LMFDB - Embedded newform 325.6.b.h.274.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [325,6,Mod(274,325)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(325, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0]))

N = Newforms(chi, 6, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("325.274");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 325 = 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	325.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$52.1247414392$
Analytic rank:	$0$
Dimension:	$18$
Coefficient field:	$\mathbb{Q}[x]/(x^{18} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{18} + 378 x^{16} + 59175 x^{14} + 4956036 x^{12} + 239061247 x^{10} + 6629044458 x^{8} + \cdots + 134659641600 $ x^18 + 378x^16 + 59175x^14 + 4956036x^12 + 239061247x^10 + 6629044458x^8 + 98240243849x^6 + 625586966104x^4 + 584919029904x^2 + 134659641600
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{7}\cdot 5^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			274.12
Root			$4.20685i$ of defining polynomial
Character	$\chi$	$=$	325.274
Dual form			325.6.b.h.274.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.20685i q^{2} +29.5845i q^{3} +21.7161 q^{4} -94.8732 q^{6} -11.5734i q^{7} +172.260i q^{8} -632.245 q^{9} +O(q^{10})$	\(q+3.20685i q^{2} +29.5845i q^{3} +21.7161 q^{4} -94.8732 q^{6} -11.5734i q^{7} +172.260i q^{8} -632.245 q^{9} -596.700 q^{11} +642.461i q^{12} -169.000i q^{13} +37.1142 q^{14} +142.504 q^{16} +2093.43i q^{17} -2027.52i q^{18} -35.4251 q^{19} +342.394 q^{21} -1913.53i q^{22} -2783.24i q^{23} -5096.22 q^{24} +541.958 q^{26} -11515.6i q^{27} -251.329i q^{28} +370.244 q^{29} +5055.06 q^{31} +5969.30i q^{32} -17653.1i q^{33} -6713.33 q^{34} -13729.9 q^{36} +4124.15i q^{37} -113.603i q^{38} +4999.79 q^{39} -18104.1 q^{41} +1098.01i q^{42} +7906.34i q^{43} -12958.0 q^{44} +8925.44 q^{46} -13185.2i q^{47} +4215.93i q^{48} +16673.1 q^{49} -61933.3 q^{51} -3670.02i q^{52} -38295.0i q^{53} +36929.0 q^{54} +1993.63 q^{56} -1048.03i q^{57} +1187.32i q^{58} -17886.6 q^{59} +7392.00 q^{61} +16210.8i q^{62} +7317.23i q^{63} -14582.5 q^{64} +56610.8 q^{66} -23357.0i q^{67} +45461.2i q^{68} +82340.9 q^{69} -33166.9 q^{71} -108910. i q^{72} +10831.7i q^{73} -13225.5 q^{74} -769.295 q^{76} +6905.85i q^{77} +16033.6i q^{78} +17755.2 q^{79} +187049. q^{81} -58057.3i q^{82} +84243.8i q^{83} +7435.46 q^{84} -25354.5 q^{86} +10953.5i q^{87} -102787. i q^{88} +46498.8 q^{89} -1955.91 q^{91} -60441.1i q^{92} +149552. i q^{93} +42283.1 q^{94} -176599. q^{96} +154213. i q^{97} +53468.0i q^{98} +377260. q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q - 182 q^{4} - 166 q^{6} - 1124 q^{9}+O(q^{10}) $ 18 * q - 182 * q^4 - 166 * q^6 - 1124 * q^9	$ 18 q - 182 q^{4} - 166 q^{6} - 1124 q^{9} - 2844 q^{11} + 684 q^{14} - 2122 q^{16} + 816 q^{19} - 7824 q^{21} + 16938 q^{24} - 1690 q^{26} + 17474 q^{29} + 1496 q^{31} + 35578 q^{34} + 1024 q^{36} + 3718 q^{39} - 57352 q^{41} + 61198 q^{44} - 24112 q^{46} + 81814 q^{49} - 62012 q^{51} + 205522 q^{54} - 46004 q^{56} + 176284 q^{59} + 56330 q^{61} + 201690 q^{64} + 85154 q^{66} + 363494 q^{69} - 141124 q^{71} + 271352 q^{74} + 92746 q^{76} + 328146 q^{79} - 139870 q^{81} + 691312 q^{84} - 589840 q^{86} + 505396 q^{89} + 4056 q^{91} + 1003212 q^{94} - 638362 q^{96} + 1515552 q^{99}+O(q^{100}) $ 18 * q - 182 * q^4 - 166 * q^6 - 1124 * q^9 - 2844 * q^11 + 684 * q^14 - 2122 * q^16 + 816 * q^19 - 7824 * q^21 + 16938 * q^24 - 1690 * q^26 + 17474 * q^29 + 1496 * q^31 + 35578 * q^34 + 1024 * q^36 + 3718 * q^39 - 57352 * q^41 + 61198 * q^44 - 24112 * q^46 + 81814 * q^49 - 62012 * q^51 + 205522 * q^54 - 46004 * q^56 + 176284 * q^59 + 56330 * q^61 + 201690 * q^64 + 85154 * q^66 + 363494 * q^69 - 141124 * q^71 + 271352 * q^74 + 92746 * q^76 + 328146 * q^79 - 139870 * q^81 + 691312 * q^84 - 589840 * q^86 + 505396 * q^89 + 4056 * q^91 + 1003212 * q^94 - 638362 * q^96 + 1515552 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/325\mathbb{Z}\right)^\times$.

$n$	$27$	$301$
$\chi(n)$	$-1$	$1$

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 $

$2$	3.20685i	0.566897i	0.958988	+	0.283448i	$0.0914784\pi$
$2$	3.20685i	0.566897i	−0.958988	+	0.283448i	$0.908522\pi$
$3$	29.5845i	1.89785i	0.315503	+	0.948925i	$0.397827\pi$
$3$	29.5845i	1.89785i	−0.315503	+	0.948925i	$0.602173\pi$
$4$	21.7161	0.678628
$5$	0	0
$5$	0	0
$6$	−94.8732	−1.07588
$7$	− 11.5734i	− 0.0892722i	−0.999003	−	0.0446361i	$-0.985787\pi$
$7$	− 11.5734i	− 0.0892722i	0.999003	−	0.0446361i	$-0.0142128\pi$
$8$	172.260i	0.951609i
$9$	−632.245	−2.60183
$10$	0	0
$11$	−596.700	−1.48687	−0.743437	−	0.668806i	$-0.766806\pi$
$11$	−596.700	−1.48687	−0.743437	+	0.668806i	$0.766806\pi$
$12$	642.461i	1.28793i
$13$	− 169.000i	− 0.277350i
$13$	− 169.000i	− 0.277350i
$14$	37.1142	0.0506081
$15$	0	0
$16$	142.504	0.139164
$17$	2093.43i	1.75686i	0.477872	+	0.878429i	$0.341408\pi$
$17$	2093.43i	1.75686i	−0.477872	+	0.878429i	$0.658592\pi$
$18$	− 2027.52i	− 1.47497i
$19$	−35.4251	−0.0225127	−0.0112563	−	0.999937i	$-0.503583\pi$
$19$	−35.4251	−0.0225127	−0.0112563	+	0.999937i	$0.503583\pi$
$20$	0	0
$21$	342.394	0.169425
$22$	− 1913.53i	− 0.842904i
$23$	− 2783.24i	− 1.09706i	−0.836130	−	0.548531i	$-0.815187\pi$
$23$	− 2783.24i	− 1.09706i	0.836130	−	0.548531i	$-0.184813\pi$
$24$	−5096.22	−1.80601
$25$	0	0
$26$	541.958	0.157229
$27$	− 11515.6i	− 3.04004i
$28$	− 251.329i	− 0.0605826i
$29$	370.244	0.0817511	0.0408755	−	0.999164i	$-0.486985\pi$
$29$	370.244	0.0817511	0.0408755	+	0.999164i	$0.486985\pi$
$30$	0	0
$31$	5055.06	0.944761	0.472381	−	0.881395i	$-0.343395\pi$
$31$	5055.06	0.944761	0.472381	+	0.881395i	$0.343395\pi$
$32$	5969.30i	1.03050i
$33$	− 17653.1i	− 2.82186i
$34$	−6713.33	−0.995957
$35$	0	0
$36$	−13729.9	−1.76568
$37$	4124.15i	0.495256i	0.968855	+	0.247628i	$0.0796511\pi$
$37$	4124.15i	0.495256i	−0.968855	+	0.247628i	$0.920349\pi$
$38$	− 113.603i	− 0.0127624i
