LMFDB - Embedded newform 2960.2.a.u.1.3

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:	$3$
Coefficient field:	3.3.892.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 8x + 10 $ x^3 - x^2 - 8*x + 10
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 370)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.59774$ 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)

$f(q)$	$=$	$q+3.34596 q^{3} -1.00000 q^{5} +2.59774 q^{7} +8.19547 q^{9} +O(q^{10})$	$q+3.34596 q^{3} -1.00000 q^{5} +2.59774 q^{7} +8.19547 q^{9} -4.74823 q^{11} +6.69193 q^{13} -3.34596 q^{15} -0.748228 q^{17} +3.34596 q^{19} +8.69193 q^{21} -1.49646 q^{23} +1.00000 q^{25} +17.3839 q^{27} +3.94370 q^{29} -7.79321 q^{31} -15.8874 q^{33} -2.59774 q^{35} -1.00000 q^{37} +22.3909 q^{39} -6.44724 q^{41} -1.94370 q^{43} -8.19547 q^{45} -1.84951 q^{47} -0.251772 q^{49} -2.50354 q^{51} +10.4472 q^{53} +4.74823 q^{55} +11.1955 q^{57} -5.84951 q^{59} +7.94370 q^{61} +21.2897 q^{63} -6.69193 q^{65} -1.84951 q^{67} -5.00709 q^{69} +3.88740 q^{71} -7.49646 q^{73} +3.34596 q^{75} -12.3346 q^{77} +16.5414 q^{79} +33.5793 q^{81} -15.2334 q^{83} +0.748228 q^{85} +13.1955 q^{87} +6.00000 q^{89} +17.3839 q^{91} -26.0758 q^{93} -3.34596 q^{95} -10.4472 q^{97} -38.9140 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{5} + q^{7} + 11 q^{9}+O(q^{10}) $ 3 * q - 3 * q^5 + q^7 + 11 * q^9	$ 3 q - 3 q^{5} + q^{7} + 11 q^{9} - 11 q^{11} + q^{17} + 6 q^{21} + 2 q^{23} + 3 q^{25} + 12 q^{27} - 5 q^{29} - 3 q^{31} - 14 q^{33} - q^{35} - 3 q^{37} + 40 q^{39} - 9 q^{41} + 11 q^{43} - 11 q^{45} - 2 q^{47} - 4 q^{49} - 14 q^{51} + 21 q^{53} + 11 q^{55} + 20 q^{57} - 14 q^{59} + 7 q^{61} + 37 q^{63} - 2 q^{67} - 28 q^{69} - 22 q^{71} - 16 q^{73} + 7 q^{77} + 26 q^{79} + 47 q^{81} - 2 q^{83} - q^{85} + 26 q^{87} + 18 q^{89} + 12 q^{91} - 18 q^{93} - 21 q^{97} - 19 q^{99}+O(q^{100}) $ 3 * q - 3 * q^5 + q^7 + 11 * q^9 - 11 * q^11 + q^17 + 6 * q^21 + 2 * q^23 + 3 * q^25 + 12 * q^27 - 5 * q^29 - 3 * q^31 - 14 * q^33 - q^35 - 3 * q^37 + 40 * q^39 - 9 * q^41 + 11 * q^43 - 11 * q^45 - 2 * q^47 - 4 * q^49 - 14 * q^51 + 21 * q^53 + 11 * q^55 + 20 * q^57 - 14 * q^59 + 7 * q^61 + 37 * q^63 - 2 * q^67 - 28 * q^69 - 22 * q^71 - 16 * q^73 + 7 * q^77 + 26 * q^79 + 47 * q^81 - 2 * q^83 - q^85 + 26 * q^87 + 18 * q^89 + 12 * q^91 - 18 * q^93 - 21 * q^97 - 19 * 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
$3$	3.34596	1.93179	0.965896	−	0.258929i	$-0.0833695\pi$
$3$	3.34596	1.93179	0.965896	+	0.258929i	$0.0833695\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	2.59774	0.981852	0.490926	−	0.871201i	$-0.336659\pi$
$7$	2.59774	0.981852	0.490926	+	0.871201i	$0.336659\pi$
$8$	0	0
$9$	8.19547	2.73182
$10$	0	0
$11$	−4.74823	−1.43164	−0.715822	−	0.698282i	$-0.753948\pi$
$11$	−4.74823	−1.43164	−0.715822	+	0.698282i	$0.753948\pi$
$12$	0	0
$13$	6.69193	1.85601	0.928003	−	0.372572i	$-0.121524\pi$
$13$	6.69193	1.85601	0.928003	+	0.372572i	$0.121524\pi$
$14$	0	0
$15$	−3.34596	−0.863924
$16$	0	0
$17$	−0.748228	−0.181472	−0.0907360	−	0.995875i	$-0.528922\pi$
$17$	−0.748228	−0.181472	−0.0907360	+	0.995875i	$0.528922\pi$
$18$	0	0
$19$	3.34596	0.767617	0.383808	−	0.923413i	$-0.374612\pi$
$19$	3.34596	0.767617	0.383808	+	0.923413i	$0.374612\pi$
$20$	0	0
$21$	8.69193	1.89673
$22$	0	0
$23$	−1.49646	−0.312033	−0.156016	−	0.987754i	$-0.549865\pi$
$23$	−1.49646	−0.312033	−0.156016	+	0.987754i	$0.549865\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	17.3839	3.34552
$28$	0	0
$29$	3.94370	0.732326	0.366163	−	0.930551i	$-0.380671\pi$
$29$	3.94370	0.732326	0.366163	+	0.930551i	$0.380671\pi$
$30$	0	0
$31$	−7.79321	−1.39970	−0.699851	−	0.714289i	$-0.746750\pi$
$31$	−7.79321	−1.39970	−0.699851	+	0.714289i	$0.746750\pi$
$32$	0	0
$33$	−15.8874	−2.76564
$34$	0	0
$35$	−2.59774	−0.439097
$36$	0	0
$37$	−1.00000	−0.164399
$37$	−1.00000	−0.164399
$38$	0	0
$39$	22.3909	3.58542
$40$	0	0
$41$	−6.44724	−1.00689	−0.503445	−	0.864027i	$-0.667934\pi$
$41$	−6.44724	−1.00689	−0.503445	+	0.864027i	$0.667934\pi$
$42$	0	0
$43$	−1.94370	−0.296411	−0.148206	−	0.988957i	$-0.547350\pi$
$43$	−1.94370	−0.296411	−0.148206	+	0.988957i	$0.547350\pi$
$44$	0	0
$45$	−8.19547	−1.22171
$46$	0	0
$47$	−1.84951	−0.269778	−0.134889	−	0.990861i	$-0.543068\pi$
$47$	−1.84951	−0.269778	−0.134889	+	0.990861i	$0.543068\pi$
$48$	0	0
$49$	−0.251772	−0.0359674
$50$	0	0
$51$	−2.50354	−0.350566
$52$	0	0
$53$	10.4472	1.43504	0.717520	−	0.696538i	$-0.245277\pi$
$53$	10.4472	1.43504	0.717520	+	0.696538i	$0.245277\pi$
$54$	0	0
$55$	4.74823	0.640251
$56$	0	0
$57$	11.1955	1.48288
$58$	0	0
$59$	−5.84951	−0.761541	−0.380770	−	0.924670i	$-0.624341\pi$
$59$	−5.84951	−0.761541	−0.380770	+	0.924670i	$0.624341\pi$
$60$	0	0
