LMFDB - Embedded newform 3825.2.a.w.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3825,2,Mod(1,3825)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3825, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("3825.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3825 = 3^{2} \cdot 5^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3825.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:	$30.5427787731$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 255)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.61803$ of defining polynomial
Character	$\chi$	$=$	3825.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.61803 q^{2} +0.618034 q^{4} -1.00000 q^{7} -2.23607 q^{8} +O(q^{10})$	$q+1.61803 q^{2} +0.618034 q^{4} -1.00000 q^{7} -2.23607 q^{8} +3.00000 q^{11} +2.00000 q^{13} -1.61803 q^{14} -4.85410 q^{16} +1.00000 q^{17} -4.23607 q^{19} +4.85410 q^{22} -0.472136 q^{23} +3.23607 q^{26} -0.618034 q^{28} +6.23607 q^{29} +2.47214 q^{31} -3.38197 q^{32} +1.61803 q^{34} +4.70820 q^{37} -6.85410 q^{38} +3.76393 q^{41} +8.47214 q^{43} +1.85410 q^{44} -0.763932 q^{46} +12.7082 q^{47} -6.00000 q^{49} +1.23607 q^{52} +3.00000 q^{53} +2.23607 q^{56} +10.0902 q^{58} +8.94427 q^{59} -10.0000 q^{61} +4.00000 q^{62} +4.23607 q^{64} +5.52786 q^{67} +0.618034 q^{68} -12.9443 q^{71} +1.76393 q^{73} +7.61803 q^{74} -2.61803 q^{76} -3.00000 q^{77} +2.00000 q^{79} +6.09017 q^{82} +4.94427 q^{83} +13.7082 q^{86} -6.70820 q^{88} +12.0000 q^{89} -2.00000 q^{91} -0.291796 q^{92} +20.5623 q^{94} -13.4164 q^{97} -9.70820 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{2} - q^{4} - 2 q^{7}+O(q^{10}) $ 2 * q + q^2 - q^4 - 2 * q^7	$ 2 q + q^{2} - q^{4} - 2 q^{7} + 6 q^{11} + 4 q^{13} - q^{14} - 3 q^{16} + 2 q^{17} - 4 q^{19} + 3 q^{22} + 8 q^{23} + 2 q^{26} + q^{28} + 8 q^{29} - 4 q^{31} - 9 q^{32} + q^{34} - 4 q^{37} - 7 q^{38} + 12 q^{41} + 8 q^{43} - 3 q^{44} - 6 q^{46} + 12 q^{47} - 12 q^{49} - 2 q^{52} + 6 q^{53} + 9 q^{58} - 20 q^{61} + 8 q^{62} + 4 q^{64} + 20 q^{67} - q^{68} - 8 q^{71} + 8 q^{73} + 13 q^{74} - 3 q^{76} - 6 q^{77} + 4 q^{79} + q^{82} - 8 q^{83} + 14 q^{86} + 24 q^{89} - 4 q^{91} - 14 q^{92} + 21 q^{94} - 6 q^{98}+O(q^{100}) $ 2 * q + q^2 - q^4 - 2 * q^7 + 6 * q^11 + 4 * q^13 - q^14 - 3 * q^16 + 2 * q^17 - 4 * q^19 + 3 * q^22 + 8 * q^23 + 2 * q^26 + q^28 + 8 * q^29 - 4 * q^31 - 9 * q^32 + q^34 - 4 * q^37 - 7 * q^38 + 12 * q^41 + 8 * q^43 - 3 * q^44 - 6 * q^46 + 12 * q^47 - 12 * q^49 - 2 * q^52 + 6 * q^53 + 9 * q^58 - 20 * q^61 + 8 * q^62 + 4 * q^64 + 20 * q^67 - q^68 - 8 * q^71 + 8 * q^73 + 13 * q^74 - 3 * q^76 - 6 * q^77 + 4 * q^79 + q^82 - 8 * q^83 + 14 * q^86 + 24 * q^89 - 4 * q^91 - 14 * q^92 + 21 * q^94 - 6 * q^98

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 $

$2$	1.61803	1.14412	0.572061	−	0.820211i	$-0.306144\pi$
$2$	1.61803	1.14412	0.572061	+	0.820211i	$0.306144\pi$
$3$	0	0
$3$	0	0
$4$	0.618034	0.309017
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−1.00000	−0.377964	−0.188982	−	0.981981i	$-0.560519\pi$
$7$	−1.00000	−0.377964	−0.188982	+	0.981981i	$0.560519\pi$
$8$	−2.23607	−0.790569
$9$	0	0
$10$	0	0
$11$	3.00000	0.904534	0.452267	−	0.891883i	$-0.350615\pi$
$11$	3.00000	0.904534	0.452267	+	0.891883i	$0.350615\pi$
$12$	0	0
$13$	2.00000	0.554700	0.277350	−	0.960769i	$-0.410544\pi$
$13$	2.00000	0.554700	0.277350	+	0.960769i	$0.410544\pi$
$14$	−1.61803	−0.432438
$15$	0	0
$16$	−4.85410	−1.21353
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	0	0
$19$	−4.23607	−0.971821	−0.485910	−	0.874009i	$-0.661512\pi$
$19$	−4.23607	−0.971821	−0.485910	+	0.874009i	$0.661512\pi$
$20$	0	0
$21$	0	0
$22$	4.85410	1.03490
$23$	−0.472136	−0.0984472	−0.0492236	−	0.998788i	$-0.515675\pi$
$23$	−0.472136	−0.0984472	−0.0492236	+	0.998788i	$0.515675\pi$
$24$	0	0
$25$	0	0
$26$	3.23607	0.634645
$27$	0	0
$28$	−0.618034	−0.116797
$29$	6.23607	1.15801	0.579004	−	0.815324i	$-0.303441\pi$
$29$	6.23607	1.15801	0.579004	+	0.815324i	$0.303441\pi$
$30$	0	0
$31$	2.47214	0.444009	0.222004	−	0.975046i	$-0.428740\pi$
$31$	2.47214	0.444009	0.222004	+	0.975046i	$0.428740\pi$
$32$	−3.38197	−0.597853
$33$	0	0
$34$	1.61803	0.277491
$35$	0	0
$36$	0	0
$37$	4.70820	0.774024	0.387012	−	0.922075i	$-0.373507\pi$
$37$	4.70820	0.774024	0.387012	+	0.922075i	$0.373507\pi$
$38$	−6.85410	−1.11188
$39$	0	0
$40$	0	0
$41$	3.76393	0.587827	0.293914	−	0.955832i	$-0.405042\pi$
$41$	3.76393	0.587827	0.293914	+	0.955832i	$0.405042\pi$
$42$	0	0
$43$	8.47214	1.29199	0.645994	−	0.763342i	$-0.276443\pi$
$43$	8.47214	1.29199	0.645994	+	0.763342i	$0.276443\pi$
$44$	1.85410	0.279516
$45$	0	0
$46$	−0.763932	−0.112636
$47$	12.7082	1.85368	0.926841	−	0.375454i	$-0.122513\pi$
$47$	12.7082	1.85368	0.926841	+	0.375454i	$0.122513\pi$
$48$	0	0
$49$	−6.00000	−0.857143
$50$	0	0
$51$	0	0
$52$	1.23607	0.171412
$53$	3.00000	0.412082	0.206041	−	0.978543i	$-0.433942\pi$
