LMFDB - Embedded newform 7225.2.a.u.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7225,2,Mod(1,7225)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7225, 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("7225.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$7225 = 5^{2} \cdot 17^{2}$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	7225.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:	$57.6919154604$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{16})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{4} - 4x^{2} + 2$ x^4 - 4*x^2 + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{29}]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 17)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$1.84776$ of defining polynomial
Character	$\chi$	$=$	7225.1

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.414214 q^{2} +2.61313 q^{3} -1.82843 q^{4} +1.08239 q^{6} +1.08239 q^{7} -1.58579 q^{8} +3.82843 q^{9} +1.08239 q^{11} -4.77791 q^{12} +1.41421 q^{13} +0.448342 q^{14} +3.00000 q^{16} +1.58579 q^{18} +4.82843 q^{19} +2.82843 q^{21} +0.448342 q^{22} -4.14386 q^{23} -4.14386 q^{24} +0.585786 q^{26} +2.16478 q^{27} -1.97908 q^{28} -4.46088 q^{29} +3.24718 q^{31} +4.41421 q^{32} +2.82843 q^{33} -7.00000 q^{36} +3.82683 q^{37} +2.00000 q^{38} +3.69552 q^{39} +8.15640 q^{41} +1.17157 q^{42} +4.82843 q^{43} -1.97908 q^{44} -1.71644 q^{46} -10.8284 q^{47} +7.83938 q^{48} -5.82843 q^{49} -2.58579 q^{52} +1.41421 q^{53} +0.896683 q^{54} -1.71644 q^{56} +12.6173 q^{57} -1.84776 q^{58} +6.00000 q^{59} +9.23880 q^{61} +1.34502 q^{62} +4.14386 q^{63} -4.17157 q^{64} +1.17157 q^{66} +6.82843 q^{67} -10.8284 q^{69} +13.0656 q^{71} -6.07107 q^{72} -5.35757 q^{73} +1.58513 q^{74} -8.82843 q^{76} +1.17157 q^{77} +1.53073 q^{78} +4.14386 q^{79} -5.82843 q^{81} +3.37849 q^{82} -0.343146 q^{83} -5.17157 q^{84} +2.00000 q^{86} -11.6569 q^{87} -1.71644 q^{88} +9.41421 q^{89} +1.53073 q^{91} +7.57675 q^{92} +8.48528 q^{93} -4.48528 q^{94} +11.5349 q^{96} -6.43996 q^{97} -2.41421 q^{98} +4.14386 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$4 q - 4 q^{2} + 4 q^{4} - 12 q^{8} + 4 q^{9} + 12 q^{16} + 12 q^{18} + 8 q^{19} + 8 q^{26} + 12 q^{32} - 28 q^{36} + 8 q^{38} + 16 q^{42} + 8 q^{43} - 32 q^{47} - 12 q^{49} - 16 q^{52} + 24 q^{59} - 28 q^{64}+ \cdots - 4 q^{98}+O(q^{100})$ 4 * q - 4 * q^2 + 4 * q^4 - 12 * q^8 + 4 * q^9 + 12 * q^16 + 12 * q^18 + 8 * q^19 + 8 * q^26 + 12 * q^32 - 28 * q^36 + 8 * q^38 + 16 * q^42 + 8 * q^43 - 32 * q^47 - 12 * q^49 - 16 * q^52 + 24 * q^59 - 28 * q^64 + 16 * q^66 + 16 * q^67 - 32 * q^69 + 4 * q^72 - 24 * q^76 + 16 * q^77 - 12 * q^81 - 24 * q^83 - 32 * q^84 + 8 * q^86 - 24 * q^87 + 32 * q^89 + 16 * q^94 - 4 * 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

)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$\alpha_n$			$\theta_n$
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$\alpha_p$			$\theta_p$

$2$	0.414214	0.292893	0.146447	−	0.989219i	$-0.453216\pi$
$2$	0.414214	0.292893	0.146447	+	0.989219i	$0.453216\pi$
$3$	2.61313	1.50869	0.754344	−	0.656479i	$-0.227955\pi$
$3$	2.61313	1.50869	0.754344	+	0.656479i	$0.227955\pi$
$4$	−1.82843	−0.914214
$5$	0	0
$5$	0	0
$6$	1.08239	0.441885
$7$	1.08239	0.409106	0.204553	−	0.978856i	$-0.434426\pi$
$7$	1.08239	0.409106	0.204553	+	0.978856i	$0.434426\pi$
$8$	−1.58579	−0.560660
$9$	3.82843	1.27614
$10$	0	0
$11$	1.08239	0.326354	0.163177	−	0.986597i	$-0.447826\pi$
$11$	1.08239	0.326354	0.163177	+	0.986597i	$0.447826\pi$
$12$	−4.77791	−1.37926
$13$	1.41421	0.392232	0.196116	−	0.980581i	$-0.437167\pi$
$13$	1.41421	0.392232	0.196116	+	0.980581i	$0.437167\pi$
$14$	0.448342	0.119824
$15$	0	0
$16$	3.00000	0.750000
$17$	0	0
$17$	0	0
$18$	1.58579	0.373773
$19$	4.82843	1.10772	0.553859	−	0.832611i	$-0.313155\pi$
$19$	4.82843	1.10772	0.553859	+	0.832611i	$0.313155\pi$
$20$	0	0
$21$	2.82843	0.617213
$22$	0.448342	0.0955867
$23$	−4.14386	−0.864054	−0.432027	−	0.901861i	$-0.642202\pi$
$23$	−4.14386	−0.864054	−0.432027	+	0.901861i	$0.642202\pi$
$24$	−4.14386	−0.845862
$25$	0	0
$26$	0.585786	0.114882
$27$	2.16478	0.416613
$28$	−1.97908	−0.374010
$29$	−4.46088	−0.828366	−0.414183	−	0.910194i	$-0.635933\pi$
$29$	−4.46088	−0.828366	−0.414183	+	0.910194i	$0.635933\pi$
$30$	0	0
$31$	3.24718	0.583210	0.291605	−	0.956539i	$-0.405811\pi$
$31$	3.24718	0.583210	0.291605	+	0.956539i	$0.405811\pi$
$32$	4.41421	0.780330
$33$	2.82843	0.492366
$34$	0	0
$35$	0	0
$36$	−7.00000	−1.16667
$37$	3.82683	0.629128	0.314564	−	0.949236i	$-0.398142\pi$
$37$	3.82683	0.629128	0.314564	+	0.949236i	$0.398142\pi$
$38$	2.00000	0.324443
$39$	3.69552	0.591756
$40$	0	0
$41$	8.15640	1.27382	0.636908	−	0.770940i	$-0.280213\pi$
$41$	8.15640	1.27382	0.636908	+	0.770940i	$0.280213\pi$
$42$	1.17157	0.180778
$43$	4.82843	0.736328	0.368164	−	0.929761i	$-0.379986\pi$
$43$	4.82843	0.736328	0.368164	+	0.929761i	$0.379986\pi$
$44$	−1.97908	−0.298357
$45$	0	0
$46$	−1.71644	−0.253076
$47$	−10.8284	−1.57949	−0.789744	−	0.613436i	$-0.789787\pi$
$47$	−10.8284	−1.57949	−0.789744	+	0.613436i	$0.789787\pi$
$48$	7.83938	1.13152
$49$	−5.82843	−0.832632
$50$	0	0
$51$	0	0
$52$	−2.58579	−0.358584
$53$	1.41421	0.194257	0.0971286	−	0.995272i	$-0.469034\pi$
$53$	1.41421	0.194257	0.0971286	+	0.995272i	$0.469034\pi$
$54$	0.896683	0.122023
$55$	0	0
