LMFDB - Embedded newform 3680.2.a.q.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3680,2,Mod(1,3680)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3680, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("3680.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3680 = 2^{5} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3680.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:	$29.3849479438$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.1573.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 7x + 2 $ x^3 - x^2 - 7*x + 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$3.06871$ of defining polynomial
Character	$\chi$	$=$	3680.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.06871 q^{3} +1.00000 q^{5} +4.06871 q^{7} +6.41697 q^{9} -3.06871 q^{11} +5.41697 q^{13} -3.06871 q^{15} +1.72045 q^{17} +5.41697 q^{19} -12.4857 q^{21} +1.00000 q^{23} +1.00000 q^{25} -10.4857 q^{27} +7.34826 q^{29} +8.06871 q^{31} +9.41697 q^{33} +4.06871 q^{35} -0.789156 q^{37} -16.6231 q^{39} +11.8579 q^{41} -4.00000 q^{43} +6.41697 q^{45} -1.44090 q^{47} +9.55438 q^{49} -5.27955 q^{51} -1.48568 q^{53} -3.06871 q^{55} -16.6231 q^{57} -9.62309 q^{59} -12.0401 q^{61} +26.1088 q^{63} +5.41697 q^{65} +0.514324 q^{67} -3.06871 q^{69} -13.1135 q^{71} +10.1374 q^{73} -3.06871 q^{75} -12.4857 q^{77} -8.00000 q^{79} +12.9266 q^{81} +9.48568 q^{83} +1.72045 q^{85} -22.5497 q^{87} -10.9714 q^{89} +22.0401 q^{91} -24.7605 q^{93} +5.41697 q^{95} -5.20612 q^{97} -19.6918 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - q^{3} + 3 q^{5} + 4 q^{7} + 6 q^{9} - q^{11} + 3 q^{13} - q^{15} + 2 q^{17} + 3 q^{19} - 16 q^{21} + 3 q^{23} + 3 q^{25} - 10 q^{27} + 17 q^{29} + 16 q^{31} + 15 q^{33} + 4 q^{35} + 9 q^{37} - 12 q^{39}+ \cdots - 13 q^{99}+O(q^{100}) $ 3 * q - q^3 + 3 * q^5 + 4 * q^7 + 6 * q^9 - q^11 + 3 * q^13 - q^15 + 2 * q^17 + 3 * q^19 - 16 * q^21 + 3 * q^23 + 3 * q^25 - 10 * q^27 + 17 * q^29 + 16 * q^31 + 15 * q^33 + 4 * q^35 + 9 * q^37 - 12 * q^39 + 16 * q^41 - 12 * q^43 + 6 * q^45 + 2 * q^47 - q^49 - 19 * q^51 + 17 * q^53 - q^55 - 12 * q^57 + 9 * q^59 + 15 * q^61 + 19 * q^63 + 3 * q^65 + 23 * q^67 - q^69 - 16 * q^71 + 14 * q^73 - q^75 - 16 * q^77 - 24 * q^79 + 11 * q^81 + 7 * q^83 + 2 * q^85 - 2 * q^87 + 10 * q^89 + 15 * q^91 - 20 * q^93 + 3 * q^95 + 9 * q^97 - 13 * 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 $

$2$	0	0
$2$	0	0
$3$	−3.06871	−1.77172	−0.885860	−	0.463953i	$-0.846431\pi$
$3$	−3.06871	−1.77172	−0.885860	+	0.463953i	$0.846431\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	4.06871	1.53783	0.768914	−	0.639353i	$-0.220798\pi$
$7$	4.06871	1.53783	0.768914	+	0.639353i	$0.220798\pi$
$8$	0	0
$9$	6.41697	2.13899
$10$	0	0
$11$	−3.06871	−0.925250	−0.462625	−	0.886554i	$-0.653092\pi$
$11$	−3.06871	−0.925250	−0.462625	+	0.886554i	$0.653092\pi$
$12$	0	0
$13$	5.41697	1.50240	0.751198	−	0.660077i	$-0.229476\pi$
$13$	5.41697	1.50240	0.751198	+	0.660077i	$0.229476\pi$
$14$	0	0
$15$	−3.06871	−0.792337
$16$	0	0
$17$	1.72045	0.417270	0.208635	−	0.977994i	$-0.433098\pi$
$17$	1.72045	0.417270	0.208635	+	0.977994i	$0.433098\pi$
$18$	0	0
$19$	5.41697	1.24274	0.621369	−	0.783518i	$-0.286577\pi$
$19$	5.41697	1.24274	0.621369	+	0.783518i	$0.286577\pi$
$20$	0	0
$21$	−12.4857	−2.72460
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−10.4857	−2.01797
$28$	0	0
$29$	7.34826	1.36454	0.682269	−	0.731101i	$-0.260993\pi$
$29$	7.34826	1.36454	0.682269	+	0.731101i	$0.260993\pi$
$30$	0	0
$31$	8.06871	1.44918	0.724591	−	0.689179i	$-0.242029\pi$
$31$	8.06871	1.44918	0.724591	+	0.689179i	$0.242029\pi$
$32$	0	0
$33$	9.41697	1.63928
$34$	0	0
$35$	4.06871	0.687737
$36$	0	0
$37$	−0.789156	−0.129736	−0.0648682	−	0.997894i	$-0.520663\pi$
$37$	−0.789156	−0.129736	−0.0648682	+	0.997894i	$0.520663\pi$
$38$	0	0
$39$	−16.6231	−2.66182
$40$	0	0
$41$	11.8579	1.85189	0.925944	−	0.377662i	$-0.123272\pi$
$41$	11.8579	1.85189	0.925944	+	0.377662i	$0.123272\pi$
$42$	0	0
$43$	−4.00000	−0.609994	−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000	−0.609994	−0.304997	+	0.952353i	$0.598656\pi$
$44$	0	0
$45$	6.41697	0.956585
$46$	0	0
$47$	−1.44090	−0.210176	−0.105088	−	0.994463i	$-0.533512\pi$
$47$	−1.44090	−0.210176	−0.105088	+	0.994463i	$0.533512\pi$
$48$	0	0
$49$	9.55438	1.36491
$50$	0	0
$51$	−5.27955	−0.739285
$52$	0	0
$53$	−1.48568	−0.204073	−0.102037	−	0.994781i	$-0.532536\pi$
$53$	−1.48568	−0.204073	−0.102037	+	0.994781i	$0.532536\pi$
$54$	0	0
$55$	−3.06871	−0.413784
$56$	0	0
$57$	−16.6231	−2.20178
$58$	0	0
$59$	−9.62309	−1.25282	−0.626410	−	0.779494i	$-0.715476\pi$
$59$	−9.62309	−1.25282	−0.626410	+	0.779494i	$0.715476\pi$
$60$	0	0
