LMFDB - Embedded newform 3330.2.a.bj.1.4

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:	$4$
Coefficient field:	4.4.54764.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 9x^{2} + 3x + 2 $ x^4 - x^3 - 9x^2 + 3x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 1110)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-2.67673$ 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)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +5.13277 q^{7} -1.00000 q^{8} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +5.13277 q^{7} -1.00000 q^{8} +1.00000 q^{10} -2.22069 q^{11} -6.60629 q^{13} -5.13277 q^{14} +1.00000 q^{16} -5.63841 q^{17} -7.51836 q^{19} -1.00000 q^{20} +2.22069 q^{22} +6.10064 q^{23} +1.00000 q^{25} +6.60629 q^{26} +5.13277 q^{28} +7.29767 q^{29} -0.967873 q^{31} -1.00000 q^{32} +5.63841 q^{34} -5.13277 q^{35} -1.00000 q^{37} +7.51836 q^{38} +1.00000 q^{40} +1.03213 q^{41} +5.29767 q^{43} -2.22069 q^{44} -6.10064 q^{46} +5.85911 q^{47} +19.3453 q^{49} -1.00000 q^{50} -6.60629 q^{52} +9.63841 q^{53} +2.22069 q^{55} -5.13277 q^{56} -7.29767 q^{58} +10.2655 q^{59} +6.32134 q^{61} +0.967873 q^{62} +1.00000 q^{64} +6.60629 q^{65} +1.49436 q^{67} -5.63841 q^{68} +5.13277 q^{70} -0.329796 q^{71} +12.8718 q^{73} +1.00000 q^{74} -7.51836 q^{76} -11.3983 q^{77} +4.00000 q^{79} -1.00000 q^{80} -1.03213 q^{82} -9.04767 q^{83} +5.63841 q^{85} -5.29767 q^{86} +2.22069 q^{88} +2.27649 q^{89} -33.9086 q^{91} +6.10064 q^{92} -5.85911 q^{94} +7.51836 q^{95} +17.5982 q^{97} -19.3453 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{2} + 4 q^{4} - 4 q^{5} + 4 q^{7} - 4 q^{8}+O(q^{10}) $ 4 * q - 4 * q^2 + 4 * q^4 - 4 * q^5 + 4 * q^7 - 4 * q^8	$ 4 q - 4 q^{2} + 4 q^{4} - 4 q^{5} + 4 q^{7} - 4 q^{8} + 4 q^{10} - 2 q^{11} - 3 q^{13} - 4 q^{14} + 4 q^{16} - 6 q^{17} + 3 q^{19} - 4 q^{20} + 2 q^{22} + q^{23} + 4 q^{25} + 3 q^{26} + 4 q^{28} + 3 q^{29} + 3 q^{31} - 4 q^{32} + 6 q^{34} - 4 q^{35} - 4 q^{37} - 3 q^{38} + 4 q^{40} + 11 q^{41} - 5 q^{43} - 2 q^{44} - q^{46} + 14 q^{49} - 4 q^{50} - 3 q^{52} + 22 q^{53} + 2 q^{55} - 4 q^{56} - 3 q^{58} + 8 q^{59} - 5 q^{61} - 3 q^{62} + 4 q^{64} + 3 q^{65} + 6 q^{67} - 6 q^{68} + 4 q^{70} + 18 q^{71} - 5 q^{73} + 4 q^{74} + 3 q^{76} + 4 q^{77} + 16 q^{79} - 4 q^{80} - 11 q^{82} + q^{83} + 6 q^{85} + 5 q^{86} + 2 q^{88} + 5 q^{89} - 13 q^{91} + q^{92} - 3 q^{95} + 7 q^{97} - 14 q^{98}+O(q^{100}) $ 4 * q - 4 * q^2 + 4 * q^4 - 4 * q^5 + 4 * q^7 - 4 * q^8 + 4 * q^10 - 2 * q^11 - 3 * q^13 - 4 * q^14 + 4 * q^16 - 6 * q^17 + 3 * q^19 - 4 * q^20 + 2 * q^22 + q^23 + 4 * q^25 + 3 * q^26 + 4 * q^28 + 3 * q^29 + 3 * q^31 - 4 * q^32 + 6 * q^34 - 4 * q^35 - 4 * q^37 - 3 * q^38 + 4 * q^40 + 11 * q^41 - 5 * q^43 - 2 * q^44 - q^46 + 14 * q^49 - 4 * q^50 - 3 * q^52 + 22 * q^53 + 2 * q^55 - 4 * q^56 - 3 * q^58 + 8 * q^59 - 5 * q^61 - 3 * q^62 + 4 * q^64 + 3 * q^65 + 6 * q^67 - 6 * q^68 + 4 * q^70 + 18 * q^71 - 5 * q^73 + 4 * q^74 + 3 * q^76 + 4 * q^77 + 16 * q^79 - 4 * q^80 - 11 * q^82 + q^83 + 6 * q^85 + 5 * q^86 + 2 * q^88 + 5 * q^89 - 13 * q^91 + q^92 - 3 * q^95 + 7 * q^97 - 14 * 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
$7$	5.13277	1.94001	0.970003	−	0.243094i	$-0.0781625\pi$
$7$	5.13277	1.94001	0.970003	+	0.243094i	$0.0781625\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	1.00000	0.316228
$11$	−2.22069	−0.669564	−0.334782	−	0.942296i	$-0.608663\pi$
$11$	−2.22069	−0.669564	−0.334782	+	0.942296i	$0.608663\pi$
$12$	0	0
$13$	−6.60629	−1.83225	−0.916127	−	0.400888i	$-0.868702\pi$
$13$	−6.60629	−1.83225	−0.916127	+	0.400888i	$0.868702\pi$
$14$	−5.13277	−1.37179
$15$	0	0
$16$	1.00000	0.250000
$17$	−5.63841	−1.36752	−0.683758	−	0.729709i	$-0.739656\pi$
$17$	−5.63841	−1.36752	−0.683758	+	0.729709i	$0.739656\pi$
$18$	0	0
$19$	−7.51836	−1.72483	−0.862415	−	0.506201i	$-0.831049\pi$
$19$	−7.51836	−1.72483	−0.862415	+	0.506201i	$0.831049\pi$
$20$	−1.00000	−0.223607
$21$	0	0
$22$	2.22069	0.473454
$23$	6.10064	1.27207	0.636036	−	0.771659i	$-0.280573\pi$
$23$	6.10064	1.27207	0.636036	+	0.771659i	$0.280573\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	6.60629	1.29560
$27$	0	0
$28$	5.13277	0.970003
$29$	7.29767	1.35514	0.677572	−	0.735457i	$-0.263032\pi$
$29$	7.29767	1.35514	0.677572	+	0.735457i	$0.263032\pi$
$30$	0	0
$31$	−0.967873	−0.173835	−0.0869176	−	0.996216i	$-0.527702\pi$
$31$	−0.967873	−0.173835	−0.0869176	+	0.996216i	$0.527702\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	5.63841	0.966980
$35$	−5.13277	−0.867597
$36$	0	0
$37$	−1.00000	−0.164399
$37$	−1.00000	−0.164399
$38$	7.51836	1.21964
$39$	0	0
$40$	1.00000	0.158114
$41$	1.03213	0.161191	0.0805955	−	0.996747i	$-0.474318\pi$
$41$	1.03213	0.161191	0.0805955	+	0.996747i	$0.474318\pi$
$42$	0	0
$43$	5.29767	0.807887	0.403944	−	0.914784i	$-0.367639\pi$
$43$	5.29767	0.807887	0.403944	+	0.914784i	$0.367639\pi$
$44$	−2.22069	−0.334782
$45$	0	0
$46$	−6.10064	−0.899491
$47$	5.85911	0.854638	0.427319	−	0.904101i	$-0.359458\pi$
$47$	5.85911	0.854638	0.427319	+	0.904101i	$0.359458\pi$
$48$	0	0
$49$	19.3453	2.76362
$50$	−1.00000	−0.141421
$51$	0	0
$52$	−6.60629	−0.916127
$53$	9.63841	1.32394	0.661969	−	0.749531i	$-0.269721\pi$
