LMFDB - Embedded newform 6960.2.a.bx.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6960,2,Mod(1,6960)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6960, 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("6960.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6960 = 2^{4} \cdot 3 \cdot 5 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6960.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:	$55.5758798068$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{17}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 435)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.56155$ of defining polynomial
Character	$\chi$	$=$	6960.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +1.00000 q^{5} -5.12311 q^{7} +1.00000 q^{9} +1.43845 q^{11} -2.00000 q^{13} -1.00000 q^{15} -7.12311 q^{17} -5.12311 q^{19} +5.12311 q^{21} -6.56155 q^{23} +1.00000 q^{25} -1.00000 q^{27} +1.00000 q^{29} -4.00000 q^{31} -1.43845 q^{33} -5.12311 q^{35} -1.68466 q^{37} +2.00000 q^{39} -1.68466 q^{41} -7.68466 q^{43} +1.00000 q^{45} +13.1231 q^{47} +19.2462 q^{49} +7.12311 q^{51} -3.43845 q^{53} +1.43845 q^{55} +5.12311 q^{57} -12.0000 q^{59} +0.876894 q^{61} -5.12311 q^{63} -2.00000 q^{65} +11.3693 q^{67} +6.56155 q^{69} +2.87689 q^{71} +1.68466 q^{73} -1.00000 q^{75} -7.36932 q^{77} +12.0000 q^{79} +1.00000 q^{81} +2.56155 q^{83} -7.12311 q^{85} -1.00000 q^{87} +12.2462 q^{89} +10.2462 q^{91} +4.00000 q^{93} -5.12311 q^{95} -5.68466 q^{97} +1.43845 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} + 2 q^{5} - 2 q^{7} + 2 q^{9} + 7 q^{11} - 4 q^{13} - 2 q^{15} - 6 q^{17} - 2 q^{19} + 2 q^{21} - 9 q^{23} + 2 q^{25} - 2 q^{27} + 2 q^{29} - 8 q^{31} - 7 q^{33} - 2 q^{35} + 9 q^{37} + 4 q^{39}+ \cdots + 7 q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 + 2 * q^5 - 2 * q^7 + 2 * q^9 + 7 * q^11 - 4 * q^13 - 2 * q^15 - 6 * q^17 - 2 * q^19 + 2 * q^21 - 9 * q^23 + 2 * q^25 - 2 * q^27 + 2 * q^29 - 8 * q^31 - 7 * q^33 - 2 * q^35 + 9 * q^37 + 4 * q^39 + 9 * q^41 - 3 * q^43 + 2 * q^45 + 18 * q^47 + 22 * q^49 + 6 * q^51 - 11 * q^53 + 7 * q^55 + 2 * q^57 - 24 * q^59 + 10 * q^61 - 2 * q^63 - 4 * q^65 - 2 * q^67 + 9 * q^69 + 14 * q^71 - 9 * q^73 - 2 * q^75 + 10 * q^77 + 24 * q^79 + 2 * q^81 + q^83 - 6 * q^85 - 2 * q^87 + 8 * q^89 + 4 * q^91 + 8 * q^93 - 2 * q^95 + q^97 + 7 * 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
$7$	−5.12311	−1.93635	−0.968176	−	0.250270i	$-0.919480\pi$
$7$	−5.12311	−1.93635	−0.968176	+	0.250270i	$0.919480\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	1.43845	0.433708	0.216854	−	0.976204i	$-0.430420\pi$
$11$	1.43845	0.433708	0.216854	+	0.976204i	$0.430420\pi$
$12$	0	0
$13$	−2.00000	−0.554700	−0.277350	−	0.960769i	$-0.589456\pi$
$13$	−2.00000	−0.554700	−0.277350	+	0.960769i	$0.589456\pi$
$14$	0	0
$15$	−1.00000	−0.258199
$16$	0	0
$17$	−7.12311	−1.72761	−0.863803	−	0.503829i	$-0.831924\pi$
$17$	−7.12311	−1.72761	−0.863803	+	0.503829i	$0.831924\pi$
$18$	0	0
$19$	−5.12311	−1.17532	−0.587661	−	0.809108i	$-0.699951\pi$
$19$	−5.12311	−1.17532	−0.587661	+	0.809108i	$0.699951\pi$
$20$	0	0
$21$	5.12311	1.11795
$22$	0	0
$23$	−6.56155	−1.36818	−0.684089	−	0.729398i	$-0.739800\pi$
$23$	−6.56155	−1.36818	−0.684089	+	0.729398i	$0.739800\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	−4.00000	−0.718421	−0.359211	−	0.933257i	$-0.616954\pi$
$31$	−4.00000	−0.718421	−0.359211	+	0.933257i	$0.616954\pi$
$32$	0	0
$33$	−1.43845	−0.250402
$34$	0	0
$35$	−5.12311	−0.865963
$36$	0	0
$37$	−1.68466	−0.276956	−0.138478	−	0.990366i	$-0.544221\pi$
$37$	−1.68466	−0.276956	−0.138478	+	0.990366i	$0.544221\pi$
$38$	0	0
$39$	2.00000	0.320256
$40$	0	0
$41$	−1.68466	−0.263099	−0.131550	−	0.991310i	$-0.541995\pi$
$41$	−1.68466	−0.263099	−0.131550	+	0.991310i	$0.541995\pi$
$42$	0	0
$43$	−7.68466	−1.17190	−0.585950	−	0.810347i	$-0.699278\pi$
$43$	−7.68466	−1.17190	−0.585950	+	0.810347i	$0.699278\pi$
$44$	0	0
$45$	1.00000	0.149071
$46$	0	0
$47$	13.1231	1.91420	0.957101	−	0.289755i	$-0.0935738\pi$
$47$	13.1231	1.91420	0.957101	+	0.289755i	$0.0935738\pi$
$48$	0	0
$49$	19.2462	2.74946
$50$	0	0
$51$	7.12311	0.997434
$52$	0	0
$53$	−3.43845	−0.472307	−0.236154	−	0.971716i	$-0.575887\pi$
$53$	−3.43845	−0.472307	−0.236154	+	0.971716i	$0.575887\pi$
$54$	0	0
$55$	1.43845	0.193960
$56$	0	0
$57$	5.12311	0.678572
$58$	0	0
$59$	−12.0000	−1.56227	−0.781133	−	0.624364i	$-0.785358\pi$
$59$	−12.0000	−1.56227	−0.781133	+	0.624364i	$0.785358\pi$
$60$	0	0
$61$	0.876894	0.112275	0.0561374	−	0.998423i	$-0.482122\pi$
$61$	0.876894	0.112275	0.0561374	+	0.998423i	$0.482122\pi$
$62$	0	0
$63$	−5.12311	−0.645451
$64$	0	0
$65$	−2.00000	−0.248069
$66$	0	0
$67$	11.3693	1.38898	0.694492	−	0.719501i	$-0.255629\pi$
