LMFDB - Embedded newform 7440.2.a.bp.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7440,2,Mod(1,7440)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7440, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("7440.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7440 = 2^{4} \cdot 3 \cdot 5 \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7440.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:	$59.4086991038$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.148.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 3x + 1 $ x^3 - x^2 - 3*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 465)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.17009$ of defining polynomial
Character	$\chi$	$=$	7440.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +1.00000 q^{5} +1.17009 q^{7} +1.00000 q^{9} -2.00000 q^{11} +0.0917087 q^{13} -1.00000 q^{15} -6.04945 q^{17} -1.17009 q^{21} +1.70928 q^{23} +1.00000 q^{25} -1.00000 q^{27} +7.95774 q^{29} -1.00000 q^{31} +2.00000 q^{33} +1.17009 q^{35} -3.35350 q^{37} -0.0917087 q^{39} -4.68035 q^{41} +0.738205 q^{43} +1.00000 q^{45} +9.12783 q^{47} -5.63090 q^{49} +6.04945 q^{51} +4.97107 q^{53} -2.00000 q^{55} -11.0361 q^{59} -3.26180 q^{61} +1.17009 q^{63} +0.0917087 q^{65} -3.85043 q^{67} -1.70928 q^{69} +1.21953 q^{71} +7.66701 q^{73} -1.00000 q^{75} -2.34017 q^{77} -3.52586 q^{79} +1.00000 q^{81} -12.5464 q^{83} -6.04945 q^{85} -7.95774 q^{87} -5.77432 q^{89} +0.107307 q^{91} +1.00000 q^{93} -12.0000 q^{97} -2.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{3} + 3 q^{5} - 2 q^{7} + 3 q^{9} - 6 q^{11} - 2 q^{13} - 3 q^{15} + 2 q^{21} - 2 q^{23} + 3 q^{25} - 3 q^{27} + 8 q^{29} - 3 q^{31} + 6 q^{33} - 2 q^{35} + 2 q^{39} + 8 q^{41} + 10 q^{43} + 3 q^{45}+ \cdots - 6 q^{99}+O(q^{100}) $ 3 * q - 3 * q^3 + 3 * q^5 - 2 * q^7 + 3 * q^9 - 6 * q^11 - 2 * q^13 - 3 * q^15 + 2 * q^21 - 2 * q^23 + 3 * q^25 - 3 * q^27 + 8 * q^29 - 3 * q^31 + 6 * q^33 - 2 * q^35 + 2 * q^39 + 8 * q^41 + 10 * q^43 + 3 * q^45 + 6 * q^47 - 13 * q^49 - 6 * q^55 - 14 * q^59 - 2 * q^61 - 2 * q^63 - 2 * q^65 + 16 * q^67 + 2 * q^69 - 20 * q^71 - 3 * q^75 + 4 * q^77 - 8 * q^79 + 3 * q^81 - 2 * q^83 - 8 * q^87 - 6 * q^89 + 12 * q^91 + 3 * q^93 - 36 * q^97 - 6 * q^99

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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	1.17009	0.442251	0.221126	−	0.975245i	$-0.429027\pi$
$7$	1.17009	0.442251	0.221126	+	0.975245i	$0.429027\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−2.00000	−0.603023	−0.301511	−	0.953463i	$-0.597491\pi$
$11$	−2.00000	−0.603023	−0.301511	+	0.953463i	$0.597491\pi$
$12$	0	0
$13$	0.0917087	0.0254354	0.0127177	−	0.999919i	$-0.495952\pi$
$13$	0.0917087	0.0254354	0.0127177	+	0.999919i	$0.495952\pi$
$14$	0	0
$15$	−1.00000	−0.258199
$16$	0	0
$17$	−6.04945	−1.46721	−0.733603	−	0.679578i	$-0.762163\pi$
$17$	−6.04945	−1.46721	−0.733603	+	0.679578i	$0.762163\pi$
$18$	0	0
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	0	0
$21$	−1.17009	−0.255334
$22$	0	0
$23$	1.70928	0.356409	0.178204	−	0.983994i	$-0.442971\pi$
$23$	1.70928	0.356409	0.178204	+	0.983994i	$0.442971\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	7.95774	1.47772	0.738858	−	0.673862i	$-0.235366\pi$
$29$	7.95774	1.47772	0.738858	+	0.673862i	$0.235366\pi$
$30$	0	0
$31$	−1.00000	−0.179605
$31$	−1.00000	−0.179605
$32$	0	0
$33$	2.00000	0.348155
$34$	0	0
$35$	1.17009	0.197781
$36$	0	0
$37$	−3.35350	−0.551313	−0.275656	−	0.961256i	$-0.588895\pi$
$37$	−3.35350	−0.551313	−0.275656	+	0.961256i	$0.588895\pi$
$38$	0	0
$39$	−0.0917087	−0.0146852
$40$	0	0
$41$	−4.68035	−0.730947	−0.365474	−	0.930822i	$-0.619093\pi$
$41$	−4.68035	−0.730947	−0.365474	+	0.930822i	$0.619093\pi$
$42$	0	0
$43$	0.738205	0.112575	0.0562876	−	0.998415i	$-0.482074\pi$
$43$	0.738205	0.112575	0.0562876	+	0.998415i	$0.482074\pi$
$44$	0	0
$45$	1.00000	0.149071
$46$	0	0
$47$	9.12783	1.33143	0.665715	−	0.746206i	$-0.268127\pi$
$47$	9.12783	1.33143	0.665715	+	0.746206i	$0.268127\pi$
$48$	0	0
$49$	−5.63090	−0.804414
$50$	0	0
$51$	6.04945	0.847092
$52$	0	0
$53$	4.97107	0.682829	0.341415	−	0.939913i	$-0.389094\pi$
$53$	4.97107	0.682829	0.341415	+	0.939913i	$0.389094\pi$
$54$	0	0
$55$	−2.00000	−0.269680
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−11.0361	−1.43678	−0.718390	−	0.695641i	$-0.755121\pi$
$59$	−11.0361	−1.43678	−0.718390	+	0.695641i	$0.755121\pi$
$60$	0	0
$61$	−3.26180	−0.417630	−0.208815	−	0.977955i	$-0.566961\pi$
$61$	−3.26180	−0.417630	−0.208815	+	0.977955i	$0.566961\pi$
$62$	0	0
$63$	1.17009	0.147417
$64$	0	0
$65$	0.0917087	0.0113751
$66$	0	0
$67$	−3.85043	−0.470405	−0.235203	−	0.971946i	$-0.575575\pi$
