LMFDB - Embedded newform 3775.2.a.x.1.25

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3775,2,Mod(1,3775)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3775, 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("3775.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3775 = 5^{2} \cdot 151 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3775.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:	$30.1435267630$
Analytic rank:	$0$
Dimension:	$44$
Twist minimal:	no (minimal twist has level 755)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.25
Character	$\chi$	$=$	3775.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.354420 q^{2} +3.10701 q^{3} -1.87439 q^{4} +1.10119 q^{6} +1.68085 q^{7} -1.37316 q^{8} +6.65351 q^{9} +O(q^{10})$	\(q+0.354420 q^{2} +3.10701 q^{3} -1.87439 q^{4} +1.10119 q^{6} +1.68085 q^{7} -1.37316 q^{8} +6.65351 q^{9} -2.85638 q^{11} -5.82374 q^{12} +1.32707 q^{13} +0.595726 q^{14} +3.26210 q^{16} -5.00845 q^{17} +2.35814 q^{18} -2.10321 q^{19} +5.22240 q^{21} -1.01236 q^{22} +4.74729 q^{23} -4.26643 q^{24} +0.470341 q^{26} +11.3515 q^{27} -3.15055 q^{28} +6.46912 q^{29} +4.14055 q^{31} +3.90248 q^{32} -8.87481 q^{33} -1.77510 q^{34} -12.4713 q^{36} +5.35424 q^{37} -0.745420 q^{38} +4.12323 q^{39} +0.954600 q^{41} +1.85093 q^{42} +8.38082 q^{43} +5.35397 q^{44} +1.68254 q^{46} +13.1017 q^{47} +10.1354 q^{48} -4.17476 q^{49} -15.5613 q^{51} -2.48744 q^{52} +2.06486 q^{53} +4.02320 q^{54} -2.30807 q^{56} -6.53470 q^{57} +2.29279 q^{58} -7.45893 q^{59} +6.84931 q^{61} +1.46750 q^{62} +11.1835 q^{63} -5.14108 q^{64} -3.14541 q^{66} +9.41746 q^{67} +9.38778 q^{68} +14.7499 q^{69} +9.54471 q^{71} -9.13635 q^{72} -12.3432 q^{73} +1.89765 q^{74} +3.94223 q^{76} -4.80114 q^{77} +1.46135 q^{78} +2.70884 q^{79} +15.3087 q^{81} +0.338330 q^{82} +10.4311 q^{83} -9.78880 q^{84} +2.97033 q^{86} +20.0996 q^{87} +3.92227 q^{88} -11.5876 q^{89} +2.23060 q^{91} -8.89826 q^{92} +12.8647 q^{93} +4.64350 q^{94} +12.1250 q^{96} +1.62011 q^{97} -1.47962 q^{98} -19.0050 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 44 q + 50 q^{4} + 16 q^{6} + 70 q^{9}+O(q^{10}) $ 44 * q + 50 * q^4 + 16 * q^6 + 70 * q^9	$ 44 q + 50 q^{4} + 16 q^{6} + 70 q^{9} + 20 q^{11} + 30 q^{14} + 58 q^{16} + 18 q^{19} + 30 q^{21} + 56 q^{24} + 68 q^{26} + 30 q^{29} + 12 q^{31} + 32 q^{34} + 106 q^{36} + 10 q^{39} + 102 q^{41} + 30 q^{44} - 4 q^{46} + 84 q^{49} + 48 q^{51} + 22 q^{54} + 110 q^{56} + 78 q^{59} + 28 q^{61} + 24 q^{64} + 48 q^{66} - 18 q^{69} + 78 q^{71} + 14 q^{74} + 34 q^{76} + 14 q^{79} + 160 q^{81} - 52 q^{84} + 76 q^{86} + 182 q^{89} + 30 q^{91} - 170 q^{94} + 130 q^{96} + 20 q^{99}+O(q^{100}) $ 44 * q + 50 * q^4 + 16 * q^6 + 70 * q^9 + 20 * q^11 + 30 * q^14 + 58 * q^16 + 18 * q^19 + 30 * q^21 + 56 * q^24 + 68 * q^26 + 30 * q^29 + 12 * q^31 + 32 * q^34 + 106 * q^36 + 10 * q^39 + 102 * q^41 + 30 * q^44 - 4 * q^46 + 84 * q^49 + 48 * q^51 + 22 * q^54 + 110 * q^56 + 78 * q^59 + 28 * q^61 + 24 * q^64 + 48 * q^66 - 18 * q^69 + 78 * q^71 + 14 * q^74 + 34 * q^76 + 14 * q^79 + 160 * q^81 - 52 * q^84 + 76 * q^86 + 182 * q^89 + 30 * q^91 - 170 * q^94 + 130 * q^96 + 20 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

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

$2$	0.354420	0.250613	0.125307	−	0.992118i	$-0.460009\pi$
$2$	0.354420	0.250613	0.125307	+	0.992118i	$0.460009\pi$
$3$	3.10701	1.79383	0.896917	−	0.442200i	$-0.145802\pi$
$3$	3.10701	1.79383	0.896917	+	0.442200i	$0.145802\pi$
$4$	−1.87439	−0.937193
$5$	0	0
$5$	0	0
$6$	1.10119	0.449558
$7$	1.68085	0.635300	0.317650	−	0.948208i	$-0.397106\pi$
$7$	1.68085	0.635300	0.317650	+	0.948208i	$0.397106\pi$
$8$	−1.37316	−0.485486
$9$	6.65351	2.21784
$10$	0	0
$11$	−2.85638	−0.861232	−0.430616	−	0.902535i	$-0.641704\pi$
$11$	−2.85638	−0.861232	−0.430616	+	0.902535i	$0.641704\pi$
$12$	−5.82374	−1.68117
$13$	1.32707	0.368063	0.184032	−	0.982920i	$-0.441085\pi$
$13$	1.32707	0.368063	0.184032	+	0.982920i	$0.441085\pi$
$14$	0.595726	0.159214
$15$	0	0
$16$	3.26210	0.815524
$17$	−5.00845	−1.21473	−0.607364	−	0.794424i	$-0.707773\pi$
$17$	−5.00845	−1.21473	−0.607364	+	0.794424i	$0.707773\pi$
$18$	2.35814	0.555819
$19$	−2.10321	−0.482510	−0.241255	−	0.970462i	$-0.577559\pi$
$19$	−2.10321	−0.482510	−0.241255	+	0.970462i	$0.577559\pi$
$20$	0	0
$21$	5.22240	1.13962
$22$	−1.01236	−0.215836
$23$	4.74729	0.989879	0.494940	−	0.868927i	$-0.335190\pi$
$23$	4.74729	0.989879	0.494940	+	0.868927i	$0.335190\pi$
$24$	−4.26643	−0.870881
$25$	0	0
$26$	0.470341	0.0922415
$27$	11.3515	2.18460
$28$	−3.15055	−0.595399
$29$	6.46912	1.20129	0.600643	−	0.799517i	$-0.294911\pi$
$29$	6.46912	1.20129	0.600643	+	0.799517i	$0.294911\pi$
$30$	0	0
$31$	4.14055	0.743665	0.371833	−	0.928300i	$-0.378730\pi$
$31$	4.14055	0.743665	0.371833	+	0.928300i	$0.378730\pi$
$32$	3.90248	0.689867
$33$	−8.87481	−1.54491
$34$	−1.77510	−0.304427
$35$	0	0
$36$	−12.4713	−2.07854
$37$	5.35424	0.880232	0.440116	−	0.897941i	$-0.354937\pi$
$37$	5.35424	0.880232	0.440116	+	0.897941i	$0.354937\pi$
$38$	−0.745420	−0.120923
$39$	4.12323	0.660244
$40$	0	0
$41$	0.954600	0.149083	0.0745417	−	0.997218i	$-0.476251\pi$
$41$	0.954600	0.149083	0.0745417	+	0.997218i	$0.476251\pi$
$42$	1.85093	0.285604
$43$	8.38082	1.27806	0.639032	−	0.769180i	$-0.279335\pi$
$43$	8.38082	1.27806	0.639032	+	0.769180i	$0.279335\pi$
$44$	5.35397	0.807141
$45$	0	0
$46$	1.68254	0.248077
$47$	13.1017	1.91108	0.955538	−	0.294866i	$-0.0952751\pi$
