LMFDB - Embedded newform 2960.2.a.bb.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2960,2,Mod(1,2960)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2960, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("2960.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2960 = 2^{4} \cdot 5 \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2960.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:	$23.6357189983$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.583504.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 8x^{3} - 2x^{2} + 15x + 8 $ x^5 - 8x^3 - 2x^2 + 15*x + 8
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1480)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-2.15365$ of defining polynomial
Character	$\chi$	$=$	2960.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.15365 q^{3} +1.00000 q^{5} +0.579974 q^{7} +6.94552 q^{9} +O(q^{10})$	$q+3.15365 q^{3} +1.00000 q^{5} +0.579974 q^{7} +6.94552 q^{9} +0.683748 q^{11} -7.07095 q^{13} +3.15365 q^{15} +7.90570 q^{17} -0.469903 q^{19} +1.82904 q^{21} +3.47256 q^{23} +1.00000 q^{25} +12.4428 q^{27} +3.47256 q^{29} -6.32001 q^{31} +2.15630 q^{33} +0.579974 q^{35} -1.00000 q^{37} -22.2993 q^{39} +0.683748 q^{41} +11.5837 q^{43} +6.94552 q^{45} -5.00376 q^{47} -6.66363 q^{49} +24.9318 q^{51} +4.50692 q^{53} +0.683748 q^{55} -1.48191 q^{57} +6.85276 q^{59} +3.71811 q^{61} +4.02822 q^{63} -7.07095 q^{65} -7.53761 q^{67} +10.9512 q^{69} +13.0218 q^{71} -15.6490 q^{73} +3.15365 q^{75} +0.396556 q^{77} +8.76169 q^{79} +18.4037 q^{81} -1.84089 q^{83} +7.90570 q^{85} +10.9512 q^{87} -4.74368 q^{89} -4.10097 q^{91} -19.9311 q^{93} -0.469903 q^{95} -11.7860 q^{97} +4.74899 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 5 q^{3} + 5 q^{5} + 5 q^{7} + 6 q^{9}+O(q^{10}) $ 5 * q + 5 * q^3 + 5 * q^5 + 5 * q^7 + 6 * q^9	$ 5 q + 5 q^{3} + 5 q^{5} + 5 q^{7} + 6 q^{9} + 7 q^{11} - 6 q^{13} + 5 q^{15} + 12 q^{19} - q^{21} + 6 q^{23} + 5 q^{25} + 17 q^{27} + 6 q^{29} + 10 q^{31} + 3 q^{33} + 5 q^{35} - 5 q^{37} - 4 q^{39} + 7 q^{41} + 22 q^{43} + 6 q^{45} + 13 q^{47} + 14 q^{49} + 10 q^{51} - 11 q^{53} + 7 q^{55} - 8 q^{57} + 10 q^{59} + 10 q^{63} - 6 q^{65} + 24 q^{67} + 26 q^{69} + 13 q^{71} - 13 q^{73} + 5 q^{75} - 19 q^{77} - 2 q^{79} + q^{81} + 13 q^{83} + 26 q^{87} + 8 q^{89} + 8 q^{91} - 20 q^{93} + 12 q^{95} - 20 q^{97} - 2 q^{99}+O(q^{100}) $ 5 * q + 5 * q^3 + 5 * q^5 + 5 * q^7 + 6 * q^9 + 7 * q^11 - 6 * q^13 + 5 * q^15 + 12 * q^19 - q^21 + 6 * q^23 + 5 * q^25 + 17 * q^27 + 6 * q^29 + 10 * q^31 + 3 * q^33 + 5 * q^35 - 5 * q^37 - 4 * q^39 + 7 * q^41 + 22 * q^43 + 6 * q^45 + 13 * q^47 + 14 * q^49 + 10 * q^51 - 11 * q^53 + 7 * q^55 - 8 * q^57 + 10 * q^59 + 10 * q^63 - 6 * q^65 + 24 * q^67 + 26 * q^69 + 13 * q^71 - 13 * q^73 + 5 * q^75 - 19 * q^77 - 2 * q^79 + q^81 + 13 * q^83 + 26 * q^87 + 8 * q^89 + 8 * q^91 - 20 * q^93 + 12 * q^95 - 20 * q^97 - 2 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	3.15365	1.82076	0.910381	−	0.413771i	$-0.135789\pi$
$3$	3.15365	1.82076	0.910381	+	0.413771i	$0.135789\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	0.579974	0.219210	0.109605	−	0.993975i	$-0.465041\pi$
$7$	0.579974	0.219210	0.109605	+	0.993975i	$0.465041\pi$
$8$	0	0
$9$	6.94552	2.31517
$10$	0	0
$11$	0.683748	0.206158	0.103079	−	0.994673i	$-0.467131\pi$
$11$	0.683748	0.206158	0.103079	+	0.994673i	$0.467131\pi$
$12$	0	0
$13$	−7.07095	−1.96113	−0.980564	−	0.196199i	$-0.937140\pi$
$13$	−7.07095	−1.96113	−0.980564	+	0.196199i	$0.937140\pi$
$14$	0	0
$15$	3.15365	0.814269
$16$	0	0
$17$	7.90570	1.91741	0.958706	−	0.284397i	$-0.0917935\pi$
$17$	7.90570	1.91741	0.958706	+	0.284397i	$0.0917935\pi$
$18$	0	0
$19$	−0.469903	−0.107803	−0.0539016	−	0.998546i	$-0.517166\pi$
$19$	−0.469903	−0.107803	−0.0539016	+	0.998546i	$0.517166\pi$
$20$	0	0
$21$	1.82904	0.399128
$22$	0	0
$23$	3.47256	0.724078	0.362039	−	0.932163i	$-0.382081\pi$
$23$	3.47256	0.724078	0.362039	+	0.932163i	$0.382081\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	12.4428	2.39462
$28$	0	0
$29$	3.47256	0.644837	0.322419	−	0.946597i	$-0.395504\pi$
$29$	3.47256	0.644837	0.322419	+	0.946597i	$0.395504\pi$
$30$	0	0
$31$	−6.32001	−1.13511	−0.567554	−	0.823336i	$-0.692110\pi$
$31$	−6.32001	−1.13511	−0.567554	+	0.823336i	$0.692110\pi$
$32$	0	0
$33$	2.15630	0.375364
$34$	0	0
$35$	0.579974	0.0980335
$36$	0	0
$37$	−1.00000	−0.164399
$37$	−1.00000	−0.164399
$38$	0	0
$39$	−22.2993	−3.57075
$40$	0	0
$41$	0.683748	0.106784	0.0533918	−	0.998574i	$-0.482997\pi$
$41$	0.683748	0.106784	0.0533918	+	0.998574i	$0.482997\pi$
$42$	0	0
$43$	11.5837	1.76650	0.883251	−	0.468900i	$-0.155349\pi$
$43$	11.5837	1.76650	0.883251	+	0.468900i	$0.155349\pi$
$44$	0	0
$45$	6.94552	1.03538
$46$	0	0
$47$	−5.00376	−0.729873	−0.364937	−	0.931032i	$-0.618909\pi$
