LMFDB - Embedded newform 6440.2.a.z.1.1

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			1.1
Root			$-3.11016$ 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)

$f(q)$	$=$	$q-3.11016 q^{3} -1.00000 q^{5} +1.00000 q^{7} +6.67310 q^{9} +O(q^{10})$	$q-3.11016 q^{3} -1.00000 q^{5} +1.00000 q^{7} +6.67310 q^{9} +4.70274 q^{11} +2.64376 q^{13} +3.11016 q^{15} -0.482419 q^{17} -6.87325 q^{19} -3.11016 q^{21} -1.00000 q^{23} +1.00000 q^{25} -11.4239 q^{27} +4.63383 q^{29} -2.89728 q^{31} -14.6263 q^{33} -1.00000 q^{35} -9.50259 q^{37} -8.22251 q^{39} -5.73017 q^{41} +5.24312 q^{43} -6.67310 q^{45} -8.14635 q^{47} +1.00000 q^{49} +1.50040 q^{51} +0.237504 q^{53} -4.70274 q^{55} +21.3769 q^{57} +8.37642 q^{59} +10.1439 q^{61} +6.67310 q^{63} -2.64376 q^{65} +5.46706 q^{67} +3.11016 q^{69} -14.6203 q^{71} -8.15244 q^{73} -3.11016 q^{75} +4.70274 q^{77} -8.48486 q^{79} +15.5110 q^{81} -14.6238 q^{83} +0.482419 q^{85} -14.4120 q^{87} +5.91656 q^{89} +2.64376 q^{91} +9.01100 q^{93} +6.87325 q^{95} +9.00270 q^{97} +31.3819 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−3.11016	−1.79565	−0.897826	−	0.440350i	$-0.854854\pi$
$3$	−3.11016	−1.79565	−0.897826	+	0.440350i	$0.854854\pi$
$4$	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
$9$	6.67310	2.22437
$10$	0	0
$11$	4.70274	1.41793	0.708965	−	0.705244i	$-0.249162\pi$
$11$	4.70274	1.41793	0.708965	+	0.705244i	$0.249162\pi$
$12$	0	0
$13$	2.64376	0.733247	0.366623	−	0.930369i	$-0.380514\pi$
$13$	2.64376	0.733247	0.366623	+	0.930369i	$0.380514\pi$
$14$	0	0
$15$	3.11016	0.803040
$16$	0	0
$17$	−0.482419	−0.117004	−0.0585018	−	0.998287i	$-0.518632\pi$
$17$	−0.482419	−0.117004	−0.0585018	+	0.998287i	$0.518632\pi$
$18$	0	0
$19$	−6.87325	−1.57683	−0.788416	−	0.615142i	$-0.789099\pi$
$19$	−6.87325	−1.57683	−0.788416	+	0.615142i	$0.789099\pi$
$20$	0	0
$21$	−3.11016	−0.678693
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−11.4239	−2.19854
$28$	0	0
$29$	4.63383	0.860481	0.430241	−	0.902714i	$-0.358429\pi$
$29$	4.63383	0.860481	0.430241	+	0.902714i	$0.358429\pi$
$30$	0	0
$31$	−2.89728	−0.520367	−0.260183	−	0.965559i	$-0.583783\pi$
$31$	−2.89728	−0.520367	−0.260183	+	0.965559i	$0.583783\pi$
$32$	0	0
$33$	−14.6263	−2.54611
$34$	0	0
$35$	−1.00000	−0.169031
$36$	0	0
$37$	−9.50259	−1.56222	−0.781108	−	0.624396i	$-0.785345\pi$
$37$	−9.50259	−1.56222	−0.781108	+	0.624396i	$0.785345\pi$
$38$	0	0
$39$	−8.22251	−1.31666
$40$	0	0
$41$	−5.73017	−0.894902	−0.447451	−	0.894308i	$-0.647668\pi$
$41$	−5.73017	−0.894902	−0.447451	+	0.894308i	$0.647668\pi$
$42$	0	0
$43$	5.24312	0.799568	0.399784	−	0.916609i	$-0.369085\pi$
$43$	5.24312	0.799568	0.399784	+	0.916609i	$0.369085\pi$
$44$	0	0
$45$	−6.67310	−0.994767
$46$	0	0
$47$	−8.14635	−1.18827	−0.594134	−	0.804366i	$-0.702505\pi$
$47$	−8.14635	−1.18827	−0.594134	+	0.804366i	$0.702505\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	1.50040	0.210098
$52$	0	0
$53$	0.237504	0.0326237	0.0163118	−	0.999867i	$-0.494808\pi$
$53$	0.237504	0.0326237	0.0163118	+	0.999867i	$0.494808\pi$
$54$	0	0
$55$	−4.70274	−0.634117
$56$	0	0
$57$	21.3769	2.83144
$58$	0	0
$59$	8.37642	1.09052	0.545258	−	0.838268i	$-0.316432\pi$
$59$	8.37642	1.09052	0.545258	+	0.838268i	$0.316432\pi$
$60$	0	0
$61$	10.1439	1.29879	0.649394	−	0.760452i	$-0.275022\pi$
$61$	10.1439	1.29879	0.649394	+	0.760452i	$0.275022\pi$
$62$	0	0
$63$	6.67310	0.840732
$64$	0	0
$65$	−2.64376	−0.327918
$66$	0	0
$67$	5.46706	0.667908	0.333954	−	0.942589i	$-0.391617\pi$
$67$	5.46706	0.667908	0.333954	+	0.942589i	$0.391617\pi$
$68$	0	0
$69$	3.11016	0.374419
$70$	0	0
$71$	−14.6203	−1.73511	−0.867557	−	0.497338i	$-0.834311\pi$
$71$	−14.6203	−1.73511	−0.867557	+	0.497338i	$0.834311\pi$
$72$	0	0
$73$	−8.15244	−0.954171	−0.477085	−	0.878857i	$-0.658307\pi$
$73$	−8.15244	−0.954171	−0.477085	+	0.878857i	$0.658307\pi$
$74$	0	0
$75$	−3.11016	−0.359130
$76$	0	0
$77$	4.70274	0.535927
$78$	0	0
$79$	−8.48486	−0.954622	−0.477311	−	0.878735i	$-0.658388\pi$
$79$	−8.48486	−0.954622	−0.477311	+	0.878735i	$0.658388\pi$
$80$	0	0
$81$	15.5110	1.72344
$82$	0	0
$83$	−14.6238	−1.60517	−0.802585	−	0.596538i	$-0.796543\pi$
$83$	−14.6238	−1.60517	−0.802585	+	0.596538i	$0.796543\pi$
$84$	0	0
$85$	0.482419	0.0523256
$86$	0	0
$87$	−14.4120	−1.54512
$88$	0	0
$89$	5.91656	0.627154	0.313577	−	0.949563i	$-0.398473\pi$
$89$	5.91656	0.627154	0.313577	+	0.949563i	$0.398473\pi$
