LMFDB - Embedded newform 1078.2.a.r.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1078,2,Mod(1,1078)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1078, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1078.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1078 = 2 \cdot 7^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1078.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:	$8.60787333789$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ of defining polynomial
Character	$\chi$	$=$	1078.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -2.82843 q^{3} +1.00000 q^{4} +2.82843 q^{6} -1.00000 q^{8} +5.00000 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -2.82843 q^{3} +1.00000 q^{4} +2.82843 q^{6} -1.00000 q^{8} +5.00000 q^{9} -1.00000 q^{11} -2.82843 q^{12} +2.82843 q^{13} +1.00000 q^{16} -2.82843 q^{17} -5.00000 q^{18} -5.65685 q^{19} +1.00000 q^{22} +8.00000 q^{23} +2.82843 q^{24} -5.00000 q^{25} -2.82843 q^{26} -5.65685 q^{27} +2.00000 q^{29} -8.48528 q^{31} -1.00000 q^{32} +2.82843 q^{33} +2.82843 q^{34} +5.00000 q^{36} +2.00000 q^{37} +5.65685 q^{38} -8.00000 q^{39} -2.82843 q^{41} -4.00000 q^{43} -1.00000 q^{44} -8.00000 q^{46} -2.82843 q^{47} -2.82843 q^{48} +5.00000 q^{50} +8.00000 q^{51} +2.82843 q^{52} +14.0000 q^{53} +5.65685 q^{54} +16.0000 q^{57} -2.00000 q^{58} +8.48528 q^{59} +8.48528 q^{61} +8.48528 q^{62} +1.00000 q^{64} -2.82843 q^{66} +4.00000 q^{67} -2.82843 q^{68} -22.6274 q^{69} -5.00000 q^{72} +14.1421 q^{73} -2.00000 q^{74} +14.1421 q^{75} -5.65685 q^{76} +8.00000 q^{78} -16.0000 q^{79} +1.00000 q^{81} +2.82843 q^{82} +16.9706 q^{83} +4.00000 q^{86} -5.65685 q^{87} +1.00000 q^{88} +11.3137 q^{89} +8.00000 q^{92} +24.0000 q^{93} +2.82843 q^{94} +2.82843 q^{96} +16.9706 q^{97} -5.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{4} - 2 q^{8} + 10 q^{9}+O(q^{10}) $ 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^8 + 10 * q^9	$ 2 q - 2 q^{2} + 2 q^{4} - 2 q^{8} + 10 q^{9} - 2 q^{11} + 2 q^{16} - 10 q^{18} + 2 q^{22} + 16 q^{23} - 10 q^{25} + 4 q^{29} - 2 q^{32} + 10 q^{36} + 4 q^{37} - 16 q^{39} - 8 q^{43} - 2 q^{44} - 16 q^{46} + 10 q^{50} + 16 q^{51} + 28 q^{53} + 32 q^{57} - 4 q^{58} + 2 q^{64} + 8 q^{67} - 10 q^{72} - 4 q^{74} + 16 q^{78} - 32 q^{79} + 2 q^{81} + 8 q^{86} + 2 q^{88} + 16 q^{92} + 48 q^{93} - 10 q^{99}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^8 + 10 * q^9 - 2 * q^11 + 2 * q^16 - 10 * q^18 + 2 * q^22 + 16 * q^23 - 10 * q^25 + 4 * q^29 - 2 * q^32 + 10 * q^36 + 4 * q^37 - 16 * q^39 - 8 * q^43 - 2 * q^44 - 16 * q^46 + 10 * q^50 + 16 * q^51 + 28 * q^53 + 32 * q^57 - 4 * q^58 + 2 * q^64 + 8 * q^67 - 10 * q^72 - 4 * q^74 + 16 * q^78 - 32 * q^79 + 2 * q^81 + 8 * q^86 + 2 * q^88 + 16 * q^92 + 48 * q^93 - 10 * 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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	−2.82843	−1.63299	−0.816497	−	0.577350i	$-0.804087\pi$
$3$	−2.82843	−1.63299	−0.816497	+	0.577350i	$0.804087\pi$
$4$	1.00000	0.500000
$5$	0	0		−	1.00000i	$-0.5\pi$
$5$	0	0			1.00000i	$0.5\pi$
$6$	2.82843	1.15470
$7$	0	0
$7$	0	0
$8$	−1.00000	−0.353553
$9$	5.00000	1.66667
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	−2.82843	−0.816497
$13$	2.82843	0.784465	0.392232	−	0.919866i	$-0.371703\pi$
$13$	2.82843	0.784465	0.392232	+	0.919866i	$0.371703\pi$
$14$	0	0
$15$	0	0
$16$	1.00000	0.250000
$17$	−2.82843	−0.685994	−0.342997	−	0.939336i	$-0.611442\pi$
$17$	−2.82843	−0.685994	−0.342997	+	0.939336i	$0.611442\pi$
$18$	−5.00000	−1.17851
$19$	−5.65685	−1.29777	−0.648886	−	0.760886i	$-0.724765\pi$
$19$	−5.65685	−1.29777	−0.648886	+	0.760886i	$0.724765\pi$
$20$	0	0
$21$	0	0
$22$	1.00000	0.213201
$23$	8.00000	1.66812	0.834058	−	0.551677i	$-0.186012\pi$
$23$	8.00000	1.66812	0.834058	+	0.551677i	$0.186012\pi$
$24$	2.82843	0.577350
$25$	−5.00000	−1.00000
$26$	−2.82843	−0.554700
$27$	−5.65685	−1.08866
$28$	0	0
$29$	2.00000	0.371391	0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000	0.371391	0.185695	+	0.982607i	$0.440546\pi$
$30$	0	0
$31$	−8.48528	−1.52400	−0.762001	−	0.647576i	$-0.775783\pi$
$31$	−8.48528	−1.52400	−0.762001	+	0.647576i	$0.775783\pi$
$32$	−1.00000	−0.176777
$33$	2.82843	0.492366
$34$	2.82843	0.485071
$35$	0	0
$36$	5.00000	0.833333
$37$	2.00000	0.328798	0.164399	−	0.986394i	$-0.447432\pi$
$37$	2.00000	0.328798	0.164399	+	0.986394i	$0.447432\pi$
$38$	5.65685	0.917663
$39$	−8.00000	−1.28103
$40$	0	0
$41$	−2.82843	−0.441726	−0.220863	−	0.975305i	$-0.570887\pi$
$41$	−2.82843	−0.441726	−0.220863	+	0.975305i	$0.570887\pi$
$42$	0	0
$43$	−4.00000	−0.609994	−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000	−0.609994	−0.304997	+	0.952353i	$0.598656\pi$
$44$	−1.00000	−0.150756
$45$	0	0
$46$	−8.00000	−1.17954
$47$	−2.82843	−0.412568	−0.206284	−	0.978492i	$-0.566137\pi$
$47$	−2.82843	−0.412568	−0.206284	+	0.978492i	$0.566137\pi$
$48$	−2.82843	−0.408248
$49$	0	0
$50$	5.00000	0.707107
$51$	8.00000	1.12022
$52$	2.82843	0.392232
$53$	14.0000	1.92305	0.961524	−	0.274721i	$-0.0885855\pi$
$53$	14.0000	1.92305	0.961524	+	0.274721i	$0.0885855\pi$
$54$	5.65685	0.769800
$55$	0	0
$56$	0	0
$57$	16.0000	2.11925
$58$	−2.00000	−0.262613
