LMFDB - Embedded newform 175.6.b.d.99.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [175,6,Mod(99,175)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(175, 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("175.99");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 175 = 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	175.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:	$28.0671684673$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{65})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 33x^{2} + 256 $ x^4 + 33*x^2 + 256
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 35)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			99.1
Root			$-4.53113i$ of defining polynomial
Character	$\chi$	$=$	175.99
Dual form			175.6.b.d.99.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.53113i q^{2} -10.5934i q^{3} +11.4689 q^{4} -48.0000 q^{6} +49.0000i q^{7} -196.963i q^{8} +130.780 q^{9} +O(q^{10})$	\(q-4.53113i q^{2} -10.5934i q^{3} +11.4689 q^{4} -48.0000 q^{6} +49.0000i q^{7} -196.963i q^{8} +130.780 q^{9} +90.5195 q^{11} -121.494i q^{12} -74.8502i q^{13} +222.025 q^{14} -525.461 q^{16} -1032.31i q^{17} -592.582i q^{18} +31.6771 q^{19} +519.076 q^{21} -410.156i q^{22} -3857.08i q^{23} -2086.51 q^{24} -339.156 q^{26} -3959.60i q^{27} +561.975i q^{28} +866.917 q^{29} +3526.76 q^{31} -3921.89i q^{32} -958.908i q^{33} -4677.54 q^{34} +1499.90 q^{36} +9531.22i q^{37} -143.533i q^{38} -792.917 q^{39} -14503.9 q^{41} -2352.00i q^{42} -9844.30i q^{43} +1038.16 q^{44} -17476.9 q^{46} +16993.5i q^{47} +5566.41i q^{48} -2401.00 q^{49} -10935.7 q^{51} -858.447i q^{52} -29621.1i q^{53} -17941.4 q^{54} +9651.19 q^{56} -335.568i q^{57} -3928.11i q^{58} -50697.4 q^{59} -2921.38 q^{61} -15980.2i q^{62} +6408.23i q^{63} -34585.3 q^{64} -4344.94 q^{66} +41086.2i q^{67} -11839.5i q^{68} -40859.5 q^{69} +61753.1 q^{71} -25758.9i q^{72} -23664.5i q^{73} +43187.2 q^{74} +363.300 q^{76} +4435.46i q^{77} +3592.81i q^{78} -45191.5 q^{79} -10166.0 q^{81} +65718.8i q^{82} +39095.9i q^{83} +5953.22 q^{84} -44605.8 q^{86} -9183.58i q^{87} -17829.0i q^{88} +41891.1 q^{89} +3667.66 q^{91} -44236.3i q^{92} -37360.4i q^{93} +76999.9 q^{94} -41546.1 q^{96} -8036.53i q^{97} +10879.2i q^{98} +11838.2 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 62 q^{4} - 192 q^{6} + 378 q^{9}+O(q^{10}) $ 4 * q + 62 * q^4 - 192 * q^6 + 378 * q^9	$ 4 q + 62 q^{4} - 192 q^{6} + 378 q^{9} - 1202 q^{11} + 98 q^{14} - 1086 q^{16} - 1260 q^{19} - 294 q^{21} - 9120 q^{24} + 2868 q^{26} - 11770 q^{29} - 792 q^{31} - 16356 q^{34} + 5274 q^{36} + 12066 q^{39} + 3548 q^{41} - 24936 q^{44} - 59072 q^{46} - 9604 q^{49} - 48822 q^{51} - 64800 q^{54} + 1470 q^{56} - 119200 q^{59} - 103692 q^{61} - 110978 q^{64} + 57696 q^{66} - 174564 q^{69} + 161488 q^{71} + 90804 q^{74} - 25120 q^{76} + 103590 q^{79} - 103356 q^{81} - 14112 q^{84} + 32808 q^{86} + 75300 q^{89} + 56546 q^{91} + 123664 q^{94} - 239712 q^{96} - 56844 q^{99}+O(q^{100}) $ 4 * q + 62 * q^4 - 192 * q^6 + 378 * q^9 - 1202 * q^11 + 98 * q^14 - 1086 * q^16 - 1260 * q^19 - 294 * q^21 - 9120 * q^24 + 2868 * q^26 - 11770 * q^29 - 792 * q^31 - 16356 * q^34 + 5274 * q^36 + 12066 * q^39 + 3548 * q^41 - 24936 * q^44 - 59072 * q^46 - 9604 * q^49 - 48822 * q^51 - 64800 * q^54 + 1470 * q^56 - 119200 * q^59 - 103692 * q^61 - 110978 * q^64 + 57696 * q^66 - 174564 * q^69 + 161488 * q^71 + 90804 * q^74 - 25120 * q^76 + 103590 * q^79 - 103356 * q^81 - 14112 * q^84 + 32808 * q^86 + 75300 * q^89 + 56546 * q^91 + 123664 * q^94 - 239712 * q^96 - 56844 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/175\mathbb{Z}\right)^\times$.

