LMFDB - Embedded newform 3330.2.a.bl.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3330,2,Mod(1,3330)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3330, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("3330.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3330 = 2 \cdot 3^{2} \cdot 5 \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3330.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:	$26.5901838731$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.23544108.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 20x^{3} + 39x^{2} + 9x - 36 $ x^5 - x^4 - 20x^3 + 39x^2 + 9*x - 36
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$1.65043$ of defining polynomial
Character	$\chi$	$=$	3330.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +4.56812 q^{7} +1.00000 q^{8} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +4.56812 q^{7} +1.00000 q^{8} +1.00000 q^{10} -3.41708 q^{11} +0.971229 q^{13} +4.56812 q^{14} +1.00000 q^{16} -2.56812 q^{17} +4.86416 q^{19} +1.00000 q^{20} -3.41708 q^{22} +1.02877 q^{23} +1.00000 q^{25} +0.971229 q^{26} +4.56812 q^{28} -3.59689 q^{29} +2.55292 q^{31} +1.00000 q^{32} -2.56812 q^{34} +4.56812 q^{35} +1.00000 q^{37} +4.86416 q^{38} +1.00000 q^{40} +7.74670 q^{41} +9.74670 q^{43} -3.41708 q^{44} +1.02877 q^{46} +4.99877 q^{47} +13.8677 q^{49} +1.00000 q^{50} +0.971229 q^{52} -6.71793 q^{53} -3.41708 q^{55} +4.56812 q^{56} -3.59689 q^{58} +6.83416 q^{59} -6.99001 q^{61} +2.55292 q^{62} +1.00000 q^{64} +0.971229 q^{65} -15.9704 q^{67} -2.56812 q^{68} +4.56812 q^{70} -8.29962 q^{71} +1.56331 q^{73} +1.00000 q^{74} +4.86416 q^{76} -15.6096 q^{77} -6.29962 q^{79} +1.00000 q^{80} +7.74670 q^{82} +7.86293 q^{83} -2.56812 q^{85} +9.74670 q^{86} -3.41708 q^{88} -17.3013 q^{89} +4.43669 q^{91} +1.02877 q^{92} +4.99877 q^{94} +4.86416 q^{95} -0.747938 q^{97} +13.8677 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 5 q^{2} + 5 q^{4} + 5 q^{5} + 3 q^{7} + 5 q^{8}+O(q^{10}) $ 5 * q + 5 * q^2 + 5 * q^4 + 5 * q^5 + 3 * q^7 + 5 * q^8	$ 5 q + 5 q^{2} + 5 q^{4} + 5 q^{5} + 3 q^{7} + 5 q^{8} + 5 q^{10} + 5 q^{11} + 5 q^{13} + 3 q^{14} + 5 q^{16} + 7 q^{17} - q^{19} + 5 q^{20} + 5 q^{22} + 5 q^{23} + 5 q^{25} + 5 q^{26} + 3 q^{28} + 2 q^{29} + 16 q^{31} + 5 q^{32} + 7 q^{34} + 3 q^{35} + 5 q^{37} - q^{38} + 5 q^{40} + 2 q^{41} + 12 q^{43} + 5 q^{44} + 5 q^{46} + 6 q^{47} + 16 q^{49} + 5 q^{50} + 5 q^{52} + 3 q^{53} + 5 q^{55} + 3 q^{56} + 2 q^{58} - 10 q^{59} + 16 q^{61} + 16 q^{62} + 5 q^{64} + 5 q^{65} + 4 q^{67} + 7 q^{68} + 3 q^{70} - 8 q^{71} - 3 q^{73} + 5 q^{74} - q^{76} + 3 q^{77} + 2 q^{79} + 5 q^{80} + 2 q^{82} - 5 q^{83} + 7 q^{85} + 12 q^{86} + 5 q^{88} - 7 q^{89} + 33 q^{91} + 5 q^{92} + 6 q^{94} - q^{95} + 14 q^{97} + 16 q^{98}+O(q^{100}) $ 5 * q + 5 * q^2 + 5 * q^4 + 5 * q^5 + 3 * q^7 + 5 * q^8 + 5 * q^10 + 5 * q^11 + 5 * q^13 + 3 * q^14 + 5 * q^16 + 7 * q^17 - q^19 + 5 * q^20 + 5 * q^22 + 5 * q^23 + 5 * q^25 + 5 * q^26 + 3 * q^28 + 2 * q^29 + 16 * q^31 + 5 * q^32 + 7 * q^34 + 3 * q^35 + 5 * q^37 - q^38 + 5 * q^40 + 2 * q^41 + 12 * q^43 + 5 * q^44 + 5 * q^46 + 6 * q^47 + 16 * q^49 + 5 * q^50 + 5 * q^52 + 3 * q^53 + 5 * q^55 + 3 * q^56 + 2 * q^58 - 10 * q^59 + 16 * q^61 + 16 * q^62 + 5 * q^64 + 5 * q^65 + 4 * q^67 + 7 * q^68 + 3 * q^70 - 8 * q^71 - 3 * q^73 + 5 * q^74 - q^76 + 3 * q^77 + 2 * q^79 + 5 * q^80 + 2 * q^82 - 5 * q^83 + 7 * q^85 + 12 * q^86 + 5 * q^88 - 7 * q^89 + 33 * q^91 + 5 * q^92 + 6 * q^94 - q^95 + 14 * q^97 + 16 * 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
$2$	1.00000	0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	4.56812	1.72659	0.863294	−	0.504701i	$-0.168397\pi$
$7$	4.56812	1.72659	0.863294	+	0.504701i	$0.168397\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	1.00000	0.316228
$11$	−3.41708	−1.03029	−0.515144	−	0.857104i	$-0.672262\pi$
$11$	−3.41708	−1.03029	−0.515144	+	0.857104i	$0.672262\pi$
$12$	0	0
$13$	0.971229	0.269370	0.134685	−	0.990888i	$-0.456998\pi$
$13$	0.971229	0.269370	0.134685	+	0.990888i	$0.456998\pi$
$14$	4.56812	1.22088
$15$	0	0
$16$	1.00000	0.250000
$17$	−2.56812	−0.622861	−0.311431	−	0.950269i	$-0.600808\pi$
$17$	−2.56812	−0.622861	−0.311431	+	0.950269i	$0.600808\pi$
$18$	0	0
$19$	4.86416	1.11592	0.557958	−	0.829869i	$-0.311585\pi$
$19$	4.86416	1.11592	0.557958	+	0.829869i	$0.311585\pi$
$20$	1.00000	0.223607
$21$	0	0
$22$	−3.41708	−0.728524
$23$	1.02877	0.214514	0.107257	−	0.994231i	$-0.465793\pi$
$23$	1.02877	0.214514	0.107257	+	0.994231i	$0.465793\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0.971229	0.190474
$27$	0	0
$28$	4.56812	0.863294
$29$	−3.59689	−0.667926	−0.333963	−	0.942586i	$-0.608386\pi$
$29$	−3.59689	−0.667926	−0.333963	+	0.942586i	$0.608386\pi$
$30$	0	0
$31$	2.55292	0.458517	0.229259	−	0.973366i	$-0.426370\pi$
$31$	2.55292	0.458517	0.229259	+	0.973366i	$0.426370\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	−2.56812	−0.440429
$35$	4.56812	0.772154
$36$	0	0
$37$	1.00000	0.164399
$37$	1.00000	0.164399
$38$	4.86416	0.789071
$39$	0	0
$40$	1.00000	0.158114
$41$	7.74670	1.20983	0.604916	−	0.796289i	$-0.293207\pi$
$41$	7.74670	1.20983	0.604916	+	0.796289i	$0.293207\pi$
$42$	0	0
$43$	9.74670	1.48636	0.743179	−	0.669093i	$-0.233317\pi$
