LMFDB - Embedded newform 525.6.d.p.274.11

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [525,6,Mod(274,525)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(525, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("525.274");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 525 = 3 \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	525.d (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:	$84.2015054018$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 322x^{10} + 38621x^{8} + 2116025x^{6} + 52188850x^{4} + 497085625x^{2} + 1556302500 $ x^12 + 322x^10 + 38621x^8 + 2116025x^6 + 52188850x^4 + 497085625*x^2 + 1556302500
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2}\cdot 5^{6} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			274.11
Root			$10.2315i$ of defining polynomial
Character	$\chi$	$=$	525.274
Dual form			525.6.d.p.274.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+9.23147i q^{2} +9.00000i q^{3} -53.2201 q^{4} -83.0833 q^{6} +49.0000i q^{7} -195.893i q^{8} -81.0000 q^{9} +O(q^{10})$	\(q+9.23147i q^{2} +9.00000i q^{3} -53.2201 q^{4} -83.0833 q^{6} +49.0000i q^{7} -195.893i q^{8} -81.0000 q^{9} +115.341 q^{11} -478.981i q^{12} +1056.86i q^{13} -452.342 q^{14} +105.336 q^{16} -1107.19i q^{17} -747.749i q^{18} -653.129 q^{19} -441.000 q^{21} +1064.77i q^{22} -1671.00i q^{23} +1763.03 q^{24} -9756.40 q^{26} -729.000i q^{27} -2607.78i q^{28} -2716.20 q^{29} -4386.51 q^{31} -5296.17i q^{32} +1038.07i q^{33} +10221.0 q^{34} +4310.83 q^{36} -15097.2i q^{37} -6029.35i q^{38} -9511.76 q^{39} -6279.56 q^{41} -4071.08i q^{42} +12089.9i q^{43} -6138.45 q^{44} +15425.7 q^{46} -3005.01i q^{47} +948.020i q^{48} -2401.00 q^{49} +9964.67 q^{51} -56246.3i q^{52} -29628.5i q^{53} +6729.74 q^{54} +9598.74 q^{56} -5878.17i q^{57} -25074.5i q^{58} -6708.74 q^{59} +3056.80 q^{61} -40493.9i q^{62} -3969.00i q^{63} +52262.1 q^{64} -9582.89 q^{66} +10472.5i q^{67} +58924.5i q^{68} +15039.0 q^{69} +2340.71 q^{71} +15867.3i q^{72} +56370.5i q^{73} +139369. q^{74} +34759.6 q^{76} +5651.70i q^{77} -87807.6i q^{78} +23527.2 q^{79} +6561.00 q^{81} -57969.6i q^{82} +42747.4i q^{83} +23470.1 q^{84} -111608. q^{86} -24445.8i q^{87} -22594.4i q^{88} +144135. q^{89} -51786.3 q^{91} +88930.5i q^{92} -39478.6i q^{93} +27740.7 q^{94} +47665.5 q^{96} -46331.0i q^{97} -22164.8i q^{98} -9342.60 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 264 q^{4} + 72 q^{6} - 972 q^{9}+O(q^{10}) $ 12 * q - 264 * q^4 + 72 * q^6 - 972 * q^9	$ 12 q - 264 q^{4} + 72 q^{6} - 972 q^{9} + 392 q^{14} + 3568 q^{16} - 520 q^{19} - 5292 q^{21} - 5454 q^{24} - 6534 q^{26} + 4244 q^{29} + 4524 q^{31} - 24778 q^{34} + 21384 q^{36} + 3816 q^{39} + 16248 q^{41} - 62322 q^{44} + 58720 q^{46} - 28812 q^{49} + 67320 q^{51} - 5832 q^{54} - 29694 q^{56} - 51316 q^{59} + 82376 q^{61} - 131894 q^{64} + 124218 q^{66} + 33336 q^{69} + 35160 q^{71} - 7942 q^{74} + 507092 q^{76} + 19472 q^{79} + 78732 q^{81} + 116424 q^{84} + 287632 q^{86} + 660720 q^{89} + 20776 q^{91} + 747360 q^{94} + 491220 q^{96}+O(q^{100}) $ 12 * q - 264 * q^4 + 72 * q^6 - 972 * q^9 + 392 * q^14 + 3568 * q^16 - 520 * q^19 - 5292 * q^21 - 5454 * q^24 - 6534 * q^26 + 4244 * q^29 + 4524 * q^31 - 24778 * q^34 + 21384 * q^36 + 3816 * q^39 + 16248 * q^41 - 62322 * q^44 + 58720 * q^46 - 28812 * q^49 + 67320 * q^51 - 5832 * q^54 - 29694 * q^56 - 51316 * q^59 + 82376 * q^61 - 131894 * q^64 + 124218 * q^66 + 33336 * q^69 + 35160 * q^71 - 7942 * q^74 + 507092 * q^76 + 19472 * q^79 + 78732 * q^81 + 116424 * q^84 + 287632 * q^86 + 660720 * q^89 + 20776 * q^91 + 747360 * q^94 + 491220 * q^96

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/525\mathbb{Z}\right)^\times$.

$n$	$127$	$176$	$451$
$\chi(n)$	$-1$	$1$	$1$

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 $

$2$	9.23147i	1.63191i	0.578116	+	0.815955i	$0.303788\pi$
$2$	9.23147i	1.63191i	−0.578116	+	0.815955i	$0.696212\pi$
$3$	9.00000i	0.577350i
$3$	9.00000i	0.577350i
$4$	−53.2201	−1.66313
$5$	0	0
$5$	0	0
$6$	−83.0833	−0.942183
$7$	49.0000i	0.377964i
$7$	49.0000i	0.377964i
$8$	− 195.893i	− 1.08216i
$9$	−81.0000	−0.333333
$10$	0	0
$11$	115.341	0.287410	0.143705	−	0.989621i	$-0.454098\pi$
$11$	115.341	0.287410	0.143705	+	0.989621i	$0.454098\pi$
$12$	− 478.981i	− 0.960207i
$13$	1056.86i	1.73444i	0.497922	+	0.867222i	$0.334097\pi$
$13$	1056.86i	1.73444i	−0.497922	+	0.867222i	$0.665903\pi$
$14$	−452.342	−0.616804
$15$	0	0
$16$	105.336	0.102867
$17$	− 1107.19i	− 0.929176i	−0.885527	−	0.464588i	$-0.846202\pi$
$17$	− 1107.19i	− 0.929176i	0.885527	−	0.464588i	$-0.153798\pi$
$18$	− 747.749i	− 0.543970i
$19$	−653.129	−0.415064	−0.207532	−	0.978228i	$-0.566543\pi$
$19$	−653.129	−0.415064	−0.207532	+	0.978228i	$0.566543\pi$
$20$	0	0
$21$	−441.000	−0.218218
$22$	1064.77i	0.469026i
$23$	− 1671.00i	− 0.658651i	−0.944216	−	0.329326i	$-0.893179\pi$
$23$	− 1671.00i	− 0.658651i	0.944216	−	0.329326i	$-0.106821\pi$
$24$	1763.03	0.624788
$25$	0	0
$26$	−9756.40	−2.83045
$27$	− 729.000i	− 0.192450i
$28$	− 2607.78i	− 0.628603i
$29$	−2716.20	−0.599745	−0.299872	−	0.953979i	$-0.596944\pi$
$29$	−2716.20	−0.599745	−0.299872	+	0.953979i	$0.596944\pi$
$30$	0	0
$31$	−4386.51	−0.819813	−0.409906	−	0.912128i	$-0.634439\pi$
$31$	−4386.51	−0.819813	−0.409906	+	0.912128i	$0.634439\pi$
$32$	− 5296.17i	− 0.914296i
$33$	1038.07i	0.165936i
$34$	10221.0	1.51633
$35$	0	0
$36$	4310.83	0.554376
$37$	− 15097.2i	− 1.81298i	−0.422230	−	0.906489i	$-0.638753\pi$
$37$	− 15097.2i	− 1.81298i	0.422230	−	0.906489i	$-0.361247\pi$
$38$	− 6029.35i	− 0.677347i
$39$	−9511.76	−1.00138
$40$	0	0
$41$	−6279.56	−0.583404	−0.291702	−	0.956509i	$-0.594222\pi$
$41$	−6279.56	−0.583404	−0.291702	+	0.956509i	$0.594222\pi$
$42$	− 4071.08i	− 0.356112i
$43$	12089.9i	0.997133i	0.866852	+	0.498566i	$0.166140\pi$
$43$	12089.9i	0.997133i	−0.866852	+	0.498566i	$0.833860\pi$
