LMFDB - Embedded newform 3680.2.a.x.1.4

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:	5.5.876604.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 9x^{3} + 8x^{2} + 18x - 16 $ x^5 - x^4 - 9x^3 + 8x^2 + 18*x - 16
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			1.4
Root			$1.63675$ 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)

$f(q)$	$=$	$q+0.706809 q^{3} +1.00000 q^{5} +5.21261 q^{7} -2.50042 q^{9} +O(q^{10})$	$q+0.706809 q^{3} +1.00000 q^{5} +5.21261 q^{7} -2.50042 q^{9} +4.73273 q^{11} -0.250999 q^{13} +0.706809 q^{15} -2.11467 q^{17} +5.84398 q^{19} +3.68431 q^{21} +1.00000 q^{23} +1.00000 q^{25} -3.88775 q^{27} -6.19011 q^{29} -3.60888 q^{31} +3.34513 q^{33} +5.21261 q^{35} +10.8474 q^{37} -0.177408 q^{39} +1.02054 q^{41} +11.6670 q^{43} -2.50042 q^{45} +0.134372 q^{47} +20.1713 q^{49} -1.49467 q^{51} -9.78309 q^{53} +4.73273 q^{55} +4.13057 q^{57} -10.3695 q^{59} +1.05036 q^{61} -13.0337 q^{63} -0.250999 q^{65} -11.2602 q^{67} +0.706809 q^{69} +10.4582 q^{71} -6.98568 q^{73} +0.706809 q^{75} +24.6698 q^{77} -6.07590 q^{79} +4.75337 q^{81} -17.6987 q^{83} -2.11467 q^{85} -4.37522 q^{87} +0.300413 q^{89} -1.30836 q^{91} -2.55079 q^{93} +5.84398 q^{95} +0.307515 q^{97} -11.8338 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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} - 3 q^{11} + 7 q^{13} - q^{15} + 9 q^{17} - q^{19} + 20 q^{21} + 5 q^{23} + 5 q^{25} - 4 q^{27} - 10 q^{29} - 21 q^{31} + 7 q^{33} - q^{35} + 8 q^{37} + 24 q^{39} - 13 q^{41} + 6 q^{43} + 6 q^{45} + 24 q^{49} + 17 q^{51} - 6 q^{53} - 3 q^{55} + 26 q^{57} - 18 q^{59} - 11 q^{61} - 4 q^{63} + 7 q^{65} - 38 q^{67} - q^{69} + 21 q^{71} - 12 q^{73} - q^{75} + 46 q^{77} + 18 q^{79} + 9 q^{81} - 20 q^{83} + 9 q^{85} + 6 q^{87} - 16 q^{89} + 3 q^{91} + 22 q^{93} - q^{95} + 29 q^{97} - 29 q^{99}+O(q^{100}) $ 5 * q - q^3 + 5 * q^5 - q^7 + 6 * q^9 - 3 * q^11 + 7 * q^13 - q^15 + 9 * q^17 - q^19 + 20 * q^21 + 5 * q^23 + 5 * q^25 - 4 * q^27 - 10 * q^29 - 21 * q^31 + 7 * q^33 - q^35 + 8 * q^37 + 24 * q^39 - 13 * q^41 + 6 * q^43 + 6 * q^45 + 24 * q^49 + 17 * q^51 - 6 * q^53 - 3 * q^55 + 26 * q^57 - 18 * q^59 - 11 * q^61 - 4 * q^63 + 7 * q^65 - 38 * q^67 - q^69 + 21 * q^71 - 12 * q^73 - q^75 + 46 * q^77 + 18 * q^79 + 9 * q^81 - 20 * q^83 + 9 * q^85 + 6 * q^87 - 16 * q^89 + 3 * q^91 + 22 * q^93 - q^95 + 29 * q^97 - 29 * 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
$3$	0.706809	0.408076	0.204038	−	0.978963i	$-0.434593\pi$
$3$	0.706809	0.408076	0.204038	+	0.978963i	$0.434593\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	5.21261	1.97018	0.985090	−	0.172041i	$-0.0550361\pi$
$7$	5.21261	1.97018	0.985090	+	0.172041i	$0.0550361\pi$
$8$	0	0
$9$	−2.50042	−0.833474
$10$	0	0
$11$	4.73273	1.42697	0.713485	−	0.700670i	$-0.247116\pi$
$11$	4.73273	1.42697	0.713485	+	0.700670i	$0.247116\pi$
$12$	0	0
$13$	−0.250999	−0.0696145	−0.0348073	−	0.999394i	$-0.511082\pi$
$13$	−0.250999	−0.0696145	−0.0348073	+	0.999394i	$0.511082\pi$
$14$	0	0
$15$	0.706809	0.182497
$16$	0	0
$17$	−2.11467	−0.512884	−0.256442	−	0.966560i	$-0.582550\pi$
$17$	−2.11467	−0.512884	−0.256442	+	0.966560i	$0.582550\pi$
$18$	0	0
$19$	5.84398	1.34070	0.670350	−	0.742045i	$-0.266144\pi$
$19$	5.84398	1.34070	0.670350	+	0.742045i	$0.266144\pi$
$20$	0	0
$21$	3.68431	0.803983
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−3.88775	−0.748197
$28$	0	0
$29$	−6.19011	−1.14947	−0.574737	−	0.818338i	$-0.694896\pi$
$29$	−6.19011	−1.14947	−0.574737	+	0.818338i	$0.694896\pi$
$30$	0	0
$31$	−3.60888	−0.648173	−0.324087	−	0.946027i	$-0.605057\pi$
$31$	−3.60888	−0.648173	−0.324087	+	0.946027i	$0.605057\pi$
$32$	0	0
$33$	3.34513	0.582313
$34$	0	0
$35$	5.21261	0.881091
$36$	0	0
$37$	10.8474	1.78330	0.891651	−	0.452724i	$-0.149548\pi$
$37$	10.8474	1.78330	0.891651	+	0.452724i	$0.149548\pi$
$38$	0	0
$39$	−0.177408	−0.0284080
$40$	0	0
$41$	1.02054	0.159382	0.0796909	−	0.996820i	$-0.474607\pi$
$41$	1.02054	0.159382	0.0796909	+	0.996820i	$0.474607\pi$
$42$	0	0
$43$	11.6670	1.77921	0.889603	−	0.456734i	$-0.150981\pi$
$43$	11.6670	1.77921	0.889603	+	0.456734i	$0.150981\pi$
$44$	0	0
$45$	−2.50042	−0.372741
$46$	0	0
$47$	0.134372	0.0196002	0.00980011	−	0.999952i	$-0.496880\pi$
$47$	0.134372	0.0196002	0.00980011	+	0.999952i	$0.496880\pi$
$48$	0	0
$49$	20.1713	2.88161
$50$	0	0
$51$	−1.49467	−0.209296
$52$	0	0
$53$	−9.78309	−1.34381	−0.671905	−	0.740637i	$-0.734524\pi$
$53$	−9.78309	−1.34381	−0.671905	+	0.740637i	$0.734524\pi$
$54$	0	0
$55$	4.73273	0.638161
$56$	0	0
$57$	4.13057	0.547108
$58$	0	0
$59$	−10.3695	−1.34999	−0.674995	−	0.737822i	$-0.735854\pi$
$59$	−10.3695	−1.34999	−0.674995	+	0.737822i	$0.735854\pi$
$60$	0	0
$61$	1.05036	0.134485	0.0672426	−	0.997737i	$-0.478580\pi$
