LMFDB - Embedded newform 7440.2.a.bs.1.1

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:	3.3.564.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 5x + 3 $ x^3 - x^2 - 5*x + 3
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			1.1
Root			$-2.08613$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} -1.00000 q^{5} -5.08613 q^{7} +1.00000 q^{9} +4.17226 q^{11} -1.08613 q^{13} -1.00000 q^{15} +0.648061 q^{17} +2.70388 q^{19} -5.08613 q^{21} -7.52420 q^{23} +1.00000 q^{25} +1.00000 q^{27} +5.90645 q^{29} -1.00000 q^{31} +4.17226 q^{33} +5.08613 q^{35} +0.913870 q^{37} -1.08613 q^{39} +2.17226 q^{41} -8.17226 q^{43} -1.00000 q^{45} +10.2281 q^{47} +18.8687 q^{49} +0.648061 q^{51} +5.52420 q^{53} -4.17226 q^{55} +2.70388 q^{57} -0.438069 q^{59} -2.00000 q^{61} -5.08613 q^{63} +1.08613 q^{65} -9.79001 q^{67} -7.52420 q^{69} -2.96969 q^{71} -1.25839 q^{73} +1.00000 q^{75} -21.2207 q^{77} -10.8203 q^{79} +1.00000 q^{81} -14.2281 q^{83} -0.648061 q^{85} +5.90645 q^{87} +18.1903 q^{89} +5.52420 q^{91} -1.00000 q^{93} -2.70388 q^{95} -13.2207 q^{97} +4.17226 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{3} - 3 q^{5} - 8 q^{7} + 3 q^{9} - 2 q^{11} + 4 q^{13} - 3 q^{15} + 4 q^{17} + 4 q^{19} - 8 q^{21} - 6 q^{23} + 3 q^{25} + 3 q^{27} - 2 q^{29} - 3 q^{31} - 2 q^{33} + 8 q^{35} + 10 q^{37} + 4 q^{39}+ \cdots - 2 q^{99}+O(q^{100}) $ 3 * q + 3 * q^3 - 3 * q^5 - 8 * q^7 + 3 * q^9 - 2 * q^11 + 4 * q^13 - 3 * q^15 + 4 * q^17 + 4 * q^19 - 8 * q^21 - 6 * q^23 + 3 * q^25 + 3 * q^27 - 2 * q^29 - 3 * q^31 - 2 * q^33 + 8 * q^35 + 10 * q^37 + 4 * q^39 - 8 * q^41 - 10 * q^43 - 3 * q^45 + 10 * q^47 + 11 * q^49 + 4 * q^51 + 2 * q^55 + 4 * q^57 + 8 * q^59 - 6 * q^61 - 8 * q^63 - 4 * q^65 - 18 * q^67 - 6 * q^69 - 10 * q^71 + 18 * q^73 + 3 * q^75 - 16 * q^77 - 20 * q^79 + 3 * q^81 - 22 * q^83 - 4 * q^85 - 2 * q^87 + 8 * q^89 - 3 * q^93 - 4 * q^95 + 8 * q^97 - 2 * 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
$7$	−5.08613	−1.92238	−0.961188	−	0.275893i	$-0.911026\pi$
$7$	−5.08613	−1.92238	−0.961188	+	0.275893i	$0.911026\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	4.17226	1.25798	0.628992	−	0.777412i	$-0.283468\pi$
$11$	4.17226	1.25798	0.628992	+	0.777412i	$0.283468\pi$
$12$	0	0
$13$	−1.08613	−0.301238	−0.150619	−	0.988592i	$-0.548127\pi$
$13$	−1.08613	−0.301238	−0.150619	+	0.988592i	$0.548127\pi$
$14$	0	0
$15$	−1.00000	−0.258199
$16$	0	0
$17$	0.648061	0.157178	0.0785889	−	0.996907i	$-0.474959\pi$
$17$	0.648061	0.157178	0.0785889	+	0.996907i	$0.474959\pi$
$18$	0	0
$19$	2.70388	0.620312	0.310156	−	0.950686i	$-0.399619\pi$
$19$	2.70388	0.620312	0.310156	+	0.950686i	$0.399619\pi$
$20$	0	0
$21$	−5.08613	−1.10988
$22$	0	0
$23$	−7.52420	−1.56890	−0.784452	−	0.620189i	$-0.787056\pi$
$23$	−7.52420	−1.56890	−0.784452	+	0.620189i	$0.787056\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	5.90645	1.09680	0.548400	−	0.836216i	$-0.315237\pi$
$29$	5.90645	1.09680	0.548400	+	0.836216i	$0.315237\pi$
$30$	0	0
$31$	−1.00000	−0.179605
$31$	−1.00000	−0.179605
$32$	0	0
$33$	4.17226	0.726297
$34$	0	0
$35$	5.08613	0.859713
$36$	0	0
$37$	0.913870	0.150239	0.0751196	−	0.997175i	$-0.476066\pi$
$37$	0.913870	0.150239	0.0751196	+	0.997175i	$0.476066\pi$
$38$	0	0
$39$	−1.08613	−0.173920
$40$	0	0
$41$	2.17226	0.339250	0.169625	−	0.985509i	$-0.445744\pi$
$41$	2.17226	0.339250	0.169625	+	0.985509i	$0.445744\pi$
$42$	0	0
$43$	−8.17226	−1.24626	−0.623129	−	0.782119i	$-0.714139\pi$
$43$	−8.17226	−1.24626	−0.623129	+	0.782119i	$0.714139\pi$
$44$	0	0
$45$	−1.00000	−0.149071
$46$	0	0
$47$	10.2281	1.49192	0.745959	−	0.665992i	$-0.231991\pi$
$47$	10.2281	1.49192	0.745959	+	0.665992i	$0.231991\pi$
$48$	0	0
$49$	18.8687	2.69553
$50$	0	0
$51$	0.648061	0.0907467
$52$	0	0
$53$	5.52420	0.758807	0.379404	−	0.925231i	$-0.376129\pi$
$53$	5.52420	0.758807	0.379404	+	0.925231i	$0.376129\pi$
$54$	0	0
$55$	−4.17226	−0.562587
$56$	0	0
$57$	2.70388	0.358137
$58$	0	0
$59$	−0.438069	−0.0570318	−0.0285159	−	0.999593i	$-0.509078\pi$
$59$	−0.438069	−0.0570318	−0.0285159	+	0.999593i	$0.509078\pi$
$60$	0	0
$61$	−2.00000	−0.256074	−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000	−0.256074	−0.128037	+	0.991769i	$0.540868\pi$
$62$	0	0
$63$	−5.08613	−0.640792
$64$	0	0
$65$	1.08613	0.134718
$66$	0	0
$67$	−9.79001	−1.19604	−0.598020	−	0.801481i	$-0.704046\pi$
$67$	−9.79001	−1.19604	−0.598020	+	0.801481i	$0.704046\pi$
$68$	0	0
$69$	−7.52420	−0.905807
$70$	0	0
$71$	−2.96969	−0.352437	−0.176219	−	0.984351i	$-0.556387\pi$
$71$	−2.96969	−0.352437	−0.176219	+	0.984351i	$0.556387\pi$
$72$	0	0
$73$	−1.25839	−0.147283	−0.0736417	−	0.997285i	$-0.523462\pi$
