LMFDB - Embedded newform 7440.2.a.bg.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7440,2,Mod(1,7440)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7440, 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("7440.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7440 = 2^{4} \cdot 3 \cdot 5 \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7440.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:	$59.4086991038$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{33}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 8 $ x^2 - x - 8
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 930)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-2.37228$ of defining polynomial
Character	$\chi$	$=$	7440.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +1.00000 q^{5} +2.37228 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +1.00000 q^{5} +2.37228 q^{7} +1.00000 q^{9} -6.37228 q^{11} -2.00000 q^{13} -1.00000 q^{15} +6.74456 q^{17} +6.37228 q^{19} -2.37228 q^{21} +2.37228 q^{23} +1.00000 q^{25} -1.00000 q^{27} +2.74456 q^{29} +1.00000 q^{31} +6.37228 q^{33} +2.37228 q^{35} +10.7446 q^{37} +2.00000 q^{39} -10.7446 q^{41} -6.37228 q^{43} +1.00000 q^{45} -4.74456 q^{47} -1.37228 q^{49} -6.74456 q^{51} -4.37228 q^{53} -6.37228 q^{55} -6.37228 q^{57} +8.74456 q^{59} -11.4891 q^{61} +2.37228 q^{63} -2.00000 q^{65} +0.744563 q^{67} -2.37228 q^{69} +2.37228 q^{71} +9.11684 q^{73} -1.00000 q^{75} -15.1168 q^{77} +10.3723 q^{79} +1.00000 q^{81} +12.0000 q^{83} +6.74456 q^{85} -2.74456 q^{87} +4.37228 q^{89} -4.74456 q^{91} -1.00000 q^{93} +6.37228 q^{95} +2.00000 q^{97} -6.37228 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} + 2 q^{5} - q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^3 + 2 * q^5 - q^7 + 2 * q^9	$ 2 q - 2 q^{3} + 2 q^{5} - q^{7} + 2 q^{9} - 7 q^{11} - 4 q^{13} - 2 q^{15} + 2 q^{17} + 7 q^{19} + q^{21} - q^{23} + 2 q^{25} - 2 q^{27} - 6 q^{29} + 2 q^{31} + 7 q^{33} - q^{35} + 10 q^{37} + 4 q^{39} - 10 q^{41} - 7 q^{43} + 2 q^{45} + 2 q^{47} + 3 q^{49} - 2 q^{51} - 3 q^{53} - 7 q^{55} - 7 q^{57} + 6 q^{59} - q^{63} - 4 q^{65} - 10 q^{67} + q^{69} - q^{71} + q^{73} - 2 q^{75} - 13 q^{77} + 15 q^{79} + 2 q^{81} + 24 q^{83} + 2 q^{85} + 6 q^{87} + 3 q^{89} + 2 q^{91} - 2 q^{93} + 7 q^{95} + 4 q^{97} - 7 q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 + 2 * q^5 - q^7 + 2 * q^9 - 7 * q^11 - 4 * q^13 - 2 * q^15 + 2 * q^17 + 7 * q^19 + q^21 - q^23 + 2 * q^25 - 2 * q^27 - 6 * q^29 + 2 * q^31 + 7 * q^33 - q^35 + 10 * q^37 + 4 * q^39 - 10 * q^41 - 7 * q^43 + 2 * q^45 + 2 * q^47 + 3 * q^49 - 2 * q^51 - 3 * q^53 - 7 * q^55 - 7 * q^57 + 6 * q^59 - q^63 - 4 * q^65 - 10 * q^67 + q^69 - q^71 + q^73 - 2 * q^75 - 13 * q^77 + 15 * q^79 + 2 * q^81 + 24 * q^83 + 2 * q^85 + 6 * q^87 + 3 * q^89 + 2 * q^91 - 2 * q^93 + 7 * q^95 + 4 * q^97 - 7 * q^99

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
$7$	2.37228	0.896638	0.448319	−	0.893874i	$-0.352023\pi$
$7$	2.37228	0.896638	0.448319	+	0.893874i	$0.352023\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−6.37228	−1.92132	−0.960658	−	0.277736i	$-0.910416\pi$
$11$	−6.37228	−1.92132	−0.960658	+	0.277736i	$0.910416\pi$
$12$	0	0
$13$	−2.00000	−0.554700	−0.277350	−	0.960769i	$-0.589456\pi$
$13$	−2.00000	−0.554700	−0.277350	+	0.960769i	$0.589456\pi$
$14$	0	0
$15$	−1.00000	−0.258199
$16$	0	0
$17$	6.74456	1.63580	0.817898	−	0.575363i	$-0.195139\pi$
$17$	6.74456	1.63580	0.817898	+	0.575363i	$0.195139\pi$
$18$	0	0
$19$	6.37228	1.46190	0.730951	−	0.682430i	$-0.239077\pi$
$19$	6.37228	1.46190	0.730951	+	0.682430i	$0.239077\pi$
$20$	0	0
$21$	−2.37228	−0.517674
$22$	0	0
$23$	2.37228	0.494655	0.247327	−	0.968932i	$-0.420448\pi$
$23$	2.37228	0.494655	0.247327	+	0.968932i	$0.420448\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	2.74456	0.509652	0.254826	−	0.966987i	$-0.417982\pi$
$29$	2.74456	0.509652	0.254826	+	0.966987i	$0.417982\pi$
$30$	0	0
$31$	1.00000	0.179605
$31$	1.00000	0.179605
$32$	0	0
$33$	6.37228	1.10927
$34$	0	0
$35$	2.37228	0.400989
$36$	0	0
$37$	10.7446	1.76640	0.883198	−	0.469001i	$-0.155386\pi$
$37$	10.7446	1.76640	0.883198	+	0.469001i	$0.155386\pi$
$38$	0	0
$39$	2.00000	0.320256
$40$	0	0
$41$	−10.7446	−1.67802	−0.839009	−	0.544117i	$-0.816865\pi$
$41$	−10.7446	−1.67802	−0.839009	+	0.544117i	$0.816865\pi$
$42$	0	0
$43$	−6.37228	−0.971764	−0.485882	−	0.874024i	$-0.661501\pi$
$43$	−6.37228	−0.971764	−0.485882	+	0.874024i	$0.661501\pi$
$44$	0	0
$45$	1.00000	0.149071
$46$	0	0
$47$	−4.74456	−0.692066	−0.346033	−	0.938222i	$-0.612471\pi$
$47$	−4.74456	−0.692066	−0.346033	+	0.938222i	$0.612471\pi$
$48$	0	0
$49$	−1.37228	−0.196040
$50$	0	0
$51$	−6.74456	−0.944428
$52$	0	0
$53$	−4.37228	−0.600579	−0.300290	−	0.953848i	$-0.597083\pi$
$53$	−4.37228	−0.600579	−0.300290	+	0.953848i	$0.597083\pi$
