LMFDB - Embedded newform 325.2.a.g.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [325,2,Mod(1,325)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(325, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("325.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 325 = 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	325.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:	$2.59513806569$
Analytic rank:	$1$
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			1.1
Root			$-1.73205$ of defining polynomial
Character	$\chi$	$=$	325.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.73205 q^{2} +0.732051 q^{3} +1.00000 q^{4} -1.26795 q^{6} -2.00000 q^{7} +1.73205 q^{8} -2.46410 q^{9} -1.26795 q^{11} +0.732051 q^{12} -1.00000 q^{13} +3.46410 q^{14} -5.00000 q^{16} -3.46410 q^{17} +4.26795 q^{18} +4.19615 q^{19} -1.46410 q^{21} +2.19615 q^{22} -4.73205 q^{23} +1.26795 q^{24} +1.73205 q^{26} -4.00000 q^{27} -2.00000 q^{28} -9.46410 q^{29} -0.196152 q^{31} +5.19615 q^{32} -0.928203 q^{33} +6.00000 q^{34} -2.46410 q^{36} +4.00000 q^{37} -7.26795 q^{38} -0.732051 q^{39} -3.46410 q^{41} +2.53590 q^{42} -10.1962 q^{43} -1.26795 q^{44} +8.19615 q^{46} -6.00000 q^{47} -3.66025 q^{48} -3.00000 q^{49} -2.53590 q^{51} -1.00000 q^{52} +10.3923 q^{53} +6.92820 q^{54} -3.46410 q^{56} +3.07180 q^{57} +16.3923 q^{58} -15.1244 q^{59} +12.3923 q^{61} +0.339746 q^{62} +4.92820 q^{63} +1.00000 q^{64} +1.60770 q^{66} +14.3923 q^{67} -3.46410 q^{68} -3.46410 q^{69} +1.26795 q^{71} -4.26795 q^{72} +4.00000 q^{73} -6.92820 q^{74} +4.19615 q^{76} +2.53590 q^{77} +1.26795 q^{78} +12.3923 q^{79} +4.46410 q^{81} +6.00000 q^{82} +6.00000 q^{83} -1.46410 q^{84} +17.6603 q^{86} -6.92820 q^{87} -2.19615 q^{88} +0.928203 q^{89} +2.00000 q^{91} -4.73205 q^{92} -0.143594 q^{93} +10.3923 q^{94} +3.80385 q^{96} -2.00000 q^{97} +5.19615 q^{98} +3.12436 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} + 2 q^{4} - 6 q^{6} - 4 q^{7} + 2 q^{9} - 6 q^{11} - 2 q^{12} - 2 q^{13} - 10 q^{16} + 12 q^{18} - 2 q^{19} + 4 q^{21} - 6 q^{22} - 6 q^{23} + 6 q^{24} - 8 q^{27} - 4 q^{28} - 12 q^{29} + 10 q^{31}+ \cdots - 18 q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 + 2 * q^4 - 6 * q^6 - 4 * q^7 + 2 * q^9 - 6 * q^11 - 2 * q^12 - 2 * q^13 - 10 * q^16 + 12 * q^18 - 2 * q^19 + 4 * q^21 - 6 * q^22 - 6 * q^23 + 6 * q^24 - 8 * q^27 - 4 * q^28 - 12 * q^29 + 10 * q^31 + 12 * q^33 + 12 * q^34 + 2 * q^36 + 8 * q^37 - 18 * q^38 + 2 * q^39 + 12 * q^42 - 10 * q^43 - 6 * q^44 + 6 * q^46 - 12 * q^47 + 10 * q^48 - 6 * q^49 - 12 * q^51 - 2 * q^52 + 20 * q^57 + 12 * q^58 - 6 * q^59 + 4 * q^61 + 18 * q^62 - 4 * q^63 + 2 * q^64 + 24 * q^66 + 8 * q^67 + 6 * q^71 - 12 * q^72 + 8 * q^73 - 2 * q^76 + 12 * q^77 + 6 * q^78 + 4 * q^79 + 2 * q^81 + 12 * q^82 + 12 * q^83 + 4 * q^84 + 18 * q^86 + 6 * q^88 - 12 * q^89 + 4 * q^91 - 6 * q^92 - 28 * q^93 + 18 * q^96 - 4 * q^97 - 18 * q^99

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	$-0.709785\pi$
$2$	−1.73205	−1.22474	−0.612372	+	0.790569i	$0.709785\pi$
$3$	0.732051	0.422650	0.211325	−	0.977416i	$-0.432222\pi$
$3$	0.732051	0.422650	0.211325	+	0.977416i	$0.432222\pi$
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	−1.26795	−0.517638
$7$	−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
$9$	−2.46410	−0.821367
$10$	0	0
$11$	−1.26795	−0.382301	−0.191151	−	0.981561i	$-0.561222\pi$
$11$	−1.26795	−0.382301	−0.191151	+	0.981561i	$0.561222\pi$
$12$	0.732051	0.211325
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	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	$0.637999\pi$
$18$	4.26795	1.00597
$19$	4.19615	0.962663	0.481332	−	0.876539i	$-0.340153\pi$
$19$	4.19615	0.962663	0.481332	+	0.876539i	$0.340153\pi$
$20$	0	0
$21$	−1.46410	−0.319493
$22$	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	$0.664228\pi$
$24$	1.26795	0.258819
$25$	0	0
$26$	1.73205	0.339683
$27$	−4.00000	−0.769800
$28$	−2.00000	−0.377964
$29$	−9.46410	−1.75744	−0.878720	−	0.477338i	$-0.841602\pi$
$29$	−9.46410	−1.75744	−0.878720	+	0.477338i	$0.841602\pi$
$30$	0	0
$31$	−0.196152	−0.0352300	−0.0176150	−	0.999845i	$-0.505607\pi$
$31$	−0.196152	−0.0352300	−0.0176150	+	0.999845i	$0.505607\pi$
$32$	5.19615	0.918559
$33$	−0.928203	−0.161579
$34$	6.00000	1.02899
$35$	0	0
$36$	−2.46410	−0.410684
$37$	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
$39$	−0.732051	−0.117222
$40$	0	0
$41$	−3.46410	−0.541002	−0.270501	−	0.962720i	$-0.587189\pi$
$41$	−3.46410	−0.541002	−0.270501	+	0.962720i	$0.587189\pi$
$42$	2.53590	0.391298
$43$	−10.1962	−1.55490	−0.777449	−	0.628946i	$-0.783487\pi$
$43$	−10.1962	−1.55490	−0.777449	+	0.628946i	$0.783487\pi$
$44$	−1.26795	−0.191151
$45$	0	0
$46$	8.19615	1.20846
$47$	−6.00000	−0.875190	−0.437595	−	0.899172i	$-0.644170\pi$
$47$	−6.00000	−0.875190	−0.437595	+	0.899172i	$0.644170\pi$
