LMFDB - Embedded newform 7728.2.a.cb.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7728,2,Mod(1,7728)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7728, 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("7728.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7728 = 2^{4} \cdot 3 \cdot 7 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7728.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:	$61.7083906820$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.2225.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 5x^{2} + 2x + 4 $ x^4 - x^3 - 5x^2 + 2x + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 3864)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$2.43828$ of defining polynomial
Character	$\chi$	$=$	7728.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +1.32719 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +1.32719 q^{5} -1.00000 q^{7} +1.00000 q^{9} -3.74301 q^{11} -5.07020 q^{13} +1.32719 q^{15} +3.38351 q^{17} +3.74301 q^{19} -1.00000 q^{21} +1.00000 q^{23} -3.23856 q^{25} +1.00000 q^{27} -8.01012 q^{29} +10.2151 q^{31} -3.74301 q^{33} -1.32719 q^{35} -0.911372 q^{37} -5.07020 q^{39} -7.38351 q^{41} +4.08408 q^{43} +1.32719 q^{45} -0.831637 q^{47} +1.00000 q^{49} +3.38351 q^{51} +10.0664 q^{53} -4.96769 q^{55} +3.74301 q^{57} -6.42594 q^{59} +2.86019 q^{61} -1.00000 q^{63} -6.72913 q^{65} -2.70058 q^{67} +1.00000 q^{69} -0.210796 q^{71} -12.8418 q^{73} -3.23856 q^{75} +3.74301 q^{77} -10.2151 q^{79} +1.00000 q^{81} +1.20126 q^{83} +4.49056 q^{85} -8.01012 q^{87} -14.8929 q^{89} +5.07020 q^{91} +10.2151 q^{93} +4.96769 q^{95} -12.8418 q^{97} -3.74301 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{3} - 3 q^{5} - 4 q^{7} + 4 q^{9}+O(q^{10}) $ 4 * q + 4 * q^3 - 3 * q^5 - 4 * q^7 + 4 * q^9	$ 4 q + 4 q^{3} - 3 q^{5} - 4 q^{7} + 4 q^{9} - 2 q^{11} + q^{13} - 3 q^{15} - 8 q^{17} + 2 q^{19} - 4 q^{21} + 4 q^{23} - q^{25} + 4 q^{27} - 10 q^{29} + 10 q^{31} - 2 q^{33} + 3 q^{35} + q^{39} - 8 q^{41} - 13 q^{43} - 3 q^{45} + 6 q^{47} + 4 q^{49} - 8 q^{51} + 5 q^{53} - 3 q^{55} + 2 q^{57} + q^{59} + 5 q^{61} - 4 q^{63} - 22 q^{65} - 3 q^{67} + 4 q^{69} - 5 q^{71} - 20 q^{73} - q^{75} + 2 q^{77} - 10 q^{79} + 4 q^{81} - 18 q^{83} + 25 q^{85} - 10 q^{87} - 31 q^{89} - q^{91} + 10 q^{93} + 3 q^{95} - 20 q^{97} - 2 q^{99}+O(q^{100}) $ 4 * q + 4 * q^3 - 3 * q^5 - 4 * q^7 + 4 * q^9 - 2 * q^11 + q^13 - 3 * q^15 - 8 * q^17 + 2 * q^19 - 4 * q^21 + 4 * q^23 - q^25 + 4 * q^27 - 10 * q^29 + 10 * q^31 - 2 * q^33 + 3 * q^35 + q^39 - 8 * q^41 - 13 * q^43 - 3 * q^45 + 6 * q^47 + 4 * q^49 - 8 * q^51 + 5 * q^53 - 3 * q^55 + 2 * q^57 + q^59 + 5 * q^61 - 4 * q^63 - 22 * q^65 - 3 * q^67 + 4 * q^69 - 5 * q^71 - 20 * q^73 - q^75 + 2 * q^77 - 10 * q^79 + 4 * q^81 - 18 * q^83 + 25 * q^85 - 10 * q^87 - 31 * q^89 - q^91 + 10 * q^93 + 3 * q^95 - 20 * q^97 - 2 * 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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	1.32719	0.593538	0.296769	−	0.954949i	$-0.404091\pi$
$5$	1.32719	0.593538	0.296769	+	0.954949i	$0.404091\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−3.74301	−1.12856	−0.564280	−	0.825583i	$-0.690846\pi$
$11$	−3.74301	−1.12856	−0.564280	+	0.825583i	$0.690846\pi$
$12$	0	0
$13$	−5.07020	−1.40622	−0.703110	−	0.711081i	$-0.748206\pi$
$13$	−5.07020	−1.40622	−0.703110	+	0.711081i	$0.748206\pi$
$14$	0	0
$15$	1.32719	0.342679
$16$	0	0
$17$	3.38351	0.820621	0.410311	−	0.911946i	$-0.365420\pi$
$17$	3.38351	0.820621	0.410311	+	0.911946i	$0.365420\pi$
$18$	0	0
$19$	3.74301	0.858705	0.429353	−	0.903137i	$-0.358742\pi$
$19$	3.74301	0.858705	0.429353	+	0.903137i	$0.358742\pi$
$20$	0	0
$21$	−1.00000	−0.218218
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	−3.23856	−0.647713
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−8.01012	−1.48744	−0.743721	−	0.668490i	$-0.766941\pi$
$29$	−8.01012	−1.48744	−0.743721	+	0.668490i	$0.766941\pi$
$30$	0	0
$31$	10.2151	1.83469	0.917347	−	0.398088i	$-0.130326\pi$
$31$	10.2151	1.83469	0.917347	+	0.398088i	$0.130326\pi$
$32$	0	0
$33$	−3.74301	−0.651574
$34$	0	0
$35$	−1.32719	−0.224336
$36$	0	0
$37$	−0.911372	−0.149829	−0.0749144	−	0.997190i	$-0.523868\pi$
$37$	−0.911372	−0.149829	−0.0749144	+	0.997190i	$0.523868\pi$
$38$	0	0
$39$	−5.07020	−0.811882
$40$	0	0
$41$	−7.38351	−1.15311	−0.576555	−	0.817058i	$-0.695603\pi$
$41$	−7.38351	−1.15311	−0.576555	+	0.817058i	$0.695603\pi$
$42$	0	0
$43$	4.08408	0.622817	0.311409	−	0.950276i	$-0.399199\pi$
$43$	4.08408	0.622817	0.311409	+	0.950276i	$0.399199\pi$
$44$	0	0
$45$	1.32719	0.197846
$46$	0	0
$47$	−0.831637	−0.121307	−0.0606534	−	0.998159i	$-0.519318\pi$
$47$	−0.831637	−0.121307	−0.0606534	+	0.998159i	$0.519318\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	3.38351	0.473786
$52$	0	0
