LMFDB - Embedded newform 2960.2.a.q.1.2

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:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 370)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.73205$ 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)

$f(q)$	$=$	$q+2.73205 q^{3} +1.00000 q^{5} +1.26795 q^{7} +4.46410 q^{9} +O(q^{10})$	$q+2.73205 q^{3} +1.00000 q^{5} +1.26795 q^{7} +4.46410 q^{9} -1.46410 q^{11} +1.46410 q^{13} +2.73205 q^{15} -1.46410 q^{17} +4.19615 q^{19} +3.46410 q^{21} +8.00000 q^{23} +1.00000 q^{25} +4.00000 q^{27} -8.92820 q^{29} +2.73205 q^{31} -4.00000 q^{33} +1.26795 q^{35} +1.00000 q^{37} +4.00000 q^{39} -2.00000 q^{41} +6.92820 q^{43} +4.46410 q^{45} +1.26795 q^{47} -5.39230 q^{49} -4.00000 q^{51} -6.00000 q^{53} -1.46410 q^{55} +11.4641 q^{57} -0.196152 q^{59} +8.92820 q^{61} +5.66025 q^{63} +1.46410 q^{65} -13.6603 q^{67} +21.8564 q^{69} +10.9282 q^{71} +12.9282 q^{73} +2.73205 q^{75} -1.85641 q^{77} -5.26795 q^{79} -2.46410 q^{81} +5.26795 q^{83} -1.46410 q^{85} -24.3923 q^{87} -2.00000 q^{89} +1.85641 q^{91} +7.46410 q^{93} +4.19615 q^{95} -2.00000 q^{97} -6.53590 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} + 2 q^{5} + 6 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^3 + 2 * q^5 + 6 * q^7 + 2 * q^9	$ 2 q + 2 q^{3} + 2 q^{5} + 6 q^{7} + 2 q^{9} + 4 q^{11} - 4 q^{13} + 2 q^{15} + 4 q^{17} - 2 q^{19} + 16 q^{23} + 2 q^{25} + 8 q^{27} - 4 q^{29} + 2 q^{31} - 8 q^{33} + 6 q^{35} + 2 q^{37} + 8 q^{39} - 4 q^{41} + 2 q^{45} + 6 q^{47} + 10 q^{49} - 8 q^{51} - 12 q^{53} + 4 q^{55} + 16 q^{57} + 10 q^{59} + 4 q^{61} - 6 q^{63} - 4 q^{65} - 10 q^{67} + 16 q^{69} + 8 q^{71} + 12 q^{73} + 2 q^{75} + 24 q^{77} - 14 q^{79} + 2 q^{81} + 14 q^{83} + 4 q^{85} - 28 q^{87} - 4 q^{89} - 24 q^{91} + 8 q^{93} - 2 q^{95} - 4 q^{97} - 20 q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 + 2 * q^5 + 6 * q^7 + 2 * q^9 + 4 * q^11 - 4 * q^13 + 2 * q^15 + 4 * q^17 - 2 * q^19 + 16 * q^23 + 2 * q^25 + 8 * q^27 - 4 * q^29 + 2 * q^31 - 8 * q^33 + 6 * q^35 + 2 * q^37 + 8 * q^39 - 4 * q^41 + 2 * q^45 + 6 * q^47 + 10 * q^49 - 8 * q^51 - 12 * q^53 + 4 * q^55 + 16 * q^57 + 10 * q^59 + 4 * q^61 - 6 * q^63 - 4 * q^65 - 10 * q^67 + 16 * q^69 + 8 * q^71 + 12 * q^73 + 2 * q^75 + 24 * q^77 - 14 * q^79 + 2 * q^81 + 14 * q^83 + 4 * q^85 - 28 * q^87 - 4 * q^89 - 24 * q^91 + 8 * q^93 - 2 * q^95 - 4 * q^97 - 20 * 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
$3$	2.73205	1.57735	0.788675	−	0.614810i	$-0.210767\pi$
$3$	2.73205	1.57735	0.788675	+	0.614810i	$0.210767\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	1.26795	0.479240	0.239620	−	0.970867i	$-0.422977\pi$
$7$	1.26795	0.479240	0.239620	+	0.970867i	$0.422977\pi$
$8$	0	0
$9$	4.46410	1.48803
$10$	0	0
$11$	−1.46410	−0.441443	−0.220722	−	0.975337i	$-0.570841\pi$
$11$	−1.46410	−0.441443	−0.220722	+	0.975337i	$0.570841\pi$
$12$	0	0
$13$	1.46410	0.406069	0.203034	−	0.979172i	$-0.434920\pi$
$13$	1.46410	0.406069	0.203034	+	0.979172i	$0.434920\pi$
$14$	0	0
$15$	2.73205	0.705412
$16$	0	0
$17$	−1.46410	−0.355097	−0.177548	−	0.984112i	$-0.556817\pi$
$17$	−1.46410	−0.355097	−0.177548	+	0.984112i	$0.556817\pi$
$18$	0	0
$19$	4.19615	0.962663	0.481332	−	0.876539i	$-0.340153\pi$
$19$	4.19615	0.962663	0.481332	+	0.876539i	$0.340153\pi$
$20$	0	0
$21$	3.46410	0.755929
$22$	0	0
$23$	8.00000	1.66812	0.834058	−	0.551677i	$-0.186012\pi$
$23$	8.00000	1.66812	0.834058	+	0.551677i	$0.186012\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	4.00000	0.769800
$28$	0	0
$29$	−8.92820	−1.65793	−0.828963	−	0.559304i	$-0.811069\pi$
$29$	−8.92820	−1.65793	−0.828963	+	0.559304i	$0.811069\pi$
$30$	0	0
$31$	2.73205	0.490691	0.245345	−	0.969436i	$-0.421099\pi$
$31$	2.73205	0.490691	0.245345	+	0.969436i	$0.421099\pi$
$32$	0	0
$33$	−4.00000	−0.696311
$34$	0	0
$35$	1.26795	0.214323
$36$	0	0
$37$	1.00000	0.164399
$37$	1.00000	0.164399
$38$	0	0
$39$	4.00000	0.640513
$40$	0	0
$41$	−2.00000	−0.312348	−0.156174	−	0.987730i	$-0.549916\pi$
$41$	−2.00000	−0.312348	−0.156174	+	0.987730i	$0.549916\pi$
$42$	0	0
$43$	6.92820	1.05654	0.528271	−	0.849076i	$-0.322841\pi$
$43$	6.92820	1.05654	0.528271	+	0.849076i	$0.322841\pi$
$44$	0	0
$45$	4.46410	0.665469
$46$	0	0
$47$	1.26795	0.184949	0.0924747	−	0.995715i	$-0.470522\pi$
$47$	1.26795	0.184949	0.0924747	+	0.995715i	$0.470522\pi$
$48$	0	0
$49$	−5.39230	−0.770329
$50$	0	0
$51$	−4.00000	−0.560112
$52$	0	0
$53$	−6.00000	−0.824163	−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000	−0.824163	−0.412082	+	0.911147i	$0.635198\pi$
$54$	0	0
$55$	−1.46410	−0.197419
$56$	0	0
$57$	11.4641	1.51846
$58$	0	0
$59$	−0.196152	−0.0255369	−0.0127684	−	0.999918i	$-0.504064\pi$
$59$	−0.196152	−0.0255369	−0.0127684	+	0.999918i	$0.504064\pi$
$60$	0	0
