LMFDB - Embedded newform 5520.2.a.ca.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5520,2,Mod(1,5520)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5520, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("5520.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5520 = 2^{4} \cdot 3 \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5520.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:	$44.0774219157$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.229.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 4x - 1 $ x^3 - 4*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 2760)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.11491$ of defining polynomial
Character	$\chi$	$=$	5520.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +1.00000 q^{5} +3.58774 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +1.00000 q^{5} +3.58774 q^{7} +1.00000 q^{9} +0.715853 q^{13} -1.00000 q^{15} -1.58774 q^{17} -3.58774 q^{21} -1.00000 q^{23} +1.00000 q^{25} -1.00000 q^{27} +5.58774 q^{29} -4.87189 q^{31} +3.58774 q^{35} +8.15604 q^{37} -0.715853 q^{39} +4.30359 q^{41} -8.45963 q^{43} +1.00000 q^{45} +7.17548 q^{47} +5.87189 q^{49} +1.58774 q^{51} +8.30359 q^{53} +2.30359 q^{59} +6.45963 q^{61} +3.58774 q^{63} +0.715853 q^{65} +6.15604 q^{67} +1.00000 q^{69} +1.01945 q^{71} -5.17548 q^{73} -1.00000 q^{75} -9.89134 q^{79} +1.00000 q^{81} -5.01945 q^{83} -1.58774 q^{85} -5.58774 q^{87} +15.7438 q^{89} +2.56829 q^{91} +4.87189 q^{93} -6.45963 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{3} + 3 q^{5} - q^{7} + 3 q^{9}+O(q^{10}) $ 3 * q - 3 * q^3 + 3 * q^5 - q^7 + 3 * q^9	$ 3 q - 3 q^{3} + 3 q^{5} - q^{7} + 3 q^{9} + 4 q^{13} - 3 q^{15} + 7 q^{17} + q^{21} - 3 q^{23} + 3 q^{25} - 3 q^{27} + 5 q^{29} - q^{31} - q^{35} + 9 q^{37} - 4 q^{39} + 3 q^{41} + 3 q^{45} - 2 q^{47} + 4 q^{49} - 7 q^{51} + 15 q^{53} - 3 q^{59} - 6 q^{61} - q^{63} + 4 q^{65} + 3 q^{67} + 3 q^{69} - 5 q^{71} + 8 q^{73} - 3 q^{75} - 8 q^{79} + 3 q^{81} - 7 q^{83} + 7 q^{85} - 5 q^{87} + 20 q^{89} + 4 q^{91} + q^{93} + 6 q^{97}+O(q^{100}) $ 3 * q - 3 * q^3 + 3 * q^5 - q^7 + 3 * q^9 + 4 * q^13 - 3 * q^15 + 7 * q^17 + q^21 - 3 * q^23 + 3 * q^25 - 3 * q^27 + 5 * q^29 - q^31 - q^35 + 9 * q^37 - 4 * q^39 + 3 * q^41 + 3 * q^45 - 2 * q^47 + 4 * q^49 - 7 * q^51 + 15 * q^53 - 3 * q^59 - 6 * q^61 - q^63 + 4 * q^65 + 3 * q^67 + 3 * q^69 - 5 * q^71 + 8 * q^73 - 3 * q^75 - 8 * q^79 + 3 * q^81 - 7 * q^83 + 7 * q^85 - 5 * q^87 + 20 * q^89 + 4 * q^91 + q^93 + 6 * 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 $

$2$	0	0
$2$	0	0
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	3.58774	1.35604	0.678019	−	0.735044i	$-0.262839\pi$
$7$	3.58774	1.35604	0.678019	+	0.735044i	$0.262839\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	0	0
$13$	0.715853	0.198542	0.0992709	−	0.995060i	$-0.468349\pi$
$13$	0.715853	0.198542	0.0992709	+	0.995060i	$0.468349\pi$
$14$	0	0
$15$	−1.00000	−0.258199
$16$	0	0
$17$	−1.58774	−0.385084	−0.192542	−	0.981289i	$-0.561673\pi$
$17$	−1.58774	−0.385084	−0.192542	+	0.981289i	$0.561673\pi$
$18$	0	0
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	0	0
$21$	−3.58774	−0.782909
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	5.58774	1.03762	0.518809	−	0.854890i	$-0.326376\pi$
$29$	5.58774	1.03762	0.518809	+	0.854890i	$0.326376\pi$
$30$	0	0
$31$	−4.87189	−0.875017	−0.437509	−	0.899214i	$-0.644139\pi$
$31$	−4.87189	−0.875017	−0.437509	+	0.899214i	$0.644139\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	3.58774	0.606439
$36$	0	0
$37$	8.15604	1.34084	0.670422	−	0.741980i	$-0.266113\pi$
$37$	8.15604	1.34084	0.670422	+	0.741980i	$0.266113\pi$
$38$	0	0
$39$	−0.715853	−0.114628
$40$	0	0
$41$	4.30359	0.672108	0.336054	−	0.941843i	$-0.390907\pi$
$41$	4.30359	0.672108	0.336054	+	0.941843i	$0.390907\pi$
$42$	0	0
$43$	−8.45963	−1.29008	−0.645041	−	0.764148i	$-0.723160\pi$
$43$	−8.45963	−1.29008	−0.645041	+	0.764148i	$0.723160\pi$
$44$	0	0
$45$	1.00000	0.149071
$46$	0	0
$47$	7.17548	1.04665	0.523326	−	0.852133i	$-0.324691\pi$
$47$	7.17548	1.04665	0.523326	+	0.852133i	$0.324691\pi$
$48$	0	0
$49$	5.87189	0.838841
$50$	0	0
$51$	1.58774	0.222328
$52$	0	0
$53$	8.30359	1.14059	0.570293	−	0.821441i	$-0.306830\pi$
$53$	8.30359	1.14059	0.570293	+	0.821441i	$0.306830\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	2.30359	0.299902	0.149951	−	0.988693i	$-0.452088\pi$
$59$	2.30359	0.299902	0.149951	+	0.988693i	$0.452088\pi$
$60$	0	0
$61$	6.45963	0.827071	0.413535	−	0.910488i	$-0.364294\pi$
$61$	6.45963	0.827071	0.413535	+	0.910488i	$0.364294\pi$
$62$	0	0
$63$	3.58774	0.452013
$64$	0	0