$61$	1.05036	0.134485	0.0672426	+	0.997737i	$0.478580\pi$
$62$	0	0
$63$	−13.0337	−1.64209
$64$	0	0
$65$	−0.250999	−0.0311326
$66$	0	0
$67$	−11.2602	−1.37565	−0.687825	−	0.725877i	$-0.741434\pi$
$67$	−11.2602	−1.37565	−0.687825	+	0.725877i	$0.741434\pi$
$68$	0	0
$69$	0.706809	0.0850898
$70$	0	0
$71$	10.4582	1.24116	0.620582	−	0.784142i	$-0.286897\pi$
$71$	10.4582	1.24116	0.620582	+	0.784142i	$0.286897\pi$
$72$	0	0
$73$	−6.98568	−0.817612	−0.408806	−	0.912621i	$-0.634055\pi$
$73$	−6.98568	−0.817612	−0.408806	+	0.912621i	$0.634055\pi$
$74$	0	0
$75$	0.706809	0.0816152
$76$	0	0
$77$	24.6698	2.81139
$78$	0	0
$79$	−6.07590	−0.683593	−0.341796	−	0.939774i	$-0.611035\pi$
$79$	−6.07590	−0.683593	−0.341796	+	0.939774i	$0.611035\pi$
$80$	0	0
$81$	4.75337	0.528153
$82$	0	0
$83$	−17.6987	−1.94269	−0.971343	−	0.237684i	$-0.923612\pi$
$83$	−17.6987	−1.94269	−0.971343	+	0.237684i	$0.923612\pi$
$84$	0	0
$85$	−2.11467	−0.229369
$86$	0	0
$87$	−4.37522	−0.469073
$88$	0	0
$89$	0.300413	0.0318438	0.0159219	−	0.999873i	$-0.494932\pi$
$89$	0.300413	0.0318438	0.0159219	+	0.999873i	$0.494932\pi$
$90$	0	0
$91$	−1.30836	−0.137153
$92$	0	0
$93$	−2.55079	−0.264504
$94$	0	0
$95$	5.84398	0.599579
$96$	0	0
$97$	0.307515	0.0312234	0.0156117	−	0.999878i	$-0.495030\pi$
$97$	0.307515	0.0312234	0.0156117	+	0.999878i	$0.495030\pi$
$98$	0	0
$99$	−11.8338	−1.18934
$100$	0	0
$101$	2.12937	0.211881	0.105940	−	0.994372i	$-0.466215\pi$
$101$	2.12937	0.211881	0.105940	+	0.994372i	$0.466215\pi$
$102$	0	0
$103$	12.4734	1.22904	0.614520	−	0.788901i	$-0.289350\pi$
$103$	12.4734	1.22904	0.614520	+	0.788901i	$0.289350\pi$
$104$	0	0
$105$	3.68431	0.359552
$106$	0	0
$107$	3.33381	0.322291	0.161146	−	0.986931i	$-0.448481\pi$
$107$	3.33381	0.322291	0.161146	+	0.986931i	$0.448481\pi$
$108$	0	0
$109$	−8.67046	−0.830479	−0.415240	−	0.909712i	$-0.636302\pi$
$109$	−8.67046	−0.830479	−0.415240	+	0.909712i	$0.636302\pi$
$110$	0	0
$111$	7.66704	0.727723
$112$	0	0
$113$	2.49258	0.234483	0.117241	−	0.993103i	$-0.462595\pi$
$113$	2.49258	0.234483	0.117241	+	0.993103i	$0.462595\pi$
$114$	0	0
$115$	1.00000	0.0932505
$116$	0	0
$117$	0.627603	0.0580219
$118$	0	0
$119$	−11.0230	−1.01047
$120$	0	0
$121$	11.3987	1.03624
$122$	0	0
$123$	0.721328	0.0650399
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	2.54715	0.226023	0.113011	−	0.993594i	$-0.463950\pi$
$127$	2.54715	0.226023	0.113011	+	0.993594i	$0.463950\pi$
$128$	0	0
$129$	8.24636	0.726052
$130$	0	0
$131$	10.3878	0.907585	0.453793	−	0.891107i	$-0.350071\pi$
$131$	10.3878	0.907585	0.453793	+	0.891107i	$0.350071\pi$
$132$	0	0
$133$	30.4623	2.64142
$134$	0	0
$135$	−3.88775	−0.334604
$136$	0	0
$137$	−2.14830	−0.183541	−0.0917706	−	0.995780i	$-0.529253\pi$
$137$	−2.14830	−0.183541	−0.0917706	+	0.995780i	$0.529253\pi$
$138$	0	0
$139$	−3.69753	−0.313620	−0.156810	−	0.987629i	$-0.550121\pi$
$139$	−3.69753	−0.313620	−0.156810	+	0.987629i	$0.550121\pi$
$140$	0	0
$141$	0.0949755	0.00799838
$142$	0	0
$143$	−1.18791	−0.0993379
$144$	0	0
$145$	−6.19011	−0.514061
$146$	0	0
$147$	14.2572	1.17592
$148$	0	0
$149$	−8.33265	−0.682638	−0.341319	−	0.939948i	$-0.610874\pi$
$149$	−8.33265	−0.682638	−0.341319	+	0.939948i	$0.610874\pi$
$150$	0	0
$151$	15.1453	1.23251	0.616255	−	0.787547i	$-0.288649\pi$
$151$	15.1453	1.23251	0.616255	+	0.787547i	$0.288649\pi$
$152$	0	0
$153$	5.28758	0.427475
$154$	0	0
$155$	−3.60888	−0.289872
$156$	0	0
$157$	−1.97518	−0.157636	−0.0788181	−	0.996889i	$-0.525115\pi$
$157$	−1.97518	−0.157636	−0.0788181	+	0.996889i	$0.525115\pi$
$158$	0	0
$159$	−6.91477	−0.548377
$160$	0	0
$161$	5.21261	0.410811
$162$	0	0
$163$	−15.7337	−1.23236	−0.616178	−	0.787607i	$-0.711320\pi$
$163$	−15.7337	−1.23236	−0.616178	+	0.787607i	$0.711320\pi$
$164$	0	0
$165$	3.34513	0.260418
$166$	0	0
$167$	−20.2115	−1.56401	−0.782005	−	0.623273i	$-0.785803\pi$
$167$	−20.2115	−1.56401	−0.782005	+	0.623273i	$0.785803\pi$
$168$	0	0
$169$	−12.9370	−0.995154
$170$	0	0
$171$	−14.6124	−1.11744
$172$	0	0
$173$	−0.579282	−0.0440420	−0.0220210	−	0.999758i	$-0.507010\pi$
$173$	−0.579282	−0.0440420	−0.0220210	+	0.999758i	$0.507010\pi$
$174$	0	0
$175$	5.21261	0.394036
$176$	0	0
$177$	−7.32923	−0.550899
$178$	0	0
$179$	6.22178	0.465038	0.232519	−	0.972592i	$-0.425303\pi$
$179$	6.22178	0.465038	0.232519	+	0.972592i	$0.425303\pi$
$180$	0	0
$181$	−11.1175	−0.826355	−0.413178	−	0.910651i	$-0.635581\pi$
$181$	−11.1175	−0.826355	−0.413178	+	0.910651i	$0.635581\pi$
$182$	0	0
$183$	0.742406	0.0548802
$184$	0	0
$185$	10.8474	0.797517
$186$	0	0
$187$	−10.0082	−0.731870
$188$	0	0
$189$	−20.2653	−1.47408
$190$	0	0
$191$	24.2812	1.75692	0.878462	−	0.477812i	$-0.158570\pi$
$191$	24.2812	1.75692	0.878462	+	0.477812i	$0.158570\pi$
$192$	0	0
$193$	−18.1931	−1.30957	−0.654785	−	0.755815i	$-0.727241\pi$
