LMFDB - Embedded newform 675.4.a.p.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [675,4,Mod(1,675)] 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("675.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(675, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 675 = 3^{3} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	675.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,-5,0,17,0,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$39.8262892539$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.1772.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 12x + 8 $ x^3 - x^2 - 12*x + 8
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 3 $
Twist minimal:	no (minimal twist has level 135)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-3.32803$ of defining polynomial
Character	$\chi$	$=$	675.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-2.12612 q^{2} -3.47962 q^{4} -30.7000 q^{7} +24.4070 q^{8} +50.1548 q^{11} +15.9592 q^{13} +65.2720 q^{14} -24.0553 q^{16} -105.668 q^{17} -21.3040 q^{19} -106.635 q^{22} -136.137 q^{23} -33.9312 q^{26} +106.824 q^{28} -224.323 q^{29} -225.982 q^{31} -144.112 q^{32} +224.663 q^{34} +416.386 q^{37} +45.2948 q^{38} +76.1411 q^{41} -31.7372 q^{43} -174.519 q^{44} +289.443 q^{46} -60.8026 q^{47} +599.493 q^{49} -55.5320 q^{52} +466.532 q^{53} -749.297 q^{56} +476.938 q^{58} +95.4239 q^{59} -357.174 q^{61} +480.464 q^{62} +498.842 q^{64} -87.8344 q^{67} +367.685 q^{68} +412.693 q^{71} +331.133 q^{73} -885.286 q^{74} +74.1296 q^{76} -1539.75 q^{77} -248.123 q^{79} -161.885 q^{82} +552.505 q^{83} +67.4771 q^{86} +1224.13 q^{88} -291.478 q^{89} -489.949 q^{91} +473.704 q^{92} +129.274 q^{94} -198.606 q^{97} -1274.59 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 5 q^{2} + 17 q^{4} + 4 q^{7} - 75 q^{8} + 5 q^{11} - 7 q^{13} + 60 q^{14} + 161 q^{16} - 155 q^{17} - 50 q^{19} + 229 q^{22} - 285 q^{23} + 185 q^{26} + 334 q^{28} + 115 q^{29} - 115 q^{31} - 775 q^{32}+ \cdots + 305 q^{98}+O(q^{100}) $ 3 * q - 5 * q^2 + 17 * q^4 + 4 * q^7 - 75 * q^8 + 5 * q^11 - 7 * q^13 + 60 * q^14 + 161 * q^16 - 155 * q^17 - 50 * q^19 + 229 * q^22 - 285 * q^23 + 185 * q^26 + 334 * q^28 + 115 * q^29 - 115 * q^31 - 775 * q^32 + 413 * q^34 + 384 * q^37 + 1150 * q^38 + 580 * q^41 + 797 * q^43 - 1415 * q^44 - 285 * q^46 + 145 * q^47 + 577 * q^49 - 825 * q^52 + 400 * q^53 - 2190 * q^56 + 59 * q^58 + 380 * q^59 - 152 * q^61 + 1005 * q^62 + 2937 * q^64 - 2 * q^67 - 475 * q^68 + 40 * q^71 + 980 * q^73 - 2720 * q^74 - 3276 * q^76 - 1950 * q^77 + 1013 * q^79 - 4 * q^82 - 270 * q^83 - 1555 * q^86 + 5193 * q^88 + 1020 * q^89 - 632 * q^91 + 1215 * q^92 + 3833 * q^94 - 720 * q^97 + 305 * q^98

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 $

$2$	−2.12612	−0.751697	−0.375848	−	0.926681i	$-0.622649\pi$
$2$	−2.12612	−0.751697	−0.375848	+	0.926681i	$0.622649\pi$
$3$	0	0
$3$	0	0
$4$	−3.47962	−0.434952
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−30.7000	−1.65765	−0.828823	−	0.559511i	$-0.810989\pi$
$7$	−30.7000	−1.65765	−0.828823	+	0.559511i	$0.810989\pi$
$8$	24.4070	1.07865
$9$	0	0
$10$	0	0
$11$	50.1548	1.37475	0.687375	−	0.726303i	$-0.258763\pi$
$11$	50.1548	1.37475	0.687375	+	0.726303i	$0.258763\pi$
$12$	0	0
$13$	15.9592	0.340484	0.170242	−	0.985402i	$-0.445545\pi$
$13$	15.9592	0.340484	0.170242	+	0.985402i	$0.445545\pi$
$14$	65.2720	1.24605
$15$	0	0
$16$	−24.0553	−0.375865
$17$	−105.668	−1.50755	−0.753774	−	0.657134i	$-0.771769\pi$
$17$	−105.668	−1.50755	−0.753774	+	0.657134i	$0.771769\pi$
$18$	0	0
$19$	−21.3040	−0.257235	−0.128618	−	0.991694i	$-0.541054\pi$
$19$	−21.3040	−0.257235	−0.128618	+	0.991694i	$0.541054\pi$
$20$	0	0
$21$	0	0
$22$	−106.635	−1.03339
$23$	−136.137	−1.23420	−0.617098	−	0.786886i	$-0.711692\pi$
$23$	−136.137	−1.23420	−0.617098	+	0.786886i	$0.711692\pi$
$24$	0	0
$25$	0	0
$26$	−33.9312	−0.255941
$27$	0	0
$28$	106.824	0.720997
$29$	−224.323	−1.43641	−0.718203	−	0.695834i	$-0.755035\pi$
$29$	−224.323	−1.43641	−0.718203	+	0.695834i	$0.755035\pi$
$30$	0	0
$31$	−225.982	−1.30928	−0.654638	−	0.755943i	$-0.727179\pi$
$31$	−225.982	−1.30928	−0.654638	+	0.755943i	$0.727179\pi$
$32$	−144.112	−0.796112
$33$	0	0
$34$	224.663	1.13322
$35$	0	0
$36$	0	0
$37$	416.386	1.85009	0.925046	−	0.379854i	$-0.124026\pi$
$37$	416.386	1.85009	0.925046	+	0.379854i	$0.124026\pi$
$38$	45.2948	0.193363
$39$	0	0
$40$	0	0
$41$	76.1411	0.290030	0.145015	−	0.989429i	$-0.453677\pi$
$41$	76.1411	0.290030	0.145015	+	0.989429i	$0.453677\pi$
$42$	0	0
$43$	−31.7372	−0.112555	−0.0562777	−	0.998415i	$-0.517923\pi$
$43$	−31.7372	−0.112555	−0.0562777	+	0.998415i	$0.517923\pi$
$44$	−174.519	−0.597950
$45$	0	0
$46$	289.443	0.927741
$47$	−60.8026	−0.188701	−0.0943507	−	0.995539i	$-0.530078\pi$
$47$	−60.8026	−0.188701	−0.0943507	+	0.995539i	$0.530078\pi$
$48$	0	0
$49$	599.493	1.74779
$50$	0	0
$51$	0	0
$52$	−55.5320	−0.148094
