LMFDB - Embedded newform 8496.2.a.bu.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8496,2,Mod(1,8496)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8496, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("8496.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8496 = 2^{4} \cdot 3^{2} \cdot 59 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8496.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:	$67.8409015573$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	5.5.246832.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 5x^{3} + 6x^{2} + 7x - 2 $ x^5 - 2x^4 - 5x^3 + 6x^2 + 7x - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 531)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.15351$ of defining polynomial
Character	$\chi$	$=$	8496.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.11101 q^{5} +1.09624 q^{7} +O(q^{10})$	$q-2.11101 q^{5} +1.09624 q^{7} -1.39257 q^{11} +6.23742 q^{13} -6.48881 q^{17} +2.66942 q^{19} -5.07756 q^{23} -0.543632 q^{25} +1.56801 q^{29} +7.25220 q^{31} -2.31417 q^{35} -2.28998 q^{37} -3.03897 q^{41} +5.01313 q^{43} +2.93882 q^{47} -5.79826 q^{49} -8.52551 q^{53} +2.93974 q^{55} +1.00000 q^{59} -6.92270 q^{61} -13.1673 q^{65} -1.04722 q^{67} +15.1100 q^{71} -0.393655 q^{73} -1.52659 q^{77} +6.27333 q^{79} -5.63407 q^{83} +13.6980 q^{85} +3.22447 q^{89} +6.83770 q^{91} -5.63517 q^{95} +2.04005 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 8 q^{5} - 2 q^{7}+O(q^{10}) $ 5 * q - 8 * q^5 - 2 * q^7	$ 5 q - 8 q^{5} - 2 q^{7} + 10 q^{11} - 4 q^{13} - 8 q^{17} + 6 q^{19} + 4 q^{23} + 7 q^{25} - 20 q^{29} + 6 q^{31} + 12 q^{35} - 8 q^{41} + 4 q^{43} - 2 q^{47} - 5 q^{49} - 20 q^{53} - 10 q^{55} + 5 q^{59} - 14 q^{61} - 2 q^{65} + 24 q^{67} + 12 q^{71} - 2 q^{73} - 2 q^{77} + 8 q^{79} - 18 q^{83} + 10 q^{85} - 4 q^{89} + 24 q^{91} - 4 q^{95} + 20 q^{97}+O(q^{100}) $ 5 * q - 8 * q^5 - 2 * q^7 + 10 * q^11 - 4 * q^13 - 8 * q^17 + 6 * q^19 + 4 * q^23 + 7 * q^25 - 20 * q^29 + 6 * q^31 + 12 * q^35 - 8 * q^41 + 4 * q^43 - 2 * q^47 - 5 * q^49 - 20 * q^53 - 10 * q^55 + 5 * q^59 - 14 * q^61 - 2 * q^65 + 24 * q^67 + 12 * q^71 - 2 * q^73 - 2 * q^77 + 8 * q^79 - 18 * q^83 + 10 * q^85 - 4 * q^89 + 24 * q^91 - 4 * q^95 + 20 * q^97

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	−2.11101	−0.944073	−0.472036	−	0.881579i	$-0.656481\pi$
$5$	−2.11101	−0.944073	−0.472036	+	0.881579i	$0.656481\pi$
$6$	0	0
$7$	1.09624	0.414339	0.207169	−	0.978305i	$-0.433575\pi$
$7$	1.09624	0.414339	0.207169	+	0.978305i	$0.433575\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−1.39257	−0.419876	−0.209938	−	0.977715i	$-0.567326\pi$
$11$	−1.39257	−0.419876	−0.209938	+	0.977715i	$0.567326\pi$
$12$	0	0
$13$	6.23742	1.72995	0.864975	−	0.501815i	$-0.167334\pi$
$13$	6.23742	1.72995	0.864975	+	0.501815i	$0.167334\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−6.48881	−1.57377	−0.786884	−	0.617101i	$-0.788307\pi$
$17$	−6.48881	−1.57377	−0.786884	+	0.617101i	$0.788307\pi$
$18$	0	0
$19$	2.66942	0.612406	0.306203	−	0.951966i	$-0.400941\pi$
$19$	2.66942	0.612406	0.306203	+	0.951966i	$0.400941\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−5.07756	−1.05874	−0.529372	−	0.848390i	$-0.677572\pi$
$23$	−5.07756	−1.05874	−0.529372	+	0.848390i	$0.677572\pi$
$24$	0	0
$25$	−0.543632	−0.108726
$26$	0	0
$27$	0	0
$28$	0	0
$29$	1.56801	0.291172	0.145586	−	0.989346i	$-0.453493\pi$
$29$	1.56801	0.291172	0.145586	+	0.989346i	$0.453493\pi$
$30$	0	0
$31$	7.25220	1.30253	0.651267	−	0.758849i	$-0.274238\pi$
$31$	7.25220	1.30253	0.651267	+	0.758849i	$0.274238\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−2.31417	−0.391166
$36$	0	0
$37$	−2.28998	−0.376470	−0.188235	−	0.982124i	$-0.560277\pi$
$37$	−2.28998	−0.376470	−0.188235	+	0.982124i	$0.560277\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−3.03897	−0.474607	−0.237303	−	0.971436i	$-0.576264\pi$
$41$	−3.03897	−0.474607	−0.237303	+	0.971436i	$0.576264\pi$
$42$	0	0
$43$	5.01313	0.764496	0.382248	−	0.924060i	$-0.375150\pi$
$43$	5.01313	0.764496	0.382248	+	0.924060i	$0.375150\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	2.93882	0.428671	0.214336	−	0.976760i	$-0.431241\pi$
$47$	2.93882	0.428671	0.214336	+	0.976760i	$0.431241\pi$
$48$	0	0
$49$	−5.79826	−0.828323
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−8.52551	−1.17107	−0.585534	−	0.810648i	$-0.699115\pi$
$53$	−8.52551	−1.17107	−0.585534	+	0.810648i	$0.699115\pi$
$54$	0	0
$55$	2.93974	0.396394
$56$	0	0
$57$	0	0
$58$	0	0
$59$	1.00000	0.130189
$59$	1.00000	0.130189
$60$	0	0
$61$	−6.92270	−0.886360	−0.443180	−	0.896433i	$-0.646150\pi$
