LMFDB - Embedded newform 3640.2.a.x.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3640,2,Mod(1,3640)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3640, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("3640.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3640 = 2^{3} \cdot 5 \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3640.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:	$29.0655463357$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.4913.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 6x^{2} + x + 1 $ x^4 - x^3 - 6*x^2 + x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.90570$ of defining polynomial
Character	$\chi$	$=$	3640.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.90570 q^{3} -1.00000 q^{5} -1.00000 q^{7} +0.631706 q^{9} +O(q^{10})$	$q-1.90570 q^{3} -1.00000 q^{5} -1.00000 q^{7} +0.631706 q^{9} +6.09896 q^{11} +1.00000 q^{13} +1.90570 q^{15} +7.44311 q^{17} +0.655849 q^{19} +1.90570 q^{21} -4.09896 q^{23} +1.00000 q^{25} +4.51327 q^{27} -8.00467 q^{29} -1.21740 q^{31} -11.6228 q^{33} +1.00000 q^{35} -1.75845 q^{37} -1.90570 q^{39} -8.81972 q^{41} +2.28756 q^{43} -0.631706 q^{45} +5.12311 q^{47} +1.00000 q^{49} -14.1844 q^{51} -1.12311 q^{53} -6.09896 q^{55} -1.24985 q^{57} +1.51919 q^{59} -8.24621 q^{61} -0.631706 q^{63} -1.00000 q^{65} +15.2545 q^{67} +7.81141 q^{69} -15.0818 q^{71} -4.49971 q^{73} -1.90570 q^{75} -6.09896 q^{77} +2.41658 q^{79} -10.4961 q^{81} +3.74650 q^{83} -7.44311 q^{85} +15.2545 q^{87} +16.1179 q^{89} -1.00000 q^{91} +2.32001 q^{93} -0.655849 q^{95} +13.1397 q^{97} +3.85275 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 3 q^{3} - 4 q^{5} - 4 q^{7} + 3 q^{9}+O(q^{10}) $ 4 * q + 3 * q^3 - 4 * q^5 - 4 * q^7 + 3 * q^9	$ 4 q + 3 q^{3} - 4 q^{5} - 4 q^{7} + 3 q^{9} + 6 q^{11} + 4 q^{13} - 3 q^{15} + 9 q^{17} + 5 q^{19} - 3 q^{21} + 2 q^{23} + 4 q^{25} + 6 q^{27} - 3 q^{29} + q^{31} - 4 q^{33} + 4 q^{35} - 11 q^{37} + 3 q^{39} - 5 q^{41} + 12 q^{43} - 3 q^{45} + 4 q^{47} + 4 q^{49} - 6 q^{51} + 12 q^{53} - 6 q^{55} + 8 q^{57} + 11 q^{59} - 3 q^{63} - 4 q^{65} + 19 q^{67} + 10 q^{69} - 8 q^{71} + 8 q^{73} + 3 q^{75} - 6 q^{77} + 13 q^{79} + 4 q^{81} + 8 q^{83} - 9 q^{85} + 19 q^{87} + 25 q^{89} - 4 q^{91} + 5 q^{93} - 5 q^{95} + 18 q^{97} + 30 q^{99}+O(q^{100}) $ 4 * q + 3 * q^3 - 4 * q^5 - 4 * q^7 + 3 * q^9 + 6 * q^11 + 4 * q^13 - 3 * q^15 + 9 * q^17 + 5 * q^19 - 3 * q^21 + 2 * q^23 + 4 * q^25 + 6 * q^27 - 3 * q^29 + q^31 - 4 * q^33 + 4 * q^35 - 11 * q^37 + 3 * q^39 - 5 * q^41 + 12 * q^43 - 3 * q^45 + 4 * q^47 + 4 * q^49 - 6 * q^51 + 12 * q^53 - 6 * q^55 + 8 * q^57 + 11 * q^59 - 3 * q^63 - 4 * q^65 + 19 * q^67 + 10 * q^69 - 8 * q^71 + 8 * q^73 + 3 * q^75 - 6 * q^77 + 13 * q^79 + 4 * q^81 + 8 * q^83 - 9 * q^85 + 19 * q^87 + 25 * q^89 - 4 * q^91 + 5 * q^93 - 5 * q^95 + 18 * q^97 + 30 * 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
$3$	−1.90570	−1.10026	−0.550129	−	0.835080i	$-0.685422\pi$
$3$	−1.90570	−1.10026	−0.550129	+	0.835080i	$0.685422\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	0.631706	0.210569
$10$	0	0
$11$	6.09896	1.83891	0.919453	−	0.393200i	$-0.128632\pi$
$11$	6.09896	1.83891	0.919453	+	0.393200i	$0.128632\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	1.90570	0.492051
$16$	0	0
$17$	7.44311	1.80522	0.902610	−	0.430459i	$-0.141648\pi$
$17$	7.44311	1.80522	0.902610	+	0.430459i	$0.141648\pi$
$18$	0	0
$19$	0.655849	0.150462	0.0752311	−	0.997166i	$-0.476031\pi$
$19$	0.655849	0.150462	0.0752311	+	0.997166i	$0.476031\pi$
$20$	0	0
$21$	1.90570	0.415859
$22$	0	0
$23$	−4.09896	−0.854693	−0.427346	−	0.904088i	$-0.640552\pi$
$23$	−4.09896	−0.854693	−0.427346	+	0.904088i	$0.640552\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	4.51327	0.868578
$28$	0	0
$29$	−8.00467	−1.48643	−0.743215	−	0.669053i	$-0.766700\pi$
$29$	−8.00467	−1.48643	−0.743215	+	0.669053i	$0.766700\pi$
$30$	0	0
$31$	−1.21740	−0.218652	−0.109326	−	0.994006i	$-0.534869\pi$
$31$	−1.21740	−0.218652	−0.109326	+	0.994006i	$0.534869\pi$
$32$	0	0
$33$	−11.6228	−2.02327
$34$	0	0
$35$	1.00000	0.169031
$36$	0	0
$37$	−1.75845	−0.289088	−0.144544	−	0.989498i	$-0.546172\pi$
$37$	−1.75845	−0.289088	−0.144544	+	0.989498i	$0.546172\pi$
$38$	0	0
$39$	−1.90570	−0.305157
$40$	0	0
$41$	−8.81972	−1.37741	−0.688704	−	0.725043i	$-0.741820\pi$
$41$	−8.81972	−1.37741	−0.688704	+	0.725043i	$0.741820\pi$
$42$	0	0
$43$	2.28756	0.348849	0.174424	−	0.984671i	$-0.444194\pi$
$43$	2.28756	0.348849	0.174424	+	0.984671i	$0.444194\pi$
$44$	0	0
$45$	−0.631706	−0.0941692
$46$	0	0
$47$	5.12311	0.747282	0.373641	−	0.927573i	$-0.378109\pi$
$47$	5.12311	0.747282	0.373641	+	0.927573i	$0.378109\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−14.1844	−1.98621
$52$	0	0
$53$	−1.12311	−0.154270	−0.0771352	−	0.997021i	$-0.524577\pi$
