LMFDB - Embedded newform 5950.2.a.bm.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5950,2,Mod(1,5950)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5950, 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("5950.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5950 = 2 \cdot 5^{2} \cdot 7 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5950.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:	$47.5109892027$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.2597.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 9x + 8 $ x^3 - x^2 - 9*x + 8
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1190)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$0.878468$ of defining polynomial
Character	$\chi$	$=$	5950.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +0.878468 q^{3} +1.00000 q^{4} +0.878468 q^{6} +1.00000 q^{7} +1.00000 q^{8} -2.22829 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} +0.878468 q^{3} +1.00000 q^{4} +0.878468 q^{6} +1.00000 q^{7} +1.00000 q^{8} -2.22829 q^{9} +5.22829 q^{11} +0.878468 q^{12} +1.00000 q^{14} +1.00000 q^{16} +1.00000 q^{17} -2.22829 q^{18} -1.22829 q^{19} +0.878468 q^{21} +5.22829 q^{22} +0.878468 q^{23} +0.878468 q^{24} -4.59289 q^{27} +1.00000 q^{28} -0.878468 q^{29} +5.47136 q^{31} +1.00000 q^{32} +4.59289 q^{33} +1.00000 q^{34} -2.22829 q^{36} -2.24306 q^{37} -1.22829 q^{38} +9.22829 q^{41} +0.878468 q^{42} +4.87847 q^{43} +5.22829 q^{44} +0.878468 q^{46} +0.878468 q^{47} +0.878468 q^{48} +1.00000 q^{49} +0.878468 q^{51} -2.87847 q^{53} -4.59289 q^{54} +1.00000 q^{56} -1.07902 q^{57} -0.878468 q^{58} +1.12153 q^{59} +5.51387 q^{61} +5.47136 q^{62} -2.22829 q^{63} +1.00000 q^{64} +4.59289 q^{66} +3.71442 q^{67} +1.00000 q^{68} +0.771706 q^{69} -8.69965 q^{71} -2.22829 q^{72} -6.00000 q^{73} -2.24306 q^{74} -1.22829 q^{76} +5.22829 q^{77} +12.2135 q^{79} +2.65017 q^{81} +9.22829 q^{82} +2.48613 q^{83} +0.878468 q^{84} +4.87847 q^{86} -0.771706 q^{87} +5.22829 q^{88} +10.2135 q^{89} +0.878468 q^{92} +4.80641 q^{93} +0.878468 q^{94} +0.878468 q^{96} -5.51387 q^{97} +1.00000 q^{98} -11.6502 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{2} + q^{3} + 3 q^{4} + q^{6} + 3 q^{7} + 3 q^{8} + 10 q^{9}+O(q^{10}) $ 3 * q + 3 * q^2 + q^3 + 3 * q^4 + q^6 + 3 * q^7 + 3 * q^8 + 10 * q^9	$ 3 q + 3 q^{2} + q^{3} + 3 q^{4} + q^{6} + 3 q^{7} + 3 q^{8} + 10 q^{9} - q^{11} + q^{12} + 3 q^{14} + 3 q^{16} + 3 q^{17} + 10 q^{18} + 13 q^{19} + q^{21} - q^{22} + q^{23} + q^{24} - 2 q^{27} + 3 q^{28} - q^{29} + 3 q^{31} + 3 q^{32} + 2 q^{33} + 3 q^{34} + 10 q^{36} - 10 q^{37} + 13 q^{38} + 11 q^{41} + q^{42} + 13 q^{43} - q^{44} + q^{46} + q^{47} + q^{48} + 3 q^{49} + q^{51} - 7 q^{53} - 2 q^{54} + 3 q^{56} + 2 q^{57} - q^{58} + 5 q^{59} + 10 q^{61} + 3 q^{62} + 10 q^{63} + 3 q^{64} + 2 q^{66} + q^{67} + 3 q^{68} + 19 q^{69} + 4 q^{71} + 10 q^{72} - 18 q^{73} - 10 q^{74} + 13 q^{76} - q^{77} + 23 q^{81} + 11 q^{82} + 14 q^{83} + q^{84} + 13 q^{86} - 19 q^{87} - q^{88} - 6 q^{89} + q^{92} - 34 q^{93} + q^{94} + q^{96} - 10 q^{97} + 3 q^{98} - 50 q^{99}+O(q^{100}) $ 3 * q + 3 * q^2 + q^3 + 3 * q^4 + q^6 + 3 * q^7 + 3 * q^8 + 10 * q^9 - q^11 + q^12 + 3 * q^14 + 3 * q^16 + 3 * q^17 + 10 * q^18 + 13 * q^19 + q^21 - q^22 + q^23 + q^24 - 2 * q^27 + 3 * q^28 - q^29 + 3 * q^31 + 3 * q^32 + 2 * q^33 + 3 * q^34 + 10 * q^36 - 10 * q^37 + 13 * q^38 + 11 * q^41 + q^42 + 13 * q^43 - q^44 + q^46 + q^47 + q^48 + 3 * q^49 + q^51 - 7 * q^53 - 2 * q^54 + 3 * q^56 + 2 * q^57 - q^58 + 5 * q^59 + 10 * q^61 + 3 * q^62 + 10 * q^63 + 3 * q^64 + 2 * q^66 + q^67 + 3 * q^68 + 19 * q^69 + 4 * q^71 + 10 * q^72 - 18 * q^73 - 10 * q^74 + 13 * q^76 - q^77 + 23 * q^81 + 11 * q^82 + 14 * q^83 + q^84 + 13 * q^86 - 19 * q^87 - q^88 - 6 * q^89 + q^92 - 34 * q^93 + q^94 + q^96 - 10 * q^97 + 3 * q^98 - 50 * 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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	0.878468	0.507184	0.253592	−	0.967311i	$-0.418388\pi$
$3$	0.878468	0.507184	0.253592	+	0.967311i	$0.418388\pi$
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	0.878468	0.358633
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	1.00000	0.353553
$9$	−2.22829	−0.742765
$10$	0	0
$11$	5.22829	1.57639	0.788195	−	0.615426i	$-0.211016\pi$
$11$	5.22829	1.57639	0.788195	+	0.615426i	$0.211016\pi$
$12$	0.878468	0.253592
$13$	0	0		−	1.00000i	$-0.5\pi$
$13$	0	0			1.00000i	$0.5\pi$
$14$	1.00000	0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	−2.22829	−0.525214
$19$	−1.22829	−0.281790	−0.140895	−	0.990025i	$-0.544998\pi$
$19$	−1.22829	−0.281790	−0.140895	+	0.990025i	$0.544998\pi$
$20$	0	0
$21$	0.878468	0.191697
$22$	5.22829	1.11468
$23$	0.878468	0.183173	0.0915866	−	0.995797i	$-0.470806\pi$
$23$	0.878468	0.183173	0.0915866	+	0.995797i	$0.470806\pi$
$24$	0.878468	0.179317
$25$	0	0
$26$	0	0
$27$	−4.59289	−0.883902
$28$	1.00000	0.188982
$29$	−0.878468	−0.163127	−0.0815637	−	0.996668i	$-0.525991\pi$
$29$	−0.878468	−0.163127	−0.0815637	+	0.996668i	$0.525991\pi$
$30$	0	0
$31$	5.47136	0.982685	0.491342	−	0.870966i	$-0.336506\pi$
$31$	5.47136	0.982685	0.491342	+	0.870966i	$0.336506\pi$
$32$	1.00000	0.176777
$33$	4.59289	0.799519
$34$	1.00000	0.171499
$35$	0	0
$36$	−2.22829	−0.371382
$37$	−2.24306	−0.368757	−0.184379	−	0.982855i	$-0.559027\pi$
$37$	−2.24306	−0.368757	−0.184379	+	0.982855i	$0.559027\pi$
$38$	−1.22829	−0.199256
$39$	0	0
$40$	0	0
$41$	9.22829	1.44122	0.720609	−	0.693342i	$-0.243862\pi$
$41$	9.22829	1.44122	0.720609	+	0.693342i	$0.243862\pi$
