LMFDB - Embedded newform 1305.2.a.k.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1305,2,Mod(1,1305)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1305, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1305.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1305 = 3^{2} \cdot 5 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1305.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:	$10.4204774638$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 435)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-0.618034$ of defining polynomial
Character	$\chi$	$=$	1305.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.618034 q^{2} -1.61803 q^{4} +1.00000 q^{5} -3.00000 q^{7} +2.23607 q^{8} +O(q^{10})$	$q-0.618034 q^{2} -1.61803 q^{4} +1.00000 q^{5} -3.00000 q^{7} +2.23607 q^{8} -0.618034 q^{10} +5.47214 q^{11} -6.23607 q^{13} +1.85410 q^{14} +1.85410 q^{16} -3.47214 q^{17} +7.70820 q^{19} -1.61803 q^{20} -3.38197 q^{22} +1.00000 q^{25} +3.85410 q^{26} +4.85410 q^{28} -1.00000 q^{29} -8.00000 q^{31} -5.61803 q^{32} +2.14590 q^{34} -3.00000 q^{35} -8.00000 q^{37} -4.76393 q^{38} +2.23607 q^{40} +4.47214 q^{41} +3.23607 q^{43} -8.85410 q^{44} -6.70820 q^{47} +2.00000 q^{49} -0.618034 q^{50} +10.0902 q^{52} +6.76393 q^{53} +5.47214 q^{55} -6.70820 q^{56} +0.618034 q^{58} -5.23607 q^{59} -5.70820 q^{61} +4.94427 q^{62} -0.236068 q^{64} -6.23607 q^{65} -11.4721 q^{67} +5.61803 q^{68} +1.85410 q^{70} -7.23607 q^{71} -8.00000 q^{73} +4.94427 q^{74} -12.4721 q^{76} -16.4164 q^{77} -6.18034 q^{79} +1.85410 q^{80} -2.76393 q^{82} +3.70820 q^{83} -3.47214 q^{85} -2.00000 q^{86} +12.2361 q^{88} -11.1803 q^{89} +18.7082 q^{91} +4.14590 q^{94} +7.70820 q^{95} +2.76393 q^{97} -1.23607 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{2} - q^{4} + 2 q^{5} - 6 q^{7}+O(q^{10}) $ 2 * q + q^2 - q^4 + 2 * q^5 - 6 * q^7	$ 2 q + q^{2} - q^{4} + 2 q^{5} - 6 q^{7} + q^{10} + 2 q^{11} - 8 q^{13} - 3 q^{14} - 3 q^{16} + 2 q^{17} + 2 q^{19} - q^{20} - 9 q^{22} + 2 q^{25} + q^{26} + 3 q^{28} - 2 q^{29} - 16 q^{31} - 9 q^{32} + 11 q^{34} - 6 q^{35} - 16 q^{37} - 14 q^{38} + 2 q^{43} - 11 q^{44} + 4 q^{49} + q^{50} + 9 q^{52} + 18 q^{53} + 2 q^{55} - q^{58} - 6 q^{59} + 2 q^{61} - 8 q^{62} + 4 q^{64} - 8 q^{65} - 14 q^{67} + 9 q^{68} - 3 q^{70} - 10 q^{71} - 16 q^{73} - 8 q^{74} - 16 q^{76} - 6 q^{77} + 10 q^{79} - 3 q^{80} - 10 q^{82} - 6 q^{83} + 2 q^{85} - 4 q^{86} + 20 q^{88} + 24 q^{91} + 15 q^{94} + 2 q^{95} + 10 q^{97} + 2 q^{98}+O(q^{100}) $ 2 * q + q^2 - q^4 + 2 * q^5 - 6 * q^7 + q^10 + 2 * q^11 - 8 * q^13 - 3 * q^14 - 3 * q^16 + 2 * q^17 + 2 * q^19 - q^20 - 9 * q^22 + 2 * q^25 + q^26 + 3 * q^28 - 2 * q^29 - 16 * q^31 - 9 * q^32 + 11 * q^34 - 6 * q^35 - 16 * q^37 - 14 * q^38 + 2 * q^43 - 11 * q^44 + 4 * q^49 + q^50 + 9 * q^52 + 18 * q^53 + 2 * q^55 - q^58 - 6 * q^59 + 2 * q^61 - 8 * q^62 + 4 * q^64 - 8 * q^65 - 14 * q^67 + 9 * q^68 - 3 * q^70 - 10 * q^71 - 16 * q^73 - 8 * q^74 - 16 * q^76 - 6 * q^77 + 10 * q^79 - 3 * q^80 - 10 * q^82 - 6 * q^83 + 2 * q^85 - 4 * q^86 + 20 * q^88 + 24 * q^91 + 15 * q^94 + 2 * q^95 + 10 * q^97 + 2 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−0.618034	−0.437016	−0.218508	−	0.975835i	$-0.570119\pi$
$2$	−0.618034	−0.437016	−0.218508	+	0.975835i	$0.570119\pi$
$3$	0	0
$3$	0	0
$4$	−1.61803	−0.809017
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−3.00000	−1.13389	−0.566947	−	0.823754i	$-0.691875\pi$
$7$	−3.00000	−1.13389	−0.566947	+	0.823754i	$0.691875\pi$
$8$	2.23607	0.790569
$9$	0	0
$10$	−0.618034	−0.195440
$11$	5.47214	1.64991	0.824956	−	0.565198i	$-0.191200\pi$
$11$	5.47214	1.64991	0.824956	+	0.565198i	$0.191200\pi$
$12$	0	0
$13$	−6.23607	−1.72957	−0.864787	−	0.502139i	$-0.832547\pi$
$13$	−6.23607	−1.72957	−0.864787	+	0.502139i	$0.832547\pi$
$14$	1.85410	0.495530
$15$	0	0
$16$	1.85410	0.463525
$17$	−3.47214	−0.842117	−0.421058	−	0.907034i	$-0.638341\pi$
$17$	−3.47214	−0.842117	−0.421058	+	0.907034i	$0.638341\pi$
$18$	0	0
$19$	7.70820	1.76838	0.884192	−	0.467124i	$-0.154710\pi$
$19$	7.70820	1.76838	0.884192	+	0.467124i	$0.154710\pi$
$20$	−1.61803	−0.361803
$21$	0	0
$22$	−3.38197	−0.721038
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	3.85410	0.755852
$27$	0	0
$28$	4.85410	0.917339
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	−8.00000	−1.43684	−0.718421	−	0.695608i	$-0.755135\pi$
$31$	−8.00000	−1.43684	−0.718421	+	0.695608i	$0.755135\pi$
$32$	−5.61803	−0.993137
$33$	0	0
$34$	2.14590	0.368018
$35$	−3.00000	−0.507093
$36$	0	0
$37$	−8.00000	−1.31519	−0.657596	−	0.753371i	$-0.728427\pi$
$37$	−8.00000	−1.31519	−0.657596	+	0.753371i	$0.728427\pi$
$38$	−4.76393	−0.772812
$39$	0	0
$40$	2.23607	0.353553
$41$	4.47214	0.698430	0.349215	−	0.937043i	$-0.386448\pi$
$41$	4.47214	0.698430	0.349215	+	0.937043i	$0.386448\pi$
$42$	0	0
$43$	3.23607	0.493496	0.246748	−	0.969080i	$-0.420638\pi$
$43$	3.23607	0.493496	0.246748	+	0.969080i	$0.420638\pi$
$44$	−8.85410	−1.33481
$45$	0	0
$46$	0	0
$47$	−6.70820	−0.978492	−0.489246	−	0.872146i	$-0.662728\pi$
$47$	−6.70820	−0.978492	−0.489246	+	0.872146i	$0.662728\pi$
$48$	0	0
$49$	2.00000	0.285714
$50$	−0.618034	−0.0874032