$65$	0.715853	0.0887906
$66$	0	0
$67$	6.15604	0.752079	0.376040	−	0.926604i	$-0.377286\pi$
$67$	6.15604	0.752079	0.376040	+	0.926604i	$0.377286\pi$
$68$	0	0
$69$	1.00000	0.120386
$70$	0	0
$71$	1.01945	0.120986	0.0604930	−	0.998169i	$-0.480733\pi$
$71$	1.01945	0.120986	0.0604930	+	0.998169i	$0.480733\pi$
$72$	0	0
$73$	−5.17548	−0.605744	−0.302872	−	0.953031i	$-0.597946\pi$
$73$	−5.17548	−0.605744	−0.302872	+	0.953031i	$0.597946\pi$
$74$	0	0
$75$	−1.00000	−0.115470
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−9.89134	−1.11286	−0.556431	−	0.830894i	$-0.687830\pi$
$79$	−9.89134	−1.11286	−0.556431	+	0.830894i	$0.687830\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−5.01945	−0.550956	−0.275478	−	0.961307i	$-0.588836\pi$
$83$	−5.01945	−0.550956	−0.275478	+	0.961307i	$0.588836\pi$
$84$	0	0
$85$	−1.58774	−0.172215
$86$	0	0
$87$	−5.58774	−0.599069
$88$	0	0
$89$	15.7438	1.66884	0.834419	−	0.551131i	$-0.185804\pi$
$89$	15.7438	1.66884	0.834419	+	0.551131i	$0.185804\pi$
$90$	0	0
$91$	2.56829	0.269230
$92$	0	0
$93$	4.87189	0.505191
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−6.45963	−0.655876	−0.327938	−	0.944699i	$-0.606354\pi$
$97$	−6.45963	−0.655876	−0.327938	+	0.944699i	$0.606354\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	5.58774	0.556001	0.278001	−	0.960581i	$-0.410328\pi$
$101$	5.58774	0.556001	0.278001	+	0.960581i	$0.410328\pi$
$102$	0	0
$103$	−5.89134	−0.580491	−0.290245	−	0.956952i	$-0.593737\pi$
$103$	−5.89134	−0.580491	−0.290245	+	0.956952i	$0.593737\pi$
$104$	0	0
$105$	−3.58774	−0.350128
$106$	0	0
$107$	0.412259	0.0398545	0.0199273	−	0.999801i	$-0.493657\pi$
$107$	0.412259	0.0398545	0.0199273	+	0.999801i	$0.493657\pi$
$108$	0	0
$109$	2.14756	0.205699	0.102849	−	0.994697i	$-0.467204\pi$
$109$	2.14756	0.205699	0.102849	+	0.994697i	$0.467204\pi$
$110$	0	0
$111$	−8.15604	−0.774137
$112$	0	0
$113$	0.980553	0.0922427	0.0461213	−	0.998936i	$-0.485314\pi$
$113$	0.980553	0.0922427	0.0461213	+	0.998936i	$0.485314\pi$
$114$	0	0
$115$	−1.00000	−0.0932505
$116$	0	0
$117$	0.715853	0.0661806
$118$	0	0
$119$	−5.69641	−0.522189
$120$	0	0
$121$	−11.0000	−1.00000
$122$	0	0
$123$	−4.30359	−0.388042
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	2.71585	0.240993	0.120496	−	0.992714i	$-0.461551\pi$
$127$	2.71585	0.240993	0.120496	+	0.992714i	$0.461551\pi$
$128$	0	0
$129$	8.45963	0.744829
$130$	0	0
$131$	2.56829	0.224393	0.112196	−	0.993686i	$-0.464211\pi$
$131$	2.56829	0.224393	0.112196	+	0.993686i	$0.464211\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−1.00000	−0.0860663
$136$	0	0
$137$	−6.45963	−0.551883	−0.275942	−	0.961174i	$-0.588990\pi$
$137$	−6.45963	−0.551883	−0.275942	+	0.961174i	$0.588990\pi$
$138$	0	0
$139$	−8.04737	−0.682569	−0.341285	−	0.939960i	$-0.610862\pi$
$139$	−8.04737	−0.682569	−0.341285	+	0.939960i	$0.610862\pi$
$140$	0	0
$141$	−7.17548	−0.604285
$142$	0	0
$143$	0	0
$144$	0	0
$145$	5.58774	0.464037
$146$	0	0
$147$	−5.87189	−0.484305
$148$	0	0
$149$	6.00000	0.491539	0.245770	−	0.969328i	$-0.420959\pi$
$149$	6.00000	0.491539	0.245770	+	0.969328i	$0.420959\pi$
$150$	0	0
$151$	0	0		−	1.00000i	$-0.5\pi$
$151$	0	0			1.00000i	$0.5\pi$
$152$	0	0
$153$	−1.58774	−0.128361
$154$	0	0
$155$	−4.87189	−0.391320
$156$	0	0
$157$	12.7632	1.01862	0.509308	−	0.860584i	$-0.329901\pi$
$157$	12.7632	1.01862	0.509308	+	0.860584i	$0.329901\pi$
$158$	0	0
$159$	−8.30359	−0.658518
$160$	0	0
$161$	−3.58774	−0.282754
$162$	0	0
$163$	16.6072	1.30078	0.650388	−	0.759602i	$-0.274606\pi$
$163$	16.6072	1.30078	0.650388	+	0.759602i	$0.274606\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	15.1755	1.17431	0.587157	−	0.809473i	$-0.300247\pi$
$167$	15.1755	1.17431	0.587157	+	0.809473i	$0.300247\pi$
$168$	0	0
$169$	−12.4876	−0.960581
$170$	0	0
$171$	0	0
$172$	0	0
$173$	17.1755	1.30583	0.652914	−	0.757432i	$-0.273546\pi$
$173$	17.1755	1.30583	0.652914	+	0.757432i	$0.273546\pi$
$174$	0	0
$175$	3.58774	0.271208
$176$	0	0
$177$	−2.30359	−0.173149
$178$	0	0
$179$	0	0		−	1.00000i	$-0.5\pi$
$179$	0	0			1.00000i	$0.5\pi$
$180$	0	0
$181$	−15.0668	−1.11991	−0.559954	−	0.828524i	$-0.689181\pi$
$181$	−15.0668	−1.11991	−0.559954	+	0.828524i	$0.689181\pi$
$182$	0	0
$183$	−6.45963	−0.477510
$184$	0	0
$185$	8.15604	0.599644
$186$	0	0
$187$	0	0
$188$	0	0
$189$	−3.58774	−0.260970
$190$	0	0
$191$	−4.60719	−0.333364	−0.166682	−	0.986011i	$-0.553305\pi$
$191$	−4.60719	−0.333364	−0.166682	+	0.986011i	$0.553305\pi$
$192$	0	0
$193$	−10.6072	−0.763522	−0.381761	−	0.924261i	$-0.624682\pi$
$193$	−10.6072	−0.763522	−0.381761	+	0.924261i	$0.624682\pi$
$194$	0	0
$195$	−0.715853	−0.0512633
$196$	0	0