$44$	−6138.45	−0.477999
$45$	0	0
$46$	15425.7	1.07486
$47$	− 3005.01i	− 0.198427i	−0.995066	−	0.0992135i	$-0.968367\pi$
$47$	− 3005.01i	− 0.198427i	0.995066	−	0.0992135i	$-0.0316327\pi$
$48$	948.020i	0.0593901i
$49$	−2401.00	−0.142857
$50$	0	0
$51$	9964.67	0.536460
$52$	− 56246.3i	− 2.88460i
$53$	− 29628.5i	− 1.44884i	−0.689361	−	0.724418i	$-0.742109\pi$
$53$	− 29628.5i	− 1.44884i	0.689361	−	0.724418i	$-0.257891\pi$
$54$	6729.74	0.314061
$55$	0	0
$56$	9598.74	0.409020
$57$	− 5878.17i	− 0.239637i
$58$	− 25074.5i	− 0.978729i
$59$	−6708.74	−0.250906	−0.125453	−	0.992100i	$-0.540038\pi$
$59$	−6708.74	−0.250906	−0.125453	+	0.992100i	$0.540038\pi$
$60$	0	0
$61$	3056.80	0.105182	0.0525912	−	0.998616i	$-0.483252\pi$
$61$	3056.80	0.105182	0.0525912	+	0.998616i	$0.483252\pi$
$62$	− 40493.9i	− 1.33786i
$63$	− 3969.00i	− 0.125988i
$64$	52262.1	1.59491
$65$	0	0
$66$	−9582.89	−0.270792
$67$	10472.5i	0.285012i	0.989794	+	0.142506i	$0.0455160\pi$
$67$	10472.5i	0.285012i	−0.989794	+	0.142506i	$0.954484\pi$
$68$	58924.5i	1.54534i
$69$	15039.0	0.380273
$70$	0	0
$71$	2340.71	0.0551063	0.0275531	−	0.999620i	$-0.491228\pi$
$71$	2340.71	0.0551063	0.0275531	+	0.999620i	$0.491228\pi$
$72$	15867.3i	0.360722i
$73$	56370.5i	1.23807i	0.785364	+	0.619035i	$0.212476\pi$
$73$	56370.5i	1.23807i	−0.785364	+	0.619035i	$0.787524\pi$
$74$	139369.	2.95861
$75$	0	0
$76$	34759.6	0.690305
$77$	5651.70i	0.108631i
$78$	− 87807.6i	− 1.63416i
$79$	23527.2	0.424134	0.212067	−	0.977255i	$-0.431981\pi$
$79$	23527.2	0.424134	0.212067	+	0.977255i	$0.431981\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	− 57969.6i	− 0.952063i
$83$	42747.4i	0.681106i	0.940225	+	0.340553i	$0.110614\pi$
$83$	42747.4i	0.681106i	−0.940225	+	0.340553i	$0.889386\pi$
$84$	23470.1	0.362924
$85$	0	0
$86$	−111608.	−1.62723
$87$	− 24445.8i	− 0.346263i
$88$	− 22594.4i	− 0.311024i
$89$	144135.	1.92883	0.964414	−	0.264396i	$-0.0851725\pi$
$89$	144135.	1.92883	0.964414	+	0.264396i	$0.0851725\pi$
$90$	0	0
$91$	−51786.3	−0.655558
$92$	88930.5i	1.09542i
$93$	− 39478.6i	− 0.473319i
$94$	27740.7	0.323815
$95$	0	0
$96$	47665.5	0.527869
$97$	− 46331.0i	− 0.499968i	−0.968250	−	0.249984i	$-0.919575\pi$
$97$	− 46331.0i	− 0.499968i	0.968250	−	0.249984i	$-0.0804254\pi$
$98$	− 22164.8i	− 0.233130i
$99$	−9342.60	−0.0958032
$100$	0	0
$101$	−148111.	−1.44472	−0.722359	−	0.691518i	$-0.756942\pi$
$101$	−148111.	−1.44472	−0.722359	+	0.691518i	$0.756942\pi$
$102$	91988.6i	0.875454i
$103$	− 127201.i	− 1.18140i	−0.806889	−	0.590702i	$-0.798851\pi$
$103$	− 127201.i	− 1.18140i	0.806889	−	0.590702i	$-0.201149\pi$
$104$	207032.	1.87695
$105$	0	0
$106$	273514.	2.36437
$107$	− 1736.04i	− 0.0146589i	−0.999973	−	0.00732945i	$-0.997667\pi$
$107$	− 1736.04i	− 0.0146589i	0.999973	−	0.00732945i	$-0.00233306\pi$
$108$	38797.4i	0.320069i
$109$	40629.1	0.327545	0.163772	−	0.986498i	$-0.447634\pi$
$109$	40629.1	0.327545	0.163772	+	0.986498i	$0.447634\pi$
$110$	0	0
$111$	135875.	1.04672
$112$	5161.44i	0.0388800i
$113$	9462.01i	0.0697087i	0.999392	+	0.0348544i	$0.0110967\pi$
$113$	9462.01i	0.0697087i	−0.999392	+	0.0348544i	$0.988903\pi$
$114$	54264.1	0.391067
$115$	0	0
$116$	144556.	0.997452
$117$	− 85605.9i	− 0.578148i
$118$	− 61931.5i	− 0.409456i
$119$	54252.1	0.351196
$120$	0	0
$121$	−147748.	−0.917396
$122$	28218.8i	0.171648i
$123$	− 56516.1i	− 0.336829i
$124$	233450.	1.36345
$125$	0	0
$126$	36639.7	0.205601
$127$	− 279435.i	− 1.53734i	−0.639643	−	0.768672i	$-0.720918\pi$
$127$	− 279435.i	− 1.53734i	0.639643	−	0.768672i	$-0.279082\pi$
$128$	312979.i	1.68846i
$129$	−108809.	−0.575695
$130$	0	0
$131$	245940.	1.25213	0.626067	−	0.779769i	$-0.284663\pi$
$131$	245940.	1.25213	0.626067	+	0.779769i	$0.284663\pi$
$132$	− 55246.0i	− 0.275973i
$133$	− 32003.3i	− 0.156880i
$134$	−96676.5	−0.465113
$135$	0	0
$136$	−216890.	−1.00552
$137$	330029.i	1.50228i	0.660144	+	0.751139i	$0.270495\pi$
$137$	330029.i	1.50228i	−0.660144	+	0.751139i	$0.729505\pi$
$138$	138832.i	0.620570i
$139$	−442941.	−1.94451	−0.972253	−	0.233931i	$-0.924841\pi$
$139$	−442941.	−1.94451	−0.972253	+	0.233931i	$0.924841\pi$
$140$	0	0
$141$	27045.1	0.114562
$142$	21608.2i	0.0899284i
$143$	121899.i	0.498496i
$144$	−8532.18	−0.0342889
$145$	0	0
$146$	−520383.	−2.02042
$147$	− 21609.0i	− 0.0824786i
$148$	803475.i	3.01521i
$149$	339444.	1.25257	0.626286	−	0.779593i	$-0.284574\pi$
$149$	339444.	1.25257	0.626286	+	0.779593i	$0.284574\pi$
$150$	0	0
$151$	284471.	1.01530	0.507651	−	0.861563i	$-0.330514\pi$
$151$	284471.	1.01530	0.507651	+	0.861563i	$0.330514\pi$
$152$	127943.i	0.449168i
$153$	89682.0i	0.309725i
$154$	−52173.5	−0.177275
$155$	0	0
$156$	506217.	1.66543
$157$	− 202866.i	− 0.656839i	−0.944532	−	0.328420i	$-0.893484\pi$
$157$	− 202866.i	− 0.656839i	0.944532	−	0.328420i	$-0.106516\pi$
$158$	217191.i	0.692148i
$159$	266656.	0.836486
$160$	0	0
$161$	81878.8	0.248947
$162$	60567.7i	0.181323i
$163$	− 111245.i	− 0.327953i	−0.986464	−	0.163977i	$-0.947568\pi$
$163$	− 111245.i	− 0.327953i	0.986464	−	0.163977i	$-0.0524322\pi$
$164$	334199.	0.970276
$165$	0	0
$166$	−394622.	−1.11150
$167$	− 164600.i	− 0.456708i	−0.973578	−	0.228354i	$-0.926666\pi$
$167$	− 164600.i	− 0.456708i	0.973578	−	0.228354i	$-0.0733344\pi$
$168$	86388.7i	0.236148i
$169$	−745666.	−2.00829
$170$	0	0
$171$	52903.5	0.138355
$172$	− 643428.i	− 1.65836i
$173$	408913.i	1.03876i	0.854543	+	0.519381i	$0.173837\pi$
$173$	408913.i	1.03876i	−0.854543	+	0.519381i	$0.826163\pi$
$174$	225671.	0.565070
$175$	0	0
$176$	12149.5	0.0295649
$177$	− 60378.7i	− 0.144861i
$178$	1.33058e6i	3.14767i
$179$	−520826.	−1.21495	−0.607477	−	0.794337i	$-0.707818\pi$
$179$	−520826.	−1.21495	−0.607477	+	0.794337i	$0.707818\pi$
$180$	0	0
$181$	7952.56	0.0180431	0.00902153	−	0.999959i	$-0.497128\pi$
$181$	7952.56	0.0180431	0.00902153	+	0.999959i	$0.497128\pi$
$182$	− 478064.i	− 1.06981i
$183$	27511.2i	0.0607271i
$184$	−327336.	−0.712769
