LMFDB - Embedded newform 7935.2.a.o.1.2

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:	$0$
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			1.2
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)

$f(q)$	$=$	$q+1.41421 q^{2} -1.00000 q^{3} -1.00000 q^{5} -1.41421 q^{6} -0.828427 q^{7} -2.82843 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.41421 q^{2} -1.00000 q^{3} -1.00000 q^{5} -1.41421 q^{6} -0.828427 q^{7} -2.82843 q^{8} +1.00000 q^{9} -1.41421 q^{10} -2.41421 q^{11} -5.41421 q^{13} -1.17157 q^{14} +1.00000 q^{15} -4.00000 q^{16} -3.41421 q^{17} +1.41421 q^{18} +1.00000 q^{19} +0.828427 q^{21} -3.41421 q^{22} +2.82843 q^{24} +1.00000 q^{25} -7.65685 q^{26} -1.00000 q^{27} +0.828427 q^{29} +1.41421 q^{30} -3.82843 q^{31} +2.41421 q^{33} -4.82843 q^{34} +0.828427 q^{35} -1.75736 q^{37} +1.41421 q^{38} +5.41421 q^{39} +2.82843 q^{40} -1.58579 q^{41} +1.17157 q^{42} -10.2426 q^{43} -1.00000 q^{45} -3.65685 q^{47} +4.00000 q^{48} -6.31371 q^{49} +1.41421 q^{50} +3.41421 q^{51} -4.24264 q^{53} -1.41421 q^{54} +2.41421 q^{55} +2.34315 q^{56} -1.00000 q^{57} +1.17157 q^{58} -0.343146 q^{59} +1.34315 q^{61} -5.41421 q^{62} -0.828427 q^{63} +8.00000 q^{64} +5.41421 q^{65} +3.41421 q^{66} +14.4853 q^{67} +1.17157 q^{70} -4.89949 q^{71} -2.82843 q^{72} -2.48528 q^{73} -2.48528 q^{74} -1.00000 q^{75} +2.00000 q^{77} +7.65685 q^{78} -12.6569 q^{79} +4.00000 q^{80} +1.00000 q^{81} -2.24264 q^{82} +13.8995 q^{83} +3.41421 q^{85} -14.4853 q^{86} -0.828427 q^{87} +6.82843 q^{88} +10.8284 q^{89} -1.41421 q^{90} +4.48528 q^{91} +3.82843 q^{93} -5.17157 q^{94} -1.00000 q^{95} +11.8995 q^{97} -8.92893 q^{98} -2.41421 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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	$-0.333333\pi$
$2$	1.41421	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$3$	−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
$7$	−0.828427	−0.313116	−0.156558	−	0.987669i	$-0.550040\pi$
$7$	−0.828427	−0.313116	−0.156558	+	0.987669i	$0.550040\pi$
$8$	−2.82843	−1.00000
$9$	1.00000	0.333333
$10$	−1.41421	−0.447214
$11$	−2.41421	−0.727913	−0.363956	−	0.931416i	$-0.618574\pi$
$11$	−2.41421	−0.727913	−0.363956	+	0.931416i	$0.618574\pi$
$12$	0	0
$13$	−5.41421	−1.50163	−0.750816	−	0.660511i	$-0.770340\pi$
$13$	−5.41421	−1.50163	−0.750816	+	0.660511i	$0.770340\pi$
$14$	−1.17157	−0.313116
$15$	1.00000	0.258199
$16$	−4.00000	−1.00000
$17$	−3.41421	−0.828068	−0.414034	−	0.910261i	$-0.635881\pi$
$17$	−3.41421	−0.828068	−0.414034	+	0.910261i	$0.635881\pi$
$18$	1.41421	0.333333
$19$	1.00000	0.229416	0.114708	−	0.993399i	$-0.463407\pi$
$19$	1.00000	0.229416	0.114708	+	0.993399i	$0.463407\pi$
$20$	0	0
$21$	0.828427	0.180778
$22$	−3.41421	−0.727913
$23$	0	0
$23$	0	0
$24$	2.82843	0.577350
$25$	1.00000	0.200000
$26$	−7.65685	−1.50163
$27$	−1.00000	−0.192450
$28$	0	0
$29$	0.828427	0.153835	0.0769175	−	0.997037i	$-0.475492\pi$
$29$	0.828427	0.153835	0.0769175	+	0.997037i	$0.475492\pi$
$30$	1.41421	0.258199
$31$	−3.82843	−0.687606	−0.343803	−	0.939042i	$-0.611715\pi$
$31$	−3.82843	−0.687606	−0.343803	+	0.939042i	$0.611715\pi$
$32$	0	0
$33$	2.41421	0.420261
$34$	−4.82843	−0.828068
$35$	0.828427	0.140030
$36$	0	0
$37$	−1.75736	−0.288908	−0.144454	−	0.989512i	$-0.546143\pi$
$37$	−1.75736	−0.288908	−0.144454	+	0.989512i	$0.546143\pi$
$38$	1.41421	0.229416
$39$	5.41421	0.866968
$40$	2.82843	0.447214
$41$	−1.58579	−0.247658	−0.123829	−	0.992304i	$-0.539517\pi$
$41$	−1.58579	−0.247658	−0.123829	+	0.992304i	$0.539517\pi$
$42$	1.17157	0.180778
$43$	−10.2426	−1.56199	−0.780994	−	0.624538i	$-0.785287\pi$
$43$	−10.2426	−1.56199	−0.780994	+	0.624538i	$0.785287\pi$
$44$	0	0
$45$	−1.00000	−0.149071
$46$	0	0
$47$	−3.65685	−0.533407	−0.266704	−	0.963779i	$-0.585934\pi$
$47$	−3.65685	−0.533407	−0.266704	+	0.963779i	$0.585934\pi$
$48$	4.00000	0.577350
$49$	−6.31371	−0.901958
$50$	1.41421	0.200000
$51$	3.41421	0.478086
$52$	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
$55$	2.41421	0.325532
$56$	2.34315	0.313116
$57$	−1.00000	−0.132453
$58$	1.17157	0.153835
$59$	−0.343146	−0.0446738	−0.0223369	−	0.999751i	$-0.507111\pi$
$59$	−0.343146	−0.0446738	−0.0223369	+	0.999751i	$0.507111\pi$
$60$	0	0
$61$	1.34315	0.171972	0.0859861	−	0.996296i	$-0.472596\pi$
$61$	1.34315	0.171972	0.0859861	+	0.996296i	$0.472596\pi$
$62$	−5.41421	−0.687606
$63$	−0.828427	−0.104372
$64$	8.00000	1.00000
$65$	5.41421	0.671551
$66$	3.41421	0.420261
$67$	14.4853	1.76966	0.884829	−	0.465915i	$-0.154275\pi$
$67$	14.4853	1.76966	0.884829	+	0.465915i	$0.154275\pi$
$68$	0	0
$69$	0	0
$70$	1.17157	0.140030
$71$	−4.89949	−0.581463	−0.290732	−	0.956805i	$-0.593899\pi$
$71$	−4.89949	−0.581463	−0.290732	+	0.956805i	$0.593899\pi$
$72$	−2.82843	−0.333333
$73$	−2.48528	−0.290880	−0.145440	−	0.989367i	$-0.546460\pi$
$73$	−2.48528	−0.290880	−0.145440	+	0.989367i	$0.546460\pi$
$74$	−2.48528	−0.288908
$75$	−1.00000	−0.115470
$76$	0	0
$77$	2.00000	0.227921
$78$	7.65685	0.866968
$79$	−12.6569	−1.42401	−0.712004	−	0.702176i	$-0.752212\pi$
$79$	−12.6569	−1.42401	−0.712004	+	0.702176i	$0.752212\pi$
$80$	4.00000	0.447214