$54$	0	0
$55$	−6.37228	−0.859238
$56$	0	0
$57$	−6.37228	−0.844029
$58$	0	0
$59$	8.74456	1.13845	0.569223	−	0.822183i	$-0.307244\pi$
$59$	8.74456	1.13845	0.569223	+	0.822183i	$0.307244\pi$
$60$	0	0
$61$	−11.4891	−1.47103	−0.735516	−	0.677507i	$-0.763060\pi$
$61$	−11.4891	−1.47103	−0.735516	+	0.677507i	$0.763060\pi$
$62$	0	0
$63$	2.37228	0.298879
$64$	0	0
$65$	−2.00000	−0.248069
$66$	0	0
$67$	0.744563	0.0909628	0.0454814	−	0.998965i	$-0.485518\pi$
$67$	0.744563	0.0909628	0.0454814	+	0.998965i	$0.485518\pi$
$68$	0	0
$69$	−2.37228	−0.285589
$70$	0	0
$71$	2.37228	0.281538	0.140769	−	0.990042i	$-0.455043\pi$
$71$	2.37228	0.281538	0.140769	+	0.990042i	$0.455043\pi$
$72$	0	0
$73$	9.11684	1.06705	0.533523	−	0.845786i	$-0.320868\pi$
$73$	9.11684	1.06705	0.533523	+	0.845786i	$0.320868\pi$
$74$	0	0
$75$	−1.00000	−0.115470
$76$	0	0
$77$	−15.1168	−1.72272
$78$	0	0
$79$	10.3723	1.16697	0.583486	−	0.812123i	$-0.301688\pi$
$79$	10.3723	1.16697	0.583486	+	0.812123i	$0.301688\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	12.0000	1.31717	0.658586	−	0.752506i	$-0.271155\pi$
$83$	12.0000	1.31717	0.658586	+	0.752506i	$0.271155\pi$
$84$	0	0
$85$	6.74456	0.731551
$86$	0	0
$87$	−2.74456	−0.294248
$88$	0	0
$89$	4.37228	0.463461	0.231730	−	0.972780i	$-0.425561\pi$
$89$	4.37228	0.463461	0.231730	+	0.972780i	$0.425561\pi$
$90$	0	0
$91$	−4.74456	−0.497365
$92$	0	0
$93$	−1.00000	−0.103695
$94$	0	0
$95$	6.37228	0.653782
$96$	0	0
$97$	2.00000	0.203069	0.101535	−	0.994832i	$-0.467625\pi$
$97$	2.00000	0.203069	0.101535	+	0.994832i	$0.467625\pi$
$98$	0	0
$99$	−6.37228	−0.640438
$100$	0	0
$101$	−9.11684	−0.907160	−0.453580	−	0.891216i	$-0.649853\pi$
$101$	−9.11684	−0.907160	−0.453580	+	0.891216i	$0.649853\pi$
$102$	0	0
$103$	8.00000	0.788263	0.394132	−	0.919054i	$-0.371045\pi$
$103$	8.00000	0.788263	0.394132	+	0.919054i	$0.371045\pi$
$104$	0	0
$105$	−2.37228	−0.231511
$106$	0	0
$107$	−6.37228	−0.616032	−0.308016	−	0.951381i	$-0.599665\pi$
$107$	−6.37228	−0.616032	−0.308016	+	0.951381i	$0.599665\pi$
$108$	0	0
$109$	−6.74456	−0.646012	−0.323006	−	0.946397i	$-0.604693\pi$
$109$	−6.74456	−0.646012	−0.323006	+	0.946397i	$0.604693\pi$
$110$	0	0
$111$	−10.7446	−1.01983
$112$	0	0
$113$	−8.37228	−0.787598	−0.393799	−	0.919197i	$-0.628839\pi$
$113$	−8.37228	−0.787598	−0.393799	+	0.919197i	$0.628839\pi$
$114$	0	0
$115$	2.37228	0.221216
$116$	0	0
$117$	−2.00000	−0.184900
$118$	0	0
$119$	16.0000	1.46672
$120$	0	0
$121$	29.6060	2.69145
$122$	0	0
$123$	10.7446	0.968805
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	9.48913	0.842024	0.421012	−	0.907055i	$-0.361675\pi$
$127$	9.48913	0.842024	0.421012	+	0.907055i	$0.361675\pi$
$128$	0	0
$129$	6.37228	0.561048
$130$	0	0
$131$	−18.2337	−1.59308	−0.796542	−	0.604583i	$-0.793340\pi$
$131$	−18.2337	−1.59308	−0.796542	+	0.604583i	$0.793340\pi$
$132$	0	0
$133$	15.1168	1.31080
$134$	0	0
$135$	−1.00000	−0.0860663
$136$	0	0
$137$	19.4891	1.66507	0.832534	−	0.553974i	$-0.186889\pi$
$137$	19.4891	1.66507	0.832534	+	0.553974i	$0.186889\pi$
$138$	0	0
$139$	−0.744563	−0.0631530	−0.0315765	−	0.999501i	$-0.510053\pi$
$139$	−0.744563	−0.0631530	−0.0315765	+	0.999501i	$0.510053\pi$
$140$	0	0
$141$	4.74456	0.399564
$142$	0	0
$143$	12.7446	1.06575
$144$	0	0
$145$	2.74456	0.227924
$146$	0	0
$147$	1.37228	0.113184
$148$	0	0
$149$	5.11684	0.419188	0.209594	−	0.977788i	$-0.432786\pi$
$149$	5.11684	0.419188	0.209594	+	0.977788i	$0.432786\pi$
$150$	0	0
$151$	8.00000	0.651031	0.325515	−	0.945537i	$-0.394462\pi$
$151$	8.00000	0.651031	0.325515	+	0.945537i	$0.394462\pi$
$152$	0	0
$153$	6.74456	0.545266
$154$	0	0
$155$	1.00000	0.0803219
$156$	0	0
$157$	3.62772	0.289523	0.144762	−	0.989467i	$-0.453758\pi$
$157$	3.62772	0.289523	0.144762	+	0.989467i	$0.453758\pi$
$158$	0	0
$159$	4.37228	0.346744
$160$	0	0
$161$	5.62772	0.443526
$162$	0	0
$163$	10.2337	0.801564	0.400782	−	0.916173i	$-0.368738\pi$
$163$	10.2337	0.801564	0.400782	+	0.916173i	$0.368738\pi$
$164$	0	0
$165$	6.37228	0.496081
$166$	0	0
$167$	18.3723	1.42169	0.710845	−	0.703349i	$-0.248313\pi$
$167$	18.3723	1.42169	0.710845	+	0.703349i	$0.248313\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	0	0
$171$	6.37228	0.487301
$172$	0	0
$173$	−18.0000	−1.36851	−0.684257	−	0.729241i	$-0.739873\pi$
$173$	−18.0000	−1.36851	−0.684257	+	0.729241i	$0.739873\pi$
$174$	0	0
$175$	2.37228	0.179328
$176$	0	0
$177$	−8.74456	−0.657282
$178$	0	0
$179$	12.0000	0.896922	0.448461	−	0.893802i	$-0.351972\pi$
$179$	12.0000	0.896922	0.448461	+	0.893802i	$0.351972\pi$
$180$	0	0
$181$	−13.8614	−1.03031	−0.515155	−	0.857097i	$-0.672266\pi$
$181$	−13.8614	−1.03031	−0.515155	+	0.857097i	$0.672266\pi$
$182$	0	0
$183$	11.4891	0.849301
$184$	0	0
$185$	10.7446	0.789956
$186$	0	0
$187$	−42.9783	−3.14288
$188$	0	0
$189$	−2.37228	−0.172558
$190$	0	0