$n$	$101$	$127$
$\chi(n)$	$1$	$-1$

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 $

$2$	− 4.53113i	− 0.800998i	−0.916297	−	0.400499i	$-0.868837\pi$
$2$	− 4.53113i	− 0.800998i	0.916297	−	0.400499i	$-0.131163\pi$
$3$	− 10.5934i	− 0.679566i	−0.940504	−	0.339783i	$-0.889646\pi$
$3$	− 10.5934i	− 0.679566i	0.940504	−	0.339783i	$-0.110354\pi$
$4$	11.4689	0.358402
$5$	0	0
$5$	0	0
$6$	−48.0000	−0.544331
$7$	49.0000i	0.377964i
$7$	49.0000i	0.377964i
$8$	− 196.963i	− 1.08808i
$9$	130.780	0.538190
$10$	0	0
$11$	90.5195	0.225559	0.112780	−	0.993620i	$-0.464025\pi$
$11$	90.5195	0.225559	0.112780	+	0.993620i	$0.464025\pi$
$12$	− 121.494i	− 0.243558i
$13$	− 74.8502i	− 0.122838i	−0.998112	−	0.0614192i	$-0.980437\pi$
$13$	− 74.8502i	− 0.122838i	0.998112	−	0.0614192i	$-0.0195627\pi$
$14$	222.025	0.302749
$15$	0	0
$16$	−525.461	−0.513146
$17$	− 1032.31i	− 0.866342i	−0.901312	−	0.433171i	$-0.857395\pi$
$17$	− 1032.31i	− 0.866342i	0.901312	−	0.433171i	$-0.142605\pi$
$18$	− 592.582i	− 0.431089i
$19$	31.6771	0.0201308	0.0100654	−	0.999949i	$-0.496796\pi$
$19$	31.6771	0.0201308	0.0100654	+	0.999949i	$0.496796\pi$
$20$	0	0
$21$	519.076	0.256852
$22$	− 410.156i	− 0.180672i
$23$	− 3857.08i	− 1.52033i	−0.649728	−	0.760167i	$-0.725117\pi$
$23$	− 3857.08i	− 1.52033i	0.649728	−	0.760167i	$-0.274883\pi$
$24$	−2086.51	−0.739421
$25$	0	0
$26$	−339.156	−0.0983934
$27$	− 3959.60i	− 1.04530i
$28$	561.975i	0.135463i
$29$	866.917	0.191418	0.0957089	−	0.995409i	$-0.469488\pi$
$29$	866.917	0.191418	0.0957089	+	0.995409i	$0.469488\pi$
$30$	0	0
$31$	3526.76	0.659131	0.329566	−	0.944133i	$-0.393098\pi$
$31$	3526.76	0.659131	0.329566	+	0.944133i	$0.393098\pi$
$32$	− 3921.89i	− 0.677049i
$33$	− 958.908i	− 0.153282i
$34$	−4677.54	−0.693938
$35$	0	0
$36$	1499.90	0.192888
$37$	9531.22i	1.14458i	0.820053	+	0.572288i	$0.193944\pi$
$37$	9531.22i	1.14458i	−0.820053	+	0.572288i	$0.806056\pi$
$38$	− 143.533i	− 0.0161247i
$39$	−792.917	−0.0834769
$40$	0	0
$41$	−14503.9	−1.34748	−0.673742	−	0.738967i	$-0.735314\pi$
$41$	−14503.9	−1.34748	−0.673742	+	0.738967i	$0.735314\pi$
$42$	− 2352.00i	− 0.205738i
$43$	− 9844.30i	− 0.811921i	−0.913891	−	0.405960i	$-0.866937\pi$
$43$	− 9844.30i	− 0.811921i	0.913891	−	0.405960i	$-0.133063\pi$
$44$	1038.16	0.0808409
$45$	0	0
$46$	−17476.9	−1.21778