$53$	10.0664	1.38273	0.691366	−	0.722505i	$-0.257009\pi$
$53$	10.0664	1.38273	0.691366	+	0.722505i	$0.257009\pi$
$54$	0	0
$55$	−4.96769	−0.669843
$56$	0	0
$57$	3.74301	0.495774
$58$	0	0
$59$	−6.42594	−0.836586	−0.418293	−	0.908312i	$-0.637372\pi$
$59$	−6.42594	−0.836586	−0.418293	+	0.908312i	$0.637372\pi$
$60$	0	0
$61$	2.86019	0.366209	0.183105	−	0.983093i	$-0.441385\pi$
$61$	2.86019	0.366209	0.183105	+	0.983093i	$0.441385\pi$
$62$	0	0
$63$	−1.00000	−0.125988
$64$	0	0
$65$	−6.72913	−0.834645
$66$	0	0
$67$	−2.70058	−0.329928	−0.164964	−	0.986300i	$-0.552751\pi$
$67$	−2.70058	−0.329928	−0.164964	+	0.986300i	$0.552751\pi$
$68$	0	0
$69$	1.00000	0.120386
$70$	0	0
$71$	−0.210796	−0.0250169	−0.0125084	−	0.999922i	$-0.503982\pi$
$71$	−0.210796	−0.0250169	−0.0125084	+	0.999922i	$0.503982\pi$
$72$	0	0
$73$	−12.8418	−1.50301	−0.751507	−	0.659725i	$-0.770673\pi$
$73$	−12.8418	−1.50301	−0.751507	+	0.659725i	$0.770673\pi$
$74$	0	0
$75$	−3.23856	−0.373957
$76$	0	0
$77$	3.74301	0.426556
$78$	0	0
$79$	−10.2151	−1.14929	−0.574647	−	0.818401i	$-0.694861\pi$
$79$	−10.2151	−1.14929	−0.574647	+	0.818401i	$0.694861\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	1.20126	0.131856	0.0659278	−	0.997824i	$-0.478999\pi$
$83$	1.20126	0.131856	0.0659278	+	0.997824i	$0.478999\pi$
$84$	0	0
$85$	4.49056	0.487070
$86$	0	0
$87$	−8.01012	−0.858775
$88$	0	0
$89$	−14.8929	−1.57865	−0.789325	−	0.613976i	$-0.789569\pi$
$89$	−14.8929	−1.57865	−0.789325	+	0.613976i	$0.789569\pi$
$90$	0	0
$91$	5.07020	0.531501
$92$	0	0
$93$	10.2151	1.05926
$94$	0	0
$95$	4.96769	0.509674
$96$	0	0
$97$	−12.8418	−1.30388	−0.651942	−	0.758269i	$-0.726045\pi$
$97$	−12.8418	−1.30388	−0.651942	+	0.758269i	$0.726045\pi$
$98$	0	0
$99$	−3.74301	−0.376187
$100$	0	0
$101$	9.10809	0.906289	0.453145	−	0.891437i	$-0.350302\pi$
$101$	9.10809	0.906289	0.453145	+	0.891437i	$0.350302\pi$
$102$	0	0
$103$	−2.02777	−0.199802	−0.0999009	−	0.994997i	$-0.531853\pi$
$103$	−2.02777	−0.199802	−0.0999009	+	0.994997i	$0.531853\pi$
$104$	0	0
$105$	−1.32719	−0.129521
$106$	0	0
$107$	14.0766	1.36083	0.680416	−	0.732826i	$-0.261799\pi$
$107$	14.0766	1.36083	0.680416	+	0.732826i	$0.261799\pi$
$108$	0	0
$109$	−8.24370	−0.789603	−0.394801	−	0.918766i	$-0.629187\pi$
$109$	−8.24370	−0.789603	−0.394801	+	0.918766i	$0.629187\pi$
$110$	0	0
$111$	−0.911372	−0.0865036
$112$	0	0
$113$	−8.51833	−0.801337	−0.400669	−	0.916223i	$-0.631222\pi$
$113$	−8.51833	−0.801337	−0.400669	+	0.916223i	$0.631222\pi$
$114$	0	0
$115$	1.32719	0.123761
$116$	0	0
$117$	−5.07020	−0.468740
$118$	0	0
$119$	−3.38351	−0.310166
$120$	0	0
$121$	3.01012	0.273648
$122$	0	0
$123$	−7.38351	−0.665749
$124$	0	0
$125$	−10.9341	−0.977980
$126$	0	0
$127$	−6.44359	−0.571776	−0.285888	−	0.958263i	$-0.592289\pi$
$127$	−6.44359	−0.571776	−0.285888	+	0.958263i	$0.592289\pi$
$128$	0	0
$129$	4.08408	0.359584
$130$	0	0
$131$	2.10251	0.183697	0.0918486	−	0.995773i	$-0.470722\pi$
$131$	2.10251	0.183697	0.0918486	+	0.995773i	$0.470722\pi$
$132$	0	0
$133$	−3.74301	−0.324560
$134$	0	0
$135$	1.32719	0.114226
$136$	0	0
$137$	−14.1874	−1.21211	−0.606055	−	0.795423i	$-0.707249\pi$
$137$	−14.1874	−1.21211	−0.606055	+	0.795423i	$0.707249\pi$
$138$	0	0
$139$	−21.3980	−1.81495	−0.907477	−	0.420103i	$-0.861994\pi$
$139$	−21.3980	−1.81495	−0.907477	+	0.420103i	$0.861994\pi$
$140$	0	0
$141$	−0.831637	−0.0700365
$142$	0	0
$143$	18.9778	1.58700
$144$	0	0
$145$	−10.6310	−0.882854
$146$	0	0
$147$	1.00000	0.0824786
$148$	0	0
$149$	15.3518	1.25767	0.628834	−	0.777540i	$-0.283533\pi$
$149$	15.3518	1.25767	0.628834	+	0.777540i	$0.283533\pi$
$150$	0	0
$151$	−2.85941	−0.232695	−0.116348	−	0.993209i	$-0.537119\pi$
$151$	−2.85941	−0.232695	−0.116348	+	0.993209i	$0.537119\pi$
$152$	0	0
$153$	3.38351	0.273540
$154$	0	0
$155$	13.5575	1.08896
$156$	0	0
$157$	4.58477	0.365905	0.182952	−	0.983122i	$-0.441435\pi$
$157$	4.58477	0.365905	0.182952	+	0.983122i	$0.441435\pi$
$158$	0	0
$159$	10.0664	0.798321
$160$	0	0
$161$	−1.00000	−0.0788110
$162$	0	0
$163$	−11.0526	−0.865703	−0.432852	−	0.901465i	$-0.642493\pi$
$163$	−11.0526	−0.865703	−0.432852	+	0.901465i	$0.642493\pi$
$164$	0	0
$165$	−4.96769	−0.386734
$166$	0	0
$167$	−22.9175	−1.77341	−0.886706	−	0.462333i	$-0.847012\pi$
$167$	−22.9175	−1.77341	−0.886706	+	0.462333i	$0.847012\pi$
$168$	0	0
$169$	12.7069	0.977457
$170$	0	0
$171$	3.74301	0.286235
$172$	0	0
$173$	−20.4873	−1.55762	−0.778808	−	0.627262i	$-0.784176\pi$
$173$	−20.4873	−1.55762	−0.778808	+	0.627262i	$0.784176\pi$
$174$	0	0
$175$	3.23856	0.244812
$176$	0	0
$177$	−6.42594	−0.483003
$178$	0	0
$179$	−15.5297	−1.16074	−0.580372	−	0.814352i	$-0.697093\pi$
$179$	−15.5297	−1.16074	−0.580372	+	0.814352i	$0.697093\pi$
$180$	0	0
$181$	−21.3417	−1.58631	−0.793157	−	0.609018i	$-0.791564\pi$
$181$	−21.3417	−1.58631	−0.793157	+	0.609018i	$0.791564\pi$
$182$	0	0