$51$	0	0
$52$	10.0902	1.39925
$53$	6.76393	0.929098	0.464549	−	0.885548i	$-0.346217\pi$
$53$	6.76393	0.929098	0.464549	+	0.885548i	$0.346217\pi$
$54$	0	0
$55$	5.47214	0.737863
$56$	−6.70820	−0.896421
$57$	0	0
$58$	0.618034	0.0811518
$59$	−5.23607	−0.681678	−0.340839	−	0.940122i	$-0.610711\pi$
$59$	−5.23607	−0.681678	−0.340839	+	0.940122i	$0.610711\pi$
$60$	0	0
$61$	−5.70820	−0.730861	−0.365430	−	0.930839i	$-0.619078\pi$
$61$	−5.70820	−0.730861	−0.365430	+	0.930839i	$0.619078\pi$
$62$	4.94427	0.627923
$63$	0	0
$64$	−0.236068	−0.0295085
$65$	−6.23607	−0.773489
$66$	0	0
$67$	−11.4721	−1.40154	−0.700772	−	0.713385i	$-0.747161\pi$
$67$	−11.4721	−1.40154	−0.700772	+	0.713385i	$0.747161\pi$
$68$	5.61803	0.681287
$69$	0	0
$70$	1.85410	0.221608
$71$	−7.23607	−0.858763	−0.429382	−	0.903123i	$-0.641268\pi$
$71$	−7.23607	−0.858763	−0.429382	+	0.903123i	$0.641268\pi$
$72$	0	0
$73$	−8.00000	−0.936329	−0.468165	−	0.883641i	$-0.655085\pi$
$73$	−8.00000	−0.936329	−0.468165	+	0.883641i	$0.655085\pi$
$74$	4.94427	0.574760
$75$	0	0
$76$	−12.4721	−1.43065
$77$	−16.4164	−1.87082
$78$	0	0
$79$	−6.18034	−0.695343	−0.347671	−	0.937616i	$-0.613027\pi$
$79$	−6.18034	−0.695343	−0.347671	+	0.937616i	$0.613027\pi$
$80$	1.85410	0.207295
$81$	0	0
$82$	−2.76393	−0.305225
$83$	3.70820	0.407028	0.203514	−	0.979072i	$-0.434764\pi$
$83$	3.70820	0.407028	0.203514	+	0.979072i	$0.434764\pi$
$84$	0	0
$85$	−3.47214	−0.376606
$86$	−2.00000	−0.215666
$87$	0	0
$88$	12.2361	1.30437
$89$	−11.1803	−1.18511	−0.592557	−	0.805529i	$-0.701881\pi$
$89$	−11.1803	−1.18511	−0.592557	+	0.805529i	$0.701881\pi$
$90$	0	0
$91$	18.7082	1.96115
$92$	0	0
$93$	0	0
$94$	4.14590	0.427617
$95$	7.70820	0.790845
$96$	0	0
$97$	2.76393	0.280635	0.140317	−	0.990107i	$-0.455188\pi$
$97$	2.76393	0.280635	0.140317	+	0.990107i	$0.455188\pi$
$98$	−1.23607	−0.124862
$99$	0	0
$100$	−1.61803	−0.161803
$101$	−4.23607	−0.421505	−0.210752	−	0.977540i	$-0.567591\pi$
$101$	−4.23607	−0.421505	−0.210752	+	0.977540i	$0.567591\pi$
$102$	0	0
$103$	7.41641	0.730760	0.365380	−	0.930858i	$-0.380939\pi$
$103$	7.41641	0.730760	0.365380	+	0.930858i	$0.380939\pi$
$104$	−13.9443	−1.36735
$105$	0	0
$106$	−4.18034	−0.406031
$107$	−7.52786	−0.727746	−0.363873	−	0.931449i	$-0.618546\pi$
$107$	−7.52786	−0.727746	−0.363873	+	0.931449i	$0.618546\pi$
$108$	0	0
$109$	−16.4164	−1.57241	−0.786203	−	0.617968i	$-0.787956\pi$
$109$	−16.4164	−1.57241	−0.786203	+	0.617968i	$0.787956\pi$
$110$	−3.38197	−0.322458
$111$	0	0
$112$	−5.56231	−0.525589
$113$	7.94427	0.747334	0.373667	−	0.927563i	$-0.378100\pi$
$113$	7.94427	0.747334	0.373667	+	0.927563i	$0.378100\pi$
$114$	0	0
$115$	0	0
$116$	1.61803	0.150231
$117$	0	0
$118$	3.23607	0.297904
$119$	10.4164	0.954871
$120$	0	0
$121$	18.9443	1.72221
$122$	3.52786	0.319398
$123$	0	0
$124$	12.9443	1.16243
$125$	1.00000	0.0894427
$126$	0	0
$127$	−6.00000	−0.532414	−0.266207	−	0.963916i	$-0.585770\pi$
$127$	−6.00000	−0.532414	−0.266207	+	0.963916i	$0.585770\pi$
$128$	11.3820	1.00603
$129$	0	0
$130$	3.85410	0.338027
$131$	−3.47214	−0.303362	−0.151681	−	0.988430i	$-0.548469\pi$
$131$	−3.47214	−0.303362	−0.151681	+	0.988430i	$0.548469\pi$
$132$	0	0
$133$	−23.1246	−2.00516
$134$	7.09017	0.612497
$135$	0	0
$136$	−7.76393	−0.665752
$137$	10.9443	0.935032	0.467516	−	0.883985i	$-0.345149\pi$
$137$	10.9443	0.935032	0.467516	+	0.883985i	$0.345149\pi$
$138$	0	0
$139$	−0.708204	−0.0600691	−0.0300345	−	0.999549i	$-0.509562\pi$
$139$	−0.708204	−0.0600691	−0.0300345	+	0.999549i	$0.509562\pi$
$140$	4.85410	0.410246
$141$	0	0
$142$	4.47214	0.375293
$143$	−34.1246	−2.85364
$144$	0	0
$145$	−1.00000	−0.0830455
$146$	4.94427	0.409191
$147$	0	0
$148$	12.9443	1.06401
$149$	−20.1803	−1.65324	−0.826619	−	0.562762i	$-0.809739\pi$
$149$	−20.1803	−1.65324	−0.826619	+	0.562762i	$0.809739\pi$
$150$	0	0
$151$	2.47214	0.201180	0.100590	−	0.994928i	$-0.467927\pi$
$151$	2.47214	0.201180	0.100590	+	0.994928i	$0.467927\pi$
$152$	17.2361	1.39803
$153$	0	0
$154$	10.1459	0.817580
$155$	−8.00000	−0.642575
$156$	0	0
$157$	−11.7082	−0.934416	−0.467208	−	0.884147i	$-0.654740\pi$
$157$	−11.7082	−0.934416	−0.467208	+	0.884147i	$0.654740\pi$
$158$	3.81966	0.303876
$159$	0	0
$160$	−5.61803	−0.444145
$161$	0	0
$162$	0	0
$163$	−15.1246	−1.18465	−0.592326	−	0.805699i	$-0.701790\pi$
$163$	−15.1246	−1.18465	−0.592326	+	0.805699i	$0.701790\pi$
$164$	−7.23607	−0.565042
$165$	0	0
$166$	−2.29180	−0.177878
$167$	17.8885	1.38426	0.692129	−	0.721774i	$-0.256673\pi$
$167$	17.8885	1.38426	0.692129	+	0.721774i	$0.256673\pi$
$168$	0	0
$169$	25.8885	1.99143
$170$	2.14590	0.164583
$171$	0	0
$172$	−5.23607	−0.399246
$173$	19.4164	1.47620	0.738101	−	0.674690i	$-0.235723\pi$
$173$	19.4164	1.47620	0.738101	+	0.674690i	$0.235723\pi$
$174$	0	0
$175$	−3.00000	−0.226779
$176$	10.1459	0.764776
$177$	0	0
$178$	6.90983	0.517914
$179$	−4.18034	−0.312453	−0.156227	−	0.987721i	$-0.549933\pi$
$179$	−4.18034	−0.312453	−0.156227	+	0.987721i	$0.549933\pi$
$180$	0	0
$181$	8.41641	0.625587	0.312793	−	0.949821i	$-0.398735\pi$
$181$	8.41641	0.625587	0.312793	+	0.949821i	$0.398735\pi$
$182$	−11.5623	−0.857055
$183$	0	0
$184$	0	0
$185$	−8.00000	−0.588172
$186$	0	0
$187$	−19.0000	−1.38942
