LMFDB - Embedded newform 3447.2.a.j.1.3

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			1.3
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)

$f(q)$	$=$	\(q-2.56319 q^{2} +4.56992 q^{4} -0.275873 q^{5} -0.724431 q^{7} -6.58718 q^{8} +0.707113 q^{10} -6.18852 q^{11} -3.89050 q^{13} +1.85685 q^{14} +7.74432 q^{16} +1.81930 q^{17} -4.40463 q^{19} -1.26072 q^{20} +15.8623 q^{22} -3.70367 q^{23} -4.92389 q^{25} +9.97208 q^{26} -3.31059 q^{28} -6.02479 q^{29} -3.23181 q^{31} -6.67577 q^{32} -4.66319 q^{34} +0.199851 q^{35} +0.897967 q^{37} +11.2899 q^{38} +1.81722 q^{40} +5.23804 q^{41} -7.55001 q^{43} -28.2810 q^{44} +9.49319 q^{46} -2.40640 q^{47} -6.47520 q^{49} +12.6209 q^{50} -17.7793 q^{52} -6.26043 q^{53} +1.70724 q^{55} +4.77196 q^{56} +15.4427 q^{58} +2.55836 q^{59} +15.3570 q^{61} +8.28373 q^{62} +1.62259 q^{64} +1.07328 q^{65} +9.09612 q^{67} +8.31403 q^{68} -0.512254 q^{70} +11.6560 q^{71} +14.9224 q^{73} -2.30166 q^{74} -20.1288 q^{76} +4.48316 q^{77} -6.96653 q^{79} -2.13645 q^{80} -13.4261 q^{82} -9.66159 q^{83} -0.501894 q^{85} +19.3521 q^{86} +40.7649 q^{88} +1.89411 q^{89} +2.81840 q^{91} -16.9255 q^{92} +6.16805 q^{94} +1.21512 q^{95} +11.2780 q^{97} +16.5971 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−2.56319	−1.81245	−0.906223	−	0.422800i	$-0.861047\pi$
$2$	−2.56319	−1.81245	−0.906223	+	0.422800i	$0.861047\pi$
$3$	0	0
$3$	0	0
$4$	4.56992	2.28496
$5$	−0.275873	−0.123374	−0.0616870	−	0.998096i	$-0.519648\pi$
$5$	−0.275873	−0.123374	−0.0616870	+	0.998096i	$0.519648\pi$
$6$	0	0
$7$	−0.724431	−0.273809	−0.136905	−	0.990584i	$-0.543715\pi$
$7$	−0.724431	−0.273809	−0.136905	+	0.990584i	$0.543715\pi$
$8$	−6.58718	−2.32892
$9$	0	0
$10$	0.707113	0.223609
$11$	−6.18852	−1.86591	−0.932955	−	0.359994i	$-0.882779\pi$
$11$	−6.18852	−1.86591	−0.932955	+	0.359994i	$0.882779\pi$
$12$	0	0
$13$	−3.89050	−1.07903	−0.539516	−	0.841975i	$-0.681393\pi$
$13$	−3.89050	−1.07903	−0.539516	+	0.841975i	$0.681393\pi$
$14$	1.85685	0.496264
$15$	0	0
$16$	7.74432	1.93608
$17$	1.81930	0.441244	0.220622	−	0.975359i	$-0.429191\pi$
$17$	1.81930	0.441244	0.220622	+	0.975359i	$0.429191\pi$
$18$	0	0
$19$	−4.40463	−1.01049	−0.505246	−	0.862975i	$-0.668598\pi$
$19$	−4.40463	−1.01049	−0.505246	+	0.862975i	$0.668598\pi$
$20$	−1.26072	−0.281905
$21$	0	0
$22$	15.8623	3.38186
$23$	−3.70367	−0.772269	−0.386134	−	0.922443i	$-0.626190\pi$
$23$	−3.70367	−0.772269	−0.386134	+	0.922443i	$0.626190\pi$
$24$	0	0
$25$	−4.92389	−0.984779
$26$	9.97208	1.95569
$27$	0	0
$28$	−3.31059	−0.625643
$29$	−6.02479	−1.11878	−0.559388	−	0.828906i	$-0.688964\pi$
$29$	−6.02479	−1.11878	−0.559388	+	0.828906i	$0.688964\pi$
$30$	0	0
$31$	−3.23181	−0.580451	−0.290225	−	0.956958i	$-0.593730\pi$
$31$	−3.23181	−0.580451	−0.290225	+	0.956958i	$0.593730\pi$
$32$	−6.67577	−1.18012
$33$	0	0
$34$	−4.66319	−0.799731
$35$	0.199851	0.0337809
$36$	0	0
$37$	0.897967	0.147625	0.0738125	−	0.997272i	$-0.476483\pi$
$37$	0.897967	0.147625	0.0738125	+	0.997272i	$0.476483\pi$
$38$	11.2899	1.83146
$39$	0	0
$40$	1.81722	0.287328
$41$	5.23804	0.818044	0.409022	−	0.912525i	$-0.365870\pi$
$41$	5.23804	0.818044	0.409022	+	0.912525i	$0.365870\pi$
$42$	0	0
$43$	−7.55001	−1.15137	−0.575683	−	0.817673i	$-0.695264\pi$
$43$	−7.55001	−1.15137	−0.575683	+	0.817673i	$0.695264\pi$
$44$	−28.2810	−4.26353
$45$	0	0
$46$	9.49319	1.39969
$47$	−2.40640	−0.351010	−0.175505	−	0.984479i	$-0.556156\pi$
$47$	−2.40640	−0.351010	−0.175505	+	0.984479i	$0.556156\pi$
$48$	0	0
$49$	−6.47520	−0.925029
$50$	12.6209	1.78486
$51$	0	0
$52$	−17.7793	−2.46554
$53$	−6.26043	−0.859937	−0.429968	−	0.902844i	$-0.641475\pi$
$53$	−6.26043	−0.859937	−0.429968	+	0.902844i	$0.641475\pi$
$54$	0	0
$55$	1.70724	0.230205
$56$	4.77196	0.637679
$57$	0	0
$58$	15.4427	2.02772
$59$	2.55836	0.333070	0.166535	−	0.986036i	$-0.446742\pi$
$59$	2.55836	0.333070	0.166535	+	0.986036i	$0.446742\pi$
$60$	0	0
$61$	15.3570	1.96627	0.983134	−	0.182889i	$-0.0585448\pi$
$61$	15.3570	1.96627	0.983134	+	0.182889i	$0.0585448\pi$
$62$	8.28373	1.05204
$63$	0	0
$64$	1.62259	0.202824
$65$	1.07328	0.133124
$66$	0	0
$67$	9.09612	1.11127	0.555634	−	0.831427i	$-0.312476\pi$
$67$	9.09612	1.11127	0.555634	+	0.831427i	$0.312476\pi$
$68$	8.31403	1.00822
$69$	0	0
$70$	−0.512254	−0.0612261
$71$	11.6560	1.38331	0.691657	−	0.722226i	$-0.256881\pi$
$71$	11.6560	1.38331	0.691657	+	0.722226i	$0.256881\pi$
$72$	0	0
$73$	14.9224	1.74654	0.873270	−	0.487236i	$-0.161995\pi$
$73$	14.9224	1.74654	0.873270	+	0.487236i	$0.161995\pi$
$74$	−2.30166	−0.267562
$75$	0	0
$76$	−20.1288	−2.30893
$77$	4.48316	0.510903
$78$	0	0
$79$	−6.96653	−0.783796	−0.391898	−	0.920009i	$-0.628181\pi$
$79$	−6.96653	−0.783796	−0.391898	+	0.920009i	$0.628181\pi$
$80$	−2.13645	−0.238862
$81$	0	0
$82$	−13.4261	−1.48266
$83$	−9.66159	−1.06050	−0.530249	−	0.847842i	$-0.677901\pi$