$67$	11.3693	1.38898	0.694492	+	0.719501i	$0.255629\pi$
$68$	0	0
$69$	6.56155	0.789918
$70$	0	0
$71$	2.87689	0.341425	0.170712	−	0.985321i	$-0.445393\pi$
$71$	2.87689	0.341425	0.170712	+	0.985321i	$0.445393\pi$
$72$	0	0
$73$	1.68466	0.197174	0.0985872	−	0.995128i	$-0.468568\pi$
$73$	1.68466	0.197174	0.0985872	+	0.995128i	$0.468568\pi$
$74$	0	0
$75$	−1.00000	−0.115470
$76$	0	0
$77$	−7.36932	−0.839812
$78$	0	0
$79$	12.0000	1.35011	0.675053	−	0.737769i	$-0.264121\pi$
$79$	12.0000	1.35011	0.675053	+	0.737769i	$0.264121\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	2.56155	0.281167	0.140583	−	0.990069i	$-0.455102\pi$
$83$	2.56155	0.281167	0.140583	+	0.990069i	$0.455102\pi$
$84$	0	0
$85$	−7.12311	−0.772609
$86$	0	0
$87$	−1.00000	−0.107211
$88$	0	0
$89$	12.2462	1.29810	0.649048	−	0.760748i	$-0.275167\pi$
$89$	12.2462	1.29810	0.649048	+	0.760748i	$0.275167\pi$
$90$	0	0
$91$	10.2462	1.07409
$92$	0	0
$93$	4.00000	0.414781
$94$	0	0
$95$	−5.12311	−0.525620
$96$	0	0
$97$	−5.68466	−0.577190	−0.288595	−	0.957451i	$-0.593188\pi$
$97$	−5.68466	−0.577190	−0.288595	+	0.957451i	$0.593188\pi$
$98$	0	0
$99$	1.43845	0.144569
$100$	0	0
$101$	−8.56155	−0.851906	−0.425953	−	0.904745i	$-0.640061\pi$
$101$	−8.56155	−0.851906	−0.425953	+	0.904745i	$0.640061\pi$
$102$	0	0
$103$	2.87689	0.283469	0.141734	−	0.989905i	$-0.454732\pi$
$103$	2.87689	0.283469	0.141734	+	0.989905i	$0.454732\pi$
$104$	0	0
$105$	5.12311	0.499964
$106$	0	0
$107$	−16.4924	−1.59438	−0.797191	−	0.603727i	$-0.793682\pi$
$107$	−16.4924	−1.59438	−0.797191	+	0.603727i	$0.793682\pi$
$108$	0	0
$109$	−5.68466	−0.544492	−0.272246	−	0.962228i	$-0.587766\pi$
$109$	−5.68466	−0.544492	−0.272246	+	0.962228i	$0.587766\pi$
$110$	0	0
$111$	1.68466	0.159901
$112$	0	0
$113$	−4.87689	−0.458780	−0.229390	−	0.973335i	$-0.573673\pi$
$113$	−4.87689	−0.458780	−0.229390	+	0.973335i	$0.573673\pi$
$114$	0	0
$115$	−6.56155	−0.611868
$116$	0	0
$117$	−2.00000	−0.184900
$118$	0	0
$119$	36.4924	3.34525
$120$	0	0
$121$	−8.93087	−0.811897
$122$	0	0
$123$	1.68466	0.151901
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−4.31534	−0.382925	−0.191462	−	0.981500i	$-0.561323\pi$
$127$	−4.31534	−0.382925	−0.191462	+	0.981500i	$0.561323\pi$
$128$	0	0
$129$	7.68466	0.676596
$130$	0	0
$131$	−10.2462	−0.895216	−0.447608	−	0.894230i	$-0.647724\pi$
$131$	−10.2462	−0.895216	−0.447608	+	0.894230i	$0.647724\pi$
$132$	0	0
$133$	26.2462	2.27584
$134$	0	0
$135$	−1.00000	−0.0860663
$136$	0	0
$137$	−15.1231	−1.29205	−0.646027	−	0.763315i	$-0.723571\pi$
$137$	−15.1231	−1.29205	−0.646027	+	0.763315i	$0.723571\pi$
$138$	0	0
$139$	17.9309	1.52088	0.760438	−	0.649410i	$-0.224984\pi$
$139$	17.9309	1.52088	0.760438	+	0.649410i	$0.224984\pi$
$140$	0	0
$141$	−13.1231	−1.10516
$142$	0	0
$143$	−2.87689	−0.240578
$144$	0	0
$145$	1.00000	0.0830455
$146$	0	0
$147$	−19.2462	−1.58740
$148$	0	0
$149$	0.246211	0.0201704	0.0100852	−	0.999949i	$-0.496790\pi$
$149$	0.246211	0.0201704	0.0100852	+	0.999949i	$0.496790\pi$
$150$	0	0
$151$	−4.31534	−0.351178	−0.175589	−	0.984464i	$-0.556183\pi$
$151$	−4.31534	−0.351178	−0.175589	+	0.984464i	$0.556183\pi$
$152$	0	0
$153$	−7.12311	−0.575869
$154$	0	0
$155$	−4.00000	−0.321288
$156$	0	0
$157$	−14.0000	−1.11732	−0.558661	−	0.829396i	$-0.688685\pi$
$157$	−14.0000	−1.11732	−0.558661	+	0.829396i	$0.688685\pi$
$158$	0	0
$159$	3.43845	0.272687
$160$	0	0
$161$	33.6155	2.64927
$162$	0	0
$163$	17.9309	1.40445	0.702227	−	0.711953i	$-0.252189\pi$
$163$	17.9309	1.40445	0.702227	+	0.711953i	$0.252189\pi$
$164$	0	0
$165$	−1.43845	−0.111983
$166$	0	0
$167$	0	0		−	1.00000i	$-0.5\pi$
$167$	0	0			1.00000i	$0.5\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	0	0
$171$	−5.12311	−0.391774
$172$	0	0
$173$	12.5616	0.955037	0.477519	−	0.878622i	$-0.341536\pi$
$173$	12.5616	0.955037	0.477519	+	0.878622i	$0.341536\pi$
$174$	0	0
$175$	−5.12311	−0.387270
$176$	0	0
$177$	12.0000	0.901975
$178$	0	0
$179$	−1.12311	−0.0839449	−0.0419724	−	0.999119i	$-0.513364\pi$
$179$	−1.12311	−0.0839449	−0.0419724	+	0.999119i	$0.513364\pi$
$180$	0	0
$181$	25.6847	1.90913	0.954563	−	0.298010i	$-0.0963228\pi$
$181$	25.6847	1.90913	0.954563	+	0.298010i	$0.0963228\pi$
$182$	0	0
$183$	−0.876894	−0.0648219
$184$	0	0
$185$	−1.68466	−0.123859
$186$	0	0
$187$	−10.2462	−0.749277
$188$	0	0
$189$	5.12311	0.372651
$190$	0	0
$191$	9.93087	0.718573	0.359286	−	0.933227i	$-0.383020\pi$
$191$	9.93087	0.718573	0.359286	+	0.933227i	$0.383020\pi$
$192$	0	0
$193$	−10.0000	−0.719816	−0.359908	−	0.932988i	$-0.617192\pi$
$193$	−10.0000	−0.719816	−0.359908	+	0.932988i	$0.617192\pi$
$194$	0	0
$195$	2.00000	0.143223
$196$	0	0
$197$	4.56155	0.324997	0.162499	−	0.986709i	$-0.448045\pi$