$61$	7.94370	1.01709	0.508543	−	0.861036i	$-0.330184\pi$
$61$	7.94370	1.01709	0.508543	+	0.861036i	$0.330184\pi$
$62$	0	0
$63$	21.2897	2.68225
$64$	0	0
$65$	−6.69193	−0.830031
$66$	0	0
$67$	−1.84951	−0.225953	−0.112977	−	0.993598i	$-0.536039\pi$
$67$	−1.84951	−0.225953	−0.112977	+	0.993598i	$0.536039\pi$
$68$	0	0
$69$	−5.00709	−0.602783
$70$	0	0
$71$	3.88740	0.461349	0.230675	−	0.973031i	$-0.425907\pi$
$71$	3.88740	0.461349	0.230675	+	0.973031i	$0.425907\pi$
$72$	0	0
$73$	−7.49646	−0.877394	−0.438697	−	0.898635i	$-0.644560\pi$
$73$	−7.49646	−0.877394	−0.438697	+	0.898635i	$0.644560\pi$
$74$	0	0
$75$	3.34596	0.386359
$76$	0	0
$77$	−12.3346	−1.40566
$78$	0	0
$79$	16.5414	1.86106	0.930528	−	0.366220i	$-0.119348\pi$
$79$	16.5414	1.86106	0.930528	+	0.366220i	$0.119348\pi$
$80$	0	0
$81$	33.5793	3.73104
$82$	0	0
$83$	−15.2334	−1.67208	−0.836039	−	0.548670i	$-0.815134\pi$
$83$	−15.2334	−1.67208	−0.836039	+	0.548670i	$0.815134\pi$
$84$	0	0
$85$	0.748228	0.0811567
$86$	0	0
$87$	13.1955	1.41470
$88$	0	0
$89$	6.00000	0.635999	0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000	0.635999	0.317999	+	0.948091i	$0.396989\pi$
$90$	0	0
$91$	17.3839	1.82232
$92$	0	0
$93$	−26.0758	−2.70393
$94$	0	0
$95$	−3.34596	−0.343289
$96$	0	0
$97$	−10.4472	−1.06076	−0.530378	−	0.847761i	$-0.677950\pi$
$97$	−10.4472	−1.06076	−0.530378	+	0.847761i	$0.677950\pi$
$98$	0	0
$99$	−38.9140	−3.91100
$100$	0	0
$101$	−12.1884	−1.21279	−0.606395	−	0.795164i	$-0.707385\pi$
$101$	−12.1884	−1.21279	−0.606395	+	0.795164i	$0.707385\pi$
$102$	0	0
$103$	1.30807	0.128888	0.0644442	−	0.997921i	$-0.479473\pi$
$103$	1.30807	0.128888	0.0644442	+	0.997921i	$0.479473\pi$
$104$	0	0
$105$	−8.69193	−0.848245
$106$	0	0
$107$	3.04498	0.294369	0.147185	−	0.989109i	$-0.452979\pi$
$107$	3.04498	0.294369	0.147185	+	0.989109i	$0.452979\pi$
$108$	0	0
$109$	−1.44015	−0.137942	−0.0689709	−	0.997619i	$-0.521972\pi$
$109$	−1.44015	−0.137942	−0.0689709	+	0.997619i	$0.521972\pi$
$110$	0	0
$111$	−3.34596	−0.317585
$112$	0	0
$113$	11.1392	1.04788	0.523942	−	0.851754i	$-0.324461\pi$
$113$	11.1392	1.04788	0.523942	+	0.851754i	$0.324461\pi$
$114$	0	0
$115$	1.49646	0.139545
$116$	0	0
$117$	54.8435	5.07028
$118$	0	0
$119$	−1.94370	−0.178179
$120$	0	0
$121$	11.5457	1.04961
$122$	0	0
$123$	−21.5722	−1.94510
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	8.84242	0.784638	0.392319	−	0.919829i	$-0.371673\pi$
$127$	8.84242	0.784638	0.392319	+	0.919829i	$0.371673\pi$
$128$	0	0
$129$	−6.50354	−0.572605
$130$	0	0
$131$	−6.15049	−0.537371	−0.268686	−	0.963228i	$-0.586589\pi$
$131$	−6.15049	−0.537371	−0.268686	+	0.963228i	$0.586589\pi$
$132$	0	0
$133$	8.69193	0.753686
$134$	0	0
$135$	−17.3839	−1.49616
$136$	0	0
$137$	10.8945	0.930779	0.465389	−	0.885106i	$-0.345914\pi$
$137$	10.8945	0.930779	0.465389	+	0.885106i	$0.345914\pi$
$138$	0	0
$139$	−1.04921	−0.0889932	−0.0444966	−	0.999010i	$-0.514168\pi$
$139$	−1.04921	−0.0889932	−0.0444966	+	0.999010i	$0.514168\pi$
$140$	0	0
$141$	−6.18838	−0.521156
$142$	0	0
$143$	−31.7748	−2.65714
$144$	0	0
$145$	−3.94370	−0.327506
$146$	0	0
$147$	−0.842420	−0.0694816
$148$	0	0
$149$	−4.18838	−0.343126	−0.171563	−	0.985173i	$-0.554882\pi$
$149$	−4.18838	−0.343126	−0.171563	+	0.985173i	$0.554882\pi$
$150$	0	0
$151$	6.69193	0.544581	0.272291	−	0.962215i	$-0.412219\pi$
$151$	6.69193	0.544581	0.272291	+	0.962215i	$0.412219\pi$
$152$	0	0
$153$	−6.13208	−0.495749
$154$	0	0
$155$	7.79321	0.625965
$156$	0	0
$157$	−3.94370	−0.314741	−0.157371	−	0.987540i	$-0.550302\pi$
$157$	−3.94370	−0.314741	−0.157371	+	0.987540i	$0.550302\pi$
$158$	0	0
$159$	34.9561	2.77220
$160$	0	0
$161$	−3.88740	−0.306370
$162$	0	0
$163$	12.1463	0.951368	0.475684	−	0.879616i	$-0.342201\pi$
$163$	12.1463	0.951368	0.475684	+	0.879616i	$0.342201\pi$
$164$	0	0
$165$	15.8874	1.23683
$166$	0	0
$167$	−17.0829	−1.32191	−0.660956	−	0.750425i	$-0.729849\pi$
$167$	−17.0829	−1.32191	−0.660956	+	0.750425i	$0.729849\pi$
$168$	0	0
$169$	31.7819	2.44476
$170$	0	0
$171$	27.4217	2.09699
$172$	0	0
$173$	−15.3417	−1.16641	−0.583205	−	0.812325i	$-0.698202\pi$
$173$	−15.3417	−1.16641	−0.583205	+	0.812325i	$0.698202\pi$
$174$	0	0
$175$	2.59774	0.196370
$176$	0	0
$177$	−19.5722	−1.47114
$178$	0	0
$179$	−2.45148	−0.183232	−0.0916161	−	0.995794i	$-0.529203\pi$
$179$	−2.45148	−0.183232	−0.0916161	+	0.995794i	$0.529203\pi$
$180$	0	0
$181$	13.1955	0.980812	0.490406	−	0.871494i	$-0.336849\pi$
$181$	13.1955	0.980812	0.490406	+	0.871494i	$0.336849\pi$
$182$	0	0
$183$	26.5793	1.96480
$184$	0	0
$185$	1.00000	0.0735215
$186$	0	0
$187$	3.55276	0.259803
$188$	0	0
$189$	45.1586	3.28481
$190$	0	0
$191$	21.7790	1.57588	0.787938	−	0.615755i	$-0.211149\pi$
$191$	21.7790	1.57588	0.787938	+	0.615755i	$0.211149\pi$
$192$	0	0
$193$	−12.9929	−0.935250	−0.467625	−	0.883927i	$-0.654890\pi$