$191$	−16.0000	−1.15772	−0.578860	−	0.815427i	$-0.696502\pi$
$191$	−16.0000	−1.15772	−0.578860	+	0.815427i	$0.696502\pi$
$192$	0	0
$193$	−7.48913	−0.539079	−0.269540	−	0.962989i	$-0.586871\pi$
$193$	−7.48913	−0.539079	−0.269540	+	0.962989i	$0.586871\pi$
$194$	0	0
$195$	2.00000	0.143223
$196$	0	0
$197$	−3.48913	−0.248590	−0.124295	−	0.992245i	$-0.539667\pi$
$197$	−3.48913	−0.248590	−0.124295	+	0.992245i	$0.539667\pi$
$198$	0	0
$199$	18.3723	1.30238	0.651188	−	0.758916i	$-0.274271\pi$
$199$	18.3723	1.30238	0.651188	+	0.758916i	$0.274271\pi$
$200$	0	0
$201$	−0.744563	−0.0525174
$202$	0	0
$203$	6.51087	0.456974
$204$	0	0
$205$	−10.7446	−0.750433
$206$	0	0
$207$	2.37228	0.164885
$208$	0	0
$209$	−40.6060	−2.80877
$210$	0	0
$211$	6.37228	0.438686	0.219343	−	0.975648i	$-0.429609\pi$
$211$	6.37228	0.438686	0.219343	+	0.975648i	$0.429609\pi$
$212$	0	0
$213$	−2.37228	−0.162546
$214$	0	0
$215$	−6.37228	−0.434586
$216$	0	0
$217$	2.37228	0.161041
$218$	0	0
$219$	−9.11684	−0.616059
$220$	0	0
$221$	−13.4891	−0.907377
$222$	0	0
$223$	−20.7446	−1.38916	−0.694579	−	0.719416i	$-0.744409\pi$
$223$	−20.7446	−1.38916	−0.694579	+	0.719416i	$0.744409\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	11.1168	0.737851	0.368925	−	0.929459i	$-0.379726\pi$
$227$	11.1168	0.737851	0.368925	+	0.929459i	$0.379726\pi$
$228$	0	0
$229$	21.1168	1.39544	0.697720	−	0.716370i	$-0.254198\pi$
$229$	21.1168	1.39544	0.697720	+	0.716370i	$0.254198\pi$
$230$	0	0
$231$	15.1168	0.994615
$232$	0	0
$233$	13.8614	0.908091	0.454045	−	0.890979i	$-0.349980\pi$
$233$	13.8614	0.908091	0.454045	+	0.890979i	$0.349980\pi$
$234$	0	0
$235$	−4.74456	−0.309501
$236$	0	0
$237$	−10.3723	−0.673752
$238$	0	0
$239$	6.51087	0.421153	0.210577	−	0.977577i	$-0.432466\pi$
$239$	6.51087	0.421153	0.210577	+	0.977577i	$0.432466\pi$
$240$	0	0
$241$	27.4891	1.77073	0.885365	−	0.464896i	$-0.153908\pi$
$241$	27.4891	1.77073	0.885365	+	0.464896i	$0.153908\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−1.37228	−0.0876718
$246$	0	0
$247$	−12.7446	−0.810917
$248$	0	0
$249$	−12.0000	−0.760469
$250$	0	0
$251$	4.00000	0.252478	0.126239	−	0.992000i	$-0.459709\pi$
$251$	4.00000	0.252478	0.126239	+	0.992000i	$0.459709\pi$
$252$	0	0
$253$	−15.1168	−0.950388
$254$	0	0
$255$	−6.74456	−0.422361
$256$	0	0
$257$	7.62772	0.475804	0.237902	−	0.971289i	$-0.423540\pi$
$257$	7.62772	0.475804	0.237902	+	0.971289i	$0.423540\pi$
$258$	0	0
$259$	25.4891	1.58382
$260$	0	0
$261$	2.74456	0.169884
$262$	0	0
$263$	−26.9783	−1.66355	−0.831775	−	0.555113i	$-0.812675\pi$
$263$	−26.9783	−1.66355	−0.831775	+	0.555113i	$0.812675\pi$
$264$	0	0
$265$	−4.37228	−0.268587
$266$	0	0
$267$	−4.37228	−0.267579
$268$	0	0
$269$	−2.00000	−0.121942	−0.0609711	−	0.998140i	$-0.519420\pi$
$269$	−2.00000	−0.121942	−0.0609711	+	0.998140i	$0.519420\pi$
$270$	0	0
$271$	31.1168	1.89021	0.945107	−	0.326762i	$-0.105957\pi$
$271$	31.1168	1.89021	0.945107	+	0.326762i	$0.105957\pi$
$272$	0	0
$273$	4.74456	0.287154
$274$	0	0
$275$	−6.37228	−0.384263
$276$	0	0
$277$	26.7446	1.60693	0.803463	−	0.595355i	$-0.202989\pi$
$277$	26.7446	1.60693	0.803463	+	0.595355i	$0.202989\pi$
$278$	0	0
$279$	1.00000	0.0598684
$280$	0	0
$281$	5.25544	0.313513	0.156757	−	0.987637i	$-0.449896\pi$
$281$	5.25544	0.313513	0.156757	+	0.987637i	$0.449896\pi$
$282$	0	0
$283$	−12.0000	−0.713326	−0.356663	−	0.934233i	$-0.616086\pi$
$283$	−12.0000	−0.713326	−0.356663	+	0.934233i	$0.616086\pi$
$284$	0	0
$285$	−6.37228	−0.377461
$286$	0	0
$287$	−25.4891	−1.50458
$288$	0	0
$289$	28.4891	1.67583
$290$	0	0
$291$	−2.00000	−0.117242
$292$	0	0
$293$	15.4891	0.904884	0.452442	−	0.891794i	$-0.350553\pi$
$293$	15.4891	0.904884	0.452442	+	0.891794i	$0.350553\pi$
$294$	0	0
$295$	8.74456	0.509128
$296$	0	0
$297$	6.37228	0.369757
$298$	0	0
$299$	−4.74456	−0.274385
$300$	0	0
$301$	−15.1168	−0.871320
$302$	0	0
$303$	9.11684	0.523749
$304$	0	0
$305$	−11.4891	−0.657865
$306$	0	0
$307$	14.9783	0.854854	0.427427	−	0.904050i	$-0.359420\pi$
$307$	14.9783	0.854854	0.427427	+	0.904050i	$0.359420\pi$
$308$	0	0
$309$	−8.00000	−0.455104
$310$	0	0
$311$	−8.00000	−0.453638	−0.226819	−	0.973937i	$-0.572833\pi$
$311$	−8.00000	−0.453638	−0.226819	+	0.973937i	$0.572833\pi$
$312$	0	0
$313$	10.0000	0.565233	0.282617	−	0.959233i	$-0.408798\pi$
$313$	10.0000	0.565233	0.282617	+	0.959233i	$0.408798\pi$
$314$	0	0
$315$	2.37228	0.133663
$316$	0	0
$317$	−13.2554	−0.744500	−0.372250	−	0.928133i	$-0.621414\pi$
$317$	−13.2554	−0.744500	−0.372250	+	0.928133i	$0.621414\pi$
$318$	0	0
$319$	−17.4891	−0.979203
$320$	0	0
$321$	6.37228	0.355666
$322$	0	0
$323$	42.9783	2.39137
$324$	0	0
$325$	−2.00000	−0.110940
$326$	0	0
$327$	6.74456	0.372975
$328$	0	0
$329$	−11.2554	−0.620532
$330$	0	0
$331$	−21.4891	−1.18115	−0.590575	−	0.806983i	$-0.701099\pi$
$331$	−21.4891	−1.18115	−0.590575	+	0.806983i	$0.701099\pi$
