LMFDB - Embedded newform 2325.2.a.bb.1.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2325,2,Mod(1,2325)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2325.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2325, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2325 = 3 \cdot 5^{2} \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2325.a (trivial)

Newform invariants

comment:select newform

sage:traces = [6,1,6,7,0,1,2,-3,6,0,7] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$18.5652184699$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.75968016.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 9x^{4} + 9x^{3} + 14x^{2} - 6x - 6 $ x^6 - x^5 - 9x^4 + 9x^3 + 14x^2 - 6x - 6
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$2.46251$ of defining polynomial
Character	$\chi$	$=$	2325.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.46251 q^{2} +1.00000 q^{3} +4.06395 q^{4} +2.46251 q^{6} +3.91086 q^{7} +5.08250 q^{8} +1.00000 q^{9} +3.77308 q^{11} +4.06395 q^{12} -4.83704 q^{13} +9.63054 q^{14} +4.38781 q^{16} -5.89630 q^{17} +2.46251 q^{18} +5.60730 q^{19} +3.91086 q^{21} +9.29125 q^{22} -5.19547 q^{23} +5.08250 q^{24} -11.9112 q^{26} +1.00000 q^{27} +15.8936 q^{28} -6.33880 q^{29} -1.00000 q^{31} +0.640010 q^{32} +3.77308 q^{33} -14.5197 q^{34} +4.06395 q^{36} +11.2913 q^{37} +13.8080 q^{38} -4.83704 q^{39} +0.663143 q^{41} +9.63054 q^{42} -3.39308 q^{43} +15.3336 q^{44} -12.7939 q^{46} -1.31684 q^{47} +4.38781 q^{48} +8.29486 q^{49} -5.89630 q^{51} -19.6575 q^{52} -10.4443 q^{53} +2.46251 q^{54} +19.8770 q^{56} +5.60730 q^{57} -15.6093 q^{58} +7.58816 q^{59} -10.3208 q^{61} -2.46251 q^{62} +3.91086 q^{63} -7.19958 q^{64} +9.29125 q^{66} -10.2874 q^{67} -23.9623 q^{68} -5.19547 q^{69} -3.68083 q^{71} +5.08250 q^{72} +12.3909 q^{73} +27.8048 q^{74} +22.7878 q^{76} +14.7560 q^{77} -11.9112 q^{78} -9.23657 q^{79} +1.00000 q^{81} +1.63300 q^{82} -0.590907 q^{83} +15.8936 q^{84} -8.35548 q^{86} -6.33880 q^{87} +19.1767 q^{88} +15.8673 q^{89} -18.9170 q^{91} -21.1141 q^{92} -1.00000 q^{93} -3.24273 q^{94} +0.640010 q^{96} +3.98585 q^{97} +20.4262 q^{98} +3.77308 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + q^{2} + 6 q^{3} + 7 q^{4} + q^{6} + 2 q^{7} - 3 q^{8} + 6 q^{9} + 7 q^{11} + 7 q^{12} + 4 q^{13} + 10 q^{14} + 17 q^{16} + q^{18} + 17 q^{19} + 2 q^{21} + 2 q^{22} - q^{23} - 3 q^{24} + 2 q^{26}+ \cdots + 7 q^{99}+O(q^{100}) $ 6 * q + q^2 + 6 * q^3 + 7 * q^4 + q^6 + 2 * q^7 - 3 * q^8 + 6 * q^9 + 7 * q^11 + 7 * q^12 + 4 * q^13 + 10 * q^14 + 17 * q^16 + q^18 + 17 * q^19 + 2 * q^21 + 2 * q^22 - q^23 - 3 * q^24 + 2 * q^26 + 6 * q^27 + 22 * q^28 - 8 * q^29 - 6 * q^31 - 35 * q^32 + 7 * q^33 - 13 * q^34 + 7 * q^36 + 14 * q^37 + 10 * q^38 + 4 * q^39 + 18 * q^41 + 10 * q^42 + 15 * q^43 + 4 * q^44 - 10 * q^46 - 13 * q^47 + 17 * q^48 + 18 * q^49 - 38 * q^52 - 13 * q^53 + q^54 + 32 * q^56 + 17 * q^57 + 23 * q^58 + 32 * q^59 - q^62 + 2 * q^63 + 29 * q^64 + 2 * q^66 + 9 * q^67 + 9 * q^68 - q^69 + q^71 - 3 * q^72 + 14 * q^73 + 20 * q^74 + 42 * q^76 - 14 * q^77 + 2 * q^78 + 13 * q^79 + 6 * q^81 - 32 * q^82 + 9 * q^83 + 22 * q^84 - 11 * q^86 - 8 * q^87 + 36 * q^88 + 34 * q^89 - 2 * q^91 - 46 * q^92 - 6 * q^93 + 19 * q^94 - 35 * q^96 + 30 * q^97 + 37 * q^98 + 7 * q^99

Coefficient data

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

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

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

$2$	2.46251	1.74126	0.870629	−	0.491941i	$-0.163712\pi$
$2$	2.46251	1.74126	0.870629	+	0.491941i	$0.163712\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	4.06395	2.03198
$5$	0	0
$5$	0	0
$6$	2.46251	1.00532
$7$	3.91086	1.47817	0.739084	−	0.673613i	$-0.235259\pi$
$7$	3.91086	1.47817	0.739084	+	0.673613i	$0.235259\pi$
$8$	5.08250	1.79694
$9$	1.00000	0.333333
$10$	0	0
$11$	3.77308	1.13763	0.568814	−	0.822466i	$-0.307403\pi$
$11$	3.77308	1.13763	0.568814	+	0.822466i	$0.307403\pi$
$12$	4.06395	1.17316
$13$	−4.83704	−1.34155	−0.670776	−	0.741660i	$-0.734039\pi$
$13$	−4.83704	−1.34155	−0.670776	+	0.741660i	$0.734039\pi$
$14$	9.63054	2.57387
$15$	0	0
$16$	4.38781	1.09695
$17$	−5.89630	−1.43006	−0.715032	−	0.699092i	$-0.753588\pi$
$17$	−5.89630	−1.43006	−0.715032	+	0.699092i	$0.753588\pi$
$18$	2.46251	0.580419
$19$	5.60730	1.28640	0.643202	−	0.765697i	$-0.277606\pi$
$19$	5.60730	1.28640	0.643202	+	0.765697i	$0.277606\pi$
$20$	0	0
$21$	3.91086	0.853421
$22$	9.29125	1.98090
$23$	−5.19547	−1.08333	−0.541665	−	0.840595i	$-0.682206\pi$
$23$	−5.19547	−1.08333	−0.541665	+	0.840595i	$0.682206\pi$
$24$	5.08250	1.03746
$25$	0	0
$26$	−11.9112	−2.33599
$27$	1.00000	0.192450
$28$	15.8936	3.00360
$29$	−6.33880	−1.17709	−0.588543	−	0.808466i	$-0.700298\pi$
$29$	−6.33880	−1.17709	−0.588543	+	0.808466i	$0.700298\pi$
$30$	0	0
$31$	−1.00000	−0.179605
$31$	−1.00000	−0.179605
$32$	0.640010	0.113139
$33$	3.77308	0.656809
$34$	−14.5197	−2.49011
$35$	0	0
$36$	4.06395	0.677325
$37$	11.2913	1.85627	0.928135	−	0.372243i	$-0.121411\pi$
$37$	11.2913	1.85627	0.928135	+	0.372243i	$0.121411\pi$
$38$	13.8080	2.23996
$39$	−4.83704	−0.774546
$40$	0	0
$41$	0.663143	0.103566	0.0517828	−	0.998658i	$-0.483510\pi$
$41$	0.663143	0.103566	0.0517828	+	0.998658i	$0.483510\pi$
$42$	9.63054	1.48602
$43$	−3.39308	−0.517439	−0.258720	−	0.965952i	$-0.583301\pi$
$43$	−3.39308	−0.517439	−0.258720	+	0.965952i	$0.583301\pi$
$44$	15.3336	2.31163
$45$	0	0
$46$	−12.7939	−1.88636
