LMFDB - Embedded newform 2960.2.a.r.1.1

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:	$1$
Dimension:	$3$
Coefficient field:	3.3.316.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 4x + 2 $ x^3 - x^2 - 4*x + 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
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			1.1
Root			$2.34292$ 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)

$f(q)$	$=$	$q-2.34292 q^{3} -1.00000 q^{5} +2.34292 q^{7} +2.48929 q^{9} +O(q^{10})$	$q-2.34292 q^{3} -1.00000 q^{5} +2.34292 q^{7} +2.48929 q^{9} +3.48929 q^{11} -4.68585 q^{13} +2.34292 q^{15} -2.29273 q^{17} -1.14637 q^{19} -5.48929 q^{21} -2.97858 q^{23} +1.00000 q^{25} +1.19656 q^{27} -2.00000 q^{29} +10.5181 q^{31} -8.17513 q^{33} -2.34292 q^{35} -1.00000 q^{37} +10.9786 q^{39} +2.17513 q^{41} -6.97858 q^{43} -2.48929 q^{45} +2.34292 q^{47} -1.51071 q^{49} +5.37169 q^{51} -0.217980 q^{53} -3.48929 q^{55} +2.68585 q^{57} -4.22533 q^{59} +0.978577 q^{61} +5.83221 q^{63} +4.68585 q^{65} -2.85363 q^{67} +6.97858 q^{69} +12.1751 q^{71} -1.82487 q^{73} -2.34292 q^{75} +8.17513 q^{77} -8.22533 q^{79} -10.2713 q^{81} +13.9070 q^{83} +2.29273 q^{85} +4.68585 q^{87} +0.393115 q^{89} -10.9786 q^{91} -24.6430 q^{93} +1.14637 q^{95} +18.3503 q^{97} +8.68585 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - q^{3} - 3 q^{5} + q^{7}+O(q^{10}) $ 3 * q - q^3 - 3 * q^5 + q^7	$ 3 q - q^{3} - 3 q^{5} + q^{7} + 3 q^{11} - 2 q^{13} + q^{15} - 4 q^{17} - 2 q^{19} - 9 q^{21} + 6 q^{23} + 3 q^{25} - q^{27} - 6 q^{29} + 6 q^{31} - 5 q^{33} - q^{35} - 3 q^{37} + 18 q^{39} - 13 q^{41} - 6 q^{43} + q^{47} - 12 q^{49} - 8 q^{51} - 11 q^{53} - 3 q^{55} - 4 q^{57} + 10 q^{59} - 12 q^{61} + 4 q^{63} + 2 q^{65} - 10 q^{67} + 6 q^{69} + 17 q^{71} - 25 q^{73} - q^{75} + 5 q^{77} - 2 q^{79} - 13 q^{81} + 15 q^{83} + 4 q^{85} + 2 q^{87} - 8 q^{89} - 18 q^{91} - 32 q^{93} + 2 q^{95} + 16 q^{97} + 14 q^{99}+O(q^{100}) $ 3 * q - q^3 - 3 * q^5 + q^7 + 3 * q^11 - 2 * q^13 + q^15 - 4 * q^17 - 2 * q^19 - 9 * q^21 + 6 * q^23 + 3 * q^25 - q^27 - 6 * q^29 + 6 * q^31 - 5 * q^33 - q^35 - 3 * q^37 + 18 * q^39 - 13 * q^41 - 6 * q^43 + q^47 - 12 * q^49 - 8 * q^51 - 11 * q^53 - 3 * q^55 - 4 * q^57 + 10 * q^59 - 12 * q^61 + 4 * q^63 + 2 * q^65 - 10 * q^67 + 6 * q^69 + 17 * q^71 - 25 * q^73 - q^75 + 5 * q^77 - 2 * q^79 - 13 * q^81 + 15 * q^83 + 4 * q^85 + 2 * q^87 - 8 * q^89 - 18 * q^91 - 32 * q^93 + 2 * q^95 + 16 * q^97 + 14 * 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
$3$	−2.34292	−1.35269	−0.676344	−	0.736586i	$-0.736437\pi$
$3$	−2.34292	−1.35269	−0.676344	+	0.736586i	$0.736437\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	2.34292	0.885542	0.442771	−	0.896635i	$-0.353996\pi$
$7$	2.34292	0.885542	0.442771	+	0.896635i	$0.353996\pi$
$8$	0	0
$9$	2.48929	0.829763
$10$	0	0
$11$	3.48929	1.05206	0.526030	−	0.850466i	$-0.323680\pi$
$11$	3.48929	1.05206	0.526030	+	0.850466i	$0.323680\pi$
$12$	0	0
$13$	−4.68585	−1.29962	−0.649810	−	0.760097i	$-0.725152\pi$
$13$	−4.68585	−1.29962	−0.649810	+	0.760097i	$0.725152\pi$
$14$	0	0
$15$	2.34292	0.604940
$16$	0	0
$17$	−2.29273	−0.556069	−0.278034	−	0.960571i	$-0.589683\pi$
$17$	−2.29273	−0.556069	−0.278034	+	0.960571i	$0.589683\pi$
$18$	0	0
$19$	−1.14637	−0.262994	−0.131497	−	0.991317i	$-0.541978\pi$
$19$	−1.14637	−0.262994	−0.131497	+	0.991317i	$0.541978\pi$
$20$	0	0
$21$	−5.48929	−1.19786
$22$	0	0
$23$	−2.97858	−0.621076	−0.310538	−	0.950561i	$-0.600509\pi$
$23$	−2.97858	−0.621076	−0.310538	+	0.950561i	$0.600509\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	1.19656	0.230278
$28$	0	0
$29$	−2.00000	−0.371391	−0.185695	−	0.982607i	$-0.559454\pi$
$29$	−2.00000	−0.371391	−0.185695	+	0.982607i	$0.559454\pi$
$30$	0	0
$31$	10.5181	1.88910	0.944549	−	0.328369i	$-0.106499\pi$
$31$	10.5181	1.88910	0.944549	+	0.328369i	$0.106499\pi$
$32$	0	0
$33$	−8.17513	−1.42311
$34$	0	0
$35$	−2.34292	−0.396026
$36$	0	0
$37$	−1.00000	−0.164399
$37$	−1.00000	−0.164399
$38$	0	0
$39$	10.9786	1.75798
$40$	0	0
$41$	2.17513	0.339699	0.169849	−	0.985470i	$-0.445672\pi$
$41$	2.17513	0.339699	0.169849	+	0.985470i	$0.445672\pi$
$42$	0	0
$43$	−6.97858	−1.06422	−0.532112	−	0.846674i	$-0.678601\pi$
$43$	−6.97858	−1.06422	−0.532112	+	0.846674i	$0.678601\pi$
$44$	0	0
$45$	−2.48929	−0.371081
$46$	0	0
$47$	2.34292	0.341750	0.170875	−	0.985293i	$-0.445340\pi$
$47$	2.34292	0.341750	0.170875	+	0.985293i	$0.445340\pi$
$48$	0	0
$49$	−1.51071	−0.215816
$50$	0	0
$51$	5.37169	0.752187
$52$	0	0
$53$	−0.217980	−0.0299419	−0.0149710	−	0.999888i	$-0.504766\pi$
$53$	−0.217980	−0.0299419	−0.0149710	+	0.999888i	$0.504766\pi$
$54$	0	0
$55$	−3.48929	−0.470496
$56$	0	0
$57$	2.68585	0.355749
$58$	0	0
$59$	−4.22533	−0.550091	−0.275045	−	0.961431i	$-0.588693\pi$
