LMFDB - Embedded newform 2960.2.a.bb.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("2960.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$23.6357189983$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.583504.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 8x^{3} - 2x^{2} + 15x + 8 $ x^5 - 8x^3 - 2x^2 + 15*x + 8
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			1.2
Root			$1.84514$ 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)

$f(q)$	$=$	$q-0.845141 q^{3} +1.00000 q^{5} +2.14219 q^{7} -2.28574 q^{9} +O(q^{10})$	$q-0.845141 q^{3} +1.00000 q^{5} +2.14219 q^{7} -2.28574 q^{9} -1.83634 q^{11} -2.76560 q^{13} -0.845141 q^{15} -6.31299 q^{17} +1.00880 q^{19} -1.81045 q^{21} +5.38831 q^{23} +1.00000 q^{25} +4.46719 q^{27} +5.38831 q^{29} +3.18704 q^{31} +1.55197 q^{33} +2.14219 q^{35} -1.00000 q^{37} +2.33732 q^{39} -1.83634 q^{41} +1.11881 q^{43} -2.28574 q^{45} +7.02338 q^{47} -2.41103 q^{49} +5.33537 q^{51} +13.9214 q^{53} -1.83634 q^{55} -0.852575 q^{57} +2.21887 q^{59} +8.69677 q^{61} -4.89648 q^{63} -2.76560 q^{65} +15.0380 q^{67} -4.55388 q^{69} +6.27408 q^{71} -7.15255 q^{73} -0.845141 q^{75} -3.93379 q^{77} +6.70065 q^{79} +3.08181 q^{81} -8.35100 q^{83} -6.31299 q^{85} -4.55388 q^{87} +2.59682 q^{89} -5.92443 q^{91} -2.69349 q^{93} +1.00880 q^{95} +13.6325 q^{97} +4.19740 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 5 q^{3} + 5 q^{5} + 5 q^{7} + 6 q^{9}+O(q^{10}) $ 5 * q + 5 * q^3 + 5 * q^5 + 5 * q^7 + 6 * q^9	$ 5 q + 5 q^{3} + 5 q^{5} + 5 q^{7} + 6 q^{9} + 7 q^{11} - 6 q^{13} + 5 q^{15} + 12 q^{19} - q^{21} + 6 q^{23} + 5 q^{25} + 17 q^{27} + 6 q^{29} + 10 q^{31} + 3 q^{33} + 5 q^{35} - 5 q^{37} - 4 q^{39} + 7 q^{41} + 22 q^{43} + 6 q^{45} + 13 q^{47} + 14 q^{49} + 10 q^{51} - 11 q^{53} + 7 q^{55} - 8 q^{57} + 10 q^{59} + 10 q^{63} - 6 q^{65} + 24 q^{67} + 26 q^{69} + 13 q^{71} - 13 q^{73} + 5 q^{75} - 19 q^{77} - 2 q^{79} + q^{81} + 13 q^{83} + 26 q^{87} + 8 q^{89} + 8 q^{91} - 20 q^{93} + 12 q^{95} - 20 q^{97} - 2 q^{99}+O(q^{100}) $ 5 * q + 5 * q^3 + 5 * q^5 + 5 * q^7 + 6 * q^9 + 7 * q^11 - 6 * q^13 + 5 * q^15 + 12 * q^19 - q^21 + 6 * q^23 + 5 * q^25 + 17 * q^27 + 6 * q^29 + 10 * q^31 + 3 * q^33 + 5 * q^35 - 5 * q^37 - 4 * q^39 + 7 * q^41 + 22 * q^43 + 6 * q^45 + 13 * q^47 + 14 * q^49 + 10 * q^51 - 11 * q^53 + 7 * q^55 - 8 * q^57 + 10 * q^59 + 10 * q^63 - 6 * q^65 + 24 * q^67 + 26 * q^69 + 13 * q^71 - 13 * q^73 + 5 * q^75 - 19 * q^77 - 2 * q^79 + q^81 + 13 * q^83 + 26 * q^87 + 8 * q^89 + 8 * q^91 - 20 * q^93 + 12 * q^95 - 20 * q^97 - 2 * 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
$3$	−0.845141	−0.487942	−0.243971	−	0.969783i	$-0.578450\pi$
$3$	−0.845141	−0.487942	−0.243971	+	0.969783i	$0.578450\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	2.14219	0.809671	0.404835	−	0.914390i	$-0.367329\pi$
$7$	2.14219	0.809671	0.404835	+	0.914390i	$0.367329\pi$
$8$	0	0
$9$	−2.28574	−0.761912
$10$	0	0
$11$	−1.83634	−0.553679	−0.276839	−	0.960916i	$-0.589287\pi$
$11$	−1.83634	−0.553679	−0.276839	+	0.960916i	$0.589287\pi$
$12$	0	0
$13$	−2.76560	−0.767040	−0.383520	−	0.923533i	$-0.625288\pi$
$13$	−2.76560	−0.767040	−0.383520	+	0.923533i	$0.625288\pi$
$14$	0	0
$15$	−0.845141	−0.218214
$16$	0	0
$17$	−6.31299	−1.53113	−0.765563	−	0.643361i	$-0.777539\pi$
$17$	−6.31299	−1.53113	−0.765563	+	0.643361i	$0.777539\pi$
$18$	0	0
$19$	1.00880	0.231434	0.115717	−	0.993282i	$-0.463083\pi$
$19$	1.00880	0.231434	0.115717	+	0.993282i	$0.463083\pi$
$20$	0	0
$21$	−1.81045	−0.395073
$22$	0	0
$23$	5.38831	1.12354	0.561771	−	0.827293i	$-0.310120\pi$
$23$	5.38831	1.12354	0.561771	+	0.827293i	$0.310120\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	4.46719	0.859711
$28$	0	0
$29$	5.38831	1.00058	0.500292	−	0.865857i	$-0.333226\pi$
$29$	5.38831	1.00058	0.500292	+	0.865857i	$0.333226\pi$
$30$	0	0
$31$	3.18704	0.572409	0.286204	−	0.958169i	$-0.407606\pi$
$31$	3.18704	0.572409	0.286204	+	0.958169i	$0.407606\pi$
$32$	0	0
$33$	1.55197	0.270163
$34$	0	0
$35$	2.14219	0.362096
$36$	0	0
$37$	−1.00000	−0.164399
$37$	−1.00000	−0.164399
$38$	0	0
$39$	2.33732	0.374271
$40$	0	0
$41$	−1.83634	−0.286789	−0.143394	−	0.989666i	$-0.545802\pi$
$41$	−1.83634	−0.286789	−0.143394	+	0.989666i	$0.545802\pi$
$42$	0	0
$43$	1.11881	0.170616	0.0853082	−	0.996355i	$-0.472813\pi$
$43$	1.11881	0.170616	0.0853082	+	0.996355i	$0.472813\pi$
$44$	0	0
$45$	−2.28574	−0.340738
$46$	0	0
$47$	7.02338	1.02447	0.512233	−	0.858847i	$-0.328819\pi$
$47$	7.02338	1.02447	0.512233	+	0.858847i	$0.328819\pi$
$48$	0	0
$49$	−2.41103	−0.344433
$50$	0	0
$51$	5.33537	0.747101
$52$	0	0
$53$	13.9214	1.91226	0.956128	−	0.292951i	$-0.0946372\pi$
$53$	13.9214	1.91226	0.956128	+	0.292951i	$0.0946372\pi$
$54$	0	0
$55$	−1.83634	−0.247613