$67$	−3.85043	−0.470405	−0.235203	+	0.971946i	$0.575575\pi$
$68$	0	0
$69$	−1.70928	−0.205773
$70$	0	0
$71$	1.21953	0.144732	0.0723661	−	0.997378i	$-0.476945\pi$
$71$	1.21953	0.144732	0.0723661	+	0.997378i	$0.476945\pi$
$72$	0	0
$73$	7.66701	0.897356	0.448678	−	0.893693i	$-0.351895\pi$
$73$	7.66701	0.897356	0.448678	+	0.893693i	$0.351895\pi$
$74$	0	0
$75$	−1.00000	−0.115470
$76$	0	0
$77$	−2.34017	−0.266687
$78$	0	0
$79$	−3.52586	−0.396690	−0.198345	−	0.980132i	$-0.563557\pi$
$79$	−3.52586	−0.396690	−0.198345	+	0.980132i	$0.563557\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−12.5464	−1.37714	−0.688572	−	0.725168i	$-0.741762\pi$
$83$	−12.5464	−1.37714	−0.688572	+	0.725168i	$0.741762\pi$
$84$	0	0
$85$	−6.04945	−0.656155
$86$	0	0
$87$	−7.95774	−0.853159
$88$	0	0
$89$	−5.77432	−0.612077	−0.306038	−	0.952019i	$-0.599004\pi$
$89$	−5.77432	−0.612077	−0.306038	+	0.952019i	$0.599004\pi$
$90$	0	0
$91$	0.107307	0.0112488
$92$	0	0
$93$	1.00000	0.103695
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−12.0000	−1.21842	−0.609208	−	0.793011i	$-0.708512\pi$
$97$	−12.0000	−1.21842	−0.609208	+	0.793011i	$0.708512\pi$
$98$	0	0
$99$	−2.00000	−0.201008
$100$	0	0
$101$	13.9421	1.38729	0.693647	−	0.720315i	$-0.256003\pi$
$101$	13.9421	1.38729	0.693647	+	0.720315i	$0.256003\pi$
$102$	0	0
$103$	−6.77205	−0.667270	−0.333635	−	0.942702i	$-0.608275\pi$
$103$	−6.77205	−0.667270	−0.333635	+	0.942702i	$0.608275\pi$
$104$	0	0
$105$	−1.17009	−0.114189
$106$	0	0
$107$	−0.630898	−0.0609912	−0.0304956	−	0.999535i	$-0.509709\pi$
$107$	−0.630898	−0.0609912	−0.0304956	+	0.999535i	$0.509709\pi$
$108$	0	0
$109$	−11.8660	−1.13656	−0.568280	−	0.822835i	$-0.692391\pi$
$109$	−11.8660	−1.13656	−0.568280	+	0.822835i	$0.692391\pi$
$110$	0	0
$111$	3.35350	0.318301
$112$	0	0
$113$	3.75872	0.353591	0.176795	−	0.984248i	$-0.443427\pi$
$113$	3.75872	0.353591	0.176795	+	0.984248i	$0.443427\pi$
$114$	0	0
$115$	1.70928	0.159391
$116$	0	0
$117$	0.0917087	0.00847848
$118$	0	0
$119$	−7.07838	−0.648874
$120$	0	0
$121$	−7.00000	−0.636364
$122$	0	0
$123$	4.68035	0.422013
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	4.18342	0.371218	0.185609	−	0.982624i	$-0.440574\pi$
$127$	4.18342	0.371218	0.185609	+	0.982624i	$0.440574\pi$
$128$	0	0
$129$	−0.738205	−0.0649953
$130$	0	0
$131$	0.695944	0.0608049	0.0304025	−	0.999538i	$-0.490321\pi$
$131$	0.695944	0.0608049	0.0304025	+	0.999538i	$0.490321\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−1.00000	−0.0860663
$136$	0	0
$137$	8.20620	0.701103	0.350552	−	0.936543i	$-0.385994\pi$
$137$	8.20620	0.701103	0.350552	+	0.936543i	$0.385994\pi$
$138$	0	0
$139$	4.49693	0.381424	0.190712	−	0.981646i	$-0.438920\pi$
$139$	4.49693	0.381424	0.190712	+	0.981646i	$0.438920\pi$
$140$	0	0
$141$	−9.12783	−0.768702
$142$	0	0
$143$	−0.183417	−0.0153381
$144$	0	0
$145$	7.95774	0.660854
$146$	0	0
$147$	5.63090	0.464429
$148$	0	0
$149$	13.3607	1.09455	0.547275	−	0.836953i	$-0.315665\pi$
$149$	13.3607	1.09455	0.547275	+	0.836953i	$0.315665\pi$
$150$	0	0
$151$	−13.8660	−1.12840	−0.564201	−	0.825638i	$-0.690816\pi$
$151$	−13.8660	−1.12840	−0.564201	+	0.825638i	$0.690816\pi$
$152$	0	0
$153$	−6.04945	−0.489069
$154$	0	0
$155$	−1.00000	−0.0803219
$156$	0	0
$157$	−20.0989	−1.60407	−0.802033	−	0.597279i	$-0.796248\pi$
$157$	−20.0989	−1.60407	−0.802033	+	0.597279i	$0.796248\pi$
$158$	0	0
$159$	−4.97107	−0.394232
$160$	0	0
$161$	2.00000	0.157622
$162$	0	0
$163$	−4.06505	−0.318399	−0.159200	−	0.987246i	$-0.550891\pi$
$163$	−4.06505	−0.318399	−0.159200	+	0.987246i	$0.550891\pi$
$164$	0	0
$165$	2.00000	0.155700
$166$	0	0
$167$	−3.91548	−0.302989	−0.151494	−	0.988458i	$-0.548409\pi$
$167$	−3.91548	−0.302989	−0.151494	+	0.988458i	$0.548409\pi$
$168$	0	0
$169$	−12.9916	−0.999353
$170$	0	0
$171$	0	0
$172$	0	0
$173$	16.0722	1.22195	0.610975	−	0.791650i	$-0.290778\pi$
$173$	16.0722	1.22195	0.610975	+	0.791650i	$0.290778\pi$
$174$	0	0
$175$	1.17009	0.0884502
$176$	0	0
$177$	11.0361	0.829525
$178$	0	0
$179$	−21.9421	−1.64003	−0.820016	−	0.572340i	$-0.806036\pi$
$179$	−21.9421	−1.64003	−0.820016	+	0.572340i	$0.806036\pi$
$180$	0	0
$181$	3.94214	0.293017	0.146509	−	0.989209i	$-0.453196\pi$
$181$	3.94214	0.293017	0.146509	+	0.989209i	$0.453196\pi$
$182$	0	0
$183$	3.26180	0.241119
$184$	0	0
$185$	−3.35350	−0.246555
$186$	0	0
$187$	12.0989	0.884759
$188$	0	0
$189$	−1.17009	−0.0851113
$190$	0	0
$191$	−23.3184	−1.68726	−0.843631	−	0.536923i	$-0.819587\pi$
$191$	−23.3184	−1.68726	−0.843631	+	0.536923i	$0.819587\pi$
$192$	0	0
$193$	−10.2557	−0.738218	−0.369109	−	0.929386i	$-0.620337\pi$
$193$	−10.2557	−0.738218	−0.369109	+	0.929386i	$0.620337\pi$
$194$	0	0
