LMFDB - Embedded newform 2325.2.a.bb.1.3

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			1.3
Root			$-0.667396$ 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)

$f(q)$	$=$	\(q-0.667396 q^{2} +1.00000 q^{3} -1.55458 q^{4} -0.667396 q^{6} +3.64222 q^{7} +2.37231 q^{8} +1.00000 q^{9} -1.15709 q^{11} -1.55458 q^{12} +5.71167 q^{13} -2.43080 q^{14} +1.52589 q^{16} -1.37855 q^{17} -0.667396 q^{18} +2.28711 q^{19} +3.64222 q^{21} +0.772236 q^{22} +5.17025 q^{23} +2.37231 q^{24} -3.81195 q^{26} +1.00000 q^{27} -5.66213 q^{28} -5.51387 q^{29} -1.00000 q^{31} -5.76300 q^{32} -1.15709 q^{33} +0.920040 q^{34} -1.55458 q^{36} +2.77224 q^{37} -1.52641 q^{38} +5.71167 q^{39} -0.122562 q^{41} -2.43080 q^{42} +1.11738 q^{43} +1.79879 q^{44} -3.45060 q^{46} -8.92527 q^{47} +1.52589 q^{48} +6.26574 q^{49} -1.37855 q^{51} -8.87926 q^{52} +3.42456 q^{53} -0.667396 q^{54} +8.64048 q^{56} +2.28711 q^{57} +3.67993 q^{58} +0.542646 q^{59} +10.4883 q^{61} +0.667396 q^{62} +3.64222 q^{63} +0.794421 q^{64} +0.772236 q^{66} -15.8274 q^{67} +2.14307 q^{68} +5.17025 q^{69} +10.7253 q^{71} +2.37231 q^{72} -8.34049 q^{73} -1.85018 q^{74} -3.55550 q^{76} -4.21437 q^{77} -3.81195 q^{78} +5.83719 q^{79} +1.00000 q^{81} +0.0817974 q^{82} +16.4935 q^{83} -5.66213 q^{84} -0.745734 q^{86} -5.51387 q^{87} -2.74498 q^{88} -8.27438 q^{89} +20.8031 q^{91} -8.03757 q^{92} -1.00000 q^{93} +5.95669 q^{94} -5.76300 q^{96} +9.97701 q^{97} -4.18173 q^{98} -1.15709 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−0.667396	−0.471920	−0.235960	−	0.971763i	$-0.575823\pi$
$2$	−0.667396	−0.471920	−0.235960	+	0.971763i	$0.575823\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	−1.55458	−0.777291
$5$	0	0
$5$	0	0
$6$	−0.667396	−0.272463
$7$	3.64222	1.37663	0.688314	−	0.725413i	$-0.258351\pi$
$7$	3.64222	1.37663	0.688314	+	0.725413i	$0.258351\pi$
$8$	2.37231	0.838740
$9$	1.00000	0.333333
$10$	0	0
$11$	−1.15709	−0.348875	−0.174438	−	0.984668i	$-0.555811\pi$
$11$	−1.15709	−0.348875	−0.174438	+	0.984668i	$0.555811\pi$
$12$	−1.55458	−0.448769
$13$	5.71167	1.58413	0.792066	−	0.610435i	$-0.209005\pi$
$13$	5.71167	1.58413	0.792066	+	0.610435i	$0.209005\pi$
$14$	−2.43080	−0.649659
$15$	0	0
$16$	1.52589	0.381473
$17$	−1.37855	−0.334348	−0.167174	−	0.985927i	$-0.553464\pi$
$17$	−1.37855	−0.334348	−0.167174	+	0.985927i	$0.553464\pi$
$18$	−0.667396	−0.157307
$19$	2.28711	0.524699	0.262349	−	0.964973i	$-0.415503\pi$
$19$	2.28711	0.524699	0.262349	+	0.964973i	$0.415503\pi$
$20$	0	0
$21$	3.64222	0.794797
$22$	0.772236	0.164641
$23$	5.17025	1.07807	0.539035	−	0.842283i	$-0.318789\pi$
$23$	5.17025	1.07807	0.539035	+	0.842283i	$0.318789\pi$
$24$	2.37231	0.484247
$25$	0	0
$26$	−3.81195	−0.747584
$27$	1.00000	0.192450
$28$	−5.66213	−1.07004
$29$	−5.51387	−1.02390	−0.511950	−	0.859015i	$-0.671077\pi$
$29$	−5.51387	−1.02390	−0.511950	+	0.859015i	$0.671077\pi$
$30$	0	0
$31$	−1.00000	−0.179605
$31$	−1.00000	−0.179605
$32$	−5.76300	−1.01876
$33$	−1.15709	−0.201423
$34$	0.920040	0.157786
$35$	0	0
$36$	−1.55458	−0.259097
$37$	2.77224	0.455753	0.227876	−	0.973690i	$-0.426822\pi$
$37$	2.77224	0.455753	0.227876	+	0.973690i	$0.426822\pi$
$38$	−1.52641	−0.247616
$39$	5.71167	0.914599
$40$	0	0
$41$	−0.122562	−0.0191410	−0.00957048	−	0.999954i	$-0.503046\pi$
$41$	−0.122562	−0.0191410	−0.00957048	+	0.999954i	$0.503046\pi$
$42$	−2.43080	−0.375081
$43$	1.11738	0.170399	0.0851993	−	0.996364i	$-0.472847\pi$
$43$	1.11738	0.170399	0.0851993	+	0.996364i	$0.472847\pi$
$44$	1.79879	0.271178
$45$	0	0
$46$	−3.45060	−0.508763
$47$	−8.92527	−1.30189	−0.650943	−	0.759127i	$-0.725626\pi$
$47$	−8.92527	−1.30189	−0.650943	+	0.759127i	$0.725626\pi$
$48$	1.52589	0.220244
$49$	6.26574	0.895106
$50$	0	0
$51$	−1.37855	−0.193036
$52$	−8.87926	−1.23133
$53$	3.42456	0.470400	0.235200	−	0.971947i	$-0.424426\pi$
$53$	3.42456	0.470400	0.235200	+	0.971947i	$0.424426\pi$
$54$	−0.667396	−0.0908211
$55$	0	0
$56$	8.64048	1.15463
$57$	2.28711	0.302935
$58$	3.67993	0.483199
$59$	0.542646	0.0706465	0.0353232	−	0.999376i	$-0.488754\pi$
$59$	0.542646	0.0706465	0.0353232	+	0.999376i	$0.488754\pi$
$60$	0	0
$61$	10.4883	1.34289	0.671444	−	0.741056i	$-0.265675\pi$
$61$	10.4883	1.34289	0.671444	+	0.741056i	$0.265675\pi$
$62$	0.667396	0.0847594
$63$	3.64222	0.458876
$64$	0.794421	0.0993026
$65$	0	0
$66$	0.772236	0.0950557
$67$	−15.8274	−1.93363	−0.966815	−	0.255477i	$-0.917767\pi$
$67$	−15.8274	−1.93363	−0.966815	+	0.255477i	$0.917767\pi$
$68$	2.14307	0.259886
$69$	5.17025	0.622424
$70$	0	0
$71$	10.7253	1.27286	0.636428	−	0.771336i	$-0.280411\pi$
$71$	10.7253	1.27286	0.636428	+	0.771336i	$0.280411\pi$
$72$	2.37231	0.279580
$73$	−8.34049	−0.976181	−0.488090	−	0.872793i	$-0.662306\pi$
$73$	−8.34049	−0.976181	−0.488090	+	0.872793i	$0.662306\pi$
$74$	−1.85018	−0.215079
$75$	0	0
$76$	−3.55550	−0.407844
$77$	−4.21437	−0.480272
$78$	−3.81195	−0.431618
$79$	5.83719	0.656735	0.328368	−	0.944550i	$-0.393502\pi$
$79$	5.83719	0.656735	0.328368	+	0.944550i	$0.393502\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0.0817974	0.00903301
$83$	16.4935	1.81039	0.905197	−	0.424993i	$-0.139723\pi$
