LMFDB - Embedded newform 3640.2.a.t.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3640,2,Mod(1,3640)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3640, 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("3640.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3640 = 2^{3} \cdot 5 \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3640.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$29.0655463357$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.24197.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 6x^{2} - x + 2 $ x^4 - 6*x^2 - x + 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.700017$ of defining polynomial
Character	$\chi$	$=$	3640.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.700017 q^{3} -1.00000 q^{5} +1.00000 q^{7} -2.50998 q^{9} +O(q^{10})$	$q-0.700017 q^{3} -1.00000 q^{5} +1.00000 q^{7} -2.50998 q^{9} +4.66704 q^{11} +1.00000 q^{13} +0.700017 q^{15} -5.15706 q^{17} -1.75703 q^{19} -0.700017 q^{21} +4.06707 q^{23} +1.00000 q^{25} +3.85708 q^{27} -7.96702 q^{29} -1.99418 q^{31} -3.26700 q^{33} -1.00000 q^{35} -8.63405 q^{37} -0.700017 q^{39} +7.22413 q^{41} -7.71415 q^{43} +2.50998 q^{45} +3.96120 q^{47} +1.00000 q^{49} +3.61003 q^{51} +10.7341 q^{53} -4.66704 q^{55} +1.22995 q^{57} +6.06125 q^{59} +11.3612 q^{61} -2.50998 q^{63} -1.00000 q^{65} -7.22413 q^{67} -2.84702 q^{69} -11.8200 q^{71} +2.24705 q^{73} -0.700017 q^{75} +4.66704 q^{77} -5.82409 q^{79} +4.82991 q^{81} -5.40003 q^{83} +5.15706 q^{85} +5.57705 q^{87} -8.98697 q^{89} +1.00000 q^{91} +1.39596 q^{93} +1.75703 q^{95} -5.97284 q^{97} -11.7142 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{5} + 4 q^{7}+O(q^{10}) $ 4 * q - 4 * q^5 + 4 * q^7	$ 4 q - 4 q^{5} + 4 q^{7} - q^{11} + 4 q^{13} - 11 q^{17} - 3 q^{19} - 9 q^{23} + 4 q^{25} + 3 q^{27} - 15 q^{29} - 6 q^{31} + q^{33} - 4 q^{35} + 2 q^{37} - 6 q^{41} - 6 q^{43} - 3 q^{47} + 4 q^{49} - 4 q^{51} - 2 q^{53} + q^{55} - 28 q^{57} - 3 q^{59} + 21 q^{61} - 4 q^{65} + 6 q^{67} - 23 q^{69} - 16 q^{71} + 15 q^{73} - q^{77} + 6 q^{79} - 8 q^{81} - 16 q^{83} + 11 q^{85} - 13 q^{87} + q^{89} + 4 q^{91} - 2 q^{93} + 3 q^{95} - 9 q^{97} - 22 q^{99}+O(q^{100}) $ 4 * q - 4 * q^5 + 4 * q^7 - q^11 + 4 * q^13 - 11 * q^17 - 3 * q^19 - 9 * q^23 + 4 * q^25 + 3 * q^27 - 15 * q^29 - 6 * q^31 + q^33 - 4 * q^35 + 2 * q^37 - 6 * q^41 - 6 * q^43 - 3 * q^47 + 4 * q^49 - 4 * q^51 - 2 * q^53 + q^55 - 28 * q^57 - 3 * q^59 + 21 * q^61 - 4 * q^65 + 6 * q^67 - 23 * q^69 - 16 * q^71 + 15 * q^73 - q^77 + 6 * q^79 - 8 * q^81 - 16 * q^83 + 11 * q^85 - 13 * q^87 + q^89 + 4 * q^91 - 2 * q^93 + 3 * q^95 - 9 * q^97 - 22 * 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
$3$	−0.700017	−0.404155	−0.202077	−	0.979370i	$-0.564769\pi$
$3$	−0.700017	−0.404155	−0.202077	+	0.979370i	$0.564769\pi$
$4$	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
$9$	−2.50998	−0.836659
$10$	0	0
$11$	4.66704	1.40716	0.703582	−	0.710614i	$-0.251583\pi$
$11$	4.66704	1.40716	0.703582	+	0.710614i	$0.251583\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	0.700017	0.180744
$16$	0	0
$17$	−5.15706	−1.25077	−0.625385	−	0.780316i	$-0.715058\pi$
$17$	−5.15706	−1.25077	−0.625385	+	0.780316i	$0.715058\pi$
$18$	0	0
$19$	−1.75703	−0.403089	−0.201545	−	0.979479i	$-0.564596\pi$
$19$	−1.75703	−0.403089	−0.201545	+	0.979479i	$0.564596\pi$
$20$	0	0
$21$	−0.700017	−0.152756
$22$	0	0
$23$	4.06707	0.848042	0.424021	−	0.905652i	$-0.360618\pi$
$23$	4.06707	0.848042	0.424021	+	0.905652i	$0.360618\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	3.85708	0.742294
$28$	0	0
$29$	−7.96702	−1.47944	−0.739719	−	0.672916i	$-0.765042\pi$
$29$	−7.96702	−1.47944	−0.739719	+	0.672916i	$0.765042\pi$
$30$	0	0
$31$	−1.99418	−0.358165	−0.179083	−	0.983834i	$-0.557313\pi$
$31$	−1.99418	−0.358165	−0.179083	+	0.983834i	$0.557313\pi$
$32$	0	0
$33$	−3.26700	−0.568712
$34$	0	0
$35$	−1.00000	−0.169031
$36$	0	0
$37$	−8.63405	−1.41943	−0.709715	−	0.704489i	$-0.751176\pi$
$37$	−8.63405	−1.41943	−0.709715	+	0.704489i	$0.751176\pi$
$38$	0	0
$39$	−0.700017	−0.112092
$40$	0	0
$41$	7.22413	1.12822	0.564110	−	0.825700i	$-0.309220\pi$
$41$	7.22413	1.12822	0.564110	+	0.825700i	$0.309220\pi$
$42$	0	0
$43$	−7.71415	−1.17640	−0.588198	−	0.808717i	$-0.700163\pi$
$43$	−7.71415	−1.17640	−0.588198	+	0.808717i	$0.700163\pi$
$44$	0	0
$45$	2.50998	0.374165
$46$	0	0
$47$	3.96120	0.577800	0.288900	−	0.957359i	$-0.406710\pi$
$47$	3.96120	0.577800	0.288900	+	0.957359i	$0.406710\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	3.61003	0.505505
$52$	0	0
$53$	10.7341	1.47444	0.737221	−	0.675651i	$-0.236138\pi$
$53$	10.7341	1.47444	0.737221	+	0.675651i	$0.236138\pi$
$54$	0	0
$55$	−4.66704	−0.629303
$56$	0	0
$57$	1.22995	0.162910
$58$	0	0
