LMFDB - Embedded newform 3447.2.a.j.1.14

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3447,2,Mod(1,3447)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3447, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("3447.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3447 = 3^{2} \cdot 383 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3447.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:	$27.5244335767$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	no (minimal twist has level 383)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.14
Character	$\chi$	$=$	3447.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+0.0582123 q^{2} -1.99661 q^{4} +3.28367 q^{5} -2.97653 q^{7} -0.232652 q^{8} +0.191150 q^{10} +6.28742 q^{11} +3.41245 q^{13} -0.173271 q^{14} +3.97968 q^{16} +2.70055 q^{17} -4.13063 q^{19} -6.55621 q^{20} +0.366005 q^{22} -1.42511 q^{23} +5.78249 q^{25} +0.198647 q^{26} +5.94297 q^{28} +4.09836 q^{29} -3.15738 q^{31} +0.696971 q^{32} +0.157205 q^{34} -9.77393 q^{35} +5.90577 q^{37} -0.240453 q^{38} -0.763953 q^{40} +5.97026 q^{41} -9.37617 q^{43} -12.5535 q^{44} -0.0829590 q^{46} -8.20132 q^{47} +1.85971 q^{49} +0.336612 q^{50} -6.81333 q^{52} +3.16697 q^{53} +20.6458 q^{55} +0.692495 q^{56} +0.238575 q^{58} +0.178110 q^{59} -10.1998 q^{61} -0.183798 q^{62} -7.91879 q^{64} +11.2054 q^{65} +14.9534 q^{67} -5.39195 q^{68} -0.568963 q^{70} -4.17716 q^{71} -0.0585921 q^{73} +0.343789 q^{74} +8.24725 q^{76} -18.7147 q^{77} +7.85351 q^{79} +13.0680 q^{80} +0.347543 q^{82} +7.20095 q^{83} +8.86772 q^{85} -0.545809 q^{86} -1.46278 q^{88} +12.9047 q^{89} -10.1572 q^{91} +2.84539 q^{92} -0.477418 q^{94} -13.5636 q^{95} +11.4365 q^{97} +0.108258 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q - 5 q^{2} + 29 q^{4} - 3 q^{5} + 17 q^{7} - 15 q^{8} + q^{10} + 28 q^{13} + 8 q^{14} + 35 q^{16} - 16 q^{17} + 13 q^{19} + 4 q^{20} + 12 q^{22} - 7 q^{23} + 67 q^{25} + 14 q^{26} + 39 q^{28} + 2 q^{29}+ \cdots + 29 q^{98}+O(q^{100}) $ 24 * q - 5 * q^2 + 29 * q^4 - 3 * q^5 + 17 * q^7 - 15 * q^8 + q^10 + 28 * q^13 + 8 * q^14 + 35 * q^16 - 16 * q^17 + 13 * q^19 + 4 * q^20 + 12 * q^22 - 7 * q^23 + 67 * q^25 + 14 * q^26 + 39 * q^28 + 2 * q^29 - 20 * q^32 - 18 * q^34 + 10 * q^35 + 35 * q^37 + 5 * q^38 - 36 * q^40 + 34 * q^43 + 36 * q^44 - 17 * q^46 + 3 * q^47 + 49 * q^49 + 20 * q^50 + 29 * q^52 + 9 * q^53 + 10 * q^55 + 47 * q^56 + 12 * q^58 + 6 * q^59 + 3 * q^61 + 58 * q^62 + 15 * q^64 - 8 * q^65 + 42 * q^67 + 23 * q^68 - 59 * q^70 + 13 * q^71 + 83 * q^73 + 45 * q^74 + 4 * q^76 + 21 * q^77 - 3 * q^79 + 93 * q^80 - 13 * q^82 + 23 * q^83 + 52 * q^85 + 43 * q^86 - 25 * q^88 + 38 * q^89 - 17 * q^91 + 61 * q^92 - 71 * q^94 + 66 * q^95 + 63 * q^97 + 29 * q^98

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 $

$2$	0.0582123	0.0411623	0.0205812	−	0.999788i	$-0.493448\pi$
$2$	0.0582123	0.0411623	0.0205812	+	0.999788i	$0.493448\pi$
$3$	0	0
$3$	0	0
$4$	−1.99661	−0.998306
$5$	3.28367	1.46850	0.734251	−	0.678878i	$-0.237534\pi$
$5$	3.28367	1.46850	0.734251	+	0.678878i	$0.237534\pi$
$6$	0	0
$7$	−2.97653	−1.12502	−0.562511	−	0.826790i	$-0.690165\pi$
$7$	−2.97653	−1.12502	−0.562511	+	0.826790i	$0.690165\pi$
$8$	−0.232652	−0.0822549
$9$	0	0
$10$	0.191150	0.0604470
$11$	6.28742	1.89573	0.947864	−	0.318674i	$-0.103237\pi$
$11$	6.28742	1.89573	0.947864	+	0.318674i	$0.103237\pi$
$12$	0	0
$13$	3.41245	0.946443	0.473222	−	0.880943i	$-0.343091\pi$
$13$	3.41245	0.946443	0.473222	+	0.880943i	$0.343091\pi$
$14$	−0.173271	−0.0463085
$15$	0	0
$16$	3.97968	0.994920
$17$	2.70055	0.654980	0.327490	−	0.944855i	$-0.393797\pi$
$17$	2.70055	0.654980	0.327490	+	0.944855i	$0.393797\pi$
$18$	0	0
$19$	−4.13063	−0.947630	−0.473815	−	0.880624i	$-0.657124\pi$
$19$	−4.13063	−0.947630	−0.473815	+	0.880624i	$0.657124\pi$
$20$	−6.55621	−1.46601
$21$	0	0
$22$	0.366005	0.0780326
$23$	−1.42511	−0.297156	−0.148578	−	0.988901i	$-0.547470\pi$
$23$	−1.42511	−0.297156	−0.148578	+	0.988901i	$0.547470\pi$
$24$	0	0
$25$	5.78249	1.15650
$26$	0.198647	0.0389578
$27$	0	0
$28$	5.94297	1.12312
$29$	4.09836	0.761047	0.380524	−	0.924771i	$-0.375744\pi$
$29$	4.09836	0.761047	0.380524	+	0.924771i	$0.375744\pi$
$30$	0	0
$31$	−3.15738	−0.567082	−0.283541	−	0.958960i	$-0.591509\pi$
$31$	−3.15738	−0.567082	−0.283541	+	0.958960i	$0.591509\pi$
$32$	0.696971	0.123208
$33$	0	0
$34$	0.157205	0.0269605
$35$	−9.77393	−1.65210
$36$	0	0
$37$	5.90577	0.970902	0.485451	−	0.874264i	$-0.338655\pi$
$37$	5.90577	0.970902	0.485451	+	0.874264i	$0.338655\pi$
$38$	−0.240453	−0.0390067
$39$	0	0
$40$	−0.763953	−0.120792
$41$	5.97026	0.932399	0.466199	−	0.884680i	$-0.345623\pi$
$41$	5.97026	0.932399	0.466199	+	0.884680i	$0.345623\pi$
$42$	0	0
$43$	−9.37617	−1.42985	−0.714926	−	0.699200i	$-0.753540\pi$
$43$	−9.37617	−1.42985	−0.714926	+	0.699200i	$0.753540\pi$
$44$	−12.5535	−1.89252
$45$	0	0
$46$	−0.0829590	−0.0122316
$47$	−8.20132	−1.19629	−0.598143	−	0.801390i	$-0.704094\pi$
$47$	−8.20132	−1.19629	−0.598143	+	0.801390i	$0.704094\pi$
$48$	0	0
$49$	1.85971	0.265673
$50$	0.336612	0.0476042
$51$	0	0
$52$	−6.81333	−0.944839
$53$	3.16697	0.435017	0.217509	−	0.976058i	$-0.430207\pi$