$83$	−9.66159	−1.06050	−0.530249	+	0.847842i	$0.677901\pi$
$84$	0	0
$85$	−0.501894	−0.0544380
$86$	19.3521	2.08679
$87$	0	0
$88$	40.7649	4.34555
$89$	1.89411	0.200775	0.100388	−	0.994948i	$-0.467992\pi$
$89$	1.89411	0.200775	0.100388	+	0.994948i	$0.467992\pi$
$90$	0	0
$91$	2.81840	0.295449
$92$	−16.9255	−1.76460
$93$	0	0
$94$	6.16805	0.636186
$95$	1.21512	0.124668
$96$	0	0
$97$	11.2780	1.14511	0.572554	−	0.819867i	$-0.305953\pi$
$97$	11.2780	1.14511	0.572554	+	0.819867i	$0.305953\pi$
$98$	16.5971	1.67656
$99$	0	0
$100$	−22.5018	−2.25018
$101$	7.40701	0.737025	0.368512	−	0.929623i	$-0.379867\pi$
$101$	7.40701	0.737025	0.368512	+	0.929623i	$0.379867\pi$
$102$	0	0
$103$	−9.70228	−0.955994	−0.477997	−	0.878361i	$-0.658637\pi$
$103$	−9.70228	−0.955994	−0.477997	+	0.878361i	$0.658637\pi$
$104$	25.6274	2.51298
$105$	0	0
$106$	16.0467	1.55859
$107$	12.0662	1.16648	0.583240	−	0.812300i	$-0.301785\pi$
$107$	12.0662	1.16648	0.583240	+	0.812300i	$0.301785\pi$
$108$	0	0
$109$	−6.44479	−0.617299	−0.308649	−	0.951176i	$-0.599877\pi$
$109$	−6.44479	−0.617299	−0.308649	+	0.951176i	$0.599877\pi$
$110$	−4.37598	−0.417233
$111$	0	0
$112$	−5.61022	−0.530116
$113$	−3.04951	−0.286874	−0.143437	−	0.989659i	$-0.545815\pi$
$113$	−3.04951	−0.286874	−0.143437	+	0.989659i	$0.545815\pi$
$114$	0	0
$115$	1.02174	0.0952779
$116$	−27.5328	−2.55636
$117$	0	0
$118$	−6.55754	−0.603671
$119$	−1.31795	−0.120817
$120$	0	0
$121$	27.2978	2.48162
$122$	−39.3629	−3.56375
$123$	0	0
$124$	−14.7691	−1.32631
$125$	2.73773	0.244870
$126$	0	0
$127$	12.7664	1.13284	0.566419	−	0.824117i	$-0.308328\pi$
$127$	12.7664	1.13284	0.566419	+	0.824117i	$0.308328\pi$
$128$	9.19253	0.812513
$129$	0	0
$130$	−2.75102	−0.241281
$131$	−12.9828	−1.13432	−0.567158	−	0.823609i	$-0.691957\pi$
$131$	−12.9828	−1.13432	−0.567158	+	0.823609i	$0.691957\pi$
$132$	0	0
$133$	3.19085	0.276682
$134$	−23.3151	−2.01411
$135$	0	0
$136$	−11.9840	−1.02762
$137$	−21.9737	−1.87734	−0.938668	−	0.344821i	$-0.887939\pi$
$137$	−21.9737	−1.87734	−0.938668	+	0.344821i	$0.887939\pi$
$138$	0	0
$139$	0.926515	0.0785860	0.0392930	−	0.999228i	$-0.487489\pi$
$139$	0.926515	0.0785860	0.0392930	+	0.999228i	$0.487489\pi$
$140$	0.913302	0.0771881
$141$	0	0
$142$	−29.8765	−2.50718
$143$	24.0765	2.01337
$144$	0	0
$145$	1.66207	0.138028
$146$	−38.2490	−3.16551
$147$	0	0
$148$	4.10364	0.337317
$149$	−1.64679	−0.134910	−0.0674552	−	0.997722i	$-0.521488\pi$
$149$	−1.64679	−0.134910	−0.0674552	+	0.997722i	$0.521488\pi$
$150$	0	0
$151$	−2.08624	−0.169776	−0.0848879	−	0.996391i	$-0.527053\pi$
$151$	−2.08624	−0.169776	−0.0848879	+	0.996391i	$0.527053\pi$
$152$	29.0141	2.35335
$153$	0	0
$154$	−11.4912	−0.925984
$155$	0.891569	0.0716125
$156$	0	0
$157$	9.43640	0.753107	0.376553	−	0.926395i	$-0.377109\pi$
$157$	9.43640	0.753107	0.376553	+	0.926395i	$0.377109\pi$
$158$	17.8565	1.42059
$159$	0	0
$160$	1.84166	0.145596
$161$	2.68305	0.211454
$162$	0	0
$163$	11.4108	0.893762	0.446881	−	0.894593i	$-0.352535\pi$
$163$	11.4108	0.893762	0.446881	+	0.894593i	$0.352535\pi$
$164$	23.9374	1.86920
$165$	0	0
$166$	24.7644	1.92209
$167$	14.3143	1.10767	0.553835	−	0.832626i	$-0.313164\pi$
$167$	14.3143	1.10767	0.553835	+	0.832626i	$0.313164\pi$
$168$	0	0
$169$	2.13602	0.164309
$170$	1.28645	0.0986660
$171$	0	0
$172$	−34.5029	−2.63082
$173$	−6.67783	−0.507706	−0.253853	−	0.967243i	$-0.581698\pi$
$173$	−6.67783	−0.507706	−0.253853	+	0.967243i	$0.581698\pi$
$174$	0	0
$175$	3.56702	0.269642
$176$	−47.9259	−3.61255
$177$	0	0
$178$	−4.85496	−0.363895
$179$	1.34485	0.100519	0.0502594	−	0.998736i	$-0.483995\pi$
$179$	1.34485	0.100519	0.0502594	+	0.998736i	$0.483995\pi$
$180$	0	0
$181$	−14.5795	−1.08369	−0.541844	−	0.840479i	$-0.682274\pi$
$181$	−14.5795	−1.08369	−0.541844	+	0.840479i	$0.682274\pi$
$182$	−7.22409	−0.535485
$183$	0	0
$184$	24.3967	1.79855
$185$	−0.247725	−0.0182131
$186$	0	0
$187$	−11.2588	−0.823321
$188$	−10.9971	−0.802043
$189$	0	0
$190$	−3.11457	−0.225955
$191$	−21.4002	−1.54846	−0.774231	−	0.632903i	$-0.781863\pi$
$191$	−21.4002	−1.54846	−0.774231	+	0.632903i	$0.781863\pi$
$192$	0	0
$193$	−2.75873	−0.198578	−0.0992889	−	0.995059i	$-0.531657\pi$
$193$	−2.75873	−0.198578	−0.0992889	+	0.995059i	$0.531657\pi$
$194$	−28.9076	−2.07544
$195$	0	0
$196$	−29.5911	−2.11365
$197$	−3.15359	−0.224684	−0.112342	−	0.993670i	$-0.535835\pi$
$197$	−3.15359	−0.224684	−0.112342	+	0.993670i	$0.535835\pi$
$198$	0	0
$199$	−22.4100	−1.58861	−0.794303	−	0.607522i	$-0.792164\pi$
$199$	−22.4100	−1.58861	−0.794303	+	0.607522i	$0.792164\pi$
$200$	32.4346	2.29347
$201$	0	0
$202$	−18.9855	−1.33582
$203$	4.36455	0.306331
$204$	0	0
$205$	−1.44503	−0.100925
$206$	24.8687	1.73269
$207$	0	0
$208$	−30.1293	−2.08909
$209$	27.2581	1.88549
$210$	0	0
$211$	7.09180	0.488220	0.244110	−	0.969748i	$-0.421504\pi$