$185$	0	0
$186$	364445.	0.772414
$187$	− 127704.i	− 0.267054i
$188$	159927.i	0.330010i
$189$	35721.0	0.0727393
$190$	0	0
$191$	−78169.1	−0.155043	−0.0775215	−	0.996991i	$-0.524701\pi$
$191$	−78169.1	−0.155043	−0.0775215	+	0.996991i	$0.524701\pi$
$192$	470359.i	0.920824i
$193$	539016.i	1.04162i	0.853673	+	0.520809i	$0.174370\pi$
$193$	539016.i	1.04162i	−0.853673	+	0.520809i	$0.825630\pi$
$194$	427703.	0.815902
$195$	0	0
$196$	127781.	0.237590
$197$	368116.i	0.675801i	0.941182	+	0.337900i	$0.109717\pi$
$197$	368116.i	0.675801i	−0.941182	+	0.337900i	$0.890283\pi$
$198$	− 86246.0i	− 0.156342i
$199$	−869159.	−1.55585	−0.777923	−	0.628360i	$-0.783727\pi$
$199$	−869159.	−1.55585	−0.777923	+	0.628360i	$0.783727\pi$
$200$	0	0
$201$	−94252.4	−0.164552
$202$	− 1.36728e6i	− 2.35765i
$203$	− 133094.i	− 0.226682i
$204$	−530321.	−0.892202
$205$	0	0
$206$	1.17426e6	1.92795
$207$	135351.i	0.219550i
$208$	111325.i	0.178417i
$209$	−75332.5	−0.119293
$210$	0	0
$211$	469698.	0.726295	0.363147	−	0.931732i	$-0.381702\pi$
$211$	469698.	0.726295	0.363147	+	0.931732i	$0.381702\pi$
$212$	1.57683e6i	2.40960i
$213$	21066.4i	0.0318156i
$214$	16026.2	0.0239220
$215$	0	0
$216$	−142806.	−0.208263
$217$	− 214939.i	− 0.309860i
$218$	375066.i	0.534523i
$219$	−507335.	−0.714800
$220$	0	0
$221$	1.17014e6	1.61160
$222$	1.25433e6i	1.70816i
$223$	− 221369.i	− 0.298094i	−0.988830	−	0.149047i	$-0.952379\pi$
$223$	− 221369.i	− 0.298094i	0.988830	−	0.149047i	$-0.0476206\pi$
$224$	259512.	0.345571
$225$	0	0
$226$	−87348.3	−0.113758
$227$	726061.i	0.935209i	0.883938	+	0.467604i	$0.154883\pi$
$227$	726061.i	0.935209i	−0.883938	+	0.467604i	$0.845117\pi$
$228$	312837.i	0.398548i
$229$	75075.5	0.0946040	0.0473020	−	0.998881i	$-0.484938\pi$
$229$	75075.5	0.0946040	0.0473020	+	0.998881i	$0.484938\pi$
$230$	0	0
$231$	−50865.3	−0.0627179
$232$	532084.i	0.649023i
$233$	− 709237.i	− 0.855857i	−0.903813	−	0.427929i	$-0.859243\pi$
$233$	− 709237.i	− 0.855857i	0.903813	−	0.427929i	$-0.140757\pi$
$234$	790268.	0.943485
$235$	0	0
$236$	357040.	0.417289
$237$	211745.i	0.244874i
$238$	500827.i	0.573119i
$239$	−1.16138e6	−1.31516	−0.657579	−	0.753386i	$-0.728419\pi$
$239$	−1.16138e6	−1.31516	−0.657579	+	0.753386i	$0.728419\pi$
$240$	0	0
$241$	−125181.	−0.138834	−0.0694172	−	0.997588i	$-0.522114\pi$
$241$	−125181.	−0.138834	−0.0694172	+	0.997588i	$0.522114\pi$
$242$	− 1.36393e6i	− 1.49711i
$243$	59049.0i	0.0641500i
$244$	−162683.	−0.174932
$245$	0	0
$246$	521726.	0.549674
$247$	− 690268.i	− 0.719905i
$248$	859285.i	0.887172i
$249$	−384727.	−0.393237
$250$	0	0
$251$	607822.	0.608965	0.304482	−	0.952518i	$-0.401516\pi$
$251$	607822.	0.608965	0.304482	+	0.952518i	$0.401516\pi$
$252$	211231.i	0.209534i
$253$	− 192734.i	− 0.189303i
$254$	2.57959e6	2.50881
$255$	0	0
$256$	−1.21687e6	−1.16050
$257$	− 1.48269e6i	− 1.40029i	−0.714002	−	0.700144i	$-0.753119\pi$
$257$	− 1.48269e6i	− 1.40029i	0.714002	−	0.700144i	$-0.246881\pi$
$258$	− 1.00447e6i	− 0.939482i
$259$	739763.	0.685241
$260$	0	0
$261$	220012.	0.199915
$262$	2.27039e6i	2.04337i
$263$	− 857027.i	− 0.764021i	−0.924158	−	0.382011i	$-0.875232\pi$
$263$	− 857027.i	− 0.764021i	0.924158	−	0.382011i	$-0.124768\pi$
$264$	203350.	0.179570
$265$	0	0
$266$	295438.	0.256013
$267$	1.29721e6i	1.11361i
$268$	− 557347.i	− 0.474011i
$269$	274811.	0.231555	0.115777	−	0.993275i	$-0.463064\pi$
$269$	274811.	0.231555	0.115777	+	0.993275i	$0.463064\pi$
$270$	0	0
$271$	1.30391e6	1.07851	0.539257	−	0.842141i	$-0.318705\pi$
$271$	1.30391e6	1.07851	0.539257	+	0.842141i	$0.318705\pi$
$272$	− 116626.i	− 0.0955813i
$273$	− 466076.i	− 0.378487i
$274$	−3.04665e6	−2.45158
$275$	0	0
$276$	−800375.	−0.632442
$277$	257759.i	0.201844i	0.994894	+	0.100922i	$0.0321792\pi$
$277$	257759.i	0.201844i	−0.994894	+	0.100922i	$0.967821\pi$
$278$	− 4.08900e6i	− 3.17326i
$279$	355307.	0.273271
$280$	0	0
$281$	1.83289e6	1.38475	0.692373	−	0.721540i	$-0.256565\pi$
$281$	1.83289e6	1.38475	0.692373	+	0.721540i	$0.256565\pi$
$282$	249666.i	0.186955i
$283$	944698.i	0.701176i	0.936530	+	0.350588i	$0.114018\pi$
$283$	944698.i	0.701176i	−0.936530	+	0.350588i	$0.885982\pi$
$284$	−124573.	−0.0916488
$285$	0	0
$286$	−1.12531e6	−0.813500
$287$	− 307699.i	− 0.220506i
$288$	428989.i	0.304765i
$289$	193997.	0.136631
$290$	0	0
$291$	416979.	0.288657
$292$	− 3.00005e6i	− 2.05907i
$293$	− 1.70371e6i	− 1.15938i	−0.814837	−	0.579690i	$-0.803174\pi$
$293$	− 1.70371e6i	− 1.15938i	0.814837	−	0.579690i	$-0.196826\pi$
$294$	199483.	0.134598
$295$	0	0
$296$	−2.95743e6	−1.96194
$297$	− 84083.4i	− 0.0553120i
$298$	3.13357e6i	2.04408i
$299$	1.76601e6	1.14239
$300$	0	0
$301$	−592407.	−0.376881
$302$	2.62609e6i	1.65688i
$303$	− 1.33300e6i	− 0.834108i
$304$	−68797.7	−0.0426963
$305$	0	0
$306$	−827897.	−0.505444
$307$	− 1.49763e6i	− 0.906897i	−0.891283	−	0.453448i	$-0.850194\pi$
$307$	− 1.49763e6i	− 0.906897i	0.891283	−	0.453448i	$-0.149806\pi$
$308$	− 300784.i	− 0.180667i
$309$	1.14481e6	0.682084
$310$	0	0
$311$	1.88566e6	1.10551	0.552754	−	0.833345i	$-0.313577\pi$
$311$	1.88566e6	1.10551	0.552754	+	0.833345i	$0.313577\pi$
$312$	1.86329e6i	1.08366i
$313$	1.18091e6i	0.681327i	0.940185	+	0.340663i	$0.110652\pi$
$313$	1.18091e6i	0.681327i	−0.940185	+	0.340663i	$0.889348\pi$
$314$	1.87275e6	1.07190
$315$	0	0
$316$	−1.25212e6	−0.705389
$317$	− 994641.i	− 0.555928i	−0.960592	−	0.277964i	$-0.910340\pi$
$317$	− 994641.i	− 0.555928i	0.960592	−	0.277964i	$-0.0896596\pi$
$318$	2.46163e6i	1.36507i
$319$	−313288.	−0.172372
$320$	0	0
$321$	15624.4	0.00846332
$322$	755862.i	0.406259i
$323$	723136.i	0.385668i
$324$	−349177.	−0.184792
$325$	0	0
$326$	1.02696e6	0.535190
$327$	365662.i	0.189108i
$328$	1.23012e6i	0.631340i
$329$	147245.	0.0749984
$330$	0	0
$331$	−3.26487e6	−1.63793	−0.818967	−	0.573841i	$-0.805453\pi$
$331$	−3.26487e6	−1.63793	−0.818967	+	0.573841i	$0.805453\pi$
$332$	− 2.27502e6i	− 1.13277i
$333$	1.22287e6i	0.604326i
