LMFDB - Embedded newform 1881.2.a.m.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1881,2,Mod(1,1881)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1881, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1881.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1881 = 3^{2} \cdot 11 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1881.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:	$15.0198606202$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.1920025.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 9x^{3} + 8x^{2} + 15x - 4 $ x^5 - x^4 - 9x^3 + 8x^2 + 15*x - 4
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 627)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.19978$ of defining polynomial
Character	$\chi$	$=$	1881.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-2.19978 q^{2} +2.83905 q^{4} -3.77215 q^{5} +0.285221 q^{7} -1.84574 q^{8} +8.29791 q^{10} -1.00000 q^{11} -4.74408 q^{13} -0.627424 q^{14} -1.61788 q^{16} -5.55383 q^{17} -1.00000 q^{19} -10.7093 q^{20} +2.19978 q^{22} +3.32406 q^{23} +9.22908 q^{25} +10.4359 q^{26} +0.809757 q^{28} -1.36073 q^{29} -7.29122 q^{31} +7.25047 q^{32} +12.2172 q^{34} -1.07589 q^{35} +2.35405 q^{37} +2.19978 q^{38} +6.96239 q^{40} -3.84097 q^{41} -0.273376 q^{43} -2.83905 q^{44} -7.31221 q^{46} -5.16503 q^{47} -6.91865 q^{49} -20.3020 q^{50} -13.4687 q^{52} -2.62973 q^{53} +3.77215 q^{55} -0.526442 q^{56} +2.99332 q^{58} -0.181257 q^{59} +4.09404 q^{61} +16.0391 q^{62} -12.7137 q^{64} +17.8953 q^{65} -13.1781 q^{67} -15.7676 q^{68} +2.36674 q^{70} +9.56720 q^{71} -8.53461 q^{73} -5.17840 q^{74} -2.83905 q^{76} -0.285221 q^{77} +7.80622 q^{79} +6.10289 q^{80} +8.44931 q^{82} -5.53053 q^{83} +20.9499 q^{85} +0.601369 q^{86} +1.84574 q^{88} +6.83811 q^{89} -1.35311 q^{91} +9.43718 q^{92} +11.3620 q^{94} +3.77215 q^{95} +8.58313 q^{97} +15.2195 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - q^{2} + 9 q^{4} - 7 q^{5} + 7 q^{7} - 3 q^{10} - 5 q^{11} + 10 q^{13} + 5 q^{14} + 13 q^{16} - 17 q^{17} - 5 q^{19} - 2 q^{20} + q^{22} + 7 q^{23} + 8 q^{25} + 8 q^{26} + 27 q^{28} - 2 q^{29} + 4 q^{31}+ \cdots + 25 q^{98}+O(q^{100}) $ 5 * q - q^2 + 9 * q^4 - 7 * q^5 + 7 * q^7 - 3 * q^10 - 5 * q^11 + 10 * q^13 + 5 * q^14 + 13 * q^16 - 17 * q^17 - 5 * q^19 - 2 * q^20 + q^22 + 7 * q^23 + 8 * q^25 + 8 * q^26 + 27 * q^28 - 2 * q^29 + 4 * q^31 + 24 * q^32 + 10 * q^34 - 8 * q^35 + 11 * q^37 + q^38 - q^41 + 6 * q^43 - 9 * q^44 - 14 * q^46 + 2 * q^47 + 14 * q^49 - 24 * q^50 + 39 * q^52 - 5 * q^53 + 7 * q^55 - 12 * q^56 + 19 * q^58 + 18 * q^59 + 19 * q^61 - 15 * q^62 - 8 * q^64 + 28 * q^65 - q^67 - 51 * q^68 - 39 * q^70 + 29 * q^71 + 14 * q^73 + 10 * q^74 - 9 * q^76 - 7 * q^77 + 3 * q^79 + 5 * q^80 - 10 * q^82 - 4 * q^83 + 15 * q^85 + 53 * q^86 - q^89 + 24 * q^91 + 62 * q^92 + 2 * q^94 + 7 * q^95 + 4 * q^97 + 25 * 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 $

$2$	−2.19978	−1.55548	−0.777741	−	0.628584i	$-0.783635\pi$
$2$	−2.19978	−1.55548	−0.777741	+	0.628584i	$0.783635\pi$
$3$	0	0
$3$	0	0
$4$	2.83905	1.41953
$5$	−3.77215	−1.68695	−0.843477	−	0.537165i	$-0.819495\pi$
$5$	−3.77215	−1.68695	−0.843477	+	0.537165i	$0.819495\pi$
$6$	0	0
$7$	0.285221	0.107803	0.0539017	−	0.998546i	$-0.482834\pi$
$7$	0.285221	0.107803	0.0539017	+	0.998546i	$0.482834\pi$
$8$	−1.84574	−0.652567
$9$	0	0
$10$	8.29791	2.62403
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−4.74408	−1.31577	−0.657885	−	0.753118i	$-0.728549\pi$
$13$	−4.74408	−1.31577	−0.657885	+	0.753118i	$0.728549\pi$
$14$	−0.627424	−0.167686
$15$	0	0
$16$	−1.61788	−0.404471
$17$	−5.55383	−1.34700	−0.673501	−	0.739186i	$-0.735210\pi$
$17$	−5.55383	−1.34700	−0.673501	+	0.739186i	$0.735210\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	−10.7093	−2.39468
$21$	0	0
$22$	2.19978	0.468996
$23$	3.32406	0.693114	0.346557	−	0.938029i	$-0.387351\pi$
$23$	3.32406	0.693114	0.346557	+	0.938029i	$0.387351\pi$
$24$	0	0
$25$	9.22908	1.84582
$26$	10.4359	2.04666
$27$	0	0
$28$	0.809757	0.153030
$29$	−1.36073	−0.252681	−0.126341	−	0.991987i	$-0.540323\pi$
$29$	−1.36073	−0.252681	−0.126341	+	0.991987i	$0.540323\pi$
$30$	0	0
$31$	−7.29122	−1.30954	−0.654771	−	0.755827i	$-0.727235\pi$
$31$	−7.29122	−1.30954	−0.654771	+	0.755827i	$0.727235\pi$
$32$	7.25047	1.28171
$33$	0	0
$34$	12.2172	2.09524
$35$	−1.07589	−0.181859
$36$	0	0
$37$	2.35405	0.387003	0.193502	−	0.981100i	$-0.438016\pi$
$37$	2.35405	0.387003	0.193502	+	0.981100i	$0.438016\pi$
$38$	2.19978	0.356852
$39$	0	0
$40$	6.96239	1.10085
$41$	−3.84097	−0.599859	−0.299930	−	0.953961i	$-0.596963\pi$
$41$	−3.84097	−0.599859	−0.299930	+	0.953961i	$0.596963\pi$
$42$	0	0
$43$	−0.273376	−0.0416895	−0.0208448	−	0.999783i	$-0.506636\pi$
$43$	−0.273376	−0.0416895	−0.0208448	+	0.999783i	$0.506636\pi$
$44$	−2.83905	−0.428003
$45$	0	0
$46$	−7.31221	−1.07813
$47$	−5.16503	−0.753397	−0.376699	−	0.926336i	$-0.622941\pi$
$47$	−5.16503	−0.753397	−0.376699	+	0.926336i	$0.622941\pi$
$48$	0	0
$49$	−6.91865	−0.988378
$50$	−20.3020	−2.87114
$51$	0	0
$52$	−13.4687	−1.86777
$53$	−2.62973	−0.361221	−0.180610	−	0.983555i	$-0.557807\pi$