$42$	0.878468	0.135551
$43$	4.87847	0.743959	0.371980	−	0.928241i	$-0.378679\pi$
$43$	4.87847	0.743959	0.371980	+	0.928241i	$0.378679\pi$
$44$	5.22829	0.788195
$45$	0	0
$46$	0.878468	0.129523
$47$	0.878468	0.128138	0.0640689	−	0.997945i	$-0.479592\pi$
$47$	0.878468	0.128138	0.0640689	+	0.997945i	$0.479592\pi$
$48$	0.878468	0.126796
$49$	1.00000	0.142857
$50$	0	0
$51$	0.878468	0.123010
$52$	0	0
$53$	−2.87847	−0.395388	−0.197694	−	0.980264i	$-0.563345\pi$
$53$	−2.87847	−0.395388	−0.197694	+	0.980264i	$0.563345\pi$
$54$	−4.59289	−0.625013
$55$	0	0
$56$	1.00000	0.133631
$57$	−1.07902	−0.142919
$58$	−0.878468	−0.115349
$59$	1.12153	0.146011	0.0730055	−	0.997332i	$-0.476741\pi$
$59$	1.12153	0.146011	0.0730055	+	0.997332i	$0.476741\pi$
$60$	0	0
$61$	5.51387	0.705979	0.352990	−	0.935627i	$-0.385165\pi$
$61$	5.51387	0.705979	0.352990	+	0.935627i	$0.385165\pi$
$62$	5.47136	0.694863
$63$	−2.22829	−0.280739
$64$	1.00000	0.125000
$65$	0	0
$66$	4.59289	0.565346
$67$	3.71442	0.453789	0.226894	−	0.973919i	$-0.427143\pi$
$67$	3.71442	0.453789	0.226894	+	0.973919i	$0.427143\pi$
$68$	1.00000	0.121268
$69$	0.771706	0.0929025
$70$	0	0
$71$	−8.69965	−1.03246	−0.516229	−	0.856450i	$-0.672665\pi$
$71$	−8.69965	−1.03246	−0.516229	+	0.856450i	$0.672665\pi$
$72$	−2.22829	−0.262607
$73$	−6.00000	−0.702247	−0.351123	−	0.936329i	$-0.614200\pi$
$73$	−6.00000	−0.702247	−0.351123	+	0.936329i	$0.614200\pi$
$74$	−2.24306	−0.260751
$75$	0	0
$76$	−1.22829	−0.140895
$77$	5.22829	0.595819
$78$	0	0
$79$	12.2135	1.37413	0.687064	−	0.726597i	$-0.258899\pi$
$79$	12.2135	1.37413	0.687064	+	0.726597i	$0.258899\pi$
$80$	0	0
$81$	2.65017	0.294464
$82$	9.22829	1.01909
$83$	2.48613	0.272888	0.136444	−	0.990648i	$-0.456433\pi$
$83$	2.48613	0.272888	0.136444	+	0.990648i	$0.456433\pi$
$84$	0.878468	0.0958487
$85$	0	0
$86$	4.87847	0.526059
$87$	−0.771706	−0.0827356
$88$	5.22829	0.557338
$89$	10.2135	1.08263	0.541316	−	0.840819i	$-0.317926\pi$
$89$	10.2135	1.08263	0.541316	+	0.840819i	$0.317926\pi$
$90$	0	0
$91$	0	0
$92$	0.878468	0.0915866
$93$	4.80641	0.498402
$94$	0.878468	0.0906071
$95$	0	0
$96$	0.878468	0.0896583
$97$	−5.51387	−0.559849	−0.279924	−	0.960022i	$-0.590309\pi$
$97$	−5.51387	−0.559849	−0.279924	+	0.960022i	$0.590309\pi$
$98$	1.00000	0.101015
$99$	−11.6502	−1.17089
$100$	0	0
$101$	6.52864	0.649624	0.324812	−	0.945779i	$-0.394699\pi$
$101$	6.52864	0.649624	0.324812	+	0.945779i	$0.394699\pi$
$102$	0.878468	0.0869813
$103$	−8.39234	−0.826922	−0.413461	−	0.910522i	$-0.635680\pi$
$103$	−8.39234	−0.826922	−0.413461	+	0.910522i	$0.635680\pi$
$104$	0	0
$105$	0	0
$106$	−2.87847	−0.279582
$107$	−6.21352	−0.600684	−0.300342	−	0.953832i	$-0.597101\pi$
$107$	−6.21352	−0.600684	−0.300342	+	0.953832i	$0.597101\pi$
$108$	−4.59289	−0.441951
$109$	19.9279	1.90875	0.954375	−	0.298609i	$-0.0965227\pi$
$109$	19.9279	1.90875	0.954375	+	0.298609i	$0.0965227\pi$
$110$	0	0
$111$	−1.97046	−0.187028
$112$	1.00000	0.0944911
$113$	−2.98523	−0.280827	−0.140413	−	0.990093i	$-0.544843\pi$
$113$	−2.98523	−0.280827	−0.140413	+	0.990093i	$0.544843\pi$
$114$	−1.07902	−0.101059
$115$	0	0
$116$	−0.878468	−0.0815637
$117$	0	0
$118$	1.12153	0.103245
$119$	1.00000	0.0916698
$120$	0	0
$121$	16.3351	1.48501
$122$	5.51387	0.499203
$123$	8.10676	0.730962
$124$	5.47136	0.491342
$125$	0	0
$126$	−2.22829	−0.198512
$127$	−10.9427	−0.971009	−0.485504	−	0.874234i	$-0.661364\pi$
$127$	−10.9427	−0.971009	−0.485504	+	0.874234i	$0.661364\pi$
$128$	1.00000	0.0883883
$129$	4.28558	0.377324
$130$	0	0
$131$	6.45659	0.564115	0.282057	−	0.959398i	$-0.408983\pi$
$131$	6.45659	0.564115	0.282057	+	0.959398i	$0.408983\pi$
$132$	4.59289	0.399760
$133$	−1.22829	−0.106507
$134$	3.71442	0.320877
$135$	0	0
$136$	1.00000	0.0857493
$137$	−16.9427	−1.44751	−0.723757	−	0.690055i	$-0.757586\pi$
$137$	−16.9427	−1.44751	−0.723757	+	0.690055i	$0.757586\pi$
$138$	0.771706	0.0656920
$139$	−10.4566	−0.886916	−0.443458	−	0.896295i	$-0.646248\pi$
$139$	−10.4566	−0.886916	−0.443458	+	0.896295i	$0.646248\pi$
$140$	0	0
$141$	0.771706	0.0649894
$142$	−8.69965	−0.730059
$143$	0	0
$144$	−2.22829	−0.185691
$145$	0	0
$146$	−6.00000	−0.496564
$147$	0.878468	0.0724548
$148$	−2.24306	−0.184379
$149$	−4.94272	−0.404923	−0.202461	−	0.979290i	$-0.564894\pi$
$149$	−4.94272	−0.404923	−0.202461	+	0.979290i	$0.564894\pi$
$150$	0	0
$151$	5.47136	0.445253	0.222626	−	0.974904i	$-0.428537\pi$
$151$	5.47136	0.445253	0.222626	+	0.974904i	$0.428537\pi$
$152$	−1.22829	−0.0996278
$153$	−2.22829	−0.180147
$154$	5.22829	0.421308
$155$	0	0
$156$	0	0
$157$	−12.0000	−0.957704	−0.478852	−	0.877896i	$-0.658947\pi$
$157$	−12.0000	−0.957704	−0.478852	+	0.877896i	$0.658947\pi$
$158$	12.2135	0.971656
$159$	−2.52864	−0.200534
$160$	0	0
$161$	0.878468	0.0692330
$162$	2.65017	0.208217
$163$	−3.54341	−0.277541	−0.138771	−	0.990325i	$-0.544315\pi$
$163$	−3.54341	−0.277541	−0.138771	+	0.990325i	$0.544315\pi$
$164$	9.22829	0.720609
$165$	0	0
$166$	2.48613	0.192961
$167$	20.6997	1.60179	0.800894	−	0.598807i	$-0.204358\pi$
$167$	20.6997	1.60179	0.800894	+	0.598807i	$0.204358\pi$
$168$	0.878468	0.0677753
$169$	−13.0000	−1.00000
$170$	0	0
$171$	2.73700	0.209304
$172$	4.87847	0.371980
$173$	7.95749	0.604996	0.302498	−	0.953150i	$-0.402179\pi$
$173$	7.95749	0.604996	0.302498	+	0.953150i	$0.402179\pi$
$174$	−0.771706	−0.0585029
$175$	0	0
$176$	5.22829	0.394097
$177$	0.985230	0.0740544
$178$	10.2135	0.765536
$179$	−18.4566	−1.37951	−0.689755	−	0.724043i	$-0.742282\pi$