$59$	8.48528	1.10469	0.552345	−	0.833616i	$-0.313733\pi$
$59$	8.48528	1.10469	0.552345	+	0.833616i	$0.313733\pi$
$60$	0	0
$61$	8.48528	1.08643	0.543214	−	0.839594i	$-0.317207\pi$
$61$	8.48528	1.08643	0.543214	+	0.839594i	$0.317207\pi$
$62$	8.48528	1.07763
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	−2.82843	−0.348155
$67$	4.00000	0.488678	0.244339	−	0.969690i	$-0.421429\pi$
$67$	4.00000	0.488678	0.244339	+	0.969690i	$0.421429\pi$
$68$	−2.82843	−0.342997
$69$	−22.6274	−2.72402
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	−5.00000	−0.589256
$73$	14.1421	1.65521	0.827606	−	0.561310i	$-0.189702\pi$
$73$	14.1421	1.65521	0.827606	+	0.561310i	$0.189702\pi$
$74$	−2.00000	−0.232495
$75$	14.1421	1.63299
$76$	−5.65685	−0.648886
$77$	0	0
$78$	8.00000	0.905822
$79$	−16.0000	−1.80014	−0.900070	−	0.435745i	$-0.856485\pi$
$79$	−16.0000	−1.80014	−0.900070	+	0.435745i	$0.856485\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	2.82843	0.312348
$83$	16.9706	1.86276	0.931381	−	0.364047i	$-0.118605\pi$
$83$	16.9706	1.86276	0.931381	+	0.364047i	$0.118605\pi$
$84$	0	0
$85$	0	0
$86$	4.00000	0.431331
$87$	−5.65685	−0.606478
$88$	1.00000	0.106600
$89$	11.3137	1.19925	0.599625	−	0.800281i	$-0.295316\pi$
$89$	11.3137	1.19925	0.599625	+	0.800281i	$0.295316\pi$
$90$	0	0
$91$	0	0
$92$	8.00000	0.834058
$93$	24.0000	2.48868
$94$	2.82843	0.291730
$95$	0	0
$96$	2.82843	0.288675
$97$	16.9706	1.72310	0.861550	−	0.507673i	$-0.169494\pi$
$97$	16.9706	1.72310	0.861550	+	0.507673i	$0.169494\pi$
$98$	0	0
$99$	−5.00000	−0.502519
$100$	−5.00000	−0.500000
$101$	2.82843	0.281439	0.140720	−	0.990050i	$-0.455058\pi$
$101$	2.82843	0.281439	0.140720	+	0.990050i	$0.455058\pi$
$102$	−8.00000	−0.792118
$103$	−14.1421	−1.39347	−0.696733	−	0.717331i	$-0.745364\pi$
$103$	−14.1421	−1.39347	−0.696733	+	0.717331i	$0.745364\pi$
$104$	−2.82843	−0.277350
$105$	0	0
$106$	−14.0000	−1.35980
$107$	12.0000	1.16008	0.580042	−	0.814587i	$-0.303036\pi$
$107$	12.0000	1.16008	0.580042	+	0.814587i	$0.303036\pi$
$108$	−5.65685	−0.544331
$109$	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
$111$	−5.65685	−0.536925
$112$	0	0
$113$	2.00000	0.188144	0.0940721	−	0.995565i	$-0.470012\pi$
$113$	2.00000	0.188144	0.0940721	+	0.995565i	$0.470012\pi$
$114$	−16.0000	−1.49854
$115$	0	0
$116$	2.00000	0.185695
$117$	14.1421	1.30744
$118$	−8.48528	−0.781133
$119$	0	0
$120$	0	0
$121$	1.00000	0.0909091
$122$	−8.48528	−0.768221
$123$	8.00000	0.721336
$124$	−8.48528	−0.762001
$125$	0	0
$126$	0	0
$127$	0	0		−	1.00000i	$-0.5\pi$
$127$	0	0			1.00000i	$0.5\pi$
$128$	−1.00000	−0.0883883
$129$	11.3137	0.996116
$130$	0	0
$131$	−11.3137	−0.988483	−0.494242	−	0.869325i	$-0.664554\pi$
$131$	−11.3137	−0.988483	−0.494242	+	0.869325i	$0.664554\pi$
$132$	2.82843	0.246183
$133$	0	0
$134$	−4.00000	−0.345547
$135$	0	0
$136$	2.82843	0.242536
$137$	−2.00000	−0.170872	−0.0854358	−	0.996344i	$-0.527228\pi$
$137$	−2.00000	−0.170872	−0.0854358	+	0.996344i	$0.527228\pi$
$138$	22.6274	1.92617
$139$	16.9706	1.43942	0.719712	−	0.694273i	$-0.244274\pi$
$139$	16.9706	1.43942	0.719712	+	0.694273i	$0.244274\pi$
$140$	0	0
$141$	8.00000	0.673722
$142$	0	0
$143$	−2.82843	−0.236525
$144$	5.00000	0.416667
$145$	0	0
$146$	−14.1421	−1.17041
$147$	0	0
$148$	2.00000	0.164399
$149$	6.00000	0.491539	0.245770	−	0.969328i	$-0.420959\pi$
$149$	6.00000	0.491539	0.245770	+	0.969328i	$0.420959\pi$
$150$	−14.1421	−1.15470
$151$	16.0000	1.30206	0.651031	−	0.759051i	$-0.274337\pi$
$151$	16.0000	1.30206	0.651031	+	0.759051i	$0.274337\pi$
$152$	5.65685	0.458831
$153$	−14.1421	−1.14332
$154$	0	0
$155$	0	0
$156$	−8.00000	−0.640513
$157$	11.3137	0.902932	0.451466	−	0.892288i	$-0.350901\pi$
$157$	11.3137	0.902932	0.451466	+	0.892288i	$0.350901\pi$
$158$	16.0000	1.27289
$159$	−39.5980	−3.14032
$160$	0	0
$161$	0	0
$162$	−1.00000	−0.0785674
$163$	−4.00000	−0.313304	−0.156652	−	0.987654i	$-0.550070\pi$
$163$	−4.00000	−0.313304	−0.156652	+	0.987654i	$0.550070\pi$
$164$	−2.82843	−0.220863
$165$	0	0
$166$	−16.9706	−1.31717
$167$	11.3137	0.875481	0.437741	−	0.899101i	$-0.355779\pi$
$167$	11.3137	0.875481	0.437741	+	0.899101i	$0.355779\pi$
$168$	0	0
$169$	−5.00000	−0.384615
$170$	0	0
$171$	−28.2843	−2.16295
$172$	−4.00000	−0.304997
$173$	−19.7990	−1.50529	−0.752645	−	0.658427i	$-0.771222\pi$
$173$	−19.7990	−1.50529	−0.752645	+	0.658427i	$0.771222\pi$
$174$	5.65685	0.428845
$175$	0	0
$176$	−1.00000	−0.0753778
$177$	−24.0000	−1.80395
$178$	−11.3137	−0.847998
$179$	20.0000	1.49487	0.747435	−	0.664335i	$-0.231285\pi$
$179$	20.0000	1.49487	0.747435	+	0.664335i	$0.231285\pi$
$180$	0	0
$181$	−5.65685	−0.420471	−0.210235	−	0.977651i	$-0.567423\pi$
$181$	−5.65685	−0.420471	−0.210235	+	0.977651i	$0.567423\pi$
$182$	0	0
$183$	−24.0000	−1.77413
$184$	−8.00000	−0.589768
$185$	0	0
$186$	−24.0000	−1.75977
$187$	2.82843	0.206835
$188$	−2.82843	−0.206284
$189$	0	0
$190$	0	0
$191$	−8.00000	−0.578860	−0.289430	−	0.957199i	$-0.593466\pi$
$191$	−8.00000	−0.578860	−0.289430	+	0.957199i	$0.593466\pi$
$192$	−2.82843	−0.204124
$193$	−14.0000	−1.00774	−0.503871	−	0.863779i	$-0.668091\pi$
$193$	−14.0000	−1.00774	−0.503871	+	0.863779i	$0.668091\pi$
$194$	−16.9706	−1.21842
$195$	0	0
