LMFDB - Embedded newform 7400.2.a.m.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7400,2,Mod(1,7400)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7400, 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("7400.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7400 = 2^{3} \cdot 5^{2} \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7400.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:	$59.0892974957$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.568.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 6x - 2 $ x^3 - x^2 - 6*x - 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1480)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$3.12489$ of defining polynomial
Character	$\chi$	$=$	7400.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.12489 q^{3} +3.12489 q^{7} +6.76491 q^{9} +O(q^{10})$	$q-3.12489 q^{3} +3.12489 q^{7} +6.76491 q^{9} +1.51514 q^{11} +5.60975 q^{19} -9.76491 q^{21} -4.00000 q^{23} -11.7649 q^{27} -1.03028 q^{29} -0.640023 q^{31} -4.73463 q^{33} -1.00000 q^{37} -5.76491 q^{41} -3.03028 q^{43} +1.18544 q^{47} +2.76491 q^{49} -8.79518 q^{53} -17.5298 q^{57} -3.67030 q^{59} -14.4995 q^{61} +21.1396 q^{63} -11.6703 q^{67} +12.4995 q^{69} -0.734633 q^{71} -7.70436 q^{73} +4.73463 q^{77} -2.57947 q^{79} +16.4693 q^{81} +12.5942 q^{83} +3.21949 q^{87} +8.56009 q^{89} +2.00000 q^{93} -6.00000 q^{97} +10.2498 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - q^{3} + q^{7} + 4 q^{9}+O(q^{10}) $ 3 * q - q^3 + q^7 + 4 * q^9	$ 3 q - q^{3} + q^{7} + 4 q^{9} + 5 q^{11} + 8 q^{19} - 13 q^{21} - 12 q^{23} - 19 q^{27} - 4 q^{29} + 6 q^{31} + 3 q^{33} - 3 q^{37} - q^{41} - 10 q^{43} - 3 q^{47} - 8 q^{49} - 11 q^{53} - 20 q^{57} - 4 q^{59} - 10 q^{61} + 22 q^{63} - 28 q^{67} + 4 q^{69} + 15 q^{71} - 5 q^{73} - 3 q^{77} + 2 q^{79} + 15 q^{81} - 5 q^{83} - 8 q^{87} - 6 q^{89} + 6 q^{93} - 18 q^{97} + 14 q^{99}+O(q^{100}) $ 3 * q - q^3 + q^7 + 4 * q^9 + 5 * q^11 + 8 * q^19 - 13 * q^21 - 12 * q^23 - 19 * q^27 - 4 * q^29 + 6 * q^31 + 3 * q^33 - 3 * q^37 - q^41 - 10 * q^43 - 3 * q^47 - 8 * q^49 - 11 * q^53 - 20 * q^57 - 4 * q^59 - 10 * q^61 + 22 * q^63 - 28 * q^67 + 4 * q^69 + 15 * q^71 - 5 * q^73 - 3 * q^77 + 2 * q^79 + 15 * q^81 - 5 * q^83 - 8 * q^87 - 6 * q^89 + 6 * q^93 - 18 * q^97 + 14 * 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
$3$	−3.12489	−1.80415	−0.902077	−	0.431576i	$-0.857958\pi$
$3$	−3.12489	−1.80415	−0.902077	+	0.431576i	$0.857958\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	3.12489	1.18110	0.590548	−	0.807003i	$-0.298912\pi$
$7$	3.12489	1.18110	0.590548	+	0.807003i	$0.298912\pi$
$8$	0	0
$9$	6.76491	2.25497
$10$	0	0
$11$	1.51514	0.456831	0.228416	−	0.973564i	$-0.426645\pi$
$11$	1.51514	0.456831	0.228416	+	0.973564i	$0.426645\pi$
$12$	0	0
$13$	0	0		−	1.00000i	$-0.5\pi$
$13$	0	0			1.00000i	$0.5\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	0	0
$19$	5.60975	1.28696	0.643482	−	0.765461i	$-0.277489\pi$
$19$	5.60975	1.28696	0.643482	+	0.765461i	$0.277489\pi$
$20$	0	0
$21$	−9.76491	−2.13088
$22$	0	0
$23$	−4.00000	−0.834058	−0.417029	−	0.908893i	$-0.636929\pi$
$23$	−4.00000	−0.834058	−0.417029	+	0.908893i	$0.636929\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−11.7649	−2.26416
$28$	0	0
$29$	−1.03028	−0.191317	−0.0956587	−	0.995414i	$-0.530496\pi$
$29$	−1.03028	−0.191317	−0.0956587	+	0.995414i	$0.530496\pi$
$30$	0	0
$31$	−0.640023	−0.114952	−0.0574758	−	0.998347i	$-0.518305\pi$
$31$	−0.640023	−0.114952	−0.0574758	+	0.998347i	$0.518305\pi$
$32$	0	0
$33$	−4.73463	−0.824194
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−1.00000	−0.164399
$37$	−1.00000	−0.164399
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−5.76491	−0.900328	−0.450164	−	0.892946i	$-0.648634\pi$
$41$	−5.76491	−0.900328	−0.450164	+	0.892946i	$0.648634\pi$
$42$	0	0
$43$	−3.03028	−0.462113	−0.231056	−	0.972940i	$-0.574218\pi$
$43$	−3.03028	−0.462113	−0.231056	+	0.972940i	$0.574218\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	1.18544	0.172914	0.0864569	−	0.996256i	$-0.472446\pi$
$47$	1.18544	0.172914	0.0864569	+	0.996256i	$0.472446\pi$
$48$	0	0
$49$	2.76491	0.394987
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−8.79518	−1.20811	−0.604056	−	0.796942i	$-0.706450\pi$
$53$	−8.79518	−1.20811	−0.604056	+	0.796942i	$0.706450\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−17.5298	−2.32188
$58$	0	0
$59$	−3.67030	−0.477832	−0.238916	−	0.971040i	$-0.576792\pi$
$59$	−3.67030	−0.477832	−0.238916	+	0.971040i	$0.576792\pi$
$60$	0	0