$179$	−18.4566	−1.37951	−0.689755	+	0.724043i	$0.742282\pi$
$180$	0	0
$181$	15.7569	1.17120	0.585602	−	0.810599i	$-0.300858\pi$
$181$	15.7569	1.17120	0.585602	+	0.810599i	$0.300858\pi$
$182$	0	0
$183$	4.84376	0.358061
$184$	0.878468	0.0647615
$185$	0	0
$186$	4.80641	0.352423
$187$	5.22829	0.382331
$188$	0.878468	0.0640689
$189$	−4.59289	−0.334084
$190$	0	0
$191$	−15.4418	−1.11733	−0.558665	−	0.829393i	$-0.688686\pi$
$191$	−15.4418	−1.11733	−0.558665	+	0.829393i	$0.688686\pi$
$192$	0.878468	0.0633980
$193$	15.7916	1.13671	0.568354	−	0.822784i	$-0.307581\pi$
$193$	15.7916	1.13671	0.568354	+	0.822784i	$0.307581\pi$
$194$	−5.51387	−0.395873
$195$	0	0
$196$	1.00000	0.0714286
$197$	9.75694	0.695153	0.347576	−	0.937652i	$-0.387005\pi$
$197$	9.75694	0.695153	0.347576	+	0.937652i	$0.387005\pi$
$198$	−11.6502	−0.827942
$199$	15.3351	1.08707	0.543537	−	0.839385i	$-0.317085\pi$
$199$	15.3351	1.08707	0.543537	+	0.839385i	$0.317085\pi$
$200$	0	0
$201$	3.26300	0.230154
$202$	6.52864	0.459354
$203$	−0.878468	−0.0616564
$204$	0.878468	0.0615051
$205$	0	0
$206$	−8.39234	−0.584722
$207$	−1.95749	−0.136055
$208$	0	0
$209$	−6.42188	−0.444211
$210$	0	0
$211$	−6.21352	−0.427757	−0.213878	−	0.976860i	$-0.568610\pi$
$211$	−6.21352	−0.427757	−0.213878	+	0.976860i	$0.568610\pi$
$212$	−2.87847	−0.197694
$213$	−7.64237	−0.523646
$214$	−6.21352	−0.424748
$215$	0	0
$216$	−4.59289	−0.312507
$217$	5.47136	0.371420
$218$	19.9279	1.34969
$219$	−5.27081	−0.356168
$220$	0	0
$221$	0	0
$222$	−1.97046	−0.132249
$223$	4.00000	0.267860	0.133930	−	0.990991i	$-0.457240\pi$
$223$	4.00000	0.267860	0.133930	+	0.990991i	$0.457240\pi$
$224$	1.00000	0.0668153
$225$	0	0
$226$	−2.98523	−0.198575
$227$	26.1710	1.73703	0.868515	−	0.495662i	$-0.165075\pi$
$227$	26.1710	1.73703	0.868515	+	0.495662i	$0.165075\pi$
$228$	−1.07902	−0.0714596
$229$	−0.392341	−0.0259266	−0.0129633	−	0.999916i	$-0.504126\pi$
$229$	−0.392341	−0.0259266	−0.0129633	+	0.999916i	$0.504126\pi$
$230$	0	0
$231$	4.59289	0.302190
$232$	−0.878468	−0.0576743
$233$	6.00000	0.393073	0.196537	−	0.980497i	$-0.437031\pi$
$233$	6.00000	0.393073	0.196537	+	0.980497i	$0.437031\pi$
$234$	0	0
$235$	0	0
$236$	1.12153	0.0730055
$237$	10.7292	0.696936
$238$	1.00000	0.0648204
$239$	−29.8984	−1.93397	−0.966984	−	0.254839i	$-0.917978\pi$
$239$	−29.8984	−1.93397	−0.966984	+	0.254839i	$0.917978\pi$
$240$	0	0
$241$	19.5781	1.26114	0.630569	−	0.776133i	$-0.282822\pi$
$241$	19.5781	1.26114	0.630569	+	0.776133i	$0.282822\pi$
$242$	16.3351	1.05006
$243$	16.1068	1.03325
$244$	5.51387	0.352990
$245$	0	0
$246$	8.10676	0.516868
$247$	0	0
$248$	5.47136	0.347432
$249$	2.18398	0.138404
$250$	0	0
$251$	−4.74217	−0.299323	−0.149661	−	0.988737i	$-0.547818\pi$
$251$	−4.74217	−0.299323	−0.149661	+	0.988737i	$0.547818\pi$
$252$	−2.22829	−0.140369
$253$	4.59289	0.288752
$254$	−10.9427	−0.686607
$255$	0	0
$256$	1.00000	0.0625000
$257$	11.3646	0.708904	0.354452	−	0.935074i	$-0.384667\pi$
$257$	11.3646	0.708904	0.354452	+	0.935074i	$0.384667\pi$
$258$	4.28558	0.266808
$259$	−2.24306	−0.139377
$260$	0	0
$261$	1.95749	0.121165
$262$	6.45659	0.398889
$263$	4.48613	0.276626	0.138313	−	0.990389i	$-0.455832\pi$
$263$	4.48613	0.276626	0.138313	+	0.990389i	$0.455832\pi$
$264$	4.59289	0.282673
$265$	0	0
$266$	−1.22829	−0.0753115
$267$	8.97226	0.549093
$268$	3.71442	0.226894
$269$	−13.0277	−0.794316	−0.397158	−	0.917750i	$-0.630003\pi$
$269$	−13.0277	−0.794316	−0.397158	+	0.917750i	$0.630003\pi$
$270$	0	0
$271$	−3.42884	−0.208287	−0.104144	−	0.994562i	$-0.533210\pi$
$271$	−3.42884	−0.208287	−0.104144	+	0.994562i	$0.533210\pi$
$272$	1.00000	0.0606339
$273$	0	0
$274$	−16.9427	−1.02355
$275$	0	0
$276$	0.771706	0.0464513
$277$	5.18578	0.311583	0.155792	−	0.987790i	$-0.450207\pi$
$277$	5.18578	0.311583	0.155792	+	0.987790i	$0.450207\pi$
$278$	−10.4566	−0.627144
$279$	−12.1918	−0.729903
$280$	0	0
$281$	0.0642469	0.00383265	0.00191632	−	0.999998i	$-0.499390\pi$
$281$	0.0642469	0.00383265	0.00191632	+	0.999998i	$0.499390\pi$
$282$	0.771706	0.0459544
$283$	−11.0277	−0.655531	−0.327766	−	0.944759i	$-0.606296\pi$
$283$	−11.0277	−0.655531	−0.327766	+	0.944759i	$0.606296\pi$
$284$	−8.69965	−0.516229
$285$	0	0
$286$	0	0
$287$	9.22829	0.544729
$288$	−2.22829	−0.131303
$289$	1.00000	0.0588235
$290$	0	0
$291$	−4.84376	−0.283946
$292$	−6.00000	−0.351123
$293$	−27.3698	−1.59896	−0.799479	−	0.600694i	$-0.794891\pi$
$293$	−27.3698	−1.59896	−0.799479	+	0.600694i	$0.794891\pi$
$294$	0.878468	0.0512333
$295$	0	0
$296$	−2.24306	−0.130375
$297$	−24.0130	−1.39337
$298$	−4.94272	−0.286324
$299$	0	0
$300$	0	0
$301$	4.87847	0.281190
$302$	5.47136	0.314841
$303$	5.73520	0.329479
$304$	−1.22829	−0.0704475
$305$	0	0
$306$	−2.22829	−0.127383
$307$	−7.75694	−0.442712	−0.221356	−	0.975193i	$-0.571048\pi$
$307$	−7.75694	−0.442712	−0.221356	+	0.975193i	$0.571048\pi$
$308$	5.22829	0.297910
$309$	−7.37240	−0.419401
$310$	0	0
$311$	23.3351	1.32321	0.661605	−	0.749853i	$-0.269876\pi$
$311$	23.3351	1.32321	0.661605	+	0.749853i	$0.269876\pi$
$312$	0	0
$313$	−24.6701	−1.39444	−0.697219	−	0.716859i	$-0.745579\pi$
$313$	−24.6701	−1.39444	−0.697219	+	0.716859i	$0.745579\pi$
$314$	−12.0000	−0.677199
$315$	0	0
$316$	12.2135	0.687064
$317$	6.45659	0.362638	0.181319	−	0.983424i	$-0.441963\pi$
$317$	6.45659	0.362638	0.181319	+	0.983424i	$0.441963\pi$
$318$	−2.52864	−0.141799
$319$	−4.59289	−0.257152
$320$	0	0
$321$	−5.45838	−0.304657
$322$	0.878468	0.0489551
$323$	−1.22829	−0.0683441
$324$	2.65017	0.147232
$325$	0	0
$326$	−3.54341	−0.196251
$327$	17.5061	0.968088