$53$	−1.12311	−0.154270	−0.0771352	+	0.997021i	$0.524577\pi$
$54$	0	0
$55$	−6.09896	−0.822384
$56$	0	0
$57$	−1.24985	−0.165547
$58$	0	0
$59$	1.51919	0.197781	0.0988906	−	0.995098i	$-0.468471\pi$
$59$	1.51919	0.197781	0.0988906	+	0.995098i	$0.468471\pi$
$60$	0	0
$61$	−8.24621	−1.05582	−0.527910	−	0.849301i	$-0.677024\pi$
$61$	−8.24621	−1.05582	−0.527910	+	0.849301i	$0.677024\pi$
$62$	0	0
$63$	−0.631706	−0.0795875
$64$	0	0
$65$	−1.00000	−0.124035
$66$	0	0
$67$	15.2545	1.86364	0.931818	−	0.362926i	$-0.118222\pi$
$67$	15.2545	1.86364	0.931818	+	0.362926i	$0.118222\pi$
$68$	0	0
$69$	7.81141	0.940383
$70$	0	0
$71$	−15.0818	−1.78988	−0.894938	−	0.446191i	$-0.852780\pi$
$71$	−15.0818	−1.78988	−0.894938	+	0.446191i	$0.852780\pi$
$72$	0	0
$73$	−4.49971	−0.526651	−0.263326	−	0.964707i	$-0.584819\pi$
$73$	−4.49971	−0.526651	−0.263326	+	0.964707i	$0.584819\pi$
$74$	0	0
$75$	−1.90570	−0.220052
$76$	0	0
$77$	−6.09896	−0.695041
$78$	0	0
$79$	2.41658	0.271887	0.135943	−	0.990717i	$-0.456594\pi$
$79$	2.41658	0.271887	0.135943	+	0.990717i	$0.456594\pi$
$80$	0	0
$81$	−10.4961	−1.16623
$82$	0	0
$83$	3.74650	0.411232	0.205616	−	0.978633i	$-0.434080\pi$
$83$	3.74650	0.411232	0.205616	+	0.978633i	$0.434080\pi$
$84$	0	0
$85$	−7.44311	−0.807319
$86$	0	0
$87$	15.2545	1.63546
$88$	0	0
$89$	16.1179	1.70849	0.854245	−	0.519871i	$-0.174020\pi$
$89$	16.1179	1.70849	0.854245	+	0.519871i	$0.174020\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	0	0
$93$	2.32001	0.240574
$94$	0	0
$95$	−0.655849	−0.0672887
$96$	0	0
$97$	13.1397	1.33414	0.667068	−	0.744996i	$-0.267549\pi$
$97$	13.1397	1.33414	0.667068	+	0.744996i	$0.267549\pi$
$98$	0	0
$99$	3.85275	0.387216
$100$	0	0
$101$	9.17139	0.912588	0.456294	−	0.889829i	$-0.349177\pi$
$101$	9.17139	0.912588	0.456294	+	0.889829i	$0.349177\pi$
$102$	0	0
$103$	11.8439	1.16701	0.583505	−	0.812109i	$-0.301681\pi$
$103$	11.8439	1.16701	0.583505	+	0.812109i	$0.301681\pi$
$104$	0	0
$105$	−1.90570	−0.185978
$106$	0	0
$107$	−1.36237	−0.131706	−0.0658528	−	0.997829i	$-0.520977\pi$
$107$	−1.36237	−0.131706	−0.0658528	+	0.997829i	$0.520977\pi$
$108$	0	0
$109$	−11.8763	−1.13754	−0.568772	−	0.822495i	$-0.692581\pi$
$109$	−11.8763	−1.13754	−0.568772	+	0.822495i	$0.692581\pi$
$110$	0	0
$111$	3.35109	0.318072
$112$	0	0
$113$	14.3865	1.35337	0.676685	−	0.736273i	$-0.263416\pi$
$113$	14.3865	1.35337	0.676685	+	0.736273i	$0.263416\pi$
$114$	0	0
$115$	4.09896	0.382230
$116$	0	0
$117$	0.631706	0.0584012
$118$	0	0
$119$	−7.44311	−0.682309
$120$	0	0
$121$	26.1973	2.38158
$122$	0	0
$123$	16.8078	1.51551
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	6.72236	0.596513	0.298256	−	0.954486i	$-0.403595\pi$
$127$	6.72236	0.596513	0.298256	+	0.954486i	$0.403595\pi$
$128$	0	0
$129$	−4.35940	−0.383824
$130$	0	0
$131$	9.12311	0.797089	0.398545	−	0.917149i	$-0.369515\pi$
$131$	9.12311	0.797089	0.398545	+	0.917149i	$0.369515\pi$
$132$	0	0
$133$	−0.655849	−0.0568693
$134$	0	0
$135$	−4.51327	−0.388440
$136$	0	0
$137$	−0.193259	−0.0165112	−0.00825561	−	0.999966i	$-0.502628\pi$
$137$	−0.193259	−0.0165112	−0.00825561	+	0.999966i	$0.502628\pi$
$138$	0	0
$139$	−2.11319	−0.179239	−0.0896193	−	0.995976i	$-0.528565\pi$
$139$	−2.11319	−0.179239	−0.0896193	+	0.995976i	$0.528565\pi$
$140$	0	0
$141$	−9.76312	−0.822203
$142$	0	0
$143$	6.09896	0.510021
$144$	0	0
$145$	8.00467	0.664751
$146$	0	0
$147$	−1.90570	−0.157180
$148$	0	0
$149$	1.12311	0.0920084	0.0460042	−	0.998941i	$-0.485351\pi$
$149$	1.12311	0.0920084	0.0460042	+	0.998941i	$0.485351\pi$
$150$	0	0
$151$	16.4584	1.33936	0.669681	−	0.742649i	$-0.266431\pi$
$151$	16.4584	1.33936	0.669681	+	0.742649i	$0.266431\pi$
$152$	0	0
$153$	4.70186	0.380123
$154$	0	0
$155$	1.21740	0.0977841
$156$	0	0
$157$	−17.0041	−1.35707	−0.678537	−	0.734566i	$-0.737386\pi$
$157$	−17.0041	−1.35707	−0.678537	+	0.734566i	$0.737386\pi$
$158$	0	0
$159$	2.14031	0.169737
$160$	0	0
$161$	4.09896	0.323043
$162$	0	0
$163$	9.68305	0.758435	0.379218	−	0.925308i	$-0.376193\pi$
$163$	9.68305	0.758435	0.379218	+	0.925308i	$0.376193\pi$
$164$	0	0
$165$	11.6228	0.904835
$166$	0	0
$167$	19.3376	1.49639	0.748196	−	0.663478i	$-0.230920\pi$
$167$	19.3376	1.49639	0.748196	+	0.663478i	$0.230920\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	0.414304	0.0316826
$172$	0	0
$173$	−15.2410	−1.15875	−0.579374	−	0.815061i	$-0.696703\pi$
$173$	−15.2410	−1.15875	−0.579374	+	0.815061i	$0.696703\pi$
$174$	0	0
$175$	−1.00000	−0.0755929
$176$	0	0
$177$	−2.89512	−0.217610
$178$	0	0
$179$	−6.81243	−0.509185	−0.254592	−	0.967048i	$-0.581941\pi$
$179$	−6.81243	−0.509185	−0.254592	+	0.967048i	$0.581941\pi$
$180$	0	0
$181$	−0.236879	−0.0176071	−0.00880356	−	0.999961i	$-0.502802\pi$
$181$	−0.236879	−0.0176071	−0.00880356	+	0.999961i	$0.502802\pi$
$182$	0	0
$183$	15.7148	1.16167