$61$	−12.0401	−1.54157	−0.770786	−	0.637094i	$-0.780136\pi$
$61$	−12.0401	−1.54157	−0.770786	+	0.637094i	$0.780136\pi$
$62$	0	0
$63$	26.1088	3.28940
$64$	0	0
$65$	5.41697	0.671892
$66$	0	0
$67$	0.514324	0.0628347	0.0314174	−	0.999506i	$-0.489998\pi$
$67$	0.514324	0.0628347	0.0314174	+	0.999506i	$0.489998\pi$
$68$	0	0
$69$	−3.06871	−0.369429
$70$	0	0
$71$	−13.1135	−1.55628	−0.778142	−	0.628088i	$-0.783838\pi$
$71$	−13.1135	−1.55628	−0.778142	+	0.628088i	$0.783838\pi$
$72$	0	0
$73$	10.1374	1.18649	0.593247	−	0.805020i	$-0.297846\pi$
$73$	10.1374	1.18649	0.593247	+	0.805020i	$0.297846\pi$
$74$	0	0
$75$	−3.06871	−0.354344
$76$	0	0
$77$	−12.4857	−1.42287
$78$	0	0
$79$	−8.00000	−0.900070	−0.450035	−	0.893011i	$-0.648589\pi$
$79$	−8.00000	−0.900070	−0.450035	+	0.893011i	$0.648589\pi$
$80$	0	0
$81$	12.9266	1.43629
$82$	0	0
$83$	9.48568	1.04119	0.520594	−	0.853804i	$-0.325711\pi$
$83$	9.48568	1.04119	0.520594	+	0.853804i	$0.325711\pi$
$84$	0	0
$85$	1.72045	0.186609
$86$	0	0
$87$	−22.5497	−2.41758
$88$	0	0
$89$	−10.9714	−1.16296	−0.581480	−	0.813560i	$-0.697526\pi$
$89$	−10.9714	−1.16296	−0.581480	+	0.813560i	$0.697526\pi$
$90$	0	0
$91$	22.0401	2.31043
$92$	0	0
$93$	−24.7605	−2.56754
$94$	0	0
$95$	5.41697	0.555769
$96$	0	0
$97$	−5.20612	−0.528602	−0.264301	−	0.964440i	$-0.585141\pi$
$97$	−5.20612	−0.528602	−0.264301	+	0.964440i	$0.585141\pi$
$98$	0	0
$99$	−19.6918	−1.97910
$100$	0	0
$101$	−14.0448	−1.39751	−0.698754	−	0.715362i	$-0.746262\pi$
$101$	−14.0448	−1.39751	−0.698754	+	0.715362i	$0.746262\pi$
$102$	0	0
$103$	11.2061	1.10417	0.552086	−	0.833787i	$-0.313832\pi$
$103$	11.2061	1.10417	0.552086	+	0.833787i	$0.313832\pi$
$104$	0	0
$105$	−12.4857	−1.21848
$106$	0	0
$107$	−12.6517	−1.22309	−0.611545	−	0.791210i	$-0.709452\pi$
$107$	−12.6517	−1.22309	−0.611545	+	0.791210i	$0.709452\pi$
$108$	0	0
$109$	−13.4170	−1.28511	−0.642556	−	0.766239i	$-0.722126\pi$
$109$	−13.4170	−1.28511	−0.642556	+	0.766239i	$0.722126\pi$
$110$	0	0
$111$	2.42169	0.229856
$112$	0	0
$113$	10.7892	1.01496	0.507479	−	0.861664i	$-0.330577\pi$
$113$	10.7892	1.01496	0.507479	+	0.861664i	$0.330577\pi$
$114$	0	0
$115$	1.00000	0.0932505
$116$	0	0
$117$	34.7605	3.21361
$118$	0	0
$119$	7.00000	0.641689
$120$	0	0
$121$	−1.58303	−0.143912
$122$	0	0
$123$	−36.3883	−3.28102
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	0.833935	0.0739998	0.0369999	−	0.999315i	$-0.488220\pi$
$127$	0.833935	0.0739998	0.0369999	+	0.999315i	$0.488220\pi$
$128$	0	0
$129$	12.2748	1.08074
$130$	0	0
$131$	0.696520	0.0608552	0.0304276	−	0.999537i	$-0.490313\pi$
$131$	0.696520	0.0608552	0.0304276	+	0.999537i	$0.490313\pi$
$132$	0	0
$133$	22.0401	1.91112
$134$	0	0
$135$	−10.4857	−0.902463
$136$	0	0
$137$	16.8579	1.44026	0.720132	−	0.693837i	$-0.244081\pi$
$137$	16.8579	1.44026	0.720132	+	0.693837i	$0.244081\pi$
$138$	0	0
$139$	14.7413	1.25034	0.625170	−	0.780488i	$-0.285029\pi$
$139$	14.7413	1.25034	0.625170	+	0.780488i	$0.285029\pi$
$140$	0	0
$141$	4.42169	0.372373
$142$	0	0
$143$	−16.6231	−1.39009
$144$	0	0
$145$	7.34826	0.610240
$146$	0	0
$147$	−29.3196	−2.41824
$148$	0	0
$149$	8.11349	0.664683	0.332341	−	0.943159i	$-0.392161\pi$
$149$	8.11349	0.664683	0.332341	+	0.943159i	$0.392161\pi$
$150$	0	0
$151$	11.9505	0.972518	0.486259	−	0.873815i	$-0.338361\pi$
$151$	11.9505	0.972518	0.486259	+	0.873815i	$0.338361\pi$
$152$	0	0
$153$	11.0401	0.892536
$154$	0	0
$155$	8.06871	0.648094
$156$	0	0
$157$	19.3483	1.54416	0.772080	−	0.635526i	$-0.219217\pi$
$157$	19.3483	1.54416	0.772080	+	0.635526i	$0.219217\pi$
$158$	0	0
$159$	4.55910	0.361560
$160$	0	0
$161$	4.06871	0.320659
$162$	0	0
$163$	8.16134	0.639246	0.319623	−	0.947545i	$-0.396444\pi$
$163$	8.16134	0.639246	0.319623	+	0.947545i	$0.396444\pi$
$164$	0	0
$165$	9.41697	0.733110
$166$	0	0
$167$	−1.39304	−0.107797	−0.0538983	−	0.998546i	$-0.517165\pi$
$167$	−1.39304	−0.107797	−0.0538983	+	0.998546i	$0.517165\pi$
$168$	0	0
$169$	16.3435	1.25720
$170$	0	0
$171$	34.7605	2.65820
$172$	0	0
$173$	−2.37219	−0.180354	−0.0901771	−	0.995926i	$-0.528743\pi$
$173$	−2.37219	−0.180354	−0.0901771	+	0.995926i	$0.528743\pi$
$174$	0	0
$175$	4.06871	0.307565
$176$	0	0
$177$	29.5305	2.21964
$178$	0	0
$179$	−18.8339	−1.40771	−0.703857	−	0.710341i	$-0.748541\pi$
$179$	−18.8339	−1.40771	−0.703857	+	0.710341i	$0.748541\pi$
$180$	0	0
$181$	−6.99528	−0.519955	−0.259978	−	0.965615i	$-0.583715\pi$
$181$	−6.99528	−0.519955	−0.259978	+	0.965615i	$0.583715\pi$
$182$	0	0
$183$	36.9474	2.73123
$184$	0	0
$185$	−0.789156	−0.0580199
$186$	0	0
$187$	−5.27955	−0.386079
$188$	0	0
$189$	−42.6632	−3.10329
$190$	0	0
$191$	−1.86258	−0.134772	−0.0673859	−	0.997727i	$-0.521466\pi$
$191$	−1.86258	−0.134772	−0.0673859	+	0.997727i	$0.521466\pi$