$90$	0	0
$91$	2.64376	0.277141
$92$	0	0
$93$	9.01100	0.934397
$94$	0	0
$95$	6.87325	0.705181
$96$	0	0
$97$	9.00270	0.914085	0.457043	−	0.889445i	$-0.348909\pi$
$97$	9.00270	0.914085	0.457043	+	0.889445i	$0.348909\pi$
$98$	0	0
$99$	31.3819	3.15400
$100$	0	0
$101$	12.0504	1.19906	0.599530	−	0.800352i	$-0.295354\pi$
$101$	12.0504	1.19906	0.599530	+	0.800352i	$0.295354\pi$
$102$	0	0
$103$	8.52945	0.840432	0.420216	−	0.907424i	$-0.361954\pi$
$103$	8.52945	0.840432	0.420216	+	0.907424i	$0.361954\pi$
$104$	0	0
$105$	3.11016	0.303521
$106$	0	0
$107$	4.47980	0.433078	0.216539	−	0.976274i	$-0.430523\pi$
$107$	4.47980	0.433078	0.216539	+	0.976274i	$0.430523\pi$
$108$	0	0
$109$	16.0479	1.53711	0.768556	−	0.639783i	$-0.220976\pi$
$109$	16.0479	1.53711	0.768556	+	0.639783i	$0.220976\pi$
$110$	0	0
$111$	29.5546	2.80520
$112$	0	0
$113$	−2.06610	−0.194363	−0.0971814	−	0.995267i	$-0.530983\pi$
$113$	−2.06610	−0.194363	−0.0971814	+	0.995267i	$0.530983\pi$
$114$	0	0
$115$	1.00000	0.0932505
$116$	0	0
$117$	17.6421	1.63101
$118$	0	0
$119$	−0.482419	−0.0442232
$120$	0	0
$121$	11.1158	1.01052
$122$	0	0
$123$	17.8218	1.60693
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−16.6477	−1.47724	−0.738622	−	0.674120i	$-0.764523\pi$
$127$	−16.6477	−1.47724	−0.738622	+	0.674120i	$0.764523\pi$
$128$	0	0
$129$	−16.3069	−1.43575
$130$	0	0
$131$	−21.5802	−1.88547	−0.942736	−	0.333539i	$-0.891757\pi$
$131$	−21.5802	−1.88547	−0.942736	+	0.333539i	$0.891757\pi$
$132$	0	0
$133$	−6.87325	−0.595986
$134$	0	0
$135$	11.4239	0.983216
$136$	0	0
$137$	−0.833186	−0.0711839	−0.0355919	−	0.999366i	$-0.511332\pi$
$137$	−0.833186	−0.0711839	−0.0355919	+	0.999366i	$0.511332\pi$
$138$	0	0
$139$	−2.57734	−0.218607	−0.109303	−	0.994008i	$-0.534862\pi$
$139$	−2.57734	−0.218607	−0.109303	+	0.994008i	$0.534862\pi$
$140$	0	0
$141$	25.3365	2.13372
$142$	0	0
$143$	12.4329	1.03969
$144$	0	0
$145$	−4.63383	−0.384819
$146$	0	0
$147$	−3.11016	−0.256522
$148$	0	0
$149$	14.9870	1.22778	0.613890	−	0.789391i	$-0.289604\pi$
$149$	14.9870	1.22778	0.613890	+	0.789391i	$0.289604\pi$
$150$	0	0
$151$	14.2359	1.15850	0.579251	−	0.815149i	$-0.303345\pi$
$151$	14.2359	1.15850	0.579251	+	0.815149i	$0.303345\pi$
$152$	0	0
$153$	−3.21923	−0.260259
$154$	0	0
$155$	2.89728	0.232715
$156$	0	0
$157$	−14.3595	−1.14601	−0.573007	−	0.819551i	$-0.694223\pi$
$157$	−14.3595	−1.14601	−0.573007	+	0.819551i	$0.694223\pi$
$158$	0	0
$159$	−0.738676	−0.0585808
$160$	0	0
$161$	−1.00000	−0.0788110
$162$	0	0
$163$	9.27766	0.726682	0.363341	−	0.931656i	$-0.381636\pi$
$163$	9.27766	0.726682	0.363341	+	0.931656i	$0.381636\pi$
$164$	0	0
$165$	14.6263	1.13865
$166$	0	0
$167$	−20.1033	−1.55564	−0.777820	−	0.628487i	$-0.783674\pi$
$167$	−20.1033	−1.55564	−0.777820	+	0.628487i	$0.783674\pi$
$168$	0	0
$169$	−6.01054	−0.462349
$170$	0	0
$171$	−45.8659	−3.50745
$172$	0	0
$173$	−16.1315	−1.22645	−0.613226	−	0.789908i	$-0.710128\pi$
$173$	−16.1315	−1.22645	−0.613226	+	0.789908i	$0.710128\pi$
$174$	0	0
$175$	1.00000	0.0755929
$176$	0	0
$177$	−26.0520	−1.95819
$178$	0	0
$179$	22.4028	1.67446	0.837232	−	0.546848i	$-0.184173\pi$
$179$	22.4028	1.67446	0.837232	+	0.546848i	$0.184173\pi$
$180$	0	0
$181$	12.2484	0.910419	0.455209	−	0.890384i	$-0.349564\pi$
$181$	12.2484	0.910419	0.455209	+	0.890384i	$0.349564\pi$
$182$	0	0
$183$	−31.5491	−2.33217
$184$	0	0
$185$	9.50259	0.698644
$186$	0	0
$187$	−2.26869	−0.165903
$188$	0	0
$189$	−11.4239	−0.830969
$190$	0	0
$191$	−4.57472	−0.331015	−0.165507	−	0.986209i	$-0.552926\pi$
$191$	−4.57472	−0.331015	−0.165507	+	0.986209i	$0.552926\pi$
$192$	0	0
$193$	−17.5765	−1.26518	−0.632590	−	0.774487i	$-0.718008\pi$
$193$	−17.5765	−1.26518	−0.632590	+	0.774487i	$0.718008\pi$
$194$	0	0
$195$	8.22251	0.588827
$196$	0	0
$197$	−11.7284	−0.835617	−0.417808	−	0.908535i	$-0.637202\pi$
$197$	−11.7284	−0.835617	−0.417808	+	0.908535i	$0.637202\pi$
$198$	0	0
$199$	12.8555	0.911302	0.455651	−	0.890158i	$-0.349406\pi$
$199$	12.8555	0.911302	0.455651	+	0.890158i	$0.349406\pi$
$200$	0	0
$201$	−17.0035	−1.19933
$202$	0	0
$203$	4.63383	0.325231
$204$	0	0
$205$	5.73017	0.400212
$206$	0	0
$207$	−6.67310	−0.463813
$208$	0	0
$209$	−32.3231	−2.23584
$210$	0	0
$211$	−1.48847	−0.102470	−0.0512352	−	0.998687i	$-0.516316\pi$
$211$	−1.48847	−0.102470	−0.0512352	+	0.998687i	$0.516316\pi$
$212$	0	0
$213$	45.4716	3.11566
$214$	0	0
$215$	−5.24312	−0.357578
$216$	0	0
$217$	−2.89728	−0.196680
