LMFDB - Embedded newform 3680.2.a.v.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3680,2,Mod(1,3680)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3680, 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("3680.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3680 = 2^{5} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3680.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:	$29.3849479438$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.1143052.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 9x^{3} + 11x^{2} + 6x - 4 $ x^5 - x^4 - 9x^3 + 11x^2 + 6*x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-2.98054$ of defining polynomial
Character	$\chi$	$=$	3680.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.80707 q^{3} -1.00000 q^{5} +4.65156 q^{7} +0.265511 q^{9} +O(q^{10})$	$q-1.80707 q^{3} -1.00000 q^{5} +4.65156 q^{7} +0.265511 q^{9} -4.38210 q^{11} -4.79250 q^{13} +1.80707 q^{15} +0.285165 q^{17} -3.30952 q^{19} -8.40570 q^{21} -1.00000 q^{23} +1.00000 q^{25} +4.94142 q^{27} -3.47320 q^{29} -2.44557 q^{31} +7.91877 q^{33} -4.65156 q^{35} +11.0610 q^{37} +8.66039 q^{39} +11.7681 q^{41} +2.24999 q^{43} -0.265511 q^{45} -8.28422 q^{47} +14.6370 q^{49} -0.515313 q^{51} +3.81591 q^{53} +4.38210 q^{55} +5.98054 q^{57} -9.43005 q^{59} -5.54005 q^{61} +1.23504 q^{63} +4.79250 q^{65} +11.2062 q^{67} +1.80707 q^{69} -15.2464 q^{71} -9.87814 q^{73} -1.80707 q^{75} -20.3836 q^{77} +12.2402 q^{79} -9.72604 q^{81} -8.15495 q^{83} -0.285165 q^{85} +6.27633 q^{87} +1.98319 q^{89} -22.2926 q^{91} +4.41932 q^{93} +3.30952 q^{95} +4.33340 q^{97} -1.16349 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - q^{3} - 5 q^{5} + q^{7} + 6 q^{9}+O(q^{10}) $ 5 * q - q^3 - 5 * q^5 + q^7 + 6 * q^9	$ 5 q - q^{3} - 5 q^{5} + q^{7} + 6 q^{9} - 9 q^{11} + 3 q^{13} + q^{15} - 21 q^{17} - 7 q^{19} + 12 q^{21} - 5 q^{23} + 5 q^{25} + 20 q^{27} - 6 q^{29} - q^{31} + 17 q^{33} - q^{35} + 8 q^{37} + 4 q^{39} + 19 q^{41} - 14 q^{43} - 6 q^{45} - 8 q^{47} + 32 q^{49} - 25 q^{51} - 6 q^{53} + 9 q^{55} + 14 q^{57} - 6 q^{59} + 23 q^{61} + 48 q^{63} - 3 q^{65} + 2 q^{67} + q^{69} + 21 q^{71} - q^{75} - 26 q^{77} + 42 q^{79} + 25 q^{81} - 28 q^{83} + 21 q^{85} + 26 q^{87} - 19 q^{91} + 46 q^{93} + 7 q^{95} - 17 q^{97} - 47 q^{99}+O(q^{100}) $ 5 * q - q^3 - 5 * q^5 + q^7 + 6 * q^9 - 9 * q^11 + 3 * q^13 + q^15 - 21 * q^17 - 7 * q^19 + 12 * q^21 - 5 * q^23 + 5 * q^25 + 20 * q^27 - 6 * q^29 - q^31 + 17 * q^33 - q^35 + 8 * q^37 + 4 * q^39 + 19 * q^41 - 14 * q^43 - 6 * q^45 - 8 * q^47 + 32 * q^49 - 25 * q^51 - 6 * q^53 + 9 * q^55 + 14 * q^57 - 6 * q^59 + 23 * q^61 + 48 * q^63 - 3 * q^65 + 2 * q^67 + q^69 + 21 * q^71 - q^75 - 26 * q^77 + 42 * q^79 + 25 * q^81 - 28 * q^83 + 21 * q^85 + 26 * q^87 - 19 * q^91 + 46 * q^93 + 7 * q^95 - 17 * q^97 - 47 * 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
$3$	−1.80707	−1.04331	−0.521657	−	0.853155i	$-0.674686\pi$
$3$	−1.80707	−1.04331	−0.521657	+	0.853155i	$0.674686\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	4.65156	1.75812	0.879062	−	0.476708i	$-0.158170\pi$
$7$	4.65156	1.75812	0.879062	+	0.476708i	$0.158170\pi$
$8$	0	0
$9$	0.265511	0.0885036
$10$	0	0
$11$	−4.38210	−1.32125	−0.660626	−	0.750715i	$-0.729709\pi$
$11$	−4.38210	−1.32125	−0.660626	+	0.750715i	$0.729709\pi$
$12$	0	0
$13$	−4.79250	−1.32920	−0.664600	−	0.747199i	$-0.731398\pi$
$13$	−4.79250	−1.32920	−0.664600	+	0.747199i	$0.731398\pi$
$14$	0	0
$15$	1.80707	0.466584
$16$	0	0
$17$	0.285165	0.0691626	0.0345813	−	0.999402i	$-0.488990\pi$
$17$	0.285165	0.0691626	0.0345813	+	0.999402i	$0.488990\pi$
$18$	0	0
$19$	−3.30952	−0.759255	−0.379628	−	0.925139i	$-0.623948\pi$
$19$	−3.30952	−0.759255	−0.379628	+	0.925139i	$0.623948\pi$
$20$	0	0
$21$	−8.40570	−1.83427
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	4.94142	0.950977
$28$	0	0
$29$	−3.47320	−0.644958	−0.322479	−	0.946577i	$-0.604516\pi$
$29$	−3.47320	−0.644958	−0.322479	+	0.946577i	$0.604516\pi$
$30$	0	0
$31$	−2.44557	−0.439237	−0.219619	−	0.975586i	$-0.570481\pi$
$31$	−2.44557	−0.439237	−0.219619	+	0.975586i	$0.570481\pi$
$32$	0	0
$33$	7.91877	1.37848
$34$	0	0
$35$	−4.65156	−0.786257
$36$	0	0
$37$	11.0610	1.81842	0.909210	−	0.416339i	$-0.136687\pi$
$37$	11.0610	1.81842	0.909210	+	0.416339i	$0.136687\pi$
$38$	0	0
$39$	8.66039	1.38677
$40$	0	0
$41$	11.7681	1.83788	0.918938	−	0.394402i	$-0.129048\pi$
$41$	11.7681	1.83788	0.918938	+	0.394402i	$0.129048\pi$
$42$	0	0
$43$	2.24999	0.343121	0.171560	−	0.985174i	$-0.445119\pi$
$43$	2.24999	0.343121	0.171560	+	0.985174i	$0.445119\pi$
$44$	0	0
$45$	−0.265511	−0.0395800
$46$	0	0
$47$	−8.28422	−1.20838	−0.604189	−	0.796841i	$-0.706503\pi$
$47$	−8.28422	−1.20838	−0.604189	+	0.796841i	$0.706503\pi$
$48$	0	0
$49$	14.6370	2.09100
$50$	0	0
$51$	−0.515313	−0.0721583
$52$	0	0