$192$	0	0
$193$	−11.7157	−0.843316	−0.421658	−	0.906755i	$-0.638552\pi$
$193$	−11.7157	−0.843316	−0.421658	+	0.906755i	$0.638552\pi$
$194$	0	0
$195$	−16.6231	−1.19040
$196$	0	0
$197$	−14.1775	−1.01010	−0.505052	−	0.863089i	$-0.668527\pi$
$197$	−14.1775	−1.01010	−0.505052	+	0.863089i	$0.668527\pi$
$198$	0	0
$199$	−20.9714	−1.48662	−0.743310	−	0.668947i	$-0.766745\pi$
$199$	−20.9714	−1.48662	−0.743310	+	0.668947i	$0.766745\pi$
$200$	0	0
$201$	−1.57831	−0.111326
$202$	0	0
$203$	29.8979	2.09842
$204$	0	0
$205$	11.8579	0.828189
$206$	0	0
$207$	6.41697	0.446010
$208$	0	0
$209$	−16.6231	−1.14984
$210$	0	0
$211$	23.1535	1.59396	0.796978	−	0.604008i	$-0.206431\pi$
$211$	23.1535	1.59396	0.796978	+	0.604008i	$0.206431\pi$
$212$	0	0
$213$	40.2415	2.75730
$214$	0	0
$215$	−4.00000	−0.272798
$216$	0	0
$217$	32.8292	2.22859
$218$	0	0
$219$	−31.1088	−2.10214
$220$	0	0
$221$	9.31961	0.626905
$222$	0	0
$223$	10.8818	0.728699	0.364349	−	0.931262i	$-0.381291\pi$
$223$	10.8818	0.728699	0.364349	+	0.931262i	$0.381291\pi$
$224$	0	0
$225$	6.41697	0.427798
$226$	0	0
$227$	19.1088	1.26829	0.634147	−	0.773213i	$-0.281352\pi$
$227$	19.1088	1.26829	0.634147	+	0.773213i	$0.281352\pi$
$228$	0	0
$229$	26.5497	1.75445	0.877226	−	0.480078i	$-0.159392\pi$
$229$	26.5497	1.75445	0.877226	+	0.480078i	$0.159392\pi$
$230$	0	0
$231$	38.3149	2.52093
$232$	0	0
$233$	−10.1374	−0.664124	−0.332062	−	0.943258i	$-0.607744\pi$
$233$	−10.1374	−0.664124	−0.332062	+	0.943258i	$0.607744\pi$
$234$	0	0
$235$	−1.44090	−0.0939937
$236$	0	0
$237$	24.5497	1.59467
$238$	0	0
$239$	−25.1535	−1.62705	−0.813524	−	0.581532i	$-0.802454\pi$
$239$	−25.1535	−1.62705	−0.813524	+	0.581532i	$0.802454\pi$
$240$	0	0
$241$	−21.6679	−1.39575	−0.697875	−	0.716219i	$-0.745871\pi$
$241$	−21.6679	−1.39575	−0.697875	+	0.716219i	$0.745871\pi$
$242$	0	0
$243$	−8.21084	−0.526726
$244$	0	0
$245$	9.55438	0.610407
$246$	0	0
$247$	29.3435	1.86708
$248$	0	0
$249$	−29.1088	−1.84469
$250$	0	0
$251$	8.52573	0.538140	0.269070	−	0.963121i	$-0.413284\pi$
$251$	8.52573	0.538140	0.269070	+	0.963121i	$0.413284\pi$
$252$	0	0
$253$	−3.06871	−0.192928
$254$	0	0
$255$	−5.27955	−0.330618
$256$	0	0
$257$	−27.5305	−1.71730	−0.858651	−	0.512560i	$-0.828697\pi$
$257$	−27.5305	−1.71730	−0.858651	+	0.512560i	$0.828697\pi$
$258$	0	0
$259$	−3.21084	−0.199512
$260$	0	0
$261$	47.1535	2.91873
$262$	0	0
$263$	−13.3883	−0.825559	−0.412780	−	0.910831i	$-0.635442\pi$
$263$	−13.3883	−0.825559	−0.412780	+	0.910831i	$0.635442\pi$
$264$	0	0
$265$	−1.48568	−0.0912643
$266$	0	0
$267$	33.6679	2.06044
$268$	0	0
$269$	−11.6231	−0.708672	−0.354336	−	0.935118i	$-0.615293\pi$
$269$	−11.6231	−0.708672	−0.354336	+	0.935118i	$0.615293\pi$
$270$	0	0
$271$	9.64702	0.586015	0.293007	−	0.956110i	$-0.405344\pi$
$271$	9.64702	0.586015	0.293007	+	0.956110i	$0.405344\pi$
$272$	0	0
$273$	−67.6345	−4.09343
$274$	0	0
$275$	−3.06871	−0.185050
$276$	0	0
$277$	−3.39304	−0.203868	−0.101934	−	0.994791i	$-0.532503\pi$
$277$	−3.39304	−0.203868	−0.101934	+	0.994791i	$0.532503\pi$
$278$	0	0
$279$	51.7766	3.09979
$280$	0	0
$281$	−2.74438	−0.163716	−0.0818579	−	0.996644i	$-0.526085\pi$
$281$	−2.74438	−0.163716	−0.0818579	+	0.996644i	$0.526085\pi$
$282$	0	0
$283$	−13.2108	−0.785303	−0.392652	−	0.919687i	$-0.628442\pi$
$283$	−13.2108	−0.785303	−0.392652	+	0.919687i	$0.628442\pi$
$284$	0	0
$285$	−16.6231	−0.984667
$286$	0	0
$287$	48.2462	2.84788
$288$	0	0
$289$	−14.0401	−0.825886
$290$	0	0
$291$	15.9761	0.936534
$292$	0	0
$293$	12.0926	0.706459	0.353230	−	0.935537i	$-0.385083\pi$
$293$	12.0926	0.706459	0.353230	+	0.935537i	$0.385083\pi$
$294$	0	0
$295$	−9.62309	−0.560278
$296$	0	0
$297$	32.1775	1.86713
$298$	0	0
$299$	5.41697	0.313271
$300$	0	0
$301$	−16.2748	−0.938066
$302$	0	0
$303$	43.0993	2.47599
$304$	0	0
$305$	−12.0401	−0.689412
$306$	0	0
$307$	19.3435	1.10399	0.551997	−	0.833846i	$-0.313866\pi$
$307$	19.3435	1.10399	0.551997	+	0.833846i	$0.313866\pi$
$308$	0	0
$309$	−34.3883	−1.95628
$310$	0	0
$311$	−21.9427	−1.24426	−0.622128	−	0.782915i	$-0.713732\pi$
$311$	−21.9427	−1.24426	−0.622128	+	0.782915i	$0.713732\pi$
$312$	0	0
$313$	9.23477	0.521980	0.260990	−	0.965341i	$-0.415951\pi$
$313$	9.23477	0.521980	0.260990	+	0.965341i	$0.415951\pi$
$314$	0	0
$315$	26.1088	1.47106
$316$	0	0
$317$	26.9953	1.51621	0.758103	−	0.652135i	$-0.226126\pi$
$317$	26.9953	1.51621	0.758103	+	0.652135i	$0.226126\pi$
$318$	0	0
$319$	−22.5497	−1.26254
$320$	0	0
$321$	38.8245	2.16697
$322$	0	0
$323$	9.31961	0.518557
$324$	0	0
$325$	5.41697	0.300479
$326$	0	0
$327$	41.1728	2.27686
$328$	0	0
$329$	−5.86258	−0.323215
$330$	0	0
$331$	9.53353	0.524010	0.262005	−	0.965066i	$-0.415616\pi$
$331$	9.53353	0.524010	0.262005	+	0.965066i	$0.415616\pi$