$218$	0	0
$219$	25.3554	1.71336
$220$	0	0
$221$	−1.27540	−0.0857926
$222$	0	0
$223$	8.05881	0.539658	0.269829	−	0.962908i	$-0.413033\pi$
$223$	8.05881	0.539658	0.269829	+	0.962908i	$0.413033\pi$
$224$	0	0
$225$	6.67310	0.444874
$226$	0	0
$227$	−23.5361	−1.56215	−0.781074	−	0.624438i	$-0.785328\pi$
$227$	−23.5361	−1.56215	−0.781074	+	0.624438i	$0.785328\pi$
$228$	0	0
$229$	0.489792	0.0323664	0.0161832	−	0.999869i	$-0.494849\pi$
$229$	0.489792	0.0323664	0.0161832	+	0.999869i	$0.494849\pi$
$230$	0	0
$231$	−14.6263	−0.962339
$232$	0	0
$233$	25.3325	1.65959	0.829793	−	0.558071i	$-0.188458\pi$
$233$	25.3325	1.65959	0.829793	+	0.558071i	$0.188458\pi$
$234$	0	0
$235$	8.14635	0.531409
$236$	0	0
$237$	26.3893	1.71417
$238$	0	0
$239$	−29.0314	−1.87788	−0.938942	−	0.344076i	$-0.888192\pi$
$239$	−29.0314	−1.87788	−0.938942	+	0.344076i	$0.888192\pi$
$240$	0	0
$241$	−2.58842	−0.166735	−0.0833675	−	0.996519i	$-0.526568\pi$
$241$	−2.58842	−0.166735	−0.0833675	+	0.996519i	$0.526568\pi$
$242$	0	0
$243$	−13.9699	−0.896167
$244$	0	0
$245$	−1.00000	−0.0638877
$246$	0	0
$247$	−18.1712	−1.15621
$248$	0	0
$249$	45.4824	2.88233
$250$	0	0
$251$	2.11021	0.133195	0.0665977	−	0.997780i	$-0.478786\pi$
$251$	2.11021	0.133195	0.0665977	+	0.997780i	$0.478786\pi$
$252$	0	0
$253$	−4.70274	−0.295659
$254$	0	0
$255$	−1.50040	−0.0939587
$256$	0	0
$257$	22.9057	1.42882	0.714409	−	0.699728i	$-0.246696\pi$
$257$	22.9057	1.42882	0.714409	+	0.699728i	$0.246696\pi$
$258$	0	0
$259$	−9.50259	−0.590462
$260$	0	0
$261$	30.9220	1.91403
$262$	0	0
$263$	14.8651	0.916619	0.458310	−	0.888793i	$-0.348455\pi$
$263$	14.8651	0.916619	0.458310	+	0.888793i	$0.348455\pi$
$264$	0	0
$265$	−0.237504	−0.0145898
$266$	0	0
$267$	−18.4015	−1.12615
$268$	0	0
$269$	25.9465	1.58199	0.790994	−	0.611823i	$-0.209564\pi$
$269$	25.9465	1.58199	0.790994	+	0.611823i	$0.209564\pi$
$270$	0	0
$271$	−20.9204	−1.27083	−0.635413	−	0.772172i	$-0.719170\pi$
$271$	−20.9204	−1.27083	−0.635413	+	0.772172i	$0.719170\pi$
$272$	0	0
$273$	−8.22251	−0.497649
$274$	0	0
$275$	4.70274	0.283586
$276$	0	0
$277$	−25.8889	−1.55551	−0.777756	−	0.628566i	$-0.783642\pi$
$277$	−25.8889	−1.55551	−0.777756	+	0.628566i	$0.783642\pi$
$278$	0	0
$279$	−19.3338	−1.15749
$280$	0	0
$281$	−14.8798	−0.887656	−0.443828	−	0.896112i	$-0.646380\pi$
$281$	−14.8798	−0.887656	−0.443828	+	0.896112i	$0.646380\pi$
$282$	0	0
$283$	−11.7934	−0.701046	−0.350523	−	0.936554i	$-0.613996\pi$
$283$	−11.7934	−0.701046	−0.350523	+	0.936554i	$0.613996\pi$
$284$	0	0
$285$	−21.3769	−1.26626
$286$	0	0
$287$	−5.73017	−0.338241
$288$	0	0
$289$	−16.7673	−0.986310
$290$	0	0
$291$	−27.9998	−1.64138
$292$	0	0
$293$	−13.3411	−0.779398	−0.389699	−	0.920942i	$-0.627421\pi$
$293$	−13.3411	−0.779398	−0.389699	+	0.920942i	$0.627421\pi$
$294$	0	0
$295$	−8.37642	−0.487694
$296$	0	0
$297$	−53.7238	−3.11737
$298$	0	0
$299$	−2.64376	−0.152892
$300$	0	0
$301$	5.24312	0.302208
$302$	0	0
$303$	−37.4787	−2.15310
$304$	0	0
$305$	−10.1439	−0.580836
$306$	0	0
$307$	−2.47120	−0.141039	−0.0705194	−	0.997510i	$-0.522466\pi$
$307$	−2.47120	−0.141039	−0.0705194	+	0.997510i	$0.522466\pi$
$308$	0	0
$309$	−26.5280	−1.50912
$310$	0	0
$311$	−30.9941	−1.75752	−0.878758	−	0.477267i	$-0.841627\pi$
$311$	−30.9941	−1.75752	−0.878758	+	0.477267i	$0.841627\pi$
$312$	0	0
$313$	13.0951	0.740180	0.370090	−	0.928996i	$-0.379327\pi$
$313$	13.0951	0.740180	0.370090	+	0.928996i	$0.379327\pi$
$314$	0	0
$315$	−6.67310	−0.375987
$316$	0	0
$317$	7.51076	0.421846	0.210923	−	0.977503i	$-0.432353\pi$
$317$	7.51076	0.421846	0.210923	+	0.977503i	$0.432353\pi$
$318$	0	0
$319$	21.7917	1.22010
$320$	0	0
$321$	−13.9329	−0.777658
$322$	0	0
$323$	3.31578	0.184495
$324$	0	0
$325$	2.64376	0.146649
$326$	0	0
$327$	−49.9116	−2.76012
$328$	0	0
$329$	−8.14635	−0.449123
$330$	0	0
$331$	2.55635	0.140510	0.0702549	−	0.997529i	$-0.477619\pi$
$331$	2.55635	0.140510	0.0702549	+	0.997529i	$0.477619\pi$
$332$	0	0
$333$	−63.4118	−3.47494
$334$	0	0
$335$	−5.46706	−0.298698
$336$	0	0
$337$	−23.8844	−1.30107	−0.650533	−	0.759478i	$-0.725454\pi$
$337$	−23.8844	−1.30107	−0.650533	+	0.759478i	$0.725454\pi$
$338$	0	0
$339$	6.42592	0.349008
$340$	0	0
$341$	−13.6251	−0.737843
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	−3.11016	−0.167445
$346$	0	0
$347$	−14.2230	−0.763530	−0.381765	−	0.924259i	$-0.624684\pi$
$347$	−14.2230	−0.763530	−0.381765	+	0.924259i	$0.624684\pi$
