LMFDB - Embedded newform 6240.2.a.bl.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6240,2,Mod(1,6240)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6240, 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("6240.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6240 = 2^{5} \cdot 3 \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6240.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:	$49.8266508613$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.618034$ of defining polynomial
Character	$\chi$	$=$	6240.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +1.00000 q^{5} -0.763932 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +1.00000 q^{5} -0.763932 q^{7} +1.00000 q^{9} -6.47214 q^{11} -1.00000 q^{13} -1.00000 q^{15} +5.23607 q^{17} -0.763932 q^{19} +0.763932 q^{21} -7.23607 q^{23} +1.00000 q^{25} -1.00000 q^{27} +9.23607 q^{29} -8.94427 q^{31} +6.47214 q^{33} -0.763932 q^{35} +0.472136 q^{37} +1.00000 q^{39} -12.4721 q^{41} +12.9443 q^{43} +1.00000 q^{45} +4.94427 q^{47} -6.41641 q^{49} -5.23607 q^{51} +0.472136 q^{53} -6.47214 q^{55} +0.763932 q^{57} +6.47214 q^{59} -6.94427 q^{61} -0.763932 q^{63} -1.00000 q^{65} +8.00000 q^{67} +7.23607 q^{69} +4.00000 q^{71} -3.70820 q^{73} -1.00000 q^{75} +4.94427 q^{77} +12.9443 q^{79} +1.00000 q^{81} -8.00000 q^{83} +5.23607 q^{85} -9.23607 q^{87} +2.94427 q^{89} +0.763932 q^{91} +8.94427 q^{93} -0.763932 q^{95} -16.6525 q^{97} -6.47214 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} + 2 q^{5} - 6 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^3 + 2 * q^5 - 6 * q^7 + 2 * q^9	$ 2 q - 2 q^{3} + 2 q^{5} - 6 q^{7} + 2 q^{9} - 4 q^{11} - 2 q^{13} - 2 q^{15} + 6 q^{17} - 6 q^{19} + 6 q^{21} - 10 q^{23} + 2 q^{25} - 2 q^{27} + 14 q^{29} + 4 q^{33} - 6 q^{35} - 8 q^{37} + 2 q^{39} - 16 q^{41} + 8 q^{43} + 2 q^{45} - 8 q^{47} + 14 q^{49} - 6 q^{51} - 8 q^{53} - 4 q^{55} + 6 q^{57} + 4 q^{59} + 4 q^{61} - 6 q^{63} - 2 q^{65} + 16 q^{67} + 10 q^{69} + 8 q^{71} + 6 q^{73} - 2 q^{75} - 8 q^{77} + 8 q^{79} + 2 q^{81} - 16 q^{83} + 6 q^{85} - 14 q^{87} - 12 q^{89} + 6 q^{91} - 6 q^{95} - 2 q^{97} - 4 q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 + 2 * q^5 - 6 * q^7 + 2 * q^9 - 4 * q^11 - 2 * q^13 - 2 * q^15 + 6 * q^17 - 6 * q^19 + 6 * q^21 - 10 * q^23 + 2 * q^25 - 2 * q^27 + 14 * q^29 + 4 * q^33 - 6 * q^35 - 8 * q^37 + 2 * q^39 - 16 * q^41 + 8 * q^43 + 2 * q^45 - 8 * q^47 + 14 * q^49 - 6 * q^51 - 8 * q^53 - 4 * q^55 + 6 * q^57 + 4 * q^59 + 4 * q^61 - 6 * q^63 - 2 * q^65 + 16 * q^67 + 10 * q^69 + 8 * q^71 + 6 * q^73 - 2 * q^75 - 8 * q^77 + 8 * q^79 + 2 * q^81 - 16 * q^83 + 6 * q^85 - 14 * q^87 - 12 * q^89 + 6 * q^91 - 6 * q^95 - 2 * q^97 - 4 * 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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−0.763932	−0.288739	−0.144370	−	0.989524i	$-0.546115\pi$
$7$	−0.763932	−0.288739	−0.144370	+	0.989524i	$0.546115\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−6.47214	−1.95142	−0.975711	−	0.219061i	$-0.929701\pi$
$11$	−6.47214	−1.95142	−0.975711	+	0.219061i	$0.929701\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	−1.00000	−0.258199
$16$	0	0
$17$	5.23607	1.26993	0.634967	−	0.772540i	$-0.281014\pi$
$17$	5.23607	1.26993	0.634967	+	0.772540i	$0.281014\pi$
$18$	0	0
$19$	−0.763932	−0.175258	−0.0876290	−	0.996153i	$-0.527929\pi$
$19$	−0.763932	−0.175258	−0.0876290	+	0.996153i	$0.527929\pi$
$20$	0	0
$21$	0.763932	0.166704
$22$	0	0
$23$	−7.23607	−1.50882	−0.754412	−	0.656401i	$-0.772078\pi$
$23$	−7.23607	−1.50882	−0.754412	+	0.656401i	$0.772078\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	9.23607	1.71509	0.857547	−	0.514405i	$-0.171987\pi$
$29$	9.23607	1.71509	0.857547	+	0.514405i	$0.171987\pi$
$30$	0	0
$31$	−8.94427	−1.60644	−0.803219	−	0.595683i	$-0.796881\pi$
$31$	−8.94427	−1.60644	−0.803219	+	0.595683i	$0.796881\pi$
$32$	0	0
$33$	6.47214	1.12665
$34$	0	0
$35$	−0.763932	−0.129128
$36$	0	0
$37$	0.472136	0.0776187	0.0388093	−	0.999247i	$-0.487644\pi$
$37$	0.472136	0.0776187	0.0388093	+	0.999247i	$0.487644\pi$
$38$	0	0
$39$	1.00000	0.160128
$40$	0	0
$41$	−12.4721	−1.94782	−0.973910	−	0.226934i	$-0.927130\pi$
$41$	−12.4721	−1.94782	−0.973910	+	0.226934i	$0.927130\pi$
$42$	0	0
$43$	12.9443	1.97398	0.986991	−	0.160773i	$-0.0513986\pi$
$43$	12.9443	1.97398	0.986991	+	0.160773i	$0.0513986\pi$
$44$	0	0
$45$	1.00000	0.149071
$46$	0	0
$47$	4.94427	0.721196	0.360598	−	0.932721i	$-0.382573\pi$
$47$	4.94427	0.721196	0.360598	+	0.932721i	$0.382573\pi$
$48$	0	0
$49$	−6.41641	−0.916630
$50$	0	0
$51$	−5.23607	−0.733196
$52$	0	0
$53$	0.472136	0.0648529	0.0324264	−	0.999474i	$-0.489677\pi$
$53$	0.472136	0.0648529	0.0324264	+	0.999474i	$0.489677\pi$
$54$	0	0
$55$	−6.47214	−0.872703
$56$	0	0
$57$	0.763932	0.101185
$58$	0	0