$196$	0	0
$197$	−10.0000	−0.712470	−0.356235	−	0.934396i	$-0.615940\pi$
$197$	−10.0000	−0.712470	−0.356235	+	0.934396i	$0.615940\pi$
$198$	5.00000	0.355335
$199$	19.7990	1.40351	0.701757	−	0.712417i	$-0.252399\pi$
$199$	19.7990	1.40351	0.701757	+	0.712417i	$0.252399\pi$
$200$	5.00000	0.353553
$201$	−11.3137	−0.798007
$202$	−2.82843	−0.199007
$203$	0	0
$204$	8.00000	0.560112
$205$	0	0
$206$	14.1421	0.985329
$207$	40.0000	2.78019
$208$	2.82843	0.196116
$209$	5.65685	0.391293
$210$	0	0
$211$	4.00000	0.275371	0.137686	−	0.990476i	$-0.456034\pi$
$211$	4.00000	0.275371	0.137686	+	0.990476i	$0.456034\pi$
$212$	14.0000	0.961524
$213$	0	0
$214$	−12.0000	−0.820303
$215$	0	0
$216$	5.65685	0.384900
$217$	0	0
$218$	−2.00000	−0.135457
$219$	−40.0000	−2.70295
$220$	0	0
$221$	−8.00000	−0.538138
$222$	5.65685	0.379663
$223$	−14.1421	−0.947027	−0.473514	−	0.880786i	$-0.657015\pi$
$223$	−14.1421	−0.947027	−0.473514	+	0.880786i	$0.657015\pi$
$224$	0	0
$225$	−25.0000	−1.66667
$226$	−2.00000	−0.133038
$227$	−11.3137	−0.750917	−0.375459	−	0.926839i	$-0.622515\pi$
$227$	−11.3137	−0.750917	−0.375459	+	0.926839i	$0.622515\pi$
$228$	16.0000	1.05963
$229$	5.65685	0.373815	0.186908	−	0.982377i	$-0.440153\pi$
$229$	5.65685	0.373815	0.186908	+	0.982377i	$0.440153\pi$
$230$	0	0
$231$	0	0
$232$	−2.00000	−0.131306
$233$	22.0000	1.44127	0.720634	−	0.693316i	$-0.243851\pi$
$233$	22.0000	1.44127	0.720634	+	0.693316i	$0.243851\pi$
$234$	−14.1421	−0.924500
$235$	0	0
$236$	8.48528	0.552345
$237$	45.2548	2.93962
$238$	0	0
$239$	−16.0000	−1.03495	−0.517477	−	0.855697i	$-0.673129\pi$
$239$	−16.0000	−1.03495	−0.517477	+	0.855697i	$0.673129\pi$
$240$	0	0
$241$	−14.1421	−0.910975	−0.455488	−	0.890242i	$-0.650535\pi$
$241$	−14.1421	−0.910975	−0.455488	+	0.890242i	$0.650535\pi$
$242$	−1.00000	−0.0642824
$243$	14.1421	0.907218
$244$	8.48528	0.543214
$245$	0	0
$246$	−8.00000	−0.510061
$247$	−16.0000	−1.01806
$248$	8.48528	0.538816
$249$	−48.0000	−3.04188
$250$	0	0
$251$	14.1421	0.892644	0.446322	−	0.894873i	$-0.352734\pi$
$251$	14.1421	0.892644	0.446322	+	0.894873i	$0.352734\pi$
$252$	0	0
$253$	−8.00000	−0.502956
$254$	0	0
$255$	0	0
$256$	1.00000	0.0625000
$257$	11.3137	0.705730	0.352865	−	0.935674i	$-0.385208\pi$
$257$	11.3137	0.705730	0.352865	+	0.935674i	$0.385208\pi$
$258$	−11.3137	−0.704361
$259$	0	0
$260$	0	0
$261$	10.0000	0.618984
$262$	11.3137	0.698963
$263$	−16.0000	−0.986602	−0.493301	−	0.869859i	$-0.664210\pi$
$263$	−16.0000	−0.986602	−0.493301	+	0.869859i	$0.664210\pi$
$264$	−2.82843	−0.174078
$265$	0	0
$266$	0	0
$267$	−32.0000	−1.95837
$268$	4.00000	0.244339
$269$	5.65685	0.344904	0.172452	−	0.985018i	$-0.444831\pi$
$269$	5.65685	0.344904	0.172452	+	0.985018i	$0.444831\pi$
$270$	0	0
$271$	16.9706	1.03089	0.515444	−	0.856923i	$-0.327627\pi$
$271$	16.9706	1.03089	0.515444	+	0.856923i	$0.327627\pi$
$272$	−2.82843	−0.171499
$273$	0	0
$274$	2.00000	0.120824
$275$	5.00000	0.301511
$276$	−22.6274	−1.36201
$277$	22.0000	1.32185	0.660926	−	0.750451i	$-0.270164\pi$
$277$	22.0000	1.32185	0.660926	+	0.750451i	$0.270164\pi$
$278$	−16.9706	−1.01783
$279$	−42.4264	−2.54000
$280$	0	0
$281$	−10.0000	−0.596550	−0.298275	−	0.954480i	$-0.596411\pi$
$281$	−10.0000	−0.596550	−0.298275	+	0.954480i	$0.596411\pi$
$282$	−8.00000	−0.476393
$283$	−11.3137	−0.672530	−0.336265	−	0.941767i	$-0.609164\pi$
$283$	−11.3137	−0.672530	−0.336265	+	0.941767i	$0.609164\pi$
$284$	0	0
$285$	0	0
$286$	2.82843	0.167248
$287$	0	0
$288$	−5.00000	−0.294628
$289$	−9.00000	−0.529412
$290$	0	0
$291$	−48.0000	−2.81381
$292$	14.1421	0.827606
$293$	−14.1421	−0.826192	−0.413096	−	0.910687i	$-0.635553\pi$
$293$	−14.1421	−0.826192	−0.413096	+	0.910687i	$0.635553\pi$
$294$	0	0
$295$	0	0
$296$	−2.00000	−0.116248
$297$	5.65685	0.328244
$298$	−6.00000	−0.347571
$299$	22.6274	1.30858
$300$	14.1421	0.816497
$301$	0	0
$302$	−16.0000	−0.920697
$303$	−8.00000	−0.459588
$304$	−5.65685	−0.324443
$305$	0	0
$306$	14.1421	0.808452
$307$	22.6274	1.29141	0.645707	−	0.763585i	$-0.276563\pi$
$307$	22.6274	1.29141	0.645707	+	0.763585i	$0.276563\pi$
$308$	0	0
$309$	40.0000	2.27552
$310$	0	0
$311$	31.1127	1.76424	0.882120	−	0.471025i	$-0.156116\pi$
$311$	31.1127	1.76424	0.882120	+	0.471025i	$0.156116\pi$
$312$	8.00000	0.452911
$313$	11.3137	0.639489	0.319744	−	0.947504i	$-0.396403\pi$
$313$	11.3137	0.639489	0.319744	+	0.947504i	$0.396403\pi$
$314$	−11.3137	−0.638470
$315$	0	0
$316$	−16.0000	−0.900070
$317$	−2.00000	−0.112331	−0.0561656	−	0.998421i	$-0.517887\pi$
$317$	−2.00000	−0.112331	−0.0561656	+	0.998421i	$0.517887\pi$
$318$	39.5980	2.22054
$319$	−2.00000	−0.111979
$320$	0	0
$321$	−33.9411	−1.89441
$322$	0	0
$323$	16.0000	0.890264
$324$	1.00000	0.0555556
$325$	−14.1421	−0.784465
$326$	4.00000	0.221540
$327$	−5.65685	−0.312825
$328$	2.82843	0.156174
$329$	0	0
$330$	0	0
$331$	−28.0000	−1.53902	−0.769510	−	0.638635i	$-0.779499\pi$
$331$	−28.0000	−1.53902	−0.769510	+	0.638635i	$0.779499\pi$
$332$	16.9706	0.931381
$333$	10.0000	0.547997
$334$	−11.3137	−0.619059
$335$	0	0
$336$	0	0
$337$	14.0000	0.762629	0.381314	−	0.924445i	$-0.375472\pi$
$337$	14.0000	0.762629	0.381314	+	0.924445i	$0.375472\pi$
$338$	5.00000	0.271964
$339$	−5.65685	−0.307238