$48$	−3.66025	−0.528312
$49$	−3.00000	−0.428571
$50$	0	0
$51$	−2.53590	−0.355097
$52$	−1.00000	−0.138675
$53$	10.3923	1.42749	0.713746	−	0.700404i	$-0.246997\pi$
$53$	10.3923	1.42749	0.713746	+	0.700404i	$0.246997\pi$
$54$	6.92820	0.942809
$55$	0	0
$56$	−3.46410	−0.462910
$57$	3.07180	0.406869
$58$	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
$63$	4.92820	0.620895
$64$	1.00000	0.125000
$65$	0	0
$66$	1.60770	0.197894
$67$	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
$69$	−3.46410	−0.417029
$70$	0	0
$71$	1.26795	0.150478	0.0752389	−	0.997166i	$-0.476028\pi$
$71$	1.26795	0.150478	0.0752389	+	0.997166i	$0.476028\pi$
$72$	−4.26795	−0.502983
$73$	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
$78$	1.26795	0.143567
$79$	12.3923	1.39424	0.697122	−	0.716953i	$-0.254464\pi$
$79$	12.3923	1.39424	0.697122	+	0.716953i	$0.254464\pi$
$80$	0	0
$81$	4.46410	0.496011
$82$	6.00000	0.662589
$83$	6.00000	0.658586	0.329293	−	0.944228i	$-0.393190\pi$
$83$	6.00000	0.658586	0.329293	+	0.944228i	$0.393190\pi$
$84$	−1.46410	−0.159747
$85$	0	0
$86$	17.6603	1.90435
$87$	−6.92820	−0.742781
$88$	−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
$91$	2.00000	0.209657
$92$	−4.73205	−0.493350
$93$	−0.143594	−0.0148900
$94$	10.3923	1.07188
$95$	0	0
$96$	3.80385	0.388229
$97$	−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
$99$	3.12436	0.314010
$100$	0	0
$101$	12.9282	1.28640	0.643202	−	0.765696i	$-0.277605\pi$
$101$	12.9282	1.28640	0.643202	+	0.765696i	$0.277605\pi$
$102$	4.39230	0.434903
$103$	−10.1962	−1.00466	−0.502328	−	0.864677i	$-0.667523\pi$
$103$	−10.1962	−1.00466	−0.502328	+	0.864677i	$0.667523\pi$
$104$	−1.73205	−0.169842
$105$	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	$0.505228\pi$
$108$	−4.00000	−0.384900
$109$	2.00000	0.191565	0.0957826	−	0.995402i	$-0.469465\pi$
$109$	2.00000	0.191565	0.0957826	+	0.995402i	$0.469465\pi$
$110$	0	0
$111$	2.92820	0.277933
$112$	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	$0.759258\pi$
$114$	−5.32051	−0.498311
$115$	0	0
$116$	−9.46410	−0.878720
$117$	2.46410	0.227806
$118$	26.1962	2.41155
$119$	6.92820	0.635107
$120$	0	0
$121$	−9.39230	−0.853846
$122$	−21.4641	−1.94327
$123$	−2.53590	−0.228654
$124$	−0.196152	−0.0176150
$125$	0	0
$126$	−8.53590	−0.760438
$127$	−5.80385	−0.515008	−0.257504	−	0.966277i	$-0.582900\pi$
$127$	−5.80385	−0.515008	−0.257504	+	0.966277i	$0.582900\pi$
$128$	−12.1244	−1.07165
$129$	−7.46410	−0.657178
$130$	0	0
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	−0.928203	−0.0807897
$133$	−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	$-0.313763\pi$
$137$	12.9282	1.10453	0.552265	+	0.833668i	$0.313763\pi$
$138$	6.00000	0.510754
$139$	−8.39230	−0.711826	−0.355913	−	0.934519i	$-0.615830\pi$
$139$	−8.39230	−0.711826	−0.355913	+	0.934519i	$0.615830\pi$
$140$	0	0
$141$	−4.39230	−0.369899
$142$	−2.19615	−0.184297
$143$	1.26795	0.106031
$144$	12.3205	1.02671
$145$	0	0
$146$	−6.92820	−0.573382
$147$	−2.19615	−0.181136
$148$	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	$-0.665292\pi$
$151$	−12.1962	−0.992509	−0.496254	+	0.868177i	$0.665292\pi$
$152$	7.26795	0.589509
$153$	8.53590	0.690086
$154$	−4.39230	−0.353942
$155$	0	0
$156$	−0.732051	−0.0586110
$157$	10.0000	0.798087	0.399043	−	0.916932i	$-0.369342\pi$
$157$	10.0000	0.798087	0.399043	+	0.916932i	$0.369342\pi$
$158$	−21.4641	−1.70759
$159$	7.60770	0.603329
$160$	0	0
$161$	9.46410	0.745876
$162$	−7.73205	−0.607487
$163$	−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	$-0.666743\pi$
$167$	−12.9282	−1.00041	−0.500207	+	0.865906i	$0.666743\pi$
$168$	−2.53590	−0.195649
$169$	1.00000	0.0769231
$170$	0	0
$171$	−10.3397	−0.790700
$172$	−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	$0.700028\pi$
$174$	12.0000	0.909718
$175$	0	0
$176$	6.33975	0.477876
$177$	−11.0718	−0.832207
$178$	−1.60770	−0.120502
$179$	−5.07180	−0.379084	−0.189542	−	0.981873i	$-0.560700\pi$
$179$	−5.07180	−0.379084	−0.189542	+	0.981873i	$0.560700\pi$
$180$	0	0
$181$	−20.3923	−1.51575	−0.757874	−	0.652401i	$-0.773762\pi$
$181$	−20.3923	−1.51575	−0.757874	+	0.652401i	$0.773762\pi$
$182$	−3.46410	−0.256776
$183$	9.07180	0.670607
$184$	−8.19615	−0.604228
$185$	0	0
$186$	0.248711	0.0182364
$187$	4.39230	0.321197
$188$	−6.00000	−0.437595
$189$	8.00000	0.581914
$190$	0	0
$191$	−18.9282	−1.36960	−0.684798	−	0.728733i	$-0.740110\pi$
$191$	−18.9282	−1.36960	−0.684798	+	0.728733i	$0.740110\pi$
$192$	0.732051	0.0528312
$193$	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	$-0.510527\pi$
$197$	−0.928203	−0.0661317	−0.0330659	+	0.999453i	$0.510527\pi$
$198$	−5.41154	−0.384582
$199$	20.0000	1.41776	0.708881	−	0.705328i	$-0.249200\pi$
$199$	20.0000	1.41776	0.708881	+	0.705328i	$0.249200\pi$
$200$	0	0
$201$	10.5359	0.743145
$202$	−22.3923	−1.57552
