LMFDB - Embedded newform 1925.2.b.h.1849.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1925,2,Mod(1849,1925)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1925, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 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("1925.1849");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1925 = 5^{2} \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1925.b (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$15.3712023891$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 3x^{2} + 1 $ x^4 + 3*x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 77)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1849.1
Root			$-0.618034i$ of defining polynomial
Character	$\chi$	$=$	1925.1849
Dual form			1925.2.b.h.1849.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.23607i q^{2} -3.23607i q^{3} -3.00000 q^{4} -7.23607 q^{6} +1.00000i q^{7} +2.23607i q^{8} -7.47214 q^{9} +O(q^{10})$	\(q-2.23607i q^{2} -3.23607i q^{3} -3.00000 q^{4} -7.23607 q^{6} +1.00000i q^{7} +2.23607i q^{8} -7.47214 q^{9} -1.00000 q^{11} +9.70820i q^{12} +1.23607i q^{13} +2.23607 q^{14} -1.00000 q^{16} +1.23607i q^{17} +16.7082i q^{18} +2.47214 q^{19} +3.23607 q^{21} +2.23607i q^{22} +6.47214i q^{23} +7.23607 q^{24} +2.76393 q^{26} +14.4721i q^{27} -3.00000i q^{28} +0.472136 q^{29} -7.23607 q^{31} +6.70820i q^{32} +3.23607i q^{33} +2.76393 q^{34} +22.4164 q^{36} +0.472136i q^{37} -5.52786i q^{38} +4.00000 q^{39} -6.76393 q^{41} -7.23607i q^{42} -8.00000i q^{43} +3.00000 q^{44} +14.4721 q^{46} +7.23607i q^{47} +3.23607i q^{48} -1.00000 q^{49} +4.00000 q^{51} -3.70820i q^{52} -8.47214i q^{53} +32.3607 q^{54} -2.23607 q^{56} -8.00000i q^{57} -1.05573i q^{58} -3.23607 q^{59} -2.76393 q^{61} +16.1803i q^{62} -7.47214i q^{63} +13.0000 q^{64} +7.23607 q^{66} +5.52786i q^{67} -3.70820i q^{68} +20.9443 q^{69} -1.52786 q^{71} -16.7082i q^{72} +5.23607i q^{73} +1.05573 q^{74} -7.41641 q^{76} -1.00000i q^{77} -8.94427i q^{78} -8.94427 q^{79} +24.4164 q^{81} +15.1246i q^{82} -15.4164i q^{83} -9.70820 q^{84} -17.8885 q^{86} -1.52786i q^{87} -2.23607i q^{88} -2.00000 q^{89} -1.23607 q^{91} -19.4164i q^{92} +23.4164i q^{93} +16.1803 q^{94} +21.7082 q^{96} -9.41641i q^{97} +2.23607i q^{98} +7.47214 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 12 q^{4} - 20 q^{6} - 12 q^{9}+O(q^{10}) $ 4 * q - 12 * q^4 - 20 * q^6 - 12 * q^9	$ 4 q - 12 q^{4} - 20 q^{6} - 12 q^{9} - 4 q^{11} - 4 q^{16} - 8 q^{19} + 4 q^{21} + 20 q^{24} + 20 q^{26} - 16 q^{29} - 20 q^{31} + 20 q^{34} + 36 q^{36} + 16 q^{39} - 36 q^{41} + 12 q^{44} + 40 q^{46} - 4 q^{49} + 16 q^{51} + 40 q^{54} - 4 q^{59} - 20 q^{61} + 52 q^{64} + 20 q^{66} + 48 q^{69} - 24 q^{71} + 40 q^{74} + 24 q^{76} + 44 q^{81} - 12 q^{84} - 8 q^{89} + 4 q^{91} + 20 q^{94} + 60 q^{96} + 12 q^{99}+O(q^{100}) $ 4 * q - 12 * q^4 - 20 * q^6 - 12 * q^9 - 4 * q^11 - 4 * q^16 - 8 * q^19 + 4 * q^21 + 20 * q^24 + 20 * q^26 - 16 * q^29 - 20 * q^31 + 20 * q^34 + 36 * q^36 + 16 * q^39 - 36 * q^41 + 12 * q^44 + 40 * q^46 - 4 * q^49 + 16 * q^51 + 40 * q^54 - 4 * q^59 - 20 * q^61 + 52 * q^64 + 20 * q^66 + 48 * q^69 - 24 * q^71 + 40 * q^74 + 24 * q^76 + 44 * q^81 - 12 * q^84 - 8 * q^89 + 4 * q^91 + 20 * q^94 + 60 * q^96 + 12 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1925\mathbb{Z}\right)^\times$.

$n$	$276$	$1002$	$1751$
$\chi(n)$	$1$	$-1$	$1$

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 $

$2$	− 2.23607i	− 1.58114i	−0.612372	−	0.790569i	$-0.709785\pi$
$2$	− 2.23607i	− 1.58114i	0.612372	−	0.790569i	$-0.290215\pi$
$3$	− 3.23607i	− 1.86834i	−0.356822	−	0.934172i	$-0.616140\pi$
$3$	− 3.23607i	− 1.86834i	0.356822	−	0.934172i	$-0.383860\pi$
$4$	−3.00000	−1.50000
$5$	0	0
$5$	0	0
$6$	−7.23607	−2.95411
$7$	1.00000i	0.377964i
$7$	1.00000i	0.377964i
$8$	2.23607i	0.790569i
$9$	−7.47214	−2.49071
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	9.70820i	2.80252i
$13$	1.23607i	0.342824i	0.985199	+	0.171412i	$0.0548329\pi$
$13$	1.23607i	0.342824i	−0.985199	+	0.171412i	$0.945167\pi$
$14$	2.23607	0.597614
$15$	0	0
$16$	−1.00000	−0.250000
$17$	1.23607i	0.299791i	0.988702	+	0.149895i	$0.0478936\pi$
$17$	1.23607i	0.299791i	−0.988702	+	0.149895i	$0.952106\pi$
$18$	16.7082i	3.93816i
$19$	2.47214	0.567147	0.283573	−	0.958951i	$-0.408480\pi$
$19$	2.47214	0.567147	0.283573	+	0.958951i	$0.408480\pi$
$20$	0	0
$21$	3.23607	0.706168
$22$	2.23607i	0.476731i
$23$	6.47214i	1.34953i	0.738031	+	0.674767i	$0.235756\pi$
$23$	6.47214i	1.34953i	−0.738031	+	0.674767i	$0.764244\pi$
$24$	7.23607	1.47706
$25$	0	0
$26$	2.76393	0.542052
$27$	14.4721i	2.78516i
$28$	− 3.00000i	− 0.566947i
$29$	0.472136	0.0876734	0.0438367	−	0.999039i	$-0.486042\pi$
$29$	0.472136	0.0876734	0.0438367	+	0.999039i	$0.486042\pi$
$30$	0	0
$31$	−7.23607	−1.29964	−0.649818	−	0.760090i	$-0.725155\pi$
$31$	−7.23607	−1.29964	−0.649818	+	0.760090i	$0.725155\pi$
$32$	6.70820i	1.18585i
$33$	3.23607i	0.563327i
$34$	2.76393	0.474010
$35$	0	0
$36$	22.4164	3.73607
$37$	0.472136i	0.0776187i	0.999247	+	0.0388093i	$0.0123565\pi$
$37$	0.472136i	0.0776187i	−0.999247	+	0.0388093i	$0.987644\pi$
$38$	− 5.52786i	− 0.896738i
$39$	4.00000	0.640513
$40$	0	0
$41$	−6.76393	−1.05635	−0.528174	−	0.849136i	$-0.677123\pi$
$41$	−6.76393	−1.05635	−0.528174	+	0.849136i	$0.677123\pi$
$42$	− 7.23607i	− 1.11655i
$43$	− 8.00000i	− 1.21999i	−0.792406	−	0.609994i	$-0.791172\pi$
$43$	− 8.00000i	− 1.21999i	0.792406	−	0.609994i	$-0.208828\pi$
$44$	3.00000	0.452267
$45$	0	0
$46$	14.4721	2.13380
$47$	7.23607i	1.05549i	0.849403	+	0.527744i	$0.176962\pi$
$47$	7.23607i	1.05549i	−0.849403	+	0.527744i	$0.823038\pi$
$48$	3.23607i	0.467086i
$49$	−1.00000	−0.142857
$50$	0	0
$51$	4.00000	0.560112
$52$	− 3.70820i	− 0.514235i
$53$	− 8.47214i	− 1.16374i	−0.813283	−	0.581869i	$-0.802322\pi$