$59$	6.47214	0.842600	0.421300	−	0.906921i	$-0.361574\pi$
$59$	6.47214	0.842600	0.421300	+	0.906921i	$0.361574\pi$
$60$	0	0
$61$	−6.94427	−0.889123	−0.444561	−	0.895748i	$-0.646640\pi$
$61$	−6.94427	−0.889123	−0.444561	+	0.895748i	$0.646640\pi$
$62$	0	0
$63$	−0.763932	−0.0962464
$64$	0	0
$65$	−1.00000	−0.124035
$66$	0	0
$67$	8.00000	0.977356	0.488678	−	0.872464i	$-0.337479\pi$
$67$	8.00000	0.977356	0.488678	+	0.872464i	$0.337479\pi$
$68$	0	0
$69$	7.23607	0.871120
$70$	0	0
$71$	4.00000	0.474713	0.237356	−	0.971423i	$-0.423719\pi$
$71$	4.00000	0.474713	0.237356	+	0.971423i	$0.423719\pi$
$72$	0	0
$73$	−3.70820	−0.434012	−0.217006	−	0.976170i	$-0.569629\pi$
$73$	−3.70820	−0.434012	−0.217006	+	0.976170i	$0.569629\pi$
$74$	0	0
$75$	−1.00000	−0.115470
$76$	0	0
$77$	4.94427	0.563452
$78$	0	0
$79$	12.9443	1.45634	0.728172	−	0.685394i	$-0.240370\pi$
$79$	12.9443	1.45634	0.728172	+	0.685394i	$0.240370\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−8.00000	−0.878114	−0.439057	−	0.898459i	$-0.644687\pi$
$83$	−8.00000	−0.878114	−0.439057	+	0.898459i	$0.644687\pi$
$84$	0	0
$85$	5.23607	0.567931
$86$	0	0
$87$	−9.23607	−0.990210
$88$	0	0
$89$	2.94427	0.312092	0.156046	−	0.987750i	$-0.450125\pi$
$89$	2.94427	0.312092	0.156046	+	0.987750i	$0.450125\pi$
$90$	0	0
$91$	0.763932	0.0800818
$92$	0	0
$93$	8.94427	0.927478
$94$	0	0
$95$	−0.763932	−0.0783778
$96$	0	0
$97$	−16.6525	−1.69080	−0.845401	−	0.534132i	$-0.820639\pi$
$97$	−16.6525	−1.69080	−0.845401	+	0.534132i	$0.820639\pi$
$98$	0	0
$99$	−6.47214	−0.650474
$100$	0	0
$101$	15.7082	1.56302	0.781512	−	0.623890i	$-0.214449\pi$
$101$	15.7082	1.56302	0.781512	+	0.623890i	$0.214449\pi$
$102$	0	0
$103$	−4.00000	−0.394132	−0.197066	−	0.980390i	$-0.563141\pi$
$103$	−4.00000	−0.394132	−0.197066	+	0.980390i	$0.563141\pi$
$104$	0	0
$105$	0.763932	0.0745521
$106$	0	0
$107$	12.0000	1.16008	0.580042	−	0.814587i	$-0.303036\pi$
$107$	12.0000	1.16008	0.580042	+	0.814587i	$0.303036\pi$
$108$	0	0
$109$	10.1803	0.975100	0.487550	−	0.873095i	$-0.337891\pi$
$109$	10.1803	0.975100	0.487550	+	0.873095i	$0.337891\pi$
$110$	0	0
$111$	−0.472136	−0.0448132
$112$	0	0
$113$	6.76393	0.636297	0.318149	−	0.948041i	$-0.396939\pi$
$113$	6.76393	0.636297	0.318149	+	0.948041i	$0.396939\pi$
$114$	0	0
$115$	−7.23607	−0.674767
$116$	0	0
$117$	−1.00000	−0.0924500
$118$	0	0
$119$	−4.00000	−0.366679
$120$	0	0
$121$	30.8885	2.80805
$122$	0	0
$123$	12.4721	1.12457
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−4.00000	−0.354943	−0.177471	−	0.984126i	$-0.556792\pi$
$127$	−4.00000	−0.354943	−0.177471	+	0.984126i	$0.556792\pi$
$128$	0	0
$129$	−12.9443	−1.13968
$130$	0	0
$131$	−10.6525	−0.930711	−0.465356	−	0.885124i	$-0.654074\pi$
$131$	−10.6525	−0.930711	−0.465356	+	0.885124i	$0.654074\pi$
$132$	0	0
$133$	0.583592	0.0506039
$134$	0	0
$135$	−1.00000	−0.0860663
$136$	0	0
$137$	8.47214	0.723823	0.361912	−	0.932212i	$-0.382124\pi$
$137$	8.47214	0.723823	0.361912	+	0.932212i	$0.382124\pi$
$138$	0	0
$139$	2.47214	0.209684	0.104842	−	0.994489i	$-0.466566\pi$
$139$	2.47214	0.209684	0.104842	+	0.994489i	$0.466566\pi$
$140$	0	0
$141$	−4.94427	−0.416383
$142$	0	0
$143$	6.47214	0.541227
$144$	0	0
$145$	9.23607	0.767014
$146$	0	0
$147$	6.41641	0.529216
$148$	0	0
$149$	4.47214	0.366372	0.183186	−	0.983078i	$-0.441359\pi$
$149$	4.47214	0.366372	0.183186	+	0.983078i	$0.441359\pi$
$150$	0	0
$151$	0.944272	0.0768438	0.0384219	−	0.999262i	$-0.487767\pi$
$151$	0.944272	0.0768438	0.0384219	+	0.999262i	$0.487767\pi$
$152$	0	0
$153$	5.23607	0.423311
$154$	0	0
$155$	−8.94427	−0.718421
$156$	0	0
$157$	−13.4164	−1.07075	−0.535373	−	0.844616i	$-0.679829\pi$
$157$	−13.4164	−1.07075	−0.535373	+	0.844616i	$0.679829\pi$
$158$	0	0
$159$	−0.472136	−0.0374428
$160$	0	0
$161$	5.52786	0.435657
$162$	0	0
$163$	1.52786	0.119672	0.0598358	−	0.998208i	$-0.480942\pi$
$163$	1.52786	0.119672	0.0598358	+	0.998208i	$0.480942\pi$
$164$	0	0
$165$	6.47214	0.503855
$166$	0	0
$167$	−3.41641	−0.264370	−0.132185	−	0.991225i	$-0.542199\pi$
$167$	−3.41641	−0.264370	−0.132185	+	0.991225i	$0.542199\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−0.763932	−0.0584193
$172$	0	0
$173$	−4.47214	−0.340010	−0.170005	−	0.985443i	$-0.554378\pi$
$173$	−4.47214	−0.340010	−0.170005	+	0.985443i	$0.554378\pi$
$174$	0	0
$175$	−0.763932	−0.0577478
$176$	0	0
$177$	−6.47214	−0.486476
$178$	0	0
$179$	16.7639	1.25300	0.626498	−	0.779423i	$-0.284488\pi$
$179$	16.7639	1.25300	0.626498	+	0.779423i	$0.284488\pi$
$180$	0	0
$181$	20.4721	1.52168	0.760841	−	0.648938i	$-0.224787\pi$
$181$	20.4721	1.52168	0.760841	+	0.648938i	$0.224787\pi$
$182$	0	0
$183$	6.94427	0.513335
$184$	0	0
$185$	0.472136	0.0347121
$186$	0	0
$187$	−33.8885	−2.47818
$188$	0	0
$189$	0.763932	0.0555679
$190$	0	0
$191$	20.9443	1.51547	0.757737	−	0.652560i	$-0.226305\pi$