$61$	−6.92270	−0.886360	−0.443180	+	0.896433i	$0.646150\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−13.1673	−1.63320
$66$	0	0
$67$	−1.04722	−0.127938	−0.0639689	−	0.997952i	$-0.520376\pi$
$67$	−1.04722	−0.127938	−0.0639689	+	0.997952i	$0.520376\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	15.1100	1.79323	0.896613	−	0.442814i	$-0.146020\pi$
$71$	15.1100	1.79323	0.896613	+	0.442814i	$0.146020\pi$
$72$	0	0
$73$	−0.393655	−0.0460738	−0.0230369	−	0.999735i	$-0.507334\pi$
$73$	−0.393655	−0.0460738	−0.0230369	+	0.999735i	$0.507334\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−1.52659	−0.173971
$78$	0	0
$79$	6.27333	0.705805	0.352902	−	0.935660i	$-0.385195\pi$
$79$	6.27333	0.705805	0.352902	+	0.935660i	$0.385195\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−5.63407	−0.618420	−0.309210	−	0.950994i	$-0.600065\pi$
$83$	−5.63407	−0.618420	−0.309210	+	0.950994i	$0.600065\pi$
$84$	0	0
$85$	13.6980	1.48575
$86$	0	0
$87$	0	0
$88$	0	0
$89$	3.22447	0.341793	0.170897	−	0.985289i	$-0.445334\pi$
$89$	3.22447	0.341793	0.170897	+	0.985289i	$0.445334\pi$
$90$	0	0
$91$	6.83770	0.716786
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−5.63517	−0.578156
$96$	0	0
$97$	2.04005	0.207136	0.103568	−	0.994622i	$-0.466974\pi$
$97$	2.04005	0.207136	0.103568	+	0.994622i	$0.466974\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	8.16783	0.812729	0.406365	−	0.913711i	$-0.366796\pi$
$101$	8.16783	0.812729	0.406365	+	0.913711i	$0.366796\pi$
$102$	0	0
$103$	−18.8669	−1.85901	−0.929504	−	0.368813i	$-0.879764\pi$
$103$	−18.8669	−1.85901	−0.929504	+	0.368813i	$0.879764\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	9.35442	0.904326	0.452163	−	0.891935i	$-0.350653\pi$
$107$	9.35442	0.904326	0.452163	+	0.891935i	$0.350653\pi$
$108$	0	0
$109$	−19.6185	−1.87911	−0.939554	−	0.342400i	$-0.888760\pi$
$109$	−19.6185	−1.87911	−0.939554	+	0.342400i	$0.888760\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	8.53075	0.802505	0.401253	−	0.915967i	$-0.368575\pi$
$113$	8.53075	0.802505	0.401253	+	0.915967i	$0.368575\pi$
$114$	0	0
$115$	10.7188	0.999531
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−7.11328	−0.652073
$120$	0	0
$121$	−9.06074	−0.823704
$122$	0	0
$123$	0	0
$124$	0	0
$125$	11.7027	1.04672
$126$	0	0
$127$	1.71474	0.152159	0.0760793	−	0.997102i	$-0.475760\pi$
$127$	1.71474	0.152159	0.0760793	+	0.997102i	$0.475760\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−3.47586	−0.303687	−0.151844	−	0.988405i	$-0.548521\pi$
$131$	−3.47586	−0.303687	−0.151844	+	0.988405i	$0.548521\pi$
$132$	0	0
$133$	2.92632	0.253744
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−16.7854	−1.43407	−0.717037	−	0.697035i	$-0.754502\pi$
$137$	−16.7854	−1.43407	−0.717037	+	0.697035i	$0.754502\pi$
$138$	0	0
$139$	−19.8309	−1.68204	−0.841018	−	0.541008i	$-0.818043\pi$
$139$	−19.8309	−1.68204	−0.841018	+	0.541008i	$0.818043\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−8.68607	−0.726365
$144$	0	0
$145$	−3.31008	−0.274887
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−18.8342	−1.54296	−0.771478	−	0.636257i	$-0.780482\pi$
$149$	−18.8342	−1.54296	−0.771478	+	0.636257i	$0.780482\pi$
$150$	0	0
$151$	−4.13992	−0.336902	−0.168451	−	0.985710i	$-0.553877\pi$
$151$	−4.13992	−0.336902	−0.168451	+	0.985710i	$0.553877\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−15.3095	−1.22969
$156$	0	0
$157$	20.2708	1.61778	0.808891	−	0.587958i	$-0.200068\pi$
$157$	20.2708	1.61778	0.808891	+	0.587958i	$0.200068\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−5.56621	−0.438679
$162$	0	0
$163$	−4.16213	−0.326004	−0.163002	−	0.986626i	$-0.552118\pi$
$163$	−4.16213	−0.326004	−0.163002	+	0.986626i	$0.552118\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−0.544372	−0.0421248	−0.0210624	−	0.999778i	$-0.506705\pi$
$167$	−0.544372	−0.0421248	−0.0210624	+	0.999778i	$0.506705\pi$
$168$	0	0
$169$	25.9055	1.99273
$170$	0	0
$171$	0	0
$172$	0	0
$173$	2.14881	0.163371	0.0816854	−	0.996658i	$-0.473970\pi$
$173$	2.14881	0.163371	0.0816854	+	0.996658i	$0.473970\pi$
$174$	0	0
$175$	−0.595950	−0.0450496
$176$	0	0
$177$	0	0
$178$	0	0
$179$	15.7304	1.17574	0.587872	−	0.808954i	$-0.299966\pi$
$179$	15.7304	1.17574	0.587872	+	0.808954i	$0.299966\pi$
$180$	0	0
$181$	−3.18613	−0.236823	−0.118412	−	0.992965i	$-0.537780\pi$
$181$	−3.18613	−0.236823	−0.118412	+	0.992965i	$0.537780\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	4.83417	0.355415
$186$	0	0
$187$	9.03614	0.660788
$188$	0	0
$189$	0	0
$190$	0	0
$191$	6.71163	0.485636	0.242818	−	0.970072i	$-0.421928\pi$
$191$	6.71163	0.485636	0.242818	+	0.970072i	$0.421928\pi$