$53$	9.63841	1.32394	0.661969	+	0.749531i	$0.269721\pi$
$54$	0	0
$55$	2.22069	0.299438
$56$	−5.13277	−0.685895
$57$	0	0
$58$	−7.29767	−0.958231
$59$	10.2655	1.33646	0.668230	−	0.743955i	$-0.267052\pi$
$59$	10.2655	1.33646	0.668230	+	0.743955i	$0.267052\pi$
$60$	0	0
$61$	6.32134	0.809365	0.404682	−	0.914457i	$-0.367382\pi$
$61$	6.32134	0.809365	0.404682	+	0.914457i	$0.367382\pi$
$62$	0.967873	0.122920
$63$	0	0
$64$	1.00000	0.125000
$65$	6.60629	0.819409
$66$	0	0
$67$	1.49436	0.182565	0.0912825	−	0.995825i	$-0.470903\pi$
$67$	1.49436	0.182565	0.0912825	+	0.995825i	$0.470903\pi$
$68$	−5.63841	−0.683758
$69$	0	0
$70$	5.13277	0.613484
$71$	−0.329796	−0.0391396	−0.0195698	−	0.999808i	$-0.506230\pi$
$71$	−0.329796	−0.0391396	−0.0195698	+	0.999808i	$0.506230\pi$
$72$	0	0
$73$	12.8718	1.50653	0.753267	−	0.657715i	$-0.228477\pi$
$73$	12.8718	1.50653	0.753267	+	0.657715i	$0.228477\pi$
$74$	1.00000	0.116248
$75$	0	0
$76$	−7.51836	−0.862415
$77$	−11.3983	−1.29896
$78$	0	0
$79$	4.00000	0.450035	0.225018	−	0.974355i	$-0.427756\pi$
$79$	4.00000	0.450035	0.225018	+	0.974355i	$0.427756\pi$
$80$	−1.00000	−0.111803
$81$	0	0
$82$	−1.03213	−0.113979
$83$	−9.04767	−0.993111	−0.496556	−	0.868005i	$-0.665402\pi$
$83$	−9.04767	−0.993111	−0.496556	+	0.868005i	$0.665402\pi$
$84$	0	0
$85$	5.63841	0.611572
$86$	−5.29767	−0.571262
$87$	0	0
$88$	2.22069	0.236727
$89$	2.27649	0.241307	0.120654	−	0.992695i	$-0.461501\pi$
$89$	2.27649	0.241307	0.120654	+	0.992695i	$0.461501\pi$
$90$	0	0
$91$	−33.9086	−3.55458
$92$	6.10064	0.636036
$93$	0	0
$94$	−5.85911	−0.604321
$95$	7.51836	0.771368
$96$	0	0
$97$	17.5982	1.78682	0.893411	−	0.449239i	$-0.148305\pi$
$97$	17.5982	1.78682	0.893411	+	0.449239i	$0.148305\pi$
$98$	−19.3453	−1.95417
$99$	0	0
$100$	1.00000	0.100000
$101$	−7.61901	−0.758120	−0.379060	−	0.925372i	$-0.623753\pi$
$101$	−7.61901	−0.758120	−0.379060	+	0.925372i	$0.623753\pi$
$102$	0	0
$103$	5.85911	0.577315	0.288657	−	0.957432i	$-0.406791\pi$
$103$	5.85911	0.577315	0.288657	+	0.957432i	$0.406791\pi$
$104$	6.60629	0.647800
$105$	0	0
$106$	−9.63841	−0.936165
$107$	−5.65926	−0.547101	−0.273551	−	0.961858i	$-0.588198\pi$
$107$	−5.65926	−0.547101	−0.273551	+	0.961858i	$0.588198\pi$
$108$	0	0
$109$	−3.16772	−0.303413	−0.151706	−	0.988426i	$-0.548477\pi$
$109$	−3.16772	−0.303413	−0.151706	+	0.988426i	$0.548477\pi$
$110$	−2.22069	−0.211735
$111$	0	0
$112$	5.13277	0.485001
$113$	0.702331	0.0660697	0.0330348	−	0.999454i	$-0.489483\pi$
$113$	0.702331	0.0660697	0.0330348	+	0.999454i	$0.489483\pi$
$114$	0	0
$115$	−6.10064	−0.568888
$116$	7.29767	0.677572
$117$	0	0
$118$	−10.2655	−0.945020
$119$	−28.9407	−2.65299
$120$	0	0
$121$	−6.06852	−0.551683
$122$	−6.32134	−0.572307
$123$	0	0
$124$	−0.967873	−0.0869176
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−19.3132	−1.71377	−0.856885	−	0.515507i	$-0.827604\pi$
$127$	−19.3132	−1.71377	−0.856885	+	0.515507i	$0.827604\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	−6.60629	−0.579410
$131$	18.5953	1.62468	0.812341	−	0.583183i	$-0.198193\pi$
$131$	18.5953	1.62468	0.812341	+	0.583183i	$0.198193\pi$
$132$	0	0
$133$	−38.5900	−3.34618
$134$	−1.49436	−0.129093
$135$	0	0
$136$	5.63841	0.483490
$137$	19.5074	1.66663	0.833316	−	0.552798i	$-0.186440\pi$
$137$	19.5074	1.66663	0.833316	+	0.552798i	$0.186440\pi$
$138$	0	0
$139$	10.9036	0.924833	0.462417	−	0.886663i	$-0.346982\pi$
$139$	10.9036	0.924833	0.462417	+	0.886663i	$0.346982\pi$
$140$	−5.13277	−0.433798
$141$	0	0
$142$	0.329796	0.0276759
$143$	14.6705	1.22681
$144$	0	0
$145$	−7.29767	−0.606038
$146$	−12.8718	−1.06528
$147$	0	0
$148$	−1.00000	−0.0821995
$149$	12.6720	1.03813	0.519065	−	0.854735i	$-0.326280\pi$
$149$	12.6720	1.03813	0.519065	+	0.854735i	$0.326280\pi$
$150$	0	0
$151$	10.5420	0.857898	0.428949	−	0.903329i	$-0.358884\pi$
$151$	10.5420	0.857898	0.428949	+	0.903329i	$0.358884\pi$
$152$	7.51836	0.609820
$153$	0	0
$154$	11.3983	0.918502
$155$	0.967873	0.0777415
$156$	0	0
$157$	−15.4990	−1.23695	−0.618476	−	0.785804i	$-0.712250\pi$
$157$	−15.4990	−1.23695	−0.618476	+	0.785804i	$0.712250\pi$
$158$	−4.00000	−0.318223
$159$	0	0
$160$	1.00000	0.0790569
$161$	31.3132	2.46783
$162$	0	0
$163$	−4.12149	−0.322820	−0.161410	−	0.986887i	$-0.551604\pi$
$163$	−4.12149	−0.322820	−0.161410	+	0.986887i	$0.551604\pi$
$164$	1.03213	0.0805955
$165$	0	0
$166$	9.04767	0.702236
$167$	−21.9474	−1.69834	−0.849169	−	0.528121i	$-0.822897\pi$
$167$	−21.9474	−1.69834	−0.849169	+	0.528121i	$0.822897\pi$
$168$	0	0
$169$	30.6430	2.35715
$170$	−5.63841	−0.432446
$171$	0	0
$172$	5.29767	0.403944
$173$	0.425841	0.0323761	0.0161880	−	0.999869i	$-0.494847\pi$
$173$	0.425841	0.0323761	0.0161880	+	0.999869i	$0.494847\pi$
$174$	0	0
$175$	5.13277	0.388001
$176$	−2.22069	−0.167391
$177$	0	0
$178$	−2.27649	−0.170630
$179$	17.5424	1.31118	0.655589	−	0.755118i	$-0.272420\pi$
$179$	17.5424	1.31118	0.655589	+	0.755118i	$0.272420\pi$
$180$	0	0
$181$	18.5311	1.37740	0.688702	−	0.725044i	$-0.258181\pi$
$181$	18.5311	1.37740	0.688702	+	0.725044i	$0.258181\pi$
$182$	33.9086	2.51347
$183$	0	0
$184$	−6.10064	−0.449746
$185$	1.00000	0.0735215
$186$	0	0
$187$	12.5212	0.915640
$188$	5.85911	0.427319
$189$	0	0
$190$	−7.51836	−0.545439
