LMFDB - Embedded newform 2960.2.a.r.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:	$1$
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			1.3
Root			$-1.81361$ 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)

$f(q)$	$=$	$q+1.81361 q^{3} -1.00000 q^{5} -1.81361 q^{7} +0.289169 q^{9} +O(q^{10})$	$q+1.81361 q^{3} -1.00000 q^{5} -1.81361 q^{7} +0.289169 q^{9} +1.28917 q^{11} +3.62721 q^{13} -1.81361 q^{15} -6.20555 q^{17} -3.10278 q^{19} -3.28917 q^{21} +1.42166 q^{23} +1.00000 q^{25} -4.91638 q^{27} -2.00000 q^{29} -4.15165 q^{31} +2.33804 q^{33} +1.81361 q^{35} -1.00000 q^{37} +6.57834 q^{39} -8.33804 q^{41} -2.57834 q^{43} -0.289169 q^{45} -1.81361 q^{47} -3.71083 q^{49} -11.2544 q^{51} +1.49472 q^{53} -1.28917 q^{55} -5.62721 q^{57} +14.3572 q^{59} -3.42166 q^{61} -0.524438 q^{63} -3.62721 q^{65} -0.897225 q^{67} +2.57834 q^{69} +1.66196 q^{71} -12.3380 q^{73} +1.81361 q^{75} -2.33804 q^{77} +10.3572 q^{79} -9.78389 q^{81} +13.1758 q^{83} +6.20555 q^{85} -3.62721 q^{87} -11.8328 q^{89} -6.57834 q^{91} -7.52946 q^{93} +3.10278 q^{95} -2.67609 q^{97} +0.372787 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	1.81361	1.04709	0.523543	−	0.851999i	$-0.324610\pi$
$3$	1.81361	1.04709	0.523543	+	0.851999i	$0.324610\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−1.81361	−0.685479	−0.342739	−	0.939431i	$-0.611355\pi$
$7$	−1.81361	−0.685479	−0.342739	+	0.939431i	$0.611355\pi$
$8$	0	0
$9$	0.289169	0.0963895
$10$	0	0
$11$	1.28917	0.388699	0.194349	−	0.980932i	$-0.437740\pi$
$11$	1.28917	0.388699	0.194349	+	0.980932i	$0.437740\pi$
$12$	0	0
$13$	3.62721	1.00601	0.503004	−	0.864284i	$-0.332228\pi$
$13$	3.62721	1.00601	0.503004	+	0.864284i	$0.332228\pi$
$14$	0	0
$15$	−1.81361	−0.468271
$16$	0	0
$17$	−6.20555	−1.50507	−0.752533	−	0.658554i	$-0.771168\pi$
$17$	−6.20555	−1.50507	−0.752533	+	0.658554i	$0.771168\pi$
$18$	0	0
$19$	−3.10278	−0.711825	−0.355913	−	0.934519i	$-0.615830\pi$
$19$	−3.10278	−0.711825	−0.355913	+	0.934519i	$0.615830\pi$
$20$	0	0
$21$	−3.28917	−0.717755
$22$	0	0
$23$	1.42166	0.296437	0.148219	−	0.988955i	$-0.452646\pi$
$23$	1.42166	0.296437	0.148219	+	0.988955i	$0.452646\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−4.91638	−0.946158
$28$	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
$31$	−4.15165	−0.745659	−0.372829	−	0.927900i	$-0.621612\pi$
$31$	−4.15165	−0.745659	−0.372829	+	0.927900i	$0.621612\pi$
$32$	0	0
$33$	2.33804	0.407001
$34$	0	0
$35$	1.81361	0.306555
$36$	0	0
$37$	−1.00000	−0.164399
$37$	−1.00000	−0.164399
$38$	0	0
$39$	6.57834	1.05338
$40$	0	0
$41$	−8.33804	−1.30218	−0.651092	−	0.758999i	$-0.725689\pi$
$41$	−8.33804	−1.30218	−0.651092	+	0.758999i	$0.725689\pi$
$42$	0	0
$43$	−2.57834	−0.393193	−0.196596	−	0.980485i	$-0.562989\pi$
$43$	−2.57834	−0.393193	−0.196596	+	0.980485i	$0.562989\pi$
$44$	0	0
$45$	−0.289169	−0.0431067
$46$	0	0
$47$	−1.81361	−0.264542	−0.132271	−	0.991214i	$-0.542227\pi$
$47$	−1.81361	−0.264542	−0.132271	+	0.991214i	$0.542227\pi$
$48$	0	0
$49$	−3.71083	−0.530119
$50$	0	0
$51$	−11.2544	−1.57593
$52$	0	0
$53$	1.49472	0.205315	0.102658	−	0.994717i	$-0.467265\pi$
$53$	1.49472	0.205315	0.102658	+	0.994717i	$0.467265\pi$
$54$	0	0
$55$	−1.28917	−0.173831
$56$	0	0
$57$	−5.62721	−0.745343
$58$	0	0
$59$	14.3572	1.86915	0.934574	−	0.355768i	$-0.115781\pi$
$59$	14.3572	1.86915	0.934574	+	0.355768i	$0.115781\pi$
$60$	0	0
$61$	−3.42166	−0.438099	−0.219050	−	0.975714i	$-0.570296\pi$
$61$	−3.42166	−0.438099	−0.219050	+	0.975714i	$0.570296\pi$
$62$	0	0
$63$	−0.524438	−0.0660730
$64$	0	0
$65$	−3.62721	−0.449900
$66$	0	0
$67$	−0.897225	−0.109613	−0.0548067	−	0.998497i	$-0.517454\pi$
$67$	−0.897225	−0.109613	−0.0548067	+	0.998497i	$0.517454\pi$
$68$	0	0
$69$	2.57834	0.310395
$70$	0	0
$71$	1.66196	0.197238	0.0986189	−	0.995125i	$-0.468558\pi$
$71$	1.66196	0.197238	0.0986189	+	0.995125i	$0.468558\pi$
$72$	0	0
$73$	−12.3380	−1.44406	−0.722029	−	0.691862i	$-0.756790\pi$
$73$	−12.3380	−1.44406	−0.722029	+	0.691862i	$0.756790\pi$
$74$	0	0
$75$	1.81361	0.209417
$76$	0	0
$77$	−2.33804	−0.266445
$78$	0	0
$79$	10.3572	1.16528	0.582638	−	0.812732i	$-0.302021\pi$
$79$	10.3572	1.16528	0.582638	+	0.812732i	$0.302021\pi$
$80$	0	0
$81$	−9.78389	−1.08710
$82$	0	0
$83$	13.1758	1.44624	0.723118	−	0.690725i	$-0.242708\pi$
$83$	13.1758	1.44624	0.723118	+	0.690725i	$0.242708\pi$
$84$	0	0
$85$	6.20555	0.673086
$86$	0	0
$87$	−3.62721	−0.388878
