LMFDB - Embedded newform 2325.2.a.bb.1.5

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			1.5
Root			$1.89294$ 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)

$f(q)$	$=$	\(q+1.89294 q^{2} +1.00000 q^{3} +1.58320 q^{4} +1.89294 q^{6} +1.92485 q^{7} -0.788968 q^{8} +1.00000 q^{9} -2.02355 q^{11} +1.58320 q^{12} +3.44035 q^{13} +3.64362 q^{14} -4.65987 q^{16} +7.60468 q^{17} +1.89294 q^{18} +0.268250 q^{19} +1.92485 q^{21} -3.83045 q^{22} +0.714890 q^{23} -0.788968 q^{24} +6.51235 q^{26} +1.00000 q^{27} +3.04743 q^{28} +5.15075 q^{29} -1.00000 q^{31} -7.24290 q^{32} -2.02355 q^{33} +14.3952 q^{34} +1.58320 q^{36} -1.83045 q^{37} +0.507780 q^{38} +3.44035 q^{39} +1.23099 q^{41} +3.64362 q^{42} +7.70545 q^{43} -3.20369 q^{44} +1.35324 q^{46} -3.33001 q^{47} -4.65987 q^{48} -3.29495 q^{49} +7.60468 q^{51} +5.44677 q^{52} +3.17210 q^{53} +1.89294 q^{54} -1.51865 q^{56} +0.268250 q^{57} +9.75003 q^{58} +7.01686 q^{59} -11.7477 q^{61} -1.89294 q^{62} +1.92485 q^{63} -4.39060 q^{64} -3.83045 q^{66} +11.3467 q^{67} +12.0398 q^{68} +0.714890 q^{69} -12.9906 q^{71} -0.788968 q^{72} +0.570221 q^{73} -3.46492 q^{74} +0.424695 q^{76} -3.89503 q^{77} +6.51235 q^{78} +14.9443 q^{79} +1.00000 q^{81} +2.33018 q^{82} -13.6842 q^{83} +3.04743 q^{84} +14.5859 q^{86} +5.15075 q^{87} +1.59652 q^{88} -1.63807 q^{89} +6.62215 q^{91} +1.13182 q^{92} -1.00000 q^{93} -6.30349 q^{94} -7.24290 q^{96} +3.13898 q^{97} -6.23713 q^{98} -2.02355 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	1.89294	1.33851	0.669254	−	0.743034i	$-0.266614\pi$
$2$	1.89294	1.33851	0.669254	+	0.743034i	$0.266614\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	1.58320	0.791602
$5$	0	0
$5$	0	0
$6$	1.89294	0.772788
$7$	1.92485	0.727525	0.363763	−	0.931492i	$-0.381492\pi$
$7$	1.92485	0.727525	0.363763	+	0.931492i	$0.381492\pi$
$8$	−0.788968	−0.278942
$9$	1.00000	0.333333
$10$	0	0
$11$	−2.02355	−0.610123	−0.305062	−	0.952333i	$-0.598677\pi$
$11$	−2.02355	−0.610123	−0.305062	+	0.952333i	$0.598677\pi$
$12$	1.58320	0.457032
$13$	3.44035	0.954180	0.477090	−	0.878854i	$-0.341691\pi$
$13$	3.44035	0.954180	0.477090	+	0.878854i	$0.341691\pi$
$14$	3.64362	0.973798
$15$	0	0
$16$	−4.65987	−1.16497
$17$	7.60468	1.84441	0.922203	−	0.386706i	$-0.126387\pi$
$17$	7.60468	1.84441	0.922203	+	0.386706i	$0.126387\pi$
$18$	1.89294	0.446169
$19$	0.268250	0.0615408	0.0307704	−	0.999526i	$-0.490204\pi$
$19$	0.268250	0.0615408	0.0307704	+	0.999526i	$0.490204\pi$
$20$	0	0
$21$	1.92485	0.420037
$22$	−3.83045	−0.816655
$23$	0.714890	0.149065	0.0745324	−	0.997219i	$-0.476254\pi$
$23$	0.714890	0.149065	0.0745324	+	0.997219i	$0.476254\pi$
$24$	−0.788968	−0.161047
$25$	0	0
$26$	6.51235	1.27718
$27$	1.00000	0.192450
$28$	3.04743	0.575910
$29$	5.15075	0.956470	0.478235	−	0.878232i	$-0.341277\pi$
$29$	5.15075	0.956470	0.478235	+	0.878232i	$0.341277\pi$
$30$	0	0
$31$	−1.00000	−0.179605
$31$	−1.00000	−0.179605
$32$	−7.24290	−1.28038
$33$	−2.02355	−0.352255
$34$	14.3952	2.46875
$35$	0	0
$36$	1.58320	0.263867
$37$	−1.83045	−0.300924	−0.150462	−	0.988616i	$-0.548076\pi$
$37$	−1.83045	−0.300924	−0.150462	+	0.988616i	$0.548076\pi$
$38$	0.507780	0.0823728
$39$	3.44035	0.550896
$40$	0	0
$41$	1.23099	0.192248	0.0961242	−	0.995369i	$-0.469355\pi$
$41$	1.23099	0.192248	0.0961242	+	0.995369i	$0.469355\pi$
$42$	3.64362	0.562222
$43$	7.70545	1.17507	0.587535	−	0.809199i	$-0.300098\pi$
$43$	7.70545	1.17507	0.587535	+	0.809199i	$0.300098\pi$
$44$	−3.20369	−0.482975
$45$	0	0
$46$	1.35324	0.199524
$47$	−3.33001	−0.485732	−0.242866	−	0.970060i	$-0.578087\pi$
$47$	−3.33001	−0.485732	−0.242866	+	0.970060i	$0.578087\pi$
$48$	−4.65987	−0.672595
$49$	−3.29495	−0.470707
$50$	0	0
$51$	7.60468	1.06487
$52$	5.44677	0.755331
$53$	3.17210	0.435721	0.217860	−	0.975980i	$-0.430092\pi$
$53$	3.17210	0.435721	0.217860	+	0.975980i	$0.430092\pi$
$54$	1.89294	0.257596
$55$	0	0
$56$	−1.51865	−0.202938
$57$	0.268250	0.0355306
$58$	9.75003	1.28024
$59$	7.01686	0.913517	0.456759	−	0.889591i	$-0.349010\pi$
$59$	7.01686	0.913517	0.456759	+	0.889591i	$0.349010\pi$
$60$	0	0
$61$	−11.7477	−1.50414	−0.752070	−	0.659083i	$-0.770945\pi$
$61$	−11.7477	−1.50414	−0.752070	+	0.659083i	$0.770945\pi$
$62$	−1.89294	−0.240403
$63$	1.92485	0.242508
$64$	−4.39060	−0.548825
$65$	0	0
$66$	−3.83045	−0.471496
$67$	11.3467	1.38622	0.693109	−	0.720833i	$-0.256241\pi$
$67$	11.3467	1.38622	0.693109	+	0.720833i	$0.256241\pi$
$68$	12.0398	1.46004
$69$	0.714890	0.0860626
$70$	0	0
$71$	−12.9906	−1.54170	−0.770851	−	0.637015i	$-0.780169\pi$
$71$	−12.9906	−1.54170	−0.770851	+	0.637015i	$0.780169\pi$
$72$	−0.788968	−0.0929808
$73$	0.570221	0.0667393	0.0333696	−	0.999443i	$-0.489376\pi$
$73$	0.570221	0.0667393	0.0333696	+	0.999443i	$0.489376\pi$
$74$	−3.46492	−0.402789
$75$	0	0
$76$	0.424695	0.0487158
$77$	−3.89503	−0.443880
$78$	6.51235	0.737379
$79$	14.9443	1.68136	0.840680	−	0.541532i	$-0.182156\pi$
$79$	14.9443	1.68136	0.840680	+	0.541532i	$0.182156\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	2.33018	0.257326
$83$	−13.6842	−1.50203	−0.751016	−	0.660284i	$-0.770436\pi$
$83$	−13.6842	−1.50203	−0.751016	+	0.660284i	$0.770436\pi$
