LMFDB - Embedded newform 7935.2.a.p.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7935,2,Mod(1,7935)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7935, 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("7935.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7935 = 3 \cdot 5 \cdot 23^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7935.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:	$63.3612940039$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ of defining polynomial
Character	$\chi$	$=$	7935.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.41421 q^{2} -1.00000 q^{3} +1.00000 q^{5} +1.41421 q^{6} -4.82843 q^{7} +2.82843 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.41421 q^{2} -1.00000 q^{3} +1.00000 q^{5} +1.41421 q^{6} -4.82843 q^{7} +2.82843 q^{8} +1.00000 q^{9} -1.41421 q^{10} -0.414214 q^{11} -2.58579 q^{13} +6.82843 q^{14} -1.00000 q^{15} -4.00000 q^{16} +0.585786 q^{17} -1.41421 q^{18} -1.00000 q^{19} +4.82843 q^{21} +0.585786 q^{22} -2.82843 q^{24} +1.00000 q^{25} +3.65685 q^{26} -1.00000 q^{27} -4.82843 q^{29} +1.41421 q^{30} +1.82843 q^{31} +0.414214 q^{33} -0.828427 q^{34} -4.82843 q^{35} +10.2426 q^{37} +1.41421 q^{38} +2.58579 q^{39} +2.82843 q^{40} -4.41421 q^{41} -6.82843 q^{42} +1.75736 q^{43} +1.00000 q^{45} +7.65685 q^{47} +4.00000 q^{48} +16.3137 q^{49} -1.41421 q^{50} -0.585786 q^{51} -4.24264 q^{53} +1.41421 q^{54} -0.414214 q^{55} -13.6569 q^{56} +1.00000 q^{57} +6.82843 q^{58} -11.6569 q^{59} -12.6569 q^{61} -2.58579 q^{62} -4.82843 q^{63} +8.00000 q^{64} -2.58579 q^{65} -0.585786 q^{66} +2.48528 q^{67} +6.82843 q^{70} +14.8995 q^{71} +2.82843 q^{72} +14.4853 q^{73} -14.4853 q^{74} -1.00000 q^{75} +2.00000 q^{77} -3.65685 q^{78} +1.34315 q^{79} -4.00000 q^{80} +1.00000 q^{81} +6.24264 q^{82} +5.89949 q^{83} +0.585786 q^{85} -2.48528 q^{86} +4.82843 q^{87} -1.17157 q^{88} -5.17157 q^{89} -1.41421 q^{90} +12.4853 q^{91} -1.82843 q^{93} -10.8284 q^{94} -1.00000 q^{95} +7.89949 q^{97} -23.0711 q^{98} -0.414214 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} + 2 q^{5} - 4 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^3 + 2 * q^5 - 4 * q^7 + 2 * q^9	$ 2 q - 2 q^{3} + 2 q^{5} - 4 q^{7} + 2 q^{9} + 2 q^{11} - 8 q^{13} + 8 q^{14} - 2 q^{15} - 8 q^{16} + 4 q^{17} - 2 q^{19} + 4 q^{21} + 4 q^{22} + 2 q^{25} - 4 q^{26} - 2 q^{27} - 4 q^{29} - 2 q^{31} - 2 q^{33} + 4 q^{34} - 4 q^{35} + 12 q^{37} + 8 q^{39} - 6 q^{41} - 8 q^{42} + 12 q^{43} + 2 q^{45} + 4 q^{47} + 8 q^{48} + 10 q^{49} - 4 q^{51} + 2 q^{55} - 16 q^{56} + 2 q^{57} + 8 q^{58} - 12 q^{59} - 14 q^{61} - 8 q^{62} - 4 q^{63} + 16 q^{64} - 8 q^{65} - 4 q^{66} - 12 q^{67} + 8 q^{70} + 10 q^{71} + 12 q^{73} - 12 q^{74} - 2 q^{75} + 4 q^{77} + 4 q^{78} + 14 q^{79} - 8 q^{80} + 2 q^{81} + 4 q^{82} - 8 q^{83} + 4 q^{85} + 12 q^{86} + 4 q^{87} - 8 q^{88} - 16 q^{89} + 8 q^{91} + 2 q^{93} - 16 q^{94} - 2 q^{95} - 4 q^{97} - 32 q^{98} + 2 q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 + 2 * q^5 - 4 * q^7 + 2 * q^9 + 2 * q^11 - 8 * q^13 + 8 * q^14 - 2 * q^15 - 8 * q^16 + 4 * q^17 - 2 * q^19 + 4 * q^21 + 4 * q^22 + 2 * q^25 - 4 * q^26 - 2 * q^27 - 4 * q^29 - 2 * q^31 - 2 * q^33 + 4 * q^34 - 4 * q^35 + 12 * q^37 + 8 * q^39 - 6 * q^41 - 8 * q^42 + 12 * q^43 + 2 * q^45 + 4 * q^47 + 8 * q^48 + 10 * q^49 - 4 * q^51 + 2 * q^55 - 16 * q^56 + 2 * q^57 + 8 * q^58 - 12 * q^59 - 14 * q^61 - 8 * q^62 - 4 * q^63 + 16 * q^64 - 8 * q^65 - 4 * q^66 - 12 * q^67 + 8 * q^70 + 10 * q^71 + 12 * q^73 - 12 * q^74 - 2 * q^75 + 4 * q^77 + 4 * q^78 + 14 * q^79 - 8 * q^80 + 2 * q^81 + 4 * q^82 - 8 * q^83 + 4 * q^85 + 12 * q^86 + 4 * q^87 - 8 * q^88 - 16 * q^89 + 8 * q^91 + 2 * q^93 - 16 * q^94 - 2 * q^95 - 4 * q^97 - 32 * q^98 + 2 * 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 $

$2$	−1.41421	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$2$	−1.41421	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	1.41421	0.577350
$7$	−4.82843	−1.82497	−0.912487	−	0.409106i	$-0.865841\pi$
$7$	−4.82843	−1.82497	−0.912487	+	0.409106i	$0.865841\pi$
$8$	2.82843	1.00000
$9$	1.00000	0.333333
$10$	−1.41421	−0.447214
$11$	−0.414214	−0.124890	−0.0624450	−	0.998048i	$-0.519890\pi$
$11$	−0.414214	−0.124890	−0.0624450	+	0.998048i	$0.519890\pi$
$12$	0	0
$13$	−2.58579	−0.717168	−0.358584	−	0.933497i	$-0.616740\pi$
$13$	−2.58579	−0.717168	−0.358584	+	0.933497i	$0.616740\pi$
$14$	6.82843	1.82497
$15$	−1.00000	−0.258199
$16$	−4.00000	−1.00000
$17$	0.585786	0.142074	0.0710370	−	0.997474i	$-0.477369\pi$
$17$	0.585786	0.142074	0.0710370	+	0.997474i	$0.477369\pi$
$18$	−1.41421	−0.333333
$19$	−1.00000	−0.229416	−0.114708	−	0.993399i	$-0.536593\pi$
$19$	−1.00000	−0.229416	−0.114708	+	0.993399i	$0.536593\pi$
$20$	0	0
$21$	4.82843	1.05365
$22$	0.585786	0.124890
$23$	0	0
$23$	0	0
$24$	−2.82843	−0.577350
$25$	1.00000	0.200000
$26$	3.65685	0.717168
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−4.82843	−0.896616	−0.448308	−	0.893879i	$-0.647973\pi$
$29$	−4.82843	−0.896616	−0.448308	+	0.893879i	$0.647973\pi$
$30$	1.41421	0.258199
$31$	1.82843	0.328395	0.164198	−	0.986427i	$-0.447497\pi$
$31$	1.82843	0.328395	0.164198	+	0.986427i	$0.447497\pi$
$32$	0	0
$33$	0.414214	0.0721053
$34$	−0.828427	−0.142074
$35$	−4.82843	−0.816153
$36$	0	0
$37$	10.2426	1.68388	0.841940	−	0.539571i	$-0.181414\pi$
$37$	10.2426	1.68388	0.841940	+	0.539571i	$0.181414\pi$
$38$	1.41421	0.229416
$39$	2.58579	0.414057
$40$	2.82843	0.447214
$41$	−4.41421	−0.689384	−0.344692	−	0.938716i	$-0.612017\pi$
$41$	−4.41421	−0.689384	−0.344692	+	0.938716i	$0.612017\pi$
$42$	−6.82843	−1.05365