$183$	2.86019	0.211431
$184$	0	0
$185$	−1.20957	−0.0889290
$186$	0	0
$187$	−12.6645	−0.926120
$188$	0	0
$189$	−1.00000	−0.0727393
$190$	0	0
$191$	2.85941	0.206899	0.103450	−	0.994635i	$-0.467012\pi$
$191$	2.85941	0.206899	0.103450	+	0.994635i	$0.467012\pi$
$192$	0	0
$193$	16.3657	1.17803	0.589013	−	0.808123i	$-0.299517\pi$
$193$	16.3657	1.17803	0.589013	+	0.808123i	$0.299517\pi$
$194$	0	0
$195$	−6.72913	−0.481883
$196$	0	0
$197$	−5.90184	−0.420489	−0.210244	−	0.977649i	$-0.567426\pi$
$197$	−5.90184	−0.420489	−0.210244	+	0.977649i	$0.567426\pi$
$198$	0	0
$199$	−1.08032	−0.0765821	−0.0382911	−	0.999267i	$-0.512191\pi$
$199$	−1.08032	−0.0765821	−0.0382911	+	0.999267i	$0.512191\pi$
$200$	0	0
$201$	−2.70058	−0.190484
$202$	0	0
$203$	8.01012	0.562200
$204$	0	0
$205$	−9.79933	−0.684415
$206$	0	0
$207$	1.00000	0.0695048
$208$	0	0
$209$	−14.0101	−0.969100
$210$	0	0
$211$	10.8418	0.746378	0.373189	−	0.927755i	$-0.378264\pi$
$211$	10.8418	0.746378	0.373189	+	0.927755i	$0.378264\pi$
$212$	0	0
$213$	−0.210796	−0.0144435
$214$	0	0
$215$	5.42036	0.369666
$216$	0	0
$217$	−10.2151	−0.693449
$218$	0	0
$219$	−12.8418	−0.867766
$220$	0	0
$221$	−17.1551	−1.15397
$222$	0	0
$223$	−5.89172	−0.394538	−0.197269	−	0.980349i	$-0.563207\pi$
$223$	−5.89172	−0.394538	−0.197269	+	0.980349i	$0.563207\pi$
$224$	0	0
$225$	−3.23856	−0.215904
$226$	0	0
$227$	17.2626	1.14576	0.572879	−	0.819640i	$-0.305827\pi$
$227$	17.2626	1.14576	0.572879	+	0.819640i	$0.305827\pi$
$228$	0	0
$229$	10.8410	0.716392	0.358196	−	0.933646i	$-0.383392\pi$
$229$	10.8410	0.716392	0.358196	+	0.933646i	$0.383392\pi$
$230$	0	0
$231$	3.74301	0.246272
$232$	0	0
$233$	−13.9766	−0.915636	−0.457818	−	0.889046i	$-0.651369\pi$
$233$	−13.9766	−0.915636	−0.457818	+	0.889046i	$0.651369\pi$
$234$	0	0
$235$	−1.10374	−0.0720002
$236$	0	0
$237$	−10.2151	−0.663545
$238$	0	0
$239$	5.89172	0.381103	0.190552	−	0.981677i	$-0.438972\pi$
$239$	5.89172	0.381103	0.190552	+	0.981677i	$0.438972\pi$
$240$	0	0
$241$	−19.0290	−1.22577	−0.612883	−	0.790174i	$-0.709990\pi$
$241$	−19.0290	−1.22577	−0.612883	+	0.790174i	$0.709990\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	1.32719	0.0847911
$246$	0	0
$247$	−18.9778	−1.20753
$248$	0	0
$249$	1.20126	0.0761269
$250$	0	0
$251$	−24.6328	−1.55481	−0.777404	−	0.629002i	$-0.783464\pi$
$251$	−24.6328	−1.55481	−0.777404	+	0.629002i	$0.783464\pi$
$252$	0	0
$253$	−3.74301	−0.235321
$254$	0	0
$255$	4.49056	0.281210
$256$	0	0
$257$	−8.12016	−0.506521	−0.253261	−	0.967398i	$-0.581503\pi$
$257$	−8.12016	−0.506521	−0.253261	+	0.967398i	$0.581503\pi$
$258$	0	0
$259$	0.911372	0.0566299
$260$	0	0
$261$	−8.01012	−0.495814
$262$	0	0
$263$	12.1505	0.749233	0.374617	−	0.927180i	$-0.377774\pi$
$263$	12.1505	0.749233	0.374617	+	0.927180i	$0.377774\pi$
$264$	0	0
$265$	13.3601	0.820704
$266$	0	0
$267$	−14.8929	−0.911433
$268$	0	0
$269$	−24.6689	−1.50409	−0.752043	−	0.659114i	$-0.770932\pi$
$269$	−24.6689	−1.50409	−0.752043	+	0.659114i	$0.770932\pi$
$270$	0	0
$271$	9.74800	0.592149	0.296074	−	0.955165i	$-0.404322\pi$
$271$	9.74800	0.592149	0.296074	+	0.955165i	$0.404322\pi$
$272$	0	0
$273$	5.07020	0.306863
$274$	0	0
$275$	12.1220	0.730983
$276$	0	0
$277$	11.0980	0.666812	0.333406	−	0.942783i	$-0.391802\pi$
$277$	11.0980	0.666812	0.333406	+	0.942783i	$0.391802\pi$
$278$	0	0
$279$	10.2151	0.611565
$280$	0	0
$281$	−7.56062	−0.451029	−0.225514	−	0.974240i	$-0.572406\pi$
$281$	−7.56062	−0.451029	−0.225514	+	0.974240i	$0.572406\pi$
$282$	0	0
$283$	21.6929	1.28951	0.644753	−	0.764391i	$-0.276960\pi$
$283$	21.6929	1.28951	0.644753	+	0.764391i	$0.276960\pi$
$284$	0	0
$285$	4.96769	0.294261
$286$	0	0
$287$	7.38351	0.435835
$288$	0	0
$289$	−5.55187	−0.326581
$290$	0	0
$291$	−12.8418	−0.752797
$292$	0	0
$293$	25.9859	1.51811	0.759057	−	0.651024i	$-0.225660\pi$
$293$	25.9859	1.51811	0.759057	+	0.651024i	$0.225660\pi$
$294$	0	0
$295$	−8.52845	−0.496546
$296$	0	0
$297$	−3.74301	−0.217191
$298$	0	0
$299$	−5.07020	−0.293217
$300$	0	0
$301$	−4.08408	−0.235403
$302$	0	0
$303$	9.10809	0.523246
$304$	0	0
$305$	3.79601	0.217359
$306$	0	0
$307$	26.2354	1.49733	0.748666	−	0.662947i	$-0.230694\pi$
$307$	26.2354	1.49733	0.748666	+	0.662947i	$0.230694\pi$
$308$	0	0
$309$	−2.02777	−0.115356
$310$	0	0
$311$	18.1462	1.02898	0.514488	−	0.857498i	$-0.327982\pi$
$311$	18.1462	1.02898	0.514488	+	0.857498i	$0.327982\pi$
$312$	0	0
$313$	0.597283	0.0337604	0.0168802	−	0.999858i	$-0.494627\pi$
$313$	0.597283	0.0337604	0.0168802	+	0.999858i	$0.494627\pi$
$314$	0	0
$315$	−1.32719	−0.0747788
$316$	0	0
$317$	3.06008	0.171871	0.0859356	−	0.996301i	$-0.472612\pi$
$317$	3.06008	0.171871	0.0859356	+	0.996301i	$0.472612\pi$
$318$	0	0
$319$	29.9820	1.67867
$320$	0	0
$321$	14.0766	0.785677
$322$	0	0
$323$	12.6645	0.704672
$324$	0	0
$325$	16.4202	0.910827
$326$	0	0
$327$	−8.24370	−0.455877