$340$	0	0
$341$	8.48528	0.459504
$342$	28.2843	1.52944
$343$	0	0
$344$	4.00000	0.215666
$345$	0	0
$346$	19.7990	1.06440
$347$	28.0000	1.50312	0.751559	−	0.659665i	$-0.229302\pi$
$347$	28.0000	1.50312	0.751559	+	0.659665i	$0.229302\pi$
$348$	−5.65685	−0.303239
$349$	−25.4558	−1.36262	−0.681310	−	0.731995i	$-0.738589\pi$
$349$	−25.4558	−1.36262	−0.681310	+	0.731995i	$0.738589\pi$
$350$	0	0
$351$	−16.0000	−0.854017
$352$	1.00000	0.0533002
$353$	11.3137	0.602168	0.301084	−	0.953598i	$-0.402652\pi$
$353$	11.3137	0.602168	0.301084	+	0.953598i	$0.402652\pi$
$354$	24.0000	1.27559
$355$	0	0
$356$	11.3137	0.599625
$357$	0	0
$358$	−20.0000	−1.05703
$359$	32.0000	1.68890	0.844448	−	0.535638i	$-0.179929\pi$
$359$	32.0000	1.68890	0.844448	+	0.535638i	$0.179929\pi$
$360$	0	0
$361$	13.0000	0.684211
$362$	5.65685	0.297318
$363$	−2.82843	−0.148454
$364$	0	0
$365$	0	0
$366$	24.0000	1.25450
$367$	8.48528	0.442928	0.221464	−	0.975169i	$-0.428916\pi$
$367$	8.48528	0.442928	0.221464	+	0.975169i	$0.428916\pi$
$368$	8.00000	0.417029
$369$	−14.1421	−0.736210
$370$	0	0
$371$	0	0
$372$	24.0000	1.24434
$373$	−10.0000	−0.517780	−0.258890	−	0.965907i	$-0.583357\pi$
$373$	−10.0000	−0.517780	−0.258890	+	0.965907i	$0.583357\pi$
$374$	−2.82843	−0.146254
$375$	0	0
$376$	2.82843	0.145865
$377$	5.65685	0.291343
$378$	0	0
$379$	−12.0000	−0.616399	−0.308199	−	0.951322i	$-0.599726\pi$
$379$	−12.0000	−0.616399	−0.308199	+	0.951322i	$0.599726\pi$
$380$	0	0
$381$	0	0
$382$	8.00000	0.409316
$383$	2.82843	0.144526	0.0722629	−	0.997386i	$-0.476978\pi$
$383$	2.82843	0.144526	0.0722629	+	0.997386i	$0.476978\pi$
$384$	2.82843	0.144338
$385$	0	0
$386$	14.0000	0.712581
$387$	−20.0000	−1.01666
$388$	16.9706	0.861550
$389$	26.0000	1.31825	0.659126	−	0.752032i	$-0.270926\pi$
$389$	26.0000	1.31825	0.659126	+	0.752032i	$0.270926\pi$
$390$	0	0
$391$	−22.6274	−1.14432
$392$	0	0
$393$	32.0000	1.61419
$394$	10.0000	0.503793
$395$	0	0
$396$	−5.00000	−0.251259
$397$	22.6274	1.13564	0.567819	−	0.823154i	$-0.307787\pi$
$397$	22.6274	1.13564	0.567819	+	0.823154i	$0.307787\pi$
$398$	−19.7990	−0.992434
$399$	0	0
$400$	−5.00000	−0.250000
$401$	30.0000	1.49813	0.749064	−	0.662497i	$-0.230503\pi$
$401$	30.0000	1.49813	0.749064	+	0.662497i	$0.230503\pi$
$402$	11.3137	0.564276
$403$	−24.0000	−1.19553
$404$	2.82843	0.140720
$405$	0	0
$406$	0	0
$407$	−2.00000	−0.0991363
$408$	−8.00000	−0.396059
$409$	−8.48528	−0.419570	−0.209785	−	0.977748i	$-0.567276\pi$
$409$	−8.48528	−0.419570	−0.209785	+	0.977748i	$0.567276\pi$
$410$	0	0
$411$	5.65685	0.279032
$412$	−14.1421	−0.696733
$413$	0	0
$414$	−40.0000	−1.96589
$415$	0	0
$416$	−2.82843	−0.138675
$417$	−48.0000	−2.35057
$418$	−5.65685	−0.276686
$419$	−25.4558	−1.24360	−0.621800	−	0.783176i	$-0.713598\pi$
$419$	−25.4558	−1.24360	−0.621800	+	0.783176i	$0.713598\pi$
$420$	0	0
$421$	14.0000	0.682318	0.341159	−	0.940006i	$-0.389181\pi$
$421$	14.0000	0.682318	0.341159	+	0.940006i	$0.389181\pi$
$422$	−4.00000	−0.194717
$423$	−14.1421	−0.687614
$424$	−14.0000	−0.679900
$425$	14.1421	0.685994
$426$	0	0
$427$	0	0
$428$	12.0000	0.580042
$429$	8.00000	0.386244
$430$	0	0
$431$	−24.0000	−1.15604	−0.578020	−	0.816023i	$-0.696174\pi$
$431$	−24.0000	−1.15604	−0.578020	+	0.816023i	$0.696174\pi$
$432$	−5.65685	−0.272166
$433$	16.9706	0.815553	0.407777	−	0.913082i	$-0.366304\pi$
$433$	16.9706	0.815553	0.407777	+	0.913082i	$0.366304\pi$
$434$	0	0
$435$	0	0
$436$	2.00000	0.0957826
$437$	−45.2548	−2.16483
$438$	40.0000	1.91127
$439$	−16.9706	−0.809961	−0.404980	−	0.914325i	$-0.632722\pi$
$439$	−16.9706	−0.809961	−0.404980	+	0.914325i	$0.632722\pi$
$440$	0	0
$441$	0	0
$442$	8.00000	0.380521
$443$	12.0000	0.570137	0.285069	−	0.958507i	$-0.407984\pi$
$443$	12.0000	0.570137	0.285069	+	0.958507i	$0.407984\pi$
$444$	−5.65685	−0.268462
$445$	0	0
$446$	14.1421	0.669650
$447$	−16.9706	−0.802680
$448$	0	0
$449$	10.0000	0.471929	0.235965	−	0.971762i	$-0.424175\pi$
$449$	10.0000	0.471929	0.235965	+	0.971762i	$0.424175\pi$
$450$	25.0000	1.17851
$451$	2.82843	0.133185
$452$	2.00000	0.0940721
$453$	−45.2548	−2.12626
$454$	11.3137	0.530979
$455$	0	0
$456$	−16.0000	−0.749269
$457$	−22.0000	−1.02912	−0.514558	−	0.857455i	$-0.672044\pi$
$457$	−22.0000	−1.02912	−0.514558	+	0.857455i	$0.672044\pi$
$458$	−5.65685	−0.264327
$459$	16.0000	0.746816
$460$	0	0
$461$	19.7990	0.922131	0.461065	−	0.887366i	$-0.347467\pi$
$461$	19.7990	0.922131	0.461065	+	0.887366i	$0.347467\pi$
$462$	0	0
$463$	−8.00000	−0.371792	−0.185896	−	0.982569i	$-0.559519\pi$
$463$	−8.00000	−0.371792	−0.185896	+	0.982569i	$0.559519\pi$
$464$	2.00000	0.0928477
$465$	0	0
$466$	−22.0000	−1.01913
$467$	25.4558	1.17796	0.588978	−	0.808149i	$-0.299530\pi$
$467$	25.4558	1.17796	0.588978	+	0.808149i	$0.299530\pi$
$468$	14.1421	0.653720
$469$	0	0
$470$	0	0
$471$	−32.0000	−1.47448
$472$	−8.48528	−0.390567
$473$	4.00000	0.183920
$474$	−45.2548	−2.07862
$475$	28.2843	1.29777
$476$	0	0
$477$	70.0000	3.20508
$478$	16.0000	0.731823
$479$	16.9706	0.775405	0.387702	−	0.921785i	$-0.373269\pi$
$479$	16.9706	0.775405	0.387702	+	0.921785i	$0.373269\pi$
$480$	0	0
$481$	5.65685	0.257930
$482$	14.1421	0.644157
$483$	0	0
$484$	1.00000	0.0454545
$485$	0	0