$43$	1.75736	0.267995	0.133997	−	0.990982i	$-0.457219\pi$
$43$	1.75736	0.267995	0.133997	+	0.990982i	$0.457219\pi$
$44$	0	0
$45$	1.00000	0.149071
$46$	0	0
$47$	7.65685	1.11687	0.558433	−	0.829549i	$-0.311403\pi$
$47$	7.65685	1.11687	0.558433	+	0.829549i	$0.311403\pi$
$48$	4.00000	0.577350
$49$	16.3137	2.33053
$50$	−1.41421	−0.200000
$51$	−0.585786	−0.0820265
$52$	0	0
$53$	−4.24264	−0.582772	−0.291386	−	0.956606i	$-0.594116\pi$
$53$	−4.24264	−0.582772	−0.291386	+	0.956606i	$0.594116\pi$
$54$	1.41421	0.192450
$55$	−0.414214	−0.0558525
$56$	−13.6569	−1.82497
$57$	1.00000	0.132453
$58$	6.82843	0.896616
$59$	−11.6569	−1.51759	−0.758797	−	0.651328i	$-0.774212\pi$
$59$	−11.6569	−1.51759	−0.758797	+	0.651328i	$0.774212\pi$
$60$	0	0
$61$	−12.6569	−1.62054	−0.810272	−	0.586054i	$-0.800681\pi$
$61$	−12.6569	−1.62054	−0.810272	+	0.586054i	$0.800681\pi$
$62$	−2.58579	−0.328395
$63$	−4.82843	−0.608325
$64$	8.00000	1.00000
$65$	−2.58579	−0.320727
$66$	−0.585786	−0.0721053
$67$	2.48528	0.303625	0.151813	−	0.988409i	$-0.451489\pi$
$67$	2.48528	0.303625	0.151813	+	0.988409i	$0.451489\pi$
$68$	0	0
$69$	0	0
$70$	6.82843	0.816153
$71$	14.8995	1.76824	0.884122	−	0.467255i	$-0.154757\pi$
$71$	14.8995	1.76824	0.884122	+	0.467255i	$0.154757\pi$
$72$	2.82843	0.333333
$73$	14.4853	1.69537	0.847687	−	0.530497i	$-0.177995\pi$
$73$	14.4853	1.69537	0.847687	+	0.530497i	$0.177995\pi$
$74$	−14.4853	−1.68388
$75$	−1.00000	−0.115470
$76$	0	0
$77$	2.00000	0.227921
$78$	−3.65685	−0.414057
$79$	1.34315	0.151116	0.0755579	−	0.997141i	$-0.475926\pi$
$79$	1.34315	0.151116	0.0755579	+	0.997141i	$0.475926\pi$
$80$	−4.00000	−0.447214
$81$	1.00000	0.111111
$82$	6.24264	0.689384
$83$	5.89949	0.647554	0.323777	−	0.946133i	$-0.395047\pi$
$83$	5.89949	0.647554	0.323777	+	0.946133i	$0.395047\pi$
$84$	0	0
$85$	0.585786	0.0635375
$86$	−2.48528	−0.267995
$87$	4.82843	0.517662
$88$	−1.17157	−0.124890
$89$	−5.17157	−0.548186	−0.274093	−	0.961703i	$-0.588378\pi$
$89$	−5.17157	−0.548186	−0.274093	+	0.961703i	$0.588378\pi$
$90$	−1.41421	−0.149071
$91$	12.4853	1.30881
$92$	0	0
$93$	−1.82843	−0.189599
$94$	−10.8284	−1.11687
$95$	−1.00000	−0.102598
$96$	0	0
$97$	7.89949	0.802072	0.401036	−	0.916062i	$-0.368650\pi$
$97$	7.89949	0.802072	0.401036	+	0.916062i	$0.368650\pi$
$98$	−23.0711	−2.33053
$99$	−0.414214	−0.0416300
$100$	0	0
$101$	−16.4142	−1.63328	−0.816638	−	0.577151i	$-0.804165\pi$
$101$	−16.4142	−1.63328	−0.816638	+	0.577151i	$0.804165\pi$
$102$	0.828427	0.0820265
$103$	−11.0711	−1.09086	−0.545432	−	0.838155i	$-0.683635\pi$
$103$	−11.0711	−1.09086	−0.545432	+	0.838155i	$0.683635\pi$
$104$	−7.31371	−0.717168
$105$	4.82843	0.471206
$106$	6.00000	0.582772
$107$	14.1421	1.36717	0.683586	−	0.729870i	$-0.260419\pi$
$107$	14.1421	1.36717	0.683586	+	0.729870i	$0.260419\pi$
$108$	0	0
$109$	−9.48528	−0.908525	−0.454263	−	0.890868i	$-0.650097\pi$
$109$	−9.48528	−0.908525	−0.454263	+	0.890868i	$0.650097\pi$
$110$	0.585786	0.0558525
$111$	−10.2426	−0.972188
$112$	19.3137	1.82497
$113$	−2.82843	−0.266076	−0.133038	−	0.991111i	$-0.542473\pi$
$113$	−2.82843	−0.266076	−0.133038	+	0.991111i	$0.542473\pi$
$114$	−1.41421	−0.132453
$115$	0	0
$116$	0	0
$117$	−2.58579	−0.239056
$118$	16.4853	1.51759
$119$	−2.82843	−0.259281
$120$	−2.82843	−0.258199
$121$	−10.8284	−0.984402
$122$	17.8995	1.62054
$123$	4.41421	0.398016
$124$	0	0
$125$	1.00000	0.0894427
$126$	6.82843	0.608325
$127$	16.0000	1.41977	0.709885	−	0.704317i	$-0.248747\pi$
$127$	16.0000	1.41977	0.709885	+	0.704317i	$0.248747\pi$
$128$	−11.3137	−1.00000
$129$	−1.75736	−0.154727
$130$	3.65685	0.320727
$131$	22.5563	1.97076	0.985379	−	0.170379i	$-0.0544990\pi$
$131$	22.5563	1.97076	0.985379	+	0.170379i	$0.0544990\pi$
$132$	0	0
$133$	4.82843	0.418678
$134$	−3.51472	−0.303625
$135$	−1.00000	−0.0860663
$136$	1.65685	0.142074
$137$	−4.58579	−0.391790	−0.195895	−	0.980625i	$-0.562761\pi$
$137$	−4.58579	−0.391790	−0.195895	+	0.980625i	$0.562761\pi$
$138$	0	0
$139$	−7.48528	−0.634893	−0.317447	−	0.948276i	$-0.602825\pi$
$139$	−7.48528	−0.634893	−0.317447	+	0.948276i	$0.602825\pi$
$140$	0	0
$141$	−7.65685	−0.644823
$142$	−21.0711	−1.76824
$143$	1.07107	0.0895672
$144$	−4.00000	−0.333333
$145$	−4.82843	−0.400979
$146$	−20.4853	−1.69537
$147$	−16.3137	−1.34553
$148$	0	0
$149$	14.4142	1.18086	0.590429	−	0.807089i	$-0.298958\pi$
$149$	14.4142	1.18086	0.590429	+	0.807089i	$0.298958\pi$
$150$	1.41421	0.115470
$151$	19.1421	1.55776	0.778882	−	0.627170i	$-0.215787\pi$
$151$	19.1421	1.55776	0.778882	+	0.627170i	$0.215787\pi$
$152$	−2.82843	−0.229416
$153$	0.585786	0.0473580
$154$	−2.82843	−0.227921
$155$	1.82843	0.146863
$156$	0	0
$157$	6.24264	0.498217	0.249108	−	0.968476i	$-0.419862\pi$
$157$	6.24264	0.498217	0.249108	+	0.968476i	$0.419862\pi$
$158$	−1.89949	−0.151116
$159$	4.24264	0.336463
$160$	0	0
$161$	0	0
$162$	−1.41421	−0.111111
$163$	20.2426	1.58553	0.792763	−	0.609530i	$-0.208642\pi$
$163$	20.2426	1.58553	0.792763	+	0.609530i	$0.208642\pi$
$164$	0	0
$165$	0.414214	0.0322465
$166$	−8.34315	−0.647554
$167$	0.727922	0.0563283	0.0281642	−	0.999603i	$-0.491034\pi$
$167$	0.727922	0.0563283	0.0281642	+	0.999603i	$0.491034\pi$
$168$	13.6569	1.05365
$169$	−6.31371	−0.485670
$170$	−0.828427	−0.0635375
$171$	−1.00000	−0.0764719
$172$	0	0
$173$	−7.55635	−0.574499	−0.287249	−	0.957856i	$-0.592741\pi$
$173$	−7.55635	−0.574499	−0.287249	+	0.957856i	$0.592741\pi$
$174$	−6.82843	−0.517662
$175$	−4.82843	−0.364995
$176$	1.65685	0.124890
$177$	11.6569	0.876183
$178$	7.31371	0.548186
$179$	5.58579	0.417501	0.208751	−	0.977969i	$-0.433060\pi$
$179$	5.58579	0.417501	0.208751	+	0.977969i	$0.433060\pi$
$180$	0	0