$61$	8.92820	1.14314	0.571570	−	0.820554i	$-0.306335\pi$
$61$	8.92820	1.14314	0.571570	+	0.820554i	$0.306335\pi$
$62$	0	0
$63$	5.66025	0.713125
$64$	0	0
$65$	1.46410	0.181599
$66$	0	0
$67$	−13.6603	−1.66887	−0.834433	−	0.551110i	$-0.814205\pi$
$67$	−13.6603	−1.66887	−0.834433	+	0.551110i	$0.814205\pi$
$68$	0	0
$69$	21.8564	2.63120
$70$	0	0
$71$	10.9282	1.29694	0.648470	−	0.761241i	$-0.275409\pi$
$71$	10.9282	1.29694	0.648470	+	0.761241i	$0.275409\pi$
$72$	0	0
$73$	12.9282	1.51313	0.756566	−	0.653917i	$-0.226876\pi$
$73$	12.9282	1.51313	0.756566	+	0.653917i	$0.226876\pi$
$74$	0	0
$75$	2.73205	0.315470
$76$	0	0
$77$	−1.85641	−0.211557
$78$	0	0
$79$	−5.26795	−0.592691	−0.296345	−	0.955081i	$-0.595768\pi$
$79$	−5.26795	−0.592691	−0.296345	+	0.955081i	$0.595768\pi$
$80$	0	0
$81$	−2.46410	−0.273789
$82$	0	0
$83$	5.26795	0.578233	0.289116	−	0.957294i	$-0.406639\pi$
$83$	5.26795	0.578233	0.289116	+	0.957294i	$0.406639\pi$
$84$	0	0
$85$	−1.46410	−0.158804
$86$	0	0
$87$	−24.3923	−2.61513
$88$	0	0
$89$	−2.00000	−0.212000	−0.106000	−	0.994366i	$-0.533804\pi$
$89$	−2.00000	−0.212000	−0.106000	+	0.994366i	$0.533804\pi$
$90$	0	0
$91$	1.85641	0.194604
$92$	0	0
$93$	7.46410	0.773991
$94$	0	0
$95$	4.19615	0.430516
$96$	0	0
$97$	−2.00000	−0.203069	−0.101535	−	0.994832i	$-0.532375\pi$
$97$	−2.00000	−0.203069	−0.101535	+	0.994832i	$0.532375\pi$
$98$	0	0
$99$	−6.53590	−0.656883
$100$	0	0
$101$	−2.53590	−0.252331	−0.126166	−	0.992009i	$-0.540267\pi$
$101$	−2.53590	−0.252331	−0.126166	+	0.992009i	$0.540267\pi$
$102$	0	0
$103$	13.4641	1.32666	0.663329	−	0.748328i	$-0.269143\pi$
$103$	13.4641	1.32666	0.663329	+	0.748328i	$0.269143\pi$
$104$	0	0
$105$	3.46410	0.338062
$106$	0	0
$107$	−6.73205	−0.650812	−0.325406	−	0.945574i	$-0.605501\pi$
$107$	−6.73205	−0.650812	−0.325406	+	0.945574i	$0.605501\pi$
$108$	0	0
$109$	2.00000	0.191565	0.0957826	−	0.995402i	$-0.469465\pi$
$109$	2.00000	0.191565	0.0957826	+	0.995402i	$0.469465\pi$
$110$	0	0
$111$	2.73205	0.259315
$112$	0	0
$113$	−17.4641	−1.64288	−0.821442	−	0.570292i	$-0.806830\pi$
$113$	−17.4641	−1.64288	−0.821442	+	0.570292i	$0.806830\pi$
$114$	0	0
$115$	8.00000	0.746004
$116$	0	0
$117$	6.53590	0.604244
$118$	0	0
$119$	−1.85641	−0.170177
$120$	0	0
$121$	−8.85641	−0.805128
$122$	0	0
$123$	−5.46410	−0.492681
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	13.6603	1.21215	0.606076	−	0.795407i	$-0.292743\pi$
$127$	13.6603	1.21215	0.606076	+	0.795407i	$0.292743\pi$
$128$	0	0
$129$	18.9282	1.66654
$130$	0	0
$131$	−12.5885	−1.09986	−0.549929	−	0.835211i	$-0.685345\pi$
$131$	−12.5885	−1.09986	−0.549929	+	0.835211i	$0.685345\pi$
$132$	0	0
$133$	5.32051	0.461347
$134$	0	0
$135$	4.00000	0.344265
$136$	0	0
$137$	2.00000	0.170872	0.0854358	−	0.996344i	$-0.472772\pi$
$137$	2.00000	0.170872	0.0854358	+	0.996344i	$0.472772\pi$
$138$	0	0
$139$	6.92820	0.587643	0.293821	−	0.955860i	$-0.405073\pi$
$139$	6.92820	0.587643	0.293821	+	0.955860i	$0.405073\pi$
$140$	0	0
$141$	3.46410	0.291730
$142$	0	0
$143$	−2.14359	−0.179256
$144$	0	0
$145$	−8.92820	−0.741447
$146$	0	0
$147$	−14.7321	−1.21508
$148$	0	0
$149$	−16.3923	−1.34291	−0.671455	−	0.741045i	$-0.734330\pi$
$149$	−16.3923	−1.34291	−0.671455	+	0.741045i	$0.734330\pi$
$150$	0	0
$151$	−8.39230	−0.682956	−0.341478	−	0.939890i	$-0.610927\pi$
$151$	−8.39230	−0.682956	−0.341478	+	0.939890i	$0.610927\pi$
$152$	0	0
$153$	−6.53590	−0.528396
$154$	0	0
$155$	2.73205	0.219444
$156$	0	0
$157$	16.9282	1.35102	0.675509	−	0.737352i	$-0.263924\pi$
$157$	16.9282	1.35102	0.675509	+	0.737352i	$0.263924\pi$
$158$	0	0
$159$	−16.3923	−1.29999
$160$	0	0
$161$	10.1436	0.799427
$162$	0	0
$163$	−23.3205	−1.82660	−0.913302	−	0.407284i	$-0.866476\pi$
$163$	−23.3205	−1.82660	−0.913302	+	0.407284i	$0.866476\pi$
$164$	0	0
$165$	−4.00000	−0.311400
$166$	0	0
$167$	−5.46410	−0.422825	−0.211412	−	0.977397i	$-0.567806\pi$
$167$	−5.46410	−0.422825	−0.211412	+	0.977397i	$0.567806\pi$
$168$	0	0
$169$	−10.8564	−0.835108
$170$	0	0
$171$	18.7321	1.43248
$172$	0	0
$173$	10.0000	0.760286	0.380143	−	0.924928i	$-0.375875\pi$
$173$	10.0000	0.760286	0.380143	+	0.924928i	$0.375875\pi$
$174$	0	0
$175$	1.26795	0.0958479
$176$	0	0
$177$	−0.535898	−0.0402806
$178$	0	0
$179$	17.6603	1.31999	0.659995	−	0.751270i	$-0.270559\pi$
$179$	17.6603	1.31999	0.659995	+	0.751270i	$0.270559\pi$
$180$	0	0
$181$	−1.46410	−0.108826	−0.0544129	−	0.998519i	$-0.517329\pi$
$181$	−1.46410	−0.108826	−0.0544129	+	0.998519i	$0.517329\pi$
$182$	0	0
$183$	24.3923	1.80313
$184$	0	0
$185$	1.00000	0.0735215
$186$	0	0
$187$	2.14359	0.156755
$188$	0	0
$189$	5.07180	0.368919
$190$	0	0
$191$	5.26795	0.381175	0.190588	−	0.981670i	$-0.438961\pi$
$191$	5.26795	0.381175	0.190588	+	0.981670i	$0.438961\pi$
$192$	0	0
$193$	−11.8564	−0.853443	−0.426721	−	0.904383i	$-0.640332\pi$