$81$	1.00000	0.111111
$82$	−2.24264	−0.247658
$83$	13.8995	1.52567	0.762834	−	0.646595i	$-0.223807\pi$
$83$	13.8995	1.52567	0.762834	+	0.646595i	$0.223807\pi$
$84$	0	0
$85$	3.41421	0.370323
$86$	−14.4853	−1.56199
$87$	−0.828427	−0.0888167
$88$	6.82843	0.727913
$89$	10.8284	1.14781	0.573905	−	0.818922i	$-0.305428\pi$
$89$	10.8284	1.14781	0.573905	+	0.818922i	$0.305428\pi$
$90$	−1.41421	−0.149071
$91$	4.48528	0.470185
$92$	0	0
$93$	3.82843	0.396989
$94$	−5.17157	−0.533407
$95$	−1.00000	−0.102598
$96$	0	0
$97$	11.8995	1.20821	0.604105	−	0.796904i	$-0.293531\pi$
$97$	11.8995	1.20821	0.604105	+	0.796904i	$0.293531\pi$
$98$	−8.92893	−0.901958
$99$	−2.41421	−0.242638
$100$	0	0
$101$	−13.5858	−1.35184	−0.675918	−	0.736977i	$-0.736253\pi$
$101$	−13.5858	−1.35184	−0.675918	+	0.736977i	$0.736253\pi$
$102$	4.82843	0.478086
$103$	−3.07107	−0.302601	−0.151301	−	0.988488i	$-0.548346\pi$
$103$	−3.07107	−0.302601	−0.151301	+	0.988488i	$0.548346\pi$
$104$	15.3137	1.50163
$105$	−0.828427	−0.0808462
$106$	−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
$109$	−7.48528	−0.716960	−0.358480	−	0.933537i	$-0.616705\pi$
$109$	−7.48528	−0.716960	−0.358480	+	0.933537i	$0.616705\pi$
$110$	3.41421	0.325532
$111$	1.75736	0.166801
$112$	3.31371	0.313116
$113$	−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
$117$	−5.41421	−0.500544
$118$	−0.485281	−0.0446738
$119$	2.82843	0.259281
$120$	−2.82843	−0.258199
$121$	−5.17157	−0.470143
$122$	1.89949	0.171972
$123$	1.58579	0.142986
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	−1.17157	−0.104372
$127$	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
$129$	10.2426	0.901814
$130$	7.65685	0.671551
$131$	−8.55635	−0.747572	−0.373786	−	0.927515i	$-0.621941\pi$
$131$	−8.55635	−0.747572	−0.373786	+	0.927515i	$0.621941\pi$
$132$	0	0
$133$	−0.828427	−0.0718337
$134$	20.4853	1.76966
$135$	1.00000	0.0860663
$136$	9.65685	0.828068
$137$	7.41421	0.633439	0.316720	−	0.948519i	$-0.397419\pi$
$137$	7.41421	0.633439	0.316720	+	0.948519i	$0.397419\pi$
$138$	0	0
$139$	9.48528	0.804531	0.402266	−	0.915523i	$-0.368223\pi$
$139$	9.48528	0.804531	0.402266	+	0.915523i	$0.368223\pi$
$140$	0	0
$141$	3.65685	0.307963
$142$	−6.92893	−0.581463
$143$	13.0711	1.09306
$144$	−4.00000	−0.333333
$145$	−0.828427	−0.0687971
$146$	−3.51472	−0.290880
$147$	6.31371	0.520746
$148$	0	0
$149$	−11.5858	−0.949145	−0.474572	−	0.880217i	$-0.657397\pi$
$149$	−11.5858	−0.949145	−0.474572	+	0.880217i	$0.657397\pi$
$150$	−1.41421	−0.115470
$151$	−9.14214	−0.743976	−0.371988	−	0.928237i	$-0.621324\pi$
$151$	−9.14214	−0.743976	−0.371988	+	0.928237i	$0.621324\pi$
$152$	−2.82843	−0.229416
$153$	−3.41421	−0.276023
$154$	2.82843	0.227921
$155$	3.82843	0.307507
$156$	0	0
$157$	2.24264	0.178982	0.0894911	−	0.995988i	$-0.471476\pi$
$157$	2.24264	0.178982	0.0894911	+	0.995988i	$0.471476\pi$
$158$	−17.8995	−1.42401
$159$	4.24264	0.336463
$160$	0	0
$161$	0	0
$162$	1.41421	0.111111
$163$	11.7574	0.920907	0.460454	−	0.887684i	$-0.347687\pi$
$163$	11.7574	0.920907	0.460454	+	0.887684i	$0.347687\pi$
$164$	0	0
$165$	−2.41421	−0.187946
$166$	19.6569	1.52567
$167$	−24.7279	−1.91350	−0.956752	−	0.290905i	$-0.906044\pi$
$167$	−24.7279	−1.91350	−0.956752	+	0.290905i	$0.906044\pi$
$168$	−2.34315	−0.180778
$169$	16.3137	1.25490
$170$	4.82843	0.370323
$171$	1.00000	0.0764719
$172$	0	0
$173$	23.5563	1.79096	0.895478	−	0.445106i	$-0.146834\pi$
$173$	23.5563	1.79096	0.895478	+	0.445106i	$0.146834\pi$
$174$	−1.17157	−0.0888167
$175$	−0.828427	−0.0626232
$176$	9.65685	0.727913
$177$	0.343146	0.0257924
$178$	15.3137	1.14781
$179$	8.41421	0.628908	0.314454	−	0.949273i	$-0.398179\pi$
$179$	8.41421	0.628908	0.314454	+	0.949273i	$0.398179\pi$
$180$	0	0
$181$	−0.656854	−0.0488236	−0.0244118	−	0.999702i	$-0.507771\pi$
$181$	−0.656854	−0.0488236	−0.0244118	+	0.999702i	$0.507771\pi$
$182$	6.34315	0.470185
$183$	−1.34315	−0.0992882
$184$	0	0
$185$	1.75736	0.129204
$186$	5.41421	0.396989
$187$	8.24264	0.602762
$188$	0	0
$189$	0.828427	0.0602592
$190$	−1.41421	−0.102598
$191$	12.0000	0.868290	0.434145	−	0.900843i	$-0.357051\pi$
$191$	12.0000	0.868290	0.434145	+	0.900843i	$0.357051\pi$
$192$	−8.00000	−0.577350
$193$	21.2132	1.52696	0.763480	−	0.645832i	$-0.223489\pi$
$193$	21.2132	1.52696	0.763480	+	0.645832i	$0.223489\pi$
$194$	16.8284	1.20821
$195$	−5.41421	−0.387720
$196$	0	0
$197$	−11.0711	−0.788781	−0.394390	−	0.918943i	$-0.629044\pi$
$197$	−11.0711	−0.788781	−0.394390	+	0.918943i	$0.629044\pi$
$198$	−3.41421	−0.242638
$199$	17.0000	1.20510	0.602549	−	0.798082i	$-0.294152\pi$
$199$	17.0000	1.20510	0.602549	+	0.798082i	$0.294152\pi$
$200$	−2.82843	−0.200000
$201$	−14.4853	−1.02171
$202$	−19.2132	−1.35184
$203$	−0.686292	−0.0481682
$204$	0	0
$205$	1.58579	0.110756
$206$	−4.34315	−0.302601
$207$	0	0
$208$	21.6569	1.50163
$209$	−2.41421	−0.166995
$210$	−1.17157	−0.0808462
$211$	−16.6569	−1.14671	−0.573353	−	0.819309i	$-0.694357\pi$
$211$	−16.6569	−1.14671	−0.573353	+	0.819309i	$0.694357\pi$
$212$	0	0
$213$	4.89949	0.335708
$214$	20.0000	1.36717
$215$	10.2426	0.698542
$216$	2.82843	0.192450
$217$	3.17157	0.215300