$53$	3.16697	0.435017	0.217509	+	0.976058i	$0.430207\pi$
$54$	0	0
$55$	20.6458	2.78388
$56$	0.692495	0.0925386
$57$	0	0
$58$	0.238575	0.0313265
$59$	0.178110	0.0231880	0.0115940	−	0.999933i	$-0.496309\pi$
$59$	0.178110	0.0231880	0.0115940	+	0.999933i	$0.496309\pi$
$60$	0	0
$61$	−10.1998	−1.30595	−0.652975	−	0.757379i	$-0.726479\pi$
$61$	−10.1998	−1.30595	−0.652975	+	0.757379i	$0.726479\pi$
$62$	−0.183798	−0.0233424
$63$	0	0
$64$	−7.91879	−0.989848
$65$	11.2054	1.38985
$66$	0	0
$67$	14.9534	1.82685	0.913424	−	0.407010i	$-0.133428\pi$
$67$	14.9534	1.82685	0.913424	+	0.407010i	$0.133428\pi$
$68$	−5.39195	−0.653870
$69$	0	0
$70$	−0.568963	−0.0680041
$71$	−4.17716	−0.495737	−0.247869	−	0.968794i	$-0.579730\pi$
$71$	−4.17716	−0.495737	−0.247869	+	0.968794i	$0.579730\pi$
$72$	0	0
$73$	−0.0585921	−0.00685769	−0.00342884	−	0.999994i	$-0.501091\pi$
$73$	−0.0585921	−0.00685769	−0.00342884	+	0.999994i	$0.501091\pi$
$74$	0.343789	0.0399646
$75$	0	0
$76$	8.24725	0.946025
$77$	−18.7147	−2.13274
$78$	0	0
$79$	7.85351	0.883589	0.441795	−	0.897116i	$-0.354342\pi$
$79$	7.85351	0.883589	0.441795	+	0.897116i	$0.354342\pi$
$80$	13.0680	1.46104
$81$	0	0
$82$	0.347543	0.0383797
$83$	7.20095	0.790407	0.395203	−	0.918594i	$-0.370674\pi$
$83$	7.20095	0.790407	0.395203	+	0.918594i	$0.370674\pi$
$84$	0	0
$85$	8.86772	0.961839
$86$	−0.545809	−0.0588561
$87$	0	0
$88$	−1.46278	−0.155933
$89$	12.9047	1.36790	0.683948	−	0.729530i	$-0.260261\pi$
$89$	12.9047	1.36790	0.683948	+	0.729530i	$0.260261\pi$
$90$	0	0
$91$	−10.1572	−1.06477
$92$	2.84539	0.296652
$93$	0	0
$94$	−0.477418	−0.0492419
$95$	−13.5636	−1.39160
$96$	0	0
$97$	11.4365	1.16120	0.580599	−	0.814190i	$-0.302818\pi$
$97$	11.4365	1.16120	0.580599	+	0.814190i	$0.302818\pi$
$98$	0.108258	0.0109357
$99$	0	0
$100$	−11.5454	−1.15454
$101$	−2.26914	−0.225788	−0.112894	−	0.993607i	$-0.536012\pi$
$101$	−2.26914	−0.225788	−0.112894	+	0.993607i	$0.536012\pi$
$102$	0	0
$103$	−7.00931	−0.690648	−0.345324	−	0.938483i	$-0.612231\pi$
$103$	−7.00931	−0.690648	−0.345324	+	0.938483i	$0.612231\pi$
$104$	−0.793913	−0.0778496
$105$	0	0
$106$	0.184357	0.0179063
$107$	−1.37392	−0.132821	−0.0664107	−	0.997792i	$-0.521155\pi$
$107$	−1.37392	−0.132821	−0.0664107	+	0.997792i	$0.521155\pi$
$108$	0	0
$109$	4.66434	0.446763	0.223382	−	0.974731i	$-0.428290\pi$
$109$	4.66434	0.446763	0.223382	+	0.974731i	$0.428290\pi$
$110$	1.20184	0.114591
$111$	0	0
$112$	−11.8456	−1.11931
$113$	5.46650	0.514245	0.257123	−	0.966379i	$-0.417226\pi$
$113$	5.46650	0.514245	0.257123	+	0.966379i	$0.417226\pi$
$114$	0	0
$115$	−4.67959	−0.436374
$116$	−8.18284	−0.759758
$117$	0	0
$118$	0.0103682	0.000954473 0
$119$	−8.03826	−0.736866
$120$	0	0
$121$	28.5317	2.59379
$122$	−0.593754	−0.0537560
$123$	0	0
$124$	6.30406	0.566121
$125$	2.56944	0.229818
$126$	0	0
$127$	12.2005	1.08262	0.541308	−	0.840825i	$-0.317929\pi$
$127$	12.2005	1.08262	0.541308	+	0.840825i	$0.317929\pi$
$128$	−1.85491	−0.163953
$129$	0	0
$130$	0.652290	0.0572096
$131$	10.9581	0.957418	0.478709	−	0.877974i	$-0.341105\pi$
$131$	10.9581	0.957418	0.478709	+	0.877974i	$0.341105\pi$
$132$	0	0
$133$	12.2949	1.06610
$134$	0.870472	0.0751973
$135$	0	0
$136$	−0.628289	−0.0538753
$137$	9.19250	0.785368	0.392684	−	0.919673i	$-0.371547\pi$
$137$	9.19250	0.785368	0.392684	+	0.919673i	$0.371547\pi$
$138$	0	0
$139$	0.449855	0.0381562	0.0190781	−	0.999818i	$-0.493927\pi$
$139$	0.449855	0.0381562	0.0190781	+	0.999818i	$0.493927\pi$
$140$	19.5147	1.64930
$141$	0	0
$142$	−0.243162	−0.0204057
$143$	21.4555	1.79420
$144$	0	0
$145$	13.4577	1.11760
$146$	−0.00341078	−0.000282279 0
$147$	0	0
$148$	−11.7915	−0.969257
$149$	−12.2889	−1.00675	−0.503374	−	0.864069i	$-0.667908\pi$
$149$	−12.2889	−1.00675	−0.503374	+	0.864069i	$0.667908\pi$
$150$	0	0
$151$	−5.30093	−0.431383	−0.215692	−	0.976462i	$-0.569201\pi$
$151$	−5.30093	−0.431383	−0.215692	+	0.976462i	$0.569201\pi$
$152$	0.960999	0.0779473
$153$	0	0
$154$	−1.08943	−0.0877884
$155$	−10.3678	−0.832761
$156$	0	0
$157$	−1.62612	−0.129779	−0.0648894	−	0.997892i	$-0.520669\pi$
$157$	−1.62612	−0.129779	−0.0648894	+	0.997892i	$0.520669\pi$
$158$	0.457171	0.0363706
$159$	0	0
$160$	2.28862	0.180931
$161$	4.24188	0.334307
$162$	0	0
$163$	−1.51551	−0.118704	−0.0593519	−	0.998237i	$-0.518903\pi$
$163$	−1.51551	−0.118704	−0.0593519	+	0.998237i	$0.518903\pi$
$164$	−11.9203	−0.930819
$165$	0	0
$166$	0.419184	0.0325350
$167$	−6.91719	−0.535268	−0.267634	−	0.963521i	$-0.586242\pi$
$167$	−6.91719	−0.535268	−0.267634	+	0.963521i	$0.586242\pi$
$168$	0	0
$169$	−1.35519	−0.104245
$170$	0.516211	0.0395915
$171$	0	0
$172$	18.7206	1.42743
$173$	17.4784	1.32886	0.664428	−	0.747352i	$-0.268675\pi$
$173$	17.4784	1.32886	0.664428	+	0.747352i	$0.268675\pi$
$174$	0	0
$175$	−17.2117	−1.30109
$176$	25.0219	1.88610
$177$	0	0
$178$	0.751213	0.0563058
$179$	5.74612	0.429485	0.214743	−	0.976671i	$-0.431109\pi$
$179$	5.74612	0.429485	0.214743	+	0.976671i	$0.431109\pi$
$180$	0	0
$181$	5.82687	0.433108	0.216554	−	0.976271i	$-0.430518\pi$
$181$	5.82687	0.433108	0.216554	+	0.976271i	$0.430518\pi$
$182$	−0.591277	−0.0438284
$183$	0	0
$184$	0.331555	0.0244425
$185$	19.3926	1.42577
$186$	0	0
