LMFDB - Embedded newform 7280.2.a.cc.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7280,2,Mod(1,7280)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7280, 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("7280.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7280 = 2^{4} \cdot 5 \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7280.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:	$58.1310926715$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	5.5.1194649.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 8x^{3} - 3x^{2} + 10x + 4 $ x^5 - 8x^3 - 3x^2 + 10*x + 4
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 3640)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.32400$ of defining polynomial
Character	$\chi$	$=$	7280.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.32400 q^{3} -1.00000 q^{5} -1.00000 q^{7} -1.24704 q^{9} +O(q^{10})$	$q-1.32400 q^{3} -1.00000 q^{5} -1.00000 q^{7} -1.24704 q^{9} -2.54610 q^{11} -1.00000 q^{13} +1.32400 q^{15} -3.92589 q^{17} +4.79313 q^{19} +1.32400 q^{21} +4.70002 q^{23} +1.00000 q^{25} +5.62306 q^{27} +4.32185 q^{29} -8.37387 q^{31} +3.37102 q^{33} +1.00000 q^{35} -1.69716 q^{37} +1.32400 q^{39} +4.84516 q^{41} +11.8188 q^{43} +1.24704 q^{45} +10.8209 q^{47} +1.00000 q^{49} +5.19786 q^{51} -9.04988 q^{53} +2.54610 q^{55} -6.34609 q^{57} +4.69502 q^{59} -5.37102 q^{61} +1.24704 q^{63} +1.00000 q^{65} -6.59289 q^{67} -6.22280 q^{69} +0.527086 q^{71} +4.27128 q^{73} -1.32400 q^{75} +2.54610 q^{77} +15.0870 q^{79} -3.70379 q^{81} -1.95013 q^{83} +3.92589 q^{85} -5.72211 q^{87} -7.91996 q^{89} +1.00000 q^{91} +11.0870 q^{93} -4.79313 q^{95} +4.32685 q^{97} +3.17508 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 5 q^{5} - 5 q^{7} + q^{9}+O(q^{10}) $ 5 * q - 5 * q^5 - 5 * q^7 + q^9	$ 5 q - 5 q^{5} - 5 q^{7} + q^{9} + 7 q^{11} - 5 q^{13} + q^{17} - 3 q^{19} + 5 q^{23} + 5 q^{25} + 9 q^{27} - 9 q^{29} - 6 q^{31} + q^{33} + 5 q^{35} - 10 q^{37} - 8 q^{41} - 6 q^{43} - q^{45} + 13 q^{47} + 5 q^{49} + 4 q^{51} - 16 q^{53} - 7 q^{55} - 10 q^{57} + q^{59} - 11 q^{61} - q^{63} + 5 q^{65} + 30 q^{67} - 15 q^{69} + 12 q^{71} + 23 q^{73} - 7 q^{77} - 2 q^{79} - 11 q^{81} + 2 q^{83} - q^{85} + 5 q^{87} - 25 q^{89} + 5 q^{91} - 22 q^{93} + 3 q^{95} - 5 q^{97} + 12 q^{99}+O(q^{100}) $ 5 * q - 5 * q^5 - 5 * q^7 + q^9 + 7 * q^11 - 5 * q^13 + q^17 - 3 * q^19 + 5 * q^23 + 5 * q^25 + 9 * q^27 - 9 * q^29 - 6 * q^31 + q^33 + 5 * q^35 - 10 * q^37 - 8 * q^41 - 6 * q^43 - q^45 + 13 * q^47 + 5 * q^49 + 4 * q^51 - 16 * q^53 - 7 * q^55 - 10 * q^57 + q^59 - 11 * q^61 - q^63 + 5 * q^65 + 30 * q^67 - 15 * q^69 + 12 * q^71 + 23 * q^73 - 7 * q^77 - 2 * q^79 - 11 * q^81 + 2 * q^83 - q^85 + 5 * q^87 - 25 * q^89 + 5 * q^91 - 22 * q^93 + 3 * q^95 - 5 * q^97 + 12 * 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
$3$	−1.32400	−0.764409	−0.382205	−	0.924078i	$-0.624835\pi$
$3$	−1.32400	−0.764409	−0.382205	+	0.924078i	$0.624835\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	−1.24704	−0.415679
$10$	0	0
$11$	−2.54610	−0.767677	−0.383839	−	0.923400i	$-0.625398\pi$
$11$	−2.54610	−0.767677	−0.383839	+	0.923400i	$0.625398\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	1.32400	0.341854
$16$	0	0
$17$	−3.92589	−0.952169	−0.476084	−	0.879400i	$-0.657944\pi$
$17$	−3.92589	−0.952169	−0.476084	+	0.879400i	$0.657944\pi$
$18$	0	0
$19$	4.79313	1.09962	0.549810	−	0.835290i	$-0.314700\pi$
$19$	4.79313	1.09962	0.549810	+	0.835290i	$0.314700\pi$
$20$	0	0
$21$	1.32400	0.288919
$22$	0	0
$23$	4.70002	0.980021	0.490010	−	0.871717i	$-0.336993\pi$
$23$	4.70002	0.980021	0.490010	+	0.871717i	$0.336993\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	5.62306	1.08216
$28$	0	0
$29$	4.32185	0.802547	0.401274	−	0.915958i	$-0.368568\pi$
$29$	4.32185	0.802547	0.401274	+	0.915958i	$0.368568\pi$
$30$	0	0
$31$	−8.37387	−1.50399	−0.751996	−	0.659168i	$-0.770909\pi$
$31$	−8.37387	−1.50399	−0.751996	+	0.659168i	$0.770909\pi$
$32$	0	0
$33$	3.37102	0.586819
$34$	0	0
$35$	1.00000	0.169031
$36$	0	0
$37$	−1.69716	−0.279012	−0.139506	−	0.990221i	$-0.544551\pi$
$37$	−1.69716	−0.279012	−0.139506	+	0.990221i	$0.544551\pi$
$38$	0	0
$39$	1.32400	0.212009
$40$	0	0
$41$	4.84516	0.756687	0.378343	−	0.925665i	$-0.376494\pi$
$41$	4.84516	0.756687	0.378343	+	0.925665i	$0.376494\pi$
$42$	0	0
$43$	11.8188	1.80235	0.901173	−	0.433460i	$-0.142707\pi$
$43$	11.8188	1.80235	0.901173	+	0.433460i	$0.142707\pi$
$44$	0	0
$45$	1.24704	0.185897
$46$	0	0
$47$	10.8209	1.57839	0.789197	−	0.614141i	$-0.210497\pi$
$47$	10.8209	1.57839	0.789197	+	0.614141i	$0.210497\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	5.19786	0.727846
$52$	0	0
$53$	−9.04988	−1.24310	−0.621548	−	0.783376i	$-0.713496\pi$
$53$	−9.04988	−1.24310	−0.621548	+	0.783376i	$0.713496\pi$