$184$	0	0
$185$	1.75845	0.129284
$186$	0	0
$187$	45.3953	3.31963
$188$	0	0
$189$	−4.51327	−0.328292
$190$	0	0
$191$	−6.92985	−0.501426	−0.250713	−	0.968061i	$-0.580665\pi$
$191$	−6.92985	−0.501426	−0.250713	+	0.968061i	$0.580665\pi$
$192$	0	0
$193$	−16.5179	−1.18899	−0.594493	−	0.804100i	$-0.702647\pi$
$193$	−16.5179	−1.18899	−0.594493	+	0.804100i	$0.702647\pi$
$194$	0	0
$195$	1.90570	0.136470
$196$	0	0
$197$	−22.6374	−1.61285	−0.806424	−	0.591338i	$-0.798600\pi$
$197$	−22.6374	−1.61285	−0.806424	+	0.591338i	$0.798600\pi$
$198$	0	0
$199$	17.2846	1.22527	0.612636	−	0.790365i	$-0.290109\pi$
$199$	17.2846	1.22527	0.612636	+	0.790365i	$0.290109\pi$
$200$	0	0
$201$	−29.0706	−2.05048
$202$	0	0
$203$	8.00467	0.561817
$204$	0	0
$205$	8.81972	0.615996
$206$	0	0
$207$	−2.58934	−0.179972
$208$	0	0
$209$	4.00000	0.276686
$210$	0	0
$211$	−18.0077	−1.23970	−0.619851	−	0.784719i	$-0.712807\pi$
$211$	−18.0077	−1.23970	−0.619851	+	0.784719i	$0.712807\pi$
$212$	0	0
$213$	28.7414	1.96933
$214$	0	0
$215$	−2.28756	−0.156010
$216$	0	0
$217$	1.21740	0.0826426
$218$	0	0
$219$	8.57511	0.579452
$220$	0	0
$221$	7.44311	0.500678
$222$	0	0
$223$	14.2145	0.951876	0.475938	−	0.879479i	$-0.342109\pi$
$223$	14.2145	0.951876	0.475938	+	0.879479i	$0.342109\pi$
$224$	0	0
$225$	0.631706	0.0421137
$226$	0	0
$227$	5.21968	0.346442	0.173221	−	0.984883i	$-0.444582\pi$
$227$	5.21968	0.346442	0.173221	+	0.984883i	$0.444582\pi$
$228$	0	0
$229$	5.45244	0.360308	0.180154	−	0.983638i	$-0.442340\pi$
$229$	5.45244	0.360308	0.180154	+	0.983638i	$0.442340\pi$
$230$	0	0
$231$	11.6228	0.764725
$232$	0	0
$233$	−2.25350	−0.147632	−0.0738158	−	0.997272i	$-0.523518\pi$
$233$	−2.25350	−0.147632	−0.0738158	+	0.997272i	$0.523518\pi$
$234$	0	0
$235$	−5.12311	−0.334195
$236$	0	0
$237$	−4.60529	−0.299145
$238$	0	0
$239$	0.476148	0.0307995	0.0153997	−	0.999881i	$-0.495098\pi$
$239$	0.476148	0.0307995	0.0153997	+	0.999881i	$0.495098\pi$
$240$	0	0
$241$	−23.8025	−1.53325	−0.766627	−	0.642092i	$-0.778067\pi$
$241$	−23.8025	−1.53325	−0.766627	+	0.642092i	$0.778067\pi$
$242$	0	0
$243$	6.46259	0.414575
$244$	0	0
$245$	−1.00000	−0.0638877
$246$	0	0
$247$	0.655849	0.0417307
$248$	0	0
$249$	−7.13972	−0.452462
$250$	0	0
$251$	18.1874	1.14798	0.573990	−	0.818862i	$-0.305395\pi$
$251$	18.1874	1.14798	0.573990	+	0.818862i	$0.305395\pi$
$252$	0	0
$253$	−24.9994	−1.57170
$254$	0	0
$255$	14.1844	0.888260
$256$	0	0
$257$	29.6155	1.84737	0.923683	−	0.383158i	$-0.125163\pi$
$257$	29.6155	1.84737	0.923683	+	0.383158i	$0.125163\pi$
$258$	0	0
$259$	1.75845	0.109265
$260$	0	0
$261$	−5.05660	−0.312995
$262$	0	0
$263$	−12.2003	−0.752304	−0.376152	−	0.926558i	$-0.622753\pi$
$263$	−12.2003	−0.752304	−0.376152	+	0.926558i	$0.622753\pi$
$264$	0	0
$265$	1.12311	0.0689918
$266$	0	0
$267$	−30.7159	−1.87978
$268$	0	0
$269$	2.92518	0.178351	0.0891757	−	0.996016i	$-0.471577\pi$
$269$	2.92518	0.178351	0.0891757	+	0.996016i	$0.471577\pi$
$270$	0	0
$271$	19.8714	1.20710	0.603551	−	0.797324i	$-0.293752\pi$
$271$	19.8714	1.20710	0.603551	+	0.797324i	$0.293752\pi$
$272$	0	0
$273$	1.90570	0.115338
$274$	0	0
$275$	6.09896	0.367781
$276$	0	0
$277$	20.6222	1.23907	0.619535	−	0.784969i	$-0.287321\pi$
$277$	20.6222	1.23907	0.619535	+	0.784969i	$0.287321\pi$
$278$	0	0
$279$	−0.769040	−0.0460412
$280$	0	0
$281$	−6.51021	−0.388366	−0.194183	−	0.980965i	$-0.562206\pi$
$281$	−6.51021	−0.388366	−0.194183	+	0.980965i	$0.562206\pi$
$282$	0	0
$283$	−13.6917	−0.813888	−0.406944	−	0.913453i	$-0.633406\pi$
$283$	−13.6917	−0.813888	−0.406944	+	0.913453i	$0.633406\pi$
$284$	0	0
$285$	1.24985	0.0740350
$286$	0	0
$287$	8.81972	0.520611
$288$	0	0
$289$	38.3999	2.25882
$290$	0	0
$291$	−25.0404	−1.46790
$292$	0	0
$293$	31.2338	1.82470	0.912349	−	0.409414i	$-0.134267\pi$
$293$	31.2338	1.82470	0.912349	+	0.409414i	$0.134267\pi$
$294$	0	0
$295$	−1.51919	−0.0884504
$296$	0	0
$297$	27.5262	1.59723
$298$	0	0
$299$	−4.09896	−0.237049
$300$	0	0
$301$	−2.28756	−0.131852
$302$	0	0
$303$	−17.4780	−1.00408
$304$	0	0
$305$	8.24621	0.472177
$306$	0	0
$307$	8.54800	0.487860	0.243930	−	0.969793i	$-0.421563\pi$
$307$	8.54800	0.487860	0.243930	+	0.969793i	$0.421563\pi$
$308$	0	0
$309$	−22.5709	−1.28401
$310$	0	0
$311$	15.8524	0.898908	0.449454	−	0.893304i	$-0.351619\pi$
$311$	15.8524	0.898908	0.449454	+	0.893304i	$0.351619\pi$
$312$	0	0
$313$	−25.2774	−1.42876	−0.714382	−	0.699756i	$-0.753292\pi$
$313$	−25.2774	−1.42876	−0.714382	+	0.699756i	$0.753292\pi$
$314$	0	0
$315$	0.631706	0.0355926
$316$	0	0
$317$	−12.0198	−0.675101	−0.337550	−	0.941307i	$-0.609598\pi$
$317$	−12.0198	−0.675101	−0.337550	+	0.941307i	$0.609598\pi$
$318$	0	0
$319$	−48.8202	−2.73340
$320$	0	0
$321$	2.59628	0.144910
$322$	0	0
$323$	4.88156	0.271617
$324$	0	0
$325$	1.00000	0.0554700
$326$	0	0
$327$	22.6327	1.25159
$328$	0	0