$47$	−5.00376	−0.729873	−0.364937	+	0.931032i	$0.618909\pi$
$48$	0	0
$49$	−6.66363	−0.951947
$50$	0	0
$51$	24.9318	3.49115
$52$	0	0
$53$	4.50692	0.619073	0.309536	−	0.950888i	$-0.399826\pi$
$53$	4.50692	0.619073	0.309536	+	0.950888i	$0.399826\pi$
$54$	0	0
$55$	0.683748	0.0921966
$56$	0	0
$57$	−1.48191	−0.196284
$58$	0	0
$59$	6.85276	0.892153	0.446077	−	0.894995i	$-0.352821\pi$
$59$	6.85276	0.892153	0.446077	+	0.894995i	$0.352821\pi$
$60$	0	0
$61$	3.71811	0.476055	0.238028	−	0.971258i	$-0.423499\pi$
$61$	3.71811	0.476055	0.238028	+	0.971258i	$0.423499\pi$
$62$	0	0
$63$	4.02822	0.507508
$64$	0	0
$65$	−7.07095	−0.877043
$66$	0	0
$67$	−7.53761	−0.920866	−0.460433	−	0.887694i	$-0.652306\pi$
$67$	−7.53761	−0.920866	−0.460433	+	0.887694i	$0.652306\pi$
$68$	0	0
$69$	10.9512	1.31837
$70$	0	0
$71$	13.0218	1.54540	0.772700	−	0.634772i	$-0.218906\pi$
$71$	13.0218	1.54540	0.772700	+	0.634772i	$0.218906\pi$
$72$	0	0
$73$	−15.6490	−1.83157	−0.915787	−	0.401665i	$-0.868432\pi$
$73$	−15.6490	−1.83157	−0.915787	+	0.401665i	$0.868432\pi$
$74$	0	0
$75$	3.15365	0.364152
$76$	0	0
$77$	0.396556	0.0451918
$78$	0	0
$79$	8.76169	0.985767	0.492884	−	0.870095i	$-0.335943\pi$
$79$	8.76169	0.985767	0.492884	+	0.870095i	$0.335943\pi$
$80$	0	0
$81$	18.4037	2.04485
$82$	0	0
$83$	−1.84089	−0.202064	−0.101032	−	0.994883i	$-0.532214\pi$
$83$	−1.84089	−0.202064	−0.101032	+	0.994883i	$0.532214\pi$
$84$	0	0
$85$	7.90570	0.857493
$86$	0	0
$87$	10.9512	1.17410
$88$	0	0
$89$	−4.74368	−0.502829	−0.251415	−	0.967879i	$-0.580896\pi$
$89$	−4.74368	−0.502829	−0.251415	+	0.967879i	$0.580896\pi$
$90$	0	0
$91$	−4.10097	−0.429898
$92$	0	0
$93$	−19.9311	−2.06676
$94$	0	0
$95$	−0.469903	−0.0482111
$96$	0	0
$97$	−11.7860	−1.19668	−0.598342	−	0.801241i	$-0.704174\pi$
$97$	−11.7860	−1.19668	−0.598342	+	0.801241i	$0.704174\pi$
$98$	0	0
$99$	4.74899	0.477291
$100$	0	0
$101$	−4.02516	−0.400519	−0.200259	−	0.979743i	$-0.564178\pi$
$101$	−4.02516	−0.400519	−0.200259	+	0.979743i	$0.564178\pi$
$102$	0	0
$103$	3.66950	0.361566	0.180783	−	0.983523i	$-0.442137\pi$
$103$	3.66950	0.361566	0.180783	+	0.983523i	$0.442137\pi$
$104$	0	0
$105$	1.82904	0.178496
$106$	0	0
$107$	15.8898	1.53612	0.768060	−	0.640378i	$-0.221222\pi$
$107$	15.8898	1.53612	0.768060	+	0.640378i	$0.221222\pi$
$108$	0	0
$109$	−9.89104	−0.947389	−0.473695	−	0.880689i	$-0.657080\pi$
$109$	−9.89104	−0.947389	−0.473695	+	0.880689i	$0.657080\pi$
$110$	0	0
$111$	−3.15365	−0.299331
$112$	0	0
$113$	−1.40145	−0.131838	−0.0659188	−	0.997825i	$-0.520998\pi$
$113$	−1.40145	−0.131838	−0.0659188	+	0.997825i	$0.520998\pi$
$114$	0	0
$115$	3.47256	0.323817
$116$	0	0
$117$	−49.1114	−4.54035
$118$	0	0
$119$	4.58510	0.420315
$120$	0	0
$121$	−10.5325	−0.957499
$122$	0	0
$123$	2.15630	0.194427
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	3.39309	0.301088	0.150544	−	0.988603i	$-0.451897\pi$
$127$	3.39309	0.301088	0.150544	+	0.988603i	$0.451897\pi$
$128$	0	0
$129$	36.5311	3.21638
$130$	0	0
$131$	−0.916889	−0.0801090	−0.0400545	−	0.999197i	$-0.512753\pi$
$131$	−0.916889	−0.0801090	−0.0400545	+	0.999197i	$0.512753\pi$
$132$	0	0
$133$	−0.272532	−0.0236315
$134$	0	0
$135$	12.4428	1.07090
$136$	0	0
$137$	−19.0510	−1.62764	−0.813818	−	0.581120i	$-0.802615\pi$
$137$	−19.0510	−1.62764	−0.813818	+	0.581120i	$0.802615\pi$
$138$	0	0
$139$	−16.1930	−1.37348	−0.686738	−	0.726905i	$-0.740958\pi$
$139$	−16.1930	−1.37348	−0.686738	+	0.726905i	$0.740958\pi$
$140$	0	0
$141$	−15.7801	−1.32893
$142$	0	0
$143$	−4.83475	−0.404302
$144$	0	0
$145$	3.47256	0.288380
$146$	0	0
$147$	−21.0148	−1.73327
$148$	0	0
$149$	−0.136340	−0.0111694	−0.00558469	−	0.999984i	$-0.501778\pi$
$149$	−0.136340	−0.0111694	−0.00558469	+	0.999984i	$0.501778\pi$
$150$	0	0
$151$	−16.1366	−1.31318	−0.656589	−	0.754249i	$-0.728001\pi$
$151$	−16.1366	−1.31318	−0.656589	+	0.754249i	$0.728001\pi$
$152$	0	0
$153$	54.9092	4.43914
$154$	0	0
$155$	−6.32001	−0.507635
$156$	0	0
$157$	−7.83110	−0.624990	−0.312495	−	0.949919i	$-0.601165\pi$
$157$	−7.83110	−0.624990	−0.312495	+	0.949919i	$0.601165\pi$
$158$	0	0
$159$	14.2133	1.12718
$160$	0	0
$161$	2.01399	0.158725
$162$	0	0
$163$	8.82333	0.691096	0.345548	−	0.938401i	$-0.387693\pi$
$163$	8.82333	0.691096	0.345548	+	0.938401i	$0.387693\pi$
$164$	0	0
$165$	2.15630	0.167868
$166$	0	0
$167$	−2.94593	−0.227963	−0.113981	−	0.993483i	$-0.536360\pi$
$167$	−2.94593	−0.227963	−0.113981	+	0.993483i	$0.536360\pi$
$168$	0	0
$169$	36.9983	2.84602
$170$	0	0
$171$	−3.26372	−0.249583
$172$	0	0
$173$	11.9423	0.907955	0.453977	−	0.891013i	$-0.350005\pi$
$173$	11.9423	0.907955	0.453977	+	0.891013i	$0.350005\pi$
$174$	0	0
$175$	0.579974	0.0438419
$176$	0	0
$177$	21.6112	1.62440
$178$	0	0
$179$	19.8265	1.48190	0.740950	−	0.671560i	$-0.234375\pi$
$179$	19.8265	1.48190	0.740950	+	0.671560i	$0.234375\pi$
$180$	0	0
$181$	−16.0274	−1.19131	−0.595653	−	0.803242i	$-0.703107\pi$