$187$	16.9795	1.24166
$188$	16.3748	1.19426
$189$	0	0
$190$	−0.789570	−0.0572814
$191$	24.0682	1.74151	0.870756	−	0.491715i	$-0.163630\pi$
$191$	24.0682	1.74151	0.870756	+	0.491715i	$0.163630\pi$
$192$	0	0
$193$	14.7149	1.05921	0.529603	−	0.848246i	$-0.322341\pi$
$193$	14.7149	1.05921	0.529603	+	0.848246i	$0.322341\pi$
$194$	0.665744	0.0477976
$195$	0	0
$196$	−3.71312	−0.265223
$197$	7.91441	0.563878	0.281939	−	0.959432i	$-0.409022\pi$
$197$	7.91441	0.563878	0.281939	+	0.959432i	$0.409022\pi$
$198$	0	0
$199$	−27.5644	−1.95399	−0.976994	−	0.213266i	$-0.931590\pi$
$199$	−27.5644	−1.95399	−0.976994	+	0.213266i	$0.931590\pi$
$200$	−1.34531	−0.0951277
$201$	0	0
$202$	−0.132092	−0.00929397
$203$	−12.1989	−0.856194
$204$	0	0
$205$	19.6044	1.36923
$206$	−0.408029	−0.0284287
$207$	0	0
$208$	13.5805	0.941635
$209$	−25.9710	−1.79645
$210$	0	0
$211$	26.3182	1.81182	0.905910	−	0.423470i	$-0.139188\pi$
$211$	26.3182	1.81182	0.905910	+	0.423470i	$0.139188\pi$
$212$	−6.32322	−0.434280
$213$	0	0
$214$	−0.0799788	−0.00546724
$215$	−30.7882	−2.09974
$216$	0	0
$217$	9.39803	0.637980
$218$	0.271522	0.0183898
$219$	0	0
$220$	−41.2217	−2.77916
$221$	9.21549	0.619901
$222$	0	0
$223$	9.05110	0.606107	0.303053	−	0.952974i	$-0.401994\pi$
$223$	9.05110	0.606107	0.303053	+	0.952974i	$0.401994\pi$
$224$	−2.07455	−0.138612
$225$	0	0
$226$	0.318218	0.0211675
$227$	−22.0060	−1.46059	−0.730296	−	0.683131i	$-0.760618\pi$
$227$	−22.0060	−1.46059	−0.730296	+	0.683131i	$0.760618\pi$
$228$	0	0
$229$	−21.7801	−1.43927	−0.719635	−	0.694352i	$-0.755691\pi$
$229$	−21.7801	−1.43927	−0.719635	+	0.694352i	$0.755691\pi$
$230$	−0.272410	−0.0179622
$231$	0	0
$232$	−0.953493	−0.0625999
$233$	−24.6379	−1.61408	−0.807041	−	0.590496i	$-0.798932\pi$
$233$	−24.6379	−1.61408	−0.807041	+	0.590496i	$0.798932\pi$
$234$	0	0
$235$	−26.9304	−1.75675
$236$	−0.355617	−0.0231487
$237$	0	0
$238$	−0.467926	−0.0303311
$239$	−12.4250	−0.803706	−0.401853	−	0.915704i	$-0.631634\pi$
$239$	−12.4250	−0.803706	−0.401853	+	0.915704i	$0.631634\pi$
$240$	0	0
$241$	4.31887	0.278203	0.139102	−	0.990278i	$-0.455579\pi$
$241$	4.31887	0.278203	0.139102	+	0.990278i	$0.455579\pi$
$242$	1.66089	0.106766
$243$	0	0
$244$	20.3650	1.30374
$245$	6.10668	0.390142
$246$	0	0
$247$	−14.0955	−0.896878
$248$	0.734571	0.0466453
$249$	0	0
$250$	0.149573	0.00945983
$251$	0.221976	0.0140110	0.00700550	−	0.999975i	$-0.497770\pi$
$251$	0.221976	0.0140110	0.00700550	+	0.999975i	$0.497770\pi$
$252$	0	0
$253$	−8.96026	−0.563327
$254$	0.710217	0.0445630
$255$	0	0
$256$	15.7296	0.983100
$257$	−23.0976	−1.44079	−0.720394	−	0.693565i	$-0.756039\pi$
$257$	−23.0976	−1.44079	−0.720394	+	0.693565i	$0.756039\pi$
$258$	0	0
$259$	−17.5787	−1.09229
$260$	−22.3727	−1.38750
$261$	0	0
$262$	0.637899	0.0394095
$263$	22.5471	1.39031	0.695156	−	0.718859i	$-0.255335\pi$
$263$	22.5471	1.39031	0.695156	+	0.718859i	$0.255335\pi$
$264$	0	0
$265$	10.3993	0.638824
$266$	0.715716	0.0438834
$267$	0	0
$268$	−29.8561	−1.82375
$269$	−13.7684	−0.839476	−0.419738	−	0.907645i	$-0.637878\pi$
$269$	−13.7684	−0.839476	−0.419738	+	0.907645i	$0.637878\pi$
$270$	0	0
$271$	12.4875	0.758560	0.379280	−	0.925282i	$-0.376172\pi$
$271$	12.4875	0.758560	0.379280	+	0.925282i	$0.376172\pi$
$272$	10.7473	0.651652
$273$	0	0
$274$	0.535117	0.0323276
$275$	36.3570	2.19241
$276$	0	0
$277$	−21.9774	−1.32050	−0.660248	−	0.751048i	$-0.729549\pi$
$277$	−21.9774	−1.32050	−0.660248	+	0.751048i	$0.729549\pi$
$278$	0.0261871	0.00157060
$279$	0	0
$280$	2.27393	0.135893
$281$	13.8550	0.826517	0.413259	−	0.910614i	$-0.364391\pi$
$281$	13.8550	0.826517	0.413259	+	0.910614i	$0.364391\pi$
$282$	0	0
$283$	−4.04186	−0.240264	−0.120132	−	0.992758i	$-0.538332\pi$
$283$	−4.04186	−0.240264	−0.120132	+	0.992758i	$0.538332\pi$
$284$	8.34016	0.494897
$285$	0	0
$286$	1.24898	0.0738534
$287$	−17.7707	−1.04897
$288$	0	0
$289$	−9.70703	−0.571002
$290$	0.783403	0.0460030
$291$	0	0
$292$	0.116986	0.00684607
$293$	10.9274	0.638386	0.319193	−	0.947690i	$-0.396588\pi$
$293$	10.9274	0.638386	0.319193	+	0.947690i	$0.396588\pi$
$294$	0	0
$295$	0.584856	0.0340516
$296$	−1.37399	−0.0798615
$297$	0	0
$298$	−0.715366	−0.0414401
$299$	−4.86311	−0.281241
$300$	0	0
$301$	27.9084	1.60861
$302$	−0.308579	−0.0177568
$303$	0	0
$304$	−16.4386	−0.942816
$305$	−33.4928	−1.91779
$306$	0	0
$307$	−7.28577	−0.415821	−0.207910	−	0.978148i	$-0.566666\pi$
$307$	−7.28577	−0.415821	−0.207910	+	0.978148i	$0.566666\pi$
$308$	37.3659	2.12912
$309$	0	0
$310$	−0.603534	−0.0342784
$311$	20.9118	1.18580	0.592899	−	0.805277i	$-0.297983\pi$
$311$	20.9118	1.18580	0.592899	+	0.805277i	$0.297983\pi$
$312$	0	0
$313$	−14.7678	−0.834728	−0.417364	−	0.908739i	$-0.637046\pi$
$313$	−14.7678	−0.834728	−0.417364	+	0.908739i	$0.637046\pi$
$314$	−0.0946604	−0.00534200
$315$	0	0
$316$	−15.6804	−0.882092
$317$	17.4469	0.979915	0.489958	−	0.871746i	$-0.337012\pi$
$317$	17.4469	0.979915	0.489958	+	0.871746i	$0.337012\pi$
$318$	0	0
$319$	25.7681	1.44274
$320$	−26.0027	−1.45359
$321$	0	0
$322$	0.246930	0.0137608
$323$	−11.1550	−0.620679
$324$	0	0
$325$	19.7325	1.09456
$326$	−0.0882212	−0.00488612
$327$	0	0
$328$	−1.38899	−0.0766944
$329$	24.4114	1.34585
$330$	0	0