$39$	4999.79	0.526369
$40$	0	0
$41$	−18104.1	−1.68197	−0.840985	−	0.541058i	$-0.818024\pi$
$41$	−18104.1	−1.68197	−0.840985	+	0.541058i	$0.818024\pi$
$42$	1098.01i	0.0960466i
$43$	7906.34i	0.652085i	0.945355	+	0.326042i	$0.105715\pi$
$43$	7906.34i	0.652085i	−0.945355	+	0.326042i	$0.894285\pi$
$44$	−12958.0	−1.00903
$45$	0	0
$46$	8925.44	0.621921
$47$	− 13185.2i	− 0.870650i	−0.900273	−	0.435325i	$-0.856634\pi$
$47$	− 13185.2i	− 0.870650i	0.900273	−	0.435325i	$-0.143366\pi$
$48$	4215.93i	0.264113i
$49$	16673.1	0.992030
$50$	0	0
$51$	−61933.3	−3.33425
$52$	− 3670.02i	− 0.188218i
$53$	− 38295.0i	− 1.87263i	−0.351160	−	0.936316i	$-0.614213\pi$
$53$	− 38295.0i	− 1.87263i	0.351160	−	0.936316i	$-0.385787\pi$
$54$	36929.0	1.72339
$55$	0	0
$56$	1993.63	0.0849522
$57$	− 1048.03i	− 0.0427256i
$58$	1187.32i	0.0463444i
$59$	−17886.6	−0.668955	−0.334477	−	0.942404i	$-0.608560\pi$
$59$	−17886.6	−0.668955	−0.334477	+	0.942404i	$0.608560\pi$
$60$	0	0
$61$	7392.00	0.254353	0.127177	−	0.991880i	$-0.459408\pi$
$61$	7392.00	0.254353	0.127177	+	0.991880i	$0.459408\pi$
$62$	16210.8i	0.535582i
$63$	7317.23i	0.232271i
$64$	−14582.5	−0.445023
$65$	0	0
$66$	56610.8	1.59970
$67$	− 23357.0i	− 0.635667i	−0.948147	−	0.317834i	$-0.897045\pi$
$67$	− 23357.0i	− 0.635667i	0.948147	−	0.317834i	$-0.102955\pi$
$68$	45461.2i	1.19225i
$69$	82340.9	2.08206
$70$	0	0
$71$	−33166.9	−0.780835	−0.390417	−	0.920638i	$-0.627669\pi$
$71$	−33166.9	−0.780835	−0.390417	+	0.920638i	$0.627669\pi$
$72$	− 108910.i	− 2.47593i
$73$	10831.7i	0.237898i	0.992900	+	0.118949i	$0.0379525\pi$
$73$	10831.7i	0.237898i	−0.992900	+	0.118949i	$0.962047\pi$
$74$	−13225.5	−0.280759
$75$	0	0
$76$	−769.295	−0.0152777
$77$	6905.85i	0.132736i
$78$	16033.6i	0.298397i
$79$	17755.2	0.320079	0.160039	−	0.987111i	$-0.448838\pi$
$79$	17755.2	0.320079	0.160039	+	0.987111i	$0.448838\pi$
$80$	0	0
$81$	187049.	3.16770
$82$	− 58057.3i	− 0.953503i
$83$	84243.8i	1.34228i	0.741331	+	0.671139i	$0.234195\pi$
$83$	84243.8i	1.34228i	−0.741331	+	0.671139i	$0.765805\pi$
$84$	7435.46	0.114977
$85$	0	0
$86$	−25354.5	−0.369665
$87$	10953.5i	0.155151i
$88$	− 102787.i	− 1.41492i
$89$	46498.8	0.622253	0.311126	−	0.950369i	$-0.399294\pi$
$89$	46498.8	0.622253	0.311126	+	0.950369i	$0.399294\pi$
$90$	0	0
$91$	−1955.91	−0.0247597
$92$	− 60441.1i	− 0.744497i
$93$	149552.i	1.79301i
$94$	42283.1	0.493568
$95$	0	0
$96$	−176599.	−1.95573
$97$	154213.i	1.66414i	0.554667	+	0.832072i	$0.312845\pi$
$97$	154213.i	1.66414i	−0.554667	+	0.832072i	$0.687155\pi$
$98$	53468.0i	0.562379i
$99$	377260.	3.86860
$100$	0	0
$101$	−110873.	−1.08149	−0.540745	−	0.841187i	$-0.681858\pi$
$101$	−110873.	−1.08149	−0.540745	+	0.841187i	$0.681858\pi$
$102$	− 198611.i	− 1.89018i
$103$	− 104407.i	− 0.969699i	−0.874598	−	0.484849i	$-0.838874\pi$
$103$	− 104407.i	− 0.969699i	0.874598	−	0.484849i	$-0.161126\pi$
$104$	29111.9	0.263929
$105$	0	0
$106$	122806.	1.06159
$107$	− 2229.04i	− 0.0188217i	−0.999956	−	0.00941084i	$-0.997004\pi$
$107$	− 2229.04i	− 0.0188217i	0.999956	−	0.00941084i	$-0.00299561\pi$
$108$	− 250075.i	− 2.06305i
$109$	47011.7	0.379000	0.189500	−	0.981881i	$-0.439313\pi$
$109$	47011.7	0.379000	0.189500	+	0.981881i	$0.439313\pi$
$110$	0	0
$111$	−122011.	−0.939921
$112$	− 1649.26i	− 0.0124235i
$113$	− 198706.i	− 1.46391i	−0.681352	−	0.731956i	$-0.738608\pi$
$113$	− 198706.i	− 1.46391i	0.681352	−	0.731956i	$-0.261392\pi$
$114$	3360.89	0.0242210
$115$	0	0
$116$	8040.27	0.0554786
$117$	106849.i	0.721618i
$118$	− 57359.5i	− 0.379228i
$119$	24228.2	0.156839
$120$	0	0
$121$	194999.	1.21079
$122$	23705.0i	0.144192i
$123$	− 535603.i	− 3.19213i
$124$	109776.	0.641142
$125$	0	0
$126$	−23465.3	−0.131674
$127$	87410.5i	0.480899i	0.970662	+	0.240450i	$0.0772949\pi$
$127$	87410.5i	0.480899i	−0.970662	+	0.240450i	$0.922705\pi$
$128$	144254.i	0.778219i
$129$	−233905.	−1.23756
$130$	0	0
$131$	−19351.5	−0.0985226	−0.0492613	−	0.998786i	$-0.515687\pi$
$131$	−19351.5	−0.0985226	−0.0492613	+	0.998786i	$0.515687\pi$
$132$	− 383356.i	− 1.91500i
$133$	409.989i	0.00200975i
$134$	74902.4	0.360358
$135$	0	0
$136$	−360614.	−1.67184
$137$	− 79960.9i	− 0.363979i	−0.983300	−	0.181989i	$-0.941746\pi$
$137$	− 79960.9i	− 0.363979i	0.983300	−	0.181989i	$-0.0582537\pi$
$138$	264055.i	1.18031i
$139$	−285962.	−1.25537	−0.627684	−	0.778468i	$-0.715997\pi$
$139$	−285962.	−1.25537	−0.627684	+	0.778468i	$0.715997\pi$
$140$	0	0
$141$	390079.	1.65236
$142$	− 106361.i	− 0.442652i
$143$	100842.i	0.412385i
$144$	−90097.7	−0.362083
$145$	0	0
$146$	−34735.8	−0.134864
$147$	493265.i	1.88272i
$148$	89560.4i	0.336095i
$149$	−506571.	−1.86928	−0.934641	−	0.355593i	$-0.884279\pi$
$149$	−506571.	−1.86928	−0.934641	+	0.355593i	$0.884279\pi$
$150$	0	0
$151$	−353322.	−1.26104	−0.630519	−	0.776174i	$-0.717158\pi$
$151$	−353322.	−1.26104	−0.630519	+	0.776174i	$0.717158\pi$
$152$	− 6102.31i	− 0.0214232i
$153$	− 1.32356e6i	− 4.57105i
$154$	−22146.0	−0.0752479
$155$	0	0
$156$	108576.	0.357209
$157$	222777.i	0.721308i	0.932700	+	0.360654i	$0.117446\pi$
$157$	222777.i	0.721308i	−0.932700	+	0.360654i	$0.882554\pi$
$158$	56938.2i	0.181452i
$159$	1.13294e6	3.55397
$160$	0	0
$161$	−32211.6	−0.0979371
$162$	599840.i	1.79576i
$163$	485830.i	1.43224i	0.697978	+	0.716120i	$0.254083\pi$
$163$	485830.i	1.43224i	−0.697978	+	0.716120i	$0.745917\pi$
$164$	−393151.	−1.14143
$165$	0	0
$166$	−270157.	−0.760933
$167$	492737.i	1.36717i	0.729869	+	0.683587i	$0.239581\pi$
$167$	492737.i	1.36717i	−0.729869	+	0.683587i	$0.760419\pi$
$168$	58980.7i	0.161226i
$169$	−28561.0	−0.0769231
$170$	0	0
$171$	22397.3	0.0585742
$172$	171695.i	0.442523i
$173$	608498.i	1.54577i	0.634549	+	0.772883i	$0.281186\pi$
$173$	608498.i	1.54577i	−0.634549	+	0.772883i	$0.718814\pi$
$174$	−35126.3	−0.0879547
$175$	0	0
$176$	−85032.3	−0.206920
$177$	− 529166.i	− 1.26958i
$178$	149115.i	0.352753i
$179$	270050.	0.629958	0.314979	−	0.949099i	$-0.398003\pi$
$179$	270050.	0.629958	0.314979	+	0.949099i	$0.398003\pi$
