LMFDB - Embedded newform 1050.6.g.o.799.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1050,6,Mod(799,1050)] 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("1050.799"); S:= CuspForms(chi, 6); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1050, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 6, names="a")

Level:	$ N $	$=$	$ 1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	1050.g (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,-32,0,72,0,0,-162,0,432] 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:	no
Analytic conductor:	$168.403010804$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 42)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			799.2
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	1050.799
Dual form			1050.6.g.o.799.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+4.00000i q^{2} -9.00000i q^{3} -16.0000 q^{4} +36.0000 q^{6} -49.0000i q^{7} -64.0000i q^{8} -81.0000 q^{9} +216.000 q^{11} +144.000i q^{12} +998.000i q^{13} +196.000 q^{14} +256.000 q^{16} -1302.00i q^{17} -324.000i q^{18} -884.000 q^{19} -441.000 q^{21} +864.000i q^{22} -2268.00i q^{23} -576.000 q^{24} -3992.00 q^{26} +729.000i q^{27} +784.000i q^{28} +1482.00 q^{29} +8360.00 q^{31} +1024.00i q^{32} -1944.00i q^{33} +5208.00 q^{34} +1296.00 q^{36} +4714.00i q^{37} -3536.00i q^{38} +8982.00 q^{39} -9786.00 q^{41} -1764.00i q^{42} +19436.0i q^{43} -3456.00 q^{44} +9072.00 q^{46} -22200.0i q^{47} -2304.00i q^{48} -2401.00 q^{49} -11718.0 q^{51} -15968.0i q^{52} +26790.0i q^{53} -2916.00 q^{54} -3136.00 q^{56} +7956.00i q^{57} +5928.00i q^{58} -28092.0 q^{59} -38866.0 q^{61} +33440.0i q^{62} +3969.00i q^{63} -4096.00 q^{64} +7776.00 q^{66} -23948.0i q^{67} +20832.0i q^{68} -20412.0 q^{69} -20628.0 q^{71} +5184.00i q^{72} +290.000i q^{73} -18856.0 q^{74} +14144.0 q^{76} -10584.0i q^{77} +35928.0i q^{78} +99544.0 q^{79} +6561.00 q^{81} -39144.0i q^{82} +19308.0i q^{83} +7056.00 q^{84} -77744.0 q^{86} -13338.0i q^{87} -13824.0i q^{88} -36390.0 q^{89} +48902.0 q^{91} +36288.0i q^{92} -75240.0i q^{93} +88800.0 q^{94} +9216.00 q^{96} +79078.0i q^{97} -9604.00i q^{98} -17496.0 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 32 q^{4} + 72 q^{6} - 162 q^{9} + 432 q^{11} + 392 q^{14} + 512 q^{16} - 1768 q^{19} - 882 q^{21} - 1152 q^{24} - 7984 q^{26} + 2964 q^{29} + 16720 q^{31} + 10416 q^{34} + 2592 q^{36} + 17964 q^{39}+ \cdots - 34992 q^{99}+O(q^{100}) $ 2 * q - 32 * q^4 + 72 * q^6 - 162 * q^9 + 432 * q^11 + 392 * q^14 + 512 * q^16 - 1768 * q^19 - 882 * q^21 - 1152 * q^24 - 7984 * q^26 + 2964 * q^29 + 16720 * q^31 + 10416 * q^34 + 2592 * q^36 + 17964 * q^39 - 19572 * q^41 - 6912 * q^44 + 18144 * q^46 - 4802 * q^49 - 23436 * q^51 - 5832 * q^54 - 6272 * q^56 - 56184 * q^59 - 77732 * q^61 - 8192 * q^64 + 15552 * q^66 - 40824 * q^69 - 41256 * q^71 - 37712 * q^74 + 28288 * q^76 + 199088 * q^79 + 13122 * q^81 + 14112 * q^84 - 155488 * q^86 - 72780 * q^89 + 97804 * q^91 + 177600 * q^94 + 18432 * q^96 - 34992 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1050\mathbb{Z}\right)^\times$.

$n$	$127$	$451$	$701$
$\chi(n)$	$-1$	$1$	$1$

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 $

$2$	4.00000i	0.707107i
$2$	4.00000i	0.707107i
$3$	− 9.00000i	− 0.577350i
$3$	− 9.00000i	− 0.577350i
$4$	−16.0000	−0.500000
$5$	0	0
$5$	0	0
$6$	36.0000	0.408248
$7$	− 49.0000i	− 0.377964i
$7$	− 49.0000i	− 0.377964i
$8$	− 64.0000i	− 0.353553i
$9$	−81.0000	−0.333333
$10$	0	0
$11$	216.000	0.538235	0.269118	−	0.963107i	$-0.413268\pi$
$11$	216.000	0.538235	0.269118	+	0.963107i	$0.413268\pi$
$12$	144.000i	0.288675i
$13$	998.000i	1.63784i	0.573906	+	0.818921i	$0.305428\pi$
$13$	998.000i	1.63784i	−0.573906	+	0.818921i	$0.694572\pi$
$14$	196.000	0.267261
$15$	0	0
$16$	256.000	0.250000
$17$	− 1302.00i	− 1.09267i	−0.837567	−	0.546335i	$-0.816023\pi$
$17$	− 1302.00i	− 1.09267i	0.837567	−	0.546335i	$-0.183977\pi$
$18$	− 324.000i	− 0.235702i
$19$	−884.000	−0.561783	−0.280891	−	0.959740i	$-0.590630\pi$
$19$	−884.000	−0.561783	−0.280891	+	0.959740i	$0.590630\pi$
$20$	0	0
$21$	−441.000	−0.218218
$22$	864.000i	0.380590i
$23$	− 2268.00i	− 0.893971i	−0.894541	−	0.446986i	$-0.852498\pi$
$23$	− 2268.00i	− 0.893971i	0.894541	−	0.446986i	$-0.147502\pi$
$24$	−576.000	−0.204124
$25$	0	0
$26$	−3992.00	−1.15813
$27$	729.000i	0.192450i
$28$	784.000i	0.188982i
$29$	1482.00	0.327230	0.163615	−	0.986524i	$-0.447685\pi$
$29$	1482.00	0.327230	0.163615	+	0.986524i	$0.447685\pi$
$30$	0	0
$31$	8360.00	1.56244	0.781218	−	0.624259i	$-0.214599\pi$
$31$	8360.00	1.56244	0.781218	+	0.624259i	$0.214599\pi$
$32$	1024.00i	0.176777i
$33$	− 1944.00i	− 0.310750i
$34$	5208.00	0.772634
$35$	0	0
$36$	1296.00	0.166667
$37$	4714.00i	0.566090i	0.959107	+	0.283045i	$0.0913445\pi$
$37$	4714.00i	0.566090i	−0.959107	+	0.283045i	$0.908655\pi$
$38$	− 3536.00i	− 0.397240i
$39$	8982.00	0.945609
$40$	0	0
$41$	−9786.00	−0.909171	−0.454585	−	0.890703i	$-0.650213\pi$
$41$	−9786.00	−0.909171	−0.454585	+	0.890703i	$0.650213\pi$
$42$	− 1764.00i	− 0.154303i
$43$	19436.0i	1.60301i	0.597989	+	0.801504i	$0.295967\pi$
$43$	19436.0i	1.60301i	−0.597989	+	0.801504i	$0.704033\pi$
$44$	−3456.00	−0.269118
$45$	0	0
$46$	9072.00	0.632133
$47$	− 22200.0i	− 1.46591i	−0.680275	−	0.732957i	$-0.738140\pi$
$47$	− 22200.0i	− 1.46591i	0.680275	−	0.732957i	$-0.261860\pi$
$48$	− 2304.00i	− 0.144338i
$49$	−2401.00	−0.142857
$50$	0	0
$51$	−11718.0	−0.630853
$52$	− 15968.0i	− 0.818921i
$53$	26790.0i	1.31004i	0.755614	+	0.655018i	$0.227339\pi$
$53$	26790.0i	1.31004i	−0.755614	+	0.655018i	$0.772661\pi$
$54$	−2916.00	−0.136083
$55$	0	0
$56$	−3136.00	−0.133631
$57$	7956.00i	0.324345i
$58$	5928.00i	0.231387i
$59$	−28092.0	−1.05064	−0.525318	−	0.850906i	$-0.676054\pi$
$59$	−28092.0	−1.05064	−0.525318	+	0.850906i	$0.676054\pi$