$197$	−9.78267	−0.696986	−0.348493	−	0.937311i	$-0.613307\pi$
$197$	−9.78267	−0.696986	−0.348493	+	0.937311i	$0.613307\pi$
$198$	0	0
$199$	−2.71585	−0.192522	−0.0962608	−	0.995356i	$-0.530688\pi$
$199$	−2.71585	−0.192522	−0.0962608	+	0.995356i	$0.530688\pi$
$200$	0	0
$201$	−6.15604	−0.434213
$202$	0	0
$203$	20.0474	1.40705
$204$	0	0
$205$	4.30359	0.300576
$206$	0	0
$207$	−1.00000	−0.0695048
$208$	0	0
$209$	0	0
$210$	0	0
$211$	13.4791	0.927938	0.463969	−	0.885851i	$-0.346425\pi$
$211$	13.4791	0.927938	0.463969	+	0.885851i	$0.346425\pi$
$212$	0	0
$213$	−1.01945	−0.0698514
$214$	0	0
$215$	−8.45963	−0.576942
$216$	0	0
$217$	−17.4791	−1.18656
$218$	0	0
$219$	5.17548	0.349727
$220$	0	0
$221$	−1.13659	−0.0764553
$222$	0	0
$223$	8.14756	0.545601	0.272800	−	0.962071i	$-0.412050\pi$
$223$	8.14756	0.545601	0.272800	+	0.962071i	$0.412050\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	−18.8106	−1.24850	−0.624252	−	0.781223i	$-0.714596\pi$
$227$	−18.8106	−1.24850	−0.624252	+	0.781223i	$0.714596\pi$
$228$	0	0
$229$	26.7717	1.76912	0.884562	−	0.466423i	$-0.154457\pi$
$229$	26.7717	1.76912	0.884562	+	0.466423i	$0.154457\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−3.74378	−0.245263	−0.122632	−	0.992452i	$-0.539133\pi$
$233$	−3.74378	−0.245263	−0.122632	+	0.992452i	$0.539133\pi$
$234$	0	0
$235$	7.17548	0.468077
$236$	0	0
$237$	9.89134	0.642511
$238$	0	0
$239$	19.6825	1.27315	0.636577	−	0.771213i	$-0.280350\pi$
$239$	19.6825	1.27315	0.636577	+	0.771213i	$0.280350\pi$
$240$	0	0
$241$	−15.7438	−1.01415	−0.507073	−	0.861903i	$-0.669273\pi$
$241$	−15.7438	−1.01415	−0.507073	+	0.861903i	$0.669273\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	5.87189	0.375141
$246$	0	0
$247$	0	0
$248$	0	0
$249$	5.01945	0.318095
$250$	0	0
$251$	−26.6630	−1.68296	−0.841478	−	0.540291i	$-0.818314\pi$
$251$	−26.6630	−1.68296	−0.841478	+	0.540291i	$0.818314\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	1.58774	0.0994282
$256$	0	0
$257$	−8.35097	−0.520919	−0.260459	−	0.965485i	$-0.583874\pi$
$257$	−8.35097	−0.520919	−0.260459	+	0.965485i	$0.583874\pi$
$258$	0	0
$259$	29.2617	1.81824
$260$	0	0
$261$	5.58774	0.345873
$262$	0	0
$263$	3.12811	0.192888	0.0964438	−	0.995338i	$-0.469253\pi$
$263$	3.12811	0.192888	0.0964438	+	0.995338i	$0.469253\pi$
$264$	0	0
$265$	8.30359	0.510086
$266$	0	0
$267$	−15.7438	−0.963504
$268$	0	0
$269$	28.7632	1.75372	0.876862	−	0.480741i	$-0.159632\pi$
$269$	28.7632	1.75372	0.876862	+	0.480741i	$0.159632\pi$
$270$	0	0
$271$	15.4402	0.937924	0.468962	−	0.883218i	$-0.344628\pi$
$271$	15.4402	0.937924	0.468962	+	0.883218i	$0.344628\pi$
$272$	0	0
$273$	−2.56829	−0.155440
$274$	0	0
$275$	0	0
$276$	0	0
$277$	22.2423	1.33641	0.668205	−	0.743977i	$-0.267063\pi$
$277$	22.2423	1.33641	0.668205	+	0.743977i	$0.267063\pi$
$278$	0	0
$279$	−4.87189	−0.291672
$280$	0	0
$281$	30.0947	1.79530	0.897651	−	0.440707i	$-0.145272\pi$
$281$	30.0947	1.79530	0.897651	+	0.440707i	$0.145272\pi$
$282$	0	0
$283$	23.8998	1.42070	0.710348	−	0.703850i	$-0.248537\pi$
$283$	23.8998	1.42070	0.710348	+	0.703850i	$0.248537\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	15.4402	0.911405
$288$	0	0
$289$	−14.4791	−0.851710
$290$	0	0
$291$	6.45963	0.378670
$292$	0	0
$293$	−10.6546	−0.622446	−0.311223	−	0.950337i	$-0.600739\pi$
$293$	−10.6546	−0.622446	−0.311223	+	0.950337i	$0.600739\pi$
$294$	0	0
$295$	2.30359	0.134120
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−0.715853	−0.0413988
$300$	0	0
$301$	−30.3510	−1.74940
$302$	0	0
$303$	−5.58774	−0.321007
$304$	0	0
$305$	6.45963	0.369877
$306$	0	0
$307$	6.56829	0.374872	0.187436	−	0.982277i	$-0.439982\pi$
$307$	6.56829	0.374872	0.187436	+	0.982277i	$0.439982\pi$
$308$	0	0
$309$	5.89134	0.335146
$310$	0	0
$311$	−22.8106	−1.29347	−0.646735	−	0.762715i	$-0.723866\pi$
$311$	−22.8106	−1.29347	−0.646735	+	0.762715i	$0.723866\pi$
$312$	0	0
$313$	12.7632	0.721420	0.360710	−	0.932678i	$-0.382534\pi$
$313$	12.7632	0.721420	0.360710	+	0.932678i	$0.382534\pi$
$314$	0	0
$315$	3.58774	0.202146
$316$	0	0
$317$	19.2144	1.07919	0.539593	−	0.841926i	$-0.318578\pi$
$317$	19.2144	1.07919	0.539593	+	0.841926i	$0.318578\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	−0.412259	−0.0230100
$322$	0	0
$323$	0	0
$324$	0	0
$325$	0.715853	0.0397084
$326$	0	0
$327$	−2.14756	−0.118760
$328$	0	0
$329$	25.7438	1.41930
$330$	0	0
$331$	−15.2229	−0.836724	−0.418362	−	0.908280i	$-0.637396\pi$
$331$	−15.2229	−0.836724	−0.418362	+	0.908280i	$0.637396\pi$
$332$	0	0
$333$	8.15604	0.446948
$334$	0	0
$335$	6.15604	0.336340
$336$	0	0
$337$	14.2423	0.775828	0.387914	−	0.921696i	$-0.373196\pi$
