LMFDB - Embedded newform 976.2.a.i.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [976,2,Mod(1,976)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(976, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("976.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 976 = 2^{4} \cdot 61 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	976.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:	$7.79339923728$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.13676.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 6x^{2} + 7x + 1 $ x^4 - x^3 - 6x^2 + 7x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 488)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$1.42957$ of defining polynomial
Character	$\chi$	$=$	976.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.22628 q^{3} -4.18262 q^{5} -0.570433 q^{7} +1.95634 q^{9} +O(q^{10})$	$q+2.22628 q^{3} -4.18262 q^{5} -0.570433 q^{7} +1.95634 q^{9} -0.473229 q^{11} -0.269945 q^{13} -9.31170 q^{15} -2.85913 q^{17} -1.04366 q^{19} -1.26995 q^{21} -8.61219 q^{23} +12.4943 q^{25} -2.32349 q^{27} -0.0972039 q^{29} -4.42069 q^{31} -1.05354 q^{33} +2.38591 q^{35} +4.22628 q^{37} -0.600975 q^{39} -9.67461 q^{41} -0.946458 q^{43} -8.18262 q^{45} -8.25816 q^{47} -6.67461 q^{49} -6.36524 q^{51} +1.08542 q^{53} +1.97934 q^{55} -2.32349 q^{57} +10.5192 q^{59} -1.00000 q^{61} -1.11596 q^{63} +1.12908 q^{65} -4.92580 q^{67} -19.1732 q^{69} +6.35536 q^{71} +14.3633 q^{73} +27.8159 q^{75} +0.269945 q^{77} -11.6976 q^{79} -11.0418 q^{81} +13.2145 q^{83} +11.9587 q^{85} -0.216403 q^{87} +8.27792 q^{89} +0.153986 q^{91} -9.84171 q^{93} +4.36524 q^{95} +17.9906 q^{97} -0.925796 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - q^{3} - 2 q^{5} - 7 q^{7} + 3 q^{9}+O(q^{10}) $ 4 * q - q^3 - 2 * q^5 - 7 * q^7 + 3 * q^9	$ 4 q - q^{3} - 2 q^{5} - 7 q^{7} + 3 q^{9} - 2 q^{11} + 4 q^{13} - 8 q^{15} - 2 q^{17} - 9 q^{19} - 15 q^{23} + 6 q^{25} - 4 q^{27} - 5 q^{29} - 17 q^{31} - 4 q^{33} + 7 q^{37} - 22 q^{39} - 15 q^{41} - 4 q^{43} - 18 q^{45} - 4 q^{47} - 3 q^{49} + 4 q^{51} - 15 q^{53} - 12 q^{55} - 4 q^{57} + 12 q^{59} - 4 q^{61} - 10 q^{65} - 3 q^{69} + q^{71} - q^{73} + 33 q^{75} - 4 q^{77} - 8 q^{79} - 20 q^{81} + 19 q^{83} + 8 q^{85} + 4 q^{87} - 6 q^{89} - 5 q^{93} - 12 q^{95} + 13 q^{97} + 16 q^{99}+O(q^{100}) $ 4 * q - q^3 - 2 * q^5 - 7 * q^7 + 3 * q^9 - 2 * q^11 + 4 * q^13 - 8 * q^15 - 2 * q^17 - 9 * q^19 - 15 * q^23 + 6 * q^25 - 4 * q^27 - 5 * q^29 - 17 * q^31 - 4 * q^33 + 7 * q^37 - 22 * q^39 - 15 * q^41 - 4 * q^43 - 18 * q^45 - 4 * q^47 - 3 * q^49 + 4 * q^51 - 15 * q^53 - 12 * q^55 - 4 * q^57 + 12 * q^59 - 4 * q^61 - 10 * q^65 - 3 * q^69 + q^71 - q^73 + 33 * q^75 - 4 * q^77 - 8 * q^79 - 20 * q^81 + 19 * q^83 + 8 * q^85 + 4 * q^87 - 6 * q^89 - 5 * q^93 - 12 * q^95 + 13 * q^97 + 16 * 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
$3$	2.22628	1.28535	0.642673	−	0.766141i	$-0.277826\pi$
$3$	2.22628	1.28535	0.642673	+	0.766141i	$0.277826\pi$
$4$	0	0
$5$	−4.18262	−1.87053	−0.935263	−	0.353955i	$-0.884837\pi$
$5$	−4.18262	−1.87053	−0.935263	+	0.353955i	$0.884837\pi$
$6$	0	0
$7$	−0.570433	−0.215603	−0.107802	−	0.994172i	$-0.534381\pi$
$7$	−0.570433	−0.215603	−0.107802	+	0.994172i	$0.534381\pi$
$8$	0	0
$9$	1.95634	0.652113
$10$	0	0
$11$	−0.473229	−0.142684	−0.0713420	−	0.997452i	$-0.522728\pi$
$11$	−0.473229	−0.142684	−0.0713420	+	0.997452i	$0.522728\pi$
$12$	0	0
$13$	−0.269945	−0.0748694	−0.0374347	−	0.999299i	$-0.511919\pi$
$13$	−0.269945	−0.0748694	−0.0374347	+	0.999299i	$0.511919\pi$
$14$	0	0
$15$	−9.31170	−2.40427
$16$	0	0
$17$	−2.85913	−0.693442	−0.346721	−	0.937968i	$-0.612705\pi$
$17$	−2.85913	−0.693442	−0.346721	+	0.937968i	$0.612705\pi$
$18$	0	0
$19$	−1.04366	−0.239432	−0.119716	−	0.992808i	$-0.538198\pi$
$19$	−1.04366	−0.239432	−0.119716	+	0.992808i	$0.538198\pi$
$20$	0	0
$21$	−1.26995	−0.277125
$22$	0	0
$23$	−8.61219	−1.79577	−0.897883	−	0.440235i	$-0.854895\pi$
$23$	−8.61219	−1.79577	−0.897883	+	0.440235i	$0.854895\pi$
$24$	0	0
$25$	12.4943	2.49886
$26$	0	0
$27$	−2.32349	−0.447155
$28$	0	0
$29$	−0.0972039	−0.0180503	−0.00902516	−	0.999959i	$-0.502873\pi$
$29$	−0.0972039	−0.0180503	−0.00902516	+	0.999959i	$0.502873\pi$
$30$	0	0
$31$	−4.42069	−0.793980	−0.396990	−	0.917823i	$-0.629945\pi$
$31$	−4.42069	−0.793980	−0.396990	+	0.917823i	$0.629945\pi$
$32$	0	0
$33$	−1.05354	−0.183398
$34$	0	0
$35$	2.38591	0.403292
$36$	0	0
$37$	4.22628	0.694797	0.347398	−	0.937718i	$-0.387065\pi$
$37$	4.22628	0.694797	0.347398	+	0.937718i	$0.387065\pi$
$38$	0	0
$39$	−0.600975	−0.0962330
$40$	0	0
$41$	−9.67461	−1.51092	−0.755460	−	0.655195i	$-0.772586\pi$
$41$	−9.67461	−1.51092	−0.755460	+	0.655195i	$0.772586\pi$
$42$	0	0
$43$	−0.946458	−0.144333	−0.0721667	−	0.997393i	$-0.522991\pi$
$43$	−0.946458	−0.144333	−0.0721667	+	0.997393i	$0.522991\pi$
$44$	0	0
