LMFDB - Embedded newform 3640.2.a.ba.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3640,2,Mod(1,3640)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3640, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("3640.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3640 = 2^{3} \cdot 5 \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3640.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:	$29.0655463357$
Analytic rank:	$0$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 3x^{6} - 11x^{5} + 24x^{4} + 33x^{3} - 41x^{2} - 31x - 4 $ x^7 - 3x^6 - 11x^5 + 24x^4 + 33x^3 - 41x^2 - 31x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-0.173825$ of defining polynomial
Character	$\chi$	$=$	3640.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.17382 q^{3} +1.00000 q^{5} +1.00000 q^{7} -1.62214 q^{9} +O(q^{10})$	$q+1.17382 q^{3} +1.00000 q^{5} +1.00000 q^{7} -1.62214 q^{9} -0.490411 q^{11} +1.00000 q^{13} +1.17382 q^{15} -7.57421 q^{17} +3.54940 q^{19} +1.17382 q^{21} -4.51918 q^{23} +1.00000 q^{25} -5.42558 q^{27} +9.36416 q^{29} +7.29385 q^{31} -0.575657 q^{33} +1.00000 q^{35} +11.3187 q^{37} +1.17382 q^{39} +11.4375 q^{41} -6.49044 q^{43} -1.62214 q^{45} +6.49642 q^{47} +1.00000 q^{49} -8.89080 q^{51} +9.10085 q^{53} -0.490411 q^{55} +4.16637 q^{57} +12.4402 q^{59} +3.09550 q^{61} -1.62214 q^{63} +1.00000 q^{65} +8.98859 q^{67} -5.30473 q^{69} +14.3391 q^{71} -13.4198 q^{73} +1.17382 q^{75} -0.490411 q^{77} -16.0994 q^{79} -1.50227 q^{81} +11.1176 q^{83} -7.57421 q^{85} +10.9919 q^{87} -3.24414 q^{89} +1.00000 q^{91} +8.56171 q^{93} +3.54940 q^{95} -8.28555 q^{97} +0.795513 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + 4 q^{3} + 7 q^{5} + 7 q^{7} + 11 q^{9}+O(q^{10}) $ 7 * q + 4 * q^3 + 7 * q^5 + 7 * q^7 + 11 * q^9	$ 7 q + 4 q^{3} + 7 q^{5} + 7 q^{7} + 11 q^{9} + 7 q^{11} + 7 q^{13} + 4 q^{15} + 3 q^{17} + 13 q^{19} + 4 q^{21} + 11 q^{23} + 7 q^{25} + 13 q^{27} + 3 q^{29} + 12 q^{31} - 11 q^{33} + 7 q^{35} - 4 q^{37} + 4 q^{39} + 6 q^{41} + 10 q^{43} + 11 q^{45} + 15 q^{47} + 7 q^{49} + 8 q^{53} + 7 q^{55} - 18 q^{57} + 13 q^{59} - q^{61} + 11 q^{63} + 7 q^{65} + 8 q^{67} - 9 q^{69} + 20 q^{71} - 17 q^{73} + 4 q^{75} + 7 q^{77} + 6 q^{79} + 39 q^{81} + 30 q^{83} + 3 q^{85} + 21 q^{87} + 5 q^{89} + 7 q^{91} - 24 q^{93} + 13 q^{95} - 9 q^{97} + 12 q^{99}+O(q^{100}) $ 7 * q + 4 * q^3 + 7 * q^5 + 7 * q^7 + 11 * q^9 + 7 * q^11 + 7 * q^13 + 4 * q^15 + 3 * q^17 + 13 * q^19 + 4 * q^21 + 11 * q^23 + 7 * q^25 + 13 * q^27 + 3 * q^29 + 12 * q^31 - 11 * q^33 + 7 * q^35 - 4 * q^37 + 4 * q^39 + 6 * q^41 + 10 * q^43 + 11 * q^45 + 15 * q^47 + 7 * q^49 + 8 * q^53 + 7 * q^55 - 18 * q^57 + 13 * q^59 - q^61 + 11 * q^63 + 7 * q^65 + 8 * q^67 - 9 * q^69 + 20 * q^71 - 17 * q^73 + 4 * q^75 + 7 * q^77 + 6 * q^79 + 39 * q^81 + 30 * q^83 + 3 * q^85 + 21 * q^87 + 5 * q^89 + 7 * q^91 - 24 * q^93 + 13 * q^95 - 9 * q^97 + 12 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	1.17382	0.677708	0.338854	−	0.940839i	$-0.389961\pi$
$3$	1.17382	0.677708	0.338854	+	0.940839i	$0.389961\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	−1.62214	−0.540712
$10$	0	0
$11$	−0.490411	−0.147864	−0.0739322	−	0.997263i	$-0.523555\pi$
$11$	−0.490411	−0.147864	−0.0739322	+	0.997263i	$0.523555\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	1.17382	0.303080
$16$	0	0
$17$	−7.57421	−1.83702	−0.918508	−	0.395402i	$-0.870605\pi$
$17$	−7.57421	−1.83702	−0.918508	+	0.395402i	$0.870605\pi$
$18$	0	0
$19$	3.54940	0.814288	0.407144	−	0.913364i	$-0.366525\pi$
$19$	3.54940	0.814288	0.407144	+	0.913364i	$0.366525\pi$
$20$	0	0
$21$	1.17382	0.256150
$22$	0	0
$23$	−4.51918	−0.942315	−0.471157	−	0.882049i	$-0.656164\pi$
$23$	−4.51918	−0.942315	−0.471157	+	0.882049i	$0.656164\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−5.42558	−1.04415
$28$	0	0
$29$	9.36416	1.73888	0.869441	−	0.494037i	$-0.164479\pi$
$29$	9.36416	1.73888	0.869441	+	0.494037i	$0.164479\pi$
$30$	0	0
$31$	7.29385	1.31001	0.655007	−	0.755623i	$-0.272666\pi$
$31$	7.29385	1.31001	0.655007	+	0.755623i	$0.272666\pi$
$32$	0	0
$33$	−0.575657	−0.100209
$34$	0	0
$35$	1.00000	0.169031
$36$	0	0
$37$	11.3187	1.86078	0.930389	−	0.366574i	$-0.119469\pi$
$37$	11.3187	1.86078	0.930389	+	0.366574i	$0.119469\pi$
$38$	0	0
$39$	1.17382	0.187962
$40$	0	0
$41$	11.4375	1.78623	0.893116	−	0.449827i	$-0.148514\pi$
$41$	11.4375	1.78623	0.893116	+	0.449827i	$0.148514\pi$
$42$	0	0
$43$	−6.49044	−0.989783	−0.494891	−	0.868955i	$-0.664792\pi$
$43$	−6.49044	−0.989783	−0.494891	+	0.868955i	$0.664792\pi$
$44$	0	0
$45$	−1.62214	−0.241814
$46$	0	0
$47$	6.49642	0.947600	0.473800	−	0.880632i	$-0.342882\pi$
$47$	6.49642	0.947600	0.473800	+	0.880632i	$0.342882\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−8.89080	−1.24496
$52$	0	0