$43$	9.74670	1.48636	0.743179	+	0.669093i	$0.233317\pi$
$44$	−3.41708	−0.515144
$45$	0	0
$46$	1.02877	0.151684
$47$	4.99877	0.729145	0.364572	−	0.931175i	$-0.381215\pi$
$47$	4.99877	0.729145	0.364572	+	0.931175i	$0.381215\pi$
$48$	0	0
$49$	13.8677	1.98111
$50$	1.00000	0.141421
$51$	0	0
$52$	0.971229	0.134685
$53$	−6.71793	−0.922779	−0.461389	−	0.887198i	$-0.652649\pi$
$53$	−6.71793	−0.922779	−0.461389	+	0.887198i	$0.652649\pi$
$54$	0	0
$55$	−3.41708	−0.460759
$56$	4.56812	0.610441
$57$	0	0
$58$	−3.59689	−0.472295
$59$	6.83416	0.889731	0.444866	−	0.895597i	$-0.353251\pi$
$59$	6.83416	0.889731	0.444866	+	0.895597i	$0.353251\pi$
$60$	0	0
$61$	−6.99001	−0.894980	−0.447490	−	0.894289i	$-0.647682\pi$
$61$	−6.99001	−0.894980	−0.447490	+	0.894289i	$0.647682\pi$
$62$	2.55292	0.324221
$63$	0	0
$64$	1.00000	0.125000
$65$	0.971229	0.120466
$66$	0	0
$67$	−15.9704	−1.95109	−0.975547	−	0.219789i	$-0.929463\pi$
$67$	−15.9704	−1.95109	−0.975547	+	0.219789i	$0.929463\pi$
$68$	−2.56812	−0.311431
$69$	0	0
$70$	4.56812	0.545995
$71$	−8.29962	−0.984984	−0.492492	−	0.870317i	$-0.663914\pi$
$71$	−8.29962	−0.984984	−0.492492	+	0.870317i	$0.663914\pi$
$72$	0	0
$73$	1.56331	0.182972	0.0914858	−	0.995806i	$-0.470838\pi$
$73$	1.56331	0.182972	0.0914858	+	0.995806i	$0.470838\pi$
$74$	1.00000	0.116248
$75$	0	0
$76$	4.86416	0.557958
$77$	−15.6096	−1.77888
$78$	0	0
$79$	−6.29962	−0.708762	−0.354381	−	0.935101i	$-0.615308\pi$
$79$	−6.29962	−0.708762	−0.354381	+	0.935101i	$0.615308\pi$
$80$	1.00000	0.111803
$81$	0	0
$82$	7.74670	0.855480
$83$	7.86293	0.863068	0.431534	−	0.902097i	$-0.357972\pi$
$83$	7.86293	0.863068	0.431534	+	0.902097i	$0.357972\pi$
$84$	0	0
$85$	−2.56812	−0.278552
$86$	9.74670	1.05101
$87$	0	0
$88$	−3.41708	−0.364262
$89$	−17.3013	−1.83393	−0.916965	−	0.398968i	$-0.869369\pi$
$89$	−17.3013	−1.83393	−0.916965	+	0.398968i	$0.869369\pi$
$90$	0	0
$91$	4.43669	0.465092
$92$	1.02877	0.107257
$93$	0	0
$94$	4.99877	0.515583
$95$	4.86416	0.499053
$96$	0	0
$97$	−0.747938	−0.0759416	−0.0379708	−	0.999279i	$-0.512089\pi$
$97$	−0.747938	−0.0759416	−0.0379708	+	0.999279i	$0.512089\pi$
$98$	13.8677	1.40085
$99$	0	0
$100$	1.00000	0.100000
$101$	18.4346	1.83431	0.917157	−	0.398526i	$-0.130478\pi$
$101$	18.4346	1.83431	0.917157	+	0.398526i	$0.130478\pi$
$102$	0	0
$103$	−2.99877	−0.295477	−0.147739	−	0.989026i	$-0.547199\pi$
$103$	−2.99877	−0.295477	−0.147739	+	0.989026i	$0.547199\pi$
$104$	0.971229	0.0952368
$105$	0	0
$106$	−6.71793	−0.652503
$107$	8.22256	0.794905	0.397452	−	0.917623i	$-0.369894\pi$
$107$	8.22256	0.794905	0.397452	+	0.917623i	$0.369894\pi$
$108$	0	0
$109$	13.7167	1.31382	0.656911	−	0.753968i	$-0.271863\pi$
$109$	13.7167	1.31382	0.656911	+	0.753968i	$0.271863\pi$
$110$	−3.41708	−0.325806
$111$	0	0
$112$	4.56812	0.431647
$113$	−3.23726	−0.304536	−0.152268	−	0.988339i	$-0.548658\pi$
$113$	−3.23726	−0.304536	−0.152268	+	0.988339i	$0.548658\pi$
$114$	0	0
$115$	1.02877	0.0959334
$116$	−3.59689	−0.333963
$117$	0	0
$118$	6.83416	0.629135
$119$	−11.7315	−1.07542
$120$	0	0
$121$	0.676423	0.0614930
$122$	−6.99001	−0.632846
$123$	0	0
$124$	2.55292	0.229259
$125$	1.00000	0.0894427
$126$	0	0
$127$	−9.21330	−0.817549	−0.408774	−	0.912636i	$-0.634044\pi$
$127$	−9.21330	−0.817549	−0.408774	+	0.912636i	$0.634044\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	0.971229	0.0851824
$131$	3.43833	0.300409	0.150204	−	0.988655i	$-0.452007\pi$
$131$	3.43833	0.300409	0.150204	+	0.988655i	$0.452007\pi$
$132$	0	0
$133$	22.2201	1.92673
$134$	−15.9704	−1.37963
$135$	0	0
$136$	−2.56812	−0.220215
$137$	−3.30085	−0.282011	−0.141005	−	0.990009i	$-0.545034\pi$
$137$	−3.30085	−0.282011	−0.141005	+	0.990009i	$0.545034\pi$
$138$	0	0
$139$	−3.06236	−0.259746	−0.129873	−	0.991531i	$-0.541457\pi$
$139$	−3.06236	−0.259746	−0.129873	+	0.991531i	$0.541457\pi$
$140$	4.56812	0.386077
$141$	0	0
$142$	−8.29962	−0.696489
$143$	−3.31876	−0.277529
$144$	0	0
$145$	−3.59689	−0.298706
$146$	1.56331	0.129380
$147$	0	0
$148$	1.00000	0.0821995
$149$	9.54293	0.781787	0.390894	−	0.920436i	$-0.372166\pi$
$149$	9.54293	0.781787	0.390894	+	0.920436i	$0.372166\pi$
$150$	0	0
$151$	8.07706	0.657302	0.328651	−	0.944451i	$-0.393406\pi$
$151$	8.07706	0.657302	0.328651	+	0.944451i	$0.393406\pi$
$152$	4.86416	0.394536
$153$	0	0
$154$	−15.6096	−1.25786
$155$	2.55292	0.205055
$156$	0	0
$157$	−21.6248	−1.72585	−0.862925	−	0.505332i	$-0.831370\pi$
$157$	−21.6248	−1.72585	−0.862925	+	0.505332i	$0.831370\pi$
$158$	−6.29962	−0.501171
$159$	0	0
$160$	1.00000	0.0790569
$161$	4.69955	0.370377
$162$	0	0
$163$	8.12585	0.636466	0.318233	−	0.948013i	$-0.396911\pi$
$163$	8.12585	0.636466	0.318233	+	0.948013i	$0.396911\pi$
$164$	7.74670	0.604916
$165$	0	0
$166$	7.86293	0.610282
$167$	−1.02877	−0.0796087	−0.0398044	−	0.999207i	$-0.512673\pi$
$167$	−1.02877	−0.0796087	−0.0398044	+	0.999207i	$0.512673\pi$
$168$	0	0
$169$	−12.0567	−0.927440
$170$	−2.56812	−0.196966
$171$	0	0
$172$	9.74670	0.743179
$173$	−7.66964	−0.583112	−0.291556	−	0.956554i	$-0.594173\pi$
$173$	−7.66964	−0.583112	−0.291556	+	0.956554i	$0.594173\pi$
$174$	0	0
$175$	4.56812	0.345318
$176$	−3.41708	−0.257572
$177$	0	0
$178$	−17.3013	−1.29678
$179$	−1.64284	−0.122792	−0.0613958	−	0.998114i	$-0.519555\pi$
$179$	−1.64284	−0.122792	−0.0613958	+	0.998114i	$0.519555\pi$