$56$	0	0
$57$	−0.852575	−0.112926
$58$	0	0
$59$	2.21887	0.288872	0.144436	−	0.989514i	$-0.453863\pi$
$59$	2.21887	0.288872	0.144436	+	0.989514i	$0.453863\pi$
$60$	0	0
$61$	8.69677	1.11351	0.556754	−	0.830678i	$-0.312047\pi$
$61$	8.69677	1.11351	0.556754	+	0.830678i	$0.312047\pi$
$62$	0	0
$63$	−4.89648	−0.616898
$64$	0	0
$65$	−2.76560	−0.343031
$66$	0	0
$67$	15.0380	1.83718	0.918590	−	0.395212i	$-0.129329\pi$
$67$	15.0380	1.83718	0.918590	+	0.395212i	$0.129329\pi$
$68$	0	0
$69$	−4.55388	−0.548223
$70$	0	0
$71$	6.27408	0.744597	0.372298	−	0.928113i	$-0.378570\pi$
$71$	6.27408	0.744597	0.372298	+	0.928113i	$0.378570\pi$
$72$	0	0
$73$	−7.15255	−0.837143	−0.418571	−	0.908184i	$-0.637469\pi$
$73$	−7.15255	−0.837143	−0.418571	+	0.908184i	$0.637469\pi$
$74$	0	0
$75$	−0.845141	−0.0975884
$76$	0	0
$77$	−3.93379	−0.448297
$78$	0	0
$79$	6.70065	0.753881	0.376941	−	0.926237i	$-0.376976\pi$
$79$	6.70065	0.753881	0.376941	+	0.926237i	$0.376976\pi$
$80$	0	0
$81$	3.08181	0.342423
$82$	0	0
$83$	−8.35100	−0.916641	−0.458321	−	0.888787i	$-0.651549\pi$
$83$	−8.35100	−0.916641	−0.458321	+	0.888787i	$0.651549\pi$
$84$	0	0
$85$	−6.31299	−0.684740
$86$	0	0
$87$	−4.55388	−0.488227
$88$	0	0
$89$	2.59682	0.275262	0.137631	−	0.990484i	$-0.456051\pi$
$89$	2.59682	0.275262	0.137631	+	0.990484i	$0.456051\pi$
$90$	0	0
$91$	−5.92443	−0.621050
$92$	0	0
$93$	−2.69349	−0.279302
$94$	0	0
$95$	1.00880	0.103500
$96$	0	0
$97$	13.6325	1.38417	0.692084	−	0.721817i	$-0.256693\pi$
$97$	13.6325	1.38417	0.692084	+	0.721817i	$0.256693\pi$
$98$	0	0
$99$	4.19740	0.421855
$100$	0	0
$101$	−4.76878	−0.474511	−0.237255	−	0.971447i	$-0.576248\pi$
$101$	−4.76878	−0.474511	−0.237255	+	0.971447i	$0.576248\pi$
$102$	0	0
$103$	−16.1572	−1.59202	−0.796008	−	0.605287i	$-0.793058\pi$
$103$	−16.1572	−1.59202	−0.796008	+	0.605287i	$0.793058\pi$
$104$	0	0
$105$	−1.81045	−0.176682
$106$	0	0
$107$	−13.6110	−1.31583	−0.657913	−	0.753094i	$-0.728561\pi$
$107$	−13.6110	−1.31583	−0.657913	+	0.753094i	$0.728561\pi$
$108$	0	0
$109$	8.57147	0.820998	0.410499	−	0.911861i	$-0.365354\pi$
$109$	8.57147	0.820998	0.410499	+	0.911861i	$0.365354\pi$
$110$	0	0
$111$	0.845141	0.0802172
$112$	0	0
$113$	−16.9228	−1.59196	−0.795981	−	0.605322i	$-0.793045\pi$
$113$	−16.9228	−1.59196	−0.795981	+	0.605322i	$0.793045\pi$
$114$	0	0
$115$	5.38831	0.502463
$116$	0	0
$117$	6.32144	0.584417
$118$	0	0
$119$	−13.5236	−1.23971
$120$	0	0
$121$	−7.62784	−0.693440
$122$	0	0
$123$	1.55197	0.139936
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	21.7938	1.93389	0.966945	−	0.254986i	$-0.0820710\pi$
$127$	21.7938	1.93389	0.966945	+	0.254986i	$0.0820710\pi$
$128$	0	0
$129$	−0.945549	−0.0832509
$130$	0	0
$131$	−13.6731	−1.19462	−0.597312	−	0.802009i	$-0.703765\pi$
$131$	−13.6731	−1.19462	−0.597312	+	0.802009i	$0.703765\pi$
$132$	0	0
$133$	2.16103	0.187385
$134$	0	0
$135$	4.46719	0.384475
$136$	0	0
$137$	−3.71290	−0.317214	−0.158607	−	0.987342i	$-0.550700\pi$
$137$	−3.71290	−0.317214	−0.158607	+	0.987342i	$0.550700\pi$
$138$	0	0
$139$	17.0560	1.44667	0.723334	−	0.690498i	$-0.242608\pi$
$139$	17.0560	1.44667	0.723334	+	0.690498i	$0.242608\pi$
$140$	0	0
$141$	−5.93574	−0.499880
$142$	0	0
$143$	5.07859	0.424693
$144$	0	0
$145$	5.38831	0.447475
$146$	0	0
$147$	2.03766	0.168063
$148$	0	0
$149$	11.5007	0.942177	0.471088	−	0.882086i	$-0.343861\pi$
$149$	11.5007	0.942177	0.471088	+	0.882086i	$0.343861\pi$
$150$	0	0
$151$	−0.736981	−0.0599747	−0.0299873	−	0.999550i	$-0.509547\pi$
$151$	−0.736981	−0.0599747	−0.0299873	+	0.999550i	$0.509547\pi$
$152$	0	0
$153$	14.4298	1.16658
$154$	0	0
$155$	3.18704	0.255989
$156$	0	0
$157$	5.81100	0.463768	0.231884	−	0.972743i	$-0.425511\pi$
$157$	5.81100	0.463768	0.231884	+	0.972743i	$0.425511\pi$
$158$	0	0
$159$	−11.7656	−0.933070
$160$	0	0
$161$	11.5428	0.909698
$162$	0	0
$163$	11.4577	0.897436	0.448718	−	0.893673i	$-0.351881\pi$
$163$	11.4577	0.897436	0.448718	+	0.893673i	$0.351881\pi$
$164$	0	0
$165$	1.55197	0.120821
$166$	0	0
$167$	19.3481	1.49720	0.748601	−	0.663021i	$-0.230726\pi$
$167$	19.3481	1.49720	0.748601	+	0.663021i	$0.230726\pi$
$168$	0	0
$169$	−5.35145	−0.411650
$170$	0	0
$171$	−2.30584	−0.176332
$172$	0	0
$173$	−14.0805	−1.07052	−0.535261	−	0.844687i	$-0.679787\pi$
$173$	−14.0805	−1.07052	−0.535261	+	0.844687i	$0.679787\pi$
$174$	0	0
$175$	2.14219	0.161934
$176$	0	0
$177$	−1.87526	−0.140953
$178$	0	0
$179$	18.3302	1.37007	0.685033	−	0.728512i	$-0.259788\pi$
$179$	18.3302	1.37007	0.685033	+	0.728512i	$0.259788\pi$
$180$	0	0
$181$	14.0722	1.04598	0.522989	−	0.852339i	$-0.324817\pi$
$181$	14.0722	1.04598	0.522989	+	0.852339i	$0.324817\pi$
$182$	0	0
$183$	−7.34999	−0.543327
$184$	0	0
$185$	−1.00000	−0.0735215
$186$	0	0
$187$	11.5928	0.847752
$188$	0	0
$189$	9.56956	0.696083
$190$	0	0