$197$	4.56155	0.324997	0.162499	+	0.986709i	$0.448045\pi$
$198$	0	0
$199$	11.0540	0.783596	0.391798	−	0.920051i	$-0.371853\pi$
$199$	11.0540	0.783596	0.391798	+	0.920051i	$0.371853\pi$
$200$	0	0
$201$	−11.3693	−0.801930
$202$	0	0
$203$	−5.12311	−0.359572
$204$	0	0
$205$	−1.68466	−0.117662
$206$	0	0
$207$	−6.56155	−0.456059
$208$	0	0
$209$	−7.36932	−0.509746
$210$	0	0
$211$	2.87689	0.198054	0.0990268	−	0.995085i	$-0.468427\pi$
$211$	2.87689	0.198054	0.0990268	+	0.995085i	$0.468427\pi$
$212$	0	0
$213$	−2.87689	−0.197122
$214$	0	0
$215$	−7.68466	−0.524089
$216$	0	0
$217$	20.4924	1.39112
$218$	0	0
$219$	−1.68466	−0.113839
$220$	0	0
$221$	14.2462	0.958304
$222$	0	0
$223$	18.2462	1.22186	0.610928	−	0.791686i	$-0.290796\pi$
$223$	18.2462	1.22186	0.610928	+	0.791686i	$0.290796\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	3.19224	0.211876	0.105938	−	0.994373i	$-0.466215\pi$
$227$	3.19224	0.211876	0.105938	+	0.994373i	$0.466215\pi$
$228$	0	0
$229$	22.0000	1.45380	0.726900	−	0.686743i	$-0.240960\pi$
$229$	22.0000	1.45380	0.726900	+	0.686743i	$0.240960\pi$
$230$	0	0
$231$	7.36932	0.484865
$232$	0	0
$233$	−10.3153	−0.675780	−0.337890	−	0.941186i	$-0.609713\pi$
$233$	−10.3153	−0.675780	−0.337890	+	0.941186i	$0.609713\pi$
$234$	0	0
$235$	13.1231	0.856057
$236$	0	0
$237$	−12.0000	−0.779484
$238$	0	0
$239$	13.1231	0.848863	0.424432	−	0.905460i	$-0.360474\pi$
$239$	13.1231	0.848863	0.424432	+	0.905460i	$0.360474\pi$
$240$	0	0
$241$	−25.0540	−1.61387	−0.806934	−	0.590641i	$-0.798875\pi$
$241$	−25.0540	−1.61387	−0.806934	+	0.590641i	$0.798875\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	19.2462	1.22960
$246$	0	0
$247$	10.2462	0.651951
$248$	0	0
$249$	−2.56155	−0.162332
$250$	0	0
$251$	−2.24621	−0.141780	−0.0708898	−	0.997484i	$-0.522584\pi$
$251$	−2.24621	−0.141780	−0.0708898	+	0.997484i	$0.522584\pi$
$252$	0	0
$253$	−9.43845	−0.593390
$254$	0	0
$255$	7.12311	0.446066
$256$	0	0
$257$	11.4384	0.713511	0.356755	−	0.934198i	$-0.383883\pi$
$257$	11.4384	0.713511	0.356755	+	0.934198i	$0.383883\pi$
$258$	0	0
$259$	8.63068	0.536285
$260$	0	0
$261$	1.00000	0.0618984
$262$	0	0
$263$	5.75379	0.354794	0.177397	−	0.984139i	$-0.443232\pi$
$263$	5.75379	0.354794	0.177397	+	0.984139i	$0.443232\pi$
$264$	0	0
$265$	−3.43845	−0.211222
$266$	0	0
$267$	−12.2462	−0.749456
$268$	0	0
$269$	11.7538	0.716641	0.358321	−	0.933599i	$-0.383349\pi$
$269$	11.7538	0.716641	0.358321	+	0.933599i	$0.383349\pi$
$270$	0	0
$271$	−17.1231	−1.04015	−0.520077	−	0.854119i	$-0.674097\pi$
$271$	−17.1231	−1.04015	−0.520077	+	0.854119i	$0.674097\pi$
$272$	0	0
$273$	−10.2462	−0.620129
$274$	0	0
$275$	1.43845	0.0867416
$276$	0	0
$277$	0.876894	0.0526875	0.0263437	−	0.999653i	$-0.491614\pi$
$277$	0.876894	0.0526875	0.0263437	+	0.999653i	$0.491614\pi$
$278$	0	0
$279$	−4.00000	−0.239474
$280$	0	0
$281$	−23.6155	−1.40878	−0.704392	−	0.709811i	$-0.748780\pi$
$281$	−23.6155	−1.40878	−0.704392	+	0.709811i	$0.748780\pi$
$282$	0	0
$283$	−6.87689	−0.408789	−0.204394	−	0.978889i	$-0.565523\pi$
$283$	−6.87689	−0.408789	−0.204394	+	0.978889i	$0.565523\pi$
$284$	0	0
$285$	5.12311	0.303467
$286$	0	0
$287$	8.63068	0.509453
$288$	0	0
$289$	33.7386	1.98463
$290$	0	0
$291$	5.68466	0.333241
$292$	0	0
$293$	−21.3693	−1.24841	−0.624204	−	0.781261i	$-0.714577\pi$
$293$	−21.3693	−1.24841	−0.624204	+	0.781261i	$0.714577\pi$
$294$	0	0
$295$	−12.0000	−0.698667
$296$	0	0
$297$	−1.43845	−0.0834672
$298$	0	0
$299$	13.1231	0.758929
$300$	0	0
$301$	39.3693	2.26921
$302$	0	0
$303$	8.56155	0.491848
$304$	0	0
$305$	0.876894	0.0502108
$306$	0	0
$307$	31.6847	1.80834	0.904169	−	0.427174i	$-0.140491\pi$
$307$	31.6847	1.80834	0.904169	+	0.427174i	$0.140491\pi$
$308$	0	0
$309$	−2.87689	−0.163661
$310$	0	0
$311$	−16.3153	−0.925158	−0.462579	−	0.886578i	$-0.653076\pi$
$311$	−16.3153	−0.925158	−0.462579	+	0.886578i	$0.653076\pi$
$312$	0	0
$313$	−21.3693	−1.20787	−0.603933	−	0.797035i	$-0.706400\pi$
$313$	−21.3693	−1.20787	−0.603933	+	0.797035i	$0.706400\pi$
$314$	0	0
$315$	−5.12311	−0.288654
$316$	0	0
$317$	4.87689	0.273914	0.136957	−	0.990577i	$-0.456268\pi$
$317$	4.87689	0.273914	0.136957	+	0.990577i	$0.456268\pi$
$318$	0	0
$319$	1.43845	0.0805376
$320$	0	0
$321$	16.4924	0.920517
$322$	0	0
$323$	36.4924	2.03049
$324$	0	0
$325$	−2.00000	−0.110940
$326$	0	0
$327$	5.68466	0.314362
$328$	0	0
$329$	−67.2311	−3.70657
$330$	0	0
$331$	−10.2462	−0.563183	−0.281591	−	0.959534i	$-0.590862\pi$
$331$	−10.2462	−0.563183	−0.281591	+	0.959534i	$0.590862\pi$
$332$	0	0
$333$	−1.68466	−0.0923187
$334$	0	0
$335$	11.3693	0.621172
$336$	0	0
$337$	−7.75379	−0.422376	−0.211188	−	0.977445i	$-0.567733\pi$
$337$	−7.75379	−0.422376	−0.211188	+	0.977445i	$0.567733\pi$
