LMFDB - Embedded newform 2960.2.a.v.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2960,2,Mod(1,2960)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2960, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("2960.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2960 = 2^{4} \cdot 5 \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2960.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:	$23.6357189983$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.286164.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 11x^{2} - 3x + 12 $ x^4 - x^3 - 11x^2 - 3x + 12
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 740)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$3.84956$ of defining polynomial
Character	$\chi$	$=$	2960.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.84956 q^{3} +1.00000 q^{5} +4.84956 q^{7} +5.11999 q^{9} +O(q^{10})$	$q+2.84956 q^{3} +1.00000 q^{5} +4.84956 q^{7} +5.11999 q^{9} -2.34191 q^{11} -4.46189 q^{13} +2.84956 q^{15} -0.461892 q^{17} +2.50765 q^{19} +13.8191 q^{21} -7.69912 q^{23} +1.00000 q^{25} +6.04102 q^{27} +7.69912 q^{29} +9.43144 q^{31} -6.67340 q^{33} +4.84956 q^{35} +1.00000 q^{37} -12.7144 q^{39} +8.34191 q^{41} -11.9391 q^{43} +5.11999 q^{45} -0.0742247 q^{47} +16.5182 q^{49} -1.31619 q^{51} +5.35721 q^{53} -2.34191 q^{55} +7.14570 q^{57} +3.73232 q^{59} +2.00000 q^{61} +24.8297 q^{63} -4.46189 q^{65} +1.19146 q^{67} -21.9391 q^{69} -4.04102 q^{71} +15.0563 q^{73} +2.84956 q^{75} -11.3572 q^{77} +8.50765 q^{79} +1.85430 q^{81} -15.0895 q^{83} -0.461892 q^{85} +21.9391 q^{87} -6.92378 q^{89} -21.6382 q^{91} +26.8754 q^{93} +2.50765 q^{95} -7.93909 q^{97} -11.9905 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 3 q^{3} + 4 q^{5} + 5 q^{7} + 13 q^{9}+O(q^{10}) $ 4 * q - 3 * q^3 + 4 * q^5 + 5 * q^7 + 13 * q^9	$ 4 q - 3 q^{3} + 4 q^{5} + 5 q^{7} + 13 q^{9} - 5 q^{11} - 6 q^{13} - 3 q^{15} + 10 q^{17} + 19 q^{21} - 2 q^{23} + 4 q^{25} - 9 q^{27} + 2 q^{29} + 4 q^{31} - 11 q^{33} + 5 q^{35} + 4 q^{37} - 2 q^{39} + 29 q^{41} - 4 q^{43} + 13 q^{45} + 9 q^{47} + q^{49} - 14 q^{51} - 3 q^{53} - 5 q^{55} + 8 q^{57} + 10 q^{59} + 8 q^{61} + 8 q^{63} - 6 q^{65} - 14 q^{67} - 44 q^{69} + 17 q^{71} + 7 q^{73} - 3 q^{75} - 21 q^{77} + 24 q^{79} + 28 q^{81} - 31 q^{83} + 10 q^{85} + 44 q^{87} - 4 q^{89} - 14 q^{91} + 18 q^{93} + 12 q^{97} + 22 q^{99}+O(q^{100}) $ 4 * q - 3 * q^3 + 4 * q^5 + 5 * q^7 + 13 * q^9 - 5 * q^11 - 6 * q^13 - 3 * q^15 + 10 * q^17 + 19 * q^21 - 2 * q^23 + 4 * q^25 - 9 * q^27 + 2 * q^29 + 4 * q^31 - 11 * q^33 + 5 * q^35 + 4 * q^37 - 2 * q^39 + 29 * q^41 - 4 * q^43 + 13 * q^45 + 9 * q^47 + q^49 - 14 * q^51 - 3 * q^53 - 5 * q^55 + 8 * q^57 + 10 * q^59 + 8 * q^61 + 8 * q^63 - 6 * q^65 - 14 * q^67 - 44 * q^69 + 17 * q^71 + 7 * q^73 - 3 * q^75 - 21 * q^77 + 24 * q^79 + 28 * q^81 - 31 * q^83 + 10 * q^85 + 44 * q^87 - 4 * q^89 - 14 * q^91 + 18 * q^93 + 12 * q^97 + 22 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	2.84956	1.64519	0.822597	−	0.568625i	$-0.192524\pi$
$3$	2.84956	1.64519	0.822597	+	0.568625i	$0.192524\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	4.84956	1.83296	0.916480	−	0.400079i	$-0.131017\pi$
$7$	4.84956	1.83296	0.916480	+	0.400079i	$0.131017\pi$
$8$	0	0
$9$	5.11999	1.70666
$10$	0	0
$11$	−2.34191	−0.706111	−0.353056	−	0.935602i	$-0.614857\pi$
$11$	−2.34191	−0.706111	−0.353056	+	0.935602i	$0.614857\pi$
$12$	0	0
$13$	−4.46189	−1.23751	−0.618753	−	0.785586i	$-0.712362\pi$
$13$	−4.46189	−1.23751	−0.618753	+	0.785586i	$0.712362\pi$
$14$	0	0
$15$	2.84956	0.735753
$16$	0	0
$17$	−0.461892	−0.112025	−0.0560126	−	0.998430i	$-0.517839\pi$
$17$	−0.461892	−0.112025	−0.0560126	+	0.998430i	$0.517839\pi$
$18$	0	0
$19$	2.50765	0.575295	0.287647	−	0.957736i	$-0.407127\pi$
$19$	2.50765	0.575295	0.287647	+	0.957736i	$0.407127\pi$
$20$	0	0
$21$	13.8191	3.01558
$22$	0	0
$23$	−7.69912	−1.60538	−0.802689	−	0.596398i	$-0.796598\pi$
$23$	−7.69912	−1.60538	−0.802689	+	0.596398i	$0.796598\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	6.04102	1.16260
$28$	0	0
$29$	7.69912	1.42969	0.714845	−	0.699283i	$-0.246497\pi$
$29$	7.69912	1.42969	0.714845	+	0.699283i	$0.246497\pi$
$30$	0	0
$31$	9.43144	1.69394	0.846968	−	0.531644i	$-0.178426\pi$
$31$	9.43144	1.69394	0.846968	+	0.531644i	$0.178426\pi$
$32$	0	0
$33$	−6.67340	−1.16169
$34$	0	0
$35$	4.84956	0.819725
$36$	0	0
$37$	1.00000	0.164399
$37$	1.00000	0.164399
$38$	0	0
$39$	−12.7144	−2.03594
$40$	0	0
$41$	8.34191	1.30279	0.651393	−	0.758740i	$-0.274185\pi$
$41$	8.34191	1.30279	0.651393	+	0.758740i	$0.274185\pi$
$42$	0	0
$43$	−11.9391	−1.82069	−0.910347	−	0.413846i	$-0.864185\pi$
$43$	−11.9391	−1.82069	−0.910347	+	0.413846i	$0.864185\pi$
$44$	0	0
$45$	5.11999	0.763242
$46$	0	0
$47$	−0.0742247	−0.0108268	−0.00541339	−	0.999985i	$-0.501723\pi$
$47$	−0.0742247	−0.0108268	−0.00541339	+	0.999985i	$0.501723\pi$