$193$	−18.1931	−1.30957	−0.654785	+	0.755815i	$0.727241\pi$
$194$	0	0
$195$	−0.177408	−0.0127045
$196$	0	0
$197$	27.7099	1.97425	0.987125	−	0.159950i	$-0.0511334\pi$
$197$	27.7099	1.97425	0.987125	+	0.159950i	$0.0511334\pi$
$198$	0	0
$199$	15.9434	1.13020	0.565098	−	0.825024i	$-0.308838\pi$
$199$	15.9434	1.13020	0.565098	+	0.825024i	$0.308838\pi$
$200$	0	0
$201$	−7.95879	−0.561370
$202$	0	0
$203$	−32.2666	−2.26467
$204$	0	0
$205$	1.02054	0.0712777
$206$	0	0
$207$	−2.50042	−0.173791
$208$	0	0
$209$	27.6579	1.91314
$210$	0	0
$211$	−6.00559	−0.413442	−0.206721	−	0.978400i	$-0.566279\pi$
$211$	−6.00559	−0.413442	−0.206721	+	0.978400i	$0.566279\pi$
$212$	0	0
$213$	7.39197	0.506489
$214$	0	0
$215$	11.6670	0.795685
$216$	0	0
$217$	−18.8117	−1.27702
$218$	0	0
$219$	−4.93754	−0.333648
$220$	0	0
$221$	0.530781	0.0357042
$222$	0	0
$223$	−2.90571	−0.194581	−0.0972903	−	0.995256i	$-0.531018\pi$
$223$	−2.90571	−0.194581	−0.0972903	+	0.995256i	$0.531018\pi$
$224$	0	0
$225$	−2.50042	−0.166695
$226$	0	0
$227$	−10.8063	−0.717240	−0.358620	−	0.933484i	$-0.616753\pi$
$227$	−10.8063	−0.717240	−0.358620	+	0.933484i	$0.616753\pi$
$228$	0	0
$229$	23.6384	1.56207	0.781034	−	0.624488i	$-0.214692\pi$
$229$	23.6384	1.56207	0.781034	+	0.624488i	$0.214692\pi$
$230$	0	0
$231$	17.4368	1.14726
$232$	0	0
$233$	13.4845	0.883400	0.441700	−	0.897163i	$-0.354375\pi$
$233$	13.4845	0.883400	0.441700	+	0.897163i	$0.354375\pi$
$234$	0	0
$235$	0.134372	0.00876548
$236$	0	0
$237$	−4.29450	−0.278958
$238$	0	0
$239$	21.6295	1.39910	0.699548	−	0.714586i	$-0.253385\pi$
$239$	21.6295	1.39910	0.699548	+	0.714586i	$0.253385\pi$
$240$	0	0
$241$	−15.0901	−0.972037	−0.486018	−	0.873949i	$-0.661551\pi$
$241$	−15.0901	−0.972037	−0.486018	+	0.873949i	$0.661551\pi$
$242$	0	0
$243$	15.0230	0.963723
$244$	0	0
$245$	20.1713	1.28869
$246$	0	0
$247$	−1.46683	−0.0933322
$248$	0	0
$249$	−12.5096	−0.792763
$250$	0	0
$251$	−2.91589	−0.184049	−0.0920245	−	0.995757i	$-0.529334\pi$
$251$	−2.91589	−0.184049	−0.0920245	+	0.995757i	$0.529334\pi$
$252$	0	0
$253$	4.73273	0.297544
$254$	0	0
$255$	−1.49467	−0.0935998
$256$	0	0
$257$	15.5613	0.970688	0.485344	−	0.874323i	$-0.338694\pi$
$257$	15.5613	0.970688	0.485344	+	0.874323i	$0.338694\pi$
$258$	0	0
$259$	56.5432	3.51342
$260$	0	0
$261$	15.4779	0.958057
$262$	0	0
$263$	−0.342206	−0.0211013	−0.0105507	−	0.999944i	$-0.503358\pi$
$263$	−0.342206	−0.0211013	−0.0105507	+	0.999944i	$0.503358\pi$
$264$	0	0
$265$	−9.78309	−0.600970
$266$	0	0
$267$	0.212335	0.0129947
$268$	0	0
$269$	−14.5603	−0.887759	−0.443880	−	0.896086i	$-0.646398\pi$
$269$	−14.5603	−0.887759	−0.443880	+	0.896086i	$0.646398\pi$
$270$	0	0
$271$	1.17019	0.0710837	0.0355419	−	0.999368i	$-0.488684\pi$
$271$	1.17019	0.0710837	0.0355419	+	0.999368i	$0.488684\pi$
$272$	0	0
$273$	−0.924758	−0.0559689
$274$	0	0
$275$	4.73273	0.285394
$276$	0	0
$277$	24.3954	1.46578	0.732888	−	0.680349i	$-0.238172\pi$
$277$	24.3954	1.46578	0.732888	+	0.680349i	$0.238172\pi$
$278$	0	0
$279$	9.02371	0.540236
$280$	0	0
$281$	2.83106	0.168887	0.0844434	−	0.996428i	$-0.473089\pi$
$281$	2.83106	0.168887	0.0844434	+	0.996428i	$0.473089\pi$
$282$	0	0
$283$	−6.81964	−0.405385	−0.202693	−	0.979242i	$-0.564969\pi$
$283$	−6.81964	−0.405385	−0.202693	+	0.979242i	$0.564969\pi$
$284$	0	0
$285$	4.13057	0.244674
$286$	0	0
$287$	5.31968	0.314011
$288$	0	0
$289$	−12.5282	−0.736950
$290$	0	0
$291$	0.217354	0.0127415
$292$	0	0
$293$	−29.0172	−1.69520	−0.847601	−	0.530634i	$-0.821954\pi$
$293$	−29.0172	−1.69520	−0.847601	+	0.530634i	$0.821954\pi$
$294$	0	0
$295$	−10.3695	−0.603734
$296$	0	0
$297$	−18.3996	−1.06765
$298$	0	0
$299$	−0.250999	−0.0145156
$300$	0	0
$301$	60.8157	3.50536
$302$	0	0
$303$	1.50506	0.0864634
$304$	0	0
$305$	1.05036	0.0601437
$306$	0	0
$307$	−9.24320	−0.527538	−0.263769	−	0.964586i	$-0.584966\pi$
$307$	−9.24320	−0.527538	−0.263769	+	0.964586i	$0.584966\pi$
$308$	0	0
$309$	8.81630	0.501542
$310$	0	0
$311$	−14.2166	−0.806151	−0.403075	−	0.915167i	$-0.632059\pi$
$311$	−14.2166	−0.806151	−0.403075	+	0.915167i	$0.632059\pi$
$312$	0	0
$313$	3.45918	0.195524	0.0977622	−	0.995210i	$-0.468832\pi$
$313$	3.45918	0.195524	0.0977622	+	0.995210i	$0.468832\pi$
$314$	0	0
$315$	−13.0337	−0.734366
$316$	0	0
$317$	15.9776	0.897393	0.448696	−	0.893684i	$-0.351888\pi$
$317$	15.9776	0.897393	0.448696	+	0.893684i	$0.351888\pi$
$318$	0	0
$319$	−29.2961	−1.64027
$320$	0	0
$321$	2.35636	0.131519
$322$	0	0
$323$	−12.3581	−0.687623
$324$	0	0
$325$	−0.250999	−0.0139229
$326$	0	0
$327$	−6.12835	−0.338899
$328$	0	0
$329$	0.700430	0.0386159
$330$	0	0
$331$	−8.63455	−0.474598	−0.237299	−	0.971437i	$-0.576262\pi$
$331$	−8.63455	−0.474598	−0.237299	+	0.971437i	$0.576262\pi$
$332$	0	0
$333$	−27.1231	−1.48634
$334$	0	0
$335$	−11.2602	−0.615209
$336$	0	0