$338$	0	0
$339$	4.87689	0.264877
$340$	0	0
$341$	−5.75379	−0.311585
$342$	0	0
$343$	−62.7386	−3.38757
$344$	0	0
$345$	6.56155	0.353262
$346$	0	0
$347$	12.1771	0.653700	0.326850	−	0.945076i	$-0.394013\pi$
$347$	12.1771	0.653700	0.326850	+	0.945076i	$0.394013\pi$
$348$	0	0
$349$	0.0691303	0.00370046	0.00185023	−	0.999998i	$-0.499411\pi$
$349$	0.0691303	0.00370046	0.00185023	+	0.999998i	$0.499411\pi$
$350$	0	0
$351$	2.00000	0.106752
$352$	0	0
$353$	30.4924	1.62295	0.811474	−	0.584389i	$-0.198666\pi$
$353$	30.4924	1.62295	0.811474	+	0.584389i	$0.198666\pi$
$354$	0	0
$355$	2.87689	0.152690
$356$	0	0
$357$	−36.4924	−1.93138
$358$	0	0
$359$	3.19224	0.168480	0.0842399	−	0.996446i	$-0.473154\pi$
$359$	3.19224	0.168480	0.0842399	+	0.996446i	$0.473154\pi$
$360$	0	0
$361$	7.24621	0.381380
$362$	0	0
$363$	8.93087	0.468749
$364$	0	0
$365$	1.68466	0.0881791
$366$	0	0
$367$	19.6847	1.02753	0.513765	−	0.857931i	$-0.328250\pi$
$367$	19.6847	1.02753	0.513765	+	0.857931i	$0.328250\pi$
$368$	0	0
$369$	−1.68466	−0.0876998
$370$	0	0
$371$	17.6155	0.914553
$372$	0	0
$373$	13.3693	0.692237	0.346118	−	0.938191i	$-0.387500\pi$
$373$	13.3693	0.692237	0.346118	+	0.938191i	$0.387500\pi$
$374$	0	0
$375$	−1.00000	−0.0516398
$376$	0	0
$377$	−2.00000	−0.103005
$378$	0	0
$379$	−8.00000	−0.410932	−0.205466	−	0.978664i	$-0.565871\pi$
$379$	−8.00000	−0.410932	−0.205466	+	0.978664i	$0.565871\pi$
$380$	0	0
$381$	4.31534	0.221082
$382$	0	0
$383$	−11.0540	−0.564832	−0.282416	−	0.959292i	$-0.591136\pi$
$383$	−11.0540	−0.564832	−0.282416	+	0.959292i	$0.591136\pi$
$384$	0	0
$385$	−7.36932	−0.375575
$386$	0	0
$387$	−7.68466	−0.390633
$388$	0	0
$389$	14.8078	0.750783	0.375392	−	0.926866i	$-0.377508\pi$
$389$	14.8078	0.750783	0.375392	+	0.926866i	$0.377508\pi$
$390$	0	0
$391$	46.7386	2.36367
$392$	0	0
$393$	10.2462	0.516853
$394$	0	0
$395$	12.0000	0.603786
$396$	0	0
$397$	24.2462	1.21688	0.608441	−	0.793599i	$-0.291795\pi$
$397$	24.2462	1.21688	0.608441	+	0.793599i	$0.291795\pi$
$398$	0	0
$399$	−26.2462	−1.31395
$400$	0	0
$401$	35.6155	1.77855	0.889277	−	0.457368i	$-0.151208\pi$
$401$	35.6155	1.77855	0.889277	+	0.457368i	$0.151208\pi$
$402$	0	0
$403$	8.00000	0.398508
$404$	0	0
$405$	1.00000	0.0496904
$406$	0	0
$407$	−2.42329	−0.120118
$408$	0	0
$409$	17.3693	0.858857	0.429429	−	0.903101i	$-0.358715\pi$
$409$	17.3693	0.858857	0.429429	+	0.903101i	$0.358715\pi$
$410$	0	0
$411$	15.1231	0.745968
$412$	0	0
$413$	61.4773	3.02510
$414$	0	0
$415$	2.56155	0.125742
$416$	0	0
$417$	−17.9309	−0.878078
$418$	0	0
$419$	27.3693	1.33708	0.668539	−	0.743677i	$-0.266920\pi$
$419$	27.3693	1.33708	0.668539	+	0.743677i	$0.266920\pi$
$420$	0	0
$421$	−10.0000	−0.487370	−0.243685	−	0.969854i	$-0.578356\pi$
$421$	−10.0000	−0.487370	−0.243685	+	0.969854i	$0.578356\pi$
$422$	0	0
$423$	13.1231	0.638067
$424$	0	0
$425$	−7.12311	−0.345521
$426$	0	0
$427$	−4.49242	−0.217404
$428$	0	0
$429$	2.87689	0.138898
$430$	0	0
$431$	21.1231	1.01746	0.508732	−	0.860925i	$-0.330114\pi$
$431$	21.1231	1.01746	0.508732	+	0.860925i	$0.330114\pi$
$432$	0	0
$433$	−4.06913	−0.195550	−0.0977750	−	0.995209i	$-0.531173\pi$
$433$	−4.06913	−0.195550	−0.0977750	+	0.995209i	$0.531173\pi$
$434$	0	0
$435$	−1.00000	−0.0479463
$436$	0	0
$437$	33.6155	1.60805
$438$	0	0
$439$	−12.4924	−0.596231	−0.298115	−	0.954530i	$-0.596358\pi$
$439$	−12.4924	−0.596231	−0.298115	+	0.954530i	$0.596358\pi$
$440$	0	0
$441$	19.2462	0.916486
$442$	0	0
$443$	−6.24621	−0.296766	−0.148383	−	0.988930i	$-0.547407\pi$
$443$	−6.24621	−0.296766	−0.148383	+	0.988930i	$0.547407\pi$
$444$	0	0
$445$	12.2462	0.580526
$446$	0	0
$447$	−0.246211	−0.0116454
$448$	0	0
$449$	11.4384	0.539814	0.269907	−	0.962886i	$-0.413007\pi$
$449$	11.4384	0.539814	0.269907	+	0.962886i	$0.413007\pi$
$450$	0	0
$451$	−2.42329	−0.114108
$452$	0	0
$453$	4.31534	0.202752
$454$	0	0
$455$	10.2462	0.480350
$456$	0	0
$457$	−8.87689	−0.415244	−0.207622	−	0.978209i	$-0.566572\pi$
$457$	−8.87689	−0.415244	−0.207622	+	0.978209i	$0.566572\pi$
$458$	0	0
$459$	7.12311	0.332478
$460$	0	0
$461$	−18.1771	−0.846591	−0.423296	−	0.905992i	$-0.639127\pi$
$461$	−18.1771	−0.846591	−0.423296	+	0.905992i	$0.639127\pi$
$462$	0	0
$463$	−25.6155	−1.19045	−0.595227	−	0.803557i	$-0.702938\pi$
$463$	−25.6155	−1.19045	−0.595227	+	0.803557i	$0.702938\pi$
$464$	0	0
$465$	4.00000	0.185496
$466$	0	0
$467$	3.36932	0.155913	0.0779567	−	0.996957i	$-0.475160\pi$
$467$	3.36932	0.155913	0.0779567	+	0.996957i	$0.475160\pi$
$468$	0	0
$469$	−58.2462	−2.68956
$470$	0	0
$471$	14.0000	0.645086
$472$	0	0
$473$	−11.0540	−0.508262
$474$	0	0
$475$	−5.12311	−0.235064
$476$	0	0
$477$	−3.43845	−0.157436
$478$	0	0
$479$	22.2462	1.01646	0.508228	−	0.861223i	$-0.330301\pi$