$59$	6.06125	0.789107	0.394554	−	0.918873i	$-0.370899\pi$
$59$	6.06125	0.789107	0.394554	+	0.918873i	$0.370899\pi$
$60$	0	0
$61$	11.3612	1.45466	0.727328	−	0.686290i	$-0.240762\pi$
$61$	11.3612	1.45466	0.727328	+	0.686290i	$0.240762\pi$
$62$	0	0
$63$	−2.50998	−0.316227
$64$	0	0
$65$	−1.00000	−0.124035
$66$	0	0
$67$	−7.22413	−0.882568	−0.441284	−	0.897368i	$-0.645477\pi$
$67$	−7.22413	−0.882568	−0.441284	+	0.897368i	$0.645477\pi$
$68$	0	0
$69$	−2.84702	−0.342740
$70$	0	0
$71$	−11.8200	−1.40278	−0.701389	−	0.712779i	$-0.747436\pi$
$71$	−11.8200	−1.40278	−0.701389	+	0.712779i	$0.747436\pi$
$72$	0	0
$73$	2.24705	0.262997	0.131499	−	0.991316i	$-0.458021\pi$
$73$	2.24705	0.262997	0.131499	+	0.991316i	$0.458021\pi$
$74$	0	0
$75$	−0.700017	−0.0808310
$76$	0	0
$77$	4.66704	0.531858
$78$	0	0
$79$	−5.82409	−0.655262	−0.327631	−	0.944806i	$-0.606250\pi$
$79$	−5.82409	−0.655262	−0.327631	+	0.944806i	$0.606250\pi$
$80$	0	0
$81$	4.82991	0.536657
$82$	0	0
$83$	−5.40003	−0.592731	−0.296365	−	0.955075i	$-0.595775\pi$
$83$	−5.40003	−0.592731	−0.296365	+	0.955075i	$0.595775\pi$
$84$	0	0
$85$	5.15706	0.559362
$86$	0	0
$87$	5.57705	0.597922
$88$	0	0
$89$	−8.98697	−0.952617	−0.476309	−	0.879278i	$-0.658025\pi$
$89$	−8.98697	−0.952617	−0.476309	+	0.879278i	$0.658025\pi$
$90$	0	0
$91$	1.00000	0.104828
$92$	0	0
$93$	1.39596	0.144754
$94$	0	0
$95$	1.75703	0.180267
$96$	0	0
$97$	−5.97284	−0.606450	−0.303225	−	0.952919i	$-0.598063\pi$
$97$	−5.97284	−0.606450	−0.303225	+	0.952919i	$0.598063\pi$
$98$	0	0
$99$	−11.7142	−1.17732
$100$	0	0
$101$	−15.0211	−1.49465	−0.747326	−	0.664458i	$-0.768662\pi$
$101$	−15.0211	−1.49465	−0.747326	+	0.664458i	$0.768662\pi$
$102$	0	0
$103$	9.09110	0.895772	0.447886	−	0.894091i	$-0.352177\pi$
$103$	9.09110	0.895772	0.447886	+	0.894091i	$0.352177\pi$
$104$	0	0
$105$	0.700017	0.0683146
$106$	0	0
$107$	−10.5142	−1.01645	−0.508224	−	0.861225i	$-0.669698\pi$
$107$	−10.5142	−1.01645	−0.508224	+	0.861225i	$0.669698\pi$
$108$	0	0
$109$	−0.705836	−0.0676069	−0.0338034	−	0.999429i	$-0.510762\pi$
$109$	−0.705836	−0.0676069	−0.0338034	+	0.999429i	$0.510762\pi$
$110$	0	0
$111$	6.04398	0.573669
$112$	0	0
$113$	−9.96120	−0.937071	−0.468535	−	0.883445i	$-0.655218\pi$
$113$	−9.96120	−0.937071	−0.468535	+	0.883445i	$0.655218\pi$
$114$	0	0
$115$	−4.06707	−0.379256
$116$	0	0
$117$	−2.50998	−0.232047
$118$	0	0
$119$	−5.15706	−0.472747
$120$	0	0
$121$	10.7812	0.980111
$122$	0	0
$123$	−5.05701	−0.455975
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−19.0870	−1.69370	−0.846850	−	0.531832i	$-0.821504\pi$
$127$	−19.0870	−1.69370	−0.846850	+	0.531832i	$0.821504\pi$
$128$	0	0
$129$	5.40003	0.475446
$130$	0	0
$131$	2.80007	0.244643	0.122321	−	0.992491i	$-0.460966\pi$
$131$	2.80007	0.244643	0.122321	+	0.992491i	$0.460966\pi$
$132$	0	0
$133$	−1.75703	−0.152353
$134$	0	0
$135$	−3.85708	−0.331964
$136$	0	0
$137$	−9.14716	−0.781495	−0.390748	−	0.920498i	$-0.627783\pi$
$137$	−9.14716	−0.781495	−0.390748	+	0.920498i	$0.627783\pi$
$138$	0	0
$139$	11.8483	1.00496	0.502479	−	0.864589i	$-0.332421\pi$
$139$	11.8483	1.00496	0.502479	+	0.864589i	$0.332421\pi$
$140$	0	0
$141$	−2.77290	−0.233521
$142$	0	0
$143$	4.66704	0.390277
$144$	0	0
$145$	7.96702	0.661625
$146$	0	0
$147$	−0.700017	−0.0577364
$148$	0	0
$149$	3.68699	0.302050	0.151025	−	0.988530i	$-0.451743\pi$
$149$	3.68699	0.302050	0.151025	+	0.988530i	$0.451743\pi$
$150$	0	0
$151$	−16.0399	−1.30531	−0.652655	−	0.757656i	$-0.726345\pi$
$151$	−16.0399	−1.30531	−0.652655	+	0.757656i	$0.726345\pi$
$152$	0	0
$153$	12.9441	1.04647
$154$	0	0
$155$	1.99418	0.160176
$156$	0	0
$157$	19.1681	1.52978	0.764889	−	0.644162i	$-0.222794\pi$
$157$	19.1681	1.52978	0.764889	+	0.644162i	$0.222794\pi$
$158$	0	0
$159$	−7.51405	−0.595903
$160$	0	0
$161$	4.06707	0.320530
$162$	0	0
$163$	2.66122	0.208442	0.104221	−	0.994554i	$-0.466765\pi$
$163$	2.66122	0.208442	0.104221	+	0.994554i	$0.466765\pi$
$164$	0	0
$165$	3.26700	0.254336
$166$	0	0
$167$	−13.8084	−1.06852	−0.534262	−	0.845319i	$-0.679411\pi$
$167$	−13.8084	−1.06852	−0.534262	+	0.845319i	$0.679411\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	4.41009	0.337248
$172$	0	0
$173$	−25.7094	−1.95465	−0.977326	−	0.211740i	$-0.932087\pi$
$173$	−25.7094	−1.95465	−0.977326	+	0.211740i	$0.932087\pi$
$174$	0	0
$175$	1.00000	0.0755929
$176$	0	0
$177$	−4.24297	−0.318922
$178$	0	0
$179$	−15.6911	−1.17281	−0.586403	−	0.810020i	$-0.699456\pi$
$179$	−15.6911	−1.17281	−0.586403	+	0.810020i	$0.699456\pi$
$180$	0	0
$181$	−2.87528	−0.213718	−0.106859	−	0.994274i	$-0.534079\pi$
$181$	−2.87528	−0.213718	−0.106859	+	0.994274i	$0.534079\pi$
$182$	0	0
$183$	−7.95305	−0.587906
$184$	0	0
$185$	8.63405	0.634788
$186$	0	0
$187$	−24.0682	−1.76004
$188$	0	0
$189$	3.85708	0.280561