$193$	−11.8564	−0.853443	−0.426721	+	0.904383i	$0.640332\pi$
$194$	0	0
$195$	4.00000	0.286446
$196$	0	0
$197$	18.7846	1.33835	0.669174	−	0.743106i	$-0.266648\pi$
$197$	18.7846	1.33835	0.669174	+	0.743106i	$0.266648\pi$
$198$	0	0
$199$	−26.0526	−1.84682	−0.923408	−	0.383819i	$-0.874609\pi$
$199$	−26.0526	−1.84682	−0.923408	+	0.383819i	$0.874609\pi$
$200$	0	0
$201$	−37.3205	−2.63239
$202$	0	0
$203$	−11.3205	−0.794544
$204$	0	0
$205$	−2.00000	−0.139686
$206$	0	0
$207$	35.7128	2.48221
$208$	0	0
$209$	−6.14359	−0.424961
$210$	0	0
$211$	−9.85641	−0.678543	−0.339272	−	0.940688i	$-0.610181\pi$
$211$	−9.85641	−0.678543	−0.339272	+	0.940688i	$0.610181\pi$
$212$	0	0
$213$	29.8564	2.04573
$214$	0	0
$215$	6.92820	0.472500
$216$	0	0
$217$	3.46410	0.235159
$218$	0	0
$219$	35.3205	2.38674
$220$	0	0
$221$	−2.14359	−0.144194
$222$	0	0
$223$	22.0526	1.47675	0.738374	−	0.674391i	$-0.235594\pi$
$223$	22.0526	1.47675	0.738374	+	0.674391i	$0.235594\pi$
$224$	0	0
$225$	4.46410	0.297607
$226$	0	0
$227$	−3.60770	−0.239451	−0.119726	−	0.992807i	$-0.538201\pi$
$227$	−3.60770	−0.239451	−0.119726	+	0.992807i	$0.538201\pi$
$228$	0	0
$229$	15.8564	1.04782	0.523910	−	0.851773i	$-0.324473\pi$
$229$	15.8564	1.04782	0.523910	+	0.851773i	$0.324473\pi$
$230$	0	0
$231$	−5.07180	−0.333700
$232$	0	0
$233$	15.0718	0.987386	0.493693	−	0.869636i	$-0.335647\pi$
$233$	15.0718	0.987386	0.493693	+	0.869636i	$0.335647\pi$
$234$	0	0
$235$	1.26795	0.0827119
$236$	0	0
$237$	−14.3923	−0.934881
$238$	0	0
$239$	17.2679	1.11697	0.558485	−	0.829514i	$-0.311383\pi$
$239$	17.2679	1.11697	0.558485	+	0.829514i	$0.311383\pi$
$240$	0	0
$241$	−8.92820	−0.575116	−0.287558	−	0.957763i	$-0.592843\pi$
$241$	−8.92820	−0.575116	−0.287558	+	0.957763i	$0.592843\pi$
$242$	0	0
$243$	−18.7321	−1.20166
$244$	0	0
$245$	−5.39230	−0.344502
$246$	0	0
$247$	6.14359	0.390907
$248$	0	0
$249$	14.3923	0.912075
$250$	0	0
$251$	−22.7321	−1.43483	−0.717417	−	0.696644i	$-0.754676\pi$
$251$	−22.7321	−1.43483	−0.717417	+	0.696644i	$0.754676\pi$
$252$	0	0
$253$	−11.7128	−0.736378
$254$	0	0
$255$	−4.00000	−0.250490
$256$	0	0
$257$	−25.4641	−1.58841	−0.794204	−	0.607652i	$-0.792112\pi$
$257$	−25.4641	−1.58841	−0.794204	+	0.607652i	$0.792112\pi$
$258$	0	0
$259$	1.26795	0.0787865
$260$	0	0
$261$	−39.8564	−2.46705
$262$	0	0
$263$	−30.0526	−1.85312	−0.926560	−	0.376147i	$-0.877249\pi$
$263$	−30.0526	−1.85312	−0.926560	+	0.376147i	$0.877249\pi$
$264$	0	0
$265$	−6.00000	−0.368577
$266$	0	0
$267$	−5.46410	−0.334398
$268$	0	0
$269$	−0.392305	−0.0239192	−0.0119596	−	0.999928i	$-0.503807\pi$
$269$	−0.392305	−0.0239192	−0.0119596	+	0.999928i	$0.503807\pi$
$270$	0	0
$271$	−16.7846	−1.01959	−0.509796	−	0.860295i	$-0.670279\pi$
$271$	−16.7846	−1.01959	−0.509796	+	0.860295i	$0.670279\pi$
$272$	0	0
$273$	5.07180	0.306959
$274$	0	0
$275$	−1.46410	−0.0882886
$276$	0	0
$277$	−26.2487	−1.57713	−0.788566	−	0.614950i	$-0.789176\pi$
$277$	−26.2487	−1.57713	−0.788566	+	0.614950i	$0.789176\pi$
$278$	0	0
$279$	12.1962	0.730165
$280$	0	0
$281$	4.92820	0.293992	0.146996	−	0.989137i	$-0.453040\pi$
$281$	4.92820	0.293992	0.146996	+	0.989137i	$0.453040\pi$
$282$	0	0
$283$	4.39230	0.261095	0.130548	−	0.991442i	$-0.458326\pi$
$283$	4.39230	0.261095	0.130548	+	0.991442i	$0.458326\pi$
$284$	0	0
$285$	11.4641	0.679075
$286$	0	0
$287$	−2.53590	−0.149689
$288$	0	0
$289$	−14.8564	−0.873906
$290$	0	0
$291$	−5.46410	−0.320311
$292$	0	0
$293$	6.00000	0.350524	0.175262	−	0.984522i	$-0.443923\pi$
$293$	6.00000	0.350524	0.175262	+	0.984522i	$0.443923\pi$
$294$	0	0
$295$	−0.196152	−0.0114204
$296$	0	0
$297$	−5.85641	−0.339823
$298$	0	0
$299$	11.7128	0.677369
$300$	0	0
$301$	8.78461	0.506336
$302$	0	0
$303$	−6.92820	−0.398015
$304$	0	0
$305$	8.92820	0.511227
$306$	0	0
$307$	12.5885	0.718461	0.359231	−	0.933249i	$-0.383039\pi$
$307$	12.5885	0.718461	0.359231	+	0.933249i	$0.383039\pi$
$308$	0	0
$309$	36.7846	2.09260
$310$	0	0
$311$	−27.1244	−1.53808	−0.769041	−	0.639200i	$-0.779266\pi$
$311$	−27.1244	−1.53808	−0.769041	+	0.639200i	$0.779266\pi$
$312$	0	0
$313$	3.85641	0.217977	0.108988	−	0.994043i	$-0.465239\pi$
$313$	3.85641	0.217977	0.108988	+	0.994043i	$0.465239\pi$
$314$	0	0
$315$	5.66025	0.318919
$316$	0	0
$317$	−31.8564	−1.78923	−0.894617	−	0.446834i	$-0.852552\pi$
$317$	−31.8564	−1.78923	−0.894617	+	0.446834i	$0.852552\pi$
$318$	0	0
$319$	13.0718	0.731880
$320$	0	0
$321$	−18.3923	−1.02656
$322$	0	0
$323$	−6.14359	−0.341839
$324$	0	0
$325$	1.46410	0.0812137
$326$	0	0
$327$	5.46410	0.302166
$328$	0	0
$329$	1.60770	0.0886351
$330$	0	0
$331$	8.87564	0.487850	0.243925	−	0.969794i	$-0.421565\pi$
$331$	8.87564	0.487850	0.243925	+	0.969794i	$0.421565\pi$
$332$	0	0
$333$	4.46410	0.244631
$334$	0	0
$335$	−13.6603	−0.746339
$336$	0	0
$337$	26.0000	1.41631	0.708155	−	0.706057i	$-0.249528\pi$