$53$	− 8.47214i	− 1.16374i	0.813283	−	0.581869i	$-0.197678\pi$
$54$	32.3607	4.40373
$55$	0	0
$56$	−2.23607	−0.298807
$57$	− 8.00000i	− 1.05963i
$58$	− 1.05573i	− 0.138624i
$59$	−3.23607	−0.421300	−0.210650	−	0.977562i	$-0.567558\pi$
$59$	−3.23607	−0.421300	−0.210650	+	0.977562i	$0.567558\pi$
$60$	0	0
$61$	−2.76393	−0.353885	−0.176943	−	0.984221i	$-0.556621\pi$
$61$	−2.76393	−0.353885	−0.176943	+	0.984221i	$0.556621\pi$
$62$	16.1803i	2.05491i
$63$	− 7.47214i	− 0.941401i
$64$	13.0000	1.62500
$65$	0	0
$66$	7.23607	0.890698
$67$	5.52786i	0.675336i	0.941265	+	0.337668i	$0.109638\pi$
$67$	5.52786i	0.675336i	−0.941265	+	0.337668i	$0.890362\pi$
$68$	− 3.70820i	− 0.449686i
$69$	20.9443	2.52139
$70$	0	0
$71$	−1.52786	−0.181324	−0.0906621	−	0.995882i	$-0.528898\pi$
$71$	−1.52786	−0.181324	−0.0906621	+	0.995882i	$0.528898\pi$
$72$	− 16.7082i	− 1.96908i
$73$	5.23607i	0.612835i	0.951897	+	0.306418i	$0.0991304\pi$
$73$	5.23607i	0.612835i	−0.951897	+	0.306418i	$0.900870\pi$
$74$	1.05573	0.122726
$75$	0	0
$76$	−7.41641	−0.850720
$77$	− 1.00000i	− 0.113961i
$78$	− 8.94427i	− 1.01274i
$79$	−8.94427	−1.00631	−0.503155	−	0.864196i	$-0.667827\pi$
$79$	−8.94427	−1.00631	−0.503155	+	0.864196i	$0.667827\pi$
$80$	0	0
$81$	24.4164	2.71293
$82$	15.1246i	1.67023i
$83$	− 15.4164i	− 1.69217i	−0.533048	−	0.846085i	$-0.678953\pi$
$83$	− 15.4164i	− 1.69217i	0.533048	−	0.846085i	$-0.321047\pi$
$84$	−9.70820	−1.05925
$85$	0	0
$86$	−17.8885	−1.92897
$87$	− 1.52786i	− 0.163804i
$88$	− 2.23607i	− 0.238366i
$89$	−2.00000	−0.212000	−0.106000	−	0.994366i	$-0.533804\pi$
$89$	−2.00000	−0.212000	−0.106000	+	0.994366i	$0.533804\pi$
$90$	0	0
$91$	−1.23607	−0.129575
$92$	− 19.4164i	− 2.02430i
$93$	23.4164i	2.42817i
$94$	16.1803	1.66887
$95$	0	0
$96$	21.7082	2.21558
$97$	− 9.41641i	− 0.956091i	−0.878335	−	0.478046i	$-0.841345\pi$
$97$	− 9.41641i	− 0.956091i	0.878335	−	0.478046i	$-0.158655\pi$
$98$	2.23607i	0.225877i
$99$	7.47214	0.750978
$100$	0	0
$101$	−9.23607	−0.919023	−0.459512	−	0.888172i	$-0.651976\pi$
$101$	−9.23607	−0.919023	−0.459512	+	0.888172i	$0.651976\pi$
$102$	− 8.94427i	− 0.885615i
$103$	5.70820i	0.562446i	0.959642	+	0.281223i	$0.0907401\pi$
$103$	5.70820i	0.562446i	−0.959642	+	0.281223i	$0.909260\pi$
$104$	−2.76393	−0.271026
$105$	0	0
$106$	−18.9443	−1.84003
$107$	− 4.00000i	− 0.386695i	−0.981130	−	0.193347i	$-0.938066\pi$
$107$	− 4.00000i	− 0.386695i	0.981130	−	0.193347i	$-0.0619344\pi$
$108$	− 43.4164i	− 4.17775i
$109$	−4.47214	−0.428353	−0.214176	−	0.976795i	$-0.568707\pi$
$109$	−4.47214	−0.428353	−0.214176	+	0.976795i	$0.568707\pi$
$110$	0	0
$111$	1.52786	0.145018
$112$	− 1.00000i	− 0.0944911i
$113$	− 2.00000i	− 0.188144i	−0.995565	−	0.0940721i	$-0.970012\pi$
$113$	− 2.00000i	− 0.188144i	0.995565	−	0.0940721i	$-0.0299884\pi$
$114$	−17.8885	−1.67542
$115$	0	0
$116$	−1.41641	−0.131510
$117$	− 9.23607i	− 0.853875i
$118$	7.23607i	0.666134i
$119$	−1.23607	−0.113310
$120$	0	0
$121$	1.00000	0.0909091
$122$	6.18034i	0.559542i
$123$	21.8885i	1.97362i
$124$	21.7082	1.94945
$125$	0	0
$126$	−16.7082	−1.48849
$127$	20.9443i	1.85850i	0.369447	+	0.929252i	$0.379547\pi$
$127$	20.9443i	1.85850i	−0.369447	+	0.929252i	$0.620453\pi$
$128$	− 15.6525i	− 1.38350i
$129$	−25.8885	−2.27936
$130$	0	0
$131$	−13.8885	−1.21345	−0.606724	−	0.794913i	$-0.707517\pi$
$131$	−13.8885	−1.21345	−0.606724	+	0.794913i	$0.707517\pi$
$132$	− 9.70820i	− 0.844991i
$133$	2.47214i	0.214361i
$134$	12.3607	1.06780
$135$	0	0
$136$	−2.76393	−0.237005
$137$	7.52786i	0.643149i	0.946884	+	0.321574i	$0.104212\pi$
$137$	7.52786i	0.643149i	−0.946884	+	0.321574i	$0.895788\pi$
$138$	− 46.8328i	− 3.98667i
$139$	10.4721	0.888235	0.444117	−	0.895969i	$-0.353517\pi$
$139$	10.4721	0.888235	0.444117	+	0.895969i	$0.353517\pi$
$140$	0	0
$141$	23.4164	1.97202
$142$	3.41641i	0.286699i
$143$	− 1.23607i	− 0.103365i
$144$	7.47214	0.622678
$145$	0	0
$146$	11.7082	0.968978
$147$	3.23607i	0.266906i
$148$	− 1.41641i	− 0.116428i
$149$	−14.0000	−1.14692	−0.573462	−	0.819232i	$-0.694400\pi$
$149$	−14.0000	−1.14692	−0.573462	+	0.819232i	$0.694400\pi$
$150$	0	0
$151$	−8.94427	−0.727875	−0.363937	−	0.931423i	$-0.618568\pi$
$151$	−8.94427	−0.727875	−0.363937	+	0.931423i	$0.618568\pi$
$152$	5.52786i	0.448369i
$153$	− 9.23607i	− 0.746692i
$154$	−2.23607	−0.180187
$155$	0	0
$156$	−12.0000	−0.960769
$157$	− 6.94427i	− 0.554213i	−0.960839	−	0.277107i	$-0.910624\pi$
$157$	− 6.94427i	− 0.554213i	0.960839	−	0.277107i	$-0.0893755\pi$
$158$	20.0000i	1.59111i
$159$	−27.4164	−2.17426
$160$	0	0
$161$	−6.47214	−0.510076
$162$	− 54.5967i	− 4.28953i
$163$	− 23.4164i	− 1.83411i	−0.398755	−	0.917057i	$-0.630558\pi$
$163$	− 23.4164i	− 1.83411i	0.398755	−	0.917057i	$-0.369442\pi$
$164$	20.2918	1.58452
$165$	0	0
$166$	−34.4721	−2.67556
$167$	− 12.9443i	− 1.00166i	−0.865546	−	0.500829i	$-0.833029\pi$
$167$	− 12.9443i	− 1.00166i	0.865546	−	0.500829i	$-0.166971\pi$
$168$	7.23607i	0.558275i
$169$	11.4721	0.882472
$170$	0	0
$171$	−18.4721	−1.41260
$172$	24.0000i	1.82998i
$173$	17.2361i	1.31043i	0.755441	+	0.655217i	$0.227423\pi$
$173$	17.2361i	1.31043i	−0.755441	+	0.655217i	$0.772577\pi$
$174$	−3.41641	−0.258997
$175$	0	0
$176$	1.00000	0.0753778
$177$	10.4721i	0.787134i
$178$	4.47214i	0.335201i
$179$	−8.94427	−0.668526	−0.334263	−	0.942480i	$-0.608487\pi$
$179$	−8.94427	−0.668526	−0.334263	+	0.942480i	$0.608487\pi$
$180$	0	0
$181$	1.41641	0.105281	0.0526404	−	0.998614i	$-0.483236\pi$
$181$	1.41641	0.105281	0.0526404	+	0.998614i	$0.483236\pi$
$182$	2.76393i	0.204876i
$183$	8.94427i	0.661180i
$184$	−14.4721	−1.06690
$185$	0	0
$186$	52.3607	3.83927
$187$	− 1.23607i	− 0.0903902i
$188$	− 21.7082i	− 1.58323i
$189$	−14.4721	−1.05269
$190$	0	0
$191$	−20.9443	−1.51547	−0.757737	−	0.652560i	$-0.773695\pi$
$191$	−20.9443	−1.51547	−0.757737	+	0.652560i	$0.773695\pi$