$190$	0	0
$191$	−15.7754	−1.14147	−0.570734	−	0.821135i	$-0.693341\pi$
$191$	−15.7754	−1.14147	−0.570734	+	0.821135i	$0.693341\pi$
$192$	0	0
$193$	−0.831302	−0.0598384	−0.0299192	−	0.999552i	$-0.509525\pi$
$193$	−0.831302	−0.0598384	−0.0299192	+	0.999552i	$0.509525\pi$
$194$	0	0
$195$	0.700017	0.0501292
$196$	0	0
$197$	−21.5400	−1.53466	−0.767330	−	0.641252i	$-0.778415\pi$
$197$	−21.5400	−1.53466	−0.767330	+	0.641252i	$0.778415\pi$
$198$	0	0
$199$	11.1622	0.791270	0.395635	−	0.918408i	$-0.370525\pi$
$199$	11.1622	0.791270	0.395635	+	0.918408i	$0.370525\pi$
$200$	0	0
$201$	5.05701	0.356694
$202$	0	0
$203$	−7.96702	−0.559175
$204$	0	0
$205$	−7.22413	−0.504555
$206$	0	0
$207$	−10.2082	−0.709522
$208$	0	0
$209$	−8.20010	−0.567213
$210$	0	0
$211$	8.58536	0.591040	0.295520	−	0.955336i	$-0.404507\pi$
$211$	8.58536	0.591040	0.295520	+	0.955336i	$0.404507\pi$
$212$	0	0
$213$	8.27421	0.566940
$214$	0	0
$215$	7.71415	0.526101
$216$	0	0
$217$	−1.99418	−0.135374
$218$	0	0
$219$	−1.57297	−0.106292
$220$	0	0
$221$	−5.15706	−0.346901
$222$	0	0
$223$	−26.9268	−1.80315	−0.901577	−	0.432619i	$-0.857590\pi$
$223$	−26.9268	−1.80315	−0.901577	+	0.432619i	$0.857590\pi$
$224$	0	0
$225$	−2.50998	−0.167332
$226$	0	0
$227$	−20.8283	−1.38243	−0.691213	−	0.722652i	$-0.742923\pi$
$227$	−20.8283	−1.38243	−0.691213	+	0.722652i	$0.742923\pi$
$228$	0	0
$229$	−7.56541	−0.499936	−0.249968	−	0.968254i	$-0.580420\pi$
$229$	−7.56541	−0.499936	−0.249968	+	0.968254i	$0.580420\pi$
$230$	0	0
$231$	−3.26700	−0.214953
$232$	0	0
$233$	9.18125	0.601484	0.300742	−	0.953706i	$-0.402766\pi$
$233$	9.18125	0.601484	0.300742	+	0.953706i	$0.402766\pi$
$234$	0	0
$235$	−3.96120	−0.258400
$236$	0	0
$237$	4.07696	0.264827
$238$	0	0
$239$	8.66814	0.560695	0.280348	−	0.959899i	$-0.409550\pi$
$239$	8.66814	0.560695	0.280348	+	0.959899i	$0.409550\pi$
$240$	0	0
$241$	5.43476	0.350084	0.175042	−	0.984561i	$-0.443994\pi$
$241$	5.43476	0.350084	0.175042	+	0.984561i	$0.443994\pi$
$242$	0	0
$243$	−14.9522	−0.959187
$244$	0	0
$245$	−1.00000	−0.0638877
$246$	0	0
$247$	−1.75703	−0.111797
$248$	0	0
$249$	3.78011	0.239555
$250$	0	0
$251$	−10.3529	−0.653470	−0.326735	−	0.945116i	$-0.605949\pi$
$251$	−10.3529	−0.653470	−0.326735	+	0.945116i	$0.605949\pi$
$252$	0	0
$253$	18.9812	1.19333
$254$	0	0
$255$	−3.61003	−0.226069
$256$	0	0
$257$	−14.8199	−0.924437	−0.462219	−	0.886766i	$-0.652946\pi$
$257$	−14.8199	−0.924437	−0.462219	+	0.886766i	$0.652946\pi$
$258$	0	0
$259$	−8.63405	−0.536494
$260$	0	0
$261$	19.9970	1.23779
$262$	0	0
$263$	19.1824	1.18283	0.591417	−	0.806366i	$-0.298569\pi$
$263$	19.1824	1.18283	0.591417	+	0.806366i	$0.298569\pi$
$264$	0	0
$265$	−10.7341	−0.659391
$266$	0	0
$267$	6.29103	0.385005
$268$	0	0
$269$	10.6868	0.651587	0.325794	−	0.945441i	$-0.394369\pi$
$269$	10.6868	0.651587	0.325794	+	0.945441i	$0.394369\pi$
$270$	0	0
$271$	22.9268	1.39271	0.696353	−	0.717700i	$-0.254805\pi$
$271$	22.9268	1.39271	0.696353	+	0.717700i	$0.254805\pi$
$272$	0	0
$273$	−0.700017	−0.0423669
$274$	0	0
$275$	4.66704	0.281433
$276$	0	0
$277$	11.6563	0.700361	0.350181	−	0.936682i	$-0.386120\pi$
$277$	11.6563	0.700361	0.350181	+	0.936682i	$0.386120\pi$
$278$	0	0
$279$	5.00535	0.299662
$280$	0	0
$281$	−10.6084	−0.632847	−0.316423	−	0.948618i	$-0.602482\pi$
$281$	−10.6084	−0.632847	−0.316423	+	0.948618i	$0.602482\pi$
$282$	0	0
$283$	−13.1582	−0.782172	−0.391086	−	0.920354i	$-0.627900\pi$
$283$	−13.1582	−0.782172	−0.391086	+	0.920354i	$0.627900\pi$
$284$	0	0
$285$	−1.22995	−0.0728558
$286$	0	0
$287$	7.22413	0.426427
$288$	0	0
$289$	9.59525	0.564427
$290$	0	0
$291$	4.18109	0.245100
$292$	0	0
$293$	20.5142	1.19845	0.599227	−	0.800579i	$-0.295475\pi$
$293$	20.5142	1.19845	0.599227	+	0.800579i	$0.295475\pi$
$294$	0	0
$295$	−6.06125	−0.352900
$296$	0	0
$297$	18.0011	1.04453
$298$	0	0
$299$	4.06707	0.235205
$300$	0	0
$301$	−7.71415	−0.444636
$302$	0	0
$303$	10.5150	0.604070
$304$	0	0
$305$	−11.3612	−0.650542
$306$	0	0
$307$	−7.28602	−0.415835	−0.207917	−	0.978146i	$-0.566669\pi$
$307$	−7.28602	−0.415835	−0.207917	+	0.978146i	$0.566669\pi$
$308$	0	0
$309$	−6.36392	−0.362031
$310$	0	0
$311$	−15.3426	−0.869996	−0.434998	−	0.900431i	$-0.643251\pi$
$311$	−15.3426	−0.869996	−0.434998	+	0.900431i	$0.643251\pi$
$312$	0	0
$313$	23.4236	1.32398	0.661990	−	0.749513i	$-0.269712\pi$
$313$	23.4236	1.32398	0.661990	+	0.749513i	$0.269712\pi$
$314$	0	0
$315$	2.50998	0.141421
$316$	0	0
$317$	−11.3810	−0.639222	−0.319611	−	0.947549i	$-0.603552\pi$
$317$	−11.3810	−0.639222	−0.319611	+	0.947549i	$0.603552\pi$
$318$	0	0
$319$	−37.1824	−2.08181
$320$	0	0
$321$	7.36013	0.410802
$322$	0	0
$323$	9.06108	0.504172
$324$	0	0
$325$	1.00000	0.0554700
$326$	0	0
$327$	0.494097	0.0273236
$328$	0	0
$329$	3.96120	0.218388
$330$	0	0
