LMFDB - Embedded newform 3775.2.a.i.1.2

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:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 755)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ of defining polynomial
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)

$f(q)$	$=$	$q+1.00000 q^{2} +2.41421 q^{3} -1.00000 q^{4} +2.41421 q^{6} +2.00000 q^{7} -3.00000 q^{8} +2.82843 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} +2.41421 q^{3} -1.00000 q^{4} +2.41421 q^{6} +2.00000 q^{7} -3.00000 q^{8} +2.82843 q^{9} +1.00000 q^{11} -2.41421 q^{12} +0.414214 q^{13} +2.00000 q^{14} -1.00000 q^{16} +4.82843 q^{17} +2.82843 q^{18} +8.65685 q^{19} +4.82843 q^{21} +1.00000 q^{22} -6.07107 q^{23} -7.24264 q^{24} +0.414214 q^{26} -0.414214 q^{27} -2.00000 q^{28} -2.17157 q^{29} +0.171573 q^{31} +5.00000 q^{32} +2.41421 q^{33} +4.82843 q^{34} -2.82843 q^{36} +1.17157 q^{37} +8.65685 q^{38} +1.00000 q^{39} -8.82843 q^{41} +4.82843 q^{42} +10.4853 q^{43} -1.00000 q^{44} -6.07107 q^{46} +11.6569 q^{47} -2.41421 q^{48} -3.00000 q^{49} +11.6569 q^{51} -0.414214 q^{52} +1.17157 q^{53} -0.414214 q^{54} -6.00000 q^{56} +20.8995 q^{57} -2.17157 q^{58} +11.8284 q^{59} -12.0000 q^{61} +0.171573 q^{62} +5.65685 q^{63} +7.00000 q^{64} +2.41421 q^{66} +4.75736 q^{67} -4.82843 q^{68} -14.6569 q^{69} -1.65685 q^{71} -8.48528 q^{72} -14.5563 q^{73} +1.17157 q^{74} -8.65685 q^{76} +2.00000 q^{77} +1.00000 q^{78} +4.82843 q^{79} -9.48528 q^{81} -8.82843 q^{82} +2.75736 q^{83} -4.82843 q^{84} +10.4853 q^{86} -5.24264 q^{87} -3.00000 q^{88} +4.82843 q^{89} +0.828427 q^{91} +6.07107 q^{92} +0.414214 q^{93} +11.6569 q^{94} +12.0711 q^{96} +18.4853 q^{97} -3.00000 q^{98} +2.82843 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 2 q^{3} - 2 q^{4} + 2 q^{6} + 4 q^{7} - 6 q^{8}+O(q^{10}) $ 2 * q + 2 * q^2 + 2 * q^3 - 2 * q^4 + 2 * q^6 + 4 * q^7 - 6 * q^8	$ 2 q + 2 q^{2} + 2 q^{3} - 2 q^{4} + 2 q^{6} + 4 q^{7} - 6 q^{8} + 2 q^{11} - 2 q^{12} - 2 q^{13} + 4 q^{14} - 2 q^{16} + 4 q^{17} + 6 q^{19} + 4 q^{21} + 2 q^{22} + 2 q^{23} - 6 q^{24} - 2 q^{26} + 2 q^{27} - 4 q^{28} - 10 q^{29} + 6 q^{31} + 10 q^{32} + 2 q^{33} + 4 q^{34} + 8 q^{37} + 6 q^{38} + 2 q^{39} - 12 q^{41} + 4 q^{42} + 4 q^{43} - 2 q^{44} + 2 q^{46} + 12 q^{47} - 2 q^{48} - 6 q^{49} + 12 q^{51} + 2 q^{52} + 8 q^{53} + 2 q^{54} - 12 q^{56} + 22 q^{57} - 10 q^{58} + 18 q^{59} - 24 q^{61} + 6 q^{62} + 14 q^{64} + 2 q^{66} + 18 q^{67} - 4 q^{68} - 18 q^{69} + 8 q^{71} + 2 q^{73} + 8 q^{74} - 6 q^{76} + 4 q^{77} + 2 q^{78} + 4 q^{79} - 2 q^{81} - 12 q^{82} + 14 q^{83} - 4 q^{84} + 4 q^{86} - 2 q^{87} - 6 q^{88} + 4 q^{89} - 4 q^{91} - 2 q^{92} - 2 q^{93} + 12 q^{94} + 10 q^{96} + 20 q^{97} - 6 q^{98}+O(q^{100}) $ 2 * q + 2 * q^2 + 2 * q^3 - 2 * q^4 + 2 * q^6 + 4 * q^7 - 6 * q^8 + 2 * q^11 - 2 * q^12 - 2 * q^13 + 4 * q^14 - 2 * q^16 + 4 * q^17 + 6 * q^19 + 4 * q^21 + 2 * q^22 + 2 * q^23 - 6 * q^24 - 2 * q^26 + 2 * q^27 - 4 * q^28 - 10 * q^29 + 6 * q^31 + 10 * q^32 + 2 * q^33 + 4 * q^34 + 8 * q^37 + 6 * q^38 + 2 * q^39 - 12 * q^41 + 4 * q^42 + 4 * q^43 - 2 * q^44 + 2 * q^46 + 12 * q^47 - 2 * q^48 - 6 * q^49 + 12 * q^51 + 2 * q^52 + 8 * q^53 + 2 * q^54 - 12 * q^56 + 22 * q^57 - 10 * q^58 + 18 * q^59 - 24 * q^61 + 6 * q^62 + 14 * q^64 + 2 * q^66 + 18 * q^67 - 4 * q^68 - 18 * q^69 + 8 * q^71 + 2 * q^73 + 8 * q^74 - 6 * q^76 + 4 * q^77 + 2 * q^78 + 4 * q^79 - 2 * q^81 - 12 * q^82 + 14 * q^83 - 4 * q^84 + 4 * q^86 - 2 * q^87 - 6 * q^88 + 4 * q^89 - 4 * q^91 - 2 * q^92 - 2 * q^93 + 12 * q^94 + 10 * q^96 + 20 * q^97 - 6 * q^98

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 $

$2$	1.00000	0.707107	0.353553	−	0.935414i	$-0.384973\pi$
$2$	1.00000	0.707107	0.353553	+	0.935414i	$0.384973\pi$
$3$	2.41421	1.39385	0.696923	−	0.717146i	$-0.254552\pi$
$3$	2.41421	1.39385	0.696923	+	0.717146i	$0.254552\pi$
$4$	−1.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	2.41421	0.985599
$7$	2.00000	0.755929	0.377964	−	0.925820i	$-0.376624\pi$
$7$	2.00000	0.755929	0.377964	+	0.925820i	$0.376624\pi$
$8$	−3.00000	−1.06066
$9$	2.82843	0.942809
$10$	0	0
$11$	1.00000	0.301511	0.150756	−	0.988571i	$-0.451829\pi$
$11$	1.00000	0.301511	0.150756	+	0.988571i	$0.451829\pi$
$12$	−2.41421	−0.696923
$13$	0.414214	0.114882	0.0574411	−	0.998349i	$-0.481706\pi$
$13$	0.414214	0.114882	0.0574411	+	0.998349i	$0.481706\pi$
$14$	2.00000	0.534522
$15$	0	0
$16$	−1.00000	−0.250000
$17$	4.82843	1.17107	0.585533	−	0.810649i	$-0.300885\pi$
$17$	4.82843	1.17107	0.585533	+	0.810649i	$0.300885\pi$
$18$	2.82843	0.666667
$19$	8.65685	1.98602	0.993009	−	0.118036i	$-0.0376599\pi$
$19$	8.65685	1.98602	0.993009	+	0.118036i	$0.0376599\pi$
$20$	0	0
$21$	4.82843	1.05365
$22$	1.00000	0.213201
$23$	−6.07107	−1.26591	−0.632953	−	0.774191i	$-0.718157\pi$
$23$	−6.07107	−1.26591	−0.632953	+	0.774191i	$0.718157\pi$
$24$	−7.24264	−1.47840
$25$	0	0
$26$	0.414214	0.0812340
$27$	−0.414214	−0.0797154
$28$	−2.00000	−0.377964
$29$	−2.17157	−0.403251	−0.201625	−	0.979463i	$-0.564622\pi$
$29$	−2.17157	−0.403251	−0.201625	+	0.979463i	$0.564622\pi$
$30$	0	0
$31$	0.171573	0.0308154	0.0154077	−	0.999881i	$-0.495095\pi$
$31$	0.171573	0.0308154	0.0154077	+	0.999881i	$0.495095\pi$
$32$	5.00000	0.883883
$33$	2.41421	0.420261
$34$	4.82843	0.828068
$35$	0	0
$36$	−2.82843	−0.471405
$37$	1.17157	0.192605	0.0963027	−	0.995352i	$-0.469298\pi$
$37$	1.17157	0.192605	0.0963027	+	0.995352i	$0.469298\pi$
$38$	8.65685	1.40433
$39$	1.00000	0.160128
$40$	0	0
$41$	−8.82843	−1.37877	−0.689384	−	0.724396i	$-0.742119\pi$
$41$	−8.82843	−1.37877	−0.689384	+	0.724396i	$0.742119\pi$