$181$	−10.6569	−0.792118	−0.396059	−	0.918225i	$-0.629622\pi$
$181$	−10.6569	−0.792118	−0.396059	+	0.918225i	$0.629622\pi$
$182$	−17.6569	−1.30881
$183$	12.6569	0.935622
$184$	0	0
$185$	10.2426	0.753054
$186$	2.58579	0.189599
$187$	−0.242641	−0.0177436
$188$	0	0
$189$	4.82843	0.351216
$190$	1.41421	0.102598
$191$	−12.0000	−0.868290	−0.434145	−	0.900843i	$-0.642949\pi$
$191$	−12.0000	−0.868290	−0.434145	+	0.900843i	$0.642949\pi$
$192$	−8.00000	−0.577350
$193$	−21.2132	−1.52696	−0.763480	−	0.645832i	$-0.776511\pi$
$193$	−21.2132	−1.52696	−0.763480	+	0.645832i	$0.776511\pi$
$194$	−11.1716	−0.802072
$195$	2.58579	0.185172
$196$	0	0
$197$	3.07107	0.218805	0.109402	−	0.993998i	$-0.465106\pi$
$197$	3.07107	0.218805	0.109402	+	0.993998i	$0.465106\pi$
$198$	0.585786	0.0416300
$199$	−17.0000	−1.20510	−0.602549	−	0.798082i	$-0.705848\pi$
$199$	−17.0000	−1.20510	−0.602549	+	0.798082i	$0.705848\pi$
$200$	2.82843	0.200000
$201$	−2.48528	−0.175298
$202$	23.2132	1.63328
$203$	23.3137	1.63630
$204$	0	0
$205$	−4.41421	−0.308302
$206$	15.6569	1.09086
$207$	0	0
$208$	10.3431	0.717168
$209$	0.414214	0.0286518
$210$	−6.82843	−0.471206
$211$	−5.34315	−0.367837	−0.183919	−	0.982941i	$-0.558878\pi$
$211$	−5.34315	−0.367837	−0.183919	+	0.982941i	$0.558878\pi$
$212$	0	0
$213$	−14.8995	−1.02090
$214$	−20.0000	−1.36717
$215$	1.75736	0.119851
$216$	−2.82843	−0.192450
$217$	−8.82843	−0.599313
$218$	13.4142	0.908525
$219$	−14.4853	−0.978825
$220$	0	0
$221$	−1.51472	−0.101891
$222$	14.4853	0.972188
$223$	16.7279	1.12018	0.560092	−	0.828430i	$-0.310766\pi$
$223$	16.7279	1.12018	0.560092	+	0.828430i	$0.310766\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	4.00000	0.266076
$227$	25.0711	1.66403	0.832013	−	0.554757i	$-0.187189\pi$
$227$	25.0711	1.66403	0.832013	+	0.554757i	$0.187189\pi$
$228$	0	0
$229$	21.6274	1.42918	0.714590	−	0.699544i	$-0.246613\pi$
$229$	21.6274	1.42918	0.714590	+	0.699544i	$0.246613\pi$
$230$	0	0
$231$	−2.00000	−0.131590
$232$	−13.6569	−0.896616
$233$	−16.8284	−1.10247	−0.551233	−	0.834351i	$-0.685843\pi$
$233$	−16.8284	−1.10247	−0.551233	+	0.834351i	$0.685843\pi$
$234$	3.65685	0.239056
$235$	7.65685	0.499478
$236$	0	0
$237$	−1.34315	−0.0872467
$238$	4.00000	0.259281
$239$	15.3848	0.995158	0.497579	−	0.867419i	$-0.334222\pi$
$239$	15.3848	0.995158	0.497579	+	0.867419i	$0.334222\pi$
$240$	4.00000	0.258199
$241$	−0.656854	−0.0423117	−0.0211559	−	0.999776i	$-0.506735\pi$
$241$	−0.656854	−0.0423117	−0.0211559	+	0.999776i	$0.506735\pi$
$242$	15.3137	0.984402
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	16.3137	1.04224
$246$	−6.24264	−0.398016
$247$	2.58579	0.164530
$248$	5.17157	0.328395
$249$	−5.89949	−0.373865
$250$	−1.41421	−0.0894427
$251$	19.3848	1.22356	0.611778	−	0.791029i	$-0.290455\pi$
$251$	19.3848	1.22356	0.611778	+	0.791029i	$0.290455\pi$
$252$	0	0
$253$	0	0
$254$	−22.6274	−1.41977
$255$	−0.585786	−0.0366834
$256$	0	0
$257$	−26.8701	−1.67611	−0.838054	−	0.545587i	$-0.816307\pi$
$257$	−26.8701	−1.67611	−0.838054	+	0.545587i	$0.816307\pi$
$258$	2.48528	0.154727
$259$	−49.4558	−3.07304
$260$	0	0
$261$	−4.82843	−0.298872
$262$	−31.8995	−1.97076
$263$	−6.72792	−0.414861	−0.207431	−	0.978250i	$-0.566510\pi$
$263$	−6.72792	−0.414861	−0.207431	+	0.978250i	$0.566510\pi$
$264$	1.17157	0.0721053
$265$	−4.24264	−0.260623
$266$	−6.82843	−0.418678
$267$	5.17157	0.316495
$268$	0	0
$269$	7.38478	0.450258	0.225129	−	0.974329i	$-0.427720\pi$
$269$	7.38478	0.450258	0.225129	+	0.974329i	$0.427720\pi$
$270$	1.41421	0.0860663
$271$	16.9706	1.03089	0.515444	−	0.856923i	$-0.327627\pi$
$271$	16.9706	1.03089	0.515444	+	0.856923i	$0.327627\pi$
$272$	−2.34315	−0.142074
$273$	−12.4853	−0.755644
$274$	6.48528	0.391790
$275$	−0.414214	−0.0249780
$276$	0	0
$277$	26.0416	1.56469	0.782345	−	0.622845i	$-0.214023\pi$
$277$	26.0416	1.56469	0.782345	+	0.622845i	$0.214023\pi$
$278$	10.5858	0.634893
$279$	1.82843	0.109465
$280$	−13.6569	−0.816153
$281$	0.0710678	0.00423955	0.00211978	−	0.999998i	$-0.499325\pi$
$281$	0.0710678	0.00423955	0.00211978	+	0.999998i	$0.499325\pi$
$282$	10.8284	0.644823
$283$	−30.8284	−1.83256	−0.916280	−	0.400539i	$-0.868823\pi$
$283$	−30.8284	−1.83256	−0.916280	+	0.400539i	$0.868823\pi$
$284$	0	0
$285$	1.00000	0.0592349
$286$	−1.51472	−0.0895672
$287$	21.3137	1.25811
$288$	0	0
$289$	−16.6569	−0.979815
$290$	6.82843	0.400979
$291$	−7.89949	−0.463077
$292$	0	0
$293$	−17.1716	−1.00317	−0.501587	−	0.865107i	$-0.667250\pi$
$293$	−17.1716	−1.00317	−0.501587	+	0.865107i	$0.667250\pi$
$294$	23.0711	1.34553
$295$	−11.6569	−0.678688
$296$	28.9706	1.68388
$297$	0.414214	0.0240351
$298$	−20.3848	−1.18086
$299$	0	0
$300$	0	0
$301$	−8.48528	−0.489083
$302$	−27.0711	−1.55776
$303$	16.4142	0.942972
$304$	4.00000	0.229416
$305$	−12.6569	−0.724729
$306$	−0.828427	−0.0473580
$307$	9.51472	0.543034	0.271517	−	0.962434i	$-0.412475\pi$
$307$	9.51472	0.543034	0.271517	+	0.962434i	$0.412475\pi$
$308$	0	0
$309$	11.0711	0.629811
$310$	−2.58579	−0.146863
$311$	−17.3137	−0.981770	−0.490885	−	0.871224i	$-0.663327\pi$
$311$	−17.3137	−0.981770	−0.490885	+	0.871224i	$0.663327\pi$
$312$	7.31371	0.414057
$313$	−16.2426	−0.918088	−0.459044	−	0.888413i	$-0.651808\pi$
$313$	−16.2426	−0.918088	−0.459044	+	0.888413i	$0.651808\pi$
$314$	−8.82843	−0.498217
$315$	−4.82843	−0.272051
$316$	0	0
$317$	19.1716	1.07678	0.538391	−	0.842695i	$-0.319032\pi$
$317$	19.1716	1.07678	0.538391	+	0.842695i	$0.319032\pi$
$318$	−6.00000	−0.336463
$319$	2.00000	0.111979
$320$	8.00000	0.447214
$321$	−14.1421	−0.789337
$322$	0	0
$323$	−0.585786	−0.0325940
$324$	0	0
$325$	−2.58579	−0.143434
$326$	−28.6274	−1.58553
$327$	9.48528	0.524537