$73$	−1.25839	−0.147283	−0.0736417	+	0.997285i	$0.523462\pi$
$74$	0	0
$75$	1.00000	0.115470
$76$	0	0
$77$	−21.2207	−2.41832
$78$	0	0
$79$	−10.8203	−1.21738	−0.608691	−	0.793408i	$-0.708305\pi$
$79$	−10.8203	−1.21738	−0.608691	+	0.793408i	$0.708305\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−14.2281	−1.56173	−0.780867	−	0.624697i	$-0.785222\pi$
$83$	−14.2281	−1.56173	−0.780867	+	0.624697i	$0.785222\pi$
$84$	0	0
$85$	−0.648061	−0.0702921
$86$	0	0
$87$	5.90645	0.633238
$88$	0	0
$89$	18.1903	1.92817	0.964086	−	0.265589i	$-0.0855663\pi$
$89$	18.1903	1.92817	0.964086	+	0.265589i	$0.0855663\pi$
$90$	0	0
$91$	5.52420	0.579093
$92$	0	0
$93$	−1.00000	−0.103695
$94$	0	0
$95$	−2.70388	−0.277412
$96$	0	0
$97$	−13.2207	−1.34235	−0.671177	−	0.741297i	$-0.734211\pi$
$97$	−13.2207	−1.34235	−0.671177	+	0.741297i	$0.734211\pi$
$98$	0	0
$99$	4.17226	0.419328
$100$	0	0
$101$	−16.8761	−1.67924	−0.839619	−	0.543175i	$-0.817222\pi$
$101$	−16.8761	−1.67924	−0.839619	+	0.543175i	$0.817222\pi$
$102$	0	0
$103$	0.493887	0.0486641	0.0243321	−	0.999704i	$-0.492254\pi$
$103$	0.493887	0.0486641	0.0243321	+	0.999704i	$0.492254\pi$
$104$	0	0
$105$	5.08613	0.496355
$106$	0	0
$107$	−9.69646	−0.937392	−0.468696	−	0.883359i	$-0.655276\pi$
$107$	−9.69646	−0.937392	−0.468696	+	0.883359i	$0.655276\pi$
$108$	0	0
$109$	−16.4003	−1.57087	−0.785434	−	0.618946i	$-0.787560\pi$
$109$	−16.4003	−1.57087	−0.785434	+	0.618946i	$0.787560\pi$
$110$	0	0
$111$	0.913870	0.0867407
$112$	0	0
$113$	12.5168	1.17748	0.588740	−	0.808323i	$-0.299624\pi$
$113$	12.5168	1.17748	0.588740	+	0.808323i	$0.299624\pi$
$114$	0	0
$115$	7.52420	0.701635
$116$	0	0
$117$	−1.08613	−0.100413
$118$	0	0
$119$	−3.29612	−0.302155
$120$	0	0
$121$	6.40776	0.582523
$122$	0	0
$123$	2.17226	0.195866
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	7.58002	0.672618	0.336309	−	0.941752i	$-0.390821\pi$
$127$	7.58002	0.672618	0.336309	+	0.941752i	$0.390821\pi$
$128$	0	0
$129$	−8.17226	−0.719527
$130$	0	0
$131$	7.67357	0.670443	0.335221	−	0.942139i	$-0.391189\pi$
$131$	7.67357	0.670443	0.335221	+	0.942139i	$0.391189\pi$
$132$	0	0
$133$	−13.7523	−1.19247
$134$	0	0
$135$	−1.00000	−0.0860663
$136$	0	0
$137$	−4.75970	−0.406648	−0.203324	−	0.979111i	$-0.565175\pi$
$137$	−4.75970	−0.406648	−0.203324	+	0.979111i	$0.565175\pi$
$138$	0	0
$139$	−10.2839	−0.872269	−0.436134	−	0.899882i	$-0.643653\pi$
$139$	−10.2839	−0.872269	−0.436134	+	0.899882i	$0.643653\pi$
$140$	0	0
$141$	10.2281	0.861360
$142$	0	0
$143$	−4.53162	−0.378953
$144$	0	0
$145$	−5.90645	−0.490504
$146$	0	0
$147$	18.8687	1.55627
$148$	0	0
$149$	−0.764504	−0.0626306	−0.0313153	−	0.999510i	$-0.509970\pi$
$149$	−0.764504	−0.0626306	−0.0313153	+	0.999510i	$0.509970\pi$
$150$	0	0
$151$	0.992582	0.0807751	0.0403876	−	0.999184i	$-0.487141\pi$
$151$	0.992582	0.0807751	0.0403876	+	0.999184i	$0.487141\pi$
$152$	0	0
$153$	0.648061	0.0523926
$154$	0	0
$155$	1.00000	0.0803219
$156$	0	0
$157$	15.9245	1.27092	0.635458	−	0.772135i	$-0.280811\pi$
$157$	15.9245	1.27092	0.635458	+	0.772135i	$0.280811\pi$
$158$	0	0
$159$	5.52420	0.438098
$160$	0	0
$161$	38.2691	3.01602
$162$	0	0
$163$	1.25839	0.0985648	0.0492824	−	0.998785i	$-0.484307\pi$
$163$	1.25839	0.0985648	0.0492824	+	0.998785i	$0.484307\pi$
$164$	0	0
$165$	−4.17226	−0.324810
$166$	0	0
$167$	−23.5800	−1.82468	−0.912338	−	0.409437i	$-0.865725\pi$
$167$	−23.5800	−1.82468	−0.912338	+	0.409437i	$0.865725\pi$
$168$	0	0
$169$	−11.8203	−0.909255
$170$	0	0
$171$	2.70388	0.206771
$172$	0	0
$173$	−20.1723	−1.53367	−0.766834	−	0.641845i	$-0.778169\pi$
$173$	−20.1723	−1.53367	−0.766834	+	0.641845i	$0.778169\pi$
$174$	0	0
$175$	−5.08613	−0.384475
$176$	0	0
$177$	−0.438069	−0.0329273
$178$	0	0
$179$	4.11164	0.307318	0.153659	−	0.988124i	$-0.450894\pi$
$179$	4.11164	0.307318	0.153659	+	0.988124i	$0.450894\pi$
$180$	0	0
$181$	10.3445	0.768902	0.384451	−	0.923145i	$-0.374391\pi$
$181$	10.3445	0.768902	0.384451	+	0.923145i	$0.374391\pi$
$182$	0	0
$183$	−2.00000	−0.147844
$184$	0	0
$185$	−0.913870	−0.0671890
$186$	0	0
$187$	2.70388	0.197727
$188$	0	0
$189$	−5.08613	−0.369962
$190$	0	0
$191$	−3.56193	−0.257732	−0.128866	−	0.991662i	$-0.541134\pi$
$191$	−3.56193	−0.257732	−0.128866	+	0.991662i	$0.541134\pi$
$192$	0	0
$193$	−3.58002	−0.257695	−0.128848	−	0.991664i	$-0.541128\pi$
$193$	−3.58002	−0.257695	−0.128848	+	0.991664i	$0.541128\pi$
$194$	0	0
$195$	1.08613	0.0777794
$196$	0	0
$197$	7.35194	0.523804	0.261902	−	0.965094i	$-0.415650\pi$
$197$	7.35194	0.523804	0.261902	+	0.965094i	$0.415650\pi$
$198$	0	0
$199$	−25.1042	−1.77959	−0.889795	−	0.456360i	$-0.849153\pi$
$199$	−25.1042	−1.77959	−0.889795	+	0.456360i	$0.849153\pi$