$192$	− 42.0689i	− 3.03606i
$193$	23.8885i	1.71954i	0.510686	+	0.859768i	$0.329392\pi$
$193$	23.8885i	1.71954i	−0.510686	+	0.859768i	$0.670608\pi$
$194$	−21.0557	−1.51171
$195$	0	0
$196$	3.00000	0.214286
$197$	− 2.00000i	− 0.142494i	−0.997459	−	0.0712470i	$-0.977302\pi$
$197$	− 2.00000i	− 0.142494i	0.997459	−	0.0712470i	$-0.0226979\pi$
$198$	− 16.7082i	− 1.18740i
$199$	−20.1803	−1.43055	−0.715273	−	0.698845i	$-0.753698\pi$
$199$	−20.1803	−1.43055	−0.715273	+	0.698845i	$0.753698\pi$
$200$	0	0
$201$	17.8885	1.26176
$202$	20.6525i	1.45310i
$203$	0.472136i	0.0331374i
$204$	−12.0000	−0.840168
$205$	0	0
$206$	12.7639	0.889305
$207$	− 48.3607i	− 3.36130i
$208$	− 1.23607i	− 0.0857059i
$209$	−2.47214	−0.171001
$210$	0	0
$211$	21.8885	1.50687	0.753435	−	0.657523i	$-0.228396\pi$
$211$	21.8885	1.50687	0.753435	+	0.657523i	$0.228396\pi$
$212$	25.4164i	1.74561i
$213$	4.94427i	0.338776i
$214$	−8.94427	−0.611418
$215$	0	0
$216$	−32.3607	−2.20187
$217$	− 7.23607i	− 0.491216i
$218$	10.0000i	0.677285i
$219$	16.9443	1.14499
$220$	0	0
$221$	−1.52786	−0.102775
$222$	− 3.41641i	− 0.229294i
$223$	− 12.1803i	− 0.815656i	−0.913059	−	0.407828i	$-0.866286\pi$
$223$	− 12.1803i	− 0.815656i	0.913059	−	0.407828i	$-0.133714\pi$
$224$	−6.70820	−0.448211
$225$	0	0
$226$	−4.47214	−0.297482
$227$	− 29.8885i	− 1.98377i	−0.127129	−	0.991886i	$-0.540576\pi$
$227$	− 29.8885i	− 1.98377i	0.127129	−	0.991886i	$-0.459424\pi$
$228$	24.0000i	1.58944i
$229$	−4.47214	−0.295527	−0.147764	−	0.989023i	$-0.547207\pi$
$229$	−4.47214	−0.295527	−0.147764	+	0.989023i	$0.547207\pi$
$230$	0	0
$231$	−3.23607	−0.212918
$232$	1.05573i	0.0693119i
$233$	17.4164i	1.14099i	0.821302	+	0.570493i	$0.193248\pi$
$233$	17.4164i	1.14099i	−0.821302	+	0.570493i	$0.806752\pi$
$234$	−20.6525	−1.35009
$235$	0	0
$236$	9.70820	0.631950
$237$	28.9443i	1.88013i
$238$	2.76393i	0.179159i
$239$	−25.8885	−1.67459	−0.837295	−	0.546751i	$-0.815864\pi$
$239$	−25.8885	−1.67459	−0.837295	+	0.546751i	$0.815864\pi$
$240$	0	0
$241$	27.1246	1.74725	0.873625	−	0.486600i	$-0.161763\pi$
$241$	27.1246	1.74725	0.873625	+	0.486600i	$0.161763\pi$
$242$	− 2.23607i	− 0.143740i
$243$	− 35.5967i	− 2.28353i
$244$	8.29180	0.530828
$245$	0	0
$246$	48.9443	3.12057
$247$	3.05573i	0.194431i
$248$	− 16.1803i	− 1.02745i
$249$	−49.8885	−3.16156
$250$	0	0
$251$	17.7082	1.11773	0.558866	−	0.829258i	$-0.311237\pi$
$251$	17.7082	1.11773	0.558866	+	0.829258i	$0.311237\pi$
$252$	22.4164i	1.41210i
$253$	− 6.47214i	− 0.406900i
$254$	46.8328	2.93855
$255$	0	0
$256$	−9.00000	−0.562500
$257$	− 6.00000i	− 0.374270i	−0.982334	−	0.187135i	$-0.940080\pi$
$257$	− 6.00000i	− 0.374270i	0.982334	−	0.187135i	$-0.0599201\pi$
$258$	57.8885i	3.60398i
$259$	−0.472136	−0.0293371
$260$	0	0
$261$	−3.52786	−0.218369
$262$	31.0557i	1.91863i
$263$	0	0	1.00000			$0$
$263$	0	0	−1.00000			$\pi$
$264$	−7.23607	−0.445349
$265$	0	0
$266$	5.52786	0.338935
$267$	6.47214i	0.396088i
$268$	− 16.5836i	− 1.01300i
$269$	13.4164	0.818013	0.409006	−	0.912532i	$-0.365875\pi$
$269$	13.4164	0.818013	0.409006	+	0.912532i	$0.365875\pi$
$270$	0	0
$271$	−1.52786	−0.0928111	−0.0464056	−	0.998923i	$-0.514777\pi$
$271$	−1.52786	−0.0928111	−0.0464056	+	0.998923i	$0.514777\pi$
$272$	− 1.23607i	− 0.0749476i
$273$	4.00000i	0.242091i
$274$	16.8328	1.01691
$275$	0	0
$276$	−62.8328	−3.78209
$277$	15.8885i	0.954650i	0.878727	+	0.477325i	$0.158394\pi$
$277$	15.8885i	0.954650i	−0.878727	+	0.477325i	$0.841606\pi$
$278$	− 23.4164i	− 1.40442i
$279$	54.0689	3.23702
$280$	0	0
$281$	−12.4721	−0.744025	−0.372013	−	0.928228i	$-0.621332\pi$
$281$	−12.4721	−0.744025	−0.372013	+	0.928228i	$0.621332\pi$
$282$	− 52.3607i	− 3.11803i
$283$	5.88854i	0.350038i	0.984565	+	0.175019i	$0.0559986\pi$
$283$	5.88854i	0.350038i	−0.984565	+	0.175019i	$0.944001\pi$
$284$	4.58359	0.271986
$285$	0	0
$286$	−2.76393	−0.163435
$287$	− 6.76393i	− 0.399262i
$288$	− 50.1246i	− 2.95362i
$289$	15.4721	0.910126
$290$	0	0
$291$	−30.4721	−1.78631
$292$	− 15.7082i	− 0.919253i
$293$	− 15.1246i	− 0.883589i	−0.897116	−	0.441795i	$-0.854342\pi$
$293$	− 15.1246i	− 0.883589i	0.897116	−	0.441795i	$-0.145658\pi$
$294$	7.23607	0.422016
$295$	0	0
$296$	−1.05573	−0.0613629
$297$	− 14.4721i	− 0.839759i
$298$	31.3050i	1.81345i
$299$	−8.00000	−0.462652
$300$	0	0
$301$	8.00000	0.461112
$302$	20.0000i	1.15087i
$303$	29.8885i	1.71705i
$304$	−2.47214	−0.141787
$305$	0	0
$306$	−20.6525	−1.18062
$307$	8.94427i	0.510477i	0.966878	+	0.255238i	$0.0821539\pi$
$307$	8.94427i	0.510477i	−0.966878	+	0.255238i	$0.917846\pi$
$308$	3.00000i	0.170941i
$309$	18.4721	1.05084
$310$	0	0
$311$	−21.7082	−1.23096	−0.615480	−	0.788153i	$-0.711038\pi$
$311$	−21.7082	−1.23096	−0.615480	+	0.788153i	$0.711038\pi$
$312$	8.94427i	0.506370i
$313$	2.94427i	0.166420i	0.996532	+	0.0832100i	$0.0265172\pi$
$313$	2.94427i	0.166420i	−0.996532	+	0.0832100i	$0.973483\pi$
$314$	−15.5279	−0.876288
$315$	0	0
$316$	26.8328	1.50946
$317$	14.0000i	0.786318i	0.919470	+	0.393159i	$0.128618\pi$
$317$	14.0000i	0.786318i	−0.919470	+	0.393159i	$0.871382\pi$
$318$	61.3050i	3.43781i
$319$	−0.472136	−0.0264345
$320$	0	0
$321$	−12.9443	−0.722479
$322$	14.4721i	0.806501i
$323$	3.05573i	0.170025i
$324$	−73.2492	−4.06940
$325$	0	0
$326$	−52.3607	−2.89999
$327$	14.4721i	0.800311i
$328$	− 15.1246i	− 0.835117i
$329$	−7.23607	−0.398937
$330$	0	0
$331$	21.8885	1.20310	0.601552	−	0.798834i	$-0.294549\pi$
$331$	21.8885	1.20310	0.601552	+	0.798834i	$0.294549\pi$
$332$	46.2492i	2.53826i
$333$	− 3.52786i	− 0.193326i
$334$	−28.9443	−1.58376
$335$	0	0
$336$	−3.23607	−0.176542
$337$	− 20.4721i	− 1.11519i	−0.830114	−	0.557594i	$-0.811725\pi$
$337$	− 20.4721i	− 1.11519i	0.830114	−	0.557594i	$-0.188275\pi$
$338$	− 25.6525i	− 1.39531i
$339$	−6.47214	−0.351518
$340$	0	0
$341$	7.23607	0.391855
$342$	41.3050i	2.23352i
$343$	− 1.00000i	− 0.0539949i