$84$	3.04743	0.332502
$85$	0	0
$86$	14.5859	1.57284
$87$	5.15075	0.552218
$88$	1.59652	0.170189
$89$	−1.63807	−0.173635	−0.0868173	−	0.996224i	$-0.527670\pi$
$89$	−1.63807	−0.173635	−0.0868173	+	0.996224i	$0.527670\pi$
$90$	0	0
$91$	6.62215	0.694190
$92$	1.13182	0.118000
$93$	−1.00000	−0.103695
$94$	−6.30349	−0.650155
$95$	0	0
$96$	−7.24290	−0.739226
$97$	3.13898	0.318715	0.159358	−	0.987221i	$-0.449058\pi$
$97$	3.13898	0.318715	0.159358	+	0.987221i	$0.449058\pi$
$98$	−6.23713	−0.630045
$99$	−2.02355	−0.203374
$100$	0	0
$101$	−6.23176	−0.620083	−0.310042	−	0.950723i	$-0.600343\pi$
$101$	−6.23176	−0.620083	−0.310042	+	0.950723i	$0.600343\pi$
$102$	14.3952	1.42533
$103$	6.31720	0.622452	0.311226	−	0.950336i	$-0.399260\pi$
$103$	6.31720	0.622452	0.311226	+	0.950336i	$0.399260\pi$
$104$	−2.71432	−0.266161
$105$	0	0
$106$	6.00457	0.583216
$107$	−4.93261	−0.476854	−0.238427	−	0.971160i	$-0.576632\pi$
$107$	−4.93261	−0.476854	−0.238427	+	0.971160i	$0.576632\pi$
$108$	1.58320	0.152344
$109$	−7.92525	−0.759102	−0.379551	−	0.925171i	$-0.623921\pi$
$109$	−7.92525	−0.759102	−0.379551	+	0.925171i	$0.623921\pi$
$110$	0	0
$111$	−1.83045	−0.173739
$112$	−8.96956	−0.847544
$113$	−18.7260	−1.76159	−0.880795	−	0.473498i	$-0.842991\pi$
$113$	−18.7260	−1.76159	−0.880795	+	0.473498i	$0.842991\pi$
$114$	0.507780	0.0475580
$115$	0	0
$116$	8.15468	0.757143
$117$	3.44035	0.318060
$118$	13.2825	1.22275
$119$	14.6379	1.34185
$120$	0	0
$121$	−6.90524	−0.627749
$122$	−22.2377	−2.01330
$123$	1.23099	0.110995
$124$	−1.58320	−0.142176
$125$	0	0
$126$	3.64362	0.324599
$127$	−0.243804	−0.0216341	−0.0108171	−	0.999941i	$-0.503443\pi$
$127$	−0.243804	−0.0216341	−0.0108171	+	0.999941i	$0.503443\pi$
$128$	6.17469	0.545770
$129$	7.70545	0.678427
$130$	0	0
$131$	4.67138	0.408140	0.204070	−	0.978956i	$-0.434583\pi$
$131$	4.67138	0.408140	0.204070	+	0.978956i	$0.434583\pi$
$132$	−3.20369	−0.278846
$133$	0.516342	0.0447725
$134$	21.4785	1.85546
$135$	0	0
$136$	−5.99985	−0.514483
$137$	−12.0181	−1.02677	−0.513386	−	0.858158i	$-0.671609\pi$
$137$	−12.0181	−1.02677	−0.513386	+	0.858158i	$0.671609\pi$
$138$	1.35324	0.115195
$139$	9.49065	0.804987	0.402493	−	0.915423i	$-0.368144\pi$
$139$	9.49065	0.804987	0.402493	+	0.915423i	$0.368144\pi$
$140$	0	0
$141$	−3.33001	−0.280437
$142$	−24.5904	−2.06358
$143$	−6.96171	−0.582168
$144$	−4.65987	−0.388323
$145$	0	0
$146$	1.07939	0.0893310
$147$	−3.29495	−0.271763
$148$	−2.89797	−0.238212
$149$	3.42903	0.280917	0.140458	−	0.990087i	$-0.455142\pi$
$149$	3.42903	0.280917	0.140458	+	0.990087i	$0.455142\pi$
$150$	0	0
$151$	−11.5823	−0.942557	−0.471278	−	0.881984i	$-0.656207\pi$
$151$	−11.5823	−0.942557	−0.471278	+	0.881984i	$0.656207\pi$
$152$	−0.211641	−0.0171663
$153$	7.60468	0.614802
$154$	−7.37304	−0.594137
$155$	0	0
$156$	5.44677	0.436091
$157$	−21.2401	−1.69514	−0.847571	−	0.530682i	$-0.821936\pi$
$157$	−21.2401	−1.69514	−0.847571	+	0.530682i	$0.821936\pi$
$158$	28.2885	2.25051
$159$	3.17210	0.251564
$160$	0	0
$161$	1.37606	0.108448
$162$	1.89294	0.148723
$163$	3.57277	0.279841	0.139920	−	0.990163i	$-0.455315\pi$
$163$	3.57277	0.279841	0.139920	+	0.990163i	$0.455315\pi$
$164$	1.94891	0.152184
$165$	0	0
$166$	−25.9032	−2.01048
$167$	−11.6839	−0.904126	−0.452063	−	0.891986i	$-0.649312\pi$
$167$	−11.6839	−0.904126	−0.452063	+	0.891986i	$0.649312\pi$
$168$	−1.51865	−0.117166
$169$	−1.16402	−0.0895397
$170$	0	0
$171$	0.268250	0.0205136
$172$	12.1993	0.930188
$173$	−17.9661	−1.36594	−0.682968	−	0.730448i	$-0.739311\pi$
$173$	−17.9661	−1.36594	−0.682968	+	0.730448i	$0.739311\pi$
$174$	9.75003	0.739148
$175$	0	0
$176$	9.42949	0.710774
$177$	7.01686	0.527420
$178$	−3.10075	−0.232411
$179$	−16.8563	−1.25990	−0.629948	−	0.776637i	$-0.716924\pi$
$179$	−16.8563	−1.25990	−0.629948	+	0.776637i	$0.716924\pi$
$180$	0	0
$181$	−11.9617	−0.889108	−0.444554	−	0.895752i	$-0.646638\pi$
$181$	−11.9617	−0.889108	−0.444554	+	0.895752i	$0.646638\pi$
$182$	12.5353	0.929179
$183$	−11.7477	−0.868416
$184$	−0.564025	−0.0415805
$185$	0	0
$186$	−1.89294	−0.138797
$187$	−15.3885	−1.12532
$188$	−5.27208	−0.384506
$189$	1.92485	0.140012
$190$	0	0
$191$	21.3972	1.54825	0.774125	−	0.633032i	$-0.218190\pi$
$191$	21.3972	1.54825	0.774125	+	0.633032i	$0.218190\pi$
$192$	−4.39060	−0.316864
$193$	12.4902	0.899064	0.449532	−	0.893264i	$-0.351591\pi$
$193$	12.4902	0.899064	0.449532	+	0.893264i	$0.351591\pi$
$194$	5.94189	0.426603
$195$	0	0
$196$	−5.21658	−0.372613
$197$	9.61876	0.685309	0.342654	−	0.939462i	$-0.388674\pi$
$197$	9.61876	0.685309	0.342654	+	0.939462i	$0.388674\pi$
$198$	−3.83045	−0.272218
$199$	16.8385	1.19365	0.596825	−	0.802371i	$-0.296429\pi$
$199$	16.8385	1.19365	0.596825	+	0.802371i	$0.296429\pi$
$200$	0	0
$201$	11.3467	0.800333
$202$	−11.7963	−0.829986
$203$	9.91442	0.695856
$204$	12.0398	0.842952
$205$	0	0
$206$	11.9581	0.833157
$207$	0.714890	0.0496883
$208$	−16.0316	−1.11159
$209$	−0.542818	−0.0375475
$210$	0	0
$211$	21.1340	1.45492	0.727461	−	0.686149i	$-0.240700\pi$
$211$	21.1340	1.45492	0.727461	+	0.686149i	$0.240700\pi$
$212$	5.02207	0.344918
$213$	−12.9906	−0.890102
$214$	−9.33711	−0.638272