$337$	14.2423	0.775828	0.387914	+	0.921696i	$0.373196\pi$
$338$	0	0
$339$	−0.980553	−0.0532563
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−4.04737	−0.218538
$344$	0	0
$345$	1.00000	0.0538382
$346$	0	0
$347$	12.0000	0.644194	0.322097	−	0.946707i	$-0.395612\pi$
$347$	12.0000	0.644194	0.322097	+	0.946707i	$0.395612\pi$
$348$	0	0
$349$	−11.4791	−0.614461	−0.307230	−	0.951635i	$-0.599402\pi$
$349$	−11.4791	−0.614461	−0.307230	+	0.951635i	$0.599402\pi$
$350$	0	0
$351$	−0.715853	−0.0382094
$352$	0	0
$353$	17.7827	0.946476	0.473238	−	0.880935i	$-0.343085\pi$
$353$	17.7827	0.946476	0.473238	+	0.880935i	$0.343085\pi$
$354$	0	0
$355$	1.01945	0.0541066
$356$	0	0
$357$	5.69641	0.301486
$358$	0	0
$359$	−20.6072	−1.08761	−0.543803	−	0.839213i	$-0.683016\pi$
$359$	−20.6072	−1.08761	−0.543803	+	0.839213i	$0.683016\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	0	0
$363$	11.0000	0.577350
$364$	0	0
$365$	−5.17548	−0.270897
$366$	0	0
$367$	10.7632	0.561836	0.280918	−	0.959732i	$-0.409361\pi$
$367$	10.7632	0.561836	0.280918	+	0.959732i	$0.409361\pi$
$368$	0	0
$369$	4.30359	0.224036
$370$	0	0
$371$	29.7911	1.54668
$372$	0	0
$373$	−11.5962	−0.600429	−0.300215	−	0.953872i	$-0.597058\pi$
$373$	−11.5962	−0.600429	−0.300215	+	0.953872i	$0.597058\pi$
$374$	0	0
$375$	−1.00000	−0.0516398
$376$	0	0
$377$	4.00000	0.206010
$378$	0	0
$379$	−0.919260	−0.0472192	−0.0236096	−	0.999721i	$-0.507516\pi$
$379$	−0.919260	−0.0472192	−0.0236096	+	0.999721i	$0.507516\pi$
$380$	0	0
$381$	−2.71585	−0.139137
$382$	0	0
$383$	3.12811	0.159839	0.0799195	−	0.996801i	$-0.474534\pi$
$383$	3.12811	0.159839	0.0799195	+	0.996801i	$0.474534\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−8.45963	−0.430027
$388$	0	0
$389$	11.4317	0.579610	0.289805	−	0.957086i	$-0.406409\pi$
$389$	11.4317	0.579610	0.289805	+	0.957086i	$0.406409\pi$
$390$	0	0
$391$	1.58774	0.0802955
$392$	0	0
$393$	−2.56829	−0.129553
$394$	0	0
$395$	−9.89134	−0.497687
$396$	0	0
$397$	−2.14756	−0.107783	−0.0538914	−	0.998547i	$-0.517162\pi$
$397$	−2.14756	−0.107783	−0.0538914	+	0.998547i	$0.517162\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−16.6461	−0.831266	−0.415633	−	0.909532i	$-0.636440\pi$
$401$	−16.6461	−0.831266	−0.415633	+	0.909532i	$0.636440\pi$
$402$	0	0
$403$	−3.48755	−0.173727
$404$	0	0
$405$	1.00000	0.0496904
$406$	0	0
$407$	0	0
$408$	0	0
$409$	2.26470	0.111982	0.0559911	−	0.998431i	$-0.482168\pi$
$409$	2.26470	0.111982	0.0559911	+	0.998431i	$0.482168\pi$
$410$	0	0
$411$	6.45963	0.318630
$412$	0	0
$413$	8.26470	0.406679
$414$	0	0
$415$	−5.01945	−0.246395
$416$	0	0
$417$	8.04737	0.394081
$418$	0	0
$419$	11.7827	0.575621	0.287811	−	0.957687i	$-0.407073\pi$
$419$	11.7827	0.575621	0.287811	+	0.957687i	$0.407073\pi$
$420$	0	0
$421$	−20.4985	−0.999037	−0.499518	−	0.866303i	$-0.666490\pi$
$421$	−20.4985	−0.999037	−0.499518	+	0.866303i	$0.666490\pi$
$422$	0	0
$423$	7.17548	0.348884
$424$	0	0
$425$	−1.58774	−0.0770168
$426$	0	0
$427$	23.1755	1.12154
$428$	0	0
$429$	0	0
$430$	0	0
$431$	16.9193	0.814972	0.407486	−	0.913211i	$-0.366406\pi$
$431$	16.9193	0.814972	0.407486	+	0.913211i	$0.366406\pi$
$432$	0	0
$433$	17.0753	0.820586	0.410293	−	0.911954i	$-0.365426\pi$
$433$	17.0753	0.820586	0.410293	+	0.911954i	$0.365426\pi$
$434$	0	0
$435$	−5.58774	−0.267912
$436$	0	0
$437$	0	0
$438$	0	0
$439$	22.0558	1.05267	0.526334	−	0.850278i	$-0.323566\pi$
$439$	22.0558	1.05267	0.526334	+	0.850278i	$0.323566\pi$
$440$	0	0
$441$	5.87189	0.279614
$442$	0	0
$443$	12.8245	0.609311	0.304656	−	0.952463i	$-0.401459\pi$
$443$	12.8245	0.609311	0.304656	+	0.952463i	$0.401459\pi$
$444$	0	0
$445$	15.7438	0.746327
$446$	0	0
$447$	−6.00000	−0.283790
$448$	0	0
$449$	10.5598	0.498349	0.249174	−	0.968459i	$-0.419841\pi$
$449$	10.5598	0.498349	0.249174	+	0.968459i	$0.419841\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	0	0
$454$	0	0
$455$	2.56829	0.120404
$456$	0	0
$457$	−6.72433	−0.314551	−0.157275	−	0.987555i	$-0.550271\pi$
$457$	−6.72433	−0.314551	−0.157275	+	0.987555i	$0.550271\pi$
$458$	0	0
$459$	1.58774	0.0741094
$460$	0	0
$461$	8.81060	0.410350	0.205175	−	0.978725i	$-0.434224\pi$
$461$	8.81060	0.410350	0.205175	+	0.978725i	$0.434224\pi$
$462$	0	0
$463$	14.1087	0.655685	0.327843	−	0.944732i	$-0.393678\pi$
$463$	14.1087	0.655685	0.327843	+	0.944732i	$0.393678\pi$
$464$	0	0
$465$	4.87189	0.225928
$466$	0	0
$467$	11.2757	0.521776	0.260888	−	0.965369i	$-0.415985\pi$
$467$	11.2757	0.521776	0.260888	+	0.965369i	$0.415985\pi$
$468$	0	0
$469$	22.0863	1.01985
$470$	0	0
$471$	−12.7632	−0.588098
$472$	0	0
$473$	0	0
$474$	0	0
$475$	0	0
$476$	0	0
$477$	8.30359	0.380195
$478$	0	0