$486$	−14.1421	−0.641500
$487$	−8.00000	−0.362515	−0.181257	−	0.983436i	$-0.558017\pi$
$487$	−8.00000	−0.362515	−0.181257	+	0.983436i	$0.558017\pi$
$488$	−8.48528	−0.384111
$489$	11.3137	0.511624
$490$	0	0
$491$	−28.0000	−1.26362	−0.631811	−	0.775122i	$-0.717688\pi$
$491$	−28.0000	−1.26362	−0.631811	+	0.775122i	$0.717688\pi$
$492$	8.00000	0.360668
$493$	−5.65685	−0.254772
$494$	16.0000	0.719874
$495$	0	0
$496$	−8.48528	−0.381000
$497$	0	0
$498$	48.0000	2.15093
$499$	−12.0000	−0.537194	−0.268597	−	0.963253i	$-0.586560\pi$
$499$	−12.0000	−0.537194	−0.268597	+	0.963253i	$0.586560\pi$
$500$	0	0
$501$	−32.0000	−1.42965
$502$	−14.1421	−0.631194
$503$	−5.65685	−0.252227	−0.126113	−	0.992016i	$-0.540250\pi$
$503$	−5.65685	−0.252227	−0.126113	+	0.992016i	$0.540250\pi$
$504$	0	0
$505$	0	0
$506$	8.00000	0.355643
$507$	14.1421	0.628074
$508$	0	0
$509$	16.9706	0.752207	0.376103	−	0.926578i	$-0.377264\pi$
$509$	16.9706	0.752207	0.376103	+	0.926578i	$0.377264\pi$
$510$	0	0
$511$	0	0
$512$	−1.00000	−0.0441942
$513$	32.0000	1.41283
$514$	−11.3137	−0.499026
$515$	0	0
$516$	11.3137	0.498058
$517$	2.82843	0.124394
$518$	0	0
$519$	56.0000	2.45813
$520$	0	0
$521$	−39.5980	−1.73482	−0.867409	−	0.497595i	$-0.834217\pi$
$521$	−39.5980	−1.73482	−0.867409	+	0.497595i	$0.834217\pi$
$522$	−10.0000	−0.437688
$523$	−33.9411	−1.48414	−0.742071	−	0.670321i	$-0.766156\pi$
$523$	−33.9411	−1.48414	−0.742071	+	0.670321i	$0.766156\pi$
$524$	−11.3137	−0.494242
$525$	0	0
$526$	16.0000	0.697633
$527$	24.0000	1.04546
$528$	2.82843	0.123091
$529$	41.0000	1.78261
$530$	0	0
$531$	42.4264	1.84115
$532$	0	0
$533$	−8.00000	−0.346518
$534$	32.0000	1.38478
$535$	0	0
$536$	−4.00000	−0.172774
$537$	−56.5685	−2.44111
$538$	−5.65685	−0.243884
$539$	0	0
$540$	0	0
$541$	−18.0000	−0.773880	−0.386940	−	0.922105i	$-0.626468\pi$
$541$	−18.0000	−0.773880	−0.386940	+	0.922105i	$0.626468\pi$
$542$	−16.9706	−0.728948
$543$	16.0000	0.686626
$544$	2.82843	0.121268
$545$	0	0
$546$	0	0
$547$	28.0000	1.19719	0.598597	−	0.801050i	$-0.295725\pi$
$547$	28.0000	1.19719	0.598597	+	0.801050i	$0.295725\pi$
$548$	−2.00000	−0.0854358
$549$	42.4264	1.81071
$550$	−5.00000	−0.213201
$551$	−11.3137	−0.481980
$552$	22.6274	0.963087
$553$	0	0
$554$	−22.0000	−0.934690
$555$	0	0
$556$	16.9706	0.719712
$557$	−18.0000	−0.762684	−0.381342	−	0.924434i	$-0.624538\pi$
$557$	−18.0000	−0.762684	−0.381342	+	0.924434i	$0.624538\pi$
$558$	42.4264	1.79605
$559$	−11.3137	−0.478519
$560$	0	0
$561$	−8.00000	−0.337760
$562$	10.0000	0.421825
$563$	−28.2843	−1.19204	−0.596020	−	0.802970i	$-0.703252\pi$
$563$	−28.2843	−1.19204	−0.596020	+	0.802970i	$0.703252\pi$
$564$	8.00000	0.336861
$565$	0	0
$566$	11.3137	0.475551
$567$	0	0
$568$	0	0
$569$	6.00000	0.251533	0.125767	−	0.992060i	$-0.459861\pi$
$569$	6.00000	0.251533	0.125767	+	0.992060i	$0.459861\pi$
$570$	0	0
$571$	4.00000	0.167395	0.0836974	−	0.996491i	$-0.473327\pi$
$571$	4.00000	0.167395	0.0836974	+	0.996491i	$0.473327\pi$
$572$	−2.82843	−0.118262
$573$	22.6274	0.945274
$574$	0	0
$575$	−40.0000	−1.66812
$576$	5.00000	0.208333
$577$	−39.5980	−1.64849	−0.824243	−	0.566237i	$-0.808399\pi$
$577$	−39.5980	−1.64849	−0.824243	+	0.566237i	$0.808399\pi$
$578$	9.00000	0.374351
$579$	39.5980	1.64564
$580$	0	0
$581$	0	0
$582$	48.0000	1.98966
$583$	−14.0000	−0.579821
$584$	−14.1421	−0.585206
$585$	0	0
$586$	14.1421	0.584206
$587$	14.1421	0.583708	0.291854	−	0.956463i	$-0.405728\pi$
$587$	14.1421	0.583708	0.291854	+	0.956463i	$0.405728\pi$
$588$	0	0
$589$	48.0000	1.97781
$590$	0	0
$591$	28.2843	1.16346
$592$	2.00000	0.0821995
$593$	−42.4264	−1.74224	−0.871122	−	0.491067i	$-0.836607\pi$
$593$	−42.4264	−1.74224	−0.871122	+	0.491067i	$0.836607\pi$
$594$	−5.65685	−0.232104
$595$	0	0
$596$	6.00000	0.245770
$597$	−56.0000	−2.29193
$598$	−22.6274	−0.925304
$599$	0	0		−	1.00000i	$-0.5\pi$
$599$	0	0			1.00000i	$0.5\pi$
$600$	−14.1421	−0.577350
$601$	−25.4558	−1.03837	−0.519183	−	0.854663i	$-0.673764\pi$
$601$	−25.4558	−1.03837	−0.519183	+	0.854663i	$0.673764\pi$
$602$	0	0
$603$	20.0000	0.814463
$604$	16.0000	0.651031
$605$	0	0
$606$	8.00000	0.324978
$607$	33.9411	1.37763	0.688814	−	0.724938i	$-0.258132\pi$
$607$	33.9411	1.37763	0.688814	+	0.724938i	$0.258132\pi$
$608$	5.65685	0.229416
$609$	0	0
$610$	0	0
$611$	−8.00000	−0.323645
$612$	−14.1421	−0.571662
$613$	−38.0000	−1.53481	−0.767403	−	0.641165i	$-0.778451\pi$
$613$	−38.0000	−1.53481	−0.767403	+	0.641165i	$0.778451\pi$
$614$	−22.6274	−0.913168
$615$	0	0
$616$	0	0
$617$	14.0000	0.563619	0.281809	−	0.959470i	$-0.409065\pi$
$617$	14.0000	0.563619	0.281809	+	0.959470i	$0.409065\pi$
$618$	−40.0000	−1.60904
$619$	2.82843	0.113684	0.0568420	−	0.998383i	$-0.481897\pi$
$619$	2.82843	0.113684	0.0568420	+	0.998383i	$0.481897\pi$
$620$	0	0
$621$	−45.2548	−1.81601
$622$	−31.1127	−1.24751
$623$	0	0
$624$	−8.00000	−0.320256
$625$	25.0000	1.00000
$626$	−11.3137	−0.452187
$627$	−16.0000	−0.638978
$628$	11.3137	0.451466
$629$	−5.65685	−0.225554
$630$	0	0
$631$	48.0000	1.91085	0.955425	−	0.295234i	$-0.0953977\pi$
$631$	48.0000	1.91085	0.955425	+	0.295234i	$0.0953977\pi$
$632$	16.0000	0.636446
$633$	−11.3137	−0.449680
$634$	2.00000	0.0794301
$635$	0	0
