LMFDB - Embedded newform 7605.2.a.be.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7605,2,Mod(1,7605)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7605, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("7605.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7605 = 3^{2} \cdot 5 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7605.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:	$60.7262307372$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 65)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.73205$ of defining polynomial
Character	$\chi$	$=$	7605.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.73205 q^{2} +1.00000 q^{4} -1.00000 q^{5} -2.00000 q^{7} -1.73205 q^{8} +O(q^{10})$	$q+1.73205 q^{2} +1.00000 q^{4} -1.00000 q^{5} -2.00000 q^{7} -1.73205 q^{8} -1.73205 q^{10} -1.26795 q^{11} -3.46410 q^{14} -5.00000 q^{16} -3.46410 q^{17} -4.19615 q^{19} -1.00000 q^{20} -2.19615 q^{22} -4.73205 q^{23} +1.00000 q^{25} -2.00000 q^{28} +9.46410 q^{29} +0.196152 q^{31} -5.19615 q^{32} -6.00000 q^{34} +2.00000 q^{35} +4.00000 q^{37} -7.26795 q^{38} +1.73205 q^{40} -3.46410 q^{41} +10.1962 q^{43} -1.26795 q^{44} -8.19615 q^{46} +6.00000 q^{47} -3.00000 q^{49} +1.73205 q^{50} +10.3923 q^{53} +1.26795 q^{55} +3.46410 q^{56} +16.3923 q^{58} -15.1244 q^{59} +12.3923 q^{61} +0.339746 q^{62} +1.00000 q^{64} +14.3923 q^{67} -3.46410 q^{68} +3.46410 q^{70} +1.26795 q^{71} +4.00000 q^{73} +6.92820 q^{74} -4.19615 q^{76} +2.53590 q^{77} +12.3923 q^{79} +5.00000 q^{80} -6.00000 q^{82} -6.00000 q^{83} +3.46410 q^{85} +17.6603 q^{86} +2.19615 q^{88} +0.928203 q^{89} -4.73205 q^{92} +10.3923 q^{94} +4.19615 q^{95} -2.00000 q^{97} -5.19615 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{4} - 2 q^{5} - 4 q^{7}+O(q^{10}) $ 2 * q + 2 * q^4 - 2 * q^5 - 4 * q^7	$ 2 q + 2 q^{4} - 2 q^{5} - 4 q^{7} - 6 q^{11} - 10 q^{16} + 2 q^{19} - 2 q^{20} + 6 q^{22} - 6 q^{23} + 2 q^{25} - 4 q^{28} + 12 q^{29} - 10 q^{31} - 12 q^{34} + 4 q^{35} + 8 q^{37} - 18 q^{38} + 10 q^{43} - 6 q^{44} - 6 q^{46} + 12 q^{47} - 6 q^{49} + 6 q^{55} + 12 q^{58} - 6 q^{59} + 4 q^{61} + 18 q^{62} + 2 q^{64} + 8 q^{67} + 6 q^{71} + 8 q^{73} + 2 q^{76} + 12 q^{77} + 4 q^{79} + 10 q^{80} - 12 q^{82} - 12 q^{83} + 18 q^{86} - 6 q^{88} - 12 q^{89} - 6 q^{92} - 2 q^{95} - 4 q^{97}+O(q^{100}) $ 2 * q + 2 * q^4 - 2 * q^5 - 4 * q^7 - 6 * q^11 - 10 * q^16 + 2 * q^19 - 2 * q^20 + 6 * q^22 - 6 * q^23 + 2 * q^25 - 4 * q^28 + 12 * q^29 - 10 * q^31 - 12 * q^34 + 4 * q^35 + 8 * q^37 - 18 * q^38 + 10 * q^43 - 6 * q^44 - 6 * q^46 + 12 * q^47 - 6 * q^49 + 6 * q^55 + 12 * q^58 - 6 * q^59 + 4 * q^61 + 18 * q^62 + 2 * q^64 + 8 * q^67 + 6 * q^71 + 8 * q^73 + 2 * q^76 + 12 * q^77 + 4 * q^79 + 10 * q^80 - 12 * q^82 - 12 * q^83 + 18 * q^86 - 6 * q^88 - 12 * q^89 - 6 * q^92 - 2 * q^95 - 4 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	1.73205	1.22474	0.612372	−	0.790569i	$-0.290215\pi$
$2$	1.73205	1.22474	0.612372	+	0.790569i	$0.290215\pi$
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−2.00000	−0.755929	−0.377964	−	0.925820i	$-0.623376\pi$
$7$	−2.00000	−0.755929	−0.377964	+	0.925820i	$0.623376\pi$
$8$	−1.73205	−0.612372
$9$	0	0
$10$	−1.73205	−0.547723
$11$	−1.26795	−0.382301	−0.191151	−	0.981561i	$-0.561222\pi$
$11$	−1.26795	−0.382301	−0.191151	+	0.981561i	$0.561222\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	−3.46410	−0.925820
$15$	0	0
$16$	−5.00000	−1.25000
$17$	−3.46410	−0.840168	−0.420084	−	0.907485i	$-0.637999\pi$
$17$	−3.46410	−0.840168	−0.420084	+	0.907485i	$0.637999\pi$
$18$	0	0
$19$	−4.19615	−0.962663	−0.481332	−	0.876539i	$-0.659847\pi$
$19$	−4.19615	−0.962663	−0.481332	+	0.876539i	$0.659847\pi$
$20$	−1.00000	−0.223607
$21$	0	0
$22$	−2.19615	−0.468221
$23$	−4.73205	−0.986701	−0.493350	−	0.869831i	$-0.664228\pi$
$23$	−4.73205	−0.986701	−0.493350	+	0.869831i	$0.664228\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	−2.00000	−0.377964
$29$	9.46410	1.75744	0.878720	−	0.477338i	$-0.158398\pi$
$29$	9.46410	1.75744	0.878720	+	0.477338i	$0.158398\pi$
$30$	0	0
$31$	0.196152	0.0352300	0.0176150	−	0.999845i	$-0.494393\pi$
$31$	0.196152	0.0352300	0.0176150	+	0.999845i	$0.494393\pi$
$32$	−5.19615	−0.918559
$33$	0	0
$34$	−6.00000	−1.02899
$35$	2.00000	0.338062
$36$	0	0
$37$	4.00000	0.657596	0.328798	−	0.944400i	$-0.393356\pi$
$37$	4.00000	0.657596	0.328798	+	0.944400i	$0.393356\pi$
$38$	−7.26795	−1.17902
$39$	0	0
$40$	1.73205	0.273861
$41$	−3.46410	−0.541002	−0.270501	−	0.962720i	$-0.587189\pi$
$41$	−3.46410	−0.541002	−0.270501	+	0.962720i	$0.587189\pi$
$42$	0	0
$43$	10.1962	1.55490	0.777449	−	0.628946i	$-0.216513\pi$
$43$	10.1962	1.55490	0.777449	+	0.628946i	$0.216513\pi$
$44$	−1.26795	−0.191151
$45$	0	0
$46$	−8.19615	−1.20846
$47$	6.00000	0.875190	0.437595	−	0.899172i	$-0.355830\pi$
$47$	6.00000	0.875190	0.437595	+	0.899172i	$0.355830\pi$
$48$	0	0
$49$	−3.00000	−0.428571
$50$	1.73205	0.244949
$51$	0	0
$52$	0	0
$53$	10.3923	1.42749	0.713746	−	0.700404i	$-0.246997\pi$
$53$	10.3923	1.42749	0.713746	+	0.700404i	$0.246997\pi$
$54$	0	0
$55$	1.26795	0.170970
$56$	3.46410	0.462910
$57$	0	0
$58$	16.3923	2.15242
$59$	−15.1244	−1.96902	−0.984512	−	0.175319i	$-0.943904\pi$
