LMFDB - Embedded newform 2960.2.a.bc.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:	$6$
Coefficient field:	6.6.693982032.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 16x^{4} + 12x^{3} + 60x^{2} - 18x - 44 $ x^6 - x^5 - 16x^4 + 12x^3 + 60x^2 - 18x - 44
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			1.2
Root			$2.45040$ of defining polynomial
Character	$\chi$	$=$	2960.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.45040 q^{3} +1.00000 q^{5} +1.57732 q^{7} +3.00447 q^{9} +O(q^{10})$	$q-2.45040 q^{3} +1.00000 q^{5} +1.57732 q^{7} +3.00447 q^{9} +2.43694 q^{11} +5.07514 q^{13} -2.45040 q^{15} +2.43247 q^{17} -3.24032 q^{19} -3.86506 q^{21} +0.280756 q^{23} +1.00000 q^{25} -0.0109469 q^{27} +10.4084 q^{29} +1.50229 q^{31} -5.97147 q^{33} +1.57732 q^{35} -1.00000 q^{37} -12.4361 q^{39} +3.14290 q^{41} -2.51196 q^{43} +3.00447 q^{45} -8.21998 q^{47} -4.51208 q^{49} -5.96053 q^{51} -5.33774 q^{53} +2.43694 q^{55} +7.94008 q^{57} -3.43007 q^{59} +5.50761 q^{61} +4.73899 q^{63} +5.07514 q^{65} +1.94124 q^{67} -0.687964 q^{69} +5.97147 q^{71} -8.31077 q^{73} -2.45040 q^{75} +3.84382 q^{77} +6.72021 q^{79} -8.98658 q^{81} -0.545247 q^{83} +2.43247 q^{85} -25.5048 q^{87} -7.33184 q^{89} +8.00509 q^{91} -3.68122 q^{93} -3.24032 q^{95} -9.99718 q^{97} +7.32170 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - q^{3} + 6 q^{5} - 8 q^{7} + 15 q^{9}+O(q^{10}) $ 6 * q - q^3 + 6 * q^5 - 8 * q^7 + 15 * q^9	$ 6 q - q^{3} + 6 q^{5} - 8 q^{7} + 15 q^{9} + 10 q^{13} - q^{15} + 3 q^{17} + 6 q^{19} + 9 q^{21} - 4 q^{23} + 6 q^{25} - 7 q^{27} + 3 q^{29} + 3 q^{31} + 9 q^{33} - 8 q^{35} - 6 q^{37} + 16 q^{39} + 10 q^{41} + 11 q^{43} + 15 q^{45} - 23 q^{47} + 8 q^{49} + 16 q^{51} + 10 q^{53} + 4 q^{57} - 6 q^{59} + q^{61} - 23 q^{63} + 10 q^{65} + 4 q^{67} - 2 q^{69} - 9 q^{71} + 11 q^{73} - q^{75} + 32 q^{77} + 14 q^{79} + 46 q^{81} - 11 q^{83} + 3 q^{85} - 32 q^{87} + 30 q^{89} + 4 q^{91} + 2 q^{93} + 6 q^{95} + 29 q^{97} + 39 q^{99}+O(q^{100}) $ 6 * q - q^3 + 6 * q^5 - 8 * q^7 + 15 * q^9 + 10 * q^13 - q^15 + 3 * q^17 + 6 * q^19 + 9 * q^21 - 4 * q^23 + 6 * q^25 - 7 * q^27 + 3 * q^29 + 3 * q^31 + 9 * q^33 - 8 * q^35 - 6 * q^37 + 16 * q^39 + 10 * q^41 + 11 * q^43 + 15 * q^45 - 23 * q^47 + 8 * q^49 + 16 * q^51 + 10 * q^53 + 4 * q^57 - 6 * q^59 + q^61 - 23 * q^63 + 10 * q^65 + 4 * q^67 - 2 * q^69 - 9 * q^71 + 11 * q^73 - q^75 + 32 * q^77 + 14 * q^79 + 46 * q^81 - 11 * q^83 + 3 * q^85 - 32 * q^87 + 30 * q^89 + 4 * q^91 + 2 * q^93 + 6 * q^95 + 29 * q^97 + 39 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	−2.45040	−1.41474	−0.707370	−	0.706843i	$-0.750119\pi$
$3$	−2.45040	−1.41474	−0.707370	+	0.706843i	$0.750119\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	1.57732	0.596169	0.298085	−	0.954539i	$-0.403652\pi$
$7$	1.57732	0.596169	0.298085	+	0.954539i	$0.403652\pi$
$8$	0	0
$9$	3.00447	1.00149
$10$	0	0
$11$	2.43694	0.734764	0.367382	−	0.930070i	$-0.380254\pi$
$11$	2.43694	0.734764	0.367382	+	0.930070i	$0.380254\pi$
$12$	0	0
$13$	5.07514	1.40759	0.703795	−	0.710403i	$-0.251487\pi$
$13$	5.07514	1.40759	0.703795	+	0.710403i	$0.251487\pi$
$14$	0	0
$15$	−2.45040	−0.632691
$16$	0	0
$17$	2.43247	0.589961	0.294980	−	0.955503i	$-0.404687\pi$
$17$	2.43247	0.589961	0.294980	+	0.955503i	$0.404687\pi$
$18$	0	0
$19$	−3.24032	−0.743380	−0.371690	−	0.928357i	$-0.621222\pi$
$19$	−3.24032	−0.743380	−0.371690	+	0.928357i	$0.621222\pi$
$20$	0	0
$21$	−3.86506	−0.843424
$22$	0	0
$23$	0.280756	0.0585416	0.0292708	−	0.999572i	$-0.490681\pi$
$23$	0.280756	0.0585416	0.0292708	+	0.999572i	$0.490681\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−0.0109469	−0.00210674
$28$	0	0
$29$	10.4084	1.93279	0.966397	−	0.257055i	$-0.0827521\pi$
$29$	10.4084	1.93279	0.966397	+	0.257055i	$0.0827521\pi$
$30$	0	0
$31$	1.50229	0.269820	0.134910	−	0.990858i	$-0.456926\pi$
$31$	1.50229	0.269820	0.134910	+	0.990858i	$0.456926\pi$
$32$	0	0
$33$	−5.97147	−1.03950
$34$	0	0
$35$	1.57732	0.266615
$36$	0	0
$37$	−1.00000	−0.164399
$37$	−1.00000	−0.164399
$38$	0	0
$39$	−12.4361	−1.99137
$40$	0	0
$41$	3.14290	0.490838	0.245419	−	0.969417i	$-0.421075\pi$
$41$	3.14290	0.490838	0.245419	+	0.969417i	$0.421075\pi$
$42$	0	0
$43$	−2.51196	−0.383070	−0.191535	−	0.981486i	$-0.561347\pi$
$43$	−2.51196	−0.383070	−0.191535	+	0.981486i	$0.561347\pi$
$44$	0	0
$45$	3.00447	0.447880
$46$	0	0
$47$	−8.21998	−1.19901	−0.599504	−	0.800372i	$-0.704635\pi$
$47$	−8.21998	−1.19901	−0.599504	+	0.800372i	$0.704635\pi$
$48$	0	0
$49$	−4.51208	−0.644582
$50$	0	0
$51$	−5.96053	−0.834641
$52$	0	0
$53$	−5.33774	−0.733195	−0.366598	−	0.930380i	$-0.619477\pi$
$53$	−5.33774	−0.733195	−0.366598	+	0.930380i	$0.619477\pi$
$54$	0	0
$55$	2.43694	0.328597
$56$	0	0
$57$	7.94008	1.05169
$58$	0	0