$180$	0	0
$181$	25.9704	1.93036	0.965182	−	0.261578i	$-0.0842429\pi$
$181$	25.9704	1.93036	0.965182	+	0.261578i	$0.0842429\pi$
$182$	4.43669	0.328869
$183$	0	0
$184$	1.02877	0.0758420
$185$	1.00000	0.0735215
$186$	0	0
$187$	8.77547	0.641726
$188$	4.99877	0.364572
$189$	0	0
$190$	4.86416	0.352883
$191$	−18.9253	−1.36939	−0.684693	−	0.728832i	$-0.740064\pi$
$191$	−18.9253	−1.36939	−0.684693	+	0.728832i	$0.740064\pi$
$192$	0	0
$193$	−2.63913	−0.189969	−0.0949845	−	0.995479i	$-0.530280\pi$
$193$	−2.63913	−0.189969	−0.0949845	+	0.995479i	$0.530280\pi$
$194$	−0.747938	−0.0536988
$195$	0	0
$196$	13.8677	0.990553
$197$	−10.9416	−0.779559	−0.389779	−	0.920908i	$-0.627449\pi$
$197$	−10.9416	−0.779559	−0.389779	+	0.920908i	$0.627449\pi$
$198$	0	0
$199$	−1.75545	−0.124441	−0.0622204	−	0.998062i	$-0.519818\pi$
$199$	−1.75545	−0.124441	−0.0622204	+	0.998062i	$0.519818\pi$
$200$	1.00000	0.0707107
$201$	0	0
$202$	18.4346	1.29706
$203$	−16.4310	−1.15323
$204$	0	0
$205$	7.74670	0.541053
$206$	−2.99877	−0.208934
$207$	0	0
$208$	0.971229	0.0673426
$209$	−16.6212	−1.14971
$210$	0	0
$211$	11.3252	0.779660	0.389830	−	0.920887i	$-0.372534\pi$
$211$	11.3252	0.779660	0.389830	+	0.920887i	$0.372534\pi$
$212$	−6.71793	−0.461389
$213$	0	0
$214$	8.22256	0.562083
$215$	9.74670	0.664720
$216$	0	0
$217$	11.6620	0.791670
$218$	13.7167	0.929012
$219$	0	0
$220$	−3.41708	−0.230379
$221$	−2.49423	−0.167780
$222$	0	0
$223$	−15.0328	−1.00667	−0.503334	−	0.864092i	$-0.667893\pi$
$223$	−15.0328	−1.00667	−0.503334	+	0.864092i	$0.667893\pi$
$224$	4.56812	0.305221
$225$	0	0
$226$	−3.23726	−0.215340
$227$	−14.3735	−0.954003	−0.477002	−	0.878902i	$-0.658276\pi$
$227$	−14.3735	−0.954003	−0.477002	+	0.878902i	$0.658276\pi$
$228$	0	0
$229$	13.2242	0.873880	0.436940	−	0.899491i	$-0.356062\pi$
$229$	13.2242	0.873880	0.436940	+	0.899491i	$0.356062\pi$
$230$	1.02877	0.0678352
$231$	0	0
$232$	−3.59689	−0.236148
$233$	0.222152	0.0145536	0.00727682	−	0.999974i	$-0.497684\pi$
$233$	0.222152	0.0145536	0.00727682	+	0.999974i	$0.497684\pi$
$234$	0	0
$235$	4.99877	0.326083
$236$	6.83416	0.444866
$237$	0	0
$238$	−11.7315	−0.760440
$239$	2.14106	0.138493	0.0692467	−	0.997600i	$-0.477940\pi$
$239$	2.14106	0.138493	0.0692467	+	0.997600i	$0.477940\pi$
$240$	0	0
$241$	28.3876	1.82860	0.914302	−	0.405033i	$-0.132740\pi$
$241$	28.3876	1.82860	0.914302	+	0.405033i	$0.132740\pi$
$242$	0.676423	0.0434821
$243$	0	0
$244$	−6.99001	−0.447490
$245$	13.8677	0.885977
$246$	0	0
$247$	4.72421	0.300595
$248$	2.55292	0.162110
$249$	0	0
$250$	1.00000	0.0632456
$251$	22.3276	1.40930	0.704652	−	0.709553i	$-0.251103\pi$
$251$	22.3276	1.40930	0.704652	+	0.709553i	$0.251103\pi$
$252$	0	0
$253$	−3.51539	−0.221011
$254$	−9.21330	−0.578094
$255$	0	0
$256$	1.00000	0.0625000
$257$	26.9392	1.68042	0.840209	−	0.542262i	$-0.182432\pi$
$257$	26.9392	1.68042	0.840209	+	0.542262i	$0.182432\pi$
$258$	0	0
$259$	4.56812	0.283849
$260$	0.971229	0.0602330
$261$	0	0
$262$	3.43833	0.212421
$263$	−16.2134	−0.999761	−0.499881	−	0.866094i	$-0.666623\pi$
$263$	−16.2134	−0.999761	−0.499881	+	0.866094i	$0.666623\pi$
$264$	0	0
$265$	−6.71793	−0.412679
$266$	22.2201	1.36240
$267$	0	0
$268$	−15.9704	−0.975547
$269$	−19.0487	−1.16142	−0.580710	−	0.814111i	$-0.697225\pi$
$269$	−19.0487	−1.16142	−0.580710	+	0.814111i	$0.697225\pi$
$270$	0	0
$271$	−1.99753	−0.121341	−0.0606707	−	0.998158i	$-0.519324\pi$
$271$	−1.99753	−0.121341	−0.0606707	+	0.998158i	$0.519324\pi$
$272$	−2.56812	−0.155715
$273$	0	0
$274$	−3.30085	−0.199412
$275$	−3.41708	−0.206058
$276$	0	0
$277$	−2.46710	−0.148234	−0.0741170	−	0.997250i	$-0.523614\pi$
$277$	−2.46710	−0.148234	−0.0741170	+	0.997250i	$0.523614\pi$
$278$	−3.06236	−0.183668
$279$	0	0
$280$	4.56812	0.272998
$281$	−15.3284	−0.914415	−0.457208	−	0.889360i	$-0.651150\pi$
$281$	−15.3284	−0.914415	−0.457208	+	0.889360i	$0.651150\pi$
$282$	0	0
$283$	−4.75710	−0.282780	−0.141390	−	0.989954i	$-0.545157\pi$
$283$	−4.75710	−0.282780	−0.141390	+	0.989954i	$0.545157\pi$
$284$	−8.29962	−0.492492
$285$	0	0
$286$	−3.31876	−0.196243
$287$	35.3879	2.08888
$288$	0	0
$289$	−10.4047	−0.612044
$290$	−3.59689	−0.211217
$291$	0	0
$292$	1.56331	0.0914858
$293$	2.17377	0.126993	0.0634964	−	0.997982i	$-0.479775\pi$
$293$	2.17377	0.126993	0.0634964	+	0.997982i	$0.479775\pi$
$294$	0	0
$295$	6.83416	0.397900
$296$	1.00000	0.0581238
$297$	0	0
$298$	9.54293	0.552807
$299$	0.999172	0.0577836
$300$	0	0
$301$	44.5241	2.56633
$302$	8.07706	0.464783
$303$	0	0
$304$	4.86416	0.278979
$305$	−6.99001	−0.400247
$306$	0	0
$307$	2.83416	0.161754	0.0808769	−	0.996724i	$-0.474228\pi$
$307$	2.83416	0.161754	0.0808769	+	0.996724i	$0.474228\pi$
$308$	−15.6096	−0.889441
$309$	0	0
$310$	2.55292	0.144996
$311$	−30.4039	−1.72405	−0.862024	−	0.506867i	$-0.830803\pi$
$311$	−30.4039	−1.72405	−0.862024	+	0.506867i	$0.830803\pi$
$312$	0	0
$313$	−27.2992	−1.54304	−0.771521	−	0.636204i	$-0.780504\pi$
$313$	−27.2992	−1.54304	−0.771521	+	0.636204i	$0.780504\pi$
$314$	−21.6248	−1.22036
$315$	0	0
$316$	−6.29962	−0.354381
$317$	29.4271	1.65279	0.826396	−	0.563090i	$-0.190388\pi$
$317$	29.4271	1.65279	0.826396	+	0.563090i	$0.190388\pi$
$318$	0	0
$319$	12.2909	0.688156
$320$	1.00000	0.0559017
$321$	0	0
$322$	4.69955	0.261896
$323$	−12.4918	−0.695060
$324$	0	0