$332$	0	0
$333$	−5.06399	−0.277505
$334$	0	0
$335$	0.514324	0.0281005
$336$	0	0
$337$	8.58303	0.467548	0.233774	−	0.972291i	$-0.424892\pi$
$337$	8.58303	0.467548	0.233774	+	0.972291i	$0.424892\pi$
$338$	0	0
$339$	−33.1088	−1.79822
$340$	0	0
$341$	−24.7605	−1.34086
$342$	0	0
$343$	10.3930	0.561171
$344$	0	0
$345$	−3.06871	−0.165214
$346$	0	0
$347$	1.20612	0.0647481	0.0323741	−	0.999476i	$-0.489693\pi$
$347$	1.20612	0.0647481	0.0323741	+	0.999476i	$0.489693\pi$
$348$	0	0
$349$	−28.4570	−1.52327	−0.761635	−	0.648006i	$-0.775603\pi$
$349$	−28.4570	−1.52327	−0.761635	+	0.648006i	$0.775603\pi$
$350$	0	0
$351$	−56.8006	−3.03179
$352$	0	0
$353$	0	0		−	1.00000i	$-0.5\pi$
$353$	0	0			1.00000i	$0.5\pi$
$354$	0	0
$355$	−13.1135	−0.695992
$356$	0	0
$357$	−21.4810	−1.13689
$358$	0	0
$359$	24.7861	1.30816	0.654080	−	0.756426i	$-0.273056\pi$
$359$	24.7861	1.30816	0.654080	+	0.756426i	$0.273056\pi$
$360$	0	0
$361$	10.3435	0.544397
$362$	0	0
$363$	4.85786	0.254972
$364$	0	0
$365$	10.1374	0.530617
$366$	0	0
$367$	11.7605	0.613893	0.306947	−	0.951727i	$-0.400693\pi$
$367$	11.7605	0.613893	0.306947	+	0.951727i	$0.400693\pi$
$368$	0	0
$369$	76.0915	3.96117
$370$	0	0
$371$	−6.04478	−0.313829
$372$	0	0
$373$	29.3836	1.52143	0.760713	−	0.649089i	$-0.224850\pi$
$373$	29.3836	1.52143	0.760713	+	0.649089i	$0.224850\pi$
$374$	0	0
$375$	−3.06871	−0.158467
$376$	0	0
$377$	39.8053	2.05008
$378$	0	0
$379$	15.0687	0.774028	0.387014	−	0.922074i	$-0.373507\pi$
$379$	15.0687	0.774028	0.387014	+	0.922074i	$0.373507\pi$
$380$	0	0
$381$	−2.55910	−0.131107
$382$	0	0
$383$	7.48568	0.382500	0.191250	−	0.981541i	$-0.438746\pi$
$383$	7.48568	0.382500	0.191250	+	0.981541i	$0.438746\pi$
$384$	0	0
$385$	−12.4857	−0.636329
$386$	0	0
$387$	−25.6679	−1.30477
$388$	0	0
$389$	28.3149	1.43562	0.717811	−	0.696238i	$-0.245144\pi$
$389$	28.3149	1.43562	0.717811	+	0.696238i	$0.245144\pi$
$390$	0	0
$391$	1.72045	0.0870068
$392$	0	0
$393$	−2.13742	−0.107818
$394$	0	0
$395$	−8.00000	−0.402524
$396$	0	0
$397$	−12.7844	−0.641632	−0.320816	−	0.947141i	$-0.603957\pi$
$397$	−12.7844	−0.641632	−0.320816	+	0.947141i	$0.603957\pi$
$398$	0	0
$399$	−67.6345	−3.38596
$400$	0	0
$401$	12.8339	0.640896	0.320448	−	0.947266i	$-0.396167\pi$
$401$	12.8339	0.640896	0.320448	+	0.947266i	$0.396167\pi$
$402$	0	0
$403$	43.7079	2.17725
$404$	0	0
$405$	12.9266	0.642326
$406$	0	0
$407$	2.42169	0.120039
$408$	0	0
$409$	−21.4617	−1.06122	−0.530608	−	0.847618i	$-0.678036\pi$
$409$	−21.4617	−1.06122	−0.530608	+	0.847618i	$0.678036\pi$
$410$	0	0
$411$	−51.7319	−2.55174
$412$	0	0
$413$	−39.1535	−1.92662
$414$	0	0
$415$	9.48568	0.465633
$416$	0	0
$417$	−45.2367	−2.21525
$418$	0	0
$419$	1.30348	0.0636792	0.0318396	−	0.999493i	$-0.489863\pi$
$419$	1.30348	0.0636792	0.0318396	+	0.999493i	$0.489863\pi$
$420$	0	0
$421$	−14.7204	−0.717431	−0.358715	−	0.933447i	$-0.616785\pi$
$421$	−14.7204	−0.717431	−0.358715	+	0.933447i	$0.616785\pi$
$422$	0	0
$423$	−9.24618	−0.449565
$424$	0	0
$425$	1.72045	0.0834540
$426$	0	0
$427$	−48.9875	−2.37067
$428$	0	0
$429$	51.0114	2.46285
$430$	0	0
$431$	29.8053	1.43567	0.717835	−	0.696213i	$-0.245133\pi$
$431$	29.8053	1.43567	0.717835	+	0.696213i	$0.245133\pi$
$432$	0	0
$433$	−16.3914	−0.787720	−0.393860	−	0.919170i	$-0.628861\pi$
$433$	−16.3914	−0.787720	−0.393860	+	0.919170i	$0.628861\pi$
$434$	0	0
$435$	−22.5497	−1.08117
$436$	0	0
$437$	5.41697	0.259129
$438$	0	0
$439$	32.9474	1.57249	0.786247	−	0.617912i	$-0.212021\pi$
$439$	32.9474	1.57249	0.786247	+	0.617912i	$0.212021\pi$
$440$	0	0
$441$	61.3102	2.91953
$442$	0	0
$443$	−13.5544	−0.643988	−0.321994	−	0.946742i	$-0.604353\pi$
$443$	−13.5544	−0.643988	−0.321994	+	0.946742i	$0.604353\pi$
$444$	0	0
$445$	−10.9714	−0.520092
$446$	0	0
$447$	−24.8979	−1.17763
$448$	0	0
$449$	−20.4810	−0.966556	−0.483278	−	0.875467i	$-0.660554\pi$
$449$	−20.4810	−0.966556	−0.483278	+	0.875467i	$0.660554\pi$
$450$	0	0
$451$	−36.3883	−1.71346
$452$	0	0
$453$	−36.6726	−1.72303
$454$	0	0
$455$	22.0401	1.03325
$456$	0	0
$457$	−6.78916	−0.317583	−0.158792	−	0.987312i	$-0.550760\pi$
$457$	−6.78916	−0.317583	−0.158792	+	0.987312i	$0.550760\pi$
$458$	0	0
$459$	−18.0401	−0.842038
$460$	0	0
$461$	−8.83394	−0.411437	−0.205719	−	0.978611i	$-0.565953\pi$
$461$	−8.83394	−0.411437	−0.205719	+	0.978611i	$0.565953\pi$
$462$	0	0
$463$	−16.9714	−0.788726	−0.394363	−	0.918955i	$-0.629035\pi$
$463$	−16.9714	−0.788726	−0.394363	+	0.918955i	$0.629035\pi$
$464$	0	0
$465$	−24.7605	−1.14824
$466$	0	0
$467$	−2.83701	−0.131281	−0.0656406	−	0.997843i	$-0.520909\pi$
$467$	−2.83701	−0.131281	−0.0656406	+	0.997843i	$0.520909\pi$
$468$	0	0
$469$	2.09264	0.0966290
$470$	0	0
$471$	−59.3742	−2.73582