$332$	0	0
$333$	10.7446	0.588798
$334$	0	0
$335$	0.744563	0.0406798
$336$	0	0
$337$	−16.9783	−0.924864	−0.462432	−	0.886655i	$-0.653023\pi$
$337$	−16.9783	−0.924864	−0.462432	+	0.886655i	$0.653023\pi$
$338$	0	0
$339$	8.37228	0.454720
$340$	0	0
$341$	−6.37228	−0.345078
$342$	0	0
$343$	−19.8614	−1.07242
$344$	0	0
$345$	−2.37228	−0.127719
$346$	0	0
$347$	32.4674	1.74294	0.871470	−	0.490449i	$-0.163167\pi$
$347$	32.4674	1.74294	0.871470	+	0.490449i	$0.163167\pi$
$348$	0	0
$349$	18.7446	1.00337	0.501687	−	0.865049i	$-0.332713\pi$
$349$	18.7446	1.00337	0.501687	+	0.865049i	$0.332713\pi$
$350$	0	0
$351$	2.00000	0.106752
$352$	0	0
$353$	−7.48913	−0.398606	−0.199303	−	0.979938i	$-0.563868\pi$
$353$	−7.48913	−0.398606	−0.199303	+	0.979938i	$0.563868\pi$
$354$	0	0
$355$	2.37228	0.125908
$356$	0	0
$357$	−16.0000	−0.846810
$358$	0	0
$359$	2.37228	0.125204	0.0626021	−	0.998039i	$-0.480060\pi$
$359$	2.37228	0.125204	0.0626021	+	0.998039i	$0.480060\pi$
$360$	0	0
$361$	21.6060	1.13716
$362$	0	0
$363$	−29.6060	−1.55391
$364$	0	0
$365$	9.11684	0.477197
$366$	0	0
$367$	−14.2337	−0.742992	−0.371496	−	0.928434i	$-0.621155\pi$
$367$	−14.2337	−0.742992	−0.371496	+	0.928434i	$0.621155\pi$
$368$	0	0
$369$	−10.7446	−0.559340
$370$	0	0
$371$	−10.3723	−0.538502
$372$	0	0
$373$	−13.8614	−0.717716	−0.358858	−	0.933392i	$-0.616834\pi$
$373$	−13.8614	−0.717716	−0.358858	+	0.933392i	$0.616834\pi$
$374$	0	0
$375$	−1.00000	−0.0516398
$376$	0	0
$377$	−5.48913	−0.282704
$378$	0	0
$379$	0.138593	0.00711906	0.00355953	−	0.999994i	$-0.498867\pi$
$379$	0.138593	0.00711906	0.00355953	+	0.999994i	$0.498867\pi$
$380$	0	0
$381$	−9.48913	−0.486143
$382$	0	0
$383$	9.48913	0.484872	0.242436	−	0.970167i	$-0.422054\pi$
$383$	9.48913	0.484872	0.242436	+	0.970167i	$0.422054\pi$
$384$	0	0
$385$	−15.1168	−0.770426
$386$	0	0
$387$	−6.37228	−0.323921
$388$	0	0
$389$	−28.9783	−1.46926	−0.734628	−	0.678470i	$-0.762643\pi$
$389$	−28.9783	−1.46926	−0.734628	+	0.678470i	$0.762643\pi$
$390$	0	0
$391$	16.0000	0.809155
$392$	0	0
$393$	18.2337	0.919768
$394$	0	0
$395$	10.3723	0.521886
$396$	0	0
$397$	38.6060	1.93758	0.968789	−	0.247887i	$-0.0797361\pi$
$397$	38.6060	1.93758	0.968789	+	0.247887i	$0.0797361\pi$
$398$	0	0
$399$	−15.1168	−0.756789
$400$	0	0
$401$	28.3723	1.41684	0.708422	−	0.705789i	$-0.249407\pi$
$401$	28.3723	1.41684	0.708422	+	0.705789i	$0.249407\pi$
$402$	0	0
$403$	−2.00000	−0.0996271
$404$	0	0
$405$	1.00000	0.0496904
$406$	0	0
$407$	−68.4674	−3.39380
$408$	0	0
$409$	24.2337	1.19828	0.599139	−	0.800645i	$-0.295510\pi$
$409$	24.2337	1.19828	0.599139	+	0.800645i	$0.295510\pi$
$410$	0	0
$411$	−19.4891	−0.961328
$412$	0	0
$413$	20.7446	1.02077
$414$	0	0
$415$	12.0000	0.589057
$416$	0	0
$417$	0.744563	0.0364614
$418$	0	0
$419$	12.0000	0.586238	0.293119	−	0.956076i	$-0.405307\pi$
$419$	12.0000	0.586238	0.293119	+	0.956076i	$0.405307\pi$
$420$	0	0
$421$	−3.48913	−0.170050	−0.0850248	−	0.996379i	$-0.527097\pi$
$421$	−3.48913	−0.170050	−0.0850248	+	0.996379i	$0.527097\pi$
$422$	0	0
$423$	−4.74456	−0.230689
$424$	0	0
$425$	6.74456	0.327159
$426$	0	0
$427$	−27.2554	−1.31898
$428$	0	0
$429$	−12.7446	−0.615313
$430$	0	0
$431$	−18.9783	−0.914150	−0.457075	−	0.889428i	$-0.651103\pi$
$431$	−18.9783	−0.914150	−0.457075	+	0.889428i	$0.651103\pi$
$432$	0	0
$433$	18.8832	0.907467	0.453733	−	0.891138i	$-0.350092\pi$
$433$	18.8832	0.907467	0.453733	+	0.891138i	$0.350092\pi$
$434$	0	0
$435$	−2.74456	−0.131592
$436$	0	0
$437$	15.1168	0.723137
$438$	0	0
$439$	−3.25544	−0.155374	−0.0776868	−	0.996978i	$-0.524753\pi$
$439$	−3.25544	−0.155374	−0.0776868	+	0.996978i	$0.524753\pi$
$440$	0	0
$441$	−1.37228	−0.0653467
$442$	0	0
$443$	17.3505	0.824349	0.412174	−	0.911105i	$-0.364769\pi$
$443$	17.3505	0.824349	0.412174	+	0.911105i	$0.364769\pi$
$444$	0	0
$445$	4.37228	0.207266
$446$	0	0
$447$	−5.11684	−0.242018
$448$	0	0
$449$	36.9783	1.74511	0.872556	−	0.488515i	$-0.162461\pi$
$449$	36.9783	1.74511	0.872556	+	0.488515i	$0.162461\pi$
$450$	0	0
$451$	68.4674	3.22400
$452$	0	0
$453$	−8.00000	−0.375873
$454$	0	0
$455$	−4.74456	−0.222429
$456$	0	0
$457$	−34.4674	−1.61232	−0.806158	−	0.591700i	$-0.798457\pi$
$457$	−34.4674	−1.61232	−0.806158	+	0.591700i	$0.798457\pi$
$458$	0	0
$459$	−6.74456	−0.314809
$460$	0	0
$461$	37.7228	1.75693	0.878463	−	0.477810i	$-0.158569\pi$
$461$	37.7228	1.75693	0.878463	+	0.477810i	$0.158569\pi$
$462$	0	0
$463$	−25.4891	−1.18458	−0.592290	−	0.805725i	$-0.701776\pi$
$463$	−25.4891	−1.18458	−0.592290	+	0.805725i	$0.701776\pi$
$464$	0	0
$465$	−1.00000	−0.0463739
$466$	0	0
$467$	−29.4891	−1.36459	−0.682297	−	0.731075i	$-0.739019\pi$
$467$	−29.4891	−1.36459	−0.682297	+	0.731075i	$0.739019\pi$
$468$	0	0
$469$	1.76631	0.0815607
$470$	0	0
$471$	−3.62772	−0.167156
$472$	0	0
$473$	40.6060	1.86706
$474$	0	0
$475$	6.37228	0.292380