$54$	0	0
$55$	2.54610	0.343316
$56$	0	0
$57$	−6.34609	−0.840560
$58$	0	0
$59$	4.69502	0.611239	0.305620	−	0.952154i	$-0.401136\pi$
$59$	4.69502	0.611239	0.305620	+	0.952154i	$0.401136\pi$
$60$	0	0
$61$	−5.37102	−0.687689	−0.343844	−	0.939027i	$-0.611729\pi$
$61$	−5.37102	−0.687689	−0.343844	+	0.939027i	$0.611729\pi$
$62$	0	0
$63$	1.24704	0.157112
$64$	0	0
$65$	1.00000	0.124035
$66$	0	0
$67$	−6.59289	−0.805450	−0.402725	−	0.915321i	$-0.631937\pi$
$67$	−6.59289	−0.805450	−0.402725	+	0.915321i	$0.631937\pi$
$68$	0	0
$69$	−6.22280	−0.749137
$70$	0	0
$71$	0.527086	0.0625536	0.0312768	−	0.999511i	$-0.490043\pi$
$71$	0.527086	0.0625536	0.0312768	+	0.999511i	$0.490043\pi$
$72$	0	0
$73$	4.27128	0.499915	0.249957	−	0.968257i	$-0.419583\pi$
$73$	4.27128	0.499915	0.249957	+	0.968257i	$0.419583\pi$
$74$	0	0
$75$	−1.32400	−0.152882
$76$	0	0
$77$	2.54610	0.290155
$78$	0	0
$79$	15.0870	1.69742	0.848708	−	0.528861i	$-0.177381\pi$
$79$	15.0870	1.69742	0.848708	+	0.528861i	$0.177381\pi$
$80$	0	0
$81$	−3.70379	−0.411532
$82$	0	0
$83$	−1.95013	−0.214055	−0.107027	−	0.994256i	$-0.534133\pi$
$83$	−1.95013	−0.214055	−0.107027	+	0.994256i	$0.534133\pi$
$84$	0	0
$85$	3.92589	0.425823
$86$	0	0
$87$	−5.72211	−0.613474
$88$	0	0
$89$	−7.91996	−0.839514	−0.419757	−	0.907636i	$-0.637885\pi$
$89$	−7.91996	−0.839514	−0.419757	+	0.907636i	$0.637885\pi$
$90$	0	0
$91$	1.00000	0.104828
$92$	0	0
$93$	11.0870	1.14967
$94$	0	0
$95$	−4.79313	−0.491765
$96$	0	0
$97$	4.32685	0.439325	0.219662	−	0.975576i	$-0.429504\pi$
$97$	4.32685	0.439325	0.219662	+	0.975576i	$0.429504\pi$
$98$	0	0
$99$	3.17508	0.319107
$100$	0	0
$101$	4.51308	0.449069	0.224534	−	0.974466i	$-0.427914\pi$
$101$	4.51308	0.449069	0.224534	+	0.974466i	$0.427914\pi$
$102$	0	0
$103$	−14.2948	−1.40851	−0.704253	−	0.709949i	$-0.748718\pi$
$103$	−14.2948	−1.40851	−0.704253	+	0.709949i	$0.748718\pi$
$104$	0	0
$105$	−1.32400	−0.129209
$106$	0	0
$107$	−4.67671	−0.452115	−0.226057	−	0.974114i	$-0.572584\pi$
$107$	−4.67671	−0.452115	−0.226057	+	0.974114i	$0.572584\pi$
$108$	0	0
$109$	−6.98815	−0.669343	−0.334671	−	0.942335i	$-0.608625\pi$
$109$	−6.98815	−0.669343	−0.334671	+	0.942335i	$0.608625\pi$
$110$	0	0
$111$	2.24704	0.213279
$112$	0	0
$113$	−2.72303	−0.256161	−0.128081	−	0.991764i	$-0.540882\pi$
$113$	−2.72303	−0.256161	−0.128081	+	0.991764i	$0.540882\pi$
$114$	0	0
$115$	−4.70002	−0.438279
$116$	0	0
$117$	1.24704	0.115289
$118$	0	0
$119$	3.92589	0.359886
$120$	0	0
$121$	−4.51739	−0.410672
$122$	0	0
$123$	−6.41497	−0.578418
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	1.05388	0.0935168	0.0467584	−	0.998906i	$-0.485111\pi$
$127$	1.05388	0.0935168	0.0467584	+	0.998906i	$0.485111\pi$
$128$	0	0
$129$	−15.6480	−1.37773
$130$	0	0
$131$	−5.40758	−0.472463	−0.236231	−	0.971697i	$-0.575912\pi$
$131$	−5.40758	−0.472463	−0.236231	+	0.971697i	$0.575912\pi$
$132$	0	0
$133$	−4.79313	−0.415617
$134$	0	0
$135$	−5.62306	−0.483956
$136$	0	0
$137$	−11.5124	−0.983570	−0.491785	−	0.870717i	$-0.663655\pi$
$137$	−11.5124	−0.983570	−0.491785	+	0.870717i	$0.663655\pi$
$138$	0	0
$139$	−10.4898	−0.889731	−0.444866	−	0.895597i	$-0.646749\pi$
$139$	−10.4898	−0.889731	−0.444866	+	0.895597i	$0.646749\pi$
$140$	0	0
$141$	−14.3268	−1.20654
$142$	0	0
$143$	2.54610	0.212915
$144$	0	0
$145$	−4.32185	−0.358910
$146$	0	0
$147$	−1.32400	−0.109201
$148$	0	0
$149$	2.77105	0.227013	0.113507	−	0.993537i	$-0.463792\pi$
$149$	2.77105	0.227013	0.113507	+	0.993537i	$0.463792\pi$
$150$	0	0
$151$	12.1266	0.986849	0.493425	−	0.869789i	$-0.335745\pi$
$151$	12.1266	0.986849	0.493425	+	0.869789i	$0.335745\pi$
$152$	0	0
$153$	4.89573	0.395796
$154$	0	0
$155$	8.37387	0.672606
$156$	0	0
$157$	6.54465	0.522320	0.261160	−	0.965296i	$-0.415895\pi$
$157$	6.54465	0.522320	0.261160	+	0.965296i	$0.415895\pi$
$158$	0	0
$159$	11.9820	0.950234
$160$	0	0
$161$	−4.70002	−0.370413
$162$	0	0
$163$	18.1736	1.42347	0.711734	−	0.702449i	$-0.247910\pi$
$163$	18.1736	1.42347	0.711734	+	0.702449i	$0.247910\pi$
$164$	0	0
$165$	−3.37102	−0.262434
$166$	0	0
$167$	16.9515	1.31175	0.655874	−	0.754870i	$-0.272300\pi$
$167$	16.9515	1.31175	0.655874	+	0.754870i	$0.272300\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−5.97721	−0.457089
$172$	0	0
$173$	−11.8202	−0.898675	−0.449338	−	0.893362i	$-0.648340\pi$
$173$	−11.8202	−0.898675	−0.449338	+	0.893362i	$0.648340\pi$
$174$	0	0
$175$	−1.00000	−0.0755929
$176$	0	0
$177$	−6.21618	−0.467237
$178$	0	0
$179$	−13.5545	−1.01311	−0.506555	−	0.862208i	$-0.669081\pi$
$179$	−13.5545	−1.01311	−0.506555	+	0.862208i	$0.669081\pi$
$180$	0	0
$181$	−13.9767	−1.03888	−0.519440	−	0.854507i	$-0.673859\pi$
$181$	−13.9767	−1.03888	−0.519440	+	0.854507i	$0.673859\pi$
$182$	0	0
$183$	7.11121	0.525676
$184$	0	0
$185$	1.69716	0.124778
$186$	0	0