$180$	0	0
$181$	−602323.	−1.36657	−0.683286	−	0.730151i	$-0.739450\pi$
$181$	−602323.	−1.36657	−0.683286	+	0.730151i	$0.739450\pi$
$182$	− 6272.30i	− 0.0140362i
$183$	218689.i	0.482724i
$184$	479440.	1.04397
$185$	0	0
$186$	−479590.	−1.01645
$187$	− 1.24915e6i	− 2.61223i
$188$	− 286332.i	− 0.590847i
$189$	−133275.	−0.271391
$190$	0	0
$191$	−138491.	−0.274688	−0.137344	−	0.990523i	$-0.543857\pi$
$191$	−138491.	−0.274688	−0.137344	+	0.990523i	$0.543857\pi$
$192$	− 431417.i	− 0.844586i
$193$	58857.2i	0.113738i	0.998382	+	0.0568691i	$0.0181118\pi$
$193$	58857.2i	0.113738i	−0.998382	+	0.0568691i	$0.981888\pi$
$194$	−494537.	−0.943398
$195$	0	0
$196$	362074.	0.673220
$197$	− 271860.i	− 0.499091i	−0.968363	−	0.249545i	$-0.919719\pi$
$197$	− 271860.i	− 0.499091i	0.968363	−	0.249545i	$-0.0802812\pi$
$198$	1.20982e6i	2.19309i
$199$	71334.8	0.127694	0.0638468	−	0.997960i	$-0.479663\pi$
$199$	71334.8	0.127694	0.0638468	+	0.997960i	$0.479663\pi$
$200$	0	0
$201$	691006.	1.20640
$202$	− 355553.i	− 0.613093i
$203$	− 4284.99i	− 0.00729810i
$204$	−1.34495e6	−2.26272
$205$	0	0
$206$	334818.	0.549719
$207$	1.75969e6i	2.85437i
$208$	− 24083.2i	− 0.0385973i
$209$	21138.1	0.0334735
$210$	0	0
$211$	480921.	0.743648	0.371824	−	0.928303i	$-0.378732\pi$
$211$	480921.	0.743648	0.371824	+	0.928303i	$0.378732\pi$
$212$	− 831618.i	− 1.27082i
$213$	− 981228.i	− 1.48191i
$214$	7148.20	0.0106699
$215$	0	0
$216$	1.98368e6	2.89292
$217$	− 58504.3i	− 0.0843409i
$218$	150759.i	0.214854i
$219$	−320452.	−0.451495
$220$	0	0
$221$	353790.	0.487265
$222$	− 391271.i	− 0.532838i
$223$	1.34684e6i	1.81365i	0.421511	+	0.906823i	$0.361500\pi$
$223$	1.34684e6i	1.81365i	−0.421511	+	0.906823i	$0.638500\pi$
$224$	69085.1	0.0919950
$225$	0	0
$226$	637221.	0.829887
$227$	− 719772.i	− 0.927108i	−0.886069	−	0.463554i	$-0.846574\pi$
$227$	− 719772.i	− 0.927108i	0.886069	−	0.463554i	$-0.153426\pi$
$228$	− 22759.2i	− 0.0289948i
$229$	568410.	0.716264	0.358132	−	0.933671i	$-0.383414\pi$
$229$	568410.	0.716264	0.358132	+	0.933671i	$0.383414\pi$
$230$	0	0
$231$	−204306.	−0.251914
$232$	63778.2i	0.0777951i
$233$	− 282144.i	− 0.340472i	−0.985403	−	0.170236i	$-0.945547\pi$
$233$	− 282144.i	− 0.340472i	0.985403	−	0.170236i	$-0.0544531\pi$
$234$	−342650.	−0.409083
$235$	0	0
$236$	−388426.	−0.453972
$237$	525278.i	0.607461i
$238$	77696.1i	0.0889113i
$239$	143244.	0.162211	0.0811056	−	0.996706i	$-0.474155\pi$
$239$	143244.	0.162211	0.0811056	+	0.996706i	$0.474155\pi$
$240$	0	0
$241$	−369914.	−0.410259	−0.205130	−	0.978735i	$-0.565762\pi$
$241$	−369914.	−0.410259	−0.205130	+	0.978735i	$0.565762\pi$
$242$	625334.i	0.686395i
$243$	2.73547e6i	2.97178i
$244$	160525.	0.172611
$245$	0	0
$246$	1.71760e6	1.80961
$247$	5986.84i	0.00624389i
$248$	870783.i	0.899043i
$249$	−2.49231e6	−2.54744
$250$	0	0
$251$	−731868.	−0.733244	−0.366622	−	0.930370i	$-0.619486\pi$
$251$	−731868.	−0.733244	−0.366622	+	0.930370i	$0.619486\pi$
$252$	158902.i	0.157626i
$253$	1.66076e6i	1.63119i
$254$	−280312.	−0.272620
$255$	0	0
$256$	−929240.	−0.886192
$257$	1.05879e6i	0.999951i	0.866040	+	0.499975i	$0.166658\pi$
$257$	1.05879e6i	0.999951i	−0.866040	+	0.499975i	$0.833342\pi$
$258$	− 750100.i	− 0.701568i
$259$	47730.4	0.0442126
$260$	0	0
$261$	−234085.	−0.212703
$262$	− 62057.3i	− 0.0558521i
$263$	− 318172.i	− 0.283644i	−0.989892	−	0.141822i	$-0.954704\pi$
$263$	− 318172.i	− 0.283644i	0.989892	−	0.141822i	$-0.0452960\pi$
$264$	3.04091e6	2.68531
$265$	0	0
$266$	−1314.77	−0.00113932
$267$	1.37565e6i	1.18094i
$268$	− 507223.i	− 0.431382i
$269$	−1.13453e6	−0.955954	−0.477977	−	0.878372i	$-0.658630\pi$
$269$	−1.13453e6	−0.955954	−0.477977	+	0.878372i	$0.658630\pi$
$270$	0	0
$271$	−1.05400e6	−0.871805	−0.435902	−	0.899994i	$-0.643571\pi$
$271$	−1.05400e6	−0.871805	−0.435902	+	0.899994i	$0.643571\pi$
$272$	298324.i	0.244492i
$273$	− 57864.6i	− 0.0469901i
$274$	256423.	0.206338
$275$	0	0
$276$	1.78812e6	1.41294
$277$	− 2.18170e6i	− 1.70843i	−0.519922	−	0.854214i	$-0.674039\pi$
$277$	− 2.18170e6i	− 1.70843i	0.519922	−	0.854214i	$-0.325961\pi$
$278$	− 917037.i	− 0.711664i
$279$	−3.19604e6	−2.45811
$280$	0	0
$281$	−2.11562e6	−1.59835	−0.799176	−	0.601097i	$-0.794730\pi$
$281$	−2.11562e6	−1.59835	−0.799176	+	0.601097i	$0.794730\pi$
$282$	1.25093e6i	0.936718i
$283$	390968.i	0.290185i	0.989418	+	0.145092i	$0.0463480\pi$
$283$	390968.i	0.290185i	−0.989418	+	0.145092i	$0.953652\pi$
$284$	−720256.	−0.529896
$285$	0	0
$286$	−323386.	−0.233779
$287$	209527.i	0.150153i
$288$	− 3.77406e6i	− 2.68119i
$289$	−2.96261e6	−2.08655
$290$	0	0
$291$	−4.56231e6	−3.15830
$292$	235223.i	0.161444i
$293$	2.44684e6i	1.66508i	0.553961	+	0.832542i	$0.313116\pi$
$293$	2.44684e6i	1.66508i	−0.553961	+	0.832542i	$0.686884\pi$
$294$	−1.58183e6	−1.06731
$295$	0	0
$296$	−710424.	−0.471290
$297$	6.87138e6i	4.52015i
$298$	− 1.62450e6i	− 1.05969i
$299$	−470368.	−0.304270
$300$	0	0
$301$	91503.3	0.0582131
$302$	− 1.13305e6i	− 0.714878i
$303$	− 3.28013e6i	− 2.05251i
$304$	−5048.23	−0.00313296
$305$	0	0
$306$	4.24447e6	2.59131
$307$	− 1.58927e6i	− 0.962390i	−0.876614	−	0.481195i	$-0.840203\pi$
$307$	− 1.58927e6i	− 0.962390i	0.876614	−	0.481195i	$-0.159797\pi$
$308$	149968.i	0.0900787i
$309$	3.08884e6	1.84034
$310$	0	0
$311$	3.15211e6	1.84799	0.923996	−	0.382403i	$-0.124903\pi$
$311$	3.15211e6	1.84799	0.923996	+	0.382403i	$0.124903\pi$
$312$	861261.i	0.500897i
$313$	− 1.58939e6i	− 0.917001i	−0.888694	−	0.458500i	$-0.848387\pi$
$313$	− 1.58939e6i	− 0.917001i	0.888694	−	0.458500i	$-0.151613\pi$
$314$	−714412.	−0.408907
$315$	0	0
$316$	385573.	0.217215
$317$	− 1.45529e6i	− 0.813396i	−0.913563	−	0.406698i	$-0.866680\pi$
$317$	− 1.45529e6i	− 0.813396i	0.913563	−	0.406698i	$-0.133320\pi$
$318$	3.63317e6i	2.01473i
$319$	−220925.	−0.121554
$320$	0	0
$321$	65945.1	0.0357207
$322$	− 103298.i	− 0.0555202i
$323$	− 74160.0i	− 0.0395516i
$324$	4.06198e6	2.14969
$325$	0	0
$326$	−1.55799e6	−0.811932
$327$	1.39082e6i	0.719285i
