LMFDB - Embedded newform 3192.2.a.v.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:	$1$
Dimension:	$3$
Coefficient field:	3.3.316.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 4x + 2 $ x^3 - x^2 - 4*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.34292$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -1.00000 q^{7} +1.00000 q^{9} +2.68585 q^{11} -6.97858 q^{13} +2.29273 q^{17} -1.00000 q^{19} -1.00000 q^{21} -4.39312 q^{23} -5.00000 q^{25} +1.00000 q^{27} +0.978577 q^{29} -7.66442 q^{31} +2.68585 q^{33} +5.66442 q^{37} -6.97858 q^{39} +7.37169 q^{41} -9.37169 q^{43} -4.68585 q^{47} +1.00000 q^{49} +2.29273 q^{51} -12.9786 q^{53} -1.00000 q^{57} -0.585462 q^{59} -6.58546 q^{61} -1.00000 q^{63} +12.0575 q^{67} -4.39312 q^{69} -1.60688 q^{71} -7.37169 q^{73} -5.00000 q^{75} -2.68585 q^{77} +14.3503 q^{79} +1.00000 q^{81} +8.35027 q^{83} +0.978577 q^{87} -15.9572 q^{89} +6.97858 q^{91} -7.66442 q^{93} +5.27131 q^{97} +2.68585 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{3} - 3 q^{7} + 3 q^{9}+O(q^{10}) $ 3 * q + 3 * q^3 - 3 * q^7 + 3 * q^9	$ 3 q + 3 q^{3} - 3 q^{7} + 3 q^{9} - 4 q^{11} - 6 q^{13} + 4 q^{17} - 3 q^{19} - 3 q^{21} - 4 q^{23} - 15 q^{25} + 3 q^{27} - 12 q^{29} + 4 q^{31} - 4 q^{33} - 10 q^{37} - 6 q^{39} - 2 q^{41} - 4 q^{43} - 2 q^{47} + 3 q^{49} + 4 q^{51} - 24 q^{53} - 3 q^{57} + 4 q^{59} - 14 q^{61} - 3 q^{63} - 4 q^{69} - 14 q^{71} + 2 q^{73} - 15 q^{75} + 4 q^{77} + 4 q^{79} + 3 q^{81} - 14 q^{83} - 12 q^{87} - 18 q^{89} + 6 q^{91} + 4 q^{93} - 2 q^{97} - 4 q^{99}+O(q^{100}) $ 3 * q + 3 * q^3 - 3 * q^7 + 3 * q^9 - 4 * q^11 - 6 * q^13 + 4 * q^17 - 3 * q^19 - 3 * q^21 - 4 * q^23 - 15 * q^25 + 3 * q^27 - 12 * q^29 + 4 * q^31 - 4 * q^33 - 10 * q^37 - 6 * q^39 - 2 * q^41 - 4 * q^43 - 2 * q^47 + 3 * q^49 + 4 * q^51 - 24 * q^53 - 3 * q^57 + 4 * q^59 - 14 * q^61 - 3 * q^63 - 4 * q^69 - 14 * q^71 + 2 * q^73 - 15 * q^75 + 4 * q^77 + 4 * q^79 + 3 * q^81 - 14 * q^83 - 12 * q^87 - 18 * q^89 + 6 * q^91 + 4 * q^93 - 2 * 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
$5$	0	0		−	1.00000i	$-0.5\pi$
$5$	0	0			1.00000i	$0.5\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	2.68585	0.809813	0.404907	−	0.914358i	$-0.367304\pi$
$11$	2.68585	0.809813	0.404907	+	0.914358i	$0.367304\pi$
$12$	0	0
$13$	−6.97858	−1.93551	−0.967755	−	0.251895i	$-0.918946\pi$
$13$	−6.97858	−1.93551	−0.967755	+	0.251895i	$0.918946\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	2.29273	0.556069	0.278034	−	0.960571i	$-0.410317\pi$
$17$	2.29273	0.556069	0.278034	+	0.960571i	$0.410317\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	−1.00000	−0.218218
$22$	0	0
$23$	−4.39312	−0.916028	−0.458014	−	0.888945i	$-0.651439\pi$
$23$	−4.39312	−0.916028	−0.458014	+	0.888945i	$0.651439\pi$
$24$	0	0
$25$	−5.00000	−1.00000
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	0.978577	0.181717	0.0908586	−	0.995864i	$-0.471039\pi$
$29$	0.978577	0.181717	0.0908586	+	0.995864i	$0.471039\pi$
$30$	0	0
$31$	−7.66442	−1.37657	−0.688286	−	0.725440i	$-0.741636\pi$
$31$	−7.66442	−1.37657	−0.688286	+	0.725440i	$0.741636\pi$
$32$	0	0
$33$	2.68585	0.467546
$34$	0	0
$35$	0	0
$36$	0	0
$37$	5.66442	0.931225	0.465613	−	0.884989i	$-0.345834\pi$
$37$	5.66442	0.931225	0.465613	+	0.884989i	$0.345834\pi$
$38$	0	0
$39$	−6.97858	−1.11747
$40$	0	0
$41$	7.37169	1.15126	0.575632	−	0.817709i	$-0.304756\pi$
$41$	7.37169	1.15126	0.575632	+	0.817709i	$0.304756\pi$
$42$	0	0
$43$	−9.37169	−1.42917	−0.714585	−	0.699549i	$-0.753384\pi$
$43$	−9.37169	−1.42917	−0.714585	+	0.699549i	$0.753384\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−4.68585	−0.683501	−0.341750	−	0.939791i	$-0.611020\pi$
$47$	−4.68585	−0.683501	−0.341750	+	0.939791i	$0.611020\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	2.29273	0.321047
$52$	0	0
$53$	−12.9786	−1.78274	−0.891372	−	0.453272i	$-0.850257\pi$
$53$	−12.9786	−1.78274	−0.891372	+	0.453272i	$0.850257\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−1.00000	−0.132453
$58$	0	0
$59$	−0.585462	−0.0762207	−0.0381103	−	0.999274i	$-0.512134\pi$
$59$	−0.585462	−0.0762207	−0.0381103	+	0.999274i	$0.512134\pi$
$60$	0	0
$61$	−6.58546	−0.843182	−0.421591	−	0.906786i	$-0.638528\pi$
$61$	−6.58546	−0.843182	−0.421591	+	0.906786i	$0.638528\pi$
$62$	0	0
$63$	−1.00000	−0.125988
$64$	0	0
$65$	0	0
$66$	0	0
$67$	12.0575	1.47306	0.736531	−	0.676403i	$-0.236462\pi$
$67$	12.0575	1.47306	0.736531	+	0.676403i	$0.236462\pi$
$68$	0	0
$69$	−4.39312	−0.528869
$70$	0	0
$71$	−1.60688	−0.190702	−0.0953511	−	0.995444i	$-0.530397\pi$
$71$	−1.60688	−0.190702	−0.0953511	+	0.995444i	$0.530397\pi$