$193$	−12.9929	−0.935250	−0.467625	+	0.883927i	$0.654890\pi$
$194$	0	0
$195$	−22.3909	−1.60345
$196$	0	0
$197$	0.616147	0.0438986	0.0219493	−	0.999759i	$-0.493013\pi$
$197$	0.616147	0.0438986	0.0219493	+	0.999759i	$0.493013\pi$
$198$	0	0
$199$	1.54852	0.109772	0.0548859	−	0.998493i	$-0.482520\pi$
$199$	1.54852	0.109772	0.0548859	+	0.998493i	$0.482520\pi$
$200$	0	0
$201$	−6.18838	−0.436495
$202$	0	0
$203$	10.2447	0.719036
$204$	0	0
$205$	6.44724	0.450295
$206$	0	0
$207$	−12.2642	−0.852418
$208$	0	0
$209$	−15.8874	−1.09895
$210$	0	0
$211$	−20.9366	−1.44134	−0.720668	−	0.693280i	$-0.756165\pi$
$211$	−20.9366	−1.44134	−0.720668	+	0.693280i	$0.756165\pi$
$212$	0	0
$213$	13.0071	0.891231
$214$	0	0
$215$	1.94370	0.132559
$216$	0	0
$217$	−20.2447	−1.37430
$218$	0	0
$219$	−25.0829	−1.69494
$220$	0	0
$221$	−5.00709	−0.336813
$222$	0	0
$223$	2.71034	0.181498	0.0907488	−	0.995874i	$-0.471074\pi$
$223$	2.71034	0.181498	0.0907488	+	0.995874i	$0.471074\pi$
$224$	0	0
$225$	8.19547	0.546365
$226$	0	0
$227$	−26.1321	−1.73445	−0.867224	−	0.497919i	$-0.834098\pi$
$227$	−26.1321	−1.73445	−0.867224	+	0.497919i	$0.834098\pi$
$228$	0	0
$229$	−24.7677	−1.63670	−0.818348	−	0.574723i	$-0.805110\pi$
$229$	−24.7677	−1.63670	−0.818348	+	0.574723i	$0.805110\pi$
$230$	0	0
$231$	−41.2713	−2.71545
$232$	0	0
$233$	−12.2783	−0.804381	−0.402190	−	0.915556i	$-0.631751\pi$
$233$	−12.2783	−0.804381	−0.402190	+	0.915556i	$0.631751\pi$
$234$	0	0
$235$	1.84951	0.120649
$236$	0	0
$237$	55.3470	3.59518
$238$	0	0
$239$	17.1771	1.11109	0.555546	−	0.831486i	$-0.312509\pi$
$239$	17.1771	1.11109	0.555546	+	0.831486i	$0.312509\pi$
$240$	0	0
$241$	−15.2713	−0.983708	−0.491854	−	0.870678i	$-0.663681\pi$
$241$	−15.2713	−0.983708	−0.491854	+	0.870678i	$0.663681\pi$
$242$	0	0
$243$	60.2036	3.86206
$244$	0	0
$245$	0.251772	0.0160851
$246$	0	0
$247$	22.3909	1.42470
$248$	0	0
$249$	−50.9703	−3.23011
$250$	0	0
$251$	−10.0379	−0.633586	−0.316793	−	0.948495i	$-0.602606\pi$
$251$	−10.0379	−0.633586	−0.316793	+	0.948495i	$0.602606\pi$
$252$	0	0
$253$	7.10552	0.446720
$254$	0	0
$255$	2.50354	0.156778
$256$	0	0
$257$	−20.9703	−1.30809	−0.654045	−	0.756456i	$-0.726929\pi$
$257$	−20.9703	−1.30809	−0.654045	+	0.756456i	$0.726929\pi$
$258$	0	0
$259$	−2.59774	−0.161415
$260$	0	0
$261$	32.3205	2.00059
$262$	0	0
$263$	−22.3725	−1.37955	−0.689775	−	0.724024i	$-0.742290\pi$
$263$	−22.3725	−1.37955	−0.689775	+	0.724024i	$0.742290\pi$
$264$	0	0
$265$	−10.4472	−0.641769
$266$	0	0
$267$	20.0758	1.22862
$268$	0	0
$269$	−4.18838	−0.255370	−0.127685	−	0.991815i	$-0.540755\pi$
$269$	−4.18838	−0.255370	−0.127685	+	0.991815i	$0.540755\pi$
$270$	0	0
$271$	12.1126	0.735788	0.367894	−	0.929868i	$-0.380079\pi$
$271$	12.1126	0.735788	0.367894	+	0.929868i	$0.380079\pi$
$272$	0	0
$273$	58.1657	3.52035
$274$	0	0
$275$	−4.74823	−0.286329
$276$	0	0
$277$	−9.30807	−0.559268	−0.279634	−	0.960107i	$-0.590213\pi$
$277$	−9.30807	−0.559268	−0.279634	+	0.960107i	$0.590213\pi$
$278$	0	0
$279$	−63.8690	−3.82374
$280$	0	0
$281$	19.2713	1.14963	0.574813	−	0.818285i	$-0.305075\pi$
$281$	19.2713	1.14963	0.574813	+	0.818285i	$0.305075\pi$
$282$	0	0
$283$	13.6848	0.813479	0.406740	−	0.913544i	$-0.366666\pi$
$283$	13.6848	0.813479	0.406740	+	0.913544i	$0.366666\pi$
$284$	0	0
$285$	−11.1955	−0.663162
$286$	0	0
$287$	−16.7482	−0.988617
$288$	0	0
$289$	−16.4402	−0.967068
$290$	0	0
$291$	−34.9561	−2.04916
$292$	0	0
$293$	−10.4472	−0.610334	−0.305167	−	0.952299i	$-0.598712\pi$
$293$	−10.4472	−0.610334	−0.305167	+	0.952299i	$0.598712\pi$
$294$	0	0
$295$	5.84951	0.340571
$296$	0	0
$297$	−82.5425	−4.78960
$298$	0	0
$299$	−10.0142	−0.579135
$300$	0	0
$301$	−5.04921	−0.291032
$302$	0	0
$303$	−40.7819	−2.34286
$304$	0	0
$305$	−7.94370	−0.454855
$306$	0	0
$307$	4.95502	0.282798	0.141399	−	0.989953i	$-0.454840\pi$
$307$	4.95502	0.282798	0.141399	+	0.989953i	$0.454840\pi$
$308$	0	0
$309$	4.37677	0.248985
$310$	0	0
$311$	−3.68060	−0.208708	−0.104354	−	0.994540i	$-0.533277\pi$
$311$	−3.68060	−0.208708	−0.104354	+	0.994540i	$0.533277\pi$
$312$	0	0
$313$	−0.992912	−0.0561227	−0.0280614	−	0.999606i	$-0.508933\pi$
$313$	−0.992912	−0.0561227	−0.0280614	+	0.999606i	$0.508933\pi$
$314$	0	0
$315$	−21.2897	−1.19954
$316$	0	0
$317$	−6.55985	−0.368438	−0.184219	−	0.982885i	$-0.558976\pi$
$317$	−6.55985	−0.368438	−0.184219	+	0.982885i	$0.558976\pi$
$318$	0	0
$319$	−18.7256	−1.04843
$320$	0	0
$321$	10.1884	0.568660
$322$	0	0
$323$	−2.50354	−0.139301
$324$	0	0
$325$	6.69193	0.371201
$326$	0	0
$327$	−4.81870	−0.266475
$328$	0	0
$329$	−4.80453	−0.264882
$330$	0	0
$331$	18.0379	0.991452	0.495726	−	0.868479i	$-0.334902\pi$
$331$	18.0379	0.991452	0.495726	+	0.868479i	$0.334902\pi$
$332$	0	0
$333$	−8.19547	−0.449109
$334$	0	0