$56$	−1.71644	−0.229369
$57$	12.6173	1.67120
$58$	−1.84776	−0.242623
$59$	6.00000	0.781133	0.390567	−	0.920575i	$-0.372279\pi$
$59$	6.00000	0.781133	0.390567	+	0.920575i	$0.372279\pi$
$60$	0	0
$61$	9.23880	1.18291	0.591453	−	0.806339i	$-0.298554\pi$
$61$	9.23880	1.18291	0.591453	+	0.806339i	$0.298554\pi$
$62$	1.34502	0.170818
$63$	4.14386	0.522077
$64$	−4.17157	−0.521447
$65$	0	0
$66$	1.17157	0.144211
$67$	6.82843	0.834225	0.417113	−	0.908855i	$-0.363042\pi$
$67$	6.82843	0.834225	0.417113	+	0.908855i	$0.363042\pi$
$68$	0	0
$69$	−10.8284	−1.30359
$70$	0	0
$71$	13.0656	1.55060	0.775302	−	0.631590i	$-0.217597\pi$
$71$	13.0656	1.55060	0.775302	+	0.631590i	$0.217597\pi$
$72$	−6.07107	−0.715482
$73$	−5.35757	−0.627056	−0.313528	−	0.949579i	$-0.601511\pi$
$73$	−5.35757	−0.627056	−0.313528	+	0.949579i	$0.601511\pi$
$74$	1.58513	0.184267
$75$	0	0
$76$	−8.82843	−1.01269
$77$	1.17157	0.133513
$78$	1.53073	0.173321
$79$	4.14386	0.466221	0.233110	−	0.972450i	$-0.425110\pi$
$79$	4.14386	0.466221	0.233110	+	0.972450i	$0.425110\pi$
$80$	0	0
$81$	−5.82843	−0.647603
$82$	3.37849	0.373092
$83$	−0.343146	−0.0376651	−0.0188326	−	0.999823i	$-0.505995\pi$
$83$	−0.343146	−0.0376651	−0.0188326	+	0.999823i	$0.505995\pi$
$84$	−5.17157	−0.564265
$85$	0	0
$86$	2.00000	0.215666
$87$	−11.6569	−1.24975
$88$	−1.71644	−0.182973
$89$	9.41421	0.997905	0.498952	−	0.866629i	$-0.333718\pi$
$89$	9.41421	0.997905	0.498952	+	0.866629i	$0.333718\pi$
$90$	0	0
$91$	1.53073	0.160464
$92$	7.57675	0.789930
$93$	8.48528	0.879883
$94$	−4.48528	−0.462621
$95$	0	0
$96$	11.5349	1.17728
$97$	−6.43996	−0.653879	−0.326939	−	0.945045i	$-0.606017\pi$
$97$	−6.43996	−0.653879	−0.326939	+	0.945045i	$0.606017\pi$
$98$	−2.41421	−0.243872
$99$	4.14386	0.416474
$100$	0	0
$101$	13.4142	1.33476	0.667382	−	0.744715i	$-0.267415\pi$
$101$	13.4142	1.33476	0.667382	+	0.744715i	$0.267415\pi$
$102$	0	0
$103$	4.48528	0.441948	0.220974	−	0.975280i	$-0.429076\pi$
$103$	4.48528	0.441948	0.220974	+	0.975280i	$0.429076\pi$
$104$	−2.24264	−0.219909
$105$	0	0
$106$	0.585786	0.0568966
$107$	−6.30864	−0.609880	−0.304940	−	0.952372i	$-0.598636\pi$
$107$	−6.30864	−0.609880	−0.304940	+	0.952372i	$0.598636\pi$
$108$	−3.95815	−0.380873
$109$	−4.27518	−0.409488	−0.204744	−	0.978816i	$-0.565636\pi$
$109$	−4.27518	−0.409488	−0.204744	+	0.978816i	$0.565636\pi$
$110$	0	0
$111$	10.0000	0.949158
$112$	3.24718	0.306829
$113$	16.1815	1.52223	0.761113	−	0.648619i	$-0.224653\pi$
$113$	16.1815	1.52223	0.761113	+	0.648619i	$0.224653\pi$
$114$	5.22625	0.489483
$115$	0	0
$116$	8.15640	0.757303
$117$	5.41421	0.500544
$118$	2.48528	0.228789
$119$	0	0
$120$	0	0
$121$	−9.82843	−0.893493
$122$	3.82683	0.346465
$123$	21.3137	1.92179
$124$	−5.93723	−0.533179
$125$	0	0
$126$	1.71644	0.152913
$127$	−17.3137	−1.53634	−0.768172	−	0.640244i	$-0.778833\pi$
$127$	−17.3137	−1.53634	−0.768172	+	0.640244i	$0.778833\pi$
$128$	−10.5563	−0.933058
$129$	12.6173	1.11089
$130$	0	0
$131$	0.185709	0.0162255	0.00811274	−	0.999967i	$-0.497418\pi$
$131$	0.185709	0.0162255	0.00811274	+	0.999967i	$0.497418\pi$
$132$	−5.17157	−0.450128
$133$	5.22625	0.453174
$134$	2.82843	0.244339
$135$	0	0
$136$	0	0
$137$	−8.72792	−0.745677	−0.372838	−	0.927896i	$-0.621615\pi$
$137$	−8.72792	−0.745677	−0.372838	+	0.927896i	$0.621615\pi$
$138$	−4.48528	−0.381813
$139$	−14.9678	−1.26955	−0.634775	−	0.772697i	$-0.718907\pi$
$139$	−14.9678	−1.26955	−0.634775	+	0.772697i	$0.718907\pi$
$140$	0	0
$141$	−28.2960	−2.38296
$142$	5.41196	0.454162
$143$	1.53073	0.128006
$144$	11.4853	0.957107
$145$	0	0
$146$	−2.21918	−0.183660
$147$	−15.2304	−1.25618
$148$	−6.99709	−0.575157
$149$	16.9706	1.39028	0.695141	−	0.718873i	$-0.255342\pi$
$149$	16.9706	1.39028	0.695141	+	0.718873i	$0.255342\pi$
$150$	0	0
$151$	12.8284	1.04396	0.521981	−	0.852957i	$-0.325193\pi$
$151$	12.8284	1.04396	0.521981	+	0.852957i	$0.325193\pi$
$152$	−7.65685	−0.621053
$153$	0	0
$154$	0.485281	0.0391051
$155$	0	0
$156$	−6.75699	−0.540992
$157$	−1.65685	−0.132231	−0.0661157	−	0.997812i	$-0.521061\pi$
$157$	−1.65685	−0.132231	−0.0661157	+	0.997812i	$0.521061\pi$
$158$	1.71644	0.136553
$159$	3.69552	0.293074
$160$	0	0
$161$	−4.48528	−0.353490
$162$	−2.41421	−0.189679
$163$	5.67459	0.444468	0.222234	−	0.974993i	$-0.428665\pi$
$163$	5.67459	0.444468	0.222234	+	0.974993i	$0.428665\pi$
$164$	−14.9134	−1.16454
$165$	0	0
$166$	−0.142136	−0.0110319
$167$	10.0042	0.774145	0.387073	−	0.922049i	$-0.373486\pi$
$167$	10.0042	0.774145	0.387073	+	0.922049i	$0.373486\pi$
$168$	−4.48528	−0.346047
$169$	−11.0000	−0.846154
$170$	0	0
$171$	18.4853	1.41360
$172$	−8.82843	−0.673161
$173$	3.37849	0.256862	0.128431	−	0.991718i	$-0.459006\pi$
$173$	3.37849	0.256862	0.128431	+	0.991718i	$0.459006\pi$
$174$	−4.82843	−0.366042
$175$	0	0
$176$	3.24718	0.244765
$177$	15.6788	1.17849
$178$	3.89949	0.292280
$179$	6.00000	0.448461	0.224231	−	0.974536i	$-0.428013\pi$
$179$	6.00000	0.448461	0.224231	+	0.974536i	$0.428013\pi$
$180$	0	0
$181$	12.4860	0.928075	0.464037	−	0.885816i	$-0.346400\pi$
$181$	12.4860	0.928075	0.464037	+	0.885816i	$0.346400\pi$
$182$	0.634051	0.0469990
$183$	24.1421	1.78464
$184$	6.57128	0.484441
$185$	0	0
$186$	3.51472	0.257712
$187$	0	0
$188$	19.7990	1.44399
$189$	2.34315	0.170439
$190$	0	0
$191$	20.0000	1.44715	0.723575	−	0.690246i	$-0.242498\pi$
$191$	20.0000	1.44715	0.723575	+	0.690246i	$0.242498\pi$