$337$	26.0000	1.41631	0.708155	+	0.706057i	$0.249528\pi$
$338$	0	0
$339$	−47.7128	−2.59140
$340$	0	0
$341$	−4.00000	−0.216612
$342$	0	0
$343$	−15.7128	−0.848412
$344$	0	0
$345$	21.8564	1.17671
$346$	0	0
$347$	−17.0718	−0.916462	−0.458231	−	0.888833i	$-0.651517\pi$
$347$	−17.0718	−0.916462	−0.458231	+	0.888833i	$0.651517\pi$
$348$	0	0
$349$	19.3205	1.03420	0.517102	−	0.855924i	$-0.327011\pi$
$349$	19.3205	1.03420	0.517102	+	0.855924i	$0.327011\pi$
$350$	0	0
$351$	5.85641	0.312592
$352$	0	0
$353$	15.8564	0.843951	0.421976	−	0.906607i	$-0.361337\pi$
$353$	15.8564	0.843951	0.421976	+	0.906607i	$0.361337\pi$
$354$	0	0
$355$	10.9282	0.580009
$356$	0	0
$357$	−5.07180	−0.268428
$358$	0	0
$359$	−8.39230	−0.442929	−0.221464	−	0.975168i	$-0.571084\pi$
$359$	−8.39230	−0.442929	−0.221464	+	0.975168i	$0.571084\pi$
$360$	0	0
$361$	−1.39230	−0.0732792
$362$	0	0
$363$	−24.1962	−1.26997
$364$	0	0
$365$	12.9282	0.676693
$366$	0	0
$367$	21.6603	1.13066	0.565328	−	0.824866i	$-0.308750\pi$
$367$	21.6603	1.13066	0.565328	+	0.824866i	$0.308750\pi$
$368$	0	0
$369$	−8.92820	−0.464784
$370$	0	0
$371$	−7.60770	−0.394972
$372$	0	0
$373$	30.7846	1.59397	0.796983	−	0.604001i	$-0.206428\pi$
$373$	30.7846	1.59397	0.796983	+	0.604001i	$0.206428\pi$
$374$	0	0
$375$	2.73205	0.141082
$376$	0	0
$377$	−13.0718	−0.673232
$378$	0	0
$379$	−11.6077	−0.596247	−0.298124	−	0.954527i	$-0.596361\pi$
$379$	−11.6077	−0.596247	−0.298124	+	0.954527i	$0.596361\pi$
$380$	0	0
$381$	37.3205	1.91199
$382$	0	0
$383$	8.00000	0.408781	0.204390	−	0.978889i	$-0.434479\pi$
$383$	8.00000	0.408781	0.204390	+	0.978889i	$0.434479\pi$
$384$	0	0
$385$	−1.85641	−0.0946112
$386$	0	0
$387$	30.9282	1.57217
$388$	0	0
$389$	15.8564	0.803952	0.401976	−	0.915650i	$-0.368324\pi$
$389$	15.8564	0.803952	0.401976	+	0.915650i	$0.368324\pi$
$390$	0	0
$391$	−11.7128	−0.592342
$392$	0	0
$393$	−34.3923	−1.73486
$394$	0	0
$395$	−5.26795	−0.265059
$396$	0	0
$397$	−22.0000	−1.10415	−0.552074	−	0.833795i	$-0.686163\pi$
$397$	−22.0000	−1.10415	−0.552074	+	0.833795i	$0.686163\pi$
$398$	0	0
$399$	14.5359	0.727705
$400$	0	0
$401$	−19.0718	−0.952400	−0.476200	−	0.879337i	$-0.657986\pi$
$401$	−19.0718	−0.952400	−0.476200	+	0.879337i	$0.657986\pi$
$402$	0	0
$403$	4.00000	0.199254
$404$	0	0
$405$	−2.46410	−0.122442
$406$	0	0
$407$	−1.46410	−0.0725728
$408$	0	0
$409$	16.9282	0.837046	0.418523	−	0.908206i	$-0.362548\pi$
$409$	16.9282	0.837046	0.418523	+	0.908206i	$0.362548\pi$
$410$	0	0
$411$	5.46410	0.269524
$412$	0	0
$413$	−0.248711	−0.0122383
$414$	0	0
$415$	5.26795	0.258593
$416$	0	0
$417$	18.9282	0.926918
$418$	0	0
$419$	34.2487	1.67316	0.836580	−	0.547846i	$-0.184552\pi$
$419$	34.2487	1.67316	0.836580	+	0.547846i	$0.184552\pi$
$420$	0	0
$421$	−27.8564	−1.35764	−0.678819	−	0.734306i	$-0.737508\pi$
$421$	−27.8564	−1.35764	−0.678819	+	0.734306i	$0.737508\pi$
$422$	0	0
$423$	5.66025	0.275211
$424$	0	0
$425$	−1.46410	−0.0710194
$426$	0	0
$427$	11.3205	0.547838
$428$	0	0
$429$	−5.85641	−0.282750
$430$	0	0
$431$	−8.19615	−0.394795	−0.197397	−	0.980324i	$-0.563249\pi$
$431$	−8.19615	−0.394795	−0.197397	+	0.980324i	$0.563249\pi$
$432$	0	0
$433$	−34.0000	−1.63394	−0.816968	−	0.576683i	$-0.804347\pi$
$433$	−34.0000	−1.63394	−0.816968	+	0.576683i	$0.804347\pi$
$434$	0	0
$435$	−24.3923	−1.16952
$436$	0	0
$437$	33.5692	1.60583
$438$	0	0
$439$	31.5167	1.50421	0.752104	−	0.659044i	$-0.229039\pi$
$439$	31.5167	1.50421	0.752104	+	0.659044i	$0.229039\pi$
$440$	0	0
$441$	−24.0718	−1.14628
$442$	0	0
$443$	39.1244	1.85885	0.929427	−	0.369006i	$-0.120302\pi$
$443$	39.1244	1.85885	0.929427	+	0.369006i	$0.120302\pi$
$444$	0	0
$445$	−2.00000	−0.0948091
$446$	0	0
$447$	−44.7846	−2.11824
$448$	0	0
$449$	33.7128	1.59101	0.795503	−	0.605950i	$-0.207207\pi$
$449$	33.7128	1.59101	0.795503	+	0.605950i	$0.207207\pi$
$450$	0	0
$451$	2.92820	0.137884
$452$	0	0
$453$	−22.9282	−1.07726
$454$	0	0
$455$	1.85641	0.0870297
$456$	0	0
$457$	−4.14359	−0.193829	−0.0969146	−	0.995293i	$-0.530897\pi$
$457$	−4.14359	−0.193829	−0.0969146	+	0.995293i	$0.530897\pi$
$458$	0	0
$459$	−5.85641	−0.273354
$460$	0	0
$461$	−26.7846	−1.24748	−0.623742	−	0.781630i	$-0.714388\pi$
$461$	−26.7846	−1.24748	−0.623742	+	0.781630i	$0.714388\pi$
$462$	0	0
$463$	−5.07180	−0.235706	−0.117853	−	0.993031i	$-0.537601\pi$
$463$	−5.07180	−0.235706	−0.117853	+	0.993031i	$0.537601\pi$
$464$	0	0
$465$	7.46410	0.346139
$466$	0	0
$467$	−37.1769	−1.72034	−0.860171	−	0.510005i	$-0.829643\pi$
$467$	−37.1769	−1.72034	−0.860171	+	0.510005i	$0.829643\pi$
$468$	0	0
$469$	−17.3205	−0.799787
$470$	0	0
$471$	46.2487	2.13103
$472$	0	0
$473$	−10.1436	−0.466403
$474$	0	0
$475$	4.19615	0.192533
$476$	0	0
$477$	−26.7846	−1.22638
$478$	0	0
$479$	34.0526	1.55590	0.777951	−	0.628325i	$-0.216259\pi$