$47$	−1.31684	−0.192081	−0.0960405	−	0.995377i	$-0.530618\pi$
$47$	−1.31684	−0.192081	−0.0960405	+	0.995377i	$0.530618\pi$
$48$	4.38781	0.633325
$49$	8.29486	1.18498
$50$	0	0
$51$	−5.89630	−0.825648
$52$	−19.6575	−2.72600
$53$	−10.4443	−1.43464	−0.717320	−	0.696744i	$-0.754632\pi$
$53$	−10.4443	−1.43464	−0.717320	+	0.696744i	$0.754632\pi$
$54$	2.46251	0.335105
$55$	0	0
$56$	19.8770	2.65617
$57$	5.60730	0.742706
$58$	−15.6093	−2.04961
$59$	7.58816	0.987895	0.493947	−	0.869492i	$-0.335554\pi$
$59$	7.58816	0.987895	0.493947	+	0.869492i	$0.335554\pi$
$60$	0	0
$61$	−10.3208	−1.32145	−0.660724	−	0.750629i	$-0.729751\pi$
$61$	−10.3208	−1.32145	−0.660724	+	0.750629i	$0.729751\pi$
$62$	−2.46251	−0.312739
$63$	3.91086	0.492723
$64$	−7.19958	−0.899948
$65$	0	0
$66$	9.29125	1.14367
$67$	−10.2874	−1.25680	−0.628400	−	0.777890i	$-0.716290\pi$
$67$	−10.2874	−1.25680	−0.628400	+	0.777890i	$0.716290\pi$
$68$	−23.9623	−2.90586
$69$	−5.19547	−0.625461
$70$	0	0
$71$	−3.68083	−0.436834	−0.218417	−	0.975856i	$-0.570089\pi$
$71$	−3.68083	−0.436834	−0.218417	+	0.975856i	$0.570089\pi$
$72$	5.08250	0.598979
$73$	12.3909	1.45025	0.725125	−	0.688618i	$-0.241782\pi$
$73$	12.3909	1.45025	0.725125	+	0.688618i	$0.241782\pi$
$74$	27.8048	3.23224
$75$	0	0
$76$	22.7878	2.61394
$77$	14.7560	1.68160
$78$	−11.9112	−1.34868
$79$	−9.23657	−1.03919	−0.519597	−	0.854411i	$-0.673918\pi$
$79$	−9.23657	−1.03919	−0.519597	+	0.854411i	$0.673918\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	1.63300	0.180334
$83$	−0.590907	−0.0648605	−0.0324302	−	0.999474i	$-0.510325\pi$
$83$	−0.590907	−0.0648605	−0.0324302	+	0.999474i	$0.510325\pi$
$84$	15.8936	1.73413
$85$	0	0
$86$	−8.35548	−0.900995
$87$	−6.33880	−0.679590
$88$	19.1767	2.04424
$89$	15.8673	1.68193	0.840964	−	0.541090i	$-0.181988\pi$
$89$	15.8673	1.68193	0.840964	+	0.541090i	$0.181988\pi$
$90$	0	0
$91$	−18.9170	−1.98304
$92$	−21.1141	−2.20130
$93$	−1.00000	−0.103695
$94$	−3.24273	−0.334462
$95$	0	0
$96$	0.640010	0.0653207
$97$	3.98585	0.404701	0.202351	−	0.979313i	$-0.435142\pi$
$97$	3.98585	0.404701	0.202351	+	0.979313i	$0.435142\pi$
$98$	20.4262	2.06335
$99$	3.77308	0.379209
$100$	0	0
$101$	−4.48372	−0.446147	−0.223073	−	0.974802i	$-0.571609\pi$
$101$	−4.48372	−0.446147	−0.223073	+	0.974802i	$0.571609\pi$
$102$	−14.5197	−1.43766
$103$	14.9600	1.47405	0.737025	−	0.675865i	$-0.236230\pi$
$103$	14.9600	1.47405	0.737025	+	0.675865i	$0.236230\pi$
$104$	−24.5842	−2.41068
$105$	0	0
$106$	−25.7193	−2.49808
$107$	1.47421	0.142517	0.0712585	−	0.997458i	$-0.477298\pi$
$107$	1.47421	0.142517	0.0712585	+	0.997458i	$0.477298\pi$
$108$	4.06395	0.391054
$109$	12.7771	1.22382	0.611911	−	0.790927i	$-0.290401\pi$
$109$	12.7771	1.22382	0.611911	+	0.790927i	$0.290401\pi$
$110$	0	0
$111$	11.2913	1.07172
$112$	17.1601	1.62148
$113$	−4.71980	−0.444002	−0.222001	−	0.975046i	$-0.571259\pi$
$113$	−4.71980	−0.444002	−0.222001	+	0.975046i	$0.571259\pi$
$114$	13.8080	1.29324
$115$	0	0
$116$	−25.7606	−2.39181
$117$	−4.83704	−0.447184
$118$	18.6859	1.72018
$119$	−23.0596	−2.11387
$120$	0	0
$121$	3.23615	0.294196
$122$	−25.4152	−2.30098
$123$	0.663143	0.0597936
$124$	−4.06395	−0.364954
$125$	0	0
$126$	9.63054	0.857957
$127$	−14.7204	−1.30622	−0.653111	−	0.757262i	$-0.726536\pi$
$127$	−14.7204	−1.30622	−0.653111	+	0.757262i	$0.726536\pi$
$128$	−19.0091	−1.68018
$129$	−3.39308	−0.298744
$130$	0	0
$131$	10.9970	0.960810	0.480405	−	0.877047i	$-0.340490\pi$
$131$	10.9970	0.960810	0.480405	+	0.877047i	$0.340490\pi$
$132$	15.3336	1.33462
$133$	21.9294	1.90152
$134$	−25.3327	−2.18841
$135$	0	0
$136$	−29.9680	−2.56973
$137$	3.29038	0.281116	0.140558	−	0.990072i	$-0.455110\pi$
$137$	3.29038	0.281116	0.140558	+	0.990072i	$0.455110\pi$
$138$	−12.7939	−1.08909
$139$	11.7607	0.997527	0.498764	−	0.866738i	$-0.333788\pi$
$139$	11.7607	0.997527	0.498764	+	0.866738i	$0.333788\pi$
$140$	0	0
$141$	−1.31684	−0.110898
$142$	−9.06408	−0.760641
$143$	−18.2505	−1.52619
$144$	4.38781	0.365651
$145$	0	0
$146$	30.5128	2.52526
$147$	8.29486	0.684149
$148$	45.8871	3.77190
$149$	−0.0294614	−0.00241357	−0.00120679	−	0.999999i	$-0.500384\pi$
$149$	−0.0294614	−0.00241357	−0.00120679	+	0.999999i	$0.500384\pi$
$150$	0	0
$151$	12.0018	0.976689	0.488345	−	0.872651i	$-0.337601\pi$
$151$	12.0018	0.976689	0.488345	+	0.872651i	$0.337601\pi$
$152$	28.4991	2.31159
$153$	−5.89630	−0.476688
$154$	36.3368	2.92810
$155$	0	0
$156$	−19.6575	−1.57386
$157$	7.34191	0.585948	0.292974	−	0.956120i	$-0.405355\pi$
$157$	7.34191	0.585948	0.292974	+	0.956120i	$0.405355\pi$
$158$	−22.7451	−1.80951
$159$	−10.4443	−0.828290
$160$	0	0
$161$	−20.3188	−1.60134
$162$	2.46251	0.193473
$163$	−11.3447	−0.888582	−0.444291	−	0.895882i	$-0.646544\pi$
$163$	−11.3447	−0.888582	−0.444291	+	0.895882i	$0.646544\pi$
$164$	2.69498	0.210443
$165$	0	0
$166$	−1.45511	−0.112939
$167$	−7.42413	−0.574496	−0.287248	−	0.957856i	$-0.592740\pi$
$167$	−7.42413	−0.574496	−0.287248	+	0.957856i	$0.592740\pi$
$168$	19.8770	1.53354
$169$	10.3969	0.799762
$170$	0	0
$171$	5.60730	0.428801
$172$	−13.7893	−1.05142
$173$	21.4708	1.63239	0.816196	−	0.577775i	$-0.196079\pi$
$173$	21.4708	1.63239	0.816196	+	0.577775i	$0.196079\pi$
$174$	−15.6093	−1.18334
$175$	0	0
$176$	16.5556	1.24792
$177$	7.58816	0.570361
$178$	39.0733	2.92867
$179$	9.85343	0.736480	0.368240	−	0.929731i	$-0.379960\pi$
$179$	9.85343	0.736480	0.368240	+	0.929731i	$0.379960\pi$
$180$	0	0
$181$	−3.95811	−0.294204	−0.147102	−	0.989121i	$-0.546995\pi$