$83$	16.4935	1.81039	0.905197	+	0.424993i	$0.139723\pi$
$84$	−5.66213	−0.617789
$85$	0	0
$86$	−0.745734	−0.0804146
$87$	−5.51387	−0.591149
$88$	−2.74498	−0.292616
$89$	−8.27438	−0.877083	−0.438541	−	0.898711i	$-0.644505\pi$
$89$	−8.27438	−0.877083	−0.438541	+	0.898711i	$0.644505\pi$
$90$	0	0
$91$	20.8031	2.18076
$92$	−8.03757	−0.837975
$93$	−1.00000	−0.103695
$94$	5.95669	0.614386
$95$	0	0
$96$	−5.76300	−0.588184
$97$	9.97701	1.01301	0.506506	−	0.862237i	$-0.330937\pi$
$97$	9.97701	1.01301	0.506506	+	0.862237i	$0.330937\pi$
$98$	−4.18173	−0.422418
$99$	−1.15709	−0.116292
$100$	0	0
$101$	−7.18097	−0.714533	−0.357266	−	0.934003i	$-0.616291\pi$
$101$	−7.18097	−0.714533	−0.357266	+	0.934003i	$0.616291\pi$
$102$	0.920040	0.0910976
$103$	−1.70411	−0.167911	−0.0839555	−	0.996470i	$-0.526755\pi$
$103$	−1.70411	−0.167911	−0.0839555	+	0.996470i	$0.526755\pi$
$104$	13.5499	1.32867
$105$	0	0
$106$	−2.28554	−0.221991
$107$	−18.1570	−1.75530	−0.877652	−	0.479299i	$-0.840891\pi$
$107$	−18.1570	−1.75530	−0.877652	+	0.479299i	$0.840891\pi$
$108$	−1.55458	−0.149590
$109$	6.50614	0.623176	0.311588	−	0.950217i	$-0.399139\pi$
$109$	6.50614	0.623176	0.311588	+	0.950217i	$0.399139\pi$
$110$	0	0
$111$	2.77224	0.263129
$112$	5.55763	0.525147
$113$	3.36042	0.316122	0.158061	−	0.987429i	$-0.449476\pi$
$113$	3.36042	0.316122	0.158061	+	0.987429i	$0.449476\pi$
$114$	−1.52641	−0.142961
$115$	0	0
$116$	8.57176	0.795868
$117$	5.71167	0.528044
$118$	−0.362160	−0.0333395
$119$	−5.02099	−0.460273
$120$	0	0
$121$	−9.66115	−0.878286
$122$	−6.99984	−0.633736
$123$	−0.122562	−0.0110510
$124$	1.55458	0.139606
$125$	0	0
$126$	−2.43080	−0.216553
$127$	10.6138	0.941823	0.470911	−	0.882181i	$-0.343925\pi$
$127$	10.6138	0.941823	0.470911	+	0.882181i	$0.343925\pi$
$128$	10.9958	0.971902
$129$	1.11738	0.0983797
$130$	0	0
$131$	14.5080	1.26757	0.633786	−	0.773508i	$-0.281500\pi$
$131$	14.5080	1.26757	0.633786	+	0.773508i	$0.281500\pi$
$132$	1.79879	0.156565
$133$	8.33014	0.722315
$134$	10.5632	0.912519
$135$	0	0
$136$	−3.27036	−0.280431
$137$	5.31941	0.454468	0.227234	−	0.973840i	$-0.427032\pi$
$137$	5.31941	0.454468	0.227234	+	0.973840i	$0.427032\pi$
$138$	−3.45060	−0.293735
$139$	−8.27443	−0.701828	−0.350914	−	0.936408i	$-0.614129\pi$
$139$	−8.27443	−0.701828	−0.350914	+	0.936408i	$0.614129\pi$
$140$	0	0
$141$	−8.92527	−0.751644
$142$	−7.15801	−0.600687
$143$	−6.60891	−0.552665
$144$	1.52589	0.127158
$145$	0	0
$146$	5.56641	0.460679
$147$	6.26574	0.516789
$148$	−4.30967	−0.354253
$149$	15.9731	1.30857	0.654284	−	0.756249i	$-0.272970\pi$
$149$	15.9731	1.30857	0.654284	+	0.756249i	$0.272970\pi$
$150$	0	0
$151$	7.39770	0.602016	0.301008	−	0.953622i	$-0.402677\pi$
$151$	7.39770	0.602016	0.301008	+	0.953622i	$0.402677\pi$
$152$	5.42574	0.440086
$153$	−1.37855	−0.111449
$154$	2.81265	0.226650
$155$	0	0
$156$	−8.87926	−0.710910
$157$	15.7243	1.25493	0.627467	−	0.778643i	$-0.284092\pi$
$157$	15.7243	1.25493	0.627467	+	0.778643i	$0.284092\pi$
$158$	−3.89572	−0.309927
$159$	3.42456	0.271585
$160$	0	0
$161$	18.8312	1.48410
$162$	−0.667396	−0.0524356
$163$	−9.68816	−0.758836	−0.379418	−	0.925225i	$-0.623876\pi$
$163$	−9.68816	−0.758836	−0.379418	+	0.925225i	$0.623876\pi$
$164$	0.190533	0.0148781
$165$	0	0
$166$	−11.0077	−0.854361
$167$	19.1075	1.47858	0.739292	−	0.673385i	$-0.235160\pi$
$167$	19.1075	1.47858	0.739292	+	0.673385i	$0.235160\pi$
$168$	8.64048	0.666628
$169$	19.6232	1.50948
$170$	0	0
$171$	2.28711	0.174900
$172$	−1.73706	−0.132449
$173$	−21.1286	−1.60638	−0.803189	−	0.595724i	$-0.796865\pi$
$173$	−21.1286	−1.60638	−0.803189	+	0.595724i	$0.796865\pi$
$174$	3.67993	0.278975
$175$	0	0
$176$	−1.76559	−0.133087
$177$	0.542646	0.0407878
$178$	5.52229	0.413913
$179$	13.0689	0.976816	0.488408	−	0.872615i	$-0.337578\pi$
$179$	13.0689	0.976816	0.488408	+	0.872615i	$0.337578\pi$
$180$	0	0
$181$	9.98243	0.741988	0.370994	−	0.928635i	$-0.379017\pi$
$181$	9.98243	0.741988	0.370994	+	0.928635i	$0.379017\pi$
$182$	−13.8839	−1.02915
$183$	10.4883	0.775316
$184$	12.2654	0.904221
$185$	0	0
$186$	0.667396	0.0489358
$187$	1.59511	0.116646
$188$	13.8751	1.01194
$189$	3.64222	0.264932
$190$	0	0
$191$	−8.69024	−0.628804	−0.314402	−	0.949290i	$-0.601804\pi$
$191$	−8.69024	−0.628804	−0.314402	+	0.949290i	$0.601804\pi$
$192$	0.794421	0.0573324
$193$	18.4643	1.32909	0.664544	−	0.747249i	$-0.268626\pi$
$193$	18.4643	1.32909	0.664544	+	0.747249i	$0.268626\pi$
$194$	−6.65862	−0.478061
$195$	0	0
$196$	−9.74061	−0.695758
$197$	1.73458	0.123584	0.0617919	−	0.998089i	$-0.480318\pi$
$197$	1.73458	0.123584	0.0617919	+	0.998089i	$0.480318\pi$
$198$	0.772236	0.0548804
$199$	13.7835	0.977084	0.488542	−	0.872540i	$-0.337529\pi$
$199$	13.7835	0.977084	0.488542	+	0.872540i	$0.337529\pi$
$200$	0	0
$201$	−15.8274	−1.11638
$202$	4.79255	0.337202
$203$	−20.0827	−1.40953
$204$	2.14307	0.150045
$205$	0	0
$206$	1.13732	0.0792406
$207$	5.17025	0.359357
$208$	8.71539	0.604304
$209$	−2.64639	−0.183054
$210$	0	0
$211$	−8.73784	−0.601538	−0.300769	−	0.953697i	$-0.597243\pi$
$211$	−8.73784	−0.601538	−0.300769	+	0.953697i	$0.597243\pi$
$212$	−5.32377	−0.365638
$213$	10.7253	0.734884
$214$	12.1179	0.828363
$215$	0	0