$211$	7.09180	0.488220	0.244110	+	0.969748i	$0.421504\pi$
$212$	−28.6097	−1.96492
$213$	0	0
$214$	−30.9278	−2.11418
$215$	2.08284	0.142049
$216$	0	0
$217$	2.34123	0.158933
$218$	16.5192	1.11882
$219$	0	0
$220$	7.80196	0.526008
$221$	−7.07798	−0.476116
$222$	0	0
$223$	−4.10459	−0.274864	−0.137432	−	0.990511i	$-0.543885\pi$
$223$	−4.10459	−0.274864	−0.137432	+	0.990511i	$0.543885\pi$
$224$	4.83613	0.323128
$225$	0	0
$226$	7.81645	0.519943
$227$	−15.7449	−1.04502	−0.522512	−	0.852632i	$-0.675005\pi$
$227$	−15.7449	−1.04502	−0.522512	+	0.852632i	$0.675005\pi$
$228$	0	0
$229$	−18.7647	−1.24000	−0.620002	−	0.784600i	$-0.712868\pi$
$229$	−18.7647	−1.24000	−0.620002	+	0.784600i	$0.712868\pi$
$230$	−2.61891	−0.172686
$231$	0	0
$232$	39.6864	2.60554
$233$	1.93072	0.126486	0.0632430	−	0.997998i	$-0.479856\pi$
$233$	1.93072	0.126486	0.0632430	+	0.997998i	$0.479856\pi$
$234$	0	0
$235$	0.663861	0.0433055
$236$	11.6915	0.761051
$237$	0	0
$238$	3.37816	0.218974
$239$	3.04733	0.197115	0.0985577	−	0.995131i	$-0.468577\pi$
$239$	3.04733	0.197115	0.0985577	+	0.995131i	$0.468577\pi$
$240$	0	0
$241$	6.43356	0.414422	0.207211	−	0.978296i	$-0.433561\pi$
$241$	6.43356	0.414422	0.207211	+	0.978296i	$0.433561\pi$
$242$	−69.9693	−4.49780
$243$	0	0
$244$	70.1804	4.49284
$245$	1.78633	0.114124
$246$	0	0
$247$	17.1362	1.09035
$248$	21.2885	1.35182
$249$	0	0
$250$	−7.01731	−0.443814
$251$	−12.1288	−0.765563	−0.382782	−	0.923839i	$-0.625034\pi$
$251$	−12.1288	−0.765563	−0.382782	+	0.923839i	$0.625034\pi$
$252$	0	0
$253$	22.9202	1.44098
$254$	−32.7227	−2.05321
$255$	0	0
$256$	−26.8073	−1.67546
$257$	22.7143	1.41688	0.708440	−	0.705772i	$-0.249399\pi$
$257$	22.7143	1.41688	0.708440	+	0.705772i	$0.249399\pi$
$258$	0	0
$259$	−0.650515	−0.0404211
$260$	4.90482	0.304184
$261$	0	0
$262$	33.2774	2.05589
$263$	−7.06731	−0.435789	−0.217895	−	0.975972i	$-0.569919\pi$
$263$	−7.06731	−0.435789	−0.217895	+	0.975972i	$0.569919\pi$
$264$	0	0
$265$	1.72708	0.106094
$266$	−8.17874	−0.501471
$267$	0	0
$268$	41.5685	2.53920
$269$	−0.0762491	−0.00464899	−0.00232449	−	0.999997i	$-0.500740\pi$
$269$	−0.0762491	−0.00464899	−0.00232449	+	0.999997i	$0.500740\pi$
$270$	0	0
$271$	−5.45496	−0.331365	−0.165683	−	0.986179i	$-0.552983\pi$
$271$	−5.45496	−0.331365	−0.165683	+	0.986179i	$0.552983\pi$
$272$	14.0892	0.854284
$273$	0	0
$274$	56.3226	3.40257
$275$	30.4716	1.83751
$276$	0	0
$277$	21.9674	1.31990	0.659948	−	0.751312i	$-0.270578\pi$
$277$	21.9674	1.31990	0.659948	+	0.751312i	$0.270578\pi$
$278$	−2.37483	−0.142433
$279$	0	0
$280$	−1.31645	−0.0786731
$281$	−9.43640	−0.562928	−0.281464	−	0.959572i	$-0.590820\pi$
$281$	−9.43640	−0.562928	−0.281464	+	0.959572i	$0.590820\pi$
$282$	0	0
$283$	−10.2835	−0.611289	−0.305644	−	0.952146i	$-0.598872\pi$
$283$	−10.2835	−0.611289	−0.305644	+	0.952146i	$0.598872\pi$
$284$	53.2671	3.16082
$285$	0	0
$286$	−61.7124	−3.64913
$287$	−3.79460	−0.223988
$288$	0	0
$289$	−13.6902	−0.805304
$290$	−4.26021	−0.250168
$291$	0	0
$292$	68.1944	3.99077
$293$	26.1438	1.52734	0.763668	−	0.645609i	$-0.223396\pi$
$293$	26.1438	1.52734	0.763668	+	0.645609i	$0.223396\pi$
$294$	0	0
$295$	−0.705781	−0.0410921
$296$	−5.91507	−0.343806
$297$	0	0
$298$	4.22103	0.244518
$299$	14.4091	0.833302
$300$	0	0
$301$	5.46946	0.315254
$302$	5.34742	0.307709
$303$	0	0
$304$	−34.1109	−1.95639
$305$	−4.23659	−0.242586
$306$	0	0
$307$	7.72432	0.440850	0.220425	−	0.975404i	$-0.429256\pi$
$307$	7.72432	0.440850	0.220425	+	0.975404i	$0.429256\pi$
$308$	20.4877	1.16739
$309$	0	0
$310$	−2.28526	−0.129794
$311$	30.3693	1.72209	0.861043	−	0.508532i	$-0.169812\pi$
$311$	30.3693	1.72209	0.861043	+	0.508532i	$0.169812\pi$
$312$	0	0
$313$	17.7502	1.00330	0.501649	−	0.865071i	$-0.332727\pi$
$313$	17.7502	1.00330	0.501649	+	0.865071i	$0.332727\pi$
$314$	−24.1872	−1.36497
$315$	0	0
$316$	−31.8365	−1.79094
$317$	23.3873	1.31356	0.656782	−	0.754081i	$-0.271917\pi$
$317$	23.3873	1.31356	0.656782	+	0.754081i	$0.271917\pi$
$318$	0	0
$319$	37.2845	2.08753
$320$	−0.447628	−0.0250232
$321$	0	0
$322$	−6.87716	−0.383249
$323$	−8.01333	−0.445873
$324$	0	0
$325$	19.1564	1.06261
$326$	−29.2480	−1.61990
$327$	0	0
$328$	−34.5039	−1.90516
$329$	1.74327	0.0961097
$330$	0	0
$331$	−28.4098	−1.56155	−0.780773	−	0.624815i	$-0.785174\pi$
$331$	−28.4098	−1.56155	−0.780773	+	0.624815i	$0.785174\pi$
$332$	−44.1527	−2.42319
$333$	0	0
$334$	−36.6901	−2.00759
$335$	−2.50937	−0.137102
$336$	0	0
$337$	−34.5819	−1.88380	−0.941899	−	0.335897i	$-0.890960\pi$
$337$	−34.5819	−1.88380	−0.941899	+	0.335897i	$0.890960\pi$
$338$	−5.47501	−0.297801
$339$	0	0
$340$	−2.29361	−0.124389
$341$	20.0001	1.08307
$342$	0	0
$343$	9.76185	0.527091
$344$	49.7332	2.68144
$345$	0	0
$346$	17.1165	0.920190
$347$	8.47863	0.455157	0.227578	−	0.973760i	$-0.426919\pi$
$347$	8.47863	0.455157	0.227578	+	0.973760i	$0.426919\pi$