$344$	17.8885	0.964486
$345$	0	0
$346$	38.5410	2.07198
$347$	3.05573i	0.164040i	0.996631	+	0.0820200i	$0.0261372\pi$
$347$	3.05573i	0.164040i	−0.996631	+	0.0820200i	$0.973863\pi$
$348$	4.58359i	0.245706i
$349$	2.76393	0.147950	0.0739749	−	0.997260i	$-0.476432\pi$
$349$	2.76393	0.147950	0.0739749	+	0.997260i	$0.476432\pi$
$350$	0	0
$351$	−17.8885	−0.954820
$352$	− 6.70820i	− 0.357548i
$353$	15.8885i	0.845662i	0.906209	+	0.422831i	$0.138964\pi$
$353$	15.8885i	0.845662i	−0.906209	+	0.422831i	$0.861036\pi$
$354$	23.4164	1.24457
$355$	0	0
$356$	6.00000	0.317999
$357$	4.00000i	0.211702i
$358$	20.0000i	1.05703i
$359$	7.05573	0.372387	0.186194	−	0.982513i	$-0.440385\pi$
$359$	7.05573	0.372387	0.186194	+	0.982513i	$0.440385\pi$
$360$	0	0
$361$	−12.8885	−0.678344
$362$	− 3.16718i	− 0.166464i
$363$	− 3.23607i	− 0.169850i
$364$	3.70820	0.194363
$365$	0	0
$366$	20.0000	1.04542
$367$	17.1246i	0.893897i	0.894560	+	0.446949i	$0.147489\pi$
$367$	17.1246i	0.893897i	−0.894560	+	0.446949i	$0.852511\pi$
$368$	− 6.47214i	− 0.337383i
$369$	50.5410	2.63106
$370$	0	0
$371$	8.47214	0.439851
$372$	− 70.2492i	− 3.64225i
$373$	− 6.00000i	− 0.310668i	−0.987862	−	0.155334i	$-0.950355\pi$
$373$	− 6.00000i	− 0.310668i	0.987862	−	0.155334i	$-0.0496454\pi$
$374$	−2.76393	−0.142920
$375$	0	0
$376$	−16.1803	−0.834437
$377$	0.583592i	0.0300565i
$378$	32.3607i	1.66445i
$379$	25.3050	1.29983	0.649914	−	0.760008i	$-0.274805\pi$
$379$	25.3050	1.29983	0.649914	+	0.760008i	$0.274805\pi$
$380$	0	0
$381$	67.7771	3.47233
$382$	46.8328i	2.39618i
$383$	− 26.6525i	− 1.36188i	−0.732340	−	0.680939i	$-0.761572\pi$
$383$	− 26.6525i	− 1.36188i	0.732340	−	0.680939i	$-0.238428\pi$
$384$	−50.6525	−2.58485
$385$	0	0
$386$	53.4164	2.71882
$387$	59.7771i	3.03864i
$388$	28.2492i	1.43414i
$389$	19.8885	1.00839	0.504195	−	0.863590i	$-0.331789\pi$
$389$	19.8885	1.00839	0.504195	+	0.863590i	$0.331789\pi$
$390$	0	0
$391$	−8.00000	−0.404577
$392$	− 2.23607i	− 0.112938i
$393$	44.9443i	2.26714i
$394$	−4.47214	−0.225303
$395$	0	0
$396$	−22.4164	−1.12647
$397$	− 0.111456i	− 0.00559383i	−0.999996	−	0.00279691i	$-0.999110\pi$
$397$	− 0.111456i	− 0.00559383i	0.999996	−	0.00279691i	$-0.000890286\pi$
$398$	45.1246i	2.26189i
$399$	8.00000	0.400501
$400$	0	0
$401$	5.05573	0.252471	0.126236	−	0.992000i	$-0.459710\pi$
$401$	5.05573	0.252471	0.126236	+	0.992000i	$0.459710\pi$
$402$	− 40.0000i	− 1.99502i
$403$	− 8.94427i	− 0.445546i
$404$	27.7082	1.37853
$405$	0	0
$406$	1.05573	0.0523949
$407$	− 0.472136i	− 0.0234029i
$408$	8.94427i	0.442807i
$409$	31.1246	1.53901	0.769507	−	0.638639i	$-0.220502\pi$
$409$	31.1246	1.53901	0.769507	+	0.638639i	$0.220502\pi$
$410$	0	0
$411$	24.3607	1.20162
$412$	− 17.1246i	− 0.843669i
$413$	− 3.23607i	− 0.159236i
$414$	−108.138	−5.31468
$415$	0	0
$416$	−8.29180	−0.406539
$417$	− 33.8885i	− 1.65953i
$418$	5.52786i	0.270377i
$419$	6.65248	0.324995	0.162497	−	0.986709i	$-0.448045\pi$
$419$	6.65248	0.324995	0.162497	+	0.986709i	$0.448045\pi$
$420$	0	0
$421$	−22.3607	−1.08979	−0.544896	−	0.838503i	$-0.683431\pi$
$421$	−22.3607	−1.08979	−0.544896	+	0.838503i	$0.683431\pi$
$422$	− 48.9443i	− 2.38257i
$423$	− 54.0689i	− 2.62892i
$424$	18.9443	0.920015
$425$	0	0
$426$	11.0557	0.535652
$427$	− 2.76393i	− 0.133756i
$428$	12.0000i	0.580042i
$429$	−4.00000	−0.193122
$430$	0	0
$431$	12.0000	0.578020	0.289010	−	0.957326i	$-0.406674\pi$
$431$	12.0000	0.578020	0.289010	+	0.957326i	$0.406674\pi$
$432$	− 14.4721i	− 0.696291i
$433$	− 0.472136i	− 0.0226894i	−0.999936	−	0.0113447i	$-0.996389\pi$
$433$	− 0.472136i	− 0.0226894i	0.999936	−	0.0113447i	$-0.00361121\pi$
$434$	−16.1803	−0.776681
$435$	0	0
$436$	13.4164	0.642529
$437$	16.0000i	0.765384i
$438$	− 37.8885i	− 1.81038i
$439$	−1.52786	−0.0729210	−0.0364605	−	0.999335i	$-0.511608\pi$
$439$	−1.52786	−0.0729210	−0.0364605	+	0.999335i	$0.511608\pi$
$440$	0	0
$441$	7.47214	0.355816
$442$	3.41641i	0.162502i
$443$	7.05573i	0.335228i	0.985853	+	0.167614i	$0.0536062\pi$
$443$	7.05573i	0.335228i	−0.985853	+	0.167614i	$0.946394\pi$
$444$	−4.58359	−0.217528
$445$	0	0
$446$	−27.2361	−1.28967
$447$	45.3050i	2.14285i
$448$	13.0000i	0.614192i
$449$	19.5279	0.921577	0.460788	−	0.887510i	$-0.347567\pi$
$449$	19.5279	0.921577	0.460788	+	0.887510i	$0.347567\pi$
$450$	0	0
$451$	6.76393	0.318501
$452$	6.00000i	0.282216i
$453$	28.9443i	1.35992i
$454$	−66.8328	−3.13662
$455$	0	0
$456$	17.8885	0.837708
$457$	24.8328i	1.16163i	0.814036	+	0.580815i	$0.197266\pi$
$457$	24.8328i	1.16163i	−0.814036	+	0.580815i	$0.802734\pi$
$458$	10.0000i	0.467269i
$459$	−17.8885	−0.834966
$460$	0	0
$461$	10.1803	0.474146	0.237073	−	0.971492i	$-0.423812\pi$
$461$	10.1803	0.474146	0.237073	+	0.971492i	$0.423812\pi$
$462$	7.23607i	0.336652i
$463$	14.4721i	0.672577i	0.941759	+	0.336289i	$0.109172\pi$
$463$	14.4721i	0.672577i	−0.941759	+	0.336289i	$0.890828\pi$
$464$	−0.472136	−0.0219184
$465$	0	0
$466$	38.9443	1.80406
$467$	− 34.0689i	− 1.57652i	−0.615342	−	0.788260i	$-0.710982\pi$
$467$	− 34.0689i	− 1.57652i	0.615342	−	0.788260i	$-0.289018\pi$
$468$	27.7082i	1.28081i
$469$	−5.52786	−0.255253
$470$	0	0
$471$	−22.4721	−1.03546
$472$	− 7.23607i	− 0.333067i
$473$	8.00000i	0.367840i
$474$	64.7214	2.97275
$475$	0	0
$476$	3.70820	0.169965
$477$	63.3050i	2.89853i
$478$	57.8885i	2.64776i
$479$	−22.4721	−1.02678	−0.513389	−	0.858156i	$-0.671610\pi$
$479$	−22.4721	−1.02678	−0.513389	+	0.858156i	$0.671610\pi$
$480$	0	0
$481$	−0.583592	−0.0266095
$482$	− 60.6525i	− 2.76264i
$483$	20.9443i	0.952997i
$484$	−3.00000	−0.136364
$485$	0	0
$486$	−79.5967	−3.61058
$487$	8.36068i	0.378859i	0.981894	+	0.189429i	$0.0606638\pi$
$487$	8.36068i	0.378859i	−0.981894	+	0.189429i	$0.939336\pi$
$488$	− 6.18034i	− 0.279771i
$489$	−75.7771	−3.42676
$490$	0	0
$491$	0	0		−	1.00000i	$-0.5\pi$
$491$	0	0			1.00000i	$0.5\pi$
$492$	− 65.6656i	− 2.96044i