$59$	−15.1244	−1.96902	−0.984512	+	0.175319i	$0.943904\pi$
$60$	0	0
$61$	12.3923	1.58667	0.793336	−	0.608784i	$-0.208342\pi$
$61$	12.3923	1.58667	0.793336	+	0.608784i	$0.208342\pi$
$62$	0.339746	0.0431478
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	0	0
$67$	14.3923	1.75830	0.879150	−	0.476545i	$-0.158111\pi$
$67$	14.3923	1.75830	0.879150	+	0.476545i	$0.158111\pi$
$68$	−3.46410	−0.420084
$69$	0	0
$70$	3.46410	0.414039
$71$	1.26795	0.150478	0.0752389	−	0.997166i	$-0.476028\pi$
$71$	1.26795	0.150478	0.0752389	+	0.997166i	$0.476028\pi$
$72$	0	0
$73$	4.00000	0.468165	0.234082	−	0.972217i	$-0.424791\pi$
$73$	4.00000	0.468165	0.234082	+	0.972217i	$0.424791\pi$
$74$	6.92820	0.805387
$75$	0	0
$76$	−4.19615	−0.481332
$77$	2.53590	0.288992
$78$	0	0
$79$	12.3923	1.39424	0.697122	−	0.716953i	$-0.254464\pi$
$79$	12.3923	1.39424	0.697122	+	0.716953i	$0.254464\pi$
$80$	5.00000	0.559017
$81$	0	0
$82$	−6.00000	−0.662589
$83$	−6.00000	−0.658586	−0.329293	−	0.944228i	$-0.606810\pi$
$83$	−6.00000	−0.658586	−0.329293	+	0.944228i	$0.606810\pi$
$84$	0	0
$85$	3.46410	0.375735
$86$	17.6603	1.90435
$87$	0	0
$88$	2.19615	0.234111
$89$	0.928203	0.0983893	0.0491947	−	0.998789i	$-0.484335\pi$
$89$	0.928203	0.0983893	0.0491947	+	0.998789i	$0.484335\pi$
$90$	0	0
$91$	0	0
$92$	−4.73205	−0.493350
$93$	0	0
$94$	10.3923	1.07188
$95$	4.19615	0.430516
$96$	0	0
$97$	−2.00000	−0.203069	−0.101535	−	0.994832i	$-0.532375\pi$
$97$	−2.00000	−0.203069	−0.101535	+	0.994832i	$0.532375\pi$
$98$	−5.19615	−0.524891
$99$	0	0
$100$	1.00000	0.100000
$101$	−12.9282	−1.28640	−0.643202	−	0.765696i	$-0.722395\pi$
$101$	−12.9282	−1.28640	−0.643202	+	0.765696i	$0.722395\pi$
$102$	0	0
$103$	10.1962	1.00466	0.502328	−	0.864677i	$-0.332477\pi$
$103$	10.1962	1.00466	0.502328	+	0.864677i	$0.332477\pi$
$104$	0	0
$105$	0	0
$106$	18.0000	1.74831
$107$	−0.339746	−0.0328445	−0.0164222	−	0.999865i	$-0.505228\pi$
$107$	−0.339746	−0.0328445	−0.0164222	+	0.999865i	$0.505228\pi$
$108$	0	0
$109$	−2.00000	−0.191565	−0.0957826	−	0.995402i	$-0.530535\pi$
$109$	−2.00000	−0.191565	−0.0957826	+	0.995402i	$0.530535\pi$
$110$	2.19615	0.209395
$111$	0	0
$112$	10.0000	0.944911
$113$	−15.4641	−1.45474	−0.727370	−	0.686245i	$-0.759258\pi$
$113$	−15.4641	−1.45474	−0.727370	+	0.686245i	$0.759258\pi$
$114$	0	0
$115$	4.73205	0.441266
$116$	9.46410	0.878720
$117$	0	0
$118$	−26.1962	−2.41155
$119$	6.92820	0.635107
$120$	0	0
$121$	−9.39230	−0.853846
$122$	21.4641	1.94327
$123$	0	0
$124$	0.196152	0.0176150
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	5.80385	0.515008	0.257504	−	0.966277i	$-0.417100\pi$
$127$	5.80385	0.515008	0.257504	+	0.966277i	$0.417100\pi$
$128$	12.1244	1.07165
$129$	0	0
$130$	0	0
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	0	0
$133$	8.39230	0.727705
$134$	24.9282	2.15347
$135$	0	0
$136$	6.00000	0.514496
$137$	−12.9282	−1.10453	−0.552265	−	0.833668i	$-0.686237\pi$
$137$	−12.9282	−1.10453	−0.552265	+	0.833668i	$0.686237\pi$
$138$	0	0
$139$	−8.39230	−0.711826	−0.355913	−	0.934519i	$-0.615830\pi$
$139$	−8.39230	−0.711826	−0.355913	+	0.934519i	$0.615830\pi$
$140$	2.00000	0.169031
$141$	0	0
$142$	2.19615	0.184297
$143$	0	0
$144$	0	0
$145$	−9.46410	−0.785951
$146$	6.92820	0.573382
$147$	0	0
$148$	4.00000	0.328798
$149$	19.8564	1.62670	0.813350	−	0.581775i	$-0.197641\pi$
$149$	19.8564	1.62670	0.813350	+	0.581775i	$0.197641\pi$
$150$	0	0
$151$	12.1962	0.992509	0.496254	−	0.868177i	$-0.334708\pi$
$151$	12.1962	0.992509	0.496254	+	0.868177i	$0.334708\pi$
$152$	7.26795	0.589509
$153$	0	0
$154$	4.39230	0.353942
$155$	−0.196152	−0.0157553
$156$	0	0
$157$	−10.0000	−0.798087	−0.399043	−	0.916932i	$-0.630658\pi$
$157$	−10.0000	−0.798087	−0.399043	+	0.916932i	$0.630658\pi$
$158$	21.4641	1.70759
$159$	0	0
$160$	5.19615	0.410792
$161$	9.46410	0.745876
$162$	0	0
$163$	−6.39230	−0.500684	−0.250342	−	0.968157i	$-0.580543\pi$
$163$	−6.39230	−0.500684	−0.250342	+	0.968157i	$0.580543\pi$
$164$	−3.46410	−0.270501
$165$	0	0
$166$	−10.3923	−0.806599
$167$	12.9282	1.00041	0.500207	−	0.865906i	$-0.333257\pi$
$167$	12.9282	1.00041	0.500207	+	0.865906i	$0.333257\pi$
$168$	0	0
$169$	0	0
$170$	6.00000	0.460179
$171$	0	0
$172$	10.1962	0.777449
$173$	−15.4641	−1.17571	−0.587857	−	0.808965i	$-0.700028\pi$
$173$	−15.4641	−1.17571	−0.587857	+	0.808965i	$0.700028\pi$
$174$	0	0
$175$	−2.00000	−0.151186
$176$	6.33975	0.477876
$177$	0	0
$178$	1.60770	0.120502
$179$	5.07180	0.379084	0.189542	−	0.981873i	$-0.439300\pi$
$179$	5.07180	0.379084	0.189542	+	0.981873i	$0.439300\pi$
$180$	0	0
$181$	−20.3923	−1.51575	−0.757874	−	0.652401i	$-0.773762\pi$
$181$	−20.3923	−1.51575	−0.757874	+	0.652401i	$0.773762\pi$
$182$	0	0
$183$	0	0
$184$	8.19615	0.604228
$185$	−4.00000	−0.294086
$186$	0	0
$187$	4.39230	0.321197
$188$	6.00000	0.437595
$189$	0	0
$190$	7.26795	0.527272
$191$	18.9282	1.36960	0.684798	−	0.728733i	$-0.259890\pi$
$191$	18.9282	1.36960	0.684798	+	0.728733i	$0.259890\pi$
$192$	0	0
$193$	10.0000	0.719816	0.359908	−	0.932988i	$-0.382808\pi$
$193$	10.0000	0.719816	0.359908	+	0.932988i	$0.382808\pi$
$194$	−3.46410	−0.248708
$195$	0	0
$196$	−3.00000	−0.214286
$197$	0.928203	0.0661317	0.0330659	−	0.999453i	$-0.489473\pi$
$197$	0.928203	0.0661317	0.0330659	+	0.999453i	$0.489473\pi$