$334$	1.51950e6	0.745306
$335$	0	0
$336$	−46453.0	−0.0224474
$337$	1.27823e6i	0.613106i	0.951854	+	0.306553i	$0.0991756\pi$
$337$	1.27823e6i	0.613106i	−0.951854	+	0.306553i	$0.900824\pi$
$338$	− 6.88359e6i	− 3.27735i
$339$	−85158.1	−0.0402464
$340$	0	0
$341$	−505943.	−0.235622
$342$	488377.i	0.225782i
$343$	− 117649.i	− 0.0539949i
$344$	2.36833e6	1.07906
$345$	0	0
$346$	−3.77487e6	−1.69517
$347$	− 4.47457e6i	− 1.99493i	−0.0711550	−	0.997465i	$-0.522668\pi$
$347$	− 4.47457e6i	− 1.99493i	0.0711550	−	0.997465i	$-0.477332\pi$
$348$	1.30101e6i	0.575879i
$349$	1.12143e6	0.492845	0.246422	−	0.969163i	$-0.420745\pi$
$349$	1.12143e6	0.492845	0.246422	+	0.969163i	$0.420745\pi$
$350$	0	0
$351$	770453.	0.333794
$352$	− 610864.i	− 0.262777i
$353$	− 2.10265e6i	− 0.898113i	−0.893503	−	0.449057i	$-0.851760\pi$
$353$	− 2.10265e6i	− 0.898113i	0.893503	−	0.449057i	$-0.148240\pi$
$354$	557384.	0.236399
$355$	0	0
$356$	−7.67086e6	−3.20789
$357$	488269.i	0.202763i
$358$	− 4.80799e6i	− 1.98269i
$359$	−4.03812e6	−1.65365	−0.826824	−	0.562461i	$-0.809855\pi$
$359$	−4.03812e6	−1.65365	−0.826824	+	0.562461i	$0.809855\pi$
$360$	0	0
$361$	−2.04952e6	−0.827722
$362$	73413.8i	0.0294447i
$363$	− 1.32973e6i	− 0.529659i
$364$	2.75607e6	1.09028
$365$	0	0
$366$	−253969.	−0.0991011
$367$	− 4.22845e6i	− 1.63876i	−0.573248	−	0.819382i	$-0.694317\pi$
$367$	− 4.22845e6i	− 1.63876i	0.573248	−	0.819382i	$-0.305683\pi$
$368$	− 176015.i	− 0.0677533i
$369$	508645.	0.194468
$370$	0	0
$371$	1.45179e6	0.547609
$372$	2.10105e6i	0.787190i
$373$	− 4.93020e6i	− 1.83482i	−0.397948	−	0.917408i	$-0.630278\pi$
$373$	− 4.93020e6i	− 1.83482i	0.397948	−	0.917408i	$-0.369722\pi$
$374$	1.17889e6	0.435808
$375$	0	0
$376$	−588659.	−0.214731
$377$	− 2.87065e6i	− 1.04022i
$378$	329757.i	0.118704i
$379$	−2.16915e6	−0.775698	−0.387849	−	0.921723i	$-0.626782\pi$
$379$	−2.16915e6	−0.775698	−0.387849	+	0.921723i	$0.626782\pi$
$380$	0	0
$381$	2.51491e6	0.887586
$382$	− 721616.i	− 0.253016i
$383$	4.85807e6i	1.69226i	0.532978	+	0.846129i	$0.321073\pi$
$383$	4.85807e6i	1.69226i	−0.532978	+	0.846129i	$0.678927\pi$
$384$	−2.81681e6	−0.974833
$385$	0	0
$386$	−4.97591e6	−1.69983
$387$	− 979285.i	− 0.332378i
$388$	2.46574e6i	0.831511i
$389$	−1.55582e6	−0.521296	−0.260648	−	0.965434i	$-0.583936\pi$
$389$	−1.55582e6	−0.521296	−0.260648	+	0.965434i	$0.583936\pi$
$390$	0	0
$391$	−1.85010e6	−0.612003
$392$	470338.i	0.154595i
$393$	2.21346e6i	0.722920i
$394$	−3.39825e6	−1.10285
$395$	0	0
$396$	497214.	0.159333
$397$	− 5.09834e6i	− 1.62350i	−0.584004	−	0.811751i	$-0.698515\pi$
$397$	− 5.09834e6i	− 1.62350i	0.584004	−	0.811751i	$-0.301485\pi$
$398$	− 8.02362e6i	− 2.53900i
$399$	288030.	0.0905744
$400$	0	0
$401$	−1.11114e6	−0.345069	−0.172535	−	0.985003i	$-0.555196\pi$
$401$	−1.11114e6	−0.345069	−0.172535	+	0.985003i	$0.555196\pi$
$402$	− 870088.i	− 0.268533i
$403$	− 4.63594e6i	− 1.42192i
$404$	7.88247e6	2.40275
$405$	0	0
$406$	1.22865e6	0.369925
$407$	− 1.74132e6i	− 0.521067i
$408$	− 1.95201e6i	− 0.580538i
$409$	−6.33105e6	−1.87140	−0.935702	−	0.352791i	$-0.885233\pi$
$409$	−6.33105e6	−1.87140	−0.935702	+	0.352791i	$0.885233\pi$
$410$	0	0
$411$	−2.97026e6	−0.867340
$412$	6.76967e6i	1.96483i
$413$	− 328728.i	− 0.0948335i
$414$	−1.24949e6	−0.358286
$415$	0	0
$416$	5.59732e6	1.58579
$417$	− 3.98647e6i	− 1.12266i
$418$	− 695430.i	− 0.194676i
$419$	4.28337e6	1.19193	0.595965	−	0.803010i	$-0.296770\pi$
$419$	4.28337e6	1.19193	0.595965	+	0.803010i	$0.296770\pi$
$420$	0	0
$421$	3.92401e6	1.07901	0.539504	−	0.841983i	$-0.318612\pi$
$421$	3.92401e6	1.07901	0.539504	+	0.841983i	$0.318612\pi$
$422$	4.33601e6i	1.18525i
$423$	243406.i	0.0661424i
$424$	−5.80400e6	−1.56788
$425$	0	0
$426$	−194474.	−0.0519202
$427$	149783.i	0.0397552i
$428$	92392.4i	0.0243796i
$429$	−1.09709e6	−0.287807
$430$	0	0
$431$	−3.14678e6	−0.815969	−0.407984	−	0.912989i	$-0.633768\pi$
$431$	−3.14678e6	−0.815969	−0.407984	+	0.912989i	$0.633768\pi$
$432$	− 76789.6i	− 0.0197967i
$433$	− 5.96601e6i	− 1.52920i	−0.644505	−	0.764600i	$-0.722937\pi$
$433$	− 5.96601e6i	− 1.52920i	0.644505	−	0.764600i	$-0.277063\pi$
$434$	1.98420e6	0.505663
$435$	0	0
$436$	−2.16228e6	−0.544749
$437$	1.09138e6i	0.273383i
$438$	− 4.68345e6i	− 1.16649i
$439$	−1.16111e6	−0.287550	−0.143775	−	0.989610i	$-0.545924\pi$
$439$	−1.16111e6	−0.287550	−0.143775	+	0.989610i	$0.545924\pi$
$440$	0	0
$441$	194481.	0.0476190
$442$	1.08021e7i	2.62999i
$443$	− 1.81040e6i	− 0.438294i	−0.975692	−	0.219147i	$-0.929673\pi$
$443$	− 1.81040e6i	− 0.438294i	0.975692	−	0.219147i	$-0.0703274\pi$
$444$	−7.23128e6	−1.74083
$445$	0	0
$446$	2.04356e6	0.486463
$447$	3.05500e6i	0.723173i
$448$	2.56085e6i	0.602821i
$449$	−5.96390e6	−1.39609	−0.698047	−	0.716052i	$-0.745947\pi$
$449$	−5.96390e6	−1.39609	−0.698047	+	0.716052i	$0.745947\pi$
$450$	0	0
$451$	−724290.	−0.167676
$452$	− 503569.i	− 0.115935i
$453$	2.56024e6i	0.586185i
$454$	−6.70261e6	−1.52618
$455$	0	0
$456$	−1.15149e6	−0.259327
$457$	5.15025e6i	1.15355i	0.816902	+	0.576776i	$0.195690\pi$
$457$	5.15025e6i	1.15355i	−0.816902	+	0.576776i	$0.804310\pi$
$458$	693057.i	0.154385i
$459$	−807138.	−0.178820
$460$	0	0
$461$	−4.76460e6	−1.04418	−0.522089	−	0.852891i	$-0.674847\pi$
$461$	−4.76460e6	−1.04418	−0.522089	+	0.852891i	$0.674847\pi$
$462$	− 469562.i	− 0.102350i
$463$	4.56442e6i	0.989539i	0.869024	+	0.494770i	$0.164748\pi$
$463$	4.56442e6i	0.989539i	−0.869024	+	0.494770i	$0.835252\pi$
$464$	−286112.	−0.0616938
$465$	0	0
$466$	6.54730e6	1.39668
$467$	1.16512e6i	0.247217i	0.992331	+	0.123609i	$0.0394467\pi$
$467$	1.16512e6i	0.247217i	−0.992331	+	0.123609i	$0.960553\pi$
$468$	4.55595e6i	0.961534i
$469$	−513152.	−0.107724
$470$	0	0
$471$	1.82579e6	0.379226
$472$	1.31419e6i	0.271522i
$473$	1.39446e6i	0.286585i
$474$	−1.95472e6	−0.399612
$475$	0	0
$476$	−2.88730e6	−0.584083
$477$	2.39990e6i	0.482945i
$478$	− 1.07212e7i	− 2.14622i