$53$	3.00000	0.412082	0.206041	+	0.978543i	$0.433942\pi$
$54$	0	0
$55$	0	0
$56$	2.23607	0.298807
$57$	0	0
$58$	10.0902	1.32490
$59$	8.94427	1.16445	0.582223	−	0.813029i	$-0.302183\pi$
$59$	8.94427	1.16445	0.582223	+	0.813029i	$0.302183\pi$
$60$	0	0
$61$	−10.0000	−1.28037	−0.640184	−	0.768221i	$-0.721142\pi$
$61$	−10.0000	−1.28037	−0.640184	+	0.768221i	$0.721142\pi$
$62$	4.00000	0.508001
$63$	0	0
$64$	4.23607	0.529508
$65$	0	0
$66$	0	0
$67$	5.52786	0.675336	0.337668	−	0.941265i	$-0.390362\pi$
$67$	5.52786	0.675336	0.337668	+	0.941265i	$0.390362\pi$
$68$	0.618034	0.0749476
$69$	0	0
$70$	0	0
$71$	−12.9443	−1.53620	−0.768101	−	0.640328i	$-0.778798\pi$
$71$	−12.9443	−1.53620	−0.768101	+	0.640328i	$0.778798\pi$
$72$	0	0
$73$	1.76393	0.206453	0.103226	−	0.994658i	$-0.467083\pi$
$73$	1.76393	0.206453	0.103226	+	0.994658i	$0.467083\pi$
$74$	7.61803	0.885578
$75$	0	0
$76$	−2.61803	−0.300309
$77$	−3.00000	−0.341882
$78$	0	0
$79$	2.00000	0.225018	0.112509	−	0.993651i	$-0.464111\pi$
$79$	2.00000	0.225018	0.112509	+	0.993651i	$0.464111\pi$
$80$	0	0
$81$	0	0
$82$	6.09017	0.672547
$83$	4.94427	0.542704	0.271352	−	0.962480i	$-0.412529\pi$
$83$	4.94427	0.542704	0.271352	+	0.962480i	$0.412529\pi$
$84$	0	0
$85$	0	0
$86$	13.7082	1.47819
$87$	0	0
$88$	−6.70820	−0.715097
$89$	12.0000	1.27200	0.635999	−	0.771690i	$-0.280588\pi$
$89$	12.0000	1.27200	0.635999	+	0.771690i	$0.280588\pi$
$90$	0	0
$91$	−2.00000	−0.209657
$92$	−0.291796	−0.0304218
$93$	0	0
$94$	20.5623	2.12084
$95$	0	0
$96$	0	0
$97$	−13.4164	−1.36223	−0.681115	−	0.732177i	$-0.738505\pi$
$97$	−13.4164	−1.36223	−0.681115	+	0.732177i	$0.738505\pi$
$98$	−9.70820	−0.980677
$99$	0	0
$100$	0	0
$101$	14.4721	1.44003	0.720016	−	0.693958i	$-0.244135\pi$
$101$	14.4721	1.44003	0.720016	+	0.693958i	$0.244135\pi$
$102$	0	0
$103$	−10.9443	−1.07837	−0.539186	−	0.842187i	$-0.681268\pi$
$103$	−10.9443	−1.07837	−0.539186	+	0.842187i	$0.681268\pi$
$104$	−4.47214	−0.438529
$105$	0	0
$106$	4.85410	0.471472
$107$	−6.47214	−0.625685	−0.312842	−	0.949805i	$-0.601281\pi$
$107$	−6.47214	−0.625685	−0.312842	+	0.949805i	$0.601281\pi$
$108$	0	0
$109$	2.47214	0.236788	0.118394	−	0.992967i	$-0.462225\pi$
$109$	2.47214	0.236788	0.118394	+	0.992967i	$0.462225\pi$
$110$	0	0
$111$	0	0
$112$	4.85410	0.458670
$113$	18.9443	1.78213	0.891064	−	0.453878i	$-0.149960\pi$
$113$	18.9443	1.78213	0.891064	+	0.453878i	$0.149960\pi$
$114$	0	0
$115$	0	0
$116$	3.85410	0.357844
$117$	0	0
$118$	14.4721	1.33227
$119$	−1.00000	−0.0916698
$120$	0	0
$121$	−2.00000	−0.181818
$122$	−16.1803	−1.46490
$123$	0	0
$124$	1.52786	0.137206
$125$	0	0
$126$	0	0
$127$	6.00000	0.532414	0.266207	−	0.963916i	$-0.414230\pi$
$127$	6.00000	0.532414	0.266207	+	0.963916i	$0.414230\pi$
$128$	13.6180	1.20368
$129$	0	0
$130$	0	0
$131$	0.944272	0.0825014	0.0412507	−	0.999149i	$-0.486866\pi$
$131$	0.944272	0.0825014	0.0412507	+	0.999149i	$0.486866\pi$
$132$	0	0
$133$	4.23607	0.367314
$134$	8.94427	0.772667
$135$	0	0
$136$	−2.23607	−0.191741
$137$	−10.4164	−0.889934	−0.444967	−	0.895547i	$-0.646785\pi$
$137$	−10.4164	−0.889934	−0.444967	+	0.895547i	$0.646785\pi$
$138$	0	0
$139$	4.94427	0.419368	0.209684	−	0.977769i	$-0.432757\pi$
$139$	4.94427	0.419368	0.209684	+	0.977769i	$0.432757\pi$
$140$	0	0
$141$	0	0
$142$	−20.9443	−1.75760
$143$	6.00000	0.501745
$144$	0	0
$145$	0	0
$146$	2.85410	0.236207
$147$	0	0
$148$	2.90983	0.239187
$149$	−10.4721	−0.857911	−0.428955	−	0.903326i	$-0.641118\pi$
$149$	−10.4721	−0.857911	−0.428955	+	0.903326i	$0.641118\pi$
$150$	0	0
$151$	18.2361	1.48403	0.742015	−	0.670383i	$-0.233870\pi$
$151$	18.2361	1.48403	0.742015	+	0.670383i	$0.233870\pi$
$152$	9.47214	0.768292
$153$	0	0
$154$	−4.85410	−0.391155
$155$	0	0
$156$	0	0
$157$	−7.41641	−0.591894	−0.295947	−	0.955204i	$-0.595635\pi$
$157$	−7.41641	−0.591894	−0.295947	+	0.955204i	$0.595635\pi$
$158$	3.23607	0.257448
$159$	0	0
$160$	0	0
$161$	0.472136	0.0372095
$162$	0	0
$163$	−10.4164	−0.815876	−0.407938	−	0.913010i	$-0.633752\pi$
$163$	−10.4164	−0.815876	−0.407938	+	0.913010i	$0.633752\pi$
$164$	2.32624	0.181649
$165$	0	0
$166$	8.00000	0.620920
$167$	9.52786	0.737288	0.368644	−	0.929571i	$-0.379822\pi$
$167$	9.52786	0.737288	0.368644	+	0.929571i	$0.379822\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	0	0
$171$	0	0
$172$	5.23607	0.399246
$173$	24.3607	1.85211	0.926054	−	0.377391i	$-0.123179\pi$
$173$	24.3607	1.85211	0.926054	+	0.377391i	$0.123179\pi$
$174$	0	0
$175$	0	0
$176$	−14.5623	−1.09768
$177$	0	0
$178$	19.4164	1.45532
$179$	10.4721	0.782724	0.391362	−	0.920237i	$-0.372004\pi$
$179$	10.4721	0.782724	0.391362	+	0.920237i	$0.372004\pi$
$180$	0	0
$181$	−21.8885	−1.62696	−0.813481	−	0.581591i	$-0.802430\pi$
$181$	−21.8885	−1.62696	−0.813481	+	0.581591i	$0.802430\pi$
$182$	−3.23607	−0.239873
$183$	0	0
$184$	1.05573	0.0778293
$185$	0	0
$186$	0	0
$187$	3.00000	0.219382
$188$	7.85410	0.572819
$189$	0	0
$190$	0	0
$191$	4.47214	0.323592	0.161796	−	0.986824i	$-0.448271\pi$