$72$	0	0
$73$	−7.37169	−0.862791	−0.431396	−	0.902163i	$-0.641979\pi$
$73$	−7.37169	−0.862791	−0.431396	+	0.902163i	$0.641979\pi$
$74$	0	0
$75$	−5.00000	−0.577350
$76$	0	0
$77$	−2.68585	−0.306081
$78$	0	0
$79$	14.3503	1.61453	0.807266	−	0.590188i	$-0.200946\pi$
$79$	14.3503	1.61453	0.807266	+	0.590188i	$0.200946\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	8.35027	0.916561	0.458281	−	0.888808i	$-0.348465\pi$
$83$	8.35027	0.916561	0.458281	+	0.888808i	$0.348465\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0.978577	0.104914
$88$	0	0
$89$	−15.9572	−1.69145	−0.845727	−	0.533615i	$-0.820833\pi$
$89$	−15.9572	−1.69145	−0.845727	+	0.533615i	$0.820833\pi$
$90$	0	0
$91$	6.97858	0.731554
$92$	0	0
$93$	−7.66442	−0.794764
$94$	0	0
$95$	0	0
$96$	0	0
$97$	5.27131	0.535220	0.267610	−	0.963527i	$-0.413766\pi$
$97$	5.27131	0.535220	0.267610	+	0.963527i	$0.413766\pi$
$98$	0	0
$99$	2.68585	0.269938
$100$	0	0
$101$	−8.00000	−0.796030	−0.398015	−	0.917379i	$-0.630301\pi$
$101$	−8.00000	−0.796030	−0.398015	+	0.917379i	$0.630301\pi$
$102$	0	0
$103$	−11.0790	−1.09164	−0.545821	−	0.837902i	$-0.683782\pi$
$103$	−11.0790	−1.09164	−0.545821	+	0.837902i	$0.683782\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−7.56404	−0.731243	−0.365622	−	0.930764i	$-0.619144\pi$
$107$	−7.56404	−0.731243	−0.365622	+	0.930764i	$0.619144\pi$
$108$	0	0
$109$	−4.87819	−0.467246	−0.233623	−	0.972327i	$-0.575058\pi$
$109$	−4.87819	−0.467246	−0.233623	+	0.972327i	$0.575058\pi$
$110$	0	0
$111$	5.66442	0.537643
$112$	0	0
$113$	10.3503	0.973671	0.486836	−	0.873494i	$-0.338151\pi$
$113$	10.3503	0.973671	0.486836	+	0.873494i	$0.338151\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−6.97858	−0.645170
$118$	0	0
$119$	−2.29273	−0.210174
$120$	0	0
$121$	−3.78623	−0.344203
$122$	0	0
$123$	7.37169	0.664683
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−0.393115	−0.0348833	−0.0174417	−	0.999848i	$-0.505552\pi$
$127$	−0.393115	−0.0348833	−0.0174417	+	0.999848i	$0.505552\pi$
$128$	0	0
$129$	−9.37169	−0.825132
$130$	0	0
$131$	−1.80765	−0.157935	−0.0789677	−	0.996877i	$-0.525162\pi$
$131$	−1.80765	−0.157935	−0.0789677	+	0.996877i	$0.525162\pi$
$132$	0	0
$133$	1.00000	0.0867110
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−14.7862	−1.26327	−0.631636	−	0.775265i	$-0.717616\pi$
$137$	−14.7862	−1.26327	−0.631636	+	0.775265i	$0.717616\pi$
$138$	0	0
$139$	−13.3717	−1.13417	−0.567086	−	0.823659i	$-0.691929\pi$
$139$	−13.3717	−1.13417	−0.567086	+	0.823659i	$0.691929\pi$
$140$	0	0
$141$	−4.68585	−0.394619
$142$	0	0
$143$	−18.7434	−1.56740
$144$	0	0
$145$	0	0
$146$	0	0
$147$	1.00000	0.0824786
$148$	0	0
$149$	5.07896	0.416085	0.208042	−	0.978120i	$-0.433291\pi$
$149$	5.07896	0.416085	0.208042	+	0.978120i	$0.433291\pi$
$150$	0	0
$151$	9.56404	0.778310	0.389155	−	0.921172i	$-0.372767\pi$
$151$	9.56404	0.778310	0.389155	+	0.921172i	$0.372767\pi$
$152$	0	0
$153$	2.29273	0.185356
$154$	0	0
$155$	0	0
$156$	0	0
$157$	15.9572	1.27352	0.636760	−	0.771062i	$-0.280274\pi$
$157$	15.9572	1.27352	0.636760	+	0.771062i	$0.280274\pi$
$158$	0	0
$159$	−12.9786	−1.02927
$160$	0	0
$161$	4.39312	0.346226
$162$	0	0
$163$	−1.37169	−0.107439	−0.0537196	−	0.998556i	$-0.517108\pi$
$163$	−1.37169	−0.107439	−0.0537196	+	0.998556i	$0.517108\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	9.37169	0.725203	0.362602	−	0.931944i	$-0.381889\pi$
$167$	9.37169	0.725203	0.362602	+	0.931944i	$0.381889\pi$
$168$	0	0
$169$	35.7005	2.74620
$170$	0	0
$171$	−1.00000	−0.0764719
$172$	0	0
$173$	−18.7862	−1.42829	−0.714145	−	0.699997i	$-0.753184\pi$
$173$	−18.7862	−1.42829	−0.714145	+	0.699997i	$0.753184\pi$
$174$	0	0
$175$	5.00000	0.377964
$176$	0	0
$177$	−0.585462	−0.0440060
$178$	0	0
$179$	−4.35027	−0.325154	−0.162577	−	0.986696i	$-0.551981\pi$
$179$	−4.35027	−0.325154	−0.162577	+	0.986696i	$0.551981\pi$
$180$	0	0
$181$	−10.9786	−0.816031	−0.408016	−	0.912975i	$-0.633779\pi$
$181$	−10.9786	−0.816031	−0.408016	+	0.912975i	$0.633779\pi$
$182$	0	0
$183$	−6.58546	−0.486811
$184$	0	0
$185$	0	0
$186$	0	0
$187$	6.15792	0.450312
$188$	0	0
$189$	−1.00000	−0.0727393
$190$	0	0
$191$	14.3503	1.03835	0.519175	−	0.854668i	$-0.326239\pi$
$191$	14.3503	1.03835	0.519175	+	0.854668i	$0.326239\pi$
$192$	0	0
$193$	−13.3288	−0.959431	−0.479716	−	0.877424i	$-0.659260\pi$
$193$	−13.3288	−0.959431	−0.479716	+	0.877424i	$0.659260\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−26.4507	−1.88453	−0.942266	−	0.334867i	$-0.891309\pi$
$197$	−26.4507	−1.88453	−0.942266	+	0.334867i	$0.891309\pi$
$198$	0	0
$199$	4.78623	0.339287	0.169643	−	0.985506i	$-0.445738\pi$