$48$	0	0
$49$	16.5182	2.35975
$50$	0	0
$51$	−1.31619	−0.184303
$52$	0	0
$53$	5.35721	0.735870	0.367935	−	0.929852i	$-0.380065\pi$
$53$	5.35721	0.735870	0.367935	+	0.929852i	$0.380065\pi$
$54$	0	0
$55$	−2.34191	−0.315783
$56$	0	0
$57$	7.14570	0.946472
$58$	0	0
$59$	3.73232	0.485906	0.242953	−	0.970038i	$-0.421884\pi$
$59$	3.73232	0.485906	0.242953	+	0.970038i	$0.421884\pi$
$60$	0	0
$61$	2.00000	0.256074	0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000	0.256074	0.128037	+	0.991769i	$0.459132\pi$
$62$	0	0
$63$	24.8297	3.12824
$64$	0	0
$65$	−4.46189	−0.553430
$66$	0	0
$67$	1.19146	0.145561	0.0727803	−	0.997348i	$-0.476813\pi$
$67$	1.19146	0.145561	0.0727803	+	0.997348i	$0.476813\pi$
$68$	0	0
$69$	−21.9391	−2.64116
$70$	0	0
$71$	−4.04102	−0.479581	−0.239791	−	0.970825i	$-0.577079\pi$
$71$	−4.04102	−0.479581	−0.239791	+	0.970825i	$0.577079\pi$
$72$	0	0
$73$	15.0563	1.76221	0.881105	−	0.472921i	$-0.156800\pi$
$73$	15.0563	1.76221	0.881105	+	0.472921i	$0.156800\pi$
$74$	0	0
$75$	2.84956	0.329039
$76$	0	0
$77$	−11.3572	−1.29427
$78$	0	0
$79$	8.50765	0.957186	0.478593	−	0.878037i	$-0.341147\pi$
$79$	8.50765	0.957186	0.478593	+	0.878037i	$0.341147\pi$
$80$	0	0
$81$	1.85430	0.206033
$82$	0	0
$83$	−15.0895	−1.65629	−0.828146	−	0.560513i	$-0.810604\pi$
$83$	−15.0895	−1.65629	−0.828146	+	0.560513i	$0.810604\pi$
$84$	0	0
$85$	−0.461892	−0.0500992
$86$	0	0
$87$	21.9391	2.35212
$88$	0	0
$89$	−6.92378	−0.733920	−0.366960	−	0.930237i	$-0.619601\pi$
$89$	−6.92378	−0.733920	−0.366960	+	0.930237i	$0.619601\pi$
$90$	0	0
$91$	−21.6382	−2.26830
$92$	0	0
$93$	26.8754	2.78685
$94$	0	0
$95$	2.50765	0.257280
$96$	0	0
$97$	−7.93909	−0.806092	−0.403046	−	0.915180i	$-0.632049\pi$
$97$	−7.93909	−0.806092	−0.403046	+	0.915180i	$0.632049\pi$
$98$	0	0
$99$	−11.9905	−1.20509
$100$	0	0
$101$	−4.35447	−0.433286	−0.216643	−	0.976251i	$-0.569511\pi$
$101$	−4.35447	−0.433286	−0.216643	+	0.976251i	$0.569511\pi$
$102$	0	0
$103$	−2.00000	−0.197066	−0.0985329	−	0.995134i	$-0.531415\pi$
$103$	−2.00000	−0.197066	−0.0985329	+	0.995134i	$0.531415\pi$
$104$	0	0
$105$	13.8191	1.34861
$106$	0	0
$107$	−8.44674	−0.816577	−0.408289	−	0.912853i	$-0.633874\pi$
$107$	−8.44674	−0.816577	−0.408289	+	0.912853i	$0.633874\pi$
$108$	0	0
$109$	−16.9544	−1.62394	−0.811968	−	0.583702i	$-0.801604\pi$
$109$	−16.9544	−1.62394	−0.811968	+	0.583702i	$0.801604\pi$
$110$	0	0
$111$	2.84956	0.270468
$112$	0	0
$113$	−7.77808	−0.731700	−0.365850	−	0.930674i	$-0.619222\pi$
$113$	−7.77808	−0.731700	−0.365850	+	0.930674i	$0.619222\pi$
$114$	0	0
$115$	−7.69912	−0.717946
$116$	0	0
$117$	−22.8448	−2.11200
$118$	0	0
$119$	−2.23997	−0.205338
$120$	0	0
$121$	−5.51548	−0.501407
$122$	0	0
$123$	23.7708	2.14334
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−11.0895	−0.984037	−0.492018	−	0.870585i	$-0.663741\pi$
$127$	−11.0895	−0.984037	−0.492018	+	0.870585i	$0.663741\pi$
$128$	0	0
$129$	−34.0211	−2.99539
$130$	0	0
$131$	−11.2830	−0.985799	−0.492900	−	0.870086i	$-0.664063\pi$
$131$	−11.2830	−0.985799	−0.492900	+	0.870086i	$0.664063\pi$
$132$	0	0
$133$	12.1610	1.05449
$134$	0	0
$135$	6.04102	0.519929
$136$	0	0
$137$	−11.4591	−0.979021	−0.489510	−	0.871997i	$-0.662825\pi$
$137$	−11.4591	−0.979021	−0.489510	+	0.871997i	$0.662825\pi$
$138$	0	0
$139$	0.443841	0.0376461	0.0188231	−	0.999823i	$-0.494008\pi$
$139$	0.443841	0.0376461	0.0188231	+	0.999823i	$0.494008\pi$
$140$	0	0
$141$	−0.211508	−0.0178122
$142$	0	0
$143$	10.4493	0.873817
$144$	0	0
$145$	7.69912	0.639377
$146$	0	0
$147$	47.0696	3.88224
$148$	0	0
$149$	17.0744	1.39879	0.699394	−	0.714736i	$-0.253453\pi$
$149$	17.0744	1.39879	0.699394	+	0.714736i	$0.253453\pi$
$150$	0	0
$151$	13.9391	1.13435	0.567173	−	0.823598i	$-0.308037\pi$
$151$	13.9391	1.13435	0.567173	+	0.823598i	$0.308037\pi$
$152$	0	0
$153$	−2.36488	−0.191189
$154$	0	0
$155$	9.43144	0.757551
$156$	0	0
$157$	−0.341906	−0.0272871	−0.0136435	−	0.999907i	$-0.504343\pi$
$157$	−0.341906	−0.0272871	−0.0136435	+	0.999907i	$0.504343\pi$
$158$	0	0
$159$	15.2657	1.21065
$160$	0	0
$161$	−37.3373	−2.94259
$162$	0	0
$163$	−7.60760	−0.595873	−0.297936	−	0.954586i	$-0.596298\pi$
$163$	−7.60760	−0.595873	−0.297936	+	0.954586i	$0.596298\pi$
$164$	0	0
$165$	−6.67340	−0.519523
$166$	0	0
$167$	5.07622	0.392809	0.196405	−	0.980523i	$-0.437073\pi$
$167$	5.07622	0.392809	0.196405	+	0.980523i	$0.437073\pi$
$168$	0	0
$169$	6.90848	0.531421
$170$	0	0
$171$	12.8391	0.981834
$172$	0	0
$173$	−1.18364	−0.0899906	−0.0449953	−	0.998987i	$-0.514327\pi$
$173$	−1.18364	−0.0899906	−0.0449953	+	0.998987i	$0.514327\pi$
$174$	0	0
$175$	4.84956	0.366592
$176$	0	0
$177$	10.6355	0.799410
$178$	0	0
$179$	−13.9059	−1.03937	−0.519687	−	0.854356i	$-0.673952\pi$
$179$	−13.9059	−1.03937	−0.519687	+	0.854356i	$0.673952\pi$
$180$	0	0
$181$	12.3545	0.918300	0.459150	−	0.888359i	$-0.348154\pi$