$191$	4.47214	0.323592	0.161796	+	0.986824i	$0.448271\pi$
$192$	0	0
$193$	17.4164	1.25366	0.626830	−	0.779156i	$-0.284352\pi$
$193$	17.4164	1.25366	0.626830	+	0.779156i	$0.284352\pi$
$194$	−21.7082	−1.55856
$195$	0	0
$196$	−3.70820	−0.264872
$197$	13.5279	0.963820	0.481910	−	0.876221i	$-0.339943\pi$
$197$	13.5279	0.963820	0.481910	+	0.876221i	$0.339943\pi$
$198$	0	0
$199$	−13.4164	−0.951064	−0.475532	−	0.879698i	$-0.657744\pi$
$199$	−13.4164	−0.951064	−0.475532	+	0.879698i	$0.657744\pi$
$200$	0	0
$201$	0	0
$202$	23.4164	1.64757
$203$	−6.23607	−0.437686
$204$	0	0
$205$	0	0
$206$	−17.7082	−1.23379
$207$	0	0
$208$	−9.70820	−0.673143
$209$	−12.7082	−0.879045
$210$	0	0
$211$	26.3607	1.81474	0.907372	−	0.420328i	$-0.138085\pi$
$211$	26.3607	1.81474	0.907372	+	0.420328i	$0.138085\pi$
$212$	1.85410	0.127340
$213$	0	0
$214$	−10.4721	−0.715860
$215$	0	0
$216$	0	0
$217$	−2.47214	−0.167820
$218$	4.00000	0.270914
$219$	0	0
$220$	0	0
$221$	2.00000	0.134535
$222$	0	0
$223$	25.4164	1.70201	0.851004	−	0.525159i	$-0.175994\pi$
$223$	25.4164	1.70201	0.851004	+	0.525159i	$0.175994\pi$
$224$	3.38197	0.225967
$225$	0	0
$226$	30.6525	2.03897
$227$	20.9443	1.39012	0.695060	−	0.718952i	$-0.255378\pi$
$227$	20.9443	1.39012	0.695060	+	0.718952i	$0.255378\pi$
$228$	0	0
$229$	10.4164	0.688336	0.344168	−	0.938908i	$-0.388161\pi$
$229$	10.4164	0.688336	0.344168	+	0.938908i	$0.388161\pi$
$230$	0	0
$231$	0	0
$232$	−13.9443	−0.915486
$233$	12.0000	0.786146	0.393073	−	0.919507i	$-0.371412\pi$
$233$	12.0000	0.786146	0.393073	+	0.919507i	$0.371412\pi$
$234$	0	0
$235$	0	0
$236$	5.52786	0.359833
$237$	0	0
$238$	−1.61803	−0.104882
$239$	8.47214	0.548017	0.274008	−	0.961727i	$-0.411650\pi$
$239$	8.47214	0.548017	0.274008	+	0.961727i	$0.411650\pi$
$240$	0	0
$241$	−12.9443	−0.833814	−0.416907	−	0.908949i	$-0.636886\pi$
$241$	−12.9443	−0.833814	−0.416907	+	0.908949i	$0.636886\pi$
$242$	−3.23607	−0.208022
$243$	0	0
$244$	−6.18034	−0.395656
$245$	0	0
$246$	0	0
$247$	−8.47214	−0.539069
$248$	−5.52786	−0.351020
$249$	0	0
$250$	0	0
$251$	−23.8885	−1.50783	−0.753916	−	0.656971i	$-0.771837\pi$
$251$	−23.8885	−1.50783	−0.753916	+	0.656971i	$0.771837\pi$
$252$	0	0
$253$	−1.41641	−0.0890488
$254$	9.70820	0.609147
$255$	0	0
$256$	13.5623	0.847644
$257$	−30.9443	−1.93025	−0.965125	−	0.261788i	$-0.915688\pi$
$257$	−30.9443	−1.93025	−0.965125	+	0.261788i	$0.915688\pi$
$258$	0	0
$259$	−4.70820	−0.292554
$260$	0	0
$261$	0	0
$262$	1.52786	0.0943918
$263$	−9.65248	−0.595197	−0.297599	−	0.954691i	$-0.596186\pi$
$263$	−9.65248	−0.595197	−0.297599	+	0.954691i	$0.596186\pi$
$264$	0	0
$265$	0	0
$266$	6.85410	0.420252
$267$	0	0
$268$	3.41641	0.208690
$269$	−15.1803	−0.925562	−0.462781	−	0.886473i	$-0.653148\pi$
$269$	−15.1803	−0.925562	−0.462781	+	0.886473i	$0.653148\pi$
$270$	0	0
$271$	−13.8885	−0.843669	−0.421834	−	0.906673i	$-0.638614\pi$
$271$	−13.8885	−0.843669	−0.421834	+	0.906673i	$0.638614\pi$
$272$	−4.85410	−0.294323
$273$	0	0
$274$	−16.8541	−1.01819
$275$	0	0
$276$	0	0
$277$	2.58359	0.155233	0.0776165	−	0.996983i	$-0.475269\pi$
$277$	2.58359	0.155233	0.0776165	+	0.996983i	$0.475269\pi$
$278$	8.00000	0.479808
$279$	0	0
$280$	0	0
$281$	−30.3607	−1.81117	−0.905583	−	0.424169i	$-0.860566\pi$
$281$	−30.3607	−1.81117	−0.905583	+	0.424169i	$0.860566\pi$
$282$	0	0
$283$	8.52786	0.506929	0.253464	−	0.967345i	$-0.418430\pi$
$283$	8.52786	0.506929	0.253464	+	0.967345i	$0.418430\pi$
$284$	−8.00000	−0.474713
$285$	0	0
$286$	9.70820	0.574058
$287$	−3.76393	−0.222178
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	0	0
$292$	1.09017	0.0637974
$293$	−26.8885	−1.57085	−0.785423	−	0.618960i	$-0.787554\pi$
$293$	−26.8885	−1.57085	−0.785423	+	0.618960i	$0.787554\pi$
$294$	0	0
$295$	0	0
$296$	−10.5279	−0.611920
$297$	0	0
$298$	−16.9443	−0.981555
$299$	−0.944272	−0.0546087
$300$	0	0
$301$	−8.47214	−0.488326
$302$	29.5066	1.69791
$303$	0	0
$304$	20.5623	1.17933
$305$	0	0
$306$	0	0
$307$	18.8328	1.07485	0.537423	−	0.843313i	$-0.319398\pi$
$307$	18.8328	1.07485	0.537423	+	0.843313i	$0.319398\pi$
$308$	−1.85410	−0.105647
$309$	0	0
$310$	0	0
$311$	−19.4721	−1.10416	−0.552082	−	0.833790i	$-0.686166\pi$
$311$	−19.4721	−1.10416	−0.552082	+	0.833790i	$0.686166\pi$
$312$	0	0
$313$	6.70820	0.379170	0.189585	−	0.981864i	$-0.439286\pi$
$313$	6.70820	0.379170	0.189585	+	0.981864i	$0.439286\pi$
$314$	−12.0000	−0.677199
$315$	0	0
$316$	1.23607	0.0695343
$317$	−20.4721	−1.14983	−0.574915	−	0.818213i	$-0.694965\pi$
$317$	−20.4721	−1.14983	−0.574915	+	0.818213i	$0.694965\pi$
$318$	0	0
$319$	18.7082	1.04746
$320$	0	0
$321$	0	0
$322$	0.763932	0.0425723
$323$	−4.23607	−0.235701
$324$	0	0
$325$	0	0
$326$	−16.8541	−0.933462
$327$	0	0
$328$	−8.41641	−0.464718
$329$	−12.7082	−0.700626
$330$	0	0
$331$	−30.1246	−1.65580	−0.827899	−	0.560877i	$-0.810464\pi$
$331$	−30.1246	−1.65580	−0.827899	+	0.560877i	$0.810464\pi$
$332$	3.05573	0.167705
$333$	0	0