$181$	−3.95811	−0.294204	−0.147102	+	0.989121i	$0.546995\pi$
$182$	−46.5833	−3.45298
$183$	−10.3208	−0.762938
$184$	−26.4060	−1.94667
$185$	0	0
$186$	−2.46251	−0.180560
$187$	−22.2472	−1.62688
$188$	−5.35158	−0.390304
$189$	3.91086	0.284474
$190$	0	0
$191$	−16.6250	−1.20294	−0.601470	−	0.798896i	$-0.705418\pi$
$191$	−16.6250	−1.20294	−0.601470	+	0.798896i	$0.705418\pi$
$192$	−7.19958	−0.519585
$193$	4.17565	0.300570	0.150285	−	0.988643i	$-0.451981\pi$
$193$	4.17565	0.300570	0.150285	+	0.988643i	$0.451981\pi$
$194$	9.81518	0.704689
$195$	0	0
$196$	33.7099	2.40785
$197$	10.0370	0.715108	0.357554	−	0.933892i	$-0.383611\pi$
$197$	10.0370	0.715108	0.357554	+	0.933892i	$0.383611\pi$
$198$	9.29125	0.660301
$199$	11.7746	0.834679	0.417339	−	0.908751i	$-0.362963\pi$
$199$	11.7746	0.834679	0.417339	+	0.908751i	$0.362963\pi$
$200$	0	0
$201$	−10.2874	−0.725614
$202$	−11.0412	−0.776856
$203$	−24.7902	−1.73993
$204$	−23.9623	−1.67770
$205$	0	0
$206$	36.8391	2.56670
$207$	−5.19547	−0.361110
$208$	−21.2240	−1.47162
$209$	21.1568	1.46345
$210$	0	0
$211$	−19.9525	−1.37359	−0.686795	−	0.726851i	$-0.740983\pi$
$211$	−19.9525	−1.37359	−0.686795	+	0.726851i	$0.740983\pi$
$212$	−42.4453	−2.91516
$213$	−3.68083	−0.252206
$214$	3.63025	0.248159
$215$	0	0
$216$	5.08250	0.345821
$217$	−3.91086	−0.265487
$218$	31.4637	2.13099
$219$	12.3909	0.837302
$220$	0	0
$221$	28.5206	1.91851
$222$	27.8048	1.86614
$223$	−17.5032	−1.17210	−0.586051	−	0.810274i	$-0.699318\pi$
$223$	−17.5032	−1.17210	−0.586051	+	0.810274i	$0.699318\pi$
$224$	2.50299	0.167238
$225$	0	0
$226$	−11.6226	−0.773121
$227$	8.55366	0.567727	0.283863	−	0.958865i	$-0.408384\pi$
$227$	8.55366	0.567727	0.283863	+	0.958865i	$0.408384\pi$
$228$	22.7878	1.50916
$229$	−18.2864	−1.20840	−0.604198	−	0.796834i	$-0.706507\pi$
$229$	−18.2864	−1.20840	−0.604198	+	0.796834i	$0.706507\pi$
$230$	0	0
$231$	14.7560	0.970874
$232$	−32.2170	−2.11515
$233$	−17.7085	−1.16012	−0.580061	−	0.814573i	$-0.696971\pi$
$233$	−17.7085	−1.16012	−0.580061	+	0.814573i	$0.696971\pi$
$234$	−11.9112	−0.778662
$235$	0	0
$236$	30.8379	2.00738
$237$	−9.23657	−0.599979
$238$	−56.7846	−3.68080
$239$	1.18501	0.0766517	0.0383258	−	0.999265i	$-0.487798\pi$
$239$	1.18501	0.0766517	0.0383258	+	0.999265i	$0.487798\pi$
$240$	0	0
$241$	18.0403	1.16208	0.581039	−	0.813876i	$-0.302646\pi$
$241$	18.0403	1.16208	0.581039	+	0.813876i	$0.302646\pi$
$242$	7.96906	0.512270
$243$	1.00000	0.0641500
$244$	−41.9434	−2.68515
$245$	0	0
$246$	1.63300	0.104116
$247$	−27.1227	−1.72578
$248$	−5.08250	−0.322739
$249$	−0.590907	−0.0374472
$250$	0	0
$251$	−2.86868	−0.181070	−0.0905348	−	0.995893i	$-0.528858\pi$
$251$	−2.86868	−0.181070	−0.0905348	+	0.995893i	$0.528858\pi$
$252$	15.8936	1.00120
$253$	−19.6029	−1.23243
$254$	−36.2491	−2.27447
$255$	0	0
$256$	−32.4108	−2.02568
$257$	10.5692	0.659292	0.329646	−	0.944105i	$-0.393071\pi$
$257$	10.5692	0.659292	0.329646	+	0.944105i	$0.393071\pi$
$258$	−8.35548	−0.520190
$259$	44.1586	2.74388
$260$	0	0
$261$	−6.33880	−0.392362
$262$	27.0802	1.67302
$263$	−27.8993	−1.72034	−0.860171	−	0.510005i	$-0.829643\pi$
$263$	−27.8993	−1.72034	−0.860171	+	0.510005i	$0.829643\pi$
$264$	19.1767	1.18024
$265$	0	0
$266$	54.0014	3.31104
$267$	15.8673	0.971062
$268$	−41.8073	−2.55379
$269$	−24.3497	−1.48463	−0.742313	−	0.670053i	$-0.766271\pi$
$269$	−24.3497	−1.48463	−0.742313	+	0.670053i	$0.766271\pi$
$270$	0	0
$271$	26.2165	1.59254	0.796268	−	0.604943i	$-0.206804\pi$
$271$	26.2165	1.59254	0.796268	+	0.604943i	$0.206804\pi$
$272$	−25.8718	−1.56871
$273$	−18.9170	−1.14491
$274$	8.10259	0.489495
$275$	0	0
$276$	−21.1141	−1.27092
$277$	5.34031	0.320868	0.160434	−	0.987047i	$-0.448711\pi$
$277$	5.34031	0.320868	0.160434	+	0.987047i	$0.448711\pi$
$278$	28.9608	1.73695
$279$	−1.00000	−0.0598684
$280$	0	0
$281$	0.305796	0.0182422	0.00912112	−	0.999958i	$-0.497097\pi$
$281$	0.305796	0.0182422	0.00912112	+	0.999958i	$0.497097\pi$
$282$	−3.24273	−0.193102
$283$	−17.0008	−1.01059	−0.505296	−	0.862946i	$-0.668617\pi$
$283$	−17.0008	−1.01059	−0.505296	+	0.862946i	$0.668617\pi$
$284$	−14.9587	−0.887637
$285$	0	0
$286$	−44.9421	−2.65748
$287$	2.59346	0.153087
$288$	0.640010	0.0377129
$289$	17.7664	1.04508
$290$	0	0
$291$	3.98585	0.233654
$292$	50.3562	2.94687
$293$	−23.8670	−1.39433	−0.697164	−	0.716912i	$-0.745555\pi$
$293$	−23.8670	−1.39433	−0.697164	+	0.716912i	$0.745555\pi$
$294$	20.4262	1.19128
$295$	0	0
$296$	57.3878	3.33560
$297$	3.77308	0.218936
$298$	−0.0725490	−0.00420265
$299$	25.1307	1.45334
$300$	0	0
$301$	−13.2699	−0.764862
$302$	29.5545	1.70067
$303$	−4.48372	−0.257583
$304$	24.6038	1.41112
$305$	0	0
$306$	−14.5197	−0.830036
$307$	−10.0221	−0.571992	−0.285996	−	0.958231i	$-0.592324\pi$
$307$	−10.0221	−0.571992	−0.285996	+	0.958231i	$0.592324\pi$
$308$	59.9677	3.41698
$309$	14.9600	0.851043
$310$	0	0
$311$	−3.00341	−0.170307	−0.0851537	−	0.996368i	$-0.527138\pi$
$311$	−3.00341	−0.170307	−0.0851537	+	0.996368i	$0.527138\pi$
$312$	−24.5842	−1.39181
$313$	−10.2325	−0.578374	−0.289187	−	0.957273i	$-0.593385\pi$
$313$	−10.2325	−0.578374	−0.289187	+	0.957273i	$0.593385\pi$
$314$	18.0795	1.02029
$315$	0	0
$316$	−37.5370	−2.11162
$317$	−23.4147	−1.31510	−0.657550	−	0.753411i	$-0.728407\pi$
$317$	−23.4147	−1.31510	−0.657550	+	0.753411i	$0.728407\pi$
$318$	−25.7193	−1.44227
$319$	−23.9168	−1.33908
$320$	0	0
$321$	1.47421	0.0822823
$322$	−50.0352	−2.78835
$323$	−33.0624	−1.83964
$324$	4.06395	0.225775
$325$	0	0
$326$	−27.9363	−1.54725