$45$	−8.18262	−1.21979
$46$	0	0
$47$	−8.25816	−1.20458	−0.602288	−	0.798279i	$-0.705744\pi$
$47$	−8.25816	−1.20458	−0.602288	+	0.798279i	$0.705744\pi$
$48$	0	0
$49$	−6.67461	−0.953515
$50$	0	0
$51$	−6.36524	−0.891312
$52$	0	0
$53$	1.08542	0.149094	0.0745468	−	0.997218i	$-0.476249\pi$
$53$	1.08542	0.149094	0.0745468	+	0.997218i	$0.476249\pi$
$54$	0	0
$55$	1.97934	0.266894
$56$	0	0
$57$	−2.32349	−0.307753
$58$	0	0
$59$	10.5192	1.36949	0.684743	−	0.728784i	$-0.259914\pi$
$59$	10.5192	1.36949	0.684743	+	0.728784i	$0.259914\pi$
$60$	0	0
$61$	−1.00000	−0.128037
$61$	−1.00000	−0.128037
$62$	0	0
$63$	−1.11596	−0.140598
$64$	0	0
$65$	1.12908	0.140045
$66$	0	0
$67$	−4.92580	−0.601782	−0.300891	−	0.953659i	$-0.597284\pi$
$67$	−4.92580	−0.601782	−0.300891	+	0.953659i	$0.597284\pi$
$68$	0	0
$69$	−19.1732	−2.30818
$70$	0	0
$71$	6.35536	0.754243	0.377121	−	0.926164i	$-0.376914\pi$
$71$	6.35536	0.754243	0.377121	+	0.926164i	$0.376914\pi$
$72$	0	0
$73$	14.3633	1.68110	0.840551	−	0.541733i	$-0.182232\pi$
$73$	14.3633	1.68110	0.840551	+	0.541733i	$0.182232\pi$
$74$	0	0
$75$	27.8159	3.21190
$76$	0	0
$77$	0.269945	0.0307631
$78$	0	0
$79$	−11.6976	−1.31608	−0.658042	−	0.752981i	$-0.728615\pi$
$79$	−11.6976	−1.31608	−0.658042	+	0.752981i	$0.728615\pi$
$80$	0	0
$81$	−11.0418	−1.22686
$82$	0	0
$83$	13.2145	1.45048	0.725240	−	0.688496i	$-0.241729\pi$
$83$	13.2145	1.45048	0.725240	+	0.688496i	$0.241729\pi$
$84$	0	0
$85$	11.9587	1.29710
$86$	0	0
$87$	−0.216403	−0.0232009
$88$	0	0
$89$	8.27792	0.877458	0.438729	−	0.898620i	$-0.355429\pi$
$89$	8.27792	0.877458	0.438729	+	0.898620i	$0.355429\pi$
$90$	0	0
$91$	0.153986	0.0161421
$92$	0	0
$93$	−9.84171	−1.02054
$94$	0	0
$95$	4.36524	0.447864
$96$	0	0
$97$	17.9906	1.82666	0.913332	−	0.407216i	$-0.133500\pi$
$97$	17.9906	1.82666	0.913332	+	0.407216i	$0.133500\pi$
$98$	0	0
$99$	−0.925796	−0.0930460
$100$	0	0
$101$	16.9568	1.68726	0.843631	−	0.536924i	$-0.180414\pi$
$101$	16.9568	1.68726	0.843631	+	0.536924i	$0.180414\pi$
$102$	0	0
$103$	8.17084	0.805096	0.402548	−	0.915399i	$-0.368125\pi$
$103$	8.17084	0.805096	0.402548	+	0.915399i	$0.368125\pi$
$104$	0	0
$105$	5.31170	0.518369
$106$	0	0
$107$	−12.0934	−1.16911	−0.584556	−	0.811353i	$-0.698731\pi$
$107$	−12.0934	−1.16911	−0.584556	+	0.811353i	$0.698731\pi$
$108$	0	0
$109$	2.16286	0.207165	0.103582	−	0.994621i	$-0.466969\pi$
$109$	2.16286	0.207165	0.103582	+	0.994621i	$0.466969\pi$
$110$	0	0
$111$	9.40890	0.893054
$112$	0	0
$113$	−14.7324	−1.38591	−0.692953	−	0.720982i	$-0.743691\pi$
$113$	−14.7324	−1.38591	−0.692953	+	0.720982i	$0.743691\pi$
$114$	0	0
$115$	36.0215	3.35902
$116$	0	0
$117$	−0.528105	−0.0488233
$118$	0	0
$119$	1.63094	0.149508
$120$	0	0
$121$	−10.7761	−0.979641
$122$	0	0
$123$	−21.5384	−1.94205
$124$	0	0
$125$	−31.3459	−2.80366
$126$	0	0
$127$	8.68449	0.770624	0.385312	−	0.922786i	$-0.374094\pi$
$127$	8.68449	0.770624	0.385312	+	0.922786i	$0.374094\pi$
$128$	0	0
$129$	−2.10708	−0.185518
$130$	0	0
$131$	−17.2704	−1.50892	−0.754460	−	0.656346i	$-0.772101\pi$
$131$	−17.2704	−1.50892	−0.754460	+	0.656346i	$0.772101\pi$
$132$	0	0
$133$	0.595339	0.0516225
$134$	0	0
$135$	9.71827	0.836415
$136$	0	0
$137$	11.5958	0.990694	0.495347	−	0.868695i	$-0.335041\pi$
$137$	11.5958	0.990694	0.495347	+	0.868695i	$0.335041\pi$
$138$	0	0
$139$	0.348394	0.0295504	0.0147752	−	0.999891i	$-0.495297\pi$
$139$	0.348394	0.0295504	0.0147752	+	0.999891i	$0.495297\pi$
$140$	0	0
$141$	−18.3850	−1.54830
$142$	0	0
$143$	0.127746	0.0106827
$144$	0	0
$145$	0.406567	0.0337636
$146$	0	0
$147$	−14.8596	−1.22560
$148$	0	0
$149$	14.8933	1.22011	0.610055	−	0.792359i	$-0.291147\pi$
$149$	14.8933	1.22011	0.610055	+	0.792359i	$0.291147\pi$
$150$	0	0
$151$	−8.93143	−0.726830	−0.363415	−	0.931627i	$-0.618389\pi$
$151$	−8.93143	−0.726830	−0.363415	+	0.931627i	$0.618389\pi$
$152$	0	0
$153$	−5.59343	−0.452202
$154$	0	0
$155$	18.4901	1.48516
$156$	0	0
$157$	16.2680	1.29833	0.649165	−	0.760647i	$-0.275118\pi$
$157$	16.2680	1.29833	0.649165	+	0.760647i	$0.275118\pi$
$158$	0	0
$159$	2.41645	0.191637
$160$	0	0
$161$	4.91268	0.387173
$162$	0	0
$163$	−15.1314	−1.18518	−0.592592	−	0.805503i	$-0.701895\pi$
$163$	−15.1314	−1.18518	−0.592592	+	0.805503i	$0.701895\pi$
$164$	0	0
$165$	4.40657	0.343051
$166$	0	0
$167$	−10.2779	−0.795329	−0.397665	−	0.917531i	$-0.630179\pi$
$167$	−10.2779	−0.795329	−0.397665	+	0.917531i	$0.630179\pi$
$168$	0	0
$169$	−12.9271	−0.994395
$170$	0	0
$171$	−2.04176	−0.156137
$172$	0	0
$173$	−19.4924	−1.48198	−0.740990	−	0.671516i	$-0.765644\pi$
$173$	−19.4924	−1.48198	−0.740990	+	0.671516i	$0.765644\pi$
$174$	0	0
$175$	−7.12717	−0.538764
$176$	0	0
$177$	23.4188	1.76026
$178$	0	0
$179$	−8.24604	−0.616338	−0.308169	−	0.951332i	$-0.599716\pi$
$179$	−8.24604	−0.616338	−0.308169	+	0.951332i	$0.599716\pi$
$180$	0	0
$181$	2.02763	0.150713	0.0753563	−	0.997157i	$-0.475991\pi$