$42$	4.82843	0.745042
$43$	10.4853	1.59899	0.799495	−	0.600672i	$-0.205100\pi$
$43$	10.4853	1.59899	0.799495	+	0.600672i	$0.205100\pi$
$44$	−1.00000	−0.150756
$45$	0	0
$46$	−6.07107	−0.895130
$47$	11.6569	1.70033	0.850163	−	0.526519i	$-0.176503\pi$
$47$	11.6569	1.70033	0.850163	+	0.526519i	$0.176503\pi$
$48$	−2.41421	−0.348462
$49$	−3.00000	−0.428571
$50$	0	0
$51$	11.6569	1.63229
$52$	−0.414214	−0.0574411
$53$	1.17157	0.160928	0.0804640	−	0.996758i	$-0.474360\pi$
$53$	1.17157	0.160928	0.0804640	+	0.996758i	$0.474360\pi$
$54$	−0.414214	−0.0563673
$55$	0	0
$56$	−6.00000	−0.801784
$57$	20.8995	2.76821
$58$	−2.17157	−0.285141
$59$	11.8284	1.53993	0.769965	−	0.638086i	$-0.220274\pi$
$59$	11.8284	1.53993	0.769965	+	0.638086i	$0.220274\pi$
$60$	0	0
$61$	−12.0000	−1.53644	−0.768221	−	0.640184i	$-0.778858\pi$
$61$	−12.0000	−1.53644	−0.768221	+	0.640184i	$0.778858\pi$
$62$	0.171573	0.0217898
$63$	5.65685	0.712697
$64$	7.00000	0.875000
$65$	0	0
$66$	2.41421	0.297169
$67$	4.75736	0.581204	0.290602	−	0.956844i	$-0.406144\pi$
$67$	4.75736	0.581204	0.290602	+	0.956844i	$0.406144\pi$
$68$	−4.82843	−0.585533
$69$	−14.6569	−1.76448
$70$	0	0
$71$	−1.65685	−0.196632	−0.0983162	−	0.995155i	$-0.531346\pi$
$71$	−1.65685	−0.196632	−0.0983162	+	0.995155i	$0.531346\pi$
$72$	−8.48528	−1.00000
$73$	−14.5563	−1.70369	−0.851846	−	0.523792i	$-0.824517\pi$
$73$	−14.5563	−1.70369	−0.851846	+	0.523792i	$0.824517\pi$
$74$	1.17157	0.136193
$75$	0	0
$76$	−8.65685	−0.993009
$77$	2.00000	0.227921
$78$	1.00000	0.113228
$79$	4.82843	0.543240	0.271620	−	0.962405i	$-0.412441\pi$
$79$	4.82843	0.543240	0.271620	+	0.962405i	$0.412441\pi$
$80$	0	0
$81$	−9.48528	−1.05392
$82$	−8.82843	−0.974937
$83$	2.75736	0.302660	0.151330	−	0.988483i	$-0.451644\pi$
$83$	2.75736	0.302660	0.151330	+	0.988483i	$0.451644\pi$
$84$	−4.82843	−0.526825
$85$	0	0
$86$	10.4853	1.13066
$87$	−5.24264	−0.562070
$88$	−3.00000	−0.319801
$89$	4.82843	0.511812	0.255906	−	0.966702i	$-0.417626\pi$
$89$	4.82843	0.511812	0.255906	+	0.966702i	$0.417626\pi$
$90$	0	0
$91$	0.828427	0.0868428
$92$	6.07107	0.632953
$93$	0.414214	0.0429519
$94$	11.6569	1.20231
$95$	0	0
$96$	12.0711	1.23200
$97$	18.4853	1.87690	0.938448	−	0.345421i	$-0.112264\pi$
$97$	18.4853	1.87690	0.938448	+	0.345421i	$0.112264\pi$
$98$	−3.00000	−0.303046
$99$	2.82843	0.284268
$100$	0	0
$101$	15.6569	1.55792	0.778958	−	0.627077i	$-0.215749\pi$
$101$	15.6569	1.55792	0.778958	+	0.627077i	$0.215749\pi$
$102$	11.6569	1.15420
$103$	9.17157	0.903702	0.451851	−	0.892093i	$-0.350764\pi$
$103$	9.17157	0.903702	0.451851	+	0.892093i	$0.350764\pi$
$104$	−1.24264	−0.121851
$105$	0	0
$106$	1.17157	0.113793
$107$	9.72792	0.940434	0.470217	−	0.882551i	$-0.344176\pi$
$107$	9.72792	0.940434	0.470217	+	0.882551i	$0.344176\pi$
$108$	0.414214	0.0398577
$109$	−14.1421	−1.35457	−0.677285	−	0.735720i	$-0.736844\pi$
$109$	−14.1421	−1.35457	−0.677285	+	0.735720i	$0.736844\pi$
$110$	0	0
$111$	2.82843	0.268462
$112$	−2.00000	−0.188982
$113$	10.1421	0.954092	0.477046	−	0.878878i	$-0.341708\pi$
$113$	10.1421	0.954092	0.477046	+	0.878878i	$0.341708\pi$
$114$	20.8995	1.95742
$115$	0	0
$116$	2.17157	0.201625
$117$	1.17157	0.108312
$118$	11.8284	1.08889
$119$	9.65685	0.885242
$120$	0	0
$121$	−10.0000	−0.909091
$122$	−12.0000	−1.08643
$123$	−21.3137	−1.92179
$124$	−0.171573	−0.0154077
$125$	0	0
$126$	5.65685	0.503953
$127$	−11.3137	−1.00393	−0.501965	−	0.864888i	$-0.667389\pi$
$127$	−11.3137	−1.00393	−0.501965	+	0.864888i	$0.667389\pi$
$128$	−3.00000	−0.265165
$129$	25.3137	2.22875
$130$	0	0
$131$	12.8284	1.12082	0.560412	−	0.828214i	$-0.310643\pi$
$131$	12.8284	1.12082	0.560412	+	0.828214i	$0.310643\pi$
$132$	−2.41421	−0.210130
$133$	17.3137	1.50129
$134$	4.75736	0.410973
$135$	0	0
$136$	−14.4853	−1.24210
$137$	−8.48528	−0.724947	−0.362473	−	0.931994i	$-0.618068\pi$
$137$	−8.48528	−0.724947	−0.362473	+	0.931994i	$0.618068\pi$
$138$	−14.6569	−1.24767
$139$	2.34315	0.198743	0.0993715	−	0.995050i	$-0.468317\pi$
$139$	2.34315	0.198743	0.0993715	+	0.995050i	$0.468317\pi$
$140$	0	0
$141$	28.1421	2.37000
$142$	−1.65685	−0.139040
$143$	0.414214	0.0346383
$144$	−2.82843	−0.235702
$145$	0	0
$146$	−14.5563	−1.20469
$147$	−7.24264	−0.597363
$148$	−1.17157	−0.0963027
$149$	−1.17157	−0.0959790	−0.0479895	−	0.998848i	$-0.515281\pi$
$149$	−1.17157	−0.0959790	−0.0479895	+	0.998848i	$0.515281\pi$
$150$	0	0
$151$	1.00000	0.0813788
$151$	1.00000	0.0813788
$152$	−25.9706	−2.10649
$153$	13.6569	1.10409
$154$	2.00000	0.161165
$155$	0	0
$156$	−1.00000	−0.0800641
$157$	−1.10051	−0.0878299	−0.0439149	−	0.999035i	$-0.513983\pi$
$157$	−1.10051	−0.0878299	−0.0439149	+	0.999035i	$0.513983\pi$
$158$	4.82843	0.384129
$159$	2.82843	0.224309
$160$	0	0
$161$	−12.1421	−0.956934
$162$	−9.48528	−0.745234
$163$	−3.58579	−0.280860	−0.140430	−	0.990091i	$-0.544849\pi$
$163$	−3.58579	−0.280860	−0.140430	+	0.990091i	$0.544849\pi$
$164$	8.82843	0.689384
$165$	0	0
$166$	2.75736	0.214013
$167$	3.51472	0.271977	0.135989	−	0.990710i	$-0.456579\pi$
$167$	3.51472	0.271977	0.135989	+	0.990710i	$0.456579\pi$
$168$	−14.4853	−1.11756
$169$	−12.8284	−0.986802
$170$	0	0
$171$	24.4853	1.87244
$172$	−10.4853	−0.799495
$173$	7.31371	0.556051	0.278025	−	0.960574i	$-0.410320\pi$
$173$	7.31371	0.556051	0.278025	+	0.960574i	$0.410320\pi$
$174$	−5.24264	−0.397444
$175$	0	0
$176$	−1.00000	−0.0753778
$177$	28.5563	2.14643
$178$	4.82843	0.361906
$179$	−15.7990	−1.18087	−0.590436	−	0.807084i	$-0.701044\pi$
$179$	−15.7990	−1.18087	−0.590436	+	0.807084i	$0.701044\pi$
$180$	0	0
$181$	−10.0000	−0.743294	−0.371647	−	0.928374i	$-0.621207\pi$
