LMFDB - Embedded newform 8024.2.a.v.1.9

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:	$18$
Coefficient field:	$\mathbb{Q}[x]/(x^{18} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{18} - 9 x^{17} - 2 x^{16} + 212 x^{15} - 289 x^{14} - 2094 x^{13} + 3933 x^{12} + 11326 x^{11} + \cdots + 1136 $ x^18 - 9x^17 - 2x^16 + 212x^15 - 289x^14 - 2094x^13 + 3933x^12 + 11326x^11 - 23166x^10 - 36429x^9 + 72042x^8 + 69272x^7 - 119982x^6 - 69890x^5 + 98783x^4 + 26872x^3 - 30932x^2 + 1600*x + 1136
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Root			$0.379802$ of defining polynomial
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)

$f(q)$	$=$	$q+0.620198 q^{3} -3.37733 q^{5} -0.896557 q^{7} -2.61535 q^{9} +O(q^{10})$	$q+0.620198 q^{3} -3.37733 q^{5} -0.896557 q^{7} -2.61535 q^{9} -0.189536 q^{11} +4.57419 q^{13} -2.09461 q^{15} +1.00000 q^{17} +2.08224 q^{19} -0.556043 q^{21} +3.42992 q^{23} +6.40637 q^{25} -3.48263 q^{27} -2.75137 q^{29} -4.80694 q^{31} -0.117550 q^{33} +3.02797 q^{35} -3.43877 q^{37} +2.83690 q^{39} -8.57341 q^{41} +3.31226 q^{43} +8.83292 q^{45} -7.35013 q^{47} -6.19618 q^{49} +0.620198 q^{51} -3.71934 q^{53} +0.640127 q^{55} +1.29140 q^{57} -1.00000 q^{59} +9.56801 q^{61} +2.34482 q^{63} -15.4486 q^{65} -6.75326 q^{67} +2.12723 q^{69} -4.45526 q^{71} +8.62477 q^{73} +3.97321 q^{75} +0.169930 q^{77} +11.8455 q^{79} +5.68615 q^{81} -15.0672 q^{83} -3.37733 q^{85} -1.70639 q^{87} -2.36786 q^{89} -4.10103 q^{91} -2.98125 q^{93} -7.03241 q^{95} +14.2649 q^{97} +0.495705 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q + 9 q^{3} + 2 q^{5} + 11 q^{7} + 31 q^{9}+O(q^{10}) $ 18 * q + 9 * q^3 + 2 * q^5 + 11 * q^7 + 31 * q^9	$ 18 q + 9 q^{3} + 2 q^{5} + 11 q^{7} + 31 q^{9} - q^{11} + 15 q^{13} + 3 q^{15} + 18 q^{17} + 26 q^{19} - 4 q^{21} + 22 q^{23} + 42 q^{25} + 45 q^{27} + 6 q^{29} + 13 q^{31} - 5 q^{33} + 4 q^{35} + 4 q^{37} + 36 q^{39} - 15 q^{41} + 12 q^{43} + 14 q^{45} + 8 q^{47} - 13 q^{49} + 9 q^{51} - 11 q^{53} + 55 q^{55} - 20 q^{57} - 18 q^{59} + 53 q^{61} + 29 q^{63} - 26 q^{65} + 2 q^{67} + 32 q^{69} + 8 q^{71} - 42 q^{73} + 72 q^{75} + 6 q^{77} - 9 q^{79} + 42 q^{81} - 4 q^{83} + 2 q^{85} + 36 q^{87} + 13 q^{89} + 68 q^{91} + q^{93} + 3 q^{95} - 56 q^{97} - 13 q^{99}+O(q^{100}) $ 18 * q + 9 * q^3 + 2 * q^5 + 11 * q^7 + 31 * q^9 - q^11 + 15 * q^13 + 3 * q^15 + 18 * q^17 + 26 * q^19 - 4 * q^21 + 22 * q^23 + 42 * q^25 + 45 * q^27 + 6 * q^29 + 13 * q^31 - 5 * q^33 + 4 * q^35 + 4 * q^37 + 36 * q^39 - 15 * q^41 + 12 * q^43 + 14 * q^45 + 8 * q^47 - 13 * q^49 + 9 * q^51 - 11 * q^53 + 55 * q^55 - 20 * q^57 - 18 * q^59 + 53 * q^61 + 29 * q^63 - 26 * q^65 + 2 * q^67 + 32 * q^69 + 8 * q^71 - 42 * q^73 + 72 * q^75 + 6 * q^77 - 9 * q^79 + 42 * q^81 - 4 * q^83 + 2 * q^85 + 36 * q^87 + 13 * q^89 + 68 * q^91 + q^93 + 3 * q^95 - 56 * q^97 - 13 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	0.620198	0.358071	0.179036	−	0.983843i	$-0.442702\pi$
$3$	0.620198	0.358071	0.179036	+	0.983843i	$0.442702\pi$
$4$	0	0
$5$	−3.37733	−1.51039	−0.755194	−	0.655501i	$-0.772458\pi$
$5$	−3.37733	−1.51039	−0.755194	+	0.655501i	$0.772458\pi$
$6$	0	0
$7$	−0.896557	−0.338867	−0.169433	−	0.985542i	$-0.554194\pi$
$7$	−0.896557	−0.338867	−0.169433	+	0.985542i	$0.554194\pi$
$8$	0	0
$9$	−2.61535	−0.871785
$10$	0	0
$11$	−0.189536	−0.0571473	−0.0285737	−	0.999592i	$-0.509097\pi$
$11$	−0.189536	−0.0571473	−0.0285737	+	0.999592i	$0.509097\pi$
$12$	0	0
$13$	4.57419	1.26865	0.634327	−	0.773065i	$-0.281277\pi$
$13$	4.57419	1.26865	0.634327	+	0.773065i	$0.281277\pi$
$14$	0	0
$15$	−2.09461	−0.540827
$16$	0	0
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	0	0
$19$	2.08224	0.477699	0.238849	−	0.971057i	$-0.423230\pi$
$19$	2.08224	0.477699	0.238849	+	0.971057i	$0.423230\pi$
$20$	0	0
$21$	−0.556043	−0.121338
$22$	0	0
$23$	3.42992	0.715188	0.357594	−	0.933877i	$-0.383597\pi$
$23$	3.42992	0.715188	0.357594	+	0.933877i	$0.383597\pi$
$24$	0	0
$25$	6.40637	1.28127
$26$	0	0
$27$	−3.48263	−0.670232
$28$	0	0
$29$	−2.75137	−0.510917	−0.255458	−	0.966820i	$-0.582226\pi$
$29$	−2.75137	−0.510917	−0.255458	+	0.966820i	$0.582226\pi$
$30$	0	0
$31$	−4.80694	−0.863352	−0.431676	−	0.902029i	$-0.642078\pi$
$31$	−4.80694	−0.863352	−0.431676	+	0.902029i	$0.642078\pi$
$32$	0	0
$33$	−0.117550	−0.0204628
$34$	0	0
$35$	3.02797	0.511821
$36$	0	0
$37$	−3.43877	−0.565331	−0.282665	−	0.959219i	$-0.591219\pi$
$37$	−3.43877	−0.565331	−0.282665	+	0.959219i	$0.591219\pi$
$38$	0	0
$39$	2.83690	0.454268
$40$	0	0
$41$	−8.57341	−1.33894	−0.669470	−	0.742839i	$-0.733479\pi$
$41$	−8.57341	−1.33894	−0.669470	+	0.742839i	$0.733479\pi$
$42$	0	0
$43$	3.31226	0.505115	0.252557	−	0.967582i	$-0.418728\pi$
$43$	3.31226	0.505115	0.252557	+	0.967582i	$0.418728\pi$
$44$	0	0
$45$	8.83292	1.31673
$46$	0	0
