LMFDB - Embedded newform 7605.2.a.be.1.1

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			1.1
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)

$f(q)$	$=$	$q-1.73205 q^{2} +1.00000 q^{4} -1.00000 q^{5} -2.00000 q^{7} +1.73205 q^{8} +O(q^{10})$	$q-1.73205 q^{2} +1.00000 q^{4} -1.00000 q^{5} -2.00000 q^{7} +1.73205 q^{8} +1.73205 q^{10} -4.73205 q^{11} +3.46410 q^{14} -5.00000 q^{16} +3.46410 q^{17} +6.19615 q^{19} -1.00000 q^{20} +8.19615 q^{22} -1.26795 q^{23} +1.00000 q^{25} -2.00000 q^{28} +2.53590 q^{29} -10.1962 q^{31} +5.19615 q^{32} -6.00000 q^{34} +2.00000 q^{35} +4.00000 q^{37} -10.7321 q^{38} -1.73205 q^{40} +3.46410 q^{41} -0.196152 q^{43} -4.73205 q^{44} +2.19615 q^{46} +6.00000 q^{47} -3.00000 q^{49} -1.73205 q^{50} -10.3923 q^{53} +4.73205 q^{55} -3.46410 q^{56} -4.39230 q^{58} +9.12436 q^{59} -8.39230 q^{61} +17.6603 q^{62} +1.00000 q^{64} -6.39230 q^{67} +3.46410 q^{68} -3.46410 q^{70} +4.73205 q^{71} +4.00000 q^{73} -6.92820 q^{74} +6.19615 q^{76} +9.46410 q^{77} -8.39230 q^{79} +5.00000 q^{80} -6.00000 q^{82} -6.00000 q^{83} -3.46410 q^{85} +0.339746 q^{86} -8.19615 q^{88} -12.9282 q^{89} -1.26795 q^{92} -10.3923 q^{94} -6.19615 q^{95} -2.00000 q^{97} +5.19615 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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	$-0.709785\pi$
$2$	−1.73205	−1.22474	−0.612372	+	0.790569i	$0.709785\pi$
$3$	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
$11$	−4.73205	−1.42677	−0.713384	−	0.700774i	$-0.752838\pi$
$11$	−4.73205	−1.42677	−0.713384	+	0.700774i	$0.752838\pi$
$12$	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	$-0.362001\pi$
$17$	3.46410	0.840168	0.420084	+	0.907485i	$0.362001\pi$
$18$	0	0
$19$	6.19615	1.42149	0.710747	−	0.703447i	$-0.248357\pi$
$19$	6.19615	1.42149	0.710747	+	0.703447i	$0.248357\pi$
$20$	−1.00000	−0.223607
$21$	0	0
$22$	8.19615	1.74743
$23$	−1.26795	−0.264386	−0.132193	−	0.991224i	$-0.542202\pi$
$23$	−1.26795	−0.264386	−0.132193	+	0.991224i	$0.542202\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	−2.00000	−0.377964
$29$	2.53590	0.470905	0.235452	−	0.971886i	$-0.424343\pi$
$29$	2.53590	0.470905	0.235452	+	0.971886i	$0.424343\pi$
$30$	0	0
$31$	−10.1962	−1.83128	−0.915642	−	0.401996i	$-0.868317\pi$
$31$	−10.1962	−1.83128	−0.915642	+	0.401996i	$0.868317\pi$
$32$	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	$0.393356\pi$
$38$	−10.7321	−1.74097
$39$	0	0
$40$	−1.73205	−0.273861
$41$	3.46410	0.541002	0.270501	−	0.962720i	$-0.412811\pi$
$41$	3.46410	0.541002	0.270501	+	0.962720i	$0.412811\pi$
$42$	0	0
$43$	−0.196152	−0.0299130	−0.0149565	−	0.999888i	$-0.504761\pi$
$43$	−0.196152	−0.0299130	−0.0149565	+	0.999888i	$0.504761\pi$
$44$	−4.73205	−0.713384
$45$	0	0
$46$	2.19615	0.323805
$47$	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	$-0.753003\pi$
$53$	−10.3923	−1.42749	−0.713746	+	0.700404i	$0.753003\pi$
$54$	0	0
$55$	4.73205	0.638070
$56$	−3.46410	−0.462910
$57$	0	0
$58$	−4.39230	−0.576738
$59$	9.12436	1.18789	0.593945	−	0.804506i	$-0.297570\pi$
$59$	9.12436	1.18789	0.593945	+	0.804506i	$0.297570\pi$
$60$	0	0
$61$	−8.39230	−1.07452	−0.537262	−	0.843415i	$-0.680541\pi$
$61$	−8.39230	−1.07452	−0.537262	+	0.843415i	$0.680541\pi$
$62$	17.6603	2.24285
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	0	0
$67$	−6.39230	−0.780944	−0.390472	−	0.920615i	$-0.627688\pi$
$67$	−6.39230	−0.780944	−0.390472	+	0.920615i	$0.627688\pi$
$68$	3.46410	0.420084
$69$	0	0
$70$	−3.46410	−0.414039
$71$	4.73205	0.561591	0.280796	−	0.959768i	$-0.409402\pi$
$71$	4.73205	0.561591	0.280796	+	0.959768i	$0.409402\pi$
$72$	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
$76$	6.19615	0.710747
$77$	9.46410	1.07853
$78$	0	0
$79$	−8.39230	−0.944208	−0.472104	−	0.881543i	$-0.656505\pi$
$79$	−8.39230	−0.944208	−0.472104	+	0.881543i	$0.656505\pi$
$80$	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
$86$	0.339746	0.0366357
$87$	0	0
$88$	−8.19615	−0.873713
$89$	−12.9282	−1.37039	−0.685193	−	0.728361i	$-0.740282\pi$
$89$	−12.9282	−1.37039	−0.685193	+	0.728361i	$0.740282\pi$
$90$	0	0
$91$	0	0
$92$	−1.26795	−0.132193
$93$	0	0
$94$	−10.3923	−1.07188
$95$	−6.19615	−0.635712
$96$	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
$101$	0.928203	0.0923597	0.0461798	−	0.998933i	$-0.485295\pi$
$101$	0.928203	0.0923597	0.0461798	+	0.998933i	$0.485295\pi$
$102$	0	0
$103$	−0.196152	−0.0193275	−0.00966374	−	0.999953i	$-0.503076\pi$
$103$	−0.196152	−0.0193275	−0.00966374	+	0.999953i	$0.503076\pi$
$104$	0	0
$105$	0	0
$106$	18.0000	1.74831
