LMFDB - Embedded newform 6440.2.a.z.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6440,2,Mod(1,6440)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6440, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("6440.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6440 = 2^{3} \cdot 5 \cdot 7 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6440.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:	$51.4236589017$
Analytic rank:	$1$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - x^{6} - 15x^{5} + 10x^{4} + 55x^{3} - 10x^{2} - 60x - 16 $ x^7 - x^6 - 15x^5 + 10x^4 + 55x^3 - 10x^2 - 60*x - 16
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$3.35792$ of defining polynomial
Character	$\chi$	$=$	6440.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.35792 q^{3} -1.00000 q^{5} +1.00000 q^{7} +8.27561 q^{9} +O(q^{10})$	$q+3.35792 q^{3} -1.00000 q^{5} +1.00000 q^{7} +8.27561 q^{9} -1.49718 q^{11} -5.92691 q^{13} -3.35792 q^{15} -7.21866 q^{17} -3.40016 q^{19} +3.35792 q^{21} -1.00000 q^{23} +1.00000 q^{25} +17.7151 q^{27} -6.23951 q^{29} -5.02791 q^{31} -5.02741 q^{33} -1.00000 q^{35} -8.37827 q^{37} -19.9021 q^{39} -8.37083 q^{41} -2.09538 q^{43} -8.27561 q^{45} +1.54864 q^{47} +1.00000 q^{49} -24.2396 q^{51} +6.30516 q^{53} +1.49718 q^{55} -11.4175 q^{57} +9.08653 q^{59} -6.19125 q^{61} +8.27561 q^{63} +5.92691 q^{65} +2.26819 q^{67} -3.35792 q^{69} -9.50858 q^{71} -0.788423 q^{73} +3.35792 q^{75} -1.49718 q^{77} +6.15171 q^{79} +34.6589 q^{81} +1.61520 q^{83} +7.21866 q^{85} -20.9518 q^{87} +9.09960 q^{89} -5.92691 q^{91} -16.8833 q^{93} +3.40016 q^{95} +0.113933 q^{97} -12.3901 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + q^{3} - 7 q^{5} + 7 q^{7} + 10 q^{9}+O(q^{10}) $ 7 * q + q^3 - 7 * q^5 + 7 * q^7 + 10 * q^9	$ 7 q + q^{3} - 7 q^{5} + 7 q^{7} + 10 q^{9} - 4 q^{11} + 2 q^{13} - q^{15} - 12 q^{17} - 12 q^{19} + q^{21} - 7 q^{23} + 7 q^{25} + 10 q^{27} + 13 q^{29} - 14 q^{31} - 25 q^{33} - 7 q^{35} - 29 q^{37} - 24 q^{39} + 5 q^{41} + 2 q^{43} - 10 q^{45} - 3 q^{47} + 7 q^{49} - 39 q^{51} - 3 q^{53} + 4 q^{55} + q^{57} + 29 q^{59} - 15 q^{61} + 10 q^{63} - 2 q^{65} + 7 q^{67} - q^{69} - 38 q^{71} - 15 q^{73} + q^{75} - 4 q^{77} - 37 q^{79} + 35 q^{81} + 7 q^{83} + 12 q^{85} - 3 q^{87} - 13 q^{89} + 2 q^{91} - 9 q^{93} + 12 q^{95} - 44 q^{97} + 10 q^{99}+O(q^{100}) $ 7 * q + q^3 - 7 * q^5 + 7 * q^7 + 10 * q^9 - 4 * q^11 + 2 * q^13 - q^15 - 12 * q^17 - 12 * q^19 + q^21 - 7 * q^23 + 7 * q^25 + 10 * q^27 + 13 * q^29 - 14 * q^31 - 25 * q^33 - 7 * q^35 - 29 * q^37 - 24 * q^39 + 5 * q^41 + 2 * q^43 - 10 * q^45 - 3 * q^47 + 7 * q^49 - 39 * q^51 - 3 * q^53 + 4 * q^55 + q^57 + 29 * q^59 - 15 * q^61 + 10 * q^63 - 2 * q^65 + 7 * q^67 - q^69 - 38 * q^71 - 15 * q^73 + q^75 - 4 * q^77 - 37 * q^79 + 35 * q^81 + 7 * q^83 + 12 * q^85 - 3 * q^87 - 13 * q^89 + 2 * q^91 - 9 * q^93 + 12 * q^95 - 44 * q^97 + 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 $

$2$	0	0
$2$	0	0
$3$	3.35792	1.93869	0.969347	−	0.245695i	$-0.0790160\pi$
$3$	3.35792	1.93869	0.969347	+	0.245695i	$0.0790160\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	8.27561	2.75854
$10$	0	0
$11$	−1.49718	−0.451417	−0.225708	−	0.974195i	$-0.572470\pi$
$11$	−1.49718	−0.451417	−0.225708	+	0.974195i	$0.572470\pi$
$12$	0	0
$13$	−5.92691	−1.64383	−0.821915	−	0.569610i	$-0.807094\pi$
$13$	−5.92691	−1.64383	−0.821915	+	0.569610i	$0.807094\pi$
$14$	0	0
$15$	−3.35792	−0.867011
$16$	0	0
$17$	−7.21866	−1.75078	−0.875391	−	0.483416i	$-0.839396\pi$
$17$	−7.21866	−1.75078	−0.875391	+	0.483416i	$0.839396\pi$
$18$	0	0
$19$	−3.40016	−0.780050	−0.390025	−	0.920804i	$-0.627534\pi$
$19$	−3.40016	−0.780050	−0.390025	+	0.920804i	$0.627534\pi$
$20$	0	0
$21$	3.35792	0.732758
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	17.7151	3.40927
$28$	0	0
$29$	−6.23951	−1.15865	−0.579324	−	0.815098i	$-0.696683\pi$
$29$	−6.23951	−1.15865	−0.579324	+	0.815098i	$0.696683\pi$
$30$	0	0
$31$	−5.02791	−0.903039	−0.451519	−	0.892261i	$-0.649118\pi$
$31$	−5.02791	−0.903039	−0.451519	+	0.892261i	$0.649118\pi$
$32$	0	0
$33$	−5.02741	−0.875159
$34$	0	0
$35$	−1.00000	−0.169031
$36$	0	0
$37$	−8.37827	−1.37738	−0.688690	−	0.725056i	$-0.741814\pi$
$37$	−8.37827	−1.37738	−0.688690	+	0.725056i	$0.741814\pi$
$38$	0	0
$39$	−19.9021	−3.18688
$40$	0	0
$41$	−8.37083	−1.30730	−0.653652	−	0.756795i	$-0.726764\pi$
$41$	−8.37083	−1.30730	−0.653652	+	0.756795i	$0.726764\pi$
$42$	0	0
$43$	−2.09538	−0.319542	−0.159771	−	0.987154i	$-0.551076\pi$
$43$	−2.09538	−0.319542	−0.159771	+	0.987154i	$0.551076\pi$
$44$	0	0
$45$	−8.27561	−1.23366
$46$	0	0
$47$	1.54864	0.225893	0.112946	−	0.993601i	$-0.463971\pi$
$47$	1.54864	0.225893	0.112946	+	0.993601i	$0.463971\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−24.2396	−3.39423