$181$	−16.0274	−1.19131	−0.595653	+	0.803242i	$0.703107\pi$
$182$	0	0
$183$	11.7256	0.866783
$184$	0	0
$185$	−1.00000	−0.0735215
$186$	0	0
$187$	5.40551	0.395290
$188$	0	0
$189$	7.21650	0.524923
$190$	0	0
$191$	−21.6093	−1.56359	−0.781796	−	0.623535i	$-0.785696\pi$
$191$	−21.6093	−1.56359	−0.781796	+	0.623535i	$0.785696\pi$
$192$	0	0
$193$	12.9345	0.931046	0.465523	−	0.885036i	$-0.345866\pi$
$193$	12.9345	0.931046	0.465523	+	0.885036i	$0.345866\pi$
$194$	0	0
$195$	−22.2993	−1.59689
$196$	0	0
$197$	4.37254	0.311531	0.155765	−	0.987794i	$-0.450216\pi$
$197$	4.37254	0.311531	0.155765	+	0.987794i	$0.450216\pi$
$198$	0	0
$199$	−9.36999	−0.664221	−0.332111	−	0.943240i	$-0.607761\pi$
$199$	−9.36999	−0.664221	−0.332111	+	0.943240i	$0.607761\pi$
$200$	0	0
$201$	−23.7710	−1.67668
$202$	0	0
$203$	2.01399	0.141355
$204$	0	0
$205$	0.683748	0.0477550
$206$	0	0
$207$	24.1187	1.67637
$208$	0	0
$209$	−0.321296	−0.0222245
$210$	0	0
$211$	18.0853	1.24505	0.622523	−	0.782601i	$-0.286108\pi$
$211$	18.0853	1.24505	0.622523	+	0.782601i	$0.286108\pi$
$212$	0	0
$213$	41.0661	2.81380
$214$	0	0
$215$	11.5837	0.790004
$216$	0	0
$217$	−3.66544	−0.248826
$218$	0	0
$219$	−49.3514	−3.33486
$220$	0	0
$221$	−55.9008	−3.76029
$222$	0	0
$223$	−25.5057	−1.70799	−0.853993	−	0.520285i	$-0.825826\pi$
$223$	−25.5057	−1.70799	−0.853993	+	0.520285i	$0.825826\pi$
$224$	0	0
$225$	6.94552	0.463035
$226$	0	0
$227$	−16.9473	−1.12483	−0.562417	−	0.826854i	$-0.690128\pi$
$227$	−16.9473	−1.12483	−0.562417	+	0.826854i	$0.690128\pi$
$228$	0	0
$229$	−9.84392	−0.650505	−0.325252	−	0.945627i	$-0.605449\pi$
$229$	−9.84392	−0.650505	−0.325252	+	0.945627i	$0.605449\pi$
$230$	0	0
$231$	1.25060	0.0822835
$232$	0	0
$233$	−20.4313	−1.33850	−0.669250	−	0.743038i	$-0.733384\pi$
$233$	−20.4313	−1.33850	−0.669250	+	0.743038i	$0.733384\pi$
$234$	0	0
$235$	−5.00376	−0.326409
$236$	0	0
$237$	27.6313	1.79485
$238$	0	0
$239$	8.68614	0.561860	0.280930	−	0.959728i	$-0.409357\pi$
$239$	8.68614	0.561860	0.280930	+	0.959728i	$0.409357\pi$
$240$	0	0
$241$	−11.0510	−0.711856	−0.355928	−	0.934513i	$-0.615835\pi$
$241$	−11.0510	−0.711856	−0.355928	+	0.934513i	$0.615835\pi$
$242$	0	0
$243$	20.7104	1.32857
$244$	0	0
$245$	−6.66363	−0.425724
$246$	0	0
$247$	3.32266	0.211416
$248$	0	0
$249$	−5.80553	−0.367910
$250$	0	0
$251$	5.38480	0.339886	0.169943	−	0.985454i	$-0.445642\pi$
$251$	5.38480	0.339886	0.169943	+	0.985454i	$0.445642\pi$
$252$	0	0
$253$	2.37435	0.149274
$254$	0	0
$255$	24.9318	1.56129
$256$	0	0
$257$	−24.8815	−1.55207	−0.776033	−	0.630692i	$-0.782771\pi$
$257$	−24.8815	−1.55207	−0.776033	+	0.630692i	$0.782771\pi$
$258$	0	0
$259$	−0.579974	−0.0360378
$260$	0	0
$261$	24.1187	1.49291
$262$	0	0
$263$	7.49167	0.461956	0.230978	−	0.972959i	$-0.425807\pi$
$263$	7.49167	0.461956	0.230978	+	0.972959i	$0.425807\pi$
$264$	0	0
$265$	4.50692	0.276858
$266$	0	0
$267$	−14.9599	−0.915532
$268$	0	0
$269$	10.4524	0.637296	0.318648	−	0.947873i	$-0.396771\pi$
$269$	10.4524	0.637296	0.318648	+	0.947873i	$0.396771\pi$
$270$	0	0
$271$	−24.2999	−1.47611	−0.738057	−	0.674738i	$-0.764257\pi$
$271$	−24.2999	−1.47611	−0.738057	+	0.674738i	$0.764257\pi$
$272$	0	0
$273$	−12.9330	−0.782742
$274$	0	0
$275$	0.683748	0.0412316
$276$	0	0
$277$	29.4037	1.76670	0.883348	−	0.468719i	$-0.155284\pi$
$277$	29.4037	1.76670	0.883348	+	0.468719i	$0.155284\pi$
$278$	0	0
$279$	−43.8957	−2.62797
$280$	0	0
$281$	−12.9823	−0.774457	−0.387228	−	0.921984i	$-0.626567\pi$
$281$	−12.9823	−0.774457	−0.387228	+	0.921984i	$0.626567\pi$
$282$	0	0
$283$	8.13321	0.483469	0.241735	−	0.970342i	$-0.422284\pi$
$283$	8.13321	0.483469	0.241735	+	0.970342i	$0.422284\pi$
$284$	0	0
$285$	−1.48191	−0.0877809
$286$	0	0
$287$	0.396556	0.0234080
$288$	0	0
$289$	45.5000	2.67647
$290$	0	0
$291$	−37.1689	−2.17888
$292$	0	0
$293$	−4.02011	−0.234857	−0.117429	−	0.993081i	$-0.537465\pi$
$293$	−4.02011	−0.234857	−0.117429	+	0.993081i	$0.537465\pi$
$294$	0	0
$295$	6.85276	0.398983
$296$	0	0
$297$	8.50773	0.493669
$298$	0	0
$299$	−24.5543	−1.42001
$300$	0	0
$301$	6.71827	0.387234
$302$	0	0
$303$	−12.6940	−0.729249
$304$	0	0
$305$	3.71811	0.212898
$306$	0	0
$307$	16.1944	0.924265	0.462132	−	0.886811i	$-0.347084\pi$
$307$	16.1944	0.924265	0.462132	+	0.886811i	$0.347084\pi$
$308$	0	0
$309$	11.5723	0.658326
$310$	0	0
$311$	18.4418	1.04574	0.522869	−	0.852413i	$-0.324862\pi$
$311$	18.4418	1.04574	0.522869	+	0.852413i	$0.324862\pi$
$312$	0	0
$313$	16.9819	0.959876	0.479938	−	0.877302i	$-0.340659\pi$
$313$	16.9819	0.959876	0.479938	+	0.877302i	$0.340659\pi$
$314$	0	0
$315$	4.02822	0.226965
$316$	0	0
$317$	−24.0855	−1.35277	−0.676387	−	0.736546i	$-0.736455\pi$
$317$	−24.0855	−1.35277	−0.676387	+	0.736546i	$0.736455\pi$
$318$	0	0
$319$	2.37435	0.132938
$320$	0	0
$321$	50.1107	2.79691
$322$	0	0
$323$	−3.71491	−0.206703