$191$	20.9443	1.51547	0.757737	+	0.652560i	$0.226305\pi$
$192$	0	0
$193$	−2.18034	−0.156944	−0.0784721	−	0.996916i	$-0.525004\pi$
$193$	−2.18034	−0.156944	−0.0784721	+	0.996916i	$0.525004\pi$
$194$	0	0
$195$	1.00000	0.0716115
$196$	0	0
$197$	−9.41641	−0.670891	−0.335446	−	0.942060i	$-0.608887\pi$
$197$	−9.41641	−0.670891	−0.335446	+	0.942060i	$0.608887\pi$
$198$	0	0
$199$	9.88854	0.700980	0.350490	−	0.936566i	$-0.386015\pi$
$199$	9.88854	0.700980	0.350490	+	0.936566i	$0.386015\pi$
$200$	0	0
$201$	−8.00000	−0.564276
$202$	0	0
$203$	−7.05573	−0.495215
$204$	0	0
$205$	−12.4721	−0.871092
$206$	0	0
$207$	−7.23607	−0.502941
$208$	0	0
$209$	4.94427	0.342002
$210$	0	0
$211$	−0.944272	−0.0650064	−0.0325032	−	0.999472i	$-0.510348\pi$
$211$	−0.944272	−0.0650064	−0.0325032	+	0.999472i	$0.510348\pi$
$212$	0	0
$213$	−4.00000	−0.274075
$214$	0	0
$215$	12.9443	0.882792
$216$	0	0
$217$	6.83282	0.463842
$218$	0	0
$219$	3.70820	0.250577
$220$	0	0
$221$	−5.23607	−0.352216
$222$	0	0
$223$	−12.1803	−0.815656	−0.407828	−	0.913059i	$-0.633714\pi$
$223$	−12.1803	−0.815656	−0.407828	+	0.913059i	$0.633714\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	21.8885	1.45279	0.726397	−	0.687276i	$-0.241193\pi$
$227$	21.8885	1.45279	0.726397	+	0.687276i	$0.241193\pi$
$228$	0	0
$229$	−2.76393	−0.182646	−0.0913229	−	0.995821i	$-0.529110\pi$
$229$	−2.76393	−0.182646	−0.0913229	+	0.995821i	$0.529110\pi$
$230$	0	0
$231$	−4.94427	−0.325309
$232$	0	0
$233$	−10.7639	−0.705169	−0.352584	−	0.935780i	$-0.614697\pi$
$233$	−10.7639	−0.705169	−0.352584	+	0.935780i	$0.614697\pi$
$234$	0	0
$235$	4.94427	0.322529
$236$	0	0
$237$	−12.9443	−0.840821
$238$	0	0
$239$	11.0557	0.715136	0.357568	−	0.933887i	$-0.383606\pi$
$239$	11.0557	0.715136	0.357568	+	0.933887i	$0.383606\pi$
$240$	0	0
$241$	10.0000	0.644157	0.322078	−	0.946713i	$-0.395619\pi$
$241$	10.0000	0.644157	0.322078	+	0.946713i	$0.395619\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−6.41641	−0.409929
$246$	0	0
$247$	0.763932	0.0486078
$248$	0	0
$249$	8.00000	0.506979
$250$	0	0
$251$	13.7082	0.865254	0.432627	−	0.901573i	$-0.357587\pi$
$251$	13.7082	0.865254	0.432627	+	0.901573i	$0.357587\pi$
$252$	0	0
$253$	46.8328	2.94435
$254$	0	0
$255$	−5.23607	−0.327895
$256$	0	0
$257$	19.7082	1.22936	0.614682	−	0.788775i	$-0.289284\pi$
$257$	19.7082	1.22936	0.614682	+	0.788775i	$0.289284\pi$
$258$	0	0
$259$	−0.360680	−0.0224116
$260$	0	0
$261$	9.23607	0.571698
$262$	0	0
$263$	−18.6525	−1.15016	−0.575080	−	0.818097i	$-0.695029\pi$
$263$	−18.6525	−1.15016	−0.575080	+	0.818097i	$0.695029\pi$
$264$	0	0
$265$	0.472136	0.0290031
$266$	0	0
$267$	−2.94427	−0.180187
$268$	0	0
$269$	−2.18034	−0.132938	−0.0664688	−	0.997789i	$-0.521173\pi$
$269$	−2.18034	−0.132938	−0.0664688	+	0.997789i	$0.521173\pi$
$270$	0	0
$271$	−21.8885	−1.32963	−0.664817	−	0.747006i	$-0.731491\pi$
$271$	−21.8885	−1.32963	−0.664817	+	0.747006i	$0.731491\pi$
$272$	0	0
$273$	−0.763932	−0.0462353
$274$	0	0
$275$	−6.47214	−0.390284
$276$	0	0
$277$	−3.52786	−0.211969	−0.105984	−	0.994368i	$-0.533799\pi$
$277$	−3.52786	−0.211969	−0.105984	+	0.994368i	$0.533799\pi$
$278$	0	0
$279$	−8.94427	−0.535480
$280$	0	0
$281$	7.88854	0.470591	0.235296	−	0.971924i	$-0.424394\pi$
$281$	7.88854	0.470591	0.235296	+	0.971924i	$0.424394\pi$
$282$	0	0
$283$	16.0000	0.951101	0.475551	−	0.879688i	$-0.342249\pi$
$283$	16.0000	0.951101	0.475551	+	0.879688i	$0.342249\pi$
$284$	0	0
$285$	0.763932	0.0452514
$286$	0	0
$287$	9.52786	0.562412
$288$	0	0
$289$	10.4164	0.612730
$290$	0	0
$291$	16.6525	0.976185
$292$	0	0
$293$	−12.4721	−0.728630	−0.364315	−	0.931276i	$-0.618697\pi$
$293$	−12.4721	−0.728630	−0.364315	+	0.931276i	$0.618697\pi$
$294$	0	0
$295$	6.47214	0.376822
$296$	0	0
$297$	6.47214	0.375551
$298$	0	0
$299$	7.23607	0.418473
$300$	0	0
$301$	−9.88854	−0.569966
$302$	0	0
$303$	−15.7082	−0.902413
$304$	0	0
$305$	−6.94427	−0.397628
$306$	0	0
$307$	3.05573	0.174400	0.0871998	−	0.996191i	$-0.472208\pi$
$307$	3.05573	0.174400	0.0871998	+	0.996191i	$0.472208\pi$
$308$	0	0
$309$	4.00000	0.227552
$310$	0	0
$311$	−20.9443	−1.18764	−0.593820	−	0.804598i	$-0.702381\pi$
$311$	−20.9443	−1.18764	−0.593820	+	0.804598i	$0.702381\pi$
$312$	0	0
$313$	21.4164	1.21053	0.605263	−	0.796025i	$-0.293068\pi$
$313$	21.4164	1.21053	0.605263	+	0.796025i	$0.293068\pi$
$314$	0	0
$315$	−0.763932	−0.0430427
$316$	0	0
$317$	−2.00000	−0.112331	−0.0561656	−	0.998421i	$-0.517887\pi$
$317$	−2.00000	−0.112331	−0.0561656	+	0.998421i	$0.517887\pi$
$318$	0	0
$319$	−59.7771	−3.34687
$320$	0	0
$321$	−12.0000	−0.669775
$322$	0	0
$323$	−4.00000	−0.222566
$324$	0	0
$325$	−1.00000	−0.0554700
$326$	0	0
$327$	−10.1803	−0.562974
$328$	0	0
$329$	−3.77709	−0.208238
$330$	0	0
$331$	−13.7082	−0.753471	−0.376736	−	0.926321i	$-0.622953\pi$