$199$	4.78623	0.339287	0.169643	+	0.985506i	$0.445738\pi$
$200$	0	0
$201$	12.0575	0.850473
$202$	0	0
$203$	−0.978577	−0.0686827
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−4.39312	−0.305343
$208$	0	0
$209$	−2.68585	−0.185784
$210$	0	0
$211$	18.6858	1.28639	0.643193	−	0.765704i	$-0.277609\pi$
$211$	18.6858	1.28639	0.643193	+	0.765704i	$0.277609\pi$
$212$	0	0
$213$	−1.60688	−0.110102
$214$	0	0
$215$	0	0
$216$	0	0
$217$	7.66442	0.520295
$218$	0	0
$219$	−7.37169	−0.498133
$220$	0	0
$221$	−16.0000	−1.07628
$222$	0	0
$223$	−22.2070	−1.48709	−0.743547	−	0.668684i	$-0.766858\pi$
$223$	−22.2070	−1.48709	−0.743547	+	0.668684i	$0.766858\pi$
$224$	0	0
$225$	−5.00000	−0.333333
$226$	0	0
$227$	2.62831	0.174447	0.0872235	−	0.996189i	$-0.472201\pi$
$227$	2.62831	0.174447	0.0872235	+	0.996189i	$0.472201\pi$
$228$	0	0
$229$	11.9572	0.790151	0.395075	−	0.918649i	$-0.370718\pi$
$229$	11.9572	0.790151	0.395075	+	0.918649i	$0.370718\pi$
$230$	0	0
$231$	−2.68585	−0.176716
$232$	0	0
$233$	−6.58546	−0.431428	−0.215714	−	0.976457i	$-0.569208\pi$
$233$	−6.58546	−0.431428	−0.215714	+	0.976457i	$0.569208\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	14.3503	0.932150
$238$	0	0
$239$	12.9786	0.839514	0.419757	−	0.907636i	$-0.362115\pi$
$239$	12.9786	0.839514	0.419757	+	0.907636i	$0.362115\pi$
$240$	0	0
$241$	6.72869	0.433433	0.216717	−	0.976235i	$-0.430465\pi$
$241$	6.72869	0.433433	0.216717	+	0.976235i	$0.430465\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	6.97858	0.444036
$248$	0	0
$249$	8.35027	0.529177
$250$	0	0
$251$	−1.80765	−0.114098	−0.0570490	−	0.998371i	$-0.518169\pi$
$251$	−1.80765	−0.114098	−0.0570490	+	0.998371i	$0.518169\pi$
$252$	0	0
$253$	−11.7992	−0.741811
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−15.3717	−0.958860	−0.479430	−	0.877580i	$-0.659157\pi$
$257$	−15.3717	−0.958860	−0.479430	+	0.877580i	$0.659157\pi$
$258$	0	0
$259$	−5.66442	−0.351970
$260$	0	0
$261$	0.978577	0.0605724
$262$	0	0
$263$	18.3503	1.13153	0.565763	−	0.824568i	$-0.308582\pi$
$263$	18.3503	1.13153	0.565763	+	0.824568i	$0.308582\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−15.9572	−0.976562
$268$	0	0
$269$	−10.5855	−0.645407	−0.322704	−	0.946500i	$-0.604592\pi$
$269$	−10.5855	−0.645407	−0.322704	+	0.946500i	$0.604592\pi$
$270$	0	0
$271$	−8.00000	−0.485965	−0.242983	−	0.970031i	$-0.578126\pi$
$271$	−8.00000	−0.485965	−0.242983	+	0.970031i	$0.578126\pi$
$272$	0	0
$273$	6.97858	0.422363
$274$	0	0
$275$	−13.4292	−0.809813
$276$	0	0
$277$	1.21377	0.0729283	0.0364642	−	0.999335i	$-0.488391\pi$
$277$	1.21377	0.0729283	0.0364642	+	0.999335i	$0.488391\pi$
$278$	0	0
$279$	−7.66442	−0.458857
$280$	0	0
$281$	−25.5640	−1.52502	−0.762511	−	0.646975i	$-0.776034\pi$
$281$	−25.5640	−1.52502	−0.762511	+	0.646975i	$0.776034\pi$
$282$	0	0
$283$	22.5426	1.34002	0.670010	−	0.742352i	$-0.266290\pi$
$283$	22.5426	1.34002	0.670010	+	0.742352i	$0.266290\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−7.37169	−0.435137
$288$	0	0
$289$	−11.7434	−0.690787
$290$	0	0
$291$	5.27131	0.309010
$292$	0	0
$293$	−19.2860	−1.12670	−0.563350	−	0.826218i	$-0.690488\pi$
$293$	−19.2860	−1.12670	−0.563350	+	0.826218i	$0.690488\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	2.68585	0.155849
$298$	0	0
$299$	30.6577	1.77298
$300$	0	0
$301$	9.37169	0.540175
$302$	0	0
$303$	−8.00000	−0.459588
$304$	0	0
$305$	0	0
$306$	0	0
$307$	4.78623	0.273165	0.136582	−	0.990629i	$-0.456388\pi$
$307$	4.78623	0.273165	0.136582	+	0.990629i	$0.456388\pi$
$308$	0	0
$309$	−11.0790	−0.630260
$310$	0	0
$311$	19.4292	1.10173	0.550865	−	0.834594i	$-0.314298\pi$
$311$	19.4292	1.10173	0.550865	+	0.834594i	$0.314298\pi$
$312$	0	0
$313$	25.3288	1.43167	0.715836	−	0.698269i	$-0.246046\pi$
$313$	25.3288	1.43167	0.715836	+	0.698269i	$0.246046\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	29.6791	1.66695	0.833473	−	0.552561i	$-0.186349\pi$
$317$	29.6791	1.66695	0.833473	+	0.552561i	$0.186349\pi$
$318$	0	0
$319$	2.62831	0.147157
$320$	0	0
$321$	−7.56404	−0.422183
$322$	0	0
$323$	−2.29273	−0.127571
$324$	0	0
$325$	34.8929	1.93551
$326$	0	0
$327$	−4.87819	−0.269765
$328$	0	0
$329$	4.68585	0.258339
$330$	0	0
$331$	−5.22846	−0.287382	−0.143691	−	0.989623i	$-0.545897\pi$
$331$	−5.22846	−0.287382	−0.143691	+	0.989623i	$0.545897\pi$
$332$	0	0
$333$	5.66442	0.310408
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−9.32885	−0.508175	−0.254087	−	0.967181i	$-0.581775\pi$
$337$	−9.32885	−0.508175	−0.254087	+	0.967181i	$0.581775\pi$
$338$	0	0
$339$	10.3503	0.562149
$340$	0	0
$341$	−20.5855	−1.11477