$181$	12.3545	0.918300	0.459150	+	0.888359i	$0.348154\pi$
$182$	0	0
$183$	5.69912	0.421291
$184$	0	0
$185$	1.00000	0.0735215
$186$	0	0
$187$	1.08171	0.0791023
$188$	0	0
$189$	29.2963	2.13099
$190$	0	0
$191$	−12.5897	−0.910959	−0.455479	−	0.890246i	$-0.650532\pi$
$191$	−12.5897	−0.910959	−0.455479	+	0.890246i	$0.650532\pi$
$192$	0	0
$193$	−8.71442	−0.627278	−0.313639	−	0.949542i	$-0.601548\pi$
$193$	−8.71442	−0.627278	−0.313639	+	0.949542i	$0.601548\pi$
$194$	0	0
$195$	−12.7144	−0.910499
$196$	0	0
$197$	−24.6639	−1.75723	−0.878616	−	0.477529i	$-0.841533\pi$
$197$	−24.6639	−1.75723	−0.878616	+	0.477529i	$0.841533\pi$
$198$	0	0
$199$	6.26768	0.444304	0.222152	−	0.975012i	$-0.428692\pi$
$199$	6.26768	0.444304	0.222152	+	0.975012i	$0.428692\pi$
$200$	0	0
$201$	3.39515	0.239475
$202$	0	0
$203$	37.3373	2.62057
$204$	0	0
$205$	8.34191	0.582624
$206$	0	0
$207$	−39.4194	−2.73984
$208$	0	0
$209$	−5.87269	−0.406222
$210$	0	0
$211$	−21.8317	−1.50295	−0.751477	−	0.659759i	$-0.770658\pi$
$211$	−21.8317	−1.50295	−0.751477	+	0.659759i	$0.770658\pi$
$212$	0	0
$213$	−11.5151	−0.789004
$214$	0	0
$215$	−11.9391	−0.814239
$216$	0	0
$217$	45.7383	3.10492
$218$	0	0
$219$	42.9039	2.89918
$220$	0	0
$221$	2.06091	0.138632
$222$	0	0
$223$	2.60959	0.174751	0.0873755	−	0.996175i	$-0.472152\pi$
$223$	2.60959	0.174751	0.0873755	+	0.996175i	$0.472152\pi$
$224$	0	0
$225$	5.11999	0.341332
$226$	0	0
$227$	1.31619	0.0873584	0.0436792	−	0.999046i	$-0.486092\pi$
$227$	1.31619	0.0873584	0.0436792	+	0.999046i	$0.486092\pi$
$228$	0	0
$229$	19.7401	1.30447	0.652233	−	0.758019i	$-0.273833\pi$
$229$	19.7401	1.30447	0.652233	+	0.758019i	$0.273833\pi$
$230$	0	0
$231$	−32.3630	−2.12933
$232$	0	0
$233$	−17.4898	−1.14579	−0.572896	−	0.819628i	$-0.694180\pi$
$233$	−17.4898	−1.14579	−0.572896	+	0.819628i	$0.694180\pi$
$234$	0	0
$235$	−0.0742247	−0.00484188
$236$	0	0
$237$	24.2431	1.57476
$238$	0	0
$239$	−16.1152	−1.04241	−0.521204	−	0.853432i	$-0.674517\pi$
$239$	−16.1152	−1.04241	−0.521204	+	0.853432i	$0.674517\pi$
$240$	0	0
$241$	14.7753	0.951763	0.475881	−	0.879509i	$-0.342129\pi$
$241$	14.7753	0.951763	0.475881	+	0.879509i	$0.342129\pi$
$242$	0	0
$243$	−12.8391	−0.823632
$244$	0	0
$245$	16.5182	1.05531
$246$	0	0
$247$	−11.1889	−0.711931
$248$	0	0
$249$	−42.9985	−2.72492
$250$	0	0
$251$	−5.04851	−0.318659	−0.159329	−	0.987225i	$-0.550933\pi$
$251$	−5.04851	−0.318659	−0.159329	+	0.987225i	$0.550933\pi$
$252$	0	0
$253$	18.0306	1.13357
$254$	0	0
$255$	−1.31619	−0.0824229
$256$	0	0
$257$	10.2525	0.639535	0.319768	−	0.947496i	$-0.396395\pi$
$257$	10.2525	0.639535	0.319768	+	0.947496i	$0.396395\pi$
$258$	0	0
$259$	4.84956	0.301337
$260$	0	0
$261$	39.4194	2.44000
$262$	0	0
$263$	9.63038	0.593835	0.296917	−	0.954903i	$-0.404041\pi$
$263$	9.63038	0.593835	0.296917	+	0.954903i	$0.404041\pi$
$264$	0	0
$265$	5.35721	0.329091
$266$	0	0
$267$	−19.7297	−1.20744
$268$	0	0
$269$	29.5592	1.80226	0.901129	−	0.433550i	$-0.142739\pi$
$269$	29.5592	1.80226	0.901129	+	0.433550i	$0.142739\pi$
$270$	0	0
$271$	−13.1227	−0.797149	−0.398575	−	0.917136i	$-0.630495\pi$
$271$	−13.1227	−0.797149	−0.398575	+	0.917136i	$0.630495\pi$
$272$	0	0
$273$	−61.6593	−3.73179
$274$	0	0
$275$	−2.34191	−0.141222
$276$	0	0
$277$	1.30363	0.0783274	0.0391637	−	0.999233i	$-0.487531\pi$
$277$	1.30363	0.0783274	0.0391637	+	0.999233i	$0.487531\pi$
$278$	0	0
$279$	48.2888	2.89098
$280$	0	0
$281$	3.93909	0.234986	0.117493	−	0.993074i	$-0.462514\pi$
$281$	3.93909	0.234986	0.117493	+	0.993074i	$0.462514\pi$
$282$	0	0
$283$	−9.31619	−0.553790	−0.276895	−	0.960900i	$-0.589305\pi$
$283$	−9.31619	−0.553790	−0.276895	+	0.960900i	$0.589305\pi$
$284$	0	0
$285$	7.14570	0.423275
$286$	0	0
$287$	40.4546	2.38796
$288$	0	0
$289$	−16.7867	−0.987450
$290$	0	0
$291$	−22.6229	−1.32618
$292$	0	0
$293$	23.2458	1.35803	0.679017	−	0.734122i	$-0.262406\pi$
$293$	23.2458	1.35803	0.679017	+	0.734122i	$0.262406\pi$
$294$	0	0
$295$	3.73232	0.217304
$296$	0	0
$297$	−14.1475	−0.820922
$298$	0	0
$299$	34.3526	1.98666
$300$	0	0
$301$	−57.8993	−3.33726
$302$	0	0
$303$	−12.4083	−0.712839
$304$	0	0
$305$	2.00000	0.114520
$306$	0	0
$307$	11.8040	0.673687	0.336844	−	0.941561i	$-0.390641\pi$
$307$	11.8040	0.673687	0.336844	+	0.941561i	$0.390641\pi$
$308$	0	0
$309$	−5.69912	−0.324211
$310$	0	0
$311$	−7.28848	−0.413292	−0.206646	−	0.978416i	$-0.566255\pi$
$311$	−7.28848	−0.413292	−0.206646	+	0.978416i	$0.566255\pi$
$312$	0	0
$313$	−6.38842	−0.361095	−0.180547	−	0.983566i	$-0.557787\pi$
$313$	−6.38842	−0.361095	−0.180547	+	0.983566i	$0.557787\pi$
$314$	0	0
$315$	24.8297	1.39899
$316$	0	0
$317$	−31.3373	−1.76008	−0.880040	−	0.474900i	$-0.842484\pi$
$317$	−31.3373	−1.76008	−0.880040	+	0.474900i	$0.842484\pi$
$318$	0	0
$319$	−18.0306	−1.00952
$320$	0	0
$321$	−24.0695	−1.34343
$322$	0	0
$323$	−1.15826	−0.0644475
$324$	0	0
$325$	−4.46189	−0.247501
$326$	0	0