$331$	−13.7082	−0.753471	−0.376736	+	0.926321i	$0.622953\pi$
$332$	0	0
$333$	0.472136	0.0258729
$334$	0	0
$335$	8.00000	0.437087
$336$	0	0
$337$	32.4721	1.76887	0.884435	−	0.466663i	$-0.154544\pi$
$337$	32.4721	1.76887	0.884435	+	0.466663i	$0.154544\pi$
$338$	0	0
$339$	−6.76393	−0.367366
$340$	0	0
$341$	57.8885	3.13484
$342$	0	0
$343$	10.2492	0.553406
$344$	0	0
$345$	7.23607	0.389577
$346$	0	0
$347$	−15.4164	−0.827596	−0.413798	−	0.910369i	$-0.635798\pi$
$347$	−15.4164	−0.827596	−0.413798	+	0.910369i	$0.635798\pi$
$348$	0	0
$349$	−23.7082	−1.26907	−0.634536	−	0.772894i	$-0.718809\pi$
$349$	−23.7082	−1.26907	−0.634536	+	0.772894i	$0.718809\pi$
$350$	0	0
$351$	1.00000	0.0533761
$352$	0	0
$353$	−4.47214	−0.238028	−0.119014	−	0.992893i	$-0.537973\pi$
$353$	−4.47214	−0.238028	−0.119014	+	0.992893i	$0.537973\pi$
$354$	0	0
$355$	4.00000	0.212298
$356$	0	0
$357$	4.00000	0.211702
$358$	0	0
$359$	−24.9443	−1.31651	−0.658254	−	0.752796i	$-0.728705\pi$
$359$	−24.9443	−1.31651	−0.658254	+	0.752796i	$0.728705\pi$
$360$	0	0
$361$	−18.4164	−0.969285
$362$	0	0
$363$	−30.8885	−1.62123
$364$	0	0
$365$	−3.70820	−0.194096
$366$	0	0
$367$	34.8328	1.81826	0.909129	−	0.416514i	$-0.136748\pi$
$367$	34.8328	1.81826	0.909129	+	0.416514i	$0.136748\pi$
$368$	0	0
$369$	−12.4721	−0.649273
$370$	0	0
$371$	−0.360680	−0.0187256
$372$	0	0
$373$	−5.41641	−0.280451	−0.140225	−	0.990120i	$-0.544783\pi$
$373$	−5.41641	−0.280451	−0.140225	+	0.990120i	$0.544783\pi$
$374$	0	0
$375$	−1.00000	−0.0516398
$376$	0	0
$377$	−9.23607	−0.475682
$378$	0	0
$379$	32.7639	1.68297	0.841485	−	0.540280i	$-0.181682\pi$
$379$	32.7639	1.68297	0.841485	+	0.540280i	$0.181682\pi$
$380$	0	0
$381$	4.00000	0.204926
$382$	0	0
$383$	37.3050	1.90619	0.953097	−	0.302665i	$-0.0978763\pi$
$383$	37.3050	1.90619	0.953097	+	0.302665i	$0.0978763\pi$
$384$	0	0
$385$	4.94427	0.251983
$386$	0	0
$387$	12.9443	0.657994
$388$	0	0
$389$	2.76393	0.140137	0.0700685	−	0.997542i	$-0.477678\pi$
$389$	2.76393	0.140137	0.0700685	+	0.997542i	$0.477678\pi$
$390$	0	0
$391$	−37.8885	−1.91611
$392$	0	0
$393$	10.6525	0.537346
$394$	0	0
$395$	12.9443	0.651297
$396$	0	0
$397$	−4.47214	−0.224450	−0.112225	−	0.993683i	$-0.535798\pi$
$397$	−4.47214	−0.224450	−0.112225	+	0.993683i	$0.535798\pi$
$398$	0	0
$399$	−0.583592	−0.0292161
$400$	0	0
$401$	5.41641	0.270483	0.135241	−	0.990813i	$-0.456819\pi$
$401$	5.41641	0.270483	0.135241	+	0.990813i	$0.456819\pi$
$402$	0	0
$403$	8.94427	0.445546
$404$	0	0
$405$	1.00000	0.0496904
$406$	0	0
$407$	−3.05573	−0.151467
$408$	0	0
$409$	16.4721	0.814495	0.407247	−	0.913318i	$-0.366489\pi$
$409$	16.4721	0.814495	0.407247	+	0.913318i	$0.366489\pi$
$410$	0	0
$411$	−8.47214	−0.417900
$412$	0	0
$413$	−4.94427	−0.243292
$414$	0	0
$415$	−8.00000	−0.392705
$416$	0	0
$417$	−2.47214	−0.121061
$418$	0	0
$419$	25.1246	1.22742	0.613709	−	0.789532i	$-0.289677\pi$
$419$	25.1246	1.22742	0.613709	+	0.789532i	$0.289677\pi$
$420$	0	0
$421$	27.7082	1.35042	0.675208	−	0.737628i	$-0.264054\pi$
$421$	27.7082	1.35042	0.675208	+	0.737628i	$0.264054\pi$
$422$	0	0
$423$	4.94427	0.240399
$424$	0	0
$425$	5.23607	0.253987
$426$	0	0
$427$	5.30495	0.256725
$428$	0	0
$429$	−6.47214	−0.312478
$430$	0	0
$431$	25.8885	1.24701	0.623504	−	0.781820i	$-0.285709\pi$
$431$	25.8885	1.24701	0.623504	+	0.781820i	$0.285709\pi$
$432$	0	0
$433$	−33.4164	−1.60589	−0.802945	−	0.596053i	$-0.796735\pi$
$433$	−33.4164	−1.60589	−0.802945	+	0.596053i	$0.796735\pi$
$434$	0	0
$435$	−9.23607	−0.442836
$436$	0	0
$437$	5.52786	0.264434
$438$	0	0
$439$	−13.8885	−0.662864	−0.331432	−	0.943479i	$-0.607532\pi$
$439$	−13.8885	−0.662864	−0.331432	+	0.943479i	$0.607532\pi$
$440$	0	0
$441$	−6.41641	−0.305543
$442$	0	0
$443$	21.5279	1.02282	0.511410	−	0.859337i	$-0.329123\pi$
$443$	21.5279	1.02282	0.511410	+	0.859337i	$0.329123\pi$
$444$	0	0
$445$	2.94427	0.139572
$446$	0	0
$447$	−4.47214	−0.211525
$448$	0	0
$449$	−24.8328	−1.17193	−0.585967	−	0.810335i	$-0.699285\pi$
$449$	−24.8328	−1.17193	−0.585967	+	0.810335i	$0.699285\pi$
$450$	0	0
$451$	80.7214	3.80102
$452$	0	0
$453$	−0.944272	−0.0443658
$454$	0	0
$455$	0.763932	0.0358137
$456$	0	0
$457$	20.2918	0.949210	0.474605	−	0.880199i	$-0.342591\pi$
$457$	20.2918	0.949210	0.474605	+	0.880199i	$0.342591\pi$
$458$	0	0
$459$	−5.23607	−0.244399
$460$	0	0
$461$	30.0000	1.39724	0.698620	−	0.715493i	$-0.253798\pi$
$461$	30.0000	1.39724	0.698620	+	0.715493i	$0.253798\pi$
$462$	0	0
$463$	5.70820	0.265283	0.132641	−	0.991164i	$-0.457654\pi$
$463$	5.70820	0.265283	0.132641	+	0.991164i	$0.457654\pi$
$464$	0	0
$465$	8.94427	0.414781
$466$	0	0
$467$	8.94427	0.413892	0.206946	−	0.978352i	$-0.433648\pi$
$467$	8.94427	0.413892	0.206946	+	0.978352i	$0.433648\pi$
$468$	0	0
$469$	−6.11146	−0.282201
$470$	0	0
$471$	13.4164	0.618195
$472$	0	0
$473$	−83.7771	−3.85207
$474$	0	0