$191$	−16.1695	−1.16998	−0.584992	−	0.811039i	$-0.698902\pi$
$191$	−16.1695	−1.16998	−0.584992	+	0.811039i	$0.698902\pi$
$192$	0	0
$193$	−11.8591	−0.853637	−0.426819	−	0.904337i	$-0.640366\pi$
$193$	−11.8591	−0.853637	−0.426819	+	0.904337i	$0.640366\pi$
$194$	−17.5982	−1.26347
$195$	0	0
$196$	19.3453	1.38181
$197$	−0.100645	−0.00717066	−0.00358533	−	0.999994i	$-0.501141\pi$
$197$	−0.100645	−0.00717066	−0.00358533	+	0.999994i	$0.501141\pi$
$198$	0	0
$199$	−2.70693	−0.191889	−0.0959446	−	0.995387i	$-0.530587\pi$
$199$	−2.70693	−0.191889	−0.0959446	+	0.995387i	$0.530587\pi$
$200$	−1.00000	−0.0707107
$201$	0	0
$202$	7.61901	0.536072
$203$	37.4573	2.62898
$204$	0	0
$205$	−1.03213	−0.0720868
$206$	−5.85911	−0.408223
$207$	0	0
$208$	−6.60629	−0.458064
$209$	16.6960	1.15489
$210$	0	0
$211$	11.8033	0.812573	0.406287	−	0.913746i	$-0.366823\pi$
$211$	11.8033	0.812573	0.406287	+	0.913746i	$0.366823\pi$
$212$	9.63841	0.661969
$213$	0	0
$214$	5.65926	0.386859
$215$	−5.29767	−0.361298
$216$	0	0
$217$	−4.96787	−0.337241
$218$	3.16772	0.214545
$219$	0	0
$220$	2.22069	0.149719
$221$	37.2490	2.50564
$222$	0	0
$223$	13.1692	0.881872	0.440936	−	0.897538i	$-0.354646\pi$
$223$	13.1692	0.881872	0.440936	+	0.897538i	$0.354646\pi$
$224$	−5.13277	−0.342948
$225$	0	0
$226$	−0.702331	−0.0467183
$227$	−5.57946	−0.370322	−0.185161	−	0.982708i	$-0.559281\pi$
$227$	−5.57946	−0.370322	−0.185161	+	0.982708i	$0.559281\pi$
$228$	0	0
$229$	−1.75990	−0.116298	−0.0581488	−	0.998308i	$-0.518520\pi$
$229$	−1.75990	−0.116298	−0.0581488	+	0.998308i	$0.518520\pi$
$230$	6.10064	0.402265
$231$	0	0
$232$	−7.29767	−0.479115
$233$	6.07664	0.398094	0.199047	−	0.979990i	$-0.436215\pi$
$233$	6.07664	0.398094	0.199047	+	0.979990i	$0.436215\pi$
$234$	0	0
$235$	−5.85911	−0.382206
$236$	10.2655	0.668230
$237$	0	0
$238$	28.9407	1.87595
$239$	−13.9403	−0.901726	−0.450863	−	0.892593i	$-0.648884\pi$
$239$	−13.9403	−0.901726	−0.450863	+	0.892593i	$0.648884\pi$
$240$	0	0
$241$	−3.89406	−0.250838	−0.125419	−	0.992104i	$-0.540028\pi$
$241$	−3.89406	−0.250838	−0.125419	+	0.992104i	$0.540028\pi$
$242$	6.06852	0.390099
$243$	0	0
$244$	6.32134	0.404682
$245$	−19.3453	−1.23593
$246$	0	0
$247$	49.6685	3.16033
$248$	0.967873	0.0614600
$249$	0	0
$250$	1.00000	0.0632456
$251$	−20.9308	−1.32114	−0.660570	−	0.750765i	$-0.729685\pi$
$251$	−20.9308	−1.32114	−0.660570	+	0.750765i	$0.729685\pi$
$252$	0	0
$253$	−13.5477	−0.851734
$254$	19.3132	1.21182
$255$	0	0
$256$	1.00000	0.0625000
$257$	4.43044	0.276363	0.138182	−	0.990407i	$-0.455874\pi$
$257$	4.43044	0.276363	0.138182	+	0.990407i	$0.455874\pi$
$258$	0	0
$259$	−5.13277	−0.318935
$260$	6.60629	0.409704
$261$	0	0
$262$	−18.5953	−1.14882
$263$	−4.20975	−0.259584	−0.129792	−	0.991541i	$-0.541431\pi$
$263$	−4.20975	−0.259584	−0.129792	+	0.991541i	$0.541431\pi$
$264$	0	0
$265$	−9.63841	−0.592083
$266$	38.5900	2.36611
$267$	0	0
$268$	1.49436	0.0912825
$269$	−12.5071	−0.762570	−0.381285	−	0.924457i	$-0.624518\pi$
$269$	−12.5071	−0.762570	−0.381285	+	0.924457i	$0.624518\pi$
$270$	0	0
$271$	−6.70693	−0.407417	−0.203709	−	0.979032i	$-0.565299\pi$
$271$	−6.70693	−0.407417	−0.203709	+	0.979032i	$0.565299\pi$
$272$	−5.63841	−0.341879
$273$	0	0
$274$	−19.5074	−1.17849
$275$	−2.22069	−0.133913
$276$	0	0
$277$	−27.0477	−1.62514	−0.812569	−	0.582866i	$-0.801931\pi$
$277$	−27.0477	−1.62514	−0.812569	+	0.582866i	$0.801931\pi$
$278$	−10.9036	−0.653956
$279$	0	0
$280$	5.13277	0.306742
$281$	1.39371	0.0831420	0.0415710	−	0.999136i	$-0.486764\pi$
$281$	1.39371	0.0831420	0.0415710	+	0.999136i	$0.486764\pi$
$282$	0	0
$283$	−15.3132	−0.910276	−0.455138	−	0.890421i	$-0.650410\pi$
$283$	−15.3132	−0.910276	−0.455138	+	0.890421i	$0.650410\pi$
$284$	−0.329796	−0.0195698
$285$	0	0
$286$	−14.6705	−0.867487
$287$	5.29767	0.312712
$288$	0	0
$289$	14.7917	0.870100
$290$	7.29767	0.428534
$291$	0	0
$292$	12.8718	0.753267
$293$	12.8672	0.751712	0.375856	−	0.926678i	$-0.377349\pi$
$293$	12.8672	0.751712	0.375856	+	0.926678i	$0.377349\pi$
$294$	0	0
$295$	−10.2655	−0.597683
$296$	1.00000	0.0581238
$297$	0	0
$298$	−12.6720	−0.734068
$299$	−40.3026	−2.33076
$300$	0	0
$301$	27.1917	1.56731
$302$	−10.5420	−0.606626
$303$	0	0
$304$	−7.51836	−0.431208
$305$	−6.32134	−0.361959
$306$	0	0
$307$	−4.08970	−0.233411	−0.116706	−	0.993167i	$-0.537233\pi$
$307$	−4.08970	−0.233411	−0.116706	+	0.993167i	$0.537233\pi$
$308$	−11.3983	−0.649479
$309$	0	0
$310$	−0.967873	−0.0549715
$311$	10.3090	0.584567	0.292283	−	0.956332i	$-0.405585\pi$
$311$	10.3090	0.584567	0.292283	+	0.956332i	$0.405585\pi$
$312$	0	0
$313$	−2.64653	−0.149591	−0.0747955	−	0.997199i	$-0.523830\pi$
$313$	−2.64653	−0.149591	−0.0747955	+	0.997199i	$0.523830\pi$
$314$	15.4990	0.874657
$315$	0	0
$316$	4.00000	0.225018
$317$	8.06885	0.453192	0.226596	−	0.973989i	$-0.427240\pi$
$317$	8.06885	0.453192	0.226596	+	0.973989i	$0.427240\pi$
$318$	0	0
$319$	−16.2059	−0.907356
$320$	−1.00000	−0.0559017
$321$	0	0
$322$	−31.3132	−1.74502
$323$	42.3916	2.35873
$324$	0	0
$325$	−6.60629	−0.366451
$326$	4.12149	0.228268
$327$	0	0
$328$	−1.03213	−0.0569897
$329$	30.0735	1.65800
$330$	0	0
$331$	19.6833	1.08189	0.540945	−	0.841058i	$-0.318067\pi$
$331$	19.6833	1.08189	0.540945	+	0.841058i	$0.318067\pi$
