LMFDB - Embedded newform 8024.2.a.bc.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8024,2,Mod(1,8024)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8024, 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("8024.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8024 = 2^{3} \cdot 17 \cdot 59 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8024.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:	$64.0719625819$
Analytic rank:	$0$
Dimension:	$33$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	8024.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.44977 q^{3} +2.91669 q^{5} -3.05603 q^{7} +8.90091 q^{9} +O(q^{10})$	$q-3.44977 q^{3} +2.91669 q^{5} -3.05603 q^{7} +8.90091 q^{9} -0.926697 q^{11} +0.486374 q^{13} -10.0619 q^{15} -1.00000 q^{17} -1.91883 q^{19} +10.5426 q^{21} +6.03687 q^{23} +3.50708 q^{25} -20.3568 q^{27} -4.39950 q^{29} +7.75169 q^{31} +3.19689 q^{33} -8.91351 q^{35} -10.3882 q^{37} -1.67788 q^{39} +10.1225 q^{41} -7.87176 q^{43} +25.9612 q^{45} +1.62814 q^{47} +2.33935 q^{49} +3.44977 q^{51} +11.4768 q^{53} -2.70289 q^{55} +6.61952 q^{57} -1.00000 q^{59} +5.75331 q^{61} -27.2015 q^{63} +1.41860 q^{65} -2.33655 q^{67} -20.8258 q^{69} +11.4554 q^{71} -11.6318 q^{73} -12.0986 q^{75} +2.83202 q^{77} +4.08954 q^{79} +43.5235 q^{81} +9.70282 q^{83} -2.91669 q^{85} +15.1773 q^{87} -16.3766 q^{89} -1.48638 q^{91} -26.7416 q^{93} -5.59663 q^{95} +4.28306 q^{97} -8.24845 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 33 q - 3 q^{3} + 3 q^{5} + 48 q^{9}+O(q^{10}) $ 33 * q - 3 * q^3 + 3 * q^5 + 48 * q^9	$ 33 q - 3 q^{3} + 3 q^{5} + 48 q^{9} - 7 q^{11} + 9 q^{13} - 7 q^{15} - 33 q^{17} + 5 q^{19} + 14 q^{21} - 19 q^{23} + 52 q^{25} - 24 q^{27} + 14 q^{29} + 19 q^{31} + 34 q^{33} + 15 q^{35} + 13 q^{37} + 26 q^{39} + 32 q^{41} + 11 q^{43} - 27 q^{47} + 93 q^{49} + 3 q^{51} - 7 q^{53} + 31 q^{55} + 6 q^{57} - 33 q^{59} + 33 q^{61} + 10 q^{63} + 21 q^{65} - 10 q^{67} + 78 q^{69} - 29 q^{71} + 30 q^{73} + 31 q^{75} + 6 q^{77} + 66 q^{79} + 97 q^{81} - 9 q^{83} - 3 q^{85} - 27 q^{87} + 68 q^{89} + 28 q^{91} + 34 q^{93} + 8 q^{95} + 62 q^{97} + 15 q^{99}+O(q^{100}) $ 33 * q - 3 * q^3 + 3 * q^5 + 48 * q^9 - 7 * q^11 + 9 * q^13 - 7 * q^15 - 33 * q^17 + 5 * q^19 + 14 * q^21 - 19 * q^23 + 52 * q^25 - 24 * q^27 + 14 * q^29 + 19 * q^31 + 34 * q^33 + 15 * q^35 + 13 * q^37 + 26 * q^39 + 32 * q^41 + 11 * q^43 - 27 * q^47 + 93 * q^49 + 3 * q^51 - 7 * q^53 + 31 * q^55 + 6 * q^57 - 33 * q^59 + 33 * q^61 + 10 * q^63 + 21 * q^65 - 10 * q^67 + 78 * q^69 - 29 * q^71 + 30 * q^73 + 31 * q^75 + 6 * q^77 + 66 * q^79 + 97 * q^81 - 9 * q^83 - 3 * q^85 - 27 * q^87 + 68 * q^89 + 28 * q^91 + 34 * q^93 + 8 * q^95 + 62 * q^97 + 15 * 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
$3$	−3.44977	−1.99173	−0.995863	−	0.0908714i	$-0.971035\pi$
$3$	−3.44977	−1.99173	−0.995863	+	0.0908714i	$0.971035\pi$
$4$	0	0
$5$	2.91669	1.30438	0.652192	−	0.758054i	$-0.273850\pi$
$5$	2.91669	1.30438	0.652192	+	0.758054i	$0.273850\pi$
$6$	0	0
$7$	−3.05603	−1.15507	−0.577536	−	0.816365i	$-0.695986\pi$
$7$	−3.05603	−1.15507	−0.577536	+	0.816365i	$0.695986\pi$
$8$	0	0
$9$	8.90091	2.96697
$10$	0	0
$11$	−0.926697	−0.279410	−0.139705	−	0.990193i	$-0.544615\pi$
$11$	−0.926697	−0.279410	−0.139705	+	0.990193i	$0.544615\pi$
$12$	0	0
$13$	0.486374	0.134896	0.0674480	−	0.997723i	$-0.478514\pi$
$13$	0.486374	0.134896	0.0674480	+	0.997723i	$0.478514\pi$
$14$	0	0
$15$	−10.0619	−2.59797
$16$	0	0
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0	0
$19$	−1.91883	−0.440209	−0.220105	−	0.975476i	$-0.570640\pi$
$19$	−1.91883	−0.440209	−0.220105	+	0.975476i	$0.570640\pi$
$20$	0	0
$21$	10.5426	2.30059
$22$	0	0
$23$	6.03687	1.25877	0.629387	−	0.777092i	$-0.283306\pi$
$23$	6.03687	1.25877	0.629387	+	0.777092i	$0.283306\pi$
$24$	0	0
$25$	3.50708	0.701417
$26$	0	0
$27$	−20.3568	−3.91766
$28$	0	0
$29$	−4.39950	−0.816967	−0.408484	−	0.912766i	$-0.633942\pi$
$29$	−4.39950	−0.816967	−0.408484	+	0.912766i	$0.633942\pi$
$30$	0	0
$31$	7.75169	1.39225	0.696123	−	0.717923i	$-0.254907\pi$
$31$	7.75169	1.39225	0.696123	+	0.717923i	$0.254907\pi$
$32$	0	0
$33$	3.19689	0.556507
$34$	0	0
$35$	−8.91351	−1.50666
$36$	0	0
$37$	−10.3882	−1.70782	−0.853909	−	0.520423i	$-0.825774\pi$
$37$	−10.3882	−1.70782	−0.853909	+	0.520423i	$0.825774\pi$
$38$	0	0
$39$	−1.67788	−0.268676
$40$	0	0
$41$	10.1225	1.58088	0.790438	−	0.612542i	$-0.209853\pi$
$41$	10.1225	1.58088	0.790438	+	0.612542i	$0.209853\pi$
$42$	0	0
$43$	−7.87176	−1.20043	−0.600216	−	0.799838i	$-0.704919\pi$
$43$	−7.87176	−1.20043	−0.600216	+	0.799838i	$0.704919\pi$
$44$	0	0
$45$	25.9612	3.87007
$46$	0	0
$47$	1.62814	0.237489	0.118745	−	0.992925i	$-0.462113\pi$
$47$	1.62814	0.237489	0.118745	+	0.992925i	$0.462113\pi$
$48$	0	0