$53$	466.532	1.20911	0.604557	−	0.796562i	$-0.293350\pi$
$53$	466.532	1.20911	0.604557	+	0.796562i	$0.293350\pi$
$54$	0	0
$55$	0	0
$56$	−749.297	−1.78802
$57$	0	0
$58$	476.938	1.07974
$59$	95.4239	0.210562	0.105281	−	0.994443i	$-0.466426\pi$
$59$	95.4239	0.210562	0.105281	+	0.994443i	$0.466426\pi$
$60$	0	0
$61$	−357.174	−0.749696	−0.374848	−	0.927086i	$-0.622305\pi$
$61$	−357.174	−0.749696	−0.374848	+	0.927086i	$0.622305\pi$
$62$	480.464	0.984178
$63$	0	0
$64$	498.842	0.974300
$65$	0	0
$66$	0	0
$67$	−87.8344	−0.160159	−0.0800797	−	0.996788i	$-0.525517\pi$
$67$	−87.8344	−0.160159	−0.0800797	+	0.996788i	$0.525517\pi$
$68$	367.685	0.655711
$69$	0	0
$70$	0	0
$71$	412.693	0.689826	0.344913	−	0.938635i	$-0.387908\pi$
$71$	412.693	0.689826	0.344913	+	0.938635i	$0.387908\pi$
$72$	0	0
$73$	331.133	0.530906	0.265453	−	0.964124i	$-0.414478\pi$
$73$	331.133	0.530906	0.265453	+	0.964124i	$0.414478\pi$
$74$	−885.286	−1.39071
$75$	0	0
$76$	74.1296	0.111885
$77$	−1539.75	−2.27885
$78$	0	0
$79$	−248.123	−0.353368	−0.176684	−	0.984268i	$-0.556537\pi$
$79$	−248.123	−0.353368	−0.176684	+	0.984268i	$0.556537\pi$
$80$	0	0
$81$	0	0
$82$	−161.885	−0.218015
$83$	552.505	0.730667	0.365333	−	0.930877i	$-0.380955\pi$
$83$	552.505	0.730667	0.365333	+	0.930877i	$0.380955\pi$
$84$	0	0
$85$	0	0
$86$	67.4771	0.0846075
$87$	0	0
$88$	1224.13	1.48287
$89$	−291.478	−0.347153	−0.173577	−	0.984820i	$-0.555532\pi$
$89$	−291.478	−0.347153	−0.173577	+	0.984820i	$0.555532\pi$
$90$	0	0
$91$	−489.949	−0.564402
$92$	473.704	0.536816
$93$	0	0
$94$	129.274	0.141846
$95$	0	0
$96$	0	0
$97$	−198.606	−0.207891	−0.103946	−	0.994583i	$-0.533147\pi$
$97$	−198.606	−0.207891	−0.103946	+	0.994583i	$0.533147\pi$
$98$	−1274.59	−1.31381
$99$	0	0
$100$	0	0
$101$	816.235	0.804143	0.402071	−	0.915608i	$-0.368290\pi$
$101$	816.235	0.804143	0.402071	+	0.915608i	$0.368290\pi$
$102$	0	0
$103$	1402.37	1.34155	0.670776	−	0.741660i	$-0.265961\pi$
$103$	1402.37	1.34155	0.670776	+	0.741660i	$0.265961\pi$
$104$	389.518	0.367263
$105$	0	0
$106$	−991.902	−0.908887
$107$	978.996	0.884515	0.442258	−	0.896888i	$-0.354178\pi$
$107$	978.996	0.884515	0.442258	+	0.896888i	$0.354178\pi$
$108$	0	0
$109$	2122.96	1.86553	0.932766	−	0.360484i	$-0.117388\pi$
$109$	2122.96	1.86553	0.932766	+	0.360484i	$0.117388\pi$
$110$	0	0
$111$	0	0
$112$	738.500	0.623051
$113$	−1794.80	−1.49416	−0.747082	−	0.664732i	$-0.768546\pi$
$113$	−1794.80	−1.49416	−0.747082	+	0.664732i	$0.768546\pi$
$114$	0	0
$115$	0	0
$116$	780.559	0.624768
$117$	0	0
$118$	−202.883	−0.158278
$119$	3244.02	2.49898
$120$	0	0
$121$	1184.50	0.889935
$122$	759.395	0.563544
$123$	0	0
$124$	786.330	0.569472
$125$	0	0
$126$	0	0
$127$	748.894	0.523257	0.261628	−	0.965169i	$-0.415741\pi$
$127$	748.894	0.523257	0.261628	+	0.965169i	$0.415741\pi$
$128$	92.2971	0.0637343
$129$	0	0
$130$	0	0
$131$	−2396.11	−1.59808	−0.799042	−	0.601275i	$-0.794660\pi$
$131$	−2396.11	−1.59808	−0.799042	+	0.601275i	$0.794660\pi$
$132$	0	0
$133$	654.033	0.426405
$134$	186.746	0.120391
$135$	0	0
$136$	−2579.05	−1.62611
$137$	−1004.48	−0.626409	−0.313205	−	0.949686i	$-0.601403\pi$
$137$	−1004.48	−0.626409	−0.313205	+	0.949686i	$0.601403\pi$
$138$	0	0
$139$	2403.02	1.46634	0.733172	−	0.680043i	$-0.238039\pi$
$139$	2403.02	1.46634	0.733172	+	0.680043i	$0.238039\pi$
$140$	0	0
$141$	0	0
$142$	−877.435	−0.518540
$143$	800.432	0.468080
$144$	0	0
$145$	0	0
$146$	−704.028	−0.399081
$147$	0	0
$148$	−1448.86	−0.804701
$149$	509.648	0.280215	0.140107	−	0.990136i	$-0.455255\pi$
$149$	509.648	0.280215	0.140107	+	0.990136i	$0.455255\pi$
$150$	0	0
$151$	1443.30	0.777840	0.388920	−	0.921272i	$-0.372848\pi$
$151$	1443.30	0.777840	0.388920	+	0.921272i	$0.372848\pi$
$152$	−519.967	−0.277466
$153$	0	0
$154$	3273.70	1.71300
$155$	0	0
$156$	0	0
$157$	2155.64	1.09579	0.547895	−	0.836547i	$-0.315429\pi$
$157$	2155.64	1.09579	0.547895	+	0.836547i	$0.315429\pi$
$158$	527.539	0.265625
$159$	0	0
$160$	0	0
$161$	4179.41	2.04586
$162$	0	0
$163$	529.909	0.254636	0.127318	−	0.991862i	$-0.459363\pi$
$163$	529.909	0.254636	0.127318	+	0.991862i	$0.459363\pi$
$164$	−264.942	−0.126149
$165$	0	0
$166$	−1174.69	−0.549240
$167$	2979.28	1.38050	0.690250	−	0.723571i	$-0.257501\pi$
$167$	2979.28	1.38050	0.690250	+	0.723571i	$0.257501\pi$
$168$	0	0
$169$	−1942.30	−0.884071
$170$	0	0
$171$	0	0
$172$	110.433	0.0489562
$173$	−1779.88	−0.782205	−0.391103	−	0.920347i	$-0.627906\pi$
$173$	−1779.88	−0.782205	−0.391103	+	0.920347i	$0.627906\pi$
$174$	0	0
$175$	0	0
$176$	−1206.49	−0.516720
$177$	0	0
$178$	619.718	0.260954
$179$	−2836.26	−1.18431	−0.592157	−	0.805823i	$-0.701724\pi$
$179$	−2836.26	−1.18431	−0.592157	+	0.805823i	$0.701724\pi$
$180$	0	0
$181$	811.890	0.333410	0.166705	−	0.986007i	$-0.446687\pi$
$181$	811.890	0.333410	0.166705	+	0.986007i	$0.446687\pi$
$182$	1041.69	0.424259
$183$	0	0
$184$	−3322.70	−1.33126
$185$	0	0
$186$	0	0
$187$	−5299.77	−2.07250
$188$	211.570	0.0820761