$47$	−7.35013	−1.07213	−0.536063	−	0.844178i	$-0.680089\pi$
$47$	−7.35013	−1.07213	−0.536063	+	0.844178i	$0.680089\pi$
$48$	0	0
$49$	−6.19618	−0.885169
$50$	0	0
$51$	0.620198	0.0868450
$52$	0	0
$53$	−3.71934	−0.510891	−0.255446	−	0.966823i	$-0.582222\pi$
$53$	−3.71934	−0.510891	−0.255446	+	0.966823i	$0.582222\pi$
$54$	0	0
$55$	0.640127	0.0863147
$56$	0	0
$57$	1.29140	0.171050
$58$	0	0
$59$	−1.00000	−0.130189
$59$	−1.00000	−0.130189
$60$	0	0
$61$	9.56801	1.22506	0.612529	−	0.790448i	$-0.290152\pi$
$61$	9.56801	1.22506	0.612529	+	0.790448i	$0.290152\pi$
$62$	0	0
$63$	2.34482	0.295419
$64$	0	0
$65$	−15.4486	−1.91616
$66$	0	0
$67$	−6.75326	−0.825042	−0.412521	−	0.910948i	$-0.635352\pi$
$67$	−6.75326	−0.825042	−0.412521	+	0.910948i	$0.635352\pi$
$68$	0	0
$69$	2.12723	0.256088
$70$	0	0
$71$	−4.45526	−0.528742	−0.264371	−	0.964421i	$-0.585164\pi$
$71$	−4.45526	−0.528742	−0.264371	+	0.964421i	$0.585164\pi$
$72$	0	0
$73$	8.62477	1.00945	0.504727	−	0.863279i	$-0.331593\pi$
$73$	8.62477	1.00945	0.504727	+	0.863279i	$0.331593\pi$
$74$	0	0
$75$	3.97321	0.458787
$76$	0	0
$77$	0.169930	0.0193653
$78$	0	0
$79$	11.8455	1.33272	0.666361	−	0.745629i	$-0.267851\pi$
$79$	11.8455	1.33272	0.666361	+	0.745629i	$0.267851\pi$
$80$	0	0
$81$	5.68615	0.631794
$82$	0	0
$83$	−15.0672	−1.65384	−0.826922	−	0.562316i	$-0.809910\pi$
$83$	−15.0672	−1.65384	−0.826922	+	0.562316i	$0.809910\pi$
$84$	0	0
$85$	−3.37733	−0.366323
$86$	0	0
$87$	−1.70639	−0.182945
$88$	0	0
$89$	−2.36786	−0.250993	−0.125497	−	0.992094i	$-0.540052\pi$
$89$	−2.36786	−0.250993	−0.125497	+	0.992094i	$0.540052\pi$
$90$	0	0
$91$	−4.10103	−0.429905
$92$	0	0
$93$	−2.98125	−0.309142
$94$	0	0
$95$	−7.03241	−0.721510
$96$	0	0
$97$	14.2649	1.44838	0.724192	−	0.689599i	$-0.242213\pi$
$97$	14.2649	1.44838	0.724192	+	0.689599i	$0.242213\pi$
$98$	0	0
$99$	0.495705	0.0498202
$100$	0	0
$101$	−12.9216	−1.28575	−0.642873	−	0.765972i	$-0.722258\pi$
$101$	−12.9216	−1.28575	−0.642873	+	0.765972i	$0.722258\pi$
$102$	0	0
$103$	8.66341	0.853631	0.426816	−	0.904339i	$-0.359635\pi$
$103$	8.66341	0.853631	0.426816	+	0.904339i	$0.359635\pi$
$104$	0	0
$105$	1.87794	0.183268
$106$	0	0
$107$	2.96675	0.286806	0.143403	−	0.989664i	$-0.454195\pi$
$107$	2.96675	0.286806	0.143403	+	0.989664i	$0.454195\pi$
$108$	0	0
$109$	−11.6544	−1.11629	−0.558145	−	0.829744i	$-0.688487\pi$
$109$	−11.6544	−1.11629	−0.558145	+	0.829744i	$0.688487\pi$
$110$	0	0
$111$	−2.13272	−0.202429
$112$	0	0
$113$	13.6665	1.28563	0.642817	−	0.766020i	$-0.277766\pi$
$113$	13.6665	1.28563	0.642817	+	0.766020i	$0.277766\pi$
$114$	0	0
$115$	−11.5840	−1.08021
$116$	0	0
$117$	−11.9631	−1.10599
$118$	0	0
$119$	−0.896557	−0.0821873
$120$	0	0
$121$	−10.9641	−0.996734
$122$	0	0
$123$	−5.31721	−0.479436
$124$	0	0
$125$	−4.74977	−0.424833
$126$	0	0
$127$	13.3387	1.18362	0.591808	−	0.806079i	$-0.298415\pi$
$127$	13.3387	1.18362	0.591808	+	0.806079i	$0.298415\pi$
$128$	0	0
$129$	2.05425	0.180867
$130$	0	0
$131$	−2.95385	−0.258079	−0.129039	−	0.991639i	$-0.541189\pi$
$131$	−2.95385	−0.258079	−0.129039	+	0.991639i	$0.541189\pi$
$132$	0	0
$133$	−1.86685	−0.161876
$134$	0	0
$135$	11.7620	1.01231
$136$	0	0
$137$	11.4226	0.975901	0.487950	−	0.872871i	$-0.337745\pi$
$137$	11.4226	0.975901	0.487950	+	0.872871i	$0.337745\pi$
$138$	0	0
$139$	5.08540	0.431338	0.215669	−	0.976466i	$-0.430807\pi$
$139$	5.08540	0.431338	0.215669	+	0.976466i	$0.430807\pi$
$140$	0	0
$141$	−4.55853	−0.383898
$142$	0	0
$143$	−0.866976	−0.0725002
$144$	0	0
$145$	9.29229	0.771683
$146$	0	0
$147$	−3.84286	−0.316954
$148$	0	0
$149$	21.5497	1.76542	0.882708	−	0.469922i	$-0.155718\pi$
$149$	21.5497	1.76542	0.882708	+	0.469922i	$0.155718\pi$
$150$	0	0
$151$	0.626274	0.0509655	0.0254827	−	0.999675i	$-0.491888\pi$
$151$	0.626274	0.0509655	0.0254827	+	0.999675i	$0.491888\pi$
$152$	0	0
$153$	−2.61535	−0.211439
$154$	0	0
$155$	16.2346	1.30400
$156$	0	0
$157$	20.2598	1.61691	0.808453	−	0.588561i	$-0.200305\pi$
$157$	20.2598	1.61691	0.808453	+	0.588561i	$0.200305\pi$
$158$	0	0
$159$	−2.30673	−0.182935
$160$	0	0
$161$	−3.07512	−0.242353
$162$	0	0
$163$	−13.3014	−1.04185	−0.520925	−	0.853603i	$-0.674413\pi$
$163$	−13.3014	−1.04185	−0.520925	+	0.853603i	$0.674413\pi$
$164$	0	0
$165$	0.397005	0.0309068
$166$	0	0
$167$	23.1786	1.79362	0.896808	−	0.442420i	$-0.145880\pi$
$167$	23.1786	1.79362	0.896808	+	0.442420i	$0.145880\pi$
$168$	0	0
$169$	7.92326	0.609481
$170$	0	0
$171$	−5.44580	−0.416450
$172$	0	0
$173$	12.4629	0.947537	0.473768	−	0.880649i	$-0.342893\pi$
$173$	12.4629	0.947537	0.473768	+	0.880649i	$0.342893\pi$
$174$	0	0
$175$	−5.74368	−0.434181
$176$	0	0
$177$	−0.620198	−0.0466169
$178$	0	0
$179$	−4.77556	−0.356942	−0.178471	−	0.983945i	$-0.557115\pi$
$179$	−4.77556	−0.356942	−0.178471	+	0.983945i	$0.557115\pi$
$180$	0	0
$181$	9.90426	0.736178	0.368089	−	0.929791i	$-0.380012\pi$