$192$	0	0
$193$	13.4512	0.968239	0.484120	−	0.875002i	$-0.339140\pi$
$193$	13.4512	0.968239	0.484120	+	0.875002i	$0.339140\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	14.5352	1.03559	0.517795	−	0.855505i	$-0.326753\pi$
$197$	14.5352	1.03559	0.517795	+	0.855505i	$0.326753\pi$
$198$	0	0
$199$	−21.8949	−1.55209	−0.776045	−	0.630677i	$-0.782777\pi$
$199$	−21.8949	−1.55209	−0.776045	+	0.630677i	$0.782777\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	1.71891	0.120644
$204$	0	0
$205$	6.41529	0.448063
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−3.71736	−0.257135
$210$	0	0
$211$	−18.8449	−1.29734	−0.648669	−	0.761071i	$-0.724674\pi$
$211$	−18.8449	−1.29734	−0.648669	+	0.761071i	$0.724674\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−10.5828	−0.721740
$216$	0	0
$217$	7.95013	0.539690
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−40.4735	−2.72254
$222$	0	0
$223$	21.6651	1.45080	0.725400	−	0.688328i	$-0.241655\pi$
$223$	21.6651	1.45080	0.725400	+	0.688328i	$0.241655\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	17.4203	1.15622	0.578112	−	0.815957i	$-0.303790\pi$
$227$	17.4203	1.15622	0.578112	+	0.815957i	$0.303790\pi$
$228$	0	0
$229$	−23.9150	−1.58035	−0.790173	−	0.612883i	$-0.790009\pi$
$229$	−23.9150	−1.58035	−0.790173	+	0.612883i	$0.790009\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−1.42711	−0.0934930	−0.0467465	−	0.998907i	$-0.514885\pi$
$233$	−1.42711	−0.0934930	−0.0467465	+	0.998907i	$0.514885\pi$
$234$	0	0
$235$	−6.20388	−0.404697
$236$	0	0
$237$	0	0
$238$	0	0
$239$	27.4489	1.77552	0.887762	−	0.460303i	$-0.152259\pi$
$239$	27.4489	1.77552	0.887762	+	0.460303i	$0.152259\pi$
$240$	0	0
$241$	−19.3569	−1.24689	−0.623444	−	0.781868i	$-0.714267\pi$
$241$	−19.3569	−1.24689	−0.623444	+	0.781868i	$0.714267\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	12.2402	0.781998
$246$	0	0
$247$	16.6503	1.05943
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−27.8353	−1.75695	−0.878474	−	0.477790i	$-0.841438\pi$
$251$	−27.8353	−1.75695	−0.878474	+	0.477790i	$0.841438\pi$
$252$	0	0
$253$	7.07087	0.444542
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−7.88924	−0.492117	−0.246059	−	0.969255i	$-0.579136\pi$
$257$	−7.88924	−0.492117	−0.246059	+	0.969255i	$0.579136\pi$
$258$	0	0
$259$	−2.51036	−0.155986
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−17.0382	−1.05062	−0.525311	−	0.850911i	$-0.676051\pi$
$263$	−17.0382	−1.05062	−0.525311	+	0.850911i	$0.676051\pi$
$264$	0	0
$265$	17.9974	1.10557
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−13.1291	−0.800494	−0.400247	−	0.916407i	$-0.631076\pi$
$269$	−13.1291	−0.800494	−0.400247	+	0.916407i	$0.631076\pi$
$270$	0	0
$271$	13.7399	0.834639	0.417319	−	0.908760i	$-0.362970\pi$
$271$	13.7399	0.834639	0.417319	+	0.908760i	$0.362970\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0.757047	0.0456517
$276$	0	0
$277$	−22.1961	−1.33363	−0.666816	−	0.745222i	$-0.732344\pi$
$277$	−22.1961	−1.33363	−0.666816	+	0.745222i	$0.732344\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−0.776826	−0.0463416	−0.0231708	−	0.999732i	$-0.507376\pi$
$281$	−0.776826	−0.0463416	−0.0231708	+	0.999732i	$0.507376\pi$
$282$	0	0
$283$	16.4114	0.975558	0.487779	−	0.872967i	$-0.337807\pi$
$283$	16.4114	0.975558	0.487779	+	0.872967i	$0.337807\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−3.33143	−0.196648
$288$	0	0
$289$	25.1047	1.47674
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−21.0273	−1.22843	−0.614214	−	0.789140i	$-0.710527\pi$
$293$	−21.0273	−1.22843	−0.614214	+	0.789140i	$0.710527\pi$
$294$	0	0
$295$	−2.11101	−0.122908
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−31.6709	−1.83157
$300$	0	0
$301$	5.49559	0.316760
$302$	0	0
$303$	0	0
$304$	0	0
$305$	14.6139	0.836789
$306$	0	0
$307$	−19.6118	−1.11930	−0.559651	−	0.828728i	$-0.689065\pi$
$307$	−19.6118	−1.11930	−0.559651	+	0.828728i	$0.689065\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−34.4576	−1.95391	−0.976956	−	0.213442i	$-0.931533\pi$
$311$	−34.4576	−1.95391	−0.976956	+	0.213442i	$0.931533\pi$
$312$	0	0
$313$	−8.20876	−0.463987	−0.231993	−	0.972717i	$-0.574525\pi$
$313$	−8.20876	−0.463987	−0.231993	+	0.972717i	$0.574525\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	11.4125	0.640989	0.320494	−	0.947250i	$-0.396151\pi$
$317$	11.4125	0.640989	0.320494	+	0.947250i	$0.396151\pi$
$318$	0	0
$319$	−2.18356	−0.122256
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−17.3213	−0.963785
$324$	0	0
$325$	−3.39086	−0.188091
$326$	0	0
$327$	0	0
$328$	0	0
$329$	3.22165	0.177615
$330$	0	0
$331$	−5.61289	−0.308512	−0.154256	−	0.988031i	$-0.549298\pi$