$493$	0.583592i	0.0262837i
$494$	6.83282	0.307423
$495$	0	0
$496$	7.23607	0.324909
$497$	− 1.52786i	− 0.0685341i
$498$	111.554i	4.99886i
$499$	−10.4721	−0.468797	−0.234399	−	0.972141i	$-0.575312\pi$
$499$	−10.4721	−0.468797	−0.234399	+	0.972141i	$0.575312\pi$
$500$	0	0
$501$	−41.8885	−1.87144
$502$	− 39.5967i	− 1.76729i
$503$	3.41641i	0.152330i	0.997095	+	0.0761650i	$0.0242676\pi$
$503$	3.41641i	0.152330i	−0.997095	+	0.0761650i	$0.975732\pi$
$504$	16.7082	0.744243
$505$	0	0
$506$	−14.4721	−0.643365
$507$	− 37.1246i	− 1.64876i
$508$	− 62.8328i	− 2.78776i
$509$	−31.5279	−1.39745	−0.698724	−	0.715391i	$-0.746248\pi$
$509$	−31.5279	−1.39745	−0.698724	+	0.715391i	$0.746248\pi$
$510$	0	0
$511$	−5.23607	−0.231630
$512$	− 11.1803i	− 0.494106i
$513$	35.7771i	1.57960i
$514$	−13.4164	−0.591772
$515$	0	0
$516$	77.6656	3.41904
$517$	− 7.23607i	− 0.318242i
$518$	1.05573i	0.0463860i
$519$	55.7771	2.44834
$520$	0	0
$521$	−14.3607	−0.629153	−0.314576	−	0.949232i	$-0.601862\pi$
$521$	−14.3607	−0.629153	−0.314576	+	0.949232i	$0.601862\pi$
$522$	7.88854i	0.345272i
$523$	− 44.0000i	− 1.92399i	−0.273075	−	0.961993i	$-0.588041\pi$
$523$	− 44.0000i	− 1.92399i	0.273075	−	0.961993i	$-0.411959\pi$
$524$	41.6656	1.82017
$525$	0	0
$526$	0	0
$527$	− 8.94427i	− 0.389619i
$528$	− 3.23607i	− 0.140832i
$529$	−18.8885	−0.821241
$530$	0	0
$531$	24.1803	1.04934
$532$	− 7.41641i	− 0.321542i
$533$	− 8.36068i	− 0.362141i
$534$	14.4721	0.626271
$535$	0	0
$536$	−12.3607	−0.533900
$537$	28.9443i	1.24904i
$538$	− 30.0000i	− 1.29339i
$539$	1.00000	0.0430730
$540$	0	0
$541$	−32.8328	−1.41159	−0.705797	−	0.708415i	$-0.749411\pi$
$541$	−32.8328	−1.41159	−0.705797	+	0.708415i	$0.749411\pi$
$542$	3.41641i	0.146747i
$543$	− 4.58359i	− 0.196701i
$544$	−8.29180	−0.355508
$545$	0	0
$546$	8.94427	0.382780
$547$	28.0000i	1.19719i	0.801050	+	0.598597i	$0.204275\pi$
$547$	28.0000i	1.19719i	−0.801050	+	0.598597i	$0.795725\pi$
$548$	− 22.5836i	− 0.964723i
$549$	20.6525	0.881426
$550$	0	0
$551$	1.16718	0.0497237
$552$	46.8328i	1.99334i
$553$	− 8.94427i	− 0.380349i
$554$	35.5279	1.50943
$555$	0	0
$556$	−31.4164	−1.33235
$557$	− 21.0557i	− 0.892160i	−0.894993	−	0.446080i	$-0.852820\pi$
$557$	− 21.0557i	− 0.892160i	0.894993	−	0.446080i	$-0.147180\pi$
$558$	− 120.902i	− 5.11818i
$559$	9.88854	0.418241
$560$	0	0
$561$	−4.00000	−0.168880
$562$	27.8885i	1.17641i
$563$	− 39.4164i	− 1.66120i	−0.556867	−	0.830602i	$-0.687997\pi$
$563$	− 39.4164i	− 1.66120i	0.556867	−	0.830602i	$-0.312003\pi$
$564$	−70.2492	−2.95803
$565$	0	0
$566$	13.1672	0.553458
$567$	24.4164i	1.02539i
$568$	− 3.41641i	− 0.143349i
$569$	−16.4721	−0.690548	−0.345274	−	0.938502i	$-0.612214\pi$
$569$	−16.4721	−0.690548	−0.345274	+	0.938502i	$0.612214\pi$
$570$	0	0
$571$	−32.9443	−1.37867	−0.689337	−	0.724440i	$-0.742098\pi$
$571$	−32.9443	−1.37867	−0.689337	+	0.724440i	$0.742098\pi$
$572$	3.70820i	0.155048i
$573$	67.7771i	2.83143i
$574$	−15.1246	−0.631289
$575$	0	0
$576$	−97.1378	−4.04741
$577$	− 28.4721i	− 1.18531i	−0.805456	−	0.592655i	$-0.798080\pi$
$577$	− 28.4721i	− 1.18531i	0.805456	−	0.592655i	$-0.201920\pi$
$578$	− 34.5967i	− 1.43903i
$579$	77.3050	3.21268
$580$	0	0
$581$	15.4164	0.639580
$582$	68.1378i	2.82440i
$583$	8.47214i	0.350880i
$584$	−11.7082	−0.484489
$585$	0	0
$586$	−33.8197	−1.39708
$587$	− 13.1246i	− 0.541711i	−0.962620	−	0.270855i	$-0.912693\pi$
$587$	− 13.1246i	− 0.541711i	0.962620	−	0.270855i	$-0.0873065\pi$
$588$	− 9.70820i	− 0.400360i
$589$	−17.8885	−0.737085
$590$	0	0
$591$	−6.47214	−0.266228
$592$	− 0.472136i	− 0.0194047i
$593$	32.2918i	1.32607i	0.748591	+	0.663033i	$0.230731\pi$
$593$	32.2918i	1.32607i	−0.748591	+	0.663033i	$0.769269\pi$
$594$	−32.3607	−1.32777
$595$	0	0
$596$	42.0000	1.72039
$597$	65.3050i	2.67275i
$598$	17.8885i	0.731517i
$599$	−3.41641	−0.139591	−0.0697953	−	0.997561i	$-0.522235\pi$
$599$	−3.41641	−0.139591	−0.0697953	+	0.997561i	$0.522235\pi$
$600$	0	0
$601$	3.12461	0.127456	0.0637278	−	0.997967i	$-0.479701\pi$
$601$	3.12461	0.127456	0.0637278	+	0.997967i	$0.479701\pi$
$602$	− 17.8885i	− 0.729083i
$603$	− 41.3050i	− 1.68207i
$604$	26.8328	1.09181
$605$	0	0
$606$	66.8328	2.71490
$607$	− 4.94427i	− 0.200682i	−0.994953	−	0.100341i	$-0.968007\pi$
$607$	− 4.94427i	− 0.200682i	0.994953	−	0.100341i	$-0.0319933\pi$
$608$	16.5836i	0.672553i
$609$	1.52786	0.0619122
$610$	0	0
$611$	−8.94427	−0.361847
$612$	27.7082i	1.12004i
$613$	47.3050i	1.91063i	0.295591	+	0.955315i	$0.404483\pi$
$613$	47.3050i	1.91063i	−0.295591	+	0.955315i	$0.595517\pi$
$614$	20.0000	0.807134
$615$	0	0
$616$	2.23607	0.0900937
$617$	33.4164i	1.34529i	0.739964	+	0.672647i	$0.234843\pi$
$617$	33.4164i	1.34529i	−0.739964	+	0.672647i	$0.765157\pi$
$618$	− 41.3050i	− 1.66153i
$619$	29.1246	1.17062	0.585308	−	0.810811i	$-0.300973\pi$
$619$	29.1246	1.17062	0.585308	+	0.810811i	$0.300973\pi$
$620$	0	0
$621$	−93.6656	−3.75867
$622$	48.5410i	1.94632i
$623$	− 2.00000i	− 0.0801283i
$624$	−4.00000	−0.160128
$625$	0	0
$626$	6.58359	0.263133
$627$	8.00000i	0.319489i
$628$	20.8328i	0.831320i
$629$	−0.583592	−0.0232693
$630$	0	0
$631$	−24.0000	−0.955425	−0.477712	−	0.878516i	$-0.658534\pi$
$631$	−24.0000	−0.955425	−0.477712	+	0.878516i	$0.658534\pi$
$632$	− 20.0000i	− 0.795557i
$633$	− 70.8328i	− 2.81535i
$634$	31.3050	1.24328
$635$	0	0
$636$	82.2492	3.26139
$637$	− 1.23607i	− 0.0489748i
$638$	1.05573i	0.0417967i
$639$	11.4164	0.451626
$640$	0	0
$641$	−24.4721	−0.966591	−0.483296	−	0.875457i	$-0.660560\pi$
$641$	−24.4721	−0.966591	−0.483296	+	0.875457i	$0.660560\pi$
$642$	28.9443i	1.14234i
$643$	29.1246i	1.14856i	0.818658	+	0.574281i	$0.194718\pi$
$643$	29.1246i	1.14856i	−0.818658	+	0.574281i	$0.805282\pi$
$644$	19.4164	0.765114
$645$	0	0
$646$	6.83282	0.268834
$647$	− 22.0689i	− 0.867617i	−0.901005	−	0.433809i	$-0.857169\pi$