$47$	16993.5i	1.12212i	0.827775	+	0.561059i	$0.189606\pi$
$47$	16993.5i	1.12212i	−0.827775	+	0.561059i	$0.810394\pi$
$48$	5566.41i	0.348716i
$49$	−2401.00	−0.142857
$50$	0	0
$51$	−10935.7	−0.588736
$52$	− 858.447i	− 0.0440256i
$53$	− 29621.1i	− 1.44848i	−0.689549	−	0.724239i	$-0.742191\pi$
$53$	− 29621.1i	− 1.44848i	0.689549	−	0.724239i	$-0.257809\pi$
$54$	−17941.4	−0.837285
$55$	0	0
$56$	9651.19	0.411255
$57$	− 335.568i	− 0.0136802i
$58$	− 3928.11i	− 0.153325i
$59$	−50697.4	−1.89607	−0.948037	−	0.318159i	$-0.896935\pi$
$59$	−50697.4	−1.89607	−0.948037	+	0.318159i	$0.896935\pi$
$60$	0	0
$61$	−2921.38	−0.100522	−0.0502612	−	0.998736i	$-0.516005\pi$
$61$	−2921.38	−0.100522	−0.0502612	+	0.998736i	$0.516005\pi$
$62$	− 15980.2i	− 0.527963i
$63$	6408.23i	0.203417i
$64$	−34585.3	−1.05546
$65$	0	0
$66$	−4344.94	−0.122779
$67$	41086.2i	1.11817i	0.829109	+	0.559086i	$0.188848\pi$
$67$	41086.2i	1.11817i	−0.829109	+	0.559086i	$0.811152\pi$
$68$	− 11839.5i	− 0.310499i
$69$	−40859.5	−1.03317
$70$	0	0
$71$	61753.1	1.45383	0.726914	−	0.686729i	$-0.240954\pi$
$71$	61753.1	1.45383	0.726914	+	0.686729i	$0.240954\pi$
$72$	− 25758.9i	− 0.585592i
$73$	− 23664.5i	− 0.519745i	−0.965643	−	0.259872i	$-0.916320\pi$
$73$	− 23664.5i	− 0.519745i	0.965643	−	0.259872i	$-0.0836805\pi$
$74$	43187.2	0.916802
$75$	0	0
$76$	363.300	0.00721493
$77$	4435.46i	0.0852533i
$78$	3592.81i	0.0668648i
$79$	−45191.5	−0.814683	−0.407341	−	0.913276i	$-0.633544\pi$
$79$	−45191.5	−0.814683	−0.407341	+	0.913276i	$0.633544\pi$
$80$	0	0
$81$	−10166.0	−0.172162
$82$	65718.8i	1.07933i
$83$	39095.9i	0.622925i	0.950259	+	0.311462i	$0.100819\pi$
$83$	39095.9i	0.622925i	−0.950259	+	0.311462i	$0.899181\pi$
$84$	5953.22	0.0920563
$85$	0	0
$86$	−44605.8	−0.650347
$87$	− 9183.58i	− 0.130081i
$88$	− 17829.0i	− 0.245426i
$89$	41891.1	0.560592	0.280296	−	0.959914i	$-0.409567\pi$
$89$	41891.1	0.560592	0.280296	+	0.959914i	$0.409567\pi$
$90$	0	0
$91$	3667.66	0.0464286
$92$	− 44236.3i	− 0.544891i
$93$	− 37360.4i	− 0.447923i
$94$	76999.9	0.898815
$95$	0	0
$96$	−41546.1	−0.460099
$97$	− 8036.53i	− 0.0867240i	−0.999059	−	0.0433620i	$-0.986193\pi$
$97$	− 8036.53i	− 0.0867240i	0.999059	−	0.0433620i	$-0.0138069\pi$
$98$	10879.2i	0.114428i
$99$	11838.2	0.121394
$100$	0	0
$101$	124979.	1.21908	0.609542	−	0.792754i	$-0.291353\pi$