$59$	−3.43007	−0.446557	−0.223278	−	0.974755i	$-0.571676\pi$
$59$	−3.43007	−0.446557	−0.223278	+	0.974755i	$0.571676\pi$
$60$	0	0
$61$	5.50761	0.705177	0.352589	−	0.935778i	$-0.385302\pi$
$61$	5.50761	0.705177	0.352589	+	0.935778i	$0.385302\pi$
$62$	0	0
$63$	4.73899	0.597057
$64$	0	0
$65$	5.07514	0.629494
$66$	0	0
$67$	1.94124	0.237160	0.118580	−	0.992944i	$-0.462166\pi$
$67$	1.94124	0.237160	0.118580	+	0.992944i	$0.462166\pi$
$68$	0	0
$69$	−0.687964	−0.0828212
$70$	0	0
$71$	5.97147	0.708684	0.354342	−	0.935116i	$-0.384705\pi$
$71$	5.97147	0.708684	0.354342	+	0.935116i	$0.384705\pi$
$72$	0	0
$73$	−8.31077	−0.972702	−0.486351	−	0.873764i	$-0.661672\pi$
$73$	−8.31077	−0.972702	−0.486351	+	0.873764i	$0.661672\pi$
$74$	0	0
$75$	−2.45040	−0.282948
$76$	0	0
$77$	3.84382	0.438044
$78$	0	0
$79$	6.72021	0.756083	0.378041	−	0.925789i	$-0.376598\pi$
$79$	6.72021	0.756083	0.378041	+	0.925789i	$0.376598\pi$
$80$	0	0
$81$	−8.98658	−0.998509
$82$	0	0
$83$	−0.545247	−0.0598486	−0.0299243	−	0.999552i	$-0.509527\pi$
$83$	−0.545247	−0.0598486	−0.0299243	+	0.999552i	$0.509527\pi$
$84$	0	0
$85$	2.43247	0.263838
$86$	0	0
$87$	−25.5048	−2.73440
$88$	0	0
$89$	−7.33184	−0.777173	−0.388587	−	0.921412i	$-0.627037\pi$
$89$	−7.33184	−0.777173	−0.388587	+	0.921412i	$0.627037\pi$
$90$	0	0
$91$	8.00509	0.839162
$92$	0	0
$93$	−3.68122	−0.381724
$94$	0	0
$95$	−3.24032	−0.332450
$96$	0	0
$97$	−9.99718	−1.01506	−0.507530	−	0.861634i	$-0.669441\pi$
$97$	−9.99718	−1.01506	−0.507530	+	0.861634i	$0.669441\pi$
$98$	0	0
$99$	7.32170	0.735858
$100$	0	0
$101$	−2.14581	−0.213516	−0.106758	−	0.994285i	$-0.534047\pi$
$101$	−2.14581	−0.213516	−0.106758	+	0.994285i	$0.534047\pi$
$102$	0	0
$103$	−1.29404	−0.127506	−0.0637529	−	0.997966i	$-0.520307\pi$
$103$	−1.29404	−0.127506	−0.0637529	+	0.997966i	$0.520307\pi$
$104$	0	0
$105$	−3.86506	−0.377191
$106$	0	0
$107$	15.0923	1.45902	0.729512	−	0.683968i	$-0.239747\pi$
$107$	15.0923	1.45902	0.729512	+	0.683968i	$0.239747\pi$
$108$	0	0
$109$	12.1366	1.16247	0.581237	−	0.813734i	$-0.302569\pi$
$109$	12.1366	1.16247	0.581237	+	0.813734i	$0.302569\pi$
$110$	0	0
$111$	2.45040	0.232582
$112$	0	0
$113$	−5.05252	−0.475301	−0.237650	−	0.971351i	$-0.576377\pi$
$113$	−5.05252	−0.475301	−0.237650	+	0.971351i	$0.576377\pi$
$114$	0	0
$115$	0.280756	0.0261806
$116$	0	0
$117$	15.2481	1.40969
$118$	0	0
$119$	3.83677	0.351716
$120$	0	0
$121$	−5.06134	−0.460121
$122$	0	0
$123$	−7.70135	−0.694408
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	8.79553	0.780477	0.390238	−	0.920714i	$-0.372393\pi$
$127$	8.79553	0.780477	0.390238	+	0.920714i	$0.372393\pi$
$128$	0	0
$129$	6.15531	0.541945
$130$	0	0
$131$	11.1368	0.973024	0.486512	−	0.873674i	$-0.338269\pi$
$131$	11.1368	0.973024	0.486512	+	0.873674i	$0.338269\pi$
$132$	0	0
$133$	−5.11100	−0.443180
$134$	0	0
$135$	−0.0109469	−0.000942162 0
$136$	0	0
$137$	17.1636	1.46638	0.733191	−	0.680023i	$-0.238030\pi$
$137$	17.1636	1.46638	0.733191	+	0.680023i	$0.238030\pi$
$138$	0	0
$139$	18.9926	1.61093	0.805466	−	0.592643i	$-0.201915\pi$
$139$	18.9926	1.61093	0.805466	+	0.592643i	$0.201915\pi$
$140$	0	0
$141$	20.1423	1.69628
$142$	0	0
$143$	12.3678	1.03425
$144$	0	0
$145$	10.4084	0.864372
$146$	0	0
$147$	11.0564	0.911916
$148$	0	0
$149$	1.02681	0.0841198	0.0420599	−	0.999115i	$-0.486608\pi$
$149$	1.02681	0.0841198	0.0420599	+	0.999115i	$0.486608\pi$
$150$	0	0
$151$	0.330357	0.0268840	0.0134420	−	0.999910i	$-0.495721\pi$
$151$	0.330357	0.0268840	0.0134420	+	0.999910i	$0.495721\pi$
$152$	0	0
$153$	7.30828	0.590839
$154$	0	0
$155$	1.50229	0.120667
$156$	0	0
$157$	10.9125	0.870915	0.435458	−	0.900209i	$-0.356587\pi$
$157$	10.9125	0.870915	0.435458	+	0.900209i	$0.356587\pi$
$158$	0	0
$159$	13.0796	1.03728
$160$	0	0
$161$	0.442840	0.0349007
$162$	0	0
$163$	−2.78401	−0.218061	−0.109030	−	0.994038i	$-0.534775\pi$
$163$	−2.78401	−0.218061	−0.109030	+	0.994038i	$0.534775\pi$
$164$	0	0
$165$	−5.97147	−0.464879
$166$	0	0
$167$	−5.84662	−0.452425	−0.226212	−	0.974078i	$-0.572634\pi$
$167$	−5.84662	−0.452425	−0.226212	+	0.974078i	$0.572634\pi$
$168$	0	0
$169$	12.7570	0.981311
$170$	0	0
$171$	−9.73543	−0.744487
$172$	0	0
$173$	−14.7373	−1.12045	−0.560227	−	0.828339i	$-0.689286\pi$
$173$	−14.7373	−1.12045	−0.560227	+	0.828339i	$0.689286\pi$
$174$	0	0
$175$	1.57732	0.119234
$176$	0	0
$177$	8.40504	0.631762
$178$	0	0
$179$	−19.1742	−1.43315	−0.716574	−	0.697511i	$-0.754291\pi$
$179$	−19.1742	−1.43315	−0.716574	+	0.697511i	$0.754291\pi$
$180$	0	0
$181$	23.0841	1.71582	0.857912	−	0.513796i	$-0.171761\pi$
$181$	23.0841	1.71582	0.857912	+	0.513796i	$0.171761\pi$
$182$	0	0
$183$	−13.4959	−0.997642
$184$	0	0
$185$	−1.00000	−0.0735215
$186$	0	0
$187$	5.92778	0.433482
$188$	0	0
$189$	−0.0172668	−0.00125597
$190$	0	0