$334$	15.4164	0.843548
$335$	0	0
$336$	0	0
$337$	−0.347524	−0.0189308	−0.00946542	−	0.999955i	$-0.503013\pi$
$337$	−0.347524	−0.0189308	−0.00946542	+	0.999955i	$0.503013\pi$
$338$	−14.5623	−0.792085
$339$	0	0
$340$	0	0
$341$	7.41641	0.401621
$342$	0	0
$343$	13.0000	0.701934
$344$	−18.9443	−1.02141
$345$	0	0
$346$	39.4164	2.11904
$347$	−1.52786	−0.0820200	−0.0410100	−	0.999159i	$-0.513058\pi$
$347$	−1.52786	−0.0820200	−0.0410100	+	0.999159i	$0.513058\pi$
$348$	0	0
$349$	−5.58359	−0.298883	−0.149441	−	0.988771i	$-0.547748\pi$
$349$	−5.58359	−0.298883	−0.149441	+	0.988771i	$0.547748\pi$
$350$	0	0
$351$	0	0
$352$	−10.1459	−0.540778
$353$	−25.4721	−1.35574	−0.677872	−	0.735179i	$-0.737098\pi$
$353$	−25.4721	−1.35574	−0.677872	+	0.735179i	$0.737098\pi$
$354$	0	0
$355$	0	0
$356$	7.41641	0.393069
$357$	0	0
$358$	16.9443	0.895533
$359$	−23.8885	−1.26079	−0.630395	−	0.776275i	$-0.717107\pi$
$359$	−23.8885	−1.26079	−0.630395	+	0.776275i	$0.717107\pi$
$360$	0	0
$361$	−1.05573	−0.0555646
$362$	−35.4164	−1.86145
$363$	0	0
$364$	−1.23607	−0.0647876
$365$	0	0
$366$	0	0
$367$	12.0000	0.626395	0.313197	−	0.949688i	$-0.398600\pi$
$367$	12.0000	0.626395	0.313197	+	0.949688i	$0.398600\pi$
$368$	2.29180	0.119468
$369$	0	0
$370$	0	0
$371$	−3.00000	−0.155752
$372$	0	0
$373$	8.47214	0.438671	0.219335	−	0.975650i	$-0.429611\pi$
$373$	8.47214	0.438671	0.219335	+	0.975650i	$0.429611\pi$
$374$	4.85410	0.251000
$375$	0	0
$376$	−28.4164	−1.46546
$377$	12.4721	0.642348
$378$	0	0
$379$	−8.36068	−0.429459	−0.214730	−	0.976674i	$-0.568887\pi$
$379$	−8.36068	−0.429459	−0.214730	+	0.976674i	$0.568887\pi$
$380$	0	0
$381$	0	0
$382$	7.23607	0.370229
$383$	−25.6525	−1.31078	−0.655390	−	0.755291i	$-0.727496\pi$
$383$	−25.6525	−1.31078	−0.655390	+	0.755291i	$0.727496\pi$
$384$	0	0
$385$	0	0
$386$	28.1803	1.43434
$387$	0	0
$388$	−8.29180	−0.420952
$389$	−16.3607	−0.829519	−0.414760	−	0.909931i	$-0.636134\pi$
$389$	−16.3607	−0.829519	−0.414760	+	0.909931i	$0.636134\pi$
$390$	0	0
$391$	−0.472136	−0.0238769
$392$	13.4164	0.677631
$393$	0	0
$394$	21.8885	1.10273
$395$	0	0
$396$	0	0
$397$	11.7639	0.590415	0.295207	−	0.955433i	$-0.404611\pi$
$397$	11.7639	0.590415	0.295207	+	0.955433i	$0.404611\pi$
$398$	−21.7082	−1.08813
$399$	0	0
$400$	0	0
$401$	−24.7082	−1.23387	−0.616934	−	0.787015i	$-0.711626\pi$
$401$	−24.7082	−1.23387	−0.616934	+	0.787015i	$0.711626\pi$
$402$	0	0
$403$	4.94427	0.246292
$404$	8.94427	0.444994
$405$	0	0
$406$	−10.0902	−0.500767
$407$	14.1246	0.700131
$408$	0	0
$409$	−17.8328	−0.881776	−0.440888	−	0.897562i	$-0.645336\pi$
$409$	−17.8328	−0.881776	−0.440888	+	0.897562i	$0.645336\pi$
$410$	0	0
$411$	0	0
$412$	−6.76393	−0.333235
$413$	−8.94427	−0.440119
$414$	0	0
$415$	0	0
$416$	−6.76393	−0.331629
$417$	0	0
$418$	−20.5623	−1.00574
$419$	21.0000	1.02592	0.512959	−	0.858413i	$-0.328549\pi$
$419$	21.0000	1.02592	0.512959	+	0.858413i	$0.328549\pi$
$420$	0	0
$421$	−22.4164	−1.09251	−0.546254	−	0.837619i	$-0.683947\pi$
$421$	−22.4164	−1.09251	−0.546254	+	0.837619i	$0.683947\pi$
$422$	42.6525	2.07629
$423$	0	0
$424$	−6.70820	−0.325779
$425$	0	0
$426$	0	0
$427$	10.0000	0.483934
$428$	−4.00000	−0.193347
$429$	0	0
$430$	0	0
$431$	18.5279	0.892456	0.446228	−	0.894919i	$-0.352767\pi$
$431$	18.5279	0.892456	0.446228	+	0.894919i	$0.352767\pi$
$432$	0	0
$433$	−6.00000	−0.288342	−0.144171	−	0.989553i	$-0.546051\pi$
$433$	−6.00000	−0.288342	−0.144171	+	0.989553i	$0.546051\pi$
$434$	−4.00000	−0.192006
$435$	0	0
$436$	1.52786	0.0731714
$437$	2.00000	0.0956730
$438$	0	0
$439$	−25.4164	−1.21306	−0.606529	−	0.795061i	$-0.707439\pi$
$439$	−25.4164	−1.21306	−0.606529	+	0.795061i	$0.707439\pi$
$440$	0	0
$441$	0	0
$442$	3.23607	0.153924
$443$	12.9443	0.615001	0.307500	−	0.951548i	$-0.400507\pi$
$443$	12.9443	0.615001	0.307500	+	0.951548i	$0.400507\pi$
$444$	0	0
$445$	0	0
$446$	41.1246	1.94731
$447$	0	0
$448$	−4.23607	−0.200135
$449$	25.4164	1.19947	0.599737	−	0.800197i	$-0.295272\pi$
$449$	25.4164	1.19947	0.599737	+	0.800197i	$0.295272\pi$
$450$	0	0
$451$	11.2918	0.531710
$452$	11.7082	0.550708
$453$	0	0
$454$	33.8885	1.59047
$455$	0	0
$456$	0	0
$457$	34.0000	1.59045	0.795226	−	0.606313i	$-0.207352\pi$
$457$	34.0000	1.59045	0.795226	+	0.606313i	$0.207352\pi$
$458$	16.8541	0.787540
$459$	0	0
$460$	0	0
$461$	14.9443	0.696024	0.348012	−	0.937490i	$-0.386857\pi$
$461$	14.9443	0.696024	0.348012	+	0.937490i	$0.386857\pi$
$462$	0	0
$463$	−23.4164	−1.08825	−0.544126	−	0.839003i	$-0.683139\pi$
$463$	−23.4164	−1.08825	−0.544126	+	0.839003i	$0.683139\pi$
$464$	−30.2705	−1.40527
$465$	0	0
$466$	19.4164	0.899448
$467$	−10.2361	−0.473669	−0.236834	−	0.971550i	$-0.576110\pi$
$467$	−10.2361	−0.473669	−0.236834	+	0.971550i	$0.576110\pi$
$468$	0	0
$469$	−5.52786	−0.255253
$470$	0	0
$471$	0	0
$472$	−20.0000	−0.920575
$473$	25.4164	1.16865
$474$	0	0
$475$	0	0
$476$	−0.618034	−0.0283275
$477$	0	0
$478$	13.7082	0.626999
