LMFDB - Embedded newform 2960.2.a.y.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2960,2,Mod(1,2960)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2960, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("2960.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2960 = 2^{4} \cdot 5 \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2960.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:	$23.6357189983$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.6397264.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 10x^{3} - 2x^{2} + 14x - 4 $ x^5 - 10x^3 - 2x^2 + 14*x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1480)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.64390$ of defining polynomial
Character	$\chi$	$=$	2960.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.64390 q^{3} -1.00000 q^{5} -1.57272 q^{7} -0.297599 q^{9} +O(q^{10})$	$q-1.64390 q^{3} -1.00000 q^{5} -1.57272 q^{7} -0.297599 q^{9} -5.49209 q^{11} -2.44304 q^{13} +1.64390 q^{15} -4.78969 q^{17} -5.38454 q^{19} +2.58540 q^{21} -5.28780 q^{23} +1.00000 q^{25} +5.42092 q^{27} +9.94494 q^{29} -6.86052 q^{31} +9.02844 q^{33} +1.57272 q^{35} +1.00000 q^{37} +4.01612 q^{39} -7.92533 q^{41} +2.94115 q^{43} +0.297599 q^{45} -2.65622 q^{47} -4.52654 q^{49} +7.87376 q^{51} +3.79570 q^{53} +5.49209 q^{55} +8.85164 q^{57} -8.37222 q^{59} +5.75145 q^{61} +0.468041 q^{63} +2.44304 q^{65} +7.38454 q^{67} +8.69260 q^{69} +1.02844 q^{71} +7.74064 q^{73} -1.64390 q^{75} +8.63754 q^{77} -5.77702 q^{79} -8.01864 q^{81} -6.67234 q^{83} +4.78969 q^{85} -16.3485 q^{87} -10.0126 q^{89} +3.84223 q^{91} +11.2780 q^{93} +5.38454 q^{95} -12.9734 q^{97} +1.63444 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 5 q^{5} - 3 q^{7} + 5 q^{9}+O(q^{10}) $ 5 * q - 5 * q^5 - 3 * q^7 + 5 * q^9	$ 5 q - 5 q^{5} - 3 q^{7} + 5 q^{9} - 3 q^{11} + 4 q^{13} + 7 q^{17} + 4 q^{19} - 10 q^{21} - 10 q^{23} + 5 q^{25} + 6 q^{27} + 19 q^{29} - 13 q^{31} + 6 q^{33} + 3 q^{35} + 5 q^{37} - 14 q^{39} + 11 q^{41} + 13 q^{43} - 5 q^{45} + 2 q^{49} + 12 q^{51} + 27 q^{53} + 3 q^{55} + 12 q^{57} - 16 q^{59} + 27 q^{61} - 37 q^{63} - 4 q^{65} + 6 q^{67} + 40 q^{69} - 34 q^{71} + 16 q^{73} + 9 q^{77} - 16 q^{79} + 9 q^{81} + 14 q^{83} - 7 q^{85} + 42 q^{87} + 38 q^{89} + 34 q^{91} + 30 q^{93} - 4 q^{95} + 5 q^{97} - 23 q^{99}+O(q^{100}) $ 5 * q - 5 * q^5 - 3 * q^7 + 5 * q^9 - 3 * q^11 + 4 * q^13 + 7 * q^17 + 4 * q^19 - 10 * q^21 - 10 * q^23 + 5 * q^25 + 6 * q^27 + 19 * q^29 - 13 * q^31 + 6 * q^33 + 3 * q^35 + 5 * q^37 - 14 * q^39 + 11 * q^41 + 13 * q^43 - 5 * q^45 + 2 * q^49 + 12 * q^51 + 27 * q^53 + 3 * q^55 + 12 * q^57 - 16 * q^59 + 27 * q^61 - 37 * q^63 - 4 * q^65 + 6 * q^67 + 40 * q^69 - 34 * q^71 + 16 * q^73 + 9 * q^77 - 16 * q^79 + 9 * q^81 + 14 * q^83 - 7 * q^85 + 42 * q^87 + 38 * q^89 + 34 * q^91 + 30 * q^93 - 4 * q^95 + 5 * q^97 - 23 * 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
$3$	−1.64390	−0.949105	−0.474552	−	0.880227i	$-0.657390\pi$
$3$	−1.64390	−0.949105	−0.474552	+	0.880227i	$0.657390\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−1.57272	−0.594433	−0.297217	−	0.954810i	$-0.596058\pi$
$7$	−1.57272	−0.594433	−0.297217	+	0.954810i	$0.596058\pi$
$8$	0	0
$9$	−0.297599	−0.0991998
$10$	0	0
$11$	−5.49209	−1.65593	−0.827964	−	0.560781i	$-0.810501\pi$
$11$	−5.49209	−1.65593	−0.827964	+	0.560781i	$0.810501\pi$
$12$	0	0
$13$	−2.44304	−0.677579	−0.338789	−	0.940862i	$-0.610017\pi$
$13$	−2.44304	−0.677579	−0.338789	+	0.940862i	$0.610017\pi$
$14$	0	0
$15$	1.64390	0.424453
$16$	0	0
$17$	−4.78969	−1.16167	−0.580835	−	0.814021i	$-0.697274\pi$
$17$	−4.78969	−1.16167	−0.580835	+	0.814021i	$0.697274\pi$
$18$	0	0
$19$	−5.38454	−1.23530	−0.617649	−	0.786454i	$-0.711915\pi$
$19$	−5.38454	−1.23530	−0.617649	+	0.786454i	$0.711915\pi$
$20$	0	0
$21$	2.58540	0.564180
$22$	0	0
$23$	−5.28780	−1.10258	−0.551291	−	0.834313i	$-0.685865\pi$
$23$	−5.28780	−1.10258	−0.551291	+	0.834313i	$0.685865\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	5.42092	1.04326
$28$	0	0
$29$	9.94494	1.84673	0.923365	−	0.383924i	$-0.125428\pi$
$29$	9.94494	1.84673	0.923365	+	0.383924i	$0.125428\pi$
$30$	0	0
$31$	−6.86052	−1.23219	−0.616093	−	0.787674i	$-0.711285\pi$
$31$	−6.86052	−1.23219	−0.616093	+	0.787674i	$0.711285\pi$
$32$	0	0
$33$	9.02844	1.57165
$34$	0	0
$35$	1.57272	0.265839
$36$	0	0
$37$	1.00000	0.164399
$37$	1.00000	0.164399
$38$	0	0
$39$	4.01612	0.643093
$40$	0	0
$41$	−7.92533	−1.23773	−0.618865	−	0.785498i	$-0.712407\pi$
$41$	−7.92533	−1.23773	−0.618865	+	0.785498i	$0.712407\pi$
$42$	0	0
$43$	2.94115	0.448521	0.224260	−	0.974529i	$-0.428003\pi$
$43$	2.94115	0.448521	0.224260	+	0.974529i	$0.428003\pi$
$44$	0	0
$45$	0.297599	0.0443635
$46$	0	0
$47$	−2.65622	−0.387450	−0.193725	−	0.981056i	$-0.562057\pi$