$325$	0.971229	0.0538741
$326$	8.12585	0.450049
$327$	0	0
$328$	7.74670	0.427740
$329$	22.8350	1.25893
$330$	0	0
$331$	−21.9305	−1.20541	−0.602705	−	0.797964i	$-0.705910\pi$
$331$	−21.9305	−1.20541	−0.602705	+	0.797964i	$0.705910\pi$
$332$	7.86293	0.431534
$333$	0	0
$334$	−1.02877	−0.0562919
$335$	−15.9704	−0.872556
$336$	0	0
$337$	29.0271	1.58121	0.790604	−	0.612328i	$-0.209767\pi$
$337$	29.0271	1.58121	0.790604	+	0.612328i	$0.209767\pi$
$338$	−12.0567	−0.655799
$339$	0	0
$340$	−2.56812	−0.139276
$341$	−8.72351	−0.472405
$342$	0	0
$343$	31.3727	1.69397
$344$	9.74670	0.525507
$345$	0	0
$346$	−7.66964	−0.412323
$347$	−12.3300	−0.661911	−0.330955	−	0.943646i	$-0.607371\pi$
$347$	−12.3300	−0.661911	−0.330955	+	0.943646i	$0.607371\pi$
$348$	0	0
$349$	1.82262	0.0975628	0.0487814	−	0.998809i	$-0.484466\pi$
$349$	1.82262	0.0975628	0.0487814	+	0.998809i	$0.484466\pi$
$350$	4.56812	0.244176
$351$	0	0
$352$	−3.41708	−0.182131
$353$	−8.04348	−0.428111	−0.214055	−	0.976822i	$-0.568667\pi$
$353$	−8.04348	−0.428111	−0.214055	+	0.976822i	$0.568667\pi$
$354$	0	0
$355$	−8.29962	−0.440498
$356$	−17.3013	−0.916965
$357$	0	0
$358$	−1.64284	−0.0868267
$359$	−5.52547	−0.291623	−0.145812	−	0.989312i	$-0.546579\pi$
$359$	−5.52547	−0.291623	−0.145812	+	0.989312i	$0.546579\pi$
$360$	0	0
$361$	4.66008	0.245267
$362$	25.9704	1.36497
$363$	0	0
$364$	4.43669	0.232546
$365$	1.56331	0.0818273
$366$	0	0
$367$	−10.4115	−0.543478	−0.271739	−	0.962371i	$-0.587599\pi$
$367$	−10.4115	−0.543478	−0.271739	+	0.962371i	$0.587599\pi$
$368$	1.02877	0.0536284
$369$	0	0
$370$	1.00000	0.0519875
$371$	−30.6883	−1.59326
$372$	0	0
$373$	20.2125	1.04656	0.523281	−	0.852160i	$-0.324708\pi$
$373$	20.2125	1.04656	0.523281	+	0.852160i	$0.324708\pi$
$374$	8.77547	0.453769
$375$	0	0
$376$	4.99877	0.257792
$377$	−3.49341	−0.179920
$378$	0	0
$379$	−15.5557	−0.799042	−0.399521	−	0.916724i	$-0.630824\pi$
$379$	−15.5557	−0.799042	−0.399521	+	0.916724i	$0.630824\pi$
$380$	4.86416	0.249526
$381$	0	0
$382$	−18.9253	−0.968302
$383$	29.5137	1.50808	0.754041	−	0.656828i	$-0.228102\pi$
$383$	29.5137	1.50808	0.754041	+	0.656828i	$0.228102\pi$
$384$	0	0
$385$	−15.6096	−0.795540
$386$	−2.63913	−0.134328
$387$	0	0
$388$	−0.747938	−0.0379708
$389$	−14.7235	−0.746512	−0.373256	−	0.927728i	$-0.621759\pi$
$389$	−14.7235	−0.746512	−0.373256	+	0.927728i	$0.621759\pi$
$390$	0	0
$391$	−2.64201	−0.133612
$392$	13.8677	0.700427
$393$	0	0
$394$	−10.9416	−0.551231
$395$	−6.29962	−0.316968
$396$	0	0
$397$	−17.7859	−0.892647	−0.446324	−	0.894872i	$-0.647267\pi$
$397$	−17.7859	−0.892647	−0.446324	+	0.894872i	$0.647267\pi$
$398$	−1.75545	−0.0879930
$399$	0	0
$400$	1.00000	0.0500000
$401$	−33.4858	−1.67220	−0.836100	−	0.548577i	$-0.815170\pi$
$401$	−33.4858	−1.67220	−0.836100	+	0.548577i	$0.815170\pi$
$402$	0	0
$403$	2.47946	0.123511
$404$	18.4346	0.917157
$405$	0	0
$406$	−16.4310	−0.815459
$407$	−3.41708	−0.169378
$408$	0	0
$409$	−30.2700	−1.49676	−0.748378	−	0.663272i	$-0.769167\pi$
$409$	−30.2700	−1.49676	−0.748378	+	0.663272i	$0.769167\pi$
$410$	7.74670	0.382582
$411$	0	0
$412$	−2.99877	−0.147739
$413$	31.2193	1.53620
$414$	0	0
$415$	7.86293	0.385976
$416$	0.971229	0.0476184
$417$	0	0
$418$	−16.6212	−0.812971
$419$	−34.5421	−1.68749	−0.843746	−	0.536743i	$-0.819655\pi$
$419$	−34.5421	−1.68749	−0.843746	+	0.536743i	$0.819655\pi$
$420$	0	0
$421$	10.6991	0.521445	0.260722	−	0.965414i	$-0.416039\pi$
$421$	10.6991	0.521445	0.260722	+	0.965414i	$0.416039\pi$
$422$	11.3252	0.551303
$423$	0	0
$424$	−6.71793	−0.326252
$425$	−2.56812	−0.124572
$426$	0	0
$427$	−31.9312	−1.54526
$428$	8.22256	0.397452
$429$	0	0
$430$	9.74670	0.470028
$431$	5.74112	0.276540	0.138270	−	0.990395i	$-0.455846\pi$
$431$	5.74112	0.276540	0.138270	+	0.990395i	$0.455846\pi$
$432$	0	0
$433$	−16.1650	−0.776841	−0.388421	−	0.921482i	$-0.626979\pi$
$433$	−16.1650	−0.776841	−0.388421	+	0.921482i	$0.626979\pi$
$434$	11.6620	0.559795
$435$	0	0
$436$	13.7167	0.656911
$437$	5.00411	0.239379
$438$	0	0
$439$	33.3279	1.59065	0.795326	−	0.606182i	$-0.207300\pi$
$439$	33.3279	1.59065	0.795326	+	0.606182i	$0.207300\pi$
$440$	−3.41708	−0.162903
$441$	0	0
$442$	−2.49423	−0.118639
$443$	11.7930	0.560303	0.280152	−	0.959956i	$-0.409615\pi$
$443$	11.7930	0.560303	0.280152	+	0.959956i	$0.409615\pi$
$444$	0	0
$445$	−17.3013	−0.820158
$446$	−15.0328	−0.711822
$447$	0	0
$448$	4.56812	0.215823
$449$	−14.8917	−0.702783	−0.351391	−	0.936229i	$-0.614291\pi$
$449$	−14.8917	−0.702783	−0.351391	+	0.936229i	$0.614291\pi$
$450$	0	0
$451$	−26.4711	−1.24647
$452$	−3.23726	−0.152268
$453$	0	0
$454$	−14.3735	−0.674582
$455$	4.43669	0.207995
$456$	0	0
$457$	−3.03793	−0.142108	−0.0710542	−	0.997472i	$-0.522636\pi$
$457$	−3.03793	−0.142108	−0.0710542	+	0.997472i	$0.522636\pi$
$458$	13.2242	0.617926
$459$	0	0
$460$	1.02877	0.0479667
$461$	−17.5465	−0.817222	−0.408611	−	0.912709i	$-0.633987\pi$
$461$	−17.5465	−0.817222	−0.408611	+	0.912709i	$0.633987\pi$
$462$	0	0
$463$	14.5522	0.676297	0.338149	−	0.941093i	$-0.390199\pi$
$463$	14.5522	0.676297	0.338149	+	0.941093i	$0.390199\pi$
$464$	−3.59689	−0.166982
$465$	0	0
$466$	0.222152	0.0102910
$467$	16.9681	0.785188	0.392594	−	0.919712i	$-0.371578\pi$
$467$	16.9681	0.785188	0.392594	+	0.919712i	$0.371578\pi$
$468$	0	0