$191$	20.9302	1.51446	0.757229	−	0.653149i	$-0.226553\pi$
$191$	20.9302	1.51446	0.757229	+	0.653149i	$0.226553\pi$
$192$	0	0
$193$	−19.5228	−1.40528	−0.702640	−	0.711545i	$-0.747996\pi$
$193$	−19.5228	−1.40528	−0.702640	+	0.711545i	$0.747996\pi$
$194$	0	0
$195$	−12.4361	−0.890570
$196$	0	0
$197$	5.84812	0.416662	0.208331	−	0.978058i	$-0.433197\pi$
$197$	5.84812	0.416662	0.208331	+	0.978058i	$0.433197\pi$
$198$	0	0
$199$	26.8260	1.90164	0.950822	−	0.309739i	$-0.100242\pi$
$199$	26.8260	1.90164	0.950822	+	0.309739i	$0.100242\pi$
$200$	0	0
$201$	−4.75682	−0.335520
$202$	0	0
$203$	16.4173	1.15227
$204$	0	0
$205$	3.14290	0.219509
$206$	0	0
$207$	0.843521	0.0586288
$208$	0	0
$209$	−7.89645	−0.546209
$210$	0	0
$211$	16.6141	1.14376	0.571882	−	0.820336i	$-0.306213\pi$
$211$	16.6141	1.14376	0.571882	+	0.820336i	$0.306213\pi$
$212$	0	0
$213$	−14.6325	−1.00260
$214$	0	0
$215$	−2.51196	−0.171314
$216$	0	0
$217$	2.36959	0.160858
$218$	0	0
$219$	20.3647	1.37612
$220$	0	0
$221$	12.3451	0.830423
$222$	0	0
$223$	−9.97198	−0.667773	−0.333887	−	0.942613i	$-0.608360\pi$
$223$	−9.97198	−0.667773	−0.333887	+	0.942613i	$0.608360\pi$
$224$	0	0
$225$	3.00447	0.200298
$226$	0	0
$227$	−15.3092	−1.01611	−0.508054	−	0.861325i	$-0.669635\pi$
$227$	−15.3092	−1.01611	−0.508054	+	0.861325i	$0.669635\pi$
$228$	0	0
$229$	23.0722	1.52465	0.762326	−	0.647194i	$-0.224058\pi$
$229$	23.0722	1.52465	0.762326	+	0.647194i	$0.224058\pi$
$230$	0	0
$231$	−9.41890	−0.619718
$232$	0	0
$233$	−21.9788	−1.43988	−0.719940	−	0.694036i	$-0.755831\pi$
$233$	−21.9788	−1.43988	−0.719940	+	0.694036i	$0.755831\pi$
$234$	0	0
$235$	−8.21998	−0.536213
$236$	0	0
$237$	−16.4672	−1.06966
$238$	0	0
$239$	−4.69833	−0.303909	−0.151955	−	0.988387i	$-0.548557\pi$
$239$	−4.69833	−0.303909	−0.151955	+	0.988387i	$0.548557\pi$
$240$	0	0
$241$	9.16357	0.590277	0.295139	−	0.955454i	$-0.404634\pi$
$241$	9.16357	0.590277	0.295139	+	0.955454i	$0.404634\pi$
$242$	0	0
$243$	22.0536	1.41474
$244$	0	0
$245$	−4.51208	−0.288266
$246$	0	0
$247$	−16.4451	−1.04637
$248$	0	0
$249$	1.33607	0.0846702
$250$	0	0
$251$	−15.8393	−0.999766	−0.499883	−	0.866093i	$-0.666624\pi$
$251$	−15.8393	−0.999766	−0.499883	+	0.866093i	$0.666624\pi$
$252$	0	0
$253$	0.684184	0.0430143
$254$	0	0
$255$	−5.96053	−0.373263
$256$	0	0
$257$	9.44136	0.588936	0.294468	−	0.955661i	$-0.404858\pi$
$257$	9.44136	0.588936	0.294468	+	0.955661i	$0.404858\pi$
$258$	0	0
$259$	−1.57732	−0.0980096
$260$	0	0
$261$	31.2717	1.93567
$262$	0	0
$263$	−14.0028	−0.863450	−0.431725	−	0.902005i	$-0.642095\pi$
$263$	−14.0028	−0.863450	−0.431725	+	0.902005i	$0.642095\pi$
$264$	0	0
$265$	−5.33774	−0.327895
$266$	0	0
$267$	17.9659	1.09950
$268$	0	0
$269$	9.80960	0.598102	0.299051	−	0.954237i	$-0.403330\pi$
$269$	9.80960	0.598102	0.299051	+	0.954237i	$0.403330\pi$
$270$	0	0
$271$	1.21008	0.0735074	0.0367537	−	0.999324i	$-0.488298\pi$
$271$	1.21008	0.0735074	0.0367537	+	0.999324i	$0.488298\pi$
$272$	0	0
$273$	−19.6157	−1.18720
$274$	0	0
$275$	2.43694	0.146953
$276$	0	0
$277$	11.2473	0.675785	0.337893	−	0.941185i	$-0.390286\pi$
$277$	11.2473	0.675785	0.337893	+	0.941185i	$0.390286\pi$
$278$	0	0
$279$	4.51359	0.270221
$280$	0	0
$281$	−5.31361	−0.316984	−0.158492	−	0.987360i	$-0.550663\pi$
$281$	−5.31361	−0.316984	−0.158492	+	0.987360i	$0.550663\pi$
$282$	0	0
$283$	−9.06505	−0.538861	−0.269431	−	0.963020i	$-0.586836\pi$
$283$	−9.06505	−0.538861	−0.269431	+	0.963020i	$0.586836\pi$
$284$	0	0
$285$	7.94008	0.470330
$286$	0	0
$287$	4.95734	0.292622
$288$	0	0
$289$	−11.0831	−0.651946
$290$	0	0
$291$	24.4971	1.43605
$292$	0	0
$293$	19.3966	1.13316	0.566581	−	0.824006i	$-0.308266\pi$
$293$	19.3966	1.13316	0.566581	+	0.824006i	$0.308266\pi$
$294$	0	0
$295$	−3.43007	−0.199706
$296$	0	0
$297$	−0.0266770	−0.00154796
$298$	0	0
$299$	1.42487	0.0824026
$300$	0	0
$301$	−3.96215	−0.228375
$302$	0	0
$303$	5.25810	0.302070
$304$	0	0
$305$	5.50761	0.315365
$306$	0	0
$307$	7.51748	0.429045	0.214523	−	0.976719i	$-0.431180\pi$
$307$	7.51748	0.429045	0.214523	+	0.976719i	$0.431180\pi$
$308$	0	0
$309$	3.17092	0.180388
$310$	0	0
$311$	11.6795	0.662284	0.331142	−	0.943581i	$-0.392566\pi$
$311$	11.6795	0.662284	0.331142	+	0.943581i	$0.392566\pi$
$312$	0	0
$313$	−5.05821	−0.285907	−0.142953	−	0.989729i	$-0.545660\pi$
$313$	−5.05821	−0.285907	−0.142953	+	0.989729i	$0.545660\pi$
$314$	0	0
$315$	4.73899	0.267012
$316$	0	0
$317$	−13.4545	−0.755678	−0.377839	−	0.925871i	$-0.623333\pi$
$317$	−13.4545	−0.755678	−0.377839	+	0.925871i	$0.623333\pi$
$318$	0	0
$319$	25.3646	1.42015
$320$	0	0
$321$	−36.9821	−2.06414
$322$	0	0
$323$	−7.88198	−0.438565
$324$	0	0
$325$	5.07514	0.281518
$326$	0	0
$327$	−29.7395	−1.64460
$328$	0	0
$329$	−12.9655	−0.714812
$330$	0	0
$331$	32.2750	1.77400	0.886998	−	0.461774i	$-0.152787\pi$