$47$	13.1017	1.91108	0.955538	+	0.294866i	$0.0952751\pi$
$48$	10.1354	1.46291
$49$	−4.17476	−0.596394
$50$	0	0
$51$	−15.5613	−2.17902
$52$	−2.48744	−0.344947
$53$	2.06486	0.283630	0.141815	−	0.989893i	$-0.454706\pi$
$53$	2.06486	0.283630	0.141815	+	0.989893i	$0.454706\pi$
$54$	4.02320	0.547489
$55$	0	0
$56$	−2.30807	−0.308429
$57$	−6.53470	−0.865542
$58$	2.29279	0.301058
$59$	−7.45893	−0.971069	−0.485535	−	0.874217i	$-0.661375\pi$
$59$	−7.45893	−0.971069	−0.485535	+	0.874217i	$0.661375\pi$
$60$	0	0
$61$	6.84931	0.876965	0.438482	−	0.898740i	$-0.355516\pi$
$61$	6.84931	0.876965	0.438482	+	0.898740i	$0.355516\pi$
$62$	1.46750	0.186372
$63$	11.1835	1.40899
$64$	−5.14108	−0.642634
$65$	0	0
$66$	−3.14541	−0.387174
$67$	9.41746	1.15053	0.575263	−	0.817969i	$-0.304900\pi$
$67$	9.41746	1.15053	0.575263	+	0.817969i	$0.304900\pi$
$68$	9.38778	1.13844
$69$	14.7499	1.77568
$70$	0	0
$71$	9.54471	1.13275	0.566374	−	0.824148i	$-0.308346\pi$
$71$	9.54471	1.13275	0.566374	+	0.824148i	$0.308346\pi$
$72$	−9.13635	−1.07673
$73$	−12.3432	−1.44466	−0.722329	−	0.691549i	$-0.756928\pi$
$73$	−12.3432	−1.44466	−0.722329	+	0.691549i	$0.756928\pi$
$74$	1.89765	0.220598
$75$	0	0
$76$	3.94223	0.452205
$77$	−4.80114	−0.547141
$78$	1.46135	0.165466
$79$	2.70884	0.304768	0.152384	−	0.988321i	$-0.451305\pi$
$79$	2.70884	0.304768	0.152384	+	0.988321i	$0.451305\pi$
$80$	0	0
$81$	15.3087	1.70097
$82$	0.338330	0.0373622
$83$	10.4311	1.14496	0.572481	−	0.819918i	$-0.305981\pi$
$83$	10.4311	1.14496	0.572481	+	0.819918i	$0.305981\pi$
$84$	−9.78880	−1.06805
$85$	0	0
$86$	2.97033	0.320299
$87$	20.0996	2.15491
$88$	3.92227	0.418116
$89$	−11.5876	−1.22828	−0.614141	−	0.789196i	$-0.710498\pi$
$89$	−11.5876	−1.22828	−0.614141	+	0.789196i	$0.710498\pi$
$90$	0	0
$91$	2.23060	0.233831
$92$	−8.89826	−0.927708
$93$	12.8647	1.33401
$94$	4.64350	0.478941
$95$	0	0
$96$	12.1250	1.23751
$97$	1.62011	0.164497	0.0822484	−	0.996612i	$-0.473790\pi$
$97$	1.62011	0.164497	0.0822484	+	0.996612i	$0.473790\pi$
$98$	−1.47962	−0.149464
$99$	−19.0050	−1.91007
$100$	0	0
$101$	−13.4438	−1.33771	−0.668854	−	0.743394i	$-0.733215\pi$
$101$	−13.4438	−1.33771	−0.668854	+	0.743394i	$0.733215\pi$
$102$	−5.51525	−0.546091
$103$	−2.65382	−0.261488	−0.130744	−	0.991416i	$-0.541737\pi$
$103$	−2.65382	−0.261488	−0.130744	+	0.991416i	$0.541737\pi$
$104$	−1.82228	−0.178690
$105$	0	0
$106$	0.731828	0.0710814
$107$	3.72460	0.360070	0.180035	−	0.983660i	$-0.442379\pi$
$107$	3.72460	0.360070	0.180035	+	0.983660i	$0.442379\pi$
$108$	−21.2771	−2.04739
$109$	−10.5019	−1.00590	−0.502948	−	0.864317i	$-0.667751\pi$
$109$	−10.5019	−1.00590	−0.502948	+	0.864317i	$0.667751\pi$
$110$	0	0
$111$	16.6357	1.57899
$112$	5.48308	0.518102
$113$	−5.47964	−0.515481	−0.257741	−	0.966214i	$-0.582978\pi$
$113$	−5.47964	−0.515481	−0.257741	+	0.966214i	$0.582978\pi$
$114$	−2.31603	−0.216916
$115$	0	0
$116$	−12.1256	−1.12584
$117$	8.82969	0.816305
$118$	−2.64359	−0.243363
$119$	−8.41844	−0.771717
$120$	0	0
$121$	−2.84108	−0.258280
$122$	2.42754	0.219779
$123$	2.96595	0.267431
$124$	−7.76099	−0.696958
$125$	0	0
$126$	3.96367	0.353112
$127$	−7.90955	−0.701859	−0.350929	−	0.936402i	$-0.614134\pi$
$127$	−7.90955	−0.701859	−0.350929	+	0.936402i	$0.614134\pi$
$128$	−9.62705	−0.850919
$129$	26.0393	2.29263
$130$	0	0
$131$	4.28090	0.374024	0.187012	−	0.982358i	$-0.440120\pi$
$131$	4.28090	0.374024	0.187012	+	0.982358i	$0.440120\pi$
$132$	16.6348	1.44788
$133$	−3.53517	−0.306538
$134$	3.33774	0.288337
$135$	0	0
$136$	6.87741	0.589733
$137$	17.9502	1.53359	0.766796	−	0.641891i	$-0.221850\pi$
$137$	17.9502	1.53359	0.766796	+	0.641891i	$0.221850\pi$
$138$	5.22766	0.445008
$139$	−17.8527	−1.51424	−0.757121	−	0.653274i	$-0.773395\pi$
$139$	−17.8527	−1.51424	−0.757121	+	0.653274i	$0.773395\pi$
$140$	0	0
$141$	40.7071	3.42815
$142$	3.38284	0.283882
$143$	−3.79062	−0.316988
$144$	21.7044	1.80870
$145$	0	0
$146$	−4.37467	−0.362050
$147$	−12.9710	−1.06983
$148$	−10.0359	−0.824948
$149$	15.7182	1.28769	0.643843	−	0.765158i	$-0.277339\pi$
$149$	15.7182	1.28769	0.643843	+	0.765158i	$0.277339\pi$
$150$	0	0
$151$	−1.00000	−0.0813788
$151$	−1.00000	−0.0813788
$152$	2.88805	0.234252
$153$	−33.3238	−2.69407
$154$	−1.70162	−0.137121
$155$	0	0
$156$	−7.72852	−0.618777
$157$	−16.1489	−1.28883	−0.644413	−	0.764678i	$-0.722898\pi$
$157$	−16.1489	−1.28883	−0.644413	+	0.764678i	$0.722898\pi$
$158$	0.960067	0.0763788
$159$	6.41554	0.508785
$160$	0	0
$161$	7.97947	0.628870
$162$	5.42571	0.426284
$163$	−12.8816	−1.00896	−0.504482	−	0.863422i	$-0.668317\pi$
$163$	−12.8816	−1.00896	−0.504482	+	0.863422i	$0.668317\pi$
$164$	−1.78929	−0.139720
$165$	0	0
$166$	3.69699	0.286942
$167$	−18.8707	−1.46026	−0.730130	−	0.683308i	$-0.760540\pi$
$167$	−18.8707	−1.46026	−0.730130	+	0.683308i	$0.760540\pi$
$168$	−7.17120	−0.553270
$169$	−11.2389	−0.864529
$170$	0	0
$171$	−13.9937	−1.07013
$172$	−15.7089	−1.19779
$173$	22.3415	1.69860	0.849298	−	0.527914i	$-0.177026\pi$
$173$	22.3415	1.69860	0.849298	+	0.527914i	$0.177026\pi$
$174$	7.12372	0.540048
$175$	0	0
$176$	−9.31780	−0.702355
$177$	−23.1750	−1.74194
$178$	−4.10688	−0.307824
$179$	5.97506	0.446597	0.223298	−	0.974750i	$-0.428318\pi$
$179$	5.97506	0.446597	0.223298	+	0.974750i	$0.428318\pi$
$180$	0	0
$181$	−23.2616	−1.72902	−0.864510	−	0.502616i	$-0.832371\pi$
$181$	−23.2616	−1.72902	−0.864510	+	0.502616i	$0.832371\pi$
$182$	0.790571	0.0586010
$183$	21.2809	1.57313
$184$	−6.51880	−0.480572