$348$	0	0
$349$	−21.8532	−1.16978	−0.584888	−	0.811114i	$-0.698862\pi$
$349$	−21.8532	−1.16978	−0.584888	+	0.811114i	$0.698862\pi$
$350$	0	0
$351$	−30.2021	−1.61207
$352$	0	0
$353$	28.7647	1.53099	0.765495	−	0.643442i	$-0.222494\pi$
$353$	28.7647	1.53099	0.765495	+	0.643442i	$0.222494\pi$
$354$	0	0
$355$	14.6203	0.775966
$356$	0	0
$357$	1.50040	0.0794096
$358$	0	0
$359$	−11.5417	−0.609150	−0.304575	−	0.952488i	$-0.598514\pi$
$359$	−11.5417	−0.609150	−0.304575	+	0.952488i	$0.598514\pi$
$360$	0	0
$361$	28.2416	1.48640
$362$	0	0
$363$	−34.5718	−1.81455
$364$	0	0
$365$	8.15244	0.426718
$366$	0	0
$367$	−19.6238	−1.02435	−0.512176	−	0.858880i	$-0.671161\pi$
$367$	−19.6238	−1.02435	−0.512176	+	0.858880i	$0.671161\pi$
$368$	0	0
$369$	−38.2380	−1.99059
$370$	0	0
$371$	0.237504	0.0123306
$372$	0	0
$373$	0.293017	0.0151718	0.00758592	−	0.999971i	$-0.497585\pi$
$373$	0.293017	0.0151718	0.00758592	+	0.999971i	$0.497585\pi$
$374$	0	0
$375$	3.11016	0.160608
$376$	0	0
$377$	12.2507	0.630945
$378$	0	0
$379$	−10.5830	−0.543615	−0.271807	−	0.962352i	$-0.587621\pi$
$379$	−10.5830	−0.543615	−0.271807	+	0.962352i	$0.587621\pi$
$380$	0	0
$381$	51.7770	2.65262
$382$	0	0
$383$	19.2302	0.982616	0.491308	−	0.870986i	$-0.336519\pi$
$383$	19.2302	0.982616	0.491308	+	0.870986i	$0.336519\pi$
$384$	0	0
$385$	−4.70274	−0.239674
$386$	0	0
$387$	34.9878	1.77853
$388$	0	0
$389$	−0.413737	−0.0209773	−0.0104887	−	0.999945i	$-0.503339\pi$
$389$	−0.413737	−0.0209773	−0.0104887	+	0.999945i	$0.503339\pi$
$390$	0	0
$391$	0.482419	0.0243970
$392$	0	0
$393$	67.1180	3.38565
$394$	0	0
$395$	8.48486	0.426920
$396$	0	0
$397$	−12.4688	−0.625791	−0.312895	−	0.949788i	$-0.601299\pi$
$397$	−12.4688	−0.625791	−0.312895	+	0.949788i	$0.601299\pi$
$398$	0	0
$399$	21.3769	1.07018
$400$	0	0
$401$	−28.9203	−1.44421	−0.722105	−	0.691783i	$-0.756825\pi$
$401$	−28.9203	−1.44421	−0.722105	+	0.691783i	$0.756825\pi$
$402$	0	0
$403$	−7.65970	−0.381557
$404$	0	0
$405$	−15.5110	−0.770747
$406$	0	0
$407$	−44.6882	−2.21511
$408$	0	0
$409$	20.5624	1.01674	0.508372	−	0.861138i	$-0.330248\pi$
$409$	20.5624	1.01674	0.508372	+	0.861138i	$0.330248\pi$
$410$	0	0
$411$	2.59134	0.127821
$412$	0	0
$413$	8.37642	0.412176
$414$	0	0
$415$	14.6238	0.717854
$416$	0	0
$417$	8.01593	0.392542
$418$	0	0
$419$	28.0458	1.37013	0.685064	−	0.728483i	$-0.259774\pi$
$419$	28.0458	1.37013	0.685064	+	0.728483i	$0.259774\pi$
$420$	0	0
$421$	−26.7776	−1.30506	−0.652530	−	0.757763i	$-0.726292\pi$
$421$	−26.7776	−1.30506	−0.652530	+	0.757763i	$0.726292\pi$
$422$	0	0
$423$	−54.3614	−2.64314
$424$	0	0
$425$	−0.482419	−0.0234007
$426$	0	0
$427$	10.1439	0.490896
$428$	0	0
$429$	−38.6684	−1.86693
$430$	0	0
$431$	21.2662	1.02436	0.512178	−	0.858880i	$-0.328839\pi$
$431$	21.2662	1.02436	0.512178	+	0.858880i	$0.328839\pi$
$432$	0	0
$433$	−23.8279	−1.14510	−0.572548	−	0.819871i	$-0.694045\pi$
$433$	−23.8279	−1.14510	−0.572548	+	0.819871i	$0.694045\pi$
$434$	0	0
$435$	14.4120	0.691001
$436$	0	0
$437$	6.87325	0.328792
$438$	0	0
$439$	−15.3425	−0.732259	−0.366130	−	0.930564i	$-0.619317\pi$
$439$	−15.3425	−0.732259	−0.366130	+	0.930564i	$0.619317\pi$
$440$	0	0
$441$	6.67310	0.317767
$442$	0	0
$443$	−9.06740	−0.430805	−0.215403	−	0.976525i	$-0.569106\pi$
$443$	−9.06740	−0.430805	−0.215403	+	0.976525i	$0.569106\pi$
$444$	0	0
$445$	−5.91656	−0.280472
$446$	0	0
$447$	−46.6119	−2.20467
$448$	0	0
$449$	19.4632	0.918527	0.459263	−	0.888300i	$-0.348113\pi$
$449$	19.4632	0.918527	0.459263	+	0.888300i	$0.348113\pi$
$450$	0	0
$451$	−26.9475	−1.26891
$452$	0	0
$453$	−44.2760	−2.08027
$454$	0	0
$455$	−2.64376	−0.123941
$456$	0	0
$457$	−33.7527	−1.57889	−0.789443	−	0.613824i	$-0.789630\pi$
$457$	−33.7527	−1.57889	−0.789443	+	0.613824i	$0.789630\pi$
$458$	0	0
$459$	5.51112	0.257237
$460$	0	0
$461$	19.3613	0.901745	0.450873	−	0.892588i	$-0.351113\pi$
$461$	19.3613	0.901745	0.450873	+	0.892588i	$0.351113\pi$
$462$	0	0
$463$	−7.90241	−0.367256	−0.183628	−	0.982996i	$-0.558784\pi$
$463$	−7.90241	−0.367256	−0.183628	+	0.982996i	$0.558784\pi$
$464$	0	0
$465$	−9.01100	−0.417875
$466$	0	0
$467$	20.7544	0.960400	0.480200	−	0.877159i	$-0.340564\pi$
$467$	20.7544	0.960400	0.480200	+	0.877159i	$0.340564\pi$
$468$	0	0
$469$	5.46706	0.252446
$470$	0	0
$471$	44.6604	2.05784
$472$	0	0
$473$	24.6570	1.13373
$474$	0	0
$475$	−6.87325	−0.315366
$476$	0	0
$477$	1.58489	0.0725671
$478$	0	0