$107$	−17.6603	−1.70728	−0.853641	−	0.520862i	$-0.825610\pi$
$107$	−17.6603	−1.70728	−0.853641	+	0.520862i	$0.825610\pi$
$108$	0	0
$109$	−2.00000	−0.191565	−0.0957826	−	0.995402i	$-0.530535\pi$
$109$	−2.00000	−0.191565	−0.0957826	+	0.995402i	$0.530535\pi$
$110$	−8.19615	−0.781472
$111$	0	0
$112$	10.0000	0.944911
$113$	−8.53590	−0.802990	−0.401495	−	0.915861i	$-0.631509\pi$
$113$	−8.53590	−0.802990	−0.401495	+	0.915861i	$0.631509\pi$
$114$	0	0
$115$	1.26795	0.118237
$116$	2.53590	0.235452
$117$	0	0
$118$	−15.8038	−1.45486
$119$	−6.92820	−0.635107
$120$	0	0
$121$	11.3923	1.03566
$122$	14.5359	1.31602
$123$	0	0
$124$	−10.1962	−0.915642
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	16.1962	1.43718	0.718588	−	0.695436i	$-0.244789\pi$
$127$	16.1962	1.43718	0.718588	+	0.695436i	$0.244789\pi$
$128$	−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
$133$	−12.3923	−1.07455
$134$	11.0718	0.956458
$135$	0	0
$136$	6.00000	0.514496
$137$	0.928203	0.0793018	0.0396509	−	0.999214i	$-0.487375\pi$
$137$	0.928203	0.0793018	0.0396509	+	0.999214i	$0.487375\pi$
$138$	0	0
$139$	12.3923	1.05110	0.525551	−	0.850762i	$-0.323859\pi$
$139$	12.3923	1.05110	0.525551	+	0.850762i	$0.323859\pi$
$140$	2.00000	0.169031
$141$	0	0
$142$	−8.19615	−0.687806
$143$	0	0
$144$	0	0
$145$	−2.53590	−0.210595
$146$	−6.92820	−0.573382
$147$	0	0
$148$	4.00000	0.328798
$149$	−7.85641	−0.643622	−0.321811	−	0.946804i	$-0.604292\pi$
$149$	−7.85641	−0.643622	−0.321811	+	0.946804i	$0.604292\pi$
$150$	0	0
$151$	1.80385	0.146795	0.0733975	−	0.997303i	$-0.476616\pi$
$151$	1.80385	0.146795	0.0733975	+	0.997303i	$0.476616\pi$
$152$	10.7321	0.870484
$153$	0	0
$154$	−16.3923	−1.32093
$155$	10.1962	0.818975
$156$	0	0
$157$	−10.0000	−0.798087	−0.399043	−	0.916932i	$-0.630658\pi$
$157$	−10.0000	−0.798087	−0.399043	+	0.916932i	$0.630658\pi$
$158$	14.5359	1.15641
$159$	0	0
$160$	−5.19615	−0.410792
$161$	2.53590	0.199857
$162$	0	0
$163$	14.3923	1.12729	0.563646	−	0.826016i	$-0.309398\pi$
$163$	14.3923	1.12729	0.563646	+	0.826016i	$0.309398\pi$
$164$	3.46410	0.270501
$165$	0	0
$166$	10.3923	0.806599
$167$	−0.928203	−0.0718265	−0.0359133	−	0.999355i	$-0.511434\pi$
$167$	−0.928203	−0.0718265	−0.0359133	+	0.999355i	$0.511434\pi$
$168$	0	0
$169$	0	0
$170$	6.00000	0.460179
$171$	0	0
$172$	−0.196152	−0.0149565
$173$	−8.53590	−0.648972	−0.324486	−	0.945890i	$-0.605191\pi$
$173$	−8.53590	−0.648972	−0.324486	+	0.945890i	$0.605191\pi$
$174$	0	0
$175$	−2.00000	−0.151186
$176$	23.6603	1.78346
$177$	0	0
$178$	22.3923	1.67837
$179$	18.9282	1.41476	0.707380	−	0.706833i	$-0.249877\pi$
$179$	18.9282	1.41476	0.707380	+	0.706833i	$0.249877\pi$
$180$	0	0
$181$	0.392305	0.0291598	0.0145799	−	0.999894i	$-0.495359\pi$
$181$	0.392305	0.0291598	0.0145799	+	0.999894i	$0.495359\pi$
$182$	0	0
$183$	0	0
$184$	−2.19615	−0.161903
$185$	−4.00000	−0.294086
$186$	0	0
$187$	−16.3923	−1.19872
$188$	6.00000	0.437595
$189$	0	0
$190$	10.7321	0.778585
$191$	5.07180	0.366982	0.183491	−	0.983021i	$-0.441260\pi$
$191$	5.07180	0.366982	0.183491	+	0.983021i	$0.441260\pi$
$192$	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
$197$	−12.9282	−0.921096	−0.460548	−	0.887635i	$-0.652347\pi$
$197$	−12.9282	−0.921096	−0.460548	+	0.887635i	$0.652347\pi$
$198$	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
$202$	−1.60770	−0.113117
$203$	−5.07180	−0.355970
$204$	0	0
$205$	−3.46410	−0.241943
$206$	0.339746	0.0236712
$207$	0	0
$208$	0	0
$209$	−29.3205	−2.02814
$210$	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
$214$	30.5885	2.09098
$215$	0.196152	0.0133775
$216$	0	0
$217$	20.3923	1.38432
$218$	3.46410	0.234619
$219$	0	0
$220$	4.73205	0.319035
$221$	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
$226$	14.7846	0.983458
$227$	−3.46410	−0.229920	−0.114960	−	0.993370i	$-0.536674\pi$
$227$	−3.46410	−0.229920	−0.114960	+	0.993370i	$0.536674\pi$
$228$	0	0
$229$	−6.39230	−0.422415	−0.211208	−	0.977441i	$-0.567740\pi$
$229$	−6.39230	−0.422415	−0.211208	+	0.977441i	$0.567740\pi$
$230$	−2.19615	−0.144810
$231$	0	0
$232$	4.39230	0.288369
$233$	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
$236$	9.12436	0.593945
$237$	0	0
$238$	12.0000	0.777844
$239$	−14.1962	−0.918273	−0.459136	−	0.888366i	$-0.651841\pi$
$239$	−14.1962	−0.918273	−0.459136	+	0.888366i	$0.651841\pi$
$240$	0	0
$241$	2.39230	0.154102	0.0770510	−	0.997027i	$-0.475450\pi$
$241$	2.39230	0.154102	0.0770510	+	0.997027i	$0.475450\pi$
$242$	−19.7321	−1.26842
$243$	0	0
$244$	−8.39230	−0.537262
$245$	3.00000	0.191663
$246$	0	0
$247$	0	0
$248$	−17.6603	−1.12143
$249$	0	0