$181$	2.02763	0.150713	0.0753563	+	0.997157i	$0.475991\pi$
$182$	0	0
$183$	−2.22628	−0.164572
$184$	0	0
$185$	−17.6769	−1.29963
$186$	0	0
$187$	1.35303	0.0989430
$188$	0	0
$189$	1.32539	0.0964082
$190$	0	0
$191$	−12.2929	−0.889486	−0.444743	−	0.895658i	$-0.646705\pi$
$191$	−12.2929	−0.889486	−0.444743	+	0.895658i	$0.646705\pi$
$192$	0	0
$193$	−13.4412	−0.967520	−0.483760	−	0.875201i	$-0.660729\pi$
$193$	−13.4412	−0.967520	−0.483760	+	0.875201i	$0.660729\pi$
$194$	0	0
$195$	2.51365	0.180006
$196$	0	0
$197$	10.4408	0.743875	0.371937	−	0.928258i	$-0.378694\pi$
$197$	10.4408	0.743875	0.371937	+	0.928258i	$0.378694\pi$
$198$	0	0
$199$	14.7765	1.04748	0.523739	−	0.851879i	$-0.324537\pi$
$199$	14.7765	1.04748	0.523739	+	0.851879i	$0.324537\pi$
$200$	0	0
$201$	−10.9662	−0.773497
$202$	0	0
$203$	0.0554483	0.00389171
$204$	0	0
$205$	40.4652	2.82621
$206$	0	0
$207$	−16.8484	−1.17104
$208$	0	0
$209$	0.493891	0.0341632
$210$	0	0
$211$	3.67227	0.252809	0.126405	−	0.991979i	$-0.459656\pi$
$211$	3.67227	0.252809	0.126405	+	0.991979i	$0.459656\pi$
$212$	0	0
$213$	14.1488	0.969463
$214$	0	0
$215$	3.95868	0.269979
$216$	0	0
$217$	2.52171	0.171185
$218$	0	0
$219$	31.9769	2.16080
$220$	0	0
$221$	0.771810	0.0519176
$222$	0	0
$223$	−17.8047	−1.19229	−0.596145	−	0.802877i	$-0.703302\pi$
$223$	−17.8047	−1.19229	−0.596145	+	0.802877i	$0.703302\pi$
$224$	0	0
$225$	24.4431	1.62954
$226$	0	0
$227$	−17.3286	−1.15014	−0.575068	−	0.818106i	$-0.695024\pi$
$227$	−17.3286	−1.15014	−0.575068	+	0.818106i	$0.695024\pi$
$228$	0	0
$229$	7.36481	0.486680	0.243340	−	0.969941i	$-0.421757\pi$
$229$	7.36481	0.486680	0.243340	+	0.969941i	$0.421757\pi$
$230$	0	0
$231$	0.600975	0.0395413
$232$	0	0
$233$	−12.0658	−0.790454	−0.395227	−	0.918583i	$-0.629334\pi$
$233$	−12.0658	−0.790454	−0.395227	+	0.918583i	$0.629334\pi$
$234$	0	0
$235$	34.5408	2.25319
$236$	0	0
$237$	−26.0422	−1.69162
$238$	0	0
$239$	−20.6845	−1.33797	−0.668984	−	0.743277i	$-0.733271\pi$
$239$	−20.6845	−1.33797	−0.668984	+	0.743277i	$0.733271\pi$
$240$	0	0
$241$	−1.35961	−0.0875800	−0.0437900	−	0.999041i	$-0.513943\pi$
$241$	−1.35961	−0.0875800	−0.0437900	+	0.999041i	$0.513943\pi$
$242$	0	0
$243$	−17.6116	−1.12979
$244$	0	0
$245$	27.9174	1.78357
$246$	0	0
$247$	0.281732	0.0179262
$248$	0	0
$249$	29.4192	1.86437
$250$	0	0
$251$	13.6132	0.859257	0.429629	−	0.903006i	$-0.358645\pi$
$251$	13.6132	0.859257	0.429629	+	0.903006i	$0.358645\pi$
$252$	0	0
$253$	4.07554	0.256227
$254$	0	0
$255$	26.6234	1.66722
$256$	0	0
$257$	−7.04834	−0.439663	−0.219832	−	0.975538i	$-0.570551\pi$
$257$	−7.04834	−0.439663	−0.219832	+	0.975538i	$0.570551\pi$
$258$	0	0
$259$	−2.41081	−0.149801
$260$	0	0
$261$	−0.190164	−0.0117708
$262$	0	0
$263$	14.1755	0.874099	0.437050	−	0.899437i	$-0.356023\pi$
$263$	14.1755	0.874099	0.437050	+	0.899437i	$0.356023\pi$
$264$	0	0
$265$	−4.53989	−0.278883
$266$	0	0
$267$	18.4290	1.12784
$268$	0	0
$269$	9.09573	0.554576	0.277288	−	0.960787i	$-0.410564\pi$
$269$	9.09573	0.554576	0.277288	+	0.960787i	$0.410564\pi$
$270$	0	0
$271$	−8.24041	−0.500569	−0.250285	−	0.968172i	$-0.580524\pi$
$271$	−8.24041	−0.500569	−0.250285	+	0.968172i	$0.580524\pi$
$272$	0	0
$273$	0.342816	0.0207482
$274$	0	0
$275$	−5.91268	−0.356548
$276$	0	0
$277$	−0.827690	−0.0497311	−0.0248655	−	0.999691i	$-0.507916\pi$
$277$	−0.827690	−0.0497311	−0.0248655	+	0.999691i	$0.507916\pi$
$278$	0	0
$279$	−8.64837	−0.517764
$280$	0	0
$281$	−21.9849	−1.31151	−0.655755	−	0.754974i	$-0.727650\pi$
$281$	−21.9849	−1.31151	−0.655755	+	0.754974i	$0.727650\pi$
$282$	0	0
$283$	−5.94836	−0.353594	−0.176797	−	0.984247i	$-0.556574\pi$
$283$	−5.94836	−0.353594	−0.176797	+	0.984247i	$0.556574\pi$
$284$	0	0
$285$	9.71827	0.575661
$286$	0	0
$287$	5.51871	0.325759
$288$	0	0
$289$	−8.82535	−0.519138
$290$	0	0
$291$	40.0521	2.34789
$292$	0	0
$293$	12.2346	0.714752	0.357376	−	0.933961i	$-0.383671\pi$
$293$	12.2346	0.714752	0.357376	+	0.933961i	$0.383671\pi$
$294$	0	0
$295$	−43.9980	−2.56166
$296$	0	0
$297$	1.09954	0.0638019
$298$	0	0
$299$	2.32482	0.134448
$300$	0	0
$301$	0.539891	0.0311188
$302$	0	0
$303$	37.7506	2.16871
$304$	0	0
$305$	4.18262	0.239496
$306$	0	0
$307$	30.7361	1.75420	0.877100	−	0.480308i	$-0.159475\pi$
$307$	30.7361	1.75420	0.877100	+	0.480308i	$0.159475\pi$
$308$	0	0
$309$	18.1906	1.03483
$310$	0	0
$311$	6.11405	0.346696	0.173348	−	0.984861i	$-0.444541\pi$
$311$	6.11405	0.346696	0.173348	+	0.984861i	$0.444541\pi$
$312$	0	0
$313$	8.97270	0.507167	0.253583	−	0.967314i	$-0.418391\pi$
$313$	8.97270	0.507167	0.253583	+	0.967314i	$0.418391\pi$
$314$	0	0
$315$	4.66764	0.262992
$316$	0	0
$317$	20.4572	1.14899	0.574497	−	0.818507i	$-0.305198\pi$
$317$	20.4572	1.14899	0.574497	+	0.818507i	$0.305198\pi$
$318$	0	0
$319$	0.0459997	0.00257549
$320$	0	0
$321$	−26.9233	−1.50271
$322$	0	0
$323$	2.98397	0.166033
$324$	0	0