$200$	0	0
$201$	−9.79001	−0.690534
$202$	0	0
$203$	−30.0410	−2.10846
$204$	0	0
$205$	−2.17226	−0.151717
$206$	0	0
$207$	−7.52420	−0.522968
$208$	0	0
$209$	11.2813	0.780343
$210$	0	0
$211$	14.4051	0.991691	0.495846	−	0.868411i	$-0.334858\pi$
$211$	14.4051	0.991691	0.495846	+	0.868411i	$0.334858\pi$
$212$	0	0
$213$	−2.96969	−0.203480
$214$	0	0
$215$	8.17226	0.557344
$216$	0	0
$217$	5.08613	0.345269
$218$	0	0
$219$	−1.25839	−0.0850342
$220$	0	0
$221$	−0.703878	−0.0473480
$222$	0	0
$223$	−11.4078	−0.763920	−0.381960	−	0.924179i	$-0.624751\pi$
$223$	−11.4078	−0.763920	−0.381960	+	0.924179i	$0.624751\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	15.3371	1.01796	0.508980	−	0.860779i	$-0.330023\pi$
$227$	15.3371	1.01796	0.508980	+	0.860779i	$0.330023\pi$
$228$	0	0
$229$	−15.8277	−1.04593	−0.522963	−	0.852355i	$-0.675174\pi$
$229$	−15.8277	−1.04593	−0.522963	+	0.852355i	$0.675174\pi$
$230$	0	0
$231$	−21.2207	−1.39622
$232$	0	0
$233$	−27.9293	−1.82971	−0.914856	−	0.403780i	$-0.867696\pi$
$233$	−27.9293	−1.82971	−0.914856	+	0.403780i	$0.867696\pi$
$234$	0	0
$235$	−10.2281	−0.667206
$236$	0	0
$237$	−10.8203	−0.702855
$238$	0	0
$239$	10.4562	0.676353	0.338176	−	0.941083i	$-0.390190\pi$
$239$	10.4562	0.676353	0.338176	+	0.941083i	$0.390190\pi$
$240$	0	0
$241$	−6.87614	−0.442931	−0.221466	−	0.975168i	$-0.571084\pi$
$241$	−6.87614	−0.442931	−0.221466	+	0.975168i	$0.571084\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−18.8687	−1.20548
$246$	0	0
$247$	−2.93676	−0.186862
$248$	0	0
$249$	−14.2281	−0.901668
$250$	0	0
$251$	−7.39292	−0.466637	−0.233318	−	0.972400i	$-0.574958\pi$
$251$	−7.39292	−0.466637	−0.233318	+	0.972400i	$0.574958\pi$
$252$	0	0
$253$	−31.3929	−1.97366
$254$	0	0
$255$	−0.648061	−0.0405831
$256$	0	0
$257$	−6.28870	−0.392279	−0.196139	−	0.980576i	$-0.562840\pi$
$257$	−6.28870	−0.392279	−0.196139	+	0.980576i	$0.562840\pi$
$258$	0	0
$259$	−4.64806	−0.288816
$260$	0	0
$261$	5.90645	0.365600
$262$	0	0
$263$	14.6284	0.902027	0.451013	−	0.892517i	$-0.351063\pi$
$263$	14.6284	0.902027	0.451013	+	0.892517i	$0.351063\pi$
$264$	0	0
$265$	−5.52420	−0.339349
$266$	0	0
$267$	18.1903	1.11323
$268$	0	0
$269$	−10.4381	−0.636420	−0.318210	−	0.948020i	$-0.603082\pi$
$269$	−10.4381	−0.636420	−0.318210	+	0.948020i	$0.603082\pi$
$270$	0	0
$271$	32.1574	1.95342	0.976712	−	0.214554i	$-0.0688297\pi$
$271$	32.1574	1.95342	0.976712	+	0.214554i	$0.0688297\pi$
$272$	0	0
$273$	5.52420	0.334340
$274$	0	0
$275$	4.17226	0.251597
$276$	0	0
$277$	−32.5397	−1.95512	−0.977560	−	0.210658i	$-0.932439\pi$
$277$	−32.5397	−1.95512	−0.977560	+	0.210658i	$0.932439\pi$
$278$	0	0
$279$	−1.00000	−0.0598684
$280$	0	0
$281$	−3.82774	−0.228344	−0.114172	−	0.993461i	$-0.536421\pi$
$281$	−3.82774	−0.228344	−0.114172	+	0.993461i	$0.536421\pi$
$282$	0	0
$283$	−6.03773	−0.358906	−0.179453	−	0.983767i	$-0.557433\pi$
$283$	−6.03773	−0.358906	−0.179453	+	0.983767i	$0.557433\pi$
$284$	0	0
$285$	−2.70388	−0.160164
$286$	0	0
$287$	−11.0484	−0.652166
$288$	0	0
$289$	−16.5800	−0.975295
$290$	0	0
$291$	−13.2207	−0.775009
$292$	0	0
$293$	9.33710	0.545479	0.272740	−	0.962088i	$-0.412070\pi$
$293$	9.33710	0.545479	0.272740	+	0.962088i	$0.412070\pi$
$294$	0	0
$295$	0.438069	0.0255054
$296$	0	0
$297$	4.17226	0.242099
$298$	0	0
$299$	8.17226	0.472614
$300$	0	0
$301$	41.5652	2.39578
$302$	0	0
$303$	−16.8761	−0.969509
$304$	0	0
$305$	2.00000	0.114520
$306$	0	0
$307$	−26.4184	−1.50778	−0.753890	−	0.657001i	$-0.771825\pi$
$307$	−26.4184	−1.50778	−0.753890	+	0.657001i	$0.771825\pi$
$308$	0	0
$309$	0.493887	0.0280962
$310$	0	0
$311$	6.36261	0.360790	0.180395	−	0.983594i	$-0.442262\pi$
$311$	6.36261	0.360790	0.180395	+	0.983594i	$0.442262\pi$
$312$	0	0
$313$	8.02289	0.453481	0.226740	−	0.973955i	$-0.427193\pi$
$313$	8.02289	0.453481	0.226740	+	0.973955i	$0.427193\pi$
$314$	0	0
$315$	5.08613	0.286571
$316$	0	0
$317$	4.75970	0.267331	0.133666	−	0.991026i	$-0.457325\pi$
$317$	4.75970	0.267331	0.133666	+	0.991026i	$0.457325\pi$
$318$	0	0
$319$	24.6433	1.37976
$320$	0	0
$321$	−9.69646	−0.541204
$322$	0	0
$323$	1.75228	0.0974993
$324$	0	0
$325$	−1.08613	−0.0602477
$326$	0	0
$327$	−16.4003	−0.906941
$328$	0	0
$329$	−52.0213	−2.86803
$330$	0	0
$331$	21.9804	1.20815	0.604075	−	0.796928i	$-0.293543\pi$
$331$	21.9804	1.20815	0.604075	+	0.796928i	$0.293543\pi$
$332$	0	0
$333$	0.913870	0.0500798
$334$	0	0
$335$	9.79001	0.534885
$336$	0	0
$337$	28.1952	1.53589	0.767944	−	0.640517i	$-0.221280\pi$
$337$	28.1952	1.53589	0.767944	+	0.640517i	$0.221280\pi$
$338$	0	0
$339$	12.5168	0.679818
$340$	0	0
$341$	−4.17226	−0.225941
$342$	0	0
$343$	−60.3659	−3.25945
$344$	0	0
$345$	7.52420	0.405089
