LMFDB - Embedded newform 2960.2.a.y.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("2960.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

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

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

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

Embedding invariants

Embedding label			1.3
Root			$0.325094$ 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)

$f(q)$	$=$	$q+0.325094 q^{3} -1.00000 q^{5} +3.82696 q^{7} -2.89431 q^{9} +O(q^{10})$	$q+0.325094 q^{3} -1.00000 q^{5} +3.82696 q^{7} -2.89431 q^{9} -1.58566 q^{11} +5.75961 q^{13} -0.325094 q^{15} -3.47997 q^{17} +2.19040 q^{19} +1.24412 q^{21} -1.34981 q^{23} +1.00000 q^{25} -1.91621 q^{27} +4.37055 q^{29} +2.47715 q^{31} -0.515490 q^{33} -3.82696 q^{35} +1.00000 q^{37} +1.87242 q^{39} +10.7185 q^{41} -7.88978 q^{43} +2.89431 q^{45} +6.71300 q^{47} +7.64565 q^{49} -1.13132 q^{51} +3.76415 q^{53} +1.58566 q^{55} +0.712085 q^{57} -8.19751 q^{59} +7.45096 q^{61} -11.0764 q^{63} -5.75961 q^{65} -0.190395 q^{67} -0.438816 q^{69} -8.51549 q^{71} +2.13470 q^{73} +0.325094 q^{75} -6.06827 q^{77} -0.408886 q^{79} +8.05999 q^{81} +4.84058 q^{83} +3.47997 q^{85} +1.42084 q^{87} +7.34417 q^{89} +22.0418 q^{91} +0.805309 q^{93} -2.19040 q^{95} +2.14494 q^{97} +4.58940 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 5 q^{5} - 3 q^{7} + 5 q^{9}+O(q^{10}) $ 5 * q - 5 * q^5 - 3 * q^7 + 5 * q^9	$ 5 q - 5 q^{5} - 3 q^{7} + 5 q^{9} - 3 q^{11} + 4 q^{13} + 7 q^{17} + 4 q^{19} - 10 q^{21} - 10 q^{23} + 5 q^{25} + 6 q^{27} + 19 q^{29} - 13 q^{31} + 6 q^{33} + 3 q^{35} + 5 q^{37} - 14 q^{39} + 11 q^{41} + 13 q^{43} - 5 q^{45} + 2 q^{49} + 12 q^{51} + 27 q^{53} + 3 q^{55} + 12 q^{57} - 16 q^{59} + 27 q^{61} - 37 q^{63} - 4 q^{65} + 6 q^{67} + 40 q^{69} - 34 q^{71} + 16 q^{73} + 9 q^{77} - 16 q^{79} + 9 q^{81} + 14 q^{83} - 7 q^{85} + 42 q^{87} + 38 q^{89} + 34 q^{91} + 30 q^{93} - 4 q^{95} + 5 q^{97} - 23 q^{99}+O(q^{100}) $ 5 * q - 5 * q^5 - 3 * q^7 + 5 * q^9 - 3 * q^11 + 4 * q^13 + 7 * q^17 + 4 * q^19 - 10 * q^21 - 10 * q^23 + 5 * q^25 + 6 * q^27 + 19 * q^29 - 13 * q^31 + 6 * q^33 + 3 * q^35 + 5 * q^37 - 14 * q^39 + 11 * q^41 + 13 * q^43 - 5 * q^45 + 2 * q^49 + 12 * q^51 + 27 * q^53 + 3 * q^55 + 12 * q^57 - 16 * q^59 + 27 * q^61 - 37 * q^63 - 4 * q^65 + 6 * q^67 + 40 * q^69 - 34 * q^71 + 16 * q^73 + 9 * q^77 - 16 * q^79 + 9 * q^81 + 14 * q^83 - 7 * q^85 + 42 * q^87 + 38 * q^89 + 34 * q^91 + 30 * q^93 - 4 * q^95 + 5 * q^97 - 23 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	0.325094	0.187693	0.0938467	−	0.995587i	$-0.470084\pi$
$3$	0.325094	0.187693	0.0938467	+	0.995587i	$0.470084\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	3.82696	1.44646	0.723228	−	0.690609i	$-0.242657\pi$
$7$	3.82696	1.44646	0.723228	+	0.690609i	$0.242657\pi$
$8$	0	0
$9$	−2.89431	−0.964771
$10$	0	0
$11$	−1.58566	−0.478095	−0.239047	−	0.971008i	$-0.576835\pi$
$11$	−1.58566	−0.478095	−0.239047	+	0.971008i	$0.576835\pi$
$12$	0	0
$13$	5.75961	1.59743	0.798715	−	0.601710i	$-0.205514\pi$
$13$	5.75961	1.59743	0.798715	+	0.601710i	$0.205514\pi$
$14$	0	0
$15$	−0.325094	−0.0839390
$16$	0	0
$17$	−3.47997	−0.844018	−0.422009	−	0.906592i	$-0.638675\pi$
$17$	−3.47997	−0.844018	−0.422009	+	0.906592i	$0.638675\pi$
$18$	0	0
$19$	2.19040	0.502511	0.251256	−	0.967921i	$-0.419157\pi$
$19$	2.19040	0.502511	0.251256	+	0.967921i	$0.419157\pi$
$20$	0	0
$21$	1.24412	0.271490
$22$	0	0
$23$	−1.34981	−0.281455	−0.140728	−	0.990048i	$-0.544944\pi$
$23$	−1.34981	−0.281455	−0.140728	+	0.990048i	$0.544944\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−1.91621	−0.368775
$28$	0	0
$29$	4.37055	0.811591	0.405795	−	0.913964i	$-0.366995\pi$
$29$	4.37055	0.811591	0.405795	+	0.913964i	$0.366995\pi$
$30$	0	0
$31$	2.47715	0.444910	0.222455	−	0.974943i	$-0.428593\pi$
$31$	2.47715	0.444910	0.222455	+	0.974943i	$0.428593\pi$
$32$	0	0
$33$	−0.515490	−0.0897352
$34$	0	0
$35$	−3.82696	−0.646875
$36$	0	0
$37$	1.00000	0.164399
$37$	1.00000	0.164399
$38$	0	0
$39$	1.87242	0.299827
$40$	0	0
$41$	10.7185	1.67394	0.836971	−	0.547248i	$-0.184324\pi$
$41$	10.7185	1.67394	0.836971	+	0.547248i	$0.184324\pi$
$42$	0	0
$43$	−7.88978	−1.20318	−0.601590	−	0.798805i	$-0.705466\pi$
$43$	−7.88978	−1.20318	−0.601590	+	0.798805i	$0.705466\pi$
$44$	0	0
$45$	2.89431	0.431459
$46$	0	0
$47$	6.71300	0.979192	0.489596	−	0.871949i	$-0.337144\pi$
$47$	6.71300	0.979192	0.489596	+	0.871949i	$0.337144\pi$
$48$	0	0
$49$	7.64565	1.09224
$50$	0	0
$51$	−1.13132	−0.158417
$52$	0	0
$53$	3.76415	0.517046	0.258523	−	0.966005i	$-0.416764\pi$
$53$	3.76415	0.517046	0.258523	+	0.966005i	$0.416764\pi$
$54$	0	0
$55$	1.58566	0.213810
$56$	0	0
$57$	0.712085	0.0943180
$58$	0	0
$59$	−8.19751	−1.06723	−0.533613	−	0.845729i	$-0.679166\pi$