$181$	−10.0000	−0.743294	−0.371647	+	0.928374i	$0.621207\pi$
$182$	0.828427	0.0614071
$183$	−28.9706	−2.14157
$184$	18.2132	1.34270
$185$	0	0
$186$	0.414214	0.0303716
$187$	4.82843	0.353090
$188$	−11.6569	−0.850163
$189$	−0.828427	−0.0602592
$190$	0	0
$191$	4.00000	0.289430	0.144715	−	0.989473i	$-0.453773\pi$
$191$	4.00000	0.289430	0.144715	+	0.989473i	$0.453773\pi$
$192$	16.8995	1.21962
$193$	10.4853	0.754747	0.377374	−	0.926061i	$-0.376827\pi$
$193$	10.4853	0.754747	0.377374	+	0.926061i	$0.376827\pi$
$194$	18.4853	1.32717
$195$	0	0
$196$	3.00000	0.214286
$197$	5.17157	0.368459	0.184230	−	0.982883i	$-0.441021\pi$
$197$	5.17157	0.368459	0.184230	+	0.982883i	$0.441021\pi$
$198$	2.82843	0.201008
$199$	−24.6274	−1.74579	−0.872896	−	0.487907i	$-0.837761\pi$
$199$	−24.6274	−1.74579	−0.872896	+	0.487907i	$0.837761\pi$
$200$	0	0
$201$	11.4853	0.810109
$202$	15.6569	1.10161
$203$	−4.34315	−0.304829
$204$	−11.6569	−0.816143
$205$	0	0
$206$	9.17157	0.639014
$207$	−17.1716	−1.19351
$208$	−0.414214	−0.0287205
$209$	8.65685	0.598807
$210$	0	0
$211$	−5.65685	−0.389434	−0.194717	−	0.980859i	$-0.562379\pi$
$211$	−5.65685	−0.389434	−0.194717	+	0.980859i	$0.562379\pi$
$212$	−1.17157	−0.0804640
$213$	−4.00000	−0.274075
$214$	9.72792	0.664987
$215$	0	0
$216$	1.24264	0.0845510
$217$	0.343146	0.0232943
$218$	−14.1421	−0.957826
$219$	−35.1421	−2.37469
$220$	0	0
$221$	2.00000	0.134535
$222$	2.82843	0.189832
$223$	−17.3137	−1.15941	−0.579706	−	0.814826i	$-0.696833\pi$
$223$	−17.3137	−1.15941	−0.579706	+	0.814826i	$0.696833\pi$
$224$	10.0000	0.668153
$225$	0	0
$226$	10.1421	0.674645
$227$	−6.00000	−0.398234	−0.199117	−	0.979976i	$-0.563807\pi$
$227$	−6.00000	−0.398234	−0.199117	+	0.979976i	$0.563807\pi$
$228$	−20.8995	−1.38410
$229$	−15.6274	−1.03269	−0.516344	−	0.856381i	$-0.672708\pi$
$229$	−15.6274	−1.03269	−0.516344	+	0.856381i	$0.672708\pi$
$230$	0	0
$231$	4.82843	0.317687
$232$	6.51472	0.427712
$233$	−26.8995	−1.76224	−0.881122	−	0.472889i	$-0.843211\pi$
$233$	−26.8995	−1.76224	−0.881122	+	0.472889i	$0.843211\pi$
$234$	1.17157	0.0765881
$235$	0	0
$236$	−11.8284	−0.769965
$237$	11.6569	0.757194
$238$	9.65685	0.625961
$239$	−4.65685	−0.301227	−0.150613	−	0.988593i	$-0.548125\pi$
$239$	−4.65685	−0.301227	−0.150613	+	0.988593i	$0.548125\pi$
$240$	0	0
$241$	−2.17157	−0.139883	−0.0699417	−	0.997551i	$-0.522281\pi$
$241$	−2.17157	−0.139883	−0.0699417	+	0.997551i	$0.522281\pi$
$242$	−10.0000	−0.642824
$243$	−21.6569	−1.38929
$244$	12.0000	0.768221
$245$	0	0
$246$	−21.3137	−1.35891
$247$	3.58579	0.228158
$248$	−0.514719	−0.0326847
$249$	6.65685	0.421861
$250$	0	0
$251$	11.4853	0.724945	0.362472	−	0.931995i	$-0.381933\pi$
$251$	11.4853	0.724945	0.362472	+	0.931995i	$0.381933\pi$
$252$	−5.65685	−0.356348
$253$	−6.07107	−0.381685
$254$	−11.3137	−0.709885
$255$	0	0
$256$	−17.0000	−1.06250
$257$	−4.07107	−0.253946	−0.126973	−	0.991906i	$-0.540526\pi$
$257$	−4.07107	−0.253946	−0.126973	+	0.991906i	$0.540526\pi$
$258$	25.3137	1.57596
$259$	2.34315	0.145596
$260$	0	0
$261$	−6.14214	−0.380189
$262$	12.8284	0.792543
$263$	17.5858	1.08439	0.542193	−	0.840254i	$-0.317594\pi$
$263$	17.5858	1.08439	0.542193	+	0.840254i	$0.317594\pi$
$264$	−7.24264	−0.445754
$265$	0	0
$266$	17.3137	1.06157
$267$	11.6569	0.713388
$268$	−4.75736	−0.290602
$269$	−17.0000	−1.03651	−0.518254	−	0.855227i	$-0.673418\pi$
$269$	−17.0000	−1.03651	−0.518254	+	0.855227i	$0.673418\pi$
$270$	0	0
$271$	−22.6274	−1.37452	−0.687259	−	0.726413i	$-0.741186\pi$
$271$	−22.6274	−1.37452	−0.687259	+	0.726413i	$0.741186\pi$
$272$	−4.82843	−0.292766
$273$	2.00000	0.121046
$274$	−8.48528	−0.512615
$275$	0	0
$276$	14.6569	0.882239
$277$	26.1421	1.57073	0.785364	−	0.619034i	$-0.212476\pi$
$277$	26.1421	1.57073	0.785364	+	0.619034i	$0.212476\pi$
$278$	2.34315	0.140533
$279$	0.485281	0.0290530
$280$	0	0
$281$	10.0000	0.596550	0.298275	−	0.954480i	$-0.403589\pi$
$281$	10.0000	0.596550	0.298275	+	0.954480i	$0.403589\pi$
$282$	28.1421	1.67584
$283$	−3.72792	−0.221602	−0.110801	−	0.993843i	$-0.535342\pi$
$283$	−3.72792	−0.221602	−0.110801	+	0.993843i	$0.535342\pi$
$284$	1.65685	0.0983162
$285$	0	0
$286$	0.414214	0.0244930
$287$	−17.6569	−1.04225
$288$	14.1421	0.833333
$289$	6.31371	0.371395
$290$	0	0
$291$	44.6274	2.61611
$292$	14.5563	0.851846
$293$	−16.4853	−0.963080	−0.481540	−	0.876424i	$-0.659922\pi$
$293$	−16.4853	−0.963080	−0.481540	+	0.876424i	$0.659922\pi$
$294$	−7.24264	−0.422399
$295$	0	0
$296$	−3.51472	−0.204289
$297$	−0.414214	−0.0240351
$298$	−1.17157	−0.0678674
$299$	−2.51472	−0.145430
$300$	0	0
$301$	20.9706	1.20872
$302$	1.00000	0.0575435
$303$	37.7990	2.17150
$304$	−8.65685	−0.496505
$305$	0	0
$306$	13.6569	0.780710
$307$	18.4853	1.05501	0.527505	−	0.849552i	$-0.323127\pi$
$307$	18.4853	1.05501	0.527505	+	0.849552i	$0.323127\pi$
$308$	−2.00000	−0.113961
$309$	22.1421	1.25962
$310$	0	0
$311$	−6.31371	−0.358018	−0.179009	−	0.983847i	$-0.557289\pi$
$311$	−6.31371	−0.358018	−0.179009	+	0.983847i	$0.557289\pi$
$312$	−3.00000	−0.169842
$313$	−22.6274	−1.27898	−0.639489	−	0.768801i	$-0.720854\pi$
$313$	−22.6274	−1.27898	−0.639489	+	0.768801i	$0.720854\pi$
$314$	−1.10051	−0.0621051
$315$	0	0
$316$	−4.82843	−0.271620
$317$	−26.5563	−1.49155	−0.745777	−	0.666196i	$-0.767921\pi$
$317$	−26.5563	−1.49155	−0.745777	+	0.666196i	$0.767921\pi$
$318$	2.82843	0.158610
$319$	−2.17157	−0.121585
$320$	0	0
$321$	23.4853	1.31082
$322$	−12.1421	−0.676655
$323$	41.7990	2.32576
$324$	9.48528	0.526960
$325$	0	0
$326$	−3.58579	−0.198598
$327$	−34.1421	−1.88806
$328$	26.4853	1.46241
$329$	23.3137	1.28533
