
$527$	−3.05573	−0.133110
$528$	0	0
$529$	−17.8147	−0.774552
$530$	0	0
$531$	11.3689	0.493370
$532$	0	0
$533$	−2.50147	−0.108351
$534$	0	0
$535$	3.36894	0.145652
$536$	0	0
$537$	4.62171	0.199442
$538$	0	0
$539$	−3.68447	−0.158701
$540$	0	0
$541$	−14.3132	−0.615373	−0.307687	−	0.951488i	$-0.599555\pi$
$541$	−14.3132	−0.615373	−0.307687	+	0.951488i	$0.599555\pi$
$542$	0	0
$543$	22.6725	0.972969
$544$	0	0
$545$	12.1349	0.519802
$546$	0	0
$547$	37.9156	1.62115	0.810576	−	0.585633i	$-0.199154\pi$
$547$	37.9156	1.62115	0.810576	+	0.585633i	$0.199154\pi$
$548$	0	0
$549$	2.55426	0.109013
$550$	0	0
$551$	1.51320	0.0644643
$552$	0	0
$553$	12.1182	0.515319
$554$	0	0
$555$	25.7643	1.09363
$556$	0	0
$557$	−3.37801	−0.143131	−0.0715654	−	0.997436i	$-0.522799\pi$
$557$	−3.37801	−0.143131	−0.0715654	+	0.997436i	$0.522799\pi$
$558$	0	0
$559$	−1.26211	−0.0533818
$560$	0	0
$561$	−8.00000	−0.337760
$562$	0	0
$563$	4.34255	0.183017	0.0915083	−	0.995804i	$-0.470831\pi$
$563$	4.34255	0.183017	0.0915083	+	0.995804i	$0.470831\pi$
$564$	0	0
$565$	−33.8411	−1.42370
$566$	0	0
$567$	1.00000	0.0419961
$568$	0	0
$569$	27.3543	1.14675	0.573375	−	0.819293i	$-0.305634\pi$
$569$	27.3543	1.14675	0.573375	+	0.819293i	$0.305634\pi$
$570$	0	0
$571$	−15.2921	−0.639956	−0.319978	−	0.947425i	$-0.603676\pi$
$571$	−15.2921	−0.639956	−0.319978	+	0.947425i	$0.603676\pi$
$572$	0	0
$573$	2.27713	0.0951284
$574$	0	0
$575$	12.4608	0.519649
$576$	0	0
$577$	6.84934	0.285142	0.142571	−	0.989785i	$-0.454463\pi$
$577$	6.84934	0.285142	0.142571	+	0.989785i	$0.454463\pi$
$578$	0	0
$579$	−8.81469	−0.366326
$580$	0	0
$581$	1.38361	0.0574018
$582$	0	0
$583$	−20.3132	−0.841287
$584$	0	0
$585$	−0.896807	−0.0370784
$586$	0	0
$587$	25.2775	1.04331	0.521657	−	0.853156i	$-0.325314\pi$
$587$	25.2775	1.04331	0.521657	+	0.853156i	$0.325314\pi$
$588$	0	0
$589$	1.40734	0.0579886
$590$	0	0
$591$	−12.3516	−0.508078
$592$	0	0
$593$	−27.2030	−1.11709	−0.558546	−	0.829473i	$-0.688641\pi$
$593$	−27.2030	−1.11709	−0.558546	+	0.829473i	$0.688641\pi$
$594$	0	0
$595$	7.02639	0.288054
$596$	0	0
$597$	−4.21171	−0.172374
$598$	0	0
$599$	30.3460	1.23990	0.619952	−	0.784640i	$-0.287152\pi$
$599$	30.3460	1.23990	0.619952	+	0.784640i	$0.287152\pi$
$600$	0	0
$601$	18.0811	0.737542	0.368771	−	0.929520i	$-0.379779\pi$
$601$	18.0811	0.737542	0.368771	+	0.929520i	$0.379779\pi$
$602$	0	0
$603$	−3.68447	−0.150043
$604$	0	0
$605$	8.33394	0.338823
$606$	0	0
$607$	−13.4948	−0.547736	−0.273868	−	0.961767i	$-0.588303\pi$