$331$	32.2750	1.77400	0.886998	+	0.461774i	$0.152787\pi$
$332$	0	0
$333$	−3.00447	−0.164644
$334$	0	0
$335$	1.94124	0.106061
$336$	0	0
$337$	16.3427	0.890245	0.445123	−	0.895470i	$-0.353160\pi$
$337$	16.3427	0.890245	0.445123	+	0.895470i	$0.353160\pi$
$338$	0	0
$339$	12.3807	0.672427
$340$	0	0
$341$	3.66099	0.198254
$342$	0	0
$343$	−18.1582	−0.980449
$344$	0	0
$345$	−0.687964	−0.0370388
$346$	0	0
$347$	23.3062	1.25114	0.625570	−	0.780168i	$-0.284866\pi$
$347$	23.3062	1.25114	0.625570	+	0.780168i	$0.284866\pi$
$348$	0	0
$349$	−31.2087	−1.67056	−0.835281	−	0.549823i	$-0.814695\pi$
$349$	−31.2087	−1.67056	−0.835281	+	0.549823i	$0.814695\pi$
$350$	0	0
$351$	−0.0555572	−0.00296542
$352$	0	0
$353$	12.7126	0.676625	0.338313	−	0.941034i	$-0.390144\pi$
$353$	12.7126	0.676625	0.338313	+	0.941034i	$0.390144\pi$
$354$	0	0
$355$	5.97147	0.316933
$356$	0	0
$357$	−9.40163	−0.497587
$358$	0	0
$359$	−12.5646	−0.663134	−0.331567	−	0.943432i	$-0.607577\pi$
$359$	−12.5646	−0.663134	−0.331567	+	0.943432i	$0.607577\pi$
$360$	0	0
$361$	−8.50034	−0.447386
$362$	0	0
$363$	12.4023	0.650952
$364$	0	0
$365$	−8.31077	−0.435005
$366$	0	0
$367$	12.6068	0.658070	0.329035	−	0.944318i	$-0.393277\pi$
$367$	12.6068	0.658070	0.329035	+	0.944318i	$0.393277\pi$
$368$	0	0
$369$	9.44273	0.491569
$370$	0	0
$371$	−8.41930	−0.437108
$372$	0	0
$373$	−19.8555	−1.02808	−0.514040	−	0.857766i	$-0.671852\pi$
$373$	−19.8555	−1.02808	−0.514040	+	0.857766i	$0.671852\pi$
$374$	0	0
$375$	−2.45040	−0.126538
$376$	0	0
$377$	52.8241	2.72058
$378$	0	0
$379$	11.9982	0.616305	0.308152	−	0.951337i	$-0.400289\pi$
$379$	11.9982	0.616305	0.308152	+	0.951337i	$0.400289\pi$
$380$	0	0
$381$	−21.5526	−1.10417
$382$	0	0
$383$	−29.3675	−1.50061	−0.750304	−	0.661093i	$-0.770093\pi$
$383$	−29.3675	−1.50061	−0.750304	+	0.661093i	$0.770093\pi$
$384$	0	0
$385$	3.84382	0.195899
$386$	0	0
$387$	−7.54711	−0.383641
$388$	0	0
$389$	−20.4371	−1.03620	−0.518102	−	0.855319i	$-0.673361\pi$
$389$	−20.4371	−1.03620	−0.518102	+	0.855319i	$0.673361\pi$
$390$	0	0
$391$	0.682930	0.0345372
$392$	0	0
$393$	−27.2896	−1.37658
$394$	0	0
$395$	6.72021	0.338130
$396$	0	0
$397$	10.4252	0.523225	0.261612	−	0.965173i	$-0.415746\pi$
$397$	10.4252	0.523225	0.261612	+	0.965173i	$0.415746\pi$
$398$	0	0
$399$	12.5240	0.626985
$400$	0	0
$401$	20.6637	1.03189	0.515947	−	0.856620i	$-0.327440\pi$
$401$	20.6637	1.03189	0.515947	+	0.856620i	$0.327440\pi$
$402$	0	0
$403$	7.62434	0.379795
$404$	0	0
$405$	−8.98658	−0.446547
$406$	0	0
$407$	−2.43694	−0.120794
$408$	0	0
$409$	−20.1072	−0.994237	−0.497118	−	0.867683i	$-0.665609\pi$
$409$	−20.1072	−0.994237	−0.497118	+	0.867683i	$0.665609\pi$
$410$	0	0
$411$	−42.0576	−2.07455
$412$	0	0
$413$	−5.41030	−0.266223
$414$	0	0
$415$	−0.545247	−0.0267651
$416$	0	0
$417$	−46.5395	−2.27905
$418$	0	0
$419$	3.60332	0.176034	0.0880169	−	0.996119i	$-0.471947\pi$
$419$	3.60332	0.176034	0.0880169	+	0.996119i	$0.471947\pi$
$420$	0	0
$421$	−32.7539	−1.59633	−0.798165	−	0.602439i	$-0.794196\pi$
$421$	−32.7539	−1.59633	−0.798165	+	0.602439i	$0.794196\pi$
$422$	0	0
$423$	−24.6967	−1.20079
$424$	0	0
$425$	2.43247	0.117992
$426$	0	0
$427$	8.68724	0.420405
$428$	0	0
$429$	−30.3061	−1.46319
$430$	0	0
$431$	−23.6183	−1.13765	−0.568826	−	0.822458i	$-0.692602\pi$
$431$	−23.6183	−1.13765	−0.568826	+	0.822458i	$0.692602\pi$
$432$	0	0
$433$	27.2138	1.30781	0.653906	−	0.756575i	$-0.273129\pi$
$433$	27.2138	1.30781	0.653906	+	0.756575i	$0.273129\pi$
$434$	0	0
$435$	−25.5048	−1.22286
$436$	0	0
$437$	−0.909738	−0.0435187
$438$	0	0
$439$	27.5222	1.31356	0.656780	−	0.754082i	$-0.271918\pi$
$439$	27.5222	1.31356	0.656780	+	0.754082i	$0.271918\pi$
$440$	0	0
$441$	−13.5564	−0.645542
$442$	0	0
$443$	8.32927	0.395736	0.197868	−	0.980229i	$-0.436598\pi$
$443$	8.32927	0.395736	0.197868	+	0.980229i	$0.436598\pi$
$444$	0	0
$445$	−7.33184	−0.347562
$446$	0	0
$447$	−2.51610	−0.119008
$448$	0	0
$449$	16.3597	0.772060	0.386030	−	0.922486i	$-0.373846\pi$
$449$	16.3597	0.772060	0.386030	+	0.922486i	$0.373846\pi$
$450$	0	0
$451$	7.65904	0.360650
$452$	0	0
$453$	−0.809507	−0.0380339
$454$	0	0
$455$	8.00509	0.375285
$456$	0	0
$457$	−16.3282	−0.763802	−0.381901	−	0.924203i	$-0.624730\pi$
$457$	−16.3282	−0.763802	−0.381901	+	0.924203i	$0.624730\pi$
$458$	0	0
$459$	−0.0266281	−0.00124289
$460$	0	0
$461$	−26.5602	−1.23703	−0.618515	−	0.785773i	$-0.712266\pi$
$461$	−26.5602	−1.23703	−0.618515	+	0.785773i	$0.712266\pi$
$462$	0	0
$463$	16.6776	0.775076	0.387538	−	0.921854i	$-0.373326\pi$
$463$	16.6776	0.775076	0.387538	+	0.921854i	$0.373326\pi$
$464$	0	0
$465$	−3.68122	−0.170712
$466$	0	0
$467$	−14.1995	−0.657073	−0.328537	−	0.944491i	$-0.606555\pi$
$467$	−14.1995	−0.657073	−0.328537	+	0.944491i	$0.606555\pi$
$468$	0	0
$469$	3.06195	0.141388
$470$	0	0