$328$	9.22829	0.509547
$329$	0.878468	0.0484315
$330$	0	0
$331$	8.91318	0.489912	0.244956	−	0.969534i	$-0.421226\pi$
$331$	8.91318	0.489912	0.244956	+	0.969534i	$0.421226\pi$
$332$	2.48613	0.136444
$333$	4.99820	0.273900
$334$	20.6997	1.13263
$335$	0	0
$336$	0.878468	0.0479244
$337$	29.5486	1.60961	0.804807	−	0.593537i	$-0.202269\pi$
$337$	29.5486	1.60961	0.804807	+	0.593537i	$0.202269\pi$
$338$	−13.0000	−0.707107
$339$	−2.62243	−0.142431
$340$	0	0
$341$	28.6059	1.54909
$342$	2.73700	0.148000
$343$	1.00000	0.0539949
$344$	4.87847	0.263029
$345$	0	0
$346$	7.95749	0.427797
$347$	0.729191	0.0391450	0.0195725	−	0.999808i	$-0.493769\pi$
$347$	0.729191	0.0391450	0.0195725	+	0.999808i	$0.493769\pi$
$348$	−0.771706	−0.0413678
$349$	−3.72740	−0.199523	−0.0997615	−	0.995011i	$-0.531808\pi$
$349$	−3.72740	−0.199523	−0.0997615	+	0.995011i	$0.531808\pi$
$350$	0	0
$351$	0	0
$352$	5.22829	0.278669
$353$	−5.92795	−0.315513	−0.157756	−	0.987478i	$-0.550426\pi$
$353$	−5.92795	−0.315513	−0.157756	+	0.987478i	$0.550426\pi$
$354$	0.985230	0.0523644
$355$	0	0
$356$	10.2135	0.541316
$357$	0.878468	0.0464935
$358$	−18.4566	−0.975461
$359$	−3.82118	−0.201674	−0.100837	−	0.994903i	$-0.532152\pi$
$359$	−3.82118	−0.201674	−0.100837	+	0.994903i	$0.532152\pi$
$360$	0	0
$361$	−17.4913	−0.920594
$362$	15.7569	0.828166
$363$	14.3498	0.753171
$364$	0	0
$365$	0	0
$366$	4.84376	0.253187
$367$	−36.9132	−1.92685	−0.963426	−	0.267974i	$-0.913646\pi$
$367$	−36.9132	−1.92685	−0.963426	+	0.267974i	$0.913646\pi$
$368$	0.878468	0.0457933
$369$	−20.5633	−1.07049
$370$	0	0
$371$	−2.87847	−0.149443
$372$	4.80641	0.249201
$373$	28.7639	1.48934	0.744669	−	0.667434i	$-0.232607\pi$
$373$	28.7639	1.48934	0.744669	+	0.667434i	$0.232607\pi$
$374$	5.22829	0.270349
$375$	0	0
$376$	0.878468	0.0453035
$377$	0	0
$378$	−4.59289	−0.236233
$379$	−30.0347	−1.54278	−0.771390	−	0.636363i	$-0.780438\pi$
$379$	−30.0347	−1.54278	−0.771390	+	0.636363i	$0.780438\pi$
$380$	0	0
$381$	−9.61283	−0.492480
$382$	−15.4418	−0.790072
$383$	17.6849	0.903655	0.451828	−	0.892105i	$-0.350772\pi$
$383$	17.6849	0.903655	0.451828	+	0.892105i	$0.350772\pi$
$384$	0.878468	0.0448291
$385$	0	0
$386$	15.7916	0.803773
$387$	−10.8707	−0.552587
$388$	−5.51387	−0.279924
$389$	25.6424	1.30012	0.650060	−	0.759883i	$-0.274744\pi$
$389$	25.6424	1.30012	0.650060	+	0.759883i	$0.274744\pi$
$390$	0	0
$391$	0.878468	0.0444260
$392$	1.00000	0.0505076
$393$	5.67191	0.286110
$394$	9.75694	0.491547
$395$	0	0
$396$	−11.6502	−0.585443
$397$	17.5486	0.880738	0.440369	−	0.897817i	$-0.354848\pi$
$397$	17.5486	0.880738	0.440369	+	0.897817i	$0.354848\pi$
$398$	15.3351	0.768677
$399$	−1.07902	−0.0540184
$400$	0	0
$401$	−3.54341	−0.176950	−0.0884748	−	0.996078i	$-0.528199\pi$
$401$	−3.54341	−0.176950	−0.0884748	+	0.996078i	$0.528199\pi$
$402$	3.26300	0.162744
$403$	0	0
$404$	6.52864	0.324812
$405$	0	0
$406$	−0.878468	−0.0435976
$407$	−11.7274	−0.581305
$408$	0.878468	0.0434907
$409$	5.42884	0.268439	0.134219	−	0.990952i	$-0.457147\pi$
$409$	5.42884	0.268439	0.134219	+	0.990952i	$0.457147\pi$
$410$	0	0
$411$	−14.8836	−0.734156
$412$	−8.39234	−0.413461
$413$	1.12153	0.0551870
$414$	−1.95749	−0.0962051
$415$	0	0
$416$	0	0
$417$	−9.18578	−0.449830
$418$	−6.42188	−0.314104
$419$	−21.6128	−1.05586	−0.527928	−	0.849289i	$-0.677031\pi$
$419$	−21.6128	−1.05586	−0.527928	+	0.849289i	$0.677031\pi$
$420$	0	0
$421$	0.243064	0.0118462	0.00592310	−	0.999982i	$-0.498115\pi$
$421$	0.243064	0.0118462	0.00592310	+	0.999982i	$0.498115\pi$
$422$	−6.21352	−0.302470
$423$	−1.95749	−0.0951762
$424$	−2.87847	−0.139791
$425$	0	0
$426$	−7.64237	−0.370274
$427$	5.51387	0.266835
$428$	−6.21352	−0.300342
$429$	0	0
$430$	0	0
$431$	−15.0277	−0.723861	−0.361931	−	0.932205i	$-0.617882\pi$
$431$	−15.0277	−0.723861	−0.361931	+	0.932205i	$0.617882\pi$
$432$	−4.59289	−0.220975
$433$	−18.4861	−0.888387	−0.444193	−	0.895931i	$-0.646510\pi$
$433$	−18.4861	−0.888387	−0.444193	+	0.895931i	$0.646510\pi$
$434$	5.47136	0.262634
$435$	0	0
$436$	19.9279	0.954375
$437$	−1.07902	−0.0516164
$438$	−5.27081	−0.251849
$439$	−24.1197	−1.15117	−0.575586	−	0.817741i	$-0.695226\pi$
$439$	−24.1197	−1.15117	−0.575586	+	0.817741i	$0.695226\pi$
$440$	0	0
$441$	−2.22829	−0.106109
$442$	0	0
$443$	−28.0347	−1.33197	−0.665985	−	0.745966i	$-0.731988\pi$
$443$	−28.0347	−1.33197	−0.665985	+	0.745966i	$0.731988\pi$
$444$	−1.97046	−0.0935139
$445$	0	0
$446$	4.00000	0.189405
$447$	−4.34202	−0.205370
$448$	1.00000	0.0472456
$449$	−39.3402	−1.85658	−0.928290	−	0.371857i	$-0.878721\pi$
$449$	−39.3402	−1.85658	−0.928290	+	0.371857i	$0.878721\pi$
$450$	0	0
$451$	48.2482	2.27192
$452$	−2.98523	−0.140413
$453$	4.80641	0.225825
$454$	26.1710	1.22827
$455$	0	0
$456$	−1.07902	−0.0505296
$457$	27.9705	1.30840	0.654201	−	0.756320i	$-0.273005\pi$
$457$	27.9705	1.30840	0.654201	+	0.756320i	$0.273005\pi$
$458$	−0.392341	−0.0183329
$459$	−4.59289	−0.214378
$460$	0	0
$461$	−19.9279	−0.928137	−0.464068	−	0.885799i	$-0.653611\pi$
$461$	−19.9279	−0.928137	−0.464068	+	0.885799i	$0.653611\pi$
$462$	4.59289	0.213681
$463$	2.94272	0.136760	0.0683798	−	0.997659i	$-0.478217\pi$
$463$	2.94272	0.136760	0.0683798	+	0.997659i	$0.478217\pi$
$464$	−0.878468	−0.0407819
$465$	0	0
$466$	6.00000	0.277945
$467$	−35.1267	−1.62547	−0.812735	−	0.582634i	$-0.802022\pi$
$467$	−35.1267	−1.62547	−0.812735	+	0.582634i	$0.802022\pi$
$468$	0	0
$469$	3.71442	0.171516
$470$	0	0
$471$	−10.5416	−0.485732
$472$	1.12153	0.0516227
$473$	25.5061	1.17277
$474$	10.7292	0.492808