$101$	124979.	1.21908	0.609542	+	0.792754i	$0.291353\pi$
$102$	49551.0i	0.471577i
$103$	− 76635.1i	− 0.711762i	−0.934531	−	0.355881i	$-0.884181\pi$
$103$	− 76635.1i	− 0.711762i	0.934531	−	0.355881i	$-0.115819\pi$
$104$	−14742.7	−0.133658
$105$	0	0
$106$	−134217.	−1.16023
$107$	− 135296.i	− 1.14242i	−0.820804	−	0.571209i	$-0.806474\pi$
$107$	− 135296.i	− 1.14242i	0.820804	−	0.571209i	$-0.193526\pi$
$108$	− 45412.1i	− 0.374638i
$109$	199508.	1.60840	0.804202	−	0.594356i	$-0.202593\pi$
$109$	199508.	1.60840	0.804202	+	0.594356i	$0.202593\pi$
$110$	0	0
$111$	100968.	0.777814
$112$	− 25747.6i	− 0.193951i
$113$	122569.i	0.902991i	0.892273	+	0.451496i	$0.149109\pi$
$113$	122569.i	0.902991i	−0.892273	+	0.451496i	$0.850891\pi$
$114$	−1520.50	−0.0109578
$115$	0	0
$116$	9942.56	0.0686046
$117$	− 9788.92i	− 0.0661104i
$118$	229716.i	1.51875i
$119$	50583.4	0.327446
$120$	0	0
$121$	−152857.	−0.949123
$122$	13237.1i	0.0805183i
$123$	153645.i	0.915704i
$124$	40448.0	0.236234
$125$	0	0
$126$	29036.5	0.162936
$127$	− 40046.2i	− 0.220319i	−0.993914	−	0.110160i	$-0.964864\pi$
$127$	− 40046.2i	− 0.220319i	0.993914	−	0.110160i	$-0.0351362\pi$
$128$	31210.2i	0.168373i
$129$	−104284.	−0.551754
$130$	0	0
$131$	211963.	1.07915	0.539576	−	0.841937i	$-0.318585\pi$
$131$	211963.	1.07915	0.539576	+	0.841937i	$0.318585\pi$
$132$	− 10997.6i	− 0.0549367i
$133$	1552.18i	0.00760873i
$134$	186167.	0.895654
$135$	0	0
$136$	−203328.	−0.942647
$137$	− 40980.5i	− 0.186542i	−0.995641	−	0.0932709i	$-0.970268\pi$
$137$	− 40980.5i	− 0.186542i	0.995641	−	0.0932709i	$-0.0297323\pi$
$138$	185140.i	0.827565i
$139$	418948.	1.83917	0.919587	−	0.392886i	$-0.128523\pi$
$139$	418948.	1.83917	0.919587	+	0.392886i	$0.128523\pi$
$140$	0	0
$141$	180019.	0.762554
$142$	− 279811.i	− 1.16451i
$143$	− 6775.40i	− 0.0277073i
$144$	−68719.9	−0.276170
$145$	0	0
$146$	−107227.	−0.416314
$147$	25434.7i	0.0970809i
$148$	109312.i	0.410218i
$149$	64962.7	0.239717	0.119858	−	0.992791i	$-0.461756\pi$
$149$	64962.7	0.239717	0.119858	+	0.992791i	$0.461756\pi$
$150$	0	0
$151$	379801.	1.35554	0.677772	−	0.735272i	$-0.262946\pi$
$151$	379801.	1.35554	0.677772	+	0.735272i	$0.262946\pi$
$152$	− 6239.22i	− 0.0219039i
$153$	− 135006.i	− 0.466256i
$154$	20097.6	0.0682878
$155$	0	0
$156$	−9093.86	−0.0299183