$335$	1.84951	0.101049
$336$	0	0
$337$	−14.8945	−0.811354	−0.405677	−	0.914016i	$-0.632964\pi$
$337$	−14.8945	−0.811354	−0.405677	+	0.914016i	$0.632964\pi$
$338$	0	0
$339$	37.2713	2.02430
$340$	0	0
$341$	37.0039	2.00387
$342$	0	0
$343$	−18.8382	−1.01717
$344$	0	0
$345$	5.00709	0.269573
$346$	0	0
$347$	9.38385	0.503752	0.251876	−	0.967760i	$-0.418953\pi$
$347$	9.38385	0.503752	0.251876	+	0.967760i	$0.418953\pi$
$348$	0	0
$349$	−32.9703	−1.76486	−0.882429	−	0.470446i	$-0.844093\pi$
$349$	−32.9703	−1.76486	−0.882429	+	0.470446i	$0.844093\pi$
$350$	0	0
$351$	116.331	6.20931
$352$	0	0
$353$	27.4260	1.45974	0.729869	−	0.683587i	$-0.239581\pi$
$353$	27.4260	1.45974	0.729869	+	0.683587i	$0.239581\pi$
$354$	0	0
$355$	−3.88740	−0.206322
$356$	0	0
$357$	−6.50354	−0.344204
$358$	0	0
$359$	25.0829	1.32382	0.661912	−	0.749582i	$-0.269745\pi$
$359$	25.0829	1.32382	0.661912	+	0.749582i	$0.269745\pi$
$360$	0	0
$361$	−7.80453	−0.410765
$362$	0	0
$363$	38.6314	2.02762
$364$	0	0
$365$	7.49646	0.392382
$366$	0	0
$367$	−2.48513	−0.129723	−0.0648614	−	0.997894i	$-0.520661\pi$
$367$	−2.48513	−0.129723	−0.0648614	+	0.997894i	$0.520661\pi$
$368$	0	0
$369$	−52.8382	−2.75065
$370$	0	0
$371$	27.1392	1.40900
$372$	0	0
$373$	−35.1586	−1.82045	−0.910223	−	0.414119i	$-0.864090\pi$
$373$	−35.1586	−1.82045	−0.910223	+	0.414119i	$0.864090\pi$
$374$	0	0
$375$	−3.34596	−0.172785
$376$	0	0
$377$	26.3909	1.35920
$378$	0	0
$379$	−24.6778	−1.26761	−0.633805	−	0.773492i	$-0.718508\pi$
$379$	−24.6778	−1.26761	−0.633805	+	0.773492i	$0.718508\pi$
$380$	0	0
$381$	29.5864	1.51576
$382$	0	0
$383$	13.3839	0.683883	0.341941	−	0.939721i	$-0.388916\pi$
$383$	13.3839	0.683883	0.341941	+	0.939721i	$0.388916\pi$
$384$	0	0
$385$	12.3346	0.628631
$386$	0	0
$387$	−15.9295	−0.809743
$388$	0	0
$389$	18.2220	0.923894	0.461947	−	0.886908i	$-0.347151\pi$
$389$	18.2220	0.923894	0.461947	+	0.886908i	$0.347151\pi$
$390$	0	0
$391$	1.11969	0.0566252
$392$	0	0
$393$	−20.5793	−1.03809
$394$	0	0
$395$	−16.5414	−0.832290
$396$	0	0
$397$	−2.22521	−0.111680	−0.0558399	−	0.998440i	$-0.517784\pi$
$397$	−2.22521	−0.111680	−0.0558399	+	0.998440i	$0.517784\pi$
$398$	0	0
$399$	29.0829	1.45596
$400$	0	0
$401$	12.3909	0.618774	0.309387	−	0.950936i	$-0.399876\pi$
$401$	12.3909	0.618774	0.309387	+	0.950936i	$0.399876\pi$
$402$	0	0
$403$	−52.1516	−2.59785
$404$	0	0
$405$	−33.5793	−1.66857
$406$	0	0
$407$	4.74823	0.235361
$408$	0	0
$409$	7.27125	0.359540	0.179770	−	0.983709i	$-0.442465\pi$
$409$	7.27125	0.359540	0.179770	+	0.983709i	$0.442465\pi$
$410$	0	0
$411$	36.4525	1.79807
$412$	0	0
$413$	−15.1955	−0.747720
$414$	0	0
$415$	15.2334	0.747776
$416$	0	0
$417$	−3.51063	−0.171916
$418$	0	0
$419$	−28.4809	−1.39138	−0.695691	−	0.718341i	$-0.744902\pi$
$419$	−28.4809	−1.39138	−0.695691	+	0.718341i	$0.744902\pi$
$420$	0	0
$421$	22.0000	1.07221	0.536107	−	0.844150i	$-0.319894\pi$
$421$	22.0000	1.07221	0.536107	+	0.844150i	$0.319894\pi$
$422$	0	0
$423$	−15.1576	−0.736987
$424$	0	0
$425$	−0.748228	−0.0362944
$426$	0	0
$427$	20.6356	0.998628
$428$	0	0
$429$	−106.317	−5.13305
$430$	0	0
$431$	−1.51487	−0.0729686	−0.0364843	−	0.999334i	$-0.511616\pi$
$431$	−1.51487	−0.0729686	−0.0364843	+	0.999334i	$0.511616\pi$
$432$	0	0
$433$	24.9929	1.20108	0.600541	−	0.799594i	$-0.294952\pi$
$433$	24.9929	1.20108	0.600541	+	0.799594i	$0.294952\pi$
$434$	0	0
$435$	−13.1955	−0.632674
$436$	0	0
$437$	−5.00709	−0.239521
$438$	0	0
$439$	5.10128	0.243471	0.121735	−	0.992563i	$-0.461154\pi$
$439$	5.10128	0.243471	0.121735	+	0.992563i	$0.461154\pi$
$440$	0	0
$441$	−2.06339	−0.0982566
$442$	0	0
$443$	−25.7369	−1.22280	−0.611399	−	0.791323i	$-0.709393\pi$
$443$	−25.7369	−1.22280	−0.611399	+	0.791323i	$0.709393\pi$
$444$	0	0
$445$	−6.00000	−0.284427
$446$	0	0
$447$	−14.0142	−0.662848
$448$	0	0
$449$	−17.7748	−0.838844	−0.419422	−	0.907791i	$-0.637767\pi$
$449$	−17.7748	−0.838844	−0.419422	+	0.907791i	$0.637767\pi$
$450$	0	0
$451$	30.6130	1.44151
$452$	0	0
$453$	22.3909	1.05202
$454$	0	0
$455$	−17.3839	−0.814968
$456$	0	0
$457$	34.3346	1.60611	0.803053	−	0.595907i	$-0.203207\pi$
$457$	34.3346	1.60611	0.803053	+	0.595907i	$0.203207\pi$
$458$	0	0
$459$	−13.0071	−0.607119
$460$	0	0
$461$	−16.9508	−0.789477	−0.394738	−	0.918794i	$-0.629165\pi$
$461$	−16.9508	−0.789477	−0.394738	+	0.918794i	$0.629165\pi$
$462$	0	0
$463$	−10.9929	−0.510884	−0.255442	−	0.966824i	$-0.582221\pi$
$463$	−10.9929	−0.510884	−0.255442	+	0.966824i	$0.582221\pi$
$464$	0	0
$465$	26.0758	1.20924
$466$	0	0
$467$	−37.4175	−1.73148	−0.865738	−	0.500498i	$-0.833150\pi$
$467$	−37.4175	−1.73148	−0.865738	+	0.500498i	$0.833150\pi$
$468$	0	0
$469$	−4.80453	−0.221853
$470$	0	0
$471$	−13.1955	−0.608015
$472$	0	0
$473$	9.22912	0.424356
$474$	0	0
$475$	3.34596	0.153523