$607$	−13.4948	−0.547736	−0.273868	+	0.961767i	$0.588303\pi$
$608$	0	0
$609$	−1.51320	−0.0613178
$610$	0	0
$611$	−0.678522	−0.0274501
$612$	0	0
$613$	−3.52786	−0.142489	−0.0712445	−	0.997459i	$-0.522697\pi$
$613$	−3.52786	−0.142489	−0.0712445	+	0.997459i	$0.522697\pi$
$614$	0	0
$615$	29.2100	1.17786
$616$	0	0
$617$	6.31321	0.254160	0.127080	−	0.991892i	$-0.459439\pi$
$617$	6.31321	0.254160	0.127080	+	0.991892i	$0.459439\pi$
$618$	0	0
$619$	11.1213	0.447004	0.223502	−	0.974704i	$-0.428251\pi$
$619$	11.1213	0.447004	0.223502	+	0.974704i	$0.428251\pi$
$620$	0	0
$621$	2.27713	0.0913780
$622$	0	0
$623$	2.34255	0.0938523
$624$	0	0
$625$	−22.4164	−0.896656
$626$	0	0
$627$	3.68447	0.147144
$628$	0	0
$629$	17.2868	0.689271
$630$	0	0
$631$	−16.7032	−0.664945	−0.332473	−	0.943113i	$-0.607883\pi$
$631$	−16.7032	−0.664945	−0.332473	+	0.943113i	$0.607883\pi$
$632$	0	0
$633$	−7.89618	−0.313845
$634$	0	0
$635$	−58.3660	−2.31618
$636$	0	0
$637$	−0.277129	−0.0109802
$638$	0	0
$639$	−12.4575	−0.492810
$640$	0	0
$641$	−21.9432	−0.866704	−0.433352	−	0.901225i	$-0.642669\pi$
$641$	−21.9432	−0.866704	−0.433352	+	0.901225i	$0.642669\pi$
$642$	0	0
$643$	−43.1754	−1.70267	−0.851335	−	0.524622i	$-0.824207\pi$
$643$	−43.1754	−1.70267	−0.851335	+	0.524622i	$0.824207\pi$
$644$	0	0
$645$	14.7379	0.580304
$646$	0	0
$647$	38.9440	1.53105	0.765523	−	0.643408i	$-0.222480\pi$
$647$	38.9440	1.53105	0.765523	+	0.643408i	$0.222480\pi$
$648$	0	0
$649$	−41.8885	−1.64427
$650$	0	0
$651$	−1.40734	−0.0551581
$652$	0	0
$653$	6.07306	0.237657	0.118829	−	0.992915i	$-0.462086\pi$
$653$	6.07306	0.237657	0.118829	+	0.992915i	$0.462086\pi$
$654$	0	0
$655$	19.3689	0.756807
$656$	0	0
$657$	2.34255	0.0913916
$658$	0	0
$659$	−16.7460	−0.652331	−0.326165	−	0.945313i	$-0.605757\pi$
$659$	−16.7460	−0.652331	−0.326165	+	0.945313i	$0.605757\pi$
$660$	0	0
$661$	40.2953	1.56730	0.783652	−	0.621200i	$-0.213355\pi$
$661$	40.2953	1.56730	0.783652	+	0.621200i	$0.213355\pi$
$662$	0	0
$663$	−0.601722	−0.0233690
$664$	0	0
$665$	−3.23607	−0.125489
$666$	0	0
$667$	−3.44574	−0.133420
$668$	0	0
$669$	20.0144	0.773801
$670$	0	0
$671$	−9.41109	−0.363311
$672$	0	0
$673$	4.60704	0.177588	0.0887942	−	0.996050i	$-0.471699\pi$
$673$	4.60704	0.177588	0.0887942	+	0.996050i	$0.471699\pi$
$674$	0	0
$675$	5.47214	0.210623
$676$	0	0
$677$	15.0751	0.579384	0.289692	−	0.957120i	$-0.406447\pi$
$677$	15.0751	0.579384	0.289692	+	0.957120i	$0.406447\pi$
$678$	0	0
$679$	9.73194	0.373477
$680$	0	0
$681$	1.23278	0.0472402