$60$	0	0
$61$	−38866.0	−1.33735	−0.668675	−	0.743555i	$-0.733138\pi$
$61$	−38866.0	−1.33735	−0.668675	+	0.743555i	$0.733138\pi$
$62$	33440.0i	1.10481i
$63$	3969.00i	0.125988i
$64$	−4096.00	−0.125000
$65$	0	0
$66$	7776.00	0.219734
$67$	− 23948.0i	− 0.651752i	−0.945413	−	0.325876i	$-0.894341\pi$
$67$	− 23948.0i	− 0.651752i	0.945413	−	0.325876i	$-0.105659\pi$
$68$	20832.0i	0.546335i
$69$	−20412.0	−0.516134
$70$	0	0
$71$	−20628.0	−0.485636	−0.242818	−	0.970072i	$-0.578072\pi$
$71$	−20628.0	−0.485636	−0.242818	+	0.970072i	$0.578072\pi$
$72$	5184.00i	0.117851i
$73$	290.000i	0.00636929i	0.999995	+	0.00318464i	$0.00101371\pi$
$73$	290.000i	0.00636929i	−0.999995	+	0.00318464i	$0.998986\pi$
$74$	−18856.0	−0.400286
$75$	0	0
$76$	14144.0	0.280891
$77$	− 10584.0i	− 0.203434i
$78$	35928.0i	0.668646i
$79$	99544.0	1.79452	0.897258	−	0.441506i	$-0.145556\pi$
$79$	99544.0	1.79452	0.897258	+	0.441506i	$0.145556\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	− 39144.0i	− 0.642881i
$83$	19308.0i	0.307639i	0.988099	+	0.153820i	$0.0491575\pi$
$83$	19308.0i	0.307639i	−0.988099	+	0.153820i	$0.950842\pi$
$84$	7056.00	0.109109
$85$	0	0
$86$	−77744.0	−1.13350
$87$	− 13338.0i	− 0.188926i
$88$	− 13824.0i	− 0.190295i
$89$	−36390.0	−0.486975	−0.243488	−	0.969904i	$-0.578292\pi$
$89$	−36390.0	−0.486975	−0.243488	+	0.969904i	$0.578292\pi$
$90$	0	0
$91$	48902.0	0.619046
$92$	36288.0i	0.446986i
$93$	− 75240.0i	− 0.902072i
$94$	88800.0	1.03656
$95$	0	0
$96$	9216.00	0.102062
$97$	79078.0i	0.853348i	0.904405	+	0.426674i	$0.140315\pi$
$97$	79078.0i	0.853348i	−0.904405	+	0.426674i	$0.859685\pi$
$98$	− 9604.00i	− 0.101015i
$99$	−17496.0	−0.179412
$100$	0	0
$101$	184626.	1.80090	0.900450	−	0.434960i	$-0.143238\pi$
$101$	184626.	1.80090	0.900450	+	0.434960i	$0.143238\pi$
$102$	− 46872.0i	− 0.446080i
$103$	64592.0i	0.599909i	0.953953	+	0.299955i	$0.0969716\pi$
$103$	64592.0i	0.599909i	−0.953953	+	0.299955i	$0.903028\pi$
$104$	63872.0	0.579065
$105$	0	0
$106$	−107160.	−0.926335
$107$	− 149592.i	− 1.26313i	−0.775322	−	0.631566i	$-0.782412\pi$
$107$	− 149592.i	− 1.26313i	0.775322	−	0.631566i	$-0.217588\pi$
$108$	− 11664.0i	− 0.0962250i
$109$	63826.0	0.514555	0.257277	−	0.966338i	$-0.417175\pi$
$109$	63826.0	0.514555	0.257277	+	0.966338i	$0.417175\pi$
$110$	0	0
$111$	42426.0	0.326832
$112$	− 12544.0i	− 0.0944911i
$113$	− 71022.0i	− 0.523235i	−0.965172	−	0.261618i	$-0.915744\pi$
$113$	− 71022.0i	− 0.523235i	0.965172	−	0.261618i	$-0.0842559\pi$
$114$	−31824.0	−0.229347
$115$	0	0
$116$	−23712.0	−0.163615
$117$	− 80838.0i	− 0.545948i
$118$	− 112368.i	− 0.742912i
$119$	−63798.0	−0.412990
$120$	0	0
$121$	−114395.	−0.710303
$122$	− 155464.i	− 0.945650i
$123$	88074.0i	0.524910i
$124$	−133760.	−0.781218
$125$	0	0
$126$	−15876.0	−0.0890871
$127$	− 269624.i	− 1.48337i	−0.670749	−	0.741685i	$-0.734027\pi$
$127$	− 269624.i	− 1.48337i	0.670749	−	0.741685i	$-0.265973\pi$
$128$	− 16384.0i	− 0.0883883i
$129$	174924.	0.925497
$130$	0	0
$131$	81180.0	0.413305	0.206653	−	0.978414i	$-0.433743\pi$
$131$	81180.0	0.413305	0.206653	+	0.978414i	$0.433743\pi$
$132$	31104.0i	0.155375i
$133$	43316.0i	0.212334i
$134$	95792.0	0.460858
$135$	0	0
$136$	−83328.0	−0.386317
$137$	260910.i	1.18765i	0.804593	+	0.593826i	$0.202383\pi$
$137$	260910.i	1.18765i	−0.804593	+	0.593826i	$0.797617\pi$
$138$	− 81648.0i	− 0.364962i
$139$	297964.	1.30806	0.654029	−	0.756470i	$-0.273078\pi$
$139$	297964.	1.30806	0.654029	+	0.756470i	$0.273078\pi$
$140$	0	0
$141$	−199800.	−0.846346
$142$	− 82512.0i	− 0.343397i
$143$	215568.i	0.881544i
$144$	−20736.0	−0.0833333
$145$	0	0
$146$	−1160.00	−0.00450377
$147$	21609.0i	0.0824786i
$148$	− 75424.0i	− 0.283045i
$149$	398970.	1.47223	0.736113	−	0.676859i	$-0.236659\pi$
$149$	398970.	1.47223	0.736113	+	0.676859i	$0.236659\pi$
$150$	0	0
$151$	−224968.	−0.802931	−0.401466	−	0.915874i	$-0.631499\pi$
$151$	−224968.	−0.802931	−0.401466	+	0.915874i	$0.631499\pi$
$152$	56576.0i	0.198620i
$153$	105462.i	0.364223i
$154$	42336.0	0.143849
$155$	0	0
$156$	−143712.	−0.472804
$157$	233218.i	0.755115i	0.925986	+	0.377557i	$0.123236\pi$
$157$	233218.i	0.755115i	−0.925986	+	0.377557i	$0.876764\pi$
$158$	398176.i	1.26891i
$159$	241110.	0.756349
$160$	0	0
$161$	−111132.	−0.337889
$162$	26244.0i	0.0785674i
$163$	466220.i	1.37443i	0.726455	+	0.687214i	$0.241166\pi$
$163$	466220.i	1.37443i	−0.726455	+	0.687214i	$0.758834\pi$
$164$	156576.	0.454585
$165$	0	0
$166$	−77232.0	−0.217534
$167$	100848.i	0.279818i	0.990164	+	0.139909i	$0.0446811\pi$
$167$	100848.i	0.279818i	−0.990164	+	0.139909i	$0.955319\pi$
$168$	28224.0i	0.0771517i
$169$	−624711.	−1.68253
$170$	0	0
$171$	71604.0	0.187261
$172$	− 310976.i	− 0.801504i
$173$	− 668838.i	− 1.69905i	−0.527550	−	0.849524i	$-0.676889\pi$
$173$	− 668838.i	− 1.69905i	0.527550	−	0.849524i	$-0.323111\pi$
$174$	53352.0	0.133591
$175$	0	0
$176$	55296.0	0.134559
$177$	252828.i	0.606585i
$178$	− 145560.i	− 0.344344i
$179$	614856.	1.43430	0.717151	−	0.696917i	$-0.245446\pi$
$179$	614856.	1.43430	0.717151	+	0.696917i	$0.245446\pi$
$180$	0	0
$181$	540686.	1.22673	0.613365	−	0.789800i	$-0.289816\pi$
$181$	540686.	1.22673	0.613365	+	0.789800i	$0.289816\pi$
$182$	195608.i	0.437732i
$183$	349794.i	0.772120i
$184$	−145152.	−0.316066
$185$	0	0
$186$	300960.	0.637862
$187$	− 281232.i	− 0.588113i
$188$	355200.i	0.732957i
$189$	35721.0	0.0727393
$190$	0	0
$191$	−41916.0	−0.0831374	−0.0415687	−	0.999136i	$-0.513236\pi$
$191$	−41916.0	−0.0831374	−0.0415687	+	0.999136i	$0.513236\pi$
$192$	36864.0i	0.0721688i
$193$	− 533998.i	− 1.03192i	−0.856612	−	0.515960i	$-0.827435\pi$
$193$	− 533998.i	− 1.03192i	0.856612	−	0.515960i	$-0.172565\pi$
$194$	−316312.	−0.603408
$195$	0	0
$196$	38416.0	0.0714286
$197$	− 824886.i	− 1.51436i	−0.653208	−	0.757179i	$-0.726577\pi$