$328$	0	0
$329$	0.831637	0.0458497
$330$	0	0
$331$	33.9088	1.86380	0.931898	−	0.362721i	$-0.118152\pi$
$331$	33.9088	1.86380	0.931898	+	0.362721i	$0.118152\pi$
$332$	0	0
$333$	−0.911372	−0.0499429
$334$	0	0
$335$	−3.58418	−0.195825
$336$	0	0
$337$	11.6675	0.635568	0.317784	−	0.948163i	$-0.397061\pi$
$337$	11.6675	0.635568	0.317784	+	0.948163i	$0.397061\pi$
$338$	0	0
$339$	−8.51833	−0.462652
$340$	0	0
$341$	−38.2354	−2.07056
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	1.32719	0.0714536
$346$	0	0
$347$	4.45948	0.239398	0.119699	−	0.992810i	$-0.461807\pi$
$347$	4.45948	0.239398	0.119699	+	0.992810i	$0.461807\pi$
$348$	0	0
$349$	2.96769	0.158857	0.0794284	−	0.996841i	$-0.474690\pi$
$349$	2.96769	0.158857	0.0794284	+	0.996841i	$0.474690\pi$
$350$	0	0
$351$	−5.07020	−0.270627
$352$	0	0
$353$	−7.15824	−0.380995	−0.190497	−	0.981688i	$-0.561010\pi$
$353$	−7.15824	−0.380995	−0.190497	+	0.981688i	$0.561010\pi$
$354$	0	0
$355$	−0.279767	−0.0148485
$356$	0	0
$357$	−3.38351	−0.179074
$358$	0	0
$359$	−21.3424	−1.12641	−0.563206	−	0.826317i	$-0.690432\pi$
$359$	−21.3424	−1.12641	−0.563206	+	0.826317i	$0.690432\pi$
$360$	0	0
$361$	−4.98988	−0.262625
$362$	0	0
$363$	3.01012	0.157990
$364$	0	0
$365$	−17.0435	−0.892096
$366$	0	0
$367$	15.1257	0.789557	0.394779	−	0.918776i	$-0.370821\pi$
$367$	15.1257	0.789557	0.394779	+	0.918776i	$0.370821\pi$
$368$	0	0
$369$	−7.38351	−0.384370
$370$	0	0
$371$	−10.0664	−0.522624
$372$	0	0
$373$	21.6506	1.12103	0.560513	−	0.828145i	$-0.310604\pi$
$373$	21.6506	1.12103	0.560513	+	0.828145i	$0.310604\pi$
$374$	0	0
$375$	−10.9341	−0.564637
$376$	0	0
$377$	40.6129	2.09167
$378$	0	0
$379$	−2.41660	−0.124132	−0.0620662	−	0.998072i	$-0.519769\pi$
$379$	−2.41660	−0.124132	−0.0620662	+	0.998072i	$0.519769\pi$
$380$	0	0
$381$	−6.44359	−0.330115
$382$	0	0
$383$	−8.92663	−0.456129	−0.228065	−	0.973646i	$-0.573240\pi$
$383$	−8.92663	−0.456129	−0.228065	+	0.973646i	$0.573240\pi$
$384$	0	0
$385$	4.96769	0.253177
$386$	0	0
$387$	4.08408	0.207606
$388$	0	0
$389$	8.97717	0.455161	0.227580	−	0.973759i	$-0.426919\pi$
$389$	8.97717	0.455161	0.227580	+	0.973759i	$0.426919\pi$
$390$	0	0
$391$	3.38351	0.171111
$392$	0	0
$393$	2.10251	0.106058
$394$	0	0
$395$	−13.5575	−0.682149
$396$	0	0
$397$	−1.42892	−0.0717155	−0.0358577	−	0.999357i	$-0.511416\pi$
$397$	−1.42892	−0.0717155	−0.0358577	+	0.999357i	$0.511416\pi$
$398$	0	0
$399$	−3.74301	−0.187385
$400$	0	0
$401$	−22.8695	−1.14205	−0.571025	−	0.820933i	$-0.693454\pi$
$401$	−22.8695	−1.14205	−0.571025	+	0.820933i	$0.693454\pi$
$402$	0	0
$403$	−51.7928	−2.57999
$404$	0	0
$405$	1.32719	0.0659487
$406$	0	0
$407$	3.41128	0.169091
$408$	0	0
$409$	17.1492	0.847971	0.423986	−	0.905669i	$-0.360631\pi$
$409$	17.1492	0.847971	0.423986	+	0.905669i	$0.360631\pi$
$410$	0	0
$411$	−14.1874	−0.699812
$412$	0	0
$413$	6.42594	0.316200
$414$	0	0
$415$	1.59430	0.0782613
$416$	0	0
$417$	−21.3980	−1.04786
$418$	0	0
$419$	16.4412	0.803205	0.401603	−	0.915814i	$-0.368453\pi$
$419$	16.4412	0.803205	0.401603	+	0.915814i	$0.368453\pi$
$420$	0	0
$421$	12.1487	0.592092	0.296046	−	0.955174i	$-0.404332\pi$
$421$	12.1487	0.592092	0.296046	+	0.955174i	$0.404332\pi$
$422$	0	0
$423$	−0.831637	−0.0404356
$424$	0	0
$425$	−10.9577	−0.531527
$426$	0	0
$427$	−2.86019	−0.138414
$428$	0	0
$429$	18.9778	0.916257
$430$	0	0
$431$	17.8004	0.857413	0.428707	−	0.903444i	$-0.358969\pi$
$431$	17.8004	0.857413	0.428707	+	0.903444i	$0.358969\pi$
$432$	0	0
$433$	30.9277	1.48629	0.743144	−	0.669131i	$-0.233334\pi$
$433$	30.9277	1.48629	0.743144	+	0.669131i	$0.233334\pi$
$434$	0	0
$435$	−10.6310	−0.509716
$436$	0	0
$437$	3.74301	0.179052
$438$	0	0
$439$	28.3571	1.35341	0.676706	−	0.736254i	$-0.263407\pi$
$439$	28.3571	1.35341	0.676706	+	0.736254i	$0.263407\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	−35.5618	−1.68959	−0.844796	−	0.535088i	$-0.820278\pi$
$443$	−35.5618	−1.68959	−0.844796	+	0.535088i	$0.820278\pi$
$444$	0	0
$445$	−19.7658	−0.936988
$446$	0	0
$447$	15.3518	0.726115
$448$	0	0
$449$	9.67761	0.456715	0.228357	−	0.973577i	$-0.426665\pi$
$449$	9.67761	0.456715	0.228357	+	0.973577i	$0.426665\pi$
$450$	0	0
$451$	27.6365	1.30135
$452$	0	0
$453$	−2.85941	−0.134347
$454$	0	0
$455$	6.72913	0.315466
$456$	0	0
$457$	0.827288	0.0386989	0.0193494	−	0.999813i	$-0.493840\pi$
$457$	0.827288	0.0386989	0.0193494	+	0.999813i	$0.493840\pi$
$458$	0	0
$459$	3.38351	0.157929
$460$	0	0
$461$	21.4170	0.997491	0.498746	−	0.866748i	$-0.333794\pi$
$461$	21.4170	0.997491	0.498746	+	0.866748i	$0.333794\pi$
$462$	0	0
$463$	−37.2427	−1.73082	−0.865408	−	0.501068i	$-0.832941\pi$
$463$	−37.2427	−1.73082	−0.865408	+	0.501068i	$0.832941\pi$
$464$	0	0
$465$	13.5575	0.628712
$466$	0	0
$467$	−2.59372	−0.120023	−0.0600114	−	0.998198i	$-0.519114\pi$
$467$	−2.59372	−0.120023	−0.0600114	+	0.998198i	$0.519114\pi$