$215$	0	0
$216$	−0.788968	−0.0536825
$217$	−1.92485	−0.130667
$218$	−15.0020	−1.01606
$219$	0.570221	0.0385320
$220$	0	0
$221$	26.1627	1.75990
$222$	−3.46492	−0.232550
$223$	4.35225	0.291448	0.145724	−	0.989325i	$-0.453449\pi$
$223$	4.35225	0.291448	0.145724	+	0.989325i	$0.453449\pi$
$224$	−13.9415	−0.931506
$225$	0	0
$226$	−35.4470	−2.35790
$227$	−9.03171	−0.599456	−0.299728	−	0.954025i	$-0.596896\pi$
$227$	−9.03171	−0.599456	−0.299728	+	0.954025i	$0.596896\pi$
$228$	0.424695	0.0281261
$229$	9.22207	0.609412	0.304706	−	0.952447i	$-0.401442\pi$
$229$	9.22207	0.609412	0.304706	+	0.952447i	$0.401442\pi$
$230$	0	0
$231$	−3.89503	−0.256274
$232$	−4.06378	−0.266800
$233$	−21.1143	−1.38324	−0.691622	−	0.722260i	$-0.743104\pi$
$233$	−21.1143	−1.38324	−0.691622	+	0.722260i	$0.743104\pi$
$234$	6.51235	0.425726
$235$	0	0
$236$	11.1091	0.723142
$237$	14.9443	0.970734
$238$	27.7086	1.79608
$239$	3.91589	0.253298	0.126649	−	0.991948i	$-0.459578\pi$
$239$	3.91589	0.253298	0.126649	+	0.991948i	$0.459578\pi$
$240$	0	0
$241$	−10.4970	−0.676172	−0.338086	−	0.941115i	$-0.609779\pi$
$241$	−10.4970	−0.676172	−0.338086	+	0.941115i	$0.609779\pi$
$242$	−13.0712	−0.840247
$243$	1.00000	0.0641500
$244$	−18.5990	−1.19068
$245$	0	0
$246$	2.33018	0.148567
$247$	0.922874	0.0587210
$248$	0.788968	0.0500995
$249$	−13.6842	−0.867198
$250$	0	0
$251$	−1.90584	−0.120296	−0.0601478	−	0.998189i	$-0.519157\pi$
$251$	−1.90584	−0.120296	−0.0601478	+	0.998189i	$0.519157\pi$
$252$	3.04743	0.191970
$253$	−1.44662	−0.0909479
$254$	−0.461506	−0.0289575
$255$	0	0
$256$	20.4695	1.27934
$257$	12.7714	0.796659	0.398329	−	0.917242i	$-0.369590\pi$
$257$	12.7714	0.796659	0.398329	+	0.917242i	$0.369590\pi$
$258$	14.5859	0.908080
$259$	−3.52334	−0.218930
$260$	0	0
$261$	5.15075	0.318823
$262$	8.84262	0.546299
$263$	−10.0322	−0.618613	−0.309307	−	0.950962i	$-0.600097\pi$
$263$	−10.0322	−0.618613	−0.309307	+	0.950962i	$0.600097\pi$
$264$	1.59652	0.0982588
$265$	0	0
$266$	0.977401	0.0599283
$267$	−1.63807	−0.100248
$268$	17.9641	1.09733
$269$	−17.1185	−1.04373	−0.521867	−	0.853027i	$-0.674764\pi$
$269$	−17.1185	−1.04373	−0.521867	+	0.853027i	$0.674764\pi$
$270$	0	0
$271$	7.56317	0.459430	0.229715	−	0.973258i	$-0.426221\pi$
$271$	7.56317	0.459430	0.229715	+	0.973258i	$0.426221\pi$
$272$	−35.4369	−2.14867
$273$	6.62215	0.400791
$274$	−22.7494	−1.37434
$275$	0	0
$276$	1.13182	0.0681273
$277$	−10.6842	−0.641951	−0.320975	−	0.947088i	$-0.604011\pi$
$277$	−10.6842	−0.641951	−0.320975	+	0.947088i	$0.604011\pi$
$278$	17.9652	1.07748
$279$	−1.00000	−0.0598684
$280$	0	0
$281$	15.7539	0.939799	0.469900	−	0.882720i	$-0.344290\pi$
$281$	15.7539	0.939799	0.469900	+	0.882720i	$0.344290\pi$
$282$	−6.30349	−0.375367
$283$	−26.5692	−1.57938	−0.789688	−	0.613509i	$-0.789757\pi$
$283$	−26.5692	−1.57938	−0.789688	+	0.613509i	$0.789757\pi$
$284$	−20.5668	−1.22041
$285$	0	0
$286$	−13.1781	−0.779236
$287$	2.36947	0.139865
$288$	−7.24290	−0.426792
$289$	40.8312	2.40183
$290$	0	0
$291$	3.13898	0.184010
$292$	0.902776	0.0528310
$293$	−23.7515	−1.38758	−0.693789	−	0.720179i	$-0.744060\pi$
$293$	−23.7515	−1.38758	−0.693789	+	0.720179i	$0.744060\pi$
$294$	−6.23713	−0.363757
$295$	0	0
$296$	1.44417	0.0839405
$297$	−2.02355	−0.117418
$298$	6.49093	0.376009
$299$	2.45947	0.142235
$300$	0	0
$301$	14.8318	0.854893
$302$	−21.9246	−1.26162
$303$	−6.23176	−0.358005
$304$	−1.25001	−0.0716931
$305$	0	0
$306$	14.3952	0.822917
$307$	17.9239	1.02297	0.511485	−	0.859292i	$-0.329095\pi$
$307$	17.9239	1.02297	0.511485	+	0.859292i	$0.329095\pi$
$308$	−6.16663	−0.351376
$309$	6.31720	0.359373
$310$	0	0
$311$	7.10858	0.403091	0.201545	−	0.979479i	$-0.435404\pi$
$311$	7.10858	0.403091	0.201545	+	0.979479i	$0.435404\pi$
$312$	−2.71432	−0.153668
$313$	−20.9587	−1.18466	−0.592328	−	0.805697i	$-0.701791\pi$
$313$	−20.9587	−1.18466	−0.592328	+	0.805697i	$0.701791\pi$
$314$	−40.2061	−2.26896
$315$	0	0
$316$	23.6598	1.33097
$317$	19.9348	1.11965	0.559824	−	0.828612i	$-0.310869\pi$
$317$	19.9348	1.11965	0.559824	+	0.828612i	$0.310869\pi$
$318$	6.00457	0.336720
$319$	−10.4228	−0.583565
$320$	0	0
$321$	−4.93261	−0.275312
$322$	2.60478	0.145159
$323$	2.03996	0.113506
$324$	1.58320	0.0879558
$325$	0	0
$326$	6.76302	0.374569
$327$	−7.92525	−0.438268
$328$	−0.971212	−0.0536262
$329$	−6.40977	−0.353382
$330$	0	0
$331$	18.4902	1.01631	0.508156	−	0.861265i	$-0.330327\pi$
$331$	18.4902	1.01631	0.508156	+	0.861265i	$0.330327\pi$
$332$	−21.6648	−1.18901
$333$	−1.83045	−0.100308
$334$	−22.1168	−1.21018
$335$	0	0
$336$	−8.96956	−0.489330
$337$	1.36343	0.0742706	0.0371353	−	0.999310i	$-0.488177\pi$
$337$	1.36343	0.0742706	0.0371353	+	0.999310i	$0.488177\pi$
$338$	−2.20341	−0.119850
$339$	−18.7260	−1.01705
$340$	0	0
$341$	2.02355	0.109581
$342$	0.507780	0.0274576
$343$	−19.8162	−1.06998
$344$	−6.07936	−0.327777
$345$	0	0
$346$	−34.0086	−1.82831
$347$	2.76556	0.148463	0.0742315	−	0.997241i	$-0.476350\pi$
$347$	2.76556	0.148463	0.0742315	+	0.997241i	$0.476350\pi$
$348$	8.15468	0.437137
$349$	6.25030	0.334571	0.167285	−	0.985909i	$-0.446500\pi$
$349$	6.25030	0.334571	0.167285	+	0.985909i	$0.446500\pi$