$475$	−0.763932	−0.0350516
$476$	0	0
$477$	0.472136	0.0216176
$478$	0	0
$479$	30.8328	1.40879	0.704394	−	0.709810i	$-0.251219\pi$
$479$	30.8328	1.40879	0.704394	+	0.709810i	$0.251219\pi$
$480$	0	0
$481$	−0.472136	−0.0215275
$482$	0	0
$483$	−5.52786	−0.251527
$484$	0	0
$485$	−16.6525	−0.756150
$486$	0	0
$487$	−3.81966	−0.173085	−0.0865427	−	0.996248i	$-0.527582\pi$
$487$	−3.81966	−0.173085	−0.0865427	+	0.996248i	$0.527582\pi$
$488$	0	0
$489$	−1.52786	−0.0690924
$490$	0	0
$491$	18.2918	0.825497	0.412749	−	0.910845i	$-0.364569\pi$
$491$	18.2918	0.825497	0.412749	+	0.910845i	$0.364569\pi$
$492$	0	0
$493$	48.3607	2.17806
$494$	0	0
$495$	−6.47214	−0.290901
$496$	0	0
$497$	−3.05573	−0.137068
$498$	0	0
$499$	−15.2361	−0.682060	−0.341030	−	0.940052i	$-0.610776\pi$
$499$	−15.2361	−0.682060	−0.341030	+	0.940052i	$0.610776\pi$
$500$	0	0
$501$	3.41641	0.152634
$502$	0	0
$503$	−15.5967	−0.695425	−0.347712	−	0.937601i	$-0.613041\pi$
$503$	−15.5967	−0.695425	−0.347712	+	0.937601i	$0.613041\pi$
$504$	0	0
$505$	15.7082	0.699006
$506$	0	0
$507$	−1.00000	−0.0444116
$508$	0	0
$509$	31.8885	1.41343	0.706717	−	0.707496i	$-0.250175\pi$
$509$	31.8885	1.41343	0.706717	+	0.707496i	$0.250175\pi$
$510$	0	0
$511$	2.83282	0.125316
$512$	0	0
$513$	0.763932	0.0337284
$514$	0	0
$515$	−4.00000	−0.176261
$516$	0	0
$517$	−32.0000	−1.40736
$518$	0	0
$519$	4.47214	0.196305
$520$	0	0
$521$	26.3607	1.15488	0.577441	−	0.816432i	$-0.304051\pi$
$521$	26.3607	1.15488	0.577441	+	0.816432i	$0.304051\pi$
$522$	0	0
$523$	−10.8328	−0.473686	−0.236843	−	0.971548i	$-0.576113\pi$
$523$	−10.8328	−0.473686	−0.236843	+	0.971548i	$0.576113\pi$
$524$	0	0
$525$	0.763932	0.0333407
$526$	0	0
$527$	−46.8328	−2.04007
$528$	0	0
$529$	29.3607	1.27655
$530$	0	0
$531$	6.47214	0.280867
$532$	0	0
$533$	12.4721	0.540228
$534$	0	0
$535$	12.0000	0.518805
$536$	0	0
$537$	−16.7639	−0.723417
$538$	0	0
$539$	41.5279	1.78873
$540$	0	0
$541$	−11.1246	−0.478284	−0.239142	−	0.970985i	$-0.576866\pi$
$541$	−11.1246	−0.478284	−0.239142	+	0.970985i	$0.576866\pi$
$542$	0	0
$543$	−20.4721	−0.878543
$544$	0	0
$545$	10.1803	0.436078
$546$	0	0
$547$	−29.8885	−1.27794	−0.638971	−	0.769231i	$-0.720640\pi$
$547$	−29.8885	−1.27794	−0.638971	+	0.769231i	$0.720640\pi$
$548$	0	0
$549$	−6.94427	−0.296374
$550$	0	0
$551$	−7.05573	−0.300584
$552$	0	0
$553$	−9.88854	−0.420504
$554$	0	0
$555$	−0.472136	−0.0200411
$556$	0	0
$557$	31.8885	1.35116	0.675580	−	0.737286i	$-0.263893\pi$
$557$	31.8885	1.35116	0.675580	+	0.737286i	$0.263893\pi$
$558$	0	0
$559$	−12.9443	−0.547484
$560$	0	0
$561$	33.8885	1.43078
$562$	0	0
$563$	25.3050	1.06648	0.533238	−	0.845965i	$-0.320975\pi$
$563$	25.3050	1.06648	0.533238	+	0.845965i	$0.320975\pi$
$564$	0	0
$565$	6.76393	0.284561
$566$	0	0
$567$	−0.763932	−0.0320821
$568$	0	0
$569$	18.3607	0.769720	0.384860	−	0.922975i	$-0.374250\pi$
$569$	18.3607	0.769720	0.384860	+	0.922975i	$0.374250\pi$
$570$	0	0
$571$	25.3050	1.05898	0.529490	−	0.848316i	$-0.322383\pi$
$571$	25.3050	1.05898	0.529490	+	0.848316i	$0.322383\pi$
$572$	0	0
$573$	−20.9443	−0.874960
$574$	0	0
$575$	−7.23607	−0.301765
$576$	0	0
$577$	41.5967	1.73170	0.865848	−	0.500308i	$-0.166780\pi$
$577$	41.5967	1.73170	0.865848	+	0.500308i	$0.166780\pi$
$578$	0	0
$579$	2.18034	0.0906118
$580$	0	0
$581$	6.11146	0.253546
$582$	0	0
$583$	−3.05573	−0.126555
$584$	0	0
$585$	−1.00000	−0.0413449
$586$	0	0
$587$	−0.944272	−0.0389743	−0.0194871	−	0.999810i	$-0.506203\pi$
$587$	−0.944272	−0.0389743	−0.0194871	+	0.999810i	$0.506203\pi$
$588$	0	0
$589$	6.83282	0.281541
$590$	0	0
$591$	9.41641	0.387339
$592$	0	0
$593$	−6.00000	−0.246390	−0.123195	−	0.992382i	$-0.539314\pi$
$593$	−6.00000	−0.246390	−0.123195	+	0.992382i	$0.539314\pi$
$594$	0	0
$595$	−4.00000	−0.163984
$596$	0	0
$597$	−9.88854	−0.404711
$598$	0	0
$599$	16.0000	0.653742	0.326871	−	0.945069i	$-0.394006\pi$
$599$	16.0000	0.653742	0.326871	+	0.945069i	$0.394006\pi$
$600$	0	0
$601$	−2.36068	−0.0962941	−0.0481471	−	0.998840i	$-0.515332\pi$
$601$	−2.36068	−0.0962941	−0.0481471	+	0.998840i	$0.515332\pi$
$602$	0	0
$603$	8.00000	0.325785
$604$	0	0
$605$	30.8885	1.25580
$606$	0	0
$607$	0.944272	0.0383268	0.0191634	−	0.999816i	$-0.493900\pi$
$607$	0.944272	0.0383268	0.0191634	+	0.999816i	$0.493900\pi$
$608$	0	0
$609$	7.05573	0.285913
$610$	0	0
$611$	−4.94427	−0.200024
$612$	0	0
$613$	−2.58359	−0.104350	−0.0521752	−	0.998638i	$-0.516615\pi$
$613$	−2.58359	−0.104350	−0.0521752	+	0.998638i	$0.516615\pi$
$614$	0	0
$615$	12.4721	0.502925
$616$	0	0
$617$	−9.41641	−0.379090	−0.189545	−	0.981872i	$-0.560701\pi$
$617$	−9.41641	−0.379090	−0.189545	+	0.981872i	$0.560701\pi$
$618$	0	0
$619$	34.6525	1.39280	0.696400	−	0.717654i	$-0.254784\pi$
$619$	34.6525	1.39280	0.696400	+	0.717654i	$0.254784\pi$
$620$	0	0
$621$	7.23607	0.290373
$622$	0	0