$468$	0	0
$469$	2.70058	0.124701
$470$	0	0
$471$	4.58477	0.211255
$472$	0	0
$473$	−15.2868	−0.702886
$474$	0	0
$475$	−12.1220	−0.556194
$476$	0	0
$477$	10.0664	0.460911
$478$	0	0
$479$	1.56829	0.0716568	0.0358284	−	0.999358i	$-0.488593\pi$
$479$	1.56829	0.0716568	0.0358284	+	0.999358i	$0.488593\pi$
$480$	0	0
$481$	4.62084	0.210692
$482$	0	0
$483$	−1.00000	−0.0455016
$484$	0	0
$485$	−17.0435	−0.773904
$486$	0	0
$487$	−7.26335	−0.329134	−0.164567	−	0.986366i	$-0.552623\pi$
$487$	−7.26335	−0.329134	−0.164567	+	0.986366i	$0.552623\pi$
$488$	0	0
$489$	−11.0526	−0.499814
$490$	0	0
$491$	−7.20308	−0.325071	−0.162535	−	0.986703i	$-0.551967\pi$
$491$	−7.20308	−0.325071	−0.162535	+	0.986703i	$0.551967\pi$
$492$	0	0
$493$	−27.1023	−1.22063
$494$	0	0
$495$	−4.96769	−0.223281
$496$	0	0
$497$	0.210796	0.00945549
$498$	0	0
$499$	9.19296	0.411533	0.205767	−	0.978601i	$-0.434031\pi$
$499$	9.19296	0.411533	0.205767	+	0.978601i	$0.434031\pi$
$500$	0	0
$501$	−22.9175	−1.02388
$502$	0	0
$503$	7.73471	0.344874	0.172437	−	0.985021i	$-0.444836\pi$
$503$	7.73471	0.344874	0.172437	+	0.985021i	$0.444836\pi$
$504$	0	0
$505$	12.0882	0.537917
$506$	0	0
$507$	12.7069	0.564335
$508$	0	0
$509$	9.59865	0.425453	0.212726	−	0.977112i	$-0.431766\pi$
$509$	9.59865	0.425453	0.212726	+	0.977112i	$0.431766\pi$
$510$	0	0
$511$	12.8418	0.568086
$512$	0	0
$513$	3.74301	0.165258
$514$	0	0
$515$	−2.69124	−0.118590
$516$	0	0
$517$	3.11283	0.136902
$518$	0	0
$519$	−20.4873	−0.899290
$520$	0	0
$521$	−41.1418	−1.80245	−0.901227	−	0.433348i	$-0.857332\pi$
$521$	−41.1418	−1.80245	−0.901227	+	0.433348i	$0.857332\pi$
$522$	0	0
$523$	4.29488	0.187802	0.0939010	−	0.995582i	$-0.470066\pi$
$523$	4.29488	0.187802	0.0939010	+	0.995582i	$0.470066\pi$
$524$	0	0
$525$	3.23856	0.141343
$526$	0	0
$527$	34.5630	1.50559
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−6.42594	−0.278862
$532$	0	0
$533$	37.4359	1.62153
$534$	0	0
$535$	18.6823	0.807706
$536$	0	0
$537$	−15.5297	−0.670155
$538$	0	0
$539$	−3.74301	−0.161223
$540$	0	0
$541$	−41.7402	−1.79455	−0.897276	−	0.441469i	$-0.854457\pi$
$541$	−41.7402	−1.79455	−0.897276	+	0.441469i	$0.854457\pi$
$542$	0	0
$543$	−21.3417	−0.915858
$544$	0	0
$545$	−10.9410	−0.468659
$546$	0	0
$547$	−21.1561	−0.904570	−0.452285	−	0.891874i	$-0.649391\pi$
$547$	−21.1561	−0.904570	−0.452285	+	0.891874i	$0.649391\pi$
$548$	0	0
$549$	2.86019	0.122070
$550$	0	0
$551$	−29.9820	−1.27727
$552$	0	0
$553$	10.2151	0.434392
$554$	0	0
$555$	−1.20957	−0.0513432
$556$	0	0
$557$	−15.0326	−0.636950	−0.318475	−	0.947931i	$-0.603171\pi$
$557$	−15.0326	−0.636950	−0.318475	+	0.947931i	$0.603171\pi$
$558$	0	0
$559$	−20.7071	−0.875818
$560$	0	0
$561$	−12.6645	−0.534696
$562$	0	0
$563$	3.93460	0.165824	0.0829118	−	0.996557i	$-0.473578\pi$
$563$	3.93460	0.165824	0.0829118	+	0.996557i	$0.473578\pi$
$564$	0	0
$565$	−11.3055	−0.475624
$566$	0	0
$567$	−1.00000	−0.0419961
$568$	0	0
$569$	30.7948	1.29098	0.645492	−	0.763767i	$-0.276652\pi$
$569$	30.7948	1.29098	0.645492	+	0.763767i	$0.276652\pi$
$570$	0	0
$571$	38.8848	1.62728	0.813639	−	0.581371i	$-0.197483\pi$
$571$	38.8848	1.62728	0.813639	+	0.581371i	$0.197483\pi$
$572$	0	0
$573$	2.85941	0.119453
$574$	0	0
$575$	−3.23856	−0.135057
$576$	0	0
$577$	18.4505	0.768106	0.384053	−	0.923311i	$-0.374528\pi$
$577$	18.4505	0.768106	0.384053	+	0.923311i	$0.374528\pi$
$578$	0	0
$579$	16.3657	0.680134
$580$	0	0
$581$	−1.20126	−0.0498367
$582$	0	0
$583$	−37.6788	−1.56050
$584$	0	0
$585$	−6.72913	−0.278215
$586$	0	0
$587$	−44.5471	−1.83866	−0.919329	−	0.393491i	$-0.871267\pi$
$587$	−44.5471	−1.83866	−0.919329	+	0.393491i	$0.871267\pi$
$588$	0	0
$589$	38.2354	1.57546
$590$	0	0
$591$	−5.90184	−0.242769
$592$	0	0
$593$	1.69261	0.0695070	0.0347535	−	0.999396i	$-0.488935\pi$
$593$	1.69261	0.0695070	0.0347535	+	0.999396i	$0.488935\pi$
$594$	0	0
$595$	−4.49056	−0.184095
$596$	0	0
$597$	−1.08032	−0.0442147
$598$	0	0
$599$	34.1180	1.39402	0.697012	−	0.717059i	$-0.254512\pi$
$599$	34.1180	1.39402	0.697012	+	0.717059i	$0.254512\pi$
$600$	0	0
$601$	44.0346	1.79621	0.898104	−	0.439783i	$-0.144945\pi$
$601$	44.0346	1.79621	0.898104	+	0.439783i	$0.144945\pi$
$602$	0	0
$603$	−2.70058	−0.109976
$604$	0	0
$605$	3.99501	0.162420
$606$	0	0
$607$	−14.9410	−0.606435	−0.303217	−	0.952921i	$-0.598061\pi$
$607$	−14.9410	−0.606435	−0.303217	+	0.952921i	$0.598061\pi$
$608$	0	0
$609$	8.01012	0.324587
$610$	0	0
$611$	4.21657	0.170584
$612$	0	0
$613$	6.47200	0.261401	0.130701	−	0.991422i	$-0.458277\pi$
$613$	6.47200	0.261401	0.130701	+	0.991422i	$0.458277\pi$
$614$	0	0
$615$	−9.79933	−0.395147
$616$	0	0
$617$	−10.0513	−0.404651	−0.202326	−	0.979318i	$-0.564850\pi$
$617$	−10.0513	−0.404651	−0.202326	+	0.979318i	$0.564850\pi$
$618$	0	0
$619$	−10.0095	−0.402315	−0.201158	−	0.979559i	$-0.564470\pi$