$328$	−12.4853	−0.689384
$329$	−36.9706	−2.03825
$330$	−0.585786	−0.0322465
$331$	3.48528	0.191568	0.0957842	−	0.995402i	$-0.469464\pi$
$331$	3.48528	0.191568	0.0957842	+	0.995402i	$0.469464\pi$
$332$	0	0
$333$	10.2426	0.561293
$334$	−1.02944	−0.0563283
$335$	2.48528	0.135785
$336$	−19.3137	−1.05365
$337$	8.97056	0.488658	0.244329	−	0.969692i	$-0.421432\pi$
$337$	8.97056	0.488658	0.244329	+	0.969692i	$0.421432\pi$
$338$	8.92893	0.485670
$339$	2.82843	0.153619
$340$	0	0
$341$	−0.757359	−0.0410133
$342$	1.41421	0.0764719
$343$	−44.9706	−2.42818
$344$	4.97056	0.267995
$345$	0	0
$346$	10.6863	0.574499
$347$	22.3848	1.20168	0.600839	−	0.799370i	$-0.294833\pi$
$347$	22.3848	1.20168	0.600839	+	0.799370i	$0.294833\pi$
$348$	0	0
$349$	−4.68629	−0.250851	−0.125426	−	0.992103i	$-0.540030\pi$
$349$	−4.68629	−0.250851	−0.125426	+	0.992103i	$0.540030\pi$
$350$	6.82843	0.364995
$351$	2.58579	0.138019
$352$	0	0
$353$	−8.34315	−0.444061	−0.222030	−	0.975040i	$-0.571268\pi$
$353$	−8.34315	−0.444061	−0.222030	+	0.975040i	$0.571268\pi$
$354$	−16.4853	−0.876183
$355$	14.8995	0.790783
$356$	0	0
$357$	2.82843	0.149696
$358$	−7.89949	−0.417501
$359$	23.6569	1.24856	0.624281	−	0.781200i	$-0.285392\pi$
$359$	23.6569	1.24856	0.624281	+	0.781200i	$0.285392\pi$
$360$	2.82843	0.149071
$361$	−18.0000	−0.947368
$362$	15.0711	0.792118
$363$	10.8284	0.568345
$364$	0	0
$365$	14.4853	0.758194
$366$	−17.8995	−0.935622
$367$	−9.21320	−0.480925	−0.240463	−	0.970658i	$-0.577299\pi$
$367$	−9.21320	−0.480925	−0.240463	+	0.970658i	$0.577299\pi$
$368$	0	0
$369$	−4.41421	−0.229795
$370$	−14.4853	−0.753054
$371$	20.4853	1.06354
$372$	0	0
$373$	−7.51472	−0.389097	−0.194549	−	0.980893i	$-0.562324\pi$
$373$	−7.51472	−0.389097	−0.194549	+	0.980893i	$0.562324\pi$
$374$	0.343146	0.0177436
$375$	−1.00000	−0.0516398
$376$	21.6569	1.11687
$377$	12.4853	0.643025
$378$	−6.82843	−0.351216
$379$	−14.4853	−0.744059	−0.372029	−	0.928221i	$-0.621338\pi$
$379$	−14.4853	−0.744059	−0.372029	+	0.928221i	$0.621338\pi$
$380$	0	0
$381$	−16.0000	−0.819705
$382$	16.9706	0.868290
$383$	−3.65685	−0.186857	−0.0934283	−	0.995626i	$-0.529783\pi$
$383$	−3.65685	−0.186857	−0.0934283	+	0.995626i	$0.529783\pi$
$384$	11.3137	0.577350
$385$	2.00000	0.101929
$386$	30.0000	1.52696
$387$	1.75736	0.0893316
$388$	0	0
$389$	28.7574	1.45806	0.729028	−	0.684484i	$-0.239972\pi$
$389$	28.7574	1.45806	0.729028	+	0.684484i	$0.239972\pi$
$390$	−3.65685	−0.185172
$391$	0	0
$392$	46.1421	2.33053
$393$	−22.5563	−1.13782
$394$	−4.34315	−0.218805
$395$	1.34315	0.0675810
$396$	0	0
$397$	10.1421	0.509019	0.254510	−	0.967070i	$-0.418086\pi$
$397$	10.1421	0.509019	0.254510	+	0.967070i	$0.418086\pi$
$398$	24.0416	1.20510
$399$	−4.82843	−0.241724
$400$	−4.00000	−0.200000
$401$	−33.0416	−1.65002	−0.825010	−	0.565118i	$-0.808831\pi$
$401$	−33.0416	−1.65002	−0.825010	+	0.565118i	$0.808831\pi$
$402$	3.51472	0.175298
$403$	−4.72792	−0.235515
$404$	0	0
$405$	1.00000	0.0496904
$406$	−32.9706	−1.63630
$407$	−4.24264	−0.210300
$408$	−1.65685	−0.0820265
$409$	−28.9411	−1.43105	−0.715523	−	0.698589i	$-0.753812\pi$
$409$	−28.9411	−1.43105	−0.715523	+	0.698589i	$0.753812\pi$
$410$	6.24264	0.308302
$411$	4.58579	0.226200
$412$	0	0
$413$	56.2843	2.76957
$414$	0	0
$415$	5.89949	0.289595
$416$	0	0
$417$	7.48528	0.366556
$418$	−0.585786	−0.0286518
$419$	−18.8995	−0.923301	−0.461650	−	0.887062i	$-0.652743\pi$
$419$	−18.8995	−0.923301	−0.461650	+	0.887062i	$0.652743\pi$
$420$	0	0
$421$	31.9706	1.55815	0.779075	−	0.626931i	$-0.215689\pi$
$421$	31.9706	1.55815	0.779075	+	0.626931i	$0.215689\pi$
$422$	7.55635	0.367837
$423$	7.65685	0.372289
$424$	−12.0000	−0.582772
$425$	0.585786	0.0284148
$426$	21.0711	1.02090
$427$	61.1127	2.95745
$428$	0	0
$429$	−1.07107	−0.0517116
$430$	−2.48528	−0.119851
$431$	−17.9289	−0.863606	−0.431803	−	0.901968i	$-0.642122\pi$
$431$	−17.9289	−0.863606	−0.431803	+	0.901968i	$0.642122\pi$
$432$	4.00000	0.192450
$433$	−4.68629	−0.225209	−0.112604	−	0.993640i	$-0.535919\pi$
$433$	−4.68629	−0.225209	−0.112604	+	0.993640i	$0.535919\pi$
$434$	12.4853	0.599313
$435$	4.82843	0.231505
$436$	0	0
$437$	0	0
$438$	20.4853	0.978825
$439$	5.51472	0.263203	0.131602	−	0.991303i	$-0.457988\pi$
$439$	5.51472	0.263203	0.131602	+	0.991303i	$0.457988\pi$
$440$	−1.17157	−0.0558525
$441$	16.3137	0.776843
$442$	2.14214	0.101891
$443$	17.0711	0.811071	0.405535	−	0.914079i	$-0.367085\pi$
$443$	17.0711	0.811071	0.405535	+	0.914079i	$0.367085\pi$
$444$	0	0
$445$	−5.17157	−0.245156
$446$	−23.6569	−1.12018
$447$	−14.4142	−0.681769
$448$	−38.6274	−1.82497
$449$	18.0000	0.849473	0.424736	−	0.905317i	$-0.360367\pi$
$449$	18.0000	0.849473	0.424736	+	0.905317i	$0.360367\pi$
$450$	−1.41421	−0.0666667
$451$	1.82843	0.0860973
$452$	0	0
$453$	−19.1421	−0.899376
$454$	−35.4558	−1.66403
$455$	12.4853	0.585319
$456$	2.82843	0.132453
$457$	−16.4853	−0.771149	−0.385574	−	0.922677i	$-0.625997\pi$
$457$	−16.4853	−0.771149	−0.385574	+	0.922677i	$0.625997\pi$
$458$	−30.5858	−1.42918
$459$	−0.585786	−0.0273422
$460$	0	0
$461$	−25.2426	−1.17567	−0.587833	−	0.808982i	$-0.700019\pi$
$461$	−25.2426	−1.17567	−0.587833	+	0.808982i	$0.700019\pi$
$462$	2.82843	0.131590
$463$	16.3431	0.759530	0.379765	−	0.925083i	$-0.376005\pi$
$463$	16.3431	0.759530	0.379765	+	0.925083i	$0.376005\pi$
$464$	19.3137	0.896616
$465$	−1.82843	−0.0847913
$466$	23.7990	1.10247
$467$	−38.7696	−1.79404	−0.897020	−	0.441989i	$-0.854273\pi$
$467$	−38.7696	−1.79404	−0.897020	+	0.441989i	$0.854273\pi$
$468$	0	0
$469$	−12.0000	−0.554109
$470$	−10.8284	−0.499478
$471$	−6.24264	−0.287646
$472$	−32.9706	−1.51759
$473$	−0.727922	−0.0334699