$218$	−10.5858	−0.716960
$219$	2.48528	0.167940
$220$	0	0
$221$	18.4853	1.24345
$222$	2.48528	0.166801
$223$	−8.72792	−0.584465	−0.292232	−	0.956347i	$-0.594398\pi$
$223$	−8.72792	−0.584465	−0.292232	+	0.956347i	$0.594398\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	−4.00000	−0.266076
$227$	−10.9289	−0.725379	−0.362689	−	0.931910i	$-0.618141\pi$
$227$	−10.9289	−0.725379	−0.362689	+	0.931910i	$0.618141\pi$
$228$	0	0
$229$	23.6274	1.56134	0.780672	−	0.624941i	$-0.214877\pi$
$229$	23.6274	1.56134	0.780672	+	0.624941i	$0.214877\pi$
$230$	0	0
$231$	−2.00000	−0.131590
$232$	−2.34315	−0.153835
$233$	−11.1716	−0.731874	−0.365937	−	0.930640i	$-0.619251\pi$
$233$	−11.1716	−0.731874	−0.365937	+	0.930640i	$0.619251\pi$
$234$	−7.65685	−0.500544
$235$	3.65685	0.238547
$236$	0	0
$237$	12.6569	0.822151
$238$	4.00000	0.259281
$239$	−21.3848	−1.38327	−0.691633	−	0.722249i	$-0.743108\pi$
$239$	−21.3848	−1.38327	−0.691633	+	0.722249i	$0.743108\pi$
$240$	−4.00000	−0.258199
$241$	−10.6569	−0.686468	−0.343234	−	0.939250i	$-0.611522\pi$
$241$	−10.6569	−0.686468	−0.343234	+	0.939250i	$0.611522\pi$
$242$	−7.31371	−0.470143
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	6.31371	0.403368
$246$	2.24264	0.142986
$247$	−5.41421	−0.344498
$248$	10.8284	0.687606
$249$	−13.8995	−0.880845
$250$	−1.41421	−0.0894427
$251$	17.3848	1.09732	0.548659	−	0.836046i	$-0.315139\pi$
$251$	17.3848	1.09732	0.548659	+	0.836046i	$0.315139\pi$
$252$	0	0
$253$	0	0
$254$	22.6274	1.41977
$255$	−3.41421	−0.213806
$256$	0	0
$257$	26.8701	1.67611	0.838054	−	0.545587i	$-0.183693\pi$
$257$	26.8701	1.67611	0.838054	+	0.545587i	$0.183693\pi$
$258$	14.4853	0.901814
$259$	1.45584	0.0904618
$260$	0	0
$261$	0.828427	0.0512784
$262$	−12.1005	−0.747572
$263$	−18.7279	−1.15481	−0.577407	−	0.816457i	$-0.695935\pi$
$263$	−18.7279	−1.15481	−0.577407	+	0.816457i	$0.695935\pi$
$264$	−6.82843	−0.420261
$265$	4.24264	0.260623
$266$	−1.17157	−0.0718337
$267$	−10.8284	−0.662689
$268$	0	0
$269$	−29.3848	−1.79162	−0.895811	−	0.444436i	$-0.853404\pi$
$269$	−29.3848	−1.79162	−0.895811	+	0.444436i	$0.853404\pi$
$270$	1.41421	0.0860663
$271$	−16.9706	−1.03089	−0.515444	−	0.856923i	$-0.672373\pi$
$271$	−16.9706	−1.03089	−0.515444	+	0.856923i	$0.672373\pi$
$272$	13.6569	0.828068
$273$	−4.48528	−0.271462
$274$	10.4853	0.633439
$275$	−2.41421	−0.145583
$276$	0	0
$277$	−22.0416	−1.32435	−0.662177	−	0.749348i	$-0.730367\pi$
$277$	−22.0416	−1.32435	−0.662177	+	0.749348i	$0.730367\pi$
$278$	13.4142	0.804531
$279$	−3.82843	−0.229202
$280$	−2.34315	−0.140030
$281$	14.0711	0.839410	0.419705	−	0.907661i	$-0.362134\pi$
$281$	14.0711	0.839410	0.419705	+	0.907661i	$0.362134\pi$
$282$	5.17157	0.307963
$283$	25.1716	1.49629	0.748147	−	0.663533i	$-0.230944\pi$
$283$	25.1716	1.49629	0.748147	+	0.663533i	$0.230944\pi$
$284$	0	0
$285$	1.00000	0.0592349
$286$	18.4853	1.09306
$287$	1.31371	0.0775458
$288$	0	0
$289$	−5.34315	−0.314303
$290$	−1.17157	−0.0687971
$291$	−11.8995	−0.697561
$292$	0	0
$293$	22.8284	1.33365	0.666825	−	0.745214i	$-0.267653\pi$
$293$	22.8284	1.33365	0.666825	+	0.745214i	$0.267653\pi$
$294$	8.92893	0.520746
$295$	0.343146	0.0199787
$296$	4.97056	0.288908
$297$	2.41421	0.140087
$298$	−16.3848	−0.949145
$299$	0	0
$300$	0	0
$301$	8.48528	0.489083
$302$	−12.9289	−0.743976
$303$	13.5858	0.780483
$304$	−4.00000	−0.229416
$305$	−1.34315	−0.0769083
$306$	−4.82843	−0.276023
$307$	26.4853	1.51159	0.755797	−	0.654806i	$-0.227249\pi$
$307$	26.4853	1.51159	0.755797	+	0.654806i	$0.227249\pi$
$308$	0	0
$309$	3.07107	0.174707
$310$	5.41421	0.307507
$311$	5.31371	0.301313	0.150656	−	0.988586i	$-0.451861\pi$
$311$	5.31371	0.301313	0.150656	+	0.988586i	$0.451861\pi$
$312$	−15.3137	−0.866968
$313$	7.75736	0.438472	0.219236	−	0.975672i	$-0.429644\pi$
$313$	7.75736	0.438472	0.219236	+	0.975672i	$0.429644\pi$
$314$	3.17157	0.178982
$315$	0.828427	0.0466766
$316$	0	0
$317$	24.8284	1.39450	0.697252	−	0.716826i	$-0.254406\pi$
$317$	24.8284	1.39450	0.697252	+	0.716826i	$0.254406\pi$
$318$	6.00000	0.336463
$319$	−2.00000	−0.111979
$320$	−8.00000	−0.447214
$321$	−14.1421	−0.789337
$322$	0	0
$323$	−3.41421	−0.189972
$324$	0	0
$325$	−5.41421	−0.300327
$326$	16.6274	0.920907
$327$	7.48528	0.413937
$328$	4.48528	0.247658
$329$	3.02944	0.167018
$330$	−3.41421	−0.187946
$331$	−13.4853	−0.741218	−0.370609	−	0.928789i	$-0.620851\pi$
$331$	−13.4853	−0.741218	−0.370609	+	0.928789i	$0.620851\pi$
$332$	0	0
$333$	−1.75736	−0.0963027
$334$	−34.9706	−1.91350
$335$	−14.4853	−0.791415
$336$	−3.31371	−0.180778
$337$	24.9706	1.36023	0.680117	−	0.733104i	$-0.261929\pi$
$337$	24.9706	1.36023	0.680117	+	0.733104i	$0.261929\pi$
$338$	23.0711	1.25490
$339$	2.82843	0.153619
$340$	0	0
$341$	9.24264	0.500517
$342$	1.41421	0.0764719
$343$	11.0294	0.595534
$344$	28.9706	1.56199
$345$	0	0
$346$	33.3137	1.79096
$347$	−14.3848	−0.772215	−0.386108	−	0.922454i	$-0.626181\pi$
$347$	−14.3848	−0.772215	−0.386108	+	0.922454i	$0.626181\pi$
$348$	0	0
$349$	−27.3137	−1.46207	−0.731035	−	0.682340i	$-0.760962\pi$
$349$	−27.3137	−1.46207	−0.731035	+	0.682340i	$0.760962\pi$
$350$	−1.17157	−0.0626232
$351$	5.41421	0.288989