$327$	12.7771	0.706574
$328$	3.37043	0.186101
$329$	−5.14999	−0.283928
$330$	0	0
$331$	20.2889	1.11518	0.557590	−	0.830116i	$-0.311726\pi$
$331$	20.2889	1.11518	0.557590	+	0.830116i	$0.311726\pi$
$332$	−2.40142	−0.131795
$333$	11.2913	0.618757
$334$	−18.2820	−1.00035
$335$	0	0
$336$	17.1601	0.936161
$337$	−15.9577	−0.869274	−0.434637	−	0.900606i	$-0.643123\pi$
$337$	−15.9577	−0.869274	−0.434637	+	0.900606i	$0.643123\pi$
$338$	25.6025	1.39259
$339$	−4.71980	−0.256345
$340$	0	0
$341$	−3.77308	−0.204324
$342$	13.8080	0.746653
$343$	5.06402	0.273431
$344$	−17.2453	−0.929806
$345$	0	0
$346$	52.8720	2.84242
$347$	6.93796	0.372449	0.186224	−	0.982507i	$-0.440375\pi$
$347$	6.93796	0.372449	0.186224	+	0.982507i	$0.440375\pi$
$348$	−25.7606	−1.38091
$349$	−9.56079	−0.511778	−0.255889	−	0.966706i	$-0.582368\pi$
$349$	−9.56079	−0.511778	−0.255889	+	0.966706i	$0.582368\pi$
$350$	0	0
$351$	−4.83704	−0.258182
$352$	2.41481	0.128710
$353$	12.0956	0.643783	0.321892	−	0.946776i	$-0.395681\pi$
$353$	12.0956	0.643783	0.321892	+	0.946776i	$0.395681\pi$
$354$	18.6859	0.993146
$355$	0	0
$356$	64.4839	3.41764
$357$	−23.0596	−1.22045
$358$	24.2642	1.28240
$359$	−28.8573	−1.52303	−0.761514	−	0.648148i	$-0.775544\pi$
$359$	−28.8573	−1.52303	−0.761514	+	0.648148i	$0.775544\pi$
$360$	0	0
$361$	12.4419	0.654835
$362$	−9.74688	−0.512284
$363$	3.23615	0.169854
$364$	−76.8778	−4.02949
$365$	0	0
$366$	−25.4152	−1.32847
$367$	23.3643	1.21961	0.609804	−	0.792552i	$-0.291248\pi$
$367$	23.3643	1.21961	0.609804	+	0.792552i	$0.291248\pi$
$368$	−22.7967	−1.18836
$369$	0.663143	0.0345219
$370$	0	0
$371$	−40.8464	−2.12064
$372$	−4.06395	−0.210706
$373$	−16.9679	−0.878563	−0.439281	−	0.898350i	$-0.644767\pi$
$373$	−16.9679	−0.878563	−0.439281	+	0.898350i	$0.644767\pi$
$374$	−54.7840	−2.83282
$375$	0	0
$376$	−6.69285	−0.345157
$377$	30.6610	1.57912
$378$	9.63054	0.495342
$379$	−23.3454	−1.19917	−0.599585	−	0.800311i	$-0.704668\pi$
$379$	−23.3454	−1.19917	−0.599585	+	0.800311i	$0.704668\pi$
$380$	0	0
$381$	−14.7204	−0.754147
$382$	−40.9391	−2.09463
$383$	−25.3541	−1.29554	−0.647768	−	0.761838i	$-0.724297\pi$
$383$	−25.3541	−1.29554	−0.647768	+	0.761838i	$0.724297\pi$
$384$	−19.0091	−0.970052
$385$	0	0
$386$	10.2826	0.523370
$387$	−3.39308	−0.172480
$388$	16.1983	0.822343
$389$	−12.6350	−0.640622	−0.320311	−	0.947312i	$-0.603787\pi$
$389$	−12.6350	−0.640622	−0.320311	+	0.947312i	$0.603787\pi$
$390$	0	0
$391$	30.6340	1.54923
$392$	42.1587	2.12933
$393$	10.9970	0.554724
$394$	24.7163	1.24519
$395$	0	0
$396$	15.3336	0.770544
$397$	25.5883	1.28424	0.642119	−	0.766605i	$-0.278055\pi$
$397$	25.5883	1.28424	0.642119	+	0.766605i	$0.278055\pi$
$398$	28.9950	1.45339
$399$	21.9294	1.09784
$400$	0	0
$401$	37.1198	1.85368	0.926838	−	0.375461i	$-0.122516\pi$
$401$	37.1198	1.85368	0.926838	+	0.375461i	$0.122516\pi$
$402$	−25.3327	−1.26348
$403$	4.83704	0.240950
$404$	−18.2216	−0.906560
$405$	0	0
$406$	−61.0460	−3.02966
$407$	42.6028	2.11174
$408$	−29.9680	−1.48364
$409$	11.2361	0.555589	0.277794	−	0.960641i	$-0.410397\pi$
$409$	11.2361	0.555589	0.277794	+	0.960641i	$0.410397\pi$
$410$	0	0
$411$	3.29038	0.162302
$412$	60.7966	2.99524
$413$	29.6763	1.46027
$414$	−12.7939	−0.628785
$415$	0	0
$416$	−3.09575	−0.151782
$417$	11.7607	0.575923
$418$	52.0989	2.54824
$419$	−26.5740	−1.29823	−0.649113	−	0.760692i	$-0.724860\pi$
$419$	−26.5740	−1.29823	−0.649113	+	0.760692i	$0.724860\pi$
$420$	0	0
$421$	21.5334	1.04948	0.524738	−	0.851264i	$-0.324163\pi$
$421$	21.5334	1.04948	0.524738	+	0.851264i	$0.324163\pi$
$422$	−49.1333	−2.39177
$423$	−1.31684	−0.0640270
$424$	−53.0834	−2.57796
$425$	0	0
$426$	−9.06408	−0.439156
$427$	−40.3634	−1.95332
$428$	5.99111	0.289591
$429$	−18.2505	−0.881144
$430$	0	0
$431$	7.20557	0.347080	0.173540	−	0.984827i	$-0.444479\pi$
$431$	7.20557	0.347080	0.173540	+	0.984827i	$0.444479\pi$
$432$	4.38781	0.211108
$433$	−12.1236	−0.582624	−0.291312	−	0.956628i	$-0.594092\pi$
$433$	−12.1236	−0.582624	−0.291312	+	0.956628i	$0.594092\pi$
$434$	−9.63054	−0.462281
$435$	0	0
$436$	51.9254	2.48678
$437$	−29.1326	−1.39360
$438$	30.5128	1.45796
$439$	38.1847	1.82245	0.911227	−	0.411905i	$-0.135136\pi$
$439$	38.1847	1.82245	0.911227	+	0.411905i	$0.135136\pi$
$440$	0	0
$441$	8.29486	0.394993
$442$	70.2323	3.34061
$443$	24.9128	1.18364	0.591822	−	0.806069i	$-0.298409\pi$
$443$	24.9128	1.18364	0.591822	+	0.806069i	$0.298409\pi$
$444$	45.8871	2.17771
$445$	0	0
$446$	−43.1018	−2.04093
$447$	−0.0294614	−0.00139348
$448$	−28.1566	−1.33027
$449$	−14.0489	−0.663007	−0.331504	−	0.943454i	$-0.607556\pi$
$449$	−14.0489	−0.663007	−0.331504	+	0.943454i	$0.607556\pi$
$450$	0	0
$451$	2.50209	0.117819
$452$	−19.1811	−0.902201
$453$	12.0018	0.563892
$454$	21.0635	0.988558
$455$	0	0
$456$	28.4991	1.33459
$457$	40.4110	1.89034	0.945172	−	0.326572i	$-0.105894\pi$
$457$	40.4110	1.89034	0.945172	+	0.326572i	$0.105894\pi$
$458$	−45.0303	−2.10413
$459$	−5.89630	−0.275216
$460$	0	0
$461$	13.7596	0.640850	0.320425	−	0.947274i	$-0.396174\pi$
$461$	13.7596	0.640850	0.320425	+	0.947274i	$0.396174\pi$
$462$	36.3368	1.69054
$463$	23.9445	1.11280	0.556398	−	0.830916i	$-0.312183\pi$
$463$	23.9445	1.11280	0.556398	+	0.830916i	$0.312183\pi$
$464$	−27.8134	−1.29121
$465$	0	0
$466$	−43.6073	−2.02007
$467$	−7.84527	−0.363036	−0.181518	−	0.983388i	$-0.558101\pi$
$467$	−7.84527	−0.363036	−0.181518	+	0.983388i	$0.558101\pi$
$468$	−19.6575	−0.908668
$469$	−40.2324	−1.85776
$470$	0	0
$471$	7.34191	0.338297
$472$	38.5669	1.77518