$52$	0	0
$53$	6.30516	0.866080	0.433040	−	0.901375i	$-0.357441\pi$
$53$	6.30516	0.866080	0.433040	+	0.901375i	$0.357441\pi$
$54$	0	0
$55$	1.49718	0.201880
$56$	0	0
$57$	−11.4175	−1.51228
$58$	0	0
$59$	9.08653	1.18297	0.591483	−	0.806318i	$-0.298543\pi$
$59$	9.08653	1.18297	0.591483	+	0.806318i	$0.298543\pi$
$60$	0	0
$61$	−6.19125	−0.792708	−0.396354	−	0.918098i	$-0.629725\pi$
$61$	−6.19125	−0.792708	−0.396354	+	0.918098i	$0.629725\pi$
$62$	0	0
$63$	8.27561	1.04263
$64$	0	0
$65$	5.92691	0.735143
$66$	0	0
$67$	2.26819	0.277103	0.138551	−	0.990355i	$-0.455755\pi$
$67$	2.26819	0.277103	0.138551	+	0.990355i	$0.455755\pi$
$68$	0	0
$69$	−3.35792	−0.404246
$70$	0	0
$71$	−9.50858	−1.12846	−0.564230	−	0.825617i	$-0.690827\pi$
$71$	−9.50858	−1.12846	−0.564230	+	0.825617i	$0.690827\pi$
$72$	0	0
$73$	−0.788423	−0.0922779	−0.0461389	−	0.998935i	$-0.514692\pi$
$73$	−0.788423	−0.0922779	−0.0461389	+	0.998935i	$0.514692\pi$
$74$	0	0
$75$	3.35792	0.387739
$76$	0	0
$77$	−1.49718	−0.170619
$78$	0	0
$79$	6.15171	0.692121	0.346061	−	0.938212i	$-0.387519\pi$
$79$	6.15171	0.692121	0.346061	+	0.938212i	$0.387519\pi$
$80$	0	0
$81$	34.6589	3.85099
$82$	0	0
$83$	1.61520	0.177291	0.0886456	−	0.996063i	$-0.471746\pi$
$83$	1.61520	0.177291	0.0886456	+	0.996063i	$0.471746\pi$
$84$	0	0
$85$	7.21866	0.782973
$86$	0	0
$87$	−20.9518	−2.24626
$88$	0	0
$89$	9.09960	0.964555	0.482278	−	0.876018i	$-0.339810\pi$
$89$	9.09960	0.964555	0.482278	+	0.876018i	$0.339810\pi$
$90$	0	0
$91$	−5.92691	−0.621309
$92$	0	0
$93$	−16.8833	−1.75072
$94$	0	0
$95$	3.40016	0.348849
$96$	0	0
$97$	0.113933	0.0115682	0.00578408	−	0.999983i	$-0.498159\pi$
$97$	0.113933	0.0115682	0.00578408	+	0.999983i	$0.498159\pi$
$98$	0	0
$99$	−12.3901	−1.24525
$100$	0	0
$101$	5.45006	0.542301	0.271151	−	0.962537i	$-0.412596\pi$
$101$	5.45006	0.542301	0.271151	+	0.962537i	$0.412596\pi$
$102$	0	0
$103$	10.9382	1.07777	0.538886	−	0.842379i	$-0.318845\pi$
$103$	10.9382	1.07777	0.538886	+	0.842379i	$0.318845\pi$
$104$	0	0
$105$	−3.35792	−0.327699
$106$	0	0
$107$	−2.24219	−0.216761	−0.108380	−	0.994110i	$-0.534566\pi$
$107$	−2.24219	−0.216761	−0.108380	+	0.994110i	$0.534566\pi$
$108$	0	0
$109$	2.80745	0.268905	0.134453	−	0.990920i	$-0.457072\pi$
$109$	2.80745	0.268905	0.134453	+	0.990920i	$0.457072\pi$
$110$	0	0
$111$	−28.1335	−2.67032
$112$	0	0
$113$	−14.3199	−1.34710	−0.673551	−	0.739141i	$-0.735232\pi$
$113$	−14.3199	−1.34710	−0.673551	+	0.739141i	$0.735232\pi$
$114$	0	0
$115$	1.00000	0.0932505
$116$	0	0
$117$	−49.0488	−4.53456
$118$	0	0
$119$	−7.21866	−0.661733
$120$	0	0
$121$	−8.75845	−0.796223
$122$	0	0
$123$	−28.1086	−2.53446
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	0.805692	0.0714936	0.0357468	−	0.999361i	$-0.488619\pi$
$127$	0.805692	0.0714936	0.0357468	+	0.999361i	$0.488619\pi$
$128$	0	0
$129$	−7.03610	−0.619494
$130$	0	0
$131$	14.9511	1.30628	0.653142	−	0.757235i	$-0.273450\pi$
$131$	14.9511	1.30628	0.653142	+	0.757235i	$0.273450\pi$
$132$	0	0
$133$	−3.40016	−0.294831
$134$	0	0
$135$	−17.7151	−1.52467
$136$	0	0
$137$	16.6324	1.42100	0.710500	−	0.703697i	$-0.248469\pi$
$137$	16.6324	1.42100	0.710500	+	0.703697i	$0.248469\pi$
$138$	0	0
$139$	4.02663	0.341535	0.170767	−	0.985311i	$-0.445375\pi$
$139$	4.02663	0.341535	0.170767	+	0.985311i	$0.445375\pi$
$140$	0	0
$141$	5.20021	0.437937
$142$	0	0
$143$	8.87365	0.742052
$144$	0	0
$145$	6.23951	0.518163
$146$	0	0
$147$	3.35792	0.276956
$148$	0	0
$149$	−16.5110	−1.35264	−0.676319	−	0.736609i	$-0.736426\pi$
$149$	−16.5110	−1.35264	−0.676319	+	0.736609i	$0.736426\pi$
$150$	0	0
$151$	17.0909	1.39084	0.695419	−	0.718605i	$-0.255219\pi$
$151$	17.0909	1.39084	0.695419	+	0.718605i	$0.255219\pi$
$152$	0	0
$153$	−59.7388	−4.82959
$154$	0	0
$155$	5.02791	0.403851
$156$	0	0
$157$	−19.3489	−1.54421	−0.772107	−	0.635493i	$-0.780797\pi$
$157$	−19.3489	−1.54421	−0.772107	+	0.635493i	$0.780797\pi$
$158$	0	0
$159$	21.1722	1.67906
$160$	0	0
$161$	−1.00000	−0.0788110
$162$	0	0
$163$	21.6017	1.69197	0.845987	−	0.533203i	$-0.179012\pi$
$163$	21.6017	1.69197	0.845987	+	0.533203i	$0.179012\pi$
$164$	0	0
$165$	5.02741	0.391383
$166$	0	0
$167$	−13.5427	−1.04797	−0.523984	−	0.851728i	$-0.675555\pi$
$167$	−13.5427	−1.04797	−0.523984	+	0.851728i	$0.675555\pi$
$168$	0	0
$169$	22.1283	1.70218
$170$	0	0
$171$	−28.1384	−2.15180
$172$	0	0
$173$	9.14645	0.695392	0.347696	−	0.937607i	$-0.386964\pi$
$173$	9.14645	0.695392	0.347696	+	0.937607i	$0.386964\pi$
$174$	0	0
$175$	1.00000	0.0755929
$176$	0	0
$177$	30.5118	2.29341
$178$	0	0
$179$	10.9088	0.815365	0.407683	−	0.913124i	$-0.366337\pi$
$179$	10.9088	0.815365	0.407683	+	0.913124i	$0.366337\pi$
$180$	0	0
$181$	9.47940	0.704598	0.352299	−	0.935887i	$-0.385400\pi$
$181$	9.47940	0.704598	0.352299	+	0.935887i	$0.385400\pi$
$182$	0	0
$183$	−20.7897	−1.53682