$181$	9.90426	0.736178	0.368089	+	0.929791i	$0.380012\pi$
$182$	0	0
$183$	5.93406	0.438658
$184$	0	0
$185$	11.6139	0.853869
$186$	0	0
$187$	−0.189536	−0.0138603
$188$	0	0
$189$	3.12238	0.227120
$190$	0	0
$191$	1.53120	0.110794	0.0553970	−	0.998464i	$-0.482358\pi$
$191$	1.53120	0.110794	0.0553970	+	0.998464i	$0.482358\pi$
$192$	0	0
$193$	2.80859	0.202167	0.101083	−	0.994878i	$-0.467769\pi$
$193$	2.80859	0.202167	0.101083	+	0.994878i	$0.467769\pi$
$194$	0	0
$195$	−9.58117	−0.686122
$196$	0	0
$197$	−3.34963	−0.238651	−0.119326	−	0.992855i	$-0.538073\pi$
$197$	−3.34963	−0.238651	−0.119326	+	0.992855i	$0.538073\pi$
$198$	0	0
$199$	2.79001	0.197779	0.0988894	−	0.995098i	$-0.468471\pi$
$199$	2.79001	0.197779	0.0988894	+	0.995098i	$0.468471\pi$
$200$	0	0
$201$	−4.18835	−0.295424
$202$	0	0
$203$	2.46676	0.173133
$204$	0	0
$205$	28.9552	2.02232
$206$	0	0
$207$	−8.97046	−0.623490
$208$	0	0
$209$	−0.394660	−0.0272992
$210$	0	0
$211$	−6.69887	−0.461169	−0.230585	−	0.973052i	$-0.574064\pi$
$211$	−6.69887	−0.461169	−0.230585	+	0.973052i	$0.574064\pi$
$212$	0	0
$213$	−2.76314	−0.189327
$214$	0	0
$215$	−11.1866	−0.762919
$216$	0	0
$217$	4.30970	0.292561
$218$	0	0
$219$	5.34906	0.361456
$220$	0	0
$221$	4.57419	0.307694
$222$	0	0
$223$	−26.7589	−1.79191	−0.895953	−	0.444149i	$-0.853506\pi$
$223$	−26.7589	−1.79191	−0.895953	+	0.444149i	$0.853506\pi$
$224$	0	0
$225$	−16.7549	−1.11700
$226$	0	0
$227$	20.5518	1.36407	0.682037	−	0.731318i	$-0.261095\pi$
$227$	20.5518	1.36407	0.682037	+	0.731318i	$0.261095\pi$
$228$	0	0
$229$	11.0034	0.727124	0.363562	−	0.931570i	$-0.381561\pi$
$229$	11.0034	0.727124	0.363562	+	0.931570i	$0.381561\pi$
$230$	0	0
$231$	0.105390	0.00693417
$232$	0	0
$233$	6.40470	0.419586	0.209793	−	0.977746i	$-0.432721\pi$
$233$	6.40470	0.419586	0.209793	+	0.977746i	$0.432721\pi$
$234$	0	0
$235$	24.8238	1.61933
$236$	0	0
$237$	7.34655	0.477210
$238$	0	0
$239$	24.8831	1.60955	0.804776	−	0.593579i	$-0.202285\pi$
$239$	24.8831	1.60955	0.804776	+	0.593579i	$0.202285\pi$
$240$	0	0
$241$	16.2910	1.04940	0.524698	−	0.851288i	$-0.324178\pi$
$241$	16.2910	1.04940	0.524698	+	0.851288i	$0.324178\pi$
$242$	0	0
$243$	13.9744	0.896460
$244$	0	0
$245$	20.9266	1.33695
$246$	0	0
$247$	9.52457	0.606034
$248$	0	0
$249$	−9.34467	−0.592194
$250$	0	0
$251$	17.6022	1.11104	0.555521	−	0.831503i	$-0.312519\pi$
$251$	17.6022	1.11104	0.555521	+	0.831503i	$0.312519\pi$
$252$	0	0
$253$	−0.650094	−0.0408711
$254$	0	0
$255$	−2.09461	−0.131170
$256$	0	0
$257$	−15.4553	−0.964073	−0.482037	−	0.876151i	$-0.660103\pi$
$257$	−15.4553	−0.964073	−0.482037	+	0.876151i	$0.660103\pi$
$258$	0	0
$259$	3.08306	0.191572
$260$	0	0
$261$	7.19581	0.445410
$262$	0	0
$263$	5.73805	0.353824	0.176912	−	0.984227i	$-0.443389\pi$
$263$	5.73805	0.353824	0.176912	+	0.984227i	$0.443389\pi$
$264$	0	0
$265$	12.5615	0.771644
$266$	0	0
$267$	−1.46854	−0.0898734
$268$	0	0
$269$	−6.16761	−0.376046	−0.188023	−	0.982165i	$-0.560208\pi$
$269$	−6.16761	−0.376046	−0.188023	+	0.982165i	$0.560208\pi$
$270$	0	0
$271$	−9.86511	−0.599263	−0.299631	−	0.954055i	$-0.596864\pi$
$271$	−9.86511	−0.599263	−0.299631	+	0.954055i	$0.596864\pi$
$272$	0	0
$273$	−2.54345	−0.153936
$274$	0	0
$275$	−1.21424	−0.0732214
$276$	0	0
$277$	15.4612	0.928973	0.464486	−	0.885580i	$-0.346239\pi$
$277$	15.4612	0.928973	0.464486	+	0.885580i	$0.346239\pi$
$278$	0	0
$279$	12.5719	0.752657
$280$	0	0
$281$	−11.2274	−0.669772	−0.334886	−	0.942259i	$-0.608698\pi$
$281$	−11.2274	−0.669772	−0.334886	+	0.942259i	$0.608698\pi$
$282$	0	0
$283$	5.37991	0.319803	0.159901	−	0.987133i	$-0.448882\pi$
$283$	5.37991	0.319803	0.159901	+	0.987133i	$0.448882\pi$
$284$	0	0
$285$	−4.36149	−0.258352
$286$	0	0
$287$	7.68655	0.453723
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	8.84707	0.518625
$292$	0	0
$293$	−1.34380	−0.0785055	−0.0392528	−	0.999229i	$-0.512498\pi$
$293$	−1.34380	−0.0785055	−0.0392528	+	0.999229i	$0.512498\pi$
$294$	0	0
$295$	3.37733	0.196636
$296$	0	0
$297$	0.660085	0.0383020
$298$	0	0
$299$	15.6891	0.907325
$300$	0	0
$301$	−2.96963	−0.171167
$302$	0	0
$303$	−8.01394	−0.460389
$304$	0	0
$305$	−32.3144	−1.85031
$306$	0	0
$307$	−9.13884	−0.521581	−0.260791	−	0.965395i	$-0.583983\pi$
$307$	−9.13884	−0.521581	−0.260791	+	0.965395i	$0.583983\pi$
$308$	0	0
$309$	5.37303	0.305661
$310$	0	0
$311$	12.1256	0.687581	0.343790	−	0.939046i	$-0.388289\pi$
$311$	12.1256	0.687581	0.343790	+	0.939046i	$0.388289\pi$
$312$	0	0
$313$	20.4585	1.15638	0.578190	−	0.815902i	$-0.303759\pi$
$313$	20.4585	1.15638	0.578190	+	0.815902i	$0.303759\pi$
$314$	0	0
$315$	−7.91922	−0.446198
$316$	0	0
$317$	17.6476	0.991190	0.495595	−	0.868554i	$-0.334950\pi$
$317$	17.6476	0.991190	0.495595	+	0.868554i	$0.334950\pi$
$318$	0	0
$319$	0.521485	0.0291975
$320$	0	0
$321$	1.83997	0.102697
$322$	0	0