$197$	− 824886.i	− 1.51436i	0.653208	−	0.757179i	$-0.273423\pi$
$198$	− 69984.0i	− 0.126863i
$199$	399544.	0.715207	0.357604	−	0.933873i	$-0.383594\pi$
$199$	399544.	0.715207	0.357604	+	0.933873i	$0.383594\pi$
$200$	0	0
$201$	−215532.	−0.376289
$202$	738504.i	1.27343i
$203$	− 72618.0i	− 0.123681i
$204$	187488.	0.315426
$205$	0	0
$206$	−258368.	−0.424200
$207$	183708.i	0.297990i
$208$	255488.i	0.409461i
$209$	−190944.	−0.302371
$210$	0	0
$211$	868868.	1.34353	0.671765	−	0.740764i	$-0.265536\pi$
$211$	868868.	1.34353	0.671765	+	0.740764i	$0.265536\pi$
$212$	− 428640.i	− 0.655018i
$213$	185652.i	0.280382i
$214$	598368.	0.893170
$215$	0	0
$216$	46656.0	0.0680414
$217$	− 409640.i	− 0.590545i
$218$	255304.i	0.363845i
$219$	2610.00	0.00367731
$220$	0	0
$221$	1.29940e6	1.78962
$222$	169704.i	0.231105i
$223$	− 626656.i	− 0.843853i	−0.906630	−	0.421927i	$-0.861354\pi$
$223$	− 626656.i	− 0.843853i	0.906630	−	0.421927i	$-0.138646\pi$
$224$	50176.0	0.0668153
$225$	0	0
$226$	284088.	0.369983
$227$	450396.i	0.580136i	0.957006	+	0.290068i	$0.0936779\pi$
$227$	450396.i	0.580136i	−0.957006	+	0.290068i	$0.906322\pi$
$228$	− 127296.i	− 0.162173i
$229$	1.06453e6	1.34143	0.670717	−	0.741714i	$-0.265987\pi$
$229$	1.06453e6	1.34143	0.670717	+	0.741714i	$0.265987\pi$
$230$	0	0
$231$	−95256.0	−0.117453
$232$	− 94848.0i	− 0.115693i
$233$	1.43618e6i	1.73308i	0.499108	+	0.866540i	$0.333661\pi$
$233$	1.43618e6i	1.73308i	−0.499108	+	0.866540i	$0.666339\pi$
$234$	323352.	0.386043
$235$	0	0
$236$	449472.	0.525318
$237$	− 895896.i	− 1.03606i
$238$	− 255192.i	− 0.292028i
$239$	997860.	1.12999	0.564995	−	0.825094i	$-0.308878\pi$
$239$	997860.	1.12999	0.564995	+	0.825094i	$0.308878\pi$
$240$	0	0
$241$	−227974.	−0.252838	−0.126419	−	0.991977i	$-0.540348\pi$
$241$	−227974.	−0.252838	−0.126419	+	0.991977i	$0.540348\pi$
$242$	− 457580.i	− 0.502260i
$243$	− 59049.0i	− 0.0641500i
$244$	621856.	0.668675
$245$	0	0
$246$	−352296.	−0.371168
$247$	− 882232.i	− 0.920111i
$248$	− 535040.i	− 0.552404i
$249$	173772.	0.177616
$250$	0	0
$251$	1.51657e6	1.51942	0.759712	−	0.650260i	$-0.225340\pi$
$251$	1.51657e6	1.51942	0.759712	+	0.650260i	$0.225340\pi$
$252$	− 63504.0i	− 0.0629941i
$253$	− 489888.i	− 0.481167i
$254$	1.07850e6	1.04890
$255$	0	0
$256$	65536.0	0.0625000
$257$	− 455886.i	− 0.430550i	−0.976553	−	0.215275i	$-0.930935\pi$
$257$	− 455886.i	− 0.430550i	0.976553	−	0.215275i	$-0.0690648\pi$
$258$	699696.i	0.654425i
$259$	230986.	0.213962
$260$	0	0
$261$	−120042.	−0.109077
$262$	324720.i	0.292251i
$263$	− 752652.i	− 0.670973i	−0.942045	−	0.335486i	$-0.891099\pi$
$263$	− 752652.i	− 0.670973i	0.942045	−	0.335486i	$-0.108901\pi$
$264$	−124416.	−0.109867
$265$	0	0
$266$	−173264.	−0.150143
$267$	327510.i	0.281155i
$268$	383168.i	0.325876i
$269$	−143682.	−0.121066	−0.0605329	−	0.998166i	$-0.519280\pi$
$269$	−143682.	−0.121066	−0.0605329	+	0.998166i	$0.519280\pi$
$270$	0	0
$271$	757496.	0.626552	0.313276	−	0.949662i	$-0.398574\pi$
$271$	757496.	0.626552	0.313276	+	0.949662i	$0.398574\pi$
$272$	− 333312.i	− 0.273167i
$273$	− 440118.i	− 0.357407i
$274$	−1.04364e6	−0.839797
$275$	0	0
$276$	326592.	0.258067
$277$	1.16214e6i	0.910035i	0.890482	+	0.455018i	$0.150367\pi$
$277$	1.16214e6i	0.910035i	−0.890482	+	0.455018i	$0.849633\pi$
$278$	1.19186e6i	0.924936i
$279$	−677160.	−0.520812
$280$	0	0
$281$	−414366.	−0.313053	−0.156527	−	0.987674i	$-0.550030\pi$
$281$	−414366.	−0.313053	−0.156527	+	0.987674i	$0.550030\pi$
$282$	− 799200.i	− 0.598457i
$283$	120428.i	0.0893843i	0.999001	+	0.0446922i	$0.0142307\pi$
$283$	120428.i	0.0893843i	−0.999001	+	0.0446922i	$0.985769\pi$
$284$	330048.	0.242818
$285$	0	0
$286$	−862272.	−0.623346
$287$	479514.i	0.343634i
$288$	− 82944.0i	− 0.0589256i
$289$	−275347.	−0.193926
$290$	0	0
$291$	711702.	0.492681
$292$	− 4640.00i	− 0.00318464i
$293$	2.20159e6i	1.49819i	0.662463	+	0.749094i	$0.269511\pi$
$293$	2.20159e6i	1.49819i	−0.662463	+	0.749094i	$0.730489\pi$
$294$	−86436.0	−0.0583212
$295$	0	0
$296$	301696.	0.200143
$297$	157464.i	0.103583i
$298$	1.59588e6i	1.04102i
$299$	2.26346e6	1.46418
$300$	0	0
$301$	952364.	0.605880
$302$	− 899872.i	− 0.567758i
$303$	− 1.66163e6i	− 1.03975i
$304$	−226304.	−0.140446
$305$	0	0
$306$	−421848.	−0.257545
$307$	− 110900.i	− 0.0671561i	−0.999436	−	0.0335781i	$-0.989310\pi$
$307$	− 110900.i	− 0.0671561i	0.999436	−	0.0335781i	$-0.0106902\pi$
$308$	169344.i	0.101717i
$309$	581328.	0.346358
$310$	0	0
$311$	−910608.	−0.533864	−0.266932	−	0.963715i	$-0.586010\pi$
$311$	−910608.	−0.533864	−0.266932	+	0.963715i	$0.586010\pi$
$312$	− 574848.i	− 0.334323i
$313$	3.12247e6i	1.80152i	0.434322	+	0.900758i	$0.356988\pi$
$313$	3.12247e6i	1.80152i	−0.434322	+	0.900758i	$0.643012\pi$
$314$	−932872.	−0.533947
$315$	0	0
$316$	−1.59270e6	−0.897258
$317$	2.76688e6i	1.54647i	0.634117	+	0.773237i	$0.281364\pi$
$317$	2.76688e6i	1.54647i	−0.634117	+	0.773237i	$0.718636\pi$
$318$	964440.i	0.534820i
$319$	320112.	0.176127
$320$	0	0
$321$	−1.34633e6	−0.729270
$322$	− 444528.i	− 0.238924i
$323$	1.15097e6i	0.613842i
$324$	−104976.	−0.0555556
$325$	0	0
$326$	−1.86488e6	−0.971867
$327$	− 574434.i	− 0.297078i
$328$	626304.i	0.321440i
$329$	−1.08780e6	−0.554063
$330$	0	0
$331$	3.22257e6	1.61671	0.808356	−	0.588694i	$-0.200358\pi$
$331$	3.22257e6	1.61671	0.808356	+	0.588694i	$0.200358\pi$
$332$	− 308928.i	− 0.153820i
$333$	− 381834.i	− 0.188697i
$334$	−403392.	−0.197861
$335$	0	0
$336$	−112896.	−0.0545545
$337$	− 1.63306e6i	− 0.783298i	−0.920115	−	0.391649i	$-0.871905\pi$
$337$	− 1.63306e6i	− 0.783298i	0.920115	−	0.391649i	$-0.128095\pi$
$338$	− 2.49884e6i	− 1.18973i
$339$	−639198.	−0.302090
$340$	0	0
$341$	1.80576e6	0.840958
$342$	286416.i	0.132413i
$343$	117649.i	0.0539949i
$344$	1.24390e6	0.566749
$345$	0	0
$346$	2.67535e6	1.20141
$347$	− 1.03642e6i	− 0.462073i	−0.972945	−	0.231036i	$-0.925788\pi$