$216$	2.37231	0.161416
$217$	−3.64222	−0.247250
$218$	−4.34217	−0.294089
$219$	−8.34049	−0.563598
$220$	0	0
$221$	−7.87384	−0.529652
$222$	−1.85018	−0.124176
$223$	−10.4409	−0.699176	−0.349588	−	0.936903i	$-0.613679\pi$
$223$	−10.4409	−0.699176	−0.349588	+	0.936903i	$0.613679\pi$
$224$	−20.9901	−1.40246
$225$	0	0
$226$	−2.24273	−0.149184
$227$	−25.1149	−1.66693	−0.833466	−	0.552571i	$-0.813647\pi$
$227$	−25.1149	−1.66693	−0.833466	+	0.552571i	$0.813647\pi$
$228$	−3.55550	−0.235469
$229$	−3.58524	−0.236919	−0.118460	−	0.992959i	$-0.537796\pi$
$229$	−3.58524	−0.236919	−0.118460	+	0.992959i	$0.537796\pi$
$230$	0	0
$231$	−4.21437	−0.277285
$232$	−13.0806	−0.858785
$233$	−18.2989	−1.19880	−0.599401	−	0.800449i	$-0.704594\pi$
$233$	−18.2989	−1.19880	−0.599401	+	0.800449i	$0.704594\pi$
$234$	−3.81195	−0.249195
$235$	0	0
$236$	−0.843588	−0.0549129
$237$	5.83719	0.379166
$238$	3.35099	0.217212
$239$	−9.65731	−0.624680	−0.312340	−	0.949970i	$-0.601113\pi$
$239$	−9.65731	−0.624680	−0.312340	+	0.949970i	$0.601113\pi$
$240$	0	0
$241$	−5.31631	−0.342454	−0.171227	−	0.985232i	$-0.554773\pi$
$241$	−5.31631	−0.342454	−0.171227	+	0.985232i	$0.554773\pi$
$242$	6.44781	0.414481
$243$	1.00000	0.0641500
$244$	−16.3049	−1.04381
$245$	0	0
$246$	0.0817974	0.00521521
$247$	13.0632	0.831192
$248$	−2.37231	−0.150642
$249$	16.4935	1.04523
$250$	0	0
$251$	−9.20850	−0.581235	−0.290618	−	0.956839i	$-0.593861\pi$
$251$	−9.20850	−0.581235	−0.290618	+	0.956839i	$0.593861\pi$
$252$	−5.66213	−0.356680
$253$	−5.98243	−0.376112
$254$	−7.08361	−0.444465
$255$	0	0
$256$	−8.92740	−0.557963
$257$	7.04525	0.439470	0.219735	−	0.975560i	$-0.429481\pi$
$257$	7.04525	0.439470	0.219735	+	0.975560i	$0.429481\pi$
$258$	−0.745734	−0.0464274
$259$	10.0971	0.627402
$260$	0	0
$261$	−5.51387	−0.341300
$262$	−9.68260	−0.598193
$263$	−4.35053	−0.268265	−0.134133	−	0.990963i	$-0.542825\pi$
$263$	−4.35053	−0.268265	−0.134133	+	0.990963i	$0.542825\pi$
$264$	−2.74498	−0.168942
$265$	0	0
$266$	−5.55951	−0.340875
$267$	−8.27438	−0.506384
$268$	24.6051	1.50299
$269$	−20.1710	−1.22985	−0.614925	−	0.788586i	$-0.710814\pi$
$269$	−20.1710	−1.22985	−0.614925	+	0.788586i	$0.710814\pi$
$270$	0	0
$271$	13.0303	0.791533	0.395767	−	0.918351i	$-0.370479\pi$
$271$	13.0303	0.791533	0.395767	+	0.918351i	$0.370479\pi$
$272$	−2.10352	−0.127545
$273$	20.8031	1.25906
$274$	−3.55015	−0.214473
$275$	0	0
$276$	−8.03757	−0.483805
$277$	−5.93472	−0.356583	−0.178292	−	0.983978i	$-0.557057\pi$
$277$	−5.93472	−0.356583	−0.178292	+	0.983978i	$0.557057\pi$
$278$	5.52232	0.331207
$279$	−1.00000	−0.0598684
$280$	0	0
$281$	−2.56333	−0.152915	−0.0764577	−	0.997073i	$-0.524361\pi$
$281$	−2.56333	−0.152915	−0.0764577	+	0.997073i	$0.524361\pi$
$282$	5.95669	0.354716
$283$	22.5046	1.33776	0.668879	−	0.743371i	$-0.266774\pi$
$283$	22.5046	1.33776	0.668879	+	0.743371i	$0.266774\pi$
$284$	−16.6733	−0.989380
$285$	0	0
$286$	4.41076	0.260814
$287$	−0.446397	−0.0263500
$288$	−5.76300	−0.339588
$289$	−15.0996	−0.888211
$290$	0	0
$291$	9.97701	0.584863
$292$	12.9660	0.758777
$293$	−9.86771	−0.576478	−0.288239	−	0.957559i	$-0.593070\pi$
$293$	−9.86771	−0.576478	−0.288239	+	0.957559i	$0.593070\pi$
$294$	−4.18173	−0.243883
$295$	0	0
$296$	6.57662	0.382258
$297$	−1.15709	−0.0671411
$298$	−10.6604	−0.617540
$299$	29.5307	1.70781
$300$	0	0
$301$	4.06973	0.234576
$302$	−4.93719	−0.284104
$303$	−7.18097	−0.412536
$304$	3.48988	0.200158
$305$	0	0
$306$	0.920040	0.0525952
$307$	13.4887	0.769843	0.384922	−	0.922949i	$-0.374229\pi$
$307$	13.4887	0.769843	0.384922	+	0.922949i	$0.374229\pi$
$308$	6.55158	0.373311
$309$	−1.70411	−0.0969435
$310$	0	0
$311$	8.33830	0.472822	0.236411	−	0.971653i	$-0.424029\pi$
$311$	8.33830	0.472822	0.236411	+	0.971653i	$0.424029\pi$
$312$	13.5499	0.767111
$313$	7.03489	0.397636	0.198818	−	0.980036i	$-0.436290\pi$
$313$	7.03489	0.397636	0.198818	+	0.980036i	$0.436290\pi$
$314$	−10.4943	−0.592229
$315$	0	0
$316$	−9.07439	−0.510474
$317$	28.9660	1.62689	0.813446	−	0.581640i	$-0.197589\pi$
$317$	28.9660	1.62689	0.813446	+	0.581640i	$0.197589\pi$
$318$	−2.28554	−0.128167
$319$	6.38004	0.357213
$320$	0	0
$321$	−18.1570	−1.01343
$322$	−12.5678	−0.700378
$323$	−3.15290	−0.175432
$324$	−1.55458	−0.0863657
$325$	0	0
$326$	6.46584	0.358110
$327$	6.50614	0.359791
$328$	−0.290756	−0.0160543
$329$	−32.5078	−1.79221
$330$	0	0
$331$	4.97223	0.273299	0.136649	−	0.990619i	$-0.456367\pi$
$331$	4.97223	0.273299	0.136649	+	0.990619i	$0.456367\pi$
$332$	−25.6405	−1.40720
$333$	2.77224	0.151918
$334$	−12.7523	−0.697773
$335$	0	0
$336$	5.55763	0.303194
$337$	3.96964	0.216240	0.108120	−	0.994138i	$-0.465517\pi$
$337$	3.96964	0.216240	0.108120	+	0.994138i	$0.465517\pi$
$338$	−13.0964	−0.712352
$339$	3.36042	0.182513
$340$	0	0
$341$	1.15709	0.0626599
$342$	−1.52641	−0.0825386
$343$	−2.67434	−0.144401
$344$	2.65077	0.142920
$345$	0	0
$346$	14.1011	0.758082
$347$	−32.8011	−1.76085	−0.880427	−	0.474182i	$-0.842744\pi$
$347$	−32.8011	−1.76085	−0.880427	+	0.474182i	$0.842744\pi$
$348$	8.57176	0.459495
$349$	36.2630	1.94111	0.970557	−	0.240872i	$-0.0774332\pi$
$349$	36.2630	1.94111	0.970557	+	0.240872i	$0.0774332\pi$