$329$	−5.12311	−0.282446
$330$	0	0
$331$	12.1472	0.667673	0.333837	−	0.942631i	$-0.391657\pi$
$331$	12.1472	0.667673	0.333837	+	0.942631i	$0.391657\pi$
$332$	0	0
$333$	−1.11083	−0.0608729
$334$	0	0
$335$	−15.2545	−0.833443
$336$	0	0
$337$	17.1491	0.934169	0.467084	−	0.884213i	$-0.345304\pi$
$337$	17.1491	0.934169	0.467084	+	0.884213i	$0.345304\pi$
$338$	0	0
$339$	−27.4164	−1.48906
$340$	0	0
$341$	−7.42489	−0.402080
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−7.81141	−0.420552
$346$	0	0
$347$	−26.8449	−1.44111	−0.720554	−	0.693398i	$-0.756113\pi$
$347$	−26.8449	−1.44111	−0.720554	+	0.693398i	$0.756113\pi$
$348$	0	0
$349$	−20.3819	−1.09102	−0.545508	−	0.838106i	$-0.683663\pi$
$349$	−20.3819	−1.09102	−0.545508	+	0.838106i	$0.683663\pi$
$350$	0	0
$351$	4.51327	0.240900
$352$	0	0
$353$	2.37719	0.126525	0.0632624	−	0.997997i	$-0.479849\pi$
$353$	2.37719	0.126525	0.0632624	+	0.997997i	$0.479849\pi$
$354$	0	0
$355$	15.0818	0.800457
$356$	0	0
$357$	14.1844	0.750716
$358$	0	0
$359$	21.6642	1.14339	0.571695	−	0.820466i	$-0.306286\pi$
$359$	21.6642	1.14339	0.571695	+	0.820466i	$0.306286\pi$
$360$	0	0
$361$	−18.5699	−0.977361
$362$	0	0
$363$	−49.9244	−2.62035
$364$	0	0
$365$	4.49971	0.235526
$366$	0	0
$367$	24.1805	1.26221	0.631106	−	0.775697i	$-0.282601\pi$
$367$	24.1805	1.26221	0.631106	+	0.775697i	$0.282601\pi$
$368$	0	0
$369$	−5.57147	−0.290039
$370$	0	0
$371$	1.12311	0.0583087
$372$	0	0
$373$	13.1324	0.679972	0.339986	−	0.940431i	$-0.389578\pi$
$373$	13.1324	0.679972	0.339986	+	0.940431i	$0.389578\pi$
$374$	0	0
$375$	1.90570	0.0984101
$376$	0	0
$377$	−8.00467	−0.412261
$378$	0	0
$379$	1.44962	0.0744618	0.0372309	−	0.999307i	$-0.488146\pi$
$379$	1.44962	0.0744618	0.0372309	+	0.999307i	$0.488146\pi$
$380$	0	0
$381$	−12.8108	−0.656318
$382$	0	0
$383$	31.3105	1.59989	0.799947	−	0.600071i	$-0.204861\pi$
$383$	31.3105	1.59989	0.799947	+	0.600071i	$0.204861\pi$
$384$	0	0
$385$	6.09896	0.310832
$386$	0	0
$387$	1.44506	0.0734566
$388$	0	0
$389$	16.6193	0.842631	0.421316	−	0.906914i	$-0.361568\pi$
$389$	16.6193	0.842631	0.421316	+	0.906914i	$0.361568\pi$
$390$	0	0
$391$	−30.5090	−1.54291
$392$	0	0
$393$	−17.3859	−0.877004
$394$	0	0
$395$	−2.41658	−0.121591
$396$	0	0
$397$	34.1801	1.71545	0.857726	−	0.514107i	$-0.171877\pi$
$397$	34.1801	1.71545	0.857726	+	0.514107i	$0.171877\pi$
$398$	0	0
$399$	1.24985	0.0625710
$400$	0	0
$401$	22.6587	1.13152	0.565760	−	0.824570i	$-0.308583\pi$
$401$	22.6587	1.13152	0.565760	+	0.824570i	$0.308583\pi$
$402$	0	0
$403$	−1.21740	−0.0606431
$404$	0	0
$405$	10.4961	0.521554
$406$	0	0
$407$	−10.7247	−0.531606
$408$	0	0
$409$	17.3495	0.857877	0.428938	−	0.903334i	$-0.358888\pi$
$409$	17.3495	0.857877	0.428938	+	0.903334i	$0.358888\pi$
$410$	0	0
$411$	0.368294	0.0181666
$412$	0	0
$413$	−1.51919	−0.0747543
$414$	0	0
$415$	−3.74650	−0.183909
$416$	0	0
$417$	4.02712	0.197209
$418$	0	0
$419$	14.0166	0.684757	0.342378	−	0.939562i	$-0.388768\pi$
$419$	14.0166	0.684757	0.342378	+	0.939562i	$0.388768\pi$
$420$	0	0
$421$	21.2383	1.03509	0.517547	−	0.855655i	$-0.326845\pi$
$421$	21.2383	1.03509	0.517547	+	0.855655i	$0.326845\pi$
$422$	0	0
$423$	3.23630	0.157354
$424$	0	0
$425$	7.44311	0.361044
$426$	0	0
$427$	8.24621	0.399062
$428$	0	0
$429$	−11.6228	−0.561155
$430$	0	0
$431$	28.3062	1.36346	0.681731	−	0.731603i	$-0.261227\pi$
$431$	28.3062	1.36346	0.681731	+	0.731603i	$0.261227\pi$
$432$	0	0
$433$	−10.6100	−0.509882	−0.254941	−	0.966957i	$-0.582056\pi$
$433$	−10.6100	−0.509882	−0.254941	+	0.966957i	$0.582056\pi$
$434$	0	0
$435$	−15.2545	−0.731398
$436$	0	0
$437$	−2.68830	−0.128599
$438$	0	0
$439$	14.8213	0.707383	0.353692	−	0.935362i	$-0.384926\pi$
$439$	14.8213	0.707383	0.353692	+	0.935362i	$0.384926\pi$
$440$	0	0
$441$	0.631706	0.0300812
$442$	0	0
$443$	25.0911	1.19211	0.596057	−	0.802942i	$-0.296733\pi$
$443$	25.0911	1.19211	0.596057	+	0.802942i	$0.296733\pi$
$444$	0	0
$445$	−16.1179	−0.764060
$446$	0	0
$447$	−2.14031	−0.101233
$448$	0	0
$449$	−9.51691	−0.449131	−0.224565	−	0.974459i	$-0.572096\pi$
$449$	−9.51691	−0.449131	−0.224565	+	0.974459i	$0.572096\pi$
$450$	0	0
$451$	−53.7911	−2.53292
$452$	0	0
$453$	−31.3648	−1.47365
$454$	0	0
$455$	1.00000	0.0468807
$456$	0	0
$457$	−33.5885	−1.57120	−0.785602	−	0.618732i	$-0.787647\pi$
$457$	−33.5885	−1.57120	−0.785602	+	0.618732i	$0.787647\pi$
$458$	0	0
$459$	33.5928	1.56798
$460$	0	0
$461$	−0.768369	−0.0357865	−0.0178933	−	0.999840i	$-0.505696\pi$
$461$	−0.768369	−0.0357865	−0.0178933	+	0.999840i	$0.505696\pi$
$462$	0	0
$463$	−11.3194	−0.526058	−0.263029	−	0.964788i	$-0.584722\pi$
$463$	−11.3194	−0.526058	−0.263029	+	0.964788i	$0.584722\pi$
$464$	0	0
$465$	−2.32001	−0.107588
$466$	0	0
$467$	4.27866	0.197993	0.0989965	−	0.995088i	$-0.468437\pi$
$467$	4.27866	0.197993	0.0989965	+	0.995088i	$0.468437\pi$