$53$	9.10085	1.25010	0.625049	−	0.780586i	$-0.285079\pi$
$53$	9.10085	1.25010	0.625049	+	0.780586i	$0.285079\pi$
$54$	0	0
$55$	−0.490411	−0.0661270
$56$	0	0
$57$	4.16637	0.551849
$58$	0	0
$59$	12.4402	1.61958	0.809788	−	0.586723i	$-0.199582\pi$
$59$	12.4402	1.61958	0.809788	+	0.586723i	$0.199582\pi$
$60$	0	0
$61$	3.09550	0.396338	0.198169	−	0.980168i	$-0.436501\pi$
$61$	3.09550	0.396338	0.198169	+	0.980168i	$0.436501\pi$
$62$	0	0
$63$	−1.62214	−0.204370
$64$	0	0
$65$	1.00000	0.124035
$66$	0	0
$67$	8.98859	1.09813	0.549066	−	0.835779i	$-0.314984\pi$
$67$	8.98859	1.09813	0.549066	+	0.835779i	$0.314984\pi$
$68$	0	0
$69$	−5.30473	−0.638614
$70$	0	0
$71$	14.3391	1.70174	0.850870	−	0.525377i	$-0.176076\pi$
$71$	14.3391	1.70174	0.850870	+	0.525377i	$0.176076\pi$
$72$	0	0
$73$	−13.4198	−1.57066	−0.785332	−	0.619075i	$-0.787508\pi$
$73$	−13.4198	−1.57066	−0.785332	+	0.619075i	$0.787508\pi$
$74$	0	0
$75$	1.17382	0.135542
$76$	0	0
$77$	−0.490411	−0.0558875
$78$	0	0
$79$	−16.0994	−1.81132	−0.905662	−	0.424000i	$-0.860626\pi$
$79$	−16.0994	−1.81132	−0.905662	+	0.424000i	$0.860626\pi$
$80$	0	0
$81$	−1.50227	−0.166919
$82$	0	0
$83$	11.1176	1.22032	0.610158	−	0.792280i	$-0.291106\pi$
$83$	11.1176	1.22032	0.610158	+	0.792280i	$0.291106\pi$
$84$	0	0
$85$	−7.57421	−0.821539
$86$	0	0
$87$	10.9919	1.17845
$88$	0	0
$89$	−3.24414	−0.343878	−0.171939	−	0.985108i	$-0.555003\pi$
$89$	−3.24414	−0.343878	−0.171939	+	0.985108i	$0.555003\pi$
$90$	0	0
$91$	1.00000	0.104828
$92$	0	0
$93$	8.56171	0.887807
$94$	0	0
$95$	3.54940	0.364161
$96$	0	0
$97$	−8.28555	−0.841270	−0.420635	−	0.907230i	$-0.638193\pi$
$97$	−8.28555	−0.841270	−0.420635	+	0.907230i	$0.638193\pi$
$98$	0	0
$99$	0.795513	0.0799521
$100$	0	0
$101$	1.28977	0.128336	0.0641682	−	0.997939i	$-0.479561\pi$
$101$	1.28977	0.128336	0.0641682	+	0.997939i	$0.479561\pi$
$102$	0	0
$103$	−5.40598	−0.532667	−0.266333	−	0.963881i	$-0.585812\pi$
$103$	−5.40598	−0.532667	−0.266333	+	0.963881i	$0.585812\pi$
$104$	0	0
$105$	1.17382	0.114554
$106$	0	0
$107$	2.68250	0.259327	0.129664	−	0.991558i	$-0.458610\pi$
$107$	2.68250	0.259327	0.129664	+	0.991558i	$0.458610\pi$
$108$	0	0
$109$	−7.23430	−0.692921	−0.346460	−	0.938065i	$-0.612616\pi$
$109$	−7.23430	−0.692921	−0.346460	+	0.938065i	$0.612616\pi$
$110$	0	0
$111$	13.2861	1.26106
$112$	0	0
$113$	−1.44315	−0.135760	−0.0678800	−	0.997693i	$-0.521624\pi$
$113$	−1.44315	−0.135760	−0.0678800	+	0.997693i	$0.521624\pi$
$114$	0	0
$115$	−4.51918	−0.421416
$116$	0	0
$117$	−1.62214	−0.149966
$118$	0	0
$119$	−7.57421	−0.694327
$120$	0	0
$121$	−10.7595	−0.978136
$122$	0	0
$123$	13.4256	1.21054
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	10.8927	0.966570	0.483285	−	0.875463i	$-0.339443\pi$
$127$	10.8927	0.966570	0.483285	+	0.875463i	$0.339443\pi$
$128$	0	0
$129$	−7.61864	−0.670784
$130$	0	0
$131$	1.90874	0.166768	0.0833839	−	0.996518i	$-0.473427\pi$
$131$	1.90874	0.166768	0.0833839	+	0.996518i	$0.473427\pi$
$132$	0	0
$133$	3.54940	0.307772
$134$	0	0
$135$	−5.42558	−0.466959
$136$	0	0
$137$	2.88764	0.246708	0.123354	−	0.992363i	$-0.460635\pi$
$137$	2.88764	0.246708	0.123354	+	0.992363i	$0.460635\pi$
$138$	0	0
$139$	−5.46525	−0.463557	−0.231778	−	0.972769i	$-0.574454\pi$
$139$	−5.46525	−0.463557	−0.231778	+	0.972769i	$0.574454\pi$
$140$	0	0
$141$	7.62566	0.642196
$142$	0	0
$143$	−0.490411	−0.0410102
$144$	0	0
$145$	9.36416	0.777651
$146$	0	0
$147$	1.17382	0.0968154
$148$	0	0
$149$	−3.02453	−0.247779	−0.123890	−	0.992296i	$-0.539537\pi$
$149$	−3.02453	−0.247779	−0.123890	+	0.992296i	$0.539537\pi$
$150$	0	0
$151$	−16.9386	−1.37844	−0.689220	−	0.724552i	$-0.742047\pi$
$151$	−16.9386	−1.37844	−0.689220	+	0.724552i	$0.742047\pi$
$152$	0	0
$153$	12.2864	0.993297
$154$	0	0
$155$	7.29385	0.585856
$156$	0	0
$157$	−1.40249	−0.111931	−0.0559655	−	0.998433i	$-0.517824\pi$
$157$	−1.40249	−0.111931	−0.0559655	+	0.998433i	$0.517824\pi$
$158$	0	0
$159$	10.6828	0.847201
$160$	0	0
$161$	−4.51918	−0.356162
$162$	0	0
$163$	13.5130	1.05842	0.529208	−	0.848492i	$-0.322489\pi$
$163$	13.5130	1.05842	0.529208	+	0.848492i	$0.322489\pi$
$164$	0	0
$165$	−0.575657	−0.0448148
$166$	0	0
$167$	−2.24464	−0.173695	−0.0868477	−	0.996222i	$-0.527679\pi$
$167$	−2.24464	−0.173695	−0.0868477	+	0.996222i	$0.527679\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−5.75760	−0.440295
$172$	0	0
$173$	11.7189	0.890969	0.445485	−	0.895290i	$-0.353031\pi$
$173$	11.7189	0.890969	0.445485	+	0.895290i	$0.353031\pi$
$174$	0	0
$175$	1.00000	0.0755929
$176$	0	0
$177$	14.6026	1.09760
$178$	0	0
$179$	13.1374	0.981933	0.490966	−	0.871179i	$-0.336644\pi$
$179$	13.1374	0.981933	0.490966	+	0.871179i	$0.336644\pi$
$180$	0	0
$181$	2.32887	0.173104	0.0865519	−	0.996247i	$-0.472415\pi$
$181$	2.32887	0.173104	0.0865519	+	0.996247i	$0.472415\pi$
$182$	0	0
$183$	3.63357	0.268601