$184$	0	0
$185$	8.37827	0.615983
$186$	0	0
$187$	10.8076	0.790332
$188$	0	0
$189$	17.7151	1.28858
$190$	0	0
$191$	15.4875	1.12063	0.560317	−	0.828278i	$-0.310679\pi$
$191$	15.4875	1.12063	0.560317	+	0.828278i	$0.310679\pi$
$192$	0	0
$193$	−10.1480	−0.730468	−0.365234	−	0.930916i	$-0.619011\pi$
$193$	−10.1480	−0.730468	−0.365234	+	0.930916i	$0.619011\pi$
$194$	0	0
$195$	19.9021	1.42522
$196$	0	0
$197$	8.00209	0.570125	0.285063	−	0.958509i	$-0.407986\pi$
$197$	8.00209	0.570125	0.285063	+	0.958509i	$0.407986\pi$
$198$	0	0
$199$	−14.9289	−1.05828	−0.529142	−	0.848533i	$-0.677486\pi$
$199$	−14.9289	−1.05828	−0.529142	+	0.848533i	$0.677486\pi$
$200$	0	0
$201$	7.61638	0.537218
$202$	0	0
$203$	−6.23951	−0.437928
$204$	0	0
$205$	8.37083	0.584644
$206$	0	0
$207$	−8.27561	−0.575195
$208$	0	0
$209$	5.09065	0.352128
$210$	0	0
$211$	6.69202	0.460698	0.230349	−	0.973108i	$-0.426013\pi$
$211$	6.69202	0.460698	0.230349	+	0.973108i	$0.426013\pi$
$212$	0	0
$213$	−31.9290	−2.18774
$214$	0	0
$215$	2.09538	0.142904
$216$	0	0
$217$	−5.02791	−0.341316
$218$	0	0
$219$	−2.64746	−0.178899
$220$	0	0
$221$	42.7843	2.87799
$222$	0	0
$223$	−24.4015	−1.63404	−0.817022	−	0.576607i	$-0.804376\pi$
$223$	−24.4015	−1.63404	−0.817022	+	0.576607i	$0.804376\pi$
$224$	0	0
$225$	8.27561	0.551707
$226$	0	0
$227$	17.4128	1.15573	0.577865	−	0.816132i	$-0.303886\pi$
$227$	17.4128	1.15573	0.577865	+	0.816132i	$0.303886\pi$
$228$	0	0
$229$	−28.8795	−1.90841	−0.954204	−	0.299155i	$-0.903295\pi$
$229$	−28.8795	−1.90841	−0.954204	+	0.299155i	$0.903295\pi$
$230$	0	0
$231$	−5.02741	−0.330779
$232$	0	0
$233$	−1.59355	−0.104397	−0.0521985	−	0.998637i	$-0.516623\pi$
$233$	−1.59355	−0.104397	−0.0521985	+	0.998637i	$0.516623\pi$
$234$	0	0
$235$	−1.54864	−0.101022
$236$	0	0
$237$	20.6569	1.34181
$238$	0	0
$239$	29.8993	1.93402	0.967012	−	0.254731i	$-0.0819868\pi$
$239$	29.8993	1.93402	0.967012	+	0.254731i	$0.0819868\pi$
$240$	0	0
$241$	−15.3508	−0.988831	−0.494415	−	0.869226i	$-0.664618\pi$
$241$	−15.3508	−0.988831	−0.494415	+	0.869226i	$0.664618\pi$
$242$	0	0
$243$	63.2365	4.05662
$244$	0	0
$245$	−1.00000	−0.0638877
$246$	0	0
$247$	20.1525	1.28227
$248$	0	0
$249$	5.42371	0.343714
$250$	0	0
$251$	22.5368	1.42251	0.711254	−	0.702935i	$-0.248128\pi$
$251$	22.5368	1.42251	0.711254	+	0.702935i	$0.248128\pi$
$252$	0	0
$253$	1.49718	0.0941269
$254$	0	0
$255$	24.2396	1.51795
$256$	0	0
$257$	2.93151	0.182863	0.0914313	−	0.995811i	$-0.470856\pi$
$257$	2.93151	0.182863	0.0914313	+	0.995811i	$0.470856\pi$
$258$	0	0
$259$	−8.37827	−0.520600
$260$	0	0
$261$	−51.6357	−3.19617
$262$	0	0
$263$	−24.4091	−1.50513	−0.752566	−	0.658517i	$-0.771184\pi$
$263$	−24.4091	−1.50513	−0.752566	+	0.658517i	$0.771184\pi$
$264$	0	0
$265$	−6.30516	−0.387323
$266$	0	0
$267$	30.5557	1.86998
$268$	0	0
$269$	12.6164	0.769234	0.384617	−	0.923076i	$-0.374334\pi$
$269$	12.6164	0.769234	0.384617	+	0.923076i	$0.374334\pi$
$270$	0	0
$271$	11.6739	0.709136	0.354568	−	0.935030i	$-0.384628\pi$
$271$	11.6739	0.709136	0.354568	+	0.935030i	$0.384628\pi$
$272$	0	0
$273$	−19.9021	−1.20453
$274$	0	0
$275$	−1.49718	−0.0902833
$276$	0	0
$277$	−2.45791	−0.147682	−0.0738408	−	0.997270i	$-0.523526\pi$
$277$	−2.45791	−0.147682	−0.0738408	+	0.997270i	$0.523526\pi$
$278$	0	0
$279$	−41.6090	−2.49106
$280$	0	0
$281$	−19.7374	−1.17743	−0.588716	−	0.808340i	$-0.700366\pi$
$281$	−19.7374	−1.17743	−0.588716	+	0.808340i	$0.700366\pi$
$282$	0	0
$283$	−5.61902	−0.334016	−0.167008	−	0.985956i	$-0.553411\pi$
$283$	−5.61902	−0.334016	−0.167008	+	0.985956i	$0.553411\pi$
$284$	0	0
$285$	11.4175	0.676312
$286$	0	0
$287$	−8.37083	−0.494115
$288$	0	0
$289$	35.1090	2.06523
$290$	0	0
$291$	0.382578	0.0224271
$292$	0	0
$293$	−24.4607	−1.42901	−0.714506	−	0.699629i	$-0.753348\pi$
$293$	−24.4607	−1.42901	−0.714506	+	0.699629i	$0.753348\pi$
$294$	0	0
$295$	−9.08653	−0.529038
$296$	0	0
$297$	−26.5226	−1.53900
$298$	0	0
$299$	5.92691	0.342762
$300$	0	0
$301$	−2.09538	−0.120776
$302$	0	0
$303$	18.3009	1.05136
$304$	0	0
$305$	6.19125	0.354510
$306$	0	0
$307$	16.9770	0.968929	0.484464	−	0.874811i	$-0.339015\pi$
$307$	16.9770	0.968929	0.484464	+	0.874811i	$0.339015\pi$
$308$	0	0
$309$	36.7295	2.08947
$310$	0	0
$311$	−19.7623	−1.12062	−0.560308	−	0.828284i	$-0.689317\pi$
$311$	−19.7623	−1.12062	−0.560308	+	0.828284i	$0.689317\pi$
$312$	0	0
$313$	−15.7740	−0.891597	−0.445799	−	0.895133i	$-0.647080\pi$
$313$	−15.7740	−0.891597	−0.445799	+	0.895133i	$0.647080\pi$
$314$	0	0
$315$	−8.27561	−0.466278
$316$	0	0
$317$	17.9297	1.00703	0.503517	−	0.863985i	$-0.332039\pi$
$317$	17.9297	1.00703	0.503517	+	0.863985i	$0.332039\pi$
$318$	0	0
$319$	9.34166	0.523033
$320$	0	0
$321$	−7.52908	−0.420232
$322$	0	0
$323$	24.5446	1.36570
$324$	0	0
$325$	−5.92691	−0.328766
$326$	0	0
$327$	9.42720	0.521325