$203$	18.9282	1.32850
$204$	−2.53590	−0.177548
$205$	0	0
$206$	17.6603	1.23045
$207$	11.6603	0.810444
$208$	5.00000	0.346688
$209$	−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
$213$	0.928203	0.0635994
$214$	0.588457	0.0402261
$215$	0	0
$216$	−6.92820	−0.471405
$217$	0.392305	0.0266314
$218$	−3.46410	−0.234619
$219$	2.92820	0.197870
$220$	0	0
$221$	3.46410	0.233021
$222$	−5.07180	−0.340397
$223$	−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	$-0.536674\pi$
$227$	−3.46410	−0.229920	−0.114960	+	0.993370i	$0.536674\pi$
$228$	3.07180	0.203435
$229$	−14.3923	−0.951070	−0.475535	−	0.879697i	$-0.657746\pi$
$229$	−14.3923	−0.951070	−0.475535	+	0.879697i	$0.657746\pi$
$230$	0	0
$231$	1.85641	0.122143
$232$	−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	$0.437031\pi$
$234$	−4.26795	−0.279005
$235$	0	0
$236$	−15.1244	−0.984512
$237$	9.07180	0.589277
$238$	−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	$-0.298190\pi$
$241$	18.3923	1.18475	0.592376	+	0.805661i	$0.298190\pi$
$242$	16.2679	1.04574
$243$	15.2679	0.979439
$244$	12.3923	0.793336
$245$	0	0
$246$	4.39230	0.280043
$247$	−4.19615	−0.266995
$248$	−0.339746	−0.0215739
$249$	4.39230	0.278351
$250$	0	0
$251$	−14.5359	−0.917498	−0.458749	−	0.888566i	$-0.651702\pi$
$251$	−14.5359	−0.917498	−0.458749	+	0.888566i	$0.651702\pi$
$252$	4.92820	0.310448
$253$	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	$0.421201\pi$
$258$	12.9282	0.804875
$259$	−8.00000	−0.497096
$260$	0	0
$261$	23.3205	1.44350
$262$	0	0
$263$	−4.73205	−0.291791	−0.145895	−	0.989300i	$-0.546606\pi$
$263$	−4.73205	−0.291791	−0.145895	+	0.989300i	$0.546606\pi$
$264$	−1.60770	−0.0989468
$265$	0	0
$266$	14.5359	0.891253
$267$	0.679492	0.0415842
$268$	14.3923	0.879150
$269$	7.85641	0.479014	0.239507	−	0.970895i	$-0.423014\pi$
$269$	7.85641	0.479014	0.239507	+	0.970895i	$0.423014\pi$
$270$	0	0
$271$	−20.9808	−1.27449	−0.637245	−	0.770661i	$-0.719926\pi$
$271$	−20.9808	−1.27449	−0.637245	+	0.770661i	$0.719926\pi$
$272$	17.3205	1.05021
$273$	1.46410	0.0886115
$274$	−22.3923	−1.35277
$275$	0	0
$276$	−3.46410	−0.208514
$277$	5.60770	0.336934	0.168467	−	0.985707i	$-0.446118\pi$
$277$	5.60770	0.336934	0.168467	+	0.985707i	$0.446118\pi$
$278$	14.5359	0.871805
$279$	0.483340	0.0289368
$280$	0	0
$281$	1.60770	0.0959071	0.0479535	−	0.998850i	$-0.484730\pi$
$281$	1.60770	0.0959071	0.0479535	+	0.998850i	$0.484730\pi$
$282$	7.60770	0.453032
$283$	−1.41154	−0.0839075	−0.0419538	−	0.999120i	$-0.513358\pi$
$283$	−1.41154	−0.0839075	−0.0419538	+	0.999120i	$0.513358\pi$
$284$	1.26795	0.0752389
$285$	0	0
$286$	−2.19615	−0.129861
$287$	6.92820	0.408959
$288$	−12.8038	−0.754474
$289$	−5.00000	−0.294118
$290$	0	0
$291$	−1.46410	−0.0858272
$292$	4.00000	0.234082
$293$	18.9282	1.10580	0.552899	−	0.833248i	$-0.313522\pi$
$293$	18.9282	1.10580	0.552899	+	0.833248i	$0.313522\pi$
$294$	3.80385	0.221845
$295$	0	0
$296$	6.92820	0.402694
$297$	5.07180	0.294295
$298$	−34.3923	−1.99229
$299$	4.73205	0.273662
$300$	0	0
$301$	20.3923	1.17539
$302$	21.1244	1.21557
$303$	9.46410	0.543698
$304$	−20.9808	−1.20333
$305$	0	0
$306$	−14.7846	−0.845180
$307$	−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
$309$	−7.46410	−0.424618
$310$	0	0
$311$	4.39230	0.249065	0.124532	−	0.992216i	$-0.460257\pi$
$311$	4.39230	0.249065	0.124532	+	0.992216i	$0.460257\pi$
$312$	−1.26795	−0.0717835
$313$	−6.39230	−0.361314	−0.180657	−	0.983546i	$-0.557822\pi$
$313$	−6.39230	−0.361314	−0.180657	+	0.983546i	$0.557822\pi$
$314$	−17.3205	−0.977453
$315$	0	0
$316$	12.3923	0.697122
$317$	−24.0000	−1.34797	−0.673987	−	0.738743i	$-0.735420\pi$
$317$	−24.0000	−1.34797	−0.673987	+	0.738743i	$0.735420\pi$
$318$	−13.1769	−0.738925
$319$	12.0000	0.671871
$320$	0	0
$321$	−0.248711	−0.0138817
$322$	−16.3923	−0.913507
$323$	−14.5359	−0.808799
$324$	4.46410	0.248006
$325$	0	0
$326$	11.0718	0.613210
$327$	1.46410	0.0809650
$328$	−6.00000	−0.331295
$329$	12.0000	0.661581
$330$	0	0
$331$	−28.5885	−1.57136	−0.785682	−	0.618631i	$-0.787688\pi$
$331$	−28.5885	−1.57136	−0.785682	+	0.618631i	$0.787688\pi$
$332$	6.00000	0.329293
$333$	−9.85641	−0.540128
$334$	22.3923	1.22525
$335$	0	0
$336$	7.32051	0.399366
$337$	5.60770	0.305471	0.152735	−	0.988267i	$-0.451192\pi$
$337$	5.60770	0.305471	0.152735	+	0.988267i	$0.451192\pi$
$338$	−1.73205	−0.0942111
$339$	−11.3205	−0.614846
$340$	0	0
$341$	0.248711	0.0134685
$342$	17.9090	0.968406
$343$	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	$0.601326\pi$
$348$	−6.92820	−0.371391
$349$	6.39230	0.342172	0.171086	−	0.985256i	$-0.445272\pi$
$349$	6.39230	0.342172	0.171086	+	0.985256i	$0.445272\pi$
$350$	0	0
$351$	4.00000	0.213504
$352$	−6.58846	−0.351166
$353$	−27.7128	−1.47500	−0.737502	−	0.675345i	$-0.763995\pi$
$353$	−27.7128	−1.47500	−0.737502	+	0.675345i	$0.763995\pi$
$354$	19.1769	1.01924