$327$	−48.3125	−2.67169
$328$	0	0
$329$	−0.359957	−0.0198451
$330$	0	0
$331$	−16.0332	−0.881264	−0.440632	−	0.897688i	$-0.645246\pi$
$331$	−16.0332	−0.881264	−0.440632	+	0.897688i	$0.645246\pi$
$332$	0	0
$333$	5.11999	0.280573
$334$	0	0
$335$	1.19146	0.0650967
$336$	0	0
$337$	25.5057	1.38938	0.694691	−	0.719308i	$-0.255541\pi$
$337$	25.5057	1.38938	0.694691	+	0.719308i	$0.255541\pi$
$338$	0	0
$339$	−22.1641	−1.20379
$340$	0	0
$341$	−22.0875	−1.19611
$342$	0	0
$343$	46.1592	2.49236
$344$	0	0
$345$	−21.9391	−1.18116
$346$	0	0
$347$	−35.0974	−1.88412	−0.942062	−	0.335438i	$-0.891116\pi$
$347$	−35.0974	−1.88412	−0.942062	+	0.335438i	$0.891116\pi$
$348$	0	0
$349$	18.7628	1.00435	0.502174	−	0.864767i	$-0.332534\pi$
$349$	18.7628	1.00435	0.502174	+	0.864767i	$0.332534\pi$
$350$	0	0
$351$	−26.9544	−1.43872
$352$	0	0
$353$	25.8782	1.37736	0.688678	−	0.725067i	$-0.258191\pi$
$353$	25.8782	1.37736	0.688678	+	0.725067i	$0.258191\pi$
$354$	0	0
$355$	−4.04102	−0.214475
$356$	0	0
$357$	−6.38293	−0.337820
$358$	0	0
$359$	3.89257	0.205442	0.102721	−	0.994710i	$-0.467245\pi$
$359$	3.89257	0.205442	0.102721	+	0.994710i	$0.467245\pi$
$360$	0	0
$361$	−12.7117	−0.669036
$362$	0	0
$363$	−15.7167	−0.824911
$364$	0	0
$365$	15.0563	0.788084
$366$	0	0
$367$	−5.76293	−0.300822	−0.150411	−	0.988624i	$-0.548060\pi$
$367$	−5.76293	−0.300822	−0.150411	+	0.988624i	$0.548060\pi$
$368$	0	0
$369$	42.7104	2.22342
$370$	0	0
$371$	25.9801	1.34882
$372$	0	0
$373$	5.42361	0.280824	0.140412	−	0.990093i	$-0.455157\pi$
$373$	5.42361	0.280824	0.140412	+	0.990093i	$0.455157\pi$
$374$	0	0
$375$	2.84956	0.147151
$376$	0	0
$377$	−34.3526	−1.76925
$378$	0	0
$379$	31.8277	1.63488	0.817439	−	0.576015i	$-0.195393\pi$
$379$	31.8277	1.63488	0.817439	+	0.576015i	$0.195393\pi$
$380$	0	0
$381$	−31.6003	−1.61893
$382$	0	0
$383$	−32.4190	−1.65654	−0.828268	−	0.560333i	$-0.810673\pi$
$383$	−32.4190	−1.65654	−0.828268	+	0.560333i	$0.810673\pi$
$384$	0	0
$385$	−11.3572	−0.578817
$386$	0	0
$387$	−61.1280	−3.10731
$388$	0	0
$389$	−3.79064	−0.192193	−0.0960965	−	0.995372i	$-0.530636\pi$
$389$	−3.79064	−0.192193	−0.0960965	+	0.995372i	$0.530636\pi$
$390$	0	0
$391$	3.55616	0.179843
$392$	0	0
$393$	−32.1515	−1.62183
$394$	0	0
$395$	8.50765	0.428066
$396$	0	0
$397$	2.64279	0.132638	0.0663189	−	0.997798i	$-0.478875\pi$
$397$	2.64279	0.132638	0.0663189	+	0.997798i	$0.478875\pi$
$398$	0	0
$399$	34.6535	1.73485
$400$	0	0
$401$	3.36762	0.168171	0.0840856	−	0.996459i	$-0.473203\pi$
$401$	3.36762	0.168171	0.0840856	+	0.996459i	$0.473203\pi$
$402$	0	0
$403$	−42.0820	−2.09626
$404$	0	0
$405$	1.85430	0.0921407
$406$	0	0
$407$	−2.34191	−0.116084
$408$	0	0
$409$	0.300882	0.0148777	0.00743884	−	0.999972i	$-0.497632\pi$
$409$	0.300882	0.0148777	0.00743884	+	0.999972i	$0.497632\pi$
$410$	0	0
$411$	−32.6535	−1.61068
$412$	0	0
$413$	18.1001	0.890648
$414$	0	0
$415$	−15.0895	−0.740716
$416$	0	0
$417$	1.26475	0.0619351
$418$	0	0
$419$	−15.8926	−0.776403	−0.388202	−	0.921575i	$-0.626904\pi$
$419$	−15.8926	−0.776403	−0.388202	+	0.921575i	$0.626904\pi$
$420$	0	0
$421$	−3.31070	−0.161354	−0.0806768	−	0.996740i	$-0.525708\pi$
$421$	−3.31070	−0.161354	−0.0806768	+	0.996740i	$0.525708\pi$
$422$	0	0
$423$	−0.380029	−0.0184777
$424$	0	0
$425$	−0.461892	−0.0224050
$426$	0	0
$427$	9.69912	0.469373
$428$	0	0
$429$	29.7760	1.43760
$430$	0	0
$431$	32.6258	1.57153	0.785765	−	0.618525i	$-0.212270\pi$
$431$	32.6258	1.57153	0.785765	+	0.618525i	$0.212270\pi$
$432$	0	0
$433$	−19.5002	−0.937118	−0.468559	−	0.883432i	$-0.655227\pi$
$433$	−19.5002	−0.937118	−0.468559	+	0.883432i	$0.655227\pi$
$434$	0	0
$435$	21.9391	1.05190
$436$	0	0
$437$	−19.3067	−0.923565
$438$	0	0
$439$	−12.8906	−0.615234	−0.307617	−	0.951510i	$-0.599532\pi$
$439$	−12.8906	−0.615234	−0.307617	+	0.951510i	$0.599532\pi$
$440$	0	0
$441$	84.5731	4.02729
$442$	0	0
$443$	21.5031	1.02164	0.510821	−	0.859687i	$-0.329342\pi$
$443$	21.5031	1.02164	0.510821	+	0.859687i	$0.329342\pi$
$444$	0	0
$445$	−6.92378	−0.328219
$446$	0	0
$447$	48.6545	2.30128
$448$	0	0
$449$	12.3829	0.584387	0.292193	−	0.956359i	$-0.405615\pi$
$449$	12.3829	0.584387	0.292193	+	0.956359i	$0.405615\pi$
$450$	0	0
$451$	−19.5360	−0.919912
$452$	0	0
$453$	39.7203	1.86622
$454$	0	0
$455$	−21.6382	−1.01441
$456$	0	0
$457$	19.4591	0.910260	0.455130	−	0.890425i	$-0.349593\pi$
$457$	19.4591	0.910260	0.455130	+	0.890425i	$0.349593\pi$
$458$	0	0
$459$	−2.79030	−0.130240
$460$	0	0
$461$	−26.8574	−1.25087	−0.625436	−	0.780275i	$-0.715079\pi$
$461$	−26.8574	−1.25087	−0.625436	+	0.780275i	$0.715079\pi$
$462$	0	0
$463$	16.5409	0.768719	0.384359	−	0.923184i	$-0.374422\pi$
$463$	16.5409	0.768719	0.384359	+	0.923184i	$0.374422\pi$
$464$	0	0
$465$	26.8754	1.24632
$466$	0	0
$467$	−6.00000	−0.277647	−0.138823	−	0.990317i	$-0.544332\pi$
$467$	−6.00000	−0.277647	−0.138823	+	0.990317i	$0.544332\pi$
$468$	0	0