$331$	−10.0671	−0.553336	−0.276668	−	0.960966i	$-0.589230\pi$
$331$	−10.0671	−0.553336	−0.276668	+	0.960966i	$0.589230\pi$
$332$	0	0
$333$	21.6713	1.18758
$334$	0	0
$335$	7.22413	0.394696
$336$	0	0
$337$	−23.2871	−1.26853	−0.634265	−	0.773116i	$-0.718697\pi$
$337$	−23.2871	−1.26853	−0.634265	+	0.773116i	$0.718697\pi$
$338$	0	0
$339$	6.97300	0.378722
$340$	0	0
$341$	−9.30691	−0.503997
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	2.84702	0.153278
$346$	0	0
$347$	26.3707	1.41565	0.707825	−	0.706387i	$-0.249676\pi$
$347$	26.3707	1.41565	0.707825	+	0.706387i	$0.249676\pi$
$348$	0	0
$349$	2.13885	0.114490	0.0572450	−	0.998360i	$-0.481768\pi$
$349$	2.13885	0.114490	0.0572450	+	0.998360i	$0.481768\pi$
$350$	0	0
$351$	3.85708	0.205875
$352$	0	0
$353$	−8.48706	−0.451720	−0.225860	−	0.974160i	$-0.572519\pi$
$353$	−8.48706	−0.451720	−0.225860	+	0.974160i	$0.572519\pi$
$354$	0	0
$355$	11.8200	0.627342
$356$	0	0
$357$	3.61003	0.191063
$358$	0	0
$359$	24.5824	1.29741	0.648704	−	0.761040i	$-0.275311\pi$
$359$	24.5824	1.29741	0.648704	+	0.761040i	$0.275311\pi$
$360$	0	0
$361$	−15.9129	−0.837519
$362$	0	0
$363$	−7.54703	−0.396116
$364$	0	0
$365$	−2.24705	−0.117616
$366$	0	0
$367$	9.60013	0.501123	0.250561	−	0.968101i	$-0.419385\pi$
$367$	9.60013	0.501123	0.250561	+	0.968101i	$0.419385\pi$
$368$	0	0
$369$	−18.1324	−0.943935
$370$	0	0
$371$	10.7341	0.557287
$372$	0	0
$373$	−12.5541	−0.650028	−0.325014	−	0.945709i	$-0.605369\pi$
$373$	−12.5541	−0.650028	−0.325014	+	0.945709i	$0.605369\pi$
$374$	0	0
$375$	0.700017	0.0361487
$376$	0	0
$377$	−7.96702	−0.410322
$378$	0	0
$379$	18.4599	0.948221	0.474111	−	0.880465i	$-0.342770\pi$
$379$	18.4599	0.948221	0.474111	+	0.880465i	$0.342770\pi$
$380$	0	0
$381$	13.3612	0.684517
$382$	0	0
$383$	−6.76127	−0.345485	−0.172742	−	0.984967i	$-0.555263\pi$
$383$	−6.76127	−0.345485	−0.172742	+	0.984967i	$0.555263\pi$
$384$	0	0
$385$	−4.66704	−0.237854
$386$	0	0
$387$	19.3623	0.984243
$388$	0	0
$389$	28.9309	1.46686	0.733428	−	0.679768i	$-0.237919\pi$
$389$	28.9309	1.46686	0.733428	+	0.679768i	$0.237919\pi$
$390$	0	0
$391$	−20.9741	−1.06071
$392$	0	0
$393$	−1.96009	−0.0988736
$394$	0	0
$395$	5.82409	0.293042
$396$	0	0
$397$	21.9340	1.10084	0.550419	−	0.834889i	$-0.314468\pi$
$397$	21.9340	1.10084	0.550419	+	0.834889i	$0.314468\pi$
$398$	0	0
$399$	1.22995	0.0615744
$400$	0	0
$401$	−17.7940	−0.888588	−0.444294	−	0.895881i	$-0.646545\pi$
$401$	−17.7940	−0.888588	−0.444294	+	0.895881i	$0.646545\pi$
$402$	0	0
$403$	−1.99418	−0.0993372
$404$	0	0
$405$	−4.82991	−0.240000
$406$	0	0
$407$	−40.2954	−1.99737
$408$	0	0
$409$	−27.0105	−1.33558	−0.667792	−	0.744348i	$-0.732761\pi$
$409$	−27.0105	−1.33558	−0.667792	+	0.744348i	$0.732761\pi$
$410$	0	0
$411$	6.40317	0.315845
$412$	0	0
$413$	6.06125	0.298255
$414$	0	0
$415$	5.40003	0.265077
$416$	0	0
$417$	−8.29400	−0.406159
$418$	0	0
$419$	−21.3810	−1.04453	−0.522266	−	0.852783i	$-0.674913\pi$
$419$	−21.3810	−1.04453	−0.522266	+	0.852783i	$0.674913\pi$
$420$	0	0
$421$	23.2127	1.13132	0.565658	−	0.824640i	$-0.308622\pi$
$421$	23.2127	1.13132	0.565658	+	0.824640i	$0.308622\pi$
$422$	0	0
$423$	−9.94252	−0.483422
$424$	0	0
$425$	−5.15706	−0.250154
$426$	0	0
$427$	11.3612	0.549809
$428$	0	0
$429$	−3.26700	−0.157732
$430$	0	0
$431$	−7.33407	−0.353270	−0.176635	−	0.984276i	$-0.556521\pi$
$431$	−7.33407	−0.353270	−0.176635	+	0.984276i	$0.556521\pi$
$432$	0	0
$433$	−33.2136	−1.59614	−0.798072	−	0.602562i	$-0.794146\pi$
$433$	−33.2136	−1.59614	−0.798072	+	0.602562i	$0.794146\pi$
$434$	0	0
$435$	−5.57705	−0.267399
$436$	0	0
$437$	−7.14594	−0.341837
$438$	0	0
$439$	−5.92556	−0.282811	−0.141406	−	0.989952i	$-0.545162\pi$
$439$	−5.92556	−0.282811	−0.141406	+	0.989952i	$0.545162\pi$
$440$	0	0
$441$	−2.50998	−0.119523
$442$	0	0
$443$	−28.2962	−1.34439	−0.672197	−	0.740373i	$-0.734649\pi$
$443$	−28.2962	−1.34439	−0.672197	+	0.740373i	$0.734649\pi$
$444$	0	0
$445$	8.98697	0.426023
$446$	0	0
$447$	−2.58095	−0.122075
$448$	0	0
$449$	20.4483	0.965013	0.482506	−	0.875892i	$-0.339727\pi$
$449$	20.4483	0.965013	0.482506	+	0.875892i	$0.339727\pi$
$450$	0	0
$451$	33.7153	1.58759
$452$	0	0
$453$	11.2282	0.527547
$454$	0	0
$455$	−1.00000	−0.0468807
$456$	0	0
$457$	−0.935426	−0.0437574	−0.0218787	−	0.999761i	$-0.506965\pi$
$457$	−0.935426	−0.0437574	−0.0218787	+	0.999761i	$0.506965\pi$
$458$	0	0
$459$	−19.8912	−0.928440
$460$	0	0
$461$	22.5670	1.05105	0.525525	−	0.850778i	$-0.323869\pi$
$461$	22.5670	1.05105	0.525525	+	0.850778i	$0.323869\pi$
$462$	0	0
$463$	−32.1110	−1.49233	−0.746164	−	0.665763i	$-0.768106\pi$
$463$	−32.1110	−1.49233	−0.746164	+	0.665763i	$0.768106\pi$
$464$	0	0
$465$	−1.39596	−0.0647361
$466$	0	0
$467$	22.0369	1.01975	0.509874	−	0.860249i	$-0.329692\pi$