$355$	0	0
$356$	0.928203	0.0491947
$357$	5.07180	0.268428
$358$	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
$363$	−6.87564	−0.360878
$364$	2.00000	0.104828
$365$	0	0
$366$	−15.7128	−0.821322
$367$	−22.1962	−1.15863	−0.579315	−	0.815104i	$-0.696680\pi$
$367$	−22.1962	−1.15863	−0.579315	+	0.815104i	$0.696680\pi$
$368$	23.6603	1.23338
$369$	8.53590	0.444361
$370$	0	0
$371$	−20.7846	−1.07908
$372$	−0.143594	−0.00744498
$373$	10.0000	0.517780	0.258890	−	0.965907i	$-0.416643\pi$
$373$	10.0000	0.517780	0.258890	+	0.965907i	$0.416643\pi$
$374$	−7.60770	−0.393385
$375$	0	0
$376$	−10.3923	−0.535942
$377$	9.46410	0.487426
$378$	−13.8564	−0.712697
$379$	−32.9808	−1.69411	−0.847054	−	0.531507i	$-0.821626\pi$
$379$	−32.9808	−1.69411	−0.847054	+	0.531507i	$0.821626\pi$
$380$	0	0
$381$	−4.24871	−0.217668
$382$	32.7846	1.67741
$383$	0.928203	0.0474290	0.0237145	−	0.999719i	$-0.492451\pi$
$383$	0.928203	0.0474290	0.0237145	+	0.999719i	$0.492451\pi$
$384$	−8.87564	−0.452933
$385$	0	0
$386$	−17.3205	−0.881591
$387$	25.1244	1.27714
$388$	−2.00000	−0.101535
$389$	6.00000	0.304212	0.152106	−	0.988364i	$-0.451394\pi$
$389$	6.00000	0.304212	0.152106	+	0.988364i	$0.451394\pi$
$390$	0	0
$391$	16.3923	0.828994
$392$	−5.19615	−0.262445
$393$	0	0
$394$	1.60770	0.0809945
$395$	0	0
$396$	3.12436	0.157005
$397$	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
$399$	−6.14359	−0.307564
$400$	0	0
$401$	−23.0718	−1.15215	−0.576075	−	0.817397i	$-0.695416\pi$
$401$	−23.0718	−1.15215	−0.576075	+	0.817397i	$0.695416\pi$
$402$	−18.2487	−0.910163
$403$	0.196152	0.00977105
$404$	12.9282	0.643202
$405$	0	0
$406$	−32.7846	−1.62707
$407$	−5.07180	−0.251400
$408$	−4.39230	−0.217451
$409$	−38.3923	−1.89838	−0.949189	−	0.314708i	$-0.898094\pi$
$409$	−38.3923	−1.89838	−0.949189	+	0.314708i	$0.898094\pi$
$410$	0	0
$411$	9.46410	0.466830
$412$	−10.1962	−0.502328
$413$	30.2487	1.48844
$414$	−20.1962	−0.992587
$415$	0	0
$416$	−5.19615	−0.254762
$417$	−6.14359	−0.300853
$418$	9.21539	0.450739
$419$	−9.46410	−0.462352	−0.231176	−	0.972912i	$-0.574257\pi$
$419$	−9.46410	−0.462352	−0.231176	+	0.972912i	$0.574257\pi$
$420$	0	0
$421$	10.7846	0.525610	0.262805	−	0.964849i	$-0.415352\pi$
$421$	10.7846	0.525610	0.262805	+	0.964849i	$0.415352\pi$
$422$	−13.8564	−0.674519
$423$	14.7846	0.718852
$424$	18.0000	0.874157
$425$	0	0
$426$	−1.60770	−0.0778931
$427$	−24.7846	−1.19941
$428$	−0.339746	−0.0164222
$429$	0.928203	0.0448141
$430$	0	0
$431$	19.5167	0.940084	0.470042	−	0.882644i	$-0.344239\pi$
$431$	19.5167	0.940084	0.470042	+	0.882644i	$0.344239\pi$
$432$	20.0000	0.962250
$433$	6.78461	0.326048	0.163024	−	0.986622i	$-0.447875\pi$
$433$	6.78461	0.326048	0.163024	+	0.986622i	$0.447875\pi$
$434$	−0.679492	−0.0326167
$435$	0	0
$436$	2.00000	0.0957826
$437$	−19.8564	−0.949861
$438$	−5.07180	−0.242340
$439$	32.0000	1.52728	0.763638	−	0.645644i	$-0.223411\pi$
$439$	32.0000	1.52728	0.763638	+	0.645644i	$0.223411\pi$
$440$	0	0
$441$	7.39230	0.352015
$442$	−6.00000	−0.285391
$443$	−34.9808	−1.66199	−0.830993	−	0.556283i	$-0.812227\pi$
$443$	−34.9808	−1.66199	−0.830993	+	0.556283i	$0.812227\pi$
$444$	2.92820	0.138966
$445$	0	0
$446$	3.46410	0.164030
$447$	14.5359	0.687524
$448$	−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
$453$	−8.92820	−0.419484
$454$	6.00000	0.281594
$455$	0	0
$456$	5.32051	0.249156
$457$	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
$459$	13.8564	0.646762
$460$	0	0
$461$	3.46410	0.161339	0.0806696	−	0.996741i	$-0.474294\pi$
$461$	3.46410	0.161339	0.0806696	+	0.996741i	$0.474294\pi$
$462$	−3.21539	−0.149593
$463$	−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	$0.844992\pi$
$468$	2.46410	0.113903
$469$	−28.7846	−1.32915
$470$	0	0
$471$	7.32051	0.337311
$472$	−26.1962	−1.20578
$473$	12.9282	0.594439
$474$	−15.7128	−0.721713
$475$	0	0
$476$	6.92820	0.317554
$477$	−25.6077	−1.17250
$478$	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
$481$	−4.00000	−0.182384
$482$	−31.8564	−1.45102
$483$	6.92820	0.315244
$484$	−9.39230	−0.426923
$485$	0	0
$486$	−26.4449	−1.19956
$487$	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
$489$	−4.67949	−0.211614
$490$	0	0
$491$	−9.46410	−0.427109	−0.213554	−	0.976931i	$-0.568504\pi$
$491$	−9.46410	−0.427109	−0.213554	+	0.976931i	$0.568504\pi$
$492$	−2.53590	−0.114327
$493$	32.7846	1.47654
$494$	7.26795	0.327000
$495$	0	0
$496$	0.980762	0.0440375
$497$	−2.53590	−0.113751
$498$	−7.60770	−0.340909
$499$	12.9808	0.581099	0.290549	−	0.956860i	$-0.406162\pi$
$499$	12.9808	0.581099	0.290549	+	0.956860i	$0.406162\pi$
$500$	0	0
$501$	−9.46410	−0.422825
$502$	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	$0.307382\pi$
$504$	8.53590	0.380219
$505$	0	0
$506$	−10.3923	−0.461994
$507$	0.732051	0.0325115
$508$	−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