$479$	34.0526	1.55590	0.777951	+	0.628325i	$0.216259\pi$
$480$	0	0
$481$	1.46410	0.0667573
$482$	0	0
$483$	27.7128	1.26098
$484$	0	0
$485$	−2.00000	−0.0908153
$486$	0	0
$487$	16.3923	0.742806	0.371403	−	0.928472i	$-0.378877\pi$
$487$	16.3923	0.742806	0.371403	+	0.928472i	$0.378877\pi$
$488$	0	0
$489$	−63.7128	−2.88119
$490$	0	0
$491$	1.07180	0.0483695	0.0241848	−	0.999708i	$-0.492301\pi$
$491$	1.07180	0.0483695	0.0241848	+	0.999708i	$0.492301\pi$
$492$	0	0
$493$	13.0718	0.588724
$494$	0	0
$495$	−6.53590	−0.293767
$496$	0	0
$497$	13.8564	0.621545
$498$	0	0
$499$	−40.5885	−1.81699	−0.908494	−	0.417897i	$-0.862767\pi$
$499$	−40.5885	−1.81699	−0.908494	+	0.417897i	$0.862767\pi$
$500$	0	0
$501$	−14.9282	−0.666943
$502$	0	0
$503$	5.07180	0.226140	0.113070	−	0.993587i	$-0.463932\pi$
$503$	5.07180	0.226140	0.113070	+	0.993587i	$0.463932\pi$
$504$	0	0
$505$	−2.53590	−0.112846
$506$	0	0
$507$	−29.6603	−1.31726
$508$	0	0
$509$	−2.00000	−0.0886484	−0.0443242	−	0.999017i	$-0.514113\pi$
$509$	−2.00000	−0.0886484	−0.0443242	+	0.999017i	$0.514113\pi$
$510$	0	0
$511$	16.3923	0.725153
$512$	0	0
$513$	16.7846	0.741059
$514$	0	0
$515$	13.4641	0.593299
$516$	0	0
$517$	−1.85641	−0.0816447
$518$	0	0
$519$	27.3205	1.19924
$520$	0	0
$521$	33.4641	1.46609	0.733044	−	0.680181i	$-0.238099\pi$
$521$	33.4641	1.46609	0.733044	+	0.680181i	$0.238099\pi$
$522$	0	0
$523$	4.78461	0.209216	0.104608	−	0.994514i	$-0.466641\pi$
$523$	4.78461	0.209216	0.104608	+	0.994514i	$0.466641\pi$
$524$	0	0
$525$	3.46410	0.151186
$526$	0	0
$527$	−4.00000	−0.174243
$528$	0	0
$529$	41.0000	1.78261
$530$	0	0
$531$	−0.875644	−0.0379997
$532$	0	0
$533$	−2.92820	−0.126835
$534$	0	0
$535$	−6.73205	−0.291052
$536$	0	0
$537$	48.2487	2.08209
$538$	0	0
$539$	7.89488	0.340057
$540$	0	0
$541$	−30.0000	−1.28980	−0.644900	−	0.764267i	$-0.723101\pi$
$541$	−30.0000	−1.28980	−0.644900	+	0.764267i	$0.723101\pi$
$542$	0	0
$543$	−4.00000	−0.171656
$544$	0	0
$545$	2.00000	0.0856706
$546$	0	0
$547$	4.00000	0.171028	0.0855138	−	0.996337i	$-0.472747\pi$
$547$	4.00000	0.171028	0.0855138	+	0.996337i	$0.472747\pi$
$548$	0	0
$549$	39.8564	1.70103
$550$	0	0
$551$	−37.4641	−1.59602
$552$	0	0
$553$	−6.67949	−0.284041
$554$	0	0
$555$	2.73205	0.115969
$556$	0	0
$557$	−32.1051	−1.36034	−0.680169	−	0.733056i	$-0.738094\pi$
$557$	−32.1051	−1.36034	−0.680169	+	0.733056i	$0.738094\pi$
$558$	0	0
$559$	10.1436	0.429028
$560$	0	0
$561$	5.85641	0.247258
$562$	0	0
$563$	−17.0718	−0.719490	−0.359745	−	0.933051i	$-0.617136\pi$
$563$	−17.0718	−0.719490	−0.359745	+	0.933051i	$0.617136\pi$
$564$	0	0
$565$	−17.4641	−0.734720
$566$	0	0
$567$	−3.12436	−0.131211
$568$	0	0
$569$	−22.0000	−0.922288	−0.461144	−	0.887325i	$-0.652561\pi$
$569$	−22.0000	−0.922288	−0.461144	+	0.887325i	$0.652561\pi$
$570$	0	0
$571$	12.0000	0.502184	0.251092	−	0.967963i	$-0.419210\pi$
$571$	12.0000	0.502184	0.251092	+	0.967963i	$0.419210\pi$
$572$	0	0
$573$	14.3923	0.601247
$574$	0	0
$575$	8.00000	0.333623
$576$	0	0
$577$	−3.60770	−0.150190	−0.0750952	−	0.997176i	$-0.523926\pi$
$577$	−3.60770	−0.150190	−0.0750952	+	0.997176i	$0.523926\pi$
$578$	0	0
$579$	−32.3923	−1.34618
$580$	0	0
$581$	6.67949	0.277112
$582$	0	0
$583$	8.78461	0.363821
$584$	0	0
$585$	6.53590	0.270226
$586$	0	0
$587$	−3.21539	−0.132713	−0.0663567	−	0.997796i	$-0.521138\pi$
$587$	−3.21539	−0.132713	−0.0663567	+	0.997796i	$0.521138\pi$
$588$	0	0
$589$	11.4641	0.472370
$590$	0	0
$591$	51.3205	2.11104
$592$	0	0
$593$	6.78461	0.278611	0.139305	−	0.990249i	$-0.455513\pi$
$593$	6.78461	0.278611	0.139305	+	0.990249i	$0.455513\pi$
$594$	0	0
$595$	−1.85641	−0.0761052
$596$	0	0
$597$	−71.1769	−2.91308
$598$	0	0
$599$	−2.53590	−0.103614	−0.0518070	−	0.998657i	$-0.516498\pi$
$599$	−2.53590	−0.103614	−0.0518070	+	0.998657i	$0.516498\pi$
$600$	0	0
$601$	48.3923	1.97396	0.986982	−	0.160833i	$-0.0514180\pi$
$601$	48.3923	1.97396	0.986982	+	0.160833i	$0.0514180\pi$
$602$	0	0
$603$	−60.9808	−2.48333
$604$	0	0
$605$	−8.85641	−0.360064
$606$	0	0
$607$	40.7846	1.65540	0.827698	−	0.561174i	$-0.189650\pi$
$607$	40.7846	1.65540	0.827698	+	0.561174i	$0.189650\pi$
$608$	0	0
$609$	−30.9282	−1.25327
$610$	0	0
$611$	1.85641	0.0751022
$612$	0	0
$613$	3.07180	0.124069	0.0620344	−	0.998074i	$-0.480241\pi$
$613$	3.07180	0.124069	0.0620344	+	0.998074i	$0.480241\pi$
$614$	0	0
$615$	−5.46410	−0.220334
$616$	0	0
$617$	12.9282	0.520470	0.260235	−	0.965545i	$-0.416200\pi$
$617$	12.9282	0.520470	0.260235	+	0.965545i	$0.416200\pi$
$618$	0	0
$619$	−9.85641	−0.396162	−0.198081	−	0.980186i	$-0.563471\pi$
$619$	−9.85641	−0.396162	−0.198081	+	0.980186i	$0.563471\pi$
$620$	0	0
$621$	32.0000	1.28412
$622$	0	0
$623$	−2.53590	−0.101599
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−16.7846	−0.670313
$628$	0	0
$629$	−1.46410	−0.0583776