$192$	−10.9008	−0.786701
$193$	−5.54328	−0.399014	−0.199507	−	0.979896i	$-0.563934\pi$
$193$	−5.54328	−0.399014	−0.199507	+	0.979896i	$0.563934\pi$
$194$	−2.66752	−0.191517
$195$	0	0
$196$	10.6569	0.761204
$197$	14.9134	1.06253	0.531267	−	0.847204i	$-0.321716\pi$
$197$	14.9134	1.06253	0.531267	+	0.847204i	$0.321716\pi$
$198$	1.71644	0.121982
$199$	−1.71644	−0.121675	−0.0608377	−	0.998148i	$-0.519377\pi$
$199$	−1.71644	−0.121675	−0.0608377	+	0.998148i	$0.519377\pi$
$200$	0	0
$201$	17.8435	1.25859
$202$	5.55635	0.390943
$203$	−4.82843	−0.338889
$204$	0	0
$205$	0	0
$206$	1.85786	0.129444
$207$	−15.8645	−1.10266
$208$	4.24264	0.294174
$209$	5.22625	0.361507
$210$	0	0
$211$	−14.9678	−1.03042	−0.515212	−	0.857063i	$-0.672287\pi$
$211$	−14.9678	−1.03042	−0.515212	+	0.857063i	$0.672287\pi$
$212$	−2.58579	−0.177593
$213$	34.1421	2.33938
$214$	−2.61313	−0.178630
$215$	0	0
$216$	−3.43289	−0.233578
$217$	3.51472	0.238595
$218$	−1.77084	−0.119936
$219$	−14.0000	−0.946032
$220$	0	0
$221$	0	0
$222$	4.14214	0.278002
$223$	−0.828427	−0.0554756	−0.0277378	−	0.999615i	$-0.508830\pi$
$223$	−0.828427	−0.0554756	−0.0277378	+	0.999615i	$0.508830\pi$
$224$	4.77791	0.319238
$225$	0	0
$226$	6.70259	0.445850
$227$	5.04054	0.334553	0.167276	−	0.985910i	$-0.446503\pi$
$227$	5.04054	0.334553	0.167276	+	0.985910i	$0.446503\pi$
$228$	−23.0698	−1.52783
$229$	−22.8284	−1.50854	−0.754272	−	0.656562i	$-0.772010\pi$
$229$	−22.8284	−1.50854	−0.754272	+	0.656562i	$0.772010\pi$
$230$	0	0
$231$	3.06147	0.201430
$232$	7.07401	0.464432
$233$	−10.1355	−0.663997	−0.331999	−	0.943280i	$-0.607723\pi$
$233$	−10.1355	−0.663997	−0.331999	+	0.943280i	$0.607723\pi$
$234$	2.24264	0.146606
$235$	0	0
$236$	−10.9706	−0.714123
$237$	10.8284	0.703382
$238$	0	0
$239$	9.17157	0.593260	0.296630	−	0.954993i	$-0.404137\pi$
$239$	9.17157	0.593260	0.296630	+	0.954993i	$0.404137\pi$
$240$	0	0
$241$	12.3003	0.792330	0.396165	−	0.918179i	$-0.370341\pi$
$241$	12.3003	0.792330	0.396165	+	0.918179i	$0.370341\pi$
$242$	−4.07107	−0.261698
$243$	−21.7248	−1.39364
$244$	−16.8925	−1.08143
$245$	0	0
$246$	8.82843	0.562880
$247$	6.82843	0.434482
$248$	−5.14933	−0.326983
$249$	−0.896683	−0.0568250
$250$	0	0
$251$	3.51472	0.221847	0.110924	−	0.993829i	$-0.464619\pi$
$251$	3.51472	0.221847	0.110924	+	0.993829i	$0.464619\pi$
$252$	−7.57675	−0.477290
$253$	−4.48528	−0.281987
$254$	−7.17157	−0.449985
$255$	0	0
$256$	3.97056	0.248160
$257$	22.1421	1.38119	0.690594	−	0.723242i	$-0.257349\pi$
$257$	22.1421	1.38119	0.690594	+	0.723242i	$0.257349\pi$
$258$	5.22625	0.325372
$259$	4.14214	0.257380
$260$	0	0
$261$	−17.0782	−1.05711
$262$	0.0769232	0.00475233
$263$	−6.48528	−0.399900	−0.199950	−	0.979806i	$-0.564078\pi$
$263$	−6.48528	−0.399900	−0.199950	+	0.979806i	$0.564078\pi$
$264$	−4.48528	−0.276050
$265$	0	0
$266$	2.16478	0.132731
$267$	24.6005	1.50553
$268$	−12.4853	−0.762660
$269$	−6.36304	−0.387961	−0.193981	−	0.981005i	$-0.562140\pi$
$269$	−6.36304	−0.387961	−0.193981	+	0.981005i	$0.562140\pi$
$270$	0	0
$271$	6.14214	0.373108	0.186554	−	0.982445i	$-0.440268\pi$
$271$	6.14214	0.373108	0.186554	+	0.982445i	$0.440268\pi$
$272$	0	0
$273$	4.00000	0.242091
$274$	−3.61522	−0.218404
$275$	0	0
$276$	19.7990	1.19176
$277$	22.0418	1.32436	0.662181	−	0.749344i	$-0.269631\pi$
$277$	22.0418	1.32436	0.662181	+	0.749344i	$0.269631\pi$
$278$	−6.19986	−0.371843
$279$	12.4316	0.744259
$280$	0	0
$281$	−17.8995	−1.06779	−0.533897	−	0.845549i	$-0.679273\pi$
$281$	−17.8995	−1.06779	−0.533897	+	0.845549i	$0.679273\pi$
$282$	−11.7206	−0.697952
$283$	22.8841	1.36032	0.680159	−	0.733065i	$-0.261911\pi$
$283$	22.8841	1.36032	0.680159	+	0.733065i	$0.261911\pi$
$284$	−23.8896	−1.41758
$285$	0	0
$286$	0.634051	0.0374922
$287$	8.82843	0.521126
$288$	16.8995	0.995812
$289$	0	0
$290$	0	0
$291$	−16.8284	−0.986500
$292$	9.79592	0.573263
$293$	23.6569	1.38205	0.691024	−	0.722832i	$-0.257160\pi$
$293$	23.6569	1.38205	0.691024	+	0.722832i	$0.257160\pi$
$294$	−6.30864	−0.367928
$295$	0	0
$296$	−6.06854	−0.352727
$297$	2.34315	0.135963
$298$	7.02944	0.407204
$299$	−5.86030	−0.338910
$300$	0	0
$301$	5.22625	0.301236
$302$	5.31371	0.305770
$303$	35.0530	2.01374
$304$	14.4853	0.830788
$305$	0	0
$306$	0	0
$307$	−2.14214	−0.122258	−0.0611291	−	0.998130i	$-0.519470\pi$
$307$	−2.14214	−0.122258	−0.0611291	+	0.998130i	$0.519470\pi$
$308$	−2.14214	−0.122060
$309$	11.7206	0.666762
$310$	0	0
$311$	4.51528	0.256038	0.128019	−	0.991772i	$-0.459138\pi$
$311$	4.51528	0.256038	0.128019	+	0.991772i	$0.459138\pi$
$312$	−5.86030	−0.331774
$313$	−12.7486	−0.720594	−0.360297	−	0.932838i	$-0.617325\pi$
$313$	−12.7486	−0.720594	−0.360297	+	0.932838i	$0.617325\pi$
$314$	−0.686292	−0.0387297
$315$	0	0
$316$	−7.57675	−0.426225
$317$	−5.80591	−0.326092	−0.163046	−	0.986618i	$-0.552132\pi$
$317$	−5.80591	−0.326092	−0.163046	+	0.986618i	$0.552132\pi$
$318$	1.53073	0.0858393
$319$	−4.82843	−0.270340
$320$	0	0
$321$	−16.4853	−0.920119
$322$	−1.85786	−0.103535
$323$	0	0
$324$	10.6569	0.592047
$325$	0	0
$326$	2.35049	0.130182
$327$	−11.1716	−0.617789
$328$	−12.9343	−0.714178
$329$	−11.7206	−0.646178
$330$	0	0
$331$	17.7990	0.978321	0.489160	−	0.872194i	$-0.337303\pi$
$331$	17.7990	0.978321	0.489160	+	0.872194i	$0.337303\pi$
$332$	0.627417	0.0344340
$333$	14.6508	0.802857
$334$	4.14386	0.226742
$335$	0	0
$336$	8.48528	0.462910