$348$	0	0
$349$	−20.9320	−1.12047	−0.560233	−	0.828335i	$-0.689288\pi$
$349$	−20.9320	−1.12047	−0.560233	+	0.828335i	$0.689288\pi$
$350$	−9.14294	−0.488711
$351$	0	0
$352$	41.3131	2.20200
$353$	27.3164	1.45390	0.726952	−	0.686688i	$-0.240936\pi$
$353$	27.3164	1.45390	0.726952	+	0.686688i	$0.240936\pi$
$354$	0	0
$355$	−3.21558	−0.170665
$356$	8.65594	0.458764
$357$	0	0
$358$	−3.44710	−0.182185
$359$	−10.4982	−0.554075	−0.277038	−	0.960859i	$-0.589353\pi$
$359$	−10.4982	−0.554075	−0.277038	+	0.960859i	$0.589353\pi$
$360$	0	0
$361$	0.400772	0.0210933
$362$	37.3700	1.96413
$363$	0	0
$364$	12.8799	0.675088
$365$	−4.11670	−0.215478
$366$	0	0
$367$	−22.5082	−1.17492	−0.587459	−	0.809254i	$-0.699872\pi$
$367$	−22.5082	−1.17492	−0.587459	+	0.809254i	$0.699872\pi$
$368$	−28.6824	−1.49517
$369$	0	0
$370$	0.634964	0.0330102
$371$	4.53525	0.235459
$372$	0	0
$373$	−17.2736	−0.894393	−0.447196	−	0.894436i	$-0.647577\pi$
$373$	−17.2736	−0.894393	−0.447196	+	0.894436i	$0.647577\pi$
$374$	28.8583	1.49223
$375$	0	0
$376$	15.8514	0.817473
$377$	23.4395	1.20719
$378$	0	0
$379$	−34.8131	−1.78823	−0.894115	−	0.447837i	$-0.852194\pi$
$379$	−34.8131	−1.78823	−0.894115	+	0.447837i	$0.852194\pi$
$380$	5.55299	0.284862
$381$	0	0
$382$	54.8526	2.80650
$383$	−1.00000	−0.0510976
$383$	−1.00000	−0.0510976
$384$	0	0
$385$	−1.23678	−0.0630322
$386$	7.07114	0.359911
$387$	0	0
$388$	51.5395	2.61652
$389$	17.9176	0.908458	0.454229	−	0.890885i	$-0.349915\pi$
$389$	17.9176	0.908458	0.454229	+	0.890885i	$0.349915\pi$
$390$	0	0
$391$	−6.73807	−0.340759
$392$	42.6533	2.15432
$393$	0	0
$394$	8.08324	0.407228
$395$	1.92188	0.0967000
$396$	0	0
$397$	30.6082	1.53618	0.768092	−	0.640340i	$-0.221206\pi$
$397$	30.6082	1.53618	0.768092	+	0.640340i	$0.221206\pi$
$398$	57.4411	2.87926
$399$	0	0
$400$	−38.1322	−1.90661
$401$	−1.98264	−0.0990081	−0.0495041	−	0.998774i	$-0.515764\pi$
$401$	−1.98264	−0.0990081	−0.0495041	+	0.998774i	$0.515764\pi$
$402$	0	0
$403$	12.5734	0.626324
$404$	33.8494	1.68407
$405$	0	0
$406$	−11.1871	−0.555208
$407$	−5.55709	−0.275455
$408$	0	0
$409$	30.0219	1.48449	0.742244	−	0.670129i	$-0.233761\pi$
$409$	30.0219	1.48449	0.742244	+	0.670129i	$0.233761\pi$
$410$	3.70388	0.182922
$411$	0	0
$412$	−44.3386	−2.18441
$413$	−1.85335	−0.0911975
$414$	0	0
$415$	2.66537	0.130838
$416$	25.9721	1.27339
$417$	0	0
$418$	−69.8677	−3.41734
$419$	−10.9723	−0.536031	−0.268016	−	0.963415i	$-0.586368\pi$
$419$	−10.9723	−0.536031	−0.268016	+	0.963415i	$0.586368\pi$
$420$	0	0
$421$	−9.53417	−0.464667	−0.232334	−	0.972636i	$-0.574636\pi$
$421$	−9.53417	−0.464667	−0.232334	+	0.972636i	$0.574636\pi$
$422$	−18.1776	−0.884872
$423$	0	0
$424$	41.2386	2.00272
$425$	−8.95802	−0.434528
$426$	0	0
$427$	−11.1251	−0.538382
$428$	55.1413	2.66536
$429$	0	0
$430$	−5.33871	−0.257455
$431$	14.8405	0.714843	0.357422	−	0.933943i	$-0.383656\pi$
$431$	14.8405	0.714843	0.357422	+	0.933943i	$0.383656\pi$
$432$	0	0
$433$	35.5848	1.71010	0.855048	−	0.518550i	$-0.173528\pi$
$433$	35.5848	1.71010	0.855048	+	0.518550i	$0.173528\pi$
$434$	−6.00099	−0.288057
$435$	0	0
$436$	−29.4521	−1.41050
$437$	16.3133	0.780371
$438$	0	0
$439$	35.8309	1.71012	0.855058	−	0.518532i	$-0.173521\pi$
$439$	35.8309	1.71012	0.855058	+	0.518532i	$0.173521\pi$
$440$	−11.2459	−0.536128
$441$	0	0
$442$	18.1422	0.862935
$443$	0.764802	0.0363369	0.0181684	−	0.999835i	$-0.494216\pi$
$443$	0.764802	0.0363369	0.0181684	+	0.999835i	$0.494216\pi$
$444$	0	0
$445$	−0.522534	−0.0247705
$446$	10.5208	0.498175
$447$	0	0
$448$	−1.17545	−0.0555350
$449$	−9.02876	−0.426094	−0.213047	−	0.977042i	$-0.568339\pi$
$449$	−9.02876	−0.426094	−0.213047	+	0.977042i	$0.568339\pi$
$450$	0	0
$451$	−32.4157	−1.52640
$452$	−13.9360	−0.655494
$453$	0	0
$454$	40.3570	1.89405
$455$	−0.777520	−0.0364507
$456$	0	0
$457$	20.8558	0.975591	0.487795	−	0.872958i	$-0.337801\pi$
$457$	20.8558	0.975591	0.487795	+	0.872958i	$0.337801\pi$
$458$	48.0973	2.24744
$459$	0	0
$460$	4.66927	0.217706
$461$	−27.8874	−1.29885	−0.649424	−	0.760427i	$-0.724990\pi$
$461$	−27.8874	−1.29885	−0.649424	+	0.760427i	$0.724990\pi$
$462$	0	0
$463$	−15.1892	−0.705901	−0.352950	−	0.935642i	$-0.614822\pi$
$463$	−15.1892	−0.705901	−0.352950	+	0.935642i	$0.614822\pi$
$464$	−46.6579	−2.16604
$465$	0	0
$466$	−4.94880	−0.229249
$467$	26.1563	1.21037	0.605184	−	0.796086i	$-0.293100\pi$
$467$	26.1563	1.21037	0.605184	+	0.796086i	$0.293100\pi$
$468$	0	0
$469$	−6.58952	−0.304276
$470$	−1.70160	−0.0784888
$471$	0	0
$472$	−16.8523	−0.775692
$473$	46.7234	2.14834
$474$	0	0
$475$	21.6879	0.995111
$476$	−6.02294	−0.276061
$477$	0	0
$478$	−7.81087	−0.357261
$479$	38.6333	1.76520	0.882599	−	0.470126i	$-0.155791\pi$
$479$	38.6333	1.76520	0.882599	+	0.470126i	$0.155791\pi$
$480$	0	0
$481$	−3.49354	−0.159292
$482$	−16.4904	−0.751117