$331$	2.79611	0.153688	0.0768441	−	0.997043i	$-0.475516\pi$
$331$	2.79611	0.153688	0.0768441	+	0.997043i	$0.475516\pi$
$332$	−14.3775	−0.789067
$333$	0	0
$334$	−0.402666	−0.0220329
$335$	49.1020	2.68273
$336$	0	0
$337$	19.3576	1.05447	0.527237	−	0.849718i	$-0.323228\pi$
$337$	19.3576	1.05447	0.527237	+	0.849718i	$0.323228\pi$
$338$	−0.0788889	−0.00429099
$339$	0	0
$340$	−17.7054	−0.960209
$341$	−19.8518	−1.07503
$342$	0	0
$343$	15.3002	0.826133
$344$	2.18139	0.117612
$345$	0	0
$346$	1.01746	0.0546988
$347$	20.7170	1.11215	0.556073	−	0.831133i	$-0.312308\pi$
$347$	20.7170	1.11215	0.556073	+	0.831133i	$0.312308\pi$
$348$	0	0
$349$	−8.46945	−0.453359	−0.226680	−	0.973969i	$-0.572787\pi$
$349$	−8.46945	−0.453359	−0.226680	+	0.973969i	$0.572787\pi$
$350$	−1.00194	−0.0535557
$351$	0	0
$352$	4.38215	0.233569
$353$	20.4242	1.08707	0.543534	−	0.839387i	$-0.317086\pi$
$353$	20.4242	1.08707	0.543534	+	0.839387i	$0.317086\pi$
$354$	0	0
$355$	−13.7164	−0.727991
$356$	−25.7657	−1.36558
$357$	0	0
$358$	0.334495	0.0176786
$359$	3.63775	0.191993	0.0959965	−	0.995382i	$-0.469396\pi$
$359$	3.63775	0.191993	0.0959965	+	0.995382i	$0.469396\pi$
$360$	0	0
$361$	−1.93793	−0.101997
$362$	0.339196	0.0178277
$363$	0	0
$364$	20.2801	1.06296
$365$	−0.192397	−0.0100705
$366$	0	0
$367$	−16.8886	−0.881575	−0.440788	−	0.897611i	$-0.645301\pi$
$367$	−16.8886	−0.881575	−0.440788	+	0.897611i	$0.645301\pi$
$368$	−5.67148	−0.295646
$369$	0	0
$370$	1.12889	0.0586881
$371$	−9.42658	−0.489404
$372$	0	0
$373$	21.2782	1.10174	0.550872	−	0.834590i	$-0.314295\pi$
$373$	21.2782	1.10174	0.550872	+	0.834590i	$0.314295\pi$
$374$	0.988416	0.0511098
$375$	0	0
$376$	1.90805	0.0984003
$377$	13.9855	0.720288
$378$	0	0
$379$	9.33457	0.479484	0.239742	−	0.970837i	$-0.422937\pi$
$379$	9.33457	0.479484	0.239742	+	0.970837i	$0.422937\pi$
$380$	27.0813	1.38924
$381$	0	0
$382$	1.40106	0.0716847
$383$	−1.00000	−0.0510976
$383$	−1.00000	−0.0510976
$384$	0	0
$385$	−61.4528	−3.13193
$386$	0.856591	0.0435994
$387$	0	0
$388$	−22.8342	−1.15923
$389$	17.3135	0.877832	0.438916	−	0.898528i	$-0.355363\pi$
$389$	17.3135	0.877832	0.438916	+	0.898528i	$0.355363\pi$
$390$	0	0
$391$	−3.84858	−0.194631
$392$	−0.432666	−0.0218529
$393$	0	0
$394$	0.460716	0.0232106
$395$	25.7883	1.29755
$396$	0	0
$397$	2.78038	0.139543	0.0697715	−	0.997563i	$-0.477773\pi$
$397$	2.78038	0.139543	0.0697715	+	0.997563i	$0.477773\pi$
$398$	−1.60459	−0.0804307
$399$	0	0
$400$	23.0125	1.15062
$401$	−10.7944	−0.539047	−0.269524	−	0.962994i	$-0.586866\pi$
$401$	−10.7944	−0.539047	−0.269524	+	0.962994i	$0.586866\pi$
$402$	0	0
$403$	−10.7744	−0.536711
$404$	4.53060	0.225406
$405$	0	0
$406$	−0.710126	−0.0352430
$407$	37.1321	1.84057
$408$	0	0
$409$	−9.90742	−0.489890	−0.244945	−	0.969537i	$-0.578770\pi$
$409$	−9.90742	−0.489890	−0.244945	+	0.969537i	$0.578770\pi$
$410$	1.14122	0.0563607
$411$	0	0
$412$	13.9949	0.689478
$413$	−0.530151	−0.0260870
$414$	0	0
$415$	23.6455	1.16071
$416$	2.37838	0.116610
$417$	0	0
$418$	−1.51183	−0.0739461
$419$	−28.7045	−1.40231	−0.701154	−	0.713010i	$-0.747331\pi$
$419$	−28.7045	−1.40231	−0.701154	+	0.713010i	$0.747331\pi$
$420$	0	0
$421$	11.8811	0.579049	0.289525	−	0.957171i	$-0.406503\pi$
$421$	11.8811	0.579049	0.289525	+	0.957171i	$0.406503\pi$
$422$	1.53204	0.0745788
$423$	0	0
$424$	−0.736803	−0.0357823
$425$	15.6159	0.757483
$426$	0	0
$427$	30.3600	1.46922
$428$	2.74317	0.132596
$429$	0	0
$430$	−1.79226	−0.0864303
$431$	−26.8892	−1.29521	−0.647604	−	0.761977i	$-0.724229\pi$
$431$	−26.8892	−1.29521	−0.647604	+	0.761977i	$0.724229\pi$
$432$	0	0
$433$	−19.5958	−0.941712	−0.470856	−	0.882210i	$-0.656055\pi$
$433$	−19.5958	−0.941712	−0.470856	+	0.882210i	$0.656055\pi$
$434$	0.547081	0.0262607
$435$	0	0
$436$	−9.31288	−0.446006
$437$	5.88659	0.281594
$438$	0	0
$439$	−7.05497	−0.336715	−0.168358	−	0.985726i	$-0.553846\pi$
$439$	−7.05497	−0.336715	−0.168358	+	0.985726i	$0.553846\pi$
$440$	−4.80329	−0.228988
$441$	0	0
$442$	0.536455	0.0255166
$443$	21.7406	1.03293	0.516463	−	0.856309i	$-0.327248\pi$
$443$	21.7406	1.03293	0.516463	+	0.856309i	$0.327248\pi$
$444$	0	0
$445$	42.3748	2.00876
$446$	0.526886	0.0249488
$447$	0	0
$448$	23.5705	1.11360
$449$	−21.2294	−1.00188	−0.500938	−	0.865483i	$-0.667012\pi$
$449$	−21.2294	−1.00188	−0.500938	+	0.865483i	$0.667012\pi$
$450$	0	0
$451$	37.5376	1.76758
$452$	−10.9145	−0.513374
$453$	0	0
$454$	−1.28102	−0.0601214
$455$	−33.3530	−1.56362
$456$	0	0
$457$	28.7223	1.34357	0.671787	−	0.740744i	$-0.265527\pi$
$457$	28.7223	1.34357	0.671787	+	0.740744i	$0.265527\pi$
$458$	−1.26787	−0.0592437
$459$	0	0
$460$	9.34332	0.435635
$461$	−20.2796	−0.944515	−0.472257	−	0.881461i	$-0.656561\pi$
$461$	−20.2796	−0.944515	−0.472257	+	0.881461i	$0.656561\pi$
$462$	0	0
$463$	9.55791	0.444194	0.222097	−	0.975025i	$-0.428710\pi$
$463$	9.55791	0.444194	0.222097	+	0.975025i	$0.428710\pi$
$464$	16.3102	0.757181
$465$	0	0
$466$	−1.43423	−0.0664393
$467$	11.2395	0.520103	0.260051	−	0.965595i	$-0.416261\pi$
$467$	11.2395	0.520103	0.260051	+	0.965595i	$0.416261\pi$
$468$	0	0
$469$	−44.5092	−2.05524
$470$	−1.56768	−0.0723118
$471$	0	0
$472$	−0.0414378	−0.00190733
$473$	−58.9519	−2.71061