$350$	0	0
$351$	3.44035	0.183632
$352$	14.6564	0.781187
$353$	7.55977	0.402366	0.201183	−	0.979554i	$-0.435521\pi$
$353$	7.55977	0.402366	0.201183	+	0.979554i	$0.435521\pi$
$354$	13.2825	0.705955
$355$	0	0
$356$	−2.59339	−0.137449
$357$	14.6379	0.774719
$358$	−31.9078	−1.68638
$359$	3.44843	0.182001	0.0910005	−	0.995851i	$-0.470994\pi$
$359$	3.44843	0.182001	0.0910005	+	0.995851i	$0.470994\pi$
$360$	0	0
$361$	−18.9280	−0.996213
$362$	−22.6428	−1.19008
$363$	−6.90524	−0.362431
$364$	10.4842	0.549522
$365$	0	0
$366$	−22.2377	−1.16238
$367$	−27.2017	−1.41992	−0.709959	−	0.704243i	$-0.751287\pi$
$367$	−27.2017	−1.41992	−0.709959	+	0.704243i	$0.751287\pi$
$368$	−3.33129	−0.173656
$369$	1.23099	0.0640828
$370$	0	0
$371$	6.10581	0.316998
$372$	−1.58320	−0.0820853
$373$	−1.93173	−0.100021	−0.0500105	−	0.998749i	$-0.515925\pi$
$373$	−1.93173	−0.100021	−0.0500105	+	0.998749i	$0.515925\pi$
$374$	−29.1294	−1.50624
$375$	0	0
$376$	2.62727	0.135491
$377$	17.7204	0.912645
$378$	3.64362	0.187407
$379$	−5.60497	−0.287908	−0.143954	−	0.989584i	$-0.545982\pi$
$379$	−5.60497	−0.287908	−0.143954	+	0.989584i	$0.545982\pi$
$380$	0	0
$381$	−0.243804	−0.0124905
$382$	40.5036	2.07234
$383$	−22.8601	−1.16810	−0.584049	−	0.811718i	$-0.698532\pi$
$383$	−22.8601	−1.16810	−0.584049	+	0.811718i	$0.698532\pi$
$384$	6.17469	0.315101
$385$	0	0
$386$	23.6431	1.20340
$387$	7.70545	0.391690
$388$	4.96965	0.252296
$389$	−37.4724	−1.89993	−0.949964	−	0.312359i	$-0.898881\pi$
$389$	−37.4724	−1.89993	−0.949964	+	0.312359i	$0.898881\pi$
$390$	0	0
$391$	5.43651	0.274936
$392$	2.59961	0.131300
$393$	4.67138	0.235640
$394$	18.2077	0.917290
$395$	0	0
$396$	−3.20369	−0.160992
$397$	12.6683	0.635804	0.317902	−	0.948124i	$-0.397022\pi$
$397$	12.6683	0.635804	0.317902	+	0.948124i	$0.397022\pi$
$398$	31.8742	1.59771
$399$	0.516342	0.0258494
$400$	0	0
$401$	−37.0834	−1.85186	−0.925929	−	0.377698i	$-0.876716\pi$
$401$	−37.0834	−1.85186	−0.925929	+	0.377698i	$0.876716\pi$
$402$	21.4785	1.07125
$403$	−3.44035	−0.171376
$404$	−9.86614	−0.490859
$405$	0	0
$406$	18.7674	0.931408
$407$	3.70401	0.183601
$408$	−5.99985	−0.297037
$409$	−17.2497	−0.852941	−0.426470	−	0.904502i	$-0.640243\pi$
$409$	−17.2497	−0.852941	−0.426470	+	0.904502i	$0.640243\pi$
$410$	0	0
$411$	−12.0181	−0.592807
$412$	10.0014	0.492734
$413$	13.5064	0.664607
$414$	1.35324	0.0665081
$415$	0	0
$416$	−24.9181	−1.22171
$417$	9.49065	0.464759
$418$	−1.02752	−0.0502576
$419$	1.26861	0.0619757	0.0309879	−	0.999520i	$-0.490135\pi$
$419$	1.26861	0.0619757	0.0309879	+	0.999520i	$0.490135\pi$
$420$	0	0
$421$	29.5671	1.44101	0.720506	−	0.693449i	$-0.243910\pi$
$421$	29.5671	1.44101	0.720506	+	0.693449i	$0.243910\pi$
$422$	40.0052	1.94742
$423$	−3.33001	−0.161911
$424$	−2.50268	−0.121541
$425$	0	0
$426$	−24.5904	−1.19141
$427$	−22.6126	−1.09430
$428$	−7.80933	−0.377478
$429$	−6.96171	−0.336115
$430$	0	0
$431$	−23.3588	−1.12515	−0.562577	−	0.826745i	$-0.690190\pi$
$431$	−23.3588	−1.12515	−0.562577	+	0.826745i	$0.690190\pi$
$432$	−4.65987	−0.224198
$433$	15.2684	0.733751	0.366875	−	0.930270i	$-0.380428\pi$
$433$	15.2684	0.733751	0.366875	+	0.930270i	$0.380428\pi$
$434$	−3.64362	−0.174899
$435$	0	0
$436$	−12.5473	−0.600906
$437$	0.191769	0.00917357
$438$	1.07939	0.0515753
$439$	13.8738	0.662161	0.331081	−	0.943602i	$-0.392587\pi$
$439$	13.8738	0.662161	0.331081	+	0.943602i	$0.392587\pi$
$440$	0	0
$441$	−3.29495	−0.156902
$442$	49.5244	2.35563
$443$	9.26968	0.440416	0.220208	−	0.975453i	$-0.429326\pi$
$443$	9.26968	0.440416	0.220208	+	0.975453i	$0.429326\pi$
$444$	−2.89797	−0.137532
$445$	0	0
$446$	8.23852	0.390105
$447$	3.42903	0.162187
$448$	−8.45124	−0.399284
$449$	14.3026	0.674982	0.337491	−	0.941329i	$-0.390422\pi$
$449$	14.3026	0.674982	0.337491	+	0.941329i	$0.390422\pi$
$450$	0	0
$451$	−2.49097	−0.117295
$452$	−29.6470	−1.39448
$453$	−11.5823	−0.544185
$454$	−17.0964	−0.802376
$455$	0	0
$456$	−0.211641	−0.00991099
$457$	36.9120	1.72667	0.863334	−	0.504633i	$-0.168372\pi$
$457$	36.9120	1.72667	0.863334	+	0.504633i	$0.168372\pi$
$458$	17.4568	0.815702
$459$	7.60468	0.354956
$460$	0	0
$461$	39.0841	1.82033	0.910164	−	0.414249i	$-0.135956\pi$
$461$	39.0841	1.82033	0.910164	+	0.414249i	$0.135956\pi$
$462$	−7.37304	−0.343025
$463$	−0.798135	−0.0370925	−0.0185462	−	0.999828i	$-0.505904\pi$
$463$	−0.798135	−0.0370925	−0.0185462	+	0.999828i	$0.505904\pi$
$464$	−24.0018	−1.11426
$465$	0	0
$466$	−39.9680	−1.85148
$467$	1.71542	0.0793801	0.0396901	−	0.999212i	$-0.487363\pi$
$467$	1.71542	0.0793801	0.0396901	+	0.999212i	$0.487363\pi$
$468$	5.44677	0.251777
$469$	21.8407	1.00851
$470$	0	0
$471$	−21.2401	−0.978691
$472$	−5.53608	−0.254819
$473$	−15.5924	−0.716938
$474$	28.2885	1.29933
$475$	0	0
$476$	23.1747	1.06221
$477$	3.17210	0.145240
$478$	7.41253	0.339041
$479$	19.9250	0.910395	0.455198	−	0.890390i	$-0.349569\pi$
$479$	19.9250	0.910395	0.455198	+	0.890390i	$0.349569\pi$
$480$	0	0
$481$	−6.29738	−0.287136
$482$	−19.8702	−0.905061
$483$	1.37606	0.0626127
$484$	−10.9324	−0.496928
$485$	0	0
$486$	1.89294	0.0858653
$487$	19.9890	0.905787	0.452893	−	0.891565i	$-0.350392\pi$