$476$	0	0
$477$	85.6201	3.92027
$478$	0	0
$479$	0.465654	0.0212763	0.0106381	−	0.999943i	$-0.496614\pi$
$479$	0.465654	0.0212763	0.0106381	+	0.999943i	$0.496614\pi$
$480$	0	0
$481$	−6.69193	−0.305126
$482$	0	0
$483$	−13.0071	−0.591843
$484$	0	0
$485$	10.4472	0.474385
$486$	0	0
$487$	−34.9561	−1.58401	−0.792006	−	0.610514i	$-0.790963\pi$
$487$	−34.9561	−1.58401	−0.792006	+	0.610514i	$0.790963\pi$
$488$	0	0
$489$	40.6409	1.83785
$490$	0	0
$491$	24.7819	1.11839	0.559195	−	0.829036i	$-0.311110\pi$
$491$	24.7819	1.11839	0.559195	+	0.829036i	$0.311110\pi$
$492$	0	0
$493$	−2.95079	−0.132897
$494$	0	0
$495$	38.9140	1.74905
$496$	0	0
$497$	10.0984	0.452976
$498$	0	0
$499$	2.33888	0.104702	0.0523512	−	0.998629i	$-0.483328\pi$
$499$	2.33888	0.104702	0.0523512	+	0.998629i	$0.483328\pi$
$500$	0	0
$501$	−57.1586	−2.55366
$502$	0	0
$503$	18.6161	0.830053	0.415026	−	0.909809i	$-0.363772\pi$
$503$	18.6161	0.830053	0.415026	+	0.909809i	$0.363772\pi$
$504$	0	0
$505$	12.1884	0.542376
$506$	0	0
$507$	106.341	4.72277
$508$	0	0
$509$	6.89448	0.305593	0.152796	−	0.988258i	$-0.451172\pi$
$509$	6.89448	0.305593	0.152796	+	0.988258i	$0.451172\pi$
$510$	0	0
$511$	−19.4738	−0.861471
$512$	0	0
$513$	58.1657	2.56808
$514$	0	0
$515$	−1.30807	−0.0576406
$516$	0	0
$517$	8.78188	0.386227
$518$	0	0
$519$	−51.3329	−2.25326
$520$	0	0
$521$	4.86083	0.212957	0.106478	−	0.994315i	$-0.466042\pi$
$521$	4.86083	0.212957	0.106478	+	0.994315i	$0.466042\pi$
$522$	0	0
$523$	−22.7677	−0.995562	−0.497781	−	0.867303i	$-0.665852\pi$
$523$	−22.7677	−0.995562	−0.497781	+	0.867303i	$0.665852\pi$
$524$	0	0
$525$	8.69193	0.379347
$526$	0	0
$527$	5.83110	0.254007
$528$	0	0
$529$	−20.7606	−0.902636
$530$	0	0
$531$	−47.9395	−2.08040
$532$	0	0
$533$	−43.1445	−1.86879
$534$	0	0
$535$	−3.04498	−0.131646
$536$	0	0
$537$	−8.20256	−0.353967
$538$	0	0
$539$	1.19547	0.0514926
$540$	0	0
$541$	8.99291	0.386636	0.193318	−	0.981136i	$-0.438075\pi$
$541$	8.99291	0.386636	0.193318	+	0.981136i	$0.438075\pi$
$542$	0	0
$543$	44.1516	1.89472
$544$	0	0
$545$	1.44015	0.0616894
$546$	0	0
$547$	18.8382	0.805463	0.402731	−	0.915318i	$-0.368061\pi$
$547$	18.8382	0.805463	0.402731	+	0.915318i	$0.368061\pi$
$548$	0	0
$549$	65.1023	2.77850
$550$	0	0
$551$	13.1955	0.562146
$552$	0	0
$553$	42.9703	1.82728
$554$	0	0
$555$	3.34596	0.142028
$556$	0	0
$557$	3.69901	0.156732	0.0783661	−	0.996925i	$-0.475030\pi$
$557$	3.69901	0.156732	0.0783661	+	0.996925i	$0.475030\pi$
$558$	0	0
$559$	−13.0071	−0.550141
$560$	0	0
$561$	11.8874	0.501886
$562$	0	0
$563$	14.3488	0.604730	0.302365	−	0.953192i	$-0.402224\pi$
$563$	14.3488	0.604730	0.302365	+	0.953192i	$0.402224\pi$
$564$	0	0
$565$	−11.1392	−0.468628
$566$	0	0
$567$	87.2302	3.66332
$568$	0	0
$569$	15.9858	0.670161	0.335080	−	0.942190i	$-0.391237\pi$
$569$	15.9858	0.670161	0.335080	+	0.942190i	$0.391237\pi$
$570$	0	0
$571$	30.2079	1.26416	0.632080	−	0.774903i	$-0.282201\pi$
$571$	30.2079	1.26416	0.632080	+	0.774903i	$0.282201\pi$
$572$	0	0
$573$	72.8718	3.04426
$574$	0	0
$575$	−1.49646	−0.0624065
$576$	0	0
$577$	−19.1813	−0.798528	−0.399264	−	0.916836i	$-0.630734\pi$
$577$	−19.1813	−0.798528	−0.399264	+	0.916836i	$0.630734\pi$
$578$	0	0
$579$	−43.4738	−1.80671
$580$	0	0
$581$	−39.5722	−1.64173
$582$	0	0
$583$	−49.6059	−2.05447
$584$	0	0
$585$	−54.8435	−2.26750
$586$	0	0
$587$	26.6130	1.09844	0.549218	−	0.835679i	$-0.314926\pi$
$587$	26.6130	1.09844	0.549218	+	0.835679i	$0.314926\pi$
$588$	0	0
$589$	−26.0758	−1.07443
$590$	0	0
$591$	2.06160	0.0848031
$592$	0	0
$593$	26.2642	1.07854	0.539270	−	0.842133i	$-0.318700\pi$
$593$	26.2642	1.07854	0.539270	+	0.842133i	$0.318700\pi$
$594$	0	0
$595$	1.94370	0.0796839
$596$	0	0
$597$	5.18130	0.212056
$598$	0	0
$599$	38.2415	1.56251	0.781253	−	0.624215i	$-0.214581\pi$
$599$	38.2415	1.56251	0.781253	+	0.624215i	$0.214581\pi$
$600$	0	0
$601$	30.2447	1.23371	0.616853	−	0.787078i	$-0.288407\pi$
$601$	30.2447	1.23371	0.616853	+	0.787078i	$0.288407\pi$
$602$	0	0
$603$	−15.1576	−0.617264
$604$	0	0
$605$	−11.5457	−0.469398
$606$	0	0
$607$	5.90157	0.239537	0.119769	−	0.992802i	$-0.461785\pi$
$607$	5.90157	0.239537	0.119769	+	0.992802i	$0.461785\pi$
$608$	0	0
$609$	34.2783	1.38903
$610$	0	0
$611$	−12.3768	−0.500710
$612$	0	0
$613$	6.15473	0.248587	0.124294	−	0.992245i	$-0.460334\pi$
$613$	6.15473	0.248587	0.124294	+	0.992245i	$0.460334\pi$
$614$	0	0
$615$	21.5722	0.869877
$616$	0	0
$617$	16.5035	0.664408	0.332204	−	0.943208i	$-0.392208\pi$
$617$	16.5035	0.664408	0.332204	+	0.943208i	$0.392208\pi$
$618$	0	0
$619$	−9.04921	−0.363719	−0.181859	−	0.983325i	$-0.558212\pi$
$619$	−9.04921	−0.363719	−0.181859	+	0.983325i	$0.558212\pi$
$620$	0	0
$621$	−26.0142	−1.04391
$622$	0	0
$623$	15.5864	0.624456
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−53.1586	−2.12295