$337$	31.9718	1.74161	0.870807	−	0.491624i	$-0.163597\pi$
$337$	31.9718	1.74161	0.870807	+	0.491624i	$0.163597\pi$
$338$	0	0
$339$	1.76178	0.0956867
$340$	0	0
$341$	−17.0798	−0.924924
$342$	0	0
$343$	68.6566	3.70711
$344$	0	0
$345$	0.706809	0.0380533
$346$	0	0
$347$	12.3122	0.660955	0.330477	−	0.943814i	$-0.392790\pi$
$347$	12.3122	0.660955	0.330477	+	0.943814i	$0.392790\pi$
$348$	0	0
$349$	−10.5094	−0.562557	−0.281279	−	0.959626i	$-0.590759\pi$
$349$	−10.5094	−0.562557	−0.281279	+	0.959626i	$0.590759\pi$
$350$	0	0
$351$	0.975819	0.0520854
$352$	0	0
$353$	−11.1250	−0.592127	−0.296063	−	0.955168i	$-0.595674\pi$
$353$	−11.1250	−0.592127	−0.296063	+	0.955168i	$0.595674\pi$
$354$	0	0
$355$	10.4582	0.555065
$356$	0	0
$357$	−7.79112	−0.412350
$358$	0	0
$359$	24.2718	1.28101	0.640507	−	0.767952i	$-0.278724\pi$
$359$	24.2718	1.28101	0.640507	+	0.767952i	$0.278724\pi$
$360$	0	0
$361$	15.1521	0.797477
$362$	0	0
$363$	8.05669	0.422867
$364$	0	0
$365$	−6.98568	−0.365647
$366$	0	0
$367$	−8.78708	−0.458682	−0.229341	−	0.973346i	$-0.573657\pi$
$367$	−8.78708	−0.458682	−0.229341	+	0.973346i	$0.573657\pi$
$368$	0	0
$369$	−2.55178	−0.132841
$370$	0	0
$371$	−50.9954	−2.64755
$372$	0	0
$373$	6.53233	0.338231	0.169116	−	0.985596i	$-0.445909\pi$
$373$	6.53233	0.338231	0.169116	+	0.985596i	$0.445909\pi$
$374$	0	0
$375$	0.706809	0.0364994
$376$	0	0
$377$	1.55371	0.0800202
$378$	0	0
$379$	−19.4920	−1.00124	−0.500620	−	0.865667i	$-0.666894\pi$
$379$	−19.4920	−1.00124	−0.500620	+	0.865667i	$0.666894\pi$
$380$	0	0
$381$	1.80034	0.0922345
$382$	0	0
$383$	−26.0604	−1.33163	−0.665813	−	0.746118i	$-0.731915\pi$
$383$	−26.0604	−1.33163	−0.665813	+	0.746118i	$0.731915\pi$
$384$	0	0
$385$	24.6698	1.25729
$386$	0	0
$387$	−29.1725	−1.48292
$388$	0	0
$389$	−14.4470	−0.732490	−0.366245	−	0.930518i	$-0.619357\pi$
$389$	−14.4470	−0.732490	−0.366245	+	0.930518i	$0.619357\pi$
$390$	0	0
$391$	−2.11467	−0.106944
$392$	0	0
$393$	7.34218	0.370364
$394$	0	0
$395$	−6.07590	−0.305712
$396$	0	0
$397$	20.9996	1.05394	0.526970	−	0.849884i	$-0.323328\pi$
$397$	20.9996	1.05394	0.526970	+	0.849884i	$0.323328\pi$
$398$	0	0
$399$	21.5310	1.07790
$400$	0	0
$401$	−28.2831	−1.41239	−0.706195	−	0.708018i	$-0.749590\pi$
$401$	−28.2831	−1.41239	−0.706195	+	0.708018i	$0.749590\pi$
$402$	0	0
$403$	0.905824	0.0451223
$404$	0	0
$405$	4.75337	0.236197
$406$	0	0
$407$	51.3378	2.54472
$408$	0	0
$409$	23.0734	1.14091	0.570454	−	0.821329i	$-0.306767\pi$
$409$	23.0734	1.14091	0.570454	+	0.821329i	$0.306767\pi$
$410$	0	0
$411$	−1.51843	−0.0748988
$412$	0	0
$413$	−54.0520	−2.65972
$414$	0	0
$415$	−17.6987	−0.868795
$416$	0	0
$417$	−2.61344	−0.127981
$418$	0	0
$419$	−27.4805	−1.34251	−0.671254	−	0.741227i	$-0.734244\pi$
$419$	−27.4805	−1.34251	−0.671254	+	0.741227i	$0.734244\pi$
$420$	0	0
$421$	−10.9524	−0.533787	−0.266893	−	0.963726i	$-0.585997\pi$
$421$	−10.9524	−0.533787	−0.266893	+	0.963726i	$0.585997\pi$
$422$	0	0
$423$	−0.335987	−0.0163363
$424$	0	0
$425$	−2.11467	−0.102577
$426$	0	0
$427$	5.47513	0.264960
$428$	0	0
$429$	−0.839624	−0.0405374
$430$	0	0
$431$	−13.7335	−0.661521	−0.330760	−	0.943715i	$-0.607305\pi$
$431$	−13.7335	−0.661521	−0.330760	+	0.943715i	$0.607305\pi$
$432$	0	0
$433$	35.5907	1.71038	0.855190	−	0.518315i	$-0.173441\pi$
$433$	35.5907	1.71038	0.855190	+	0.518315i	$0.173441\pi$
$434$	0	0
$435$	−4.37522	−0.209776
$436$	0	0
$437$	5.84398	0.279555
$438$	0	0
$439$	−32.1827	−1.53600	−0.767999	−	0.640451i	$-0.778748\pi$
$439$	−32.1827	−1.53600	−0.767999	+	0.640451i	$0.778748\pi$
$440$	0	0
$441$	−50.4366	−2.40174
$442$	0	0
$443$	−32.5678	−1.54734	−0.773670	−	0.633588i	$-0.781581\pi$
$443$	−32.5678	−1.54734	−0.773670	+	0.633588i	$0.781581\pi$
$444$	0	0
$445$	0.300413	0.0142410
$446$	0	0
$447$	−5.88959	−0.278568
$448$	0	0
$449$	−16.9995	−0.802257	−0.401128	−	0.916022i	$-0.631382\pi$
$449$	−16.9995	−0.802257	−0.401128	+	0.916022i	$0.631382\pi$
$450$	0	0
$451$	4.82994	0.227433
$452$	0	0
$453$	10.7049	0.502958
$454$	0	0
$455$	−1.30836	−0.0613368
$456$	0	0
$457$	16.0768	0.752040	0.376020	−	0.926611i	$-0.377292\pi$
$457$	16.0768	0.752040	0.376020	+	0.926611i	$0.377292\pi$
$458$	0	0
$459$	8.22131	0.383738
$460$	0	0
$461$	14.2977	0.665910	0.332955	−	0.942943i	$-0.391954\pi$
$461$	14.2977	0.665910	0.332955	+	0.942943i	$0.391954\pi$
$462$	0	0
$463$	32.5700	1.51365	0.756827	−	0.653615i	$-0.226748\pi$
$463$	32.5700	1.51365	0.756827	+	0.653615i	$0.226748\pi$
$464$	0	0
$465$	−2.55079	−0.118290
$466$	0	0
$467$	4.31563	0.199704	0.0998518	−	0.995002i	$-0.468163\pi$
$467$	4.31563	0.199704	0.0998518	+	0.995002i	$0.468163\pi$
$468$	0	0
$469$	−58.6948	−2.71028
$470$	0	0
$471$	−1.39607	−0.0643276
$472$	0	0
$473$	55.2169	2.53887
$474$	0	0
$475$	5.84398	0.268140
$476$	0	0
$477$	24.4618	1.12003
$478$	0	0