$157$	281546.i	0.911590i	0.890085	+	0.455795i	$0.150645\pi$
$157$	281546.i	0.911590i	−0.890085	+	0.455795i	$0.849355\pi$
$158$	204768.i	0.652559i
$159$	−313788.	−0.984337
$160$	0	0
$161$	188997.	0.574632
$162$	46063.3i	0.137901i
$163$	− 382587.i	− 1.12788i	−0.825817	−	0.563938i	$-0.809286\pi$
$163$	− 382587.i	− 1.12788i	0.825817	−	0.563938i	$-0.190714\pi$
$164$	−166343.	−0.482941
$165$	0	0
$166$	177148.	0.498961
$167$	388635.i	1.07833i	0.842201	+	0.539164i	$0.181260\pi$
$167$	388635.i	1.07833i	−0.842201	+	0.539164i	$0.818740\pi$
$168$	− 102239.i	− 0.279475i
$169$	365690.	0.984911
$170$	0	0
$171$	4142.73	0.0108342
$172$	− 112903.i	− 0.290994i
$173$	− 389693.i	− 0.989936i	−0.868911	−	0.494968i	$-0.835180\pi$
$173$	− 389693.i	− 0.989936i	0.868911	−	0.494968i	$-0.164820\pi$
$174$	−41612.0	−0.104195
$175$	0	0
$176$	−47564.5	−0.115745
$177$	537057.i	1.28851i
$178$	− 189814.i	− 0.449033i
$179$	149812.	0.349473	0.174737	−	0.984615i	$-0.444093\pi$
$179$	149812.	0.349473	0.174737	+	0.984615i	$0.444093\pi$
$180$	0	0
$181$	−821019.	−1.86276	−0.931380	−	0.364049i	$-0.881394\pi$
$181$	−821019.	−1.86276	−0.931380	+	0.364049i	$0.881394\pi$
$182$	− 16618.6i	− 0.0371892i
$183$	30947.3i	0.0683117i
$184$	−759702.	−1.65424
$185$	0	0
$186$	−169285.	−0.358786
$187$	− 93444.5i	− 0.195411i
$188$	194897.i	0.402170i
$189$	194020.	0.395087
$190$	0	0
$191$	286619.	0.568487	0.284244	−	0.958752i	$-0.408258\pi$
$191$	286619.	0.568487	0.284244	+	0.958752i	$0.408258\pi$
$192$	366376.i	0.717255i
$193$	993573.i	1.92002i	0.279959	+	0.960012i	$0.409679\pi$
$193$	993573.i	1.92002i	−0.279959	+	0.960012i	$0.590321\pi$
$194$	−36414.6	−0.0694657
$195$	0	0
$196$	−27536.8	−0.0512003
$197$	38209.0i	0.0701456i	0.999385	+	0.0350728i	$0.0111663\pi$
$197$	38209.0i	0.0701456i	−0.999385	+	0.0350728i	$0.988834\pi$
$198$	− 53640.2i	− 0.0972361i
$199$	−730487.	−1.30761	−0.653807	−	0.756661i	$-0.726829\pi$
$199$	−730487.	−1.30761	−0.653807	+	0.756661i	$0.726829\pi$
$200$	0	0
$201$	435242.	0.759872
$202$	− 566296.i	− 0.976484i
$203$	42478.9i	0.0723491i
$204$	−125420.	−0.211004
$205$	0	0
$206$	−347244.	−0.570120
$207$	− 504429.i	− 0.818228i
$208$	39330.9i	0.0630340i
$209$	2867.39	0.00454069
$210$	0	0
$211$	194286.	0.300424	0.150212	−	0.988654i	$-0.452004\pi$
$211$	194286.	0.300424	0.150212	+	0.988654i	$0.452004\pi$