$191$	7.63049	0.552123	0.276062	−	0.961140i	$-0.410971\pi$
$191$	7.63049	0.552123	0.276062	+	0.961140i	$0.410971\pi$
$192$	0	0
$193$	3.18819	0.229491	0.114745	−	0.993395i	$-0.463395\pi$
$193$	3.18819	0.229491	0.114745	+	0.993395i	$0.463395\pi$
$194$	0	0
$195$	2.33732	0.167379
$196$	0	0
$197$	−1.65654	−0.118023	−0.0590116	−	0.998257i	$-0.518795\pi$
$197$	−1.65654	−0.118023	−0.0590116	+	0.998257i	$0.518795\pi$
$198$	0	0
$199$	−17.0245	−1.20683	−0.603417	−	0.797426i	$-0.706195\pi$
$199$	−17.0245	−1.20683	−0.603417	+	0.797426i	$0.706195\pi$
$200$	0	0
$201$	−12.7092	−0.896438
$202$	0	0
$203$	11.5428	0.810144
$204$	0	0
$205$	−1.83634	−0.128256
$206$	0	0
$207$	−12.3163	−0.856040
$208$	0	0
$209$	−1.85250	−0.128140
$210$	0	0
$211$	10.2460	0.705365	0.352682	−	0.935743i	$-0.385270\pi$
$211$	10.2460	0.705365	0.352682	+	0.935743i	$0.385270\pi$
$212$	0	0
$213$	−5.30248	−0.363320
$214$	0	0
$215$	1.11881	0.0763020
$216$	0	0
$217$	6.82723	0.463463
$218$	0	0
$219$	6.04491	0.408477
$220$	0	0
$221$	17.4592	1.17443
$222$	0	0
$223$	−26.8081	−1.79520	−0.897600	−	0.440810i	$-0.854691\pi$
$223$	−26.8081	−1.79520	−0.897600	+	0.440810i	$0.854691\pi$
$224$	0	0
$225$	−2.28574	−0.152382
$226$	0	0
$227$	10.0644	0.667995	0.333997	−	0.942574i	$-0.391602\pi$
$227$	10.0644	0.667995	0.333997	+	0.942574i	$0.391602\pi$
$228$	0	0
$229$	21.0635	1.39192	0.695959	−	0.718081i	$-0.254979\pi$
$229$	21.0635	1.39192	0.695959	+	0.718081i	$0.254979\pi$
$230$	0	0
$231$	3.32461	0.218743
$232$	0	0
$233$	17.2123	1.12762	0.563809	−	0.825905i	$-0.309336\pi$
$233$	17.2123	1.12762	0.563809	+	0.825905i	$0.309336\pi$
$234$	0	0
$235$	7.02338	0.458155
$236$	0	0
$237$	−5.66299	−0.367851
$238$	0	0
$239$	1.78270	0.115313	0.0576566	−	0.998336i	$-0.481637\pi$
$239$	1.78270	0.115313	0.0576566	+	0.998336i	$0.481637\pi$
$240$	0	0
$241$	4.28710	0.276156	0.138078	−	0.990421i	$-0.455907\pi$
$241$	4.28710	0.276156	0.138078	+	0.990421i	$0.455907\pi$
$242$	0	0
$243$	−16.0061	−1.02679
$244$	0	0
$245$	−2.41103	−0.154035
$246$	0	0
$247$	−2.78993	−0.177519
$248$	0	0
$249$	7.05777	0.447268
$250$	0	0
$251$	−3.55745	−0.224544	−0.112272	−	0.993677i	$-0.535813\pi$
$251$	−3.55745	−0.224544	−0.112272	+	0.993677i	$0.535813\pi$
$252$	0	0
$253$	−9.89480	−0.622081
$254$	0	0
$255$	5.33537	0.334114
$256$	0	0
$257$	−18.2643	−1.13930	−0.569649	−	0.821888i	$-0.692921\pi$
$257$	−18.2643	−1.13930	−0.569649	+	0.821888i	$0.692921\pi$
$258$	0	0
$259$	−2.14219	−0.133109
$260$	0	0
$261$	−12.3163	−0.762358
$262$	0	0
$263$	−0.734714	−0.0453044	−0.0226522	−	0.999743i	$-0.507211\pi$
$263$	−0.734714	−0.0453044	−0.0226522	+	0.999743i	$0.507211\pi$
$264$	0	0
$265$	13.9214	0.855186
$266$	0	0
$267$	−2.19468	−0.134312
$268$	0	0
$269$	10.6357	0.648469	0.324235	−	0.945977i	$-0.394893\pi$
$269$	10.6357	0.648469	0.324235	+	0.945977i	$0.394893\pi$
$270$	0	0
$271$	8.23324	0.500133	0.250067	−	0.968229i	$-0.419547\pi$
$271$	8.23324	0.500133	0.250067	+	0.968229i	$0.419547\pi$
$272$	0	0
$273$	5.00698	0.303036
$274$	0	0
$275$	−1.83634	−0.110736
$276$	0	0
$277$	14.0818	0.846094	0.423047	−	0.906108i	$-0.360961\pi$
$277$	14.0818	0.846094	0.423047	+	0.906108i	$0.360961\pi$
$278$	0	0
$279$	−7.28473	−0.436125
$280$	0	0
$281$	17.3533	1.03521	0.517606	−	0.855619i	$-0.326823\pi$
$281$	17.3533	1.03521	0.517606	+	0.855619i	$0.326823\pi$
$282$	0	0
$283$	−3.21736	−0.191252	−0.0956260	−	0.995417i	$-0.530485\pi$
$283$	−3.21736	−0.191252	−0.0956260	+	0.995417i	$0.530485\pi$
$284$	0	0
$285$	−0.852575	−0.0505022
$286$	0	0
$287$	−3.93379	−0.232205
$288$	0	0
$289$	22.8539	1.34435
$290$	0	0
$291$	−11.5214	−0.675394
$292$	0	0
$293$	5.78773	0.338123	0.169061	−	0.985606i	$-0.445926\pi$
$293$	5.78773	0.338123	0.169061	+	0.985606i	$0.445926\pi$
$294$	0	0
$295$	2.21887	0.129188
$296$	0	0
$297$	−8.20330	−0.476004
$298$	0	0
$299$	−14.9019	−0.861800
$300$	0	0
$301$	2.39669	0.138143
$302$	0	0
$303$	4.03029	0.231534
$304$	0	0
$305$	8.69677	0.497976
$306$	0	0
$307$	11.0617	0.631325	0.315662	−	0.948872i	$-0.397773\pi$
$307$	11.0617	0.631325	0.315662	+	0.948872i	$0.397773\pi$
$308$	0	0
$309$	13.6551	0.776811
$310$	0	0
$311$	10.1319	0.574527	0.287264	−	0.957852i	$-0.407254\pi$
$311$	10.1319	0.574527	0.287264	+	0.957852i	$0.407254\pi$
$312$	0	0
$313$	5.24683	0.296568	0.148284	−	0.988945i	$-0.452625\pi$
$313$	5.24683	0.296568	0.148284	+	0.988945i	$0.452625\pi$
$314$	0	0
$315$	−4.89648	−0.275885
$316$	0	0
$317$	−33.3242	−1.87167	−0.935836	−	0.352437i	$-0.885353\pi$
$317$	−33.3242	−1.87167	−0.935836	+	0.352437i	$0.885353\pi$
$318$	0	0
$319$	−9.89480	−0.554002
$320$	0	0
$321$	11.5032	0.642047
$322$	0	0
$323$	−6.36852	−0.354354
$324$	0	0
$325$	−2.76560	−0.153408
$326$	0	0
$327$	−7.24410	−0.400600
$328$	0	0
$329$	15.0454	0.829480
$330$	0	0
$331$	−17.6010	−0.967438	−0.483719	−	0.875223i	$-0.660714\pi$