$53$	−2.62973	−0.361221	−0.180610	+	0.983555i	$0.557807\pi$
$54$	0	0
$55$	3.77215	0.508636
$56$	−0.526442	−0.0703488
$57$	0	0
$58$	2.99332	0.393042
$59$	−0.181257	−0.0235976	−0.0117988	−	0.999930i	$-0.503756\pi$
$59$	−0.181257	−0.0235976	−0.0117988	+	0.999930i	$0.503756\pi$
$60$	0	0
$61$	4.09404	0.524188	0.262094	−	0.965042i	$-0.415587\pi$
$61$	4.09404	0.524188	0.262094	+	0.965042i	$0.415587\pi$
$62$	16.0391	2.03697
$63$	0	0
$64$	−12.7137	−1.58921
$65$	17.8953	2.21964
$66$	0	0
$67$	−13.1781	−1.60996	−0.804981	−	0.593301i	$-0.797824\pi$
$67$	−13.1781	−1.60996	−0.804981	+	0.593301i	$0.797824\pi$
$68$	−15.7676	−1.91211
$69$	0	0
$70$	2.36674	0.282879
$71$	9.56720	1.13542	0.567709	−	0.823229i	$-0.307830\pi$
$71$	9.56720	1.13542	0.567709	+	0.823229i	$0.307830\pi$
$72$	0	0
$73$	−8.53461	−0.998901	−0.499450	−	0.866342i	$-0.666465\pi$
$73$	−8.53461	−0.998901	−0.499450	+	0.866342i	$0.666465\pi$
$74$	−5.17840	−0.601977
$75$	0	0
$76$	−2.83905	−0.325662
$77$	−0.285221	−0.0325039
$78$	0	0
$79$	7.80622	0.878268	0.439134	−	0.898422i	$-0.355285\pi$
$79$	7.80622	0.878268	0.439134	+	0.898422i	$0.355285\pi$
$80$	6.10289	0.682324
$81$	0	0
$82$	8.44931	0.933071
$83$	−5.53053	−0.607054	−0.303527	−	0.952823i	$-0.598164\pi$
$83$	−5.53053	−0.607054	−0.303527	+	0.952823i	$0.598164\pi$
$84$	0	0
$85$	20.9499	2.27233
$86$	0.601369	0.0648473
$87$	0	0
$88$	1.84574	0.196756
$89$	6.83811	0.724839	0.362419	−	0.932015i	$-0.381951\pi$
$89$	6.83811	0.724839	0.362419	+	0.932015i	$0.381951\pi$
$90$	0	0
$91$	−1.35311	−0.141844
$92$	9.43718	0.983894
$93$	0	0
$94$	11.3620	1.17190
$95$	3.77215	0.387014
$96$	0	0
$97$	8.58313	0.871485	0.435742	−	0.900071i	$-0.356486\pi$
$97$	8.58313	0.871485	0.435742	+	0.900071i	$0.356486\pi$
$98$	15.2195	1.53741
$99$	0	0
$100$	26.2019	2.62019
$101$	15.2232	1.51477	0.757385	−	0.652969i	$-0.226477\pi$
$101$	15.2232	1.51477	0.757385	+	0.652969i	$0.226477\pi$
$102$	0	0
$103$	−13.0844	−1.28924	−0.644620	−	0.764503i	$-0.722984\pi$
$103$	−13.0844	−1.28924	−0.644620	+	0.764503i	$0.722984\pi$
$104$	8.75632	0.858627
$105$	0	0
$106$	5.78483	0.561873
$107$	−13.3367	−1.28931	−0.644656	−	0.764473i	$-0.722999\pi$
$107$	−13.3367	−1.28931	−0.644656	+	0.764473i	$0.722999\pi$
$108$	0	0
$109$	−8.72078	−0.835300	−0.417650	−	0.908608i	$-0.637146\pi$
$109$	−8.72078	−0.835300	−0.417650	+	0.908608i	$0.637146\pi$
$110$	−8.29791	−0.791175
$111$	0	0
$112$	−0.461454	−0.0436033
$113$	11.4610	1.07816	0.539081	−	0.842254i	$-0.318772\pi$
$113$	11.4610	1.07816	0.539081	+	0.842254i	$0.318772\pi$
$114$	0	0
$115$	−12.5388	−1.16925
$116$	−3.86319	−0.358688
$117$	0	0
$118$	0.398726	0.0367057
$119$	−1.58407	−0.145211
$120$	0	0
$121$	1.00000	0.0909091
$122$	−9.00600	−0.815365
$123$	0	0
$124$	−20.7002	−1.85893
$125$	−15.9527	−1.42685
$126$	0	0
$127$	15.0627	1.33660	0.668298	−	0.743893i	$-0.267023\pi$
$127$	15.0627	1.33660	0.668298	+	0.743893i	$0.267023\pi$
$128$	13.4665	1.19028
$129$	0	0
$130$	−39.3659	−3.45262
$131$	7.32559	0.640040	0.320020	−	0.947411i	$-0.396310\pi$
$131$	7.32559	0.640040	0.320020	+	0.947411i	$0.396310\pi$
$132$	0	0
$133$	−0.285221	−0.0247318
$134$	28.9890	2.50427
$135$	0	0
$136$	10.2509	0.879009
$137$	16.7182	1.42833	0.714165	−	0.699977i	$-0.246807\pi$
$137$	16.7182	1.42833	0.714165	+	0.699977i	$0.246807\pi$
$138$	0	0
$139$	14.6532	1.24287	0.621433	−	0.783467i	$-0.286551\pi$
$139$	14.6532	1.24287	0.621433	+	0.783467i	$0.286551\pi$
$140$	−3.05452	−0.258154
$141$	0	0
$142$	−21.0458	−1.76612
$143$	4.74408	0.396720
$144$	0	0
$145$	5.13288	0.426262
$146$	18.7743	1.55377
$147$	0	0
$148$	6.68327	0.549361
$149$	−16.2379	−1.33026	−0.665132	−	0.746726i	$-0.731625\pi$
$149$	−16.2379	−1.33026	−0.665132	+	0.746726i	$0.731625\pi$
$150$	0	0
$151$	−6.62713	−0.539308	−0.269654	−	0.962957i	$-0.586909\pi$
$151$	−6.62713	−0.539308	−0.269654	+	0.962957i	$0.586909\pi$
$152$	1.84574	0.149709
$153$	0	0
$154$	0.627424	0.0505593
$155$	27.5036	2.20914
$156$	0	0
$157$	18.2284	1.45478	0.727392	−	0.686222i	$-0.240732\pi$
$157$	18.2284	1.45478	0.727392	+	0.686222i	$0.240732\pi$
$158$	−17.1720	−1.36613
$159$	0	0
$160$	−27.3498	−2.16219
$161$	0.948091	0.0747200
$162$	0	0
$163$	−9.09973	−0.712746	−0.356373	−	0.934344i	$-0.615987\pi$
$163$	−9.09973	−0.712746	−0.356373	+	0.934344i	$0.615987\pi$
$164$	−10.9047	−0.851516
$165$	0	0
$166$	12.1660	0.944262
$167$	−13.8907	−1.07490	−0.537448	−	0.843297i	$-0.680612\pi$
$167$	−13.8907	−1.07490	−0.537448	+	0.843297i	$0.680612\pi$
$168$	0	0
$169$	9.50626	0.731250
$170$	−46.0852	−3.53457
$171$	0	0
$172$	−0.776130	−0.0591794
$173$	−25.4759	−1.93690	−0.968449	−	0.249211i	$-0.919829\pi$
$173$	−25.4759	−1.93690	−0.968449	+	0.249211i	$0.919829\pi$
$174$	0	0
$175$	2.63233	0.198985
$176$	1.61788	0.121952
$177$	0	0
$178$	−15.0424	−1.12747
$179$	23.7149	1.77254	0.886269	−	0.463172i	$-0.153289\pi$
$179$	23.7149	1.77254	0.886269	+	0.463172i	$0.153289\pi$
$180$	0	0
$181$	21.1449	1.57169	0.785843	−	0.618425i	$-0.212229\pi$
$181$	21.1449	1.57169	0.785843	+	0.618425i	$0.212229\pi$
$182$	2.97655	0.220636
$183$	0	0
$184$	−6.13534	−0.452303
$185$	−8.87981	−0.652857