$325$	−3.37279	−0.187088
$326$	0	0
$327$	4.81514	0.266278
$328$	0	0
$329$	4.71073	0.259711
$330$	0	0
$331$	−4.60655	−0.253199	−0.126600	−	0.991954i	$-0.540406\pi$
$331$	−4.60655	−0.253199	−0.126600	+	0.991954i	$0.540406\pi$
$332$	0	0
$333$	8.26804	0.453086
$334$	0	0
$335$	20.6027	1.12565
$336$	0	0
$337$	30.3417	1.65282	0.826408	−	0.563072i	$-0.190381\pi$
$337$	30.3417	1.65282	0.826408	+	0.563072i	$0.190381\pi$
$338$	0	0
$339$	−32.7985	−1.78137
$340$	0	0
$341$	2.09200	0.113288
$342$	0	0
$343$	7.80045	0.421184
$344$	0	0
$345$	80.1941	4.31751
$346$	0	0
$347$	−33.9089	−1.82032	−0.910162	−	0.414253i	$-0.864043\pi$
$347$	−33.9089	−1.82032	−0.910162	+	0.414253i	$0.864043\pi$
$348$	0	0
$349$	17.4188	0.932406	0.466203	−	0.884678i	$-0.345622\pi$
$349$	17.4188	0.932406	0.466203	+	0.884678i	$0.345622\pi$
$350$	0	0
$351$	0.627215	0.0334782
$352$	0	0
$353$	−6.96230	−0.370566	−0.185283	−	0.982685i	$-0.559320\pi$
$353$	−6.96230	−0.370566	−0.185283	+	0.982685i	$0.559320\pi$
$354$	0	0
$355$	−26.5821	−1.41083
$356$	0	0
$357$	3.63094	0.192170
$358$	0	0
$359$	35.3525	1.86583	0.932917	−	0.360092i	$-0.117255\pi$
$359$	35.3525	1.86583	0.932917	+	0.360092i	$0.117255\pi$
$360$	0	0
$361$	−17.9108	−0.942672
$362$	0	0
$363$	−23.9906	−1.25918
$364$	0	0
$365$	−60.0764	−3.14454
$366$	0	0
$367$	−31.0544	−1.62103	−0.810513	−	0.585720i	$-0.800812\pi$
$367$	−31.0544	−1.62103	−0.810513	+	0.585720i	$0.800812\pi$
$368$	0	0
$369$	−18.9268	−0.985290
$370$	0	0
$371$	−0.619158	−0.0321451
$372$	0	0
$373$	−9.46841	−0.490256	−0.245128	−	0.969491i	$-0.578830\pi$
$373$	−9.46841	−0.490256	−0.245128	+	0.969491i	$0.578830\pi$
$374$	0	0
$375$	−69.7849	−3.60368
$376$	0	0
$377$	0.0262398	0.00135142
$378$	0	0
$379$	5.00234	0.256953	0.128476	−	0.991713i	$-0.458991\pi$
$379$	5.00234	0.256953	0.128476	+	0.991713i	$0.458991\pi$
$380$	0	0
$381$	19.3341	0.990517
$382$	0	0
$383$	−23.5413	−1.20291	−0.601453	−	0.798908i	$-0.705411\pi$
$383$	−23.5413	−1.20291	−0.601453	+	0.798908i	$0.705411\pi$
$384$	0	0
$385$	−1.12908	−0.0575432
$386$	0	0
$387$	−1.85159	−0.0941217
$388$	0	0
$389$	−37.9214	−1.92269	−0.961346	−	0.275343i	$-0.911208\pi$
$389$	−37.9214	−1.92269	−0.961346	+	0.275343i	$0.911208\pi$
$390$	0	0
$391$	24.6234	1.24526
$392$	0	0
$393$	−38.4488	−1.93948
$394$	0	0
$395$	48.9267	2.46177
$396$	0	0
$397$	−31.9712	−1.60459	−0.802295	−	0.596928i	$-0.796388\pi$
$397$	−31.9712	−1.60459	−0.802295	+	0.596928i	$0.796388\pi$
$398$	0	0
$399$	1.32539	0.0663527
$400$	0	0
$401$	12.1033	0.604409	0.302204	−	0.953243i	$-0.402278\pi$
$401$	12.1033	0.604409	0.302204	+	0.953243i	$0.402278\pi$
$402$	0	0
$403$	1.19335	0.0594448
$404$	0	0
$405$	46.1835	2.29488
$406$	0	0
$407$	−2.00000	−0.0991363
$408$	0	0
$409$	−23.9849	−1.18598	−0.592989	−	0.805211i	$-0.702052\pi$
$409$	−23.9849	−1.18598	−0.592989	+	0.805211i	$0.702052\pi$
$410$	0	0
$411$	25.8155	1.27338
$412$	0	0
$413$	−6.00051	−0.295266
$414$	0	0
$415$	−55.2712	−2.71316
$416$	0	0
$417$	0.775623	0.0379824
$418$	0	0
$419$	−2.04600	−0.0999536	−0.0499768	−	0.998750i	$-0.515915\pi$
$419$	−2.04600	−0.0999536	−0.0499768	+	0.998750i	$0.515915\pi$
$420$	0	0
$421$	−13.1732	−0.642021	−0.321011	−	0.947076i	$-0.604023\pi$
$421$	−13.1732	−0.642021	−0.321011	+	0.947076i	$0.604023\pi$
$422$	0	0
$423$	−16.1558	−0.785519
$424$	0	0
$425$	−35.7229	−1.73282
$426$	0	0
$427$	0.570433	0.0276052
$428$	0	0
$429$	0.284399	0.0137309
$430$	0	0
$431$	−20.0224	−0.964446	−0.482223	−	0.876048i	$-0.660171\pi$
$431$	−20.0224	−0.964446	−0.482223	+	0.876048i	$0.660171\pi$
$432$	0	0
$433$	−16.5446	−0.795081	−0.397541	−	0.917585i	$-0.630136\pi$
$433$	−16.5446	−0.795081	−0.397541	+	0.917585i	$0.630136\pi$
$434$	0	0
$435$	0.905134	0.0433978
$436$	0	0
$437$	8.98821	0.429965
$438$	0	0
$439$	−10.9051	−0.520474	−0.260237	−	0.965545i	$-0.583801\pi$
$439$	−10.9051	−0.520474	−0.260237	+	0.965545i	$0.583801\pi$
$440$	0	0
$441$	−13.0578	−0.621799
$442$	0	0
$443$	−2.98215	−0.141686	−0.0708430	−	0.997487i	$-0.522569\pi$
$443$	−2.98215	−0.141686	−0.0708430	+	0.997487i	$0.522569\pi$
$444$	0	0
$445$	−34.6234	−1.64131
$446$	0	0
$447$	33.1568	1.56826
$448$	0	0
$449$	5.31404	0.250785	0.125392	−	0.992107i	$-0.459981\pi$
$449$	5.31404	0.250785	0.125392	+	0.992107i	$0.459981\pi$
$450$	0	0
$451$	4.57830	0.215584
$452$	0	0
$453$	−19.8839	−0.934227
$454$	0	0
$455$	−0.644064	−0.0301942
$456$	0	0
$457$	0.732290	0.0342551	0.0171275	−	0.999853i	$-0.494548\pi$
$457$	0.732290	0.0342551	0.0171275	+	0.999853i	$0.494548\pi$
$458$	0	0
$459$	6.64316	0.310076
$460$	0	0
$461$	−6.02825	−0.280764	−0.140382	−	0.990097i	$-0.544833\pi$
$461$	−6.02825	−0.280764	−0.140382	+	0.990097i	$0.544833\pi$
$462$	0	0
$463$	−10.8816	−0.505709	−0.252855	−	0.967504i	$-0.581369\pi$
$463$	−10.8816	−0.505709	−0.252855	+	0.967504i	$0.581369\pi$
$464$	0	0
$465$	41.1642	1.90894
$466$	0	0
$467$	33.9540	1.57120	0.785601	−	0.618733i	$-0.212354\pi$