$468$	0	0
$469$	−15.2545	−0.704388
$470$	0	0
$471$	32.4047	1.49313
$472$	0	0
$473$	13.9517	0.641500
$474$	0	0
$475$	0.655849	0.0300924
$476$	0	0
$477$	−0.709473	−0.0324845
$478$	0	0
$479$	−24.3665	−1.11333	−0.556666	−	0.830736i	$-0.687920\pi$
$479$	−24.3665	−1.11333	−0.556666	+	0.830736i	$0.687920\pi$
$480$	0	0
$481$	−1.75845	−0.0801786
$482$	0	0
$483$	−7.81141	−0.355431
$484$	0	0
$485$	−13.1397	−0.596644
$486$	0	0
$487$	−3.11844	−0.141310	−0.0706550	−	0.997501i	$-0.522509\pi$
$487$	−3.11844	−0.141310	−0.0706550	+	0.997501i	$0.522509\pi$
$488$	0	0
$489$	−18.4530	−0.834475
$490$	0	0
$491$	−13.8690	−0.625900	−0.312950	−	0.949770i	$-0.601317\pi$
$491$	−13.8690	−0.625900	−0.312950	+	0.949770i	$0.601317\pi$
$492$	0	0
$493$	−59.5796	−2.68333
$494$	0	0
$495$	−3.85275	−0.173168
$496$	0	0
$497$	15.0818	0.676509
$498$	0	0
$499$	−24.7695	−1.10883	−0.554417	−	0.832239i	$-0.687059\pi$
$499$	−24.7695	−1.10883	−0.554417	+	0.832239i	$0.687059\pi$
$500$	0	0
$501$	−36.8518	−1.64642
$502$	0	0
$503$	−11.6735	−0.520495	−0.260248	−	0.965542i	$-0.583804\pi$
$503$	−11.6735	−0.520495	−0.260248	+	0.965542i	$0.583804\pi$
$504$	0	0
$505$	−9.17139	−0.408122
$506$	0	0
$507$	−1.90570	−0.0846353
$508$	0	0
$509$	18.0560	0.800319	0.400159	−	0.916446i	$-0.368955\pi$
$509$	18.0560	0.800319	0.400159	+	0.916446i	$0.368955\pi$
$510$	0	0
$511$	4.49971	0.199055
$512$	0	0
$513$	2.96002	0.130688
$514$	0	0
$515$	−11.8439	−0.521903
$516$	0	0
$517$	31.2456	1.37418
$518$	0	0
$519$	29.0448	1.27492
$520$	0	0
$521$	33.8763	1.48415	0.742074	−	0.670318i	$-0.233842\pi$
$521$	33.8763	1.48415	0.742074	+	0.670318i	$0.233842\pi$
$522$	0	0
$523$	−0.550384	−0.0240666	−0.0120333	−	0.999928i	$-0.503830\pi$
$523$	−0.550384	−0.0240666	−0.0120333	+	0.999928i	$0.503830\pi$
$524$	0	0
$525$	1.90570	0.0831717
$526$	0	0
$527$	−9.06126	−0.394715
$528$	0	0
$529$	−6.19851	−0.269500
$530$	0	0
$531$	0.959679	0.0416465
$532$	0	0
$533$	−8.81972	−0.382024
$534$	0	0
$535$	1.36237	0.0589006
$536$	0	0
$537$	12.9825	0.560235
$538$	0	0
$539$	6.09896	0.262701
$540$	0	0
$541$	19.1584	0.823683	0.411842	−	0.911255i	$-0.364886\pi$
$541$	19.1584	0.823683	0.411842	+	0.911255i	$0.364886\pi$
$542$	0	0
$543$	0.451422	0.0193724
$544$	0	0
$545$	11.8763	0.508725
$546$	0	0
$547$	−14.3525	−0.613667	−0.306833	−	0.951763i	$-0.599269\pi$
$547$	−14.3525	−0.613667	−0.306833	+	0.951763i	$0.599269\pi$
$548$	0	0
$549$	−5.20918	−0.222322
$550$	0	0
$551$	−5.24985	−0.223651
$552$	0	0
$553$	−2.41658	−0.102763
$554$	0	0
$555$	−3.35109	−0.142246
$556$	0	0
$557$	25.2261	1.06886	0.534431	−	0.845212i	$-0.320526\pi$
$557$	25.2261	1.06886	0.534431	+	0.845212i	$0.320526\pi$
$558$	0	0
$559$	2.28756	0.0967533
$560$	0	0
$561$	−86.5099	−3.65245
$562$	0	0
$563$	39.1697	1.65080	0.825402	−	0.564545i	$-0.190948\pi$
$563$	39.1697	1.65080	0.825402	+	0.564545i	$0.190948\pi$
$564$	0	0
$565$	−14.3865	−0.605245
$566$	0	0
$567$	10.4961	0.440793
$568$	0	0
$569$	−1.56986	−0.0658120	−0.0329060	−	0.999458i	$-0.510476\pi$
$569$	−1.56986	−0.0658120	−0.0329060	+	0.999458i	$0.510476\pi$
$570$	0	0
$571$	−3.21501	−0.134544	−0.0672720	−	0.997735i	$-0.521430\pi$
$571$	−3.21501	−0.134544	−0.0672720	+	0.997735i	$0.521430\pi$
$572$	0	0
$573$	13.2062	0.551698
$574$	0	0
$575$	−4.09896	−0.170939
$576$	0	0
$577$	9.84308	0.409773	0.204886	−	0.978786i	$-0.434318\pi$
$577$	9.84308	0.409773	0.204886	+	0.978786i	$0.434318\pi$
$578$	0	0
$579$	31.4783	1.30819
$580$	0	0
$581$	−3.74650	−0.155431
$582$	0	0
$583$	−6.84978	−0.283689
$584$	0	0
$585$	−0.631706	−0.0261178
$586$	0	0
$587$	−43.4119	−1.79180	−0.895900	−	0.444256i	$-0.853468\pi$
$587$	−43.4119	−1.79180	−0.895900	+	0.444256i	$0.853468\pi$
$588$	0	0
$589$	−0.798432	−0.0328988
$590$	0	0
$591$	43.1402	1.77455
$592$	0	0
$593$	9.97288	0.409537	0.204769	−	0.978810i	$-0.434356\pi$
$593$	9.97288	0.409537	0.204769	+	0.978810i	$0.434356\pi$
$594$	0	0
$595$	7.44311	0.305138
$596$	0	0
$597$	−32.9393	−1.34812
$598$	0	0
$599$	25.0960	1.02539	0.512697	−	0.858570i	$-0.328646\pi$
$599$	25.0960	1.02539	0.512697	+	0.858570i	$0.328646\pi$
$600$	0	0
$601$	−31.3859	−1.28026	−0.640129	−	0.768267i	$-0.721119\pi$
$601$	−31.3859	−1.28026	−0.640129	+	0.768267i	$0.721119\pi$
$602$	0	0
$603$	9.63637	0.392423
$604$	0	0
$605$	−26.1973	−1.06507
$606$	0	0
$607$	−29.7192	−1.20626	−0.603132	−	0.797642i	$-0.706081\pi$
$607$	−29.7192	−1.20626	−0.603132	+	0.797642i	$0.706081\pi$
$608$	0	0
$609$	−15.2545	−0.618144
$610$	0	0
$611$	5.12311	0.207259
$612$	0	0
$613$	−39.7725	−1.60639	−0.803197	−	0.595713i	$-0.796869\pi$
$613$	−39.7725	−1.60639	−0.803197	+	0.595713i	$0.796869\pi$
$614$	0	0
$615$	−16.8078	−0.677754
$616$	0	0
$617$	47.2566	1.90248	0.951239	−	0.308455i	$-0.0998120\pi$
$617$	47.2566	1.90248	0.951239	+	0.308455i	$0.0998120\pi$
$618$	0	0
$619$	−43.7253	−1.75747	−0.878734	−	0.477312i	$-0.841611\pi$