$479$	22.2462	1.01646	0.508228	+	0.861223i	$0.330301\pi$
$480$	0	0
$481$	3.36932	0.153628
$482$	0	0
$483$	−33.6155	−1.52956
$484$	0	0
$485$	−5.68466	−0.258127
$486$	0	0
$487$	−34.2462	−1.55184	−0.775922	−	0.630829i	$-0.782715\pi$
$487$	−34.2462	−1.55184	−0.775922	+	0.630829i	$0.782715\pi$
$488$	0	0
$489$	−17.9309	−0.810862
$490$	0	0
$491$	−6.73863	−0.304110	−0.152055	−	0.988372i	$-0.548589\pi$
$491$	−6.73863	−0.304110	−0.152055	+	0.988372i	$0.548589\pi$
$492$	0	0
$493$	−7.12311	−0.320809
$494$	0	0
$495$	1.43845	0.0646534
$496$	0	0
$497$	−14.7386	−0.661118
$498$	0	0
$499$	−42.7386	−1.91324	−0.956622	−	0.291332i	$-0.905902\pi$
$499$	−42.7386	−1.91324	−0.956622	+	0.291332i	$0.905902\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	39.3693	1.75539	0.877696	−	0.479219i	$-0.159080\pi$
$503$	39.3693	1.75539	0.877696	+	0.479219i	$0.159080\pi$
$504$	0	0
$505$	−8.56155	−0.380984
$506$	0	0
$507$	9.00000	0.399704
$508$	0	0
$509$	18.4924	0.819662	0.409831	−	0.912161i	$-0.365588\pi$
$509$	18.4924	0.819662	0.409831	+	0.912161i	$0.365588\pi$
$510$	0	0
$511$	−8.63068	−0.381799
$512$	0	0
$513$	5.12311	0.226191
$514$	0	0
$515$	2.87689	0.126771
$516$	0	0
$517$	18.8769	0.830205
$518$	0	0
$519$	−12.5616	−0.551391
$520$	0	0
$521$	−0.246211	−0.0107867	−0.00539336	−	0.999985i	$-0.501717\pi$
$521$	−0.246211	−0.0107867	−0.00539336	+	0.999985i	$0.501717\pi$
$522$	0	0
$523$	−18.7386	−0.819383	−0.409692	−	0.912224i	$-0.634364\pi$
$523$	−18.7386	−0.819383	−0.409692	+	0.912224i	$0.634364\pi$
$524$	0	0
$525$	5.12311	0.223591
$526$	0	0
$527$	28.4924	1.24115
$528$	0	0
$529$	20.0540	0.871912
$530$	0	0
$531$	−12.0000	−0.520756
$532$	0	0
$533$	3.36932	0.145941
$534$	0	0
$535$	−16.4924	−0.713030
$536$	0	0
$537$	1.12311	0.0484656
$538$	0	0
$539$	27.6847	1.19246
$540$	0	0
$541$	−40.7386	−1.75149	−0.875745	−	0.482773i	$-0.839629\pi$
$541$	−40.7386	−1.75149	−0.875745	+	0.482773i	$0.839629\pi$
$542$	0	0
$543$	−25.6847	−1.10223
$544$	0	0
$545$	−5.68466	−0.243504
$546$	0	0
$547$	42.7386	1.82737	0.913686	−	0.406421i	$-0.133223\pi$
$547$	42.7386	1.82737	0.913686	+	0.406421i	$0.133223\pi$
$548$	0	0
$549$	0.876894	0.0374249
$550$	0	0
$551$	−5.12311	−0.218252
$552$	0	0
$553$	−61.4773	−2.61428
$554$	0	0
$555$	1.68466	0.0715098
$556$	0	0
$557$	−20.4233	−0.865363	−0.432681	−	0.901547i	$-0.642432\pi$
$557$	−20.4233	−0.865363	−0.432681	+	0.901547i	$0.642432\pi$
$558$	0	0
$559$	15.3693	0.650053
$560$	0	0
$561$	10.2462	0.432595
$562$	0	0
$563$	−9.12311	−0.384493	−0.192247	−	0.981347i	$-0.561577\pi$
$563$	−9.12311	−0.384493	−0.192247	+	0.981347i	$0.561577\pi$
$564$	0	0
$565$	−4.87689	−0.205172
$566$	0	0
$567$	−5.12311	−0.215150
$568$	0	0
$569$	−16.2462	−0.681077	−0.340538	−	0.940231i	$-0.610609\pi$
$569$	−16.2462	−0.681077	−0.340538	+	0.940231i	$0.610609\pi$
$570$	0	0
$571$	−39.0540	−1.63436	−0.817179	−	0.576384i	$-0.804463\pi$
$571$	−39.0540	−1.63436	−0.817179	+	0.576384i	$0.804463\pi$
$572$	0	0
$573$	−9.93087	−0.414868
$574$	0	0
$575$	−6.56155	−0.273636
$576$	0	0
$577$	−30.4924	−1.26942	−0.634708	−	0.772752i	$-0.718880\pi$
$577$	−30.4924	−1.26942	−0.634708	+	0.772752i	$0.718880\pi$
$578$	0	0
$579$	10.0000	0.415586
$580$	0	0
$581$	−13.1231	−0.544438
$582$	0	0
$583$	−4.94602	−0.204843
$584$	0	0
$585$	−2.00000	−0.0826898
$586$	0	0
$587$	−24.4924	−1.01091	−0.505455	−	0.862853i	$-0.668675\pi$
$587$	−24.4924	−1.01091	−0.505455	+	0.862853i	$0.668675\pi$
$588$	0	0
$589$	20.4924	0.844376
$590$	0	0
$591$	−4.56155	−0.187637
$592$	0	0
$593$	36.2462	1.48845	0.744227	−	0.667927i	$-0.232818\pi$
$593$	36.2462	1.48845	0.744227	+	0.667927i	$0.232818\pi$
$594$	0	0
$595$	36.4924	1.49604
$596$	0	0
$597$	−11.0540	−0.452409
$598$	0	0
$599$	20.9848	0.857418	0.428709	−	0.903443i	$-0.358969\pi$
$599$	20.9848	0.857418	0.428709	+	0.903443i	$0.358969\pi$
$600$	0	0
$601$	41.3693	1.68749	0.843745	−	0.536745i	$-0.180346\pi$
$601$	41.3693	1.68749	0.843745	+	0.536745i	$0.180346\pi$
$602$	0	0
$603$	11.3693	0.462994
$604$	0	0
$605$	−8.93087	−0.363091
$606$	0	0
$607$	16.0000	0.649420	0.324710	−	0.945814i	$-0.394733\pi$
$607$	16.0000	0.649420	0.324710	+	0.945814i	$0.394733\pi$
$608$	0	0
$609$	5.12311	0.207599
$610$	0	0
$611$	−26.2462	−1.06181
$612$	0	0
$613$	−2.00000	−0.0807792	−0.0403896	−	0.999184i	$-0.512860\pi$
$613$	−2.00000	−0.0807792	−0.0403896	+	0.999184i	$0.512860\pi$
$614$	0	0
$615$	1.68466	0.0679320
$616$	0	0
$617$	8.24621	0.331980	0.165990	−	0.986127i	$-0.446918\pi$
$617$	8.24621	0.331980	0.165990	+	0.986127i	$0.446918\pi$
$618$	0	0
$619$	−29.1231	−1.17056	−0.585278	−	0.810833i	$-0.699015\pi$
$619$	−29.1231	−1.17056	−0.585278	+	0.810833i	$0.699015\pi$
$620$	0	0
$621$	6.56155	0.263306
$622$	0	0
$623$	−62.7386	−2.51357