$195$	−0.0917087	−0.00656740
$196$	0	0
$197$	23.9916	1.70933	0.854665	−	0.519180i	$-0.173763\pi$
$197$	23.9916	1.70933	0.854665	+	0.519180i	$0.173763\pi$
$198$	0	0
$199$	16.1217	1.14284	0.571418	−	0.820659i	$-0.306394\pi$
$199$	16.1217	1.14284	0.571418	+	0.820659i	$0.306394\pi$
$200$	0	0
$201$	3.85043	0.271589
$202$	0	0
$203$	9.31124	0.653521
$204$	0	0
$205$	−4.68035	−0.326890
$206$	0	0
$207$	1.70928	0.118803
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−18.4391	−1.26940	−0.634699	−	0.772759i	$-0.718876\pi$
$211$	−18.4391	−1.26940	−0.634699	+	0.772759i	$0.718876\pi$
$212$	0	0
$213$	−1.21953	−0.0835611
$214$	0	0
$215$	0.738205	0.0503451
$216$	0	0
$217$	−1.17009	−0.0794306
$218$	0	0
$219$	−7.66701	−0.518089
$220$	0	0
$221$	−0.554787	−0.0373190
$222$	0	0
$223$	15.2762	1.02297	0.511484	−	0.859293i	$-0.329096\pi$
$223$	15.2762	1.02297	0.511484	+	0.859293i	$0.329096\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	19.7815	1.31294	0.656472	−	0.754350i	$-0.272048\pi$
$227$	19.7815	1.31294	0.656472	+	0.754350i	$0.272048\pi$
$228$	0	0
$229$	25.4329	1.68066	0.840328	−	0.542079i	$-0.182363\pi$
$229$	25.4329	1.68066	0.840328	+	0.542079i	$0.182363\pi$
$230$	0	0
$231$	2.34017	0.153972
$232$	0	0
$233$	−16.2062	−1.06170	−0.530852	−	0.847465i	$-0.678128\pi$
$233$	−16.2062	−1.06170	−0.530852	+	0.847465i	$0.678128\pi$
$234$	0	0
$235$	9.12783	0.595434
$236$	0	0
$237$	3.52586	0.229029
$238$	0	0
$239$	−19.7587	−1.27809	−0.639043	−	0.769171i	$-0.720669\pi$
$239$	−19.7587	−1.27809	−0.639043	+	0.769171i	$0.720669\pi$
$240$	0	0
$241$	−8.65368	−0.557433	−0.278716	−	0.960373i	$-0.589909\pi$
$241$	−8.65368	−0.557433	−0.278716	+	0.960373i	$0.589909\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−5.63090	−0.359745
$246$	0	0
$247$	0	0
$248$	0	0
$249$	12.5464	0.795094
$250$	0	0
$251$	18.9360	1.19523	0.597615	−	0.801783i	$-0.296115\pi$
$251$	18.9360	1.19523	0.597615	+	0.801783i	$0.296115\pi$
$252$	0	0
$253$	−3.41855	−0.214922
$254$	0	0
$255$	6.04945	0.378831
$256$	0	0
$257$	−13.1012	−0.817228	−0.408614	−	0.912707i	$-0.633988\pi$
$257$	−13.1012	−0.817228	−0.408614	+	0.912707i	$0.633988\pi$
$258$	0	0
$259$	−3.92389	−0.243819
$260$	0	0
$261$	7.95774	0.492572
$262$	0	0
$263$	−8.49693	−0.523943	−0.261972	−	0.965076i	$-0.584373\pi$
$263$	−8.49693	−0.523943	−0.261972	+	0.965076i	$0.584373\pi$
$264$	0	0
$265$	4.97107	0.305370
$266$	0	0
$267$	5.77432	0.353383
$268$	0	0
$269$	21.8010	1.32923	0.664615	−	0.747186i	$-0.268596\pi$
$269$	21.8010	1.32923	0.664615	+	0.747186i	$0.268596\pi$
$270$	0	0
$271$	−10.3402	−0.628121	−0.314060	−	0.949403i	$-0.601689\pi$
$271$	−10.3402	−0.628121	−0.314060	+	0.949403i	$0.601689\pi$
$272$	0	0
$273$	−0.107307	−0.00649452
$274$	0	0
$275$	−2.00000	−0.120605
$276$	0	0
$277$	−0.646496	−0.0388442	−0.0194221	−	0.999811i	$-0.506183\pi$
$277$	−0.646496	−0.0388442	−0.0194221	+	0.999811i	$0.506183\pi$
$278$	0	0
$279$	−1.00000	−0.0598684
$280$	0	0
$281$	−17.3874	−1.03724	−0.518621	−	0.855004i	$-0.673555\pi$
$281$	−17.3874	−1.03724	−0.518621	+	0.855004i	$0.673555\pi$
$282$	0	0
$283$	−2.92881	−0.174100	−0.0870498	−	0.996204i	$-0.527744\pi$
$283$	−2.92881	−0.174100	−0.0870498	+	0.996204i	$0.527744\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−5.47641	−0.323262
$288$	0	0
$289$	19.5958	1.15270
$290$	0	0
$291$	12.0000	0.703452
$292$	0	0
$293$	−31.9916	−1.86897	−0.934484	−	0.356004i	$-0.884139\pi$
$293$	−31.9916	−1.86897	−0.934484	+	0.356004i	$0.884139\pi$
$294$	0	0
$295$	−11.0361	−0.642548
$296$	0	0
$297$	2.00000	0.116052
$298$	0	0
$299$	0.156755	0.00906540
$300$	0	0
$301$	0.863763	0.0497865
$302$	0	0
$303$	−13.9421	−0.800955
$304$	0	0
$305$	−3.26180	−0.186770
$306$	0	0
$307$	5.66701	0.323434	0.161717	−	0.986837i	$-0.448297\pi$
$307$	5.66701	0.323434	0.161717	+	0.986837i	$0.448297\pi$
$308$	0	0
$309$	6.77205	0.385249
$310$	0	0
$311$	16.2401	0.920889	0.460444	−	0.887689i	$-0.347690\pi$
$311$	16.2401	0.920889	0.460444	+	0.887689i	$0.347690\pi$
$312$	0	0
$313$	7.42574	0.419728	0.209864	−	0.977731i	$-0.432698\pi$
$313$	7.42574	0.419728	0.209864	+	0.977731i	$0.432698\pi$
$314$	0	0
$315$	1.17009	0.0659269
$316$	0	0
$317$	−13.2534	−0.744384	−0.372192	−	0.928156i	$-0.621394\pi$
$317$	−13.2534	−0.744384	−0.372192	+	0.928156i	$0.621394\pi$
$318$	0	0
$319$	−15.9155	−0.891096
$320$	0	0
$321$	0.630898	0.0352133
$322$	0	0
$323$	0	0
$324$	0	0
$325$	0.0917087	0.00508709
$326$	0	0
$327$	11.8660	0.656193
$328$	0	0
$329$	10.6803	0.588827
$330$	0	0
$331$	−3.62475	−0.199235	−0.0996173	−	0.995026i	$-0.531762\pi$
$331$	−3.62475	−0.199235	−0.0996173	+	0.995026i	$0.531762\pi$
$332$	0	0
$333$	−3.35350	−0.183771
$334$	0	0
$335$	−3.85043	−0.210372
$336$	0	0