$198$	0	0
$199$	20.0000	1.41776	0.708881	−	0.705328i	$-0.249200\pi$
$199$	20.0000	1.41776	0.708881	+	0.705328i	$0.249200\pi$
$200$	−1.73205	−0.122474
$201$	0	0
$202$	−22.3923	−1.57552
$203$	−18.9282	−1.32850
$204$	0	0
$205$	3.46410	0.241943
$206$	17.6603	1.23045
$207$	0	0
$208$	0	0
$209$	5.32051	0.368027
$210$	0	0
$211$	8.00000	0.550743	0.275371	−	0.961338i	$-0.411199\pi$
$211$	8.00000	0.550743	0.275371	+	0.961338i	$0.411199\pi$
$212$	10.3923	0.713746
$213$	0	0
$214$	−0.588457	−0.0402261
$215$	−10.1962	−0.695372
$216$	0	0
$217$	−0.392305	−0.0266314
$218$	−3.46410	−0.234619
$219$	0	0
$220$	1.26795	0.0854851
$221$	0	0
$222$	0	0
$223$	−2.00000	−0.133930	−0.0669650	−	0.997755i	$-0.521332\pi$
$223$	−2.00000	−0.133930	−0.0669650	+	0.997755i	$0.521332\pi$
$224$	10.3923	0.694365
$225$	0	0
$226$	−26.7846	−1.78169
$227$	3.46410	0.229920	0.114960	−	0.993370i	$-0.463326\pi$
$227$	3.46410	0.229920	0.114960	+	0.993370i	$0.463326\pi$
$228$	0	0
$229$	14.3923	0.951070	0.475535	−	0.879697i	$-0.342254\pi$
$229$	14.3923	0.951070	0.475535	+	0.879697i	$0.342254\pi$
$230$	8.19615	0.540438
$231$	0	0
$232$	−16.3923	−1.07621
$233$	6.00000	0.393073	0.196537	−	0.980497i	$-0.437031\pi$
$233$	6.00000	0.393073	0.196537	+	0.980497i	$0.437031\pi$
$234$	0	0
$235$	−6.00000	−0.391397
$236$	−15.1244	−0.984512
$237$	0	0
$238$	12.0000	0.777844
$239$	−3.80385	−0.246050	−0.123025	−	0.992404i	$-0.539260\pi$
$239$	−3.80385	−0.246050	−0.123025	+	0.992404i	$0.539260\pi$
$240$	0	0
$241$	−18.3923	−1.18475	−0.592376	−	0.805661i	$-0.701810\pi$
$241$	−18.3923	−1.18475	−0.592376	+	0.805661i	$0.701810\pi$
$242$	−16.2679	−1.04574
$243$	0	0
$244$	12.3923	0.793336
$245$	3.00000	0.191663
$246$	0	0
$247$	0	0
$248$	−0.339746	−0.0215739
$249$	0	0
$250$	−1.73205	−0.109545
$251$	14.5359	0.917498	0.458749	−	0.888566i	$-0.348298\pi$
$251$	14.5359	0.917498	0.458749	+	0.888566i	$0.348298\pi$
$252$	0	0
$253$	6.00000	0.377217
$254$	10.0526	0.630754
$255$	0	0
$256$	19.0000	1.18750
$257$	7.85641	0.490069	0.245035	−	0.969514i	$-0.421201\pi$
$257$	7.85641	0.490069	0.245035	+	0.969514i	$0.421201\pi$
$258$	0	0
$259$	−8.00000	−0.497096
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−4.73205	−0.291791	−0.145895	−	0.989300i	$-0.546606\pi$
$263$	−4.73205	−0.291791	−0.145895	+	0.989300i	$0.546606\pi$
$264$	0	0
$265$	−10.3923	−0.638394
$266$	14.5359	0.891253
$267$	0	0
$268$	14.3923	0.879150
$269$	−7.85641	−0.479014	−0.239507	−	0.970895i	$-0.576986\pi$
$269$	−7.85641	−0.479014	−0.239507	+	0.970895i	$0.576986\pi$
$270$	0	0
$271$	20.9808	1.27449	0.637245	−	0.770661i	$-0.280074\pi$
$271$	20.9808	1.27449	0.637245	+	0.770661i	$0.280074\pi$
$272$	17.3205	1.05021
$273$	0	0
$274$	−22.3923	−1.35277
$275$	−1.26795	−0.0764602
$276$	0	0
$277$	−5.60770	−0.336934	−0.168467	−	0.985707i	$-0.553882\pi$
$277$	−5.60770	−0.336934	−0.168467	+	0.985707i	$0.553882\pi$
$278$	−14.5359	−0.871805
$279$	0	0
$280$	−3.46410	−0.207020
$281$	1.60770	0.0959071	0.0479535	−	0.998850i	$-0.484730\pi$
$281$	1.60770	0.0959071	0.0479535	+	0.998850i	$0.484730\pi$
$282$	0	0
$283$	1.41154	0.0839075	0.0419538	−	0.999120i	$-0.486642\pi$
$283$	1.41154	0.0839075	0.0419538	+	0.999120i	$0.486642\pi$
$284$	1.26795	0.0752389
$285$	0	0
$286$	0	0
$287$	6.92820	0.408959
$288$	0	0
$289$	−5.00000	−0.294118
$290$	−16.3923	−0.962589
$291$	0	0
$292$	4.00000	0.234082
$293$	−18.9282	−1.10580	−0.552899	−	0.833248i	$-0.686478\pi$
$293$	−18.9282	−1.10580	−0.552899	+	0.833248i	$0.686478\pi$
$294$	0	0
$295$	15.1244	0.880574
$296$	−6.92820	−0.402694
$297$	0	0
$298$	34.3923	1.99229
$299$	0	0
$300$	0	0
$301$	−20.3923	−1.17539
$302$	21.1244	1.21557
$303$	0	0
$304$	20.9808	1.20333
$305$	−12.3923	−0.709581
$306$	0	0
$307$	−22.7846	−1.30039	−0.650193	−	0.759769i	$-0.725312\pi$
$307$	−22.7846	−1.30039	−0.650193	+	0.759769i	$0.725312\pi$
$308$	2.53590	0.144496
$309$	0	0
$310$	−0.339746	−0.0192963
$311$	−4.39230	−0.249065	−0.124532	−	0.992216i	$-0.539743\pi$
$311$	−4.39230	−0.249065	−0.124532	+	0.992216i	$0.539743\pi$
$312$	0	0
$313$	6.39230	0.361314	0.180657	−	0.983546i	$-0.442178\pi$
$313$	6.39230	0.361314	0.180657	+	0.983546i	$0.442178\pi$
$314$	−17.3205	−0.977453
$315$	0	0
$316$	12.3923	0.697122
$317$	24.0000	1.34797	0.673987	−	0.738743i	$-0.264580\pi$
$317$	24.0000	1.34797	0.673987	+	0.738743i	$0.264580\pi$
$318$	0	0
$319$	−12.0000	−0.671871
$320$	−1.00000	−0.0559017
$321$	0	0
$322$	16.3923	0.913507
$323$	14.5359	0.808799
$324$	0	0
$325$	0	0
$326$	−11.0718	−0.613210
$327$	0	0
$328$	6.00000	0.331295
$329$	−12.0000	−0.661581
$330$	0	0
$331$	28.5885	1.57136	0.785682	−	0.618631i	$-0.212312\pi$
$331$	28.5885	1.57136	0.785682	+	0.618631i	$0.212312\pi$
$332$	−6.00000	−0.329293
$333$	0	0
$334$	22.3923	1.22525
$335$	−14.3923	−0.786336
$336$	0	0
$337$	−5.60770	−0.305471	−0.152735	−	0.988267i	$-0.548808\pi$
$337$	−5.60770	−0.305471	−0.152735	+	0.988267i	$0.548808\pi$
$338$	0	0
$339$	0	0
$340$	3.46410	0.187867
$341$	−0.248711	−0.0134685
$342$	0	0
$343$	20.0000	1.07990
$344$	−17.6603	−0.952177
$345$	0	0
$346$	−26.7846	−1.43995
$347$	−11.6603	−0.625955	−0.312978	−	0.949761i	$-0.601326\pi$
$347$	−11.6603	−0.625955	−0.312978	+	0.949761i	$0.601326\pi$
$348$	0	0
$349$	−6.39230	−0.342172	−0.171086	−	0.985256i	$-0.554728\pi$