$467$	33.9540	1.57120	0.785601	+	0.618733i	$0.212354\pi$
$468$	0	0
$469$	2.80984	0.129746
$470$	0	0
$471$	36.2173	1.66880
$472$	0	0
$473$	0.447891	0.0205941
$474$	0	0
$475$	−13.0398	−0.598309
$476$	0	0
$477$	2.12344	0.0972258
$478$	0	0
$479$	29.6778	1.35601	0.678007	−	0.735056i	$-0.262844\pi$
$479$	29.6778	1.35601	0.678007	+	0.735056i	$0.262844\pi$
$480$	0	0
$481$	−1.14087	−0.0520190
$482$	0	0
$483$	10.9370	0.497651
$484$	0	0
$485$	−75.2477	−3.41682
$486$	0	0
$487$	16.4092	0.743573	0.371787	−	0.928318i	$-0.378745\pi$
$487$	16.4092	0.743573	0.371787	+	0.928318i	$0.378745\pi$
$488$	0	0
$489$	−33.6868	−1.52337
$490$	0	0
$491$	−16.7666	−0.756666	−0.378333	−	0.925669i	$-0.623503\pi$
$491$	−16.7666	−0.756666	−0.378333	+	0.925669i	$0.623503\pi$
$492$	0	0
$493$	0.277919	0.0125168
$494$	0	0
$495$	3.87225	0.174045
$496$	0	0
$497$	−3.62531	−0.162617
$498$	0	0
$499$	33.8094	1.51352	0.756758	−	0.653696i	$-0.226782\pi$
$499$	33.8094	1.51352	0.756758	+	0.653696i	$0.226782\pi$
$500$	0	0
$501$	−22.8816	−1.02227
$502$	0	0
$503$	−6.80006	−0.303200	−0.151600	−	0.988442i	$-0.548442\pi$
$503$	−6.80006	−0.303200	−0.151600	+	0.988442i	$0.548442\pi$
$504$	0	0
$505$	−70.9237	−3.15607
$506$	0	0
$507$	−28.7795	−1.27814
$508$	0	0
$509$	−19.4882	−0.863798	−0.431899	−	0.901922i	$-0.642156\pi$
$509$	−19.4882	−0.863798	−0.431899	+	0.901922i	$0.642156\pi$
$510$	0	0
$511$	−8.19332	−0.362451
$512$	0	0
$513$	2.42494	0.107064
$514$	0	0
$515$	−34.1755	−1.50595
$516$	0	0
$517$	3.90800	0.171874
$518$	0	0
$519$	−43.3956	−1.90486
$520$	0	0
$521$	−31.7445	−1.39075	−0.695376	−	0.718646i	$-0.744762\pi$
$521$	−31.7445	−1.39075	−0.695376	+	0.718646i	$0.744762\pi$
$522$	0	0
$523$	−8.82252	−0.385782	−0.192891	−	0.981220i	$-0.561786\pi$
$523$	−8.82252	−0.385782	−0.192891	+	0.981220i	$0.561786\pi$
$524$	0	0
$525$	−15.8671	−0.692497
$526$	0	0
$527$	12.6393	0.550579
$528$	0	0
$529$	51.1698	2.22477
$530$	0	0
$531$	20.5792	0.893060
$532$	0	0
$533$	2.61162	0.113122
$534$	0	0
$535$	50.5821	2.18685
$536$	0	0
$537$	−18.3580	−0.792207
$538$	0	0
$539$	3.15862	0.136051
$540$	0	0
$541$	6.18115	0.265748	0.132874	−	0.991133i	$-0.457579\pi$
$541$	6.18115	0.265748	0.132874	+	0.991133i	$0.457579\pi$
$542$	0	0
$543$	4.51408	0.193718
$544$	0	0
$545$	−9.04643	−0.387507
$546$	0	0
$547$	−10.2835	−0.439691	−0.219845	−	0.975535i	$-0.570555\pi$
$547$	−10.2835	−0.439691	−0.219845	+	0.975535i	$0.570555\pi$
$548$	0	0
$549$	−1.95634	−0.0834945
$550$	0	0
$551$	0.101448	0.00432183
$552$	0	0
$553$	6.67270	0.283752
$554$	0	0
$555$	−39.3539	−1.67048
$556$	0	0
$557$	28.0835	1.18994	0.594968	−	0.803749i	$-0.297165\pi$
$557$	28.0835	1.18994	0.594968	+	0.803749i	$0.297165\pi$
$558$	0	0
$559$	0.255492	0.0108062
$560$	0	0
$561$	3.01222	0.127176
$562$	0	0
$563$	25.4037	1.07064	0.535319	−	0.844650i	$-0.320191\pi$
$563$	25.4037	1.07064	0.535319	+	0.844650i	$0.320191\pi$
$564$	0	0
$565$	61.6200	2.59237
$566$	0	0
$567$	6.29858	0.264516
$568$	0	0
$569$	25.3046	1.06082	0.530412	−	0.847740i	$-0.322037\pi$
$569$	25.3046	1.06082	0.530412	+	0.847740i	$0.322037\pi$
$570$	0	0
$571$	−10.2027	−0.426970	−0.213485	−	0.976946i	$-0.568481\pi$
$571$	−10.2027	−0.426970	−0.213485	+	0.976946i	$0.568481\pi$
$572$	0	0
$573$	−27.3676	−1.14330
$574$	0	0
$575$	−107.603	−4.48737
$576$	0	0
$577$	19.0937	0.794882	0.397441	−	0.917628i	$-0.369898\pi$
$577$	19.0937	0.794882	0.397441	+	0.917628i	$0.369898\pi$
$578$	0	0
$579$	−29.9239	−1.24360
$580$	0	0
$581$	−7.53798	−0.312728
$582$	0	0
$583$	−0.513651	−0.0212733
$584$	0	0
$585$	2.20886	0.0913252
$586$	0	0
$587$	−10.6930	−0.441346	−0.220673	−	0.975348i	$-0.570825\pi$
$587$	−10.6930	−0.441346	−0.220673	+	0.975348i	$0.570825\pi$
$588$	0	0
$589$	4.61371	0.190105
$590$	0	0
$591$	23.2441	0.956136
$592$	0	0
$593$	4.60946	0.189288	0.0946440	−	0.995511i	$-0.469829\pi$
$593$	4.60946	0.189288	0.0946440	+	0.995511i	$0.469829\pi$
$594$	0	0
$595$	−6.82162	−0.279659
$596$	0	0
$597$	32.8966	1.34637
$598$	0	0
$599$	−32.0864	−1.31102	−0.655508	−	0.755188i	$-0.727545\pi$
$599$	−32.0864	−1.31102	−0.655508	+	0.755188i	$0.727545\pi$
$600$	0	0
$601$	−14.4765	−0.590507	−0.295254	−	0.955419i	$-0.595404\pi$
$601$	−14.4765	−0.590507	−0.295254	+	0.955419i	$0.595404\pi$
$602$	0	0
$603$	−9.63652	−0.392430
$604$	0	0
$605$	45.0722	1.83244
$606$	0	0
$607$	−12.5859	−0.510846	−0.255423	−	0.966829i	$-0.582215\pi$
$607$	−12.5859	−0.510846	−0.255423	+	0.966829i	$0.582215\pi$
$608$	0	0
$609$	0.123444	0.00500219
$610$	0	0
$611$	2.22925	0.0901859
$612$	0	0
$613$	−31.3775	−1.26732	−0.633662	−	0.773610i	$-0.718449\pi$
$613$	−31.3775	−1.26732	−0.633662	+	0.773610i	$0.718449\pi$
$614$	0	0
$615$	90.0870	3.63266
$616$	0	0
$617$	20.2826	0.816547	0.408273	−	0.912860i	$-0.366131\pi$
$617$	20.2826	0.816547	0.408273	+	0.912860i	$0.366131\pi$
$618$	0	0
$619$	−7.37096	−0.296264	−0.148132	−	0.988968i	$-0.547326\pi$