$636$	−39.5980	−1.57016
$637$	0	0
$638$	2.00000	0.0791808
$639$	0	0
$640$	0	0
$641$	−26.0000	−1.02694	−0.513469	−	0.858108i	$-0.671640\pi$
$641$	−26.0000	−1.02694	−0.513469	+	0.858108i	$0.671640\pi$
$642$	33.9411	1.33955
$643$	25.4558	1.00388	0.501940	−	0.864902i	$-0.332620\pi$
$643$	25.4558	1.00388	0.501940	+	0.864902i	$0.332620\pi$
$644$	0	0
$645$	0	0
$646$	−16.0000	−0.629512
$647$	14.1421	0.555985	0.277992	−	0.960583i	$-0.410331\pi$
$647$	14.1421	0.555985	0.277992	+	0.960583i	$0.410331\pi$
$648$	−1.00000	−0.0392837
$649$	−8.48528	−0.333076
$650$	14.1421	0.554700
$651$	0	0
$652$	−4.00000	−0.156652
$653$	18.0000	0.704394	0.352197	−	0.935926i	$-0.385435\pi$
$653$	18.0000	0.704394	0.352197	+	0.935926i	$0.385435\pi$
$654$	5.65685	0.221201
$655$	0	0
$656$	−2.82843	−0.110432
$657$	70.7107	2.75869
$658$	0	0
$659$	−28.0000	−1.09073	−0.545363	−	0.838200i	$-0.683608\pi$
$659$	−28.0000	−1.09073	−0.545363	+	0.838200i	$0.683608\pi$
$660$	0	0
$661$	0	0		−	1.00000i	$-0.5\pi$
$661$	0	0			1.00000i	$0.5\pi$
$662$	28.0000	1.08825
$663$	22.6274	0.878776
$664$	−16.9706	−0.658586
$665$	0	0
$666$	−10.0000	−0.387492
$667$	16.0000	0.619522
$668$	11.3137	0.437741
$669$	40.0000	1.54649
$670$	0	0
$671$	−8.48528	−0.327571
$672$	0	0
$673$	14.0000	0.539660	0.269830	−	0.962908i	$-0.413032\pi$
$673$	14.0000	0.539660	0.269830	+	0.962908i	$0.413032\pi$
$674$	−14.0000	−0.539260
$675$	28.2843	1.08866
$676$	−5.00000	−0.192308
$677$	2.82843	0.108705	0.0543526	−	0.998522i	$-0.482690\pi$
$677$	2.82843	0.108705	0.0543526	+	0.998522i	$0.482690\pi$
$678$	5.65685	0.217250
$679$	0	0
$680$	0	0
$681$	32.0000	1.22624
$682$	−8.48528	−0.324918
$683$	28.0000	1.07139	0.535695	−	0.844411i	$-0.320050\pi$
$683$	28.0000	1.07139	0.535695	+	0.844411i	$0.320050\pi$
$684$	−28.2843	−1.08148
$685$	0	0
$686$	0	0
$687$	−16.0000	−0.610438
$688$	−4.00000	−0.152499
$689$	39.5980	1.50856
$690$	0	0
$691$	8.48528	0.322795	0.161398	−	0.986889i	$-0.448400\pi$
$691$	8.48528	0.322795	0.161398	+	0.986889i	$0.448400\pi$
$692$	−19.7990	−0.752645
$693$	0	0
$694$	−28.0000	−1.06287
$695$	0	0
$696$	5.65685	0.214423
$697$	8.00000	0.303022
$698$	25.4558	0.963518
$699$	−62.2254	−2.35358
$700$	0	0
$701$	18.0000	0.679851	0.339925	−	0.940452i	$-0.389598\pi$
$701$	18.0000	0.679851	0.339925	+	0.940452i	$0.389598\pi$
$702$	16.0000	0.603881
$703$	−11.3137	−0.426705
$704$	−1.00000	−0.0376889
$705$	0	0
$706$	−11.3137	−0.425797
$707$	0	0
$708$	−24.0000	−0.901975
$709$	26.0000	0.976450	0.488225	−	0.872718i	$-0.337644\pi$
$709$	26.0000	0.976450	0.488225	+	0.872718i	$0.337644\pi$
$710$	0	0
$711$	−80.0000	−3.00023
$712$	−11.3137	−0.423999
$713$	−67.8823	−2.54221
$714$	0	0
$715$	0	0
$716$	20.0000	0.747435
$717$	45.2548	1.69007
$718$	−32.0000	−1.19423
$719$	−19.7990	−0.738378	−0.369189	−	0.929354i	$-0.620364\pi$
$719$	−19.7990	−0.738378	−0.369189	+	0.929354i	$0.620364\pi$
$720$	0	0
$721$	0	0
$722$	−13.0000	−0.483810
$723$	40.0000	1.48762
$724$	−5.65685	−0.210235
$725$	−10.0000	−0.371391
$726$	2.82843	0.104973
$727$	−19.7990	−0.734304	−0.367152	−	0.930161i	$-0.619667\pi$
$727$	−19.7990	−0.734304	−0.367152	+	0.930161i	$0.619667\pi$
$728$	0	0
$729$	−43.0000	−1.59259
$730$	0	0
$731$	11.3137	0.418453
$732$	−24.0000	−0.887066
$733$	−48.0833	−1.77600	−0.887998	−	0.459848i	$-0.847904\pi$
$733$	−48.0833	−1.77600	−0.887998	+	0.459848i	$0.847904\pi$
$734$	−8.48528	−0.313197
$735$	0	0
$736$	−8.00000	−0.294884
$737$	−4.00000	−0.147342
$738$	14.1421	0.520579
$739$	−20.0000	−0.735712	−0.367856	−	0.929883i	$-0.619908\pi$
$739$	−20.0000	−0.735712	−0.367856	+	0.929883i	$0.619908\pi$
$740$	0	0
$741$	45.2548	1.66248
$742$	0	0
$743$	−8.00000	−0.293492	−0.146746	−	0.989174i	$-0.546880\pi$
$743$	−8.00000	−0.293492	−0.146746	+	0.989174i	$0.546880\pi$
$744$	−24.0000	−0.879883
$745$	0	0
$746$	10.0000	0.366126
$747$	84.8528	3.10460
$748$	2.82843	0.103418
$749$	0	0
$750$	0	0
$751$	−32.0000	−1.16770	−0.583848	−	0.811863i	$-0.698454\pi$
$751$	−32.0000	−1.16770	−0.583848	+	0.811863i	$0.698454\pi$
$752$	−2.82843	−0.103142
$753$	−40.0000	−1.45768
$754$	−5.65685	−0.206010
$755$	0	0
$756$	0	0
$757$	10.0000	0.363456	0.181728	−	0.983349i	$-0.441831\pi$
$757$	10.0000	0.363456	0.181728	+	0.983349i	$0.441831\pi$
$758$	12.0000	0.435860
$759$	22.6274	0.821323
$760$	0	0
$761$	−8.48528	−0.307591	−0.153796	−	0.988103i	$-0.549150\pi$
$761$	−8.48528	−0.307591	−0.153796	+	0.988103i	$0.549150\pi$
$762$	0	0
$763$	0	0
$764$	−8.00000	−0.289430
$765$	0	0
$766$	−2.82843	−0.102195
$767$	24.0000	0.866590
$768$	−2.82843	−0.102062
$769$	2.82843	0.101996	0.0509978	−	0.998699i	$-0.483760\pi$
$769$	2.82843	0.101996	0.0509978	+	0.998699i	$0.483760\pi$
$770$	0	0
$771$	−32.0000	−1.15245
$772$	−14.0000	−0.503871
$773$	−28.2843	−1.01731	−0.508657	−	0.860969i	$-0.669858\pi$
$773$	−28.2843	−1.01731	−0.508657	+	0.860969i	$0.669858\pi$
$774$	20.0000	0.718885
$775$	42.4264	1.52400
$776$	−16.9706	−0.609208
$777$	0	0
$778$	−26.0000	−0.932145
$779$	16.0000	0.573259
$780$	0	0
$781$	0	0
$782$	22.6274	0.809155
$783$	−11.3137	−0.404319
$784$	0	0
$785$	0	0
$786$	−32.0000	−1.14140
$787$	39.5980	1.41152	0.705758	−	0.708453i	$-0.250607\pi$
$787$	39.5980	1.41152	0.705758	+	0.708453i	$0.250607\pi$
$788$	−10.0000	−0.356235