$47$	−2.65622	−0.387450	−0.193725	+	0.981056i	$0.562057\pi$
$48$	0	0
$49$	−4.52654	−0.646649
$50$	0	0
$51$	7.87376	1.10255
$52$	0	0
$53$	3.79570	0.521380	0.260690	−	0.965423i	$-0.416050\pi$
$53$	3.79570	0.521380	0.260690	+	0.965423i	$0.416050\pi$
$54$	0	0
$55$	5.49209	0.740554
$56$	0	0
$57$	8.85164	1.17243
$58$	0	0
$59$	−8.37222	−1.08997	−0.544985	−	0.838446i	$-0.683465\pi$
$59$	−8.37222	−1.08997	−0.544985	+	0.838446i	$0.683465\pi$
$60$	0	0
$61$	5.75145	0.736398	0.368199	−	0.929747i	$-0.379975\pi$
$61$	5.75145	0.736398	0.368199	+	0.929747i	$0.379975\pi$
$62$	0	0
$63$	0.468041	0.0589677
$64$	0	0
$65$	2.44304	0.303022
$66$	0	0
$67$	7.38454	0.902165	0.451083	−	0.892482i	$-0.351038\pi$
$67$	7.38454	0.902165	0.451083	+	0.892482i	$0.351038\pi$
$68$	0	0
$69$	8.69260	1.04647
$70$	0	0
$71$	1.02844	0.122053	0.0610267	−	0.998136i	$-0.480563\pi$
$71$	1.02844	0.122053	0.0610267	+	0.998136i	$0.480563\pi$
$72$	0	0
$73$	7.74064	0.905974	0.452987	−	0.891517i	$-0.350358\pi$
$73$	7.74064	0.905974	0.452987	+	0.891517i	$0.350358\pi$
$74$	0	0
$75$	−1.64390	−0.189821
$76$	0	0
$77$	8.63754	0.984339
$78$	0	0
$79$	−5.77702	−0.649965	−0.324983	−	0.945720i	$-0.605358\pi$
$79$	−5.77702	−0.649965	−0.324983	+	0.945720i	$0.605358\pi$
$80$	0	0
$81$	−8.01864	−0.890960
$82$	0	0
$83$	−6.67234	−0.732384	−0.366192	−	0.930539i	$-0.619339\pi$
$83$	−6.67234	−0.732384	−0.366192	+	0.930539i	$0.619339\pi$
$84$	0	0
$85$	4.78969	0.519515
$86$	0	0
$87$	−16.3485	−1.75274
$88$	0	0
$89$	−10.0126	−1.06134	−0.530668	−	0.847580i	$-0.678059\pi$
$89$	−10.0126	−1.06134	−0.530668	+	0.847580i	$0.678059\pi$
$90$	0	0
$91$	3.84223	0.402775
$92$	0	0
$93$	11.2780	1.16947
$94$	0	0
$95$	5.38454	0.552442
$96$	0	0
$97$	−12.9734	−1.31725	−0.658624	−	0.752473i	$-0.728861\pi$
$97$	−12.9734	−1.31725	−0.658624	+	0.752473i	$0.728861\pi$
$98$	0	0
$99$	1.63444	0.164268
$100$	0	0
$101$	−3.27799	−0.326172	−0.163086	−	0.986612i	$-0.552145\pi$
$101$	−3.27799	−0.326172	−0.163086	+	0.986612i	$0.552145\pi$
$102$	0	0
$103$	−7.91453	−0.779842	−0.389921	−	0.920848i	$-0.627498\pi$
$103$	−7.91453	−0.779842	−0.389921	+	0.920848i	$0.627498\pi$
$104$	0	0
$105$	−2.58540	−0.252309
$106$	0	0
$107$	8.39687	0.811756	0.405878	−	0.913927i	$-0.366966\pi$
$107$	8.39687	0.811756	0.405878	+	0.913927i	$0.366966\pi$
$108$	0	0
$109$	−13.0708	−1.25195	−0.625977	−	0.779842i	$-0.715300\pi$
$109$	−13.0708	−1.25195	−0.625977	+	0.779842i	$0.715300\pi$
$110$	0	0
$111$	−1.64390	−0.156032
$112$	0	0
$113$	10.5296	0.990545	0.495273	−	0.868738i	$-0.335068\pi$
$113$	10.5296	0.990545	0.495273	+	0.868738i	$0.335068\pi$
$114$	0	0
$115$	5.28780	0.493090
$116$	0	0
$117$	0.727049	0.0672157
$118$	0	0
$119$	7.53286	0.690536
$120$	0	0
$121$	19.1631	1.74210
$122$	0	0
$123$	13.0284	1.17473
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−0.647690	−0.0574732	−0.0287366	−	0.999587i	$-0.509148\pi$
$127$	−0.647690	−0.0574732	−0.0287366	+	0.999587i	$0.509148\pi$
$128$	0	0
$129$	−4.83495	−0.425693
$130$	0	0
$131$	−5.93479	−0.518525	−0.259262	−	0.965807i	$-0.583479\pi$
$131$	−5.93479	−0.518525	−0.259262	+	0.965807i	$0.583479\pi$
$132$	0	0
$133$	8.46839	0.734303
$134$	0	0
$135$	−5.42092	−0.466558
$136$	0	0
$137$	3.74064	0.319585	0.159792	−	0.987151i	$-0.448918\pi$
$137$	3.74064	0.319585	0.159792	+	0.987151i	$0.448918\pi$
$138$	0	0
$139$	19.7641	1.67637	0.838183	−	0.545388i	$-0.183618\pi$
$139$	19.7641	1.67637	0.838183	+	0.545388i	$0.183618\pi$
$140$	0	0
$141$	4.36656	0.367731
$142$	0	0
$143$	13.4174	1.12202
$144$	0	0
$145$	−9.94494	−0.825882
$146$	0	0
$147$	7.44118	0.613738
$148$	0	0
$149$	12.7346	1.04326	0.521631	−	0.853171i	$-0.325324\pi$
$149$	12.7346	1.04326	0.521631	+	0.853171i	$0.325324\pi$
$150$	0	0
$151$	4.69260	0.381878	0.190939	−	0.981602i	$-0.438847\pi$
$151$	4.69260	0.381878	0.190939	+	0.981602i	$0.438847\pi$
$152$	0	0
$153$	1.42541	0.115238
$154$	0	0
$155$	6.86052	0.551050
$156$	0	0
$157$	−9.51674	−0.759519	−0.379759	−	0.925085i	$-0.623993\pi$
$157$	−9.51674	−0.759519	−0.379759	+	0.925085i	$0.623993\pi$
$158$	0	0
$159$	−6.23975	−0.494844
$160$	0	0
$161$	8.31624	0.655411
$162$	0	0
$163$	13.3947	1.04915	0.524577	−	0.851363i	$-0.324224\pi$
$163$	13.3947	1.04915	0.524577	+	0.851363i	$0.324224\pi$
$164$	0	0
$165$	−9.02844	−0.702863
$166$	0	0
$167$	13.3193	1.03068	0.515340	−	0.856986i	$-0.327666\pi$
$167$	13.3193	1.03068	0.515340	+	0.856986i	$0.327666\pi$
$168$	0	0
$169$	−7.03153	−0.540887
$170$	0	0
$171$	1.60244	0.122541
$172$	0	0
$173$	−18.4351	−1.40159	−0.700796	−	0.713362i	$-0.747172\pi$
$173$	−18.4351	−1.40159	−0.700796	+	0.713362i	$0.747172\pi$
$174$	0	0
$175$	−1.57272	−0.118887
$176$	0	0
$177$	13.7631	1.03450
$178$	0	0
$179$	−10.5053	−0.785206	−0.392603	−	0.919708i	$-0.628425\pi$
$179$	−10.5053	−0.785206	−0.392603	+	0.919708i	$0.628425\pi$
$180$	0	0
$181$	19.3298	1.43678	0.718388	−	0.695643i	$-0.244880\pi$