$469$	−72.9547	−3.36874
$470$	4.99877	0.230576
$471$	0	0
$472$	6.83416	0.314568
$473$	−33.3052	−1.53138
$474$	0	0
$475$	4.86416	0.223183
$476$	−11.7315	−0.537712
$477$	0	0
$478$	2.14106	0.0979297
$479$	−26.7267	−1.22117	−0.610587	−	0.791949i	$-0.709066\pi$
$479$	−26.7267	−1.22117	−0.610587	+	0.791949i	$0.709066\pi$
$480$	0	0
$481$	0.971229	0.0442842
$482$	28.3876	1.29302
$483$	0	0
$484$	0.676423	0.0307465
$485$	−0.747938	−0.0339621
$486$	0	0
$487$	−32.1054	−1.45484	−0.727418	−	0.686195i	$-0.759280\pi$
$487$	−32.1054	−1.45484	−0.727418	+	0.686195i	$0.759280\pi$
$488$	−6.99001	−0.316423
$489$	0	0
$490$	13.8677	0.626481
$491$	−38.0588	−1.71757	−0.858784	−	0.512338i	$-0.828780\pi$
$491$	−38.0588	−1.71757	−0.858784	+	0.512338i	$0.828780\pi$
$492$	0	0
$493$	9.23726	0.416025
$494$	4.72421	0.212552
$495$	0	0
$496$	2.55292	0.114629
$497$	−37.9137	−1.70066
$498$	0	0
$499$	−32.5421	−1.45678	−0.728392	−	0.685160i	$-0.759732\pi$
$499$	−32.5421	−1.45678	−0.728392	+	0.685160i	$0.759732\pi$
$500$	1.00000	0.0447214
$501$	0	0
$502$	22.3276	0.996528
$503$	16.1549	0.720313	0.360156	−	0.932892i	$-0.382723\pi$
$503$	16.1549	0.720313	0.360156	+	0.932892i	$0.382723\pi$
$504$	0	0
$505$	18.4346	0.820330
$506$	−3.51539	−0.156278
$507$	0	0
$508$	−9.21330	−0.408774
$509$	−12.5021	−0.554144	−0.277072	−	0.960849i	$-0.589364\pi$
$509$	−12.5021	−0.554144	−0.277072	+	0.960849i	$0.589364\pi$
$510$	0	0
$511$	7.14139	0.315916
$512$	1.00000	0.0441942
$513$	0	0
$514$	26.9392	1.18824
$515$	−2.99877	−0.132141
$516$	0	0
$517$	−17.0812	−0.751229
$518$	4.56812	0.200712
$519$	0	0
$520$	0.971229	0.0425912
$521$	1.29912	0.0569153	0.0284576	−	0.999595i	$-0.490940\pi$
$521$	1.29912	0.0569153	0.0284576	+	0.999595i	$0.490940\pi$
$522$	0	0
$523$	26.3995	1.15437	0.577184	−	0.816614i	$-0.304151\pi$
$523$	26.3995	1.15437	0.577184	+	0.816614i	$0.304151\pi$
$524$	3.43833	0.150204
$525$	0	0
$526$	−16.2134	−0.706938
$527$	−6.55620	−0.285593
$528$	0	0
$529$	−21.9416	−0.953984
$530$	−6.71793	−0.291808
$531$	0	0
$532$	22.2201	0.963363
$533$	7.52382	0.325893
$534$	0	0
$535$	8.22256	0.355492
$536$	−15.9704	−0.689816
$537$	0	0
$538$	−19.0487	−0.821248
$539$	−47.3872	−2.04111
$540$	0	0
$541$	10.3600	0.445413	0.222706	−	0.974886i	$-0.428511\pi$
$541$	10.3600	0.445413	0.222706	+	0.974886i	$0.428511\pi$
$542$	−1.99753	−0.0858013
$543$	0	0
$544$	−2.56812	−0.110107
$545$	13.7167	0.587559
$546$	0	0
$547$	31.4051	1.34279	0.671393	−	0.741101i	$-0.265696\pi$
$547$	31.4051	1.34279	0.671393	+	0.741101i	$0.265696\pi$
$548$	−3.30085	−0.141005
$549$	0	0
$550$	−3.41708	−0.145705
$551$	−17.4959	−0.745349
$552$	0	0
$553$	−28.7774	−1.22374
$554$	−2.46710	−0.104817
$555$	0	0
$556$	−3.06236	−0.129873
$557$	40.6872	1.72397	0.861986	−	0.506932i	$-0.169221\pi$
$557$	40.6872	1.72397	0.861986	+	0.506932i	$0.169221\pi$
$558$	0	0
$559$	9.46628	0.400381
$560$	4.56812	0.193038
$561$	0	0
$562$	−15.3284	−0.646589
$563$	−22.9777	−0.968394	−0.484197	−	0.874959i	$-0.660888\pi$
$563$	−22.9777	−0.968394	−0.484197	+	0.874959i	$0.660888\pi$
$564$	0	0
$565$	−3.23726	−0.136193
$566$	−4.75710	−0.199956
$567$	0	0
$568$	−8.29962	−0.348244
$569$	19.5129	0.818024	0.409012	−	0.912529i	$-0.365873\pi$
$569$	19.5129	0.818024	0.409012	+	0.912529i	$0.365873\pi$
$570$	0	0
$571$	−44.0843	−1.84487	−0.922436	−	0.386150i	$-0.873805\pi$
$571$	−44.0843	−1.84487	−0.922436	+	0.386150i	$0.873805\pi$
$572$	−3.31876	−0.138764
$573$	0	0
$574$	35.3879	1.47706
$575$	1.02877	0.0429027
$576$	0	0
$577$	1.03165	0.0429481	0.0214740	−	0.999769i	$-0.493164\pi$
$577$	1.03165	0.0429481	0.0214740	+	0.999769i	$0.493164\pi$
$578$	−10.4047	−0.432780
$579$	0	0
$580$	−3.59689	−0.149353
$581$	35.9188	1.49016
$582$	0	0
$583$	22.9557	0.950728
$584$	1.56331	0.0646902
$585$	0	0
$586$	2.17377	0.0897975
$587$	21.7214	0.896539	0.448269	−	0.893899i	$-0.352041\pi$
$587$	21.7214	0.896539	0.448269	+	0.893899i	$0.352041\pi$
$588$	0	0
$589$	12.4178	0.511666
$590$	6.83416	0.281358
$591$	0	0
$592$	1.00000	0.0410997
$593$	−9.95763	−0.408911	−0.204455	−	0.978876i	$-0.565542\pi$
$593$	−9.95763	−0.408911	−0.204455	+	0.978876i	$0.565542\pi$
$594$	0	0
$595$	−11.7315	−0.480944
$596$	9.54293	0.390894
$597$	0	0
$598$	0.999172	0.0408592
$599$	11.9704	0.489097	0.244549	−	0.969637i	$-0.421360\pi$
$599$	11.9704	0.489097	0.244549	+	0.969637i	$0.421360\pi$
$600$	0	0
$601$	15.3447	0.625925	0.312962	−	0.949766i	$-0.398679\pi$
$601$	15.3447	0.625925	0.312962	+	0.949766i	$0.398679\pi$
$602$	44.5241	1.81467
$603$	0	0
$604$	8.07706	0.328651
$605$	0.676423	0.0275005
$606$	0	0
$607$	−26.0471	−1.05722	−0.528609	−	0.848866i	$-0.677286\pi$
$607$	−26.0471	−1.05722	−0.528609	+	0.848866i	$0.677286\pi$
$608$	4.86416	0.197268
$609$	0	0
$610$	−6.99001	−0.283017
$611$	4.85494	0.196410
$612$	0	0
$613$	−1.83897	−0.0742753	−0.0371376	−	0.999310i	$-0.511824\pi$
$613$	−1.83897	−0.0742753	−0.0371376	+	0.999310i	$0.511824\pi$
$614$	2.83416	0.114377
$615$	0	0
$616$	−15.6096	−0.628930
$617$	46.2326	1.86125	0.930627	−	0.365969i	$-0.119262\pi$
$617$	46.2326	1.86125	0.930627	+	0.365969i	$0.119262\pi$
$618$	0	0
$619$	−9.17991	−0.368972	−0.184486	−	0.982835i	$-0.559062\pi$
$619$	−9.17991	−0.368972	−0.184486	+	0.982835i	$0.559062\pi$
$620$	2.55292	0.102528
$621$	0	0
$622$	−30.4039	−1.21909
$623$	−79.0343	−3.16644
$624$	0	0