$187$	9.99571	0.730958
$188$	0	0
$189$	−5.62306	−0.409017
$190$	0	0
$191$	16.2160	1.17335	0.586673	−	0.809824i	$-0.300438\pi$
$191$	16.2160	1.17335	0.586673	+	0.809824i	$0.300438\pi$
$192$	0	0
$193$	−9.37009	−0.674474	−0.337237	−	0.941420i	$-0.609492\pi$
$193$	−9.37009	−0.674474	−0.337237	+	0.941420i	$0.609492\pi$
$194$	0	0
$195$	−1.32400	−0.0948133
$196$	0	0
$197$	−1.21064	−0.0862546	−0.0431273	−	0.999070i	$-0.513732\pi$
$197$	−1.21064	−0.0862546	−0.0431273	+	0.999070i	$0.513732\pi$
$198$	0	0
$199$	−17.5938	−1.24719	−0.623596	−	0.781746i	$-0.714329\pi$
$199$	−17.5938	−1.24719	−0.623596	+	0.781746i	$0.714329\pi$
$200$	0	0
$201$	8.72896	0.615694
$202$	0	0
$203$	−4.32185	−0.303334
$204$	0	0
$205$	−4.84516	−0.338401
$206$	0	0
$207$	−5.86109	−0.407374
$208$	0	0
$209$	−12.2038	−0.844154
$210$	0	0
$211$	−23.5677	−1.62247	−0.811235	−	0.584721i	$-0.801204\pi$
$211$	−23.5677	−1.62247	−0.811235	+	0.584721i	$0.801204\pi$
$212$	0	0
$213$	−0.697859	−0.0478165
$214$	0	0
$215$	−11.8188	−0.806034
$216$	0	0
$217$	8.37387	0.568456
$218$	0	0
$219$	−5.65515	−0.382139
$220$	0	0
$221$	3.92589	0.264084
$222$	0	0
$223$	−0.376715	−0.0252267	−0.0126134	−	0.999920i	$-0.504015\pi$
$223$	−0.376715	−0.0252267	−0.0126134	+	0.999920i	$0.504015\pi$
$224$	0	0
$225$	−1.24704	−0.0831358
$226$	0	0
$227$	3.59998	0.238939	0.119469	−	0.992838i	$-0.461881\pi$
$227$	3.59998	0.238939	0.119469	+	0.992838i	$0.461881\pi$
$228$	0	0
$229$	−17.2219	−1.13805	−0.569027	−	0.822319i	$-0.692680\pi$
$229$	−17.2219	−1.13805	−0.569027	+	0.822319i	$0.692680\pi$
$230$	0	0
$231$	−3.37102	−0.221797
$232$	0	0
$233$	6.88265	0.450897	0.225449	−	0.974255i	$-0.427615\pi$
$233$	6.88265	0.450897	0.225449	+	0.974255i	$0.427615\pi$
$234$	0	0
$235$	−10.8209	−0.705879
$236$	0	0
$237$	−19.9751	−1.29752
$238$	0	0
$239$	−3.52849	−0.228239	−0.114119	−	0.993467i	$-0.536405\pi$
$239$	−3.52849	−0.228239	−0.114119	+	0.993467i	$0.536405\pi$
$240$	0	0
$241$	4.02563	0.259314	0.129657	−	0.991559i	$-0.458612\pi$
$241$	4.02563	0.259314	0.129657	+	0.991559i	$0.458612\pi$
$242$	0	0
$243$	−11.9654	−0.767579
$244$	0	0
$245$	−1.00000	−0.0638877
$246$	0	0
$247$	−4.79313	−0.304980
$248$	0	0
$249$	2.58196	0.163625
$250$	0	0
$251$	−18.1231	−1.14392	−0.571959	−	0.820282i	$-0.693816\pi$
$251$	−18.1231	−1.14392	−0.571959	+	0.820282i	$0.693816\pi$
$252$	0	0
$253$	−11.9667	−0.752340
$254$	0	0
$255$	−5.19786	−0.325503
$256$	0	0
$257$	13.4423	0.838511	0.419255	−	0.907868i	$-0.362291\pi$
$257$	13.4423	0.838511	0.419255	+	0.907868i	$0.362291\pi$
$258$	0	0
$259$	1.69716	0.105457
$260$	0	0
$261$	−5.38950	−0.333602
$262$	0	0
$263$	27.5514	1.69889	0.849446	−	0.527675i	$-0.176936\pi$
$263$	27.5514	1.69889	0.849446	+	0.527675i	$0.176936\pi$
$264$	0	0
$265$	9.04988	0.555930
$266$	0	0
$267$	10.4860	0.641732
$268$	0	0
$269$	−16.5131	−1.00682	−0.503410	−	0.864048i	$-0.667922\pi$
$269$	−16.5131	−1.00682	−0.503410	+	0.864048i	$0.667922\pi$
$270$	0	0
$271$	0.0458648	0.00278609	0.00139305	−	0.999999i	$-0.499557\pi$
$271$	0.0458648	0.00278609	0.00139305	+	0.999999i	$0.499557\pi$
$272$	0	0
$273$	−1.32400	−0.0801318
$274$	0	0
$275$	−2.54610	−0.153535
$276$	0	0
$277$	25.2375	1.51638	0.758188	−	0.652036i	$-0.226085\pi$
$277$	25.2375	1.51638	0.758188	+	0.652036i	$0.226085\pi$
$278$	0	0
$279$	10.4425	0.625178
$280$	0	0
$281$	16.4437	0.980952	0.490476	−	0.871455i	$-0.336823\pi$
$281$	16.4437	0.980952	0.490476	+	0.871455i	$0.336823\pi$
$282$	0	0
$283$	−7.36038	−0.437529	−0.218765	−	0.975778i	$-0.570203\pi$
$283$	−7.36038	−0.437529	−0.218765	+	0.975778i	$0.570203\pi$
$284$	0	0
$285$	6.34609	0.375910
$286$	0	0
$287$	−4.84516	−0.286001
$288$	0	0
$289$	−1.58737	−0.0933745
$290$	0	0
$291$	−5.72872	−0.335824
$292$	0	0
$293$	−20.8442	−1.21773	−0.608866	−	0.793273i	$-0.708375\pi$
$293$	−20.8442	−1.21773	−0.608866	+	0.793273i	$0.708375\pi$
$294$	0	0
$295$	−4.69502	−0.273354
$296$	0	0
$297$	−14.3168	−0.830748
$298$	0	0
$299$	−4.70002	−0.271809
$300$	0	0
$301$	−11.8188	−0.681223
$302$	0	0
$303$	−5.97530	−0.343272
$304$	0	0
$305$	5.37102	0.307544
$306$	0	0
$307$	0.504070	0.0287688	0.0143844	−	0.999897i	$-0.495421\pi$
$307$	0.504070	0.0287688	0.0143844	+	0.999897i	$0.495421\pi$
$308$	0	0
$309$	18.9262	1.07667
$310$	0	0
$311$	15.6860	0.889473	0.444736	−	0.895662i	$-0.353297\pi$
$311$	15.6860	0.889473	0.444736	+	0.895662i	$0.353297\pi$
$312$	0	0
$313$	0.227797	0.0128758	0.00643791	−	0.999979i	$-0.497951\pi$
$313$	0.227797	0.0128758	0.00643791	+	0.999979i	$0.497951\pi$
$314$	0	0
$315$	−1.24704	−0.0702625
$316$	0	0
$317$	5.76349	0.323710	0.161855	−	0.986815i	$-0.448252\pi$
$317$	5.76349	0.323710	0.161855	+	0.986815i	$0.448252\pi$
$318$	0	0
$319$	−11.0038	−0.616097
$320$	0	0
$321$	6.19194	0.345600
$322$	0	0
$323$	−18.8173	−1.04702
$324$	0	0
$325$	−1.00000	−0.0554700
$326$	0	0