$471$	−26.7401	−1.23212
$472$	0	0
$473$	−6.12149	−0.281466
$474$	0	0
$475$	−3.24032	−0.148676
$476$	0	0
$477$	−16.0371	−0.734287
$478$	0	0
$479$	9.87433	0.451170	0.225585	−	0.974224i	$-0.427571\pi$
$479$	9.87433	0.451170	0.225585	+	0.974224i	$0.427571\pi$
$480$	0	0
$481$	−5.07514	−0.231406
$482$	0	0
$483$	−1.08514	−0.0493754
$484$	0	0
$485$	−9.99718	−0.453949
$486$	0	0
$487$	−29.8891	−1.35440	−0.677202	−	0.735797i	$-0.736808\pi$
$487$	−29.8891	−1.35440	−0.677202	+	0.735797i	$0.736808\pi$
$488$	0	0
$489$	6.82195	0.308499
$490$	0	0
$491$	9.74330	0.439709	0.219855	−	0.975533i	$-0.429442\pi$
$491$	9.74330	0.439709	0.219855	+	0.975533i	$0.429442\pi$
$492$	0	0
$493$	25.3182	1.14027
$494$	0	0
$495$	7.32170	0.329086
$496$	0	0
$497$	9.41890	0.422495
$498$	0	0
$499$	24.9740	1.11799	0.558996	−	0.829171i	$-0.311187\pi$
$499$	24.9740	1.11799	0.558996	+	0.829171i	$0.311187\pi$
$500$	0	0
$501$	14.3266	0.640064
$502$	0	0
$503$	9.10652	0.406039	0.203020	−	0.979175i	$-0.434924\pi$
$503$	9.10652	0.406039	0.203020	+	0.979175i	$0.434924\pi$
$504$	0	0
$505$	−2.14581	−0.0954873
$506$	0	0
$507$	−31.2599	−1.38830
$508$	0	0
$509$	22.4896	0.996832	0.498416	−	0.866938i	$-0.333915\pi$
$509$	22.4896	0.996832	0.498416	+	0.866938i	$0.333915\pi$
$510$	0	0
$511$	−13.1087	−0.579895
$512$	0	0
$513$	0.0354715	0.00156611
$514$	0	0
$515$	−1.29404	−0.0570223
$516$	0	0
$517$	−20.0316	−0.880988
$518$	0	0
$519$	36.1122	1.58515
$520$	0	0
$521$	−21.1871	−0.928224	−0.464112	−	0.885777i	$-0.653626\pi$
$521$	−21.1871	−0.928224	−0.464112	+	0.885777i	$0.653626\pi$
$522$	0	0
$523$	−18.5582	−0.811493	−0.405747	−	0.913986i	$-0.632988\pi$
$523$	−18.5582	−0.811493	−0.405747	+	0.913986i	$0.632988\pi$
$524$	0	0
$525$	−3.86506	−0.168685
$526$	0	0
$527$	3.65428	0.159183
$528$	0	0
$529$	−22.9212	−0.996573
$530$	0	0
$531$	−10.3055	−0.447222
$532$	0	0
$533$	15.9506	0.690898
$534$	0	0
$535$	15.0923	0.652495
$536$	0	0
$537$	46.9845	2.02753
$538$	0	0
$539$	−10.9956	−0.473616
$540$	0	0
$541$	−36.0519	−1.54999	−0.774996	−	0.631966i	$-0.782248\pi$
$541$	−36.0519	−1.54999	−0.774996	+	0.631966i	$0.782248\pi$
$542$	0	0
$543$	−56.5652	−2.42745
$544$	0	0
$545$	12.1366	0.519874
$546$	0	0
$547$	−15.0752	−0.644569	−0.322284	−	0.946643i	$-0.604451\pi$
$547$	−15.0752	−0.644569	−0.322284	+	0.946643i	$0.604451\pi$
$548$	0	0
$549$	16.5474	0.706227
$550$	0	0
$551$	−33.7266	−1.43680
$552$	0	0
$553$	10.5999	0.450753
$554$	0	0
$555$	2.45040	0.104014
$556$	0	0
$557$	26.5210	1.12373	0.561866	−	0.827228i	$-0.310084\pi$
$557$	26.5210	1.12373	0.561866	+	0.827228i	$0.310084\pi$
$558$	0	0
$559$	−12.7486	−0.539206
$560$	0	0
$561$	−14.5254	−0.613264
$562$	0	0
$563$	−31.1038	−1.31087	−0.655435	−	0.755252i	$-0.727515\pi$
$563$	−31.1038	−1.31087	−0.655435	+	0.755252i	$0.727515\pi$
$564$	0	0
$565$	−5.05252	−0.212561
$566$	0	0
$567$	−14.1747	−0.595280
$568$	0	0
$569$	−9.71180	−0.407140	−0.203570	−	0.979060i	$-0.565254\pi$
$569$	−9.71180	−0.407140	−0.203570	+	0.979060i	$0.565254\pi$
$570$	0	0
$571$	28.5300	1.19395	0.596973	−	0.802262i	$-0.296370\pi$
$571$	28.5300	1.19395	0.596973	+	0.802262i	$0.296370\pi$
$572$	0	0
$573$	−51.2875	−2.14256
$574$	0	0
$575$	0.280756	0.0117083
$576$	0	0
$577$	17.5344	0.729965	0.364982	−	0.931014i	$-0.381075\pi$
$577$	17.5344	0.729965	0.364982	+	0.931014i	$0.381075\pi$
$578$	0	0
$579$	47.8387	1.98811
$580$	0	0
$581$	−0.860026	−0.0356799
$582$	0	0
$583$	−13.0077	−0.538725
$584$	0	0
$585$	15.2481	0.630431
$586$	0	0
$587$	−27.5828	−1.13847	−0.569233	−	0.822176i	$-0.692760\pi$
$587$	−27.5828	−1.13847	−0.569233	+	0.822176i	$0.692760\pi$
$588$	0	0
$589$	−4.86790	−0.200578
$590$	0	0
$591$	−14.3303	−0.589468
$592$	0	0
$593$	10.7416	0.441103	0.220552	−	0.975375i	$-0.429214\pi$
$593$	10.7416	0.441103	0.220552	+	0.975375i	$0.429214\pi$
$594$	0	0
$595$	3.83677	0.157292
$596$	0	0
$597$	−65.7344	−2.69033
$598$	0	0
$599$	6.89575	0.281753	0.140876	−	0.990027i	$-0.455008\pi$
$599$	6.89575	0.281753	0.140876	+	0.990027i	$0.455008\pi$
$600$	0	0
$601$	−25.8631	−1.05498	−0.527488	−	0.849563i	$-0.676866\pi$
$601$	−25.8631	−1.05498	−0.527488	+	0.849563i	$0.676866\pi$
$602$	0	0
$603$	5.83240	0.237513
$604$	0	0
$605$	−5.06134	−0.205773
$606$	0	0
$607$	31.7601	1.28910	0.644551	−	0.764561i	$-0.277044\pi$
$607$	31.7601	1.28910	0.644551	+	0.764561i	$0.277044\pi$
$608$	0	0
$609$	−40.2291	−1.63017
$610$	0	0
$611$	−41.7176	−1.68771
$612$	0	0
$613$	13.0180	0.525793	0.262896	−	0.964824i	$-0.415322\pi$
$613$	13.0180	0.525793	0.262896	+	0.964824i	$0.415322\pi$
$614$	0	0
$615$	−7.70135	−0.310549
$616$	0	0
$617$	21.7820	0.876911	0.438455	−	0.898753i	$-0.355526\pi$
$617$	21.7820	0.876911	0.438455	+	0.898753i	$0.355526\pi$
$618$	0	0
$619$	−11.9956	−0.482143	−0.241072	−	0.970507i	$-0.577499\pi$
$619$	−11.9956	−0.482143	−0.241072	+	0.970507i	$0.577499\pi$