$188$	10.8541	0.791617
$189$	0	0
$190$	−4.76393	−0.345612
$191$	−12.0000	−0.868290	−0.434145	−	0.900843i	$-0.642949\pi$
$191$	−12.0000	−0.868290	−0.434145	+	0.900843i	$0.642949\pi$
$192$	0	0
$193$	−6.00000	−0.431889	−0.215945	−	0.976406i	$-0.569283\pi$
$193$	−6.00000	−0.431889	−0.215945	+	0.976406i	$0.569283\pi$
$194$	−1.70820	−0.122642
$195$	0	0
$196$	−3.23607	−0.231148
$197$	−14.9443	−1.06474	−0.532368	−	0.846513i	$-0.678698\pi$
$197$	−14.9443	−1.06474	−0.532368	+	0.846513i	$0.678698\pi$
$198$	0	0
$199$	20.7082	1.46797	0.733983	−	0.679168i	$-0.237659\pi$
$199$	20.7082	1.46797	0.733983	+	0.679168i	$0.237659\pi$
$200$	2.23607	0.158114
$201$	0	0
$202$	2.61803	0.184204
$203$	3.00000	0.210559
$204$	0	0
$205$	4.47214	0.312348
$206$	−4.58359	−0.319354
$207$	0	0
$208$	−11.5623	−0.801702
$209$	42.1803	2.91768
$210$	0	0
$211$	−27.8885	−1.91993	−0.959963	−	0.280126i	$-0.909624\pi$
$211$	−27.8885	−1.91993	−0.959963	+	0.280126i	$0.909624\pi$
$212$	−10.9443	−0.751656
$213$	0	0
$214$	4.65248	0.318037
$215$	3.23607	0.220698
$216$	0	0
$217$	24.0000	1.62923
$218$	10.1459	0.687167
$219$	0	0
$220$	−8.85410	−0.596943
$221$	21.6525	1.45650
$222$	0	0
$223$	−13.0000	−0.870544	−0.435272	−	0.900299i	$-0.643348\pi$
$223$	−13.0000	−0.870544	−0.435272	+	0.900299i	$0.643348\pi$
$224$	16.8541	1.12611
$225$	0	0
$226$	−4.90983	−0.326597
$227$	5.81966	0.386264	0.193132	−	0.981173i	$-0.438135\pi$
$227$	5.81966	0.386264	0.193132	+	0.981173i	$0.438135\pi$
$228$	0	0
$229$	9.41641	0.622254	0.311127	−	0.950368i	$-0.399294\pi$
$229$	9.41641	0.622254	0.311127	+	0.950368i	$0.399294\pi$
$230$	0	0
$231$	0	0
$232$	−2.23607	−0.146805
$233$	−1.41641	−0.0927920	−0.0463960	−	0.998923i	$-0.514774\pi$
$233$	−1.41641	−0.0927920	−0.0463960	+	0.998923i	$0.514774\pi$
$234$	0	0
$235$	−6.70820	−0.437595
$236$	8.47214	0.551489
$237$	0	0
$238$	−6.43769	−0.417294
$239$	−11.8885	−0.769006	−0.384503	−	0.923124i	$-0.625627\pi$
$239$	−11.8885	−0.769006	−0.384503	+	0.923124i	$0.625627\pi$
$240$	0	0
$241$	7.00000	0.450910	0.225455	−	0.974254i	$-0.427613\pi$
$241$	7.00000	0.450910	0.225455	+	0.974254i	$0.427613\pi$
$242$	−11.7082	−0.752632
$243$	0	0
$244$	9.23607	0.591279
$245$	2.00000	0.127775
$246$	0	0
$247$	−48.0689	−3.05855
$248$	−17.8885	−1.13592
$249$	0	0
$250$	−0.618034	−0.0390879
$251$	24.8885	1.57095	0.785475	−	0.618893i	$-0.212418\pi$
$251$	24.8885	1.57095	0.785475	+	0.618893i	$0.212418\pi$
$252$	0	0
$253$	0	0
$254$	3.70820	0.232673
$255$	0	0
$256$	−6.56231	−0.410144
$257$	9.70820	0.605581	0.302791	−	0.953057i	$-0.402082\pi$
$257$	9.70820	0.605581	0.302791	+	0.953057i	$0.402082\pi$
$258$	0	0
$259$	24.0000	1.49129
$260$	10.0902	0.625766
$261$	0	0
$262$	2.14590	0.132574
$263$	−24.0000	−1.47990	−0.739952	−	0.672660i	$-0.765152\pi$
$263$	−24.0000	−1.47990	−0.739952	+	0.672660i	$0.765152\pi$
$264$	0	0
$265$	6.76393	0.415505
$266$	14.2918	0.876286
$267$	0	0
$268$	18.5623	1.13387
$269$	30.2361	1.84353	0.921763	−	0.387754i	$-0.126749\pi$
$269$	30.2361	1.84353	0.921763	+	0.387754i	$0.126749\pi$
$270$	0	0
$271$	14.3607	0.872349	0.436175	−	0.899862i	$-0.356333\pi$
$271$	14.3607	0.872349	0.436175	+	0.899862i	$0.356333\pi$
$272$	−6.43769	−0.390343
$273$	0	0
$274$	−6.76393	−0.408624
$275$	5.47214	0.329982
$276$	0	0
$277$	8.70820	0.523225	0.261613	−	0.965173i	$-0.415746\pi$
$277$	8.70820	0.523225	0.261613	+	0.965173i	$0.415746\pi$
$278$	0.437694	0.0262511
$279$	0	0
$280$	−6.70820	−0.400892
$281$	23.1246	1.37950	0.689749	−	0.724048i	$-0.257721\pi$
$281$	23.1246	1.37950	0.689749	+	0.724048i	$0.257721\pi$
$282$	0	0
$283$	18.4721	1.09805	0.549027	−	0.835804i	$-0.314998\pi$
$283$	18.4721	1.09805	0.549027	+	0.835804i	$0.314998\pi$
$284$	11.7082	0.694754
$285$	0	0
$286$	21.0902	1.24709
$287$	−13.4164	−0.791946
$288$	0	0
$289$	−4.94427	−0.290840
$290$	0.618034	0.0362922
$291$	0	0
$292$	12.9443	0.757506
$293$	17.9443	1.04832	0.524158	−	0.851621i	$-0.324380\pi$
$293$	17.9443	1.04832	0.524158	+	0.851621i	$0.324380\pi$
$294$	0	0
$295$	−5.23607	−0.304856
$296$	−17.8885	−1.03975
$297$	0	0
$298$	12.4721	0.722491
$299$	0	0
$300$	0	0
$301$	−9.70820	−0.559572
$302$	−1.52786	−0.0879187
$303$	0	0
$304$	14.2918	0.819691
$305$	−5.70820	−0.326851
$306$	0	0
$307$	24.9443	1.42364	0.711822	−	0.702360i	$-0.247870\pi$
$307$	24.9443	1.42364	0.711822	+	0.702360i	$0.247870\pi$
$308$	26.5623	1.51353
$309$	0	0
$310$	4.94427	0.280816
$311$	−23.4721	−1.33098	−0.665491	−	0.746406i	$-0.731778\pi$
$311$	−23.4721	−1.33098	−0.665491	+	0.746406i	$0.731778\pi$
$312$	0	0
$313$	1.18034	0.0667168	0.0333584	−	0.999443i	$-0.489380\pi$
$313$	1.18034	0.0667168	0.0333584	+	0.999443i	$0.489380\pi$
$314$	7.23607	0.408355
$315$	0	0
$316$	10.0000	0.562544
$317$	28.4164	1.59602	0.798012	−	0.602641i	$-0.205885\pi$
$317$	28.4164	1.59602	0.798012	+	0.602641i	$0.205885\pi$
$318$	0	0
$319$	−5.47214	−0.306381
$320$	−0.236068	−0.0131966
$321$	0	0
$322$	0	0
$323$	−26.7639	−1.48919
$324$	0	0
$325$	−6.23607	−0.345915
$326$	9.34752	0.517711
$327$	0	0
$328$	10.0000	0.552158
$329$	20.1246	1.10951
$330$	0	0
$331$	−15.8885	−0.873313	−0.436657	−	0.899628i	$-0.643838\pi$
$331$	−15.8885	−0.873313	−0.436657	+	0.899628i	$0.643838\pi$