$337$	13.3268	0.725959	0.362980	−	0.931797i	$-0.381759\pi$
$337$	13.3268	0.725959	0.362980	+	0.931797i	$0.381759\pi$
$338$	0	0
$339$	−3.75872	−0.204146
$340$	0	0
$341$	2.00000	0.108306
$342$	0	0
$343$	−14.7792	−0.798004
$344$	0	0
$345$	−1.70928	−0.0920243
$346$	0	0
$347$	−16.2290	−0.871218	−0.435609	−	0.900136i	$-0.643467\pi$
$347$	−16.2290	−0.871218	−0.435609	+	0.900136i	$0.643467\pi$
$348$	0	0
$349$	2.38962	0.127913	0.0639567	−	0.997953i	$-0.479628\pi$
$349$	2.38962	0.127913	0.0639567	+	0.997953i	$0.479628\pi$
$350$	0	0
$351$	−0.0917087	−0.00489505
$352$	0	0
$353$	−36.2062	−1.92706	−0.963531	−	0.267597i	$-0.913770\pi$
$353$	−36.2062	−1.92706	−0.963531	+	0.267597i	$0.913770\pi$
$354$	0	0
$355$	1.21953	0.0647262
$356$	0	0
$357$	7.07838	0.374627
$358$	0	0
$359$	−0.382433	−0.0201841	−0.0100920	−	0.999949i	$-0.503212\pi$
$359$	−0.382433	−0.0201841	−0.0100920	+	0.999949i	$0.503212\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	0	0
$363$	7.00000	0.367405
$364$	0	0
$365$	7.66701	0.401310
$366$	0	0
$367$	7.63317	0.398448	0.199224	−	0.979954i	$-0.436158\pi$
$367$	7.63317	0.398448	0.199224	+	0.979954i	$0.436158\pi$
$368$	0	0
$369$	−4.68035	−0.243649
$370$	0	0
$371$	5.81658	0.301982
$372$	0	0
$373$	−17.5486	−0.908634	−0.454317	−	0.890840i	$-0.650117\pi$
$373$	−17.5486	−0.908634	−0.454317	+	0.890840i	$0.650117\pi$
$374$	0	0
$375$	−1.00000	−0.0516398
$376$	0	0
$377$	0.729794	0.0375863
$378$	0	0
$379$	−7.68649	−0.394828	−0.197414	−	0.980320i	$-0.563254\pi$
$379$	−7.68649	−0.394828	−0.197414	+	0.980320i	$0.563254\pi$
$380$	0	0
$381$	−4.18342	−0.214323
$382$	0	0
$383$	5.92389	0.302697	0.151348	−	0.988480i	$-0.451638\pi$
$383$	5.92389	0.302697	0.151348	+	0.988480i	$0.451638\pi$
$384$	0	0
$385$	−2.34017	−0.119266
$386$	0	0
$387$	0.738205	0.0375251
$388$	0	0
$389$	−28.7636	−1.45837	−0.729187	−	0.684314i	$-0.760102\pi$
$389$	−28.7636	−1.45837	−0.729187	+	0.684314i	$0.760102\pi$
$390$	0	0
$391$	−10.3402	−0.522925
$392$	0	0
$393$	−0.695944	−0.0351057
$394$	0	0
$395$	−3.52586	−0.177405
$396$	0	0
$397$	9.41855	0.472704	0.236352	−	0.971668i	$-0.424048\pi$
$397$	9.41855	0.472704	0.236352	+	0.971668i	$0.424048\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	13.9721	0.697734	0.348867	−	0.937172i	$-0.386566\pi$
$401$	13.9721	0.697734	0.348867	+	0.937172i	$0.386566\pi$
$402$	0	0
$403$	−0.0917087	−0.00456834
$404$	0	0
$405$	1.00000	0.0496904
$406$	0	0
$407$	6.70701	0.332454
$408$	0	0
$409$	4.73820	0.234289	0.117145	−	0.993115i	$-0.462626\pi$
$409$	4.73820	0.234289	0.117145	+	0.993115i	$0.462626\pi$
$410$	0	0
$411$	−8.20620	−0.404782
$412$	0	0
$413$	−12.9132	−0.635418
$414$	0	0
$415$	−12.5464	−0.615877
$416$	0	0
$417$	−4.49693	−0.220216
$418$	0	0
$419$	−30.3123	−1.48085	−0.740426	−	0.672138i	$-0.765376\pi$
$419$	−30.3123	−1.48085	−0.740426	+	0.672138i	$0.765376\pi$
$420$	0	0
$421$	−33.9916	−1.65665	−0.828324	−	0.560249i	$-0.810706\pi$
$421$	−33.9916	−1.65665	−0.828324	+	0.560249i	$0.810706\pi$
$422$	0	0
$423$	9.12783	0.443810
$424$	0	0
$425$	−6.04945	−0.293441
$426$	0	0
$427$	−3.81658	−0.184697
$428$	0	0
$429$	0.183417	0.00885548
$430$	0	0
$431$	−16.6381	−0.801428	−0.400714	−	0.916203i	$-0.631238\pi$
$431$	−16.6381	−0.801428	−0.400714	+	0.916203i	$0.631238\pi$
$432$	0	0
$433$	−32.8976	−1.58096	−0.790479	−	0.612489i	$-0.790168\pi$
$433$	−32.8976	−1.58096	−0.790479	+	0.612489i	$0.790168\pi$
$434$	0	0
$435$	−7.95774	−0.381544
$436$	0	0
$437$	0	0
$438$	0	0
$439$	28.5113	1.36077	0.680385	−	0.732855i	$-0.261813\pi$
$439$	28.5113	1.36077	0.680385	+	0.732855i	$0.261813\pi$
$440$	0	0
$441$	−5.63090	−0.268138
$442$	0	0
$443$	20.0494	0.952578	0.476289	−	0.879289i	$-0.341982\pi$
$443$	20.0494	0.952578	0.476289	+	0.879289i	$0.341982\pi$
$444$	0	0
$445$	−5.77432	−0.273729
$446$	0	0
$447$	−13.3607	−0.631939
$448$	0	0
$449$	−31.7308	−1.49747	−0.748735	−	0.662869i	$-0.769339\pi$
$449$	−31.7308	−1.49747	−0.748735	+	0.662869i	$0.769339\pi$
$450$	0	0
$451$	9.36069	0.440778
$452$	0	0
$453$	13.8660	0.651483
$454$	0	0
$455$	0.107307	0.00503064
$456$	0	0
$457$	−38.7864	−1.81435	−0.907176	−	0.420751i	$-0.861767\pi$
$457$	−38.7864	−1.81435	−0.907176	+	0.420751i	$0.861767\pi$
$458$	0	0
$459$	6.04945	0.282364
$460$	0	0
$461$	−0.324575	−0.0151169	−0.00755847	−	0.999971i	$-0.502406\pi$
$461$	−0.324575	−0.0151169	−0.00755847	+	0.999971i	$0.502406\pi$
$462$	0	0
$463$	−16.8371	−0.782486	−0.391243	−	0.920287i	$-0.627955\pi$
$463$	−16.8371	−0.782486	−0.391243	+	0.920287i	$0.627955\pi$
$464$	0	0
$465$	1.00000	0.0463739
$466$	0	0
$467$	31.0700	1.43775	0.718873	−	0.695141i	$-0.244658\pi$
$467$	31.0700	1.43775	0.718873	+	0.695141i	$0.244658\pi$
$468$	0	0
$469$	−4.50534	−0.208037
$470$	0	0
$471$	20.0989	0.926108
$472$	0	0