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	7.36932	0.294302
$628$	0	0
$629$	12.0000	0.478471
$630$	0	0
$631$	−32.0000	−1.27390	−0.636950	−	0.770905i	$-0.719804\pi$
$631$	−32.0000	−1.27390	−0.636950	+	0.770905i	$0.719804\pi$
$632$	0	0
$633$	−2.87689	−0.114346
$634$	0	0
$635$	−4.31534	−0.171249
$636$	0	0
$637$	−38.4924	−1.52513
$638$	0	0
$639$	2.87689	0.113808
$640$	0	0
$641$	19.4384	0.767773	0.383886	−	0.923380i	$-0.374585\pi$
$641$	19.4384	0.767773	0.383886	+	0.923380i	$0.374585\pi$
$642$	0	0
$643$	18.7386	0.738980	0.369490	−	0.929235i	$-0.379532\pi$
$643$	18.7386	0.738980	0.369490	+	0.929235i	$0.379532\pi$
$644$	0	0
$645$	7.68466	0.302583
$646$	0	0
$647$	−5.93087	−0.233167	−0.116583	−	0.993181i	$-0.537194\pi$
$647$	−5.93087	−0.233167	−0.116583	+	0.993181i	$0.537194\pi$
$648$	0	0
$649$	−17.2614	−0.677568
$650$	0	0
$651$	−20.4924	−0.803161
$652$	0	0
$653$	12.2462	0.479231	0.239616	−	0.970868i	$-0.422979\pi$
$653$	12.2462	0.479231	0.239616	+	0.970868i	$0.422979\pi$
$654$	0	0
$655$	−10.2462	−0.400353
$656$	0	0
$657$	1.68466	0.0657248
$658$	0	0
$659$	−6.56155	−0.255602	−0.127801	−	0.991800i	$-0.540792\pi$
$659$	−6.56155	−0.255602	−0.127801	+	0.991800i	$0.540792\pi$
$660$	0	0
$661$	1.05398	0.0409949	0.0204974	−	0.999790i	$-0.493475\pi$
$661$	1.05398	0.0409949	0.0204974	+	0.999790i	$0.493475\pi$
$662$	0	0
$663$	−14.2462	−0.553277
$664$	0	0
$665$	26.2462	1.01778
$666$	0	0
$667$	−6.56155	−0.254064
$668$	0	0
$669$	−18.2462	−0.705439
$670$	0	0
$671$	1.26137	0.0486945
$672$	0	0
$673$	−6.00000	−0.231283	−0.115642	−	0.993291i	$-0.536892\pi$
$673$	−6.00000	−0.231283	−0.115642	+	0.993291i	$0.536892\pi$
$674$	0	0
$675$	−1.00000	−0.0384900
$676$	0	0
$677$	13.8617	0.532750	0.266375	−	0.963869i	$-0.414174\pi$
$677$	13.8617	0.532750	0.266375	+	0.963869i	$0.414174\pi$
$678$	0	0
$679$	29.1231	1.11764
$680$	0	0
$681$	−3.19224	−0.122327
$682$	0	0
$683$	21.4384	0.820319	0.410160	−	0.912014i	$-0.365473\pi$
$683$	21.4384	0.820319	0.410160	+	0.912014i	$0.365473\pi$
$684$	0	0
$685$	−15.1231	−0.577824
$686$	0	0
$687$	−22.0000	−0.839352
$688$	0	0
$689$	6.87689	0.261989
$690$	0	0
$691$	4.00000	0.152167	0.0760836	−	0.997101i	$-0.475758\pi$
$691$	4.00000	0.152167	0.0760836	+	0.997101i	$0.475758\pi$
$692$	0	0
$693$	−7.36932	−0.279937
$694$	0	0
$695$	17.9309	0.680157
$696$	0	0
$697$	12.0000	0.454532
$698$	0	0
$699$	10.3153	0.390162
$700$	0	0
$701$	3.75379	0.141779	0.0708893	−	0.997484i	$-0.477416\pi$
$701$	3.75379	0.141779	0.0708893	+	0.997484i	$0.477416\pi$
$702$	0	0
$703$	8.63068	0.325512
$704$	0	0
$705$	−13.1231	−0.494245
$706$	0	0
$707$	43.8617	1.64959
$708$	0	0
$709$	−12.4233	−0.466567	−0.233283	−	0.972409i	$-0.574947\pi$
$709$	−12.4233	−0.466567	−0.233283	+	0.972409i	$0.574947\pi$
$710$	0	0
$711$	12.0000	0.450035
$712$	0	0
$713$	26.2462	0.982928
$714$	0	0
$715$	−2.87689	−0.107590
$716$	0	0
$717$	−13.1231	−0.490091
$718$	0	0
$719$	4.49242	0.167539	0.0837695	−	0.996485i	$-0.473304\pi$
$719$	4.49242	0.167539	0.0837695	+	0.996485i	$0.473304\pi$
$720$	0	0
$721$	−14.7386	−0.548895
$722$	0	0
$723$	25.0540	0.931767
$724$	0	0
$725$	1.00000	0.0371391
$726$	0	0
$727$	38.7386	1.43674	0.718368	−	0.695663i	$-0.244889\pi$
$727$	38.7386	1.43674	0.718368	+	0.695663i	$0.244889\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	54.7386	2.02458
$732$	0	0
$733$	−22.9848	−0.848965	−0.424482	−	0.905436i	$-0.639544\pi$
$733$	−22.9848	−0.848965	−0.424482	+	0.905436i	$0.639544\pi$
$734$	0	0
$735$	−19.2462	−0.709907
$736$	0	0
$737$	16.3542	0.602413
$738$	0	0
$739$	24.0000	0.882854	0.441427	−	0.897297i	$-0.354472\pi$
$739$	24.0000	0.882854	0.441427	+	0.897297i	$0.354472\pi$
$740$	0	0
$741$	−10.2462	−0.376404
$742$	0	0
$743$	−34.8769	−1.27951	−0.639755	−	0.768579i	$-0.720964\pi$
$743$	−34.8769	−1.27951	−0.639755	+	0.768579i	$0.720964\pi$
$744$	0	0
$745$	0.246211	0.00902048
$746$	0	0
$747$	2.56155	0.0937223
$748$	0	0
$749$	84.4924	3.08729
$750$	0	0
$751$	−48.4924	−1.76951	−0.884757	−	0.466053i	$-0.845676\pi$
$751$	−48.4924	−1.76951	−0.884757	+	0.466053i	$0.845676\pi$
$752$	0	0
$753$	2.24621	0.0818565
$754$	0	0
$755$	−4.31534	−0.157051
$756$	0	0
$757$	5.68466	0.206612	0.103306	−	0.994650i	$-0.467058\pi$
$757$	5.68466	0.206612	0.103306	+	0.994650i	$0.467058\pi$
$758$	0	0
$759$	9.43845	0.342594
$760$	0	0
$761$	−24.8769	−0.901787	−0.450893	−	0.892578i	$-0.648895\pi$
$761$	−24.8769	−0.901787	−0.450893	+	0.892578i	$0.648895\pi$
$762$	0	0
$763$	29.1231	1.05433
$764$	0	0
$765$	−7.12311	−0.257536
$766$	0	0
$767$	24.0000	0.866590
$768$	0	0
$769$	43.6155	1.57282	0.786408	−	0.617707i	$-0.211938\pi$
$769$	43.6155	1.57282	0.786408	+	0.617707i	$0.211938\pi$
$770$	0	0
$771$	−11.4384	−0.411946
$772$	0	0