$59$	−4.22533	−0.550091	−0.275045	+	0.961431i	$0.588693\pi$
$60$	0	0
$61$	0.978577	0.125294	0.0626470	−	0.998036i	$-0.480046\pi$
$61$	0.978577	0.125294	0.0626470	+	0.998036i	$0.480046\pi$
$62$	0	0
$63$	5.83221	0.734790
$64$	0	0
$65$	4.68585	0.581208
$66$	0	0
$67$	−2.85363	−0.348627	−0.174313	−	0.984690i	$-0.555771\pi$
$67$	−2.85363	−0.348627	−0.174313	+	0.984690i	$0.555771\pi$
$68$	0	0
$69$	6.97858	0.840122
$70$	0	0
$71$	12.1751	1.44492	0.722461	−	0.691411i	$-0.243011\pi$
$71$	12.1751	1.44492	0.722461	+	0.691411i	$0.243011\pi$
$72$	0	0
$73$	−1.82487	−0.213584	−0.106792	−	0.994281i	$-0.534058\pi$
$73$	−1.82487	−0.213584	−0.106792	+	0.994281i	$0.534058\pi$
$74$	0	0
$75$	−2.34292	−0.270537
$76$	0	0
$77$	8.17513	0.931643
$78$	0	0
$79$	−8.22533	−0.925422	−0.462711	−	0.886509i	$-0.653123\pi$
$79$	−8.22533	−0.925422	−0.462711	+	0.886509i	$0.653123\pi$
$80$	0	0
$81$	−10.2713	−1.14126
$82$	0	0
$83$	13.9070	1.52649	0.763244	−	0.646111i	$-0.223606\pi$
$83$	13.9070	1.52649	0.763244	+	0.646111i	$0.223606\pi$
$84$	0	0
$85$	2.29273	0.248682
$86$	0	0
$87$	4.68585	0.502375
$88$	0	0
$89$	0.393115	0.0416701	0.0208351	−	0.999783i	$-0.493368\pi$
$89$	0.393115	0.0416701	0.0208351	+	0.999783i	$0.493368\pi$
$90$	0	0
$91$	−10.9786	−1.15087
$92$	0	0
$93$	−24.6430	−2.55536
$94$	0	0
$95$	1.14637	0.117615
$96$	0	0
$97$	18.3503	1.86319	0.931594	−	0.363501i	$-0.118419\pi$
$97$	18.3503	1.86319	0.931594	+	0.363501i	$0.118419\pi$
$98$	0	0
$99$	8.68585	0.872960
$100$	0	0
$101$	8.07475	0.803468	0.401734	−	0.915756i	$-0.368408\pi$
$101$	8.07475	0.803468	0.401734	+	0.915756i	$0.368408\pi$
$102$	0	0
$103$	0.685846	0.0675784	0.0337892	−	0.999429i	$-0.489243\pi$
$103$	0.685846	0.0675784	0.0337892	+	0.999429i	$0.489243\pi$
$104$	0	0
$105$	5.48929	0.535700
$106$	0	0
$107$	−6.85363	−0.662566	−0.331283	−	0.943531i	$-0.607481\pi$
$107$	−6.85363	−0.662566	−0.331283	+	0.943531i	$0.607481\pi$
$108$	0	0
$109$	−13.5640	−1.29920	−0.649600	−	0.760276i	$-0.725063\pi$
$109$	−13.5640	−1.29920	−0.649600	+	0.760276i	$0.725063\pi$
$110$	0	0
$111$	2.34292	0.222380
$112$	0	0
$113$	−12.6858	−1.19338	−0.596692	−	0.802470i	$-0.703519\pi$
$113$	−12.6858	−1.19338	−0.596692	+	0.802470i	$0.703519\pi$
$114$	0	0
$115$	2.97858	0.277754
$116$	0	0
$117$	−11.6644	−1.07838
$118$	0	0
$119$	−5.37169	−0.492422
$120$	0	0
$121$	1.17513	0.106830
$122$	0	0
$123$	−5.09617	−0.459506
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−20.3001	−1.80134	−0.900670	−	0.434503i	$-0.856924\pi$
$127$	−20.3001	−1.80134	−0.900670	+	0.434503i	$0.856924\pi$
$128$	0	0
$129$	16.3503	1.43956
$130$	0	0
$131$	−19.6890	−1.72023	−0.860117	−	0.510097i	$-0.829610\pi$
$131$	−19.6890	−1.72023	−0.860117	+	0.510097i	$0.829610\pi$
$132$	0	0
$133$	−2.68585	−0.232892
$134$	0	0
$135$	−1.19656	−0.102983
$136$	0	0
$137$	−2.58546	−0.220891	−0.110445	−	0.993882i	$-0.535228\pi$
$137$	−2.58546	−0.220891	−0.110445	+	0.993882i	$0.535228\pi$
$138$	0	0
$139$	−19.3288	−1.63945	−0.819726	−	0.572756i	$-0.805874\pi$
$139$	−19.3288	−1.63945	−0.819726	+	0.572756i	$0.805874\pi$
$140$	0	0
$141$	−5.48929	−0.462281
$142$	0	0
$143$	−16.3503	−1.36728
$144$	0	0
$145$	2.00000	0.166091
$146$	0	0
$147$	3.53948	0.291931
$148$	0	0
$149$	−9.09617	−0.745188	−0.372594	−	0.927995i	$-0.621532\pi$
$149$	−9.09617	−0.745188	−0.372594	+	0.927995i	$0.621532\pi$
$150$	0	0
$151$	6.87819	0.559739	0.279870	−	0.960038i	$-0.409709\pi$
$151$	6.87819	0.559739	0.279870	+	0.960038i	$0.409709\pi$
$152$	0	0
$153$	−5.70727	−0.461405
$154$	0	0
$155$	−10.5181	−0.844831
$156$	0	0
$157$	−7.78202	−0.621073	−0.310536	−	0.950561i	$-0.600509\pi$
$157$	−7.78202	−0.621073	−0.310536	+	0.950561i	$0.600509\pi$
$158$	0	0
$159$	0.510711	0.0405021
$160$	0	0
$161$	−6.97858	−0.549989
$162$	0	0
$163$	−6.72869	−0.527032	−0.263516	−	0.964655i	$-0.584882\pi$
$163$	−6.72869	−0.527032	−0.263516	+	0.964655i	$0.584882\pi$
$164$	0	0
$165$	8.17513	0.636433
$166$	0	0
$167$	−6.29273	−0.486946	−0.243473	−	0.969908i	$-0.578287\pi$
$167$	−6.29273	−0.486946	−0.243473	+	0.969908i	$0.578287\pi$
$168$	0	0
$169$	8.95715	0.689012
$170$	0	0
$171$	−2.85363	−0.218223
$172$	0	0
$173$	−15.5468	−1.18200	−0.591002	−	0.806670i	$-0.701267\pi$
$173$	−15.5468	−1.18200	−0.591002	+	0.806670i	$0.701267\pi$
$174$	0	0
$175$	2.34292	0.177108
$176$	0	0
$177$	9.89962	0.744101
$178$	0	0
$179$	3.53948	0.264553	0.132277	−	0.991213i	$-0.457771\pi$
$179$	3.53948	0.264553	0.132277	+	0.991213i	$0.457771\pi$
$180$	0	0
$181$	−8.86098	−0.658632	−0.329316	−	0.944220i	$-0.606818\pi$
$181$	−8.86098	−0.658632	−0.329316	+	0.944220i	$0.606818\pi$
$182$	0	0
$183$	−2.29273	−0.169484
$184$	0	0
$185$	1.00000	0.0735215
$186$	0	0
$187$	−8.00000	−0.585018
$188$	0	0
$189$	2.80344	0.203920
$190$	0	0
$191$	−2.51806	−0.182200	−0.0911001	−	0.995842i	$-0.529038\pi$