$479$	10.9582	0.500691	0.250345	−	0.968157i	$-0.419456\pi$
$479$	10.9582	0.500691	0.250345	+	0.968157i	$0.419456\pi$
$480$	0	0
$481$	5.83852	0.266214
$482$	0	0
$483$	3.58774	0.163248
$484$	0	0
$485$	−6.45963	−0.293317
$486$	0	0
$487$	−24.7717	−1.12251	−0.561256	−	0.827642i	$-0.689682\pi$
$487$	−24.7717	−1.12251	−0.561256	+	0.827642i	$0.689682\pi$
$488$	0	0
$489$	−16.6072	−0.751003
$490$	0	0
$491$	0.559817	0.0252642	0.0126321	−	0.999920i	$-0.495979\pi$
$491$	0.559817	0.0252642	0.0126321	+	0.999920i	$0.495979\pi$
$492$	0	0
$493$	−8.87189	−0.399570
$494$	0	0
$495$	0	0
$496$	0	0
$497$	3.65751	0.164062
$498$	0	0
$499$	−5.77419	−0.258488	−0.129244	−	0.991613i	$-0.541255\pi$
$499$	−5.77419	−0.258488	−0.129244	+	0.991613i	$0.541255\pi$
$500$	0	0
$501$	−15.1755	−0.677991
$502$	0	0
$503$	5.69641	0.253990	0.126995	−	0.991903i	$-0.459467\pi$
$503$	5.69641	0.253990	0.126995	+	0.991903i	$0.459467\pi$
$504$	0	0
$505$	5.58774	0.248651
$506$	0	0
$507$	12.4876	0.554592
$508$	0	0
$509$	−31.3789	−1.39084	−0.695422	−	0.718601i	$-0.744783\pi$
$509$	−31.3789	−1.39084	−0.695422	+	0.718601i	$0.744783\pi$
$510$	0	0
$511$	−18.5683	−0.821413
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−5.89134	−0.259603
$516$	0	0
$517$	0	0
$518$	0	0
$519$	−17.1755	−0.753920
$520$	0	0
$521$	13.1755	0.577228	0.288614	−	0.957445i	$-0.406806\pi$
$521$	13.1755	0.577228	0.288614	+	0.957445i	$0.406806\pi$
$522$	0	0
$523$	24.4596	1.06954	0.534772	−	0.844996i	$-0.320397\pi$
$523$	24.4596	1.06954	0.534772	+	0.844996i	$0.320397\pi$
$524$	0	0
$525$	−3.58774	−0.156582
$526$	0	0
$527$	7.73530	0.336955
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	2.30359	0.0999675
$532$	0	0
$533$	3.08074	0.133442
$534$	0	0
$535$	0.412259	0.0178235
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−27.8385	−1.19687	−0.598436	−	0.801171i	$-0.704211\pi$
$541$	−27.8385	−1.19687	−0.598436	+	0.801171i	$0.704211\pi$
$542$	0	0
$543$	15.0668	0.646579
$544$	0	0
$545$	2.14756	0.0919913
$546$	0	0
$547$	−1.13659	−0.0485970	−0.0242985	−	0.999705i	$-0.507735\pi$
$547$	−1.13659	−0.0485970	−0.0242985	+	0.999705i	$0.507735\pi$
$548$	0	0
$549$	6.45963	0.275690
$550$	0	0
$551$	0	0
$552$	0	0
$553$	−35.4876	−1.50908
$554$	0	0
$555$	−8.15604	−0.346204
$556$	0	0
$557$	−11.4791	−0.486384	−0.243192	−	0.969978i	$-0.578195\pi$
$557$	−11.4791	−0.486384	−0.243192	+	0.969978i	$0.578195\pi$
$558$	0	0
$559$	−6.05585	−0.256135
$560$	0	0
$561$	0	0
$562$	0	0
$563$	31.5877	1.33126	0.665632	−	0.746280i	$-0.268162\pi$
$563$	31.5877	1.33126	0.665632	+	0.746280i	$0.268162\pi$
$564$	0	0
$565$	0.980553	0.0412522
$566$	0	0
$567$	3.58774	0.150671
$568$	0	0
$569$	13.0807	0.548373	0.274187	−	0.961677i	$-0.411591\pi$
$569$	13.0807	0.548373	0.274187	+	0.961677i	$0.411591\pi$
$570$	0	0
$571$	43.5823	1.82386	0.911931	−	0.410343i	$-0.134591\pi$
$571$	43.5823	1.82386	0.911931	+	0.410343i	$0.134591\pi$
$572$	0	0
$573$	4.60719	0.192468
$574$	0	0
$575$	−1.00000	−0.0417029
$576$	0	0
$577$	7.52645	0.313330	0.156665	−	0.987652i	$-0.449926\pi$
$577$	7.52645	0.313330	0.156665	+	0.987652i	$0.449926\pi$
$578$	0	0
$579$	10.6072	0.440820
$580$	0	0
$581$	−18.0085	−0.747118
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0.715853	0.0295969
$586$	0	0
$587$	−6.95815	−0.287194	−0.143597	−	0.989636i	$-0.545867\pi$
$587$	−6.95815	−0.287194	−0.143597	+	0.989636i	$0.545867\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	9.78267	0.402405
$592$	0	0
$593$	6.00000	0.246390	0.123195	−	0.992382i	$-0.460686\pi$
$593$	6.00000	0.246390	0.123195	+	0.992382i	$0.460686\pi$
$594$	0	0
$595$	−5.69641	−0.233530
$596$	0	0
$597$	2.71585	0.111152
$598$	0	0
$599$	−29.5962	−1.20927	−0.604634	−	0.796503i	$-0.706681\pi$
$599$	−29.5962	−1.20927	−0.604634	+	0.796503i	$0.706681\pi$
$600$	0	0
$601$	−10.7771	−0.439609	−0.219804	−	0.975544i	$-0.570542\pi$
$601$	−10.7771	−0.439609	−0.219804	+	0.975544i	$0.570542\pi$
$602$	0	0
$603$	6.15604	0.250693
$604$	0	0
$605$	−11.0000	−0.447214
$606$	0	0
$607$	5.28415	0.214477	0.107238	−	0.994233i	$-0.465799\pi$
$607$	5.28415	0.214477	0.107238	+	0.994233i	$0.465799\pi$
$608$	0	0
$609$	−20.0474	−0.812360
$610$	0	0
$611$	5.13659	0.207804
$612$	0	0
$613$	−33.9472	−1.37111	−0.685557	−	0.728019i	$-0.740441\pi$
$613$	−33.9472	−1.37111	−0.685557	+	0.728019i	$0.740441\pi$
$614$	0	0
$615$	−4.30359	−0.173538
$616$	0	0
$617$	1.37041	0.0551707	0.0275854	−	0.999619i	$-0.491218\pi$
$617$	1.37041	0.0551707	0.0275854	+	0.999619i	$0.491218\pi$
$618$	0	0
$619$	−6.35097	−0.255267	−0.127633	−	0.991821i	$-0.540738\pi$
$619$	−6.35097	−0.255267	−0.127633	+	0.991821i	$0.540738\pi$
$620$	0	0
$621$	1.00000	0.0401286
$622$	0	0
$623$	56.4846	2.26301