$331$	−5.61289	−0.308512	−0.154256	+	0.988031i	$0.549298\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	2.21068	0.120783
$336$	0	0
$337$	5.94854	0.324038	0.162019	−	0.986788i	$-0.448199\pi$
$337$	5.94854	0.324038	0.162019	+	0.986788i	$0.448199\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−10.0992	−0.546903
$342$	0	0
$343$	−14.0299	−0.757545
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−12.6802	−0.680708	−0.340354	−	0.940297i	$-0.610547\pi$
$347$	−12.6802	−0.680708	−0.340354	+	0.940297i	$0.610547\pi$
$348$	0	0
$349$	−0.624399	−0.0334233	−0.0167117	−	0.999860i	$-0.505320\pi$
$349$	−0.624399	−0.0334233	−0.0167117	+	0.999860i	$0.505320\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−13.4958	−0.718311	−0.359155	−	0.933278i	$-0.616935\pi$
$353$	−13.4958	−0.718311	−0.359155	+	0.933278i	$0.616935\pi$
$354$	0	0
$355$	−31.8974	−1.69294
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−9.47711	−0.500183	−0.250091	−	0.968222i	$-0.580461\pi$
$359$	−9.47711	−0.500183	−0.250091	+	0.968222i	$0.580461\pi$
$360$	0	0
$361$	−11.8742	−0.624959
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0.831009	0.0434970
$366$	0	0
$367$	−31.9206	−1.66624	−0.833120	−	0.553092i	$-0.813448\pi$
$367$	−31.9206	−1.66624	−0.833120	+	0.553092i	$0.813448\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−9.34598	−0.485219
$372$	0	0
$373$	−34.6247	−1.79280	−0.896398	−	0.443249i	$-0.853826\pi$
$373$	−34.6247	−1.79280	−0.896398	+	0.443249i	$0.853826\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	9.78032	0.503712
$378$	0	0
$379$	−19.1592	−0.984141	−0.492071	−	0.870555i	$-0.663760\pi$
$379$	−19.1592	−0.984141	−0.492071	+	0.870555i	$0.663760\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	12.0369	0.615055	0.307527	−	0.951539i	$-0.400498\pi$
$383$	12.0369	0.615055	0.307527	+	0.951539i	$0.400498\pi$
$384$	0	0
$385$	3.22265	0.164241
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−35.7718	−1.81370	−0.906851	−	0.421452i	$-0.861521\pi$
$389$	−35.7718	−1.81370	−0.906851	+	0.421452i	$0.861521\pi$
$390$	0	0
$391$	32.9473	1.66622
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−13.2431	−0.666331
$396$	0	0
$397$	−6.07608	−0.304950	−0.152475	−	0.988307i	$-0.548724\pi$
$397$	−6.07608	−0.304950	−0.152475	+	0.988307i	$0.548724\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	12.6527	0.631846	0.315923	−	0.948785i	$-0.397686\pi$
$401$	12.6527	0.631846	0.315923	+	0.948785i	$0.397686\pi$
$402$	0	0
$403$	45.2350	2.25332
$404$	0	0
$405$	0	0
$406$	0	0
$407$	3.18896	0.158071
$408$	0	0
$409$	27.6114	1.36530	0.682649	−	0.730747i	$-0.260828\pi$
$409$	27.6114	1.36530	0.682649	+	0.730747i	$0.260828\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	1.09624	0.0539423
$414$	0	0
$415$	11.8936	0.583833
$416$	0	0
$417$	0	0
$418$	0	0
$419$	37.3028	1.82236	0.911179	−	0.412010i	$-0.135173\pi$
$419$	37.3028	1.82236	0.911179	+	0.412010i	$0.135173\pi$
$420$	0	0
$421$	−17.1206	−0.834408	−0.417204	−	0.908813i	$-0.636990\pi$
$421$	−17.1206	−0.834408	−0.417204	+	0.908813i	$0.636990\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	3.52752	0.171110
$426$	0	0
$427$	−7.58892	−0.367254
$428$	0	0
$429$	0	0
$430$	0	0
$431$	6.95204	0.334868	0.167434	−	0.985883i	$-0.446452\pi$
$431$	6.95204	0.334868	0.167434	+	0.985883i	$0.446452\pi$
$432$	0	0
$433$	0.780679	0.0375171	0.0187585	−	0.999824i	$-0.494029\pi$
$433$	0.780679	0.0375171	0.0187585	+	0.999824i	$0.494029\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−13.5541	−0.648381
$438$	0	0
$439$	−29.3855	−1.40249	−0.701246	−	0.712919i	$-0.747373\pi$
$439$	−29.3855	−1.40249	−0.701246	+	0.712919i	$0.747373\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−25.0656	−1.19090	−0.595452	−	0.803391i	$-0.703027\pi$
$443$	−25.0656	−1.19090	−0.595452	+	0.803391i	$0.703027\pi$
$444$	0	0
$445$	−6.80690	−0.322678
$446$	0	0
$447$	0	0
$448$	0	0
$449$	34.3303	1.62014	0.810072	−	0.586330i	$-0.199428\pi$
$449$	34.3303	1.62014	0.810072	+	0.586330i	$0.199428\pi$
$450$	0	0
$451$	4.23198	0.199276
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−14.4345	−0.676698
$456$	0	0
$457$	−2.23763	−0.104672	−0.0523360	−	0.998630i	$-0.516667\pi$
$457$	−2.23763	−0.104672	−0.0523360	+	0.998630i	$0.516667\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−35.3414	−1.64601	−0.823006	−	0.568033i	$-0.807705\pi$
$461$	−35.3414	−1.64601	−0.823006	+	0.568033i	$0.807705\pi$
$462$	0	0
$463$	5.69731	0.264776	0.132388	−	0.991198i	$-0.457735\pi$
$463$	5.69731	0.264776	0.132388	+	0.991198i	$0.457735\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−28.5160	−1.31956	−0.659781	−	0.751458i	$-0.729351\pi$