$332$	−6.00000	−0.329293
$333$	0	0
$334$	−11.0557	−0.604943
$335$	−11.4721	−0.626790
$336$	0	0
$337$	−20.4721	−1.11519	−0.557594	−	0.830114i	$-0.688275\pi$
$337$	−20.4721	−1.11519	−0.557594	+	0.830114i	$0.688275\pi$
$338$	−16.0000	−0.870285
$339$	0	0
$340$	5.61803	0.304681
$341$	−43.7771	−2.37066
$342$	0	0
$343$	15.0000	0.809924
$344$	7.23607	0.390143
$345$	0	0
$346$	−12.0000	−0.645124
$347$	−17.5279	−0.940945	−0.470473	−	0.882415i	$-0.655917\pi$
$347$	−17.5279	−0.940945	−0.470473	+	0.882415i	$0.655917\pi$
$348$	0	0
$349$	−18.9443	−1.01406	−0.507032	−	0.861927i	$-0.669257\pi$
$349$	−18.9443	−1.01406	−0.507032	+	0.861927i	$0.669257\pi$
$350$	1.85410	0.0991059
$351$	0	0
$352$	−30.7426	−1.63859
$353$	−3.52786	−0.187769	−0.0938846	−	0.995583i	$-0.529928\pi$
$353$	−3.52786	−0.187769	−0.0938846	+	0.995583i	$0.529928\pi$
$354$	0	0
$355$	−7.23607	−0.384051
$356$	18.0902	0.958777
$357$	0	0
$358$	2.58359	0.136547
$359$	4.58359	0.241913	0.120956	−	0.992658i	$-0.461404\pi$
$359$	4.58359	0.241913	0.120956	+	0.992658i	$0.461404\pi$
$360$	0	0
$361$	40.4164	2.12718
$362$	−5.20163	−0.273391
$363$	0	0
$364$	−30.2705	−1.58661
$365$	−8.00000	−0.418739
$366$	0	0
$367$	15.4164	0.804730	0.402365	−	0.915479i	$-0.368188\pi$
$367$	15.4164	0.804730	0.402365	+	0.915479i	$0.368188\pi$
$368$	0	0
$369$	0	0
$370$	4.94427	0.257040
$371$	−20.2918	−1.05350
$372$	0	0
$373$	15.8885	0.822678	0.411339	−	0.911483i	$-0.365061\pi$
$373$	15.8885	0.822678	0.411339	+	0.911483i	$0.365061\pi$
$374$	11.7426	0.607198
$375$	0	0
$376$	−15.0000	−0.773566
$377$	6.23607	0.321174
$378$	0	0
$379$	6.94427	0.356703	0.178352	−	0.983967i	$-0.442924\pi$
$379$	6.94427	0.356703	0.178352	+	0.983967i	$0.442924\pi$
$380$	−12.4721	−0.639807
$381$	0	0
$382$	7.41641	0.379456
$383$	20.0689	1.02547	0.512736	−	0.858546i	$-0.328632\pi$
$383$	20.0689	1.02547	0.512736	+	0.858546i	$0.328632\pi$
$384$	0	0
$385$	−16.4164	−0.836658
$386$	3.70820	0.188743
$387$	0	0
$388$	−4.47214	−0.227038
$389$	−22.2361	−1.12741	−0.563707	−	0.825975i	$-0.690625\pi$
$389$	−22.2361	−1.12741	−0.563707	+	0.825975i	$0.690625\pi$
$390$	0	0
$391$	0	0
$392$	4.47214	0.225877
$393$	0	0
$394$	9.23607	0.465306
$395$	−6.18034	−0.310967
$396$	0	0
$397$	−6.58359	−0.330421	−0.165211	−	0.986258i	$-0.552830\pi$
$397$	−6.58359	−0.330421	−0.165211	+	0.986258i	$0.552830\pi$
$398$	−12.7984	−0.641525
$399$	0	0
$400$	1.85410	0.0927051
$401$	36.6525	1.83034	0.915169	−	0.403071i	$-0.132057\pi$
$401$	36.6525	1.83034	0.915169	+	0.403071i	$0.132057\pi$
$402$	0	0
$403$	49.8885	2.48513
$404$	6.85410	0.341004
$405$	0	0
$406$	−1.85410	−0.0920175
$407$	−43.7771	−2.16995
$408$	0	0
$409$	8.65248	0.427837	0.213919	−	0.976851i	$-0.431377\pi$
$409$	8.65248	0.427837	0.213919	+	0.976851i	$0.431377\pi$
$410$	−2.76393	−0.136501
$411$	0	0
$412$	−12.0000	−0.591198
$413$	15.7082	0.772950
$414$	0	0
$415$	3.70820	0.182029
$416$	35.0344	1.71770
$417$	0	0
$418$	−26.0689	−1.27507
$419$	−34.3607	−1.67863	−0.839315	−	0.543646i	$-0.817043\pi$
$419$	−34.3607	−1.67863	−0.839315	+	0.543646i	$0.817043\pi$
$420$	0	0
$421$	−1.81966	−0.0886848	−0.0443424	−	0.999016i	$-0.514119\pi$
$421$	−1.81966	−0.0886848	−0.0443424	+	0.999016i	$0.514119\pi$
$422$	17.2361	0.839039
$423$	0	0
$424$	15.1246	0.734516
$425$	−3.47214	−0.168423
$426$	0	0
$427$	17.1246	0.828718
$428$	12.1803	0.588759
$429$	0	0
$430$	−2.00000	−0.0964486
$431$	−34.4721	−1.66046	−0.830232	−	0.557418i	$-0.811792\pi$
$431$	−34.4721	−1.66046	−0.830232	+	0.557418i	$0.811792\pi$
$432$	0	0
$433$	−4.65248	−0.223584	−0.111792	−	0.993732i	$-0.535659\pi$
$433$	−4.65248	−0.223584	−0.111792	+	0.993732i	$0.535659\pi$
$434$	−14.8328	−0.711998
$435$	0	0
$436$	26.5623	1.27210
$437$	0	0
$438$	0	0
$439$	−10.1246	−0.483221	−0.241611	−	0.970373i	$-0.577676\pi$
$439$	−10.1246	−0.483221	−0.241611	+	0.970373i	$0.577676\pi$
$440$	12.2361	0.583332
$441$	0	0
$442$	−13.3820	−0.636515
$443$	−23.7639	−1.12906	−0.564529	−	0.825413i	$-0.690942\pi$
$443$	−23.7639	−1.12906	−0.564529	+	0.825413i	$0.690942\pi$
$444$	0	0
$445$	−11.1803	−0.529999
$446$	8.03444	0.380442
$447$	0	0
$448$	0.708204	0.0334595
$449$	−1.76393	−0.0832451	−0.0416225	−	0.999133i	$-0.513253\pi$
$449$	−1.76393	−0.0832451	−0.0416225	+	0.999133i	$0.513253\pi$
$450$	0	0
$451$	24.4721	1.15235
$452$	−12.8541	−0.604606
$453$	0	0
$454$	−3.59675	−0.168804
$455$	18.7082	0.877054
$456$	0	0
$457$	6.12461	0.286497	0.143249	−	0.989687i	$-0.454245\pi$
$457$	6.12461	0.286497	0.143249	+	0.989687i	$0.454245\pi$
$458$	−5.81966	−0.271935
$459$	0	0
$460$	0	0
$461$	−2.36068	−0.109948	−0.0549739	−	0.998488i	$-0.517508\pi$
$461$	−2.36068	−0.109948	−0.0549739	+	0.998488i	$0.517508\pi$
$462$	0	0
$463$	−2.52786	−0.117480	−0.0587399	−	0.998273i	$-0.518708\pi$
$463$	−2.52786	−0.117480	−0.0587399	+	0.998273i	$0.518708\pi$
$464$	−1.85410	−0.0860745
$465$	0	0
$466$	0.875388	0.0405516
$467$	12.9443	0.598989	0.299495	−	0.954098i	$-0.403182\pi$
$467$	12.9443	0.598989	0.299495	+	0.954098i	$0.403182\pi$
$468$	0	0
$469$	34.4164	1.58920
$470$	4.14590	0.191236
$471$	0	0
$472$	−11.7082	−0.538914
$473$	17.7082	0.814224
$474$	0	0
$475$	7.70820	0.353677
$476$	−16.8541	−0.772506
$477$	0	0
$478$	7.34752	0.336068