$332$	−9.04767	−0.496556
$333$	0	0
$334$	21.9474	1.20091
$335$	−1.49436	−0.0816456
$336$	0	0
$337$	−23.4891	−1.27953	−0.639765	−	0.768570i	$-0.720968\pi$
$337$	−23.4891	−1.27953	−0.639765	+	0.768570i	$0.720968\pi$
$338$	−30.6430	−1.66676
$339$	0	0
$340$	5.63841	0.305786
$341$	2.14935	0.116394
$342$	0	0
$343$	63.3658	3.42143
$344$	−5.29767	−0.285631
$345$	0	0
$346$	−0.425841	−0.0228933
$347$	−18.7069	−1.00424	−0.502120	−	0.864798i	$-0.667447\pi$
$347$	−18.7069	−1.00424	−0.502120	+	0.864798i	$0.667447\pi$
$348$	0	0
$349$	16.1123	0.862470	0.431235	−	0.902240i	$-0.358078\pi$
$349$	16.1123	0.862470	0.431235	+	0.902240i	$0.358078\pi$
$350$	−5.13277	−0.274358
$351$	0	0
$352$	2.22069	0.118363
$353$	−2.37320	−0.126313	−0.0631565	−	0.998004i	$-0.520117\pi$
$353$	−2.37320	−0.126313	−0.0631565	+	0.998004i	$0.520117\pi$
$354$	0	0
$355$	0.329796	0.0175038
$356$	2.27649	0.120654
$357$	0	0
$358$	−17.5424	−0.927143
$359$	18.5953	0.981424	0.490712	−	0.871322i	$-0.336737\pi$
$359$	18.5953	0.981424	0.490712	+	0.871322i	$0.336737\pi$
$360$	0	0
$361$	37.5258	1.97504
$362$	−18.5311	−0.973972
$363$	0	0
$364$	−33.9086	−1.77729
$365$	−12.8718	−0.673742
$366$	0	0
$367$	−29.9520	−1.56348	−0.781740	−	0.623605i	$-0.785668\pi$
$367$	−29.9520	−1.56348	−0.781740	+	0.623605i	$0.785668\pi$
$368$	6.10064	0.318018
$369$	0	0
$370$	−1.00000	−0.0519875
$371$	49.4718	2.56845
$372$	0	0
$373$	−6.00000	−0.310668	−0.155334	−	0.987862i	$-0.549645\pi$
$373$	−6.00000	−0.310668	−0.155334	+	0.987862i	$0.549645\pi$
$374$	−12.5212	−0.647455
$375$	0	0
$376$	−5.85911	−0.302160
$377$	−48.2105	−2.48297
$378$	0	0
$379$	23.3185	1.19779	0.598896	−	0.800827i	$-0.295606\pi$
$379$	23.3185	1.19779	0.598896	+	0.800827i	$0.295606\pi$
$380$	7.51836	0.385684
$381$	0	0
$382$	16.1695	0.827304
$383$	16.6063	0.848542	0.424271	−	0.905535i	$-0.360530\pi$
$383$	16.6063	0.848542	0.424271	+	0.905535i	$0.360530\pi$
$384$	0	0
$385$	11.3983	0.580912
$386$	11.8591	0.603613
$387$	0	0
$388$	17.5982	0.893411
$389$	−15.5795	−0.789910	−0.394955	−	0.918701i	$-0.629240\pi$
$389$	−15.5795	−0.789910	−0.394955	+	0.918701i	$0.629240\pi$
$390$	0	0
$391$	−34.3980	−1.73958
$392$	−19.3453	−0.977087
$393$	0	0
$394$	0.100645	0.00507042
$395$	−4.00000	−0.201262
$396$	0	0
$397$	−23.2126	−1.16501	−0.582503	−	0.812829i	$-0.697926\pi$
$397$	−23.2126	−1.16501	−0.582503	+	0.812829i	$0.697926\pi$
$398$	2.70693	0.135686
$399$	0	0
$400$	1.00000	0.0500000
$401$	−5.72351	−0.285818	−0.142909	−	0.989736i	$-0.545646\pi$
$401$	−5.72351	−0.285818	−0.142909	+	0.989736i	$0.545646\pi$
$402$	0	0
$403$	6.39405	0.318510
$404$	−7.61901	−0.379060
$405$	0	0
$406$	−37.4573	−1.85897
$407$	2.22069	0.110076
$408$	0	0
$409$	28.0254	1.38577	0.692885	−	0.721049i	$-0.256340\pi$
$409$	28.0254	1.38577	0.692885	+	0.721049i	$0.256340\pi$
$410$	1.03213	0.0509731
$411$	0	0
$412$	5.85911	0.288657
$413$	52.6907	2.59274
$414$	0	0
$415$	9.04767	0.444133
$416$	6.60629	0.323900
$417$	0	0
$418$	−16.6960	−0.816627
$419$	−16.2415	−0.793451	−0.396726	−	0.917937i	$-0.629854\pi$
$419$	−16.2415	−0.793451	−0.396726	+	0.917937i	$0.629854\pi$
$420$	0	0
$421$	6.03495	0.294126	0.147063	−	0.989127i	$-0.453018\pi$
$421$	6.03495	0.294126	0.147063	+	0.989127i	$0.453018\pi$
$422$	−11.8033	−0.574576
$423$	0	0
$424$	−9.63841	−0.468083
$425$	−5.63841	−0.273503
$426$	0	0
$427$	32.4460	1.57017
$428$	−5.65926	−0.273551
$429$	0	0
$430$	5.29767	0.255476
$431$	33.2924	1.60364	0.801819	−	0.597568i	$-0.203866\pi$
$431$	33.2924	1.60364	0.801819	+	0.597568i	$0.203866\pi$
$432$	0	0
$433$	8.67054	0.416680	0.208340	−	0.978057i	$-0.433194\pi$
$433$	8.67054	0.416680	0.208340	+	0.978057i	$0.433194\pi$
$434$	4.96787	0.238466
$435$	0	0
$436$	−3.16772	−0.151706
$437$	−45.8669	−2.19411
$438$	0	0
$439$	32.8874	1.56963	0.784814	−	0.619731i	$-0.212758\pi$
$439$	32.8874	1.56963	0.784814	+	0.619731i	$0.212758\pi$
$440$	−2.22069	−0.105867
$441$	0	0
$442$	−37.2490	−1.77175
$443$	−14.8185	−0.704049	−0.352025	−	0.935991i	$-0.614507\pi$
$443$	−14.8185	−0.704049	−0.352025	+	0.935991i	$0.614507\pi$
$444$	0	0
$445$	−2.27649	−0.107916
$446$	−13.1692	−0.623578
$447$	0	0
$448$	5.13277	0.242501
$449$	−9.22882	−0.435535	−0.217767	−	0.976001i	$-0.569877\pi$
$449$	−9.22882	−0.435535	−0.217767	+	0.976001i	$0.569877\pi$
$450$	0	0
$451$	−2.29204	−0.107928
$452$	0.702331	0.0330348
$453$	0	0
$454$	5.57946	0.261857
$455$	33.9086	1.58966
$456$	0	0
$457$	−36.4753	−1.70624	−0.853121	−	0.521713i	$-0.825293\pi$
$457$	−36.4753	−1.70624	−0.853121	+	0.521713i	$0.825293\pi$
$458$	1.75990	0.0822348
$459$	0	0
$460$	−6.10064	−0.284444
$461$	−26.6642	−1.24188	−0.620938	−	0.783860i	$-0.713248\pi$
$461$	−26.6642	−1.24188	−0.620938	+	0.783860i	$0.713248\pi$
$462$	0	0
$463$	6.25249	0.290578	0.145289	−	0.989389i	$-0.453589\pi$
$463$	6.25249	0.290578	0.145289	+	0.989389i	$0.453589\pi$
$464$	7.29767	0.338786
$465$	0	0
$466$	−6.07664	−0.281495
$467$	−16.1748	−0.748480	−0.374240	−	0.927332i	$-0.622096\pi$
$467$	−16.1748	−0.748480	−0.374240	+	0.927332i	$0.622096\pi$
$468$	0	0
$469$	7.67020	0.354177
$470$	5.85911	0.270260
$471$	0	0
$472$	−10.2655	−0.472510
$473$	−11.7645	−0.540932
$474$	0	0
$475$	−7.51836	−0.344966
$476$	−28.9407	−1.32649
$477$	0	0
$478$	13.9403	0.637617