$625$	1.00000	0.0400000
$626$	−27.2992	−1.09110
$627$	0	0
$628$	−21.6248	−0.862925
$629$	−2.56812	−0.102398
$630$	0	0
$631$	43.0379	1.71331	0.856655	−	0.515889i	$-0.172538\pi$
$631$	43.0379	1.71331	0.856655	+	0.515889i	$0.172538\pi$
$632$	−6.29962	−0.250585
$633$	0	0
$634$	29.4271	1.16870
$635$	−9.21330	−0.365619
$636$	0	0
$637$	13.4687	0.533651
$638$	12.2909	0.486600
$639$	0	0
$640$	1.00000	0.0395285
$641$	−4.81578	−0.190212	−0.0951059	−	0.995467i	$-0.530319\pi$
$641$	−4.81578	−0.190212	−0.0951059	+	0.995467i	$0.530319\pi$
$642$	0	0
$643$	−9.60247	−0.378685	−0.189342	−	0.981911i	$-0.560636\pi$
$643$	−9.60247	−0.378685	−0.189342	+	0.981911i	$0.560636\pi$
$644$	4.69955	0.185188
$645$	0	0
$646$	−12.4918	−0.491482
$647$	−33.6288	−1.32209	−0.661043	−	0.750348i	$-0.729886\pi$
$647$	−33.6288	−1.32209	−0.661043	+	0.750348i	$0.729886\pi$
$648$	0	0
$649$	−23.3528	−0.916679
$650$	0.971229	0.0380947
$651$	0	0
$652$	8.12585	0.318233
$653$	−13.6085	−0.532541	−0.266271	−	0.963898i	$-0.585792\pi$
$653$	−13.6085	−0.532541	−0.266271	+	0.963898i	$0.585792\pi$
$654$	0	0
$655$	3.43833	0.134347
$656$	7.74670	0.302458
$657$	0	0
$658$	22.8350	0.890200
$659$	−9.21290	−0.358884	−0.179442	−	0.983769i	$-0.557429\pi$
$659$	−9.21290	−0.358884	−0.179442	+	0.983769i	$0.557429\pi$
$660$	0	0
$661$	−12.9033	−0.501881	−0.250941	−	0.968002i	$-0.580740\pi$
$661$	−12.9033	−0.501881	−0.250941	+	0.968002i	$0.580740\pi$
$662$	−21.9305	−0.852353
$663$	0	0
$664$	7.86293	0.305141
$665$	22.2201	0.861658
$666$	0	0
$667$	−3.70038	−0.143279
$668$	−1.02877	−0.0398044
$669$	0	0
$670$	−15.9704	−0.616990
$671$	23.8854	0.922086
$672$	0	0
$673$	4.76919	0.183839	0.0919194	−	0.995766i	$-0.470700\pi$
$673$	4.76919	0.183839	0.0919194	+	0.995766i	$0.470700\pi$
$674$	29.0271	1.11808
$675$	0	0
$676$	−12.0567	−0.463720
$677$	42.6724	1.64003	0.820017	−	0.572339i	$-0.193964\pi$
$677$	42.6724	1.64003	0.820017	+	0.572339i	$0.193964\pi$
$678$	0	0
$679$	−3.41667	−0.131120
$680$	−2.56812	−0.0984830
$681$	0	0
$682$	−8.72351	−0.334041
$683$	12.0141	0.459706	0.229853	−	0.973225i	$-0.426176\pi$
$683$	12.0141	0.459706	0.229853	+	0.973225i	$0.426176\pi$
$684$	0	0
$685$	−3.30085	−0.126119
$686$	31.3727	1.19781
$687$	0	0
$688$	9.74670	0.371590
$689$	−6.52465	−0.248569
$690$	0	0
$691$	−32.3464	−1.23051	−0.615257	−	0.788327i	$-0.710948\pi$
$691$	−32.3464	−1.23051	−0.615257	+	0.788327i	$0.710948\pi$
$692$	−7.66964	−0.291556
$693$	0	0
$694$	−12.3300	−0.468042
$695$	−3.06236	−0.116162
$696$	0	0
$697$	−19.8945	−0.753557
$698$	1.82262	0.0689873
$699$	0	0
$700$	4.56812	0.172659
$701$	43.9983	1.66179	0.830897	−	0.556426i	$-0.187828\pi$
$701$	43.9983	1.66179	0.830897	+	0.556426i	$0.187828\pi$
$702$	0	0
$703$	4.86416	0.183455
$704$	−3.41708	−0.128786
$705$	0	0
$706$	−8.04348	−0.302720
$707$	84.2116	3.16710
$708$	0	0
$709$	−7.23008	−0.271531	−0.135766	−	0.990741i	$-0.543349\pi$
$709$	−7.23008	−0.271531	−0.135766	+	0.990741i	$0.543349\pi$
$710$	−8.29962	−0.311479
$711$	0	0
$712$	−17.3013	−0.648392
$713$	2.62637	0.0983582
$714$	0	0
$715$	−3.31876	−0.124115
$716$	−1.64284	−0.0613958
$717$	0	0
$718$	−5.52547	−0.206209
$719$	−18.0454	−0.672982	−0.336491	−	0.941687i	$-0.609240\pi$
$719$	−18.0454	−0.672982	−0.336491	+	0.941687i	$0.609240\pi$
$720$	0	0
$721$	−13.6987	−0.510167
$722$	4.66008	0.173430
$723$	0	0
$724$	25.9704	0.965182
$725$	−3.59689	−0.133585
$726$	0	0
$727$	−11.0659	−0.410413	−0.205206	−	0.978719i	$-0.565787\pi$
$727$	−11.0659	−0.410413	−0.205206	+	0.978719i	$0.565787\pi$
$728$	4.43669	0.164435
$729$	0	0
$730$	1.56331	0.0578607
$731$	−25.0307	−0.925795
$732$	0	0
$733$	−21.8869	−0.808411	−0.404205	−	0.914668i	$-0.632452\pi$
$733$	−21.8869	−0.808411	−0.404205	+	0.914668i	$0.632452\pi$
$734$	−10.4115	−0.384297
$735$	0	0
$736$	1.02877	0.0379210
$737$	54.5721	2.01019
$738$	0	0
$739$	7.26521	0.267255	0.133627	−	0.991032i	$-0.457337\pi$
$739$	7.26521	0.267255	0.133627	+	0.991032i	$0.457337\pi$
$740$	1.00000	0.0367607
$741$	0	0
$742$	−30.6883	−1.12660
$743$	−15.4941	−0.568425	−0.284212	−	0.958761i	$-0.591732\pi$
$743$	−15.4941	−0.568425	−0.284212	+	0.958761i	$0.591732\pi$
$744$	0	0
$745$	9.54293	0.349626
$746$	20.2125	0.740031
$747$	0	0
$748$	8.77547	0.320863
$749$	37.5617	1.37247
$750$	0	0
$751$	−30.9013	−1.12761	−0.563803	−	0.825910i	$-0.690662\pi$
$751$	−30.9013	−1.12761	−0.563803	+	0.825910i	$0.690662\pi$
$752$	4.99877	0.182286
$753$	0	0
$754$	−3.49341	−0.127222
$755$	8.07706	0.293954
$756$	0	0
$757$	−38.2601	−1.39059	−0.695294	−	0.718725i	$-0.744726\pi$
$757$	−38.2601	−1.39059	−0.695294	+	0.718725i	$0.744726\pi$
$758$	−15.5557	−0.565008
$759$	0	0
$760$	4.86416	0.176442
$761$	7.92408	0.287248	0.143624	−	0.989632i	$-0.454124\pi$
$761$	7.92408	0.287248	0.143624	+	0.989632i	$0.454124\pi$
$762$	0	0
$763$	62.6595	2.26843
$764$	−18.9253	−0.684693
$765$	0	0
$766$	29.5137	1.06637
$767$	6.63753	0.239667
$768$	0	0
$769$	−33.6108	−1.21204	−0.606018	−	0.795451i	$-0.707234\pi$
$769$	−33.6108	−1.21204	−0.606018	+	0.795451i	$0.707234\pi$
$770$	−15.6096	−0.562532
$771$	0	0
$772$	−2.63913	−0.0949845
$773$	25.9812	0.934478	0.467239	−	0.884131i	$-0.345249\pi$
$773$	25.9812	0.934478	0.467239	+	0.884131i	$0.345249\pi$
$774$	0	0
$775$	2.55292	0.0917034
$776$	−0.747938	−0.0268494
$777$	0	0
$778$	−14.7235	−0.527863