$186$	0	0
$187$	5.55383	0.406136
$188$	−14.6638	−1.06947
$189$	0	0
$190$	−8.29791	−0.601994
$191$	4.50161	0.325725	0.162863	−	0.986649i	$-0.447927\pi$
$191$	4.50161	0.325725	0.162863	+	0.986649i	$0.447927\pi$
$192$	0	0
$193$	27.0229	1.94515	0.972575	−	0.232587i	$-0.0747192\pi$
$193$	27.0229	1.94515	0.972575	+	0.232587i	$0.0747192\pi$
$194$	−18.8810	−1.35558
$195$	0	0
$196$	−19.6424	−1.40303
$197$	−8.86605	−0.631680	−0.315840	−	0.948813i	$-0.602286\pi$
$197$	−8.86605	−0.631680	−0.315840	+	0.948813i	$0.602286\pi$
$198$	0	0
$199$	20.3981	1.44598	0.722992	−	0.690857i	$-0.242766\pi$
$199$	20.3981	1.44598	0.722992	+	0.690857i	$0.242766\pi$
$200$	−17.0345	−1.20452
$201$	0	0
$202$	−33.4879	−2.35620
$203$	−0.388109	−0.0272399
$204$	0	0
$205$	14.4887	1.01194
$206$	28.7828	2.00539
$207$	0	0
$208$	7.67536	0.532190
$209$	1.00000	0.0691714
$210$	0	0
$211$	−14.9039	−1.02603	−0.513015	−	0.858380i	$-0.671471\pi$
$211$	−14.9039	−1.02603	−0.513015	+	0.858380i	$0.671471\pi$
$212$	−7.46593	−0.512763
$213$	0	0
$214$	29.3380	2.00550
$215$	1.03122	0.0703283
$216$	0	0
$217$	−2.07961	−0.141173
$218$	19.1838	1.29929
$219$	0	0
$220$	10.7093	0.722022
$221$	26.3478	1.77235
$222$	0	0
$223$	24.2633	1.62479	0.812396	−	0.583107i	$-0.198163\pi$
$223$	24.2633	1.62479	0.812396	+	0.583107i	$0.198163\pi$
$224$	2.06798	0.138173
$225$	0	0
$226$	−25.2118	−1.67706
$227$	12.7303	0.844941	0.422470	−	0.906377i	$-0.361163\pi$
$227$	12.7303	0.844941	0.422470	+	0.906377i	$0.361163\pi$
$228$	0	0
$229$	20.8791	1.37973	0.689865	−	0.723938i	$-0.257670\pi$
$229$	20.8791	1.37973	0.689865	+	0.723938i	$0.257670\pi$
$230$	27.5827	1.81875
$231$	0	0
$232$	2.51155	0.164891
$233$	−13.3620	−0.875371	−0.437685	−	0.899128i	$-0.644202\pi$
$233$	−13.3620	−0.875371	−0.437685	+	0.899128i	$0.644202\pi$
$234$	0	0
$235$	19.4833	1.27095
$236$	−0.514598	−0.0334975
$237$	0	0
$238$	3.48461	0.225874
$239$	8.33045	0.538852	0.269426	−	0.963021i	$-0.413166\pi$
$239$	8.33045	0.538852	0.269426	+	0.963021i	$0.413166\pi$
$240$	0	0
$241$	16.3940	1.05603	0.528014	−	0.849235i	$-0.322937\pi$
$241$	16.3940	1.05603	0.528014	+	0.849235i	$0.322937\pi$
$242$	−2.19978	−0.141408
$243$	0	0
$244$	11.6232	0.744099
$245$	26.0982	1.66735
$246$	0	0
$247$	4.74408	0.301858
$248$	13.4577	0.854564
$249$	0	0
$250$	35.0925	2.21945
$251$	−19.6236	−1.23863	−0.619315	−	0.785143i	$-0.712589\pi$
$251$	−19.6236	−1.23863	−0.619315	+	0.785143i	$0.712589\pi$
$252$	0	0
$253$	−3.32406	−0.208982
$254$	−33.1347	−2.07905
$255$	0	0
$256$	−4.19595	−0.262247
$257$	−6.90418	−0.430671	−0.215335	−	0.976540i	$-0.569084\pi$
$257$	−6.90418	−0.430671	−0.215335	+	0.976540i	$0.569084\pi$
$258$	0	0
$259$	0.671423	0.0417202
$260$	50.8058	3.15084
$261$	0	0
$262$	−16.1147	−0.995571
$263$	1.56529	0.0965201	0.0482601	−	0.998835i	$-0.484632\pi$
$263$	1.56529	0.0965201	0.0482601	+	0.998835i	$0.484632\pi$
$264$	0	0
$265$	9.91971	0.609363
$266$	0.627424	0.0384698
$267$	0	0
$268$	−37.4133	−2.28538
$269$	11.2801	0.687758	0.343879	−	0.939014i	$-0.388259\pi$
$269$	11.2801	0.687758	0.343879	+	0.939014i	$0.388259\pi$
$270$	0	0
$271$	−24.8371	−1.50875	−0.754374	−	0.656445i	$-0.772059\pi$
$271$	−24.8371	−1.50875	−0.754374	+	0.656445i	$0.772059\pi$
$272$	8.98545	0.544823
$273$	0	0
$274$	−36.7764	−2.22174
$275$	−9.22908	−0.556535
$276$	0	0
$277$	28.6191	1.71956	0.859778	−	0.510668i	$-0.170602\pi$
$277$	28.6191	1.71956	0.859778	+	0.510668i	$0.170602\pi$
$278$	−32.2339	−1.93326
$279$	0	0
$280$	1.98582	0.118675
$281$	12.0016	0.715956	0.357978	−	0.933730i	$-0.383466\pi$
$281$	12.0016	0.715956	0.357978	+	0.933730i	$0.383466\pi$
$282$	0	0
$283$	−23.7337	−1.41082	−0.705411	−	0.708798i	$-0.749238\pi$
$283$	−23.7337	−1.41082	−0.705411	+	0.708798i	$0.749238\pi$
$284$	27.1618	1.61176
$285$	0	0
$286$	−10.4359	−0.617090
$287$	−1.09553	−0.0646668
$288$	0	0
$289$	13.8451	0.814415
$290$	−11.2912	−0.663044
$291$	0	0
$292$	−24.2302	−1.41797
$293$	−7.29885	−0.426403	−0.213202	−	0.977008i	$-0.568389\pi$
$293$	−7.29885	−0.426403	−0.213202	+	0.977008i	$0.568389\pi$
$294$	0	0
$295$	0.683728	0.0398082
$296$	−4.34495	−0.252545
$297$	0	0
$298$	35.7200	2.06920
$299$	−15.7696	−0.911979
$300$	0	0
$301$	−0.0779726	−0.00449427
$302$	14.5783	0.838884
$303$	0	0
$304$	1.61788	0.0927919
$305$	−15.4433	−0.884281
$306$	0	0
$307$	7.39396	0.421996	0.210998	−	0.977487i	$-0.432329\pi$
$307$	7.39396	0.421996	0.210998	+	0.977487i	$0.432329\pi$
$308$	−0.809757	−0.0461402
$309$	0	0
$310$	−60.5019	−3.43628
$311$	−10.2577	−0.581664	−0.290832	−	0.956774i	$-0.593932\pi$
$311$	−10.2577	−0.581664	−0.290832	+	0.956774i	$0.593932\pi$
$312$	0	0
$313$	−6.92802	−0.391595	−0.195797	−	0.980644i	$-0.562729\pi$
$313$	−6.92802	−0.391595	−0.195797	+	0.980644i	$0.562729\pi$
$314$	−40.0985	−2.26289
$315$	0	0
$316$	22.1623	1.24673
$317$	21.3336	1.19822	0.599108	−	0.800668i	$-0.295522\pi$
$317$	21.3336	1.19822	0.599108	+	0.800668i	$0.295522\pi$
$318$	0	0
$319$	1.36073	0.0761863
$320$	47.9579	2.68093
$321$	0	0
$322$	−2.08560	−0.116226
$323$	5.55383	0.309024
$324$	0	0
$325$	−43.7835	−2.42867
$326$	20.0175	1.10866
$327$	0	0
$328$	7.08943	0.391448