$331$	−17.6010	−0.967438	−0.483719	+	0.875223i	$0.660714\pi$
$332$	0	0
$333$	2.28574	0.125258
$334$	0	0
$335$	15.0380	0.821612
$336$	0	0
$337$	14.3624	0.782369	0.391184	−	0.920312i	$-0.372065\pi$
$337$	14.3624	0.782369	0.391184	+	0.920312i	$0.372065\pi$
$338$	0	0
$339$	14.3021	0.776785
$340$	0	0
$341$	−5.85250	−0.316930
$342$	0	0
$343$	−20.1602	−1.08855
$344$	0	0
$345$	−4.55388	−0.245173
$346$	0	0
$347$	3.69663	0.198446	0.0992228	−	0.995065i	$-0.468364\pi$
$347$	3.69663	0.198446	0.0992228	+	0.995065i	$0.468364\pi$
$348$	0	0
$349$	17.7954	0.952566	0.476283	−	0.879292i	$-0.341984\pi$
$349$	17.7954	0.952566	0.476283	+	0.879292i	$0.341984\pi$
$350$	0	0
$351$	−12.3545	−0.659433
$352$	0	0
$353$	−22.8156	−1.21435	−0.607177	−	0.794567i	$-0.707698\pi$
$353$	−22.8156	−1.21435	−0.607177	+	0.794567i	$0.707698\pi$
$354$	0	0
$355$	6.27408	0.332994
$356$	0	0
$357$	11.4294	0.604906
$358$	0	0
$359$	−2.54296	−0.134212	−0.0671060	−	0.997746i	$-0.521377\pi$
$359$	−2.54296	−0.134212	−0.0671060	+	0.997746i	$0.521377\pi$
$360$	0	0
$361$	−17.9823	−0.946438
$362$	0	0
$363$	6.44660	0.338359
$364$	0	0
$365$	−7.15255	−0.374382
$366$	0	0
$367$	−21.9523	−1.14590	−0.572949	−	0.819591i	$-0.694201\pi$
$367$	−21.9523	−1.14590	−0.572949	+	0.819591i	$0.694201\pi$
$368$	0	0
$369$	4.19740	0.218508
$370$	0	0
$371$	29.8223	1.54830
$372$	0	0
$373$	8.70267	0.450607	0.225304	−	0.974289i	$-0.427663\pi$
$373$	8.70267	0.450607	0.225304	+	0.974289i	$0.427663\pi$
$374$	0	0
$375$	−0.845141	−0.0436429
$376$	0	0
$377$	−14.9019	−0.767488
$378$	0	0
$379$	−10.2501	−0.526510	−0.263255	−	0.964726i	$-0.584796\pi$
$379$	−10.2501	−0.526510	−0.263255	+	0.964726i	$0.584796\pi$
$380$	0	0
$381$	−18.4188	−0.943626
$382$	0	0
$383$	−12.8650	−0.657371	−0.328685	−	0.944439i	$-0.606606\pi$
$383$	−12.8650	−0.657371	−0.328685	+	0.944439i	$0.606606\pi$
$384$	0	0
$385$	−3.93379	−0.200485
$386$	0	0
$387$	−2.55730	−0.129995
$388$	0	0
$389$	−14.7739	−0.749067	−0.374533	−	0.927213i	$-0.622197\pi$
$389$	−14.7739	−0.749067	−0.374533	+	0.927213i	$0.622197\pi$
$390$	0	0
$391$	−34.0164	−1.72028
$392$	0	0
$393$	11.5557	0.582908
$394$	0	0
$395$	6.70065	0.337146
$396$	0	0
$397$	−25.6937	−1.28953	−0.644764	−	0.764382i	$-0.723044\pi$
$397$	−25.6937	−1.28953	−0.644764	+	0.764382i	$0.723044\pi$
$398$	0	0
$399$	−1.82637	−0.0914331
$400$	0	0
$401$	−16.3311	−0.815538	−0.407769	−	0.913085i	$-0.633693\pi$
$401$	−16.3311	−0.815538	−0.407769	+	0.913085i	$0.633693\pi$
$402$	0	0
$403$	−8.81407	−0.439060
$404$	0	0
$405$	3.08181	0.153136
$406$	0	0
$407$	1.83634	0.0910242
$408$	0	0
$409$	36.5506	1.80731	0.903655	−	0.428261i	$-0.140874\pi$
$409$	36.5506	1.80731	0.903655	+	0.428261i	$0.140874\pi$
$410$	0	0
$411$	3.13792	0.154782
$412$	0	0
$413$	4.75323	0.233891
$414$	0	0
$415$	−8.35100	−0.409934
$416$	0	0
$417$	−14.4147	−0.705891
$418$	0	0
$419$	32.4089	1.58328	0.791640	−	0.610988i	$-0.209228\pi$
$419$	32.4089	1.58328	0.791640	+	0.610988i	$0.209228\pi$
$420$	0	0
$421$	33.5350	1.63440	0.817199	−	0.576356i	$-0.195526\pi$
$421$	33.5350	1.63440	0.817199	+	0.576356i	$0.195526\pi$
$422$	0	0
$423$	−16.0536	−0.780553
$424$	0	0
$425$	−6.31299	−0.306225
$426$	0	0
$427$	18.6301	0.901574
$428$	0	0
$429$	−4.29213	−0.207226
$430$	0	0
$431$	−9.59342	−0.462099	−0.231049	−	0.972942i	$-0.574216\pi$
$431$	−9.59342	−0.462099	−0.231049	+	0.972942i	$0.574216\pi$
$432$	0	0
$433$	−18.0805	−0.868894	−0.434447	−	0.900697i	$-0.643056\pi$
$433$	−18.0805	−0.868894	−0.434447	+	0.900697i	$0.643056\pi$
$434$	0	0
$435$	−4.55388	−0.218342
$436$	0	0
$437$	5.43571	0.260025
$438$	0	0
$439$	−8.71527	−0.415957	−0.207979	−	0.978133i	$-0.566688\pi$
$439$	−8.71527	−0.415957	−0.207979	+	0.978133i	$0.566688\pi$
$440$	0	0
$441$	5.51099	0.262428
$442$	0	0
$443$	−2.08265	−0.0989495	−0.0494748	−	0.998775i	$-0.515755\pi$
$443$	−2.08265	−0.0989495	−0.0494748	+	0.998775i	$0.515755\pi$
$444$	0	0
$445$	2.59682	0.123101
$446$	0	0
$447$	−9.71973	−0.459728
$448$	0	0
$449$	0.135770	0.00640740	0.00320370	−	0.999995i	$-0.498980\pi$
$449$	0.135770	0.00640740	0.00320370	+	0.999995i	$0.498980\pi$
$450$	0	0
$451$	3.37216	0.158789
$452$	0	0
$453$	0.622853	0.0292642
$454$	0	0
$455$	−5.92443	−0.277742
$456$	0	0
$457$	−33.8739	−1.58455	−0.792277	−	0.610161i	$-0.791105\pi$
$457$	−33.8739	−1.58455	−0.792277	+	0.610161i	$0.791105\pi$
$458$	0	0
$459$	−28.2014	−1.31633
$460$	0	0
$461$	−25.2123	−1.17426	−0.587128	−	0.809494i	$-0.699741\pi$
$461$	−25.2123	−1.17426	−0.587128	+	0.809494i	$0.699741\pi$
$462$	0	0
$463$	9.33652	0.433905	0.216953	−	0.976182i	$-0.430388\pi$
$463$	9.33652	0.433905	0.216953	+	0.976182i	$0.430388\pi$
$464$	0	0
$465$	−2.69349	−0.124908
$466$	0	0
$467$	−25.5507	−1.18235	−0.591173	−	0.806545i	$-0.701335\pi$
$467$	−25.5507	−1.18235	−0.591173	+	0.806545i	$0.701335\pi$