$779$	37.6812	1.35007
$780$	0	0
$781$	28.3604	1.01482
$782$	−2.64201	−0.0944781
$783$	0	0
$784$	13.8677	0.495276
$785$	−21.6248	−0.771824
$786$	0	0
$787$	−16.6897	−0.594922	−0.297461	−	0.954734i	$-0.596140\pi$
$787$	−16.6897	−0.594922	−0.297461	+	0.954734i	$0.596140\pi$
$788$	−10.9416	−0.389779
$789$	0	0
$790$	−6.29962	−0.224130
$791$	−14.7882	−0.525808
$792$	0	0
$793$	−6.78890	−0.241081
$794$	−17.7859	−0.631197
$795$	0	0
$796$	−1.75545	−0.0622204
$797$	−0.474525	−0.0168085	−0.00840427	−	0.999965i	$-0.502675\pi$
$797$	−0.474525	−0.0168085	−0.00840427	+	0.999965i	$0.502675\pi$
$798$	0	0
$799$	−12.8374	−0.454156
$800$	1.00000	0.0353553
$801$	0	0
$802$	−33.4858	−1.18242
$803$	−5.34195	−0.188513
$804$	0	0
$805$	4.69955	0.165638
$806$	2.47946	0.0873354
$807$	0	0
$808$	18.4346	0.648528
$809$	−27.1716	−0.955303	−0.477651	−	0.878549i	$-0.658512\pi$
$809$	−27.1716	−0.955303	−0.477651	+	0.878549i	$0.658512\pi$
$810$	0	0
$811$	−19.8249	−0.696146	−0.348073	−	0.937467i	$-0.613164\pi$
$811$	−19.8249	−0.696146	−0.348073	+	0.937467i	$0.613164\pi$
$812$	−16.4310	−0.576617
$813$	0	0
$814$	−3.41708	−0.119769
$815$	8.12585	0.284636
$816$	0	0
$817$	47.4095	1.65865
$818$	−30.2700	−1.05837
$819$	0	0
$820$	7.74670	0.270527
$821$	40.2450	1.40456	0.702279	−	0.711902i	$-0.252166\pi$
$821$	40.2450	1.40456	0.702279	+	0.711902i	$0.252166\pi$
$822$	0	0
$823$	−9.93257	−0.346227	−0.173114	−	0.984902i	$-0.555383\pi$
$823$	−9.93257	−0.346227	−0.173114	+	0.984902i	$0.555383\pi$
$824$	−2.99877	−0.104467
$825$	0	0
$826$	31.2193	1.08626
$827$	−7.50975	−0.261140	−0.130570	−	0.991439i	$-0.541681\pi$
$827$	−7.50975	−0.261140	−0.130570	+	0.991439i	$0.541681\pi$
$828$	0	0
$829$	15.4826	0.537733	0.268866	−	0.963178i	$-0.413351\pi$
$829$	15.4826	0.537733	0.268866	+	0.963178i	$0.413351\pi$
$830$	7.86293	0.272926
$831$	0	0
$832$	0.971229	0.0336713
$833$	−35.6141	−1.23395
$834$	0	0
$835$	−1.02877	−0.0356021
$836$	−16.6212	−0.574857
$837$	0	0
$838$	−34.5421	−1.19324
$839$	−4.72154	−0.163006	−0.0815028	−	0.996673i	$-0.525972\pi$
$839$	−4.72154	−0.163006	−0.0815028	+	0.996673i	$0.525972\pi$
$840$	0	0
$841$	−16.0624	−0.553874
$842$	10.6991	0.368717
$843$	0	0
$844$	11.3252	0.389830
$845$	−12.0567	−0.414764
$846$	0	0
$847$	3.08998	0.106173
$848$	−6.71793	−0.230695
$849$	0	0
$850$	−2.56812	−0.0880859
$851$	1.02877	0.0352658
$852$	0	0
$853$	10.4846	0.358986	0.179493	−	0.983759i	$-0.442554\pi$
$853$	10.4846	0.358986	0.179493	+	0.983759i	$0.442554\pi$
$854$	−31.9312	−1.09266
$855$	0	0
$856$	8.22256	0.281041
$857$	−16.5665	−0.565900	−0.282950	−	0.959135i	$-0.591313\pi$
$857$	−16.5665	−0.565900	−0.282950	+	0.959135i	$0.591313\pi$
$858$	0	0
$859$	−6.70676	−0.228832	−0.114416	−	0.993433i	$-0.536500\pi$
$859$	−6.70676	−0.228832	−0.114416	+	0.993433i	$0.536500\pi$
$860$	9.74670	0.332360
$861$	0	0
$862$	5.74112	0.195543
$863$	6.65751	0.226624	0.113312	−	0.993559i	$-0.463854\pi$
$863$	6.65751	0.226624	0.113312	+	0.993559i	$0.463854\pi$
$864$	0	0
$865$	−7.66964	−0.260776
$866$	−16.1650	−0.549310
$867$	0	0
$868$	11.6620	0.395835
$869$	21.5263	0.730229
$870$	0	0
$871$	−15.5109	−0.525567
$872$	13.7167	0.464506
$873$	0	0
$874$	5.00411	0.169267
$875$	4.56812	0.154431
$876$	0	0
$877$	−35.9444	−1.21376	−0.606879	−	0.794795i	$-0.707579\pi$
$877$	−35.9444	−1.21376	−0.606879	+	0.794795i	$0.707579\pi$
$878$	33.3279	1.12476
$879$	0	0
$880$	−3.41708	−0.115190
$881$	31.8018	1.07143	0.535715	−	0.844399i	$-0.320042\pi$
$881$	31.8018	1.07143	0.535715	+	0.844399i	$0.320042\pi$
$882$	0	0
$883$	−33.2325	−1.11836	−0.559181	−	0.829045i	$-0.688884\pi$
$883$	−33.2325	−1.11836	−0.559181	+	0.829045i	$0.688884\pi$
$884$	−2.49423	−0.0838902
$885$	0	0
$886$	11.7930	0.396194
$887$	−12.0543	−0.404745	−0.202373	−	0.979309i	$-0.564865\pi$
$887$	−12.0543	−0.404745	−0.202373	+	0.979309i	$0.564865\pi$
$888$	0	0
$889$	−42.0875	−1.41157
$890$	−17.3013	−0.579940
$891$	0	0
$892$	−15.0328	−0.503334
$893$	24.3148	0.813664
$894$	0	0
$895$	−1.64284	−0.0549140
$896$	4.56812	0.152610
$897$	0	0
$898$	−14.8917	−0.496942
$899$	−9.18257	−0.306256
$900$	0	0
$901$	17.2525	0.574763
$902$	−26.4711	−0.881391
$903$	0	0
$904$	−3.23726	−0.107670
$905$	25.9704	0.863285
$906$	0	0
$907$	34.8146	1.15600	0.578001	−	0.816036i	$-0.303833\pi$
$907$	34.8146	1.15600	0.578001	+	0.816036i	$0.303833\pi$
$908$	−14.3735	−0.477002
$909$	0	0
$910$	4.43669	0.147075
$911$	5.67325	0.187963	0.0939816	−	0.995574i	$-0.470041\pi$
$911$	5.67325	0.187963	0.0939816	+	0.995574i	$0.470041\pi$
$912$	0	0
$913$	−26.8682	−0.889209
$914$	−3.03793	−0.100486
$915$	0	0
$916$	13.2242	0.436940
$917$	15.7067	0.518682
$918$	0	0
$919$	−43.2346	−1.42618	−0.713090	−	0.701073i	$-0.752705\pi$
$919$	−43.2346	−1.42618	−0.713090	+	0.701073i	$0.752705\pi$
$920$	1.02877	0.0339176
$921$	0	0
$922$	−17.5465	−0.577863
$923$	−8.06083	−0.265325
$924$	0	0
$925$	1.00000	0.0328798
$926$	14.5522	0.478214
$927$	0	0
$928$	−3.59689	−0.118074
$929$	−45.4575	−1.49141	−0.745707	−	0.666275i	$-0.767888\pi$
$929$	−45.4575	−1.49141	−0.745707	+	0.666275i	$0.767888\pi$
$930$	0	0
$931$	67.4549	2.21075
$932$	0.222152	0.00727682
$933$	0	0
$934$	16.9681	0.555212
$935$	8.77547	0.286989
$936$	0	0
$937$	47.5222	1.55248	0.776241	−	0.630437i	$-0.217124\pi$
$937$	47.5222	1.55248	0.776241	+	0.630437i	$0.217124\pi$