$53$	3.81591	0.524155	0.262078	−	0.965047i	$-0.415592\pi$
$53$	3.81591	0.524155	0.262078	+	0.965047i	$0.415592\pi$
$54$	0	0
$55$	4.38210	0.590882
$56$	0	0
$57$	5.98054	0.792141
$58$	0	0
$59$	−9.43005	−1.22769	−0.613844	−	0.789427i	$-0.710378\pi$
$59$	−9.43005	−1.22769	−0.613844	+	0.789427i	$0.710378\pi$
$60$	0	0
$61$	−5.54005	−0.709331	−0.354665	−	0.934993i	$-0.615405\pi$
$61$	−5.54005	−0.709331	−0.354665	+	0.934993i	$0.615405\pi$
$62$	0	0
$63$	1.23504	0.155600
$64$	0	0
$65$	4.79250	0.594436
$66$	0	0
$67$	11.2062	1.36905	0.684526	−	0.728988i	$-0.260009\pi$
$67$	11.2062	1.36905	0.684526	+	0.728988i	$0.260009\pi$
$68$	0	0
$69$	1.80707	0.217546
$70$	0	0
$71$	−15.2464	−1.80942	−0.904709	−	0.426030i	$-0.859912\pi$
$71$	−15.2464	−1.80942	−0.904709	+	0.426030i	$0.859912\pi$
$72$	0	0
$73$	−9.87814	−1.15615	−0.578075	−	0.815984i	$-0.696196\pi$
$73$	−9.87814	−1.15615	−0.578075	+	0.815984i	$0.696196\pi$
$74$	0	0
$75$	−1.80707	−0.208663
$76$	0	0
$77$	−20.3836	−2.32293
$78$	0	0
$79$	12.2402	1.37713	0.688566	−	0.725174i	$-0.258241\pi$
$79$	12.2402	1.37713	0.688566	+	0.725174i	$0.258241\pi$
$80$	0	0
$81$	−9.72604	−1.08067
$82$	0	0
$83$	−8.15495	−0.895122	−0.447561	−	0.894254i	$-0.647707\pi$
$83$	−8.15495	−0.895122	−0.447561	+	0.894254i	$0.647707\pi$
$84$	0	0
$85$	−0.285165	−0.0309305
$86$	0	0
$87$	6.27633	0.672893
$88$	0	0
$89$	1.98319	0.210217	0.105109	−	0.994461i	$-0.466481\pi$
$89$	1.98319	0.210217	0.105109	+	0.994461i	$0.466481\pi$
$90$	0	0
$91$	−22.2926	−2.33690
$92$	0	0
$93$	4.41932	0.458262
$94$	0	0
$95$	3.30952	0.339549
$96$	0	0
$97$	4.33340	0.439990	0.219995	−	0.975501i	$-0.429396\pi$
$97$	4.33340	0.439990	0.219995	+	0.975501i	$0.429396\pi$
$98$	0	0
$99$	−1.16349	−0.116936
$100$	0	0
$101$	17.9203	1.78313	0.891566	−	0.452890i	$-0.149607\pi$
$101$	17.9203	1.78313	0.891566	+	0.452890i	$0.149607\pi$
$102$	0	0
$103$	−17.6351	−1.73764	−0.868819	−	0.495130i	$-0.835120\pi$
$103$	−17.6351	−1.73764	−0.868819	+	0.495130i	$0.835120\pi$
$104$	0	0
$105$	8.40570	0.820312
$106$	0	0
$107$	17.0535	1.64862	0.824312	−	0.566136i	$-0.191562\pi$
$107$	17.0535	1.64862	0.824312	+	0.566136i	$0.191562\pi$
$108$	0	0
$109$	11.3174	1.08401	0.542006	−	0.840375i	$-0.317665\pi$
$109$	11.3174	1.08401	0.542006	+	0.840375i	$0.317665\pi$
$110$	0	0
$111$	−19.9881	−1.89718
$112$	0	0
$113$	4.82783	0.454164	0.227082	−	0.973876i	$-0.427081\pi$
$113$	4.82783	0.454164	0.227082	+	0.973876i	$0.427081\pi$
$114$	0	0
$115$	1.00000	0.0932505
$116$	0	0
$117$	−1.27246	−0.117639
$118$	0	0
$119$	1.32646	0.121596
$120$	0	0
$121$	8.20280	0.745709
$122$	0	0
$123$	−21.2659	−1.91748
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	7.09524	0.629601	0.314800	−	0.949158i	$-0.398062\pi$
$127$	7.09524	0.629601	0.314800	+	0.949158i	$0.398062\pi$
$128$	0	0
$129$	−4.06590	−0.357983
$130$	0	0
$131$	11.0081	0.961781	0.480890	−	0.876781i	$-0.340314\pi$
$131$	11.0081	0.961781	0.480890	+	0.876781i	$0.340314\pi$
$132$	0	0
$133$	−15.3944	−1.33486
$134$	0	0
$135$	−4.94142	−0.425290
$136$	0	0
$137$	−2.82861	−0.241665	−0.120832	−	0.992673i	$-0.538556\pi$
$137$	−2.82861	−0.241665	−0.120832	+	0.992673i	$0.538556\pi$
$138$	0	0
$139$	3.14536	0.266786	0.133393	−	0.991063i	$-0.457413\pi$
$139$	3.14536	0.266786	0.133393	+	0.991063i	$0.457413\pi$
$140$	0	0
$141$	14.9702	1.26072
$142$	0	0
$143$	21.0012	1.75621
$144$	0	0
$145$	3.47320	0.288434
$146$	0	0
$147$	−26.4501	−2.18157
$148$	0	0
$149$	15.6949	1.28578	0.642888	−	0.765960i	$-0.277736\pi$
$149$	15.6949	1.28578	0.642888	+	0.765960i	$0.277736\pi$
$150$	0	0
$151$	−2.48412	−0.202155	−0.101077	−	0.994879i	$-0.532229\pi$
$151$	−2.48412	−0.202155	−0.101077	+	0.994879i	$0.532229\pi$
$152$	0	0
$153$	0.0757143	0.00612114
$154$	0	0
$155$	2.44557	0.196433
$156$	0	0
$157$	1.66989	0.133271	0.0666357	−	0.997777i	$-0.478773\pi$
$157$	1.66989	0.133271	0.0666357	+	0.997777i	$0.478773\pi$
$158$	0	0
$159$	−6.89562	−0.546858
$160$	0	0
$161$	−4.65156	−0.366594
$162$	0	0
$163$	4.92836	0.386019	0.193009	−	0.981197i	$-0.438175\pi$
$163$	4.92836	0.386019	0.193009	+	0.981197i	$0.438175\pi$
$164$	0	0
$165$	−7.91877	−0.616476
$166$	0	0
$167$	7.34993	0.568755	0.284377	−	0.958712i	$-0.408213\pi$
$167$	7.34993	0.568755	0.284377	+	0.958712i	$0.408213\pi$
$168$	0	0
$169$	9.96805	0.766773
$170$	0	0
$171$	−0.878712	−0.0671968
$172$	0	0
$173$	25.4824	1.93739	0.968696	−	0.248249i	$-0.0798551\pi$
$173$	25.4824	1.93739	0.968696	+	0.248249i	$0.0798551\pi$
$174$	0	0
$175$	4.65156	0.351625
$176$	0	0
$177$	17.0408	1.28086
$178$	0	0
$179$	−7.78168	−0.581630	−0.290815	−	0.956779i	$-0.593926\pi$
$179$	−7.78168	−0.581630	−0.290815	+	0.956779i	$0.593926\pi$
$180$	0	0
$181$	23.7922	1.76846	0.884231	−	0.467050i	$-0.154683\pi$
$181$	23.7922	1.76846	0.884231	+	0.467050i	$0.154683\pi$
$182$	0	0
$183$	10.0113	0.740054
$184$	0	0
$185$	−11.0610	−0.813222