$59$	−8.19751	−1.06723	−0.533613	+	0.845729i	$0.679166\pi$
$60$	0	0
$61$	7.45096	0.953998	0.476999	−	0.878904i	$-0.341725\pi$
$61$	7.45096	0.953998	0.476999	+	0.878904i	$0.341725\pi$
$62$	0	0
$63$	−11.0764	−1.39550
$64$	0	0
$65$	−5.75961	−0.714392
$66$	0	0
$67$	−0.190395	−0.0232605	−0.0116302	−	0.999932i	$-0.503702\pi$
$67$	−0.190395	−0.0232605	−0.0116302	+	0.999932i	$0.503702\pi$
$68$	0	0
$69$	−0.438816	−0.0528272
$70$	0	0
$71$	−8.51549	−1.01060	−0.505301	−	0.862943i	$-0.668619\pi$
$71$	−8.51549	−1.01060	−0.505301	+	0.862943i	$0.668619\pi$
$72$	0	0
$73$	2.13470	0.249848	0.124924	−	0.992166i	$-0.460131\pi$
$73$	2.13470	0.249848	0.124924	+	0.992166i	$0.460131\pi$
$74$	0	0
$75$	0.325094	0.0375387
$76$	0	0
$77$	−6.06827	−0.691543
$78$	0	0
$79$	−0.408886	−0.0460033	−0.0230016	−	0.999735i	$-0.507322\pi$
$79$	−0.408886	−0.0460033	−0.0230016	+	0.999735i	$0.507322\pi$
$80$	0	0
$81$	8.05999	0.895555
$82$	0	0
$83$	4.84058	0.531323	0.265662	−	0.964066i	$-0.414410\pi$
$83$	4.84058	0.531323	0.265662	+	0.964066i	$0.414410\pi$
$84$	0	0
$85$	3.47997	0.377456
$86$	0	0
$87$	1.42084	0.152330
$88$	0	0
$89$	7.34417	0.778480	0.389240	−	0.921136i	$-0.372738\pi$
$89$	7.34417	0.778480	0.389240	+	0.921136i	$0.372738\pi$
$90$	0	0
$91$	22.0418	2.31061
$92$	0	0
$93$	0.805309	0.0835066
$94$	0	0
$95$	−2.19040	−0.224730
$96$	0	0
$97$	2.14494	0.217786	0.108893	−	0.994053i	$-0.465269\pi$
$97$	2.14494	0.217786	0.108893	+	0.994053i	$0.465269\pi$
$98$	0	0
$99$	4.58940	0.461252
$100$	0	0
$101$	7.19469	0.715899	0.357949	−	0.933741i	$-0.383476\pi$
$101$	7.19469	0.715899	0.357949	+	0.933741i	$0.383476\pi$
$102$	0	0
$103$	18.0347	1.77701	0.888507	−	0.458863i	$-0.151743\pi$
$103$	18.0347	1.77701	0.888507	+	0.458863i	$0.151743\pi$
$104$	0	0
$105$	−1.24412	−0.121414
$106$	0	0
$107$	−6.57830	−0.635949	−0.317974	−	0.948099i	$-0.603003\pi$
$107$	−6.57830	−0.635949	−0.317974	+	0.948099i	$0.603003\pi$
$108$	0	0
$109$	16.3724	1.56819	0.784095	−	0.620641i	$-0.213127\pi$
$109$	16.3724	1.56819	0.784095	+	0.620641i	$0.213127\pi$
$110$	0	0
$111$	0.325094	0.0308566
$112$	0	0
$113$	−19.3033	−1.81590	−0.907952	−	0.419075i	$-0.862355\pi$
$113$	−19.3033	−1.81590	−0.907952	+	0.419075i	$0.862355\pi$
$114$	0	0
$115$	1.34981	0.125871
$116$	0	0
$117$	−16.6701	−1.54115
$118$	0	0
$119$	−13.3177	−1.22083
$120$	0	0
$121$	−8.48568	−0.771425
$122$	0	0
$123$	3.48451	0.314188
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−3.93523	−0.349195	−0.174598	−	0.984640i	$-0.555862\pi$
$127$	−3.93523	−0.349195	−0.174598	+	0.984640i	$0.555862\pi$
$128$	0	0
$129$	−2.56492	−0.225829
$130$	0	0
$131$	17.6330	1.54060	0.770299	−	0.637683i	$-0.220107\pi$
$131$	17.6330	1.54060	0.770299	+	0.637683i	$0.220107\pi$
$132$	0	0
$133$	8.38256	0.726860
$134$	0	0
$135$	1.91621	0.164921
$136$	0	0
$137$	−1.86530	−0.159363	−0.0796817	−	0.996820i	$-0.525390\pi$
$137$	−1.86530	−0.159363	−0.0796817	+	0.996820i	$0.525390\pi$
$138$	0	0
$139$	4.10679	0.348334	0.174167	−	0.984716i	$-0.444277\pi$
$139$	4.10679	0.348334	0.174167	+	0.984716i	$0.444277\pi$
$140$	0	0
$141$	2.18236	0.183788
$142$	0	0
$143$	−9.13279	−0.763723
$144$	0	0
$145$	−4.37055	−0.362954
$146$	0	0
$147$	2.48556	0.205005
$148$	0	0
$149$	5.85052	0.479294	0.239647	−	0.970860i	$-0.422968\pi$
$149$	5.85052	0.479294	0.239647	+	0.970860i	$0.422968\pi$
$150$	0	0
$151$	−4.43882	−0.361226	−0.180613	−	0.983554i	$-0.557808\pi$
$151$	−4.43882	−0.361226	−0.180613	+	0.983554i	$0.557808\pi$
$152$	0	0
$153$	10.0721	0.814284
$154$	0	0
$155$	−2.47715	−0.198970
$156$	0	0
$157$	9.19016	0.733454	0.366727	−	0.930329i	$-0.380478\pi$
$157$	9.19016	0.733454	0.366727	+	0.930329i	$0.380478\pi$
$158$	0	0
$159$	1.22370	0.0970461
$160$	0	0
$161$	−5.16568	−0.407112
$162$	0	0
$163$	23.8131	1.86519	0.932593	−	0.360929i	$-0.117540\pi$
$163$	23.8131	1.86519	0.932593	+	0.360929i	$0.117540\pi$
$164$	0	0
$165$	0.515490	0.0401308
$166$	0	0
$167$	−17.8233	−1.37921	−0.689606	−	0.724185i	$-0.742216\pi$
$167$	−17.8233	−1.37921	−0.689606	+	0.724185i	$0.742216\pi$
$168$	0	0
$169$	20.1732	1.55178
$170$	0	0
$171$	−6.33969	−0.484808
$172$	0	0
$173$	20.9645	1.59390	0.796952	−	0.604042i	$-0.206444\pi$
$173$	20.9645	1.59390	0.796952	+	0.604042i	$0.206444\pi$
$174$	0	0
$175$	3.82696	0.289291
$176$	0	0
$177$	−2.66497	−0.200311
$178$	0	0
$179$	−6.93149	−0.518084	−0.259042	−	0.965866i	$-0.583407\pi$
$179$	−6.93149	−0.518084	−0.259042	+	0.965866i	$0.583407\pi$
$180$	0	0
$181$	17.6392	1.31111	0.655554	−	0.755148i	$-0.272435\pi$
$181$	17.6392	1.31111	0.655554	+	0.755148i	$0.272435\pi$
$182$	0	0
$183$	2.42227	0.179059
$184$	0	0
$185$	−1.00000	−0.0735215
$186$	0	0
$187$	5.51806	0.403520
$188$	0	0
$189$	−7.33326	−0.533416
$190$	0	0
$191$	−15.9582	−1.15470	−0.577349	−	0.816498i	$-0.695913\pi$
$191$	−15.9582	−1.15470	−0.577349	+	0.816498i	$0.695913\pi$