$474$	0	0
$475$	−23.8853	−1.09593
$476$	16.0493	0.735618
$477$	0	0
$478$	−0.723288	−0.0330824
$479$	−2.22275	−0.101560	−0.0507801	−	0.998710i	$-0.516171\pi$
$479$	−2.22275	−0.101560	−0.0507801	+	0.998710i	$0.516171\pi$
$480$	0	0
$481$	20.1531	0.918904
$482$	0.251412	0.0114515
$483$	0	0
$484$	−56.9666	−2.58939
$485$	37.5536	1.70522
$486$	0	0
$487$	−4.06739	−0.184311	−0.0921554	−	0.995745i	$-0.529376\pi$
$487$	−4.06739	−0.184311	−0.0921554	+	0.995745i	$0.529376\pi$
$488$	2.37300	0.107421
$489$	0	0
$490$	0.355484	0.0160591
$491$	−9.69539	−0.437547	−0.218773	−	0.975776i	$-0.570206\pi$
$491$	−9.69539	−0.437547	−0.218773	+	0.975776i	$0.570206\pi$
$492$	0	0
$493$	11.0678	0.498470
$494$	−0.820535	−0.0369176
$495$	0	0
$496$	−12.5654	−0.564201
$497$	12.4334	0.557715
$498$	0	0
$499$	−28.6798	−1.28388	−0.641942	−	0.766753i	$-0.721871\pi$
$499$	−28.6798	−1.28388	−0.641942	+	0.766753i	$0.721871\pi$
$500$	−5.13017	−0.229428
$501$	0	0
$502$	0.0129217	0.000576725 0
$503$	0.700587	0.0312376	0.0156188	−	0.999878i	$-0.495028\pi$
$503$	0.700587	0.0312376	0.0156188	+	0.999878i	$0.495028\pi$
$504$	0	0
$505$	−7.45112	−0.331571
$506$	−0.521598	−0.0231879
$507$	0	0
$508$	−24.3596	−1.08078
$509$	−21.4938	−0.952695	−0.476348	−	0.879257i	$-0.658040\pi$
$509$	−21.4938	−0.952695	−0.476348	+	0.879257i	$0.658040\pi$
$510$	0	0
$511$	0.174401	0.00771505
$512$	4.62548	0.204419
$513$	0	0
$514$	−1.34456	−0.0593062
$515$	−23.0163	−1.01422
$516$	0	0
$517$	−51.5651	−2.26783
$518$	−1.02330	−0.0449610
$519$	0	0
$520$	−2.60695	−0.114322
$521$	−15.5853	−0.682805	−0.341403	−	0.939917i	$-0.610902\pi$
$521$	−15.5853	−0.682805	−0.341403	+	0.939917i	$0.610902\pi$
$522$	0	0
$523$	40.1479	1.75555	0.877773	−	0.479077i	$-0.159029\pi$
$523$	40.1479	1.75555	0.877773	+	0.479077i	$0.159029\pi$
$524$	−21.8792	−0.955795
$525$	0	0
$526$	1.31252	0.0572285
$527$	−8.52667	−0.371427
$528$	0	0
$529$	−20.9691	−0.911698
$530$	0.605367	0.0262955
$531$	0	0
$532$	−24.5482	−1.06430
$533$	20.3732	0.882462
$534$	0	0
$535$	−4.51148	−0.195048
$536$	−3.47894	−0.150267
$537$	0	0
$538$	−0.801493	−0.0345548
$539$	11.6928	0.503645
$540$	0	0
$541$	13.4726	0.579231	0.289615	−	0.957143i	$-0.406473\pi$
$541$	13.4726	0.579231	0.289615	+	0.957143i	$0.406473\pi$
$542$	0.726925	0.0312241
$543$	0	0
$544$	1.88220	0.0806988
$545$	15.3162	0.656073
$546$	0	0
$547$	37.1134	1.58686	0.793428	−	0.608664i	$-0.208294\pi$
$547$	37.1134	1.58686	0.793428	+	0.608664i	$0.208294\pi$
$548$	−18.3538	−0.784037
$549$	0	0
$550$	2.11642	0.0902446
$551$	−16.9288	−0.721191
$552$	0	0
$553$	−23.3762	−0.994057
$554$	−1.27936	−0.0543547
$555$	0	0
$556$	−0.898186	−0.0380916
$557$	−33.9301	−1.43766	−0.718832	−	0.695184i	$-0.755323\pi$
$557$	−33.9301	−1.43766	−0.718832	+	0.695184i	$0.755323\pi$
$558$	0	0
$559$	−31.9957	−1.35327
$560$	−38.8971	−1.64370
$561$	0	0
$562$	0.806529	0.0340214
$563$	8.93503	0.376567	0.188283	−	0.982115i	$-0.439708\pi$
$563$	8.93503	0.376567	0.188283	+	0.982115i	$0.439708\pi$
$564$	0	0
$565$	17.9502	0.755170
$566$	−0.235286	−0.00988981
$567$	0	0
$568$	0.971824	0.0407768
$569$	0.332203	0.0139267	0.00696334	−	0.999976i	$-0.497783\pi$
$569$	0.332203	0.0139267	0.00696334	+	0.999976i	$0.497783\pi$
$570$	0	0
$571$	23.9910	1.00399	0.501997	−	0.864869i	$-0.332599\pi$
$571$	23.9910	1.00399	0.501997	+	0.864869i	$0.332599\pi$
$572$	−42.8383	−1.79116
$573$	0	0
$574$	−1.03447	−0.0431780
$575$	−8.24068	−0.343660
$576$	0	0
$577$	−3.29929	−0.137351	−0.0686756	−	0.997639i	$-0.521877\pi$
$577$	−3.29929	−0.137351	−0.0686756	+	0.997639i	$0.521877\pi$
$578$	−0.565069	−0.0235038
$579$	0	0
$580$	−26.8697	−1.11571
$581$	−21.4338	−0.889225
$582$	0	0
$583$	19.9121	0.824675
$584$	0.0136316	0.000564079 0
$585$	0	0
$586$	0.636110	0.0262775
$587$	−38.2342	−1.57809	−0.789047	−	0.614333i	$-0.789425\pi$
$587$	−38.2342	−1.57809	−0.789047	+	0.614333i	$0.789425\pi$
$588$	0	0
$589$	13.0420	0.537384
$590$	0.0340458	0.00140164
$591$	0	0
$592$	23.5031	0.965970
$593$	−14.6870	−0.603121	−0.301561	−	0.953447i	$-0.597508\pi$
$593$	−14.6870	−0.603121	−0.301561	+	0.953447i	$0.597508\pi$
$594$	0	0
$595$	−26.3950	−1.08209
$596$	24.5362	1.00504
$597$	0	0
$598$	−0.283093	−0.0115765
$599$	−35.9103	−1.46725	−0.733627	−	0.679552i	$-0.762174\pi$
$599$	−35.9103	−1.46725	−0.733627	+	0.679552i	$0.762174\pi$
$600$	0	0
$601$	−22.5470	−0.919713	−0.459857	−	0.887993i	$-0.652099\pi$
$601$	−22.5470	−0.919713	−0.459857	+	0.887993i	$0.652099\pi$
$602$	1.62461	0.0662143
$603$	0	0
$604$	10.5839	0.430653
$605$	93.6886	3.80898
$606$	0	0
$607$	10.5257	0.427225	0.213612	−	0.976919i	$-0.431477\pi$
$607$	10.5257	0.427225	0.213612	+	0.976919i	$0.431477\pi$
$608$	−2.87892	−0.116756
$609$	0	0
$610$	−1.94969	−0.0789407
$611$	−27.9866	−1.13222
$612$	0	0
$613$	−22.4808	−0.907989	−0.453995	−	0.891004i	$-0.650001\pi$
$613$	−22.4808	−0.907989	−0.453995	+	0.891004i	$0.650001\pi$
$614$	−0.424121	−0.0171161
$615$	0	0
$616$	4.35401	0.175428
$617$	−48.7813	−1.96386	−0.981931	−	0.189240i	$-0.939398\pi$
$617$	−48.7813	−1.96386	−0.981931	+	0.189240i	$0.939398\pi$
$618$	0	0
$619$	46.5798	1.87220	0.936101	−	0.351732i	$-0.114407\pi$
$619$	46.5798	1.87220	0.936101	+	0.351732i	$0.114407\pi$