$186$	0	0
$187$	−1.24962	−0.0913813
$188$	0	0
$189$	22.9853	1.67193
$190$	0	0
$191$	−5.81402	−0.420688	−0.210344	−	0.977627i	$-0.567458\pi$
$191$	−5.81402	−0.420688	−0.210344	+	0.977627i	$0.567458\pi$
$192$	0	0
$193$	−8.53422	−0.614306	−0.307153	−	0.951660i	$-0.599376\pi$
$193$	−8.53422	−0.614306	−0.307153	+	0.951660i	$0.599376\pi$
$194$	0	0
$195$	−8.66039	−0.620184
$196$	0	0
$197$	−21.9792	−1.56595	−0.782977	−	0.622051i	$-0.786300\pi$
$197$	−21.9792	−1.56595	−0.782977	+	0.622051i	$0.786300\pi$
$198$	0	0
$199$	26.6774	1.89111	0.945556	−	0.325459i	$-0.105519\pi$
$199$	26.6774	1.89111	0.945556	+	0.325459i	$0.105519\pi$
$200$	0	0
$201$	−20.2504	−1.42835
$202$	0	0
$203$	−16.1558	−1.13391
$204$	0	0
$205$	−11.7681	−0.821923
$206$	0	0
$207$	−0.265511	−0.0184543
$208$	0	0
$209$	14.5026	1.00317
$210$	0	0
$211$	21.5831	1.48584	0.742921	−	0.669379i	$-0.233440\pi$
$211$	21.5831	1.48584	0.742921	+	0.669379i	$0.233440\pi$
$212$	0	0
$213$	27.5514	1.88779
$214$	0	0
$215$	−2.24999	−0.153448
$216$	0	0
$217$	−11.3757	−0.772233
$218$	0	0
$219$	17.8505	1.20623
$220$	0	0
$221$	−1.36665	−0.0919310
$222$	0	0
$223$	−5.43756	−0.364126	−0.182063	−	0.983287i	$-0.558278\pi$
$223$	−5.43756	−0.364126	−0.182063	+	0.983287i	$0.558278\pi$
$224$	0	0
$225$	0.265511	0.0177007
$226$	0	0
$227$	−18.0875	−1.20051	−0.600256	−	0.799808i	$-0.704935\pi$
$227$	−18.0875	−1.20051	−0.600256	+	0.799808i	$0.704935\pi$
$228$	0	0
$229$	−4.62834	−0.305849	−0.152925	−	0.988238i	$-0.548869\pi$
$229$	−4.62834	−0.305849	−0.152925	+	0.988238i	$0.548869\pi$
$230$	0	0
$231$	36.8346	2.42354
$232$	0	0
$233$	7.38793	0.484000	0.242000	−	0.970276i	$-0.422197\pi$
$233$	7.38793	0.484000	0.242000	+	0.970276i	$0.422197\pi$
$234$	0	0
$235$	8.28422	0.540403
$236$	0	0
$237$	−22.1190	−1.43678
$238$	0	0
$239$	6.83735	0.442272	0.221136	−	0.975243i	$-0.429024\pi$
$239$	6.83735	0.442272	0.221136	+	0.975243i	$0.429024\pi$
$240$	0	0
$241$	7.95130	0.512188	0.256094	−	0.966652i	$-0.417564\pi$
$241$	7.95130	0.512188	0.256094	+	0.966652i	$0.417564\pi$
$242$	0	0
$243$	2.75139	0.176502
$244$	0	0
$245$	−14.6370	−0.935123
$246$	0	0
$247$	15.8609	1.00920
$248$	0	0
$249$	14.7366	0.933893
$250$	0	0
$251$	−29.2227	−1.84452	−0.922262	−	0.386566i	$-0.873661\pi$
$251$	−29.2227	−1.84452	−0.922262	+	0.386566i	$0.873661\pi$
$252$	0	0
$253$	4.38210	0.275500
$254$	0	0
$255$	0.515313	0.0322702
$256$	0	0
$257$	1.49980	0.0935548	0.0467774	−	0.998905i	$-0.485105\pi$
$257$	1.49980	0.0935548	0.0467774	+	0.998905i	$0.485105\pi$
$258$	0	0
$259$	51.4509	3.19701
$260$	0	0
$261$	−0.922173	−0.0570810
$262$	0	0
$263$	−21.5869	−1.33110	−0.665552	−	0.746352i	$-0.731804\pi$
$263$	−21.5869	−1.33110	−0.665552	+	0.746352i	$0.731804\pi$
$264$	0	0
$265$	−3.81591	−0.234409
$266$	0	0
$267$	−3.58376	−0.219322
$268$	0	0
$269$	−7.49276	−0.456842	−0.228421	−	0.973562i	$-0.573356\pi$
$269$	−7.49276	−0.456842	−0.228421	+	0.973562i	$0.573356\pi$
$270$	0	0
$271$	12.7175	0.772531	0.386265	−	0.922388i	$-0.373765\pi$
$271$	12.7175	0.772531	0.386265	+	0.922388i	$0.373765\pi$
$272$	0	0
$273$	40.2843	2.43812
$274$	0	0
$275$	−4.38210	−0.264251
$276$	0	0
$277$	27.9099	1.67695	0.838473	−	0.544943i	$-0.183449\pi$
$277$	27.9099	1.67695	0.838473	+	0.544943i	$0.183449\pi$
$278$	0	0
$279$	−0.649325	−0.0388741
$280$	0	0
$281$	21.7961	1.30024	0.650122	−	0.759830i	$-0.274718\pi$
$281$	21.7961	1.30024	0.650122	+	0.759830i	$0.274718\pi$
$282$	0	0
$283$	23.5235	1.39833	0.699163	−	0.714962i	$-0.253556\pi$
$283$	23.5235	1.39833	0.699163	+	0.714962i	$0.253556\pi$
$284$	0	0
$285$	−5.98054	−0.354256
$286$	0	0
$287$	54.7402	3.23121
$288$	0	0
$289$	−16.9187	−0.995217
$290$	0	0
$291$	−7.83076	−0.459047
$292$	0	0
$293$	−0.675631	−0.0394708	−0.0197354	−	0.999805i	$-0.506282\pi$
$293$	−0.675631	−0.0394708	−0.0197354	+	0.999805i	$0.506282\pi$
$294$	0	0
$295$	9.43005	0.549039
$296$	0	0
$297$	−21.6538	−1.25648
$298$	0	0
$299$	4.79250	0.277157
$300$	0	0
$301$	10.4660	0.603249
$302$	0	0
$303$	−32.3832	−1.86037
$304$	0	0
$305$	5.54005	0.317222
$306$	0	0
$307$	−3.23138	−0.184425	−0.0922123	−	0.995739i	$-0.529394\pi$
$307$	−3.23138	−0.184425	−0.0922123	+	0.995739i	$0.529394\pi$
$308$	0	0
$309$	31.8679	1.81290
$310$	0	0
$311$	21.8033	1.23635	0.618176	−	0.786039i	$-0.287872\pi$
$311$	21.8033	1.23635	0.618176	+	0.786039i	$0.287872\pi$
$312$	0	0
$313$	28.9311	1.63528	0.817641	−	0.575729i	$-0.195282\pi$
$313$	28.9311	1.63528	0.817641	+	0.575729i	$0.195282\pi$
$314$	0	0
$315$	−1.23504	−0.0695865
$316$	0	0
$317$	−15.2017	−0.853810	−0.426905	−	0.904296i	$-0.640396\pi$
$317$	−15.2017	−0.853810	−0.426905	+	0.904296i	$0.640396\pi$
$318$	0	0
$319$	15.2199	0.852152
$320$	0	0
$321$	−30.8169	−1.72003
$322$	0	0
$323$	−0.943758	−0.0525121
$324$	0	0
$325$	−4.79250	−0.265840
$326$	0	0
$327$	−20.4514	−1.13096
$328$	0	0
$329$	−38.5345	−2.12448
$330$	0	0