$192$	0	0
$193$	8.34264	0.600517	0.300258	−	0.953858i	$-0.402927\pi$
$193$	8.34264	0.600517	0.300258	+	0.953858i	$0.402927\pi$
$194$	0	0
$195$	−1.87242	−0.134087
$196$	0	0
$197$	−13.0946	−0.932955	−0.466478	−	0.884533i	$-0.654477\pi$
$197$	−13.0946	−0.932955	−0.466478	+	0.884533i	$0.654477\pi$
$198$	0	0
$199$	−26.1156	−1.85128	−0.925641	−	0.378402i	$-0.876474\pi$
$199$	−26.1156	−1.85128	−0.925641	+	0.378402i	$0.876474\pi$
$200$	0	0
$201$	−0.0618964	−0.00436583
$202$	0	0
$203$	16.7259	1.17393
$204$	0	0
$205$	−10.7185	−0.748609
$206$	0	0
$207$	3.90678	0.271540
$208$	0	0
$209$	−3.47322	−0.240248
$210$	0	0
$211$	17.7551	1.22231	0.611155	−	0.791511i	$-0.290705\pi$
$211$	17.7551	1.22231	0.611155	+	0.791511i	$0.290705\pi$
$212$	0	0
$213$	−2.76834	−0.189683
$214$	0	0
$215$	7.88978	0.538078
$216$	0	0
$217$	9.47997	0.643543
$218$	0	0
$219$	0.693979	0.0468947
$220$	0	0
$221$	−20.0433	−1.34826
$222$	0	0
$223$	0.408077	0.0273268	0.0136634	−	0.999907i	$-0.495651\pi$
$223$	0.408077	0.0273268	0.0136634	+	0.999907i	$0.495651\pi$
$224$	0	0
$225$	−2.89431	−0.192954
$226$	0	0
$227$	23.8916	1.58574	0.792871	−	0.609390i	$-0.208585\pi$
$227$	23.8916	1.58574	0.792871	+	0.609390i	$0.208585\pi$
$228$	0	0
$229$	−11.5083	−0.760491	−0.380246	−	0.924886i	$-0.624161\pi$
$229$	−11.5083	−0.760491	−0.380246	+	0.924886i	$0.624161\pi$
$230$	0	0
$231$	−1.97276	−0.129798
$232$	0	0
$233$	8.39734	0.550128	0.275064	−	0.961426i	$-0.411301\pi$
$233$	8.39734	0.550128	0.275064	+	0.961426i	$0.411301\pi$
$234$	0	0
$235$	−6.71300	−0.437908
$236$	0	0
$237$	−0.132927	−0.00863450
$238$	0	0
$239$	10.4337	0.674901	0.337451	−	0.941343i	$-0.390435\pi$
$239$	10.4337	0.674901	0.337451	+	0.941343i	$0.390435\pi$
$240$	0	0
$241$	−1.10372	−0.0710969	−0.0355484	−	0.999368i	$-0.511318\pi$
$241$	−1.10372	−0.0710969	−0.0355484	+	0.999368i	$0.511318\pi$
$242$	0	0
$243$	8.36888	0.536864
$244$	0	0
$245$	−7.64565	−0.488463
$246$	0	0
$247$	12.6158	0.802726
$248$	0	0
$249$	1.57365	0.0997258
$250$	0	0
$251$	4.46544	0.281856	0.140928	−	0.990020i	$-0.454991\pi$
$251$	4.46544	0.281856	0.140928	+	0.990020i	$0.454991\pi$
$252$	0	0
$253$	2.14034	0.134562
$254$	0	0
$255$	1.13132	0.0708460
$256$	0	0
$257$	−1.49929	−0.0935230	−0.0467615	−	0.998906i	$-0.514890\pi$
$257$	−1.49929	−0.0935230	−0.0467615	+	0.998906i	$0.514890\pi$
$258$	0	0
$259$	3.82696	0.237796
$260$	0	0
$261$	−12.6497	−0.782999
$262$	0	0
$263$	−24.7827	−1.52817	−0.764085	−	0.645116i	$-0.776809\pi$
$263$	−24.7827	−1.52817	−0.764085	+	0.645116i	$0.776809\pi$
$264$	0	0
$265$	−3.76415	−0.231230
$266$	0	0
$267$	2.38755	0.146116
$268$	0	0
$269$	−28.6338	−1.74583	−0.872917	−	0.487868i	$-0.837775\pi$
$269$	−28.6338	−1.74583	−0.872917	+	0.487868i	$0.837775\pi$
$270$	0	0
$271$	6.69055	0.406422	0.203211	−	0.979135i	$-0.434862\pi$
$271$	6.69055	0.406422	0.203211	+	0.979135i	$0.434862\pi$
$272$	0	0
$273$	7.16568	0.433687
$274$	0	0
$275$	−1.58566	−0.0956189
$276$	0	0
$277$	−21.1475	−1.27063	−0.635314	−	0.772254i	$-0.719129\pi$
$277$	−21.1475	−1.27063	−0.635314	+	0.772254i	$0.719129\pi$
$278$	0	0
$279$	−7.16966	−0.429236
$280$	0	0
$281$	25.1732	1.50170	0.750852	−	0.660470i	$-0.229643\pi$
$281$	25.1732	1.50170	0.750852	+	0.660470i	$0.229643\pi$
$282$	0	0
$283$	19.1788	1.14006	0.570031	−	0.821623i	$-0.306931\pi$
$283$	19.1788	1.14006	0.570031	+	0.821623i	$0.306931\pi$
$284$	0	0
$285$	−0.712085	−0.0421803
$286$	0	0
$287$	41.0191	2.42128
$288$	0	0
$289$	−4.88978	−0.287634
$290$	0	0
$291$	0.697308	0.0408769
$292$	0	0
$293$	31.2305	1.82451	0.912253	−	0.409627i	$-0.134341\pi$
$293$	31.2305	1.82451	0.912253	+	0.409627i	$0.134341\pi$
$294$	0	0
$295$	8.19751	0.477278
$296$	0	0
$297$	3.03846	0.176309
$298$	0	0
$299$	−7.77439	−0.449605
$300$	0	0
$301$	−30.1939	−1.74035
$302$	0	0
$303$	2.33895	0.134369
$304$	0	0
$305$	−7.45096	−0.426641
$306$	0	0
$307$	7.97866	0.455366	0.227683	−	0.973735i	$-0.426885\pi$
$307$	7.97866	0.455366	0.227683	+	0.973735i	$0.426885\pi$
$308$	0	0
$309$	5.86299	0.333534
$310$	0	0
$311$	−1.55573	−0.0882174	−0.0441087	−	0.999027i	$-0.514045\pi$
$311$	−1.55573	−0.0882174	−0.0441087	+	0.999027i	$0.514045\pi$
$312$	0	0
$313$	−10.2985	−0.582104	−0.291052	−	0.956707i	$-0.594005\pi$
$313$	−10.2985	−0.582104	−0.291052	+	0.956707i	$0.594005\pi$
$314$	0	0
$315$	11.0764	0.624086
$316$	0	0
$317$	10.3342	0.580429	0.290214	−	0.956962i	$-0.406273\pi$
$317$	10.3342	0.580429	0.290214	+	0.956962i	$0.406273\pi$
$318$	0	0
$319$	−6.93021	−0.388017
$320$	0	0
$321$	−2.13857	−0.119363
$322$	0	0
$323$	−7.62252	−0.424128
$324$	0	0
$325$	5.75961	0.319486
$326$	0	0
$327$	5.32257	0.294339
$328$	0	0
$329$	25.6904	1.41636
$330$	0	0
$331$	1.20321	0.0661342	0.0330671	−	0.999453i	$-0.489472\pi$
$331$	1.20321	0.0661342	0.0330671	+	0.999453i	$0.489472\pi$
$332$	0	0