$483$	0	0
$484$	124.749	5.67039
$485$	−3.11129	−0.141276
$486$	0	0
$487$	−22.2807	−1.00963	−0.504817	−	0.863226i	$-0.668440\pi$
$487$	−22.2807	−1.00963	−0.504817	+	0.863226i	$0.668440\pi$
$488$	−101.160	−4.57928
$489$	0	0
$490$	−4.57870	−0.206844
$491$	4.56067	0.205820	0.102910	−	0.994691i	$-0.467185\pi$
$491$	4.56067	0.205820	0.102910	+	0.994691i	$0.467185\pi$
$492$	0	0
$493$	−10.9609	−0.493653
$494$	−43.9233	−1.97620
$495$	0	0
$496$	−25.0282	−1.12380
$497$	−8.44398	−0.378764
$498$	0	0
$499$	39.3596	1.76198	0.880989	−	0.473137i	$-0.156879\pi$
$499$	39.3596	1.76198	0.880989	+	0.473137i	$0.156879\pi$
$500$	12.5112	0.559518
$501$	0	0
$502$	31.0884	1.38754
$503$	20.5894	0.918037	0.459018	−	0.888427i	$-0.348201\pi$
$503$	20.5894	0.918037	0.459018	+	0.888427i	$0.348201\pi$
$504$	0	0
$505$	−2.04339	−0.0909297
$506$	−58.7488	−2.61170
$507$	0	0
$508$	58.3416	2.58849
$509$	−15.4099	−0.683033	−0.341517	−	0.939876i	$-0.610941\pi$
$509$	−15.4099	−0.683033	−0.341517	+	0.939876i	$0.610941\pi$
$510$	0	0
$511$	−10.8103	−0.478219
$512$	50.3271	2.22417
$513$	0	0
$514$	−58.2210	−2.56802
$515$	2.67659	0.117945
$516$	0	0
$517$	14.8921	0.654953
$518$	1.66739	0.0732610
$519$	0	0
$520$	−7.06991	−0.310036
$521$	−44.5290	−1.95085	−0.975425	−	0.220333i	$-0.929286\pi$
$521$	−44.5290	−1.95085	−0.975425	+	0.220333i	$0.929286\pi$
$522$	0	0
$523$	14.8949	0.651308	0.325654	−	0.945489i	$-0.394415\pi$
$523$	14.8949	0.651308	0.325654	+	0.945489i	$0.394415\pi$
$524$	−59.3306	−2.59187
$525$	0	0
$526$	18.1148	0.789844
$527$	−5.87962	−0.256120
$528$	0	0
$529$	−9.28283	−0.403601
$530$	−4.42683	−0.192289
$531$	0	0
$532$	14.5819	0.632207
$533$	−20.3786	−0.882695
$534$	0	0
$535$	−3.32872	−0.143913
$536$	−59.9178	−2.58805
$537$	0	0
$538$	0.195440	0.00842604
$539$	40.0719	1.72602
$540$	0	0
$541$	18.5092	0.795775	0.397887	−	0.917434i	$-0.369743\pi$
$541$	18.5092	0.795775	0.397887	+	0.917434i	$0.369743\pi$
$542$	13.9821	0.600582
$543$	0	0
$544$	−12.1452	−0.520721
$545$	1.77794	0.0761586
$546$	0	0
$547$	−17.7546	−0.759132	−0.379566	−	0.925165i	$-0.623927\pi$
$547$	−17.7546	−0.759132	−0.379566	+	0.925165i	$0.623927\pi$
$548$	−100.418	−4.28964
$549$	0	0
$550$	−78.1044	−3.33038
$551$	26.5370	1.13051
$552$	0	0
$553$	5.04677	0.214611
$554$	−56.3066	−2.39224
$555$	0	0
$556$	4.23410	0.179566
$557$	13.2531	0.561551	0.280776	−	0.959773i	$-0.409408\pi$
$557$	13.2531	0.561551	0.280776	+	0.959773i	$0.409408\pi$
$558$	0	0
$559$	29.3733	1.24236
$560$	1.54771	0.0654026
$561$	0	0
$562$	24.1872	1.02028
$563$	1.13349	0.0477710	0.0238855	−	0.999715i	$-0.492396\pi$
$563$	1.13349	0.0477710	0.0238855	+	0.999715i	$0.492396\pi$
$564$	0	0
$565$	0.841276	0.0353927
$566$	26.3584	1.10793
$567$	0	0
$568$	−76.7803	−3.22163
$569$	−29.6200	−1.24173	−0.620867	−	0.783916i	$-0.713219\pi$
$569$	−29.6200	−1.24173	−0.620867	+	0.783916i	$0.713219\pi$
$570$	0	0
$571$	−26.4317	−1.10613	−0.553065	−	0.833138i	$-0.686542\pi$
$571$	−26.4317	−1.10613	−0.553065	+	0.833138i	$0.686542\pi$
$572$	110.027	4.60048
$573$	0	0
$574$	9.72625	0.405966
$575$	18.2365	0.760514
$576$	0	0
$577$	31.8889	1.32755	0.663776	−	0.747931i	$-0.268953\pi$
$577$	31.8889	1.32755	0.663776	+	0.747931i	$0.268953\pi$
$578$	35.0904	1.45957
$579$	0	0
$580$	7.59555	0.315388
$581$	6.99916	0.290374
$582$	0	0
$583$	38.7428	1.60456
$584$	−98.2968	−4.06755
$585$	0	0
$586$	−67.0114	−2.76821
$587$	23.5267	0.971050	0.485525	−	0.874223i	$-0.338629\pi$
$587$	23.5267	0.971050	0.485525	+	0.874223i	$0.338629\pi$
$588$	0	0
$589$	14.2349	0.586540
$590$	1.80905	0.0744773
$591$	0	0
$592$	6.95414	0.285814
$593$	21.3096	0.875078	0.437539	−	0.899199i	$-0.355850\pi$
$593$	21.3096	0.875078	0.437539	+	0.899199i	$0.355850\pi$
$594$	0	0
$595$	0.363588	0.0149056
$596$	−7.52570	−0.308265
$597$	0	0
$598$	−36.9333	−1.51031
$599$	−17.4219	−0.711841	−0.355920	−	0.934516i	$-0.615833\pi$
$599$	−17.4219	−0.711841	−0.355920	+	0.934516i	$0.615833\pi$
$600$	0	0
$601$	35.7937	1.46006	0.730028	−	0.683417i	$-0.239507\pi$
$601$	35.7937	1.46006	0.730028	+	0.683417i	$0.239507\pi$
$602$	−14.0192	−0.571382
$603$	0	0
$604$	−9.53395	−0.387931
$605$	−7.53071	−0.306167
$606$	0	0
$607$	33.2173	1.34825	0.674124	−	0.738618i	$-0.264521\pi$
$607$	33.2173	1.34825	0.674124	+	0.738618i	$0.264521\pi$
$608$	29.4043	1.19250
$609$	0	0
$610$	10.8592	0.439674
$611$	9.36212	0.378751
$612$	0	0
$613$	−38.1375	−1.54036	−0.770180	−	0.637826i	$-0.779834\pi$
$613$	−38.1375	−1.54036	−0.770180	+	0.637826i	$0.779834\pi$
$614$	−19.7989	−0.799017
$615$	0	0
$616$	−29.5313	−1.18985
$617$	−12.9584	−0.521687	−0.260844	−	0.965381i	$-0.584001\pi$
$617$	−12.9584	−0.521687	−0.260844	+	0.965381i	$0.584001\pi$
$618$	0	0
$619$	−18.7069	−0.751892	−0.375946	−	0.926641i	$-0.622682\pi$
$619$	−18.7069	−0.751892	−0.375946	+	0.926641i	$0.622682\pi$
$620$	4.07440	0.163632
$621$	0	0
$622$	−77.8422	−3.12119