$337$	−34.4734	−1.87788	−0.938942	−	0.344075i	$-0.888192\pi$
$337$	−34.4734	−1.87788	−0.938942	+	0.344075i	$0.888192\pi$
$338$	−4.55635	−0.247833
$339$	42.2843	2.29657
$340$	0	0
$341$	3.51472	0.190333
$342$	7.65685	0.414035
$343$	−13.8854	−0.749741
$344$	−7.65685	−0.412830
$345$	0	0
$346$	1.39942	0.0752332
$347$	−3.88123	−0.208355	−0.104178	−	0.994559i	$-0.533221\pi$
$347$	−3.88123	−0.208355	−0.104178	+	0.994559i	$0.533221\pi$
$348$	21.3137	1.14253
$349$	4.24264	0.227103	0.113552	−	0.993532i	$-0.463777\pi$
$349$	4.24264	0.227103	0.113552	+	0.993532i	$0.463777\pi$
$350$	0	0
$351$	3.06147	0.163409
$352$	4.77791	0.254663
$353$	14.0000	0.745145	0.372572	−	0.928003i	$-0.378476\pi$
$353$	14.0000	0.745145	0.372572	+	0.928003i	$0.378476\pi$
$354$	6.49435	0.345171
$355$	0	0
$356$	−17.2132	−0.912298
$357$	0	0
$358$	2.48528	0.131351
$359$	23.1716	1.22295	0.611474	−	0.791264i	$-0.290577\pi$
$359$	23.1716	1.22295	0.611474	+	0.791264i	$0.290577\pi$
$360$	0	0
$361$	4.31371	0.227037
$362$	5.17186	0.271827
$363$	−25.6829	−1.34800
$364$	−2.79884	−0.146699
$365$	0	0
$366$	10.0000	0.522708
$367$	−26.3170	−1.37373	−0.686867	−	0.726783i	$-0.741015\pi$
$367$	−26.3170	−1.37373	−0.686867	+	0.726783i	$0.741015\pi$
$368$	−12.4316	−0.648041
$369$	31.2262	1.62557
$370$	0	0
$371$	1.53073	0.0794717
$372$	−15.5147	−0.804401
$373$	−19.5563	−1.01259	−0.506295	−	0.862361i	$-0.668985\pi$
$373$	−19.5563	−1.01259	−0.506295	+	0.862361i	$0.668985\pi$
$374$	0	0
$375$	0	0
$376$	17.1716	0.885556
$377$	−6.30864	−0.324912
$378$	0.970563	0.0499204
$379$	−1.08239	−0.0555988	−0.0277994	−	0.999614i	$-0.508850\pi$
$379$	−1.08239	−0.0555988	−0.0277994	+	0.999614i	$0.508850\pi$
$380$	0	0
$381$	−45.2429	−2.31786
$382$	8.28427	0.423860
$383$	−5.51472	−0.281789	−0.140894	−	0.990025i	$-0.544998\pi$
$383$	−5.51472	−0.281789	−0.140894	+	0.990025i	$0.544998\pi$
$384$	−27.5851	−1.40769
$385$	0	0
$386$	−2.29610	−0.116868
$387$	18.4853	0.939660
$388$	11.7750	0.597785
$389$	16.1421	0.818439	0.409219	−	0.912436i	$-0.365801\pi$
$389$	16.1421	0.818439	0.409219	+	0.912436i	$0.365801\pi$
$390$	0	0
$391$	0	0
$392$	9.24264	0.466824
$393$	0.485281	0.0244792
$394$	6.17733	0.311209
$395$	0	0
$396$	−7.57675	−0.380746
$397$	−27.2680	−1.36854	−0.684272	−	0.729227i	$-0.739880\pi$
$397$	−27.2680	−1.36854	−0.684272	+	0.729227i	$0.739880\pi$
$398$	−0.710974	−0.0356379
$399$	13.6569	0.683698
$400$	0	0
$401$	17.0782	0.852843	0.426422	−	0.904525i	$-0.359774\pi$
$401$	17.0782	0.852843	0.426422	+	0.904525i	$0.359774\pi$
$402$	7.39104	0.368631
$403$	4.59220	0.228754
$404$	−24.5269	−1.22026
$405$	0	0
$406$	−2.00000	−0.0992583
$407$	4.14214	0.205318
$408$	0	0
$409$	19.3137	0.955001	0.477501	−	0.878631i	$-0.341543\pi$
$409$	19.3137	0.955001	0.477501	+	0.878631i	$0.341543\pi$
$410$	0	0
$411$	−22.8072	−1.12499
$412$	−8.20101	−0.404035
$413$	6.49435	0.319566
$414$	−6.57128	−0.322961
$415$	0	0
$416$	6.24264	0.306071
$417$	−39.1127	−1.91536
$418$	2.16478	0.105883
$419$	26.9510	1.31664	0.658322	−	0.752737i	$-0.271267\pi$
$419$	26.9510	1.31664	0.658322	+	0.752737i	$0.271267\pi$
$420$	0	0
$421$	17.4142	0.848717	0.424358	−	0.905494i	$-0.360500\pi$
$421$	17.4142	0.848717	0.424358	+	0.905494i	$0.360500\pi$
$422$	−6.19986	−0.301804
$423$	−41.4558	−2.01565
$424$	−2.24264	−0.108912
$425$	0	0
$426$	14.1421	0.685189
$427$	10.0000	0.483934
$428$	11.5349	0.557560
$429$	4.00000	0.193122
$430$	0	0
$431$	−39.8309	−1.91859	−0.959294	−	0.282408i	$-0.908867\pi$
$431$	−39.8309	−1.91859	−0.959294	+	0.282408i	$0.908867\pi$
$432$	6.49435	0.312460
$433$	−15.1716	−0.729099	−0.364550	−	0.931184i	$-0.618777\pi$
$433$	−15.1716	−0.729099	−0.364550	+	0.931184i	$0.618777\pi$
$434$	1.45584	0.0698828
$435$	0	0
$436$	7.81685	0.374359
$437$	−20.0083	−0.957128
$438$	−5.79899	−0.277086
$439$	−10.9008	−0.520269	−0.260134	−	0.965572i	$-0.583767\pi$
$439$	−10.9008	−0.520269	−0.260134	+	0.965572i	$0.583767\pi$
$440$	0	0
$441$	−22.3137	−1.06256
$442$	0	0
$443$	−15.7990	−0.750633	−0.375316	−	0.926897i	$-0.622466\pi$
$443$	−15.7990	−0.750633	−0.375316	+	0.926897i	$0.622466\pi$
$444$	−18.2843	−0.867733
$445$	0	0
$446$	−0.343146	−0.0162484
$447$	44.3462	2.09750
$448$	−4.51528	−0.213327
$449$	18.7946	0.886973	0.443486	−	0.896281i	$-0.353741\pi$
$449$	18.7946	0.886973	0.443486	+	0.896281i	$0.353741\pi$
$450$	0	0
$451$	8.82843	0.415714
$452$	−29.5867	−1.39164
$453$	33.5223	1.57501
$454$	2.08786	0.0979882
$455$	0	0
$456$	−20.0083	−0.936976
$457$	−18.8284	−0.880757	−0.440378	−	0.897812i	$-0.645156\pi$
$457$	−18.8284	−0.880757	−0.440378	+	0.897812i	$0.645156\pi$
$458$	−9.45584	−0.441843
$459$	0	0
$460$	0	0
$461$	24.0416	1.11973	0.559865	−	0.828584i	$-0.310853\pi$
$461$	24.0416	1.11973	0.559865	+	0.828584i	$0.310853\pi$
$462$	1.26810	0.0589974
$463$	30.6274	1.42338	0.711688	−	0.702495i	$-0.247931\pi$
$463$	30.6274	1.42338	0.711688	+	0.702495i	$0.247931\pi$
$464$	−13.3827	−0.621274
$465$	0	0
$466$	−4.19825	−0.194480
$467$	12.6274	0.584327	0.292164	−	0.956368i	$-0.405625\pi$
$467$	12.6274	0.584327	0.292164	+	0.956368i	$0.405625\pi$
$468$	−9.89949	−0.457604
$469$	7.39104	0.341286
$470$	0	0
$471$	−4.32957	−0.199496
$472$	−9.51472	−0.437950
$473$	5.22625	0.240303
$474$	4.48528	0.206016
$475$	0	0
$476$	0	0
$477$	5.41421	0.247900
$478$	3.79899	0.173762
$479$	−34.6047	−1.58113	−0.790564	−	0.612379i	$-0.790213\pi$
$479$	−34.6047	−1.58113	−0.790564	+	0.612379i	$0.790213\pi$