$628$	0	0
$629$	0.748228	0.0298338
$630$	0	0
$631$	0.507780	0.0202144	0.0101072	−	0.999949i	$-0.496783\pi$
$631$	0.507780	0.0202144	0.0101072	+	0.999949i	$0.496783\pi$
$632$	0	0
$633$	−70.0531	−2.78436
$634$	0	0
$635$	−8.84242	−0.350901
$636$	0	0
$637$	−1.68484	−0.0667558
$638$	0	0
$639$	31.8590	1.26032
$640$	0	0
$641$	−40.6356	−1.60501	−0.802505	−	0.596645i	$-0.796500\pi$
$641$	−40.6356	−1.60501	−0.802505	+	0.596645i	$0.796500\pi$
$642$	0	0
$643$	9.53011	0.375831	0.187915	−	0.982185i	$-0.439827\pi$
$643$	9.53011	0.375831	0.187915	+	0.982185i	$0.439827\pi$
$644$	0	0
$645$	6.50354	0.256077
$646$	0	0
$647$	21.0071	0.825874	0.412937	−	0.910760i	$-0.364503\pi$
$647$	21.0071	0.825874	0.412937	+	0.910760i	$0.364503\pi$
$648$	0	0
$649$	27.7748	1.09026
$650$	0	0
$651$	−67.7380	−2.65486
$652$	0	0
$653$	−45.5496	−1.78249	−0.891247	−	0.453519i	$-0.850168\pi$
$653$	−45.5496	−1.78249	−0.891247	+	0.453519i	$0.850168\pi$
$654$	0	0
$655$	6.15049	0.240320
$656$	0	0
$657$	−61.4370	−2.39689
$658$	0	0
$659$	−9.38385	−0.365543	−0.182772	−	0.983155i	$-0.558507\pi$
$659$	−9.38385	−0.365543	−0.182772	+	0.983155i	$0.558507\pi$
$660$	0	0
$661$	4.05630	0.157772	0.0788859	−	0.996884i	$-0.474864\pi$
$661$	4.05630	0.157772	0.0788859	+	0.996884i	$0.474864\pi$
$662$	0	0
$663$	−16.7535	−0.650653
$664$	0	0
$665$	−8.69193	−0.337058
$666$	0	0
$667$	−5.90157	−0.228510
$668$	0	0
$669$	9.06869	0.350616
$670$	0	0
$671$	−37.7185	−1.45611
$672$	0	0
$673$	7.60906	0.293308	0.146654	−	0.989188i	$-0.453150\pi$
$673$	7.60906	0.293308	0.146654	+	0.989188i	$0.453150\pi$
$674$	0	0
$675$	17.3839	0.669105
$676$	0	0
$677$	28.7677	1.10563	0.552816	−	0.833303i	$-0.313553\pi$
$677$	28.7677	1.10563	0.552816	+	0.833303i	$0.313553\pi$
$678$	0	0
$679$	−27.1392	−1.04151
$680$	0	0
$681$	−87.4370	−3.35059
$682$	0	0
$683$	−0.0704767	−0.00269671	−0.00134836	−	0.999999i	$-0.500429\pi$
$683$	−0.0704767	−0.00269671	−0.00134836	+	0.999999i	$0.500429\pi$
$684$	0	0
$685$	−10.8945	−0.416257
$686$	0	0
$687$	−82.8718	−3.16176
$688$	0	0
$689$	69.9122	2.66344
$690$	0	0
$691$	−10.6498	−0.405137	−0.202569	−	0.979268i	$-0.564929\pi$
$691$	−10.6498	−0.405137	−0.202569	+	0.979268i	$0.564929\pi$
$692$	0	0
$693$	−101.088	−3.84002
$694$	0	0
$695$	1.04921	0.0397990
$696$	0	0
$697$	4.82401	0.182722
$698$	0	0
$699$	−41.0829	−1.55390
$700$	0	0
$701$	−7.98582	−0.301620	−0.150810	−	0.988563i	$-0.548188\pi$
$701$	−7.98582	−0.301620	−0.150810	+	0.988563i	$0.548188\pi$
$702$	0	0
$703$	−3.34596	−0.126195
$704$	0	0
$705$	6.18838	0.233068
$706$	0	0
$707$	−31.6622	−1.19078
$708$	0	0
$709$	−37.2149	−1.39764	−0.698818	−	0.715299i	$-0.746290\pi$
$709$	−37.2149	−1.39764	−0.698818	+	0.715299i	$0.746290\pi$
$710$	0	0
$711$	135.565	5.08408
$712$	0	0
$713$	11.6622	0.436752
$714$	0	0
$715$	31.7748	1.18831
$716$	0	0
$717$	57.4738	2.14640
$718$	0	0
$719$	−30.1742	−1.12531	−0.562654	−	0.826692i	$-0.690220\pi$
$719$	−30.1742	−1.12531	−0.562654	+	0.826692i	$0.690220\pi$
$720$	0	0
$721$	3.39803	0.126549
$722$	0	0
$723$	−51.0970	−1.90032
$724$	0	0
$725$	3.94370	0.146465
$726$	0	0
$727$	18.7677	0.696056	0.348028	−	0.937484i	$-0.386851\pi$
$727$	18.7677	0.696056	0.348028	+	0.937484i	$0.386851\pi$
$728$	0	0
$729$	100.701	3.72967
$730$	0	0
$731$	1.45433	0.0537903
$732$	0	0
$733$	26.1094	0.964374	0.482187	−	0.876068i	$-0.339843\pi$
$733$	26.1094	0.964374	0.482187	+	0.876068i	$0.339843\pi$
$734$	0	0
$735$	0.842420	0.0310731
$736$	0	0
$737$	8.78188	0.323485
$738$	0	0
$739$	−42.6130	−1.56754	−0.783772	−	0.621049i	$-0.786707\pi$
$739$	−42.6130	−1.56754	−0.783772	+	0.621049i	$0.786707\pi$
$740$	0	0
$741$	74.9193	2.75223
$742$	0	0
$743$	35.4922	1.30208	0.651042	−	0.759042i	$-0.274332\pi$
$743$	35.4922	1.30208	0.651042	+	0.759042i	$0.274332\pi$
$744$	0	0
$745$	4.18838	0.153450
$746$	0	0
$747$	−124.845	−4.56782
$748$	0	0
$749$	7.91005	0.289027
$750$	0	0
$751$	−21.5722	−0.787182	−0.393591	−	0.919286i	$-0.628767\pi$
$751$	−21.5722	−0.787182	−0.393591	+	0.919286i	$0.628767\pi$
$752$	0	0
$753$	−33.5864	−1.22396
$754$	0	0
$755$	−6.69193	−0.243544
$756$	0	0
$757$	−23.7890	−0.864625	−0.432312	−	0.901724i	$-0.642302\pi$
$757$	−23.7890	−0.864625	−0.432312	+	0.901724i	$0.642302\pi$
$758$	0	0
$759$	23.7748	0.862970
$760$	0	0
$761$	−18.6724	−0.676876	−0.338438	−	0.940989i	$-0.609898\pi$
$761$	−18.6724	−0.676876	−0.338438	+	0.940989i	$0.609898\pi$
$762$	0	0
$763$	−3.74114	−0.135438
$764$	0	0
$765$	6.13208	0.221706
$766$	0	0
$767$	−39.1445	−1.41342
$768$	0	0
$769$	−2.48937	−0.0897689	−0.0448845	−	0.998992i	$-0.514292\pi$
$769$	−2.48937	−0.0897689	−0.0448845	+	0.998992i	$0.514292\pi$
$770$	0	0
$771$	−70.1657	−2.52696
$772$	0	0
$773$	0.950786	0.0341974	0.0170987	−	0.999854i	$-0.494557\pi$