$346$	0	0
$347$	−17.1090	−0.918461	−0.459230	−	0.888317i	$-0.651875\pi$
$347$	−17.1090	−0.918461	−0.459230	+	0.888317i	$0.651875\pi$
$348$	0	0
$349$	−2.53643	−0.135772	−0.0678859	−	0.997693i	$-0.521625\pi$
$349$	−2.53643	−0.135772	−0.0678859	+	0.997693i	$0.521625\pi$
$350$	0	0
$351$	−1.08613	−0.0579733
$352$	0	0
$353$	−25.2255	−1.34262	−0.671308	−	0.741178i	$-0.734267\pi$
$353$	−25.2255	−1.34262	−0.671308	+	0.741178i	$0.734267\pi$
$354$	0	0
$355$	2.96969	0.157615
$356$	0	0
$357$	−3.29612	−0.174449
$358$	0	0
$359$	−29.7704	−1.57122	−0.785610	−	0.618722i	$-0.787651\pi$
$359$	−29.7704	−1.57122	−0.785610	+	0.618722i	$0.787651\pi$
$360$	0	0
$361$	−11.6890	−0.615213
$362$	0	0
$363$	6.40776	0.336320
$364$	0	0
$365$	1.25839	0.0658672
$366$	0	0
$367$	12.3445	0.644379	0.322189	−	0.946675i	$-0.395581\pi$
$367$	12.3445	0.644379	0.322189	+	0.946675i	$0.395581\pi$
$368$	0	0
$369$	2.17226	0.113083
$370$	0	0
$371$	−28.0968	−1.45871
$372$	0	0
$373$	34.3807	1.78016	0.890082	−	0.455800i	$-0.150647\pi$
$373$	34.3807	1.78016	0.890082	+	0.455800i	$0.150647\pi$
$374$	0	0
$375$	−1.00000	−0.0516398
$376$	0	0
$377$	−6.41518	−0.330398
$378$	0	0
$379$	−22.3297	−1.14700	−0.573499	−	0.819206i	$-0.694415\pi$
$379$	−22.3297	−1.14700	−0.573499	+	0.819206i	$0.694415\pi$
$380$	0	0
$381$	7.58002	0.388336
$382$	0	0
$383$	−15.4126	−0.787545	−0.393773	−	0.919208i	$-0.628830\pi$
$383$	−15.4126	−0.787545	−0.393773	+	0.919208i	$0.628830\pi$
$384$	0	0
$385$	21.2207	1.08150
$386$	0	0
$387$	−8.17226	−0.415419
$388$	0	0
$389$	−14.9549	−0.758241	−0.379121	−	0.925347i	$-0.623773\pi$
$389$	−14.9549	−0.758241	−0.379121	+	0.925347i	$0.623773\pi$
$390$	0	0
$391$	−4.87614	−0.246597
$392$	0	0
$393$	7.67357	0.387080
$394$	0	0
$395$	10.8203	0.544429
$396$	0	0
$397$	−33.0484	−1.65865	−0.829326	−	0.558765i	$-0.811275\pi$
$397$	−33.0484	−1.65865	−0.829326	+	0.558765i	$0.811275\pi$
$398$	0	0
$399$	−13.7523	−0.688475
$400$	0	0
$401$	3.56193	0.177874	0.0889372	−	0.996037i	$-0.471653\pi$
$401$	3.56193	0.177874	0.0889372	+	0.996037i	$0.471653\pi$
$402$	0	0
$403$	1.08613	0.0541040
$404$	0	0
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	3.81290	0.188999
$408$	0	0
$409$	−13.6555	−0.675220	−0.337610	−	0.941286i	$-0.609618\pi$
$409$	−13.6555	−0.675220	−0.337610	+	0.941286i	$0.609618\pi$
$410$	0	0
$411$	−4.75970	−0.234778
$412$	0	0
$413$	2.22808	0.109637
$414$	0	0
$415$	14.2281	0.698429
$416$	0	0
$417$	−10.2839	−0.503605
$418$	0	0
$419$	17.7948	0.869334	0.434667	−	0.900591i	$-0.356866\pi$
$419$	17.7948	0.869334	0.434667	+	0.900591i	$0.356866\pi$
$420$	0	0
$421$	8.70869	0.424435	0.212218	−	0.977222i	$-0.431931\pi$
$421$	8.70869	0.424435	0.212218	+	0.977222i	$0.431931\pi$
$422$	0	0
$423$	10.2281	0.497306
$424$	0	0
$425$	0.648061	0.0314356
$426$	0	0
$427$	10.1723	0.492270
$428$	0	0
$429$	−4.53162	−0.218789
$430$	0	0
$431$	16.2658	0.783496	0.391748	−	0.920072i	$-0.371870\pi$
$431$	16.2658	0.783496	0.391748	+	0.920072i	$0.371870\pi$
$432$	0	0
$433$	−7.85063	−0.377277	−0.188639	−	0.982047i	$-0.560408\pi$
$433$	−7.85063	−0.377277	−0.188639	+	0.982047i	$0.560408\pi$
$434$	0	0
$435$	−5.90645	−0.283193
$436$	0	0
$437$	−20.3445	−0.973210
$438$	0	0
$439$	−16.2233	−0.774294	−0.387147	−	0.922018i	$-0.626539\pi$
$439$	−16.2233	−0.774294	−0.387147	+	0.922018i	$0.626539\pi$
$440$	0	0
$441$	18.8687	0.898510
$442$	0	0
$443$	−20.7449	−0.985618	−0.492809	−	0.870138i	$-0.664030\pi$
$443$	−20.7449	−0.985618	−0.492809	+	0.870138i	$0.664030\pi$
$444$	0	0
$445$	−18.1903	−0.862305
$446$	0	0
$447$	−0.764504	−0.0361598
$448$	0	0
$449$	−24.7826	−1.16956	−0.584781	−	0.811191i	$-0.698820\pi$
$449$	−24.7826	−1.16956	−0.584781	+	0.811191i	$0.698820\pi$
$450$	0	0
$451$	9.06324	0.426771
$452$	0	0
$453$	0.992582	0.0466356
$454$	0	0
$455$	−5.52420	−0.258978
$456$	0	0
$457$	27.8868	1.30449	0.652245	−	0.758008i	$-0.273827\pi$
$457$	27.8868	1.30449	0.652245	+	0.758008i	$0.273827\pi$
$458$	0	0
$459$	0.648061	0.0302489
$460$	0	0
$461$	−29.4110	−1.36981	−0.684904	−	0.728634i	$-0.740156\pi$
$461$	−29.4110	−1.36981	−0.684904	+	0.728634i	$0.740156\pi$
$462$	0	0
$463$	25.1239	1.16760	0.583802	−	0.811896i	$-0.301564\pi$
$463$	25.1239	1.16760	0.583802	+	0.811896i	$0.301564\pi$
$464$	0	0
$465$	1.00000	0.0463739
$466$	0	0
$467$	34.8055	1.61061	0.805303	−	0.592864i	$-0.202003\pi$
$467$	34.8055	1.61061	0.805303	+	0.592864i	$0.202003\pi$
$468$	0	0
$469$	49.7933	2.29924
$470$	0	0
$471$	15.9245	0.733764
$472$	0	0
$473$	−34.0968	−1.56777
$474$	0	0
$475$	2.70388	0.124062
$476$	0	0
$477$	5.52420	0.252936
$478$	0	0
$479$	−35.7704	−1.63439	−0.817195	−	0.576362i	$-0.804472\pi$
$479$	−35.7704	−1.63439	−0.817195	+	0.576362i	$0.804472\pi$