$473$	−12.8024	−0.588653
$474$	−22.7451	−1.04472
$475$	0	0
$476$	−93.7133	−4.29534
$477$	−10.4443	−0.478213
$478$	2.91809	0.133470
$479$	36.2715	1.65729	0.828644	−	0.559776i	$-0.189113\pi$
$479$	36.2715	1.65729	0.828644	+	0.559776i	$0.189113\pi$
$480$	0	0
$481$	−54.6162	−2.49028
$482$	44.4244	2.02348
$483$	−20.3188	−0.924536
$484$	13.1516	0.597799
$485$	0	0
$486$	2.46251	0.111702
$487$	−5.83307	−0.264322	−0.132161	−	0.991228i	$-0.542192\pi$
$487$	−5.83307	−0.264322	−0.132161	+	0.991228i	$0.542192\pi$
$488$	−52.4557	−2.37456
$489$	−11.3447	−0.513023
$490$	0	0
$491$	3.76466	0.169897	0.0849484	−	0.996385i	$-0.472927\pi$
$491$	3.76466	0.169897	0.0849484	+	0.996385i	$0.472927\pi$
$492$	2.69498	0.121499
$493$	37.3755	1.68331
$494$	−66.7900	−3.00502
$495$	0	0
$496$	−4.38781	−0.197018
$497$	−14.3952	−0.645714
$498$	−1.45511	−0.0652052
$499$	11.2661	0.504339	0.252169	−	0.967683i	$-0.418856\pi$
$499$	11.2661	0.504339	0.252169	+	0.967683i	$0.418856\pi$
$500$	0	0
$501$	−7.42413	−0.331686
$502$	−7.06416	−0.315289
$503$	36.7533	1.63875	0.819374	−	0.573259i	$-0.194321\pi$
$503$	36.7533	1.63875	0.819374	+	0.573259i	$0.194321\pi$
$504$	19.8770	0.885391
$505$	0	0
$506$	−48.2724	−2.14597
$507$	10.3969	0.461743
$508$	−59.8229	−2.65421
$509$	36.8000	1.63113	0.815565	−	0.578666i	$-0.196427\pi$
$509$	36.8000	1.63113	0.815565	+	0.578666i	$0.196427\pi$
$510$	0	0
$511$	48.4593	2.14371
$512$	−41.7939	−1.84704
$513$	5.60730	0.247569
$514$	26.0269	1.14800
$515$	0	0
$516$	−13.7893	−0.607040
$517$	−4.96855	−0.218517
$518$	108.741	4.77780
$519$	21.4708	0.942462
$520$	0	0
$521$	−8.57486	−0.375672	−0.187836	−	0.982200i	$-0.560147\pi$
$521$	−8.57486	−0.375672	−0.187836	+	0.982200i	$0.560147\pi$
$522$	−15.6093	−0.683203
$523$	6.23174	0.272495	0.136247	−	0.990675i	$-0.456496\pi$
$523$	6.23174	0.272495	0.136247	+	0.990675i	$0.456496\pi$
$524$	44.6912	1.95234
$525$	0	0
$526$	−68.7022	−2.99556
$527$	5.89630	0.256847
$528$	16.5556	0.720488
$529$	3.99288	0.173603
$530$	0	0
$531$	7.58816	0.329298
$532$	89.1201	3.86385
$533$	−3.20765	−0.138939
$534$	39.0733	1.69087
$535$	0	0
$536$	−52.2855	−2.25839
$537$	9.85343	0.425207
$538$	−59.9613	−2.58512
$539$	31.2972	1.34807
$540$	0	0
$541$	11.9562	0.514037	0.257018	−	0.966407i	$-0.417260\pi$
$541$	11.9562	0.514037	0.257018	+	0.966407i	$0.417260\pi$
$542$	64.5583	2.77302
$543$	−3.95811	−0.169859
$544$	−3.77369	−0.161796
$545$	0	0
$546$	−46.5833	−1.99358
$547$	36.2669	1.55066	0.775330	−	0.631557i	$-0.217584\pi$
$547$	36.2669	1.55066	0.775330	+	0.631557i	$0.217584\pi$
$548$	13.3719	0.571221
$549$	−10.3208	−0.440483
$550$	0	0
$551$	−35.5436	−1.51421
$552$	−26.4060	−1.12391
$553$	−36.1230	−1.53610
$554$	13.1506	0.558714
$555$	0	0
$556$	47.7948	2.02695
$557$	36.6763	1.55403	0.777013	−	0.629484i	$-0.216734\pi$
$557$	36.6763	1.55403	0.777013	+	0.629484i	$0.216734\pi$
$558$	−2.46251	−0.104246
$559$	16.4124	0.694172
$560$	0	0
$561$	−22.2472	−0.939279
$562$	0.753025	0.0317644
$563$	11.7391	0.494742	0.247371	−	0.968921i	$-0.420433\pi$
$563$	11.7391	0.494742	0.247371	+	0.968921i	$0.420433\pi$
$564$	−5.35158	−0.225342
$565$	0	0
$566$	−41.8646	−1.75970
$567$	3.91086	0.164241
$568$	−18.7078	−0.784963
$569$	0.948286	0.0397542	0.0198771	−	0.999802i	$-0.493673\pi$
$569$	0.948286	0.0397542	0.0198771	+	0.999802i	$0.493673\pi$
$570$	0	0
$571$	41.3896	1.73210	0.866051	−	0.499956i	$-0.166651\pi$
$571$	41.3896	1.73210	0.866051	+	0.499956i	$0.166651\pi$
$572$	−74.1693	−3.10117
$573$	−16.6250	−0.694518
$574$	6.38643	0.266564
$575$	0	0
$576$	−7.19958	−0.299983
$577$	11.6673	0.485716	0.242858	−	0.970062i	$-0.421915\pi$
$577$	11.6673	0.485716	0.242858	+	0.970062i	$0.421915\pi$
$578$	43.7499	1.81976
$579$	4.17565	0.173534
$580$	0	0
$581$	−2.31096	−0.0958747
$582$	9.81518	0.406852
$583$	−39.4074	−1.63209
$584$	62.9770	2.60601
$585$	0	0
$586$	−58.7728	−2.42788
$587$	−22.5741	−0.931735	−0.465867	−	0.884855i	$-0.654258\pi$
$587$	−22.5741	−0.931735	−0.465867	+	0.884855i	$0.654258\pi$
$588$	33.7099	1.39017
$589$	−5.60730	−0.231045
$590$	0	0
$591$	10.0370	0.412868
$592$	49.5438	2.03624
$593$	−1.50814	−0.0619317	−0.0309659	−	0.999520i	$-0.509858\pi$
$593$	−1.50814	−0.0619317	−0.0309659	+	0.999520i	$0.509858\pi$
$594$	9.29125	0.381225
$595$	0	0
$596$	−0.119730	−0.00490433
$597$	11.7746	0.481902
$598$	61.8845	2.53064
$599$	2.77147	0.113239	0.0566196	−	0.998396i	$-0.481968\pi$
$599$	2.77147	0.113239	0.0566196	+	0.998396i	$0.481968\pi$
$600$	0	0
$601$	−18.0452	−0.736078	−0.368039	−	0.929810i	$-0.619971\pi$
$601$	−18.0452	−0.736078	−0.368039	+	0.929810i	$0.619971\pi$
$602$	−32.6772	−1.33182
$603$	−10.2874	−0.418933
$604$	48.7746	1.98461
$605$	0	0
$606$	−11.0412	−0.448518
$607$	−11.3139	−0.459219	−0.229609	−	0.973283i	$-0.573745\pi$
$607$	−11.3139	−0.459219	−0.229609	+	0.973283i	$0.573745\pi$
$608$	3.58873	0.145542
$609$	−24.7902	−1.00455
$610$	0	0
$611$	6.36961	0.257687
$612$	−23.9623	−0.968619
$613$	1.32044	0.0533319	0.0266659	−	0.999644i	$-0.491511\pi$
$613$	1.32044	0.0533319	0.0266659	+	0.999644i	$0.491511\pi$
$614$	−24.6795	−0.995985
$615$	0	0
$616$	74.9975	3.02174
$617$	−31.4084	−1.26446	−0.632228	−	0.774783i	$-0.717859\pi$
$617$	−31.4084	−1.26446	−0.632228	+	0.774783i	$0.717859\pi$
$618$	36.8391	1.48189
$619$	−24.2173	−0.973377	−0.486689	−	0.873576i	$-0.661795\pi$
$619$	−24.2173	−0.973377	−0.486689	+	0.873576i	$0.661795\pi$
$620$	0	0
$621$	−5.19547	−0.208487
$622$	−7.39591	−0.296549
$623$	62.0548	2.48617
$624$	−21.2240	−0.849639