$323$	2.08224	0.115859
$324$	0	0
$325$	29.3040	1.62549
$326$	0	0
$327$	−7.22804	−0.399711
$328$	0	0
$329$	6.58981	0.363308
$330$	0	0
$331$	−29.5414	−1.62374	−0.811872	−	0.583835i	$-0.801551\pi$
$331$	−29.5414	−1.62374	−0.811872	+	0.583835i	$0.801551\pi$
$332$	0	0
$333$	8.99361	0.492847
$334$	0	0
$335$	22.8080	1.24613
$336$	0	0
$337$	−32.2634	−1.75750	−0.878751	−	0.477281i	$-0.841622\pi$
$337$	−32.2634	−1.75750	−0.878751	+	0.477281i	$0.841622\pi$
$338$	0	0
$339$	8.47591	0.460348
$340$	0	0
$341$	0.911090	0.0493383
$342$	0	0
$343$	11.8311	0.638821
$344$	0	0
$345$	−7.18435	−0.386793
$346$	0	0
$347$	34.2933	1.84096	0.920481	−	0.390787i	$-0.127797\pi$
$347$	34.2933	1.84096	0.920481	+	0.390787i	$0.127797\pi$
$348$	0	0
$349$	−5.38601	−0.288306	−0.144153	−	0.989555i	$-0.546046\pi$
$349$	−5.38601	−0.288306	−0.144153	+	0.989555i	$0.546046\pi$
$350$	0	0
$351$	−15.9302	−0.850293
$352$	0	0
$353$	−6.93777	−0.369260	−0.184630	−	0.982808i	$-0.559109\pi$
$353$	−6.93777	−0.369260	−0.184630	+	0.982808i	$0.559109\pi$
$354$	0	0
$355$	15.0469	0.798605
$356$	0	0
$357$	−0.556043	−0.0294289
$358$	0	0
$359$	4.67984	0.246993	0.123496	−	0.992345i	$-0.460589\pi$
$359$	4.67984	0.246993	0.123496	+	0.992345i	$0.460589\pi$
$360$	0	0
$361$	−14.6643	−0.771804
$362$	0	0
$363$	−6.79989	−0.356902
$364$	0	0
$365$	−29.1287	−1.52467
$366$	0	0
$367$	27.7674	1.44945	0.724724	−	0.689039i	$-0.241967\pi$
$367$	27.7674	1.44945	0.724724	+	0.689039i	$0.241967\pi$
$368$	0	0
$369$	22.4225	1.16727
$370$	0	0
$371$	3.33461	0.173124
$372$	0	0
$373$	20.1602	1.04386	0.521928	−	0.852990i	$-0.325213\pi$
$373$	20.1602	1.04386	0.521928	+	0.852990i	$0.325213\pi$
$374$	0	0
$375$	−2.94580	−0.152120
$376$	0	0
$377$	−12.5853	−0.648176
$378$	0	0
$379$	37.9837	1.95109	0.975545	−	0.219800i	$-0.0705405\pi$
$379$	37.9837	1.95109	0.975545	+	0.219800i	$0.0705405\pi$
$380$	0	0
$381$	8.27261	0.423819
$382$	0	0
$383$	−35.0963	−1.79334	−0.896668	−	0.442705i	$-0.854019\pi$
$383$	−35.0963	−1.79334	−0.896668	+	0.442705i	$0.854019\pi$
$384$	0	0
$385$	−0.573911	−0.0292492
$386$	0	0
$387$	−8.66273	−0.440351
$388$	0	0
$389$	21.3036	1.08013	0.540067	−	0.841622i	$-0.318399\pi$
$389$	21.3036	1.08013	0.540067	+	0.841622i	$0.318399\pi$
$390$	0	0
$391$	3.42992	0.173458
$392$	0	0
$393$	−1.83197	−0.0924106
$394$	0	0
$395$	−40.0062	−2.01293
$396$	0	0
$397$	1.21077	0.0607666	0.0303833	−	0.999538i	$-0.490327\pi$
$397$	1.21077	0.0607666	0.0303833	+	0.999538i	$0.490327\pi$
$398$	0	0
$399$	−1.15781	−0.0579632
$400$	0	0
$401$	−34.9212	−1.74388	−0.871941	−	0.489610i	$-0.837139\pi$
$401$	−34.9212	−1.74388	−0.871941	+	0.489610i	$0.837139\pi$
$402$	0	0
$403$	−21.9879	−1.09529
$404$	0	0
$405$	−19.2040	−0.954254
$406$	0	0
$407$	0.651772	0.0323072
$408$	0	0
$409$	−3.88952	−0.192324	−0.0961621	−	0.995366i	$-0.530657\pi$
$409$	−3.88952	−0.192324	−0.0961621	+	0.995366i	$0.530657\pi$
$410$	0	0
$411$	7.08429	0.349442
$412$	0	0
$413$	0.896557	0.0441167
$414$	0	0
$415$	50.8871	2.49795
$416$	0	0
$417$	3.15396	0.154450
$418$	0	0
$419$	31.1554	1.52204	0.761020	−	0.648728i	$-0.224699\pi$
$419$	31.1554	1.52204	0.761020	+	0.648728i	$0.224699\pi$
$420$	0	0
$421$	−9.79618	−0.477436	−0.238718	−	0.971089i	$-0.576727\pi$
$421$	−9.79618	−0.477436	−0.238718	+	0.971089i	$0.576727\pi$
$422$	0	0
$423$	19.2232	0.934664
$424$	0	0
$425$	6.40637	0.310755
$426$	0	0
$427$	−8.57827	−0.415132
$428$	0	0
$429$	−0.537696	−0.0259602
$430$	0	0
$431$	11.4566	0.551846	0.275923	−	0.961180i	$-0.411017\pi$
$431$	11.4566	0.551846	0.275923	+	0.961180i	$0.411017\pi$
$432$	0	0
$433$	12.5526	0.603239	0.301620	−	0.953428i	$-0.402473\pi$
$433$	12.5526	0.603239	0.301620	+	0.953428i	$0.402473\pi$
$434$	0	0
$435$	5.76306	0.276317
$436$	0	0
$437$	7.14191	0.341644
$438$	0	0
$439$	−14.3345	−0.684150	−0.342075	−	0.939673i	$-0.611130\pi$
$439$	−14.3345	−0.684150	−0.342075	+	0.939673i	$0.611130\pi$
$440$	0	0
$441$	16.2052	0.771677
$442$	0	0
$443$	−23.4673	−1.11496	−0.557482	−	0.830189i	$-0.688232\pi$
$443$	−23.4673	−1.11496	−0.557482	+	0.830189i	$0.688232\pi$
$444$	0	0
$445$	7.99706	0.379097
$446$	0	0
$447$	13.3650	0.632145
$448$	0	0
$449$	−26.3405	−1.24309	−0.621543	−	0.783380i	$-0.713494\pi$
$449$	−26.3405	−1.24309	−0.621543	+	0.783380i	$0.713494\pi$
$450$	0	0
$451$	1.62497	0.0765169
$452$	0	0
$453$	0.388414	0.0182493
$454$	0	0
$455$	13.8505	0.649323
$456$	0	0
$457$	24.8142	1.16076	0.580379	−	0.814347i	$-0.302904\pi$
$457$	24.8142	1.16076	0.580379	+	0.814347i	$0.302904\pi$
$458$	0	0
$459$	−3.48263	−0.162555
$460$	0	0
$461$	−39.6818	−1.84817	−0.924083	−	0.382193i	$-0.875169\pi$
$461$	−39.6818	−1.84817	−0.924083	+	0.382193i	$0.875169\pi$
$462$	0	0
$463$	−26.0030	−1.20846	−0.604232	−	0.796809i	$-0.706520\pi$
$463$	−26.0030	−1.20846	−0.604232	+	0.796809i	$0.706520\pi$
$464$	0	0
$465$	10.0687	0.466924
$466$	0	0