$330$	0	0
$331$	−28.0000	−1.53902	−0.769510	−	0.638635i	$-0.779499\pi$
$331$	−28.0000	−1.53902	−0.769510	+	0.638635i	$0.779499\pi$
$332$	−2.75736	−0.151330
$333$	3.31371	0.181590
$334$	3.51472	0.192317
$335$	0	0
$336$	−4.82843	−0.263412
$337$	−21.7279	−1.18360	−0.591798	−	0.806087i	$-0.701582\pi$
$337$	−21.7279	−1.18360	−0.591798	+	0.806087i	$0.701582\pi$
$338$	−12.8284	−0.697774
$339$	24.4853	1.32986
$340$	0	0
$341$	0.171573	0.00929119
$342$	24.4853	1.32401
$343$	−20.0000	−1.07990
$344$	−31.4558	−1.69599
$345$	0	0
$346$	7.31371	0.393187
$347$	25.4558	1.36654	0.683271	−	0.730165i	$-0.260557\pi$
$347$	25.4558	1.36654	0.683271	+	0.730165i	$0.260557\pi$
$348$	5.24264	0.281035
$349$	−14.5147	−0.776955	−0.388478	−	0.921458i	$-0.626999\pi$
$349$	−14.5147	−0.776955	−0.388478	+	0.921458i	$0.626999\pi$
$350$	0	0
$351$	−0.171573	−0.00915788
$352$	5.00000	0.266501
$353$	4.75736	0.253209	0.126604	−	0.991953i	$-0.459592\pi$
$353$	4.75736	0.253209	0.126604	+	0.991953i	$0.459592\pi$
$354$	28.5563	1.51775
$355$	0	0
$356$	−4.82843	−0.255906
$357$	23.3137	1.23389
$358$	−15.7990	−0.835003
$359$	−12.1421	−0.640837	−0.320419	−	0.947276i	$-0.603824\pi$
$359$	−12.1421	−0.640837	−0.320419	+	0.947276i	$0.603824\pi$
$360$	0	0
$361$	55.9411	2.94427
$362$	−10.0000	−0.525588
$363$	−24.1421	−1.26713
$364$	−0.828427	−0.0434214
$365$	0	0
$366$	−28.9706	−1.51432
$367$	−13.3137	−0.694970	−0.347485	−	0.937686i	$-0.612964\pi$
$367$	−13.3137	−0.694970	−0.347485	+	0.937686i	$0.612964\pi$
$368$	6.07107	0.316476
$369$	−24.9706	−1.29992
$370$	0	0
$371$	2.34315	0.121650
$372$	−0.414214	−0.0214760
$373$	−25.8701	−1.33950	−0.669750	−	0.742586i	$-0.733599\pi$
$373$	−25.8701	−1.33950	−0.669750	+	0.742586i	$0.733599\pi$
$374$	4.82843	0.249672
$375$	0	0
$376$	−34.9706	−1.80347
$377$	−0.899495	−0.0463263
$378$	−0.828427	−0.0426097
$379$	6.00000	0.308199	0.154100	−	0.988055i	$-0.450752\pi$
$379$	6.00000	0.308199	0.154100	+	0.988055i	$0.450752\pi$
$380$	0	0
$381$	−27.3137	−1.39932
$382$	4.00000	0.204658
$383$	−4.34315	−0.221924	−0.110962	−	0.993825i	$-0.535393\pi$
$383$	−4.34315	−0.221924	−0.110962	+	0.993825i	$0.535393\pi$
$384$	−7.24264	−0.369599
$385$	0	0
$386$	10.4853	0.533687
$387$	29.6569	1.50754
$388$	−18.4853	−0.938448
$389$	−31.9411	−1.61948	−0.809740	−	0.586789i	$-0.800392\pi$
$389$	−31.9411	−1.61948	−0.809740	+	0.586789i	$0.800392\pi$
$390$	0	0
$391$	−29.3137	−1.48246
$392$	9.00000	0.454569
$393$	30.9706	1.56226
$394$	5.17157	0.260540
$395$	0	0
$396$	−2.82843	−0.142134
$397$	29.6569	1.48843	0.744217	−	0.667937i	$-0.232823\pi$
$397$	29.6569	1.48843	0.744217	+	0.667937i	$0.232823\pi$
$398$	−24.6274	−1.23446
$399$	41.7990	2.09257
$400$	0	0
$401$	−34.9706	−1.74635	−0.873173	−	0.487410i	$-0.837942\pi$
$401$	−34.9706	−1.74635	−0.873173	+	0.487410i	$0.837942\pi$
$402$	11.4853	0.572834
$403$	0.0710678	0.00354014
$404$	−15.6569	−0.778958
$405$	0	0
$406$	−4.34315	−0.215547
$407$	1.17157	0.0580727
$408$	−34.9706	−1.73130
$409$	−14.0000	−0.692255	−0.346128	−	0.938187i	$-0.612504\pi$
$409$	−14.0000	−0.692255	−0.346128	+	0.938187i	$0.612504\pi$
$410$	0	0
$411$	−20.4853	−1.01046
$412$	−9.17157	−0.451851
$413$	23.6569	1.16408
$414$	−17.1716	−0.843937
$415$	0	0
$416$	2.07107	0.101542
$417$	5.65685	0.277017
$418$	8.65685	0.423421
$419$	9.79899	0.478712	0.239356	−	0.970932i	$-0.423064\pi$
$419$	9.79899	0.478712	0.239356	+	0.970932i	$0.423064\pi$
$420$	0	0
$421$	−30.8284	−1.50249	−0.751243	−	0.660026i	$-0.770545\pi$
$421$	−30.8284	−1.50249	−0.751243	+	0.660026i	$0.770545\pi$
$422$	−5.65685	−0.275371
$423$	32.9706	1.60308
$424$	−3.51472	−0.170690
$425$	0	0
$426$	−4.00000	−0.193801
$427$	−24.0000	−1.16144
$428$	−9.72792	−0.470217
$429$	1.00000	0.0482805
$430$	0	0
$431$	8.82843	0.425250	0.212625	−	0.977134i	$-0.431799\pi$
$431$	8.82843	0.425250	0.212625	+	0.977134i	$0.431799\pi$
$432$	0.414214	0.0199289
$433$	−5.17157	−0.248530	−0.124265	−	0.992249i	$-0.539657\pi$
$433$	−5.17157	−0.248530	−0.124265	+	0.992249i	$0.539657\pi$
$434$	0.343146	0.0164715
$435$	0	0
$436$	14.1421	0.677285
$437$	−52.5563	−2.51411
$438$	−35.1421	−1.67916
$439$	19.2843	0.920388	0.460194	−	0.887818i	$-0.347780\pi$
$439$	19.2843	0.920388	0.460194	+	0.887818i	$0.347780\pi$
$440$	0	0
$441$	−8.48528	−0.404061
$442$	2.00000	0.0951303
$443$	30.0000	1.42534	0.712672	−	0.701498i	$-0.247485\pi$
$443$	30.0000	1.42534	0.712672	+	0.701498i	$0.247485\pi$
$444$	−2.82843	−0.134231
$445$	0	0
$446$	−17.3137	−0.819828
$447$	−2.82843	−0.133780
$448$	14.0000	0.661438
$449$	−14.4853	−0.683603	−0.341801	−	0.939772i	$-0.611037\pi$
$449$	−14.4853	−0.683603	−0.341801	+	0.939772i	$0.611037\pi$
$450$	0	0
$451$	−8.82843	−0.415714
$452$	−10.1421	−0.477046
$453$	2.41421	0.113430
$454$	−6.00000	−0.281594
$455$	0	0
$456$	−62.6985	−2.93613
$457$	−5.51472	−0.257968	−0.128984	−	0.991647i	$-0.541172\pi$
$457$	−5.51472	−0.257968	−0.128984	+	0.991647i	$0.541172\pi$
$458$	−15.6274	−0.730221
$459$	−2.00000	−0.0933520
$460$	0	0
$461$	−0.343146	−0.0159819	−0.00799095	−	0.999968i	$-0.502544\pi$
$461$	−0.343146	−0.0159819	−0.00799095	+	0.999968i	$0.502544\pi$
$462$	4.82843	0.224639
$463$	22.1421	1.02903	0.514516	−	0.857481i	$-0.327972\pi$
$463$	22.1421	1.02903	0.514516	+	0.857481i	$0.327972\pi$
$464$	2.17157	0.100813
$465$	0	0
$466$	−26.8995	−1.24610
$467$	5.31371	0.245889	0.122945	−	0.992414i	$-0.460766\pi$
$467$	5.31371	0.245889	0.122945	+	0.992414i	$0.460766\pi$
$468$	−1.17157	−0.0541560
$469$	9.51472	0.439349
$470$	0	0
$471$	−2.65685	−0.122421
$472$	−35.4853	−1.63334
$473$	10.4853	0.482114
$474$	11.6569	0.535417
$475$	0	0
$476$	−9.65685	−0.442621