$479$	9.15981e6	1.82410	0.912048	−	0.410084i	$-0.134500\pi$
$479$	9.15981e6	1.82410	0.912048	+	0.410084i	$0.134500\pi$
$480$	0	0
$481$	1.59557e7	3.14451
$482$	− 1.15561e6i	− 0.226565i
$483$	736909.i	0.143730i
$484$	7.86314e6	1.52575
$485$	0	0
$486$	−545109.	−0.104687
$487$	8.61453e6i	1.64592i	0.568098	+	0.822961i	$0.307679\pi$
$487$	8.61453e6i	1.64592i	−0.568098	+	0.822961i	$0.692321\pi$
$488$	− 598806.i	− 0.113825i
$489$	1.00121e6	0.189344
$490$	0	0
$491$	−8.42942e6	−1.57795	−0.788977	−	0.614423i	$-0.789389\pi$
$491$	−8.42942e6	−1.57795	−0.788977	+	0.614423i	$0.789389\pi$
$492$	3.00779e6i	0.560189i
$493$	3.00734e6i	0.557269i
$494$	6.37219e6	1.17482
$495$	0	0
$496$	−462055.	−0.0843314
$497$	114695.i	0.0208282i
$498$	− 3.55159e6i	− 0.641727i
$499$	−3.81339e6	−0.685582	−0.342791	−	0.939412i	$-0.611372\pi$
$499$	−3.81339e6	−0.685582	−0.342791	+	0.939412i	$0.611372\pi$
$500$	0	0
$501$	1.48140e6	0.263681
$502$	5.61109e6i	0.993776i
$503$	8.58671e6i	1.51324i	0.653857	+	0.756618i	$0.273150\pi$
$503$	8.58671e6i	1.51324i	−0.653857	+	0.756618i	$0.726850\pi$
$504$	−777498.	−0.136340
$505$	0	0
$506$	1.77922e6	0.308925
$507$	− 6.71099e6i	− 1.15949i
$508$	1.48715e7i	2.55680i
$509$	−2.65821e6	−0.454772	−0.227386	−	0.973805i	$-0.573018\pi$
$509$	−2.65821e6	−0.454772	−0.227386	+	0.973805i	$0.573018\pi$
$510$	0	0
$511$	−2.76216e6	−0.467946
$512$	− 1.21818e6i	− 0.205369i
$513$	476131.i	0.0798791i
$514$	1.36874e7	2.28514
$515$	0	0
$516$	5.79085e6	0.957454
$517$	− 346600.i	− 0.0570298i
$518$	6.82911e6i	1.11825i
$519$	−3.68022e6	−0.599730
$520$	0	0
$521$	−208330.	−0.0336246	−0.0168123	−	0.999859i	$-0.505352\pi$
$521$	−208330.	−0.0336246	−0.0168123	+	0.999859i	$0.505352\pi$
$522$	2.03104e6i	0.326243i
$523$	− 5.25405e6i	− 0.839924i	−0.907542	−	0.419962i	$-0.862043\pi$
$523$	− 5.25405e6i	− 0.839924i	0.907542	−	0.419962i	$-0.137957\pi$
$524$	−1.30890e7	−2.08246
$525$	0	0
$526$	7.91163e6	1.24681
$527$	4.85668e6i	0.761750i
$528$	109345.i	0.0170693i
$529$	3.64412e6	0.566178
$530$	0	0
$531$	543408.	0.0836353
$532$	1.70322e6i	0.260911i
$533$	− 6.63663e6i	− 1.01188i
$534$	−1.19752e7	−1.81731
$535$	0	0
$536$	2.05148e6	0.308430
$537$	− 4.68743e6i	− 0.701454i
$538$	2.53691e6i	0.377876i
$539$	−276933.	−0.0410585
$540$	0	0
$541$	−5.26838e6	−0.773898	−0.386949	−	0.922101i	$-0.626471\pi$
$541$	−5.26838e6	−0.773898	−0.386949	+	0.922101i	$0.626471\pi$
$542$	1.20371e7i	1.76004i
$543$	71573.0i	0.0104172i
$544$	−5.86384e6	−0.849542
$545$	0	0
$546$	4.30257e6	0.617656
$547$	− 1.16069e7i	− 1.65862i	−0.558789	−	0.829310i	$-0.688734\pi$
$547$	− 1.16069e7i	− 1.65862i	0.558789	−	0.829310i	$-0.311266\pi$
$548$	− 1.75642e7i	− 2.49848i
$549$	−247601.	−0.0350608
$550$	0	0
$551$	1.77403e6	0.248933
$552$	− 2.94602e6i	− 0.411518i
$553$	1.15283e6i	0.160308i
$554$	−2.37950e6	−0.329390
$555$	0	0
$556$	2.35734e7	3.23396
$557$	− 1.31114e7i	− 1.79065i	−0.445409	−	0.895327i	$-0.646942\pi$
$557$	− 1.31114e7i	− 1.79065i	0.445409	−	0.895327i	$-0.353058\pi$
$558$	3.28001e6i	0.445953i
$559$	−1.27774e7	−1.72947
$560$	0	0
$561$	1.14933e6	0.154184
$562$	1.69203e7i	2.25978i
$563$	7.18218e6i	0.954960i	0.878643	+	0.477480i	$0.158450\pi$
$563$	7.18218e6i	0.954960i	−0.878643	+	0.477480i	$0.841550\pi$
$564$	−1.43934e6	−0.190531
$565$	0	0
$566$	−8.72096e6	−1.14426
$567$	321489.i	0.0419961i
$568$	− 458527.i	− 0.0596341i
$569$	−1.87452e6	−0.242722	−0.121361	−	0.992608i	$-0.538726\pi$
$569$	−1.87452e6	−0.242722	−0.121361	+	0.992608i	$0.538726\pi$
$570$	0	0
$571$	8.33432e6	1.06974	0.534872	−	0.844933i	$-0.320360\pi$
$571$	8.33432e6	1.06974	0.534872	+	0.844933i	$0.320360\pi$
$572$	− 6.48750e6i	− 0.829062i
$573$	− 703522.i	− 0.0895141i
$574$	2.84051e6	0.359846
$575$	0	0
$576$	−4.23323e6	−0.531638
$577$	− 9.62870e6i	− 1.20400i	−0.798494	−	0.602002i	$-0.794370\pi$
$577$	− 9.62870e6i	− 1.20400i	0.798494	−	0.602002i	$-0.205630\pi$
$578$	1.79088e6i	0.222970i
$579$	−4.85115e6	−0.601379
$580$	0	0
$581$	−2.09462e6	−0.257434
$582$	3.84933e6i	0.471061i
$583$	− 3.41737e6i	− 0.416409i
$584$	1.10426e7	1.33980
$585$	0	0
$586$	1.57277e7	1.89200
$587$	883712.i	0.105856i	0.998598	+	0.0529280i	$0.0168554\pi$
$587$	883712.i	0.105856i	−0.998598	+	0.0529280i	$0.983145\pi$
$588$	1.15003e6i	0.137172i
$589$	2.86496e6	0.340275
$590$	0	0
$591$	−3.31304e6	−0.390174
$592$	− 1.59027e6i	− 0.186495i
$593$	1.02593e7i	1.19807i	0.800725	+	0.599033i	$0.204448\pi$
$593$	1.02593e7i	1.19807i	−0.800725	+	0.599033i	$0.795552\pi$
$594$	776214.	0.0902642
$595$	0	0
$596$	−1.80653e7	−2.08319
$597$	− 7.82243e6i	− 0.898268i
$598$	1.63029e7i	1.86428i
$599$	3.88028e6	0.441871	0.220936	−	0.975288i	$-0.429089\pi$
$599$	3.88028e6	0.441871	0.220936	+	0.975288i	$0.429089\pi$
$600$	0	0
$601$	−9.91233e6	−1.11941	−0.559706	−	0.828691i	$-0.689086\pi$
$601$	−9.91233e6	−1.11941	−0.559706	+	0.828691i	$0.689086\pi$
$602$	− 5.46879e6i	− 0.615035i
$603$	− 848271.i	− 0.0950039i
$604$	−1.51396e7	−1.68858
$605$	0	0
$606$	1.23055e7	1.36119
$607$	1.06131e7i	1.16915i	0.811341	+	0.584574i	$0.198738\pi$
$607$	1.06131e7i	1.16915i	−0.811341	+	0.584574i	$0.801262\pi$
$608$	3.45908e6i	0.379491i
$609$	1.19784e6	0.130875
$610$	0	0
$611$	3.17588e6	0.344161
$612$	− 4.77289e6i	− 0.515113i
$613$	4.28735e6i	0.460827i	0.973093	+	0.230413i	$0.0740079\pi$
$613$	4.28735e6i	0.460827i	−0.973093	+	0.230413i	$0.925992\pi$
$614$	1.38253e7	1.47997
$615$	0	0
$616$	1.10713e6	0.117556
$617$	− 1.69434e7i	− 1.79179i	−0.444269	−	0.895894i	$-0.646536\pi$
$617$	− 1.69434e7i	− 1.79179i	0.444269	−	0.895894i	$-0.353464\pi$
$618$	1.05683e7i	1.11310i
$619$	6.23958e6	0.654529	0.327265	−	0.944933i	$-0.393873\pi$
$619$	6.23958e6	0.654529	0.327265	+	0.944933i	$0.393873\pi$
$620$	0	0
$621$	−1.21816e6	−0.126758
$622$	1.74074e7i	1.80409i
$623$	7.06260e6i	0.729029i
$624$	−1.00193e6	−0.103009
$625$	0	0
$626$	−1.09015e7	−1.11186
$627$	− 677992.i	− 0.0688741i
$628$	1.07965e7i	1.09241i
$629$	−1.67154e7	−1.68458