$620$	20.7005	0.831350
$621$	0	0
$622$	1.21732	0.0488102
$623$	−38.4112	−1.53891
$624$	0	0
$625$	−20.4753	−0.819010
$626$	−0.859671	−0.0343593
$627$	0	0
$628$	3.24674	0.129559
$629$	15.9488	0.635921
$630$	0	0
$631$	−28.6283	−1.13967	−0.569837	−	0.821758i	$-0.692994\pi$
$631$	−28.6283	−1.13967	−0.569837	+	0.821758i	$0.692994\pi$
$632$	−1.82714	−0.0726796
$633$	0	0
$634$	1.01562	0.0403356
$635$	40.0623	1.58982
$636$	0	0
$637$	6.34618	0.251445
$638$	1.50002	0.0593865
$639$	0	0
$640$	−6.09092	−0.240765
$641$	23.5231	0.929108	0.464554	−	0.885545i	$-0.346215\pi$
$641$	23.5231	0.929108	0.464554	+	0.885545i	$0.346215\pi$
$642$	0	0
$643$	2.86793	0.113100	0.0565501	−	0.998400i	$-0.481990\pi$
$643$	2.86793	0.113100	0.0565501	+	0.998400i	$0.481990\pi$
$644$	−8.46938	−0.333740
$645$	0	0
$646$	−0.649356	−0.0255486
$647$	−37.8982	−1.48993	−0.744966	−	0.667102i	$-0.767535\pi$
$647$	−37.8982	−1.48993	−0.744966	+	0.667102i	$0.767535\pi$
$648$	0	0
$649$	1.11986	0.0439582
$650$	1.14867	0.0450546
$651$	0	0
$652$	3.02588	0.118503
$653$	16.2458	0.635748	0.317874	−	0.948133i	$-0.397031\pi$
$653$	16.2458	0.635748	0.317874	+	0.948133i	$0.397031\pi$
$654$	0	0
$655$	35.9829	1.40597
$656$	23.7597	0.927662
$657$	0	0
$658$	1.42105	0.0553982
$659$	−24.4071	−0.950767	−0.475384	−	0.879779i	$-0.657691\pi$
$659$	−24.4071	−0.950767	−0.475384	+	0.879779i	$0.657691\pi$
$660$	0	0
$661$	9.10610	0.354186	0.177093	−	0.984194i	$-0.443331\pi$
$661$	9.10610	0.354186	0.177093	+	0.984194i	$0.443331\pi$
$662$	0.162768	0.00632617
$663$	0	0
$664$	−1.67532	−0.0650148
$665$	40.3725	1.56558
$666$	0	0
$667$	−5.84062	−0.226150
$668$	13.8109	0.534361
$669$	0	0
$670$	2.85834	0.110427
$671$	−64.1304	−2.47573
$672$	0	0
$673$	−0.633421	−0.0244166	−0.0122083	−	0.999925i	$-0.503886\pi$
$673$	−0.633421	−0.0244166	−0.0122083	+	0.999925i	$0.503886\pi$
$674$	1.12685	0.0434046
$675$	0	0
$676$	2.70579	0.104069
$677$	10.9591	0.421193	0.210596	−	0.977573i	$-0.432459\pi$
$677$	10.9591	0.421193	0.210596	+	0.977573i	$0.432459\pi$
$678$	0	0
$679$	−34.0410	−1.30637
$680$	−2.06309	−0.0791160
$681$	0	0
$682$	−1.15562	−0.0442509
$683$	−23.3336	−0.892836	−0.446418	−	0.894825i	$-0.647301\pi$
$683$	−23.3336	−0.892836	−0.446418	+	0.894825i	$0.647301\pi$
$684$	0	0
$685$	30.1851	1.15331
$686$	0.890661	0.0340056
$687$	0	0
$688$	−37.3141	−1.42259
$689$	10.8071	0.411719
$690$	0	0
$691$	2.24682	0.0854732	0.0427366	−	0.999086i	$-0.486392\pi$
$691$	2.24682	0.0854732	0.0427366	+	0.999086i	$0.486392\pi$
$692$	−34.8975	−1.32660
$693$	0	0
$694$	1.20598	0.0457785
$695$	1.47718	0.0560325
$696$	0	0
$697$	16.1230	0.610702
$698$	−0.493026	−0.0186613
$699$	0	0
$700$	34.3652	1.29888
$701$	8.34327	0.315121	0.157560	−	0.987509i	$-0.449637\pi$
$701$	8.34327	0.315121	0.157560	+	0.987509i	$0.449637\pi$
$702$	0	0
$703$	−24.3945	−0.920057
$704$	−49.7887	−1.87648
$705$	0	0
$706$	1.18894	0.0447463
$707$	6.75417	0.254017
$708$	0	0
$709$	−6.23940	−0.234326	−0.117163	−	0.993113i	$-0.537380\pi$
$709$	−6.23940	−0.234326	−0.117163	+	0.993113i	$0.537380\pi$
$710$	−0.798464	−0.0299658
$711$	0	0
$712$	−3.00231	−0.112516
$713$	4.49961	0.168512
$714$	0	0
$715$	70.4528	2.63479
$716$	−11.4728	−0.428758
$717$	0	0
$718$	0.211762	0.00790288
$719$	1.00232	0.0373801	0.0186901	−	0.999825i	$-0.494050\pi$
$719$	1.00232	0.0373801	0.0186901	+	0.999825i	$0.494050\pi$
$720$	0	0
$721$	20.8634	0.776994
$722$	−0.112812	−0.00419842
$723$	0	0
$724$	−11.6340	−0.432374
$725$	23.6987	0.880149
$726$	0	0
$727$	−10.3508	−0.383889	−0.191944	−	0.981406i	$-0.561479\pi$
$727$	−10.3508	−0.383889	−0.191944	+	0.981406i	$0.561479\pi$
$728$	2.36310	0.0875825
$729$	0	0
$730$	−0.0111999	−0.000414527 0
$731$	−25.3208	−0.936524
$732$	0	0
$733$	−24.2474	−0.895599	−0.447800	−	0.894134i	$-0.647792\pi$
$733$	−24.2474	−0.895599	−0.447800	+	0.894134i	$0.647792\pi$
$734$	−0.983122	−0.0362877
$735$	0	0
$736$	−0.993260	−0.0366120
$737$	94.0183	3.46321
$738$	0	0
$739$	−35.0291	−1.28856	−0.644282	−	0.764788i	$-0.722844\pi$
$739$	−35.0291	−1.28856	−0.644282	+	0.764788i	$0.722844\pi$
$740$	−38.7195	−1.42336
$741$	0	0
$742$	−0.548743	−0.0201450
$743$	28.7141	1.05342	0.526710	−	0.850045i	$-0.323426\pi$
$743$	28.7141	1.05342	0.526710	+	0.850045i	$0.323426\pi$
$744$	0	0
$745$	−40.3527	−1.47841
$746$	1.23865	0.0453504
$747$	0	0
$748$	−33.9015	−1.23956
$749$	4.08950	0.149427
$750$	0	0
$751$	−38.3453	−1.39924	−0.699620	−	0.714516i	$-0.746647\pi$
$751$	−38.3453	−1.39924	−0.699620	+	0.714516i	$0.746647\pi$
$752$	−32.6386	−1.19021
$753$	0	0
$754$	0.814126	0.0296487
$755$	−17.4065	−0.633487
$756$	0	0
$757$	−4.95780	−0.180194	−0.0900972	−	0.995933i	$-0.528718\pi$
$757$	−4.95780	−0.180194	−0.0900972	+	0.995933i	$0.528718\pi$
$758$	0.543387	0.0197367
$759$	0	0
$760$	3.15560	0.114466
$761$	−0.398692	−0.0144526	−0.00722629	−	0.999974i	$-0.502300\pi$
$761$	−0.398692	−0.0144526	−0.00722629	+	0.999974i	$0.502300\pi$
$762$	0	0
$763$	−13.8835	−0.502618
$764$	−48.0548	−1.73856
$765$	0	0
$766$	−0.0582123	−0.00210330
$767$	0.607793	0.0219461
$768$	0	0
$769$	20.5817	0.742196	0.371098	−	0.928594i	$-0.378981\pi$
$769$	20.5817	0.742196	0.371098	+	0.928594i	$0.378981\pi$
$770$	−3.57731	−0.128917
$771$	0	0
$772$	−29.3800	−1.05741