$181$	19.3298	1.43678	0.718388	+	0.695643i	$0.244880\pi$
$182$	0	0
$183$	−9.45479	−0.698919
$184$	0	0
$185$	−1.00000	−0.0735215
$186$	0	0
$187$	26.3054	1.92364
$188$	0	0
$189$	−8.52560	−0.620146
$190$	0	0
$191$	−20.9908	−1.51884	−0.759422	−	0.650598i	$-0.774518\pi$
$191$	−20.9908	−1.51884	−0.759422	+	0.650598i	$0.774518\pi$
$192$	0	0
$193$	23.9684	1.72528	0.862641	−	0.505818i	$-0.168809\pi$
$193$	23.9684	1.72528	0.862641	+	0.505818i	$0.168809\pi$
$194$	0	0
$195$	−4.01612	−0.287600
$196$	0	0
$197$	−21.3200	−1.51899	−0.759495	−	0.650514i	$-0.774554\pi$
$197$	−21.3200	−1.51899	−0.759495	+	0.650514i	$0.774554\pi$
$198$	0	0
$199$	16.0644	1.13878	0.569388	−	0.822069i	$-0.307180\pi$
$199$	16.0644	1.13878	0.569388	+	0.822069i	$0.307180\pi$
$200$	0	0
$201$	−12.1394	−0.856250
$202$	0	0
$203$	−15.6406	−1.09776
$204$	0	0
$205$	7.92533	0.553529
$206$	0	0
$207$	1.57364	0.109376
$208$	0	0
$209$	29.5724	2.04557
$210$	0	0
$211$	1.31821	0.0907492	0.0453746	−	0.998970i	$-0.485552\pi$
$211$	1.31821	0.0907492	0.0453746	+	0.998970i	$0.485552\pi$
$212$	0	0
$213$	−1.69065	−0.115841
$214$	0	0
$215$	−2.94115	−0.200585
$216$	0	0
$217$	10.7897	0.732452
$218$	0	0
$219$	−12.7248	−0.859864
$220$	0	0
$221$	11.7014	0.787123
$222$	0	0
$223$	21.4965	1.43951	0.719755	−	0.694228i	$-0.244254\pi$
$223$	21.4965	1.43951	0.719755	+	0.694228i	$0.244254\pi$
$224$	0	0
$225$	−0.297599	−0.0198400
$226$	0	0
$227$	−21.9569	−1.45733	−0.728664	−	0.684871i	$-0.759858\pi$
$227$	−21.9569	−1.45733	−0.728664	+	0.684871i	$0.759858\pi$
$228$	0	0
$229$	−13.6521	−0.902158	−0.451079	−	0.892484i	$-0.648961\pi$
$229$	−13.6521	−0.902158	−0.451079	+	0.892484i	$0.648961\pi$
$230$	0	0
$231$	−14.1992	−0.934241
$232$	0	0
$233$	−4.00689	−0.262500	−0.131250	−	0.991349i	$-0.541899\pi$
$233$	−4.00689	−0.262500	−0.131250	+	0.991349i	$0.541899\pi$
$234$	0	0
$235$	2.65622	0.173273
$236$	0	0
$237$	9.49683	0.616885
$238$	0	0
$239$	−10.8955	−0.704774	−0.352387	−	0.935854i	$-0.614630\pi$
$239$	−10.8955	−0.704774	−0.352387	+	0.935854i	$0.614630\pi$
$240$	0	0
$241$	−25.7975	−1.66176	−0.830882	−	0.556448i	$-0.812164\pi$
$241$	−25.7975	−1.66176	−0.830882	+	0.556448i	$0.812164\pi$
$242$	0	0
$243$	−3.08093	−0.197642
$244$	0	0
$245$	4.52654	0.289190
$246$	0	0
$247$	13.1547	0.837012
$248$	0	0
$249$	10.9686	0.695109
$250$	0	0
$251$	29.3972	1.85553	0.927766	−	0.373162i	$-0.121726\pi$
$251$	29.3972	1.85553	0.927766	+	0.373162i	$0.121726\pi$
$252$	0	0
$253$	29.0411	1.82580
$254$	0	0
$255$	−7.87376	−0.493074
$256$	0	0
$257$	1.44684	0.0902512	0.0451256	−	0.998981i	$-0.485631\pi$
$257$	1.44684	0.0902512	0.0451256	+	0.998981i	$0.485631\pi$
$258$	0	0
$259$	−1.57272	−0.0977242
$260$	0	0
$261$	−2.95961	−0.183195
$262$	0	0
$263$	12.8374	0.791589	0.395795	−	0.918339i	$-0.370469\pi$
$263$	12.8374	0.791589	0.395795	+	0.918339i	$0.370469\pi$
$264$	0	0
$265$	−3.79570	−0.233168
$266$	0	0
$267$	16.4597	1.00732
$268$	0	0
$269$	−6.99469	−0.426474	−0.213237	−	0.977001i	$-0.568401\pi$
$269$	−6.99469	−0.426474	−0.213237	+	0.977001i	$0.568401\pi$
$270$	0	0
$271$	−1.90190	−0.115532	−0.0577662	−	0.998330i	$-0.518398\pi$
$271$	−1.90190	−0.115532	−0.0577662	+	0.998330i	$0.518398\pi$
$272$	0	0
$273$	−6.31624	−0.382276
$274$	0	0
$275$	−5.49209	−0.331186
$276$	0	0
$277$	−24.9820	−1.50102	−0.750510	−	0.660859i	$-0.770192\pi$
$277$	−24.9820	−1.50102	−0.750510	+	0.660859i	$0.770192\pi$
$278$	0	0
$279$	2.04169	0.122233
$280$	0	0
$281$	−2.03153	−0.121191	−0.0605956	−	0.998162i	$-0.519300\pi$
$281$	−2.03153	−0.121191	−0.0605956	+	0.998162i	$0.519300\pi$
$282$	0	0
$283$	13.2689	0.788753	0.394377	−	0.918949i	$-0.370961\pi$
$283$	13.2689	0.788753	0.394377	+	0.918949i	$0.370961\pi$
$284$	0	0
$285$	−8.85164	−0.524326
$286$	0	0
$287$	12.4644	0.735747
$288$	0	0
$289$	5.94115	0.349479
$290$	0	0
$291$	21.3269	1.25021
$292$	0	0
$293$	7.86915	0.459721	0.229860	−	0.973224i	$-0.426173\pi$
$293$	7.86915	0.459721	0.229860	+	0.973224i	$0.426173\pi$
$294$	0	0
$295$	8.37222	0.487449
$296$	0	0
$297$	−29.7722	−1.72756
$298$	0	0
$299$	12.9183	0.747086
$300$	0	0
$301$	−4.62561	−0.266616
$302$	0	0
$303$	5.38869	0.309572
$304$	0	0
$305$	−5.75145	−0.329327
$306$	0	0
$307$	16.6827	0.952133	0.476066	−	0.879409i	$-0.342062\pi$
$307$	16.6827	0.952133	0.476066	+	0.879409i	$0.342062\pi$
$308$	0	0
$309$	13.0107	0.740152
$310$	0	0
$311$	−19.9617	−1.13192	−0.565962	−	0.824431i	$-0.691495\pi$
$311$	−19.9617	−1.13192	−0.565962	+	0.824431i	$0.691495\pi$
$312$	0	0
$313$	25.7337	1.45455	0.727276	−	0.686345i	$-0.240786\pi$
$313$	25.7337	1.45455	0.727276	+	0.686345i	$0.240786\pi$
$314$	0	0
$315$	−0.468041	−0.0263711
$316$	0	0
$317$	11.6667	0.655266	0.327633	−	0.944805i	$-0.393749\pi$
$317$	11.6667	0.655266	0.327633	+	0.944805i	$0.393749\pi$
$318$	0	0
$319$	−54.6185	−3.05805
$320$	0	0
$321$	−13.8036	−0.770441
$322$	0	0
$323$	25.7903	1.43501