$189$	0	0
$190$	0	0
$191$	1148.46	0.435077	0.217539	−	0.976052i	$-0.430197\pi$
$191$	1148.46	0.435077	0.217539	+	0.976052i	$0.430197\pi$
$192$	0	0
$193$	2150.93	0.802215	0.401107	−	0.916031i	$-0.368625\pi$
$193$	2150.93	0.802215	0.401107	+	0.916031i	$0.368625\pi$
$194$	422.261	0.156271
$195$	0	0
$196$	−2086.00	−0.760206
$197$	5057.51	1.82910	0.914551	−	0.404471i	$-0.132544\pi$
$197$	5057.51	1.82910	0.914551	+	0.404471i	$0.132544\pi$
$198$	0	0
$199$	−3554.12	−1.26605	−0.633027	−	0.774130i	$-0.718188\pi$
$199$	−3554.12	−1.26605	−0.633027	+	0.774130i	$0.718188\pi$
$200$	0	0
$201$	0	0
$202$	−1735.41	−0.604471
$203$	6886.73	2.38105
$204$	0	0
$205$	0	0
$206$	−2981.61	−1.00844
$207$	0	0
$208$	−383.905	−0.127976
$209$	−1068.50	−0.353634
$210$	0	0
$211$	−107.909	−0.0352075	−0.0176038	−	0.999845i	$-0.505604\pi$
$211$	−107.909	−0.0352075	−0.0176038	+	0.999845i	$0.505604\pi$
$212$	−1623.35	−0.525907
$213$	0	0
$214$	−2081.46	−0.664887
$215$	0	0
$216$	0	0
$217$	6937.65	2.17032
$218$	−4513.67	−1.40231
$219$	0	0
$220$	0	0
$221$	−1686.38	−0.513296
$222$	0	0
$223$	1942.03	0.583174	0.291587	−	0.956544i	$-0.405817\pi$
$223$	1942.03	0.583174	0.291587	+	0.956544i	$0.405817\pi$
$224$	4424.24	1.31967
$225$	0	0
$226$	3815.96	1.12316
$227$	−3443.99	−1.00698	−0.503492	−	0.864000i	$-0.667952\pi$
$227$	−3443.99	−1.00698	−0.503492	+	0.864000i	$0.667952\pi$
$228$	0	0
$229$	2069.39	0.597159	0.298579	−	0.954385i	$-0.403487\pi$
$229$	2069.39	0.597159	0.298579	+	0.954385i	$0.403487\pi$
$230$	0	0
$231$	0	0
$232$	−5475.07	−1.54938
$233$	2769.24	0.778622	0.389311	−	0.921106i	$-0.372713\pi$
$233$	2769.24	0.778622	0.389311	+	0.921106i	$0.372713\pi$
$234$	0	0
$235$	0	0
$236$	−332.038	−0.0915842
$237$	0	0
$238$	−6897.17	−1.87848
$239$	−2943.84	−0.796742	−0.398371	−	0.917224i	$-0.630424\pi$
$239$	−2943.84	−0.796742	−0.398371	+	0.917224i	$0.630424\pi$
$240$	0	0
$241$	4474.54	1.19598	0.597988	−	0.801505i	$-0.295967\pi$
$241$	4474.54	1.19598	0.597988	+	0.801505i	$0.295967\pi$
$242$	−2518.40	−0.668961
$243$	0	0
$244$	1242.83	0.326082
$245$	0	0
$246$	0	0
$247$	−339.995	−0.0875845
$248$	−5515.55	−1.41225
$249$	0	0
$250$	0	0
$251$	−2121.41	−0.533474	−0.266737	−	0.963769i	$-0.585945\pi$
$251$	−2121.41	−0.533474	−0.266737	+	0.963769i	$0.585945\pi$
$252$	0	0
$253$	−6827.92	−1.69671
$254$	−1592.24	−0.393330
$255$	0	0
$256$	−4186.97	−1.02221
$257$	−4548.24	−1.10393	−0.551967	−	0.833866i	$-0.686123\pi$
$257$	−4548.24	−1.10393	−0.551967	+	0.833866i	$0.686123\pi$
$258$	0	0
$259$	−12783.1	−3.06680
$260$	0	0
$261$	0	0
$262$	5094.42	1.20127
$263$	3499.84	0.820567	0.410284	−	0.911958i	$-0.365430\pi$
$263$	3499.84	0.820567	0.410284	+	0.911958i	$0.365430\pi$
$264$	0	0
$265$	0	0
$266$	−1390.55	−0.320527
$267$	0	0
$268$	305.630	0.0696616
$269$	2594.25	0.588008	0.294004	−	0.955804i	$-0.405012\pi$
$269$	2594.25	0.588008	0.294004	+	0.955804i	$0.405012\pi$
$270$	0	0
$271$	8518.20	1.90939	0.954693	−	0.297593i	$-0.0961838\pi$
$271$	8518.20	1.90939	0.954693	+	0.297593i	$0.0961838\pi$
$272$	2541.89	0.566634
$273$	0	0
$274$	2135.63	0.470870
$275$	0	0
$276$	0	0
$277$	1890.50	0.410068	0.205034	−	0.978755i	$-0.434269\pi$
$277$	1890.50	0.410068	0.205034	+	0.978755i	$0.434269\pi$
$278$	−5109.12	−1.10225
$279$	0	0
$280$	0	0
$281$	3138.68	0.666328	0.333164	−	0.942869i	$-0.391884\pi$
$281$	3138.68	0.666328	0.333164	+	0.942869i	$0.391884\pi$
$282$	0	0
$283$	−4069.09	−0.854708	−0.427354	−	0.904084i	$-0.640554\pi$
$283$	−4069.09	−0.854708	−0.427354	+	0.904084i	$0.640554\pi$
$284$	−1436.01	−0.300041
$285$	0	0
$286$	−1701.81	−0.351854
$287$	−2337.54	−0.480768
$288$	0	0
$289$	6252.77	1.27270
$290$	0	0
$291$	0	0
$292$	−1152.22	−0.230919
$293$	−3637.95	−0.725362	−0.362681	−	0.931913i	$-0.618139\pi$
$293$	−3637.95	−0.725362	−0.362681	+	0.931913i	$0.618139\pi$
$294$	0	0
$295$	0	0
$296$	10162.7	1.99560
$297$	0	0
$298$	−1083.57	−0.210637
$299$	−2172.64	−0.420224
$300$	0	0
$301$	974.334	0.186577
$302$	−3068.62	−0.584700
$303$	0	0
$304$	512.474	0.0966856
$305$	0	0
$306$	0	0
$307$	6829.07	1.26956	0.634781	−	0.772692i	$-0.281090\pi$
$307$	6829.07	1.26956	0.634781	+	0.772692i	$0.281090\pi$
$308$	5357.75	0.991189
$309$	0	0
$310$	0	0
$311$	6601.03	1.20357	0.601785	−	0.798659i	$-0.294457\pi$
$311$	6601.03	1.20357	0.601785	+	0.798659i	$0.294457\pi$
$312$	0	0
$313$	−2766.59	−0.499607	−0.249803	−	0.968297i	$-0.580366\pi$
$313$	−2766.59	−0.499607	−0.249803	+	0.968297i	$0.580366\pi$
$314$	−4583.15	−0.823701
$315$	0	0
$316$	863.373	0.153698
$317$	−4564.41	−0.808716	−0.404358	−	0.914601i	$-0.632505\pi$
$317$	−4564.41	−0.808716	−0.404358	+	0.914601i	$0.632505\pi$
$318$	0	0
$319$	−11250.9	−1.97470
$320$	0	0
$321$	0	0
$322$	−8885.92	−1.53787
$323$	2251.15	0.387794
$324$	0	0
$325$	0	0
$326$	−1126.65	−0.191409
$327$	0	0
$328$	1858.38	0.312841
$329$	1866.64	0.312800
$330$	0	0
$331$	−4665.62	−0.774760	−0.387380	−	0.921920i	$-0.626620\pi$