$474$	1.89949	0.0872467
$475$	−1.00000	−0.0458831
$476$	0	0
$477$	−4.24264	−0.194257
$478$	−21.7574	−0.995158
$479$	−19.2426	−0.879219	−0.439609	−	0.898189i	$-0.644883\pi$
$479$	−19.2426	−0.879219	−0.439609	+	0.898189i	$0.644883\pi$
$480$	0	0
$481$	−26.4853	−1.20762
$482$	0.928932	0.0423117
$483$	0	0
$484$	0	0
$485$	7.89949	0.358698
$486$	1.41421	0.0641500
$487$	13.8995	0.629846	0.314923	−	0.949117i	$-0.398021\pi$
$487$	13.8995	0.629846	0.314923	+	0.949117i	$0.398021\pi$
$488$	−35.7990	−1.62054
$489$	−20.2426	−0.915404
$490$	−23.0711	−1.04224
$491$	15.3137	0.691098	0.345549	−	0.938401i	$-0.387693\pi$
$491$	15.3137	0.691098	0.345549	+	0.938401i	$0.387693\pi$
$492$	0	0
$493$	−2.82843	−0.127386
$494$	−3.65685	−0.164530
$495$	−0.414214	−0.0186175
$496$	−7.31371	−0.328395
$497$	−71.9411	−3.22700
$498$	8.34315	0.373865
$499$	−33.4853	−1.49901	−0.749504	−	0.662000i	$-0.769708\pi$
$499$	−33.4853	−1.49901	−0.749504	+	0.662000i	$0.769708\pi$
$500$	0	0
$501$	−0.727922	−0.0325212
$502$	−27.4142	−1.22356
$503$	−21.5147	−0.959294	−0.479647	−	0.877462i	$-0.659235\pi$
$503$	−21.5147	−0.959294	−0.479647	+	0.877462i	$0.659235\pi$
$504$	−13.6569	−0.608325
$505$	−16.4142	−0.730423
$506$	0	0
$507$	6.31371	0.280402
$508$	0	0
$509$	−14.8284	−0.657258	−0.328629	−	0.944459i	$-0.606587\pi$
$509$	−14.8284	−0.657258	−0.328629	+	0.944459i	$0.606587\pi$
$510$	0.828427	0.0366834
$511$	−69.9411	−3.09401
$512$	22.6274	1.00000
$513$	1.00000	0.0441511
$514$	38.0000	1.67611
$515$	−11.0711	−0.487850
$516$	0	0
$517$	−3.17157	−0.139486
$518$	69.9411	3.07304
$519$	7.55635	0.331687
$520$	−7.31371	−0.320727
$521$	34.8995	1.52897	0.764487	−	0.644639i	$-0.222992\pi$
$521$	34.8995	1.52897	0.764487	+	0.644639i	$0.222992\pi$
$522$	6.82843	0.298872
$523$	−16.2426	−0.710241	−0.355121	−	0.934821i	$-0.615560\pi$
$523$	−16.2426	−0.710241	−0.355121	+	0.934821i	$0.615560\pi$
$524$	0	0
$525$	4.82843	0.210730
$526$	9.51472	0.414861
$527$	1.07107	0.0466564
$528$	−1.65685	−0.0721053
$529$	0	0
$530$	6.00000	0.260623
$531$	−11.6569	−0.505864
$532$	0	0
$533$	11.4142	0.494404
$534$	−7.31371	−0.316495
$535$	14.1421	0.611418
$536$	7.02944	0.303625
$537$	−5.58579	−0.241044
$538$	−10.4437	−0.450258
$539$	−6.75736	−0.291060
$540$	0	0
$541$	25.1421	1.08094	0.540472	−	0.841362i	$-0.318246\pi$
$541$	25.1421	1.08094	0.540472	+	0.841362i	$0.318246\pi$
$542$	−24.0000	−1.03089
$543$	10.6569	0.457329
$544$	0	0
$545$	−9.48528	−0.406305
$546$	17.6569	0.755644
$547$	−31.2132	−1.33458	−0.667290	−	0.744798i	$-0.732546\pi$
$547$	−31.2132	−1.33458	−0.667290	+	0.744798i	$0.732546\pi$
$548$	0	0
$549$	−12.6569	−0.540181
$550$	0.585786	0.0249780
$551$	4.82843	0.205698
$552$	0	0
$553$	−6.48528	−0.275782
$554$	−36.8284	−1.56469
$555$	−10.2426	−0.434776
$556$	0	0
$557$	−32.8701	−1.39275	−0.696375	−	0.717679i	$-0.745205\pi$
$557$	−32.8701	−1.39275	−0.696375	+	0.717679i	$0.745205\pi$
$558$	−2.58579	−0.109465
$559$	−4.54416	−0.192197
$560$	19.3137	0.816153
$561$	0.242641	0.0102443
$562$	−0.100505	−0.00423955
$563$	19.8995	0.838664	0.419332	−	0.907833i	$-0.362264\pi$
$563$	19.8995	0.838664	0.419332	+	0.907833i	$0.362264\pi$
$564$	0	0
$565$	−2.82843	−0.118993
$566$	43.5980	1.83256
$567$	−4.82843	−0.202775
$568$	42.1421	1.76824
$569$	40.3553	1.69178	0.845892	−	0.533354i	$-0.179069\pi$
$569$	40.3553	1.69178	0.845892	+	0.533354i	$0.179069\pi$
$570$	−1.41421	−0.0592349
$571$	−32.4558	−1.35823	−0.679117	−	0.734030i	$-0.737637\pi$
$571$	−32.4558	−1.35823	−0.679117	+	0.734030i	$0.737637\pi$
$572$	0	0
$573$	12.0000	0.501307
$574$	−30.1421	−1.25811
$575$	0	0
$576$	8.00000	0.333333
$577$	39.6985	1.65267	0.826335	−	0.563179i	$-0.190422\pi$
$577$	39.6985	1.65267	0.826335	+	0.563179i	$0.190422\pi$
$578$	23.5563	0.979815
$579$	21.2132	0.881591
$580$	0	0
$581$	−28.4853	−1.18177
$582$	11.1716	0.463077
$583$	1.75736	0.0727824
$584$	40.9706	1.69537
$585$	−2.58579	−0.106909
$586$	24.2843	1.00317
$587$	−10.5858	−0.436922	−0.218461	−	0.975846i	$-0.570104\pi$
$587$	−10.5858	−0.436922	−0.218461	+	0.975846i	$0.570104\pi$
$588$	0	0
$589$	−1.82843	−0.0753390
$590$	16.4853	0.678688
$591$	−3.07107	−0.126327
$592$	−40.9706	−1.68388
$593$	−33.5980	−1.37970	−0.689852	−	0.723951i	$-0.742324\pi$
$593$	−33.5980	−1.37970	−0.689852	+	0.723951i	$0.742324\pi$
$594$	−0.585786	−0.0240351
$595$	−2.82843	−0.115954
$596$	0	0
$597$	17.0000	0.695764
$598$	0	0
$599$	−21.0416	−0.859738	−0.429869	−	0.902891i	$-0.641440\pi$
$599$	−21.0416	−0.859738	−0.429869	+	0.902891i	$0.641440\pi$
$600$	−2.82843	−0.115470
$601$	14.1716	0.578071	0.289035	−	0.957318i	$-0.406666\pi$
$601$	14.1716	0.578071	0.289035	+	0.957318i	$0.406666\pi$
$602$	12.0000	0.489083
$603$	2.48528	0.101208
$604$	0	0
$605$	−10.8284	−0.440238
$606$	−23.2132	−0.942972
$607$	−31.3553	−1.27267	−0.636337	−	0.771411i	$-0.719551\pi$
$607$	−31.3553	−1.27267	−0.636337	+	0.771411i	$0.719551\pi$
$608$	0	0
$609$	−23.3137	−0.944719
$610$	17.8995	0.724729
$611$	−19.7990	−0.800981
$612$	0	0
$613$	−44.7696	−1.80823	−0.904113	−	0.427294i	$-0.859467\pi$
$613$	−44.7696	−1.80823	−0.904113	+	0.427294i	$0.859467\pi$
$614$	−13.4558	−0.543034
$615$	4.41421	0.177998
$616$	5.65685	0.227921
$617$	−22.3848	−0.901177	−0.450589	−	0.892732i	$-0.648786\pi$
$617$	−22.3848	−0.901177	−0.450589	+	0.892732i	$0.648786\pi$
$618$	−15.6569	−0.629811
$619$	−6.65685	−0.267562	−0.133781	−	0.991011i	$-0.542712\pi$
$619$	−6.65685	−0.267562	−0.133781	+	0.991011i	$0.542712\pi$
$620$	0	0
$621$	0	0
$622$	24.4853	0.981770
$623$	24.9706	1.00042
$624$	−10.3431	−0.414057
$625$	1.00000	0.0400000
$626$	22.9706	0.918088
$627$	−0.414214	−0.0165421
$628$	0	0
$629$	6.00000	0.239236
$630$	6.82843	0.272051