$191$	−2.51806	−0.182200	−0.0911001	+	0.995842i	$0.529038\pi$
$192$	0	0
$193$	−21.9143	−1.57743	−0.788713	−	0.614761i	$-0.789252\pi$
$193$	−21.9143	−1.57743	−0.788713	+	0.614761i	$0.789252\pi$
$194$	0	0
$195$	−10.9786	−0.786192
$196$	0	0
$197$	−21.3545	−1.52144	−0.760722	−	0.649078i	$-0.775155\pi$
$197$	−21.3545	−1.52144	−0.760722	+	0.649078i	$0.775155\pi$
$198$	0	0
$199$	12.1249	0.859514	0.429757	−	0.902944i	$-0.358599\pi$
$199$	12.1249	0.859514	0.429757	+	0.902944i	$0.358599\pi$
$200$	0	0
$201$	6.68585	0.471583
$202$	0	0
$203$	−4.68585	−0.328882
$204$	0	0
$205$	−2.17513	−0.151918
$206$	0	0
$207$	−7.41454	−0.515346
$208$	0	0
$209$	−4.00000	−0.276686
$210$	0	0
$211$	13.1109	0.902589	0.451295	−	0.892375i	$-0.350962\pi$
$211$	13.1109	0.902589	0.451295	+	0.892375i	$0.350962\pi$
$212$	0	0
$213$	−28.5254	−1.95453
$214$	0	0
$215$	6.97858	0.475935
$216$	0	0
$217$	24.6430	1.67288
$218$	0	0
$219$	4.27552	0.288913
$220$	0	0
$221$	10.7434	0.722678
$222$	0	0
$223$	−26.3576	−1.76504	−0.882518	−	0.470278i	$-0.844154\pi$
$223$	−26.3576	−1.76504	−0.882518	+	0.470278i	$0.844154\pi$
$224$	0	0
$225$	2.48929	0.165953
$226$	0	0
$227$	−5.70727	−0.378805	−0.189402	−	0.981900i	$-0.560655\pi$
$227$	−5.70727	−0.378805	−0.189402	+	0.981900i	$0.560655\pi$
$228$	0	0
$229$	−12.5682	−0.830533	−0.415267	−	0.909700i	$-0.636312\pi$
$229$	−12.5682	−0.830533	−0.415267	+	0.909700i	$0.636312\pi$
$230$	0	0
$231$	−19.1537	−1.26022
$232$	0	0
$233$	−10.7862	−0.706629	−0.353315	−	0.935505i	$-0.614946\pi$
$233$	−10.7862	−0.706629	−0.353315	+	0.935505i	$0.614946\pi$
$234$	0	0
$235$	−2.34292	−0.152835
$236$	0	0
$237$	19.2713	1.25181
$238$	0	0
$239$	2.50337	0.161929	0.0809646	−	0.996717i	$-0.474200\pi$
$239$	2.50337	0.161929	0.0809646	+	0.996717i	$0.474200\pi$
$240$	0	0
$241$	7.95715	0.512565	0.256283	−	0.966602i	$-0.417502\pi$
$241$	7.95715	0.512565	0.256283	+	0.966602i	$0.417502\pi$
$242$	0	0
$243$	20.4752	1.31349
$244$	0	0
$245$	1.51071	0.0965158
$246$	0	0
$247$	5.37169	0.341793
$248$	0	0
$249$	−32.5829	−2.06486
$250$	0	0
$251$	19.7894	1.24909	0.624547	−	0.780987i	$-0.285284\pi$
$251$	19.7894	1.24909	0.624547	+	0.780987i	$0.285284\pi$
$252$	0	0
$253$	−10.3931	−0.653410
$254$	0	0
$255$	−5.37169	−0.336388
$256$	0	0
$257$	−13.0361	−0.813171	−0.406585	−	0.913613i	$-0.633281\pi$
$257$	−13.0361	−0.813171	−0.406585	+	0.913613i	$0.633281\pi$
$258$	0	0
$259$	−2.34292	−0.145582
$260$	0	0
$261$	−4.97858	−0.308166
$262$	0	0
$263$	14.7936	0.912211	0.456106	−	0.889926i	$-0.349244\pi$
$263$	14.7936	0.912211	0.456106	+	0.889926i	$0.349244\pi$
$264$	0	0
$265$	0.217980	0.0133904
$266$	0	0
$267$	−0.921039	−0.0563666
$268$	0	0
$269$	16.6858	1.01735	0.508677	−	0.860957i	$-0.330135\pi$
$269$	16.6858	1.01735	0.508677	+	0.860957i	$0.330135\pi$
$270$	0	0
$271$	12.1751	0.739587	0.369793	−	0.929114i	$-0.379428\pi$
$271$	12.1751	0.739587	0.369793	+	0.929114i	$0.379428\pi$
$272$	0	0
$273$	25.7220	1.55676
$274$	0	0
$275$	3.48929	0.210412
$276$	0	0
$277$	5.70727	0.342917	0.171458	−	0.985191i	$-0.445152\pi$
$277$	5.70727	0.342917	0.171458	+	0.985191i	$0.445152\pi$
$278$	0	0
$279$	26.1825	1.56750
$280$	0	0
$281$	−14.3503	−0.856065	−0.428033	−	0.903763i	$-0.640793\pi$
$281$	−14.3503	−0.856065	−0.428033	+	0.903763i	$0.640793\pi$
$282$	0	0
$283$	−9.12181	−0.542235	−0.271118	−	0.962546i	$-0.587393\pi$
$283$	−9.12181	−0.542235	−0.271118	+	0.962546i	$0.587393\pi$
$284$	0	0
$285$	−2.68585	−0.159096
$286$	0	0
$287$	5.09617	0.300818
$288$	0	0
$289$	−11.7434	−0.690787
$290$	0	0
$291$	−42.9933	−2.52031
$292$	0	0
$293$	7.95715	0.464862	0.232431	−	0.972613i	$-0.425332\pi$
$293$	7.95715	0.464862	0.232431	+	0.972613i	$0.425332\pi$
$294$	0	0
$295$	4.22533	0.246008
$296$	0	0
$297$	4.17513	0.242266
$298$	0	0
$299$	13.9572	0.807163
$300$	0	0
$301$	−16.3503	−0.942414
$302$	0	0
$303$	−18.9185	−1.08684
$304$	0	0
$305$	−0.978577	−0.0560332
$306$	0	0
$307$	−18.1077	−1.03346	−0.516731	−	0.856148i	$-0.672851\pi$
$307$	−18.1077	−1.03346	−0.516731	+	0.856148i	$0.672851\pi$
$308$	0	0
$309$	−1.60688	−0.0914125
$310$	0	0
$311$	−6.18248	−0.350576	−0.175288	−	0.984517i	$-0.556086\pi$
$311$	−6.18248	−0.350576	−0.175288	+	0.984517i	$0.556086\pi$
$312$	0	0
$313$	15.9572	0.901952	0.450976	−	0.892536i	$-0.351076\pi$
$313$	15.9572	0.901952	0.450976	+	0.892536i	$0.351076\pi$
$314$	0	0
$315$	−5.83221	−0.328608
$316$	0	0
$317$	11.3717	0.638698	0.319349	−	0.947637i	$-0.396536\pi$
$317$	11.3717	0.638698	0.319349	+	0.947637i	$0.396536\pi$
$318$	0	0
$319$	−6.97858	−0.390725
$320$	0	0
$321$	16.0575	0.896244
$322$	0	0
$323$	2.62831	0.146243
$324$	0	0
$325$	−4.68585	−0.259924
$326$	0	0
$327$	31.7795	1.75741
$328$	0	0
$329$	5.48929	0.302634
$330$	0	0
$331$	−15.8751	−0.872572	−0.436286	−	0.899808i	$-0.643706\pi$
$331$	−15.8751	−0.872572	−0.436286	+	0.899808i	$0.643706\pi$