$328$	− 3.11861e6i	− 1.60058i
$329$	−152598.	−0.0777248
$330$	0	0
$331$	3.21928e6	1.61506	0.807530	−	0.589826i	$-0.200804\pi$
$331$	3.21928e6	1.61506	0.807530	+	0.589826i	$0.200804\pi$
$332$	1.82945e6i	0.910908i
$333$	− 2.60747e6i	− 1.28857i
$334$	−1.58013e6	−0.775046
$335$	0	0
$336$	48792.7	0.0235780
$337$	− 3.07320e6i	− 1.47406i	−0.675859	−	0.737031i	$-0.736227\pi$
$337$	− 3.07320e6i	− 1.47406i	0.675859	−	0.737031i	$-0.263773\pi$
$338$	− 91590.9i	− 0.0436074i
$339$	5.87863e6	2.77829
$340$	0	0
$341$	−3.01635e6	−1.40474
$342$	71824.9i	0.0332055i
$343$	− 387478.i	− 0.177833i
$344$	−1.36194e6	−0.620530
$345$	0	0
$346$	−1.95136e6	−0.876289
$347$	− 679230.i	− 0.302826i	−0.988471	−	0.151413i	$-0.951618\pi$
$347$	− 679230.i	− 0.302826i	0.988471	−	0.151413i	$-0.0483823\pi$
$348$	237868.i	0.105290i
$349$	−3.00182e6	−1.31923	−0.659616	−	0.751603i	$-0.729281\pi$
$349$	−3.00182e6	−1.31923	−0.659616	+	0.751603i	$0.729281\pi$
$350$	0	0
$351$	−1.94614e6	−0.843154
$352$	− 3.56188e6i	− 1.53222i
$353$	− 2.13910e6i	− 0.913682i	−0.889548	−	0.456841i	$-0.848981\pi$
$353$	− 2.13910e6i	− 0.913682i	0.889548	−	0.456841i	$-0.151019\pi$
$354$	1.69696e6	0.719718
$355$	0	0
$356$	1.00977e6	0.422278
$357$	716779.i	0.297656i
$358$	866010.i	0.357121i
$359$	−4.04695e6	−1.65726	−0.828631	−	0.559795i	$-0.810880\pi$
$359$	−4.04695e6	−1.65726	−0.828631	+	0.559795i	$0.810880\pi$
$360$	0	0
$361$	−2.47484e6	−0.999493
$362$	− 1.93156e6i	− 0.774705i
$363$	5.76897e6i	2.29790i
$364$	−42474.7	−0.0168026
$365$	0	0
$366$	−701303.	−0.273655
$367$	1.59021e6i	0.616295i	0.951339	+	0.308148i	$0.0997090\pi$
$367$	1.59021e6i	0.616295i	−0.951339	+	0.308148i	$0.900291\pi$
$368$	− 396624.i	− 0.152672i
$369$	1.14463e7	4.37620
$370$	0	0
$371$	−443204.	−0.167174
$372$	3.24768e6i	1.21679i
$373$	3.36023e6i	1.25054i	0.780409	+	0.625270i	$0.215011\pi$
$373$	3.36023e6i	1.25054i	−0.780409	+	0.625270i	$0.784989\pi$
$374$	4.00584e6	1.48086
$375$	0	0
$376$	2.27128e6	0.828518
$377$	− 62571.3i	− 0.0226737i
$378$	− 427394.i	− 0.153850i
$379$	−849174.	−0.303668	−0.151834	−	0.988406i	$-0.548518\pi$
$379$	−849174.	−0.303668	−0.151834	+	0.988406i	$0.548518\pi$
$380$	0	0
$381$	−2.58600e6	−0.912674
$382$	− 444121.i	− 0.155720i
$383$	4.04395e6i	1.40867i	0.709868	+	0.704335i	$0.248755\pi$
$383$	4.04395e6i	1.40867i	−0.709868	+	0.704335i	$0.751245\pi$
$384$	−4.26768e6	−1.47694
$385$	0	0
$386$	−188746.	−0.0644777
$387$	− 4.99874e6i	− 1.69662i
$388$	3.34890e6i	1.12934i
$389$	2.94042e6	0.985226	0.492613	−	0.870249i	$-0.336042\pi$
$389$	2.94042e6	0.985226	0.492613	+	0.870249i	$0.336042\pi$
$390$	0	0
$391$	5.82653e6	1.92738
$392$	2.87209e6i	0.944025i
$393$	− 572504.i	− 0.186981i
$394$	871814.	0.282933
$395$	0	0
$396$	8.19263e6	2.62534
$397$	− 1.24060e6i	− 0.395053i	−0.980298	−	0.197526i	$-0.936709\pi$
$397$	− 1.24060e6i	− 0.395053i	0.980298	−	0.197526i	$-0.0632908\pi$
$398$	228760.i	0.0723890i
$399$	−12129.3	−0.00381421
$400$	0	0
$401$	−972004.	−0.301861	−0.150931	−	0.988544i	$-0.548227\pi$
$401$	−972004.	−0.301861	−0.150931	+	0.988544i	$0.548227\pi$
$402$	2.21595e6i	0.683904i
$403$	− 854305.i	− 0.262030i
$404$	−2.40773e6	−0.733930
$405$	0	0
$406$	13741.3	0.00413727
$407$	− 2.46088e6i	− 0.736383i
$408$	− 1.06686e7i	− 3.17290i
$409$	−571971.	−0.169070	−0.0845348	−	0.996421i	$-0.526940\pi$
$409$	−571971.	−0.169070	−0.0845348	+	0.996421i	$0.526940\pi$
$410$	0	0
$411$	2.36561e6	0.690777
$412$	− 2.26731e6i	− 0.658065i
$413$	207009.i	0.0597191i
$414$	−5.64306e6	−1.61813
$415$	0	0
$416$	1.00881e6	0.285809
$417$	− 8.46005e6i	− 2.38250i
$418$	67786.9i	0.0189760i
$419$	4.06416e6	1.13093	0.565465	−	0.824772i	$-0.308697\pi$
$419$	4.06416e6	1.13093	0.565465	+	0.824772i	$0.308697\pi$
$420$	0	0
$421$	−3.73588e6	−1.02728	−0.513639	−	0.858007i	$-0.671703\pi$
$421$	−3.73588e6	−1.02728	−0.513639	+	0.858007i	$0.671703\pi$
$422$	1.54224e6i	0.421572i
$423$	8.33631e6i	2.26528i
$424$	6.59668e6	1.78201
$425$	0	0
$426$	3.14665e6	0.840088
$427$	− 85550.6i	− 0.0227067i
$428$	− 48406.0i	− 0.0127729i
$429$	−2.98337e6	−0.782644
$430$	0	0
$431$	−1.53689e6	−0.398520	−0.199260	−	0.979947i	$-0.563854\pi$
$431$	−1.53689e6	−0.398520	−0.199260	+	0.979947i	$0.563854\pi$
$432$	− 1.64103e6i	− 0.423065i
$433$	− 112160.i	− 0.0287487i	−0.999897	−	0.0143744i	$-0.995424\pi$
$433$	− 112160.i	− 0.0287487i	0.999897	−	0.0143744i	$-0.00457566\pi$
$434$	187615.	0.0478126
$435$	0	0
$436$	1.02091e6	0.257200
$437$	98596.5i	0.0246978i
$438$	− 1.02764e6i	− 0.255951i
$439$	164279.	0.0406837	0.0203418	−	0.999793i	$-0.493525\pi$
$439$	164279.	0.0406837	0.0203418	+	0.999793i	$0.493525\pi$
$440$	0	0
$441$	−1.05415e7	−2.58110
$442$	1.13455e6i	0.276229i
$443$	− 4.62345e6i	− 1.11933i	−0.828720	−	0.559664i	$-0.810930\pi$
$443$	− 4.62345e6i	− 1.11933i	0.828720	−	0.559664i	$-0.189070\pi$
$444$	−2.64960e6	−0.637857
$445$	0	0
$446$	−4.31910e6	−1.02815
$447$	− 1.49867e7i	− 3.54761i
$448$	168769.i	0.0397282i
$449$	−421554.	−0.0986820	−0.0493410	−	0.998782i	$-0.515712\pi$
$449$	−421554.	−0.0986820	−0.0493410	+	0.998782i	$0.515712\pi$
$450$	0	0
$451$	1.08027e7	2.50088
$452$	− 4.31512e6i	− 0.993452i
$453$	− 1.04529e7i	− 2.39326i
$454$	2.30820e6	0.525574
$455$	0	0
$456$	180534.	0.0406581
$457$	5.34433e6i	1.19702i	0.801114	+	0.598512i	$0.204241\pi$
$457$	5.34433e6i	1.19702i	−0.801114	+	0.598512i	$0.795759\pi$
$458$	1.82281e6i	0.406047i
$459$	2.41072e7	5.34091
$460$	0	0
$461$	−4.95164e6	−1.08517	−0.542584	−	0.840002i	$-0.682554\pi$
$461$	−4.95164e6	−1.08517	−0.542584	+	0.840002i	$0.682554\pi$
$462$	− 655180.i	− 0.142809i
$463$	3.87726e6i	0.840567i	0.907393	+	0.420283i	$0.138069\pi$
$463$	3.87726e6i	0.840567i	−0.907393	+	0.420283i	$0.861931\pi$
$464$	52761.5	0.0113768
$465$	0	0
$466$	904795.	0.193013
$467$	− 1.64031e6i	− 0.348043i	−0.984742	−	0.174022i	$-0.944324\pi$
$467$	− 1.64031e6i	− 0.348043i	0.984742	−	0.174022i	$-0.0556763\pi$
$468$	2.32035e6i	0.489711i
$469$	−270320.	−0.0567474
$470$	0	0
$471$	−6.59075e6	−1.36893