$329$	−1.47317	−0.0812187
$330$	0	0
$331$	25.2867	1.38988	0.694942	−	0.719065i	$-0.255430\pi$
$331$	25.2867	1.38988	0.694942	+	0.719065i	$0.255430\pi$
$332$	−15.7015	−0.861730
$333$	0	0
$334$	30.5566	1.67198
$335$	49.7097	2.71593
$336$	0	0
$337$	−21.2229	−1.15608	−0.578041	−	0.816007i	$-0.696183\pi$
$337$	−21.2229	−1.15608	−0.578041	+	0.816007i	$0.696183\pi$
$338$	−20.9117	−1.13745
$339$	0	0
$340$	59.4778	3.22564
$341$	7.29122	0.394842
$342$	0	0
$343$	−3.96989	−0.214354
$344$	0.504581	0.0272052
$345$	0	0
$346$	56.0416	3.01281
$347$	−19.8663	−1.06648	−0.533240	−	0.845964i	$-0.679026\pi$
$347$	−19.8663	−1.06648	−0.533240	+	0.845964i	$0.679026\pi$
$348$	0	0
$349$	−22.2628	−1.19170	−0.595851	−	0.803095i	$-0.703185\pi$
$349$	−22.2628	−1.19170	−0.595851	+	0.803095i	$0.703185\pi$
$350$	−5.79055	−0.309518
$351$	0	0
$352$	−7.25047	−0.386451
$353$	−30.5355	−1.62524	−0.812619	−	0.582795i	$-0.801959\pi$
$353$	−30.5355	−1.62524	−0.812619	+	0.582795i	$0.801959\pi$
$354$	0	0
$355$	−36.0889	−1.91540
$356$	19.4138	1.02893
$357$	0	0
$358$	−52.1677	−2.75715
$359$	20.2416	1.06831	0.534156	−	0.845386i	$-0.320630\pi$
$359$	20.2416	1.06831	0.534156	+	0.845386i	$0.320630\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	−46.5142	−2.44473
$363$	0	0
$364$	−3.84155	−0.201352
$365$	32.1938	1.68510
$366$	0	0
$367$	−18.9033	−0.986743	−0.493371	−	0.869819i	$-0.664236\pi$
$367$	−18.9033	−0.986743	−0.493371	+	0.869819i	$0.664236\pi$
$368$	−5.37794	−0.280344
$369$	0	0
$370$	19.5337	1.01551
$371$	−0.750053	−0.0389408
$372$	0	0
$373$	−23.4933	−1.21644	−0.608219	−	0.793769i	$-0.708116\pi$
$373$	−23.4933	−1.21644	−0.608219	+	0.793769i	$0.708116\pi$
$374$	−12.2172	−0.631738
$375$	0	0
$376$	9.53329	0.491642
$377$	6.45541	0.332471
$378$	0	0
$379$	−21.3654	−1.09747	−0.548734	−	0.835997i	$-0.684890\pi$
$379$	−21.3654	−1.09747	−0.548734	+	0.835997i	$0.684890\pi$
$380$	10.7093	0.549377
$381$	0	0
$382$	−9.90258	−0.506660
$383$	24.1653	1.23479	0.617395	−	0.786653i	$-0.288188\pi$
$383$	24.1653	1.23479	0.617395	+	0.786653i	$0.288188\pi$
$384$	0	0
$385$	1.07589	0.0548326
$386$	−59.4446	−3.02565
$387$	0	0
$388$	24.3680	1.23710
$389$	18.5741	0.941745	0.470873	−	0.882201i	$-0.343939\pi$
$389$	18.5741	0.941745	0.470873	+	0.882201i	$0.343939\pi$
$390$	0	0
$391$	−18.4613	−0.933626
$392$	12.7700	0.644983
$393$	0	0
$394$	19.5034	0.982567
$395$	−29.4462	−1.48160
$396$	0	0
$397$	−31.5368	−1.58279	−0.791394	−	0.611307i	$-0.790644\pi$
$397$	−31.5368	−1.58279	−0.791394	+	0.611307i	$0.790644\pi$
$398$	−44.8715	−2.24920
$399$	0	0
$400$	−14.9316	−0.746578
$401$	−13.2073	−0.659539	−0.329770	−	0.944061i	$-0.606971\pi$
$401$	−13.2073	−0.659539	−0.329770	+	0.944061i	$0.606971\pi$
$402$	0	0
$403$	34.5901	1.72306
$404$	43.2196	2.15026
$405$	0	0
$406$	0.853756	0.0423712
$407$	−2.35405	−0.116686
$408$	0	0
$409$	−6.85652	−0.339033	−0.169517	−	0.985527i	$-0.554221\pi$
$409$	−6.85652	−0.339033	−0.169517	+	0.985527i	$0.554221\pi$
$410$	−31.8720	−1.57405
$411$	0	0
$412$	−37.1472	−1.83011
$413$	−0.0516982	−0.00254390
$414$	0	0
$415$	20.8620	1.02407
$416$	−34.3968	−1.68644
$417$	0	0
$418$	−2.19978	−0.107595
$419$	−35.4076	−1.72978	−0.864888	−	0.501964i	$-0.832611\pi$
$419$	−35.4076	−1.72978	−0.864888	+	0.501964i	$0.832611\pi$
$420$	0	0
$421$	20.1569	0.982386	0.491193	−	0.871051i	$-0.336561\pi$
$421$	20.1569	0.982386	0.491193	+	0.871051i	$0.336561\pi$
$422$	32.7855	1.59597
$423$	0	0
$424$	4.85378	0.235721
$425$	−51.2568	−2.48632
$426$	0	0
$427$	1.16770	0.0565092
$428$	−37.8637	−1.83021
$429$	0	0
$430$	−2.26845	−0.109394
$431$	9.40051	0.452807	0.226403	−	0.974034i	$-0.427303\pi$
$431$	9.40051	0.452807	0.226403	+	0.974034i	$0.427303\pi$
$432$	0	0
$433$	33.7828	1.62350	0.811748	−	0.584008i	$-0.198516\pi$
$433$	33.7828	1.62350	0.811748	+	0.584008i	$0.198516\pi$
$434$	4.57469	0.219592
$435$	0	0
$436$	−24.7588	−1.18573
$437$	−3.32406	−0.159011
$438$	0	0
$439$	36.8924	1.76078	0.880388	−	0.474254i	$-0.157282\pi$
$439$	36.8924	1.76078	0.880388	+	0.474254i	$0.157282\pi$
$440$	−6.96239	−0.331919
$441$	0	0
$442$	−57.9595	−2.75685
$443$	−2.62320	−0.124632	−0.0623161	−	0.998056i	$-0.519849\pi$
$443$	−2.62320	−0.124632	−0.0623161	+	0.998056i	$0.519849\pi$
$444$	0	0
$445$	−25.7944	−1.22277
$446$	−53.3741	−2.52733
$447$	0	0
$448$	−3.62621	−0.171322
$449$	18.4382	0.870151	0.435075	−	0.900394i	$-0.356722\pi$
$449$	18.4382	0.870151	0.435075	+	0.900394i	$0.356722\pi$
$450$	0	0
$451$	3.84097	0.180864
$452$	32.5385	1.53048
$453$	0	0
$454$	−28.0039	−1.31429
$455$	5.10412	0.239285
$456$	0	0
$457$	−32.3809	−1.51472	−0.757358	−	0.652999i	$-0.773510\pi$
$457$	−32.3809	−1.51472	−0.757358	+	0.652999i	$0.773510\pi$
$458$	−45.9296	−2.14615
$459$	0	0
$460$	−35.5984	−1.65978
$461$	8.54990	0.398208	0.199104	−	0.979978i	$-0.436197\pi$
$461$	8.54990	0.398208	0.199104	+	0.979978i	$0.436197\pi$
$462$	0	0
$463$	35.1511	1.63361	0.816806	−	0.576913i	$-0.195743\pi$
$463$	35.1511	1.63361	0.816806	+	0.576913i	$0.195743\pi$
$464$	2.20150	0.102202
$465$	0	0
$466$	29.3934	1.36162
$467$	−19.6171	−0.907770	−0.453885	−	0.891060i	$-0.649962\pi$
$467$	−19.6171	−0.907770	−0.453885	+	0.891060i	$0.649962\pi$