$61$	−14.4995	−1.85648	−0.928238	−	0.371987i	$-0.878677\pi$
$61$	−14.4995	−1.85648	−0.928238	+	0.371987i	$0.878677\pi$
$62$	0	0
$63$	21.1396	2.66333
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−11.6703	−1.42575	−0.712877	−	0.701289i	$-0.752608\pi$
$67$	−11.6703	−1.42575	−0.712877	+	0.701289i	$0.752608\pi$
$68$	0	0
$69$	12.4995	1.50477
$70$	0	0
$71$	−0.734633	−0.0871849	−0.0435924	−	0.999049i	$-0.513880\pi$
$71$	−0.734633	−0.0871849	−0.0435924	+	0.999049i	$0.513880\pi$
$72$	0	0
$73$	−7.70436	−0.901727	−0.450863	−	0.892593i	$-0.648884\pi$
$73$	−7.70436	−0.901727	−0.450863	+	0.892593i	$0.648884\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	4.73463	0.539561
$78$	0	0
$79$	−2.57947	−0.290213	−0.145107	−	0.989416i	$-0.546353\pi$
$79$	−2.57947	−0.290213	−0.145107	+	0.989416i	$0.546353\pi$
$80$	0	0
$81$	16.4693	1.82992
$82$	0	0
$83$	12.5942	1.38239	0.691194	−	0.722669i	$-0.257085\pi$
$83$	12.5942	1.38239	0.691194	+	0.722669i	$0.257085\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	3.21949	0.345166
$88$	0	0
$89$	8.56009	0.907368	0.453684	−	0.891163i	$-0.350109\pi$
$89$	8.56009	0.907368	0.453684	+	0.891163i	$0.350109\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	2.00000	0.207390
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−6.00000	−0.609208	−0.304604	−	0.952479i	$-0.598524\pi$
$97$	−6.00000	−0.609208	−0.304604	+	0.952479i	$0.598524\pi$
$98$	0	0
$99$	10.2498	1.03014
$100$	0	0
$101$	−15.2947	−1.52188	−0.760941	−	0.648821i	$-0.775262\pi$
$101$	−15.2947	−1.52188	−0.760941	+	0.648821i	$0.775262\pi$
$102$	0	0
$103$	14.7493	1.45329	0.726646	−	0.687012i	$-0.241078\pi$
$103$	14.7493	1.45329	0.726646	+	0.687012i	$0.241078\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	4.95035	0.478568	0.239284	−	0.970950i	$-0.423087\pi$
$107$	4.95035	0.478568	0.239284	+	0.970950i	$0.423087\pi$
$108$	0	0
$109$	−18.4995	−1.77193	−0.885967	−	0.463748i	$-0.846504\pi$
$109$	−18.4995	−1.77193	−0.885967	+	0.463748i	$0.846504\pi$
$110$	0	0
$111$	3.12489	0.296601
$112$	0	0
$113$	17.4693	1.64337	0.821685	−	0.569942i	$-0.193034\pi$
$113$	17.4693	1.64337	0.821685	+	0.569942i	$0.193034\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−8.70436	−0.791305
$122$	0	0
$123$	18.0147	1.62433
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−17.5639	−1.55854	−0.779271	−	0.626687i	$-0.784410\pi$
$127$	−17.5639	−1.55854	−0.779271	+	0.626687i	$0.784410\pi$
$128$	0	0
$129$	9.46927	0.833722
$130$	0	0
$131$	0.640023	0.0559191	0.0279596	−	0.999609i	$-0.491099\pi$
$131$	0.640023	0.0559191	0.0279596	+	0.999609i	$0.491099\pi$
$132$	0	0
$133$	17.5298	1.52003
$134$	0	0
$135$	0	0
$136$	0	0
$137$	10.4995	0.897036	0.448518	−	0.893774i	$-0.351952\pi$
$137$	10.4995	0.897036	0.448518	+	0.893774i	$0.351952\pi$
$138$	0	0
$139$	−14.5601	−1.23497	−0.617486	−	0.786582i	$-0.711849\pi$
$139$	−14.5601	−1.23497	−0.617486	+	0.786582i	$0.711849\pi$
$140$	0	0
$141$	−3.70436	−0.311963
$142$	0	0
$143$	0	0
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−8.64002	−0.712617
$148$	0	0
$149$	7.76491	0.636126	0.318063	−	0.948070i	$-0.396968\pi$
$149$	7.76491	0.636126	0.318063	+	0.948070i	$0.396968\pi$
$150$	0	0
$151$	21.7796	1.77240	0.886199	−	0.463305i	$-0.153337\pi$
$151$	21.7796	1.77240	0.886199	+	0.463305i	$0.153337\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	19.3553	1.54472	0.772360	−	0.635185i	$-0.219076\pi$
$157$	19.3553	1.54472	0.772360	+	0.635185i	$0.219076\pi$
$158$	0	0
$159$	27.4839	2.17962
$160$	0	0
$161$	−12.4995	−0.985102
$162$	0	0
$163$	−11.2195	−0.878779	−0.439389	−	0.898297i	$-0.644805\pi$
$163$	−11.2195	−0.878779	−0.439389	+	0.898297i	$0.644805\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	12.1892	0.943230	0.471615	−	0.881805i	$-0.343671\pi$
$167$	12.1892	0.943230	0.471615	+	0.881805i	$0.343671\pi$
$168$	0	0
$169$	−13.0000	−1.00000
$170$	0	0
$171$	37.9494	2.90207
$172$	0	0
$173$	−6.23509	−0.474045	−0.237023	−	0.971504i	$-0.576172\pi$
$173$	−6.23509	−0.474045	−0.237023	+	0.971504i	$0.576172\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	11.4693	0.862083
$178$	0	0
$179$	0.450805	0.0336947	0.0168474	−	0.999858i	$-0.494637\pi$
$179$	0.450805	0.0336947	0.0168474	+	0.999858i	$0.494637\pi$
$180$	0	0
$181$	−7.64380	−0.568160	−0.284080	−	0.958801i	$-0.591688\pi$
$181$	−7.64380	−0.568160	−0.284080	+	0.958801i	$0.591688\pi$
$182$	0	0
$183$	45.3094	3.34937
$184$	0	0
$185$	0	0
$186$	0	0
$187$	0	0
$188$	0	0
$189$	−36.7640	−2.67419
$190$	0	0
$191$	18.1093	1.31034	0.655171	−	0.755481i	$-0.272597\pi$