$647$	− 22.0689i	− 0.867617i	0.901005	−	0.433809i	$-0.142831\pi$
$648$	54.5967i	2.14476i
$649$	3.23607	0.127027
$650$	0	0
$651$	−23.4164	−0.917761
$652$	70.2492i	2.75117i
$653$	− 42.9443i	− 1.68054i	−0.542169	−	0.840270i	$-0.682397\pi$
$653$	− 42.9443i	− 1.68054i	0.542169	−	0.840270i	$-0.317603\pi$
$654$	32.3607	1.26540
$655$	0	0
$656$	6.76393	0.264087
$657$	− 39.1246i	− 1.52640i
$658$	16.1803i	0.630775i
$659$	−17.8885	−0.696839	−0.348419	−	0.937339i	$-0.613281\pi$
$659$	−17.8885	−0.696839	−0.348419	+	0.937339i	$0.613281\pi$
$660$	0	0
$661$	12.8328	0.499139	0.249569	−	0.968357i	$-0.419711\pi$
$661$	12.8328	0.499139	0.249569	+	0.968357i	$0.419711\pi$
$662$	− 48.9443i	− 1.90227i
$663$	4.94427i	0.192020i
$664$	34.4721	1.33778
$665$	0	0
$666$	−7.88854	−0.305675
$667$	3.05573i	0.118318i
$668$	38.8328i	1.50249i
$669$	−39.4164	−1.52393
$670$	0	0
$671$	2.76393	0.106700
$672$	21.7082i	0.837412i
$673$	− 5.41641i	− 0.208787i	−0.994536	−	0.104394i	$-0.966710\pi$
$673$	− 5.41641i	− 0.208787i	0.994536	−	0.104394i	$-0.0332902\pi$
$674$	−45.7771	−1.76327
$675$	0	0
$676$	−34.4164	−1.32371
$677$	3.70820i	0.142518i	0.997458	+	0.0712589i	$0.0227017\pi$
$677$	3.70820i	0.142518i	−0.997458	+	0.0712589i	$0.977298\pi$
$678$	14.4721i	0.555799i
$679$	9.41641	0.361369
$680$	0	0
$681$	−96.7214	−3.70637
$682$	− 16.1803i	− 0.619577i
$683$	− 29.8885i	− 1.14365i	−0.820374	−	0.571827i	$-0.806235\pi$
$683$	− 29.8885i	− 1.14365i	0.820374	−	0.571827i	$-0.193765\pi$
$684$	55.4164	2.11890
$685$	0	0
$686$	−2.23607	−0.0853735
$687$	14.4721i	0.552146i
$688$	8.00000i	0.304997i
$689$	10.4721	0.398957
$690$	0	0
$691$	−48.5410	−1.84659	−0.923294	−	0.384095i	$-0.874514\pi$
$691$	−48.5410	−1.84659	−0.923294	+	0.384095i	$0.874514\pi$
$692$	− 51.7082i	− 1.96565i
$693$	7.47214i	0.283843i
$694$	6.83282	0.259370
$695$	0	0
$696$	3.41641	0.129499
$697$	− 8.36068i	− 0.316683i
$698$	− 6.18034i	− 0.233929i
$699$	56.3607	2.13176
$700$	0	0
$701$	−24.4721	−0.924300	−0.462150	−	0.886802i	$-0.652922\pi$
$701$	−24.4721	−0.924300	−0.462150	+	0.886802i	$0.652922\pi$
$702$	40.0000i	1.50970i
$703$	1.16718i	0.0440212i
$704$	−13.0000	−0.489956
$705$	0	0
$706$	35.5279	1.33711
$707$	− 9.23607i	− 0.347358i
$708$	− 31.4164i	− 1.18070i
$709$	−2.94427	−0.110574	−0.0552872	−	0.998470i	$-0.517607\pi$
$709$	−2.94427	−0.110574	−0.0552872	+	0.998470i	$0.517607\pi$
$710$	0	0
$711$	66.8328	2.50643
$712$	− 4.47214i	− 0.167600i
$713$	− 46.8328i	− 1.75390i
$714$	8.94427	0.334731
$715$	0	0
$716$	26.8328	1.00279
$717$	83.7771i	3.12871i
$718$	− 15.7771i	− 0.588796i
$719$	−33.4853	−1.24879	−0.624395	−	0.781108i	$-0.714655\pi$
$719$	−33.4853	−1.24879	−0.624395	+	0.781108i	$0.714655\pi$
$720$	0	0
$721$	−5.70820	−0.212585
$722$	28.8197i	1.07256i
$723$	− 87.7771i	− 3.26447i
$724$	−4.24922	−0.157921
$725$	0	0
$726$	−7.23607	−0.268556
$727$	− 51.0132i	− 1.89197i	−0.324206	−	0.945987i	$-0.605097\pi$
$727$	− 51.0132i	− 1.89197i	0.324206	−	0.945987i	$-0.394903\pi$
$728$	− 2.76393i	− 0.102438i
$729$	−41.9443	−1.55349
$730$	0	0
$731$	9.88854	0.365741
$732$	− 26.8328i	− 0.991769i
$733$	− 13.2361i	− 0.488885i	−0.969664	−	0.244443i	$-0.921395\pi$
$733$	− 13.2361i	− 0.488885i	0.969664	−	0.244443i	$-0.0786050\pi$
$734$	38.2918	1.41338
$735$	0	0
$736$	−43.4164	−1.60035
$737$	− 5.52786i	− 0.203621i
$738$	− 113.013i	− 4.16007i
$739$	−7.05573	−0.259549	−0.129775	−	0.991544i	$-0.541425\pi$
$739$	−7.05573	−0.259549	−0.129775	+	0.991544i	$0.541425\pi$
$740$	0	0
$741$	9.88854	0.363265
$742$	− 18.9443i	− 0.695466i
$743$	33.8885i	1.24325i	0.783315	+	0.621625i	$0.213527\pi$
$743$	33.8885i	1.24325i	−0.783315	+	0.621625i	$0.786473\pi$
$744$	−52.3607	−1.91964
$745$	0	0
$746$	−13.4164	−0.491210
$747$	115.193i	4.21471i
$748$	3.70820i	0.135585i
$749$	4.00000	0.146157
$750$	0	0
$751$	38.4721	1.40387	0.701934	−	0.712242i	$-0.252320\pi$
$751$	38.4721	1.40387	0.701934	+	0.712242i	$0.252320\pi$
$752$	− 7.23607i	− 0.263872i
$753$	− 57.3050i	− 2.08831i
$754$	1.30495	0.0475235
$755$	0	0
$756$	43.4164	1.57904
$757$	− 19.8885i	− 0.722861i	−0.932399	−	0.361431i	$-0.882288\pi$
$757$	− 19.8885i	− 0.722861i	0.932399	−	0.361431i	$-0.117712\pi$
$758$	− 56.5836i	− 2.05521i
$759$	−20.9443	−0.760229
$760$	0	0
$761$	17.5967	0.637882	0.318941	−	0.947775i	$-0.396673\pi$
$761$	17.5967	0.637882	0.318941	+	0.947775i	$0.396673\pi$
$762$	− 151.554i	− 5.49023i
$763$	− 4.47214i	− 0.161902i
$764$	62.8328	2.27321
$765$	0	0
$766$	−59.5967	−2.15332
$767$	− 4.00000i	− 0.144432i
$768$	29.1246i	1.05094i
$769$	−31.7082	−1.14343	−0.571714	−	0.820453i	$-0.693721\pi$
$769$	−31.7082	−1.14343	−0.571714	+	0.820453i	$0.693721\pi$
$770$	0	0
$771$	−19.4164	−0.699265
$772$	− 71.6656i	− 2.57930i
$773$	− 6.36068i	− 0.228778i	−0.993436	−	0.114389i	$-0.963509\pi$
$773$	− 6.36068i	− 0.228778i	0.993436	−	0.114389i	$-0.0364910\pi$
$774$	133.666	4.80451
$775$	0	0
$776$	21.0557	0.755857
$777$	1.52786i	0.0548118i
$778$	− 44.4721i	− 1.59440i
$779$	−16.7214	−0.599105
$780$	0	0
$781$	1.52786	0.0546713
$782$	17.8885i	0.639693i
$783$	6.83282i	0.244185i
$784$	1.00000	0.0357143
$785$	0	0
$786$	100.498	3.58466
$787$	− 16.5836i	− 0.591141i	−0.955321	−	0.295571i	$-0.904490\pi$
$787$	− 16.5836i	− 0.591141i	0.955321	−	0.295571i	$-0.0955098\pi$
$788$	6.00000i	0.213741i
$789$	0	0
$790$	0	0
$791$	2.00000	0.0711118
$792$	16.7082i	0.593700i
$793$	− 3.41641i	− 0.121320i
$794$	−0.249224	−0.00884461
$795$	0	0
$796$	60.5410	2.14582
$797$	2.94427i	0.104291i	0.998639	+	0.0521457i	$0.0166060\pi$
$797$	2.94427i	0.104291i	−0.998639	+	0.0521457i	$0.983394\pi$
$798$	− 17.8885i	− 0.633248i
$799$	−8.94427	−0.316426
$800$	0	0
$801$	14.9443	0.528030
$802$	− 11.3050i	− 0.399192i
$803$	− 5.23607i	− 0.184777i
$804$	−53.6656	−1.89264
$805$	0	0
$806$	−20.0000	−0.704470
$807$	− 43.4164i	− 1.52833i
$808$	− 20.6525i	− 0.726552i
$809$	−38.9443	−1.36921	−0.684604	−	0.728915i	$-0.740025\pi$