$352$	0	0
$353$	−19.6569	−1.04623	−0.523114	−	0.852262i	$-0.675230\pi$
$353$	−19.6569	−1.04623	−0.523114	+	0.852262i	$0.675230\pi$
$354$	0.485281	0.0257924
$355$	4.89949	0.260038
$356$	0	0
$357$	−2.82843	−0.149696
$358$	11.8995	0.628908
$359$	−12.3431	−0.651446	−0.325723	−	0.945465i	$-0.605608\pi$
$359$	−12.3431	−0.651446	−0.325723	+	0.945465i	$0.605608\pi$
$360$	2.82843	0.149071
$361$	−18.0000	−0.947368
$362$	−0.928932	−0.0488236
$363$	5.17157	0.271437
$364$	0	0
$365$	2.48528	0.130086
$366$	−1.89949	−0.0992882
$367$	−33.2132	−1.73372	−0.866858	−	0.498556i	$-0.833864\pi$
$367$	−33.2132	−1.73372	−0.866858	+	0.498556i	$0.833864\pi$
$368$	0	0
$369$	−1.58579	−0.0825527
$370$	2.48528	0.129204
$371$	3.51472	0.182475
$372$	0	0
$373$	24.4853	1.26780	0.633900	−	0.773415i	$-0.281453\pi$
$373$	24.4853	1.26780	0.633900	+	0.773415i	$0.281453\pi$
$374$	11.6569	0.602762
$375$	1.00000	0.0516398
$376$	10.3431	0.533407
$377$	−4.48528	−0.231004
$378$	1.17157	0.0602592
$379$	−2.48528	−0.127660	−0.0638302	−	0.997961i	$-0.520332\pi$
$379$	−2.48528	−0.127660	−0.0638302	+	0.997961i	$0.520332\pi$
$380$	0	0
$381$	−16.0000	−0.819705
$382$	16.9706	0.868290
$383$	−7.65685	−0.391247	−0.195623	−	0.980679i	$-0.562673\pi$
$383$	−7.65685	−0.391247	−0.195623	+	0.980679i	$0.562673\pi$
$384$	−11.3137	−0.577350
$385$	−2.00000	−0.101929
$386$	30.0000	1.52696
$387$	−10.2426	−0.520663
$388$	0	0
$389$	−37.2426	−1.88828	−0.944138	−	0.329549i	$-0.893103\pi$
$389$	−37.2426	−1.88828	−0.944138	+	0.329549i	$0.893103\pi$
$390$	−7.65685	−0.387720
$391$	0	0
$392$	17.8579	0.901958
$393$	8.55635	0.431611
$394$	−15.6569	−0.788781
$395$	12.6569	0.636835
$396$	0	0
$397$	−18.1421	−0.910528	−0.455264	−	0.890357i	$-0.650455\pi$
$397$	−18.1421	−0.910528	−0.455264	+	0.890357i	$0.650455\pi$
$398$	24.0416	1.20510
$399$	0.828427	0.0414732
$400$	−4.00000	−0.200000
$401$	−15.0416	−0.751143	−0.375572	−	0.926793i	$-0.622554\pi$
$401$	−15.0416	−0.751143	−0.375572	+	0.926793i	$0.622554\pi$
$402$	−20.4853	−1.02171
$403$	20.7279	1.03253
$404$	0	0
$405$	−1.00000	−0.0496904
$406$	−0.970563	−0.0481682
$407$	4.24264	0.210300
$408$	−9.65685	−0.478086
$409$	38.9411	1.92551	0.962757	−	0.270367i	$-0.0871450\pi$
$409$	38.9411	1.92551	0.962757	+	0.270367i	$0.0871450\pi$
$410$	2.24264	0.110756
$411$	−7.41421	−0.365716
$412$	0	0
$413$	0.284271	0.0139881
$414$	0	0
$415$	−13.8995	−0.682299
$416$	0	0
$417$	−9.48528	−0.464496
$418$	−3.41421	−0.166995
$419$	−0.899495	−0.0439432	−0.0219716	−	0.999759i	$-0.506994\pi$
$419$	−0.899495	−0.0439432	−0.0219716	+	0.999759i	$0.506994\pi$
$420$	0	0
$421$	1.97056	0.0960394	0.0480197	−	0.998846i	$-0.484709\pi$
$421$	1.97056	0.0960394	0.0480197	+	0.998846i	$0.484709\pi$
$422$	−23.5563	−1.14671
$423$	−3.65685	−0.177802
$424$	12.0000	0.582772
$425$	−3.41421	−0.165614
$426$	6.92893	0.335708
$427$	−1.11270	−0.0538472
$428$	0	0
$429$	−13.0711	−0.631077
$430$	14.4853	0.698542
$431$	32.0711	1.54481	0.772404	−	0.635131i	$-0.219054\pi$
$431$	32.0711	1.54481	0.772404	+	0.635131i	$0.219054\pi$
$432$	4.00000	0.192450
$433$	27.3137	1.31261	0.656307	−	0.754494i	$-0.272118\pi$
$433$	27.3137	1.31261	0.656307	+	0.754494i	$0.272118\pi$
$434$	4.48528	0.215300
$435$	0.828427	0.0397200
$436$	0	0
$437$	0	0
$438$	3.51472	0.167940
$439$	22.4853	1.07316	0.536582	−	0.843848i	$-0.319715\pi$
$439$	22.4853	1.07316	0.536582	+	0.843848i	$0.319715\pi$
$440$	−6.82843	−0.325532
$441$	−6.31371	−0.300653
$442$	26.1421	1.24345
$443$	2.92893	0.139158	0.0695789	−	0.997576i	$-0.477834\pi$
$443$	2.92893	0.139158	0.0695789	+	0.997576i	$0.477834\pi$
$444$	0	0
$445$	−10.8284	−0.513317
$446$	−12.3431	−0.584465
$447$	11.5858	0.547989
$448$	−6.62742	−0.313116
$449$	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
$451$	3.82843	0.180274
$452$	0	0
$453$	9.14214	0.429535
$454$	−15.4558	−0.725379
$455$	−4.48528	−0.210273
$456$	2.82843	0.132453
$457$	−0.485281	−0.0227005	−0.0113503	−	0.999936i	$-0.503613\pi$
$457$	−0.485281	−0.0227005	−0.0113503	+	0.999936i	$0.503613\pi$
$458$	33.4142	1.56134
$459$	3.41421	0.159362
$460$	0	0
$461$	−16.7574	−0.780468	−0.390234	−	0.920716i	$-0.627606\pi$
$461$	−16.7574	−0.780468	−0.390234	+	0.920716i	$0.627606\pi$
$462$	−2.82843	−0.131590
$463$	27.6569	1.28532	0.642662	−	0.766150i	$-0.277830\pi$
$463$	27.6569	1.28532	0.642662	+	0.766150i	$0.277830\pi$
$464$	−3.31371	−0.153835
$465$	−3.82843	−0.177539
$466$	−15.7990	−0.731874
$467$	−34.7696	−1.60894	−0.804472	−	0.593991i	$-0.797551\pi$
$467$	−34.7696	−1.60894	−0.804472	+	0.593991i	$0.797551\pi$
$468$	0	0
$469$	−12.0000	−0.554109
$470$	5.17157	0.238547
$471$	−2.24264	−0.103335
$472$	0.970563	0.0446738
$473$	24.7279	1.13699
$474$	17.8995	0.822151
$475$	1.00000	0.0458831
$476$	0	0
$477$	−4.24264	−0.194257
$478$	−30.2426	−1.38327
$479$	10.7574	0.491516	0.245758	−	0.969331i	$-0.420963\pi$
$479$	10.7574	0.491516	0.245758	+	0.969331i	$0.420963\pi$
$480$	0	0
$481$	9.51472	0.433834
$482$	−15.0711	−0.686468
$483$	0	0
$484$	0	0
$485$	−11.8995	−0.540328
$486$	−1.41421	−0.0641500
$487$	−5.89949	−0.267332	−0.133666	−	0.991026i	$-0.542675\pi$
$487$	−5.89949	−0.267332	−0.133666	+	0.991026i	$0.542675\pi$