$331$	5.13558	0.282277	0.141138	−	0.989990i	$-0.454924\pi$
$331$	5.13558	0.282277	0.141138	+	0.989990i	$0.454924\pi$
$332$	0	0
$333$	2.93682	0.160937
$334$	0	0
$335$	−11.2062	−0.612259
$336$	0	0
$337$	2.54759	0.138776	0.0693879	−	0.997590i	$-0.477895\pi$
$337$	2.54759	0.138776	0.0693879	+	0.997590i	$0.477895\pi$
$338$	0	0
$339$	−8.72424	−0.473836
$340$	0	0
$341$	10.7167	0.580344
$342$	0	0
$343$	35.5239	1.91811
$344$	0	0
$345$	−1.80707	−0.0972895
$346$	0	0
$347$	18.2462	0.979509	0.489755	−	0.871860i	$-0.337086\pi$
$347$	18.2462	0.979509	0.489755	+	0.871860i	$0.337086\pi$
$348$	0	0
$349$	0.262585	0.0140558	0.00702792	−	0.999975i	$-0.497763\pi$
$349$	0.262585	0.0140558	0.00702792	+	0.999975i	$0.497763\pi$
$350$	0	0
$351$	−23.6818	−1.26404
$352$	0	0
$353$	−5.75461	−0.306287	−0.153144	−	0.988204i	$-0.548940\pi$
$353$	−5.75461	−0.306287	−0.153144	+	0.988204i	$0.548940\pi$
$354$	0	0
$355$	15.2464	0.809197
$356$	0	0
$357$	−2.39701	−0.126863
$358$	0	0
$359$	16.1677	0.853296	0.426648	−	0.904418i	$-0.359694\pi$
$359$	16.1677	0.853296	0.426648	+	0.904418i	$0.359694\pi$
$360$	0	0
$361$	−8.04710	−0.423531
$362$	0	0
$363$	−14.8231	−0.778009
$364$	0	0
$365$	9.87814	0.517046
$366$	0	0
$367$	−9.25214	−0.482958	−0.241479	−	0.970406i	$-0.577632\pi$
$367$	−9.25214	−0.482958	−0.241479	+	0.970406i	$0.577632\pi$
$368$	0	0
$369$	3.12457	0.162659
$370$	0	0
$371$	17.7499	0.921530
$372$	0	0
$373$	15.4970	0.802403	0.401202	−	0.915990i	$-0.368593\pi$
$373$	15.4970	0.802403	0.401202	+	0.915990i	$0.368593\pi$
$374$	0	0
$375$	1.80707	0.0933168
$376$	0	0
$377$	16.6453	0.857278
$378$	0	0
$379$	−21.4839	−1.10356	−0.551778	−	0.833991i	$-0.686050\pi$
$379$	−21.4839	−1.10356	−0.551778	+	0.833991i	$0.686050\pi$
$380$	0	0
$381$	−12.8216	−0.656871
$382$	0	0
$383$	12.3758	0.632373	0.316187	−	0.948697i	$-0.397597\pi$
$383$	12.3758	0.632373	0.316187	+	0.948697i	$0.397597\pi$
$384$	0	0
$385$	20.3836	1.03884
$386$	0	0
$387$	0.597397	0.0303674
$388$	0	0
$389$	−13.3944	−0.679123	−0.339562	−	0.940584i	$-0.610279\pi$
$389$	−13.3944	−0.679123	−0.339562	+	0.940584i	$0.610279\pi$
$390$	0	0
$391$	−0.285165	−0.0144214
$392$	0	0
$393$	−19.8924	−1.00344
$394$	0	0
$395$	−12.2402	−0.615872
$396$	0	0
$397$	−37.1195	−1.86297	−0.931487	−	0.363775i	$-0.881488\pi$
$397$	−37.1195	−1.86297	−0.931487	+	0.363775i	$0.881488\pi$
$398$	0	0
$399$	27.8188	1.39268
$400$	0	0
$401$	−4.40938	−0.220194	−0.110097	−	0.993921i	$-0.535116\pi$
$401$	−4.40938	−0.220194	−0.110097	+	0.993921i	$0.535116\pi$
$402$	0	0
$403$	11.7204	0.583834
$404$	0	0
$405$	9.72604	0.483291
$406$	0	0
$407$	−48.4705	−2.40259
$408$	0	0
$409$	−10.0066	−0.494793	−0.247397	−	0.968914i	$-0.579575\pi$
$409$	−10.0066	−0.494793	−0.247397	+	0.968914i	$0.579575\pi$
$410$	0	0
$411$	5.11151	0.252132
$412$	0	0
$413$	−43.8644	−2.15843
$414$	0	0
$415$	8.15495	0.400311
$416$	0	0
$417$	−5.68389	−0.278341
$418$	0	0
$419$	−11.9583	−0.584203	−0.292101	−	0.956387i	$-0.594354\pi$
$419$	−11.9583	−0.584203	−0.292101	+	0.956387i	$0.594354\pi$
$420$	0	0
$421$	26.4678	1.28996	0.644982	−	0.764198i	$-0.276865\pi$
$421$	26.4678	1.28996	0.644982	+	0.764198i	$0.276865\pi$
$422$	0	0
$423$	−2.19955	−0.106946
$424$	0	0
$425$	0.285165	0.0138325
$426$	0	0
$427$	−25.7699	−1.24709
$428$	0	0
$429$	−37.9507	−1.83228
$430$	0	0
$431$	12.8006	0.616582	0.308291	−	0.951292i	$-0.400243\pi$
$431$	12.8006	0.616582	0.308291	+	0.951292i	$0.400243\pi$
$432$	0	0
$433$	12.5437	0.602812	0.301406	−	0.953496i	$-0.402544\pi$
$433$	12.5437	0.602812	0.301406	+	0.953496i	$0.402544\pi$
$434$	0	0
$435$	−6.27633	−0.300927
$436$	0	0
$437$	3.30952	0.158316
$438$	0	0
$439$	37.7389	1.80118	0.900590	−	0.434670i	$-0.143135\pi$
$439$	37.7389	1.80118	0.900590	+	0.434670i	$0.143135\pi$
$440$	0	0
$441$	3.88628	0.185061
$442$	0	0
$443$	−3.89055	−0.184846	−0.0924228	−	0.995720i	$-0.529461\pi$
$443$	−3.89055	−0.184846	−0.0924228	+	0.995720i	$0.529461\pi$
$444$	0	0
$445$	−1.98319	−0.0940120
$446$	0	0
$447$	−28.3618	−1.34147
$448$	0	0
$449$	−4.10031	−0.193506	−0.0967528	−	0.995308i	$-0.530846\pi$
$449$	−4.10031	−0.193506	−0.0967528	+	0.995308i	$0.530846\pi$
$450$	0	0
$451$	−51.5692	−2.42830
$452$	0	0
$453$	4.48898	0.210911
$454$	0	0
$455$	22.2926	1.04509
$456$	0	0
$457$	14.8783	0.695979	0.347990	−	0.937498i	$-0.386864\pi$
$457$	14.8783	0.695979	0.347990	+	0.937498i	$0.386864\pi$
$458$	0	0
$459$	1.40912	0.0657720
$460$	0	0
$461$	1.90588	0.0887655	0.0443828	−	0.999015i	$-0.485868\pi$
$461$	1.90588	0.0887655	0.0443828	+	0.999015i	$0.485868\pi$
$462$	0	0
$463$	−21.2271	−0.986509	−0.493255	−	0.869885i	$-0.664193\pi$
$463$	−21.2271	−0.986509	−0.493255	+	0.869885i	$0.664193\pi$
$464$	0	0
$465$	−4.41932	−0.204941
$466$	0	0
$467$	12.0609	0.558114	0.279057	−	0.960275i	$-0.409978\pi$
$467$	12.0609	0.558114	0.279057	+	0.960275i	$0.409978\pi$
$468$	0	0
$469$	52.1262	2.40696