$324$	0	0
$325$	−2.44304	−0.135516
$326$	0	0
$327$	21.4870	1.18824
$328$	0	0
$329$	4.17750	0.230313
$330$	0	0
$331$	3.23530	0.177828	0.0889142	−	0.996039i	$-0.471660\pi$
$331$	3.23530	0.177828	0.0889142	+	0.996039i	$0.471660\pi$
$332$	0	0
$333$	−0.297599	−0.0163083
$334$	0	0
$335$	−7.38454	−0.403461
$336$	0	0
$337$	−25.4428	−1.38596	−0.692978	−	0.720959i	$-0.743702\pi$
$337$	−25.4428	−1.38596	−0.692978	+	0.720959i	$0.743702\pi$
$338$	0	0
$339$	−17.3097	−0.940131
$340$	0	0
$341$	37.6786	2.04041
$342$	0	0
$343$	18.1281	0.978823
$344$	0	0
$345$	−8.69260	−0.467994
$346$	0	0
$347$	−25.7286	−1.38118	−0.690592	−	0.723244i	$-0.742650\pi$
$347$	−25.7286	−1.38118	−0.690592	+	0.723244i	$0.742650\pi$
$348$	0	0
$349$	−1.89280	−0.101319	−0.0506596	−	0.998716i	$-0.516132\pi$
$349$	−1.89280	−0.101319	−0.0506596	+	0.998716i	$0.516132\pi$
$350$	0	0
$351$	−13.2435	−0.706888
$352$	0	0
$353$	8.68179	0.462085	0.231043	−	0.972944i	$-0.425786\pi$
$353$	8.68179	0.462085	0.231043	+	0.972944i	$0.425786\pi$
$354$	0	0
$355$	−1.02844	−0.0545839
$356$	0	0
$357$	−12.3832	−0.655391
$358$	0	0
$359$	−20.4586	−1.07976	−0.539881	−	0.841741i	$-0.681531\pi$
$359$	−20.4586	−1.07976	−0.539881	+	0.841741i	$0.681531\pi$
$360$	0	0
$361$	9.99329	0.525963
$362$	0	0
$363$	−31.5021	−1.65343
$364$	0	0
$365$	−7.74064	−0.405164
$366$	0	0
$367$	11.1891	0.584068	0.292034	−	0.956408i	$-0.405668\pi$
$367$	11.1891	0.584068	0.292034	+	0.956408i	$0.405668\pi$
$368$	0	0
$369$	2.35857	0.122782
$370$	0	0
$371$	−5.96959	−0.309926
$372$	0	0
$373$	5.30740	0.274807	0.137403	−	0.990515i	$-0.456124\pi$
$373$	5.30740	0.274807	0.137403	+	0.990515i	$0.456124\pi$
$374$	0	0
$375$	1.64390	0.0848905
$376$	0	0
$377$	−24.2959	−1.25130
$378$	0	0
$379$	−13.0291	−0.669262	−0.334631	−	0.942349i	$-0.608612\pi$
$379$	−13.0291	−0.669262	−0.334631	+	0.942349i	$0.608612\pi$
$380$	0	0
$381$	1.06474	0.0545481
$382$	0	0
$383$	23.7337	1.21273	0.606367	−	0.795185i	$-0.292626\pi$
$383$	23.7337	1.21273	0.606367	+	0.795185i	$0.292626\pi$
$384$	0	0
$385$	−8.63754	−0.440210
$386$	0	0
$387$	−0.875284	−0.0444932
$388$	0	0
$389$	−21.8526	−1.10797	−0.553985	−	0.832527i	$-0.686894\pi$
$389$	−21.8526	−1.10797	−0.553985	+	0.832527i	$0.686894\pi$
$390$	0	0
$391$	25.3269	1.28084
$392$	0	0
$393$	9.75619	0.492134
$394$	0	0
$395$	5.77702	0.290673
$396$	0	0
$397$	−22.0126	−1.10478	−0.552391	−	0.833585i	$-0.686284\pi$
$397$	−22.0126	−1.10478	−0.552391	+	0.833585i	$0.686284\pi$
$398$	0	0
$399$	−13.9212	−0.696930
$400$	0	0
$401$	−38.9802	−1.94658	−0.973290	−	0.229578i	$-0.926265\pi$
$401$	−38.9802	−1.94658	−0.973290	+	0.229578i	$0.926265\pi$
$402$	0	0
$403$	16.7606	0.834903
$404$	0	0
$405$	8.01864	0.398449
$406$	0	0
$407$	−5.49209	−0.272233
$408$	0	0
$409$	−36.8229	−1.82078	−0.910388	−	0.413755i	$-0.864217\pi$
$409$	−36.8229	−1.82078	−0.910388	+	0.413755i	$0.864217\pi$
$410$	0	0
$411$	−6.14924	−0.303319
$412$	0	0
$413$	13.1672	0.647914
$414$	0	0
$415$	6.67234	0.327532
$416$	0	0
$417$	−32.4901	−1.59105
$418$	0	0
$419$	−18.6610	−0.911648	−0.455824	−	0.890070i	$-0.650655\pi$
$419$	−18.6610	−0.911648	−0.455824	+	0.890070i	$0.650655\pi$
$420$	0	0
$421$	2.30291	0.112237	0.0561186	−	0.998424i	$-0.482128\pi$
$421$	2.30291	0.112237	0.0561186	+	0.998424i	$0.482128\pi$
$422$	0	0
$423$	0.790490	0.0384349
$424$	0	0
$425$	−4.78969	−0.232334
$426$	0	0
$427$	−9.04543	−0.437739
$428$	0	0
$429$	−22.0569	−1.06492
$430$	0	0
$431$	33.6226	1.61954	0.809772	−	0.586744i	$-0.199591\pi$
$431$	33.6226	1.61954	0.809772	+	0.586744i	$0.199591\pi$
$432$	0	0
$433$	15.3447	0.737418	0.368709	−	0.929545i	$-0.379800\pi$
$433$	15.3447	0.737418	0.368709	+	0.929545i	$0.379800\pi$
$434$	0	0
$435$	16.3485	0.783849
$436$	0	0
$437$	28.4724	1.36202
$438$	0	0
$439$	−36.3833	−1.73648	−0.868239	−	0.496146i	$-0.834748\pi$
$439$	−36.3833	−1.73648	−0.868239	+	0.496146i	$0.834748\pi$
$440$	0	0
$441$	1.34710	0.0641475
$442$	0	0
$443$	31.3804	1.49093	0.745463	−	0.666547i	$-0.232229\pi$
$443$	31.3804	1.49093	0.745463	+	0.666547i	$0.232229\pi$
$444$	0	0
$445$	10.0126	0.474644
$446$	0	0
$447$	−20.9344	−0.990165
$448$	0	0
$449$	19.3447	0.912932	0.456466	−	0.889741i	$-0.349115\pi$
$449$	19.3447	0.912932	0.456466	+	0.889741i	$0.349115\pi$
$450$	0	0
$451$	43.5267	2.04959
$452$	0	0
$453$	−7.71415	−0.362442
$454$	0	0
$455$	−3.84223	−0.180127
$456$	0	0
$457$	−19.5490	−0.914462	−0.457231	−	0.889348i	$-0.651159\pi$
$457$	−19.5490	−0.914462	−0.457231	+	0.889348i	$0.651159\pi$
$458$	0	0
$459$	−25.9645	−1.21192
$460$	0	0
$461$	−8.89489	−0.414276	−0.207138	−	0.978312i	$-0.566415\pi$
$461$	−8.89489	−0.414276	−0.207138	+	0.978312i	$0.566415\pi$
$462$	0	0
$463$	−6.38629	−0.296796	−0.148398	−	0.988928i	$-0.547412\pi$
$463$	−6.38629	−0.296796	−0.148398	+	0.988928i	$0.547412\pi$
$464$	0	0
$465$	−11.2780	−0.523004
$466$	0	0
$467$	29.1226	1.34763	0.673817	−	0.738898i	$-0.264654\pi$