$479$	6.47214	0.295719	0.147860	−	0.989008i	$-0.452762\pi$
$479$	6.47214	0.295719	0.147860	+	0.989008i	$0.452762\pi$
$480$	0	0
$481$	49.8885	2.27472
$482$	−4.32624	−0.197055
$483$	0	0
$484$	−30.6525	−1.39329
$485$	2.76393	0.125504
$486$	0	0
$487$	−12.0000	−0.543772	−0.271886	−	0.962329i	$-0.587647\pi$
$487$	−12.0000	−0.543772	−0.271886	+	0.962329i	$0.587647\pi$
$488$	−12.7639	−0.577796
$489$	0	0
$490$	−1.23607	−0.0558399
$491$	−25.8885	−1.16833	−0.584167	−	0.811634i	$-0.698579\pi$
$491$	−25.8885	−1.16833	−0.584167	+	0.811634i	$0.698579\pi$
$492$	0	0
$493$	3.47214	0.156377
$494$	29.7082	1.33664
$495$	0	0
$496$	−14.8328	−0.666013
$497$	21.7082	0.973746
$498$	0	0
$499$	−23.7639	−1.06382	−0.531910	−	0.846801i	$-0.678525\pi$
$499$	−23.7639	−1.06382	−0.531910	+	0.846801i	$0.678525\pi$
$500$	−1.61803	−0.0723607
$501$	0	0
$502$	−15.3820	−0.686531
$503$	30.5967	1.36424	0.682121	−	0.731240i	$-0.261058\pi$
$503$	30.5967	1.36424	0.682121	+	0.731240i	$0.261058\pi$
$504$	0	0
$505$	−4.23607	−0.188503
$506$	0	0
$507$	0	0
$508$	9.70820	0.430732
$509$	−6.18034	−0.273939	−0.136969	−	0.990575i	$-0.543736\pi$
$509$	−6.18034	−0.273939	−0.136969	+	0.990575i	$0.543736\pi$
$510$	0	0
$511$	24.0000	1.06170
$512$	−18.7082	−0.826794
$513$	0	0
$514$	−6.00000	−0.264649
$515$	7.41641	0.326806
$516$	0	0
$517$	−36.7082	−1.61442
$518$	−14.8328	−0.651717
$519$	0	0
$520$	−13.9443	−0.611497
$521$	20.7639	0.909684	0.454842	−	0.890572i	$-0.349696\pi$
$521$	20.7639	0.909684	0.454842	+	0.890572i	$0.349696\pi$
$522$	0	0
$523$	−29.8328	−1.30450	−0.652249	−	0.758005i	$-0.726174\pi$
$523$	−29.8328	−1.30450	−0.652249	+	0.758005i	$0.726174\pi$
$524$	5.61803	0.245425
$525$	0	0
$526$	14.8328	0.646741
$527$	27.7771	1.20999
$528$	0	0
$529$	−23.0000	−1.00000
$530$	−4.18034	−0.181582
$531$	0	0
$532$	37.4164	1.62221
$533$	−27.8885	−1.20799
$534$	0	0
$535$	−7.52786	−0.325458
$536$	−25.6525	−1.10802
$537$	0	0
$538$	−18.6869	−0.805650
$539$	10.9443	0.471403
$540$	0	0
$541$	38.3607	1.64925	0.824627	−	0.565677i	$-0.191385\pi$
$541$	38.3607	1.64925	0.824627	+	0.565677i	$0.191385\pi$
$542$	−8.87539	−0.381231
$543$	0	0
$544$	19.5066	0.836338
$545$	−16.4164	−0.703202
$546$	0	0
$547$	27.4721	1.17462	0.587312	−	0.809361i	$-0.300186\pi$
$547$	27.4721	1.17462	0.587312	+	0.809361i	$0.300186\pi$
$548$	−17.7082	−0.756457
$549$	0	0
$550$	−3.38197	−0.144208
$551$	−7.70820	−0.328381
$552$	0	0
$553$	18.5410	0.788444
$554$	−5.38197	−0.228658
$555$	0	0
$556$	1.14590	0.0485969
$557$	−2.76393	−0.117112	−0.0585558	−	0.998284i	$-0.518650\pi$
$557$	−2.76393	−0.117112	−0.0585558	+	0.998284i	$0.518650\pi$
$558$	0	0
$559$	−20.1803	−0.853537
$560$	−5.56231	−0.235050
$561$	0	0
$562$	−14.2918	−0.602863
$563$	−44.1246	−1.85963	−0.929815	−	0.368026i	$-0.880034\pi$
$563$	−44.1246	−1.85963	−0.929815	+	0.368026i	$0.880034\pi$
$564$	0	0
$565$	7.94427	0.334218
$566$	−11.4164	−0.479867
$567$	0	0
$568$	−16.1803	−0.678912
$569$	5.18034	0.217171	0.108586	−	0.994087i	$-0.465368\pi$
$569$	5.18034	0.217171	0.108586	+	0.994087i	$0.465368\pi$
$570$	0	0
$571$	−20.0000	−0.836974	−0.418487	−	0.908223i	$-0.637439\pi$
$571$	−20.0000	−0.836974	−0.418487	+	0.908223i	$0.637439\pi$
$572$	55.2148	2.30865
$573$	0	0
$574$	8.29180	0.346093
$575$	0	0
$576$	0	0
$577$	−29.3050	−1.21998	−0.609991	−	0.792409i	$-0.708827\pi$
$577$	−29.3050	−1.21998	−0.609991	+	0.792409i	$0.708827\pi$
$578$	3.05573	0.127102
$579$	0	0
$580$	1.61803	0.0671852
$581$	−11.1246	−0.461527
$582$	0	0
$583$	37.0132	1.53293
$584$	−17.8885	−0.740233
$585$	0	0
$586$	−11.0902	−0.458131
$587$	1.81966	0.0751054	0.0375527	−	0.999295i	$-0.488044\pi$
$587$	1.81966	0.0751054	0.0375527	+	0.999295i	$0.488044\pi$
$588$	0	0
$589$	−61.6656	−2.54089
$590$	3.23607	0.133227
$591$	0	0
$592$	−14.8328	−0.609625
$593$	24.6525	1.01236	0.506178	−	0.862429i	$-0.331058\pi$
$593$	24.6525	1.01236	0.506178	+	0.862429i	$0.331058\pi$
$594$	0	0
$595$	10.4164	0.427031
$596$	32.6525	1.33750
$597$	0	0
$598$	0	0
$599$	8.88854	0.363176	0.181588	−	0.983375i	$-0.441876\pi$
$599$	8.88854	0.363176	0.181588	+	0.983375i	$0.441876\pi$
$600$	0	0
$601$	8.11146	0.330873	0.165437	−	0.986220i	$-0.447097\pi$
$601$	8.11146	0.330873	0.165437	+	0.986220i	$0.447097\pi$
$602$	6.00000	0.244542
$603$	0	0
$604$	−4.00000	−0.162758
$605$	18.9443	0.770194
$606$	0	0
$607$	40.6525	1.65003	0.825017	−	0.565109i	$-0.191166\pi$
$607$	40.6525	1.65003	0.825017	+	0.565109i	$0.191166\pi$
$608$	−43.3050	−1.75625
$609$	0	0
$610$	3.52786	0.142839
$611$	41.8328	1.69237
$612$	0	0
$613$	7.29180	0.294513	0.147256	−	0.989098i	$-0.452956\pi$
$613$	7.29180	0.294513	0.147256	+	0.989098i	$0.452956\pi$
$614$	−15.4164	−0.622156
$615$	0	0
$616$	−36.7082	−1.47902
$617$	−43.3050	−1.74339	−0.871696	−	0.490047i	$-0.836980\pi$
$617$	−43.3050	−1.74339	−0.871696	+	0.490047i	$0.836980\pi$
$618$	0	0
$619$	−3.52786	−0.141797	−0.0708984	−	0.997484i	$-0.522587\pi$
$619$	−3.52786	−0.141797	−0.0708984	+	0.997484i	$0.522587\pi$
$620$	12.9443	0.519854
$621$	0	0
$622$	14.5066	0.581661
$623$	33.5410	1.34379
$624$	0	0
$625$	1.00000	0.0400000
$626$	−0.729490	−0.0291563
$627$	0	0
$628$	18.9443	0.755959
$629$	27.7771	1.10755
$630$	0	0
$631$	−38.4853	−1.53208	−0.766038	−	0.642796i	$-0.777774\pi$