$349$	−6.39230	−0.342172	−0.171086	+	0.985256i	$0.554728\pi$
$350$	−3.46410	−0.185164
$351$	0	0
$352$	6.58846	0.351166
$353$	27.7128	1.47500	0.737502	−	0.675345i	$-0.236005\pi$
$353$	27.7128	1.47500	0.737502	+	0.675345i	$0.236005\pi$
$354$	0	0
$355$	−1.26795	−0.0672958
$356$	0.928203	0.0491947
$357$	0	0
$358$	8.78461	0.464281
$359$	8.19615	0.432576	0.216288	−	0.976330i	$-0.430605\pi$
$359$	8.19615	0.432576	0.216288	+	0.976330i	$0.430605\pi$
$360$	0	0
$361$	−1.39230	−0.0732792
$362$	−35.3205	−1.85640
$363$	0	0
$364$	0	0
$365$	−4.00000	−0.209370
$366$	0	0
$367$	22.1962	1.15863	0.579315	−	0.815104i	$-0.303320\pi$
$367$	22.1962	1.15863	0.579315	+	0.815104i	$0.303320\pi$
$368$	23.6603	1.23338
$369$	0	0
$370$	−6.92820	−0.360180
$371$	−20.7846	−1.07908
$372$	0	0
$373$	−10.0000	−0.517780	−0.258890	−	0.965907i	$-0.583357\pi$
$373$	−10.0000	−0.517780	−0.258890	+	0.965907i	$0.583357\pi$
$374$	7.60770	0.393385
$375$	0	0
$376$	−10.3923	−0.535942
$377$	0	0
$378$	0	0
$379$	32.9808	1.69411	0.847054	−	0.531507i	$-0.178374\pi$
$379$	32.9808	1.69411	0.847054	+	0.531507i	$0.178374\pi$
$380$	4.19615	0.215258
$381$	0	0
$382$	32.7846	1.67741
$383$	−0.928203	−0.0474290	−0.0237145	−	0.999719i	$-0.507549\pi$
$383$	−0.928203	−0.0474290	−0.0237145	+	0.999719i	$0.507549\pi$
$384$	0	0
$385$	−2.53590	−0.129241
$386$	17.3205	0.881591
$387$	0	0
$388$	−2.00000	−0.101535
$389$	−6.00000	−0.304212	−0.152106	−	0.988364i	$-0.548606\pi$
$389$	−6.00000	−0.304212	−0.152106	+	0.988364i	$0.548606\pi$
$390$	0	0
$391$	16.3923	0.828994
$392$	5.19615	0.262445
$393$	0	0
$394$	1.60770	0.0809945
$395$	−12.3923	−0.623525
$396$	0	0
$397$	12.7846	0.641641	0.320821	−	0.947140i	$-0.396041\pi$
$397$	12.7846	0.641641	0.320821	+	0.947140i	$0.396041\pi$
$398$	34.6410	1.73640
$399$	0	0
$400$	−5.00000	−0.250000
$401$	−23.0718	−1.15215	−0.576075	−	0.817397i	$-0.695416\pi$
$401$	−23.0718	−1.15215	−0.576075	+	0.817397i	$0.695416\pi$
$402$	0	0
$403$	0	0
$404$	−12.9282	−0.643202
$405$	0	0
$406$	−32.7846	−1.62707
$407$	−5.07180	−0.251400
$408$	0	0
$409$	38.3923	1.89838	0.949189	−	0.314708i	$-0.101906\pi$
$409$	38.3923	1.89838	0.949189	+	0.314708i	$0.101906\pi$
$410$	6.00000	0.296319
$411$	0	0
$412$	10.1962	0.502328
$413$	30.2487	1.48844
$414$	0	0
$415$	6.00000	0.294528
$416$	0	0
$417$	0	0
$418$	9.21539	0.450739
$419$	9.46410	0.462352	0.231176	−	0.972912i	$-0.425743\pi$
$419$	9.46410	0.462352	0.231176	+	0.972912i	$0.425743\pi$
$420$	0	0
$421$	−10.7846	−0.525610	−0.262805	−	0.964849i	$-0.584648\pi$
$421$	−10.7846	−0.525610	−0.262805	+	0.964849i	$0.584648\pi$
$422$	13.8564	0.674519
$423$	0	0
$424$	−18.0000	−0.874157
$425$	−3.46410	−0.168034
$426$	0	0
$427$	−24.7846	−1.19941
$428$	−0.339746	−0.0164222
$429$	0	0
$430$	−17.6603	−0.851653
$431$	19.5167	0.940084	0.470042	−	0.882644i	$-0.344239\pi$
$431$	19.5167	0.940084	0.470042	+	0.882644i	$0.344239\pi$
$432$	0	0
$433$	−6.78461	−0.326048	−0.163024	−	0.986622i	$-0.552125\pi$
$433$	−6.78461	−0.326048	−0.163024	+	0.986622i	$0.552125\pi$
$434$	−0.679492	−0.0326167
$435$	0	0
$436$	−2.00000	−0.0957826
$437$	19.8564	0.949861
$438$	0	0
$439$	32.0000	1.52728	0.763638	−	0.645644i	$-0.223411\pi$
$439$	32.0000	1.52728	0.763638	+	0.645644i	$0.223411\pi$
$440$	−2.19615	−0.104697
$441$	0	0
$442$	0	0
$443$	−34.9808	−1.66199	−0.830993	−	0.556283i	$-0.812227\pi$
$443$	−34.9808	−1.66199	−0.830993	+	0.556283i	$0.812227\pi$
$444$	0	0
$445$	−0.928203	−0.0440011
$446$	−3.46410	−0.164030
$447$	0	0
$448$	−2.00000	−0.0944911
$449$	27.4641	1.29611	0.648056	−	0.761593i	$-0.275582\pi$
$449$	27.4641	1.29611	0.648056	+	0.761593i	$0.275582\pi$
$450$	0	0
$451$	4.39230	0.206826
$452$	−15.4641	−0.727370
$453$	0	0
$454$	6.00000	0.281594
$455$	0	0
$456$	0	0
$457$	30.7846	1.44004	0.720022	−	0.693952i	$-0.244132\pi$
$457$	30.7846	1.44004	0.720022	+	0.693952i	$0.244132\pi$
$458$	24.9282	1.16482
$459$	0	0
$460$	4.73205	0.220633
$461$	3.46410	0.161339	0.0806696	−	0.996741i	$-0.474294\pi$
$461$	3.46410	0.161339	0.0806696	+	0.996741i	$0.474294\pi$
$462$	0	0
$463$	−18.3923	−0.854763	−0.427381	−	0.904071i	$-0.640564\pi$
$463$	−18.3923	−0.854763	−0.427381	+	0.904071i	$0.640564\pi$
$464$	−47.3205	−2.19680
$465$	0	0
$466$	10.3923	0.481414
$467$	−38.1962	−1.76751	−0.883754	−	0.467953i	$-0.844992\pi$
$467$	−38.1962	−1.76751	−0.883754	+	0.467953i	$0.844992\pi$
$468$	0	0
$469$	−28.7846	−1.32915
$470$	−10.3923	−0.479361
$471$	0	0
$472$	26.1962	1.20578
$473$	−12.9282	−0.594439
$474$	0	0
$475$	−4.19615	−0.192533
$476$	6.92820	0.317554
$477$	0	0
$478$	−6.58846	−0.301349
$479$	18.3397	0.837964	0.418982	−	0.907994i	$-0.362387\pi$
$479$	18.3397	0.837964	0.418982	+	0.907994i	$0.362387\pi$
$480$	0	0
$481$	0	0
$482$	−31.8564	−1.45102
$483$	0	0
$484$	−9.39230	−0.426923
$485$	2.00000	0.0908153
$486$	0	0
$487$	5.60770	0.254109	0.127054	−	0.991896i	$-0.459448\pi$
$487$	5.60770	0.254109	0.127054	+	0.991896i	$0.459448\pi$
$488$	−21.4641	−0.971634
$489$	0	0
$490$	5.19615	0.234738
$491$	9.46410	0.427109	0.213554	−	0.976931i	$-0.431496\pi$
$491$	9.46410	0.427109	0.213554	+	0.976931i	$0.431496\pi$
$492$	0	0
$493$	−32.7846	−1.47654
$494$	0	0
$495$	0	0
$496$	−0.980762	−0.0440375
$497$	−2.53590	−0.113751
$498$	0	0