$476$	0	0
$477$	−4.37228	−0.200193
$478$	0	0
$479$	19.8614	0.907491	0.453745	−	0.891131i	$-0.350088\pi$
$479$	19.8614	0.907491	0.453745	+	0.891131i	$0.350088\pi$
$480$	0	0
$481$	−21.4891	−0.979820
$482$	0	0
$483$	−5.62772	−0.256070
$484$	0	0
$485$	2.00000	0.0908153
$486$	0	0
$487$	14.5109	0.657550	0.328775	−	0.944408i	$-0.393364\pi$
$487$	14.5109	0.657550	0.328775	+	0.944408i	$0.393364\pi$
$488$	0	0
$489$	−10.2337	−0.462783
$490$	0	0
$491$	12.6060	0.568899	0.284450	−	0.958691i	$-0.408189\pi$
$491$	12.6060	0.568899	0.284450	+	0.958691i	$0.408189\pi$
$492$	0	0
$493$	18.5109	0.833688
$494$	0	0
$495$	−6.37228	−0.286413
$496$	0	0
$497$	5.62772	0.252438
$498$	0	0
$499$	−10.5109	−0.470531	−0.235266	−	0.971931i	$-0.575596\pi$
$499$	−10.5109	−0.470531	−0.235266	+	0.971931i	$0.575596\pi$
$500$	0	0
$501$	−18.3723	−0.820813
$502$	0	0
$503$	17.4891	0.779802	0.389901	−	0.920857i	$-0.372509\pi$
$503$	17.4891	0.779802	0.389901	+	0.920857i	$0.372509\pi$
$504$	0	0
$505$	−9.11684	−0.405694
$506$	0	0
$507$	9.00000	0.399704
$508$	0	0
$509$	−38.7446	−1.71732	−0.858661	−	0.512543i	$-0.828703\pi$
$509$	−38.7446	−1.71732	−0.858661	+	0.512543i	$0.828703\pi$
$510$	0	0
$511$	21.6277	0.956754
$512$	0	0
$513$	−6.37228	−0.281343
$514$	0	0
$515$	8.00000	0.352522
$516$	0	0
$517$	30.2337	1.32968
$518$	0	0
$519$	18.0000	0.790112
$520$	0	0
$521$	−8.97825	−0.393344	−0.196672	−	0.980469i	$-0.563013\pi$
$521$	−8.97825	−0.393344	−0.196672	+	0.980469i	$0.563013\pi$
$522$	0	0
$523$	−15.8614	−0.693571	−0.346785	−	0.937944i	$-0.612727\pi$
$523$	−15.8614	−0.693571	−0.346785	+	0.937944i	$0.612727\pi$
$524$	0	0
$525$	−2.37228	−0.103535
$526$	0	0
$527$	6.74456	0.293798
$528$	0	0
$529$	−17.3723	−0.755317
$530$	0	0
$531$	8.74456	0.379482
$532$	0	0
$533$	21.4891	0.930797
$534$	0	0
$535$	−6.37228	−0.275498
$536$	0	0
$537$	−12.0000	−0.517838
$538$	0	0
$539$	8.74456	0.376655
$540$	0	0
$541$	7.48913	0.321983	0.160991	−	0.986956i	$-0.448531\pi$
$541$	7.48913	0.321983	0.160991	+	0.986956i	$0.448531\pi$
$542$	0	0
$543$	13.8614	0.594850
$544$	0	0
$545$	−6.74456	−0.288905
$546$	0	0
$547$	−8.74456	−0.373890	−0.186945	−	0.982370i	$-0.559859\pi$
$547$	−8.74456	−0.373890	−0.186945	+	0.982370i	$0.559859\pi$
$548$	0	0
$549$	−11.4891	−0.490344
$550$	0	0
$551$	17.4891	0.745062
$552$	0	0
$553$	24.6060	1.04635
$554$	0	0
$555$	−10.7446	−0.456081
$556$	0	0
$557$	29.1168	1.23372	0.616860	−	0.787073i	$-0.288404\pi$
$557$	29.1168	1.23372	0.616860	+	0.787073i	$0.288404\pi$
$558$	0	0
$559$	12.7446	0.539038
$560$	0	0
$561$	42.9783	1.81454
$562$	0	0
$563$	−38.9783	−1.64274	−0.821369	−	0.570398i	$-0.806789\pi$
$563$	−38.9783	−1.64274	−0.821369	+	0.570398i	$0.806789\pi$
$564$	0	0
$565$	−8.37228	−0.352225
$566$	0	0
$567$	2.37228	0.0996265
$568$	0	0
$569$	4.37228	0.183296	0.0916478	−	0.995791i	$-0.470787\pi$
$569$	4.37228	0.183296	0.0916478	+	0.995791i	$0.470787\pi$
$570$	0	0
$571$	15.2554	0.638420	0.319210	−	0.947684i	$-0.396582\pi$
$571$	15.2554	0.638420	0.319210	+	0.947684i	$0.396582\pi$
$572$	0	0
$573$	16.0000	0.668410
$574$	0	0
$575$	2.37228	0.0989310
$576$	0	0
$577$	32.2337	1.34191	0.670953	−	0.741500i	$-0.265885\pi$
$577$	32.2337	1.34191	0.670953	+	0.741500i	$0.265885\pi$
$578$	0	0
$579$	7.48913	0.311237
$580$	0	0
$581$	28.4674	1.18103
$582$	0	0
$583$	27.8614	1.15390
$584$	0	0
$585$	−2.00000	−0.0826898
$586$	0	0
$587$	−12.0000	−0.495293	−0.247647	−	0.968850i	$-0.579657\pi$
$587$	−12.0000	−0.495293	−0.247647	+	0.968850i	$0.579657\pi$
$588$	0	0
$589$	6.37228	0.262565
$590$	0	0
$591$	3.48913	0.143523
$592$	0	0
$593$	−0.978251	−0.0401719	−0.0200860	−	0.999798i	$-0.506394\pi$
$593$	−0.978251	−0.0401719	−0.0200860	+	0.999798i	$0.506394\pi$
$594$	0	0
$595$	16.0000	0.655936
$596$	0	0
$597$	−18.3723	−0.751927
$598$	0	0
$599$	7.11684	0.290786	0.145393	−	0.989374i	$-0.453555\pi$
$599$	7.11684	0.290786	0.145393	+	0.989374i	$0.453555\pi$
$600$	0	0
$601$	35.4891	1.44763	0.723816	−	0.689993i	$-0.242387\pi$
$601$	35.4891	1.44763	0.723816	+	0.689993i	$0.242387\pi$
$602$	0	0
$603$	0.744563	0.0303209
$604$	0	0
$605$	29.6060	1.20365
$606$	0	0
$607$	10.3723	0.420998	0.210499	−	0.977594i	$-0.432491\pi$
$607$	10.3723	0.420998	0.210499	+	0.977594i	$0.432491\pi$
$608$	0	0
$609$	−6.51087	−0.263834
$610$	0	0
$611$	9.48913	0.383889
$612$	0	0
$613$	12.5109	0.505309	0.252655	−	0.967557i	$-0.418696\pi$
$613$	12.5109	0.505309	0.252655	+	0.967557i	$0.418696\pi$
$614$	0	0
$615$	10.7446	0.433263
$616$	0	0
$617$	25.1168	1.01117	0.505583	−	0.862778i	$-0.331277\pi$
$617$	25.1168	1.01117	0.505583	+	0.862778i	$0.331277\pi$
$618$	0	0
$619$	18.2337	0.732874	0.366437	−	0.930443i	$-0.380578\pi$
$619$	18.2337	0.732874	0.366437	+	0.930443i	$0.380578\pi$
$620$	0	0
$621$	−2.37228	−0.0951964
$622$	0	0
$623$	10.3723	0.415557
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	40.6060	1.62165