$328$	0	0
$329$	1.54864	0.0853794
$330$	0	0
$331$	−9.62108	−0.528823	−0.264411	−	0.964410i	$-0.585178\pi$
$331$	−9.62108	−0.528823	−0.264411	+	0.964410i	$0.585178\pi$
$332$	0	0
$333$	−69.3353	−3.79955
$334$	0	0
$335$	−2.26819	−0.123924
$336$	0	0
$337$	−1.33510	−0.0727278	−0.0363639	−	0.999339i	$-0.511578\pi$
$337$	−1.33510	−0.0727278	−0.0363639	+	0.999339i	$0.511578\pi$
$338$	0	0
$339$	−48.0850	−2.61162
$340$	0	0
$341$	7.52768	0.407647
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	3.35792	0.180784
$346$	0	0
$347$	13.1116	0.703870	0.351935	−	0.936025i	$-0.385524\pi$
$347$	13.1116	0.703870	0.351935	+	0.936025i	$0.385524\pi$
$348$	0	0
$349$	−25.0759	−1.34228	−0.671142	−	0.741329i	$-0.734196\pi$
$349$	−25.0759	−1.34228	−0.671142	+	0.741329i	$0.734196\pi$
$350$	0	0
$351$	−104.996	−5.60425
$352$	0	0
$353$	−27.7619	−1.47761	−0.738807	−	0.673917i	$-0.764611\pi$
$353$	−27.7619	−1.47761	−0.738807	+	0.673917i	$0.764611\pi$
$354$	0	0
$355$	9.50858	0.504663
$356$	0	0
$357$	−24.2396	−1.28290
$358$	0	0
$359$	−29.5678	−1.56053	−0.780263	−	0.625451i	$-0.784915\pi$
$359$	−29.5678	−1.56053	−0.780263	+	0.625451i	$0.784915\pi$
$360$	0	0
$361$	−7.43891	−0.391522
$362$	0	0
$363$	−29.4102	−1.54363
$364$	0	0
$365$	0.788423	0.0412679
$366$	0	0
$367$	2.12494	0.110921	0.0554604	−	0.998461i	$-0.482337\pi$
$367$	2.12494	0.110921	0.0554604	+	0.998461i	$0.482337\pi$
$368$	0	0
$369$	−69.2738	−3.60625
$370$	0	0
$371$	6.30516	0.327348
$372$	0	0
$373$	19.8587	1.02824	0.514121	−	0.857717i	$-0.328118\pi$
$373$	19.8587	1.02824	0.514121	+	0.857717i	$0.328118\pi$
$374$	0	0
$375$	−3.35792	−0.173402
$376$	0	0
$377$	36.9810	1.90462
$378$	0	0
$379$	−25.1572	−1.29224	−0.646119	−	0.763237i	$-0.723609\pi$
$379$	−25.1572	−1.29224	−0.646119	+	0.763237i	$0.723609\pi$
$380$	0	0
$381$	2.70545	0.138604
$382$	0	0
$383$	−30.2153	−1.54393	−0.771965	−	0.635665i	$-0.780726\pi$
$383$	−30.2153	−1.54393	−0.771965	+	0.635665i	$0.780726\pi$
$384$	0	0
$385$	1.49718	0.0763033
$386$	0	0
$387$	−17.3405	−0.881468
$388$	0	0
$389$	−9.45089	−0.479179	−0.239589	−	0.970874i	$-0.577013\pi$
$389$	−9.45089	−0.479179	−0.239589	+	0.970874i	$0.577013\pi$
$390$	0	0
$391$	7.21866	0.365063
$392$	0	0
$393$	50.2046	2.53249
$394$	0	0
$395$	−6.15171	−0.309526
$396$	0	0
$397$	35.4506	1.77922	0.889608	−	0.456724i	$-0.150977\pi$
$397$	35.4506	1.77922	0.889608	+	0.456724i	$0.150977\pi$
$398$	0	0
$399$	−11.4175	−0.571588
$400$	0	0
$401$	−27.7477	−1.38566	−0.692828	−	0.721103i	$-0.743635\pi$
$401$	−27.7477	−1.38566	−0.692828	+	0.721103i	$0.743635\pi$
$402$	0	0
$403$	29.8000	1.48444
$404$	0	0
$405$	−34.6589	−1.72221
$406$	0	0
$407$	12.5438	0.621772
$408$	0	0
$409$	−29.6713	−1.46715	−0.733576	−	0.679608i	$-0.762150\pi$
$409$	−29.6713	−1.46715	−0.733576	+	0.679608i	$0.762150\pi$
$410$	0	0
$411$	55.8502	2.75489
$412$	0	0
$413$	9.08653	0.447119
$414$	0	0
$415$	−1.61520	−0.0792871
$416$	0	0
$417$	13.5211	0.662131
$418$	0	0
$419$	−18.6648	−0.911834	−0.455917	−	0.890022i	$-0.650689\pi$
$419$	−18.6648	−0.911834	−0.455917	+	0.890022i	$0.650689\pi$
$420$	0	0
$421$	0.800365	0.0390074	0.0195037	−	0.999810i	$-0.493791\pi$
$421$	0.800365	0.0390074	0.0195037	+	0.999810i	$0.493791\pi$
$422$	0	0
$423$	12.8160	0.623133
$424$	0	0
$425$	−7.21866	−0.350156
$426$	0	0
$427$	−6.19125	−0.299616
$428$	0	0
$429$	29.7970	1.43861
$430$	0	0
$431$	32.0094	1.54184	0.770919	−	0.636934i	$-0.219797\pi$
$431$	32.0094	1.54184	0.770919	+	0.636934i	$0.219797\pi$
$432$	0	0
$433$	−9.30312	−0.447079	−0.223540	−	0.974695i	$-0.571761\pi$
$433$	−9.30312	−0.447079	−0.223540	+	0.974695i	$0.571761\pi$
$434$	0	0
$435$	20.9518	1.00456
$436$	0	0
$437$	3.40016	0.162652
$438$	0	0
$439$	8.91135	0.425315	0.212658	−	0.977127i	$-0.431788\pi$
$439$	8.91135	0.425315	0.212658	+	0.977127i	$0.431788\pi$
$440$	0	0
$441$	8.27561	0.394077
$442$	0	0
$443$	9.54837	0.453657	0.226828	−	0.973935i	$-0.427164\pi$
$443$	9.54837	0.453657	0.226828	+	0.973935i	$0.427164\pi$
$444$	0	0
$445$	−9.09960	−0.431362
$446$	0	0
$447$	−55.4427	−2.62235
$448$	0	0
$449$	16.2927	0.768898	0.384449	−	0.923146i	$-0.374391\pi$
$449$	16.2927	0.768898	0.384449	+	0.923146i	$0.374391\pi$
$450$	0	0
$451$	12.5326	0.590139
$452$	0	0
$453$	57.3898	2.69641
$454$	0	0
$455$	5.92691	0.277858
$456$	0	0
$457$	−33.6852	−1.57572	−0.787862	−	0.615851i	$-0.788812\pi$
$457$	−33.6852	−1.57572	−0.787862	+	0.615851i	$0.788812\pi$
$458$	0	0
$459$	−127.879	−5.96888
$460$	0	0
$461$	−39.7868	−1.85305	−0.926527	−	0.376228i	$-0.877221\pi$
$461$	−39.7868	−1.85305	−0.926527	+	0.376228i	$0.877221\pi$
$462$	0	0
$463$	12.2457	0.569104	0.284552	−	0.958661i	$-0.408155\pi$
$463$	12.2457	0.569104	0.284552	+	0.958661i	$0.408155\pi$
$464$	0	0
$465$	16.8833	0.782944
$466$	0	0
$467$	−25.0953	−1.16127	−0.580637	−	0.814163i	$-0.697196\pi$
$467$	−25.0953	−1.16127	−0.580637	+	0.814163i	$0.697196\pi$