$184$	0	0
$185$	11.3187	0.832165
$186$	0	0
$187$	3.71448	0.271629
$188$	0	0
$189$	−5.42558	−0.394653
$190$	0	0
$191$	−6.83427	−0.494510	−0.247255	−	0.968950i	$-0.579529\pi$
$191$	−6.83427	−0.494510	−0.247255	+	0.968950i	$0.579529\pi$
$192$	0	0
$193$	−23.0969	−1.66255	−0.831274	−	0.555863i	$-0.812388\pi$
$193$	−23.0969	−1.66255	−0.831274	+	0.555863i	$0.812388\pi$
$194$	0	0
$195$	1.17382	0.0840593
$196$	0	0
$197$	−26.0282	−1.85443	−0.927216	−	0.374527i	$-0.877805\pi$
$197$	−26.0282	−1.85443	−0.927216	+	0.374527i	$0.877805\pi$
$198$	0	0
$199$	−11.0550	−0.783666	−0.391833	−	0.920036i	$-0.628159\pi$
$199$	−11.0550	−0.783666	−0.391833	+	0.920036i	$0.628159\pi$
$200$	0	0
$201$	10.5510	0.744212
$202$	0	0
$203$	9.36416	0.657235
$204$	0	0
$205$	11.4375	0.798827
$206$	0	0
$207$	7.33073	0.509521
$208$	0	0
$209$	−1.74066	−0.120404
$210$	0	0
$211$	−5.73756	−0.394990	−0.197495	−	0.980304i	$-0.563281\pi$
$211$	−5.73756	−0.394990	−0.197495	+	0.980304i	$0.563281\pi$
$212$	0	0
$213$	16.8316	1.15328
$214$	0	0
$215$	−6.49044	−0.442644
$216$	0	0
$217$	7.29385	0.495139
$218$	0	0
$219$	−15.7524	−1.06445
$220$	0	0
$221$	−7.57421	−0.509497
$222$	0	0
$223$	5.67225	0.379842	0.189921	−	0.981799i	$-0.439177\pi$
$223$	5.67225	0.379842	0.189921	+	0.981799i	$0.439177\pi$
$224$	0	0
$225$	−1.62214	−0.108142
$226$	0	0
$227$	−7.12962	−0.473210	−0.236605	−	0.971606i	$-0.576035\pi$
$227$	−7.12962	−0.473210	−0.236605	+	0.971606i	$0.576035\pi$
$228$	0	0
$229$	−13.4017	−0.885607	−0.442803	−	0.896619i	$-0.646016\pi$
$229$	−13.4017	−0.885607	−0.442803	+	0.896619i	$0.646016\pi$
$230$	0	0
$231$	−0.575657	−0.0378754
$232$	0	0
$233$	−23.6228	−1.54758	−0.773790	−	0.633443i	$-0.781641\pi$
$233$	−23.6228	−1.54758	−0.773790	+	0.633443i	$0.781641\pi$
$234$	0	0
$235$	6.49642	0.423780
$236$	0	0
$237$	−18.8979	−1.22755
$238$	0	0
$239$	−18.5430	−1.19945	−0.599723	−	0.800208i	$-0.704723\pi$
$239$	−18.5430	−1.19945	−0.599723	+	0.800208i	$0.704723\pi$
$240$	0	0
$241$	2.18362	0.140659	0.0703295	−	0.997524i	$-0.477595\pi$
$241$	2.18362	0.140659	0.0703295	+	0.997524i	$0.477595\pi$
$242$	0	0
$243$	14.5133	0.931030
$244$	0	0
$245$	1.00000	0.0638877
$246$	0	0
$247$	3.54940	0.225843
$248$	0	0
$249$	13.0501	0.827018
$250$	0	0
$251$	12.4310	0.784639	0.392320	−	0.919829i	$-0.371673\pi$
$251$	12.4310	0.784639	0.392320	+	0.919829i	$0.371673\pi$
$252$	0	0
$253$	2.21626	0.139335
$254$	0	0
$255$	−8.89080	−0.556763
$256$	0	0
$257$	−26.5427	−1.65569	−0.827844	−	0.560958i	$-0.810433\pi$
$257$	−26.5427	−1.65569	−0.827844	+	0.560958i	$0.810433\pi$
$258$	0	0
$259$	11.3187	0.703308
$260$	0	0
$261$	−15.1899	−0.940234
$262$	0	0
$263$	0.633804	0.0390820	0.0195410	−	0.999809i	$-0.493780\pi$
$263$	0.633804	0.0390820	0.0195410	+	0.999809i	$0.493780\pi$
$264$	0	0
$265$	9.10085	0.559061
$266$	0	0
$267$	−3.80805	−0.233049
$268$	0	0
$269$	−10.8629	−0.662321	−0.331161	−	0.943574i	$-0.607440\pi$
$269$	−10.8629	−0.662321	−0.331161	+	0.943574i	$0.607440\pi$
$270$	0	0
$271$	21.7618	1.32193	0.660967	−	0.750415i	$-0.270146\pi$
$271$	21.7618	1.32193	0.660967	+	0.750415i	$0.270146\pi$
$272$	0	0
$273$	1.17382	0.0710431
$274$	0	0
$275$	−0.490411	−0.0295729
$276$	0	0
$277$	−5.28089	−0.317298	−0.158649	−	0.987335i	$-0.550714\pi$
$277$	−5.28089	−0.317298	−0.158649	+	0.987335i	$0.550714\pi$
$278$	0	0
$279$	−11.8316	−0.708340
$280$	0	0
$281$	31.6370	1.88730	0.943652	−	0.330940i	$-0.107366\pi$
$281$	31.6370	1.88730	0.943652	+	0.330940i	$0.107366\pi$
$282$	0	0
$283$	−17.6704	−1.05040	−0.525198	−	0.850980i	$-0.676009\pi$
$283$	−17.6704	−1.05040	−0.525198	+	0.850980i	$0.676009\pi$
$284$	0	0
$285$	4.16637	0.246795
$286$	0	0
$287$	11.4375	0.675132
$288$	0	0
$289$	40.3687	2.37463
$290$	0	0
$291$	−9.72579	−0.570136
$292$	0	0
$293$	19.8004	1.15675	0.578376	−	0.815770i	$-0.303687\pi$
$293$	19.8004	1.15675	0.578376	+	0.815770i	$0.303687\pi$
$294$	0	0
$295$	12.4402	0.724296
$296$	0	0
$297$	2.66076	0.154393
$298$	0	0
$299$	−4.51918	−0.261351
$300$	0	0
$301$	−6.49044	−0.374103
$302$	0	0
$303$	1.51396	0.0869747
$304$	0	0
$305$	3.09550	0.177248
$306$	0	0
$307$	10.7866	0.615621	0.307811	−	0.951448i	$-0.400404\pi$
$307$	10.7866	0.615621	0.307811	+	0.951448i	$0.400404\pi$
$308$	0	0
$309$	−6.34567	−0.360993
$310$	0	0
$311$	13.7557	0.780016	0.390008	−	0.920811i	$-0.372472\pi$
$311$	13.7557	0.780016	0.390008	+	0.920811i	$0.372472\pi$
$312$	0	0
$313$	18.1411	1.02540	0.512699	−	0.858569i	$-0.328646\pi$
$313$	18.1411	1.02540	0.512699	+	0.858569i	$0.328646\pi$
$314$	0	0
$315$	−1.62214	−0.0913970
$316$	0	0
$317$	−9.52650	−0.535061	−0.267531	−	0.963549i	$-0.586208\pi$
$317$	−9.52650	−0.535061	−0.267531	+	0.963549i	$0.586208\pi$
$318$	0	0
$319$	−4.59229	−0.257119
$320$	0	0
$321$	3.14879	0.175748
$322$	0	0
$323$	−26.8839	−1.49586
$324$	0	0
$325$	1.00000	0.0554700
$326$	0	0
$327$	−8.49181	−0.469598