$324$	0	0
$325$	−7.07095	−0.392226
$326$	0	0
$327$	−31.1929	−1.72497
$328$	0	0
$329$	−2.90205	−0.159995
$330$	0	0
$331$	−17.7771	−0.977115	−0.488558	−	0.872532i	$-0.662477\pi$
$331$	−17.7771	−0.977115	−0.488558	+	0.872532i	$0.662477\pi$
$332$	0	0
$333$	−6.94552	−0.380612
$334$	0	0
$335$	−7.53761	−0.411824
$336$	0	0
$337$	−18.9569	−1.03265	−0.516325	−	0.856393i	$-0.672700\pi$
$337$	−18.9569	−1.03265	−0.516325	+	0.856393i	$0.672700\pi$
$338$	0	0
$339$	−4.41969	−0.240045
$340$	0	0
$341$	−4.32130	−0.234011
$342$	0	0
$343$	−7.92455	−0.427886
$344$	0	0
$345$	10.9512	0.589594
$346$	0	0
$347$	−14.3989	−0.772972	−0.386486	−	0.922295i	$-0.626311\pi$
$347$	−14.3989	−0.772972	−0.386486	+	0.922295i	$0.626311\pi$
$348$	0	0
$349$	−6.83821	−0.366041	−0.183021	−	0.983109i	$-0.558587\pi$
$349$	−6.83821	−0.366041	−0.183021	+	0.983109i	$0.558587\pi$
$350$	0	0
$351$	−87.9823	−4.69615
$352$	0	0
$353$	19.1496	1.01923	0.509614	−	0.860403i	$-0.329788\pi$
$353$	19.1496	1.01923	0.509614	+	0.860403i	$0.329788\pi$
$354$	0	0
$355$	13.0218	0.691124
$356$	0	0
$357$	14.4598	0.765294
$358$	0	0
$359$	21.9926	1.16073	0.580363	−	0.814358i	$-0.302911\pi$
$359$	21.9926	1.16073	0.580363	+	0.814358i	$0.302911\pi$
$360$	0	0
$361$	−18.7792	−0.988378
$362$	0	0
$363$	−33.2158	−1.74338
$364$	0	0
$365$	−15.6490	−0.819105
$366$	0	0
$367$	−14.1132	−0.736701	−0.368351	−	0.929687i	$-0.620077\pi$
$367$	−14.1132	−0.736701	−0.368351	+	0.929687i	$0.620077\pi$
$368$	0	0
$369$	4.74899	0.247222
$370$	0	0
$371$	2.61390	0.135707
$372$	0	0
$373$	−13.5386	−0.701003	−0.350501	−	0.936562i	$-0.613989\pi$
$373$	−13.5386	−0.701003	−0.350501	+	0.936562i	$0.613989\pi$
$374$	0	0
$375$	3.15365	0.162854
$376$	0	0
$377$	−24.5543	−1.26461
$378$	0	0
$379$	30.5153	1.56746	0.783732	−	0.621100i	$-0.213314\pi$
$379$	30.5153	1.56746	0.783732	+	0.621100i	$0.213314\pi$
$380$	0	0
$381$	10.7006	0.548209
$382$	0	0
$383$	−29.5348	−1.50916	−0.754579	−	0.656209i	$-0.772159\pi$
$383$	−29.5348	−1.50916	−0.754579	+	0.656209i	$0.772159\pi$
$384$	0	0
$385$	0.396556	0.0202104
$386$	0	0
$387$	80.4550	4.08976
$388$	0	0
$389$	−23.1560	−1.17406	−0.587029	−	0.809566i	$-0.699703\pi$
$389$	−23.1560	−1.17406	−0.587029	+	0.809566i	$0.699703\pi$
$390$	0	0
$391$	27.4530	1.38836
$392$	0	0
$393$	−2.89155	−0.145859
$394$	0	0
$395$	8.76169	0.440849
$396$	0	0
$397$	21.6875	1.08847	0.544233	−	0.838934i	$-0.316821\pi$
$397$	21.6875	1.08847	0.544233	+	0.838934i	$0.316821\pi$
$398$	0	0
$399$	−0.859470	−0.0430273
$400$	0	0
$401$	10.8476	0.541702	0.270851	−	0.962621i	$-0.412695\pi$
$401$	10.8476	0.541702	0.270851	+	0.962621i	$0.412695\pi$
$402$	0	0
$403$	44.6885	2.22609
$404$	0	0
$405$	18.4037	0.914486
$406$	0	0
$407$	−0.683748	−0.0338921
$408$	0	0
$409$	−6.35026	−0.314000	−0.157000	−	0.987599i	$-0.550182\pi$
$409$	−6.35026	−0.314000	−0.157000	+	0.987599i	$0.550182\pi$
$410$	0	0
$411$	−60.0802	−2.96354
$412$	0	0
$413$	3.97442	0.195569
$414$	0	0
$415$	−1.84089	−0.0903657
$416$	0	0
$417$	−51.0672	−2.50077
$418$	0	0
$419$	−8.57892	−0.419108	−0.209554	−	0.977797i	$-0.567201\pi$
$419$	−8.57892	−0.419108	−0.209554	+	0.977797i	$0.567201\pi$
$420$	0	0
$421$	9.92683	0.483804	0.241902	−	0.970301i	$-0.422229\pi$
$421$	9.92683	0.483804	0.241902	+	0.970301i	$0.422229\pi$
$422$	0	0
$423$	−34.7537	−1.68978
$424$	0	0
$425$	7.90570	0.383483
$426$	0	0
$427$	2.15641	0.104356
$428$	0	0
$429$	−15.2471	−0.736137
$430$	0	0
$431$	16.9499	0.816450	0.408225	−	0.912881i	$-0.366148\pi$
$431$	16.9499	0.816450	0.408225	+	0.912881i	$0.366148\pi$
$432$	0	0
$433$	7.94228	0.381682	0.190841	−	0.981621i	$-0.438879\pi$
$433$	7.94228	0.381682	0.190841	+	0.981621i	$0.438879\pi$
$434$	0	0
$435$	10.9512	0.525071
$436$	0	0
$437$	−1.63177	−0.0780579
$438$	0	0
$439$	27.8957	1.33139	0.665696	−	0.746223i	$-0.268135\pi$
$439$	27.8957	1.33139	0.665696	+	0.746223i	$0.268135\pi$
$440$	0	0
$441$	−46.2824	−2.20392
$442$	0	0
$443$	15.7978	0.750574	0.375287	−	0.926909i	$-0.377544\pi$
$443$	15.7978	0.750574	0.375287	+	0.926909i	$0.377544\pi$
$444$	0	0
$445$	−4.74368	−0.224872
$446$	0	0
$447$	−0.429968	−0.0203368
$448$	0	0
$449$	15.4854	0.730800	0.365400	−	0.930851i	$-0.380932\pi$
$449$	15.4854	0.730800	0.365400	+	0.930851i	$0.380932\pi$
$450$	0	0
$451$	0.467512	0.0220143
$452$	0	0
$453$	−50.8892	−2.39098
$454$	0	0
$455$	−4.10097	−0.192256
$456$	0	0
$457$	2.83358	0.132549	0.0662746	−	0.997801i	$-0.478889\pi$
$457$	2.83358	0.132549	0.0662746	+	0.997801i	$0.478889\pi$
$458$	0	0
$459$	98.3689	4.59147
$460$	0	0
$461$	12.4313	0.578983	0.289492	−	0.957181i	$-0.406514\pi$
$461$	12.4313	0.578983	0.289492	+	0.957181i	$0.406514\pi$
$462$	0	0
$463$	5.18198	0.240827	0.120413	−	0.992724i	$-0.461578\pi$
$463$	5.18198	0.240827	0.120413	+	0.992724i	$0.461578\pi$
$464$	0	0
$465$	−19.9311	−0.924283
$466$	0	0
$467$	4.23327	0.195892	0.0979462	−	0.995192i	$-0.468773\pi$