$49$	2.33935	0.334193
$50$	0	0
$51$	3.44977	0.483064
$52$	0	0
$53$	11.4768	1.57646	0.788230	−	0.615381i	$-0.210998\pi$
$53$	11.4768	1.57646	0.788230	+	0.615381i	$0.210998\pi$
$54$	0	0
$55$	−2.70289	−0.364457
$56$	0	0
$57$	6.61952	0.876776
$58$	0	0
$59$	−1.00000	−0.130189
$59$	−1.00000	−0.130189
$60$	0	0
$61$	5.75331	0.736636	0.368318	−	0.929700i	$-0.379934\pi$
$61$	5.75331	0.736636	0.368318	+	0.929700i	$0.379934\pi$
$62$	0	0
$63$	−27.2015	−3.42707
$64$	0	0
$65$	1.41860	0.175956
$66$	0	0
$67$	−2.33655	−0.285455	−0.142728	−	0.989762i	$-0.545587\pi$
$67$	−2.33655	−0.285455	−0.142728	+	0.989762i	$0.545587\pi$
$68$	0	0
$69$	−20.8258	−2.50713
$70$	0	0
$71$	11.4554	1.35950	0.679752	−	0.733442i	$-0.262087\pi$
$71$	11.4554	1.35950	0.679752	+	0.733442i	$0.262087\pi$
$72$	0	0
$73$	−11.6318	−1.36140	−0.680700	−	0.732563i	$-0.738324\pi$
$73$	−11.6318	−1.36140	−0.680700	+	0.732563i	$0.738324\pi$
$74$	0	0
$75$	−12.0986	−1.39703
$76$	0	0
$77$	2.83202	0.322738
$78$	0	0
$79$	4.08954	0.460109	0.230055	−	0.973178i	$-0.426110\pi$
$79$	4.08954	0.460109	0.230055	+	0.973178i	$0.426110\pi$
$80$	0	0
$81$	43.5235	4.83594
$82$	0	0
$83$	9.70282	1.06502	0.532511	−	0.846423i	$-0.321248\pi$
$83$	9.70282	1.06502	0.532511	+	0.846423i	$0.321248\pi$
$84$	0	0
$85$	−2.91669	−0.316360
$86$	0	0
$87$	15.1773	1.62717
$88$	0	0
$89$	−16.3766	−1.73592	−0.867958	−	0.496638i	$-0.834568\pi$
$89$	−16.3766	−1.73592	−0.867958	+	0.496638i	$0.834568\pi$
$90$	0	0
$91$	−1.48638	−0.155815
$92$	0	0
$93$	−26.7416	−2.77297
$94$	0	0
$95$	−5.59663	−0.574202
$96$	0	0
$97$	4.28306	0.434879	0.217439	−	0.976074i	$-0.430230\pi$
$97$	4.28306	0.434879	0.217439	+	0.976074i	$0.430230\pi$
$98$	0	0
$99$	−8.24845	−0.829000
$100$	0	0
$101$	3.74034	0.372177	0.186089	−	0.982533i	$-0.440419\pi$
$101$	3.74034	0.372177	0.186089	+	0.982533i	$0.440419\pi$
$102$	0	0
$103$	−5.71816	−0.563427	−0.281714	−	0.959499i	$-0.590903\pi$
$103$	−5.71816	−0.563427	−0.281714	+	0.959499i	$0.590903\pi$
$104$	0	0
$105$	30.7495	3.00085
$106$	0	0
$107$	−11.0083	−1.06421	−0.532107	−	0.846677i	$-0.678600\pi$
$107$	−11.0083	−1.06421	−0.532107	+	0.846677i	$0.678600\pi$
$108$	0	0
$109$	14.4611	1.38512	0.692559	−	0.721361i	$-0.256483\pi$
$109$	14.4611	1.38512	0.692559	+	0.721361i	$0.256483\pi$
$110$	0	0
$111$	35.8371	3.40150
$112$	0	0
$113$	−14.9558	−1.40692	−0.703462	−	0.710733i	$-0.748363\pi$
$113$	−14.9558	−1.40692	−0.703462	+	0.710733i	$0.748363\pi$
$114$	0	0
$115$	17.6077	1.64193
$116$	0	0
$117$	4.32917	0.400232
$118$	0	0
$119$	3.05603	0.280146
$120$	0	0
$121$	−10.1412	−0.921930
$122$	0	0
$123$	−34.9204	−3.14867
$124$	0	0
$125$	−4.35438	−0.389467
$126$	0	0
$127$	10.9796	0.974285	0.487142	−	0.873323i	$-0.338039\pi$
$127$	10.9796	0.974285	0.487142	+	0.873323i	$0.338039\pi$
$128$	0	0
$129$	27.1558	2.39093
$130$	0	0
$131$	−3.45417	−0.301792	−0.150896	−	0.988550i	$-0.548216\pi$
$131$	−3.45417	−0.301792	−0.150896	+	0.988550i	$0.548216\pi$
$132$	0	0
$133$	5.86401	0.508474
$134$	0	0
$135$	−59.3744	−5.11014
$136$	0	0
$137$	12.3120	1.05188	0.525940	−	0.850521i	$-0.323713\pi$
$137$	12.3120	1.05188	0.525940	+	0.850521i	$0.323713\pi$
$138$	0	0
$139$	−8.32055	−0.705740	−0.352870	−	0.935672i	$-0.614794\pi$
$139$	−8.32055	−0.705740	−0.352870	+	0.935672i	$0.614794\pi$
$140$	0	0
$141$	−5.61672	−0.473013
$142$	0	0
$143$	−0.450721	−0.0376912
$144$	0	0
$145$	−12.8320	−1.06564
$146$	0	0
$147$	−8.07021	−0.665620
$148$	0	0
$149$	18.0534	1.47899	0.739496	−	0.673161i	$-0.235064\pi$
$149$	18.0534	1.47899	0.739496	+	0.673161i	$0.235064\pi$
$150$	0	0
$151$	3.60820	0.293631	0.146816	−	0.989164i	$-0.453098\pi$
$151$	3.60820	0.293631	0.146816	+	0.989164i	$0.453098\pi$
$152$	0	0
$153$	−8.90091	−0.719596
$154$	0	0
$155$	22.6093	1.81602
$156$	0	0
$157$	3.61306	0.288354	0.144177	−	0.989552i	$-0.453947\pi$
$157$	3.61306	0.288354	0.144177	+	0.989552i	$0.453947\pi$
$158$	0	0
$159$	−39.5923	−3.13987
$160$	0	0
$161$	−18.4489	−1.45398
$162$	0	0
$163$	−22.6111	−1.77104	−0.885519	−	0.464603i	$-0.846197\pi$
$163$	−22.6111	−1.77104	−0.885519	+	0.464603i	$0.846197\pi$
$164$	0	0
$165$	9.32434	0.725899
$166$	0	0
$167$	7.14673	0.553030	0.276515	−	0.961010i	$-0.410820\pi$
$167$	7.14673	0.553030	0.276515	+	0.961010i	$0.410820\pi$
$168$	0	0
$169$	−12.7634	−0.981803
$170$	0	0
$171$	−17.0793	−1.30609
$172$	0	0
$173$	13.1582	1.00040	0.500201	−	0.865909i	$-0.333260\pi$
$173$	13.1582	1.00040	0.500201	+	0.865909i	$0.333260\pi$
$174$	0	0
$175$	−10.7178	−0.810187
$176$	0	0
$177$	3.44977	0.259301
$178$	0	0
$179$	−5.48445	−0.409927	−0.204964	−	0.978770i	$-0.565708\pi$
$179$	−5.48445	−0.409927	−0.204964	+	0.978770i	$0.565708\pi$
$180$	0	0
$181$	−15.2677	−1.13484	−0.567419	−	0.823429i	$-0.692058\pi$
$181$	−15.2677	−1.13484	−0.567419	+	0.823429i	$0.692058\pi$
$182$	0	0
$183$	−19.8476	−1.46718