$328$	0	0
$329$	6.49642	0.358159
$330$	0	0
$331$	32.9407	1.81058	0.905291	−	0.424792i	$-0.139653\pi$
$331$	32.9407	1.81058	0.905291	+	0.424792i	$0.139653\pi$
$332$	0	0
$333$	−18.3604	−1.00614
$334$	0	0
$335$	8.98859	0.491099
$336$	0	0
$337$	−22.1763	−1.20802	−0.604011	−	0.796976i	$-0.706432\pi$
$337$	−22.1763	−1.20802	−0.604011	+	0.796976i	$0.706432\pi$
$338$	0	0
$339$	−1.69400	−0.0920056
$340$	0	0
$341$	−3.57699	−0.193705
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	−5.30473	−0.285597
$346$	0	0
$347$	9.54503	0.512404	0.256202	−	0.966623i	$-0.417529\pi$
$347$	9.54503	0.512404	0.256202	+	0.966623i	$0.417529\pi$
$348$	0	0
$349$	1.44046	0.0771058	0.0385529	−	0.999257i	$-0.487725\pi$
$349$	1.44046	0.0771058	0.0385529	+	0.999257i	$0.487725\pi$
$350$	0	0
$351$	−5.42558	−0.289596
$352$	0	0
$353$	−18.0835	−0.962487	−0.481243	−	0.876587i	$-0.659815\pi$
$353$	−18.0835	−0.962487	−0.481243	+	0.876587i	$0.659815\pi$
$354$	0	0
$355$	14.3391	0.761041
$356$	0	0
$357$	−8.89080	−0.470551
$358$	0	0
$359$	27.1519	1.43302	0.716511	−	0.697575i	$-0.245738\pi$
$359$	27.1519	1.43302	0.716511	+	0.697575i	$0.245738\pi$
$360$	0	0
$361$	−6.40177	−0.336935
$362$	0	0
$363$	−12.6298	−0.662891
$364$	0	0
$365$	−13.4198	−0.702422
$366$	0	0
$367$	4.33159	0.226107	0.113053	−	0.993589i	$-0.463937\pi$
$367$	4.33159	0.226107	0.113053	+	0.993589i	$0.463937\pi$
$368$	0	0
$369$	−18.5531	−0.965836
$370$	0	0
$371$	9.10085	0.472493
$372$	0	0
$373$	−12.2767	−0.635662	−0.317831	−	0.948147i	$-0.602955\pi$
$373$	−12.2767	−0.635662	−0.317831	+	0.948147i	$0.602955\pi$
$374$	0	0
$375$	1.17382	0.0606161
$376$	0	0
$377$	9.36416	0.482279
$378$	0	0
$379$	−32.6175	−1.67545	−0.837725	−	0.546092i	$-0.816115\pi$
$379$	−32.6175	−1.67545	−0.837725	+	0.546092i	$0.816115\pi$
$380$	0	0
$381$	12.7861	0.655052
$382$	0	0
$383$	8.13675	0.415769	0.207884	−	0.978153i	$-0.433342\pi$
$383$	8.13675	0.415769	0.207884	+	0.978153i	$0.433342\pi$
$384$	0	0
$385$	−0.490411	−0.0249937
$386$	0	0
$387$	10.5284	0.535187
$388$	0	0
$389$	18.7551	0.950923	0.475462	−	0.879736i	$-0.342281\pi$
$389$	18.7551	0.950923	0.475462	+	0.879736i	$0.342281\pi$
$390$	0	0
$391$	34.2293	1.73105
$392$	0	0
$393$	2.24053	0.113020
$394$	0	0
$395$	−16.0994	−0.810049
$396$	0	0
$397$	−24.5632	−1.23279	−0.616396	−	0.787437i	$-0.711408\pi$
$397$	−24.5632	−1.23279	−0.616396	+	0.787437i	$0.711408\pi$
$398$	0	0
$399$	4.16637	0.208579
$400$	0	0
$401$	−9.74863	−0.486823	−0.243412	−	0.969923i	$-0.578267\pi$
$401$	−9.74863	−0.486823	−0.243412	+	0.969923i	$0.578267\pi$
$402$	0	0
$403$	7.29385	0.363333
$404$	0	0
$405$	−1.50227	−0.0746484
$406$	0	0
$407$	−5.55080	−0.275143
$408$	0	0
$409$	3.52479	0.174290	0.0871449	−	0.996196i	$-0.472226\pi$
$409$	3.52479	0.174290	0.0871449	+	0.996196i	$0.472226\pi$
$410$	0	0
$411$	3.38959	0.167196
$412$	0	0
$413$	12.4402	0.612142
$414$	0	0
$415$	11.1176	0.545742
$416$	0	0
$417$	−6.41525	−0.314156
$418$	0	0
$419$	14.4777	0.707281	0.353640	−	0.935381i	$-0.384944\pi$
$419$	14.4777	0.707281	0.353640	+	0.935381i	$0.384944\pi$
$420$	0	0
$421$	−10.7510	−0.523973	−0.261986	−	0.965072i	$-0.584378\pi$
$421$	−10.7510	−0.523973	−0.261986	+	0.965072i	$0.584378\pi$
$422$	0	0
$423$	−10.5381	−0.512379
$424$	0	0
$425$	−7.57421	−0.367403
$426$	0	0
$427$	3.09550	0.149802
$428$	0	0
$429$	−0.575657	−0.0277930
$430$	0	0
$431$	26.2258	1.26325	0.631626	−	0.775274i	$-0.282388\pi$
$431$	26.2258	1.26325	0.631626	+	0.775274i	$0.282388\pi$
$432$	0	0
$433$	−14.7581	−0.709227	−0.354613	−	0.935013i	$-0.615388\pi$
$433$	−14.7581	−0.709227	−0.354613	+	0.935013i	$0.615388\pi$
$434$	0	0
$435$	10.9919	0.527021
$436$	0	0
$437$	−16.0404	−0.767315
$438$	0	0
$439$	−10.1335	−0.483643	−0.241822	−	0.970321i	$-0.577745\pi$
$439$	−10.1335	−0.483643	−0.241822	+	0.970321i	$0.577745\pi$
$440$	0	0
$441$	−1.62214	−0.0772445
$442$	0	0
$443$	−26.9131	−1.27868	−0.639341	−	0.768924i	$-0.720793\pi$
$443$	−26.9131	−1.27868	−0.639341	+	0.768924i	$0.720793\pi$
$444$	0	0
$445$	−3.24414	−0.153787
$446$	0	0
$447$	−3.55027	−0.167922
$448$	0	0
$449$	−10.5949	−0.500003	−0.250001	−	0.968245i	$-0.580431\pi$
$449$	−10.5949	−0.500003	−0.250001	+	0.968245i	$0.580431\pi$
$450$	0	0
$451$	−5.60906	−0.264120
$452$	0	0
$453$	−19.8829	−0.934181
$454$	0	0
$455$	1.00000	0.0468807
$456$	0	0
$457$	35.6730	1.66871	0.834357	−	0.551224i	$-0.185839\pi$
$457$	35.6730	1.66871	0.834357	+	0.551224i	$0.185839\pi$
$458$	0	0
$459$	41.0945	1.91813
$460$	0	0
$461$	16.6738	0.776578	0.388289	−	0.921538i	$-0.373066\pi$
$461$	16.6738	0.776578	0.388289	+	0.921538i	$0.373066\pi$
$462$	0	0
$463$	21.8428	1.01512	0.507560	−	0.861617i	$-0.330548\pi$
$463$	21.8428	1.01512	0.507560	+	0.861617i	$0.330548\pi$
$464$	0	0
$465$	8.56171	0.397040
$466$	0	0
$467$	0.940737	0.0435321	0.0217661	−	0.999763i	$-0.493071\pi$