$630$	0	0
$631$	−1.71539e7	−1.71510	−0.857552	−	0.514397i	$-0.828016\pi$
$631$	−1.71539e7	−1.71510	−0.857552	+	0.514397i	$0.828016\pi$
$632$	− 4.60881e6i	− 0.458983i
$633$	4.22729e6i	0.419326i
$634$	9.18201e6	0.907224
$635$	0	0
$636$	−1.41915e7	−1.39118
$637$	− 2.53753e6i	− 0.247778i
$638$	− 2.89211e6i	− 0.281296i
$639$	−189597.	−0.0183688
$640$	0	0
$641$	−9.88678e6	−0.950407	−0.475203	−	0.879876i	$-0.657626\pi$
$641$	−9.88678e6	−0.950407	−0.475203	+	0.879876i	$0.657626\pi$
$642$	144236.i	0.0138114i
$643$	− 2.46320e6i	− 0.234948i	−0.993076	−	0.117474i	$-0.962520\pi$
$643$	− 2.46320e6i	− 0.234948i	0.993076	−	0.117474i	$-0.0374797\pi$
$644$	−4.35760e6	−0.414030
$645$	0	0
$646$	−6.67561e6	−0.629375
$647$	− 882523.i	− 0.0828830i	−0.999141	−	0.0414415i	$-0.986805\pi$
$647$	− 882523.i	− 0.0828830i	0.999141	−	0.0414415i	$-0.0131950\pi$
$648$	− 1.28525e6i	− 0.120241i
$649$	−773791.	−0.0721128
$650$	0	0
$651$	1.93445e6	0.178898
$652$	5.92047e6i	0.545428i
$653$	− 1.86617e7i	− 1.71264i	−0.516442	−	0.856322i	$-0.672744\pi$
$653$	− 1.86617e7i	− 1.71264i	0.516442	−	0.856322i	$-0.327256\pi$
$654$	−3.37559e6	−0.308607
$655$	0	0
$656$	−661461.	−0.0600129
$657$	− 4.56601e6i	− 0.412690i
$658$	1.35929e6i	0.122391i
$659$	1.24055e7	1.11276	0.556381	−	0.830927i	$-0.312189\pi$
$659$	1.24055e7	1.11276	0.556381	+	0.830927i	$0.312189\pi$
$660$	0	0
$661$	−8.12571e6	−0.723366	−0.361683	−	0.932301i	$-0.617798\pi$
$661$	−8.12571e6	−0.723366	−0.361683	+	0.932301i	$0.617798\pi$
$662$	− 3.01396e7i	− 2.67296i
$663$	1.05313e7i	0.930460i
$664$	8.37391e6	0.737069
$665$	0	0
$666$	−1.12889e7	−0.986205
$667$	4.53876e6i	0.395023i
$668$	8.76003e6i	0.759564i
$669$	1.99232e6	0.172105
$670$	0	0
$671$	352574.	0.0302304
$672$	2.33561e6i	0.199516i
$673$	− 1.81600e7i	− 1.54553i	−0.634691	−	0.772766i	$-0.718873\pi$
$673$	− 1.81600e7i	− 1.54553i	0.634691	−	0.772766i	$-0.281127\pi$
$674$	−1.18000e7	−1.00053
$675$	0	0
$676$	3.96844e7	3.34005
$677$	2.04138e7i	1.71180i	0.517145	+	0.855898i	$0.326995\pi$
$677$	2.04138e7i	1.71180i	−0.517145	+	0.855898i	$0.673005\pi$
$678$	− 786135.i	− 0.0656784i
$679$	2.27022e6	0.188970
$680$	0	0
$681$	−6.53455e6	−0.539943
$682$	− 4.67060e6i	− 0.384514i
$683$	8.44161e6i	0.692426i	0.938156	+	0.346213i	$0.112533\pi$
$683$	8.44161e6i	0.692426i	−0.938156	+	0.346213i	$0.887467\pi$
$684$	−2.81553e6	−0.230102
$685$	0	0
$686$	1.08607e6	0.0881148
$687$	675679.i	0.0546196i
$688$	1.27350e6i	0.102572i
$689$	3.13132e7	2.51292
$690$	0	0
$691$	−1.61933e7	−1.29015	−0.645075	−	0.764119i	$-0.723174\pi$
$691$	−1.61933e7	−1.29015	−0.645075	+	0.764119i	$0.723174\pi$
$692$	− 2.17624e7i	− 1.72759i
$693$	− 457788.i	− 0.0362102i
$694$	4.13069e7	3.25555
$695$	0	0
$696$	−4.78875e6	−0.374713
$697$	6.95264e6i	0.542086i
$698$	1.03525e7i	0.804278i
$699$	6.38313e6	0.494129
$700$	0	0
$701$	−9.12040e6	−0.701001	−0.350501	−	0.936562i	$-0.613989\pi$
$701$	−9.12040e6	−0.701001	−0.350501	+	0.936562i	$0.613989\pi$
$702$	7.11241e6i	0.544721i
$703$	9.86043e6i	0.752502i
$704$	6.02796e6	0.458394
$705$	0	0
$706$	1.94106e7	1.46564
$707$	− 7.25742e6i	− 0.546052i
$708$	3.21336e6i	0.240922i
$709$	5.58486e6	0.417250	0.208625	−	0.977996i	$-0.433101\pi$
$709$	5.58486e6	0.417250	0.208625	+	0.977996i	$0.433101\pi$
$710$	0	0
$711$	−1.90570e6	−0.141378
$712$	− 2.82349e7i	− 2.08731i
$713$	7.32983e6i	0.539971i
$714$	−4.50744e6	−0.330891
$715$	0	0
$716$	2.77184e7	2.02062
$717$	− 1.04524e7i	− 0.759307i
$718$	− 3.72778e7i	− 2.69860i
$719$	−1.04372e7	−0.752944	−0.376472	−	0.926428i	$-0.622863\pi$
$719$	−1.04372e7	−0.752944	−0.376472	+	0.926428i	$0.622863\pi$
$720$	0	0
$721$	6.23287e6	0.446529
$722$	− 1.89201e7i	− 1.35077i
$723$	− 1.12663e6i	− 0.0801560i
$724$	−423236.	−0.0300079
$725$	0	0
$726$	1.22753e7	0.864355
$727$	1.83852e7i	1.29013i	0.764129	+	0.645064i	$0.223169\pi$
$727$	1.83852e7i	1.29013i	−0.764129	+	0.645064i	$0.776831\pi$
$728$	1.01446e7i	0.709422i
$729$	−531441.	−0.0370370
$730$	0	0
$731$	1.33858e7	0.926512
$732$	− 1.46415e6i	− 0.100997i
$733$	950020.i	0.0653090i	0.999467	+	0.0326545i	$0.0103961\pi$
$733$	950020.i	0.0653090i	−0.999467	+	0.0326545i	$0.989604\pi$
$734$	3.90348e7	2.67431
$735$	0	0
$736$	−8.84987e6	−0.602202
$737$	1.20790e6i	0.0819151i
$738$	4.69554e6i	0.317354i
$739$	1.66516e7	1.12162	0.560809	−	0.827945i	$-0.310490\pi$
$739$	1.66516e7	1.12162	0.560809	+	0.827945i	$0.310490\pi$
$740$	0	0
$741$	6.21241e6	0.415638
$742$	1.34022e7i	0.893648i
$743$	3.93319e6i	0.261381i	0.991423	+	0.130690i	$0.0417193\pi$
$743$	3.93319e6i	0.261381i	−0.991423	+	0.130690i	$0.958281\pi$
$744$	−7.73356e6	−0.512209
$745$	0	0
$746$	4.55130e7	2.99425
$747$	− 3.46254e6i	− 0.227035i
$748$	6.79640e6i	0.444145i
$749$	85066.1	0.00554054
$750$	0	0
$751$	−2.07892e7	−1.34505	−0.672526	−	0.740074i	$-0.734790\pi$
$751$	−2.07892e7	−1.34505	−0.672526	+	0.740074i	$0.734790\pi$
$752$	− 316534.i	− 0.0204115i
$753$	5.47040e6i	0.351586i
$754$	2.65003e7	1.69755
$755$	0	0
$756$	−1.90108e6	−0.120975
$757$	1.32199e7i	0.838471i	0.907878	+	0.419235i	$0.137702\pi$
$757$	1.32199e7i	0.838471i	−0.907878	+	0.419235i	$0.862298\pi$
$758$	− 2.00245e7i	− 1.26587i
$759$	1.73461e6	0.109294
$760$	0	0
$761$	1.05925e7	0.663034	0.331517	−	0.943449i	$-0.392440\pi$
$761$	1.05925e7	0.663034	0.331517	+	0.943449i	$0.392440\pi$
$762$	2.32164e7i	1.44846i
$763$	1.99082e6i	0.123800i
$764$	4.16017e6	0.257856
$765$	0	0
$766$	−4.48471e7	−2.76161
$767$	− 7.09022e6i	− 0.435182i
$768$	− 1.09518e7i	− 0.670014i
$769$	−2.36257e7	−1.44069	−0.720343	−	0.693618i	$-0.756016\pi$
$769$	−2.36257e7	−1.44069	−0.720343	+	0.693618i	$0.756016\pi$
$770$	0	0
$771$	1.33442e7	0.808456
$772$	− 2.86865e7i	− 1.73234i
$773$	− 4.74893e6i	− 0.285856i	−0.989733	−	0.142928i	$-0.954348\pi$
$773$	− 4.74893e6i	− 0.285856i	0.989733	−	0.142928i	$-0.0456517\pi$
$774$	9.04024e6	0.542410
$775$	0	0
$776$	−9.07590e6	−0.541048
$777$	6.65787e6i	0.395624i
$778$	− 1.43625e7i	− 0.850708i