$468$	0	0
$469$	2.26819	0.104735
$470$	0	0
$471$	−64.9722	−2.99376
$472$	0	0
$473$	3.13716	0.144247
$474$	0	0
$475$	−3.40016	−0.156010
$476$	0	0
$477$	52.1790	2.38911
$478$	0	0
$479$	−15.1889	−0.694000	−0.347000	−	0.937865i	$-0.612800\pi$
$479$	−15.1889	−0.694000	−0.347000	+	0.937865i	$0.612800\pi$
$480$	0	0
$481$	49.6573	2.26418
$482$	0	0
$483$	−3.35792	−0.152791
$484$	0	0
$485$	−0.113933	−0.00517344
$486$	0	0
$487$	−31.5839	−1.43120	−0.715601	−	0.698509i	$-0.753847\pi$
$487$	−31.5839	−1.43120	−0.715601	+	0.698509i	$0.753847\pi$
$488$	0	0
$489$	72.5367	3.28022
$490$	0	0
$491$	40.8944	1.84554	0.922770	−	0.385352i	$-0.125920\pi$
$491$	40.8944	1.84554	0.922770	+	0.385352i	$0.125920\pi$
$492$	0	0
$493$	45.0409	2.02854
$494$	0	0
$495$	12.3901	0.556892
$496$	0	0
$497$	−9.50858	−0.426518
$498$	0	0
$499$	−35.2075	−1.57610	−0.788052	−	0.615608i	$-0.788910\pi$
$499$	−35.2075	−1.57610	−0.788052	+	0.615608i	$0.788910\pi$
$500$	0	0
$501$	−45.4754	−2.03169
$502$	0	0
$503$	−23.4372	−1.04501	−0.522507	−	0.852635i	$-0.675003\pi$
$503$	−23.4372	−1.04501	−0.522507	+	0.852635i	$0.675003\pi$
$504$	0	0
$505$	−5.45006	−0.242525
$506$	0	0
$507$	74.3050	3.30000
$508$	0	0
$509$	34.2770	1.51930	0.759650	−	0.650332i	$-0.225370\pi$
$509$	34.2770	1.51930	0.759650	+	0.650332i	$0.225370\pi$
$510$	0	0
$511$	−0.788423	−0.0348778
$512$	0	0
$513$	−60.2341	−2.65940
$514$	0	0
$515$	−10.9382	−0.481994
$516$	0	0
$517$	−2.31860	−0.101972
$518$	0	0
$519$	30.7130	1.34815
$520$	0	0
$521$	31.7379	1.39046	0.695232	−	0.718785i	$-0.255302\pi$
$521$	31.7379	1.39046	0.695232	+	0.718785i	$0.255302\pi$
$522$	0	0
$523$	−16.4712	−0.720234	−0.360117	−	0.932907i	$-0.617263\pi$
$523$	−16.4712	−0.720234	−0.360117	+	0.932907i	$0.617263\pi$
$524$	0	0
$525$	3.35792	0.146552
$526$	0	0
$527$	36.2947	1.58102
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	75.1966	3.26325
$532$	0	0
$533$	49.6132	2.14899
$534$	0	0
$535$	2.24219	0.0969383
$536$	0	0
$537$	36.6310	1.58074
$538$	0	0
$539$	−1.49718	−0.0644881
$540$	0	0
$541$	5.98180	0.257178	0.128589	−	0.991698i	$-0.458955\pi$
$541$	5.98180	0.257178	0.128589	+	0.991698i	$0.458955\pi$
$542$	0	0
$543$	31.8310	1.36600
$544$	0	0
$545$	−2.80745	−0.120258
$546$	0	0
$547$	−20.5580	−0.878996	−0.439498	−	0.898243i	$-0.644844\pi$
$547$	−20.5580	−0.878996	−0.439498	+	0.898243i	$0.644844\pi$
$548$	0	0
$549$	−51.2364	−2.18671
$550$	0	0
$551$	21.2153	0.903803
$552$	0	0
$553$	6.15171	0.261597
$554$	0	0
$555$	28.1335	1.19420
$556$	0	0
$557$	−4.58691	−0.194354	−0.0971768	−	0.995267i	$-0.530981\pi$
$557$	−4.58691	−0.194354	−0.0971768	+	0.995267i	$0.530981\pi$
$558$	0	0
$559$	12.4191	0.525273
$560$	0	0
$561$	36.2911	1.53221
$562$	0	0
$563$	4.30676	0.181508	0.0907542	−	0.995873i	$-0.471072\pi$
$563$	4.30676	0.181508	0.0907542	+	0.995873i	$0.471072\pi$
$564$	0	0
$565$	14.3199	0.602442
$566$	0	0
$567$	34.6589	1.45554
$568$	0	0
$569$	−40.5945	−1.70181	−0.850906	−	0.525319i	$-0.823946\pi$
$569$	−40.5945	−1.70181	−0.850906	+	0.525319i	$0.823946\pi$
$570$	0	0
$571$	11.8763	0.497007	0.248503	−	0.968631i	$-0.420061\pi$
$571$	11.8763	0.497007	0.248503	+	0.968631i	$0.420061\pi$
$572$	0	0
$573$	52.0057	2.17257
$574$	0	0
$575$	−1.00000	−0.0417029
$576$	0	0
$577$	−25.2613	−1.05164	−0.525820	−	0.850596i	$-0.676242\pi$
$577$	−25.2613	−1.05164	−0.525820	+	0.850596i	$0.676242\pi$
$578$	0	0
$579$	−34.0761	−1.41615
$580$	0	0
$581$	1.61520	0.0670098
$582$	0	0
$583$	−9.43996	−0.390963
$584$	0	0
$585$	49.0488	2.02792
$586$	0	0
$587$	9.78848	0.404014	0.202007	−	0.979384i	$-0.435254\pi$
$587$	9.78848	0.404014	0.202007	+	0.979384i	$0.435254\pi$
$588$	0	0
$589$	17.0957	0.704415
$590$	0	0
$591$	26.8703	1.10530
$592$	0	0
$593$	−5.84271	−0.239931	−0.119966	−	0.992778i	$-0.538278\pi$
$593$	−5.84271	−0.239931	−0.119966	+	0.992778i	$0.538278\pi$
$594$	0	0
$595$	7.21866	0.295936
$596$	0	0
$597$	−50.1301	−2.05169
$598$	0	0
$599$	−13.0077	−0.531482	−0.265741	−	0.964045i	$-0.585617\pi$
$599$	−13.0077	−0.531482	−0.265741	+	0.964045i	$0.585617\pi$
$600$	0	0
$601$	12.4429	0.507558	0.253779	−	0.967262i	$-0.418326\pi$
$601$	12.4429	0.507558	0.253779	+	0.967262i	$0.418326\pi$
$602$	0	0
$603$	18.7706	0.764399
$604$	0	0
$605$	8.75845	0.356082
$606$	0	0
$607$	−39.0038	−1.58312	−0.791558	−	0.611094i	$-0.790730\pi$
$607$	−39.0038	−1.58312	−0.791558	+	0.611094i	$0.790730\pi$
$608$	0	0
$609$	−20.9518	−0.849008
$610$	0	0
$611$	−9.17867	−0.371329
$612$	0	0
$613$	−1.41784	−0.0572660	−0.0286330	−	0.999590i	$-0.509115\pi$
$613$	−1.41784	−0.0572660	−0.0286330	+	0.999590i	$0.509115\pi$
$614$	0	0
$615$	28.1086	1.13345
$616$	0	0
$617$	30.2514	1.21787	0.608937	−	0.793218i	$-0.291596\pi$
$617$	30.2514	1.21787	0.608937	+	0.793218i	$0.291596\pi$
$618$	0	0
$619$	20.4451	0.821759	0.410880	−	0.911690i	$-0.365222\pi$