$250$	1.73205	0.109545
$251$	21.4641	1.35480	0.677401	−	0.735614i	$-0.263106\pi$
$251$	21.4641	1.35480	0.677401	+	0.735614i	$0.263106\pi$
$252$	0	0
$253$	6.00000	0.377217
$254$	−28.0526	−1.76017
$255$	0	0
$256$	19.0000	1.18750
$257$	−19.8564	−1.23861	−0.619304	−	0.785151i	$-0.712585\pi$
$257$	−19.8564	−1.23861	−0.619304	+	0.785151i	$0.712585\pi$
$258$	0	0
$259$	−8.00000	−0.497096
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−1.26795	−0.0781851	−0.0390925	−	0.999236i	$-0.512447\pi$
$263$	−1.26795	−0.0781851	−0.0390925	+	0.999236i	$0.512447\pi$
$264$	0	0
$265$	10.3923	0.638394
$266$	21.4641	1.31605
$267$	0	0
$268$	−6.39230	−0.390472
$269$	19.8564	1.21067	0.605333	−	0.795972i	$-0.293040\pi$
$269$	19.8564	1.21067	0.605333	+	0.795972i	$0.293040\pi$
$270$	0	0
$271$	−30.9808	−1.88195	−0.940974	−	0.338480i	$-0.890087\pi$
$271$	−30.9808	−1.88195	−0.940974	+	0.338480i	$0.890087\pi$
$272$	−17.3205	−1.05021
$273$	0	0
$274$	−1.60770	−0.0971244
$275$	−4.73205	−0.285353
$276$	0	0
$277$	−26.3923	−1.58576	−0.792880	−	0.609378i	$-0.791419\pi$
$277$	−26.3923	−1.58576	−0.792880	+	0.609378i	$0.791419\pi$
$278$	−21.4641	−1.28733
$279$	0	0
$280$	3.46410	0.207020
$281$	22.3923	1.33581	0.667906	−	0.744245i	$-0.267191\pi$
$281$	22.3923	1.33581	0.667906	+	0.744245i	$0.267191\pi$
$282$	0	0
$283$	32.5885	1.93718	0.968591	−	0.248658i	$-0.0799895\pi$
$283$	32.5885	1.93718	0.968591	+	0.248658i	$0.0799895\pi$
$284$	4.73205	0.280796
$285$	0	0
$286$	0	0
$287$	−6.92820	−0.408959
$288$	0	0
$289$	−5.00000	−0.294118
$290$	4.39230	0.257925
$291$	0	0
$292$	4.00000	0.234082
$293$	−5.07180	−0.296298	−0.148149	−	0.988965i	$-0.547331\pi$
$293$	−5.07180	−0.296298	−0.148149	+	0.988965i	$0.547331\pi$
$294$	0	0
$295$	−9.12436	−0.531241
$296$	6.92820	0.402694
$297$	0	0
$298$	13.6077	0.788273
$299$	0	0
$300$	0	0
$301$	0.392305	0.0226121
$302$	−3.12436	−0.179786
$303$	0	0
$304$	−30.9808	−1.77687
$305$	8.39230	0.480542
$306$	0	0
$307$	18.7846	1.07209	0.536047	−	0.844188i	$-0.319917\pi$
$307$	18.7846	1.07209	0.536047	+	0.844188i	$0.319917\pi$
$308$	9.46410	0.539267
$309$	0	0
$310$	−17.6603	−1.00304
$311$	16.3923	0.929522	0.464761	−	0.885436i	$-0.346140\pi$
$311$	16.3923	0.929522	0.464761	+	0.885436i	$0.346140\pi$
$312$	0	0
$313$	−14.3923	−0.813501	−0.406751	−	0.913539i	$-0.633338\pi$
$313$	−14.3923	−0.813501	−0.406751	+	0.913539i	$0.633338\pi$
$314$	17.3205	0.977453
$315$	0	0
$316$	−8.39230	−0.472104
$317$	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
$322$	−4.39230	−0.244774
$323$	21.4641	1.19429
$324$	0	0
$325$	0	0
$326$	−24.9282	−1.38065
$327$	0	0
$328$	6.00000	0.331295
$329$	−12.0000	−0.661581
$330$	0	0
$331$	−2.58846	−0.142274	−0.0711372	−	0.997467i	$-0.522663\pi$
$331$	−2.58846	−0.142274	−0.0711372	+	0.997467i	$0.522663\pi$
$332$	−6.00000	−0.329293
$333$	0	0
$334$	1.60770	0.0879692
$335$	6.39230	0.349249
$336$	0	0
$337$	−26.3923	−1.43768	−0.718840	−	0.695175i	$-0.755327\pi$
$337$	−26.3923	−1.43768	−0.718840	+	0.695175i	$0.755327\pi$
$338$	0	0
$339$	0	0
$340$	−3.46410	−0.187867
$341$	48.2487	2.61281
$342$	0	0
$343$	20.0000	1.07990
$344$	−0.339746	−0.0183179
$345$	0	0
$346$	14.7846	0.794826
$347$	5.66025	0.303858	0.151929	−	0.988391i	$-0.451451\pi$
$347$	5.66025	0.303858	0.151929	+	0.988391i	$0.451451\pi$
$348$	0	0
$349$	14.3923	0.770402	0.385201	−	0.922833i	$-0.374132\pi$
$349$	14.3923	0.770402	0.385201	+	0.922833i	$0.374132\pi$
$350$	3.46410	0.185164
$351$	0	0
$352$	−24.5885	−1.31057
$353$	−27.7128	−1.47500	−0.737502	−	0.675345i	$-0.763995\pi$
$353$	−27.7128	−1.47500	−0.737502	+	0.675345i	$0.763995\pi$
$354$	0	0
$355$	−4.73205	−0.251151
$356$	−12.9282	−0.685193
$357$	0	0
$358$	−32.7846	−1.73272
$359$	−2.19615	−0.115908	−0.0579542	−	0.998319i	$-0.518458\pi$
$359$	−2.19615	−0.115908	−0.0579542	+	0.998319i	$0.518458\pi$
$360$	0	0
$361$	19.3923	1.02065
$362$	−0.679492	−0.0357133
$363$	0	0
$364$	0	0
$365$	−4.00000	−0.209370
$366$	0	0
$367$	11.8038	0.616156	0.308078	−	0.951361i	$-0.400314\pi$
$367$	11.8038	0.616156	0.308078	+	0.951361i	$0.400314\pi$
$368$	6.33975	0.330482
$369$	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	$0.583357\pi$
$374$	28.3923	1.46813
$375$	0	0
$376$	10.3923	0.535942
$377$	0	0
$378$	0	0
$379$	−18.9808	−0.974976	−0.487488	−	0.873130i	$-0.662087\pi$
$379$	−18.9808	−0.974976	−0.487488	+	0.873130i	$0.662087\pi$
$380$	−6.19615	−0.317856
$381$	0	0
$382$	−8.78461	−0.449460
$383$	12.9282	0.660600	0.330300	−	0.943876i	$-0.392850\pi$
$383$	12.9282	0.660600	0.330300	+	0.943876i	$0.392850\pi$
$384$	0	0
$385$	−9.46410	−0.482335