$467$	22.0369	1.01975	0.509874	+	0.860249i	$0.329692\pi$
$468$	0	0
$469$	−7.22413	−0.333579
$470$	0	0
$471$	−13.4180	−0.618267
$472$	0	0
$473$	−36.0022	−1.65538
$474$	0	0
$475$	−1.75703	−0.0806179
$476$	0	0
$477$	−26.9424	−1.23361
$478$	0	0
$479$	8.41306	0.384403	0.192201	−	0.981356i	$-0.438437\pi$
$479$	8.41306	0.384403	0.192201	+	0.981356i	$0.438437\pi$
$480$	0	0
$481$	−8.63405	−0.393679
$482$	0	0
$483$	−2.84702	−0.129544
$484$	0	0
$485$	5.97284	0.271213
$486$	0	0
$487$	−14.9389	−0.676947	−0.338473	−	0.940976i	$-0.609911\pi$
$487$	−14.9389	−0.676947	−0.338473	+	0.940976i	$0.609911\pi$
$488$	0	0
$489$	−1.86289	−0.0842430
$490$	0	0
$491$	5.41167	0.244225	0.122113	−	0.992516i	$-0.461033\pi$
$491$	5.41167	0.244225	0.122113	+	0.992516i	$0.461033\pi$
$492$	0	0
$493$	41.0864	1.85044
$494$	0	0
$495$	11.7142	0.526512
$496$	0	0
$497$	−11.8200	−0.530200
$498$	0	0
$499$	3.47874	0.155730	0.0778649	−	0.996964i	$-0.475190\pi$
$499$	3.47874	0.155730	0.0778649	+	0.996964i	$0.475190\pi$
$500$	0	0
$501$	9.66610	0.431849
$502$	0	0
$503$	−22.8799	−1.02016	−0.510082	−	0.860126i	$-0.670385\pi$
$503$	−22.8799	−1.02016	−0.510082	+	0.860126i	$0.670385\pi$
$504$	0	0
$505$	15.0211	0.668428
$506$	0	0
$507$	−0.700017	−0.0310888
$508$	0	0
$509$	29.3854	1.30249	0.651243	−	0.758869i	$-0.274248\pi$
$509$	29.3854	1.30249	0.651243	+	0.758869i	$0.274248\pi$
$510$	0	0
$511$	2.24705	0.0994036
$512$	0	0
$513$	−6.77698	−0.299211
$514$	0	0
$515$	−9.09110	−0.400602
$516$	0	0
$517$	18.4871	0.813060
$518$	0	0
$519$	17.9970	0.789982
$520$	0	0
$521$	9.49426	0.415951	0.207976	−	0.978134i	$-0.433313\pi$
$521$	9.49426	0.415951	0.207976	+	0.978134i	$0.433313\pi$
$522$	0	0
$523$	28.5634	1.24899	0.624495	−	0.781029i	$-0.285305\pi$
$523$	28.5634	1.24899	0.624495	+	0.781029i	$0.285305\pi$
$524$	0	0
$525$	−0.700017	−0.0305512
$526$	0	0
$527$	10.2841	0.447983
$528$	0	0
$529$	−6.45895	−0.280824
$530$	0	0
$531$	−15.2136	−0.660214
$532$	0	0
$533$	7.22413	0.312912
$534$	0	0
$535$	10.5142	0.454569
$536$	0	0
$537$	10.9840	0.473995
$538$	0	0
$539$	4.66704	0.201023
$540$	0	0
$541$	34.3625	1.47736	0.738680	−	0.674057i	$-0.235450\pi$
$541$	34.3625	1.47736	0.738680	+	0.674057i	$0.235450\pi$
$542$	0	0
$543$	2.01275	0.0863752
$544$	0	0
$545$	0.705836	0.0302347
$546$	0	0
$547$	24.2881	1.03848	0.519241	−	0.854628i	$-0.326215\pi$
$547$	24.2881	1.03848	0.519241	+	0.854628i	$0.326215\pi$
$548$	0	0
$549$	−28.5164	−1.21705
$550$	0	0
$551$	13.9983	0.596346
$552$	0	0
$553$	−5.82409	−0.247666
$554$	0	0
$555$	−6.04398	−0.256553
$556$	0	0
$557$	−28.6524	−1.21404	−0.607021	−	0.794686i	$-0.707636\pi$
$557$	−28.6524	−1.21404	−0.607021	+	0.794686i	$0.707636\pi$
$558$	0	0
$559$	−7.71415	−0.326274
$560$	0	0
$561$	16.8481	0.711328
$562$	0	0
$563$	−11.4157	−0.481116	−0.240558	−	0.970635i	$-0.577330\pi$
$563$	−11.4157	−0.481116	−0.240558	+	0.970635i	$0.577330\pi$
$564$	0	0
$565$	9.96120	0.419071
$566$	0	0
$567$	4.82991	0.202837
$568$	0	0
$569$	31.9223	1.33825	0.669126	−	0.743149i	$-0.266669\pi$
$569$	31.9223	1.33825	0.669126	+	0.743149i	$0.266669\pi$
$570$	0	0
$571$	34.5755	1.44694	0.723469	−	0.690357i	$-0.242546\pi$
$571$	34.5755	1.44694	0.723469	+	0.690357i	$0.242546\pi$
$572$	0	0
$573$	11.0430	0.461330
$574$	0	0
$575$	4.06707	0.169608
$576$	0	0
$577$	13.7223	0.571267	0.285633	−	0.958339i	$-0.407796\pi$
$577$	13.7223	0.571267	0.285633	+	0.958339i	$0.407796\pi$
$578$	0	0
$579$	0.581925	0.0241840
$580$	0	0
$581$	−5.40003	−0.224031
$582$	0	0
$583$	50.0964	2.07478
$584$	0	0
$585$	2.50998	0.103775
$586$	0	0
$587$	−20.4281	−0.843159	−0.421580	−	0.906791i	$-0.638524\pi$
$587$	−20.4281	−0.843159	−0.421580	+	0.906791i	$0.638524\pi$
$588$	0	0
$589$	3.50383	0.144373
$590$	0	0
$591$	15.0784	0.620240
$592$	0	0
$593$	−29.9142	−1.22843	−0.614216	−	0.789138i	$-0.710527\pi$
$593$	−29.9142	−1.22843	−0.614216	+	0.789138i	$0.710527\pi$
$594$	0	0
$595$	5.15706	0.211419
$596$	0	0
$597$	−7.81375	−0.319796
$598$	0	0
$599$	−36.7550	−1.50177	−0.750884	−	0.660434i	$-0.770372\pi$
$599$	−36.7550	−1.50177	−0.750884	+	0.660434i	$0.770372\pi$
$600$	0	0
$601$	34.9543	1.42582	0.712908	−	0.701257i	$-0.247378\pi$
$601$	34.9543	1.42582	0.712908	+	0.701257i	$0.247378\pi$
$602$	0	0
$603$	18.1324	0.738408
$604$	0	0
$605$	−10.7812	−0.438319
$606$	0	0
$607$	38.9528	1.58104	0.790522	−	0.612434i	$-0.209809\pi$
$607$	38.9528	1.58104	0.790522	+	0.612434i	$0.209809\pi$
$608$	0	0
$609$	5.57705	0.225993
$610$	0	0
$611$	3.96120	0.160253
$612$	0	0
$613$	9.32482	0.376626	0.188313	−	0.982109i	$-0.439698\pi$
$613$	9.32482	0.376626	0.188313	+	0.982109i	$0.439698\pi$
$614$	0	0
$615$	5.05701	0.203918
$616$	0	0
$617$	7.63358	0.307316	0.153658	−	0.988124i	$-0.450895\pi$
$617$	7.63358	0.307316	0.153658	+	0.988124i	$0.450895\pi$
$618$	0	0