$467$	−28.5160	−1.31956	−0.659781	+	0.751458i	$0.729351\pi$
$468$	0	0
$469$	−1.14800	−0.0530096
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−6.98115	−0.320994
$474$	0	0
$475$	−1.45118	−0.0665847
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−8.67563	−0.396400	−0.198200	−	0.980162i	$-0.563510\pi$
$479$	−8.67563	−0.396400	−0.198200	+	0.980162i	$0.563510\pi$
$480$	0	0
$481$	−14.2836	−0.651274
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−4.30656	−0.195551
$486$	0	0
$487$	−21.5198	−0.975156	−0.487578	−	0.873080i	$-0.662119\pi$
$487$	−21.5198	−0.975156	−0.487578	+	0.873080i	$0.662119\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−13.1169	−0.591957	−0.295979	−	0.955195i	$-0.595646\pi$
$491$	−13.1169	−0.591957	−0.295979	+	0.955195i	$0.595646\pi$
$492$	0	0
$493$	−10.1745	−0.458236
$494$	0	0
$495$	0	0
$496$	0	0
$497$	16.5642	0.743004
$498$	0	0
$499$	27.5880	1.23501	0.617505	−	0.786567i	$-0.288144\pi$
$499$	27.5880	1.23501	0.617505	+	0.786567i	$0.288144\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−28.2351	−1.25894	−0.629470	−	0.777025i	$-0.716728\pi$
$503$	−28.2351	−1.25894	−0.629470	+	0.777025i	$0.716728\pi$
$504$	0	0
$505$	−17.2424	−0.767276
$506$	0	0
$507$	0	0
$508$	0	0
$509$	21.1253	0.936363	0.468181	−	0.883632i	$-0.344909\pi$
$509$	21.1253	0.936363	0.468181	+	0.883632i	$0.344909\pi$
$510$	0	0
$511$	−0.431539	−0.0190902
$512$	0	0
$513$	0	0
$514$	0	0
$515$	39.8282	1.75504
$516$	0	0
$517$	−4.09252	−0.179989
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−8.65236	−0.379067	−0.189533	−	0.981874i	$-0.560697\pi$
$521$	−8.65236	−0.379067	−0.189533	+	0.981874i	$0.560697\pi$
$522$	0	0
$523$	21.6819	0.948085	0.474043	−	0.880502i	$-0.342794\pi$
$523$	21.6819	0.948085	0.474043	+	0.880502i	$0.342794\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−47.0581	−2.04988
$528$	0	0
$529$	2.78158	0.120938
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−18.9553	−0.821046
$534$	0	0
$535$	−19.7473	−0.853749
$536$	0	0
$537$	0	0
$538$	0	0
$539$	8.07450	0.347793
$540$	0	0
$541$	−33.6531	−1.44686	−0.723430	−	0.690398i	$-0.757436\pi$
$541$	−33.6531	−1.44686	−0.723430	+	0.690398i	$0.757436\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	41.4148	1.77402
$546$	0	0
$547$	0.162107	0.00693119	0.00346559	−	0.999994i	$-0.498897\pi$
$547$	0.162107	0.00693119	0.00346559	+	0.999994i	$0.498897\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	4.18566	0.178315
$552$	0	0
$553$	6.87706	0.292442
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−30.2877	−1.28333	−0.641666	−	0.766984i	$-0.721757\pi$
$557$	−30.2877	−1.28333	−0.641666	+	0.766984i	$0.721757\pi$
$558$	0	0
$559$	31.2690	1.32254
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−18.7555	−0.790450	−0.395225	−	0.918584i	$-0.629333\pi$
$563$	−18.7555	−0.790450	−0.395225	+	0.918584i	$0.629333\pi$
$564$	0	0
$565$	−18.0085	−0.757624
$566$	0	0
$567$	0	0
$568$	0	0
$569$	4.23290	0.177452	0.0887261	−	0.996056i	$-0.471720\pi$
$569$	4.23290	0.177452	0.0887261	+	0.996056i	$0.471720\pi$
$570$	0	0
$571$	−13.7100	−0.573746	−0.286873	−	0.957969i	$-0.592616\pi$
$571$	−13.7100	−0.573746	−0.286873	+	0.957969i	$0.592616\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	2.76032	0.115113
$576$	0	0
$577$	20.1181	0.837529	0.418765	−	0.908095i	$-0.362463\pi$
$577$	20.1181	0.837529	0.418765	+	0.908095i	$0.362463\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−6.17628	−0.256235
$582$	0	0
$583$	11.8724	0.491704
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−12.9025	−0.532544	−0.266272	−	0.963898i	$-0.585792\pi$
$587$	−12.9025	−0.532544	−0.266272	+	0.963898i	$0.585792\pi$
$588$	0	0
$589$	19.3591	0.797679
$590$	0	0
$591$	0	0
$592$	0	0
$593$	22.1016	0.907603	0.453802	−	0.891103i	$-0.350067\pi$
$593$	22.1016	0.907603	0.453802	+	0.891103i	$0.350067\pi$
$594$	0	0
$595$	15.0162	0.615605
$596$	0	0
$597$	0	0
$598$	0	0
$599$	44.4096	1.81453	0.907263	−	0.420564i	$-0.138168\pi$
$599$	44.4096	1.81453	0.907263	+	0.420564i	$0.138168\pi$
$600$	0	0
$601$	−19.5324	−0.796741	−0.398371	−	0.917225i	$-0.630424\pi$
$601$	−19.5324	−0.796741	−0.398371	+	0.917225i	$0.630424\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	19.1273	0.777636
$606$	0	0
$607$	24.0502	0.976167	0.488084	−	0.872797i	$-0.337696\pi$
$607$	24.0502	0.976167	0.488084	+	0.872797i	$0.337696\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	18.3307	0.741579
$612$	0	0
$613$	−3.46718	−0.140038	−0.0700190	−	0.997546i	$-0.522306\pi$
$613$	−3.46718	−0.140038	−0.0700190	+	0.997546i	$0.522306\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	31.8929	1.28396	0.641980	−	0.766722i	$-0.278113\pi$