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−12.9497	−0.516337
$630$	0	0
$631$	−5.57926	−0.222107	−0.111053	−	0.993814i	$-0.535422\pi$
$631$	−5.57926	−0.222107	−0.111053	+	0.993814i	$0.535422\pi$
$632$	0	0
$633$	−13.4791	−0.535745
$634$	0	0
$635$	2.71585	0.107775
$636$	0	0
$637$	4.20341	0.166545
$638$	0	0
$639$	1.01945	0.0403287
$640$	0	0
$641$	−26.0947	−1.03068	−0.515340	−	0.856986i	$-0.672334\pi$
$641$	−26.0947	−1.03068	−0.515340	+	0.856986i	$0.672334\pi$
$642$	0	0
$643$	31.0753	1.22549	0.612745	−	0.790281i	$-0.290065\pi$
$643$	31.0753	1.22549	0.612745	+	0.790281i	$0.290065\pi$
$644$	0	0
$645$	8.45963	0.333098
$646$	0	0
$647$	2.86341	0.112572	0.0562862	−	0.998415i	$-0.482074\pi$
$647$	2.86341	0.112572	0.0562862	+	0.998415i	$0.482074\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	17.4791	0.685059
$652$	0	0
$653$	15.2313	0.596048	0.298024	−	0.954558i	$-0.403672\pi$
$653$	15.2313	0.596048	0.298024	+	0.954558i	$0.403672\pi$
$654$	0	0
$655$	2.56829	0.100352
$656$	0	0
$657$	−5.17548	−0.201915
$658$	0	0
$659$	−20.7019	−0.806433	−0.403216	−	0.915105i	$-0.632108\pi$
$659$	−20.7019	−0.806433	−0.403216	+	0.915105i	$0.632108\pi$
$660$	0	0
$661$	33.9472	1.32039	0.660196	−	0.751093i	$-0.270473\pi$
$661$	33.9472	1.32039	0.660196	+	0.751093i	$0.270473\pi$
$662$	0	0
$663$	1.13659	0.0441415
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−5.58774	−0.216358
$668$	0	0
$669$	−8.14756	−0.315003
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−44.0558	−1.69823	−0.849114	−	0.528209i	$-0.822864\pi$
$673$	−44.0558	−1.69823	−0.849114	+	0.528209i	$0.822864\pi$
$674$	0	0
$675$	−1.00000	−0.0384900
$676$	0	0
$677$	−28.0085	−1.07645	−0.538227	−	0.842800i	$-0.680906\pi$
$677$	−28.0085	−1.07645	−0.538227	+	0.842800i	$0.680906\pi$
$678$	0	0
$679$	−23.1755	−0.889393
$680$	0	0
$681$	18.8106	0.720824
$682$	0	0
$683$	3.90526	0.149430	0.0747152	−	0.997205i	$-0.476195\pi$
$683$	3.90526	0.149430	0.0747152	+	0.997205i	$0.476195\pi$
$684$	0	0
$685$	−6.45963	−0.246810
$686$	0	0
$687$	−26.7717	−1.02140
$688$	0	0
$689$	5.94415	0.226454
$690$	0	0
$691$	−48.4068	−1.84148	−0.920741	−	0.390174i	$-0.872415\pi$
$691$	−48.4068	−1.84148	−0.920741	+	0.390174i	$0.872415\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−8.04737	−0.305254
$696$	0	0
$697$	−6.83299	−0.258818
$698$	0	0
$699$	3.74378	0.141603
$700$	0	0
$701$	−19.2144	−0.725717	−0.362858	−	0.931844i	$-0.618199\pi$
$701$	−19.2144	−0.725717	−0.362858	+	0.931844i	$0.618199\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	−7.17548	−0.270244
$706$	0	0
$707$	20.0474	0.753959
$708$	0	0
$709$	14.4596	0.543043	0.271521	−	0.962432i	$-0.412473\pi$
$709$	14.4596	0.543043	0.271521	+	0.962432i	$0.412473\pi$
$710$	0	0
$711$	−9.89134	−0.370954
$712$	0	0
$713$	4.87189	0.182454
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−19.6825	−0.735056
$718$	0	0
$719$	−34.2338	−1.27671	−0.638353	−	0.769744i	$-0.720384\pi$
$719$	−34.2338	−1.27671	−0.638353	+	0.769744i	$0.720384\pi$
$720$	0	0
$721$	−21.1366	−0.787168
$722$	0	0
$723$	15.7438	0.585517
$724$	0	0
$725$	5.58774	0.207524
$726$	0	0
$727$	−22.4512	−0.832667	−0.416334	−	0.909212i	$-0.636685\pi$
$727$	−22.4512	−0.832667	−0.416334	+	0.909212i	$0.636685\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	13.4317	0.496790
$732$	0	0
$733$	−5.27567	−0.194861	−0.0974306	−	0.995242i	$-0.531062\pi$
$733$	−5.27567	−0.194861	−0.0974306	+	0.995242i	$0.531062\pi$
$734$	0	0
$735$	−5.87189	−0.216588
$736$	0	0
$737$	0	0
$738$	0	0
$739$	3.34544	0.123064	0.0615320	−	0.998105i	$-0.480401\pi$
$739$	3.34544	0.123064	0.0615320	+	0.998105i	$0.480401\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	13.1366	0.481935	0.240967	−	0.970533i	$-0.422535\pi$
$743$	13.1366	0.481935	0.240967	+	0.970533i	$0.422535\pi$
$744$	0	0
$745$	6.00000	0.219823
$746$	0	0
$747$	−5.01945	−0.183652
$748$	0	0
$749$	1.47908	0.0540443
$750$	0	0
$751$	−11.2453	−0.410345	−0.205173	−	0.978726i	$-0.565776\pi$
$751$	−11.2453	−0.410345	−0.205173	+	0.978726i	$0.565776\pi$
$752$	0	0
$753$	26.6630	0.971655
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−17.2926	−0.628511	−0.314256	−	0.949338i	$-0.601755\pi$
$757$	−17.2926	−0.628511	−0.314256	+	0.949338i	$0.601755\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−11.6964	−0.423994	−0.211997	−	0.977270i	$-0.567997\pi$
$761$	−11.6964	−0.423994	−0.211997	+	0.977270i	$0.567997\pi$
$762$	0	0
$763$	7.70488	0.278936
$764$	0	0
$765$	−1.58774	−0.0574049
$766$	0	0
$767$	1.64903	0.0595432
$768$	0	0
$769$	−37.5653	−1.35464	−0.677320	−	0.735688i	$-0.736859\pi$
$769$	−37.5653	−1.35464	−0.677320	+	0.735688i	$0.736859\pi$
$770$	0	0