$619$	−43.7253	−1.75747	−0.878734	+	0.477312i	$0.841611\pi$
$620$	0	0
$621$	−18.4997	−0.742368
$622$	0	0
$623$	−16.1179	−0.645748
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−7.62281	−0.304426
$628$	0	0
$629$	−13.0884	−0.521868
$630$	0	0
$631$	15.7457	0.626826	0.313413	−	0.949617i	$-0.398528\pi$
$631$	15.7457	0.626826	0.313413	+	0.949617i	$0.398528\pi$
$632$	0	0
$633$	34.3174	1.36399
$634$	0	0
$635$	−6.72236	−0.266769
$636$	0	0
$637$	1.00000	0.0396214
$638$	0	0
$639$	−9.52724	−0.376892
$640$	0	0
$641$	42.6467	1.68444	0.842222	−	0.539131i	$-0.181247\pi$
$641$	42.6467	1.68444	0.842222	+	0.539131i	$0.181247\pi$
$642$	0	0
$643$	21.7996	0.859691	0.429846	−	0.902902i	$-0.358568\pi$
$643$	21.7996	0.859691	0.429846	+	0.902902i	$0.358568\pi$
$644$	0	0
$645$	4.35940	0.171651
$646$	0	0
$647$	−6.02823	−0.236994	−0.118497	−	0.992954i	$-0.537808\pi$
$647$	−6.02823	−0.236994	−0.118497	+	0.992954i	$0.537808\pi$
$648$	0	0
$649$	9.26546	0.363701
$650$	0	0
$651$	−2.32001	−0.0909283
$652$	0	0
$653$	23.1397	0.905527	0.452764	−	0.891631i	$-0.350438\pi$
$653$	23.1397	0.905527	0.452764	+	0.891631i	$0.350438\pi$
$654$	0	0
$655$	−9.12311	−0.356469
$656$	0	0
$657$	−2.84249	−0.110896
$658$	0	0
$659$	4.64891	0.181096	0.0905478	−	0.995892i	$-0.471138\pi$
$659$	4.64891	0.181096	0.0905478	+	0.995892i	$0.471138\pi$
$660$	0	0
$661$	35.4148	1.37748	0.688739	−	0.725010i	$-0.258165\pi$
$661$	35.4148	1.37748	0.688739	+	0.725010i	$0.258165\pi$
$662$	0	0
$663$	−14.1844	−0.550875
$664$	0	0
$665$	0.655849	0.0254327
$666$	0	0
$667$	32.8108	1.27044
$668$	0	0
$669$	−27.0887	−1.04731
$670$	0	0
$671$	−50.2933	−1.94155
$672$	0	0
$673$	−32.3886	−1.24849	−0.624244	−	0.781230i	$-0.714593\pi$
$673$	−32.3886	−1.24849	−0.624244	+	0.781230i	$0.714593\pi$
$674$	0	0
$675$	4.51327	0.173716
$676$	0	0
$677$	−5.21239	−0.200329	−0.100164	−	0.994971i	$-0.531937\pi$
$677$	−5.21239	−0.200329	−0.100164	+	0.994971i	$0.531937\pi$
$678$	0	0
$679$	−13.1397	−0.504256
$680$	0	0
$681$	−9.94716	−0.381176
$682$	0	0
$683$	−6.92358	−0.264923	−0.132462	−	0.991188i	$-0.542288\pi$
$683$	−6.92358	−0.264923	−0.132462	+	0.991188i	$0.542288\pi$
$684$	0	0
$685$	0.193259	0.00738404
$686$	0	0
$687$	−10.3907	−0.396432
$688$	0	0
$689$	−1.12311	−0.0427869
$690$	0	0
$691$	−44.3592	−1.68750	−0.843751	−	0.536734i	$-0.819658\pi$
$691$	−44.3592	−1.68750	−0.843751	+	0.536734i	$0.819658\pi$
$692$	0	0
$693$	−3.85275	−0.146354
$694$	0	0
$695$	2.11319	0.0801579
$696$	0	0
$697$	−65.6461	−2.48652
$698$	0	0
$699$	4.29450	0.162433
$700$	0	0
$701$	−38.6960	−1.46153	−0.730764	−	0.682630i	$-0.760836\pi$
$701$	−38.6960	−1.46153	−0.730764	+	0.682630i	$0.760836\pi$
$702$	0	0
$703$	−1.15328	−0.0434968
$704$	0	0
$705$	9.76312	0.367700
$706$	0	0
$707$	−9.17139	−0.344926
$708$	0	0
$709$	0.299276	0.0112396	0.00561978	−	0.999984i	$-0.498211\pi$
$709$	0.299276	0.0112396	0.00561978	+	0.999984i	$0.498211\pi$
$710$	0	0
$711$	1.52657	0.0572508
$712$	0	0
$713$	4.99009	0.186880
$714$	0	0
$715$	−6.09896	−0.228088
$716$	0	0
$717$	−0.907397	−0.0338874
$718$	0	0
$719$	−7.26341	−0.270880	−0.135440	−	0.990786i	$-0.543245\pi$
$719$	−7.26341	−0.270880	−0.135440	+	0.990786i	$0.543245\pi$
$720$	0	0
$721$	−11.8439	−0.441088
$722$	0	0
$723$	45.3605	1.68698
$724$	0	0
$725$	−8.00467	−0.297286
$726$	0	0
$727$	−22.6954	−0.841724	−0.420862	−	0.907125i	$-0.638272\pi$
$727$	−22.6954	−0.841724	−0.420862	+	0.907125i	$0.638272\pi$
$728$	0	0
$729$	19.1724	0.710089
$730$	0	0
$731$	17.0265	0.629749
$732$	0	0
$733$	−30.7188	−1.13462	−0.567312	−	0.823503i	$-0.692017\pi$
$733$	−30.7188	−1.13462	−0.567312	+	0.823503i	$0.692017\pi$
$734$	0	0
$735$	1.90570	0.0702929
$736$	0	0
$737$	93.0367	3.42705
$738$	0	0
$739$	−30.9084	−1.13699	−0.568493	−	0.822688i	$-0.692473\pi$
$739$	−30.9084	−1.13699	−0.568493	+	0.822688i	$0.692473\pi$
$740$	0	0
$741$	−1.24985	−0.0459145
$742$	0	0
$743$	−40.5168	−1.48642	−0.743208	−	0.669060i	$-0.766697\pi$
$743$	−40.5168	−1.48642	−0.743208	+	0.669060i	$0.766697\pi$
$744$	0	0
$745$	−1.12311	−0.0411474
$746$	0	0
$747$	2.36669	0.0865926
$748$	0	0
$749$	1.36237	0.0497801
$750$	0	0
$751$	24.2238	0.883938	0.441969	−	0.897030i	$-0.354280\pi$
$751$	24.2238	0.883938	0.441969	+	0.897030i	$0.354280\pi$
$752$	0	0
$753$	−34.6598	−1.26307
$754$	0	0
$755$	−16.4584	−0.598981
$756$	0	0
$757$	20.5528	0.747003	0.373502	−	0.927630i	$-0.378157\pi$
$757$	20.5528	0.747003	0.373502	+	0.927630i	$0.378157\pi$
$758$	0	0
$759$	47.6415	1.72928
$760$	0	0
$761$	−38.2768	−1.38753	−0.693767	−	0.720200i	$-0.744050\pi$
$761$	−38.2768	−1.38753	−0.693767	+	0.720200i	$0.744050\pi$
$762$	0	0
$763$	11.8763	0.429951
$764$	0	0
$765$	−4.70186	−0.169996
$766$	0	0
$767$	1.51919	0.0548546
$768$	0	0
$769$	7.49913	0.270425	0.135213	−	0.990817i	$-0.456828\pi$
$769$	7.49913	0.270425	0.135213	+	0.990817i	$0.456828\pi$