$327$	9.25227	0.511652
$328$	0	0
$329$	−10.8209	−0.596577
$330$	0	0
$331$	−0.800209	−0.0439835	−0.0219917	−	0.999758i	$-0.507001\pi$
$331$	−0.800209	−0.0439835	−0.0219917	+	0.999758i	$0.507001\pi$
$332$	0	0
$333$	2.11642	0.115979
$334$	0	0
$335$	6.59289	0.360208
$336$	0	0
$337$	20.1051	1.09519	0.547596	−	0.836743i	$-0.315543\pi$
$337$	20.1051	1.09519	0.547596	+	0.836743i	$0.315543\pi$
$338$	0	0
$339$	3.60528	0.195812
$340$	0	0
$341$	21.3207	1.15458
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	6.22280	0.335024
$346$	0	0
$347$	34.1048	1.83084	0.915420	−	0.402500i	$-0.131859\pi$
$347$	34.1048	1.83084	0.915420	+	0.402500i	$0.131859\pi$
$348$	0	0
$349$	−29.4240	−1.57503	−0.787516	−	0.616294i	$-0.788633\pi$
$349$	−29.4240	−1.57503	−0.787516	+	0.616294i	$0.788633\pi$
$350$	0	0
$351$	−5.62306	−0.300137
$352$	0	0
$353$	29.1592	1.55199	0.775995	−	0.630740i	$-0.217248\pi$
$353$	29.1592	1.55199	0.775995	+	0.630740i	$0.217248\pi$
$354$	0	0
$355$	−0.527086	−0.0279748
$356$	0	0
$357$	−5.19786	−0.275100
$358$	0	0
$359$	−14.1148	−0.744949	−0.372474	−	0.928042i	$-0.621491\pi$
$359$	−14.1148	−0.744949	−0.372474	+	0.928042i	$0.621491\pi$
$360$	0	0
$361$	3.97413	0.209165
$362$	0	0
$363$	5.98100	0.313921
$364$	0	0
$365$	−4.27128	−0.223569
$366$	0	0
$367$	−35.9708	−1.87766	−0.938831	−	0.344377i	$-0.888090\pi$
$367$	−35.9708	−1.87766	−0.938831	+	0.344377i	$0.888090\pi$
$368$	0	0
$369$	−6.04209	−0.314539
$370$	0	0
$371$	9.04988	0.469846
$372$	0	0
$373$	−13.9634	−0.722997	−0.361498	−	0.932373i	$-0.617735\pi$
$373$	−13.9634	−0.722997	−0.361498	+	0.932373i	$0.617735\pi$
$374$	0	0
$375$	1.32400	0.0683708
$376$	0	0
$377$	−4.32185	−0.222587
$378$	0	0
$379$	6.02872	0.309674	0.154837	−	0.987940i	$-0.450515\pi$
$379$	6.02872	0.309674	0.154837	+	0.987940i	$0.450515\pi$
$380$	0	0
$381$	−1.39533	−0.0714851
$382$	0	0
$383$	29.2034	1.49222	0.746112	−	0.665820i	$-0.231918\pi$
$383$	29.2034	1.49222	0.746112	+	0.665820i	$0.231918\pi$
$384$	0	0
$385$	−2.54610	−0.129761
$386$	0	0
$387$	−14.7384	−0.749197
$388$	0	0
$389$	−33.4461	−1.69579	−0.847893	−	0.530167i	$-0.822129\pi$
$389$	−33.4461	−1.69579	−0.847893	+	0.530167i	$0.822129\pi$
$390$	0	0
$391$	−18.4518	−0.933145
$392$	0	0
$393$	7.15961	0.361155
$394$	0	0
$395$	−15.0870	−0.759108
$396$	0	0
$397$	−13.4133	−0.673193	−0.336597	−	0.941649i	$-0.609276\pi$
$397$	−13.4133	−0.673193	−0.336597	+	0.941649i	$0.609276\pi$
$398$	0	0
$399$	6.34609	0.317702
$400$	0	0
$401$	7.29785	0.364437	0.182219	−	0.983258i	$-0.441672\pi$
$401$	7.29785	0.364437	0.182219	+	0.983258i	$0.441672\pi$
$402$	0	0
$403$	8.37387	0.417132
$404$	0	0
$405$	3.70379	0.184043
$406$	0	0
$407$	4.32114	0.214191
$408$	0	0
$409$	25.0879	1.24052	0.620258	−	0.784398i	$-0.287028\pi$
$409$	25.0879	1.24052	0.620258	+	0.784398i	$0.287028\pi$
$410$	0	0
$411$	15.2423	0.751850
$412$	0	0
$413$	−4.69502	−0.231027
$414$	0	0
$415$	1.95013	0.0957282
$416$	0	0
$417$	13.8884	0.680119
$418$	0	0
$419$	10.2727	0.501853	0.250927	−	0.968006i	$-0.419265\pi$
$419$	10.2727	0.501853	0.250927	+	0.968006i	$0.419265\pi$
$420$	0	0
$421$	7.32870	0.357179	0.178590	−	0.983924i	$-0.442847\pi$
$421$	7.32870	0.357179	0.178590	+	0.983924i	$0.442847\pi$
$422$	0	0
$423$	−13.4941	−0.656105
$424$	0	0
$425$	−3.92589	−0.190434
$426$	0	0
$427$	5.37102	0.259922
$428$	0	0
$429$	−3.37102	−0.162754
$430$	0	0
$431$	−7.89235	−0.380161	−0.190081	−	0.981768i	$-0.560875\pi$
$431$	−7.89235	−0.380161	−0.190081	+	0.981768i	$0.560875\pi$
$432$	0	0
$433$	−18.2100	−0.875118	−0.437559	−	0.899190i	$-0.644157\pi$
$433$	−18.2100	−0.875118	−0.437559	+	0.899190i	$0.644157\pi$
$434$	0	0
$435$	5.72211	0.274354
$436$	0	0
$437$	22.5278	1.07765
$438$	0	0
$439$	−29.0262	−1.38534	−0.692672	−	0.721253i	$-0.743567\pi$
$439$	−29.0262	−1.38534	−0.692672	+	0.721253i	$0.743567\pi$
$440$	0	0
$441$	−1.24704	−0.0593827
$442$	0	0
$443$	−31.0549	−1.47546	−0.737731	−	0.675095i	$-0.764103\pi$
$443$	−31.0549	−1.47546	−0.737731	+	0.675095i	$0.764103\pi$
$444$	0	0
$445$	7.91996	0.375442
$446$	0	0
$447$	−3.66886	−0.173531
$448$	0	0
$449$	−18.4090	−0.868774	−0.434387	−	0.900726i	$-0.643035\pi$
$449$	−18.4090	−0.868774	−0.434387	+	0.900726i	$0.643035\pi$
$450$	0	0
$451$	−12.3362	−0.580891
$452$	0	0
$453$	−16.0556	−0.754357
$454$	0	0
$455$	−1.00000	−0.0468807
$456$	0	0
$457$	−36.8958	−1.72591	−0.862957	−	0.505278i	$-0.831390\pi$
$457$	−36.8958	−1.72591	−0.862957	+	0.505278i	$0.831390\pi$
$458$	0	0
$459$	−22.0755	−1.03040
$460$	0	0
$461$	−2.22641	−0.103694	−0.0518470	−	0.998655i	$-0.516511\pi$
$461$	−2.22641	−0.103694	−0.0518470	+	0.998655i	$0.516511\pi$
$462$	0	0
$463$	−26.9564	−1.25277	−0.626384	−	0.779514i	$-0.715466\pi$
$463$	−26.9564	−1.25277	−0.626384	+	0.779514i	$0.715466\pi$
$464$	0	0
$465$	−11.0870	−0.514146
$466$	0	0