$479$	−7.72351	−0.352896	−0.176448	−	0.984310i	$-0.556461\pi$
$479$	−7.72351	−0.352896	−0.176448	+	0.984310i	$0.556461\pi$
$480$	0	0
$481$	6.60629	0.301221
$482$	3.89406	0.177369
$483$	0	0
$484$	−6.06852	−0.275842
$485$	−17.5982	−0.799091
$486$	0	0
$487$	35.8429	1.62420	0.812098	−	0.583522i	$-0.198326\pi$
$487$	35.8429	1.62420	0.812098	+	0.583522i	$0.198326\pi$
$488$	−6.32134	−0.286154
$489$	0	0
$490$	19.3453	0.873934
$491$	−2.74718	−0.123978	−0.0619892	−	0.998077i	$-0.519744\pi$
$491$	−2.74718	−0.123978	−0.0619892	+	0.998077i	$0.519744\pi$
$492$	0	0
$493$	−41.1473	−1.85318
$494$	−49.6685	−2.23469
$495$	0	0
$496$	−0.967873	−0.0434588
$497$	−1.69277	−0.0759310
$498$	0	0
$499$	−3.07698	−0.137744	−0.0688722	−	0.997625i	$-0.521940\pi$
$499$	−3.07698	−0.137744	−0.0688722	+	0.997625i	$0.521940\pi$
$500$	−1.00000	−0.0447214
$501$	0	0
$502$	20.9308	0.934187
$503$	−8.88278	−0.396063	−0.198032	−	0.980196i	$-0.563455\pi$
$503$	−8.88278	−0.396063	−0.198032	+	0.980196i	$0.563455\pi$
$504$	0	0
$505$	7.61901	0.339041
$506$	13.5477	0.602267
$507$	0	0
$508$	−19.3132	−0.856885
$509$	−2.32836	−0.103203	−0.0516013	−	0.998668i	$-0.516432\pi$
$509$	−2.32836	−0.103203	−0.0516013	+	0.998668i	$0.516432\pi$
$510$	0	0
$511$	66.0682	2.92268
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	−4.43044	−0.195418
$515$	−5.85911	−0.258183
$516$	0	0
$517$	−13.0113	−0.572236
$518$	5.13277	0.225521
$519$	0	0
$520$	−6.60629	−0.289705
$521$	26.3344	1.15373	0.576865	−	0.816839i	$-0.304276\pi$
$521$	26.3344	1.15373	0.576865	+	0.816839i	$0.304276\pi$
$522$	0	0
$523$	10.9887	0.480503	0.240252	−	0.970711i	$-0.422770\pi$
$523$	10.9887	0.480503	0.240252	+	0.970711i	$0.422770\pi$
$524$	18.5953	0.812341
$525$	0	0
$526$	4.20975	0.183554
$527$	5.45727	0.237722
$528$	0	0
$529$	14.2179	0.618168
$530$	9.63841	0.418666
$531$	0	0
$532$	−38.5900	−1.67309
$533$	−6.81852	−0.295343
$534$	0	0
$535$	5.65926	0.244671
$536$	−1.49436	−0.0645465
$537$	0	0
$538$	12.5071	0.539219
$539$	−42.9601	−1.85042
$540$	0	0
$541$	−6.61934	−0.284588	−0.142294	−	0.989824i	$-0.545448\pi$
$541$	−6.61934	−0.284588	−0.142294	+	0.989824i	$0.545448\pi$
$542$	6.70693	0.288087
$543$	0	0
$544$	5.63841	0.241745
$545$	3.16772	0.135690
$546$	0	0
$547$	−23.1801	−0.991110	−0.495555	−	0.868577i	$-0.665035\pi$
$547$	−23.1801	−0.991110	−0.495555	+	0.868577i	$0.665035\pi$
$548$	19.5074	0.833316
$549$	0	0
$550$	2.22069	0.0946907
$551$	−54.8665	−2.33739
$552$	0	0
$553$	20.5311	0.873071
$554$	27.0477	1.14915
$555$	0	0
$556$	10.9036	0.462417
$557$	−11.3411	−0.480537	−0.240268	−	0.970706i	$-0.577235\pi$
$557$	−11.3411	−0.480537	−0.240268	+	0.970706i	$0.577235\pi$
$558$	0	0
$559$	−34.9979	−1.48025
$560$	−5.13277	−0.216899
$561$	0	0
$562$	−1.39371	−0.0587903
$563$	−33.3705	−1.40640	−0.703198	−	0.710994i	$-0.748245\pi$
$563$	−33.3705	−1.40640	−0.703198	+	0.710994i	$0.748245\pi$
$564$	0	0
$565$	−0.702331	−0.0295473
$566$	15.3132	0.643663
$567$	0	0
$568$	0.329796	0.0138379
$569$	0.500343	0.0209755	0.0104877	−	0.999945i	$-0.496662\pi$
$569$	0.500343	0.0209755	0.0104877	+	0.999945i	$0.496662\pi$
$570$	0	0
$571$	27.5463	1.15278	0.576388	−	0.817176i	$-0.304462\pi$
$571$	27.5463	1.15278	0.576388	+	0.817176i	$0.304462\pi$
$572$	14.6705	0.613406
$573$	0	0
$574$	−5.29767	−0.221120
$575$	6.10064	0.254414
$576$	0	0
$577$	12.3902	0.515810	0.257905	−	0.966170i	$-0.416968\pi$
$577$	12.3902	0.515810	0.257905	+	0.966170i	$0.416968\pi$
$578$	−14.7917	−0.615253
$579$	0	0
$580$	−7.29767	−0.303019
$581$	−46.4396	−1.92664
$582$	0	0
$583$	−21.4040	−0.886462
$584$	−12.8718	−0.532640
$585$	0	0
$586$	−12.8672	−0.531540
$587$	−0.808273	−0.0333610	−0.0166805	−	0.999861i	$-0.505310\pi$
$587$	−0.808273	−0.0333610	−0.0166805	+	0.999861i	$0.505310\pi$
$588$	0	0
$589$	7.27682	0.299836
$590$	10.2655	0.422626
$591$	0	0
$592$	−1.00000	−0.0410997
$593$	24.4121	1.00248	0.501242	−	0.865307i	$-0.332877\pi$
$593$	24.4121	1.00248	0.501242	+	0.865307i	$0.332877\pi$
$594$	0	0
$595$	28.9407	1.18645
$596$	12.6720	0.519065
$597$	0	0
$598$	40.3026	1.64810
$599$	29.3440	1.19896	0.599481	−	0.800389i	$-0.295374\pi$
$599$	29.3440	1.19896	0.599481	+	0.800389i	$0.295374\pi$
$600$	0	0
$601$	−9.55791	−0.389875	−0.194938	−	0.980816i	$-0.562450\pi$
$601$	−9.55791	−0.389875	−0.194938	+	0.980816i	$0.562450\pi$
$602$	−27.1917	−1.10825
$603$	0	0
$604$	10.5420	0.428949
$605$	6.06852	0.246720
$606$	0	0
$607$	−31.4912	−1.27819	−0.639094	−	0.769129i	$-0.720690\pi$
$607$	−31.4912	−1.27819	−0.639094	+	0.769129i	$0.720690\pi$
$608$	7.51836	0.304910
$609$	0	0
$610$	6.32134	0.255944
$611$	−38.7069	−1.56591
$612$	0	0
$613$	6.41489	0.259095	0.129548	−	0.991573i	$-0.458648\pi$
$613$	6.41489	0.259095	0.129548	+	0.991573i	$0.458648\pi$
$614$	4.08970	0.165047
$615$	0	0
$616$	11.3983	0.459251
$617$	38.8153	1.56265	0.781323	−	0.624127i	$-0.214545\pi$
$617$	38.8153	1.56265	0.781323	+	0.624127i	$0.214545\pi$
$618$	0	0
$619$	−0.00460032	−0.000184902 0	−9.24512e−5	−	1.00000i	$-0.500029\pi$
$619$	−0.00460032	−0.000184902 0	−9.24512e−5		1.00000i	$0.500029\pi$
$620$	0.967873	0.0388707
$621$	0	0
$622$	−10.3090	−0.413351
$623$	11.6847	0.468138
$624$	0	0
$625$	1.00000	0.0400000
$626$	2.64653	0.105777
$627$	0	0