$468$	0	0
$469$	32.2141	1.48751
$470$	0	0
$471$	−4.91111	−0.226292
$472$	0	0
$473$	−2.05451	−0.0944666
$474$	0	0
$475$	1.00880	0.0462867
$476$	0	0
$477$	−31.8207	−1.45697
$478$	0	0
$479$	−0.374490	−0.0171109	−0.00855545	−	0.999963i	$-0.502723\pi$
$479$	−0.374490	−0.0171109	−0.00855545	+	0.999963i	$0.502723\pi$
$480$	0	0
$481$	2.76560	0.126101
$482$	0	0
$483$	−9.75527	−0.443880
$484$	0	0
$485$	13.6325	0.619019
$486$	0	0
$487$	−24.5780	−1.11374	−0.556868	−	0.830601i	$-0.687997\pi$
$487$	−24.5780	−1.11374	−0.556868	+	0.830601i	$0.687997\pi$
$488$	0	0
$489$	−9.68336	−0.437897
$490$	0	0
$491$	9.99671	0.451145	0.225573	−	0.974226i	$-0.427575\pi$
$491$	9.99671	0.451145	0.225573	+	0.974226i	$0.427575\pi$
$492$	0	0
$493$	−34.0164	−1.53202
$494$	0	0
$495$	4.19740	0.188659
$496$	0	0
$497$	13.4403	0.602878
$498$	0	0
$499$	0.296002	0.0132509	0.00662543	−	0.999978i	$-0.497891\pi$
$499$	0.296002	0.0132509	0.00662543	+	0.999978i	$0.497891\pi$
$500$	0	0
$501$	−16.3519	−0.730547
$502$	0	0
$503$	−0.184424	−0.00822306	−0.00411153	−	0.999992i	$-0.501309\pi$
$503$	−0.184424	−0.00822306	−0.00411153	+	0.999992i	$0.501309\pi$
$504$	0	0
$505$	−4.76878	−0.212208
$506$	0	0
$507$	4.52273	0.200862
$508$	0	0
$509$	0.0720659	0.00319427	0.00159713	−	0.999999i	$-0.499492\pi$
$509$	0.0720659	0.00319427	0.00159713	+	0.999999i	$0.499492\pi$
$510$	0	0
$511$	−15.3221	−0.677810
$512$	0	0
$513$	4.50649	0.198966
$514$	0	0
$515$	−16.1572	−0.711971
$516$	0	0
$517$	−12.8973	−0.567225
$518$	0	0
$519$	11.9000	0.522352
$520$	0	0
$521$	−14.4605	−0.633524	−0.316762	−	0.948505i	$-0.602596\pi$
$521$	−14.4605	−0.633524	−0.316762	+	0.948505i	$0.602596\pi$
$522$	0	0
$523$	26.2877	1.14948	0.574741	−	0.818336i	$-0.305103\pi$
$523$	26.2877	1.14948	0.574741	+	0.818336i	$0.305103\pi$
$524$	0	0
$525$	−1.81045	−0.0790145
$526$	0	0
$527$	−20.1197	−0.876430
$528$	0	0
$529$	6.03392	0.262344
$530$	0	0
$531$	−5.07175	−0.220095
$532$	0	0
$533$	5.07859	0.219978
$534$	0	0
$535$	−13.6110	−0.588455
$536$	0	0
$537$	−15.4916	−0.668513
$538$	0	0
$539$	4.42749	0.190705
$540$	0	0
$541$	23.9637	1.03028	0.515140	−	0.857106i	$-0.327740\pi$
$541$	23.9637	1.03028	0.515140	+	0.857106i	$0.327740\pi$
$542$	0	0
$543$	−11.8930	−0.510377
$544$	0	0
$545$	8.57147	0.367162
$546$	0	0
$547$	15.1064	0.645902	0.322951	−	0.946416i	$-0.395325\pi$
$547$	15.1064	0.645902	0.322951	+	0.946416i	$0.395325\pi$
$548$	0	0
$549$	−19.8785	−0.848395
$550$	0	0
$551$	5.43571	0.231569
$552$	0	0
$553$	14.3540	0.610396
$554$	0	0
$555$	0.845141	0.0358742
$556$	0	0
$557$	−26.9078	−1.14012	−0.570060	−	0.821603i	$-0.693080\pi$
$557$	−26.9078	−1.14012	−0.570060	+	0.821603i	$0.693080\pi$
$558$	0	0
$559$	−3.09417	−0.130869
$560$	0	0
$561$	−9.79757	−0.413654
$562$	0	0
$563$	−31.1217	−1.31162	−0.655812	−	0.754924i	$-0.727674\pi$
$563$	−31.1217	−1.31162	−0.655812	+	0.754924i	$0.727674\pi$
$564$	0	0
$565$	−16.9228	−0.711947
$566$	0	0
$567$	6.60181	0.277250
$568$	0	0
$569$	29.3657	1.23107	0.615537	−	0.788108i	$-0.288939\pi$
$569$	29.3657	1.23107	0.615537	+	0.788108i	$0.288939\pi$
$570$	0	0
$571$	4.29028	0.179542	0.0897712	−	0.995962i	$-0.471386\pi$
$571$	4.29028	0.179542	0.0897712	+	0.995962i	$0.471386\pi$
$572$	0	0
$573$	−6.44884	−0.269404
$574$	0	0
$575$	5.38831	0.224708
$576$	0	0
$577$	6.48002	0.269767	0.134883	−	0.990861i	$-0.456934\pi$
$577$	6.48002	0.269767	0.134883	+	0.990861i	$0.456934\pi$
$578$	0	0
$579$	−2.69447	−0.111978
$580$	0	0
$581$	−17.8894	−0.742178
$582$	0	0
$583$	−25.5645	−1.05877
$584$	0	0
$585$	6.32144	0.261359
$586$	0	0
$587$	39.1520	1.61598	0.807988	−	0.589199i	$-0.200557\pi$
$587$	39.1520	1.61598	0.807988	+	0.589199i	$0.200557\pi$
$588$	0	0
$589$	3.21507	0.132475
$590$	0	0
$591$	1.40001	0.0575885
$592$	0	0
$593$	4.73414	0.194408	0.0972040	−	0.995264i	$-0.469010\pi$
$593$	4.73414	0.194408	0.0972040	+	0.995264i	$0.469010\pi$
$594$	0	0
$595$	−13.5236	−0.554414
$596$	0	0
$597$	14.3881	0.588865
$598$	0	0
$599$	−6.48495	−0.264968	−0.132484	−	0.991185i	$-0.542295\pi$
$599$	−6.48495	−0.264968	−0.132484	+	0.991185i	$0.542295\pi$
$600$	0	0
$601$	39.3000	1.60308	0.801540	−	0.597941i	$-0.204014\pi$
$601$	39.3000	1.60308	0.801540	+	0.597941i	$0.204014\pi$
$602$	0	0
$603$	−34.3728	−1.39977
$604$	0	0
$605$	−7.62784	−0.310116
$606$	0	0
$607$	13.5099	0.548350	0.274175	−	0.961680i	$-0.411595\pi$
$607$	13.5099	0.548350	0.274175	+	0.961680i	$0.411595\pi$
$608$	0	0
$609$	−9.75527	−0.395304
$610$	0	0
$611$	−19.4239	−0.785806
$612$	0	0
$613$	−41.8193	−1.68907	−0.844533	−	0.535504i	$-0.820122\pi$
$613$	−41.8193	−1.68907	−0.844533	+	0.535504i	$0.820122\pi$
$614$	0	0
$615$	1.55197	0.0625814
$616$	0	0
$617$	−28.7404	−1.15704	−0.578522	−	0.815667i	$-0.696370\pi$
$617$	−28.7404	−1.15704	−0.578522	+	0.815667i	$0.696370\pi$