$682$	0	0
$683$	43.9907	1.68326	0.841628	−	0.540058i	$-0.181598\pi$
$683$	43.9907	1.68326	0.841628	+	0.540058i	$0.181598\pi$
$684$	0	0
$685$	−52.3713	−2.00101
$686$	0	0
$687$	−15.2868	−0.583229
$688$	0	0
$689$	−1.52786	−0.0582070
$690$	0	0
$691$	−5.26211	−0.200180	−0.100090	−	0.994978i	$-0.531913\pi$
$691$	−5.26211	−0.200180	−0.100090	+	0.994978i	$0.531913\pi$
$692$	0	0
$693$	−3.68447	−0.139961
$694$	0	0
$695$	−27.8358	−1.05587
$696$	0	0
$697$	19.5988	0.742357
$698$	0	0
$699$	14.6017	0.552288
$700$	0	0
$701$	0.559257	0.0211229	0.0105614	−	0.999944i	$-0.496638\pi$
$701$	0.559257	0.0211229	0.0105614	+	0.999944i	$0.496638\pi$
$702$	0	0
$703$	−7.96160	−0.300277
$704$	0	0
$705$	7.92320	0.298405
$706$	0	0
$707$	−1.70820	−0.0642436
$708$	0	0
$709$	37.7296	1.41697	0.708483	−	0.705728i	$-0.249380\pi$
$709$	37.7296	1.41697	0.708483	+	0.705728i	$0.249380\pi$
$710$	0	0
$711$	12.1182	0.454468
$712$	0	0
$713$	−3.20470	−0.120017
$714$	0	0
$715$	3.30426	0.123572
$716$	0	0
$717$	−10.6197	−0.396599
$718$	0	0
$719$	−37.4121	−1.39523	−0.697617	−	0.716471i	$-0.745756\pi$
$719$	−37.4121	−1.39523	−0.697617	+	0.716471i	$0.745756\pi$
$720$	0	0
$721$	−9.96160	−0.370989
$722$	0	0
$723$	8.32085	0.309456
$724$	0	0
$725$	−8.28042	−0.307527
$726$	0	0
$727$	−3.23810	−0.120094	−0.0600472	−	0.998196i	$-0.519125\pi$
$727$	−3.23810	−0.120094	−0.0600472	+	0.998196i	$0.519125\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	9.88854	0.365741
$732$	0	0
$733$	−26.0475	−0.962085	−0.481043	−	0.876697i	$-0.659742\pi$
$733$	−26.0475	−0.962085	−0.481043	+	0.876697i	$0.659742\pi$
$734$	0	0
$735$	3.23607	0.119364
$736$	0	0
$737$	13.5753	0.500054
$738$	0	0
$739$	−47.9954	−1.76554	−0.882769	−	0.469807i	$-0.844323\pi$
$739$	−47.9954	−1.76554	−0.882769	+	0.469807i	$0.844323\pi$
$740$	0	0
$741$	0.277129	0.0101806
$742$	0	0
$743$	−52.2705	−1.91762	−0.958809	−	0.284052i	$-0.908321\pi$
$743$	−52.2705	−1.91762	−0.958809	+	0.284052i	$0.908321\pi$
$744$	0	0
$745$	5.06854	0.185697
$746$	0	0
$747$	1.38361	0.0506237
$748$	0	0
$749$	1.04106	0.0380395
$750$	0	0
$751$	47.4578	1.73176	0.865880	−	0.500252i	$-0.166759\pi$
$751$	47.4578	1.73176	0.865880	+	0.500252i	$0.166759\pi$
$752$	0	0
$753$	−3.80828	−0.138781
$754$	0	0
$755$	−47.2153	−1.71834
$756$	0	0
$757$	−11.0211	−0.400568	−0.200284	−	0.979738i	$-0.564186\pi$
$757$	−11.0211	−0.400568	−0.200284	+	0.979738i	$0.564186\pi$
$758$	0	0
$759$	−8.39001	−0.304538
$760$	0	0
$761$	−53.0041	−1.92140	−0.960698	−	0.277594i	$-0.910463\pi$