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−19.3864	−1.04072	−0.520358	−	0.853948i	$-0.674201\pi$
$347$	−19.3864	−1.04072	−0.520358	+	0.853948i	$0.674201\pi$
$348$	0	0
$349$	7.95715	0.425937	0.212968	−	0.977059i	$-0.431687\pi$
$349$	7.95715	0.425937	0.212968	+	0.977059i	$0.431687\pi$
$350$	0	0
$351$	−6.97858	−0.372489
$352$	0	0
$353$	33.6216	1.78950	0.894748	−	0.446571i	$-0.147355\pi$
$353$	33.6216	1.78950	0.894748	+	0.446571i	$0.147355\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−2.29273	−0.121344
$358$	0	0
$359$	11.7220	0.618661	0.309331	−	0.950955i	$-0.399895\pi$
$359$	11.7220	0.618661	0.309331	+	0.950955i	$0.399895\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−3.78623	−0.198726
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−11.2138	−0.585354	−0.292677	−	0.956211i	$-0.594546\pi$
$367$	−11.2138	−0.585354	−0.292677	+	0.956211i	$0.594546\pi$
$368$	0	0
$369$	7.37169	0.383755
$370$	0	0
$371$	12.9786	0.673814
$372$	0	0
$373$	−18.4507	−0.955339	−0.477669	−	0.878540i	$-0.658518\pi$
$373$	−18.4507	−0.955339	−0.477669	+	0.878540i	$0.658518\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−6.82908	−0.351715
$378$	0	0
$379$	11.9425	0.613443	0.306722	−	0.951799i	$-0.400768\pi$
$379$	11.9425	0.613443	0.306722	+	0.951799i	$0.400768\pi$
$380$	0	0
$381$	−0.393115	−0.0201399
$382$	0	0
$383$	3.21377	0.164216	0.0821080	−	0.996623i	$-0.473835\pi$
$383$	3.21377	0.164216	0.0821080	+	0.996623i	$0.473835\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−9.37169	−0.476390
$388$	0	0
$389$	24.2070	1.22735	0.613673	−	0.789560i	$-0.289691\pi$
$389$	24.2070	1.22735	0.613673	+	0.789560i	$0.289691\pi$
$390$	0	0
$391$	−10.0722	−0.509375
$392$	0	0
$393$	−1.80765	−0.0911840
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−6.20077	−0.311208	−0.155604	−	0.987820i	$-0.549732\pi$
$397$	−6.20077	−0.311208	−0.155604	+	0.987820i	$0.549732\pi$
$398$	0	0
$399$	1.00000	0.0500626
$400$	0	0
$401$	5.17935	0.258644	0.129322	−	0.991603i	$-0.458720\pi$
$401$	5.17935	0.258644	0.129322	+	0.991603i	$0.458720\pi$
$402$	0	0
$403$	53.4868	2.66437
$404$	0	0
$405$	0	0
$406$	0	0
$407$	15.2138	0.754119
$408$	0	0
$409$	−12.6002	−0.623038	−0.311519	−	0.950240i	$-0.600838\pi$
$409$	−12.6002	−0.623038	−0.311519	+	0.950240i	$0.600838\pi$
$410$	0	0
$411$	−14.7862	−0.729351
$412$	0	0
$413$	0.585462	0.0288087
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−13.3717	−0.654815
$418$	0	0
$419$	25.1365	1.22800	0.613999	−	0.789307i	$-0.289560\pi$
$419$	25.1365	1.22800	0.613999	+	0.789307i	$0.289560\pi$
$420$	0	0
$421$	−20.4935	−0.998792	−0.499396	−	0.866374i	$-0.666445\pi$
$421$	−20.4935	−0.998792	−0.499396	+	0.866374i	$0.666445\pi$
$422$	0	0
$423$	−4.68585	−0.227834
$424$	0	0
$425$	−11.4637	−0.556069
$426$	0	0
$427$	6.58546	0.318693
$428$	0	0
$429$	−18.7434	−0.904939
$430$	0	0
$431$	34.3074	1.65253	0.826265	−	0.563281i	$-0.190461\pi$
$431$	34.3074	1.65253	0.826265	+	0.563281i	$0.190461\pi$
$432$	0	0
$433$	−34.5573	−1.66072	−0.830359	−	0.557229i	$-0.811865\pi$
$433$	−34.5573	−1.66072	−0.830359	+	0.557229i	$0.811865\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	4.39312	0.210151
$438$	0	0
$439$	−4.92104	−0.234868	−0.117434	−	0.993081i	$-0.537467\pi$
$439$	−4.92104	−0.234868	−0.117434	+	0.993081i	$0.537467\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	−12.4422	−0.591148	−0.295574	−	0.955320i	$-0.595511\pi$
$443$	−12.4422	−0.591148	−0.295574	+	0.955320i	$0.595511\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	5.07896	0.240227
$448$	0	0
$449$	−27.6069	−1.30285	−0.651425	−	0.758713i	$-0.725828\pi$
$449$	−27.6069	−1.30285	−0.651425	+	0.758713i	$0.725828\pi$
$450$	0	0
$451$	19.7992	0.932309
$452$	0	0
$453$	9.56404	0.449358
$454$	0	0
$455$	0	0
$456$	0	0
$457$	23.5725	1.10267	0.551337	−	0.834283i	$-0.314118\pi$
$457$	23.5725	1.10267	0.551337	+	0.834283i	$0.314118\pi$
$458$	0	0
$459$	2.29273	0.107016
$460$	0	0
$461$	−3.41454	−0.159031	−0.0795154	−	0.996834i	$-0.525337\pi$
$461$	−3.41454	−0.159031	−0.0795154	+	0.996834i	$0.525337\pi$
$462$	0	0
$463$	34.7434	1.61466	0.807331	−	0.590099i	$-0.200911\pi$
$463$	34.7434	1.61466	0.807331	+	0.590099i	$0.200911\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−16.3503	−0.756600	−0.378300	−	0.925683i	$-0.623491\pi$
$467$	−16.3503	−0.756600	−0.378300	+	0.925683i	$0.623491\pi$
$468$	0	0
$469$	−12.0575	−0.556765
$470$	0	0
$471$	15.9572	0.735267
$472$	0	0
$473$	−25.1709	−1.15736
$474$	0	0
$475$	5.00000	0.229416
$476$	0	0
$477$	−12.9786	−0.594248
$478$	0	0
$479$	−12.6002	−0.575716	−0.287858	−	0.957673i	$-0.592943\pi$