$631$	28.4558	1.13281	0.566405	−	0.824127i	$-0.308334\pi$
$631$	28.4558	1.13281	0.566405	+	0.824127i	$0.308334\pi$
$632$	3.79899	0.151116
$633$	5.34315	0.212371
$634$	−27.1127	−1.07678
$635$	16.0000	0.634941
$636$	0	0
$637$	−42.1838	−1.67138
$638$	−2.82843	−0.111979
$639$	14.8995	0.589415
$640$	−11.3137	−0.447214
$641$	10.0711	0.397783	0.198892	−	0.980021i	$-0.436266\pi$
$641$	10.0711	0.397783	0.198892	+	0.980021i	$0.436266\pi$
$642$	20.0000	0.789337
$643$	24.0000	0.946468	0.473234	−	0.880937i	$-0.343087\pi$
$643$	24.0000	0.946468	0.473234	+	0.880937i	$0.343087\pi$
$644$	0	0
$645$	−1.75736	−0.0691960
$646$	0.828427	0.0325940
$647$	−4.24264	−0.166795	−0.0833977	−	0.996516i	$-0.526577\pi$
$647$	−4.24264	−0.166795	−0.0833977	+	0.996516i	$0.526577\pi$
$648$	2.82843	0.111111
$649$	4.82843	0.189532
$650$	3.65685	0.143434
$651$	8.82843	0.346013
$652$	0	0
$653$	15.1127	0.591406	0.295703	−	0.955280i	$-0.404446\pi$
$653$	15.1127	0.591406	0.295703	+	0.955280i	$0.404446\pi$
$654$	−13.4142	−0.524537
$655$	22.5563	0.881349
$656$	17.6569	0.689384
$657$	14.4853	0.565125
$658$	52.2843	2.03825
$659$	−46.3553	−1.80575	−0.902874	−	0.429906i	$-0.858547\pi$
$659$	−46.3553	−1.80575	−0.902874	+	0.429906i	$0.858547\pi$
$660$	0	0
$661$	−25.0000	−0.972387	−0.486194	−	0.873851i	$-0.661615\pi$
$661$	−25.0000	−0.972387	−0.486194	+	0.873851i	$0.661615\pi$
$662$	−4.92893	−0.191568
$663$	1.51472	0.0588268
$664$	16.6863	0.647554
$665$	4.82843	0.187238
$666$	−14.4853	−0.561293
$667$	0	0
$668$	0	0
$669$	−16.7279	−0.646739
$670$	−3.51472	−0.135785
$671$	5.24264	0.202390
$672$	0	0
$673$	18.0000	0.693849	0.346925	−	0.937893i	$-0.387226\pi$
$673$	18.0000	0.693849	0.346925	+	0.937893i	$0.387226\pi$
$674$	−12.6863	−0.488658
$675$	−1.00000	−0.0384900
$676$	0	0
$677$	−36.0000	−1.38359	−0.691796	−	0.722093i	$-0.743180\pi$
$677$	−36.0000	−1.38359	−0.691796	+	0.722093i	$0.743180\pi$
$678$	−4.00000	−0.153619
$679$	−38.1421	−1.46376
$680$	1.65685	0.0635375
$681$	−25.0711	−0.960725
$682$	1.07107	0.0410133
$683$	10.2426	0.391924	0.195962	−	0.980612i	$-0.437217\pi$
$683$	10.2426	0.391924	0.195962	+	0.980612i	$0.437217\pi$
$684$	0	0
$685$	−4.58579	−0.175214
$686$	63.5980	2.42818
$687$	−21.6274	−0.825137
$688$	−7.02944	−0.267995
$689$	10.9706	0.417945
$690$	0	0
$691$	−7.68629	−0.292400	−0.146200	−	0.989255i	$-0.546704\pi$
$691$	−7.68629	−0.292400	−0.146200	+	0.989255i	$0.546704\pi$
$692$	0	0
$693$	2.00000	0.0759737
$694$	−31.6569	−1.20168
$695$	−7.48528	−0.283933
$696$	13.6569	0.517662
$697$	−2.58579	−0.0979436
$698$	6.62742	0.250851
$699$	16.8284	0.636510
$700$	0	0
$701$	−17.3137	−0.653930	−0.326965	−	0.945036i	$-0.606026\pi$
$701$	−17.3137	−0.653930	−0.326965	+	0.945036i	$0.606026\pi$
$702$	−3.65685	−0.138019
$703$	−10.2426	−0.386309
$704$	−3.31371	−0.124890
$705$	−7.65685	−0.288374
$706$	11.7990	0.444061
$707$	79.2548	2.98068
$708$	0	0
$709$	−31.9411	−1.19957	−0.599787	−	0.800160i	$-0.704748\pi$
$709$	−31.9411	−1.19957	−0.599787	+	0.800160i	$0.704748\pi$
$710$	−21.0711	−0.790783
$711$	1.34315	0.0503719
$712$	−14.6274	−0.548186
$713$	0	0
$714$	−4.00000	−0.149696
$715$	1.07107	0.0400557
$716$	0	0
$717$	−15.3848	−0.574555
$718$	−33.4558	−1.24856
$719$	0	0		−	1.00000i	$-0.5\pi$
$719$	0	0			1.00000i	$0.5\pi$
$720$	−4.00000	−0.149071
$721$	53.4558	1.99080
$722$	25.4558	0.947368
$723$	0.656854	0.0244287
$724$	0	0
$725$	−4.82843	−0.179323
$726$	−15.3137	−0.568345
$727$	−25.9411	−0.962103	−0.481052	−	0.876692i	$-0.659745\pi$
$727$	−25.9411	−0.962103	−0.481052	+	0.876692i	$0.659745\pi$
$728$	35.3137	1.30881
$729$	1.00000	0.0370370
$730$	−20.4853	−0.758194
$731$	1.02944	0.0380751
$732$	0	0
$733$	20.4853	0.756641	0.378321	−	0.925675i	$-0.376502\pi$
$733$	20.4853	0.756641	0.378321	+	0.925675i	$0.376502\pi$
$734$	13.0294	0.480925
$735$	−16.3137	−0.601740
$736$	0	0
$737$	−1.02944	−0.0379198
$738$	6.24264	0.229795
$739$	−44.1127	−1.62271	−0.811356	−	0.584552i	$-0.801270\pi$
$739$	−44.1127	−1.62271	−0.811356	+	0.584552i	$0.801270\pi$
$740$	0	0
$741$	−2.58579	−0.0949912
$742$	−28.9706	−1.06354
$743$	−17.7990	−0.652982	−0.326491	−	0.945200i	$-0.605866\pi$
$743$	−17.7990	−0.652982	−0.326491	+	0.945200i	$0.605866\pi$
$744$	−5.17157	−0.189599
$745$	14.4142	0.528096
$746$	10.6274	0.389097
$747$	5.89949	0.215851
$748$	0	0
$749$	−68.2843	−2.49505
$750$	1.41421	0.0516398
$751$	−5.51472	−0.201235	−0.100617	−	0.994925i	$-0.532082\pi$
$751$	−5.51472	−0.201235	−0.100617	+	0.994925i	$0.532082\pi$
$752$	−30.6274	−1.11687
$753$	−19.3848	−0.706421
$754$	−17.6569	−0.643025
$755$	19.1421	0.696654
$756$	0	0
$757$	22.9706	0.834879	0.417440	−	0.908705i	$-0.362928\pi$
$757$	22.9706	0.834879	0.417440	+	0.908705i	$0.362928\pi$
$758$	20.4853	0.744059
$759$	0	0
$760$	−2.82843	−0.102598
$761$	4.07107	0.147576	0.0737880	−	0.997274i	$-0.476491\pi$
$761$	4.07107	0.147576	0.0737880	+	0.997274i	$0.476491\pi$
$762$	22.6274	0.819705
$763$	45.7990	1.65803
$764$	0	0
$765$	0.585786	0.0211792
$766$	5.17157	0.186857
$767$	30.1421	1.08837
$768$	0	0
$769$	−15.2843	−0.551165	−0.275582	−	0.961277i	$-0.588871\pi$
$769$	−15.2843	−0.551165	−0.275582	+	0.961277i	$0.588871\pi$
$770$	−2.82843	−0.101929
$771$	26.8701	0.967701
$772$	0	0
$773$	−42.7696	−1.53831	−0.769157	−	0.639060i	$-0.779324\pi$
$773$	−42.7696	−1.53831	−0.769157	+	0.639060i	$0.779324\pi$
$774$	−2.48528	−0.0893316
$775$	1.82843	0.0656790
$776$	22.3431	0.802072
$777$	49.4558	1.77422
$778$	−40.6690	−1.45806
$779$	4.41421	0.158156
$780$	0	0
$781$	−6.17157	−0.220836
$782$	0	0
$783$	4.82843	0.172554
$784$	−65.2548	−2.33053
$785$	6.24264	0.222809
$786$	31.8995	1.13782
$787$	−53.5980	−1.91056	−0.955281	−	0.295700i	$-0.904447\pi$