$470$	0	0
$471$	−3.01760	−0.139044
$472$	0	0
$473$	−9.85970	−0.453349
$474$	0	0
$475$	−3.30952	−0.151851
$476$	0	0
$477$	1.01316	0.0463896
$478$	0	0
$479$	8.96710	0.409717	0.204859	−	0.978792i	$-0.434327\pi$
$479$	8.96710	0.409717	0.204859	+	0.978792i	$0.434327\pi$
$480$	0	0
$481$	−53.0099	−2.41704
$482$	0	0
$483$	8.40570	0.382473
$484$	0	0
$485$	−4.33340	−0.196769
$486$	0	0
$487$	−23.3799	−1.05945	−0.529723	−	0.848171i	$-0.677704\pi$
$487$	−23.3799	−1.05945	−0.529723	+	0.848171i	$0.677704\pi$
$488$	0	0
$489$	−8.90590	−0.402739
$490$	0	0
$491$	−13.2821	−0.599412	−0.299706	−	0.954032i	$-0.596889\pi$
$491$	−13.2821	−0.599412	−0.299706	+	0.954032i	$0.596889\pi$
$492$	0	0
$493$	−0.990435	−0.0446070
$494$	0	0
$495$	1.16349	0.0522952
$496$	0	0
$497$	−70.9196	−3.18118
$498$	0	0
$499$	−18.9496	−0.848301	−0.424150	−	0.905592i	$-0.639427\pi$
$499$	−18.9496	−0.848301	−0.424150	+	0.905592i	$0.639427\pi$
$500$	0	0
$501$	−13.2819	−0.593390
$502$	0	0
$503$	30.1400	1.34388	0.671938	−	0.740607i	$-0.265462\pi$
$503$	30.1400	1.34388	0.671938	+	0.740607i	$0.265462\pi$
$504$	0	0
$505$	−17.9203	−0.797441
$506$	0	0
$507$	−18.0130	−0.799985
$508$	0	0
$509$	8.77973	0.389155	0.194577	−	0.980887i	$-0.437667\pi$
$509$	8.77973	0.389155	0.194577	+	0.980887i	$0.437667\pi$
$510$	0	0
$511$	−45.9487	−2.03265
$512$	0	0
$513$	−16.3537	−0.722034
$514$	0	0
$515$	17.6351	0.777095
$516$	0	0
$517$	36.3023	1.59657
$518$	0	0
$519$	−46.0486	−2.02131
$520$	0	0
$521$	37.6420	1.64912	0.824562	−	0.565771i	$-0.191421\pi$
$521$	37.6420	1.64912	0.824562	+	0.565771i	$0.191421\pi$
$522$	0	0
$523$	−20.7013	−0.905204	−0.452602	−	0.891713i	$-0.649504\pi$
$523$	−20.7013	−0.905204	−0.452602	+	0.891713i	$0.649504\pi$
$524$	0	0
$525$	−8.40570	−0.366855
$526$	0	0
$527$	−0.697390	−0.0303788
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−2.50378	−0.108655
$532$	0	0
$533$	−56.3988	−2.44290
$534$	0	0
$535$	−17.0535	−0.737287
$536$	0	0
$537$	14.0621	0.606823
$538$	0	0
$539$	−64.1407	−2.76274
$540$	0	0
$541$	22.2767	0.957751	0.478875	−	0.877883i	$-0.341045\pi$
$541$	22.2767	0.957751	0.478875	+	0.877883i	$0.341045\pi$
$542$	0	0
$543$	−42.9943	−1.84506
$544$	0	0
$545$	−11.3174	−0.484785
$546$	0	0
$547$	−33.4571	−1.43052	−0.715262	−	0.698857i	$-0.753692\pi$
$547$	−33.4571	−1.43052	−0.715262	+	0.698857i	$0.753692\pi$
$548$	0	0
$549$	−1.47094	−0.0627783
$550$	0	0
$551$	11.4946	0.489687
$552$	0	0
$553$	56.9361	2.42117
$554$	0	0
$555$	19.9881	0.848446
$556$	0	0
$557$	19.7297	0.835974	0.417987	−	0.908453i	$-0.362736\pi$
$557$	19.7297	0.835974	0.417987	+	0.908453i	$0.362736\pi$
$558$	0	0
$559$	−10.7831	−0.456076
$560$	0	0
$561$	2.25816	0.0953394
$562$	0	0
$563$	−8.44680	−0.355990	−0.177995	−	0.984031i	$-0.556961\pi$
$563$	−8.44680	−0.355990	−0.177995	+	0.984031i	$0.556961\pi$
$564$	0	0
$565$	−4.82783	−0.203108
$566$	0	0
$567$	−45.2412	−1.89995
$568$	0	0
$569$	27.8345	1.16688	0.583442	−	0.812155i	$-0.301706\pi$
$569$	27.8345	1.16688	0.583442	+	0.812155i	$0.301706\pi$
$570$	0	0
$571$	17.8837	0.748409	0.374204	−	0.927346i	$-0.377916\pi$
$571$	17.8837	0.748409	0.374204	+	0.927346i	$0.377916\pi$
$572$	0	0
$573$	10.5064	0.438909
$574$	0	0
$575$	−1.00000	−0.0417029
$576$	0	0
$577$	−8.19106	−0.340998	−0.170499	−	0.985358i	$-0.554538\pi$
$577$	−8.19106	−0.340998	−0.170499	+	0.985358i	$0.554538\pi$
$578$	0	0
$579$	15.4219	0.640914
$580$	0	0
$581$	−37.9332	−1.57373
$582$	0	0
$583$	−16.7217	−0.692542
$584$	0	0
$585$	1.27246	0.0526097
$586$	0	0
$587$	−39.9336	−1.64824	−0.824119	−	0.566417i	$-0.808329\pi$
$587$	−39.9336	−1.64824	−0.824119	+	0.566417i	$0.808329\pi$
$588$	0	0
$589$	8.09365	0.333493
$590$	0	0
$591$	39.7180	1.63378
$592$	0	0
$593$	−17.4592	−0.716963	−0.358482	−	0.933537i	$-0.616705\pi$
$593$	−17.4592	−0.716963	−0.358482	+	0.933537i	$0.616705\pi$
$594$	0	0
$595$	−1.32646	−0.0543796
$596$	0	0
$597$	−48.2080	−1.97302
$598$	0	0
$599$	−9.83913	−0.402016	−0.201008	−	0.979590i	$-0.564422\pi$
$599$	−9.83913	−0.402016	−0.201008	+	0.979590i	$0.564422\pi$
$600$	0	0
$601$	20.1621	0.822428	0.411214	−	0.911539i	$-0.365105\pi$
$601$	20.1621	0.822428	0.411214	+	0.911539i	$0.365105\pi$
$602$	0	0
$603$	2.97536	0.121166
$604$	0	0
$605$	−8.20280	−0.333491
$606$	0	0
$607$	−7.64121	−0.310147	−0.155074	−	0.987903i	$-0.549561\pi$
$607$	−7.64121	−0.310147	−0.155074	+	0.987903i	$0.549561\pi$
$608$	0	0
$609$	29.1947	1.18303
$610$	0	0
$611$	39.7021	1.60618
$612$	0	0
$613$	−0.518216	−0.0209305	−0.0104653	−	0.999945i	$-0.503331\pi$
$613$	−0.518216	−0.0209305	−0.0104653	+	0.999945i	$0.503331\pi$
$614$	0	0
$615$	21.2659	0.857524
$616$	0	0
$617$	−22.5090	−0.906180	−0.453090	−	0.891465i	$-0.649678\pi$
$617$	−22.5090	−0.906180	−0.453090	+	0.891465i	$0.649678\pi$
$618$	0	0
$619$	−15.7750	−0.634052	−0.317026	−	0.948417i	$-0.602684\pi$
$619$	−15.7750	−0.634052	−0.317026	+	0.948417i	$0.602684\pi$