$185$	0	0
$186$	4.55952	0.334321
$187$	14.3061	1.04616
$188$	−24.5576	−1.79105
$189$	19.0801	1.38788
$190$	0	0
$191$	27.0139	1.95466	0.977328	−	0.211731i	$-0.0679100\pi$
$191$	27.0139	1.95466	0.977328	+	0.211731i	$0.0679100\pi$
$192$	−15.9734	−1.15278
$193$	−6.21287	−0.447212	−0.223606	−	0.974680i	$-0.571783\pi$
$193$	−6.21287	−0.447212	−0.223606	+	0.974680i	$0.571783\pi$
$194$	0.574198	0.0412250
$195$	0	0
$196$	7.82511	0.558936
$197$	−5.00008	−0.356241	−0.178121	−	0.984009i	$-0.557002\pi$
$197$	−5.00008	−0.356241	−0.178121	+	0.984009i	$0.557002\pi$
$198$	−6.73575	−0.478689
$199$	7.93954	0.562819	0.281409	−	0.959588i	$-0.409198\pi$
$199$	7.93954	0.562819	0.281409	+	0.959588i	$0.409198\pi$
$200$	0	0
$201$	29.2601	2.06385
$202$	−4.76476	−0.335247
$203$	10.8736	0.763177
$204$	29.1679	2.04216
$205$	0	0
$206$	−0.940567	−0.0655324
$207$	31.5862	2.19539
$208$	4.32904	0.300165
$209$	6.00757	0.415553
$210$	0	0
$211$	4.51361	0.310730	0.155365	−	0.987857i	$-0.450345\pi$
$211$	4.51361	0.310730	0.155365	+	0.987857i	$0.450345\pi$
$212$	−3.87034	−0.265816
$213$	29.6555	2.03196
$214$	1.32007	0.0902383
$215$	0	0
$216$	−15.5874	−1.06059
$217$	6.95963	0.472450
$218$	−3.72207	−0.252091
$219$	−38.3503	−2.59148
$220$	0	0
$221$	−6.64658	−0.447097
$222$	5.89603	0.395715
$223$	−23.4845	−1.57264	−0.786320	−	0.617819i	$-0.788016\pi$
$223$	−23.4845	−1.57264	−0.786320	+	0.617819i	$0.788016\pi$
$224$	6.55946	0.438272
$225$	0	0
$226$	−1.94210	−0.129186
$227$	0.585872	0.0388857	0.0194429	−	0.999811i	$-0.493811\pi$
$227$	0.585872	0.0388857	0.0194429	+	0.999811i	$0.493811\pi$
$228$	12.2485	0.811180
$229$	21.7200	1.43530	0.717650	−	0.696404i	$-0.245218\pi$
$229$	21.7200	1.43530	0.717650	+	0.696404i	$0.245218\pi$
$230$	0	0
$231$	−14.9172	−0.981479
$232$	−8.88315	−0.583207
$233$	−10.0081	−0.655653	−0.327827	−	0.944738i	$-0.606316\pi$
$233$	−10.0081	−0.655653	−0.327827	+	0.944738i	$0.606316\pi$
$234$	3.12942	0.204577
$235$	0	0
$236$	13.9809	0.910080
$237$	8.41639	0.546703
$238$	−2.98367	−0.193402
$239$	6.00231	0.388257	0.194129	−	0.980976i	$-0.437812\pi$
$239$	6.00231	0.388257	0.194129	+	0.980976i	$0.437812\pi$
$240$	0	0
$241$	−11.8694	−0.764577	−0.382289	−	0.924043i	$-0.624864\pi$
$241$	−11.8694	−0.764577	−0.382289	+	0.924043i	$0.624864\pi$
$242$	−1.00693	−0.0647282
$243$	13.5098	0.866653
$244$	−12.8383	−0.821885
$245$	0	0
$246$	1.05119	0.0670216
$247$	−2.79111	−0.177594
$248$	−5.68565	−0.361039
$249$	32.4095	2.05387
$250$	0	0
$251$	24.9540	1.57508	0.787541	−	0.616262i	$-0.211354\pi$
$251$	24.9540	1.57508	0.787541	+	0.616262i	$0.211354\pi$
$252$	−20.9623	−1.32050
$253$	−13.5601	−0.852516
$254$	−2.80330	−0.175895
$255$	0	0
$256$	6.87013	0.429383
$257$	18.4958	1.15374	0.576869	−	0.816837i	$-0.304274\pi$
$257$	18.4958	1.15374	0.576869	+	0.816837i	$0.304274\pi$
$258$	9.22886	0.574564
$259$	8.99966	0.559211
$260$	0	0
$261$	43.0424	2.66426
$262$	1.51724	0.0937353
$263$	8.30801	0.512294	0.256147	−	0.966638i	$-0.417547\pi$
$263$	8.30801	0.512294	0.256147	+	0.966638i	$0.417547\pi$
$264$	12.1865	0.750030
$265$	0	0
$266$	−1.25294	−0.0768225
$267$	−36.0028	−2.20333
$268$	−17.6520	−1.07827
$269$	2.19475	0.133816	0.0669081	−	0.997759i	$-0.478687\pi$
$269$	2.19475	0.133816	0.0669081	+	0.997759i	$0.478687\pi$
$270$	0	0
$271$	18.1150	1.10041	0.550205	−	0.835029i	$-0.314549\pi$
$271$	18.1150	1.10041	0.550205	+	0.835029i	$0.314549\pi$
$272$	−16.3381	−0.990640
$273$	6.93051	0.419453
$274$	6.36193	0.384338
$275$	0	0
$276$	−27.6470	−1.66415
$277$	13.4680	0.809215	0.404607	−	0.914491i	$-0.367408\pi$
$277$	13.4680	0.809215	0.404607	+	0.914491i	$0.367408\pi$
$278$	−6.32735	−0.379489
$279$	27.5492	1.64933
$280$	0	0
$281$	−21.2240	−1.26612	−0.633058	−	0.774104i	$-0.718201\pi$
$281$	−21.2240	−1.26612	−0.633058	+	0.774104i	$0.718201\pi$
$282$	14.4274	0.859140
$283$	−8.61296	−0.511987	−0.255994	−	0.966678i	$-0.582403\pi$
$283$	−8.61296	−0.511987	−0.255994	+	0.966678i	$0.582403\pi$
$284$	−17.8905	−1.06160
$285$	0	0
$286$	−1.34347	−0.0794413
$287$	1.60453	0.0947127
$288$	25.9652	1.53001
$289$	8.08460	0.475565
$290$	0	0
$291$	5.03368	0.295080
$292$	23.1359	1.35392
$293$	1.22724	0.0716960	0.0358480	−	0.999357i	$-0.488587\pi$
$293$	1.22724	0.0716960	0.0358480	+	0.999357i	$0.488587\pi$
$294$	−4.59719	−0.268114
$295$	0	0
$296$	−7.35224	−0.427340
$297$	−32.4242	−1.88145
$298$	5.57085	0.322711
$299$	6.30000	0.364338
$300$	0	0
$301$	14.0869	0.811954
$302$	−0.354420	−0.0203946
$303$	−41.7700	−2.39962
$304$	−6.86087	−0.393498
$305$	0	0
$306$	−11.8106	−0.675169
$307$	10.2649	0.585849	0.292924	−	0.956136i	$-0.405372\pi$
$307$	10.2649	0.585849	0.292924	+	0.956136i	$0.405372\pi$
$308$	8.99919	0.512776
$309$	−8.24544	−0.469067
$310$	0	0
$311$	−17.5802	−0.996884	−0.498442	−	0.866923i	$-0.666094\pi$
$311$	−17.5802	−0.996884	−0.498442	+	0.866923i	$0.666094\pi$
$312$	−5.66185	−0.320539
$313$	23.4489	1.32541	0.662705	−	0.748880i	$-0.269408\pi$
$313$	23.4489	1.32541	0.662705	+	0.748880i	$0.269408\pi$
$314$	−5.72351	−0.322996
$315$	0	0
$316$	−5.07741	−0.285626
$317$	−25.7647	−1.44709	−0.723546	−	0.690276i	$-0.757489\pi$
$317$	−25.7647	−1.44709	−0.723546	+	0.690276i	$0.757489\pi$
$318$	2.27380	0.127508
$319$	−18.4783	−1.03459
$320$	0	0
$321$	11.5724	0.645906
$322$	2.82809	0.157603
$323$	10.5338	0.586118
$324$	−28.6944	−1.59413
$325$	0	0
$326$	−4.56550	−0.252860
$327$	−32.6294	−1.80441
$328$	−1.31082	−0.0723779
$329$	22.0219	1.21411
$330$	0	0
$331$	8.72271	0.479444	0.239722	−	0.970842i	$-0.422944\pi$