$477$	3.31371	0.151724
$478$	−4.65685	−0.213000
$479$	−1.85786	−0.0848880	−0.0424440	−	0.999099i	$-0.513514\pi$
$479$	−1.85786	−0.0848880	−0.0424440	+	0.999099i	$0.513514\pi$
$480$	0	0
$481$	0.485281	0.0221269
$482$	−2.17157	−0.0989124
$483$	−29.3137	−1.33382
$484$	10.0000	0.454545
$485$	0	0
$486$	−21.6569	−0.982375
$487$	−3.17157	−0.143718	−0.0718588	−	0.997415i	$-0.522893\pi$
$487$	−3.17157	−0.143718	−0.0718588	+	0.997415i	$0.522893\pi$
$488$	36.0000	1.62964
$489$	−8.65685	−0.391476
$490$	0	0
$491$	37.6569	1.69943	0.849715	−	0.527242i	$-0.176774\pi$
$491$	37.6569	1.69943	0.849715	+	0.527242i	$0.176774\pi$
$492$	21.3137	0.960896
$493$	−10.4853	−0.472233
$494$	3.58579	0.161332
$495$	0	0
$496$	−0.171573	−0.00770385
$497$	−3.31371	−0.148640
$498$	6.65685	0.298301
$499$	18.4853	0.827515	0.413757	−	0.910387i	$-0.364216\pi$
$499$	18.4853	0.827515	0.413757	+	0.910387i	$0.364216\pi$
$500$	0	0
$501$	8.48528	0.379094
$502$	11.4853	0.512613
$503$	−10.1421	−0.452215	−0.226108	−	0.974102i	$-0.572600\pi$
$503$	−10.1421	−0.452215	−0.226108	+	0.974102i	$0.572600\pi$
$504$	−16.9706	−0.755929
$505$	0	0
$506$	−6.07107	−0.269892
$507$	−30.9706	−1.37545
$508$	11.3137	0.501965
$509$	21.6569	0.959923	0.479962	−	0.877290i	$-0.340651\pi$
$509$	21.6569	0.959923	0.479962	+	0.877290i	$0.340651\pi$
$510$	0	0
$511$	−29.1127	−1.28787
$512$	−11.0000	−0.486136
$513$	−3.58579	−0.158316
$514$	−4.07107	−0.179567
$515$	0	0
$516$	−25.3137	−1.11437
$517$	11.6569	0.512668
$518$	2.34315	0.102952
$519$	17.6569	0.775050
$520$	0	0
$521$	25.9706	1.13779	0.568896	−	0.822410i	$-0.307371\pi$
$521$	25.9706	1.13779	0.568896	+	0.822410i	$0.307371\pi$
$522$	−6.14214	−0.268834
$523$	16.5563	0.723959	0.361979	−	0.932186i	$-0.382101\pi$
$523$	16.5563	0.723959	0.361979	+	0.932186i	$0.382101\pi$
$524$	−12.8284	−0.560412
$525$	0	0
$526$	17.5858	0.766777
$527$	0.828427	0.0360869
$528$	−2.41421	−0.105065
$529$	13.8579	0.602516
$530$	0	0
$531$	33.4558	1.45186
$532$	−17.3137	−0.750644
$533$	−3.65685	−0.158396
$534$	11.6569	0.504441
$535$	0	0
$536$	−14.2721	−0.616460
$537$	−38.1421	−1.64595
$538$	−17.0000	−0.732922
$539$	−3.00000	−0.129219
$540$	0	0
$541$	18.1127	0.778726	0.389363	−	0.921084i	$-0.372695\pi$
$541$	18.1127	0.778726	0.389363	+	0.921084i	$0.372695\pi$
$542$	−22.6274	−0.971931
$543$	−24.1421	−1.03604
$544$	24.1421	1.03509
$545$	0	0
$546$	2.00000	0.0855921
$547$	28.2843	1.20935	0.604674	−	0.796473i	$-0.293303\pi$
$547$	28.2843	1.20935	0.604674	+	0.796473i	$0.293303\pi$
$548$	8.48528	0.362473
$549$	−33.9411	−1.44857
$550$	0	0
$551$	−18.7990	−0.800864
$552$	43.9706	1.87151
$553$	9.65685	0.410651
$554$	26.1421	1.11067
$555$	0	0
$556$	−2.34315	−0.0993715
$557$	−23.2426	−0.984822	−0.492411	−	0.870363i	$-0.663884\pi$
$557$	−23.2426	−0.984822	−0.492411	+	0.870363i	$0.663884\pi$
$558$	0.485281	0.0205436
$559$	4.34315	0.183695
$560$	0	0
$561$	11.6569	0.492153
$562$	10.0000	0.421825
$563$	9.45584	0.398516	0.199258	−	0.979947i	$-0.436147\pi$
$563$	9.45584	0.398516	0.199258	+	0.979947i	$0.436147\pi$
$564$	−28.1421	−1.18500
$565$	0	0
$566$	−3.72792	−0.156696
$567$	−18.9706	−0.796689
$568$	4.97056	0.208560
$569$	−14.9706	−0.627599	−0.313799	−	0.949489i	$-0.601602\pi$
$569$	−14.9706	−0.627599	−0.313799	+	0.949489i	$0.601602\pi$
$570$	0	0
$571$	10.1716	0.425667	0.212834	−	0.977088i	$-0.431731\pi$
$571$	10.1716	0.425667	0.212834	+	0.977088i	$0.431731\pi$
$572$	−0.414214	−0.0173191
$573$	9.65685	0.403421
$574$	−17.6569	−0.736983
$575$	0	0
$576$	19.7990	0.824958
$577$	−35.3137	−1.47013	−0.735064	−	0.677997i	$-0.762848\pi$
$577$	−35.3137	−1.47013	−0.735064	+	0.677997i	$0.762848\pi$
$578$	6.31371	0.262616
$579$	25.3137	1.05200
$580$	0	0
$581$	5.51472	0.228789
$582$	44.6274	1.84987
$583$	1.17157	0.0485216
$584$	43.6690	1.80704
$585$	0	0
$586$	−16.4853	−0.681001
$587$	−6.68629	−0.275973	−0.137986	−	0.990434i	$-0.544063\pi$
$587$	−6.68629	−0.275973	−0.137986	+	0.990434i	$0.544063\pi$
$588$	7.24264	0.298681
$589$	1.48528	0.0612000
$590$	0	0
$591$	12.4853	0.513576
$592$	−1.17157	−0.0481513
$593$	−36.0711	−1.48126	−0.740631	−	0.671912i	$-0.765473\pi$
$593$	−36.0711	−1.48126	−0.740631	+	0.671912i	$0.765473\pi$
$594$	−0.414214	−0.0169954
$595$	0	0
$596$	1.17157	0.0479895
$597$	−59.4558	−2.43337
$598$	−2.51472	−0.102834
$599$	38.2843	1.56425	0.782126	−	0.623120i	$-0.214135\pi$
$599$	38.2843	1.56425	0.782126	+	0.623120i	$0.214135\pi$
$600$	0	0
$601$	−39.9706	−1.63043	−0.815217	−	0.579156i	$-0.803382\pi$
$601$	−39.9706	−1.63043	−0.815217	+	0.579156i	$0.803382\pi$
$602$	20.9706	0.854696
$603$	13.4558	0.547964
$604$	−1.00000	−0.0406894
$605$	0	0
$606$	37.7990	1.53548
$607$	−24.6274	−0.999596	−0.499798	−	0.866142i	$-0.666592\pi$
$607$	−24.6274	−0.999596	−0.499798	+	0.866142i	$0.666592\pi$
$608$	43.2843	1.75541
$609$	−10.4853	−0.424885
$610$	0	0
$611$	4.82843	0.195337
$612$	−13.6569	−0.552046
$613$	31.1127	1.25663	0.628315	−	0.777959i	$-0.283745\pi$
$613$	31.1127	1.25663	0.628315	+	0.777959i	$0.283745\pi$
$614$	18.4853	0.746005
$615$	0	0
$616$	−6.00000	−0.241747
$617$	23.0416	0.927621	0.463811	−	0.885934i	$-0.346482\pi$
$617$	23.0416	0.927621	0.463811	+	0.885934i	$0.346482\pi$
$618$	22.1421	0.890687
$619$	−31.7990	−1.27811	−0.639055	−	0.769161i	$-0.720674\pi$
$619$	−31.7990	−1.27811	−0.639055	+	0.769161i	$0.720674\pi$
$620$	0	0
$621$	2.51472	0.100912
$622$	−6.31371	−0.253157
$623$	9.65685	0.386894
$624$	−1.00000	−0.0400320
$625$	0	0
$626$	−22.6274	−0.904373
$627$	20.8995	0.834645
$628$	1.10051	0.0439149
$629$	5.65685	0.225554
$630$	0	0
$631$	13.5147	0.538012	0.269006	−	0.963138i	$-0.413305\pi$