$499$	−12.9808	−0.581099	−0.290549	−	0.956860i	$-0.593838\pi$
$499$	−12.9808	−0.581099	−0.290549	+	0.956860i	$0.593838\pi$
$500$	−1.00000	−0.0447214
$501$	0	0
$502$	25.1769	1.12370
$503$	25.5167	1.13773	0.568866	−	0.822430i	$-0.307382\pi$
$503$	25.5167	1.13773	0.568866	+	0.822430i	$0.307382\pi$
$504$	0	0
$505$	12.9282	0.575297
$506$	10.3923	0.461994
$507$	0	0
$508$	5.80385	0.257504
$509$	32.5359	1.44213	0.721064	−	0.692868i	$-0.243653\pi$
$509$	32.5359	1.44213	0.721064	+	0.692868i	$0.243653\pi$
$510$	0	0
$511$	−8.00000	−0.353899
$512$	8.66025	0.382733
$513$	0	0
$514$	13.6077	0.600210
$515$	−10.1962	−0.449296
$516$	0	0
$517$	−7.60770	−0.334586
$518$	−13.8564	−0.608816
$519$	0	0
$520$	0	0
$521$	7.60770	0.333299	0.166650	−	0.986016i	$-0.446705\pi$
$521$	7.60770	0.333299	0.166650	+	0.986016i	$0.446705\pi$
$522$	0	0
$523$	−13.8038	−0.603600	−0.301800	−	0.953371i	$-0.597588\pi$
$523$	−13.8038	−0.603600	−0.301800	+	0.953371i	$0.597588\pi$
$524$	0	0
$525$	0	0
$526$	−8.19615	−0.357369
$527$	−0.679492	−0.0295991
$528$	0	0
$529$	−0.607695	−0.0264215
$530$	−18.0000	−0.781870
$531$	0	0
$532$	8.39230	0.363853
$533$	0	0
$534$	0	0
$535$	0.339746	0.0146885
$536$	−24.9282	−1.07673
$537$	0	0
$538$	−13.6077	−0.586669
$539$	3.80385	0.163843
$540$	0	0
$541$	5.60770	0.241094	0.120547	−	0.992708i	$-0.461535\pi$
$541$	5.60770	0.241094	0.120547	+	0.992708i	$0.461535\pi$
$542$	36.3397	1.56093
$543$	0	0
$544$	18.0000	0.771744
$545$	2.00000	0.0856706
$546$	0	0
$547$	−1.80385	−0.0771270	−0.0385635	−	0.999256i	$-0.512278\pi$
$547$	−1.80385	−0.0771270	−0.0385635	+	0.999256i	$0.512278\pi$
$548$	−12.9282	−0.552265
$549$	0	0
$550$	−2.19615	−0.0936443
$551$	−39.7128	−1.69182
$552$	0	0
$553$	−24.7846	−1.05395
$554$	−9.71281	−0.412658
$555$	0	0
$556$	−8.39230	−0.355913
$557$	−25.8564	−1.09557	−0.547786	−	0.836619i	$-0.684529\pi$
$557$	−25.8564	−1.09557	−0.547786	+	0.836619i	$0.684529\pi$
$558$	0	0
$559$	0	0
$560$	−10.0000	−0.422577
$561$	0	0
$562$	2.78461	0.117462
$563$	−16.0526	−0.676535	−0.338267	−	0.941050i	$-0.609841\pi$
$563$	−16.0526	−0.676535	−0.338267	+	0.941050i	$0.609841\pi$
$564$	0	0
$565$	15.4641	0.650580
$566$	2.44486	0.102765
$567$	0	0
$568$	−2.19615	−0.0921485
$569$	9.46410	0.396756	0.198378	−	0.980126i	$-0.436433\pi$
$569$	9.46410	0.396756	0.198378	+	0.980126i	$0.436433\pi$
$570$	0	0
$571$	15.6077	0.653162	0.326581	−	0.945169i	$-0.394103\pi$
$571$	15.6077	0.653162	0.326581	+	0.945169i	$0.394103\pi$
$572$	0	0
$573$	0	0
$574$	12.0000	0.500870
$575$	−4.73205	−0.197340
$576$	0	0
$577$	4.00000	0.166522	0.0832611	−	0.996528i	$-0.473466\pi$
$577$	4.00000	0.166522	0.0832611	+	0.996528i	$0.473466\pi$
$578$	−8.66025	−0.360219
$579$	0	0
$580$	−9.46410	−0.392975
$581$	12.0000	0.497844
$582$	0	0
$583$	−13.1769	−0.545732
$584$	−6.92820	−0.286691
$585$	0	0
$586$	−32.7846	−1.35432
$587$	−15.4641	−0.638272	−0.319136	−	0.947709i	$-0.603393\pi$
$587$	−15.4641	−0.638272	−0.319136	+	0.947709i	$0.603393\pi$
$588$	0	0
$589$	−0.823085	−0.0339146
$590$	26.1962	1.07848
$591$	0	0
$592$	−20.0000	−0.821995
$593$	−14.7846	−0.607131	−0.303566	−	0.952811i	$-0.598177\pi$
$593$	−14.7846	−0.607131	−0.303566	+	0.952811i	$0.598177\pi$
$594$	0	0
$595$	−6.92820	−0.284029
$596$	19.8564	0.813350
$597$	0	0
$598$	0	0
$599$	28.3923	1.16008	0.580039	−	0.814589i	$-0.303037\pi$
$599$	28.3923	1.16008	0.580039	+	0.814589i	$0.303037\pi$
$600$	0	0
$601$	−39.5692	−1.61406	−0.807031	−	0.590509i	$-0.798927\pi$
$601$	−39.5692	−1.61406	−0.807031	+	0.590509i	$0.798927\pi$
$602$	−35.3205	−1.43956
$603$	0	0
$604$	12.1962	0.496254
$605$	9.39230	0.381851
$606$	0	0
$607$	−26.9808	−1.09512	−0.547558	−	0.836768i	$-0.684442\pi$
$607$	−26.9808	−1.09512	−0.547558	+	0.836768i	$0.684442\pi$
$608$	21.8038	0.884263
$609$	0	0
$610$	−21.4641	−0.869056
$611$	0	0
$612$	0	0
$613$	−26.0000	−1.05013	−0.525065	−	0.851062i	$-0.675959\pi$
$613$	−26.0000	−1.05013	−0.525065	+	0.851062i	$0.675959\pi$
$614$	−39.4641	−1.59264
$615$	0	0
$616$	−4.39230	−0.176971
$617$	−21.7128	−0.874125	−0.437062	−	0.899431i	$-0.643981\pi$
$617$	−21.7128	−0.874125	−0.437062	+	0.899431i	$0.643981\pi$
$618$	0	0
$619$	44.9808	1.80793	0.903965	−	0.427607i	$-0.140643\pi$
$619$	44.9808	1.80793	0.903965	+	0.427607i	$0.140643\pi$
$620$	−0.196152	−0.00787767
$621$	0	0
$622$	−7.60770	−0.305041
$623$	−1.85641	−0.0743754
$624$	0	0
$625$	1.00000	0.0400000
$626$	11.0718	0.442518
$627$	0	0
$628$	−10.0000	−0.399043
$629$	−13.8564	−0.552491
$630$	0	0
$631$	−16.1962	−0.644759	−0.322379	−	0.946611i	$-0.604483\pi$
$631$	−16.1962	−0.644759	−0.322379	+	0.946611i	$0.604483\pi$
$632$	−21.4641	−0.853796
$633$	0	0
$634$	41.5692	1.65092
$635$	−5.80385	−0.230319
$636$	0	0
$637$	0	0
$638$	−20.7846	−0.822871
$639$	0	0
$640$	−12.1244	−0.479257
$641$	0.928203	0.0366618	0.0183309	−	0.999832i	$-0.494165\pi$
$641$	0.928203	0.0366618	0.0183309	+	0.999832i	$0.494165\pi$
$642$	0	0
$643$	−34.7846	−1.37177	−0.685886	−	0.727709i	$-0.740585\pi$
$643$	−34.7846	−1.37177	−0.685886	+	0.727709i	$0.740585\pi$
$644$	9.46410	0.372938
$645$	0	0
$646$	25.1769	0.990572
$647$	16.0526	0.631091	0.315546	−	0.948910i	$-0.397812\pi$
$647$	16.0526	0.631091	0.315546	+	0.948910i	$0.397812\pi$
$648$	0	0
$649$	19.1769	0.752760
$650$	0	0
$651$	0	0
$652$	−6.39230	−0.250342