$467$	0.940737	0.0435321	0.0217661	+	0.999763i	$0.493071\pi$
$468$	0	0
$469$	8.98859	0.415055
$470$	0	0
$471$	−1.64628	−0.0758565
$472$	0	0
$473$	3.18298	0.146354
$474$	0	0
$475$	3.54940	0.162858
$476$	0	0
$477$	−14.7628	−0.675943
$478$	0	0
$479$	−28.9975	−1.32493	−0.662465	−	0.749093i	$-0.730490\pi$
$479$	−28.9975	−1.32493	−0.662465	+	0.749093i	$0.730490\pi$
$480$	0	0
$481$	11.3187	0.516087
$482$	0	0
$483$	−5.30473	−0.241374
$484$	0	0
$485$	−8.28555	−0.376227
$486$	0	0
$487$	15.4938	0.702090	0.351045	−	0.936359i	$-0.385826\pi$
$487$	15.4938	0.702090	0.351045	+	0.936359i	$0.385826\pi$
$488$	0	0
$489$	15.8619	0.717298
$490$	0	0
$491$	18.3139	0.826496	0.413248	−	0.910618i	$-0.364394\pi$
$491$	18.3139	0.826496	0.413248	+	0.910618i	$0.364394\pi$
$492$	0	0
$493$	−70.9262	−3.19435
$494$	0	0
$495$	0.795513	0.0357556
$496$	0	0
$497$	14.3391	0.643197
$498$	0	0
$499$	−20.1303	−0.901154	−0.450577	−	0.892737i	$-0.648782\pi$
$499$	−20.1303	−0.901154	−0.450577	+	0.892737i	$0.648782\pi$
$500$	0	0
$501$	−2.63481	−0.117715
$502$	0	0
$503$	−11.0803	−0.494046	−0.247023	−	0.969010i	$-0.579452\pi$
$503$	−11.0803	−0.494046	−0.247023	+	0.969010i	$0.579452\pi$
$504$	0	0
$505$	1.28977	0.0573938
$506$	0	0
$507$	1.17382	0.0521314
$508$	0	0
$509$	24.1482	1.07035	0.535176	−	0.844741i	$-0.320245\pi$
$509$	24.1482	1.07035	0.535176	+	0.844741i	$0.320245\pi$
$510$	0	0
$511$	−13.4198	−0.593655
$512$	0	0
$513$	−19.2575	−0.850241
$514$	0	0
$515$	−5.40598	−0.238216
$516$	0	0
$517$	−3.18592	−0.140116
$518$	0	0
$519$	13.7559	0.603817
$520$	0	0
$521$	−29.7715	−1.30431	−0.652157	−	0.758084i	$-0.726136\pi$
$521$	−29.7715	−1.30431	−0.652157	+	0.758084i	$0.726136\pi$
$522$	0	0
$523$	11.3562	0.496573	0.248287	−	0.968687i	$-0.420133\pi$
$523$	11.3562	0.496573	0.248287	+	0.968687i	$0.420133\pi$
$524$	0	0
$525$	1.17382	0.0512299
$526$	0	0
$527$	−55.2452	−2.40652
$528$	0	0
$529$	−2.57699	−0.112043
$530$	0	0
$531$	−20.1797	−0.875724
$532$	0	0
$533$	11.4375	0.495411
$534$	0	0
$535$	2.68250	0.115975
$536$	0	0
$537$	15.4210	0.665464
$538$	0	0
$539$	−0.490411	−0.0211235
$540$	0	0
$541$	9.95659	0.428067	0.214034	−	0.976826i	$-0.431340\pi$
$541$	9.95659	0.428067	0.214034	+	0.976826i	$0.431340\pi$
$542$	0	0
$543$	2.73369	0.117314
$544$	0	0
$545$	−7.23430	−0.309884
$546$	0	0
$547$	23.3150	0.996877	0.498438	−	0.866925i	$-0.333907\pi$
$547$	23.3150	0.996877	0.498438	+	0.866925i	$0.333907\pi$
$548$	0	0
$549$	−5.02132	−0.214305
$550$	0	0
$551$	33.2371	1.41595
$552$	0	0
$553$	−16.0994	−0.684616
$554$	0	0
$555$	13.2861	0.563965
$556$	0	0
$557$	11.3303	0.480082	0.240041	−	0.970763i	$-0.422839\pi$
$557$	11.3303	0.480082	0.240041	+	0.970763i	$0.422839\pi$
$558$	0	0
$559$	−6.49044	−0.274516
$560$	0	0
$561$	4.36015	0.184085
$562$	0	0
$563$	−15.9701	−0.673059	−0.336530	−	0.941673i	$-0.609253\pi$
$563$	−15.9701	−0.673059	−0.336530	+	0.941673i	$0.609253\pi$
$564$	0	0
$565$	−1.44315	−0.0607137
$566$	0	0
$567$	−1.50227	−0.0630894
$568$	0	0
$569$	6.28116	0.263320	0.131660	−	0.991295i	$-0.457969\pi$
$569$	6.28116	0.263320	0.131660	+	0.991295i	$0.457969\pi$
$570$	0	0
$571$	23.9437	1.00201	0.501006	−	0.865444i	$-0.332963\pi$
$571$	23.9437	1.00201	0.501006	+	0.865444i	$0.332963\pi$
$572$	0	0
$573$	−8.02223	−0.335134
$574$	0	0
$575$	−4.51918	−0.188463
$576$	0	0
$577$	3.31434	0.137978	0.0689888	−	0.997617i	$-0.478023\pi$
$577$	3.31434	0.137978	0.0689888	+	0.997617i	$0.478023\pi$
$578$	0	0
$579$	−27.1117	−1.12672
$580$	0	0
$581$	11.1176	0.461236
$582$	0	0
$583$	−4.46316	−0.184845
$584$	0	0
$585$	−1.62214	−0.0670670
$586$	0	0
$587$	44.7840	1.84844	0.924218	−	0.381866i	$-0.124718\pi$
$587$	44.7840	1.84844	0.924218	+	0.381866i	$0.124718\pi$
$588$	0	0
$589$	25.8888	1.06673
$590$	0	0
$591$	−30.5525	−1.25676
$592$	0	0
$593$	4.25049	0.174547	0.0872734	−	0.996184i	$-0.472185\pi$
$593$	4.25049	0.174547	0.0872734	+	0.996184i	$0.472185\pi$
$594$	0	0
$595$	−7.57421	−0.310512
$596$	0	0
$597$	−12.9766	−0.531097
$598$	0	0
$599$	−30.7922	−1.25813	−0.629067	−	0.777351i	$-0.716563\pi$
$599$	−30.7922	−1.25813	−0.629067	+	0.777351i	$0.716563\pi$
$600$	0	0
$601$	−37.8127	−1.54241	−0.771207	−	0.636584i	$-0.780347\pi$
$601$	−37.8127	−1.54241	−0.771207	+	0.636584i	$0.780347\pi$
$602$	0	0
$603$	−14.5807	−0.593772
$604$	0	0
$605$	−10.7595	−0.437436
$606$	0	0
$607$	−27.4956	−1.11601	−0.558006	−	0.829837i	$-0.688433\pi$
$607$	−27.4956	−1.11601	−0.558006	+	0.829837i	$0.688433\pi$
$608$	0	0
$609$	10.9919	0.445414
$610$	0	0
$611$	6.49642	0.262817
$612$	0	0
$613$	6.97627	0.281769	0.140884	−	0.990026i	$-0.455005\pi$
$613$	6.97627	0.281769	0.140884	+	0.990026i	$0.455005\pi$
$614$	0	0
$615$	13.4256	0.541372
$616$	0	0
$617$	15.6063	0.628286	0.314143	−	0.949376i	$-0.398283\pi$
$617$	15.6063	0.628286	0.314143	+	0.949376i	$0.398283\pi$
$618$	0	0
$619$	−25.9583	−1.04335	−0.521677	−	0.853143i	$-0.674693\pi$