$467$	29.1226	1.34763	0.673817	+	0.738898i	$0.264654\pi$
$468$	0	0
$469$	−11.6138	−0.536277
$470$	0	0
$471$	15.6446	0.720863
$472$	0	0
$473$	−16.1531	−0.742718
$474$	0	0
$475$	−5.38454	−0.247060
$476$	0	0
$477$	−1.12960	−0.0517208
$478$	0	0
$479$	−23.5711	−1.07699	−0.538497	−	0.842628i	$-0.681008\pi$
$479$	−23.5711	−1.07699	−0.538497	+	0.842628i	$0.681008\pi$
$480$	0	0
$481$	−2.44304	−0.111393
$482$	0	0
$483$	−13.6710	−0.622054
$484$	0	0
$485$	12.9734	0.589091
$486$	0	0
$487$	−23.0935	−1.04647	−0.523233	−	0.852189i	$-0.675274\pi$
$487$	−23.0935	−1.04647	−0.523233	+	0.852189i	$0.675274\pi$
$488$	0	0
$489$	−22.0195	−0.995757
$490$	0	0
$491$	−36.6843	−1.65554	−0.827770	−	0.561068i	$-0.810391\pi$
$491$	−36.6843	−1.65554	−0.827770	+	0.561068i	$0.810391\pi$
$492$	0	0
$493$	−47.6332	−2.14529
$494$	0	0
$495$	−1.63444	−0.0734628
$496$	0	0
$497$	−1.61745	−0.0725526
$498$	0	0
$499$	9.90969	0.443619	0.221809	−	0.975090i	$-0.428804\pi$
$499$	9.90969	0.443619	0.221809	+	0.975090i	$0.428804\pi$
$500$	0	0
$501$	−21.8956	−0.978224
$502$	0	0
$503$	−40.9241	−1.82471	−0.912357	−	0.409396i	$-0.865739\pi$
$503$	−40.9241	−1.82471	−0.912357	+	0.409396i	$0.865739\pi$
$504$	0	0
$505$	3.27799	0.145869
$506$	0	0
$507$	11.5591	0.513359
$508$	0	0
$509$	−27.3244	−1.21113	−0.605566	−	0.795795i	$-0.707053\pi$
$509$	−27.3244	−1.21113	−0.605566	+	0.795795i	$0.707053\pi$
$510$	0	0
$511$	−12.1739	−0.538541
$512$	0	0
$513$	−29.1892	−1.28873
$514$	0	0
$515$	7.91453	0.348756
$516$	0	0
$517$	14.5882	0.641589
$518$	0	0
$519$	30.3054	1.33026
$520$	0	0
$521$	11.7423	0.514441	0.257221	−	0.966353i	$-0.417193\pi$
$521$	11.7423	0.514441	0.257221	+	0.966353i	$0.417193\pi$
$522$	0	0
$523$	38.8980	1.70089	0.850446	−	0.526063i	$-0.176332\pi$
$523$	38.8980	1.70089	0.850446	+	0.526063i	$0.176332\pi$
$524$	0	0
$525$	2.58540	0.112836
$526$	0	0
$527$	32.8598	1.43139
$528$	0	0
$529$	4.96079	0.215686
$530$	0	0
$531$	2.49157	0.108125
$532$	0	0
$533$	19.3619	0.838659
$534$	0	0
$535$	−8.39687	−0.363028
$536$	0	0
$537$	17.2697	0.745243
$538$	0	0
$539$	24.8602	1.07080
$540$	0	0
$541$	7.35166	0.316072	0.158036	−	0.987433i	$-0.449484\pi$
$541$	7.35166	0.316072	0.158036	+	0.987433i	$0.449484\pi$
$542$	0	0
$543$	−31.7763	−1.36365
$544$	0	0
$545$	13.0708	0.559891
$546$	0	0
$547$	1.41561	0.0605269	0.0302635	−	0.999542i	$-0.490365\pi$
$547$	1.41561	0.0605269	0.0302635	+	0.999542i	$0.490365\pi$
$548$	0	0
$549$	−1.71163	−0.0730505
$550$	0	0
$551$	−53.5489	−2.28126
$552$	0	0
$553$	9.08565	0.386361
$554$	0	0
$555$	1.64390	0.0697796
$556$	0	0
$557$	22.1514	0.938583	0.469292	−	0.883043i	$-0.344509\pi$
$557$	22.1514	0.938583	0.469292	+	0.883043i	$0.344509\pi$
$558$	0	0
$559$	−7.18536	−0.303908
$560$	0	0
$561$	−43.2434	−1.82574
$562$	0	0
$563$	18.2731	0.770119	0.385060	−	0.922892i	$-0.374181\pi$
$563$	18.2731	0.770119	0.385060	+	0.922892i	$0.374181\pi$
$564$	0	0
$565$	−10.5296	−0.442985
$566$	0	0
$567$	12.6111	0.529616
$568$	0	0
$569$	28.1916	1.18185	0.590926	−	0.806726i	$-0.298763\pi$
$569$	28.1916	1.18185	0.590926	+	0.806726i	$0.298763\pi$
$570$	0	0
$571$	32.2220	1.34845	0.674224	−	0.738527i	$-0.264478\pi$
$571$	32.2220	1.34845	0.674224	+	0.738527i	$0.264478\pi$
$572$	0	0
$573$	34.5068	1.44154
$574$	0	0
$575$	−5.28780	−0.220516
$576$	0	0
$577$	−0.610043	−0.0253964	−0.0126982	−	0.999919i	$-0.504042\pi$
$577$	−0.610043	−0.0253964	−0.0126982	+	0.999919i	$0.504042\pi$
$578$	0	0
$579$	−39.4016	−1.63747
$580$	0	0
$581$	10.4937	0.435354
$582$	0	0
$583$	−20.8464	−0.863368
$584$	0	0
$585$	−0.727049	−0.0300598
$586$	0	0
$587$	−8.29350	−0.342310	−0.171155	−	0.985244i	$-0.554750\pi$
$587$	−8.29350	−0.342310	−0.171155	+	0.985244i	$0.554750\pi$
$588$	0	0
$589$	36.9407	1.52212
$590$	0	0
$591$	35.0480	1.44168
$592$	0	0
$593$	−33.2631	−1.36595	−0.682975	−	0.730442i	$-0.739314\pi$
$593$	−33.2631	−1.36595	−0.682975	+	0.730442i	$0.739314\pi$
$594$	0	0
$595$	−7.53286	−0.308817
$596$	0	0
$597$	−26.4083	−1.08082
$598$	0	0
$599$	−1.56606	−0.0639875	−0.0319938	−	0.999488i	$-0.510186\pi$
$599$	−1.56606	−0.0639875	−0.0319938	+	0.999488i	$0.510186\pi$
$600$	0	0
$601$	44.4575	1.81346	0.906729	−	0.421713i	$-0.138571\pi$
$601$	44.4575	1.81346	0.906729	+	0.421713i	$0.138571\pi$
$602$	0	0
$603$	−2.19764	−0.0894946
$604$	0	0
$605$	−19.1631	−0.779090
$606$	0	0
$607$	−14.6856	−0.596071	−0.298035	−	0.954555i	$-0.596331\pi$
$607$	−14.6856	−0.596071	−0.298035	+	0.954555i	$0.596331\pi$
$608$	0	0
$609$	25.7116	1.04189
$610$	0	0
$611$	6.48927	0.262528
$612$	0	0
$613$	−35.2011	−1.42176	−0.710880	−	0.703314i	$-0.751703\pi$
$613$	−35.2011	−1.42176	−0.710880	+	0.703314i	$0.751703\pi$
$614$	0	0
$615$	−13.0284	−0.525357
$616$	0	0
$617$	8.37017	0.336970	0.168485	−	0.985704i	$-0.446112\pi$
$617$	8.37017	0.336970	0.168485	+	0.985704i	$0.446112\pi$
$618$	0	0