$619$	−10.0095	−0.402315	−0.201158	+	0.979559i	$0.564470\pi$
$620$	0	0
$621$	1.00000	0.0401286
$622$	0	0
$623$	14.8929	0.596673
$624$	0	0
$625$	1.68111	0.0672445
$626$	0	0
$627$	−14.0101	−0.559510
$628$	0	0
$629$	−3.08364	−0.122953
$630$	0	0
$631$	−12.9251	−0.514539	−0.257269	−	0.966340i	$-0.582823\pi$
$631$	−12.9251	−0.514539	−0.257269	+	0.966340i	$0.582823\pi$
$632$	0	0
$633$	10.8418	0.430921
$634$	0	0
$635$	−8.55187	−0.339371
$636$	0	0
$637$	−5.07020	−0.200889
$638$	0	0
$639$	−0.210796	−0.00833896
$640$	0	0
$641$	−13.3676	−0.527989	−0.263994	−	0.964524i	$-0.585040\pi$
$641$	−13.3676	−0.527989	−0.263994	+	0.964524i	$0.585040\pi$
$642$	0	0
$643$	−25.5994	−1.00954	−0.504772	−	0.863253i	$-0.668423\pi$
$643$	−25.5994	−1.00954	−0.504772	+	0.863253i	$0.668423\pi$
$644$	0	0
$645$	5.42036	0.213427
$646$	0	0
$647$	−15.9953	−0.628839	−0.314419	−	0.949284i	$-0.601810\pi$
$647$	−15.9953	−0.628839	−0.314419	+	0.949284i	$0.601810\pi$
$648$	0	0
$649$	24.0524	0.944138
$650$	0	0
$651$	−10.2151	−0.400363
$652$	0	0
$653$	17.9397	0.702036	0.351018	−	0.936369i	$-0.385836\pi$
$653$	17.9397	0.702036	0.351018	+	0.936369i	$0.385836\pi$
$654$	0	0
$655$	2.79043	0.109031
$656$	0	0
$657$	−12.8418	−0.501005
$658$	0	0
$659$	25.8366	1.00645	0.503225	−	0.864155i	$-0.332147\pi$
$659$	25.8366	1.00645	0.503225	+	0.864155i	$0.332147\pi$
$660$	0	0
$661$	5.27587	0.205207	0.102604	−	0.994722i	$-0.467283\pi$
$661$	5.27587	0.205207	0.102604	+	0.994722i	$0.467283\pi$
$662$	0	0
$663$	−17.1551	−0.666248
$664$	0	0
$665$	−4.96769	−0.192639
$666$	0	0
$667$	−8.01012	−0.310153
$668$	0	0
$669$	−5.89172	−0.227787
$670$	0	0
$671$	−10.7057	−0.413289
$672$	0	0
$673$	−49.9947	−1.92715	−0.963577	−	0.267431i	$-0.913825\pi$
$673$	−49.9947	−1.92715	−0.963577	+	0.267431i	$0.913825\pi$
$674$	0	0
$675$	−3.23856	−0.124652
$676$	0	0
$677$	−24.9637	−0.959434	−0.479717	−	0.877423i	$-0.659261\pi$
$677$	−24.9637	−0.959434	−0.479717	+	0.877423i	$0.659261\pi$
$678$	0	0
$679$	12.8418	0.492822
$680$	0	0
$681$	17.2626	0.661503
$682$	0	0
$683$	−26.8418	−1.02707	−0.513536	−	0.858068i	$-0.671665\pi$
$683$	−26.8418	−1.02707	−0.513536	+	0.858068i	$0.671665\pi$
$684$	0	0
$685$	−18.8294	−0.719433
$686$	0	0
$687$	10.8410	0.413609
$688$	0	0
$689$	−51.0389	−1.94443
$690$	0	0
$691$	−4.02602	−0.153157	−0.0765785	−	0.997064i	$-0.524400\pi$
$691$	−4.02602	−0.153157	−0.0765785	+	0.997064i	$0.524400\pi$
$692$	0	0
$693$	3.74301	0.142185
$694$	0	0
$695$	−28.3992	−1.07724
$696$	0	0
$697$	−24.9822	−0.946267
$698$	0	0
$699$	−13.9766	−0.528643
$700$	0	0
$701$	−18.0109	−0.680262	−0.340131	−	0.940378i	$-0.610472\pi$
$701$	−18.0109	−0.680262	−0.340131	+	0.940378i	$0.610472\pi$
$702$	0	0
$703$	−3.41128	−0.128659
$704$	0	0
$705$	−1.10374	−0.0415693
$706$	0	0
$707$	−9.10809	−0.342545
$708$	0	0
$709$	46.0104	1.72796	0.863978	−	0.503530i	$-0.167965\pi$
$709$	46.0104	1.72796	0.863978	+	0.503530i	$0.167965\pi$
$710$	0	0
$711$	−10.2151	−0.383098
$712$	0	0
$713$	10.2151	0.382560
$714$	0	0
$715$	25.1872	0.941947
$716$	0	0
$717$	5.89172	0.220030
$718$	0	0
$719$	16.1521	0.602371	0.301186	−	0.953566i	$-0.402618\pi$
$719$	16.1521	0.602371	0.301186	+	0.953566i	$0.402618\pi$
$720$	0	0
$721$	2.02777	0.0755180
$722$	0	0
$723$	−19.0290	−0.707696
$724$	0	0
$725$	25.9413	0.963435
$726$	0	0
$727$	−24.7771	−0.918933	−0.459467	−	0.888195i	$-0.651959\pi$
$727$	−24.7771	−0.918933	−0.459467	+	0.888195i	$0.651959\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	13.8185	0.511097
$732$	0	0
$733$	44.5111	1.64405	0.822027	−	0.569449i	$-0.192843\pi$
$733$	44.5111	1.64405	0.822027	+	0.569449i	$0.192843\pi$
$734$	0	0
$735$	1.32719	0.0489542
$736$	0	0
$737$	10.1083	0.372343
$738$	0	0
$739$	−33.4846	−1.23175	−0.615875	−	0.787844i	$-0.711198\pi$
$739$	−33.4846	−1.23175	−0.615875	+	0.787844i	$0.711198\pi$
$740$	0	0
$741$	−18.9778	−0.697167
$742$	0	0
$743$	18.5588	0.680857	0.340429	−	0.940270i	$-0.389428\pi$
$743$	18.5588	0.680857	0.340429	+	0.940270i	$0.389428\pi$
$744$	0	0
$745$	20.3748	0.746473
$746$	0	0
$747$	1.20126	0.0439519
$748$	0	0
$749$	−14.0766	−0.514346
$750$	0	0
$751$	−29.4651	−1.07520	−0.537598	−	0.843201i	$-0.680668\pi$
$751$	−29.4651	−1.07520	−0.537598	+	0.843201i	$0.680668\pi$
$752$	0	0
$753$	−24.6328	−0.897669
$754$	0	0
$755$	−3.79498	−0.138113
$756$	0	0
$757$	4.68610	0.170319	0.0851597	−	0.996367i	$-0.472860\pi$
$757$	4.68610	0.170319	0.0851597	+	0.996367i	$0.472860\pi$
$758$	0	0
$759$	−3.74301	−0.135863
$760$	0	0
$761$	0.936750	0.0339572	0.0169786	−	0.999856i	$-0.494595\pi$
$761$	0.936750	0.0339572	0.0169786	+	0.999856i	$0.494595\pi$
$762$	0	0
$763$	8.24370	0.298442
$764$	0	0
$765$	4.49056	0.162357
$766$	0	0
$767$	32.5808	1.17643
$768$	0	0
$769$	−18.7024	−0.674426	−0.337213	−	0.941428i	$-0.609484\pi$
$769$	−18.7024	−0.674426	−0.337213	+	0.941428i	$0.609484\pi$
$770$	0	0
$771$	−8.12016	−0.292440