$333$	−2.89431	−0.158607
$334$	0	0
$335$	0.190395	0.0104024
$336$	0	0
$337$	−11.0094	−0.599719	−0.299860	−	0.953983i	$-0.596940\pi$
$337$	−11.0094	−0.599719	−0.299860	+	0.953983i	$0.596940\pi$
$338$	0	0
$339$	−6.27540	−0.340833
$340$	0	0
$341$	−3.92792	−0.212709
$342$	0	0
$343$	2.47089	0.133416
$344$	0	0
$345$	0.438816	0.0236251
$346$	0	0
$347$	−17.5663	−0.943011	−0.471505	−	0.881863i	$-0.656289\pi$
$347$	−17.5663	−0.943011	−0.471505	+	0.881863i	$0.656289\pi$
$348$	0	0
$349$	−9.68294	−0.518316	−0.259158	−	0.965835i	$-0.583445\pi$
$349$	−9.68294	−0.518316	−0.259158	+	0.965835i	$0.583445\pi$
$350$	0	0
$351$	−11.0366	−0.589091
$352$	0	0
$353$	−7.75508	−0.412761	−0.206381	−	0.978472i	$-0.566168\pi$
$353$	−7.75508	−0.412761	−0.206381	+	0.978472i	$0.566168\pi$
$354$	0	0
$355$	8.51549	0.451955
$356$	0	0
$357$	−4.32952	−0.229143
$358$	0	0
$359$	−13.8381	−0.730345	−0.365172	−	0.930940i	$-0.618990\pi$
$359$	−13.8381	−0.730345	−0.365172	+	0.930940i	$0.618990\pi$
$360$	0	0
$361$	−14.2022	−0.747483
$362$	0	0
$363$	−2.75865	−0.144791
$364$	0	0
$365$	−2.13470	−0.111735
$366$	0	0
$367$	−7.28007	−0.380016	−0.190008	−	0.981783i	$-0.560851\pi$
$367$	−7.28007	−0.380016	−0.190008	+	0.981783i	$0.560851\pi$
$368$	0	0
$369$	−31.0226	−1.61497
$370$	0	0
$371$	14.4053	0.747884
$372$	0	0
$373$	14.4388	0.747614	0.373807	−	0.927507i	$-0.378052\pi$
$373$	14.4388	0.747614	0.373807	+	0.927507i	$0.378052\pi$
$374$	0	0
$375$	−0.325094	−0.0167878
$376$	0	0
$377$	25.1727	1.29646
$378$	0	0
$379$	−26.4025	−1.35621	−0.678103	−	0.734967i	$-0.737198\pi$
$379$	−26.4025	−1.35621	−0.678103	+	0.734967i	$0.737198\pi$
$380$	0	0
$381$	−1.27932	−0.0655417
$382$	0	0
$383$	−12.2985	−0.628423	−0.314211	−	0.949353i	$-0.601740\pi$
$383$	−12.2985	−0.628423	−0.314211	+	0.949353i	$0.601740\pi$
$384$	0	0
$385$	6.06827	0.309268
$386$	0	0
$387$	22.8355	1.16079
$388$	0	0
$389$	−2.73317	−0.138577	−0.0692886	−	0.997597i	$-0.522073\pi$
$389$	−2.73317	−0.138577	−0.0692886	+	0.997597i	$0.522073\pi$
$390$	0	0
$391$	4.69731	0.237553
$392$	0	0
$393$	5.73237	0.289160
$394$	0	0
$395$	0.408886	0.0205733
$396$	0	0
$397$	−4.65583	−0.233670	−0.116835	−	0.993151i	$-0.537275\pi$
$397$	−4.65583	−0.233670	−0.116835	+	0.993151i	$0.537275\pi$
$398$	0	0
$399$	2.72512	0.136427
$400$	0	0
$401$	28.6702	1.43172	0.715861	−	0.698243i	$-0.246035\pi$
$401$	28.6702	1.43172	0.715861	+	0.698243i	$0.246035\pi$
$402$	0	0
$403$	14.2674	0.710712
$404$	0	0
$405$	−8.05999	−0.400504
$406$	0	0
$407$	−1.58566	−0.0785983
$408$	0	0
$409$	−31.9966	−1.58213	−0.791064	−	0.611733i	$-0.790473\pi$
$409$	−31.9966	−1.58213	−0.791064	+	0.611733i	$0.790473\pi$
$410$	0	0
$411$	−0.606399	−0.0299115
$412$	0	0
$413$	−31.3716	−1.54369
$414$	0	0
$415$	−4.84058	−0.237615
$416$	0	0
$417$	1.33510	0.0653799
$418$	0	0
$419$	6.09617	0.297818	0.148909	−	0.988851i	$-0.452424\pi$
$419$	6.09617	0.297818	0.148909	+	0.988851i	$0.452424\pi$
$420$	0	0
$421$	−16.7395	−0.815834	−0.407917	−	0.913019i	$-0.633745\pi$
$421$	−16.7395	−0.815834	−0.407917	+	0.913019i	$0.633745\pi$
$422$	0	0
$423$	−19.4295	−0.944696
$424$	0	0
$425$	−3.47997	−0.168804
$426$	0	0
$427$	28.5146	1.37992
$428$	0	0
$429$	−2.96902	−0.143346
$430$	0	0
$431$	−21.2911	−1.02556	−0.512778	−	0.858521i	$-0.671384\pi$
$431$	−21.2911	−1.02556	−0.512778	+	0.858521i	$0.671384\pi$
$432$	0	0
$433$	−7.68117	−0.369133	−0.184567	−	0.982820i	$-0.559088\pi$
$433$	−7.68117	−0.369133	−0.184567	+	0.982820i	$0.559088\pi$
$434$	0	0
$435$	−1.42084	−0.0681241
$436$	0	0
$437$	−2.95662	−0.141434
$438$	0	0
$439$	−21.8068	−1.04078	−0.520392	−	0.853928i	$-0.674214\pi$
$439$	−21.8068	−1.04078	−0.520392	+	0.853928i	$0.674214\pi$
$440$	0	0
$441$	−22.1289	−1.05376
$442$	0	0
$443$	−14.3250	−0.680600	−0.340300	−	0.940317i	$-0.610529\pi$
$443$	−14.3250	−0.680600	−0.340300	+	0.940317i	$0.610529\pi$
$444$	0	0
$445$	−7.34417	−0.348147
$446$	0	0
$447$	1.90197	0.0899602
$448$	0	0
$449$	−3.68117	−0.173725	−0.0868625	−	0.996220i	$-0.527684\pi$
$449$	−3.68117	−0.173725	−0.0868625	+	0.996220i	$0.527684\pi$
$450$	0	0
$451$	−16.9958	−0.800303
$452$	0	0
$453$	−1.44303	−0.0677997
$454$	0	0
$455$	−22.0418	−1.03334
$456$	0	0
$457$	3.44532	0.161165	0.0805826	−	0.996748i	$-0.474322\pi$
$457$	3.44532	0.161165	0.0805826	+	0.996748i	$0.474322\pi$
$458$	0	0
$459$	6.66836	0.311252
$460$	0	0
$461$	1.74649	0.0813419	0.0406710	−	0.999173i	$-0.487050\pi$
$461$	1.74649	0.0813419	0.0406710	+	0.999173i	$0.487050\pi$
$462$	0	0
$463$	31.0788	1.44436	0.722178	−	0.691708i	$-0.243141\pi$
$463$	31.0788	1.44436	0.722178	+	0.691708i	$0.243141\pi$
$464$	0	0
$465$	−0.805309	−0.0373453
$466$	0	0
$467$	8.46146	0.391550	0.195775	−	0.980649i	$-0.437278\pi$
$467$	8.46146	0.391550	0.195775	+	0.980649i	$0.437278\pi$
$468$	0	0
$469$	−0.728635	−0.0336452
$470$	0	0
$471$	2.98767	0.137664