$619$	−7.37096	−0.296264	−0.148132	+	0.988968i	$0.547326\pi$
$620$	0	0
$621$	20.0103	0.802986
$622$	0	0
$623$	−4.72200	−0.189183
$624$	0	0
$625$	68.6365	2.74546
$626$	0	0
$627$	1.09954	0.0439115
$628$	0	0
$629$	−12.0835	−0.481801
$630$	0	0
$631$	18.0943	0.720322	0.360161	−	0.932890i	$-0.382722\pi$
$631$	18.0943	0.720322	0.360161	+	0.932890i	$0.382722\pi$
$632$	0	0
$633$	8.17551	0.324947
$634$	0	0
$635$	−36.3239	−1.44147
$636$	0	0
$637$	1.80178	0.0713891
$638$	0	0
$639$	12.4332	0.491851
$640$	0	0
$641$	−12.9485	−0.511434	−0.255717	−	0.966752i	$-0.582311\pi$
$641$	−12.9485	−0.511434	−0.255717	+	0.966752i	$0.582311\pi$
$642$	0	0
$643$	−2.30416	−0.0908672	−0.0454336	−	0.998967i	$-0.514467\pi$
$643$	−2.30416	−0.0908672	−0.0454336	+	0.998967i	$0.514467\pi$
$644$	0	0
$645$	8.81313	0.347017
$646$	0	0
$647$	28.8169	1.13291	0.566455	−	0.824093i	$-0.308315\pi$
$647$	28.8169	1.13291	0.566455	+	0.824093i	$0.308315\pi$
$648$	0	0
$649$	−4.97800	−0.195404
$650$	0	0
$651$	5.61404	0.220031
$652$	0	0
$653$	37.5797	1.47061	0.735304	−	0.677737i	$-0.237039\pi$
$653$	37.5797	1.47061	0.735304	+	0.677737i	$0.237039\pi$
$654$	0	0
$655$	72.2354	2.82247
$656$	0	0
$657$	28.0995	1.09627
$658$	0	0
$659$	−6.12301	−0.238519	−0.119259	−	0.992863i	$-0.538052\pi$
$659$	−6.12301	−0.238519	−0.119259	+	0.992863i	$0.538052\pi$
$660$	0	0
$661$	10.0319	0.390195	0.195097	−	0.980784i	$-0.437498\pi$
$661$	10.0319	0.390195	0.195097	+	0.980784i	$0.437498\pi$
$662$	0	0
$663$	1.71827	0.0667320
$664$	0	0
$665$	−2.49008	−0.0965611
$666$	0	0
$667$	0.837138	0.0324141
$668$	0	0
$669$	−39.6383	−1.53250
$670$	0	0
$671$	0.473229	0.0182688
$672$	0	0
$673$	−5.65718	−0.218068	−0.109034	−	0.994038i	$-0.534776\pi$
$673$	−5.65718	−0.218068	−0.109034	+	0.994038i	$0.534776\pi$
$674$	0	0
$675$	−29.0304	−1.11738
$676$	0	0
$677$	40.2582	1.54725	0.773623	−	0.633646i	$-0.218442\pi$
$677$	40.2582	1.54725	0.773623	+	0.633646i	$0.218442\pi$
$678$	0	0
$679$	−10.2624	−0.393835
$680$	0	0
$681$	−38.5783	−1.47832
$682$	0	0
$683$	−34.5460	−1.32186	−0.660932	−	0.750446i	$-0.729839\pi$
$683$	−34.5460	−1.32186	−0.660932	+	0.750446i	$0.729839\pi$
$684$	0	0
$685$	−48.5007	−1.85312
$686$	0	0
$687$	16.3962	0.625552
$688$	0	0
$689$	−0.293004	−0.0111625
$690$	0	0
$691$	6.99342	0.266042	0.133021	−	0.991113i	$-0.457532\pi$
$691$	6.99342	0.266042	0.133021	+	0.991113i	$0.457532\pi$
$692$	0	0
$693$	0.528105	0.0200610
$694$	0	0
$695$	−1.45720	−0.0552747
$696$	0	0
$697$	27.6610	1.04773
$698$	0	0
$699$	−26.8618	−1.01601
$700$	0	0
$701$	−6.98961	−0.263994	−0.131997	−	0.991250i	$-0.542139\pi$
$701$	−6.98961	−0.263994	−0.131997	+	0.991250i	$0.542139\pi$
$702$	0	0
$703$	−4.41081	−0.166357
$704$	0	0
$705$	76.8975	2.89613
$706$	0	0
$707$	−9.67270	−0.363779
$708$	0	0
$709$	2.13332	0.0801187	0.0400593	−	0.999197i	$-0.487245\pi$
$709$	2.13332	0.0801187	0.0400593	+	0.999197i	$0.487245\pi$
$710$	0	0
$711$	−22.8845	−0.858235
$712$	0	0
$713$	38.0718	1.42580
$714$	0	0
$715$	−0.534313	−0.0199822
$716$	0	0
$717$	−46.0495	−1.71975
$718$	0	0
$719$	−37.0544	−1.38190	−0.690948	−	0.722904i	$-0.742807\pi$
$719$	−37.0544	−1.38190	−0.690948	+	0.722904i	$0.742807\pi$
$720$	0	0
$721$	−4.66091	−0.173581
$722$	0	0
$723$	−3.02687	−0.112571
$724$	0	0
$725$	−1.21450	−0.0451053
$726$	0	0
$727$	41.6356	1.54418	0.772090	−	0.635513i	$-0.219212\pi$
$727$	41.6356	1.54418	0.772090	+	0.635513i	$0.219212\pi$
$728$	0	0
$729$	−6.08318	−0.225303
$730$	0	0
$731$	2.70605	0.100087
$732$	0	0
$733$	14.0840	0.520205	0.260103	−	0.965581i	$-0.416244\pi$
$733$	14.0840	0.520205	0.260103	+	0.965581i	$0.416244\pi$
$734$	0	0
$735$	62.1519	2.29251
$736$	0	0
$737$	2.33103	0.0858646
$738$	0	0
$739$	−33.6141	−1.23651	−0.618257	−	0.785976i	$-0.712161\pi$
$739$	−33.6141	−1.23651	−0.618257	+	0.785976i	$0.712161\pi$
$740$	0	0
$741$	0.627215	0.0230413
$742$	0	0
$743$	−12.7985	−0.469533	−0.234766	−	0.972052i	$-0.575433\pi$
$743$	−12.7985	−0.469533	−0.234766	+	0.972052i	$0.575433\pi$
$744$	0	0
$745$	−62.2932	−2.28225
$746$	0	0
$747$	25.8520	0.945876
$748$	0	0
$749$	6.89847	0.252065
$750$	0	0
$751$	−3.84692	−0.140376	−0.0701880	−	0.997534i	$-0.522360\pi$
$751$	−3.84692	−0.140376	−0.0701880	+	0.997534i	$0.522360\pi$
$752$	0	0
$753$	30.3068	1.10444
$754$	0	0
$755$	37.3568	1.35955
$756$	0	0
$757$	−20.5564	−0.747133	−0.373567	−	0.927603i	$-0.621865\pi$
$757$	−20.5564	−0.747133	−0.373567	+	0.927603i	$0.621865\pi$
$758$	0	0
$759$	9.07330	0.329340
$760$	0	0
$761$	45.9389	1.66528	0.832642	−	0.553811i	$-0.186827\pi$
$761$	45.9389	1.66528	0.832642	+	0.553811i	$0.186827\pi$
$762$	0	0
$763$	−1.23377	−0.0446654
$764$	0	0
$765$	23.3952	0.845856
$766$	0	0
$767$	−2.83962	−0.102533
$768$	0	0
$769$	16.9276	0.610423	0.305212	−	0.952285i	$-0.401273\pi$
$769$	16.9276	0.610423	0.305212	+	0.952285i	$0.401273\pi$
$770$	0	0
$771$	−15.6916	−0.565119
$772$	0	0
$773$	−45.3539	−1.63127	−0.815633	−	0.578570i	$-0.803611\pi$