$479$	20.9724	0.958255	0.479128	−	0.877745i	$-0.340953\pi$
$479$	20.9724	0.958255	0.479128	+	0.877745i	$0.340953\pi$
$480$	0	0
$481$	−25.1226	−1.14549
$482$	0	0
$483$	3.11016	0.141517
$484$	0	0
$485$	−9.00270	−0.408791
$486$	0	0
$487$	20.6545	0.935945	0.467973	−	0.883743i	$-0.344985\pi$
$487$	20.6545	0.935945	0.467973	+	0.883743i	$0.344985\pi$
$488$	0	0
$489$	−28.8550	−1.30487
$490$	0	0
$491$	−14.1069	−0.636637	−0.318318	−	0.947984i	$-0.603118\pi$
$491$	−14.1069	−0.636637	−0.318318	+	0.947984i	$0.603118\pi$
$492$	0	0
$493$	−2.23545	−0.100679
$494$	0	0
$495$	−31.3819	−1.41051
$496$	0	0
$497$	−14.6203	−0.655811
$498$	0	0
$499$	11.1951	0.501162	0.250581	−	0.968096i	$-0.419378\pi$
$499$	11.1951	0.501162	0.250581	+	0.968096i	$0.419378\pi$
$500$	0	0
$501$	62.5245	2.79339
$502$	0	0
$503$	−10.4815	−0.467349	−0.233674	−	0.972315i	$-0.575075\pi$
$503$	−10.4815	−0.467349	−0.233674	+	0.972315i	$0.575075\pi$
$504$	0	0
$505$	−12.0504	−0.536236
$506$	0	0
$507$	18.6938	0.830219
$508$	0	0
$509$	35.5530	1.57586	0.787930	−	0.615765i	$-0.211153\pi$
$509$	35.5530	1.57586	0.787930	+	0.615765i	$0.211153\pi$
$510$	0	0
$511$	−8.15244	−0.360643
$512$	0	0
$513$	78.5196	3.46673
$514$	0	0
$515$	−8.52945	−0.375852
$516$	0	0
$517$	−38.3102	−1.68488
$518$	0	0
$519$	50.1714	2.20228
$520$	0	0
$521$	29.5101	1.29286	0.646430	−	0.762973i	$-0.276261\pi$
$521$	29.5101	1.29286	0.646430	+	0.762973i	$0.276261\pi$
$522$	0	0
$523$	−0.651063	−0.0284690	−0.0142345	−	0.999899i	$-0.504531\pi$
$523$	−0.651063	−0.0284690	−0.0142345	+	0.999899i	$0.504531\pi$
$524$	0	0
$525$	−3.11016	−0.135739
$526$	0	0
$527$	1.39770	0.0608848
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	55.8967	2.42571
$532$	0	0
$533$	−15.1492	−0.656184
$534$	0	0
$535$	−4.47980	−0.193679
$536$	0	0
$537$	−69.6763	−3.00675
$538$	0	0
$539$	4.70274	0.202561
$540$	0	0
$541$	−25.6197	−1.10148	−0.550738	−	0.834678i	$-0.685654\pi$
$541$	−25.6197	−1.10148	−0.550738	+	0.834678i	$0.685654\pi$
$542$	0	0
$543$	−38.0946	−1.63480
$544$	0	0
$545$	−16.0479	−0.687417
$546$	0	0
$547$	−20.5101	−0.876947	−0.438473	−	0.898744i	$-0.644481\pi$
$547$	−20.5101	−0.876947	−0.438473	+	0.898744i	$0.644481\pi$
$548$	0	0
$549$	67.6910	2.88898
$550$	0	0
$551$	−31.8495	−1.35683
$552$	0	0
$553$	−8.48486	−0.360813
$554$	0	0
$555$	−29.5546	−1.25452
$556$	0	0
$557$	11.2800	0.477948	0.238974	−	0.971026i	$-0.423189\pi$
$557$	11.2800	0.477948	0.238974	+	0.971026i	$0.423189\pi$
$558$	0	0
$559$	13.8615	0.586280
$560$	0	0
$561$	7.05599	0.297904
$562$	0	0
$563$	−30.2785	−1.27609	−0.638043	−	0.770001i	$-0.720256\pi$
$563$	−30.2785	−1.27609	−0.638043	+	0.770001i	$0.720256\pi$
$564$	0	0
$565$	2.06610	0.0869217
$566$	0	0
$567$	15.5110	0.651400
$568$	0	0
$569$	−28.6026	−1.19909	−0.599543	−	0.800343i	$-0.704651\pi$
$569$	−28.6026	−1.19909	−0.599543	+	0.800343i	$0.704651\pi$
$570$	0	0
$571$	−8.83682	−0.369809	−0.184905	−	0.982756i	$-0.559198\pi$
$571$	−8.83682	−0.369809	−0.184905	+	0.982756i	$0.559198\pi$
$572$	0	0
$573$	14.2281	0.594388
$574$	0	0
$575$	−1.00000	−0.0417029
$576$	0	0
$577$	7.42461	0.309090	0.154545	−	0.987986i	$-0.450609\pi$
$577$	7.42461	0.309090	0.154545	+	0.987986i	$0.450609\pi$
$578$	0	0
$579$	54.6656	2.27182
$580$	0	0
$581$	−14.6238	−0.606697
$582$	0	0
$583$	1.11692	0.0462581
$584$	0	0
$585$	−17.6421	−0.729410
$586$	0	0
$587$	−14.6878	−0.606232	−0.303116	−	0.952954i	$-0.598027\pi$
$587$	−14.6878	−0.606232	−0.303116	+	0.952954i	$0.598027\pi$
$588$	0	0
$589$	19.9137	0.820531
$590$	0	0
$591$	36.4773	1.50048
$592$	0	0
$593$	23.7609	0.975744	0.487872	−	0.872915i	$-0.337773\pi$
$593$	23.7609	0.975744	0.487872	+	0.872915i	$0.337773\pi$
$594$	0	0
$595$	0.482419	0.0197772
$596$	0	0
$597$	−39.9827	−1.63638
$598$	0	0
$599$	13.2614	0.541845	0.270923	−	0.962601i	$-0.412671\pi$
$599$	13.2614	0.541845	0.270923	+	0.962601i	$0.412671\pi$
$600$	0	0
$601$	25.4348	1.03751	0.518754	−	0.854924i	$-0.326396\pi$
$601$	25.4348	1.03751	0.518754	+	0.854924i	$0.326396\pi$
$602$	0	0
$603$	36.4823	1.48567
$604$	0	0
$605$	−11.1158	−0.451920
$606$	0	0
$607$	−10.0772	−0.409021	−0.204510	−	0.978864i	$-0.565560\pi$
$607$	−10.0772	−0.409021	−0.204510	+	0.978864i	$0.565560\pi$
$608$	0	0
$609$	−14.4120	−0.584002
$610$	0	0
$611$	−21.5370	−0.871293
$612$	0	0
$613$	−26.9697	−1.08930	−0.544649	−	0.838664i	$-0.683337\pi$
$613$	−26.9697	−1.08930	−0.544649	+	0.838664i	$0.683337\pi$
$614$	0	0
$615$	−17.8218	−0.718642
$616$	0	0