$386$	−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
$391$	−4.39230	−0.222128
$392$	−5.19615	−0.262445
$393$	0	0
$394$	22.3923	1.12811
$395$	8.39230	0.422263
$396$	0	0
$397$	−28.7846	−1.44466	−0.722329	−	0.691549i	$-0.756928\pi$
$397$	−28.7846	−1.44466	−0.722329	+	0.691549i	$0.756928\pi$
$398$	−34.6410	−1.73640
$399$	0	0
$400$	−5.00000	−0.250000
$401$	−36.9282	−1.84411	−0.922053	−	0.387063i	$-0.873490\pi$
$401$	−36.9282	−1.84411	−0.922053	+	0.387063i	$0.873490\pi$
$402$	0	0
$403$	0	0
$404$	0.928203	0.0461798
$405$	0	0
$406$	8.78461	0.435973
$407$	−18.9282	−0.938236
$408$	0	0
$409$	17.6077	0.870644	0.435322	−	0.900275i	$-0.356634\pi$
$409$	17.6077	0.870644	0.435322	+	0.900275i	$0.356634\pi$
$410$	6.00000	0.296319
$411$	0	0
$412$	−0.196152	−0.00966374
$413$	−18.2487	−0.897960
$414$	0	0
$415$	6.00000	0.294528
$416$	0	0
$417$	0	0
$418$	50.7846	2.48396
$419$	2.53590	0.123887	0.0619434	−	0.998080i	$-0.480270\pi$
$419$	2.53590	0.123887	0.0619434	+	0.998080i	$0.480270\pi$
$420$	0	0
$421$	30.7846	1.50035	0.750175	−	0.661239i	$-0.229969\pi$
$421$	30.7846	1.50035	0.750175	+	0.661239i	$0.229969\pi$
$422$	−13.8564	−0.674519
$423$	0	0
$424$	−18.0000	−0.874157
$425$	3.46410	0.168034
$426$	0	0
$427$	16.7846	0.812264
$428$	−17.6603	−0.853641
$429$	0	0
$430$	−0.339746	−0.0163840
$431$	−25.5167	−1.22909	−0.614547	−	0.788880i	$-0.710661\pi$
$431$	−25.5167	−1.22909	−0.614547	+	0.788880i	$0.710661\pi$
$432$	0	0
$433$	34.7846	1.67164	0.835821	−	0.549002i	$-0.184992\pi$
$433$	34.7846	1.67164	0.835821	+	0.549002i	$0.184992\pi$
$434$	−35.3205	−1.69544
$435$	0	0
$436$	−2.00000	−0.0957826
$437$	−7.85641	−0.375823
$438$	0	0
$439$	32.0000	1.52728	0.763638	−	0.645644i	$-0.223411\pi$
$439$	32.0000	1.52728	0.763638	+	0.645644i	$0.223411\pi$
$440$	8.19615	0.390736
$441$	0	0
$442$	0	0
$443$	16.9808	0.806780	0.403390	−	0.915028i	$-0.367832\pi$
$443$	16.9808	0.806780	0.403390	+	0.915028i	$0.367832\pi$
$444$	0	0
$445$	12.9282	0.612856
$446$	3.46410	0.164030
$447$	0	0
$448$	−2.00000	−0.0944911
$449$	20.5359	0.969149	0.484574	−	0.874750i	$-0.338974\pi$
$449$	20.5359	0.969149	0.484574	+	0.874750i	$0.338974\pi$
$450$	0	0
$451$	−16.3923	−0.771883
$452$	−8.53590	−0.401495
$453$	0	0
$454$	6.00000	0.281594
$455$	0	0
$456$	0	0
$457$	−10.7846	−0.504483	−0.252241	−	0.967664i	$-0.581168\pi$
$457$	−10.7846	−0.504483	−0.252241	+	0.967664i	$0.581168\pi$
$458$	11.0718	0.517351
$459$	0	0
$460$	1.26795	0.0591184
$461$	−3.46410	−0.161339	−0.0806696	−	0.996741i	$-0.525706\pi$
$461$	−3.46410	−0.161339	−0.0806696	+	0.996741i	$0.525706\pi$
$462$	0	0
$463$	2.39230	0.111180	0.0555899	−	0.998454i	$-0.482296\pi$
$463$	2.39230	0.111180	0.0555899	+	0.998454i	$0.482296\pi$
$464$	−12.6795	−0.588631
$465$	0	0
$466$	−10.3923	−0.481414
$467$	−27.8038	−1.28661	−0.643304	−	0.765611i	$-0.722437\pi$
$467$	−27.8038	−1.28661	−0.643304	+	0.765611i	$0.722437\pi$
$468$	0	0
$469$	12.7846	0.590338
$470$	10.3923	0.479361
$471$	0	0
$472$	15.8038	0.727431
$473$	0.928203	0.0426788
$474$	0	0
$475$	6.19615	0.284299
$476$	−6.92820	−0.317554
$477$	0	0
$478$	24.5885	1.12465
$479$	35.6603	1.62936	0.814679	−	0.579912i	$-0.196913\pi$
$479$	35.6603	1.62936	0.814679	+	0.579912i	$0.196913\pi$
$480$	0	0
$481$	0	0
$482$	−4.14359	−0.188736
$483$	0	0
$484$	11.3923	0.517832
$485$	2.00000	0.0908153
$486$	0	0
$487$	26.3923	1.19595	0.597975	−	0.801515i	$-0.295972\pi$
$487$	26.3923	1.19595	0.597975	+	0.801515i	$0.295972\pi$
$488$	−14.5359	−0.658009
$489$	0	0
$490$	−5.19615	−0.234738
$491$	2.53590	0.114443	0.0572217	−	0.998361i	$-0.481776\pi$
$491$	2.53590	0.114443	0.0572217	+	0.998361i	$0.481776\pi$
$492$	0	0
$493$	8.78461	0.395639
$494$	0	0
$495$	0	0
$496$	50.9808	2.28910
$497$	−9.46410	−0.424523
$498$	0	0
$499$	38.9808	1.74502	0.872509	−	0.488598i	$-0.162491\pi$
$499$	38.9808	1.74502	0.872509	+	0.488598i	$0.162491\pi$
$500$	−1.00000	−0.0447214
$501$	0	0
$502$	−37.1769	−1.65929
$503$	−19.5167	−0.870205	−0.435102	−	0.900381i	$-0.643288\pi$
$503$	−19.5167	−0.870205	−0.435102	+	0.900381i	$0.643288\pi$
$504$	0	0
$505$	−0.928203	−0.0413045
$506$	−10.3923	−0.461994
$507$	0	0
$508$	16.1962	0.718588
$509$	39.4641	1.74922	0.874608	−	0.484831i	$-0.161119\pi$
$509$	39.4641	1.74922	0.874608	+	0.484831i	$0.161119\pi$
$510$	0	0
$511$	−8.00000	−0.353899
$512$	−8.66025	−0.382733
$513$	0	0
$514$	34.3923	1.51698
$515$	0.196152	0.00864351
$516$	0	0
$517$	−28.3923	−1.24869
$518$	13.8564	0.608816
$519$	0	0
$520$	0	0
$521$	28.3923	1.24389	0.621945	−	0.783061i	$-0.286343\pi$
$521$	28.3923	1.24389	0.621945	+	0.783061i	$0.286343\pi$
$522$	0	0