$472$	− 3.08113e6i	− 0.636583i
$473$	− 4.71771e6i	− 0.969568i
$474$	−1.68449e6	−0.344368
$475$	0	0
$476$	526141.	0.106435
$477$	2.42118e7i	4.87227i
$478$	459361.i	0.0919570i
$479$	−2.63694e6	−0.525123	−0.262561	−	0.964915i	$-0.584567\pi$
$479$	−2.63694e6	−0.525123	−0.262561	+	0.964915i	$0.584567\pi$
$480$	0	0
$481$	696981.	0.137359
$482$	− 1.18626e6i	− 0.232575i
$483$	− 952965.i	− 0.185870i
$484$	4.23463e6	0.821678
$485$	0	0
$486$	−8.77225e6	−1.68469
$487$	− 3.47150e6i	− 0.663277i	−0.943407	−	0.331639i	$-0.892399\pi$
$487$	− 3.47150e6i	− 0.663277i	0.943407	−	0.331639i	$-0.107601\pi$
$488$	1.27334e6i	0.242045i
$489$	−1.43731e7	−2.71817
$490$	0	0
$491$	−2.74272e6	−0.513425	−0.256713	−	0.966488i	$-0.582639\pi$
$491$	−2.74272e6	−0.513425	−0.256713	+	0.966488i	$0.582639\pi$
$492$	− 1.16312e7i	− 2.16627i
$493$	775082.i	0.143625i
$494$	−19198.9	−0.00353964
$495$	0	0
$496$	720368.	0.131477
$497$	383854.i	0.0697068i
$498$	− 7.99248e6i	− 1.44414i
$499$	1.00565e7	1.80799	0.903996	−	0.427540i	$-0.140620\pi$
$499$	1.00565e7	1.80799	0.903996	+	0.427540i	$0.140620\pi$
$500$	0	0
$501$	−1.45774e7	−2.59469
$502$	− 2.34699e6i	− 0.415674i
$503$	5.52516e6i	0.973699i	0.873486	+	0.486850i	$0.161854\pi$
$503$	5.52516e6i	0.973699i	−0.873486	+	0.486850i	$0.838146\pi$
$504$	−1.26046e6	−0.221031
$505$	0	0
$506$	−5.32580e6	−0.924717
$507$	− 844964.i	− 0.145988i
$508$	1.89821e6i	0.326352i
$509$	6.59944e6	1.12905	0.564524	−	0.825416i	$-0.309060\pi$
$509$	6.59944e6	1.12905	0.564524	+	0.825416i	$0.309060\pi$
$510$	0	0
$511$	125360.	0.0212377
$512$	1.63618e6i	0.275839i
$513$	407943.i	0.0684393i
$514$	−3.39540e6	−0.566869
$515$	0	0
$516$	−5.07951e6	−0.839842
$517$	7.86763e6i	1.29455i
$518$	153064.i	0.0250640i
$519$	−1.80021e7	−2.93363
$520$	0	0
$521$	−653159.	−0.105420	−0.0527102	−	0.998610i	$-0.516786\pi$
$521$	−653159.	−0.105420	−0.0527102	+	0.998610i	$0.516786\pi$
$522$	− 750677.i	− 0.120580i
$523$	3.60676e6i	0.576584i	0.957543	+	0.288292i	$0.0930874\pi$
$523$	3.60676e6i	0.576584i	−0.957543	+	0.288292i	$0.906913\pi$
$524$	−420239.	−0.0668602
$525$	0	0
$526$	1.02033e6	0.160797
$527$	1.05824e7i	1.65981i
$528$	− 2.51564e6i	− 0.392703i
$529$	−1.31008e6	−0.203544
$530$	0	0
$531$	1.13087e7	1.74051
$532$	8903.36i	0.00136388i
$533$	3.05960e6i	0.466495i
$534$	−4.41149e6	−0.669472
$535$	0	0
$536$	4.02346e6	0.604906
$537$	7.98931e6i	1.19557i
$538$	− 3.63828e6i	− 0.541927i
$539$	−9.94881e6	−1.47502
$540$	0	0
$541$	−1.92001e6	−0.282040	−0.141020	−	0.990007i	$-0.545038\pi$
$541$	−1.92001e6	−0.282040	−0.141020	+	0.990007i	$0.545038\pi$
$542$	− 3.38004e6i	− 0.494223i
$543$	− 1.78194e7i	− 2.59355i
$544$	−1.24963e7	−1.81044
$545$	0	0
$546$	185563.	0.0266385
$547$	7.41583e6i	1.05972i	0.848085	+	0.529860i	$0.177756\pi$
$547$	7.41583e6i	1.05972i	−0.848085	+	0.529860i	$0.822244\pi$
$548$	− 1.73644e6i	− 0.247006i
$549$	−4.67356e6	−0.661784
$550$	0	0
$551$	−13115.9	−0.00184043
$552$	1.41840e7i	1.98130i
$553$	− 205488.i	− 0.0285741i
$554$	6.99640e6	0.968502
$555$	0	0
$556$	−6.20998e6	−0.851928
$557$	− 3.52870e6i	− 0.481922i	−0.970535	−	0.240961i	$-0.922537\pi$
$557$	− 3.52870e6i	− 0.481922i	0.970535	−	0.240961i	$-0.0774626\pi$
$558$	− 1.02492e7i	− 1.39349i
$559$	1.33617e6	0.180856
$560$	0	0
$561$	3.69556e7	4.95761
$562$	− 6.78449e6i	− 0.906100i
$563$	− 4.31809e6i	− 0.574144i	−0.957909	−	0.287072i	$-0.907318\pi$
$563$	− 4.31809e6i	− 0.574144i	0.957909	−	0.287072i	$-0.0926818\pi$
$564$	8.47100e6	1.12134
$565$	0	0
$566$	−1.25378e6	−0.164505
$567$	− 2.16480e6i	− 0.282787i
$568$	− 5.71332e6i	− 0.743049i
$569$	3.99812e6	0.517696	0.258848	−	0.965918i	$-0.416657\pi$
$569$	3.99812e6	0.517696	0.258848	+	0.965918i	$0.416657\pi$
$570$	0	0
$571$	298969.	0.0383739	0.0191869	−	0.999816i	$-0.493892\pi$
$571$	298969.	0.0383739	0.0191869	+	0.999816i	$0.493892\pi$
$572$	2.18990e6i	0.279856i
$573$	− 4.09720e6i	− 0.521316i
$574$	−671921.	−0.0851213
$575$	0	0
$576$	9.21972e6	1.15787
$577$	− 2.22363e6i	− 0.278050i	−0.990289	−	0.139025i	$-0.955603\pi$
$577$	− 2.22363e6i	− 0.278050i	0.990289	−	0.139025i	$-0.0443969\pi$
$578$	− 9.50064e6i	− 1.18286i
$579$	−1.74126e6	−0.215858
$580$	0	0
$581$	974988.	0.119828
$582$	− 1.46307e7i	− 1.79043i
$583$	2.28506e7i	2.78437i
$584$	−1.86587e6	−0.226386
$585$	0	0
$586$	−7.84665e6	−0.943931
$587$	− 3.25215e6i	− 0.389560i	−0.980847	−	0.194780i	$-0.937601\pi$
$587$	− 3.25215e6i	− 0.389560i	0.980847	−	0.194780i	$-0.0623993\pi$
$588$	1.07118e7i	1.27767i
$589$	−179076.	−0.0212691
$590$	0	0
$591$	8.04285e6	0.947199
$592$	587709.i	0.0689220i
$593$	1.23502e7i	1.44224i	0.692810	+	0.721121i	$0.256373\pi$
$593$	1.23502e7i	1.44224i	−0.692810	+	0.721121i	$0.743627\pi$
$594$	−2.20355e7	−2.56246
$595$	0	0
$596$	−1.10007e7	−1.26855
$597$	2.11041e6i	0.242343i
$598$	− 1.50840e6i	− 0.172490i
$599$	1.26026e7	1.43513	0.717567	−	0.696490i	$-0.245256\pi$
$599$	1.26026e7	1.43513	0.717567	+	0.696490i	$0.245256\pi$
$600$	0	0
$601$	1.10475e7	1.24760	0.623802	−	0.781582i	$-0.285587\pi$
$601$	1.10475e7	1.24760	0.623802	+	0.781582i	$0.285587\pi$
$602$	293437.i	0.0330008i
$603$	1.47673e7i	1.65390i
$604$	−7.67277e6	−0.855776
$605$	0	0
$606$	1.05189e7	1.16356
$607$	1.47315e7i	1.62284i	0.584462	+	0.811421i	$0.301306\pi$
$607$	1.47315e7i	1.62284i	−0.584462	+	0.811421i	$0.698694\pi$
$608$	− 211463.i	− 0.0231993i
$609$	126769.	0.0138507
$610$	0	0
$611$	−2.22831e6	−0.241475
$612$	− 2.87426e7i	− 3.10204i
$613$	− 6.38520e6i	− 0.686315i	−0.939278	−	0.343157i	$-0.888504\pi$
$613$	− 6.38520e6i	− 0.686315i	0.939278	−	0.343157i	$-0.111496\pi$
$614$	5.09655e6	0.545576
$615$	0	0
$616$	−1.18960e6	−0.126313
$617$	1.69836e7i	1.79604i	0.439954	+	0.898020i	$0.354995\pi$
$617$	1.69836e7i	1.79604i	−0.439954	+	0.898020i	$0.645005\pi$
$618$	9.90544e6i	1.04328i
$619$	1.06421e7	1.11635	0.558175	−	0.829723i	$-0.311502\pi$
$619$	1.06421e7	1.11635	0.558175	+	0.829723i	$0.311502\pi$
$620$	0	0
$621$	−3.20508e7	−3.33511
$622$	1.01083e7i	1.04762i
$623$	− 538150.i	− 0.0555499i