$772$	0	0
$773$	−5.35827	−0.192724	−0.0963618	−	0.995346i	$-0.530721\pi$
$773$	−5.35827	−0.192724	−0.0963618	+	0.995346i	$0.530721\pi$
$774$	0	0
$775$	−33.0824	−1.18835
$776$	0	0
$777$	0.911372	0.0326953
$778$	0	0
$779$	−27.6365	−0.990182
$780$	0	0
$781$	0.789011	0.0282330
$782$	0	0
$783$	−8.01012	−0.286258
$784$	0	0
$785$	6.08487	0.217178
$786$	0	0
$787$	−26.0308	−0.927897	−0.463948	−	0.885862i	$-0.653568\pi$
$787$	−26.0308	−0.927897	−0.463948	+	0.885862i	$0.653568\pi$
$788$	0	0
$789$	12.1505	0.432570
$790$	0	0
$791$	8.51833	0.302877
$792$	0	0
$793$	−14.5017	−0.514971
$794$	0	0
$795$	13.3601	0.473834
$796$	0	0
$797$	28.7343	1.01782	0.508910	−	0.860820i	$-0.330049\pi$
$797$	28.7343	1.01782	0.508910	+	0.860820i	$0.330049\pi$
$798$	0	0
$799$	−2.81385	−0.0995469
$800$	0	0
$801$	−14.8929	−0.526216
$802$	0	0
$803$	48.0668	1.69624
$804$	0	0
$805$	−1.32719	−0.0467773
$806$	0	0
$807$	−24.6689	−0.868385
$808$	0	0
$809$	−31.7817	−1.11738	−0.558692	−	0.829375i	$-0.688697\pi$
$809$	−31.7817	−1.11738	−0.558692	+	0.829375i	$0.688697\pi$
$810$	0	0
$811$	44.6848	1.56909	0.784547	−	0.620069i	$-0.212896\pi$
$811$	44.6848	1.56909	0.784547	+	0.620069i	$0.212896\pi$
$812$	0	0
$813$	9.74800	0.341877
$814$	0	0
$815$	−14.6689	−0.513828
$816$	0	0
$817$	15.2868	0.534816
$818$	0	0
$819$	5.07020	0.177167
$820$	0	0
$821$	30.3581	1.05951	0.529753	−	0.848152i	$-0.322284\pi$
$821$	30.3581	1.05951	0.529753	+	0.848152i	$0.322284\pi$
$822$	0	0
$823$	14.1004	0.491508	0.245754	−	0.969332i	$-0.420964\pi$
$823$	14.1004	0.491508	0.245754	+	0.969332i	$0.420964\pi$
$824$	0	0
$825$	12.1220	0.422033
$826$	0	0
$827$	−51.9559	−1.80668	−0.903342	−	0.428922i	$-0.858893\pi$
$827$	−51.9559	−1.80668	−0.903342	+	0.428922i	$0.858893\pi$
$828$	0	0
$829$	42.8782	1.48922	0.744611	−	0.667498i	$-0.232635\pi$
$829$	42.8782	1.48922	0.744611	+	0.667498i	$0.232635\pi$
$830$	0	0
$831$	11.0980	0.384984
$832$	0	0
$833$	3.38351	0.117232
$834$	0	0
$835$	−30.4160	−1.05259
$836$	0	0
$837$	10.2151	0.353087
$838$	0	0
$839$	23.3157	0.804948	0.402474	−	0.915431i	$-0.368150\pi$
$839$	23.3157	0.804948	0.402474	+	0.915431i	$0.368150\pi$
$840$	0	0
$841$	35.1621	1.21249
$842$	0	0
$843$	−7.56062	−0.260402
$844$	0	0
$845$	16.8645	0.580158
$846$	0	0
$847$	−3.01012	−0.103429
$848$	0	0
$849$	21.6929	0.744497
$850$	0	0
$851$	−0.911372	−0.0312414
$852$	0	0
$853$	−14.7024	−0.503400	−0.251700	−	0.967805i	$-0.580990\pi$
$853$	−14.7024	−0.503400	−0.251700	+	0.967805i	$0.580990\pi$
$854$	0	0
$855$	4.96769	0.169891
$856$	0	0
$857$	30.3835	1.03788	0.518940	−	0.854811i	$-0.326327\pi$
$857$	30.3835	1.03788	0.518940	+	0.854811i	$0.326327\pi$
$858$	0	0
$859$	24.0202	0.819560	0.409780	−	0.912184i	$-0.365605\pi$
$859$	24.0202	0.819560	0.409780	+	0.912184i	$0.365605\pi$
$860$	0	0
$861$	7.38351	0.251629
$862$	0	0
$863$	−4.21657	−0.143534	−0.0717668	−	0.997421i	$-0.522864\pi$
$863$	−4.21657	−0.143534	−0.0717668	+	0.997421i	$0.522864\pi$
$864$	0	0
$865$	−27.1905	−0.924505
$866$	0	0
$867$	−5.55187	−0.188551
$868$	0	0
$869$	38.2354	1.29705
$870$	0	0
$871$	13.6925	0.463952
$872$	0	0
$873$	−12.8418	−0.434628
$874$	0	0
$875$	10.9341	0.369642
$876$	0	0
$877$	56.1035	1.89448	0.947241	−	0.320522i	$-0.103858\pi$
$877$	56.1035	1.89448	0.947241	+	0.320522i	$0.103858\pi$
$878$	0	0
$879$	25.9859	0.876483
$880$	0	0
$881$	−49.7605	−1.67647	−0.838237	−	0.545307i	$-0.816413\pi$
$881$	−49.7605	−1.67647	−0.838237	+	0.545307i	$0.816413\pi$
$882$	0	0
$883$	37.7548	1.27055	0.635275	−	0.772286i	$-0.280887\pi$
$883$	37.7548	1.27055	0.635275	+	0.772286i	$0.280887\pi$
$884$	0	0
$885$	−8.52845	−0.286681
$886$	0	0
$887$	24.9385	0.837353	0.418676	−	0.908135i	$-0.362494\pi$
$887$	24.9385	0.837353	0.418676	+	0.908135i	$0.362494\pi$
$888$	0	0
$889$	6.44359	0.216111
$890$	0	0
$891$	−3.74301	−0.125396
$892$	0	0
$893$	−3.11283	−0.104167
$894$	0	0
$895$	−20.6109	−0.688945
$896$	0	0
$897$	−5.07020	−0.169289
$898$	0	0
$899$	−81.8246	−2.72900
$900$	0	0
$901$	34.0599	1.13470
$902$	0	0
$903$	−4.08408	−0.135910
$904$	0	0
$905$	−28.3245	−0.941537
$906$	0	0
$907$	−52.2735	−1.73571	−0.867857	−	0.496814i	$-0.834503\pi$
$907$	−52.2735	−1.73571	−0.867857	+	0.496814i	$0.834503\pi$
$908$	0	0
$909$	9.10809	0.302096
$910$	0	0
$911$	10.4494	0.346203	0.173101	−	0.984904i	$-0.444621\pi$
$911$	10.4494	0.346203	0.173101	+	0.984904i	$0.444621\pi$
$912$	0	0
$913$	−4.49634	−0.148807
$914$	0	0
$915$	3.79601	0.125492
$916$	0	0
$917$	−2.10251	−0.0694310
$918$	0	0
$919$	7.14281	0.235620	0.117810	−	0.993036i	$-0.462413\pi$
$919$	7.14281	0.235620	0.117810	+	0.993036i	$0.462413\pi$
$920$	0	0
$921$	26.2354	0.864486
$922$	0	0
$923$	1.06878	0.0351793
$924$	0	0
$925$	2.95154	0.0970460
$926$	0	0
$927$	−2.02777	−0.0666006
$928$	0	0
$929$	−59.6248	−1.95623	−0.978113	−	0.208073i	$-0.933281\pi$
$929$	−59.6248	−1.95623	−0.978113	+	0.208073i	$0.933281\pi$