$630$	0	0
$631$	20.9808	0.835231	0.417615	−	0.908624i	$-0.362866\pi$
$631$	20.9808	0.835231	0.417615	+	0.908624i	$0.362866\pi$
$632$	0	0
$633$	−26.9282	−1.07030
$634$	0	0
$635$	13.6603	0.542091
$636$	0	0
$637$	−7.89488	−0.312807
$638$	0	0
$639$	48.7846	1.92989
$640$	0	0
$641$	−13.4641	−0.531800	−0.265900	−	0.964001i	$-0.585669\pi$
$641$	−13.4641	−0.531800	−0.265900	+	0.964001i	$0.585669\pi$
$642$	0	0
$643$	−41.4641	−1.63518	−0.817592	−	0.575798i	$-0.804692\pi$
$643$	−41.4641	−1.63518	−0.817592	+	0.575798i	$0.804692\pi$
$644$	0	0
$645$	18.9282	0.745297
$646$	0	0
$647$	−19.7128	−0.774991	−0.387495	−	0.921872i	$-0.626660\pi$
$647$	−19.7128	−0.774991	−0.387495	+	0.921872i	$0.626660\pi$
$648$	0	0
$649$	0.287187	0.0112731
$650$	0	0
$651$	9.46410	0.370927
$652$	0	0
$653$	−18.0000	−0.704394	−0.352197	−	0.935926i	$-0.614565\pi$
$653$	−18.0000	−0.704394	−0.352197	+	0.935926i	$0.614565\pi$
$654$	0	0
$655$	−12.5885	−0.491872
$656$	0	0
$657$	57.7128	2.25159
$658$	0	0
$659$	−6.92820	−0.269884	−0.134942	−	0.990853i	$-0.543085\pi$
$659$	−6.92820	−0.269884	−0.134942	+	0.990853i	$0.543085\pi$
$660$	0	0
$661$	−44.9282	−1.74750	−0.873752	−	0.486371i	$-0.838320\pi$
$661$	−44.9282	−1.74750	−0.873752	+	0.486371i	$0.838320\pi$
$662$	0	0
$663$	−5.85641	−0.227444
$664$	0	0
$665$	5.32051	0.206320
$666$	0	0
$667$	−71.4256	−2.76561
$668$	0	0
$669$	60.2487	2.32935
$670$	0	0
$671$	−13.0718	−0.504631
$672$	0	0
$673$	−19.0718	−0.735164	−0.367582	−	0.929991i	$-0.619814\pi$
$673$	−19.0718	−0.735164	−0.367582	+	0.929991i	$0.619814\pi$
$674$	0	0
$675$	4.00000	0.153960
$676$	0	0
$677$	−31.8564	−1.22434	−0.612171	−	0.790726i	$-0.709703\pi$
$677$	−31.8564	−1.22434	−0.612171	+	0.790726i	$0.709703\pi$
$678$	0	0
$679$	−2.53590	−0.0973188
$680$	0	0
$681$	−9.85641	−0.377698
$682$	0	0
$683$	−36.7846	−1.40752	−0.703762	−	0.710436i	$-0.748498\pi$
$683$	−36.7846	−1.40752	−0.703762	+	0.710436i	$0.748498\pi$
$684$	0	0
$685$	2.00000	0.0764161
$686$	0	0
$687$	43.3205	1.65278
$688$	0	0
$689$	−8.78461	−0.334667
$690$	0	0
$691$	−20.3923	−0.775760	−0.387880	−	0.921710i	$-0.626792\pi$
$691$	−20.3923	−0.775760	−0.387880	+	0.921710i	$0.626792\pi$
$692$	0	0
$693$	−8.28719	−0.314804
$694$	0	0
$695$	6.92820	0.262802
$696$	0	0
$697$	2.92820	0.110914
$698$	0	0
$699$	41.1769	1.55745
$700$	0	0
$701$	15.8564	0.598888	0.299444	−	0.954114i	$-0.403199\pi$
$701$	15.8564	0.598888	0.299444	+	0.954114i	$0.403199\pi$
$702$	0	0
$703$	4.19615	0.158261
$704$	0	0
$705$	3.46410	0.130466
$706$	0	0
$707$	−3.21539	−0.120927
$708$	0	0
$709$	16.1436	0.606285	0.303143	−	0.952945i	$-0.401964\pi$
$709$	16.1436	0.606285	0.303143	+	0.952945i	$0.401964\pi$
$710$	0	0
$711$	−23.5167	−0.881944
$712$	0	0
$713$	21.8564	0.818529
$714$	0	0
$715$	−2.14359	−0.0801659
$716$	0	0
$717$	47.1769	1.76185
$718$	0	0
$719$	−8.39230	−0.312980	−0.156490	−	0.987680i	$-0.550018\pi$
$719$	−8.39230	−0.312980	−0.156490	+	0.987680i	$0.550018\pi$
$720$	0	0
$721$	17.0718	0.635787
$722$	0	0
$723$	−24.3923	−0.907160
$724$	0	0
$725$	−8.92820	−0.331585
$726$	0	0
$727$	32.7846	1.21591	0.607957	−	0.793970i	$-0.291989\pi$
$727$	32.7846	1.21591	0.607957	+	0.793970i	$0.291989\pi$
$728$	0	0
$729$	−43.7846	−1.62165
$730$	0	0
$731$	−10.1436	−0.375174
$732$	0	0
$733$	0.143594	0.00530375	0.00265187	−	0.999996i	$-0.499156\pi$
$733$	0.143594	0.00530375	0.00265187	+	0.999996i	$0.499156\pi$
$734$	0	0
$735$	−14.7321	−0.543400
$736$	0	0
$737$	20.0000	0.736709
$738$	0	0
$739$	6.92820	0.254858	0.127429	−	0.991848i	$-0.459327\pi$
$739$	6.92820	0.254858	0.127429	+	0.991848i	$0.459327\pi$
$740$	0	0
$741$	16.7846	0.616598
$742$	0	0
$743$	7.12436	0.261367	0.130684	−	0.991424i	$-0.458283\pi$
$743$	7.12436	0.261367	0.130684	+	0.991424i	$0.458283\pi$
$744$	0	0
$745$	−16.3923	−0.600568
$746$	0	0
$747$	23.5167	0.860430
$748$	0	0
$749$	−8.53590	−0.311895
$750$	0	0
$751$	−0.392305	−0.0143154	−0.00715770	−	0.999974i	$-0.502278\pi$
$751$	−0.392305	−0.0143154	−0.00715770	+	0.999974i	$0.502278\pi$
$752$	0	0
$753$	−62.1051	−2.26324
$754$	0	0
$755$	−8.39230	−0.305427
$756$	0	0
$757$	−53.7128	−1.95223	−0.976113	−	0.217265i	$-0.930286\pi$
$757$	−53.7128	−1.95223	−0.976113	+	0.217265i	$0.930286\pi$
$758$	0	0
$759$	−32.0000	−1.16153
$760$	0	0
$761$	10.0000	0.362500	0.181250	−	0.983437i	$-0.441986\pi$
$761$	10.0000	0.362500	0.181250	+	0.983437i	$0.441986\pi$
$762$	0	0
$763$	2.53590	0.0918057
$764$	0	0
$765$	−6.53590	−0.236306
$766$	0	0
$767$	−0.287187	−0.0103697
$768$	0	0
$769$	20.9282	0.754690	0.377345	−	0.926073i	$-0.376837\pi$
$769$	20.9282	0.754690	0.377345	+	0.926073i	$0.376837\pi$
$770$	0	0
$771$	−69.5692	−2.50547
$772$	0	0
$773$	−38.7846	−1.39499	−0.697493	−	0.716592i	$-0.745701\pi$
$773$	−38.7846	−1.39499	−0.697493	+	0.716592i	$0.745701\pi$
$774$	0	0
$775$	2.73205	0.0981382
$776$	0	0