$623$	−2.24922	−0.0901132
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−4.94427	−0.197455
$628$	0	0
$629$	2.47214	0.0985705
$630$	0	0
$631$	−23.4164	−0.932192	−0.466096	−	0.884734i	$-0.654340\pi$
$631$	−23.4164	−0.932192	−0.466096	+	0.884734i	$0.654340\pi$
$632$	0	0
$633$	0.944272	0.0375314
$634$	0	0
$635$	−4.00000	−0.158735
$636$	0	0
$637$	6.41641	0.254227
$638$	0	0
$639$	4.00000	0.158238
$640$	0	0
$641$	26.0000	1.02694	0.513469	−	0.858108i	$-0.328360\pi$
$641$	26.0000	1.02694	0.513469	+	0.858108i	$0.328360\pi$
$642$	0	0
$643$	8.00000	0.315489	0.157745	−	0.987480i	$-0.449578\pi$
$643$	8.00000	0.315489	0.157745	+	0.987480i	$0.449578\pi$
$644$	0	0
$645$	−12.9443	−0.509680
$646$	0	0
$647$	−29.7082	−1.16795	−0.583975	−	0.811772i	$-0.698503\pi$
$647$	−29.7082	−1.16795	−0.583975	+	0.811772i	$0.698503\pi$
$648$	0	0
$649$	−41.8885	−1.64427
$650$	0	0
$651$	−6.83282	−0.267799
$652$	0	0
$653$	26.0000	1.01746	0.508729	−	0.860927i	$-0.330115\pi$
$653$	26.0000	1.01746	0.508729	+	0.860927i	$0.330115\pi$
$654$	0	0
$655$	−10.6525	−0.416227
$656$	0	0
$657$	−3.70820	−0.144671
$658$	0	0
$659$	26.2918	1.02418	0.512091	−	0.858931i	$-0.328871\pi$
$659$	26.2918	1.02418	0.512091	+	0.858931i	$0.328871\pi$
$660$	0	0
$661$	−27.1246	−1.05503	−0.527513	−	0.849547i	$-0.676875\pi$
$661$	−27.1246	−1.05503	−0.527513	+	0.849547i	$0.676875\pi$
$662$	0	0
$663$	5.23607	0.203352
$664$	0	0
$665$	0.583592	0.0226307
$666$	0	0
$667$	−66.8328	−2.58778
$668$	0	0
$669$	12.1803	0.470919
$670$	0	0
$671$	44.9443	1.73505
$672$	0	0
$673$	5.05573	0.194884	0.0974420	−	0.995241i	$-0.468934\pi$
$673$	5.05573	0.194884	0.0974420	+	0.995241i	$0.468934\pi$
$674$	0	0
$675$	−1.00000	−0.0384900
$676$	0	0
$677$	−30.0000	−1.15299	−0.576497	−	0.817099i	$-0.695581\pi$
$677$	−30.0000	−1.15299	−0.576497	+	0.817099i	$0.695581\pi$
$678$	0	0
$679$	12.7214	0.488201
$680$	0	0
$681$	−21.8885	−0.838771
$682$	0	0
$683$	49.8885	1.90893	0.954466	−	0.298320i	$-0.0964261\pi$
$683$	49.8885	1.90893	0.954466	+	0.298320i	$0.0964261\pi$
$684$	0	0
$685$	8.47214	0.323704
$686$	0	0
$687$	2.76393	0.105451
$688$	0	0
$689$	−0.472136	−0.0179869
$690$	0	0
$691$	−21.7082	−0.825819	−0.412909	−	0.910772i	$-0.635487\pi$
$691$	−21.7082	−0.825819	−0.412909	+	0.910772i	$0.635487\pi$
$692$	0	0
$693$	4.94427	0.187817
$694$	0	0
$695$	2.47214	0.0937735
$696$	0	0
$697$	−65.3050	−2.47360
$698$	0	0
$699$	10.7639	0.407129
$700$	0	0
$701$	18.7639	0.708704	0.354352	−	0.935112i	$-0.384701\pi$
$701$	18.7639	0.708704	0.354352	+	0.935112i	$0.384701\pi$
$702$	0	0
$703$	−0.360680	−0.0136033
$704$	0	0
$705$	−4.94427	−0.186212
$706$	0	0
$707$	−12.0000	−0.451306
$708$	0	0
$709$	−44.6525	−1.67696	−0.838479	−	0.544933i	$-0.816555\pi$
$709$	−44.6525	−1.67696	−0.838479	+	0.544933i	$0.816555\pi$
$710$	0	0
$711$	12.9443	0.485448
$712$	0	0
$713$	64.7214	2.42383
$714$	0	0
$715$	6.47214	0.242044
$716$	0	0
$717$	−11.0557	−0.412884
$718$	0	0
$719$	−25.5279	−0.952029	−0.476014	−	0.879438i	$-0.657919\pi$
$719$	−25.5279	−0.952029	−0.476014	+	0.879438i	$0.657919\pi$
$720$	0	0
$721$	3.05573	0.113801
$722$	0	0
$723$	−10.0000	−0.371904
$724$	0	0
$725$	9.23607	0.343019
$726$	0	0
$727$	5.52786	0.205017	0.102509	−	0.994732i	$-0.467313\pi$
$727$	5.52786	0.205017	0.102509	+	0.994732i	$0.467313\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	67.7771	2.50683
$732$	0	0
$733$	−7.52786	−0.278048	−0.139024	−	0.990289i	$-0.544397\pi$
$733$	−7.52786	−0.278048	−0.139024	+	0.990289i	$0.544397\pi$
$734$	0	0
$735$	6.41641	0.236673
$736$	0	0
$737$	−51.7771	−1.90723
$738$	0	0
$739$	−20.1803	−0.742346	−0.371173	−	0.928564i	$-0.621044\pi$
$739$	−20.1803	−0.742346	−0.371173	+	0.928564i	$0.621044\pi$
$740$	0	0
$741$	−0.763932	−0.0280637
$742$	0	0
$743$	20.5836	0.755139	0.377569	−	0.925981i	$-0.376760\pi$
$743$	20.5836	0.755139	0.377569	+	0.925981i	$0.376760\pi$
$744$	0	0
$745$	4.47214	0.163846
$746$	0	0
$747$	−8.00000	−0.292705
$748$	0	0
$749$	−9.16718	−0.334962
$750$	0	0
$751$	−20.0000	−0.729810	−0.364905	−	0.931045i	$-0.618899\pi$
$751$	−20.0000	−0.729810	−0.364905	+	0.931045i	$0.618899\pi$
$752$	0	0
$753$	−13.7082	−0.499555
$754$	0	0
$755$	0.944272	0.0343656
$756$	0	0
$757$	−26.9443	−0.979306	−0.489653	−	0.871917i	$-0.662877\pi$
$757$	−26.9443	−0.979306	−0.489653	+	0.871917i	$0.662877\pi$
$758$	0	0
$759$	−46.8328	−1.69992
$760$	0	0
$761$	−2.58359	−0.0936551	−0.0468276	−	0.998903i	$-0.514911\pi$
$761$	−2.58359	−0.0936551	−0.0468276	+	0.998903i	$0.514911\pi$
$762$	0	0
$763$	−7.77709	−0.281549
$764$	0	0
$765$	5.23607	0.189310
$766$	0	0
$767$	−6.47214	−0.233695
$768$	0	0
$769$	32.8328	1.18398	0.591991	−	0.805945i	$-0.298342\pi$
$769$	32.8328	1.18398	0.591991	+	0.805945i	$0.298342\pi$
$770$	0	0
$771$	−19.7082	−0.709774
$772$	0	0
$773$	−46.3607	−1.66748	−0.833739	−	0.552159i	$-0.813804\pi$