$88$	0	0
$89$	−11.8328	−1.25427	−0.627135	−	0.778910i	$-0.715773\pi$
$89$	−11.8328	−1.25427	−0.627135	+	0.778910i	$0.715773\pi$
$90$	0	0
$91$	−6.57834	−0.689597
$92$	0	0
$93$	−7.52946	−0.780769
$94$	0	0
$95$	3.10278	0.318338
$96$	0	0
$97$	−2.67609	−0.271716	−0.135858	−	0.990728i	$-0.543379\pi$
$97$	−2.67609	−0.271716	−0.135858	+	0.990728i	$0.543379\pi$
$98$	0	0
$99$	0.372787	0.0374665
$100$	0	0
$101$	13.7003	1.36323	0.681614	−	0.731712i	$-0.261278\pi$
$101$	13.7003	1.36323	0.681614	+	0.731712i	$0.261278\pi$
$102$	0	0
$103$	−7.62721	−0.751532	−0.375766	−	0.926715i	$-0.622620\pi$
$103$	−7.62721	−0.751532	−0.375766	+	0.926715i	$0.622620\pi$
$104$	0	0
$105$	3.28917	0.320990
$106$	0	0
$107$	−4.89722	−0.473433	−0.236716	−	0.971579i	$-0.576071\pi$
$107$	−4.89722	−0.473433	−0.236716	+	0.971579i	$0.576071\pi$
$108$	0	0
$109$	−16.9894	−1.62729	−0.813646	−	0.581360i	$-0.802521\pi$
$109$	−16.9894	−1.62729	−0.813646	+	0.581360i	$0.802521\pi$
$110$	0	0
$111$	−1.81361	−0.172140
$112$	0	0
$113$	−4.37279	−0.411357	−0.205679	−	0.978620i	$-0.565940\pi$
$113$	−4.37279	−0.411357	−0.205679	+	0.978620i	$0.565940\pi$
$114$	0	0
$115$	−1.42166	−0.132571
$116$	0	0
$117$	1.04888	0.0969686
$118$	0	0
$119$	11.2544	1.03169
$120$	0	0
$121$	−9.33804	−0.848913
$122$	0	0
$123$	−15.1219	−1.36350
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−7.34307	−0.651592	−0.325796	−	0.945440i	$-0.605632\pi$
$127$	−7.34307	−0.651592	−0.325796	+	0.945440i	$0.605632\pi$
$128$	0	0
$129$	−4.67609	−0.411707
$130$	0	0
$131$	−20.6705	−1.80599	−0.902997	−	0.429647i	$-0.858638\pi$
$131$	−20.6705	−1.80599	−0.902997	+	0.429647i	$0.858638\pi$
$132$	0	0
$133$	5.62721	0.487941
$134$	0	0
$135$	4.91638	0.423135
$136$	0	0
$137$	−10.4111	−0.889480	−0.444740	−	0.895660i	$-0.646704\pi$
$137$	−10.4111	−0.889480	−0.444740	+	0.895660i	$0.646704\pi$
$138$	0	0
$139$	6.09775	0.517205	0.258602	−	0.965984i	$-0.416738\pi$
$139$	6.09775	0.517205	0.258602	+	0.965984i	$0.416738\pi$
$140$	0	0
$141$	−3.28917	−0.276998
$142$	0	0
$143$	4.67609	0.391034
$144$	0	0
$145$	2.00000	0.166091
$146$	0	0
$147$	−6.72999	−0.555080
$148$	0	0
$149$	−19.1219	−1.56653	−0.783265	−	0.621688i	$-0.786447\pi$
$149$	−19.1219	−1.56653	−0.783265	+	0.621688i	$0.786447\pi$
$150$	0	0
$151$	18.6167	1.51500	0.757501	−	0.652834i	$-0.226420\pi$
$151$	18.6167	1.51500	0.757501	+	0.652834i	$0.226420\pi$
$152$	0	0
$153$	−1.79445	−0.145073
$154$	0	0
$155$	4.15165	0.333469
$156$	0	0
$157$	−9.49472	−0.757761	−0.378881	−	0.925446i	$-0.623691\pi$
$157$	−9.49472	−0.757761	−0.378881	+	0.925446i	$0.623691\pi$
$158$	0	0
$159$	2.71083	0.214983
$160$	0	0
$161$	−2.57834	−0.203201
$162$	0	0
$163$	−7.21611	−0.565210	−0.282605	−	0.959236i	$-0.591198\pi$
$163$	−7.21611	−0.565210	−0.282605	+	0.959236i	$0.591198\pi$
$164$	0	0
$165$	−2.33804	−0.182017
$166$	0	0
$167$	−10.2056	−0.789729	−0.394865	−	0.918739i	$-0.629208\pi$
$167$	−10.2056	−0.789729	−0.394865	+	0.918739i	$0.629208\pi$
$168$	0	0
$169$	0.156674	0.0120519
$170$	0	0
$171$	−0.897225	−0.0686125
$172$	0	0
$173$	11.5925	0.881359	0.440680	−	0.897664i	$-0.354737\pi$
$173$	11.5925	0.881359	0.440680	+	0.897664i	$0.354737\pi$
$174$	0	0
$175$	−1.81361	−0.137096
$176$	0	0
$177$	26.0383	1.95716
$178$	0	0
$179$	−6.72999	−0.503023	−0.251511	−	0.967854i	$-0.580928\pi$
$179$	−6.72999	−0.503023	−0.251511	+	0.967854i	$0.580928\pi$
$180$	0	0
$181$	9.96526	0.740712	0.370356	−	0.928890i	$-0.379236\pi$
$181$	9.96526	0.740712	0.370356	+	0.928890i	$0.379236\pi$
$182$	0	0
$183$	−6.20555	−0.458727
$184$	0	0
$185$	1.00000	0.0735215
$186$	0	0
$187$	−8.00000	−0.585018
$188$	0	0
$189$	8.91638	0.648571
$190$	0	0
$191$	12.1517	0.879263	0.439631	−	0.898178i	$-0.355109\pi$
$191$	12.1517	0.879263	0.439631	+	0.898178i	$0.355109\pi$
$192$	0	0
$193$	−4.31335	−0.310482	−0.155241	−	0.987877i	$-0.549615\pi$
$193$	−4.31335	−0.310482	−0.155241	+	0.987877i	$0.549615\pi$
$194$	0	0
$195$	−6.57834	−0.471085
$196$	0	0
$197$	25.8363	1.84076	0.920381	−	0.391022i	$-0.127878\pi$
$197$	25.8363	1.84076	0.920381	+	0.391022i	$0.127878\pi$
$198$	0	0
$199$	9.68111	0.686276	0.343138	−	0.939285i	$-0.388510\pi$
$199$	9.68111	0.686276	0.343138	+	0.939285i	$0.388510\pi$
$200$	0	0
$201$	−1.62721	−0.114775
$202$	0	0
$203$	3.62721	0.254580
$204$	0	0
$205$	8.33804	0.582354
$206$	0	0
$207$	0.411100	0.0285734
$208$	0	0
$209$	−4.00000	−0.276686
$210$	0	0
$211$	−10.6030	−0.729943	−0.364971	−	0.931019i	$-0.618921\pi$