$472$	0	0
$473$	12.5105	0.575234
$474$	0	0
$475$	2.19040	0.100502
$476$	0	0
$477$	−10.8946	−0.498831
$478$	0	0
$479$	−21.1265	−0.965296	−0.482648	−	0.875814i	$-0.660325\pi$
$479$	−21.1265	−0.965296	−0.482648	+	0.875814i	$0.660325\pi$
$480$	0	0
$481$	5.75961	0.262616
$482$	0	0
$483$	−1.67933	−0.0764123
$484$	0	0
$485$	−2.14494	−0.0973967
$486$	0	0
$487$	39.3183	1.78168	0.890841	−	0.454316i	$-0.150116\pi$
$487$	39.3183	1.78168	0.890841	+	0.454316i	$0.150116\pi$
$488$	0	0
$489$	7.74151	0.350083
$490$	0	0
$491$	−18.5025	−0.835005	−0.417503	−	0.908676i	$-0.637095\pi$
$491$	−18.5025	−0.835005	−0.417503	+	0.908676i	$0.637095\pi$
$492$	0	0
$493$	−15.2094	−0.684997
$494$	0	0
$495$	−4.58940	−0.206278
$496$	0	0
$497$	−32.5885	−1.46179
$498$	0	0
$499$	−43.4821	−1.94652	−0.973262	−	0.229697i	$-0.926226\pi$
$499$	−43.4821	−1.94652	−0.973262	+	0.229697i	$0.926226\pi$
$500$	0	0
$501$	−5.79427	−0.258869
$502$	0	0
$503$	−15.2788	−0.681247	−0.340624	−	0.940200i	$-0.610638\pi$
$503$	−15.2788	−0.681247	−0.340624	+	0.940200i	$0.610638\pi$
$504$	0	0
$505$	−7.19469	−0.320160
$506$	0	0
$507$	6.55818	0.291259
$508$	0	0
$509$	31.6882	1.40455	0.702277	−	0.711904i	$-0.252167\pi$
$509$	31.6882	1.40455	0.702277	+	0.711904i	$0.252167\pi$
$510$	0	0
$511$	8.16942	0.361394
$512$	0	0
$513$	−4.19725	−0.185313
$514$	0	0
$515$	−18.0347	−0.794705
$516$	0	0
$517$	−10.6445	−0.468147
$518$	0	0
$519$	6.81546	0.299165
$520$	0	0
$521$	29.8245	1.30663	0.653316	−	0.757085i	$-0.273377\pi$
$521$	29.8245	1.30663	0.653316	+	0.757085i	$0.273377\pi$
$522$	0	0
$523$	−17.7814	−0.777526	−0.388763	−	0.921338i	$-0.627097\pi$
$523$	−17.7814	−0.777526	−0.388763	+	0.921338i	$0.627097\pi$
$524$	0	0
$525$	1.24412	0.0542981
$526$	0	0
$527$	−8.62043	−0.375512
$528$	0	0
$529$	−21.1780	−0.920783
$530$	0	0
$531$	23.7262	1.02963
$532$	0	0
$533$	61.7342	2.67400
$534$	0	0
$535$	6.57830	0.284405
$536$	0	0
$537$	−2.25339	−0.0972410
$538$	0	0
$539$	−12.1234	−0.522192
$540$	0	0
$541$	14.7520	0.634238	0.317119	−	0.948386i	$-0.397284\pi$
$541$	14.7520	0.634238	0.317119	+	0.948386i	$0.397284\pi$
$542$	0	0
$543$	5.73439	0.246086
$544$	0	0
$545$	−16.3724	−0.701316
$546$	0	0
$547$	3.52763	0.150831	0.0754154	−	0.997152i	$-0.475972\pi$
$547$	3.52763	0.150831	0.0754154	+	0.997152i	$0.475972\pi$
$548$	0	0
$549$	−21.5654	−0.920390
$550$	0	0
$551$	9.57323	0.407833
$552$	0	0
$553$	−1.56479	−0.0665417
$554$	0	0
$555$	−0.325094	−0.0137995
$556$	0	0
$557$	−30.2003	−1.27963	−0.639813	−	0.768531i	$-0.720988\pi$
$557$	−30.2003	−1.27963	−0.639813	+	0.768531i	$0.720988\pi$
$558$	0	0
$559$	−45.4421	−1.92200
$560$	0	0
$561$	1.79389	0.0757381
$562$	0	0
$563$	−41.0573	−1.73036	−0.865179	−	0.501463i	$-0.832795\pi$
$563$	−41.0573	−1.73036	−0.865179	+	0.501463i	$0.832795\pi$
$564$	0	0
$565$	19.3033	0.812097
$566$	0	0
$567$	30.8453	1.29538
$568$	0	0
$569$	−37.3784	−1.56698	−0.783492	−	0.621402i	$-0.786564\pi$
$569$	−37.3784	−1.56698	−0.783492	+	0.621402i	$0.786564\pi$
$570$	0	0
$571$	−12.9731	−0.542909	−0.271454	−	0.962451i	$-0.587505\pi$
$571$	−12.9731	−0.542909	−0.271454	+	0.962451i	$0.587505\pi$
$572$	0	0
$573$	−5.18793	−0.216729
$574$	0	0
$575$	−1.34981	−0.0562910
$576$	0	0
$577$	15.5317	0.646593	0.323296	−	0.946298i	$-0.395209\pi$
$577$	15.5317	0.646593	0.323296	+	0.946298i	$0.395209\pi$
$578$	0	0
$579$	2.71215	0.112713
$580$	0	0
$581$	18.5247	0.768536
$582$	0	0
$583$	−5.96867	−0.247197
$584$	0	0
$585$	16.6701	0.689225
$586$	0	0
$587$	−27.7802	−1.14661	−0.573306	−	0.819341i	$-0.694339\pi$
$587$	−27.7802	−1.14661	−0.573306	+	0.819341i	$0.694339\pi$
$588$	0	0
$589$	5.42594	0.223572
$590$	0	0
$591$	−4.25700	−0.175109
$592$	0	0
$593$	−1.29511	−0.0531840	−0.0265920	−	0.999646i	$-0.508465\pi$
$593$	−1.29511	−0.0531840	−0.0265920	+	0.999646i	$0.508465\pi$
$594$	0	0
$595$	13.3177	0.545974
$596$	0	0
$597$	−8.49002	−0.347473
$598$	0	0
$599$	6.61388	0.270236	0.135118	−	0.990830i	$-0.456859\pi$
$599$	6.61388	0.270236	0.135118	+	0.990830i	$0.456859\pi$
$600$	0	0
$601$	−5.76421	−0.235127	−0.117564	−	0.993065i	$-0.537508\pi$
$601$	−5.76421	−0.235127	−0.117564	+	0.993065i	$0.537508\pi$
$602$	0	0
$603$	0.551063	0.0224410
$604$	0	0
$605$	8.48568	0.344992
$606$	0	0
$607$	24.8720	1.00952	0.504761	−	0.863259i	$-0.331580\pi$
$607$	24.8720	1.00952	0.504761	+	0.863259i	$0.331580\pi$
$608$	0	0
$609$	5.43751	0.220339
$610$	0	0
$611$	38.6643	1.56419
$612$	0	0
$613$	−10.0630	−0.406441	−0.203220	−	0.979133i	$-0.565141\pi$
$613$	−10.0630	−0.406441	−0.203220	+	0.979133i	$0.565141\pi$
$614$	0	0
$615$	−3.48451	−0.140509
$616$	0	0
$617$	48.0422	1.93411	0.967053	−	0.254575i	$-0.0819356\pi$
$617$	48.0422	1.93411	0.967053	+	0.254575i	$0.0819356\pi$
$618$	0	0
$619$	9.46520	0.380438	0.190219	−	0.981742i	$-0.439080\pi$
$619$	9.46520	0.380438	0.190219	+	0.981742i	$0.439080\pi$