$479$	−12.6002	−0.575716	−0.287858	+	0.957673i	$0.592943\pi$
$480$	0	0
$481$	−39.5296	−1.80240
$482$	0	0
$483$	4.39312	0.199894
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−25.7648	−1.16751	−0.583757	−	0.811928i	$-0.698418\pi$
$487$	−25.7648	−1.16751	−0.583757	+	0.811928i	$0.698418\pi$
$488$	0	0
$489$	−1.37169	−0.0620301
$490$	0	0
$491$	−20.0575	−0.905184	−0.452592	−	0.891718i	$-0.649501\pi$
$491$	−20.0575	−0.905184	−0.452592	+	0.891718i	$0.649501\pi$
$492$	0	0
$493$	2.24361	0.101047
$494$	0	0
$495$	0	0
$496$	0	0
$497$	1.60688	0.0720786
$498$	0	0
$499$	29.2860	1.31102	0.655511	−	0.755186i	$-0.272453\pi$
$499$	29.2860	1.31102	0.655511	+	0.755186i	$0.272453\pi$
$500$	0	0
$501$	9.37169	0.418696
$502$	0	0
$503$	19.4292	0.866307	0.433153	−	0.901320i	$-0.357401\pi$
$503$	19.4292	0.866307	0.433153	+	0.901320i	$0.357401\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	35.7005	1.58552
$508$	0	0
$509$	20.0722	0.889686	0.444843	−	0.895609i	$-0.353259\pi$
$509$	20.0722	0.889686	0.444843	+	0.895609i	$0.353259\pi$
$510$	0	0
$511$	7.37169	0.326104
$512$	0	0
$513$	−1.00000	−0.0441511
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−12.5855	−0.553508
$518$	0	0
$519$	−18.7862	−0.824624
$520$	0	0
$521$	−39.4868	−1.72995	−0.864973	−	0.501818i	$-0.832665\pi$
$521$	−39.4868	−1.72995	−0.864973	+	0.501818i	$0.832665\pi$
$522$	0	0
$523$	−31.1281	−1.36114	−0.680568	−	0.732685i	$-0.738267\pi$
$523$	−31.1281	−1.36114	−0.680568	+	0.732685i	$0.738267\pi$
$524$	0	0
$525$	5.00000	0.218218
$526$	0	0
$527$	−17.5725	−0.765468
$528$	0	0
$529$	−3.70054	−0.160893
$530$	0	0
$531$	−0.585462	−0.0254069
$532$	0	0
$533$	−51.4439	−2.22828
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−4.35027	−0.187728
$538$	0	0
$539$	2.68585	0.115688
$540$	0	0
$541$	22.7862	0.979657	0.489828	−	0.871819i	$-0.337059\pi$
$541$	22.7862	0.979657	0.489828	+	0.871819i	$0.337059\pi$
$542$	0	0
$543$	−10.9786	−0.471136
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−2.30115	−0.0983902	−0.0491951	−	0.998789i	$-0.515666\pi$
$547$	−2.30115	−0.0983902	−0.0491951	+	0.998789i	$0.515666\pi$
$548$	0	0
$549$	−6.58546	−0.281061
$550$	0	0
$551$	−0.978577	−0.0416888
$552$	0	0
$553$	−14.3503	−0.610236
$554$	0	0
$555$	0	0
$556$	0	0
$557$	27.7367	1.17524	0.587620	−	0.809137i	$-0.300065\pi$
$557$	27.7367	1.17524	0.587620	+	0.809137i	$0.300065\pi$
$558$	0	0
$559$	65.4011	2.76617
$560$	0	0
$561$	6.15792	0.259988
$562$	0	0
$563$	22.5426	0.950058	0.475029	−	0.879970i	$-0.342438\pi$
$563$	22.5426	0.950058	0.475029	+	0.879970i	$0.342438\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−1.00000	−0.0419961
$568$	0	0
$569$	−6.43596	−0.269810	−0.134905	−	0.990859i	$-0.543073\pi$
$569$	−6.43596	−0.269810	−0.134905	+	0.990859i	$0.543073\pi$
$570$	0	0
$571$	−7.91431	−0.331204	−0.165602	−	0.986193i	$-0.552957\pi$
$571$	−7.91431	−0.331204	−0.165602	+	0.986193i	$0.552957\pi$
$572$	0	0
$573$	14.3503	0.599491
$574$	0	0
$575$	21.9656	0.916028
$576$	0	0
$577$	28.7434	1.19660	0.598301	−	0.801271i	$-0.295843\pi$
$577$	28.7434	1.19660	0.598301	+	0.801271i	$0.295843\pi$
$578$	0	0
$579$	−13.3288	−0.553928
$580$	0	0
$581$	−8.35027	−0.346428
$582$	0	0
$583$	−34.8585	−1.44369
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−38.1067	−1.57283	−0.786415	−	0.617699i	$-0.788065\pi$
$587$	−38.1067	−1.57283	−0.786415	+	0.617699i	$0.788065\pi$
$588$	0	0
$589$	7.66442	0.315807
$590$	0	0
$591$	−26.4507	−1.08803
$592$	0	0
$593$	−20.4507	−0.839808	−0.419904	−	0.907569i	$-0.637936\pi$
$593$	−20.4507	−0.839808	−0.419904	+	0.907569i	$0.637936\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	4.78623	0.195887
$598$	0	0
$599$	−2.30742	−0.0942788	−0.0471394	−	0.998888i	$-0.515010\pi$
$599$	−2.30742	−0.0942788	−0.0471394	+	0.998888i	$0.515010\pi$
$600$	0	0
$601$	39.3435	1.60486	0.802428	−	0.596749i	$-0.203541\pi$
$601$	39.3435	1.60486	0.802428	+	0.596749i	$0.203541\pi$
$602$	0	0
$603$	12.0575	0.491021
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−29.7073	−1.20578	−0.602890	−	0.797824i	$-0.705984\pi$
$607$	−29.7073	−1.20578	−0.602890	+	0.797824i	$0.705984\pi$
$608$	0	0
$609$	−0.978577	−0.0396539
$610$	0	0
$611$	32.7005	1.32292
$612$	0	0
$613$	4.62831	0.186936	0.0934678	−	0.995622i	$-0.470205\pi$
$613$	4.62831	0.186936	0.0934678	+	0.995622i	$0.470205\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−31.2860	−1.25953	−0.629763	−	0.776787i	$-0.716848\pi$
$617$	−31.2860	−1.25953	−0.629763	+	0.776787i	$0.716848\pi$
$618$	0	0
$619$	40.1151	1.61236	0.806181	−	0.591670i	$-0.201531\pi$
$619$	40.1151	1.61236	0.806181	+	0.591670i	$0.201531\pi$
$620$	0	0