$479$	−6.11146	−0.279240	−0.139620	−	0.990205i	$-0.544588\pi$
$479$	−6.11146	−0.279240	−0.139620	+	0.990205i	$0.544588\pi$
$480$	0	0
$481$	9.41641	0.429351
$482$	−20.9443	−0.953985
$483$	0	0
$484$	−1.23607	−0.0561849
$485$	0	0
$486$	0	0
$487$	14.8328	0.672139	0.336070	−	0.941837i	$-0.390902\pi$
$487$	14.8328	0.672139	0.336070	+	0.941837i	$0.390902\pi$
$488$	22.3607	1.01222
$489$	0	0
$490$	0	0
$491$	−20.8328	−0.940172	−0.470086	−	0.882621i	$-0.655777\pi$
$491$	−20.8328	−0.940172	−0.470086	+	0.882621i	$0.655777\pi$
$492$	0	0
$493$	6.23607	0.280858
$494$	−13.7082	−0.616761
$495$	0	0
$496$	−12.0000	−0.538816
$497$	12.9443	0.580630
$498$	0	0
$499$	6.36068	0.284743	0.142372	−	0.989813i	$-0.454527\pi$
$499$	6.36068	0.284743	0.142372	+	0.989813i	$0.454527\pi$
$500$	0	0
$501$	0	0
$502$	−38.6525	−1.72514
$503$	−1.88854	−0.0842060	−0.0421030	−	0.999113i	$-0.513406\pi$
$503$	−1.88854	−0.0842060	−0.0421030	+	0.999113i	$0.513406\pi$
$504$	0	0
$505$	0	0
$506$	−2.29180	−0.101883
$507$	0	0
$508$	3.70820	0.164525
$509$	36.9443	1.63753	0.818763	−	0.574132i	$-0.194660\pi$
$509$	36.9443	1.63753	0.818763	+	0.574132i	$0.194660\pi$
$510$	0	0
$511$	−1.76393	−0.0780318
$512$	−5.29180	−0.233867
$513$	0	0
$514$	−50.0689	−2.20844
$515$	0	0
$516$	0	0
$517$	38.1246	1.67672
$518$	−7.61803	−0.334717
$519$	0	0
$520$	0	0
$521$	20.2361	0.886558	0.443279	−	0.896384i	$-0.353815\pi$
$521$	20.2361	0.886558	0.443279	+	0.896384i	$0.353815\pi$
$522$	0	0
$523$	−40.8328	−1.78549	−0.892747	−	0.450558i	$-0.851225\pi$
$523$	−40.8328	−1.78549	−0.892747	+	0.450558i	$0.851225\pi$
$524$	0.583592	0.0254943
$525$	0	0
$526$	−15.6180	−0.680979
$527$	2.47214	0.107688
$528$	0	0
$529$	−22.7771	−0.990308
$530$	0	0
$531$	0	0
$532$	2.61803	0.113506
$533$	7.52786	0.326068
$534$	0	0
$535$	0	0
$536$	−12.3607	−0.533900
$537$	0	0
$538$	−24.5623	−1.05896
$539$	−18.0000	−0.775315
$540$	0	0
$541$	23.4164	1.00675	0.503375	−	0.864068i	$-0.332091\pi$
$541$	23.4164	1.00675	0.503375	+	0.864068i	$0.332091\pi$
$542$	−22.4721	−0.965261
$543$	0	0
$544$	−3.38197	−0.145001
$545$	0	0
$546$	0	0
$547$	−0.416408	−0.0178043	−0.00890216	−	0.999960i	$-0.502834\pi$
$547$	−0.416408	−0.0178043	−0.00890216	+	0.999960i	$0.502834\pi$
$548$	−6.43769	−0.275005
$549$	0	0
$550$	0	0
$551$	−26.4164	−1.12538
$552$	0	0
$553$	−2.00000	−0.0850487
$554$	4.18034	0.177606
$555$	0	0
$556$	3.05573	0.129592
$557$	8.11146	0.343693	0.171847	−	0.985124i	$-0.445027\pi$
$557$	8.11146	0.343693	0.171847	+	0.985124i	$0.445027\pi$
$558$	0	0
$559$	16.9443	0.716666
$560$	0	0
$561$	0	0
$562$	−49.1246	−2.07220
$563$	20.1246	0.848151	0.424076	−	0.905627i	$-0.360599\pi$
$563$	20.1246	0.848151	0.424076	+	0.905627i	$0.360599\pi$
$564$	0	0
$565$	0	0
$566$	13.7984	0.579989
$567$	0	0
$568$	28.9443	1.21447
$569$	−8.11146	−0.340050	−0.170025	−	0.985440i	$-0.554385\pi$
$569$	−8.11146	−0.340050	−0.170025	+	0.985440i	$0.554385\pi$
$570$	0	0
$571$	40.8328	1.70880	0.854400	−	0.519616i	$-0.173925\pi$
$571$	40.8328	1.70880	0.854400	+	0.519616i	$0.173925\pi$
$572$	3.70820	0.155048
$573$	0	0
$574$	−6.09017	−0.254199
$575$	0	0
$576$	0	0
$577$	33.3050	1.38650	0.693252	−	0.720696i	$-0.256177\pi$
$577$	33.3050	1.38650	0.693252	+	0.720696i	$0.256177\pi$
$578$	1.61803	0.0673013
$579$	0	0
$580$	0	0
$581$	−4.94427	−0.205123
$582$	0	0
$583$	9.00000	0.372742
$584$	−3.94427	−0.163215
$585$	0	0
$586$	−43.5066	−1.79724
$587$	−2.23607	−0.0922924	−0.0461462	−	0.998935i	$-0.514694\pi$
$587$	−2.23607	−0.0922924	−0.0461462	+	0.998935i	$0.514694\pi$
$588$	0	0
$589$	−10.4721	−0.431497
$590$	0	0
$591$	0	0
$592$	−22.8541	−0.939298
$593$	−17.4721	−0.717495	−0.358747	−	0.933435i	$-0.616796\pi$
$593$	−17.4721	−0.717495	−0.358747	+	0.933435i	$0.616796\pi$
$594$	0	0
$595$	0	0
$596$	−6.47214	−0.265109
$597$	0	0
$598$	−1.52786	−0.0624790
$599$	−4.47214	−0.182727	−0.0913633	−	0.995818i	$-0.529122\pi$
$599$	−4.47214	−0.182727	−0.0913633	+	0.995818i	$0.529122\pi$
$600$	0	0
$601$	−0.583592	−0.0238052	−0.0119026	−	0.999929i	$-0.503789\pi$
$601$	−0.583592	−0.0238052	−0.0119026	+	0.999929i	$0.503789\pi$
$602$	−13.7082	−0.558705
$603$	0	0
$604$	11.2705	0.458591
$605$	0	0
$606$	0	0
$607$	−46.8885	−1.90315	−0.951574	−	0.307421i	$-0.900534\pi$
$607$	−46.8885	−1.90315	−0.951574	+	0.307421i	$0.900534\pi$
$608$	14.3262	0.581006
$609$	0	0
$610$	0	0
$611$	25.4164	1.02824
$612$	0	0
$613$	−42.9443	−1.73450	−0.867251	−	0.497870i	$-0.834115\pi$
$613$	−42.9443	−1.73450	−0.867251	+	0.497870i	$0.834115\pi$
$614$	30.4721	1.22976
$615$	0	0
$616$	6.70820	0.270281
$617$	11.8885	0.478615	0.239307	−	0.970944i	$-0.423080\pi$
$617$	11.8885	0.478615	0.239307	+	0.970944i	$0.423080\pi$
$618$	0	0
$619$	−7.88854	−0.317067	−0.158534	−	0.987354i	$-0.550677\pi$
$619$	−7.88854	−0.317067	−0.158534	+	0.987354i	$0.550677\pi$
$620$	0	0
$621$	0	0
$622$	−31.5066	−1.26330
$623$	−12.0000	−0.480770
$624$	0	0
$625$	0	0
$626$	10.8541	0.433817
$627$	0	0
$628$	−4.58359	−0.182905
$629$	4.70820	0.187728