$809$	−38.9443	−1.36921	−0.684604	+	0.728915i	$0.740025\pi$
$810$	0	0
$811$	18.8328	0.661310	0.330655	−	0.943752i	$-0.392730\pi$
$811$	18.8328	0.661310	0.330655	+	0.943752i	$0.392730\pi$
$812$	− 1.41641i	− 0.0497062i
$813$	4.94427i	0.173403i
$814$	−1.05573	−0.0370033
$815$	0	0
$816$	−4.00000	−0.140028
$817$	− 19.7771i	− 0.691913i
$818$	− 69.5967i	− 2.43339i
$819$	9.23607	0.322734
$820$	0	0
$821$	−8.83282	−0.308267	−0.154134	−	0.988050i	$-0.549259\pi$
$821$	−8.83282	−0.308267	−0.154134	+	0.988050i	$0.549259\pi$
$822$	− 54.4721i	− 1.89993i
$823$	49.8885i	1.73901i	0.493928	+	0.869503i	$0.335561\pi$
$823$	49.8885i	1.73901i	−0.493928	+	0.869503i	$0.664439\pi$
$824$	−12.7639	−0.444653
$825$	0	0
$826$	−7.23607	−0.251775
$827$	4.94427i	0.171929i	0.996298	+	0.0859646i	$0.0273972\pi$
$827$	4.94427i	0.171929i	−0.996298	+	0.0859646i	$0.972603\pi$
$828$	145.082i	5.04195i
$829$	16.8328	0.584628	0.292314	−	0.956322i	$-0.405575\pi$
$829$	16.8328	0.584628	0.292314	+	0.956322i	$0.405575\pi$
$830$	0	0
$831$	51.4164	1.78362
$832$	16.0689i	0.557088i
$833$	− 1.23607i	− 0.0428272i
$834$	−75.7771	−2.62395
$835$	0	0
$836$	7.41641	0.256502
$837$	− 104.721i	− 3.61970i
$838$	− 14.8754i	− 0.513862i
$839$	14.0689	0.485712	0.242856	−	0.970062i	$-0.421916\pi$
$839$	14.0689	0.485712	0.242856	+	0.970062i	$0.421916\pi$
$840$	0	0
$841$	−28.7771	−0.992313
$842$	50.0000i	1.72311i
$843$	40.3607i	1.39010i
$844$	−65.6656	−2.26030
$845$	0	0
$846$	−120.902	−4.15669
$847$	1.00000i	0.0343604i
$848$	8.47214i	0.290934i
$849$	19.0557	0.653991
$850$	0	0
$851$	−3.05573	−0.104749
$852$	− 14.8328i	− 0.508164i
$853$	− 0.652476i	− 0.0223403i	−0.999938	−	0.0111702i	$-0.996444\pi$
$853$	− 0.652476i	− 0.0223403i	0.999938	−	0.0111702i	$-0.00355565\pi$
$854$	−6.18034	−0.211487
$855$	0	0
$856$	8.94427	0.305709
$857$	10.7639i	0.367689i	0.982955	+	0.183844i	$0.0588543\pi$
$857$	10.7639i	0.367689i	−0.982955	+	0.183844i	$0.941146\pi$
$858$	8.94427i	0.305352i
$859$	−40.5410	−1.38324	−0.691621	−	0.722261i	$-0.743103\pi$
$859$	−40.5410	−1.38324	−0.691621	+	0.722261i	$0.743103\pi$
$860$	0	0
$861$	−21.8885	−0.745960
$862$	− 26.8328i	− 0.913929i
$863$	20.9443i	0.712951i	0.934305	+	0.356476i	$0.116022\pi$
$863$	20.9443i	0.712951i	−0.934305	+	0.356476i	$0.883978\pi$
$864$	−97.0820	−3.30280
$865$	0	0
$866$	−1.05573	−0.0358751
$867$	− 50.0689i	− 1.70043i
$868$	21.7082i	0.736824i
$869$	8.94427	0.303414
$870$	0	0
$871$	−6.83282	−0.231521
$872$	− 10.0000i	− 0.338643i
$873$	70.3607i	2.38135i
$874$	35.7771	1.21018
$875$	0	0
$876$	−50.8328	−1.71748
$877$	41.4164i	1.39853i	0.714861	+	0.699266i	$0.246490\pi$
$877$	41.4164i	1.39853i	−0.714861	+	0.699266i	$0.753510\pi$
$878$	3.41641i	0.115298i
$879$	−48.9443	−1.65085
$880$	0	0
$881$	29.4164	0.991064	0.495532	−	0.868590i	$-0.334973\pi$
$881$	29.4164	0.991064	0.495532	+	0.868590i	$0.334973\pi$
$882$	− 16.7082i	− 0.562594i
$883$	− 8.94427i	− 0.300999i	−0.988610	−	0.150499i	$-0.951912\pi$
$883$	− 8.94427i	− 0.300999i	0.988610	−	0.150499i	$-0.0480881\pi$
$884$	4.58359	0.154163
$885$	0	0
$886$	15.7771	0.530042
$887$	40.3607i	1.35518i	0.735440	+	0.677589i	$0.236975\pi$
$887$	40.3607i	1.35518i	−0.735440	+	0.677589i	$0.763025\pi$
$888$	3.41641i	0.114647i
$889$	−20.9443	−0.702448
$890$	0	0
$891$	−24.4164	−0.817980
$892$	36.5410i	1.22348i
$893$	17.8885i	0.598617i
$894$	101.305	3.38814
$895$	0	0
$896$	15.6525	0.522913
$897$	25.8885i	0.864393i
$898$	− 43.6656i	− 1.45714i
$899$	−3.41641	−0.113944
$900$	0	0
$901$	10.4721	0.348877
$902$	− 15.1246i	− 0.503594i
$903$	− 25.8885i	− 0.861517i
$904$	4.47214	0.148741
$905$	0	0
$906$	64.7214	2.15022
$907$	13.5279i	0.449185i	0.974453	+	0.224593i	$0.0721052\pi$
$907$	13.5279i	0.449185i	−0.974453	+	0.224593i	$0.927895\pi$
$908$	89.6656i	2.97566i
$909$	69.0132	2.28902
$910$	0	0
$911$	33.5279	1.11083	0.555414	−	0.831574i	$-0.312560\pi$
$911$	33.5279	1.11083	0.555414	+	0.831574i	$0.312560\pi$
$912$	8.00000i	0.264906i
$913$	15.4164i	0.510209i
$914$	55.5279	1.83670
$915$	0	0
$916$	13.4164	0.443291
$917$	− 13.8885i	− 0.458640i
$918$	40.0000i	1.32020i
$919$	−6.11146	−0.201598	−0.100799	−	0.994907i	$-0.532140\pi$
$919$	−6.11146	−0.201598	−0.100799	+	0.994907i	$0.532140\pi$
$920$	0	0
$921$	28.9443	0.953746
$922$	− 22.7639i	− 0.749690i
$923$	− 1.88854i	− 0.0621622i
$924$	9.70820	0.319376
$925$	0	0
$926$	32.3607	1.06344
$927$	− 42.6525i	− 1.40089i
$928$	3.16718i	0.103968i
$929$	−28.2492	−0.926827	−0.463413	−	0.886142i	$-0.653376\pi$
$929$	−28.2492	−0.926827	−0.463413	+	0.886142i	$0.653376\pi$
$930$	0	0
$931$	−2.47214	−0.0810210
$932$	− 52.2492i	− 1.71148i
$933$	70.2492i	2.29986i
$934$	−76.1803	−2.49270
$935$	0	0
$936$	20.6525	0.675047
$937$	20.6525i	0.674687i	0.941382	+	0.337343i	$0.109528\pi$
$937$	20.6525i	0.674687i	−0.941382	+	0.337343i	$0.890472\pi$
$938$	12.3607i	0.403591i
$939$	9.52786	0.310930
$940$	0	0
$941$	−41.5967	−1.35602	−0.678008	−	0.735055i	$-0.737156\pi$
$941$	−41.5967	−1.35602	−0.678008	+	0.735055i	$0.737156\pi$
$942$	50.2492i	1.63721i
$943$	− 43.7771i	− 1.42558i
$944$	3.23607	0.105325
$945$	0	0
$946$	17.8885	0.581607
$947$	− 58.8328i	− 1.91181i	−0.293677	−	0.955905i	$-0.594879\pi$
$947$	− 58.8328i	− 1.91181i	0.293677	−	0.955905i	$-0.405121\pi$
$948$	− 86.8328i	− 2.82020i
$949$	−6.47214	−0.210094
$950$	0	0
$951$	45.3050	1.46911
$952$	− 2.76393i	− 0.0895796i
$953$	− 5.05573i	− 0.163771i	−0.996642	−	0.0818855i	$-0.973906\pi$
$953$	− 5.05573i	− 0.163771i	0.996642	−	0.0818855i	$-0.0260942\pi$
$954$	141.554	4.58299
$955$	0	0
$956$	77.6656	2.51189
$957$	1.52786i	0.0493888i
$958$	50.2492i	1.62348i
$959$	−7.52786	−0.243087
$960$	0	0
$961$	21.3607	0.689054
$962$	1.30495i	0.0420733i
$963$	29.8885i	0.963145i
$964$	−81.3738	−2.62087
$965$	0	0
$966$	46.8328	1.50682
$967$	21.8885i	0.703888i	0.936021	+	0.351944i	$0.114479\pi$