$469$	5.77808	0.266807
$470$	0	0
$471$	−0.974282	−0.0448925
$472$	0	0
$473$	27.9602	1.28561
$474$	0	0
$475$	2.50765	0.115059
$476$	0	0
$477$	27.4288	1.25588
$478$	0	0
$479$	17.8144	0.813959	0.406980	−	0.913437i	$-0.366582\pi$
$479$	17.8144	0.813959	0.406980	+	0.913437i	$0.366582\pi$
$480$	0	0
$481$	−4.46189	−0.203445
$482$	0	0
$483$	−106.395	−4.84114
$484$	0	0
$485$	−7.93909	−0.360495
$486$	0	0
$487$	31.1638	1.41216	0.706082	−	0.708130i	$-0.250461\pi$
$487$	31.1638	1.41216	0.706082	+	0.708130i	$0.250461\pi$
$488$	0	0
$489$	−21.6783	−0.980326
$490$	0	0
$491$	0.923784	0.0416898	0.0208449	−	0.999783i	$-0.493364\pi$
$491$	0.923784	0.0416898	0.0208449	+	0.999783i	$0.493364\pi$
$492$	0	0
$493$	−3.55616	−0.160161
$494$	0	0
$495$	−11.9905	−0.538934
$496$	0	0
$497$	−19.5972	−0.879054
$498$	0	0
$499$	25.4566	1.13959	0.569796	−	0.821786i	$-0.307022\pi$
$499$	25.4566	1.13959	0.569796	+	0.821786i	$0.307022\pi$
$500$	0	0
$501$	14.4650	0.646248
$502$	0	0
$503$	37.5718	1.67524	0.837622	−	0.546250i	$-0.183945\pi$
$503$	37.5718	1.67524	0.837622	+	0.546250i	$0.183945\pi$
$504$	0	0
$505$	−4.35447	−0.193771
$506$	0	0
$507$	19.6861	0.874291
$508$	0	0
$509$	−6.28100	−0.278400	−0.139200	−	0.990264i	$-0.544453\pi$
$509$	−6.28100	−0.278400	−0.139200	+	0.990264i	$0.544453\pi$
$510$	0	0
$511$	73.0166	3.23006
$512$	0	0
$513$	15.1488	0.668835
$514$	0	0
$515$	−2.00000	−0.0881305
$516$	0	0
$517$	0.173827	0.00764491
$518$	0	0
$519$	−3.37286	−0.148052
$520$	0	0
$521$	2.59993	0.113905	0.0569525	−	0.998377i	$-0.481862\pi$
$521$	2.59993	0.113905	0.0569525	+	0.998377i	$0.481862\pi$
$522$	0	0
$523$	−16.0609	−0.702295	−0.351147	−	0.936320i	$-0.614208\pi$
$523$	−16.0609	−0.702295	−0.351147	+	0.936320i	$0.614208\pi$
$524$	0	0
$525$	13.8191	0.603115
$526$	0	0
$527$	−4.35630	−0.189764
$528$	0	0
$529$	36.2764	1.57724
$530$	0	0
$531$	19.1094	0.829278
$532$	0	0
$533$	−37.2207	−1.61221
$534$	0	0
$535$	−8.44674	−0.365184
$536$	0	0
$537$	−39.6256	−1.70997
$538$	0	0
$539$	−38.6841	−1.66624
$540$	0	0
$541$	30.8935	1.32821	0.664107	−	0.747637i	$-0.268812\pi$
$541$	30.8935	1.32821	0.664107	+	0.747637i	$0.268812\pi$
$542$	0	0
$543$	35.2048	1.51078
$544$	0	0
$545$	−16.9544	−0.726246
$546$	0	0
$547$	−2.14845	−0.0918611	−0.0459305	−	0.998945i	$-0.514625\pi$
$547$	−2.14845	−0.0918611	−0.0459305	+	0.998945i	$0.514625\pi$
$548$	0	0
$549$	10.2400	0.437031
$550$	0	0
$551$	19.3067	0.822494
$552$	0	0
$553$	41.2584	1.75448
$554$	0	0
$555$	2.84956	0.120957
$556$	0	0
$557$	−11.3802	−0.482194	−0.241097	−	0.970501i	$-0.577507\pi$
$557$	−11.3802	−0.482194	−0.241097	+	0.970501i	$0.577507\pi$
$558$	0	0
$559$	53.2709	2.25312
$560$	0	0
$561$	3.08239	0.130139
$562$	0	0
$563$	−17.2859	−0.728513	−0.364257	−	0.931299i	$-0.618677\pi$
$563$	−17.2859	−0.728513	−0.364257	+	0.931299i	$0.618677\pi$
$564$	0	0
$565$	−7.77808	−0.327226
$566$	0	0
$567$	8.99252	0.377650
$568$	0	0
$569$	17.4898	0.733209	0.366604	−	0.930377i	$-0.380520\pi$
$569$	17.4898	0.733209	0.366604	+	0.930377i	$0.380520\pi$
$570$	0	0
$571$	−16.7248	−0.699913	−0.349956	−	0.936766i	$-0.613804\pi$
$571$	−16.7248	−0.699913	−0.349956	+	0.936766i	$0.613804\pi$
$572$	0	0
$573$	−35.8751	−1.49870
$574$	0	0
$575$	−7.69912	−0.321075
$576$	0	0
$577$	−25.8907	−1.07785	−0.538923	−	0.842355i	$-0.681169\pi$
$577$	−25.8907	−1.07785	−0.538923	+	0.842355i	$0.681169\pi$
$578$	0	0
$579$	−24.8323	−1.03199
$580$	0	0
$581$	−73.1776	−3.03592
$582$	0	0
$583$	−12.5461	−0.519606
$584$	0	0
$585$	−22.8448	−0.944517
$586$	0	0
$587$	−24.3829	−1.00639	−0.503196	−	0.864173i	$-0.667842\pi$
$587$	−24.3829	−1.00639	−0.503196	+	0.864173i	$0.667842\pi$
$588$	0	0
$589$	23.6508	0.974513
$590$	0	0
$591$	−70.2813	−2.89099
$592$	0	0
$593$	17.9192	0.735853	0.367927	−	0.929855i	$-0.380068\pi$
$593$	17.9192	0.735853	0.367927	+	0.929855i	$0.380068\pi$
$594$	0	0
$595$	−2.23997	−0.0918299
$596$	0	0
$597$	17.8601	0.730966
$598$	0	0
$599$	−27.6887	−1.13133	−0.565665	−	0.824635i	$-0.691380\pi$
$599$	−27.6887	−1.13133	−0.565665	+	0.824635i	$0.691380\pi$
$600$	0	0
$601$	−33.7172	−1.37535	−0.687676	−	0.726018i	$-0.741369\pi$
$601$	−33.7172	−1.37535	−0.687676	+	0.726018i	$0.741369\pi$
$602$	0	0
$603$	6.10028	0.248423
$604$	0	0
$605$	−5.51548	−0.224236
$606$	0	0
$607$	18.6535	0.757123	0.378561	−	0.925576i	$-0.376419\pi$
$607$	18.6535	0.757123	0.378561	+	0.925576i	$0.376419\pi$
$608$	0	0
$609$	106.395	4.31134
$610$	0	0
$611$	0.331182	0.0133982
$612$	0	0
$613$	20.5155	0.828612	0.414306	−	0.910138i	$-0.364024\pi$
$613$	20.5155	0.828612	0.414306	+	0.910138i	$0.364024\pi$
$614$	0	0
$615$	23.7708	0.958529
$616$	0	0
$617$	28.7915	1.15910	0.579552	−	0.814935i	$-0.303228\pi$
$617$	28.7915	1.15910	0.579552	+	0.814935i	$0.303228\pi$
$618$	0	0
$619$	48.3630	1.94387	0.971937	−	0.235240i	$-0.0755875\pi$
$619$	48.3630	1.94387	0.971937	+	0.235240i	$0.0755875\pi$
$620$	0	0
$621$	−46.5106	−1.86640