$617$	11.9150	0.479680	0.239840	−	0.970812i	$-0.422905\pi$
$617$	11.9150	0.479680	0.239840	+	0.970812i	$0.422905\pi$
$618$	0	0
$619$	−16.0867	−0.646577	−0.323289	−	0.946300i	$-0.604788\pi$
$619$	−16.0867	−0.646577	−0.323289	+	0.946300i	$0.604788\pi$
$620$	0	0
$621$	11.4239	0.458427
$622$	0	0
$623$	5.91656	0.237042
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	100.530	4.01479
$628$	0	0
$629$	4.58423	0.182785
$630$	0	0
$631$	9.15470	0.364443	0.182221	−	0.983258i	$-0.441671\pi$
$631$	9.15470	0.364443	0.182221	+	0.983258i	$0.441671\pi$
$632$	0	0
$633$	4.62938	0.184001
$634$	0	0
$635$	16.6477	0.660643
$636$	0	0
$637$	2.64376	0.104750
$638$	0	0
$639$	−97.5629	−3.85953
$640$	0	0
$641$	−11.3087	−0.446667	−0.223333	−	0.974742i	$-0.571694\pi$
$641$	−11.3087	−0.446667	−0.223333	+	0.974742i	$0.571694\pi$
$642$	0	0
$643$	−3.30783	−0.130448	−0.0652240	−	0.997871i	$-0.520776\pi$
$643$	−3.30783	−0.130448	−0.0652240	+	0.997871i	$0.520776\pi$
$644$	0	0
$645$	16.3069	0.642085
$646$	0	0
$647$	−33.5241	−1.31797	−0.658985	−	0.752157i	$-0.729014\pi$
$647$	−33.5241	−1.31797	−0.658985	+	0.752157i	$0.729014\pi$
$648$	0	0
$649$	39.3921	1.54628
$650$	0	0
$651$	9.01100	0.353169
$652$	0	0
$653$	23.6135	0.924069	0.462034	−	0.886862i	$-0.347120\pi$
$653$	23.6135	0.924069	0.462034	+	0.886862i	$0.347120\pi$
$654$	0	0
$655$	21.5802	0.843209
$656$	0	0
$657$	−54.4021	−2.12243
$658$	0	0
$659$	−19.7336	−0.768711	−0.384356	−	0.923185i	$-0.625576\pi$
$659$	−19.7336	−0.768711	−0.384356	+	0.923185i	$0.625576\pi$
$660$	0	0
$661$	−27.3181	−1.06255	−0.531276	−	0.847199i	$-0.678287\pi$
$661$	−27.3181	−1.06255	−0.531276	+	0.847199i	$0.678287\pi$
$662$	0	0
$663$	3.96669	0.154054
$664$	0	0
$665$	6.87325	0.266533
$666$	0	0
$667$	−4.63383	−0.179423
$668$	0	0
$669$	−25.0642	−0.969037
$670$	0	0
$671$	47.7040	1.84159
$672$	0	0
$673$	−4.93301	−0.190154	−0.0950768	−	0.995470i	$-0.530310\pi$
$673$	−4.93301	−0.190154	−0.0950768	+	0.995470i	$0.530310\pi$
$674$	0	0
$675$	−11.4239	−0.439708
$676$	0	0
$677$	−27.7088	−1.06493	−0.532467	−	0.846451i	$-0.678735\pi$
$677$	−27.7088	−1.06493	−0.532467	+	0.846451i	$0.678735\pi$
$678$	0	0
$679$	9.00270	0.345492
$680$	0	0
$681$	73.2012	2.80507
$682$	0	0
$683$	37.5449	1.43662	0.718308	−	0.695726i	$-0.244917\pi$
$683$	37.5449	1.43662	0.718308	+	0.695726i	$0.244917\pi$
$684$	0	0
$685$	0.833186	0.0318344
$686$	0	0
$687$	−1.52333	−0.0581187
$688$	0	0
$689$	0.627904	0.0239212
$690$	0	0
$691$	−6.84549	−0.260415	−0.130207	−	0.991487i	$-0.541564\pi$
$691$	−6.84549	−0.260415	−0.130207	+	0.991487i	$0.541564\pi$
$692$	0	0
$693$	31.3819	1.19210
$694$	0	0
$695$	2.57734	0.0977639
$696$	0	0
$697$	2.76434	0.104707
$698$	0	0
$699$	−78.7881	−2.98004
$700$	0	0
$701$	27.3152	1.03168	0.515840	−	0.856685i	$-0.327480\pi$
$701$	27.3152	1.03168	0.515840	+	0.856685i	$0.327480\pi$
$702$	0	0
$703$	65.3137	2.46335
$704$	0	0
$705$	−25.3365	−0.954226
$706$	0	0
$707$	12.0504	0.453202
$708$	0	0
$709$	46.0233	1.72844	0.864220	−	0.503114i	$-0.167812\pi$
$709$	46.0233	1.72844	0.864220	+	0.503114i	$0.167812\pi$
$710$	0	0
$711$	−56.6204	−2.12343
$712$	0	0
$713$	2.89728	0.108504
$714$	0	0
$715$	−12.4329	−0.464965
$716$	0	0
$717$	90.2923	3.37203
$718$	0	0
$719$	−21.8789	−0.815945	−0.407972	−	0.912994i	$-0.633764\pi$
$719$	−21.8789	−0.815945	−0.407972	+	0.912994i	$0.633764\pi$
$720$	0	0
$721$	8.52945	0.317653
$722$	0	0
$723$	8.05041	0.299398
$724$	0	0
$725$	4.63383	0.172096
$726$	0	0
$727$	−2.49631	−0.0925832	−0.0462916	−	0.998928i	$-0.514740\pi$
$727$	−2.49631	−0.0925832	−0.0462916	+	0.998928i	$0.514740\pi$
$728$	0	0
$729$	−3.08446	−0.114239
$730$	0	0
$731$	−2.52938	−0.0935524
$732$	0	0
$733$	−27.5174	−1.01638	−0.508189	−	0.861245i	$-0.669685\pi$
$733$	−27.5174	−1.01638	−0.508189	+	0.861245i	$0.669685\pi$
$734$	0	0
$735$	3.11016	0.114720
$736$	0	0
$737$	25.7102	0.947047
$738$	0	0
$739$	−36.1905	−1.33129	−0.665645	−	0.746269i	$-0.731843\pi$
$739$	−36.1905	−1.33129	−0.665645	+	0.746269i	$0.731843\pi$
$740$	0	0
$741$	56.5154	2.07615
$742$	0	0
$743$	41.7638	1.53217	0.766083	−	0.642741i	$-0.222203\pi$
$743$	41.7638	1.53217	0.766083	+	0.642741i	$0.222203\pi$
$744$	0	0
$745$	−14.9870	−0.549080
$746$	0	0
$747$	−97.5861	−3.57049
$748$	0	0
$749$	4.47980	0.163688
$750$	0	0
$751$	−37.8958	−1.38284	−0.691418	−	0.722455i	$-0.743014\pi$
$751$	−37.8958	−1.38284	−0.691418	+	0.722455i	$0.743014\pi$
$752$	0	0
$753$	−6.56310	−0.239173
$754$	0	0