$211$	−10.6030	−0.729943	−0.364971	+	0.931019i	$0.618921\pi$
$212$	0	0
$213$	3.01413	0.206525
$214$	0	0
$215$	2.57834	0.175841
$216$	0	0
$217$	7.52946	0.511133
$218$	0	0
$219$	−22.3764	−1.51205
$220$	0	0
$221$	−22.5089	−1.51411
$222$	0	0
$223$	11.5386	0.772680	0.386340	−	0.922356i	$-0.373739\pi$
$223$	11.5386	0.772680	0.386340	+	0.922356i	$0.373739\pi$
$224$	0	0
$225$	0.289169	0.0192779
$226$	0	0
$227$	−1.79445	−0.119102	−0.0595509	−	0.998225i	$-0.518967\pi$
$227$	−1.79445	−0.119102	−0.0595509	+	0.998225i	$0.518967\pi$
$228$	0	0
$229$	10.1708	0.672106	0.336053	−	0.941843i	$-0.390908\pi$
$229$	10.1708	0.672106	0.336053	+	0.941843i	$0.390908\pi$
$230$	0	0
$231$	−4.24029	−0.278991
$232$	0	0
$233$	13.6655	0.895258	0.447629	−	0.894219i	$-0.352268\pi$
$233$	13.6655	0.895258	0.447629	+	0.894219i	$0.352268\pi$
$234$	0	0
$235$	1.81361	0.118307
$236$	0	0
$237$	18.7839	1.22014
$238$	0	0
$239$	21.5733	1.39546	0.697731	−	0.716360i	$-0.254193\pi$
$239$	21.5733	1.39546	0.697731	+	0.716360i	$0.254193\pi$
$240$	0	0
$241$	−0.843326	−0.0543234	−0.0271617	−	0.999631i	$-0.508647\pi$
$241$	−0.843326	−0.0543234	−0.0271617	+	0.999631i	$0.508647\pi$
$242$	0	0
$243$	−2.99498	−0.192128
$244$	0	0
$245$	3.71083	0.237076
$246$	0	0
$247$	−11.2544	−0.716102
$248$	0	0
$249$	23.8958	1.51433
$250$	0	0
$251$	4.63224	0.292384	0.146192	−	0.989256i	$-0.453298\pi$
$251$	4.63224	0.292384	0.146192	+	0.989256i	$0.453298\pi$
$252$	0	0
$253$	1.83276	0.115225
$254$	0	0
$255$	11.2544	0.704779
$256$	0	0
$257$	16.3033	1.01697	0.508486	−	0.861070i	$-0.330205\pi$
$257$	16.3033	1.01697	0.508486	+	0.861070i	$0.330205\pi$
$258$	0	0
$259$	1.81361	0.112692
$260$	0	0
$261$	−0.578337	−0.0357982
$262$	0	0
$263$	−26.5280	−1.63579	−0.817894	−	0.575370i	$-0.804858\pi$
$263$	−26.5280	−1.63579	−0.817894	+	0.575370i	$0.804858\pi$
$264$	0	0
$265$	−1.49472	−0.0918198
$266$	0	0
$267$	−21.4600	−1.31333
$268$	0	0
$269$	8.37279	0.510498	0.255249	−	0.966875i	$-0.417843\pi$
$269$	8.37279	0.510498	0.255249	+	0.966875i	$0.417843\pi$
$270$	0	0
$271$	1.66196	0.100957	0.0504783	−	0.998725i	$-0.483925\pi$
$271$	1.66196	0.100957	0.0504783	+	0.998725i	$0.483925\pi$
$272$	0	0
$273$	−11.9305	−0.722068
$274$	0	0
$275$	1.28917	0.0777398
$276$	0	0
$277$	1.79445	0.107818	0.0539090	−	0.998546i	$-0.482832\pi$
$277$	1.79445	0.107818	0.0539090	+	0.998546i	$0.482832\pi$
$278$	0	0
$279$	−1.20053	−0.0718737
$280$	0	0
$281$	6.67609	0.398262	0.199131	−	0.979973i	$-0.436188\pi$
$281$	6.67609	0.398262	0.199131	+	0.979973i	$0.436188\pi$
$282$	0	0
$283$	2.61665	0.155544	0.0777719	−	0.996971i	$-0.475219\pi$
$283$	2.61665	0.155544	0.0777719	+	0.996971i	$0.475219\pi$
$284$	0	0
$285$	5.62721	0.333327
$286$	0	0
$287$	15.1219	0.892619
$288$	0	0
$289$	21.5089	1.26523
$290$	0	0
$291$	−4.85337	−0.284510
$292$	0	0
$293$	−0.843326	−0.0492676	−0.0246338	−	0.999697i	$-0.507842\pi$
$293$	−0.843326	−0.0492676	−0.0246338	+	0.999697i	$0.507842\pi$
$294$	0	0
$295$	−14.3572	−0.835909
$296$	0	0
$297$	−6.33804	−0.367771
$298$	0	0
$299$	5.15667	0.298218
$300$	0	0
$301$	4.67609	0.269525
$302$	0	0
$303$	24.8469	1.42742
$304$	0	0
$305$	3.42166	0.195924
$306$	0	0
$307$	14.9008	0.850433	0.425217	−	0.905092i	$-0.360198\pi$
$307$	14.9008	0.850433	0.425217	+	0.905092i	$0.360198\pi$
$308$	0	0
$309$	−13.8328	−0.786918
$310$	0	0
$311$	21.2005	1.20217	0.601086	−	0.799185i	$-0.294735\pi$
$311$	21.2005	1.20217	0.601086	+	0.799185i	$0.294735\pi$
$312$	0	0
$313$	7.15667	0.404519	0.202260	−	0.979332i	$-0.435172\pi$
$313$	7.15667	0.404519	0.202260	+	0.979332i	$0.435172\pi$
$314$	0	0
$315$	0.524438	0.0295487
$316$	0	0
$317$	−5.25443	−0.295118	−0.147559	−	0.989053i	$-0.547142\pi$
$317$	−5.25443	−0.295118	−0.147559	+	0.989053i	$0.547142\pi$
$318$	0	0
$319$	−2.57834	−0.144359
$320$	0	0
$321$	−8.88164	−0.495725
$322$	0	0
$323$	19.2544	1.07134
$324$	0	0
$325$	3.62721	0.201202
$326$	0	0
$327$	−30.8122	−1.70392
$328$	0	0
$329$	3.28917	0.181338
$330$	0	0
$331$	−18.3189	−1.00690	−0.503449	−	0.864025i	$-0.667936\pi$
$331$	−18.3189	−1.00690	−0.503449	+	0.864025i	$0.667936\pi$
$332$	0	0
$333$	−0.289169	−0.0158463
$334$	0	0
$335$	0.897225	0.0490206
$336$	0	0
$337$	−8.74914	−0.476596	−0.238298	−	0.971192i	$-0.576590\pi$
$337$	−8.74914	−0.476596	−0.238298	+	0.971192i	$0.576590\pi$
$338$	0	0
$339$	−7.93051	−0.430726
$340$	0	0
$341$	−5.35218	−0.289837
$342$	0	0
$343$	19.4252	1.04886
$344$	0	0
$345$	−2.57834	−0.138813
$346$	0	0