$620$	0	0
$621$	−4.94142	−0.198292
$622$	0	0
$623$	9.22490	0.369588
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−26.2073	−1.04662
$628$	0	0
$629$	3.15421	0.125767
$630$	0	0
$631$	36.1022	1.43721	0.718603	−	0.695421i	$-0.244782\pi$
$631$	36.1022	1.43721	0.718603	+	0.695421i	$0.244782\pi$
$632$	0	0
$633$	−39.0022	−1.55020
$634$	0	0
$635$	−7.09524	−0.281566
$636$	0	0
$637$	−70.1477	−2.77935
$638$	0	0
$639$	−4.04809	−0.160140
$640$	0	0
$641$	−2.17328	−0.0858395	−0.0429198	−	0.999079i	$-0.513666\pi$
$641$	−2.17328	−0.0858395	−0.0429198	+	0.999079i	$0.513666\pi$
$642$	0	0
$643$	15.5752	0.614227	0.307113	−	0.951673i	$-0.400637\pi$
$643$	15.5752	0.614227	0.307113	+	0.951673i	$0.400637\pi$
$644$	0	0
$645$	4.06590	0.160095
$646$	0	0
$647$	8.47998	0.333382	0.166691	−	0.986009i	$-0.446692\pi$
$647$	8.47998	0.333382	0.166691	+	0.986009i	$0.446692\pi$
$648$	0	0
$649$	41.3234	1.62209
$650$	0	0
$651$	20.5567	0.805682
$652$	0	0
$653$	−16.3027	−0.637973	−0.318986	−	0.947759i	$-0.603342\pi$
$653$	−16.3027	−0.637973	−0.318986	+	0.947759i	$0.603342\pi$
$654$	0	0
$655$	−11.0081	−0.430121
$656$	0	0
$657$	−2.62275	−0.102323
$658$	0	0
$659$	−0.674301	−0.0262670	−0.0131335	−	0.999914i	$-0.504181\pi$
$659$	−0.674301	−0.0262670	−0.0131335	+	0.999914i	$0.504181\pi$
$660$	0	0
$661$	3.55427	0.138245	0.0691225	−	0.997608i	$-0.477980\pi$
$661$	3.55427	0.138245	0.0691225	+	0.997608i	$0.477980\pi$
$662$	0	0
$663$	2.46964	0.0959128
$664$	0	0
$665$	15.3944	0.596970
$666$	0	0
$667$	3.47320	0.134483
$668$	0	0
$669$	9.82607	0.379898
$670$	0	0
$671$	24.2770	0.937205
$672$	0	0
$673$	14.4874	0.558448	0.279224	−	0.960226i	$-0.409923\pi$
$673$	14.4874	0.558448	0.279224	+	0.960226i	$0.409923\pi$
$674$	0	0
$675$	4.94142	0.190195
$676$	0	0
$677$	13.4408	0.516571	0.258285	−	0.966069i	$-0.416843\pi$
$677$	13.4408	0.516571	0.258285	+	0.966069i	$0.416843\pi$
$678$	0	0
$679$	20.1570	0.773556
$680$	0	0
$681$	32.6855	1.25251
$682$	0	0
$683$	−12.6694	−0.484783	−0.242391	−	0.970179i	$-0.577932\pi$
$683$	−12.6694	−0.484783	−0.242391	+	0.970179i	$0.577932\pi$
$684$	0	0
$685$	2.82861	0.108076
$686$	0	0
$687$	8.36374	0.319097
$688$	0	0
$689$	−18.2877	−0.696707
$690$	0	0
$691$	47.0038	1.78811	0.894054	−	0.447960i	$-0.147849\pi$
$691$	47.0038	1.78811	0.894054	+	0.447960i	$0.147849\pi$
$692$	0	0
$693$	−5.41206	−0.205587
$694$	0	0
$695$	−3.14536	−0.119310
$696$	0	0
$697$	3.35586	0.127112
$698$	0	0
$699$	−13.3505	−0.504963
$700$	0	0
$701$	37.3330	1.41005	0.705024	−	0.709184i	$-0.250936\pi$
$701$	37.3330	1.41005	0.705024	+	0.709184i	$0.250936\pi$
$702$	0	0
$703$	−36.6066	−1.38064
$704$	0	0
$705$	−14.9702	−0.563810
$706$	0	0
$707$	83.3571	3.13497
$708$	0	0
$709$	38.0927	1.43060	0.715301	−	0.698817i	$-0.246290\pi$
$709$	38.0927	1.43060	0.715301	+	0.698817i	$0.246290\pi$
$710$	0	0
$711$	3.24991	0.121881
$712$	0	0
$713$	2.44557	0.0915873
$714$	0	0
$715$	−21.0012	−0.785401
$716$	0	0
$717$	−12.3556	−0.461428
$718$	0	0
$719$	25.5206	0.951757	0.475879	−	0.879511i	$-0.342130\pi$
$719$	25.5206	0.951757	0.475879	+	0.879511i	$0.342130\pi$
$720$	0	0
$721$	−82.0307	−3.05498
$722$	0	0
$723$	−14.3686	−0.534373
$724$	0	0
$725$	−3.47320	−0.128992
$726$	0	0
$727$	41.6377	1.54426	0.772129	−	0.635465i	$-0.219192\pi$
$727$	41.6377	1.54426	0.772129	+	0.635465i	$0.219192\pi$
$728$	0	0
$729$	24.2061	0.896524
$730$	0	0
$731$	0.641619	0.0237311
$732$	0	0
$733$	26.6076	0.982773	0.491386	−	0.870942i	$-0.336490\pi$
$733$	26.6076	0.982773	0.491386	+	0.870942i	$0.336490\pi$
$734$	0	0
$735$	26.4501	0.975626
$736$	0	0
$737$	−49.1066	−1.80886
$738$	0	0
$739$	0.749723	0.0275790	0.0137895	−	0.999905i	$-0.495611\pi$
$739$	0.749723	0.0275790	0.0137895	+	0.999905i	$0.495611\pi$
$740$	0	0
$741$	−28.6617	−1.05291
$742$	0	0
$743$	−27.3965	−1.00508	−0.502540	−	0.864554i	$-0.667601\pi$
$743$	−27.3965	−1.00508	−0.502540	+	0.864554i	$0.667601\pi$
$744$	0	0
$745$	−15.6949	−0.575017
$746$	0	0
$747$	−2.16523	−0.0792214
$748$	0	0
$749$	79.3253	2.89848
$750$	0	0
$751$	7.46662	0.272461	0.136230	−	0.990677i	$-0.456501\pi$
$751$	7.46662	0.272461	0.136230	+	0.990677i	$0.456501\pi$
$752$	0	0
$753$	52.8076	1.92442
$754$	0	0
$755$	2.48412	0.0904063
$756$	0	0
$757$	20.1426	0.732096	0.366048	−	0.930596i	$-0.380711\pi$
$757$	20.1426	0.732096	0.366048	+	0.930596i	$0.380711\pi$
$758$	0	0
$759$	−7.91877	−0.287433
$760$	0	0
$761$	−49.9203	−1.80961	−0.904804	−	0.425828i	$-0.859983\pi$
$761$	−49.9203	−1.80961	−0.904804	+	0.425828i	$0.859983\pi$
$762$	0	0
$763$	52.6436	1.90583
$764$	0	0
$765$	−0.0757143	−0.00273746
$766$	0	0
$767$	45.1935	1.63184
$768$	0	0
$769$	−31.7567	−1.14517	−0.572587	−	0.819844i	$-0.694060\pi$
$769$	−31.7567	−1.14517	−0.572587	+	0.819844i	$0.694060\pi$
$770$	0	0
$771$	−2.71024	−0.0976070
$772$	0	0
$773$	−28.3113	−1.01829	−0.509143	−	0.860682i	$-0.670038\pi$