$331$	8.72271	0.479444	0.239722	+	0.970842i	$0.422944\pi$
$332$	−19.5519	−1.07305
$333$	35.6245	1.95221
$334$	−6.68817	−0.365960
$335$	0	0
$336$	17.0360	0.929389
$337$	−22.9000	−1.24744	−0.623721	−	0.781647i	$-0.714380\pi$
$337$	−22.9000	−1.24744	−0.623721	+	0.781647i	$0.714380\pi$
$338$	−3.98329	−0.216662
$339$	−17.0253	−0.924688
$340$	0	0
$341$	−11.8270	−0.640468
$342$	−4.95967	−0.268188
$343$	−18.7830	−1.01419
$344$	−11.5082	−0.620482
$345$	0	0
$346$	7.91830	0.425690
$347$	−15.9674	−0.857174	−0.428587	−	0.903501i	$-0.640988\pi$
$347$	−15.9674	−0.857174	−0.428587	+	0.903501i	$0.640988\pi$
$348$	−37.6745	−2.01956
$349$	29.5353	1.58099	0.790495	−	0.612468i	$-0.209823\pi$
$349$	29.5353	1.58099	0.790495	+	0.612468i	$0.209823\pi$
$350$	0	0
$351$	15.0643	0.804071
$352$	−11.1470	−0.594135
$353$	19.4597	1.03574	0.517868	−	0.855461i	$-0.326726\pi$
$353$	19.4597	1.03574	0.517868	+	0.855461i	$0.326726\pi$
$354$	−8.21368	−0.436552
$355$	0	0
$356$	21.7196	1.15114
$357$	−26.1562	−1.38433
$358$	2.11768	0.111923
$359$	1.73712	0.0916819	0.0458410	−	0.998949i	$-0.485403\pi$
$359$	1.73712	0.0916819	0.0458410	+	0.998949i	$0.485403\pi$
$360$	0	0
$361$	−14.5765	−0.767185
$362$	−8.24438	−0.433315
$363$	−8.82725	−0.463311
$364$	−4.18101	−0.219145
$365$	0	0
$366$	7.54238	0.394246
$367$	13.1464	0.686234	0.343117	−	0.939293i	$-0.388517\pi$
$367$	13.1464	0.686234	0.343117	+	0.939293i	$0.388517\pi$
$368$	15.4861	0.807270
$369$	6.35144	0.330643
$370$	0	0
$371$	3.47071	0.180190
$372$	−24.1135	−1.25023
$373$	−24.4846	−1.26776	−0.633882	−	0.773429i	$-0.718540\pi$
$373$	−24.4846	−1.26776	−0.633882	+	0.773429i	$0.718540\pi$
$374$	5.07036	0.262182
$375$	0	0
$376$	−17.9907	−0.927801
$377$	8.58499	0.442149
$378$	6.76238	0.347820
$379$	−15.2162	−0.781606	−0.390803	−	0.920474i	$-0.627803\pi$
$379$	−15.2162	−0.781606	−0.390803	+	0.920474i	$0.627803\pi$
$380$	0	0
$381$	−24.5750	−1.25902
$382$	9.57427	0.489862
$383$	−22.2112	−1.13494	−0.567469	−	0.823395i	$-0.692077\pi$
$383$	−22.2112	−1.13494	−0.567469	+	0.823395i	$0.692077\pi$
$384$	−29.9114	−1.52641
$385$	0	0
$386$	−2.20197	−0.112077
$387$	55.7619	2.83454
$388$	−3.03670	−0.154165
$389$	−16.9753	−0.860683	−0.430341	−	0.902666i	$-0.641607\pi$
$389$	−16.9753	−0.860683	−0.430341	+	0.902666i	$0.641607\pi$
$390$	0	0
$391$	−23.7766	−1.20243
$392$	5.73262	0.289541
$393$	13.3008	0.670937
$394$	−1.77213	−0.0892786
$395$	0	0
$396$	35.6227	1.79011
$397$	4.29941	0.215781	0.107891	−	0.994163i	$-0.465590\pi$
$397$	4.29941	0.215781	0.107891	+	0.994163i	$0.465590\pi$
$398$	2.81393	0.141050
$399$	−10.9838	−0.549879
$400$	0	0
$401$	−11.6387	−0.581211	−0.290605	−	0.956843i	$-0.593857\pi$
$401$	−11.6387	−0.581211	−0.290605	+	0.956843i	$0.593857\pi$
$402$	10.3704	0.517228
$403$	5.49481	0.273716
$404$	25.1989	1.25369
$405$	0	0
$406$	3.85382	0.191262
$407$	−15.2938	−0.758084
$408$	21.3682	1.05788
$409$	24.6720	1.21995	0.609976	−	0.792420i	$-0.291179\pi$
$409$	24.6720	1.21995	0.609976	+	0.792420i	$0.291179\pi$
$410$	0	0
$411$	55.7716	2.75101
$412$	4.97428	0.245065
$413$	−12.5373	−0.616920
$414$	11.1948	0.550194
$415$	0	0
$416$	5.17886	0.253915
$417$	−55.4684	−2.71630
$418$	2.12921	0.104143
$419$	−25.3614	−1.23898	−0.619492	−	0.785003i	$-0.712661\pi$
$419$	−25.3614	−1.23898	−0.619492	+	0.785003i	$0.712661\pi$
$420$	0	0
$421$	−2.50380	−0.122028	−0.0610138	−	0.998137i	$-0.519433\pi$
$421$	−2.50380	−0.122028	−0.0610138	+	0.998137i	$0.519433\pi$
$422$	1.59972	0.0778730
$423$	87.1722	4.23846
$424$	−2.83538	−0.137698
$425$	0	0
$426$	10.5105	0.509236
$427$	11.5126	0.557136
$428$	−6.98134	−0.337456
$429$	−11.7775	−0.568624
$430$	0	0
$431$	−35.9592	−1.73209	−0.866046	−	0.499964i	$-0.833346\pi$
$431$	−35.9592	−1.73209	−0.866046	+	0.499964i	$0.833346\pi$
$432$	37.0297	1.78159
$433$	−14.8435	−0.713335	−0.356667	−	0.934231i	$-0.616087\pi$
$433$	−14.8435	−0.713335	−0.356667	+	0.934231i	$0.616087\pi$
$434$	2.46663	0.118402
$435$	0	0
$436$	19.6845	0.942718
$437$	−9.98456	−0.477626
$438$	−13.5921	−0.649458
$439$	−20.8936	−0.997198	−0.498599	−	0.866833i	$-0.666152\pi$
$439$	−20.8936	−0.997198	−0.498599	+	0.866833i	$0.666152\pi$
$440$	0	0
$441$	−27.7768	−1.32271
$442$	−2.35568	−0.112048
$443$	18.0170	0.856013	0.428006	−	0.903776i	$-0.359216\pi$
$443$	18.0170	0.856013	0.428006	+	0.903776i	$0.359216\pi$
$444$	−31.1817	−1.47982
$445$	0	0
$446$	−8.32340	−0.394124
$447$	48.8366	2.30989
$448$	−8.64135	−0.408266
$449$	−17.3310	−0.817900	−0.408950	−	0.912557i	$-0.634105\pi$
$449$	−17.3310	−0.817900	−0.408950	+	0.912557i	$0.634105\pi$
$450$	0	0
$451$	−2.72670	−0.128395
$452$	10.2710	0.483106
$453$	−3.10701	−0.145980
$454$	0.207645	0.00974527
$455$	0	0
$456$	8.97319	0.420208
$457$	−14.6790	−0.686654	−0.343327	−	0.939216i	$-0.611554\pi$
$457$	−14.6790	−0.686654	−0.343327	+	0.939216i	$0.611554\pi$
$458$	7.69802	0.359705
$459$	−56.8535	−2.65369
$460$	0	0
$461$	−2.88478	−0.134357	−0.0671787	−	0.997741i	$-0.521400\pi$
$461$	−2.88478	−0.134357	−0.0671787	+	0.997741i	$0.521400\pi$
$462$	−5.28695	−0.245971
$463$	15.8013	0.734350	0.367175	−	0.930152i	$-0.380325\pi$
$463$	15.8013	0.734350	0.367175	+	0.930152i	$0.380325\pi$
$464$	21.1029	0.979678
$465$	0	0
$466$	−3.54708	−0.164315
$467$	−12.8097	−0.592763	−0.296381	−	0.955070i	$-0.595780\pi$
$467$	−12.8097	−0.592763	−0.296381	+	0.955070i	$0.595780\pi$
$468$	−16.5502	−0.765035
$469$	15.8293	0.730929
$470$	0	0
$471$	−50.1749	−2.31194
$472$	10.2423	0.471440
$473$	−23.9388	−1.10071