$473$	−1.47641	−0.0678854
$474$	0	0
$475$	0	0
$476$	0	0
$477$	4.97107	0.227610
$478$	0	0
$479$	30.3123	1.38500	0.692502	−	0.721416i	$-0.256508\pi$
$479$	30.3123	1.38500	0.692502	+	0.721416i	$0.256508\pi$
$480$	0	0
$481$	−0.307546	−0.0140229
$482$	0	0
$483$	−2.00000	−0.0910032
$484$	0	0
$485$	−12.0000	−0.544892
$486$	0	0
$487$	−5.92777	−0.268613	−0.134306	−	0.990940i	$-0.542881\pi$
$487$	−5.92777	−0.268613	−0.134306	+	0.990940i	$0.542881\pi$
$488$	0	0
$489$	4.06505	0.183828
$490$	0	0
$491$	11.1194	0.501812	0.250906	−	0.968011i	$-0.419271\pi$
$491$	11.1194	0.501812	0.250906	+	0.968011i	$0.419271\pi$
$492$	0	0
$493$	−48.1399	−2.16811
$494$	0	0
$495$	−2.00000	−0.0898933
$496$	0	0
$497$	1.42696	0.0640080
$498$	0	0
$499$	0.482553	0.0216020	0.0108010	−	0.999942i	$-0.496562\pi$
$499$	0.482553	0.0216020	0.0108010	+	0.999942i	$0.496562\pi$
$500$	0	0
$501$	3.91548	0.174931
$502$	0	0
$503$	−28.1795	−1.25646	−0.628232	−	0.778026i	$-0.716221\pi$
$503$	−28.1795	−1.25646	−0.628232	+	0.778026i	$0.716221\pi$
$504$	0	0
$505$	13.9421	0.620417
$506$	0	0
$507$	12.9916	0.576977
$508$	0	0
$509$	−30.8359	−1.36678	−0.683388	−	0.730055i	$-0.739494\pi$
$509$	−30.8359	−1.36678	−0.683388	+	0.730055i	$0.739494\pi$
$510$	0	0
$511$	8.97107	0.396857
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−6.77205	−0.298412
$516$	0	0
$517$	−18.2557	−0.802883
$518$	0	0
$519$	−16.0722	−0.705493
$520$	0	0
$521$	−29.0472	−1.27258	−0.636290	−	0.771450i	$-0.719532\pi$
$521$	−29.0472	−1.27258	−0.636290	+	0.771450i	$0.719532\pi$
$522$	0	0
$523$	−16.5958	−0.725685	−0.362842	−	0.931851i	$-0.618194\pi$
$523$	−16.5958	−0.725685	−0.362842	+	0.931851i	$0.618194\pi$
$524$	0	0
$525$	−1.17009	−0.0510668
$526$	0	0
$527$	6.04945	0.263518
$528$	0	0
$529$	−20.0784	−0.872973
$530$	0	0
$531$	−11.0361	−0.478927
$532$	0	0
$533$	−0.429229	−0.0185920
$534$	0	0
$535$	−0.630898	−0.0272761
$536$	0	0
$537$	21.9421	0.946873
$538$	0	0
$539$	11.2618	0.485080
$540$	0	0
$541$	−11.6370	−0.500315	−0.250158	−	0.968205i	$-0.580482\pi$
$541$	−11.6370	−0.500315	−0.250158	+	0.968205i	$0.580482\pi$
$542$	0	0
$543$	−3.94214	−0.169173
$544$	0	0
$545$	−11.8660	−0.508285
$546$	0	0
$547$	12.9821	0.555076	0.277538	−	0.960715i	$-0.410482\pi$
$547$	12.9821	0.555076	0.277538	+	0.960715i	$0.410482\pi$
$548$	0	0
$549$	−3.26180	−0.139210
$550$	0	0
$551$	0	0
$552$	0	0
$553$	−4.12556	−0.175437
$554$	0	0
$555$	3.35350	0.142348
$556$	0	0
$557$	45.9337	1.94627	0.973137	−	0.230225i	$-0.0739464\pi$
$557$	45.9337	1.94627	0.973137	+	0.230225i	$0.0739464\pi$
$558$	0	0
$559$	0.0676998	0.00286340
$560$	0	0
$561$	−12.0989	−0.510816
$562$	0	0
$563$	20.2017	0.851399	0.425699	−	0.904865i	$-0.360028\pi$
$563$	20.2017	0.851399	0.425699	+	0.904865i	$0.360028\pi$
$564$	0	0
$565$	3.75872	0.158131
$566$	0	0
$567$	1.17009	0.0491390
$568$	0	0
$569$	21.1883	0.888261	0.444131	−	0.895962i	$-0.353513\pi$
$569$	21.1883	0.888261	0.444131	+	0.895962i	$0.353513\pi$
$570$	0	0
$571$	−0.299135	−0.0125184	−0.00625921	−	0.999980i	$-0.501992\pi$
$571$	−0.299135	−0.0125184	−0.00625921	+	0.999980i	$0.501992\pi$
$572$	0	0
$573$	23.3184	0.974141
$574$	0	0
$575$	1.70928	0.0712817
$576$	0	0
$577$	19.7009	0.820158	0.410079	−	0.912050i	$-0.365501\pi$
$577$	19.7009	0.820158	0.410079	+	0.912050i	$0.365501\pi$
$578$	0	0
$579$	10.2557	0.426210
$580$	0	0
$581$	−14.6803	−0.609043
$582$	0	0
$583$	−9.94214	−0.411761
$584$	0	0
$585$	0.0917087	0.00379169
$586$	0	0
$587$	13.6598	0.563801	0.281901	−	0.959444i	$-0.409035\pi$
$587$	13.6598	0.563801	0.281901	+	0.959444i	$0.409035\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	−23.9916	−0.986882
$592$	0	0
$593$	36.4846	1.49824	0.749122	−	0.662432i	$-0.230476\pi$
$593$	36.4846	1.49824	0.749122	+	0.662432i	$0.230476\pi$
$594$	0	0
$595$	−7.07838	−0.290185
$596$	0	0
$597$	−16.1217	−0.659817
$598$	0	0
$599$	−37.1038	−1.51602	−0.758010	−	0.652242i	$-0.773828\pi$
$599$	−37.1038	−1.51602	−0.758010	+	0.652242i	$0.773828\pi$
$600$	0	0
$601$	−26.8638	−1.09580	−0.547898	−	0.836545i	$-0.684572\pi$
$601$	−26.8638	−1.09580	−0.547898	+	0.836545i	$0.684572\pi$
$602$	0	0
$603$	−3.85043	−0.156802
$604$	0	0
$605$	−7.00000	−0.284590
$606$	0	0
$607$	−6.88777	−0.279566	−0.139783	−	0.990182i	$-0.544641\pi$
$607$	−6.88777	−0.279566	−0.139783	+	0.990182i	$0.544641\pi$
$608$	0	0
$609$	−9.31124	−0.377311
$610$	0	0
$611$	0.837101	0.0338655
$612$	0	0
$613$	28.4729	1.15001	0.575005	−	0.818150i	$-0.305000\pi$
$613$	28.4729	1.15001	0.575005	+	0.818150i	$0.305000\pi$
$614$	0	0
$615$	4.68035	0.188730
$616$	0	0
$617$	5.20394	0.209503	0.104751	−	0.994498i	$-0.466595\pi$
$617$	5.20394	0.209503	0.104751	+	0.994498i	$0.466595\pi$
$618$	0	0