$184$	0	0
$185$	−30.2993	−2.22765
$186$	0	0
$187$	0.926697	0.0677668
$188$	0	0
$189$	62.2110	4.52519
$190$	0	0
$191$	−1.85368	−0.134128	−0.0670639	−	0.997749i	$-0.521363\pi$
$191$	−1.85368	−0.134128	−0.0670639	+	0.997749i	$0.521363\pi$
$192$	0	0
$193$	0.890371	0.0640903	0.0320452	−	0.999486i	$-0.489798\pi$
$193$	0.890371	0.0640903	0.0320452	+	0.999486i	$0.489798\pi$
$194$	0	0
$195$	−4.89385	−0.350456
$196$	0	0
$197$	12.7013	0.904929	0.452465	−	0.891782i	$-0.350545\pi$
$197$	12.7013	0.904929	0.452465	+	0.891782i	$0.350545\pi$
$198$	0	0
$199$	−13.6350	−0.966560	−0.483280	−	0.875466i	$-0.660555\pi$
$199$	−13.6350	−0.966560	−0.483280	+	0.875466i	$0.660555\pi$
$200$	0	0
$201$	8.06057	0.568549
$202$	0	0
$203$	13.4450	0.943656
$204$	0	0
$205$	29.5243	2.06207
$206$	0	0
$207$	53.7336	3.73475
$208$	0	0
$209$	1.77817	0.122999
$210$	0	0
$211$	−6.91809	−0.476261	−0.238130	−	0.971233i	$-0.576535\pi$
$211$	−6.91809	−0.476261	−0.238130	+	0.971233i	$0.576535\pi$
$212$	0	0
$213$	−39.5184	−2.70776
$214$	0	0
$215$	−22.9595	−1.56582
$216$	0	0
$217$	−23.6894	−1.60814
$218$	0	0
$219$	40.1270	2.71153
$220$	0	0
$221$	−0.486374	−0.0327171
$222$	0	0
$223$	7.80605	0.522732	0.261366	−	0.965240i	$-0.415827\pi$
$223$	7.80605	0.522732	0.261366	+	0.965240i	$0.415827\pi$
$224$	0	0
$225$	31.2162	2.08108
$226$	0	0
$227$	16.2527	1.07873	0.539365	−	0.842072i	$-0.318664\pi$
$227$	16.2527	1.07873	0.539365	+	0.842072i	$0.318664\pi$
$228$	0	0
$229$	18.1816	1.20148	0.600738	−	0.799446i	$-0.294874\pi$
$229$	18.1816	1.20148	0.600738	+	0.799446i	$0.294874\pi$
$230$	0	0
$231$	−9.76981	−0.642806
$232$	0	0
$233$	−5.73290	−0.375575	−0.187787	−	0.982210i	$-0.560132\pi$
$233$	−5.73290	−0.375575	−0.187787	+	0.982210i	$0.560132\pi$
$234$	0	0
$235$	4.74879	0.309777
$236$	0	0
$237$	−14.1080	−0.916411
$238$	0	0
$239$	20.8759	1.35035	0.675175	−	0.737657i	$-0.264068\pi$
$239$	20.8759	1.35035	0.675175	+	0.737657i	$0.264068\pi$
$240$	0	0
$241$	2.22347	0.143226	0.0716130	−	0.997432i	$-0.477185\pi$
$241$	2.22347	0.143226	0.0716130	+	0.997432i	$0.477185\pi$
$242$	0	0
$243$	−89.0755	−5.71420
$244$	0	0
$245$	6.82316	0.435915
$246$	0	0
$247$	−0.933269	−0.0593825
$248$	0	0
$249$	−33.4725	−2.12123
$250$	0	0
$251$	4.15676	0.262372	0.131186	−	0.991358i	$-0.458121\pi$
$251$	4.15676	0.262372	0.131186	+	0.991358i	$0.458121\pi$
$252$	0	0
$253$	−5.59435	−0.351714
$254$	0	0
$255$	10.0619	0.630101
$256$	0	0
$257$	19.2712	1.20211	0.601053	−	0.799210i	$-0.294748\pi$
$257$	19.2712	1.20211	0.601053	+	0.799210i	$0.294748\pi$
$258$	0	0
$259$	31.7468	1.97265
$260$	0	0
$261$	−39.1596	−2.42392
$262$	0	0
$263$	−4.45800	−0.274892	−0.137446	−	0.990509i	$-0.543889\pi$
$263$	−4.45800	−0.274892	−0.137446	+	0.990509i	$0.543889\pi$
$264$	0	0
$265$	33.4743	2.05631
$266$	0	0
$267$	56.4955	3.45747
$268$	0	0
$269$	−27.1672	−1.65641	−0.828205	−	0.560425i	$-0.810638\pi$
$269$	−27.1672	−1.65641	−0.828205	+	0.560425i	$0.810638\pi$
$270$	0	0
$271$	−9.63991	−0.585583	−0.292792	−	0.956176i	$-0.594584\pi$
$271$	−9.63991	−0.585583	−0.292792	+	0.956176i	$0.594584\pi$
$272$	0	0
$273$	5.12766	0.310340
$274$	0	0
$275$	−3.25000	−0.195983
$276$	0	0
$277$	−14.8313	−0.891129	−0.445564	−	0.895250i	$-0.646997\pi$
$277$	−14.8313	−0.891129	−0.445564	+	0.895250i	$0.646997\pi$
$278$	0	0
$279$	68.9971	4.13075
$280$	0	0
$281$	27.1244	1.61811	0.809054	−	0.587734i	$-0.199980\pi$
$281$	27.1244	1.61811	0.809054	+	0.587734i	$0.199980\pi$
$282$	0	0
$283$	−5.57510	−0.331405	−0.165703	−	0.986176i	$-0.552989\pi$
$283$	−5.57510	−0.331405	−0.165703	+	0.986176i	$0.552989\pi$
$284$	0	0
$285$	19.3071	1.14365
$286$	0	0
$287$	−30.9348	−1.82603
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	−14.7756	−0.866159
$292$	0	0
$293$	−11.8366	−0.691502	−0.345751	−	0.938326i	$-0.612376\pi$
$293$	−11.8366	−0.691502	−0.345751	+	0.938326i	$0.612376\pi$
$294$	0	0
$295$	−2.91669	−0.169816
$296$	0	0
$297$	18.8646	1.09463
$298$	0	0
$299$	2.93618	0.169804
$300$	0	0
$301$	24.0564	1.38659
$302$	0	0
$303$	−12.9033	−0.741275
$304$	0	0
$305$	16.7806	0.960856
$306$	0	0
$307$	6.93782	0.395962	0.197981	−	0.980206i	$-0.436562\pi$
$307$	6.93782	0.395962	0.197981	+	0.980206i	$0.436562\pi$
$308$	0	0
$309$	19.7263	1.12219
$310$	0	0
$311$	24.2032	1.37244	0.686218	−	0.727396i	$-0.259270\pi$
$311$	24.2032	1.37244	0.686218	+	0.727396i	$0.259270\pi$
$312$	0	0
$313$	−14.6509	−0.828116	−0.414058	−	0.910251i	$-0.635889\pi$
$313$	−14.6509	−0.828116	−0.414058	+	0.910251i	$0.635889\pi$
$314$	0	0
$315$	−79.3383	−4.47021
$316$	0	0
$317$	−26.2606	−1.47494	−0.737470	−	0.675380i	$-0.763980\pi$
$317$	−26.2606	−1.47494	−0.737470	+	0.675380i	$0.763980\pi$
$318$	0	0
$319$	4.07701	0.228269
$320$	0	0
$321$	37.9761	2.11962
$322$	0	0
$323$	1.91883	0.106766
$324$	0	0
$325$	1.70575	0.0946183
$326$	0	0