$513$	−16.7846	−0.741059
$514$	−13.6077	−0.600210
$515$	0	0
$516$	−7.46410	−0.328589
$517$	7.60770	0.334586
$518$	13.8564	0.608816
$519$	−11.3205	−0.496915
$520$	0	0
$521$	−7.60770	−0.333299	−0.166650	−	0.986016i	$-0.553295\pi$
$521$	−7.60770	−0.333299	−0.166650	+	0.986016i	$0.553295\pi$
$522$	−40.3923	−1.76792
$523$	13.8038	0.603600	0.301800	−	0.953371i	$-0.402412\pi$
$523$	13.8038	0.603600	0.301800	+	0.953371i	$0.402412\pi$
$524$	0	0
$525$	0	0
$526$	8.19615	0.357369
$527$	0.679492	0.0295991
$528$	4.64102	0.201974
$529$	−0.607695	−0.0264215
$530$	0	0
$531$	37.2679	1.61729
$532$	−8.39230	−0.363853
$533$	3.46410	0.150047
$534$	−1.17691	−0.0509301
$535$	0	0
$536$	24.9282	1.07673
$537$	−3.71281	−0.160220
$538$	−13.6077	−0.586669
$539$	3.80385	0.163843
$540$	0	0
$541$	−5.60770	−0.241094	−0.120547	−	0.992708i	$-0.538465\pi$
$541$	−5.60770	−0.241094	−0.120547	+	0.992708i	$0.538465\pi$
$542$	36.3397	1.56093
$543$	−14.9282	−0.640631
$544$	−18.0000	−0.771744
$545$	0	0
$546$	−2.53590	−0.108526
$547$	1.80385	0.0771270	0.0385635	−	0.999256i	$-0.487722\pi$
$547$	1.80385	0.0771270	0.0385635	+	0.999256i	$0.487722\pi$
$548$	12.9282	0.552265
$549$	−30.5359	−1.30324
$550$	0	0
$551$	−39.7128	−1.69182
$552$	−6.00000	−0.255377
$553$	−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	$-0.315471\pi$
$557$	25.8564	1.09557	0.547786	+	0.836619i	$0.315471\pi$
$558$	−0.837169	−0.0354402
$559$	10.1962	0.431251
$560$	0	0
$561$	3.21539	0.135754
$562$	−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	$0.609841\pi$
$564$	−4.39230	−0.184949
$565$	0	0
$566$	2.44486	0.102765
$567$	−8.92820	−0.374949
$568$	2.19615	0.0921485
$569$	−9.46410	−0.396756	−0.198378	−	0.980126i	$-0.563567\pi$
$569$	−9.46410	−0.396756	−0.198378	+	0.980126i	$0.563567\pi$
$570$	0	0
$571$	15.6077	0.653162	0.326581	−	0.945169i	$-0.394103\pi$
$571$	15.6077	0.653162	0.326581	+	0.945169i	$0.394103\pi$
$572$	1.26795	0.0530156
$573$	−13.8564	−0.578860
$574$	−12.0000	−0.500870
$575$	0	0
$576$	−2.46410	−0.102671
$577$	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
$579$	7.32051	0.304230
$580$	0	0
$581$	−12.0000	−0.497844
$582$	2.53590	0.105116
$583$	−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	$-0.396607\pi$
$587$	15.4641	0.638272	0.319136	+	0.947709i	$0.396607\pi$
$588$	−2.19615	−0.0905678
$589$	−0.823085	−0.0339146
$590$	0	0
$591$	−0.679492	−0.0279506
$592$	−20.0000	−0.821995
$593$	14.7846	0.607131	0.303566	−	0.952811i	$-0.401823\pi$
$593$	14.7846	0.607131	0.303566	+	0.952811i	$0.401823\pi$
$594$	−8.78461	−0.360437
$595$	0	0
$596$	19.8564	0.813350
$597$	14.6410	0.599217
$598$	−8.19615	−0.335166
$599$	−28.3923	−1.16008	−0.580039	−	0.814589i	$-0.696963\pi$
$599$	−28.3923	−1.16008	−0.580039	+	0.814589i	$0.696963\pi$
$600$	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
$603$	−35.4641	−1.44421
$604$	−12.1962	−0.496254
$605$	0	0
$606$	−16.3923	−0.665892
$607$	26.9808	1.09512	0.547558	−	0.836768i	$-0.315558\pi$
$607$	26.9808	1.09512	0.547558	+	0.836768i	$0.315558\pi$
$608$	21.8038	0.884263
$609$	13.8564	0.561490
$610$	0	0
$611$	6.00000	0.242734
$612$	8.53590	0.345043
$613$	−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	$-0.356019\pi$
$617$	21.7128	0.874125	0.437062	+	0.899431i	$0.356019\pi$
$618$	12.9282	0.520049
$619$	−44.9808	−1.80793	−0.903965	−	0.427607i	$-0.859357\pi$
$619$	−44.9808	−1.80793	−0.903965	+	0.427607i	$0.859357\pi$
$620$	0	0
$621$	18.9282	0.759563
$622$	−7.60770	−0.305041
$623$	−1.85641	−0.0743754
$624$	3.66025	0.146527
$625$	0	0
$626$	11.0718	0.442518
$627$	−3.89488	−0.155547
$628$	10.0000	0.399043
$629$	−13.8564	−0.552491
$630$	0	0
$631$	16.1962	0.644759	0.322379	−	0.946611i	$-0.395517\pi$
$631$	16.1962	0.644759	0.322379	+	0.946611i	$0.395517\pi$
$632$	21.4641	0.853796
$633$	5.85641	0.232771
$634$	41.5692	1.65092
$635$	0	0
$636$	7.60770	0.301665
$637$	3.00000	0.118864
$638$	−20.7846	−0.822871
$639$	−3.12436	−0.123598
$640$	0	0
$641$	−0.928203	−0.0366618	−0.0183309	−	0.999832i	$-0.505835\pi$
$641$	−0.928203	−0.0366618	−0.0183309	+	0.999832i	$0.505835\pi$
$642$	0.430781	0.0170016
$643$	−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	$0.397812\pi$
$648$	7.73205	0.303744
$649$	19.1769	0.752760
$650$	0	0
$651$	0.287187	0.0112557
$652$	−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	$0.372986\pi$
$654$	−2.53590	−0.0991615
$655$	0	0
$656$	17.3205	0.676252
$657$	−9.85641	−0.384535
$658$	−20.7846	−0.810268
$659$	−14.5359	−0.566238	−0.283119	−	0.959085i	$-0.591369\pi$
$659$	−14.5359	−0.566238	−0.283119	+	0.959085i	$0.591369\pi$
$660$	0	0
$661$	−30.7846	−1.19738	−0.598691	−	0.800980i	$-0.704312\pi$
$661$	−30.7846	−1.19738	−0.598691	+	0.800980i	$0.704312\pi$
$662$	49.5167	1.92452
$663$	2.53590	0.0984861
$664$	10.3923	0.403300
$665$	0	0
$666$	17.0718	0.661519
$667$	44.7846	1.73407
$668$	−12.9282	−0.500207