$773$	−46.3607	−1.66748	−0.833739	+	0.552159i	$0.813804\pi$
$774$	0	0
$775$	−8.94427	−0.321288
$776$	0	0
$777$	0.360680	0.0129393
$778$	0	0
$779$	9.52786	0.341371
$780$	0	0
$781$	−25.8885	−0.926365
$782$	0	0
$783$	−9.23607	−0.330070
$784$	0	0
$785$	−13.4164	−0.478852
$786$	0	0
$787$	−32.3607	−1.15353	−0.576767	−	0.816909i	$-0.695686\pi$
$787$	−32.3607	−1.15353	−0.576767	+	0.816909i	$0.695686\pi$
$788$	0	0
$789$	18.6525	0.664046
$790$	0	0
$791$	−5.16718	−0.183724
$792$	0	0
$793$	6.94427	0.246598
$794$	0	0
$795$	−0.472136	−0.0167449
$796$	0	0
$797$	−18.5836	−0.658265	−0.329132	−	0.944284i	$-0.606756\pi$
$797$	−18.5836	−0.658265	−0.329132	+	0.944284i	$0.606756\pi$
$798$	0	0
$799$	25.8885	0.915871
$800$	0	0
$801$	2.94427	0.104031
$802$	0	0
$803$	24.0000	0.846942
$804$	0	0
$805$	5.52786	0.194832
$806$	0	0
$807$	2.18034	0.0767516
$808$	0	0
$809$	−46.7214	−1.64264	−0.821318	−	0.570471i	$-0.806761\pi$
$809$	−46.7214	−1.64264	−0.821318	+	0.570471i	$0.806761\pi$
$810$	0	0
$811$	−5.70820	−0.200442	−0.100221	−	0.994965i	$-0.531955\pi$
$811$	−5.70820	−0.200442	−0.100221	+	0.994965i	$0.531955\pi$
$812$	0	0
$813$	21.8885	0.767665
$814$	0	0
$815$	1.52786	0.0535187
$816$	0	0
$817$	−9.88854	−0.345956
$818$	0	0
$819$	0.763932	0.0266939
$820$	0	0
$821$	51.3050	1.79056	0.895278	−	0.445509i	$-0.146977\pi$
$821$	51.3050	1.79056	0.895278	+	0.445509i	$0.146977\pi$
$822$	0	0
$823$	−47.7771	−1.66540	−0.832702	−	0.553721i	$-0.813207\pi$
$823$	−47.7771	−1.66540	−0.832702	+	0.553721i	$0.813207\pi$
$824$	0	0
$825$	6.47214	0.225331
$826$	0	0
$827$	−3.05573	−0.106258	−0.0531290	−	0.998588i	$-0.516919\pi$
$827$	−3.05573	−0.106258	−0.0531290	+	0.998588i	$0.516919\pi$
$828$	0	0
$829$	−43.8885	−1.52431	−0.762156	−	0.647393i	$-0.775859\pi$
$829$	−43.8885	−1.52431	−0.762156	+	0.647393i	$0.775859\pi$
$830$	0	0
$831$	3.52786	0.122380
$832$	0	0
$833$	−33.5967	−1.16406
$834$	0	0
$835$	−3.41641	−0.118230
$836$	0	0
$837$	8.94427	0.309159
$838$	0	0
$839$	17.8885	0.617581	0.308791	−	0.951130i	$-0.400076\pi$
$839$	17.8885	0.617581	0.308791	+	0.951130i	$0.400076\pi$
$840$	0	0
$841$	56.3050	1.94155
$842$	0	0
$843$	−7.88854	−0.271696
$844$	0	0
$845$	1.00000	0.0344010
$846$	0	0
$847$	−23.5967	−0.810794
$848$	0	0
$849$	−16.0000	−0.549119
$850$	0	0
$851$	−3.41641	−0.117113
$852$	0	0
$853$	−17.4164	−0.596326	−0.298163	−	0.954515i	$-0.596374\pi$
$853$	−17.4164	−0.596326	−0.298163	+	0.954515i	$0.596374\pi$
$854$	0	0
$855$	−0.763932	−0.0261259
$856$	0	0
$857$	−19.1246	−0.653284	−0.326642	−	0.945148i	$-0.605917\pi$
$857$	−19.1246	−0.653284	−0.326642	+	0.945148i	$0.605917\pi$
$858$	0	0
$859$	−16.9443	−0.578131	−0.289066	−	0.957309i	$-0.593345\pi$
$859$	−16.9443	−0.578131	−0.289066	+	0.957309i	$0.593345\pi$
$860$	0	0
$861$	−9.52786	−0.324709
$862$	0	0
$863$	−27.4164	−0.933265	−0.466633	−	0.884451i	$-0.654533\pi$
$863$	−27.4164	−0.933265	−0.466633	+	0.884451i	$0.654533\pi$
$864$	0	0
$865$	−4.47214	−0.152057
$866$	0	0
$867$	−10.4164	−0.353760
$868$	0	0
$869$	−83.7771	−2.84194
$870$	0	0
$871$	−8.00000	−0.271070
$872$	0	0
$873$	−16.6525	−0.563601
$874$	0	0
$875$	−0.763932	−0.0258256
$876$	0	0
$877$	−38.0000	−1.28317	−0.641584	−	0.767052i	$-0.721723\pi$
$877$	−38.0000	−1.28317	−0.641584	+	0.767052i	$0.721723\pi$
$878$	0	0
$879$	12.4721	0.420675
$880$	0	0
$881$	6.58359	0.221807	0.110903	−	0.993831i	$-0.464626\pi$
$881$	6.58359	0.221807	0.110903	+	0.993831i	$0.464626\pi$
$882$	0	0
$883$	−21.8885	−0.736608	−0.368304	−	0.929705i	$-0.620061\pi$
$883$	−21.8885	−0.736608	−0.368304	+	0.929705i	$0.620061\pi$
$884$	0	0
$885$	−6.47214	−0.217558
$886$	0	0
$887$	−14.0689	−0.472387	−0.236193	−	0.971706i	$-0.575900\pi$
$887$	−14.0689	−0.472387	−0.236193	+	0.971706i	$0.575900\pi$
$888$	0	0
$889$	3.05573	0.102486
$890$	0	0
$891$	−6.47214	−0.216825
$892$	0	0
$893$	−3.77709	−0.126395
$894$	0	0
$895$	16.7639	0.560356
$896$	0	0
$897$	−7.23607	−0.241605
$898$	0	0
$899$	−82.6099	−2.75519
$900$	0	0
$901$	2.47214	0.0823588
$902$	0	0
$903$	9.88854	0.329070
$904$	0	0
$905$	20.4721	0.680517
$906$	0	0
$907$	−24.0000	−0.796907	−0.398453	−	0.917189i	$-0.630453\pi$
$907$	−24.0000	−0.796907	−0.398453	+	0.917189i	$0.630453\pi$
$908$	0	0
$909$	15.7082	0.521008
$910$	0	0
$911$	−54.4721	−1.80474	−0.902371	−	0.430960i	$-0.858175\pi$
$911$	−54.4721	−1.80474	−0.902371	+	0.430960i	$0.858175\pi$
$912$	0	0
$913$	51.7771	1.71357
$914$	0	0
$915$	6.94427	0.229571
$916$	0	0
$917$	8.13777	0.268733
$918$	0	0
$919$	−15.7771	−0.520438	−0.260219	−	0.965550i	$-0.583795\pi$
$919$	−15.7771	−0.520438	−0.260219	+	0.965550i	$0.583795\pi$
$920$	0	0
$921$	−3.05573	−0.100690
$922$	0	0
$923$	−4.00000	−0.131662
$924$	0	0
$925$	0.472136	0.0155237
$926$	0	0
$927$	−4.00000	−0.131377
$928$	0	0
$929$	47.8885	1.57117	0.785586	−	0.618752i	$-0.212362\pi$
$929$	47.8885	1.57117	0.785586	+	0.618752i	$0.212362\pi$