$623$	−1.37215	−0.0549742
$624$	0	0
$625$	23.8642	0.954568
$626$	−45.4969	−1.81842
$627$	0	0
$628$	43.1236	1.72082
$629$	1.63367	0.0651386
$630$	0	0
$631$	−38.4480	−1.53059	−0.765296	−	0.643679i	$-0.777407\pi$
$631$	−38.4480	−1.53059	−0.765296	+	0.643679i	$0.777407\pi$
$632$	45.8898	1.82540
$633$	0	0
$634$	−59.9461	−2.38076
$635$	−3.52191	−0.139763
$636$	0	0
$637$	25.1918	0.998135
$638$	−95.5672	−3.78354
$639$	0	0
$640$	−2.53597	−0.100243
$641$	−4.47012	−0.176559	−0.0882795	−	0.996096i	$-0.528137\pi$
$641$	−4.47012	−0.176559	−0.0882795	+	0.996096i	$0.528137\pi$
$642$	0	0
$643$	14.7760	0.582707	0.291354	−	0.956615i	$-0.405894\pi$
$643$	14.7760	0.582707	0.291354	+	0.956615i	$0.405894\pi$
$644$	12.2613	0.483164
$645$	0	0
$646$	20.5396	0.808121
$647$	−40.7337	−1.60141	−0.800703	−	0.599061i	$-0.795541\pi$
$647$	−40.7337	−1.60141	−0.800703	+	0.599061i	$0.795541\pi$
$648$	0	0
$649$	−15.8324	−0.621478
$650$	−49.1015	−1.92592
$651$	0	0
$652$	52.1464	2.04221
$653$	−25.4310	−0.995191	−0.497595	−	0.867409i	$-0.665784\pi$
$653$	−25.4310	−0.995191	−0.497595	+	0.867409i	$0.665784\pi$
$654$	0	0
$655$	3.58161	0.139945
$656$	40.5650	1.58380
$657$	0	0
$658$	−4.46833	−0.174194
$659$	−16.2534	−0.633141	−0.316571	−	0.948569i	$-0.602531\pi$
$659$	−16.2534	−0.633141	−0.316571	+	0.948569i	$0.602531\pi$
$660$	0	0
$661$	1.45508	0.0565959	0.0282980	−	0.999600i	$-0.490991\pi$
$661$	1.45508	0.0565959	0.0282980	+	0.999600i	$0.490991\pi$
$662$	72.8197	2.83022
$663$	0	0
$664$	63.6426	2.46981
$665$	−0.880269	−0.0341354
$666$	0	0
$667$	22.3138	0.863995
$668$	65.4150	2.53098
$669$	0	0
$670$	6.43198	0.248489
$671$	−95.0374	−3.66888
$672$	0	0
$673$	36.4263	1.40413	0.702066	−	0.712112i	$-0.252261\pi$
$673$	36.4263	1.40413	0.702066	+	0.712112i	$0.252261\pi$
$674$	88.6399	3.41428
$675$	0	0
$676$	9.76143	0.375439
$677$	−30.9540	−1.18966	−0.594829	−	0.803852i	$-0.702780\pi$
$677$	−30.9540	−1.18966	−0.594829	+	0.803852i	$0.702780\pi$
$678$	0	0
$679$	−8.17013	−0.313541
$680$	3.30606	0.126782
$681$	0	0
$682$	−51.2641	−1.96300
$683$	−9.50895	−0.363850	−0.181925	−	0.983312i	$-0.558233\pi$
$683$	−9.50895	−0.363850	−0.181925	+	0.983312i	$0.558233\pi$
$684$	0	0
$685$	6.06193	0.231615
$686$	−25.0214	−0.955323
$687$	0	0
$688$	−58.4697	−2.22913
$689$	24.3562	0.927899
$690$	0	0
$691$	−7.73998	−0.294443	−0.147221	−	0.989104i	$-0.547033\pi$
$691$	−7.73998	−0.294443	−0.147221	+	0.989104i	$0.547033\pi$
$692$	−30.5172	−1.16009
$693$	0	0
$694$	−21.7323	−0.824946
$695$	−0.255600	−0.00969546
$696$	0	0
$697$	9.52954	0.360957
$698$	53.6526	2.03078
$699$	0	0
$700$	16.3010	0.616120
$701$	25.3947	0.959146	0.479573	−	0.877502i	$-0.340792\pi$
$701$	25.3947	0.959146	0.479573	+	0.877502i	$0.340792\pi$
$702$	0	0
$703$	−3.95521	−0.149174
$704$	−10.0414	−0.378451
$705$	0	0
$706$	−70.0169	−2.63512
$707$	−5.36587	−0.201804
$708$	0	0
$709$	−9.84224	−0.369633	−0.184816	−	0.982773i	$-0.559169\pi$
$709$	−9.84224	−0.369633	−0.184816	+	0.982773i	$0.559169\pi$
$710$	8.24212	0.309321
$711$	0	0
$712$	−12.4768	−0.467590
$713$	11.9696	0.448264
$714$	0	0
$715$	−6.64204	−0.248398
$716$	6.14585	0.229681
$717$	0	0
$718$	26.9089	1.00423
$719$	−7.06619	−0.263524	−0.131762	−	0.991281i	$-0.542064\pi$
$719$	−7.06619	−0.263524	−0.131762	+	0.991281i	$0.542064\pi$
$720$	0	0
$721$	7.02863	0.261760
$722$	−1.02725	−0.0382304
$723$	0	0
$724$	−66.6273	−2.47618
$725$	29.6654	1.10175
$726$	0	0
$727$	49.6710	1.84220	0.921098	−	0.389331i	$-0.127294\pi$
$727$	49.6710	1.84220	0.921098	+	0.389331i	$0.127294\pi$
$728$	−18.5653	−0.688076
$729$	0	0
$730$	10.5519	0.390542
$731$	−13.7357	−0.508033
$732$	0	0
$733$	−31.9212	−1.17904	−0.589519	−	0.807755i	$-0.700683\pi$
$733$	−31.9212	−1.17904	−0.589519	+	0.807755i	$0.700683\pi$
$734$	57.6926	2.12947
$735$	0	0
$736$	24.7248	0.911369
$737$	−56.2916	−2.07353
$738$	0	0
$739$	−21.1540	−0.778161	−0.389081	−	0.921204i	$-0.627207\pi$
$739$	−21.1540	−0.778161	−0.389081	+	0.921204i	$0.627207\pi$
$740$	−1.13208	−0.0416161
$741$	0	0
$742$	−11.6247	−0.426756
$743$	16.7999	0.616328	0.308164	−	0.951333i	$-0.400285\pi$
$743$	16.7999	0.616328	0.308164	+	0.951333i	$0.400285\pi$
$744$	0	0
$745$	0.454305	0.0166444
$746$	44.2754	1.62104
$747$	0	0
$748$	−51.4516	−1.88126
$749$	−8.74110	−0.319393
$750$	0	0
$751$	41.0779	1.49895	0.749477	−	0.662030i	$-0.230305\pi$
$751$	41.0779	1.49895	0.749477	+	0.662030i	$0.230305\pi$
$752$	−18.6359	−0.679583
$753$	0	0
$754$	−60.0797	−2.18797
$755$	0.575537	0.0209459
$756$	0	0
$757$	−22.9744	−0.835019	−0.417509	−	0.908673i	$-0.637097\pi$
$757$	−22.9744	−0.835019	−0.417509	+	0.908673i	$0.637097\pi$
$758$	89.2325	3.24107
$759$	0	0
$760$	−8.00419	−0.290343
$761$	29.1691	1.05738	0.528690	−	0.848815i	$-0.322683\pi$
$761$	29.1691	1.05738	0.528690	+	0.848815i	$0.322683\pi$
$762$	0	0
$763$	4.66880	0.169022
$764$	−97.7970	−3.53817