$327$	−49.8873	−2.75878
$328$	0	0
$329$	−4.97566	−0.274317
$330$	0	0
$331$	30.7461	1.68996	0.844979	−	0.534799i	$-0.179613\pi$
$331$	30.7461	1.68996	0.844979	+	0.534799i	$0.179613\pi$
$332$	0	0
$333$	−92.4648	−5.06704
$334$	0	0
$335$	−6.81500	−0.372343
$336$	0	0
$337$	−13.9490	−0.759851	−0.379925	−	0.925017i	$-0.624050\pi$
$337$	−13.9490	−0.759851	−0.379925	+	0.925017i	$0.624050\pi$
$338$	0	0
$339$	51.5941	2.80221
$340$	0	0
$341$	−7.18347	−0.389007
$342$	0	0
$343$	14.2431	0.769056
$344$	0	0
$345$	−60.7425	−3.27026
$346$	0	0
$347$	7.65919	0.411167	0.205583	−	0.978640i	$-0.434091\pi$
$347$	7.65919	0.411167	0.205583	+	0.978640i	$0.434091\pi$
$348$	0	0
$349$	−21.4462	−1.14799	−0.573993	−	0.818860i	$-0.694606\pi$
$349$	−21.4462	−1.14799	−0.573993	+	0.818860i	$0.694606\pi$
$350$	0	0
$351$	−9.90101	−0.528477
$352$	0	0
$353$	33.8974	1.80417	0.902087	−	0.431554i	$-0.142035\pi$
$353$	33.8974	1.80417	0.902087	+	0.431554i	$0.142035\pi$
$354$	0	0
$355$	33.4118	1.77331
$356$	0	0
$357$	−10.5426	−0.557974
$358$	0	0
$359$	−15.1324	−0.798657	−0.399328	−	0.916808i	$-0.630757\pi$
$359$	−15.1324	−0.798657	−0.399328	+	0.916808i	$0.630757\pi$
$360$	0	0
$361$	−15.3181	−0.806216
$362$	0	0
$363$	34.9849	1.83623
$364$	0	0
$365$	−33.9264	−1.77579
$366$	0	0
$367$	−24.2979	−1.26834	−0.634169	−	0.773194i	$-0.718658\pi$
$367$	−24.2979	−1.26834	−0.634169	+	0.773194i	$0.718658\pi$
$368$	0	0
$369$	90.0998	4.69041
$370$	0	0
$371$	−35.0735	−1.82092
$372$	0	0
$373$	0.632038	0.0327257	0.0163628	−	0.999866i	$-0.494791\pi$
$373$	0.632038	0.0327257	0.0163628	+	0.999866i	$0.494791\pi$
$374$	0	0
$375$	15.0216	0.775712
$376$	0	0
$377$	−2.13980	−0.110206
$378$	0	0
$379$	−1.30393	−0.0669786	−0.0334893	−	0.999439i	$-0.510662\pi$
$379$	−1.30393	−0.0669786	−0.0334893	+	0.999439i	$0.510662\pi$
$380$	0	0
$381$	−37.8772	−1.94051
$382$	0	0
$383$	−10.5963	−0.541446	−0.270723	−	0.962657i	$-0.587263\pi$
$383$	−10.5963	−0.541446	−0.270723	+	0.962657i	$0.587263\pi$
$384$	0	0
$385$	8.26012	0.420975
$386$	0	0
$387$	−70.0658	−3.56165
$388$	0	0
$389$	5.03584	0.255327	0.127664	−	0.991818i	$-0.459252\pi$
$389$	5.03584	0.255327	0.127664	+	0.991818i	$0.459252\pi$
$390$	0	0
$391$	−6.03687	−0.305298
$392$	0	0
$393$	11.9161	0.601087
$394$	0	0
$395$	11.9279	0.600159
$396$	0	0
$397$	4.89310	0.245578	0.122789	−	0.992433i	$-0.460816\pi$
$397$	4.89310	0.245578	0.122789	+	0.992433i	$0.460816\pi$
$398$	0	0
$399$	−20.2295	−1.01274
$400$	0	0
$401$	1.50670	0.0752409	0.0376204	−	0.999292i	$-0.488022\pi$
$401$	1.50670	0.0752409	0.0376204	+	0.999292i	$0.488022\pi$
$402$	0	0
$403$	3.77022	0.187808
$404$	0	0
$405$	126.944	6.30792
$406$	0	0
$407$	9.62676	0.477181
$408$	0	0
$409$	−1.84649	−0.0913032	−0.0456516	−	0.998957i	$-0.514536\pi$
$409$	−1.84649	−0.0913032	−0.0456516	+	0.998957i	$0.514536\pi$
$410$	0	0
$411$	−42.4734	−2.09506
$412$	0	0
$413$	3.05603	0.150378
$414$	0	0
$415$	28.3001	1.38920
$416$	0	0
$417$	28.7040	1.40564
$418$	0	0
$419$	−14.5877	−0.712656	−0.356328	−	0.934361i	$-0.615971\pi$
$419$	−14.5877	−0.712656	−0.356328	+	0.934361i	$0.615971\pi$
$420$	0	0
$421$	36.4305	1.77552	0.887758	−	0.460310i	$-0.152262\pi$
$421$	36.4305	1.77552	0.887758	+	0.460310i	$0.152262\pi$
$422$	0	0
$423$	14.4920	0.704623
$424$	0	0
$425$	−3.50708	−0.170119
$426$	0	0
$427$	−17.5823	−0.850868
$428$	0	0
$429$	1.55489	0.0750706
$430$	0	0
$431$	16.4890	0.794248	0.397124	−	0.917765i	$-0.370008\pi$
$431$	16.4890	0.794248	0.397124	+	0.917765i	$0.370008\pi$
$432$	0	0
$433$	−8.13320	−0.390857	−0.195428	−	0.980718i	$-0.562610\pi$
$433$	−8.13320	−0.390857	−0.195428	+	0.980718i	$0.562610\pi$
$434$	0	0
$435$	44.2674	2.12246
$436$	0	0
$437$	−11.5837	−0.554125
$438$	0	0
$439$	27.9365	1.33334	0.666668	−	0.745354i	$-0.267720\pi$
$439$	27.9365	1.33334	0.666668	+	0.745354i	$0.267720\pi$
$440$	0	0
$441$	20.8223	0.991540
$442$	0	0
$443$	28.3873	1.34872	0.674361	−	0.738401i	$-0.264419\pi$
$443$	28.3873	1.34872	0.674361	+	0.738401i	$0.264419\pi$
$444$	0	0
$445$	−47.7654	−2.26430
$446$	0	0
$447$	−62.2801	−2.94575
$448$	0	0
$449$	11.1305	0.525282	0.262641	−	0.964894i	$-0.415407\pi$
$449$	11.1305	0.525282	0.262641	+	0.964894i	$0.415407\pi$
$450$	0	0
$451$	−9.38053	−0.441712
$452$	0	0
$453$	−12.4475	−0.584833
$454$	0	0
$455$	−4.33530	−0.203242
$456$	0	0
$457$	1.25750	0.0588235	0.0294117	−	0.999567i	$-0.490637\pi$
$457$	1.25750	0.0588235	0.0294117	+	0.999567i	$0.490637\pi$
$458$	0	0
$459$	20.3568	0.950173
$460$	0	0
$461$	12.2433	0.570227	0.285113	−	0.958494i	$-0.407969\pi$
$461$	12.2433	0.570227	0.285113	+	0.958494i	$0.407969\pi$
$462$	0	0
$463$	24.5371	1.14033	0.570167	−	0.821529i	$-0.306878\pi$
$463$	24.5371	1.14033	0.570167	+	0.821529i	$0.306878\pi$
$464$	0	0
$465$	−77.9968	−3.61702
$466$	0	0
$467$	35.9521	1.66366	0.831832	−	0.555028i	$-0.187292\pi$