$347$	− 1.03642e6i	− 0.462073i	0.972945	−	0.231036i	$-0.0742117\pi$
$348$	213408.i	0.0944632i
$349$	4.22999e6	1.85898	0.929491	−	0.368844i	$-0.120246\pi$
$349$	4.22999e6	1.85898	0.929491	+	0.368844i	$0.120246\pi$
$350$	0	0
$351$	−727542.	−0.315203
$352$	221184.i	0.0951474i
$353$	238806.i	0.102002i	0.998699	+	0.0510010i	$0.0162412\pi$
$353$	238806.i	0.102002i	−0.998699	+	0.0510010i	$0.983759\pi$
$354$	−1.01131e6	−0.428921
$355$	0	0
$356$	582240.	0.243488
$357$	574182.i	0.238440i
$358$	2.45942e6i	1.01421i
$359$	2.66428e6	1.09105	0.545523	−	0.838096i	$-0.316331\pi$
$359$	2.66428e6	1.09105	0.545523	+	0.838096i	$0.316331\pi$
$360$	0	0
$361$	−1.69464e6	−0.684400
$362$	2.16274e6i	0.867429i
$363$	1.02956e6i	0.410094i
$364$	−782432.	−0.309523
$365$	0	0
$366$	−1.39918e6	−0.545971
$367$	1.71083e6i	0.663044i	0.943448	+	0.331522i	$0.107562\pi$
$367$	1.71083e6i	0.663044i	−0.943448	+	0.331522i	$0.892438\pi$
$368$	− 580608.i	− 0.223493i
$369$	792666.	0.303057
$370$	0	0
$371$	1.31271e6	0.495147
$372$	1.20384e6i	0.451036i
$373$	− 3.96649e6i	− 1.47616i	−0.674712	−	0.738081i	$-0.735732\pi$
$373$	− 3.96649e6i	− 1.47616i	0.674712	−	0.738081i	$-0.264268\pi$
$374$	1.12493e6	0.415859
$375$	0	0
$376$	−1.42080e6	−0.518279
$377$	1.47904e6i	0.535951i
$378$	142884.i	0.0514344i
$379$	−828668.	−0.296335	−0.148167	−	0.988962i	$-0.547337\pi$
$379$	−828668.	−0.296335	−0.148167	+	0.988962i	$0.547337\pi$
$380$	0	0
$381$	−2.42662e6	−0.856424
$382$	− 167664.i	− 0.0587870i
$383$	− 2.55686e6i	− 0.890657i	−0.895367	−	0.445329i	$-0.853087\pi$
$383$	− 2.55686e6i	− 0.890657i	0.895367	−	0.445329i	$-0.146913\pi$
$384$	−147456.	−0.0510310
$385$	0	0
$386$	2.13599e6	0.729678
$387$	− 1.57432e6i	− 0.534336i
$388$	− 1.26525e6i	− 0.426674i
$389$	−2.91785e6	−0.977664	−0.488832	−	0.872378i	$-0.662577\pi$
$389$	−2.91785e6	−0.977664	−0.488832	+	0.872378i	$0.662577\pi$
$390$	0	0
$391$	−2.95294e6	−0.976815
$392$	153664.i	0.0505076i
$393$	− 730620.i	− 0.238622i
$394$	3.29954e6	1.07081
$395$	0	0
$396$	279936.	0.0897059
$397$	− 2.50715e6i	− 0.798370i	−0.916870	−	0.399185i	$-0.869293\pi$
$397$	− 2.50715e6i	− 0.798370i	0.916870	−	0.399185i	$-0.130707\pi$
$398$	1.59818e6i	0.505728i
$399$	389844.	0.122591
$400$	0	0
$401$	990666.	0.307657	0.153828	−	0.988098i	$-0.450840\pi$
$401$	990666.	0.307657	0.153828	+	0.988098i	$0.450840\pi$
$402$	− 862128.i	− 0.266077i
$403$	8.34328e6i	2.55902i
$404$	−2.95402e6	−0.900450
$405$	0	0
$406$	290472.	0.0874559
$407$	1.01822e6i	0.304689i
$408$	749952.i	0.223040i
$409$	−4.51824e6	−1.33555	−0.667777	−	0.744362i	$-0.732754\pi$
$409$	−4.51824e6	−1.33555	−0.667777	+	0.744362i	$0.732754\pi$
$410$	0	0
$411$	2.34819e6	0.685691
$412$	− 1.03347e6i	− 0.299955i
$413$	1.37651e6i	0.397103i
$414$	−734832.	−0.210711
$415$	0	0
$416$	−1.02195e6	−0.289532
$417$	− 2.68168e6i	− 0.755207i
$418$	− 763776.i	− 0.213809i
$419$	−605220.	−0.168414	−0.0842070	−	0.996448i	$-0.526836\pi$
$419$	−605220.	−0.168414	−0.0842070	+	0.996448i	$0.526836\pi$
$420$	0	0
$421$	4.49893e6	1.23710	0.618549	−	0.785746i	$-0.287721\pi$
$421$	4.49893e6	1.23710	0.618549	+	0.785746i	$0.287721\pi$
$422$	3.47547e6i	0.950020i
$423$	1.79820e6i	0.488638i
$424$	1.71456e6	0.463167
$425$	0	0
$426$	−742608.	−0.198260
$427$	1.90443e6i	0.505471i
$428$	2.39347e6i	0.631566i
$429$	1.94011e6	0.508960
$430$	0	0
$431$	5.37594e6	1.39400	0.696998	−	0.717074i	$-0.254519\pi$
$431$	5.37594e6	1.39400	0.696998	+	0.717074i	$0.254519\pi$
$432$	186624.i	0.0481125i
$433$	− 1.98561e6i	− 0.508950i	−0.967079	−	0.254475i	$-0.918097\pi$
$433$	− 1.98561e6i	− 0.508950i	0.967079	−	0.254475i	$-0.0819027\pi$
$434$	1.63856e6	0.417578
$435$	0	0
$436$	−1.02122e6	−0.257277
$437$	2.00491e6i	0.502217i
$438$	10440.0i	0.00260025i
$439$	−3.38727e6	−0.838859	−0.419429	−	0.907788i	$-0.637770\pi$
$439$	−3.38727e6	−0.838859	−0.419429	+	0.907788i	$0.637770\pi$
$440$	0	0
$441$	194481.	0.0476190
$442$	5.19758e6i	1.26545i
$443$	2.14094e6i	0.518318i	0.965835	+	0.259159i	$0.0834453\pi$
$443$	2.14094e6i	0.518318i	−0.965835	+	0.259159i	$0.916555\pi$
$444$	−678816.	−0.163416
$445$	0	0
$446$	2.50662e6	0.596695
$447$	− 3.59073e6i	− 0.849990i
$448$	200704.i	0.0472456i
$449$	6.97808e6	1.63350	0.816752	−	0.576990i	$-0.195773\pi$
$449$	6.97808e6	1.63350	0.816752	+	0.576990i	$0.195773\pi$
$450$	0	0
$451$	−2.11378e6	−0.489348
$452$	1.13635e6i	0.261618i
$453$	2.02471e6i	0.463573i
$454$	−1.80158e6	−0.410218
$455$	0	0
$456$	509184.	0.114673
$457$	5.17999e6i	1.16021i	0.814540	+	0.580107i	$0.196989\pi$
$457$	5.17999e6i	1.16021i	−0.814540	+	0.580107i	$0.803011\pi$
$458$	4.25812e6i	0.948537i
$459$	949158.	0.210284
$460$	0	0
$461$	−7.83001e6	−1.71597	−0.857985	−	0.513674i	$-0.828284\pi$
$461$	−7.83001e6	−1.71597	−0.857985	+	0.513674i	$0.828284\pi$
$462$	− 381024.i	− 0.0830515i
$463$	165320.i	0.0358404i	0.999839	+	0.0179202i	$0.00570448\pi$
$463$	165320.i	0.0358404i	−0.999839	+	0.0179202i	$0.994296\pi$
$464$	379392.	0.0818075
$465$	0	0
$466$	−5.74471e6	−1.22547
$467$	1.79329e6i	0.380504i	0.981735	+	0.190252i	$0.0609304\pi$
$467$	1.79329e6i	0.380504i	−0.981735	+	0.190252i	$0.939070\pi$
$468$	1.29341e6i	0.272974i
$469$	−1.17345e6	−0.246339
$470$	0	0
$471$	2.09896e6	0.435966
$472$	1.79789e6i	0.371456i
$473$	4.19818e6i	0.862795i
$474$	3.58358e6	0.732608
$475$	0	0
$476$	1.02077e6	0.206495
$477$	− 2.16999e6i	− 0.436678i
$478$	3.99144e6i	0.799024i
$479$	6.59657e6	1.31365	0.656824	−	0.754044i	$-0.271899\pi$
$479$	6.59657e6	1.31365	0.656824	+	0.754044i	$0.271899\pi$
$480$	0	0
$481$	−4.70457e6	−0.927166
$482$	− 911896.i	− 0.178784i
$483$	1.00019e6i	0.195080i
$484$	1.83032e6	0.355151
$485$	0	0
$486$	236196.	0.0453609
$487$	5.97393e6i	1.14140i	0.821159	+	0.570700i	$0.193328\pi$
$487$	5.97393e6i	1.14140i	−0.821159	+	0.570700i	$0.806672\pi$
$488$	2.48742e6i	0.472825i
$489$	4.19598e6	0.793526