$622$	0	0
$623$	−33.5773	−1.34525
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−16.7346	−0.668314
$628$	0	0
$629$	−0.461892	−0.0184168
$630$	0	0
$631$	−28.1404	−1.12025	−0.560125	−	0.828408i	$-0.689247\pi$
$631$	−28.1404	−1.12025	−0.560125	+	0.828408i	$0.689247\pi$
$632$	0	0
$633$	−62.2106	−2.47265
$634$	0	0
$635$	−11.0895	−0.440075
$636$	0	0
$637$	−73.7025	−2.92020
$638$	0	0
$639$	−20.6900	−0.818483
$640$	0	0
$641$	−13.7221	−0.541990	−0.270995	−	0.962581i	$-0.587353\pi$
$641$	−13.7221	−0.541990	−0.270995	+	0.962581i	$0.587353\pi$
$642$	0	0
$643$	10.7199	0.422752	0.211376	−	0.977405i	$-0.432206\pi$
$643$	10.7199	0.422752	0.211376	+	0.977405i	$0.432206\pi$
$644$	0	0
$645$	−34.0211	−1.33958
$646$	0	0
$647$	23.3012	0.916066	0.458033	−	0.888935i	$-0.348554\pi$
$647$	23.3012	0.916066	0.458033	+	0.888935i	$0.348554\pi$
$648$	0	0
$649$	−8.74074	−0.343104
$650$	0	0
$651$	130.334	5.10819
$652$	0	0
$653$	18.7659	0.734365	0.367182	−	0.930149i	$-0.380322\pi$
$653$	18.7659	0.734365	0.367182	+	0.930149i	$0.380322\pi$
$654$	0	0
$655$	−11.2830	−0.440863
$656$	0	0
$657$	77.0882	3.00750
$658$	0	0
$659$	−13.4086	−0.522327	−0.261163	−	0.965295i	$-0.584106\pi$
$659$	−13.4086	−0.522327	−0.261163	+	0.965295i	$0.584106\pi$
$660$	0	0
$661$	26.3581	1.02521	0.512606	−	0.858624i	$-0.328680\pi$
$661$	26.3581	1.02521	0.512606	+	0.858624i	$0.328680\pi$
$662$	0	0
$663$	5.87269	0.228076
$664$	0	0
$665$	12.1610	0.471584
$666$	0	0
$667$	−59.2764	−2.29519
$668$	0	0
$669$	7.43617	0.287499
$670$	0	0
$671$	−4.68381	−0.180817
$672$	0	0
$673$	15.5666	0.600047	0.300024	−	0.953932i	$-0.403005\pi$
$673$	15.5666	0.600047	0.300024	+	0.953932i	$0.403005\pi$
$674$	0	0
$675$	6.04102	0.232519
$676$	0	0
$677$	40.2201	1.54578	0.772892	−	0.634538i	$-0.218810\pi$
$677$	40.2201	1.54578	0.772892	+	0.634538i	$0.218810\pi$
$678$	0	0
$679$	−38.5011	−1.47754
$680$	0	0
$681$	3.75055	0.143722
$682$	0	0
$683$	42.2097	1.61511	0.807554	−	0.589794i	$-0.200791\pi$
$683$	42.2097	1.61511	0.807554	+	0.589794i	$0.200791\pi$
$684$	0	0
$685$	−11.4591	−0.437831
$686$	0	0
$687$	56.2507	2.14610
$688$	0	0
$689$	−23.9033	−0.910643
$690$	0	0
$691$	−7.78666	−0.296218	−0.148109	−	0.988971i	$-0.547319\pi$
$691$	−7.78666	−0.296218	−0.148109	+	0.988971i	$0.547319\pi$
$692$	0	0
$693$	−58.1488	−2.20889
$694$	0	0
$695$	0.443841	0.0168358
$696$	0	0
$697$	−3.85306	−0.145945
$698$	0	0
$699$	−49.8381	−1.88505
$700$	0	0
$701$	13.4552	0.508194	0.254097	−	0.967179i	$-0.418222\pi$
$701$	13.4552	0.508194	0.254097	+	0.967179i	$0.418222\pi$
$702$	0	0
$703$	2.50765	0.0945779
$704$	0	0
$705$	−0.211508	−0.00796584
$706$	0	0
$707$	−21.1172	−0.794196
$708$	0	0
$709$	−34.7753	−1.30602	−0.653008	−	0.757351i	$-0.726493\pi$
$709$	−34.7753	−1.30602	−0.653008	+	0.757351i	$0.726493\pi$
$710$	0	0
$711$	43.5591	1.63359
$712$	0	0
$713$	−72.6137	−2.71941
$714$	0	0
$715$	10.4493	0.390783
$716$	0	0
$717$	−45.9213	−1.71496
$718$	0	0
$719$	−7.38751	−0.275508	−0.137754	−	0.990466i	$-0.543988\pi$
$719$	−7.38751	−0.275508	−0.137754	+	0.990466i	$0.543988\pi$
$720$	0	0
$721$	−9.69912	−0.361214
$722$	0	0
$723$	42.1032	1.56583
$724$	0	0
$725$	7.69912	0.285938
$726$	0	0
$727$	14.8323	0.550098	0.275049	−	0.961430i	$-0.411306\pi$
$727$	14.8323	0.550098	0.275049	+	0.961430i	$0.411306\pi$
$728$	0	0
$729$	−42.1488	−1.56107
$730$	0	0
$731$	5.51457	0.203964
$732$	0	0
$733$	−7.35172	−0.271542	−0.135771	−	0.990740i	$-0.543351\pi$
$733$	−7.35172	−0.271542	−0.135771	+	0.990740i	$0.543351\pi$
$734$	0	0
$735$	47.0696	1.73619
$736$	0	0
$737$	−2.79030	−0.102782
$738$	0	0
$739$	−23.3823	−0.860133	−0.430066	−	0.902797i	$-0.641510\pi$
$739$	−23.3823	−0.860133	−0.430066	+	0.902797i	$0.641510\pi$
$740$	0	0
$741$	−31.8834	−1.17126
$742$	0	0
$743$	7.24196	0.265682	0.132841	−	0.991137i	$-0.457590\pi$
$743$	7.24196	0.265682	0.132841	+	0.991137i	$0.457590\pi$
$744$	0	0
$745$	17.0744	0.625557
$746$	0	0
$747$	−77.2582	−2.82673
$748$	0	0
$749$	−40.9630	−1.49675
$750$	0	0
$751$	−0.551266	−0.0201160	−0.0100580	−	0.999949i	$-0.503202\pi$
$751$	−0.551266	−0.0201160	−0.0100580	+	0.999949i	$0.503202\pi$
$752$	0	0
$753$	−14.3860	−0.524256
$754$	0	0
$755$	13.9391	0.507295
$756$	0	0
$757$	−31.9908	−1.16273	−0.581363	−	0.813644i	$-0.697481\pi$
$757$	−31.9908	−1.16273	−0.581363	+	0.813644i	$0.697481\pi$
$758$	0	0
$759$	51.3793	1.86495
$760$	0	0
$761$	10.6679	0.386711	0.193356	−	0.981129i	$-0.438063\pi$
$761$	10.6679	0.386711	0.193356	+	0.981129i	$0.438063\pi$
$762$	0	0
$763$	−82.2213	−2.97661
$764$	0	0
$765$	−2.36488	−0.0855024
$766$	0	0
$767$	−16.6532	−0.601312
$768$	0	0
$769$	−16.5314	−0.596137	−0.298068	−	0.954545i	$-0.596342\pi$
$769$	−16.5314	−0.596137	−0.298068	+	0.954545i	$0.596342\pi$
$770$	0	0
$771$	29.2152	1.05216
$772$	0	0
$773$	45.8889	1.65051	0.825255	−	0.564761i	$-0.191032\pi$
$773$	45.8889	1.65051	0.825255	+	0.564761i	$0.191032\pi$
$774$	0	0