$467$	6.45999	0.298933	0.149466	−	0.988767i	$-0.452244\pi$
$467$	6.45999	0.298933	0.149466	+	0.988767i	$0.452244\pi$
$468$	0	0
$469$	6.05468	0.279579
$470$	0	0
$471$	12.5651	0.578968
$472$	0	0
$473$	−0.627793	−0.0288660
$474$	0	0
$475$	13.3396	0.612063
$476$	0	0
$477$	9.72741	0.445387
$478$	0	0
$479$	−31.2516	−1.42792	−0.713961	−	0.700186i	$-0.753100\pi$
$479$	−31.2516	−1.42792	−0.713961	+	0.700186i	$0.753100\pi$
$480$	0	0
$481$	−15.7296	−0.717209
$482$	0	0
$483$	−1.90718	−0.0867798
$484$	0	0
$485$	−48.1774	−2.18762
$486$	0	0
$487$	27.1954	1.23234	0.616170	−	0.787613i	$-0.288684\pi$
$487$	27.1954	1.23234	0.616170	+	0.787613i	$0.288684\pi$
$488$	0	0
$489$	−8.24952	−0.373056
$490$	0	0
$491$	14.4141	0.650498	0.325249	−	0.945628i	$-0.394552\pi$
$491$	14.4141	0.650498	0.325249	+	0.945628i	$0.394552\pi$
$492$	0	0
$493$	−2.75137	−0.123916
$494$	0	0
$495$	−1.67416	−0.0752479
$496$	0	0
$497$	3.99439	0.179173
$498$	0	0
$499$	15.1251	0.677090	0.338545	−	0.940950i	$-0.390065\pi$
$499$	15.1251	0.677090	0.338545	+	0.940950i	$0.390065\pi$
$500$	0	0
$501$	14.3753	0.642242
$502$	0	0
$503$	11.8496	0.528345	0.264173	−	0.964475i	$-0.414901\pi$
$503$	11.8496	0.528345	0.264173	+	0.964475i	$0.414901\pi$
$504$	0	0
$505$	43.6405	1.94198
$506$	0	0
$507$	4.91399	0.218238
$508$	0	0
$509$	−21.8618	−0.969009	−0.484504	−	0.874789i	$-0.661000\pi$
$509$	−21.8618	−0.969009	−0.484504	+	0.874789i	$0.661000\pi$
$510$	0	0
$511$	−7.73260	−0.342070
$512$	0	0
$513$	−7.25167	−0.320169
$514$	0	0
$515$	−29.2592	−1.28931
$516$	0	0
$517$	1.39312	0.0612692
$518$	0	0
$519$	7.72946	0.339286
$520$	0	0
$521$	1.63823	0.0717722	0.0358861	−	0.999356i	$-0.488575\pi$
$521$	1.63823	0.0717722	0.0358861	+	0.999356i	$0.488575\pi$
$522$	0	0
$523$	17.6021	0.769687	0.384844	−	0.922982i	$-0.374255\pi$
$523$	17.6021	0.769687	0.384844	+	0.922982i	$0.374255\pi$
$524$	0	0
$525$	−3.56222	−0.155468
$526$	0	0
$527$	−4.80694	−0.209394
$528$	0	0
$529$	−11.2357	−0.488507
$530$	0	0
$531$	2.61535	0.113497
$532$	0	0
$533$	−39.2164	−1.69865
$534$	0	0
$535$	−10.0197	−0.433189
$536$	0	0
$537$	−2.96179	−0.127811
$538$	0	0
$539$	1.17440	0.0505851
$540$	0	0
$541$	9.22019	0.396407	0.198204	−	0.980161i	$-0.436489\pi$
$541$	9.22019	0.396407	0.198204	+	0.980161i	$0.436489\pi$
$542$	0	0
$543$	6.14260	0.263604
$544$	0	0
$545$	39.3608	1.68603
$546$	0	0
$547$	22.7933	0.974573	0.487286	−	0.873242i	$-0.337987\pi$
$547$	22.7933	0.974573	0.487286	+	0.873242i	$0.337987\pi$
$548$	0	0
$549$	−25.0238	−1.06799
$550$	0	0
$551$	−5.72901	−0.244064
$552$	0	0
$553$	−10.6202	−0.451615
$554$	0	0
$555$	7.20290	0.305746
$556$	0	0
$557$	−1.04801	−0.0444055	−0.0222027	−	0.999753i	$-0.507068\pi$
$557$	−1.04801	−0.0444055	−0.0222027	+	0.999753i	$0.507068\pi$
$558$	0	0
$559$	15.1509	0.640815
$560$	0	0
$561$	−0.117550	−0.00496296
$562$	0	0
$563$	−5.12577	−0.216026	−0.108013	−	0.994150i	$-0.534449\pi$
$563$	−5.12577	−0.216026	−0.108013	+	0.994150i	$0.534449\pi$
$564$	0	0
$565$	−46.1562	−1.94181
$566$	0	0
$567$	−5.09796	−0.214094
$568$	0	0
$569$	−38.4194	−1.61062	−0.805312	−	0.592851i	$-0.798002\pi$
$569$	−38.4194	−1.61062	−0.805312	+	0.592851i	$0.798002\pi$
$570$	0	0
$571$	33.8278	1.41565	0.707825	−	0.706388i	$-0.249677\pi$
$571$	33.8278	1.41565	0.707825	+	0.706388i	$0.249677\pi$
$572$	0	0
$573$	0.949649	0.0396722
$574$	0	0
$575$	21.9733	0.916351
$576$	0	0
$577$	−3.99444	−0.166291	−0.0831454	−	0.996537i	$-0.526497\pi$
$577$	−3.99444	−0.166291	−0.0831454	+	0.996537i	$0.526497\pi$
$578$	0	0
$579$	1.74188	0.0723901
$580$	0	0
$581$	13.5086	0.560433
$582$	0	0
$583$	0.704951	0.0291961
$584$	0	0
$585$	40.4035	1.67048
$586$	0	0
$587$	22.9826	0.948594	0.474297	−	0.880365i	$-0.342702\pi$
$587$	22.9826	0.948594	0.474297	+	0.880365i	$0.342702\pi$
$588$	0	0
$589$	−10.0092	−0.412422
$590$	0	0
$591$	−2.07743	−0.0854542
$592$	0	0
$593$	−3.67822	−0.151047	−0.0755233	−	0.997144i	$-0.524063\pi$
$593$	−3.67822	−0.151047	−0.0755233	+	0.997144i	$0.524063\pi$
$594$	0	0
$595$	3.02797	0.124135
$596$	0	0
$597$	1.73036	0.0708189
$598$	0	0
$599$	19.6087	0.801189	0.400594	−	0.916255i	$-0.368804\pi$
$599$	19.6087	0.801189	0.400594	+	0.916255i	$0.368804\pi$
$600$	0	0
$601$	1.03932	0.0423949	0.0211974	−	0.999775i	$-0.493252\pi$
$601$	1.03932	0.0423949	0.0211974	+	0.999775i	$0.493252\pi$
$602$	0	0
$603$	17.6622	0.719259
$604$	0	0
$605$	37.0293	1.50546
$606$	0	0
$607$	−23.9072	−0.970365	−0.485182	−	0.874413i	$-0.661247\pi$
$607$	−23.9072	−0.970365	−0.485182	+	0.874413i	$0.661247\pi$
$608$	0	0
$609$	1.52988	0.0619939
$610$	0	0
$611$	−33.6209	−1.36016
$612$	0	0
$613$	31.6251	1.27733	0.638663	−	0.769487i	$-0.279488\pi$
$613$	31.6251	1.27733	0.638663	+	0.769487i	$0.279488\pi$
$614$	0	0
$615$	17.9580	0.724135
$616$	0	0
$617$	27.9296	1.12440	0.562201	−	0.827001i	$-0.309955\pi$
$617$	27.9296	1.12440	0.562201	+	0.827001i	$0.309955\pi$