$490$	0	0
$491$	381264.	0.0713710	0.0356855	−	0.999363i	$-0.488639\pi$
$491$	381264.	0.0713710	0.0356855	+	0.999363i	$0.488639\pi$
$492$	− 1.40918e6i	− 0.262455i
$493$	− 1.92956e6i	− 0.357554i
$494$	3.52893e6	0.650617
$495$	0	0
$496$	2.14016e6	0.390609
$497$	1.01077e6i	0.183553i
$498$	695088.i	0.125593i
$499$	−1.54351e6	−0.277497	−0.138748	−	0.990328i	$-0.544308\pi$
$499$	−1.54351e6	−0.277497	−0.138748	+	0.990328i	$0.544308\pi$
$500$	0	0
$501$	907632.	0.161553
$502$	6.06629e6i	1.07439i
$503$	− 4.02300e6i	− 0.708974i	−0.935061	−	0.354487i	$-0.884656\pi$
$503$	− 4.02300e6i	− 0.708974i	0.935061	−	0.354487i	$-0.115344\pi$
$504$	254016.	0.0445435
$505$	0	0
$506$	1.95955e6	0.340236
$507$	5.62240e6i	0.971408i
$508$	4.31398e6i	0.741685i
$509$	1.94715e6	0.333123	0.166562	−	0.986031i	$-0.446734\pi$
$509$	1.94715e6	0.333123	0.166562	+	0.986031i	$0.446734\pi$
$510$	0	0
$511$	14210.0	0.00240736
$512$	262144.i	0.0441942i
$513$	− 644436.i	− 0.108115i
$514$	1.82354e6	0.304445
$515$	0	0
$516$	−2.79878e6	−0.462749
$517$	− 4.79520e6i	− 0.789006i
$518$	923944.i	0.151294i
$519$	−6.01954e6	−0.980946
$520$	0	0
$521$	7.38569e6	1.19206	0.596028	−	0.802963i	$-0.296745\pi$
$521$	7.38569e6	1.19206	0.596028	+	0.802963i	$0.296745\pi$
$522$	− 480168.i	− 0.0771289i
$523$	− 329740.i	− 0.0527130i	−0.999653	−	0.0263565i	$-0.991610\pi$
$523$	− 329740.i	− 0.0527130i	0.999653	−	0.0263565i	$-0.00839050\pi$
$524$	−1.29888e6	−0.206653
$525$	0	0
$526$	3.01061e6	0.474449
$527$	− 1.08847e7i	− 1.70722i
$528$	− 497664.i	− 0.0776875i
$529$	1.29252e6	0.200816
$530$	0	0
$531$	2.27545e6	0.350212
$532$	− 693056.i	− 0.106167i
$533$	− 9.76643e6i	− 1.48908i
$534$	−1.31004e6	−0.198807
$535$	0	0
$536$	−1.53267e6	−0.230429
$537$	− 5.53370e6i	− 0.828095i
$538$	− 574728.i	− 0.0856065i
$539$	−518616.	−0.0768907
$540$	0	0
$541$	87086.0	0.0127925	0.00639625	−	0.999980i	$-0.497964\pi$
$541$	87086.0	0.0127925	0.00639625	+	0.999980i	$0.497964\pi$
$542$	3.02998e6i	0.443039i
$543$	− 4.86617e6i	− 0.708252i
$544$	1.33325e6	0.193158
$545$	0	0
$546$	1.76047e6	0.252725
$547$	− 6.91531e6i	− 0.988196i	−0.869406	−	0.494098i	$-0.835498\pi$
$547$	− 6.91531e6i	− 0.988196i	0.869406	−	0.494098i	$-0.164502\pi$
$548$	− 4.17456e6i	− 0.593826i
$549$	3.14815e6	0.445784
$550$	0	0
$551$	−1.31009e6	−0.183832
$552$	1.30637e6i	0.182481i
$553$	− 4.87766e6i	− 0.678263i
$554$	−4.64855e6	−0.643492
$555$	0	0
$556$	−4.76742e6	−0.654029
$557$	1.52258e6i	0.207942i	0.994580	+	0.103971i	$0.0331549\pi$
$557$	1.52258e6i	0.207942i	−0.994580	+	0.103971i	$0.966845\pi$
$558$	− 2.70864e6i	− 0.368270i
$559$	−1.93971e7	−2.62548
$560$	0	0
$561$	−2.53109e6	−0.339547
$562$	− 1.65746e6i	− 0.221362i
$563$	− 7.86462e6i	− 1.04570i	−0.852425	−	0.522850i	$-0.824869\pi$
$563$	− 7.86462e6i	− 1.04570i	0.852425	−	0.522850i	$-0.175131\pi$
$564$	3.19680e6	0.423173
$565$	0	0
$566$	−481712.	−0.0632043
$567$	− 321489.i	− 0.0419961i
$568$	1.32019e6i	0.171698i
$569$	1.46321e6	0.189464	0.0947321	−	0.995503i	$-0.469801\pi$
$569$	1.46321e6	0.189464	0.0947321	+	0.995503i	$0.469801\pi$
$570$	0	0
$571$	9.19855e6	1.18067	0.590336	−	0.807158i	$-0.298995\pi$
$571$	9.19855e6	1.18067	0.590336	+	0.807158i	$0.298995\pi$
$572$	− 3.44909e6i	− 0.440772i
$573$	377244.i	0.0479994i
$574$	−1.91806e6	−0.242986
$575$	0	0
$576$	331776.	0.0416667
$577$	− 3.28939e6i	− 0.411317i	−0.978624	−	0.205658i	$-0.934066\pi$
$577$	− 3.28939e6i	− 0.411317i	0.978624	−	0.205658i	$-0.0659336\pi$
$578$	− 1.10139e6i	− 0.137126i
$579$	−4.80598e6	−0.595780
$580$	0	0
$581$	946092.	0.116277
$582$	2.84681e6i	0.348378i
$583$	5.78664e6i	0.705107i
$584$	18560.0	0.00225188
$585$	0	0
$586$	−8.80634e6	−1.05938
$587$	− 5.12929e6i	− 0.614416i	−0.951642	−	0.307208i	$-0.900605\pi$
$587$	− 5.12929e6i	− 0.614416i	0.951642	−	0.307208i	$-0.0993947\pi$
$588$	− 345744.i	− 0.0412393i
$589$	−7.39024e6	−0.877749
$590$	0	0
$591$	−7.42397e6	−0.874315
$592$	1.20678e6i	0.141522i
$593$	− 2.75433e6i	− 0.321647i	−0.986983	−	0.160823i	$-0.948585\pi$
$593$	− 2.75433e6i	− 0.321647i	0.986983	−	0.160823i	$-0.0514150\pi$
$594$	−629856.	−0.0732445
$595$	0	0
$596$	−6.38352e6	−0.736113
$597$	− 3.59590e6i	− 0.412925i
$598$	9.05386e6i	1.03533i
$599$	9.88616e6	1.12580	0.562899	−	0.826525i	$-0.309686\pi$
$599$	9.88616e6	1.12580	0.562899	+	0.826525i	$0.309686\pi$
$600$	0	0
$601$	1.37039e7	1.54760	0.773798	−	0.633433i	$-0.218355\pi$
$601$	1.37039e7	1.54760	0.773798	+	0.633433i	$0.218355\pi$
$602$	3.80946e6i	0.428422i
$603$	1.93979e6i	0.217251i
$604$	3.59949e6	0.401466
$605$	0	0
$606$	6.64654e6	0.735214
$607$	7.85310e6i	0.865107i	0.901608	+	0.432553i	$0.142387\pi$
$607$	7.85310e6i	0.865107i	−0.901608	+	0.432553i	$0.857613\pi$
$608$	− 905216.i	− 0.0993101i
$609$	−653562.	−0.0714075
$610$	0	0
$611$	2.21556e7	2.40094
$612$	− 1.68739e6i	− 0.182112i
$613$	1.46977e7i	1.57978i	0.613246	+	0.789892i	$0.289864\pi$
$613$	1.46977e7i	1.57978i	−0.613246	+	0.789892i	$0.710136\pi$
$614$	443600.	0.0474865
$615$	0	0
$616$	−677376.	−0.0719247
$617$	− 6.28370e6i	− 0.664511i	−0.943189	−	0.332256i	$-0.892190\pi$
$617$	− 6.28370e6i	− 0.664511i	0.943189	−	0.332256i	$-0.107810\pi$
$618$	2.32531e6i	0.244912i
$619$	2.26692e6	0.237799	0.118900	−	0.992906i	$-0.462063\pi$
$619$	2.26692e6	0.237799	0.118900	+	0.992906i	$0.462063\pi$
$620$	0	0
$621$	1.65337e6	0.172045
$622$	− 3.64243e6i	− 0.377499i
$623$	1.78311e6i	0.184059i
$624$	2.29939e6	0.236402
$625$	0	0
$626$	−1.24899e7	−1.27386
$627$	1.71850e6i	0.174574i
$628$	− 3.73149e6i	− 0.377557i
$629$	6.13763e6	0.618549
$630$	0	0
$631$	−1.17477e7	−1.17457	−0.587285	−	0.809380i	$-0.699803\pi$
$631$	−1.17477e7	−1.17457	−0.587285	+	0.809380i	$0.699803\pi$
$632$	− 6.37082e6i	− 0.634457i
$633$	− 7.81981e6i	− 0.775688i
$634$	−1.10675e7	−1.09352
$635$	0	0
$636$	−3.85776e6	−0.378175
$637$	− 2.39620e6i	− 0.233978i