$775$	9.43144	0.338787
$776$	0	0
$777$	13.8191	0.495758
$778$	0	0
$779$	20.9186	0.749487
$780$	0	0
$781$	9.46370	0.338638
$782$	0	0
$783$	46.5106	1.66215
$784$	0	0
$785$	−0.341906	−0.0122032
$786$	0	0
$787$	39.5545	1.40997	0.704983	−	0.709224i	$-0.250955\pi$
$787$	39.5545	1.40997	0.704983	+	0.709224i	$0.250955\pi$
$788$	0	0
$789$	27.4423	0.976973
$790$	0	0
$791$	−37.7203	−1.34118
$792$	0	0
$793$	−8.92378	−0.316893
$794$	0	0
$795$	15.2657	0.541418
$796$	0	0
$797$	−51.1365	−1.81135	−0.905675	−	0.423973i	$-0.860635\pi$
$797$	−51.1365	−1.81135	−0.905675	+	0.423973i	$0.860635\pi$
$798$	0	0
$799$	0.0342838	0.00121287
$800$	0	0
$801$	−35.4497	−1.25255
$802$	0	0
$803$	−35.2605	−1.24432
$804$	0	0
$805$	−37.3373	−1.31597
$806$	0	0
$807$	84.2308	2.96506
$808$	0	0
$809$	10.6838	0.375623	0.187811	−	0.982205i	$-0.439861\pi$
$809$	10.6838	0.375623	0.187811	+	0.982205i	$0.439861\pi$
$810$	0	0
$811$	40.8430	1.43419	0.717096	−	0.696975i	$-0.245471\pi$
$811$	40.8430	1.43419	0.717096	+	0.696975i	$0.245471\pi$
$812$	0	0
$813$	−37.3940	−1.31146
$814$	0	0
$815$	−7.60760	−0.266482
$816$	0	0
$817$	−29.9391	−1.04744
$818$	0	0
$819$	−110.787	−3.87122
$820$	0	0
$821$	−27.6486	−0.964943	−0.482472	−	0.875912i	$-0.660261\pi$
$821$	−27.6486	−0.964943	−0.482472	+	0.875912i	$0.660261\pi$
$822$	0	0
$823$	−43.2181	−1.50649	−0.753244	−	0.657741i	$-0.771512\pi$
$823$	−43.2181	−1.50649	−0.753244	+	0.657741i	$0.771512\pi$
$824$	0	0
$825$	−6.67340	−0.232338
$826$	0	0
$827$	−24.2345	−0.842715	−0.421358	−	0.906895i	$-0.638446\pi$
$827$	−24.2345	−0.842715	−0.421358	+	0.906895i	$0.638446\pi$
$828$	0	0
$829$	4.05144	0.140712	0.0703561	−	0.997522i	$-0.477586\pi$
$829$	4.05144	0.140712	0.0703561	+	0.997522i	$0.477586\pi$
$830$	0	0
$831$	3.71476	0.128864
$832$	0	0
$833$	−7.62963	−0.264351
$834$	0	0
$835$	5.07622	0.175670
$836$	0	0
$837$	56.9755	1.96936
$838$	0	0
$839$	−50.6746	−1.74948	−0.874742	−	0.484590i	$-0.838969\pi$
$839$	−50.6746	−1.74948	−0.874742	+	0.484590i	$0.838969\pi$
$840$	0	0
$841$	30.2764	1.04401
$842$	0	0
$843$	11.2247	0.386598
$844$	0	0
$845$	6.90848	0.237659
$846$	0	0
$847$	−26.7476	−0.919059
$848$	0	0
$849$	−26.5470	−0.911092
$850$	0	0
$851$	−7.69912	−0.263922
$852$	0	0
$853$	−18.5923	−0.636588	−0.318294	−	0.947992i	$-0.603110\pi$
$853$	−18.5923	−0.636588	−0.318294	+	0.947992i	$0.603110\pi$
$854$	0	0
$855$	12.8391	0.439090
$856$	0	0
$857$	7.33732	0.250638	0.125319	−	0.992116i	$-0.460005\pi$
$857$	7.33732	0.250638	0.125319	+	0.992116i	$0.460005\pi$
$858$	0	0
$859$	−2.55909	−0.0873150	−0.0436575	−	0.999047i	$-0.513901\pi$
$859$	−2.55909	−0.0873150	−0.0436575	+	0.999047i	$0.513901\pi$
$860$	0	0
$861$	115.278	3.92865
$862$	0	0
$863$	−11.1970	−0.381149	−0.190574	−	0.981673i	$-0.561035\pi$
$863$	−11.1970	−0.381149	−0.190574	+	0.981673i	$0.561035\pi$
$864$	0	0
$865$	−1.18364	−0.0402450
$866$	0	0
$867$	−47.8346	−1.62455
$868$	0	0
$869$	−19.9241	−0.675880
$870$	0	0
$871$	−5.31619	−0.180132
$872$	0	0
$873$	−40.6480	−1.37573
$874$	0	0
$875$	4.84956	0.163945
$876$	0	0
$877$	−38.2666	−1.29217	−0.646086	−	0.763265i	$-0.723595\pi$
$877$	−38.2666	−1.29217	−0.646086	+	0.763265i	$0.723595\pi$
$878$	0	0
$879$	66.2403	2.23423
$880$	0	0
$881$	−2.94582	−0.0992471	−0.0496236	−	0.998768i	$-0.515802\pi$
$881$	−2.94582	−0.0992471	−0.0496236	+	0.998768i	$0.515802\pi$
$882$	0	0
$883$	−4.05144	−0.136342	−0.0681709	−	0.997674i	$-0.521716\pi$
$883$	−4.05144	−0.136342	−0.0681709	+	0.997674i	$0.521716\pi$
$884$	0	0
$885$	10.6355	0.357507
$886$	0	0
$887$	−24.1260	−0.810071	−0.405035	−	0.914301i	$-0.632741\pi$
$887$	−24.1260	−0.810071	−0.405035	+	0.914301i	$0.632741\pi$
$888$	0	0
$889$	−53.7793	−1.80370
$890$	0	0
$891$	−4.34259	−0.145482
$892$	0	0
$893$	−0.186130	−0.00622859
$894$	0	0
$895$	−13.9059	−0.464823
$896$	0	0
$897$	97.8898	3.26845
$898$	0	0
$899$	72.6137	2.42180
$900$	0	0
$901$	−2.47445	−0.0824359
$902$	0	0
$903$	−164.988	−5.49044
$904$	0	0
$905$	12.3545	0.410676
$906$	0	0
$907$	−3.08171	−0.102326	−0.0511632	−	0.998690i	$-0.516293\pi$
$907$	−3.08171	−0.102326	−0.0511632	+	0.998690i	$0.516293\pi$
$908$	0	0
$909$	−22.2948	−0.739472
$910$	0	0
$911$	37.6619	1.24780	0.623898	−	0.781506i	$-0.285548\pi$
$911$	37.6619	1.24780	0.623898	+	0.781506i	$0.285548\pi$
$912$	0	0
$913$	35.3383	1.16953
$914$	0	0
$915$	5.69912	0.188407
$916$	0	0
$917$	−54.7175	−1.80693
$918$	0	0
$919$	−26.4624	−0.872914	−0.436457	−	0.899725i	$-0.643767\pi$
$919$	−26.4624	−0.872914	−0.436457	+	0.899725i	$0.643767\pi$
$920$	0	0
$921$	33.6361	1.10835
$922$	0	0
$923$	18.0306	0.593485
$924$	0	0
$925$	1.00000	0.0328798
$926$	0	0
$927$	−10.2400	−0.336325
$928$	0	0
$929$	36.2250	1.18850	0.594252	−	0.804279i	$-0.297448\pi$
$929$	36.2250	1.18850	0.594252	+	0.804279i	$0.297448\pi$
$930$	0	0
$931$	41.4220	1.35755
$932$	0	0
$933$	−20.7689	−0.679945
$934$	0	0
$935$	1.08171	0.0353756
$936$	0	0
$937$	43.1745	1.41045	0.705224	−	0.708984i	$-0.250846\pi$