$480$	0	0
$481$	5.41196	0.246764
$482$	5.09494	0.232068
$483$	−11.7206	−0.533306
$484$	17.9706	0.816844
$485$	0	0
$486$	−8.99869	−0.408189
$487$	−4.40649	−0.199677	−0.0998386	−	0.995004i	$-0.531833\pi$
$487$	−4.40649	−0.199677	−0.0998386	+	0.995004i	$0.531833\pi$
$488$	−14.6508	−0.663209
$489$	14.8284	0.670565
$490$	0	0
$491$	−25.1127	−1.13332	−0.566660	−	0.823952i	$-0.691765\pi$
$491$	−25.1127	−1.13332	−0.566660	+	0.823952i	$0.691765\pi$
$492$	−38.9706	−1.75693
$493$	0	0
$494$	2.82843	0.127257
$495$	0	0
$496$	9.74153	0.437408
$497$	14.1421	0.634361
$498$	−0.371418	−0.0166437
$499$	37.0321	1.65778	0.828892	−	0.559408i	$-0.188972\pi$
$499$	37.0321	1.65778	0.828892	+	0.559408i	$0.188972\pi$
$500$	0	0
$501$	26.1421	1.16794
$502$	1.45584	0.0649775
$503$	−14.9678	−0.667380	−0.333690	−	0.942683i	$-0.608294\pi$
$503$	−14.9678	−0.667380	−0.333690	+	0.942683i	$0.608294\pi$
$504$	−6.57128	−0.292708
$505$	0	0
$506$	−1.85786	−0.0825921
$507$	−28.7444	−1.27658
$508$	31.6569	1.40455
$509$	3.02944	0.134277	0.0671387	−	0.997744i	$-0.478613\pi$
$509$	3.02944	0.134277	0.0671387	+	0.997744i	$0.478613\pi$
$510$	0	0
$511$	−5.79899	−0.256532
$512$	22.7574	1.00574
$513$	10.4525	0.461489
$514$	9.17157	0.404541
$515$	0	0
$516$	−23.0698	−1.01559
$517$	−11.7206	−0.515472
$518$	1.71573	0.0753848
$519$	8.82843	0.387525
$520$	0	0
$521$	−3.11586	−0.136508	−0.0682542	−	0.997668i	$-0.521743\pi$
$521$	−3.11586	−0.136508	−0.0682542	+	0.997668i	$0.521743\pi$
$522$	−7.07401	−0.309621
$523$	−6.82843	−0.298586	−0.149293	−	0.988793i	$-0.547700\pi$
$523$	−6.82843	−0.298586	−0.149293	+	0.988793i	$0.547700\pi$
$524$	−0.339556	−0.0148336
$525$	0	0
$526$	−2.68629	−0.117128
$527$	0	0
$528$	8.48528	0.369274
$529$	−5.82843	−0.253410
$530$	0	0
$531$	22.9706	0.996838
$532$	−9.55582	−0.414297
$533$	11.5349	0.499632
$534$	10.1899	0.440959
$535$	0	0
$536$	−10.8284	−0.467717
$537$	15.6788	0.676588
$538$	−2.63566	−0.113631
$539$	−6.30864	−0.271733
$540$	0	0
$541$	18.3463	0.788768	0.394384	−	0.918946i	$-0.370958\pi$
$541$	18.3463	0.788768	0.394384	+	0.918946i	$0.370958\pi$
$542$	2.54416	0.109281
$543$	32.6274	1.40018
$544$	0	0
$545$	0	0
$546$	1.65685	0.0709068
$547$	−24.7862	−1.05978	−0.529891	−	0.848065i	$-0.677767\pi$
$547$	−24.7862	−1.05978	−0.529891	+	0.848065i	$0.677767\pi$
$548$	15.9584	0.681708
$549$	35.3701	1.50956
$550$	0	0
$551$	−21.5391	−0.917595
$552$	17.1716	0.730871
$553$	4.48528	0.190734
$554$	9.13001	0.387897
$555$	0	0
$556$	27.3675	1.16064
$557$	28.2426	1.19668	0.598340	−	0.801243i	$-0.295827\pi$
$557$	28.2426	1.19668	0.598340	+	0.801243i	$0.295827\pi$
$558$	5.14933	0.217988
$559$	6.82843	0.288812
$560$	0	0
$561$	0	0
$562$	−7.41421	−0.312750
$563$	−38.7696	−1.63394	−0.816971	−	0.576679i	$-0.804348\pi$
$563$	−38.7696	−1.63394	−0.816971	+	0.576679i	$0.804348\pi$
$564$	51.7373	2.17853
$565$	0	0
$566$	9.47890	0.398428
$567$	−6.30864	−0.264938
$568$	−20.7193	−0.869362
$569$	−36.0416	−1.51094	−0.755472	−	0.655181i	$-0.772592\pi$
$569$	−36.0416	−1.51094	−0.755472	+	0.655181i	$0.772592\pi$
$570$	0	0
$571$	−47.2220	−1.97618	−0.988089	−	0.153883i	$-0.950822\pi$
$571$	−47.2220	−1.97618	−0.988089	+	0.153883i	$0.950822\pi$
$572$	−2.79884	−0.117025
$573$	52.2625	2.18330
$574$	3.65685	0.152634
$575$	0	0
$576$	−15.9706	−0.665440
$577$	−12.9289	−0.538238	−0.269119	−	0.963107i	$-0.586733\pi$
$577$	−12.9289	−0.538238	−0.269119	+	0.963107i	$0.586733\pi$
$578$	0	0
$579$	−14.4853	−0.601988
$580$	0	0
$581$	−0.371418	−0.0154090
$582$	−6.97056	−0.288939
$583$	1.53073	0.0633965
$584$	8.49596	0.351565
$585$	0	0
$586$	9.79899	0.404793
$587$	22.6863	0.936363	0.468182	−	0.883632i	$-0.344909\pi$
$587$	22.6863	0.936363	0.468182	+	0.883632i	$0.344909\pi$
$588$	27.8477	1.14842
$589$	15.6788	0.646032
$590$	0	0
$591$	38.9706	1.60303
$592$	11.4805	0.471846
$593$	−27.0711	−1.11168	−0.555838	−	0.831291i	$-0.687602\pi$
$593$	−27.0711	−1.11168	−0.555838	+	0.831291i	$0.687602\pi$
$594$	0.970563	0.0398227
$595$	0	0
$596$	−31.0294	−1.27102
$597$	−4.48528	−0.183570
$598$	−2.42742	−0.0992645
$599$	34.6274	1.41484	0.707419	−	0.706794i	$-0.249859\pi$
$599$	34.6274	1.41484	0.707419	+	0.706794i	$0.249859\pi$
$600$	0	0
$601$	−20.3253	−0.829088	−0.414544	−	0.910029i	$-0.636059\pi$
$601$	−20.3253	−0.829088	−0.414544	+	0.910029i	$0.636059\pi$
$602$	2.16478	0.0882300
$603$	26.1421	1.06459
$604$	−23.4558	−0.954405
$605$	0	0
$606$	14.5194	0.589812
$607$	34.3421	1.39390	0.696951	−	0.717119i	$-0.254540\pi$
$607$	34.3421	1.39390	0.696951	+	0.717119i	$0.254540\pi$
$608$	21.3137	0.864385
$609$	−12.6173	−0.511278
$610$	0	0
$611$	−15.3137	−0.619526
$612$	0	0
$613$	17.3137	0.699294	0.349647	−	0.936881i	$-0.386302\pi$
$613$	17.3137	0.699294	0.349647	+	0.936881i	$0.386302\pi$
$614$	−0.887302	−0.0358086
$615$	0	0
$616$	−1.85786	−0.0748555
$617$	−3.37849	−0.136013	−0.0680065	−	0.997685i	$-0.521664\pi$
$617$	−3.37849	−0.136013	−0.0680065	+	0.997685i	$0.521664\pi$
$618$	4.85483	0.195290
$619$	−9.63274	−0.387173	−0.193586	−	0.981083i	$-0.562012\pi$
$619$	−9.63274	−0.387173	−0.193586	+	0.981083i	$0.562012\pi$
$620$	0	0
$621$	−8.97056	−0.359976
$622$	1.87029	0.0749918
$623$	10.1899	0.408249
$624$	11.0866	0.443817
$625$	0	0
$626$	−5.28064	−0.211057
$627$	13.6569	0.545402
$628$	3.02944	0.120888
$629$	0	0
$630$	0	0
$631$	6.68629	0.266177	0.133089	−	0.991104i	$-0.457511\pi$