$770$	0	0
$771$	−56.4384	−2.03258
$772$	0	0
$773$	6.57511	0.236490	0.118245	−	0.992984i	$-0.462273\pi$
$773$	6.57511	0.236490	0.118245	+	0.992984i	$0.462273\pi$
$774$	0	0
$775$	−1.21740	−0.0437304
$776$	0	0
$777$	−3.35109	−0.120220
$778$	0	0
$779$	−5.78440	−0.207248
$780$	0	0
$781$	−91.9831	−3.29141
$782$	0	0
$783$	−36.1272	−1.29108
$784$	0	0
$785$	17.0041	0.606902
$786$	0	0
$787$	18.1635	0.647460	0.323730	−	0.946150i	$-0.395063\pi$
$787$	18.1635	0.647460	0.323730	+	0.946150i	$0.395063\pi$
$788$	0	0
$789$	23.2502	0.827728
$790$	0	0
$791$	−14.3865	−0.511526
$792$	0	0
$793$	−8.24621	−0.292832
$794$	0	0
$795$	−2.14031	−0.0759088
$796$	0	0
$797$	41.2066	1.45961	0.729806	−	0.683655i	$-0.239611\pi$
$797$	41.2066	1.45961	0.729806	+	0.683655i	$0.239611\pi$
$798$	0	0
$799$	38.1319	1.34901
$800$	0	0
$801$	10.1817	0.359754
$802$	0	0
$803$	−27.4436	−0.968462
$804$	0	0
$805$	−4.09896	−0.144469
$806$	0	0
$807$	−5.57453	−0.196233
$808$	0	0
$809$	35.8779	1.26140	0.630700	−	0.776027i	$-0.282768\pi$
$809$	35.8779	1.26140	0.630700	+	0.776027i	$0.282768\pi$
$810$	0	0
$811$	−50.4875	−1.77286	−0.886428	−	0.462866i	$-0.846821\pi$
$811$	−50.4875	−1.77286	−0.886428	+	0.462866i	$0.846821\pi$
$812$	0	0
$813$	−37.8690	−1.32812
$814$	0	0
$815$	−9.68305	−0.339183
$816$	0	0
$817$	1.50029	0.0524886
$818$	0	0
$819$	−0.631706	−0.0220736
$820$	0	0
$821$	−18.6327	−0.650287	−0.325143	−	0.945665i	$-0.605413\pi$
$821$	−18.6327	−0.650287	−0.325143	+	0.945665i	$0.605413\pi$
$822$	0	0
$823$	51.0416	1.77920	0.889600	−	0.456741i	$-0.150983\pi$
$823$	51.0416	1.77920	0.889600	+	0.456741i	$0.150983\pi$
$824$	0	0
$825$	−11.6228	−0.404654
$826$	0	0
$827$	−31.7923	−1.10553	−0.552763	−	0.833339i	$-0.686427\pi$
$827$	−31.7923	−1.10553	−0.552763	+	0.833339i	$0.686427\pi$
$828$	0	0
$829$	−45.1074	−1.56664	−0.783322	−	0.621616i	$-0.786476\pi$
$829$	−45.1074	−1.56664	−0.783322	+	0.621616i	$0.786476\pi$
$830$	0	0
$831$	−39.2999	−1.36330
$832$	0	0
$833$	7.44311	0.257889
$834$	0	0
$835$	−19.3376	−0.669207
$836$	0	0
$837$	−5.49446	−0.189916
$838$	0	0
$839$	18.4267	0.636160	0.318080	−	0.948064i	$-0.396962\pi$
$839$	18.4267	0.636160	0.318080	+	0.948064i	$0.396962\pi$
$840$	0	0
$841$	35.0747	1.20947
$842$	0	0
$843$	12.4065	0.427303
$844$	0	0
$845$	−1.00000	−0.0344010
$846$	0	0
$847$	−26.1973	−0.900151
$848$	0	0
$849$	26.0923	0.895487
$850$	0	0
$851$	7.20784	0.247082
$852$	0	0
$853$	−3.52875	−0.120822	−0.0604110	−	0.998174i	$-0.519241\pi$
$853$	−3.52875	−0.120822	−0.0604110	+	0.998174i	$0.519241\pi$
$854$	0	0
$855$	−0.414304	−0.0141689
$856$	0	0
$857$	17.4945	0.597599	0.298800	−	0.954316i	$-0.403414\pi$
$857$	17.4945	0.597599	0.298800	+	0.954316i	$0.403414\pi$
$858$	0	0
$859$	−49.1040	−1.67541	−0.837703	−	0.546126i	$-0.816102\pi$
$859$	−49.1040	−1.67541	−0.837703	+	0.546126i	$0.816102\pi$
$860$	0	0
$861$	−16.8078	−0.572807
$862$	0	0
$863$	51.6610	1.75856	0.879281	−	0.476303i	$-0.158023\pi$
$863$	51.6610	1.75856	0.879281	+	0.476303i	$0.158023\pi$
$864$	0	0
$865$	15.2410	0.518208
$866$	0	0
$867$	−73.1789	−2.48529
$868$	0	0
$869$	14.7386	0.499974
$870$	0	0
$871$	15.2545	0.516880
$872$	0	0
$873$	8.30044	0.280927
$874$	0	0
$875$	1.00000	0.0338062
$876$	0	0
$877$	4.79377	0.161874	0.0809370	−	0.996719i	$-0.474209\pi$
$877$	4.79377	0.161874	0.0809370	+	0.996719i	$0.474209\pi$
$878$	0	0
$879$	−59.5223	−2.00764
$880$	0	0
$881$	4.55073	0.153318	0.0766590	−	0.997057i	$-0.475575\pi$
$881$	4.55073	0.153318	0.0766590	+	0.997057i	$0.475575\pi$
$882$	0	0
$883$	−7.19495	−0.242129	−0.121065	−	0.992645i	$-0.538631\pi$
$883$	−7.19495	−0.242129	−0.121065	+	0.992645i	$0.538631\pi$
$884$	0	0
$885$	2.89512	0.0973183
$886$	0	0
$887$	13.6627	0.458749	0.229374	−	0.973338i	$-0.426332\pi$
$887$	13.6627	0.458749	0.229374	+	0.973338i	$0.426332\pi$
$888$	0	0
$889$	−6.72236	−0.225461
$890$	0	0
$891$	−64.0151	−2.14459
$892$	0	0
$893$	3.35999	0.112438
$894$	0	0
$895$	6.81243	0.227714
$896$	0	0
$897$	7.81141	0.260815
$898$	0	0
$899$	9.74490	0.325011
$900$	0	0
$901$	−8.35940	−0.278492
$902$	0	0
$903$	4.35940	0.145072
$904$	0	0
$905$	0.236879	0.00787414
$906$	0	0
$907$	−31.6193	−1.04990	−0.524950	−	0.851133i	$-0.675916\pi$
$907$	−31.6193	−1.04990	−0.524950	+	0.851133i	$0.675916\pi$
$908$	0	0
$909$	5.79362	0.192162
$910$	0	0
$911$	16.2255	0.537574	0.268787	−	0.963200i	$-0.413377\pi$
$911$	16.2255	0.537574	0.268787	+	0.963200i	$0.413377\pi$
$912$	0	0
$913$	22.8498	0.756217
$914$	0	0
$915$	−15.7148	−0.519516
$916$	0	0
$917$	−9.12311	−0.301271
$918$	0	0
$919$	26.1661	0.863141	0.431571	−	0.902079i	$-0.357960\pi$
$919$	26.1661	0.863141	0.431571	+	0.902079i	$0.357960\pi$
$920$	0	0
$921$	−16.2899	−0.536772
$922$	0	0
$923$	−15.0818	−0.496422
$924$	0	0
$925$	−1.75845	−0.0578176
$926$	0	0
$927$	7.48184	0.245736
$928$	0	0
$929$	−40.6342	−1.33316	−0.666582	−	0.745432i	$-0.732243\pi$