$618$	0	0
$619$	33.1568	1.33268	0.666342	−	0.745646i	$-0.267859\pi$
$619$	33.1568	1.33268	0.666342	+	0.745646i	$0.267859\pi$
$620$	0	0
$621$	24.0706	0.965921
$622$	0	0
$623$	5.56287	0.222872
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	1.56562	0.0625249
$628$	0	0
$629$	6.31299	0.251716
$630$	0	0
$631$	24.7412	0.984933	0.492466	−	0.870332i	$-0.336095\pi$
$631$	24.7412	0.984933	0.492466	+	0.870332i	$0.336095\pi$
$632$	0	0
$633$	−8.65932	−0.344177
$634$	0	0
$635$	21.7938	0.864862
$636$	0	0
$637$	6.66795	0.264194
$638$	0	0
$639$	−14.3409	−0.567317
$640$	0	0
$641$	−42.9105	−1.69486	−0.847431	−	0.530906i	$-0.821852\pi$
$641$	−42.9105	−1.69486	−0.847431	+	0.530906i	$0.821852\pi$
$642$	0	0
$643$	15.4422	0.608981	0.304490	−	0.952515i	$-0.401514\pi$
$643$	15.4422	0.608981	0.304490	+	0.952515i	$0.401514\pi$
$644$	0	0
$645$	−0.945549	−0.0372309
$646$	0	0
$647$	−24.1910	−0.951045	−0.475523	−	0.879703i	$-0.657741\pi$
$647$	−24.1910	−0.951045	−0.475523	+	0.879703i	$0.657741\pi$
$648$	0	0
$649$	−4.07461	−0.159942
$650$	0	0
$651$	−5.76997	−0.226143
$652$	0	0
$653$	−38.3080	−1.49911	−0.749554	−	0.661943i	$-0.769732\pi$
$653$	−38.3080	−1.49911	−0.749554	+	0.661943i	$0.769732\pi$
$654$	0	0
$655$	−13.6731	−0.534252
$656$	0	0
$657$	16.3489	0.637830
$658$	0	0
$659$	35.4170	1.37965	0.689826	−	0.723975i	$-0.257687\pi$
$659$	35.4170	1.37965	0.689826	+	0.723975i	$0.257687\pi$
$660$	0	0
$661$	35.5089	1.38114	0.690568	−	0.723267i	$-0.257360\pi$
$661$	35.5089	1.38114	0.690568	+	0.723267i	$0.257360\pi$
$662$	0	0
$663$	−14.7555	−0.573056
$664$	0	0
$665$	2.16103	0.0838012
$666$	0	0
$667$	29.0339	1.12420
$668$	0	0
$669$	22.6566	0.875954
$670$	0	0
$671$	−15.9703	−0.616525
$672$	0	0
$673$	−39.0126	−1.50382	−0.751912	−	0.659263i	$-0.770868\pi$
$673$	−39.0126	−1.50382	−0.751912	+	0.659263i	$0.770868\pi$
$674$	0	0
$675$	4.46719	0.171942
$676$	0	0
$677$	21.5982	0.830085	0.415042	−	0.909802i	$-0.363767\pi$
$677$	21.5982	0.830085	0.415042	+	0.909802i	$0.363767\pi$
$678$	0	0
$679$	29.2033	1.12072
$680$	0	0
$681$	−8.50579	−0.325943
$682$	0	0
$683$	3.01886	0.115513	0.0577566	−	0.998331i	$-0.481605\pi$
$683$	3.01886	0.115513	0.0577566	+	0.998331i	$0.481605\pi$
$684$	0	0
$685$	−3.71290	−0.141863
$686$	0	0
$687$	−17.8017	−0.679176
$688$	0	0
$689$	−38.5011	−1.46678
$690$	0	0
$691$	−15.0805	−0.573689	−0.286844	−	0.957977i	$-0.592606\pi$
$691$	−15.0805	−0.573689	−0.286844	+	0.957977i	$0.592606\pi$
$692$	0	0
$693$	8.99162	0.341563
$694$	0	0
$695$	17.0560	0.646970
$696$	0	0
$697$	11.5928	0.439110
$698$	0	0
$699$	−14.5468	−0.550212
$700$	0	0
$701$	−0.0862835	−0.00325888	−0.00162944	−	0.999999i	$-0.500519\pi$
$701$	−0.0862835	−0.00325888	−0.00162944	+	0.999999i	$0.500519\pi$
$702$	0	0
$703$	−1.00880	−0.0380475
$704$	0	0
$705$	−5.93574	−0.223553
$706$	0	0
$707$	−10.2156	−0.384198
$708$	0	0
$709$	1.57483	0.0591439	0.0295719	−	0.999563i	$-0.490586\pi$
$709$	1.57483	0.0591439	0.0295719	+	0.999563i	$0.490586\pi$
$710$	0	0
$711$	−15.3159	−0.574392
$712$	0	0
$713$	17.1728	0.643125
$714$	0	0
$715$	5.07859	0.189929
$716$	0	0
$717$	−1.50663	−0.0562662
$718$	0	0
$719$	43.4396	1.62002	0.810011	−	0.586415i	$-0.199461\pi$
$719$	43.4396	1.62002	0.810011	+	0.586415i	$0.199461\pi$
$720$	0	0
$721$	−34.6117	−1.28901
$722$	0	0
$723$	−3.62320	−0.134748
$724$	0	0
$725$	5.38831	0.200117
$726$	0	0
$727$	25.9917	0.963978	0.481989	−	0.876177i	$-0.339914\pi$
$727$	25.9917	0.963978	0.481989	+	0.876177i	$0.339914\pi$
$728$	0	0
$729$	4.28201	0.158593
$730$	0	0
$731$	−7.06302	−0.261235
$732$	0	0
$733$	−18.7369	−0.692062	−0.346031	−	0.938223i	$-0.612471\pi$
$733$	−18.7369	−0.692062	−0.346031	+	0.938223i	$0.612471\pi$
$734$	0	0
$735$	2.03766	0.0751603
$736$	0	0
$737$	−27.6149	−1.01721
$738$	0	0
$739$	−0.662458	−0.0243689	−0.0121845	−	0.999926i	$-0.503879\pi$
$739$	−0.662458	−0.0243689	−0.0121845	+	0.999926i	$0.503879\pi$
$740$	0	0
$741$	2.35788	0.0866189
$742$	0	0
$743$	7.98058	0.292779	0.146390	−	0.989227i	$-0.453235\pi$
$743$	7.98058	0.292779	0.146390	+	0.989227i	$0.453235\pi$
$744$	0	0
$745$	11.5007	0.421354
$746$	0	0
$747$	19.0882	0.698400
$748$	0	0
$749$	−29.1573	−1.06539
$750$	0	0
$751$	20.1243	0.734345	0.367173	−	0.930153i	$-0.380326\pi$
$751$	20.1243	0.734345	0.367173	+	0.930153i	$0.380326\pi$
$752$	0	0
$753$	3.00655	0.109565
$754$	0	0
$755$	−0.736981	−0.0268215
$756$	0	0
$757$	−16.7104	−0.607351	−0.303676	−	0.952775i	$-0.598214\pi$
$757$	−16.7104	−0.607351	−0.303676	+	0.952775i	$0.598214\pi$
$758$	0	0
$759$	8.36250	0.303539
$760$	0	0
$761$	−18.2084	−0.660053	−0.330027	−	0.943972i	$-0.607058\pi$
$761$	−18.2084	−0.660053	−0.330027	+	0.943972i	$0.607058\pi$
$762$	0	0
$763$	18.3617	0.664738
$764$	0	0
$765$	14.4298	0.521712
$766$	0	0
$767$	−6.13650	−0.221576
$768$	0	0