$467$	4.23327	0.195892	0.0979462	+	0.995192i	$0.468773\pi$
$468$	0	0
$469$	−4.37162	−0.201863
$470$	0	0
$471$	−24.6966	−1.13796
$472$	0	0
$473$	7.92036	0.364178
$474$	0	0
$475$	−0.469903	−0.0215606
$476$	0	0
$477$	31.3029	1.43326
$478$	0	0
$479$	26.3556	1.20422	0.602110	−	0.798413i	$-0.294327\pi$
$479$	26.3556	1.20422	0.602110	+	0.798413i	$0.294327\pi$
$480$	0	0
$481$	7.07095	0.322407
$482$	0	0
$483$	6.35143	0.289000
$484$	0	0
$485$	−11.7860	−0.535174
$486$	0	0
$487$	38.3170	1.73631	0.868155	−	0.496293i	$-0.165306\pi$
$487$	38.3170	1.73631	0.868155	+	0.496293i	$0.165306\pi$
$488$	0	0
$489$	27.8257	1.25832
$490$	0	0
$491$	−17.3990	−0.785207	−0.392604	−	0.919708i	$-0.628426\pi$
$491$	−17.3990	−0.785207	−0.392604	+	0.919708i	$0.628426\pi$
$492$	0	0
$493$	27.4530	1.23642
$494$	0	0
$495$	4.74899	0.213451
$496$	0	0
$497$	7.55229	0.338767
$498$	0	0
$499$	18.2907	0.818804	0.409402	−	0.912354i	$-0.365737\pi$
$499$	18.2907	0.818804	0.409402	+	0.912354i	$0.365737\pi$
$500$	0	0
$501$	−9.29042	−0.415065
$502$	0	0
$503$	5.30184	0.236398	0.118199	−	0.992990i	$-0.462288\pi$
$503$	5.30184	0.236398	0.118199	+	0.992990i	$0.462288\pi$
$504$	0	0
$505$	−4.02516	−0.179117
$506$	0	0
$507$	116.680	5.18193
$508$	0	0
$509$	15.2634	0.676539	0.338270	−	0.941049i	$-0.390158\pi$
$509$	15.2634	0.676539	0.338270	+	0.941049i	$0.390158\pi$
$510$	0	0
$511$	−9.07600	−0.401499
$512$	0	0
$513$	−5.84691	−0.258147
$514$	0	0
$515$	3.66950	0.161697
$516$	0	0
$517$	−3.42131	−0.150469
$518$	0	0
$519$	37.6618	1.65317
$520$	0	0
$521$	−29.8966	−1.30979	−0.654897	−	0.755718i	$-0.727288\pi$
$521$	−29.8966	−1.30979	−0.654897	+	0.755718i	$0.727288\pi$
$522$	0	0
$523$	11.9033	0.520496	0.260248	−	0.965542i	$-0.416196\pi$
$523$	11.9033	0.520496	0.260248	+	0.965542i	$0.416196\pi$
$524$	0	0
$525$	1.82904	0.0798257
$526$	0	0
$527$	−49.9641	−2.17647
$528$	0	0
$529$	−10.9414	−0.475711
$530$	0	0
$531$	47.5960	2.06549
$532$	0	0
$533$	−4.83475	−0.209416
$534$	0	0
$535$	15.8898	0.686974
$536$	0	0
$537$	62.5258	2.69819
$538$	0	0
$539$	−4.55624	−0.196251
$540$	0	0
$541$	31.9349	1.37299	0.686493	−	0.727136i	$-0.259149\pi$
$541$	31.9349	1.37299	0.686493	+	0.727136i	$0.259149\pi$
$542$	0	0
$543$	−50.5448	−2.16908
$544$	0	0
$545$	−9.89104	−0.423685
$546$	0	0
$547$	−40.0812	−1.71375	−0.856874	−	0.515526i	$-0.827597\pi$
$547$	−40.0812	−1.71375	−0.856874	+	0.515526i	$0.827597\pi$
$548$	0	0
$549$	25.8242	1.10215
$550$	0	0
$551$	−1.63177	−0.0695155
$552$	0	0
$553$	5.08156	0.216090
$554$	0	0
$555$	−3.15365	−0.133865
$556$	0	0
$557$	37.7313	1.59873	0.799363	−	0.600848i	$-0.205170\pi$
$557$	37.7313	1.59873	0.799363	+	0.600848i	$0.205170\pi$
$558$	0	0
$559$	−81.9080	−3.46434
$560$	0	0
$561$	17.0471	0.719728
$562$	0	0
$563$	4.96163	0.209108	0.104554	−	0.994519i	$-0.466659\pi$
$563$	4.96163	0.209108	0.104554	+	0.994519i	$0.466659\pi$
$564$	0	0
$565$	−1.40145	−0.0589596
$566$	0	0
$567$	10.6737	0.448251
$568$	0	0
$569$	4.11427	0.172479	0.0862395	−	0.996274i	$-0.472515\pi$
$569$	4.11427	0.172479	0.0862395	+	0.996274i	$0.472515\pi$
$570$	0	0
$571$	−16.0968	−0.673629	−0.336814	−	0.941571i	$-0.609350\pi$
$571$	−16.0968	−0.673629	−0.336814	+	0.941571i	$0.609350\pi$
$572$	0	0
$573$	−68.1481	−2.84693
$574$	0	0
$575$	3.47256	0.144816
$576$	0	0
$577$	−44.6022	−1.85681	−0.928406	−	0.371567i	$-0.878821\pi$
$577$	−44.6022	−1.85681	−0.928406	+	0.371567i	$0.878821\pi$
$578$	0	0
$579$	40.7909	1.69521
$580$	0	0
$581$	−1.06767	−0.0442944
$582$	0	0
$583$	3.08160	0.127627
$584$	0	0
$585$	−49.1114	−2.03051
$586$	0	0
$587$	−38.0335	−1.56981	−0.784905	−	0.619617i	$-0.787288\pi$
$587$	−38.0335	−1.56981	−0.784905	+	0.619617i	$0.787288\pi$
$588$	0	0
$589$	2.96979	0.122368
$590$	0	0
$591$	13.7895	0.567223
$592$	0	0
$593$	−0.0972908	−0.00399525	−0.00199763	−	0.999998i	$-0.500636\pi$
$593$	−0.0972908	−0.00399525	−0.00199763	+	0.999998i	$0.500636\pi$
$594$	0	0
$595$	4.58510	0.187971
$596$	0	0
$597$	−29.5497	−1.20939
$598$	0	0
$599$	1.13685	0.0464505	0.0232253	−	0.999730i	$-0.492607\pi$
$599$	1.13685	0.0464505	0.0232253	+	0.999730i	$0.492607\pi$
$600$	0	0
$601$	27.8392	1.13559	0.567793	−	0.823171i	$-0.307797\pi$
$601$	27.8392	1.13559	0.567793	+	0.823171i	$0.307797\pi$
$602$	0	0
$603$	−52.3526	−2.13196
$604$	0	0
$605$	−10.5325	−0.428207
$606$	0	0
$607$	−37.6061	−1.52638	−0.763191	−	0.646172i	$-0.776369\pi$
$607$	−37.6061	−1.52638	−0.763191	+	0.646172i	$0.776369\pi$
$608$	0	0
$609$	6.35143	0.257373
$610$	0	0
$611$	35.3813	1.43137
$612$	0	0
$613$	17.3580	0.701085	0.350542	−	0.936547i	$-0.385997\pi$
$613$	17.3580	0.701085	0.350542	+	0.936547i	$0.385997\pi$
$614$	0	0
$615$	2.15630	0.0869505
$616$	0	0
$617$	−4.75638	−0.191485	−0.0957423	−	0.995406i	$-0.530522\pi$
$617$	−4.75638	−0.191485	−0.0957423	+	0.995406i	$0.530522\pi$
$618$	0	0
$619$	−41.2861	−1.65943	−0.829713	−	0.558190i	$-0.811496\pi$