$191$	18.1093	1.31034	0.655171	+	0.755481i	$0.272597\pi$
$192$	0	0
$193$	−17.5298	−1.26182	−0.630912	−	0.775854i	$-0.717319\pi$
$193$	−17.5298	−1.26182	−0.630912	+	0.775854i	$0.717319\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	6.73463	0.479823	0.239911	−	0.970795i	$-0.422882\pi$
$197$	6.73463	0.479823	0.239911	+	0.970795i	$0.422882\pi$
$198$	0	0
$199$	11.8595	0.840699	0.420349	−	0.907362i	$-0.361907\pi$
$199$	11.8595	0.840699	0.420349	+	0.907362i	$0.361907\pi$
$200$	0	0
$201$	36.4683	2.57228
$202$	0	0
$203$	−3.21949	−0.225964
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−27.0596	−1.88077
$208$	0	0
$209$	8.49954	0.587926
$210$	0	0
$211$	−21.8255	−1.50253	−0.751263	−	0.660003i	$-0.770555\pi$
$211$	−21.8255	−1.50253	−0.751263	+	0.660003i	$0.770555\pi$
$212$	0	0
$213$	2.29564	0.157295
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−2.00000	−0.135769
$218$	0	0
$219$	24.0752	1.62685
$220$	0	0
$221$	0	0
$222$	0	0
$223$	6.62534	0.443666	0.221833	−	0.975085i	$-0.428796\pi$
$223$	6.62534	0.443666	0.221833	+	0.975085i	$0.428796\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−17.2800	−1.14692	−0.573458	−	0.819235i	$-0.694399\pi$
$227$	−17.2800	−1.14692	−0.573458	+	0.819235i	$0.694399\pi$
$228$	0	0
$229$	11.3553	0.750378	0.375189	−	0.926948i	$-0.377578\pi$
$229$	11.3553	0.750378	0.375189	+	0.926948i	$0.377578\pi$
$230$	0	0
$231$	−14.7952	−0.973452
$232$	0	0
$233$	6.00000	0.393073	0.196537	−	0.980497i	$-0.437031\pi$
$233$	6.00000	0.393073	0.196537	+	0.980497i	$0.437031\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	8.06055	0.523589
$238$	0	0
$239$	23.6997	1.53300	0.766502	−	0.642242i	$-0.221996\pi$
$239$	23.6997	1.53300	0.766502	+	0.642242i	$0.221996\pi$
$240$	0	0
$241$	−27.5904	−1.77725	−0.888626	−	0.458633i	$-0.848339\pi$
$241$	−27.5904	−1.77725	−0.888626	+	0.458633i	$0.848339\pi$
$242$	0	0
$243$	−16.1698	−1.03730
$244$	0	0
$245$	0	0
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−39.3553	−2.49404
$250$	0	0
$251$	3.04965	0.192492	0.0962462	−	0.995358i	$-0.469316\pi$
$251$	3.04965	0.192492	0.0962462	+	0.995358i	$0.469316\pi$
$252$	0	0
$253$	−6.06055	−0.381024
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−25.4693	−1.58873	−0.794365	−	0.607441i	$-0.792196\pi$
$257$	−25.4693	−1.58873	−0.794365	+	0.607441i	$0.792196\pi$
$258$	0	0
$259$	−3.12489	−0.194171
$260$	0	0
$261$	−6.96972	−0.431415
$262$	0	0
$263$	−0.0946093	−0.00583386	−0.00291693	−	0.999996i	$-0.500928\pi$
$263$	−0.0946093	−0.00583386	−0.00291693	+	0.999996i	$0.500928\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−26.7493	−1.63703
$268$	0	0
$269$	−17.5904	−1.07250	−0.536252	−	0.844058i	$-0.680160\pi$
$269$	−17.5904	−1.07250	−0.536252	+	0.844058i	$0.680160\pi$
$270$	0	0
$271$	6.32500	0.384217	0.192108	−	0.981374i	$-0.438468\pi$
$271$	6.32500	0.384217	0.192108	+	0.981374i	$0.438468\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−13.5904	−0.816566	−0.408283	−	0.912855i	$-0.633872\pi$
$277$	−13.5904	−0.816566	−0.408283	+	0.912855i	$0.633872\pi$
$278$	0	0
$279$	−4.32970	−0.259212
$280$	0	0
$281$	18.9991	1.13339	0.566695	−	0.823928i	$-0.308222\pi$
$281$	18.9991	1.13339	0.566695	+	0.823928i	$0.308222\pi$
$282$	0	0
$283$	−7.34060	−0.436353	−0.218177	−	0.975909i	$-0.570011\pi$
$283$	−7.34060	−0.436353	−0.218177	+	0.975909i	$0.570011\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−18.0147	−1.06337
$288$	0	0
$289$	−17.0000	−1.00000
$290$	0	0
$291$	18.7493	1.09910
$292$	0	0
$293$	10.0000	0.584206	0.292103	−	0.956387i	$-0.405645\pi$
$293$	10.0000	0.584206	0.292103	+	0.956387i	$0.405645\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−17.8255	−1.03434
$298$	0	0
$299$	0	0
$300$	0	0
$301$	−9.46927	−0.545799
$302$	0	0
$303$	47.7943	2.74571
$304$	0	0
$305$	0	0
$306$	0	0
$307$	4.40493	0.251403	0.125701	−	0.992068i	$-0.459882\pi$
$307$	4.40493	0.251403	0.125701	+	0.992068i	$0.459882\pi$
$308$	0	0
$309$	−46.0899	−2.62196
$310$	0	0
$311$	−24.1698	−1.37055	−0.685273	−	0.728286i	$-0.740317\pi$
$311$	−24.1698	−1.37055	−0.685273	+	0.728286i	$0.740317\pi$
$312$	0	0
$313$	−14.9991	−0.847798	−0.423899	−	0.905709i	$-0.639339\pi$
$313$	−14.9991	−0.847798	−0.423899	+	0.905709i	$0.639339\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−24.5601	−1.37943	−0.689716	−	0.724080i	$-0.742265\pi$
$317$	−24.5601	−1.37943	−0.689716	+	0.724080i	$0.742265\pi$
$318$	0	0
$319$	−1.56101	−0.0873998
$320$	0	0
$321$	−15.4693	−0.863410
$322$	0	0
$323$	0	0
$324$	0	0
$325$	0	0
$326$	0	0
$327$	57.8089	3.19684
$328$	0	0