$474$	2.98294	0.137011
$475$	0	0
$476$	15.7794	0.723248
$477$	13.7386	0.629046
$478$	2.12734	0.0973024
$479$	1.69879	0.0776199	0.0388099	−	0.999247i	$-0.487643\pi$
$479$	1.69879	0.0776199	0.0388099	+	0.999247i	$0.487643\pi$
$480$	0	0
$481$	7.10546	0.323981
$482$	−4.20677	−0.191613
$483$	24.7923	1.12809
$484$	5.32527	0.242058
$485$	0	0
$486$	4.78814	0.217194
$487$	−2.07121	−0.0938556	−0.0469278	−	0.998898i	$-0.514943\pi$
$487$	−2.07121	−0.0938556	−0.0469278	+	0.998898i	$0.514943\pi$
$488$	−9.40521	−0.425754
$489$	−40.0233	−1.80991
$490$	0	0
$491$	−11.8900	−0.536590	−0.268295	−	0.963337i	$-0.586460\pi$
$491$	−11.8900	−0.536590	−0.268295	+	0.963337i	$0.586460\pi$
$492$	−5.55934	−0.250634
$493$	−32.4003	−1.45924
$494$	−0.989226	−0.0445074
$495$	0	0
$496$	13.5069	0.606477
$497$	16.0432	0.719635
$498$	11.4866	0.514727
$499$	−7.75056	−0.346963	−0.173481	−	0.984837i	$-0.555502\pi$
$499$	−7.75056	−0.346963	−0.173481	+	0.984837i	$0.555502\pi$
$500$	0	0
$501$	−58.6315	−2.61946
$502$	8.84420	0.394736
$503$	−20.3334	−0.906621	−0.453310	−	0.891353i	$-0.649757\pi$
$503$	−20.3334	−0.906621	−0.453310	+	0.891353i	$0.649757\pi$
$504$	−15.3568	−0.684046
$505$	0	0
$506$	−4.80597	−0.213652
$507$	−34.9193	−1.55082
$508$	14.8255	0.657777
$509$	41.5158	1.84016	0.920078	−	0.391736i	$-0.128125\pi$
$509$	41.5158	1.84016	0.920078	+	0.391736i	$0.128125\pi$
$510$	0	0
$511$	−20.7470	−0.917791
$512$	21.6890	0.958528
$513$	−23.8746	−1.05409
$514$	6.55529	0.289142
$515$	0	0
$516$	−48.8077	−2.14864
$517$	−37.4234	−1.64588
$518$	3.18966	0.140146
$519$	69.4154	3.04700
$520$	0	0
$521$	−17.7886	−0.779331	−0.389665	−	0.920957i	$-0.627409\pi$
$521$	−17.7886	−0.779331	−0.389665	+	0.920957i	$0.627409\pi$
$522$	15.2551	0.667698
$523$	−12.2432	−0.535357	−0.267679	−	0.963508i	$-0.586257\pi$
$523$	−12.2432	−0.535357	−0.267679	+	0.963508i	$0.586257\pi$
$524$	−8.02406	−0.350533
$525$	0	0
$526$	2.94453	0.128387
$527$	−20.7378	−0.903351
$528$	−28.9505	−1.25991
$529$	−0.463190	−0.0201387
$530$	0	0
$531$	−49.6281	−2.15367
$532$	6.62628	0.287286
$533$	1.26682	0.0548722
$534$	−12.7601	−0.552184
$535$	0	0
$536$	−12.9317	−0.558564
$537$	18.5646	0.801120
$538$	0.777864	0.0335361
$539$	11.9247	0.513634
$540$	0	0
$541$	−37.2486	−1.60144	−0.800721	−	0.599037i	$-0.795550\pi$
$541$	−37.2486	−1.60144	−0.800721	+	0.599037i	$0.795550\pi$
$542$	6.42034	0.275777
$543$	−72.2740	−3.10157
$544$	−19.5454	−0.838001
$545$	0	0
$546$	2.45631	0.105120
$547$	−30.4695	−1.30278	−0.651392	−	0.758742i	$-0.725814\pi$
$547$	−30.4695	−1.30278	−0.651392	+	0.758742i	$0.725814\pi$
$548$	−33.6457	−1.43727
$549$	45.5720	1.94497
$550$	0	0
$551$	−13.6059	−0.579632
$552$	−20.2540	−0.862067
$553$	4.55314	0.193619
$554$	4.77334	0.202800
$555$	0	0
$556$	33.4628	1.41914
$557$	38.6449	1.63744	0.818719	−	0.574195i	$-0.194685\pi$
$557$	38.6449	1.63744	0.818719	+	0.574195i	$0.194685\pi$
$558$	9.76400	0.413343
$559$	11.1220	0.470409
$560$	0	0
$561$	44.4491	1.87664
$562$	−7.52221	−0.317305
$563$	−4.39305	−0.185145	−0.0925725	−	0.995706i	$-0.529509\pi$
$563$	−4.39305	−0.185145	−0.0925725	+	0.995706i	$0.529509\pi$
$564$	−76.3008	−3.21284
$565$	0	0
$566$	−3.05261	−0.128311
$567$	25.7316	1.08062
$568$	−13.1064	−0.549933
$569$	23.1092	0.968787	0.484393	−	0.874850i	$-0.339040\pi$
$569$	23.1092	0.968787	0.484393	+	0.874850i	$0.339040\pi$
$570$	0	0
$571$	−31.0268	−1.29843	−0.649215	−	0.760605i	$-0.724903\pi$
$571$	−31.0268	−1.29843	−0.649215	+	0.760605i	$0.724903\pi$
$572$	7.10510	0.297079
$573$	83.9324	3.50633
$574$	0.568680	0.0237362
$575$	0	0
$576$	−34.2062	−1.42526
$577$	36.1216	1.50376	0.751882	−	0.659298i	$-0.229146\pi$
$577$	36.1216	1.50376	0.751882	+	0.659298i	$0.229146\pi$
$578$	2.86535	0.119183
$579$	−19.3035	−0.802224
$580$	0	0
$581$	17.5331	0.727394
$582$	1.78404	0.0739508
$583$	−5.89803	−0.244271
$584$	16.9492	0.701361
$585$	0	0
$586$	0.434958	0.0179680
$587$	38.4423	1.58668	0.793342	−	0.608776i	$-0.208339\pi$
$587$	38.4423	1.58668	0.793342	+	0.608776i	$0.208339\pi$
$588$	24.3127	1.00264
$589$	−8.70845	−0.358826
$590$	0	0
$591$	−15.5353	−0.639037
$592$	17.4661	0.717851
$593$	−34.5596	−1.41919	−0.709595	−	0.704609i	$-0.751122\pi$
$593$	−34.5596	−1.41919	−0.709595	+	0.704609i	$0.751122\pi$
$594$	−11.4918	−0.471515
$595$	0	0
$596$	−29.4620	−1.20681
$597$	24.6682	1.00960
$598$	2.23285	0.0913080
$599$	25.2949	1.03352	0.516761	−	0.856130i	$-0.327137\pi$
$599$	25.2949	1.03352	0.516761	+	0.856130i	$0.327137\pi$
$600$	0	0
$601$	34.3891	1.40276	0.701381	−	0.712787i	$-0.252567\pi$
$601$	34.3891	1.40276	0.701381	+	0.712787i	$0.252567\pi$
$602$	4.99267	0.203486
$603$	62.6592	2.55168
$604$	1.87439	0.0762677
$605$	0	0
$606$	−14.8041	−0.601377
$607$	6.16562	0.250255	0.125127	−	0.992141i	$-0.460066\pi$
$607$	6.16562	0.250255	0.125127	+	0.992141i	$0.460066\pi$
$608$	−8.20773	−0.332867
$609$	33.7844	1.36901
$610$	0	0
$611$	17.3869	0.703398
$612$	62.4617	2.52486
$613$	17.5586	0.709184	0.354592	−	0.935021i	$-0.384620\pi$
$613$	17.5586	0.709184	0.354592	+	0.935021i	$0.384620\pi$
$614$	3.63809	0.146821
$615$	0	0
$616$	6.59274	0.265629
$617$	−30.2115	−1.21627	−0.608135	−	0.793834i	$-0.708082\pi$
$617$	−30.2115	−1.21627	−0.608135	+	0.793834i	$0.708082\pi$
$618$	−2.92235	−0.117554
$619$	32.6243	1.31128	0.655641	−	0.755072i	$-0.272398\pi$
$619$	32.6243	1.31128	0.655641	+	0.755072i	$0.272398\pi$
$620$	0	0
$621$	53.8889	2.16249
$622$	−6.23080	−0.249832
$623$	−19.4770	−0.780328
$624$	13.4504	0.538445
$625$	0	0