$475$	0	0
$476$	1.00000	0.0458349
$477$	6.41407	0.293680
$478$	−29.8984	−1.36752
$479$	−16.0000	−0.731059	−0.365529	−	0.930800i	$-0.619112\pi$
$479$	−16.0000	−0.731059	−0.365529	+	0.930800i	$0.619112\pi$
$480$	0	0
$481$	0	0
$482$	19.5781	0.891759
$483$	0.771706	0.0351138
$484$	16.3351	0.742503
$485$	0	0
$486$	16.1068	0.730618
$487$	−24.6276	−1.11598	−0.557991	−	0.829847i	$-0.688428\pi$
$487$	−24.6276	−1.11598	−0.557991	+	0.829847i	$0.688428\pi$
$488$	5.51387	0.249601
$489$	−3.11277	−0.140765
$490$	0	0
$491$	3.02774	0.136640	0.0683201	−	0.997663i	$-0.478236\pi$
$491$	3.02774	0.136640	0.0683201	+	0.997663i	$0.478236\pi$
$492$	8.10676	0.365481
$493$	−0.878468	−0.0395642
$494$	0	0
$495$	0	0
$496$	5.47136	0.245671
$497$	−8.69965	−0.390233
$498$	2.18398	0.0978667
$499$	29.1692	1.30579	0.652897	−	0.757447i	$-0.273554\pi$
$499$	29.1692	1.30579	0.652897	+	0.757447i	$0.273554\pi$
$500$	0	0
$501$	18.1840	0.812400
$502$	−4.74217	−0.211653
$503$	−1.97046	−0.0878585	−0.0439292	−	0.999035i	$-0.513988\pi$
$503$	−1.97046	−0.0878585	−0.0439292	+	0.999035i	$0.513988\pi$
$504$	−2.22829	−0.0992561
$505$	0	0
$506$	4.59289	0.204179
$507$	−11.4201	−0.507184
$508$	−10.9427	−0.485504
$509$	20.5208	0.909570	0.454785	−	0.890601i	$-0.349716\pi$
$509$	20.5208	0.909570	0.454785	+	0.890601i	$0.349716\pi$
$510$	0	0
$511$	−6.00000	−0.265424
$512$	1.00000	0.0441942
$513$	5.64142	0.249075
$514$	11.3646	0.501271
$515$	0	0
$516$	4.28558	0.188662
$517$	4.59289	0.201995
$518$	−2.24306	−0.0985546
$519$	6.99040	0.306844
$520$	0	0
$521$	−6.79344	−0.297626	−0.148813	−	0.988865i	$-0.547545\pi$
$521$	−6.79344	−0.297626	−0.148813	+	0.988865i	$0.547545\pi$
$522$	1.95749	0.0856768
$523$	−11.7569	−0.514095	−0.257047	−	0.966399i	$-0.582750\pi$
$523$	−11.7569	−0.514095	−0.257047	+	0.966399i	$0.582750\pi$
$524$	6.45659	0.282057
$525$	0	0
$526$	4.48613	0.195604
$527$	5.47136	0.238336
$528$	4.59289	0.199880
$529$	−22.2283	−0.966448
$530$	0	0
$531$	−2.49910	−0.108452
$532$	−1.22829	−0.0532533
$533$	0	0
$534$	8.97226	0.388267
$535$	0	0
$536$	3.71442	0.160439
$537$	−16.2135	−0.699665
$538$	−13.0277	−0.561666
$539$	5.22829	0.225199
$540$	0	0
$541$	−38.8707	−1.67118	−0.835590	−	0.549353i	$-0.814874\pi$
$541$	−38.8707	−1.67118	−0.835590	+	0.549353i	$0.814874\pi$
$542$	−3.42884	−0.147281
$543$	13.8420	0.594016
$544$	1.00000	0.0428746
$545$	0	0
$546$	0	0
$547$	−20.1840	−0.863005	−0.431502	−	0.902112i	$-0.642016\pi$
$547$	−20.1840	−0.863005	−0.431502	+	0.902112i	$0.642016\pi$
$548$	−16.9427	−0.723757
$549$	−12.2865	−0.524376
$550$	0	0
$551$	1.07902	0.0459677
$552$	0.771706	0.0328460
$553$	12.2135	0.519372
$554$	5.18578	0.220323
$555$	0	0
$556$	−10.4566	−0.443458
$557$	−17.0148	−0.720939	−0.360469	−	0.932771i	$-0.617384\pi$
$557$	−17.0148	−0.720939	−0.360469	+	0.932771i	$0.617384\pi$
$558$	−12.1918	−0.516120
$559$	0	0
$560$	0	0
$561$	4.59289	0.193912
$562$	0.0642469	0.00271009
$563$	17.0277	0.717634	0.358817	−	0.933408i	$-0.383180\pi$
$563$	17.0277	0.717634	0.358817	+	0.933408i	$0.383180\pi$
$564$	0.771706	0.0324947
$565$	0	0
$566$	−11.0277	−0.463531
$567$	2.65017	0.111297
$568$	−8.69965	−0.365029
$569$	8.31512	0.348588	0.174294	−	0.984694i	$-0.444236\pi$
$569$	8.31512	0.348588	0.174294	+	0.984694i	$0.444236\pi$
$570$	0	0
$571$	−4.25604	−0.178110	−0.0890548	−	0.996027i	$-0.528385\pi$
$571$	−4.25604	−0.178110	−0.0890548	+	0.996027i	$0.528385\pi$
$572$	0	0
$573$	−13.5651	−0.566692
$574$	9.22829	0.385182
$575$	0	0
$576$	−2.22829	−0.0928456
$577$	−18.4141	−0.766588	−0.383294	−	0.923626i	$-0.625210\pi$
$577$	−18.4141	−0.766588	−0.383294	+	0.923626i	$0.625210\pi$
$578$	1.00000	0.0415945
$579$	13.8725	0.576520
$580$	0	0
$581$	2.48613	0.103142
$582$	−4.84376	−0.200780
$583$	−15.0495	−0.623286
$584$	−6.00000	−0.248282
$585$	0	0
$586$	−27.3698	−1.13063
$587$	−7.05728	−0.291285	−0.145643	−	0.989337i	$-0.546525\pi$
$587$	−7.05728	−0.291285	−0.145643	+	0.989337i	$0.546525\pi$
$588$	0.878468	0.0362274
$589$	−6.72043	−0.276911
$590$	0	0
$591$	8.57116	0.352570
$592$	−2.24306	−0.0921894
$593$	3.79164	0.155704	0.0778521	−	0.996965i	$-0.475194\pi$
$593$	3.79164	0.155704	0.0778521	+	0.996965i	$0.475194\pi$
$594$	−24.0130	−0.985264
$595$	0	0
$596$	−4.94272	−0.202461
$597$	13.4714	0.551346
$598$	0	0
$599$	22.3628	0.913719	0.456860	−	0.889539i	$-0.348974\pi$
$599$	22.3628	0.913719	0.456860	+	0.889539i	$0.348974\pi$
$600$	0	0
$601$	18.6276	0.759836	0.379918	−	0.925020i	$-0.375952\pi$
$601$	18.6276	0.759836	0.379918	+	0.925020i	$0.375952\pi$
$602$	4.87847	0.198832
$603$	−8.27682	−0.337058
$604$	5.47136	0.222626
$605$	0	0
$606$	5.73520	0.232977
$607$	19.7274	0.800710	0.400355	−	0.916360i	$-0.368887\pi$
$607$	19.7274	0.800710	0.400355	+	0.916360i	$0.368887\pi$
$608$	−1.22829	−0.0498139
$609$	−0.771706	−0.0312711
$610$	0	0
$611$	0	0
$612$	−2.22829	−0.0900734
$613$	−33.4548	−1.35123	−0.675613	−	0.737256i	$-0.736121\pi$
$613$	−33.4548	−1.35123	−0.675613	+	0.737256i	$0.736121\pi$
$614$	−7.75694	−0.313044
$615$	0	0
$616$	5.22829	0.210654
$617$	−39.0070	−1.57036	−0.785181	−	0.619267i	$-0.787430\pi$
$617$	−39.0070	−1.57036	−0.785181	+	0.619267i	$0.787430\pi$
$618$	−7.37240	−0.296562
$619$	−18.7292	−0.752790	−0.376395	−	0.926459i	$-0.622836\pi$
$619$	−18.7292	−0.752790	−0.376395	+	0.926459i	$0.622836\pi$
$620$	0	0
$621$	−4.03471	−0.161907
$622$	23.3351	0.935650
$623$	10.2135	0.409196
$624$	0	0
$625$	0	0
$626$	−24.6701	−0.986016
$627$	−5.64142	−0.225297
$628$	−12.0000	−0.478852
$629$	−2.24306	−0.0894368
$630$	0	0
$631$	−14.8707	−0.591992	−0.295996	−	0.955189i	$-0.595651\pi$