$329$	3.70436	0.204228
$330$	0	0
$331$	−16.1698	−0.888775	−0.444387	−	0.895835i	$-0.646579\pi$
$331$	−16.1698	−0.888775	−0.444387	+	0.895835i	$0.646579\pi$
$332$	0	0
$333$	−6.76491	−0.370715
$334$	0	0
$335$	0	0
$336$	0	0
$337$	27.8548	1.51735	0.758674	−	0.651470i	$-0.225847\pi$
$337$	27.8548	1.51735	0.758674	+	0.651470i	$0.225847\pi$
$338$	0	0
$339$	−54.5895	−2.96489
$340$	0	0
$341$	−0.969724	−0.0525135
$342$	0	0
$343$	−13.2342	−0.714578
$344$	0	0
$345$	0	0
$346$	0	0
$347$	4.49954	0.241548	0.120774	−	0.992680i	$-0.461462\pi$
$347$	4.49954	0.241548	0.120774	+	0.992680i	$0.461462\pi$
$348$	0	0
$349$	−35.5298	−1.90187	−0.950934	−	0.309395i	$-0.899874\pi$
$349$	−35.5298	−1.90187	−0.950934	+	0.309395i	$0.899874\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−16.0606	−0.854817	−0.427408	−	0.904059i	$-0.640573\pi$
$353$	−16.0606	−0.854817	−0.427408	+	0.904059i	$0.640573\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−5.51514	−0.291078	−0.145539	−	0.989353i	$-0.546492\pi$
$359$	−5.51514	−0.291078	−0.145539	+	0.989353i	$0.546492\pi$
$360$	0	0
$361$	12.4693	0.656277
$362$	0	0
$363$	27.2001	1.42764
$364$	0	0
$365$	0	0
$366$	0	0
$367$	5.60975	0.292826	0.146413	−	0.989224i	$-0.453227\pi$
$367$	5.60975	0.292826	0.146413	+	0.989224i	$0.453227\pi$
$368$	0	0
$369$	−38.9991	−2.03021
$370$	0	0
$371$	−27.4839	−1.42690
$372$	0	0
$373$	19.8255	1.02652	0.513262	−	0.858232i	$-0.328437\pi$
$373$	19.8255	1.02652	0.513262	+	0.858232i	$0.328437\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	17.5151	0.899692	0.449846	−	0.893106i	$-0.351479\pi$
$379$	17.5151	0.899692	0.449846	+	0.893106i	$0.351479\pi$
$380$	0	0
$381$	54.8851	2.81185
$382$	0	0
$383$	−11.5298	−0.589146	−0.294573	−	0.955629i	$-0.595177\pi$
$383$	−11.5298	−0.589146	−0.294573	+	0.955629i	$0.595177\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−20.4995	−1.04205
$388$	0	0
$389$	28.0899	1.42422	0.712108	−	0.702070i	$-0.247741\pi$
$389$	28.0899	1.42422	0.712108	+	0.702070i	$0.247741\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	−2.00000	−0.100887
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−1.26537	−0.0635070	−0.0317535	−	0.999496i	$-0.510109\pi$
$397$	−1.26537	−0.0635070	−0.0317535	+	0.999496i	$0.510109\pi$
$398$	0	0
$399$	−54.7787	−2.74236
$400$	0	0
$401$	−4.43899	−0.221673	−0.110836	−	0.993839i	$-0.535353\pi$
$401$	−4.43899	−0.221673	−0.110836	+	0.993839i	$0.535353\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−1.51514	−0.0751026
$408$	0	0
$409$	−0.909172	−0.0449556	−0.0224778	−	0.999747i	$-0.507156\pi$
$409$	−0.909172	−0.0449556	−0.0224778	+	0.999747i	$0.507156\pi$
$410$	0	0
$411$	−32.8099	−1.61839
$412$	0	0
$413$	−11.4693	−0.564366
$414$	0	0
$415$	0	0
$416$	0	0
$417$	45.4986	2.22808
$418$	0	0
$419$	−19.6050	−0.957769	−0.478885	−	0.877878i	$-0.658959\pi$
$419$	−19.6050	−0.957769	−0.478885	+	0.877878i	$0.658959\pi$
$420$	0	0
$421$	0.909172	0.0443103	0.0221552	−	0.999755i	$-0.492947\pi$
$421$	0.909172	0.0443103	0.0221552	+	0.999755i	$0.492947\pi$
$422$	0	0
$423$	8.01938	0.389915
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−45.3094	−2.19268
$428$	0	0
$429$	0	0
$430$	0	0
$431$	13.3288	0.642025	0.321012	−	0.947075i	$-0.395977\pi$
$431$	13.3288	0.642025	0.321012	+	0.947075i	$0.395977\pi$
$432$	0	0
$433$	11.8255	0.568295	0.284148	−	0.958781i	$-0.408289\pi$
$433$	11.8255	0.568295	0.284148	+	0.958781i	$0.408289\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−22.4390	−1.07340
$438$	0	0
$439$	−15.5492	−0.742123	−0.371061	−	0.928608i	$-0.621006\pi$
$439$	−15.5492	−0.742123	−0.371061	+	0.928608i	$0.621006\pi$
$440$	0	0
$441$	18.7044	0.890684
$442$	0	0
$443$	−27.4646	−1.30488	−0.652441	−	0.757840i	$-0.726255\pi$
$443$	−27.4646	−1.30488	−0.652441	+	0.757840i	$0.726255\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−24.2645	−1.14767
$448$	0	0
$449$	14.0000	0.660701	0.330350	−	0.943858i	$-0.392833\pi$
$449$	14.0000	0.660701	0.330350	+	0.943858i	$0.392833\pi$
$450$	0	0
$451$	−8.73463	−0.411298
$452$	0	0
$453$	−68.0587	−3.19768
$454$	0	0
$455$	0	0
$456$	0	0
$457$	36.9385	1.72791	0.863956	−	0.503568i	$-0.167980\pi$
$457$	36.9385	1.72791	0.863956	+	0.503568i	$0.167980\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−21.1807	−0.986485	−0.493243	−	0.869892i	$-0.664189\pi$
$461$	−21.1807	−0.986485	−0.493243	+	0.869892i	$0.664189\pi$
$462$	0	0
$463$	−6.08991	−0.283022	−0.141511	−	0.989937i	$-0.545196\pi$
$463$	−6.08991	−0.283022	−0.141511	+	0.989937i	$0.545196\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−14.6282	−0.676913	−0.338456	−	0.940982i	$-0.609905\pi$