$761$	−53.0041	−1.92140	−0.960698	+	0.277594i	$0.910463\pi$
$762$	0	0
$763$	3.74989	0.135755
$764$	0	0
$765$	7.02639	0.254040
$766$	0	0
$767$	−3.15066	−0.113764
$768$	0	0
$769$	−46.4775	−1.67602	−0.838010	−	0.545655i	$-0.816281\pi$
$769$	−46.4775	−1.67602	−0.838010	+	0.545655i	$0.816281\pi$
$770$	0	0
$771$	5.83576	0.210170
$772$	0	0
$773$	41.7296	1.50091	0.750455	−	0.660921i	$-0.229834\pi$
$773$	41.7296	1.50091	0.750455	+	0.660921i	$0.229834\pi$
$774$	0	0
$775$	−7.70117	−0.276634
$776$	0	0
$777$	7.96160	0.285621
$778$	0	0
$779$	−9.02639	−0.323404
$780$	0	0
$781$	45.8992	1.64240
$782$	0	0
$783$	−1.51320	−0.0540772
$784$	0	0
$785$	−61.0564	−2.17920
$786$	0	0
$787$	−2.46556	−0.0878877	−0.0439438	−	0.999034i	$-0.513992\pi$
$787$	−2.46556	−0.0878877	−0.0439438	+	0.999034i	$0.513992\pi$
$788$	0	0
$789$	0.490092	0.0174477
$790$	0	0
$791$	−10.4575	−0.371825
$792$	0	0
$793$	−0.707858	−0.0251368
$794$	0	0
$795$	17.8411	0.632758
$796$	0	0
$797$	−29.3936	−1.04118	−0.520588	−	0.853808i	$-0.674287\pi$
$797$	−29.3936	−1.04118	−0.520588	+	0.853808i	$0.674287\pi$
$798$	0	0
$799$	5.31616	0.188072
$800$	0	0
$801$	2.34255	0.0827699
$802$	0	0
$803$	−8.63106	−0.304583
$804$	0	0
$805$	7.36894	0.259721
$806$	0	0
$807$	17.2381	0.606810
$808$	0	0
$809$	−39.6475	−1.39393	−0.696966	−	0.717104i	$-0.745467\pi$
$809$	−39.6475	−1.39393	−0.696966	+	0.717104i	$0.745467\pi$
$810$	0	0
$811$	−29.0751	−1.02097	−0.510483	−	0.859888i	$-0.670533\pi$
$811$	−29.0751	−1.02097	−0.510483	+	0.859888i	$0.670533\pi$
$812$	0	0
$813$	9.88854	0.346806
$814$	0	0
$815$	−25.8885	−0.906836
$816$	0	0
$817$	−4.55426	−0.159333
$818$	0	0
$819$	−0.277129	−0.00968365
$820$	0	0
$821$	14.5980	0.509473	0.254736	−	0.967011i	$-0.418011\pi$
$821$	14.5980	0.509473	0.254736	+	0.967011i	$0.418011\pi$
$822$	0	0
$823$	6.57827	0.229304	0.114652	−	0.993406i	$-0.463425\pi$
$823$	6.57827	0.229304	0.114652	+	0.993406i	$0.463425\pi$
$824$	0	0
$825$	−20.1619	−0.701948
$826$	0	0
$827$	16.0609	0.558491	0.279246	−	0.960220i	$-0.409916\pi$
$827$	16.0609	0.558491	0.279246	+	0.960220i	$0.409916\pi$
$828$	0	0
$829$	27.0443	0.939289	0.469645	−	0.882856i	$-0.344382\pi$
$829$	27.0443	0.939289	0.469645	+	0.882856i	$0.344382\pi$
$830$	0	0
$831$	−6.76596	−0.234709
$832$	0	0
$833$	2.17127	0.0752302
$834$	0	0
$835$	−24.9549	−0.863600
$836$	0	0
$837$	−1.40734	−0.0486449
$838$	0	0
$839$	−23.4230	−0.808651	−0.404326	−	0.914615i	$-0.632494\pi$
$839$	−23.4230	−0.808651	−0.404326	+	0.914615i	$0.632494\pi$
$840$	0	0