$618$	0	0
$619$	−21.2528	−0.854223	−0.427111	−	0.904199i	$-0.640469\pi$
$619$	−21.2528	−0.854223	−0.427111	+	0.904199i	$0.640469\pi$
$620$	0	0
$621$	−11.9451	−0.479342
$622$	0	0
$623$	2.12293	0.0850533
$624$	0	0
$625$	−15.9903	−0.639611
$626$	0	0
$627$	−0.244767	−0.00977506
$628$	0	0
$629$	−3.43877	−0.137113
$630$	0	0
$631$	40.5826	1.61557	0.807783	−	0.589479i	$-0.200667\pi$
$631$	40.5826	1.61557	0.807783	+	0.589479i	$0.200667\pi$
$632$	0	0
$633$	−4.15462	−0.165131
$634$	0	0
$635$	−45.0491	−1.78772
$636$	0	0
$637$	−28.3426	−1.12297
$638$	0	0
$639$	11.6521	0.460949
$640$	0	0
$641$	−43.4305	−1.71540	−0.857701	−	0.514149i	$-0.828108\pi$
$641$	−43.4305	−1.71540	−0.857701	+	0.514149i	$0.828108\pi$
$642$	0	0
$643$	−18.0425	−0.711527	−0.355763	−	0.934576i	$-0.615779\pi$
$643$	−18.0425	−0.711527	−0.355763	+	0.934576i	$0.615779\pi$
$644$	0	0
$645$	−6.93790	−0.273180
$646$	0	0
$647$	−35.7809	−1.40669	−0.703346	−	0.710848i	$-0.748311\pi$
$647$	−35.7809	−1.40669	−0.703346	+	0.710848i	$0.748311\pi$
$648$	0	0
$649$	0.189536	0.00743995
$650$	0	0
$651$	2.67287	0.104758
$652$	0	0
$653$	−26.5231	−1.03793	−0.518965	−	0.854796i	$-0.673682\pi$
$653$	−26.5231	−1.03793	−0.518965	+	0.854796i	$0.673682\pi$
$654$	0	0
$655$	9.97612	0.389799
$656$	0	0
$657$	−22.5568	−0.880026
$658$	0	0
$659$	7.81636	0.304482	0.152241	−	0.988343i	$-0.451351\pi$
$659$	7.81636	0.304482	0.152241	+	0.988343i	$0.451351\pi$
$660$	0	0
$661$	7.93963	0.308816	0.154408	−	0.988007i	$-0.450653\pi$
$661$	7.93963	0.308816	0.154408	+	0.988007i	$0.450653\pi$
$662$	0	0
$663$	2.83690	0.110176
$664$	0	0
$665$	6.30496	0.244496
$666$	0	0
$667$	−9.43698	−0.365401
$668$	0	0
$669$	−16.5958	−0.641630
$670$	0	0
$671$	−1.81349	−0.0700088
$672$	0	0
$673$	8.74300	0.337018	0.168509	−	0.985700i	$-0.446105\pi$
$673$	8.74300	0.337018	0.168509	+	0.985700i	$0.446105\pi$
$674$	0	0
$675$	−22.3110	−0.858751
$676$	0	0
$677$	−33.2635	−1.27842	−0.639210	−	0.769032i	$-0.720739\pi$
$677$	−33.2635	−1.27842	−0.639210	+	0.769032i	$0.720739\pi$
$678$	0	0
$679$	−12.7893	−0.490809
$680$	0	0
$681$	12.7462	0.488435
$682$	0	0
$683$	27.6558	1.05822	0.529110	−	0.848553i	$-0.322526\pi$
$683$	27.6558	1.05822	0.529110	+	0.848553i	$0.322526\pi$
$684$	0	0
$685$	−38.5780	−1.47399
$686$	0	0
$687$	6.82427	0.260362
$688$	0	0
$689$	−17.0130	−0.648144
$690$	0	0
$691$	−24.9467	−0.949018	−0.474509	−	0.880251i	$-0.657374\pi$
$691$	−24.9467	−0.949018	−0.474509	+	0.880251i	$0.657374\pi$
$692$	0	0
$693$	−0.444428	−0.0168824
$694$	0	0
$695$	−17.1751	−0.651488
$696$	0	0
$697$	−8.57341	−0.324741
$698$	0	0
$699$	3.97218	0.150242
$700$	0	0
$701$	31.3961	1.18582	0.592908	−	0.805270i	$-0.297980\pi$
$701$	31.3961	1.18582	0.592908	+	0.805270i	$0.297980\pi$
$702$	0	0
$703$	−7.16035	−0.270058
$704$	0	0
$705$	15.3957	0.579835
$706$	0	0
$707$	11.5850	0.435697
$708$	0	0
$709$	9.09140	0.341435	0.170717	−	0.985320i	$-0.445391\pi$
$709$	9.09140	0.341435	0.170717	+	0.985320i	$0.445391\pi$
$710$	0	0
$711$	−30.9802	−1.16185
$712$	0	0
$713$	−16.4874	−0.617459
$714$	0	0
$715$	2.92807	0.109503
$716$	0	0
$717$	15.4324	0.576334
$718$	0	0
$719$	−17.5737	−0.655390	−0.327695	−	0.944784i	$-0.606272\pi$
$719$	−17.5737	−0.655390	−0.327695	+	0.944784i	$0.606272\pi$
$720$	0	0
$721$	−7.76724	−0.289267
$722$	0	0
$723$	10.1036	0.375759
$724$	0	0
$725$	−17.6263	−0.654624
$726$	0	0
$727$	47.4929	1.76141	0.880706	−	0.473662i	$-0.157068\pi$
$727$	47.4929	1.76141	0.880706	+	0.473662i	$0.157068\pi$
$728$	0	0
$729$	−8.39153	−0.310797
$730$	0	0
$731$	3.31226	0.122508
$732$	0	0
$733$	25.8516	0.954849	0.477424	−	0.878673i	$-0.341570\pi$
$733$	25.8516	0.954849	0.477424	+	0.878673i	$0.341570\pi$
$734$	0	0
$735$	12.9786	0.478723
$736$	0	0
$737$	1.27999	0.0471489
$738$	0	0
$739$	13.9185	0.512001	0.256000	−	0.966677i	$-0.417595\pi$
$739$	13.9185	0.512001	0.256000	+	0.966677i	$0.417595\pi$
$740$	0	0
$741$	5.90712	0.217003
$742$	0	0
$743$	46.6759	1.71237	0.856186	−	0.516668i	$-0.172828\pi$
$743$	46.6759	1.71237	0.856186	+	0.516668i	$0.172828\pi$
$744$	0	0
$745$	−72.7803	−2.66646
$746$	0	0
$747$	39.4062	1.44180
$748$	0	0
$749$	−2.65986	−0.0971891
$750$	0	0
$751$	−4.86355	−0.177474	−0.0887368	−	0.996055i	$-0.528283\pi$
$751$	−4.86355	−0.177474	−0.0887368	+	0.996055i	$0.528283\pi$
$752$	0	0
$753$	10.9168	0.397832
$754$	0	0
$755$	−2.11514	−0.0769776
$756$	0	0
$757$	−39.2234	−1.42560	−0.712799	−	0.701369i	$-0.752573\pi$
$757$	−39.2234	−1.42560	−0.712799	+	0.701369i	$0.752573\pi$
$758$	0	0
$759$	−0.403187	−0.0146348
$760$	0	0
$761$	8.14711	0.295333	0.147666	−	0.989037i	$-0.452824\pi$
$761$	8.14711	0.295333	0.147666	+	0.989037i	$0.452824\pi$
$762$	0	0
$763$	10.4488	0.378274
$764$	0	0
$765$	8.83292	0.319355
$766$	0	0
$767$	−4.57419	−0.165165
$768$	0	0
$769$	−41.5947	−1.49994	−0.749972	−	0.661470i	$-0.769933\pi$