$669$	−1.46410	−0.0566054
$670$	0	0
$671$	−15.7128	−0.606586
$672$	−7.60770	−0.293473
$673$	−6.39230	−0.246405	−0.123203	−	0.992382i	$-0.539317\pi$
$673$	−6.39230	−0.246405	−0.123203	+	0.992382i	$0.539317\pi$
$674$	−9.71281	−0.374124
$675$	0	0
$676$	1.00000	0.0384615
$677$	10.3923	0.399409	0.199704	−	0.979856i	$-0.436002\pi$
$677$	10.3923	0.399409	0.199704	+	0.979856i	$0.436002\pi$
$678$	19.6077	0.753029
$679$	4.00000	0.153506
$680$	0	0
$681$	−2.53590	−0.0971758
$682$	−0.430781	−0.0164954
$683$	−39.4641	−1.51005	−0.755026	−	0.655695i	$-0.772376\pi$
$683$	−39.4641	−1.51005	−0.755026	+	0.655695i	$0.772376\pi$
$684$	−10.3397	−0.395350
$685$	0	0
$686$	−34.6410	−1.32260
$687$	−10.5359	−0.401970
$688$	50.9808	1.94362
$689$	−10.3923	−0.395915
$690$	0	0
$691$	45.7654	1.74100	0.870498	−	0.492171i	$-0.163797\pi$
$691$	45.7654	1.74100	0.870498	+	0.492171i	$0.163797\pi$
$692$	−15.4641	−0.587857
$693$	−6.24871	−0.237369
$694$	20.1962	0.766635
$695$	0	0
$696$	−12.0000	−0.454859
$697$	12.0000	0.454532
$698$	−11.0718	−0.419074
$699$	4.39230	0.166132
$700$	0	0
$701$	42.0000	1.58632	0.793159	−	0.609015i	$-0.208435\pi$
$701$	42.0000	1.58632	0.793159	+	0.609015i	$0.208435\pi$
$702$	−6.92820	−0.261488
$703$	16.7846	0.633044
$704$	−1.26795	−0.0477876
$705$	0	0
$706$	48.0000	1.80650
$707$	−25.8564	−0.972430
$708$	−11.0718	−0.416104
$709$	9.60770	0.360825	0.180412	−	0.983591i	$-0.442257\pi$
$709$	9.60770	0.360825	0.180412	+	0.983591i	$0.442257\pi$
$710$	0	0
$711$	−30.5359	−1.14519
$712$	1.60770	0.0602509
$713$	0.928203	0.0347615
$714$	−8.78461	−0.328756
$715$	0	0
$716$	−5.07180	−0.189542
$717$	−2.78461	−0.103993
$718$	−14.1962	−0.529796
$719$	1.85641	0.0692323	0.0346161	−	0.999401i	$-0.488979\pi$
$719$	1.85641	0.0692323	0.0346161	+	0.999401i	$0.488979\pi$
$720$	0	0
$721$	20.3923	0.759449
$722$	2.41154	0.0897483
$723$	13.4641	0.500735
$724$	−20.3923	−0.757874
$725$	0	0
$726$	11.9090	0.441983
$727$	−13.4115	−0.497407	−0.248703	−	0.968580i	$-0.580004\pi$
$727$	−13.4115	−0.497407	−0.248703	+	0.968580i	$0.580004\pi$
$728$	3.46410	0.128388
$729$	−2.21539	−0.0820515
$730$	0	0
$731$	35.3205	1.30638
$732$	9.07180	0.335303
$733$	−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
$738$	−14.7846	−0.544229
$739$	−7.80385	−0.287069	−0.143535	−	0.989645i	$-0.545847\pi$
$739$	−7.80385	−0.287069	−0.143535	+	0.989645i	$0.545847\pi$
$740$	0	0
$741$	−3.07180	−0.112845
$742$	36.0000	1.32160
$743$	43.8564	1.60894	0.804468	−	0.593996i	$-0.202451\pi$
$743$	43.8564	1.60894	0.804468	+	0.593996i	$0.202451\pi$
$744$	−0.248711	−0.00911820
$745$	0	0
$746$	−17.3205	−0.634149
$747$	−14.7846	−0.540941
$748$	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
$753$	−10.6410	−0.387780
$754$	−16.3923	−0.596973
$755$	0	0
$756$	8.00000	0.290957
$757$	−18.3923	−0.668480	−0.334240	−	0.942488i	$-0.608480\pi$
$757$	−18.3923	−0.668480	−0.334240	+	0.942488i	$0.608480\pi$
$758$	57.1244	2.07485
$759$	4.39230	0.159431
$760$	0	0
$761$	7.85641	0.284795	0.142397	−	0.989810i	$-0.454519\pi$
$761$	7.85641	0.284795	0.142397	+	0.989810i	$0.454519\pi$
$762$	7.35898	0.266588
$763$	−4.00000	−0.144810
$764$	−18.9282	−0.684798
$765$	0	0
$766$	−1.60770	−0.0580884
$767$	15.1244	0.546109
$768$	13.9090	0.501897
$769$	−6.78461	−0.244659	−0.122330	−	0.992490i	$-0.539037\pi$
$769$	−6.78461	−0.244659	−0.122330	+	0.992490i	$0.539037\pi$
$770$	0	0
$771$	5.75129	0.207128
$772$	10.0000	0.359908
$773$	6.92820	0.249190	0.124595	−	0.992208i	$-0.460237\pi$
$773$	6.92820	0.249190	0.124595	+	0.992208i	$0.460237\pi$
$774$	−43.5167	−1.56417
$775$	0	0
$776$	−3.46410	−0.124354
$777$	−5.85641	−0.210097
$778$	−10.3923	−0.372582
$779$	−14.5359	−0.520803
$780$	0	0
$781$	−1.60770	−0.0575279
$782$	−28.3923	−1.01531
$783$	37.8564	1.35288
$784$	15.0000	0.535714
$785$	0	0
$786$	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
$789$	−3.46410	−0.123325
$790$	0	0
$791$	30.9282	1.09968
$792$	5.41154	0.192291
$793$	−12.3923	−0.440064
$794$	−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	$0.669341\pi$
$798$	10.6410	0.376688
$799$	20.7846	0.735307
$800$	0	0
$801$	−2.28719	−0.0808138
$802$	39.9615	1.41109
$803$	−5.07180	−0.178980
$804$	10.5359	0.371572
$805$	0	0
$806$	−0.339746	−0.0119670
$807$	5.75129	0.202455
$808$	22.3923	0.787759
$809$	−9.46410	−0.332740	−0.166370	−	0.986063i	$-0.553205\pi$
$809$	−9.46410	−0.332740	−0.166370	+	0.986063i	$0.553205\pi$
$810$	0	0
$811$	28.1962	0.990101	0.495050	−	0.868864i	$-0.335150\pi$
$811$	28.1962	0.990101	0.495050	+	0.868864i	$0.335150\pi$
$812$	18.9282	0.664250
$813$	−15.3590	−0.538663
$814$	8.78461	0.307900
$815$	0	0
$816$	12.6795	0.443871
$817$	−42.7846	−1.49684
$818$	66.4974	2.32503
$819$	−4.92820	−0.172205
$820$	0	0
$821$	−40.6410	−1.41838	−0.709191	−	0.705017i	$-0.750939\pi$
$821$	−40.6410	−1.41838	−0.709191	+	0.705017i	$0.750939\pi$
$822$	−16.3923	−0.571747