$619$	−41.2861	−1.65943	−0.829713	+	0.558190i	$0.811496\pi$
$620$	0	0
$621$	43.2083	1.73389
$622$	0	0
$623$	−2.75121	−0.110225
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−1.01325	−0.0404655
$628$	0	0
$629$	−7.90570	−0.315221
$630$	0	0
$631$	0.205913	0.00819726	0.00409863	−	0.999992i	$-0.498695\pi$
$631$	0.205913	0.00819726	0.00409863	+	0.999992i	$0.498695\pi$
$632$	0	0
$633$	57.0349	2.26693
$634$	0	0
$635$	3.39309	0.134651
$636$	0	0
$637$	47.1182	1.86689
$638$	0	0
$639$	90.4429	3.57787
$640$	0	0
$641$	5.81866	0.229823	0.114912	−	0.993376i	$-0.463342\pi$
$641$	5.81866	0.229823	0.114912	+	0.993376i	$0.463342\pi$
$642$	0	0
$643$	−15.3510	−0.605384	−0.302692	−	0.953088i	$-0.597885\pi$
$643$	−15.3510	−0.605384	−0.302692	+	0.953088i	$0.597885\pi$
$644$	0	0
$645$	36.5311	1.43841
$646$	0	0
$647$	25.7278	1.01147	0.505733	−	0.862690i	$-0.331222\pi$
$647$	25.7278	1.01147	0.505733	+	0.862690i	$0.331222\pi$
$648$	0	0
$649$	4.68556	0.183924
$650$	0	0
$651$	−11.5595	−0.453054
$652$	0	0
$653$	−8.75258	−0.342515	−0.171257	−	0.985226i	$-0.554783\pi$
$653$	−8.75258	−0.342515	−0.171257	+	0.985226i	$0.554783\pi$
$654$	0	0
$655$	−0.916889	−0.0358258
$656$	0	0
$657$	−108.690	−4.24041
$658$	0	0
$659$	5.23970	0.204110	0.102055	−	0.994779i	$-0.467458\pi$
$659$	5.23970	0.204110	0.102055	+	0.994779i	$0.467458\pi$
$660$	0	0
$661$	−4.49606	−0.174876	−0.0874382	−	0.996170i	$-0.527868\pi$
$661$	−4.49606	−0.174876	−0.0874382	+	0.996170i	$0.527868\pi$
$662$	0	0
$663$	−176.292	−6.84660
$664$	0	0
$665$	−0.272532	−0.0105683
$666$	0	0
$667$	12.0586	0.466912
$668$	0	0
$669$	−80.4360	−3.10983
$670$	0	0
$671$	2.54225	0.0981426
$672$	0	0
$673$	9.56573	0.368732	0.184366	−	0.982858i	$-0.440977\pi$
$673$	9.56573	0.368732	0.184366	+	0.982858i	$0.440977\pi$
$674$	0	0
$675$	12.4428	0.478923
$676$	0	0
$677$	−41.4612	−1.59348	−0.796741	−	0.604321i	$-0.793445\pi$
$677$	−41.4612	−1.59348	−0.796741	+	0.604321i	$0.793445\pi$
$678$	0	0
$679$	−6.83556	−0.262325
$680$	0	0
$681$	−53.4460	−2.04805
$682$	0	0
$683$	11.1219	0.425569	0.212785	−	0.977099i	$-0.431747\pi$
$683$	11.1219	0.425569	0.212785	+	0.977099i	$0.431747\pi$
$684$	0	0
$685$	−19.0510	−0.727901
$686$	0	0
$687$	−31.0443	−1.18441
$688$	0	0
$689$	−31.8682	−1.21408
$690$	0	0
$691$	−8.38555	−0.319001	−0.159501	−	0.987198i	$-0.550988\pi$
$691$	−8.38555	−0.319001	−0.159501	+	0.987198i	$0.550988\pi$
$692$	0	0
$693$	2.75429	0.104627
$694$	0	0
$695$	−16.1930	−0.614237
$696$	0	0
$697$	5.40551	0.204748
$698$	0	0
$699$	−64.4332	−2.43709
$700$	0	0
$701$	−51.7038	−1.95283	−0.976413	−	0.215910i	$-0.930728\pi$
$701$	−51.7038	−1.95283	−0.976413	+	0.215910i	$0.930728\pi$
$702$	0	0
$703$	0.469903	0.0177227
$704$	0	0
$705$	−15.7801	−0.594313
$706$	0	0
$707$	−2.33449	−0.0877975
$708$	0	0
$709$	−36.9434	−1.38744	−0.693720	−	0.720245i	$-0.744029\pi$
$709$	−36.9434	−1.38744	−0.693720	+	0.720245i	$0.744029\pi$
$710$	0	0
$711$	60.8545	2.28222
$712$	0	0
$713$	−21.9466	−0.821906
$714$	0	0
$715$	−4.83475	−0.180809
$716$	0	0
$717$	27.3931	1.02301
$718$	0	0
$719$	−41.9193	−1.56333	−0.781663	−	0.623701i	$-0.785628\pi$
$719$	−41.9193	−1.56333	−0.781663	+	0.623701i	$0.785628\pi$
$720$	0	0
$721$	2.12821	0.0792588
$722$	0	0
$723$	−34.8510	−1.29612
$724$	0	0
$725$	3.47256	0.128967
$726$	0	0
$727$	−27.6971	−1.02723	−0.513615	−	0.858021i	$-0.671694\pi$
$727$	−27.6971	−1.02723	−0.513615	+	0.858021i	$0.671694\pi$
$728$	0	0
$729$	10.1023	0.374161
$730$	0	0
$731$	91.5775	3.38712
$732$	0	0
$733$	−1.69178	−0.0624872	−0.0312436	−	0.999512i	$-0.509947\pi$
$733$	−1.69178	−0.0624872	−0.0312436	+	0.999512i	$0.509947\pi$
$734$	0	0
$735$	−21.0148	−0.775141
$736$	0	0
$737$	−5.15383	−0.189844
$738$	0	0
$739$	41.9571	1.54342	0.771708	−	0.635977i	$-0.219403\pi$
$739$	41.9571	1.54342	0.771708	+	0.635977i	$0.219403\pi$
$740$	0	0
$741$	10.4785	0.384938
$742$	0	0
$743$	0.905681	0.0332262	0.0166131	−	0.999862i	$-0.494712\pi$
$743$	0.905681	0.0332262	0.0166131	+	0.999862i	$0.494712\pi$
$744$	0	0
$745$	−0.136340	−0.00499510
$746$	0	0
$747$	−12.7859	−0.467813
$748$	0	0
$749$	9.21565	0.336732
$750$	0	0
$751$	−13.1660	−0.480433	−0.240216	−	0.970719i	$-0.577218\pi$
$751$	−13.1660	−0.480433	−0.240216	+	0.970719i	$0.577218\pi$
$752$	0	0
$753$	16.9818	0.618851
$754$	0	0
$755$	−16.1366	−0.587271
$756$	0	0
$757$	37.5239	1.36383	0.681915	−	0.731431i	$-0.261147\pi$
$757$	37.5239	1.36383	0.681915	+	0.731431i	$0.261147\pi$
$758$	0	0
$759$	7.48788	0.271793
$760$	0	0
$761$	46.4057	1.68221	0.841103	−	0.540876i	$-0.181907\pi$
$761$	46.4057	1.68221	0.841103	+	0.540876i	$0.181907\pi$
$762$	0	0
$763$	−5.73655	−0.207677
$764$	0	0
$765$	54.9092	1.98524
$766$	0	0
$767$	−48.4555	−1.74963
$768$	0	0
$769$	−26.4629	−0.954276	−0.477138	−	0.878828i	$-0.658326\pi$
$769$	−26.4629	−0.954276	−0.477138	+	0.878828i	$0.658326\pi$
$770$	0	0