$631$	−38.4853	−1.53208	−0.766038	+	0.642796i	$0.777774\pi$
$632$	−13.8197	−0.549717
$633$	0	0
$634$	−17.5623	−0.697488
$635$	−6.00000	−0.238103
$636$	0	0
$637$	−12.4721	−0.494164
$638$	3.38197	0.133893
$639$	0	0
$640$	11.3820	0.449912
$641$	−46.2361	−1.82621	−0.913107	−	0.407719i	$-0.866324\pi$
$641$	−46.2361	−1.82621	−0.913107	+	0.407719i	$0.866324\pi$
$642$	0	0
$643$	−35.8328	−1.41311	−0.706554	−	0.707659i	$-0.749751\pi$
$643$	−35.8328	−1.41311	−0.706554	+	0.707659i	$0.749751\pi$
$644$	0	0
$645$	0	0
$646$	16.5410	0.650798
$647$	−11.2361	−0.441735	−0.220868	−	0.975304i	$-0.570889\pi$
$647$	−11.2361	−0.441735	−0.220868	+	0.975304i	$0.570889\pi$
$648$	0	0
$649$	−28.6525	−1.12471
$650$	3.85410	0.151170
$651$	0	0
$652$	24.4721	0.958403
$653$	−8.88854	−0.347836	−0.173918	−	0.984760i	$-0.555643\pi$
$653$	−8.88854	−0.347836	−0.173918	+	0.984760i	$0.555643\pi$
$654$	0	0
$655$	−3.47214	−0.135668
$656$	8.29180	0.323740
$657$	0	0
$658$	−12.4377	−0.484872
$659$	21.0000	0.818044	0.409022	−	0.912525i	$-0.365870\pi$
$659$	21.0000	0.818044	0.409022	+	0.912525i	$0.365870\pi$
$660$	0	0
$661$	−37.8328	−1.47153	−0.735763	−	0.677239i	$-0.763176\pi$
$661$	−37.8328	−1.47153	−0.735763	+	0.677239i	$0.763176\pi$
$662$	9.81966	0.381652
$663$	0	0
$664$	8.29180	0.321784
$665$	−23.1246	−0.896734
$666$	0	0
$667$	0	0
$668$	−28.9443	−1.11989
$669$	0	0
$670$	7.09017	0.273917
$671$	−31.2361	−1.20586
$672$	0	0
$673$	−33.2918	−1.28330	−0.641652	−	0.766996i	$-0.721751\pi$
$673$	−33.2918	−1.28330	−0.641652	+	0.766996i	$0.721751\pi$
$674$	12.6525	0.487355
$675$	0	0
$676$	−41.8885	−1.61110
$677$	20.8885	0.802812	0.401406	−	0.915900i	$-0.368522\pi$
$677$	20.8885	0.802812	0.401406	+	0.915900i	$0.368522\pi$
$678$	0	0
$679$	−8.29180	−0.318210
$680$	−7.76393	−0.297733
$681$	0	0
$682$	27.0557	1.03602
$683$	−7.41641	−0.283781	−0.141890	−	0.989882i	$-0.545318\pi$
$683$	−7.41641	−0.283781	−0.141890	+	0.989882i	$0.545318\pi$
$684$	0	0
$685$	10.9443	0.418159
$686$	−9.27051	−0.353950
$687$	0	0
$688$	6.00000	0.228748
$689$	−42.1803	−1.60694
$690$	0	0
$691$	−34.7082	−1.32036	−0.660181	−	0.751106i	$-0.729520\pi$
$691$	−34.7082	−1.32036	−0.660181	+	0.751106i	$0.729520\pi$
$692$	−31.4164	−1.19427
$693$	0	0
$694$	10.8328	0.411208
$695$	−0.708204	−0.0268637
$696$	0	0
$697$	−15.5279	−0.588160
$698$	11.7082	0.443162
$699$	0	0
$700$	4.85410	0.183468
$701$	5.12461	0.193554	0.0967770	−	0.995306i	$-0.469147\pi$
$701$	5.12461	0.193554	0.0967770	+	0.995306i	$0.469147\pi$
$702$	0	0
$703$	−61.6656	−2.32576
$704$	−1.29180	−0.0486864
$705$	0	0
$706$	2.18034	0.0820582
$707$	12.7082	0.477941
$708$	0	0
$709$	19.8885	0.746930	0.373465	−	0.927644i	$-0.378170\pi$
$709$	19.8885	0.746930	0.373465	+	0.927644i	$0.378170\pi$
$710$	4.47214	0.167836
$711$	0	0
$712$	−25.0000	−0.936915
$713$	0	0
$714$	0	0
$715$	−34.1246	−1.27619
$716$	6.76393	0.252780
$717$	0	0
$718$	−2.83282	−0.105720
$719$	11.2361	0.419035	0.209517	−	0.977805i	$-0.432811\pi$
$719$	11.2361	0.419035	0.209517	+	0.977805i	$0.432811\pi$
$720$	0	0
$721$	−22.2492	−0.828604
$722$	−24.9787	−0.929611
$723$	0	0
$724$	−13.6180	−0.506110
$725$	−1.00000	−0.0371391
$726$	0	0
$727$	2.11146	0.0783096	0.0391548	−	0.999233i	$-0.487533\pi$
$727$	2.11146	0.0783096	0.0391548	+	0.999233i	$0.487533\pi$
$728$	41.8328	1.55043
$729$	0	0
$730$	4.94427	0.182996
$731$	−11.2361	−0.415581
$732$	0	0
$733$	−14.2918	−0.527880	−0.263940	−	0.964539i	$-0.585022\pi$
$733$	−14.2918	−0.527880	−0.263940	+	0.964539i	$0.585022\pi$
$734$	−9.52786	−0.351680
$735$	0	0
$736$	0	0
$737$	−62.7771	−2.31242
$738$	0	0
$739$	41.4853	1.52606	0.763031	−	0.646362i	$-0.223711\pi$
$739$	41.4853	1.52606	0.763031	+	0.646362i	$0.223711\pi$
$740$	12.9443	0.475841
$741$	0	0
$742$	12.5410	0.460395
$743$	10.8197	0.396935	0.198467	−	0.980107i	$-0.436404\pi$
$743$	10.8197	0.396935	0.198467	+	0.980107i	$0.436404\pi$
$744$	0	0
$745$	−20.1803	−0.739350
$746$	−9.81966	−0.359523
$747$	0	0
$748$	30.7426	1.12406
$749$	22.5836	0.825186
$750$	0	0
$751$	43.4164	1.58429	0.792144	−	0.610335i	$-0.208965\pi$
$751$	43.4164	1.58429	0.792144	+	0.610335i	$0.208965\pi$
$752$	−12.4377	−0.453556
$753$	0	0
$754$	−3.85410	−0.140358
$755$	2.47214	0.0899702
$756$	0	0
$757$	−34.8328	−1.26602	−0.633010	−	0.774144i	$-0.718181\pi$
$757$	−34.8328	−1.26602	−0.633010	+	0.774144i	$0.718181\pi$
$758$	−4.29180	−0.155885
$759$	0	0
$760$	17.2361	0.625218
$761$	−33.0132	−1.19673	−0.598363	−	0.801225i	$-0.704182\pi$
$761$	−33.0132	−1.19673	−0.598363	+	0.801225i	$0.704182\pi$
$762$	0	0
$763$	49.2492	1.78294
$764$	19.4164	0.702461
$765$	0	0
$766$	−12.4033	−0.448148
$767$	32.6525	1.17901
$768$	0	0
$769$	34.6525	1.24960	0.624800	−	0.780785i	$-0.285180\pi$
$769$	34.6525	1.24960	0.624800	+	0.780785i	$0.285180\pi$
$770$	10.1459	0.365633
$771$	0	0
$772$	9.70820	0.349406
$773$	52.2492	1.87927	0.939637	−	0.342173i	$-0.111163\pi$
$773$	52.2492	1.87927	0.939637	+	0.342173i	$0.111163\pi$
$774$	0	0
$775$	−8.00000	−0.287368
$776$	6.18034	0.221861
$777$	0	0
$778$	13.7426	0.492698
$779$	34.4721	1.23509
$780$	0	0
$781$	−39.5967	−1.41688
$782$	0	0
$783$	0	0
$784$	3.70820	0.132436
$785$	−11.7082	−0.417884
$786$	0	0