$841$	−26.7102	−0.921043
$842$	0	0
$843$	8.11617	0.279536
$844$	0	0
$845$	−41.8204	−1.43866
$846$	0	0
$847$	2.57533	0.0884894
$848$	0	0
$849$	−17.2868	−0.593282
$850$	0	0
$851$	18.1296	0.621474
$852$	0	0
$853$	9.38707	0.321407	0.160704	−	0.987003i	$-0.448624\pi$
$853$	9.38707	0.321407	0.160704	+	0.987003i	$0.448624\pi$
$854$	0	0
$855$	−3.23607	−0.110671
$856$	0	0
$857$	43.3924	1.48226	0.741128	−	0.671364i	$-0.234291\pi$
$857$	43.3924	1.48226	0.741128	+	0.671364i	$0.234291\pi$
$858$	0	0
$859$	2.83282	0.0966544	0.0483272	−	0.998832i	$-0.484611\pi$
$859$	2.83282	0.0966544	0.0483272	+	0.998832i	$0.484611\pi$
$860$	0	0
$861$	9.02639	0.307619
$862$	0	0
$863$	−31.9432	−1.08736	−0.543679	−	0.839293i	$-0.682969\pi$
$863$	−31.9432	−1.08736	−0.543679	+	0.839293i	$0.682969\pi$
$864$	0	0
$865$	−3.32148	−0.112934
$866$	0	0
$867$	−12.2856	−0.417240
$868$	0	0
$869$	−44.6492	−1.51462
$870$	0	0
$871$	1.02107	0.0345977
$872$	0	0
$873$	9.73194	0.329376
$874$	0	0
$875$	1.52786	0.0516512
$876$	0	0
$877$	−15.1997	−0.513257	−0.256629	−	0.966510i	$-0.582612\pi$
$877$	−15.1997	−0.513257	−0.256629	+	0.966510i	$0.582612\pi$
$878$	0	0
$879$	−17.8939	−0.603545
$880$	0	0
$881$	−8.64341	−0.291204	−0.145602	−	0.989343i	$-0.546512\pi$
$881$	−8.64341	−0.291204	−0.145602	+	0.989343i	$0.546512\pi$
$882$	0	0
$883$	52.8088	1.77716	0.888579	−	0.458724i	$-0.151693\pi$
$883$	52.8088	1.77716	0.888579	+	0.458724i	$0.151693\pi$
$884$	0	0
$885$	36.7907	1.23670
$886$	0	0
$887$	−33.8411	−1.13627	−0.568136	−	0.822935i	$-0.692335\pi$
$887$	−33.8411	−1.13627	−0.568136	+	0.822935i	$0.692335\pi$
$888$	0	0
$889$	−18.0361	−0.604911
$890$	0	0
$891$	−3.68447	−0.123434
$892$	0	0
$893$	−2.44840	−0.0819327
$894$	0	0
$895$	14.9562	0.499930
$896$	0	0
$897$	−0.631057	−0.0210704
$898$	0	0
$899$	2.12959	0.0710257
$900$	0	0
$901$	11.9707	0.398801
$902$	0	0
$903$	4.55426	0.151556
$904$	0	0
$905$	73.3696	2.43889
$906$	0	0
$907$	37.8615	1.25717	0.628586	−	0.777740i	$-0.283634\pi$
$907$	37.8615	1.25717	0.628586	+	0.777740i	$0.283634\pi$
$908$	0	0
$909$	−1.70820	−0.0566575
$910$	0	0
$911$	9.40174	0.311494	0.155747	−	0.987797i	$-0.450222\pi$
$911$	9.40174	0.311494	0.155747	+	0.987797i	$0.450222\pi$
$912$	0	0
$913$	−5.09787	−0.168715
$914$	0	0
$915$	8.26575	0.273257
$916$	0	0
$917$	5.98533	0.197653
$918$	0	0
$919$	−22.7379	−0.750054	−0.375027	−	0.927014i	$-0.622367\pi$
$919$	−22.7379	−0.750054	−0.375027	+	0.927014i	$0.622367\pi$
$920$	0	0
$921$	−19.2381	−0.633917
$922$	0	0
$923$	3.45232	0.113634
$924$	0	0