$769$	−41.5947	−1.49994	−0.749972	+	0.661470i	$0.769933\pi$
$770$	0	0
$771$	−9.58532	−0.345207
$772$	0	0
$773$	47.2373	1.69901	0.849504	−	0.527583i	$-0.176902\pi$
$773$	47.2373	1.69901	0.849504	+	0.527583i	$0.176902\pi$
$774$	0	0
$775$	−30.7950	−1.10619
$776$	0	0
$777$	1.91211	0.0685964
$778$	0	0
$779$	−17.8519	−0.639610
$780$	0	0
$781$	0.844433	0.0302162
$782$	0	0
$783$	9.58201	0.342433
$784$	0	0
$785$	−68.4240	−2.44216
$786$	0	0
$787$	5.22680	0.186315	0.0931577	−	0.995651i	$-0.470304\pi$
$787$	5.22680	0.186315	0.0931577	+	0.995651i	$0.470304\pi$
$788$	0	0
$789$	3.55873	0.126694
$790$	0	0
$791$	−12.2528	−0.435659
$792$	0	0
$793$	43.7660	1.55417
$794$	0	0
$795$	7.79059	0.276304
$796$	0	0
$797$	−14.3628	−0.508755	−0.254378	−	0.967105i	$-0.581871\pi$
$797$	−14.3628	−0.508755	−0.254378	+	0.967105i	$0.581871\pi$
$798$	0	0
$799$	−7.35013	−0.260029
$800$	0	0
$801$	6.19281	0.218812
$802$	0	0
$803$	−1.63471	−0.0576876
$804$	0	0
$805$	10.3857	0.366048
$806$	0	0
$807$	−3.82514	−0.134651
$808$	0	0
$809$	−32.2324	−1.13323	−0.566616	−	0.823982i	$-0.691748\pi$
$809$	−32.2324	−1.13323	−0.566616	+	0.823982i	$0.691748\pi$
$810$	0	0
$811$	−8.87761	−0.311735	−0.155868	−	0.987778i	$-0.549817\pi$
$811$	−8.87761	−0.311735	−0.155868	+	0.987778i	$0.549817\pi$
$812$	0	0
$813$	−6.11832	−0.214579
$814$	0	0
$815$	44.9234	1.57360
$816$	0	0
$817$	6.89692	0.241293
$818$	0	0
$819$	10.7256	0.374784
$820$	0	0
$821$	−40.4221	−1.41074	−0.705370	−	0.708839i	$-0.749219\pi$
$821$	−40.4221	−1.41074	−0.705370	+	0.708839i	$0.749219\pi$
$822$	0	0
$823$	37.4782	1.30641	0.653204	−	0.757182i	$-0.273424\pi$
$823$	37.4782	1.30641	0.653204	+	0.757182i	$0.273424\pi$
$824$	0	0
$825$	−0.753068	−0.0262185
$826$	0	0
$827$	−11.5563	−0.401854	−0.200927	−	0.979606i	$-0.564395\pi$
$827$	−11.5563	−0.401854	−0.200927	+	0.979606i	$0.564395\pi$
$828$	0	0
$829$	56.2831	1.95479	0.977396	−	0.211416i	$-0.0678076\pi$
$829$	56.2831	1.95479	0.977396	+	0.211416i	$0.0678076\pi$
$830$	0	0
$831$	9.58899	0.332638
$832$	0	0
$833$	−6.19618	−0.214685
$834$	0	0
$835$	−78.2819	−2.70906
$836$	0	0
$837$	16.7408	0.578647
$838$	0	0
$839$	21.2794	0.734645	0.367323	−	0.930094i	$-0.380275\pi$
$839$	21.2794	0.734645	0.367323	+	0.930094i	$0.380275\pi$
$840$	0	0
$841$	−21.4300	−0.738964
$842$	0	0
$843$	−6.96322	−0.239826
$844$	0	0
$845$	−26.7595	−0.920554
$846$	0	0
$847$	9.82992	0.337760
$848$	0	0
$849$	3.33661	0.114512
$850$	0	0
$851$	−11.7947	−0.404318
$852$	0	0
$853$	−5.09347	−0.174397	−0.0871986	−	0.996191i	$-0.527791\pi$
$853$	−5.09347	−0.174397	−0.0871986	+	0.996191i	$0.527791\pi$
$854$	0	0
$855$	18.3923	0.629002
$856$	0	0
$857$	19.3384	0.660587	0.330293	−	0.943878i	$-0.392852\pi$
$857$	19.3384	0.660587	0.330293	+	0.943878i	$0.392852\pi$
$858$	0	0
$859$	33.2710	1.13519	0.567595	−	0.823308i	$-0.307874\pi$
$859$	33.2710	1.13519	0.567595	+	0.823308i	$0.307874\pi$
$860$	0	0
$861$	4.76718	0.162465
$862$	0	0
$863$	14.5293	0.494584	0.247292	−	0.968941i	$-0.420459\pi$
$863$	14.5293	0.494584	0.247292	+	0.968941i	$0.420459\pi$
$864$	0	0
$865$	−42.0914	−1.43115
$866$	0	0
$867$	0.620198	0.0210630
$868$	0	0
$869$	−2.24515	−0.0761615
$870$	0	0
$871$	−30.8907	−1.04669
$872$	0	0
$873$	−37.3078	−1.26268
$874$	0	0
$875$	4.25845	0.143962
$876$	0	0
$877$	−3.06361	−0.103451	−0.0517254	−	0.998661i	$-0.516472\pi$
$877$	−3.06361	−0.103451	−0.0517254	+	0.998661i	$0.516472\pi$
$878$	0	0
$879$	−0.833421	−0.0281106
$880$	0	0
$881$	−49.9112	−1.68155	−0.840775	−	0.541385i	$-0.817900\pi$
$881$	−49.9112	−1.68155	−0.840775	+	0.541385i	$0.817900\pi$
$882$	0	0
$883$	−32.4972	−1.09362	−0.546808	−	0.837258i	$-0.684157\pi$
$883$	−32.4972	−1.09362	−0.546808	+	0.837258i	$0.684157\pi$
$884$	0	0
$885$	2.09461	0.0704097
$886$	0	0
$887$	48.5815	1.63121	0.815603	−	0.578612i	$-0.196405\pi$
$887$	48.5815	1.63121	0.815603	+	0.578612i	$0.196405\pi$
$888$	0	0
$889$	−11.9589	−0.401088
$890$	0	0
$891$	−1.07773	−0.0361053
$892$	0	0
$893$	−15.3047	−0.512153
$894$	0	0
$895$	16.1287	0.539122
$896$	0	0
$897$	9.73036	0.324887
$898$	0	0
$899$	13.2257	0.441101
$900$	0	0
$901$	−3.71934	−0.123909
$902$	0	0
$903$	−1.84176	−0.0612899
$904$	0	0
$905$	−33.4500	−1.11191
$906$	0	0
$907$	50.0009	1.66025	0.830126	−	0.557575i	$-0.188268\pi$
$907$	50.0009	1.66025	0.830126	+	0.557575i	$0.188268\pi$
$908$	0	0
$909$	33.7946	1.12089
$910$	0	0
$911$	32.1778	1.06610	0.533049	−	0.846085i	$-0.321046\pi$
$911$	32.1778	1.06610	0.533049	+	0.846085i	$0.321046\pi$
$912$	0	0
$913$	2.85579	0.0945128
$914$	0	0
$915$	−20.0413	−0.662545
$916$	0	0
$917$	2.64829	0.0874543
$918$	0	0
$919$	−28.1310	−0.927954	−0.463977	−	0.885847i	$-0.653578\pi$
$919$	−28.1310	−0.927954	−0.463977	+	0.885847i	$0.653578\pi$
$920$	0	0
$921$	−5.66789	−0.186763
$922$	0	0
$923$	−20.3792	−0.670790
$924$	0	0
$925$	−22.0301	−0.724344
$926$	0	0
$927$	−22.6579	−0.744183
$928$	0	0