$620$	0	0
$621$	−0.00307341	−0.000123332 0
$622$	0	0
$623$	−11.5646	−0.463327
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	19.3495	0.772744
$628$	0	0
$629$	−2.43247	−0.0969889
$630$	0	0
$631$	−14.3634	−0.571798	−0.285899	−	0.958260i	$-0.592292\pi$
$631$	−14.3634	−0.571798	−0.285899	+	0.958260i	$0.592292\pi$
$632$	0	0
$633$	−40.7113	−1.61813
$634$	0	0
$635$	8.79553	0.349040
$636$	0	0
$637$	−22.8994	−0.907308
$638$	0	0
$639$	17.9411	0.709739
$640$	0	0
$641$	45.8138	1.80954	0.904768	−	0.425905i	$-0.140044\pi$
$641$	45.8138	1.80954	0.904768	+	0.425905i	$0.140044\pi$
$642$	0	0
$643$	−11.2550	−0.443855	−0.221928	−	0.975063i	$-0.571235\pi$
$643$	−11.2550	−0.443855	−0.221928	+	0.975063i	$0.571235\pi$
$644$	0	0
$645$	6.15531	0.242365
$646$	0	0
$647$	−41.9262	−1.64829	−0.824145	−	0.566379i	$-0.808344\pi$
$647$	−41.9262	−1.64829	−0.824145	+	0.566379i	$0.808344\pi$
$648$	0	0
$649$	−8.35886	−0.328114
$650$	0	0
$651$	−5.80644	−0.227572
$652$	0	0
$653$	−16.1592	−0.632359	−0.316179	−	0.948699i	$-0.602400\pi$
$653$	−16.1592	−0.632359	−0.316179	+	0.948699i	$0.602400\pi$
$654$	0	0
$655$	11.1368	0.435150
$656$	0	0
$657$	−24.9694	−0.974150
$658$	0	0
$659$	0.0533611	0.00207865	0.00103933	−	0.999999i	$-0.499669\pi$
$659$	0.0533611	0.00207865	0.00103933	+	0.999999i	$0.499669\pi$
$660$	0	0
$661$	−5.76135	−0.224090	−0.112045	−	0.993703i	$-0.535740\pi$
$661$	−5.76135	−0.224090	−0.112045	+	0.993703i	$0.535740\pi$
$662$	0	0
$663$	−30.2505	−1.17483
$664$	0	0
$665$	−5.11100	−0.198196
$666$	0	0
$667$	2.92222	0.113149
$668$	0	0
$669$	24.4354	0.944725
$670$	0	0
$671$	13.4217	0.518139
$672$	0	0
$673$	36.1319	1.39278	0.696390	−	0.717663i	$-0.254788\pi$
$673$	36.1319	1.39278	0.696390	+	0.717663i	$0.254788\pi$
$674$	0	0
$675$	−0.0109469	−0.000421348 0
$676$	0	0
$677$	8.98925	0.345485	0.172742	−	0.984967i	$-0.444737\pi$
$677$	8.98925	0.345485	0.172742	+	0.984967i	$0.444737\pi$
$678$	0	0
$679$	−15.7687	−0.605147
$680$	0	0
$681$	37.5137	1.43753
$682$	0	0
$683$	33.9958	1.30081	0.650406	−	0.759586i	$-0.274599\pi$
$683$	33.9958	1.30081	0.650406	+	0.759586i	$0.274599\pi$
$684$	0	0
$685$	17.1636	0.655786
$686$	0	0
$687$	−56.5360	−2.15698
$688$	0	0
$689$	−27.0898	−1.03204
$690$	0	0
$691$	−31.2138	−1.18743	−0.593714	−	0.804676i	$-0.702339\pi$
$691$	−31.2138	−1.18743	−0.593714	+	0.804676i	$0.702339\pi$
$692$	0	0
$693$	11.5486	0.438696
$694$	0	0
$695$	18.9926	0.720430
$696$	0	0
$697$	7.64500	0.289575
$698$	0	0
$699$	53.8569	2.03706
$700$	0	0
$701$	−17.0796	−0.645087	−0.322544	−	0.946555i	$-0.604538\pi$
$701$	−17.0796	−0.645087	−0.322544	+	0.946555i	$0.604538\pi$
$702$	0	0
$703$	3.24032	0.122211
$704$	0	0
$705$	20.1423	0.758601
$706$	0	0
$707$	−3.38462	−0.127292
$708$	0	0
$709$	26.8360	1.00785	0.503923	−	0.863748i	$-0.331889\pi$
$709$	26.8360	1.00785	0.503923	+	0.863748i	$0.331889\pi$
$710$	0	0
$711$	20.1907	0.757209
$712$	0	0
$713$	0.421777	0.0157957
$714$	0	0
$715$	12.3678	0.462529
$716$	0	0
$717$	11.5128	0.429953
$718$	0	0
$719$	−53.4910	−1.99488	−0.997439	−	0.0715165i	$-0.977216\pi$
$719$	−53.4910	−1.99488	−0.997439	+	0.0715165i	$0.977216\pi$
$720$	0	0
$721$	−2.04111	−0.0760150
$722$	0	0
$723$	−22.4544	−0.835089
$724$	0	0
$725$	10.4084	0.386559
$726$	0	0
$727$	−11.6425	−0.431796	−0.215898	−	0.976416i	$-0.569268\pi$
$727$	−11.6425	−0.431796	−0.215898	+	0.976416i	$0.569268\pi$
$728$	0	0
$729$	−27.0804	−1.00298
$730$	0	0
$731$	−6.11027	−0.225996
$732$	0	0
$733$	−35.8395	−1.32376	−0.661881	−	0.749608i	$-0.730242\pi$
$733$	−35.8395	−1.32376	−0.661881	+	0.749608i	$0.730242\pi$
$734$	0	0
$735$	11.0564	0.407821
$736$	0	0
$737$	4.73068	0.174257
$738$	0	0
$739$	13.2373	0.486942	0.243471	−	0.969908i	$-0.421714\pi$
$739$	13.2373	0.486942	0.243471	+	0.969908i	$0.421714\pi$
$740$	0	0
$741$	40.2970	1.48035
$742$	0	0
$743$	−13.0288	−0.477979	−0.238989	−	0.971022i	$-0.576816\pi$
$743$	−13.0288	−0.477979	−0.238989	+	0.971022i	$0.576816\pi$
$744$	0	0
$745$	1.02681	0.0376195
$746$	0	0
$747$	−1.63818	−0.0599377
$748$	0	0
$749$	23.8053	0.869825
$750$	0	0
$751$	−7.03126	−0.256574	−0.128287	−	0.991737i	$-0.540948\pi$
$751$	−7.03126	−0.256574	−0.128287	+	0.991737i	$0.540948\pi$
$752$	0	0
$753$	38.8126	1.41441
$754$	0	0
$755$	0.330357	0.0120229
$756$	0	0
$757$	14.5077	0.527290	0.263645	−	0.964620i	$-0.415075\pi$
$757$	14.5077	0.527290	0.263645	+	0.964620i	$0.415075\pi$
$758$	0	0
$759$	−1.67653	−0.0608540
$760$	0	0
$761$	−13.9239	−0.504739	−0.252370	−	0.967631i	$-0.581210\pi$
$761$	−13.9239	−0.504739	−0.252370	+	0.967631i	$0.581210\pi$
$762$	0	0
$763$	19.1432	0.693031
$764$	0	0
$765$	7.30828	0.264231
$766$	0	0
$767$	−17.4081	−0.628569
$768$	0	0
$769$	12.1006	0.436358	0.218179	−	0.975909i	$-0.429988\pi$
$769$	12.1006	0.436358	0.218179	+	0.975909i	$0.429988\pi$
$770$	0	0