$487$	19.9890	0.905787	0.452893	+	0.891565i	$0.350392\pi$
$488$	9.26857	0.419569
$489$	3.57277	0.161566
$490$	0	0
$491$	−30.9805	−1.39813	−0.699065	−	0.715058i	$-0.746400\pi$
$491$	−30.9805	−1.39813	−0.699065	+	0.715058i	$0.746400\pi$
$492$	1.94891	0.0878635
$493$	39.1698	1.76412
$494$	1.74694	0.0785985
$495$	0	0
$496$	4.65987	0.209234
$497$	−25.0050	−1.12163
$498$	−25.9032	−1.16075
$499$	−6.32240	−0.283029	−0.141515	−	0.989936i	$-0.545197\pi$
$499$	−6.32240	−0.283029	−0.141515	+	0.989936i	$0.545197\pi$
$500$	0	0
$501$	−11.6839	−0.521997
$502$	−3.60763	−0.161017
$503$	−12.6172	−0.562571	−0.281286	−	0.959624i	$-0.590761\pi$
$503$	−12.6172	−0.562571	−0.281286	+	0.959624i	$0.590761\pi$
$504$	−1.51865	−0.0676459
$505$	0	0
$506$	−2.73835	−0.121734
$507$	−1.16402	−0.0516958
$508$	−0.385992	−0.0171256
$509$	−11.5920	−0.513805	−0.256902	−	0.966437i	$-0.582702\pi$
$509$	−11.5920	−0.513805	−0.256902	+	0.966437i	$0.582702\pi$
$510$	0	0
$511$	1.09759	0.0485545
$512$	26.3980	1.16664
$513$	0.268250	0.0118435
$514$	24.1754	1.06633
$515$	0	0
$516$	12.1993	0.537044
$517$	6.73844	0.296356
$518$	−6.66946	−0.293039
$519$	−17.9661	−0.788623
$520$	0	0
$521$	14.3456	0.628491	0.314246	−	0.949342i	$-0.398248\pi$
$521$	14.3456	0.628491	0.314246	+	0.949342i	$0.398248\pi$
$522$	9.75003	0.426747
$523$	−23.3611	−1.02151	−0.510755	−	0.859727i	$-0.670634\pi$
$523$	−23.3611	−1.02151	−0.510755	+	0.859727i	$0.670634\pi$
$524$	7.39575	0.323085
$525$	0	0
$526$	−18.9903	−0.828018
$527$	−7.60468	−0.331265
$528$	9.42949	0.410366
$529$	−22.4889	−0.977780
$530$	0	0
$531$	7.01686	0.304506
$532$	0.817474	0.0354420
$533$	4.23503	0.183440
$534$	−3.10075	−0.134183
$535$	0	0
$536$	−8.95217	−0.386675
$537$	−16.8563	−0.727401
$538$	−32.4043	−1.39705
$539$	6.66750	0.287189
$540$	0	0
$541$	17.2433	0.741349	0.370674	−	0.928763i	$-0.379127\pi$
$541$	17.2433	0.741349	0.370674	+	0.928763i	$0.379127\pi$
$542$	14.3166	0.614950
$543$	−11.9617	−0.513327
$544$	−55.0800	−2.36153
$545$	0	0
$546$	12.5353	0.536462
$547$	−11.7986	−0.504472	−0.252236	−	0.967666i	$-0.581166\pi$
$547$	−11.7986	−0.504472	−0.252236	+	0.967666i	$0.581166\pi$
$548$	−19.0270	−0.812795
$549$	−11.7477	−0.501380
$550$	0	0
$551$	1.38169	0.0588619
$552$	−0.564025	−0.0240065
$553$	28.7655	1.22323
$554$	−20.2245	−0.859256
$555$	0	0
$556$	15.0256	0.637229
$557$	4.77368	0.202267	0.101134	−	0.994873i	$-0.467753\pi$
$557$	4.77368	0.202267	0.101134	+	0.994873i	$0.467753\pi$
$558$	−1.89294	−0.0801343
$559$	26.5094	1.12123
$560$	0	0
$561$	−15.3885	−0.649701
$562$	29.8211	1.25793
$563$	15.2955	0.644629	0.322314	−	0.946633i	$-0.395539\pi$
$563$	15.2955	0.644629	0.322314	+	0.946633i	$0.395539\pi$
$564$	−5.27208	−0.221995
$565$	0	0
$566$	−50.2938	−2.11401
$567$	1.92485	0.0808361
$568$	10.2492	0.430046
$569$	1.55943	0.0653747	0.0326873	−	0.999466i	$-0.489593\pi$
$569$	1.55943	0.0653747	0.0326873	+	0.999466i	$0.489593\pi$
$570$	0	0
$571$	36.3781	1.52238	0.761189	−	0.648531i	$-0.224616\pi$
$571$	36.3781	1.52238	0.761189	+	0.648531i	$0.224616\pi$
$572$	−11.0218	−0.460845
$573$	21.3972	0.893883
$574$	4.48526	0.187211
$575$	0	0
$576$	−4.39060	−0.182942
$577$	−13.6972	−0.570221	−0.285110	−	0.958495i	$-0.592030\pi$
$577$	−13.6972	−0.570221	−0.285110	+	0.958495i	$0.592030\pi$
$578$	77.2908	3.21487
$579$	12.4902	0.519075
$580$	0	0
$581$	−26.3400	−1.09277
$582$	5.94189	0.246299
$583$	−6.41890	−0.265844
$584$	−0.449886	−0.0186164
$585$	0	0
$586$	−44.9601	−1.85728
$587$	−11.2055	−0.462499	−0.231250	−	0.972894i	$-0.574281\pi$
$587$	−11.2055	−0.462499	−0.231250	+	0.972894i	$0.574281\pi$
$588$	−5.21658	−0.215128
$589$	−0.268250	−0.0110531
$590$	0	0
$591$	9.61876	0.395663
$592$	8.52966	0.350567
$593$	−1.57118	−0.0645205	−0.0322602	−	0.999480i	$-0.510271\pi$
$593$	−1.57118	−0.0645205	−0.0322602	+	0.999480i	$0.510271\pi$
$594$	−3.83045	−0.157165
$595$	0	0
$596$	5.42885	0.222374
$597$	16.8385	0.689154
$598$	4.65561	0.190382
$599$	26.6448	1.08868	0.544338	−	0.838866i	$-0.316781\pi$
$599$	26.6448	1.08868	0.544338	+	0.838866i	$0.316781\pi$
$600$	0	0
$601$	−4.91310	−0.200410	−0.100205	−	0.994967i	$-0.531950\pi$
$601$	−4.91310	−0.200410	−0.100205	+	0.994967i	$0.531950\pi$
$602$	28.0757	1.14428
$603$	11.3467	0.462073
$604$	−18.3372	−0.746130
$605$	0	0
$606$	−11.7963	−0.479193
$607$	14.0184	0.568987	0.284494	−	0.958678i	$-0.408175\pi$
$607$	14.0184	0.568987	0.284494	+	0.958678i	$0.408175\pi$
$608$	−1.94291	−0.0787954
$609$	9.91442	0.401753
$610$	0	0
$611$	−11.4564	−0.463476
$612$	12.0398	0.486678
$613$	20.2034	0.816008	0.408004	−	0.912980i	$-0.366225\pi$
$613$	20.2034	0.816008	0.408004	+	0.912980i	$0.366225\pi$
$614$	33.9288	1.36925
$615$	0	0
$616$	3.07306	0.123817
$617$	46.1203	1.85673	0.928367	−	0.371665i	$-0.121213\pi$
$617$	46.1203	1.85673	0.928367	+	0.371665i	$0.121213\pi$
$618$	11.9581	0.481023
$619$	34.5262	1.38772	0.693862	−	0.720108i	$-0.255908\pi$
$619$	34.5262	1.38772	0.693862	+	0.720108i	$0.255908\pi$
$620$	0	0
$621$	0.714890	0.0286875
$622$	13.4561	0.539540
$623$	−3.15303	−0.126323
$624$	−16.0316	−0.641777
$625$	0	0
$626$	−39.6735	−1.58567
$627$	−0.542818	−0.0216781
$628$	−33.6273	−1.34188
$629$	−13.9200	−0.555026
$630$	0	0