$779$	4.10137e6	0.242150
$780$	0	0
$781$	269979.	0.0158381
$782$	− 1.70792e7i	− 0.998734i
$783$	1.98011e6i	0.115421i
$784$	−252911.	−0.0146952
$785$	0	0
$786$	−2.04335e7	−1.17974
$787$	310955.i	0.0178962i	0.999960	+	0.00894811i	$0.00284831\pi$
$787$	310955.i	0.0178962i	−0.999960	+	0.00894811i	$0.997152\pi$
$788$	− 1.95912e7i	− 1.12394i
$789$	7.71325e6	0.441108
$790$	0	0
$791$	−463638.	−0.0263474
$792$	1.83015e6i	0.103675i
$793$	3.23062e6i	0.182433i
$794$	4.70652e7	2.64941
$795$	0	0
$796$	4.62567e7	2.58757
$797$	1.43413e7i	0.799731i	0.916574	+	0.399866i	$0.130943\pi$
$797$	1.43413e7i	0.799731i	−0.916574	+	0.399866i	$0.869057\pi$
$798$	2.65894e6i	0.147809i
$799$	−3.32710e6	−0.184374
$800$	0	0
$801$	−1.16749e7	−0.642943
$802$	− 1.02574e7i	− 0.563122i
$803$	6.50182e6i	0.355833i
$804$	5.01612e6	0.273670
$805$	0	0
$806$	4.27965e7	2.32044
$807$	2.47330e6i	0.133688i
$808$	2.90138e7i	1.56342i
$809$	−1.28777e7	−0.691780	−0.345890	−	0.938275i	$-0.612423\pi$
$809$	−1.28777e7	−0.691780	−0.345890	+	0.938275i	$0.612423\pi$
$810$	0	0
$811$	184994.	0.00987655	0.00493828	−	0.999988i	$-0.498428\pi$
$811$	184994.	0.00987655	0.00493828	+	0.999988i	$0.498428\pi$
$812$	7.08326e6i	0.377002i
$813$	1.17352e7i	0.622681i
$814$	1.60750e7	0.850334
$815$	0	0
$816$	1.04963e6	0.0551839
$817$	− 7.89629e6i	− 0.413874i
$818$	− 5.84449e7i	− 3.05396i
$819$	4.19469e6	0.218519
$820$	0	0
$821$	−8.53546e6	−0.441946	−0.220973	−	0.975280i	$-0.570923\pi$
$821$	−8.53546e6	−0.441946	−0.220973	+	0.975280i	$0.570923\pi$
$822$	− 2.74198e7i	− 1.41542i
$823$	− 1.24082e7i	− 0.638572i	−0.947658	−	0.319286i	$-0.896557\pi$
$823$	− 1.24082e7i	− 0.638572i	0.947658	−	0.319286i	$-0.103443\pi$
$824$	−2.49178e7	−1.27847
$825$	0	0
$826$	3.03465e6	0.154760
$827$	− 3.13008e7i	− 1.59145i	−0.605660	−	0.795723i	$-0.707091\pi$
$827$	− 3.13008e7i	− 1.59145i	0.605660	−	0.795723i	$-0.292909\pi$
$828$	− 7.20337e6i	− 0.365141i
$829$	−1.61781e7	−0.817600	−0.408800	−	0.912624i	$-0.634053\pi$
$829$	−1.61781e7	−0.817600	−0.408800	+	0.912624i	$0.634053\pi$
$830$	0	0
$831$	−2.31983e6	−0.116534
$832$	5.52339e7i	2.76629i
$833$	2.65835e6i	0.132739i
$834$	3.68010e7	1.83208
$835$	0	0
$836$	4.00920e6	0.198400
$837$	3.19776e6i	0.157773i
$838$	3.95419e7i	1.94512i
$839$	−1.23824e7	−0.607294	−0.303647	−	0.952785i	$-0.598204\pi$
$839$	−1.23824e7	−0.607294	−0.303647	+	0.952785i	$0.598204\pi$
$840$	0	0
$841$	−1.31334e7	−0.640306
$842$	3.62244e7i	1.76084i
$843$	1.64960e7i	0.799483i
$844$	−2.49974e7	−1.20792
$845$	0	0
$846$	−2.24699e6	−0.107938
$847$	− 7.23963e6i	− 0.346743i
$848$	− 3.12093e6i	− 0.149037i
$849$	−8.50228e6	−0.404824
$850$	0	0
$851$	−2.52274e7	−1.19412
$852$	− 1.12115e6i	− 0.0529135i
$853$	2.90262e7i	1.36589i	0.730468	+	0.682947i	$0.239302\pi$
$853$	2.90262e7i	1.36589i	−0.730468	+	0.682947i	$0.760698\pi$
$854$	−1.38272e6	−0.0648769
$855$	0	0
$856$	−340078.	−0.0158633
$857$	2.95624e7i	1.37495i	0.726208	+	0.687475i	$0.241281\pi$
$857$	2.95624e7i	1.37495i	−0.726208	+	0.687475i	$0.758719\pi$
$858$	− 1.01278e7i	− 0.469674i
$859$	−1.38347e7	−0.639716	−0.319858	−	0.947465i	$-0.603635\pi$
$859$	−1.38347e7	−0.639716	−0.319858	+	0.947465i	$0.603635\pi$
$860$	0	0
$861$	2.76929e6	0.127309
$862$	− 2.90494e7i	− 1.33159i
$863$	2.46236e7i	1.12545i	0.826645	+	0.562723i	$0.190246\pi$
$863$	2.46236e7i	1.12545i	−0.826645	+	0.562723i	$0.809754\pi$
$864$	−3.86090e6	−0.175956
$865$	0	0
$866$	5.50750e7	2.49551
$867$	1.74597e6i	0.0788842i
$868$	1.14391e7i	0.515337i
$869$	2.71365e6	0.121900
$870$	0	0
$871$	−1.10680e7	−0.494337
$872$	− 7.95894e6i	− 0.354457i
$873$	3.75281e6i	0.166656i
$874$	−1.00750e7	−0.446136
$875$	0	0
$876$	2.70004e7	1.18880
$877$	− 9.33000e6i	− 0.409621i	−0.978802	−	0.204811i	$-0.934342\pi$
$877$	− 9.33000e6i	− 0.409621i	0.978802	−	0.204811i	$-0.0656579\pi$
$878$	− 1.07188e7i	− 0.469255i
$879$	1.53333e7	0.669368
$880$	0	0
$881$	−1.19978e7	−0.520788	−0.260394	−	0.965502i	$-0.583853\pi$
$881$	−1.19978e7	−0.520788	−0.260394	+	0.965502i	$0.583853\pi$
$882$	1.79535e6i	0.0777100i
$883$	− 3.35916e7i	− 1.44987i	−0.688818	−	0.724934i	$-0.741870\pi$
$883$	− 3.35916e7i	− 1.44987i	0.688818	−	0.724934i	$-0.258130\pi$
$884$	−6.22751e7	−2.68030
$885$	0	0
$886$	1.67127e7	0.715255
$887$	− 1.36404e6i	− 0.0582128i	−0.999576	−	0.0291064i	$-0.990734\pi$
$887$	− 1.36404e6i	− 0.0582128i	0.999576	−	0.0291064i	$-0.00926617\pi$
$888$	− 2.66169e7i	− 1.13273i
$889$	1.36923e7	0.581062
$890$	0	0
$891$	756751.	0.0319344
$892$	1.17813e7i	0.495769i
$893$	1.96266e6i	0.0823600i
$894$	−2.82021e7	−1.18015
$895$	0	0
$896$	−1.53360e7	−0.638178
$897$	1.58941e7i	0.659561i
$898$	− 5.50556e7i	− 2.27830i
$899$	1.19146e7	0.491678
$900$	0	0
$901$	−3.28042e7	−1.34622
$902$	− 6.68626e6i	− 0.273632i
$903$	− 5.33166e6i	− 0.217592i
$904$	1.85354e6	0.0754363
$905$	0	0
$906$	−2.36348e7	−0.956601
$907$	− 4.08257e7i	− 1.64784i	−0.566705	−	0.823921i	$-0.691782\pi$
$907$	− 4.08257e7i	− 1.64784i	0.566705	−	0.823921i	$-0.308218\pi$
$908$	− 3.86410e7i	− 1.55537i
$909$	1.19970e7	0.481573
$910$	0	0
$911$	−4.16530e7	−1.66284	−0.831421	−	0.555643i	$-0.812472\pi$
$911$	−4.16530e7	−1.66284	−0.831421	+	0.555643i	$0.812472\pi$
$912$	− 619180.i	− 0.0246507i
$913$	4.93052e6i	0.195756i
$914$	−4.75444e7	−1.88249
$915$	0	0
$916$	−3.99552e6	−0.157338
$917$	1.20511e7i	0.473262i
$918$	− 7.45108e6i	− 0.291818i
$919$	−3.68906e6	−0.144088	−0.0720438	−	0.997401i	$-0.522952\pi$
$919$	−3.68906e6	−0.144088	−0.0720438	+	0.997401i	$0.522952\pi$
$920$	0	0
$921$	1.34786e7	0.523597
$922$	− 4.39843e7i	− 1.70400i
$923$	2.47381e6i	0.0955787i
$924$	2.70706e6	0.104308
$925$	0	0
$926$	−4.21363e7	−1.61484
$927$	1.03033e7i	0.393802i
$928$	1.43854e7i	0.548344i
$929$	−2.31197e7	−0.878908	−0.439454	−	0.898265i	$-0.644828\pi$
$929$	−2.31197e7	−0.878908	−0.439454	+	0.898265i	$0.644828\pi$
$930$	0	0
$931$	1.56816e6	0.0592949
$932$	3.77456e7i	1.42340i
$933$	1.69709e7i	0.638265i