$347$	−15.7350	−0.844700	−0.422350	−	0.906433i	$-0.638795\pi$
$347$	−15.7350	−0.844700	−0.422350	+	0.906433i	$0.638795\pi$
$348$	0	0
$349$	5.52946	0.295985	0.147993	−	0.988988i	$-0.452719\pi$
$349$	5.52946	0.295985	0.147993	+	0.988988i	$0.452719\pi$
$350$	0	0
$351$	−17.8328	−0.951842
$352$	0	0
$353$	−3.83276	−0.203997	−0.101999	−	0.994785i	$-0.532524\pi$
$353$	−3.83276	−0.203997	−0.101999	+	0.994785i	$0.532524\pi$
$354$	0	0
$355$	−1.66196	−0.0882074
$356$	0	0
$357$	20.4111	1.08027
$358$	0	0
$359$	−8.95469	−0.472611	−0.236305	−	0.971679i	$-0.575937\pi$
$359$	−8.95469	−0.472611	−0.236305	+	0.971679i	$0.575937\pi$
$360$	0	0
$361$	−9.37279	−0.493305
$362$	0	0
$363$	−16.9355	−0.888885
$364$	0	0
$365$	12.3380	0.645803
$366$	0	0
$367$	−9.53500	−0.497723	−0.248861	−	0.968539i	$-0.580056\pi$
$367$	−9.53500	−0.497723	−0.248861	+	0.968539i	$0.580056\pi$
$368$	0	0
$369$	−2.41110	−0.125517
$370$	0	0
$371$	−2.71083	−0.140739
$372$	0	0
$373$	−31.5230	−1.63220	−0.816099	−	0.577912i	$-0.803868\pi$
$373$	−31.5230	−1.63220	−0.816099	+	0.577912i	$0.803868\pi$
$374$	0	0
$375$	−1.81361	−0.0936542
$376$	0	0
$377$	−7.25443	−0.373622
$378$	0	0
$379$	16.5436	0.849787	0.424894	−	0.905243i	$-0.360311\pi$
$379$	16.5436	0.849787	0.424894	+	0.905243i	$0.360311\pi$
$380$	0	0
$381$	−13.3174	−0.682273
$382$	0	0
$383$	−17.7633	−0.907661	−0.453831	−	0.891088i	$-0.649943\pi$
$383$	−17.7633	−0.907661	−0.453831	+	0.891088i	$0.649943\pi$
$384$	0	0
$385$	2.33804	0.119158
$386$	0	0
$387$	−0.745574	−0.0378997
$388$	0	0
$389$	−36.6550	−1.85848	−0.929240	−	0.369476i	$-0.879537\pi$
$389$	−36.6550	−1.85848	−0.929240	+	0.369476i	$0.879537\pi$
$390$	0	0
$391$	−8.82220	−0.446158
$392$	0	0
$393$	−37.4882	−1.89103
$394$	0	0
$395$	−10.3572	−0.521127
$396$	0	0
$397$	14.7008	0.737811	0.368906	−	0.929467i	$-0.379733\pi$
$397$	14.7008	0.737811	0.368906	+	0.929467i	$0.379733\pi$
$398$	0	0
$399$	10.2056	0.510917
$400$	0	0
$401$	34.6066	1.72817	0.864086	−	0.503345i	$-0.167897\pi$
$401$	34.6066	1.72817	0.864086	+	0.503345i	$0.167897\pi$
$402$	0	0
$403$	−15.0589	−0.750138
$404$	0	0
$405$	9.78389	0.486165
$406$	0	0
$407$	−1.28917	−0.0639017
$408$	0	0
$409$	24.6550	1.21911	0.609555	−	0.792744i	$-0.291348\pi$
$409$	24.6550	1.21911	0.609555	+	0.792744i	$0.291348\pi$
$410$	0	0
$411$	−18.8816	−0.931363
$412$	0	0
$413$	−26.0383	−1.28126
$414$	0	0
$415$	−13.1758	−0.646776
$416$	0	0
$417$	11.0589	0.541558
$418$	0	0
$419$	18.0347	0.881055	0.440527	−	0.897739i	$-0.354791\pi$
$419$	18.0347	0.881055	0.440527	+	0.897739i	$0.354791\pi$
$420$	0	0
$421$	2.67609	0.130425	0.0652123	−	0.997871i	$-0.479228\pi$
$421$	2.67609	0.130425	0.0652123	+	0.997871i	$0.479228\pi$
$422$	0	0
$423$	−0.524438	−0.0254990
$424$	0	0
$425$	−6.20555	−0.301013
$426$	0	0
$427$	6.20555	0.300308
$428$	0	0
$429$	8.48059	0.409447
$430$	0	0
$431$	−3.88666	−0.187214	−0.0936070	−	0.995609i	$-0.529840\pi$
$431$	−3.88666	−0.187214	−0.0936070	+	0.995609i	$0.529840\pi$
$432$	0	0
$433$	24.7491	1.18937	0.594684	−	0.803960i	$-0.297277\pi$
$433$	24.7491	1.18937	0.594684	+	0.803960i	$0.297277\pi$
$434$	0	0
$435$	3.62721	0.173912
$436$	0	0
$437$	−4.41110	−0.211012
$438$	0	0
$439$	−10.9950	−0.524762	−0.262381	−	0.964964i	$-0.584508\pi$
$439$	−10.9950	−0.524762	−0.262381	+	0.964964i	$0.584508\pi$
$440$	0	0
$441$	−1.07306	−0.0510979
$442$	0	0
$443$	24.4303	1.16072	0.580358	−	0.814361i	$-0.302913\pi$
$443$	24.4303	1.16072	0.580358	+	0.814361i	$0.302913\pi$
$444$	0	0
$445$	11.8328	0.560927
$446$	0	0
$447$	−34.6797	−1.64029
$448$	0	0
$449$	15.0872	0.712008	0.356004	−	0.934484i	$-0.384139\pi$
$449$	15.0872	0.712008	0.356004	+	0.934484i	$0.384139\pi$
$450$	0	0
$451$	−10.7491	−0.506157
$452$	0	0
$453$	33.7633	1.58634
$454$	0	0
$455$	6.57834	0.308397
$456$	0	0
$457$	−17.6655	−0.826358	−0.413179	−	0.910650i	$-0.635582\pi$
$457$	−17.6655	−0.826358	−0.413179	+	0.910650i	$0.635582\pi$
$458$	0	0
$459$	30.5089	1.42403
$460$	0	0
$461$	−17.6655	−0.822766	−0.411383	−	0.911463i	$-0.634954\pi$
$461$	−17.6655	−0.822766	−0.411383	+	0.911463i	$0.634954\pi$
$462$	0	0
$463$	3.51941	0.163561	0.0817805	−	0.996650i	$-0.473939\pi$
$463$	3.51941	0.163561	0.0817805	+	0.996650i	$0.473939\pi$
$464$	0	0
$465$	7.52946	0.349170
$466$	0	0
$467$	41.6061	1.92530	0.962650	−	0.270749i	$-0.0872712\pi$
$467$	41.6061	1.92530	0.962650	+	0.270749i	$0.0872712\pi$
$468$	0	0
$469$	1.62721	0.0751377
$470$	0	0
$471$	−17.2197	−0.793441
$472$	0	0
$473$	−3.32391	−0.152834
$474$	0	0