$765$	0	0
$766$	2.56319	0.0926116
$767$	−9.95330	−0.359393
$768$	0	0
$769$	41.5608	1.49872	0.749361	−	0.662162i	$-0.230361\pi$
$769$	41.5608	1.49872	0.749361	+	0.662162i	$0.230361\pi$
$770$	3.17010	0.114242
$771$	0	0
$772$	−12.6072	−0.453742
$773$	−23.7462	−0.854091	−0.427046	−	0.904230i	$-0.640446\pi$
$773$	−23.7462	−0.854091	−0.427046	+	0.904230i	$0.640446\pi$
$774$	0	0
$775$	15.9131	0.571615
$776$	−74.2902	−2.66686
$777$	0	0
$778$	−45.9261	−1.64653
$779$	−23.0716	−0.826627
$780$	0	0
$781$	−72.1335	−2.58114
$782$	17.2709	0.617607
$783$	0	0
$784$	−50.1460	−1.79093
$785$	−2.60325	−0.0929138
$786$	0	0
$787$	−41.5434	−1.48086	−0.740431	−	0.672132i	$-0.765379\pi$
$787$	−41.5434	−1.48086	−0.740431	+	0.672132i	$0.765379\pi$
$788$	−14.4117	−0.513394
$789$	0	0
$790$	−4.92612	−0.175264
$791$	2.20916	0.0785486
$792$	0	0
$793$	−59.7466	−2.12166
$794$	−78.4546	−2.78425
$795$	0	0
$796$	−102.412	−3.62990
$797$	−10.1496	−0.359518	−0.179759	−	0.983711i	$-0.557532\pi$
$797$	−10.1496	−0.359518	−0.179759	+	0.983711i	$0.557532\pi$
$798$	0	0
$799$	−4.37796	−0.154881
$800$	32.8708	1.16216
$801$	0	0
$802$	5.08186	0.179447
$803$	−92.3479	−3.25889
$804$	0	0
$805$	−0.740181	−0.0260880
$806$	−32.2279	−1.13518
$807$	0	0
$808$	−48.7913	−1.71647
$809$	−53.7807	−1.89083	−0.945414	−	0.325872i	$-0.894342\pi$
$809$	−53.7807	−1.89083	−0.945414	+	0.325872i	$0.894342\pi$
$810$	0	0
$811$	17.0106	0.597323	0.298661	−	0.954359i	$-0.403460\pi$
$811$	17.0106	0.597323	0.298661	+	0.954359i	$0.403460\pi$
$812$	19.9456	0.699954
$813$	0	0
$814$	14.2438	0.499247
$815$	−3.14793	−0.110267
$816$	0	0
$817$	33.2550	1.16345
$818$	−76.9517	−2.69055
$819$	0	0
$820$	−6.60367	−0.230610
$821$	17.4774	0.609967	0.304983	−	0.952358i	$-0.401349\pi$
$821$	17.4774	0.609967	0.304983	+	0.952358i	$0.401349\pi$
$822$	0	0
$823$	−1.53419	−0.0534783	−0.0267392	−	0.999642i	$-0.508512\pi$
$823$	−1.53419	−0.0534783	−0.0267392	+	0.999642i	$0.508512\pi$
$824$	63.9106	2.22643
$825$	0	0
$826$	4.75049	0.165291
$827$	−53.8676	−1.87316	−0.936579	−	0.350455i	$-0.886027\pi$
$827$	−53.8676	−1.87316	−0.936579	+	0.350455i	$0.886027\pi$
$828$	0	0
$829$	−41.7587	−1.45034	−0.725170	−	0.688570i	$-0.758239\pi$
$829$	−41.7587	−1.45034	−0.725170	+	0.688570i	$0.758239\pi$
$830$	−6.83183	−0.237136
$831$	0	0
$832$	−6.31269	−0.218853
$833$	−11.7803	−0.408163
$834$	0	0
$835$	−3.94891	−0.136658
$836$	124.568	4.30826
$837$	0	0
$838$	28.1240	0.971527
$839$	20.3586	0.702857	0.351429	−	0.936215i	$-0.385696\pi$
$839$	20.3586	0.702857	0.351429	+	0.936215i	$0.385696\pi$
$840$	0	0
$841$	7.29810	0.251659
$842$	24.4379	0.842184
$843$	0	0
$844$	32.4090	1.11556
$845$	−0.589269	−0.0202715
$846$	0	0
$847$	−19.7754	−0.679490
$848$	−48.4828	−1.66491
$849$	0	0
$850$	22.9611	0.787558
$851$	−3.32577	−0.114006
$852$	0	0
$853$	−0.434982	−0.0148935	−0.00744675	−	0.999972i	$-0.502370\pi$
$853$	−0.434982	−0.0148935	−0.00744675	+	0.999972i	$0.502370\pi$
$854$	28.5157	0.975788
$855$	0	0
$856$	−79.4819	−2.71664
$857$	10.4022	0.355331	0.177666	−	0.984091i	$-0.443145\pi$
$857$	10.4022	0.355331	0.177666	+	0.984091i	$0.443145\pi$
$858$	0	0
$859$	−17.3006	−0.590290	−0.295145	−	0.955452i	$-0.595368\pi$
$859$	−17.3006	−0.590290	−0.295145	+	0.955452i	$0.595368\pi$
$860$	9.51841	0.324575
$861$	0	0
$862$	−38.0390	−1.29561
$863$	34.0045	1.15752	0.578762	−	0.815496i	$-0.303536\pi$
$863$	34.0045	1.15752	0.578762	+	0.815496i	$0.303536\pi$
$864$	0	0
$865$	1.84223	0.0626378
$866$	−91.2104	−3.09945
$867$	0	0
$868$	10.6992	0.363155
$869$	43.1125	1.46249
$870$	0	0
$871$	−35.3885	−1.19909
$872$	42.4529	1.43764
$873$	0	0
$874$	−41.8140	−1.41438
$875$	−1.98330	−0.0670477
$876$	0	0
$877$	6.36395	0.214895	0.107448	−	0.994211i	$-0.465732\pi$
$877$	6.36395	0.214895	0.107448	+	0.994211i	$0.465732\pi$
$878$	−91.8413	−3.09949
$879$	0	0
$880$	13.2214	0.445695
$881$	−24.5237	−0.826226	−0.413113	−	0.910680i	$-0.635559\pi$
$881$	−24.5237	−0.826226	−0.413113	+	0.910680i	$0.635559\pi$
$882$	0	0
$883$	11.6868	0.393294	0.196647	−	0.980474i	$-0.436995\pi$
$883$	11.6868	0.393294	0.196647	+	0.980474i	$0.436995\pi$
$884$	−32.3458	−1.08791
$885$	0	0
$886$	−1.96033	−0.0658586
$887$	−40.5816	−1.36260	−0.681298	−	0.732006i	$-0.738584\pi$
$887$	−40.5816	−1.36260	−0.681298	+	0.732006i	$0.738584\pi$
$888$	0	0
$889$	−9.24840	−0.310181
$890$	1.33935	0.0448951
$891$	0	0
$892$	−18.7576	−0.628052
$893$	10.5993	0.354692
$894$	0	0
$895$	−0.371007	−0.0124014
$896$	−6.65936	−0.222474
$897$	0	0
$898$	23.1424	0.772272
$899$	19.4710	0.649394
$900$	0	0
$901$	−11.3896	−0.379442
$902$	83.0874	2.76651
$903$	0	0
$904$	20.0876	0.668105
$905$	4.02209	0.133699
$906$	0	0
$907$	−43.4327	−1.44216	−0.721079	−	0.692853i	$-0.756354\pi$
$907$	−43.4327	−1.44216	−0.721079	+	0.692853i	$0.756354\pi$
$908$	−71.9528	−2.38784
$909$	0	0
$910$	1.99293	0.0660649