$631$	−37.0446	−1.47472	−0.737360	−	0.675500i	$-0.763928\pi$
$631$	−37.0446	−1.47472	−0.737360	+	0.675500i	$0.763928\pi$
$632$	−11.7905	−0.469003
$633$	21.1340	0.840000
$634$	37.7352	1.49866
$635$	0	0
$636$	5.02207	0.199138
$637$	−11.3358	−0.449140
$638$	−19.7297	−0.781106
$639$	−12.9906	−0.513901
$640$	0	0
$641$	−29.8914	−1.18064	−0.590320	−	0.807169i	$-0.700998\pi$
$641$	−29.8914	−1.18064	−0.590320	+	0.807169i	$0.700998\pi$
$642$	−9.33711	−0.368506
$643$	36.1477	1.42553	0.712763	−	0.701405i	$-0.247443\pi$
$643$	36.1477	1.42553	0.712763	+	0.701405i	$0.247443\pi$
$644$	2.17858	0.0858479
$645$	0	0
$646$	3.86151	0.151929
$647$	−25.0553	−0.985025	−0.492512	−	0.870305i	$-0.663921\pi$
$647$	−25.0553	−0.985025	−0.492512	+	0.870305i	$0.663921\pi$
$648$	−0.788968	−0.0309936
$649$	−14.1990	−0.557358
$650$	0	0
$651$	−1.92485	−0.0754408
$652$	5.65642	0.221522
$653$	−32.1230	−1.25707	−0.628536	−	0.777780i	$-0.716346\pi$
$653$	−32.1230	−1.25707	−0.628536	+	0.777780i	$0.716346\pi$
$654$	−15.0020	−0.586624
$655$	0	0
$656$	−5.73626	−0.223963
$657$	0.570221	0.0222464
$658$	−12.1333	−0.473004
$659$	12.5550	0.489073	0.244536	−	0.969640i	$-0.421364\pi$
$659$	12.5550	0.489073	0.244536	+	0.969640i	$0.421364\pi$
$660$	0	0
$661$	20.5455	0.799127	0.399563	−	0.916706i	$-0.369162\pi$
$661$	20.5455	0.799127	0.399563	+	0.916706i	$0.369162\pi$
$662$	35.0007	1.36034
$663$	26.1627	1.01608
$664$	10.7964	0.418980
$665$	0	0
$666$	−3.46492	−0.134263
$667$	3.68222	0.142576
$668$	−18.4980	−0.715708
$669$	4.35225	0.168268
$670$	0	0
$671$	23.7721	0.917711
$672$	−13.9415	−0.537805
$673$	−17.5886	−0.677992	−0.338996	−	0.940788i	$-0.610087\pi$
$673$	−17.5886	−0.677992	−0.338996	+	0.940788i	$0.610087\pi$
$674$	2.58088	0.0994118
$675$	0	0
$676$	−1.84287	−0.0708798
$677$	30.8887	1.18715	0.593574	−	0.804779i	$-0.297716\pi$
$677$	30.8887	1.18715	0.593574	+	0.804779i	$0.297716\pi$
$678$	−35.4470	−1.36133
$679$	6.04207	0.231873
$680$	0	0
$681$	−9.03171	−0.346096
$682$	3.83045	0.146675
$683$	7.11535	0.272261	0.136131	−	0.990691i	$-0.456533\pi$
$683$	7.11535	0.272261	0.136131	+	0.990691i	$0.456533\pi$
$684$	0.424695	0.0162386
$685$	0	0
$686$	−37.5109	−1.43217
$687$	9.22207	0.351844
$688$	−35.9064	−1.36892
$689$	10.9131	0.415756
$690$	0	0
$691$	6.59785	0.250994	0.125497	−	0.992094i	$-0.459947\pi$
$691$	6.59785	0.250994	0.125497	+	0.992094i	$0.459947\pi$
$692$	−28.4440	−1.08128
$693$	−3.89503	−0.147960
$694$	5.23502	0.198719
$695$	0	0
$696$	−4.06378	−0.154037
$697$	9.36129	0.354584
$698$	11.8314	0.447826
$699$	−21.1143	−0.798616
$700$	0	0
$701$	−40.7475	−1.53901	−0.769505	−	0.638640i	$-0.779497\pi$
$701$	−40.7475	−1.53901	−0.769505	+	0.638640i	$0.779497\pi$
$702$	6.51235	0.245793
$703$	−0.491018	−0.0185191
$704$	8.88459	0.334851
$705$	0	0
$706$	14.3101	0.538570
$707$	−11.9952	−0.451126
$708$	11.1091	0.417506
$709$	−5.18210	−0.194618	−0.0973089	−	0.995254i	$-0.531023\pi$
$709$	−5.18210	−0.194618	−0.0973089	+	0.995254i	$0.531023\pi$
$710$	0	0
$711$	14.9443	0.560453
$712$	1.29238	0.0484340
$713$	−0.714890	−0.0267728
$714$	27.7086	1.03697
$715$	0	0
$716$	−26.6869	−0.997336
$717$	3.91589	0.146242
$718$	6.52765	0.243610
$719$	−49.7640	−1.85588	−0.927942	−	0.372724i	$-0.878424\pi$
$719$	−49.7640	−1.85588	−0.927942	+	0.372724i	$0.878424\pi$
$720$	0	0
$721$	12.1597	0.452850
$722$	−35.8296	−1.33344
$723$	−10.4970	−0.390388
$724$	−18.9378	−0.703819
$725$	0	0
$726$	−13.0712	−0.485117
$727$	27.4425	1.01779	0.508893	−	0.860830i	$-0.330055\pi$
$727$	27.4425	1.01779	0.508893	+	0.860830i	$0.330055\pi$
$728$	−5.22467	−0.193639
$729$	1.00000	0.0370370
$730$	0	0
$731$	58.5975	2.16731
$732$	−18.5990	−0.687440
$733$	−37.7306	−1.39361	−0.696806	−	0.717260i	$-0.745396\pi$
$733$	−37.7306	−1.39361	−0.696806	+	0.717260i	$0.745396\pi$
$734$	−51.4911	−1.90057
$735$	0	0
$736$	−5.17788	−0.190859
$737$	−22.9606	−0.845764
$738$	2.33018	0.0857753
$739$	15.6884	0.577107	0.288554	−	0.957464i	$-0.406826\pi$
$739$	15.6884	0.577107	0.288554	+	0.957464i	$0.406826\pi$
$740$	0	0
$741$	0.922874	0.0339026
$742$	11.5579	0.424304
$743$	−1.93922	−0.0711432	−0.0355716	−	0.999367i	$-0.511325\pi$
$743$	−1.93922	−0.0711432	−0.0355716	+	0.999367i	$0.511325\pi$
$744$	0.788968	0.0289250
$745$	0	0
$746$	−3.65663	−0.133879
$747$	−13.6842	−0.500677
$748$	−24.3631	−0.890802
$749$	−9.49454	−0.346923
$750$	0	0
$751$	21.5760	0.787321	0.393660	−	0.919256i	$-0.371209\pi$
$751$	21.5760	0.787321	0.393660	+	0.919256i	$0.371209\pi$
$752$	15.5174	0.565862
$753$	−1.90584	−0.0694527
$754$	33.5435	1.22158
$755$	0	0
$756$	3.04743	0.110834
$757$	48.5442	1.76437	0.882184	−	0.470905i	$-0.156072\pi$
$757$	48.5442	1.76437	0.882184	+	0.470905i	$0.156072\pi$
$758$	−10.6098	−0.385367
$759$	−1.44662	−0.0525088
$760$	0	0
$761$	34.3418	1.24489	0.622445	−	0.782664i	$-0.286140\pi$
$761$	34.3418	1.24489	0.622445	+	0.782664i	$0.286140\pi$
$762$	−0.461506	−0.0167186
$763$	−15.2549	−0.552266
$764$	33.8762	1.22560
$765$	0	0
$766$	−43.2727	−1.56351
$767$	24.1404	0.871660
$768$	20.4695	0.738629
$769$	−36.7179	−1.32408	−0.662041	−	0.749468i	$-0.730310\pi$
$769$	−36.7179	−1.32408	−0.662041	+	0.749468i	$0.730310\pi$
$770$	0	0