$628$	0	0
$629$	72.4674	2.88946
$630$	0	0
$631$	−5.35053	−0.213001	−0.106501	−	0.994313i	$-0.533965\pi$
$631$	−5.35053	−0.213001	−0.106501	+	0.994313i	$0.533965\pi$
$632$	0	0
$633$	−6.37228	−0.253275
$634$	0	0
$635$	9.48913	0.376564
$636$	0	0
$637$	2.74456	0.108744
$638$	0	0
$639$	2.37228	0.0938460
$640$	0	0
$641$	18.0000	0.710957	0.355479	−	0.934684i	$-0.384318\pi$
$641$	18.0000	0.710957	0.355479	+	0.934684i	$0.384318\pi$
$642$	0	0
$643$	−38.0951	−1.50232	−0.751162	−	0.660118i	$-0.770506\pi$
$643$	−38.0951	−1.50232	−0.751162	+	0.660118i	$0.770506\pi$
$644$	0	0
$645$	6.37228	0.250908
$646$	0	0
$647$	35.5842	1.39896	0.699480	−	0.714652i	$-0.253415\pi$
$647$	35.5842	1.39896	0.699480	+	0.714652i	$0.253415\pi$
$648$	0	0
$649$	−55.7228	−2.18731
$650$	0	0
$651$	−2.37228	−0.0929770
$652$	0	0
$653$	−4.97825	−0.194814	−0.0974070	−	0.995245i	$-0.531055\pi$
$653$	−4.97825	−0.194814	−0.0974070	+	0.995245i	$0.531055\pi$
$654$	0	0
$655$	−18.2337	−0.712449
$656$	0	0
$657$	9.11684	0.355682
$658$	0	0
$659$	−36.0000	−1.40236	−0.701180	−	0.712984i	$-0.747343\pi$
$659$	−36.0000	−1.40236	−0.701180	+	0.712984i	$0.747343\pi$
$660$	0	0
$661$	40.9783	1.59387	0.796935	−	0.604066i	$-0.206454\pi$
$661$	40.9783	1.59387	0.796935	+	0.604066i	$0.206454\pi$
$662$	0	0
$663$	13.4891	0.523874
$664$	0	0
$665$	15.1168	0.586206
$666$	0	0
$667$	6.51087	0.252102
$668$	0	0
$669$	20.7446	0.802031
$670$	0	0
$671$	73.2119	2.82632
$672$	0	0
$673$	−23.4891	−0.905439	−0.452720	−	0.891653i	$-0.649546\pi$
$673$	−23.4891	−0.905439	−0.452720	+	0.891653i	$0.649546\pi$
$674$	0	0
$675$	−1.00000	−0.0384900
$676$	0	0
$677$	−1.39403	−0.0535770	−0.0267885	−	0.999641i	$-0.508528\pi$
$677$	−1.39403	−0.0535770	−0.0267885	+	0.999641i	$0.508528\pi$
$678$	0	0
$679$	4.74456	0.182080
$680$	0	0
$681$	−11.1168	−0.425998
$682$	0	0
$683$	−15.8614	−0.606920	−0.303460	−	0.952844i	$-0.598142\pi$
$683$	−15.8614	−0.606920	−0.303460	+	0.952844i	$0.598142\pi$
$684$	0	0
$685$	19.4891	0.744641
$686$	0	0
$687$	−21.1168	−0.805658
$688$	0	0
$689$	8.74456	0.333141
$690$	0	0
$691$	47.8614	1.82073	0.910367	−	0.413802i	$-0.135799\pi$
$691$	47.8614	1.82073	0.910367	+	0.413802i	$0.135799\pi$
$692$	0	0
$693$	−15.1168	−0.574241
$694$	0	0
$695$	−0.744563	−0.0282429
$696$	0	0
$697$	−72.4674	−2.74490
$698$	0	0
$699$	−13.8614	−0.524287
$700$	0	0
$701$	5.39403	0.203730	0.101865	−	0.994798i	$-0.467519\pi$
$701$	5.39403	0.203730	0.101865	+	0.994798i	$0.467519\pi$
$702$	0	0
$703$	68.4674	2.58230
$704$	0	0
$705$	4.74456	0.178691
$706$	0	0
$707$	−21.6277	−0.813394
$708$	0	0
$709$	22.8832	0.859395	0.429697	−	0.902973i	$-0.358620\pi$
$709$	22.8832	0.859395	0.429697	+	0.902973i	$0.358620\pi$
$710$	0	0
$711$	10.3723	0.388991
$712$	0	0
$713$	2.37228	0.0888426
$714$	0	0
$715$	12.7446	0.476620
$716$	0	0
$717$	−6.51087	−0.243153
$718$	0	0
$719$	−27.2554	−1.01646	−0.508228	−	0.861222i	$-0.669699\pi$
$719$	−27.2554	−1.01646	−0.508228	+	0.861222i	$0.669699\pi$
$720$	0	0
$721$	18.9783	0.706787
$722$	0	0
$723$	−27.4891	−1.02233
$724$	0	0
$725$	2.74456	0.101930
$726$	0	0
$727$	−13.6277	−0.505424	−0.252712	−	0.967542i	$-0.581323\pi$
$727$	−13.6277	−0.505424	−0.252712	+	0.967542i	$0.581323\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−42.9783	−1.58961
$732$	0	0
$733$	−34.0000	−1.25582	−0.627909	−	0.778287i	$-0.716089\pi$
$733$	−34.0000	−1.25582	−0.627909	+	0.778287i	$0.716089\pi$
$734$	0	0
$735$	1.37228	0.0506174
$736$	0	0
$737$	−4.74456	−0.174768
$738$	0	0
$739$	−22.9783	−0.845269	−0.422634	−	0.906300i	$-0.638895\pi$
$739$	−22.9783	−0.845269	−0.422634	+	0.906300i	$0.638895\pi$
$740$	0	0
$741$	12.7446	0.468183
$742$	0	0
$743$	29.6277	1.08694	0.543468	−	0.839430i	$-0.317111\pi$
$743$	29.6277	1.08694	0.543468	+	0.839430i	$0.317111\pi$
$744$	0	0
$745$	5.11684	0.187467
$746$	0	0
$747$	12.0000	0.439057
$748$	0	0
$749$	−15.1168	−0.552357
$750$	0	0
$751$	27.2554	0.994565	0.497283	−	0.867589i	$-0.334331\pi$
$751$	27.2554	0.994565	0.497283	+	0.867589i	$0.334331\pi$
$752$	0	0
$753$	−4.00000	−0.145768
$754$	0	0
$755$	8.00000	0.291150
$756$	0	0
$757$	−26.0000	−0.944986	−0.472493	−	0.881334i	$-0.656646\pi$
$757$	−26.0000	−0.944986	−0.472493	+	0.881334i	$0.656646\pi$
$758$	0	0
$759$	15.1168	0.548707
$760$	0	0
$761$	−18.1386	−0.657523	−0.328762	−	0.944413i	$-0.606631\pi$
$761$	−18.1386	−0.657523	−0.328762	+	0.944413i	$0.606631\pi$
$762$	0	0
$763$	−16.0000	−0.579239
$764$	0	0
$765$	6.74456	0.243850
$766$	0	0
$767$	−17.4891	−0.631496
$768$	0	0
$769$	−3.62772	−0.130819	−0.0654094	−	0.997859i	$-0.520835\pi$
$769$	−3.62772	−0.130819	−0.0654094	+	0.997859i	$0.520835\pi$
$770$	0	0
$771$	−7.62772	−0.274706
$772$	0	0
$773$	−18.6060	−0.669210	−0.334605	−	0.942358i	$-0.608603\pi$
$773$	−18.6060	−0.669210	−0.334605	+	0.942358i	$0.608603\pi$
$774$	0	0
$775$	1.00000	0.0359211
$776$	0	0
$777$	−25.4891	−0.914417