$823$	46.5885	1.62397	0.811986	−	0.583677i	$-0.198387\pi$
$823$	46.5885	1.62397	0.811986	+	0.583677i	$0.198387\pi$
$824$	−17.6603	−0.615224
$825$	0	0
$826$	−52.3923	−1.82296
$827$	−18.0000	−0.625921	−0.312961	−	0.949766i	$-0.601321\pi$
$827$	−18.0000	−0.625921	−0.312961	+	0.949766i	$0.601321\pi$
$828$	11.6603	0.405222
$829$	−20.3923	−0.708254	−0.354127	−	0.935197i	$-0.615222\pi$
$829$	−20.3923	−0.708254	−0.354127	+	0.935197i	$0.615222\pi$
$830$	0	0
$831$	4.10512	0.142405
$832$	−1.00000	−0.0346688
$833$	10.3923	0.360072
$834$	10.6410	0.368468
$835$	0	0
$836$	−5.32051	−0.184014
$837$	0.784610	0.0271201
$838$	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
$843$	1.17691	0.0405351
$844$	8.00000	0.275371
$845$	0	0
$846$	−25.6077	−0.880411
$847$	18.7846	0.645447
$848$	−51.9615	−1.78437
$849$	−1.03332	−0.0354635
$850$	0	0
$851$	−18.9282	−0.648850
$852$	0.928203	0.0317997
$853$	−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	$0.801876\pi$
$858$	−1.60770	−0.0548858
$859$	45.1769	1.54142	0.770708	−	0.637188i	$-0.219903\pi$
$859$	45.1769	1.54142	0.770708	+	0.637188i	$0.219903\pi$
$860$	0	0
$861$	5.07180	0.172846
$862$	−33.8038	−1.15136
$863$	−2.78461	−0.0947892	−0.0473946	−	0.998876i	$-0.515092\pi$
$863$	−2.78461	−0.0947892	−0.0473946	+	0.998876i	$0.515092\pi$
$864$	−20.7846	−0.707107
$865$	0	0
$866$	−11.7513	−0.399325
$867$	−3.66025	−0.124309
$868$	0.392305	0.0133157
$869$	−15.7128	−0.533021
$870$	0	0
$871$	−14.3923	−0.487665
$872$	3.46410	0.117309
$873$	4.92820	0.166794
$874$	34.3923	1.16334
$875$	0	0
$876$	2.92820	0.0989348
$877$	−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
$879$	13.8564	0.467365
$880$	0	0
$881$	−12.6795	−0.427183	−0.213591	−	0.976923i	$-0.568516\pi$
$881$	−12.6795	−0.427183	−0.213591	+	0.976923i	$0.568516\pi$
$882$	−12.8038	−0.431128
$883$	−34.1962	−1.15079	−0.575396	−	0.817875i	$-0.695152\pi$
$883$	−34.1962	−1.15079	−0.575396	+	0.817875i	$0.695152\pi$
$884$	3.46410	0.116510
$885$	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	$0.597208\pi$
$888$	5.07180	0.170198
$889$	11.6077	0.389310
$890$	0	0
$891$	−5.66025	−0.189626
$892$	−2.00000	−0.0669650
$893$	−25.1769	−0.842513
$894$	−25.1769	−0.842042
$895$	0	0
$896$	24.2487	0.810093
$897$	3.46410	0.115663
$898$	−47.5692	−1.58741
$899$	1.85641	0.0619146
$900$	0	0
$901$	−36.0000	−1.19933
$902$	−7.60770	−0.253309
$903$	14.9282	0.496779
$904$	−26.7846	−0.890843
$905$	0	0
$906$	15.4641	0.513760
$907$	−39.7654	−1.32039	−0.660194	−	0.751095i	$-0.729526\pi$
$907$	−39.7654	−1.32039	−0.660194	+	0.751095i	$0.729526\pi$
$908$	−3.46410	−0.114960
$909$	−31.8564	−1.05661
$910$	0	0
$911$	−36.0000	−1.19273	−0.596367	−	0.802712i	$-0.703390\pi$
$911$	−36.0000	−1.19273	−0.596367	+	0.802712i	$0.703390\pi$
$912$	−15.3590	−0.508587
$913$	−7.60770	−0.251778
$914$	−53.3205	−1.76369
$915$	0	0
$916$	−14.3923	−0.475535
$917$	0	0
$918$	−24.0000	−0.792118
$919$	−53.1769	−1.75414	−0.877072	−	0.480358i	$-0.840507\pi$
$919$	−53.1769	−1.75414	−0.877072	+	0.480358i	$0.840507\pi$
$920$	0	0
$921$	−16.6795	−0.549608
$922$	−6.00000	−0.197599
$923$	−1.26795	−0.0417351
$924$	1.85641	0.0610713
$925$	0	0
$926$	31.8564	1.04687
$927$	25.1244	0.825192
$928$	−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
$933$	3.21539	0.105267
$934$	66.1577	2.16475
$935$	0	0
$936$	4.26795	0.139502
$937$	6.78461	0.221644	0.110822	−	0.993840i	$-0.464652\pi$
$937$	6.78461	0.221644	0.110822	+	0.993840i	$0.464652\pi$
$938$	49.8564	1.62787
$939$	−4.67949	−0.152709
$940$	0	0
$941$	−31.1769	−1.01634	−0.508169	−	0.861257i	$-0.669678\pi$
$941$	−31.1769	−1.01634	−0.508169	+	0.861257i	$0.669678\pi$
$942$	−12.6795	−0.413120
$943$	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	$-0.345927\pi$
$947$	28.6410	0.930708	0.465354	+	0.885125i	$0.345927\pi$
$948$	9.07180	0.294638
$949$	−4.00000	−0.129845
$950$	0	0
$951$	−17.5692	−0.569721
$952$	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	$0.432851\pi$
$954$	44.3538	1.43601
$955$	0	0
$956$	−3.80385	−0.123025
$957$	8.78461	0.283966
$958$	−31.7654	−1.02629
$959$	−25.8564	−0.834947
$960$	0	0
$961$	−30.9615	−0.998759
$962$	6.92820	0.223374
$963$	0.837169	0.0269774
$964$	18.3923	0.592376
$965$	0	0
$966$	−12.0000	−0.386094
$967$	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
$969$	−10.6410	−0.341839
$970$	0	0
$971$	5.07180	0.162762	0.0813809	−	0.996683i	$-0.474067\pi$
$971$	5.07180	0.162762	0.0813809	+	0.996683i	$0.474067\pi$
$972$	15.2679	0.489720
$973$	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	$-0.719108\pi$
$977$	−39.7128	−1.27053	−0.635263	+	0.772296i	$0.719108\pi$
$978$	8.10512	0.259173
$979$	−1.17691	−0.0376144
$980$	0	0
$981$	−4.92820	−0.157345
$982$	16.3923	0.523099
$983$	13.6077	0.434018	0.217009	−	0.976170i	$-0.430370\pi$
$983$	13.6077	0.434018	0.217009	+	0.976170i	$0.430370\pi$