$619$	−32.4517	−1.30434	−0.652172	−	0.758071i	$-0.726142\pi$
$619$	−32.4517	−1.30434	−0.652172	+	0.758071i	$0.726142\pi$
$620$	0	0
$621$	15.6870	0.629497
$622$	0	0
$623$	−8.98697	−0.360055
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	5.74021	0.229242
$628$	0	0
$629$	44.5263	1.77538
$630$	0	0
$631$	−29.0222	−1.15535	−0.577677	−	0.816265i	$-0.696041\pi$
$631$	−29.0222	−1.15535	−0.577677	+	0.816265i	$0.696041\pi$
$632$	0	0
$633$	−6.00989	−0.238872
$634$	0	0
$635$	19.0870	0.757445
$636$	0	0
$637$	1.00000	0.0396214
$638$	0	0
$639$	29.6680	1.17365
$640$	0	0
$641$	−39.7765	−1.57108	−0.785539	−	0.618812i	$-0.787614\pi$
$641$	−39.7765	−1.57108	−0.785539	+	0.618812i	$0.787614\pi$
$642$	0	0
$643$	−14.8742	−0.586580	−0.293290	−	0.956023i	$-0.594750\pi$
$643$	−14.8742	−0.586580	−0.293290	+	0.956023i	$0.594750\pi$
$644$	0	0
$645$	−5.40003	−0.212626
$646$	0	0
$647$	11.5788	0.455209	0.227605	−	0.973754i	$-0.426911\pi$
$647$	11.5788	0.455209	0.227605	+	0.973754i	$0.426911\pi$
$648$	0	0
$649$	28.2881	1.11040
$650$	0	0
$651$	1.39596	0.0547120
$652$	0	0
$653$	13.5458	0.530089	0.265044	−	0.964236i	$-0.414613\pi$
$653$	13.5458	0.530089	0.265044	+	0.964236i	$0.414613\pi$
$654$	0	0
$655$	−2.80007	−0.109408
$656$	0	0
$657$	−5.64004	−0.220039
$658$	0	0
$659$	7.25129	0.282470	0.141235	−	0.989976i	$-0.454893\pi$
$659$	7.25129	0.282470	0.141235	+	0.989976i	$0.454893\pi$
$660$	0	0
$661$	35.1907	1.36876	0.684380	−	0.729126i	$-0.260073\pi$
$661$	35.1907	1.36876	0.684380	+	0.729126i	$0.260073\pi$
$662$	0	0
$663$	3.61003	0.140202
$664$	0	0
$665$	1.75703	0.0681345
$666$	0	0
$667$	−32.4024	−1.25463
$668$	0	0
$669$	18.8492	0.728753
$670$	0	0
$671$	53.0233	2.04694
$672$	0	0
$673$	−12.0985	−0.466363	−0.233181	−	0.972433i	$-0.574914\pi$
$673$	−12.0985	−0.466363	−0.233181	+	0.972433i	$0.574914\pi$
$674$	0	0
$675$	3.85708	0.148459
$676$	0	0
$677$	−28.4638	−1.09395	−0.546976	−	0.837148i	$-0.684221\pi$
$677$	−28.4638	−1.09395	−0.546976	+	0.837148i	$0.684221\pi$
$678$	0	0
$679$	−5.97284	−0.229217
$680$	0	0
$681$	14.5802	0.558714
$682$	0	0
$683$	−19.3039	−0.738645	−0.369322	−	0.929301i	$-0.620410\pi$
$683$	−19.3039	−0.738645	−0.369322	+	0.929301i	$0.620410\pi$
$684$	0	0
$685$	9.14716	0.349495
$686$	0	0
$687$	5.29591	0.202052
$688$	0	0
$689$	10.7341	0.408937
$690$	0	0
$691$	31.6350	1.20345	0.601726	−	0.798703i	$-0.294480\pi$
$691$	31.6350	1.20345	0.601726	+	0.798703i	$0.294480\pi$
$692$	0	0
$693$	−11.7142	−0.444984
$694$	0	0
$695$	−11.8483	−0.449431
$696$	0	0
$697$	−37.2552	−1.41114
$698$	0	0
$699$	−6.42703	−0.243093
$700$	0	0
$701$	31.3792	1.18517	0.592587	−	0.805506i	$-0.298107\pi$
$701$	31.3792	1.18517	0.592587	+	0.805506i	$0.298107\pi$
$702$	0	0
$703$	15.1703	0.572157
$704$	0	0
$705$	2.77290	0.104434
$706$	0	0
$707$	−15.0211	−0.564925
$708$	0	0
$709$	17.3647	0.652146	0.326073	−	0.945345i	$-0.394274\pi$
$709$	17.3647	0.652146	0.326073	+	0.945345i	$0.394274\pi$
$710$	0	0
$711$	14.6183	0.548231
$712$	0	0
$713$	−8.11047	−0.303739
$714$	0	0
$715$	−4.66704	−0.174537
$716$	0	0
$717$	−6.06784	−0.226608
$718$	0	0
$719$	−14.4797	−0.540001	−0.270000	−	0.962860i	$-0.587024\pi$
$719$	−14.4797	−0.540001	−0.270000	+	0.962860i	$0.587024\pi$
$720$	0	0
$721$	9.09110	0.338570
$722$	0	0
$723$	−3.80442	−0.141488
$724$	0	0
$725$	−7.96702	−0.295888
$726$	0	0
$727$	4.82141	0.178816	0.0894081	−	0.995995i	$-0.471502\pi$
$727$	4.82141	0.178816	0.0894081	+	0.995995i	$0.471502\pi$
$728$	0	0
$729$	−4.02292	−0.148997
$730$	0	0
$731$	39.7823	1.47140
$732$	0	0
$733$	21.5257	0.795070	0.397535	−	0.917587i	$-0.369866\pi$
$733$	21.5257	0.795070	0.397535	+	0.917587i	$0.369866\pi$
$734$	0	0
$735$	0.700017	0.0258205
$736$	0	0
$737$	−33.7153	−1.24192
$738$	0	0
$739$	15.6563	0.575928	0.287964	−	0.957641i	$-0.407022\pi$
$739$	15.6563	0.575928	0.287964	+	0.957641i	$0.407022\pi$
$740$	0	0
$741$	1.22995	0.0451832
$742$	0	0
$743$	−41.6336	−1.52739	−0.763694	−	0.645578i	$-0.776616\pi$
$743$	−41.6336	−1.52739	−0.763694	+	0.645578i	$0.776616\pi$
$744$	0	0
$745$	−3.68699	−0.135081
$746$	0	0
$747$	13.5540	0.495913
$748$	0	0
$749$	−10.5142	−0.384181
$750$	0	0
$751$	24.6856	0.900790	0.450395	−	0.892829i	$-0.351283\pi$
$751$	24.6856	0.900790	0.450395	+	0.892829i	$0.351283\pi$
$752$	0	0
$753$	7.24721	0.264103
$754$	0	0
$755$	16.0399	0.583752
$756$	0	0
$757$	−53.3825	−1.94022	−0.970109	−	0.242669i	$-0.921977\pi$
$757$	−53.3825	−1.94022	−0.970109	+	0.242669i	$0.921977\pi$
$758$	0	0
$759$	−13.2871	−0.482292
$760$	0	0
$761$	33.5799	1.21727	0.608635	−	0.793450i	$-0.291717\pi$
$761$	33.5799	1.21727	0.608635	+	0.793450i	$0.291717\pi$
$762$	0	0
$763$	−0.705836	−0.0255530
$764$	0	0
$765$	−12.9441	−0.467995
$766$	0	0
$767$	6.06125	0.218859
$768$	0	0
$769$	26.3897	0.951636	0.475818	−	0.879544i	$-0.342152\pi$