$480$	0	0
$481$	−0.992582	−0.0452578
$482$	0	0
$483$	38.2691	1.74130
$484$	0	0
$485$	13.2207	0.600319
$486$	0	0
$487$	21.5652	0.977212	0.488606	−	0.872505i	$-0.337506\pi$
$487$	21.5652	0.977212	0.488606	+	0.872505i	$0.337506\pi$
$488$	0	0
$489$	1.25839	0.0569064
$490$	0	0
$491$	−26.7497	−1.20720	−0.603598	−	0.797289i	$-0.706267\pi$
$491$	−26.7497	−1.20720	−0.603598	+	0.797289i	$0.706267\pi$
$492$	0	0
$493$	3.82774	0.172393
$494$	0	0
$495$	−4.17226	−0.187529
$496$	0	0
$497$	15.1042	0.677517
$498$	0	0
$499$	10.9368	0.489597	0.244798	−	0.969574i	$-0.421278\pi$
$499$	10.9368	0.489597	0.244798	+	0.969574i	$0.421278\pi$
$500$	0	0
$501$	−23.5800	−1.05348
$502$	0	0
$503$	18.2281	0.812750	0.406375	−	0.913706i	$-0.366793\pi$
$503$	18.2281	0.812750	0.406375	+	0.913706i	$0.366793\pi$
$504$	0	0
$505$	16.8761	0.750978
$506$	0	0
$507$	−11.8203	−0.524959
$508$	0	0
$509$	10.6103	0.470295	0.235147	−	0.971960i	$-0.424443\pi$
$509$	10.6103	0.470295	0.235147	+	0.971960i	$0.424443\pi$
$510$	0	0
$511$	6.40034	0.283134
$512$	0	0
$513$	2.70388	0.119379
$514$	0	0
$515$	−0.493887	−0.0217633
$516$	0	0
$517$	42.6742	1.87681
$518$	0	0
$519$	−20.1723	−0.885464
$520$	0	0
$521$	−25.6768	−1.12492	−0.562461	−	0.826824i	$-0.690145\pi$
$521$	−25.6768	−1.12492	−0.562461	+	0.826824i	$0.690145\pi$
$522$	0	0
$523$	−20.1574	−0.881423	−0.440711	−	0.897649i	$-0.645274\pi$
$523$	−20.1574	−0.881423	−0.440711	+	0.897649i	$0.645274\pi$
$524$	0	0
$525$	−5.08613	−0.221977
$526$	0	0
$527$	−0.648061	−0.0282300
$528$	0	0
$529$	33.6136	1.46146
$530$	0	0
$531$	−0.438069	−0.0190106
$532$	0	0
$533$	−2.35936	−0.102195
$534$	0	0
$535$	9.69646	0.419215
$536$	0	0
$537$	4.11164	0.177430
$538$	0	0
$539$	78.7252	3.39094
$540$	0	0
$541$	30.4610	1.30962	0.654810	−	0.755794i	$-0.272749\pi$
$541$	30.4610	1.30962	0.654810	+	0.755794i	$0.272749\pi$
$542$	0	0
$543$	10.3445	0.443926
$544$	0	0
$545$	16.4003	0.702513
$546$	0	0
$547$	11.8506	0.506697	0.253348	−	0.967375i	$-0.418468\pi$
$547$	11.8506	0.506697	0.253348	+	0.967375i	$0.418468\pi$
$548$	0	0
$549$	−2.00000	−0.0853579
$550$	0	0
$551$	15.9703	0.680359
$552$	0	0
$553$	55.0336	2.34027
$554$	0	0
$555$	−0.913870	−0.0387916
$556$	0	0
$557$	−3.95902	−0.167749	−0.0838745	−	0.996476i	$-0.526729\pi$
$557$	−3.95902	−0.167749	−0.0838745	+	0.996476i	$0.526729\pi$
$558$	0	0
$559$	8.87614	0.375421
$560$	0	0
$561$	2.70388	0.114158
$562$	0	0
$563$	13.6965	0.577237	0.288618	−	0.957444i	$-0.406804\pi$
$563$	13.6965	0.577237	0.288618	+	0.957444i	$0.406804\pi$
$564$	0	0
$565$	−12.5168	−0.526585
$566$	0	0
$567$	−5.08613	−0.213597
$568$	0	0
$569$	21.4865	0.900760	0.450380	−	0.892837i	$-0.351289\pi$
$569$	21.4865	0.900760	0.450380	+	0.892837i	$0.351289\pi$
$570$	0	0
$571$	3.46838	0.145147	0.0725736	−	0.997363i	$-0.476879\pi$
$571$	3.46838	0.145147	0.0725736	+	0.997363i	$0.476879\pi$
$572$	0	0
$573$	−3.56193	−0.148802
$574$	0	0
$575$	−7.52420	−0.313781
$576$	0	0
$577$	16.9516	0.705704	0.352852	−	0.935679i	$-0.385212\pi$
$577$	16.9516	0.705704	0.352852	+	0.935679i	$0.385212\pi$
$578$	0	0
$579$	−3.58002	−0.148780
$580$	0	0
$581$	72.3659	3.00224
$582$	0	0
$583$	23.0484	0.954567
$584$	0	0
$585$	1.08613	0.0449060
$586$	0	0
$587$	−6.97294	−0.287804	−0.143902	−	0.989592i	$-0.545965\pi$
$587$	−6.97294	−0.287804	−0.143902	+	0.989592i	$0.545965\pi$
$588$	0	0
$589$	−2.70388	−0.111411
$590$	0	0
$591$	7.35194	0.302418
$592$	0	0
$593$	−27.9097	−1.14611	−0.573057	−	0.819515i	$-0.694243\pi$
$593$	−27.9097	−1.14611	−0.573057	+	0.819515i	$0.694243\pi$
$594$	0	0
$595$	3.29612	0.135128
$596$	0	0
$597$	−25.1042	−1.02745
$598$	0	0
$599$	−30.4381	−1.24367	−0.621833	−	0.783150i	$-0.713612\pi$
$599$	−30.4381	−1.24367	−0.621833	+	0.783150i	$0.713612\pi$
$600$	0	0
$601$	−25.7013	−1.04838	−0.524188	−	0.851602i	$-0.675631\pi$
$601$	−25.7013	−1.04838	−0.524188	+	0.851602i	$0.675631\pi$
$602$	0	0
$603$	−9.79001	−0.398680
$604$	0	0
$605$	−6.40776	−0.260512
$606$	0	0
$607$	22.7119	0.921849	0.460925	−	0.887439i	$-0.347518\pi$
$607$	22.7119	0.921849	0.460925	+	0.887439i	$0.347518\pi$
$608$	0	0
$609$	−30.0410	−1.21732
$610$	0	0
$611$	−11.1090	−0.449423
$612$	0	0
$613$	30.4939	1.23164	0.615818	−	0.787888i	$-0.288826\pi$
$613$	30.4939	1.23164	0.615818	+	0.787888i	$0.288826\pi$
$614$	0	0
$615$	−2.17226	−0.0875940
$616$	0	0
$617$	10.6890	0.430325	0.215162	−	0.976578i	$-0.430972\pi$
$617$	10.6890	0.430325	0.215162	+	0.976578i	$0.430972\pi$
$618$	0	0
$619$	12.3397	0.495975	0.247987	−	0.968763i	$-0.420231\pi$
$619$	12.3397	0.495975	0.247987	+	0.968763i	$0.420231\pi$
$620$	0	0
$621$	−7.52420	−0.301936
$622$	0	0
$623$	−92.5185	−3.70667