$638$	1.28045e6i	0.124540i
$639$	1.67087e6	0.161879
$640$	0	0
$641$	5.93231e6	0.570268	0.285134	−	0.958488i	$-0.407962\pi$
$641$	5.93231e6	0.570268	0.285134	+	0.958488i	$0.407962\pi$
$642$	− 5.38531e6i	− 0.515672i
$643$	− 6.94443e6i	− 0.662383i	−0.943564	−	0.331191i	$-0.892549\pi$
$643$	− 6.94443e6i	− 0.662383i	0.943564	−	0.331191i	$-0.107451\pi$
$644$	1.77811e6	0.168945
$645$	0	0
$646$	−4.60387e6	−0.434052
$647$	4.97050e6i	0.466809i	0.972380	+	0.233404i	$0.0749866\pi$
$647$	4.97050e6i	0.466809i	−0.972380	+	0.233404i	$0.925013\pi$
$648$	− 419904.i	− 0.0392837i
$649$	−6.06787e6	−0.565490
$650$	0	0
$651$	−3.68676e6	−0.340951
$652$	− 7.45952e6i	− 0.687214i
$653$	− 1.83355e7i	− 1.68271i	−0.540484	−	0.841354i	$-0.681759\pi$
$653$	− 1.83355e7i	− 1.68271i	0.540484	−	0.841354i	$-0.318241\pi$
$654$	2.29774e6	0.210066
$655$	0	0
$656$	−2.50522e6	−0.227293
$657$	− 23490.0i	− 0.00212310i
$658$	− 4.35120e6i	− 0.391782i
$659$	−9.01402e6	−0.808546	−0.404273	−	0.914638i	$-0.632475\pi$
$659$	−9.01402e6	−0.808546	−0.404273	+	0.914638i	$0.632475\pi$
$660$	0	0
$661$	699398.	0.0622617	0.0311308	−	0.999515i	$-0.490089\pi$
$661$	699398.	0.0622617	0.0311308	+	0.999515i	$0.490089\pi$
$662$	1.28903e7i	1.14319i
$663$	− 1.16946e7i	− 1.03324i
$664$	1.23571e6	0.108767
$665$	0	0
$666$	1.52734e6	0.133429
$667$	− 3.36118e6i	− 0.292534i
$668$	− 1.61357e6i	− 0.139909i
$669$	−5.63990e6	−0.487199
$670$	0	0
$671$	−8.39506e6	−0.719809
$672$	− 451584.i	− 0.0385758i
$673$	− 5.80603e6i	− 0.494130i	−0.968999	−	0.247065i	$-0.920534\pi$
$673$	− 5.80603e6i	− 0.494130i	0.968999	−	0.247065i	$-0.0794662\pi$
$674$	6.53223e6	0.553875
$675$	0	0
$676$	9.99538e6	0.841264
$677$	− 985074.i	− 0.0826033i	−0.999147	−	0.0413016i	$-0.986850\pi$
$677$	− 985074.i	− 0.0826033i	0.999147	−	0.0413016i	$-0.0131505\pi$
$678$	− 2.55679e6i	− 0.213610i
$679$	3.87482e6	0.322535
$680$	0	0
$681$	4.05356e6	0.334942
$682$	7.22304e6i	0.594647i
$683$	− 1.88208e7i	− 1.54379i	−0.635752	−	0.771894i	$-0.719310\pi$
$683$	− 1.88208e7i	− 1.54379i	0.635752	−	0.771894i	$-0.280690\pi$
$684$	−1.14566e6	−0.0936304
$685$	0	0
$686$	−470596.	−0.0381802
$687$	− 9.58077e6i	− 0.774477i
$688$	4.97562e6i	0.400752i
$689$	−2.67364e7	−2.14563
$690$	0	0
$691$	−1.93385e7	−1.54073	−0.770366	−	0.637601i	$-0.779927\pi$
$691$	−1.93385e7	−1.54073	−0.770366	+	0.637601i	$0.779927\pi$
$692$	1.07014e7i	0.849524i
$693$	857304.i	0.0678113i
$694$	4.14566e6	0.326735
$695$	0	0
$696$	−853632.	−0.0667956
$697$	1.27414e7i	0.993423i
$698$	1.69199e7i	1.31450i
$699$	1.29256e7	1.00059
$700$	0	0
$701$	−1.41489e6	−0.108750	−0.0543748	−	0.998521i	$-0.517317\pi$
$701$	−1.41489e6	−0.108750	−0.0543748	+	0.998521i	$0.517317\pi$
$702$	− 2.91017e6i	− 0.222882i
$703$	− 4.16718e6i	− 0.318019i
$704$	−884736.	−0.0672794
$705$	0	0
$706$	−955224.	−0.0721263
$707$	− 9.04667e6i	− 0.680676i
$708$	− 4.04525e6i	− 0.303293i
$709$	754906.	0.0563998	0.0281999	−	0.999602i	$-0.491023\pi$
$709$	754906.	0.0563998	0.0281999	+	0.999602i	$0.491023\pi$
$710$	0	0
$711$	−8.06306e6	−0.598172
$712$	2.32896e6i	0.172172i
$713$	− 1.89605e7i	− 1.39677i
$714$	−2.29673e6	−0.168603
$715$	0	0
$716$	−9.83770e6	−0.717151
$717$	− 8.98074e6i	− 0.652400i
$718$	1.06571e7i	0.771486i
$719$	−1.08854e6	−0.0785279	−0.0392639	−	0.999229i	$-0.512501\pi$
$719$	−1.08854e6	−0.0785279	−0.0392639	+	0.999229i	$0.512501\pi$
$720$	0	0
$721$	3.16501e6	0.226744
$722$	− 6.77857e6i	− 0.483944i
$723$	2.05177e6i	0.145976i
$724$	−8.65098e6	−0.613365
$725$	0	0
$726$	−4.11822e6	−0.289980
$727$	755392.i	0.0530074i	0.999649	+	0.0265037i	$0.00843737\pi$
$727$	755392.i	0.0530074i	−0.999649	+	0.0265037i	$0.991563\pi$
$728$	− 3.12973e6i	− 0.218866i
$729$	−531441.	−0.0370370
$730$	0	0
$731$	2.53057e7	1.75156
$732$	− 5.59670e6i	− 0.386060i
$733$	1.56369e6i	0.107495i	0.998555	+	0.0537477i	$0.0171167\pi$
$733$	1.56369e6i	0.107495i	−0.998555	+	0.0537477i	$0.982883\pi$
$734$	−6.84333e6	−0.468843
$735$	0	0
$736$	2.32243e6	0.158033
$737$	− 5.17277e6i	− 0.350796i
$738$	3.17066e6i	0.214294i
$739$	1.05544e7	0.710922	0.355461	−	0.934691i	$-0.384324\pi$
$739$	1.05544e7	0.710922	0.355461	+	0.934691i	$0.384324\pi$
$740$	0	0
$741$	−7.94009e6	−0.531227
$742$	5.25084e6i	0.350122i
$743$	1.73678e7i	1.15418i	0.816682	+	0.577088i	$0.195811\pi$
$743$	1.73678e7i	1.15418i	−0.816682	+	0.577088i	$0.804189\pi$
$744$	−4.81536e6	−0.318931
$745$	0	0
$746$	1.58660e7	1.04380
$747$	− 1.56395e6i	− 0.102546i
$748$	4.49971e6i	0.294056i
$749$	−7.33001e6	−0.477419
$750$	0	0
$751$	−2.80181e7	−1.81276	−0.906378	−	0.422467i	$-0.861164\pi$
$751$	−2.80181e7	−1.81276	−0.906378	+	0.422467i	$0.861164\pi$
$752$	− 5.68320e6i	− 0.366478i
$753$	− 1.36491e7i	− 0.877239i
$754$	−5.91614e6	−0.378975
$755$	0	0
$756$	−571536.	−0.0363696
$757$	1.01979e7i	0.646801i	0.946262	+	0.323401i	$0.104826\pi$
$757$	1.01979e7i	0.646801i	−0.946262	+	0.323401i	$0.895174\pi$
$758$	− 3.31467e6i	− 0.209540i
$759$	−4.40899e6	−0.277802
$760$	0	0
$761$	2.57535e6	0.161204	0.0806018	−	0.996746i	$-0.474316\pi$
$761$	2.57535e6	0.161204	0.0806018	+	0.996746i	$0.474316\pi$
$762$	− 9.70646e6i	− 0.605583i
$763$	− 3.12747e6i	− 0.194483i
$764$	670656.	0.0415687
$765$	0	0
$766$	1.02275e7	0.629790
$767$	− 2.80358e7i	− 1.72078i
$768$	− 589824.i	− 0.0360844i
$769$	−971234.	−0.0592254	−0.0296127	−	0.999561i	$-0.509427\pi$
$769$	−971234.	−0.0592254	−0.0296127	+	0.999561i	$0.509427\pi$
$770$	0	0
$771$	−4.10297e6	−0.248578
$772$	8.54397e6i	0.515960i
$773$	− 1.72921e7i	− 1.04088i	−0.853899	−	0.520439i	$-0.825768\pi$
$773$	− 1.72921e7i	− 1.04088i	0.853899	−	0.520439i	$-0.174232\pi$
$774$	6.29726e6	0.377833
$775$	0	0
$776$	5.06099e6	0.301704
$777$	− 2.07887e6i	− 0.123531i
$778$	− 1.16714e7i	− 0.691313i
$779$	8.65082e6	0.510756
$780$	0	0
$781$	−4.45565e6	−0.261387
$782$	− 1.18117e7i	− 0.690712i
$783$	1.08038e6i	0.0629755i