$332$	0	0
$333$	−2.48929	−0.136412
$334$	0	0
$335$	2.85363	0.155911
$336$	0	0
$337$	9.58967	0.522383	0.261191	−	0.965287i	$-0.415885\pi$
$337$	9.58967	0.522383	0.261191	+	0.965287i	$0.415885\pi$
$338$	0	0
$339$	29.7220	1.61428
$340$	0	0
$341$	36.7005	1.98745
$342$	0	0
$343$	−19.9399	−1.07666
$344$	0	0
$345$	−6.97858	−0.375714
$346$	0	0
$347$	−28.9357	−1.55335	−0.776676	−	0.629901i	$-0.783096\pi$
$347$	−28.9357	−1.55335	−0.776676	+	0.629901i	$0.783096\pi$
$348$	0	0
$349$	22.6430	1.21205	0.606026	−	0.795445i	$-0.292763\pi$
$349$	22.6430	1.21205	0.606026	+	0.795445i	$0.292763\pi$
$350$	0	0
$351$	−5.60688	−0.299273
$352$	0	0
$353$	8.39312	0.446720	0.223360	−	0.974736i	$-0.428297\pi$
$353$	8.39312	0.446720	0.223360	+	0.974736i	$0.428297\pi$
$354$	0	0
$355$	−12.1751	−0.646189
$356$	0	0
$357$	12.5855	0.666093
$358$	0	0
$359$	13.2969	0.701786	0.350893	−	0.936416i	$-0.385878\pi$
$359$	13.2969	0.701786	0.350893	+	0.936416i	$0.385878\pi$
$360$	0	0
$361$	−17.6858	−0.930834
$362$	0	0
$363$	−2.75325	−0.144508
$364$	0	0
$365$	1.82487	0.0955178
$366$	0	0
$367$	−6.60375	−0.344713	−0.172356	−	0.985035i	$-0.555138\pi$
$367$	−6.60375	−0.344713	−0.172356	+	0.985035i	$0.555138\pi$
$368$	0	0
$369$	5.41454	0.281870
$370$	0	0
$371$	−0.510711	−0.0265148
$372$	0	0
$373$	33.2688	1.72259	0.861296	−	0.508103i	$-0.169653\pi$
$373$	33.2688	1.72259	0.861296	+	0.508103i	$0.169653\pi$
$374$	0	0
$375$	2.34292	0.120988
$376$	0	0
$377$	9.37169	0.482667
$378$	0	0
$379$	2.11760	0.108774	0.0543868	−	0.998520i	$-0.482680\pi$
$379$	2.11760	0.108774	0.0543868	+	0.998520i	$0.482680\pi$
$380$	0	0
$381$	47.5615	2.43665
$382$	0	0
$383$	32.1151	1.64100	0.820502	−	0.571644i	$-0.193694\pi$
$383$	32.1151	1.64100	0.820502	+	0.571644i	$0.193694\pi$
$384$	0	0
$385$	−8.17513	−0.416643
$386$	0	0
$387$	−17.3717	−0.883053
$388$	0	0
$389$	−8.77781	−0.445053	−0.222526	−	0.974927i	$-0.571430\pi$
$389$	−8.77781	−0.445053	−0.222526	+	0.974927i	$0.571430\pi$
$390$	0	0
$391$	6.82908	0.345361
$392$	0	0
$393$	46.1298	2.32694
$394$	0	0
$395$	8.22533	0.413861
$396$	0	0
$397$	−34.4397	−1.72848	−0.864240	−	0.503080i	$-0.832200\pi$
$397$	−34.4397	−1.72848	−0.864240	+	0.503080i	$0.832200\pi$
$398$	0	0
$399$	6.29273	0.315031
$400$	0	0
$401$	−24.0722	−1.20211	−0.601055	−	0.799208i	$-0.705253\pi$
$401$	−24.0722	−1.20211	−0.601055	+	0.799208i	$0.705253\pi$
$402$	0	0
$403$	−49.2860	−2.45511
$404$	0	0
$405$	10.2713	0.510385
$406$	0	0
$407$	−3.48929	−0.172958
$408$	0	0
$409$	−3.22219	−0.159327	−0.0796636	−	0.996822i	$-0.525385\pi$
$409$	−3.22219	−0.159327	−0.0796636	+	0.996822i	$0.525385\pi$
$410$	0	0
$411$	6.05754	0.298796
$412$	0	0
$413$	−9.89962	−0.487128
$414$	0	0
$415$	−13.9070	−0.682666
$416$	0	0
$417$	45.2860	2.21767
$418$	0	0
$419$	36.8610	1.80078	0.900388	−	0.435087i	$-0.143282\pi$
$419$	36.8610	1.80078	0.900388	+	0.435087i	$0.143282\pi$
$420$	0	0
$421$	−18.3503	−0.894337	−0.447169	−	0.894450i	$-0.647568\pi$
$421$	−18.3503	−0.894337	−0.447169	+	0.894450i	$0.647568\pi$
$422$	0	0
$423$	5.83221	0.283572
$424$	0	0
$425$	−2.29273	−0.111214
$426$	0	0
$427$	2.29273	0.110953
$428$	0	0
$429$	38.3074	1.84950
$430$	0	0
$431$	−2.41767	−0.116455	−0.0582276	−	0.998303i	$-0.518545\pi$
$431$	−2.41767	−0.116455	−0.0582276	+	0.998303i	$0.518545\pi$
$432$	0	0
$433$	6.41033	0.308061	0.154030	−	0.988066i	$-0.450775\pi$
$433$	6.41033	0.308061	0.154030	+	0.988066i	$0.450775\pi$
$434$	0	0
$435$	−4.68585	−0.224669
$436$	0	0
$437$	3.41454	0.163340
$438$	0	0
$439$	12.4752	0.595409	0.297705	−	0.954658i	$-0.403779\pi$
$439$	12.4752	0.595409	0.297705	+	0.954658i	$0.403779\pi$
$440$	0	0
$441$	−3.76060	−0.179076
$442$	0	0
$443$	8.53527	0.405523	0.202761	−	0.979228i	$-0.435008\pi$
$443$	8.53527	0.405523	0.202761	+	0.979228i	$0.435008\pi$
$444$	0	0
$445$	−0.393115	−0.0186354
$446$	0	0
$447$	21.3116	1.00801
$448$	0	0
$449$	−13.7648	−0.649601	−0.324801	−	0.945782i	$-0.605297\pi$
$449$	−13.7648	−0.649601	−0.324801	+	0.945782i	$0.605297\pi$
$450$	0	0
$451$	7.58967	0.357384
$452$	0	0
$453$	−16.1151	−0.757152
$454$	0	0
$455$	10.9786	0.514684
$456$	0	0
$457$	6.78623	0.317446	0.158723	−	0.987323i	$-0.449262\pi$
$457$	6.78623	0.317446	0.158723	+	0.987323i	$0.449262\pi$
$458$	0	0
$459$	−2.74338	−0.128050
$460$	0	0
$461$	6.78623	0.316066	0.158033	−	0.987434i	$-0.449485\pi$
$461$	6.78623	0.316066	0.158033	+	0.987434i	$0.449485\pi$
$462$	0	0
$463$	−26.3074	−1.22261	−0.611305	−	0.791395i	$-0.709355\pi$
$463$	−26.3074	−1.22261	−0.611305	+	0.791395i	$0.709355\pi$
$464$	0	0
$465$	24.6430	1.14279
$466$	0	0
$467$	26.4422	1.22360	0.611800	−	0.791012i	$-0.290446\pi$
$467$	26.4422	1.22360	0.611800	+	0.791012i	$0.290446\pi$
$468$	0	0
$469$	−6.68585	−0.308724
$470$	0	0
$471$	18.2327	0.840117
$472$	0	0
$473$	−24.3503	−1.11963
$474$	0	0