$631$	6.68629	0.266177	0.133089	+	0.991104i	$0.457511\pi$
$632$	−6.57128	−0.261391
$633$	−39.1127	−1.55459
$634$	−2.40489	−0.0955102
$635$	0	0
$636$	−6.75699	−0.267932
$637$	−8.24264	−0.326585
$638$	−2.00000	−0.0791808
$639$	50.0208	1.97879
$640$	0	0
$641$	−14.9903	−0.592082	−0.296041	−	0.955175i	$-0.595666\pi$
$641$	−14.9903	−0.592082	−0.296041	+	0.955175i	$0.595666\pi$
$642$	−6.82843	−0.269497
$643$	−40.0936	−1.58114	−0.790568	−	0.612374i	$-0.790215\pi$
$643$	−40.0936	−1.58114	−0.790568	+	0.612374i	$0.790215\pi$
$644$	8.20101	0.323165
$645$	0	0
$646$	0	0
$647$	−2.82843	−0.111197	−0.0555985	−	0.998453i	$-0.517707\pi$
$647$	−2.82843	−0.111197	−0.0555985	+	0.998453i	$0.517707\pi$
$648$	9.24264	0.363085
$649$	6.49435	0.254926
$650$	0	0
$651$	9.18440	0.359965
$652$	−10.3756	−0.406339
$653$	−17.7122	−0.693133	−0.346566	−	0.938025i	$-0.612652\pi$
$653$	−17.7122	−0.693133	−0.346566	+	0.938025i	$0.612652\pi$
$654$	−4.62742	−0.180946
$655$	0	0
$656$	24.4692	0.955362
$657$	−20.5111	−0.800213
$658$	−4.85483	−0.189261
$659$	8.48528	0.330540	0.165270	−	0.986248i	$-0.447151\pi$
$659$	8.48528	0.330540	0.165270	+	0.986248i	$0.447151\pi$
$660$	0	0
$661$	−41.2132	−1.60301	−0.801504	−	0.597990i	$-0.795966\pi$
$661$	−41.2132	−1.60301	−0.801504	+	0.597990i	$0.795966\pi$
$662$	7.37258	0.286544
$663$	0	0
$664$	0.544156	0.0211173
$665$	0	0
$666$	6.06854	0.235151
$667$	18.4853	0.715753
$668$	−18.2919	−0.707734
$669$	−2.16478	−0.0836954
$670$	0	0
$671$	10.0000	0.386046
$672$	12.4853	0.481630
$673$	0.317025	0.0122204	0.00611021	−	0.999981i	$-0.498055\pi$
$673$	0.317025	0.0122204	0.00611021	+	0.999981i	$0.498055\pi$
$674$	−14.2793	−0.550020
$675$	0	0
$676$	20.1127	0.773565
$677$	43.9204	1.68800	0.843999	−	0.536344i	$-0.180195\pi$
$677$	43.9204	1.68800	0.843999	+	0.536344i	$0.180195\pi$
$678$	17.5147	0.672649
$679$	−6.97056	−0.267506
$680$	0	0
$681$	13.1716	0.504736
$682$	1.45584	0.0557472
$683$	31.2806	1.19692	0.598459	−	0.801153i	$-0.295780\pi$
$683$	31.2806	1.19692	0.598459	+	0.801153i	$0.295780\pi$
$684$	−33.7990	−1.29234
$685$	0	0
$686$	−5.75152	−0.219594
$687$	−59.6536	−2.27593
$688$	14.4853	0.552246
$689$	2.00000	0.0761939
$690$	0	0
$691$	−40.7276	−1.54935	−0.774676	−	0.632358i	$-0.782087\pi$
$691$	−40.7276	−1.54935	−0.774676	+	0.632358i	$0.782087\pi$
$692$	−6.17733	−0.234827
$693$	4.48528	0.170382
$694$	−1.60766	−0.0610258
$695$	0	0
$696$	18.4853	0.700683
$697$	0	0
$698$	1.75736	0.0665170
$699$	−26.4853	−1.00177
$700$	0	0
$701$	−21.6985	−0.819540	−0.409770	−	0.912189i	$-0.634391\pi$
$701$	−21.6985	−0.819540	−0.409770	+	0.912189i	$0.634391\pi$
$702$	1.26810	0.0478614
$703$	18.4776	0.696896
$704$	−4.51528	−0.170176
$705$	0	0
$706$	5.79899	0.218248
$707$	14.5194	0.546060
$708$	−28.6675	−1.07739
$709$	−11.5893	−0.435245	−0.217622	−	0.976033i	$-0.569830\pi$
$709$	−11.5893	−0.435245	−0.217622	+	0.976033i	$0.569830\pi$
$710$	0	0
$711$	15.8645	0.594964
$712$	−14.9289	−0.559485
$713$	−13.4558	−0.503925
$714$	0	0
$715$	0	0
$716$	−10.9706	−0.409989
$717$	23.9665	0.895044
$718$	9.59798	0.358193
$719$	−14.0711	−0.524763	−0.262382	−	0.964964i	$-0.584508\pi$
$719$	−14.0711	−0.524763	−0.262382	+	0.964964i	$0.584508\pi$
$720$	0	0
$721$	4.85483	0.180803
$722$	1.78680	0.0664977
$723$	32.1421	1.19538
$724$	−22.8297	−0.848459
$725$	0	0
$726$	−10.6382	−0.394821
$727$	−19.1127	−0.708851	−0.354425	−	0.935084i	$-0.615324\pi$
$727$	−19.1127	−0.708851	−0.354425	+	0.935084i	$0.615324\pi$
$728$	−2.42742	−0.0899661
$729$	−39.2843	−1.45497
$730$	0	0
$731$	0	0
$732$	−44.1421	−1.63154
$733$	12.0416	0.444768	0.222384	−	0.974959i	$-0.428616\pi$
$733$	12.0416	0.444768	0.222384	+	0.974959i	$0.428616\pi$
$734$	−10.9008	−0.402358
$735$	0	0
$736$	−18.2919	−0.674248
$737$	7.39104	0.272252
$738$	12.9343	0.476119
$739$	−34.2843	−1.26117	−0.630584	−	0.776121i	$-0.717184\pi$
$739$	−34.2843	−1.26117	−0.630584	+	0.776121i	$0.717184\pi$
$740$	0	0
$741$	17.8435	0.655499
$742$	0.634051	0.0232767
$743$	−49.2780	−1.80783	−0.903917	−	0.427708i	$-0.859321\pi$
$743$	−49.2780	−1.80783	−0.903917	+	0.427708i	$0.859321\pi$
$744$	−13.4558	−0.493315
$745$	0	0
$746$	−8.10051	−0.296581
$747$	−1.31371	−0.0480661
$748$	0	0
$749$	−6.82843	−0.249505
$750$	0	0
$751$	−26.2082	−0.956350	−0.478175	−	0.878265i	$-0.658702\pi$
$751$	−26.2082	−0.956350	−0.478175	+	0.878265i	$0.658702\pi$
$752$	−32.4853	−1.18462
$753$	9.18440	0.334698
$754$	−2.61313	−0.0951644
$755$	0	0
$756$	−4.28427	−0.155817
$757$	−53.4558	−1.94289	−0.971443	−	0.237274i	$-0.923746\pi$
$757$	−53.4558	−1.94289	−0.971443	+	0.237274i	$0.923746\pi$
$758$	−0.448342	−0.0162845
$759$	−11.7206	−0.425431
$760$	0	0
$761$	−21.6985	−0.786569	−0.393285	−	0.919417i	$-0.628661\pi$
$761$	−21.6985	−0.786569	−0.393285	+	0.919417i	$0.628661\pi$
$762$	−18.7402	−0.678887
$763$	−4.62742	−0.167524
$764$	−36.5685	−1.32300
$765$	0	0
$766$	−2.28427	−0.0825341
$767$	8.48528	0.306386
$768$	10.3756	0.374397
$769$	−12.7279	−0.458981	−0.229490	−	0.973311i	$-0.573706\pi$
$769$	−12.7279	−0.458981	−0.229490	+	0.973311i	$0.573706\pi$
$770$	0	0
$771$	57.8602	2.08378
$772$	10.1355	0.364784
$773$	−4.82843	−0.173666	−0.0868332	−	0.996223i	$-0.527675\pi$
$773$	−4.82843	−0.173666	−0.0868332	+	0.996223i	$0.527675\pi$
$774$	7.65685	0.275220
$775$	0	0
$776$	10.2124	0.366604
$777$	10.8239	0.388306
$778$	6.68629	0.239715
$779$	39.3826	1.41103
$780$	0	0