$619$	−14.0494	−0.564695	−0.282348	−	0.959312i	$-0.591113\pi$
$619$	−14.0494	−0.564695	−0.282348	+	0.959312i	$0.591113\pi$
$620$	0	0
$621$	−1.70928	−0.0685909
$622$	0	0
$623$	−6.75646	−0.270692
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	20.2868	0.808890
$630$	0	0
$631$	3.15061	0.125424	0.0627120	−	0.998032i	$-0.480025\pi$
$631$	3.15061	0.125424	0.0627120	+	0.998032i	$0.480025\pi$
$632$	0	0
$633$	18.4391	0.732887
$634$	0	0
$635$	4.18342	0.166014
$636$	0	0
$637$	−0.516403	−0.0204606
$638$	0	0
$639$	1.21953	0.0482441
$640$	0	0
$641$	4.19448	0.165672	0.0828360	−	0.996563i	$-0.473602\pi$
$641$	4.19448	0.165672	0.0828360	+	0.996563i	$0.473602\pi$
$642$	0	0
$643$	38.0722	1.50142	0.750711	−	0.660631i	$-0.229711\pi$
$643$	38.0722	1.50142	0.750711	+	0.660631i	$0.229711\pi$
$644$	0	0
$645$	−0.738205	−0.0290668
$646$	0	0
$647$	5.20847	0.204766	0.102383	−	0.994745i	$-0.467353\pi$
$647$	5.20847	0.204766	0.102383	+	0.994745i	$0.467353\pi$
$648$	0	0
$649$	22.0722	0.866411
$650$	0	0
$651$	1.17009	0.0458593
$652$	0	0
$653$	25.6658	1.00438	0.502190	−	0.864757i	$-0.332528\pi$
$653$	25.6658	1.00438	0.502190	+	0.864757i	$0.332528\pi$
$654$	0	0
$655$	0.695944	0.0271928
$656$	0	0
$657$	7.66701	0.299119
$658$	0	0
$659$	11.9721	0.466367	0.233184	−	0.972433i	$-0.425086\pi$
$659$	11.9721	0.466367	0.233184	+	0.972433i	$0.425086\pi$
$660$	0	0
$661$	36.9504	1.43720	0.718601	−	0.695422i	$-0.244783\pi$
$661$	36.9504	1.43720	0.718601	+	0.695422i	$0.244783\pi$
$662$	0	0
$663$	0.554787	0.0215462
$664$	0	0
$665$	0	0
$666$	0	0
$667$	13.6020	0.526670
$668$	0	0
$669$	−15.2762	−0.590611
$670$	0	0
$671$	6.52359	0.251840
$672$	0	0
$673$	−20.8853	−0.805070	−0.402535	−	0.915405i	$-0.631871\pi$
$673$	−20.8853	−0.805070	−0.402535	+	0.915405i	$0.631871\pi$
$674$	0	0
$675$	−1.00000	−0.0384900
$676$	0	0
$677$	−31.1110	−1.19569	−0.597847	−	0.801611i	$-0.703977\pi$
$677$	−31.1110	−1.19569	−0.597847	+	0.801611i	$0.703977\pi$
$678$	0	0
$679$	−14.0410	−0.538846
$680$	0	0
$681$	−19.7815	−0.758029
$682$	0	0
$683$	−27.1545	−1.03904	−0.519519	−	0.854459i	$-0.673889\pi$
$683$	−27.1545	−1.03904	−0.519519	+	0.854459i	$0.673889\pi$
$684$	0	0
$685$	8.20620	0.313543
$686$	0	0
$687$	−25.4329	−0.970327
$688$	0	0
$689$	0.455891	0.0173680
$690$	0	0
$691$	−42.3234	−1.61006	−0.805028	−	0.593237i	$-0.797850\pi$
$691$	−42.3234	−1.61006	−0.805028	+	0.593237i	$0.797850\pi$
$692$	0	0
$693$	−2.34017	−0.0888958
$694$	0	0
$695$	4.49693	0.170578
$696$	0	0
$697$	28.3135	1.07245
$698$	0	0
$699$	16.2062	0.612975
$700$	0	0
$701$	−8.69472	−0.328395	−0.164198	−	0.986427i	$-0.552503\pi$
$701$	−8.69472	−0.328395	−0.164198	+	0.986427i	$0.552503\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	−9.12783	−0.343774
$706$	0	0
$707$	16.3135	0.613533
$708$	0	0
$709$	−31.3074	−1.17577	−0.587886	−	0.808943i	$-0.700040\pi$
$709$	−31.3074	−1.17577	−0.587886	+	0.808943i	$0.700040\pi$
$710$	0	0
$711$	−3.52586	−0.132230
$712$	0	0
$713$	−1.70928	−0.0640129
$714$	0	0
$715$	−0.183417	−0.00685942
$716$	0	0
$717$	19.7587	0.737903
$718$	0	0
$719$	−20.5236	−0.765401	−0.382700	−	0.923873i	$-0.625006\pi$
$719$	−20.5236	−0.765401	−0.382700	+	0.923873i	$0.625006\pi$
$720$	0	0
$721$	−7.92389	−0.295101
$722$	0	0
$723$	8.65368	0.321834
$724$	0	0
$725$	7.95774	0.295543
$726$	0	0
$727$	−31.7971	−1.17929	−0.589645	−	0.807663i	$-0.700732\pi$
$727$	−31.7971	−1.17929	−0.589645	+	0.807663i	$0.700732\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−4.46573	−0.165171
$732$	0	0
$733$	10.1978	0.376664	0.188332	−	0.982105i	$-0.439692\pi$
$733$	10.1978	0.376664	0.188332	+	0.982105i	$0.439692\pi$
$734$	0	0
$735$	5.63090	0.207699
$736$	0	0
$737$	7.70086	0.283665
$738$	0	0
$739$	−41.3835	−1.52232	−0.761158	−	0.648567i	$-0.775369\pi$
$739$	−41.3835	−1.52232	−0.761158	+	0.648567i	$0.775369\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	23.8987	0.876757	0.438378	−	0.898791i	$-0.355553\pi$
$743$	23.8987	0.876757	0.438378	+	0.898791i	$0.355553\pi$
$744$	0	0
$745$	13.3607	0.489498
$746$	0	0
$747$	−12.5464	−0.459048
$748$	0	0
$749$	−0.738205	−0.0269734
$750$	0	0
$751$	−51.6475	−1.88465	−0.942323	−	0.334706i	$-0.891363\pi$
$751$	−51.6475	−1.88465	−0.942323	+	0.334706i	$0.891363\pi$
$752$	0	0
$753$	−18.9360	−0.690066
$754$	0	0
$755$	−13.8660	−0.504637
$756$	0	0
$757$	−0.687534	−0.0249888	−0.0124944	−	0.999922i	$-0.503977\pi$
$757$	−0.687534	−0.0249888	−0.0124944	+	0.999922i	$0.503977\pi$
$758$	0	0
$759$	3.41855	0.124086
$760$	0	0
$761$	−32.1555	−1.16564	−0.582819	−	0.812602i	$-0.698050\pi$
$761$	−32.1555	−1.16564	−0.582819	+	0.812602i	$0.698050\pi$
$762$	0	0
$763$	−13.8843	−0.502645
$764$	0	0
$765$	−6.04945	−0.218718
$766$	0	0
$767$	−1.01211	−0.0365451
$768$	0	0