$468$	0	0
$469$	−3.75867	−0.173559
$470$	−42.8590	−1.97694
$471$	0	0
$472$	0.334553	0.0153990
$473$	0.273376	0.0125699
$474$	0	0
$475$	−9.22908	−0.423459
$476$	−4.49725	−0.206131
$477$	0	0
$478$	−18.3252	−0.838175
$479$	−3.69611	−0.168880	−0.0844399	−	0.996429i	$-0.526910\pi$
$479$	−3.69611	−0.168880	−0.0844399	+	0.996429i	$0.526910\pi$
$480$	0	0
$481$	−11.1678	−0.509207
$482$	−36.0632	−1.64263
$483$	0	0
$484$	2.83905	0.129048
$485$	−32.3768	−1.47016
$486$	0	0
$487$	2.13850	0.0969046	0.0484523	−	0.998825i	$-0.484571\pi$
$487$	2.13850	0.0969046	0.0484523	+	0.998825i	$0.484571\pi$
$488$	−7.55652	−0.342068
$489$	0	0
$490$	−57.4103	−2.59353
$491$	4.53598	0.204706	0.102353	−	0.994748i	$-0.467363\pi$
$491$	4.53598	0.204706	0.102353	+	0.994748i	$0.467363\pi$
$492$	0	0
$493$	7.55727	0.340363
$494$	−10.4359	−0.469535
$495$	0	0
$496$	11.7963	0.529672
$497$	2.72876	0.122402
$498$	0	0
$499$	−31.0692	−1.39085	−0.695425	−	0.718599i	$-0.744784\pi$
$499$	−31.0692	−1.39085	−0.695425	+	0.718599i	$0.744784\pi$
$500$	−45.2906	−2.02546
$501$	0	0
$502$	43.1676	1.92667
$503$	−20.4190	−0.910440	−0.455220	−	0.890379i	$-0.650439\pi$
$503$	−20.4190	−0.910440	−0.455220	+	0.890379i	$0.650439\pi$
$504$	0	0
$505$	−57.4243	−2.55535
$506$	7.31221	0.325068
$507$	0	0
$508$	42.7637	1.89733
$509$	−24.6556	−1.09284	−0.546421	−	0.837511i	$-0.684010\pi$
$509$	−24.6556	−1.09284	−0.546421	+	0.837511i	$0.684010\pi$
$510$	0	0
$511$	−2.43425	−0.107685
$512$	−17.7028	−0.782360
$513$	0	0
$514$	15.1877	0.669901
$515$	49.3561	2.17489
$516$	0	0
$517$	5.16503	0.227158
$518$	−1.47699	−0.0648951
$519$	0	0
$520$	−33.0301	−1.44847
$521$	29.7031	1.30132	0.650658	−	0.759371i	$-0.274493\pi$
$521$	29.7031	1.30132	0.650658	+	0.759371i	$0.274493\pi$
$522$	0	0
$523$	−11.0994	−0.485341	−0.242670	−	0.970109i	$-0.578023\pi$
$523$	−11.0994	−0.485341	−0.242670	+	0.970109i	$0.578023\pi$
$524$	20.7978	0.908554
$525$	0	0
$526$	−3.44331	−0.150135
$527$	40.4942	1.76396
$528$	0	0
$529$	−11.9506	−0.519593
$530$	−21.8212	−0.947854
$531$	0	0
$532$	−0.809757	−0.0351074
$533$	18.2219	0.789277
$534$	0	0
$535$	50.3081	2.17501
$536$	24.3233	1.05061
$537$	0	0
$538$	−24.8137	−1.06980
$539$	6.91865	0.298007
$540$	0	0
$541$	−13.3404	−0.573549	−0.286775	−	0.957998i	$-0.592583\pi$
$541$	−13.3404	−0.573549	−0.286775	+	0.957998i	$0.592583\pi$
$542$	54.6363	2.34683
$543$	0	0
$544$	−40.2679	−1.72647
$545$	32.8961	1.40911
$546$	0	0
$547$	27.4886	1.17533	0.587665	−	0.809104i	$-0.300047\pi$
$547$	27.4886	1.17533	0.587665	+	0.809104i	$0.300047\pi$
$548$	47.4638	2.02755
$549$	0	0
$550$	20.3020	0.865680
$551$	1.36073	0.0579691
$552$	0	0
$553$	2.22650	0.0946802
$554$	−62.9559	−2.67474
$555$	0	0
$556$	41.6012	1.76428
$557$	−13.5987	−0.576194	−0.288097	−	0.957601i	$-0.593023\pi$
$557$	−13.5987	−0.576194	−0.288097	+	0.957601i	$0.593023\pi$
$558$	0	0
$559$	1.29692	0.0548538
$560$	1.74067	0.0735567
$561$	0	0
$562$	−26.4010	−1.11366
$563$	−28.6585	−1.20781	−0.603907	−	0.797055i	$-0.706390\pi$
$563$	−28.6585	−1.20781	−0.603907	+	0.797055i	$0.706390\pi$
$564$	0	0
$565$	−43.2326	−1.81881
$566$	52.2090	2.19451
$567$	0	0
$568$	−17.6585	−0.740936
$569$	−7.63179	−0.319941	−0.159971	−	0.987122i	$-0.551140\pi$
$569$	−7.63179	−0.319941	−0.159971	+	0.987122i	$0.551140\pi$
$570$	0	0
$571$	21.3321	0.892721	0.446361	−	0.894853i	$-0.352720\pi$
$571$	21.3321	0.892721	0.446361	+	0.894853i	$0.352720\pi$
$572$	13.4687	0.563154
$573$	0	0
$574$	2.40992	0.100588
$575$	30.6780	1.27936
$576$	0	0
$577$	−16.0695	−0.668983	−0.334491	−	0.942399i	$-0.608565\pi$
$577$	−16.0695	−0.668983	−0.334491	+	0.942399i	$0.608565\pi$
$578$	−30.4561	−1.26681
$579$	0	0
$580$	14.5725	0.605091
$581$	−1.57742	−0.0654425
$582$	0	0
$583$	2.62973	0.108912
$584$	15.7527	0.651849
$585$	0	0
$586$	16.0559	0.663263
$587$	5.44519	0.224747	0.112373	−	0.993666i	$-0.464155\pi$
$587$	5.44519	0.224747	0.112373	+	0.993666i	$0.464155\pi$
$588$	0	0
$589$	7.29122	0.300430
$590$	−1.50405	−0.0619209
$591$	0	0
$592$	−3.80857	−0.156531
$593$	−5.86311	−0.240769	−0.120385	−	0.992727i	$-0.538413\pi$
$593$	−5.86311	−0.240769	−0.120385	+	0.992727i	$0.538413\pi$
$594$	0	0
$595$	5.97534	0.244965
$596$	−46.1004	−1.88835
$597$	0	0
$598$	34.6897	1.41857
$599$	4.92480	0.201222	0.100611	−	0.994926i	$-0.467920\pi$
$599$	4.92480	0.201222	0.100611	+	0.994926i	$0.467920\pi$
$600$	0	0
$601$	16.3561	0.667180	0.333590	−	0.942718i	$-0.391740\pi$
$601$	16.3561	0.667180	0.333590	+	0.942718i	$0.391740\pi$
$602$	0.171523	0.00699076
$603$	0	0
$604$	−18.8148	−0.765562
$605$	−3.77215	−0.153360
$606$	0	0
$607$	1.40051	0.0568449	0.0284224	−	0.999596i	$-0.490952\pi$
$607$	1.40051	0.0568449	0.0284224	+	0.999596i	$0.490952\pi$
$608$	−7.25047	−0.294045
$609$	0	0
$610$	33.9720	1.37548
$611$	24.5033	0.991297
$612$	0	0
$613$	−3.61611	−0.146053	−0.0730267	−	0.997330i	$-0.523266\pi$
$613$	−3.61611	−0.146053	−0.0730267	+	0.997330i	$0.523266\pi$
$614$	−16.2651	−0.656407
$615$	0	0
$616$	0.526442	0.0212110
$617$	27.2201	1.09584	0.547920	−	0.836531i	$-0.315420\pi$
$617$	27.2201	1.09584	0.547920	+	0.836531i	$0.315420\pi$
$618$	0	0