$523$	−24.1962	−1.05802	−0.529012	−	0.848614i	$-0.677437\pi$
$523$	−24.1962	−1.05802	−0.529012	+	0.848614i	$0.677437\pi$
$524$	0	0
$525$	0	0
$526$	2.19615	0.0957568
$527$	−35.3205	−1.53859
$528$	0	0
$529$	−21.3923	−0.930100
$530$	−18.0000	−0.781870
$531$	0	0
$532$	−12.3923	−0.537275
$533$	0	0
$534$	0	0
$535$	17.6603	0.763519
$536$	−11.0718	−0.478229
$537$	0	0
$538$	−34.3923	−1.48276
$539$	14.1962	0.611472
$540$	0	0
$541$	26.3923	1.13469	0.567347	−	0.823479i	$-0.307970\pi$
$541$	26.3923	1.13469	0.567347	+	0.823479i	$0.307970\pi$
$542$	53.6603	2.30491
$543$	0	0
$544$	18.0000	0.771744
$545$	2.00000	0.0856706
$546$	0	0
$547$	−12.1962	−0.521470	−0.260735	−	0.965410i	$-0.583965\pi$
$547$	−12.1962	−0.521470	−0.260735	+	0.965410i	$0.583965\pi$
$548$	0.928203	0.0396509
$549$	0	0
$550$	8.19615	0.349485
$551$	15.7128	0.669388
$552$	0	0
$553$	16.7846	0.713754
$554$	45.7128	1.94215
$555$	0	0
$556$	12.3923	0.525551
$557$	1.85641	0.0786585	0.0393292	−	0.999226i	$-0.487478\pi$
$557$	1.85641	0.0786585	0.0393292	+	0.999226i	$0.487478\pi$
$558$	0	0
$559$	0	0
$560$	−10.0000	−0.422577
$561$	0	0
$562$	−38.7846	−1.63603
$563$	22.0526	0.929405	0.464702	−	0.885467i	$-0.346161\pi$
$563$	22.0526	0.929405	0.464702	+	0.885467i	$0.346161\pi$
$564$	0	0
$565$	8.53590	0.359108
$566$	−56.4449	−2.37255
$567$	0	0
$568$	8.19615	0.343903
$569$	2.53590	0.106310	0.0531552	−	0.998586i	$-0.483072\pi$
$569$	2.53590	0.106310	0.0531552	+	0.998586i	$0.483072\pi$
$570$	0	0
$571$	36.3923	1.52297	0.761485	−	0.648182i	$-0.224470\pi$
$571$	36.3923	1.52297	0.761485	+	0.648182i	$0.224470\pi$
$572$	0	0
$573$	0	0
$574$	12.0000	0.500870
$575$	−1.26795	−0.0528771
$576$	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
$580$	−2.53590	−0.105297
$581$	12.0000	0.497844
$582$	0	0
$583$	49.1769	2.03670
$584$	6.92820	0.286691
$585$	0	0
$586$	8.78461	0.362889
$587$	−8.53590	−0.352314	−0.176157	−	0.984362i	$-0.556367\pi$
$587$	−8.53590	−0.352314	−0.176157	+	0.984362i	$0.556367\pi$
$588$	0	0
$589$	−63.1769	−2.60316
$590$	15.8038	0.650634
$591$	0	0
$592$	−20.0000	−0.821995
$593$	26.7846	1.09991	0.549956	−	0.835194i	$-0.314644\pi$
$593$	26.7846	1.09991	0.549956	+	0.835194i	$0.314644\pi$
$594$	0	0
$595$	6.92820	0.284029
$596$	−7.85641	−0.321811
$597$	0	0
$598$	0	0
$599$	7.60770	0.310842	0.155421	−	0.987848i	$-0.450327\pi$
$599$	7.60770	0.310842	0.155421	+	0.987848i	$0.450327\pi$
$600$	0	0
$601$	43.5692	1.77723	0.888613	−	0.458658i	$-0.151670\pi$
$601$	43.5692	1.77723	0.888613	+	0.458658i	$0.151670\pi$
$602$	−0.679492	−0.0276940
$603$	0	0
$604$	1.80385	0.0733975
$605$	−11.3923	−0.463163
$606$	0	0
$607$	24.9808	1.01394	0.506969	−	0.861964i	$-0.330766\pi$
$607$	24.9808	1.01394	0.506969	+	0.861964i	$0.330766\pi$
$608$	32.1962	1.30573
$609$	0	0
$610$	−14.5359	−0.588541
$611$	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	$0.675959\pi$
$614$	−32.5359	−1.31304
$615$	0	0
$616$	16.3923	0.660465
$617$	33.7128	1.35723	0.678613	−	0.734496i	$-0.262581\pi$
$617$	33.7128	1.35723	0.678613	+	0.734496i	$0.262581\pi$
$618$	0	0
$619$	−6.98076	−0.280581	−0.140290	−	0.990110i	$-0.544804\pi$
$619$	−6.98076	−0.280581	−0.140290	+	0.990110i	$0.544804\pi$
$620$	10.1962	0.409487
$621$	0	0
$622$	−28.3923	−1.13843
$623$	25.8564	1.03592
$624$	0	0
$625$	1.00000	0.0400000
$626$	24.9282	0.996331
$627$	0	0
$628$	−10.0000	−0.399043
$629$	13.8564	0.552491
$630$	0	0
$631$	−5.80385	−0.231048	−0.115524	−	0.993305i	$-0.536855\pi$
$631$	−5.80385	−0.231048	−0.115524	+	0.993305i	$0.536855\pi$
$632$	−14.5359	−0.578207
$633$	0	0
$634$	−41.5692	−1.65092
$635$	−16.1962	−0.642725
$636$	0	0
$637$	0	0
$638$	20.7846	0.822871
$639$	0	0
$640$	12.1244	0.479257
$641$	−12.9282	−0.510633	−0.255317	−	0.966857i	$-0.582180\pi$
$641$	−12.9282	−0.510633	−0.255317	+	0.966857i	$0.582180\pi$
$642$	0	0
$643$	6.78461	0.267559	0.133779	−	0.991011i	$-0.457289\pi$
$643$	6.78461	0.267559	0.133779	+	0.991011i	$0.457289\pi$
$644$	2.53590	0.0999284
$645$	0	0
$646$	−37.1769	−1.46271
$647$	−22.0526	−0.866976	−0.433488	−	0.901159i	$-0.642717\pi$
$647$	−22.0526	−0.866976	−0.433488	+	0.901159i	$0.642717\pi$
$648$	0	0
$649$	−43.1769	−1.69484
$650$	0	0
$651$	0	0
$652$	14.3923	0.563646
$653$	−7.85641	−0.307445	−0.153722	−	0.988114i	$-0.549126\pi$
$653$	−7.85641	−0.307445	−0.153722	+	0.988114i	$0.549126\pi$
$654$	0	0
$655$	0	0
$656$	−17.3205	−0.676252
$657$	0	0
$658$	20.7846	0.810268
$659$	21.4641	0.836123	0.418061	−	0.908419i	$-0.362710\pi$
$659$	21.4641	0.836123	0.418061	+	0.908419i	$0.362710\pi$
$660$	0	0
$661$	−10.7846	−0.419473	−0.209736	−	0.977758i	$-0.567261\pi$