$350$	0	0
$351$	5.71167	0.304866
$352$	6.66830	0.355422
$353$	−30.9661	−1.64816	−0.824080	−	0.566473i	$-0.808307\pi$
$353$	−30.9661	−1.64816	−0.824080	+	0.566473i	$0.808307\pi$
$354$	−0.362160	−0.0192486
$355$	0	0
$356$	12.8632	0.681749
$357$	−5.02099	−0.265739
$358$	−8.72213	−0.460979
$359$	−11.6777	−0.616324	−0.308162	−	0.951334i	$-0.599714\pi$
$359$	−11.6777	−0.616324	−0.308162	+	0.951334i	$0.599714\pi$
$360$	0	0
$361$	−13.7691	−0.724691
$362$	−6.66223	−0.350159
$363$	−9.66115	−0.507079
$364$	−32.3402	−1.69509
$365$	0	0
$366$	−6.99984	−0.365887
$367$	2.52502	0.131805	0.0659026	−	0.997826i	$-0.479007\pi$
$367$	2.52502	0.131805	0.0659026	+	0.997826i	$0.479007\pi$
$368$	7.88924	0.411255
$369$	−0.122562	−0.00638032
$370$	0	0
$371$	12.4730	0.647566
$372$	1.55458	0.0806014
$373$	15.0961	0.781646	0.390823	−	0.920466i	$-0.372191\pi$
$373$	15.0961	0.781646	0.390823	+	0.920466i	$0.372191\pi$
$374$	−1.06457	−0.0550475
$375$	0	0
$376$	−21.1736	−1.09194
$377$	−31.4934	−1.62199
$378$	−2.43080	−0.125027
$379$	−19.1633	−0.984354	−0.492177	−	0.870495i	$-0.663799\pi$
$379$	−19.1633	−0.984354	−0.492177	+	0.870495i	$0.663799\pi$
$380$	0	0
$381$	10.6138	0.543761
$382$	5.79983	0.296745
$383$	−2.27917	−0.116460	−0.0582301	−	0.998303i	$-0.518546\pi$
$383$	−2.27917	−0.116460	−0.0582301	+	0.998303i	$0.518546\pi$
$384$	10.9958	0.561128
$385$	0	0
$386$	−12.3230	−0.627223
$387$	1.11738	0.0567995
$388$	−15.5101	−0.787405
$389$	−14.5865	−0.739563	−0.369782	−	0.929119i	$-0.620568\pi$
$389$	−14.5865	−0.739563	−0.369782	+	0.929119i	$0.620568\pi$
$390$	0	0
$391$	−7.12745	−0.360451
$392$	14.8643	0.750761
$393$	14.5080	0.731833
$394$	−1.15765	−0.0583217
$395$	0	0
$396$	1.79879	0.0903926
$397$	33.0931	1.66089	0.830447	−	0.557098i	$-0.188085\pi$
$397$	33.0931	1.66089	0.830447	+	0.557098i	$0.188085\pi$
$398$	−9.19903	−0.461106
$399$	8.33014	0.417029
$400$	0	0
$401$	10.6123	0.529953	0.264977	−	0.964255i	$-0.414636\pi$
$401$	10.6123	0.529953	0.264977	+	0.964255i	$0.414636\pi$
$402$	10.5632	0.526843
$403$	−5.71167	−0.284519
$404$	11.1634	0.555400
$405$	0	0
$406$	13.4031	0.665185
$407$	−3.20772	−0.159001
$408$	−3.27036	−0.161907
$409$	−28.1971	−1.39426	−0.697128	−	0.716947i	$-0.745539\pi$
$409$	−28.1971	−1.39426	−0.697128	+	0.716947i	$0.745539\pi$
$410$	0	0
$411$	5.31941	0.262387
$412$	2.64918	0.130516
$413$	1.97643	0.0972540
$414$	−3.45060	−0.169588
$415$	0	0
$416$	−32.9164	−1.61386
$417$	−8.27443	−0.405201
$418$	1.76619	0.0863871
$419$	30.2128	1.47599	0.737996	−	0.674805i	$-0.235772\pi$
$419$	30.2128	1.47599	0.737996	+	0.674805i	$0.235772\pi$
$420$	0	0
$421$	−18.4246	−0.897959	−0.448980	−	0.893542i	$-0.648212\pi$
$421$	−18.4246	−0.897959	−0.448980	+	0.893542i	$0.648212\pi$
$422$	5.83160	0.283878
$423$	−8.92527	−0.433962
$424$	8.12414	0.394543
$425$	0	0
$426$	−7.15801	−0.346807
$427$	38.2006	1.84866
$428$	28.2266	1.36438
$429$	−6.60891	−0.319081
$430$	0	0
$431$	−13.6428	−0.657152	−0.328576	−	0.944478i	$-0.606569\pi$
$431$	−13.6428	−0.657152	−0.328576	+	0.944478i	$0.606569\pi$
$432$	1.52589	0.0734145
$433$	−33.6141	−1.61539	−0.807696	−	0.589599i	$-0.799286\pi$
$433$	−33.6141	−1.61539	−0.807696	+	0.589599i	$0.799286\pi$
$434$	2.43080	0.116682
$435$	0	0
$436$	−10.1143	−0.484389
$437$	11.8249	0.565662
$438$	5.56641	0.265973
$439$	−21.9603	−1.04811	−0.524054	−	0.851685i	$-0.675581\pi$
$439$	−21.9603	−1.04811	−0.524054	+	0.851685i	$0.675581\pi$
$440$	0	0
$441$	6.26574	0.298369
$442$	5.25497	0.249953
$443$	−5.44404	−0.258654	−0.129327	−	0.991602i	$-0.541282\pi$
$443$	−5.44404	−0.258654	−0.129327	+	0.991602i	$0.541282\pi$
$444$	−4.30967	−0.204528
$445$	0	0
$446$	6.96824	0.329956
$447$	15.9731	0.755502
$448$	2.89345	0.136703
$449$	36.8483	1.73898	0.869489	−	0.493952i	$-0.164448\pi$
$449$	36.8483	1.73898	0.869489	+	0.493952i	$0.164448\pi$
$450$	0	0
$451$	0.141815	0.00667781
$452$	−5.22406	−0.245719
$453$	7.39770	0.347574
$454$	16.7616	0.786659
$455$	0	0
$456$	5.42574	0.254084
$457$	−5.03603	−0.235575	−0.117788	−	0.993039i	$-0.537580\pi$
$457$	−5.03603	−0.235575	−0.117788	+	0.993039i	$0.537580\pi$
$458$	2.39277	0.111807
$459$	−1.37855	−0.0643453
$460$	0	0
$461$	−22.5649	−1.05095	−0.525476	−	0.850808i	$-0.676113\pi$
$461$	−22.5649	−1.05095	−0.525476	+	0.850808i	$0.676113\pi$
$462$	2.81265	0.130856
$463$	7.92517	0.368314	0.184157	−	0.982897i	$-0.441045\pi$
$463$	7.92517	0.368314	0.184157	+	0.982897i	$0.441045\pi$
$464$	−8.41357	−0.390590
$465$	0	0
$466$	12.2126	0.565739
$467$	−10.7488	−0.497397	−0.248698	−	0.968581i	$-0.580003\pi$
$467$	−10.7488	−0.497397	−0.248698	+	0.968581i	$0.580003\pi$
$468$	−8.87926	−0.410444
$469$	−57.6470	−2.66189
$470$	0	0
$471$	15.7243	0.724536
$472$	1.28733	0.0592540
$473$	−1.29291	−0.0594479
$474$	−3.89572	−0.178936
$475$	0	0
$476$	7.80554	0.357766
$477$	3.42456	0.156800
$478$	6.44525	0.294799
$479$	−31.1215	−1.42198	−0.710989	−	0.703203i	$-0.751752\pi$
$479$	−31.1215	−1.42198	−0.710989	+	0.703203i	$0.751752\pi$
$480$	0	0
$481$	15.8341	0.721973
$482$	3.54809	0.161611
$483$	18.8312	0.856847
$484$	15.0190	0.682684
$485$	0	0
$486$	−0.667396	−0.0302737
$487$	22.1648	1.00438	0.502192	−	0.864756i	$-0.332527\pi$