$773$	0.950786	0.0341974	0.0170987	+	0.999854i	$0.494557\pi$
$774$	0	0
$775$	−7.79321	−0.279940
$776$	0	0
$777$	−8.69193	−0.311821
$778$	0	0
$779$	−21.5722	−0.772906
$780$	0	0
$781$	−18.4582	−0.660488
$782$	0	0
$783$	68.5567	2.45002
$784$	0	0
$785$	3.94370	0.140757
$786$	0	0
$787$	29.4359	1.04928	0.524639	−	0.851325i	$-0.324200\pi$
$787$	29.4359	1.04928	0.524639	+	0.851325i	$0.324200\pi$
$788$	0	0
$789$	−74.8577	−2.66500
$790$	0	0
$791$	28.9366	1.02887
$792$	0	0
$793$	53.1586	1.88772
$794$	0	0
$795$	−34.9561	−1.23976
$796$	0	0
$797$	−14.1742	−0.502076	−0.251038	−	0.967977i	$-0.580772\pi$
$797$	−14.1742	−0.502076	−0.251038	+	0.967977i	$0.580772\pi$
$798$	0	0
$799$	1.38385	0.0489572
$800$	0	0
$801$	49.1728	1.73744
$802$	0	0
$803$	35.5949	1.25612
$804$	0	0
$805$	3.88740	0.137013
$806$	0	0
$807$	−14.0142	−0.493322
$808$	0	0
$809$	−32.9929	−1.15997	−0.579985	−	0.814627i	$-0.696941\pi$
$809$	−32.9929	−1.15997	−0.579985	+	0.814627i	$0.696941\pi$
$810$	0	0
$811$	−40.5567	−1.42414	−0.712069	−	0.702110i	$-0.752242\pi$
$811$	−40.5567	−1.42414	−0.712069	+	0.702110i	$0.752242\pi$
$812$	0	0
$813$	40.5283	1.42139
$814$	0	0
$815$	−12.1463	−0.425465
$816$	0	0
$817$	−6.50354	−0.227530
$818$	0	0
$819$	142.469	4.97826
$820$	0	0
$821$	−38.8661	−1.35644	−0.678219	−	0.734860i	$-0.737248\pi$
$821$	−38.8661	−1.35644	−0.678219	+	0.734860i	$0.737248\pi$
$822$	0	0
$823$	49.6243	1.72979	0.864897	−	0.501949i	$-0.167384\pi$
$823$	49.6243	1.72979	0.864897	+	0.501949i	$0.167384\pi$
$824$	0	0
$825$	−15.8874	−0.553128
$826$	0	0
$827$	30.0195	1.04388	0.521940	−	0.852982i	$-0.325209\pi$
$827$	30.0195	1.04388	0.521940	+	0.852982i	$0.325209\pi$
$828$	0	0
$829$	10.5598	0.366759	0.183379	−	0.983042i	$-0.441296\pi$
$829$	10.5598	0.366759	0.183379	+	0.983042i	$0.441296\pi$
$830$	0	0
$831$	−31.1445	−1.08039
$832$	0	0
$833$	0.188383	0.00652708
$834$	0	0
$835$	17.0829	0.591177
$836$	0	0
$837$	−135.476	−4.68273
$838$	0	0
$839$	−29.3839	−1.01444	−0.507222	−	0.861816i	$-0.669327\pi$
$839$	−29.3839	−1.01444	−0.507222	+	0.861816i	$0.669327\pi$
$840$	0	0
$841$	−13.4472	−0.463698
$842$	0	0
$843$	64.4809	2.22084
$844$	0	0
$845$	−31.7819	−1.09333
$846$	0	0
$847$	29.9926	1.03056
$848$	0	0
$849$	45.7890	1.57147
$850$	0	0
$851$	1.49646	0.0512979
$852$	0	0
$853$	6.22521	0.213147	0.106573	−	0.994305i	$-0.466012\pi$
$853$	6.22521	0.213147	0.106573	+	0.994305i	$0.466012\pi$
$854$	0	0
$855$	−27.4217	−0.937804
$856$	0	0
$857$	−3.65119	−0.124722	−0.0623611	−	0.998054i	$-0.519863\pi$
$857$	−3.65119	−0.124722	−0.0623611	+	0.998054i	$0.519863\pi$
$858$	0	0
$859$	36.3162	1.23909	0.619547	−	0.784960i	$-0.287316\pi$
$859$	36.3162	1.23909	0.619547	+	0.784960i	$0.287316\pi$
$860$	0	0
$861$	−56.0390	−1.90980
$862$	0	0
$863$	−24.5751	−0.836546	−0.418273	−	0.908321i	$-0.637364\pi$
$863$	−24.5751	−0.836546	−0.418273	+	0.908321i	$0.637364\pi$
$864$	0	0
$865$	15.3417	0.521634
$866$	0	0
$867$	−55.0082	−1.86817
$868$	0	0
$869$	−78.5425	−2.66437
$870$	0	0
$871$	−12.3768	−0.419371
$872$	0	0
$873$	−85.6201	−2.89780
$874$	0	0
$875$	−2.59774	−0.0878195
$876$	0	0
$877$	20.0563	0.677253	0.338627	−	0.940921i	$-0.390038\pi$
$877$	20.0563	0.677253	0.338627	+	0.940921i	$0.390038\pi$
$878$	0	0
$879$	−34.9561	−1.17904
$880$	0	0
$881$	−15.0266	−0.506258	−0.253129	−	0.967433i	$-0.581460\pi$
$881$	−15.0266	−0.506258	−0.253129	+	0.967433i	$0.581460\pi$
$882$	0	0
$883$	6.62145	0.222830	0.111415	−	0.993774i	$-0.464462\pi$
$883$	6.62145	0.222830	0.111415	+	0.993774i	$0.464462\pi$
$884$	0	0
$885$	19.5722	0.657914
$886$	0	0
$887$	37.5538	1.26093	0.630467	−	0.776216i	$-0.282863\pi$
$887$	37.5538	1.26093	0.630467	+	0.776216i	$0.282863\pi$
$888$	0	0
$889$	22.9703	0.770398
$890$	0	0
$891$	−159.442	−5.34152
$892$	0	0
$893$	−6.18838	−0.207086
$894$	0	0
$895$	2.45148	0.0819439
$896$	0	0
$897$	−33.5071	−1.11877
$898$	0	0
$899$	−30.7341	−1.02504
$900$	0	0
$901$	−7.81692	−0.260419
$902$	0	0
$903$	−16.8945	−0.562213
$904$	0	0
$905$	−13.1955	−0.438632
$906$	0	0
$907$	−50.4667	−1.67572	−0.837860	−	0.545885i	$-0.816193\pi$
$907$	−50.4667	−1.67572	−0.837860	+	0.545885i	$0.816193\pi$
$908$	0	0
$909$	−99.8895	−3.31313
$910$	0	0
$911$	−15.9395	−0.528098	−0.264049	−	0.964509i	$-0.585058\pi$
$911$	−15.9395	−0.528098	−0.264049	+	0.964509i	$0.585058\pi$
$912$	0	0
$913$	72.3315	2.39382
$914$	0	0
$915$	−26.5793	−0.878685
$916$	0	0
$917$	−15.9774	−0.527619
$918$	0	0
$919$	−32.5414	−1.07344	−0.536721	−	0.843760i	$-0.680337\pi$
$919$	−32.5414	−1.07344	−0.536721	+	0.843760i	$0.680337\pi$
$920$	0	0
$921$	16.5793	0.546307
$922$	0	0
$923$	26.0142	0.856267
$924$	0	0
$925$	−1.00000	−0.0328798
$926$	0	0
$927$	10.7203	0.352100
$928$	0	0
$929$	−34.1094	−1.11909	−0.559547	−	0.828799i	$-0.689025\pi$
$929$	−34.1094	−1.11909	−0.559547	+	0.828799i	$0.689025\pi$
$930$	0	0