$771$	−78.4677	−2.82594
$772$	0	0
$773$	−6.19093	−0.222672	−0.111336	−	0.993783i	$-0.535513\pi$
$773$	−6.19093	−0.222672	−0.111336	+	0.993783i	$0.535513\pi$
$774$	0	0
$775$	−6.32001	−0.227021
$776$	0	0
$777$	−1.82904	−0.0656163
$778$	0	0
$779$	−0.321296	−0.0115116
$780$	0	0
$781$	8.90361	0.318596
$782$	0	0
$783$	43.2083	1.54414
$784$	0	0
$785$	−7.83110	−0.279504
$786$	0	0
$787$	−20.1656	−0.718826	−0.359413	−	0.933179i	$-0.617023\pi$
$787$	−20.1656	−0.718826	−0.359413	+	0.933179i	$0.617023\pi$
$788$	0	0
$789$	23.6261	0.841112
$790$	0	0
$791$	−0.812806	−0.0289001
$792$	0	0
$793$	−26.2906	−0.933606
$794$	0	0
$795$	14.2133	0.504092
$796$	0	0
$797$	−34.4292	−1.21955	−0.609773	−	0.792576i	$-0.708739\pi$
$797$	−34.4292	−1.21955	−0.609773	+	0.792576i	$0.708739\pi$
$798$	0	0
$799$	−39.5582	−1.39947
$800$	0	0
$801$	−32.9473	−1.16414
$802$	0	0
$803$	−10.7000	−0.377593
$804$	0	0
$805$	2.01399	0.0709839
$806$	0	0
$807$	32.9633	1.16036
$808$	0	0
$809$	20.1067	0.706915	0.353458	−	0.935451i	$-0.385006\pi$
$809$	20.1067	0.706915	0.353458	+	0.935451i	$0.385006\pi$
$810$	0	0
$811$	16.2982	0.572307	0.286154	−	0.958184i	$-0.407623\pi$
$811$	16.2982	0.572307	0.286154	+	0.958184i	$0.407623\pi$
$812$	0	0
$813$	−76.6334	−2.68765
$814$	0	0
$815$	8.82333	0.309068
$816$	0	0
$817$	−5.44324	−0.190435
$818$	0	0
$819$	−28.4833	−0.995289
$820$	0	0
$821$	31.2634	1.09110	0.545550	−	0.838078i	$-0.316321\pi$
$821$	31.2634	1.09110	0.545550	+	0.838078i	$0.316321\pi$
$822$	0	0
$823$	−27.1994	−0.948112	−0.474056	−	0.880495i	$-0.657211\pi$
$823$	−27.1994	−0.948112	−0.474056	+	0.880495i	$0.657211\pi$
$824$	0	0
$825$	2.15630	0.0750728
$826$	0	0
$827$	−0.707380	−0.0245980	−0.0122990	−	0.999924i	$-0.503915\pi$
$827$	−0.707380	−0.0245980	−0.0122990	+	0.999924i	$0.503915\pi$
$828$	0	0
$829$	8.60732	0.298944	0.149472	−	0.988766i	$-0.452243\pi$
$829$	8.60732	0.298944	0.149472	+	0.988766i	$0.452243\pi$
$830$	0	0
$831$	92.7289	3.21673
$832$	0	0
$833$	−52.6806	−1.82528
$834$	0	0
$835$	−2.94593	−0.101948
$836$	0	0
$837$	−78.6386	−2.71815
$838$	0	0
$839$	34.3578	1.18616	0.593081	−	0.805143i	$-0.297911\pi$
$839$	34.3578	1.18616	0.593081	+	0.805143i	$0.297911\pi$
$840$	0	0
$841$	−16.9414	−0.584185
$842$	0	0
$843$	−40.9415	−1.41010
$844$	0	0
$845$	36.9983	1.27278
$846$	0	0
$847$	−6.10857	−0.209893
$848$	0	0
$849$	25.6493	0.880282
$850$	0	0
$851$	−3.47256	−0.119038
$852$	0	0
$853$	−8.60030	−0.294469	−0.147234	−	0.989102i	$-0.547037\pi$
$853$	−8.60030	−0.294469	−0.147234	+	0.989102i	$0.547037\pi$
$854$	0	0
$855$	−3.26372	−0.111617
$856$	0	0
$857$	−34.0578	−1.16339	−0.581697	−	0.813406i	$-0.697611\pi$
$857$	−34.0578	−1.16339	−0.581697	+	0.813406i	$0.697611\pi$
$858$	0	0
$859$	38.5279	1.31455	0.657277	−	0.753649i	$-0.271708\pi$
$859$	38.5279	1.31455	0.657277	+	0.753649i	$0.271708\pi$
$860$	0	0
$861$	1.25060	0.0426203
$862$	0	0
$863$	32.6268	1.11063	0.555315	−	0.831640i	$-0.312598\pi$
$863$	32.6268	1.11063	0.555315	+	0.831640i	$0.312598\pi$
$864$	0	0
$865$	11.9423	0.406050
$866$	0	0
$867$	143.491	4.87322
$868$	0	0
$869$	5.99079	0.203224
$870$	0	0
$871$	53.2981	1.80594
$872$	0	0
$873$	−81.8597	−2.77053
$874$	0	0
$875$	0.579974	0.0196067
$876$	0	0
$877$	35.1206	1.18594	0.592969	−	0.805225i	$-0.297956\pi$
$877$	35.1206	1.18594	0.592969	+	0.805225i	$0.297956\pi$
$878$	0	0
$879$	−12.6780	−0.427619
$880$	0	0
$881$	21.7832	0.733896	0.366948	−	0.930241i	$-0.380403\pi$
$881$	21.7832	0.733896	0.366948	+	0.930241i	$0.380403\pi$
$882$	0	0
$883$	14.5994	0.491310	0.245655	−	0.969357i	$-0.420997\pi$
$883$	14.5994	0.491310	0.245655	+	0.969357i	$0.420997\pi$
$884$	0	0
$885$	21.6112	0.726453
$886$	0	0
$887$	−45.0633	−1.51308	−0.756538	−	0.653949i	$-0.773111\pi$
$887$	−45.0633	−1.51308	−0.756538	+	0.653949i	$0.773111\pi$
$888$	0	0
$889$	1.96790	0.0660014
$890$	0	0
$891$	12.5835	0.421562
$892$	0	0
$893$	2.35128	0.0786827
$894$	0	0
$895$	19.8265	0.662726
$896$	0	0
$897$	−77.4356	−2.58550
$898$	0	0
$899$	−21.9466	−0.731960
$900$	0	0
$901$	35.6303	1.18702
$902$	0	0
$903$	21.1871	0.705062
$904$	0	0
$905$	−16.0274	−0.532768
$906$	0	0
$907$	−24.0077	−0.797164	−0.398582	−	0.917133i	$-0.630498\pi$
$907$	−24.0077	−0.797164	−0.398582	+	0.917133i	$0.630498\pi$
$908$	0	0
$909$	−27.9568	−0.927270
$910$	0	0
$911$	23.0249	0.762848	0.381424	−	0.924400i	$-0.375434\pi$
$911$	23.0249	0.762848	0.381424	+	0.924400i	$0.375434\pi$
$912$	0	0
$913$	−1.25871	−0.0416571
$914$	0	0
$915$	11.7256	0.387637
$916$	0	0
$917$	−0.531772	−0.0175607
$918$	0	0
$919$	10.0302	0.330867	0.165434	−	0.986221i	$-0.447098\pi$
$919$	10.0302	0.330867	0.165434	+	0.986221i	$0.447098\pi$
$920$	0	0
$921$	51.0716	1.68287
$922$	0	0
$923$	−92.0763	−3.03073
$924$	0	0
$925$	−1.00000	−0.0328798
$926$	0	0
$927$	25.4865	0.837088
$928$	0	0
$929$	−0.361502	−0.0118605	−0.00593024	−	0.999982i	$-0.501888\pi$