$784$	−614656.	−0.0357143
$785$	0	0
$786$	2.92248e6	0.168731
$787$	1.65515e7i	0.952576i	0.879289	+	0.476288i	$0.158018\pi$
$787$	1.65515e7i	0.952576i	−0.879289	+	0.476288i	$0.841982\pi$
$788$	1.31982e7i	0.757179i
$789$	−6.77387e6	−0.387386
$790$	0	0
$791$	−3.48008e6	−0.197764
$792$	1.11974e6i	0.0634316i
$793$	− 3.87883e7i	− 2.19037i
$794$	1.00286e7	0.564533
$795$	0	0
$796$	−6.39270e6	−0.357604
$797$	− 2.91057e6i	− 0.162305i	−0.996702	−	0.0811526i	$-0.974140\pi$
$797$	− 2.91057e6i	− 0.162305i	0.996702	−	0.0811526i	$-0.0258601\pi$
$798$	1.55938e6i	0.0866849i
$799$	−2.89044e7	−1.60176
$800$	0	0
$801$	2.94759e6	0.162325
$802$	3.96266e6i	0.217546i
$803$	62640.0i	0.00342817i
$804$	3.44851e6	0.188145
$805$	0	0
$806$	−3.33731e7	−1.80950
$807$	1.29314e6i	0.0698974i
$808$	− 1.18161e7i	− 0.636714i
$809$	1.16252e7	0.624496	0.312248	−	0.950001i	$-0.398918\pi$
$809$	1.16252e7	0.624496	0.312248	+	0.950001i	$0.398918\pi$
$810$	0	0
$811$	3.09020e7	1.64981	0.824906	−	0.565270i	$-0.191228\pi$
$811$	3.09020e7	1.64981	0.824906	+	0.565270i	$0.191228\pi$
$812$	1.16189e6i	0.0618407i
$813$	− 6.81746e6i	− 0.361740i
$814$	−4.07290e6	−0.215448
$815$	0	0
$816$	−2.99981e6	−0.157713
$817$	− 1.71814e7i	− 0.900542i
$818$	− 1.80730e7i	− 0.944379i
$819$	−3.96106e6	−0.206349
$820$	0	0
$821$	−2.22870e7	−1.15397	−0.576984	−	0.816755i	$-0.695771\pi$
$821$	−2.22870e7	−1.15397	−0.576984	+	0.816755i	$0.695771\pi$
$822$	9.39276e6i	0.484857i
$823$	− 1.64895e7i	− 0.848610i	−0.905519	−	0.424305i	$-0.860518\pi$
$823$	− 1.64895e7i	− 0.848610i	0.905519	−	0.424305i	$-0.139482\pi$
$824$	4.13389e6	0.212100
$825$	0	0
$826$	−5.50603e6	−0.280795
$827$	2.37457e7i	1.20732i	0.797244	+	0.603658i	$0.206291\pi$
$827$	2.37457e7i	1.20732i	−0.797244	+	0.603658i	$0.793709\pi$
$828$	− 2.93933e6i	− 0.148995i
$829$	−2.60865e7	−1.31835	−0.659173	−	0.751991i	$-0.729094\pi$
$829$	−2.60865e7	−1.31835	−0.659173	+	0.751991i	$0.729094\pi$
$830$	0	0
$831$	1.04592e7	0.525409
$832$	− 4.08781e6i	− 0.204730i
$833$	3.12610e6i	0.156096i
$834$	1.07267e7	0.534012
$835$	0	0
$836$	3.05510e6	0.151186
$837$	6.09444e6i	0.300691i
$838$	− 2.42088e6i	− 0.119087i
$839$	−1.00872e7	−0.494729	−0.247365	−	0.968922i	$-0.579565\pi$
$839$	−1.00872e7	−0.494729	−0.247365	+	0.968922i	$0.579565\pi$
$840$	0	0
$841$	−1.83148e7	−0.892920
$842$	1.79957e7i	0.874761i
$843$	3.72929e6i	0.180741i
$844$	−1.39019e7	−0.671765
$845$	0	0
$846$	−7.19280e6	−0.345519
$847$	5.60536e6i	0.268469i
$848$	6.85824e6i	0.327509i
$849$	1.08385e6	0.0516061
$850$	0	0
$851$	1.06914e7	0.506068
$852$	− 2.97043e6i	− 0.140191i
$853$	− 2.43630e7i	− 1.14646i	−0.819395	−	0.573229i	$-0.805691\pi$
$853$	− 2.43630e7i	− 1.14646i	0.819395	−	0.573229i	$-0.194309\pi$
$854$	−7.61774e6	−0.357422
$855$	0	0
$856$	−9.57389e6	−0.446585
$857$	− 2.45612e6i	− 0.114234i	−0.998367	−	0.0571172i	$-0.981809\pi$
$857$	− 2.45612e6i	− 0.114234i	0.998367	−	0.0571172i	$-0.0181909\pi$
$858$	7.76045e6i	0.359889i
$859$	−8.62982e6	−0.399042	−0.199521	−	0.979894i	$-0.563939\pi$
$859$	−8.62982e6	−0.399042	−0.199521	+	0.979894i	$0.563939\pi$
$860$	0	0
$861$	4.31563e6	0.198397
$862$	2.15038e7i	0.985703i
$863$	1.05199e7i	0.480824i	0.970671	+	0.240412i	$0.0772825\pi$
$863$	1.05199e7i	0.480824i	−0.970671	+	0.240412i	$0.922717\pi$
$864$	−746496.	−0.0340207
$865$	0	0
$866$	7.94246e6	0.359882
$867$	2.47812e6i	0.111963i
$868$	6.55424e6i	0.295273i
$869$	2.15015e7	0.965872
$870$	0	0
$871$	2.39001e7	1.06747
$872$	− 4.08486e6i	− 0.181922i
$873$	− 6.40532e6i	− 0.284449i
$874$	−8.01965e6	−0.355121
$875$	0	0
$876$	−41760.0	−0.00183865
$877$	1.14540e7i	0.502872i	0.967874	+	0.251436i	$0.0809028\pi$
$877$	1.14540e7i	0.502872i	−0.967874	+	0.251436i	$0.919097\pi$
$878$	− 1.35491e7i	− 0.593163i
$879$	1.98143e7	0.864980
$880$	0	0
$881$	−1.18134e7	−0.512786	−0.256393	−	0.966573i	$-0.582534\pi$
$881$	−1.18134e7	−0.512786	−0.256393	+	0.966573i	$0.582534\pi$
$882$	777924.i	0.0336718i
$883$	4.63221e6i	0.199934i	0.994991	+	0.0999670i	$0.0318737\pi$
$883$	4.63221e6i	0.199934i	−0.994991	+	0.0999670i	$0.968126\pi$
$884$	−2.07903e7	−0.894810
$885$	0	0
$886$	−8.56378e6	−0.366506
$887$	− 4.47728e7i	− 1.91075i	−0.295388	−	0.955377i	$-0.595449\pi$
$887$	− 4.47728e7i	− 1.91075i	0.295388	−	0.955377i	$-0.404551\pi$
$888$	− 2.71526e6i	− 0.115553i
$889$	−1.32116e7	−0.560661
$890$	0	0
$891$	1.41718e6	0.0598039
$892$	1.00265e7i	0.421927i
$893$	1.96248e7i	0.823525i
$894$	1.43629e7	0.601034
$895$	0	0
$896$	−802816.	−0.0334077
$897$	− 2.03712e7i	− 0.845347i
$898$	2.79123e7i	1.15506i
$899$	1.23895e7	0.511276
$900$	0	0
$901$	3.48806e7	1.43144
$902$	− 8.45510e6i	− 0.346021i
$903$	− 8.57128e6i	− 0.349805i
$904$	−4.54541e6	−0.184992
$905$	0	0
$906$	−8.09885e6	−0.327795
$907$	2.08357e7i	0.840986i	0.907296	+	0.420493i	$0.138143\pi$
$907$	2.08357e7i	0.840986i	−0.907296	+	0.420493i	$0.861857\pi$
$908$	− 7.20634e6i	− 0.290068i
$909$	−1.49547e7	−0.600300
$910$	0	0
$911$	5.27869e6	0.210732	0.105366	−	0.994434i	$-0.466399\pi$
$911$	5.27869e6	0.210732	0.105366	+	0.994434i	$0.466399\pi$
$912$	2.03674e6i	0.0810863i
$913$	4.17053e6i	0.165582i
$914$	−2.07200e7	−0.820396
$915$	0	0
$916$	−1.70325e7	−0.670717
$917$	− 3.97782e6i	− 0.156215i
$918$	3.79663e6i	0.148693i
$919$	−2.51286e7	−0.981477	−0.490738	−	0.871307i	$-0.663273\pi$
$919$	−2.51286e7	−0.981477	−0.490738	+	0.871307i	$0.663273\pi$
$920$	0	0
$921$	−998100.	−0.0387726
$922$	− 3.13200e7i	− 1.21337i
$923$	− 2.05867e7i	− 0.795396i
$924$	1.52410e6	0.0587263
$925$	0	0
$926$	−661280.	−0.0253430
$927$	− 5.23195e6i	− 0.199970i
$928$	1.51757e6i	0.0578467i
$929$	−1.38042e7	−0.524774	−0.262387	−	0.964963i	$-0.584510\pi$
$929$	−1.38042e7	−0.524774	−0.262387	+	0.964963i	$0.584510\pi$
$930$	0	0
$931$	2.12248e6	0.0802547
$932$	− 2.29788e7i	− 0.866540i
$933$	8.19547e6i	0.308226i
$934$	−7.17317e6	−0.269057
$935$	0	0