$771$	8.35097	0.300753
$772$	0	0
$773$	33.7827	1.21508	0.607539	−	0.794290i	$-0.292157\pi$
$773$	33.7827	1.21508	0.607539	+	0.794290i	$0.292157\pi$
$774$	0	0
$775$	−4.87189	−0.175003
$776$	0	0
$777$	−29.2617	−1.04976
$778$	0	0
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	−5.58774	−0.199690
$784$	0	0
$785$	12.7632	0.455539
$786$	0	0
$787$	38.6406	1.37739	0.688695	−	0.725051i	$-0.258184\pi$
$787$	38.6406	1.37739	0.688695	+	0.725051i	$0.258184\pi$
$788$	0	0
$789$	−3.12811	−0.111364
$790$	0	0
$791$	3.51797	0.125085
$792$	0	0
$793$	4.62414	0.164208
$794$	0	0
$795$	−8.30359	−0.294498
$796$	0	0
$797$	−33.7353	−1.19497	−0.597483	−	0.801882i	$-0.703832\pi$
$797$	−33.7353	−1.19497	−0.597483	+	0.801882i	$0.703832\pi$
$798$	0	0
$799$	−11.3928	−0.403049
$800$	0	0
$801$	15.7438	0.556279
$802$	0	0
$803$	0	0
$804$	0	0
$805$	−3.58774	−0.126451
$806$	0	0
$807$	−28.7632	−1.01251
$808$	0	0
$809$	38.0474	1.33767	0.668837	−	0.743409i	$-0.266792\pi$
$809$	38.0474	1.33767	0.668837	+	0.743409i	$0.266792\pi$
$810$	0	0
$811$	−24.1421	−0.847744	−0.423872	−	0.905722i	$-0.639329\pi$
$811$	−24.1421	−0.847744	−0.423872	+	0.905722i	$0.639329\pi$
$812$	0	0
$813$	−15.4402	−0.541511
$814$	0	0
$815$	16.6072	0.581724
$816$	0	0
$817$	0	0
$818$	0	0
$819$	2.56829	0.0897435
$820$	0	0
$821$	−4.18645	−0.146108	−0.0730541	−	0.997328i	$-0.523275\pi$
$821$	−4.18645	−0.146108	−0.0730541	+	0.997328i	$0.523275\pi$
$822$	0	0
$823$	−9.59622	−0.334503	−0.167252	−	0.985914i	$-0.553489\pi$
$823$	−9.59622	−0.334503	−0.167252	+	0.985914i	$0.553489\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	36.2897	1.26192	0.630958	−	0.775817i	$-0.282662\pi$
$827$	36.2897	1.26192	0.630958	+	0.775817i	$0.282662\pi$
$828$	0	0
$829$	−49.9248	−1.73396	−0.866980	−	0.498343i	$-0.833942\pi$
$829$	−49.9248	−1.73396	−0.866980	+	0.498343i	$0.833942\pi$
$830$	0	0
$831$	−22.2423	−0.771577
$832$	0	0
$833$	−9.32304	−0.323024
$834$	0	0
$835$	15.1755	0.525169
$836$	0	0
$837$	4.87189	0.168397
$838$	0	0
$839$	−49.3091	−1.70234	−0.851170	−	0.524890i	$-0.824106\pi$
$839$	−49.3091	−1.70234	−0.851170	+	0.524890i	$0.824106\pi$
$840$	0	0
$841$	2.22285	0.0766502
$842$	0	0
$843$	−30.0947	−1.03652
$844$	0	0
$845$	−12.4876	−0.429585
$846$	0	0
$847$	−39.4652	−1.35604
$848$	0	0
$849$	−23.8998	−0.820239
$850$	0	0
$851$	−8.15604	−0.279585
$852$	0	0
$853$	28.2982	0.968910	0.484455	−	0.874816i	$-0.339018\pi$
$853$	28.2982	0.968910	0.484455	+	0.874816i	$0.339018\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−20.8634	−0.712681	−0.356340	−	0.934356i	$-0.615976\pi$
$857$	−20.8634	−0.712681	−0.356340	+	0.934356i	$0.615976\pi$
$858$	0	0
$859$	5.77419	0.197013	0.0985065	−	0.995136i	$-0.468593\pi$
$859$	5.77419	0.197013	0.0985065	+	0.995136i	$0.468593\pi$
$860$	0	0
$861$	−15.4402	−0.526200
$862$	0	0
$863$	12.6072	0.429154	0.214577	−	0.976707i	$-0.431163\pi$
$863$	12.6072	0.429154	0.214577	+	0.976707i	$0.431163\pi$
$864$	0	0
$865$	17.1755	0.583984
$866$	0	0
$867$	14.4791	0.491735
$868$	0	0
$869$	0	0
$870$	0	0
$871$	4.40682	0.149319
$872$	0	0
$873$	−6.45963	−0.218625
$874$	0	0
$875$	3.58774	0.121288
$876$	0	0
$877$	2.16451	0.0730904	0.0365452	−	0.999332i	$-0.488365\pi$
$877$	2.16451	0.0730904	0.0365452	+	0.999332i	$0.488365\pi$
$878$	0	0
$879$	10.6546	0.359369
$880$	0	0
$881$	32.4288	1.09255	0.546276	−	0.837605i	$-0.316045\pi$
$881$	32.4288	1.09255	0.546276	+	0.837605i	$0.316045\pi$
$882$	0	0
$883$	4.82452	0.162358	0.0811790	−	0.996700i	$-0.474131\pi$
$883$	4.82452	0.162358	0.0811790	+	0.996700i	$0.474131\pi$
$884$	0	0
$885$	−2.30359	−0.0774345
$886$	0	0
$887$	47.7996	1.60495	0.802477	−	0.596683i	$-0.203515\pi$
$887$	47.7996	1.60495	0.802477	+	0.596683i	$0.203515\pi$
$888$	0	0
$889$	9.74378	0.326796
$890$	0	0
$891$	0	0
$892$	0	0
$893$	0	0
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0.715853	0.0239016
$898$	0	0
$899$	−27.2229	−0.907933
$900$	0	0
$901$	−13.1840	−0.439221
$902$	0	0
$903$	30.3510	1.01002
$904$	0	0
$905$	−15.0668	−0.500838
$906$	0	0
$907$	−21.6266	−0.718101	−0.359050	−	0.933318i	$-0.616899\pi$
$907$	−21.6266	−0.718101	−0.359050	+	0.933318i	$0.616899\pi$
$908$	0	0
$909$	5.58774	0.185334
$910$	0	0
$911$	−46.3510	−1.53568	−0.767838	−	0.640644i	$-0.778667\pi$
$911$	−46.3510	−1.53568	−0.767838	+	0.640644i	$0.778667\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	−6.45963	−0.213549
$916$	0	0
$917$	9.21438	0.304286
$918$	0	0
$919$	−50.8106	−1.67609	−0.838043	−	0.545603i	$-0.816300\pi$
$919$	−50.8106	−1.67609	−0.838043	+	0.545603i	$0.816300\pi$
$920$	0	0
$921$	−6.56829	−0.216433
$922$	0	0
$923$	0.729774	0.0240208
$924$	0	0
$925$	8.15604	0.268169
$926$	0	0