$331$	−4665.62	−0.774760	−0.387380	+	0.921920i	$0.626620\pi$
$332$	−1922.51	−0.317805
$333$	0	0
$334$	−6334.30	−1.03772
$335$	0	0
$336$	0	0
$337$	3807.06	0.615382	0.307691	−	0.951486i	$-0.400444\pi$
$337$	3807.06	0.615382	0.307691	+	0.951486i	$0.400444\pi$
$338$	4129.57	0.664553
$339$	0	0
$340$	0	0
$341$	−11334.1	−1.79993
$342$	0	0
$343$	−7874.33	−1.23957
$344$	−774.612	−0.121408
$345$	0	0
$346$	3784.23	0.587981
$347$	−2311.26	−0.357565	−0.178783	−	0.983889i	$-0.557216\pi$
$347$	−2311.26	−0.357565	−0.178783	+	0.983889i	$0.557216\pi$
$348$	0	0
$349$	−10873.2	−1.66771	−0.833854	−	0.551984i	$-0.813871\pi$
$349$	−10873.2	−1.66771	−0.833854	+	0.551984i	$0.813871\pi$
$350$	0	0
$351$	0	0
$352$	−7227.89	−1.09445
$353$	2893.74	0.436312	0.218156	−	0.975914i	$-0.429996\pi$
$353$	2893.74	0.436312	0.218156	+	0.975914i	$0.429996\pi$
$354$	0	0
$355$	0	0
$356$	1014.23	0.150995
$357$	0	0
$358$	6030.23	0.890245
$359$	−9875.47	−1.45183	−0.725915	−	0.687784i	$-0.758584\pi$
$359$	−9875.47	−1.45183	−0.725915	+	0.687784i	$0.758584\pi$
$360$	0	0
$361$	−6405.14	−0.933830
$362$	−1726.17	−0.250624
$363$	0	0
$364$	1704.83	0.245488
$365$	0	0
$366$	0	0
$367$	−10722.2	−1.52505	−0.762523	−	0.646961i	$-0.776040\pi$
$367$	−10722.2	−1.52505	−0.762523	+	0.646961i	$0.776040\pi$
$368$	3274.82	0.463891
$369$	0	0
$370$	0	0
$371$	−14322.5	−2.00428
$372$	0	0
$373$	−4175.90	−0.579678	−0.289839	−	0.957075i	$-0.593602\pi$
$373$	−4175.90	−0.579678	−0.289839	+	0.957075i	$0.593602\pi$
$374$	11267.9	1.55789
$375$	0	0
$376$	−1484.01	−0.203543
$377$	−3580.03	−0.489074
$378$	0	0
$379$	1715.14	0.232457	0.116228	−	0.993223i	$-0.462920\pi$
$379$	1715.14	0.232457	0.116228	+	0.993223i	$0.462920\pi$
$380$	0	0
$381$	0	0
$382$	−2441.77	−0.327046
$383$	−5139.06	−0.685624	−0.342812	−	0.939404i	$-0.611379\pi$
$383$	−5139.06	−0.685624	−0.342812	+	0.939404i	$0.611379\pi$
$384$	0	0
$385$	0	0
$386$	−4573.14	−0.603022
$387$	0	0
$388$	691.074	0.0904226
$389$	8996.53	1.17260	0.586301	−	0.810093i	$-0.300584\pi$
$389$	8996.53	1.17260	0.586301	+	0.810093i	$0.300584\pi$
$390$	0	0
$391$	14385.3	1.86061
$392$	14631.8	1.88525
$393$	0	0
$394$	−10752.9	−1.37493
$395$	0	0
$396$	0	0
$397$	105.823	0.0133781	0.00668905	−	0.999978i	$-0.497871\pi$
$397$	105.823	0.0133781	0.00668905	+	0.999978i	$0.497871\pi$
$398$	7556.48	0.951688
$399$	0	0
$400$	0	0
$401$	14481.8	1.80346	0.901730	−	0.432300i	$-0.142298\pi$
$401$	14481.8	1.80346	0.901730	+	0.432300i	$0.142298\pi$
$402$	0	0
$403$	−3606.50	−0.445788
$404$	−2840.18	−0.349763
$405$	0	0
$406$	−14642.0	−1.78983
$407$	20883.8	2.54341
$408$	0	0
$409$	−861.065	−0.104100	−0.0520500	−	0.998644i	$-0.516576\pi$
$409$	−861.065	−0.104100	−0.0520500	+	0.998644i	$0.516576\pi$
$410$	0	0
$411$	0	0
$412$	−4879.72	−0.583511
$413$	−2929.52	−0.349037
$414$	0	0
$415$	0	0
$416$	−2299.91	−0.271064
$417$	0	0
$418$	2271.75	0.265825
$419$	−13017.7	−1.51780	−0.758900	−	0.651207i	$-0.774263\pi$
$419$	−13017.7	−1.51780	−0.758900	+	0.651207i	$0.774263\pi$
$420$	0	0
$421$	14546.6	1.68398	0.841991	−	0.539492i	$-0.181384\pi$
$421$	14546.6	1.68398	0.841991	+	0.539492i	$0.181384\pi$
$422$	229.428	0.0264654
$423$	0	0
$424$	11386.7	1.30421
$425$	0	0
$426$	0	0
$427$	10965.3	1.24273
$428$	−3406.53	−0.384722
$429$	0	0
$430$	0	0
$431$	−3539.94	−0.395622	−0.197811	−	0.980240i	$-0.563383\pi$
$431$	−3539.94	−0.395622	−0.197811	+	0.980240i	$0.563383\pi$
$432$	0	0
$433$	−669.471	−0.0743019	−0.0371509	−	0.999310i	$-0.511828\pi$
$433$	−669.471	−0.0743019	−0.0371509	+	0.999310i	$0.511828\pi$
$434$	−14750.3	−1.63142
$435$	0	0
$436$	−7387.09	−0.811416
$437$	2900.26	0.317478
$438$	0	0
$439$	12568.5	1.36643	0.683216	−	0.730216i	$-0.260581\pi$
$439$	12568.5	1.36643	0.683216	+	0.730216i	$0.260581\pi$
$440$	0	0
$441$	0	0
$442$	3585.45	0.385843
$443$	−11060.3	−1.18621	−0.593106	−	0.805124i	$-0.702099\pi$
$443$	−11060.3	−1.18621	−0.593106	+	0.805124i	$0.702099\pi$
$444$	0	0
$445$	0	0
$446$	−4128.98	−0.438370
$447$	0	0
$448$	−15314.5	−1.61504
$449$	18553.9	1.95014	0.975072	−	0.221891i	$-0.0712228\pi$
$449$	18553.9	1.95014	0.975072	+	0.221891i	$0.0712228\pi$
$450$	0	0
$451$	3818.84	0.398719
$452$	6245.21	0.649889
$453$	0	0
$454$	7322.33	0.756947
$455$	0	0
$456$	0	0
$457$	6802.26	0.696272	0.348136	−	0.937444i	$-0.386815\pi$
$457$	6802.26	0.696272	0.348136	+	0.937444i	$0.386815\pi$
$458$	−4399.78	−0.448882
$459$	0	0
$460$	0	0
$461$	14894.4	1.50477	0.752386	−	0.658722i	$-0.228903\pi$
$461$	14894.4	1.50477	0.752386	+	0.658722i	$0.228903\pi$
$462$	0	0
$463$	14288.7	1.43423	0.717117	−	0.696952i	$-0.245461\pi$
$463$	14288.7	1.43423	0.717117	+	0.696952i	$0.245461\pi$
$464$	5396.17	0.539895
$465$	0	0
$466$	−5887.74	−0.585288
$467$	−13115.1	−1.29956	−0.649780	−	0.760122i	$-0.725139\pi$
$467$	−13115.1	−1.29956	−0.649780	+	0.760122i	$0.725139\pi$
$468$	0	0
$469$	2696.52	0.265488
$470$	0	0
$471$	0	0
$472$	2329.01	0.227122
$473$	−1591.77	−0.154735
$474$	0	0
$475$	0	0
$476$	−11287.9	−1.08694