$475$	−1.14637	−0.0525989
$476$	0	0
$477$	−0.542616	−0.0248447
$478$	0	0
$479$	−36.8255	−1.68260	−0.841300	−	0.540569i	$-0.818209\pi$
$479$	−36.8255	−1.68260	−0.841300	+	0.540569i	$0.818209\pi$
$480$	0	0
$481$	4.68585	0.213656
$482$	0	0
$483$	16.3503	0.743963
$484$	0	0
$485$	−18.3503	−0.833243
$486$	0	0
$487$	−3.07896	−0.139521	−0.0697605	−	0.997564i	$-0.522224\pi$
$487$	−3.07896	−0.139521	−0.0697605	+	0.997564i	$0.522224\pi$
$488$	0	0
$489$	15.7648	0.712909
$490$	0	0
$491$	34.6577	1.56408	0.782040	−	0.623228i	$-0.214179\pi$
$491$	34.6577	1.56408	0.782040	+	0.623228i	$0.214179\pi$
$492$	0	0
$493$	4.58546	0.206519
$494$	0	0
$495$	−8.68585	−0.390400
$496$	0	0
$497$	28.5254	1.27954
$498$	0	0
$499$	−25.0607	−1.12187	−0.560935	−	0.827860i	$-0.689558\pi$
$499$	−25.0607	−1.12187	−0.560935	+	0.827860i	$0.689558\pi$
$500$	0	0
$501$	14.7434	0.658686
$502$	0	0
$503$	1.75639	0.0783134	0.0391567	−	0.999233i	$-0.487533\pi$
$503$	1.75639	0.0783134	0.0391567	+	0.999233i	$0.487533\pi$
$504$	0	0
$505$	−8.07475	−0.359322
$506$	0	0
$507$	−20.9859	−0.932018
$508$	0	0
$509$	30.9101	1.37007	0.685033	−	0.728512i	$-0.259788\pi$
$509$	30.9101	1.37007	0.685033	+	0.728512i	$0.259788\pi$
$510$	0	0
$511$	−4.27552	−0.189138
$512$	0	0
$513$	−1.37169	−0.0605617
$514$	0	0
$515$	−0.685846	−0.0302220
$516$	0	0
$517$	8.17513	0.359542
$518$	0	0
$519$	36.4250	1.59888
$520$	0	0
$521$	−17.2113	−0.754039	−0.377019	−	0.926205i	$-0.623051\pi$
$521$	−17.2113	−0.754039	−0.377019	+	0.926205i	$0.623051\pi$
$522$	0	0
$523$	−13.6069	−0.594988	−0.297494	−	0.954724i	$-0.596151\pi$
$523$	−13.6069	−0.594988	−0.297494	+	0.954724i	$0.596151\pi$
$524$	0	0
$525$	−5.48929	−0.239572
$526$	0	0
$527$	−24.1151	−1.05047
$528$	0	0
$529$	−14.1281	−0.614264
$530$	0	0
$531$	−10.5181	−0.456445
$532$	0	0
$533$	−10.1923	−0.441480
$534$	0	0
$535$	6.85363	0.296308
$536$	0	0
$537$	−8.29273	−0.357858
$538$	0	0
$539$	−5.27131	−0.227051
$540$	0	0
$541$	−8.97858	−0.386019	−0.193010	−	0.981197i	$-0.561825\pi$
$541$	−8.97858	−0.386019	−0.193010	+	0.981197i	$0.561825\pi$
$542$	0	0
$543$	20.7606	0.890922
$544$	0	0
$545$	13.5640	0.581020
$546$	0	0
$547$	−24.4998	−1.04753	−0.523767	−	0.851861i	$-0.675474\pi$
$547$	−24.4998	−1.04753	−0.523767	+	0.851861i	$0.675474\pi$
$548$	0	0
$549$	2.43596	0.103964
$550$	0	0
$551$	2.29273	0.0976736
$552$	0	0
$553$	−19.2713	−0.819499
$554$	0	0
$555$	−2.34292	−0.0994515
$556$	0	0
$557$	−11.5149	−0.487903	−0.243951	−	0.969787i	$-0.578444\pi$
$557$	−11.5149	−0.487903	−0.243951	+	0.969787i	$0.578444\pi$
$558$	0	0
$559$	32.7005	1.38309
$560$	0	0
$561$	18.7434	0.791346
$562$	0	0
$563$	2.39312	0.100858	0.0504289	−	0.998728i	$-0.483941\pi$
$563$	2.39312	0.100858	0.0504289	+	0.998728i	$0.483941\pi$
$564$	0	0
$565$	12.6858	0.533698
$566$	0	0
$567$	−24.0649	−1.01063
$568$	0	0
$569$	−0.192347	−0.00806360	−0.00403180	−	0.999992i	$-0.501283\pi$
$569$	−0.192347	−0.00806360	−0.00403180	+	0.999992i	$0.501283\pi$
$570$	0	0
$571$	−17.7476	−0.742714	−0.371357	−	0.928490i	$-0.621107\pi$
$571$	−17.7476	−0.742714	−0.371357	+	0.928490i	$0.621107\pi$
$572$	0	0
$573$	5.89962	0.246460
$574$	0	0
$575$	−2.97858	−0.124215
$576$	0	0
$577$	3.66442	0.152552	0.0762760	−	0.997087i	$-0.475697\pi$
$577$	3.66442	0.152552	0.0762760	+	0.997087i	$0.475697\pi$
$578$	0	0
$579$	51.3435	2.13376
$580$	0	0
$581$	32.5829	1.35177
$582$	0	0
$583$	−0.760597	−0.0315007
$584$	0	0
$585$	11.6644	0.482265
$586$	0	0
$587$	−3.41454	−0.140933	−0.0704665	−	0.997514i	$-0.522449\pi$
$587$	−3.41454	−0.140933	−0.0704665	+	0.997514i	$0.522449\pi$
$588$	0	0
$589$	−12.0575	−0.496822
$590$	0	0
$591$	50.0319	2.05804
$592$	0	0
$593$	−9.23940	−0.379417	−0.189708	−	0.981840i	$-0.560754\pi$
$593$	−9.23940	−0.379417	−0.189708	+	0.981840i	$0.560754\pi$
$594$	0	0
$595$	5.37169	0.220218
$596$	0	0
$597$	−28.4078	−1.16265
$598$	0	0
$599$	41.8824	1.71127	0.855634	−	0.517581i	$-0.173167\pi$
$599$	41.8824	1.71127	0.855634	+	0.517581i	$0.173167\pi$
$600$	0	0
$601$	−24.8866	−1.01515	−0.507573	−	0.861609i	$-0.669457\pi$
$601$	−24.8866	−1.01515	−0.507573	+	0.861609i	$0.669457\pi$
$602$	0	0
$603$	−7.10352	−0.289278
$604$	0	0
$605$	−1.17513	−0.0477760
$606$	0	0
$607$	−25.0852	−1.01818	−0.509089	−	0.860714i	$-0.670018\pi$
$607$	−25.0852	−1.01818	−0.509089	+	0.860714i	$0.670018\pi$
$608$	0	0
$609$	10.9786	0.444874
$610$	0	0
$611$	−10.9786	−0.444146
$612$	0	0
$613$	18.2902	0.738735	0.369367	−	0.929283i	$-0.379574\pi$
$613$	18.2902	0.738735	0.369367	+	0.929283i	$0.379574\pi$
$614$	0	0
$615$	5.09617	0.205498
$616$	0	0
$617$	19.1109	0.769375	0.384687	−	0.923047i	$-0.374309\pi$
$617$	19.1109	0.769375	0.384687	+	0.923047i	$0.374309\pi$
$618$	0	0
$619$	23.1537	0.930626	0.465313	−	0.885146i	$-0.345942\pi$
$619$	23.1537	0.930626	0.465313	+	0.885146i	$0.345942\pi$
$620$	0	0