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	11.2813	0.450531
$628$	0	0
$629$	0.592243	0.0236143
$630$	0	0
$631$	23.1600	0.921986	0.460993	−	0.887404i	$-0.347493\pi$
$631$	23.1600	0.921986	0.460993	+	0.887404i	$0.347493\pi$
$632$	0	0
$633$	14.4051	0.572553
$634$	0	0
$635$	−7.58002	−0.300804
$636$	0	0
$637$	−20.4939	−0.811997
$638$	0	0
$639$	−2.96969	−0.117479
$640$	0	0
$641$	1.63739	0.0646731	0.0323366	−	0.999477i	$-0.489705\pi$
$641$	1.63739	0.0646731	0.0323366	+	0.999477i	$0.489705\pi$
$642$	0	0
$643$	0.531618	0.0209650	0.0104825	−	0.999945i	$-0.496663\pi$
$643$	0.531618	0.0209650	0.0104825	+	0.999945i	$0.496663\pi$
$644$	0	0
$645$	8.17226	0.321782
$646$	0	0
$647$	9.86391	0.387790	0.193895	−	0.981022i	$-0.437888\pi$
$647$	9.86391	0.387790	0.193895	+	0.981022i	$0.437888\pi$
$648$	0	0
$649$	−1.82774	−0.0717451
$650$	0	0
$651$	5.08613	0.199341
$652$	0	0
$653$	−43.8081	−1.71434	−0.857172	−	0.515031i	$-0.827780\pi$
$653$	−43.8081	−1.71434	−0.857172	+	0.515031i	$0.827780\pi$
$654$	0	0
$655$	−7.67357	−0.299831
$656$	0	0
$657$	−1.25839	−0.0490945
$658$	0	0
$659$	−34.0639	−1.32694	−0.663470	−	0.748203i	$-0.730917\pi$
$659$	−34.0639	−1.32694	−0.663470	+	0.748203i	$0.730917\pi$
$660$	0	0
$661$	21.3567	0.830681	0.415341	−	0.909666i	$-0.363662\pi$
$661$	21.3567	0.830681	0.415341	+	0.909666i	$0.363662\pi$
$662$	0	0
$663$	−0.703878	−0.0273364
$664$	0	0
$665$	13.7523	0.533290
$666$	0	0
$667$	−44.4413	−1.72077
$668$	0	0
$669$	−11.4078	−0.441049
$670$	0	0
$671$	−8.34452	−0.322137
$672$	0	0
$673$	28.8990	1.11398	0.556988	−	0.830521i	$-0.311957\pi$
$673$	28.8990	1.11398	0.556988	+	0.830521i	$0.311957\pi$
$674$	0	0
$675$	1.00000	0.0384900
$676$	0	0
$677$	−27.8539	−1.07051	−0.535256	−	0.844690i	$-0.679785\pi$
$677$	−27.8539	−1.07051	−0.535256	+	0.844690i	$0.679785\pi$
$678$	0	0
$679$	67.2420	2.58051
$680$	0	0
$681$	15.3371	0.587719
$682$	0	0
$683$	−15.7113	−0.601176	−0.300588	−	0.953754i	$-0.597183\pi$
$683$	−15.7113	−0.601176	−0.300588	+	0.953754i	$0.597183\pi$
$684$	0	0
$685$	4.75970	0.181859
$686$	0	0
$687$	−15.8277	−0.603866
$688$	0	0
$689$	−6.00000	−0.228582
$690$	0	0
$691$	−7.92454	−0.301464	−0.150732	−	0.988575i	$-0.548163\pi$
$691$	−7.92454	−0.301464	−0.150732	+	0.988575i	$0.548163\pi$
$692$	0	0
$693$	−21.2207	−0.806106
$694$	0	0
$695$	10.2839	0.390090
$696$	0	0
$697$	1.40776	0.0533226
$698$	0	0
$699$	−27.9293	−1.05638
$700$	0	0
$701$	34.0968	1.28782	0.643909	−	0.765102i	$-0.277311\pi$
$701$	34.0968	1.28782	0.643909	+	0.765102i	$0.277311\pi$
$702$	0	0
$703$	2.47099	0.0931953
$704$	0	0
$705$	−10.2281	−0.385212
$706$	0	0
$707$	85.8342	3.22813
$708$	0	0
$709$	−46.8826	−1.76071	−0.880357	−	0.474311i	$-0.842697\pi$
$709$	−46.8826	−1.76071	−0.880357	+	0.474311i	$0.842697\pi$
$710$	0	0
$711$	−10.8203	−0.405794
$712$	0	0
$713$	7.52420	0.281783
$714$	0	0
$715$	4.53162	0.169473
$716$	0	0
$717$	10.4562	0.390492
$718$	0	0
$719$	4.51678	0.168448	0.0842238	−	0.996447i	$-0.473159\pi$
$719$	4.51678	0.168448	0.0842238	+	0.996447i	$0.473159\pi$
$720$	0	0
$721$	−2.51197	−0.0935508
$722$	0	0
$723$	−6.87614	−0.255726
$724$	0	0
$725$	5.90645	0.219360
$726$	0	0
$727$	−4.72677	−0.175306	−0.0876531	−	0.996151i	$-0.527937\pi$
$727$	−4.72677	−0.175306	−0.0876531	+	0.996151i	$0.527937\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−5.29612	−0.195884
$732$	0	0
$733$	−1.87875	−0.0693932	−0.0346966	−	0.999398i	$-0.511046\pi$
$733$	−1.87875	−0.0693932	−0.0346966	+	0.999398i	$0.511046\pi$
$734$	0	0
$735$	−18.8687	−0.695983
$736$	0	0
$737$	−40.8465	−1.50460
$738$	0	0
$739$	16.1526	0.594184	0.297092	−	0.954849i	$-0.403983\pi$
$739$	16.1526	0.594184	0.297092	+	0.954849i	$0.403983\pi$
$740$	0	0
$741$	−2.93676	−0.107885
$742$	0	0
$743$	23.3175	0.855435	0.427717	−	0.903913i	$-0.359318\pi$
$743$	23.3175	0.855435	0.427717	+	0.903913i	$0.359318\pi$
$744$	0	0
$745$	0.764504	0.0280092
$746$	0	0
$747$	−14.2281	−0.520578
$748$	0	0
$749$	49.3175	1.80202
$750$	0	0
$751$	3.46838	0.126563	0.0632815	−	0.997996i	$-0.479843\pi$
$751$	3.46838	0.126563	0.0632815	+	0.997996i	$0.479843\pi$
$752$	0	0
$753$	−7.39292	−0.269413
$754$	0	0
$755$	−0.992582	−0.0361237
$756$	0	0
$757$	41.9326	1.52407	0.762033	−	0.647538i	$-0.224201\pi$
$757$	41.9326	1.52407	0.762033	+	0.647538i	$0.224201\pi$
$758$	0	0
$759$	−31.3929	−1.13949
$760$	0	0
$761$	−13.1877	−0.478055	−0.239028	−	0.971013i	$-0.576829\pi$
$761$	−13.1877	−0.478055	−0.239028	+	0.971013i	$0.576829\pi$
$762$	0	0
$763$	83.4143	3.01980
$764$	0	0
$765$	−0.648061	−0.0234307
$766$	0	0
$767$	0.475800	0.0171802
$768$	0	0
$769$	−47.6816	−1.71944	−0.859722	−	0.510763i	$-0.829363\pi$
$769$	−47.6816	−1.71944	−0.859722	+	0.510763i	$0.829363\pi$
$770$	0	0