$619$	23.2650	0.935098	0.467549	−	0.883967i	$-0.345137\pi$
$619$	23.2650	0.935098	0.467549	+	0.883967i	$0.345137\pi$
$620$	0	0
$621$	−28.6647	−1.15027
$622$	0	0
$623$	15.7471	0.630893
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−48.6140	−1.94146
$628$	0	0
$629$	−4.78969	−0.190978
$630$	0	0
$631$	−31.2132	−1.24258	−0.621288	−	0.783582i	$-0.713390\pi$
$631$	−31.2132	−1.24258	−0.621288	+	0.783582i	$0.713390\pi$
$632$	0	0
$633$	−2.16700	−0.0861305
$634$	0	0
$635$	0.647690	0.0257028
$636$	0	0
$637$	11.0585	0.438156
$638$	0	0
$639$	−0.306063	−0.0121077
$640$	0	0
$641$	−12.3115	−0.486275	−0.243137	−	0.969992i	$-0.578177\pi$
$641$	−12.3115	−0.486275	−0.243137	+	0.969992i	$0.578177\pi$
$642$	0	0
$643$	−34.4344	−1.35796	−0.678979	−	0.734157i	$-0.737577\pi$
$643$	−34.4344	−1.35796	−0.678979	+	0.734157i	$0.737577\pi$
$644$	0	0
$645$	4.83495	0.190376
$646$	0	0
$647$	−20.7907	−0.817366	−0.408683	−	0.912676i	$-0.634012\pi$
$647$	−20.7907	−0.817366	−0.408683	+	0.912676i	$0.634012\pi$
$648$	0	0
$649$	45.9810	1.80491
$650$	0	0
$651$	−17.7372	−0.695174
$652$	0	0
$653$	−4.52630	−0.177128	−0.0885638	−	0.996071i	$-0.528228\pi$
$653$	−4.52630	−0.177128	−0.0885638	+	0.996071i	$0.528228\pi$
$654$	0	0
$655$	5.93479	0.231891
$656$	0	0
$657$	−2.30361	−0.0898724
$658$	0	0
$659$	−19.2347	−0.749276	−0.374638	−	0.927171i	$-0.622233\pi$
$659$	−19.2347	−0.749276	−0.374638	+	0.927171i	$0.622233\pi$
$660$	0	0
$661$	32.3809	1.25947	0.629735	−	0.776810i	$-0.283163\pi$
$661$	32.3809	1.25947	0.629735	+	0.776810i	$0.283163\pi$
$662$	0	0
$663$	−19.2360	−0.747063
$664$	0	0
$665$	−8.46839	−0.328390
$666$	0	0
$667$	−52.5868	−2.03617
$668$	0	0
$669$	−35.3380	−1.36625
$670$	0	0
$671$	−31.5875	−1.21942
$672$	0	0
$673$	33.7013	1.29909	0.649545	−	0.760323i	$-0.274959\pi$
$673$	33.7013	1.29909	0.649545	+	0.760323i	$0.274959\pi$
$674$	0	0
$675$	5.42092	0.208651
$676$	0	0
$677$	0.717245	0.0275660	0.0137830	−	0.999905i	$-0.495613\pi$
$677$	0.717245	0.0275660	0.0137830	+	0.999905i	$0.495613\pi$
$678$	0	0
$679$	20.4035	0.783016
$680$	0	0
$681$	36.0949	1.38316
$682$	0	0
$683$	−21.4971	−0.822565	−0.411282	−	0.911508i	$-0.634919\pi$
$683$	−21.4971	−0.822565	−0.411282	+	0.911508i	$0.634919\pi$
$684$	0	0
$685$	−3.74064	−0.142923
$686$	0	0
$687$	22.4427	0.856243
$688$	0	0
$689$	−9.27307	−0.353276
$690$	0	0
$691$	−15.3573	−0.584220	−0.292110	−	0.956385i	$-0.594357\pi$
$691$	−15.3573	−0.584220	−0.292110	+	0.956385i	$0.594357\pi$
$692$	0	0
$693$	−2.57053	−0.0976462
$694$	0	0
$695$	−19.7641	−0.749694
$696$	0	0
$697$	37.9599	1.43783
$698$	0	0
$699$	6.58691	0.249140
$700$	0	0
$701$	−43.6963	−1.65039	−0.825193	−	0.564851i	$-0.808934\pi$
$701$	−43.6963	−1.65039	−0.825193	+	0.564851i	$0.808934\pi$
$702$	0	0
$703$	−5.38454	−0.203082
$704$	0	0
$705$	−4.36656	−0.164454
$706$	0	0
$707$	5.15537	0.193888
$708$	0	0
$709$	30.1538	1.13245	0.566224	−	0.824251i	$-0.308404\pi$
$709$	30.1538	1.13245	0.566224	+	0.824251i	$0.308404\pi$
$710$	0	0
$711$	1.71924	0.0644764
$712$	0	0
$713$	36.2770	1.35859
$714$	0	0
$715$	−13.4174	−0.501783
$716$	0	0
$717$	17.9112	0.668905
$718$	0	0
$719$	−29.1138	−1.08576	−0.542881	−	0.839810i	$-0.682667\pi$
$719$	−29.1138	−1.08576	−0.542881	+	0.839810i	$0.682667\pi$
$720$	0	0
$721$	12.4474	0.463564
$722$	0	0
$723$	42.4085	1.57719
$724$	0	0
$725$	9.94494	0.369346
$726$	0	0
$727$	11.6325	0.431427	0.215713	−	0.976457i	$-0.430792\pi$
$727$	11.6325	0.431427	0.215713	+	0.976457i	$0.430792\pi$
$728$	0	0
$729$	29.1206	1.07854
$730$	0	0
$731$	−14.0872	−0.521034
$732$	0	0
$733$	−42.3769	−1.56523	−0.782614	−	0.622507i	$-0.786114\pi$
$733$	−42.3769	−1.56523	−0.782614	+	0.622507i	$0.786114\pi$
$734$	0	0
$735$	−7.44118	−0.274472
$736$	0	0
$737$	−40.5566	−1.49392
$738$	0	0
$739$	−3.10885	−0.114361	−0.0571804	−	0.998364i	$-0.518211\pi$
$739$	−3.10885	−0.114361	−0.0571804	+	0.998364i	$0.518211\pi$
$740$	0	0
$741$	−21.6249	−0.794412
$742$	0	0
$743$	−51.7873	−1.89989	−0.949945	−	0.312416i	$-0.898862\pi$
$743$	−51.7873	−1.89989	−0.949945	+	0.312416i	$0.898862\pi$
$744$	0	0
$745$	−12.7346	−0.466561
$746$	0	0
$747$	1.98568	0.0726524
$748$	0	0
$749$	−13.2059	−0.482535
$750$	0	0
$751$	−29.2516	−1.06741	−0.533703	−	0.845672i	$-0.679200\pi$
$751$	−29.2516	−1.06741	−0.533703	+	0.845672i	$0.679200\pi$
$752$	0	0
$753$	−48.3259	−1.76110
$754$	0	0
$755$	−4.69260	−0.170781
$756$	0	0
$757$	49.4043	1.79563	0.897814	−	0.440375i	$-0.145154\pi$
$757$	49.4043	1.79563	0.897814	+	0.440375i	$0.145154\pi$
$758$	0	0
$759$	−47.7405	−1.73287
$760$	0	0
$761$	21.5105	0.779753	0.389877	−	0.920867i	$-0.372518\pi$
$761$	21.5105	0.779753	0.389877	+	0.920867i	$0.372518\pi$
$762$	0	0
$763$	20.5567	0.744203
$764$	0	0
$765$	−1.42541	−0.0515358
$766$	0	0
$767$	20.4537	0.738540
$768$	0	0
$769$	−13.0164	−0.469384	−0.234692	−	0.972070i	$-0.575408\pi$
$769$	−13.0164	−0.469384	−0.234692	+	0.972070i	$0.575408\pi$