$787$	−53.5980	−1.91056	−0.955281	+	0.295700i	$0.904447\pi$
$788$	0	0
$789$	6.72792	0.239520
$790$	−1.89949	−0.0675810
$791$	13.6569	0.485582
$792$	−1.17157	−0.0416300
$793$	32.7279	1.16220
$794$	−14.3431	−0.509019
$795$	4.24264	0.150471
$796$	0	0
$797$	2.48528	0.0880332	0.0440166	−	0.999031i	$-0.485985\pi$
$797$	2.48528	0.0880332	0.0440166	+	0.999031i	$0.485985\pi$
$798$	6.82843	0.241724
$799$	4.48528	0.158678
$800$	0	0
$801$	−5.17157	−0.182729
$802$	46.7279	1.65002
$803$	−6.00000	−0.211735
$804$	0	0
$805$	0	0
$806$	6.68629	0.235515
$807$	−7.38478	−0.259956
$808$	−46.4264	−1.63328
$809$	−22.4142	−0.788042	−0.394021	−	0.919101i	$-0.628916\pi$
$809$	−22.4142	−0.788042	−0.394021	+	0.919101i	$0.628916\pi$
$810$	−1.41421	−0.0496904
$811$	34.3137	1.20492	0.602459	−	0.798150i	$-0.294188\pi$
$811$	34.3137	1.20492	0.602459	+	0.798150i	$0.294188\pi$
$812$	0	0
$813$	−16.9706	−0.595184
$814$	6.00000	0.210300
$815$	20.2426	0.709069
$816$	2.34315	0.0820265
$817$	−1.75736	−0.0614822
$818$	40.9289	1.43105
$819$	12.4853	0.436271
$820$	0	0
$821$	−0.556349	−0.0194167	−0.00970836	−	0.999953i	$-0.503090\pi$
$821$	−0.556349	−0.0194167	−0.00970836	+	0.999953i	$0.503090\pi$
$822$	−6.48528	−0.226200
$823$	−13.6152	−0.474597	−0.237298	−	0.971437i	$-0.576262\pi$
$823$	−13.6152	−0.474597	−0.237298	+	0.971437i	$0.576262\pi$
$824$	−31.3137	−1.09086
$825$	0.414214	0.0144211
$826$	−79.5980	−2.76957
$827$	−41.3553	−1.43807	−0.719033	−	0.694976i	$-0.755415\pi$
$827$	−41.3553	−1.43807	−0.719033	+	0.694976i	$0.755415\pi$
$828$	0	0
$829$	−1.97056	−0.0684405	−0.0342202	−	0.999414i	$-0.510895\pi$
$829$	−1.97056	−0.0684405	−0.0342202	+	0.999414i	$0.510895\pi$
$830$	−8.34315	−0.289595
$831$	−26.0416	−0.903374
$832$	−20.6863	−0.717168
$833$	9.55635	0.331108
$834$	−10.5858	−0.366556
$835$	0.727922	0.0251908
$836$	0	0
$837$	−1.82843	−0.0631997
$838$	26.7279	0.923301
$839$	−6.55635	−0.226350	−0.113175	−	0.993575i	$-0.536102\pi$
$839$	−6.55635	−0.226350	−0.113175	+	0.993575i	$0.536102\pi$
$840$	13.6569	0.471206
$841$	−5.68629	−0.196079
$842$	−45.2132	−1.55815
$843$	−0.0710678	−0.00244771
$844$	0	0
$845$	−6.31371	−0.217198
$846$	−10.8284	−0.372289
$847$	52.2843	1.79651
$848$	16.9706	0.582772
$849$	30.8284	1.05803
$850$	−0.828427	−0.0284148
$851$	0	0
$852$	0	0
$853$	12.0000	0.410872	0.205436	−	0.978671i	$-0.434139\pi$
$853$	12.0000	0.410872	0.205436	+	0.978671i	$0.434139\pi$
$854$	−86.4264	−2.95745
$855$	−1.00000	−0.0341993
$856$	40.0000	1.36717
$857$	29.3553	1.00276	0.501380	−	0.865227i	$-0.332826\pi$
$857$	29.3553	1.00276	0.501380	+	0.865227i	$0.332826\pi$
$858$	1.51472	0.0517116
$859$	−10.9706	−0.374311	−0.187155	−	0.982330i	$-0.559927\pi$
$859$	−10.9706	−0.374311	−0.187155	+	0.982330i	$0.559927\pi$
$860$	0	0
$861$	−21.3137	−0.726369
$862$	25.3553	0.863606
$863$	−14.5858	−0.496506	−0.248253	−	0.968695i	$-0.579856\pi$
$863$	−14.5858	−0.496506	−0.248253	+	0.968695i	$0.579856\pi$
$864$	0	0
$865$	−7.55635	−0.256924
$866$	6.62742	0.225209
$867$	16.6569	0.565696
$868$	0	0
$869$	−0.556349	−0.0188729
$870$	−6.82843	−0.231505
$871$	−6.42641	−0.217750
$872$	−26.8284	−0.908525
$873$	7.89949	0.267357
$874$	0	0
$875$	−4.82843	−0.163231
$876$	0	0
$877$	15.7574	0.532088	0.266044	−	0.963961i	$-0.414283\pi$
$877$	15.7574	0.532088	0.266044	+	0.963961i	$0.414283\pi$
$878$	−7.79899	−0.263203
$879$	17.1716	0.579183
$880$	1.65685	0.0558525
$881$	20.8284	0.701728	0.350864	−	0.936427i	$-0.385888\pi$
$881$	20.8284	0.701728	0.350864	+	0.936427i	$0.385888\pi$
$882$	−23.0711	−0.776843
$883$	3.79899	0.127846	0.0639231	−	0.997955i	$-0.479639\pi$
$883$	3.79899	0.127846	0.0639231	+	0.997955i	$0.479639\pi$
$884$	0	0
$885$	11.6569	0.391841
$886$	−24.1421	−0.811071
$887$	−36.1421	−1.21353	−0.606767	−	0.794880i	$-0.707534\pi$
$887$	−36.1421	−1.21353	−0.606767	+	0.794880i	$0.707534\pi$
$888$	−28.9706	−0.972188
$889$	−77.2548	−2.59104
$890$	7.31371	0.245156
$891$	−0.414214	−0.0138767
$892$	0	0
$893$	−7.65685	−0.256227
$894$	20.3848	0.681769
$895$	5.58579	0.186712
$896$	54.6274	1.82497
$897$	0	0
$898$	−25.4558	−0.849473
$899$	−8.82843	−0.294445
$900$	0	0
$901$	−2.48528	−0.0827967
$902$	−2.58579	−0.0860973
$903$	8.48528	0.282372
$904$	−8.00000	−0.266076
$905$	−10.6569	−0.354246
$906$	27.0711	0.899376
$907$	−4.87006	−0.161708	−0.0808538	−	0.996726i	$-0.525765\pi$
$907$	−4.87006	−0.161708	−0.0808538	+	0.996726i	$0.525765\pi$
$908$	0	0
$909$	−16.4142	−0.544425
$910$	−17.6569	−0.585319
$911$	−35.5269	−1.17706	−0.588530	−	0.808476i	$-0.700293\pi$
$911$	−35.5269	−1.17706	−0.588530	+	0.808476i	$0.700293\pi$
$912$	−4.00000	−0.132453
$913$	−2.44365	−0.0808730
$914$	23.3137	0.771149
$915$	12.6569	0.418423
$916$	0	0
$917$	−108.912	−3.59658
$918$	0.828427	0.0273422
$919$	−11.0000	−0.362857	−0.181428	−	0.983404i	$-0.558072\pi$
$919$	−11.0000	−0.362857	−0.181428	+	0.983404i	$0.558072\pi$
$920$	0	0
$921$	−9.51472	−0.313521
$922$	35.6985	1.17567
$923$	−38.5269	−1.26813
$924$	0	0
$925$	10.2426	0.336776
$926$	−23.1127	−0.759530
$927$	−11.0711	−0.363622
$928$	0	0
$929$	18.8995	0.620072	0.310036	−	0.950725i	$-0.399659\pi$
$929$	18.8995	0.620072	0.310036	+	0.950725i	$0.399659\pi$
$930$	2.58579	0.0847913
$931$	−16.3137	−0.534660
$932$	0	0
$933$	17.3137	0.566825
$934$	54.8284	1.79404
$935$	−0.242641	−0.00793520
$936$	−7.31371	−0.239056
$937$	−44.3848	−1.44999	−0.724994	−	0.688755i	$-0.758157\pi$
$937$	−44.3848	−1.44999	−0.724994	+	0.688755i	$0.758157\pi$
$938$	16.9706	0.554109
$939$	16.2426	0.530059
$940$	0	0
$941$	12.8995	0.420512	0.210256	−	0.977646i	$-0.432570\pi$
$941$	12.8995	0.420512	0.210256	+	0.977646i	$0.432570\pi$
$942$	8.82843	0.287646