$967$	21.8885i	0.703888i	−0.936021	+	0.351944i	$0.885521\pi$
$968$	2.23607i	0.0718699i
$969$	9.88854	0.317666
$970$	0	0
$971$	−29.1246	−0.934653	−0.467327	−	0.884085i	$-0.654783\pi$
$971$	−29.1246	−0.934653	−0.467327	+	0.884085i	$0.654783\pi$
$972$	106.790i	3.42530i
$973$	10.4721i	0.335721i
$974$	18.6950	0.599028
$975$	0	0
$976$	2.76393	0.0884713
$977$	5.05573i	0.161747i	0.996724	+	0.0808735i	$0.0257710\pi$
$977$	5.05573i	0.161747i	−0.996724	+	0.0808735i	$0.974229\pi$
$978$	169.443i	5.41818i
$979$	2.00000	0.0639203
$980$	0	0
$981$	33.4164	1.06690
$982$	0	0
$983$	− 44.1803i	− 1.40913i	−0.709637	−	0.704567i	$-0.751141\pi$
$983$	− 44.1803i	− 1.40913i	0.709637	−	0.704567i	$-0.248859\pi$
$984$	−48.9443	−1.56029
$985$	0	0
$986$	1.30495	0.0415581
$987$	23.4164i	0.745352i
$988$	− 9.16718i	− 0.291647i
$989$	51.7771	1.64642
$990$	0	0
$991$	−26.2492	−0.833834	−0.416917	−	0.908945i	$-0.636889\pi$
$991$	−26.2492	−0.833834	−0.416917	+	0.908945i	$0.636889\pi$
$992$	− 48.5410i	− 1.54118i
$993$	− 70.8328i	− 2.24781i
$994$	−3.41641	−0.108362
$995$	0	0
$996$	149.666	4.74234
$997$	32.6525i	1.03411i	0.855951	+	0.517057i	$0.172973\pi$
$997$	32.6525i	1.03411i	−0.855951	+	0.517057i	$0.827027\pi$
$998$	23.4164i	0.741233i
$999$	−6.83282	−0.216181

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1925.2.b.h.1849.1		4
5.2	odd	4		1925.2.a.r.1.2		2
5.3	odd	4		77.2.a.d.1.1	✓	2
5.4	even	2	inner	1925.2.b.h.1849.4		4
15.8	even	4		693.2.a.h.1.2		2
20.3	even	4		1232.2.a.m.1.1		2
35.3	even	12		539.2.e.j.177.2		4
35.13	even	4		539.2.a.f.1.1		2
35.18	odd	12		539.2.e.i.177.2		4
35.23	odd	12		539.2.e.i.67.2		4
35.33	even	12		539.2.e.j.67.2		4
40.3	even	4		4928.2.a.bv.1.2		2
40.13	odd	4		4928.2.a.bm.1.1		2
55.3	odd	20		847.2.f.a.372.1		4
55.8	even	20		847.2.f.m.372.1		4
55.13	even	20		847.2.f.b.323.1		4
55.18	even	20		847.2.f.m.148.1		4
55.28	even	20		847.2.f.b.729.1		4
55.38	odd	20		847.2.f.n.729.1		4
55.43	even	4		847.2.a.f.1.2		2
55.48	odd	20		847.2.f.a.148.1		4
55.53	odd	20		847.2.f.n.323.1		4
105.83	odd	4		4851.2.a.y.1.2		2
140.83	odd	4		8624.2.a.ce.1.2		2
165.98	odd	4		7623.2.a.bl.1.1		2
385.153	odd	4		5929.2.a.m.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
77.2.a.d.1.1	✓	2	5.3	odd	4
539.2.a.f.1.1		2	35.13	even	4
539.2.e.i.67.2		4	35.23	odd	12
539.2.e.i.177.2		4	35.18	odd	12
539.2.e.j.67.2		4	35.33	even	12
539.2.e.j.177.2		4	35.3	even	12
693.2.a.h.1.2		2	15.8	even	4
847.2.a.f.1.2		2	55.43	even	4
847.2.f.a.148.1		4	55.48	odd	20
847.2.f.a.372.1		4	55.3	odd	20
847.2.f.b.323.1		4	55.13	even	20
847.2.f.b.729.1		4	55.28	even	20
847.2.f.m.148.1		4	55.18	even	20
847.2.f.m.372.1		4	55.8	even	20
847.2.f.n.323.1		4	55.53	odd	20
847.2.f.n.729.1		4	55.38	odd	20
1232.2.a.m.1.1		2	20.3	even	4
1925.2.a.r.1.2		2	5.2	odd	4
1925.2.b.h.1849.1		4	1.1	even	1	trivial
1925.2.b.h.1849.4		4	5.4	even	2	inner
4851.2.a.y.1.2		2	105.83	odd	4
4928.2.a.bm.1.1		2	40.13	odd	4
4928.2.a.bv.1.2		2	40.3	even	4
5929.2.a.m.1.2		2	385.153	odd	4
7623.2.a.bl.1.1		2	165.98	odd	4
8624.2.a.ce.1.2		2	140.83	odd	4

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:	\( N \)	\(=\)	\( 1925 = 5^{2} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1925.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.23607i q^{2} -3.23607i q^{3} -3.00000 q^{4} -7.23607 q^{6} +1.00000i q^{7} +2.23607i q^{8} -7.47214 q^{9} +O(q^{10})\)	\(q-2.23607i q^{2} -3.23607i q^{3} -3.00000 q^{4} -7.23607 q^{6} +1.00000i q^{7} +2.23607i q^{8} -7.47214 q^{9} -1.00000 q^{11} +9.70820i q^{12} +1.23607i q^{13} +2.23607 q^{14} -1.00000 q^{16} +1.23607i q^{17} +16.7082i q^{18} +2.47214 q^{19} +3.23607 q^{21} +2.23607i q^{22} +6.47214i q^{23} +7.23607 q^{24} +2.76393 q^{26} +14.4721i q^{27} -3.00000i q^{28} +0.472136 q^{29} -7.23607 q^{31} +6.70820i q^{32} +3.23607i q^{33} +2.76393 q^{34} +22.4164 q^{36} +0.472136i q^{37} -5.52786i q^{38} +4.00000 q^{39} -6.76393 q^{41} -7.23607i q^{42} -8.00000i q^{43} +3.00000 q^{44} +14.4721 q^{46} +7.23607i q^{47} +3.23607i q^{48} -1.00000 q^{49} +4.00000 q^{51} -3.70820i q^{52} -8.47214i q^{53} +32.3607 q^{54} -2.23607 q^{56} -8.00000i q^{57} -1.05573i q^{58} -3.23607 q^{59} -2.76393 q^{61} +16.1803i q^{62} -7.47214i q^{63} +13.0000 q^{64} +7.23607 q^{66} +5.52786i q^{67} -3.70820i q^{68} +20.9443 q^{69} -1.52786 q^{71} -16.7082i q^{72} +5.23607i q^{73} +1.05573 q^{74} -7.41641 q^{76} -1.00000i q^{77} -8.94427i q^{78} -8.94427 q^{79} +24.4164 q^{81} +15.1246i q^{82} -15.4164i q^{83} -9.70820 q^{84} -17.8885 q^{86} -1.52786i q^{87} -2.23607i q^{88} -2.00000 q^{89} -1.23607 q^{91} -19.4164i q^{92} +23.4164i q^{93} +16.1803 q^{94} +21.7082 q^{96} -9.41641i q^{97} +2.23607i q^{98} +7.47214 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 12 q^{4} - 20 q^{6} - 12 q^{9}+O(q^{10}) \) 4 * q - 12 * q^4 - 20 * q^6 - 12 * q^9	\( 4 q - 12 q^{4} - 20 q^{6} - 12 q^{9} - 4 q^{11} - 4 q^{16} - 8 q^{19} + 4 q^{21} + 20 q^{24} + 20 q^{26} - 16 q^{29} - 20 q^{31} + 20 q^{34} + 36 q^{36} + 16 q^{39} - 36 q^{41} + 12 q^{44} + 40 q^{46} - 4 q^{49} + 16 q^{51} + 40 q^{54} - 4 q^{59} - 20 q^{61} + 52 q^{64} + 20 q^{66} + 48 q^{69} - 24 q^{71} + 40 q^{74} + 24 q^{76} + 44 q^{81} - 12 q^{84} - 8 q^{89} + 4 q^{91} + 20 q^{94} + 60 q^{96} + 12 q^{99}+O(q^{100}) \) 4 * q - 12 * q^4 - 20 * q^6 - 12 * q^9 - 4 * q^11 - 4 * q^16 - 8 * q^19 + 4 * q^21 + 20 * q^24 + 20 * q^26 - 16 * q^29 - 20 * q^31 + 20 * q^34 + 36 * q^36 + 16 * q^39 - 36 * q^41 + 12 * q^44 + 40 * q^46 - 4 * q^49 + 16 * q^51 + 40 * q^54 - 4 * q^59 - 20 * q^61 + 52 * q^64 + 20 * q^66 + 48 * q^69 - 24 * q^71 + 40 * q^74 + 24 * q^76 + 44 * q^81 - 12 * q^84 - 8 * q^89 + 4 * q^91 + 20 * q^94 + 60 * q^96 + 12 * q^99

Introduction

L-functions

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Database

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