$479$	19.0248	0.869265	0.434632	−	0.900608i	$-0.356878\pi$
$479$	19.0248	0.869265	0.434632	+	0.900608i	$0.356878\pi$
$480$	0	0
$481$	−2.72268	−0.124144
$482$	0	0
$483$	3.68431	0.167642
$484$	0	0
$485$	0.307515	0.0139635
$486$	0	0
$487$	−4.21675	−0.191079	−0.0955396	−	0.995426i	$-0.530458\pi$
$487$	−4.21675	−0.191079	−0.0955396	+	0.995426i	$0.530458\pi$
$488$	0	0
$489$	−11.1207	−0.502895
$490$	0	0
$491$	−35.6441	−1.60860	−0.804299	−	0.594225i	$-0.797459\pi$
$491$	−35.6441	−1.60860	−0.804299	+	0.594225i	$0.797459\pi$
$492$	0	0
$493$	13.0901	0.589547
$494$	0	0
$495$	−11.8338	−0.531890
$496$	0	0
$497$	54.5146	2.44531
$498$	0	0
$499$	16.5025	0.738754	0.369377	−	0.929280i	$-0.379571\pi$
$499$	16.5025	0.738754	0.369377	+	0.929280i	$0.379571\pi$
$500$	0	0
$501$	−14.2856	−0.638235
$502$	0	0
$503$	22.4530	1.00113	0.500564	−	0.865699i	$-0.333126\pi$
$503$	22.4530	1.00113	0.500564	+	0.865699i	$0.333126\pi$
$504$	0	0
$505$	2.12937	0.0947559
$506$	0	0
$507$	−9.14398	−0.406099
$508$	0	0
$509$	12.8789	0.570848	0.285424	−	0.958401i	$-0.407866\pi$
$509$	12.8789	0.570848	0.285424	+	0.958401i	$0.407866\pi$
$510$	0	0
$511$	−36.4136	−1.61084
$512$	0	0
$513$	−22.7199	−1.00311
$514$	0	0
$515$	12.4734	0.549643
$516$	0	0
$517$	0.635947	0.0279689
$518$	0	0
$519$	−0.409441	−0.0179725
$520$	0	0
$521$	20.6355	0.904058	0.452029	−	0.892003i	$-0.350700\pi$
$521$	20.6355	0.904058	0.452029	+	0.892003i	$0.350700\pi$
$522$	0	0
$523$	16.6823	0.729467	0.364734	−	0.931112i	$-0.381160\pi$
$523$	16.6823	0.729467	0.364734	+	0.931112i	$0.381160\pi$
$524$	0	0
$525$	3.68431	0.160797
$526$	0	0
$527$	7.63160	0.332438
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	25.9281	1.12518
$532$	0	0
$533$	−0.256155	−0.0110953
$534$	0	0
$535$	3.33381	0.144133
$536$	0	0
$537$	4.39761	0.189771
$538$	0	0
$539$	95.4650	4.11197
$540$	0	0
$541$	−21.1313	−0.908506	−0.454253	−	0.890873i	$-0.650094\pi$
$541$	−21.1313	−0.908506	−0.454253	+	0.890873i	$0.650094\pi$
$542$	0	0
$543$	−7.85792	−0.337216
$544$	0	0
$545$	−8.67046	−0.371402
$546$	0	0
$547$	21.7265	0.928959	0.464480	−	0.885584i	$-0.346241\pi$
$547$	21.7265	0.928959	0.464480	+	0.885584i	$0.346241\pi$
$548$	0	0
$549$	−2.62635	−0.112090
$550$	0	0
$551$	−36.1749	−1.54110
$552$	0	0
$553$	−31.6713	−1.34680
$554$	0	0
$555$	7.66704	0.325448
$556$	0	0
$557$	−44.8821	−1.90172	−0.950858	−	0.309628i	$-0.899795\pi$
$557$	−44.8821	−1.90172	−0.950858	+	0.309628i	$0.899795\pi$
$558$	0	0
$559$	−2.92841	−0.123859
$560$	0	0
$561$	−7.07386	−0.298659
$562$	0	0
$563$	−19.5584	−0.824287	−0.412143	−	0.911119i	$-0.635220\pi$
$563$	−19.5584	−0.824287	−0.412143	+	0.911119i	$0.635220\pi$
$564$	0	0
$565$	2.49258	0.104864
$566$	0	0
$567$	24.7775	1.04056
$568$	0	0
$569$	−20.5392	−0.861047	−0.430524	−	0.902579i	$-0.641671\pi$
$569$	−20.5392	−0.861047	−0.430524	+	0.902579i	$0.641671\pi$
$570$	0	0
$571$	−12.8879	−0.539340	−0.269670	−	0.962953i	$-0.586915\pi$
$571$	−12.8879	−0.539340	−0.269670	+	0.962953i	$0.586915\pi$
$572$	0	0
$573$	17.1621	0.716959
$574$	0	0
$575$	1.00000	0.0417029
$576$	0	0
$577$	18.2883	0.761351	0.380675	−	0.924709i	$-0.375692\pi$
$577$	18.2883	0.761351	0.380675	+	0.924709i	$0.375692\pi$
$578$	0	0
$579$	−12.8591	−0.534404
$580$	0	0
$581$	−92.2564	−3.82744
$582$	0	0
$583$	−46.3007	−1.91758
$584$	0	0
$585$	0.627603	0.0259482
$586$	0	0
$587$	−42.1035	−1.73780	−0.868898	−	0.494991i	$-0.835171\pi$
$587$	−42.1035	−1.73780	−0.868898	+	0.494991i	$0.835171\pi$
$588$	0	0
$589$	−21.0902	−0.869006
$590$	0	0
$591$	19.5856	0.805644
$592$	0	0
$593$	34.5902	1.42045	0.710225	−	0.703975i	$-0.248593\pi$
$593$	34.5902	1.42045	0.710225	+	0.703975i	$0.248593\pi$
$594$	0	0
$595$	−11.0230	−0.451897
$596$	0	0
$597$	11.2689	0.461206
$598$	0	0
$599$	−10.4714	−0.427849	−0.213925	−	0.976850i	$-0.568625\pi$
$599$	−10.4714	−0.427849	−0.213925	+	0.976850i	$0.568625\pi$
$600$	0	0
$601$	−41.2450	−1.68242	−0.841210	−	0.540708i	$-0.818156\pi$
$601$	−41.2450	−1.68242	−0.841210	+	0.540708i	$0.818156\pi$
$602$	0	0
$603$	28.1552	1.14657
$604$	0	0
$605$	11.3987	0.463423
$606$	0	0
$607$	−17.5933	−0.714091	−0.357045	−	0.934087i	$-0.616216\pi$
$607$	−17.5933	−0.714091	−0.357045	+	0.934087i	$0.616216\pi$
$608$	0	0
$609$	−22.8063	−0.924159
$610$	0	0
$611$	−0.0337273	−0.00136446
$612$	0	0
$613$	20.8066	0.840370	0.420185	−	0.907439i	$-0.361965\pi$
$613$	20.8066	0.840370	0.420185	+	0.907439i	$0.361965\pi$
$614$	0	0
$615$	0.721328	0.0290867
$616$	0	0
$617$	−35.4765	−1.42823	−0.714115	−	0.700028i	$-0.753171\pi$
$617$	−35.4765	−1.42823	−0.714115	+	0.700028i	$0.753171\pi$
$618$	0	0
$619$	−20.2405	−0.813533	−0.406766	−	0.913532i	$-0.633344\pi$
$619$	−20.2405	−0.813533	−0.406766	+	0.913532i	$0.633344\pi$
$620$	0	0
$621$	−3.88775	−0.156010
$622$	0	0
$623$	1.56594	0.0627379