$625$	0	0
$626$	−25.1976	−1.00710
$627$	21.1568	0.844922
$628$	29.8372	1.19063
$629$	−66.5766	−2.65458
$630$	0	0
$631$	−16.7012	−0.664863	−0.332432	−	0.943127i	$-0.607869\pi$
$631$	−16.7012	−0.664863	−0.332432	+	0.943127i	$0.607869\pi$
$632$	−46.9449	−1.86737
$633$	−19.9525	−0.793042
$634$	−57.6589	−2.28993
$635$	0	0
$636$	−42.4453	−1.68307
$637$	−40.1225	−1.58971
$638$	−58.8954	−2.33169
$639$	−3.68083	−0.145611
$640$	0	0
$641$	45.6730	1.80398	0.901988	−	0.431762i	$-0.142108\pi$
$641$	45.6730	1.80398	0.901988	+	0.431762i	$0.142108\pi$
$642$	3.63025	0.143275
$643$	36.4701	1.43824	0.719120	−	0.694886i	$-0.244545\pi$
$643$	36.4701	1.43824	0.719120	+	0.694886i	$0.244545\pi$
$644$	−82.5745	−3.25389
$645$	0	0
$646$	−81.4164	−3.20329
$647$	−3.16329	−0.124362	−0.0621808	−	0.998065i	$-0.519806\pi$
$647$	−3.16329	−0.124362	−0.0621808	+	0.998065i	$0.519806\pi$
$648$	5.08250	0.199660
$649$	28.6308	1.12386
$650$	0	0
$651$	−3.91086	−0.153279
$652$	−46.1042	−1.80558
$653$	−34.2753	−1.34130	−0.670648	−	0.741776i	$-0.733984\pi$
$653$	−34.2753	−1.34130	−0.670648	+	0.741776i	$0.733984\pi$
$654$	31.4637	1.23033
$655$	0	0
$656$	2.90974	0.113606
$657$	12.3909	0.483416
$658$	−12.6819	−0.494392
$659$	22.2146	0.865357	0.432679	−	0.901548i	$-0.357568\pi$
$659$	22.2146	0.865357	0.432679	+	0.901548i	$0.357568\pi$
$660$	0	0
$661$	−22.4547	−0.873387	−0.436693	−	0.899610i	$-0.643851\pi$
$661$	−22.4547	−0.873387	−0.436693	+	0.899610i	$0.643851\pi$
$662$	49.9617	1.94182
$663$	28.5206	1.10765
$664$	−3.00329	−0.116550
$665$	0	0
$666$	27.8048	1.07741
$667$	32.9330	1.27517
$668$	−30.1713	−1.16736
$669$	−17.5032	−0.676713
$670$	0	0
$671$	−38.9414	−1.50332
$672$	2.50299	0.0965550
$673$	−9.90870	−0.381952	−0.190976	−	0.981595i	$-0.561165\pi$
$673$	−9.90870	−0.381952	−0.190976	+	0.981595i	$0.561165\pi$
$674$	−39.2961	−1.51363
$675$	0	0
$676$	42.2526	1.62510
$677$	−16.1570	−0.620963	−0.310482	−	0.950579i	$-0.600490\pi$
$677$	−16.1570	−0.620963	−0.310482	+	0.950579i	$0.600490\pi$
$678$	−11.6226	−0.446362
$679$	15.5881	0.598216
$680$	0	0
$681$	8.55366	0.327777
$682$	−9.29125	−0.355780
$683$	−14.6765	−0.561583	−0.280791	−	0.959769i	$-0.590597\pi$
$683$	−14.6765	−0.561583	−0.280791	+	0.959769i	$0.590597\pi$
$684$	22.7878	0.871314
$685$	0	0
$686$	12.4702	0.476114
$687$	−18.2864	−0.697668
$688$	−14.8882	−0.567606
$689$	50.5196	1.92465
$690$	0	0
$691$	−4.78243	−0.181932	−0.0909662	−	0.995854i	$-0.528996\pi$
$691$	−4.78243	−0.181932	−0.0909662	+	0.995854i	$0.528996\pi$
$692$	87.2562	3.31698
$693$	14.7560	0.560535
$694$	17.0848	0.648529
$695$	0	0
$696$	−32.2170	−1.22118
$697$	−3.91009	−0.148105
$698$	−23.5435	−0.891136
$699$	−17.7085	−0.669796
$700$	0	0
$701$	14.9186	0.563467	0.281734	−	0.959493i	$-0.409091\pi$
$701$	14.9186	0.563467	0.281734	+	0.959493i	$0.409091\pi$
$702$	−11.9112	−0.449561
$703$	63.3135	2.38791
$704$	−27.1646	−1.02381
$705$	0	0
$706$	29.7855	1.12099
$707$	−17.5352	−0.659480
$708$	30.8379	1.15896
$709$	31.0264	1.16522	0.582611	−	0.812751i	$-0.302031\pi$
$709$	31.0264	1.16522	0.582611	+	0.812751i	$0.302031\pi$
$710$	0	0
$711$	−9.23657	−0.346398
$712$	80.6455	3.02232
$713$	5.19547	0.194572
$714$	−56.7846	−2.12511
$715$	0	0
$716$	40.0439	1.49651
$717$	1.18501	0.0442549
$718$	−71.0613	−2.65198
$719$	22.5881	0.842394	0.421197	−	0.906969i	$-0.361610\pi$
$719$	22.5881	0.842394	0.421197	+	0.906969i	$0.361610\pi$
$720$	0	0
$721$	58.5064	2.17889
$722$	30.6382	1.14024
$723$	18.0403	0.670926
$724$	−16.0856	−0.597815
$725$	0	0
$726$	7.96906	0.295759
$727$	−30.6620	−1.13719	−0.568596	−	0.822617i	$-0.692513\pi$
$727$	−30.6620	−1.13719	−0.568596	+	0.822617i	$0.692513\pi$
$728$	−96.1457	−3.56340
$729$	1.00000	0.0370370
$730$	0	0
$731$	20.0066	0.739971
$732$	−41.9434	−1.55027
$733$	−17.6648	−0.652463	−0.326232	−	0.945290i	$-0.605779\pi$
$733$	−17.6648	−0.652463	−0.326232	+	0.945290i	$0.605779\pi$
$734$	57.5349	2.12365
$735$	0	0
$736$	−3.32515	−0.122567
$737$	−38.8150	−1.42977
$738$	1.63300	0.0601114
$739$	34.0146	1.25125	0.625623	−	0.780126i	$-0.284845\pi$
$739$	34.0146	1.25125	0.625623	+	0.780126i	$0.284845\pi$
$740$	0	0
$741$	−27.1227	−0.996378
$742$	−100.585	−3.69258
$743$	1.10230	0.0404394	0.0202197	−	0.999796i	$-0.493563\pi$
$743$	1.10230	0.0404394	0.0202197	+	0.999796i	$0.493563\pi$
$744$	−5.08250	−0.186334
$745$	0	0
$746$	−41.7835	−1.52980
$747$	−0.590907	−0.0216202
$748$	−90.4117	−3.30578
$749$	5.76543	0.210664
$750$	0	0
$751$	29.9534	1.09302	0.546508	−	0.837454i	$-0.315957\pi$
$751$	29.9534	1.09302	0.546508	+	0.837454i	$0.315957\pi$
$752$	−5.77804	−0.210704
$753$	−2.86868	−0.104541
$754$	75.5030	2.74966
$755$	0	0
$756$	15.8936	0.578043
$757$	20.3970	0.741340	0.370670	−	0.928765i	$-0.379128\pi$
$757$	20.3970	0.741340	0.370670	+	0.928765i	$0.379128\pi$
$758$	−57.4882	−2.08806
$759$	−19.6029	−0.711541
$760$	0	0
$761$	−42.3063	−1.53360	−0.766802	−	0.641884i	$-0.778153\pi$
$761$	−42.3063	−1.53360	−0.766802	+	0.641884i	$0.778153\pi$
$762$	−36.2491	−1.31316
$763$	49.9694	1.80901
$764$	−67.5630	−2.44435
$765$	0	0
$766$	−62.4348	−2.25586
$767$	−36.7042	−1.32531
$768$	−32.4108	−1.16953
$769$	−2.54358	−0.0917239	−0.0458620	−	0.998948i	$-0.514603\pi$
$769$	−2.54358	−0.0917239	−0.0458620	+	0.998948i	$0.514603\pi$
$770$	0	0
$771$	10.5692	0.380642
$772$	16.9697	0.610751
$773$	−43.4158	−1.56156	−0.780778	−	0.624808i	$-0.785177\pi$
$773$	−43.4158	−1.56156	−0.780778	+	0.624808i	$0.785177\pi$
$774$	−8.35548	−0.300332
$775$	0	0
$776$	20.2581	0.727222
$777$	44.1586	1.58418