$630$	0	0
$631$	8.12461	0.323436	0.161718	−	0.986837i	$-0.448297\pi$
$631$	8.12461	0.323436	0.161718	+	0.986837i	$0.448297\pi$
$632$	−4.47214	−0.177892
$633$	0	0
$634$	−33.1246	−1.31555
$635$	0	0
$636$	0	0
$637$	−12.0000	−0.475457
$638$	30.2705	1.19842
$639$	0	0
$640$	0	0
$641$	23.2918	0.919971	0.459985	−	0.887927i	$-0.347855\pi$
$641$	23.2918	0.919971	0.459985	+	0.887927i	$0.347855\pi$
$642$	0	0
$643$	49.2492	1.94220	0.971100	−	0.238673i	$-0.0767125\pi$
$643$	49.2492	1.94220	0.971100	+	0.238673i	$0.0767125\pi$
$644$	0.291796	0.0114984
$645$	0	0
$646$	−6.85410	−0.269671
$647$	36.9443	1.45243	0.726215	−	0.687468i	$-0.241278\pi$
$647$	36.9443	1.45243	0.726215	+	0.687468i	$0.241278\pi$
$648$	0	0
$649$	26.8328	1.05328
$650$	0	0
$651$	0	0
$652$	−6.43769	−0.252120
$653$	33.3050	1.30332	0.651662	−	0.758510i	$-0.274072\pi$
$653$	33.3050	1.30332	0.651662	+	0.758510i	$0.274072\pi$
$654$	0	0
$655$	0	0
$656$	−18.2705	−0.713344
$657$	0	0
$658$	−20.5623	−0.801602
$659$	−26.8328	−1.04526	−0.522629	−	0.852560i	$-0.675049\pi$
$659$	−26.8328	−1.04526	−0.522629	+	0.852560i	$0.675049\pi$
$660$	0	0
$661$	−1.58359	−0.0615946	−0.0307973	−	0.999526i	$-0.509805\pi$
$661$	−1.58359	−0.0615946	−0.0307973	+	0.999526i	$0.509805\pi$
$662$	−48.7426	−1.89444
$663$	0	0
$664$	−11.0557	−0.429045
$665$	0	0
$666$	0	0
$667$	−2.94427	−0.114003
$668$	5.88854	0.227835
$669$	0	0
$670$	0	0
$671$	−30.0000	−1.15814
$672$	0	0
$673$	−3.52786	−0.135989	−0.0679946	−	0.997686i	$-0.521660\pi$
$673$	−3.52786	−0.135989	−0.0679946	+	0.997686i	$0.521660\pi$
$674$	−0.562306	−0.0216592
$675$	0	0
$676$	−5.56231	−0.213935
$677$	13.0557	0.501772	0.250886	−	0.968017i	$-0.419278\pi$
$677$	13.0557	0.501772	0.250886	+	0.968017i	$0.419278\pi$
$678$	0	0
$679$	13.4164	0.514874
$680$	0	0
$681$	0	0
$682$	12.0000	0.459504
$683$	−29.7771	−1.13939	−0.569694	−	0.821857i	$-0.692938\pi$
$683$	−29.7771	−1.13939	−0.569694	+	0.821857i	$0.692938\pi$
$684$	0	0
$685$	0	0
$686$	21.0344	0.803099
$687$	0	0
$688$	−41.1246	−1.56786
$689$	6.00000	0.228582
$690$	0	0
$691$	−46.2492	−1.75940	−0.879702	−	0.475526i	$-0.842258\pi$
$691$	−46.2492	−1.75940	−0.879702	+	0.475526i	$0.842258\pi$
$692$	15.0557	0.572333
$693$	0	0
$694$	−2.47214	−0.0938410
$695$	0	0
$696$	0	0
$697$	3.76393	0.142569
$698$	−9.03444	−0.341959
$699$	0	0
$700$	0	0
$701$	18.0000	0.679851	0.339925	−	0.940452i	$-0.389598\pi$
$701$	18.0000	0.679851	0.339925	+	0.940452i	$0.389598\pi$
$702$	0	0
$703$	−19.9443	−0.752212
$704$	12.7082	0.478958
$705$	0	0
$706$	−41.2148	−1.55114
$707$	−14.4721	−0.544281
$708$	0	0
$709$	−37.4164	−1.40520	−0.702601	−	0.711584i	$-0.747978\pi$
$709$	−37.4164	−1.40520	−0.702601	+	0.711584i	$0.747978\pi$
$710$	0	0
$711$	0	0
$712$	−26.8328	−1.00560
$713$	−1.16718	−0.0437114
$714$	0	0
$715$	0	0
$716$	6.47214	0.241875
$717$	0	0
$718$	−38.6525	−1.44250
$719$	−28.3050	−1.05560	−0.527798	−	0.849370i	$-0.676982\pi$
$719$	−28.3050	−1.05560	−0.527798	+	0.849370i	$0.676982\pi$
$720$	0	0
$721$	10.9443	0.407586
$722$	−1.70820	−0.0635728
$723$	0	0
$724$	−13.5279	−0.502759
$725$	0	0
$726$	0	0
$727$	27.3050	1.01268	0.506342	−	0.862333i	$-0.330997\pi$
$727$	27.3050	1.01268	0.506342	+	0.862333i	$0.330997\pi$
$728$	4.47214	0.165748
$729$	0	0
$730$	0	0
$731$	8.47214	0.313353
$732$	0	0
$733$	−14.4721	−0.534541	−0.267270	−	0.963622i	$-0.586122\pi$
$733$	−14.4721	−0.534541	−0.267270	+	0.963622i	$0.586122\pi$
$734$	19.4164	0.716673
$735$	0	0
$736$	1.59675	0.0588569
$737$	16.5836	0.610864
$738$	0	0
$739$	14.8328	0.545634	0.272817	−	0.962066i	$-0.412045\pi$
$739$	14.8328	0.545634	0.272817	+	0.962066i	$0.412045\pi$
$740$	0	0
$741$	0	0
$742$	−4.85410	−0.178200
$743$	−19.8885	−0.729640	−0.364820	−	0.931078i	$-0.618869\pi$
$743$	−19.8885	−0.729640	−0.364820	+	0.931078i	$0.618869\pi$
$744$	0	0
$745$	0	0
$746$	13.7082	0.501893
$747$	0	0
$748$	1.85410	0.0677927
$749$	6.47214	0.236487
$750$	0	0
$751$	4.11146	0.150029	0.0750146	−	0.997182i	$-0.476100\pi$
$751$	4.11146	0.150029	0.0750146	+	0.997182i	$0.476100\pi$
$752$	−61.6869	−2.24949
$753$	0	0
$754$	20.1803	0.734925
$755$	0	0
$756$	0	0
$757$	4.58359	0.166593	0.0832967	−	0.996525i	$-0.473455\pi$
$757$	4.58359	0.166593	0.0832967	+	0.996525i	$0.473455\pi$
$758$	−13.5279	−0.491354
$759$	0	0
$760$	0	0
$761$	5.05573	0.183270	0.0916350	−	0.995793i	$-0.470791\pi$
$761$	5.05573	0.183270	0.0916350	+	0.995793i	$0.470791\pi$
$762$	0	0
$763$	−2.47214	−0.0894973
$764$	2.76393	0.0999956
$765$	0	0
$766$	−41.5066	−1.49969
$767$	17.8885	0.645918
$768$	0	0
$769$	14.0557	0.506863	0.253431	−	0.967353i	$-0.418441\pi$
$769$	14.0557	0.506863	0.253431	+	0.967353i	$0.418441\pi$
$770$	0	0
$771$	0	0
$772$	10.7639	0.387402
$773$	20.7771	0.747300	0.373650	−	0.927570i	$-0.378106\pi$
$773$	20.7771	0.747300	0.373650	+	0.927570i	$0.378106\pi$
$774$	0	0
$775$	0	0
$776$	30.0000	1.07694
$777$	0	0
$778$	−26.4721	−0.949072
$779$	−15.9443	−0.571263
$780$	0	0
$781$	−38.8328	−1.38955