$631$	13.5147	0.538012	0.269006	+	0.963138i	$0.413305\pi$
$632$	−14.4853	−0.576194
$633$	−13.6569	−0.542811
$634$	−26.5563	−1.05469
$635$	0	0
$636$	−2.82843	−0.112154
$637$	−1.24264	−0.0492352
$638$	−2.17157	−0.0859734
$639$	−4.68629	−0.185387
$640$	0	0
$641$	37.2843	1.47264	0.736320	−	0.676633i	$-0.236562\pi$
$641$	37.2843	1.47264	0.736320	+	0.676633i	$0.236562\pi$
$642$	23.4853	0.926890
$643$	18.0000	0.709851	0.354925	−	0.934895i	$-0.384506\pi$
$643$	18.0000	0.709851	0.354925	+	0.934895i	$0.384506\pi$
$644$	12.1421	0.478467
$645$	0	0
$646$	41.7990	1.64456
$647$	−35.3137	−1.38833	−0.694163	−	0.719818i	$-0.744225\pi$
$647$	−35.3137	−1.38833	−0.694163	+	0.719818i	$0.744225\pi$
$648$	28.4558	1.11785
$649$	11.8284	0.464306
$650$	0	0
$651$	0.828427	0.0324686
$652$	3.58579	0.140430
$653$	19.6569	0.769232	0.384616	−	0.923077i	$-0.374334\pi$
$653$	19.6569	0.769232	0.384616	+	0.923077i	$0.374334\pi$
$654$	−34.1421	−1.33506
$655$	0	0
$656$	8.82843	0.344692
$657$	−41.1716	−1.60626
$658$	23.3137	0.908863
$659$	5.97056	0.232580	0.116290	−	0.993215i	$-0.462900\pi$
$659$	5.97056	0.232580	0.116290	+	0.993215i	$0.462900\pi$
$660$	0	0
$661$	3.79899	0.147764	0.0738818	−	0.997267i	$-0.476461\pi$
$661$	3.79899	0.147764	0.0738818	+	0.997267i	$0.476461\pi$
$662$	−28.0000	−1.08825
$663$	4.82843	0.187521
$664$	−8.27208	−0.321019
$665$	0	0
$666$	3.31371	0.128404
$667$	13.1838	0.510477
$668$	−3.51472	−0.135989
$669$	−41.7990	−1.61604
$670$	0	0
$671$	−12.0000	−0.463255
$672$	24.1421	0.931303
$673$	19.3137	0.744489	0.372244	−	0.928135i	$-0.378588\pi$
$673$	19.3137	0.744489	0.372244	+	0.928135i	$0.378588\pi$
$674$	−21.7279	−0.836928
$675$	0	0
$676$	12.8284	0.493401
$677$	−29.4558	−1.13208	−0.566040	−	0.824378i	$-0.691525\pi$
$677$	−29.4558	−1.13208	−0.566040	+	0.824378i	$0.691525\pi$
$678$	24.4853	0.940352
$679$	36.9706	1.41880
$680$	0	0
$681$	−14.4853	−0.555077
$682$	0.171573	0.00656986
$683$	25.2426	0.965883	0.482941	−	0.875653i	$-0.339568\pi$
$683$	25.2426	0.965883	0.482941	+	0.875653i	$0.339568\pi$
$684$	−24.4853	−0.936218
$685$	0	0
$686$	−20.0000	−0.763604
$687$	−37.7279	−1.43941
$688$	−10.4853	−0.399748
$689$	0.485281	0.0184877
$690$	0	0
$691$	−7.17157	−0.272819	−0.136410	−	0.990653i	$-0.543556\pi$
$691$	−7.17157	−0.272819	−0.136410	+	0.990653i	$0.543556\pi$
$692$	−7.31371	−0.278025
$693$	5.65685	0.214886
$694$	25.4558	0.966291
$695$	0	0
$696$	15.7279	0.596165
$697$	−42.6274	−1.61463
$698$	−14.5147	−0.549390
$699$	−64.9411	−2.45630
$700$	0	0
$701$	2.68629	0.101460	0.0507299	−	0.998712i	$-0.483845\pi$
$701$	2.68629	0.101460	0.0507299	+	0.998712i	$0.483845\pi$
$702$	−0.171573	−0.00647560
$703$	10.1421	0.382518
$704$	7.00000	0.263822
$705$	0	0
$706$	4.75736	0.179046
$707$	31.3137	1.17767
$708$	−28.5563	−1.07321
$709$	−36.6569	−1.37668	−0.688339	−	0.725390i	$-0.741660\pi$
$709$	−36.6569	−1.37668	−0.688339	+	0.725390i	$0.741660\pi$
$710$	0	0
$711$	13.6569	0.512172
$712$	−14.4853	−0.542859
$713$	−1.04163	−0.0390094
$714$	23.3137	0.872494
$715$	0	0
$716$	15.7990	0.590436
$717$	−11.2426	−0.419864
$718$	−12.1421	−0.453140
$719$	16.1421	0.602000	0.301000	−	0.953624i	$-0.402680\pi$
$719$	16.1421	0.602000	0.301000	+	0.953624i	$0.402680\pi$
$720$	0	0
$721$	18.3431	0.683134
$722$	55.9411	2.08191
$723$	−5.24264	−0.194976
$724$	10.0000	0.371647
$725$	0	0
$726$	−24.1421	−0.895999
$727$	−5.65685	−0.209801	−0.104901	−	0.994483i	$-0.533452\pi$
$727$	−5.65685	−0.209801	−0.104901	+	0.994483i	$0.533452\pi$
$728$	−2.48528	−0.0921107
$729$	−23.8284	−0.882534
$730$	0	0
$731$	50.6274	1.87252
$732$	28.9706	1.07078
$733$	−45.1838	−1.66890	−0.834450	−	0.551083i	$-0.814215\pi$
$733$	−45.1838	−1.66890	−0.834450	+	0.551083i	$0.814215\pi$
$734$	−13.3137	−0.491418
$735$	0	0
$736$	−30.3553	−1.11891
$737$	4.75736	0.175240
$738$	−24.9706	−0.919179
$739$	12.6274	0.464507	0.232254	−	0.972655i	$-0.425390\pi$
$739$	12.6274	0.464507	0.232254	+	0.972655i	$0.425390\pi$
$740$	0	0
$741$	8.65685	0.318017
$742$	2.34315	0.0860196
$743$	−20.8284	−0.764121	−0.382060	−	0.924137i	$-0.624785\pi$
$743$	−20.8284	−0.764121	−0.382060	+	0.924137i	$0.624785\pi$
$744$	−1.24264	−0.0455574
$745$	0	0
$746$	−25.8701	−0.947170
$747$	7.79899	0.285350
$748$	−4.82843	−0.176545
$749$	19.4558	0.710901
$750$	0	0
$751$	11.8579	0.432700	0.216350	−	0.976316i	$-0.430585\pi$
$751$	11.8579	0.432700	0.216350	+	0.976316i	$0.430585\pi$
$752$	−11.6569	−0.425082
$753$	27.7279	1.01046
$754$	−0.899495	−0.0327577
$755$	0	0
$756$	0.828427	0.0301296
$757$	11.5147	0.418510	0.209255	−	0.977861i	$-0.432896\pi$
$757$	11.5147	0.418510	0.209255	+	0.977861i	$0.432896\pi$
$758$	6.00000	0.217930
$759$	−14.6569	−0.532010
$760$	0	0
$761$	−11.4558	−0.415274	−0.207637	−	0.978206i	$-0.566577\pi$
$761$	−11.4558	−0.415274	−0.207637	+	0.978206i	$0.566577\pi$
$762$	−27.3137	−0.989471
$763$	−28.2843	−1.02396
$764$	−4.00000	−0.144715
$765$	0	0
$766$	−4.34315	−0.156924
$767$	4.89949	0.176911
$768$	−41.0416	−1.48096
$769$	40.7696	1.47019	0.735094	−	0.677965i	$-0.237138\pi$
$769$	40.7696	1.47019	0.735094	+	0.677965i	$0.237138\pi$
$770$	0	0
$771$	−9.82843	−0.353962
$772$	−10.4853	−0.377374
$773$	25.8579	0.930043	0.465021	−	0.885299i	$-0.346047\pi$
$773$	25.8579	0.930043	0.465021	+	0.885299i	$0.346047\pi$
$774$	29.6569	1.06599
$775$	0	0
$776$	−55.4558	−1.99075
$777$	5.65685	0.202939
$778$	−31.9411	−1.14514
$779$	−76.4264	−2.73826
$780$	0	0
$781$	−1.65685	−0.0592869
$782$	−29.3137	−1.04826
$783$	0.899495	0.0321453
$784$	3.00000	0.107143
$785$	0	0
$786$	30.9706	1.10468
$787$	−14.3431	−0.511278	−0.255639	−	0.966772i	$-0.582286\pi$