$620$	0	0
$621$	2.58652	0.103793
$622$	0	0
$623$	28.1059	1.12604
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−1.12913	−0.0450929
$628$	0	0
$629$	−3.47997	−0.138756
$630$	0	0
$631$	−42.2366	−1.68141	−0.840706	−	0.541492i	$-0.817860\pi$
$631$	−42.2366	−1.68141	−0.840706	+	0.541492i	$0.817860\pi$
$632$	0	0
$633$	5.77208	0.229419
$634$	0	0
$635$	3.93523	0.156165
$636$	0	0
$637$	44.0360	1.74477
$638$	0	0
$639$	24.6465	0.975000
$640$	0	0
$641$	−4.55291	−0.179829	−0.0899145	−	0.995949i	$-0.528659\pi$
$641$	−4.55291	−0.179829	−0.0899145	+	0.995949i	$0.528659\pi$
$642$	0	0
$643$	27.8825	1.09958	0.549790	−	0.835303i	$-0.314708\pi$
$643$	27.8825	1.09958	0.549790	+	0.835303i	$0.314708\pi$
$644$	0	0
$645$	2.56492	0.100994
$646$	0	0
$647$	−20.2517	−0.796178	−0.398089	−	0.917347i	$-0.630326\pi$
$647$	−20.2517	−0.796178	−0.398089	+	0.917347i	$0.630326\pi$
$648$	0	0
$649$	12.9985	0.510235
$650$	0	0
$651$	3.08189	0.120789
$652$	0	0
$653$	−26.2513	−1.02729	−0.513645	−	0.858003i	$-0.671705\pi$
$653$	−26.2513	−1.02729	−0.513645	+	0.858003i	$0.671705\pi$
$654$	0	0
$655$	−17.6330	−0.688976
$656$	0	0
$657$	−6.17849	−0.241046
$658$	0	0
$659$	−27.8904	−1.08646	−0.543229	−	0.839585i	$-0.682798\pi$
$659$	−27.8904	−1.08646	−0.543229	+	0.839585i	$0.682798\pi$
$660$	0	0
$661$	36.5279	1.42077	0.710386	−	0.703812i	$-0.248520\pi$
$661$	36.5279	1.42077	0.710386	+	0.703812i	$0.248520\pi$
$662$	0	0
$663$	−6.51597	−0.253059
$664$	0	0
$665$	−8.38256	−0.325062
$666$	0	0
$667$	−5.89942	−0.228426
$668$	0	0
$669$	0.132663	0.00512907
$670$	0	0
$671$	−11.8147	−0.456101
$672$	0	0
$673$	−40.8738	−1.57557	−0.787785	−	0.615950i	$-0.788772\pi$
$673$	−40.8738	−1.57557	−0.787785	+	0.615950i	$0.788772\pi$
$674$	0	0
$675$	−1.91621	−0.0737549
$676$	0	0
$677$	−23.2146	−0.892211	−0.446105	−	0.894980i	$-0.647189\pi$
$677$	−23.2146	−0.892211	−0.446105	+	0.894980i	$0.647189\pi$
$678$	0	0
$679$	8.20861	0.315018
$680$	0	0
$681$	7.76703	0.297633
$682$	0	0
$683$	10.2792	0.393321	0.196661	−	0.980472i	$-0.436990\pi$
$683$	10.2792	0.393321	0.196661	+	0.980472i	$0.436990\pi$
$684$	0	0
$685$	1.86530	0.0712695
$686$	0	0
$687$	−3.74129	−0.142739
$688$	0	0
$689$	21.6801	0.825944
$690$	0	0
$691$	−15.1026	−0.574529	−0.287265	−	0.957851i	$-0.592746\pi$
$691$	−15.1026	−0.574529	−0.287265	+	0.957851i	$0.592746\pi$
$692$	0	0
$693$	17.5635	0.667181
$694$	0	0
$695$	−4.10679	−0.155780
$696$	0	0
$697$	−37.3000	−1.41284
$698$	0	0
$699$	2.72993	0.103255
$700$	0	0
$701$	3.00900	0.113648	0.0568242	−	0.998384i	$-0.481903\pi$
$701$	3.00900	0.113648	0.0568242	+	0.998384i	$0.481903\pi$
$702$	0	0
$703$	2.19040	0.0826123
$704$	0	0
$705$	−2.18236	−0.0821924
$706$	0	0
$707$	27.5338	1.03552
$708$	0	0
$709$	24.4075	0.916642	0.458321	−	0.888787i	$-0.348451\pi$
$709$	24.4075	0.916642	0.458321	+	0.888787i	$0.348451\pi$
$710$	0	0
$711$	1.18344	0.0443826
$712$	0	0
$713$	−3.34369	−0.125222
$714$	0	0
$715$	9.13279	0.341547
$716$	0	0
$717$	3.39195	0.126675
$718$	0	0
$719$	−2.68872	−0.100272	−0.0501361	−	0.998742i	$-0.515966\pi$
$719$	−2.68872	−0.100272	−0.0501361	+	0.998742i	$0.515966\pi$
$720$	0	0
$721$	69.0182	2.57037
$722$	0	0
$723$	−0.358813	−0.0133444
$724$	0	0
$725$	4.37055	0.162318
$726$	0	0
$727$	−3.58068	−0.132800	−0.0664001	−	0.997793i	$-0.521151\pi$
$727$	−3.58068	−0.132800	−0.0664001	+	0.997793i	$0.521151\pi$
$728$	0	0
$729$	−21.4593	−0.794789
$730$	0	0
$731$	27.4562	1.01551
$732$	0	0
$733$	13.3136	0.491748	0.245874	−	0.969302i	$-0.420925\pi$
$733$	13.3136	0.491748	0.245874	+	0.969302i	$0.420925\pi$
$734$	0	0
$735$	−2.48556	−0.0916812
$736$	0	0
$737$	0.301902	0.0111207
$738$	0	0
$739$	−7.25614	−0.266921	−0.133461	−	0.991054i	$-0.542609\pi$
$739$	−7.25614	−0.266921	−0.133461	+	0.991054i	$0.542609\pi$
$740$	0	0
$741$	4.10134	0.150666
$742$	0	0
$743$	−41.2631	−1.51380	−0.756899	−	0.653532i	$-0.773287\pi$
$743$	−41.2631	−1.51380	−0.756899	+	0.653532i	$0.773287\pi$
$744$	0	0
$745$	−5.85052	−0.214347
$746$	0	0
$747$	−14.0102	−0.512605
$748$	0	0
$749$	−25.1749	−0.919872
$750$	0	0
$751$	30.2365	1.10335	0.551673	−	0.834060i	$-0.313989\pi$
$751$	30.2365	1.10335	0.551673	+	0.834060i	$0.313989\pi$
$752$	0	0
$753$	1.45169	0.0529025
$754$	0	0
$755$	4.43882	0.161545
$756$	0	0
$757$	31.7494	1.15395	0.576975	−	0.816762i	$-0.304233\pi$
$757$	31.7494	1.15395	0.576975	+	0.816762i	$0.304233\pi$
$758$	0	0
$759$	0.695814	0.0252564
$760$	0	0
$761$	−4.70533	−0.170568	−0.0852841	−	0.996357i	$-0.527180\pi$
$761$	−4.70533	−0.170568	−0.0852841	+	0.996357i	$0.527180\pi$
$762$	0	0
$763$	62.6565	2.26832
$764$	0	0
$765$	−10.0721	−0.364159
$766$	0	0
$767$	−47.2145	−1.70482
$768$	0	0
$769$	−0.916159	−0.0330375	−0.0165188	−	0.999864i	$-0.505258\pi$
$769$	−0.916159	−0.0330375	−0.0165188	+	0.999864i	$0.505258\pi$
$770$	0	0
$771$	−0.487410	−0.0175536
$772$	0	0