$467$	−14.6282	−0.676913	−0.338456	+	0.940982i	$0.609905\pi$
$468$	0	0
$469$	−36.4683	−1.68395
$470$	0	0
$471$	−60.4830	−2.78691
$472$	0	0
$473$	−4.59129	−0.211108
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−59.4986	−2.72425
$478$	0	0
$479$	19.6703	0.898759	0.449379	−	0.893341i	$-0.351645\pi$
$479$	19.6703	0.898759	0.449379	+	0.893341i	$0.351645\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	39.0596	1.77727
$484$	0	0
$485$	0	0
$486$	0	0
$487$	4.65940	0.211138	0.105569	−	0.994412i	$-0.466334\pi$
$487$	4.65940	0.211138	0.105569	+	0.994412i	$0.466334\pi$
$488$	0	0
$489$	35.0596	1.58545
$490$	0	0
$491$	37.6197	1.69775	0.848877	−	0.528590i	$-0.177279\pi$
$491$	37.6197	1.69775	0.848877	+	0.528590i	$0.177279\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−2.29564	−0.102974
$498$	0	0
$499$	−0.359060	−0.0160737	−0.00803686	−	0.999968i	$-0.502558\pi$
$499$	−0.359060	−0.0160737	−0.00803686	+	0.999968i	$0.502558\pi$
$500$	0	0
$501$	−38.0899	−1.70173
$502$	0	0
$503$	−8.49954	−0.378976	−0.189488	−	0.981883i	$-0.560683\pi$
$503$	−8.49954	−0.378976	−0.189488	+	0.981883i	$0.560683\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	40.6235	1.80415
$508$	0	0
$509$	−16.1745	−0.716924	−0.358462	−	0.933544i	$-0.616699\pi$
$509$	−16.1745	−0.716924	−0.358462	+	0.933544i	$0.616699\pi$
$510$	0	0
$511$	−24.0752	−1.06503
$512$	0	0
$513$	−65.9982	−2.91389
$514$	0	0
$515$	0	0
$516$	0	0
$517$	1.79610	0.0789925
$518$	0	0
$519$	19.4839	0.855250
$520$	0	0
$521$	−11.2947	−0.494831	−0.247415	−	0.968909i	$-0.579581\pi$
$521$	−11.2947	−0.494831	−0.247415	+	0.968909i	$0.579581\pi$
$522$	0	0
$523$	11.0303	0.482320	0.241160	−	0.970485i	$-0.422472\pi$
$523$	11.0303	0.482320	0.241160	+	0.970485i	$0.422472\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−7.00000	−0.304348
$530$	0	0
$531$	−24.8292	−1.07750
$532$	0	0
$533$	0	0
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−1.40871	−0.0607905
$538$	0	0
$539$	4.18922	0.180442
$540$	0	0
$541$	18.9991	0.816834	0.408417	−	0.912795i	$-0.366081\pi$
$541$	18.9991	0.816834	0.408417	+	0.912795i	$0.366081\pi$
$542$	0	0
$543$	23.8860	1.02505
$544$	0	0
$545$	0	0
$546$	0	0
$547$	33.9982	1.45366	0.726828	−	0.686819i	$-0.240994\pi$
$547$	33.9982	1.45366	0.726828	+	0.686819i	$0.240994\pi$
$548$	0	0
$549$	−98.0881	−4.18630
$550$	0	0
$551$	−5.77959	−0.246219
$552$	0	0
$553$	−8.06055	−0.342770
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−19.8789	−0.842296	−0.421148	−	0.906992i	$-0.638373\pi$
$557$	−19.8789	−0.842296	−0.421148	+	0.906992i	$0.638373\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−28.4995	−1.20111	−0.600556	−	0.799583i	$-0.705054\pi$
$563$	−28.4995	−1.20111	−0.600556	+	0.799583i	$0.705054\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	51.4646	2.16131
$568$	0	0
$569$	38.4995	1.61398	0.806992	−	0.590562i	$-0.201094\pi$
$569$	38.4995	1.61398	0.806992	+	0.590562i	$0.201094\pi$
$570$	0	0
$571$	21.3553	0.893691	0.446845	−	0.894611i	$-0.352547\pi$
$571$	21.3553	0.893691	0.446845	+	0.894611i	$0.352547\pi$
$572$	0	0
$573$	−56.5895	−2.36406
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−26.9385	−1.12147	−0.560733	−	0.827997i	$-0.689481\pi$
$577$	−26.9385	−1.12147	−0.560733	+	0.827997i	$0.689481\pi$
$578$	0	0
$579$	54.7787	2.27652
$580$	0	0
$581$	39.3553	1.63273
$582$	0	0
$583$	−13.3259	−0.551903
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−24.1505	−0.996796	−0.498398	−	0.866948i	$-0.666078\pi$
$587$	−24.1505	−0.996796	−0.498398	+	0.866948i	$0.666078\pi$
$588$	0	0
$589$	−3.59037	−0.147939
$590$	0	0
$591$	−21.0450	−0.865674
$592$	0	0
$593$	−7.58325	−0.311407	−0.155703	−	0.987804i	$-0.549764\pi$
$593$	−7.58325	−0.311407	−0.155703	+	0.987804i	$0.549764\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−37.0596	−1.51675
$598$	0	0
$599$	−38.5142	−1.57365	−0.786824	−	0.617177i	$-0.788276\pi$
$599$	−38.5142	−1.57365	−0.786824	+	0.617177i	$0.788276\pi$
$600$	0	0
$601$	−23.4087	−0.954861	−0.477431	−	0.878669i	$-0.658432\pi$
$601$	−23.4087	−0.954861	−0.477431	+	0.878669i	$0.658432\pi$
$602$	0	0
$603$	−78.9485	−3.21503
$604$	0	0
$605$	0	0
$606$	0	0
$607$	15.5005	0.629144	0.314572	−	0.949234i	$-0.398139\pi$
$607$	15.5005	0.629144	0.314572	+	0.949234i	$0.398139\pi$
$608$	0	0
$609$	10.0606	0.407674
$610$	0	0
$611$	0	0
$612$	0	0
$613$	−22.2645	−0.899253	−0.449626	−	0.893217i	$-0.648443\pi$
$613$	−22.2645	−0.899253	−0.449626	+	0.893217i	$0.648443\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	46.3232	1.86490	0.932450	−	0.361298i	$-0.117666\pi$