$487$	22.1648	1.00438	0.502192	+	0.864756i	$0.332527\pi$
$488$	24.8815	1.12633
$489$	−9.68816	−0.438114
$490$	0	0
$491$	38.3744	1.73181	0.865905	−	0.500208i	$-0.166743\pi$
$491$	38.3744	1.73181	0.865905	+	0.500208i	$0.166743\pi$
$492$	0.190533	0.00858988
$493$	7.60116	0.342339
$494$	−8.71834	−0.392256
$495$	0	0
$496$	−1.52589	−0.0685146
$497$	39.0638	1.75225
$498$	−11.0077	−0.493266
$499$	−3.14013	−0.140572	−0.0702858	−	0.997527i	$-0.522391\pi$
$499$	−3.14013	−0.140572	−0.0702858	+	0.997527i	$0.522391\pi$
$500$	0	0
$501$	19.1075	0.853660
$502$	6.14572	0.274297
$503$	26.4210	1.17805	0.589027	−	0.808114i	$-0.299511\pi$
$503$	26.4210	1.17805	0.589027	+	0.808114i	$0.299511\pi$
$504$	8.64048	0.384878
$505$	0	0
$506$	3.99265	0.177495
$507$	19.6232	0.871496
$508$	−16.5000	−0.732070
$509$	−27.5147	−1.21957	−0.609785	−	0.792567i	$-0.708744\pi$
$509$	−27.5147	−1.21957	−0.609785	+	0.792567i	$0.708744\pi$
$510$	0	0
$511$	−30.3779	−1.34384
$512$	−16.0335	−0.708588
$513$	2.28711	0.100978
$514$	−4.70197	−0.207395
$515$	0	0
$516$	−1.73706	−0.0764697
$517$	10.3273	0.454196
$518$	−6.73875	−0.296084
$519$	−21.1286	−0.927443
$520$	0	0
$521$	30.2824	1.32670	0.663349	−	0.748310i	$-0.269135\pi$
$521$	30.2824	1.32670	0.663349	+	0.748310i	$0.269135\pi$
$522$	3.67993	0.161066
$523$	−42.8200	−1.87239	−0.936193	−	0.351486i	$-0.885676\pi$
$523$	−42.8200	−1.87239	−0.936193	+	0.351486i	$0.885676\pi$
$524$	−22.5539	−0.985273
$525$	0	0
$526$	2.90353	0.126600
$527$	1.37855	0.0600507
$528$	−1.76559	−0.0768376
$529$	3.73144	0.162236
$530$	0	0
$531$	0.542646	0.0235488
$532$	−12.9499	−0.561449
$533$	−0.700034	−0.0303218
$534$	5.52229	0.238973
$535$	0	0
$536$	−37.5477	−1.62181
$537$	13.0689	0.563965
$538$	13.4621	0.580391
$539$	−7.25001	−0.312280
$540$	0	0
$541$	−10.9585	−0.471142	−0.235571	−	0.971857i	$-0.575696\pi$
$541$	−10.9585	−0.471142	−0.235571	+	0.971857i	$0.575696\pi$
$542$	−8.69636	−0.373541
$543$	9.98243	0.428387
$544$	7.94460	0.340622
$545$	0	0
$546$	−13.8839	−0.594177
$547$	11.8536	0.506822	0.253411	−	0.967359i	$-0.418448\pi$
$547$	11.8536	0.506822	0.253411	+	0.967359i	$0.418448\pi$
$548$	−8.26947	−0.353254
$549$	10.4883	0.447629
$550$	0	0
$551$	−12.6108	−0.537239
$552$	12.2654	0.522052
$553$	21.2603	0.904080
$554$	3.96081	0.168279
$555$	0	0
$556$	12.8633	0.545525
$557$	−31.2361	−1.32351	−0.661757	−	0.749718i	$-0.730189\pi$
$557$	−31.2361	−1.32351	−0.661757	+	0.749718i	$0.730189\pi$
$558$	0.667396	0.0282531
$559$	6.38210	0.269934
$560$	0	0
$561$	1.59511	0.0673455
$562$	1.71075	0.0721638
$563$	−16.1583	−0.680990	−0.340495	−	0.940246i	$-0.610595\pi$
$563$	−16.1583	−0.680990	−0.340495	+	0.940246i	$0.610595\pi$
$564$	13.8751	0.584246
$565$	0	0
$566$	−15.0195	−0.631315
$567$	3.64222	0.152959
$568$	25.4437	1.06760
$569$	−8.39847	−0.352082	−0.176041	−	0.984383i	$-0.556329\pi$
$569$	−8.39847	−0.352082	−0.176041	+	0.984383i	$0.556329\pi$
$570$	0	0
$571$	−5.39179	−0.225640	−0.112820	−	0.993615i	$-0.535988\pi$
$571$	−5.39179	−0.225640	−0.112820	+	0.993615i	$0.535988\pi$
$572$	10.2741	0.429582
$573$	−8.69024	−0.363040
$574$	0.297924	0.0124351
$575$	0	0
$576$	0.794421	0.0331009
$577$	−14.7206	−0.612828	−0.306414	−	0.951898i	$-0.599129\pi$
$577$	−14.7206	−0.612828	−0.306414	+	0.951898i	$0.599129\pi$
$578$	10.0774	0.419165
$579$	18.4643	0.767349
$580$	0	0
$581$	60.0728	2.49224
$582$	−6.65862	−0.276008
$583$	−3.96252	−0.164111
$584$	−19.7863	−0.818761
$585$	0	0
$586$	6.58567	0.272052
$587$	−41.7243	−1.72215	−0.861073	−	0.508481i	$-0.830207\pi$
$587$	−41.7243	−1.72215	−0.861073	+	0.508481i	$0.830207\pi$
$588$	−9.74061	−0.401696
$589$	−2.28711	−0.0942387
$590$	0	0
$591$	1.73458	0.0713511
$592$	4.23013	0.173857
$593$	−40.4139	−1.65960	−0.829801	−	0.558059i	$-0.811546\pi$
$593$	−40.4139	−1.65960	−0.829801	+	0.558059i	$0.811546\pi$
$594$	0.772236	0.0316852
$595$	0	0
$596$	−24.8315	−1.01714
$597$	13.7835	0.564120
$598$	−19.7087	−0.805949
$599$	38.8888	1.58895	0.794477	−	0.607294i	$-0.207745\pi$
$599$	38.8888	1.58895	0.794477	+	0.607294i	$0.207745\pi$
$600$	0	0
$601$	13.9675	0.569747	0.284873	−	0.958565i	$-0.408048\pi$
$601$	13.9675	0.569747	0.284873	+	0.958565i	$0.408048\pi$
$602$	−2.71612	−0.110701
$603$	−15.8274	−0.644543
$604$	−11.5003	−0.467942
$605$	0	0
$606$	4.79255	0.194684
$607$	−40.8060	−1.65626	−0.828132	−	0.560534i	$-0.810596\pi$
$607$	−40.8060	−1.65626	−0.828132	+	0.560534i	$0.810596\pi$
$608$	−13.1806	−0.534544
$609$	−20.0827	−0.813792
$610$	0	0
$611$	−50.9782	−2.06236
$612$	2.14307	0.0866286
$613$	−26.4962	−1.07017	−0.535086	−	0.844798i	$-0.679721\pi$
$613$	−26.4962	−1.07017	−0.535086	+	0.844798i	$0.679721\pi$
$614$	−9.00233	−0.363304
$615$	0	0
$616$	−9.99780	−0.402823
$617$	28.7271	1.15651	0.578255	−	0.815856i	$-0.303734\pi$
$617$	28.7271	1.15651	0.578255	+	0.815856i	$0.303734\pi$
$618$	1.13732	0.0457496
$619$	−44.0519	−1.77060	−0.885299	−	0.465023i	$-0.846046\pi$
$619$	−44.0519	−1.77060	−0.885299	+	0.465023i	$0.846046\pi$
$620$	0	0
$621$	5.17025	0.207475
$622$	−5.56495	−0.223134
$623$	−30.1371	−1.20742
$624$	8.71539	0.348895
$625$	0	0
$626$	−4.69506	−0.187652
$627$	−2.64639	−0.105687
$628$	−24.4447	−0.975449
$629$	−3.82167	−0.152380