$488$	−3.79899	−0.171972
$489$	−11.7574	−0.531686
$490$	8.92893	0.403368
$491$	−7.31371	−0.330063	−0.165032	−	0.986288i	$-0.552773\pi$
$491$	−7.31371	−0.330063	−0.165032	+	0.986288i	$0.552773\pi$
$492$	0	0
$493$	−2.82843	−0.127386
$494$	−7.65685	−0.344498
$495$	2.41421	0.108511
$496$	15.3137	0.687606
$497$	4.05887	0.182065
$498$	−19.6569	−0.880845
$499$	−16.5147	−0.739300	−0.369650	−	0.929171i	$-0.620522\pi$
$499$	−16.5147	−0.739300	−0.369650	+	0.929171i	$0.620522\pi$
$500$	0	0
$501$	24.7279	1.10476
$502$	24.5858	1.09732
$503$	38.4853	1.71597	0.857987	−	0.513671i	$-0.171715\pi$
$503$	38.4853	1.71597	0.857987	+	0.513671i	$0.171715\pi$
$504$	2.34315	0.104372
$505$	13.5858	0.604560
$506$	0	0
$507$	−16.3137	−0.724517
$508$	0	0
$509$	−9.17157	−0.406523	−0.203261	−	0.979125i	$-0.565154\pi$
$509$	−9.17157	−0.406523	−0.203261	+	0.979125i	$0.565154\pi$
$510$	−4.82843	−0.213806
$511$	2.05887	0.0910792
$512$	−22.6274	−1.00000
$513$	−1.00000	−0.0441511
$514$	38.0000	1.67611
$515$	3.07107	0.135327
$516$	0	0
$517$	8.82843	0.388274
$518$	2.05887	0.0904618
$519$	−23.5563	−1.03401
$520$	−15.3137	−0.671551
$521$	−15.1005	−0.661565	−0.330783	−	0.943707i	$-0.607313\pi$
$521$	−15.1005	−0.661565	−0.330783	+	0.943707i	$0.607313\pi$
$522$	1.17157	0.0512784
$523$	7.75736	0.339206	0.169603	−	0.985512i	$-0.445752\pi$
$523$	7.75736	0.339206	0.169603	+	0.985512i	$0.445752\pi$
$524$	0	0
$525$	0.828427	0.0361555
$526$	−26.4853	−1.15481
$527$	13.0711	0.569385
$528$	−9.65685	−0.420261
$529$	0	0
$530$	6.00000	0.260623
$531$	−0.343146	−0.0148913
$532$	0	0
$533$	8.58579	0.371892
$534$	−15.3137	−0.662689
$535$	−14.1421	−0.611418
$536$	−40.9706	−1.76966
$537$	−8.41421	−0.363100
$538$	−41.5563	−1.79162
$539$	15.2426	0.656547
$540$	0	0
$541$	−3.14214	−0.135091	−0.0675455	−	0.997716i	$-0.521517\pi$
$541$	−3.14214	−0.135091	−0.0675455	+	0.997716i	$0.521517\pi$
$542$	−24.0000	−1.03089
$543$	0.656854	0.0281883
$544$	0	0
$545$	7.48528	0.320634
$546$	−6.34315	−0.271462
$547$	11.2132	0.479442	0.239721	−	0.970842i	$-0.422944\pi$
$547$	11.2132	0.479442	0.239721	+	0.970842i	$0.422944\pi$
$548$	0	0
$549$	1.34315	0.0573241
$550$	−3.41421	−0.145583
$551$	0.828427	0.0352922
$552$	0	0
$553$	10.4853	0.445880
$554$	−31.1716	−1.32435
$555$	−1.75736	−0.0745957
$556$	0	0
$557$	−20.8701	−0.884293	−0.442146	−	0.896943i	$-0.645783\pi$
$557$	−20.8701	−0.884293	−0.442146	+	0.896943i	$0.645783\pi$
$558$	−5.41421	−0.229202
$559$	55.4558	2.34553
$560$	−3.31371	−0.140030
$561$	−8.24264	−0.348005
$562$	19.8995	0.839410
$563$	−0.100505	−0.00423578	−0.00211789	−	0.999998i	$-0.500674\pi$
$563$	−0.100505	−0.00423578	−0.00211789	+	0.999998i	$0.500674\pi$
$564$	0	0
$565$	2.82843	0.118993
$566$	35.5980	1.49629
$567$	−0.828427	−0.0347907
$568$	13.8579	0.581463
$569$	30.3553	1.27256	0.636281	−	0.771457i	$-0.280472\pi$
$569$	30.3553	1.27256	0.636281	+	0.771457i	$0.280472\pi$
$570$	1.41421	0.0592349
$571$	−18.4558	−0.772353	−0.386177	−	0.922425i	$-0.626204\pi$
$571$	−18.4558	−0.772353	−0.386177	+	0.922425i	$0.626204\pi$
$572$	0	0
$573$	−12.0000	−0.501307
$574$	1.85786	0.0775458
$575$	0	0
$576$	8.00000	0.333333
$577$	−19.6985	−0.820059	−0.410029	−	0.912072i	$-0.634482\pi$
$577$	−19.6985	−0.820059	−0.410029	+	0.912072i	$0.634482\pi$
$578$	−7.55635	−0.314303
$579$	−21.2132	−0.881591
$580$	0	0
$581$	−11.5147	−0.477711
$582$	−16.8284	−0.697561
$583$	10.2426	0.424207
$584$	7.02944	0.290880
$585$	5.41421	0.223850
$586$	32.2843	1.33365
$587$	−13.4142	−0.553664	−0.276832	−	0.960918i	$-0.589285\pi$
$587$	−13.4142	−0.553664	−0.276832	+	0.960918i	$0.589285\pi$
$588$	0	0
$589$	−3.82843	−0.157748
$590$	0.485281	0.0199787
$591$	11.0711	0.455403
$592$	7.02944	0.288908
$593$	45.5980	1.87248	0.936242	−	0.351355i	$-0.114279\pi$
$593$	45.5980	1.87248	0.936242	+	0.351355i	$0.114279\pi$
$594$	3.41421	0.140087
$595$	−2.82843	−0.115954
$596$	0	0
$597$	−17.0000	−0.695764
$598$	0	0
$599$	27.0416	1.10489	0.552446	−	0.833549i	$-0.313695\pi$
$599$	27.0416	1.10489	0.552446	+	0.833549i	$0.313695\pi$
$600$	2.82843	0.115470
$601$	19.8284	0.808818	0.404409	−	0.914578i	$-0.367477\pi$
$601$	19.8284	0.808818	0.404409	+	0.914578i	$0.367477\pi$
$602$	12.0000	0.489083
$603$	14.4853	0.589886
$604$	0	0
$605$	5.17157	0.210254
$606$	19.2132	0.780483
$607$	39.3553	1.59738	0.798692	−	0.601740i	$-0.205526\pi$
$607$	39.3553	1.59738	0.798692	+	0.601740i	$0.205526\pi$
$608$	0	0
$609$	0.686292	0.0278099
$610$	−1.89949	−0.0769083
$611$	19.7990	0.800981
$612$	0	0
$613$	−28.7696	−1.16199	−0.580996	−	0.813907i	$-0.697337\pi$
$613$	−28.7696	−1.16199	−0.580996	+	0.813907i	$0.697337\pi$
$614$	37.4558	1.51159
$615$	−1.58579	−0.0639451
$616$	−5.65685	−0.227921
$617$	−14.3848	−0.579109	−0.289555	−	0.957161i	$-0.593507\pi$
$617$	−14.3848	−0.579109	−0.289555	+	0.957161i	$0.593507\pi$
$618$	4.34315	0.174707
$619$	−4.65685	−0.187175	−0.0935874	−	0.995611i	$-0.529833\pi$
$619$	−4.65685	−0.187175	−0.0935874	+	0.995611i	$0.529833\pi$
$620$	0	0
$621$	0	0
$622$	7.51472	0.301313
$623$	−8.97056	−0.359398
$624$	−21.6569	−0.866968
$625$	1.00000	0.0400000
$626$	10.9706	0.438472
$627$	2.41421	0.0964144
$628$	0	0