$467$	35.9521	1.66366	0.831832	+	0.555028i	$0.187292\pi$
$468$	0	0
$469$	7.14059	0.329722
$470$	0	0
$471$	−12.4642	−0.574321
$472$	0	0
$473$	7.29474	0.335412
$474$	0	0
$475$	−6.72949	−0.308770
$476$	0	0
$477$	102.154	4.67731
$478$	0	0
$479$	33.5743	1.53405	0.767025	−	0.641618i	$-0.221736\pi$
$479$	33.5743	1.53405	0.767025	+	0.641618i	$0.221736\pi$
$480$	0	0
$481$	−5.05258	−0.230378
$482$	0	0
$483$	63.6444	2.89592
$484$	0	0
$485$	12.4924	0.567249
$486$	0	0
$487$	42.6443	1.93240	0.966199	−	0.257796i	$-0.0829964\pi$
$487$	42.6443	1.93240	0.966199	+	0.257796i	$0.0829964\pi$
$488$	0	0
$489$	78.0031	3.52742
$490$	0	0
$491$	40.0541	1.80762	0.903808	−	0.427938i	$-0.140760\pi$
$491$	40.0541	1.80762	0.903808	+	0.427938i	$0.140760\pi$
$492$	0	0
$493$	4.39950	0.198144
$494$	0	0
$495$	−24.0582	−1.08133
$496$	0	0
$497$	−35.0081	−1.57033
$498$	0	0
$499$	9.85680	0.441251	0.220625	−	0.975359i	$-0.429190\pi$
$499$	9.85680	0.441251	0.220625	+	0.975359i	$0.429190\pi$
$500$	0	0
$501$	−24.6546	−1.10148
$502$	0	0
$503$	36.3178	1.61933	0.809665	−	0.586893i	$-0.199649\pi$
$503$	36.3178	1.61933	0.809665	+	0.586893i	$0.199649\pi$
$504$	0	0
$505$	10.9094	0.485462
$506$	0	0
$507$	44.0309	1.95548
$508$	0	0
$509$	40.1732	1.78064	0.890322	−	0.455331i	$-0.150479\pi$
$509$	40.1732	1.78064	0.890322	+	0.455331i	$0.150479\pi$
$510$	0	0
$511$	35.5472	1.57251
$512$	0	0
$513$	39.0612	1.72459
$514$	0	0
$515$	−16.6781	−0.734925
$516$	0	0
$517$	−1.50880	−0.0663568
$518$	0	0
$519$	−45.3928	−1.99252
$520$	0	0
$521$	23.9249	1.04817	0.524085	−	0.851666i	$-0.324408\pi$
$521$	23.9249	1.04817	0.524085	+	0.851666i	$0.324408\pi$
$522$	0	0
$523$	27.8536	1.21795	0.608977	−	0.793188i	$-0.291580\pi$
$523$	27.8536	1.21795	0.608977	+	0.793188i	$0.291580\pi$
$524$	0	0
$525$	36.9738	1.61367
$526$	0	0
$527$	−7.75169	−0.337669
$528$	0	0
$529$	13.4438	0.584514
$530$	0	0
$531$	−8.90091	−0.386267
$532$	0	0
$533$	4.92334	0.213254
$534$	0	0
$535$	−32.1078	−1.38814
$536$	0	0
$537$	18.9201	0.816463
$538$	0	0
$539$	−2.16787	−0.0933767
$540$	0	0
$541$	30.1570	1.29655	0.648275	−	0.761406i	$-0.275491\pi$
$541$	30.1570	1.29655	0.648275	+	0.761406i	$0.275491\pi$
$542$	0	0
$543$	52.6700	2.26029
$544$	0	0
$545$	42.1784	1.80673
$546$	0	0
$547$	18.5779	0.794333	0.397166	−	0.917747i	$-0.369994\pi$
$547$	18.5779	0.794333	0.397166	+	0.917747i	$0.369994\pi$
$548$	0	0
$549$	51.2097	2.18558
$550$	0	0
$551$	8.44189	0.359637
$552$	0	0
$553$	−12.4978	−0.531460
$554$	0	0
$555$	104.526	4.43686
$556$	0	0
$557$	45.9661	1.94764	0.973822	−	0.227311i	$-0.0729934\pi$
$557$	45.9661	1.94764	0.973822	+	0.227311i	$0.0729934\pi$
$558$	0	0
$559$	−3.82862	−0.161933
$560$	0	0
$561$	−3.19689	−0.134973
$562$	0	0
$563$	17.7432	0.747785	0.373892	−	0.927472i	$-0.378023\pi$
$563$	17.7432	0.747785	0.373892	+	0.927472i	$0.378023\pi$
$564$	0	0
$565$	−43.6215	−1.83517
$566$	0	0
$567$	−133.009	−5.58586
$568$	0	0
$569$	−10.2266	−0.428721	−0.214361	−	0.976755i	$-0.568767\pi$
$569$	−10.2266	−0.428721	−0.214361	+	0.976755i	$0.568767\pi$
$570$	0	0
$571$	−8.42385	−0.352527	−0.176264	−	0.984343i	$-0.556401\pi$
$571$	−8.42385	−0.352527	−0.176264	+	0.984343i	$0.556401\pi$
$572$	0	0
$573$	6.39478	0.267146
$574$	0	0
$575$	21.1718	0.882926
$576$	0	0
$577$	3.39880	0.141494	0.0707469	−	0.997494i	$-0.477462\pi$
$577$	3.39880	0.141494	0.0707469	+	0.997494i	$0.477462\pi$
$578$	0	0
$579$	−3.07158	−0.127650
$580$	0	0
$581$	−29.6522	−1.23018
$582$	0	0
$583$	−10.6355	−0.440478
$584$	0	0
$585$	12.6269	0.522056
$586$	0	0
$587$	4.54110	0.187431	0.0937157	−	0.995599i	$-0.470126\pi$
$587$	4.54110	0.187431	0.0937157	+	0.995599i	$0.470126\pi$
$588$	0	0
$589$	−14.8742	−0.612879
$590$	0	0
$591$	−43.8165	−1.80237
$592$	0	0
$593$	3.96269	0.162728	0.0813641	−	0.996684i	$-0.474072\pi$
$593$	3.96269	0.162728	0.0813641	+	0.996684i	$0.474072\pi$
$594$	0	0
$595$	8.91351	0.365418
$596$	0	0
$597$	47.0376	1.92512
$598$	0	0
$599$	21.6235	0.883514	0.441757	−	0.897135i	$-0.354355\pi$
$599$	21.6235	0.883514	0.441757	+	0.897135i	$0.354355\pi$
$600$	0	0
$601$	−20.7365	−0.845860	−0.422930	−	0.906162i	$-0.638998\pi$
$601$	−20.7365	−0.845860	−0.422930	+	0.906162i	$0.638998\pi$
$602$	0	0
$603$	−20.7974	−0.846937
$604$	0	0
$605$	−29.5788	−1.20255
$606$	0	0
$607$	17.5042	0.710475	0.355237	−	0.934776i	$-0.384400\pi$
$607$	17.5042	0.710475	0.355237	+	0.934776i	$0.384400\pi$
$608$	0	0
$609$	−46.3823	−1.87950
$610$	0	0
$611$	0.791887	0.0320363
$612$	0	0
$613$	−21.3789	−0.863487	−0.431744	−	0.901996i	$-0.642101\pi$
$613$	−21.3789	−0.863487	−0.431744	+	0.901996i	$0.642101\pi$
$614$	0	0
$615$	−101.852	−4.10707
$616$	0	0
$617$	16.5152	0.664876	0.332438	−	0.943125i	$-0.392129\pi$
$617$	16.5152	0.664876	0.332438	+	0.943125i	$0.392129\pi$
$618$	0	0
$619$	−36.2586	−1.45736	−0.728679	−	0.684856i	$-0.759865\pi$