$773$	−28.3113	−1.01829	−0.509143	+	0.860682i	$0.670038\pi$
$774$	0	0
$775$	−2.44557	−0.0878475
$776$	0	0
$777$	−92.9756	−3.33548
$778$	0	0
$779$	−38.9469	−1.39542
$780$	0	0
$781$	66.8114	2.39070
$782$	0	0
$783$	−17.1626	−0.613340
$784$	0	0
$785$	−1.66989	−0.0596008
$786$	0	0
$787$	−14.2157	−0.506735	−0.253367	−	0.967370i	$-0.581538\pi$
$787$	−14.2157	−0.506735	−0.253367	+	0.967370i	$0.581538\pi$
$788$	0	0
$789$	39.0090	1.38876
$790$	0	0
$791$	22.4569	0.798477
$792$	0	0
$793$	26.5507	0.942842
$794$	0	0
$795$	6.89562	0.244563
$796$	0	0
$797$	−4.27296	−0.151356	−0.0756780	−	0.997132i	$-0.524112\pi$
$797$	−4.27296	−0.151356	−0.0756780	+	0.997132i	$0.524112\pi$
$798$	0	0
$799$	−2.36237	−0.0835746
$800$	0	0
$801$	0.526557	0.0186050
$802$	0	0
$803$	43.2870	1.52757
$804$	0	0
$805$	4.65156	0.163946
$806$	0	0
$807$	13.5400	0.476629
$808$	0	0
$809$	2.97019	0.104426	0.0522132	−	0.998636i	$-0.483372\pi$
$809$	2.97019	0.104426	0.0522132	+	0.998636i	$0.483372\pi$
$810$	0	0
$811$	−24.4381	−0.858139	−0.429070	−	0.903271i	$-0.641158\pi$
$811$	−24.4381	−0.858139	−0.429070	+	0.903271i	$0.641158\pi$
$812$	0	0
$813$	−22.9814	−0.805992
$814$	0	0
$815$	−4.92836	−0.172633
$816$	0	0
$817$	−7.44639	−0.260516
$818$	0	0
$819$	−5.91892	−0.206824
$820$	0	0
$821$	−7.88814	−0.275298	−0.137649	−	0.990481i	$-0.543955\pi$
$821$	−7.88814	−0.275298	−0.137649	+	0.990481i	$0.543955\pi$
$822$	0	0
$823$	14.3154	0.499005	0.249503	−	0.968374i	$-0.419733\pi$
$823$	14.3154	0.499005	0.249503	+	0.968374i	$0.419733\pi$
$824$	0	0
$825$	7.91877	0.275696
$826$	0	0
$827$	−22.4641	−0.781153	−0.390576	−	0.920570i	$-0.627724\pi$
$827$	−22.4641	−0.781153	−0.390576	+	0.920570i	$0.627724\pi$
$828$	0	0
$829$	−42.8763	−1.48915	−0.744577	−	0.667536i	$-0.767349\pi$
$829$	−42.8763	−1.48915	−0.744577	+	0.667536i	$0.767349\pi$
$830$	0	0
$831$	−50.4353	−1.74958
$832$	0	0
$833$	4.17395	0.144619
$834$	0	0
$835$	−7.34993	−0.254355
$836$	0	0
$837$	−12.0846	−0.417704
$838$	0	0
$839$	20.1216	0.694673	0.347337	−	0.937741i	$-0.387086\pi$
$839$	20.1216	0.694673	0.347337	+	0.937741i	$0.387086\pi$
$840$	0	0
$841$	−16.9369	−0.584030
$842$	0	0
$843$	−39.3871	−1.35656
$844$	0	0
$845$	−9.96805	−0.342911
$846$	0	0
$847$	38.1558	1.31105
$848$	0	0
$849$	−42.5086	−1.45889
$850$	0	0
$851$	−11.0610	−0.379167
$852$	0	0
$853$	−48.2488	−1.65201	−0.826004	−	0.563664i	$-0.809391\pi$
$853$	−48.2488	−1.65201	−0.826004	+	0.563664i	$0.809391\pi$
$854$	0	0
$855$	0.878712	0.0300513
$856$	0	0
$857$	−35.6234	−1.21687	−0.608436	−	0.793603i	$-0.708203\pi$
$857$	−35.6234	−1.21687	−0.608436	+	0.793603i	$0.708203\pi$
$858$	0	0
$859$	−30.4312	−1.03830	−0.519149	−	0.854684i	$-0.673751\pi$
$859$	−30.4312	−1.03830	−0.519149	+	0.854684i	$0.673751\pi$
$860$	0	0
$861$	−98.9195	−3.37117
$862$	0	0
$863$	13.6755	0.465520	0.232760	−	0.972534i	$-0.425224\pi$
$863$	13.6755	0.465520	0.232760	+	0.972534i	$0.425224\pi$
$864$	0	0
$865$	−25.4824	−0.866428
$866$	0	0
$867$	30.5733	1.03832
$868$	0	0
$869$	−53.6378	−1.81954
$870$	0	0
$871$	−53.7056	−1.81974
$872$	0	0
$873$	1.15056	0.0389407
$874$	0	0
$875$	−4.65156	−0.157251
$876$	0	0
$877$	−28.4382	−0.960291	−0.480145	−	0.877189i	$-0.659416\pi$
$877$	−28.4382	−0.960291	−0.480145	+	0.877189i	$0.659416\pi$
$878$	0	0
$879$	1.22091	0.0411804
$880$	0	0
$881$	14.0086	0.471963	0.235982	−	0.971758i	$-0.424170\pi$
$881$	14.0086	0.471963	0.235982	+	0.971758i	$0.424170\pi$
$882$	0	0
$883$	13.5629	0.456429	0.228214	−	0.973611i	$-0.426711\pi$
$883$	13.5629	0.456429	0.228214	+	0.973611i	$0.426711\pi$
$884$	0	0
$885$	−17.0408	−0.572820
$886$	0	0
$887$	−1.73459	−0.0582418	−0.0291209	−	0.999576i	$-0.509271\pi$
$887$	−1.73459	−0.0582418	−0.0291209	+	0.999576i	$0.509271\pi$
$888$	0	0
$889$	33.0039	1.10692
$890$	0	0
$891$	42.6205	1.42784
$892$	0	0
$893$	27.4168	0.917467
$894$	0	0
$895$	7.78168	0.260113
$896$	0	0
$897$	−8.66039	−0.289162
$898$	0	0
$899$	8.49396	0.283289
$900$	0	0
$901$	1.08816	0.0362520
$902$	0	0
$903$	−18.9128	−0.629378
$904$	0	0
$905$	−23.7922	−0.790880
$906$	0	0
$907$	−18.2428	−0.605743	−0.302871	−	0.953031i	$-0.597945\pi$
$907$	−18.2428	−0.605743	−0.302871	+	0.953031i	$0.597945\pi$
$908$	0	0
$909$	4.75802	0.157814
$910$	0	0
$911$	−16.9469	−0.561475	−0.280738	−	0.959785i	$-0.590579\pi$
$911$	−16.9469	−0.561475	−0.280738	+	0.959785i	$0.590579\pi$
$912$	0	0
$913$	35.7358	1.18268
$914$	0	0
$915$	−10.0113	−0.330962
$916$	0	0
$917$	51.2047	1.69093
$918$	0	0
$919$	−28.5266	−0.941007	−0.470503	−	0.882398i	$-0.655928\pi$
$919$	−28.5266	−0.941007	−0.470503	+	0.882398i	$0.655928\pi$
$920$	0	0
$921$	5.83933	0.192413
$922$	0	0
$923$	73.0685	2.40508
$924$	0	0
$925$	11.0610	0.363684
$926$	0	0
$927$	−4.68231	−0.153787
$928$	0	0
$929$	−19.7391	−0.647620	−0.323810	−	0.946122i	$-0.604964\pi$
$929$	−19.7391	−0.647620	−0.323810	+	0.946122i	$0.604964\pi$
$930$	0	0