$925$	43.5670	1.43247
$926$	0	0
$927$	−9.96160	−0.327182
$928$	0	0
$929$	40.8505	1.34026	0.670131	−	0.742243i	$-0.266238\pi$
$929$	40.8505	1.34026	0.670131	+	0.742243i	$0.266238\pi$
$930$	0	0
$931$	−1.00000	−0.0327737
$932$	0	0
$933$	−1.08478	−0.0355142
$934$	0	0
$935$	−25.8885	−0.846646
$936$	0	0
$937$	6.25917	0.204478	0.102239	−	0.994760i	$-0.467399\pi$
$937$	6.25917	0.204478	0.102239	+	0.994760i	$0.467399\pi$
$938$	0	0
$939$	6.31321	0.206024
$940$	0	0
$941$	51.8117	1.68901	0.844507	−	0.535544i	$-0.179893\pi$
$941$	51.8117	1.68901	0.844507	+	0.535544i	$0.179893\pi$
$942$	0	0
$943$	20.5543	0.669339
$944$	0	0
$945$	3.23607	0.105269
$946$	0	0
$947$	−12.4979	−0.406127	−0.203064	−	0.979166i	$-0.565090\pi$
$947$	−12.4979	−0.406127	−0.203064	+	0.979166i	$0.565090\pi$
$948$	0	0
$949$	−0.649187	−0.0210735
$950$	0	0
$951$	26.1149	0.846834
$952$	0	0
$953$	−2.65105	−0.0858758	−0.0429379	−	0.999078i	$-0.513672\pi$
$953$	−2.65105	−0.0858758	−0.0429379	+	0.999078i	$0.513672\pi$
$954$	0	0
$955$	7.36894	0.238453
$956$	0	0
$957$	5.57533	0.180225
$958$	0	0
$959$	−16.1836	−0.522597
$960$	0	0
$961$	−29.0194	−0.936109
$962$	0	0
$963$	1.04106	0.0335477
$964$	0	0
$965$	−28.5249	−0.918250
$966$	0	0
$967$	48.7032	1.56619	0.783095	−	0.621902i	$-0.213640\pi$
$967$	48.7032	1.56619	0.783095	+	0.621902i	$0.213640\pi$
$968$	0	0
$969$	−2.17127	−0.0697514
$970$	0	0
$971$	10.0528	0.322609	0.161305	−	0.986905i	$-0.448430\pi$
$971$	10.0528	0.322609	0.161305	+	0.986905i	$0.448430\pi$
$972$	0	0
$973$	−8.60172	−0.275759
$974$	0	0
$975$	−1.51648	−0.0485664
$976$	0	0
$977$	−43.2234	−1.38284	−0.691420	−	0.722453i	$-0.743015\pi$
$977$	−43.2234	−1.38284	−0.691420	+	0.722453i	$0.743015\pi$
$978$	0	0
$979$	−8.63106	−0.275850
$980$	0	0
$981$	3.74989	0.119725
$982$	0	0
$983$	−1.02107	−0.0325671	−0.0162836	−	0.999867i	$-0.505183\pi$
$983$	−1.02107	−0.0325671	−0.0162836	+	0.999867i	$0.505183\pi$
$984$	0	0
$985$	−39.9707	−1.27357
$986$	0	0
$987$	2.44840	0.0779335
$988$	0	0
$989$	10.3706	0.329767
$990$	0	0
$991$	34.5070	1.09615	0.548075	−	0.836429i	$-0.315361\pi$
$991$	34.5070	1.09615	0.548075	+	0.836429i	$0.315361\pi$
$992$	0	0
$993$	−33.1062	−1.05059
$994$	0	0
$995$	−13.6294	−0.432080
$996$	0	0
$997$	−5.92320	−0.187590	−0.0937948	−	0.995592i	$-0.529900\pi$
$997$	−5.92320	−0.187590	−0.0937948	+	0.995592i	$0.529900\pi$
$998$	0	0
$999$	7.96160	0.251894

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3192.2.a.z.1.3	✓	4
3.2	odd	2		9576.2.a.ch.1.2		4