$777$	3.46410	0.124274
$778$	0	0
$779$	−8.39230	−0.300686
$780$	0	0
$781$	−16.0000	−0.572525
$782$	0	0
$783$	−35.7128	−1.27627
$784$	0	0
$785$	16.9282	0.604193
$786$	0	0
$787$	−11.8038	−0.420762	−0.210381	−	0.977620i	$-0.567470\pi$
$787$	−11.8038	−0.420762	−0.210381	+	0.977620i	$0.567470\pi$
$788$	0	0
$789$	−82.1051	−2.92302
$790$	0	0
$791$	−22.1436	−0.787336
$792$	0	0
$793$	13.0718	0.464193
$794$	0	0
$795$	−16.3923	−0.581375
$796$	0	0
$797$	17.4641	0.618610	0.309305	−	0.950963i	$-0.399904\pi$
$797$	17.4641	0.618610	0.309305	+	0.950963i	$0.399904\pi$
$798$	0	0
$799$	−1.85641	−0.0656749
$800$	0	0
$801$	−8.92820	−0.315463
$802$	0	0
$803$	−18.9282	−0.667962
$804$	0	0
$805$	10.1436	0.357515
$806$	0	0
$807$	−1.07180	−0.0377290
$808$	0	0
$809$	−39.5692	−1.39118	−0.695590	−	0.718439i	$-0.744857\pi$
$809$	−39.5692	−1.39118	−0.695590	+	0.718439i	$0.744857\pi$
$810$	0	0
$811$	14.1436	0.496649	0.248324	−	0.968677i	$-0.420120\pi$
$811$	14.1436	0.496649	0.248324	+	0.968677i	$0.420120\pi$
$812$	0	0
$813$	−45.8564	−1.60825
$814$	0	0
$815$	−23.3205	−0.816882
$816$	0	0
$817$	29.0718	1.01709
$818$	0	0
$819$	8.28719	0.289578
$820$	0	0
$821$	30.0000	1.04701	0.523504	−	0.852023i	$-0.324625\pi$
$821$	30.0000	1.04701	0.523504	+	0.852023i	$0.324625\pi$
$822$	0	0
$823$	10.4449	0.364085	0.182043	−	0.983291i	$-0.441729\pi$
$823$	10.4449	0.364085	0.182043	+	0.983291i	$0.441729\pi$
$824$	0	0
$825$	−4.00000	−0.139262
$826$	0	0
$827$	−4.39230	−0.152735	−0.0763677	−	0.997080i	$-0.524332\pi$
$827$	−4.39230	−0.152735	−0.0763677	+	0.997080i	$0.524332\pi$
$828$	0	0
$829$	31.5692	1.09644	0.548222	−	0.836333i	$-0.315305\pi$
$829$	31.5692	1.09644	0.548222	+	0.836333i	$0.315305\pi$
$830$	0	0
$831$	−71.7128	−2.48769
$832$	0	0
$833$	7.89488	0.273541
$834$	0	0
$835$	−5.46410	−0.189093
$836$	0	0
$837$	10.9282	0.377734
$838$	0	0
$839$	−32.7846	−1.13185	−0.565925	−	0.824457i	$-0.691481\pi$
$839$	−32.7846	−1.13185	−0.565925	+	0.824457i	$0.691481\pi$
$840$	0	0
$841$	50.7128	1.74872
$842$	0	0
$843$	13.4641	0.463728
$844$	0	0
$845$	−10.8564	−0.373472
$846$	0	0
$847$	−11.2295	−0.385849
$848$	0	0
$849$	12.0000	0.411839
$850$	0	0
$851$	8.00000	0.274236
$852$	0	0
$853$	−30.0000	−1.02718	−0.513590	−	0.858036i	$-0.671685\pi$
$853$	−30.0000	−1.02718	−0.513590	+	0.858036i	$0.671685\pi$
$854$	0	0
$855$	18.7321	0.640623
$856$	0	0
$857$	−4.14359	−0.141542	−0.0707712	−	0.997493i	$-0.522546\pi$
$857$	−4.14359	−0.141542	−0.0707712	+	0.997493i	$0.522546\pi$
$858$	0	0
$859$	14.4449	0.492852	0.246426	−	0.969162i	$-0.420744\pi$
$859$	14.4449	0.492852	0.246426	+	0.969162i	$0.420744\pi$
$860$	0	0
$861$	−6.92820	−0.236113
$862$	0	0
$863$	−38.4449	−1.30868	−0.654339	−	0.756201i	$-0.727053\pi$
$863$	−38.4449	−1.30868	−0.654339	+	0.756201i	$0.727053\pi$
$864$	0	0
$865$	10.0000	0.340010
$866$	0	0
$867$	−40.5885	−1.37846
$868$	0	0
$869$	7.71281	0.261639
$870$	0	0
$871$	−20.0000	−0.677674
$872$	0	0
$873$	−8.92820	−0.302174
$874$	0	0
$875$	1.26795	0.0428645
$876$	0	0
$877$	−50.7846	−1.71487	−0.857437	−	0.514589i	$-0.827945\pi$
$877$	−50.7846	−1.71487	−0.857437	+	0.514589i	$0.827945\pi$
$878$	0	0
$879$	16.3923	0.552899
$880$	0	0
$881$	47.3205	1.59427	0.797134	−	0.603802i	$-0.206348\pi$
$881$	47.3205	1.59427	0.797134	+	0.603802i	$0.206348\pi$
$882$	0	0
$883$	34.2487	1.15256	0.576280	−	0.817252i	$-0.304504\pi$
$883$	34.2487	1.15256	0.576280	+	0.817252i	$0.304504\pi$
$884$	0	0
$885$	−0.535898	−0.0180140
$886$	0	0
$887$	−20.1962	−0.678120	−0.339060	−	0.940765i	$-0.610109\pi$
$887$	−20.1962	−0.678120	−0.339060	+	0.940765i	$0.610109\pi$
$888$	0	0
$889$	17.3205	0.580911
$890$	0	0
$891$	3.60770	0.120862
$892$	0	0
$893$	5.32051	0.178044
$894$	0	0
$895$	17.6603	0.590317
$896$	0	0
$897$	32.0000	1.06845
$898$	0	0
$899$	−24.3923	−0.813529
$900$	0	0
$901$	8.78461	0.292658
$902$	0	0
$903$	24.0000	0.798670
$904$	0	0
$905$	−1.46410	−0.0486684
$906$	0	0
$907$	−5.75129	−0.190968	−0.0954842	−	0.995431i	$-0.530440\pi$
$907$	−5.75129	−0.190968	−0.0954842	+	0.995431i	$0.530440\pi$
$908$	0	0
$909$	−11.3205	−0.375478
$910$	0	0
$911$	27.9090	0.924665	0.462333	−	0.886707i	$-0.347013\pi$
$911$	27.9090	0.924665	0.462333	+	0.886707i	$0.347013\pi$
$912$	0	0
$913$	−7.71281	−0.255257
$914$	0	0
$915$	24.3923	0.806385
$916$	0	0
$917$	−15.9615	−0.527096
$918$	0	0
$919$	13.2679	0.437669	0.218835	−	0.975762i	$-0.429774\pi$
$919$	13.2679	0.437669	0.218835	+	0.975762i	$0.429774\pi$
$920$	0	0
$921$	34.3923	1.13326
$922$	0	0
$923$	16.0000	0.526646
$924$	0	0
$925$	1.00000	0.0328798
$926$	0	0
$927$	60.1051	1.97411
$928$	0	0
$929$	−15.8564	−0.520232	−0.260116	−	0.965577i	$-0.583761\pi$
$929$	−15.8564	−0.520232	−0.260116	+	0.965577i	$0.583761\pi$
$930$	0	0
$931$	−22.6269	−0.741568
$932$	0	0
$933$	−74.1051	−2.42609
$934$	0	0
$935$	2.14359	0.0701030