$930$	0	0
$931$	3.74301	0.122672
$932$	0	0
$933$	18.1462	0.594079
$934$	0	0
$935$	−16.8082	−0.549688
$936$	0	0
$937$	38.6177	1.26158	0.630792	−	0.775952i	$-0.282730\pi$
$937$	38.6177	1.26158	0.630792	+	0.775952i	$0.282730\pi$
$938$	0	0
$939$	0.597283	0.0194916
$940$	0	0
$941$	−57.9022	−1.88756	−0.943779	−	0.330576i	$-0.892757\pi$
$941$	−57.9022	−1.88756	−0.943779	+	0.330576i	$0.892757\pi$
$942$	0	0
$943$	−7.38351	−0.240440
$944$	0	0
$945$	−1.32719	−0.0431735
$946$	0	0
$947$	−34.9823	−1.13677	−0.568386	−	0.822762i	$-0.692432\pi$
$947$	−34.9823	−1.13677	−0.568386	+	0.822762i	$0.692432\pi$
$948$	0	0
$949$	65.1103	2.11357
$950$	0	0
$951$	3.06008	0.0992298
$952$	0	0
$953$	54.8953	1.77823	0.889117	−	0.457681i	$-0.151320\pi$
$953$	54.8953	1.77823	0.889117	+	0.457681i	$0.151320\pi$
$954$	0	0
$955$	3.79498	0.122803
$956$	0	0
$957$	29.9820	0.969179
$958$	0	0
$959$	14.1874	0.458134
$960$	0	0
$961$	73.3492	2.36610
$962$	0	0
$963$	14.0766	0.453611
$964$	0	0
$965$	21.7204	0.699204
$966$	0	0
$967$	46.6096	1.49886	0.749432	−	0.662081i	$-0.230327\pi$
$967$	46.6096	1.49886	0.749432	+	0.662081i	$0.230327\pi$
$968$	0	0
$969$	12.6645	0.406843
$970$	0	0
$971$	−15.4409	−0.495521	−0.247760	−	0.968821i	$-0.579695\pi$
$971$	−15.4409	−0.495521	−0.247760	+	0.968821i	$0.579695\pi$
$972$	0	0
$973$	21.3980	0.685988
$974$	0	0
$975$	16.4202	0.525866
$976$	0	0
$977$	35.6040	1.13907	0.569537	−	0.821966i	$-0.307122\pi$
$977$	35.6040	1.13907	0.569537	+	0.821966i	$0.307122\pi$
$978$	0	0
$979$	55.7444	1.78160
$980$	0	0
$981$	−8.24370	−0.263201
$982$	0	0
$983$	−9.59009	−0.305877	−0.152938	−	0.988236i	$-0.548874\pi$
$983$	−9.59009	−0.305877	−0.152938	+	0.988236i	$0.548874\pi$
$984$	0	0
$985$	−7.83287	−0.249576
$986$	0	0
$987$	0.831637	0.0264713
$988$	0	0
$989$	4.08408	0.129866
$990$	0	0
$991$	−15.9792	−0.507596	−0.253798	−	0.967257i	$-0.581680\pi$
$991$	−15.9792	−0.507596	−0.253798	+	0.967257i	$0.581680\pi$
$992$	0	0
$993$	33.9088	1.07606
$994$	0	0
$995$	−1.43380	−0.0454544
$996$	0	0
$997$	−40.2809	−1.27571	−0.637855	−	0.770156i	$-0.720178\pi$
$997$	−40.2809	−1.27571	−0.637855	+	0.770156i	$0.720178\pi$
$998$	0	0
$999$	−0.911372	−0.0288345

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7728.2.a.cb.1.4		4
4.3	odd	2		3864.2.a.r.1.4	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3864.2.a.r.1.4	✓	4	4.3	odd	2
7728.2.a.cb.1.4		4	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:	\( N \)	\(=\)	\( 7728 = 2^{4} \cdot 3 \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7728.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +1.32719 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +1.32719 q^{5} -1.00000 q^{7} +1.00000 q^{9} -3.74301 q^{11} -5.07020 q^{13} +1.32719 q^{15} +3.38351 q^{17} +3.74301 q^{19} -1.00000 q^{21} +1.00000 q^{23} -3.23856 q^{25} +1.00000 q^{27} -8.01012 q^{29} +10.2151 q^{31} -3.74301 q^{33} -1.32719 q^{35} -0.911372 q^{37} -5.07020 q^{39} -7.38351 q^{41} +4.08408 q^{43} +1.32719 q^{45} -0.831637 q^{47} +1.00000 q^{49} +3.38351 q^{51} +10.0664 q^{53} -4.96769 q^{55} +3.74301 q^{57} -6.42594 q^{59} +2.86019 q^{61} -1.00000 q^{63} -6.72913 q^{65} -2.70058 q^{67} +1.00000 q^{69} -0.210796 q^{71} -12.8418 q^{73} -3.23856 q^{75} +3.74301 q^{77} -10.2151 q^{79} +1.00000 q^{81} +1.20126 q^{83} +4.49056 q^{85} -8.01012 q^{87} -14.8929 q^{89} +5.07020 q^{91} +10.2151 q^{93} +4.96769 q^{95} -12.8418 q^{97} -3.74301 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{3} - 3 q^{5} - 4 q^{7} + 4 q^{9}+O(q^{10}) \) 4 * q + 4 * q^3 - 3 * q^5 - 4 * q^7 + 4 * q^9	\( 4 q + 4 q^{3} - 3 q^{5} - 4 q^{7} + 4 q^{9} - 2 q^{11} + q^{13} - 3 q^{15} - 8 q^{17} + 2 q^{19} - 4 q^{21} + 4 q^{23} - q^{25} + 4 q^{27} - 10 q^{29} + 10 q^{31} - 2 q^{33} + 3 q^{35} + q^{39} - 8 q^{41} - 13 q^{43} - 3 q^{45} + 6 q^{47} + 4 q^{49} - 8 q^{51} + 5 q^{53} - 3 q^{55} + 2 q^{57} + q^{59} + 5 q^{61} - 4 q^{63} - 22 q^{65} - 3 q^{67} + 4 q^{69} - 5 q^{71} - 20 q^{73} - q^{75} + 2 q^{77} - 10 q^{79} + 4 q^{81} - 18 q^{83} + 25 q^{85} - 10 q^{87} - 31 q^{89} - q^{91} + 10 q^{93} + 3 q^{95} - 20 q^{97} - 2 q^{99}+O(q^{100}) \) 4 * q + 4 * q^3 - 3 * q^5 - 4 * q^7 + 4 * q^9 - 2 * q^11 + q^13 - 3 * q^15 - 8 * q^17 + 2 * q^19 - 4 * q^21 + 4 * q^23 - q^25 + 4 * q^27 - 10 * q^29 + 10 * q^31 - 2 * q^33 + 3 * q^35 + q^39 - 8 * q^41 - 13 * q^43 - 3 * q^45 + 6 * q^47 + 4 * q^49 - 8 * q^51 + 5 * q^53 - 3 * q^55 + 2 * q^57 + q^59 + 5 * q^61 - 4 * q^63 - 22 * q^65 - 3 * q^67 + 4 * q^69 - 5 * q^71 - 20 * q^73 - q^75 + 2 * q^77 - 10 * q^79 + 4 * q^81 - 18 * q^83 + 25 * q^85 - 10 * q^87 - 31 * q^89 - q^91 + 10 * q^93 + 3 * q^95 - 20 * q^97 - 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

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Groups

Database

Properties

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