$782$	−0.763932	−0.0273182
$783$	0	0
$784$	29.1246	1.04016
$785$	0	0
$786$	0	0
$787$	19.5836	0.698080	0.349040	−	0.937108i	$-0.386508\pi$
$787$	19.5836	0.698080	0.349040	+	0.937108i	$0.386508\pi$
$788$	8.36068	0.297837
$789$	0	0
$790$	0	0
$791$	−18.9443	−0.673581
$792$	0	0
$793$	−20.0000	−0.710221
$794$	19.0344	0.675507
$795$	0	0
$796$	−8.29180	−0.293895
$797$	−39.0000	−1.38145	−0.690725	−	0.723117i	$-0.742709\pi$
$797$	−39.0000	−1.38145	−0.690725	+	0.723117i	$0.742709\pi$
$798$	0	0
$799$	12.7082	0.449584
$800$	0	0
$801$	0	0
$802$	−39.9787	−1.41170
$803$	5.29180	0.186743
$804$	0	0
$805$	0	0
$806$	8.00000	0.281788
$807$	0	0
$808$	−32.3607	−1.13844
$809$	−7.52786	−0.264666	−0.132333	−	0.991205i	$-0.542247\pi$
$809$	−7.52786	−0.264666	−0.132333	+	0.991205i	$0.542247\pi$
$810$	0	0
$811$	2.00000	0.0702295	0.0351147	−	0.999383i	$-0.488820\pi$
$811$	2.00000	0.0702295	0.0351147	+	0.999383i	$0.488820\pi$
$812$	−3.85410	−0.135252
$813$	0	0
$814$	22.8541	0.801036
$815$	0	0
$816$	0	0
$817$	−35.8885	−1.25558
$818$	−28.8541	−1.00886
$819$	0	0
$820$	0	0
$821$	−7.52786	−0.262724	−0.131362	−	0.991334i	$-0.541935\pi$
$821$	−7.52786	−0.262724	−0.131362	+	0.991334i	$0.541935\pi$
$822$	0	0
$823$	−7.00000	−0.244005	−0.122002	−	0.992530i	$-0.538932\pi$
$823$	−7.00000	−0.244005	−0.122002	+	0.992530i	$0.538932\pi$
$824$	24.4721	0.852527
$825$	0	0
$826$	−14.4721	−0.503550
$827$	0.111456	0.00387571	0.00193786	−	0.999998i	$-0.499383\pi$
$827$	0.111456	0.00387571	0.00193786	+	0.999998i	$0.499383\pi$
$828$	0	0
$829$	21.3607	0.741887	0.370944	−	0.928655i	$-0.379034\pi$
$829$	21.3607	0.741887	0.370944	+	0.928655i	$0.379034\pi$
$830$	0	0
$831$	0	0
$832$	8.47214	0.293718
$833$	−6.00000	−0.207888
$834$	0	0
$835$	0	0
$836$	−7.85410	−0.271640
$837$	0	0
$838$	33.9787	1.17378
$839$	16.4164	0.566757	0.283379	−	0.959008i	$-0.408545\pi$
$839$	16.4164	0.566757	0.283379	+	0.959008i	$0.408545\pi$
$840$	0	0
$841$	9.88854	0.340984
$842$	−36.2705	−1.24996
$843$	0	0
$844$	16.2918	0.560787
$845$	0	0
$846$	0	0
$847$	2.00000	0.0687208
$848$	−14.5623	−0.500072
$849$	0	0
$850$	0	0
$851$	−2.22291	−0.0762005
$852$	0	0
$853$	24.2492	0.830278	0.415139	−	0.909758i	$-0.363733\pi$
$853$	24.2492	0.830278	0.415139	+	0.909758i	$0.363733\pi$
$854$	16.1803	0.553680
$855$	0	0
$856$	14.4721	0.494647
$857$	−54.3607	−1.85693	−0.928463	−	0.371426i	$-0.878869\pi$
$857$	−54.3607	−1.85693	−0.928463	+	0.371426i	$0.878869\pi$
$858$	0	0
$859$	37.1803	1.26858	0.634288	−	0.773097i	$-0.281293\pi$
$859$	37.1803	1.26858	0.634288	+	0.773097i	$0.281293\pi$
$860$	0	0
$861$	0	0
$862$	29.9787	1.02108
$863$	32.2361	1.09733	0.548664	−	0.836043i	$-0.315137\pi$
$863$	32.2361	1.09733	0.548664	+	0.836043i	$0.315137\pi$
$864$	0	0
$865$	0	0
$866$	−9.70820	−0.329898
$867$	0	0
$868$	−1.52786	−0.0518591
$869$	6.00000	0.203536
$870$	0	0
$871$	11.0557	0.374609
$872$	−5.52786	−0.187197
$873$	0	0
$874$	3.23607	0.109462
$875$	0	0
$876$	0	0
$877$	−54.1246	−1.82766	−0.913829	−	0.406099i	$-0.866889\pi$
$877$	−54.1246	−1.82766	−0.913829	+	0.406099i	$0.866889\pi$
$878$	−41.1246	−1.38789
$879$	0	0
$880$	0	0
$881$	−7.06888	−0.238157	−0.119078	−	0.992885i	$-0.537994\pi$
$881$	−7.06888	−0.238157	−0.119078	+	0.992885i	$0.537994\pi$
$882$	0	0
$883$	56.8328	1.91258	0.956288	−	0.292426i	$-0.0944624\pi$
$883$	56.8328	1.91258	0.956288	+	0.292426i	$0.0944624\pi$
$884$	1.23607	0.0415735
$885$	0	0
$886$	20.9443	0.703637
$887$	31.3050	1.05112	0.525559	−	0.850757i	$-0.323856\pi$
$887$	31.3050	1.05112	0.525559	+	0.850757i	$0.323856\pi$
$888$	0	0
$889$	−6.00000	−0.201234
$890$	0	0
$891$	0	0
$892$	15.7082	0.525950
$893$	−53.8328	−1.80145
$894$	0	0
$895$	0	0
$896$	−13.6180	−0.454947
$897$	0	0
$898$	41.1246	1.37235
$899$	15.4164	0.514166
$900$	0	0
$901$	3.00000	0.0999445
$902$	18.2705	0.608341
$903$	0	0
$904$	−42.3607	−1.40890
$905$	0	0
$906$	0	0
$907$	−13.5836	−0.451036	−0.225518	−	0.974239i	$-0.572407\pi$
$907$	−13.5836	−0.451036	−0.225518	+	0.974239i	$0.572407\pi$
$908$	12.9443	0.429571
$909$	0	0
$910$	0	0
$911$	49.2492	1.63170	0.815850	−	0.578264i	$-0.196270\pi$
$911$	49.2492	1.63170	0.815850	+	0.578264i	$0.196270\pi$
$912$	0	0
$913$	14.8328	0.490895
$914$	55.0132	1.81967
$915$	0	0
$916$	6.43769	0.212707
$917$	−0.944272	−0.0311826
$918$	0	0
$919$	−18.7082	−0.617127	−0.308563	−	0.951204i	$-0.599848\pi$
$919$	−18.7082	−0.617127	−0.308563	+	0.951204i	$0.599848\pi$
$920$	0	0
$921$	0	0
$922$	24.1803	0.796337
$923$	−25.8885	−0.852132
$924$	0	0
$925$	0	0
$926$	−37.8885	−1.24509
$927$	0	0
$928$	−21.0902	−0.692319
$929$	32.4853	1.06581	0.532904	−	0.846176i	$-0.321101\pi$
$929$	32.4853	1.06581	0.532904	+	0.846176i	$0.321101\pi$
$930$	0	0
$931$	25.4164	0.832989
$932$	7.41641	0.242933
$933$	0	0
$934$	−16.5623	−0.541935
$935$	0	0
$936$	0	0
$937$	25.8885	0.845742	0.422871	−	0.906190i	$-0.361022\pi$
$937$	25.8885	0.845742	0.422871	+	0.906190i	$0.361022\pi$
$938$	−8.94427	−0.292041
$939$	0	0
$940$	0	0