$617$	31.8929	1.28396	0.641980	+	0.766722i	$0.278113\pi$
$618$	0	0
$619$	−8.81235	−0.354198	−0.177099	−	0.984193i	$-0.556671\pi$
$619$	−8.81235	−0.354198	−0.177099	+	0.984193i	$0.556671\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	3.53479	0.141618
$624$	0	0
$625$	−21.9863	−0.879452
$626$	0	0
$627$	0	0
$628$	0	0
$629$	14.8592	0.592476
$630$	0	0
$631$	−4.11493	−0.163813	−0.0819065	−	0.996640i	$-0.526101\pi$
$631$	−4.11493	−0.163813	−0.0819065	+	0.996640i	$0.526101\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−3.61984	−0.143649
$636$	0	0
$637$	−36.1662	−1.43296
$638$	0	0
$639$	0	0
$640$	0	0
$641$	43.6648	1.72465	0.862327	−	0.506351i	$-0.169006\pi$
$641$	43.6648	1.72465	0.862327	+	0.506351i	$0.169006\pi$
$642$	0	0
$643$	25.0167	0.986561	0.493281	−	0.869870i	$-0.335798\pi$
$643$	25.0167	0.986561	0.493281	+	0.869870i	$0.335798\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	0.907084	0.0356612	0.0178306	−	0.999841i	$-0.494324\pi$
$647$	0.907084	0.0356612	0.0178306	+	0.999841i	$0.494324\pi$
$648$	0	0
$649$	−1.39257	−0.0546633
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−14.5632	−0.569902	−0.284951	−	0.958542i	$-0.591977\pi$
$653$	−14.5632	−0.569902	−0.284951	+	0.958542i	$0.591977\pi$
$654$	0	0
$655$	7.33758	0.286703
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−29.6707	−1.15581	−0.577904	−	0.816105i	$-0.696129\pi$
$659$	−29.6707	−1.15581	−0.577904	+	0.816105i	$0.696129\pi$
$660$	0	0
$661$	34.3832	1.33735	0.668677	−	0.743553i	$-0.266861\pi$
$661$	34.3832	1.33735	0.668677	+	0.743553i	$0.266861\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−6.17748	−0.239553
$666$	0	0
$667$	−7.96164	−0.308276
$668$	0	0
$669$	0	0
$670$	0	0
$671$	9.64036	0.372162
$672$	0	0
$673$	16.3217	0.629155	0.314577	−	0.949232i	$-0.398137\pi$
$673$	16.3217	0.629155	0.314577	+	0.949232i	$0.398137\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−30.0619	−1.15537	−0.577687	−	0.816258i	$-0.696045\pi$
$677$	−30.0619	−1.15537	−0.577687	+	0.816258i	$0.696045\pi$
$678$	0	0
$679$	2.23638	0.0858243
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−24.5873	−0.940808	−0.470404	−	0.882451i	$-0.655892\pi$
$683$	−24.5873	−0.940808	−0.470404	+	0.882451i	$0.655892\pi$
$684$	0	0
$685$	35.4342	1.35387
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−53.1772	−2.02589
$690$	0	0
$691$	−10.7921	−0.410549	−0.205275	−	0.978704i	$-0.565809\pi$
$691$	−10.7921	−0.410549	−0.205275	+	0.978704i	$0.565809\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	41.8633	1.58796
$696$	0	0
$697$	19.7193	0.746921
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−26.8572	−1.01438	−0.507192	−	0.861833i	$-0.669317\pi$
$701$	−26.8572	−1.01438	−0.507192	+	0.861833i	$0.669317\pi$
$702$	0	0
$703$	−6.11290	−0.230552
$704$	0	0
$705$	0	0
$706$	0	0
$707$	8.95388	0.336745
$708$	0	0
$709$	39.0954	1.46826	0.734129	−	0.679010i	$-0.237591\pi$
$709$	39.0954	1.46826	0.734129	+	0.679010i	$0.237591\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−36.8234	−1.37905
$714$	0	0
$715$	18.3364	0.685742
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−41.0530	−1.53102	−0.765510	−	0.643424i	$-0.777513\pi$
$719$	−41.0530	−1.53102	−0.765510	+	0.643424i	$0.777513\pi$
$720$	0	0
$721$	−20.6826	−0.770259
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−0.852419	−0.0316580
$726$	0	0
$727$	28.2244	1.04678	0.523392	−	0.852092i	$-0.324666\pi$
$727$	28.2244	1.04678	0.523392	+	0.852092i	$0.324666\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−32.5293	−1.20314
$732$	0	0
$733$	18.0881	0.668099	0.334049	−	0.942556i	$-0.391585\pi$
$733$	18.0881	0.668099	0.334049	+	0.942556i	$0.391585\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	1.45832	0.0537181
$738$	0	0
$739$	1.41118	0.0519113	0.0259556	−	0.999663i	$-0.491737\pi$
$739$	1.41118	0.0519113	0.0259556	+	0.999663i	$0.491737\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	26.5689	0.974719	0.487359	−	0.873202i	$-0.337960\pi$
$743$	26.5689	0.974719	0.487359	+	0.873202i	$0.337960\pi$
$744$	0	0
$745$	39.7591	1.45666
$746$	0	0
$747$	0	0
$748$	0	0
$749$	10.2547	0.374697
$750$	0	0
$751$	−19.6508	−0.717066	−0.358533	−	0.933517i	$-0.616723\pi$
$751$	−19.6508	−0.717066	−0.358533	+	0.933517i	$0.616723\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	8.73942	0.318060
$756$	0	0
$757$	23.8108	0.865418	0.432709	−	0.901534i	$-0.357558\pi$
$757$	23.8108	0.865418	0.432709	+	0.901534i	$0.357558\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	32.7124	1.18582	0.592911	−	0.805268i	$-0.297979\pi$
$761$	32.7124	1.18582	0.592911	+	0.805268i	$0.297979\pi$