$769$	26.3897	0.951636	0.475818	+	0.879544i	$0.342152\pi$
$770$	0	0
$771$	10.3741	0.373616
$772$	0	0
$773$	27.9937	1.00686	0.503432	−	0.864035i	$-0.332070\pi$
$773$	27.9937	1.00686	0.503432	+	0.864035i	$0.332070\pi$
$774$	0	0
$775$	−1.99418	−0.0716331
$776$	0	0
$777$	6.04398	0.216827
$778$	0	0
$779$	−12.6930	−0.454773
$780$	0	0
$781$	−55.1645	−1.97394
$782$	0	0
$783$	−30.7294	−1.09818
$784$	0	0
$785$	−19.1681	−0.684137
$786$	0	0
$787$	38.5905	1.37560	0.687802	−	0.725898i	$-0.258576\pi$
$787$	38.5905	1.37560	0.687802	+	0.725898i	$0.258576\pi$
$788$	0	0
$789$	−13.4280	−0.478048
$790$	0	0
$791$	−9.96120	−0.354179
$792$	0	0
$793$	11.3612	0.403449
$794$	0	0
$795$	7.51405	0.266496
$796$	0	0
$797$	−2.96575	−0.105052	−0.0525261	−	0.998620i	$-0.516727\pi$
$797$	−2.96575	−0.105052	−0.0525261	+	0.998620i	$0.516727\pi$
$798$	0	0
$799$	−20.4281	−0.722695
$800$	0	0
$801$	22.5571	0.797016
$802$	0	0
$803$	10.4871	0.370080
$804$	0	0
$805$	−4.06707	−0.143345
$806$	0	0
$807$	−7.48095	−0.263342
$808$	0	0
$809$	−26.4915	−0.931390	−0.465695	−	0.884945i	$-0.654196\pi$
$809$	−26.4915	−0.931390	−0.465695	+	0.884945i	$0.654196\pi$
$810$	0	0
$811$	1.94219	0.0681994	0.0340997	−	0.999418i	$-0.489144\pi$
$811$	1.94219	0.0681994	0.0340997	+	0.999418i	$0.489144\pi$
$812$	0	0
$813$	−16.0492	−0.562869
$814$	0	0
$815$	−2.66122	−0.0932183
$816$	0	0
$817$	13.5540	0.474193
$818$	0	0
$819$	−2.50998	−0.0877057
$820$	0	0
$821$	33.7308	1.17721	0.588606	−	0.808420i	$-0.299677\pi$
$821$	33.7308	1.17721	0.588606	+	0.808420i	$0.299677\pi$
$822$	0	0
$823$	−4.96730	−0.173149	−0.0865746	−	0.996245i	$-0.527592\pi$
$823$	−4.96730	−0.173149	−0.0865746	+	0.996245i	$0.527592\pi$
$824$	0	0
$825$	−3.26700	−0.113742
$826$	0	0
$827$	−27.8681	−0.969068	−0.484534	−	0.874772i	$-0.661011\pi$
$827$	−27.8681	−0.969068	−0.484534	+	0.874772i	$0.661011\pi$
$828$	0	0
$829$	13.7918	0.479007	0.239504	−	0.970895i	$-0.423015\pi$
$829$	13.7918	0.479007	0.239504	+	0.970895i	$0.423015\pi$
$830$	0	0
$831$	−8.15963	−0.283054
$832$	0	0
$833$	−5.15706	−0.178681
$834$	0	0
$835$	13.8084	0.477859
$836$	0	0
$837$	−7.69170	−0.265864
$838$	0	0
$839$	−52.9978	−1.82969	−0.914843	−	0.403809i	$-0.867686\pi$
$839$	−52.9978	−1.82969	−0.914843	+	0.403809i	$0.867686\pi$
$840$	0	0
$841$	34.4734	1.18874
$842$	0	0
$843$	7.42609	0.255768
$844$	0	0
$845$	−1.00000	−0.0344010
$846$	0	0
$847$	10.7812	0.370447
$848$	0	0
$849$	9.21093	0.316118
$850$	0	0
$851$	−35.1153	−1.20374
$852$	0	0
$853$	31.2879	1.07128	0.535639	−	0.844447i	$-0.320071\pi$
$853$	31.2879	1.07128	0.535639	+	0.844447i	$0.320071\pi$
$854$	0	0
$855$	−4.41009	−0.150822
$856$	0	0
$857$	−3.58345	−0.122408	−0.0612041	−	0.998125i	$-0.519494\pi$
$857$	−3.58345	−0.122408	−0.0612041	+	0.998125i	$0.519494\pi$
$858$	0	0
$859$	10.3296	0.352443	0.176221	−	0.984351i	$-0.443613\pi$
$859$	10.3296	0.352443	0.176221	+	0.984351i	$0.443613\pi$
$860$	0	0
$861$	−5.05701	−0.172342
$862$	0	0
$863$	35.8287	1.21962	0.609811	−	0.792547i	$-0.291245\pi$
$863$	35.8287	1.21962	0.609811	+	0.792547i	$0.291245\pi$
$864$	0	0
$865$	25.7094	0.874147
$866$	0	0
$867$	−6.71684	−0.228116
$868$	0	0
$869$	−27.1813	−0.922061
$870$	0	0
$871$	−7.22413	−0.244780
$872$	0	0
$873$	14.9917	0.507392
$874$	0	0
$875$	−1.00000	−0.0338062
$876$	0	0
$877$	7.77288	0.262471	0.131236	−	0.991351i	$-0.458106\pi$
$877$	7.77288	0.262471	0.131236	+	0.991351i	$0.458106\pi$
$878$	0	0
$879$	−14.3603	−0.484361
$880$	0	0
$881$	−8.92256	−0.300609	−0.150304	−	0.988640i	$-0.548025\pi$
$881$	−8.92256	−0.300609	−0.150304	+	0.988640i	$0.548025\pi$
$882$	0	0
$883$	−10.8485	−0.365080	−0.182540	−	0.983198i	$-0.558432\pi$
$883$	−10.8485	−0.365080	−0.182540	+	0.983198i	$0.558432\pi$
$884$	0	0
$885$	4.24297	0.142626
$886$	0	0
$887$	−50.0490	−1.68048	−0.840241	−	0.542213i	$-0.817587\pi$
$887$	−50.0490	−1.68048	−0.840241	+	0.542213i	$0.817587\pi$
$888$	0	0
$889$	−19.0870	−0.640158
$890$	0	0
$891$	22.5414	0.755165
$892$	0	0
$893$	−6.95993	−0.232905
$894$	0	0
$895$	15.6911	0.524494
$896$	0	0
$897$	−2.84702	−0.0950591
$898$	0	0
$899$	15.8877	0.529883
$900$	0	0
$901$	−55.3564	−1.84419
$902$	0	0
$903$	5.40003	0.179702
$904$	0	0
$905$	2.87528	0.0955776
$906$	0	0
$907$	42.4307	1.40889	0.704444	−	0.709760i	$-0.251197\pi$
$907$	42.4307	1.40889	0.704444	+	0.709760i	$0.251197\pi$
$908$	0	0
$909$	37.7025	1.25051
$910$	0	0
$911$	31.9653	1.05906	0.529528	−	0.848292i	$-0.322369\pi$
$911$	31.9653	1.05906	0.529528	+	0.848292i	$0.322369\pi$
$912$	0	0
$913$	−25.2021	−0.834069
$914$	0	0
$915$	7.95305	0.262920
$916$	0	0
$917$	2.80007	0.0924663
$918$	0	0
$919$	−12.2458	−0.403953	−0.201976	−	0.979390i	$-0.564736\pi$
$919$	−12.2458	−0.403953	−0.201976	+	0.979390i	$0.564736\pi$
$920$	0	0
$921$	5.10033	0.168062
$922$	0	0
$923$	−11.8200	−0.389061
$924$	0	0