$778$	0	0
$779$	−68.4674	−2.45310
$780$	0	0
$781$	−15.1168	−0.540923
$782$	0	0
$783$	−2.74456	−0.0980827
$784$	0	0
$785$	3.62772	0.129479
$786$	0	0
$787$	27.1168	0.966611	0.483306	−	0.875452i	$-0.339436\pi$
$787$	27.1168	0.966611	0.483306	+	0.875452i	$0.339436\pi$
$788$	0	0
$789$	26.9783	0.960451
$790$	0	0
$791$	−19.8614	−0.706190
$792$	0	0
$793$	22.9783	0.815982
$794$	0	0
$795$	4.37228	0.155069
$796$	0	0
$797$	−43.4891	−1.54046	−0.770232	−	0.637764i	$-0.779860\pi$
$797$	−43.4891	−1.54046	−0.770232	+	0.637764i	$0.779860\pi$
$798$	0	0
$799$	−32.0000	−1.13208
$800$	0	0
$801$	4.37228	0.154487
$802$	0	0
$803$	−58.0951	−2.05013
$804$	0	0
$805$	5.62772	0.198351
$806$	0	0
$807$	2.00000	0.0704033
$808$	0	0
$809$	−30.6060	−1.07605	−0.538024	−	0.842929i	$-0.680829\pi$
$809$	−30.6060	−1.07605	−0.538024	+	0.842929i	$0.680829\pi$
$810$	0	0
$811$	46.3723	1.62835	0.814176	−	0.580619i	$-0.197189\pi$
$811$	46.3723	1.62835	0.814176	+	0.580619i	$0.197189\pi$
$812$	0	0
$813$	−31.1168	−1.09132
$814$	0	0
$815$	10.2337	0.358470
$816$	0	0
$817$	−40.6060	−1.42062
$818$	0	0
$819$	−4.74456	−0.165788
$820$	0	0
$821$	26.7446	0.933392	0.466696	−	0.884418i	$-0.345444\pi$
$821$	26.7446	0.933392	0.466696	+	0.884418i	$0.345444\pi$
$822$	0	0
$823$	28.7446	1.00197	0.500986	−	0.865455i	$-0.332971\pi$
$823$	28.7446	1.00197	0.500986	+	0.865455i	$0.332971\pi$
$824$	0	0
$825$	6.37228	0.221854
$826$	0	0
$827$	−5.48913	−0.190876	−0.0954378	−	0.995435i	$-0.530425\pi$
$827$	−5.48913	−0.190876	−0.0954378	+	0.995435i	$0.530425\pi$
$828$	0	0
$829$	32.0951	1.11471	0.557354	−	0.830275i	$-0.311816\pi$
$829$	32.0951	1.11471	0.557354	+	0.830275i	$0.311816\pi$
$830$	0	0
$831$	−26.7446	−0.927759
$832$	0	0
$833$	−9.25544	−0.320682
$834$	0	0
$835$	18.3723	0.635799
$836$	0	0
$837$	−1.00000	−0.0345651
$838$	0	0
$839$	11.8614	0.409501	0.204751	−	0.978814i	$-0.434362\pi$
$839$	11.8614	0.409501	0.204751	+	0.978814i	$0.434362\pi$
$840$	0	0
$841$	−21.4674	−0.740254
$842$	0	0
$843$	−5.25544	−0.181007
$844$	0	0
$845$	−9.00000	−0.309609
$846$	0	0
$847$	70.2337	2.41326
$848$	0	0
$849$	12.0000	0.411839
$850$	0	0
$851$	25.4891	0.873756
$852$	0	0
$853$	24.0951	0.825000	0.412500	−	0.910958i	$-0.364656\pi$
$853$	24.0951	0.825000	0.412500	+	0.910958i	$0.364656\pi$
$854$	0	0
$855$	6.37228	0.217927
$856$	0	0
$857$	−15.4891	−0.529098	−0.264549	−	0.964372i	$-0.585223\pi$
$857$	−15.4891	−0.529098	−0.264549	+	0.964372i	$0.585223\pi$
$858$	0	0
$859$	−5.48913	−0.187287	−0.0936433	−	0.995606i	$-0.529851\pi$
$859$	−5.48913	−0.187287	−0.0936433	+	0.995606i	$0.529851\pi$
$860$	0	0
$861$	25.4891	0.868667
$862$	0	0
$863$	−45.3505	−1.54375	−0.771875	−	0.635774i	$-0.780681\pi$
$863$	−45.3505	−1.54375	−0.771875	+	0.635774i	$0.780681\pi$
$864$	0	0
$865$	−18.0000	−0.612018
$866$	0	0
$867$	−28.4891	−0.967541
$868$	0	0
$869$	−66.0951	−2.24212
$870$	0	0
$871$	−1.48913	−0.0504571
$872$	0	0
$873$	2.00000	0.0676897
$874$	0	0
$875$	2.37228	0.0801977
$876$	0	0
$877$	0.978251	0.0330332	0.0165166	−	0.999864i	$-0.494742\pi$
$877$	0.978251	0.0330332	0.0165166	+	0.999864i	$0.494742\pi$
$878$	0	0
$879$	−15.4891	−0.522435
$880$	0	0
$881$	−48.9783	−1.65012	−0.825060	−	0.565046i	$-0.808859\pi$
$881$	−48.9783	−1.65012	−0.825060	+	0.565046i	$0.808859\pi$
$882$	0	0
$883$	−16.1386	−0.543107	−0.271553	−	0.962423i	$-0.587537\pi$
$883$	−16.1386	−0.543107	−0.271553	+	0.962423i	$0.587537\pi$
$884$	0	0
$885$	−8.74456	−0.293945
$886$	0	0
$887$	19.2554	0.646534	0.323267	−	0.946308i	$-0.395219\pi$
$887$	19.2554	0.646534	0.323267	+	0.946308i	$0.395219\pi$
$888$	0	0
$889$	22.5109	0.754991
$890$	0	0
$891$	−6.37228	−0.213479
$892$	0	0
$893$	−30.2337	−1.01173
$894$	0	0
$895$	12.0000	0.401116
$896$	0	0
$897$	4.74456	0.158416
$898$	0	0
$899$	2.74456	0.0915363
$900$	0	0
$901$	−29.4891	−0.982425
$902$	0	0
$903$	15.1168	0.503057
$904$	0	0
$905$	−13.8614	−0.460769
$906$	0	0
$907$	−5.48913	−0.182263	−0.0911317	−	0.995839i	$-0.529048\pi$
$907$	−5.48913	−0.182263	−0.0911317	+	0.995839i	$0.529048\pi$
$908$	0	0
$909$	−9.11684	−0.302387
$910$	0	0
$911$	25.4891	0.844492	0.422246	−	0.906481i	$-0.361242\pi$
$911$	25.4891	0.844492	0.422246	+	0.906481i	$0.361242\pi$
$912$	0	0
$913$	−76.4674	−2.53070
$914$	0	0
$915$	11.4891	0.379819
$916$	0	0
$917$	−43.2554	−1.42842
$918$	0	0
$919$	−22.2337	−0.733422	−0.366711	−	0.930335i	$-0.619516\pi$
$919$	−22.2337	−0.733422	−0.366711	+	0.930335i	$0.619516\pi$
$920$	0	0
$921$	−14.9783	−0.493550
$922$	0	0
$923$	−4.74456	−0.156169
$924$	0	0
$925$	10.7446	0.353279
$926$	0	0
$927$	8.00000	0.262754
$928$	0	0
$929$	1.11684	0.0366425	0.0183212	−	0.999832i	$-0.494168\pi$
$929$	1.11684	0.0366425	0.0183212	+	0.999832i	$0.494168\pi$
$930$	0	0
$931$	−8.74456	−0.286591
$932$	0	0
$933$	8.00000	0.261908
$934$	0	0
$935$	−42.9783	−1.40554
$936$	0	0
$937$	−42.7446	−1.39640	−0.698202	−	0.715901i	$-0.746016\pi$