$773$	−16.0700	−0.577998	−0.288999	−	0.957329i	$-0.593322\pi$
$773$	−16.0700	−0.577998	−0.288999	+	0.957329i	$0.593322\pi$
$774$	0	0
$775$	−18.2575	−0.655829
$776$	−2.66072	−0.0955142
$777$	0	0
$778$	1.00786	0.0361336
$779$	−24.6609	−0.883569
$780$	0	0
$781$	−26.2635	−0.939783
$782$	−0.224035	−0.00801147
$783$	0	0
$784$	7.40106	0.264324
$785$	−5.33965	−0.190580
$786$	0	0
$787$	−10.6688	−0.380301	−0.190151	−	0.981755i	$-0.560898\pi$
$787$	−10.6688	−0.380301	−0.190151	+	0.981755i	$0.560898\pi$
$788$	−15.8020	−0.562923
$789$	0	0
$790$	1.50120	0.0534103
$791$	−16.2712	−0.578537
$792$	0	0
$793$	−34.8063	−1.23601
$794$	0.161852	0.00574392
$795$	0	0
$796$	55.0354	1.95068
$797$	−7.43926	−0.263512	−0.131756	−	0.991282i	$-0.542062\pi$
$797$	−7.43926	−0.263512	−0.131756	+	0.991282i	$0.542062\pi$
$798$	0	0
$799$	−22.1481	−0.783543
$800$	4.03023	0.142490
$801$	0	0
$802$	−0.628368	−0.0221884
$803$	−0.368393	−0.0130003
$804$	0	0
$805$	13.9289	0.490930
$806$	−0.627203	−0.0220923
$807$	0	0
$808$	0.527921	0.0185722
$809$	−32.1034	−1.12870	−0.564348	−	0.825537i	$-0.690872\pi$
$809$	−32.1034	−1.12870	−0.564348	+	0.825537i	$0.690872\pi$
$810$	0	0
$811$	−42.2868	−1.48489	−0.742445	−	0.669907i	$-0.766334\pi$
$811$	−42.2868	−1.48489	−0.742445	+	0.669907i	$0.766334\pi$
$812$	24.3564	0.854744
$813$	0	0
$814$	2.16154	0.0757621
$815$	−4.97643	−0.174317
$816$	0	0
$817$	38.7294	1.35497
$818$	−0.576734	−0.0201650
$819$	0	0
$820$	−39.1423	−1.36691
$821$	6.87318	0.239876	0.119938	−	0.992781i	$-0.461730\pi$
$821$	6.87318	0.239876	0.119938	+	0.992781i	$0.461730\pi$
$822$	0	0
$823$	−31.2390	−1.08892	−0.544461	−	0.838786i	$-0.683266\pi$
$823$	−31.2390	−1.08892	−0.544461	+	0.838786i	$0.683266\pi$
$824$	1.63073	0.0568092
$825$	0	0
$826$	−0.0308613	−0.00107380
$827$	−26.0816	−0.906947	−0.453474	−	0.891270i	$-0.649815\pi$
$827$	−26.0816	−0.906947	−0.453474	+	0.891270i	$0.649815\pi$
$828$	0	0
$829$	53.0257	1.84166	0.920829	−	0.389967i	$-0.127514\pi$
$829$	53.0257	1.84166	0.920829	+	0.389967i	$0.127514\pi$
$830$	1.37646	0.0477777
$831$	0	0
$832$	−27.0225	−0.936835
$833$	5.02225	0.174011
$834$	0	0
$835$	−22.7138	−0.786043
$836$	51.8540	1.79341
$837$	0	0
$838$	−1.67096	−0.0577223
$839$	8.35260	0.288364	0.144182	−	0.989551i	$-0.453945\pi$
$839$	8.35260	0.288364	0.144182	+	0.989551i	$0.453945\pi$
$840$	0	0
$841$	−12.2034	−0.420807
$842$	0.691626	0.0238350
$843$	0	0
$844$	−52.5472	−1.80875
$845$	−4.45000	−0.153085
$846$	0	0
$847$	−84.9253	−2.91807
$848$	12.6035	0.432807
$849$	0	0
$850$	0.909038	0.0311798
$851$	−8.41637	−0.288509
$852$	0	0
$853$	54.2075	1.85603	0.928014	−	0.372544i	$-0.121514\pi$
$853$	54.2075	1.85603	0.928014	+	0.372544i	$0.121514\pi$
$854$	1.76732	0.0604766
$855$	0	0
$856$	0.319644	0.0109252
$857$	34.9843	1.19504	0.597521	−	0.801854i	$-0.296153\pi$
$857$	34.9843	1.19504	0.597521	+	0.801854i	$0.296153\pi$
$858$	0	0
$859$	−18.1201	−0.618250	−0.309125	−	0.951021i	$-0.600036\pi$
$859$	−18.1201	−0.618250	−0.309125	+	0.951021i	$0.600036\pi$
$860$	61.4722	2.09618
$861$	0	0
$862$	−1.56528	−0.0533138
$863$	−8.08060	−0.275067	−0.137533	−	0.990497i	$-0.543917\pi$
$863$	−8.08060	−0.275067	−0.137533	+	0.990497i	$0.543917\pi$
$864$	0	0
$865$	57.3932	1.95143
$866$	−1.14072	−0.0387631
$867$	0	0
$868$	−18.7642	−0.636899
$869$	49.3783	1.67505
$870$	0	0
$871$	51.0277	1.72901
$872$	−1.08517	−0.0367485
$873$	0	0
$874$	0.342672	0.0115911
$875$	−7.64801	−0.258550
$876$	0	0
$877$	0.893449	0.0301696	0.0150848	−	0.999886i	$-0.495198\pi$
$877$	0.893449	0.0301696	0.0150848	+	0.999886i	$0.495198\pi$
$878$	−0.410686	−0.0138600
$879$	0	0
$880$	82.1637	2.76974
$881$	3.00730	0.101318	0.0506592	−	0.998716i	$-0.483868\pi$
$881$	3.00730	0.101318	0.0506592	+	0.998716i	$0.483868\pi$
$882$	0	0
$883$	−1.84700	−0.0621565	−0.0310782	−	0.999517i	$-0.509894\pi$
$883$	−1.84700	−0.0621565	−0.0310782	+	0.999517i	$0.509894\pi$
$884$	−18.3998	−0.618851
$885$	0	0
$886$	1.26557	0.0425177
$887$	−39.0563	−1.31138	−0.655690	−	0.755030i	$-0.727622\pi$
$887$	−39.0563	−1.31138	−0.655690	+	0.755030i	$0.727622\pi$
$888$	0	0
$889$	−36.3150	−1.21797
$890$	2.46674	0.0826852
$891$	0	0
$892$	−18.0715	−0.605080
$893$	33.8766	1.13364
$894$	0	0
$895$	18.8684	0.630700
$896$	5.52120	0.184450
$897$	0	0
$898$	−1.23581	−0.0412396
$899$	−12.9401	−0.431576
$900$	0	0
$901$	8.55257	0.284928
$902$	2.18515	0.0727575
$903$	0	0
$904$	−1.27179	−0.0422992
$905$	19.1335	0.636020
$906$	0	0
$907$	−25.7704	−0.855691	−0.427845	−	0.903852i	$-0.640727\pi$
$907$	−25.7704	−0.855691	−0.427845	+	0.903852i	$0.640727\pi$
$908$	43.9375	1.45812
$909$	0	0
$910$	−1.94156	−0.0643621
$911$	48.8267	1.61770	0.808850	−	0.588015i	$-0.200091\pi$
$911$	48.8267	1.61770	0.808850	+	0.588015i	$0.200091\pi$
$912$	0	0
$913$	45.2754	1.49840
$914$	1.67199	0.0553046
$915$	0	0
$916$	43.4864	1.43683
$917$	−32.6172	−1.07712
$918$	0	0
$919$	−26.7476	−0.882321	−0.441160	−	0.897428i	$-0.645433\pi$
$919$	−26.7476	−0.882321	−0.441160	+	0.897428i	$0.645433\pi$
$920$	1.08872	0.0358939
$921$	0	0
$922$	−1.18052	−0.0388784
$923$	−14.2543	−0.469187
$924$	0	0
$925$	34.1500	1.12285
$926$	0.556388	0.0182841
$927$	0	0
$928$	2.85644	0.0937672