$769$	−34.2206	−1.23403	−0.617013	−	0.786953i	$-0.711657\pi$
$769$	−34.2206	−1.23403	−0.617013	+	0.786953i	$0.711657\pi$
$770$	0	0
$771$	13.1012	0.471827
$772$	0	0
$773$	9.20781	0.331182	0.165591	−	0.986195i	$-0.447047\pi$
$773$	9.20781	0.331182	0.165591	+	0.986195i	$0.447047\pi$
$774$	0	0
$775$	−1.00000	−0.0359211
$776$	0	0
$777$	3.92389	0.140769
$778$	0	0
$779$	0	0
$780$	0	0
$781$	−2.43907	−0.0872768
$782$	0	0
$783$	−7.95774	−0.284386
$784$	0	0
$785$	−20.0989	−0.717360
$786$	0	0
$787$	35.4908	1.26511	0.632555	−	0.774515i	$-0.282006\pi$
$787$	35.4908	1.26511	0.632555	+	0.774515i	$0.282006\pi$
$788$	0	0
$789$	8.49693	0.302499
$790$	0	0
$791$	4.39803	0.156376
$792$	0	0
$793$	−0.299135	−0.0106226
$794$	0	0
$795$	−4.97107	−0.176306
$796$	0	0
$797$	6.89884	0.244369	0.122185	−	0.992507i	$-0.461010\pi$
$797$	6.89884	0.244369	0.122185	+	0.992507i	$0.461010\pi$
$798$	0	0
$799$	−55.2183	−1.95348
$800$	0	0
$801$	−5.77432	−0.204026
$802$	0	0
$803$	−15.3340	−0.541126
$804$	0	0
$805$	2.00000	0.0704907
$806$	0	0
$807$	−21.8010	−0.767431
$808$	0	0
$809$	56.2278	1.97686	0.988432	−	0.151668i	$-0.0484644\pi$
$809$	56.2278	1.97686	0.988432	+	0.151668i	$0.0484644\pi$
$810$	0	0
$811$	−34.7070	−1.21873	−0.609364	−	0.792891i	$-0.708575\pi$
$811$	−34.7070	−1.21873	−0.609364	+	0.792891i	$0.708575\pi$
$812$	0	0
$813$	10.3402	0.362646
$814$	0	0
$815$	−4.06505	−0.142392
$816$	0	0
$817$	0	0
$818$	0	0
$819$	0.107307	0.00374962
$820$	0	0
$821$	31.2651	1.09116	0.545580	−	0.838059i	$-0.316309\pi$
$821$	31.2651	1.09116	0.545580	+	0.838059i	$0.316309\pi$
$822$	0	0
$823$	44.5523	1.55300	0.776499	−	0.630119i	$-0.216994\pi$
$823$	44.5523	1.55300	0.776499	+	0.630119i	$0.216994\pi$
$824$	0	0
$825$	2.00000	0.0696311
$826$	0	0
$827$	2.97107	0.103314	0.0516571	−	0.998665i	$-0.483550\pi$
$827$	2.97107	0.103314	0.0516571	+	0.998665i	$0.483550\pi$
$828$	0	0
$829$	−30.2823	−1.05175	−0.525874	−	0.850562i	$-0.676262\pi$
$829$	−30.2823	−1.05175	−0.525874	+	0.850562i	$0.676262\pi$
$830$	0	0
$831$	0.646496	0.0224267
$832$	0	0
$833$	34.0638	1.18024
$834$	0	0
$835$	−3.91548	−0.135501
$836$	0	0
$837$	1.00000	0.0345651
$838$	0	0
$839$	47.8687	1.65261	0.826305	−	0.563223i	$-0.190439\pi$
$839$	47.8687	1.65261	0.826305	+	0.563223i	$0.190439\pi$
$840$	0	0
$841$	34.3256	1.18364
$842$	0	0
$843$	17.3874	0.598852
$844$	0	0
$845$	−12.9916	−0.446924
$846$	0	0
$847$	−8.19061	−0.281433
$848$	0	0
$849$	2.92881	0.100517
$850$	0	0
$851$	−5.73206	−0.196493
$852$	0	0
$853$	−47.8531	−1.63846	−0.819229	−	0.573466i	$-0.805598\pi$
$853$	−47.8531	−1.63846	−0.819229	+	0.573466i	$0.805598\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	3.13170	0.106977	0.0534884	−	0.998568i	$-0.482966\pi$
$857$	3.13170	0.106977	0.0534884	+	0.998568i	$0.482966\pi$
$858$	0	0
$859$	18.4703	0.630197	0.315099	−	0.949059i	$-0.397962\pi$
$859$	18.4703	0.630197	0.315099	+	0.949059i	$0.397962\pi$
$860$	0	0
$861$	5.47641	0.186636
$862$	0	0
$863$	47.9130	1.63098	0.815489	−	0.578772i	$-0.196468\pi$
$863$	47.9130	1.63098	0.815489	+	0.578772i	$0.196468\pi$
$864$	0	0
$865$	16.0722	0.546472
$866$	0	0
$867$	−19.5958	−0.665509
$868$	0	0
$869$	7.05172	0.239213
$870$	0	0
$871$	−0.353118	−0.0119650
$872$	0	0
$873$	−12.0000	−0.406138
$874$	0	0
$875$	1.17009	0.0395561
$876$	0	0
$877$	44.8104	1.51314	0.756571	−	0.653912i	$-0.226873\pi$
$877$	44.8104	1.51314	0.756571	+	0.653912i	$0.226873\pi$
$878$	0	0
$879$	31.9916	1.07905
$880$	0	0
$881$	−24.4501	−0.823746	−0.411873	−	0.911241i	$-0.635125\pi$
$881$	−24.4501	−0.823746	−0.411873	+	0.911241i	$0.635125\pi$
$882$	0	0
$883$	−39.8043	−1.33952	−0.669761	−	0.742577i	$-0.733603\pi$
$883$	−39.8043	−1.33952	−0.669761	+	0.742577i	$0.733603\pi$
$884$	0	0
$885$	11.0361	0.370975
$886$	0	0
$887$	−16.9588	−0.569420	−0.284710	−	0.958614i	$-0.591897\pi$
$887$	−16.9588	−0.569420	−0.284710	+	0.958614i	$0.591897\pi$
$888$	0	0
$889$	4.89496	0.164172
$890$	0	0
$891$	−2.00000	−0.0670025
$892$	0	0
$893$	0	0
$894$	0	0
$895$	−21.9421	−0.733445
$896$	0	0
$897$	−0.156755	−0.00523391
$898$	0	0
$899$	−7.95774	−0.265405
$900$	0	0
$901$	−30.0722	−1.00185
$902$	0	0
$903$	−0.863763	−0.0287442
$904$	0	0
$905$	3.94214	0.131041
$906$	0	0
$907$	3.72487	0.123682	0.0618412	−	0.998086i	$-0.480303\pi$
$907$	3.72487	0.123682	0.0618412	+	0.998086i	$0.480303\pi$
$908$	0	0
$909$	13.9421	0.462432
$910$	0	0
$911$	56.5523	1.87366	0.936831	−	0.349781i	$-0.113744\pi$
$911$	56.5523	1.87366	0.936831	+	0.349781i	$0.113744\pi$
$912$	0	0
$913$	25.0928	0.830449
$914$	0	0
$915$	3.26180	0.107832
$916$	0	0
$917$	0.814315	0.0268911
$918$	0	0
$919$	40.9770	1.35171	0.675854	−	0.737036i	$-0.263775\pi$
$919$	40.9770	1.35171	0.675854	+	0.737036i	$0.263775\pi$
$920$	0	0