$619$	−25.9583	−1.04335	−0.521677	+	0.853143i	$0.674693\pi$
$620$	0	0
$621$	24.5192	0.983921
$622$	0	0
$623$	−3.24414	−0.129974
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−2.04323	−0.0815989
$628$	0	0
$629$	−85.7300	−3.41828
$630$	0	0
$631$	−19.6072	−0.780548	−0.390274	−	0.920699i	$-0.627620\pi$
$631$	−19.6072	−0.780548	−0.390274	+	0.920699i	$0.627620\pi$
$632$	0	0
$633$	−6.73489	−0.267688
$634$	0	0
$635$	10.8927	0.432263
$636$	0	0
$637$	1.00000	0.0396214
$638$	0	0
$639$	−23.2600	−0.920150
$640$	0	0
$641$	30.8215	1.21738	0.608689	−	0.793409i	$-0.291696\pi$
$641$	30.8215	1.21738	0.608689	+	0.793409i	$0.291696\pi$
$642$	0	0
$643$	46.6367	1.83917	0.919585	−	0.392891i	$-0.128525\pi$
$643$	46.6367	1.83917	0.919585	+	0.392891i	$0.128525\pi$
$644$	0	0
$645$	−7.61864	−0.299984
$646$	0	0
$647$	−38.2987	−1.50568	−0.752839	−	0.658204i	$-0.771316\pi$
$647$	−38.2987	−1.50568	−0.752839	+	0.658204i	$0.771316\pi$
$648$	0	0
$649$	−6.10081	−0.239478
$650$	0	0
$651$	8.56171	0.335560
$652$	0	0
$653$	3.75489	0.146940	0.0734700	−	0.997297i	$-0.476593\pi$
$653$	3.75489	0.146940	0.0734700	+	0.997297i	$0.476593\pi$
$654$	0	0
$655$	1.90874	0.0745808
$656$	0	0
$657$	21.7687	0.849276
$658$	0	0
$659$	26.3697	1.02722	0.513608	−	0.858025i	$-0.328308\pi$
$659$	26.3697	1.02722	0.513608	+	0.858025i	$0.328308\pi$
$660$	0	0
$661$	3.14396	0.122286	0.0611429	−	0.998129i	$-0.480525\pi$
$661$	3.14396	0.122286	0.0611429	+	0.998129i	$0.480525\pi$
$662$	0	0
$663$	−8.89080	−0.345290
$664$	0	0
$665$	3.54940	0.137640
$666$	0	0
$667$	−42.3184	−1.63857
$668$	0	0
$669$	6.65823	0.257422
$670$	0	0
$671$	−1.51807	−0.0586043
$672$	0	0
$673$	22.6340	0.872478	0.436239	−	0.899831i	$-0.356310\pi$
$673$	22.6340	0.872478	0.436239	+	0.899831i	$0.356310\pi$
$674$	0	0
$675$	−5.42558	−0.208831
$676$	0	0
$677$	22.9117	0.880569	0.440285	−	0.897858i	$-0.354878\pi$
$677$	22.9117	0.880569	0.440285	+	0.897858i	$0.354878\pi$
$678$	0	0
$679$	−8.28555	−0.317970
$680$	0	0
$681$	−8.36893	−0.320698
$682$	0	0
$683$	23.1527	0.885913	0.442956	−	0.896543i	$-0.353930\pi$
$683$	23.1527	0.885913	0.442956	+	0.896543i	$0.353930\pi$
$684$	0	0
$685$	2.88764	0.110331
$686$	0	0
$687$	−15.7312	−0.600183
$688$	0	0
$689$	9.10085	0.346715
$690$	0	0
$691$	−12.7767	−0.486047	−0.243024	−	0.970020i	$-0.578139\pi$
$691$	−12.7767	−0.486047	−0.243024	+	0.970020i	$0.578139\pi$
$692$	0	0
$693$	0.795513	0.0302190
$694$	0	0
$695$	−5.46525	−0.207309
$696$	0	0
$697$	−86.6298	−3.28134
$698$	0	0
$699$	−27.7290	−1.04881
$700$	0	0
$701$	12.1501	0.458904	0.229452	−	0.973320i	$-0.426307\pi$
$701$	12.1501	0.458904	0.229452	+	0.973320i	$0.426307\pi$
$702$	0	0
$703$	40.1745	1.51521
$704$	0	0
$705$	7.62566	0.287199
$706$	0	0
$707$	1.28977	0.0485066
$708$	0	0
$709$	−32.0334	−1.20304	−0.601520	−	0.798858i	$-0.705438\pi$
$709$	−32.0334	−1.20304	−0.601520	+	0.798858i	$0.705438\pi$
$710$	0	0
$711$	26.1154	0.979405
$712$	0	0
$713$	−32.9623	−1.23445
$714$	0	0
$715$	−0.490411	−0.0183403
$716$	0	0
$717$	−21.7662	−0.812874
$718$	0	0
$719$	−35.4751	−1.32300	−0.661499	−	0.749946i	$-0.730080\pi$
$719$	−35.4751	−1.32300	−0.661499	+	0.749946i	$0.730080\pi$
$720$	0	0
$721$	−5.40598	−0.201329
$722$	0	0
$723$	2.56318	0.0953257
$724$	0	0
$725$	9.36416	0.347776
$726$	0	0
$727$	−20.9654	−0.777564	−0.388782	−	0.921330i	$-0.627104\pi$
$727$	−20.9654	−0.777564	−0.388782	+	0.921330i	$0.627104\pi$
$728$	0	0
$729$	21.5429	0.797886
$730$	0	0
$731$	49.1600	1.81825
$732$	0	0
$733$	−43.3040	−1.59947	−0.799735	−	0.600353i	$-0.795027\pi$
$733$	−43.3040	−1.59947	−0.799735	+	0.600353i	$0.795027\pi$
$734$	0	0
$735$	1.17382	0.0432972
$736$	0	0
$737$	−4.40810	−0.162375
$738$	0	0
$739$	−21.1996	−0.779839	−0.389920	−	0.920849i	$-0.627497\pi$
$739$	−21.1996	−0.779839	−0.389920	+	0.920849i	$0.627497\pi$
$740$	0	0
$741$	4.16637	0.153055
$742$	0	0
$743$	42.7127	1.56698	0.783489	−	0.621405i	$-0.213438\pi$
$743$	42.7127	1.56698	0.783489	+	0.621405i	$0.213438\pi$
$744$	0	0
$745$	−3.02453	−0.110810
$746$	0	0
$747$	−18.0343	−0.659839
$748$	0	0
$749$	2.68250	0.0980166
$750$	0	0
$751$	26.0223	0.949566	0.474783	−	0.880103i	$-0.342527\pi$
$751$	26.0223	0.949566	0.474783	+	0.880103i	$0.342527\pi$
$752$	0	0
$753$	14.5918	0.531756
$754$	0	0
$755$	−16.9386	−0.616458
$756$	0	0
$757$	−29.6654	−1.07821	−0.539103	−	0.842240i	$-0.681237\pi$
$757$	−29.6654	−1.07821	−0.539103	+	0.842240i	$0.681237\pi$
$758$	0	0
$759$	2.60150	0.0944284
$760$	0	0
$761$	9.60345	0.348125	0.174062	−	0.984735i	$-0.444311\pi$
$761$	9.60345	0.348125	0.174062	+	0.984735i	$0.444311\pi$
$762$	0	0
$763$	−7.23430	−0.261899
$764$	0	0
$765$	12.2864	0.444216
$766$	0	0
$767$	12.4402	0.449190
$768$	0	0
$769$	43.1168	1.55483	0.777416	−	0.628987i	$-0.216530\pi$
$769$	43.1168	1.55483	0.777416	+	0.628987i	$0.216530\pi$
$770$	0	0
$771$	−31.1565	−1.12207
$772$	0	0