$653$	19.8564	0.777041	0.388521	−	0.921440i	$-0.372986\pi$
$653$	19.8564	0.777041	0.388521	+	0.921440i	$0.372986\pi$
$654$	0	0
$655$	0	0
$656$	17.3205	0.676252
$657$	0	0
$658$	−20.7846	−0.810268
$659$	14.5359	0.566238	0.283119	−	0.959085i	$-0.408631\pi$
$659$	14.5359	0.566238	0.283119	+	0.959085i	$0.408631\pi$
$660$	0	0
$661$	30.7846	1.19738	0.598691	−	0.800980i	$-0.295688\pi$
$661$	30.7846	1.19738	0.598691	+	0.800980i	$0.295688\pi$
$662$	49.5167	1.92452
$663$	0	0
$664$	10.3923	0.403300
$665$	−8.39230	−0.325440
$666$	0	0
$667$	−44.7846	−1.73407
$668$	12.9282	0.500207
$669$	0	0
$670$	−24.9282	−0.963061
$671$	−15.7128	−0.606586
$672$	0	0
$673$	6.39230	0.246405	0.123203	−	0.992382i	$-0.460683\pi$
$673$	6.39230	0.246405	0.123203	+	0.992382i	$0.460683\pi$
$674$	−9.71281	−0.374124
$675$	0	0
$676$	0	0
$677$	10.3923	0.399409	0.199704	−	0.979856i	$-0.436002\pi$
$677$	10.3923	0.399409	0.199704	+	0.979856i	$0.436002\pi$
$678$	0	0
$679$	4.00000	0.153506
$680$	−6.00000	−0.230089
$681$	0	0
$682$	−0.430781	−0.0164954
$683$	39.4641	1.51005	0.755026	−	0.655695i	$-0.227624\pi$
$683$	39.4641	1.51005	0.755026	+	0.655695i	$0.227624\pi$
$684$	0	0
$685$	12.9282	0.493961
$686$	34.6410	1.32260
$687$	0	0
$688$	−50.9808	−1.94362
$689$	0	0
$690$	0	0
$691$	−45.7654	−1.74100	−0.870498	−	0.492171i	$-0.836203\pi$
$691$	−45.7654	−1.74100	−0.870498	+	0.492171i	$0.836203\pi$
$692$	−15.4641	−0.587857
$693$	0	0
$694$	−20.1962	−0.766635
$695$	8.39230	0.318338
$696$	0	0
$697$	12.0000	0.454532
$698$	−11.0718	−0.419074
$699$	0	0
$700$	−2.00000	−0.0755929
$701$	−42.0000	−1.58632	−0.793159	−	0.609015i	$-0.791565\pi$
$701$	−42.0000	−1.58632	−0.793159	+	0.609015i	$0.791565\pi$
$702$	0	0
$703$	−16.7846	−0.633044
$704$	−1.26795	−0.0477876
$705$	0	0
$706$	48.0000	1.80650
$707$	25.8564	0.972430
$708$	0	0
$709$	−9.60770	−0.360825	−0.180412	−	0.983591i	$-0.557743\pi$
$709$	−9.60770	−0.360825	−0.180412	+	0.983591i	$0.557743\pi$
$710$	−2.19615	−0.0824201
$711$	0	0
$712$	−1.60770	−0.0602509
$713$	−0.928203	−0.0347615
$714$	0	0
$715$	0	0
$716$	5.07180	0.189542
$717$	0	0
$718$	14.1962	0.529796
$719$	−1.85641	−0.0692323	−0.0346161	−	0.999401i	$-0.511021\pi$
$719$	−1.85641	−0.0692323	−0.0346161	+	0.999401i	$0.511021\pi$
$720$	0	0
$721$	−20.3923	−0.759449
$722$	−2.41154	−0.0897483
$723$	0	0
$724$	−20.3923	−0.757874
$725$	9.46410	0.351488
$726$	0	0
$727$	13.4115	0.497407	0.248703	−	0.968580i	$-0.419996\pi$
$727$	13.4115	0.497407	0.248703	+	0.968580i	$0.419996\pi$
$728$	0	0
$729$	0	0
$730$	−6.92820	−0.256424
$731$	−35.3205	−1.30638
$732$	0	0
$733$	−38.0000	−1.40356	−0.701781	−	0.712393i	$-0.747612\pi$
$733$	−38.0000	−1.40356	−0.701781	+	0.712393i	$0.747612\pi$
$734$	38.4449	1.41903
$735$	0	0
$736$	24.5885	0.906343
$737$	−18.2487	−0.672200
$738$	0	0
$739$	7.80385	0.287069	0.143535	−	0.989645i	$-0.454153\pi$
$739$	7.80385	0.287069	0.143535	+	0.989645i	$0.454153\pi$
$740$	−4.00000	−0.147043
$741$	0	0
$742$	−36.0000	−1.32160
$743$	−43.8564	−1.60894	−0.804468	−	0.593996i	$-0.797549\pi$
$743$	−43.8564	−1.60894	−0.804468	+	0.593996i	$0.797549\pi$
$744$	0	0
$745$	−19.8564	−0.727482
$746$	−17.3205	−0.634149
$747$	0	0
$748$	4.39230	0.160599
$749$	0.679492	0.0248281
$750$	0	0
$751$	15.6077	0.569533	0.284766	−	0.958597i	$-0.408084\pi$
$751$	15.6077	0.569533	0.284766	+	0.958597i	$0.408084\pi$
$752$	−30.0000	−1.09399
$753$	0	0
$754$	0	0
$755$	−12.1962	−0.443863
$756$	0	0
$757$	18.3923	0.668480	0.334240	−	0.942488i	$-0.391520\pi$
$757$	18.3923	0.668480	0.334240	+	0.942488i	$0.391520\pi$
$758$	57.1244	2.07485
$759$	0	0
$760$	−7.26795	−0.263636
$761$	7.85641	0.284795	0.142397	−	0.989810i	$-0.454519\pi$
$761$	7.85641	0.284795	0.142397	+	0.989810i	$0.454519\pi$
$762$	0	0
$763$	4.00000	0.144810
$764$	18.9282	0.684798
$765$	0	0
$766$	−1.60770	−0.0580884
$767$	0	0
$768$	0	0
$769$	6.78461	0.244659	0.122330	−	0.992490i	$-0.460963\pi$
$769$	6.78461	0.244659	0.122330	+	0.992490i	$0.460963\pi$
$770$	−4.39230	−0.158288
$771$	0	0
$772$	10.0000	0.359908
$773$	−6.92820	−0.249190	−0.124595	−	0.992208i	$-0.539763\pi$
$773$	−6.92820	−0.249190	−0.124595	+	0.992208i	$0.539763\pi$
$774$	0	0
$775$	0.196152	0.00704600
$776$	3.46410	0.124354
$777$	0	0
$778$	−10.3923	−0.372582
$779$	14.5359	0.520803
$780$	0	0
$781$	−1.60770	−0.0575279
$782$	28.3923	1.01531
$783$	0	0
$784$	15.0000	0.535714
$785$	10.0000	0.356915
$786$	0	0
$787$	51.5692	1.83824	0.919122	−	0.393973i	$-0.128900\pi$
$787$	51.5692	1.83824	0.919122	+	0.393973i	$0.128900\pi$
$788$	0.928203	0.0330659
$789$	0	0
$790$	−21.4641	−0.763658
$791$	30.9282	1.09968
$792$	0	0
$793$	0	0
$794$	22.1436	0.785847
$795$	0	0
$796$	20.0000	0.708881
$797$	−28.6410	−1.01452	−0.507258	−	0.861794i	$-0.669341\pi$
$797$	−28.6410	−1.01452	−0.507258	+	0.861794i	$0.669341\pi$
$798$	0	0
$799$	−20.7846	−0.735307
$800$	−5.19615	−0.183712
$801$	0	0
$802$	−39.9615	−1.41109
$803$	−5.07180	−0.178980
$804$	0	0
$805$	−9.46410	−0.333566
$806$	0	0
$807$	0	0
$808$	22.3923	0.787759
$809$	9.46410	0.332740	0.166370	−	0.986063i	$-0.446795\pi$
$809$	9.46410	0.332740	0.166370	+	0.986063i	$0.446795\pi$
$810$	0	0
$811$	−28.1962	−0.990101	−0.495050	−	0.868864i	$-0.664850\pi$
$811$	−28.1962	−0.990101	−0.495050	+	0.868864i	$0.664850\pi$
$812$	−18.9282	−0.664250