$773$	−45.3539	−1.63127	−0.815633	+	0.578570i	$0.803611\pi$
$774$	0	0
$775$	−55.2335	−1.98405
$776$	0	0
$777$	−5.36715	−0.192545
$778$	0	0
$779$	10.0970	0.361763
$780$	0	0
$781$	−3.00754	−0.107618
$782$	0	0
$783$	0.225852	0.00807129
$784$	0	0
$785$	−68.0430	−2.42856
$786$	0	0
$787$	−3.21121	−0.114467	−0.0572337	−	0.998361i	$-0.518228\pi$
$787$	−3.21121	−0.114467	−0.0572337	+	0.998361i	$0.518228\pi$
$788$	0	0
$789$	31.5587	1.12352
$790$	0	0
$791$	8.40384	0.298806
$792$	0	0
$793$	0.269945	0.00958604
$794$	0	0
$795$	−10.1071	−0.358461
$796$	0	0
$797$	−44.6913	−1.58305	−0.791524	−	0.611138i	$-0.790712\pi$
$797$	−44.6913	−1.58305	−0.791524	+	0.611138i	$0.790712\pi$
$798$	0	0
$799$	23.6112	0.835304
$800$	0	0
$801$	16.1944	0.572201
$802$	0	0
$803$	−6.79715	−0.239866
$804$	0	0
$805$	−20.5479	−0.724217
$806$	0	0
$807$	20.2497	0.712822
$808$	0	0
$809$	33.4308	1.17536	0.587682	−	0.809092i	$-0.300040\pi$
$809$	33.4308	1.17536	0.587682	+	0.809092i	$0.300040\pi$
$810$	0	0
$811$	−48.0355	−1.68675	−0.843377	−	0.537323i	$-0.819436\pi$
$811$	−48.0355	−1.68675	−0.843377	+	0.537323i	$0.819436\pi$
$812$	0	0
$813$	−18.3455	−0.643404
$814$	0	0
$815$	63.2890	2.21692
$816$	0	0
$817$	0.987782	0.0345581
$818$	0	0
$819$	0.301248	0.0105265
$820$	0	0
$821$	2.75000	0.0959757	0.0479878	−	0.998848i	$-0.484719\pi$
$821$	2.75000	0.0959757	0.0479878	+	0.998848i	$0.484719\pi$
$822$	0	0
$823$	43.6614	1.52194	0.760971	−	0.648786i	$-0.224723\pi$
$823$	43.6614	1.52194	0.760971	+	0.648786i	$0.224723\pi$
$824$	0	0
$825$	−13.1633	−0.458287
$826$	0	0
$827$	12.5022	0.434744	0.217372	−	0.976089i	$-0.430252\pi$
$827$	12.5022	0.434744	0.217372	+	0.976089i	$0.430252\pi$
$828$	0	0
$829$	15.0164	0.521540	0.260770	−	0.965401i	$-0.416024\pi$
$829$	15.0164	0.521540	0.260770	+	0.965401i	$0.416024\pi$
$830$	0	0
$831$	−1.84267	−0.0639216
$832$	0	0
$833$	19.0836	0.661207
$834$	0	0
$835$	42.9886	1.48768
$836$	0	0
$837$	10.2714	0.355032
$838$	0	0
$839$	−46.0073	−1.58835	−0.794175	−	0.607689i	$-0.792097\pi$
$839$	−46.0073	−1.58835	−0.794175	+	0.607689i	$0.792097\pi$
$840$	0	0
$841$	−28.9906	−0.999674
$842$	0	0
$843$	−48.9447	−1.68574
$844$	0	0
$845$	54.0693	1.86004
$846$	0	0
$847$	6.14702	0.211214
$848$	0	0
$849$	−13.2427	−0.454490
$850$	0	0
$851$	−36.3975	−1.24769
$852$	0	0
$853$	−44.1722	−1.51243	−0.756214	−	0.654325i	$-0.772953\pi$
$853$	−44.1722	−1.51243	−0.756214	+	0.654325i	$0.772953\pi$
$854$	0	0
$855$	8.53989	0.292058
$856$	0	0
$857$	−1.16901	−0.0399327	−0.0199663	−	0.999801i	$-0.506356\pi$
$857$	−1.16901	−0.0399327	−0.0199663	+	0.999801i	$0.506356\pi$
$858$	0	0
$859$	−12.5004	−0.426508	−0.213254	−	0.976997i	$-0.568406\pi$
$859$	−12.5004	−0.426508	−0.213254	+	0.976997i	$0.568406\pi$
$860$	0	0
$861$	12.2862	0.418713
$862$	0	0
$863$	−28.3332	−0.964473	−0.482236	−	0.876041i	$-0.660175\pi$
$863$	−28.3332	−0.964473	−0.482236	+	0.876041i	$0.660175\pi$
$864$	0	0
$865$	81.5294	2.77208
$866$	0	0
$867$	−19.6477	−0.667272
$868$	0	0
$869$	5.53565	0.187784
$870$	0	0
$871$	1.32970	0.0450550
$872$	0	0
$873$	35.1956	1.19119
$874$	0	0
$875$	17.8807	0.604479
$876$	0	0
$877$	−25.8897	−0.874234	−0.437117	−	0.899405i	$-0.644000\pi$
$877$	−25.8897	−0.874234	−0.437117	+	0.899405i	$0.644000\pi$
$878$	0	0
$879$	27.2377	0.918703
$880$	0	0
$881$	13.7210	0.462273	0.231137	−	0.972921i	$-0.425756\pi$
$881$	13.7210	0.462273	0.231137	+	0.972921i	$0.425756\pi$
$882$	0	0
$883$	−41.3981	−1.39316	−0.696579	−	0.717480i	$-0.745295\pi$
$883$	−41.3981	−1.39316	−0.696579	+	0.717480i	$0.745295\pi$
$884$	0	0
$885$	−97.9519	−3.29262
$886$	0	0
$887$	−50.9633	−1.71118	−0.855590	−	0.517655i	$-0.826805\pi$
$887$	−50.9633	−1.71118	−0.855590	+	0.517655i	$0.826805\pi$
$888$	0	0
$889$	−4.95392	−0.166149
$890$	0	0
$891$	5.22528	0.175053
$892$	0	0
$893$	8.61873	0.288415
$894$	0	0
$895$	34.4901	1.15288
$896$	0	0
$897$	5.17571	0.172812
$898$	0	0
$899$	0.429709	0.0143316
$900$	0	0
$901$	−3.10335	−0.103388
$902$	0	0
$903$	1.20195	0.0399984
$904$	0	0
$905$	−8.48082	−0.281912
$906$	0	0
$907$	31.5614	1.04798	0.523989	−	0.851725i	$-0.324443\pi$
$907$	31.5614	1.04798	0.523989	+	0.851725i	$0.324443\pi$
$908$	0	0
$909$	33.1732	1.10028
$910$	0	0
$911$	−2.98319	−0.0988376	−0.0494188	−	0.998778i	$-0.515737\pi$
$911$	−2.98319	−0.0988376	−0.0494188	+	0.998778i	$0.515737\pi$
$912$	0	0
$913$	−6.25348	−0.206960
$914$	0	0
$915$	9.31170	0.307835
$916$	0	0
$917$	9.85159	0.325328
$918$	0	0
$919$	15.7971	0.521098	0.260549	−	0.965461i	$-0.416096\pi$
$919$	15.7971	0.521098	0.260549	+	0.965461i	$0.416096\pi$
$920$	0	0
$921$	68.4272	2.25475
$922$	0	0
$923$	−1.71560	−0.0564697
$924$	0	0
$925$	52.8045	1.73620
$926$	0	0
$927$	15.9849	0.525014
$928$	0	0
$929$	51.8164	1.70004	0.850021	−	0.526749i	$-0.176589\pi$
$929$	51.8164	1.70004	0.850021	+	0.526749i	$0.176589\pi$
$930$	0	0
$931$	6.96603	0.228303
$932$	0	0