$630$	0	0
$631$	−0.777435	−0.0309492	−0.0154746	−	0.999880i	$-0.504926\pi$
$631$	−0.777435	−0.0309492	−0.0154746	+	0.999880i	$0.504926\pi$
$632$	13.8476	0.550830
$633$	−8.73784	−0.347298
$634$	−19.3318	−0.767764
$635$	0	0
$636$	−5.32377	−0.211101
$637$	35.7878	1.41797
$638$	−4.25801	−0.168576
$639$	10.7253	0.424286
$640$	0	0
$641$	12.6585	0.499981	0.249991	−	0.968248i	$-0.419572\pi$
$641$	12.6585	0.499981	0.249991	+	0.968248i	$0.419572\pi$
$642$	12.1179	0.478256
$643$	−46.7414	−1.84330	−0.921651	−	0.388021i	$-0.873159\pi$
$643$	−46.7414	−1.84330	−0.921651	+	0.388021i	$0.873159\pi$
$644$	−29.2746	−1.15358
$645$	0	0
$646$	2.10423	0.0827899
$647$	−19.0171	−0.747639	−0.373820	−	0.927501i	$-0.621952\pi$
$647$	−19.0171	−0.747639	−0.373820	+	0.927501i	$0.621952\pi$
$648$	2.37231	0.0931933
$649$	−0.627890	−0.0246468
$650$	0	0
$651$	−3.64222	−0.142750
$652$	15.0611	0.589836
$653$	23.6344	0.924885	0.462442	−	0.886649i	$-0.346973\pi$
$653$	23.6344	0.924885	0.462442	+	0.886649i	$0.346973\pi$
$654$	−4.34217	−0.169792
$655$	0	0
$656$	−0.187016	−0.00730176
$657$	−8.34049	−0.325394
$658$	21.6956	0.845781
$659$	0.736021	0.0286713	0.0143356	−	0.999897i	$-0.495437\pi$
$659$	0.736021	0.0286713	0.0143356	+	0.999897i	$0.495437\pi$
$660$	0	0
$661$	23.1277	0.899561	0.449781	−	0.893139i	$-0.351502\pi$
$661$	23.1277	0.899561	0.449781	+	0.893139i	$0.351502\pi$
$662$	−3.31845	−0.128975
$663$	−7.87384	−0.305794
$664$	39.1277	1.51845
$665$	0	0
$666$	−1.85018	−0.0716930
$667$	−28.5081	−1.10384
$668$	−29.7042	−1.14929
$669$	−10.4409	−0.403670
$670$	0	0
$671$	−12.1359	−0.468500
$672$	−20.9901	−0.809711
$673$	−32.9126	−1.26869	−0.634344	−	0.773051i	$-0.718730\pi$
$673$	−32.9126	−1.26869	−0.634344	+	0.773051i	$0.718730\pi$
$674$	−2.64932	−0.102048
$675$	0	0
$676$	−30.5059	−1.17330
$677$	4.83090	0.185667	0.0928333	−	0.995682i	$-0.470408\pi$
$677$	4.83090	0.185667	0.0928333	+	0.995682i	$0.470408\pi$
$678$	−2.24273	−0.0861317
$679$	36.3384	1.39454
$680$	0	0
$681$	−25.1149	−0.962403
$682$	−0.772236	−0.0295705
$683$	−36.8203	−1.40889	−0.704444	−	0.709759i	$-0.748804\pi$
$683$	−36.8203	−1.40889	−0.704444	+	0.709759i	$0.748804\pi$
$684$	−3.55550	−0.135948
$685$	0	0
$686$	1.78484	0.0681456
$687$	−3.58524	−0.136785
$688$	1.70500	0.0650025
$689$	19.5600	0.745176
$690$	0	0
$691$	25.2787	0.961646	0.480823	−	0.876818i	$-0.340338\pi$
$691$	25.2787	0.961646	0.480823	+	0.876818i	$0.340338\pi$
$692$	32.8462	1.24862
$693$	−4.21437	−0.160091
$694$	21.8913	0.830983
$695$	0	0
$696$	−13.0806	−0.495820
$697$	0.168958	0.00639975
$698$	−24.2018	−0.916051
$699$	−18.2989	−0.692128
$700$	0	0
$701$	50.8414	1.92025	0.960127	−	0.279565i	$-0.0901901\pi$
$701$	50.8414	1.92025	0.960127	+	0.279565i	$0.0901901\pi$
$702$	−3.81195	−0.143873
$703$	6.34041	0.239133
$704$	−0.919215	−0.0346442
$705$	0	0
$706$	20.6667	0.777800
$707$	−26.1546	−0.983646
$708$	−0.843588	−0.0317040
$709$	−37.1976	−1.39698	−0.698492	−	0.715618i	$-0.746145\pi$
$709$	−37.1976	−1.39698	−0.698492	+	0.715618i	$0.746145\pi$
$710$	0	0
$711$	5.83719	0.218912
$712$	−19.6294	−0.735644
$713$	−5.17025	−0.193627
$714$	3.35099	0.125407
$715$	0	0
$716$	−20.3167	−0.759270
$717$	−9.65731	−0.360659
$718$	7.79364	0.290856
$719$	35.3863	1.31969	0.659843	−	0.751403i	$-0.270623\pi$
$719$	35.3863	1.31969	0.659843	+	0.751403i	$0.270623\pi$
$720$	0	0
$721$	−6.20674	−0.231151
$722$	9.18946	0.341996
$723$	−5.31631	−0.197716
$724$	−15.5185	−0.576741
$725$	0	0
$726$	6.44781	0.239301
$727$	−40.7826	−1.51254	−0.756272	−	0.654257i	$-0.772981\pi$
$727$	−40.7826	−1.51254	−0.756272	+	0.654257i	$0.772981\pi$
$728$	49.3516	1.82909
$729$	1.00000	0.0370370
$730$	0	0
$731$	−1.54036	−0.0569724
$732$	−16.3049	−0.602647
$733$	26.8011	0.989922	0.494961	−	0.868915i	$-0.335182\pi$
$733$	26.8011	0.989922	0.494961	+	0.868915i	$0.335182\pi$
$734$	−1.68519	−0.0622015
$735$	0	0
$736$	−29.7961	−1.09830
$737$	18.3138	0.674596
$738$	0.0817974	0.00301100
$739$	−8.28484	−0.304763	−0.152381	−	0.988322i	$-0.548694\pi$
$739$	−8.28484	−0.304763	−0.152381	+	0.988322i	$0.548694\pi$
$740$	0	0
$741$	13.0632	0.479889
$742$	−8.32443	−0.305599
$743$	−24.8614	−0.912077	−0.456039	−	0.889960i	$-0.650732\pi$
$743$	−24.8614	−0.912077	−0.456039	+	0.889960i	$0.650732\pi$
$744$	−2.37231	−0.0869733
$745$	0	0
$746$	−10.0751	−0.368874
$747$	16.4935	0.603464
$748$	−2.47973	−0.0906677
$749$	−66.1317	−2.41640
$750$	0	0
$751$	17.0776	0.623172	0.311586	−	0.950218i	$-0.399140\pi$
$751$	17.0776	0.623172	0.311586	+	0.950218i	$0.399140\pi$
$752$	−13.6190	−0.496634
$753$	−9.20850	−0.335576
$754$	21.0186	0.765451
$755$	0	0
$756$	−5.66213	−0.205930
$757$	−22.6984	−0.824988	−0.412494	−	0.910960i	$-0.635342\pi$
$757$	−22.6984	−0.824988	−0.412494	+	0.910960i	$0.635342\pi$
$758$	12.7895	0.464536
$759$	−5.98243	−0.217149
$760$	0	0
$761$	4.34342	0.157449	0.0787244	−	0.996896i	$-0.474915\pi$
$761$	4.34342	0.157449	0.0787244	+	0.996896i	$0.474915\pi$
$762$	−7.08361	−0.256612
$763$	23.6968	0.857881
$764$	13.5097	0.488764
$765$	0	0
$766$	1.52111	0.0549599
$767$	3.09942	0.111913
$768$	−8.92740	−0.322140
$769$	2.57132	0.0927242	0.0463621	−	0.998925i	$-0.485237\pi$