$773$	18.7690	0.675074	0.337537	−	0.941312i	$-0.390406\pi$
$773$	18.7690	0.675074	0.337537	+	0.941312i	$0.390406\pi$
$774$	0	0
$775$	2.47715	0.0889820
$776$	0	0
$777$	1.24412	0.0446327
$778$	0	0
$779$	23.4777	0.841174
$780$	0	0
$781$	13.5027	0.483164
$782$	0	0
$783$	−8.37488	−0.299294
$784$	0	0
$785$	−9.19016	−0.328011
$786$	0	0
$787$	−53.7975	−1.91767	−0.958836	−	0.283960i	$-0.908352\pi$
$787$	−53.7975	−1.91767	−0.958836	+	0.283960i	$0.908352\pi$
$788$	0	0
$789$	−8.05673	−0.286827
$790$	0	0
$791$	−73.8731	−2.62663
$792$	0	0
$793$	42.9147	1.52394
$794$	0	0
$795$	−1.22370	−0.0434003
$796$	0	0
$797$	45.2478	1.60276	0.801380	−	0.598156i	$-0.204100\pi$
$797$	45.2478	1.60276	0.801380	+	0.598156i	$0.204100\pi$
$798$	0	0
$799$	−23.3611	−0.826456
$800$	0	0
$801$	−21.2563	−0.751055
$802$	0	0
$803$	−3.38491	−0.119451
$804$	0	0
$805$	5.16568	0.182066
$806$	0	0
$807$	−9.30870	−0.327682
$808$	0	0
$809$	−10.0837	−0.354523	−0.177262	−	0.984164i	$-0.556724\pi$
$809$	−10.0837	−0.354523	−0.177262	+	0.984164i	$0.556724\pi$
$810$	0	0
$811$	23.0527	0.809490	0.404745	−	0.914430i	$-0.367360\pi$
$811$	23.0527	0.809490	0.404745	+	0.914430i	$0.367360\pi$
$812$	0	0
$813$	2.17506	0.0762827
$814$	0	0
$815$	−23.8131	−0.834137
$816$	0	0
$817$	−17.2817	−0.604611
$818$	0	0
$819$	−63.7960	−2.22921
$820$	0	0
$821$	−7.33215	−0.255894	−0.127947	−	0.991781i	$-0.540839\pi$
$821$	−7.33215	−0.255894	−0.127947	+	0.991781i	$0.540839\pi$
$822$	0	0
$823$	−21.8172	−0.760501	−0.380250	−	0.924884i	$-0.624162\pi$
$823$	−21.8172	−0.760501	−0.380250	+	0.924884i	$0.624162\pi$
$824$	0	0
$825$	−0.515490	−0.0179470
$826$	0	0
$827$	−21.3136	−0.741145	−0.370573	−	0.928803i	$-0.620839\pi$
$827$	−21.3136	−0.741145	−0.370573	+	0.928803i	$0.620839\pi$
$828$	0	0
$829$	−49.5761	−1.72185	−0.860925	−	0.508732i	$-0.830114\pi$
$829$	−49.5761	−1.72185	−0.860925	+	0.508732i	$0.830114\pi$
$830$	0	0
$831$	−6.87493	−0.238489
$832$	0	0
$833$	−26.6067	−0.921867
$834$	0	0
$835$	17.8233	0.616802
$836$	0	0
$837$	−4.74674	−0.164071
$838$	0	0
$839$	35.3886	1.22175	0.610875	−	0.791727i	$-0.290818\pi$
$839$	35.3886	1.22175	0.610875	+	0.791727i	$0.290818\pi$
$840$	0	0
$841$	−9.89830	−0.341321
$842$	0	0
$843$	8.18365	0.281860
$844$	0	0
$845$	−20.1732	−0.693978
$846$	0	0
$847$	−32.4744	−1.11583
$848$	0	0
$849$	6.23492	0.213982
$850$	0	0
$851$	−1.34981	−0.0462709
$852$	0	0
$853$	−8.92866	−0.305711	−0.152856	−	0.988249i	$-0.548847\pi$
$853$	−8.92866	−0.305711	−0.152856	+	0.988249i	$0.548847\pi$
$854$	0	0
$855$	6.33969	0.216813
$856$	0	0
$857$	−51.3056	−1.75257	−0.876283	−	0.481797i	$-0.839984\pi$
$857$	−51.3056	−1.75257	−0.876283	+	0.481797i	$0.839984\pi$
$858$	0	0
$859$	6.13273	0.209246	0.104623	−	0.994512i	$-0.466636\pi$
$859$	6.13273	0.209246	0.104623	+	0.994512i	$0.466636\pi$
$860$	0	0
$861$	13.3351	0.454459
$862$	0	0
$863$	5.92388	0.201651	0.100826	−	0.994904i	$-0.467852\pi$
$863$	5.92388	0.201651	0.100826	+	0.994904i	$0.467852\pi$
$864$	0	0
$865$	−20.9645	−0.712816
$866$	0	0
$867$	−1.58964	−0.0539870
$868$	0	0
$869$	0.648354	0.0219939
$870$	0	0
$871$	−1.09660	−0.0371570
$872$	0	0
$873$	−6.20813	−0.210113
$874$	0	0
$875$	−3.82696	−0.129375
$876$	0	0
$877$	−49.5092	−1.67181	−0.835904	−	0.548876i	$-0.815056\pi$
$877$	−49.5092	−1.67181	−0.835904	+	0.548876i	$0.815056\pi$
$878$	0	0
$879$	10.1529	0.342448
$880$	0	0
$881$	5.35275	0.180339	0.0901694	−	0.995926i	$-0.471259\pi$
$881$	5.35275	0.180339	0.0901694	+	0.995926i	$0.471259\pi$
$882$	0	0
$883$	−37.2522	−1.25364	−0.626819	−	0.779165i	$-0.715643\pi$
$883$	−37.2522	−1.25364	−0.626819	+	0.779165i	$0.715643\pi$
$884$	0	0
$885$	2.66497	0.0895819
$886$	0	0
$887$	27.4696	0.922339	0.461170	−	0.887312i	$-0.347430\pi$
$887$	27.4696	0.922339	0.461170	+	0.887312i	$0.347430\pi$
$888$	0	0
$889$	−15.0600	−0.505096
$890$	0	0
$891$	−12.7804	−0.428160
$892$	0	0
$893$	14.7041	0.492055
$894$	0	0
$895$	6.93149	0.231694
$896$	0	0
$897$	−2.52741	−0.0843878
$898$	0	0
$899$	10.8265	0.361085
$900$	0	0
$901$	−13.0991	−0.436396
$902$	0	0
$903$	−9.81587	−0.326652
$904$	0	0
$905$	−17.6392	−0.586345
$906$	0	0
$907$	−49.5281	−1.64455	−0.822276	−	0.569089i	$-0.807296\pi$
$907$	−49.5281	−1.64455	−0.822276	+	0.569089i	$0.807296\pi$
$908$	0	0
$909$	−20.8237	−0.690678
$910$	0	0
$911$	−23.3923	−0.775020	−0.387510	−	0.921865i	$-0.626665\pi$
$911$	−23.3923	−0.775020	−0.387510	+	0.921865i	$0.626665\pi$
$912$	0	0
$913$	−7.67552	−0.254023
$914$	0	0
$915$	−2.42227	−0.0800777
$916$	0	0
$917$	67.4807	2.22841
$918$	0	0
$919$	−60.2518	−1.98752	−0.993761	−	0.111531i	$-0.964425\pi$
$919$	−60.2518	−1.98752	−0.993761	+	0.111531i	$0.964425\pi$
$920$	0	0
$921$	2.59382	0.0854692
$922$	0	0
$923$	−49.0459	−1.61437
$924$	0	0
$925$	1.00000	0.0328798
$926$	0	0
$927$	−52.1981	−1.71441
$928$	0	0