$475$	−3.10278	−0.142365
$476$	0	0
$477$	0.432226	0.0197903
$478$	0	0
$479$	7.67107	0.350500	0.175250	−	0.984524i	$-0.443927\pi$
$479$	7.67107	0.350500	0.175250	+	0.984524i	$0.443927\pi$
$480$	0	0
$481$	−3.62721	−0.165387
$482$	0	0
$483$	−4.67609	−0.212769
$484$	0	0
$485$	2.67609	0.121515
$486$	0	0
$487$	17.4600	0.791187	0.395594	−	0.918426i	$-0.370539\pi$
$487$	17.4600	0.791187	0.395594	+	0.918426i	$0.370539\pi$
$488$	0	0
$489$	−13.0872	−0.591823
$490$	0	0
$491$	−16.1955	−0.730893	−0.365446	−	0.930832i	$-0.619084\pi$
$491$	−16.1955	−0.730893	−0.365446	+	0.930832i	$0.619084\pi$
$492$	0	0
$493$	12.4111	0.558968
$494$	0	0
$495$	−0.372787	−0.0167555
$496$	0	0
$497$	−3.01413	−0.135202
$498$	0	0
$499$	−9.41612	−0.421524	−0.210762	−	0.977537i	$-0.567594\pi$
$499$	−9.41612	−0.421524	−0.210762	+	0.977537i	$0.567594\pi$
$500$	0	0
$501$	−18.5089	−0.826915
$502$	0	0
$503$	25.2333	1.12510	0.562549	−	0.826764i	$-0.309821\pi$
$503$	25.2333	1.12510	0.562549	+	0.826764i	$0.309821\pi$
$504$	0	0
$505$	−13.7003	−0.609654
$506$	0	0
$507$	0.284145	0.0126193
$508$	0	0
$509$	39.4736	1.74964	0.874818	−	0.484451i	$-0.160981\pi$
$509$	39.4736	1.74964	0.874818	+	0.484451i	$0.160981\pi$
$510$	0	0
$511$	22.3764	0.989872
$512$	0	0
$513$	15.2544	0.673499
$514$	0	0
$515$	7.62721	0.336095
$516$	0	0
$517$	−2.33804	−0.102827
$518$	0	0
$519$	21.0242	0.922859
$520$	0	0
$521$	22.6413	0.991935	0.495968	−	0.868341i	$-0.334813\pi$
$521$	22.6413	0.991935	0.495968	+	0.868341i	$0.334813\pi$
$522$	0	0
$523$	−25.8328	−1.12959	−0.564794	−	0.825232i	$-0.691044\pi$
$523$	−25.8328	−1.12959	−0.564794	+	0.825232i	$0.691044\pi$
$524$	0	0
$525$	−3.28917	−0.143551
$526$	0	0
$527$	25.7633	1.12227
$528$	0	0
$529$	−20.9789	−0.912125
$530$	0	0
$531$	4.15165	0.180166
$532$	0	0
$533$	−30.2439	−1.31001
$534$	0	0
$535$	4.89722	0.211725
$536$	0	0
$537$	−12.2056	−0.526708
$538$	0	0
$539$	−4.78389	−0.206057
$540$	0	0
$541$	−4.57834	−0.196838	−0.0984190	−	0.995145i	$-0.531379\pi$
$541$	−4.57834	−0.196838	−0.0984190	+	0.995145i	$0.531379\pi$
$542$	0	0
$543$	18.0731	0.775589
$544$	0	0
$545$	16.9894	0.727748
$546$	0	0
$547$	−14.7244	−0.629572	−0.314786	−	0.949163i	$-0.601933\pi$
$547$	−14.7244	−0.629572	−0.314786	+	0.949163i	$0.601933\pi$
$548$	0	0
$549$	−0.989437	−0.0422282
$550$	0	0
$551$	6.20555	0.264365
$552$	0	0
$553$	−18.7839	−0.798772
$554$	0	0
$555$	1.81361	0.0769833
$556$	0	0
$557$	12.4494	0.527499	0.263749	−	0.964591i	$-0.415041\pi$
$557$	12.4494	0.527499	0.263749	+	0.964591i	$0.415041\pi$
$558$	0	0
$559$	−9.35218	−0.395555
$560$	0	0
$561$	−14.5089	−0.612564
$562$	0	0
$563$	−9.83276	−0.414402	−0.207201	−	0.978298i	$-0.566435\pi$
$563$	−9.83276	−0.414402	−0.207201	+	0.978298i	$0.566435\pi$
$564$	0	0
$565$	4.37279	0.183965
$566$	0	0
$567$	17.7441	0.745183
$568$	0	0
$569$	−20.2439	−0.848667	−0.424333	−	0.905506i	$-0.639492\pi$
$569$	−20.2439	−0.848667	−0.424333	+	0.905506i	$0.639492\pi$
$570$	0	0
$571$	41.6691	1.74380	0.871899	−	0.489686i	$-0.162889\pi$
$571$	41.6691	1.74380	0.871899	+	0.489686i	$0.162889\pi$
$572$	0	0
$573$	22.0383	0.920664
$574$	0	0
$575$	1.42166	0.0592874
$576$	0	0
$577$	−9.04888	−0.376710	−0.188355	−	0.982101i	$-0.560315\pi$
$577$	−9.04888	−0.376710	−0.188355	+	0.982101i	$0.560315\pi$
$578$	0	0
$579$	−7.82272	−0.325101
$580$	0	0
$581$	−23.8958	−0.991364
$582$	0	0
$583$	1.92694	0.0798059
$584$	0	0
$585$	−1.04888	−0.0433657
$586$	0	0
$587$	4.41110	0.182066	0.0910328	−	0.995848i	$-0.470983\pi$
$587$	4.41110	0.182066	0.0910328	+	0.995848i	$0.470983\pi$
$588$	0	0
$589$	12.8816	0.530779
$590$	0	0
$591$	46.8569	1.92744
$592$	0	0
$593$	−11.9269	−0.489781	−0.244890	−	0.969551i	$-0.578752\pi$
$593$	−11.9269	−0.489781	−0.244890	+	0.969551i	$0.578752\pi$
$594$	0	0
$595$	−11.2544	−0.461386
$596$	0	0
$597$	17.5577	0.718590
$598$	0	0
$599$	27.4564	1.12184	0.560919	−	0.827871i	$-0.310448\pi$
$599$	27.4564	1.12184	0.560919	+	0.827871i	$0.310448\pi$
$600$	0	0
$601$	15.7038	0.640573	0.320286	−	0.947321i	$-0.396221\pi$
$601$	15.7038	0.640573	0.320286	+	0.947321i	$0.396221\pi$
$602$	0	0
$603$	−0.259449	−0.0105656
$604$	0	0
$605$	9.33804	0.379645
$606$	0	0
$607$	−23.1355	−0.939043	−0.469521	−	0.882921i	$-0.655574\pi$
$607$	−23.1355	−0.939043	−0.469521	+	0.882921i	$0.655574\pi$
$608$	0	0
$609$	6.57834	0.266568
$610$	0	0
$611$	−6.57834	−0.266131
$612$	0	0
$613$	−42.1013	−1.70046	−0.850228	−	0.526414i	$-0.823536\pi$
$613$	−42.1013	−1.70046	−0.850228	+	0.526414i	$0.823536\pi$