$762$	0	0
$763$	−21.5065	−0.778588
$764$	0	0
$765$	0	0
$766$	0	0
$767$	6.23742	0.225220
$768$	0	0
$769$	14.2572	0.514127	0.257064	−	0.966394i	$-0.417245\pi$
$769$	14.2572	0.514127	0.257064	+	0.966394i	$0.417245\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−13.7948	−0.496166	−0.248083	−	0.968739i	$-0.579801\pi$
$773$	−13.7948	−0.496166	−0.248083	+	0.968739i	$0.579801\pi$
$774$	0	0
$775$	−3.94253	−0.141620
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−8.11227	−0.290652
$780$	0	0
$781$	−21.0418	−0.752934
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−42.7918	−1.52730
$786$	0	0
$787$	−24.1930	−0.862388	−0.431194	−	0.902259i	$-0.641907\pi$
$787$	−24.1930	−0.862388	−0.431194	+	0.902259i	$0.641907\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	9.35173	0.332509
$792$	0	0
$793$	−43.1798	−1.53336
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−19.4755	−0.689859	−0.344929	−	0.938629i	$-0.612097\pi$
$797$	−19.4755	−0.689859	−0.344929	+	0.938629i	$0.612097\pi$
$798$	0	0
$799$	−19.0695	−0.674629
$800$	0	0
$801$	0	0
$802$	0	0
$803$	0.548193	0.0193453
$804$	0	0
$805$	11.7503	0.414145
$806$	0	0
$807$	0	0
$808$	0	0
$809$	19.6358	0.690357	0.345179	−	0.938537i	$-0.387818\pi$
$809$	19.6358	0.690357	0.345179	+	0.938537i	$0.387818\pi$
$810$	0	0
$811$	42.1136	1.47881	0.739405	−	0.673261i	$-0.235107\pi$
$811$	42.1136	1.47881	0.739405	+	0.673261i	$0.235107\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	8.78631	0.307771
$816$	0	0
$817$	13.3821	0.468182
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−20.0009	−0.698037	−0.349019	−	0.937116i	$-0.613485\pi$
$821$	−20.0009	−0.698037	−0.349019	+	0.937116i	$0.613485\pi$
$822$	0	0
$823$	−16.9437	−0.590620	−0.295310	−	0.955401i	$-0.595423\pi$
$823$	−16.9437	−0.590620	−0.295310	+	0.955401i	$0.595423\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	33.5188	1.16556	0.582782	−	0.812628i	$-0.301964\pi$
$827$	33.5188	1.16556	0.582782	+	0.812628i	$0.301964\pi$
$828$	0	0
$829$	35.0043	1.21575	0.607876	−	0.794032i	$-0.292022\pi$
$829$	35.0043	1.21575	0.607876	+	0.794032i	$0.292022\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	37.6238	1.30359
$834$	0	0
$835$	1.14918	0.0397689
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−24.4827	−0.845237	−0.422619	−	0.906308i	$-0.638889\pi$
$839$	−24.4827	−0.845237	−0.422619	+	0.906308i	$0.638889\pi$
$840$	0	0
$841$	−26.5414	−0.915219
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−54.6867	−1.88128
$846$	0	0
$847$	−9.93273	−0.341293
$848$	0	0
$849$	0	0
$850$	0	0
$851$	11.6275	0.398585
$852$	0	0
$853$	−56.8488	−1.94647	−0.973233	−	0.229821i	$-0.926186\pi$
$853$	−56.8488	−1.94647	−0.973233	+	0.229821i	$0.926186\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	43.6475	1.49097	0.745485	−	0.666523i	$-0.232218\pi$
$857$	43.6475	1.49097	0.745485	+	0.666523i	$0.232218\pi$
$858$	0	0
$859$	35.0392	1.19552	0.597761	−	0.801674i	$-0.296057\pi$
$859$	35.0392	1.19552	0.597761	+	0.801674i	$0.296057\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−30.4689	−1.03717	−0.518587	−	0.855025i	$-0.673542\pi$
$863$	−30.4689	−1.03717	−0.518587	+	0.855025i	$0.673542\pi$
$864$	0	0
$865$	−4.53616	−0.154234
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−8.73607	−0.296351
$870$	0	0
$871$	−6.53193	−0.221326
$872$	0	0
$873$	0	0
$874$	0	0
$875$	12.8289	0.433696
$876$	0	0
$877$	17.9852	0.607315	0.303658	−	0.952781i	$-0.401792\pi$
$877$	17.9852	0.607315	0.303658	+	0.952781i	$0.401792\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	27.8993	0.939950	0.469975	−	0.882680i	$-0.344263\pi$
$881$	27.8993	0.939950	0.469975	+	0.882680i	$0.344263\pi$
$882$	0	0
$883$	−19.7435	−0.664423	−0.332211	−	0.943205i	$-0.607795\pi$
$883$	−19.7435	−0.664423	−0.332211	+	0.943205i	$0.607795\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−34.0507	−1.14331	−0.571655	−	0.820494i	$-0.693698\pi$
$887$	−34.0507	−1.14331	−0.571655	+	0.820494i	$0.693698\pi$
$888$	0	0
$889$	1.87976	0.0630452
$890$	0	0
$891$	0	0
$892$	0	0
$893$	7.84494	0.262521
$894$	0	0
$895$	−33.2070	−1.10999
$896$	0	0
$897$	0	0
$898$	0	0
$899$	11.3715	0.379261
$900$	0	0
$901$	55.3204	1.84299
$902$	0	0
$903$	0	0
$904$	0	0
$905$	6.72596	0.223579
$906$	0	0
$907$	53.9117	1.79011	0.895054	−	0.445957i	$-0.147136\pi$
$907$	53.9117	1.79011	0.895054	+	0.445957i	$0.147136\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−1.45383	−0.0481675	−0.0240837	−	0.999710i	$-0.507667\pi$
$911$	−1.45383	−0.0481675	−0.0240837	+	0.999710i	$0.507667\pi$
$912$	0	0
$913$	7.84585	0.259660
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−3.81037	−0.125829