$617$	46.3232	1.86490	0.932450	+	0.361298i	$0.117666\pi$
$618$	0	0
$619$	−26.7034	−1.07330	−0.536651	−	0.843804i	$-0.680311\pi$
$619$	−26.7034	−1.07330	−0.536651	+	0.843804i	$0.680311\pi$
$620$	0	0
$621$	47.0596	1.88844
$622$	0	0
$623$	26.7493	1.07169
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−26.5601	−1.06071
$628$	0	0
$629$	0	0
$630$	0	0
$631$	−9.29942	−0.370204	−0.185102	−	0.982719i	$-0.559262\pi$
$631$	−9.29942	−0.370204	−0.185102	+	0.982719i	$0.559262\pi$
$632$	0	0
$633$	68.2021	2.71079
$634$	0	0
$635$	0	0
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−4.96972	−0.196599
$640$	0	0
$641$	−21.1131	−0.833916	−0.416958	−	0.908926i	$-0.636904\pi$
$641$	−21.1131	−0.833916	−0.416958	+	0.908926i	$0.636904\pi$
$642$	0	0
$643$	17.9007	0.705934	0.352967	−	0.935636i	$-0.385173\pi$
$643$	17.9007	0.705934	0.352967	+	0.935636i	$0.385173\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	32.6500	1.28360	0.641802	−	0.766870i	$-0.278187\pi$
$647$	32.6500	1.28360	0.641802	+	0.766870i	$0.278187\pi$
$648$	0	0
$649$	−5.56101	−0.218289
$650$	0	0
$651$	6.24977	0.244948
$652$	0	0
$653$	42.0587	1.64588	0.822942	−	0.568125i	$-0.192331\pi$
$653$	42.0587	1.64588	0.822942	+	0.568125i	$0.192331\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−52.1193	−2.03337
$658$	0	0
$659$	−0.613528	−0.0238997	−0.0119498	−	0.999929i	$-0.503804\pi$
$659$	−0.613528	−0.0238997	−0.0119498	+	0.999929i	$0.503804\pi$
$660$	0	0
$661$	−41.6803	−1.62118	−0.810588	−	0.585617i	$-0.800852\pi$
$661$	−41.6803	−1.62118	−0.810588	+	0.585617i	$0.800852\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	4.12110	0.159570
$668$	0	0
$669$	−20.7034	−0.800441
$670$	0	0
$671$	−21.9688	−0.848096
$672$	0	0
$673$	−36.2039	−1.39556	−0.697779	−	0.716313i	$-0.745828\pi$
$673$	−36.2039	−1.39556	−0.697779	+	0.716313i	$0.745828\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−41.6732	−1.60163	−0.800815	−	0.598912i	$-0.795600\pi$
$677$	−41.6732	−1.60163	−0.800815	+	0.598912i	$0.795600\pi$
$678$	0	0
$679$	−18.7493	−0.719533
$680$	0	0
$681$	53.9982	2.06921
$682$	0	0
$683$	11.5005	0.440053	0.220026	−	0.975494i	$-0.429386\pi$
$683$	11.5005	0.440053	0.220026	+	0.975494i	$0.429386\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−35.4839	−1.35380
$688$	0	0
$689$	0	0
$690$	0	0
$691$	44.7181	1.70116	0.850579	−	0.525848i	$-0.176252\pi$
$691$	44.7181	1.70116	0.850579	+	0.525848i	$0.176252\pi$
$692$	0	0
$693$	32.0294	1.21669
$694$	0	0
$695$	0	0
$696$	0	0
$697$	0	0
$698$	0	0
$699$	−18.7493	−0.709164
$700$	0	0
$701$	41.9083	1.58285	0.791426	−	0.611264i	$-0.209339\pi$
$701$	41.9083	1.58285	0.791426	+	0.611264i	$0.209339\pi$
$702$	0	0
$703$	−5.60975	−0.211576
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−47.7943	−1.79749
$708$	0	0
$709$	35.0284	1.31552	0.657760	−	0.753227i	$-0.271504\pi$
$709$	35.0284	1.31552	0.657760	+	0.753227i	$0.271504\pi$
$710$	0	0
$711$	−17.4499	−0.654422
$712$	0	0
$713$	2.56009	0.0958763
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−74.0587	−2.76577
$718$	0	0
$719$	13.8936	0.518143	0.259071	−	0.965858i	$-0.416583\pi$
$719$	13.8936	0.518143	0.259071	+	0.965858i	$0.416583\pi$
$720$	0	0
$721$	46.0899	1.71648
$722$	0	0
$723$	86.2167	3.20644
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−25.0284	−0.928254	−0.464127	−	0.885769i	$-0.653632\pi$
$727$	−25.0284	−0.928254	−0.464127	+	0.885769i	$0.653632\pi$
$728$	0	0
$729$	1.12110	0.0415224
$730$	0	0
$731$	0	0
$732$	0	0
$733$	−17.7649	−0.656162	−0.328081	−	0.944650i	$-0.606402\pi$
$733$	−17.7649	−0.656162	−0.328081	+	0.944650i	$0.606402\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−17.6821	−0.651329
$738$	0	0
$739$	43.7943	1.61100	0.805499	−	0.592597i	$-0.201897\pi$
$739$	43.7943	1.61100	0.805499	+	0.592597i	$0.201897\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−19.4646	−0.714086	−0.357043	−	0.934088i	$-0.616215\pi$
$743$	−19.4646	−0.714086	−0.357043	+	0.934088i	$0.616215\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	85.1983	3.11724
$748$	0	0
$749$	15.4693	0.565235
$750$	0	0
$751$	−13.5151	−0.493174	−0.246587	−	0.969121i	$-0.579309\pi$
$751$	−13.5151	−0.493174	−0.246587	+	0.969121i	$0.579309\pi$
$752$	0	0
$753$	−9.52982	−0.347286
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−40.7106	−1.47965	−0.739825	−	0.672799i	$-0.765092\pi$
$757$	−40.7106	−1.47965	−0.739825	+	0.672799i	$0.765092\pi$
$758$	0	0
$759$	18.9385	0.687425
$760$	0	0
$761$	−4.17454	−0.151327	−0.0756635	−	0.997133i	$-0.524107\pi$