$614$	0	0
$615$	15.1219	0.609775
$616$	0	0
$617$	−4.60303	−0.185311	−0.0926556	−	0.995698i	$-0.529536\pi$
$617$	−4.60303	−0.185311	−0.0926556	+	0.995698i	$0.529536\pi$
$618$	0	0
$619$	8.24029	0.331205	0.165603	−	0.986193i	$-0.447043\pi$
$619$	8.24029	0.331205	0.165603	+	0.986193i	$0.447043\pi$
$620$	0	0
$621$	−6.98944	−0.280476
$622$	0	0
$623$	21.4600	0.859776
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−7.25443	−0.289714
$628$	0	0
$629$	6.20555	0.247431
$630$	0	0
$631$	−17.2772	−0.687793	−0.343896	−	0.939008i	$-0.611747\pi$
$631$	−17.2772	−0.687793	−0.343896	+	0.939008i	$0.611747\pi$
$632$	0	0
$633$	−19.2297	−0.764313
$634$	0	0
$635$	7.34307	0.291401
$636$	0	0
$637$	−13.4600	−0.533304
$638$	0	0
$639$	0.480585	0.0190117
$640$	0	0
$641$	12.2091	0.482231	0.241116	−	0.970496i	$-0.422487\pi$
$641$	12.2091	0.482231	0.241116	+	0.970496i	$0.422487\pi$
$642$	0	0
$643$	−21.1184	−0.832827	−0.416413	−	0.909175i	$-0.636713\pi$
$643$	−21.1184	−0.832827	−0.416413	+	0.909175i	$0.636713\pi$
$644$	0	0
$645$	4.67609	0.184121
$646$	0	0
$647$	−28.8222	−1.13312	−0.566559	−	0.824021i	$-0.691726\pi$
$647$	−28.8222	−1.13312	−0.566559	+	0.824021i	$0.691726\pi$
$648$	0	0
$649$	18.5089	0.726536
$650$	0	0
$651$	13.6555	0.535200
$652$	0	0
$653$	−30.8716	−1.20810	−0.604049	−	0.796947i	$-0.706447\pi$
$653$	−30.8716	−1.20810	−0.604049	+	0.796947i	$0.706447\pi$
$654$	0	0
$655$	20.6705	0.807665
$656$	0	0
$657$	−3.56777	−0.139192
$658$	0	0
$659$	−31.5713	−1.22984	−0.614922	−	0.788588i	$-0.710813\pi$
$659$	−31.5713	−1.22984	−0.614922	+	0.788588i	$0.710813\pi$
$660$	0	0
$661$	−15.0872	−0.586824	−0.293412	−	0.955986i	$-0.594791\pi$
$661$	−15.0872	−0.586824	−0.293412	+	0.955986i	$0.594791\pi$
$662$	0	0
$663$	−40.8222	−1.58540
$664$	0	0
$665$	−5.62721	−0.218214
$666$	0	0
$667$	−2.84333	−0.110094
$668$	0	0
$669$	20.9264	0.809062
$670$	0	0
$671$	−4.41110	−0.170289
$672$	0	0
$673$	−7.39697	−0.285132	−0.142566	−	0.989785i	$-0.545535\pi$
$673$	−7.39697	−0.285132	−0.142566	+	0.989785i	$0.545535\pi$
$674$	0	0
$675$	−4.91638	−0.189232
$676$	0	0
$677$	11.3970	0.438021	0.219011	−	0.975723i	$-0.429717\pi$
$677$	11.3970	0.438021	0.219011	+	0.975723i	$0.429717\pi$
$678$	0	0
$679$	4.85337	0.186255
$680$	0	0
$681$	−3.25443	−0.124710
$682$	0	0
$683$	1.75614	0.0671967	0.0335984	−	0.999435i	$-0.489303\pi$
$683$	1.75614	0.0671967	0.0335984	+	0.999435i	$0.489303\pi$
$684$	0	0
$685$	10.4111	0.397788
$686$	0	0
$687$	18.4458	0.703753
$688$	0	0
$689$	5.42166	0.206549
$690$	0	0
$691$	−39.2233	−1.49212	−0.746061	−	0.665877i	$-0.768058\pi$
$691$	−39.2233	−1.49212	−0.746061	+	0.665877i	$0.768058\pi$
$692$	0	0
$693$	−0.676089	−0.0256825
$694$	0	0
$695$	−6.09775	−0.231301
$696$	0	0
$697$	51.7422	1.95987
$698$	0	0
$699$	24.7839	0.937413
$700$	0	0
$701$	−0.167237	−0.00631645	−0.00315823	−	0.999995i	$-0.501005\pi$
$701$	−0.167237	−0.00631645	−0.00315823	+	0.999995i	$0.501005\pi$
$702$	0	0
$703$	3.10278	0.117023
$704$	0	0
$705$	3.28917	0.123877
$706$	0	0
$707$	−24.8469	−0.934464
$708$	0	0
$709$	−15.6172	−0.586515	−0.293258	−	0.956033i	$-0.594739\pi$
$709$	−15.6172	−0.586515	−0.293258	+	0.956033i	$0.594739\pi$
$710$	0	0
$711$	2.99498	0.112320
$712$	0	0
$713$	−5.90225	−0.221041
$714$	0	0
$715$	−4.67609	−0.174876
$716$	0	0
$717$	39.1255	1.46117
$718$	0	0
$719$	40.9547	1.52735	0.763676	−	0.645599i	$-0.223392\pi$
$719$	40.9547	1.52735	0.763676	+	0.645599i	$0.223392\pi$
$720$	0	0
$721$	13.8328	0.515159
$722$	0	0
$723$	−1.52946	−0.0568813
$724$	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
$729$	23.9200	0.885924
$730$	0	0
$731$	16.0000	0.591781
$732$	0	0
$733$	5.01413	0.185201	0.0926006	−	0.995703i	$-0.470482\pi$
$733$	5.01413	0.185201	0.0926006	+	0.995703i	$0.470482\pi$
$734$	0	0
$735$	6.72999	0.248239
$736$	0	0
$737$	−1.15667	−0.0426066
$738$	0	0
$739$	36.4358	1.34031	0.670156	−	0.742220i	$-0.266227\pi$
$739$	36.4358	1.34031	0.670156	+	0.742220i	$0.266227\pi$
$740$	0	0
$741$	−20.4111	−0.749821
$742$	0	0
$743$	6.66698	0.244588	0.122294	−	0.992494i	$-0.460975\pi$
$743$	6.66698	0.244588	0.122294	+	0.992494i	$0.460975\pi$
$744$	0	0
$745$	19.1219	0.700573
$746$	0	0
$747$	3.81004	0.139402
$748$	0	0
$749$	8.88164	0.324528
$750$	0	0
$751$	31.1986	1.13845	0.569226	−	0.822181i	$-0.307243\pi$
$751$	31.1986	1.13845	0.569226	+	0.822181i	$0.307243\pi$
$752$	0	0
$753$	8.40105	0.306151
$754$	0	0
$755$	−18.6167	−0.677529
$756$	0	0