$661$	−10.7846	−0.419473	−0.209736	+	0.977758i	$0.567261\pi$
$662$	4.48334	0.174250
$663$	0	0
$664$	−10.3923	−0.403300
$665$	12.3923	0.480553
$666$	0	0
$667$	−3.21539	−0.124500
$668$	−0.928203	−0.0359133
$669$	0	0
$670$	−11.0718	−0.427741
$671$	39.7128	1.53310
$672$	0	0
$673$	−14.3923	−0.554783	−0.277391	−	0.960757i	$-0.589470\pi$
$673$	−14.3923	−0.554783	−0.277391	+	0.960757i	$0.589470\pi$
$674$	45.7128	1.76079
$675$	0	0
$676$	0	0
$677$	−10.3923	−0.399409	−0.199704	−	0.979856i	$-0.563998\pi$
$677$	−10.3923	−0.399409	−0.199704	+	0.979856i	$0.563998\pi$
$678$	0	0
$679$	4.00000	0.153506
$680$	−6.00000	−0.230089
$681$	0	0
$682$	−83.5692	−3.20003
$683$	32.5359	1.24495	0.622476	−	0.782639i	$-0.286127\pi$
$683$	32.5359	1.24495	0.622476	+	0.782639i	$0.286127\pi$
$684$	0	0
$685$	−0.928203	−0.0354648
$686$	−34.6410	−1.32260
$687$	0	0
$688$	0.980762	0.0373912
$689$	0	0
$690$	0	0
$691$	47.7654	1.81708	0.908540	−	0.417797i	$-0.137198\pi$
$691$	47.7654	1.81708	0.908540	+	0.417797i	$0.137198\pi$
$692$	−8.53590	−0.324486
$693$	0	0
$694$	−9.80385	−0.372149
$695$	−12.3923	−0.470067
$696$	0	0
$697$	12.0000	0.454532
$698$	−24.9282	−0.943546
$699$	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
$703$	24.7846	0.934769
$704$	−4.73205	−0.178346
$705$	0	0
$706$	48.0000	1.80650
$707$	−1.85641	−0.0698174
$708$	0	0
$709$	−30.3923	−1.14141	−0.570703	−	0.821156i	$-0.693329\pi$
$709$	−30.3923	−1.14141	−0.570703	+	0.821156i	$0.693329\pi$
$710$	8.19615	0.307596
$711$	0	0
$712$	−22.3923	−0.839187
$713$	12.9282	0.484165
$714$	0	0
$715$	0	0
$716$	18.9282	0.707380
$717$	0	0
$718$	3.80385	0.141958
$719$	25.8564	0.964281	0.482141	−	0.876094i	$-0.339859\pi$
$719$	25.8564	0.964281	0.482141	+	0.876094i	$0.339859\pi$
$720$	0	0
$721$	0.392305	0.0146102
$722$	−33.5885	−1.25003
$723$	0	0
$724$	0.392305	0.0145799
$725$	2.53590	0.0941809
$726$	0	0
$727$	44.5885	1.65369	0.826847	−	0.562427i	$-0.190132\pi$
$727$	44.5885	1.65369	0.826847	+	0.562427i	$0.190132\pi$
$728$	0	0
$729$	0	0
$730$	6.92820	0.256424
$731$	−0.679492	−0.0251319
$732$	0	0
$733$	−38.0000	−1.40356	−0.701781	−	0.712393i	$-0.747612\pi$
$733$	−38.0000	−1.40356	−0.701781	+	0.712393i	$0.747612\pi$
$734$	−20.4449	−0.754634
$735$	0	0
$736$	−6.58846	−0.242854
$737$	30.2487	1.11423
$738$	0	0
$739$	18.1962	0.669356	0.334678	−	0.942332i	$-0.391372\pi$
$739$	18.1962	0.669356	0.334678	+	0.942332i	$0.391372\pi$
$740$	−4.00000	−0.147043
$741$	0	0
$742$	−36.0000	−1.32160
$743$	−16.1436	−0.592251	−0.296126	−	0.955149i	$-0.595695\pi$
$743$	−16.1436	−0.592251	−0.296126	+	0.955149i	$0.595695\pi$
$744$	0	0
$745$	7.85641	0.287836
$746$	17.3205	0.634149
$747$	0	0
$748$	−16.3923	−0.599362
$749$	35.3205	1.29058
$750$	0	0
$751$	36.3923	1.32797	0.663987	−	0.747744i	$-0.268863\pi$
$751$	36.3923	1.32797	0.663987	+	0.747744i	$0.268863\pi$
$752$	−30.0000	−1.09399
$753$	0	0
$754$	0	0
$755$	−1.80385	−0.0656487
$756$	0	0
$757$	−2.39230	−0.0869498	−0.0434749	−	0.999055i	$-0.513843\pi$
$757$	−2.39230	−0.0869498	−0.0434749	+	0.999055i	$0.513843\pi$
$758$	32.8756	1.19410
$759$	0	0
$760$	−10.7321	−0.389292
$761$	−19.8564	−0.719794	−0.359897	−	0.932992i	$-0.617188\pi$
$761$	−19.8564	−0.719794	−0.359897	+	0.932992i	$0.617188\pi$
$762$	0	0
$763$	4.00000	0.144810
$764$	5.07180	0.183491
$765$	0	0
$766$	−22.3923	−0.809067
$767$	0	0
$768$	0	0
$769$	−34.7846	−1.25437	−0.627183	−	0.778872i	$-0.715792\pi$
$769$	−34.7846	−1.25437	−0.627183	+	0.778872i	$0.715792\pi$
$770$	16.3923	0.590738
$771$	0	0
$772$	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	$0.460237\pi$
$774$	0	0
$775$	−10.1962	−0.366257
$776$	−3.46410	−0.124354
$777$	0	0
$778$	10.3923	0.372582
$779$	21.4641	0.769031
$780$	0	0
$781$	−22.3923	−0.801260
$782$	7.60770	0.272051
$783$	0	0
$784$	15.0000	0.535714
$785$	10.0000	0.356915
$786$	0	0
$787$	−31.5692	−1.12532	−0.562661	−	0.826688i	$-0.690222\pi$
$787$	−31.5692	−1.12532	−0.562661	+	0.826688i	$0.690222\pi$
$788$	−12.9282	−0.460548
$789$	0	0
$790$	−14.5359	−0.517164
$791$	17.0718	0.607003
$792$	0	0
$793$	0	0
$794$	49.8564	1.76934
$795$	0	0
$796$	20.0000	0.708881
$797$	40.6410	1.43958	0.719789	−	0.694193i	$-0.244238\pi$
$797$	40.6410	1.43958	0.719789	+	0.694193i	$0.244238\pi$
$798$	0	0
$799$	20.7846	0.735307
$800$	5.19615	0.183712
$801$	0	0
$802$	63.9615	2.25856
$803$	−18.9282	−0.667962
$804$	0	0
$805$	−2.53590	−0.0893787
$806$	0	0
$807$	0	0
$808$	1.60770	0.0565585
$809$	2.53590	0.0891574	0.0445787	−	0.999006i	$-0.485805\pi$
$809$	2.53590	0.0891574	0.0445787	+	0.999006i	$0.485805\pi$
$810$	0	0
$811$	−17.8038	−0.625178	−0.312589	−	0.949889i	$-0.601196\pi$