$931$	−48.4413	−1.58760
$932$	0	0
$933$	−39.4002	−1.28990
$934$	0	0
$935$	1.24962	0.0408670
$936$	0	0
$937$	−46.4289	−1.51677	−0.758383	−	0.651809i	$-0.774010\pi$
$937$	−46.4289	−1.51677	−0.758383	+	0.651809i	$0.774010\pi$
$938$	0	0
$939$	−52.2806	−1.70611
$940$	0	0
$941$	−54.6408	−1.78124	−0.890619	−	0.454751i	$-0.849728\pi$
$941$	−54.6408	−1.78124	−0.890619	+	0.454751i	$0.849728\pi$
$942$	0	0
$943$	−11.7681	−0.383224
$944$	0	0
$945$	−22.9853	−0.747712
$946$	0	0
$947$	23.3385	0.758399	0.379200	−	0.925315i	$-0.376199\pi$
$947$	23.3385	0.758399	0.379200	+	0.925315i	$0.376199\pi$
$948$	0	0
$949$	47.3410	1.53675
$950$	0	0
$951$	27.4705	0.890792
$952$	0	0
$953$	34.6233	1.12156	0.560779	−	0.827966i	$-0.310502\pi$
$953$	34.6233	1.12156	0.560779	+	0.827966i	$0.310502\pi$
$954$	0	0
$955$	5.81402	0.188137
$956$	0	0
$957$	−27.5035	−0.889062
$958$	0	0
$959$	−13.1575	−0.424876
$960$	0	0
$961$	−25.0192	−0.807071
$962$	0	0
$963$	4.52789	0.145909
$964$	0	0
$965$	8.53422	0.274726
$966$	0	0
$967$	22.2831	0.716576	0.358288	−	0.933611i	$-0.383361\pi$
$967$	22.2831	0.716576	0.358288	+	0.933611i	$0.383361\pi$
$968$	0	0
$969$	1.70544	0.0547866
$970$	0	0
$971$	40.8981	1.31248	0.656242	−	0.754551i	$-0.272145\pi$
$971$	40.8981	1.31248	0.656242	+	0.754551i	$0.272145\pi$
$972$	0	0
$973$	14.6308	0.469042
$974$	0	0
$975$	8.66039	0.277355
$976$	0	0
$977$	9.11906	0.291744	0.145872	−	0.989303i	$-0.453401\pi$
$977$	9.11906	0.291744	0.145872	+	0.989303i	$0.453401\pi$
$978$	0	0
$979$	−8.69052	−0.277750
$980$	0	0
$981$	3.00489	0.0959389
$982$	0	0
$983$	1.55459	0.0495838	0.0247919	−	0.999693i	$-0.492108\pi$
$983$	1.55459	0.0495838	0.0247919	+	0.999693i	$0.492108\pi$
$984$	0	0
$985$	21.9792	0.700316
$986$	0	0
$987$	69.6347	2.21650
$988$	0	0
$989$	−2.24999	−0.0715456
$990$	0	0
$991$	−19.6635	−0.624631	−0.312316	−	0.949978i	$-0.601105\pi$
$991$	−19.6635	−0.624631	−0.312316	+	0.949978i	$0.601105\pi$
$992$	0	0
$993$	−9.28036	−0.294503
$994$	0	0
$995$	−26.6774	−0.845731
$996$	0	0
$997$	21.0904	0.667941	0.333970	−	0.942584i	$-0.391611\pi$
$997$	21.0904	0.667941	0.333970	+	0.942584i	$0.391611\pi$
$998$	0	0
$999$	54.6571	1.72927

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3680.2.a.v.1.2	✓	5
4.3	odd	2		3680.2.a.z.1.4	yes	5
8.3	odd	2		7360.2.a.cn.1.2		5
8.5	even	2		7360.2.a.cr.1.4		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3680.2.a.v.1.2	✓	5	1.1	even	1	trivial
3680.2.a.z.1.4	yes	5	4.3	odd	2
7360.2.a.cn.1.2		5	8.3	odd	2
7360.2.a.cr.1.4		5	8.5	even	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:	\( N \)	\(=\)	\( 3680 = 2^{5} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3680.a (trivial)

\(f(q)\)	\(=\)	\(q-1.80707 q^{3} -1.00000 q^{5} +4.65156 q^{7} +0.265511 q^{9} +O(q^{10})\)	\(q-1.80707 q^{3} -1.00000 q^{5} +4.65156 q^{7} +0.265511 q^{9} -4.38210 q^{11} -4.79250 q^{13} +1.80707 q^{15} +0.285165 q^{17} -3.30952 q^{19} -8.40570 q^{21} -1.00000 q^{23} +1.00000 q^{25} +4.94142 q^{27} -3.47320 q^{29} -2.44557 q^{31} +7.91877 q^{33} -4.65156 q^{35} +11.0610 q^{37} +8.66039 q^{39} +11.7681 q^{41} +2.24999 q^{43} -0.265511 q^{45} -8.28422 q^{47} +14.6370 q^{49} -0.515313 q^{51} +3.81591 q^{53} +4.38210 q^{55} +5.98054 q^{57} -9.43005 q^{59} -5.54005 q^{61} +1.23504 q^{63} +4.79250 q^{65} +11.2062 q^{67} +1.80707 q^{69} -15.2464 q^{71} -9.87814 q^{73} -1.80707 q^{75} -20.3836 q^{77} +12.2402 q^{79} -9.72604 q^{81} -8.15495 q^{83} -0.285165 q^{85} +6.27633 q^{87} +1.98319 q^{89} -22.2926 q^{91} +4.41932 q^{93} +3.30952 q^{95} +4.33340 q^{97} -1.16349 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - q^{3} - 5 q^{5} + q^{7} + 6 q^{9}+O(q^{10}) \) 5 * q - q^3 - 5 * q^5 + q^7 + 6 * q^9	\( 5 q - q^{3} - 5 q^{5} + q^{7} + 6 q^{9} - 9 q^{11} + 3 q^{13} + q^{15} - 21 q^{17} - 7 q^{19} + 12 q^{21} - 5 q^{23} + 5 q^{25} + 20 q^{27} - 6 q^{29} - q^{31} + 17 q^{33} - q^{35} + 8 q^{37} + 4 q^{39} + 19 q^{41} - 14 q^{43} - 6 q^{45} - 8 q^{47} + 32 q^{49} - 25 q^{51} - 6 q^{53} + 9 q^{55} + 14 q^{57} - 6 q^{59} + 23 q^{61} + 48 q^{63} - 3 q^{65} + 2 q^{67} + q^{69} + 21 q^{71} - q^{75} - 26 q^{77} + 42 q^{79} + 25 q^{81} - 28 q^{83} + 21 q^{85} + 26 q^{87} - 19 q^{91} + 46 q^{93} + 7 q^{95} - 17 q^{97} - 47 q^{99}+O(q^{100}) \) 5 * q - q^3 - 5 * q^5 + q^7 + 6 * q^9 - 9 * q^11 + 3 * q^13 + q^15 - 21 * q^17 - 7 * q^19 + 12 * q^21 - 5 * q^23 + 5 * q^25 + 20 * q^27 - 6 * q^29 - q^31 + 17 * q^33 - q^35 + 8 * q^37 + 4 * q^39 + 19 * q^41 - 14 * q^43 - 6 * q^45 - 8 * q^47 + 32 * q^49 - 25 * q^51 - 6 * q^53 + 9 * q^55 + 14 * q^57 - 6 * q^59 + 23 * q^61 + 48 * q^63 - 3 * q^65 + 2 * q^67 + q^69 + 21 * q^71 - q^75 - 26 * q^77 + 42 * q^79 + 25 * q^81 - 28 * q^83 + 21 * q^85 + 26 * q^87 - 19 * q^91 + 46 * q^93 + 7 * q^95 - 17 * q^97 - 47 * q^99

Introduction

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