$757$	−51.9789	−1.88920	−0.944602	−	0.328218i	$-0.893552\pi$
$757$	−51.9789	−1.88920	−0.944602	+	0.328218i	$0.893552\pi$
$758$	0	0
$759$	3.32391	0.120650
$760$	0	0
$761$	−15.1320	−0.548534	−0.274267	−	0.961654i	$-0.588435\pi$
$761$	−15.1320	−0.548534	−0.274267	+	0.961654i	$0.588435\pi$
$762$	0	0
$763$	30.8122	1.11547
$764$	0	0
$765$	1.79445	0.0648785
$766$	0	0
$767$	52.0766	1.88038
$768$	0	0
$769$	40.2933	1.45301	0.726506	−	0.687160i	$-0.241143\pi$
$769$	40.2933	1.45301	0.726506	+	0.687160i	$0.241143\pi$
$770$	0	0
$771$	29.5678	1.06486
$772$	0	0
$773$	−31.6691	−1.13906	−0.569529	−	0.821971i	$-0.692874\pi$
$773$	−31.6691	−1.13906	−0.569529	+	0.821971i	$0.692874\pi$
$774$	0	0
$775$	−4.15165	−0.149132
$776$	0	0
$777$	3.28917	0.117998
$778$	0	0
$779$	25.8711	0.926928
$780$	0	0
$781$	2.14254	0.0766661
$782$	0	0
$783$	9.83276	0.351394
$784$	0	0
$785$	9.49472	0.338881
$786$	0	0
$787$	−39.9880	−1.42542	−0.712709	−	0.701460i	$-0.752532\pi$
$787$	−39.9880	−1.42542	−0.712709	+	0.701460i	$0.752532\pi$
$788$	0	0
$789$	−48.1114	−1.71281
$790$	0	0
$791$	7.93051	0.281977
$792$	0	0
$793$	−12.4111	−0.440731
$794$	0	0
$795$	−2.71083	−0.0961433
$796$	0	0
$797$	5.98995	0.212175	0.106088	−	0.994357i	$-0.466168\pi$
$797$	5.98995	0.212175	0.106088	+	0.994357i	$0.466168\pi$
$798$	0	0
$799$	11.2544	0.398153
$800$	0	0
$801$	−3.42166	−0.120899
$802$	0	0
$803$	−15.9058	−0.561304
$804$	0	0
$805$	2.57834	0.0908744
$806$	0	0
$807$	15.1849	0.534535
$808$	0	0
$809$	27.3522	0.961651	0.480826	−	0.876816i	$-0.340337\pi$
$809$	27.3522	0.961651	0.480826	+	0.876816i	$0.340337\pi$
$810$	0	0
$811$	−23.9753	−0.841887	−0.420943	−	0.907087i	$-0.638301\pi$
$811$	−23.9753	−0.841887	−0.420943	+	0.907087i	$0.638301\pi$
$812$	0	0
$813$	3.01413	0.105710
$814$	0	0
$815$	7.21611	0.252769
$816$	0	0
$817$	8.00000	0.279885
$818$	0	0
$819$	−1.90225	−0.0664699
$820$	0	0
$821$	42.1779	1.47202	0.736010	−	0.676970i	$-0.236707\pi$
$821$	42.1779	1.47202	0.736010	+	0.676970i	$0.236707\pi$
$822$	0	0
$823$	−9.20053	−0.320710	−0.160355	−	0.987059i	$-0.551264\pi$
$823$	−9.20053	−0.320710	−0.160355	+	0.987059i	$0.551264\pi$
$824$	0	0
$825$	2.33804	0.0814003
$826$	0	0
$827$	9.31386	0.323875	0.161937	−	0.986801i	$-0.448226\pi$
$827$	9.31386	0.323875	0.161937	+	0.986801i	$0.448226\pi$
$828$	0	0
$829$	0.167237	0.00580838	0.00290419	−	0.999996i	$-0.499076\pi$
$829$	0.167237	0.00580838	0.00290419	+	0.999996i	$0.499076\pi$
$830$	0	0
$831$	3.25443	0.112895
$832$	0	0
$833$	23.0278	0.797864
$834$	0	0
$835$	10.2056	0.353178
$836$	0	0
$837$	20.4111	0.705511
$838$	0	0
$839$	−36.2721	−1.25225	−0.626126	−	0.779721i	$-0.715361\pi$
$839$	−36.2721	−1.25225	−0.626126	+	0.779721i	$0.715361\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	0	0
$843$	12.1078	0.417015
$844$	0	0
$845$	−0.156674	−0.00538976
$846$	0	0
$847$	16.9355	0.581912
$848$	0	0
$849$	4.74557	0.162868
$850$	0	0
$851$	−1.42166	−0.0487340
$852$	0	0
$853$	−12.3133	−0.421601	−0.210801	−	0.977529i	$-0.567607\pi$
$853$	−12.3133	−0.421601	−0.210801	+	0.977529i	$0.567607\pi$
$854$	0	0
$855$	0.897225	0.0306844
$856$	0	0
$857$	−30.8716	−1.05455	−0.527277	−	0.849694i	$-0.676787\pi$
$857$	−30.8716	−1.05455	−0.527277	+	0.849694i	$0.676787\pi$
$858$	0	0
$859$	−20.5628	−0.701592	−0.350796	−	0.936452i	$-0.614089\pi$
$859$	−20.5628	−0.701592	−0.350796	+	0.936452i	$0.614089\pi$
$860$	0	0
$861$	27.4252	0.934649
$862$	0	0
$863$	41.0816	1.39844	0.699218	−	0.714909i	$-0.253532\pi$
$863$	41.0816	1.39844	0.699218	+	0.714909i	$0.253532\pi$
$864$	0	0
$865$	−11.5925	−0.394156
$866$	0	0
$867$	39.0086	1.32480
$868$	0	0
$869$	13.3522	0.452942
$870$	0	0
$871$	−3.25443	−0.110272
$872$	0	0
$873$	−0.773841	−0.0261905
$874$	0	0
$875$	1.81361	0.0613111
$876$	0	0
$877$	−49.5266	−1.67239	−0.836196	−	0.548430i	$-0.815226\pi$
$877$	−49.5266	−1.67239	−0.836196	+	0.548430i	$0.815226\pi$
$878$	0	0
$879$	−1.52946	−0.0515874
$880$	0	0
$881$	50.9583	1.71683	0.858414	−	0.512958i	$-0.171450\pi$
$881$	50.9583	1.71683	0.858414	+	0.512958i	$0.171450\pi$
$882$	0	0
$883$	−39.3421	−1.32397	−0.661984	−	0.749518i	$-0.730285\pi$
$883$	−39.3421	−1.32397	−0.661984	+	0.749518i	$0.730285\pi$
$884$	0	0
$885$	−26.0383	−0.875268
$886$	0	0
$887$	42.4202	1.42433	0.712166	−	0.702011i	$-0.247714\pi$
$887$	42.4202	1.42433	0.712166	+	0.702011i	$0.247714\pi$
$888$	0	0
$889$	13.3174	0.446652
$890$	0	0
$891$	−12.6131	−0.422554
$892$	0	0
$893$	5.62721	0.188308
$894$	0	0