$769$	2.57132	0.0927242	0.0463621	+	0.998925i	$0.485237\pi$
$770$	0	0
$771$	7.04525	0.253728
$772$	−28.7042	−1.03309
$773$	12.4442	0.447588	0.223794	−	0.974636i	$-0.428156\pi$
$773$	12.4442	0.447588	0.223794	+	0.974636i	$0.428156\pi$
$774$	−0.745734	−0.0268049
$775$	0	0
$776$	23.6686	0.849653
$777$	10.0971	0.362231
$778$	9.73495	0.349015
$779$	−0.280313	−0.0100432
$780$	0	0
$781$	−12.4101	−0.444068
$782$	4.75683	0.170104
$783$	−5.51387	−0.197050
$784$	9.56084	0.341459
$785$	0	0
$786$	−9.68260	−0.345367
$787$	37.3565	1.33161	0.665807	−	0.746124i	$-0.268087\pi$
$787$	37.3565	1.33161	0.665807	+	0.746124i	$0.268087\pi$
$788$	−2.69655	−0.0960606
$789$	−4.35053	−0.154883
$790$	0	0
$791$	12.2394	0.435183
$792$	−2.74498	−0.0975385
$793$	59.9056	2.12731
$794$	−22.0862	−0.783809
$795$	0	0
$796$	−21.4275	−0.759479
$797$	−42.1378	−1.49260	−0.746299	−	0.665611i	$-0.768171\pi$
$797$	−42.1378	−1.49260	−0.746299	+	0.665611i	$0.768171\pi$
$798$	−5.55951	−0.196804
$799$	12.3040	0.435283
$800$	0	0
$801$	−8.27438	−0.292361
$802$	−7.08261	−0.250096
$803$	9.65069	0.340565
$804$	24.6051	0.867754
$805$	0	0
$806$	3.81195	0.134270
$807$	−20.1710	−0.710054
$808$	−17.0355	−0.599307
$809$	2.15676	0.0758277	0.0379138	−	0.999281i	$-0.487929\pi$
$809$	2.15676	0.0758277	0.0379138	+	0.999281i	$0.487929\pi$
$810$	0	0
$811$	12.8658	0.451780	0.225890	−	0.974153i	$-0.427471\pi$
$811$	12.8658	0.451780	0.225890	+	0.974153i	$0.427471\pi$
$812$	31.2202	1.09561
$813$	13.0303	0.456992
$814$	2.14082	0.0750358
$815$	0	0
$816$	−2.10352	−0.0736380
$817$	2.55557	0.0894079
$818$	18.8186	0.657977
$819$	20.8031	0.726921
$820$	0	0
$821$	13.9740	0.487696	0.243848	−	0.969813i	$-0.421590\pi$
$821$	13.9740	0.487696	0.243848	+	0.969813i	$0.421590\pi$
$822$	−3.55015	−0.123826
$823$	45.9522	1.60179	0.800897	−	0.598802i	$-0.204357\pi$
$823$	45.9522	1.60179	0.800897	+	0.598802i	$0.204357\pi$
$824$	−4.04269	−0.140834
$825$	0	0
$826$	−1.31906	−0.0458961
$827$	6.37379	0.221638	0.110819	−	0.993841i	$-0.464653\pi$
$827$	6.37379	0.221638	0.110819	+	0.993841i	$0.464653\pi$
$828$	−8.03757	−0.279325
$829$	−21.0176	−0.729970	−0.364985	−	0.931013i	$-0.618926\pi$
$829$	−21.0176	−0.729970	−0.364985	+	0.931013i	$0.618926\pi$
$830$	0	0
$831$	−5.93472	−0.205873
$832$	4.53747	0.157308
$833$	−8.63765	−0.299277
$834$	5.52232	0.191222
$835$	0	0
$836$	4.11403	0.142287
$837$	−1.00000	−0.0345651
$838$	−20.1639	−0.696551
$839$	20.6247	0.712045	0.356022	−	0.934477i	$-0.384133\pi$
$839$	20.6247	0.712045	0.356022	+	0.934477i	$0.384133\pi$
$840$	0	0
$841$	1.40275	0.0483708
$842$	12.2965	0.423765
$843$	−2.56333	−0.0882857
$844$	13.5837	0.467570
$845$	0	0
$846$	5.95669	0.204795
$847$	−35.1880	−1.20907
$848$	5.22551	0.179445
$849$	22.5046	0.772355
$850$	0	0
$851$	14.3331	0.491334
$852$	−16.6733	−0.571219
$853$	6.65175	0.227752	0.113876	−	0.993495i	$-0.463673\pi$
$853$	6.65175	0.227752	0.113876	+	0.993495i	$0.463673\pi$
$854$	−25.4949	−0.872418
$855$	0	0
$856$	−43.0741	−1.47224
$857$	−26.3351	−0.899590	−0.449795	−	0.893132i	$-0.648503\pi$
$857$	−26.3351	−0.899590	−0.449795	+	0.893132i	$0.648503\pi$
$858$	4.41076	0.150581
$859$	26.8919	0.917541	0.458770	−	0.888555i	$-0.348290\pi$
$859$	26.8919	0.917541	0.458770	+	0.888555i	$0.348290\pi$
$860$	0	0
$861$	−0.446397	−0.0152132
$862$	9.10517	0.310123
$863$	−5.04588	−0.171764	−0.0858818	−	0.996305i	$-0.527371\pi$
$863$	−5.04588	−0.171764	−0.0858818	+	0.996305i	$0.527371\pi$
$864$	−5.76300	−0.196061
$865$	0	0
$866$	22.4339	0.762336
$867$	−15.0996	−0.512809
$868$	5.66213	0.192185
$869$	−6.75414	−0.229119
$870$	0	0
$871$	−90.4012	−3.06313
$872$	15.4346	0.522682
$873$	9.97701	0.337671
$874$	−7.89190	−0.266947
$875$	0	0
$876$	12.9660	0.438080
$877$	1.50014	0.0506560	0.0253280	−	0.999679i	$-0.491937\pi$
$877$	1.50014	0.0506560	0.0253280	+	0.999679i	$0.491937\pi$
$878$	14.6562	0.494623
$879$	−9.86771	−0.332830
$880$	0	0
$881$	−52.2328	−1.75977	−0.879884	−	0.475188i	$-0.842380\pi$
$881$	−52.2328	−1.75977	−0.879884	+	0.475188i	$0.842380\pi$
$882$	−4.18173	−0.140806
$883$	39.4859	1.32881	0.664403	−	0.747375i	$-0.268686\pi$
$883$	39.4859	1.32881	0.664403	+	0.747375i	$0.268686\pi$
$884$	12.2405	0.411694
$885$	0	0
$886$	3.63333	0.122064
$887$	48.5387	1.62977	0.814885	−	0.579623i	$-0.196800\pi$
$887$	48.5387	1.62977	0.814885	+	0.579623i	$0.196800\pi$
$888$	6.57662	0.220697
$889$	38.6578	1.29654
$890$	0	0
$891$	−1.15709	−0.0387639
$892$	16.2313	0.543464
$893$	−20.4131	−0.683097
$894$	−10.6604	−0.356537
$895$	0	0
$896$	40.0491	1.33795
$897$	29.5307	0.986003
$898$	−24.5924	−0.820659
$899$	5.51387	0.183898
$900$	0	0
$901$	−4.72094	−0.157277
$902$	−0.0946468	−0.00315139
$903$	4.06973	0.135432
$904$	7.97198	0.265144
$905$	0	0
$906$	−4.93719	−0.164027
$907$	−47.3682	−1.57284	−0.786418	−	0.617694i	$-0.788067\pi$
$907$	−47.3682	−1.57284	−0.786418	+	0.617694i	$0.788067\pi$
$908$	39.0431	1.29569
$909$	−7.18097	−0.238178
$910$	0	0
$911$	−28.7118	−0.951264	−0.475632	−	0.879644i	$-0.657781\pi$
$911$	−28.7118	−0.951264	−0.475632	+	0.879644i	$0.657781\pi$
$912$	3.48988	0.115562
$913$	−19.0844	−0.631602
$914$	3.36102	0.111173
$915$	0	0
$916$	5.57355	0.184155
$917$	52.8414	1.74498