$477$	0	0
$478$	6258.96	0.598908
$479$	7919.79	0.755458	0.377729	−	0.925916i	$-0.376705\pi$
$479$	7919.79	0.755458	0.377729	+	0.925916i	$0.376705\pi$
$480$	0	0
$481$	6645.20	0.629927
$482$	−9513.40	−0.899011
$483$	0	0
$484$	−4121.62	−0.387079
$485$	0	0
$486$	0	0
$487$	17003.1	1.58210	0.791051	−	0.611751i	$-0.209534\pi$
$487$	17003.1	1.58210	0.791051	+	0.611751i	$0.209534\pi$
$488$	−8717.56	−0.808659
$489$	0	0
$490$	0	0
$491$	6391.74	0.587485	0.293743	−	0.955885i	$-0.405099\pi$
$491$	6391.74	0.587485	0.293743	+	0.955885i	$0.405099\pi$
$492$	0	0
$493$	23703.8	2.16545
$494$	722.870	0.0658370
$495$	0	0
$496$	5436.07	0.492111
$497$	−12669.7	−1.14349
$498$	0	0
$499$	−2674.12	−0.239900	−0.119950	−	0.992780i	$-0.538273\pi$
$499$	−2674.12	−0.239900	−0.119950	+	0.992780i	$0.538273\pi$
$500$	0	0
$501$	0	0
$502$	4510.36	0.401011
$503$	−1263.13	−0.111969	−0.0559843	−	0.998432i	$-0.517830\pi$
$503$	−1263.13	−0.111969	−0.0559843	+	0.998432i	$0.517830\pi$
$504$	0	0
$505$	0	0
$506$	14517.0	1.27541
$507$	0	0
$508$	−2605.86	−0.227592
$509$	1380.84	0.120245	0.0601226	−	0.998191i	$-0.480851\pi$
$509$	1380.84	0.120245	0.0601226	+	0.998191i	$0.480851\pi$
$510$	0	0
$511$	−10165.8	−0.880055
$512$	8163.62	0.704657
$513$	0	0
$514$	9670.09	0.829824
$515$	0	0
$516$	0	0
$517$	−3049.54	−0.259417
$518$	27178.3	2.30530
$519$	0	0
$520$	0	0
$521$	2689.80	0.226185	0.113092	−	0.993584i	$-0.463924\pi$
$521$	2689.80	0.226185	0.113092	+	0.993584i	$0.463924\pi$
$522$	0	0
$523$	−7144.18	−0.597310	−0.298655	−	0.954361i	$-0.596538\pi$
$523$	−7144.18	−0.597310	−0.298655	+	0.954361i	$0.596538\pi$
$524$	8337.54	0.695090
$525$	0	0
$526$	−7441.07	−0.616818
$527$	23879.1	1.97379
$528$	0	0
$529$	6366.25	0.523239
$530$	0	0
$531$	0	0
$532$	−2275.78	−0.185466
$533$	1215.15	0.0987508
$534$	0	0
$535$	0	0
$536$	−2143.78	−0.172756
$537$	0	0
$538$	−5515.68	−0.442004
$539$	30067.4	2.40278
$540$	0	0
$541$	−3310.57	−0.263091	−0.131546	−	0.991310i	$-0.541994\pi$
$541$	−3310.57	−0.263091	−0.131546	+	0.991310i	$0.541994\pi$
$542$	−18110.7	−1.43528
$543$	0	0
$544$	15228.0	1.20018
$545$	0	0
$546$	0	0
$547$	−2286.52	−0.178729	−0.0893643	−	0.995999i	$-0.528484\pi$
$547$	−2286.52	−0.178729	−0.0893643	+	0.995999i	$0.528484\pi$
$548$	3495.19	0.272458
$549$	0	0
$550$	0	0
$551$	4778.98	0.369494
$552$	0	0
$553$	7617.39	0.585758
$554$	−4019.42	−0.308247
$555$	0	0
$556$	−8361.60	−0.637789
$557$	13846.6	1.05332	0.526658	−	0.850077i	$-0.323445\pi$
$557$	13846.6	1.05332	0.526658	+	0.850077i	$0.323445\pi$
$558$	0	0
$559$	−506.502	−0.0383233
$560$	0	0
$561$	0	0
$562$	−6673.21	−0.500876
$563$	−16164.4	−1.21003	−0.605017	−	0.796213i	$-0.706834\pi$
$563$	−16164.4	−1.21003	−0.605017	+	0.796213i	$0.706834\pi$
$564$	0	0
$565$	0	0
$566$	8651.37	0.642481
$567$	0	0
$568$	10072.6	0.744080
$569$	−496.334	−0.0365684	−0.0182842	−	0.999833i	$-0.505820\pi$
$569$	−496.334	−0.0365684	−0.0182842	+	0.999833i	$0.505820\pi$
$570$	0	0
$571$	5971.18	0.437629	0.218815	−	0.975766i	$-0.429781\pi$
$571$	5971.18	0.437629	0.218815	+	0.975766i	$0.429781\pi$
$572$	−2785.20	−0.203592
$573$	0	0
$574$	4969.88	0.361392
$575$	0	0
$576$	0	0
$577$	−9571.82	−0.690607	−0.345304	−	0.938491i	$-0.612224\pi$
$577$	−9571.82	−0.690607	−0.345304	+	0.938491i	$0.612224\pi$
$578$	−13294.1	−0.956684
$579$	0	0
$580$	0	0
$581$	−16961.9	−1.21119
$582$	0	0
$583$	23398.8	1.66223
$584$	8081.97	0.572662
$585$	0	0
$586$	7734.71	0.545253
$587$	5322.18	0.374225	0.187112	−	0.982339i	$-0.440087\pi$
$587$	5322.18	0.374225	0.187112	+	0.982339i	$0.440087\pi$
$588$	0	0
$589$	4814.31	0.336791
$590$	0	0
$591$	0	0
$592$	−10016.3	−0.695385
$593$	−11066.5	−0.766354	−0.383177	−	0.923675i	$-0.625170\pi$
$593$	−11066.5	−0.766354	−0.383177	+	0.923675i	$0.625170\pi$
$594$	0	0
$595$	0	0
$596$	−1773.38	−0.121880
$597$	0	0
$598$	4619.29	0.315881
$599$	−18385.0	−1.25407	−0.627037	−	0.778989i	$-0.715733\pi$
$599$	−18385.0	−1.25407	−0.627037	+	0.778989i	$0.715733\pi$
$600$	0	0
$601$	−617.830	−0.0419331	−0.0209666	−	0.999780i	$-0.506674\pi$
$601$	−617.830	−0.0419331	−0.0209666	+	0.999780i	$0.506674\pi$
$602$	−2071.55	−0.140249
$603$	0	0
$604$	−5022.12	−0.338323
$605$	0	0
$606$	0	0
$607$	−13839.8	−0.925436	−0.462718	−	0.886506i	$-0.653126\pi$
$607$	−13839.8	−0.925436	−0.462718	+	0.886506i	$0.653126\pi$
$608$	3070.15	0.204788
$609$	0	0
$610$	0	0
$611$	−970.362	−0.0642498
$612$	0	0
$613$	25655.4	1.69039	0.845196	−	0.534457i	$-0.179484\pi$
$613$	25655.4	1.69039	0.845196	+	0.534457i	$0.179484\pi$
$614$	−14519.4	−0.954326
$615$	0	0
$616$	−37580.8	−2.45808
$617$	4218.26	0.275236	0.137618	−	0.990485i	$-0.456055\pi$
$617$	4218.26	0.275236	0.137618	+	0.990485i	$0.456055\pi$
$618$	0	0
$619$	12182.6	0.791047	0.395524	−	0.918456i	$-0.370563\pi$
$619$	12182.6	0.791047	0.395524	+	0.918456i	$0.370563\pi$
$620$	0	0
$621$	0	0
$622$	−14034.6	−0.904719
$623$	8948.40	0.575458
$624$	0	0
$625$	0	0
$626$	5882.10	0.375553
$627$	0	0
$628$	−7500.80	−0.476616