$789$	45.2548	1.61111
$790$	0	0
$791$	0	0
$792$	5.00000	0.177667
$793$	24.0000	0.852265
$794$	−22.6274	−0.803017
$795$	0	0
$796$	19.7990	0.701757
$797$	11.3137	0.400752	0.200376	−	0.979719i	$-0.435784\pi$
$797$	11.3137	0.400752	0.200376	+	0.979719i	$0.435784\pi$
$798$	0	0
$799$	8.00000	0.283020
$800$	5.00000	0.176777
$801$	56.5685	1.99875
$802$	−30.0000	−1.05934
$803$	−14.1421	−0.499065
$804$	−11.3137	−0.399004
$805$	0	0
$806$	24.0000	0.845364
$807$	−16.0000	−0.563227
$808$	−2.82843	−0.0995037
$809$	−6.00000	−0.210949	−0.105474	−	0.994422i	$-0.533636\pi$
$809$	−6.00000	−0.210949	−0.105474	+	0.994422i	$0.533636\pi$
$810$	0	0
$811$	22.6274	0.794556	0.397278	−	0.917698i	$-0.369955\pi$
$811$	22.6274	0.794556	0.397278	+	0.917698i	$0.369955\pi$
$812$	0	0
$813$	−48.0000	−1.68343
$814$	2.00000	0.0701000
$815$	0	0
$816$	8.00000	0.280056
$817$	22.6274	0.791633
$818$	8.48528	0.296681
$819$	0	0
$820$	0	0
$821$	6.00000	0.209401	0.104701	−	0.994504i	$-0.466612\pi$
$821$	6.00000	0.209401	0.104701	+	0.994504i	$0.466612\pi$
$822$	−5.65685	−0.197305
$823$	32.0000	1.11545	0.557725	−	0.830026i	$-0.311674\pi$
$823$	32.0000	1.11545	0.557725	+	0.830026i	$0.311674\pi$
$824$	14.1421	0.492665
$825$	−14.1421	−0.492366
$826$	0	0
$827$	−12.0000	−0.417281	−0.208640	−	0.977992i	$-0.566904\pi$
$827$	−12.0000	−0.417281	−0.208640	+	0.977992i	$0.566904\pi$
$828$	40.0000	1.39010
$829$	−11.3137	−0.392941	−0.196471	−	0.980510i	$-0.562948\pi$
$829$	−11.3137	−0.392941	−0.196471	+	0.980510i	$0.562948\pi$
$830$	0	0
$831$	−62.2254	−2.15858
$832$	2.82843	0.0980581
$833$	0	0
$834$	48.0000	1.66210
$835$	0	0
$836$	5.65685	0.195646
$837$	48.0000	1.65912
$838$	25.4558	0.879358
$839$	2.82843	0.0976481	0.0488241	−	0.998807i	$-0.484453\pi$
$839$	2.82843	0.0976481	0.0488241	+	0.998807i	$0.484453\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	−14.0000	−0.482472
$843$	28.2843	0.974162
$844$	4.00000	0.137686
$845$	0	0
$846$	14.1421	0.486217
$847$	0	0
$848$	14.0000	0.480762
$849$	32.0000	1.09824
$850$	−14.1421	−0.485071
$851$	16.0000	0.548473
$852$	0	0
$853$	−8.48528	−0.290531	−0.145265	−	0.989393i	$-0.546404\pi$
$853$	−8.48528	−0.290531	−0.145265	+	0.989393i	$0.546404\pi$
$854$	0	0
$855$	0	0
$856$	−12.0000	−0.410152
$857$	−42.4264	−1.44926	−0.724629	−	0.689139i	$-0.757989\pi$
$857$	−42.4264	−1.44926	−0.724629	+	0.689139i	$0.757989\pi$
$858$	−8.00000	−0.273115
$859$	25.4558	0.868542	0.434271	−	0.900782i	$-0.357006\pi$
$859$	25.4558	0.868542	0.434271	+	0.900782i	$0.357006\pi$
$860$	0	0
$861$	0	0
$862$	24.0000	0.817443
$863$	−40.0000	−1.36162	−0.680808	−	0.732462i	$-0.738371\pi$
$863$	−40.0000	−1.36162	−0.680808	+	0.732462i	$0.738371\pi$
$864$	5.65685	0.192450
$865$	0	0
$866$	−16.9706	−0.576683
$867$	25.4558	0.864526
$868$	0	0
$869$	16.0000	0.542763
$870$	0	0
$871$	11.3137	0.383350
$872$	−2.00000	−0.0677285
$873$	84.8528	2.87183
$874$	45.2548	1.53077
$875$	0	0
$876$	−40.0000	−1.35147
$877$	−46.0000	−1.55331	−0.776655	−	0.629926i	$-0.783085\pi$
$877$	−46.0000	−1.55331	−0.776655	+	0.629926i	$0.783085\pi$
$878$	16.9706	0.572729
$879$	40.0000	1.34917
$880$	0	0
$881$	5.65685	0.190584	0.0952921	−	0.995449i	$-0.469621\pi$
$881$	5.65685	0.190584	0.0952921	+	0.995449i	$0.469621\pi$
$882$	0	0
$883$	36.0000	1.21150	0.605748	−	0.795656i	$-0.292874\pi$
$883$	36.0000	1.21150	0.605748	+	0.795656i	$0.292874\pi$
$884$	−8.00000	−0.269069
$885$	0	0
$886$	−12.0000	−0.403148
$887$	−28.2843	−0.949693	−0.474846	−	0.880069i	$-0.657496\pi$
$887$	−28.2843	−0.949693	−0.474846	+	0.880069i	$0.657496\pi$
$888$	5.65685	0.189832
$889$	0	0
$890$	0	0
$891$	−1.00000	−0.0335013
$892$	−14.1421	−0.473514
$893$	16.0000	0.535420
$894$	16.9706	0.567581
$895$	0	0
$896$	0	0
$897$	−64.0000	−2.13690
$898$	−10.0000	−0.333704
$899$	−16.9706	−0.566000
$900$	−25.0000	−0.833333
$901$	−39.5980	−1.31920
$902$	−2.82843	−0.0941763
$903$	0	0
$904$	−2.00000	−0.0665190
$905$	0	0
$906$	45.2548	1.50349
$907$	28.0000	0.929725	0.464862	−	0.885383i	$-0.346104\pi$
$907$	28.0000	0.929725	0.464862	+	0.885383i	$0.346104\pi$
$908$	−11.3137	−0.375459
$909$	14.1421	0.469065
$910$	0	0
$911$	−48.0000	−1.59031	−0.795155	−	0.606406i	$-0.792611\pi$
$911$	−48.0000	−1.59031	−0.795155	+	0.606406i	$0.792611\pi$
$912$	16.0000	0.529813
$913$	−16.9706	−0.561644
$914$	22.0000	0.727695
$915$	0	0
$916$	5.65685	0.186908
$917$	0	0
$918$	−16.0000	−0.528079
$919$	32.0000	1.05558	0.527791	−	0.849374i	$-0.323020\pi$
$919$	32.0000	1.05558	0.527791	+	0.849374i	$0.323020\pi$
$920$	0	0
$921$	−64.0000	−2.10887
$922$	−19.7990	−0.652045
$923$	0	0
$924$	0	0
$925$	−10.0000	−0.328798
$926$	8.00000	0.262896
$927$	−70.7107	−2.32244
$928$	−2.00000	−0.0656532
$929$	28.2843	0.927977	0.463988	−	0.885841i	$-0.346418\pi$
$929$	28.2843	0.927977	0.463988	+	0.885841i	$0.346418\pi$
$930$	0	0
$931$	0	0
$932$	22.0000	0.720634
$933$	−88.0000	−2.88099
$934$	−25.4558	−0.832941
$935$	0	0
$936$	−14.1421	−0.462250
$937$	−42.4264	−1.38601	−0.693005	−	0.720933i	$-0.743714\pi$
$937$	−42.4264	−1.38601	−0.693005	+	0.720933i	$0.743714\pi$
$938$	0	0
$939$	−32.0000	−1.04428
$940$	0	0
$941$	−2.82843	−0.0922041	−0.0461020	−	0.998937i	$-0.514680\pi$
$941$	−2.82843	−0.0922041	−0.0461020	+	0.998937i	$0.514680\pi$