$467$	13.3532	0.617914	0.308957	−	0.951076i	$-0.400020\pi$
$467$	13.3532	0.617914	0.308957	+	0.951076i	$0.400020\pi$
$468$	0	0
$469$	6.59289	0.304432
$470$	0	0
$471$	−8.66508	−0.399266
$472$	0	0
$473$	−30.0917	−1.38362
$474$	0	0
$475$	4.79313	0.219924
$476$	0	0
$477$	11.2855	0.516729
$478$	0	0
$479$	−34.1331	−1.55958	−0.779790	−	0.626041i	$-0.784674\pi$
$479$	−34.1331	−1.55958	−0.779790	+	0.626041i	$0.784674\pi$
$480$	0	0
$481$	1.69716	0.0773840
$482$	0	0
$483$	6.22280	0.283147
$484$	0	0
$485$	−4.32685	−0.196472
$486$	0	0
$487$	−25.0103	−1.13333	−0.566663	−	0.823950i	$-0.691766\pi$
$487$	−25.0103	−1.13333	−0.566663	+	0.823950i	$0.691766\pi$
$488$	0	0
$489$	−24.0618	−1.08811
$490$	0	0
$491$	−21.5227	−0.971305	−0.485653	−	0.874152i	$-0.661418\pi$
$491$	−21.5227	−0.971305	−0.485653	+	0.874152i	$0.661418\pi$
$492$	0	0
$493$	−16.9671	−0.764160
$494$	0	0
$495$	−3.17508	−0.142709
$496$	0	0
$497$	−0.527086	−0.0236430
$498$	0	0
$499$	−7.70432	−0.344893	−0.172446	−	0.985019i	$-0.555167\pi$
$499$	−7.70432	−0.344893	−0.172446	+	0.985019i	$0.555167\pi$
$500$	0	0
$501$	−22.4437	−1.00271
$502$	0	0
$503$	1.57835	0.0703754	0.0351877	−	0.999381i	$-0.488797\pi$
$503$	1.57835	0.0703754	0.0351877	+	0.999381i	$0.488797\pi$
$504$	0	0
$505$	−4.51308	−0.200830
$506$	0	0
$507$	−1.32400	−0.0588007
$508$	0	0
$509$	−10.2566	−0.454617	−0.227309	−	0.973823i	$-0.572993\pi$
$509$	−10.2566	−0.454617	−0.227309	+	0.973823i	$0.572993\pi$
$510$	0	0
$511$	−4.27128	−0.188950
$512$	0	0
$513$	26.9521	1.18996
$514$	0	0
$515$	14.2948	0.629903
$516$	0	0
$517$	−27.5511	−1.21170
$518$	0	0
$519$	15.6499	0.686955
$520$	0	0
$521$	27.6343	1.21068	0.605340	−	0.795967i	$-0.293037\pi$
$521$	27.6343	1.21068	0.605340	+	0.795967i	$0.293037\pi$
$522$	0	0
$523$	−22.8725	−1.00015	−0.500073	−	0.865983i	$-0.666694\pi$
$523$	−22.8725	−1.00015	−0.500073	+	0.865983i	$0.666694\pi$
$524$	0	0
$525$	1.32400	0.0577839
$526$	0	0
$527$	32.8749	1.43205
$528$	0	0
$529$	−0.909858	−0.0395590
$530$	0	0
$531$	−5.85486	−0.254079
$532$	0	0
$533$	−4.84516	−0.209867
$534$	0	0
$535$	4.67671	0.202192
$536$	0	0
$537$	17.9461	0.774430
$538$	0	0
$539$	−2.54610	−0.109668
$540$	0	0
$541$	36.3440	1.56255	0.781276	−	0.624186i	$-0.214569\pi$
$541$	36.3440	1.56255	0.781276	+	0.624186i	$0.214569\pi$
$542$	0	0
$543$	18.5051	0.794129
$544$	0	0
$545$	6.98815	0.299339
$546$	0	0
$547$	14.7090	0.628913	0.314456	−	0.949272i	$-0.398178\pi$
$547$	14.7090	0.628913	0.314456	+	0.949272i	$0.398178\pi$
$548$	0	0
$549$	6.69786	0.285858
$550$	0	0
$551$	20.7152	0.882497
$552$	0	0
$553$	−15.0870	−0.641563
$554$	0	0
$555$	−2.24704	−0.0953814
$556$	0	0
$557$	−36.7791	−1.55838	−0.779191	−	0.626787i	$-0.784370\pi$
$557$	−36.7791	−1.55838	−0.779191	+	0.626787i	$0.784370\pi$
$558$	0	0
$559$	−11.8188	−0.499881
$560$	0	0
$561$	−13.2343	−0.558751
$562$	0	0
$563$	−4.92804	−0.207692	−0.103846	−	0.994593i	$-0.533115\pi$
$563$	−4.92804	−0.207692	−0.103846	+	0.994593i	$0.533115\pi$
$564$	0	0
$565$	2.72303	0.114559
$566$	0	0
$567$	3.70379	0.155545
$568$	0	0
$569$	7.05627	0.295814	0.147907	−	0.989001i	$-0.452746\pi$
$569$	7.05627	0.295814	0.147907	+	0.989001i	$0.452746\pi$
$570$	0	0
$571$	10.4183	0.435994	0.217997	−	0.975949i	$-0.430048\pi$
$571$	10.4183	0.435994	0.217997	+	0.975949i	$0.430048\pi$
$572$	0	0
$573$	−21.4698	−0.896916
$574$	0	0
$575$	4.70002	0.196004
$576$	0	0
$577$	28.3146	1.17875	0.589376	−	0.807859i	$-0.299374\pi$
$577$	28.3146	1.17875	0.589376	+	0.807859i	$0.299374\pi$
$578$	0	0
$579$	12.4060	0.515574
$580$	0	0
$581$	1.95013	0.0809051
$582$	0	0
$583$	23.0419	0.954297
$584$	0	0
$585$	−1.24704	−0.0515586
$586$	0	0
$587$	−23.5510	−0.972056	−0.486028	−	0.873943i	$-0.661555\pi$
$587$	−23.5510	−0.972056	−0.486028	+	0.873943i	$0.661555\pi$
$588$	0	0
$589$	−40.1371	−1.65382
$590$	0	0
$591$	1.60288	0.0659338
$592$	0	0
$593$	3.39617	0.139464	0.0697321	−	0.997566i	$-0.477786\pi$
$593$	3.39617	0.139464	0.0697321	+	0.997566i	$0.477786\pi$
$594$	0	0
$595$	−3.92589	−0.160946
$596$	0	0
$597$	23.2941	0.953365
$598$	0	0
$599$	−9.15921	−0.374235	−0.187118	−	0.982338i	$-0.559915\pi$
$599$	−9.15921	−0.374235	−0.187118	+	0.982338i	$0.559915\pi$
$600$	0	0
$601$	30.1707	1.23069	0.615344	−	0.788259i	$-0.289017\pi$
$601$	30.1707	1.23069	0.615344	+	0.788259i	$0.289017\pi$
$602$	0	0
$603$	8.22158	0.334809
$604$	0	0
$605$	4.51739	0.183658
$606$	0	0
$607$	−20.4832	−0.831388	−0.415694	−	0.909504i	$-0.636461\pi$
$607$	−20.4832	−0.831388	−0.415694	+	0.909504i	$0.636461\pi$
$608$	0	0
$609$	5.72211	0.231871
$610$	0	0
$611$	−10.8209	−0.437768
$612$	0	0
$613$	7.15493	0.288985	0.144493	−	0.989506i	$-0.453845\pi$
$613$	7.15493	0.288985	0.144493	+	0.989506i	$0.453845\pi$
$614$	0	0
$615$	6.41497	0.258676
$616$	0	0
$617$	−30.7640	−1.23851	−0.619256	−	0.785190i	$-0.712565\pi$
$617$	−30.7640	−1.23851	−0.619256	+	0.785190i	$0.712565\pi$