$629$	−43998.8	−2.78910
$630$	0	0
$631$	16673.2	1.05190	0.525951	−	0.850515i	$-0.323709\pi$
$631$	16673.2	1.05190	0.525951	+	0.850515i	$0.323709\pi$
$632$	−6055.95	−0.381159
$633$	0	0
$634$	9704.48	0.607909
$635$	0	0
$636$	0	0
$637$	9567.44	0.595096
$638$	23920.7	1.48437
$639$	0	0
$640$	0	0
$641$	−26270.8	−1.61878	−0.809388	−	0.587275i	$-0.800201\pi$
$641$	−26270.8	−1.61878	−0.809388	+	0.587275i	$0.800201\pi$
$642$	0	0
$643$	−11556.3	−0.708767	−0.354384	−	0.935100i	$-0.615309\pi$
$643$	−11556.3	−0.708767	−0.354384	+	0.935100i	$0.615309\pi$
$644$	−14542.7	−0.889851
$645$	0	0
$646$	−4786.22	−0.291504
$647$	−25395.2	−1.54310	−0.771552	−	0.636166i	$-0.780519\pi$
$647$	−25395.2	−1.54310	−0.771552	+	0.636166i	$0.780519\pi$
$648$	0	0
$649$	4785.96	0.289469
$650$	0	0
$651$	0	0
$652$	−1843.88	−0.110754
$653$	−6182.84	−0.370526	−0.185263	−	0.982689i	$-0.559314\pi$
$653$	−6182.84	−0.370526	−0.185263	+	0.982689i	$0.559314\pi$
$654$	0	0
$655$	0	0
$656$	−1831.60	−0.109012
$657$	0	0
$658$	−3968.70	−0.235131
$659$	8885.32	0.525224	0.262612	−	0.964901i	$-0.415416\pi$
$659$	8885.32	0.525224	0.262612	+	0.964901i	$0.415416\pi$
$660$	0	0
$661$	−7171.60	−0.422001	−0.211001	−	0.977486i	$-0.567672\pi$
$661$	−7171.60	−0.422001	−0.211001	+	0.977486i	$0.567672\pi$
$662$	9919.66	0.582385
$663$	0	0
$664$	13485.0	0.788133
$665$	0	0
$666$	0	0
$667$	30538.7	1.77281
$668$	−10366.7	−0.600451
$669$	0	0
$670$	0	0
$671$	−17914.0	−1.03064
$672$	0	0
$673$	4250.52	0.243455	0.121728	−	0.992564i	$-0.461157\pi$
$673$	4250.52	0.243455	0.121728	+	0.992564i	$0.461157\pi$
$674$	−8094.27	−0.462581
$675$	0	0
$676$	6758.47	0.384528
$677$	2233.65	0.126804	0.0634019	−	0.997988i	$-0.479805\pi$
$677$	2233.65	0.126804	0.0634019	+	0.997988i	$0.479805\pi$
$678$	0	0
$679$	6097.23	0.344610
$680$	0	0
$681$	0	0
$682$	24097.6	1.35300
$683$	−6071.62	−0.340153	−0.170076	−	0.985431i	$-0.554401\pi$
$683$	−6071.62	−0.340153	−0.170076	+	0.985431i	$0.554401\pi$
$684$	0	0
$685$	0	0
$686$	16741.8	0.931784
$687$	0	0
$688$	763.450	0.0423056
$689$	7445.49	0.411684
$690$	0	0
$691$	−765.784	−0.0421589	−0.0210794	−	0.999778i	$-0.506710\pi$
$691$	−765.784	−0.0421589	−0.0210794	+	0.999778i	$0.506710\pi$
$692$	6193.29	0.340222
$693$	0	0
$694$	4914.02	0.268780
$695$	0	0
$696$	0	0
$697$	−8045.70	−0.437235
$698$	23117.8	1.25361
$699$	0	0
$700$	0	0
$701$	−23564.3	−1.26963	−0.634814	−	0.772665i	$-0.718923\pi$
$701$	−23564.3	−1.26963	−0.634814	+	0.772665i	$0.718923\pi$
$702$	0	0
$703$	−8870.67	−0.475909
$704$	25019.3	1.33942
$705$	0	0
$706$	−6152.43	−0.327974
$707$	−25058.4	−1.33298
$708$	0	0
$709$	24172.3	1.28041	0.640204	−	0.768205i	$-0.278850\pi$
$709$	24172.3	1.28041	0.640204	+	0.768205i	$0.278850\pi$
$710$	0	0
$711$	0	0
$712$	−7114.13	−0.374457
$713$	30764.5	1.61590
$714$	0	0
$715$	0	0
$716$	9869.11	0.515120
$717$	0	0
$718$	20996.4	1.09134
$719$	−12829.0	−0.665424	−0.332712	−	0.943028i	$-0.607964\pi$
$719$	−12829.0	−0.665424	−0.332712	+	0.943028i	$0.607964\pi$
$720$	0	0
$721$	−43052.9	−2.22382
$722$	13618.1	0.701957
$723$	0	0
$724$	−2825.06	−0.145018
$725$	0	0
$726$	0	0
$727$	24724.7	1.26133	0.630666	−	0.776054i	$-0.282782\pi$
$727$	24724.7	1.26133	0.630666	+	0.776054i	$0.282782\pi$
$728$	−11958.2	−0.608792
$729$	0	0
$730$	0	0
$731$	3353.62	0.169683
$732$	0	0
$733$	20172.2	1.01648	0.508239	−	0.861216i	$-0.330297\pi$
$733$	20172.2	1.01648	0.508239	+	0.861216i	$0.330297\pi$
$734$	22796.6	1.14637
$735$	0	0
$736$	19618.9	0.982559
$737$	−4405.32	−0.220179
$738$	0	0
$739$	−16452.8	−0.818979	−0.409489	−	0.912315i	$-0.634293\pi$
$739$	−16452.8	−0.818979	−0.409489	+	0.912315i	$0.634293\pi$
$740$	0	0
$741$	0	0
$742$	30451.4	1.50661
$743$	20864.6	1.03021	0.515107	−	0.857126i	$-0.327752\pi$
$743$	20864.6	1.03021	0.515107	+	0.857126i	$0.327752\pi$
$744$	0	0
$745$	0	0
$746$	8878.47	0.435742
$747$	0	0
$748$	18441.2	0.901438
$749$	−30055.2	−1.46621
$750$	0	0
$751$	15525.7	0.754381	0.377191	−	0.926136i	$-0.376890\pi$
$751$	15525.7	0.754381	0.377191	+	0.926136i	$0.376890\pi$
$752$	1462.63	0.0709262
$753$	0	0
$754$	7611.56	0.367635
$755$	0	0
$756$	0	0
$757$	−30105.7	−1.44546	−0.722729	−	0.691132i	$-0.757112\pi$
$757$	−30105.7	−1.44546	−0.722729	+	0.691132i	$0.757112\pi$
$758$	−3646.60	−0.174737
$759$	0	0
$760$	0	0
$761$	21739.9	1.03557	0.517786	−	0.855510i	$-0.326756\pi$
$761$	21739.9	1.03557	0.517786	+	0.855510i	$0.326756\pi$
$762$	0	0
$763$	−65175.0	−3.09239
$764$	−3996.21	−0.189238
$765$	0	0
$766$	10926.3	0.515381
$767$	1522.89	0.0716929
$768$	0	0
$769$	1942.22	0.0910772	0.0455386	−	0.998963i	$-0.485500\pi$
$769$	1942.22	0.0910772	0.0455386	+	0.998963i	$0.485500\pi$
$770$	0	0
$771$	0	0
$772$	−7484.42	−0.348925
$773$	7921.02	0.368563	0.184282	−	0.982873i	$-0.441004\pi$
$773$	7921.02	0.368563	0.184282	+	0.982873i	$0.441004\pi$
$774$	0	0
$775$	0	0
$776$	−4847.40	−0.224241
$777$	0	0
$778$	−19127.7	−0.881441
$779$	−1622.11	−0.0746060
$780$	0	0