$755$	−14.2359	−0.518098
$756$	0	0
$757$	−1.41938	−0.0515883	−0.0257941	−	0.999667i	$-0.508211\pi$
$757$	−1.41938	−0.0515883	−0.0257941	+	0.999667i	$0.508211\pi$
$758$	0	0
$759$	14.6263	0.530900
$760$	0	0
$761$	−26.0520	−0.944383	−0.472191	−	0.881496i	$-0.656537\pi$
$761$	−26.0520	−0.944383	−0.472191	+	0.881496i	$0.656537\pi$
$762$	0	0
$763$	16.0479	0.580974
$764$	0	0
$765$	3.21923	0.116391
$766$	0	0
$767$	22.1452	0.799617
$768$	0	0
$769$	−5.77591	−0.208285	−0.104142	−	0.994562i	$-0.533210\pi$
$769$	−5.77591	−0.208285	−0.104142	+	0.994562i	$0.533210\pi$
$770$	0	0
$771$	−71.2404	−2.56566
$772$	0	0
$773$	40.0015	1.43875	0.719376	−	0.694620i	$-0.244428\pi$
$773$	40.0015	1.43875	0.719376	+	0.694620i	$0.244428\pi$
$774$	0	0
$775$	−2.89728	−0.104073
$776$	0	0
$777$	29.5546	1.06027
$778$	0	0
$779$	39.3849	1.41111
$780$	0	0
$781$	−68.7556	−2.46027
$782$	0	0
$783$	−52.9366	−1.89180
$784$	0	0
$785$	14.3595	0.512513
$786$	0	0
$787$	−28.7624	−1.02527	−0.512635	−	0.858607i	$-0.671331\pi$
$787$	−28.7624	−1.02527	−0.512635	+	0.858607i	$0.671331\pi$
$788$	0	0
$789$	−46.2327	−1.64593
$790$	0	0
$791$	−2.06610	−0.0734622
$792$	0	0
$793$	26.8179	0.952332
$794$	0	0
$795$	0.738676	0.0261981
$796$	0	0
$797$	34.1922	1.21115	0.605575	−	0.795788i	$-0.292943\pi$
$797$	34.1922	1.21115	0.605575	+	0.795788i	$0.292943\pi$
$798$	0	0
$799$	3.92995	0.139032
$800$	0	0
$801$	39.4818	1.39502
$802$	0	0
$803$	−38.3388	−1.35295
$804$	0	0
$805$	1.00000	0.0352454
$806$	0	0
$807$	−80.6979	−2.84070
$808$	0	0
$809$	−11.7196	−0.412038	−0.206019	−	0.978548i	$-0.566051\pi$
$809$	−11.7196	−0.412038	−0.206019	+	0.978548i	$0.566051\pi$
$810$	0	0
$811$	40.0042	1.40474	0.702369	−	0.711813i	$-0.252125\pi$
$811$	40.0042	1.40474	0.702369	+	0.711813i	$0.252125\pi$
$812$	0	0
$813$	65.0660	2.28196
$814$	0	0
$815$	−9.27766	−0.324982
$816$	0	0
$817$	−36.0373	−1.26078
$818$	0	0
$819$	17.6421	0.616464
$820$	0	0
$821$	−39.1969	−1.36798	−0.683991	−	0.729491i	$-0.739757\pi$
$821$	−39.1969	−1.36798	−0.683991	+	0.729491i	$0.739757\pi$
$822$	0	0
$823$	−34.2278	−1.19311	−0.596553	−	0.802574i	$-0.703464\pi$
$823$	−34.2278	−1.19311	−0.596553	+	0.802574i	$0.703464\pi$
$824$	0	0
$825$	−14.6263	−0.509222
$826$	0	0
$827$	4.04556	0.140678	0.0703390	−	0.997523i	$-0.477592\pi$
$827$	4.04556	0.140678	0.0703390	+	0.997523i	$0.477592\pi$
$828$	0	0
$829$	−47.3155	−1.64333	−0.821667	−	0.569968i	$-0.806956\pi$
$829$	−47.3155	−1.64333	−0.821667	+	0.569968i	$0.806956\pi$
$830$	0	0
$831$	80.5186	2.79316
$832$	0	0
$833$	−0.482419	−0.0167148
$834$	0	0
$835$	20.1033	0.695703
$836$	0	0
$837$	33.0983	1.14405
$838$	0	0
$839$	−16.9780	−0.586146	−0.293073	−	0.956090i	$-0.594678\pi$
$839$	−16.9780	−0.586146	−0.293073	+	0.956090i	$0.594678\pi$
$840$	0	0
$841$	−7.52760	−0.259572
$842$	0	0
$843$	46.2787	1.59392
$844$	0	0
$845$	6.01054	0.206769
$846$	0	0
$847$	11.1158	0.381942
$848$	0	0
$849$	36.6794	1.25883
$850$	0	0
$851$	9.50259	0.325745
$852$	0	0
$853$	−56.8635	−1.94697	−0.973484	−	0.228755i	$-0.926535\pi$
$853$	−56.8635	−1.94697	−0.973484	+	0.228755i	$0.926535\pi$
$854$	0	0
$855$	45.8659	1.56858
$856$	0	0
$857$	−37.3780	−1.27681	−0.638404	−	0.769701i	$-0.720405\pi$
$857$	−37.3780	−1.27681	−0.638404	+	0.769701i	$0.720405\pi$
$858$	0	0
$859$	−28.3328	−0.966704	−0.483352	−	0.875426i	$-0.660581\pi$
$859$	−28.3328	−0.966704	−0.483352	+	0.875426i	$0.660581\pi$
$860$	0	0
$861$	17.8218	0.607364
$862$	0	0
$863$	5.95879	0.202840	0.101420	−	0.994844i	$-0.467661\pi$
$863$	5.95879	0.202840	0.101420	+	0.994844i	$0.467661\pi$
$864$	0	0
$865$	16.1315	0.548486
$866$	0	0
$867$	52.1489	1.77107
$868$	0	0
$869$	−39.9021	−1.35359
$870$	0	0
$871$	14.4536	0.489742
$872$	0	0
$873$	60.0759	2.03326
$874$	0	0
$875$	−1.00000	−0.0338062
$876$	0	0
$877$	−1.61517	−0.0545405	−0.0272702	−	0.999628i	$-0.508681\pi$
$877$	−1.61517	−0.0545405	−0.0272702	+	0.999628i	$0.508681\pi$
$878$	0	0
$879$	41.4931	1.39953
$880$	0	0
$881$	−24.3705	−0.821062	−0.410531	−	0.911847i	$-0.634657\pi$
$881$	−24.3705	−0.821062	−0.410531	+	0.911847i	$0.634657\pi$
$882$	0	0
$883$	29.7942	1.00265	0.501327	−	0.865258i	$-0.332845\pi$
$883$	29.7942	1.00265	0.501327	+	0.865258i	$0.332845\pi$
$884$	0	0
$885$	26.0520	0.875728
$886$	0	0
$887$	26.3301	0.884079	0.442039	−	0.896996i	$-0.354255\pi$
$887$	26.3301	0.884079	0.442039	+	0.896996i	$0.354255\pi$
$888$	0	0
$889$	−16.6477	−0.558346
$890$	0	0
$891$	72.9442	2.44372
$892$	0	0