$619$	−36.2586	−1.45736	−0.728679	+	0.684856i	$0.759865\pi$
$620$	0	0
$621$	−122.891	−4.93146
$622$	0	0
$623$	50.0474	2.00511
$624$	0	0
$625$	−30.2358	−1.20943
$626$	0	0
$627$	−6.13429	−0.244980
$628$	0	0
$629$	10.3882	0.414207
$630$	0	0
$631$	28.7595	1.14490	0.572449	−	0.819940i	$-0.305994\pi$
$631$	28.7595	1.14490	0.572449	+	0.819940i	$0.305994\pi$
$632$	0	0
$633$	23.8658	0.948581
$634$	0	0
$635$	32.0242	1.27084
$636$	0	0
$637$	1.13780	0.0450812
$638$	0	0
$639$	101.963	4.03361
$640$	0	0
$641$	−13.9026	−0.549118	−0.274559	−	0.961570i	$-0.588532\pi$
$641$	−13.9026	−0.549118	−0.274559	+	0.961570i	$0.588532\pi$
$642$	0	0
$643$	−33.7078	−1.32931	−0.664653	−	0.747152i	$-0.731421\pi$
$643$	−33.7078	−1.32931	−0.664653	+	0.747152i	$0.731421\pi$
$644$	0	0
$645$	79.2049	3.11869
$646$	0	0
$647$	13.7125	0.539093	0.269547	−	0.962987i	$-0.413126\pi$
$647$	13.7125	0.539093	0.269547	+	0.962987i	$0.413126\pi$
$648$	0	0
$649$	0.926697	0.0363760
$650$	0	0
$651$	81.7231	3.20298
$652$	0	0
$653$	9.64929	0.377606	0.188803	−	0.982015i	$-0.439539\pi$
$653$	9.64929	0.377606	0.188803	+	0.982015i	$0.439539\pi$
$654$	0	0
$655$	−10.0747	−0.393652
$656$	0	0
$657$	−103.534	−4.03923
$658$	0	0
$659$	−15.7550	−0.613728	−0.306864	−	0.951753i	$-0.599280\pi$
$659$	−15.7550	−0.613728	−0.306864	+	0.951753i	$0.599280\pi$
$660$	0	0
$661$	12.4509	0.484285	0.242142	−	0.970241i	$-0.422150\pi$
$661$	12.4509	0.484285	0.242142	+	0.970241i	$0.422150\pi$
$662$	0	0
$663$	1.67788	0.0651634
$664$	0	0
$665$	17.1035	0.663245
$666$	0	0
$667$	−26.5592	−1.02838
$668$	0	0
$669$	−26.9291	−1.04114
$670$	0	0
$671$	−5.33158	−0.205823
$672$	0	0
$673$	−35.4929	−1.36815	−0.684075	−	0.729412i	$-0.739794\pi$
$673$	−35.4929	−1.36815	−0.684075	+	0.729412i	$0.739794\pi$
$674$	0	0
$675$	−71.3929	−2.74791
$676$	0	0
$677$	29.0575	1.11677	0.558385	−	0.829582i	$-0.311421\pi$
$677$	29.0575	1.11677	0.558385	+	0.829582i	$0.311421\pi$
$678$	0	0
$679$	−13.0892	−0.502317
$680$	0	0
$681$	−56.0680	−2.14853
$682$	0	0
$683$	1.49076	0.0570425	0.0285213	−	0.999593i	$-0.490920\pi$
$683$	1.49076	0.0570425	0.0285213	+	0.999593i	$0.490920\pi$
$684$	0	0
$685$	35.9102	1.37206
$686$	0	0
$687$	−62.7224	−2.39301
$688$	0	0
$689$	5.58202	0.212658
$690$	0	0
$691$	15.1725	0.577188	0.288594	−	0.957452i	$-0.406812\pi$
$691$	15.1725	0.577188	0.288594	+	0.957452i	$0.406812\pi$
$692$	0	0
$693$	25.2075	0.957555
$694$	0	0
$695$	−24.2685	−0.920555
$696$	0	0
$697$	−10.1225	−0.383419
$698$	0	0
$699$	19.7772	0.748042
$700$	0	0
$701$	−15.4218	−0.582472	−0.291236	−	0.956651i	$-0.594067\pi$
$701$	−15.4218	−0.582472	−0.291236	+	0.956651i	$0.594067\pi$
$702$	0	0
$703$	19.9333	0.751797
$704$	0	0
$705$	−16.3822	−0.616991
$706$	0	0
$707$	−11.4306	−0.429892
$708$	0	0
$709$	−29.5488	−1.10973	−0.554865	−	0.831940i	$-0.687230\pi$
$709$	−29.5488	−1.10973	−0.554865	+	0.831940i	$0.687230\pi$
$710$	0	0
$711$	36.4006	1.36513
$712$	0	0
$713$	46.7960	1.75252
$714$	0	0
$715$	−1.31462	−0.0491638
$716$	0	0
$717$	−72.0171	−2.68953
$718$	0	0
$719$	19.4816	0.726540	0.363270	−	0.931684i	$-0.381660\pi$
$719$	19.4816	0.726540	0.363270	+	0.931684i	$0.381660\pi$
$720$	0	0
$721$	17.4749	0.650799
$722$	0	0
$723$	−7.67044	−0.285267
$724$	0	0
$725$	−15.4294	−0.573034
$726$	0	0
$727$	23.4861	0.871053	0.435526	−	0.900176i	$-0.356562\pi$
$727$	23.4861	0.871053	0.435526	+	0.900176i	$0.356562\pi$
$728$	0	0
$729$	176.720	6.54518
$730$	0	0
$731$	7.87176	0.291148
$732$	0	0
$733$	16.6300	0.614242	0.307121	−	0.951671i	$-0.400634\pi$
$733$	16.6300	0.614242	0.307121	+	0.951671i	$0.400634\pi$
$734$	0	0
$735$	−23.5383	−0.868224
$736$	0	0
$737$	2.16528	0.0797590
$738$	0	0
$739$	−12.4717	−0.458780	−0.229390	−	0.973335i	$-0.573673\pi$
$739$	−12.4717	−0.458780	−0.229390	+	0.973335i	$0.573673\pi$
$740$	0	0
$741$	3.21956	0.118274
$742$	0	0
$743$	11.1079	0.407508	0.203754	−	0.979022i	$-0.434686\pi$
$743$	11.1079	0.407508	0.203754	+	0.979022i	$0.434686\pi$
$744$	0	0
$745$	52.6562	1.92917
$746$	0	0
$747$	86.3639	3.15989
$748$	0	0
$749$	33.6418	1.22924
$750$	0	0
$751$	18.9615	0.691915	0.345958	−	0.938250i	$-0.387554\pi$
$751$	18.9615	0.691915	0.345958	+	0.938250i	$0.387554\pi$
$752$	0	0
$753$	−14.3399	−0.522574
$754$	0	0
$755$	10.5240	0.383008
$756$	0	0
$757$	7.39191	0.268664	0.134332	−	0.990936i	$-0.457111\pi$
$757$	7.39191	0.268664	0.134332	+	0.990936i	$0.457111\pi$
$758$	0	0
$759$	19.2992	0.700517
$760$	0	0
$761$	51.1688	1.85487	0.927434	−	0.373987i	$-0.122010\pi$
$761$	51.1688	1.85487	0.927434	+	0.373987i	$0.122010\pi$
$762$	0	0
$763$	−44.1935	−1.59991
$764$	0	0
$765$	−25.9612	−0.938629
$766$	0	0
$767$	−0.486374	−0.0175620
$768$	0	0
$769$	−14.0127	−0.505310	−0.252655	−	0.967556i	$-0.581304\pi$
$769$	−14.0127	−0.505310	−0.252655	+	0.967556i	$0.581304\pi$
$770$	0	0
$771$	−66.4812	−2.39426