$781$	20698.5	0.948338
$782$	−30585.0	−1.39861
$783$	0	0
$784$	−14421.0	−0.656933
$785$	0	0
$786$	0	0
$787$	1361.40	0.0616630	0.0308315	−	0.999525i	$-0.490184\pi$
$787$	1361.40	0.0616630	0.0308315	+	0.999525i	$0.490184\pi$
$788$	−17598.2	−0.795571
$789$	0	0
$790$	0	0
$791$	55100.4	2.47679
$792$	0	0
$793$	−5700.22	−0.255260
$794$	−224.992	−0.0100563
$795$	0	0
$796$	12367.0	0.550672
$797$	−27917.5	−1.24076	−0.620381	−	0.784300i	$-0.713022\pi$
$797$	−27917.5	−1.24076	−0.620381	+	0.784300i	$0.713022\pi$
$798$	0	0
$799$	6424.90	0.284476
$800$	0	0
$801$	0	0
$802$	−30790.1	−1.35565
$803$	16607.9	0.729863
$804$	0	0
$805$	0	0
$806$	7667.84	0.335097
$807$	0	0
$808$	19921.9	0.867388
$809$	5275.87	0.229283	0.114641	−	0.993407i	$-0.463428\pi$
$809$	5275.87	0.229283	0.114641	+	0.993407i	$0.463428\pi$
$810$	0	0
$811$	−23769.1	−1.02916	−0.514579	−	0.857443i	$-0.672052\pi$
$811$	−23769.1	−1.02916	−0.514579	+	0.857443i	$0.672052\pi$
$812$	−23963.2	−1.03564
$813$	0	0
$814$	−44401.4	−1.91188
$815$	0	0
$816$	0	0
$817$	676.129	0.0289532
$818$	1830.73	0.0782516
$819$	0	0
$820$	0	0
$821$	−34209.3	−1.45422	−0.727108	−	0.686523i	$-0.759136\pi$
$821$	−34209.3	−1.45422	−0.727108	+	0.686523i	$0.759136\pi$
$822$	0	0
$823$	−1240.13	−0.0525252	−0.0262626	−	0.999655i	$-0.508361\pi$
$823$	−1240.13	−0.0525252	−0.0262626	+	0.999655i	$0.508361\pi$
$824$	34227.8	1.44706
$825$	0	0
$826$	6228.50	0.262370
$827$	26971.0	1.13407	0.567034	−	0.823694i	$-0.308091\pi$
$827$	26971.0	1.13407	0.567034	+	0.823694i	$0.308091\pi$
$828$	0	0
$829$	−26743.1	−1.12042	−0.560208	−	0.828352i	$-0.689279\pi$
$829$	−26743.1	−1.12042	−0.560208	+	0.828352i	$0.689279\pi$
$830$	0	0
$831$	0	0
$832$	7961.13	0.331734
$833$	−63347.3	−2.63488
$834$	0	0
$835$	0	0
$836$	3717.96	0.153814
$837$	0	0
$838$	27677.3	1.14093
$839$	−5273.14	−0.216983	−0.108492	−	0.994097i	$-0.534602\pi$
$839$	−5273.14	−0.216983	−0.108492	+	0.994097i	$0.534602\pi$
$840$	0	0
$841$	25931.9	1.06326
$842$	−30927.7	−1.26584
$843$	0	0
$844$	375.483	0.0153136
$845$	0	0
$846$	0	0
$847$	−36364.3	−1.47520
$848$	−11222.6	−0.454464
$849$	0	0
$850$	0	0
$851$	−56685.5	−2.28338
$852$	0	0
$853$	−5043.46	−0.202444	−0.101222	−	0.994864i	$-0.532275\pi$
$853$	−5043.46	−0.202444	−0.101222	+	0.994864i	$0.532275\pi$
$854$	−23313.5	−0.934157
$855$	0	0
$856$	23894.4	0.954081
$857$	14478.5	0.577103	0.288552	−	0.957464i	$-0.406826\pi$
$857$	14478.5	0.577103	0.288552	+	0.957464i	$0.406826\pi$
$858$	0	0
$859$	−7873.88	−0.312751	−0.156376	−	0.987698i	$-0.549981\pi$
$859$	−7873.88	−0.312751	−0.156376	+	0.987698i	$0.549981\pi$
$860$	0	0
$861$	0	0
$862$	7526.35	0.297388
$863$	−24166.0	−0.953209	−0.476604	−	0.879118i	$-0.658133\pi$
$863$	−24166.0	−0.953209	−0.476604	+	0.879118i	$0.658133\pi$
$864$	0	0
$865$	0	0
$866$	1423.38	0.0558525
$867$	0	0
$868$	−24140.4	−0.943983
$869$	−12444.6	−0.485792
$870$	0	0
$871$	−1401.77	−0.0545317
$872$	51815.2	2.01225
$873$	0	0
$874$	−6166.29	−0.238647
$875$	0	0
$876$	0	0
$877$	−6411.82	−0.246878	−0.123439	−	0.992352i	$-0.539392\pi$
$877$	−6411.82	−0.246878	−0.123439	+	0.992352i	$0.539392\pi$
$878$	−26722.2	−1.02714
$879$	0	0
$880$	0	0
$881$	22908.5	0.876059	0.438029	−	0.898961i	$-0.355677\pi$
$881$	22908.5	0.876059	0.438029	+	0.898961i	$0.355677\pi$
$882$	0	0
$883$	−39198.4	−1.49392	−0.746960	−	0.664869i	$-0.768487\pi$
$883$	−39198.4	−1.49392	−0.746960	+	0.664869i	$0.768487\pi$
$884$	5867.97	0.223259
$885$	0	0
$886$	23515.6	0.891672
$887$	−5422.19	−0.205253	−0.102626	−	0.994720i	$-0.532725\pi$
$887$	−5422.19	−0.205253	−0.102626	+	0.994720i	$0.532725\pi$
$888$	0	0
$889$	−22991.1	−0.867375
$890$	0	0
$891$	0	0
$892$	−6757.51	−0.253653
$893$	1295.34	0.0485406
$894$	0	0
$895$	0	0
$896$	−2833.53	−0.105649
$897$	0	0
$898$	−39447.9	−1.46592
$899$	50693.0	1.88065
$900$	0	0
$901$	−49297.6	−1.82280
$902$	−8119.32	−0.299716
$903$	0	0
$904$	−43805.7	−1.61168
$905$	0	0
$906$	0	0
$907$	30727.0	1.12489	0.562444	−	0.826836i	$-0.309861\pi$
$907$	30727.0	1.12489	0.562444	+	0.826836i	$0.309861\pi$
$908$	11983.8	0.437990
$909$	0	0
$910$	0	0
$911$	4101.34	0.149159	0.0745793	−	0.997215i	$-0.476239\pi$
$911$	4101.34	0.149159	0.0745793	+	0.997215i	$0.476239\pi$
$912$	0	0
$913$	27710.8	1.00448
$914$	−14462.4	−0.523385
$915$	0	0
$916$	−7200.69	−0.259735
$917$	73560.7	2.64906
$918$	0	0
$919$	−50926.3	−1.82797	−0.913984	−	0.405750i	$-0.867010\pi$
$919$	−50926.3	−1.82797	−0.913984	+	0.405750i	$0.867010\pi$
$920$	0	0
$921$	0	0
$922$	−31667.2	−1.13113
$923$	6586.26	0.234875
$924$	0	0
$925$	0	0
$926$	−30379.4	−1.07811
$927$	0	0
$928$	32327.6	1.14354
$929$	22040.8	0.778402	0.389201	−	0.921153i	$-0.372751\pi$
$929$	22040.8	0.778402	0.389201	+	0.921153i	$0.372751\pi$
$930$	0	0
$931$	−12771.6	−0.449593
$932$	−9635.89	−0.338663
$933$	0	0
$934$	27884.3	0.976875
$935$	0	0
$936$	0	0
$937$	−28111.2	−0.980097	−0.490049	−	0.871695i	$-0.663021\pi$
$937$	−28111.2	−0.980097	−0.490049	+	0.871695i	$0.663021\pi$