$937$	−42.7446	−1.39640	−0.698202	+	0.715901i	$0.746016\pi$
$938$	0	0
$939$	−10.0000	−0.326338
$940$	0	0
$941$	37.7228	1.22973	0.614864	−	0.788633i	$-0.289211\pi$
$941$	37.7228	1.22973	0.614864	+	0.788633i	$0.289211\pi$
$942$	0	0
$943$	−25.4891	−0.830040
$944$	0	0
$945$	−2.37228	−0.0771703
$946$	0	0
$947$	−8.74456	−0.284160	−0.142080	−	0.989855i	$-0.545379\pi$
$947$	−8.74456	−0.284160	−0.142080	+	0.989855i	$0.545379\pi$
$948$	0	0
$949$	−18.2337	−0.591891
$950$	0	0
$951$	13.2554	0.429837
$952$	0	0
$953$	−45.7228	−1.48111	−0.740554	−	0.671997i	$-0.765437\pi$
$953$	−45.7228	−1.48111	−0.740554	+	0.671997i	$0.765437\pi$
$954$	0	0
$955$	−16.0000	−0.517748
$956$	0	0
$957$	17.4891	0.565343
$958$	0	0
$959$	46.2337	1.49296
$960$	0	0
$961$	1.00000	0.0322581
$962$	0	0
$963$	−6.37228	−0.205344
$964$	0	0
$965$	−7.48913	−0.241083
$966$	0	0
$967$	−8.00000	−0.257263	−0.128631	−	0.991692i	$-0.541058\pi$
$967$	−8.00000	−0.257263	−0.128631	+	0.991692i	$0.541058\pi$
$968$	0	0
$969$	−42.9783	−1.38066
$970$	0	0
$971$	−54.7011	−1.75544	−0.877720	−	0.479173i	$-0.840937\pi$
$971$	−54.7011	−1.75544	−0.877720	+	0.479173i	$0.840937\pi$
$972$	0	0
$973$	−1.76631	−0.0566254
$974$	0	0
$975$	2.00000	0.0640513
$976$	0	0
$977$	−46.0000	−1.47167	−0.735835	−	0.677161i	$-0.763210\pi$
$977$	−46.0000	−1.47167	−0.735835	+	0.677161i	$0.763210\pi$
$978$	0	0
$979$	−27.8614	−0.890454
$980$	0	0
$981$	−6.74456	−0.215337
$982$	0	0
$983$	−42.9783	−1.37079	−0.685397	−	0.728170i	$-0.740371\pi$
$983$	−42.9783	−1.37079	−0.685397	+	0.728170i	$0.740371\pi$
$984$	0	0
$985$	−3.48913	−0.111173
$986$	0	0
$987$	11.2554	0.358265
$988$	0	0
$989$	−15.1168	−0.480688
$990$	0	0
$991$	7.39403	0.234879	0.117440	−	0.993080i	$-0.462531\pi$
$991$	7.39403	0.234879	0.117440	+	0.993080i	$0.462531\pi$
$992$	0	0
$993$	21.4891	0.681937
$994$	0	0
$995$	18.3723	0.582440
$996$	0	0
$997$	6.00000	0.190022	0.0950110	−	0.995476i	$-0.469711\pi$
$997$	6.00000	0.190022	0.0950110	+	0.995476i	$0.469711\pi$
$998$	0	0
$999$	−10.7446	−0.339943

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7440.2.a.bg.1.2		2
4.3	odd	2		930.2.a.r.1.1	✓	2
12.11	even	2		2790.2.a.bd.1.1		2
20.3	even	4		4650.2.d.bh.3349.2		4
20.7	even	4		4650.2.d.bh.3349.3		4
20.19	odd	2		4650.2.a.by.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
930.2.a.r.1.1	✓	2	4.3	odd	2
2790.2.a.bd.1.1		2	12.11	even	2
4650.2.a.by.1.2		2	20.19	odd	2
4650.2.d.bh.3349.2		4	20.3	even	4
4650.2.d.bh.3349.3		4	20.7	even	4
7440.2.a.bg.1.2		2	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 7440 = 2^{4} \cdot 3 \cdot 5 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7440.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +1.00000 q^{5} +2.37228 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +1.00000 q^{5} +2.37228 q^{7} +1.00000 q^{9} -6.37228 q^{11} -2.00000 q^{13} -1.00000 q^{15} +6.74456 q^{17} +6.37228 q^{19} -2.37228 q^{21} +2.37228 q^{23} +1.00000 q^{25} -1.00000 q^{27} +2.74456 q^{29} +1.00000 q^{31} +6.37228 q^{33} +2.37228 q^{35} +10.7446 q^{37} +2.00000 q^{39} -10.7446 q^{41} -6.37228 q^{43} +1.00000 q^{45} -4.74456 q^{47} -1.37228 q^{49} -6.74456 q^{51} -4.37228 q^{53} -6.37228 q^{55} -6.37228 q^{57} +8.74456 q^{59} -11.4891 q^{61} +2.37228 q^{63} -2.00000 q^{65} +0.744563 q^{67} -2.37228 q^{69} +2.37228 q^{71} +9.11684 q^{73} -1.00000 q^{75} -15.1168 q^{77} +10.3723 q^{79} +1.00000 q^{81} +12.0000 q^{83} +6.74456 q^{85} -2.74456 q^{87} +4.37228 q^{89} -4.74456 q^{91} -1.00000 q^{93} +6.37228 q^{95} +2.00000 q^{97} -6.37228 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} + 2 q^{5} - q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^3 + 2 * q^5 - q^7 + 2 * q^9	\( 2 q - 2 q^{3} + 2 q^{5} - q^{7} + 2 q^{9} - 7 q^{11} - 4 q^{13} - 2 q^{15} + 2 q^{17} + 7 q^{19} + q^{21} - q^{23} + 2 q^{25} - 2 q^{27} - 6 q^{29} + 2 q^{31} + 7 q^{33} - q^{35} + 10 q^{37} + 4 q^{39} - 10 q^{41} - 7 q^{43} + 2 q^{45} + 2 q^{47} + 3 q^{49} - 2 q^{51} - 3 q^{53} - 7 q^{55} - 7 q^{57} + 6 q^{59} - q^{63} - 4 q^{65} - 10 q^{67} + q^{69} - q^{71} + q^{73} - 2 q^{75} - 13 q^{77} + 15 q^{79} + 2 q^{81} + 24 q^{83} + 2 q^{85} + 6 q^{87} + 3 q^{89} + 2 q^{91} - 2 q^{93} + 7 q^{95} + 4 q^{97} - 7 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 + 2 * q^5 - q^7 + 2 * q^9 - 7 * q^11 - 4 * q^13 - 2 * q^15 + 2 * q^17 + 7 * q^19 + q^21 - q^23 + 2 * q^25 - 2 * q^27 - 6 * q^29 + 2 * q^31 + 7 * q^33 - q^35 + 10 * q^37 + 4 * q^39 - 10 * q^41 - 7 * q^43 + 2 * q^45 + 2 * q^47 + 3 * q^49 - 2 * q^51 - 3 * q^53 - 7 * q^55 - 7 * q^57 + 6 * q^59 - q^63 - 4 * q^65 - 10 * q^67 + q^69 - q^71 + q^73 - 2 * q^75 - 13 * q^77 + 15 * q^79 + 2 * q^81 + 24 * q^83 + 2 * q^85 + 6 * q^87 + 3 * q^89 + 2 * q^91 - 2 * q^93 + 7 * q^95 + 4 * q^97 - 7 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more