$628$	−15.4990	−0.618476
$629$	5.63841	0.224818
$630$	0	0
$631$	34.5999	1.37740	0.688701	−	0.725046i	$-0.258181\pi$
$631$	34.5999	1.37740	0.688701	+	0.725046i	$0.258181\pi$
$632$	−4.00000	−0.159111
$633$	0	0
$634$	−8.06885	−0.320455
$635$	19.3132	0.766422
$636$	0	0
$637$	−127.801	−5.06365
$638$	16.2059	0.641597
$639$	0	0
$640$	1.00000	0.0395285
$641$	40.8711	1.61431	0.807156	−	0.590338i	$-0.201005\pi$
$641$	40.8711	1.61431	0.807156	+	0.590338i	$0.201005\pi$
$642$	0	0
$643$	11.4788	0.452680	0.226340	−	0.974048i	$-0.427324\pi$
$643$	11.4788	0.452680	0.226340	+	0.974048i	$0.427324\pi$
$644$	31.3132	1.23391
$645$	0	0
$646$	−42.3916	−1.66788
$647$	17.7490	0.697783	0.348892	−	0.937163i	$-0.386558\pi$
$647$	17.7490	0.697783	0.348892	+	0.937163i	$0.386558\pi$
$648$	0	0
$649$	−22.7966	−0.894846
$650$	6.60629	0.259120
$651$	0	0
$652$	−4.12149	−0.161410
$653$	−21.2768	−0.832626	−0.416313	−	0.909221i	$-0.636678\pi$
$653$	−21.2768	−0.832626	−0.416313	+	0.909221i	$0.636678\pi$
$654$	0	0
$655$	−18.5953	−0.726580
$656$	1.03213	0.0402978
$657$	0	0
$658$	−30.0735	−1.17239
$659$	−23.5548	−0.917563	−0.458781	−	0.888549i	$-0.651714\pi$
$659$	−23.5548	−0.917563	−0.458781	+	0.888549i	$0.651714\pi$
$660$	0	0
$661$	1.34357	0.0522587	0.0261294	−	0.999659i	$-0.491682\pi$
$661$	1.34357	0.0522587	0.0261294	+	0.999659i	$0.491682\pi$
$662$	−19.6833	−0.765012
$663$	0	0
$664$	9.04767	0.351118
$665$	38.5900	1.49646
$666$	0	0
$667$	44.5205	1.72384
$668$	−21.9474	−0.849169
$669$	0	0
$670$	1.49436	0.0577321
$671$	−14.0378	−0.541922
$672$	0	0
$673$	17.5562	0.676742	0.338371	−	0.941013i	$-0.390124\pi$
$673$	17.5562	0.676742	0.338371	+	0.941013i	$0.390124\pi$
$674$	23.4891	0.904765
$675$	0	0
$676$	30.6430	1.17858
$677$	19.2073	0.738196	0.369098	−	0.929391i	$-0.379667\pi$
$677$	19.2073	0.738196	0.369098	+	0.929391i	$0.379667\pi$
$678$	0	0
$679$	90.3274	3.46645
$680$	−5.63841	−0.216223
$681$	0	0
$682$	−2.14935	−0.0823029
$683$	34.7532	1.32980	0.664898	−	0.746935i	$-0.268475\pi$
$683$	34.7532	1.32980	0.664898	+	0.746935i	$0.268475\pi$
$684$	0	0
$685$	−19.5074	−0.745340
$686$	−63.3658	−2.41932
$687$	0	0
$688$	5.29767	0.201972
$689$	−63.6741	−2.42579
$690$	0	0
$691$	24.3344	0.925724	0.462862	−	0.886430i	$-0.346823\pi$
$691$	24.3344	0.925724	0.462862	+	0.886430i	$0.346823\pi$
$692$	0.425841	0.0161880
$693$	0	0
$694$	18.7069	0.710105
$695$	−10.9036	−0.413598
$696$	0	0
$697$	−5.81955	−0.220431
$698$	−16.1123	−0.609858
$699$	0	0
$700$	5.13277	0.194001
$701$	8.81354	0.332883	0.166441	−	0.986051i	$-0.446772\pi$
$701$	8.81354	0.332883	0.166441	+	0.986051i	$0.446772\pi$
$702$	0	0
$703$	7.51836	0.283560
$704$	−2.22069	−0.0836956
$705$	0	0
$706$	2.37320	0.0893167
$707$	−39.1066	−1.47076
$708$	0	0
$709$	−9.07238	−0.340720	−0.170360	−	0.985382i	$-0.554493\pi$
$709$	−9.07238	−0.340720	−0.170360	+	0.985382i	$0.554493\pi$
$710$	−0.329796	−0.0123770
$711$	0	0
$712$	−2.27649	−0.0853151
$713$	−5.90465	−0.221131
$714$	0	0
$715$	−14.6705	−0.548647
$716$	17.5424	0.655589
$717$	0	0
$718$	−18.5953	−0.693972
$719$	−22.8665	−0.852778	−0.426389	−	0.904540i	$-0.640214\pi$
$719$	−22.8665	−0.852778	−0.426389	+	0.904540i	$0.640214\pi$
$720$	0	0
$721$	30.0735	1.11999
$722$	−37.5258	−1.39657
$723$	0	0
$724$	18.5311	0.688702
$725$	7.29767	0.271029
$726$	0	0
$727$	−2.86407	−0.106222	−0.0531112	−	0.998589i	$-0.516914\pi$
$727$	−2.86407	−0.106222	−0.0531112	+	0.998589i	$0.516914\pi$
$728$	33.9086	1.25673
$729$	0	0
$730$	12.8718	0.476408
$731$	−29.8704	−1.10480
$732$	0	0
$733$	−21.6360	−0.799144	−0.399572	−	0.916702i	$-0.630841\pi$
$733$	−21.6360	−0.799144	−0.399572	+	0.916702i	$0.630841\pi$
$734$	29.9520	1.10555
$735$	0	0
$736$	−6.10064	−0.224873
$737$	−3.31851	−0.122239
$738$	0	0
$739$	16.7278	0.615341	0.307671	−	0.951493i	$-0.400451\pi$
$739$	16.7278	0.615341	0.307671	+	0.951493i	$0.400451\pi$
$740$	1.00000	0.0367607
$741$	0	0
$742$	−49.4718	−1.81617
$743$	21.6624	0.794717	0.397358	−	0.917663i	$-0.369927\pi$
$743$	21.6624	0.794717	0.397358	+	0.917663i	$0.369927\pi$
$744$	0	0
$745$	−12.6720	−0.464265
$746$	6.00000	0.219676
$747$	0	0
$748$	12.5212	0.457820
$749$	−29.0477	−1.06138
$750$	0	0
$751$	24.8129	0.905435	0.452717	−	0.891654i	$-0.350455\pi$
$751$	24.8129	0.905435	0.452717	+	0.891654i	$0.350455\pi$
$752$	5.85911	0.213660
$753$	0	0
$754$	48.2105	1.75572
$755$	−10.5420	−0.383664
$756$	0	0
$757$	1.97281	0.0717030	0.0358515	−	0.999357i	$-0.488586\pi$
$757$	1.97281	0.0717030	0.0358515	+	0.999357i	$0.488586\pi$
$758$	−23.3185	−0.846967
$759$	0	0
$760$	−7.51836	−0.272720
$761$	−25.4735	−0.923414	−0.461707	−	0.887032i	$-0.652763\pi$
$761$	−25.4735	−0.923414	−0.461707	+	0.887032i	$0.652763\pi$
$762$	0	0
$763$	−16.2592	−0.588622
$764$	−16.1695	−0.584992
$765$	0	0
$766$	−16.6063	−0.600009
$767$	−67.8171	−2.44873
$768$	0	0
$769$	−41.9675	−1.51339	−0.756694	−	0.653770i	$-0.773187\pi$
$769$	−41.9675	−1.51339	−0.756694	+	0.653770i	$0.773187\pi$
$770$	−11.3983	−0.410767
$771$	0	0
$772$	−11.8591	−0.426819
$773$	−46.0410	−1.65598	−0.827990	−	0.560743i	$-0.810515\pi$
$773$	−46.0410	−1.65598	−0.827990	+	0.560743i	$0.810515\pi$
$774$	0	0
$775$	−0.967873	−0.0347670
$776$	−17.5982	−0.631737
$777$	0	0
$778$	15.5795	0.558551
$779$	−7.75990	−0.278027