$937$	43.1745	1.41045	0.705224	+	0.708984i	$0.250846\pi$
$938$	0	0
$939$	−18.2042	−0.594071
$940$	0	0
$941$	42.4830	1.38491	0.692454	−	0.721462i	$-0.256530\pi$
$941$	42.4830	1.38491	0.692454	+	0.721462i	$0.256530\pi$
$942$	0	0
$943$	−64.2253	−2.09146
$944$	0	0
$945$	29.2963	0.953009
$946$	0	0
$947$	−31.0310	−1.00837	−0.504185	−	0.863596i	$-0.668207\pi$
$947$	−31.0310	−1.00837	−0.504185	+	0.863596i	$0.668207\pi$
$948$	0	0
$949$	−67.1797	−2.18075
$950$	0	0
$951$	−89.2975	−2.89567
$952$	0	0
$953$	−51.4447	−1.66646	−0.833229	−	0.552927i	$-0.813511\pi$
$953$	−51.4447	−1.66646	−0.833229	+	0.552927i	$0.813511\pi$
$954$	0	0
$955$	−12.5897	−0.407393
$956$	0	0
$957$	−51.3793	−1.66086
$958$	0	0
$959$	−55.5718	−1.79451
$960$	0	0
$961$	57.9520	1.86942
$962$	0	0
$963$	−43.2472	−1.39362
$964$	0	0
$965$	−8.71442	−0.280527
$966$	0	0
$967$	−48.1947	−1.54984	−0.774919	−	0.632061i	$-0.782209\pi$
$967$	−48.1947	−1.54984	−0.774919	+	0.632061i	$0.782209\pi$
$968$	0	0
$969$	−3.30054	−0.106029
$970$	0	0
$971$	6.03061	0.193532	0.0967658	−	0.995307i	$-0.469150\pi$
$971$	6.03061	0.193532	0.0967658	+	0.995307i	$0.469150\pi$
$972$	0	0
$973$	2.15243	0.0690038
$974$	0	0
$975$	−12.7144	−0.407187
$976$	0	0
$977$	52.2042	1.67016	0.835080	−	0.550129i	$-0.185421\pi$
$977$	52.2042	1.67016	0.835080	+	0.550129i	$0.185421\pi$
$978$	0	0
$979$	16.2149	0.518229
$980$	0	0
$981$	−86.8063	−2.77151
$982$	0	0
$983$	14.3142	0.456552	0.228276	−	0.973596i	$-0.426691\pi$
$983$	14.3142	0.456552	0.228276	+	0.973596i	$0.426691\pi$
$984$	0	0
$985$	−24.6639	−0.785858
$986$	0	0
$987$	−1.02572	−0.0326490
$988$	0	0
$989$	91.9205	2.92290
$990$	0	0
$991$	11.8848	0.377532	0.188766	−	0.982022i	$-0.439551\pi$
$991$	11.8848	0.377532	0.188766	+	0.982022i	$0.439551\pi$
$992$	0	0
$993$	−45.6875	−1.44985
$994$	0	0
$995$	6.26768	0.198699
$996$	0	0
$997$	51.0310	1.61617	0.808083	−	0.589068i	$-0.200505\pi$
$997$	51.0310	1.61617	0.808083	+	0.589068i	$0.200505\pi$
$998$	0	0
$999$	6.04102	0.191130

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2960.2.a.v.1.4		4
4.3	odd	2		740.2.a.f.1.1	✓	4
12.11	even	2		6660.2.a.r.1.1		4
20.3	even	4		3700.2.d.i.149.2		8
20.7	even	4		3700.2.d.i.149.7		8
20.19	odd	2		3700.2.a.j.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
740.2.a.f.1.1	✓	4	4.3	odd	2
2960.2.a.v.1.4		4	1.1	even	1	trivial
3700.2.a.j.1.4		4	20.19	odd	2
3700.2.d.i.149.2		8	20.3	even	4
3700.2.d.i.149.7		8	20.7	even	4
6660.2.a.r.1.1		4	12.11	even	2

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:	\( N \)	\(=\)	\( 2960 = 2^{4} \cdot 5 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2960.a (trivial)

\(f(q)\)	\(=\)	\(q+2.84956 q^{3} +1.00000 q^{5} +4.84956 q^{7} +5.11999 q^{9} +O(q^{10})\)	\(q+2.84956 q^{3} +1.00000 q^{5} +4.84956 q^{7} +5.11999 q^{9} -2.34191 q^{11} -4.46189 q^{13} +2.84956 q^{15} -0.461892 q^{17} +2.50765 q^{19} +13.8191 q^{21} -7.69912 q^{23} +1.00000 q^{25} +6.04102 q^{27} +7.69912 q^{29} +9.43144 q^{31} -6.67340 q^{33} +4.84956 q^{35} +1.00000 q^{37} -12.7144 q^{39} +8.34191 q^{41} -11.9391 q^{43} +5.11999 q^{45} -0.0742247 q^{47} +16.5182 q^{49} -1.31619 q^{51} +5.35721 q^{53} -2.34191 q^{55} +7.14570 q^{57} +3.73232 q^{59} +2.00000 q^{61} +24.8297 q^{63} -4.46189 q^{65} +1.19146 q^{67} -21.9391 q^{69} -4.04102 q^{71} +15.0563 q^{73} +2.84956 q^{75} -11.3572 q^{77} +8.50765 q^{79} +1.85430 q^{81} -15.0895 q^{83} -0.461892 q^{85} +21.9391 q^{87} -6.92378 q^{89} -21.6382 q^{91} +26.8754 q^{93} +2.50765 q^{95} -7.93909 q^{97} -11.9905 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 3 q^{3} + 4 q^{5} + 5 q^{7} + 13 q^{9}+O(q^{10}) \) 4 * q - 3 * q^3 + 4 * q^5 + 5 * q^7 + 13 * q^9	\( 4 q - 3 q^{3} + 4 q^{5} + 5 q^{7} + 13 q^{9} - 5 q^{11} - 6 q^{13} - 3 q^{15} + 10 q^{17} + 19 q^{21} - 2 q^{23} + 4 q^{25} - 9 q^{27} + 2 q^{29} + 4 q^{31} - 11 q^{33} + 5 q^{35} + 4 q^{37} - 2 q^{39} + 29 q^{41} - 4 q^{43} + 13 q^{45} + 9 q^{47} + q^{49} - 14 q^{51} - 3 q^{53} - 5 q^{55} + 8 q^{57} + 10 q^{59} + 8 q^{61} + 8 q^{63} - 6 q^{65} - 14 q^{67} - 44 q^{69} + 17 q^{71} + 7 q^{73} - 3 q^{75} - 21 q^{77} + 24 q^{79} + 28 q^{81} - 31 q^{83} + 10 q^{85} + 44 q^{87} - 4 q^{89} - 14 q^{91} + 18 q^{93} + 12 q^{97} + 22 q^{99}+O(q^{100}) \) 4 * q - 3 * q^3 + 4 * q^5 + 5 * q^7 + 13 * q^9 - 5 * q^11 - 6 * q^13 - 3 * q^15 + 10 * q^17 + 19 * q^21 - 2 * q^23 + 4 * q^25 - 9 * q^27 + 2 * q^29 + 4 * q^31 - 11 * q^33 + 5 * q^35 + 4 * q^37 - 2 * q^39 + 29 * q^41 - 4 * q^43 + 13 * q^45 + 9 * q^47 + q^49 - 14 * q^51 - 3 * q^53 - 5 * q^55 + 8 * q^57 + 10 * q^59 + 8 * q^61 + 8 * q^63 - 6 * q^65 - 14 * q^67 - 44 * q^69 + 17 * q^71 + 7 * q^73 - 3 * q^75 - 21 * q^77 + 24 * q^79 + 28 * q^81 - 31 * q^83 + 10 * q^85 + 44 * q^87 - 4 * q^89 - 14 * q^91 + 18 * q^93 + 12 * q^97 + 22 * q^99

Introduction

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