$472$	0	0
$473$	12.2748	0.564397
$474$	0	0
$475$	5.41697	0.248548
$476$	0	0
$477$	−9.53353	−0.436510
$478$	0	0
$479$	9.24618	0.422469	0.211234	−	0.977435i	$-0.432252\pi$
$479$	9.24618	0.422469	0.211234	+	0.977435i	$0.432252\pi$
$480$	0	0
$481$	−4.27483	−0.194916
$482$	0	0
$483$	−12.4857	−0.568118
$484$	0	0
$485$	−5.20612	−0.236398
$486$	0	0
$487$	−3.44090	−0.155922	−0.0779609	−	0.996956i	$-0.524841\pi$
$487$	−3.44090	−0.155922	−0.0779609	+	0.996956i	$0.524841\pi$
$488$	0	0
$489$	−25.0448	−1.13256
$490$	0	0
$491$	−15.0734	−0.680254	−0.340127	−	0.940379i	$-0.610470\pi$
$491$	−15.0734	−0.680254	−0.340127	+	0.940379i	$0.610470\pi$
$492$	0	0
$493$	12.6423	0.569380
$494$	0	0
$495$	−19.6918	−0.885081
$496$	0	0
$497$	−53.3549	−2.39330
$498$	0	0
$499$	−14.7319	−0.659489	−0.329744	−	0.944070i	$-0.606963\pi$
$499$	−14.7319	−0.659489	−0.329744	+	0.944070i	$0.606963\pi$
$500$	0	0
$501$	4.27483	0.190985
$502$	0	0
$503$	29.9219	1.33415	0.667075	−	0.744991i	$-0.267546\pi$
$503$	29.9219	1.33415	0.667075	+	0.744991i	$0.267546\pi$
$504$	0	0
$505$	−14.0448	−0.624984
$506$	0	0
$507$	−50.1535	−2.22740
$508$	0	0
$509$	−22.7444	−1.00813	−0.504063	−	0.863667i	$-0.668162\pi$
$509$	−22.7444	−1.00813	−0.504063	+	0.863667i	$0.668162\pi$
$510$	0	0
$511$	41.2462	1.82462
$512$	0	0
$513$	−56.8006	−2.50781
$514$	0	0
$515$	11.2061	0.493801
$516$	0	0
$517$	4.42169	0.194466
$518$	0	0
$519$	7.27955	0.319537
$520$	0	0
$521$	−23.5783	−1.03298	−0.516492	−	0.856292i	$-0.672763\pi$
$521$	−23.5783	−1.03298	−0.516492	+	0.856292i	$0.672763\pi$
$522$	0	0
$523$	−11.3035	−0.494267	−0.247133	−	0.968981i	$-0.579489\pi$
$523$	−11.3035	−0.494267	−0.247133	+	0.968981i	$0.579489\pi$
$524$	0	0
$525$	−12.4857	−0.544920
$526$	0	0
$527$	13.8818	0.604700
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−61.7511	−2.67977
$532$	0	0
$533$	64.2337	2.78227
$534$	0	0
$535$	−12.6517	−0.546982
$536$	0	0
$537$	57.7958	2.49407
$538$	0	0
$539$	−29.3196	−1.26289
$540$	0	0
$541$	11.8147	0.507955	0.253977	−	0.967210i	$-0.418261\pi$
$541$	11.8147	0.507955	0.253977	+	0.967210i	$0.418261\pi$
$542$	0	0
$543$	21.4665	0.921214
$544$	0	0
$545$	−13.4170	−0.574720
$546$	0	0
$547$	16.5736	0.708636	0.354318	−	0.935125i	$-0.384713\pi$
$547$	16.5736	0.708636	0.354318	+	0.935125i	$0.384713\pi$
$548$	0	0
$549$	−77.2607	−3.29741
$550$	0	0
$551$	39.8053	1.69576
$552$	0	0
$553$	−32.5497	−1.38415
$554$	0	0
$555$	2.42169	0.102795
$556$	0	0
$557$	−11.5752	−0.490458	−0.245229	−	0.969465i	$-0.578863\pi$
$557$	−11.5752	−0.490458	−0.245229	+	0.969465i	$0.578863\pi$
$558$	0	0
$559$	−21.6679	−0.916453
$560$	0	0
$561$	16.2014	0.684024
$562$	0	0
$563$	−14.1822	−0.597708	−0.298854	−	0.954299i	$-0.596604\pi$
$563$	−14.1822	−0.597708	−0.298854	+	0.954299i	$0.596604\pi$
$564$	0	0
$565$	10.7892	0.453903
$566$	0	0
$567$	52.5944	2.20876
$568$	0	0
$569$	25.0609	1.05061	0.525304	−	0.850915i	$-0.323952\pi$
$569$	25.0609	1.05061	0.525304	+	0.850915i	$0.323952\pi$
$570$	0	0
$571$	−23.0687	−0.965395	−0.482698	−	0.875787i	$-0.660343\pi$
$571$	−23.0687	−0.965395	−0.482698	+	0.875787i	$0.660343\pi$
$572$	0	0
$573$	5.71573	0.238778
$574$	0	0
$575$	1.00000	0.0417029
$576$	0	0
$577$	11.3930	0.474298	0.237149	−	0.971473i	$-0.423787\pi$
$577$	11.3930	0.474298	0.237149	+	0.971473i	$0.423787\pi$
$578$	0	0
$579$	35.9521	1.49412
$580$	0	0
$581$	38.5944	1.60117
$582$	0	0
$583$	4.55910	0.188819
$584$	0	0
$585$	34.7605	1.43717
$586$	0	0
$587$	23.9922	0.990264	0.495132	−	0.868818i	$-0.335120\pi$
$587$	23.9922	0.990264	0.495132	+	0.868818i	$0.335120\pi$
$588$	0	0
$589$	43.7079	1.80095
$590$	0	0
$591$	43.5065	1.78962
$592$	0	0
$593$	21.1088	0.866833	0.433417	−	0.901194i	$-0.357308\pi$
$593$	21.1088	0.866833	0.433417	+	0.901194i	$0.357308\pi$
$594$	0	0
$595$	7.00000	0.286972
$596$	0	0
$597$	64.3549	2.63387
$598$	0	0
$599$	24.8740	1.01632	0.508162	−	0.861262i	$-0.330325\pi$
$599$	24.8740	1.01632	0.508162	+	0.861262i	$0.330325\pi$
$600$	0	0
$601$	−25.5513	−1.04226	−0.521130	−	0.853477i	$-0.674489\pi$
$601$	−25.5513	−1.04226	−0.521130	+	0.853477i	$0.674489\pi$
$602$	0	0
$603$	3.30040	0.134403
$604$	0	0
$605$	−1.58303	−0.0643594
$606$	0	0
$607$	−22.9714	−0.932378	−0.466189	−	0.884685i	$-0.654373\pi$
$607$	−22.9714	−0.932378	−0.466189	+	0.884685i	$0.654373\pi$
$608$	0	0
$609$	−91.7480	−3.71782
$610$	0	0
$611$	−7.80529	−0.315768
$612$	0	0
$613$	−21.0609	−0.850642	−0.425321	−	0.905043i	$-0.639839\pi$
$613$	−21.0609	−0.850642	−0.425321	+	0.905043i	$0.639839\pi$
$614$	0	0
$615$	−36.3883	−1.46732
$616$	0	0
$617$	−45.5257	−1.83280	−0.916399	−	0.400267i	$-0.868917\pi$
$617$	−45.5257	−1.83280	−0.916399	+	0.400267i	$0.868917\pi$
$618$	0	0
$619$	−0.113487	−0.00456145	−0.00228072	−	0.999997i	$-0.500726\pi$