$925$	−8.63405	−0.283886
$926$	0	0
$927$	−22.8184	−0.749456
$928$	0	0
$929$	−7.68355	−0.252089	−0.126045	−	0.992025i	$-0.540228\pi$
$929$	−7.68355	−0.252089	−0.126045	+	0.992025i	$0.540228\pi$
$930$	0	0
$931$	−1.75703	−0.0575842
$932$	0	0
$933$	10.7400	0.351613
$934$	0	0
$935$	24.0682	0.787113
$936$	0	0
$937$	−42.7361	−1.39613	−0.698064	−	0.716035i	$-0.745955\pi$
$937$	−42.7361	−1.39613	−0.698064	+	0.716035i	$0.745955\pi$
$938$	0	0
$939$	−16.3969	−0.535093
$940$	0	0
$941$	25.5026	0.831361	0.415681	−	0.909511i	$-0.363543\pi$
$941$	25.5026	0.831361	0.415681	+	0.909511i	$0.363543\pi$
$942$	0	0
$943$	29.3810	0.956778
$944$	0	0
$945$	−3.85708	−0.125471
$946$	0	0
$947$	34.2829	1.11404	0.557022	−	0.830498i	$-0.311944\pi$
$947$	34.2829	1.11404	0.557022	+	0.830498i	$0.311944\pi$
$948$	0	0
$949$	2.24705	0.0729423
$950$	0	0
$951$	7.96690	0.258345
$952$	0	0
$953$	2.54248	0.0823591	0.0411796	−	0.999152i	$-0.486888\pi$
$953$	2.54248	0.0823591	0.0411796	+	0.999152i	$0.486888\pi$
$954$	0	0
$955$	15.7754	0.510480
$956$	0	0
$957$	26.0283	0.841374
$958$	0	0
$959$	−9.14716	−0.295377
$960$	0	0
$961$	−27.0232	−0.871718
$962$	0	0
$963$	26.3904	0.850420
$964$	0	0
$965$	0.831302	0.0267606
$966$	0	0
$967$	35.8354	1.15239	0.576195	−	0.817312i	$-0.304537\pi$
$967$	35.8354	1.15239	0.576195	+	0.817312i	$0.304537\pi$
$968$	0	0
$969$	−6.34291	−0.203764
$970$	0	0
$971$	4.25054	0.136406	0.0682032	−	0.997671i	$-0.478273\pi$
$971$	4.25054	0.136406	0.0682032	+	0.997671i	$0.478273\pi$
$972$	0	0
$973$	11.8483	0.379839
$974$	0	0
$975$	−0.700017	−0.0224185
$976$	0	0
$977$	54.7899	1.75288	0.876442	−	0.481508i	$-0.159911\pi$
$977$	54.7899	1.75288	0.876442	+	0.481508i	$0.159911\pi$
$978$	0	0
$979$	−41.9425	−1.34049
$980$	0	0
$981$	1.77163	0.0565639
$982$	0	0
$983$	43.6306	1.39160	0.695800	−	0.718235i	$-0.255050\pi$
$983$	43.6306	1.39160	0.695800	+	0.718235i	$0.255050\pi$
$984$	0	0
$985$	21.5400	0.686321
$986$	0	0
$987$	−2.77290	−0.0882625
$988$	0	0
$989$	−31.3740	−0.997634
$990$	0	0
$991$	−0.200682	−0.00637487	−0.00318743	−	0.999995i	$-0.501015\pi$
$991$	−0.200682	−0.00637487	−0.00318743	+	0.999995i	$0.501015\pi$
$992$	0	0
$993$	7.04711	0.223633
$994$	0	0
$995$	−11.1622	−0.353867
$996$	0	0
$997$	5.67205	0.179636	0.0898178	−	0.995958i	$-0.471372\pi$
$997$	5.67205	0.179636	0.0898178	+	0.995958i	$0.471372\pi$
$998$	0	0
$999$	−33.3022	−1.05363

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3640.2.a.t.1.2	✓	4
4.3	odd	2		7280.2.a.bv.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3640.2.a.t.1.2	✓	4	1.1	even	1	trivial
7280.2.a.bv.1.3		4	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 3640 = 2^{3} \cdot 5 \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3640.a (trivial)

\(f(q)\)	\(=\)	\(q-0.700017 q^{3} -1.00000 q^{5} +1.00000 q^{7} -2.50998 q^{9} +O(q^{10})\)	\(q-0.700017 q^{3} -1.00000 q^{5} +1.00000 q^{7} -2.50998 q^{9} +4.66704 q^{11} +1.00000 q^{13} +0.700017 q^{15} -5.15706 q^{17} -1.75703 q^{19} -0.700017 q^{21} +4.06707 q^{23} +1.00000 q^{25} +3.85708 q^{27} -7.96702 q^{29} -1.99418 q^{31} -3.26700 q^{33} -1.00000 q^{35} -8.63405 q^{37} -0.700017 q^{39} +7.22413 q^{41} -7.71415 q^{43} +2.50998 q^{45} +3.96120 q^{47} +1.00000 q^{49} +3.61003 q^{51} +10.7341 q^{53} -4.66704 q^{55} +1.22995 q^{57} +6.06125 q^{59} +11.3612 q^{61} -2.50998 q^{63} -1.00000 q^{65} -7.22413 q^{67} -2.84702 q^{69} -11.8200 q^{71} +2.24705 q^{73} -0.700017 q^{75} +4.66704 q^{77} -5.82409 q^{79} +4.82991 q^{81} -5.40003 q^{83} +5.15706 q^{85} +5.57705 q^{87} -8.98697 q^{89} +1.00000 q^{91} +1.39596 q^{93} +1.75703 q^{95} -5.97284 q^{97} -11.7142 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{5} + 4 q^{7}+O(q^{10}) \) 4 * q - 4 * q^5 + 4 * q^7	\( 4 q - 4 q^{5} + 4 q^{7} - q^{11} + 4 q^{13} - 11 q^{17} - 3 q^{19} - 9 q^{23} + 4 q^{25} + 3 q^{27} - 15 q^{29} - 6 q^{31} + q^{33} - 4 q^{35} + 2 q^{37} - 6 q^{41} - 6 q^{43} - 3 q^{47} + 4 q^{49} - 4 q^{51} - 2 q^{53} + q^{55} - 28 q^{57} - 3 q^{59} + 21 q^{61} - 4 q^{65} + 6 q^{67} - 23 q^{69} - 16 q^{71} + 15 q^{73} - q^{77} + 6 q^{79} - 8 q^{81} - 16 q^{83} + 11 q^{85} - 13 q^{87} + q^{89} + 4 q^{91} - 2 q^{93} + 3 q^{95} - 9 q^{97} - 22 q^{99}+O(q^{100}) \) 4 * q - 4 * q^5 + 4 * q^7 - q^11 + 4 * q^13 - 11 * q^17 - 3 * q^19 - 9 * q^23 + 4 * q^25 + 3 * q^27 - 15 * q^29 - 6 * q^31 + q^33 - 4 * q^35 + 2 * q^37 - 6 * q^41 - 6 * q^43 - 3 * q^47 + 4 * q^49 - 4 * q^51 - 2 * q^53 + q^55 - 28 * q^57 - 3 * q^59 + 21 * q^61 - 4 * q^65 + 6 * q^67 - 23 * q^69 - 16 * q^71 + 15 * q^73 - q^77 + 6 * q^79 - 8 * q^81 - 16 * q^83 + 11 * q^85 - 13 * q^87 + q^89 + 4 * q^91 - 2 * q^93 + 3 * q^95 - 9 * q^97 - 22 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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