$787$	1.88854	0.0673193	0.0336597	−	0.999433i	$-0.489284\pi$
$787$	1.88854	0.0673193	0.0336597	+	0.999433i	$0.489284\pi$
$788$	24.1803	0.861389
$789$	0	0
$790$	3.81966	0.135897
$791$	−23.8328	−0.847397
$792$	0	0
$793$	35.5967	1.26408
$794$	4.06888	0.144399
$795$	0	0
$796$	−33.5066	−1.18761
$797$	−6.94427	−0.245979	−0.122989	−	0.992408i	$-0.539248\pi$
$797$	−6.94427	−0.245979	−0.122989	+	0.992408i	$0.539248\pi$
$798$	0	0
$799$	23.2918	0.824005
$800$	−5.61803	−0.198627
$801$	0	0
$802$	−22.6525	−0.799887
$803$	−43.7771	−1.54486
$804$	0	0
$805$	0	0
$806$	−30.8328	−1.08604
$807$	0	0
$808$	−9.47214	−0.333229
$809$	−44.2361	−1.55526	−0.777629	−	0.628724i	$-0.783578\pi$
$809$	−44.2361	−1.55526	−0.777629	+	0.628724i	$0.783578\pi$
$810$	0	0
$811$	−22.7082	−0.797393	−0.398696	−	0.917083i	$-0.630537\pi$
$811$	−22.7082	−0.797393	−0.398696	+	0.917083i	$0.630537\pi$
$812$	−4.85410	−0.170346
$813$	0	0
$814$	27.0557	0.948303
$815$	−15.1246	−0.529792
$816$	0	0
$817$	24.9443	0.872690
$818$	−5.34752	−0.186972
$819$	0	0
$820$	−7.23607	−0.252694
$821$	10.5836	0.369370	0.184685	−	0.982798i	$-0.440874\pi$
$821$	10.5836	0.369370	0.184685	+	0.982798i	$0.440874\pi$
$822$	0	0
$823$	−28.7639	−1.00265	−0.501324	−	0.865260i	$-0.667153\pi$
$823$	−28.7639	−1.00265	−0.501324	+	0.865260i	$0.667153\pi$
$824$	16.5836	0.577717
$825$	0	0
$826$	−9.70820	−0.337792
$827$	7.41641	0.257894	0.128947	−	0.991652i	$-0.458840\pi$
$827$	7.41641	0.257894	0.128947	+	0.991652i	$0.458840\pi$
$828$	0	0
$829$	−20.0000	−0.694629	−0.347314	−	0.937749i	$-0.612906\pi$
$829$	−20.0000	−0.694629	−0.347314	+	0.937749i	$0.612906\pi$
$830$	−2.29180	−0.0795494
$831$	0	0
$832$	1.47214	0.0510371
$833$	−6.94427	−0.240605
$834$	0	0
$835$	17.8885	0.619059
$836$	−68.2492	−2.36045
$837$	0	0
$838$	21.2361	0.733588
$839$	−14.8885	−0.514010	−0.257005	−	0.966410i	$-0.582736\pi$
$839$	−14.8885	−0.514010	−0.257005	+	0.966410i	$0.582736\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	1.12461	0.0387567
$843$	0	0
$844$	45.1246	1.55325
$845$	25.8885	0.890593
$846$	0	0
$847$	−56.8328	−1.95280
$848$	12.5410	0.430660
$849$	0	0
$850$	2.14590	0.0736037
$851$	0	0
$852$	0	0
$853$	−28.7639	−0.984858	−0.492429	−	0.870353i	$-0.663891\pi$
$853$	−28.7639	−0.984858	−0.492429	+	0.870353i	$0.663891\pi$
$854$	−10.5836	−0.362163
$855$	0	0
$856$	−16.8328	−0.575334
$857$	23.8885	0.816017	0.408009	−	0.912978i	$-0.366223\pi$
$857$	23.8885	0.816017	0.408009	+	0.912978i	$0.366223\pi$
$858$	0	0
$859$	−15.7082	−0.535957	−0.267979	−	0.963425i	$-0.586356\pi$
$859$	−15.7082	−0.535957	−0.267979	+	0.963425i	$0.586356\pi$
$860$	−5.23607	−0.178548
$861$	0	0
$862$	21.3050	0.725650
$863$	−15.5967	−0.530919	−0.265460	−	0.964122i	$-0.585524\pi$
$863$	−15.5967	−0.530919	−0.265460	+	0.964122i	$0.585524\pi$
$864$	0	0
$865$	19.4164	0.660178
$866$	2.87539	0.0977097
$867$	0	0
$868$	−38.8328	−1.31807
$869$	−33.8197	−1.14725
$870$	0	0
$871$	71.5410	2.42407
$872$	−36.7082	−1.24310
$873$	0	0
$874$	0	0
$875$	−3.00000	−0.101419
$876$	0	0
$877$	48.8328	1.64897	0.824484	−	0.565886i	$-0.191466\pi$
$877$	48.8328	1.64897	0.824484	+	0.565886i	$0.191466\pi$
$878$	6.25735	0.211175
$879$	0	0
$880$	10.1459	0.342018
$881$	−12.7082	−0.428150	−0.214075	−	0.976817i	$-0.568674\pi$
$881$	−12.7082	−0.428150	−0.214075	+	0.976817i	$0.568674\pi$
$882$	0	0
$883$	−32.0000	−1.07689	−0.538443	−	0.842662i	$-0.680987\pi$
$883$	−32.0000	−1.07689	−0.538443	+	0.842662i	$0.680987\pi$
$884$	−35.0344	−1.17834
$885$	0	0
$886$	14.6869	0.493417
$887$	44.5967	1.49741	0.748706	−	0.662902i	$-0.230675\pi$
$887$	44.5967	1.49741	0.748706	+	0.662902i	$0.230675\pi$
$888$	0	0
$889$	18.0000	0.603701
$890$	6.90983	0.231618
$891$	0	0
$892$	21.0344	0.704285
$893$	−51.7082	−1.73035
$894$	0	0
$895$	−4.18034	−0.139733
$896$	−34.1459	−1.14073
$897$	0	0
$898$	1.09017	0.0363794
$899$	8.00000	0.266815
$900$	0	0
$901$	−23.4853	−0.782409
$902$	−15.1246	−0.503594
$903$	0	0
$904$	17.7639	0.590820
$905$	8.41641	0.279771
$906$	0	0
$907$	26.7639	0.888682	0.444341	−	0.895858i	$-0.353438\pi$
$907$	26.7639	0.888682	0.444341	+	0.895858i	$0.353438\pi$
$908$	−9.41641	−0.312494
$909$	0	0
$910$	−11.5623	−0.383287
$911$	18.0557	0.598213	0.299106	−	0.954220i	$-0.403311\pi$
$911$	18.0557	0.598213	0.299106	+	0.954220i	$0.403311\pi$
$912$	0	0
$913$	20.2918	0.671560
$914$	−3.78522	−0.125204
$915$	0	0
$916$	−15.2361	−0.503414
$917$	10.4164	0.343980
$918$	0	0
$919$	20.1246	0.663850	0.331925	−	0.943306i	$-0.392302\pi$
$919$	20.1246	0.663850	0.331925	+	0.943306i	$0.392302\pi$
$920$	0	0
$921$	0	0
$922$	1.45898	0.0480490
$923$	45.1246	1.48529
$924$	0	0
$925$	−8.00000	−0.263038
$926$	1.56231	0.0513406
$927$	0	0
$928$	5.61803	0.184421
$929$	56.8328	1.86462	0.932312	−	0.361655i	$-0.117788\pi$
$929$	56.8328	1.86462	0.932312	+	0.361655i	$0.117788\pi$
$930$	0	0
$931$	15.4164	0.505252
$932$	2.29180	0.0750703
$933$	0	0
$934$	−8.00000	−0.261768
$935$	−19.0000	−0.621366
$936$	0	0
$937$	19.7639	0.645660	0.322830	−	0.946457i	$-0.395366\pi$
$937$	19.7639	0.645660	0.322830	+	0.946457i	$0.395366\pi$
$938$	−21.2705	−0.694507
$939$	0	0
$940$	10.8541	0.354022
$941$	46.0689	1.50180	0.750901	−	0.660414i	$-0.229619\pi$
$941$	46.0689	1.50180	0.750901	+	0.660414i	$0.229619\pi$