$941$	12.4721	0.406580	0.203290	−	0.979119i	$-0.434837\pi$
$941$	12.4721	0.406580	0.203290	+	0.979119i	$0.434837\pi$
$942$	0	0
$943$	−1.77709	−0.0578699
$944$	−43.4164	−1.41308
$945$	0	0
$946$	41.1246	1.33708
$947$	28.4721	0.925220	0.462610	−	0.886562i	$-0.346913\pi$
$947$	28.4721	0.925220	0.462610	+	0.886562i	$0.346913\pi$
$948$	0	0
$949$	3.52786	0.114519
$950$	0	0
$951$	0	0
$952$	2.23607	0.0724714
$953$	−39.3607	−1.27502	−0.637509	−	0.770443i	$-0.720035\pi$
$953$	−39.3607	−1.27502	−0.637509	+	0.770443i	$0.720035\pi$
$954$	0	0
$955$	0	0
$956$	5.23607	0.169347
$957$	0	0
$958$	−9.88854	−0.319484
$959$	10.4164	0.336363
$960$	0	0
$961$	−24.8885	−0.802856
$962$	15.2361	0.491231
$963$	0	0
$964$	−8.00000	−0.257663
$965$	0	0
$966$	0	0
$967$	46.0000	1.47926	0.739630	−	0.673014i	$-0.235000\pi$
$967$	46.0000	1.47926	0.739630	+	0.673014i	$0.235000\pi$
$968$	4.47214	0.143740
$969$	0	0
$970$	0	0
$971$	21.3050	0.683708	0.341854	−	0.939753i	$-0.388945\pi$
$971$	21.3050	0.683708	0.341854	+	0.939753i	$0.388945\pi$
$972$	0	0
$973$	−4.94427	−0.158506
$974$	24.0000	0.769010
$975$	0	0
$976$	48.5410	1.55376
$977$	−13.0557	−0.417690	−0.208845	−	0.977949i	$-0.566970\pi$
$977$	−13.0557	−0.417690	−0.208845	+	0.977949i	$0.566970\pi$
$978$	0	0
$979$	36.0000	1.15056
$980$	0	0
$981$	0	0
$982$	−33.7082	−1.07567
$983$	33.5279	1.06937	0.534686	−	0.845051i	$-0.320430\pi$
$983$	33.5279	1.06937	0.534686	+	0.845051i	$0.320430\pi$
$984$	0	0
$985$	0	0
$986$	10.0902	0.321336
$987$	0	0
$988$	−5.23607	−0.166582
$989$	−4.00000	−0.127193
$990$	0	0
$991$	−24.0000	−0.762385	−0.381193	−	0.924496i	$-0.624487\pi$
$991$	−24.0000	−0.762385	−0.381193	+	0.924496i	$0.624487\pi$
$992$	−8.36068	−0.265452
$993$	0	0
$994$	20.9443	0.664312
$995$	0	0
$996$	0	0
$997$	−34.5967	−1.09569	−0.547845	−	0.836580i	$-0.684552\pi$
$997$	−34.5967	−1.09569	−0.547845	+	0.836580i	$0.684552\pi$
$998$	10.2918	0.325781
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3825.2.a.w.1.2		2
3.2	odd	2		1275.2.a.i.1.1		2
5.2	odd	4		765.2.b.b.154.4		4
5.3	odd	4		765.2.b.b.154.1		4
5.4	even	2		3825.2.a.q.1.1		2
15.2	even	4		255.2.b.b.154.1	✓	4
15.8	even	4		255.2.b.b.154.4	yes	4
15.14	odd	2		1275.2.a.m.1.2		2
60.23	odd	4		4080.2.m.n.2449.1		4
60.47	odd	4		4080.2.m.n.2449.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
255.2.b.b.154.1	✓	4	15.2	even	4
255.2.b.b.154.4	yes	4	15.8	even	4
765.2.b.b.154.1		4	5.3	odd	4
765.2.b.b.154.4		4	5.2	odd	4
1275.2.a.i.1.1		2	3.2	odd	2
1275.2.a.m.1.2		2	15.14	odd	2
3825.2.a.q.1.1		2	5.4	even	2
3825.2.a.w.1.2		2	1.1	even	1	trivial
4080.2.m.n.2449.1		4	60.23	odd	4
4080.2.m.n.2449.3		4	60.47	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 3825 = 3^{2} \cdot 5^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3825.a (trivial)

\(f(q)\)	\(=\)	\(q+1.61803 q^{2} +0.618034 q^{4} -1.00000 q^{7} -2.23607 q^{8} +O(q^{10})\)	\(q+1.61803 q^{2} +0.618034 q^{4} -1.00000 q^{7} -2.23607 q^{8} +3.00000 q^{11} +2.00000 q^{13} -1.61803 q^{14} -4.85410 q^{16} +1.00000 q^{17} -4.23607 q^{19} +4.85410 q^{22} -0.472136 q^{23} +3.23607 q^{26} -0.618034 q^{28} +6.23607 q^{29} +2.47214 q^{31} -3.38197 q^{32} +1.61803 q^{34} +4.70820 q^{37} -6.85410 q^{38} +3.76393 q^{41} +8.47214 q^{43} +1.85410 q^{44} -0.763932 q^{46} +12.7082 q^{47} -6.00000 q^{49} +1.23607 q^{52} +3.00000 q^{53} +2.23607 q^{56} +10.0902 q^{58} +8.94427 q^{59} -10.0000 q^{61} +4.00000 q^{62} +4.23607 q^{64} +5.52786 q^{67} +0.618034 q^{68} -12.9443 q^{71} +1.76393 q^{73} +7.61803 q^{74} -2.61803 q^{76} -3.00000 q^{77} +2.00000 q^{79} +6.09017 q^{82} +4.94427 q^{83} +13.7082 q^{86} -6.70820 q^{88} +12.0000 q^{89} -2.00000 q^{91} -0.291796 q^{92} +20.5623 q^{94} -13.4164 q^{97} -9.70820 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{2} - q^{4} - 2 q^{7}+O(q^{10}) \) 2 * q + q^2 - q^4 - 2 * q^7	\( 2 q + q^{2} - q^{4} - 2 q^{7} + 6 q^{11} + 4 q^{13} - q^{14} - 3 q^{16} + 2 q^{17} - 4 q^{19} + 3 q^{22} + 8 q^{23} + 2 q^{26} + q^{28} + 8 q^{29} - 4 q^{31} - 9 q^{32} + q^{34} - 4 q^{37} - 7 q^{38} + 12 q^{41} + 8 q^{43} - 3 q^{44} - 6 q^{46} + 12 q^{47} - 12 q^{49} - 2 q^{52} + 6 q^{53} + 9 q^{58} - 20 q^{61} + 8 q^{62} + 4 q^{64} + 20 q^{67} - q^{68} - 8 q^{71} + 8 q^{73} + 13 q^{74} - 3 q^{76} - 6 q^{77} + 4 q^{79} + q^{82} - 8 q^{83} + 14 q^{86} + 24 q^{89} - 4 q^{91} - 14 q^{92} + 21 q^{94} - 6 q^{98}+O(q^{100}) \) 2 * q + q^2 - q^4 - 2 * q^7 + 6 * q^11 + 4 * q^13 - q^14 - 3 * q^16 + 2 * q^17 - 4 * q^19 + 3 * q^22 + 8 * q^23 + 2 * q^26 + q^28 + 8 * q^29 - 4 * q^31 - 9 * q^32 + q^34 - 4 * q^37 - 7 * q^38 + 12 * q^41 + 8 * q^43 - 3 * q^44 - 6 * q^46 + 12 * q^47 - 12 * q^49 - 2 * q^52 + 6 * q^53 + 9 * q^58 - 20 * q^61 + 8 * q^62 + 4 * q^64 + 20 * q^67 - q^68 - 8 * q^71 + 8 * q^73 + 13 * q^74 - 3 * q^76 - 6 * q^77 + 4 * q^79 + q^82 - 8 * q^83 + 14 * q^86 + 24 * q^89 - 4 * q^91 - 14 * q^92 + 21 * q^94 - 6 * q^98

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