$629$	6.00000	0.239236
$630$	1.17157	0.0466766
$631$	22.4558	0.893953	0.446977	−	0.894546i	$-0.352501\pi$
$631$	22.4558	0.893953	0.446977	+	0.894546i	$0.352501\pi$
$632$	35.7990	1.42401
$633$	16.6569	0.662051
$634$	35.1127	1.39450
$635$	−16.0000	−0.634941
$636$	0	0
$637$	34.1838	1.35441
$638$	−2.82843	−0.111979
$639$	−4.89949	−0.193821
$640$	−11.3137	−0.447214
$641$	4.07107	0.160798	0.0803988	−	0.996763i	$-0.474381\pi$
$641$	4.07107	0.160798	0.0803988	+	0.996763i	$0.474381\pi$
$642$	−20.0000	−0.789337
$643$	−24.0000	−0.946468	−0.473234	−	0.880937i	$-0.656913\pi$
$643$	−24.0000	−0.946468	−0.473234	+	0.880937i	$0.656913\pi$
$644$	0	0
$645$	−10.2426	−0.403304
$646$	−4.82843	−0.189972
$647$	4.24264	0.166795	0.0833977	−	0.996516i	$-0.473423\pi$
$647$	4.24264	0.166795	0.0833977	+	0.996516i	$0.473423\pi$
$648$	−2.82843	−0.111111
$649$	0.828427	0.0325186
$650$	−7.65685	−0.300327
$651$	−3.17157	−0.124304
$652$	0	0
$653$	−47.1127	−1.84366	−0.921831	−	0.387592i	$-0.873307\pi$
$653$	−47.1127	−1.84366	−0.921831	+	0.387592i	$0.873307\pi$
$654$	10.5858	0.413937
$655$	8.55635	0.334324
$656$	6.34315	0.247658
$657$	−2.48528	−0.0969601
$658$	4.28427	0.167018
$659$	−24.3553	−0.948749	−0.474375	−	0.880323i	$-0.657326\pi$
$659$	−24.3553	−0.948749	−0.474375	+	0.880323i	$0.657326\pi$
$660$	0	0
$661$	25.0000	0.972387	0.486194	−	0.873851i	$-0.338385\pi$
$661$	25.0000	0.972387	0.486194	+	0.873851i	$0.338385\pi$
$662$	−19.0711	−0.741218
$663$	−18.4853	−0.717909
$664$	−39.3137	−1.52567
$665$	0.828427	0.0321250
$666$	−2.48528	−0.0963027
$667$	0	0
$668$	0	0
$669$	8.72792	0.337441
$670$	−20.4853	−0.791415
$671$	−3.24264	−0.125181
$672$	0	0
$673$	18.0000	0.693849	0.346925	−	0.937893i	$-0.387226\pi$
$673$	18.0000	0.693849	0.346925	+	0.937893i	$0.387226\pi$
$674$	35.3137	1.36023
$675$	−1.00000	−0.0384900
$676$	0	0
$677$	36.0000	1.38359	0.691796	−	0.722093i	$-0.256820\pi$
$677$	36.0000	1.38359	0.691796	+	0.722093i	$0.256820\pi$
$678$	4.00000	0.153619
$679$	−9.85786	−0.378310
$680$	−9.65685	−0.370323
$681$	10.9289	0.418798
$682$	13.0711	0.500517
$683$	1.75736	0.0672435	0.0336217	−	0.999435i	$-0.489296\pi$
$683$	1.75736	0.0672435	0.0336217	+	0.999435i	$0.489296\pi$
$684$	0	0
$685$	−7.41421	−0.283283
$686$	15.5980	0.595534
$687$	−23.6274	−0.901442
$688$	40.9706	1.56199
$689$	22.9706	0.875109
$690$	0	0
$691$	−30.3137	−1.15319	−0.576594	−	0.817031i	$-0.695619\pi$
$691$	−30.3137	−1.15319	−0.576594	+	0.817031i	$0.695619\pi$
$692$	0	0
$693$	2.00000	0.0759737
$694$	−20.3431	−0.772215
$695$	−9.48528	−0.359797
$696$	2.34315	0.0888167
$697$	5.41421	0.205078
$698$	−38.6274	−1.46207
$699$	11.1716	0.422548
$700$	0	0
$701$	−5.31371	−0.200696	−0.100348	−	0.994952i	$-0.531996\pi$
$701$	−5.31371	−0.200696	−0.100348	+	0.994952i	$0.531996\pi$
$702$	7.65685	0.288989
$703$	−1.75736	−0.0662801
$704$	−19.3137	−0.727913
$705$	−3.65685	−0.137725
$706$	−27.7990	−1.04623
$707$	11.2548	0.423282
$708$	0	0
$709$	−35.9411	−1.34980	−0.674899	−	0.737910i	$-0.735813\pi$
$709$	−35.9411	−1.34980	−0.674899	+	0.737910i	$0.735813\pi$
$710$	6.92893	0.260038
$711$	−12.6569	−0.474669
$712$	−30.6274	−1.14781
$713$	0	0
$714$	−4.00000	−0.149696
$715$	−13.0711	−0.488830
$716$	0	0
$717$	21.3848	0.798629
$718$	−17.4558	−0.651446
$719$	0	0		−	1.00000i	$-0.5\pi$
$719$	0	0			1.00000i	$0.5\pi$
$720$	4.00000	0.149071
$721$	2.54416	0.0947493
$722$	−25.4558	−0.947368
$723$	10.6569	0.396333
$724$	0	0
$725$	0.828427	0.0307670
$726$	7.31371	0.271437
$727$	−41.9411	−1.55551	−0.777755	−	0.628567i	$-0.783642\pi$
$727$	−41.9411	−1.55551	−0.777755	+	0.628567i	$0.783642\pi$
$728$	−12.6863	−0.470185
$729$	1.00000	0.0370370
$730$	3.51472	0.130086
$731$	34.9706	1.29343
$732$	0	0
$733$	−3.51472	−0.129819	−0.0649095	−	0.997891i	$-0.520676\pi$
$733$	−3.51472	−0.129819	−0.0649095	+	0.997891i	$0.520676\pi$
$734$	−46.9706	−1.73372
$735$	−6.31371	−0.232885
$736$	0	0
$737$	−34.9706	−1.28816
$738$	−2.24264	−0.0825527
$739$	18.1127	0.666286	0.333143	−	0.942876i	$-0.391891\pi$
$739$	18.1127	0.666286	0.333143	+	0.942876i	$0.391891\pi$
$740$	0	0
$741$	5.41421	0.198896
$742$	4.97056	0.182475
$743$	−21.7990	−0.799727	−0.399864	−	0.916575i	$-0.630943\pi$
$743$	−21.7990	−0.799727	−0.399864	+	0.916575i	$0.630943\pi$
$744$	−10.8284	−0.396989
$745$	11.5858	0.424470
$746$	34.6274	1.26780
$747$	13.8995	0.508556
$748$	0	0
$749$	−11.7157	−0.428083
$750$	1.41421	0.0516398
$751$	22.4853	0.820500	0.410250	−	0.911973i	$-0.365442\pi$
$751$	22.4853	0.820500	0.410250	+	0.911973i	$0.365442\pi$
$752$	14.6274	0.533407
$753$	−17.3848	−0.633536
$754$	−6.34315	−0.231004
$755$	9.14214	0.332716
$756$	0	0
$757$	10.9706	0.398732	0.199366	−	0.979925i	$-0.436112\pi$
$757$	10.9706	0.398732	0.199366	+	0.979925i	$0.436112\pi$
$758$	−3.51472	−0.127660
$759$	0	0
$760$	2.82843	0.102598
$761$	−10.0711	−0.365076	−0.182538	−	0.983199i	$-0.558431\pi$
$761$	−10.0711	−0.365076	−0.182538	+	0.983199i	$0.558431\pi$
$762$	−22.6274	−0.819705
$763$	6.20101	0.224492
$764$	0	0
$765$	3.41421	0.123441
$766$	−10.8284	−0.391247
$767$	1.85786	0.0670836
$768$	0	0
$769$	−41.2843	−1.48875	−0.744374	−	0.667762i	$-0.767252\pi$