$619$	20.4451	0.821759	0.410880	+	0.911690i	$0.365222\pi$
$620$	0	0
$621$	−17.7151	−0.710881
$622$	0	0
$623$	9.09960	0.364568
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	17.0940	0.682668
$628$	0	0
$629$	60.4798	2.41149
$630$	0	0
$631$	−0.298128	−0.0118683	−0.00593415	−	0.999982i	$-0.501889\pi$
$631$	−0.298128	−0.0118683	−0.00593415	+	0.999982i	$0.501889\pi$
$632$	0	0
$633$	22.4713	0.893152
$634$	0	0
$635$	−0.805692	−0.0319729
$636$	0	0
$637$	−5.92691	−0.234833
$638$	0	0
$639$	−78.6893	−3.11290
$640$	0	0
$641$	29.2807	1.15652	0.578258	−	0.815854i	$-0.303733\pi$
$641$	29.2807	1.15652	0.578258	+	0.815854i	$0.303733\pi$
$642$	0	0
$643$	13.1214	0.517457	0.258728	−	0.965950i	$-0.416697\pi$
$643$	13.1214	0.517457	0.258728	+	0.965950i	$0.416697\pi$
$644$	0	0
$645$	7.03610	0.277046
$646$	0	0
$647$	−19.4460	−0.764501	−0.382251	−	0.924059i	$-0.624851\pi$
$647$	−19.4460	−0.764501	−0.382251	+	0.924059i	$0.624851\pi$
$648$	0	0
$649$	−13.6042	−0.534010
$650$	0	0
$651$	−16.8833	−0.661708
$652$	0	0
$653$	−39.2141	−1.53457	−0.767283	−	0.641309i	$-0.778392\pi$
$653$	−39.2141	−1.53457	−0.767283	+	0.641309i	$0.778392\pi$
$654$	0	0
$655$	−14.9511	−0.584188
$656$	0	0
$657$	−6.52468	−0.254552
$658$	0	0
$659$	−33.5034	−1.30511	−0.652554	−	0.757742i	$-0.726303\pi$
$659$	−33.5034	−1.30511	−0.652554	+	0.757742i	$0.726303\pi$
$660$	0	0
$661$	−9.35422	−0.363837	−0.181918	−	0.983314i	$-0.558231\pi$
$661$	−9.35422	−0.363837	−0.181918	+	0.983314i	$0.558231\pi$
$662$	0	0
$663$	143.666	5.57954
$664$	0	0
$665$	3.40016	0.131853
$666$	0	0
$667$	6.23951	0.241595
$668$	0	0
$669$	−81.9381	−3.16791
$670$	0	0
$671$	9.26941	0.357842
$672$	0	0
$673$	5.27239	0.203236	0.101618	−	0.994824i	$-0.467598\pi$
$673$	5.27239	0.203236	0.101618	+	0.994824i	$0.467598\pi$
$674$	0	0
$675$	17.7151	0.681853
$676$	0	0
$677$	17.6657	0.678947	0.339474	−	0.940616i	$-0.389751\pi$
$677$	17.6657	0.678947	0.339474	+	0.940616i	$0.389751\pi$
$678$	0	0
$679$	0.113933	0.00437235
$680$	0	0
$681$	58.4709	2.24061
$682$	0	0
$683$	20.0748	0.768139	0.384070	−	0.923304i	$-0.374522\pi$
$683$	20.0748	0.768139	0.384070	+	0.923304i	$0.374522\pi$
$684$	0	0
$685$	−16.6324	−0.635491
$686$	0	0
$687$	−96.9749	−3.69982
$688$	0	0
$689$	−37.3701	−1.42369
$690$	0	0
$691$	−30.8060	−1.17192	−0.585959	−	0.810341i	$-0.699282\pi$
$691$	−30.8060	−1.17192	−0.585959	+	0.810341i	$0.699282\pi$
$692$	0	0
$693$	−12.3901	−0.470660
$694$	0	0
$695$	−4.02663	−0.152739
$696$	0	0
$697$	60.4262	2.28880
$698$	0	0
$699$	−5.35102	−0.202394
$700$	0	0
$701$	6.35841	0.240154	0.120077	−	0.992765i	$-0.461686\pi$
$701$	6.35841	0.240154	0.120077	+	0.992765i	$0.461686\pi$
$702$	0	0
$703$	28.4875	1.07442
$704$	0	0
$705$	−5.20021	−0.195851
$706$	0	0
$707$	5.45006	0.204971
$708$	0	0
$709$	−9.04813	−0.339810	−0.169905	−	0.985460i	$-0.554346\pi$
$709$	−9.04813	−0.339810	−0.169905	+	0.985460i	$0.554346\pi$
$710$	0	0
$711$	50.9091	1.90924
$712$	0	0
$713$	5.02791	0.188297
$714$	0	0
$715$	−8.87365	−0.331856
$716$	0	0
$717$	100.399	3.74948
$718$	0	0
$719$	−17.3739	−0.647938	−0.323969	−	0.946068i	$-0.605017\pi$
$719$	−17.3739	−0.647938	−0.323969	+	0.946068i	$0.605017\pi$
$720$	0	0
$721$	10.9382	0.407359
$722$	0	0
$723$	−51.5467	−1.91704
$724$	0	0
$725$	−6.23951	−0.231729
$726$	0	0
$727$	−18.3250	−0.679637	−0.339819	−	0.940491i	$-0.610366\pi$
$727$	−18.3250	−0.679637	−0.339819	+	0.940491i	$0.610366\pi$
$728$	0	0
$729$	108.366	4.01357
$730$	0	0
$731$	15.1258	0.559448
$732$	0	0
$733$	−33.8369	−1.24979	−0.624897	−	0.780707i	$-0.714859\pi$
$733$	−33.8369	−1.24979	−0.624897	+	0.780707i	$0.714859\pi$
$734$	0	0
$735$	−3.35792	−0.123859
$736$	0	0
$737$	−3.39588	−0.125089
$738$	0	0
$739$	−0.494432	−0.0181880	−0.00909399	−	0.999959i	$-0.502895\pi$
$739$	−0.494432	−0.0181880	−0.00909399	+	0.999959i	$0.502895\pi$
$740$	0	0
$741$	67.6703	2.48593
$742$	0	0
$743$	−48.2268	−1.76927	−0.884635	−	0.466285i	$-0.845592\pi$
$743$	−48.2268	−1.76927	−0.884635	+	0.466285i	$0.845592\pi$
$744$	0	0
$745$	16.5110	0.604918
$746$	0	0
$747$	13.3668	0.489064
$748$	0	0
$749$	−2.24219	−0.0819278
$750$	0	0
$751$	−17.3396	−0.632732	−0.316366	−	0.948637i	$-0.602463\pi$
$751$	−17.3396	−0.632732	−0.316366	+	0.948637i	$0.602463\pi$
$752$	0	0
$753$	75.6766	2.75781
$754$	0	0
$755$	−17.0909	−0.622002
$756$	0	0
$757$	1.70860	0.0621000	0.0310500	−	0.999518i	$-0.490115\pi$
$757$	1.70860	0.0621000	0.0310500	+	0.999518i	$0.490115\pi$
$758$	0	0
$759$	5.02741	0.182483
$760$	0	0
$761$	1.62739	0.0589930	0.0294965	−	0.999565i	$-0.490610\pi$
$761$	1.62739	0.0589930	0.0294965	+	0.999565i	$0.490610\pi$
$762$	0	0
$763$	2.80745	0.101637
$764$	0	0
$765$	59.7388	2.15986
$766$	0	0
$767$	−53.8551	−1.94459
$768$	0	0
$769$	17.7016	0.638337	0.319168	−	0.947698i	$-0.396596\pi$
$769$	17.7016	0.638337	0.319168	+	0.947698i	$0.396596\pi$