$769$	−37.9839	−1.36973	−0.684866	−	0.728669i	$-0.740139\pi$
$769$	−37.9839	−1.36973	−0.684866	+	0.728669i	$0.740139\pi$
$770$	0	0
$771$	15.4359	0.555912
$772$	0	0
$773$	−20.8631	−0.750395	−0.375198	−	0.926945i	$-0.622425\pi$
$773$	−20.8631	−0.750395	−0.375198	+	0.926945i	$0.622425\pi$
$774$	0	0
$775$	3.18704	0.114482
$776$	0	0
$777$	1.81045	0.0649495
$778$	0	0
$779$	−1.85250	−0.0663726
$780$	0	0
$781$	−11.5214	−0.412267
$782$	0	0
$783$	24.0706	0.860214
$784$	0	0
$785$	5.81100	0.207404
$786$	0	0
$787$	0.979835	0.0349273	0.0174637	−	0.999847i	$-0.494441\pi$
$787$	0.979835	0.0349273	0.0174637	+	0.999847i	$0.494441\pi$
$788$	0	0
$789$	0.620936	0.0221059
$790$	0	0
$791$	−36.2518	−1.28897
$792$	0	0
$793$	−24.0518	−0.854104
$794$	0	0
$795$	−11.7656	−0.417282
$796$	0	0
$797$	−6.78822	−0.240451	−0.120226	−	0.992747i	$-0.538362\pi$
$797$	−6.78822	−0.240451	−0.120226	+	0.992747i	$0.538362\pi$
$798$	0	0
$799$	−44.3386	−1.56859
$800$	0	0
$801$	−5.93565	−0.209726
$802$	0	0
$803$	13.1345	0.463508
$804$	0	0
$805$	11.5428	0.406829
$806$	0	0
$807$	−8.98865	−0.316416
$808$	0	0
$809$	46.3317	1.62894	0.814469	−	0.580207i	$-0.197029\pi$
$809$	46.3317	1.62894	0.814469	+	0.580207i	$0.197029\pi$
$810$	0	0
$811$	7.08317	0.248724	0.124362	−	0.992237i	$-0.460312\pi$
$811$	7.08317	0.248724	0.124362	+	0.992237i	$0.460312\pi$
$812$	0	0
$813$	−6.95824	−0.244036
$814$	0	0
$815$	11.4577	0.401346
$816$	0	0
$817$	1.12865	0.0394864
$818$	0	0
$819$	13.5417	0.473185
$820$	0	0
$821$	16.0721	0.560919	0.280459	−	0.959866i	$-0.409513\pi$
$821$	16.0721	0.560919	0.280459	+	0.959866i	$0.409513\pi$
$822$	0	0
$823$	50.4423	1.75831	0.879155	−	0.476536i	$-0.158108\pi$
$823$	50.4423	1.75831	0.879155	+	0.476536i	$0.158108\pi$
$824$	0	0
$825$	1.55197	0.0540326
$826$	0	0
$827$	41.1787	1.43193	0.715963	−	0.698139i	$-0.245988\pi$
$827$	41.1787	1.43193	0.715963	+	0.698139i	$0.245988\pi$
$828$	0	0
$829$	−14.8454	−0.515601	−0.257801	−	0.966198i	$-0.582998\pi$
$829$	−14.8454	−0.515601	−0.257801	+	0.966198i	$0.582998\pi$
$830$	0	0
$831$	−11.9011	−0.412845
$832$	0	0
$833$	15.2208	0.527371
$834$	0	0
$835$	19.3481	0.669569
$836$	0	0
$837$	14.2371	0.492106
$838$	0	0
$839$	−32.5974	−1.12539	−0.562693	−	0.826666i	$-0.690235\pi$
$839$	−32.5974	−1.12539	−0.562693	+	0.826666i	$0.690235\pi$
$840$	0	0
$841$	0.0339222	0.00116973
$842$	0	0
$843$	−14.6660	−0.505124
$844$	0	0
$845$	−5.35145	−0.184096
$846$	0	0
$847$	−16.3403	−0.561458
$848$	0	0
$849$	2.71912	0.0933200
$850$	0	0
$851$	−5.38831	−0.184709
$852$	0	0
$853$	−1.67681	−0.0574130	−0.0287065	−	0.999588i	$-0.509139\pi$
$853$	−1.67681	−0.0574130	−0.0287065	+	0.999588i	$0.509139\pi$
$854$	0	0
$855$	−2.30584	−0.0788582
$856$	0	0
$857$	−11.4908	−0.392518	−0.196259	−	0.980552i	$-0.562879\pi$
$857$	−11.4908	−0.392518	−0.196259	+	0.980552i	$0.562879\pi$
$858$	0	0
$859$	−20.8250	−0.710539	−0.355269	−	0.934764i	$-0.615611\pi$
$859$	−20.8250	−0.710539	−0.355269	+	0.934764i	$0.615611\pi$
$860$	0	0
$861$	3.32461	0.113302
$862$	0	0
$863$	40.0967	1.36491	0.682454	−	0.730928i	$-0.260913\pi$
$863$	40.0967	1.36491	0.682454	+	0.730928i	$0.260913\pi$
$864$	0	0
$865$	−14.0805	−0.478752
$866$	0	0
$867$	−19.3148	−0.655963
$868$	0	0
$869$	−12.3047	−0.417408
$870$	0	0
$871$	−41.5890	−1.40919
$872$	0	0
$873$	−31.1603	−1.05462
$874$	0	0
$875$	2.14219	0.0724192
$876$	0	0
$877$	37.5875	1.26924	0.634619	−	0.772825i	$-0.281157\pi$
$877$	37.5875	1.26924	0.634619	+	0.772825i	$0.281157\pi$
$878$	0	0
$879$	−4.89145	−0.164984
$880$	0	0
$881$	−27.9518	−0.941721	−0.470860	−	0.882208i	$-0.656056\pi$
$881$	−27.9518	−0.941721	−0.470860	+	0.882208i	$0.656056\pi$
$882$	0	0
$883$	−37.7187	−1.26933	−0.634667	−	0.772786i	$-0.718863\pi$
$883$	−37.7187	−1.26933	−0.634667	+	0.772786i	$0.718863\pi$
$884$	0	0
$885$	−1.87526	−0.0630360
$886$	0	0
$887$	−29.7382	−0.998510	−0.499255	−	0.866455i	$-0.666393\pi$
$887$	−29.7382	−0.998510	−0.499255	+	0.866455i	$0.666393\pi$
$888$	0	0
$889$	46.6865	1.56581
$890$	0	0
$891$	−5.65926	−0.189592
$892$	0	0
$893$	7.08516	0.237096
$894$	0	0
$895$	18.3302	0.612712
$896$	0	0
$897$	12.5942	0.420509
$898$	0	0
$899$	17.1728	0.572743
$900$	0	0
$901$	−87.8859	−2.92790
$902$	0	0
$903$	−2.02554	−0.0674058
$904$	0	0
$905$	14.0722	0.467776
$906$	0	0
$907$	31.5583	1.04788	0.523939	−	0.851756i	$-0.324462\pi$
$907$	31.5583	1.04788	0.523939	+	0.851756i	$0.324462\pi$
$908$	0	0
$909$	10.9002	0.361536
$910$	0	0
$911$	21.0754	0.698258	0.349129	−	0.937075i	$-0.386478\pi$
$911$	21.0754	0.698258	0.349129	+	0.937075i	$0.386478\pi$
$912$	0	0
$913$	15.3353	0.507525
$914$	0	0
$915$	−7.34999	−0.242983
$916$	0	0
$917$	−29.2904	−0.967253
$918$	0	0
$919$	−23.3636	−0.770693	−0.385346	−	0.922772i	$-0.625918\pi$
$919$	−23.3636	−0.770693	−0.385346	+	0.922772i	$0.625918\pi$
$920$	0	0