$621$	−4.39312	−0.176290
$622$	0	0
$623$	15.9572	0.639310
$624$	0	0
$625$	25.0000	1.00000
$626$	0	0
$627$	−2.68585	−0.107262
$628$	0	0
$629$	12.9870	0.517826
$630$	0	0
$631$	33.9572	1.35181	0.675906	−	0.736987i	$-0.263752\pi$
$631$	33.9572	1.35181	0.675906	+	0.736987i	$0.263752\pi$
$632$	0	0
$633$	18.6858	0.742696
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−6.97858	−0.276501
$638$	0	0
$639$	−1.60688	−0.0635674
$640$	0	0
$641$	47.1365	1.86178	0.930890	−	0.365300i	$-0.119034\pi$
$641$	47.1365	1.86178	0.930890	+	0.365300i	$0.119034\pi$
$642$	0	0
$643$	−7.61531	−0.300318	−0.150159	−	0.988662i	$-0.547979\pi$
$643$	−7.61531	−0.300318	−0.150159	+	0.988662i	$0.547979\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	35.9290	1.41251	0.706257	−	0.707955i	$-0.250382\pi$
$647$	35.9290	1.41251	0.706257	+	0.707955i	$0.250382\pi$
$648$	0	0
$649$	−1.57246	−0.0617245
$650$	0	0
$651$	7.66442	0.300392
$652$	0	0
$653$	−27.7367	−1.08542	−0.542710	−	0.839920i	$-0.682602\pi$
$653$	−27.7367	−1.08542	−0.542710	+	0.839920i	$0.682602\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−7.37169	−0.287597
$658$	0	0
$659$	6.39312	0.249040	0.124520	−	0.992217i	$-0.460261\pi$
$659$	6.39312	0.249040	0.124520	+	0.992217i	$0.460261\pi$
$660$	0	0
$661$	17.1365	0.666533	0.333266	−	0.942833i	$-0.391849\pi$
$661$	17.1365	0.666533	0.333266	+	0.942833i	$0.391849\pi$
$662$	0	0
$663$	−16.0000	−0.621389
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−4.29900	−0.166458
$668$	0	0
$669$	−22.2070	−0.858574
$670$	0	0
$671$	−17.6875	−0.682820
$672$	0	0
$673$	39.8715	1.53693	0.768466	−	0.639891i	$-0.221020\pi$
$673$	39.8715	1.53693	0.768466	+	0.639891i	$0.221020\pi$
$674$	0	0
$675$	−5.00000	−0.192450
$676$	0	0
$677$	−23.9572	−0.920748	−0.460374	−	0.887725i	$-0.652285\pi$
$677$	−23.9572	−0.920748	−0.460374	+	0.887725i	$0.652285\pi$
$678$	0	0
$679$	−5.27131	−0.202294
$680$	0	0
$681$	2.62831	0.100717
$682$	0	0
$683$	−17.7220	−0.678112	−0.339056	−	0.940766i	$-0.610108\pi$
$683$	−17.7220	−0.678112	−0.339056	+	0.940766i	$0.610108\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	11.9572	0.456194
$688$	0	0
$689$	90.5720	3.45052
$690$	0	0
$691$	−1.25662	−0.0478039	−0.0239020	−	0.999714i	$-0.507609\pi$
$691$	−1.25662	−0.0478039	−0.0239020	+	0.999714i	$0.507609\pi$
$692$	0	0
$693$	−2.68585	−0.102027
$694$	0	0
$695$	0	0
$696$	0	0
$697$	16.9013	0.640183
$698$	0	0
$699$	−6.58546	−0.249085
$700$	0	0
$701$	−22.9210	−0.865716	−0.432858	−	0.901462i	$-0.642495\pi$
$701$	−22.9210	−0.865716	−0.432858	+	0.901462i	$0.642495\pi$
$702$	0	0
$703$	−5.66442	−0.213638
$704$	0	0
$705$	0	0
$706$	0	0
$707$	8.00000	0.300871
$708$	0	0
$709$	7.75639	0.291297	0.145649	−	0.989336i	$-0.453473\pi$
$709$	7.75639	0.291297	0.145649	+	0.989336i	$0.453473\pi$
$710$	0	0
$711$	14.3503	0.538177
$712$	0	0
$713$	33.6707	1.26098
$714$	0	0
$715$	0	0
$716$	0	0
$717$	12.9786	0.484694
$718$	0	0
$719$	−34.3565	−1.28128	−0.640641	−	0.767840i	$-0.721331\pi$
$719$	−34.3565	−1.28128	−0.640641	+	0.767840i	$0.721331\pi$
$720$	0	0
$721$	11.0790	0.412602
$722$	0	0
$723$	6.72869	0.250243
$724$	0	0
$725$	−4.89289	−0.181717
$726$	0	0
$727$	16.4998	0.611943	0.305971	−	0.952041i	$-0.401019\pi$
$727$	16.4998	0.611943	0.305971	+	0.952041i	$0.401019\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−21.4868	−0.794717
$732$	0	0
$733$	3.57246	0.131952	0.0659759	−	0.997821i	$-0.478984\pi$
$733$	3.57246	0.131952	0.0659759	+	0.997821i	$0.478984\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	32.3847	1.19291
$738$	0	0
$739$	−6.74338	−0.248059	−0.124030	−	0.992279i	$-0.539582\pi$
$739$	−6.74338	−0.248059	−0.124030	+	0.992279i	$0.539582\pi$
$740$	0	0
$741$	6.97858	0.256364
$742$	0	0
$743$	36.2646	1.33042	0.665209	−	0.746657i	$-0.268342\pi$
$743$	36.2646	1.33042	0.665209	+	0.746657i	$0.268342\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	8.35027	0.305520
$748$	0	0
$749$	7.56404	0.276384
$750$	0	0
$751$	45.5934	1.66373	0.831864	−	0.554980i	$-0.187274\pi$
$751$	45.5934	1.66373	0.831864	+	0.554980i	$0.187274\pi$
$752$	0	0
$753$	−1.80765	−0.0658745
$754$	0	0
$755$	0	0
$756$	0	0
$757$	24.0722	0.874920	0.437460	−	0.899238i	$-0.355878\pi$
$757$	24.0722	0.874920	0.437460	+	0.899238i	$0.355878\pi$
$758$	0	0
$759$	−11.7992	−0.428285
$760$	0	0
$761$	40.3650	1.46323	0.731614	−	0.681719i	$-0.238767\pi$
$761$	40.3650	1.46323	0.731614	+	0.681719i	$0.238767\pi$
$762$	0	0
$763$	4.87819	0.176602
$764$	0	0
$765$	0	0
$766$	0	0
$767$	4.08569	0.147526
$768$	0	0
$769$	50.4998	1.82107	0.910534	−	0.413434i	$-0.135671\pi$
$769$	50.4998	1.82107	0.910534	+	0.413434i	$0.135671\pi$