$619$	−0.113487	−0.00456145	−0.00228072	+	0.999997i	$0.500726\pi$
$620$	0	0
$621$	−10.4857	−0.420776
$622$	0	0
$623$	−44.6392	−1.78843
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	51.0114	2.03720
$628$	0	0
$629$	−1.35770	−0.0541351
$630$	0	0
$631$	14.2270	0.566367	0.283183	−	0.959066i	$-0.408609\pi$
$631$	14.2270	0.566367	0.283183	+	0.959066i	$0.408609\pi$
$632$	0	0
$633$	−71.0515	−2.82404
$634$	0	0
$635$	0.833935	0.0330937
$636$	0	0
$637$	51.7558	2.05064
$638$	0	0
$639$	−84.1488	−3.32888
$640$	0	0
$641$	3.81473	0.150673	0.0753363	−	0.997158i	$-0.475997\pi$
$641$	3.81473	0.150673	0.0753363	+	0.997158i	$0.475997\pi$
$642$	0	0
$643$	46.4092	1.83020	0.915100	−	0.403228i	$-0.132112\pi$
$643$	46.4092	1.83020	0.915100	+	0.403228i	$0.132112\pi$
$644$	0	0
$645$	12.2748	0.483321
$646$	0	0
$647$	−43.7574	−1.72028	−0.860141	−	0.510056i	$-0.829625\pi$
$647$	−43.7574	−1.72028	−0.860141	+	0.510056i	$0.829625\pi$
$648$	0	0
$649$	29.5305	1.15917
$650$	0	0
$651$	−100.743	−3.94844
$652$	0	0
$653$	22.0562	0.863125	0.431563	−	0.902083i	$-0.357962\pi$
$653$	22.0562	0.863125	0.431563	+	0.902083i	$0.357962\pi$
$654$	0	0
$655$	0.696520	0.0272153
$656$	0	0
$657$	65.0515	2.53790
$658$	0	0
$659$	6.88179	0.268077	0.134038	−	0.990976i	$-0.457205\pi$
$659$	6.88179	0.268077	0.134038	+	0.990976i	$0.457205\pi$
$660$	0	0
$661$	12.7844	0.497257	0.248628	−	0.968599i	$-0.420020\pi$
$661$	12.7844	0.497257	0.248628	+	0.968599i	$0.420020\pi$
$662$	0	0
$663$	−28.5992	−1.11070
$664$	0	0
$665$	22.0401	0.854677
$666$	0	0
$667$	7.34826	0.284526
$668$	0	0
$669$	−33.3930	−1.29105
$670$	0	0
$671$	36.9474	1.42634
$672$	0	0
$673$	−38.6392	−1.48943	−0.744716	−	0.667381i	$-0.767415\pi$
$673$	−38.6392	−1.48943	−0.744716	+	0.667381i	$0.767415\pi$
$674$	0	0
$675$	−10.4857	−0.403594
$676$	0	0
$677$	−31.3388	−1.20445	−0.602224	−	0.798327i	$-0.705719\pi$
$677$	−31.3388	−1.20445	−0.602224	+	0.798327i	$0.705719\pi$
$678$	0	0
$679$	−21.1822	−0.812898
$680$	0	0
$681$	−58.6392	−2.24706
$682$	0	0
$683$	−33.5065	−1.28209	−0.641046	−	0.767503i	$-0.721499\pi$
$683$	−33.5065	−1.28209	−0.641046	+	0.767503i	$0.721499\pi$
$684$	0	0
$685$	16.8579	0.644106
$686$	0	0
$687$	−81.4732	−3.10839
$688$	0	0
$689$	−8.04786	−0.306599
$690$	0	0
$691$	17.1661	0.653028	0.326514	−	0.945192i	$-0.394126\pi$
$691$	17.1661	0.653028	0.326514	+	0.945192i	$0.394126\pi$
$692$	0	0
$693$	−80.1202	−3.04351
$694$	0	0
$695$	14.7413	0.559169
$696$	0	0
$697$	20.4008	0.772737
$698$	0	0
$699$	31.1088	1.17664
$700$	0	0
$701$	−14.8579	−0.561174	−0.280587	−	0.959829i	$-0.590529\pi$
$701$	−14.8579	−0.561174	−0.280587	+	0.959829i	$0.590529\pi$
$702$	0	0
$703$	−4.27483	−0.161228
$704$	0	0
$705$	4.42169	0.166530
$706$	0	0
$707$	−57.1441	−2.14913
$708$	0	0
$709$	−1.35298	−0.0508123	−0.0254061	−	0.999677i	$-0.508088\pi$
$709$	−1.35298	−0.0508123	−0.0254061	+	0.999677i	$0.508088\pi$
$710$	0	0
$711$	−51.3357	−1.92524
$712$	0	0
$713$	8.06871	0.302175
$714$	0	0
$715$	−16.6231	−0.621668
$716$	0	0
$717$	77.1889	2.88267
$718$	0	0
$719$	−4.15827	−0.155077	−0.0775386	−	0.996989i	$-0.524706\pi$
$719$	−4.15827	−0.155077	−0.0775386	+	0.996989i	$0.524706\pi$
$720$	0	0
$721$	45.5944	1.69803
$722$	0	0
$723$	66.4924	2.47288
$724$	0	0
$725$	7.34826	0.272908
$726$	0	0
$727$	−23.8006	−0.882714	−0.441357	−	0.897332i	$-0.645503\pi$
$727$	−23.8006	−0.882714	−0.441357	+	0.897332i	$0.645503\pi$
$728$	0	0
$729$	−13.5830	−0.503075
$730$	0	0
$731$	−6.88179	−0.254532
$732$	0	0
$733$	−16.6423	−0.614697	−0.307349	−	0.951597i	$-0.599442\pi$
$733$	−16.6423	−0.614697	−0.307349	+	0.951597i	$0.599442\pi$
$734$	0	0
$735$	−29.3196	−1.08147
$736$	0	0
$737$	−1.57831	−0.0581379
$738$	0	0
$739$	−42.9171	−1.57873	−0.789366	−	0.613923i	$-0.789591\pi$
$739$	−42.9171	−1.57873	−0.789366	+	0.613923i	$0.789591\pi$
$740$	0	0
$741$	−90.0467	−3.30795
$742$	0	0
$743$	50.9858	1.87049	0.935244	−	0.354002i	$-0.115180\pi$
$743$	50.9858	1.87049	0.935244	+	0.354002i	$0.115180\pi$
$744$	0	0
$745$	8.11349	0.297255
$746$	0	0
$747$	60.8693	2.22709
$748$	0	0
$749$	−51.4762	−1.88090
$750$	0	0
$751$	39.8053	1.45252	0.726258	−	0.687422i	$-0.241258\pi$
$751$	39.8053	1.45252	0.726258	+	0.687422i	$0.241258\pi$
$752$	0	0
$753$	−26.1630	−0.953432
$754$	0	0
$755$	11.9505	0.434923
$756$	0	0
$757$	21.3388	0.775573	0.387786	−	0.921749i	$-0.373240\pi$
$757$	21.3388	0.775573	0.387786	+	0.921749i	$0.373240\pi$
$758$	0	0
$759$	9.41697	0.341814
$760$	0	0
$761$	33.0306	1.19736	0.598679	−	0.800989i	$-0.295692\pi$
$761$	33.0306	1.19736	0.598679	+	0.800989i	$0.295692\pi$
$762$	0	0
$763$	−54.5897	−1.97628
$764$	0	0
$765$	11.0401	0.399154
$766$	0	0
$767$	−52.1280	−1.88223
$768$	0	0