$621$	−3.56404	−0.143020
$622$	0	0
$623$	0.921039	0.0369006
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	9.37169	0.374269
$628$	0	0
$629$	2.29273	0.0914172
$630$	0	0
$631$	42.3832	1.68725	0.843625	−	0.536932i	$-0.180417\pi$
$631$	42.3832	1.68725	0.843625	+	0.536932i	$0.180417\pi$
$632$	0	0
$633$	−30.7178	−1.22092
$634$	0	0
$635$	20.3001	0.805584
$636$	0	0
$637$	7.07896	0.280479
$638$	0	0
$639$	30.3074	1.19894
$640$	0	0
$641$	−26.6686	−1.05335	−0.526674	−	0.850067i	$-0.676561\pi$
$641$	−26.6686	−1.05335	−0.526674	+	0.850067i	$0.676561\pi$
$642$	0	0
$643$	−46.0575	−1.81633	−0.908166	−	0.418610i	$-0.862517\pi$
$643$	−46.0575	−1.81633	−0.908166	+	0.418610i	$0.862517\pi$
$644$	0	0
$645$	−16.3503	−0.643791
$646$	0	0
$647$	−13.1709	−0.517802	−0.258901	−	0.965904i	$-0.583360\pi$
$647$	−13.1709	−0.517802	−0.258901	+	0.965904i	$0.583360\pi$
$648$	0	0
$649$	−14.7434	−0.578728
$650$	0	0
$651$	−57.7367	−2.26288
$652$	0	0
$653$	41.0080	1.60477	0.802383	−	0.596810i	$-0.203565\pi$
$653$	41.0080	1.60477	0.802383	+	0.596810i	$0.203565\pi$
$654$	0	0
$655$	19.6890	0.769312
$656$	0	0
$657$	−4.54262	−0.177224
$658$	0	0
$659$	2.41875	0.0942211	0.0471105	−	0.998890i	$-0.484999\pi$
$659$	2.41875	0.0942211	0.0471105	+	0.998890i	$0.484999\pi$
$660$	0	0
$661$	13.7648	0.535389	0.267694	−	0.963504i	$-0.413738\pi$
$661$	13.7648	0.535389	0.267694	+	0.963504i	$0.413738\pi$
$662$	0	0
$663$	−25.1709	−0.977558
$664$	0	0
$665$	2.68585	0.104153
$666$	0	0
$667$	5.95715	0.230662
$668$	0	0
$669$	61.7539	2.38754
$670$	0	0
$671$	3.41454	0.131817
$672$	0	0
$673$	−31.1109	−1.19924	−0.599618	−	0.800286i	$-0.704681\pi$
$673$	−31.1109	−1.19924	−0.599618	+	0.800286i	$0.704681\pi$
$674$	0	0
$675$	1.19656	0.0460555
$676$	0	0
$677$	35.1109	1.34942	0.674710	−	0.738083i	$-0.264269\pi$
$677$	35.1109	1.34942	0.674710	+	0.738083i	$0.264269\pi$
$678$	0	0
$679$	42.9933	1.64993
$680$	0	0
$681$	13.3717	0.512404
$682$	0	0
$683$	21.8077	0.834447	0.417223	−	0.908804i	$-0.363003\pi$
$683$	21.8077	0.834447	0.417223	+	0.908804i	$0.363003\pi$
$684$	0	0
$685$	2.58546	0.0987854
$686$	0	0
$687$	29.4464	1.12345
$688$	0	0
$689$	1.02142	0.0389131
$690$	0	0
$691$	31.1940	1.18668	0.593339	−	0.804953i	$-0.297810\pi$
$691$	31.1940	1.18668	0.593339	+	0.804953i	$0.297810\pi$
$692$	0	0
$693$	20.3503	0.773043
$694$	0	0
$695$	19.3288	0.733185
$696$	0	0
$697$	−4.98700	−0.188896
$698$	0	0
$699$	25.2713	0.955849
$700$	0	0
$701$	−12.3931	−0.468082	−0.234041	−	0.972227i	$-0.575195\pi$
$701$	−12.3931	−0.468082	−0.234041	+	0.972227i	$0.575195\pi$
$702$	0	0
$703$	1.14637	0.0432360
$704$	0	0
$705$	5.48929	0.206739
$706$	0	0
$707$	18.9185	0.711504
$708$	0	0
$709$	39.6363	1.48857	0.744286	−	0.667861i	$-0.232790\pi$
$709$	39.6363	1.48857	0.744286	+	0.667861i	$0.232790\pi$
$710$	0	0
$711$	−20.4752	−0.767880
$712$	0	0
$713$	−31.3288	−1.17327
$714$	0	0
$715$	16.3503	0.611465
$716$	0	0
$717$	−5.86519	−0.219040
$718$	0	0
$719$	18.7031	0.697506	0.348753	−	0.937215i	$-0.386605\pi$
$719$	18.7031	0.697506	0.348753	+	0.937215i	$0.386605\pi$
$720$	0	0
$721$	1.60688	0.0598435
$722$	0	0
$723$	−18.6430	−0.693341
$724$	0	0
$725$	−2.00000	−0.0742781
$726$	0	0
$727$	−4.00000	−0.148352	−0.0741759	−	0.997245i	$-0.523633\pi$
$727$	−4.00000	−0.148352	−0.0741759	+	0.997245i	$0.523633\pi$
$728$	0	0
$729$	−17.1579	−0.635479
$730$	0	0
$731$	16.0000	0.591781
$732$	0	0
$733$	−26.5254	−0.979738	−0.489869	−	0.871796i	$-0.662955\pi$
$733$	−26.5254	−0.979738	−0.489869	+	0.871796i	$0.662955\pi$
$734$	0	0
$735$	−3.53948	−0.130556
$736$	0	0
$737$	−9.95715	−0.366776
$738$	0	0
$739$	0.496019	0.0182463	0.00912317	−	0.999958i	$-0.497096\pi$
$739$	0.496019	0.0182463	0.00912317	+	0.999958i	$0.497096\pi$
$740$	0	0
$741$	−12.5855	−0.462338
$742$	0	0
$743$	40.6503	1.49132	0.745658	−	0.666329i	$-0.232135\pi$
$743$	40.6503	1.49132	0.745658	+	0.666329i	$0.232135\pi$
$744$	0	0
$745$	9.09617	0.333258
$746$	0	0
$747$	34.6184	1.26662
$748$	0	0
$749$	−16.0575	−0.586730
$750$	0	0
$751$	−11.1046	−0.405212	−0.202606	−	0.979260i	$-0.564941\pi$
$751$	−11.1046	−0.405212	−0.202606	+	0.979260i	$0.564941\pi$
$752$	0	0
$753$	−46.3650	−1.68963
$754$	0	0
$755$	−6.87819	−0.250323
$756$	0	0
$757$	−45.1281	−1.64021	−0.820104	−	0.572215i	$-0.806084\pi$
$757$	−45.1281	−1.64021	−0.820104	+	0.572215i	$0.806084\pi$
$758$	0	0
$759$	24.3503	0.883859
$760$	0	0
$761$	−52.0466	−1.88669	−0.943344	−	0.331817i	$-0.892338\pi$
$761$	−52.0466	−1.88669	−0.943344	+	0.331817i	$0.892338\pi$
$762$	0	0
$763$	−31.7795	−1.15050
$764$	0	0
$765$	5.70727	0.206347
$766$	0	0
$767$	19.7992	0.714909
$768$	0	0
$769$	−35.9865	−1.29771	−0.648854	−	0.760913i	$-0.724751\pi$
$769$	−35.9865	−1.29771	−0.648854	+	0.760913i	$0.724751\pi$
$770$	0	0
$771$	30.5426	1.09997
$772$	0	0
$773$	27.7476	0.998012	0.499006	−	0.866599i	$-0.333699\pi$