$624$	712492.	0.0732518
$625$	0	0
$626$	5.09694e6	0.519845
$627$	625362.i	0.0635276i
$628$	4.83784e6i	0.489500i
$629$	−8.63362e6	−0.870095
$630$	0	0
$631$	8.93725e6	0.893574	0.446787	−	0.894640i	$-0.352568\pi$
$631$	8.93725e6	0.893574	0.446787	+	0.894640i	$0.352568\pi$
$632$	3.05850e6i	0.304590i
$633$	1.42278e7i	1.41133i
$634$	4.66691e6	0.461111
$635$	0	0
$636$	2.46030e7	2.41183
$637$	− 2.81775e6i	− 0.275140i
$638$	− 708473.i	− 0.0689083i
$639$	2.09696e7	2.03160
$640$	0	0
$641$	−1.12700e7	−1.08337	−0.541686	−	0.840581i	$-0.682214\pi$
$641$	−1.12700e7	−1.08337	−0.541686	+	0.840581i	$0.682214\pi$
$642$	211476.i	0.0202499i
$643$	− 1.16584e7i	− 1.11202i	−0.831177	−	0.556008i	$-0.812332\pi$
$643$	− 1.16584e7i	− 1.11202i	0.831177	−	0.556008i	$-0.187668\pi$
$644$	−699510.	−0.0664629
$645$	0	0
$646$	237820.	0.0224217
$647$	8.09860e6i	0.760587i	0.924866	+	0.380294i	$0.124177\pi$
$647$	8.09860e6i	0.760587i	−0.924866	+	0.380294i	$0.875823\pi$
$648$	3.22211e7i	3.01441i
$649$	1.06729e7	0.994651
$650$	0	0
$651$	1.73082e6	0.160066
$652$	1.05503e7i	0.971958i
$653$	2.50550e6i	0.229938i	0.993369	+	0.114969i	$0.0366769\pi$
$653$	2.50550e6i	0.229938i	−0.993369	+	0.114969i	$0.963323\pi$
$654$	−4.46015e6	−0.407760
$655$	0	0
$656$	−2.57992e6	−0.234070
$657$	− 6.84831e6i	− 0.618971i
$658$	− 489360.i	− 0.0440619i
$659$	2.14090e7	1.92036	0.960182	−	0.279376i	$-0.0901274\pi$
$659$	2.14090e7	1.92036	0.960182	+	0.279376i	$0.0901274\pi$
$660$	0	0
$661$	1.59525e7	1.42012	0.710060	−	0.704141i	$-0.248668\pi$
$661$	1.59525e7	1.42012	0.710060	+	0.704141i	$0.248668\pi$
$662$	1.03238e7i	0.915572i
$663$	1.04667e7i	0.924755i
$664$	−1.45118e7	−1.27732
$665$	0	0
$666$	8.36177e6	0.730488
$667$	− 1.03048e6i	− 0.0896860i
$668$	1.07003e7i	0.927803i
$669$	−3.98455e7	−3.44203
$670$	0	0
$671$	−4.41080e6	−0.378191
$672$	2.04385e6i	0.174593i
$673$	− 9.71969e6i	− 0.827208i	−0.910457	−	0.413604i	$-0.864270\pi$
$673$	− 9.71969e6i	− 0.827208i	0.910457	−	0.413604i	$-0.135730\pi$
$674$	9.85529e6	0.835641
$675$	0	0
$676$	−620234.	−0.0522022
$677$	1.01440e7i	0.850628i	0.905046	+	0.425314i	$0.139836\pi$
$677$	1.01440e7i	0.850628i	−0.905046	+	0.425314i	$0.860164\pi$
$678$	1.88519e7i	1.57500i
$679$	1.78477e6	0.148562
$680$	0	0
$681$	2.12941e7	1.75951
$682$	− 9.67300e6i	− 0.796343i
$683$	6.41414e6i	0.526122i	0.964779	+	0.263061i	$0.0847321\pi$
$683$	6.41414e6i	0.526122i	−0.964779	+	0.263061i	$0.915268\pi$
$684$	486383.	0.0397501
$685$	0	0
$686$	1.24259e6	0.100813
$687$	1.68161e7i	1.35936i
$688$	1.12669e6i	0.0907471i
$689$	−6.47185e6	−0.519374
$690$	0	0
$691$	−8.38877e6	−0.668349	−0.334175	−	0.942511i	$-0.608458\pi$
$691$	−8.38877e6	−0.668349	−0.334175	+	0.942511i	$0.608458\pi$
$692$	1.32142e7i	1.04900i
$693$	− 4.36619e6i	− 0.345358i
$694$	2.17819e6	0.171671
$695$	0	0
$696$	−1.88685e6	−0.147643
$697$	− 3.78998e7i	− 2.95498i
$698$	− 9.62639e6i	− 0.747868i
$699$	8.34712e6	0.646165
$700$	0	0
$701$	−5.05150e6	−0.388263	−0.194131	−	0.980976i	$-0.562189\pi$
$701$	−5.05150e6	−0.388263	−0.194131	+	0.980976i	$0.562189\pi$
$702$	− 6.24099e6i	− 0.477981i
$703$	− 146098.i	− 0.0111495i
$704$	8.70138e6	0.661693
$705$	0	0
$706$	6.85979e6	0.517963
$707$	1.28318e6i	0.0965470i
$708$	− 1.14914e7i	− 0.861570i
$709$	−1.37386e7	−1.02642	−0.513212	−	0.858262i	$-0.671545\pi$
$709$	−1.37386e7	−1.02642	−0.513212	+	0.858262i	$0.671545\pi$
$710$	0	0
$711$	−1.12256e7	−0.832791
$712$	8.00986e6i	0.592141i
$713$	− 1.40694e7i	− 1.03646i
$714$	−2.29860e6	−0.168740
$715$	0	0
$716$	5.86443e6	0.427507
$717$	4.23780e6i	0.307852i
$718$	− 1.29780e7i	− 0.939496i
$719$	8.31655e6	0.599958	0.299979	−	0.953946i	$-0.403020\pi$
$719$	8.31655e6	0.599958	0.299979	+	0.953946i	$0.403020\pi$
$720$	0	0
$721$	−1.20835e6	−0.0865671
$722$	− 7.93646e6i	− 0.566609i
$723$	− 1.09437e7i	− 0.778610i
$724$	−1.30801e7	−0.927395
$725$	0	0
$726$	−1.85002e7	−1.30267
$727$	4.72548e6i	0.331596i	0.986160	+	0.165798i	$0.0530200\pi$
$727$	4.72548e6i	0.331596i	−0.986160	+	0.165798i	$0.946980\pi$
$728$	− 336924.i	− 0.0235615i
$729$	−3.54746e7	−2.47229
$730$	0	0
$731$	−1.65514e7	−1.14562
$732$	4.74907e6i	0.327590i
$733$	− 1.07477e6i	− 0.0738847i	−0.999317	−	0.0369423i	$-0.988238\pi$
$733$	− 1.07477e6i	− 0.0738847i	0.999317	−	0.0369423i	$-0.0117618\pi$
$734$	−5.09956e6	−0.349376
$735$	0	0
$736$	1.66140e7	1.13052
$737$	1.39371e7i	0.945157i
$738$	3.67064e7i	2.48086i
$739$	2.04180e7	1.37531	0.687657	−	0.726036i	$-0.258639\pi$
$739$	2.04180e7	1.37531	0.687657	+	0.726036i	$0.258639\pi$
$740$	0	0
$741$	−177118.	−0.0118500
$742$	− 1.42129e6i	− 0.0947703i
$743$	2.55138e7i	1.69552i	0.530377	+	0.847762i	$0.322050\pi$
$743$	2.55138e7i	1.69552i	−0.530377	+	0.847762i	$0.677950\pi$
$744$	−2.57617e7	−1.70625
$745$	0	0
$746$	−1.07758e7	−0.708926
$747$	− 5.32627e7i	− 3.49238i
$748$	− 2.71267e7i	− 1.77273i
$749$	−25797.6	−0.00168025
$750$	0	0
$751$	2.09344e7	1.35444	0.677220	−	0.735780i	$-0.263184\pi$
$751$	2.09344e7	1.35444	0.677220	+	0.735780i	$0.263184\pi$
$752$	− 1.87896e6i	− 0.121163i
$753$	− 2.16520e7i	− 1.39159i
$754$	200657.	0.0128536
$755$	0	0
$756$	−2.89422e6	−0.184173
$757$	− 6.00641e6i	− 0.380956i	−0.981691	−	0.190478i	$-0.938996\pi$
$757$	− 6.00641e6i	− 0.380956i	0.981691	−	0.190478i	$-0.0610039\pi$
$758$	− 2.72317e6i	− 0.172148i
$759$	−4.91328e7	−3.09576
$760$	0	0
$761$	−4.48972e6	−0.281033	−0.140517	−	0.990078i	$-0.544876\pi$
$761$	−4.48972e6	−0.281033	−0.140517	+	0.990078i	$0.544876\pi$
$762$	− 8.29291e6i	− 0.517392i
$763$	− 544085.i	− 0.0338342i
$764$	−3.00749e6	−0.186411
$765$	0	0
$766$	−1.29684e7	−0.798570
$767$	3.02283e6i	0.185535i
$768$	− 2.74911e7i	− 1.68186i
$769$	−2.20361e7	−1.34375	−0.671877	−	0.740663i	$-0.734512\pi$
$769$	−2.20361e7	−1.34375	−0.671877	+	0.740663i	$0.734512\pi$
$770$	0	0
$771$	−3.13239e7	−1.89776
$772$	1.27815e6i	0.0771859i
$773$	9.09786e6i	0.547634i	0.961782	+	0.273817i	$0.0882863\pi$
$773$	9.09786e6i	0.547634i	−0.961782	+	0.273817i	$0.911714\pi$
$774$	1.60302e7	0.961806
$775$	0	0