$769$	−13.9104	−0.501623	−0.250812	−	0.968036i	$-0.580697\pi$
$769$	−13.9104	−0.501623	−0.250812	+	0.968036i	$0.580697\pi$
$770$	0	0
$771$	84.4829	3.04258
$772$	0	0
$773$	−6.14686	−0.221087	−0.110544	−	0.993871i	$-0.535259\pi$
$773$	−6.14686	−0.221087	−0.110544	+	0.993871i	$0.535259\pi$
$774$	0	0
$775$	8.06871	0.289837
$776$	0	0
$777$	9.85314	0.353480
$778$	0	0
$779$	64.2337	2.30141
$780$	0	0
$781$	40.2415	1.43995
$782$	0	0
$783$	−77.0515	−2.75359
$784$	0	0
$785$	19.3483	0.690569
$786$	0	0
$787$	47.1057	1.67914	0.839568	−	0.543254i	$-0.182808\pi$
$787$	47.1057	1.67914	0.839568	+	0.543254i	$0.182808\pi$
$788$	0	0
$789$	41.0848	1.46266
$790$	0	0
$791$	43.8979	1.56083
$792$	0	0
$793$	−65.2206	−2.31605
$794$	0	0
$795$	4.55910	0.161695
$796$	0	0
$797$	−46.1728	−1.63552	−0.817761	−	0.575557i	$-0.804785\pi$
$797$	−46.1728	−1.63552	−0.817761	+	0.575557i	$0.804785\pi$
$798$	0	0
$799$	−2.47899	−0.0877002
$800$	0	0
$801$	−70.4028	−2.48756
$802$	0	0
$803$	−31.1088	−1.09780
$804$	0	0
$805$	4.06871	0.143403
$806$	0	0
$807$	35.6679	1.25557
$808$	0	0
$809$	0.966630	0.0339849	0.0169925	−	0.999856i	$-0.494591\pi$
$809$	0.966630	0.0339849	0.0169925	+	0.999856i	$0.494591\pi$
$810$	0	0
$811$	−30.3196	−1.06467	−0.532333	−	0.846535i	$-0.678684\pi$
$811$	−30.3196	−1.06467	−0.532333	+	0.846535i	$0.678684\pi$
$812$	0	0
$813$	−29.6039	−1.03825
$814$	0	0
$815$	8.16134	0.285879
$816$	0	0
$817$	−21.6679	−0.758063
$818$	0	0
$819$	141.430	4.94198
$820$	0	0
$821$	7.39304	0.258019	0.129009	−	0.991643i	$-0.458820\pi$
$821$	7.39304	0.258019	0.129009	+	0.991643i	$0.458820\pi$
$822$	0	0
$823$	−49.6106	−1.72932	−0.864658	−	0.502361i	$-0.832465\pi$
$823$	−49.6106	−1.72932	−0.864658	+	0.502361i	$0.832465\pi$
$824$	0	0
$825$	9.41697	0.327857
$826$	0	0
$827$	31.9552	1.11119	0.555596	−	0.831452i	$-0.312490\pi$
$827$	31.9552	1.11119	0.555596	+	0.831452i	$0.312490\pi$
$828$	0	0
$829$	−43.4284	−1.50833	−0.754165	−	0.656685i	$-0.771958\pi$
$829$	−43.4284	−1.50833	−0.754165	+	0.656685i	$0.771958\pi$
$830$	0	0
$831$	10.4122	0.361197
$832$	0	0
$833$	16.4378	0.569537
$834$	0	0
$835$	−1.39304	−0.0482081
$836$	0	0
$837$	−84.6059	−2.92441
$838$	0	0
$839$	−19.9521	−0.688824	−0.344412	−	0.938819i	$-0.611922\pi$
$839$	−19.9521	−0.688824	−0.344412	+	0.938819i	$0.611922\pi$
$840$	0	0
$841$	24.9969	0.861963
$842$	0	0
$843$	8.42169	0.290058
$844$	0	0
$845$	16.3435	0.562235
$846$	0	0
$847$	−6.44090	−0.221312
$848$	0	0
$849$	40.5402	1.39134
$850$	0	0
$851$	−0.789156	−0.0270519
$852$	0	0
$853$	54.1936	1.85555	0.927777	−	0.373136i	$-0.121717\pi$
$853$	54.1936	1.85555	0.927777	+	0.373136i	$0.121717\pi$
$854$	0	0
$855$	34.7605	1.18878
$856$	0	0
$857$	20.8818	0.713308	0.356654	−	0.934236i	$-0.383917\pi$
$857$	20.8818	0.713308	0.356654	+	0.934236i	$0.383917\pi$
$858$	0	0
$859$	56.7224	1.93534	0.967672	−	0.252212i	$-0.0811581\pi$
$859$	56.7224	1.93534	0.967672	+	0.252212i	$0.0811581\pi$
$860$	0	0
$861$	−148.053	−5.04565
$862$	0	0
$863$	23.3452	0.794679	0.397340	−	0.917672i	$-0.369933\pi$
$863$	23.3452	0.794679	0.397340	+	0.917672i	$0.369933\pi$
$864$	0	0
$865$	−2.37219	−0.0806568
$866$	0	0
$867$	43.0848	1.46324
$868$	0	0
$869$	24.5497	0.832790
$870$	0	0
$871$	2.78608	0.0944027
$872$	0	0
$873$	−33.4075	−1.13067
$874$	0	0
$875$	4.06871	0.137547
$876$	0	0
$877$	−26.1775	−0.883951	−0.441975	−	0.897027i	$-0.645722\pi$
$877$	−26.1775	−0.883951	−0.441975	+	0.897027i	$0.645722\pi$
$878$	0	0
$879$	−37.1088	−1.25165
$880$	0	0
$881$	−15.8147	−0.532812	−0.266406	−	0.963861i	$-0.585836\pi$
$881$	−15.8147	−0.532812	−0.266406	+	0.963861i	$0.585836\pi$
$882$	0	0
$883$	3.83866	0.129181	0.0645905	−	0.997912i	$-0.479426\pi$
$883$	3.83866	0.129181	0.0645905	+	0.997912i	$0.479426\pi$
$884$	0	0
$885$	29.5305	0.992655
$886$	0	0
$887$	−13.6679	−0.458922	−0.229461	−	0.973318i	$-0.573696\pi$
$887$	−13.6679	−0.458922	−0.229461	+	0.973318i	$0.573696\pi$
$888$	0	0
$889$	3.39304	0.113799
$890$	0	0
$891$	−39.6679	−1.32892
$892$	0	0
$893$	−7.80529	−0.261194
$894$	0	0
$895$	−18.8339	−0.629549
$896$	0	0
$897$	−16.6231	−0.555029
$898$	0	0
$899$	59.2910	1.97746
$900$	0	0
$901$	−2.55603	−0.0851536
$902$	0	0
$903$	49.9427	1.66199
$904$	0	0
$905$	−6.99528	−0.232531
$906$	0	0
$907$	−37.4762	−1.24438	−0.622189	−	0.782867i	$-0.713756\pi$
$907$	−37.4762	−1.24438	−0.622189	+	0.782867i	$0.713756\pi$
$908$	0	0
$909$	−90.1249	−2.98925
$910$	0	0
$911$	−31.1983	−1.03365	−0.516823	−	0.856092i	$-0.672886\pi$
$911$	−31.1983	−1.03365	−0.516823	+	0.856092i	$0.672886\pi$
$912$	0	0
$913$	−29.1088	−0.963360
$914$	0	0
$915$	36.9474	1.22144
$916$	0	0
$917$	2.83394	0.0935848
$918$	0	0
$919$	−12.0896	−0.398798	−0.199399	−	0.979918i	$-0.563899\pi$
$919$	−12.0896	−0.398798	−0.199399	+	0.979918i	$0.563899\pi$