$626$	8.31077	0.332165
$627$	18.6656	0.745432
$628$	30.2694	1.20788
$629$	−26.8165	−1.06924
$630$	0	0
$631$	−7.54376	−0.300313	−0.150156	−	0.988662i	$-0.547978\pi$
$631$	−7.54376	−0.300313	−0.150156	+	0.988662i	$0.547978\pi$
$632$	−3.71967	−0.147961
$633$	14.0238	0.557398
$634$	−9.13154	−0.362660
$635$	0	0
$636$	−12.0252	−0.476830
$637$	−5.54020	−0.219511
$638$	−6.54908	−0.259281
$639$	63.5059	2.51225
$640$	0	0
$641$	7.80533	0.308292	0.154146	−	0.988048i	$-0.450737\pi$
$641$	7.80533	0.308292	0.154146	+	0.988048i	$0.450737\pi$
$642$	4.10148	0.161873
$643$	14.3699	0.566692	0.283346	−	0.959018i	$-0.408555\pi$
$643$	14.3699	0.566692	0.283346	+	0.959018i	$0.408555\pi$
$644$	−14.9566	−0.589373
$645$	0	0
$646$	3.73340	0.146889
$647$	−6.09736	−0.239712	−0.119856	−	0.992791i	$-0.538243\pi$
$647$	−6.09736	−0.239712	−0.119856	+	0.992791i	$0.538243\pi$
$648$	−21.0213	−0.825795
$649$	21.3055	0.836316
$650$	0	0
$651$	21.6236	0.847497
$652$	24.1451	0.945595
$653$	−34.6912	−1.35757	−0.678786	−	0.734337i	$-0.737493\pi$
$653$	−34.6912	−1.35757	−0.678786	+	0.734337i	$0.737493\pi$
$654$	−11.5645	−0.452208
$655$	0	0
$656$	3.11400	0.121581
$657$	−82.1254	−3.20402
$658$	7.80501	0.304271
$659$	−8.89965	−0.346681	−0.173341	−	0.984862i	$-0.555456\pi$
$659$	−8.89965	−0.346681	−0.173341	+	0.984862i	$0.555456\pi$
$660$	0	0
$661$	37.3947	1.45449	0.727244	−	0.686379i	$-0.240801\pi$
$661$	37.3947	1.45449	0.727244	+	0.686379i	$0.240801\pi$
$662$	3.09151	0.120155
$663$	−20.6510	−0.802018
$664$	−14.3236	−0.555863
$665$	0	0
$666$	12.6261	0.489250
$667$	30.7108	1.18913
$668$	35.3710	1.36855
$669$	−72.9667	−2.82106
$670$	0	0
$671$	−19.5643	−0.755270
$672$	20.3803	0.786187
$673$	−2.59305	−0.0999547	−0.0499774	−	0.998750i	$-0.515915\pi$
$673$	−2.59305	−0.0999547	−0.0499774	+	0.998750i	$0.515915\pi$
$674$	−8.11623	−0.312625
$675$	0	0
$676$	21.0660	0.810231
$677$	3.50429	0.134681	0.0673403	−	0.997730i	$-0.478549\pi$
$677$	3.50429	0.134681	0.0673403	+	0.997730i	$0.478549\pi$
$678$	−6.03411	−0.231739
$679$	2.72315	0.104505
$680$	0	0
$681$	1.82031	0.0697545
$682$	−4.19173	−0.160510
$683$	−21.7431	−0.831977	−0.415988	−	0.909370i	$-0.636564\pi$
$683$	−21.7431	−0.831977	−0.415988	+	0.909370i	$0.636564\pi$
$684$	26.2297	1.00292
$685$	0	0
$686$	−6.65709	−0.254169
$687$	67.4843	2.57469
$688$	27.3391	1.04229
$689$	2.74022	0.104394
$690$	0	0
$691$	19.8296	0.754354	0.377177	−	0.926141i	$-0.376895\pi$
$691$	19.8296	0.754354	0.377177	+	0.926141i	$0.376895\pi$
$692$	−41.8767	−1.59191
$693$	−31.9444	−1.21347
$694$	−5.65916	−0.214819
$695$	0	0
$696$	−27.6000	−1.04618
$697$	−4.78107	−0.181096
$698$	10.4679	0.396217
$699$	−31.0953	−1.17613
$700$	0	0
$701$	45.7729	1.72882	0.864409	−	0.502790i	$-0.167693\pi$
$701$	45.7729	1.72882	0.864409	+	0.502790i	$0.167693\pi$
$702$	5.33908	0.201511
$703$	−11.2611	−0.424720
$704$	14.6849	0.553457
$705$	0	0
$706$	6.89692	0.259569
$707$	−22.5969	−0.849846
$708$	43.4388	1.63253
$709$	−50.4288	−1.89389	−0.946947	−	0.321389i	$-0.895850\pi$
$709$	−50.4288	−1.89389	−0.946947	+	0.321389i	$0.895850\pi$
$710$	0	0
$711$	18.0233	0.675926
$712$	15.9116	0.596314
$713$	19.6564	0.736139
$714$	−9.27028	−0.346931
$715$	0	0
$716$	−11.1996	−0.418547
$717$	18.6493	0.696469
$718$	0.615672	0.0229767
$719$	33.4164	1.24622	0.623111	−	0.782134i	$-0.285869\pi$
$719$	33.4164	1.24622	0.623111	+	0.782134i	$0.285869\pi$
$720$	0	0
$721$	−4.46066	−0.166124
$722$	−5.16621	−0.192266
$723$	−36.8784	−1.37152
$724$	43.6012	1.62043
$725$	0	0
$726$	−3.12856	−0.116112
$727$	18.1067	0.671541	0.335770	−	0.941944i	$-0.391003\pi$
$727$	18.1067	0.671541	0.335770	+	0.941944i	$0.391003\pi$
$728$	−3.06298	−0.113521
$729$	−3.95107	−0.146336
$730$	0	0
$731$	−41.9750	−1.55250
$732$	−39.8886	−1.47433
$733$	27.7585	1.02528	0.512641	−	0.858603i	$-0.328667\pi$
$733$	27.7585	1.02528	0.512641	+	0.858603i	$0.328667\pi$
$734$	4.65934	0.171979
$735$	0	0
$736$	18.5262	0.682885
$737$	−26.8999	−0.990870
$738$	2.25108	0.0828634
$739$	14.7326	0.541948	0.270974	−	0.962587i	$-0.412654\pi$
$739$	14.7326	0.541948	0.270974	+	0.962587i	$0.412654\pi$
$740$	0	0
$741$	−8.67201	−0.318574
$742$	1.23009	0.0451580
$743$	−43.9630	−1.61285	−0.806423	−	0.591339i	$-0.798600\pi$
$743$	−43.9630	−1.61285	−0.806423	+	0.591339i	$0.798600\pi$
$744$	−17.6654	−0.647644
$745$	0	0
$746$	−8.67784	−0.317718
$747$	69.4035	2.53934
$748$	−26.8151	−0.980457
$749$	6.26047	0.228753
$750$	0	0
$751$	−25.1273	−0.916908	−0.458454	−	0.888718i	$-0.651597\pi$
$751$	−25.1273	−0.916908	−0.458454	+	0.888718i	$0.651597\pi$
$752$	42.7390	1.55853
$753$	77.5323	2.82544
$754$	3.04269	0.110808
$755$	0	0
$756$	−35.7635	−1.30071
$757$	6.51730	0.236875	0.118438	−	0.992961i	$-0.462211\pi$
$757$	6.51730	0.236875	0.118438	+	0.992961i	$0.462211\pi$
$758$	−5.39295	−0.195881
$759$	−42.1313	−1.52927
$760$	0	0
$761$	37.9930	1.37725	0.688623	−	0.725120i	$-0.258216\pi$
$761$	37.9930	1.37725	0.688623	+	0.725120i	$0.258216\pi$
$762$	−8.70989	−0.315526
$763$	−17.6520	−0.639045
$764$	−50.6345	−1.83189
$765$	0	0
$766$	−7.87209	−0.284430
$767$	−9.89853	−0.357415
$768$	21.3456	0.770242
$769$	21.8616	0.788350	0.394175	−	0.919035i	$-0.371030\pi$
$769$	21.8616	0.788350	0.394175	+	0.919035i	$0.371030\pi$
$770$	0	0
$771$	57.4667	2.06961
$772$	11.6453	0.419124
$773$	−22.4601	−0.807832	−0.403916	−	0.914796i	$-0.632351\pi$
$773$	−22.4601	−0.807832	−0.403916	+	0.914796i	$0.632351\pi$
$774$	19.7632	0.710372
$775$	0	0
$776$	−2.22467	−0.0798609
$777$	27.9620	1.00313