$770$	0	0
$771$	9.84377	0.354515
$772$	0	0
$773$	−24.7630	−0.890662	−0.445331	−	0.895366i	$-0.646914\pi$
$773$	−24.7630	−0.890662	−0.445331	+	0.895366i	$0.646914\pi$
$774$	0	0
$775$	−5.02791	−0.180608
$776$	0	0
$777$	−28.1335	−1.00929
$778$	0	0
$779$	28.4622	1.01976
$780$	0	0
$781$	14.2361	0.509406
$782$	0	0
$783$	−110.533	−3.95014
$784$	0	0
$785$	19.3489	0.690593
$786$	0	0
$787$	−31.7695	−1.13246	−0.566230	−	0.824247i	$-0.691599\pi$
$787$	−31.7695	−1.13246	−0.566230	+	0.824247i	$0.691599\pi$
$788$	0	0
$789$	−81.9639	−2.91799
$790$	0	0
$791$	−14.3199	−0.509157
$792$	0	0
$793$	36.6950	1.30308
$794$	0	0
$795$	−21.1722	−0.750901
$796$	0	0
$797$	49.7518	1.76230	0.881149	−	0.472839i	$-0.156771\pi$
$797$	49.7518	1.76230	0.881149	+	0.472839i	$0.156771\pi$
$798$	0	0
$799$	−11.1791	−0.395489
$800$	0	0
$801$	75.3047	2.66076
$802$	0	0
$803$	1.18041	0.0416558
$804$	0	0
$805$	1.00000	0.0352454
$806$	0	0
$807$	42.3647	1.49131
$808$	0	0
$809$	−3.68704	−0.129630	−0.0648148	−	0.997897i	$-0.520646\pi$
$809$	−3.68704	−0.129630	−0.0648148	+	0.997897i	$0.520646\pi$
$810$	0	0
$811$	−45.7426	−1.60624	−0.803120	−	0.595817i	$-0.796828\pi$
$811$	−45.7426	−1.60624	−0.803120	+	0.595817i	$0.796828\pi$
$812$	0	0
$813$	39.1999	1.37480
$814$	0	0
$815$	−21.6017	−0.756674
$816$	0	0
$817$	7.12462	0.249259
$818$	0	0
$819$	−49.0488	−1.71390
$820$	0	0
$821$	−35.0500	−1.22325	−0.611627	−	0.791147i	$-0.709484\pi$
$821$	−35.0500	−1.22325	−0.611627	+	0.791147i	$0.709484\pi$
$822$	0	0
$823$	25.8388	0.900684	0.450342	−	0.892856i	$-0.351302\pi$
$823$	25.8388	0.900684	0.450342	+	0.892856i	$0.351302\pi$
$824$	0	0
$825$	−5.02741	−0.175032
$826$	0	0
$827$	11.3064	0.393161	0.196581	−	0.980488i	$-0.437016\pi$
$827$	11.3064	0.393161	0.196581	+	0.980488i	$0.437016\pi$
$828$	0	0
$829$	−9.26534	−0.321799	−0.160899	−	0.986971i	$-0.551439\pi$
$829$	−9.26534	−0.321799	−0.160899	+	0.986971i	$0.551439\pi$
$830$	0	0
$831$	−8.25346	−0.286309
$832$	0	0
$833$	−7.21866	−0.250112
$834$	0	0
$835$	13.5427	0.468665
$836$	0	0
$837$	−89.0697	−3.07870
$838$	0	0
$839$	41.8544	1.44497	0.722486	−	0.691385i	$-0.242999\pi$
$839$	41.8544	1.44497	0.722486	+	0.691385i	$0.242999\pi$
$840$	0	0
$841$	9.93145	0.342464
$842$	0	0
$843$	−66.2764	−2.28268
$844$	0	0
$845$	−22.1283	−0.761236
$846$	0	0
$847$	−8.75845	−0.300944
$848$	0	0
$849$	−18.8682	−0.647555
$850$	0	0
$851$	8.37827	0.287203
$852$	0	0
$853$	3.25319	0.111387	0.0556935	−	0.998448i	$-0.482263\pi$
$853$	3.25319	0.111387	0.0556935	+	0.998448i	$0.482263\pi$
$854$	0	0
$855$	28.1384	0.962313
$856$	0	0
$857$	−22.3394	−0.763100	−0.381550	−	0.924348i	$-0.624610\pi$
$857$	−22.3394	−0.763100	−0.381550	+	0.924348i	$0.624610\pi$
$858$	0	0
$859$	−13.0193	−0.444213	−0.222106	−	0.975022i	$-0.571293\pi$
$859$	−13.0193	−0.444213	−0.222106	+	0.975022i	$0.571293\pi$
$860$	0	0
$861$	−28.1086	−0.957937
$862$	0	0
$863$	−21.5216	−0.732605	−0.366302	−	0.930496i	$-0.619376\pi$
$863$	−21.5216	−0.732605	−0.366302	+	0.930496i	$0.619376\pi$
$864$	0	0
$865$	−9.14645	−0.310989
$866$	0	0
$867$	117.893	4.00386
$868$	0	0
$869$	−9.21021	−0.312435
$870$	0	0
$871$	−13.4433	−0.455510
$872$	0	0
$873$	0.942867	0.0319112
$874$	0	0
$875$	−1.00000	−0.0338062
$876$	0	0
$877$	50.3101	1.69885	0.849426	−	0.527707i	$-0.176948\pi$
$877$	50.3101	1.69885	0.849426	+	0.527707i	$0.176948\pi$
$878$	0	0
$879$	−82.1371	−2.77042
$880$	0	0
$881$	4.63997	0.156325	0.0781623	−	0.996941i	$-0.475095\pi$
$881$	4.63997	0.156325	0.0781623	+	0.996941i	$0.475095\pi$
$882$	0	0
$883$	−36.6683	−1.23398	−0.616992	−	0.786969i	$-0.711649\pi$
$883$	−36.6683	−1.23398	−0.616992	+	0.786969i	$0.711649\pi$
$884$	0	0
$885$	−30.5118	−1.02564
$886$	0	0
$887$	−26.2509	−0.881420	−0.440710	−	0.897649i	$-0.645273\pi$
$887$	−26.2509	−0.881420	−0.440710	+	0.897649i	$0.645273\pi$
$888$	0	0
$889$	0.805692	0.0270220
$890$	0	0
$891$	−51.8906	−1.73840
$892$	0	0
$893$	−5.26563	−0.176208
$894$	0	0
$895$	−10.9088	−0.364642
$896$	0	0
$897$	19.9021	0.664511
$898$	0	0
$899$	31.3717	1.04630
$900$	0	0
$901$	−45.5148	−1.51632
$902$	0	0
$903$	−7.03610	−0.234147
$904$	0	0
$905$	−9.47940	−0.315106
$906$	0	0
$907$	−39.1797	−1.30094	−0.650470	−	0.759532i	$-0.725428\pi$
$907$	−39.1797	−1.30094	−0.650470	+	0.759532i	$0.725428\pi$
$908$	0	0
$909$	45.1026	1.49596
$910$	0	0
$911$	−16.9625	−0.561993	−0.280996	−	0.959709i	$-0.590665\pi$
$911$	−16.9625	−0.561993	−0.280996	+	0.959709i	$0.590665\pi$
$912$	0	0
$913$	−2.41824	−0.0800322
$914$	0	0
$915$	20.7897	0.687286
$916$	0	0
$917$	14.9511	0.493729
$918$	0	0
$919$	29.7325	0.980784	0.490392	−	0.871502i	$-0.336854\pi$
$919$	29.7325	0.980784	0.490392	+	0.871502i	$0.336854\pi$
$920$	0	0
$921$	57.0074	1.87846
$922$	0	0
$923$	56.3565	1.85500
$924$	0	0
$925$	−8.37827	−0.275476
$926$	0	0
$927$	90.5202	2.97307
$928$	0	0
$929$	53.0579	1.74077	0.870386	−	0.492369i	$-0.163869\pi$