$631$	−14.8707	−0.591992	−0.295996	+	0.955189i	$0.595651\pi$
$632$	12.2135	0.485828
$633$	−5.45838	−0.216951
$634$	6.45659	0.256424
$635$	0	0
$636$	−2.52864	−0.100267
$637$	0	0
$638$	−4.59289	−0.181834
$639$	19.3854	0.766874
$640$	0	0
$641$	−48.8836	−1.93079	−0.965394	−	0.260797i	$-0.916015\pi$
$641$	−48.8836	−1.93079	−0.965394	+	0.260797i	$0.916015\pi$
$642$	−5.45838	−0.215425
$643$	25.8264	1.01849	0.509246	−	0.860621i	$-0.329924\pi$
$643$	25.8264	1.01849	0.509246	+	0.860621i	$0.329924\pi$
$644$	0.878468	0.0346165
$645$	0	0
$646$	−1.22829	−0.0483266
$647$	12.9132	0.507669	0.253835	−	0.967248i	$-0.418308\pi$
$647$	12.9132	0.507669	0.253835	+	0.967248i	$0.418308\pi$
$648$	2.65017	0.104109
$649$	5.86370	0.230170
$650$	0	0
$651$	4.80641	0.188378
$652$	−3.54341	−0.138771
$653$	−42.1840	−1.65079	−0.825393	−	0.564558i	$-0.809047\pi$
$653$	−42.1840	−1.65079	−0.825393	+	0.564558i	$0.809047\pi$
$654$	17.5061	0.684541
$655$	0	0
$656$	9.22829	0.360304
$657$	13.3698	0.521604
$658$	0.878468	0.0342463
$659$	11.3698	0.442903	0.221452	−	0.975171i	$-0.428921\pi$
$659$	11.3698	0.442903	0.221452	+	0.975171i	$0.428921\pi$
$660$	0	0
$661$	10.4436	0.406209	0.203105	−	0.979157i	$-0.434897\pi$
$661$	10.4436	0.406209	0.203105	+	0.979157i	$0.434897\pi$
$662$	8.91318	0.346420
$663$	0	0
$664$	2.48613	0.0964805
$665$	0	0
$666$	4.99820	0.193677
$667$	−0.771706	−0.0298806
$668$	20.6997	0.800894
$669$	3.51387	0.135854
$670$	0	0
$671$	28.8281	1.11290
$672$	0.878468	0.0338876
$673$	9.44182	0.363955	0.181978	−	0.983303i	$-0.441750\pi$
$673$	9.44182	0.363955	0.181978	+	0.983303i	$0.441750\pi$
$674$	29.5486	1.13817
$675$	0	0
$676$	−13.0000	−0.500000
$677$	−9.60766	−0.369252	−0.184626	−	0.982809i	$-0.559107\pi$
$677$	−9.60766	−0.369252	−0.184626	+	0.982809i	$0.559107\pi$
$678$	−2.62243	−0.100714
$679$	−5.51387	−0.211603
$680$	0	0
$681$	22.9904	0.880994
$682$	28.6059	1.09538
$683$	43.8854	1.67923	0.839615	−	0.543182i	$-0.182781\pi$
$683$	43.8854	1.67923	0.839615	+	0.543182i	$0.182781\pi$
$684$	2.73700	0.104652
$685$	0	0
$686$	1.00000	0.0381802
$687$	−0.344659	−0.0131496
$688$	4.87847	0.185990
$689$	0	0
$690$	0	0
$691$	2.67011	0.101576	0.0507879	−	0.998709i	$-0.483827\pi$
$691$	2.67011	0.101576	0.0507879	+	0.998709i	$0.483827\pi$
$692$	7.95749	0.302498
$693$	−11.6502	−0.442554
$694$	0.729191	0.0276797
$695$	0	0
$696$	−0.771706	−0.0292514
$697$	9.22829	0.349547
$698$	−3.72740	−0.141084
$699$	5.27081	0.199360
$700$	0	0
$701$	22.2135	0.838993	0.419497	−	0.907757i	$-0.362207\pi$
$701$	22.2135	0.838993	0.419497	+	0.907757i	$0.362207\pi$
$702$	0	0
$703$	2.75514	0.103912
$704$	5.22829	0.197049
$705$	0	0
$706$	−5.92795	−0.223101
$707$	6.52864	0.245535
$708$	0.985230	0.0370272
$709$	17.6631	0.663353	0.331677	−	0.943393i	$-0.392386\pi$
$709$	17.6631	0.663353	0.331677	+	0.943393i	$0.392386\pi$
$710$	0	0
$711$	−27.2153	−1.02065
$712$	10.2135	0.382768
$713$	4.80641	0.180002
$714$	0.878468	0.0328758
$715$	0	0
$716$	−18.4566	−0.689755
$717$	−26.2648	−0.980877
$718$	−3.82118	−0.142605
$719$	−4.49910	−0.167788	−0.0838941	−	0.996475i	$-0.526736\pi$
$719$	−4.49910	−0.167788	−0.0838941	+	0.996475i	$0.526736\pi$
$720$	0	0
$721$	−8.39234	−0.312547
$722$	−17.4913	−0.650959
$723$	17.1988	0.639629
$724$	15.7569	0.585602
$725$	0	0
$726$	14.3498	0.532572
$727$	42.2778	1.56800	0.783998	−	0.620764i	$-0.213177\pi$
$727$	42.2778	1.56800	0.783998	+	0.620764i	$0.213177\pi$
$728$	0	0
$729$	6.19875	0.229583
$730$	0	0
$731$	4.87847	0.180437
$732$	4.84376	0.179031
$733$	−39.3698	−1.45416	−0.727078	−	0.686555i	$-0.759122\pi$
$733$	−39.3698	−1.45416	−0.727078	+	0.686555i	$0.759122\pi$
$734$	−36.9132	−1.36249
$735$	0	0
$736$	0.878468	0.0323808
$737$	19.4201	0.715348
$738$	−20.5633	−0.756947
$739$	−4.69965	−0.172879	−0.0864397	−	0.996257i	$-0.527549\pi$
$739$	−4.69965	−0.172879	−0.0864397	+	0.996257i	$0.527549\pi$
$740$	0	0
$741$	0	0
$742$	−2.87847	−0.105672
$743$	21.3993	0.785064	0.392532	−	0.919738i	$-0.371599\pi$
$743$	21.3993	0.785064	0.392532	+	0.919738i	$0.371599\pi$
$744$	4.80641	0.176212
$745$	0	0
$746$	28.7639	1.05312
$747$	−5.53982	−0.202692
$748$	5.22829	0.191165
$749$	−6.21352	−0.227037
$750$	0	0
$751$	−46.3125	−1.68997	−0.844983	−	0.534793i	$-0.820390\pi$
$751$	−46.3125	−1.68997	−0.844983	+	0.534793i	$0.820390\pi$
$752$	0.878468	0.0320344
$753$	−4.16584	−0.151812
$754$	0	0
$755$	0	0
$756$	−4.59289	−0.167042
$757$	−27.8264	−1.01137	−0.505683	−	0.862719i	$-0.668759\pi$
$757$	−27.8264	−1.01137	−0.505683	+	0.862719i	$0.668759\pi$
$758$	−30.0347	−1.09091
$759$	4.03471	0.146451
$760$	0	0
$761$	0.114570	0.00415316	0.00207658	−	0.999998i	$-0.499339\pi$
$761$	0.114570	0.00415316	0.00207658	+	0.999998i	$0.499339\pi$
$762$	−9.61283	−0.348236
$763$	19.9279	0.721440
$764$	−15.4418	−0.558665
$765$	0	0
$766$	17.6849	0.638981
$767$	0	0
$768$	0.878468	0.0316990
$769$	38.9982	1.40631	0.703156	−	0.711036i	$-0.251774\pi$
$769$	38.9982	1.40631	0.703156	+	0.711036i	$0.251774\pi$
$770$	0	0
$771$	9.98343	0.359545
$772$	15.7916	0.568354
$773$	14.0555	0.505541	0.252770	−	0.967526i	$-0.418658\pi$
$773$	14.0555	0.505541	0.252770	+	0.967526i	$0.418658\pi$
$774$	−10.8707	−0.390738
$775$	0	0
$776$	−5.51387	−0.197936
$777$	−1.97046	−0.0706899
$778$	25.6424	0.919323
$779$	−11.3351	−0.406121
$780$	0	0
$781$	−45.4843	−1.62756
$782$	0.878468	0.0314140
$783$	4.03471	0.144189
$784$	1.00000	0.0357143
$785$	0	0
$786$	5.67191	0.202310
$787$	−35.7144	−1.27308	−0.636541	−	0.771243i	$-0.719635\pi$
$787$	−35.7144	−1.27308	−0.636541	+	0.771243i	$0.719635\pi$