$921$	−9.34869	−0.308050
$922$	0	0
$923$	−17.3516	−0.571135
$924$	0	0
$925$	−1.00000	−0.0328798
$926$	0	0
$927$	36.9311	1.21298
$928$	0	0
$929$	−47.9449	−1.57302	−0.786511	−	0.617577i	$-0.788114\pi$
$929$	−47.9449	−1.57302	−0.786511	+	0.617577i	$0.788114\pi$
$930$	0	0
$931$	−2.43224	−0.0797135
$932$	0	0
$933$	−8.56288	−0.280336
$934$	0	0
$935$	11.5928	0.379126
$936$	0	0
$937$	41.2487	1.34754	0.673768	−	0.738943i	$-0.264675\pi$
$937$	41.2487	1.34754	0.673768	+	0.738943i	$0.264675\pi$
$938$	0	0
$939$	−4.43431	−0.144708
$940$	0	0
$941$	−33.3222	−1.08627	−0.543136	−	0.839644i	$-0.682763\pi$
$941$	−33.3222	−1.08627	−0.543136	+	0.839644i	$0.682763\pi$
$942$	0	0
$943$	−9.89480	−0.322219
$944$	0	0
$945$	9.56956	0.311298
$946$	0	0
$947$	17.4307	0.566421	0.283211	−	0.959058i	$-0.408600\pi$
$947$	17.4307	0.566421	0.283211	+	0.959058i	$0.408600\pi$
$948$	0	0
$949$	19.7811	0.642122
$950$	0	0
$951$	28.1636	0.913267
$952$	0	0
$953$	−38.6625	−1.25240	−0.626200	−	0.779663i	$-0.715391\pi$
$953$	−38.6625	−1.25240	−0.626200	+	0.779663i	$0.715391\pi$
$954$	0	0
$955$	7.63049	0.246917
$956$	0	0
$957$	8.36250	0.270321
$958$	0	0
$959$	−7.95373	−0.256839
$960$	0	0
$961$	−20.8428	−0.672348
$962$	0	0
$963$	31.1112	1.00254
$964$	0	0
$965$	3.18819	0.102631
$966$	0	0
$967$	−34.3441	−1.10443	−0.552216	−	0.833701i	$-0.686218\pi$
$967$	−34.3441	−1.10443	−0.552216	+	0.833701i	$0.686218\pi$
$968$	0	0
$969$	5.38230	0.172904
$970$	0	0
$971$	13.8590	0.444758	0.222379	−	0.974960i	$-0.428618\pi$
$971$	13.8590	0.444758	0.222379	+	0.974960i	$0.428618\pi$
$972$	0	0
$973$	36.5371	1.17133
$974$	0	0
$975$	2.33732	0.0748542
$976$	0	0
$977$	14.6157	0.467598	0.233799	−	0.972285i	$-0.424884\pi$
$977$	14.6157	0.467598	0.233799	+	0.972285i	$0.424884\pi$
$978$	0	0
$979$	−4.76865	−0.152407
$980$	0	0
$981$	−19.5921	−0.625529
$982$	0	0
$983$	−47.4125	−1.51223	−0.756113	−	0.654442i	$-0.772904\pi$
$983$	−47.4125	−1.51223	−0.756113	+	0.654442i	$0.772904\pi$
$984$	0	0
$985$	−1.65654	−0.0527816
$986$	0	0
$987$	−12.7155	−0.404738
$988$	0	0
$989$	6.02848	0.191694
$990$	0	0
$991$	30.7010	0.975250	0.487625	−	0.873053i	$-0.337863\pi$
$991$	30.7010	0.975250	0.487625	+	0.873053i	$0.337863\pi$
$992$	0	0
$993$	14.8753	0.472054
$994$	0	0
$995$	−17.0245	−0.539713
$996$	0	0
$997$	37.7699	1.19619	0.598093	−	0.801427i	$-0.295925\pi$
$997$	37.7699	1.19619	0.598093	+	0.801427i	$0.295925\pi$
$998$	0	0
$999$	−4.46719	−0.141336

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1480.2.a.g.1.4	✓	5	4.3	odd	2
2960.2.a.bb.1.2		5	1.1	even	1	trivial
7400.2.a.r.1.2		5	20.19	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-0.845141 q^{3} +1.00000 q^{5} +2.14219 q^{7} -2.28574 q^{9} +O(q^{10})\)	\(q-0.845141 q^{3} +1.00000 q^{5} +2.14219 q^{7} -2.28574 q^{9} -1.83634 q^{11} -2.76560 q^{13} -0.845141 q^{15} -6.31299 q^{17} +1.00880 q^{19} -1.81045 q^{21} +5.38831 q^{23} +1.00000 q^{25} +4.46719 q^{27} +5.38831 q^{29} +3.18704 q^{31} +1.55197 q^{33} +2.14219 q^{35} -1.00000 q^{37} +2.33732 q^{39} -1.83634 q^{41} +1.11881 q^{43} -2.28574 q^{45} +7.02338 q^{47} -2.41103 q^{49} +5.33537 q^{51} +13.9214 q^{53} -1.83634 q^{55} -0.852575 q^{57} +2.21887 q^{59} +8.69677 q^{61} -4.89648 q^{63} -2.76560 q^{65} +15.0380 q^{67} -4.55388 q^{69} +6.27408 q^{71} -7.15255 q^{73} -0.845141 q^{75} -3.93379 q^{77} +6.70065 q^{79} +3.08181 q^{81} -8.35100 q^{83} -6.31299 q^{85} -4.55388 q^{87} +2.59682 q^{89} -5.92443 q^{91} -2.69349 q^{93} +1.00880 q^{95} +13.6325 q^{97} +4.19740 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 5 q^{3} + 5 q^{5} + 5 q^{7} + 6 q^{9}+O(q^{10}) \) 5 * q + 5 * q^3 + 5 * q^5 + 5 * q^7 + 6 * q^9	\( 5 q + 5 q^{3} + 5 q^{5} + 5 q^{7} + 6 q^{9} + 7 q^{11} - 6 q^{13} + 5 q^{15} + 12 q^{19} - q^{21} + 6 q^{23} + 5 q^{25} + 17 q^{27} + 6 q^{29} + 10 q^{31} + 3 q^{33} + 5 q^{35} - 5 q^{37} - 4 q^{39} + 7 q^{41} + 22 q^{43} + 6 q^{45} + 13 q^{47} + 14 q^{49} + 10 q^{51} - 11 q^{53} + 7 q^{55} - 8 q^{57} + 10 q^{59} + 10 q^{63} - 6 q^{65} + 24 q^{67} + 26 q^{69} + 13 q^{71} - 13 q^{73} + 5 q^{75} - 19 q^{77} - 2 q^{79} + q^{81} + 13 q^{83} + 26 q^{87} + 8 q^{89} + 8 q^{91} - 20 q^{93} + 12 q^{95} - 20 q^{97} - 2 q^{99}+O(q^{100}) \) 5 * q + 5 * q^3 + 5 * q^5 + 5 * q^7 + 6 * q^9 + 7 * q^11 - 6 * q^13 + 5 * q^15 + 12 * q^19 - q^21 + 6 * q^23 + 5 * q^25 + 17 * q^27 + 6 * q^29 + 10 * q^31 + 3 * q^33 + 5 * q^35 - 5 * q^37 - 4 * q^39 + 7 * q^41 + 22 * q^43 + 6 * q^45 + 13 * q^47 + 14 * q^49 + 10 * q^51 - 11 * q^53 + 7 * q^55 - 8 * q^57 + 10 * q^59 + 10 * q^63 - 6 * q^65 + 24 * q^67 + 26 * q^69 + 13 * q^71 - 13 * q^73 + 5 * q^75 - 19 * q^77 - 2 * q^79 + q^81 + 13 * q^83 + 26 * q^87 + 8 * q^89 + 8 * q^91 - 20 * q^93 + 12 * q^95 - 20 * q^97 - 2 * q^99

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