$771$	−23.1351	−0.833191
$772$	0	0
$773$	−40.9345	−1.47231	−0.736156	−	0.676812i	$-0.763361\pi$
$773$	−40.9345	−1.47231	−0.736156	+	0.676812i	$0.763361\pi$
$774$	0	0
$775$	1.50229	0.0539639
$776$	0	0
$777$	3.86506	0.138658
$778$	0	0
$779$	−10.1840	−0.364879
$780$	0	0
$781$	14.5521	0.520715
$782$	0	0
$783$	−0.113940	−0.00407189
$784$	0	0
$785$	10.9125	0.389485
$786$	0	0
$787$	−44.4327	−1.58385	−0.791927	−	0.610616i	$-0.790922\pi$
$787$	−44.4327	−1.58385	−0.791927	+	0.610616i	$0.790922\pi$
$788$	0	0
$789$	34.3125	1.22156
$790$	0	0
$791$	−7.96941	−0.283360
$792$	0	0
$793$	27.9519	0.992601
$794$	0	0
$795$	13.0796	0.463886
$796$	0	0
$797$	−39.8692	−1.41224	−0.706119	−	0.708093i	$-0.749556\pi$
$797$	−39.8692	−1.41224	−0.706119	+	0.708093i	$0.749556\pi$
$798$	0	0
$799$	−19.9949	−0.707367
$800$	0	0
$801$	−22.0283	−0.778331
$802$	0	0
$803$	−20.2528	−0.714706
$804$	0	0
$805$	0.442840	0.0156081
$806$	0	0
$807$	−24.0375	−0.846159
$808$	0	0
$809$	38.3286	1.34756	0.673781	−	0.738931i	$-0.264669\pi$
$809$	38.3286	1.34756	0.673781	+	0.738931i	$0.264669\pi$
$810$	0	0
$811$	16.7489	0.588134	0.294067	−	0.955785i	$-0.404991\pi$
$811$	16.7489	0.588134	0.294067	+	0.955785i	$0.404991\pi$
$812$	0	0
$813$	−2.96519	−0.103994
$814$	0	0
$815$	−2.78401	−0.0975197
$816$	0	0
$817$	8.13955	0.284767
$818$	0	0
$819$	24.0510	0.840412
$820$	0	0
$821$	13.9789	0.487868	0.243934	−	0.969792i	$-0.421562\pi$
$821$	13.9789	0.487868	0.243934	+	0.969792i	$0.421562\pi$
$822$	0	0
$823$	49.0366	1.70931	0.854654	−	0.519198i	$-0.173769\pi$
$823$	49.0366	1.70931	0.854654	+	0.519198i	$0.173769\pi$
$824$	0	0
$825$	−5.97147	−0.207900
$826$	0	0
$827$	−16.6602	−0.579332	−0.289666	−	0.957128i	$-0.593544\pi$
$827$	−16.6602	−0.579332	−0.289666	+	0.957128i	$0.593544\pi$
$828$	0	0
$829$	49.1851	1.70827	0.854135	−	0.520052i	$-0.174087\pi$
$829$	49.1851	1.70827	0.854135	+	0.520052i	$0.174087\pi$
$830$	0	0
$831$	−27.5604	−0.956061
$832$	0	0
$833$	−10.9755	−0.380278
$834$	0	0
$835$	−5.84662	−0.202331
$836$	0	0
$837$	−0.0164455	−0.000568439 0
$838$	0	0
$839$	−53.2547	−1.83856	−0.919279	−	0.393607i	$-0.871227\pi$
$839$	−53.2547	−1.83856	−0.919279	+	0.393607i	$0.871227\pi$
$840$	0	0
$841$	79.3350	2.73569
$842$	0	0
$843$	13.0205	0.448449
$844$	0	0
$845$	12.7570	0.438855
$846$	0	0
$847$	−7.98332	−0.274310
$848$	0	0
$849$	22.2130	0.762349
$850$	0	0
$851$	−0.280756	−0.00962418
$852$	0	0
$853$	40.9356	1.40161	0.700804	−	0.713354i	$-0.252825\pi$
$853$	40.9356	1.40161	0.700804	+	0.713354i	$0.252825\pi$
$854$	0	0
$855$	−9.73543	−0.332945
$856$	0	0
$857$	−16.6783	−0.569719	−0.284859	−	0.958569i	$-0.591947\pi$
$857$	−16.6783	−0.569719	−0.284859	+	0.958569i	$0.591947\pi$
$858$	0	0
$859$	5.58757	0.190645	0.0953226	−	0.995446i	$-0.469612\pi$
$859$	5.58757	0.190645	0.0953226	+	0.995446i	$0.469612\pi$
$860$	0	0
$861$	−12.1475	−0.413984
$862$	0	0
$863$	−53.0809	−1.80689	−0.903447	−	0.428699i	$-0.858972\pi$
$863$	−53.0809	−1.80689	−0.903447	+	0.428699i	$0.858972\pi$
$864$	0	0
$865$	−14.7373	−0.501082
$866$	0	0
$867$	27.1580	0.922335
$868$	0	0
$869$	16.3767	0.555543
$870$	0	0
$871$	9.85207	0.333825
$872$	0	0
$873$	−30.0362	−1.01657
$874$	0	0
$875$	1.57732	0.0533230
$876$	0	0
$877$	−28.1603	−0.950907	−0.475453	−	0.879741i	$-0.657716\pi$
$877$	−28.1603	−0.950907	−0.475453	+	0.879741i	$0.657716\pi$
$878$	0	0
$879$	−47.5295	−1.60313
$880$	0	0
$881$	47.2415	1.59161	0.795803	−	0.605555i	$-0.207049\pi$
$881$	47.2415	1.59161	0.795803	+	0.605555i	$0.207049\pi$
$882$	0	0
$883$	21.7989	0.733592	0.366796	−	0.930301i	$-0.380455\pi$
$883$	21.7989	0.733592	0.366796	+	0.930301i	$0.380455\pi$
$884$	0	0
$885$	8.40504	0.282532
$886$	0	0
$887$	−17.0518	−0.572543	−0.286272	−	0.958149i	$-0.592416\pi$
$887$	−17.0518	−0.572543	−0.286272	+	0.958149i	$0.592416\pi$
$888$	0	0
$889$	13.8733	0.465296
$890$	0	0
$891$	−21.8997	−0.733668
$892$	0	0
$893$	26.6354	0.891318
$894$	0	0
$895$	−19.1742	−0.640923
$896$	0	0
$897$	−3.49151	−0.116578
$898$	0	0
$899$	15.6365	0.521505
$900$	0	0
$901$	−12.9839	−0.432556
$902$	0	0
$903$	9.70887	0.323091
$904$	0	0
$905$	23.0841	0.767340
$906$	0	0
$907$	33.3541	1.10750	0.553752	−	0.832682i	$-0.313196\pi$
$907$	33.3541	1.10750	0.553752	+	0.832682i	$0.313196\pi$
$908$	0	0
$909$	−6.44702	−0.213834
$910$	0	0
$911$	38.5501	1.27722	0.638611	−	0.769530i	$-0.279509\pi$
$911$	38.5501	1.27722	0.638611	+	0.769530i	$0.279509\pi$
$912$	0	0
$913$	−1.32873	−0.0439746
$914$	0	0
$915$	−13.4959	−0.446159
$916$	0	0
$917$	17.5662	0.580087
$918$	0	0
$919$	−42.4437	−1.40009	−0.700044	−	0.714100i	$-0.746836\pi$
$919$	−42.4437	−1.40009	−0.700044	+	0.714100i	$0.746836\pi$
$920$	0	0
$921$	−18.4209	−0.606988
$922$	0	0
$923$	30.3061	0.997536
$924$	0	0
$925$	−1.00000	−0.0328798
$926$	0	0
$927$	−3.88791	−0.127696
$928$	0	0