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	19.5489	0.780707
$628$	0	0
$629$	−22.9387	−0.914626
$630$	0	0
$631$	7.40468	0.294776	0.147388	−	0.989079i	$-0.452913\pi$
$631$	7.40468	0.294776	0.147388	+	0.989079i	$0.452913\pi$
$632$	0	0
$633$	−4.24480	−0.168716
$634$	0	0
$635$	2.54715	0.101080
$636$	0	0
$637$	−5.06296	−0.200602
$638$	0	0
$639$	−26.1500	−1.03448
$640$	0	0
$641$	16.0313	0.633200	0.316600	−	0.948559i	$-0.397459\pi$
$641$	16.0313	0.633200	0.316600	+	0.948559i	$0.397459\pi$
$642$	0	0
$643$	3.77825	0.149000	0.0744998	−	0.997221i	$-0.476264\pi$
$643$	3.77825	0.149000	0.0744998	+	0.997221i	$0.476264\pi$
$644$	0	0
$645$	8.24636	0.324700
$646$	0	0
$647$	43.1818	1.69765	0.848826	−	0.528672i	$-0.177310\pi$
$647$	43.1818	1.69765	0.848826	+	0.528672i	$0.177310\pi$
$648$	0	0
$649$	−49.0759	−1.92640
$650$	0	0
$651$	−13.2962	−0.521121
$652$	0	0
$653$	−29.8519	−1.16819	−0.584097	−	0.811684i	$-0.698551\pi$
$653$	−29.8519	−1.16819	−0.584097	+	0.811684i	$0.698551\pi$
$654$	0	0
$655$	10.3878	0.405885
$656$	0	0
$657$	17.4671	0.681458
$658$	0	0
$659$	15.9212	0.620202	0.310101	−	0.950704i	$-0.399637\pi$
$659$	15.9212	0.620202	0.310101	+	0.950704i	$0.399637\pi$
$660$	0	0
$661$	−27.1121	−1.05454	−0.527269	−	0.849699i	$-0.676784\pi$
$661$	−27.1121	−1.05454	−0.527269	+	0.849699i	$0.676784\pi$
$662$	0	0
$663$	0.375160	0.0145700
$664$	0	0
$665$	30.4623	1.18128
$666$	0	0
$667$	−6.19011	−0.239682
$668$	0	0
$669$	−2.05378	−0.0794037
$670$	0	0
$671$	4.97108	0.191907
$672$	0	0
$673$	−35.6258	−1.37327	−0.686637	−	0.727001i	$-0.740914\pi$
$673$	−35.6258	−1.37327	−0.686637	+	0.727001i	$0.740914\pi$
$674$	0	0
$675$	−3.88775	−0.149639
$676$	0	0
$677$	−48.4246	−1.86111	−0.930554	−	0.366154i	$-0.880674\pi$
$677$	−48.4246	−1.86111	−0.930554	+	0.366154i	$0.880674\pi$
$678$	0	0
$679$	1.60295	0.0615156
$680$	0	0
$681$	−7.63800	−0.292689
$682$	0	0
$683$	−14.8433	−0.567965	−0.283982	−	0.958829i	$-0.591656\pi$
$683$	−14.8433	−0.567965	−0.283982	+	0.958829i	$0.591656\pi$
$684$	0	0
$685$	−2.14830	−0.0820821
$686$	0	0
$687$	16.7078	0.637443
$688$	0	0
$689$	2.45554	0.0935488
$690$	0	0
$691$	−24.3369	−0.925820	−0.462910	−	0.886405i	$-0.653195\pi$
$691$	−24.3369	−0.925820	−0.462910	+	0.886405i	$0.653195\pi$
$692$	0	0
$693$	−61.6850	−2.34322
$694$	0	0
$695$	−3.69753	−0.140255
$696$	0	0
$697$	−2.15811	−0.0817443
$698$	0	0
$699$	9.53098	0.360495
$700$	0	0
$701$	19.0235	0.718509	0.359255	−	0.933240i	$-0.383031\pi$
$701$	19.0235	0.718509	0.359255	+	0.933240i	$0.383031\pi$
$702$	0	0
$703$	63.3919	2.39087
$704$	0	0
$705$	0.0949755	0.00357698
$706$	0	0
$707$	11.0996	0.417443
$708$	0	0
$709$	−10.7681	−0.404405	−0.202202	−	0.979344i	$-0.564810\pi$
$709$	−10.7681	−0.404405	−0.202202	+	0.979344i	$0.564810\pi$
$710$	0	0
$711$	15.1923	0.569757
$712$	0	0
$713$	−3.60888	−0.135154
$714$	0	0
$715$	−1.18791	−0.0444253
$716$	0	0
$717$	15.2879	0.570937
$718$	0	0
$719$	−17.3605	−0.647436	−0.323718	−	0.946154i	$-0.604933\pi$
$719$	−17.3605	−0.647436	−0.323718	+	0.946154i	$0.604933\pi$
$720$	0	0
$721$	65.0189	2.42143
$722$	0	0
$723$	−10.6658	−0.396665
$724$	0	0
$725$	−6.19011	−0.229895
$726$	0	0
$727$	4.28344	0.158864	0.0794321	−	0.996840i	$-0.474689\pi$
$727$	4.28344	0.158864	0.0794321	+	0.996840i	$0.474689\pi$
$728$	0	0
$729$	−3.64176	−0.134880
$730$	0	0
$731$	−24.6720	−0.912526
$732$	0	0
$733$	−29.7449	−1.09865	−0.549327	−	0.835607i	$-0.685116\pi$
$733$	−29.7449	−1.09865	−0.549327	+	0.835607i	$0.685116\pi$
$734$	0	0
$735$	14.2572	0.525885
$736$	0	0
$737$	−53.2913	−1.96301
$738$	0	0
$739$	−19.9696	−0.734592	−0.367296	−	0.930104i	$-0.619716\pi$
$739$	−19.9696	−0.734592	−0.367296	+	0.930104i	$0.619716\pi$
$740$	0	0
$741$	−1.03677	−0.0380867
$742$	0	0
$743$	12.3943	0.454703	0.227351	−	0.973813i	$-0.426993\pi$
$743$	12.3943	0.454703	0.227351	+	0.973813i	$0.426993\pi$
$744$	0	0
$745$	−8.33265	−0.305285
$746$	0	0
$747$	44.2542	1.61918
$748$	0	0
$749$	17.3778	0.634972
$750$	0	0
$751$	−35.0772	−1.27999	−0.639993	−	0.768381i	$-0.721063\pi$
$751$	−35.0772	−1.27999	−0.639993	+	0.768381i	$0.721063\pi$
$752$	0	0
$753$	−2.06097	−0.0751060
$754$	0	0
$755$	15.1453	0.551195
$756$	0	0
$757$	−2.91074	−0.105793	−0.0528963	−	0.998600i	$-0.516845\pi$
$757$	−2.91074	−0.105793	−0.0528963	+	0.998600i	$0.516845\pi$
$758$	0	0
$759$	3.34513	0.121421
$760$	0	0
$761$	20.6494	0.748542	0.374271	−	0.927319i	$-0.377893\pi$
$761$	20.6494	0.748542	0.374271	+	0.927319i	$0.377893\pi$
$762$	0	0
$763$	−45.1957	−1.63619
$764$	0	0
$765$	5.28758	0.191173
$766$	0	0
$767$	2.60272	0.0939790
$768$	0	0
$769$	−31.0441	−1.11948	−0.559739	−	0.828669i	$-0.689099\pi$
$769$	−31.0441	−1.11948	−0.559739	+	0.828669i	$0.689099\pi$
$770$	0	0
$771$	10.9989	0.396115
$772$	0	0
$773$	−1.64944	−0.0593262	−0.0296631	−	0.999560i	$-0.509443\pi$