$772$	0	0
$773$	−19.9972	−0.719248	−0.359624	−	0.933097i	$-0.617095\pi$
$773$	−19.9972	−0.719248	−0.359624	+	0.933097i	$0.617095\pi$
$774$	0	0
$775$	27.1858	0.976544
$776$	0	0
$777$	−109.519	−3.92898
$778$	0	0
$779$	−19.4234	−0.695917
$780$	0	0
$781$	−10.6157	−0.379859
$782$	0	0
$783$	89.5597	3.20060
$784$	0	0
$785$	10.5382	0.376124
$786$	0	0
$787$	9.98925	0.356078	0.178039	−	0.984023i	$-0.443025\pi$
$787$	9.98925	0.356078	0.178039	+	0.984023i	$0.443025\pi$
$788$	0	0
$789$	15.3791	0.547510
$790$	0	0
$791$	45.7055	1.62510
$792$	0	0
$793$	2.79826	0.0993692
$794$	0	0
$795$	−115.478	−4.09560
$796$	0	0
$797$	16.6672	0.590383	0.295192	−	0.955438i	$-0.404617\pi$
$797$	16.6672	0.590383	0.295192	+	0.955438i	$0.404617\pi$
$798$	0	0
$799$	−1.62814	−0.0575996
$800$	0	0
$801$	−145.767	−5.15041
$802$	0	0
$803$	10.7792	0.380388
$804$	0	0
$805$	−53.8097	−1.89654
$806$	0	0
$807$	93.7204	3.29912
$808$	0	0
$809$	−52.5035	−1.84592	−0.922962	−	0.384890i	$-0.874239\pi$
$809$	−52.5035	−1.84592	−0.922962	+	0.384890i	$0.874239\pi$
$810$	0	0
$811$	−7.83221	−0.275026	−0.137513	−	0.990500i	$-0.543911\pi$
$811$	−7.83221	−0.275026	−0.137513	+	0.990500i	$0.543911\pi$
$812$	0	0
$813$	33.2555	1.16632
$814$	0	0
$815$	−65.9496	−2.31011
$816$	0	0
$817$	15.1046	0.528442
$818$	0	0
$819$	−13.2301	−0.462297
$820$	0	0
$821$	12.9428	0.451707	0.225853	−	0.974161i	$-0.427483\pi$
$821$	12.9428	0.451707	0.225853	+	0.974161i	$0.427483\pi$
$822$	0	0
$823$	−43.4679	−1.51520	−0.757599	−	0.652721i	$-0.773628\pi$
$823$	−43.4679	−1.51520	−0.757599	+	0.652721i	$0.773628\pi$
$824$	0	0
$825$	11.2118	0.390343
$826$	0	0
$827$	11.1498	0.387715	0.193858	−	0.981030i	$-0.437900\pi$
$827$	11.1498	0.387715	0.193858	+	0.981030i	$0.437900\pi$
$828$	0	0
$829$	24.2934	0.843743	0.421872	−	0.906656i	$-0.361373\pi$
$829$	24.2934	0.843743	0.421872	+	0.906656i	$0.361373\pi$
$830$	0	0
$831$	51.1647	1.77488
$832$	0	0
$833$	−2.33935	−0.0810536
$834$	0	0
$835$	20.8448	0.721364
$836$	0	0
$837$	−157.799	−5.45435
$838$	0	0
$839$	−17.6091	−0.607935	−0.303967	−	0.952682i	$-0.598311\pi$
$839$	−17.6091	−0.607935	−0.303967	+	0.952682i	$0.598311\pi$
$840$	0	0
$841$	−9.64437	−0.332565
$842$	0	0
$843$	−93.5731	−3.22283
$844$	0	0
$845$	−37.2270	−1.28065
$846$	0	0
$847$	30.9920	1.06490
$848$	0	0
$849$	19.2328	0.660068
$850$	0	0
$851$	−62.7125	−2.14976
$852$	0	0
$853$	−23.9293	−0.819323	−0.409661	−	0.912238i	$-0.634353\pi$
$853$	−23.9293	−0.819323	−0.409661	+	0.912238i	$0.634353\pi$
$854$	0	0
$855$	−49.8151	−1.70364
$856$	0	0
$857$	10.5437	0.360166	0.180083	−	0.983651i	$-0.442363\pi$
$857$	10.5437	0.360166	0.180083	+	0.983651i	$0.442363\pi$
$858$	0	0
$859$	0.453499	0.0154732	0.00773659	−	0.999970i	$-0.497537\pi$
$859$	0.453499	0.0154732	0.00773659	+	0.999970i	$0.497537\pi$
$860$	0	0
$861$	106.718	3.63694
$862$	0	0
$863$	22.2539	0.757532	0.378766	−	0.925492i	$-0.376348\pi$
$863$	22.2539	0.757532	0.378766	+	0.925492i	$0.376348\pi$
$864$	0	0
$865$	38.3785	1.30491
$866$	0	0
$867$	−3.44977	−0.117160
$868$	0	0
$869$	−3.78977	−0.128559
$870$	0	0
$871$	−1.13644	−0.0385068
$872$	0	0
$873$	38.1231	1.29027
$874$	0	0
$875$	13.3071	0.449863
$876$	0	0
$877$	−36.7436	−1.24074	−0.620372	−	0.784307i	$-0.713018\pi$
$877$	−36.7436	−1.24074	−0.620372	+	0.784307i	$0.713018\pi$
$878$	0	0
$879$	40.8336	1.37728
$880$	0	0
$881$	21.5379	0.725630	0.362815	−	0.931861i	$-0.381816\pi$
$881$	21.5379	0.725630	0.362815	+	0.931861i	$0.381816\pi$
$882$	0	0
$883$	−45.6665	−1.53680	−0.768400	−	0.639970i	$-0.778947\pi$
$883$	−45.6665	−1.53680	−0.768400	+	0.639970i	$0.778947\pi$
$884$	0	0
$885$	10.0619	0.338227
$886$	0	0
$887$	−55.7732	−1.87268	−0.936340	−	0.351095i	$-0.885809\pi$
$887$	−55.7732	−1.87268	−0.936340	+	0.351095i	$0.885809\pi$
$888$	0	0
$889$	−33.5541	−1.12537
$890$	0	0
$891$	−40.3331	−1.35121
$892$	0	0
$893$	−3.12413	−0.104545
$894$	0	0
$895$	−15.9965	−0.534703
$896$	0	0
$897$	−10.1291	−0.338202
$898$	0	0
$899$	−34.1036	−1.13742
$900$	0	0
$901$	−11.4768	−0.382348
$902$	0	0
$903$	−82.9889	−2.76170
$904$	0	0
$905$	−44.5311	−1.48026
$906$	0	0
$907$	−15.1892	−0.504350	−0.252175	−	0.967682i	$-0.581146\pi$
$907$	−15.1892	−0.504350	−0.252175	+	0.967682i	$0.581146\pi$
$908$	0	0
$909$	33.2924	1.10424
$910$	0	0
$911$	−40.0177	−1.32584	−0.662922	−	0.748688i	$-0.730684\pi$
$911$	−40.0177	−1.32584	−0.662922	+	0.748688i	$0.730684\pi$
$912$	0	0
$913$	−8.99157	−0.297578
$914$	0	0
$915$	−57.8893	−1.91376
$916$	0	0
$917$	10.5561	0.348592
$918$	0	0
$919$	−38.3276	−1.26431	−0.632155	−	0.774842i	$-0.717830\pi$
$919$	−38.3276	−1.26431	−0.632155	+	0.774842i	$0.717830\pi$
$920$	0	0
$921$	−23.9339	−0.788648
$922$	0	0
$923$	5.57160	0.183392
$924$	0	0
$925$	−36.4324	−1.19789
$926$	0	0
$927$	−50.8968	−1.67167
$928$	0	0
$929$	43.0413	1.41214	0.706069	−	0.708143i	$-0.250467\pi$