$787$	−14.3431	−0.511278	−0.255639	+	0.966772i	$0.582286\pi$
$788$	−5.17157	−0.184230
$789$	42.4558	1.51147
$790$	0	0
$791$	20.2843	0.721226
$792$	−8.48528	−0.301511
$793$	−4.97056	−0.176510
$794$	29.6569	1.05248
$795$	0	0
$796$	24.6274	0.872896
$797$	2.34315	0.0829985	0.0414992	−	0.999139i	$-0.486787\pi$
$797$	2.34315	0.0829985	0.0414992	+	0.999139i	$0.486787\pi$
$798$	41.7990	1.47967
$799$	56.2843	1.99119
$800$	0	0
$801$	13.6569	0.482541
$802$	−34.9706	−1.23485
$803$	−14.5563	−0.513682
$804$	−11.4853	−0.405055
$805$	0	0
$806$	0.0710678	0.00250326
$807$	−41.0416	−1.44473
$808$	−46.9706	−1.65242
$809$	−50.4264	−1.77290	−0.886449	−	0.462826i	$-0.846835\pi$
$809$	−50.4264	−1.77290	−0.886449	+	0.462826i	$0.846835\pi$
$810$	0	0
$811$	−28.6274	−1.00524	−0.502622	−	0.864506i	$-0.667631\pi$
$811$	−28.6274	−1.00524	−0.502622	+	0.864506i	$0.667631\pi$
$812$	4.34315	0.152415
$813$	−54.6274	−1.91587
$814$	1.17157	0.0410636
$815$	0	0
$816$	−11.6569	−0.408072
$817$	90.7696	3.17562
$818$	−14.0000	−0.489499
$819$	2.34315	0.0818761
$820$	0	0
$821$	20.3431	0.709981	0.354990	−	0.934870i	$-0.384484\pi$
$821$	20.3431	0.709981	0.354990	+	0.934870i	$0.384484\pi$
$822$	−20.4853	−0.714506
$823$	8.34315	0.290824	0.145412	−	0.989371i	$-0.453549\pi$
$823$	8.34315	0.290824	0.145412	+	0.989371i	$0.453549\pi$
$824$	−27.5147	−0.958521
$825$	0	0
$826$	23.6569	0.823127
$827$	31.1127	1.08189	0.540947	−	0.841057i	$-0.318066\pi$
$827$	31.1127	1.08189	0.540947	+	0.841057i	$0.318066\pi$
$828$	17.1716	0.596753
$829$	3.34315	0.116112	0.0580561	−	0.998313i	$-0.481510\pi$
$829$	3.34315	0.116112	0.0580561	+	0.998313i	$0.481510\pi$
$830$	0	0
$831$	63.1127	2.18936
$832$	2.89949	0.100522
$833$	−14.4853	−0.501885
$834$	5.65685	0.195881
$835$	0	0
$836$	−8.65685	−0.299404
$837$	−0.0710678	−0.00245646
$838$	9.79899	0.338500
$839$	12.6863	0.437979	0.218990	−	0.975727i	$-0.429724\pi$
$839$	12.6863	0.437979	0.218990	+	0.975727i	$0.429724\pi$
$840$	0	0
$841$	−24.2843	−0.837389
$842$	−30.8284	−1.06242
$843$	24.1421	0.831499
$844$	5.65685	0.194717
$845$	0	0
$846$	32.9706	1.13355
$847$	−20.0000	−0.687208
$848$	−1.17157	−0.0402320
$849$	−9.00000	−0.308879
$850$	0	0
$851$	−7.11270	−0.243820
$852$	4.00000	0.137038
$853$	47.7990	1.63661	0.818303	−	0.574787i	$-0.194915\pi$
$853$	47.7990	1.63661	0.818303	+	0.574787i	$0.194915\pi$
$854$	−24.0000	−0.821263
$855$	0	0
$856$	−29.1838	−0.997481
$857$	−2.07107	−0.0707463	−0.0353732	−	0.999374i	$-0.511262\pi$
$857$	−2.07107	−0.0707463	−0.0353732	+	0.999374i	$0.511262\pi$
$858$	1.00000	0.0341394
$859$	15.3137	0.522497	0.261248	−	0.965272i	$-0.415866\pi$
$859$	15.3137	0.522497	0.261248	+	0.965272i	$0.415866\pi$
$860$	0	0
$861$	−42.6274	−1.45274
$862$	8.82843	0.300697
$863$	−39.2548	−1.33625	−0.668125	−	0.744049i	$-0.732903\pi$
$863$	−39.2548	−1.33625	−0.668125	+	0.744049i	$0.732903\pi$
$864$	−2.07107	−0.0704592
$865$	0	0
$866$	−5.17157	−0.175737
$867$	15.2426	0.517667
$868$	−0.343146	−0.0116471
$869$	4.82843	0.163793
$870$	0	0
$871$	1.97056	0.0667700
$872$	42.4264	1.43674
$873$	52.2843	1.76955
$874$	−52.5563	−1.77775
$875$	0	0
$876$	35.1421	1.18734
$877$	21.1005	0.712513	0.356257	−	0.934388i	$-0.384053\pi$
$877$	21.1005	0.712513	0.356257	+	0.934388i	$0.384053\pi$
$878$	19.2843	0.650813
$879$	−39.7990	−1.34239
$880$	0	0
$881$	33.1716	1.11758	0.558789	−	0.829310i	$-0.311266\pi$
$881$	33.1716	1.11758	0.558789	+	0.829310i	$0.311266\pi$
$882$	−8.48528	−0.285714
$883$	−0.485281	−0.0163310	−0.00816551	−	0.999967i	$-0.502599\pi$
$883$	−0.485281	−0.0163310	−0.00816551	+	0.999967i	$0.502599\pi$
$884$	−2.00000	−0.0672673
$885$	0	0
$886$	30.0000	1.00787
$887$	23.8701	0.801478	0.400739	−	0.916192i	$-0.368753\pi$
$887$	23.8701	0.801478	0.400739	+	0.916192i	$0.368753\pi$
$888$	−8.48528	−0.284747
$889$	−22.6274	−0.758899
$890$	0	0
$891$	−9.48528	−0.317769
$892$	17.3137	0.579706
$893$	100.912	3.37688
$894$	−2.82843	−0.0945968
$895$	0	0
$896$	−6.00000	−0.200446
$897$	−6.07107	−0.202707
$898$	−14.4853	−0.483380
$899$	−0.372583	−0.0124263
$900$	0	0
$901$	5.65685	0.188457
$902$	−8.82843	−0.293954
$903$	50.6274	1.68477
$904$	−30.4264	−1.01197
$905$	0	0
$906$	2.41421	0.0802069
$907$	−53.5980	−1.77969	−0.889846	−	0.456261i	$-0.849188\pi$
$907$	−53.5980	−1.77969	−0.889846	+	0.456261i	$0.849188\pi$
$908$	6.00000	0.199117
$909$	44.2843	1.46882
$910$	0	0
$911$	−23.8284	−0.789471	−0.394736	−	0.918795i	$-0.629164\pi$
$911$	−23.8284	−0.789471	−0.394736	+	0.918795i	$0.629164\pi$
$912$	−20.8995	−0.692051
$913$	2.75736	0.0912553
$914$	−5.51472	−0.182411
$915$	0	0
$916$	15.6274	0.516344
$917$	25.6569	0.847264
$918$	−2.00000	−0.0660098
$919$	20.0000	0.659739	0.329870	−	0.944027i	$-0.392995\pi$
$919$	20.0000	0.659739	0.329870	+	0.944027i	$0.392995\pi$
$920$	0	0
$921$	44.6274	1.47052
$922$	−0.343146	−0.0113009
$923$	−0.686292	−0.0225896
$924$	−4.82843	−0.158844
$925$	0	0
$926$	22.1421	0.727636
$927$	25.9411	0.852018
$928$	−10.8579	−0.356427
$929$	−10.0000	−0.328089	−0.164045	−	0.986453i	$-0.552454\pi$
$929$	−10.0000	−0.328089	−0.164045	+	0.986453i	$0.552454\pi$
$930$	0	0
$931$	−25.9706	−0.851151
$932$	26.8995	0.881122
$933$	−15.2426	−0.499022
$934$	5.31371	0.173870
$935$	0	0
$936$	−3.51472	−0.114882
$937$	−22.3431	−0.729919	−0.364959	−	0.931023i	$-0.618917\pi$
$937$	−22.3431	−0.729919	−0.364959	+	0.931023i	$0.618917\pi$
$938$	9.51472	0.310667
$939$	−54.6274	−1.78270
$940$	0	0
$941$	21.1716	0.690174	0.345087	−	0.938571i	$-0.387849\pi$
$941$	21.1716	0.690174	0.345087	+	0.938571i	$0.387849\pi$
$942$	−2.65685	−0.0865650
$943$	53.5980	1.74539
$944$	−11.8284	−0.384983
$945$	0	0
$946$	10.4853	0.340906