$771$	−6.28870	−0.226482
$772$	0	0
$773$	30.6236	1.10145	0.550727	−	0.834685i	$-0.314350\pi$
$773$	30.6236	1.10145	0.550727	+	0.834685i	$0.314350\pi$
$774$	0	0
$775$	−1.00000	−0.0359211
$776$	0	0
$777$	−4.64806	−0.166748
$778$	0	0
$779$	5.87353	0.210441
$780$	0	0
$781$	−12.3903	−0.443360
$782$	0	0
$783$	5.90645	0.211079
$784$	0	0
$785$	−15.9245	−0.568371
$786$	0	0
$787$	−26.9219	−0.959663	−0.479832	−	0.877361i	$-0.659302\pi$
$787$	−26.9219	−0.959663	−0.479832	+	0.877361i	$0.659302\pi$
$788$	0	0
$789$	14.6284	0.520785
$790$	0	0
$791$	−63.6620	−2.26356
$792$	0	0
$793$	2.17226	0.0771392
$794$	0	0
$795$	−5.52420	−0.195923
$796$	0	0
$797$	−23.9197	−0.847280	−0.423640	−	0.905831i	$-0.639248\pi$
$797$	−23.9197	−0.847280	−0.423640	+	0.905831i	$0.639248\pi$
$798$	0	0
$799$	6.62842	0.234497
$800$	0	0
$801$	18.1903	0.642724
$802$	0	0
$803$	−5.25033	−0.185280
$804$	0	0
$805$	−38.2691	−1.34881
$806$	0	0
$807$	−10.4381	−0.367437
$808$	0	0
$809$	34.8336	1.22468	0.612342	−	0.790593i	$-0.290228\pi$
$809$	34.8336	1.22468	0.612342	+	0.790593i	$0.290228\pi$
$810$	0	0
$811$	37.8277	1.32831	0.664156	−	0.747594i	$-0.268791\pi$
$811$	37.8277	1.32831	0.664156	+	0.747594i	$0.268791\pi$
$812$	0	0
$813$	32.1574	1.12781
$814$	0	0
$815$	−1.25839	−0.0440795
$816$	0	0
$817$	−22.0968	−0.773069
$818$	0	0
$819$	5.52420	0.193031
$820$	0	0
$821$	−31.6439	−1.10438	−0.552190	−	0.833718i	$-0.686208\pi$
$821$	−31.6439	−1.10438	−0.552190	+	0.833718i	$0.686208\pi$
$822$	0	0
$823$	−16.9729	−0.591639	−0.295820	−	0.955244i	$-0.595593\pi$
$823$	−16.9729	−0.591639	−0.295820	+	0.955244i	$0.595593\pi$
$824$	0	0
$825$	4.17226	0.145259
$826$	0	0
$827$	25.0532	0.871185	0.435593	−	0.900144i	$-0.356539\pi$
$827$	25.0532	0.871185	0.435593	+	0.900144i	$0.356539\pi$
$828$	0	0
$829$	28.7497	0.998517	0.499259	−	0.866453i	$-0.333606\pi$
$829$	28.7497	0.998517	0.499259	+	0.866453i	$0.333606\pi$
$830$	0	0
$831$	−32.5397	−1.12879
$832$	0	0
$833$	12.2281	0.423678
$834$	0	0
$835$	23.5800	0.816020
$836$	0	0
$837$	−1.00000	−0.0345651
$838$	0	0
$839$	7.83099	0.270356	0.135178	−	0.990821i	$-0.456839\pi$
$839$	7.83099	0.270356	0.135178	+	0.990821i	$0.456839\pi$
$840$	0	0
$841$	5.88617	0.202971
$842$	0	0
$843$	−3.82774	−0.131834
$844$	0	0
$845$	11.8203	0.406631
$846$	0	0
$847$	−32.5907	−1.11983
$848$	0	0
$849$	−6.03773	−0.207214
$850$	0	0
$851$	−6.87614	−0.235711
$852$	0	0
$853$	35.1600	1.20386	0.601928	−	0.798550i	$-0.294399\pi$
$853$	35.1600	1.20386	0.601928	+	0.798550i	$0.294399\pi$
$854$	0	0
$855$	−2.70388	−0.0924707
$856$	0	0
$857$	10.6136	0.362553	0.181276	−	0.983432i	$-0.441977\pi$
$857$	10.6136	0.362553	0.181276	+	0.983432i	$0.441977\pi$
$858$	0	0
$859$	−19.8491	−0.677242	−0.338621	−	0.940923i	$-0.609960\pi$
$859$	−19.8491	−0.677242	−0.338621	+	0.940923i	$0.609960\pi$
$860$	0	0
$861$	−11.0484	−0.376528
$862$	0	0
$863$	9.03356	0.307506	0.153753	−	0.988109i	$-0.450864\pi$
$863$	9.03356	0.307506	0.153753	+	0.988109i	$0.450864\pi$
$864$	0	0
$865$	20.1723	0.685877
$866$	0	0
$867$	−16.5800	−0.563087
$868$	0	0
$869$	−45.1452	−1.53145
$870$	0	0
$871$	10.6332	0.360293
$872$	0	0
$873$	−13.2207	−0.447452
$874$	0	0
$875$	5.08613	0.171943
$876$	0	0
$877$	33.6406	1.13596	0.567982	−	0.823041i	$-0.307724\pi$
$877$	33.6406	1.13596	0.567982	+	0.823041i	$0.307724\pi$
$878$	0	0
$879$	9.33710	0.314933
$880$	0	0
$881$	7.48647	0.252226	0.126113	−	0.992016i	$-0.459750\pi$
$881$	7.48647	0.252226	0.126113	+	0.992016i	$0.459750\pi$
$882$	0	0
$883$	−17.7374	−0.596912	−0.298456	−	0.954423i	$-0.596472\pi$
$883$	−17.7374	−0.596912	−0.298456	+	0.954423i	$0.596472\pi$
$884$	0	0
$885$	0.438069	0.0147255
$886$	0	0
$887$	−15.8687	−0.532819	−0.266410	−	0.963860i	$-0.585837\pi$
$887$	−15.8687	−0.532819	−0.266410	+	0.963860i	$0.585837\pi$
$888$	0	0
$889$	−38.5530	−1.29302
$890$	0	0
$891$	4.17226	0.139776
$892$	0	0
$893$	27.6555	0.925455
$894$	0	0
$895$	−4.11164	−0.137437
$896$	0	0
$897$	8.17226	0.272864
$898$	0	0
$899$	−5.90645	−0.196991
$900$	0	0
$901$	3.58002	0.119268
$902$	0	0
$903$	41.5652	1.38320
$904$	0	0
$905$	−10.3445	−0.343864
$906$	0	0
$907$	−8.55451	−0.284048	−0.142024	−	0.989863i	$-0.545361\pi$
$907$	−8.55451	−0.284048	−0.142024	+	0.989863i	$0.545361\pi$
$908$	0	0
$909$	−16.8761	−0.559746
$910$	0	0
$911$	13.5800	0.449926	0.224963	−	0.974367i	$-0.427774\pi$
$911$	13.5800	0.449926	0.224963	+	0.974367i	$0.427774\pi$
$912$	0	0
$913$	−59.3632	−1.96464
$914$	0	0
$915$	2.00000	0.0661180
$916$	0	0
$917$	−39.0288	−1.28884
$918$	0	0
$919$	−52.3148	−1.72571	−0.862854	−	0.505454i	$-0.831325\pi$
$919$	−52.3148	−1.72571	−0.862854	+	0.505454i	$0.831325\pi$
$920$	0	0
$921$	−26.4184	−0.870517
$922$	0	0
$923$	3.22547	0.106168