$781$	14.1421	0.506045
$782$	0	0
$783$	−9.65685	−0.345108
$784$	−17.4853	−0.624474
$785$	0	0
$786$	0.201010	0.00716979
$787$	−49.6494	−1.76981	−0.884905	−	0.465772i	$-0.845777\pi$
$787$	−49.6494	−1.76981	−0.884905	+	0.465772i	$0.845777\pi$
$788$	−27.2680	−0.971384
$789$	−16.9469	−0.603324
$790$	0	0
$791$	17.5147	0.622752
$792$	−6.57128	−0.233500
$793$	13.0656	0.463974
$794$	−11.2948	−0.400837
$795$	0	0
$796$	3.13839	0.111237
$797$	−17.2132	−0.609723	−0.304861	−	0.952397i	$-0.598610\pi$
$797$	−17.2132	−0.609723	−0.304861	+	0.952397i	$0.598610\pi$
$798$	5.65685	0.200250
$799$	0	0
$800$	0	0
$801$	36.0416	1.27347
$802$	7.07401	0.249792
$803$	−5.79899	−0.204642
$804$	−32.6256	−1.15062
$805$	0	0
$806$	1.90215	0.0670004
$807$	−16.6274	−0.585313
$808$	−21.2721	−0.748349
$809$	−29.7724	−1.04674	−0.523371	−	0.852105i	$-0.675326\pi$
$809$	−29.7724	−1.04674	−0.523371	+	0.852105i	$0.675326\pi$
$810$	0	0
$811$	44.2693	1.55451	0.777253	−	0.629189i	$-0.216613\pi$
$811$	44.2693	1.55451	0.777253	+	0.629189i	$0.216613\pi$
$812$	8.82843	0.309817
$813$	16.0502	0.562904
$814$	1.71573	0.0601363
$815$	0	0
$816$	0	0
$817$	23.3137	0.815643
$818$	8.00000	0.279713
$819$	5.86030	0.204776
$820$	0	0
$821$	14.3881	0.502149	0.251074	−	0.967968i	$-0.419216\pi$
$821$	14.3881	0.502149	0.251074	+	0.967968i	$0.419216\pi$
$822$	−9.44703	−0.329503
$823$	23.5181	0.819791	0.409895	−	0.912133i	$-0.365565\pi$
$823$	23.5181	0.819791	0.409895	+	0.912133i	$0.365565\pi$
$824$	−7.11270	−0.247783
$825$	0	0
$826$	2.69005	0.0935988
$827$	−34.7135	−1.20711	−0.603553	−	0.797323i	$-0.706249\pi$
$827$	−34.7135	−1.20711	−0.603553	+	0.797323i	$0.706249\pi$
$828$	29.0070	1.00806
$829$	−13.9411	−0.484195	−0.242098	−	0.970252i	$-0.577835\pi$
$829$	−13.9411	−0.484195	−0.242098	+	0.970252i	$0.577835\pi$
$830$	0	0
$831$	57.5980	1.99805
$832$	−5.89949	−0.204528
$833$	0	0
$834$	−16.2010	−0.560995
$835$	0	0
$836$	−9.55582	−0.330495
$837$	7.02944	0.242973
$838$	11.1635	0.385636
$839$	−3.88123	−0.133995	−0.0669974	−	0.997753i	$-0.521342\pi$
$839$	−3.88123	−0.133995	−0.0669974	+	0.997753i	$0.521342\pi$
$840$	0	0
$841$	−9.10051	−0.313811
$842$	7.21320	0.248583
$843$	−46.7736	−1.61097
$844$	27.3675	0.942028
$845$	0	0
$846$	−17.1716	−0.590371
$847$	−10.6382	−0.365533
$848$	4.24264	0.145693
$849$	59.7990	2.05230
$850$	0	0
$851$	−15.8579	−0.543601
$852$	−62.4264	−2.13869
$853$	−42.5754	−1.45775	−0.728877	−	0.684645i	$-0.759957\pi$
$853$	−42.5754	−1.45775	−0.728877	+	0.684645i	$0.759957\pi$
$854$	4.14214	0.141741
$855$	0	0
$856$	10.0042	0.341935
$857$	−3.82683	−0.130722	−0.0653611	−	0.997862i	$-0.520820\pi$
$857$	−3.82683	−0.130722	−0.0653611	+	0.997862i	$0.520820\pi$
$858$	1.65685	0.0565641
$859$	−1.02944	−0.0351239	−0.0175620	−	0.999846i	$-0.505590\pi$
$859$	−1.02944	−0.0351239	−0.0175620	+	0.999846i	$0.505590\pi$
$860$	0	0
$861$	23.0698	0.786216
$862$	−16.4985	−0.561942
$863$	−34.6274	−1.17873	−0.589365	−	0.807867i	$-0.700622\pi$
$863$	−34.6274	−1.17873	−0.589365	+	0.807867i	$0.700622\pi$
$864$	9.55582	0.325096
$865$	0	0
$866$	−6.28427	−0.213548
$867$	0	0
$868$	−6.42641	−0.218126
$869$	4.48528	0.152153
$870$	0	0
$871$	9.65685	0.327210
$872$	6.77952	0.229583
$873$	−24.6549	−0.834443
$874$	−8.28772	−0.280336
$875$	0	0
$876$	25.5980	0.864876
$877$	37.6436	1.27113	0.635567	−	0.772045i	$-0.280766\pi$
$877$	37.6436	1.27113	0.635567	+	0.772045i	$0.280766\pi$
$878$	−4.51528	−0.152383
$879$	61.8183	2.08508
$880$	0	0
$881$	−18.5320	−0.624358	−0.312179	−	0.950023i	$-0.601059\pi$
$881$	−18.5320	−0.624358	−0.312179	+	0.950023i	$0.601059\pi$
$882$	−9.24264	−0.311216
$883$	8.00000	0.269221	0.134611	−	0.990899i	$-0.457022\pi$
$883$	8.00000	0.269221	0.134611	+	0.990899i	$0.457022\pi$
$884$	0	0
$885$	0	0
$886$	−6.54416	−0.219855
$887$	−48.8615	−1.64061	−0.820304	−	0.571927i	$-0.806196\pi$
$887$	−48.8615	−1.64061	−0.820304	+	0.571927i	$0.806196\pi$
$888$	−15.8579	−0.532155
$889$	−18.7402	−0.628527
$890$	0	0
$891$	−6.30864	−0.211348
$892$	1.51472	0.0507165
$893$	−52.2843	−1.74963
$894$	18.3688	0.614345
$895$	0	0
$896$	−11.4261	−0.381720
$897$	−15.3137	−0.511310
$898$	7.78498	0.259788
$899$	−14.4853	−0.483111
$900$	0	0
$901$	0	0
$902$	3.65685	0.121760
$903$	13.6569	0.454472
$904$	−25.6604	−0.853452
$905$	0	0
$906$	13.8854	0.461311
$907$	25.0489	0.831734	0.415867	−	0.909425i	$-0.363478\pi$
$907$	25.0489	0.831734	0.415867	+	0.909425i	$0.363478\pi$
$908$	−9.21627	−0.305852
$909$	51.3553	1.70335
$910$	0	0
$911$	−8.47343	−0.280737	−0.140369	−	0.990099i	$-0.544829\pi$
$911$	−8.47343	−0.280737	−0.140369	+	0.990099i	$0.544829\pi$
$912$	37.8519	1.25340
$913$	−0.371418	−0.0122922
$914$	−7.79899	−0.257968
$915$	0	0
$916$	41.7401	1.37913
$917$	0.201010	0.00663794
$918$	0	0
$919$	−3.31371	−0.109309	−0.0546546	−	0.998505i	$-0.517406\pi$
$919$	−3.31371	−0.109309	−0.0546546	+	0.998505i	$0.517406\pi$
$920$	0	0
$921$	−5.59767	−0.184450
$922$	9.95837	0.327961
$923$	18.4776	0.608197
$924$	−5.59767	−0.184150
$925$	0	0
$926$	12.6863	0.416897
$927$	17.1716	0.563988
$928$	−19.6913	−0.646399
$929$	−20.9594	−0.687656	−0.343828	−	0.939033i	$-0.611724\pi$
$929$	−20.9594	−0.687656	−0.343828	+	0.939033i	$0.611724\pi$
$930$	0	0
$931$	−28.1421	−0.922321
$932$	18.5320	0.607035
$933$	11.7990	0.386282
$934$	5.23045	0.171145
$935$	0	0
$936$	−8.58579	−0.280635
$937$	−3.55635	−0.116181	−0.0580904	−	0.998311i	$-0.518501\pi$