$911$	31.0767	1.02962	0.514809	−	0.857305i	$-0.327863\pi$
$911$	31.0767	1.02962	0.514809	+	0.857305i	$0.327863\pi$
$912$	0	0
$913$	59.7909	1.97879
$914$	−53.4572	−1.76821
$915$	0	0
$916$	−85.7530	−2.83336
$917$	9.40518	0.310586
$918$	0	0
$919$	23.1418	0.763377	0.381688	−	0.924291i	$-0.375343\pi$
$919$	23.1418	0.763377	0.381688	+	0.924291i	$0.375343\pi$
$920$	−6.73039	−0.221894
$921$	0	0
$922$	71.4807	2.35409
$923$	−45.3478	−1.49264
$924$	0	0
$925$	−4.42150	−0.145378
$926$	38.9327	1.27941
$927$	0	0
$928$	40.2201	1.32029
$929$	−5.69579	−0.186873	−0.0934364	−	0.995625i	$-0.529785\pi$
$929$	−5.69579	−0.186873	−0.0934364	+	0.995625i	$0.529785\pi$
$930$	0	0
$931$	28.5209	0.934734
$932$	8.82325	0.289015
$933$	0	0
$934$	−67.0433	−2.19372
$935$	3.10598	0.101576
$936$	0	0
$937$	51.4163	1.67970	0.839848	−	0.542822i	$-0.182644\pi$
$937$	51.4163	1.67970	0.839848	+	0.542822i	$0.182644\pi$
$938$	16.8901	0.551483
$939$	0	0
$940$	3.03379	0.0989513
$941$	29.2397	0.953186	0.476593	−	0.879124i	$-0.341872\pi$
$941$	29.2397	0.953186	0.476593	+	0.879124i	$0.341872\pi$
$942$	0	0
$943$	−19.4000	−0.631750
$944$	19.8127	0.644849
$945$	0	0
$946$	−119.761	−3.89376
$947$	−8.37646	−0.272198	−0.136099	−	0.990695i	$-0.543457\pi$
$947$	−8.37646	−0.272198	−0.136099	+	0.990695i	$0.543457\pi$
$948$	0	0
$949$	−58.0558	−1.88457
$950$	−55.5902	−1.80358
$951$	0	0
$952$	8.68160	0.281372
$953$	24.8856	0.806124	0.403062	−	0.915173i	$-0.367946\pi$
$953$	24.8856	0.806124	0.403062	+	0.915173i	$0.367946\pi$
$954$	0	0
$955$	5.90372	0.191040
$956$	13.9260	0.450401
$957$	0	0
$958$	−99.0242	−3.19933
$959$	15.9184	0.514032
$960$	0	0
$961$	−20.5554	−0.663077
$962$	8.95460	0.288708
$963$	0	0
$964$	29.4008	0.946937
$965$	0.761058	0.0244993
$966$	0	0
$967$	38.4531	1.23657	0.618284	−	0.785955i	$-0.287828\pi$
$967$	38.4531	1.23657	0.618284	+	0.785955i	$0.287828\pi$
$968$	−179.815	−5.77949
$969$	0	0
$970$	7.97482	0.256056
$971$	50.7811	1.62964	0.814822	−	0.579711i	$-0.196834\pi$
$971$	50.7811	1.62964	0.814822	+	0.579711i	$0.196834\pi$
$972$	0	0
$973$	−0.671196	−0.0215176
$974$	57.1095	1.82991
$975$	0	0
$976$	118.930	3.80685
$977$	−40.1821	−1.28554	−0.642770	−	0.766059i	$-0.722215\pi$
$977$	−40.1821	−1.28554	−0.642770	+	0.766059i	$0.722215\pi$
$978$	0	0
$979$	−11.7217	−0.374629
$980$	8.16338	0.260770
$981$	0	0
$982$	−11.6898	−0.373038
$983$	10.3943	0.331526	0.165763	−	0.986166i	$-0.446991\pi$
$983$	10.3943	0.331526	0.165763	+	0.986166i	$0.446991\pi$
$984$	0	0
$985$	0.869990	0.0277202
$986$	28.0948	0.894719
$987$	0	0
$988$	78.3112	2.49141
$989$	27.9627	0.889163
$990$	0	0
$991$	−0.755443	−0.0239974	−0.0119987	−	0.999928i	$-0.503819\pi$
$991$	−0.755443	−0.0239974	−0.0119987	+	0.999928i	$0.503819\pi$
$992$	21.5748	0.685001
$993$	0	0
$994$	21.6435	0.686490
$995$	6.18232	0.195993
$996$	0	0
$997$	−11.6501	−0.368962	−0.184481	−	0.982836i	$-0.559060\pi$
$997$	−11.6501	−0.368962	−0.184481	+	0.982836i	$0.559060\pi$
$998$	−100.886	−3.19349
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3447.2.a.j.1.3		24
3.2	odd	2		383.2.a.c.1.22	✓	24
12.11	even	2		6128.2.a.p.1.14		24
15.14	odd	2		9575.2.a.e.1.3		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
383.2.a.c.1.22	✓	24	3.2	odd	2
3447.2.a.j.1.3		24	1.1	even	1	trivial
6128.2.a.p.1.14		24	12.11	even	2
9575.2.a.e.1.3		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)

\(f(q)\)	\(=\)	\(q-2.56319 q^{2} +4.56992 q^{4} -0.275873 q^{5} -0.724431 q^{7} -6.58718 q^{8} +0.707113 q^{10} -6.18852 q^{11} -3.89050 q^{13} +1.85685 q^{14} +7.74432 q^{16} +1.81930 q^{17} -4.40463 q^{19} -1.26072 q^{20} +15.8623 q^{22} -3.70367 q^{23} -4.92389 q^{25} +9.97208 q^{26} -3.31059 q^{28} -6.02479 q^{29} -3.23181 q^{31} -6.67577 q^{32} -4.66319 q^{34} +0.199851 q^{35} +0.897967 q^{37} +11.2899 q^{38} +1.81722 q^{40} +5.23804 q^{41} -7.55001 q^{43} -28.2810 q^{44} +9.49319 q^{46} -2.40640 q^{47} -6.47520 q^{49} +12.6209 q^{50} -17.7793 q^{52} -6.26043 q^{53} +1.70724 q^{55} +4.77196 q^{56} +15.4427 q^{58} +2.55836 q^{59} +15.3570 q^{61} +8.28373 q^{62} +1.62259 q^{64} +1.07328 q^{65} +9.09612 q^{67} +8.31403 q^{68} -0.512254 q^{70} +11.6560 q^{71} +14.9224 q^{73} -2.30166 q^{74} -20.1288 q^{76} +4.48316 q^{77} -6.96653 q^{79} -2.13645 q^{80} -13.4261 q^{82} -9.66159 q^{83} -0.501894 q^{85} +19.3521 q^{86} +40.7649 q^{88} +1.89411 q^{89} +2.81840 q^{91} -16.9255 q^{92} +6.16805 q^{94} +1.21512 q^{95} +11.2780 q^{97} +16.5971 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 5 q^{2} + 29 q^{4} - 3 q^{5} + 17 q^{7} - 15 q^{8} + q^{10} + 28 q^{13} + 8 q^{14} + 35 q^{16} - 16 q^{17} + 13 q^{19} + 4 q^{20} + 12 q^{22} - 7 q^{23} + 67 q^{25} + 14 q^{26} + 39 q^{28} + 2 q^{29}+ \cdots + 29 q^{98}+O(q^{100}) \) 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

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