$947$	−16.0711	−0.522240	−0.261120	−	0.965306i	$-0.584092\pi$
$947$	−16.0711	−0.522240	−0.261120	+	0.965306i	$0.584092\pi$
$948$	−11.6569	−0.378597
$949$	−6.02944	−0.195724
$950$	0	0
$951$	−64.1127	−2.07900
$952$	−28.9706	−0.938941
$953$	−49.5980	−1.60664	−0.803318	−	0.595550i	$-0.796934\pi$
$953$	−49.5980	−1.60664	−0.803318	+	0.595550i	$0.796934\pi$
$954$	3.31371	0.107285
$955$	0	0
$956$	4.65685	0.150613
$957$	−5.24264	−0.169471
$958$	−1.85786	−0.0600249
$959$	−16.9706	−0.548008
$960$	0	0
$961$	−30.9706	−0.999050
$962$	0.485281	0.0156461
$963$	27.5147	0.886649
$964$	2.17157	0.0699417
$965$	0	0
$966$	−29.3137	−0.943153
$967$	28.4142	0.913740	0.456870	−	0.889534i	$-0.348971\pi$
$967$	28.4142	0.913740	0.456870	+	0.889534i	$0.348971\pi$
$968$	30.0000	0.964237
$969$	100.912	3.24175
$970$	0	0
$971$	−40.4853	−1.29923	−0.649617	−	0.760261i	$-0.725071\pi$
$971$	−40.4853	−1.29923	−0.649617	+	0.760261i	$0.725071\pi$
$972$	21.6569	0.694644
$973$	4.68629	0.150236
$974$	−3.17157	−0.101624
$975$	0	0
$976$	12.0000	0.384111
$977$	−5.17157	−0.165453	−0.0827266	−	0.996572i	$-0.526363\pi$
$977$	−5.17157	−0.165453	−0.0827266	+	0.996572i	$0.526363\pi$
$978$	−8.65685	−0.276816
$979$	4.82843	0.154317
$980$	0	0
$981$	−40.0000	−1.27710
$982$	37.6569	1.20168
$983$	−35.9411	−1.14634	−0.573172	−	0.819435i	$-0.694287\pi$
$983$	−35.9411	−1.14634	−0.573172	+	0.819435i	$0.694287\pi$
$984$	63.9411	2.03837
$985$	0	0
$986$	−10.4853	−0.333919
$987$	56.2843	1.79155
$988$	−3.58579	−0.114079
$989$	−63.6569	−2.02417
$990$	0	0
$991$	−12.6569	−0.402058	−0.201029	−	0.979585i	$-0.564429\pi$
$991$	−12.6569	−0.402058	−0.201029	+	0.979585i	$0.564429\pi$
$992$	0.857864	0.0272372
$993$	−67.5980	−2.14516
$994$	−3.31371	−0.105104
$995$	0	0
$996$	−6.65685	−0.210930
$997$	42.9706	1.36089	0.680446	−	0.732799i	$-0.261786\pi$
$997$	42.9706	1.36089	0.680446	+	0.732799i	$0.261786\pi$
$998$	18.4853	0.585141
$999$	−0.485281	−0.0153536

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3775.2.a.i.1.2		2
5.4	even	2		755.2.a.g.1.1	✓	2
15.14	odd	2		6795.2.a.t.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
755.2.a.g.1.1	✓	2	5.4	even	2
3775.2.a.i.1.2		2	1.1	even	1	trivial
6795.2.a.t.1.1		2	15.14	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +2.41421 q^{3} -1.00000 q^{4} +2.41421 q^{6} +2.00000 q^{7} -3.00000 q^{8} +2.82843 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} +2.41421 q^{3} -1.00000 q^{4} +2.41421 q^{6} +2.00000 q^{7} -3.00000 q^{8} +2.82843 q^{9} +1.00000 q^{11} -2.41421 q^{12} +0.414214 q^{13} +2.00000 q^{14} -1.00000 q^{16} +4.82843 q^{17} +2.82843 q^{18} +8.65685 q^{19} +4.82843 q^{21} +1.00000 q^{22} -6.07107 q^{23} -7.24264 q^{24} +0.414214 q^{26} -0.414214 q^{27} -2.00000 q^{28} -2.17157 q^{29} +0.171573 q^{31} +5.00000 q^{32} +2.41421 q^{33} +4.82843 q^{34} -2.82843 q^{36} +1.17157 q^{37} +8.65685 q^{38} +1.00000 q^{39} -8.82843 q^{41} +4.82843 q^{42} +10.4853 q^{43} -1.00000 q^{44} -6.07107 q^{46} +11.6569 q^{47} -2.41421 q^{48} -3.00000 q^{49} +11.6569 q^{51} -0.414214 q^{52} +1.17157 q^{53} -0.414214 q^{54} -6.00000 q^{56} +20.8995 q^{57} -2.17157 q^{58} +11.8284 q^{59} -12.0000 q^{61} +0.171573 q^{62} +5.65685 q^{63} +7.00000 q^{64} +2.41421 q^{66} +4.75736 q^{67} -4.82843 q^{68} -14.6569 q^{69} -1.65685 q^{71} -8.48528 q^{72} -14.5563 q^{73} +1.17157 q^{74} -8.65685 q^{76} +2.00000 q^{77} +1.00000 q^{78} +4.82843 q^{79} -9.48528 q^{81} -8.82843 q^{82} +2.75736 q^{83} -4.82843 q^{84} +10.4853 q^{86} -5.24264 q^{87} -3.00000 q^{88} +4.82843 q^{89} +0.828427 q^{91} +6.07107 q^{92} +0.414214 q^{93} +11.6569 q^{94} +12.0711 q^{96} +18.4853 q^{97} -3.00000 q^{98} +2.82843 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{3} - 2 q^{4} + 2 q^{6} + 4 q^{7} - 6 q^{8}+O(q^{10}) \) 2 * q + 2 * q^2 + 2 * q^3 - 2 * q^4 + 2 * q^6 + 4 * q^7 - 6 * q^8	\( 2 q + 2 q^{2} + 2 q^{3} - 2 q^{4} + 2 q^{6} + 4 q^{7} - 6 q^{8} + 2 q^{11} - 2 q^{12} - 2 q^{13} + 4 q^{14} - 2 q^{16} + 4 q^{17} + 6 q^{19} + 4 q^{21} + 2 q^{22} + 2 q^{23} - 6 q^{24} - 2 q^{26} + 2 q^{27} - 4 q^{28} - 10 q^{29} + 6 q^{31} + 10 q^{32} + 2 q^{33} + 4 q^{34} + 8 q^{37} + 6 q^{38} + 2 q^{39} - 12 q^{41} + 4 q^{42} + 4 q^{43} - 2 q^{44} + 2 q^{46} + 12 q^{47} - 2 q^{48} - 6 q^{49} + 12 q^{51} + 2 q^{52} + 8 q^{53} + 2 q^{54} - 12 q^{56} + 22 q^{57} - 10 q^{58} + 18 q^{59} - 24 q^{61} + 6 q^{62} + 14 q^{64} + 2 q^{66} + 18 q^{67} - 4 q^{68} - 18 q^{69} + 8 q^{71} + 2 q^{73} + 8 q^{74} - 6 q^{76} + 4 q^{77} + 2 q^{78} + 4 q^{79} - 2 q^{81} - 12 q^{82} + 14 q^{83} - 4 q^{84} + 4 q^{86} - 2 q^{87} - 6 q^{88} + 4 q^{89} - 4 q^{91} - 2 q^{92} - 2 q^{93} + 12 q^{94} + 10 q^{96} + 20 q^{97} - 6 q^{98}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^3 - 2 * q^4 + 2 * q^6 + 4 * q^7 - 6 * q^8 + 2 * q^11 - 2 * q^12 - 2 * q^13 + 4 * q^14 - 2 * q^16 + 4 * q^17 + 6 * q^19 + 4 * q^21 + 2 * q^22 + 2 * q^23 - 6 * q^24 - 2 * q^26 + 2 * q^27 - 4 * q^28 - 10 * q^29 + 6 * q^31 + 10 * q^32 + 2 * q^33 + 4 * q^34 + 8 * q^37 + 6 * q^38 + 2 * q^39 - 12 * q^41 + 4 * q^42 + 4 * q^43 - 2 * q^44 + 2 * q^46 + 12 * q^47 - 2 * q^48 - 6 * q^49 + 12 * q^51 + 2 * q^52 + 8 * q^53 + 2 * q^54 - 12 * q^56 + 22 * q^57 - 10 * q^58 + 18 * q^59 - 24 * q^61 + 6 * q^62 + 14 * q^64 + 2 * q^66 + 18 * q^67 - 4 * q^68 - 18 * q^69 + 8 * q^71 + 2 * q^73 + 8 * q^74 - 6 * q^76 + 4 * q^77 + 2 * q^78 + 4 * q^79 - 2 * q^81 - 12 * q^82 + 14 * q^83 - 4 * q^84 + 4 * q^86 - 2 * q^87 - 6 * q^88 + 4 * q^89 - 4 * q^91 - 2 * q^92 - 2 * q^93 + 12 * q^94 + 10 * q^96 + 20 * q^97 - 6 * q^98

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