$773$	27.7476	0.998012	0.499006	+	0.866599i	$0.333699\pi$
$774$	0	0
$775$	10.5181	0.377820
$776$	0	0
$777$	5.48929	0.196927
$778$	0	0
$779$	−2.49350	−0.0893389
$780$	0	0
$781$	42.4826	1.52015
$782$	0	0
$783$	−2.39312	−0.0855230
$784$	0	0
$785$	7.78202	0.277752
$786$	0	0
$787$	21.8725	0.779672	0.389836	−	0.920884i	$-0.372532\pi$
$787$	21.8725	0.779672	0.389836	+	0.920884i	$0.372532\pi$
$788$	0	0
$789$	−34.6602	−1.23394
$790$	0	0
$791$	−29.7220	−1.05679
$792$	0	0
$793$	−4.58546	−0.162835
$794$	0	0
$795$	−0.510711	−0.0181131
$796$	0	0
$797$	−40.9504	−1.45054	−0.725269	−	0.688465i	$-0.758285\pi$
$797$	−40.9504	−1.45054	−0.725269	+	0.688465i	$0.758285\pi$
$798$	0	0
$799$	−5.37169	−0.190037
$800$	0	0
$801$	0.978577	0.0345763
$802$	0	0
$803$	−6.36748	−0.224704
$804$	0	0
$805$	6.97858	0.245963
$806$	0	0
$807$	−39.0937	−1.37616
$808$	0	0
$809$	−14.7005	−0.516843	−0.258422	−	0.966032i	$-0.583202\pi$
$809$	−14.7005	−0.516843	−0.258422	+	0.966032i	$0.583202\pi$
$810$	0	0
$811$	−52.0894	−1.82911	−0.914554	−	0.404464i	$-0.867458\pi$
$811$	−52.0894	−1.82911	−0.914554	+	0.404464i	$0.867458\pi$
$812$	0	0
$813$	−28.5254	−1.00043
$814$	0	0
$815$	6.72869	0.235696
$816$	0	0
$817$	8.00000	0.279885
$818$	0	0
$819$	−27.3288	−0.954947
$820$	0	0
$821$	−50.4910	−1.76215	−0.881074	−	0.472979i	$-0.843179\pi$
$821$	−50.4910	−1.76215	−0.881074	+	0.472979i	$0.843179\pi$
$822$	0	0
$823$	18.1825	0.633801	0.316901	−	0.948459i	$-0.397358\pi$
$823$	18.1825	0.633801	0.316901	+	0.948459i	$0.397358\pi$
$824$	0	0
$825$	−8.17513	−0.284622
$826$	0	0
$827$	−16.6002	−0.577244	−0.288622	−	0.957443i	$-0.593197\pi$
$827$	−16.6002	−0.577244	−0.288622	+	0.957443i	$0.593197\pi$
$828$	0	0
$829$	12.3931	0.430431	0.215215	−	0.976567i	$-0.430955\pi$
$829$	12.3931	0.430431	0.215215	+	0.976567i	$0.430955\pi$
$830$	0	0
$831$	−13.3717	−0.463859
$832$	0	0
$833$	3.46365	0.120009
$834$	0	0
$835$	6.29273	0.217769
$836$	0	0
$837$	12.5855	0.435017
$838$	0	0
$839$	46.8585	1.61773	0.808867	−	0.587992i	$-0.200081\pi$
$839$	46.8585	1.61773	0.808867	+	0.587992i	$0.200081\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	0	0
$843$	33.6216	1.15799
$844$	0	0
$845$	−8.95715	−0.308135
$846$	0	0
$847$	2.75325	0.0946028
$848$	0	0
$849$	21.3717	0.733475
$850$	0	0
$851$	2.97858	0.102104
$852$	0	0
$853$	−29.9143	−1.02425	−0.512123	−	0.858912i	$-0.671141\pi$
$853$	−29.9143	−1.02425	−0.512123	+	0.858912i	$0.671141\pi$
$854$	0	0
$855$	2.85363	0.0975922
$856$	0	0
$857$	41.0080	1.40081	0.700403	−	0.713748i	$-0.253004\pi$
$857$	41.0080	1.40081	0.700403	+	0.713748i	$0.253004\pi$
$858$	0	0
$859$	1.93260	0.0659393	0.0329697	−	0.999456i	$-0.489504\pi$
$859$	1.93260	0.0659393	0.0329697	+	0.999456i	$0.489504\pi$
$860$	0	0
$861$	−11.9399	−0.406912
$862$	0	0
$863$	32.2744	1.09863	0.549317	−	0.835614i	$-0.314888\pi$
$863$	32.2744	1.09863	0.549317	+	0.835614i	$0.314888\pi$
$864$	0	0
$865$	15.5468	0.528608
$866$	0	0
$867$	27.5138	0.934419
$868$	0	0
$869$	−28.7005	−0.973599
$870$	0	0
$871$	13.3717	0.453083
$872$	0	0
$873$	45.6791	1.54600
$874$	0	0
$875$	−2.34292	−0.0792053
$876$	0	0
$877$	50.2302	1.69615	0.848076	−	0.529875i	$-0.177761\pi$
$877$	50.2302	1.69615	0.848076	+	0.529875i	$0.177761\pi$
$878$	0	0
$879$	−18.6430	−0.628813
$880$	0	0
$881$	−6.25831	−0.210848	−0.105424	−	0.994427i	$-0.533620\pi$
$881$	−6.25831	−0.210848	−0.105424	+	0.994427i	$0.533620\pi$
$882$	0	0
$883$	49.6510	1.67089	0.835444	−	0.549576i	$-0.185211\pi$
$883$	49.6510	1.67089	0.835444	+	0.549576i	$0.185211\pi$
$884$	0	0
$885$	−9.89962	−0.332772
$886$	0	0
$887$	−20.4152	−0.685474	−0.342737	−	0.939431i	$-0.611354\pi$
$887$	−20.4152	−0.685474	−0.342737	+	0.939431i	$0.611354\pi$
$888$	0	0
$889$	−47.5615	−1.59516
$890$	0	0
$891$	−35.8396	−1.20067
$892$	0	0
$893$	−2.68585	−0.0898784
$894$	0	0
$895$	−3.53948	−0.118312
$896$	0	0
$897$	−32.7005	−1.09184
$898$	0	0
$899$	−21.0361	−0.701594
$900$	0	0
$901$	0.499771	0.0166498
$902$	0	0
$903$	38.3074	1.27479
$904$	0	0
$905$	8.86098	0.294549
$906$	0	0
$907$	44.1642	1.46645	0.733224	−	0.679987i	$-0.238015\pi$
$907$	44.1642	1.46645	0.733224	+	0.679987i	$0.238015\pi$
$908$	0	0
$909$	20.1004	0.666688
$910$	0	0
$911$	27.9901	0.927355	0.463677	−	0.886004i	$-0.346530\pi$
$911$	27.9901	0.927355	0.463677	+	0.886004i	$0.346530\pi$
$912$	0	0
$913$	48.5254	1.60596
$914$	0	0
$915$	2.29273	0.0757953
$916$	0	0
$917$	−46.1298	−1.52334
$918$	0	0
$919$	−53.2102	−1.75524	−0.877621	−	0.479355i	$-0.840870\pi$
$919$	−53.2102	−1.75524	−0.877621	+	0.479355i	$0.840870\pi$
$920$	0	0
$921$	42.4250	1.39795
$922$	0	0
$923$	−57.0508	−1.87785
$924$	0	0
$925$	−1.00000	−0.0328798
$926$	0	0
$927$	1.70727	0.0560741
$928$	0	0
$929$	22.2990	0.731607	0.365803	−	0.930692i	$-0.380794\pi$
$929$	22.2990	0.731607	0.365803	+	0.930692i	$0.380794\pi$