$933$	13.6116	0.445624
$934$	0	0
$935$	−5.65919	−0.185075
$936$	0	0
$937$	−7.97366	−0.260488	−0.130244	−	0.991482i	$-0.541576\pi$
$937$	−7.97366	−0.260488	−0.130244	+	0.991482i	$0.541576\pi$
$938$	0	0
$939$	19.9758	0.651885
$940$	0	0
$941$	−4.19326	−0.136696	−0.0683482	−	0.997662i	$-0.521773\pi$
$941$	−4.19326	−0.136696	−0.0683482	+	0.997662i	$0.521773\pi$
$942$	0	0
$943$	83.3195	2.71326
$944$	0	0
$945$	−5.54362	−0.180334
$946$	0	0
$947$	22.7558	0.739465	0.369732	−	0.929138i	$-0.379449\pi$
$947$	22.7558	0.739465	0.369732	+	0.929138i	$0.379449\pi$
$948$	0	0
$949$	−3.87732	−0.125863
$950$	0	0
$951$	45.5436	1.47685
$952$	0	0
$953$	−48.4514	−1.56950	−0.784748	−	0.619815i	$-0.787208\pi$
$953$	−48.4514	−1.56950	−0.784748	+	0.619815i	$0.787208\pi$
$954$	0	0
$955$	51.4167	1.66381
$956$	0	0
$957$	0.102408	0.00331039
$958$	0	0
$959$	−6.61461	−0.213597
$960$	0	0
$961$	−11.4575	−0.369596
$962$	0	0
$963$	−23.6588	−0.762393
$964$	0	0
$965$	56.2195	1.80977
$966$	0	0
$967$	−15.2244	−0.489583	−0.244791	−	0.969576i	$-0.578720\pi$
$967$	−15.2244	−0.489583	−0.244791	+	0.969576i	$0.578720\pi$
$968$	0	0
$969$	6.64316	0.213409
$970$	0	0
$971$	24.8733	0.798221	0.399110	−	0.916903i	$-0.369319\pi$
$971$	24.8733	0.798221	0.399110	+	0.916903i	$0.369319\pi$
$972$	0	0
$973$	−0.198735	−0.00637116
$974$	0	0
$975$	−7.50878	−0.240473
$976$	0	0
$977$	−23.7371	−0.759416	−0.379708	−	0.925106i	$-0.623976\pi$
$977$	−23.7371	−0.759416	−0.379708	+	0.925106i	$0.623976\pi$
$978$	0	0
$979$	−3.91735	−0.125199
$980$	0	0
$981$	4.23129	0.135095
$982$	0	0
$983$	−10.0281	−0.319846	−0.159923	−	0.987130i	$-0.551125\pi$
$983$	−10.0281	−0.319846	−0.159923	+	0.987130i	$0.551125\pi$
$984$	0	0
$985$	−43.6698	−1.39144
$986$	0	0
$987$	10.4874	0.333818
$988$	0	0
$989$	8.15108	0.259189
$990$	0	0
$991$	31.0602	0.986661	0.493330	−	0.869842i	$-0.335779\pi$
$991$	31.0602	0.986661	0.493330	+	0.869842i	$0.335779\pi$
$992$	0	0
$993$	−10.2555	−0.325448
$994$	0	0
$995$	−61.8044	−1.95933
$996$	0	0
$997$	27.8940	0.883411	0.441705	−	0.897160i	$-0.354374\pi$
$997$	27.8940	0.883411	0.441705	+	0.897160i	$0.354374\pi$
$998$	0	0
$999$	−9.81972	−0.310682

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	976.2.a.i.1.4		4
3.2	odd	2		8784.2.a.bv.1.4		4
4.3	odd	2		488.2.a.c.1.1	✓	4
8.3	odd	2		3904.2.a.y.1.4		4
8.5	even	2		3904.2.a.bf.1.1		4
12.11	even	2		4392.2.a.n.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
488.2.a.c.1.1	✓	4	4.3	odd	2
976.2.a.i.1.4		4	1.1	even	1	trivial
3904.2.a.y.1.4		4	8.3	odd	2
3904.2.a.bf.1.1		4	8.5	even	2
4392.2.a.n.1.4		4	12.11	even	2
8784.2.a.bv.1.4		4	3.2	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:	\( N \)	\(=\)	\( 976 = 2^{4} \cdot 61 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	976.a (trivial)

\(f(q)\)	\(=\)	\(q+2.22628 q^{3} -4.18262 q^{5} -0.570433 q^{7} +1.95634 q^{9} +O(q^{10})\)	\(q+2.22628 q^{3} -4.18262 q^{5} -0.570433 q^{7} +1.95634 q^{9} -0.473229 q^{11} -0.269945 q^{13} -9.31170 q^{15} -2.85913 q^{17} -1.04366 q^{19} -1.26995 q^{21} -8.61219 q^{23} +12.4943 q^{25} -2.32349 q^{27} -0.0972039 q^{29} -4.42069 q^{31} -1.05354 q^{33} +2.38591 q^{35} +4.22628 q^{37} -0.600975 q^{39} -9.67461 q^{41} -0.946458 q^{43} -8.18262 q^{45} -8.25816 q^{47} -6.67461 q^{49} -6.36524 q^{51} +1.08542 q^{53} +1.97934 q^{55} -2.32349 q^{57} +10.5192 q^{59} -1.00000 q^{61} -1.11596 q^{63} +1.12908 q^{65} -4.92580 q^{67} -19.1732 q^{69} +6.35536 q^{71} +14.3633 q^{73} +27.8159 q^{75} +0.269945 q^{77} -11.6976 q^{79} -11.0418 q^{81} +13.2145 q^{83} +11.9587 q^{85} -0.216403 q^{87} +8.27792 q^{89} +0.153986 q^{91} -9.84171 q^{93} +4.36524 q^{95} +17.9906 q^{97} -0.925796 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - q^{3} - 2 q^{5} - 7 q^{7} + 3 q^{9}+O(q^{10}) \) 4 * q - q^3 - 2 * q^5 - 7 * q^7 + 3 * q^9	\( 4 q - q^{3} - 2 q^{5} - 7 q^{7} + 3 q^{9} - 2 q^{11} + 4 q^{13} - 8 q^{15} - 2 q^{17} - 9 q^{19} - 15 q^{23} + 6 q^{25} - 4 q^{27} - 5 q^{29} - 17 q^{31} - 4 q^{33} + 7 q^{37} - 22 q^{39} - 15 q^{41} - 4 q^{43} - 18 q^{45} - 4 q^{47} - 3 q^{49} + 4 q^{51} - 15 q^{53} - 12 q^{55} - 4 q^{57} + 12 q^{59} - 4 q^{61} - 10 q^{65} - 3 q^{69} + q^{71} - q^{73} + 33 q^{75} - 4 q^{77} - 8 q^{79} - 20 q^{81} + 19 q^{83} + 8 q^{85} + 4 q^{87} - 6 q^{89} - 5 q^{93} - 12 q^{95} + 13 q^{97} + 16 q^{99}+O(q^{100}) \) 4 * q - q^3 - 2 * q^5 - 7 * q^7 + 3 * q^9 - 2 * q^11 + 4 * q^13 - 8 * q^15 - 2 * q^17 - 9 * q^19 - 15 * q^23 + 6 * q^25 - 4 * q^27 - 5 * q^29 - 17 * q^31 - 4 * q^33 + 7 * q^37 - 22 * q^39 - 15 * q^41 - 4 * q^43 - 18 * q^45 - 4 * q^47 - 3 * q^49 + 4 * q^51 - 15 * q^53 - 12 * q^55 - 4 * q^57 + 12 * q^59 - 4 * q^61 - 10 * q^65 - 3 * q^69 + q^71 - q^73 + 33 * q^75 - 4 * q^77 - 8 * q^79 - 20 * q^81 + 19 * q^83 + 8 * q^85 + 4 * q^87 - 6 * q^89 - 5 * q^93 - 12 * q^95 + 13 * q^97 + 16 * q^99

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