$780$	0	0
$781$	0.732376	0.0262065
$782$	34.3980	1.23007
$783$	0	0
$784$	19.3453	0.690905
$785$	15.4990	0.553182
$786$	0	0
$787$	−8.04801	−0.286881	−0.143440	−	0.989659i	$-0.545816\pi$
$787$	−8.04801	−0.286881	−0.143440	+	0.989659i	$0.545816\pi$
$788$	−0.100645	−0.00358533
$789$	0	0
$790$	4.00000	0.142314
$791$	3.60490	0.128176
$792$	0	0
$793$	−41.7606	−1.48296
$794$	23.2126	0.823783
$795$	0	0
$796$	−2.70693	−0.0959446
$797$	26.2662	0.930397	0.465198	−	0.885206i	$-0.345983\pi$
$797$	26.2662	0.930397	0.465198	+	0.885206i	$0.345983\pi$
$798$	0	0
$799$	−33.0361	−1.16873
$800$	−1.00000	−0.0353553
$801$	0	0
$802$	5.72351	0.202104
$803$	−28.5844	−1.00872
$804$	0	0
$805$	−31.3132	−1.10365
$806$	−6.39405	−0.225221
$807$	0	0
$808$	7.61901	0.268036
$809$	−13.3301	−0.468662	−0.234331	−	0.972157i	$-0.575290\pi$
$809$	−13.3301	−0.468662	−0.234331	+	0.972157i	$0.575290\pi$
$810$	0	0
$811$	−30.4251	−1.06837	−0.534186	−	0.845367i	$-0.679382\pi$
$811$	−30.4251	−1.06837	−0.534186	+	0.845367i	$0.679382\pi$
$812$	37.4573	1.31449
$813$	0	0
$814$	−2.22069	−0.0778353
$815$	4.12149	0.144369
$816$	0	0
$817$	−39.8298	−1.39347
$818$	−28.0254	−0.979887
$819$	0	0
$820$	−1.03213	−0.0360434
$821$	26.2895	0.917512	0.458756	−	0.888562i	$-0.348295\pi$
$821$	26.2895	0.917512	0.458756	+	0.888562i	$0.348295\pi$
$822$	0	0
$823$	24.4050	0.850705	0.425352	−	0.905028i	$-0.360150\pi$
$823$	24.4050	0.850705	0.425352	+	0.905028i	$0.360150\pi$
$824$	−5.85911	−0.204112
$825$	0	0
$826$	−52.6907	−1.83334
$827$	37.5887	1.30709	0.653543	−	0.756890i	$-0.273282\pi$
$827$	37.5887	1.30709	0.653543	+	0.756890i	$0.273282\pi$
$828$	0	0
$829$	36.4481	1.26589	0.632947	−	0.774195i	$-0.281845\pi$
$829$	36.4481	1.26589	0.632947	+	0.774195i	$0.281845\pi$
$830$	−9.04767	−0.314049
$831$	0	0
$832$	−6.60629	−0.229032
$833$	−109.077	−3.77929
$834$	0	0
$835$	21.9474	0.759520
$836$	16.6960	0.577443
$837$	0	0
$838$	16.2415	0.561055
$839$	43.3277	1.49584	0.747919	−	0.663790i	$-0.231053\pi$
$839$	43.3277	1.49584	0.747919	+	0.663790i	$0.231053\pi$
$840$	0	0
$841$	24.2560	0.836413
$842$	−6.03495	−0.207978
$843$	0	0
$844$	11.8033	0.406287
$845$	−30.6430	−1.05415
$846$	0	0
$847$	−31.1483	−1.07027
$848$	9.63841	0.330984
$849$	0	0
$850$	5.63841	0.193396
$851$	−6.10064	−0.209127
$852$	0	0
$853$	17.6430	0.604085	0.302043	−	0.953294i	$-0.402332\pi$
$853$	17.6430	0.604085	0.302043	+	0.953294i	$0.402332\pi$
$854$	−32.4460	−1.11028
$855$	0	0
$856$	5.65926	0.193429
$857$	−35.6222	−1.21683	−0.608415	−	0.793619i	$-0.708194\pi$
$857$	−35.6222	−1.21683	−0.608415	+	0.793619i	$0.708194\pi$
$858$	0	0
$859$	−5.49292	−0.187416	−0.0937080	−	0.995600i	$-0.529872\pi$
$859$	−5.49292	−0.187416	−0.0937080	+	0.995600i	$0.529872\pi$
$860$	−5.29767	−0.180649
$861$	0	0
$862$	−33.2924	−1.13394
$863$	1.12501	0.0382958	0.0191479	−	0.999817i	$-0.493905\pi$
$863$	1.12501	0.0382958	0.0191479	+	0.999817i	$0.493905\pi$
$864$	0	0
$865$	−0.425841	−0.0144790
$866$	−8.67054	−0.294637
$867$	0	0
$868$	−4.96787	−0.168621
$869$	−8.88278	−0.301328
$870$	0	0
$871$	−9.87216	−0.334506
$872$	3.16772	0.107273
$873$	0	0
$874$	45.8669	1.55147
$875$	−5.13277	−0.173519
$876$	0	0
$877$	31.6367	1.06829	0.534147	−	0.845392i	$-0.320633\pi$
$877$	31.6367	1.06829	0.534147	+	0.845392i	$0.320633\pi$
$878$	−32.8874	−1.10990
$879$	0	0
$880$	2.22069	0.0748596
$881$	24.8012	0.835575	0.417787	−	0.908545i	$-0.362806\pi$
$881$	24.8012	0.835575	0.417787	+	0.908545i	$0.362806\pi$
$882$	0	0
$883$	−27.7980	−0.935478	−0.467739	−	0.883867i	$-0.654931\pi$
$883$	−27.7980	−0.935478	−0.467739	+	0.883867i	$0.654931\pi$
$884$	37.2490	1.25282
$885$	0	0
$886$	14.8185	0.497838
$887$	28.5226	0.957696	0.478848	−	0.877898i	$-0.341054\pi$
$887$	28.5226	0.957696	0.478848	+	0.877898i	$0.341054\pi$
$888$	0	0
$889$	−99.1303	−3.32472
$890$	2.27649	0.0763081
$891$	0	0
$892$	13.1692	0.440936
$893$	−44.0509	−1.47411
$894$	0	0
$895$	−17.5424	−0.586377
$896$	−5.13277	−0.171474
$897$	0	0
$898$	9.22882	0.307970
$899$	−7.06322	−0.235572
$900$	0	0
$901$	−54.3453	−1.81051
$902$	2.29204	0.0763165
$903$	0	0
$904$	−0.702331	−0.0233592
$905$	−18.5311	−0.615994
$906$	0	0
$907$	41.1854	1.36754	0.683769	−	0.729698i	$-0.260340\pi$
$907$	41.1854	1.36754	0.683769	+	0.729698i	$0.260340\pi$
$908$	−5.57946	−0.185161
$909$	0	0
$910$	−33.9086	−1.12406
$911$	−21.3440	−0.707157	−0.353578	−	0.935405i	$-0.615035\pi$
$911$	−21.3440	−0.707157	−0.353578	+	0.935405i	$0.615035\pi$
$912$	0	0
$913$	20.0921	0.664952
$914$	36.4753	1.20650
$915$	0	0
$916$	−1.75990	−0.0581488
$917$	95.4456	3.15189
$918$	0	0
$919$	36.9089	1.21751	0.608756	−	0.793358i	$-0.291669\pi$
$919$	36.9089	1.21751	0.608756	+	0.793358i	$0.291669\pi$
$920$	6.10064	0.201132
$921$	0	0
$922$	26.6642	0.878138
$923$	2.17873	0.0717137
$924$	0	0
$925$	−1.00000	−0.0328798
$926$	−6.25249	−0.205469
$927$	0	0
$928$	−7.29767	−0.239558
$929$	−21.0350	−0.690136	−0.345068	−	0.938578i	$-0.612144\pi$
$929$	−21.0350	−0.690136	−0.345068	+	0.938578i	$0.612144\pi$
$930$	0	0
$931$	−145.445	−4.76678
$932$	6.07664	0.199047
$933$	0	0
$934$	16.1748	0.529255
$935$	−12.5212	−0.409487
$936$	0	0
$937$	9.67652	0.316118	0.158059	−	0.987430i	$-0.449476\pi$
$937$	9.67652	0.316118	0.158059	+	0.987430i	$0.449476\pi$