$778$	−31.1139	−1.11549
$779$	3.71845	0.133227
$780$	0	0
$781$	−13.8881	−0.496955
$782$	75.4366	2.69761
$783$	−6.33880	−0.226530
$784$	36.3962	1.29987
$785$	0	0
$786$	27.0802	0.965917
$787$	−11.5860	−0.412997	−0.206498	−	0.978447i	$-0.566207\pi$
$787$	−11.5860	−0.412997	−0.206498	+	0.978447i	$0.566207\pi$
$788$	40.7900	1.45308
$789$	−27.8993	−0.993240
$790$	0	0
$791$	−18.4585	−0.656309
$792$	19.1767	0.681415
$793$	49.9223	1.77279
$794$	63.0114	2.23619
$795$	0	0
$796$	47.8514	1.69605
$797$	37.0975	1.31406	0.657030	−	0.753864i	$-0.271812\pi$
$797$	37.0975	1.31406	0.657030	+	0.753864i	$0.271812\pi$
$798$	54.0014	1.91163
$799$	7.76449	0.274688
$800$	0	0
$801$	15.8673	0.560643
$802$	91.4079	3.22773
$803$	46.7520	1.64984
$804$	−41.8073	−1.47443
$805$	0	0
$806$	11.9112	0.419556
$807$	−24.3497	−0.857149
$808$	−22.7885	−0.801697
$809$	−22.2275	−0.781479	−0.390739	−	0.920501i	$-0.627781\pi$
$809$	−22.2275	−0.781479	−0.390739	+	0.920501i	$0.627781\pi$
$810$	0	0
$811$	4.80625	0.168770	0.0843852	−	0.996433i	$-0.473107\pi$
$811$	4.80625	0.168770	0.0843852	+	0.996433i	$0.473107\pi$
$812$	−100.746	−3.53550
$813$	26.2165	0.919452
$814$	104.910	3.67709
$815$	0	0
$816$	−25.8718	−0.905696
$817$	−19.0260	−0.665636
$818$	27.6690	0.967423
$819$	−18.9170	−0.661013
$820$	0	0
$821$	54.7778	1.91176	0.955880	−	0.293758i	$-0.0949058\pi$
$821$	54.7778	1.91176	0.955880	+	0.293758i	$0.0949058\pi$
$822$	8.10259	0.282610
$823$	40.1760	1.40045	0.700224	−	0.713924i	$-0.253084\pi$
$823$	40.1760	1.40045	0.700224	+	0.713924i	$0.253084\pi$
$824$	76.0341	2.64877
$825$	0	0
$826$	73.0781	2.54271
$827$	−1.84956	−0.0643154	−0.0321577	−	0.999483i	$-0.510238\pi$
$827$	−1.84956	−0.0643154	−0.0321577	+	0.999483i	$0.510238\pi$
$828$	−21.1141	−0.733767
$829$	12.8593	0.446622	0.223311	−	0.974747i	$-0.428313\pi$
$829$	12.8593	0.446622	0.223311	+	0.974747i	$0.428313\pi$
$830$	0	0
$831$	5.34031	0.185253
$832$	34.8246	1.20733
$833$	−48.9090	−1.69460
$834$	28.9608	1.00283
$835$	0	0
$836$	85.9803	2.97369
$837$	−1.00000	−0.0345651
$838$	−65.4388	−2.26054
$839$	−49.4500	−1.70720	−0.853601	−	0.520927i	$-0.825586\pi$
$839$	−49.4500	−1.70720	−0.853601	+	0.520927i	$0.825586\pi$
$840$	0	0
$841$	11.1803	0.385529
$842$	53.0263	1.82741
$843$	0.305796	0.0105322
$844$	−81.0862	−2.79110
$845$	0	0
$846$	−3.24273	−0.111487
$847$	12.6562	0.434871
$848$	−45.8278	−1.57373
$849$	−17.0008	−0.583465
$850$	0	0
$851$	−58.6633	−2.01095
$852$	−14.9587	−0.512477
$853$	17.9548	0.614762	0.307381	−	0.951587i	$-0.400547\pi$
$853$	17.9548	0.614762	0.307381	+	0.951587i	$0.400547\pi$
$854$	−99.3953	−3.40124
$855$	0	0
$856$	7.49267	0.256094
$857$	28.9951	0.990452	0.495226	−	0.868764i	$-0.335085\pi$
$857$	28.9951	0.990452	0.495226	+	0.868764i	$0.335085\pi$
$858$	−44.9421	−1.53430
$859$	−39.3399	−1.34226	−0.671130	−	0.741339i	$-0.734191\pi$
$859$	−39.3399	−1.34226	−0.671130	+	0.741339i	$0.734191\pi$
$860$	0	0
$861$	2.59346	0.0883850
$862$	17.7438	0.604356
$863$	25.2944	0.861032	0.430516	−	0.902583i	$-0.358332\pi$
$863$	25.2944	0.861032	0.430516	+	0.902583i	$0.358332\pi$
$864$	0.640010	0.0217736
$865$	0	0
$866$	−29.8545	−1.01450
$867$	17.7664	0.603378
$868$	−15.8936	−0.539463
$869$	−34.8503	−1.18222
$870$	0	0
$871$	49.7603	1.68606
$872$	64.9395	2.19913
$873$	3.98585	0.134900
$874$	−71.7392	−2.42662
$875$	0	0
$876$	50.3562	1.70138
$877$	45.7315	1.54424	0.772121	−	0.635476i	$-0.219196\pi$
$877$	45.7315	1.54424	0.772121	+	0.635476i	$0.219196\pi$
$878$	94.0301	3.17336
$879$	−23.8670	−0.805016
$880$	0	0
$881$	−7.69529	−0.259261	−0.129630	−	0.991562i	$-0.541379\pi$
$881$	−7.69529	−0.259261	−0.129630	+	0.991562i	$0.541379\pi$
$882$	20.4262	0.687785
$883$	−5.44217	−0.183144	−0.0915719	−	0.995798i	$-0.529189\pi$
$883$	−5.44217	−0.183144	−0.0915719	+	0.995798i	$0.529189\pi$
$884$	115.906	3.89836
$885$	0	0
$886$	61.3481	2.06103
$887$	45.0909	1.51400	0.757002	−	0.653412i	$-0.226663\pi$
$887$	45.0909	1.51400	0.757002	+	0.653412i	$0.226663\pi$
$888$	57.3878	1.92581
$889$	−57.5694	−1.93081
$890$	0	0
$891$	3.77308	0.126403
$892$	−71.1322	−2.38168
$893$	−7.38393	−0.247094
$894$	−0.0725490	−0.00242640
$895$	0	0
$896$	−74.3419	−2.48359
$897$	25.1307	0.839088
$898$	−34.5955	−1.15447
$899$	6.33880	0.211411
$900$	0	0
$901$	61.5830	2.05163
$902$	6.16143	0.205153
$903$	−13.2699	−0.441593
$904$	−23.9884	−0.797843
$905$	0	0
$906$	29.5545	0.981881
$907$	25.3635	0.842180	0.421090	−	0.907019i	$-0.361648\pi$
$907$	25.3635	0.842180	0.421090	+	0.907019i	$0.361648\pi$
$908$	34.7617	1.15361
$909$	−4.48372	−0.148716
$910$	0	0
$911$	−2.49514	−0.0826676	−0.0413338	−	0.999145i	$-0.513161\pi$
$911$	−2.49514	−0.0826676	−0.0413338	+	0.999145i	$0.513161\pi$
$912$	24.6038	0.814712
$913$	−2.22954	−0.0737871
$914$	99.5124	3.29158
$915$	0	0
$916$	−74.3149	−2.45543
$917$	43.0077	1.42024
$918$	−14.5197	−0.479222
$919$	17.7435	0.585306	0.292653	−	0.956219i	$-0.405462\pi$
$919$	17.7435	0.585306	0.292653	+	0.956219i	$0.405462\pi$
$920$	0	0
$921$	−10.0221	−0.330240
$922$	33.8832	1.11588
$923$	17.8043	0.586036
$924$	59.9677	1.97279
$925$	0	0
$926$	58.9636	1.93766
$927$	14.9600	0.491350
$928$	−4.05689	−0.133174
$929$	28.5183	0.935654	0.467827	−	0.883820i	$-0.345037\pi$
$929$	28.5183	0.935654	0.467827	+	0.883820i	$0.345037\pi$
$930$	0	0
$931$	46.5118	1.52436
$932$	−71.9665	−2.35734
$933$	−3.00341	−0.0983271
$934$	−19.3191	−0.632139
$935$	0	0
$936$	−24.5842	−0.803561
$937$	−7.18881	−0.234848	−0.117424	−	0.993082i	$-0.537464\pi$