$773$	−1.64944	−0.0593262	−0.0296631	+	0.999560i	$0.509443\pi$
$774$	0	0
$775$	−3.60888	−0.129635
$776$	0	0
$777$	39.9652	1.43374
$778$	0	0
$779$	5.96402	0.213683
$780$	0	0
$781$	49.4959	1.77110
$782$	0	0
$783$	24.0656	0.860034
$784$	0	0
$785$	−1.97518	−0.0704971
$786$	0	0
$787$	23.0488	0.821600	0.410800	−	0.911725i	$-0.365249\pi$
$787$	23.0488	0.821600	0.410800	+	0.911725i	$0.365249\pi$
$788$	0	0
$789$	−0.241874	−0.00861095
$790$	0	0
$791$	12.9929	0.461973
$792$	0	0
$793$	−0.263640	−0.00936213
$794$	0	0
$795$	−6.91477	−0.245242
$796$	0	0
$797$	7.69319	0.272507	0.136253	−	0.990674i	$-0.456494\pi$
$797$	7.69319	0.272507	0.136253	+	0.990674i	$0.456494\pi$
$798$	0	0
$799$	−0.284154	−0.0100526
$800$	0	0
$801$	−0.751160	−0.0265409
$802$	0	0
$803$	−33.0613	−1.16671
$804$	0	0
$805$	5.21261	0.183720
$806$	0	0
$807$	−10.2914	−0.362273
$808$	0	0
$809$	−8.02914	−0.282290	−0.141145	−	0.989989i	$-0.545078\pi$
$809$	−8.02914	−0.282290	−0.141145	+	0.989989i	$0.545078\pi$
$810$	0	0
$811$	10.5287	0.369712	0.184856	−	0.982766i	$-0.440818\pi$
$811$	10.5287	0.369712	0.184856	+	0.982766i	$0.440818\pi$
$812$	0	0
$813$	0.827098	0.0290076
$814$	0	0
$815$	−15.7337	−0.551127
$816$	0	0
$817$	68.1819	2.38538
$818$	0	0
$819$	3.27145	0.114314
$820$	0	0
$821$	−17.5562	−0.612715	−0.306357	−	0.951917i	$-0.599110\pi$
$821$	−17.5562	−0.612715	−0.306357	+	0.951917i	$0.599110\pi$
$822$	0	0
$823$	−26.7749	−0.933316	−0.466658	−	0.884438i	$-0.654542\pi$
$823$	−26.7749	−0.933316	−0.466658	+	0.884438i	$0.654542\pi$
$824$	0	0
$825$	3.34513	0.116463
$826$	0	0
$827$	11.8652	0.412592	0.206296	−	0.978490i	$-0.433859\pi$
$827$	11.8652	0.412592	0.206296	+	0.978490i	$0.433859\pi$
$828$	0	0
$829$	49.0687	1.70423	0.852113	−	0.523358i	$-0.175321\pi$
$829$	49.0687	1.70423	0.852113	+	0.523358i	$0.175321\pi$
$830$	0	0
$831$	17.2429	0.598149
$832$	0	0
$833$	−42.6556	−1.47793
$834$	0	0
$835$	−20.2115	−0.699446
$836$	0	0
$837$	14.0304	0.484961
$838$	0	0
$839$	18.1610	0.626989	0.313494	−	0.949590i	$-0.398500\pi$
$839$	18.1610	0.626989	0.313494	+	0.949590i	$0.398500\pi$
$840$	0	0
$841$	9.31748	0.321292
$842$	0	0
$843$	2.00102	0.0689186
$844$	0	0
$845$	−12.9370	−0.445046
$846$	0	0
$847$	59.4169	2.04159
$848$	0	0
$849$	−4.82018	−0.165428
$850$	0	0
$851$	10.8474	0.371844
$852$	0	0
$853$	−35.3535	−1.21048	−0.605241	−	0.796042i	$-0.706923\pi$
$853$	−35.3535	−1.21048	−0.605241	+	0.796042i	$0.706923\pi$
$854$	0	0
$855$	−14.6124	−0.499734
$856$	0	0
$857$	−26.2249	−0.895827	−0.447913	−	0.894077i	$-0.647833\pi$
$857$	−26.2249	−0.895827	−0.447913	+	0.894077i	$0.647833\pi$
$858$	0	0
$859$	−34.7616	−1.18605	−0.593026	−	0.805184i	$-0.702067\pi$
$859$	−34.7616	−1.18605	−0.593026	+	0.805184i	$0.702067\pi$
$860$	0	0
$861$	3.76000	0.128140
$862$	0	0
$863$	−18.7005	−0.636571	−0.318286	−	0.947995i	$-0.603107\pi$
$863$	−18.7005	−0.636571	−0.318286	+	0.947995i	$0.603107\pi$
$864$	0	0
$865$	−0.579282	−0.0196962
$866$	0	0
$867$	−8.85501	−0.300732
$868$	0	0
$869$	−28.7556	−0.975466
$870$	0	0
$871$	2.82629	0.0957652
$872$	0	0
$873$	−0.768916	−0.0260239
$874$	0	0
$875$	5.21261	0.176218
$876$	0	0
$877$	22.1856	0.749155	0.374577	−	0.927196i	$-0.377788\pi$
$877$	22.1856	0.749155	0.374577	+	0.927196i	$0.377788\pi$
$878$	0	0
$879$	−20.5096	−0.691772
$880$	0	0
$881$	−34.8580	−1.17440	−0.587198	−	0.809443i	$-0.699769\pi$
$881$	−34.8580	−1.17440	−0.587198	+	0.809443i	$0.699769\pi$
$882$	0	0
$883$	42.7441	1.43845	0.719227	−	0.694775i	$-0.244496\pi$
$883$	42.7441	1.43845	0.719227	+	0.694775i	$0.244496\pi$
$884$	0	0
$885$	−7.32923	−0.246369
$886$	0	0
$887$	47.9301	1.60934	0.804668	−	0.593725i	$-0.202343\pi$
$887$	47.9301	1.60934	0.804668	+	0.593725i	$0.202343\pi$
$888$	0	0
$889$	13.2773	0.445305
$890$	0	0
$891$	22.4964	0.753658
$892$	0	0
$893$	0.785269	0.0262780
$894$	0	0
$895$	6.22178	0.207971
$896$	0	0
$897$	−0.177408	−0.00592348
$898$	0	0
$899$	22.3394	0.745059
$900$	0	0
$901$	20.6880	0.689219
$902$	0	0
$903$	42.9850	1.43045
$904$	0	0
$905$	−11.1175	−0.369557
$906$	0	0
$907$	4.88310	0.162141	0.0810704	−	0.996708i	$-0.474166\pi$
$907$	4.88310	0.162141	0.0810704	+	0.996708i	$0.474166\pi$
$908$	0	0
$909$	−5.32433	−0.176597
$910$	0	0
$911$	2.83697	0.0939930	0.0469965	−	0.998895i	$-0.485035\pi$
$911$	2.83697	0.0939930	0.0469965	+	0.998895i	$0.485035\pi$
$912$	0	0
$913$	−83.7631	−2.77215
$914$	0	0
$915$	0.742406	0.0245432
$916$	0	0
$917$	54.1475	1.78811
$918$	0	0
$919$	−13.2430	−0.436847	−0.218424	−	0.975854i	$-0.570091\pi$
$919$	−13.2430	−0.436847	−0.218424	+	0.975854i	$0.570091\pi$
$920$	0	0
$921$	−6.53318	−0.215275
$922$	0	0
$923$	−2.62500	−0.0864030
$924$	0	0
$925$	10.8474	0.356660
$926$	0	0
$927$	−31.1887	−1.02437
$928$	0	0
$929$	−57.7862	−1.89590	−0.947952	−	0.318413i	$-0.896850\pi$
$929$	−57.7862	−1.89590	−0.947952	+	0.318413i	$0.896850\pi$