$942$	32.0000	1.04262
$943$	−22.6274	−0.736850
$944$	8.48528	0.276172
$945$	0	0
$946$	−4.00000	−0.130051
$947$	−4.00000	−0.129983	−0.0649913	−	0.997886i	$-0.520702\pi$
$947$	−4.00000	−0.129983	−0.0649913	+	0.997886i	$0.520702\pi$
$948$	45.2548	1.46981
$949$	40.0000	1.29845
$950$	−28.2843	−0.917663
$951$	5.65685	0.183436
$952$	0	0
$953$	−22.0000	−0.712650	−0.356325	−	0.934362i	$-0.615970\pi$
$953$	−22.0000	−0.712650	−0.356325	+	0.934362i	$0.615970\pi$
$954$	−70.0000	−2.26633
$955$	0	0
$956$	−16.0000	−0.517477
$957$	5.65685	0.182860
$958$	−16.9706	−0.548294
$959$	0	0
$960$	0	0
$961$	41.0000	1.32258
$962$	−5.65685	−0.182384
$963$	60.0000	1.93347
$964$	−14.1421	−0.455488
$965$	0	0
$966$	0	0
$967$	16.0000	0.514525	0.257263	−	0.966342i	$-0.417179\pi$
$967$	16.0000	0.514525	0.257263	+	0.966342i	$0.417179\pi$
$968$	−1.00000	−0.0321412
$969$	−45.2548	−1.45379
$970$	0	0
$971$	36.7696	1.17999	0.589996	−	0.807406i	$-0.299129\pi$
$971$	36.7696	1.17999	0.589996	+	0.807406i	$0.299129\pi$
$972$	14.1421	0.453609
$973$	0	0
$974$	8.00000	0.256337
$975$	40.0000	1.28103
$976$	8.48528	0.271607
$977$	−18.0000	−0.575871	−0.287936	−	0.957650i	$-0.592969\pi$
$977$	−18.0000	−0.575871	−0.287936	+	0.957650i	$0.592969\pi$
$978$	−11.3137	−0.361773
$979$	−11.3137	−0.361588
$980$	0	0
$981$	10.0000	0.319275
$982$	28.0000	0.893516
$983$	−2.82843	−0.0902128	−0.0451064	−	0.998982i	$-0.514363\pi$
$983$	−2.82843	−0.0902128	−0.0451064	+	0.998982i	$0.514363\pi$
$984$	−8.00000	−0.255031
$985$	0	0
$986$	5.65685	0.180151
$987$	0	0
$988$	−16.0000	−0.509028
$989$	−32.0000	−1.01754
$990$	0	0
$991$	−40.0000	−1.27064	−0.635321	−	0.772248i	$-0.719132\pi$
$991$	−40.0000	−1.27064	−0.635321	+	0.772248i	$0.719132\pi$
$992$	8.48528	0.269408
$993$	79.1960	2.51321
$994$	0	0
$995$	0	0
$996$	−48.0000	−1.52094
$997$	−42.4264	−1.34366	−0.671829	−	0.740706i	$-0.734491\pi$
$997$	−42.4264	−1.34366	−0.671829	+	0.740706i	$0.734491\pi$
$998$	12.0000	0.379853
$999$	−11.3137	−0.357950

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1078.2.a.r.1.1	✓	2
3.2	odd	2		9702.2.a.dn.1.2		2
4.3	odd	2		8624.2.a.by.1.2		2
7.2	even	3		1078.2.e.r.67.2		4
7.3	odd	6		1078.2.e.r.177.1		4
7.4	even	3		1078.2.e.r.177.2		4
7.5	odd	6		1078.2.e.r.67.1		4
7.6	odd	2	inner	1078.2.a.r.1.2	yes	2
21.20	even	2		9702.2.a.dn.1.1		2
28.27	even	2		8624.2.a.by.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1078.2.a.r.1.1	✓	2	1.1	even	1	trivial
1078.2.a.r.1.2	yes	2	7.6	odd	2	inner
1078.2.e.r.67.1		4	7.5	odd	6
1078.2.e.r.67.2		4	7.2	even	3
1078.2.e.r.177.1		4	7.3	odd	6
1078.2.e.r.177.2		4	7.4	even	3
8624.2.a.by.1.1		2	28.27	even	2
8624.2.a.by.1.2		2	4.3	odd	2
9702.2.a.dn.1.1		2	21.20	even	2
9702.2.a.dn.1.2		2	3.2	odd	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:	\( N \)	\(=\)	\( 1078 = 2 \cdot 7^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1078.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -2.82843 q^{3} +1.00000 q^{4} +2.82843 q^{6} -1.00000 q^{8} +5.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -2.82843 q^{3} +1.00000 q^{4} +2.82843 q^{6} -1.00000 q^{8} +5.00000 q^{9} -1.00000 q^{11} -2.82843 q^{12} +2.82843 q^{13} +1.00000 q^{16} -2.82843 q^{17} -5.00000 q^{18} -5.65685 q^{19} +1.00000 q^{22} +8.00000 q^{23} +2.82843 q^{24} -5.00000 q^{25} -2.82843 q^{26} -5.65685 q^{27} +2.00000 q^{29} -8.48528 q^{31} -1.00000 q^{32} +2.82843 q^{33} +2.82843 q^{34} +5.00000 q^{36} +2.00000 q^{37} +5.65685 q^{38} -8.00000 q^{39} -2.82843 q^{41} -4.00000 q^{43} -1.00000 q^{44} -8.00000 q^{46} -2.82843 q^{47} -2.82843 q^{48} +5.00000 q^{50} +8.00000 q^{51} +2.82843 q^{52} +14.0000 q^{53} +5.65685 q^{54} +16.0000 q^{57} -2.00000 q^{58} +8.48528 q^{59} +8.48528 q^{61} +8.48528 q^{62} +1.00000 q^{64} -2.82843 q^{66} +4.00000 q^{67} -2.82843 q^{68} -22.6274 q^{69} -5.00000 q^{72} +14.1421 q^{73} -2.00000 q^{74} +14.1421 q^{75} -5.65685 q^{76} +8.00000 q^{78} -16.0000 q^{79} +1.00000 q^{81} +2.82843 q^{82} +16.9706 q^{83} +4.00000 q^{86} -5.65685 q^{87} +1.00000 q^{88} +11.3137 q^{89} +8.00000 q^{92} +24.0000 q^{93} +2.82843 q^{94} +2.82843 q^{96} +16.9706 q^{97} -5.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{8} + 10 q^{9}+O(q^{10}) \) 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^8 + 10 * q^9	\( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{8} + 10 q^{9} - 2 q^{11} + 2 q^{16} - 10 q^{18} + 2 q^{22} + 16 q^{23} - 10 q^{25} + 4 q^{29} - 2 q^{32} + 10 q^{36} + 4 q^{37} - 16 q^{39} - 8 q^{43} - 2 q^{44} - 16 q^{46} + 10 q^{50} + 16 q^{51} + 28 q^{53} + 32 q^{57} - 4 q^{58} + 2 q^{64} + 8 q^{67} - 10 q^{72} - 4 q^{74} + 16 q^{78} - 32 q^{79} + 2 q^{81} + 8 q^{86} + 2 q^{88} + 16 q^{92} + 48 q^{93} - 10 q^{99}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^8 + 10 * q^9 - 2 * q^11 + 2 * q^16 - 10 * q^18 + 2 * q^22 + 16 * q^23 - 10 * q^25 + 4 * q^29 - 2 * q^32 + 10 * q^36 + 4 * q^37 - 16 * q^39 - 8 * q^43 - 2 * q^44 - 16 * q^46 + 10 * q^50 + 16 * q^51 + 28 * q^53 + 32 * q^57 - 4 * q^58 + 2 * q^64 + 8 * q^67 - 10 * q^72 - 4 * q^74 + 16 * q^78 - 32 * q^79 + 2 * q^81 + 8 * q^86 + 2 * q^88 + 16 * q^92 + 48 * q^93 - 10 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more