$619$	−47.7931	−1.92097	−0.960483	−	0.278339i	$-0.910216\pi$
$619$	−47.7931	−1.92097	−0.960483	+	0.278339i	$0.910216\pi$
$620$	78.0841	3.13593
$621$	0	0
$622$	22.5648	0.904768
$623$	1.95037	0.0781400
$624$	0	0
$625$	14.0305	0.561221
$626$	15.2402	0.609119
$627$	0	0
$628$	51.7514	2.06510
$629$	−13.0740	−0.521294
$630$	0	0
$631$	−7.46936	−0.297351	−0.148675	−	0.988886i	$-0.547501\pi$
$631$	−7.46936	−0.297351	−0.148675	+	0.988886i	$0.547501\pi$
$632$	−14.4082	−0.573129
$633$	0	0
$634$	−46.9294	−1.86381
$635$	−56.8186	−2.25478
$636$	0	0
$637$	32.8226	1.30048
$638$	−2.99332	−0.118507
$639$	0	0
$640$	−50.7975	−2.00795
$641$	−14.3855	−0.568193	−0.284096	−	0.958796i	$-0.591694\pi$
$641$	−14.3855	−0.568193	−0.284096	+	0.958796i	$0.591694\pi$
$642$	0	0
$643$	−8.85612	−0.349251	−0.174626	−	0.984635i	$-0.555872\pi$
$643$	−8.85612	−0.349251	−0.174626	+	0.984635i	$0.555872\pi$
$644$	2.69168	0.106067
$645$	0	0
$646$	−12.2172	−0.480681
$647$	16.8325	0.661755	0.330877	−	0.943674i	$-0.392655\pi$
$647$	16.8325	0.661755	0.330877	+	0.943674i	$0.392655\pi$
$648$	0	0
$649$	0.181257	0.00711496
$650$	96.3142	3.77775
$651$	0	0
$652$	−25.8346	−1.01176
$653$	−6.09828	−0.238644	−0.119322	−	0.992856i	$-0.538072\pi$
$653$	−6.09828	−0.238644	−0.119322	+	0.992856i	$0.538072\pi$
$654$	0	0
$655$	−27.6332	−1.07972
$656$	6.21424	0.242625
$657$	0	0
$658$	3.24067	0.126334
$659$	−11.4878	−0.447500	−0.223750	−	0.974647i	$-0.571830\pi$
$659$	−11.4878	−0.447500	−0.223750	+	0.974647i	$0.571830\pi$
$660$	0	0
$661$	−3.06328	−0.119148	−0.0595739	−	0.998224i	$-0.518974\pi$
$661$	−3.06328	−0.119148	−0.0595739	+	0.998224i	$0.518974\pi$
$662$	−55.6254	−2.16194
$663$	0	0
$664$	10.2079	0.396143
$665$	1.07589	0.0417214
$666$	0	0
$667$	−4.52315	−0.175137
$668$	−39.4365	−1.52584
$669$	0	0
$670$	−109.351	−4.22459
$671$	−4.09404	−0.158049
$672$	0	0
$673$	8.72170	0.336197	0.168098	−	0.985770i	$-0.446237\pi$
$673$	8.72170	0.336197	0.168098	+	0.985770i	$0.446237\pi$
$674$	46.6857	1.79827
$675$	0	0
$676$	26.9888	1.03803
$677$	−46.1571	−1.77396	−0.886981	−	0.461806i	$-0.847201\pi$
$677$	−46.1571	−1.77396	−0.886981	+	0.461806i	$0.847201\pi$
$678$	0	0
$679$	2.44809	0.0939489
$680$	−38.6679	−1.48285
$681$	0	0
$682$	−16.0391	−0.614170
$683$	3.72309	0.142460	0.0712301	−	0.997460i	$-0.477308\pi$
$683$	3.72309	0.142460	0.0712301	+	0.997460i	$0.477308\pi$
$684$	0	0
$685$	−63.0634	−2.40953
$686$	8.73290	0.333424
$687$	0	0
$688$	0.442291	0.0168622
$689$	12.4756	0.475283
$690$	0	0
$691$	26.7843	1.01892	0.509461	−	0.860494i	$-0.329845\pi$
$691$	26.7843	1.01892	0.509461	+	0.860494i	$0.329845\pi$
$692$	−72.3275	−2.74948
$693$	0	0
$694$	43.7017	1.65889
$695$	−55.2739	−2.09666
$696$	0	0
$697$	21.3321	0.808012
$698$	48.9735	1.85367
$699$	0	0
$700$	7.47331	0.282465
$701$	35.5284	1.34189	0.670944	−	0.741508i	$-0.265889\pi$
$701$	35.5284	1.34189	0.670944	+	0.741508i	$0.265889\pi$
$702$	0	0
$703$	−2.35405	−0.0887846
$704$	12.7137	0.479166
$705$	0	0
$706$	67.1714	2.52803
$707$	4.34198	0.163297
$708$	0	0
$709$	−47.6498	−1.78952	−0.894762	−	0.446543i	$-0.852655\pi$
$709$	−47.6498	−1.78952	−0.894762	+	0.446543i	$0.852655\pi$
$710$	79.3877	2.97937
$711$	0	0
$712$	−12.6214	−0.473005
$713$	−24.2365	−0.907663
$714$	0	0
$715$	−17.8953	−0.669248
$716$	67.3280	2.51616
$717$	0	0
$718$	−44.5272	−1.66174
$719$	10.3758	0.386951	0.193475	−	0.981105i	$-0.438024\pi$
$719$	10.3758	0.386951	0.193475	+	0.981105i	$0.438024\pi$
$720$	0	0
$721$	−3.73193	−0.138984
$722$	−2.19978	−0.0818675
$723$	0	0
$724$	60.0315	2.23105
$725$	−12.5583	−0.466404
$726$	0	0
$727$	3.91731	0.145285	0.0726425	−	0.997358i	$-0.476857\pi$
$727$	3.91731	0.145285	0.0726425	+	0.997358i	$0.476857\pi$
$728$	2.49748	0.0925629
$729$	0	0
$730$	−70.8194	−2.62114
$731$	1.51829	0.0561559
$732$	0	0
$733$	29.4989	1.08957	0.544784	−	0.838577i	$-0.316612\pi$
$733$	29.4989	1.08957	0.544784	+	0.838577i	$0.316612\pi$
$734$	41.5831	1.53486
$735$	0	0
$736$	24.1010	0.888374
$737$	13.1781	0.485422
$738$	0	0
$739$	−5.82902	−0.214424	−0.107212	−	0.994236i	$-0.534192\pi$
$739$	−5.82902	−0.214424	−0.107212	+	0.994236i	$0.534192\pi$
$740$	−25.2103	−0.926747
$741$	0	0
$742$	1.64995	0.0605717
$743$	13.4613	0.493848	0.246924	−	0.969035i	$-0.420580\pi$
$743$	13.4613	0.493848	0.246924	+	0.969035i	$0.420580\pi$
$744$	0	0
$745$	61.2519	2.24410
$746$	51.6803	1.89215
$747$	0	0
$748$	15.7676	0.576522
$749$	−3.80392	−0.138992
$750$	0	0
$751$	15.7502	0.574732	0.287366	−	0.957821i	$-0.407220\pi$
$751$	15.7502	0.574732	0.287366	+	0.957821i	$0.407220\pi$
$752$	8.35641	0.304727
$753$	0	0
$754$	−14.2005	−0.517152
$755$	24.9985	0.909788
$756$	0	0
$757$	19.0154	0.691126	0.345563	−	0.938396i	$-0.387688\pi$
$757$	19.0154	0.691126	0.345563	+	0.938396i	$0.387688\pi$
$758$	46.9993	1.70709
$759$	0	0
$760$	−6.96239	−0.252552
$761$	23.2403	0.842459	0.421229	−	0.906954i	$-0.361599\pi$
$761$	23.2403	0.842459	0.421229	+	0.906954i	$0.361599\pi$
$762$	0	0
$763$	−2.48735	−0.0900481
$764$	12.7803	0.462376
$765$	0	0
$766$	−53.1585	−1.92070
$767$	0.859897	0.0310491
$768$	0	0
$769$	−39.4802	−1.42369	−0.711847	−	0.702335i	$-0.752141\pi$