$920$	0	0
$921$	−59.3597	−1.95597
$922$	0	0
$923$	−71.0353	−2.33816
$924$	0	0
$925$	−0.789156	−0.0259473
$926$	0	0
$927$	71.9093	2.36181
$928$	0	0
$929$	−49.5752	−1.62651	−0.813255	−	0.581907i	$-0.802307\pi$
$929$	−49.5752	−1.62651	−0.813255	+	0.581907i	$0.802307\pi$
$930$	0	0
$931$	51.7558	1.69623
$932$	0	0
$933$	67.3357	2.20447
$934$	0	0
$935$	−5.27955	−0.172660
$936$	0	0
$937$	27.4648	0.897237	0.448618	−	0.893723i	$-0.351916\pi$
$937$	27.4648	0.897237	0.448618	+	0.893723i	$0.351916\pi$
$938$	0	0
$939$	−28.3388	−0.924802
$940$	0	0
$941$	2.17747	0.0709836	0.0354918	−	0.999370i	$-0.488700\pi$
$941$	2.17747	0.0709836	0.0354918	+	0.999370i	$0.488700\pi$
$942$	0	0
$943$	11.8579	0.386145
$944$	0	0
$945$	−42.6632	−1.38783
$946$	0	0
$947$	−18.3082	−0.594937	−0.297468	−	0.954732i	$-0.596142\pi$
$947$	−18.3082	−0.594937	−0.297468	+	0.954732i	$0.596142\pi$
$948$	0	0
$949$	54.9141	1.78259
$950$	0	0
$951$	−82.8406	−2.68629
$952$	0	0
$953$	27.6184	0.894647	0.447323	−	0.894372i	$-0.352377\pi$
$953$	27.6184	0.894647	0.447323	+	0.894372i	$0.352377\pi$
$954$	0	0
$955$	−1.86258	−0.0602718
$956$	0	0
$957$	69.1983	2.23686
$958$	0	0
$959$	68.5897	2.21488
$960$	0	0
$961$	34.1040	1.10013
$962$	0	0
$963$	−81.1858	−2.61618
$964$	0	0
$965$	−11.7157	−0.377143
$966$	0	0
$967$	−19.2556	−0.619219	−0.309610	−	0.950864i	$-0.600198\pi$
$967$	−19.2556	−0.619219	−0.309610	+	0.950864i	$0.600198\pi$
$968$	0	0
$969$	−28.5992	−0.918737
$970$	0	0
$971$	−20.7460	−0.665771	−0.332886	−	0.942967i	$-0.608022\pi$
$971$	−20.7460	−0.665771	−0.332886	+	0.942967i	$0.608022\pi$
$972$	0	0
$973$	59.9780	1.92281
$974$	0	0
$975$	−16.6231	−0.532365
$976$	0	0
$977$	23.1775	0.741513	0.370757	−	0.928730i	$-0.379098\pi$
$977$	23.1775	0.741513	0.370757	+	0.928730i	$0.379098\pi$
$978$	0	0
$979$	33.6679	1.07603
$980$	0	0
$981$	−86.0962	−2.74884
$982$	0	0
$983$	−35.7110	−1.13900	−0.569502	−	0.821990i	$-0.692864\pi$
$983$	−35.7110	−1.13900	−0.569502	+	0.821990i	$0.692864\pi$
$984$	0	0
$985$	−14.1775	−0.451732
$986$	0	0
$987$	17.9906	0.572646
$988$	0	0
$989$	−4.00000	−0.127193
$990$	0	0
$991$	−44.8292	−1.42405	−0.712023	−	0.702156i	$-0.752221\pi$
$991$	−44.8292	−1.42405	−0.712023	+	0.702156i	$0.752221\pi$
$992$	0	0
$993$	−29.2556	−0.928399
$994$	0	0
$995$	−20.9714	−0.664837
$996$	0	0
$997$	−29.5210	−0.934940	−0.467470	−	0.884009i	$-0.654834\pi$
$997$	−29.5210	−0.934940	−0.467470	+	0.884009i	$0.654834\pi$
$998$	0	0
$999$	8.27483	0.261804

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3680.2.a.q.1.1	✓	3
4.3	odd	2		3680.2.a.r.1.3	yes	3
8.3	odd	2		7360.2.a.bx.1.1		3
8.5	even	2		7360.2.a.cd.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3680.2.a.q.1.1	✓	3	1.1	even	1	trivial
3680.2.a.r.1.3	yes	3	4.3	odd	2
7360.2.a.bx.1.1		3	8.3	odd	2
7360.2.a.cd.1.3		3	8.5	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:	\( N \)	\(=\)	\( 3680 = 2^{5} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3680.a (trivial)

\(f(q)\)	\(=\)	\(q-3.06871 q^{3} +1.00000 q^{5} +4.06871 q^{7} +6.41697 q^{9} -3.06871 q^{11} +5.41697 q^{13} -3.06871 q^{15} +1.72045 q^{17} +5.41697 q^{19} -12.4857 q^{21} +1.00000 q^{23} +1.00000 q^{25} -10.4857 q^{27} +7.34826 q^{29} +8.06871 q^{31} +9.41697 q^{33} +4.06871 q^{35} -0.789156 q^{37} -16.6231 q^{39} +11.8579 q^{41} -4.00000 q^{43} +6.41697 q^{45} -1.44090 q^{47} +9.55438 q^{49} -5.27955 q^{51} -1.48568 q^{53} -3.06871 q^{55} -16.6231 q^{57} -9.62309 q^{59} -12.0401 q^{61} +26.1088 q^{63} +5.41697 q^{65} +0.514324 q^{67} -3.06871 q^{69} -13.1135 q^{71} +10.1374 q^{73} -3.06871 q^{75} -12.4857 q^{77} -8.00000 q^{79} +12.9266 q^{81} +9.48568 q^{83} +1.72045 q^{85} -22.5497 q^{87} -10.9714 q^{89} +22.0401 q^{91} -24.7605 q^{93} +5.41697 q^{95} -5.20612 q^{97} -19.6918 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - q^{3} + 3 q^{5} + 4 q^{7} + 6 q^{9} - q^{11} + 3 q^{13} - q^{15} + 2 q^{17} + 3 q^{19} - 16 q^{21} + 3 q^{23} + 3 q^{25} - 10 q^{27} + 17 q^{29} + 16 q^{31} + 15 q^{33} + 4 q^{35} + 9 q^{37} - 12 q^{39}+ \cdots - 13 q^{99}+O(q^{100}) \) 3 * q - q^3 + 3 * q^5 + 4 * q^7 + 6 * q^9 - q^11 + 3 * q^13 - q^15 + 2 * q^17 + 3 * q^19 - 16 * q^21 + 3 * q^23 + 3 * q^25 - 10 * q^27 + 17 * q^29 + 16 * q^31 + 15 * q^33 + 4 * q^35 + 9 * q^37 - 12 * q^39 + 16 * q^41 - 12 * q^43 + 6 * q^45 + 2 * q^47 - q^49 - 19 * q^51 + 17 * q^53 - q^55 - 12 * q^57 + 9 * q^59 + 15 * q^61 + 19 * q^63 + 3 * q^65 + 23 * q^67 - q^69 - 16 * q^71 + 14 * q^73 - q^75 - 16 * q^77 - 24 * q^79 + 11 * q^81 + 7 * q^83 + 2 * q^85 - 2 * q^87 + 10 * q^89 + 15 * q^91 - 20 * q^93 + 3 * q^95 + 9 * q^97 - 13 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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