$788$	9.75694	0.347576
$789$	3.94092	0.140300
$790$	0	0
$791$	−2.98523	−0.106143
$792$	−11.6502	−0.413971
$793$	0	0
$794$	17.5486	0.622776
$795$	0	0
$796$	15.3351	0.543537
$797$	11.5139	0.407842	0.203921	−	0.978987i	$-0.434631\pi$
$797$	11.5139	0.407842	0.203921	+	0.978987i	$0.434631\pi$
$798$	−1.07902	−0.0381968
$799$	0.878468	0.0310780
$800$	0	0
$801$	−22.7587	−0.804140
$802$	−3.54341	−0.125122
$803$	−31.3698	−1.10701
$804$	3.26300	0.115077
$805$	0	0
$806$	0	0
$807$	−11.4445	−0.402864
$808$	6.52864	0.229677
$809$	18.6997	0.657445	0.328722	−	0.944427i	$-0.393382\pi$
$809$	18.6997	0.657445	0.328722	+	0.944427i	$0.393382\pi$
$810$	0	0
$811$	34.0990	1.19738	0.598688	−	0.800982i	$-0.295689\pi$
$811$	34.0990	1.19738	0.598688	+	0.800982i	$0.295689\pi$
$812$	−0.878468	−0.0308282
$813$	−3.01213	−0.105640
$814$	−11.7274	−0.411045
$815$	0	0
$816$	0.878468	0.0307525
$817$	−5.99219	−0.209640
$818$	5.42884	0.189815
$819$	0	0
$820$	0	0
$821$	18.7551	0.654559	0.327279	−	0.944928i	$-0.393868\pi$
$821$	18.7551	0.654559	0.327279	+	0.944928i	$0.393868\pi$
$822$	−14.8836	−0.519126
$823$	−27.6553	−0.964005	−0.482002	−	0.876170i	$-0.660090\pi$
$823$	−27.6553	−0.964005	−0.482002	+	0.876170i	$0.660090\pi$
$824$	−8.39234	−0.292361
$825$	0	0
$826$	1.12153	0.0390231
$827$	−8.02954	−0.279214	−0.139607	−	0.990207i	$-0.544584\pi$
$827$	−8.02954	−0.279214	−0.139607	+	0.990207i	$0.544584\pi$
$828$	−1.95749	−0.0680273
$829$	−49.2682	−1.71115	−0.855577	−	0.517675i	$-0.826797\pi$
$829$	−49.2682	−1.71115	−0.855577	+	0.517675i	$0.826797\pi$
$830$	0	0
$831$	4.55554	0.158030
$832$	0	0
$833$	1.00000	0.0346479
$834$	−9.18578	−0.318078
$835$	0	0
$836$	−6.42188	−0.222105
$837$	−25.1293	−0.868597
$838$	−21.6128	−0.746603
$839$	38.0052	1.31208	0.656042	−	0.754724i	$-0.272229\pi$
$839$	38.0052	1.31208	0.656042	+	0.754724i	$0.272229\pi$
$840$	0	0
$841$	−28.2283	−0.973389
$842$	0.243064	0.00837653
$843$	0.0564389	0.00194386
$844$	−6.21352	−0.213878
$845$	0	0
$846$	−1.95749	−0.0672997
$847$	16.3351	0.561279
$848$	−2.87847	−0.0988470
$849$	−9.68752	−0.332475
$850$	0	0
$851$	−1.97046	−0.0675465
$852$	−7.64237	−0.261823
$853$	−41.5486	−1.42260	−0.711298	−	0.702890i	$-0.751892\pi$
$853$	−41.5486	−1.42260	−0.711298	+	0.702890i	$0.751892\pi$
$854$	5.51387	0.188681
$855$	0	0
$856$	−6.21352	−0.212374
$857$	12.0295	0.410921	0.205461	−	0.978665i	$-0.434131\pi$
$857$	12.0295	0.410921	0.205461	+	0.978665i	$0.434131\pi$
$858$	0	0
$859$	−8.09895	−0.276333	−0.138166	−	0.990409i	$-0.544121\pi$
$859$	−8.09895	−0.276333	−0.138166	+	0.990409i	$0.544121\pi$
$860$	0	0
$861$	8.10676	0.276278
$862$	−15.0277	−0.511847
$863$	36.1285	1.22983	0.614914	−	0.788594i	$-0.289191\pi$
$863$	36.1285	1.22983	0.614914	+	0.788594i	$0.289191\pi$
$864$	−4.59289	−0.156253
$865$	0	0
$866$	−18.4861	−0.628184
$867$	0.878468	0.0298343
$868$	5.47136	0.185710
$869$	63.8559	2.16616
$870$	0	0
$871$	0	0
$872$	19.9279	0.674845
$873$	12.2865	0.415836
$874$	−1.07902	−0.0364983
$875$	0	0
$876$	−5.27081	−0.178084
$877$	17.9705	0.606819	0.303410	−	0.952860i	$-0.401875\pi$
$877$	17.9705	0.606819	0.303410	+	0.952860i	$0.401875\pi$
$878$	−24.1197	−0.814002
$879$	−24.0435	−0.810966
$880$	0	0
$881$	4.49129	0.151316	0.0756578	−	0.997134i	$-0.475894\pi$
$881$	4.49129	0.151316	0.0756578	+	0.997134i	$0.475894\pi$
$882$	−2.22829	−0.0750306
$883$	−11.9409	−0.401844	−0.200922	−	0.979607i	$-0.564394\pi$
$883$	−11.9409	−0.401844	−0.200922	+	0.979607i	$0.564394\pi$
$884$	0	0
$885$	0	0
$886$	−28.0347	−0.941844
$887$	−29.6719	−0.996285	−0.498143	−	0.867095i	$-0.665984\pi$
$887$	−29.6719	−0.996285	−0.498143	+	0.867095i	$0.665984\pi$
$888$	−1.97046	−0.0661243
$889$	−10.9427	−0.367007
$890$	0	0
$891$	13.8559	0.464190
$892$	4.00000	0.133930
$893$	−1.07902	−0.0361079
$894$	−4.34202	−0.145219
$895$	0	0
$896$	1.00000	0.0334077
$897$	0	0
$898$	−39.3402	−1.31280
$899$	−4.80641	−0.160303
$900$	0	0
$901$	−2.87847	−0.0958957
$902$	48.2482	1.60649
$903$	4.28558	0.142615
$904$	−2.98523	−0.0992873
$905$	0	0
$906$	4.80641	0.159682
$907$	6.57116	0.218192	0.109096	−	0.994031i	$-0.465204\pi$
$907$	6.57116	0.218192	0.109096	+	0.994031i	$0.465204\pi$
$908$	26.1710	0.868515
$909$	−14.5477	−0.482518
$910$	0	0
$911$	−19.2413	−0.637492	−0.318746	−	0.947840i	$-0.603262\pi$
$911$	−19.2413	−0.637492	−0.318746	+	0.947840i	$0.603262\pi$
$912$	−1.07902	−0.0357298
$913$	12.9982	0.430178
$914$	27.9705	0.925181
$915$	0	0
$916$	−0.392341	−0.0129633
$917$	6.45659	0.213215
$918$	−4.59289	−0.151588
$919$	27.4635	0.905939	0.452969	−	0.891526i	$-0.350365\pi$
$919$	27.4635	0.905939	0.452969	+	0.891526i	$0.350365\pi$
$920$	0	0
$921$	−6.81422	−0.224536
$922$	−19.9279	−0.656292
$923$	0	0
$924$	4.59289	0.151095
$925$	0	0
$926$	2.94272	0.0967036
$927$	18.7006	0.614208
$928$	−0.878468	−0.0288371
$929$	−25.4548	−0.835145	−0.417572	−	0.908644i	$-0.637119\pi$
$929$	−25.4548	−0.835145	−0.417572	+	0.908644i	$0.637119\pi$
$930$	0	0
$931$	−1.22829	−0.0402557
$932$	6.00000	0.196537
$933$	20.4991	0.671110
$934$	−35.1267	−1.14938
$935$	0	0
$936$	0	0
$937$	20.1493	0.658248	0.329124	−	0.944287i	$-0.393247\pi$
$937$	20.1493	0.658248	0.329124	+	0.944287i	$0.393247\pi$
$938$	3.71442	0.121280
$939$	−21.6719	−0.707236
$940$	0	0
$941$	−30.0000	−0.977972	−0.488986	−	0.872292i	$-0.662633\pi$
$941$	−30.0000	−0.977972	−0.488986	+	0.872292i	$0.662633\pi$
$942$	−10.5416	−0.343464
$943$	8.10676	0.263992
$944$	1.12153	0.0365028
$945$	0	0
$946$	25.5061	0.829274
$947$	12.8142	0.416406	0.208203	−	0.978086i	$-0.433238\pi$