$929$	−40.6342	−1.33316	−0.666582	+	0.745432i	$0.732243\pi$
$930$	0	0
$931$	0.655849	0.0214946
$932$	0	0
$933$	−30.2100	−0.989031
$934$	0	0
$935$	−45.3953	−1.48458
$936$	0	0
$937$	23.9169	0.781330	0.390665	−	0.920533i	$-0.372245\pi$
$937$	23.9169	0.781330	0.390665	+	0.920533i	$0.372245\pi$
$938$	0	0
$939$	48.1712	1.57201
$940$	0	0
$941$	−4.51315	−0.147125	−0.0735623	−	0.997291i	$-0.523437\pi$
$941$	−4.51315	−0.147125	−0.0735623	+	0.997291i	$0.523437\pi$
$942$	0	0
$943$	36.1517	1.17726
$944$	0	0
$945$	4.51327	0.146817
$946$	0	0
$947$	10.8999	0.354199	0.177100	−	0.984193i	$-0.443329\pi$
$947$	10.8999	0.354199	0.177100	+	0.984193i	$0.443329\pi$
$948$	0	0
$949$	−4.49971	−0.146067
$950$	0	0
$951$	22.9062	0.742785
$952$	0	0
$953$	17.7797	0.575942	0.287971	−	0.957639i	$-0.407019\pi$
$953$	17.7797	0.575942	0.287971	+	0.957639i	$0.407019\pi$
$954$	0	0
$955$	6.92985	0.224245
$956$	0	0
$957$	93.0367	3.00745
$958$	0	0
$959$	0.193259	0.00624065
$960$	0	0
$961$	−29.5179	−0.952191
$962$	0	0
$963$	−0.860620	−0.0277331
$964$	0	0
$965$	16.5179	0.531731
$966$	0	0
$967$	−12.1085	−0.389384	−0.194692	−	0.980864i	$-0.562371\pi$
$967$	−12.1085	−0.389384	−0.194692	+	0.980864i	$0.562371\pi$
$968$	0	0
$969$	−9.30281	−0.298849
$970$	0	0
$971$	−31.8540	−1.02224	−0.511121	−	0.859509i	$-0.670770\pi$
$971$	−31.8540	−1.02224	−0.511121	+	0.859509i	$0.670770\pi$
$972$	0	0
$973$	2.11319	0.0677458
$974$	0	0
$975$	−1.90570	−0.0610314
$976$	0	0
$977$	21.3372	0.682638	0.341319	−	0.939948i	$-0.389126\pi$
$977$	21.3372	0.682638	0.341319	+	0.939948i	$0.389126\pi$
$978$	0	0
$979$	98.3022	3.14175
$980$	0	0
$981$	−7.50234	−0.239531
$982$	0	0
$983$	32.8618	1.04813	0.524065	−	0.851678i	$-0.324415\pi$
$983$	32.8618	1.04813	0.524065	+	0.851678i	$0.324415\pi$
$984$	0	0
$985$	22.6374	0.721287
$986$	0	0
$987$	9.76312	0.310764
$988$	0	0
$989$	−9.37660	−0.298159
$990$	0	0
$991$	1.71644	0.0545245	0.0272623	−	0.999628i	$-0.491321\pi$
$991$	1.71644	0.0545245	0.0272623	+	0.999628i	$0.491321\pi$
$992$	0	0
$993$	−23.1491	−0.734613
$994$	0	0
$995$	−17.2846	−0.547958
$996$	0	0
$997$	40.6616	1.28777	0.643883	−	0.765124i	$-0.277322\pi$
$997$	40.6616	1.28777	0.643883	+	0.765124i	$0.277322\pi$
$998$	0	0
$999$	−7.93637	−0.251096

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3640.2.a.x.1.1	✓	4
4.3	odd	2		7280.2.a.br.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3640.2.a.x.1.1	✓	4	1.1	even	1	trivial
7280.2.a.br.1.4		4	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 3640 = 2^{3} \cdot 5 \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3640.a (trivial)

\(f(q)\)	\(=\)	\(q-1.90570 q^{3} -1.00000 q^{5} -1.00000 q^{7} +0.631706 q^{9} +O(q^{10})\)	\(q-1.90570 q^{3} -1.00000 q^{5} -1.00000 q^{7} +0.631706 q^{9} +6.09896 q^{11} +1.00000 q^{13} +1.90570 q^{15} +7.44311 q^{17} +0.655849 q^{19} +1.90570 q^{21} -4.09896 q^{23} +1.00000 q^{25} +4.51327 q^{27} -8.00467 q^{29} -1.21740 q^{31} -11.6228 q^{33} +1.00000 q^{35} -1.75845 q^{37} -1.90570 q^{39} -8.81972 q^{41} +2.28756 q^{43} -0.631706 q^{45} +5.12311 q^{47} +1.00000 q^{49} -14.1844 q^{51} -1.12311 q^{53} -6.09896 q^{55} -1.24985 q^{57} +1.51919 q^{59} -8.24621 q^{61} -0.631706 q^{63} -1.00000 q^{65} +15.2545 q^{67} +7.81141 q^{69} -15.0818 q^{71} -4.49971 q^{73} -1.90570 q^{75} -6.09896 q^{77} +2.41658 q^{79} -10.4961 q^{81} +3.74650 q^{83} -7.44311 q^{85} +15.2545 q^{87} +16.1179 q^{89} -1.00000 q^{91} +2.32001 q^{93} -0.655849 q^{95} +13.1397 q^{97} +3.85275 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 3 q^{3} - 4 q^{5} - 4 q^{7} + 3 q^{9}+O(q^{10}) \) 4 * q + 3 * q^3 - 4 * q^5 - 4 * q^7 + 3 * q^9	\( 4 q + 3 q^{3} - 4 q^{5} - 4 q^{7} + 3 q^{9} + 6 q^{11} + 4 q^{13} - 3 q^{15} + 9 q^{17} + 5 q^{19} - 3 q^{21} + 2 q^{23} + 4 q^{25} + 6 q^{27} - 3 q^{29} + q^{31} - 4 q^{33} + 4 q^{35} - 11 q^{37} + 3 q^{39} - 5 q^{41} + 12 q^{43} - 3 q^{45} + 4 q^{47} + 4 q^{49} - 6 q^{51} + 12 q^{53} - 6 q^{55} + 8 q^{57} + 11 q^{59} - 3 q^{63} - 4 q^{65} + 19 q^{67} + 10 q^{69} - 8 q^{71} + 8 q^{73} + 3 q^{75} - 6 q^{77} + 13 q^{79} + 4 q^{81} + 8 q^{83} - 9 q^{85} + 19 q^{87} + 25 q^{89} - 4 q^{91} + 5 q^{93} - 5 q^{95} + 18 q^{97} + 30 q^{99}+O(q^{100}) \) 4 * q + 3 * q^3 - 4 * q^5 - 4 * q^7 + 3 * q^9 + 6 * q^11 + 4 * q^13 - 3 * q^15 + 9 * q^17 + 5 * q^19 - 3 * q^21 + 2 * q^23 + 4 * q^25 + 6 * q^27 - 3 * q^29 + q^31 - 4 * q^33 + 4 * q^35 - 11 * q^37 + 3 * q^39 - 5 * q^41 + 12 * q^43 - 3 * q^45 + 4 * q^47 + 4 * q^49 - 6 * q^51 + 12 * q^53 - 6 * q^55 + 8 * q^57 + 11 * q^59 - 3 * q^63 - 4 * q^65 + 19 * q^67 + 10 * q^69 - 8 * q^71 + 8 * q^73 + 3 * q^75 - 6 * q^77 + 13 * q^79 + 4 * q^81 + 8 * q^83 - 9 * q^85 + 19 * q^87 + 25 * q^89 - 4 * q^91 + 5 * q^93 - 5 * q^95 + 18 * q^97 + 30 * q^99

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