$770$	0	0
$771$	−15.3717	−0.553598
$772$	0	0
$773$	−27.2860	−0.981409	−0.490705	−	0.871326i	$-0.663261\pi$
$773$	−27.2860	−0.981409	−0.490705	+	0.871326i	$0.663261\pi$
$774$	0	0
$775$	38.3221	1.37657
$776$	0	0
$777$	−5.66442	−0.203210
$778$	0	0
$779$	−7.37169	−0.264118
$780$	0	0
$781$	−4.31585	−0.154433
$782$	0	0
$783$	0.978577	0.0349715
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−13.2860	−0.473595	−0.236797	−	0.971559i	$-0.576098\pi$
$787$	−13.2860	−0.473595	−0.236797	+	0.971559i	$0.576098\pi$
$788$	0	0
$789$	18.3503	0.653287
$790$	0	0
$791$	−10.3503	−0.368013
$792$	0	0
$793$	45.9572	1.63199
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−26.7862	−0.948817	−0.474408	−	0.880305i	$-0.657338\pi$
$797$	−26.7862	−0.948817	−0.474408	+	0.880305i	$0.657338\pi$
$798$	0	0
$799$	−10.7434	−0.380074
$800$	0	0
$801$	−15.9572	−0.563818
$802$	0	0
$803$	−19.7992	−0.698700
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−10.5855	−0.372626
$808$	0	0
$809$	14.5855	0.512798	0.256399	−	0.966571i	$-0.417464\pi$
$809$	14.5855	0.512798	0.256399	+	0.966571i	$0.417464\pi$
$810$	0	0
$811$	2.04285	0.0717340	0.0358670	−	0.999357i	$-0.488581\pi$
$811$	2.04285	0.0717340	0.0358670	+	0.999357i	$0.488581\pi$
$812$	0	0
$813$	−8.00000	−0.280572
$814$	0	0
$815$	0	0
$816$	0	0
$817$	9.37169	0.327874
$818$	0	0
$819$	6.97858	0.243851
$820$	0	0
$821$	16.0920	0.561613	0.280807	−	0.959764i	$-0.409398\pi$
$821$	16.0920	0.561613	0.280807	+	0.959764i	$0.409398\pi$
$822$	0	0
$823$	6.82908	0.238047	0.119023	−	0.992891i	$-0.462024\pi$
$823$	6.82908	0.238047	0.119023	+	0.992891i	$0.462024\pi$
$824$	0	0
$825$	−13.4292	−0.467546
$826$	0	0
$827$	33.4355	1.16267	0.581333	−	0.813666i	$-0.302531\pi$
$827$	33.4355	1.16267	0.581333	+	0.813666i	$0.302531\pi$
$828$	0	0
$829$	31.4783	1.09329	0.546644	−	0.837365i	$-0.315905\pi$
$829$	31.4783	1.09329	0.546644	+	0.837365i	$0.315905\pi$
$830$	0	0
$831$	1.21377	0.0421052
$832$	0	0
$833$	2.29273	0.0794384
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−7.66442	−0.264921
$838$	0	0
$839$	−18.0722	−0.623923	−0.311961	−	0.950095i	$-0.600986\pi$
$839$	−18.0722	−0.623923	−0.311961	+	0.950095i	$0.600986\pi$
$840$	0	0
$841$	−28.0424	−0.966979
$842$	0	0
$843$	−25.5640	−0.880472
$844$	0	0
$845$	0	0
$846$	0	0
$847$	3.78623	0.130096
$848$	0	0
$849$	22.5426	0.773661
$850$	0	0
$851$	−24.8845	−0.853028
$852$	0	0
$853$	−10.4998	−0.359505	−0.179753	−	0.983712i	$-0.557530\pi$
$853$	−10.4998	−0.359505	−0.179753	+	0.983712i	$0.557530\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−22.6148	−0.772508	−0.386254	−	0.922392i	$-0.626231\pi$
$857$	−22.6148	−0.772508	−0.386254	+	0.922392i	$0.626231\pi$
$858$	0	0
$859$	−16.9870	−0.579589	−0.289794	−	0.957089i	$-0.593587\pi$
$859$	−16.9870	−0.579589	−0.289794	+	0.957089i	$0.593587\pi$
$860$	0	0
$861$	−7.37169	−0.251227
$862$	0	0
$863$	35.9656	1.22428	0.612141	−	0.790748i	$-0.290308\pi$
$863$	35.9656	1.22428	0.612141	+	0.790748i	$0.290308\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−11.7434	−0.398826
$868$	0	0
$869$	38.5426	1.30747
$870$	0	0
$871$	−84.1445	−2.85113
$872$	0	0
$873$	5.27131	0.178407
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−23.4208	−0.790865	−0.395432	−	0.918495i	$-0.629405\pi$
$877$	−23.4208	−0.790865	−0.395432	+	0.918495i	$0.629405\pi$
$878$	0	0
$879$	−19.2860	−0.650501
$880$	0	0
$881$	−37.0361	−1.24778	−0.623889	−	0.781513i	$-0.714448\pi$
$881$	−37.0361	−1.24778	−0.623889	+	0.781513i	$0.714448\pi$
$882$	0	0
$883$	−2.54262	−0.0855658	−0.0427829	−	0.999084i	$-0.513622\pi$
$883$	−2.54262	−0.0855658	−0.0427829	+	0.999084i	$0.513622\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−1.37169	−0.0460569	−0.0230285	−	0.999735i	$-0.507331\pi$
$887$	−1.37169	−0.0460569	−0.0230285	+	0.999735i	$0.507331\pi$
$888$	0	0
$889$	0.393115	0.0131847
$890$	0	0
$891$	2.68585	0.0899792
$892$	0	0
$893$	4.68585	0.156806
$894$	0	0
$895$	0	0
$896$	0	0
$897$	30.6577	1.02363
$898$	0	0
$899$	−7.50023	−0.250147
$900$	0	0
$901$	−29.7564	−0.991329
$902$	0	0
$903$	9.37169	0.311870
$904$	0	0
$905$	0	0
$906$	0	0
$907$	11.0705	0.367591	0.183796	−	0.982964i	$-0.441162\pi$
$907$	11.0705	0.367591	0.183796	+	0.982964i	$0.441162\pi$
$908$	0	0
$909$	−8.00000	−0.265343
$910$	0	0
$911$	−37.0508	−1.22755	−0.613774	−	0.789482i	$-0.710349\pi$
$911$	−37.0508	−1.22755	−0.613774	+	0.789482i	$0.710349\pi$
$912$	0	0
$913$	22.4275	0.742243
$914$	0	0
$915$	0	0
$916$	0	0
$917$	1.80765	0.0596940
$918$	0	0
$919$	−29.4868	−0.972679	−0.486339	−	0.873770i	$-0.661668\pi$
$919$	−29.4868	−0.972679	−0.486339	+	0.873770i	$0.661668\pi$
$920$	0	0