$942$	0	0
$943$	0	0
$944$	−9.70820	−0.315975
$945$	0	0
$946$	−10.9443	−0.355829
$947$	12.7082	0.412961	0.206481	−	0.978451i	$-0.433799\pi$
$947$	12.7082	0.412961	0.206481	+	0.978451i	$0.433799\pi$
$948$	0	0
$949$	49.8885	1.61945
$950$	−4.76393	−0.154562
$951$	0	0
$952$	23.2918	0.754891
$953$	−19.0132	−0.615897	−0.307948	−	0.951403i	$-0.599642\pi$
$953$	−19.0132	−0.615897	−0.307948	+	0.951403i	$0.599642\pi$
$954$	0	0
$955$	−12.0000	−0.388311
$956$	19.2361	0.622139
$957$	0	0
$958$	−4.00000	−0.129234
$959$	−32.8328	−1.06023
$960$	0	0
$961$	33.0000	1.06452
$962$	−30.8328	−0.994090
$963$	0	0
$964$	−11.3262	−0.364794
$965$	−6.00000	−0.193147
$966$	0	0
$967$	−35.1246	−1.12953	−0.564766	−	0.825251i	$-0.691033\pi$
$967$	−35.1246	−1.12953	−0.564766	+	0.825251i	$0.691033\pi$
$968$	42.3607	1.36152
$969$	0	0
$970$	−1.70820	−0.0548471
$971$	23.0557	0.739894	0.369947	−	0.929053i	$-0.379376\pi$
$971$	23.0557	0.739894	0.369947	+	0.929053i	$0.379376\pi$
$972$	0	0
$973$	2.12461	0.0681119
$974$	7.41641	0.237637
$975$	0	0
$976$	−10.5836	−0.338773
$977$	22.4721	0.718947	0.359474	−	0.933155i	$-0.382956\pi$
$977$	22.4721	0.718947	0.359474	+	0.933155i	$0.382956\pi$
$978$	0	0
$979$	−61.1803	−1.95533
$980$	−3.23607	−0.103372
$981$	0	0
$982$	16.0000	0.510581
$983$	−37.5279	−1.19695	−0.598476	−	0.801140i	$-0.704227\pi$
$983$	−37.5279	−1.19695	−0.598476	+	0.801140i	$0.704227\pi$
$984$	0	0
$985$	−14.9443	−0.476164
$986$	−2.14590	−0.0683393
$987$	0	0
$988$	77.7771	2.47442
$989$	0	0
$990$	0	0
$991$	22.4853	0.714269	0.357134	−	0.934053i	$-0.383754\pi$
$991$	22.4853	0.714269	0.357134	+	0.934053i	$0.383754\pi$
$992$	44.9443	1.42698
$993$	0	0
$994$	−13.4164	−0.425543
$995$	20.7082	0.656494
$996$	0	0
$997$	7.41641	0.234880	0.117440	−	0.993080i	$-0.462531\pi$
$997$	7.41641	0.234880	0.117440	+	0.993080i	$0.462531\pi$
$998$	14.6869	0.464906
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1305.2.a.k.1.1		2
3.2	odd	2		435.2.a.e.1.2	✓	2
5.4	even	2		6525.2.a.s.1.2		2
12.11	even	2		6960.2.a.bu.1.2		2
15.2	even	4		2175.2.c.j.349.3		4
15.8	even	4		2175.2.c.j.349.2		4
15.14	odd	2		2175.2.a.q.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
435.2.a.e.1.2	✓	2	3.2	odd	2
1305.2.a.k.1.1		2	1.1	even	1	trivial
2175.2.a.q.1.1		2	15.14	odd	2
2175.2.c.j.349.2		4	15.8	even	4
2175.2.c.j.349.3		4	15.2	even	4
6525.2.a.s.1.2		2	5.4	even	2
6960.2.a.bu.1.2		2	12.11	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:	\( N \)	\(=\)	\( 1305 = 3^{2} \cdot 5 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1305.a (trivial)

\(f(q)\)	\(=\)	\(q-0.618034 q^{2} -1.61803 q^{4} +1.00000 q^{5} -3.00000 q^{7} +2.23607 q^{8} +O(q^{10})\)	\(q-0.618034 q^{2} -1.61803 q^{4} +1.00000 q^{5} -3.00000 q^{7} +2.23607 q^{8} -0.618034 q^{10} +5.47214 q^{11} -6.23607 q^{13} +1.85410 q^{14} +1.85410 q^{16} -3.47214 q^{17} +7.70820 q^{19} -1.61803 q^{20} -3.38197 q^{22} +1.00000 q^{25} +3.85410 q^{26} +4.85410 q^{28} -1.00000 q^{29} -8.00000 q^{31} -5.61803 q^{32} +2.14590 q^{34} -3.00000 q^{35} -8.00000 q^{37} -4.76393 q^{38} +2.23607 q^{40} +4.47214 q^{41} +3.23607 q^{43} -8.85410 q^{44} -6.70820 q^{47} +2.00000 q^{49} -0.618034 q^{50} +10.0902 q^{52} +6.76393 q^{53} +5.47214 q^{55} -6.70820 q^{56} +0.618034 q^{58} -5.23607 q^{59} -5.70820 q^{61} +4.94427 q^{62} -0.236068 q^{64} -6.23607 q^{65} -11.4721 q^{67} +5.61803 q^{68} +1.85410 q^{70} -7.23607 q^{71} -8.00000 q^{73} +4.94427 q^{74} -12.4721 q^{76} -16.4164 q^{77} -6.18034 q^{79} +1.85410 q^{80} -2.76393 q^{82} +3.70820 q^{83} -3.47214 q^{85} -2.00000 q^{86} +12.2361 q^{88} -11.1803 q^{89} +18.7082 q^{91} +4.14590 q^{94} +7.70820 q^{95} +2.76393 q^{97} -1.23607 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{2} - q^{4} + 2 q^{5} - 6 q^{7}+O(q^{10}) \) 2 * q + q^2 - q^4 + 2 * q^5 - 6 * q^7	\( 2 q + q^{2} - q^{4} + 2 q^{5} - 6 q^{7} + q^{10} + 2 q^{11} - 8 q^{13} - 3 q^{14} - 3 q^{16} + 2 q^{17} + 2 q^{19} - q^{20} - 9 q^{22} + 2 q^{25} + q^{26} + 3 q^{28} - 2 q^{29} - 16 q^{31} - 9 q^{32} + 11 q^{34} - 6 q^{35} - 16 q^{37} - 14 q^{38} + 2 q^{43} - 11 q^{44} + 4 q^{49} + q^{50} + 9 q^{52} + 18 q^{53} + 2 q^{55} - q^{58} - 6 q^{59} + 2 q^{61} - 8 q^{62} + 4 q^{64} - 8 q^{65} - 14 q^{67} + 9 q^{68} - 3 q^{70} - 10 q^{71} - 16 q^{73} - 8 q^{74} - 16 q^{76} - 6 q^{77} + 10 q^{79} - 3 q^{80} - 10 q^{82} - 6 q^{83} + 2 q^{85} - 4 q^{86} + 20 q^{88} + 24 q^{91} + 15 q^{94} + 2 q^{95} + 10 q^{97} + 2 q^{98}+O(q^{100}) \) 2 * q + q^2 - q^4 + 2 * q^5 - 6 * q^7 + q^10 + 2 * q^11 - 8 * q^13 - 3 * q^14 - 3 * q^16 + 2 * q^17 + 2 * q^19 - q^20 - 9 * q^22 + 2 * q^25 + q^26 + 3 * q^28 - 2 * q^29 - 16 * q^31 - 9 * q^32 + 11 * q^34 - 6 * q^35 - 16 * q^37 - 14 * q^38 + 2 * q^43 - 11 * q^44 + 4 * q^49 + q^50 + 9 * q^52 + 18 * q^53 + 2 * q^55 - q^58 - 6 * q^59 + 2 * q^61 - 8 * q^62 + 4 * q^64 - 8 * q^65 - 14 * q^67 + 9 * q^68 - 3 * q^70 - 10 * q^71 - 16 * q^73 - 8 * q^74 - 16 * q^76 - 6 * q^77 + 10 * q^79 - 3 * q^80 - 10 * q^82 - 6 * q^83 + 2 * q^85 - 4 * q^86 + 20 * q^88 + 24 * q^91 + 15 * q^94 + 2 * q^95 + 10 * q^97 + 2 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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