$769$	−41.2843	−1.48875	−0.744374	+	0.667762i	$0.767252\pi$
$770$	−2.82843	−0.101929
$771$	−26.8701	−0.967701
$772$	0	0
$773$	−30.7696	−1.10670	−0.553352	−	0.832948i	$-0.686652\pi$
$773$	−30.7696	−1.10670	−0.553352	+	0.832948i	$0.686652\pi$
$774$	−14.4853	−0.520663
$775$	−3.82843	−0.137521
$776$	−33.6569	−1.20821
$777$	−1.45584	−0.0522281
$778$	−52.6690	−1.88828
$779$	−1.58579	−0.0568167
$780$	0	0
$781$	11.8284	0.423254
$782$	0	0
$783$	−0.828427	−0.0296056
$784$	25.2548	0.901958
$785$	−2.24264	−0.0800433
$786$	12.1005	0.431611
$787$	−25.5980	−0.912469	−0.456235	−	0.889859i	$-0.650802\pi$
$787$	−25.5980	−0.912469	−0.456235	+	0.889859i	$0.650802\pi$
$788$	0	0
$789$	18.7279	0.666732
$790$	17.8995	0.636835
$791$	2.34315	0.0833127
$792$	6.82843	0.242638
$793$	−7.27208	−0.258239
$794$	−25.6569	−0.910528
$795$	−4.24264	−0.150471
$796$	0	0
$797$	14.4853	0.513095	0.256547	−	0.966532i	$-0.417415\pi$
$797$	14.4853	0.513095	0.256547	+	0.966532i	$0.417415\pi$
$798$	1.17157	0.0414732
$799$	12.4853	0.441698
$800$	0	0
$801$	10.8284	0.382604
$802$	−21.2721	−0.751143
$803$	6.00000	0.211735
$804$	0	0
$805$	0	0
$806$	29.3137	1.03253
$807$	29.3848	1.03439
$808$	38.4264	1.35184
$809$	−19.5858	−0.688600	−0.344300	−	0.938860i	$-0.611884\pi$
$809$	−19.5858	−0.688600	−0.344300	+	0.938860i	$0.611884\pi$
$810$	−1.41421	−0.0496904
$811$	11.6863	0.410361	0.205181	−	0.978724i	$-0.434222\pi$
$811$	11.6863	0.410361	0.205181	+	0.978724i	$0.434222\pi$
$812$	0	0
$813$	16.9706	0.595184
$814$	6.00000	0.210300
$815$	−11.7574	−0.411842
$816$	−13.6569	−0.478086
$817$	−10.2426	−0.358345
$818$	55.0711	1.92551
$819$	4.48528	0.156728
$820$	0	0
$821$	30.5563	1.06642	0.533212	−	0.845982i	$-0.320985\pi$
$821$	30.5563	1.06642	0.533212	+	0.845982i	$0.320985\pi$
$822$	−10.4853	−0.365716
$823$	−50.3848	−1.75630	−0.878151	−	0.478383i	$-0.841223\pi$
$823$	−50.3848	−1.75630	−0.878151	+	0.478383i	$0.841223\pi$
$824$	8.68629	0.302601
$825$	2.41421	0.0840521
$826$	0.402020	0.0139881
$827$	−29.3553	−1.02079	−0.510393	−	0.859942i	$-0.670500\pi$
$827$	−29.3553	−1.02079	−0.510393	+	0.859942i	$0.670500\pi$
$828$	0	0
$829$	31.9706	1.11038	0.555192	−	0.831722i	$-0.312645\pi$
$829$	31.9706	1.11038	0.555192	+	0.831722i	$0.312645\pi$
$830$	−19.6569	−0.682299
$831$	22.0416	0.764616
$832$	−43.3137	−1.50163
$833$	21.5563	0.746883
$834$	−13.4142	−0.464496
$835$	24.7279	0.855745
$836$	0	0
$837$	3.82843	0.132330
$838$	−1.27208	−0.0439432
$839$	−24.5563	−0.847779	−0.423890	−	0.905714i	$-0.639336\pi$
$839$	−24.5563	−0.847779	−0.423890	+	0.905714i	$0.639336\pi$
$840$	2.34315	0.0808462
$841$	−28.3137	−0.976335
$842$	2.78680	0.0960394
$843$	−14.0711	−0.484633
$844$	0	0
$845$	−16.3137	−0.561209
$846$	−5.17157	−0.177802
$847$	4.28427	0.147209
$848$	16.9706	0.582772
$849$	−25.1716	−0.863886
$850$	−4.82843	−0.165614
$851$	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	$0.434139\pi$
$854$	−1.57359	−0.0538472
$855$	−1.00000	−0.0341993
$856$	−40.0000	−1.36717
$857$	−41.3553	−1.41267	−0.706336	−	0.707877i	$-0.749653\pi$
$857$	−41.3553	−1.41267	−0.706336	+	0.707877i	$0.749653\pi$
$858$	−18.4853	−0.631077
$859$	22.9706	0.783745	0.391873	−	0.920019i	$-0.371827\pi$
$859$	22.9706	0.783745	0.391873	+	0.920019i	$0.371827\pi$
$860$	0	0
$861$	−1.31371	−0.0447711
$862$	45.3553	1.54481
$863$	−17.4142	−0.592787	−0.296393	−	0.955066i	$-0.595784\pi$
$863$	−17.4142	−0.592787	−0.296393	+	0.955066i	$0.595784\pi$
$864$	0	0
$865$	−23.5563	−0.800940
$866$	38.6274	1.31261
$867$	5.34315	0.181463
$868$	0	0
$869$	30.5563	1.03655
$870$	1.17157	0.0397200
$871$	−78.4264	−2.65738
$872$	21.1716	0.716960
$873$	11.8995	0.402737
$874$	0	0
$875$	0.828427	0.0280059
$876$	0	0
$877$	24.2426	0.818616	0.409308	−	0.912396i	$-0.365770\pi$
$877$	24.2426	0.818616	0.409308	+	0.912396i	$0.365770\pi$
$878$	31.7990	1.07316
$879$	−22.8284	−0.769984
$880$	−9.65685	−0.325532
$881$	−15.1716	−0.511143	−0.255572	−	0.966790i	$-0.582264\pi$
$881$	−15.1716	−0.511143	−0.255572	+	0.966790i	$0.582264\pi$
$882$	−8.92893	−0.300653
$883$	−35.7990	−1.20473	−0.602366	−	0.798220i	$-0.705775\pi$
$883$	−35.7990	−1.20473	−0.602366	+	0.798220i	$0.705775\pi$
$884$	0	0
$885$	−0.343146	−0.0115347
$886$	4.14214	0.139158
$887$	−7.85786	−0.263841	−0.131921	−	0.991260i	$-0.542114\pi$
$887$	−7.85786	−0.263841	−0.131921	+	0.991260i	$0.542114\pi$
$888$	−4.97056	−0.166801
$889$	−13.2548	−0.444553
$890$	−15.3137	−0.513317
$891$	−2.41421	−0.0808792
$892$	0	0
$893$	−3.65685	−0.122372
$894$	16.3848	0.547989
$895$	−8.41421	−0.281256
$896$	−9.37258	−0.313116
$897$	0	0
$898$	25.4558	0.849473
$899$	−3.17157	−0.105778
$900$	0	0
$901$	14.4853	0.482575
$902$	5.41421	0.180274
$903$	−8.48528	−0.282372
$904$	8.00000	0.266076
$905$	0.656854	0.0218346
$906$	12.9289	0.429535
$907$	−48.8701	−1.62270	−0.811352	−	0.584558i	$-0.801268\pi$
$907$	−48.8701	−1.62270	−0.811352	+	0.584558i	$0.801268\pi$
$908$	0	0
$909$	−13.5858	−0.450612
$910$	−6.34315	−0.210273
$911$	−29.5269	−0.978270	−0.489135	−	0.872208i	$-0.662688\pi$
$911$	−29.5269	−0.978270	−0.489135	+	0.872208i	$0.662688\pi$
$912$	4.00000	0.132453
$913$	−33.5563	−1.11055
$914$	−0.686292	−0.0227005