$773$	−1.50552	−0.0541497	−0.0270749	−	0.999633i	$-0.508619\pi$
$773$	−1.50552	−0.0541497	−0.0270749	+	0.999633i	$0.508619\pi$
$774$	0	0
$775$	7.29385	0.262003
$776$	0	0
$777$	13.2861	0.476637
$778$	0	0
$779$	40.5961	1.45451
$780$	0	0
$781$	−7.03206	−0.251627
$782$	0	0
$783$	−50.8060	−1.81566
$784$	0	0
$785$	−1.40249	−0.0500571
$786$	0	0
$787$	−9.54613	−0.340283	−0.170141	−	0.985420i	$-0.554422\pi$
$787$	−9.54613	−0.340283	−0.170141	+	0.985420i	$0.554422\pi$
$788$	0	0
$789$	0.743974	0.0264862
$790$	0	0
$791$	−1.44315	−0.0513125
$792$	0	0
$793$	3.09550	0.109924
$794$	0	0
$795$	10.6828	0.378880
$796$	0	0
$797$	19.9882	0.708017	0.354009	−	0.935242i	$-0.384818\pi$
$797$	19.9882	0.708017	0.354009	+	0.935242i	$0.384818\pi$
$798$	0	0
$799$	−49.2053	−1.74076
$800$	0	0
$801$	5.26243	0.185939
$802$	0	0
$803$	6.58120	0.232245
$804$	0	0
$805$	−4.51918	−0.159280
$806$	0	0
$807$	−12.7511	−0.448860
$808$	0	0
$809$	−31.0068	−1.09014	−0.545070	−	0.838391i	$-0.683497\pi$
$809$	−31.0068	−1.09014	−0.545070	+	0.838391i	$0.683497\pi$
$810$	0	0
$811$	43.5519	1.52931	0.764657	−	0.644438i	$-0.222909\pi$
$811$	43.5519	1.52931	0.764657	+	0.644438i	$0.222909\pi$
$812$	0	0
$813$	25.5445	0.895885
$814$	0	0
$815$	13.5130	0.473338
$816$	0	0
$817$	−23.0372	−0.805968
$818$	0	0
$819$	−1.62214	−0.0566820
$820$	0	0
$821$	−40.1350	−1.40072	−0.700361	−	0.713789i	$-0.746978\pi$
$821$	−40.1350	−1.40072	−0.700361	+	0.713789i	$0.746978\pi$
$822$	0	0
$823$	8.63794	0.301100	0.150550	−	0.988602i	$-0.451896\pi$
$823$	8.63794	0.301100	0.150550	+	0.988602i	$0.451896\pi$
$824$	0	0
$825$	−0.575657	−0.0200418
$826$	0	0
$827$	−14.5902	−0.507352	−0.253676	−	0.967289i	$-0.581640\pi$
$827$	−14.5902	−0.507352	−0.253676	+	0.967289i	$0.581640\pi$
$828$	0	0
$829$	−7.81279	−0.271349	−0.135675	−	0.990753i	$-0.543320\pi$
$829$	−7.81279	−0.271349	−0.135675	+	0.990753i	$0.543320\pi$
$830$	0	0
$831$	−6.19884	−0.215036
$832$	0	0
$833$	−7.57421	−0.262431
$834$	0	0
$835$	−2.24464	−0.0776789
$836$	0	0
$837$	−39.5734	−1.36786
$838$	0	0
$839$	16.9237	0.584270	0.292135	−	0.956377i	$-0.405634\pi$
$839$	16.9237	0.584270	0.292135	+	0.956377i	$0.405634\pi$
$840$	0	0
$841$	58.6876	2.02371
$842$	0	0
$843$	37.1363	1.27904
$844$	0	0
$845$	1.00000	0.0344010
$846$	0	0
$847$	−10.7595	−0.369701
$848$	0	0
$849$	−20.7419	−0.711861
$850$	0	0
$851$	−51.1511	−1.75344
$852$	0	0
$853$	−49.2624	−1.68671	−0.843356	−	0.537355i	$-0.819423\pi$
$853$	−49.2624	−1.68671	−0.843356	+	0.537355i	$0.819423\pi$
$854$	0	0
$855$	−5.75760	−0.196906
$856$	0	0
$857$	−15.2779	−0.521881	−0.260941	−	0.965355i	$-0.584033\pi$
$857$	−15.2779	−0.521881	−0.260941	+	0.965355i	$0.584033\pi$
$858$	0	0
$859$	−22.5662	−0.769948	−0.384974	−	0.922927i	$-0.625790\pi$
$859$	−22.5662	−0.769948	−0.384974	+	0.922927i	$0.625790\pi$
$860$	0	0
$861$	13.4256	0.457542
$862$	0	0
$863$	−24.6025	−0.837479	−0.418739	−	0.908106i	$-0.637528\pi$
$863$	−24.6025	−0.837479	−0.418739	+	0.908106i	$0.637528\pi$
$864$	0	0
$865$	11.7189	0.398453
$866$	0	0
$867$	47.3858	1.60931
$868$	0	0
$869$	7.89533	0.267831
$870$	0	0
$871$	8.98859	0.304567
$872$	0	0
$873$	13.4403	0.454885
$874$	0	0
$875$	1.00000	0.0338062
$876$	0	0
$877$	−12.8126	−0.432649	−0.216325	−	0.976321i	$-0.569407\pi$
$877$	−12.8126	−0.432649	−0.216325	+	0.976321i	$0.569407\pi$
$878$	0	0
$879$	23.2422	0.783940
$880$	0	0
$881$	24.5471	0.827014	0.413507	−	0.910501i	$-0.364304\pi$
$881$	24.5471	0.827014	0.413507	+	0.910501i	$0.364304\pi$
$882$	0	0
$883$	12.8088	0.431049	0.215524	−	0.976498i	$-0.430854\pi$
$883$	12.8088	0.431049	0.215524	+	0.976498i	$0.430854\pi$
$884$	0	0
$885$	14.6026	0.490861
$886$	0	0
$887$	38.3936	1.28913	0.644566	−	0.764549i	$-0.277038\pi$
$887$	38.3936	1.28913	0.644566	+	0.764549i	$0.277038\pi$
$888$	0	0
$889$	10.8927	0.365329
$890$	0	0
$891$	0.736730	0.0246814
$892$	0	0
$893$	23.0584	0.771619
$894$	0	0
$895$	13.1374	0.439134
$896$	0	0
$897$	−5.30473	−0.177120
$898$	0	0
$899$	68.3008	2.27796
$900$	0	0
$901$	−68.9318	−2.29645
$902$	0	0
$903$	−7.61864	−0.253532
$904$	0	0
$905$	2.32887	0.0774143
$906$	0	0
$907$	18.1165	0.601547	0.300773	−	0.953696i	$-0.402755\pi$
$907$	18.1165	0.601547	0.300773	+	0.953696i	$0.402755\pi$
$908$	0	0
$909$	−2.09217	−0.0693931
$910$	0	0
$911$	39.3726	1.30447	0.652236	−	0.758016i	$-0.273831\pi$
$911$	39.3726	1.30447	0.652236	+	0.758016i	$0.273831\pi$
$912$	0	0
$913$	−5.45219	−0.180441
$914$	0	0
$915$	3.63357	0.120122
$916$	0	0
$917$	1.90874	0.0630323
$918$	0	0
$919$	24.5622	0.810232	0.405116	−	0.914265i	$-0.367231\pi$
$919$	24.5622	0.810232	0.405116	+	0.914265i	$0.367231\pi$
$920$	0	0
$921$	12.6615	0.417211
$922$	0	0
$923$	14.3391	0.471978
$924$	0	0
$925$	11.3187	0.372156
$926$	0	0
$927$	8.76923	0.288019
$928$	0	0
$929$	−15.8864	−0.521216	−0.260608	−	0.965445i	$-0.583923\pi$
$929$	−15.8864	−0.521216	−0.260608	+	0.965445i	$0.583923\pi$