$778$	−6.01640	−0.215698
$779$	−2.00772	−0.0719342
$780$	0	0
$781$	−27.2633	−0.975559
$782$	−8.42691	−0.301346
$783$	73.4343	2.62433
$784$	−13.6185	−0.486374
$785$	0	0
$786$	4.71408	0.168145
$787$	−41.9740	−1.49621	−0.748106	−	0.663579i	$-0.769037\pi$
$787$	−41.9740	−1.49621	−0.748106	+	0.663579i	$0.769037\pi$
$788$	9.37208	0.333867
$789$	25.8131	0.918970
$790$	0	0
$791$	−9.21043	−0.327485
$792$	26.0969	0.927313
$793$	9.08953	0.322779
$794$	1.52380	0.0540776
$795$	0	0
$796$	−14.8818	−0.527470
$797$	−2.39391	−0.0847968	−0.0423984	−	0.999101i	$-0.513500\pi$
$797$	−2.39391	−0.0847968	−0.0423984	+	0.999101i	$0.513500\pi$
$798$	−3.89289	−0.137807
$799$	−65.6192	−2.32144
$800$	0	0
$801$	−77.0982	−2.72413
$802$	−4.12501	−0.145659
$803$	35.2568	1.24419
$804$	−54.8448	−1.93423
$805$	0	0
$806$	1.94747	0.0685968
$807$	6.81911	0.240044
$808$	18.4605	0.649438
$809$	−8.29152	−0.291514	−0.145757	−	0.989320i	$-0.546562\pi$
$809$	−8.29152	−0.291514	−0.145757	+	0.989320i	$0.546562\pi$
$810$	0	0
$811$	−8.29661	−0.291333	−0.145667	−	0.989334i	$-0.546533\pi$
$811$	−8.29661	−0.291333	−0.145667	+	0.989334i	$0.546533\pi$
$812$	−20.3813	−0.715244
$813$	56.2836	1.97395
$814$	−5.42042	−0.189986
$815$	0	0
$816$	−50.7625	−1.77704
$817$	−17.6266	−0.616678
$818$	8.74426	0.305736
$819$	14.8413	0.518599
$820$	0	0
$821$	5.02653	0.175427	0.0877135	−	0.996146i	$-0.472044\pi$
$821$	5.02653	0.175427	0.0877135	+	0.996146i	$0.472044\pi$
$822$	19.7666	0.689439
$823$	45.8233	1.59730	0.798650	−	0.601796i	$-0.205548\pi$
$823$	45.8233	1.59730	0.798650	+	0.601796i	$0.205548\pi$
$824$	3.64412	0.126949
$825$	0	0
$826$	−4.44347	−0.154608
$827$	4.52431	0.157326	0.0786628	−	0.996901i	$-0.474935\pi$
$827$	4.52431	0.157326	0.0786628	+	0.996901i	$0.474935\pi$
$828$	−59.2047	−2.05751
$829$	−39.4692	−1.37082	−0.685410	−	0.728157i	$-0.740377\pi$
$829$	−39.4692	−1.37082	−0.685410	+	0.728157i	$0.740377\pi$
$830$	0	0
$831$	41.8453	1.45160
$832$	−6.82258	−0.236530
$833$	20.9091	0.724457
$834$	−19.6591	−0.680740
$835$	0	0
$836$	−11.2605	−0.389453
$837$	47.0015	1.62461
$838$	−8.98858	−0.310505
$839$	10.6688	0.368329	0.184164	−	0.982895i	$-0.441042\pi$
$839$	10.6688	0.368329	0.184164	+	0.982895i	$0.441042\pi$
$840$	0	0
$841$	12.8496	0.443088
$842$	−0.887396	−0.0305817
$843$	−65.9432	−2.27120
$844$	−8.46026	−0.291214
$845$	0	0
$846$	30.8956	1.06221
$847$	−4.77541	−0.164085
$848$	6.73577	0.231307
$849$	−26.7606	−0.918420
$850$	0	0
$851$	25.4182	0.871324
$852$	−55.5859	−1.90434
$853$	−35.0169	−1.19896	−0.599478	−	0.800391i	$-0.704625\pi$
$853$	−35.0169	−1.19896	−0.599478	+	0.800391i	$0.704625\pi$
$854$	4.08031	0.139625
$855$	0	0
$856$	−5.11447	−0.174809
$857$	34.6531	1.18373	0.591864	−	0.806038i	$-0.298392\pi$
$857$	34.6531	1.18373	0.591864	+	0.806038i	$0.298392\pi$
$858$	−4.17419	−0.142504
$859$	4.83017	0.164803	0.0824016	−	0.996599i	$-0.473741\pi$
$859$	4.83017	0.164803	0.0824016	+	0.996599i	$0.473741\pi$
$860$	0	0
$861$	4.98531	0.169899
$862$	−12.7447	−0.434085
$863$	−35.7925	−1.21839	−0.609196	−	0.793020i	$-0.708508\pi$
$863$	−35.7925	−1.21839	−0.609196	+	0.793020i	$0.708508\pi$
$864$	44.2990	1.50708
$865$	0	0
$866$	−5.26085	−0.178771
$867$	25.1189	0.853084
$868$	−13.0450	−0.442777
$869$	−7.73748	−0.262476
$870$	0	0
$871$	12.4976	0.423467
$872$	14.4207	0.488348
$873$	10.7794	0.364827
$874$	−3.53873	−0.119699
$875$	0	0
$876$	71.8834	2.42871
$877$	−12.6008	−0.425500	−0.212750	−	0.977107i	$-0.568242\pi$
$877$	−12.6008	−0.425500	−0.212750	+	0.977107i	$0.568242\pi$
$878$	−7.40512	−0.249911
$879$	3.81304	0.128611
$880$	0	0
$881$	−6.48325	−0.218426	−0.109213	−	0.994018i	$-0.534833\pi$
$881$	−6.48325	−0.218426	−0.109213	+	0.994018i	$0.534833\pi$
$882$	−9.84467	−0.331487
$883$	5.61321	0.188900	0.0944498	−	0.995530i	$-0.469891\pi$
$883$	5.61321	0.188900	0.0944498	+	0.995530i	$0.469891\pi$
$884$	12.4583	0.419016
$885$	0	0
$886$	6.38558	0.214528
$887$	−30.9825	−1.04029	−0.520145	−	0.854078i	$-0.674122\pi$
$887$	−30.9825	−1.04029	−0.520145	+	0.854078i	$0.674122\pi$
$888$	−22.8435	−0.766577
$889$	−13.2947	−0.445891
$890$	0	0
$891$	−43.7275	−1.46493
$892$	44.0191	1.47387
$893$	−27.5556	−0.922113
$894$	17.3087	0.578889
$895$	0	0
$896$	−16.1816	−0.540589
$897$	19.5742	0.653562
$898$	−6.14245	−0.204976
$899$	26.7857	0.893355
$900$	0	0
$901$	−10.3418	−0.344534
$902$	−0.966399	−0.0321776
$903$	43.7681	1.45651
$904$	7.52443	0.250259
$905$	0	0
$906$	−1.10119	−0.0365845
$907$	8.22700	0.273173	0.136587	−	0.990628i	$-0.456387\pi$
$907$	8.22700	0.273173	0.136587	+	0.990628i	$0.456387\pi$
$908$	−1.09815	−0.0364434
$909$	−89.4485	−2.96682
$910$	0	0
$911$	41.1332	1.36280	0.681402	−	0.731909i	$-0.261370\pi$
$911$	41.1332	1.36280	0.681402	+	0.731909i	$0.261370\pi$
$912$	−21.3168	−0.705870
$913$	−29.7952	−0.986078
$914$	−5.20253	−0.172084
$915$	0	0
$916$	−40.7117	−1.34515
$917$	7.19553	0.237617
$918$	−20.1500	−0.665050
$919$	20.1916	0.666060	0.333030	−	0.942916i	$-0.391929\pi$
$919$	20.1916	0.666060	0.333030	+	0.942916i	$0.391929\pi$
$920$	0	0
$921$	31.8932	1.05092
$922$	−1.02242	−0.0336717
$923$	12.6665	0.416923
$924$	27.9606	0.919835
$925$	0	0
$926$	5.60031	0.184038
$927$	−17.6572	−0.579939
$928$	25.2456	0.828727
$929$	39.2502	1.28776	0.643879	−	0.765127i	$-0.277324\pi$
$929$	39.2502	1.28776	0.643879	+	0.765127i	$0.277324\pi$
$930$	0	0
$931$	8.78039	0.287766
$932$	18.7591	0.614474
$933$	−54.6220	−1.78824
$934$	−4.54002	−0.148554
$935$	0	0
$936$	−12.1246	−0.396305