$921$	4.78623	0.157712
$922$	0	0
$923$	11.2138	0.369106
$924$	0	0
$925$	−28.3221	−0.931225
$926$	0	0
$927$	−11.0790	−0.363881
$928$	0	0
$929$	−37.5359	−1.23151	−0.615756	−	0.787937i	$-0.711149\pi$
$929$	−37.5359	−1.23151	−0.615756	+	0.787937i	$0.711149\pi$
$930$	0	0
$931$	−1.00000	−0.0327737
$932$	0	0
$933$	19.4292	0.636084
$934$	0	0
$935$	0	0
$936$	0	0
$937$	2.78623	0.0910222	0.0455111	−	0.998964i	$-0.485508\pi$
$937$	2.78623	0.0910222	0.0455111	+	0.998964i	$0.485508\pi$
$938$	0	0
$939$	25.3288	0.826576
$940$	0	0
$941$	−53.3288	−1.73847	−0.869235	−	0.494399i	$-0.835388\pi$
$941$	−53.3288	−1.73847	−0.869235	+	0.494399i	$0.835388\pi$
$942$	0	0
$943$	−32.3847	−1.05459
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−40.8438	−1.32724	−0.663622	−	0.748068i	$-0.730982\pi$
$947$	−40.8438	−1.32724	−0.663622	+	0.748068i	$0.730982\pi$
$948$	0	0
$949$	51.4439	1.66994
$950$	0	0
$951$	29.6791	0.962411
$952$	0	0
$953$	19.0508	0.617116	0.308558	−	0.951205i	$-0.400154\pi$
$953$	19.0508	0.617116	0.308558	+	0.951205i	$0.400154\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	2.62831	0.0849611
$958$	0	0
$959$	14.7862	0.477472
$960$	0	0
$961$	27.7434	0.894948
$962$	0	0
$963$	−7.56404	−0.243748
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−0.384694	−0.0123709	−0.00618546	−	0.999981i	$-0.501969\pi$
$967$	−0.384694	−0.0123709	−0.00618546	+	0.999981i	$0.501969\pi$
$968$	0	0
$969$	−2.29273	−0.0736531
$970$	0	0
$971$	4.49977	0.144405	0.0722023	−	0.997390i	$-0.476997\pi$
$971$	4.49977	0.144405	0.0722023	+	0.997390i	$0.476997\pi$
$972$	0	0
$973$	13.3717	0.428677
$974$	0	0
$975$	34.8929	1.11747
$976$	0	0
$977$	35.4355	1.13368	0.566841	−	0.823827i	$-0.308165\pi$
$977$	35.4355	1.13368	0.566841	+	0.823827i	$0.308165\pi$
$978$	0	0
$979$	−42.8585	−1.36976
$980$	0	0
$981$	−4.87819	−0.155749
$982$	0	0
$983$	−1.17092	−0.0373467	−0.0186733	−	0.999826i	$-0.505944\pi$
$983$	−1.17092	−0.0373467	−0.0186733	+	0.999826i	$0.505944\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	4.68585	0.149152
$988$	0	0
$989$	41.1709	1.30916
$990$	0	0
$991$	25.7648	0.818446	0.409223	−	0.912434i	$-0.365800\pi$
$991$	25.7648	0.818446	0.409223	+	0.912434i	$0.365800\pi$
$992$	0	0
$993$	−5.22846	−0.165920
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−25.4439	−0.805817	−0.402909	−	0.915240i	$-0.632001\pi$
$997$	−25.4439	−0.805817	−0.402909	+	0.915240i	$0.632001\pi$
$998$	0	0
$999$	5.66442	0.179214

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3192.2.a.v.1.3	✓	3
3.2	odd	2		9576.2.a.cc.1.1		3
4.3	odd	2		6384.2.a.bv.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3192.2.a.v.1.3	✓	3	1.1	even	1	trivial
6384.2.a.bv.1.1		3	4.3	odd	2
9576.2.a.cc.1.1		3	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -1.00000 q^{7} +1.00000 q^{9} +2.68585 q^{11} -6.97858 q^{13} +2.29273 q^{17} -1.00000 q^{19} -1.00000 q^{21} -4.39312 q^{23} -5.00000 q^{25} +1.00000 q^{27} +0.978577 q^{29} -7.66442 q^{31} +2.68585 q^{33} +5.66442 q^{37} -6.97858 q^{39} +7.37169 q^{41} -9.37169 q^{43} -4.68585 q^{47} +1.00000 q^{49} +2.29273 q^{51} -12.9786 q^{53} -1.00000 q^{57} -0.585462 q^{59} -6.58546 q^{61} -1.00000 q^{63} +12.0575 q^{67} -4.39312 q^{69} -1.60688 q^{71} -7.37169 q^{73} -5.00000 q^{75} -2.68585 q^{77} +14.3503 q^{79} +1.00000 q^{81} +8.35027 q^{83} +0.978577 q^{87} -15.9572 q^{89} +6.97858 q^{91} -7.66442 q^{93} +5.27131 q^{97} +2.68585 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{3} - 3 q^{7} + 3 q^{9}+O(q^{10}) \) 3 * q + 3 * q^3 - 3 * q^7 + 3 * q^9	\( 3 q + 3 q^{3} - 3 q^{7} + 3 q^{9} - 4 q^{11} - 6 q^{13} + 4 q^{17} - 3 q^{19} - 3 q^{21} - 4 q^{23} - 15 q^{25} + 3 q^{27} - 12 q^{29} + 4 q^{31} - 4 q^{33} - 10 q^{37} - 6 q^{39} - 2 q^{41} - 4 q^{43} - 2 q^{47} + 3 q^{49} + 4 q^{51} - 24 q^{53} - 3 q^{57} + 4 q^{59} - 14 q^{61} - 3 q^{63} - 4 q^{69} - 14 q^{71} + 2 q^{73} - 15 q^{75} + 4 q^{77} + 4 q^{79} + 3 q^{81} - 14 q^{83} - 12 q^{87} - 18 q^{89} + 6 q^{91} + 4 q^{93} - 2 q^{97} - 4 q^{99}+O(q^{100}) \) 3 * q + 3 * q^3 - 3 * q^7 + 3 * q^9 - 4 * q^11 - 6 * q^13 + 4 * q^17 - 3 * q^19 - 3 * q^21 - 4 * q^23 - 15 * q^25 + 3 * q^27 - 12 * q^29 + 4 * q^31 - 4 * q^33 - 10 * q^37 - 6 * q^39 - 2 * q^41 - 4 * q^43 - 2 * q^47 + 3 * q^49 + 4 * q^51 - 24 * q^53 - 3 * q^57 + 4 * q^59 - 14 * q^61 - 3 * q^63 - 4 * q^69 - 14 * q^71 + 2 * q^73 - 15 * q^75 + 4 * q^77 + 4 * q^79 + 3 * q^81 - 14 * q^83 - 12 * q^87 - 18 * q^89 + 6 * q^91 + 4 * q^93 - 2 * q^97 - 4 * q^99

Introduction

L-functions

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Database

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