$929$	22.9570	0.753196	0.376598	−	0.926377i	$-0.377094\pi$
$929$	22.9570	0.753196	0.376598	+	0.926377i	$0.377094\pi$
$930$	0	0
$931$	−12.9019	−0.422844
$932$	0	0
$933$	7.52028	0.246203
$934$	0	0
$935$	0.640127	0.0209344
$936$	0	0
$937$	7.74122	0.252895	0.126447	−	0.991973i	$-0.459643\pi$
$937$	7.74122	0.252895	0.126447	+	0.991973i	$0.459643\pi$
$938$	0	0
$939$	12.6883	0.414067
$940$	0	0
$941$	48.6550	1.58611	0.793054	−	0.609151i	$-0.208490\pi$
$941$	48.6550	1.58611	0.793054	+	0.609151i	$0.208490\pi$
$942$	0	0
$943$	−29.4061	−0.957594
$944$	0	0
$945$	−10.5453	−0.343039
$946$	0	0
$947$	−3.44821	−0.112052	−0.0560259	−	0.998429i	$-0.517843\pi$
$947$	−3.44821	−0.112052	−0.0560259	+	0.998429i	$0.517843\pi$
$948$	0	0
$949$	39.4514	1.28065
$950$	0	0
$951$	10.9450	0.354917
$952$	0	0
$953$	−34.6044	−1.12094	−0.560472	−	0.828173i	$-0.689380\pi$
$953$	−34.6044	−1.12094	−0.560472	+	0.828173i	$0.689380\pi$
$954$	0	0
$955$	−5.17138	−0.167342
$956$	0	0
$957$	0.323424	0.0104548
$958$	0	0
$959$	−10.2410	−0.330700
$960$	0	0
$961$	−7.89331	−0.254623
$962$	0	0
$963$	−7.75910	−0.250033
$964$	0	0
$965$	−9.48554	−0.305350
$966$	0	0
$967$	−9.08681	−0.292212	−0.146106	−	0.989269i	$-0.546674\pi$
$967$	−9.08681	−0.292212	−0.146106	+	0.989269i	$0.546674\pi$
$968$	0	0
$969$	1.29140	0.0414857
$970$	0	0
$971$	28.6387	0.919059	0.459529	−	0.888163i	$-0.348018\pi$
$971$	28.6387	0.919059	0.459529	+	0.888163i	$0.348018\pi$
$972$	0	0
$973$	−4.55936	−0.146166
$974$	0	0
$975$	18.1743	0.582042
$976$	0	0
$977$	−16.5284	−0.528789	−0.264395	−	0.964415i	$-0.585172\pi$
$977$	−16.5284	−0.528789	−0.264395	+	0.964415i	$0.585172\pi$
$978$	0	0
$979$	0.448796	0.0143436
$980$	0	0
$981$	30.4804	0.973165
$982$	0	0
$983$	19.5899	0.624821	0.312411	−	0.949947i	$-0.398864\pi$
$983$	19.5899	0.624821	0.312411	+	0.949947i	$0.398864\pi$
$984$	0	0
$985$	11.3128	0.360456
$986$	0	0
$987$	4.08699	0.130090
$988$	0	0
$989$	11.3608	0.361252
$990$	0	0
$991$	2.71115	0.0861225	0.0430612	−	0.999072i	$-0.486289\pi$
$991$	2.71115	0.0861225	0.0430612	+	0.999072i	$0.486289\pi$
$992$	0	0
$993$	−18.3215	−0.581416
$994$	0	0
$995$	−9.42280	−0.298723
$996$	0	0
$997$	4.45021	0.140940	0.0704698	−	0.997514i	$-0.477550\pi$
$997$	4.45021	0.140940	0.0704698	+	0.997514i	$0.477550\pi$
$998$	0	0
$999$	11.9760	0.378903

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8024.2.a.v.1.9	✓	18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8024.2.a.v.1.9	✓	18	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)

\(f(q)\)	\(=\)	\(q+0.620198 q^{3} -3.37733 q^{5} -0.896557 q^{7} -2.61535 q^{9} +O(q^{10})\)	\(q+0.620198 q^{3} -3.37733 q^{5} -0.896557 q^{7} -2.61535 q^{9} -0.189536 q^{11} +4.57419 q^{13} -2.09461 q^{15} +1.00000 q^{17} +2.08224 q^{19} -0.556043 q^{21} +3.42992 q^{23} +6.40637 q^{25} -3.48263 q^{27} -2.75137 q^{29} -4.80694 q^{31} -0.117550 q^{33} +3.02797 q^{35} -3.43877 q^{37} +2.83690 q^{39} -8.57341 q^{41} +3.31226 q^{43} +8.83292 q^{45} -7.35013 q^{47} -6.19618 q^{49} +0.620198 q^{51} -3.71934 q^{53} +0.640127 q^{55} +1.29140 q^{57} -1.00000 q^{59} +9.56801 q^{61} +2.34482 q^{63} -15.4486 q^{65} -6.75326 q^{67} +2.12723 q^{69} -4.45526 q^{71} +8.62477 q^{73} +3.97321 q^{75} +0.169930 q^{77} +11.8455 q^{79} +5.68615 q^{81} -15.0672 q^{83} -3.37733 q^{85} -1.70639 q^{87} -2.36786 q^{89} -4.10103 q^{91} -2.98125 q^{93} -7.03241 q^{95} +14.2649 q^{97} +0.495705 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q + 9 q^{3} + 2 q^{5} + 11 q^{7} + 31 q^{9}+O(q^{10}) \) 18 * q + 9 * q^3 + 2 * q^5 + 11 * q^7 + 31 * q^9	\( 18 q + 9 q^{3} + 2 q^{5} + 11 q^{7} + 31 q^{9} - q^{11} + 15 q^{13} + 3 q^{15} + 18 q^{17} + 26 q^{19} - 4 q^{21} + 22 q^{23} + 42 q^{25} + 45 q^{27} + 6 q^{29} + 13 q^{31} - 5 q^{33} + 4 q^{35} + 4 q^{37} + 36 q^{39} - 15 q^{41} + 12 q^{43} + 14 q^{45} + 8 q^{47} - 13 q^{49} + 9 q^{51} - 11 q^{53} + 55 q^{55} - 20 q^{57} - 18 q^{59} + 53 q^{61} + 29 q^{63} - 26 q^{65} + 2 q^{67} + 32 q^{69} + 8 q^{71} - 42 q^{73} + 72 q^{75} + 6 q^{77} - 9 q^{79} + 42 q^{81} - 4 q^{83} + 2 q^{85} + 36 q^{87} + 13 q^{89} + 68 q^{91} + q^{93} + 3 q^{95} - 56 q^{97} - 13 q^{99}+O(q^{100}) \) 18 * q + 9 * q^3 + 2 * q^5 + 11 * q^7 + 31 * q^9 - q^11 + 15 * q^13 + 3 * q^15 + 18 * q^17 + 26 * q^19 - 4 * q^21 + 22 * q^23 + 42 * q^25 + 45 * q^27 + 6 * q^29 + 13 * q^31 - 5 * q^33 + 4 * q^35 + 4 * q^37 + 36 * q^39 - 15 * q^41 + 12 * q^43 + 14 * q^45 + 8 * q^47 - 13 * q^49 + 9 * q^51 - 11 * q^53 + 55 * q^55 - 20 * q^57 - 18 * q^59 + 53 * q^61 + 29 * q^63 - 26 * q^65 + 2 * q^67 + 32 * q^69 + 8 * q^71 - 42 * q^73 + 72 * q^75 + 6 * q^77 - 9 * q^79 + 42 * q^81 - 4 * q^83 + 2 * q^85 + 36 * q^87 + 13 * q^89 + 68 * q^91 + q^93 + 3 * q^95 - 56 * q^97 - 13 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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