$811$	−17.8038	−0.625178	−0.312589	+	0.949889i	$0.601196\pi$
$812$	−5.07180	−0.177985
$813$	0	0
$814$	32.7846	1.14910
$815$	−14.3923	−0.504140
$816$	0	0
$817$	−1.21539	−0.0425211
$818$	−30.4974	−1.06632
$819$	0	0
$820$	−3.46410	−0.120972
$821$	28.6410	0.999578	0.499789	−	0.866147i	$-0.333411\pi$
$821$	28.6410	0.999578	0.499789	+	0.866147i	$0.333411\pi$
$822$	0	0
$823$	−15.4115	−0.537213	−0.268606	−	0.963250i	$-0.586563\pi$
$823$	−15.4115	−0.537213	−0.268606	+	0.963250i	$0.586563\pi$
$824$	−0.339746	−0.0118356
$825$	0	0
$826$	31.6077	1.09977
$827$	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
$829$	0.392305	0.0136253	0.00681266	−	0.999977i	$-0.497831\pi$
$829$	0.392305	0.0136253	0.00681266	+	0.999977i	$0.497831\pi$
$830$	−10.3923	−0.360722
$831$	0	0
$832$	0	0
$833$	−10.3923	−0.360072
$834$	0	0
$835$	0.928203	0.0321218
$836$	−29.3205	−1.01407
$837$	0	0
$838$	−4.39230	−0.151730
$839$	0.339746	0.0117293	0.00586467	−	0.999983i	$-0.498133\pi$
$839$	0.339746	0.0117293	0.00586467	+	0.999983i	$0.498133\pi$
$840$	0	0
$841$	−22.5692	−0.778249
$842$	−53.3205	−1.83755
$843$	0	0
$844$	8.00000	0.275371
$845$	0	0
$846$	0	0
$847$	−22.7846	−0.782888
$848$	51.9615	1.78437
$849$	0	0
$850$	−6.00000	−0.205798
$851$	−5.07180	−0.173859
$852$	0	0
$853$	−8.00000	−0.273915	−0.136957	−	0.990577i	$-0.543732\pi$
$853$	−8.00000	−0.273915	−0.136957	+	0.990577i	$0.543732\pi$
$854$	−29.0718	−0.994816
$855$	0	0
$856$	−30.5885	−1.04549
$857$	35.5692	1.21502	0.607511	−	0.794311i	$-0.292168\pi$
$857$	35.5692	1.21502	0.607511	+	0.794311i	$0.292168\pi$
$858$	0	0
$859$	−17.1769	−0.586069	−0.293034	−	0.956102i	$-0.594665\pi$
$859$	−17.1769	−0.586069	−0.293034	+	0.956102i	$0.594665\pi$
$860$	0.196152	0.00668874
$861$	0	0
$862$	44.1962	1.50533
$863$	−38.7846	−1.32024	−0.660122	−	0.751159i	$-0.729495\pi$
$863$	−38.7846	−1.32024	−0.660122	+	0.751159i	$0.729495\pi$
$864$	0	0
$865$	8.53590	0.290229
$866$	−60.2487	−2.04733
$867$	0	0
$868$	20.3923	0.692160
$869$	39.7128	1.34716
$870$	0	0
$871$	0	0
$872$	−3.46410	−0.117309
$873$	0	0
$874$	13.6077	0.460287
$875$	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
$880$	−23.6603	−0.797587
$881$	47.3205	1.59427	0.797134	−	0.603802i	$-0.206348\pi$
$881$	47.3205	1.59427	0.797134	+	0.603802i	$0.206348\pi$
$882$	0	0
$883$	23.8038	0.801063	0.400532	−	0.916283i	$-0.368825\pi$
$883$	23.8038	0.801063	0.400532	+	0.916283i	$0.368825\pi$
$884$	0	0
$885$	0	0
$886$	−29.4115	−0.988100
$887$	47.9090	1.60863	0.804313	−	0.594206i	$-0.202534\pi$
$887$	47.9090	1.60863	0.804313	+	0.594206i	$0.202534\pi$
$888$	0	0
$889$	−32.3923	−1.08640
$890$	−22.3923	−0.750592
$891$	0	0
$892$	−2.00000	−0.0669650
$893$	37.1769	1.24408
$894$	0	0
$895$	−18.9282	−0.632700
$896$	24.2487	0.810093
$897$	0	0
$898$	−35.5692	−1.18696
$899$	−25.8564	−0.862359
$900$	0	0
$901$	−36.0000	−1.19933
$902$	28.3923	0.945360
$903$	0	0
$904$	−14.7846	−0.491729
$905$	−0.392305	−0.0130407
$906$	0	0
$907$	−53.7654	−1.78525	−0.892625	−	0.450800i	$-0.851139\pi$
$907$	−53.7654	−1.78525	−0.892625	+	0.450800i	$0.851139\pi$
$908$	−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
$913$	28.3923	0.939648
$914$	18.6795	0.617863
$915$	0	0
$916$	−6.39230	−0.211208
$917$	0	0
$918$	0	0
$919$	9.17691	0.302718	0.151359	−	0.988479i	$-0.451635\pi$
$919$	9.17691	0.302718	0.151359	+	0.988479i	$0.451635\pi$
$920$	2.19615	0.0724050
$921$	0	0
$922$	6.00000	0.197599
$923$	0	0
$924$	0	0
$925$	4.00000	0.131519
$926$	−4.14359	−0.136167
$927$	0	0
$928$	13.1769	0.432553
$929$	44.5359	1.46118	0.730588	−	0.682819i	$-0.239246\pi$
$929$	44.5359	1.46118	0.730588	+	0.682819i	$0.239246\pi$
$930$	0	0
$931$	−18.5885	−0.609212
$932$	6.00000	0.196537
$933$	0	0
$934$	48.1577	1.57577
$935$	16.3923	0.536086
$936$	0	0
$937$	34.7846	1.13636	0.568182	−	0.822903i	$-0.307647\pi$
$937$	34.7846	1.13636	0.568182	+	0.822903i	$0.307647\pi$
$938$	−22.1436	−0.723014
$939$	0	0
$940$	−6.00000	−0.195698
$941$	31.1769	1.01634	0.508169	−	0.861257i	$-0.330322\pi$
$941$	31.1769	1.01634	0.508169	+	0.861257i	$0.330322\pi$
$942$	0	0
$943$	−4.39230	−0.143033
$944$	−45.6218	−1.48486
$945$	0	0
$946$	−1.60770	−0.0522707
$947$	40.6410	1.32066	0.660328	−	0.750978i	$-0.270417\pi$
$947$	40.6410	1.32066	0.660328	+	0.750978i	$0.270417\pi$
$948$	0	0
$949$	0	0
$950$	−10.7321	−0.348194
$951$	0	0
$952$	−12.0000	−0.388922
$953$	−0.928203	−0.0300675	−0.0150337	−	0.999887i	$-0.504786\pi$
$953$	−0.928203	−0.0300675	−0.0150337	+	0.999887i	$0.504786\pi$
$954$	0	0
$955$	−5.07180	−0.164119
$956$	−14.1962	−0.459136
$957$	0	0
$958$	−61.7654	−1.99555