$618$	0	0
$619$	−14.9971	−0.602786	−0.301393	−	0.953500i	$-0.597452\pi$
$619$	−14.9971	−0.602786	−0.301393	+	0.953500i	$0.597452\pi$
$620$	0	0
$621$	26.4284	1.06054
$622$	0	0
$623$	7.91996	0.317307
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	16.1578	0.645279
$628$	0	0
$629$	6.66288	0.265667
$630$	0	0
$631$	−42.5258	−1.69293	−0.846464	−	0.532447i	$-0.821273\pi$
$631$	−42.5258	−1.69293	−0.846464	+	0.532447i	$0.821273\pi$
$632$	0	0
$633$	31.2036	1.24023
$634$	0	0
$635$	−1.05388	−0.0418220
$636$	0	0
$637$	−1.00000	−0.0396214
$638$	0	0
$639$	−0.657295	−0.0260022
$640$	0	0
$641$	−1.93897	−0.0765848	−0.0382924	−	0.999267i	$-0.512192\pi$
$641$	−1.93897	−0.0765848	−0.0382924	+	0.999267i	$0.512192\pi$
$642$	0	0
$643$	−10.7539	−0.424093	−0.212046	−	0.977260i	$-0.568013\pi$
$643$	−10.7539	−0.424093	−0.212046	+	0.977260i	$0.568013\pi$
$644$	0	0
$645$	15.6480	0.616139
$646$	0	0
$647$	0.201400	0.00791785	0.00395892	−	0.999992i	$-0.498740\pi$
$647$	0.201400	0.00791785	0.00395892	+	0.999992i	$0.498740\pi$
$648$	0	0
$649$	−11.9540	−0.469234
$650$	0	0
$651$	−11.0870	−0.434533
$652$	0	0
$653$	17.7520	0.694691	0.347346	−	0.937737i	$-0.387083\pi$
$653$	17.7520	0.694691	0.347346	+	0.937737i	$0.387083\pi$
$654$	0	0
$655$	5.40758	0.211292
$656$	0	0
$657$	−5.32644	−0.207804
$658$	0	0
$659$	28.7235	1.11891	0.559454	−	0.828861i	$-0.311011\pi$
$659$	28.7235	1.11891	0.559454	+	0.828861i	$0.311011\pi$
$660$	0	0
$661$	−27.6076	−1.07381	−0.536906	−	0.843642i	$-0.680407\pi$
$661$	−27.6076	−1.07381	−0.536906	+	0.843642i	$0.680407\pi$
$662$	0	0
$663$	−5.19786	−0.201868
$664$	0	0
$665$	4.79313	0.185870
$666$	0	0
$667$	20.3128	0.786513
$668$	0	0
$669$	0.498770	0.0192835
$670$	0	0
$671$	13.6751	0.527923
$672$	0	0
$673$	21.1668	0.815919	0.407960	−	0.913000i	$-0.366240\pi$
$673$	21.1668	0.815919	0.407960	+	0.913000i	$0.366240\pi$
$674$	0	0
$675$	5.62306	0.216432
$676$	0	0
$677$	−8.12016	−0.312083	−0.156042	−	0.987750i	$-0.549873\pi$
$677$	−8.12016	−0.312083	−0.156042	+	0.987750i	$0.549873\pi$
$678$	0	0
$679$	−4.32685	−0.166049
$680$	0	0
$681$	−4.76635	−0.182647
$682$	0	0
$683$	−17.9886	−0.688316	−0.344158	−	0.938912i	$-0.611836\pi$
$683$	−17.9886	−0.688316	−0.344158	+	0.938912i	$0.611836\pi$
$684$	0	0
$685$	11.5124	0.439866
$686$	0	0
$687$	22.8017	0.869938
$688$	0	0
$689$	9.04988	0.344773
$690$	0	0
$691$	−34.2442	−1.30271	−0.651356	−	0.758773i	$-0.725799\pi$
$691$	−34.2442	−1.30271	−0.651356	+	0.758773i	$0.725799\pi$
$692$	0	0
$693$	−3.17508	−0.120611
$694$	0	0
$695$	10.4898	0.397900
$696$	0	0
$697$	−19.0216	−0.720493
$698$	0	0
$699$	−9.11260	−0.344670
$700$	0	0
$701$	1.63131	0.0616138	0.0308069	−	0.999525i	$-0.490192\pi$
$701$	1.63131	0.0616138	0.0308069	+	0.999525i	$0.490192\pi$
$702$	0	0
$703$	−8.13473	−0.306807
$704$	0	0
$705$	14.3268	0.539580
$706$	0	0
$707$	−4.51308	−0.169732
$708$	0	0
$709$	17.7093	0.665088	0.332544	−	0.943088i	$-0.392093\pi$
$709$	17.7093	0.665088	0.332544	+	0.943088i	$0.392093\pi$
$710$	0	0
$711$	−18.8140	−0.705580
$712$	0	0
$713$	−39.3573	−1.47394
$714$	0	0
$715$	−2.54610	−0.0952186
$716$	0	0
$717$	4.67170	0.174468
$718$	0	0
$719$	17.3710	0.647828	0.323914	−	0.946087i	$-0.395001\pi$
$719$	17.3710	0.647828	0.323914	+	0.946087i	$0.395001\pi$
$720$	0	0
$721$	14.2948	0.532365
$722$	0	0
$723$	−5.32992	−0.198222
$724$	0	0
$725$	4.32185	0.160509
$726$	0	0
$727$	−12.3161	−0.456777	−0.228389	−	0.973570i	$-0.573346\pi$
$727$	−12.3161	−0.456777	−0.228389	+	0.973570i	$0.573346\pi$
$728$	0	0
$729$	26.9535	0.998276
$730$	0	0
$731$	−46.3992	−1.71614
$732$	0	0
$733$	38.2101	1.41132	0.705661	−	0.708549i	$-0.250650\pi$
$733$	38.2101	1.41132	0.705661	+	0.708549i	$0.250650\pi$
$734$	0	0
$735$	1.32400	0.0488363
$736$	0	0
$737$	16.7862	0.618326
$738$	0	0
$739$	20.3072	0.747011	0.373505	−	0.927628i	$-0.378156\pi$
$739$	20.3072	0.747011	0.373505	+	0.927628i	$0.378156\pi$
$740$	0	0
$741$	6.34609	0.233129
$742$	0	0
$743$	18.4200	0.675763	0.337882	−	0.941189i	$-0.390290\pi$
$743$	18.4200	0.675763	0.337882	+	0.941189i	$0.390290\pi$
$744$	0	0
$745$	−2.77105	−0.101523
$746$	0	0
$747$	2.43189	0.0889780
$748$	0	0
$749$	4.67671	0.170883
$750$	0	0
$751$	21.5352	0.785832	0.392916	−	0.919574i	$-0.371466\pi$
$751$	21.5352	0.785832	0.392916	+	0.919574i	$0.371466\pi$
$752$	0	0
$753$	23.9948	0.874421
$754$	0	0
$755$	−12.1266	−0.441332
$756$	0	0
$757$	−18.6537	−0.677980	−0.338990	−	0.940790i	$-0.610085\pi$
$757$	−18.6537	−0.677980	−0.338990	+	0.940790i	$0.610085\pi$
$758$	0	0
$759$	15.8438	0.575095
$760$	0	0
$761$	12.1471	0.440330	0.220165	−	0.975463i	$-0.429340\pi$
$761$	12.1471	0.440330	0.220165	+	0.975463i	$0.429340\pi$
$762$	0	0
$763$	6.98815	0.252988
$764$	0	0
$765$	−4.89573	−0.177006
$766$	0	0
$767$	−4.69502	−0.169527
$768$	0	0