$776$	−2.65646e7	−1.58361
$777$	1.41208e6i	0.0839088i
$778$	9.42950e6i	0.558521i
$779$	641341.	0.0378656
$780$	0	0
$781$	1.97907e7	1.16100
$782$	1.86848e7i	1.09263i
$783$	− 4.26360e6i	− 0.248526i
$784$	2.37598e6	0.138055
$785$	0	0
$786$	1.83594e6	0.105999
$787$	1.65992e7i	0.955325i	0.878544	+	0.477662i	$0.158516\pi$
$787$	1.65992e7i	0.955325i	−0.878544	+	0.477662i	$0.841484\pi$
$788$	− 5.90374e6i	− 0.338697i
$789$	9.41298e6	0.538313
$790$	0	0
$791$	−2.29971e6	−0.130687
$792$	6.49867e7i	3.68139i
$793$	− 1.24925e6i	− 0.0705449i
$794$	3.97841e6	0.223954
$795$	0	0
$796$	1.54911e6	0.0866564
$797$	2.41936e7i	1.34913i	0.738213	+	0.674567i	$0.235670\pi$
$797$	2.41936e7i	1.34913i	−0.738213	+	0.674567i	$0.764330\pi$
$798$	− 38897.0i	− 0.00216226i
$799$	2.76024e7	1.52961
$800$	0	0
$801$	−2.93986e7	−1.61900
$802$	− 3.11707e6i	− 0.171124i
$803$	− 6.46329e6i	− 0.353724i
$804$	1.50059e7	0.818697
$805$	0	0
$806$	2.73963e6	0.148544
$807$	− 3.35647e7i	− 1.81426i
$808$	− 1.90989e7i	− 1.02916i
$809$	1.25176e7	0.672433	0.336216	−	0.941785i	$-0.390853\pi$
$809$	1.25176e7	0.672433	0.336216	+	0.941785i	$0.390853\pi$
$810$	0	0
$811$	−1.21034e7	−0.646183	−0.323091	−	0.946368i	$-0.604722\pi$
$811$	−1.21034e7	−0.646183	−0.323091	+	0.946368i	$0.604722\pi$
$812$	− 93053.3i	− 0.00495270i
$813$	− 3.11822e7i	− 1.65455i
$814$	7.89166e6	0.417453
$815$	0	0
$816$	−8.82576e6	−0.464010
$817$	− 280083.i	− 0.0146802i
$818$	− 1.83423e6i	− 0.0958450i
$819$	1.23661e6	0.0644205
$820$	0	0
$821$	9.24362e6	0.478613	0.239306	−	0.970944i	$-0.423080\pi$
$821$	9.24362e6	0.478613	0.239306	+	0.970944i	$0.423080\pi$
$822$	7.58615e6i	0.391599i
$823$	2.93437e7i	1.51013i	0.655648	+	0.755067i	$0.272396\pi$
$823$	2.93437e7i	1.51013i	−0.655648	+	0.755067i	$0.727604\pi$
$824$	1.79851e7	0.922774
$825$	0	0
$826$	−663846.	−0.0338545
$827$	− 1.74083e6i	− 0.0885099i	−0.999020	−	0.0442550i	$-0.985909\pi$
$827$	− 1.74083e6i	− 0.0885099i	0.999020	−	0.0442550i	$-0.0140914\pi$
$828$	3.82136e7i	1.93706i
$829$	−2.00227e7	−1.01190	−0.505948	−	0.862564i	$-0.668857\pi$
$829$	−2.00227e7	−1.01190	−0.505948	+	0.862564i	$0.668857\pi$
$830$	0	0
$831$	6.45447e7	3.24234
$832$	2.46444e6i	0.123427i
$833$	3.49039e7i	1.74286i
$834$	2.71301e7	1.35063
$835$	0	0
$836$	459038.	0.0227160
$837$	− 5.82123e7i	− 2.87211i
$838$	1.30332e7i	0.641121i
$839$	−1.34556e7	−0.659931	−0.329966	−	0.943993i	$-0.607037\pi$
$839$	−1.34556e7	−0.659931	−0.329966	+	0.943993i	$0.607037\pi$
$840$	0	0
$841$	−2.03741e7	−0.993317
$842$	− 1.19804e7i	− 0.582360i
$843$	− 6.25897e7i	− 3.03343i
$844$	1.04437e7	0.504661
$845$	0	0
$846$	−2.67333e7	−1.28418
$847$	− 2.25681e6i	− 0.108090i
$848$	− 5.45720e6i	− 0.260604i
$849$	−1.15666e7	−0.550727
$850$	0	0
$851$	1.14785e7	0.543326
$852$	− 2.13084e7i	− 1.00566i
$853$	− 2.46799e7i	− 1.16137i	−0.814129	−	0.580685i	$-0.802785\pi$
$853$	− 2.46799e7i	− 1.16137i	0.814129	−	0.580685i	$-0.197215\pi$
$854$	274348.	0.0128723
$855$	0	0
$856$	383973.	0.0179109
$857$	1.53970e6i	0.0716118i	0.999359	+	0.0358059i	$0.0113998\pi$
$857$	1.53970e6i	0.0716118i	−0.999359	+	0.0358059i	$0.988600\pi$
$858$	− 9.56723e6i	− 0.443678i
$859$	−2.43418e7	−1.12556	−0.562781	−	0.826606i	$-0.690268\pi$
$859$	−2.43418e7	−1.12556	−0.562781	+	0.826606i	$0.690268\pi$
$860$	0	0
$861$	−6.19875e6	−0.284968
$862$	− 4.92858e6i	− 0.225920i
$863$	2.27132e7i	1.03813i	0.854735	+	0.519064i	$0.173720\pi$
$863$	2.27132e7i	1.03813i	−0.854735	+	0.519064i	$0.826280\pi$
$864$	6.87403e7	3.13276
$865$	0	0
$866$	359681.	0.0162976
$867$	− 8.76474e7i	− 3.95996i
$868$	− 1.27049e6i	− 0.0572361i
$869$	−1.05945e7	−0.475917
$870$	0	0
$871$	−3.94733e6	−0.176302
$872$	8.09821e6i	0.360660i
$873$	− 9.75003e7i	− 4.32982i
$874$	−316184.	−0.0140011
$875$	0	0
$876$	−6.95897e6	−0.306397
$877$	− 1.86650e7i	− 0.819464i	−0.912206	−	0.409732i	$-0.865622\pi$
$877$	− 1.86650e7i	− 0.819464i	0.912206	−	0.409732i	$-0.134378\pi$
$878$	526817.i	0.0230634i
$879$	−7.23886e7	−3.16008
$880$	0	0
$881$	2.91491e7	1.26528	0.632639	−	0.774447i	$-0.281972\pi$
$881$	2.91491e7	1.26528	0.632639	+	0.774447i	$0.281972\pi$
$882$	− 3.38049e7i	− 1.46322i
$883$	837238.i	0.0361366i	0.999837	+	0.0180683i	$0.00575163\pi$
$883$	837238.i	0.0361366i	−0.999837	+	0.0180683i	$0.994248\pi$
$884$	7.68295e6	0.330672
$885$	0	0
$886$	1.48267e7	0.634543
$887$	2.00532e7i	0.855806i	0.903825	+	0.427903i	$0.140748\pi$
$887$	2.00532e7i	0.855806i	−0.903825	+	0.427903i	$0.859252\pi$
$888$	− 2.10176e7i	− 0.894437i
$889$	1.01164e6	0.0429309
$890$	0	0
$891$	−1.11612e8	−4.70997
$892$	2.92480e7i	1.23079i
$893$	467088.i	0.0196006i
$894$	4.80600e7	2.01113
$895$	0	0
$896$	1.66951e6	0.0694733
$897$	− 1.39156e7i	− 0.577459i
$898$	− 1.35186e6i	− 0.0559425i
$899$	1.87161e6	0.0772353
$900$	0	0
$901$	8.01680e7	3.28995
$902$	3.46428e7i	1.41774i
$903$	2.70708e6i	0.110480i
$904$	3.42290e7	1.39307
$905$	0	0
$906$	3.35208e7	1.35673
$907$	− 2.26638e7i	− 0.914775i	−0.889267	−	0.457388i	$-0.848785\pi$
$907$	− 2.26638e7i	− 0.914775i	0.889267	−	0.457388i	$-0.151215\pi$
$908$	− 1.56306e7i	− 0.629162i
$909$	7.00989e7	2.81386
$910$	0	0
$911$	−1.78322e7	−0.711885	−0.355942	−	0.934508i	$-0.615840\pi$
$911$	−1.78322e7	−0.711885	−0.355942	+	0.934508i	$0.615840\pi$
$912$	− 149350.i	− 0.00594589i
$913$	− 5.02682e7i	− 1.99580i
$914$	−1.71385e7	−0.678589
$915$	0	0
$916$	1.23436e7	0.486077
$917$	223962.i	0.00879533i
$918$	7.73083e7i	3.02775i
$919$	1.61471e7	0.630677	0.315338	−	0.948979i	$-0.397882\pi$
$919$	1.61471e7	0.630677	0.315338	+	0.948979i	$0.397882\pi$
$920$	0	0
$921$	4.70178e7	1.82647
$922$	− 1.58792e7i	− 0.615178i
$923$	5.60521e6i	0.216565i
$924$	−4.43674e6	−0.170956
$925$	0	0
$926$	−1.24338e7	−0.476514
$927$	6.60109e7i	2.52299i
$928$	2.21010e6i	0.0842445i
$929$	4.41295e7	1.67761	0.838803	−	0.544435i	$-0.183256\pi$
$929$	4.41295e7	1.67761	0.838803	+	0.544435i	$0.183256\pi$
$930$	0	0
$931$	−590644.	−0.0223332
$932$	− 6.12708e6i	− 0.231054i
$933$	9.32536e7i	3.50721i
$934$	5.26022e6	0.197305