$924$	0	0
$925$	0.913870	0.0300479
$926$	0	0
$927$	0.493887	0.0162214
$928$	0	0
$929$	−5.37483	−0.176343	−0.0881713	−	0.996105i	$-0.528102\pi$
$929$	−5.37483	−0.176343	−0.0881713	+	0.996105i	$0.528102\pi$
$930$	0	0
$931$	51.0187	1.67207
$932$	0	0
$933$	6.36261	0.208302
$934$	0	0
$935$	−2.70388	−0.0884263
$936$	0	0
$937$	58.9433	1.92559	0.962796	−	0.270228i	$-0.0870992\pi$
$937$	58.9433	1.92559	0.962796	+	0.270228i	$0.0870992\pi$
$938$	0	0
$939$	8.02289	0.261817
$940$	0	0
$941$	−4.62517	−0.150776	−0.0753881	−	0.997154i	$-0.524020\pi$
$941$	−4.62517	−0.150776	−0.0753881	+	0.997154i	$0.524020\pi$
$942$	0	0
$943$	−16.3445	−0.532251
$944$	0	0
$945$	5.08613	0.165452
$946$	0	0
$947$	23.8591	0.775317	0.387658	−	0.921803i	$-0.373284\pi$
$947$	23.8591	0.775317	0.387658	+	0.921803i	$0.373284\pi$
$948$	0	0
$949$	1.36678	0.0443674
$950$	0	0
$951$	4.75970	0.154344
$952$	0	0
$953$	−15.8081	−0.512074	−0.256037	−	0.966667i	$-0.582417\pi$
$953$	−15.8081	−0.512074	−0.256037	+	0.966667i	$0.582417\pi$
$954$	0	0
$955$	3.56193	0.115261
$956$	0	0
$957$	24.6433	0.796603
$958$	0	0
$959$	24.2084	0.781731
$960$	0	0
$961$	1.00000	0.0322581
$962$	0	0
$963$	−9.69646	−0.312464
$964$	0	0
$965$	3.58002	0.115245
$966$	0	0
$967$	−7.90320	−0.254150	−0.127075	−	0.991893i	$-0.540559\pi$
$967$	−7.90320	−0.254150	−0.127075	+	0.991893i	$0.540559\pi$
$968$	0	0
$969$	1.75228	0.0562913
$970$	0	0
$971$	44.3478	1.42319	0.711594	−	0.702591i	$-0.247974\pi$
$971$	44.3478	1.42319	0.711594	+	0.702591i	$0.247974\pi$
$972$	0	0
$973$	52.3052	1.67683
$974$	0	0
$975$	−1.08613	−0.0347840
$976$	0	0
$977$	23.3781	0.747931	0.373966	−	0.927443i	$-0.377998\pi$
$977$	23.3781	0.747931	0.373966	+	0.927443i	$0.377998\pi$
$978$	0	0
$979$	75.8949	2.42561
$980$	0	0
$981$	−16.4003	−0.523623
$982$	0	0
$983$	35.7374	1.13985	0.569924	−	0.821698i	$-0.306973\pi$
$983$	35.7374	1.13985	0.569924	+	0.821698i	$0.306973\pi$
$984$	0	0
$985$	−7.35194	−0.234252
$986$	0	0
$987$	−52.0213	−1.65586
$988$	0	0
$989$	61.4897	1.95526
$990$	0	0
$991$	−61.4143	−1.95089	−0.975444	−	0.220247i	$-0.929314\pi$
$991$	−61.4143	−1.95089	−0.975444	+	0.220247i	$0.929314\pi$
$992$	0	0
$993$	21.9804	0.697526
$994$	0	0
$995$	25.1042	0.795857
$996$	0	0
$997$	52.8220	1.67289	0.836445	−	0.548051i	$-0.184630\pi$
$997$	52.8220	1.67289	0.836445	+	0.548051i	$0.184630\pi$
$998$	0	0
$999$	0.913870	0.0289136

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7440.2.a.bs.1.1		3
4.3	odd	2		465.2.a.e.1.1	✓	3
12.11	even	2		1395.2.a.j.1.3		3
20.3	even	4		2325.2.c.k.1024.5		6
20.7	even	4		2325.2.c.k.1024.2		6
20.19	odd	2		2325.2.a.r.1.3		3
60.59	even	2		6975.2.a.bf.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
465.2.a.e.1.1	✓	3	4.3	odd	2
1395.2.a.j.1.3		3	12.11	even	2
2325.2.a.r.1.3		3	20.19	odd	2
2325.2.c.k.1024.2		6	20.7	even	4
2325.2.c.k.1024.5		6	20.3	even	4
6975.2.a.bf.1.1		3	60.59	even	2
7440.2.a.bs.1.1		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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -1.00000 q^{5} -5.08613 q^{7} +1.00000 q^{9} +4.17226 q^{11} -1.08613 q^{13} -1.00000 q^{15} +0.648061 q^{17} +2.70388 q^{19} -5.08613 q^{21} -7.52420 q^{23} +1.00000 q^{25} +1.00000 q^{27} +5.90645 q^{29} -1.00000 q^{31} +4.17226 q^{33} +5.08613 q^{35} +0.913870 q^{37} -1.08613 q^{39} +2.17226 q^{41} -8.17226 q^{43} -1.00000 q^{45} +10.2281 q^{47} +18.8687 q^{49} +0.648061 q^{51} +5.52420 q^{53} -4.17226 q^{55} +2.70388 q^{57} -0.438069 q^{59} -2.00000 q^{61} -5.08613 q^{63} +1.08613 q^{65} -9.79001 q^{67} -7.52420 q^{69} -2.96969 q^{71} -1.25839 q^{73} +1.00000 q^{75} -21.2207 q^{77} -10.8203 q^{79} +1.00000 q^{81} -14.2281 q^{83} -0.648061 q^{85} +5.90645 q^{87} +18.1903 q^{89} +5.52420 q^{91} -1.00000 q^{93} -2.70388 q^{95} -13.2207 q^{97} +4.17226 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{3} - 3 q^{5} - 8 q^{7} + 3 q^{9} - 2 q^{11} + 4 q^{13} - 3 q^{15} + 4 q^{17} + 4 q^{19} - 8 q^{21} - 6 q^{23} + 3 q^{25} + 3 q^{27} - 2 q^{29} - 3 q^{31} - 2 q^{33} + 8 q^{35} + 10 q^{37} + 4 q^{39}+ \cdots - 2 q^{99}+O(q^{100}) \) 3 * q + 3 * q^3 - 3 * q^5 - 8 * q^7 + 3 * q^9 - 2 * q^11 + 4 * q^13 - 3 * q^15 + 4 * q^17 + 4 * q^19 - 8 * q^21 - 6 * q^23 + 3 * q^25 + 3 * q^27 - 2 * q^29 - 3 * q^31 - 2 * q^33 + 8 * q^35 + 10 * q^37 + 4 * q^39 - 8 * q^41 - 10 * q^43 - 3 * q^45 + 10 * q^47 + 11 * q^49 + 4 * q^51 + 2 * q^55 + 4 * q^57 + 8 * q^59 - 6 * q^61 - 8 * q^63 - 4 * q^65 - 18 * q^67 - 6 * q^69 - 10 * q^71 + 18 * q^73 + 3 * q^75 - 16 * q^77 - 20 * q^79 + 3 * q^81 - 22 * q^83 - 4 * q^85 - 2 * q^87 + 8 * q^89 - 3 * q^93 - 4 * q^95 + 8 * q^97 - 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more