$893$	55.9919	1.87370
$894$	0	0
$895$	−22.4028	−0.748843
$896$	0	0
$897$	8.22251	0.274542
$898$	0	0
$899$	−13.4255	−0.447766
$900$	0	0
$901$	−0.114576	−0.00381709
$902$	0	0
$903$	−16.3069	−0.542661
$904$	0	0
$905$	−12.2484	−0.407152
$906$	0	0
$907$	−22.5702	−0.749431	−0.374716	−	0.927140i	$-0.622260\pi$
$907$	−22.5702	−0.749431	−0.374716	+	0.927140i	$0.622260\pi$
$908$	0	0
$909$	80.4136	2.66715
$910$	0	0
$911$	48.4478	1.60515	0.802573	−	0.596553i	$-0.203464\pi$
$911$	48.4478	1.60515	0.802573	+	0.596553i	$0.203464\pi$
$912$	0	0
$913$	−68.7719	−2.27602
$914$	0	0
$915$	31.5491	1.04298
$916$	0	0
$917$	−21.5802	−0.712642
$918$	0	0
$919$	16.4923	0.544030	0.272015	−	0.962293i	$-0.412310\pi$
$919$	16.4923	0.544030	0.272015	+	0.962293i	$0.412310\pi$
$920$	0	0
$921$	7.68583	0.253257
$922$	0	0
$923$	−38.6526	−1.27227
$924$	0	0
$925$	−9.50259	−0.312443
$926$	0	0
$927$	56.9179	1.86943
$928$	0	0
$929$	−49.8519	−1.63559	−0.817793	−	0.575512i	$-0.804803\pi$
$929$	−49.8519	−1.63559	−0.817793	+	0.575512i	$0.804803\pi$
$930$	0	0
$931$	−6.87325	−0.225262
$932$	0	0
$933$	96.3968	3.15589
$934$	0	0
$935$	2.26869	0.0741941
$936$	0	0
$937$	−24.7346	−0.808045	−0.404022	−	0.914749i	$-0.632388\pi$
$937$	−24.7346	−0.808045	−0.404022	+	0.914749i	$0.632388\pi$
$938$	0	0
$939$	−40.7280	−1.32911
$940$	0	0
$941$	−50.3856	−1.64252	−0.821262	−	0.570551i	$-0.806730\pi$
$941$	−50.3856	−1.64252	−0.821262	+	0.570551i	$0.806730\pi$
$942$	0	0
$943$	5.73017	0.186600
$944$	0	0
$945$	11.4239	0.371621
$946$	0	0
$947$	39.5925	1.28658	0.643291	−	0.765622i	$-0.277569\pi$
$947$	39.5925	1.28658	0.643291	+	0.765622i	$0.277569\pi$
$948$	0	0
$949$	−21.5531	−0.699643
$950$	0	0
$951$	−23.3597	−0.757489
$952$	0	0
$953$	16.0060	0.518484	0.259242	−	0.965812i	$-0.416527\pi$
$953$	16.0060	0.518484	0.259242	+	0.965812i	$0.416527\pi$
$954$	0	0
$955$	4.57472	0.148034
$956$	0	0
$957$	−67.7757	−2.19088
$958$	0	0
$959$	−0.833186	−0.0269050
$960$	0	0
$961$	−22.6058	−0.729219
$962$	0	0
$963$	29.8942	0.963326
$964$	0	0
$965$	17.5765	0.565806
$966$	0	0
$967$	9.12935	0.293580	0.146790	−	0.989168i	$-0.453106\pi$
$967$	9.12935	0.293580	0.146790	+	0.989168i	$0.453106\pi$
$968$	0	0
$969$	−10.3126	−0.331289
$970$	0	0
$971$	1.93148	0.0619842	0.0309921	−	0.999520i	$-0.490133\pi$
$971$	1.93148	0.0619842	0.0309921	+	0.999520i	$0.490133\pi$
$972$	0	0
$973$	−2.57734	−0.0826256
$974$	0	0
$975$	−8.22251	−0.263331
$976$	0	0
$977$	−42.1045	−1.34704	−0.673521	−	0.739168i	$-0.735219\pi$
$977$	−42.1045	−1.34704	−0.673521	+	0.739168i	$0.735219\pi$
$978$	0	0
$979$	27.8241	0.889261
$980$	0	0
$981$	107.089	3.41910
$982$	0	0
$983$	−33.6571	−1.07349	−0.536747	−	0.843743i	$-0.680347\pi$
$983$	−33.6571	−1.07349	−0.536747	+	0.843743i	$0.680347\pi$
$984$	0	0
$985$	11.7284	0.373699
$986$	0	0
$987$	25.3365	0.806469
$988$	0	0
$989$	−5.24312	−0.166721
$990$	0	0
$991$	−39.2645	−1.24728	−0.623639	−	0.781713i	$-0.714346\pi$
$991$	−39.2645	−1.24728	−0.623639	+	0.781713i	$0.714346\pi$
$992$	0	0
$993$	−7.95067	−0.252307
$994$	0	0
$995$	−12.8555	−0.407547
$996$	0	0
$997$	57.1351	1.80949	0.904743	−	0.425957i	$-0.140063\pi$
$997$	57.1351	1.80949	0.904743	+	0.425957i	$0.140063\pi$
$998$	0	0
$999$	108.557	3.43459

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6440.2.a.z.1.1	✓	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)

\(f(q)\)	\(=\)	\(q-3.11016 q^{3} -1.00000 q^{5} +1.00000 q^{7} +6.67310 q^{9} +O(q^{10})\)	\(q-3.11016 q^{3} -1.00000 q^{5} +1.00000 q^{7} +6.67310 q^{9} +4.70274 q^{11} +2.64376 q^{13} +3.11016 q^{15} -0.482419 q^{17} -6.87325 q^{19} -3.11016 q^{21} -1.00000 q^{23} +1.00000 q^{25} -11.4239 q^{27} +4.63383 q^{29} -2.89728 q^{31} -14.6263 q^{33} -1.00000 q^{35} -9.50259 q^{37} -8.22251 q^{39} -5.73017 q^{41} +5.24312 q^{43} -6.67310 q^{45} -8.14635 q^{47} +1.00000 q^{49} +1.50040 q^{51} +0.237504 q^{53} -4.70274 q^{55} +21.3769 q^{57} +8.37642 q^{59} +10.1439 q^{61} +6.67310 q^{63} -2.64376 q^{65} +5.46706 q^{67} +3.11016 q^{69} -14.6203 q^{71} -8.15244 q^{73} -3.11016 q^{75} +4.70274 q^{77} -8.48486 q^{79} +15.5110 q^{81} -14.6238 q^{83} +0.482419 q^{85} -14.4120 q^{87} +5.91656 q^{89} +2.64376 q^{91} +9.01100 q^{93} +6.87325 q^{95} +9.00270 q^{97} +31.3819 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + q^{3} - 7 q^{5} + 7 q^{7} + 10 q^{9}+O(q^{10}) \) 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

Related objects

Downloads

Learn more