$929$	−0.361502	−0.0118605	−0.00593024	+	0.999982i	$0.501888\pi$
$930$	0	0
$931$	3.13126	0.102623
$932$	0	0
$933$	58.1590	1.90404
$934$	0	0
$935$	5.40551	0.176779
$936$	0	0
$937$	45.6702	1.49198	0.745990	−	0.665957i	$-0.231977\pi$
$937$	45.6702	1.49198	0.745990	+	0.665957i	$0.231977\pi$
$938$	0	0
$939$	53.5551	1.74771
$940$	0	0
$941$	−9.90879	−0.323017	−0.161509	−	0.986871i	$-0.551636\pi$
$941$	−9.90879	−0.323017	−0.161509	+	0.986871i	$0.551636\pi$
$942$	0	0
$943$	2.37435	0.0773196
$944$	0	0
$945$	7.21650	0.234753
$946$	0	0
$947$	−15.9299	−0.517651	−0.258825	−	0.965924i	$-0.583335\pi$
$947$	−15.9299	−0.517651	−0.258825	+	0.965924i	$0.583335\pi$
$948$	0	0
$949$	110.653	3.59195
$950$	0	0
$951$	−75.9571	−2.46308
$952$	0	0
$953$	3.95709	0.128183	0.0640913	−	0.997944i	$-0.479585\pi$
$953$	3.95709	0.128183	0.0640913	+	0.997944i	$0.479585\pi$
$954$	0	0
$955$	−21.6093	−0.699259
$956$	0	0
$957$	7.48788	0.242049
$958$	0	0
$959$	−11.0491	−0.356793
$960$	0	0
$961$	8.94253	0.288469
$962$	0	0
$963$	110.363	3.55638
$964$	0	0
$965$	12.9345	0.416376
$966$	0	0
$967$	−13.2033	−0.424588	−0.212294	−	0.977206i	$-0.568093\pi$
$967$	−13.2033	−0.424588	−0.212294	+	0.977206i	$0.568093\pi$
$968$	0	0
$969$	−11.7155	−0.376357
$970$	0	0
$971$	−32.1047	−1.03029	−0.515144	−	0.857103i	$-0.672262\pi$
$971$	−32.1047	−1.03029	−0.515144	+	0.857103i	$0.672262\pi$
$972$	0	0
$973$	−9.39154	−0.301079
$974$	0	0
$975$	−22.2993	−0.714149
$976$	0	0
$977$	−50.0220	−1.60035	−0.800174	−	0.599769i	$-0.795259\pi$
$977$	−50.0220	−1.60035	−0.800174	+	0.599769i	$0.795259\pi$
$978$	0	0
$979$	−3.24348	−0.103662
$980$	0	0
$981$	−68.6984	−2.19337
$982$	0	0
$983$	25.6681	0.818687	0.409343	−	0.912380i	$-0.365758\pi$
$983$	25.6681	0.818687	0.409343	+	0.912380i	$0.365758\pi$
$984$	0	0
$985$	4.37254	0.139321
$986$	0	0
$987$	−9.15206	−0.291313
$988$	0	0
$989$	40.2252	1.27909
$990$	0	0
$991$	−14.2126	−0.451477	−0.225739	−	0.974188i	$-0.572480\pi$
$991$	−14.2126	−0.451477	−0.225739	+	0.974188i	$0.572480\pi$
$992$	0	0
$993$	−56.0626	−1.77909
$994$	0	0
$995$	−9.36999	−0.297049
$996$	0	0
$997$	47.3052	1.49817	0.749086	−	0.662473i	$-0.230493\pi$
$997$	47.3052	1.49817	0.749086	+	0.662473i	$0.230493\pi$
$998$	0	0
$999$	−12.4428	−0.393672

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2960.2.a.bb.1.5		5
4.3	odd	2		1480.2.a.g.1.1	✓	5
20.19	odd	2		7400.2.a.r.1.5		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1480.2.a.g.1.1	✓	5	4.3	odd	2
2960.2.a.bb.1.5		5	1.1	even	1	trivial
7400.2.a.r.1.5		5	20.19	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2960 = 2^{4} \cdot 5 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2960.a (trivial)

\(f(q)\)	\(=\)	\(q+3.15365 q^{3} +1.00000 q^{5} +0.579974 q^{7} +6.94552 q^{9} +O(q^{10})\)	\(q+3.15365 q^{3} +1.00000 q^{5} +0.579974 q^{7} +6.94552 q^{9} +0.683748 q^{11} -7.07095 q^{13} +3.15365 q^{15} +7.90570 q^{17} -0.469903 q^{19} +1.82904 q^{21} +3.47256 q^{23} +1.00000 q^{25} +12.4428 q^{27} +3.47256 q^{29} -6.32001 q^{31} +2.15630 q^{33} +0.579974 q^{35} -1.00000 q^{37} -22.2993 q^{39} +0.683748 q^{41} +11.5837 q^{43} +6.94552 q^{45} -5.00376 q^{47} -6.66363 q^{49} +24.9318 q^{51} +4.50692 q^{53} +0.683748 q^{55} -1.48191 q^{57} +6.85276 q^{59} +3.71811 q^{61} +4.02822 q^{63} -7.07095 q^{65} -7.53761 q^{67} +10.9512 q^{69} +13.0218 q^{71} -15.6490 q^{73} +3.15365 q^{75} +0.396556 q^{77} +8.76169 q^{79} +18.4037 q^{81} -1.84089 q^{83} +7.90570 q^{85} +10.9512 q^{87} -4.74368 q^{89} -4.10097 q^{91} -19.9311 q^{93} -0.469903 q^{95} -11.7860 q^{97} +4.74899 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 5 q^{3} + 5 q^{5} + 5 q^{7} + 6 q^{9}+O(q^{10}) \) 5 * q + 5 * q^3 + 5 * q^5 + 5 * q^7 + 6 * q^9	\( 5 q + 5 q^{3} + 5 q^{5} + 5 q^{7} + 6 q^{9} + 7 q^{11} - 6 q^{13} + 5 q^{15} + 12 q^{19} - q^{21} + 6 q^{23} + 5 q^{25} + 17 q^{27} + 6 q^{29} + 10 q^{31} + 3 q^{33} + 5 q^{35} - 5 q^{37} - 4 q^{39} + 7 q^{41} + 22 q^{43} + 6 q^{45} + 13 q^{47} + 14 q^{49} + 10 q^{51} - 11 q^{53} + 7 q^{55} - 8 q^{57} + 10 q^{59} + 10 q^{63} - 6 q^{65} + 24 q^{67} + 26 q^{69} + 13 q^{71} - 13 q^{73} + 5 q^{75} - 19 q^{77} - 2 q^{79} + q^{81} + 13 q^{83} + 26 q^{87} + 8 q^{89} + 8 q^{91} - 20 q^{93} + 12 q^{95} - 20 q^{97} - 2 q^{99}+O(q^{100}) \) 5 * q + 5 * q^3 + 5 * q^5 + 5 * q^7 + 6 * q^9 + 7 * q^11 - 6 * q^13 + 5 * q^15 + 12 * q^19 - q^21 + 6 * q^23 + 5 * q^25 + 17 * q^27 + 6 * q^29 + 10 * q^31 + 3 * q^33 + 5 * q^35 - 5 * q^37 - 4 * q^39 + 7 * q^41 + 22 * q^43 + 6 * q^45 + 13 * q^47 + 14 * q^49 + 10 * q^51 - 11 * q^53 + 7 * q^55 - 8 * q^57 + 10 * q^59 + 10 * q^63 - 6 * q^65 + 24 * q^67 + 26 * q^69 + 13 * q^71 - 13 * q^73 + 5 * q^75 - 19 * q^77 - 2 * q^79 + q^81 + 13 * q^83 + 26 * q^87 + 8 * q^89 + 8 * q^91 - 20 * q^93 + 12 * q^95 - 20 * q^97 - 2 * q^99

Introduction

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