$938$	−5733.12	−0.199566
$939$	0	0
$940$	0	0
$941$	52811.3	1.82954	0.914770	−	0.403974i	$-0.132371\pi$
$941$	52811.3	1.82954	0.914770	+	0.403974i	$0.132371\pi$
$942$	0	0
$943$	−10365.6	−0.357954
$944$	−2295.45	−0.0791427
$945$	0	0
$946$	3384.30	0.116314
$947$	1826.23	0.0626657	0.0313329	−	0.999509i	$-0.490025\pi$
$947$	1826.23	0.0626657	0.0313329	+	0.999509i	$0.490025\pi$
$948$	0	0
$949$	5284.63	0.180765
$950$	0	0
$951$	0	0
$952$	79176.9	2.69552
$953$	41164.2	1.39920	0.699601	−	0.714534i	$-0.253361\pi$
$953$	41164.2	1.39920	0.699601	+	0.714534i	$0.253361\pi$
$954$	0	0
$955$	0	0
$956$	10243.4	0.346545
$957$	0	0
$958$	−16838.4	−0.567875
$959$	30837.4	1.03837
$960$	0	0
$961$	21276.8	0.714202
$962$	−14128.5	−0.473514
$963$	0	0
$964$	−15569.7	−0.520192
$965$	0	0
$966$	0	0
$967$	7088.78	0.235739	0.117870	−	0.993029i	$-0.462394\pi$
$967$	7088.78	0.235739	0.117870	+	0.993029i	$0.462394\pi$
$968$	28910.2	0.959927
$969$	0	0
$970$	0	0
$971$	2355.90	0.0778625	0.0389312	−	0.999242i	$-0.487605\pi$
$971$	2355.90	0.0778625	0.0389312	+	0.999242i	$0.487605\pi$
$972$	0	0
$973$	−73773.0	−2.43068
$974$	−36150.6	−1.18926
$975$	0	0
$976$	8591.95	0.281784
$977$	16795.1	0.549972	0.274986	−	0.961448i	$-0.411327\pi$
$977$	16795.1	0.549972	0.274986	+	0.961448i	$0.411327\pi$
$978$	0	0
$979$	−14619.0	−0.477249
$980$	0	0
$981$	0	0
$982$	−13589.6	−0.441611
$983$	14733.8	0.478063	0.239032	−	0.971012i	$-0.423170\pi$
$983$	14733.8	0.478063	0.239032	+	0.971012i	$0.423170\pi$
$984$	0	0
$985$	0	0
$986$	−50397.2	−1.62776
$987$	0	0
$988$	1183.05	0.0380950
$989$	4320.61	0.138915
$990$	0	0
$991$	36809.2	1.17990	0.589950	−	0.807440i	$-0.299147\pi$
$991$	36809.2	1.17990	0.589950	+	0.807440i	$0.299147\pi$
$992$	32566.6	1.04233
$993$	0	0
$994$	26937.3	0.859556
$995$	0	0
$996$	0	0
$997$	21210.0	0.673749	0.336875	−	0.941550i	$-0.390630\pi$
$997$	21210.0	0.673749	0.336875	+	0.941550i	$0.390630\pi$
$998$	5685.49	0.180332
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	675.4.a.p.1.2		3
3.2	odd	2		675.4.a.s.1.2		3
5.2	odd	4		675.4.b.n.649.3		6
5.3	odd	4		675.4.b.n.649.4		6
5.4	even	2		135.4.a.h.1.2	yes	3
15.2	even	4		675.4.b.m.649.4		6
15.8	even	4		675.4.b.m.649.3		6
15.14	odd	2		135.4.a.e.1.2	✓	3
20.19	odd	2		2160.4.a.bq.1.1		3
45.4	even	6		405.4.e.q.136.2		6
45.14	odd	6		405.4.e.v.136.2		6
45.29	odd	6		405.4.e.v.271.2		6
45.34	even	6		405.4.e.q.271.2		6
60.59	even	2		2160.4.a.bi.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
135.4.a.e.1.2	✓	3	15.14	odd	2
135.4.a.h.1.2	yes	3	5.4	even	2
405.4.e.q.136.2		6	45.4	even	6
405.4.e.q.271.2		6	45.34	even	6
405.4.e.v.136.2		6	45.14	odd	6
405.4.e.v.271.2		6	45.29	odd	6
675.4.a.p.1.2		3	1.1	even	1	trivial
675.4.a.s.1.2		3	3.2	odd	2
675.4.b.m.649.3		6	15.8	even	4
675.4.b.m.649.4		6	15.2	even	4
675.4.b.n.649.3		6	5.2	odd	4
675.4.b.n.649.4		6	5.3	odd	4
2160.4.a.bi.1.1		3	60.59	even	2
2160.4.a.bq.1.1		3	20.19	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 675 = 3^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	675.a (trivial)

\(f(q)\)	\(=\)	\(q-2.12612 q^{2} -3.47962 q^{4} -30.7000 q^{7} +24.4070 q^{8} +50.1548 q^{11} +15.9592 q^{13} +65.2720 q^{14} -24.0553 q^{16} -105.668 q^{17} -21.3040 q^{19} -106.635 q^{22} -136.137 q^{23} -33.9312 q^{26} +106.824 q^{28} -224.323 q^{29} -225.982 q^{31} -144.112 q^{32} +224.663 q^{34} +416.386 q^{37} +45.2948 q^{38} +76.1411 q^{41} -31.7372 q^{43} -174.519 q^{44} +289.443 q^{46} -60.8026 q^{47} +599.493 q^{49} -55.5320 q^{52} +466.532 q^{53} -749.297 q^{56} +476.938 q^{58} +95.4239 q^{59} -357.174 q^{61} +480.464 q^{62} +498.842 q^{64} -87.8344 q^{67} +367.685 q^{68} +412.693 q^{71} +331.133 q^{73} -885.286 q^{74} +74.1296 q^{76} -1539.75 q^{77} -248.123 q^{79} -161.885 q^{82} +552.505 q^{83} +67.4771 q^{86} +1224.13 q^{88} -291.478 q^{89} -489.949 q^{91} +473.704 q^{92} +129.274 q^{94} -198.606 q^{97} -1274.59 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 5 q^{2} + 17 q^{4} + 4 q^{7} - 75 q^{8} + 5 q^{11} - 7 q^{13} + 60 q^{14} + 161 q^{16} - 155 q^{17} - 50 q^{19} + 229 q^{22} - 285 q^{23} + 185 q^{26} + 334 q^{28} + 115 q^{29} - 115 q^{31} - 775 q^{32}+ \cdots + 305 q^{98}+O(q^{100}) \) 3 * q - 5 * q^2 + 17 * q^4 + 4 * q^7 - 75 * q^8 + 5 * q^11 - 7 * q^13 + 60 * q^14 + 161 * q^16 - 155 * q^17 - 50 * q^19 + 229 * q^22 - 285 * q^23 + 185 * q^26 + 334 * q^28 + 115 * q^29 - 115 * q^31 - 775 * q^32 + 413 * q^34 + 384 * q^37 + 1150 * q^38 + 580 * q^41 + 797 * q^43 - 1415 * q^44 - 285 * q^46 + 145 * q^47 + 577 * q^49 - 825 * q^52 + 400 * q^53 - 2190 * q^56 + 59 * q^58 + 380 * q^59 - 152 * q^61 + 1005 * q^62 + 2937 * q^64 - 2 * q^67 - 475 * q^68 + 40 * q^71 + 980 * q^73 - 2720 * q^74 - 3276 * q^76 - 1950 * q^77 + 1013 * q^79 - 4 * q^82 - 270 * q^83 - 1555 * q^86 + 5193 * q^88 + 1020 * q^89 - 632 * q^91 + 1215 * q^92 + 3833 * q^94 - 720 * q^97 + 305 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more