$895$	6.72999	0.224959
$896$	0	0
$897$	9.35218	0.312260
$898$	0	0
$899$	8.30330	0.276931
$900$	0	0
$901$	−9.27555	−0.309013
$902$	0	0
$903$	8.48059	0.282216
$904$	0	0
$905$	−9.96526	−0.331256
$906$	0	0
$907$	21.6756	0.719726	0.359863	−	0.933005i	$-0.382824\pi$
$907$	21.6756	0.719726	0.359863	+	0.933005i	$0.382824\pi$
$908$	0	0
$909$	3.96169	0.131401
$910$	0	0
$911$	−19.4444	−0.644221	−0.322111	−	0.946702i	$-0.604392\pi$
$911$	−19.4444	−0.644221	−0.322111	+	0.946702i	$0.604392\pi$
$912$	0	0
$913$	16.9859	0.562150
$914$	0	0
$915$	6.20555	0.205149
$916$	0	0
$917$	37.4882	1.23797
$918$	0	0
$919$	−48.8167	−1.61031	−0.805157	−	0.593062i	$-0.797919\pi$
$919$	−48.8167	−1.61031	−0.805157	+	0.593062i	$0.797919\pi$
$920$	0	0
$921$	27.0242	0.890477
$922$	0	0
$923$	6.02827	0.198423
$924$	0	0
$925$	−1.00000	−0.0328798
$926$	0	0
$927$	−2.20555	−0.0724398
$928$	0	0
$929$	44.8011	1.46988	0.734938	−	0.678135i	$-0.237211\pi$
$929$	44.8011	1.46988	0.734938	+	0.678135i	$0.237211\pi$
$930$	0	0
$931$	11.5139	0.377352
$932$	0	0
$933$	38.4494	1.25878
$934$	0	0
$935$	8.00000	0.261628
$936$	0	0
$937$	−37.2297	−1.21624	−0.608121	−	0.793844i	$-0.708077\pi$
$937$	−37.2297	−1.21624	−0.608121	+	0.793844i	$0.708077\pi$
$938$	0	0
$939$	12.9794	0.423566
$940$	0	0
$941$	12.1572	0.396313	0.198157	−	0.980170i	$-0.436505\pi$
$941$	12.1572	0.396313	0.198157	+	0.980170i	$0.436505\pi$
$942$	0	0
$943$	−11.8539	−0.386016
$944$	0	0
$945$	−8.91638	−0.290050
$946$	0	0
$947$	−28.9300	−0.940099	−0.470049	−	0.882640i	$-0.655764\pi$
$947$	−28.9300	−0.940099	−0.470049	+	0.882640i	$0.655764\pi$
$948$	0	0
$949$	−44.7527	−1.45273
$950$	0	0
$951$	−9.52946	−0.309014
$952$	0	0
$953$	−48.4842	−1.57056	−0.785278	−	0.619143i	$-0.787480\pi$
$953$	−48.4842	−1.57056	−0.785278	+	0.619143i	$0.787480\pi$
$954$	0	0
$955$	−12.1517	−0.393218
$956$	0	0
$957$	−4.67609	−0.151156
$958$	0	0
$959$	18.8816	0.609720
$960$	0	0
$961$	−13.7638	−0.443993
$962$	0	0
$963$	−1.41612	−0.0456339
$964$	0	0
$965$	4.31335	0.138852
$966$	0	0
$967$	15.6272	0.502537	0.251269	−	0.967917i	$-0.419152\pi$
$967$	15.6272	0.502537	0.251269	+	0.967917i	$0.419152\pi$
$968$	0	0
$969$	34.9200	1.12179
$970$	0	0
$971$	44.6832	1.43395	0.716977	−	0.697097i	$-0.245525\pi$
$971$	44.6832	1.43395	0.716977	+	0.697097i	$0.245525\pi$
$972$	0	0
$973$	−11.0589	−0.354533
$974$	0	0
$975$	6.57834	0.210675
$976$	0	0
$977$	9.51941	0.304553	0.152277	−	0.988338i	$-0.451340\pi$
$977$	9.51941	0.304553	0.152277	+	0.988338i	$0.451340\pi$
$978$	0	0
$979$	−15.2544	−0.487534
$980$	0	0
$981$	−4.91281	−0.156854
$982$	0	0
$983$	−8.77187	−0.279779	−0.139890	−	0.990167i	$-0.544675\pi$
$983$	−8.77187	−0.279779	−0.139890	+	0.990167i	$0.544675\pi$
$984$	0	0
$985$	−25.8363	−0.823214
$986$	0	0
$987$	5.96526	0.189876
$988$	0	0
$989$	−3.66553	−0.116557
$990$	0	0
$991$	−18.5033	−0.587777	−0.293889	−	0.955840i	$-0.594949\pi$
$991$	−18.5033	−0.587777	−0.293889	+	0.955840i	$0.594949\pi$
$992$	0	0
$993$	−33.2233	−1.05431
$994$	0	0
$995$	−9.68111	−0.306912
$996$	0	0
$997$	38.4877	1.21892	0.609459	−	0.792817i	$-0.291387\pi$
$997$	38.4877	1.21892	0.609459	+	0.792817i	$0.291387\pi$
$998$	0	0
$999$	4.91638	0.155547

Twists

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

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

\(f(q)\)	\(=\)	\(q+1.81361 q^{3} -1.00000 q^{5} -1.81361 q^{7} +0.289169 q^{9} +O(q^{10})\)	\(q+1.81361 q^{3} -1.00000 q^{5} -1.81361 q^{7} +0.289169 q^{9} +1.28917 q^{11} +3.62721 q^{13} -1.81361 q^{15} -6.20555 q^{17} -3.10278 q^{19} -3.28917 q^{21} +1.42166 q^{23} +1.00000 q^{25} -4.91638 q^{27} -2.00000 q^{29} -4.15165 q^{31} +2.33804 q^{33} +1.81361 q^{35} -1.00000 q^{37} +6.57834 q^{39} -8.33804 q^{41} -2.57834 q^{43} -0.289169 q^{45} -1.81361 q^{47} -3.71083 q^{49} -11.2544 q^{51} +1.49472 q^{53} -1.28917 q^{55} -5.62721 q^{57} +14.3572 q^{59} -3.42166 q^{61} -0.524438 q^{63} -3.62721 q^{65} -0.897225 q^{67} +2.57834 q^{69} +1.66196 q^{71} -12.3380 q^{73} +1.81361 q^{75} -2.33804 q^{77} +10.3572 q^{79} -9.78389 q^{81} +13.1758 q^{83} +6.20555 q^{85} -3.62721 q^{87} -11.8328 q^{89} -6.57834 q^{91} -7.52946 q^{93} +3.10278 q^{95} -2.67609 q^{97} +0.372787 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - q^{3} - 3 q^{5} + q^{7}+O(q^{10}) \) 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

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Database

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