$813$	0	0
$814$	−8.78461	−0.307900
$815$	6.39230	0.223913
$816$	0	0
$817$	−42.7846	−1.49684
$818$	66.4974	2.32503
$819$	0	0
$820$	3.46410	0.120972
$821$	−40.6410	−1.41838	−0.709191	−	0.705017i	$-0.750939\pi$
$821$	−40.6410	−1.41838	−0.709191	+	0.705017i	$0.750939\pi$
$822$	0	0
$823$	−46.5885	−1.62397	−0.811986	−	0.583677i	$-0.801613\pi$
$823$	−46.5885	−1.62397	−0.811986	+	0.583677i	$0.801613\pi$
$824$	−17.6603	−0.615224
$825$	0	0
$826$	52.3923	1.82296
$827$	18.0000	0.625921	0.312961	−	0.949766i	$-0.398679\pi$
$827$	18.0000	0.625921	0.312961	+	0.949766i	$0.398679\pi$
$828$	0	0
$829$	−20.3923	−0.708254	−0.354127	−	0.935197i	$-0.615222\pi$
$829$	−20.3923	−0.708254	−0.354127	+	0.935197i	$0.615222\pi$
$830$	10.3923	0.360722
$831$	0	0
$832$	0	0
$833$	10.3923	0.360072
$834$	0	0
$835$	−12.9282	−0.447399
$836$	5.32051	0.184014
$837$	0	0
$838$	16.3923	0.566263
$839$	17.6603	0.609700	0.304850	−	0.952400i	$-0.401394\pi$
$839$	17.6603	0.609700	0.304850	+	0.952400i	$0.401394\pi$
$840$	0	0
$841$	60.5692	2.08859
$842$	−18.6795	−0.643738
$843$	0	0
$844$	8.00000	0.275371
$845$	0	0
$846$	0	0
$847$	18.7846	0.645447
$848$	−51.9615	−1.78437
$849$	0	0
$850$	−6.00000	−0.205798
$851$	−18.9282	−0.648850
$852$	0	0
$853$	−8.00000	−0.273915	−0.136957	−	0.990577i	$-0.543732\pi$
$853$	−8.00000	−0.273915	−0.136957	+	0.990577i	$0.543732\pi$
$854$	−42.9282	−1.46897
$855$	0	0
$856$	0.588457	0.0201131
$857$	−47.5692	−1.62493	−0.812467	−	0.583007i	$-0.801876\pi$
$857$	−47.5692	−1.62493	−0.812467	+	0.583007i	$0.801876\pi$
$858$	0	0
$859$	45.1769	1.54142	0.770708	−	0.637188i	$-0.219903\pi$
$859$	45.1769	1.54142	0.770708	+	0.637188i	$0.219903\pi$
$860$	−10.1962	−0.347686
$861$	0	0
$862$	33.8038	1.15136
$863$	2.78461	0.0947892	0.0473946	−	0.998876i	$-0.484908\pi$
$863$	2.78461	0.0947892	0.0473946	+	0.998876i	$0.484908\pi$
$864$	0	0
$865$	15.4641	0.525795
$866$	−11.7513	−0.399325
$867$	0	0
$868$	−0.392305	−0.0133157
$869$	−15.7128	−0.533021
$870$	0	0
$871$	0	0
$872$	3.46410	0.117309
$873$	0	0
$874$	34.3923	1.16334
$875$	2.00000	0.0676123
$876$	0	0
$877$	−2.00000	−0.0675352	−0.0337676	−	0.999430i	$-0.510751\pi$
$877$	−2.00000	−0.0675352	−0.0337676	+	0.999430i	$0.510751\pi$
$878$	55.4256	1.87052
$879$	0	0
$880$	−6.33975	−0.213713
$881$	12.6795	0.427183	0.213591	−	0.976923i	$-0.431484\pi$
$881$	12.6795	0.427183	0.213591	+	0.976923i	$0.431484\pi$
$882$	0	0
$883$	34.1962	1.15079	0.575396	−	0.817875i	$-0.304848\pi$
$883$	34.1962	1.15079	0.575396	+	0.817875i	$0.304848\pi$
$884$	0	0
$885$	0	0
$886$	−60.5885	−2.03551
$887$	−17.9090	−0.601324	−0.300662	−	0.953731i	$-0.597208\pi$
$887$	−17.9090	−0.601324	−0.300662	+	0.953731i	$0.597208\pi$
$888$	0	0
$889$	−11.6077	−0.389310
$890$	−1.60770	−0.0538901
$891$	0	0
$892$	−2.00000	−0.0669650
$893$	−25.1769	−0.842513
$894$	0	0
$895$	−5.07180	−0.169531
$896$	−24.2487	−0.810093
$897$	0	0
$898$	47.5692	1.58741
$899$	1.85641	0.0619146
$900$	0	0
$901$	−36.0000	−1.19933
$902$	7.60770	0.253309
$903$	0	0
$904$	26.7846	0.890843
$905$	20.3923	0.677863
$906$	0	0
$907$	39.7654	1.32039	0.660194	−	0.751095i	$-0.270474\pi$
$907$	39.7654	1.32039	0.660194	+	0.751095i	$0.270474\pi$
$908$	3.46410	0.114960
$909$	0	0
$910$	0	0
$911$	36.0000	1.19273	0.596367	−	0.802712i	$-0.296610\pi$
$911$	36.0000	1.19273	0.596367	+	0.802712i	$0.296610\pi$
$912$	0	0
$913$	7.60770	0.251778
$914$	53.3205	1.76369
$915$	0	0
$916$	14.3923	0.475535
$917$	0	0
$918$	0	0
$919$	−53.1769	−1.75414	−0.877072	−	0.480358i	$-0.840507\pi$
$919$	−53.1769	−1.75414	−0.877072	+	0.480358i	$0.840507\pi$
$920$	−8.19615	−0.270219
$921$	0	0
$922$	6.00000	0.197599
$923$	0	0
$924$	0	0
$925$	4.00000	0.131519
$926$	−31.8564	−1.04687
$927$	0	0
$928$	−49.1769	−1.61431
$929$	51.4641	1.68848	0.844241	−	0.535963i	$-0.180052\pi$
$929$	51.4641	1.68848	0.844241	+	0.535963i	$0.180052\pi$
$930$	0	0
$931$	12.5885	0.412570
$932$	6.00000	0.196537
$933$	0	0
$934$	−66.1577	−2.16475
$935$	−4.39230	−0.143644
$936$	0	0
$937$	−6.78461	−0.221644	−0.110822	−	0.993840i	$-0.535348\pi$
$937$	−6.78461	−0.221644	−0.110822	+	0.993840i	$0.535348\pi$
$938$	−49.8564	−1.62787
$939$	0	0
$940$	−6.00000	−0.195698
$941$	−31.1769	−1.01634	−0.508169	−	0.861257i	$-0.669678\pi$
$941$	−31.1769	−1.01634	−0.508169	+	0.861257i	$0.669678\pi$
$942$	0	0
$943$	16.3923	0.533807
$944$	75.6218	2.46128
$945$	0	0
$946$	−22.3923	−0.728037
$947$	−28.6410	−0.930708	−0.465354	−	0.885125i	$-0.654073\pi$
$947$	−28.6410	−0.930708	−0.465354	+	0.885125i	$0.654073\pi$
$948$	0	0
$949$	0	0
$950$	−7.26795	−0.235803
$951$	0	0
$952$	−12.0000	−0.388922
$953$	12.9282	0.418786	0.209393	−	0.977832i	$-0.432851\pi$
$953$	12.9282	0.418786	0.209393	+	0.977832i	$0.432851\pi$
$954$	0	0
$955$	−18.9282	−0.612502
$956$	−3.80385	−0.123025
$957$	0	0
$958$	31.7654	1.02629
$959$	25.8564	0.834947
$960$	0	0
$961$	−30.9615	−0.998759
$962$	0	0
$963$	0	0
$964$	−18.3923	−0.592376
$965$	−10.0000	−0.321911
$966$	0	0
$967$	29.6077	0.952119	0.476060	−	0.879413i	$-0.342065\pi$
$967$	29.6077	0.952119	0.476060	+	0.879413i	$0.342065\pi$
$968$	16.2679	0.522872
$969$	0	0
$970$	3.46410	0.111226
$971$	−5.07180	−0.162762	−0.0813809	−	0.996683i	$-0.525933\pi$
$971$	−5.07180	−0.162762	−0.0813809	+	0.996683i	$0.525933\pi$
$972$	0	0