$769$	−39.4802	−1.42369	−0.711847	+	0.702335i	$0.752141\pi$
$770$	−2.36674	−0.0852912
$771$	0	0
$772$	76.7195	2.76119
$773$	0.583500	0.0209870	0.0104935	−	0.999945i	$-0.496660\pi$
$773$	0.583500	0.0209870	0.0104935	+	0.999945i	$0.496660\pi$
$774$	0	0
$775$	−67.2913	−2.41718
$776$	−15.8422	−0.568702
$777$	0	0
$778$	−40.8591	−1.46487
$779$	3.84097	0.137617
$780$	0	0
$781$	−9.56720	−0.342341
$782$	40.6108	1.45224
$783$	0	0
$784$	11.1936	0.399770
$785$	−68.7601	−2.45415
$786$	0	0
$787$	24.4941	0.873121	0.436560	−	0.899675i	$-0.356197\pi$
$787$	24.4941	0.873121	0.436560	+	0.899675i	$0.356197\pi$
$788$	−25.1712	−0.896686
$789$	0	0
$790$	64.7753	2.30460
$791$	3.26892	0.116229
$792$	0	0
$793$	−19.4224	−0.689711
$794$	69.3742	2.46200
$795$	0	0
$796$	57.9113	2.05261
$797$	5.85756	0.207485	0.103743	−	0.994604i	$-0.466918\pi$
$797$	5.85756	0.207485	0.103743	+	0.994604i	$0.466918\pi$
$798$	0	0
$799$	28.6857	1.01483
$800$	66.9152	2.36581
$801$	0	0
$802$	29.0531	1.02590
$803$	8.53461	0.301180
$804$	0	0
$805$	−3.57634	−0.126049
$806$	−76.0908	−2.68019
$807$	0	0
$808$	−28.0981	−0.988488
$809$	5.99724	0.210852	0.105426	−	0.994427i	$-0.466379\pi$
$809$	5.99724	0.210852	0.105426	+	0.994427i	$0.466379\pi$
$810$	0	0
$811$	0.774595	0.0271997	0.0135998	−	0.999908i	$-0.495671\pi$
$811$	0.774595	0.0271997	0.0135998	+	0.999908i	$0.495671\pi$
$812$	−1.10186	−0.0386678
$813$	0	0
$814$	5.17840	0.181503
$815$	34.3255	1.20237
$816$	0	0
$817$	0.273376	0.00956423
$818$	15.0829	0.527360
$819$	0	0
$820$	41.1342	1.43647
$821$	−54.7986	−1.91248	−0.956242	−	0.292576i	$-0.905488\pi$
$821$	−54.7986	−1.91248	−0.956242	+	0.292576i	$0.905488\pi$
$822$	0	0
$823$	29.0580	1.01290	0.506448	−	0.862270i	$-0.330958\pi$
$823$	29.0580	1.01290	0.506448	+	0.862270i	$0.330958\pi$
$824$	24.1503	0.841315
$825$	0	0
$826$	0.113725	0.00395700
$827$	31.0825	1.08084	0.540422	−	0.841394i	$-0.318265\pi$
$827$	31.0825	1.08084	0.540422	+	0.841394i	$0.318265\pi$
$828$	0	0
$829$	33.2504	1.15483	0.577417	−	0.816450i	$-0.304061\pi$
$829$	33.2504	1.15483	0.577417	+	0.816450i	$0.304061\pi$
$830$	−45.8918	−1.59293
$831$	0	0
$832$	60.3148	2.09104
$833$	38.4250	1.33135
$834$	0	0
$835$	52.3978	1.81330
$836$	2.83905	0.0981907
$837$	0	0
$838$	77.8892	2.69064
$839$	−38.9887	−1.34604	−0.673020	−	0.739625i	$-0.735003\pi$
$839$	−38.9887	−1.34604	−0.673020	+	0.739625i	$0.735003\pi$
$840$	0	0
$841$	−27.1484	−0.936152
$842$	−44.3408	−1.52808
$843$	0	0
$844$	−42.3131	−1.45648
$845$	−35.8590	−1.23359
$846$	0	0
$847$	0.285221	0.00980030
$848$	4.25459	0.146103
$849$	0	0
$850$	112.754	3.86743
$851$	7.82499	0.268237
$852$	0	0
$853$	−37.6341	−1.28857	−0.644283	−	0.764787i	$-0.722844\pi$
$853$	−37.6341	−1.28857	−0.644283	+	0.764787i	$0.722844\pi$
$854$	−2.56870	−0.0878991
$855$	0	0
$856$	24.6161	0.841362
$857$	−15.4016	−0.526110	−0.263055	−	0.964781i	$-0.584730\pi$
$857$	−15.4016	−0.526110	−0.263055	+	0.964781i	$0.584730\pi$
$858$	0	0
$859$	29.1781	0.995545	0.497773	−	0.867308i	$-0.334151\pi$
$859$	29.1781	0.995545	0.497773	+	0.867308i	$0.334151\pi$
$860$	2.92768	0.0998329
$861$	0	0
$862$	−20.6791	−0.704333
$863$	17.3099	0.589236	0.294618	−	0.955615i	$-0.404808\pi$
$863$	17.3099	0.589236	0.294618	+	0.955615i	$0.404808\pi$
$864$	0	0
$865$	96.0989	3.26746
$866$	−74.3148	−2.52532
$867$	0	0
$868$	−5.90412	−0.200399
$869$	−7.80622	−0.264808
$870$	0	0
$871$	62.5179	2.11834
$872$	16.0963	0.545089
$873$	0	0
$874$	7.31221	0.247339
$875$	−4.55004	−0.153820
$876$	0	0
$877$	−35.0158	−1.18240	−0.591199	−	0.806526i	$-0.701345\pi$
$877$	−35.0158	−1.18240	−0.591199	+	0.806526i	$0.701345\pi$
$878$	−81.1552	−2.73886
$879$	0	0
$880$	−6.10289	−0.205728
$881$	58.9596	1.98640	0.993199	−	0.116429i	$-0.0371448\pi$
$881$	58.9596	1.98640	0.993199	+	0.116429i	$0.0371448\pi$
$882$	0	0
$883$	3.76986	0.126866	0.0634329	−	0.997986i	$-0.479795\pi$
$883$	3.76986	0.126866	0.0634329	+	0.997986i	$0.479795\pi$
$884$	74.8028	2.51589
$885$	0	0
$886$	5.77048	0.193863
$887$	18.8100	0.631577	0.315789	−	0.948830i	$-0.397731\pi$
$887$	18.8100	0.631577	0.315789	+	0.948830i	$0.397731\pi$
$888$	0	0
$889$	4.29619	0.144090
$890$	56.7420	1.90200
$891$	0	0
$892$	68.8848	2.30643
$893$	5.16503	0.172841
$894$	0	0
$895$	−89.4562	−2.99019
$896$	3.84092	0.128316
$897$	0	0
$898$	−40.5600	−1.35350
$899$	9.92140	0.330897
$900$	0	0
$901$	14.6051	0.486565
$902$	−8.44931	−0.281331
$903$	0	0
$904$	−21.1540	−0.703573
$905$	−79.7616	−2.65136
$906$	0	0
$907$	−50.6212	−1.68085	−0.840425	−	0.541928i	$-0.817695\pi$
$907$	−50.6212	−1.68085	−0.840425	+	0.541928i	$0.817695\pi$
$908$	36.1420	1.19942
$909$	0	0
$910$	−11.2280	−0.372204
$911$	38.8951	1.28865	0.644326	−	0.764751i	$-0.277138\pi$
$911$	38.8951	1.28865	0.644326	+	0.764751i	$0.277138\pi$
$912$	0	0
$913$	5.53053	0.183034
$914$	71.2311	2.35612
$915$	0	0
$916$	59.2769	1.95856
$917$	2.08941	0.0689984
$918$	0	0
$919$	27.9238	0.921120	0.460560	−	0.887628i	$-0.347649\pi$
$919$	27.9238	0.921120	0.460560	+	0.887628i	$0.347649\pi$
$920$	23.1434	0.763015
$921$	0	0
$922$	−18.8079	−0.619406
$923$	−45.3875	−1.49395
$924$	0	0
$925$	21.7257	0.714337