$771$	12.7714	0.459951
$772$	19.7745	0.711701
$773$	37.7497	1.35776	0.678880	−	0.734249i	$-0.262466\pi$
$773$	37.7497	1.35776	0.678880	+	0.734249i	$0.262466\pi$
$774$	14.5859	0.524280
$775$	0	0
$776$	−2.47656	−0.0889032
$777$	−3.52334	−0.126399
$778$	−70.9329	−2.54307
$779$	0.330213	0.0118311
$780$	0	0
$781$	26.2872	0.940629
$782$	10.2910	0.368004
$783$	5.15075	0.184073
$784$	15.3541	0.548359
$785$	0	0
$786$	8.84262	0.315406
$787$	−41.6668	−1.48526	−0.742630	−	0.669702i	$-0.766422\pi$
$787$	−41.6668	−1.48526	−0.742630	+	0.669702i	$0.766422\pi$
$788$	15.2285	0.542492
$789$	−10.0322	−0.357156
$790$	0	0
$791$	−36.0447	−1.28160
$792$	1.59652	0.0567298
$793$	−40.4162	−1.43522
$794$	23.9803	0.851028
$795$	0	0
$796$	26.6588	0.944896
$797$	22.4918	0.796702	0.398351	−	0.917233i	$-0.369583\pi$
$797$	22.4918	0.796702	0.398351	+	0.917233i	$0.369583\pi$
$798$	0.977401	0.0345996
$799$	−25.3237	−0.895887
$800$	0	0
$801$	−1.63807	−0.0578782
$802$	−70.1965	−2.47873
$803$	−1.15387	−0.0407192
$804$	17.9641	0.633545
$805$	0	0
$806$	−6.51235	−0.229388
$807$	−17.1185	−0.602601
$808$	4.91666	0.172967
$809$	−50.8508	−1.78782	−0.893909	−	0.448249i	$-0.852048\pi$
$809$	−50.8508	−1.78782	−0.893909	+	0.448249i	$0.852048\pi$
$810$	0	0
$811$	47.5404	1.66937	0.834685	−	0.550727i	$-0.185649\pi$
$811$	47.5404	1.66937	0.834685	+	0.550727i	$0.185649\pi$
$812$	15.6965	0.550841
$813$	7.56317	0.265252
$814$	7.01144	0.245751
$815$	0	0
$816$	−35.4369	−1.24054
$817$	2.06699	0.0723148
$818$	−32.6525	−1.14167
$819$	6.62215	0.231397
$820$	0	0
$821$	34.3319	1.19819	0.599096	−	0.800677i	$-0.295527\pi$
$821$	34.3319	1.19819	0.599096	+	0.800677i	$0.295527\pi$
$822$	−22.7494	−0.793477
$823$	20.3428	0.709105	0.354552	−	0.935036i	$-0.384633\pi$
$823$	20.3428	0.709105	0.354552	+	0.935036i	$0.384633\pi$
$824$	−4.98407	−0.173628
$825$	0	0
$826$	25.5668	0.889581
$827$	−36.0102	−1.25220	−0.626099	−	0.779744i	$-0.715349\pi$
$827$	−36.0102	−1.25220	−0.626099	+	0.779744i	$0.715349\pi$
$828$	1.13182	0.0393333
$829$	0.457819	0.0159007	0.00795036	−	0.999968i	$-0.497469\pi$
$829$	0.457819	0.0159007	0.00795036	+	0.999968i	$0.497469\pi$
$830$	0	0
$831$	−10.6842	−0.370631
$832$	−15.1052	−0.523678
$833$	−25.0571	−0.868175
$834$	17.9652	0.622084
$835$	0	0
$836$	−0.859391	−0.0297227
$837$	−1.00000	−0.0345651
$838$	2.40140	0.0829549
$839$	50.5045	1.74361	0.871804	−	0.489855i	$-0.162950\pi$
$839$	50.5045	1.74361	0.871804	+	0.489855i	$0.162950\pi$
$840$	0	0
$841$	−2.46979	−0.0851653
$842$	55.9686	1.92880
$843$	15.7539	0.542593
$844$	33.4594	1.15172
$845$	0	0
$846$	−6.30349	−0.216718
$847$	−13.2916	−0.456704
$848$	−14.7816	−0.507601
$849$	−26.5692	−0.911853
$850$	0	0
$851$	−1.30857	−0.0448572
$852$	−20.5668	−0.704607
$853$	29.3700	1.00561	0.502804	−	0.864400i	$-0.332302\pi$
$853$	29.3700	1.00561	0.502804	+	0.864400i	$0.332302\pi$
$854$	−42.8042	−1.46473
$855$	0	0
$856$	3.89167	0.133015
$857$	−27.5152	−0.939901	−0.469950	−	0.882693i	$-0.655728\pi$
$857$	−27.5152	−0.939901	−0.469950	+	0.882693i	$0.655728\pi$
$858$	−13.1781	−0.449892
$859$	6.12446	0.208964	0.104482	−	0.994527i	$-0.466682\pi$
$859$	6.12446	0.208964	0.104482	+	0.994527i	$0.466682\pi$
$860$	0	0
$861$	2.36947	0.0807514
$862$	−44.2167	−1.50603
$863$	−24.0692	−0.819324	−0.409662	−	0.912237i	$-0.634353\pi$
$863$	−24.0692	−0.819324	−0.409662	+	0.912237i	$0.634353\pi$
$864$	−7.24290	−0.246409
$865$	0	0
$866$	28.9020	0.982131
$867$	40.8312	1.38670
$868$	−3.04743	−0.103437
$869$	−30.2405	−1.02584
$870$	0	0
$871$	39.0365	1.32270
$872$	6.25277	0.211746
$873$	3.13898	0.106238
$874$	0.363007	0.0122789
$875$	0	0
$876$	0.902776	0.0305020
$877$	−28.2686	−0.954562	−0.477281	−	0.878751i	$-0.658378\pi$
$877$	−28.2686	−0.954562	−0.477281	+	0.878751i	$0.658378\pi$
$878$	26.2622	0.886308
$879$	−23.7515	−0.801118
$880$	0	0
$881$	32.7825	1.10447	0.552235	−	0.833688i	$-0.313775\pi$
$881$	32.7825	1.10447	0.552235	+	0.833688i	$0.313775\pi$
$882$	−6.23713	−0.210015
$883$	7.87793	0.265114	0.132557	−	0.991175i	$-0.457681\pi$
$883$	7.87793	0.265114	0.132557	+	0.991175i	$0.457681\pi$
$884$	41.4209	1.39314
$885$	0	0
$886$	17.5469	0.589499
$887$	3.02131	0.101446	0.0507228	−	0.998713i	$-0.483847\pi$
$887$	3.02131	0.101446	0.0507228	+	0.998713i	$0.483847\pi$
$888$	1.44417	0.0484631
$889$	−0.469287	−0.0157394
$890$	0	0
$891$	−2.02355	−0.0677915
$892$	6.89049	0.230711
$893$	−0.893275	−0.0298923
$894$	6.49093	0.217089
$895$	0	0
$896$	11.8853	0.397062
$897$	2.45947	0.0821192
$898$	27.0739	0.903469
$899$	−5.15075	−0.171787
$900$	0	0
$901$	24.1228	0.803646
$902$	−4.71524	−0.157000
$903$	14.8318	0.493573
$904$	14.7742	0.491382
$905$	0	0
$906$	−21.9246	−0.728396
$907$	48.0192	1.59445	0.797226	−	0.603681i	$-0.206300\pi$
$907$	48.0192	1.59445	0.797226	+	0.603681i	$0.206300\pi$
$908$	−14.2990	−0.474530
$909$	−6.23176	−0.206694
$910$	0	0
$911$	−35.5860	−1.17902	−0.589509	−	0.807762i	$-0.700679\pi$
$911$	−35.5860	−1.17902	−0.589509	+	0.807762i	$0.700679\pi$
$912$	−1.25001	−0.0413920
$913$	27.6906	0.916424
$914$	69.8719	2.31116
$915$	0	0
$916$	14.6004	0.482411
$917$	8.99171	0.296932
$918$	14.3952	0.475111
$919$	32.6713	1.07773	0.538864	−	0.842393i	$-0.318854\pi$