$936$	−5.17363e6	−0.193022
$937$	4.73307e7i	1.76114i	0.473915	+	0.880570i	$0.342840\pi$
$937$	4.73307e7i	1.76114i	−0.473915	+	0.880570i	$0.657160\pi$
$938$	− 4.69381e6i	− 0.174188i
$939$	2.81023e7	1.04011
$940$	0	0
$941$	−3.25570e7	−1.19859	−0.599295	−	0.800528i	$-0.704552\pi$
$941$	−3.25570e7	−1.19859	−0.599295	+	0.800528i	$0.704552\pi$
$942$	8.39585e6i	0.308274i
$943$	2.21946e7i	0.812773i
$944$	−7.19155e6	−0.262659
$945$	0	0
$946$	−1.67927e7	−0.610088
$947$	5.27117e6i	0.190999i	0.995429	+	0.0954997i	$0.0304449\pi$
$947$	5.27117e6i	0.190999i	−0.995429	+	0.0954997i	$0.969555\pi$
$948$	1.43343e7i	0.518032i
$949$	−289420.	−0.0104319
$950$	0	0
$951$	2.49019e7	0.892857
$952$	4.08307e6i	0.146014i
$953$	8.20579e6i	0.292677i	0.989235	+	0.146338i	$0.0467488\pi$
$953$	8.20579e6i	0.292677i	−0.989235	+	0.146338i	$0.953251\pi$
$954$	8.67996e6	0.308778
$955$	0	0
$956$	−1.59658e7	−0.564995
$957$	− 2.88101e6i	− 0.101687i
$958$	2.63863e7i	0.928890i
$959$	1.27846e7	0.448890
$960$	0	0
$961$	4.12604e7	1.44120
$962$	− 1.88183e7i	− 0.655605i
$963$	1.21170e7i	0.421044i
$964$	3.64758e6	0.126419
$965$	0	0
$966$	−4.00075e6	−0.137943
$967$	1.18118e7i	0.406210i	0.979157	+	0.203105i	$0.0651032\pi$
$967$	1.18118e7i	0.406210i	−0.979157	+	0.203105i	$0.934897\pi$
$968$	7.32128e6i	0.251130i
$969$	1.03587e7	0.354402
$970$	0	0
$971$	−3.67702e7	−1.25155	−0.625774	−	0.780004i	$-0.715217\pi$
$971$	−3.67702e7	−1.25155	−0.625774	+	0.780004i	$0.715217\pi$
$972$	944784.i	0.0320750i
$973$	− 1.46002e7i	− 0.494399i
$974$	−2.38957e7	−0.807091
$975$	0	0
$976$	−9.94970e6	−0.334338
$977$	1.85183e7i	0.620674i	0.950627	+	0.310337i	$0.100442\pi$
$977$	1.85183e7i	0.620674i	−0.950627	+	0.310337i	$0.899558\pi$
$978$	1.67839e7i	0.561108i
$979$	−7.86024e6	−0.262107
$980$	0	0
$981$	−5.16991e6	−0.171518
$982$	1.52506e6i	0.0504670i
$983$	2.72169e7i	0.898370i	0.893439	+	0.449185i	$0.148286\pi$
$983$	2.72169e7i	0.898370i	−0.893439	+	0.449185i	$0.851714\pi$
$984$	5.63674e6	0.185584
$985$	0	0
$986$	7.71826e6	0.252829
$987$	9.79020e6i	0.319889i
$988$	1.41157e7i	0.460056i
$989$	4.40808e7	1.43304
$990$	0	0
$991$	1.63398e7	0.528522	0.264261	−	0.964451i	$-0.414872\pi$
$991$	1.63398e7	0.528522	0.264261	+	0.964451i	$0.414872\pi$
$992$	8.56064e6i	0.276202i
$993$	− 2.90031e7i	− 0.933409i
$994$	−4.04309e6	−0.129792
$995$	0	0
$996$	−2.78035e6	−0.0888079
$997$	3.02062e7i	0.962406i	0.876609	+	0.481203i	$0.159800\pi$
$997$	3.02062e7i	0.962406i	−0.876609	+	0.481203i	$0.840200\pi$
$998$	− 6.17403e6i	− 0.196220i
$999$	−3.43651e6	−0.108944

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1050.6.g.o.799.2		2
5.2	odd	4		42.6.a.a.1.1	✓	1
5.3	odd	4		1050.6.a.n.1.1		1
5.4	even	2	inner	1050.6.g.o.799.1		2
15.2	even	4		126.6.a.k.1.1		1
20.7	even	4		336.6.a.j.1.1		1
35.2	odd	12		294.6.e.r.67.1		2
35.12	even	12		294.6.e.h.67.1		2
35.17	even	12		294.6.e.h.79.1		2
35.27	even	4		294.6.a.h.1.1		1
35.32	odd	12		294.6.e.r.79.1		2
60.47	odd	4		1008.6.a.x.1.1		1
105.62	odd	4		882.6.a.o.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
42.6.a.a.1.1	✓	1	5.2	odd	4
126.6.a.k.1.1		1	15.2	even	4
294.6.a.h.1.1		1	35.27	even	4
294.6.e.h.67.1		2	35.12	even	12
294.6.e.h.79.1		2	35.17	even	12
294.6.e.r.67.1		2	35.2	odd	12
294.6.e.r.79.1		2	35.32	odd	12
336.6.a.j.1.1		1	20.7	even	4
882.6.a.o.1.1		1	105.62	odd	4
1008.6.a.x.1.1		1	60.47	odd	4
1050.6.a.n.1.1		1	5.3	odd	4
1050.6.g.o.799.1		2	5.4	even	2	inner
1050.6.g.o.799.2		2	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	1050.g (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+4.00000i q^{2} -9.00000i q^{3} -16.0000 q^{4} +36.0000 q^{6} -49.0000i q^{7} -64.0000i q^{8} -81.0000 q^{9} +216.000 q^{11} +144.000i q^{12} +998.000i q^{13} +196.000 q^{14} +256.000 q^{16} -1302.00i q^{17} -324.000i q^{18} -884.000 q^{19} -441.000 q^{21} +864.000i q^{22} -2268.00i q^{23} -576.000 q^{24} -3992.00 q^{26} +729.000i q^{27} +784.000i q^{28} +1482.00 q^{29} +8360.00 q^{31} +1024.00i q^{32} -1944.00i q^{33} +5208.00 q^{34} +1296.00 q^{36} +4714.00i q^{37} -3536.00i q^{38} +8982.00 q^{39} -9786.00 q^{41} -1764.00i q^{42} +19436.0i q^{43} -3456.00 q^{44} +9072.00 q^{46} -22200.0i q^{47} -2304.00i q^{48} -2401.00 q^{49} -11718.0 q^{51} -15968.0i q^{52} +26790.0i q^{53} -2916.00 q^{54} -3136.00 q^{56} +7956.00i q^{57} +5928.00i q^{58} -28092.0 q^{59} -38866.0 q^{61} +33440.0i q^{62} +3969.00i q^{63} -4096.00 q^{64} +7776.00 q^{66} -23948.0i q^{67} +20832.0i q^{68} -20412.0 q^{69} -20628.0 q^{71} +5184.00i q^{72} +290.000i q^{73} -18856.0 q^{74} +14144.0 q^{76} -10584.0i q^{77} +35928.0i q^{78} +99544.0 q^{79} +6561.00 q^{81} -39144.0i q^{82} +19308.0i q^{83} +7056.00 q^{84} -77744.0 q^{86} -13338.0i q^{87} -13824.0i q^{88} -36390.0 q^{89} +48902.0 q^{91} +36288.0i q^{92} -75240.0i q^{93} +88800.0 q^{94} +9216.00 q^{96} +79078.0i q^{97} -9604.00i q^{98} -17496.0 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 32 q^{4} + 72 q^{6} - 162 q^{9} + 432 q^{11} + 392 q^{14} + 512 q^{16} - 1768 q^{19} - 882 q^{21} - 1152 q^{24} - 7984 q^{26} + 2964 q^{29} + 16720 q^{31} + 10416 q^{34} + 2592 q^{36} + 17964 q^{39}+ \cdots - 34992 q^{99}+O(q^{100}) \) 2 * q - 32 * q^4 + 72 * q^6 - 162 * q^9 + 432 * q^11 + 392 * q^14 + 512 * q^16 - 1768 * q^19 - 882 * q^21 - 1152 * q^24 - 7984 * q^26 + 2964 * q^29 + 16720 * q^31 + 10416 * q^34 + 2592 * q^36 + 17964 * q^39 - 19572 * q^41 - 6912 * q^44 + 18144 * q^46 - 4802 * q^49 - 23436 * q^51 - 5832 * q^54 - 6272 * q^56 - 56184 * q^59 - 77732 * q^61 - 8192 * q^64 + 15552 * q^66 - 40824 * q^69 - 41256 * q^71 - 37712 * q^74 + 28288 * q^76 + 199088 * q^79 + 13122 * q^81 + 14112 * q^84 - 155488 * q^86 - 72780 * q^89 + 97804 * q^91 + 177600 * q^94 + 18432 * q^96 - 34992 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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