$761$	−4.17454	−0.151327	−0.0756635	+	0.997133i	$0.524107\pi$
$762$	0	0
$763$	−57.8089	−2.09282
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−7.09083	−0.255702	−0.127851	−	0.991793i	$-0.540808\pi$
$769$	−7.09083	−0.255702	−0.127851	+	0.991793i	$0.540808\pi$
$770$	0	0
$771$	79.5885	2.86631
$772$	0	0
$773$	−1.26537	−0.0455121	−0.0227560	−	0.999741i	$-0.507244\pi$
$773$	−1.26537	−0.0455121	−0.0227560	+	0.999741i	$0.507244\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	9.76491	0.350314
$778$	0	0
$779$	−32.3397	−1.15869
$780$	0	0
$781$	−1.11307	−0.0398288
$782$	0	0
$783$	12.1211	0.433173
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−27.3141	−0.973643	−0.486821	−	0.873502i	$-0.661844\pi$
$787$	−27.3141	−0.973643	−0.486821	+	0.873502i	$0.661844\pi$
$788$	0	0
$789$	0.295643	0.0105252
$790$	0	0
$791$	54.5895	1.94098
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−24.9991	−0.885513	−0.442756	−	0.896642i	$-0.645999\pi$
$797$	−24.9991	−0.885513	−0.442756	+	0.896642i	$0.645999\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	57.9083	2.04609
$802$	0	0
$803$	−11.6732	−0.411937
$804$	0	0
$805$	0	0
$806$	0	0
$807$	54.9679	1.93496
$808$	0	0
$809$	11.4693	0.403238	0.201619	−	0.979464i	$-0.435380\pi$
$809$	11.4693	0.403238	0.201619	+	0.979464i	$0.435380\pi$
$810$	0	0
$811$	−20.8851	−0.733375	−0.366687	−	0.930344i	$-0.619508\pi$
$811$	−20.8851	−0.733375	−0.366687	+	0.930344i	$0.619508\pi$
$812$	0	0
$813$	−19.7649	−0.693186
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−16.9991	−0.594723
$818$	0	0
$819$	0	0
$820$	0	0
$821$	12.7034	0.443353	0.221677	−	0.975120i	$-0.428847\pi$
$821$	12.7034	0.443353	0.221677	+	0.975120i	$0.428847\pi$
$822$	0	0
$823$	20.2909	0.707298	0.353649	−	0.935378i	$-0.384941\pi$
$823$	20.2909	0.707298	0.353649	+	0.935378i	$0.384941\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−23.7190	−0.824792	−0.412396	−	0.911005i	$-0.635308\pi$
$827$	−23.7190	−0.824792	−0.412396	+	0.911005i	$0.635308\pi$
$828$	0	0
$829$	52.5895	1.82651	0.913254	−	0.407392i	$-0.133562\pi$
$829$	52.5895	1.82651	0.913254	+	0.407392i	$0.133562\pi$
$830$	0	0
$831$	42.4683	1.47321
$832$	0	0
$833$	0	0
$834$	0	0
$835$	0	0
$836$	0	0
$837$	7.52982	0.260269
$838$	0	0
$839$	34.5601	1.19315	0.596573	−	0.802558i	$-0.296528\pi$
$839$	34.5601	1.19315	0.596573	+	0.802558i	$0.296528\pi$
$840$	0	0
$841$	−27.9385	−0.963398
$842$	0	0
$843$	−59.3700	−2.04481
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−27.2001	−0.934607
$848$	0	0
$849$	22.9385	0.787248
$850$	0	0
$851$	4.00000	0.137118
$852$	0	0
$853$	31.7115	1.08578	0.542890	−	0.839804i	$-0.317330\pi$
$853$	31.7115	1.08578	0.542890	+	0.839804i	$0.317330\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−7.93945	−0.271206	−0.135603	−	0.990763i	$-0.543297\pi$
$857$	−7.93945	−0.271206	−0.135603	+	0.990763i	$0.543297\pi$
$858$	0	0
$859$	30.8898	1.05395	0.526973	−	0.849882i	$-0.323327\pi$
$859$	30.8898	1.05395	0.526973	+	0.849882i	$0.323327\pi$
$860$	0	0
$861$	56.2938	1.91849
$862$	0	0
$863$	20.7081	0.704913	0.352457	−	0.935828i	$-0.385346\pi$
$863$	20.7081	0.704913	0.352457	+	0.935828i	$0.385346\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	53.1231	1.80415
$868$	0	0
$869$	−3.90826	−0.132578
$870$	0	0
$871$	0	0
$872$	0	0
$873$	−40.5895	−1.37374
$874$	0	0
$875$	0	0
$876$	0	0
$877$	39.1202	1.32099	0.660497	−	0.750828i	$-0.270345\pi$
$877$	39.1202	1.32099	0.660497	+	0.750828i	$0.270345\pi$
$878$	0	0
$879$	−31.2489	−1.05400
$880$	0	0
$881$	12.3491	0.416051	0.208026	−	0.978123i	$-0.433296\pi$
$881$	12.3491	0.416051	0.208026	+	0.978123i	$0.433296\pi$
$882$	0	0
$883$	−35.2195	−1.18523	−0.592615	−	0.805486i	$-0.701904\pi$
$883$	−35.2195	−1.18523	−0.592615	+	0.805486i	$0.701904\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−51.8043	−1.73942	−0.869708	−	0.493566i	$-0.835693\pi$
$887$	−51.8043	−1.73942	−0.869708	+	0.493566i	$0.835693\pi$
$888$	0	0
$889$	−54.8851	−1.84079
$890$	0	0
$891$	24.9532	0.835964
$892$	0	0
$893$	6.65001	0.222534
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	0.659401	0.0219923
$900$	0	0
$901$	0	0
$902$	0	0
$903$	29.5904	0.984706
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−55.8695	−1.85512	−0.927558	−	0.373679i	$-0.878096\pi$
$907$	−55.8695	−1.85512	−0.927558	+	0.373679i	$0.878096\pi$
$908$	0	0
$909$	−103.467	−3.43180
$910$	0	0
$911$	33.2607	1.10198	0.550988	−	0.834513i	$-0.314251\pi$
$911$	33.2607	1.10198	0.550988	+	0.834513i	$0.314251\pi$
$912$	0	0
$913$	19.0819	0.631518
$914$	0	0