$929$	−24.2392	−0.795262	−0.397631	−	0.917545i	$-0.630168\pi$
$929$	−24.2392	−0.795262	−0.397631	+	0.917545i	$0.630168\pi$
$930$	0	0
$931$	14.6206	0.479170
$932$	0	0
$933$	−28.6195	−0.936959
$934$	0	0
$935$	5.92778	0.193859
$936$	0	0
$937$	58.2087	1.90160	0.950798	−	0.309812i	$-0.100266\pi$
$937$	58.2087	1.90160	0.950798	+	0.309812i	$0.100266\pi$
$938$	0	0
$939$	12.3946	0.404484
$940$	0	0
$941$	−46.0691	−1.50181	−0.750905	−	0.660411i	$-0.770382\pi$
$941$	−46.0691	−1.50181	−0.750905	+	0.660411i	$0.770382\pi$
$942$	0	0
$943$	0.882386	0.0287344
$944$	0	0
$945$	−0.0172668	−0.000561688 0
$946$	0	0
$947$	27.6363	0.898058	0.449029	−	0.893517i	$-0.351770\pi$
$947$	27.6363	0.898058	0.449029	+	0.893517i	$0.351770\pi$
$948$	0	0
$949$	−42.1783	−1.36917
$950$	0	0
$951$	32.9688	1.06909
$952$	0	0
$953$	22.3788	0.724921	0.362460	−	0.931999i	$-0.381937\pi$
$953$	22.3788	0.724921	0.362460	+	0.931999i	$0.381937\pi$
$954$	0	0
$955$	20.9302	0.677286
$956$	0	0
$957$	−62.1536	−2.00914
$958$	0	0
$959$	27.0724	0.874212
$960$	0	0
$961$	−28.7431	−0.927197
$962$	0	0
$963$	45.3442	1.46120
$964$	0	0
$965$	−19.5228	−0.628461
$966$	0	0
$967$	1.14098	0.0366916	0.0183458	−	0.999832i	$-0.494160\pi$
$967$	1.14098	0.0366916	0.0183458	+	0.999832i	$0.494160\pi$
$968$	0	0
$969$	19.3140	0.620455
$970$	0	0
$971$	−17.0069	−0.545778	−0.272889	−	0.962046i	$-0.587979\pi$
$971$	−17.0069	−0.545778	−0.272889	+	0.962046i	$0.587979\pi$
$972$	0	0
$973$	29.9573	0.960387
$974$	0	0
$975$	−12.4361	−0.398275
$976$	0	0
$977$	−25.1382	−0.804241	−0.402121	−	0.915587i	$-0.631727\pi$
$977$	−25.1382	−0.804241	−0.402121	+	0.915587i	$0.631727\pi$
$978$	0	0
$979$	−17.8672	−0.571039
$980$	0	0
$981$	36.4640	1.16421
$982$	0	0
$983$	18.5100	0.590378	0.295189	−	0.955439i	$-0.404617\pi$
$983$	18.5100	0.590378	0.295189	+	0.955439i	$0.404617\pi$
$984$	0	0
$985$	5.84812	0.186337
$986$	0	0
$987$	31.7707	1.01127
$988$	0	0
$989$	−0.705247	−0.0224256
$990$	0	0
$991$	43.7310	1.38916	0.694580	−	0.719415i	$-0.255590\pi$
$991$	43.7310	1.38916	0.694580	+	0.719415i	$0.255590\pi$
$992$	0	0
$993$	−79.0867	−2.50974
$994$	0	0
$995$	26.8260	0.850441
$996$	0	0
$997$	−27.2065	−0.861639	−0.430819	−	0.902438i	$-0.641775\pi$
$997$	−27.2065	−0.861639	−0.430819	+	0.902438i	$0.641775\pi$
$998$	0	0
$999$	0.0109469	0.000346346 0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2960.2.a.bc.1.2		6
4.3	odd	2		1480.2.a.k.1.5	✓	6
20.19	odd	2		7400.2.a.s.1.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1480.2.a.k.1.5	✓	6	4.3	odd	2
2960.2.a.bc.1.2		6	1.1	even	1	trivial
7400.2.a.s.1.2		6	20.19	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-2.45040 q^{3} +1.00000 q^{5} +1.57732 q^{7} +3.00447 q^{9} +O(q^{10})\)	\(q-2.45040 q^{3} +1.00000 q^{5} +1.57732 q^{7} +3.00447 q^{9} +2.43694 q^{11} +5.07514 q^{13} -2.45040 q^{15} +2.43247 q^{17} -3.24032 q^{19} -3.86506 q^{21} +0.280756 q^{23} +1.00000 q^{25} -0.0109469 q^{27} +10.4084 q^{29} +1.50229 q^{31} -5.97147 q^{33} +1.57732 q^{35} -1.00000 q^{37} -12.4361 q^{39} +3.14290 q^{41} -2.51196 q^{43} +3.00447 q^{45} -8.21998 q^{47} -4.51208 q^{49} -5.96053 q^{51} -5.33774 q^{53} +2.43694 q^{55} +7.94008 q^{57} -3.43007 q^{59} +5.50761 q^{61} +4.73899 q^{63} +5.07514 q^{65} +1.94124 q^{67} -0.687964 q^{69} +5.97147 q^{71} -8.31077 q^{73} -2.45040 q^{75} +3.84382 q^{77} +6.72021 q^{79} -8.98658 q^{81} -0.545247 q^{83} +2.43247 q^{85} -25.5048 q^{87} -7.33184 q^{89} +8.00509 q^{91} -3.68122 q^{93} -3.24032 q^{95} -9.99718 q^{97} +7.32170 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - q^{3} + 6 q^{5} - 8 q^{7} + 15 q^{9}+O(q^{10}) \) 6 * q - q^3 + 6 * q^5 - 8 * q^7 + 15 * q^9	\( 6 q - q^{3} + 6 q^{5} - 8 q^{7} + 15 q^{9} + 10 q^{13} - q^{15} + 3 q^{17} + 6 q^{19} + 9 q^{21} - 4 q^{23} + 6 q^{25} - 7 q^{27} + 3 q^{29} + 3 q^{31} + 9 q^{33} - 8 q^{35} - 6 q^{37} + 16 q^{39} + 10 q^{41} + 11 q^{43} + 15 q^{45} - 23 q^{47} + 8 q^{49} + 16 q^{51} + 10 q^{53} + 4 q^{57} - 6 q^{59} + q^{61} - 23 q^{63} + 10 q^{65} + 4 q^{67} - 2 q^{69} - 9 q^{71} + 11 q^{73} - q^{75} + 32 q^{77} + 14 q^{79} + 46 q^{81} - 11 q^{83} + 3 q^{85} - 32 q^{87} + 30 q^{89} + 4 q^{91} + 2 q^{93} + 6 q^{95} + 29 q^{97} + 39 q^{99}+O(q^{100}) \) 6 * q - q^3 + 6 * q^5 - 8 * q^7 + 15 * q^9 + 10 * q^13 - q^15 + 3 * q^17 + 6 * q^19 + 9 * q^21 - 4 * q^23 + 6 * q^25 - 7 * q^27 + 3 * q^29 + 3 * q^31 + 9 * q^33 - 8 * q^35 - 6 * q^37 + 16 * q^39 + 10 * q^41 + 11 * q^43 + 15 * q^45 - 23 * q^47 + 8 * q^49 + 16 * q^51 + 10 * q^53 + 4 * q^57 - 6 * q^59 + q^61 - 23 * q^63 + 10 * q^65 + 4 * q^67 - 2 * q^69 - 9 * q^71 + 11 * q^73 - q^75 + 32 * q^77 + 14 * q^79 + 46 * q^81 - 11 * q^83 + 3 * q^85 - 32 * q^87 + 30 * q^89 + 4 * q^91 + 2 * q^93 + 6 * q^95 + 29 * q^97 + 39 * q^99

Introduction

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