$770$	0	0
$771$	−2.37845	−0.0856578
$772$	0	0
$773$	19.8665	0.714547	0.357273	−	0.934000i	$-0.383706\pi$
$773$	19.8665	0.714547	0.357273	+	0.934000i	$0.383706\pi$
$774$	0	0
$775$	−6.86052	−0.246437
$776$	0	0
$777$	2.58540	0.0927505
$778$	0	0
$779$	42.6743	1.52897
$780$	0	0
$781$	−5.64829	−0.202112
$782$	0	0
$783$	53.9107	1.92661
$784$	0	0
$785$	9.51674	0.339667
$786$	0	0
$787$	−28.2579	−1.00729	−0.503643	−	0.863912i	$-0.668007\pi$
$787$	−28.2579	−1.00729	−0.503643	+	0.863912i	$0.668007\pi$
$788$	0	0
$789$	−21.1034	−0.751301
$790$	0	0
$791$	−16.5602	−0.588813
$792$	0	0
$793$	−14.0510	−0.498967
$794$	0	0
$795$	6.23975	0.221301
$796$	0	0
$797$	−23.4460	−0.830501	−0.415251	−	0.909707i	$-0.636306\pi$
$797$	−23.4460	−0.830501	−0.415251	+	0.909707i	$0.636306\pi$
$798$	0	0
$799$	12.7225	0.450089
$800$	0	0
$801$	2.97975	0.105284
$802$	0	0
$803$	−42.5123	−1.50023
$804$	0	0
$805$	−8.31624	−0.293109
$806$	0	0
$807$	11.4986	0.404768
$808$	0	0
$809$	26.3155	0.925205	0.462603	−	0.886566i	$-0.346916\pi$
$809$	26.3155	0.925205	0.462603	+	0.886566i	$0.346916\pi$
$810$	0	0
$811$	5.74134	0.201606	0.100803	−	0.994906i	$-0.467859\pi$
$811$	5.74134	0.201606	0.100803	+	0.994906i	$0.467859\pi$
$812$	0	0
$813$	3.12654	0.109652
$814$	0	0
$815$	−13.3947	−0.469196
$816$	0	0
$817$	−15.8367	−0.554057
$818$	0	0
$819$	−1.14345	−0.0399552
$820$	0	0
$821$	−0.597202	−0.0208425	−0.0104212	−	0.999946i	$-0.503317\pi$
$821$	−0.597202	−0.0208425	−0.0104212	+	0.999946i	$0.503317\pi$
$822$	0	0
$823$	2.22602	0.0775942	0.0387971	−	0.999247i	$-0.487647\pi$
$823$	2.22602	0.0775942	0.0387971	+	0.999247i	$0.487647\pi$
$824$	0	0
$825$	9.02844	0.314330
$826$	0	0
$827$	34.3769	1.19540	0.597702	−	0.801719i	$-0.296081\pi$
$827$	34.3769	1.19540	0.597702	+	0.801719i	$0.296081\pi$
$828$	0	0
$829$	−14.7502	−0.512296	−0.256148	−	0.966638i	$-0.582453\pi$
$829$	−14.7502	−0.512296	−0.256148	+	0.966638i	$0.582453\pi$
$830$	0	0
$831$	41.0678	1.42463
$832$	0	0
$833$	21.6808	0.751193
$834$	0	0
$835$	−13.3193	−0.460934
$836$	0	0
$837$	−37.1903	−1.28548
$838$	0	0
$839$	4.48089	0.154698	0.0773488	−	0.997004i	$-0.475354\pi$
$839$	4.48089	0.154698	0.0773488	+	0.997004i	$0.475354\pi$
$840$	0	0
$841$	69.9018	2.41041
$842$	0	0
$843$	3.33963	0.115023
$844$	0	0
$845$	7.03153	0.241892
$846$	0	0
$847$	−30.1382	−1.03556
$848$	0	0
$849$	−21.8127	−0.748610
$850$	0	0
$851$	−5.28780	−0.181263
$852$	0	0
$853$	−32.6710	−1.11863	−0.559317	−	0.828954i	$-0.688936\pi$
$853$	−32.6710	−1.11863	−0.559317	+	0.828954i	$0.688936\pi$
$854$	0	0
$855$	−1.60244	−0.0548022
$856$	0	0
$857$	2.65948	0.0908463	0.0454231	−	0.998968i	$-0.485536\pi$
$857$	2.65948	0.0908463	0.0454231	+	0.998968i	$0.485536\pi$
$858$	0	0
$859$	−28.9331	−0.987184	−0.493592	−	0.869694i	$-0.664316\pi$
$859$	−28.9331	−0.987184	−0.493592	+	0.869694i	$0.664316\pi$
$860$	0	0
$861$	−20.4901	−0.698301
$862$	0	0
$863$	15.4333	0.525356	0.262678	−	0.964884i	$-0.415394\pi$
$863$	15.4333	0.525356	0.262678	+	0.964884i	$0.415394\pi$
$864$	0	0
$865$	18.4351	0.626811
$866$	0	0
$867$	−9.76664	−0.331693
$868$	0	0
$869$	31.7279	1.07630
$870$	0	0
$871$	−18.0408	−0.611288
$872$	0	0
$873$	3.86087	0.130671
$874$	0	0
$875$	1.57272	0.0531677
$876$	0	0
$877$	11.3806	0.384295	0.192147	−	0.981366i	$-0.438455\pi$
$877$	11.3806	0.384295	0.192147	+	0.981366i	$0.438455\pi$
$878$	0	0
$879$	−12.9361	−0.436323
$880$	0	0
$881$	−12.6663	−0.426737	−0.213368	−	0.976972i	$-0.568443\pi$
$881$	−12.6663	−0.426737	−0.213368	+	0.976972i	$0.568443\pi$
$882$	0	0
$883$	−15.6674	−0.527249	−0.263625	−	0.964625i	$-0.584918\pi$
$883$	−15.6674	−0.527249	−0.263625	+	0.964625i	$0.584918\pi$
$884$	0	0
$885$	−13.7631	−0.462641
$886$	0	0
$887$	−31.3190	−1.05159	−0.525795	−	0.850612i	$-0.676232\pi$
$887$	−31.3190	−1.05159	−0.525795	+	0.850612i	$0.676232\pi$
$888$	0	0
$889$	1.01864	0.0341640
$890$	0	0
$891$	44.0391	1.47537
$892$	0	0
$893$	14.3025	0.478616
$894$	0	0
$895$	10.5053	0.351155
$896$	0	0
$897$	−21.2364	−0.709063
$898$	0	0
$899$	−68.2274	−2.27551
$900$	0	0
$901$	−18.1803	−0.605672
$902$	0	0
$903$	7.60403	0.253046
$904$	0	0
$905$	−19.3298	−0.642545
$906$	0	0
$907$	37.8815	1.25783	0.628917	−	0.777472i	$-0.283498\pi$
$907$	37.8815	1.25783	0.628917	+	0.777472i	$0.283498\pi$
$908$	0	0
$909$	0.975529	0.0323562
$910$	0	0
$911$	37.1590	1.23113	0.615566	−	0.788086i	$-0.288928\pi$
$911$	37.1590	1.23113	0.615566	+	0.788086i	$0.288928\pi$
$912$	0	0
$913$	36.6451	1.21278
$914$	0	0
$915$	9.45479	0.312566
$916$	0	0
$917$	9.33377	0.308228
$918$	0	0
$919$	51.4992	1.69880	0.849400	−	0.527749i	$-0.176964\pi$
$919$	51.4992	1.69880	0.849400	+	0.527749i	$0.176964\pi$
$920$	0	0
$921$	−27.4247	−0.903674
$922$	0	0
$923$	−2.51252	−0.0827007
$924$	0	0
$925$	1.00000	0.0328798
$926$	0	0
$927$	2.35536	0.0773601
$928$	0	0
$929$	−29.9099	−0.981312	−0.490656	−	0.871353i	$-0.663243\pi$