$947$	12.8142	0.416406	0.208203	+	0.978086i	$0.433238\pi$
$948$	10.7292	0.348468
$949$	0	0
$950$	0	0
$951$	5.67191	0.183924
$952$	1.00000	0.0324102
$953$	−12.3281	−0.399346	−0.199673	−	0.979863i	$-0.563988\pi$
$953$	−12.3281	−0.399346	−0.199673	+	0.979863i	$0.563988\pi$
$954$	6.41407	0.207663
$955$	0	0
$956$	−29.8984	−0.966984
$957$	−4.03471	−0.130424
$958$	−16.0000	−0.516937
$959$	−16.9427	−0.547109
$960$	0	0
$961$	−1.06425	−0.0343305
$962$	0	0
$963$	13.8456	0.446167
$964$	19.5781	0.630569
$965$	0	0
$966$	0.771706	0.0248292
$967$	−29.6979	−0.955019	−0.477509	−	0.878627i	$-0.658460\pi$
$967$	−29.6979	−0.955019	−0.477509	+	0.878627i	$0.658460\pi$
$968$	16.3351	0.525029
$969$	−1.07902	−0.0346630
$970$	0	0
$971$	−28.0860	−0.901322	−0.450661	−	0.892695i	$-0.648812\pi$
$971$	−28.0860	−0.901322	−0.450661	+	0.892695i	$0.648812\pi$
$972$	16.1068	0.516625
$973$	−10.4566	−0.335223
$974$	−24.6276	−0.789119
$975$	0	0
$976$	5.51387	0.176495
$977$	−6.63024	−0.212120	−0.106060	−	0.994360i	$-0.533824\pi$
$977$	−6.63024	−0.212120	−0.106060	+	0.994360i	$0.533824\pi$
$978$	−3.11277	−0.0995356
$979$	53.3993	1.70665
$980$	0	0
$981$	−44.4053	−1.41775
$982$	3.02774	0.0966192
$983$	46.6701	1.48855	0.744273	−	0.667876i	$-0.232796\pi$
$983$	46.6701	1.48855	0.744273	+	0.667876i	$0.232796\pi$
$984$	8.10676	0.258434
$985$	0	0
$986$	−0.878468	−0.0279761
$987$	0.771706	0.0245637
$988$	0	0
$989$	4.28558	0.136273
$990$	0	0
$991$	−3.02774	−0.0961795	−0.0480897	−	0.998843i	$-0.515313\pi$
$991$	−3.02774	−0.0961795	−0.0480897	+	0.998843i	$0.515313\pi$
$992$	5.47136	0.173716
$993$	7.82994	0.248476
$994$	−8.69965	−0.275936
$995$	0	0
$996$	2.18398	0.0692022
$997$	8.11457	0.256991	0.128496	−	0.991710i	$-0.458985\pi$
$997$	8.11457	0.256991	0.128496	+	0.991710i	$0.458985\pi$
$998$	29.1692	0.923335
$999$	10.3021	0.325945

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5950.2.a.bm.1.2		3
5.4	even	2		1190.2.a.j.1.2	✓	3
20.19	odd	2		9520.2.a.z.1.2		3
35.34	odd	2		8330.2.a.bu.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1190.2.a.j.1.2	✓	3	5.4	even	2
5950.2.a.bm.1.2		3	1.1	even	1	trivial
8330.2.a.bu.1.2		3	35.34	odd	2
9520.2.a.z.1.2		3	20.19	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:	\( N \)	\(=\)	\( 5950 = 2 \cdot 5^{2} \cdot 7 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5950.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +0.878468 q^{3} +1.00000 q^{4} +0.878468 q^{6} +1.00000 q^{7} +1.00000 q^{8} -2.22829 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} +0.878468 q^{3} +1.00000 q^{4} +0.878468 q^{6} +1.00000 q^{7} +1.00000 q^{8} -2.22829 q^{9} +5.22829 q^{11} +0.878468 q^{12} +1.00000 q^{14} +1.00000 q^{16} +1.00000 q^{17} -2.22829 q^{18} -1.22829 q^{19} +0.878468 q^{21} +5.22829 q^{22} +0.878468 q^{23} +0.878468 q^{24} -4.59289 q^{27} +1.00000 q^{28} -0.878468 q^{29} +5.47136 q^{31} +1.00000 q^{32} +4.59289 q^{33} +1.00000 q^{34} -2.22829 q^{36} -2.24306 q^{37} -1.22829 q^{38} +9.22829 q^{41} +0.878468 q^{42} +4.87847 q^{43} +5.22829 q^{44} +0.878468 q^{46} +0.878468 q^{47} +0.878468 q^{48} +1.00000 q^{49} +0.878468 q^{51} -2.87847 q^{53} -4.59289 q^{54} +1.00000 q^{56} -1.07902 q^{57} -0.878468 q^{58} +1.12153 q^{59} +5.51387 q^{61} +5.47136 q^{62} -2.22829 q^{63} +1.00000 q^{64} +4.59289 q^{66} +3.71442 q^{67} +1.00000 q^{68} +0.771706 q^{69} -8.69965 q^{71} -2.22829 q^{72} -6.00000 q^{73} -2.24306 q^{74} -1.22829 q^{76} +5.22829 q^{77} +12.2135 q^{79} +2.65017 q^{81} +9.22829 q^{82} +2.48613 q^{83} +0.878468 q^{84} +4.87847 q^{86} -0.771706 q^{87} +5.22829 q^{88} +10.2135 q^{89} +0.878468 q^{92} +4.80641 q^{93} +0.878468 q^{94} +0.878468 q^{96} -5.51387 q^{97} +1.00000 q^{98} -11.6502 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{2} + q^{3} + 3 q^{4} + q^{6} + 3 q^{7} + 3 q^{8} + 10 q^{9}+O(q^{10}) \) 3 * q + 3 * q^2 + q^3 + 3 * q^4 + q^6 + 3 * q^7 + 3 * q^8 + 10 * q^9	\( 3 q + 3 q^{2} + q^{3} + 3 q^{4} + q^{6} + 3 q^{7} + 3 q^{8} + 10 q^{9} - q^{11} + q^{12} + 3 q^{14} + 3 q^{16} + 3 q^{17} + 10 q^{18} + 13 q^{19} + q^{21} - q^{22} + q^{23} + q^{24} - 2 q^{27} + 3 q^{28} - q^{29} + 3 q^{31} + 3 q^{32} + 2 q^{33} + 3 q^{34} + 10 q^{36} - 10 q^{37} + 13 q^{38} + 11 q^{41} + q^{42} + 13 q^{43} - q^{44} + q^{46} + q^{47} + q^{48} + 3 q^{49} + q^{51} - 7 q^{53} - 2 q^{54} + 3 q^{56} + 2 q^{57} - q^{58} + 5 q^{59} + 10 q^{61} + 3 q^{62} + 10 q^{63} + 3 q^{64} + 2 q^{66} + q^{67} + 3 q^{68} + 19 q^{69} + 4 q^{71} + 10 q^{72} - 18 q^{73} - 10 q^{74} + 13 q^{76} - q^{77} + 23 q^{81} + 11 q^{82} + 14 q^{83} + q^{84} + 13 q^{86} - 19 q^{87} - q^{88} - 6 q^{89} + q^{92} - 34 q^{93} + q^{94} + q^{96} - 10 q^{97} + 3 q^{98} - 50 q^{99}+O(q^{100}) \) 3 * q + 3 * q^2 + q^3 + 3 * q^4 + q^6 + 3 * q^7 + 3 * q^8 + 10 * q^9 - q^11 + q^12 + 3 * q^14 + 3 * q^16 + 3 * q^17 + 10 * q^18 + 13 * q^19 + q^21 - q^22 + q^23 + q^24 - 2 * q^27 + 3 * q^28 - q^29 + 3 * q^31 + 3 * q^32 + 2 * q^33 + 3 * q^34 + 10 * q^36 - 10 * q^37 + 13 * q^38 + 11 * q^41 + q^42 + 13 * q^43 - q^44 + q^46 + q^47 + q^48 + 3 * q^49 + q^51 - 7 * q^53 - 2 * q^54 + 3 * q^56 + 2 * q^57 - q^58 + 5 * q^59 + 10 * q^61 + 3 * q^62 + 10 * q^63 + 3 * q^64 + 2 * q^66 + q^67 + 3 * q^68 + 19 * q^69 + 4 * q^71 + 10 * q^72 - 18 * q^73 - 10 * q^74 + 13 * q^76 - q^77 + 23 * q^81 + 11 * q^82 + 14 * q^83 + q^84 + 13 * q^86 - 19 * q^87 - q^88 - 6 * q^89 + q^92 - 34 * q^93 + q^94 + q^96 - 10 * q^97 + 3 * q^98 - 50 * q^99

Introduction

L-functions

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