$773$	24.7386	0.889787	0.444893	−	0.895584i	$-0.353242\pi$
$773$	24.7386	0.889787	0.444893	+	0.895584i	$0.353242\pi$
$774$	0	0
$775$	−4.00000	−0.143684
$776$	0	0
$777$	−8.63068	−0.309624
$778$	0	0
$779$	8.63068	0.309226
$780$	0	0
$781$	4.13826	0.148079
$782$	0	0
$783$	−1.00000	−0.0357371
$784$	0	0
$785$	−14.0000	−0.499681
$786$	0	0
$787$	31.2311	1.11327	0.556633	−	0.830758i	$-0.312093\pi$
$787$	31.2311	1.11327	0.556633	+	0.830758i	$0.312093\pi$
$788$	0	0
$789$	−5.75379	−0.204840
$790$	0	0
$791$	24.9848	0.888359
$792$	0	0
$793$	−1.75379	−0.0622789
$794$	0	0
$795$	3.43845	0.121949
$796$	0	0
$797$	38.4924	1.36347	0.681736	−	0.731598i	$-0.261225\pi$
$797$	38.4924	1.36347	0.681736	+	0.731598i	$0.261225\pi$
$798$	0	0
$799$	−93.4773	−3.30699
$800$	0	0
$801$	12.2462	0.432699
$802$	0	0
$803$	2.42329	0.0855161
$804$	0	0
$805$	33.6155	1.18479
$806$	0	0
$807$	−11.7538	−0.413753
$808$	0	0
$809$	−22.1771	−0.779705	−0.389852	−	0.920877i	$-0.627474\pi$
$809$	−22.1771	−0.779705	−0.389852	+	0.920877i	$0.627474\pi$
$810$	0	0
$811$	−44.1771	−1.55127	−0.775634	−	0.631183i	$-0.782569\pi$
$811$	−44.1771	−1.55127	−0.775634	+	0.631183i	$0.782569\pi$
$812$	0	0
$813$	17.1231	0.600534
$814$	0	0
$815$	17.9309	0.628091
$816$	0	0
$817$	39.3693	1.37736
$818$	0	0
$819$	10.2462	0.358032
$820$	0	0
$821$	55.6155	1.94100	0.970498	−	0.241111i	$-0.0775116\pi$
$821$	55.6155	1.94100	0.970498	+	0.241111i	$0.0775116\pi$
$822$	0	0
$823$	−16.0000	−0.557725	−0.278862	−	0.960331i	$-0.589957\pi$
$823$	−16.0000	−0.557725	−0.278862	+	0.960331i	$0.589957\pi$
$824$	0	0
$825$	−1.43845	−0.0500803
$826$	0	0
$827$	−53.6155	−1.86439	−0.932197	−	0.361951i	$-0.882111\pi$
$827$	−53.6155	−1.86439	−0.932197	+	0.361951i	$0.882111\pi$
$828$	0	0
$829$	−4.24621	−0.147477	−0.0737385	−	0.997278i	$-0.523493\pi$
$829$	−4.24621	−0.147477	−0.0737385	+	0.997278i	$0.523493\pi$
$830$	0	0
$831$	−0.876894	−0.0304191
$832$	0	0
$833$	−137.093	−4.74998
$834$	0	0
$835$	0	0
$836$	0	0
$837$	4.00000	0.138260
$838$	0	0
$839$	−42.7386	−1.47550	−0.737751	−	0.675073i	$-0.764112\pi$
$839$	−42.7386	−1.47550	−0.737751	+	0.675073i	$0.764112\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	23.6155	0.813362
$844$	0	0
$845$	−9.00000	−0.309609
$846$	0	0
$847$	45.7538	1.57212
$848$	0	0
$849$	6.87689	0.236014
$850$	0	0
$851$	11.0540	0.378925
$852$	0	0
$853$	19.4384	0.665560	0.332780	−	0.943005i	$-0.392013\pi$
$853$	19.4384	0.665560	0.332780	+	0.943005i	$0.392013\pi$
$854$	0	0
$855$	−5.12311	−0.175207
$856$	0	0
$857$	−15.4384	−0.527367	−0.263684	−	0.964609i	$-0.584938\pi$
$857$	−15.4384	−0.527367	−0.263684	+	0.964609i	$0.584938\pi$
$858$	0	0
$859$	23.3693	0.797351	0.398675	−	0.917092i	$-0.369470\pi$
$859$	23.3693	0.797351	0.398675	+	0.917092i	$0.369470\pi$
$860$	0	0
$861$	−8.63068	−0.294133
$862$	0	0
$863$	−40.9848	−1.39514	−0.697570	−	0.716516i	$-0.745735\pi$
$863$	−40.9848	−1.39514	−0.697570	+	0.716516i	$0.745735\pi$
$864$	0	0
$865$	12.5616	0.427106
$866$	0	0
$867$	−33.7386	−1.14582
$868$	0	0
$869$	17.2614	0.585552
$870$	0	0
$871$	−22.7386	−0.770469
$872$	0	0
$873$	−5.68466	−0.192397
$874$	0	0
$875$	−5.12311	−0.173193
$876$	0	0
$877$	8.87689	0.299751	0.149876	−	0.988705i	$-0.452113\pi$
$877$	8.87689	0.299751	0.149876	+	0.988705i	$0.452113\pi$
$878$	0	0
$879$	21.3693	0.720769
$880$	0	0
$881$	−8.06913	−0.271856	−0.135928	−	0.990719i	$-0.543402\pi$
$881$	−8.06913	−0.271856	−0.135928	+	0.990719i	$0.543402\pi$
$882$	0	0
$883$	42.7386	1.43827	0.719135	−	0.694871i	$-0.244538\pi$
$883$	42.7386	1.43827	0.719135	+	0.694871i	$0.244538\pi$
$884$	0	0
$885$	12.0000	0.403376
$886$	0	0
$887$	43.8617	1.47273	0.736367	−	0.676583i	$-0.236540\pi$
$887$	43.8617	1.47273	0.736367	+	0.676583i	$0.236540\pi$
$888$	0	0
$889$	22.1080	0.741477
$890$	0	0
$891$	1.43845	0.0481898
$892$	0	0
$893$	−67.2311	−2.24980
$894$	0	0
$895$	−1.12311	−0.0375413
$896$	0	0
$897$	−13.1231	−0.438168
$898$	0	0
$899$	−4.00000	−0.133407
$900$	0	0
$901$	24.4924	0.815961
$902$	0	0
$903$	−39.3693	−1.31013
$904$	0	0
$905$	25.6847	0.853787
$906$	0	0
$907$	−31.0540	−1.03113	−0.515565	−	0.856850i	$-0.672418\pi$
$907$	−31.0540	−1.03113	−0.515565	+	0.856850i	$0.672418\pi$
$908$	0	0
$909$	−8.56155	−0.283969
$910$	0	0
$911$	18.5616	0.614972	0.307486	−	0.951553i	$-0.400512\pi$
$911$	18.5616	0.614972	0.307486	+	0.951553i	$0.400512\pi$
$912$	0	0
$913$	3.68466	0.121944
$914$	0	0
$915$	−0.876894	−0.0289892
$916$	0	0
$917$	52.4924	1.73345
$918$	0	0
$919$	−30.7386	−1.01397	−0.506987	−	0.861954i	$-0.669241\pi$
$919$	−30.7386	−1.01397	−0.506987	+	0.861954i	$0.669241\pi$
$920$	0	0
$921$	−31.6847	−1.04404
$922$	0	0
$923$	−5.75379	−0.189388
$924$	0	0
$925$	−1.68466	−0.0553912
$926$	0	0
$927$	2.87689	0.0944896
$928$	0	0