$929$	38.6663	1.26860	0.634300	−	0.773087i	$-0.281288\pi$
$929$	38.6663	1.26860	0.634300	+	0.773087i	$0.281288\pi$
$930$	0	0
$931$	16.7470	0.548861
$932$	0	0
$933$	−0.505759	−0.0165578
$934$	0	0
$935$	−5.51806	−0.180460
$936$	0	0
$937$	−21.9551	−0.717242	−0.358621	−	0.933483i	$-0.616753\pi$
$937$	−21.9551	−0.717242	−0.358621	+	0.933483i	$0.616753\pi$
$938$	0	0
$939$	−3.34798	−0.109257
$940$	0	0
$941$	−45.3812	−1.47938	−0.739692	−	0.672946i	$-0.765029\pi$
$941$	−45.3812	−1.47938	−0.739692	+	0.672946i	$0.765029\pi$
$942$	0	0
$943$	−14.4679	−0.471139
$944$	0	0
$945$	7.33326	0.238551
$946$	0	0
$947$	7.45391	0.242220	0.121110	−	0.992639i	$-0.461355\pi$
$947$	7.45391	0.242220	0.121110	+	0.992639i	$0.461355\pi$
$948$	0	0
$949$	12.2950	0.399114
$950$	0	0
$951$	3.35960	0.108943
$952$	0	0
$953$	36.9969	1.19845	0.599223	−	0.800582i	$-0.295476\pi$
$953$	36.9969	1.19845	0.599223	+	0.800582i	$0.295476\pi$
$954$	0	0
$955$	15.9582	0.516396
$956$	0	0
$957$	−2.25297	−0.0728283
$958$	0	0
$959$	−7.13844	−0.230512
$960$	0	0
$961$	−24.8637	−0.802055
$962$	0	0
$963$	19.0397	0.613545
$964$	0	0
$965$	−8.34264	−0.268559
$966$	0	0
$967$	−27.8626	−0.896001	−0.448001	−	0.894033i	$-0.647864\pi$
$967$	−27.8626	−0.896001	−0.448001	+	0.894033i	$0.647864\pi$
$968$	0	0
$969$	−2.47804	−0.0796061
$970$	0	0
$971$	11.7618	0.377455	0.188728	−	0.982029i	$-0.439564\pi$
$971$	11.7618	0.377455	0.188728	+	0.982029i	$0.439564\pi$
$972$	0	0
$973$	15.7166	0.503850
$974$	0	0
$975$	1.87242	0.0599654
$976$	0	0
$977$	−7.44532	−0.238197	−0.119098	−	0.992882i	$-0.538000\pi$
$977$	−7.44532	−0.238197	−0.119098	+	0.992882i	$0.538000\pi$
$978$	0	0
$979$	−11.6454	−0.372187
$980$	0	0
$981$	−47.3868	−1.51294
$982$	0	0
$983$	48.9817	1.56227	0.781137	−	0.624360i	$-0.214640\pi$
$983$	48.9817	1.56227	0.781137	+	0.624360i	$0.214640\pi$
$984$	0	0
$985$	13.0946	0.417230
$986$	0	0
$987$	8.35181	0.265841
$988$	0	0
$989$	10.6497	0.338641
$990$	0	0
$991$	−33.3005	−1.05783	−0.528913	−	0.848676i	$-0.677400\pi$
$991$	−33.3005	−1.05783	−0.528913	+	0.848676i	$0.677400\pi$
$992$	0	0
$993$	0.391156	0.0124129
$994$	0	0
$995$	26.1156	0.827919
$996$	0	0
$997$	−38.5742	−1.22166	−0.610829	−	0.791762i	$-0.709164\pi$
$997$	−38.5742	−1.22166	−0.610829	+	0.791762i	$0.709164\pi$
$998$	0	0
$999$	−1.91621	−0.0606262

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2960.2.a.y.1.3		5
4.3	odd	2		1480.2.a.i.1.3	✓	5
20.19	odd	2		7400.2.a.p.1.3		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1480.2.a.i.1.3	✓	5	4.3	odd	2
2960.2.a.y.1.3		5	1.1	even	1	trivial
7400.2.a.p.1.3		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)

\(f(q)\)	\(=\)	\(q+0.325094 q^{3} -1.00000 q^{5} +3.82696 q^{7} -2.89431 q^{9} +O(q^{10})\)	\(q+0.325094 q^{3} -1.00000 q^{5} +3.82696 q^{7} -2.89431 q^{9} -1.58566 q^{11} +5.75961 q^{13} -0.325094 q^{15} -3.47997 q^{17} +2.19040 q^{19} +1.24412 q^{21} -1.34981 q^{23} +1.00000 q^{25} -1.91621 q^{27} +4.37055 q^{29} +2.47715 q^{31} -0.515490 q^{33} -3.82696 q^{35} +1.00000 q^{37} +1.87242 q^{39} +10.7185 q^{41} -7.88978 q^{43} +2.89431 q^{45} +6.71300 q^{47} +7.64565 q^{49} -1.13132 q^{51} +3.76415 q^{53} +1.58566 q^{55} +0.712085 q^{57} -8.19751 q^{59} +7.45096 q^{61} -11.0764 q^{63} -5.75961 q^{65} -0.190395 q^{67} -0.438816 q^{69} -8.51549 q^{71} +2.13470 q^{73} +0.325094 q^{75} -6.06827 q^{77} -0.408886 q^{79} +8.05999 q^{81} +4.84058 q^{83} +3.47997 q^{85} +1.42084 q^{87} +7.34417 q^{89} +22.0418 q^{91} +0.805309 q^{93} -2.19040 q^{95} +2.14494 q^{97} +4.58940 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 5 q^{5} - 3 q^{7} + 5 q^{9}+O(q^{10}) \) 5 * q - 5 * q^5 - 3 * q^7 + 5 * q^9	\( 5 q - 5 q^{5} - 3 q^{7} + 5 q^{9} - 3 q^{11} + 4 q^{13} + 7 q^{17} + 4 q^{19} - 10 q^{21} - 10 q^{23} + 5 q^{25} + 6 q^{27} + 19 q^{29} - 13 q^{31} + 6 q^{33} + 3 q^{35} + 5 q^{37} - 14 q^{39} + 11 q^{41} + 13 q^{43} - 5 q^{45} + 2 q^{49} + 12 q^{51} + 27 q^{53} + 3 q^{55} + 12 q^{57} - 16 q^{59} + 27 q^{61} - 37 q^{63} - 4 q^{65} + 6 q^{67} + 40 q^{69} - 34 q^{71} + 16 q^{73} + 9 q^{77} - 16 q^{79} + 9 q^{81} + 14 q^{83} - 7 q^{85} + 42 q^{87} + 38 q^{89} + 34 q^{91} + 30 q^{93} - 4 q^{95} + 5 q^{97} - 23 q^{99}+O(q^{100}) \) 5 * q - 5 * q^5 - 3 * q^7 + 5 * q^9 - 3 * q^11 + 4 * q^13 + 7 * q^17 + 4 * q^19 - 10 * q^21 - 10 * q^23 + 5 * q^25 + 6 * q^27 + 19 * q^29 - 13 * q^31 + 6 * q^33 + 3 * q^35 + 5 * q^37 - 14 * q^39 + 11 * q^41 + 13 * q^43 - 5 * q^45 + 2 * q^49 + 12 * q^51 + 27 * q^53 + 3 * q^55 + 12 * q^57 - 16 * q^59 + 27 * q^61 - 37 * q^63 - 4 * q^65 + 6 * q^67 + 40 * q^69 - 34 * q^71 + 16 * q^73 + 9 * q^77 - 16 * q^79 + 9 * q^81 + 14 * q^83 - 7 * q^85 + 42 * q^87 + 38 * q^89 + 34 * q^91 + 30 * q^93 - 4 * q^95 + 5 * q^97 - 23 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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