4.3	odd	2		6384.2.a.by.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3192.2.a.z.1.3	✓	4	1.1	even	1	trivial
6384.2.a.by.1.4		4	4.3	odd	2
9576.2.a.ch.1.2		4	3.2	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:	\( N \)	\(=\)	\( 3192 = 2^{3} \cdot 3 \cdot 7 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3192.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +3.23607 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +3.23607 q^{5} +1.00000 q^{7} +1.00000 q^{9} -3.68447 q^{11} -0.277129 q^{13} +3.23607 q^{15} +2.17127 q^{17} -1.00000 q^{19} +1.00000 q^{21} +2.27713 q^{23} +5.47214 q^{25} +1.00000 q^{27} -1.51320 q^{29} -1.40734 q^{31} -3.68447 q^{33} +3.23607 q^{35} +7.96160 q^{37} -0.277129 q^{39} +9.02639 q^{41} +4.55426 q^{43} +3.23607 q^{45} +2.44840 q^{47} +1.00000 q^{49} +2.17127 q^{51} +5.51320 q^{53} -11.9232 q^{55} -1.00000 q^{57} +11.3689 q^{59} +2.55426 q^{61} +1.00000 q^{63} -0.896807 q^{65} -3.68447 q^{67} +2.27713 q^{69} -12.4575 q^{71} +2.34255 q^{73} +5.47214 q^{75} -3.68447 q^{77} +12.1182 q^{79} +1.00000 q^{81} +1.38361 q^{83} +7.02639 q^{85} -1.51320 q^{87} +2.34255 q^{89} -0.277129 q^{91} -1.40734 q^{93} -3.23607 q^{95} +9.73194 q^{97} -3.68447 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{3} + 4 q^{5} + 4 q^{7} + 4 q^{9}+O(q^{10}) \) 4 * q + 4 * q^3 + 4 * q^5 + 4 * q^7 + 4 * q^9	\( 4 q + 4 q^{3} + 4 q^{5} + 4 q^{7} + 4 q^{9} + 4 q^{11} + 6 q^{13} + 4 q^{15} + 6 q^{17} - 4 q^{19} + 4 q^{21} + 2 q^{23} + 4 q^{25} + 4 q^{27} + 10 q^{29} + 6 q^{31} + 4 q^{33} + 4 q^{35} + 6 q^{37} + 6 q^{39} + 4 q^{41} + 4 q^{43} + 4 q^{45} + 4 q^{49} + 6 q^{51} + 6 q^{53} + 4 q^{55} - 4 q^{57} + 8 q^{59} - 4 q^{61} + 4 q^{63} + 16 q^{65} + 4 q^{67} + 2 q^{69} + 2 q^{71} + 4 q^{73} + 4 q^{75} + 4 q^{77} - 14 q^{79} + 4 q^{81} + 2 q^{83} - 4 q^{85} + 10 q^{87} + 4 q^{89} + 6 q^{91} + 6 q^{93} - 4 q^{95} + 4 q^{97} + 4 q^{99}+O(q^{100}) \) 4 * q + 4 * q^3 + 4 * q^5 + 4 * q^7 + 4 * q^9 + 4 * q^11 + 6 * q^13 + 4 * q^15 + 6 * q^17 - 4 * q^19 + 4 * q^21 + 2 * q^23 + 4 * q^25 + 4 * q^27 + 10 * q^29 + 6 * q^31 + 4 * q^33 + 4 * q^35 + 6 * q^37 + 6 * q^39 + 4 * q^41 + 4 * q^43 + 4 * q^45 + 4 * q^49 + 6 * q^51 + 6 * q^53 + 4 * q^55 - 4 * q^57 + 8 * q^59 - 4 * q^61 + 4 * q^63 + 16 * q^65 + 4 * q^67 + 2 * q^69 + 2 * q^71 + 4 * q^73 + 4 * q^75 + 4 * q^77 - 14 * q^79 + 4 * q^81 + 2 * q^83 - 4 * q^85 + 10 * q^87 + 4 * q^89 + 6 * q^91 + 6 * q^93 - 4 * q^95 + 4 * q^97 + 4 * q^99

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