$938$	−7.67020	−0.250441
$939$	0	0
$940$	−5.85911	−0.191103
$941$	48.1332	1.56910	0.784548	−	0.620068i	$-0.212895\pi$
$941$	48.1332	1.56910	0.784548	+	0.620068i	$0.212895\pi$
$942$	0	0
$943$	6.29664	0.205047
$944$	10.2655	0.334115
$945$	0	0
$946$	11.7645	0.382497
$947$	17.7871	0.578002	0.289001	−	0.957329i	$-0.406677\pi$
$947$	17.7871	0.578002	0.289001	+	0.957329i	$0.406677\pi$
$948$	0	0
$949$	−85.0350	−2.76035
$950$	7.51836	0.243928
$951$	0	0
$952$	28.9407	0.937973
$953$	−30.0442	−0.973226	−0.486613	−	0.873618i	$-0.661768\pi$
$953$	−30.0442	−0.973226	−0.486613	+	0.873618i	$0.661768\pi$
$954$	0	0
$955$	16.1695	0.523233
$956$	−13.9403	−0.450863
$957$	0	0
$958$	7.72351	0.249535
$959$	100.127	3.23327
$960$	0	0
$961$	−30.0632	−0.969781
$962$	−6.60629	−0.212995
$963$	0	0
$964$	−3.89406	−0.125419
$965$	11.8591	0.381758
$966$	0	0
$967$	15.7052	0.505044	0.252522	−	0.967591i	$-0.418740\pi$
$967$	15.7052	0.505044	0.252522	+	0.967591i	$0.418740\pi$
$968$	6.06852	0.195050
$969$	0	0
$970$	17.5982	0.565043
$971$	−36.8693	−1.18319	−0.591597	−	0.806234i	$-0.701502\pi$
$971$	−36.8693	−1.18319	−0.591597	+	0.806234i	$0.701502\pi$
$972$	0	0
$973$	55.9658	1.79418
$974$	−35.8429	−1.14848
$975$	0	0
$976$	6.32134	0.202341
$977$	−31.8404	−1.01866	−0.509332	−	0.860570i	$-0.670107\pi$
$977$	−31.8404	−1.01866	−0.509332	+	0.860570i	$0.670107\pi$
$978$	0	0
$979$	−5.05539	−0.161571
$980$	−19.3453	−0.617964
$981$	0	0
$982$	2.74718	0.0876660
$983$	−11.8708	−0.378618	−0.189309	−	0.981918i	$-0.560625\pi$
$983$	−11.8708	−0.378618	−0.189309	+	0.981918i	$0.560625\pi$
$984$	0	0
$985$	0.100645	0.00320682
$986$	41.1473	1.31040
$987$	0	0
$988$	49.6685	1.58016
$989$	32.3192	1.02769
$990$	0	0
$991$	−13.9990	−0.444691	−0.222346	−	0.974968i	$-0.571371\pi$
$991$	−13.9990	−0.444691	−0.222346	+	0.974968i	$0.571371\pi$
$992$	0.967873	0.0307300
$993$	0	0
$994$	1.69277	0.0536913
$995$	2.70693	0.0858155
$996$	0	0
$997$	−26.4050	−0.836255	−0.418127	−	0.908388i	$-0.637313\pi$
$997$	−26.4050	−0.836255	−0.418127	+	0.908388i	$0.637313\pi$
$998$	3.07698	0.0974000
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3330.2.a.bj.1.4		4
3.2	odd	2		1110.2.a.s.1.4	✓	4
12.11	even	2		8880.2.a.cg.1.1		4
15.14	odd	2		5550.2.a.cj.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1110.2.a.s.1.4	✓	4	3.2	odd	2
3330.2.a.bj.1.4		4	1.1	even	1	trivial
5550.2.a.cj.1.1		4	15.14	odd	2
8880.2.a.cg.1.1		4	12.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +5.13277 q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +5.13277 q^{7} -1.00000 q^{8} +1.00000 q^{10} -2.22069 q^{11} -6.60629 q^{13} -5.13277 q^{14} +1.00000 q^{16} -5.63841 q^{17} -7.51836 q^{19} -1.00000 q^{20} +2.22069 q^{22} +6.10064 q^{23} +1.00000 q^{25} +6.60629 q^{26} +5.13277 q^{28} +7.29767 q^{29} -0.967873 q^{31} -1.00000 q^{32} +5.63841 q^{34} -5.13277 q^{35} -1.00000 q^{37} +7.51836 q^{38} +1.00000 q^{40} +1.03213 q^{41} +5.29767 q^{43} -2.22069 q^{44} -6.10064 q^{46} +5.85911 q^{47} +19.3453 q^{49} -1.00000 q^{50} -6.60629 q^{52} +9.63841 q^{53} +2.22069 q^{55} -5.13277 q^{56} -7.29767 q^{58} +10.2655 q^{59} +6.32134 q^{61} +0.967873 q^{62} +1.00000 q^{64} +6.60629 q^{65} +1.49436 q^{67} -5.63841 q^{68} +5.13277 q^{70} -0.329796 q^{71} +12.8718 q^{73} +1.00000 q^{74} -7.51836 q^{76} -11.3983 q^{77} +4.00000 q^{79} -1.00000 q^{80} -1.03213 q^{82} -9.04767 q^{83} +5.63841 q^{85} -5.29767 q^{86} +2.22069 q^{88} +2.27649 q^{89} -33.9086 q^{91} +6.10064 q^{92} -5.85911 q^{94} +7.51836 q^{95} +17.5982 q^{97} -19.3453 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{2} + 4 q^{4} - 4 q^{5} + 4 q^{7} - 4 q^{8}+O(q^{10}) \) 4 * q - 4 * q^2 + 4 * q^4 - 4 * q^5 + 4 * q^7 - 4 * q^8	\( 4 q - 4 q^{2} + 4 q^{4} - 4 q^{5} + 4 q^{7} - 4 q^{8} + 4 q^{10} - 2 q^{11} - 3 q^{13} - 4 q^{14} + 4 q^{16} - 6 q^{17} + 3 q^{19} - 4 q^{20} + 2 q^{22} + q^{23} + 4 q^{25} + 3 q^{26} + 4 q^{28} + 3 q^{29} + 3 q^{31} - 4 q^{32} + 6 q^{34} - 4 q^{35} - 4 q^{37} - 3 q^{38} + 4 q^{40} + 11 q^{41} - 5 q^{43} - 2 q^{44} - q^{46} + 14 q^{49} - 4 q^{50} - 3 q^{52} + 22 q^{53} + 2 q^{55} - 4 q^{56} - 3 q^{58} + 8 q^{59} - 5 q^{61} - 3 q^{62} + 4 q^{64} + 3 q^{65} + 6 q^{67} - 6 q^{68} + 4 q^{70} + 18 q^{71} - 5 q^{73} + 4 q^{74} + 3 q^{76} + 4 q^{77} + 16 q^{79} - 4 q^{80} - 11 q^{82} + q^{83} + 6 q^{85} + 5 q^{86} + 2 q^{88} + 5 q^{89} - 13 q^{91} + q^{92} - 3 q^{95} + 7 q^{97} - 14 q^{98}+O(q^{100}) \) 4 * q - 4 * q^2 + 4 * q^4 - 4 * q^5 + 4 * q^7 - 4 * q^8 + 4 * q^10 - 2 * q^11 - 3 * q^13 - 4 * q^14 + 4 * q^16 - 6 * q^17 + 3 * q^19 - 4 * q^20 + 2 * q^22 + q^23 + 4 * q^25 + 3 * q^26 + 4 * q^28 + 3 * q^29 + 3 * q^31 - 4 * q^32 + 6 * q^34 - 4 * q^35 - 4 * q^37 - 3 * q^38 + 4 * q^40 + 11 * q^41 - 5 * q^43 - 2 * q^44 - q^46 + 14 * q^49 - 4 * q^50 - 3 * q^52 + 22 * q^53 + 2 * q^55 - 4 * q^56 - 3 * q^58 + 8 * q^59 - 5 * q^61 - 3 * q^62 + 4 * q^64 + 3 * q^65 + 6 * q^67 - 6 * q^68 + 4 * q^70 + 18 * q^71 - 5 * q^73 + 4 * q^74 + 3 * q^76 + 4 * q^77 + 16 * q^79 - 4 * q^80 - 11 * q^82 + q^83 + 6 * q^85 + 5 * q^86 + 2 * q^88 + 5 * q^89 - 13 * q^91 + q^92 - 3 * q^95 + 7 * q^97 - 14 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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