$930$	0	0
$931$	117.880	3.86337
$932$	0	0
$933$	−10.0484	−0.328971
$934$	0	0
$935$	−10.0082	−0.327302
$936$	0	0
$937$	−26.8345	−0.876646	−0.438323	−	0.898818i	$-0.644427\pi$
$937$	−26.8345	−0.876646	−0.438323	+	0.898818i	$0.644427\pi$
$938$	0	0
$939$	2.44498	0.0797888
$940$	0	0
$941$	−13.8892	−0.452776	−0.226388	−	0.974037i	$-0.572692\pi$
$941$	−13.8892	−0.452776	−0.226388	+	0.974037i	$0.572692\pi$
$942$	0	0
$943$	1.02054	0.0332334
$944$	0	0
$945$	−20.2653	−0.659230
$946$	0	0
$947$	15.4571	0.502289	0.251144	−	0.967950i	$-0.419193\pi$
$947$	15.4571	0.502289	0.251144	+	0.967950i	$0.419193\pi$
$948$	0	0
$949$	1.75340	0.0569177
$950$	0	0
$951$	11.2931	0.366205
$952$	0	0
$953$	46.9863	1.52203	0.761017	−	0.648731i	$-0.224700\pi$
$953$	46.9863	1.52203	0.761017	+	0.648731i	$0.224700\pi$
$954$	0	0
$955$	24.2812	0.785721
$956$	0	0
$957$	−20.7067	−0.669354
$958$	0	0
$959$	−11.1982	−0.361609
$960$	0	0
$961$	−17.9760	−0.579871
$962$	0	0
$963$	−8.33593	−0.268621
$964$	0	0
$965$	−18.1931	−0.585658
$966$	0	0
$967$	−11.0807	−0.356331	−0.178165	−	0.984001i	$-0.557016\pi$
$967$	−11.0807	−0.356331	−0.178165	+	0.984001i	$0.557016\pi$
$968$	0	0
$969$	−8.73481	−0.280603
$970$	0	0
$971$	−28.3401	−0.909478	−0.454739	−	0.890625i	$-0.650267\pi$
$971$	−28.3401	−0.909478	−0.454739	+	0.890625i	$0.650267\pi$
$972$	0	0
$973$	−19.2738	−0.617888
$974$	0	0
$975$	−0.177408	−0.00568161
$976$	0	0
$977$	−11.9491	−0.382286	−0.191143	−	0.981562i	$-0.561219\pi$
$977$	−11.9491	−0.382286	−0.191143	+	0.981562i	$0.561219\pi$
$978$	0	0
$979$	1.42177	0.0454401
$980$	0	0
$981$	21.6798	0.692183
$982$	0	0
$983$	−51.9566	−1.65716	−0.828579	−	0.559873i	$-0.810850\pi$
$983$	−51.9566	−1.65716	−0.828579	+	0.559873i	$0.810850\pi$
$984$	0	0
$985$	27.7099	0.882912
$986$	0	0
$987$	0.495070	0.0157582
$988$	0	0
$989$	11.6670	0.370990
$990$	0	0
$991$	−40.7230	−1.29361	−0.646804	−	0.762656i	$-0.723895\pi$
$991$	−40.7230	−1.29361	−0.646804	+	0.762656i	$0.723895\pi$
$992$	0	0
$993$	−6.10298	−0.193672
$994$	0	0
$995$	15.9434	0.505439
$996$	0	0
$997$	−14.4242	−0.456820	−0.228410	−	0.973565i	$-0.573353\pi$
$997$	−14.4242	−0.456820	−0.228410	+	0.973565i	$0.573353\pi$
$998$	0	0
$999$	−42.1719	−1.33426

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3680.2.a.x.1.4	✓	5
4.3	odd	2		3680.2.a.ba.1.2	yes	5
8.3	odd	2		7360.2.a.cl.1.4		5
8.5	even	2		7360.2.a.cq.1.2		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3680.2.a.x.1.4	✓	5	1.1	even	1	trivial
3680.2.a.ba.1.2	yes	5	4.3	odd	2
7360.2.a.cl.1.4		5	8.3	odd	2
7360.2.a.cq.1.2		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)

\(f(q)\)	\(=\)	\(q+0.706809 q^{3} +1.00000 q^{5} +5.21261 q^{7} -2.50042 q^{9} +O(q^{10})\)	\(q+0.706809 q^{3} +1.00000 q^{5} +5.21261 q^{7} -2.50042 q^{9} +4.73273 q^{11} -0.250999 q^{13} +0.706809 q^{15} -2.11467 q^{17} +5.84398 q^{19} +3.68431 q^{21} +1.00000 q^{23} +1.00000 q^{25} -3.88775 q^{27} -6.19011 q^{29} -3.60888 q^{31} +3.34513 q^{33} +5.21261 q^{35} +10.8474 q^{37} -0.177408 q^{39} +1.02054 q^{41} +11.6670 q^{43} -2.50042 q^{45} +0.134372 q^{47} +20.1713 q^{49} -1.49467 q^{51} -9.78309 q^{53} +4.73273 q^{55} +4.13057 q^{57} -10.3695 q^{59} +1.05036 q^{61} -13.0337 q^{63} -0.250999 q^{65} -11.2602 q^{67} +0.706809 q^{69} +10.4582 q^{71} -6.98568 q^{73} +0.706809 q^{75} +24.6698 q^{77} -6.07590 q^{79} +4.75337 q^{81} -17.6987 q^{83} -2.11467 q^{85} -4.37522 q^{87} +0.300413 q^{89} -1.30836 q^{91} -2.55079 q^{93} +5.84398 q^{95} +0.307515 q^{97} -11.8338 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - q^{3} + 5 q^{5} - q^{7} + 6 q^{9}+O(q^{10}) \) 5 * q - q^3 + 5 * q^5 - q^7 + 6 * q^9	\( 5 q - q^{3} + 5 q^{5} - q^{7} + 6 q^{9} - 3 q^{11} + 7 q^{13} - q^{15} + 9 q^{17} - q^{19} + 20 q^{21} + 5 q^{23} + 5 q^{25} - 4 q^{27} - 10 q^{29} - 21 q^{31} + 7 q^{33} - q^{35} + 8 q^{37} + 24 q^{39} - 13 q^{41} + 6 q^{43} + 6 q^{45} + 24 q^{49} + 17 q^{51} - 6 q^{53} - 3 q^{55} + 26 q^{57} - 18 q^{59} - 11 q^{61} - 4 q^{63} + 7 q^{65} - 38 q^{67} - q^{69} + 21 q^{71} - 12 q^{73} - q^{75} + 46 q^{77} + 18 q^{79} + 9 q^{81} - 20 q^{83} + 9 q^{85} + 6 q^{87} - 16 q^{89} + 3 q^{91} + 22 q^{93} - q^{95} + 29 q^{97} - 29 q^{99}+O(q^{100}) \) 5 * q - q^3 + 5 * q^5 - q^7 + 6 * q^9 - 3 * q^11 + 7 * q^13 - q^15 + 9 * q^17 - q^19 + 20 * q^21 + 5 * q^23 + 5 * q^25 - 4 * q^27 - 10 * q^29 - 21 * q^31 + 7 * q^33 - q^35 + 8 * q^37 + 24 * q^39 - 13 * q^41 + 6 * q^43 + 6 * q^45 + 24 * q^49 + 17 * q^51 - 6 * q^53 - 3 * q^55 + 26 * q^57 - 18 * q^59 - 11 * q^61 - 4 * q^63 + 7 * q^65 - 38 * q^67 - q^69 + 21 * q^71 - 12 * q^73 - q^75 + 46 * q^77 + 18 * q^79 + 9 * q^81 - 20 * q^83 + 9 * q^85 + 6 * q^87 - 16 * q^89 + 3 * q^91 + 22 * q^93 - q^95 + 29 * q^97 - 29 * q^99

Introduction

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