$934$	−1.07558e7	−0.403436
$935$	0	0
$936$	−1.67696e7	−0.625651
$937$	1.37315e7i	0.510937i	0.966817	+	0.255469i	$0.0822298\pi$
$937$	1.37315e7i	0.510937i	−0.966817	+	0.255469i	$0.917770\pi$
$938$	− 4.73715e6i	− 0.175796i
$939$	−1.06282e7	−0.393364
$940$	0	0
$941$	9.22651e6	0.339675	0.169837	−	0.985472i	$-0.445676\pi$
$941$	9.22651e6	0.339675	0.169837	+	0.985472i	$0.445676\pi$
$942$	1.68547e7i	0.618863i
$943$	1.04931e7i	0.384260i
$944$	−706668.	−0.0258099
$945$	0	0
$946$	−1.28729e7	−0.467681
$947$	− 1.14581e7i	− 0.415182i	−0.978216	−	0.207591i	$-0.933438\pi$
$947$	− 1.14581e7i	− 0.415182i	0.978216	−	0.207591i	$-0.0665624\pi$
$948$	− 1.12691e7i	− 0.407256i
$949$	−5.95759e7	−2.14736
$950$	0	0
$951$	8.95177e6	0.320965
$952$	− 1.06276e7i	− 0.380052i
$953$	3.41960e7i	1.21967i	0.792528	+	0.609836i	$0.208765\pi$
$953$	3.41960e7i	1.21967i	−0.792528	+	0.609836i	$0.791235\pi$
$954$	−2.21547e7	−0.788123
$955$	0	0
$956$	6.18085e7	2.18728
$957$	− 2.81960e6i	− 0.0995193i
$958$	8.45585e7i	2.97676i
$959$	−1.61714e7	−0.567807
$960$	0	0
$961$	−9.38771e6	−0.327907
$962$	1.47294e8i	5.13155i
$963$	140620.i	0.00488630i
$964$	6.66216e6	0.230899
$965$	0	0
$966$	−6.80276e6	−0.234554
$967$	3.18624e7i	1.09575i	0.836559	+	0.547876i	$0.184564\pi$
$967$	3.18624e7i	1.09575i	−0.836559	+	0.547876i	$0.815436\pi$
$968$	2.89427e7i	0.992773i
$969$	−6.50822e6	−0.222665
$970$	0	0
$971$	−1.58702e6	−0.0540176	−0.0270088	−	0.999635i	$-0.508598\pi$
$971$	−1.58702e6	−0.0540176	−0.0270088	+	0.999635i	$0.508598\pi$
$972$	− 3.14259e6i	− 0.106690i
$973$	− 2.17041e7i	− 0.734954i
$974$	−7.95248e7	−2.68600
$975$	0	0
$976$	321990.	0.0108198
$977$	− 5.22296e7i	− 1.75057i	−0.483603	−	0.875287i	$-0.660672\pi$
$977$	− 5.22296e7i	− 1.75057i	0.483603	−	0.875287i	$-0.339328\pi$
$978$	9.24260e6i	0.308992i
$979$	1.66246e7	0.554364
$980$	0	0
$981$	−3.29095e6	−0.109182
$982$	− 7.78160e7i	− 2.57508i
$983$	1.96281e7i	0.647878i	0.946078	+	0.323939i	$0.105007\pi$
$983$	1.96281e7i	0.647878i	−0.946078	+	0.323939i	$0.894993\pi$
$984$	−1.10711e7	−0.364504
$985$	0	0
$986$	−2.77621e7	−0.909412
$987$	1.32521e6i	0.0433003i
$988$	3.67361e7i	1.19729i
$989$	2.02022e7	0.656763
$990$	0	0
$991$	2.84622e7	0.920628	0.460314	−	0.887756i	$-0.347737\pi$
$991$	2.84622e7	0.920628	0.460314	+	0.887756i	$0.347737\pi$
$992$	2.32317e7i	0.749551i
$993$	− 2.93839e7i	− 0.945662i
$994$	−1.05880e6	−0.0339898
$995$	0	0
$996$	2.04752e7	0.654003
$997$	− 3.84279e7i	− 1.22436i	−0.790719	−	0.612179i	$-0.790293\pi$
$997$	− 3.84279e7i	− 1.22436i	0.790719	−	0.612179i	$-0.209707\pi$
$998$	− 3.52032e7i	− 1.11881i
$999$	−1.10059e7	−0.348908

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	525.6.d.p.274.11		12
5.2	odd	4		525.6.a.s.1.1	yes	6
5.3	odd	4		525.6.a.r.1.6	✓	6
5.4	even	2	inner	525.6.d.p.274.2		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
525.6.a.r.1.6	✓	6	5.3	odd	4
525.6.a.s.1.1	yes	6	5.2	odd	4
525.6.d.p.274.2		12	5.4	even	2	inner
525.6.d.p.274.11		12	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:	\( N \)	\(=\)	\( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	525.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+9.23147i q^{2} +9.00000i q^{3} -53.2201 q^{4} -83.0833 q^{6} +49.0000i q^{7} -195.893i q^{8} -81.0000 q^{9} +O(q^{10})\)	\(q+9.23147i q^{2} +9.00000i q^{3} -53.2201 q^{4} -83.0833 q^{6} +49.0000i q^{7} -195.893i q^{8} -81.0000 q^{9} +115.341 q^{11} -478.981i q^{12} +1056.86i q^{13} -452.342 q^{14} +105.336 q^{16} -1107.19i q^{17} -747.749i q^{18} -653.129 q^{19} -441.000 q^{21} +1064.77i q^{22} -1671.00i q^{23} +1763.03 q^{24} -9756.40 q^{26} -729.000i q^{27} -2607.78i q^{28} -2716.20 q^{29} -4386.51 q^{31} -5296.17i q^{32} +1038.07i q^{33} +10221.0 q^{34} +4310.83 q^{36} -15097.2i q^{37} -6029.35i q^{38} -9511.76 q^{39} -6279.56 q^{41} -4071.08i q^{42} +12089.9i q^{43} -6138.45 q^{44} +15425.7 q^{46} -3005.01i q^{47} +948.020i q^{48} -2401.00 q^{49} +9964.67 q^{51} -56246.3i q^{52} -29628.5i q^{53} +6729.74 q^{54} +9598.74 q^{56} -5878.17i q^{57} -25074.5i q^{58} -6708.74 q^{59} +3056.80 q^{61} -40493.9i q^{62} -3969.00i q^{63} +52262.1 q^{64} -9582.89 q^{66} +10472.5i q^{67} +58924.5i q^{68} +15039.0 q^{69} +2340.71 q^{71} +15867.3i q^{72} +56370.5i q^{73} +139369. q^{74} +34759.6 q^{76} +5651.70i q^{77} -87807.6i q^{78} +23527.2 q^{79} +6561.00 q^{81} -57969.6i q^{82} +42747.4i q^{83} +23470.1 q^{84} -111608. q^{86} -24445.8i q^{87} -22594.4i q^{88} +144135. q^{89} -51786.3 q^{91} +88930.5i q^{92} -39478.6i q^{93} +27740.7 q^{94} +47665.5 q^{96} -46331.0i q^{97} -22164.8i q^{98} -9342.60 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 264 q^{4} + 72 q^{6} - 972 q^{9}+O(q^{10}) \) 12 * q - 264 * q^4 + 72 * q^6 - 972 * q^9	\( 12 q - 264 q^{4} + 72 q^{6} - 972 q^{9} + 392 q^{14} + 3568 q^{16} - 520 q^{19} - 5292 q^{21} - 5454 q^{24} - 6534 q^{26} + 4244 q^{29} + 4524 q^{31} - 24778 q^{34} + 21384 q^{36} + 3816 q^{39} + 16248 q^{41} - 62322 q^{44} + 58720 q^{46} - 28812 q^{49} + 67320 q^{51} - 5832 q^{54} - 29694 q^{56} - 51316 q^{59} + 82376 q^{61} - 131894 q^{64} + 124218 q^{66} + 33336 q^{69} + 35160 q^{71} - 7942 q^{74} + 507092 q^{76} + 19472 q^{79} + 78732 q^{81} + 116424 q^{84} + 287632 q^{86} + 660720 q^{89} + 20776 q^{91} + 747360 q^{94} + 491220 q^{96}+O(q^{100}) \) 12 * q - 264 * q^4 + 72 * q^6 - 972 * q^9 + 392 * q^14 + 3568 * q^16 - 520 * q^19 - 5292 * q^21 - 5454 * q^24 - 6534 * q^26 + 4244 * q^29 + 4524 * q^31 - 24778 * q^34 + 21384 * q^36 + 3816 * q^39 + 16248 * q^41 - 62322 * q^44 + 58720 * q^46 - 28812 * q^49 + 67320 * q^51 - 5832 * q^54 - 29694 * q^56 - 51316 * q^59 + 82376 * q^61 - 131894 * q^64 + 124218 * q^66 + 33336 * q^69 + 35160 * q^71 - 7942 * q^74 + 507092 * q^76 + 19472 * q^79 + 78732 * q^81 + 116424 * q^84 + 287632 * q^86 + 660720 * q^89 + 20776 * q^91 + 747360 * q^94 + 491220 * q^96

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more