$929$	43.0413	1.41214	0.706069	+	0.708143i	$0.250467\pi$
$930$	0	0
$931$	−4.48881	−0.147115
$932$	0	0
$933$	−83.4953	−2.73351
$934$	0	0
$935$	2.70289	0.0883939
$936$	0	0
$937$	51.1076	1.66961	0.834806	−	0.550544i	$-0.185580\pi$
$937$	51.1076	1.66961	0.834806	+	0.550544i	$0.185580\pi$
$938$	0	0
$939$	50.5421	1.64938
$940$	0	0
$941$	−6.13936	−0.200137	−0.100069	−	0.994981i	$-0.531906\pi$
$941$	−6.13936	−0.200137	−0.100069	+	0.994981i	$0.531906\pi$
$942$	0	0
$943$	61.1085	1.98997
$944$	0	0
$945$	181.450	5.90258
$946$	0	0
$947$	−43.7836	−1.42278	−0.711388	−	0.702800i	$-0.751933\pi$
$947$	−43.7836	−1.42278	−0.711388	+	0.702800i	$0.751933\pi$
$948$	0	0
$949$	−5.65741	−0.183647
$950$	0	0
$951$	90.5929	2.93768
$952$	0	0
$953$	−34.9575	−1.13238	−0.566192	−	0.824273i	$-0.691584\pi$
$953$	−34.9575	−1.13238	−0.566192	+	0.824273i	$0.691584\pi$
$954$	0	0
$955$	−5.40662	−0.174954
$956$	0	0
$957$	−14.0647	−0.454648
$958$	0	0
$959$	−37.6258	−1.21500
$960$	0	0
$961$	29.0887	0.938346
$962$	0	0
$963$	−97.9840	−3.15749
$964$	0	0
$965$	2.59694	0.0835984
$966$	0	0
$967$	15.3327	0.493065	0.246532	−	0.969135i	$-0.420709\pi$
$967$	15.3327	0.493065	0.246532	+	0.969135i	$0.420709\pi$
$968$	0	0
$969$	−6.61952	−0.212649
$970$	0	0
$971$	7.69288	0.246876	0.123438	−	0.992352i	$-0.460608\pi$
$971$	7.69288	0.246876	0.123438	+	0.992352i	$0.460608\pi$
$972$	0	0
$973$	25.4279	0.815181
$974$	0	0
$975$	−5.88446	−0.188454
$976$	0	0
$977$	43.7302	1.39905	0.699527	−	0.714606i	$-0.253394\pi$
$977$	43.7302	1.39905	0.699527	+	0.714606i	$0.253394\pi$
$978$	0	0
$979$	15.1761	0.485031
$980$	0	0
$981$	128.717	4.10961
$982$	0	0
$983$	39.0850	1.24662	0.623309	−	0.781976i	$-0.285788\pi$
$983$	39.0850	1.24662	0.623309	+	0.781976i	$0.285788\pi$
$984$	0	0
$985$	37.0457	1.18037
$986$	0	0
$987$	17.1649	0.546365
$988$	0	0
$989$	−47.5208	−1.51107
$990$	0	0
$991$	−13.7550	−0.436942	−0.218471	−	0.975843i	$-0.570107\pi$
$991$	−13.7550	−0.436942	−0.218471	+	0.975843i	$0.570107\pi$
$992$	0	0
$993$	−106.067	−3.36593
$994$	0	0
$995$	−39.7691	−1.26077
$996$	0	0
$997$	5.58869	0.176996	0.0884978	−	0.996076i	$-0.471793\pi$
$997$	5.58869	0.176996	0.0884978	+	0.996076i	$0.471793\pi$
$998$	0	0
$999$	211.471	6.69065

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8024.2.a.bc.1.1	✓	33

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8024.2.a.bc.1.1	✓	33	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 8024 = 2^{3} \cdot 17 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8024.a (trivial)

\(f(q)\)	\(=\)	\(q-3.44977 q^{3} +2.91669 q^{5} -3.05603 q^{7} +8.90091 q^{9} +O(q^{10})\)	\(q-3.44977 q^{3} +2.91669 q^{5} -3.05603 q^{7} +8.90091 q^{9} -0.926697 q^{11} +0.486374 q^{13} -10.0619 q^{15} -1.00000 q^{17} -1.91883 q^{19} +10.5426 q^{21} +6.03687 q^{23} +3.50708 q^{25} -20.3568 q^{27} -4.39950 q^{29} +7.75169 q^{31} +3.19689 q^{33} -8.91351 q^{35} -10.3882 q^{37} -1.67788 q^{39} +10.1225 q^{41} -7.87176 q^{43} +25.9612 q^{45} +1.62814 q^{47} +2.33935 q^{49} +3.44977 q^{51} +11.4768 q^{53} -2.70289 q^{55} +6.61952 q^{57} -1.00000 q^{59} +5.75331 q^{61} -27.2015 q^{63} +1.41860 q^{65} -2.33655 q^{67} -20.8258 q^{69} +11.4554 q^{71} -11.6318 q^{73} -12.0986 q^{75} +2.83202 q^{77} +4.08954 q^{79} +43.5235 q^{81} +9.70282 q^{83} -2.91669 q^{85} +15.1773 q^{87} -16.3766 q^{89} -1.48638 q^{91} -26.7416 q^{93} -5.59663 q^{95} +4.28306 q^{97} -8.24845 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 33 q - 3 q^{3} + 3 q^{5} + 48 q^{9}+O(q^{10}) \) 33 * q - 3 * q^3 + 3 * q^5 + 48 * q^9	\( 33 q - 3 q^{3} + 3 q^{5} + 48 q^{9} - 7 q^{11} + 9 q^{13} - 7 q^{15} - 33 q^{17} + 5 q^{19} + 14 q^{21} - 19 q^{23} + 52 q^{25} - 24 q^{27} + 14 q^{29} + 19 q^{31} + 34 q^{33} + 15 q^{35} + 13 q^{37} + 26 q^{39} + 32 q^{41} + 11 q^{43} - 27 q^{47} + 93 q^{49} + 3 q^{51} - 7 q^{53} + 31 q^{55} + 6 q^{57} - 33 q^{59} + 33 q^{61} + 10 q^{63} + 21 q^{65} - 10 q^{67} + 78 q^{69} - 29 q^{71} + 30 q^{73} + 31 q^{75} + 6 q^{77} + 66 q^{79} + 97 q^{81} - 9 q^{83} - 3 q^{85} - 27 q^{87} + 68 q^{89} + 28 q^{91} + 34 q^{93} + 8 q^{95} + 62 q^{97} + 15 q^{99}+O(q^{100}) \) 33 * q - 3 * q^3 + 3 * q^5 + 48 * q^9 - 7 * q^11 + 9 * q^13 - 7 * q^15 - 33 * q^17 + 5 * q^19 + 14 * q^21 - 19 * q^23 + 52 * q^25 - 24 * q^27 + 14 * q^29 + 19 * q^31 + 34 * q^33 + 15 * q^35 + 13 * q^37 + 26 * q^39 + 32 * q^41 + 11 * q^43 - 27 * q^47 + 93 * q^49 + 3 * q^51 - 7 * q^53 + 31 * q^55 + 6 * q^57 - 33 * q^59 + 33 * q^61 + 10 * q^63 + 21 * q^65 - 10 * q^67 + 78 * q^69 - 29 * q^71 + 30 * q^73 + 31 * q^75 + 6 * q^77 + 66 * q^79 + 97 * q^81 - 9 * q^83 - 3 * q^85 - 27 * q^87 + 68 * q^89 + 28 * q^91 + 34 * q^93 + 8 * q^95 + 62 * q^97 + 15 * q^99

Introduction

L-functions

Modular forms

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