$915$	0	0
$916$	0	0
$917$	2.00000	0.0660458
$918$	0	0
$919$	4.98910	0.164575	0.0822876	−	0.996609i	$-0.473777\pi$
$919$	4.98910	0.164575	0.0822876	+	0.996609i	$0.473777\pi$
$920$	0	0
$921$	−13.7649	−0.453569
$922$	0	0
$923$	0	0
$924$	0	0
$925$	0	0
$926$	0	0
$927$	99.7778	3.27713
$928$	0	0
$929$	30.0000	0.984268	0.492134	−	0.870519i	$-0.336217\pi$
$929$	30.0000	0.984268	0.492134	+	0.870519i	$0.336217\pi$
$930$	0	0
$931$	15.5104	0.508334
$932$	0	0
$933$	75.5280	2.47268
$934$	0	0
$935$	0	0
$936$	0	0
$937$	58.8851	1.92369	0.961846	−	0.273591i	$-0.0882115\pi$
$937$	58.8851	1.92369	0.961846	+	0.273591i	$0.0882115\pi$
$938$	0	0
$939$	46.8704	1.52956
$940$	0	0
$941$	19.6509	0.640602	0.320301	−	0.947316i	$-0.396216\pi$
$941$	19.6509	0.640602	0.320301	+	0.947316i	$0.396216\pi$
$942$	0	0
$943$	23.0596	0.750925
$944$	0	0
$945$	0	0
$946$	0	0
$947$	32.3397	1.05090	0.525449	−	0.850825i	$-0.323897\pi$
$947$	32.3397	1.05090	0.525449	+	0.850825i	$0.323897\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	76.7475	2.48871
$952$	0	0
$953$	48.8245	1.58158	0.790791	−	0.612086i	$-0.209669\pi$
$953$	48.8245	1.58158	0.790791	+	0.612086i	$0.209669\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	4.87798	0.157683
$958$	0	0
$959$	32.8099	1.05949
$960$	0	0
$961$	−30.5904	−0.986786
$962$	0	0
$963$	33.4886	1.07916
$964$	0	0
$965$	0	0
$966$	0	0
$967$	14.9915	0.482095	0.241047	−	0.970513i	$-0.422509\pi$
$967$	14.9915	0.482095	0.241047	+	0.970513i	$0.422509\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−18.4390	−0.591735	−0.295868	−	0.955229i	$-0.595609\pi$
$971$	−18.4390	−0.591735	−0.295868	+	0.955229i	$0.595609\pi$
$972$	0	0
$973$	−45.4986	−1.45862
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−29.5298	−0.944743	−0.472371	−	0.881400i	$-0.656602\pi$
$977$	−29.5298	−0.944743	−0.472371	+	0.881400i	$0.656602\pi$
$978$	0	0
$979$	12.9697	0.414514
$980$	0	0
$981$	−125.148	−3.99566
$982$	0	0
$983$	27.9348	0.890980	0.445490	−	0.895287i	$-0.353029\pi$
$983$	27.9348	0.890980	0.445490	+	0.895287i	$0.353029\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−11.5757	−0.368458
$988$	0	0
$989$	12.1211	0.385429
$990$	0	0
$991$	−3.85952	−0.122602	−0.0613008	−	0.998119i	$-0.519525\pi$
$991$	−3.85952	−0.122602	−0.0613008	+	0.998119i	$0.519525\pi$
$992$	0	0
$993$	50.5289	1.60349
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−18.6500	−0.590652	−0.295326	−	0.955397i	$-0.595428\pi$
$997$	−18.6500	−0.590652	−0.295326	+	0.955397i	$0.595428\pi$
$998$	0	0
$999$	11.7649	0.372225

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7400.2.a.m.1.1		3
5.4	even	2		1480.2.a.f.1.3	✓	3
20.19	odd	2		2960.2.a.s.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1480.2.a.f.1.3	✓	3	5.4	even	2
2960.2.a.s.1.1		3	20.19	odd	2
7400.2.a.m.1.1		3	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:	\( N \)	\(=\)	\( 7400 = 2^{3} \cdot 5^{2} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7400.a (trivial)

\(f(q)\)	\(=\)	\(q-3.12489 q^{3} +3.12489 q^{7} +6.76491 q^{9} +O(q^{10})\)	\(q-3.12489 q^{3} +3.12489 q^{7} +6.76491 q^{9} +1.51514 q^{11} +5.60975 q^{19} -9.76491 q^{21} -4.00000 q^{23} -11.7649 q^{27} -1.03028 q^{29} -0.640023 q^{31} -4.73463 q^{33} -1.00000 q^{37} -5.76491 q^{41} -3.03028 q^{43} +1.18544 q^{47} +2.76491 q^{49} -8.79518 q^{53} -17.5298 q^{57} -3.67030 q^{59} -14.4995 q^{61} +21.1396 q^{63} -11.6703 q^{67} +12.4995 q^{69} -0.734633 q^{71} -7.70436 q^{73} +4.73463 q^{77} -2.57947 q^{79} +16.4693 q^{81} +12.5942 q^{83} +3.21949 q^{87} +8.56009 q^{89} +2.00000 q^{93} -6.00000 q^{97} +10.2498 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - q^{3} + q^{7} + 4 q^{9}+O(q^{10}) \) 3 * q - q^3 + q^7 + 4 * q^9	\( 3 q - q^{3} + q^{7} + 4 q^{9} + 5 q^{11} + 8 q^{19} - 13 q^{21} - 12 q^{23} - 19 q^{27} - 4 q^{29} + 6 q^{31} + 3 q^{33} - 3 q^{37} - q^{41} - 10 q^{43} - 3 q^{47} - 8 q^{49} - 11 q^{53} - 20 q^{57} - 4 q^{59} - 10 q^{61} + 22 q^{63} - 28 q^{67} + 4 q^{69} + 15 q^{71} - 5 q^{73} - 3 q^{77} + 2 q^{79} + 15 q^{81} - 5 q^{83} - 8 q^{87} - 6 q^{89} + 6 q^{93} - 18 q^{97} + 14 q^{99}+O(q^{100}) \) 3 * q - q^3 + q^7 + 4 * q^9 + 5 * q^11 + 8 * q^19 - 13 * q^21 - 12 * q^23 - 19 * q^27 - 4 * q^29 + 6 * q^31 + 3 * q^33 - 3 * q^37 - q^41 - 10 * q^43 - 3 * q^47 - 8 * q^49 - 11 * q^53 - 20 * q^57 - 4 * q^59 - 10 * q^61 + 22 * q^63 - 28 * q^67 + 4 * q^69 + 15 * q^71 - 5 * q^73 - 3 * q^77 + 2 * q^79 + 15 * q^81 - 5 * q^83 - 8 * q^87 - 6 * q^89 + 6 * q^93 - 18 * q^97 + 14 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more