$769$	−19.8123	−0.714451	−0.357225	−	0.934018i	$-0.616277\pi$
$769$	−19.8123	−0.714451	−0.357225	+	0.934018i	$0.616277\pi$
$770$	0	0
$771$	−17.7976	−0.640965
$772$	0	0
$773$	−50.7217	−1.82433	−0.912167	−	0.409819i	$-0.865592\pi$
$773$	−50.7217	−1.82433	−0.912167	+	0.409819i	$0.865592\pi$
$774$	0	0
$775$	−8.37387	−0.300798
$776$	0	0
$777$	−2.24704	−0.0806120
$778$	0	0
$779$	23.2235	0.832068
$780$	0	0
$781$	−1.34201	−0.0480210
$782$	0	0
$783$	24.3020	0.868482
$784$	0	0
$785$	−6.54465	−0.233588
$786$	0	0
$787$	3.14961	0.112272	0.0561358	−	0.998423i	$-0.482122\pi$
$787$	3.14961	0.112272	0.0561358	+	0.998423i	$0.482122\pi$
$788$	0	0
$789$	−36.4779	−1.29865
$790$	0	0
$791$	2.72303	0.0968198
$792$	0	0
$793$	5.37102	0.190731
$794$	0	0
$795$	−11.9820	−0.424958
$796$	0	0
$797$	−3.38503	−0.119904	−0.0599520	−	0.998201i	$-0.519095\pi$
$797$	−3.38503	−0.119904	−0.0599520	+	0.998201i	$0.519095\pi$
$798$	0	0
$799$	−42.4818	−1.50290
$800$	0	0
$801$	9.87648	0.348968
$802$	0	0
$803$	−10.8751	−0.383773
$804$	0	0
$805$	4.70002	0.165654
$806$	0	0
$807$	21.8632	0.769623
$808$	0	0
$809$	−4.89543	−0.172114	−0.0860571	−	0.996290i	$-0.527427\pi$
$809$	−4.89543	−0.172114	−0.0860571	+	0.996290i	$0.527427\pi$
$810$	0	0
$811$	21.6544	0.760389	0.380194	−	0.924907i	$-0.375857\pi$
$811$	21.6544	0.760389	0.380194	+	0.924907i	$0.375857\pi$
$812$	0	0
$813$	−0.0607248	−0.00212971
$814$	0	0
$815$	−18.1736	−0.636595
$816$	0	0
$817$	56.6490	1.98190
$818$	0	0
$819$	−1.24704	−0.0435750
$820$	0	0
$821$	7.35956	0.256850	0.128425	−	0.991719i	$-0.459008\pi$
$821$	7.35956	0.256850	0.128425	+	0.991719i	$0.459008\pi$
$822$	0	0
$823$	−36.8948	−1.28607	−0.643035	−	0.765837i	$-0.722325\pi$
$823$	−36.8948	−1.28607	−0.643035	+	0.765837i	$0.722325\pi$
$824$	0	0
$825$	3.37102	0.117364
$826$	0	0
$827$	50.2841	1.74855	0.874275	−	0.485431i	$-0.161337\pi$
$827$	50.2841	1.74855	0.874275	+	0.485431i	$0.161337\pi$
$828$	0	0
$829$	−27.7254	−0.962944	−0.481472	−	0.876461i	$-0.659898\pi$
$829$	−27.7254	−0.962944	−0.481472	+	0.876461i	$0.659898\pi$
$830$	0	0
$831$	−33.4144	−1.15913
$832$	0	0
$833$	−3.92589	−0.136024
$834$	0	0
$835$	−16.9515	−0.586632
$836$	0	0
$837$	−47.0868	−1.62756
$838$	0	0
$839$	−39.6877	−1.37017	−0.685086	−	0.728462i	$-0.740235\pi$
$839$	−39.6877	−1.37017	−0.685086	+	0.728462i	$0.740235\pi$
$840$	0	0
$841$	−10.3216	−0.355918
$842$	0	0
$843$	−21.7714	−0.749848
$844$	0	0
$845$	−1.00000	−0.0344010
$846$	0	0
$847$	4.51739	0.155219
$848$	0	0
$849$	9.74511	0.334451
$850$	0	0
$851$	−7.97669	−0.273438
$852$	0	0
$853$	0.599513	0.0205269	0.0102635	−	0.999947i	$-0.496733\pi$
$853$	0.599513	0.0205269	0.0102635	+	0.999947i	$0.496733\pi$
$854$	0	0
$855$	5.97721	0.204416
$856$	0	0
$857$	−37.7333	−1.28894	−0.644472	−	0.764628i	$-0.722923\pi$
$857$	−37.7333	−1.28894	−0.644472	+	0.764628i	$0.722923\pi$
$858$	0	0
$859$	−14.5805	−0.497481	−0.248740	−	0.968570i	$-0.580017\pi$
$859$	−14.5805	−0.497481	−0.248740	+	0.968570i	$0.580017\pi$
$860$	0	0
$861$	6.41497	0.218621
$862$	0	0
$863$	35.9402	1.22342	0.611709	−	0.791083i	$-0.290482\pi$
$863$	35.9402	1.22342	0.611709	+	0.791083i	$0.290482\pi$
$864$	0	0
$865$	11.8202	0.401900
$866$	0	0
$867$	2.10166	0.0713763
$868$	0	0
$869$	−38.4129	−1.30307
$870$	0	0
$871$	6.59289	0.223392
$872$	0	0
$873$	−5.39574	−0.182618
$874$	0	0
$875$	1.00000	0.0338062
$876$	0	0
$877$	43.3174	1.46272	0.731362	−	0.681989i	$-0.238885\pi$
$877$	43.3174	1.46272	0.731362	+	0.681989i	$0.238885\pi$
$878$	0	0
$879$	27.5977	0.930846
$880$	0	0
$881$	−33.5858	−1.13154	−0.565768	−	0.824565i	$-0.691420\pi$
$881$	−33.5858	−1.13154	−0.565768	+	0.824565i	$0.691420\pi$
$882$	0	0
$883$	23.6436	0.795670	0.397835	−	0.917457i	$-0.369762\pi$
$883$	23.6436	0.795670	0.397835	+	0.917457i	$0.369762\pi$
$884$	0	0
$885$	6.21618	0.208955
$886$	0	0
$887$	−12.7330	−0.427534	−0.213767	−	0.976885i	$-0.568573\pi$
$887$	−12.7330	−0.427534	−0.213767	+	0.976885i	$0.568573\pi$
$888$	0	0
$889$	−1.05388	−0.0353460
$890$	0	0
$891$	9.43021	0.315924
$892$	0	0
$893$	51.8661	1.73563
$894$	0	0
$895$	13.5545	0.453076
$896$	0	0
$897$	6.22280	0.207773
$898$	0	0
$899$	−36.1906	−1.20702
$900$	0	0
$901$	35.5288	1.18364
$902$	0	0
$903$	15.6480	0.520733
$904$	0	0
$905$	13.9767	0.464601
$906$	0	0
$907$	17.7514	0.589424	0.294712	−	0.955586i	$-0.404776\pi$
$907$	17.7514	0.589424	0.294712	+	0.955586i	$0.404776\pi$
$908$	0	0
$909$	−5.62798	−0.186668
$910$	0	0
$911$	−7.76256	−0.257185	−0.128593	−	0.991698i	$-0.541046\pi$
$911$	−7.76256	−0.257185	−0.128593	+	0.991698i	$0.541046\pi$
$912$	0	0
$913$	4.96522	0.164325
$914$	0	0
$915$	−7.11121	−0.235089
$916$	0	0
$917$	5.40758	0.178574
$918$	0	0
$919$	−36.5271	−1.20492	−0.602459	−	0.798150i	$-0.705812\pi$
$919$	−36.5271	−1.20492	−0.602459	+	0.798150i	$0.705812\pi$
$920$	0	0
$921$	−0.667387	−0.0219911
$922$	0	0