$930$	0	0
$931$	4.90170	0.160647
$932$	0	0
$933$	20.9443	0.685685
$934$	0	0
$935$	−33.8885	−1.10827
$936$	0	0
$937$	−54.0000	−1.76410	−0.882052	−	0.471153i	$-0.843838\pi$
$937$	−54.0000	−1.76410	−0.882052	+	0.471153i	$0.843838\pi$
$938$	0	0
$939$	−21.4164	−0.698898
$940$	0	0
$941$	15.5279	0.506194	0.253097	−	0.967441i	$-0.418551\pi$
$941$	15.5279	0.506194	0.253097	+	0.967441i	$0.418551\pi$
$942$	0	0
$943$	90.2492	2.93892
$944$	0	0
$945$	0.763932	0.0248507
$946$	0	0
$947$	−1.88854	−0.0613694	−0.0306847	−	0.999529i	$-0.509769\pi$
$947$	−1.88854	−0.0613694	−0.0306847	+	0.999529i	$0.509769\pi$
$948$	0	0
$949$	3.70820	0.120373
$950$	0	0
$951$	2.00000	0.0648544
$952$	0	0
$953$	42.1803	1.36636	0.683178	−	0.730252i	$-0.260597\pi$
$953$	42.1803	1.36636	0.683178	+	0.730252i	$0.260597\pi$
$954$	0	0
$955$	20.9443	0.677741
$956$	0	0
$957$	59.7771	1.93232
$958$	0	0
$959$	−6.47214	−0.208996
$960$	0	0
$961$	49.0000	1.58065
$962$	0	0
$963$	12.0000	0.386695
$964$	0	0
$965$	−2.18034	−0.0701876
$966$	0	0
$967$	−0.763932	−0.0245664	−0.0122832	−	0.999925i	$-0.503910\pi$
$967$	−0.763932	−0.0245664	−0.0122832	+	0.999925i	$0.503910\pi$
$968$	0	0
$969$	4.00000	0.128499
$970$	0	0
$971$	60.5410	1.94285	0.971427	−	0.237339i	$-0.0762752\pi$
$971$	60.5410	1.94285	0.971427	+	0.237339i	$0.0762752\pi$
$972$	0	0
$973$	−1.88854	−0.0605439
$974$	0	0
$975$	1.00000	0.0320256
$976$	0	0
$977$	13.7771	0.440768	0.220384	−	0.975413i	$-0.429269\pi$
$977$	13.7771	0.440768	0.220384	+	0.975413i	$0.429269\pi$
$978$	0	0
$979$	−19.0557	−0.609024
$980$	0	0
$981$	10.1803	0.325033
$982$	0	0
$983$	−19.4164	−0.619287	−0.309644	−	0.950853i	$-0.600210\pi$
$983$	−19.4164	−0.619287	−0.309644	+	0.950853i	$0.600210\pi$
$984$	0	0
$985$	−9.41641	−0.300032
$986$	0	0
$987$	3.77709	0.120226
$988$	0	0
$989$	−93.6656	−2.97839
$990$	0	0
$991$	27.0557	0.859454	0.429727	−	0.902959i	$-0.358610\pi$
$991$	27.0557	0.859454	0.429727	+	0.902959i	$0.358610\pi$
$992$	0	0
$993$	13.7082	0.435017
$994$	0	0
$995$	9.88854	0.313488
$996$	0	0
$997$	−16.4721	−0.521678	−0.260839	−	0.965382i	$-0.583999\pi$
$997$	−16.4721	−0.521678	−0.260839	+	0.965382i	$0.583999\pi$
$998$	0	0
$999$	−0.472136	−0.0149377

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6240.2.a.bl.1.2	✓	2
4.3	odd	2		6240.2.a.bv.1.1	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6240.2.a.bl.1.2	✓	2	1.1	even	1	trivial
6240.2.a.bv.1.1	yes	2	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:	\( N \)	\(=\)	\( 6240 = 2^{5} \cdot 3 \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6240.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +1.00000 q^{5} -0.763932 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +1.00000 q^{5} -0.763932 q^{7} +1.00000 q^{9} -6.47214 q^{11} -1.00000 q^{13} -1.00000 q^{15} +5.23607 q^{17} -0.763932 q^{19} +0.763932 q^{21} -7.23607 q^{23} +1.00000 q^{25} -1.00000 q^{27} +9.23607 q^{29} -8.94427 q^{31} +6.47214 q^{33} -0.763932 q^{35} +0.472136 q^{37} +1.00000 q^{39} -12.4721 q^{41} +12.9443 q^{43} +1.00000 q^{45} +4.94427 q^{47} -6.41641 q^{49} -5.23607 q^{51} +0.472136 q^{53} -6.47214 q^{55} +0.763932 q^{57} +6.47214 q^{59} -6.94427 q^{61} -0.763932 q^{63} -1.00000 q^{65} +8.00000 q^{67} +7.23607 q^{69} +4.00000 q^{71} -3.70820 q^{73} -1.00000 q^{75} +4.94427 q^{77} +12.9443 q^{79} +1.00000 q^{81} -8.00000 q^{83} +5.23607 q^{85} -9.23607 q^{87} +2.94427 q^{89} +0.763932 q^{91} +8.94427 q^{93} -0.763932 q^{95} -16.6525 q^{97} -6.47214 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} + 2 q^{5} - 6 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^3 + 2 * q^5 - 6 * q^7 + 2 * q^9	\( 2 q - 2 q^{3} + 2 q^{5} - 6 q^{7} + 2 q^{9} - 4 q^{11} - 2 q^{13} - 2 q^{15} + 6 q^{17} - 6 q^{19} + 6 q^{21} - 10 q^{23} + 2 q^{25} - 2 q^{27} + 14 q^{29} + 4 q^{33} - 6 q^{35} - 8 q^{37} + 2 q^{39} - 16 q^{41} + 8 q^{43} + 2 q^{45} - 8 q^{47} + 14 q^{49} - 6 q^{51} - 8 q^{53} - 4 q^{55} + 6 q^{57} + 4 q^{59} + 4 q^{61} - 6 q^{63} - 2 q^{65} + 16 q^{67} + 10 q^{69} + 8 q^{71} + 6 q^{73} - 2 q^{75} - 8 q^{77} + 8 q^{79} + 2 q^{81} - 16 q^{83} + 6 q^{85} - 14 q^{87} - 12 q^{89} + 6 q^{91} - 6 q^{95} - 2 q^{97} - 4 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 + 2 * q^5 - 6 * q^7 + 2 * q^9 - 4 * q^11 - 2 * q^13 - 2 * q^15 + 6 * q^17 - 6 * q^19 + 6 * q^21 - 10 * q^23 + 2 * q^25 - 2 * q^27 + 14 * q^29 + 4 * q^33 - 6 * q^35 - 8 * q^37 + 2 * q^39 - 16 * q^41 + 8 * q^43 + 2 * q^45 - 8 * q^47 + 14 * q^49 - 6 * q^51 - 8 * q^53 - 4 * q^55 + 6 * q^57 + 4 * q^59 + 4 * q^61 - 6 * q^63 - 2 * q^65 + 16 * q^67 + 10 * q^69 + 8 * q^71 + 6 * q^73 - 2 * q^75 - 8 * q^77 + 8 * q^79 + 2 * q^81 - 16 * q^83 + 6 * q^85 - 14 * q^87 - 12 * q^89 + 6 * q^91 - 6 * q^95 - 2 * q^97 - 4 * q^99

Introduction

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