$937$	−7.18881	−0.234848	−0.117424	+	0.993082i	$0.537464\pi$
$938$	−99.0728	−3.23484
$939$	−10.2325	−0.333924
$940$	0	0
$941$	−4.41233	−0.143838	−0.0719189	−	0.997410i	$-0.522912\pi$
$941$	−4.41233	−0.143838	−0.0719189	+	0.997410i	$0.522912\pi$
$942$	18.0795	0.589063
$943$	−3.44534	−0.112196
$944$	33.2954	1.08367
$945$	0	0
$946$	−31.5259	−1.02500
$947$	−46.3158	−1.50506	−0.752530	−	0.658558i	$-0.771167\pi$
$947$	−46.3158	−1.50506	−0.752530	+	0.658558i	$0.771167\pi$
$948$	−37.5370	−1.21914
$949$	−59.9354	−1.94558
$950$	0	0
$951$	−23.4147	−0.759274
$952$	−117.201	−3.79850
$953$	−56.1358	−1.81842	−0.909209	−	0.416341i	$-0.863312\pi$
$953$	−56.1358	−1.81842	−0.909209	+	0.416341i	$0.863312\pi$
$954$	−25.7193	−0.832693
$955$	0	0
$956$	4.81581	0.155754
$957$	−23.9168	−0.773120
$958$	89.3189	2.88576
$959$	12.8682	0.415537
$960$	0	0
$961$	1.00000	0.0322581
$962$	−134.493	−4.33622
$963$	1.47421	0.0475057
$964$	73.3149	2.36132
$965$	0	0
$966$	−50.0352	−1.60985
$967$	−21.4773	−0.690664	−0.345332	−	0.938481i	$-0.612234\pi$
$967$	−21.4773	−0.690664	−0.345332	+	0.938481i	$0.612234\pi$
$968$	16.4478	0.528651
$969$	−33.0624	−1.06212
$970$	0	0
$971$	23.1016	0.741366	0.370683	−	0.928759i	$-0.379124\pi$
$971$	23.1016	0.741366	0.370683	+	0.928759i	$0.379124\pi$
$972$	4.06395	0.130351
$973$	45.9944	1.47451
$974$	−14.3640	−0.460252
$975$	0	0
$976$	−45.2859	−1.44957
$977$	14.3103	0.457827	0.228913	−	0.973447i	$-0.426483\pi$
$977$	14.3103	0.457827	0.228913	+	0.973447i	$0.426483\pi$
$978$	−27.9363	−0.893305
$979$	59.8686	1.91341
$980$	0	0
$981$	12.7771	0.407941
$982$	9.27052	0.295834
$983$	−32.1775	−1.02630	−0.513152	−	0.858298i	$-0.671522\pi$
$983$	−32.1775	−1.02630	−0.513152	+	0.858298i	$0.671522\pi$
$984$	3.37043	0.107445
$985$	0	0
$986$	92.0374	2.93107
$987$	−5.14999	−0.163926
$988$	−110.226	−3.50674
$989$	17.6286	0.560557
$990$	0	0
$991$	−50.8207	−1.61437	−0.807186	−	0.590298i	$-0.799010\pi$
$991$	−50.8207	−1.61437	−0.807186	+	0.590298i	$0.799010\pi$
$992$	−0.640010	−0.0203203
$993$	20.2889	0.643850
$994$	−35.4484	−1.12435
$995$	0	0
$996$	−2.40142	−0.0760919
$997$	−39.1814	−1.24089	−0.620444	−	0.784251i	$-0.713048\pi$
$997$	−39.1814	−1.24089	−0.620444	+	0.784251i	$0.713048\pi$
$998$	27.7428	0.878183
$999$	11.2913	0.357239

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2325.2.a.bb.1.6	yes	6
3.2	odd	2		6975.2.a.ca.1.1		6
5.2	odd	4		2325.2.c.r.1024.11		12
5.3	odd	4		2325.2.c.r.1024.2		12
5.4	even	2		2325.2.a.y.1.1	✓	6
15.14	odd	2		6975.2.a.cc.1.6		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2325.2.a.y.1.1	✓	6	5.4	even	2
2325.2.a.bb.1.6	yes	6	1.1	even	1	trivial
2325.2.c.r.1024.2		12	5.3	odd	4
2325.2.c.r.1024.11		12	5.2	odd	4
6975.2.a.ca.1.1		6	3.2	odd	2
6975.2.a.cc.1.6		6	15.14	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2325 = 3 \cdot 5^{2} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2325.a (trivial)

\(f(q)\)	\(=\)	\(q+2.46251 q^{2} +1.00000 q^{3} +4.06395 q^{4} +2.46251 q^{6} +3.91086 q^{7} +5.08250 q^{8} +1.00000 q^{9} +3.77308 q^{11} +4.06395 q^{12} -4.83704 q^{13} +9.63054 q^{14} +4.38781 q^{16} -5.89630 q^{17} +2.46251 q^{18} +5.60730 q^{19} +3.91086 q^{21} +9.29125 q^{22} -5.19547 q^{23} +5.08250 q^{24} -11.9112 q^{26} +1.00000 q^{27} +15.8936 q^{28} -6.33880 q^{29} -1.00000 q^{31} +0.640010 q^{32} +3.77308 q^{33} -14.5197 q^{34} +4.06395 q^{36} +11.2913 q^{37} +13.8080 q^{38} -4.83704 q^{39} +0.663143 q^{41} +9.63054 q^{42} -3.39308 q^{43} +15.3336 q^{44} -12.7939 q^{46} -1.31684 q^{47} +4.38781 q^{48} +8.29486 q^{49} -5.89630 q^{51} -19.6575 q^{52} -10.4443 q^{53} +2.46251 q^{54} +19.8770 q^{56} +5.60730 q^{57} -15.6093 q^{58} +7.58816 q^{59} -10.3208 q^{61} -2.46251 q^{62} +3.91086 q^{63} -7.19958 q^{64} +9.29125 q^{66} -10.2874 q^{67} -23.9623 q^{68} -5.19547 q^{69} -3.68083 q^{71} +5.08250 q^{72} +12.3909 q^{73} +27.8048 q^{74} +22.7878 q^{76} +14.7560 q^{77} -11.9112 q^{78} -9.23657 q^{79} +1.00000 q^{81} +1.63300 q^{82} -0.590907 q^{83} +15.8936 q^{84} -8.35548 q^{86} -6.33880 q^{87} +19.1767 q^{88} +15.8673 q^{89} -18.9170 q^{91} -21.1141 q^{92} -1.00000 q^{93} -3.24273 q^{94} +0.640010 q^{96} +3.98585 q^{97} +20.4262 q^{98} +3.77308 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + q^{2} + 6 q^{3} + 7 q^{4} + q^{6} + 2 q^{7} - 3 q^{8} + 6 q^{9} + 7 q^{11} + 7 q^{12} + 4 q^{13} + 10 q^{14} + 17 q^{16} + q^{18} + 17 q^{19} + 2 q^{21} + 2 q^{22} - q^{23} - 3 q^{24} + 2 q^{26}+ \cdots + 7 q^{99}+O(q^{100}) \) 6 * q + q^2 + 6 * q^3 + 7 * q^4 + q^6 + 2 * q^7 - 3 * q^8 + 6 * q^9 + 7 * q^11 + 7 * q^12 + 4 * q^13 + 10 * q^14 + 17 * q^16 + q^18 + 17 * q^19 + 2 * q^21 + 2 * q^22 - q^23 - 3 * q^24 + 2 * q^26 + 6 * q^27 + 22 * q^28 - 8 * q^29 - 6 * q^31 - 35 * q^32 + 7 * q^33 - 13 * q^34 + 7 * q^36 + 14 * q^37 + 10 * q^38 + 4 * q^39 + 18 * q^41 + 10 * q^42 + 15 * q^43 + 4 * q^44 - 10 * q^46 - 13 * q^47 + 17 * q^48 + 18 * q^49 - 38 * q^52 - 13 * q^53 + q^54 + 32 * q^56 + 17 * q^57 + 23 * q^58 + 32 * q^59 - q^62 + 2 * q^63 + 29 * q^64 + 2 * q^66 + 9 * q^67 + 9 * q^68 - q^69 + q^71 - 3 * q^72 + 14 * q^73 + 20 * q^74 + 42 * q^76 - 14 * q^77 + 2 * q^78 + 13 * q^79 + 6 * q^81 - 32 * q^82 + 9 * q^83 + 22 * q^84 - 11 * q^86 - 8 * q^87 + 36 * q^88 + 34 * q^89 - 2 * q^91 - 46 * q^92 - 6 * q^93 + 19 * q^94 - 35 * q^96 + 30 * q^97 + 37 * q^98 + 7 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

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Database

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