$915$	1.34315	0.0444030
$916$	0	0
$917$	7.08831	0.234077
$918$	4.82843	0.159362
$919$	11.0000	0.362857	0.181428	−	0.983404i	$-0.441928\pi$
$919$	11.0000	0.362857	0.181428	+	0.983404i	$0.441928\pi$
$920$	0	0
$921$	−26.4853	−0.872720
$922$	−23.6985	−0.780468
$923$	26.5269	0.873144
$924$	0	0
$925$	−1.75736	−0.0577816
$926$	39.1127	1.28532
$927$	−3.07107	−0.100867
$928$	0	0
$929$	−0.899495	−0.0295115	−0.0147557	−	0.999891i	$-0.504697\pi$
$929$	−0.899495	−0.0295115	−0.0147557	+	0.999891i	$0.504697\pi$
$930$	−5.41421	−0.177539
$931$	−6.31371	−0.206923
$932$	0	0
$933$	−5.31371	−0.173963
$934$	−49.1716	−1.60894
$935$	−8.24264	−0.269563
$936$	15.3137	0.500544
$937$	7.61522	0.248779	0.124389	−	0.992233i	$-0.460303\pi$
$937$	7.61522	0.248779	0.124389	+	0.992233i	$0.460303\pi$
$938$	−16.9706	−0.554109
$939$	−7.75736	−0.253152
$940$	0	0
$941$	6.89949	0.224917	0.112459	−	0.993656i	$-0.464127\pi$
$941$	6.89949	0.224917	0.112459	+	0.993656i	$0.464127\pi$
$942$	−3.17157	−0.103335
$943$	0	0
$944$	1.37258	0.0446738
$945$	−0.828427	−0.0269487
$946$	34.9706	1.13699
$947$	41.0711	1.33463	0.667315	−	0.744775i	$-0.267443\pi$
$947$	41.0711	1.33463	0.667315	+	0.744775i	$0.267443\pi$
$948$	0	0
$949$	13.4558	0.436795
$950$	1.41421	0.0458831
$951$	−24.8284	−0.805117
$952$	−8.00000	−0.259281
$953$	−0.343146	−0.0111156	−0.00555779	−	0.999985i	$-0.501769\pi$
$953$	−0.343146	−0.0111156	−0.00555779	+	0.999985i	$0.501769\pi$
$954$	−6.00000	−0.194257
$955$	−12.0000	−0.388311
$956$	0	0
$957$	2.00000	0.0646508
$958$	15.2132	0.491516
$959$	−6.14214	−0.198340
$960$	8.00000	0.258199
$961$	−16.3431	−0.527198
$962$	13.4558	0.433834
$963$	14.1421	0.455724
$964$	0	0
$965$	−21.2132	−0.682877
$966$	0	0
$967$	−36.9289	−1.18755	−0.593777	−	0.804630i	$-0.702364\pi$
$967$	−36.9289	−1.18755	−0.593777	+	0.804630i	$0.702364\pi$
$968$	14.6274	0.470143
$969$	3.41421	0.109680
$970$	−16.8284	−0.540328
$971$	−20.4853	−0.657404	−0.328702	−	0.944434i	$-0.606611\pi$
$971$	−20.4853	−0.657404	−0.328702	+	0.944434i	$0.606611\pi$
$972$	0	0
$973$	−7.85786	−0.251912
$974$	−8.34315	−0.267332
$975$	5.41421	0.173394
$976$	−5.37258	−0.171972
$977$	13.5563	0.433706	0.216853	−	0.976204i	$-0.430421\pi$
$977$	13.5563	0.433706	0.216853	+	0.976204i	$0.430421\pi$
$978$	−16.6274	−0.531686
$979$	−26.1421	−0.835506
$980$	0	0
$981$	−7.48528	−0.238987
$982$	−10.3431	−0.330063
$983$	24.8701	0.793232	0.396616	−	0.917985i	$-0.370185\pi$
$983$	24.8701	0.793232	0.396616	+	0.917985i	$0.370185\pi$
$984$	−4.48528	−0.142986
$985$	11.0711	0.352754
$986$	−4.00000	−0.127386
$987$	−3.02944	−0.0964281
$988$	0	0
$989$	0	0
$990$	3.41421	0.108511
$991$	32.6274	1.03644	0.518222	−	0.855246i	$-0.326594\pi$
$991$	32.6274	1.03644	0.518222	+	0.855246i	$0.326594\pi$
$992$	0	0
$993$	13.4853	0.427942
$994$	5.74012	0.182065
$995$	−17.0000	−0.538936
$996$	0	0
$997$	25.8995	0.820245	0.410123	−	0.912030i	$-0.365486\pi$
$997$	25.8995	0.820245	0.410123	+	0.912030i	$0.365486\pi$
$998$	−23.3553	−0.739300
$999$	1.75736	0.0556004

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7935.2.a.o.1.2	✓	2
23.22	odd	2		7935.2.a.p.1.2	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7935.2.a.o.1.2	✓	2	1.1	even	1	trivial
7935.2.a.p.1.2	yes	2	23.22	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:	\( N \)	\(=\)	\( 7935 = 3 \cdot 5 \cdot 23^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7935.a (trivial)

\(f(q)\)	\(=\)	\(q+1.41421 q^{2} -1.00000 q^{3} -1.00000 q^{5} -1.41421 q^{6} -0.828427 q^{7} -2.82843 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.41421 q^{2} -1.00000 q^{3} -1.00000 q^{5} -1.41421 q^{6} -0.828427 q^{7} -2.82843 q^{8} +1.00000 q^{9} -1.41421 q^{10} -2.41421 q^{11} -5.41421 q^{13} -1.17157 q^{14} +1.00000 q^{15} -4.00000 q^{16} -3.41421 q^{17} +1.41421 q^{18} +1.00000 q^{19} +0.828427 q^{21} -3.41421 q^{22} +2.82843 q^{24} +1.00000 q^{25} -7.65685 q^{26} -1.00000 q^{27} +0.828427 q^{29} +1.41421 q^{30} -3.82843 q^{31} +2.41421 q^{33} -4.82843 q^{34} +0.828427 q^{35} -1.75736 q^{37} +1.41421 q^{38} +5.41421 q^{39} +2.82843 q^{40} -1.58579 q^{41} +1.17157 q^{42} -10.2426 q^{43} -1.00000 q^{45} -3.65685 q^{47} +4.00000 q^{48} -6.31371 q^{49} +1.41421 q^{50} +3.41421 q^{51} -4.24264 q^{53} -1.41421 q^{54} +2.41421 q^{55} +2.34315 q^{56} -1.00000 q^{57} +1.17157 q^{58} -0.343146 q^{59} +1.34315 q^{61} -5.41421 q^{62} -0.828427 q^{63} +8.00000 q^{64} +5.41421 q^{65} +3.41421 q^{66} +14.4853 q^{67} +1.17157 q^{70} -4.89949 q^{71} -2.82843 q^{72} -2.48528 q^{73} -2.48528 q^{74} -1.00000 q^{75} +2.00000 q^{77} +7.65685 q^{78} -12.6569 q^{79} +4.00000 q^{80} +1.00000 q^{81} -2.24264 q^{82} +13.8995 q^{83} +3.41421 q^{85} -14.4853 q^{86} -0.828427 q^{87} +6.82843 q^{88} +10.8284 q^{89} -1.41421 q^{90} +4.48528 q^{91} +3.82843 q^{93} -5.17157 q^{94} -1.00000 q^{95} +11.8995 q^{97} -8.92893 q^{98} -2.41421 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} - 2 q^{5} + 4 q^{7} + 2 q^{9}+O(q^{10}) \) 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

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Database

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