$937$	−32.5608	−1.06371	−0.531857	−	0.846834i	$-0.678506\pi$
$937$	−32.5608	−1.06371	−0.531857	+	0.846834i	$0.678506\pi$
$938$	5.61022	0.183180
$939$	72.8560	2.37757
$940$	0	0
$941$	−13.9619	−0.455144	−0.227572	−	0.973761i	$-0.573079\pi$
$941$	−13.9619	−0.455144	−0.227572	+	0.973761i	$0.573079\pi$
$942$	−17.7830	−0.579402
$943$	4.53177	0.147575
$944$	−24.3317	−0.791930
$945$	0	0
$946$	−8.48441	−0.275852
$947$	−3.72045	−0.120898	−0.0604492	−	0.998171i	$-0.519253\pi$
$947$	−3.72045	−0.120898	−0.0604492	+	0.998171i	$0.519253\pi$
$948$	−15.7756	−0.512366
$949$	−16.3803	−0.531726
$950$	0	0
$951$	−80.0513	−2.59584
$952$	11.5599	0.374658
$953$	−44.9628	−1.45649	−0.728244	−	0.685318i	$-0.759663\pi$
$953$	−44.9628	−1.45649	−0.728244	+	0.685318i	$0.759663\pi$
$954$	4.86923	0.157647
$955$	0	0
$956$	−11.2507	−0.363872
$957$	−57.4123	−1.85587
$958$	0.602087	0.0194526
$959$	30.1716	0.974291
$960$	0	0
$961$	−13.8558	−0.446962
$962$	2.51832	0.0811939
$963$	24.7817	0.798578
$964$	22.2479	0.716557
$965$	0	0
$966$	8.78689	0.282714
$967$	4.49060	0.144408	0.0722040	−	0.997390i	$-0.476997\pi$
$967$	4.49060	0.144408	0.0722040	+	0.997390i	$0.476997\pi$
$968$	3.90125	0.125391
$969$	32.7287	1.05140
$970$	0	0
$971$	43.2685	1.38855	0.694277	−	0.719708i	$-0.255724\pi$
$971$	43.2685	1.38855	0.694277	+	0.719708i	$0.255724\pi$
$972$	−25.3225	−0.812221
$973$	−30.0076	−0.961998
$974$	−0.734080	−0.0235214
$975$	0	0
$976$	22.3431	0.715186
$977$	−37.2679	−1.19231	−0.596153	−	0.802871i	$-0.703305\pi$
$977$	−37.2679	−1.19231	−0.596153	+	0.802871i	$0.703305\pi$
$978$	−14.1851	−0.453588
$979$	33.0986	1.05784
$980$	0	0
$981$	−69.8743	−2.23091
$982$	−4.21407	−0.134476
$983$	−3.65208	−0.116483	−0.0582416	−	0.998303i	$-0.518549\pi$
$983$	−3.65208	−0.116483	−0.0582416	+	0.998303i	$0.518549\pi$
$984$	−4.07273	−0.129834
$985$	0	0
$986$	−11.4833	−0.365704
$987$	68.4223	2.17791
$988$	5.23162	0.166440
$989$	39.7862	1.26513
$990$	0	0
$991$	8.44442	0.268246	0.134123	−	0.990965i	$-0.457178\pi$
$991$	8.44442	0.268246	0.134123	+	0.990965i	$0.457178\pi$
$992$	16.1584	0.513030
$993$	27.1016	0.860042
$994$	5.68603	0.180350
$995$	0	0
$996$	−60.7480	−1.92487
$997$	−51.9929	−1.64663	−0.823315	−	0.567584i	$-0.807878\pi$
$997$	−51.9929	−1.64663	−0.823315	+	0.567584i	$0.807878\pi$
$998$	−2.74696	−0.0869534
$999$	60.7787	1.92295

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3775.2.a.x.1.25		44
5.2	odd	4		755.2.b.d.454.25	yes	44
5.3	odd	4		755.2.b.d.454.20	✓	44
5.4	even	2	inner	3775.2.a.x.1.20		44

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
755.2.b.d.454.20	✓	44	5.3	odd	4
755.2.b.d.454.25	yes	44	5.2	odd	4
3775.2.a.x.1.20		44	5.4	even	2	inner
3775.2.a.x.1.25		44	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 3775 = 5^{2} \cdot 151 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3775.a (trivial)

\(f(q)\)	\(=\)	\(q+0.354420 q^{2} +3.10701 q^{3} -1.87439 q^{4} +1.10119 q^{6} +1.68085 q^{7} -1.37316 q^{8} +6.65351 q^{9} +O(q^{10})\)	\(q+0.354420 q^{2} +3.10701 q^{3} -1.87439 q^{4} +1.10119 q^{6} +1.68085 q^{7} -1.37316 q^{8} +6.65351 q^{9} -2.85638 q^{11} -5.82374 q^{12} +1.32707 q^{13} +0.595726 q^{14} +3.26210 q^{16} -5.00845 q^{17} +2.35814 q^{18} -2.10321 q^{19} +5.22240 q^{21} -1.01236 q^{22} +4.74729 q^{23} -4.26643 q^{24} +0.470341 q^{26} +11.3515 q^{27} -3.15055 q^{28} +6.46912 q^{29} +4.14055 q^{31} +3.90248 q^{32} -8.87481 q^{33} -1.77510 q^{34} -12.4713 q^{36} +5.35424 q^{37} -0.745420 q^{38} +4.12323 q^{39} +0.954600 q^{41} +1.85093 q^{42} +8.38082 q^{43} +5.35397 q^{44} +1.68254 q^{46} +13.1017 q^{47} +10.1354 q^{48} -4.17476 q^{49} -15.5613 q^{51} -2.48744 q^{52} +2.06486 q^{53} +4.02320 q^{54} -2.30807 q^{56} -6.53470 q^{57} +2.29279 q^{58} -7.45893 q^{59} +6.84931 q^{61} +1.46750 q^{62} +11.1835 q^{63} -5.14108 q^{64} -3.14541 q^{66} +9.41746 q^{67} +9.38778 q^{68} +14.7499 q^{69} +9.54471 q^{71} -9.13635 q^{72} -12.3432 q^{73} +1.89765 q^{74} +3.94223 q^{76} -4.80114 q^{77} +1.46135 q^{78} +2.70884 q^{79} +15.3087 q^{81} +0.338330 q^{82} +10.4311 q^{83} -9.78880 q^{84} +2.97033 q^{86} +20.0996 q^{87} +3.92227 q^{88} -11.5876 q^{89} +2.23060 q^{91} -8.89826 q^{92} +12.8647 q^{93} +4.64350 q^{94} +12.1250 q^{96} +1.62011 q^{97} -1.47962 q^{98} -19.0050 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q + 50 q^{4} + 16 q^{6} + 70 q^{9}+O(q^{10}) \) 44 * q + 50 * q^4 + 16 * q^6 + 70 * q^9	\( 44 q + 50 q^{4} + 16 q^{6} + 70 q^{9} + 20 q^{11} + 30 q^{14} + 58 q^{16} + 18 q^{19} + 30 q^{21} + 56 q^{24} + 68 q^{26} + 30 q^{29} + 12 q^{31} + 32 q^{34} + 106 q^{36} + 10 q^{39} + 102 q^{41} + 30 q^{44} - 4 q^{46} + 84 q^{49} + 48 q^{51} + 22 q^{54} + 110 q^{56} + 78 q^{59} + 28 q^{61} + 24 q^{64} + 48 q^{66} - 18 q^{69} + 78 q^{71} + 14 q^{74} + 34 q^{76} + 14 q^{79} + 160 q^{81} - 52 q^{84} + 76 q^{86} + 182 q^{89} + 30 q^{91} - 170 q^{94} + 130 q^{96} + 20 q^{99}+O(q^{100}) \) 44 * q + 50 * q^4 + 16 * q^6 + 70 * q^9 + 20 * q^11 + 30 * q^14 + 58 * q^16 + 18 * q^19 + 30 * q^21 + 56 * q^24 + 68 * q^26 + 30 * q^29 + 12 * q^31 + 32 * q^34 + 106 * q^36 + 10 * q^39 + 102 * q^41 + 30 * q^44 - 4 * q^46 + 84 * q^49 + 48 * q^51 + 22 * q^54 + 110 * q^56 + 78 * q^59 + 28 * q^61 + 24 * q^64 + 48 * q^66 - 18 * q^69 + 78 * q^71 + 14 * q^74 + 34 * q^76 + 14 * q^79 + 160 * q^81 - 52 * q^84 + 76 * q^86 + 182 * q^89 + 30 * q^91 - 170 * q^94 + 130 * q^96 + 20 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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