$921$	−5.66701	−0.186734
$922$	0	0
$923$	0.111842	0.00368132
$924$	0	0
$925$	−3.35350	−0.110263
$926$	0	0
$927$	−6.77205	−0.222423
$928$	0	0
$929$	40.2713	1.32126	0.660628	−	0.750713i	$-0.270290\pi$
$929$	40.2713	1.32126	0.660628	+	0.750713i	$0.270290\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−16.2401	−0.531675
$934$	0	0
$935$	12.0989	0.395676
$936$	0	0
$937$	−28.9405	−0.945446	−0.472723	−	0.881211i	$-0.656729\pi$
$937$	−28.9405	−0.945446	−0.472723	+	0.881211i	$0.656729\pi$
$938$	0	0
$939$	−7.42574	−0.242330
$940$	0	0
$941$	45.7431	1.49118	0.745592	−	0.666403i	$-0.232167\pi$
$941$	45.7431	1.49118	0.745592	+	0.666403i	$0.232167\pi$
$942$	0	0
$943$	−8.00000	−0.260516
$944$	0	0
$945$	−1.17009	−0.0380629
$946$	0	0
$947$	57.7093	1.87530	0.937650	−	0.347582i	$-0.112997\pi$
$947$	57.7093	1.87530	0.937650	+	0.347582i	$0.112997\pi$
$948$	0	0
$949$	0.703132	0.0228246
$950$	0	0
$951$	13.2534	0.429770
$952$	0	0
$953$	−3.89269	−0.126097	−0.0630483	−	0.998010i	$-0.520082\pi$
$953$	−3.89269	−0.126097	−0.0630483	+	0.998010i	$0.520082\pi$
$954$	0	0
$955$	−23.3184	−0.754567
$956$	0	0
$957$	15.9155	0.514474
$958$	0	0
$959$	9.60197	0.310064
$960$	0	0
$961$	1.00000	0.0322581
$962$	0	0
$963$	−0.630898	−0.0203304
$964$	0	0
$965$	−10.2557	−0.330141
$966$	0	0
$967$	−23.1317	−0.743865	−0.371933	−	0.928260i	$-0.621305\pi$
$967$	−23.1317	−0.743865	−0.371933	+	0.928260i	$0.621305\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	3.51414	0.112774	0.0563870	−	0.998409i	$-0.482042\pi$
$971$	3.51414	0.112774	0.0563870	+	0.998409i	$0.482042\pi$
$972$	0	0
$973$	5.26180	0.168685
$974$	0	0
$975$	−0.0917087	−0.00293703
$976$	0	0
$977$	37.5174	1.20029	0.600145	−	0.799891i	$-0.295110\pi$
$977$	37.5174	1.20029	0.600145	+	0.799891i	$0.295110\pi$
$978$	0	0
$979$	11.5486	0.369096
$980$	0	0
$981$	−11.8660	−0.378853
$982$	0	0
$983$	−56.2434	−1.79388	−0.896942	−	0.442147i	$-0.854217\pi$
$983$	−56.2434	−1.79388	−0.896942	+	0.442147i	$0.854217\pi$
$984$	0	0
$985$	23.9916	0.764436
$986$	0	0
$987$	−10.6803	−0.339959
$988$	0	0
$989$	1.26180	0.0401228
$990$	0	0
$991$	−61.4284	−1.95134	−0.975669	−	0.219251i	$-0.929639\pi$
$991$	−61.4284	−1.95134	−0.975669	+	0.219251i	$0.929639\pi$
$992$	0	0
$993$	3.62475	0.115028
$994$	0	0
$995$	16.1217	0.511092
$996$	0	0
$997$	49.6307	1.57182	0.785910	−	0.618340i	$-0.212195\pi$
$997$	49.6307	1.57182	0.785910	+	0.618340i	$0.212195\pi$
$998$	0	0
$999$	3.35350	0.106100

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7440.2.a.bp.1.3		3
4.3	odd	2		465.2.a.f.1.3	✓	3
12.11	even	2		1395.2.a.i.1.1		3
20.3	even	4		2325.2.c.o.1024.1		6
20.7	even	4		2325.2.c.o.1024.6		6
20.19	odd	2		2325.2.a.q.1.1		3
60.59	even	2		6975.2.a.be.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
465.2.a.f.1.3	✓	3	4.3	odd	2
1395.2.a.i.1.1		3	12.11	even	2
2325.2.a.q.1.1		3	20.19	odd	2
2325.2.c.o.1024.1		6	20.3	even	4
2325.2.c.o.1024.6		6	20.7	even	4
6975.2.a.be.1.3		3	60.59	even	2
7440.2.a.bp.1.3		3	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 7440 = 2^{4} \cdot 3 \cdot 5 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7440.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +1.00000 q^{5} +1.17009 q^{7} +1.00000 q^{9} -2.00000 q^{11} +0.0917087 q^{13} -1.00000 q^{15} -6.04945 q^{17} -1.17009 q^{21} +1.70928 q^{23} +1.00000 q^{25} -1.00000 q^{27} +7.95774 q^{29} -1.00000 q^{31} +2.00000 q^{33} +1.17009 q^{35} -3.35350 q^{37} -0.0917087 q^{39} -4.68035 q^{41} +0.738205 q^{43} +1.00000 q^{45} +9.12783 q^{47} -5.63090 q^{49} +6.04945 q^{51} +4.97107 q^{53} -2.00000 q^{55} -11.0361 q^{59} -3.26180 q^{61} +1.17009 q^{63} +0.0917087 q^{65} -3.85043 q^{67} -1.70928 q^{69} +1.21953 q^{71} +7.66701 q^{73} -1.00000 q^{75} -2.34017 q^{77} -3.52586 q^{79} +1.00000 q^{81} -12.5464 q^{83} -6.04945 q^{85} -7.95774 q^{87} -5.77432 q^{89} +0.107307 q^{91} +1.00000 q^{93} -12.0000 q^{97} -2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{3} + 3 q^{5} - 2 q^{7} + 3 q^{9} - 6 q^{11} - 2 q^{13} - 3 q^{15} + 2 q^{21} - 2 q^{23} + 3 q^{25} - 3 q^{27} + 8 q^{29} - 3 q^{31} + 6 q^{33} - 2 q^{35} + 2 q^{39} + 8 q^{41} + 10 q^{43} + 3 q^{45}+ \cdots - 6 q^{99}+O(q^{100}) \) 3 * q - 3 * q^3 + 3 * q^5 - 2 * q^7 + 3 * q^9 - 6 * q^11 - 2 * q^13 - 3 * q^15 + 2 * q^21 - 2 * q^23 + 3 * q^25 - 3 * q^27 + 8 * q^29 - 3 * q^31 + 6 * q^33 - 2 * q^35 + 2 * q^39 + 8 * q^41 + 10 * q^43 + 3 * q^45 + 6 * q^47 - 13 * q^49 - 6 * q^55 - 14 * q^59 - 2 * q^61 - 2 * q^63 - 2 * q^65 + 16 * q^67 + 2 * q^69 - 20 * q^71 - 3 * q^75 + 4 * q^77 - 8 * q^79 + 3 * q^81 - 2 * q^83 - 8 * q^87 - 6 * q^89 + 12 * q^91 + 3 * q^93 - 36 * q^97 - 6 * q^99

Introduction

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