$930$	0	0
$931$	3.54940	0.116327
$932$	0	0
$933$	16.1468	0.528623
$934$	0	0
$935$	3.71448	0.121476
$936$	0	0
$937$	−16.6164	−0.542833	−0.271417	−	0.962462i	$-0.587492\pi$
$937$	−16.6164	−0.542833	−0.271417	+	0.962462i	$0.587492\pi$
$938$	0	0
$939$	21.2945	0.694920
$940$	0	0
$941$	−12.4663	−0.406390	−0.203195	−	0.979138i	$-0.565133\pi$
$941$	−12.4663	−0.406390	−0.203195	+	0.979138i	$0.565133\pi$
$942$	0	0
$943$	−51.6880	−1.68319
$944$	0	0
$945$	−5.42558	−0.176494
$946$	0	0
$947$	−26.9677	−0.876333	−0.438167	−	0.898894i	$-0.644372\pi$
$947$	−26.9677	−0.876333	−0.438167	+	0.898894i	$0.644372\pi$
$948$	0	0
$949$	−13.4198	−0.435624
$950$	0	0
$951$	−11.1824	−0.362615
$952$	0	0
$953$	−33.1971	−1.07536	−0.537679	−	0.843149i	$-0.680699\pi$
$953$	−33.1971	−1.07536	−0.537679	+	0.843149i	$0.680699\pi$
$954$	0	0
$955$	−6.83427	−0.221152
$956$	0	0
$957$	−5.39054	−0.174251
$958$	0	0
$959$	2.88764	0.0932468
$960$	0	0
$961$	22.2003	0.716138
$962$	0	0
$963$	−4.35139	−0.140221
$964$	0	0
$965$	−23.0969	−0.743514
$966$	0	0
$967$	51.6717	1.66165	0.830825	−	0.556533i	$-0.187869\pi$
$967$	51.6717	1.66165	0.830825	+	0.556533i	$0.187869\pi$
$968$	0	0
$969$	−31.5570	−1.01376
$970$	0	0
$971$	21.6907	0.696089	0.348044	−	0.937478i	$-0.386846\pi$
$971$	21.6907	0.696089	0.348044	+	0.937478i	$0.386846\pi$
$972$	0	0
$973$	−5.46525	−0.175208
$974$	0	0
$975$	1.17382	0.0375925
$976$	0	0
$977$	−39.2645	−1.25618	−0.628091	−	0.778140i	$-0.716163\pi$
$977$	−39.2645	−1.25618	−0.628091	+	0.778140i	$0.716163\pi$
$978$	0	0
$979$	1.59096	0.0508473
$980$	0	0
$981$	11.7350	0.374670
$982$	0	0
$983$	17.5027	0.558248	0.279124	−	0.960255i	$-0.409956\pi$
$983$	17.5027	0.558248	0.279124	+	0.960255i	$0.409956\pi$
$984$	0	0
$985$	−26.0282	−0.829327
$986$	0	0
$987$	7.62566	0.242727
$988$	0	0
$989$	29.3315	0.932687
$990$	0	0
$991$	−4.65843	−0.147980	−0.0739900	−	0.997259i	$-0.523573\pi$
$991$	−4.65843	−0.147980	−0.0739900	+	0.997259i	$0.523573\pi$
$992$	0	0
$993$	38.6666	1.22705
$994$	0	0
$995$	−11.0550	−0.350466
$996$	0	0
$997$	−24.6426	−0.780438	−0.390219	−	0.920722i	$-0.627601\pi$
$997$	−24.6426	−0.780438	−0.390219	+	0.920722i	$0.627601\pi$
$998$	0	0
$999$	−61.4103	−1.94294

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3640.2.a.ba.1.4	✓	7
4.3	odd	2		7280.2.a.cf.1.4		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3640.2.a.ba.1.4	✓	7	1.1	even	1	trivial
7280.2.a.cf.1.4		7	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 3640 = 2^{3} \cdot 5 \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3640.a (trivial)

\(f(q)\)	\(=\)	\(q+1.17382 q^{3} +1.00000 q^{5} +1.00000 q^{7} -1.62214 q^{9} +O(q^{10})\)	\(q+1.17382 q^{3} +1.00000 q^{5} +1.00000 q^{7} -1.62214 q^{9} -0.490411 q^{11} +1.00000 q^{13} +1.17382 q^{15} -7.57421 q^{17} +3.54940 q^{19} +1.17382 q^{21} -4.51918 q^{23} +1.00000 q^{25} -5.42558 q^{27} +9.36416 q^{29} +7.29385 q^{31} -0.575657 q^{33} +1.00000 q^{35} +11.3187 q^{37} +1.17382 q^{39} +11.4375 q^{41} -6.49044 q^{43} -1.62214 q^{45} +6.49642 q^{47} +1.00000 q^{49} -8.89080 q^{51} +9.10085 q^{53} -0.490411 q^{55} +4.16637 q^{57} +12.4402 q^{59} +3.09550 q^{61} -1.62214 q^{63} +1.00000 q^{65} +8.98859 q^{67} -5.30473 q^{69} +14.3391 q^{71} -13.4198 q^{73} +1.17382 q^{75} -0.490411 q^{77} -16.0994 q^{79} -1.50227 q^{81} +11.1176 q^{83} -7.57421 q^{85} +10.9919 q^{87} -3.24414 q^{89} +1.00000 q^{91} +8.56171 q^{93} +3.54940 q^{95} -8.28555 q^{97} +0.795513 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 4 q^{3} + 7 q^{5} + 7 q^{7} + 11 q^{9}+O(q^{10}) \) 7 * q + 4 * q^3 + 7 * q^5 + 7 * q^7 + 11 * q^9	\( 7 q + 4 q^{3} + 7 q^{5} + 7 q^{7} + 11 q^{9} + 7 q^{11} + 7 q^{13} + 4 q^{15} + 3 q^{17} + 13 q^{19} + 4 q^{21} + 11 q^{23} + 7 q^{25} + 13 q^{27} + 3 q^{29} + 12 q^{31} - 11 q^{33} + 7 q^{35} - 4 q^{37} + 4 q^{39} + 6 q^{41} + 10 q^{43} + 11 q^{45} + 15 q^{47} + 7 q^{49} + 8 q^{53} + 7 q^{55} - 18 q^{57} + 13 q^{59} - q^{61} + 11 q^{63} + 7 q^{65} + 8 q^{67} - 9 q^{69} + 20 q^{71} - 17 q^{73} + 4 q^{75} + 7 q^{77} + 6 q^{79} + 39 q^{81} + 30 q^{83} + 3 q^{85} + 21 q^{87} + 5 q^{89} + 7 q^{91} - 24 q^{93} + 13 q^{95} - 9 q^{97} + 12 q^{99}+O(q^{100}) \) 7 * q + 4 * q^3 + 7 * q^5 + 7 * q^7 + 11 * q^9 + 7 * q^11 + 7 * q^13 + 4 * q^15 + 3 * q^17 + 13 * q^19 + 4 * q^21 + 11 * q^23 + 7 * q^25 + 13 * q^27 + 3 * q^29 + 12 * q^31 - 11 * q^33 + 7 * q^35 - 4 * q^37 + 4 * q^39 + 6 * q^41 + 10 * q^43 + 11 * q^45 + 15 * q^47 + 7 * q^49 + 8 * q^53 + 7 * q^55 - 18 * q^57 + 13 * q^59 - q^61 + 11 * q^63 + 7 * q^65 + 8 * q^67 - 9 * q^69 + 20 * q^71 - 17 * q^73 + 4 * q^75 + 7 * q^77 + 6 * q^79 + 39 * q^81 + 30 * q^83 + 3 * q^85 + 21 * q^87 + 5 * q^89 + 7 * q^91 - 24 * q^93 + 13 * q^95 - 9 * q^97 + 12 * q^99

Introduction

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