$935$	0	0
$936$	−1.84058e7	−0.686698
$937$	2.76747e7i	1.02976i	0.857263	+	0.514878i	$0.172163\pi$
$937$	2.76747e7i	1.02976i	−0.857263	+	0.514878i	$0.827837\pi$
$938$	− 866876.i	− 0.0321699i
$939$	4.70214e7	1.74033
$940$	0	0
$941$	−9.84362e6	−0.362394	−0.181197	−	0.983447i	$-0.557997\pi$
$941$	−9.84362e6	−0.362394	−0.181197	+	0.983447i	$0.557997\pi$
$942$	− 2.11356e7i	− 0.776044i
$943$	5.03882e7i	1.84522i
$944$	−2.54891e6	−0.0930947
$945$	0	0
$946$	1.51290e7	0.549645
$947$	1.20978e7i	0.438360i	0.975684	+	0.219180i	$0.0703382\pi$
$947$	1.20978e7i	0.438360i	−0.975684	+	0.219180i	$0.929662\pi$
$948$	1.14070e7i	0.412240i
$949$	1.83056e6	0.0659810
$950$	0	0
$951$	4.30541e7	1.54370
$952$	4.17353e6i	0.149249i
$953$	2.22761e7i	0.794524i	0.917705	+	0.397262i	$0.130040\pi$
$953$	2.22761e7i	0.794524i	−0.917705	+	0.397262i	$0.869960\pi$
$954$	−7.76437e7	−2.76207
$955$	0	0
$956$	3.11070e6	0.110081
$957$	− 6.53596e6i	− 0.230690i
$958$	− 8.45627e6i	− 0.297690i
$959$	−925420.	−0.0324932
$960$	0	0
$961$	−3.07551e6	−0.107426
$962$	2.23511e6i	0.0778685i
$963$	1.40930e6i	0.0489708i
$964$	−8.03309e6	−0.278413
$965$	0	0
$966$	3.05602e6	0.105369
$967$	− 1.59432e7i	− 0.548288i	−0.961689	−	0.274144i	$-0.911605\pi$
$967$	− 1.59432e7i	− 0.548288i	0.961689	−	0.274144i	$-0.0883945\pi$
$968$	3.35905e7i	1.15220i
$969$	2.19399e6	0.0750629
$970$	0	0
$971$	1.41319e6	0.0481008	0.0240504	−	0.999711i	$-0.492344\pi$
$971$	1.41319e6	0.0481008	0.0240504	+	0.999711i	$0.492344\pi$
$972$	5.94038e7i	2.01673i
$973$	3.30955e6i	0.112069i
$974$	1.11326e7	0.376010
$975$	0	0
$976$	1.05339e6	0.0353969
$977$	− 3.55699e7i	− 1.19219i	−0.802913	−	0.596096i	$-0.796718\pi$
$977$	− 3.55699e7i	− 1.19219i	0.802913	−	0.596096i	$-0.203282\pi$
$978$	− 4.60923e7i	− 1.54092i
$979$	−2.77458e7	−0.925211
$980$	0	0
$981$	−2.97229e7	−0.986095
$982$	− 8.79548e6i	− 0.291059i
$983$	− 4.80377e7i	− 1.58562i	−0.609471	−	0.792808i	$-0.708618\pi$
$983$	− 4.80377e7i	− 1.58562i	0.609471	−	0.792808i	$-0.291382\pi$
$984$	9.22627e7	3.03765
$985$	0	0
$986$	−2.48557e6	−0.0814206
$987$	− 4.51455e6i	− 0.147510i
$988$	130011.i	0.00423728i
$989$	2.20052e7	0.715377
$990$	0	0
$991$	−4.57999e7	−1.48143	−0.740714	−	0.671821i	$-0.765512\pi$
$991$	−4.57999e7	−1.48143	−0.740714	+	0.671821i	$0.765512\pi$
$992$	3.01752e7i	0.973577i
$993$	9.52409e7i	3.06514i
$994$	−1.23096e6	−0.0395166
$995$	0	0
$996$	−5.41233e7	−1.72877
$997$	− 1.03492e7i	− 0.329738i	−0.986315	−	0.164869i	$-0.947280\pi$
$997$	− 1.03492e7i	− 0.329738i	0.986315	−	0.164869i	$-0.0527202\pi$
$998$	3.22498e7i	1.02494i
$999$	4.74922e7	1.50560

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	325.6.b.h.274.12		18
5.2	odd	4		325.6.a.i.1.4	yes	9
5.3	odd	4		325.6.a.h.1.6	✓	9
5.4	even	2	inner	325.6.b.h.274.7		18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
325.6.a.h.1.6	✓	9	5.3	odd	4
325.6.a.i.1.4	yes	9	5.2	odd	4
325.6.b.h.274.7		18	5.4	even	2	inner
325.6.b.h.274.12		18	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 325 = 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	325.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+3.20685i q^{2} +29.5845i q^{3} +21.7161 q^{4} -94.8732 q^{6} -11.5734i q^{7} +172.260i q^{8} -632.245 q^{9} +O(q^{10})\)	\(q+3.20685i q^{2} +29.5845i q^{3} +21.7161 q^{4} -94.8732 q^{6} -11.5734i q^{7} +172.260i q^{8} -632.245 q^{9} -596.700 q^{11} +642.461i q^{12} -169.000i q^{13} +37.1142 q^{14} +142.504 q^{16} +2093.43i q^{17} -2027.52i q^{18} -35.4251 q^{19} +342.394 q^{21} -1913.53i q^{22} -2783.24i q^{23} -5096.22 q^{24} +541.958 q^{26} -11515.6i q^{27} -251.329i q^{28} +370.244 q^{29} +5055.06 q^{31} +5969.30i q^{32} -17653.1i q^{33} -6713.33 q^{34} -13729.9 q^{36} +4124.15i q^{37} -113.603i q^{38} +4999.79 q^{39} -18104.1 q^{41} +1098.01i q^{42} +7906.34i q^{43} -12958.0 q^{44} +8925.44 q^{46} -13185.2i q^{47} +4215.93i q^{48} +16673.1 q^{49} -61933.3 q^{51} -3670.02i q^{52} -38295.0i q^{53} +36929.0 q^{54} +1993.63 q^{56} -1048.03i q^{57} +1187.32i q^{58} -17886.6 q^{59} +7392.00 q^{61} +16210.8i q^{62} +7317.23i q^{63} -14582.5 q^{64} +56610.8 q^{66} -23357.0i q^{67} +45461.2i q^{68} +82340.9 q^{69} -33166.9 q^{71} -108910. i q^{72} +10831.7i q^{73} -13225.5 q^{74} -769.295 q^{76} +6905.85i q^{77} +16033.6i q^{78} +17755.2 q^{79} +187049. q^{81} -58057.3i q^{82} +84243.8i q^{83} +7435.46 q^{84} -25354.5 q^{86} +10953.5i q^{87} -102787. i q^{88} +46498.8 q^{89} -1955.91 q^{91} -60441.1i q^{92} +149552. i q^{93} +42283.1 q^{94} -176599. q^{96} +154213. i q^{97} +53468.0i q^{98} +377260. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q - 182 q^{4} - 166 q^{6} - 1124 q^{9}+O(q^{10}) \) 18 * q - 182 * q^4 - 166 * q^6 - 1124 * q^9	\( 18 q - 182 q^{4} - 166 q^{6} - 1124 q^{9} - 2844 q^{11} + 684 q^{14} - 2122 q^{16} + 816 q^{19} - 7824 q^{21} + 16938 q^{24} - 1690 q^{26} + 17474 q^{29} + 1496 q^{31} + 35578 q^{34} + 1024 q^{36} + 3718 q^{39} - 57352 q^{41} + 61198 q^{44} - 24112 q^{46} + 81814 q^{49} - 62012 q^{51} + 205522 q^{54} - 46004 q^{56} + 176284 q^{59} + 56330 q^{61} + 201690 q^{64} + 85154 q^{66} + 363494 q^{69} - 141124 q^{71} + 271352 q^{74} + 92746 q^{76} + 328146 q^{79} - 139870 q^{81} + 691312 q^{84} - 589840 q^{86} + 505396 q^{89} + 4056 q^{91} + 1003212 q^{94} - 638362 q^{96} + 1515552 q^{99}+O(q^{100}) \) 18 * q - 182 * q^4 - 166 * q^6 - 1124 * q^9 - 2844 * q^11 + 684 * q^14 - 2122 * q^16 + 816 * q^19 - 7824 * q^21 + 16938 * q^24 - 1690 * q^26 + 17474 * q^29 + 1496 * q^31 + 35578 * q^34 + 1024 * q^36 + 3718 * q^39 - 57352 * q^41 + 61198 * q^44 - 24112 * q^46 + 81814 * q^49 - 62012 * q^51 + 205522 * q^54 - 46004 * q^56 + 176284 * q^59 + 56330 * q^61 + 201690 * q^64 + 85154 * q^66 + 363494 * q^69 - 141124 * q^71 + 271352 * q^74 + 92746 * q^76 + 328146 * q^79 - 139870 * q^81 + 691312 * q^84 - 589840 * q^86 + 505396 * q^89 + 4056 * q^91 + 1003212 * q^94 - 638362 * q^96 + 1515552 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more