$929$	−29.9099	−0.981312	−0.490656	+	0.871353i	$0.663243\pi$
$930$	0	0
$931$	24.3734	0.798805
$932$	0	0
$933$	32.8150	1.07432
$934$	0	0
$935$	−26.3054	−0.860280
$936$	0	0
$937$	−23.5086	−0.767994	−0.383997	−	0.923334i	$-0.625453\pi$
$937$	−23.5086	−0.767994	−0.383997	+	0.923334i	$0.625453\pi$
$938$	0	0
$939$	−42.3035	−1.38052
$940$	0	0
$941$	22.0570	0.719038	0.359519	−	0.933138i	$-0.382941\pi$
$941$	22.0570	0.719038	0.359519	+	0.933138i	$0.382941\pi$
$942$	0	0
$943$	41.9075	1.36470
$944$	0	0
$945$	8.52560	0.277338
$946$	0	0
$947$	−21.3359	−0.693323	−0.346661	−	0.937990i	$-0.612685\pi$
$947$	−21.3359	−0.693323	−0.346661	+	0.937990i	$0.612685\pi$
$948$	0	0
$949$	−18.9107	−0.613868
$950$	0	0
$951$	−19.1788	−0.621916
$952$	0	0
$953$	39.7882	1.28887	0.644433	−	0.764661i	$-0.277093\pi$
$953$	39.7882	1.28887	0.644433	+	0.764661i	$0.277093\pi$
$954$	0	0
$955$	20.9908	0.679248
$956$	0	0
$957$	89.7873	2.90241
$958$	0	0
$959$	−5.88299	−0.189972
$960$	0	0
$961$	16.0667	0.518281
$962$	0	0
$963$	−2.49890	−0.0805260
$964$	0	0
$965$	−23.9684	−0.771569
$966$	0	0
$967$	26.1284	0.840232	0.420116	−	0.907470i	$-0.361989\pi$
$967$	26.1284	0.840232	0.420116	+	0.907470i	$0.361989\pi$
$968$	0	0
$969$	−42.3966	−1.36198
$970$	0	0
$971$	24.5470	0.787752	0.393876	−	0.919164i	$-0.371134\pi$
$971$	24.5470	0.787752	0.393876	+	0.919164i	$0.371134\pi$
$972$	0	0
$973$	−31.0834	−0.996488
$974$	0	0
$975$	4.01612	0.128619
$976$	0	0
$977$	15.5490	0.497456	0.248728	−	0.968573i	$-0.419988\pi$
$977$	15.5490	0.497456	0.248728	+	0.968573i	$0.419988\pi$
$978$	0	0
$979$	54.9903	1.75750
$980$	0	0
$981$	3.88986	0.124194
$982$	0	0
$983$	−33.9619	−1.08322	−0.541608	−	0.840631i	$-0.682184\pi$
$983$	−33.9619	−1.08322	−0.541608	+	0.840631i	$0.682184\pi$
$984$	0	0
$985$	21.3200	0.679313
$986$	0	0
$987$	−6.86738	−0.218591
$988$	0	0
$989$	−15.5522	−0.494531
$990$	0	0
$991$	26.6172	0.845523	0.422761	−	0.906241i	$-0.361061\pi$
$991$	26.6172	0.845523	0.422761	+	0.906241i	$0.361061\pi$
$992$	0	0
$993$	−5.31851	−0.168778
$994$	0	0
$995$	−16.0644	−0.509276
$996$	0	0
$997$	7.89607	0.250071	0.125036	−	0.992152i	$-0.460096\pi$
$997$	7.89607	0.250071	0.125036	+	0.992152i	$0.460096\pi$
$998$	0	0
$999$	5.42092	0.171510

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2960.2.a.y.1.2		5
4.3	odd	2		1480.2.a.i.1.4	✓	5
20.19	odd	2		7400.2.a.p.1.2		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1480.2.a.i.1.4	✓	5	4.3	odd	2
2960.2.a.y.1.2		5	1.1	even	1	trivial
7400.2.a.p.1.2		5	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:	\( N \)	\(=\)	\( 2960 = 2^{4} \cdot 5 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2960.a (trivial)

\(f(q)\)	\(=\)	\(q-1.64390 q^{3} -1.00000 q^{5} -1.57272 q^{7} -0.297599 q^{9} +O(q^{10})\)	\(q-1.64390 q^{3} -1.00000 q^{5} -1.57272 q^{7} -0.297599 q^{9} -5.49209 q^{11} -2.44304 q^{13} +1.64390 q^{15} -4.78969 q^{17} -5.38454 q^{19} +2.58540 q^{21} -5.28780 q^{23} +1.00000 q^{25} +5.42092 q^{27} +9.94494 q^{29} -6.86052 q^{31} +9.02844 q^{33} +1.57272 q^{35} +1.00000 q^{37} +4.01612 q^{39} -7.92533 q^{41} +2.94115 q^{43} +0.297599 q^{45} -2.65622 q^{47} -4.52654 q^{49} +7.87376 q^{51} +3.79570 q^{53} +5.49209 q^{55} +8.85164 q^{57} -8.37222 q^{59} +5.75145 q^{61} +0.468041 q^{63} +2.44304 q^{65} +7.38454 q^{67} +8.69260 q^{69} +1.02844 q^{71} +7.74064 q^{73} -1.64390 q^{75} +8.63754 q^{77} -5.77702 q^{79} -8.01864 q^{81} -6.67234 q^{83} +4.78969 q^{85} -16.3485 q^{87} -10.0126 q^{89} +3.84223 q^{91} +11.2780 q^{93} +5.38454 q^{95} -12.9734 q^{97} +1.63444 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 5 q^{5} - 3 q^{7} + 5 q^{9}+O(q^{10}) \) 5 * q - 5 * q^5 - 3 * q^7 + 5 * q^9	\( 5 q - 5 q^{5} - 3 q^{7} + 5 q^{9} - 3 q^{11} + 4 q^{13} + 7 q^{17} + 4 q^{19} - 10 q^{21} - 10 q^{23} + 5 q^{25} + 6 q^{27} + 19 q^{29} - 13 q^{31} + 6 q^{33} + 3 q^{35} + 5 q^{37} - 14 q^{39} + 11 q^{41} + 13 q^{43} - 5 q^{45} + 2 q^{49} + 12 q^{51} + 27 q^{53} + 3 q^{55} + 12 q^{57} - 16 q^{59} + 27 q^{61} - 37 q^{63} - 4 q^{65} + 6 q^{67} + 40 q^{69} - 34 q^{71} + 16 q^{73} + 9 q^{77} - 16 q^{79} + 9 q^{81} + 14 q^{83} - 7 q^{85} + 42 q^{87} + 38 q^{89} + 34 q^{91} + 30 q^{93} - 4 q^{95} + 5 q^{97} - 23 q^{99}+O(q^{100}) \) 5 * q - 5 * q^5 - 3 * q^7 + 5 * q^9 - 3 * q^11 + 4 * q^13 + 7 * q^17 + 4 * q^19 - 10 * q^21 - 10 * q^23 + 5 * q^25 + 6 * q^27 + 19 * q^29 - 13 * q^31 + 6 * q^33 + 3 * q^35 + 5 * q^37 - 14 * q^39 + 11 * q^41 + 13 * q^43 - 5 * q^45 + 2 * q^49 + 12 * q^51 + 27 * q^53 + 3 * q^55 + 12 * q^57 - 16 * q^59 + 27 * q^61 - 37 * q^63 - 4 * q^65 + 6 * q^67 + 40 * q^69 - 34 * q^71 + 16 * q^73 + 9 * q^77 - 16 * q^79 + 9 * q^81 + 14 * q^83 - 7 * q^85 + 42 * q^87 + 38 * q^89 + 34 * q^91 + 30 * q^93 - 4 * q^95 + 5 * q^97 - 23 * q^99

Introduction

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