$929$	−24.4021	−0.800609	−0.400304	−	0.916382i	$-0.631096\pi$
$929$	−24.4021	−0.800609	−0.400304	+	0.916382i	$0.631096\pi$
$930$	0	0
$931$	−7.68178	−0.251760
$932$	49.1923	1.61135
$933$	0	0
$934$	0.654278	0.0214086
$935$	55.7551	1.82339
$936$	0	0
$937$	−4.63607	−0.151454	−0.0757270	−	0.997129i	$-0.524128\pi$
$937$	−4.63607	−0.151454	−0.0757270	+	0.997129i	$0.524128\pi$
$938$	−2.59098	−0.0845986
$939$	0	0
$940$	53.7696	1.75377
$941$	3.44151	0.112190	0.0560950	−	0.998425i	$-0.482135\pi$
$941$	3.44151	0.112190	0.0560950	+	0.998425i	$0.482135\pi$
$942$	0	0
$943$	−8.50828	−0.277068
$944$	0.708823	0.0230702
$945$	0	0
$946$	−3.43173	−0.111575
$947$	53.5501	1.74015	0.870073	−	0.492923i	$-0.164072\pi$
$947$	53.5501	1.74015	0.870073	+	0.492923i	$0.164072\pi$
$948$	0	0
$949$	−0.199943	−0.00649041
$950$	−1.39042	−0.0451112
$951$	0	0
$952$	1.87012	0.0606109
$953$	12.5380	0.406146	0.203073	−	0.979164i	$-0.434907\pi$
$953$	12.5380	0.406146	0.203073	+	0.979164i	$0.434907\pi$
$954$	0	0
$955$	79.0319	2.55741
$956$	24.8079	0.802345
$957$	0	0
$958$	−0.129392	−0.00418045
$959$	−27.3617	−0.883556
$960$	0	0
$961$	−21.0310	−0.678418
$962$	1.17316	0.0378242
$963$	0	0
$964$	−8.62311	−0.277732
$965$	48.3190	1.55544
$966$	0	0
$967$	−13.9055	−0.447171	−0.223585	−	0.974684i	$-0.571776\pi$
$967$	−13.9055	−0.447171	−0.223585	+	0.974684i	$0.571776\pi$
$968$	−6.63795	−0.213352
$969$	0	0
$970$	2.18608	0.0701909
$971$	32.9568	1.05763	0.528817	−	0.848736i	$-0.322636\pi$
$971$	32.9568	1.05763	0.528817	+	0.848736i	$0.322636\pi$
$972$	0	0
$973$	−1.33901	−0.0429266
$974$	−0.236772	−0.00758666
$975$	0	0
$976$	−40.5919	−1.29932
$977$	54.6024	1.74689	0.873443	−	0.486926i	$-0.161882\pi$
$977$	54.6024	1.74689	0.873443	+	0.486926i	$0.161882\pi$
$978$	0	0
$979$	81.1374	2.59316
$980$	−12.1927	−0.389481
$981$	0	0
$982$	−0.564391	−0.0180104
$983$	−21.2851	−0.678890	−0.339445	−	0.940626i	$-0.610239\pi$
$983$	−21.2851	−0.678890	−0.339445	+	0.940626i	$0.610239\pi$
$984$	0	0
$985$	25.9883	0.828057
$986$	0.644285	0.0205182
$987$	0	0
$988$	28.1433	0.895359
$989$	13.3621	0.424889
$990$	0	0
$991$	−5.08207	−0.161437	−0.0807187	−	0.996737i	$-0.525722\pi$
$991$	−5.08207	−0.161437	−0.0807187	+	0.996737i	$0.525722\pi$
$992$	−2.20060	−0.0698692
$993$	0	0
$994$	0.723778	0.0229569
$995$	−90.5124	−2.86944
$996$	0	0
$997$	−42.2654	−1.33856	−0.669279	−	0.743011i	$-0.733397\pi$
$997$	−42.2654	−1.33856	−0.669279	+	0.743011i	$0.733397\pi$
$998$	−1.66952	−0.0528477
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3447.2.a.j.1.14		24
3.2	odd	2		383.2.a.c.1.11	✓	24
12.11	even	2		6128.2.a.p.1.24		24
15.14	odd	2		9575.2.a.e.1.14		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
383.2.a.c.1.11	✓	24	3.2	odd	2
3447.2.a.j.1.14		24	1.1	even	1	trivial
6128.2.a.p.1.24		24	12.11	even	2
9575.2.a.e.1.14		24	15.14	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:	\( N \)	\(=\)	\( 3447 = 3^{2} \cdot 383 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3447.a (trivial)

\(f(q)\)	\(=\)	\(q+0.0582123 q^{2} -1.99661 q^{4} +3.28367 q^{5} -2.97653 q^{7} -0.232652 q^{8} +0.191150 q^{10} +6.28742 q^{11} +3.41245 q^{13} -0.173271 q^{14} +3.97968 q^{16} +2.70055 q^{17} -4.13063 q^{19} -6.55621 q^{20} +0.366005 q^{22} -1.42511 q^{23} +5.78249 q^{25} +0.198647 q^{26} +5.94297 q^{28} +4.09836 q^{29} -3.15738 q^{31} +0.696971 q^{32} +0.157205 q^{34} -9.77393 q^{35} +5.90577 q^{37} -0.240453 q^{38} -0.763953 q^{40} +5.97026 q^{41} -9.37617 q^{43} -12.5535 q^{44} -0.0829590 q^{46} -8.20132 q^{47} +1.85971 q^{49} +0.336612 q^{50} -6.81333 q^{52} +3.16697 q^{53} +20.6458 q^{55} +0.692495 q^{56} +0.238575 q^{58} +0.178110 q^{59} -10.1998 q^{61} -0.183798 q^{62} -7.91879 q^{64} +11.2054 q^{65} +14.9534 q^{67} -5.39195 q^{68} -0.568963 q^{70} -4.17716 q^{71} -0.0585921 q^{73} +0.343789 q^{74} +8.24725 q^{76} -18.7147 q^{77} +7.85351 q^{79} +13.0680 q^{80} +0.347543 q^{82} +7.20095 q^{83} +8.86772 q^{85} -0.545809 q^{86} -1.46278 q^{88} +12.9047 q^{89} -10.1572 q^{91} +2.84539 q^{92} -0.477418 q^{94} -13.5636 q^{95} +11.4365 q^{97} +0.108258 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 5 q^{2} + 29 q^{4} - 3 q^{5} + 17 q^{7} - 15 q^{8} + q^{10} + 28 q^{13} + 8 q^{14} + 35 q^{16} - 16 q^{17} + 13 q^{19} + 4 q^{20} + 12 q^{22} - 7 q^{23} + 67 q^{25} + 14 q^{26} + 39 q^{28} + 2 q^{29}+ \cdots + 29 q^{98}+O(q^{100}) \) 24 * q - 5 * q^2 + 29 * q^4 - 3 * q^5 + 17 * q^7 - 15 * q^8 + q^10 + 28 * q^13 + 8 * q^14 + 35 * q^16 - 16 * q^17 + 13 * q^19 + 4 * q^20 + 12 * q^22 - 7 * q^23 + 67 * q^25 + 14 * q^26 + 39 * q^28 + 2 * q^29 - 20 * q^32 - 18 * q^34 + 10 * q^35 + 35 * q^37 + 5 * q^38 - 36 * q^40 + 34 * q^43 + 36 * q^44 - 17 * q^46 + 3 * q^47 + 49 * q^49 + 20 * q^50 + 29 * q^52 + 9 * q^53 + 10 * q^55 + 47 * q^56 + 12 * q^58 + 6 * q^59 + 3 * q^61 + 58 * q^62 + 15 * q^64 - 8 * q^65 + 42 * q^67 + 23 * q^68 - 59 * q^70 + 13 * q^71 + 83 * q^73 + 45 * q^74 + 4 * q^76 + 21 * q^77 - 3 * q^79 + 93 * q^80 - 13 * q^82 + 23 * q^83 + 52 * q^85 + 43 * q^86 - 25 * q^88 + 38 * q^89 - 17 * q^91 + 61 * q^92 - 71 * q^94 + 66 * q^95 + 63 * q^97 + 29 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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