$984$	−4.39230	−0.140022
$985$	0	0
$986$	−56.7846	−1.80839
$987$	8.78461	0.279617
$988$	−4.19615	−0.133497
$989$	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
$993$	−20.9282	−0.664136
$994$	4.39230	0.139315
$995$	0	0
$996$	4.39230	0.139176
$997$	−54.3923	−1.72262	−0.861311	−	0.508078i	$-0.830356\pi$
$997$	−54.3923	−1.72262	−0.861311	+	0.508078i	$0.830356\pi$
$998$	−22.4833	−0.711698
$999$	−16.0000	−0.506218

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	325.2.a.g.1.1		2
3.2	odd	2		2925.2.a.z.1.2		2
4.3	odd	2		5200.2.a.ca.1.1		2
5.2	odd	4		325.2.b.e.274.2		4
5.3	odd	4		325.2.b.e.274.3		4
5.4	even	2		65.2.a.c.1.2	✓	2
13.12	even	2		4225.2.a.w.1.2		2
15.2	even	4		2925.2.c.v.2224.3		4
15.8	even	4		2925.2.c.v.2224.2		4
15.14	odd	2		585.2.a.k.1.1		2
20.19	odd	2		1040.2.a.h.1.2		2
35.34	odd	2		3185.2.a.k.1.2		2
40.19	odd	2		4160.2.a.bj.1.1		2
40.29	even	2		4160.2.a.y.1.2		2
55.54	odd	2		7865.2.a.h.1.1		2
60.59	even	2		9360.2.a.cm.1.2		2
65.4	even	6		845.2.e.f.146.2		4
65.9	even	6		845.2.e.e.146.1		4
65.19	odd	12		845.2.m.a.361.2		4
65.24	odd	12		845.2.m.a.316.2		4
65.29	even	6		845.2.e.e.191.1		4
65.34	odd	4		845.2.c.e.506.3		4
65.44	odd	4		845.2.c.e.506.1		4
65.49	even	6		845.2.e.f.191.2		4
65.54	odd	12		845.2.m.c.316.2		4
65.59	odd	12		845.2.m.c.361.2		4
65.64	even	2		845.2.a.d.1.1		2
195.194	odd	2		7605.2.a.be.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
65.2.a.c.1.2	✓	2	5.4	even	2
325.2.a.g.1.1		2	1.1	even	1	trivial
325.2.b.e.274.2		4	5.2	odd	4
325.2.b.e.274.3		4	5.3	odd	4
585.2.a.k.1.1		2	15.14	odd	2
845.2.a.d.1.1		2	65.64	even	2
845.2.c.e.506.1		4	65.44	odd	4
845.2.c.e.506.3		4	65.34	odd	4
845.2.e.e.146.1		4	65.9	even	6
845.2.e.e.191.1		4	65.29	even	6
845.2.e.f.146.2		4	65.4	even	6
845.2.e.f.191.2		4	65.49	even	6
845.2.m.a.316.2		4	65.24	odd	12
845.2.m.a.361.2		4	65.19	odd	12
845.2.m.c.316.2		4	65.54	odd	12
845.2.m.c.361.2		4	65.59	odd	12
1040.2.a.h.1.2		2	20.19	odd	2
2925.2.a.z.1.2		2	3.2	odd	2
2925.2.c.v.2224.2		4	15.8	even	4
2925.2.c.v.2224.3		4	15.2	even	4
3185.2.a.k.1.2		2	35.34	odd	2
4160.2.a.y.1.2		2	40.29	even	2
4160.2.a.bj.1.1		2	40.19	odd	2
4225.2.a.w.1.2		2	13.12	even	2
5200.2.a.ca.1.1		2	4.3	odd	2
7605.2.a.be.1.2		2	195.194	odd	2
7865.2.a.h.1.1		2	55.54	odd	2
9360.2.a.cm.1.2		2	60.59	even	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:	\( N \)	\(=\)	\( 325 = 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	325.a (trivial)

\(f(q)\)	\(=\)	\(q-1.73205 q^{2} +0.732051 q^{3} +1.00000 q^{4} -1.26795 q^{6} -2.00000 q^{7} +1.73205 q^{8} -2.46410 q^{9} -1.26795 q^{11} +0.732051 q^{12} -1.00000 q^{13} +3.46410 q^{14} -5.00000 q^{16} -3.46410 q^{17} +4.26795 q^{18} +4.19615 q^{19} -1.46410 q^{21} +2.19615 q^{22} -4.73205 q^{23} +1.26795 q^{24} +1.73205 q^{26} -4.00000 q^{27} -2.00000 q^{28} -9.46410 q^{29} -0.196152 q^{31} +5.19615 q^{32} -0.928203 q^{33} +6.00000 q^{34} -2.46410 q^{36} +4.00000 q^{37} -7.26795 q^{38} -0.732051 q^{39} -3.46410 q^{41} +2.53590 q^{42} -10.1962 q^{43} -1.26795 q^{44} +8.19615 q^{46} -6.00000 q^{47} -3.66025 q^{48} -3.00000 q^{49} -2.53590 q^{51} -1.00000 q^{52} +10.3923 q^{53} +6.92820 q^{54} -3.46410 q^{56} +3.07180 q^{57} +16.3923 q^{58} -15.1244 q^{59} +12.3923 q^{61} +0.339746 q^{62} +4.92820 q^{63} +1.00000 q^{64} +1.60770 q^{66} +14.3923 q^{67} -3.46410 q^{68} -3.46410 q^{69} +1.26795 q^{71} -4.26795 q^{72} +4.00000 q^{73} -6.92820 q^{74} +4.19615 q^{76} +2.53590 q^{77} +1.26795 q^{78} +12.3923 q^{79} +4.46410 q^{81} +6.00000 q^{82} +6.00000 q^{83} -1.46410 q^{84} +17.6603 q^{86} -6.92820 q^{87} -2.19615 q^{88} +0.928203 q^{89} +2.00000 q^{91} -4.73205 q^{92} -0.143594 q^{93} +10.3923 q^{94} +3.80385 q^{96} -2.00000 q^{97} +5.19615 q^{98} +3.12436 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} + 2 q^{4} - 6 q^{6} - 4 q^{7} + 2 q^{9} - 6 q^{11} - 2 q^{12} - 2 q^{13} - 10 q^{16} + 12 q^{18} - 2 q^{19} + 4 q^{21} - 6 q^{22} - 6 q^{23} + 6 q^{24} - 8 q^{27} - 4 q^{28} - 12 q^{29} + 10 q^{31}+ \cdots - 18 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 + 2 * q^4 - 6 * q^6 - 4 * q^7 + 2 * q^9 - 6 * q^11 - 2 * q^12 - 2 * q^13 - 10 * q^16 + 12 * q^18 - 2 * q^19 + 4 * q^21 - 6 * q^22 - 6 * q^23 + 6 * q^24 - 8 * q^27 - 4 * q^28 - 12 * q^29 + 10 * q^31 + 12 * q^33 + 12 * q^34 + 2 * q^36 + 8 * q^37 - 18 * q^38 + 2 * q^39 + 12 * q^42 - 10 * q^43 - 6 * q^44 + 6 * q^46 - 12 * q^47 + 10 * q^48 - 6 * q^49 - 12 * q^51 - 2 * q^52 + 20 * q^57 + 12 * q^58 - 6 * q^59 + 4 * q^61 + 18 * q^62 - 4 * q^63 + 2 * q^64 + 24 * q^66 + 8 * q^67 + 6 * q^71 - 12 * q^72 + 8 * q^73 - 2 * q^76 + 12 * q^77 + 6 * q^78 + 4 * q^79 + 2 * q^81 + 12 * q^82 + 12 * q^83 + 4 * q^84 + 18 * q^86 + 6 * q^88 - 12 * q^89 + 4 * q^91 - 6 * q^92 - 28 * q^93 + 18 * q^96 - 4 * q^97 - 18 * q^99

Introduction

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