$918$	0.920040	0.0303659
$919$	−3.19601	−0.105427	−0.0527133	−	0.998610i	$-0.516787\pi$
$919$	−3.19601	−0.105427	−0.0527133	+	0.998610i	$0.516787\pi$
$920$	0	0
$921$	13.4887	0.444469
$922$	15.0597	0.495966
$923$	61.2593	2.01637
$924$	6.55158	0.215531
$925$	0	0
$926$	−5.28923	−0.173815
$927$	−1.70411	−0.0559703
$928$	31.7764	1.04311
$929$	2.28411	0.0749392	0.0374696	−	0.999298i	$-0.488070\pi$
$929$	2.28411	0.0749392	0.0374696	+	0.999298i	$0.488070\pi$
$930$	0	0
$931$	14.3304	0.469661
$932$	28.4472	0.931818
$933$	8.33830	0.272984
$934$	7.17373	0.234732
$935$	0	0
$936$	13.5499	0.442892
$937$	−9.90513	−0.323587	−0.161793	−	0.986825i	$-0.551728\pi$
$937$	−9.90513	−0.323587	−0.161793	+	0.986825i	$0.551728\pi$
$938$	38.4734	1.25620
$939$	7.03489	0.229575
$940$	0	0
$941$	−8.74045	−0.284930	−0.142465	−	0.989800i	$-0.545503\pi$
$941$	−8.74045	−0.284930	−0.142465	+	0.989800i	$0.545503\pi$
$942$	−10.4943	−0.341923
$943$	−0.633676	−0.0206353
$944$	0.828019	0.0269497
$945$	0	0
$946$	0.862880	0.0280547
$947$	55.8934	1.81629	0.908146	−	0.418654i	$-0.137498\pi$
$947$	55.8934	1.81629	0.908146	+	0.418654i	$0.137498\pi$
$948$	−9.07439	−0.294723
$949$	−47.6381	−1.54640
$950$	0	0
$951$	28.9660	0.939287
$952$	−11.9114	−0.386049
$953$	−36.3033	−1.17598	−0.587989	−	0.808869i	$-0.700080\pi$
$953$	−36.3033	−1.17598	−0.587989	+	0.808869i	$0.700080\pi$
$954$	−2.28554	−0.0739971
$955$	0	0
$956$	15.0131	0.485558
$957$	6.38004	0.206237
$958$	20.7704	0.671060
$959$	19.3744	0.625634
$960$	0	0
$961$	1.00000	0.0322581
$962$	−10.5676	−0.340714
$963$	−18.1570	−0.585101
$964$	8.26465	0.266186
$965$	0	0
$966$	−12.5678	−0.404363
$967$	−25.7135	−0.826890	−0.413445	−	0.910529i	$-0.635675\pi$
$967$	−25.7135	−0.826890	−0.413445	+	0.910529i	$0.635675\pi$
$968$	−22.9193	−0.736653
$969$	−3.15290	−0.101286
$970$	0	0
$971$	33.1226	1.06296	0.531478	−	0.847072i	$-0.321637\pi$
$971$	33.1226	1.06296	0.531478	+	0.847072i	$0.321637\pi$
$972$	−1.55458	−0.0498633
$973$	−30.1373	−0.966156
$974$	−14.7927	−0.473989
$975$	0	0
$976$	16.0040	0.512275
$977$	−22.6735	−0.725388	−0.362694	−	0.931908i	$-0.618143\pi$
$977$	−22.6735	−0.725388	−0.362694	+	0.931908i	$0.618143\pi$
$978$	6.46584	0.206755
$979$	9.57419	0.305992
$980$	0	0
$981$	6.50614	0.207725
$982$	−25.6109	−0.817276
$983$	9.35725	0.298450	0.149225	−	0.988803i	$-0.452322\pi$
$983$	9.35725	0.298450	0.149225	+	0.988803i	$0.452322\pi$
$984$	−0.290756	−0.00926895
$985$	0	0
$986$	−5.07298	−0.161557
$987$	−32.5078	−1.03473
$988$	−20.3078	−0.646079
$989$	5.77712	0.183702
$990$	0	0
$991$	12.9858	0.412506	0.206253	−	0.978499i	$-0.433873\pi$
$991$	12.9858	0.412506	0.206253	+	0.978499i	$0.433873\pi$
$992$	5.76300	0.182976
$993$	4.97223	0.157789
$994$	−26.0710	−0.826922
$995$	0	0
$996$	−25.6405	−0.812449
$997$	3.74266	0.118531	0.0592656	−	0.998242i	$-0.481124\pi$
$997$	3.74266	0.118531	0.0592656	+	0.998242i	$0.481124\pi$
$998$	2.09571	0.0663385
$999$	2.77224	0.0877097

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2325.2.a.bb.1.3	yes	6
3.2	odd	2		6975.2.a.ca.1.4		6
5.2	odd	4		2325.2.c.r.1024.6		12
5.3	odd	4		2325.2.c.r.1024.7		12
5.4	even	2		2325.2.a.y.1.4	✓	6
15.14	odd	2		6975.2.a.cc.1.3		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2325.2.a.y.1.4	✓	6	5.4	even	2
2325.2.a.bb.1.3	yes	6	1.1	even	1	trivial
2325.2.c.r.1024.6		12	5.2	odd	4
2325.2.c.r.1024.7		12	5.3	odd	4
6975.2.a.ca.1.4		6	3.2	odd	2
6975.2.a.cc.1.3		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)

\(f(q)\)	\(=\)	\(q-0.667396 q^{2} +1.00000 q^{3} -1.55458 q^{4} -0.667396 q^{6} +3.64222 q^{7} +2.37231 q^{8} +1.00000 q^{9} -1.15709 q^{11} -1.55458 q^{12} +5.71167 q^{13} -2.43080 q^{14} +1.52589 q^{16} -1.37855 q^{17} -0.667396 q^{18} +2.28711 q^{19} +3.64222 q^{21} +0.772236 q^{22} +5.17025 q^{23} +2.37231 q^{24} -3.81195 q^{26} +1.00000 q^{27} -5.66213 q^{28} -5.51387 q^{29} -1.00000 q^{31} -5.76300 q^{32} -1.15709 q^{33} +0.920040 q^{34} -1.55458 q^{36} +2.77224 q^{37} -1.52641 q^{38} +5.71167 q^{39} -0.122562 q^{41} -2.43080 q^{42} +1.11738 q^{43} +1.79879 q^{44} -3.45060 q^{46} -8.92527 q^{47} +1.52589 q^{48} +6.26574 q^{49} -1.37855 q^{51} -8.87926 q^{52} +3.42456 q^{53} -0.667396 q^{54} +8.64048 q^{56} +2.28711 q^{57} +3.67993 q^{58} +0.542646 q^{59} +10.4883 q^{61} +0.667396 q^{62} +3.64222 q^{63} +0.794421 q^{64} +0.772236 q^{66} -15.8274 q^{67} +2.14307 q^{68} +5.17025 q^{69} +10.7253 q^{71} +2.37231 q^{72} -8.34049 q^{73} -1.85018 q^{74} -3.55550 q^{76} -4.21437 q^{77} -3.81195 q^{78} +5.83719 q^{79} +1.00000 q^{81} +0.0817974 q^{82} +16.4935 q^{83} -5.66213 q^{84} -0.745734 q^{86} -5.51387 q^{87} -2.74498 q^{88} -8.27438 q^{89} +20.8031 q^{91} -8.03757 q^{92} -1.00000 q^{93} +5.95669 q^{94} -5.76300 q^{96} +9.97701 q^{97} -4.18173 q^{98} -1.15709 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + q^{2} + 6 q^{3} + 7 q^{4} + q^{6} + 2 q^{7} - 3 q^{8} + 6 q^{9} + 7 q^{11} + 7 q^{12} + 4 q^{13} + 10 q^{14} + 17 q^{16} + q^{18} + 17 q^{19} + 2 q^{21} + 2 q^{22} - q^{23} - 3 q^{24} + 2 q^{26}+ \cdots + 7 q^{99}+O(q^{100}) \) 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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