$938$	−72.9547	−2.38206
$939$	0	0
$940$	4.99877	0.163042
$941$	18.5824	0.605769	0.302885	−	0.953027i	$-0.402050\pi$
$941$	18.5824	0.605769	0.302885	+	0.953027i	$0.402050\pi$
$942$	0	0
$943$	7.96959	0.259525
$944$	6.83416	0.222433
$945$	0	0
$946$	−33.3052	−1.08285
$947$	12.0141	0.390405	0.195202	−	0.980763i	$-0.437464\pi$
$947$	12.0141	0.390405	0.195202	+	0.980763i	$0.437464\pi$
$948$	0	0
$949$	1.51833	0.0492871
$950$	4.86416	0.157814
$951$	0	0
$952$	−11.7315	−0.380220
$953$	34.1350	1.10574	0.552871	−	0.833267i	$-0.313532\pi$
$953$	34.1350	1.10574	0.552871	+	0.833267i	$0.313532\pi$
$954$	0	0
$955$	−18.9253	−0.612408
$956$	2.14106	0.0692467
$957$	0	0
$958$	−26.7267	−0.863500
$959$	−15.0787	−0.486917
$960$	0	0
$961$	−24.4826	−0.789762
$962$	0.971229	0.0313137
$963$	0	0
$964$	28.3876	0.914302
$965$	−2.63913	−0.0849567
$966$	0	0
$967$	19.3288	0.621572	0.310786	−	0.950480i	$-0.399408\pi$
$967$	19.3288	0.621572	0.310786	+	0.950480i	$0.399408\pi$
$968$	0.676423	0.0217411
$969$	0	0
$970$	−0.747938	−0.0240148
$971$	16.7079	0.536182	0.268091	−	0.963394i	$-0.413607\pi$
$971$	16.7079	0.536182	0.268091	+	0.963394i	$0.413607\pi$
$972$	0	0
$973$	−13.9892	−0.448474
$974$	−32.1054	−1.02872
$975$	0	0
$976$	−6.99001	−0.223745
$977$	26.4539	0.846334	0.423167	−	0.906052i	$-0.360918\pi$
$977$	26.4539	0.846334	0.423167	+	0.906052i	$0.360918\pi$
$978$	0	0
$979$	59.1198	1.88948
$980$	13.8677	0.442989
$981$	0	0
$982$	−38.0588	−1.21450
$983$	−47.7091	−1.52168	−0.760842	−	0.648937i	$-0.775214\pi$
$983$	−47.7091	−1.52168	−0.760842	+	0.648937i	$0.775214\pi$
$984$	0	0
$985$	−10.9416	−0.348629
$986$	9.23726	0.294174
$987$	0	0
$988$	4.72421	0.150297
$989$	10.0271	0.318844
$990$	0	0
$991$	−15.3846	−0.488708	−0.244354	−	0.969686i	$-0.578576\pi$
$991$	−15.3846	−0.488708	−0.244354	+	0.969686i	$0.578576\pi$
$992$	2.55292	0.0810552
$993$	0	0
$994$	−37.9137	−1.20255
$995$	−1.75545	−0.0556517
$996$	0	0
$997$	−25.0966	−0.794816	−0.397408	−	0.917642i	$-0.630090\pi$
$997$	−25.0966	−0.794816	−0.397408	+	0.917642i	$0.630090\pi$
$998$	−32.5421	−1.03010
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3330.2.a.bl.1.5	yes	5
3.2	odd	2		3330.2.a.bk.1.5	✓	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3330.2.a.bk.1.5	✓	5	3.2	odd	2
3330.2.a.bl.1.5	yes	5	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 3330 = 2 \cdot 3^{2} \cdot 5 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3330.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +4.56812 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +4.56812 q^{7} +1.00000 q^{8} +1.00000 q^{10} -3.41708 q^{11} +0.971229 q^{13} +4.56812 q^{14} +1.00000 q^{16} -2.56812 q^{17} +4.86416 q^{19} +1.00000 q^{20} -3.41708 q^{22} +1.02877 q^{23} +1.00000 q^{25} +0.971229 q^{26} +4.56812 q^{28} -3.59689 q^{29} +2.55292 q^{31} +1.00000 q^{32} -2.56812 q^{34} +4.56812 q^{35} +1.00000 q^{37} +4.86416 q^{38} +1.00000 q^{40} +7.74670 q^{41} +9.74670 q^{43} -3.41708 q^{44} +1.02877 q^{46} +4.99877 q^{47} +13.8677 q^{49} +1.00000 q^{50} +0.971229 q^{52} -6.71793 q^{53} -3.41708 q^{55} +4.56812 q^{56} -3.59689 q^{58} +6.83416 q^{59} -6.99001 q^{61} +2.55292 q^{62} +1.00000 q^{64} +0.971229 q^{65} -15.9704 q^{67} -2.56812 q^{68} +4.56812 q^{70} -8.29962 q^{71} +1.56331 q^{73} +1.00000 q^{74} +4.86416 q^{76} -15.6096 q^{77} -6.29962 q^{79} +1.00000 q^{80} +7.74670 q^{82} +7.86293 q^{83} -2.56812 q^{85} +9.74670 q^{86} -3.41708 q^{88} -17.3013 q^{89} +4.43669 q^{91} +1.02877 q^{92} +4.99877 q^{94} +4.86416 q^{95} -0.747938 q^{97} +13.8677 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 5 q^{2} + 5 q^{4} + 5 q^{5} + 3 q^{7} + 5 q^{8}+O(q^{10}) \) 5 * q + 5 * q^2 + 5 * q^4 + 5 * q^5 + 3 * q^7 + 5 * q^8	\( 5 q + 5 q^{2} + 5 q^{4} + 5 q^{5} + 3 q^{7} + 5 q^{8} + 5 q^{10} + 5 q^{11} + 5 q^{13} + 3 q^{14} + 5 q^{16} + 7 q^{17} - q^{19} + 5 q^{20} + 5 q^{22} + 5 q^{23} + 5 q^{25} + 5 q^{26} + 3 q^{28} + 2 q^{29} + 16 q^{31} + 5 q^{32} + 7 q^{34} + 3 q^{35} + 5 q^{37} - q^{38} + 5 q^{40} + 2 q^{41} + 12 q^{43} + 5 q^{44} + 5 q^{46} + 6 q^{47} + 16 q^{49} + 5 q^{50} + 5 q^{52} + 3 q^{53} + 5 q^{55} + 3 q^{56} + 2 q^{58} - 10 q^{59} + 16 q^{61} + 16 q^{62} + 5 q^{64} + 5 q^{65} + 4 q^{67} + 7 q^{68} + 3 q^{70} - 8 q^{71} - 3 q^{73} + 5 q^{74} - q^{76} + 3 q^{77} + 2 q^{79} + 5 q^{80} + 2 q^{82} - 5 q^{83} + 7 q^{85} + 12 q^{86} + 5 q^{88} - 7 q^{89} + 33 q^{91} + 5 q^{92} + 6 q^{94} - q^{95} + 14 q^{97} + 16 q^{98}+O(q^{100}) \) 5 * q + 5 * q^2 + 5 * q^4 + 5 * q^5 + 3 * q^7 + 5 * q^8 + 5 * q^10 + 5 * q^11 + 5 * q^13 + 3 * q^14 + 5 * q^16 + 7 * q^17 - q^19 + 5 * q^20 + 5 * q^22 + 5 * q^23 + 5 * q^25 + 5 * q^26 + 3 * q^28 + 2 * q^29 + 16 * q^31 + 5 * q^32 + 7 * q^34 + 3 * q^35 + 5 * q^37 - q^38 + 5 * q^40 + 2 * q^41 + 12 * q^43 + 5 * q^44 + 5 * q^46 + 6 * q^47 + 16 * q^49 + 5 * q^50 + 5 * q^52 + 3 * q^53 + 5 * q^55 + 3 * q^56 + 2 * q^58 - 10 * q^59 + 16 * q^61 + 16 * q^62 + 5 * q^64 + 5 * q^65 + 4 * q^67 + 7 * q^68 + 3 * q^70 - 8 * q^71 - 3 * q^73 + 5 * q^74 - q^76 + 3 * q^77 + 2 * q^79 + 5 * q^80 + 2 * q^82 - 5 * q^83 + 7 * q^85 + 12 * q^86 + 5 * q^88 - 7 * q^89 + 33 * q^91 + 5 * q^92 + 6 * q^94 - q^95 + 14 * q^97 + 16 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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