$927$	−5.89134	−0.193497
$928$	0	0
$929$	8.91078	0.292353	0.146177	−	0.989259i	$-0.453303\pi$
$929$	8.91078	0.292353	0.146177	+	0.989259i	$0.453303\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	22.8106	0.746785
$934$	0	0
$935$	0	0
$936$	0	0
$937$	3.81355	0.124583	0.0622916	−	0.998058i	$-0.480159\pi$
$937$	3.81355	0.124583	0.0622916	+	0.998058i	$0.480159\pi$
$938$	0	0
$939$	−12.7632	−0.416512
$940$	0	0
$941$	−37.4876	−1.22206	−0.611030	−	0.791608i	$-0.709244\pi$
$941$	−37.4876	−1.22206	−0.611030	+	0.791608i	$0.709244\pi$
$942$	0	0
$943$	−4.30359	−0.140144
$944$	0	0
$945$	−3.58774	−0.116709
$946$	0	0
$947$	−34.6461	−1.12585	−0.562923	−	0.826509i	$-0.690323\pi$
$947$	−34.6461	−1.12585	−0.562923	+	0.826509i	$0.690323\pi$
$948$	0	0
$949$	−3.70488	−0.120266
$950$	0	0
$951$	−19.2144	−0.623069
$952$	0	0
$953$	13.3230	0.431576	0.215788	−	0.976440i	$-0.430768\pi$
$953$	13.3230	0.431576	0.215788	+	0.976440i	$0.430768\pi$
$954$	0	0
$955$	−4.60719	−0.149085
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−23.1755	−0.748375
$960$	0	0
$961$	−7.26470	−0.234345
$962$	0	0
$963$	0.412259	0.0132848
$964$	0	0
$965$	−10.6072	−0.341457
$966$	0	0
$967$	−22.9332	−0.737481	−0.368741	−	0.929532i	$-0.620211\pi$
$967$	−22.9332	−0.737481	−0.368741	+	0.929532i	$0.620211\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−39.5653	−1.26971	−0.634856	−	0.772630i	$-0.718941\pi$
$971$	−39.5653	−1.26971	−0.634856	+	0.772630i	$0.718941\pi$
$972$	0	0
$973$	−28.8719	−0.925590
$974$	0	0
$975$	−0.715853	−0.0229256
$976$	0	0
$977$	−47.1142	−1.50732	−0.753658	−	0.657267i	$-0.771713\pi$
$977$	−47.1142	−1.50732	−0.753658	+	0.657267i	$0.771713\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	2.14756	0.0685663
$982$	0	0
$983$	47.3006	1.50866	0.754328	−	0.656498i	$-0.227963\pi$
$983$	47.3006	1.50866	0.754328	+	0.656498i	$0.227963\pi$
$984$	0	0
$985$	−9.78267	−0.311702
$986$	0	0
$987$	−25.7438	−0.819433
$988$	0	0
$989$	8.45963	0.269001
$990$	0	0
$991$	18.3983	0.584442	0.292221	−	0.956351i	$-0.405606\pi$
$991$	18.3983	0.584442	0.292221	+	0.956351i	$0.405606\pi$
$992$	0	0
$993$	15.2229	0.483083
$994$	0	0
$995$	−2.71585	−0.0860983
$996$	0	0
$997$	−17.4178	−0.551627	−0.275813	−	0.961211i	$-0.588947\pi$
$997$	−17.4178	−0.551627	−0.275813	+	0.961211i	$0.588947\pi$
$998$	0	0
$999$	−8.15604	−0.258046

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5520.2.a.ca.1.3		3
4.3	odd	2		2760.2.a.t.1.1	✓	3
12.11	even	2		8280.2.a.bh.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2760.2.a.t.1.1	✓	3	4.3	odd	2
5520.2.a.ca.1.3		3	1.1	even	1	trivial
8280.2.a.bh.1.1		3	12.11	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:	\( N \)	\(=\)	\( 5520 = 2^{4} \cdot 3 \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5520.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +1.00000 q^{5} +3.58774 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +1.00000 q^{5} +3.58774 q^{7} +1.00000 q^{9} +0.715853 q^{13} -1.00000 q^{15} -1.58774 q^{17} -3.58774 q^{21} -1.00000 q^{23} +1.00000 q^{25} -1.00000 q^{27} +5.58774 q^{29} -4.87189 q^{31} +3.58774 q^{35} +8.15604 q^{37} -0.715853 q^{39} +4.30359 q^{41} -8.45963 q^{43} +1.00000 q^{45} +7.17548 q^{47} +5.87189 q^{49} +1.58774 q^{51} +8.30359 q^{53} +2.30359 q^{59} +6.45963 q^{61} +3.58774 q^{63} +0.715853 q^{65} +6.15604 q^{67} +1.00000 q^{69} +1.01945 q^{71} -5.17548 q^{73} -1.00000 q^{75} -9.89134 q^{79} +1.00000 q^{81} -5.01945 q^{83} -1.58774 q^{85} -5.58774 q^{87} +15.7438 q^{89} +2.56829 q^{91} +4.87189 q^{93} -6.45963 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{3} + 3 q^{5} - q^{7} + 3 q^{9}+O(q^{10}) \) 3 * q - 3 * q^3 + 3 * q^5 - q^7 + 3 * q^9	\( 3 q - 3 q^{3} + 3 q^{5} - q^{7} + 3 q^{9} + 4 q^{13} - 3 q^{15} + 7 q^{17} + q^{21} - 3 q^{23} + 3 q^{25} - 3 q^{27} + 5 q^{29} - q^{31} - q^{35} + 9 q^{37} - 4 q^{39} + 3 q^{41} + 3 q^{45} - 2 q^{47} + 4 q^{49} - 7 q^{51} + 15 q^{53} - 3 q^{59} - 6 q^{61} - q^{63} + 4 q^{65} + 3 q^{67} + 3 q^{69} - 5 q^{71} + 8 q^{73} - 3 q^{75} - 8 q^{79} + 3 q^{81} - 7 q^{83} + 7 q^{85} - 5 q^{87} + 20 q^{89} + 4 q^{91} + q^{93} + 6 q^{97}+O(q^{100}) \) 3 * q - 3 * q^3 + 3 * q^5 - q^7 + 3 * q^9 + 4 * q^13 - 3 * q^15 + 7 * q^17 + q^21 - 3 * q^23 + 3 * q^25 - 3 * q^27 + 5 * q^29 - q^31 - q^35 + 9 * q^37 - 4 * q^39 + 3 * q^41 + 3 * q^45 - 2 * q^47 + 4 * q^49 - 7 * q^51 + 15 * q^53 - 3 * q^59 - 6 * q^61 - q^63 + 4 * q^65 + 3 * q^67 + 3 * q^69 - 5 * q^71 + 8 * q^73 - 3 * q^75 - 8 * q^79 + 3 * q^81 - 7 * q^83 + 7 * q^85 - 5 * q^87 + 20 * q^89 + 4 * q^91 + q^93 + 6 * q^97

Introduction

L-functions

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