$931$	−0.842420	−0.0276092
$932$	0	0
$933$	−12.3152	−0.403180
$934$	0	0
$935$	−3.55276	−0.116188
$936$	0	0
$937$	0.0141751	0.000463082 0	0.000231541	−	1.00000i	$-0.499926\pi$
$937$	0.0141751	0.000463082 0	0.000231541		1.00000i	$0.499926\pi$
$938$	0	0
$939$	−3.32225	−0.108417
$940$	0	0
$941$	11.1813	0.364500	0.182250	−	0.983252i	$-0.441662\pi$
$941$	11.1813	0.364500	0.182250	+	0.983252i	$0.441662\pi$
$942$	0	0
$943$	9.64802	0.314183
$944$	0	0
$945$	−45.1586	−1.46901
$946$	0	0
$947$	53.6427	1.74315	0.871577	−	0.490259i	$-0.163098\pi$
$947$	53.6427	1.74315	0.871577	+	0.490259i	$0.163098\pi$
$948$	0	0
$949$	−50.1657	−1.62845
$950$	0	0
$951$	−21.9490	−0.711745
$952$	0	0
$953$	−9.88740	−0.320284	−0.160142	−	0.987094i	$-0.551195\pi$
$953$	−9.88740	−0.320284	−0.160142	+	0.987094i	$0.551195\pi$
$954$	0	0
$955$	−21.7790	−0.704753
$956$	0	0
$957$	−62.6551	−2.02535
$958$	0	0
$959$	28.3010	0.913886
$960$	0	0
$961$	29.7341	0.959163
$962$	0	0
$963$	24.9550	0.804164
$964$	0	0
$965$	12.9929	0.418257
$966$	0	0
$967$	−51.6254	−1.66016	−0.830080	−	0.557644i	$-0.811705\pi$
$967$	−51.6254	−1.66016	−0.830080	+	0.557644i	$0.811705\pi$
$968$	0	0
$969$	−8.37677	−0.269100
$970$	0	0
$971$	−20.5598	−0.659797	−0.329898	−	0.944016i	$-0.607014\pi$
$971$	−20.5598	−0.659797	−0.329898	+	0.944016i	$0.607014\pi$
$972$	0	0
$973$	−2.72558	−0.0873781
$974$	0	0
$975$	22.3909	0.717084
$976$	0	0
$977$	38.4472	1.23004	0.615018	−	0.788513i	$-0.289149\pi$
$977$	38.4472	1.23004	0.615018	+	0.788513i	$0.289149\pi$
$978$	0	0
$979$	−28.4894	−0.910524
$980$	0	0
$981$	−11.8027	−0.376833
$982$	0	0
$983$	−11.8973	−0.379466	−0.189733	−	0.981836i	$-0.560762\pi$
$983$	−11.8973	−0.379466	−0.189733	+	0.981836i	$0.560762\pi$
$984$	0	0
$985$	−0.616147	−0.0196321
$986$	0	0
$987$	−16.0758	−0.511698
$988$	0	0
$989$	2.90866	0.0924900
$990$	0	0
$991$	19.7932	0.628752	0.314376	−	0.949299i	$-0.398205\pi$
$991$	19.7932	0.628752	0.314376	+	0.949299i	$0.398205\pi$
$992$	0	0
$993$	60.3541	1.91528
$994$	0	0
$995$	−1.54852	−0.0490914
$996$	0	0
$997$	5.88740	0.186456	0.0932279	−	0.995645i	$-0.470281\pi$
$997$	5.88740	0.186456	0.0932279	+	0.995645i	$0.470281\pi$
$998$	0	0
$999$	−17.3839	−0.550001

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2960.2.a.u.1.3		3
4.3	odd	2		370.2.a.g.1.1	✓	3
12.11	even	2		3330.2.a.bg.1.1		3
20.3	even	4		1850.2.b.o.149.1		6
20.7	even	4		1850.2.b.o.149.6		6
20.19	odd	2		1850.2.a.z.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
370.2.a.g.1.1	✓	3	4.3	odd	2
1850.2.a.z.1.3		3	20.19	odd	2
1850.2.b.o.149.1		6	20.3	even	4
1850.2.b.o.149.6		6	20.7	even	4
2960.2.a.u.1.3		3	1.1	even	1	trivial
3330.2.a.bg.1.1		3	12.11	even	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)

\(f(q)\)	\(=\)	\(q+3.34596 q^{3} -1.00000 q^{5} +2.59774 q^{7} +8.19547 q^{9} +O(q^{10})\)	\(q+3.34596 q^{3} -1.00000 q^{5} +2.59774 q^{7} +8.19547 q^{9} -4.74823 q^{11} +6.69193 q^{13} -3.34596 q^{15} -0.748228 q^{17} +3.34596 q^{19} +8.69193 q^{21} -1.49646 q^{23} +1.00000 q^{25} +17.3839 q^{27} +3.94370 q^{29} -7.79321 q^{31} -15.8874 q^{33} -2.59774 q^{35} -1.00000 q^{37} +22.3909 q^{39} -6.44724 q^{41} -1.94370 q^{43} -8.19547 q^{45} -1.84951 q^{47} -0.251772 q^{49} -2.50354 q^{51} +10.4472 q^{53} +4.74823 q^{55} +11.1955 q^{57} -5.84951 q^{59} +7.94370 q^{61} +21.2897 q^{63} -6.69193 q^{65} -1.84951 q^{67} -5.00709 q^{69} +3.88740 q^{71} -7.49646 q^{73} +3.34596 q^{75} -12.3346 q^{77} +16.5414 q^{79} +33.5793 q^{81} -15.2334 q^{83} +0.748228 q^{85} +13.1955 q^{87} +6.00000 q^{89} +17.3839 q^{91} -26.0758 q^{93} -3.34596 q^{95} -10.4472 q^{97} -38.9140 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{5} + q^{7} + 11 q^{9}+O(q^{10}) \) 3 * q - 3 * q^5 + q^7 + 11 * q^9	\( 3 q - 3 q^{5} + q^{7} + 11 q^{9} - 11 q^{11} + q^{17} + 6 q^{21} + 2 q^{23} + 3 q^{25} + 12 q^{27} - 5 q^{29} - 3 q^{31} - 14 q^{33} - q^{35} - 3 q^{37} + 40 q^{39} - 9 q^{41} + 11 q^{43} - 11 q^{45} - 2 q^{47} - 4 q^{49} - 14 q^{51} + 21 q^{53} + 11 q^{55} + 20 q^{57} - 14 q^{59} + 7 q^{61} + 37 q^{63} - 2 q^{67} - 28 q^{69} - 22 q^{71} - 16 q^{73} + 7 q^{77} + 26 q^{79} + 47 q^{81} - 2 q^{83} - q^{85} + 26 q^{87} + 18 q^{89} + 12 q^{91} - 18 q^{93} - 21 q^{97} - 19 q^{99}+O(q^{100}) \) 3 * q - 3 * q^5 + q^7 + 11 * q^9 - 11 * q^11 + q^17 + 6 * q^21 + 2 * q^23 + 3 * q^25 + 12 * q^27 - 5 * q^29 - 3 * q^31 - 14 * q^33 - q^35 - 3 * q^37 + 40 * q^39 - 9 * q^41 + 11 * q^43 - 11 * q^45 - 2 * q^47 - 4 * q^49 - 14 * q^51 + 21 * q^53 + 11 * q^55 + 20 * q^57 - 14 * q^59 + 7 * q^61 + 37 * q^63 - 2 * q^67 - 28 * q^69 - 22 * q^71 - 16 * q^73 + 7 * q^77 + 26 * q^79 + 47 * q^81 - 2 * q^83 - q^85 + 26 * q^87 + 18 * q^89 + 12 * q^91 - 18 * q^93 - 21 * q^97 - 19 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more