$918$	0	0
$919$	−51.8918	−1.71175	−0.855877	−	0.517180i	$-0.826982\pi$
$919$	−51.8918	−1.71175	−0.855877	+	0.517180i	$0.826982\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	94.2475	3.10219
$924$	0	0
$925$	1.24490	0.0409322
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−13.8429	−0.454172	−0.227086	−	0.973875i	$-0.572920\pi$
$929$	−13.8429	−0.454172	−0.227086	+	0.973875i	$0.572920\pi$
$930$	0	0
$931$	−15.4780	−0.507270
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−19.0754	−0.623832
$936$	0	0
$937$	−21.1929	−0.692343	−0.346171	−	0.938171i	$-0.612518\pi$
$937$	−21.1929	−0.692343	−0.346171	+	0.938171i	$0.612518\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−56.0903	−1.82849	−0.914246	−	0.405160i	$-0.867216\pi$
$941$	−56.0903	−1.82849	−0.914246	+	0.405160i	$0.867216\pi$
$942$	0	0
$943$	15.4305	0.502487
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−4.13650	−0.134418	−0.0672091	−	0.997739i	$-0.521409\pi$
$947$	−4.13650	−0.134418	−0.0672091	+	0.997739i	$0.521409\pi$
$948$	0	0
$949$	−2.45539	−0.0797053
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−47.9570	−1.55348	−0.776740	−	0.629822i	$-0.783128\pi$
$953$	−47.9570	−1.55348	−0.776740	+	0.629822i	$0.783128\pi$
$954$	0	0
$955$	−14.1683	−0.458476
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−18.4008	−0.594192
$960$	0	0
$961$	21.5944	0.696592
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−28.3957	−0.914089
$966$	0	0
$967$	21.7913	0.700760	0.350380	−	0.936608i	$-0.386052\pi$
$967$	21.7913	0.700760	0.350380	+	0.936608i	$0.386052\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−35.8613	−1.15084	−0.575422	−	0.817857i	$-0.695162\pi$
$971$	−35.8613	−1.15084	−0.575422	+	0.817857i	$0.695162\pi$
$972$	0	0
$973$	−21.7394	−0.696933
$974$	0	0
$975$	0	0
$976$	0	0
$977$	31.7008	1.01420	0.507100	−	0.861887i	$-0.330717\pi$
$977$	31.7008	1.01420	0.507100	+	0.861887i	$0.330717\pi$
$978$	0	0
$979$	−4.49031	−0.143511
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−25.0289	−0.798297	−0.399148	−	0.916886i	$-0.630694\pi$
$983$	−25.0289	−0.798297	−0.399148	+	0.916886i	$0.630694\pi$
$984$	0	0
$985$	−30.6840	−0.977672
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−25.4545	−0.809405
$990$	0	0
$991$	16.4805	0.523520	0.261760	−	0.965133i	$-0.415697\pi$
$991$	16.4805	0.523520	0.261760	+	0.965133i	$0.415697\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	46.2205	1.46529
$996$	0	0
$997$	20.6217	0.653097	0.326549	−	0.945180i	$-0.394114\pi$
$997$	20.6217	0.653097	0.326549	+	0.945180i	$0.394114\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8496.2.a.bu.1.3		5
3.2	odd	2		8496.2.a.bz.1.3		5
4.3	odd	2		531.2.a.e.1.2	✓	5
12.11	even	2		531.2.a.g.1.4	yes	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
531.2.a.e.1.2	✓	5	4.3	odd	2
531.2.a.g.1.4	yes	5	12.11	even	2
8496.2.a.bu.1.3		5	1.1	even	1	trivial
8496.2.a.bz.1.3		5	3.2	odd	2

Overview	Random
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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:	\( N \)	\(=\)	\( 8496 = 2^{4} \cdot 3^{2} \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8496.a (trivial)

\(f(q)\)	\(=\)	\(q-2.11101 q^{5} +1.09624 q^{7} +O(q^{10})\)	\(q-2.11101 q^{5} +1.09624 q^{7} -1.39257 q^{11} +6.23742 q^{13} -6.48881 q^{17} +2.66942 q^{19} -5.07756 q^{23} -0.543632 q^{25} +1.56801 q^{29} +7.25220 q^{31} -2.31417 q^{35} -2.28998 q^{37} -3.03897 q^{41} +5.01313 q^{43} +2.93882 q^{47} -5.79826 q^{49} -8.52551 q^{53} +2.93974 q^{55} +1.00000 q^{59} -6.92270 q^{61} -13.1673 q^{65} -1.04722 q^{67} +15.1100 q^{71} -0.393655 q^{73} -1.52659 q^{77} +6.27333 q^{79} -5.63407 q^{83} +13.6980 q^{85} +3.22447 q^{89} +6.83770 q^{91} -5.63517 q^{95} +2.04005 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 8 q^{5} - 2 q^{7}+O(q^{10}) \) 5 * q - 8 * q^5 - 2 * q^7	\( 5 q - 8 q^{5} - 2 q^{7} + 10 q^{11} - 4 q^{13} - 8 q^{17} + 6 q^{19} + 4 q^{23} + 7 q^{25} - 20 q^{29} + 6 q^{31} + 12 q^{35} - 8 q^{41} + 4 q^{43} - 2 q^{47} - 5 q^{49} - 20 q^{53} - 10 q^{55} + 5 q^{59} - 14 q^{61} - 2 q^{65} + 24 q^{67} + 12 q^{71} - 2 q^{73} - 2 q^{77} + 8 q^{79} - 18 q^{83} + 10 q^{85} - 4 q^{89} + 24 q^{91} - 4 q^{95} + 20 q^{97}+O(q^{100}) \) 5 * q - 8 * q^5 - 2 * q^7 + 10 * q^11 - 4 * q^13 - 8 * q^17 + 6 * q^19 + 4 * q^23 + 7 * q^25 - 20 * q^29 + 6 * q^31 + 12 * q^35 - 8 * q^41 + 4 * q^43 - 2 * q^47 - 5 * q^49 - 20 * q^53 - 10 * q^55 + 5 * q^59 - 14 * q^61 - 2 * q^65 + 24 * q^67 + 12 * q^71 - 2 * q^73 - 2 * q^77 + 8 * q^79 - 18 * q^83 + 10 * q^85 - 4 * q^89 + 24 * q^91 - 4 * q^95 + 20 * q^97

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