$930$	0	0
$931$	1.73183	0.0567584
$932$	0	0
$933$	14.4851	0.474220
$934$	0	0
$935$	8.00000	0.261628
$936$	0	0
$937$	−48.7178	−1.59154	−0.795770	−	0.605599i	$-0.792933\pi$
$937$	−48.7178	−1.59154	−0.795770	+	0.605599i	$0.792933\pi$
$938$	0	0
$939$	−37.3864	−1.22006
$940$	0	0
$941$	−22.5573	−0.735347	−0.367674	−	0.929955i	$-0.619846\pi$
$941$	−22.5573	−0.735347	−0.367674	+	0.929955i	$0.619846\pi$
$942$	0	0
$943$	−6.47881	−0.210979
$944$	0	0
$945$	−2.80344	−0.0911960
$946$	0	0
$947$	−34.7925	−1.13060	−0.565302	−	0.824884i	$-0.691240\pi$
$947$	−34.7925	−1.13060	−0.565302	+	0.824884i	$0.691240\pi$
$948$	0	0
$949$	8.55104	0.277578
$950$	0	0
$951$	−26.6430	−0.863958
$952$	0	0
$953$	−43.3461	−1.40412	−0.702058	−	0.712119i	$-0.747735\pi$
$953$	−43.3461	−1.40412	−0.702058	+	0.712119i	$0.747735\pi$
$954$	0	0
$955$	2.51806	0.0814824
$956$	0	0
$957$	16.3503	0.528529
$958$	0	0
$959$	−6.05754	−0.195608
$960$	0	0
$961$	79.6295	2.56869
$962$	0	0
$963$	−17.0607	−0.549773
$964$	0	0
$965$	21.9143	0.705447
$966$	0	0
$967$	7.31415	0.235207	0.117604	−	0.993061i	$-0.462479\pi$
$967$	7.31415	0.235207	0.117604	+	0.993061i	$0.462479\pi$
$968$	0	0
$969$	−6.15792	−0.197821
$970$	0	0
$971$	−46.2730	−1.48497	−0.742486	−	0.669862i	$-0.766353\pi$
$971$	−46.2730	−1.48497	−0.742486	+	0.669862i	$0.766353\pi$
$972$	0	0
$973$	−45.2860	−1.45180
$974$	0	0
$975$	10.9786	0.351596
$976$	0	0
$977$	−20.3074	−0.649692	−0.324846	−	0.945767i	$-0.605312\pi$
$977$	−20.3074	−0.649692	−0.324846	+	0.945767i	$0.605312\pi$
$978$	0	0
$979$	1.37169	0.0438395
$980$	0	0
$981$	−33.7648	−1.07803
$982$	0	0
$983$	52.6012	1.67772	0.838859	−	0.544348i	$-0.183223\pi$
$983$	52.6012	1.67772	0.838859	+	0.544348i	$0.183223\pi$
$984$	0	0
$985$	21.3545	0.680410
$986$	0	0
$987$	−12.8610	−0.409370
$988$	0	0
$989$	20.7862	0.660964
$990$	0	0
$991$	−5.29587	−0.168229	−0.0841144	−	0.996456i	$-0.526806\pi$
$991$	−5.29587	−0.168229	−0.0841144	+	0.996456i	$0.526806\pi$
$992$	0	0
$993$	37.1940	1.18032
$994$	0	0
$995$	−12.1249	−0.384387
$996$	0	0
$997$	−1.61531	−0.0511573	−0.0255786	−	0.999673i	$-0.508143\pi$
$997$	−1.61531	−0.0511573	−0.0255786	+	0.999673i	$0.508143\pi$
$998$	0	0
$999$	−1.19656	−0.0378574

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2960.2.a.r.1.1		3
4.3	odd	2		1480.2.a.e.1.3	✓	3
20.19	odd	2		7400.2.a.l.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1480.2.a.e.1.3	✓	3	4.3	odd	2
2960.2.a.r.1.1		3	1.1	even	1	trivial
7400.2.a.l.1.1		3	20.19	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2960 = 2^{4} \cdot 5 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2960.a (trivial)

\(f(q)\)	\(=\)	\(q-2.34292 q^{3} -1.00000 q^{5} +2.34292 q^{7} +2.48929 q^{9} +O(q^{10})\)	\(q-2.34292 q^{3} -1.00000 q^{5} +2.34292 q^{7} +2.48929 q^{9} +3.48929 q^{11} -4.68585 q^{13} +2.34292 q^{15} -2.29273 q^{17} -1.14637 q^{19} -5.48929 q^{21} -2.97858 q^{23} +1.00000 q^{25} +1.19656 q^{27} -2.00000 q^{29} +10.5181 q^{31} -8.17513 q^{33} -2.34292 q^{35} -1.00000 q^{37} +10.9786 q^{39} +2.17513 q^{41} -6.97858 q^{43} -2.48929 q^{45} +2.34292 q^{47} -1.51071 q^{49} +5.37169 q^{51} -0.217980 q^{53} -3.48929 q^{55} +2.68585 q^{57} -4.22533 q^{59} +0.978577 q^{61} +5.83221 q^{63} +4.68585 q^{65} -2.85363 q^{67} +6.97858 q^{69} +12.1751 q^{71} -1.82487 q^{73} -2.34292 q^{75} +8.17513 q^{77} -8.22533 q^{79} -10.2713 q^{81} +13.9070 q^{83} +2.29273 q^{85} +4.68585 q^{87} +0.393115 q^{89} -10.9786 q^{91} -24.6430 q^{93} +1.14637 q^{95} +18.3503 q^{97} +8.68585 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - q^{3} - 3 q^{5} + q^{7}+O(q^{10}) \) 3 * q - q^3 - 3 * q^5 + q^7	\( 3 q - q^{3} - 3 q^{5} + q^{7} + 3 q^{11} - 2 q^{13} + q^{15} - 4 q^{17} - 2 q^{19} - 9 q^{21} + 6 q^{23} + 3 q^{25} - q^{27} - 6 q^{29} + 6 q^{31} - 5 q^{33} - q^{35} - 3 q^{37} + 18 q^{39} - 13 q^{41} - 6 q^{43} + q^{47} - 12 q^{49} - 8 q^{51} - 11 q^{53} - 3 q^{55} - 4 q^{57} + 10 q^{59} - 12 q^{61} + 4 q^{63} + 2 q^{65} - 10 q^{67} + 6 q^{69} + 17 q^{71} - 25 q^{73} - q^{75} + 5 q^{77} - 2 q^{79} - 13 q^{81} + 15 q^{83} + 4 q^{85} + 2 q^{87} - 8 q^{89} - 18 q^{91} - 32 q^{93} + 2 q^{95} + 16 q^{97} + 14 q^{99}+O(q^{100}) \) 3 * q - q^3 - 3 * q^5 + q^7 + 3 * q^11 - 2 * q^13 + q^15 - 4 * q^17 - 2 * q^19 - 9 * q^21 + 6 * q^23 + 3 * q^25 - q^27 - 6 * q^29 + 6 * q^31 - 5 * q^33 - q^35 - 3 * q^37 + 18 * q^39 - 13 * q^41 - 6 * q^43 + q^47 - 12 * q^49 - 8 * q^51 - 11 * q^53 - 3 * q^55 - 4 * q^57 + 10 * q^59 - 12 * q^61 + 4 * q^63 + 2 * q^65 - 10 * q^67 + 6 * q^69 + 17 * q^71 - 25 * q^73 - q^75 + 5 * q^77 - 2 * q^79 - 13 * q^81 + 15 * q^83 + 4 * q^85 + 2 * q^87 - 8 * q^89 - 18 * q^91 - 32 * q^93 + 2 * q^95 + 16 * q^97 + 14 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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