$943$	0	0
$944$	46.6274	1.51759
$945$	4.82843	0.157069
$946$	1.02944	0.0334699
$947$	26.9289	0.875073	0.437536	−	0.899201i	$-0.355851\pi$
$947$	26.9289	0.875073	0.437536	+	0.899201i	$0.355851\pi$
$948$	0	0
$949$	−37.4558	−1.21587
$950$	1.41421	0.0458831
$951$	−19.1716	−0.621681
$952$	−8.00000	−0.259281
$953$	11.6569	0.377603	0.188801	−	0.982015i	$-0.439540\pi$
$953$	11.6569	0.377603	0.188801	+	0.982015i	$0.439540\pi$
$954$	6.00000	0.194257
$955$	−12.0000	−0.388311
$956$	0	0
$957$	−2.00000	−0.0646508
$958$	27.2132	0.879219
$959$	22.1421	0.715007
$960$	−8.00000	−0.258199
$961$	−27.6569	−0.892157
$962$	37.4558	1.20762
$963$	14.1421	0.455724
$964$	0	0
$965$	−21.2132	−0.682877
$966$	0	0
$967$	−51.0711	−1.64233	−0.821167	−	0.570687i	$-0.806677\pi$
$967$	−51.0711	−1.64233	−0.821167	+	0.570687i	$0.806677\pi$
$968$	−30.6274	−0.984402
$969$	0.585786	0.0188182
$970$	−11.1716	−0.358698
$971$	3.51472	0.112793	0.0563963	−	0.998408i	$-0.482039\pi$
$971$	3.51472	0.112793	0.0563963	+	0.998408i	$0.482039\pi$
$972$	0	0
$973$	36.1421	1.15866
$974$	−19.6569	−0.629846
$975$	2.58579	0.0828114
$976$	50.6274	1.62054
$977$	17.5563	0.561677	0.280839	−	0.959755i	$-0.409387\pi$
$977$	17.5563	0.561677	0.280839	+	0.959755i	$0.409387\pi$
$978$	28.6274	0.915404
$979$	2.14214	0.0684630
$980$	0	0
$981$	−9.48528	−0.302842
$982$	−21.6569	−0.691098
$983$	28.8701	0.920812	0.460406	−	0.887709i	$-0.347704\pi$
$983$	28.8701	0.920812	0.460406	+	0.887709i	$0.347704\pi$
$984$	12.4853	0.398016
$985$	3.07107	0.0978524
$986$	4.00000	0.127386
$987$	36.9706	1.17679
$988$	0	0
$989$	0	0
$990$	0.585786	0.0186175
$991$	−12.6274	−0.401123	−0.200562	−	0.979681i	$-0.564277\pi$
$991$	−12.6274	−0.401123	−0.200562	+	0.979681i	$0.564277\pi$
$992$	0	0
$993$	−3.48528	−0.110602
$994$	101.740	3.22700
$995$	−17.0000	−0.538936
$996$	0	0
$997$	6.10051	0.193205	0.0966025	−	0.995323i	$-0.469202\pi$
$997$	6.10051	0.193205	0.0966025	+	0.995323i	$0.469202\pi$
$998$	47.3553	1.49901
$999$	−10.2426	−0.324063

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7935.2.a.p.1.1	yes	2
23.22	odd	2		7935.2.a.o.1.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7935.2.a.o.1.1	✓	2	23.22	odd	2
7935.2.a.p.1.1	yes	2	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 7935 = 3 \cdot 5 \cdot 23^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7935.a (trivial)

\(f(q)\)	\(=\)	\(q-1.41421 q^{2} -1.00000 q^{3} +1.00000 q^{5} +1.41421 q^{6} -4.82843 q^{7} +2.82843 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.41421 q^{2} -1.00000 q^{3} +1.00000 q^{5} +1.41421 q^{6} -4.82843 q^{7} +2.82843 q^{8} +1.00000 q^{9} -1.41421 q^{10} -0.414214 q^{11} -2.58579 q^{13} +6.82843 q^{14} -1.00000 q^{15} -4.00000 q^{16} +0.585786 q^{17} -1.41421 q^{18} -1.00000 q^{19} +4.82843 q^{21} +0.585786 q^{22} -2.82843 q^{24} +1.00000 q^{25} +3.65685 q^{26} -1.00000 q^{27} -4.82843 q^{29} +1.41421 q^{30} +1.82843 q^{31} +0.414214 q^{33} -0.828427 q^{34} -4.82843 q^{35} +10.2426 q^{37} +1.41421 q^{38} +2.58579 q^{39} +2.82843 q^{40} -4.41421 q^{41} -6.82843 q^{42} +1.75736 q^{43} +1.00000 q^{45} +7.65685 q^{47} +4.00000 q^{48} +16.3137 q^{49} -1.41421 q^{50} -0.585786 q^{51} -4.24264 q^{53} +1.41421 q^{54} -0.414214 q^{55} -13.6569 q^{56} +1.00000 q^{57} +6.82843 q^{58} -11.6569 q^{59} -12.6569 q^{61} -2.58579 q^{62} -4.82843 q^{63} +8.00000 q^{64} -2.58579 q^{65} -0.585786 q^{66} +2.48528 q^{67} +6.82843 q^{70} +14.8995 q^{71} +2.82843 q^{72} +14.4853 q^{73} -14.4853 q^{74} -1.00000 q^{75} +2.00000 q^{77} -3.65685 q^{78} +1.34315 q^{79} -4.00000 q^{80} +1.00000 q^{81} +6.24264 q^{82} +5.89949 q^{83} +0.585786 q^{85} -2.48528 q^{86} +4.82843 q^{87} -1.17157 q^{88} -5.17157 q^{89} -1.41421 q^{90} +12.4853 q^{91} -1.82843 q^{93} -10.8284 q^{94} -1.00000 q^{95} +7.89949 q^{97} -23.0711 q^{98} -0.414214 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} + 2 q^{5} - 4 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^3 + 2 * q^5 - 4 * q^7 + 2 * q^9	\( 2 q - 2 q^{3} + 2 q^{5} - 4 q^{7} + 2 q^{9} + 2 q^{11} - 8 q^{13} + 8 q^{14} - 2 q^{15} - 8 q^{16} + 4 q^{17} - 2 q^{19} + 4 q^{21} + 4 q^{22} + 2 q^{25} - 4 q^{26} - 2 q^{27} - 4 q^{29} - 2 q^{31} - 2 q^{33} + 4 q^{34} - 4 q^{35} + 12 q^{37} + 8 q^{39} - 6 q^{41} - 8 q^{42} + 12 q^{43} + 2 q^{45} + 4 q^{47} + 8 q^{48} + 10 q^{49} - 4 q^{51} + 2 q^{55} - 16 q^{56} + 2 q^{57} + 8 q^{58} - 12 q^{59} - 14 q^{61} - 8 q^{62} - 4 q^{63} + 16 q^{64} - 8 q^{65} - 4 q^{66} - 12 q^{67} + 8 q^{70} + 10 q^{71} + 12 q^{73} - 12 q^{74} - 2 q^{75} + 4 q^{77} + 4 q^{78} + 14 q^{79} - 8 q^{80} + 2 q^{81} + 4 q^{82} - 8 q^{83} + 4 q^{85} + 12 q^{86} + 4 q^{87} - 8 q^{88} - 16 q^{89} + 8 q^{91} + 2 q^{93} - 16 q^{94} - 2 q^{95} - 4 q^{97} - 32 q^{98} + 2 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 + 2 * q^5 - 4 * q^7 + 2 * q^9 + 2 * q^11 - 8 * q^13 + 8 * q^14 - 2 * q^15 - 8 * q^16 + 4 * q^17 - 2 * q^19 + 4 * q^21 + 4 * q^22 + 2 * q^25 - 4 * q^26 - 2 * q^27 - 4 * q^29 - 2 * q^31 - 2 * q^33 + 4 * q^34 - 4 * q^35 + 12 * q^37 + 8 * q^39 - 6 * q^41 - 8 * q^42 + 12 * q^43 + 2 * q^45 + 4 * q^47 + 8 * q^48 + 10 * q^49 - 4 * q^51 + 2 * q^55 - 16 * q^56 + 2 * q^57 + 8 * q^58 - 12 * q^59 - 14 * q^61 - 8 * q^62 - 4 * q^63 + 16 * q^64 - 8 * q^65 - 4 * q^66 - 12 * q^67 + 8 * q^70 + 10 * q^71 + 12 * q^73 - 12 * q^74 - 2 * q^75 + 4 * q^77 + 4 * q^78 + 14 * q^79 - 8 * q^80 + 2 * q^81 + 4 * q^82 - 8 * q^83 + 4 * q^85 + 12 * q^86 + 4 * q^87 - 8 * q^88 - 16 * q^89 + 8 * q^91 + 2 * q^93 - 16 * q^94 - 2 * q^95 - 4 * q^97 - 32 * q^98 + 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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