$929$	53.0579	1.74077	0.870386	+	0.492369i	$0.163869\pi$
$930$	0	0
$931$	−3.40016	−0.111436
$932$	0	0
$933$	−66.3601	−2.17253
$934$	0	0
$935$	−10.8076	−0.353447
$936$	0	0
$937$	0.401035	0.0131012	0.00655062	−	0.999979i	$-0.497915\pi$
$937$	0.401035	0.0131012	0.00655062	+	0.999979i	$0.497915\pi$
$938$	0	0
$939$	−52.9677	−1.72853
$940$	0	0
$941$	−5.27695	−0.172024	−0.0860119	−	0.996294i	$-0.527412\pi$
$941$	−5.27695	−0.172024	−0.0860119	+	0.996294i	$0.527412\pi$
$942$	0	0
$943$	8.37083	0.272592
$944$	0	0
$945$	−17.7151	−0.576271
$946$	0	0
$947$	39.4911	1.28329	0.641645	−	0.767002i	$-0.278252\pi$
$947$	39.4911	1.28329	0.641645	+	0.767002i	$0.278252\pi$
$948$	0	0
$949$	4.67291	0.151689
$950$	0	0
$951$	60.2066	1.95233
$952$	0	0
$953$	46.8858	1.51878	0.759389	−	0.650637i	$-0.225498\pi$
$953$	46.8858	1.51878	0.759389	+	0.650637i	$0.225498\pi$
$954$	0	0
$955$	−15.4875	−0.501163
$956$	0	0
$957$	31.3685	1.01400
$958$	0	0
$959$	16.6324	0.537088
$960$	0	0
$961$	−5.72016	−0.184521
$962$	0	0
$963$	−18.5555	−0.597942
$964$	0	0
$965$	10.1480	0.326675
$966$	0	0
$967$	14.1433	0.454817	0.227409	−	0.973799i	$-0.426975\pi$
$967$	14.1433	0.454817	0.227409	+	0.973799i	$0.426975\pi$
$968$	0	0
$969$	82.4187	2.64767
$970$	0	0
$971$	12.1121	0.388696	0.194348	−	0.980933i	$-0.437741\pi$
$971$	12.1121	0.388696	0.194348	+	0.980933i	$0.437741\pi$
$972$	0	0
$973$	4.02663	0.129088
$974$	0	0
$975$	−19.9021	−0.637377
$976$	0	0
$977$	60.1594	1.92467	0.962336	−	0.271864i	$-0.0876401\pi$
$977$	60.1594	1.92467	0.962336	+	0.271864i	$0.0876401\pi$
$978$	0	0
$979$	−13.6237	−0.435416
$980$	0	0
$981$	23.2334	0.741785
$982$	0	0
$983$	12.9032	0.411550	0.205775	−	0.978599i	$-0.434029\pi$
$983$	12.9032	0.411550	0.205775	+	0.978599i	$0.434029\pi$
$984$	0	0
$985$	−8.00209	−0.254968
$986$	0	0
$987$	5.20021	0.165525
$988$	0	0
$989$	2.09538	0.0666291
$990$	0	0
$991$	−42.1002	−1.33736	−0.668678	−	0.743552i	$-0.733140\pi$
$991$	−42.1002	−1.33736	−0.668678	+	0.743552i	$0.733140\pi$
$992$	0	0
$993$	−32.3068	−1.02523
$994$	0	0
$995$	14.9289	0.473279
$996$	0	0
$997$	−21.7841	−0.689910	−0.344955	−	0.938619i	$-0.612106\pi$
$997$	−21.7841	−0.689910	−0.344955	+	0.938619i	$0.612106\pi$
$998$	0	0
$999$	−148.422	−4.69585

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6440.2.a.z.1.7	✓	7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6440.2.a.z.1.7	✓	7	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 6440 = 2^{3} \cdot 5 \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6440.a (trivial)

\(f(q)\)	\(=\)	\(q+3.35792 q^{3} -1.00000 q^{5} +1.00000 q^{7} +8.27561 q^{9} +O(q^{10})\)	\(q+3.35792 q^{3} -1.00000 q^{5} +1.00000 q^{7} +8.27561 q^{9} -1.49718 q^{11} -5.92691 q^{13} -3.35792 q^{15} -7.21866 q^{17} -3.40016 q^{19} +3.35792 q^{21} -1.00000 q^{23} +1.00000 q^{25} +17.7151 q^{27} -6.23951 q^{29} -5.02791 q^{31} -5.02741 q^{33} -1.00000 q^{35} -8.37827 q^{37} -19.9021 q^{39} -8.37083 q^{41} -2.09538 q^{43} -8.27561 q^{45} +1.54864 q^{47} +1.00000 q^{49} -24.2396 q^{51} +6.30516 q^{53} +1.49718 q^{55} -11.4175 q^{57} +9.08653 q^{59} -6.19125 q^{61} +8.27561 q^{63} +5.92691 q^{65} +2.26819 q^{67} -3.35792 q^{69} -9.50858 q^{71} -0.788423 q^{73} +3.35792 q^{75} -1.49718 q^{77} +6.15171 q^{79} +34.6589 q^{81} +1.61520 q^{83} +7.21866 q^{85} -20.9518 q^{87} +9.09960 q^{89} -5.92691 q^{91} -16.8833 q^{93} +3.40016 q^{95} +0.113933 q^{97} -12.3901 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + q^{3} - 7 q^{5} + 7 q^{7} + 10 q^{9}+O(q^{10}) \) 7 * q + q^3 - 7 * q^5 + 7 * q^7 + 10 * q^9	\( 7 q + q^{3} - 7 q^{5} + 7 q^{7} + 10 q^{9} - 4 q^{11} + 2 q^{13} - q^{15} - 12 q^{17} - 12 q^{19} + q^{21} - 7 q^{23} + 7 q^{25} + 10 q^{27} + 13 q^{29} - 14 q^{31} - 25 q^{33} - 7 q^{35} - 29 q^{37} - 24 q^{39} + 5 q^{41} + 2 q^{43} - 10 q^{45} - 3 q^{47} + 7 q^{49} - 39 q^{51} - 3 q^{53} + 4 q^{55} + q^{57} + 29 q^{59} - 15 q^{61} + 10 q^{63} - 2 q^{65} + 7 q^{67} - q^{69} - 38 q^{71} - 15 q^{73} + q^{75} - 4 q^{77} - 37 q^{79} + 35 q^{81} + 7 q^{83} + 12 q^{85} - 3 q^{87} - 13 q^{89} + 2 q^{91} - 9 q^{93} + 12 q^{95} - 44 q^{97} + 10 q^{99}+O(q^{100}) \) 7 * q + q^3 - 7 * q^5 + 7 * q^7 + 10 * q^9 - 4 * q^11 + 2 * q^13 - q^15 - 12 * q^17 - 12 * q^19 + q^21 - 7 * q^23 + 7 * q^25 + 10 * q^27 + 13 * q^29 - 14 * q^31 - 25 * q^33 - 7 * q^35 - 29 * q^37 - 24 * q^39 + 5 * q^41 + 2 * q^43 - 10 * q^45 - 3 * q^47 + 7 * q^49 - 39 * q^51 - 3 * q^53 + 4 * q^55 + q^57 + 29 * q^59 - 15 * q^61 + 10 * q^63 - 2 * q^65 + 7 * q^67 - q^69 - 38 * q^71 - 15 * q^73 + q^75 - 4 * q^77 - 37 * q^79 + 35 * q^81 + 7 * q^83 + 12 * q^85 - 3 * q^87 - 13 * q^89 + 2 * q^91 - 9 * q^93 + 12 * q^95 - 44 * q^97 + 10 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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