$937$	−3.55635	−0.116181	−0.0580904	+	0.998311i	$0.518501\pi$
$938$	3.06147	0.0999605
$939$	−33.3137	−1.08715
$940$	0	0
$941$	18.6858	0.609141	0.304570	−	0.952490i	$-0.401487\pi$
$941$	18.6858	0.609141	0.304570	+	0.952490i	$0.401487\pi$
$942$	−1.79337	−0.0584310
$943$	−33.7990	−1.10065
$944$	18.0000	0.585850
$945$	0	0
$946$	2.16478	0.0703832
$947$	−14.8590	−0.482852	−0.241426	−	0.970419i	$-0.577615\pi$
$947$	−14.8590	−0.482852	−0.241426	+	0.970419i	$0.577615\pi$
$948$	−19.7990	−0.643041
$949$	−7.57675	−0.245952
$950$	0	0
$951$	−15.1716	−0.491972
$952$	0	0
$953$	9.69848	0.314165	0.157082	−	0.987586i	$-0.449791\pi$
$953$	9.69848	0.314165	0.157082	+	0.987586i	$0.449791\pi$
$954$	2.24264	0.0726082
$955$	0	0
$956$	−16.7696	−0.542366
$957$	−12.6173	−0.407859
$958$	−14.3337	−0.463102
$959$	−9.44703	−0.305061
$960$	0	0
$961$	−20.4558	−0.659866
$962$	2.24171	0.0722756
$963$	−24.1522	−0.778293
$964$	−22.4901	−0.724358
$965$	0	0
$966$	−4.85483	−0.156202
$967$	32.3431	1.04009	0.520043	−	0.854140i	$-0.325916\pi$
$967$	32.3431	1.04009	0.520043	+	0.854140i	$0.325916\pi$
$968$	15.5858	0.500946
$969$	0	0
$970$	0	0
$971$	−55.7401	−1.78879	−0.894393	−	0.447283i	$-0.852392\pi$
$971$	−55.7401	−1.78879	−0.894393	+	0.447283i	$0.852392\pi$
$972$	39.7222	1.27409
$973$	−16.2010	−0.519381
$974$	−1.82523	−0.0584841
$975$	0	0
$976$	27.7164	0.887180
$977$	1.61522	0.0516756	0.0258378	−	0.999666i	$-0.491775\pi$
$977$	1.61522	0.0516756	0.0258378	+	0.999666i	$0.491775\pi$
$978$	6.14214	0.196404
$979$	10.1899	0.325670
$980$	0	0
$981$	−16.3672	−0.522564
$982$	−10.4020	−0.331942
$983$	−23.5181	−0.750112	−0.375056	−	0.927002i	$-0.622377\pi$
$983$	−23.5181	−0.750112	−0.375056	+	0.927002i	$0.622377\pi$
$984$	−33.7990	−1.07747
$985$	0	0
$986$	0	0
$987$	−30.6274	−0.974881
$988$	−12.4853	−0.397210
$989$	−20.0083	−0.636228
$990$	0	0
$991$	−32.0685	−1.01869	−0.509345	−	0.860563i	$-0.670112\pi$
$991$	−32.0685	−1.01869	−0.509345	+	0.860563i	$0.670112\pi$
$992$	14.3337	0.455096
$993$	46.5110	1.47598
$994$	5.85786	0.185800
$995$	0	0
$996$	1.63952	0.0519502
$997$	−7.70806	−0.244117	−0.122058	−	0.992523i	$-0.538950\pi$
$997$	−7.70806	−0.244117	−0.122058	+	0.992523i	$0.538950\pi$
$998$	15.3392	0.485554
$999$	8.28427	0.262103

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7225.2.a.u.1.4		4
5.4	even	2		289.2.a.f.1.1		4
15.14	odd	2		2601.2.a.bb.1.4		4
17.3	odd	16		425.2.m.a.26.1		4
17.6	odd	16		425.2.m.a.376.1		4
17.16	even	2	inner	7225.2.a.u.1.3		4
20.19	odd	2		4624.2.a.bp.1.4		4
85.3	even	16		425.2.n.a.349.1		4
85.4	even	4		289.2.b.b.288.4		4
85.9	even	8		289.2.c.c.251.3		8
85.14	odd	16		289.2.d.a.179.1		4
85.19	even	8		289.2.c.c.38.1		8
85.23	even	16		425.2.n.b.274.1		4
85.24	odd	16		289.2.d.c.134.1		4
85.29	odd	16		289.2.d.b.110.1		4
85.37	even	16		425.2.n.b.349.1		4
85.39	odd	16		289.2.d.c.110.1		4
85.44	odd	16		289.2.d.b.134.1		4
85.49	even	8		289.2.c.c.38.2		8
85.54	odd	16		17.2.d.a.9.1	yes	4
85.57	even	16		425.2.n.a.274.1		4
85.59	even	8		289.2.c.c.251.4		8
85.64	even	4		289.2.b.b.288.3		4
85.74	odd	16		17.2.d.a.2.1	✓	4
85.79	odd	16		289.2.d.a.155.1		4
85.84	even	2		289.2.a.f.1.2		4
255.74	even	16		153.2.l.c.19.1		4
255.224	even	16		153.2.l.c.145.1		4
255.254	odd	2		2601.2.a.bb.1.3		4
340.139	even	16		272.2.v.d.145.1		4
340.159	even	16		272.2.v.d.257.1		4
340.339	odd	2		4624.2.a.bp.1.1		4
595.54	even	48		833.2.v.a.655.1		8
595.74	odd	48		833.2.v.b.716.1		8
595.139	even	16		833.2.l.a.638.1		4
595.159	even	48		833.2.v.a.410.1		8
595.244	even	16		833.2.l.a.393.1		4
595.394	odd	48		833.2.v.b.655.1		8
595.479	even	48		833.2.v.a.128.1		8
595.499	odd	48		833.2.v.b.410.1		8
595.564	odd	48		833.2.v.b.128.1		8
595.584	even	48		833.2.v.a.716.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
17.2.d.a.2.1	✓	4	85.74	odd	16
17.2.d.a.9.1	yes	4	85.54	odd	16
153.2.l.c.19.1		4	255.74	even	16
153.2.l.c.145.1		4	255.224	even	16
272.2.v.d.145.1		4	340.139	even	16
272.2.v.d.257.1		4	340.159	even	16
289.2.a.f.1.1		4	5.4	even	2
289.2.a.f.1.2		4	85.84	even	2
289.2.b.b.288.3		4	85.64	even	4
289.2.b.b.288.4		4	85.4	even	4
289.2.c.c.38.1		8	85.19	even	8
289.2.c.c.38.2		8	85.49	even	8
289.2.c.c.251.3		8	85.9	even	8
289.2.c.c.251.4		8	85.59	even	8
289.2.d.a.155.1		4	85.79	odd	16
289.2.d.a.179.1		4	85.14	odd	16
289.2.d.b.110.1		4	85.29	odd	16
289.2.d.b.134.1		4	85.44	odd	16
289.2.d.c.110.1		4	85.39	odd	16
289.2.d.c.134.1		4	85.24	odd	16
425.2.m.a.26.1		4	17.3	odd	16
425.2.m.a.376.1		4	17.6	odd	16
425.2.n.a.274.1		4	85.57	even	16
425.2.n.a.349.1		4	85.3	even	16
425.2.n.b.274.1		4	85.23	even	16
425.2.n.b.349.1		4	85.37	even	16
833.2.l.a.393.1		4	595.244	even	16
833.2.l.a.638.1		4	595.139	even	16
833.2.v.a.128.1		8	595.479	even	48
833.2.v.a.410.1		8	595.159	even	48
833.2.v.a.655.1		8	595.54	even	48
833.2.v.a.716.1		8	595.584	even	48
833.2.v.b.128.1		8	595.564	odd	48
833.2.v.b.410.1		8	595.499	odd	48
833.2.v.b.655.1		8	595.394	odd	48
833.2.v.b.716.1		8	595.74	odd	48
2601.2.a.bb.1.3		4	255.254	odd	2
2601.2.a.bb.1.4		4	15.14	odd	2
4624.2.a.bp.1.1		4	340.339	odd	2
4624.2.a.bp.1.4		4	20.19	odd	2
7225.2.a.u.1.3		4	17.16	even	2	inner
7225.2.a.u.1.4		4	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	7225.2.a.u.1.4

Level	$7225$
Weight	$2$
Character	7225.1
Self dual	yes
Analytic conductor	$57.692$
Analytic rank	$0$
Dimension	$4$
Inner twists	$2$