$929$	4.87689	0.160006	0.0800029	−	0.996795i	$-0.474507\pi$
$929$	4.87689	0.160006	0.0800029	+	0.996795i	$0.474507\pi$
$930$	0	0
$931$	−98.6004	−3.23150
$932$	0	0
$933$	16.3153	0.534140
$934$	0	0
$935$	−10.2462	−0.335087
$936$	0	0
$937$	39.1231	1.27810	0.639048	−	0.769167i	$-0.279328\pi$
$937$	39.1231	1.27810	0.639048	+	0.769167i	$0.279328\pi$
$938$	0	0
$939$	21.3693	0.697361
$940$	0	0
$941$	−0.738634	−0.0240788	−0.0120394	−	0.999928i	$-0.503832\pi$
$941$	−0.738634	−0.0240788	−0.0120394	+	0.999928i	$0.503832\pi$
$942$	0	0
$943$	11.0540	0.359967
$944$	0	0
$945$	5.12311	0.166655
$946$	0	0
$947$	3.36932	0.109488	0.0547440	−	0.998500i	$-0.482566\pi$
$947$	3.36932	0.109488	0.0547440	+	0.998500i	$0.482566\pi$
$948$	0	0
$949$	−3.36932	−0.109373
$950$	0	0
$951$	−4.87689	−0.158144
$952$	0	0
$953$	−34.4924	−1.11732	−0.558660	−	0.829397i	$-0.688684\pi$
$953$	−34.4924	−1.11732	−0.558660	+	0.829397i	$0.688684\pi$
$954$	0	0
$955$	9.93087	0.321355
$956$	0	0
$957$	−1.43845	−0.0464984
$958$	0	0
$959$	77.4773	2.50187
$960$	0	0
$961$	−15.0000	−0.483871
$962$	0	0
$963$	−16.4924	−0.531461
$964$	0	0
$965$	−10.0000	−0.321911
$966$	0	0
$967$	−27.0540	−0.869997	−0.434999	−	0.900431i	$-0.643251\pi$
$967$	−27.0540	−0.869997	−0.434999	+	0.900431i	$0.643251\pi$
$968$	0	0
$969$	−36.4924	−1.17231
$970$	0	0
$971$	44.6695	1.43351	0.716756	−	0.697324i	$-0.245626\pi$
$971$	44.6695	1.43351	0.716756	+	0.697324i	$0.245626\pi$
$972$	0	0
$973$	−91.8617	−2.94495
$974$	0	0
$975$	2.00000	0.0640513
$976$	0	0
$977$	−7.43845	−0.237977	−0.118989	−	0.992896i	$-0.537965\pi$
$977$	−7.43845	−0.237977	−0.118989	+	0.992896i	$0.537965\pi$
$978$	0	0
$979$	17.6155	0.562995
$980$	0	0
$981$	−5.68466	−0.181497
$982$	0	0
$983$	−27.8617	−0.888651	−0.444326	−	0.895865i	$-0.646557\pi$
$983$	−27.8617	−0.888651	−0.444326	+	0.895865i	$0.646557\pi$
$984$	0	0
$985$	4.56155	0.145343
$986$	0	0
$987$	67.2311	2.13999
$988$	0	0
$989$	50.4233	1.60337
$990$	0	0
$991$	−4.94602	−0.157116	−0.0785578	−	0.996910i	$-0.525032\pi$
$991$	−4.94602	−0.157116	−0.0785578	+	0.996910i	$0.525032\pi$
$992$	0	0
$993$	10.2462	0.325154
$994$	0	0
$995$	11.0540	0.350435
$996$	0	0
$997$	54.0388	1.71143	0.855713	−	0.517450i	$-0.173119\pi$
$997$	54.0388	1.71143	0.855713	+	0.517450i	$0.173119\pi$
$998$	0	0
$999$	1.68466	0.0533002

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6960.2.a.bx.1.1		2
4.3	odd	2		435.2.a.h.1.1	✓	2
12.11	even	2		1305.2.a.i.1.2		2
20.3	even	4		2175.2.c.h.349.3		4
20.7	even	4		2175.2.c.h.349.2		4
20.19	odd	2		2175.2.a.m.1.2		2
60.59	even	2		6525.2.a.bc.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
435.2.a.h.1.1	✓	2	4.3	odd	2
1305.2.a.i.1.2		2	12.11	even	2
2175.2.a.m.1.2		2	20.19	odd	2
2175.2.c.h.349.2		4	20.7	even	4
2175.2.c.h.349.3		4	20.3	even	4
6525.2.a.bc.1.1		2	60.59	even	2
6960.2.a.bx.1.1		2	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:	\( N \)	\(=\)	\( 6960 = 2^{4} \cdot 3 \cdot 5 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6960.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +1.00000 q^{5} -5.12311 q^{7} +1.00000 q^{9} +1.43845 q^{11} -2.00000 q^{13} -1.00000 q^{15} -7.12311 q^{17} -5.12311 q^{19} +5.12311 q^{21} -6.56155 q^{23} +1.00000 q^{25} -1.00000 q^{27} +1.00000 q^{29} -4.00000 q^{31} -1.43845 q^{33} -5.12311 q^{35} -1.68466 q^{37} +2.00000 q^{39} -1.68466 q^{41} -7.68466 q^{43} +1.00000 q^{45} +13.1231 q^{47} +19.2462 q^{49} +7.12311 q^{51} -3.43845 q^{53} +1.43845 q^{55} +5.12311 q^{57} -12.0000 q^{59} +0.876894 q^{61} -5.12311 q^{63} -2.00000 q^{65} +11.3693 q^{67} +6.56155 q^{69} +2.87689 q^{71} +1.68466 q^{73} -1.00000 q^{75} -7.36932 q^{77} +12.0000 q^{79} +1.00000 q^{81} +2.56155 q^{83} -7.12311 q^{85} -1.00000 q^{87} +12.2462 q^{89} +10.2462 q^{91} +4.00000 q^{93} -5.12311 q^{95} -5.68466 q^{97} +1.43845 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} + 2 q^{5} - 2 q^{7} + 2 q^{9} + 7 q^{11} - 4 q^{13} - 2 q^{15} - 6 q^{17} - 2 q^{19} + 2 q^{21} - 9 q^{23} + 2 q^{25} - 2 q^{27} + 2 q^{29} - 8 q^{31} - 7 q^{33} - 2 q^{35} + 9 q^{37} + 4 q^{39}+ \cdots + 7 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 + 2 * q^5 - 2 * q^7 + 2 * q^9 + 7 * q^11 - 4 * q^13 - 2 * q^15 - 6 * q^17 - 2 * q^19 + 2 * q^21 - 9 * q^23 + 2 * q^25 - 2 * q^27 + 2 * q^29 - 8 * q^31 - 7 * q^33 - 2 * q^35 + 9 * q^37 + 4 * q^39 + 9 * q^41 - 3 * q^43 + 2 * q^45 + 18 * q^47 + 22 * q^49 + 6 * q^51 - 11 * q^53 + 7 * q^55 + 2 * q^57 - 24 * q^59 + 10 * q^61 - 2 * q^63 - 4 * q^65 - 2 * q^67 + 9 * q^69 + 14 * q^71 - 9 * q^73 - 2 * q^75 + 10 * q^77 + 24 * q^79 + 2 * q^81 + q^83 - 6 * q^85 - 2 * q^87 + 8 * q^89 + 4 * q^91 + 8 * q^93 - 2 * q^95 + q^97 + 7 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more