$936$	0	0
$937$	3.85641	0.125983	0.0629917	−	0.998014i	$-0.479936\pi$
$937$	3.85641	0.125983	0.0629917	+	0.998014i	$0.479936\pi$
$938$	0	0
$939$	10.5359	0.343826
$940$	0	0
$941$	28.3923	0.925563	0.462781	−	0.886472i	$-0.346852\pi$
$941$	28.3923	0.925563	0.462781	+	0.886472i	$0.346852\pi$
$942$	0	0
$943$	−16.0000	−0.521032
$944$	0	0
$945$	5.07180	0.164986
$946$	0	0
$947$	45.1769	1.46805	0.734026	−	0.679121i	$-0.237639\pi$
$947$	45.1769	1.46805	0.734026	+	0.679121i	$0.237639\pi$
$948$	0	0
$949$	18.9282	0.614435
$950$	0	0
$951$	−87.0333	−2.82225
$952$	0	0
$953$	−11.8564	−0.384067	−0.192033	−	0.981388i	$-0.561508\pi$
$953$	−11.8564	−0.384067	−0.192033	+	0.981388i	$0.561508\pi$
$954$	0	0
$955$	5.26795	0.170467
$956$	0	0
$957$	35.7128	1.15443
$958$	0	0
$959$	2.53590	0.0818884
$960$	0	0
$961$	−23.5359	−0.759223
$962$	0	0
$963$	−30.0526	−0.968430
$964$	0	0
$965$	−11.8564	−0.381671
$966$	0	0
$967$	23.6077	0.759172	0.379586	−	0.925156i	$-0.376066\pi$
$967$	23.6077	0.759172	0.379586	+	0.925156i	$0.376066\pi$
$968$	0	0
$969$	−16.7846	−0.539199
$970$	0	0
$971$	−14.9282	−0.479069	−0.239534	−	0.970888i	$-0.576995\pi$
$971$	−14.9282	−0.479069	−0.239534	+	0.970888i	$0.576995\pi$
$972$	0	0
$973$	8.78461	0.281622
$974$	0	0
$975$	4.00000	0.128103
$976$	0	0
$977$	−34.0000	−1.08776	−0.543878	−	0.839164i	$-0.683045\pi$
$977$	−34.0000	−1.08776	−0.543878	+	0.839164i	$0.683045\pi$
$978$	0	0
$979$	2.92820	0.0935858
$980$	0	0
$981$	8.92820	0.285056
$982$	0	0
$983$	38.4449	1.22620	0.613100	−	0.790005i	$-0.289922\pi$
$983$	38.4449	1.22620	0.613100	+	0.790005i	$0.289922\pi$
$984$	0	0
$985$	18.7846	0.598527
$986$	0	0
$987$	4.39230	0.139809
$988$	0	0
$989$	55.4256	1.76243
$990$	0	0
$991$	−5.94744	−0.188927	−0.0944633	−	0.995528i	$-0.530114\pi$
$991$	−5.94744	−0.188927	−0.0944633	+	0.995528i	$0.530114\pi$
$992$	0	0
$993$	24.2487	0.769510
$994$	0	0
$995$	−26.0526	−0.825922
$996$	0	0
$997$	−4.14359	−0.131229	−0.0656145	−	0.997845i	$-0.520901\pi$
$997$	−4.14359	−0.131229	−0.0656145	+	0.997845i	$0.520901\pi$
$998$	0	0
$999$	4.00000	0.126554

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2960.2.a.q.1.2		2
4.3	odd	2		370.2.a.e.1.1	✓	2
12.11	even	2		3330.2.a.bd.1.2		2
20.3	even	4		1850.2.b.l.149.3		4
20.7	even	4		1850.2.b.l.149.2		4
20.19	odd	2		1850.2.a.x.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
370.2.a.e.1.1	✓	2	4.3	odd	2
1850.2.a.x.1.2		2	20.19	odd	2
1850.2.b.l.149.2		4	20.7	even	4
1850.2.b.l.149.3		4	20.3	even	4
2960.2.a.q.1.2		2	1.1	even	1	trivial
3330.2.a.bd.1.2		2	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)

\(f(q)\)	\(=\)	\(q+2.73205 q^{3} +1.00000 q^{5} +1.26795 q^{7} +4.46410 q^{9} +O(q^{10})\)	\(q+2.73205 q^{3} +1.00000 q^{5} +1.26795 q^{7} +4.46410 q^{9} -1.46410 q^{11} +1.46410 q^{13} +2.73205 q^{15} -1.46410 q^{17} +4.19615 q^{19} +3.46410 q^{21} +8.00000 q^{23} +1.00000 q^{25} +4.00000 q^{27} -8.92820 q^{29} +2.73205 q^{31} -4.00000 q^{33} +1.26795 q^{35} +1.00000 q^{37} +4.00000 q^{39} -2.00000 q^{41} +6.92820 q^{43} +4.46410 q^{45} +1.26795 q^{47} -5.39230 q^{49} -4.00000 q^{51} -6.00000 q^{53} -1.46410 q^{55} +11.4641 q^{57} -0.196152 q^{59} +8.92820 q^{61} +5.66025 q^{63} +1.46410 q^{65} -13.6603 q^{67} +21.8564 q^{69} +10.9282 q^{71} +12.9282 q^{73} +2.73205 q^{75} -1.85641 q^{77} -5.26795 q^{79} -2.46410 q^{81} +5.26795 q^{83} -1.46410 q^{85} -24.3923 q^{87} -2.00000 q^{89} +1.85641 q^{91} +7.46410 q^{93} +4.19615 q^{95} -2.00000 q^{97} -6.53590 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} + 2 q^{5} + 6 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 + 2 * q^5 + 6 * q^7 + 2 * q^9	\( 2 q + 2 q^{3} + 2 q^{5} + 6 q^{7} + 2 q^{9} + 4 q^{11} - 4 q^{13} + 2 q^{15} + 4 q^{17} - 2 q^{19} + 16 q^{23} + 2 q^{25} + 8 q^{27} - 4 q^{29} + 2 q^{31} - 8 q^{33} + 6 q^{35} + 2 q^{37} + 8 q^{39} - 4 q^{41} + 2 q^{45} + 6 q^{47} + 10 q^{49} - 8 q^{51} - 12 q^{53} + 4 q^{55} + 16 q^{57} + 10 q^{59} + 4 q^{61} - 6 q^{63} - 4 q^{65} - 10 q^{67} + 16 q^{69} + 8 q^{71} + 12 q^{73} + 2 q^{75} + 24 q^{77} - 14 q^{79} + 2 q^{81} + 14 q^{83} + 4 q^{85} - 28 q^{87} - 4 q^{89} - 24 q^{91} + 8 q^{93} - 2 q^{95} - 4 q^{97} - 20 q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 + 2 * q^5 + 6 * q^7 + 2 * q^9 + 4 * q^11 - 4 * q^13 + 2 * q^15 + 4 * q^17 - 2 * q^19 + 16 * q^23 + 2 * q^25 + 8 * q^27 - 4 * q^29 + 2 * q^31 - 8 * q^33 + 6 * q^35 + 2 * q^37 + 8 * q^39 - 4 * q^41 + 2 * q^45 + 6 * q^47 + 10 * q^49 - 8 * q^51 - 12 * q^53 + 4 * q^55 + 16 * q^57 + 10 * q^59 + 4 * q^61 - 6 * q^63 - 4 * q^65 - 10 * q^67 + 16 * q^69 + 8 * q^71 + 12 * q^73 + 2 * q^75 + 24 * q^77 - 14 * q^79 + 2 * q^81 + 14 * q^83 + 4 * q^85 - 28 * q^87 - 4 * q^89 - 24 * q^91 + 8 * q^93 - 2 * q^95 - 4 * q^97 - 20 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more