$959$	−1.85641	−0.0599465
$960$	0	0
$961$	72.9615	2.35360
$962$	0	0
$963$	0	0
$964$	2.39230	0.0770510
$965$	−10.0000	−0.321911
$966$	0	0
$967$	50.3923	1.62051	0.810254	−	0.586079i	$-0.199329\pi$
$967$	50.3923	1.62051	0.810254	+	0.586079i	$0.199329\pi$
$968$	19.7321	0.634212
$969$	0	0
$970$	−3.46410	−0.111226
$971$	−18.9282	−0.607435	−0.303717	−	0.952762i	$-0.598228\pi$
$971$	−18.9282	−0.607435	−0.303717	+	0.952762i	$0.598228\pi$
$972$	0	0
$973$	−24.7846	−0.794558
$974$	−45.7128	−1.46473
$975$	0	0
$976$	41.9615	1.34316
$977$	−15.7128	−0.502697	−0.251349	−	0.967897i	$-0.580874\pi$
$977$	−15.7128	−0.502697	−0.251349	+	0.967897i	$0.580874\pi$
$978$	0	0
$979$	61.1769	1.95522
$980$	3.00000	0.0958315
$981$	0	0
$982$	−4.39230	−0.140164
$983$	−34.3923	−1.09694	−0.548472	−	0.836169i	$-0.684790\pi$
$983$	−34.3923	−1.09694	−0.548472	+	0.836169i	$0.684790\pi$
$984$	0	0
$985$	12.9282	0.411927
$986$	−15.2154	−0.484557
$987$	0	0
$988$	0	0
$989$	0.248711	0.00790856
$990$	0	0
$991$	8.00000	0.254128	0.127064	−	0.991894i	$-0.459445\pi$
$991$	8.00000	0.254128	0.127064	+	0.991894i	$0.459445\pi$
$992$	−52.9808	−1.68214
$993$	0	0
$994$	16.3923	0.519932
$995$	−20.0000	−0.634043
$996$	0	0
$997$	33.6077	1.06437	0.532183	−	0.846629i	$-0.321372\pi$
$997$	33.6077	1.06437	0.532183	+	0.846629i	$0.321372\pi$
$998$	−67.5167	−2.13720
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7605.2.a.be.1.1		2
3.2	odd	2		845.2.a.d.1.2		2
13.12	even	2		585.2.a.k.1.2		2
15.14	odd	2		4225.2.a.w.1.1		2
39.2	even	12		845.2.m.c.316.1		4
39.5	even	4		845.2.c.e.506.2		4
39.8	even	4		845.2.c.e.506.4		4
39.11	even	12		845.2.m.a.316.1		4
39.17	odd	6		845.2.e.e.146.2		4
39.20	even	12		845.2.m.c.361.1		4
39.23	odd	6		845.2.e.e.191.2		4
39.29	odd	6		845.2.e.f.191.1		4
39.32	even	12		845.2.m.a.361.1		4
39.35	odd	6		845.2.e.f.146.1		4
39.38	odd	2		65.2.a.c.1.1	✓	2
52.51	odd	2		9360.2.a.cm.1.1		2
65.12	odd	4		2925.2.c.v.2224.4		4
65.38	odd	4		2925.2.c.v.2224.1		4
65.64	even	2		2925.2.a.z.1.1		2
156.155	even	2		1040.2.a.h.1.1		2
195.38	even	4		325.2.b.e.274.4		4
195.77	even	4		325.2.b.e.274.1		4
195.194	odd	2		325.2.a.g.1.2		2
273.272	even	2		3185.2.a.k.1.1		2
312.77	odd	2		4160.2.a.y.1.1		2
312.155	even	2		4160.2.a.bj.1.2		2
429.428	even	2		7865.2.a.h.1.2		2
780.779	even	2		5200.2.a.ca.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
65.2.a.c.1.1	✓	2	39.38	odd	2
325.2.a.g.1.2		2	195.194	odd	2
325.2.b.e.274.1		4	195.77	even	4
325.2.b.e.274.4		4	195.38	even	4
585.2.a.k.1.2		2	13.12	even	2
845.2.a.d.1.2		2	3.2	odd	2
845.2.c.e.506.2		4	39.5	even	4
845.2.c.e.506.4		4	39.8	even	4
845.2.e.e.146.2		4	39.17	odd	6
845.2.e.e.191.2		4	39.23	odd	6
845.2.e.f.146.1		4	39.35	odd	6
845.2.e.f.191.1		4	39.29	odd	6
845.2.m.a.316.1		4	39.11	even	12
845.2.m.a.361.1		4	39.32	even	12
845.2.m.c.316.1		4	39.2	even	12
845.2.m.c.361.1		4	39.20	even	12
1040.2.a.h.1.1		2	156.155	even	2
2925.2.a.z.1.1		2	65.64	even	2
2925.2.c.v.2224.1		4	65.38	odd	4
2925.2.c.v.2224.4		4	65.12	odd	4
3185.2.a.k.1.1		2	273.272	even	2
4160.2.a.y.1.1		2	312.77	odd	2
4160.2.a.bj.1.2		2	312.155	even	2
4225.2.a.w.1.1		2	15.14	odd	2
5200.2.a.ca.1.2		2	780.779	even	2
7605.2.a.be.1.1		2	1.1	even	1	trivial
7865.2.a.h.1.2		2	429.428	even	2
9360.2.a.cm.1.1		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)

\(f(q)\)	\(=\)	\(q-1.73205 q^{2} +1.00000 q^{4} -1.00000 q^{5} -2.00000 q^{7} +1.73205 q^{8} +O(q^{10})\)	\(q-1.73205 q^{2} +1.00000 q^{4} -1.00000 q^{5} -2.00000 q^{7} +1.73205 q^{8} +1.73205 q^{10} -4.73205 q^{11} +3.46410 q^{14} -5.00000 q^{16} +3.46410 q^{17} +6.19615 q^{19} -1.00000 q^{20} +8.19615 q^{22} -1.26795 q^{23} +1.00000 q^{25} -2.00000 q^{28} +2.53590 q^{29} -10.1962 q^{31} +5.19615 q^{32} -6.00000 q^{34} +2.00000 q^{35} +4.00000 q^{37} -10.7321 q^{38} -1.73205 q^{40} +3.46410 q^{41} -0.196152 q^{43} -4.73205 q^{44} +2.19615 q^{46} +6.00000 q^{47} -3.00000 q^{49} -1.73205 q^{50} -10.3923 q^{53} +4.73205 q^{55} -3.46410 q^{56} -4.39230 q^{58} +9.12436 q^{59} -8.39230 q^{61} +17.6603 q^{62} +1.00000 q^{64} -6.39230 q^{67} +3.46410 q^{68} -3.46410 q^{70} +4.73205 q^{71} +4.00000 q^{73} -6.92820 q^{74} +6.19615 q^{76} +9.46410 q^{77} -8.39230 q^{79} +5.00000 q^{80} -6.00000 q^{82} -6.00000 q^{83} -3.46410 q^{85} +0.339746 q^{86} -8.19615 q^{88} -12.9282 q^{89} -1.26795 q^{92} -10.3923 q^{94} -6.19615 q^{95} -2.00000 q^{97} +5.19615 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{4} - 2 q^{5} - 4 q^{7}+O(q^{10}) \) 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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more