$923$	−0.527086	−0.0173492
$924$	0	0
$925$	−1.69716	−0.0558024
$926$	0	0
$927$	17.8261	0.585486
$928$	0	0
$929$	−51.2973	−1.68301	−0.841505	−	0.540249i	$-0.818330\pi$
$929$	−51.2973	−1.68301	−0.841505	+	0.540249i	$0.818330\pi$
$930$	0	0
$931$	4.79313	0.157089
$932$	0	0
$933$	−20.7682	−0.679921
$934$	0	0
$935$	−9.99571	−0.326895
$936$	0	0
$937$	16.6195	0.542935	0.271467	−	0.962448i	$-0.412491\pi$
$937$	16.6195	0.542935	0.271467	+	0.962448i	$0.412491\pi$
$938$	0	0
$939$	−0.301602	−0.00984240
$940$	0	0
$941$	−2.23375	−0.0728180	−0.0364090	−	0.999337i	$-0.511592\pi$
$941$	−2.23375	−0.0728180	−0.0364090	+	0.999337i	$0.511592\pi$
$942$	0	0
$943$	22.7723	0.741569
$944$	0	0
$945$	5.62306	0.182918
$946$	0	0
$947$	40.7174	1.32314	0.661568	−	0.749885i	$-0.269891\pi$
$947$	40.7174	1.32314	0.661568	+	0.749885i	$0.269891\pi$
$948$	0	0
$949$	−4.27128	−0.138651
$950$	0	0
$951$	−7.63084	−0.247447
$952$	0	0
$953$	31.6803	1.02623	0.513113	−	0.858321i	$-0.328492\pi$
$953$	31.6803	1.02623	0.513113	+	0.858321i	$0.328492\pi$
$954$	0	0
$955$	−16.2160	−0.524736
$956$	0	0
$957$	14.5690	0.470950
$958$	0	0
$959$	11.5124	0.371754
$960$	0	0
$961$	39.1217	1.26199
$962$	0	0
$963$	5.83203	0.187934
$964$	0	0
$965$	9.37009	0.301634
$966$	0	0
$967$	37.8987	1.21874	0.609370	−	0.792886i	$-0.291422\pi$
$967$	37.8987	1.21874	0.609370	+	0.792886i	$0.291422\pi$
$968$	0	0
$969$	24.9141	0.800355
$970$	0	0
$971$	−13.9824	−0.448716	−0.224358	−	0.974507i	$-0.572029\pi$
$971$	−13.9824	−0.448716	−0.224358	+	0.974507i	$0.572029\pi$
$972$	0	0
$973$	10.4898	0.336287
$974$	0	0
$975$	1.32400	0.0424018
$976$	0	0
$977$	41.7478	1.33563	0.667816	−	0.744327i	$-0.267229\pi$
$977$	41.7478	1.33563	0.667816	+	0.744327i	$0.267229\pi$
$978$	0	0
$979$	20.1650	0.644476
$980$	0	0
$981$	8.71447	0.278232
$982$	0	0
$983$	32.4144	1.03386	0.516929	−	0.856028i	$-0.327075\pi$
$983$	32.4144	1.03386	0.516929	+	0.856028i	$0.327075\pi$
$984$	0	0
$985$	1.21064	0.0385742
$986$	0	0
$987$	14.3268	0.456029
$988$	0	0
$989$	55.5484	1.76634
$990$	0	0
$991$	22.5373	0.715922	0.357961	−	0.933737i	$-0.383472\pi$
$991$	22.5373	0.715922	0.357961	+	0.933737i	$0.383472\pi$
$992$	0	0
$993$	1.05947	0.0336213
$994$	0	0
$995$	17.5938	0.557762
$996$	0	0
$997$	−34.0690	−1.07898	−0.539488	−	0.841994i	$-0.681382\pi$
$997$	−34.0690	−1.07898	−0.539488	+	0.841994i	$0.681382\pi$
$998$	0	0
$999$	−9.54325	−0.301935

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7280.2.a.cc.1.2		5
4.3	odd	2		3640.2.a.y.1.4	✓	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3640.2.a.y.1.4	✓	5	4.3	odd	2
7280.2.a.cc.1.2		5	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:	\( N \)	\(=\)	\( 7280 = 2^{4} \cdot 5 \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7280.a (trivial)

\(f(q)\)	\(=\)	\(q-1.32400 q^{3} -1.00000 q^{5} -1.00000 q^{7} -1.24704 q^{9} +O(q^{10})\)	\(q-1.32400 q^{3} -1.00000 q^{5} -1.00000 q^{7} -1.24704 q^{9} -2.54610 q^{11} -1.00000 q^{13} +1.32400 q^{15} -3.92589 q^{17} +4.79313 q^{19} +1.32400 q^{21} +4.70002 q^{23} +1.00000 q^{25} +5.62306 q^{27} +4.32185 q^{29} -8.37387 q^{31} +3.37102 q^{33} +1.00000 q^{35} -1.69716 q^{37} +1.32400 q^{39} +4.84516 q^{41} +11.8188 q^{43} +1.24704 q^{45} +10.8209 q^{47} +1.00000 q^{49} +5.19786 q^{51} -9.04988 q^{53} +2.54610 q^{55} -6.34609 q^{57} +4.69502 q^{59} -5.37102 q^{61} +1.24704 q^{63} +1.00000 q^{65} -6.59289 q^{67} -6.22280 q^{69} +0.527086 q^{71} +4.27128 q^{73} -1.32400 q^{75} +2.54610 q^{77} +15.0870 q^{79} -3.70379 q^{81} -1.95013 q^{83} +3.92589 q^{85} -5.72211 q^{87} -7.91996 q^{89} +1.00000 q^{91} +11.0870 q^{93} -4.79313 q^{95} +4.32685 q^{97} +3.17508 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 5 q^{5} - 5 q^{7} + q^{9}+O(q^{10}) \) 5 * q - 5 * q^5 - 5 * q^7 + q^9	\( 5 q - 5 q^{5} - 5 q^{7} + q^{9} + 7 q^{11} - 5 q^{13} + q^{17} - 3 q^{19} + 5 q^{23} + 5 q^{25} + 9 q^{27} - 9 q^{29} - 6 q^{31} + q^{33} + 5 q^{35} - 10 q^{37} - 8 q^{41} - 6 q^{43} - q^{45} + 13 q^{47} + 5 q^{49} + 4 q^{51} - 16 q^{53} - 7 q^{55} - 10 q^{57} + q^{59} - 11 q^{61} - q^{63} + 5 q^{65} + 30 q^{67} - 15 q^{69} + 12 q^{71} + 23 q^{73} - 7 q^{77} - 2 q^{79} - 11 q^{81} + 2 q^{83} - q^{85} + 5 q^{87} - 25 q^{89} + 5 q^{91} - 22 q^{93} + 3 q^{95} - 5 q^{97} + 12 q^{99}+O(q^{100}) \) 5 * q - 5 * q^5 - 5 * q^7 + q^9 + 7 * q^11 - 5 * q^13 + q^17 - 3 * q^19 + 5 * q^23 + 5 * q^25 + 9 * q^27 - 9 * q^29 - 6 * q^31 + q^33 + 5 * q^35 - 10 * q^37 - 8 * q^41 - 6 * q^43 - q^45 + 13 * q^47 + 5 * q^49 + 4 * q^51 - 16 * q^53 - 7 * q^55 - 10 * q^57 + q^59 - 11 * q^61 - q^63 + 5 * q^65 + 30 * q^67 - 15 * q^69 + 12 * q^71 + 23 * q^73 - 7 * q^77 - 2 * q^79 - 11 * q^81 + 2 * q^83 - q^85 + 5 * q^87 - 25 * q^89 + 5 * q^91 - 22 * q^93 + 3 * q^95 - 5 * q^97 + 12 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more