$973$	16.7846	0.538090
$974$	9.71281	0.311219
$975$	0	0
$976$	−61.9615	−1.98334
$977$	39.7128	1.27053	0.635263	−	0.772296i	$-0.280892\pi$
$977$	39.7128	1.27053	0.635263	+	0.772296i	$0.280892\pi$
$978$	0	0
$979$	−1.17691	−0.0376144
$980$	3.00000	0.0958315
$981$	0	0
$982$	16.3923	0.523099
$983$	−13.6077	−0.434018	−0.217009	−	0.976170i	$-0.569630\pi$
$983$	−13.6077	−0.434018	−0.217009	+	0.976170i	$0.569630\pi$
$984$	0	0
$985$	−0.928203	−0.0295750
$986$	−56.7846	−1.80839
$987$	0	0
$988$	0	0
$989$	−48.2487	−1.53422
$990$	0	0
$991$	8.00000	0.254128	0.127064	−	0.991894i	$-0.459445\pi$
$991$	8.00000	0.254128	0.127064	+	0.991894i	$0.459445\pi$
$992$	−1.01924	−0.0323608
$993$	0	0
$994$	−4.39230	−0.139315
$995$	−20.0000	−0.634043
$996$	0	0
$997$	54.3923	1.72262	0.861311	−	0.508078i	$-0.169644\pi$
$997$	54.3923	1.72262	0.861311	+	0.508078i	$0.169644\pi$
$998$	−22.4833	−0.711698
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7605.2.a.be.1.2		2
3.2	odd	2		845.2.a.d.1.1		2
13.12	even	2		585.2.a.k.1.1		2
15.14	odd	2		4225.2.a.w.1.2		2
39.2	even	12		845.2.m.a.316.2		4
39.5	even	4		845.2.c.e.506.3		4
39.8	even	4		845.2.c.e.506.1		4
39.11	even	12		845.2.m.c.316.2		4
39.17	odd	6		845.2.e.e.146.1		4
39.20	even	12		845.2.m.a.361.2		4
39.23	odd	6		845.2.e.e.191.1		4
39.29	odd	6		845.2.e.f.191.2		4
39.32	even	12		845.2.m.c.361.2		4
39.35	odd	6		845.2.e.f.146.2		4
39.38	odd	2		65.2.a.c.1.2	✓	2
52.51	odd	2		9360.2.a.cm.1.2		2
65.12	odd	4		2925.2.c.v.2224.2		4
65.38	odd	4		2925.2.c.v.2224.3		4
65.64	even	2		2925.2.a.z.1.2		2
156.155	even	2		1040.2.a.h.1.2		2
195.38	even	4		325.2.b.e.274.2		4
195.77	even	4		325.2.b.e.274.3		4
195.194	odd	2		325.2.a.g.1.1		2
273.272	even	2		3185.2.a.k.1.2		2
312.77	odd	2		4160.2.a.y.1.2		2
312.155	even	2		4160.2.a.bj.1.1		2
429.428	even	2		7865.2.a.h.1.1		2
780.779	even	2		5200.2.a.ca.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
65.2.a.c.1.2	✓	2	39.38	odd	2
325.2.a.g.1.1		2	195.194	odd	2
325.2.b.e.274.2		4	195.38	even	4
325.2.b.e.274.3		4	195.77	even	4
585.2.a.k.1.1		2	13.12	even	2
845.2.a.d.1.1		2	3.2	odd	2
845.2.c.e.506.1		4	39.8	even	4
845.2.c.e.506.3		4	39.5	even	4
845.2.e.e.146.1		4	39.17	odd	6
845.2.e.e.191.1		4	39.23	odd	6
845.2.e.f.146.2		4	39.35	odd	6
845.2.e.f.191.2		4	39.29	odd	6
845.2.m.a.316.2		4	39.2	even	12
845.2.m.a.361.2		4	39.20	even	12
845.2.m.c.316.2		4	39.11	even	12
845.2.m.c.361.2		4	39.32	even	12
1040.2.a.h.1.2		2	156.155	even	2
2925.2.a.z.1.2		2	65.64	even	2
2925.2.c.v.2224.2		4	65.12	odd	4
2925.2.c.v.2224.3		4	65.38	odd	4
3185.2.a.k.1.2		2	273.272	even	2
4160.2.a.y.1.2		2	312.77	odd	2
4160.2.a.bj.1.1		2	312.155	even	2
4225.2.a.w.1.2		2	15.14	odd	2
5200.2.a.ca.1.1		2	780.779	even	2
7605.2.a.be.1.2		2	1.1	even	1	trivial
7865.2.a.h.1.1		2	429.428	even	2
9360.2.a.cm.1.2		2	52.51	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 7605 = 3^{2} \cdot 5 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7605.a (trivial)

\(f(q)\)	\(=\)	\(q+1.73205 q^{2} +1.00000 q^{4} -1.00000 q^{5} -2.00000 q^{7} -1.73205 q^{8} +O(q^{10})\)	\(q+1.73205 q^{2} +1.00000 q^{4} -1.00000 q^{5} -2.00000 q^{7} -1.73205 q^{8} -1.73205 q^{10} -1.26795 q^{11} -3.46410 q^{14} -5.00000 q^{16} -3.46410 q^{17} -4.19615 q^{19} -1.00000 q^{20} -2.19615 q^{22} -4.73205 q^{23} +1.00000 q^{25} -2.00000 q^{28} +9.46410 q^{29} +0.196152 q^{31} -5.19615 q^{32} -6.00000 q^{34} +2.00000 q^{35} +4.00000 q^{37} -7.26795 q^{38} +1.73205 q^{40} -3.46410 q^{41} +10.1962 q^{43} -1.26795 q^{44} -8.19615 q^{46} +6.00000 q^{47} -3.00000 q^{49} +1.73205 q^{50} +10.3923 q^{53} +1.26795 q^{55} +3.46410 q^{56} +16.3923 q^{58} -15.1244 q^{59} +12.3923 q^{61} +0.339746 q^{62} +1.00000 q^{64} +14.3923 q^{67} -3.46410 q^{68} +3.46410 q^{70} +1.26795 q^{71} +4.00000 q^{73} +6.92820 q^{74} -4.19615 q^{76} +2.53590 q^{77} +12.3923 q^{79} +5.00000 q^{80} -6.00000 q^{82} -6.00000 q^{83} +3.46410 q^{85} +17.6603 q^{86} +2.19615 q^{88} +0.928203 q^{89} -4.73205 q^{92} +10.3923 q^{94} +4.19615 q^{95} -2.00000 q^{97} -5.19615 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{4} - 2 q^{5} - 4 q^{7}+O(q^{10}) \) 2 * q + 2 * q^4 - 2 * q^5 - 4 * q^7	\( 2 q + 2 q^{4} - 2 q^{5} - 4 q^{7} - 6 q^{11} - 10 q^{16} + 2 q^{19} - 2 q^{20} + 6 q^{22} - 6 q^{23} + 2 q^{25} - 4 q^{28} + 12 q^{29} - 10 q^{31} - 12 q^{34} + 4 q^{35} + 8 q^{37} - 18 q^{38} + 10 q^{43} - 6 q^{44} - 6 q^{46} + 12 q^{47} - 6 q^{49} + 6 q^{55} + 12 q^{58} - 6 q^{59} + 4 q^{61} + 18 q^{62} + 2 q^{64} + 8 q^{67} + 6 q^{71} + 8 q^{73} + 2 q^{76} + 12 q^{77} + 4 q^{79} + 10 q^{80} - 12 q^{82} - 12 q^{83} + 18 q^{86} - 6 q^{88} - 12 q^{89} - 6 q^{92} - 2 q^{95} - 4 q^{97}+O(q^{100}) \) 2 * q + 2 * q^4 - 2 * q^5 - 4 * q^7 - 6 * q^11 - 10 * q^16 + 2 * q^19 - 2 * q^20 + 6 * q^22 - 6 * q^23 + 2 * q^25 - 4 * q^28 + 12 * q^29 - 10 * q^31 - 12 * q^34 + 4 * q^35 + 8 * q^37 - 18 * q^38 + 10 * q^43 - 6 * q^44 - 6 * q^46 + 12 * q^47 - 6 * q^49 + 6 * q^55 + 12 * q^58 - 6 * q^59 + 4 * q^61 + 18 * q^62 + 2 * q^64 + 8 * q^67 + 6 * q^71 + 8 * q^73 + 2 * q^76 + 12 * q^77 + 4 * q^79 + 10 * q^80 - 12 * q^82 - 12 * q^83 + 18 * q^86 - 6 * q^88 - 12 * q^89 - 6 * q^92 - 2 * q^95 - 4 * q^97

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