$926$	−77.3249	−2.54106
$927$	0	0
$928$	−9.86594	−0.323865
$929$	−7.81430	−0.256379	−0.128189	−	0.991750i	$-0.540917\pi$
$929$	−7.81430	−0.256379	−0.128189	+	0.991750i	$0.540917\pi$
$930$	0	0
$931$	6.91865	0.226750
$932$	−37.9353	−1.24261
$933$	0	0
$934$	43.1533	1.41202
$935$	−20.9499	−0.685134
$936$	0	0
$937$	−36.8471	−1.20374	−0.601871	−	0.798593i	$-0.705578\pi$
$937$	−36.8471	−1.20374	−0.601871	+	0.798593i	$0.705578\pi$
$938$	8.26826	0.269968
$939$	0	0
$940$	55.3140	1.80414
$941$	20.9485	0.682900	0.341450	−	0.939900i	$-0.389082\pi$
$941$	20.9485	0.682900	0.341450	+	0.939900i	$0.389082\pi$
$942$	0	0
$943$	−12.7676	−0.415771
$944$	0.293252	0.00954455
$945$	0	0
$946$	−0.601369	−0.0195522
$947$	−18.9097	−0.614481	−0.307241	−	0.951632i	$-0.599406\pi$
$947$	−18.9097	−0.614481	−0.307241	+	0.951632i	$0.599406\pi$
$948$	0	0
$949$	40.4888	1.31432
$950$	20.3020	0.658684
$951$	0	0
$952$	2.92377	0.0947600
$953$	59.9999	1.94359	0.971794	−	0.235830i	$-0.0757807\pi$
$953$	59.9999	1.94359	0.971794	+	0.235830i	$0.0757807\pi$
$954$	0	0
$955$	−16.9807	−0.549484
$956$	23.6506	0.764915
$957$	0	0
$958$	8.13066	0.262690
$959$	4.76837	0.153979
$960$	0	0
$961$	22.1620	0.714902
$962$	24.5667	0.792063
$963$	0	0
$964$	46.5434	1.49906
$965$	−101.934	−3.28138
$966$	0	0
$967$	−13.4653	−0.433015	−0.216508	−	0.976281i	$-0.569467\pi$
$967$	−13.4653	−0.433015	−0.216508	+	0.976281i	$0.569467\pi$
$968$	−1.84574	−0.0593242
$969$	0	0
$970$	71.2220	2.28680
$971$	16.1615	0.518647	0.259324	−	0.965791i	$-0.416500\pi$
$971$	16.1615	0.518647	0.259324	+	0.965791i	$0.416500\pi$
$972$	0	0
$973$	4.17939	0.133985
$974$	−4.70423	−0.150733
$975$	0	0
$976$	−6.62367	−0.212019
$977$	23.9441	0.766038	0.383019	−	0.923740i	$-0.374884\pi$
$977$	23.9441	0.766038	0.383019	+	0.923740i	$0.374884\pi$
$978$	0	0
$979$	−6.83811	−0.218547
$980$	74.0940	2.36685
$981$	0	0
$982$	−9.97819	−0.318417
$983$	−2.99043	−0.0953800	−0.0476900	−	0.998862i	$-0.515186\pi$
$983$	−2.99043	−0.0953800	−0.0476900	+	0.998862i	$0.515186\pi$
$984$	0	0
$985$	33.4440	1.06562
$986$	−16.6244	−0.529428
$987$	0	0
$988$	13.4687	0.428496
$989$	−0.908719	−0.0288956
$990$	0	0
$991$	−36.7899	−1.16867	−0.584334	−	0.811513i	$-0.698644\pi$
$991$	−36.7899	−1.16867	−0.584334	+	0.811513i	$0.698644\pi$
$992$	−52.8648	−1.67846
$993$	0	0
$994$	−6.00269	−0.190394
$995$	−76.9446	−2.43931
$996$	0	0
$997$	−31.3403	−0.992559	−0.496279	−	0.868163i	$-0.665301\pi$
$997$	−31.3403	−0.992559	−0.496279	+	0.868163i	$0.665301\pi$
$998$	68.3456	2.16344
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1881.2.a.m.1.2		5
3.2	odd	2		627.2.a.j.1.4	✓	5
33.32	even	2		6897.2.a.s.1.2		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
627.2.a.j.1.4	✓	5	3.2	odd	2
1881.2.a.m.1.2		5	1.1	even	1	trivial
6897.2.a.s.1.2		5	33.32	even	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:	\( N \)	\(=\)	\( 1881 = 3^{2} \cdot 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1881.a (trivial)

\(f(q)\)	\(=\)	\(q-2.19978 q^{2} +2.83905 q^{4} -3.77215 q^{5} +0.285221 q^{7} -1.84574 q^{8} +8.29791 q^{10} -1.00000 q^{11} -4.74408 q^{13} -0.627424 q^{14} -1.61788 q^{16} -5.55383 q^{17} -1.00000 q^{19} -10.7093 q^{20} +2.19978 q^{22} +3.32406 q^{23} +9.22908 q^{25} +10.4359 q^{26} +0.809757 q^{28} -1.36073 q^{29} -7.29122 q^{31} +7.25047 q^{32} +12.2172 q^{34} -1.07589 q^{35} +2.35405 q^{37} +2.19978 q^{38} +6.96239 q^{40} -3.84097 q^{41} -0.273376 q^{43} -2.83905 q^{44} -7.31221 q^{46} -5.16503 q^{47} -6.91865 q^{49} -20.3020 q^{50} -13.4687 q^{52} -2.62973 q^{53} +3.77215 q^{55} -0.526442 q^{56} +2.99332 q^{58} -0.181257 q^{59} +4.09404 q^{61} +16.0391 q^{62} -12.7137 q^{64} +17.8953 q^{65} -13.1781 q^{67} -15.7676 q^{68} +2.36674 q^{70} +9.56720 q^{71} -8.53461 q^{73} -5.17840 q^{74} -2.83905 q^{76} -0.285221 q^{77} +7.80622 q^{79} +6.10289 q^{80} +8.44931 q^{82} -5.53053 q^{83} +20.9499 q^{85} +0.601369 q^{86} +1.84574 q^{88} +6.83811 q^{89} -1.35311 q^{91} +9.43718 q^{92} +11.3620 q^{94} +3.77215 q^{95} +8.58313 q^{97} +15.2195 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - q^{2} + 9 q^{4} - 7 q^{5} + 7 q^{7} - 3 q^{10} - 5 q^{11} + 10 q^{13} + 5 q^{14} + 13 q^{16} - 17 q^{17} - 5 q^{19} - 2 q^{20} + q^{22} + 7 q^{23} + 8 q^{25} + 8 q^{26} + 27 q^{28} - 2 q^{29} + 4 q^{31}+ \cdots + 25 q^{98}+O(q^{100}) \) 5 * q - q^2 + 9 * q^4 - 7 * q^5 + 7 * q^7 - 3 * q^10 - 5 * q^11 + 10 * q^13 + 5 * q^14 + 13 * q^16 - 17 * q^17 - 5 * q^19 - 2 * q^20 + q^22 + 7 * q^23 + 8 * q^25 + 8 * q^26 + 27 * q^28 - 2 * q^29 + 4 * q^31 + 24 * q^32 + 10 * q^34 - 8 * q^35 + 11 * q^37 + q^38 - q^41 + 6 * q^43 - 9 * q^44 - 14 * q^46 + 2 * q^47 + 14 * q^49 - 24 * q^50 + 39 * q^52 - 5 * q^53 + 7 * q^55 - 12 * q^56 + 19 * q^58 + 18 * q^59 + 19 * q^61 - 15 * q^62 - 8 * q^64 + 28 * q^65 - q^67 - 51 * q^68 - 39 * q^70 + 29 * q^71 + 14 * q^73 + 10 * q^74 - 9 * q^76 - 7 * q^77 + 3 * q^79 + 5 * q^80 - 10 * q^82 - 4 * q^83 + 15 * q^85 + 53 * q^86 - q^89 + 24 * q^91 + 62 * q^92 + 2 * q^94 + 7 * q^95 + 4 * q^97 + 25 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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