$919$	32.6713	1.07773	0.538864	+	0.842393i	$0.318854\pi$
$920$	0	0
$921$	17.9239	0.590613
$922$	73.9836	2.43652
$923$	−44.6922	−1.47106
$924$	−6.16663	−0.202867
$925$	0	0
$926$	−1.51082	−0.0496485
$927$	6.31720	0.207484
$928$	−37.3064	−1.22464
$929$	58.8643	1.93128	0.965638	−	0.259891i	$-0.0836868\pi$
$929$	58.8643	1.93128	0.965638	+	0.259891i	$0.0836868\pi$
$930$	0	0
$931$	−0.883871	−0.0289677
$932$	−33.4282	−1.09498
$933$	7.10858	0.232725
$934$	3.24718	0.106251
$935$	0	0
$936$	−2.71432	−0.0887205
$937$	28.5511	0.932724	0.466362	−	0.884594i	$-0.345564\pi$
$937$	28.5511	0.932724	0.466362	+	0.884594i	$0.345564\pi$
$938$	41.3430	1.34990
$939$	−20.9587	−0.683961
$940$	0	0
$941$	−23.6080	−0.769597	−0.384799	−	0.923001i	$-0.625729\pi$
$941$	−23.6080	−0.769597	−0.384799	+	0.923001i	$0.625729\pi$
$942$	−40.2061	−1.30998
$943$	0.880022	0.0286575
$944$	−32.6977	−1.06422
$945$	0	0
$946$	−29.5154	−0.959627
$947$	−22.2266	−0.722268	−0.361134	−	0.932514i	$-0.617610\pi$
$947$	−22.2266	−0.722268	−0.361134	+	0.932514i	$0.617610\pi$
$948$	23.6598	0.768435
$949$	1.96176	0.0636813
$950$	0	0
$951$	19.9348	0.646429
$952$	−11.5488	−0.374299
$953$	−46.5383	−1.50752	−0.753762	−	0.657148i	$-0.771763\pi$
$953$	−46.5383	−1.50752	−0.753762	+	0.657148i	$0.771763\pi$
$954$	6.00457	0.194405
$955$	0	0
$956$	6.19966	0.200511
$957$	−10.4228	−0.336921
$958$	37.7167	1.21857
$959$	−23.1330	−0.747003
$960$	0	0
$961$	1.00000	0.0322581
$962$	−11.9205	−0.384333
$963$	−4.93261	−0.158951
$964$	−16.6189	−0.535259
$965$	0	0
$966$	2.60478	0.0838076
$967$	41.0489	1.32004	0.660021	−	0.751247i	$-0.270547\pi$
$967$	41.0489	1.32004	0.660021	+	0.751247i	$0.270547\pi$
$968$	5.44802	0.175106
$969$	2.03996	0.0655329
$970$	0	0
$971$	−28.9627	−0.929457	−0.464729	−	0.885453i	$-0.653848\pi$
$971$	−28.9627	−0.929457	−0.464729	+	0.885453i	$0.653848\pi$
$972$	1.58320	0.0507813
$973$	18.2681	0.585648
$974$	37.8378	1.21240
$975$	0	0
$976$	54.7429	1.75228
$977$	−24.3217	−0.778119	−0.389060	−	0.921213i	$-0.627200\pi$
$977$	−24.3217	−0.778119	−0.389060	+	0.921213i	$0.627200\pi$
$978$	6.76302	0.216257
$979$	3.31471	0.105938
$980$	0	0
$981$	−7.92525	−0.253034
$982$	−58.6441	−1.87141
$983$	51.7467	1.65046	0.825232	−	0.564794i	$-0.191044\pi$
$983$	51.7467	1.65046	0.825232	+	0.564794i	$0.191044\pi$
$984$	−0.971212	−0.0309611
$985$	0	0
$986$	74.1459	2.36129
$987$	−6.40977	−0.204025
$988$	1.46110	0.0464837
$989$	5.50855	0.175162
$990$	0	0
$991$	−50.5874	−1.60696	−0.803482	−	0.595330i	$-0.797021\pi$
$991$	−50.5874	−1.60696	−0.803482	+	0.595330i	$0.797021\pi$
$992$	7.24290	0.229962
$993$	18.4902	0.586768
$994$	−47.3328	−1.50131
$995$	0	0
$996$	−21.6648	−0.686476
$997$	55.5169	1.75824	0.879118	−	0.476604i	$-0.158132\pi$
$997$	55.5169	1.75824	0.879118	+	0.476604i	$0.158132\pi$
$998$	−11.9679	−0.378837
$999$	−1.83045	−0.0579129

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2325.2.a.bb.1.5	yes	6
3.2	odd	2		6975.2.a.ca.1.2		6
5.2	odd	4		2325.2.c.r.1024.10		12
5.3	odd	4		2325.2.c.r.1024.3		12
5.4	even	2		2325.2.a.y.1.2	✓	6
15.14	odd	2		6975.2.a.cc.1.5		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2325.2.a.y.1.2	✓	6	5.4	even	2
2325.2.a.bb.1.5	yes	6	1.1	even	1	trivial
2325.2.c.r.1024.3		12	5.3	odd	4
2325.2.c.r.1024.10		12	5.2	odd	4
6975.2.a.ca.1.2		6	3.2	odd	2
6975.2.a.cc.1.5		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)

\(f(q)\)	\(=\)	\(q+1.89294 q^{2} +1.00000 q^{3} +1.58320 q^{4} +1.89294 q^{6} +1.92485 q^{7} -0.788968 q^{8} +1.00000 q^{9} -2.02355 q^{11} +1.58320 q^{12} +3.44035 q^{13} +3.64362 q^{14} -4.65987 q^{16} +7.60468 q^{17} +1.89294 q^{18} +0.268250 q^{19} +1.92485 q^{21} -3.83045 q^{22} +0.714890 q^{23} -0.788968 q^{24} +6.51235 q^{26} +1.00000 q^{27} +3.04743 q^{28} +5.15075 q^{29} -1.00000 q^{31} -7.24290 q^{32} -2.02355 q^{33} +14.3952 q^{34} +1.58320 q^{36} -1.83045 q^{37} +0.507780 q^{38} +3.44035 q^{39} +1.23099 q^{41} +3.64362 q^{42} +7.70545 q^{43} -3.20369 q^{44} +1.35324 q^{46} -3.33001 q^{47} -4.65987 q^{48} -3.29495 q^{49} +7.60468 q^{51} +5.44677 q^{52} +3.17210 q^{53} +1.89294 q^{54} -1.51865 q^{56} +0.268250 q^{57} +9.75003 q^{58} +7.01686 q^{59} -11.7477 q^{61} -1.89294 q^{62} +1.92485 q^{63} -4.39060 q^{64} -3.83045 q^{66} +11.3467 q^{67} +12.0398 q^{68} +0.714890 q^{69} -12.9906 q^{71} -0.788968 q^{72} +0.570221 q^{73} -3.46492 q^{74} +0.424695 q^{76} -3.89503 q^{77} +6.51235 q^{78} +14.9443 q^{79} +1.00000 q^{81} +2.33018 q^{82} -13.6842 q^{83} +3.04743 q^{84} +14.5859 q^{86} +5.15075 q^{87} +1.59652 q^{88} -1.63807 q^{89} +6.62215 q^{91} +1.13182 q^{92} -1.00000 q^{93} -6.30349 q^{94} -7.24290 q^{96} +3.13898 q^{97} -6.23713 q^{98} -2.02355 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + q^{2} + 6 q^{3} + 7 q^{4} + q^{6} + 2 q^{7} - 3 q^{8} + 6 q^{9} + 7 q^{11} + 7 q^{12} + 4 q^{13} + 10 q^{14} + 17 q^{16} + q^{18} + 17 q^{19} + 2 q^{21} + 2 q^{22} - q^{23} - 3 q^{24} + 2 q^{26}+ \cdots + 7 q^{99}+O(q^{100}) \) 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

Fields

Representations

Groups

Database

Properties

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