$212$	− 339721.i	− 0.519138i
$213$	− 654175.i	− 0.987972i
$214$	−613043.	−0.915075
$215$	0	0
$216$	−779894.	−1.13737
$217$	172811.i	0.249128i
$218$	− 903999.i	− 1.28833i
$219$	−250687.	−0.353201
$220$	0	0
$221$	−77268.8	−0.106420
$222$	− 457499.i	− 0.623028i
$223$	1.13569e6i	1.52931i	0.644438	+	0.764656i	$0.277091\pi$
$223$	1.13569e6i	1.52931i	−0.644438	+	0.764656i	$0.722909\pi$
$224$	192172.	0.255900
$225$	0	0
$226$	555375.	0.723294
$227$	143806.i	0.185231i	0.995702	+	0.0926155i	$0.0295227\pi$
$227$	143806.i	0.185231i	−0.995702	+	0.0926155i	$0.970477\pi$
$228$	− 3848.58i	− 0.00490302i
$229$	3832.50	0.00482940	0.00241470	−	0.999997i	$-0.499231\pi$
$229$	3832.50	0.00482940	0.00241470	+	0.999997i	$0.499231\pi$
$230$	0	0
$231$	46986.5	0.0579353
$232$	− 170751.i	− 0.208277i
$233$	− 1.35599e6i	− 1.63631i	−0.574995	−	0.818157i	$-0.694996\pi$
$233$	− 1.35599e6i	− 1.63631i	0.574995	−	0.818157i	$-0.305004\pi$
$234$	−44354.8	−0.0529543
$235$	0	0
$236$	−581442.	−0.679557
$237$	478731.i	0.553631i
$238$	− 229200.i	− 0.262284i
$239$	478372.	0.541716	0.270858	−	0.962619i	$-0.412693\pi$
$239$	478372.	0.541716	0.270858	+	0.962619i	$0.412693\pi$
$240$	0	0
$241$	−1.31082e6	−1.45379	−0.726894	−	0.686750i	$-0.759037\pi$
$241$	−1.31082e6	−1.45379	−0.726894	+	0.686750i	$0.759037\pi$
$242$	692616.i	0.760246i
$243$	− 854490.i	− 0.928307i
$244$	−33504.9	−0.0360275
$245$	0	0
$246$	696185.	0.733477
$247$	− 2371.04i	− 0.00247284i
$248$	− 694642.i	− 0.717186i
$249$	414157.	0.423318
$250$	0	0
$251$	340113.	0.340753	0.170376	−	0.985379i	$-0.445502\pi$
$251$	340113.	0.340753	0.170376	+	0.985379i	$0.445502\pi$
$252$	73495.1i	0.0729050i
$253$	− 349141.i	− 0.342925i
$254$	−181454.	−0.176475
$255$	0	0
$256$	−965313.	−0.920594
$257$	58839.6i	0.0555696i	0.999614	+	0.0277848i	$0.00884531\pi$
$257$	58839.6i	0.0555696i	−0.999614	+	0.0277848i	$0.991155\pi$
$258$	472526.i	0.441954i
$259$	−467030.	−0.432609
$260$	0	0
$261$	113376.	0.103019
$262$	− 960433.i	− 0.864398i
$263$	− 1380.79i	− 0.00123095i	−1.00000		0.000615473i	$-0.999804\pi$
$263$	− 1380.79i	− 0.00123095i	1.00000		0.000615473i	$-0.000195911\pi$
$264$	−188869.	−0.166783
$265$	0	0
$266$	7033.11	0.00609458
$267$	− 443769.i	− 0.380959i
$268$	471212.i	0.400756i
$269$	−886839.	−0.747247	−0.373624	−	0.927580i	$-0.621885\pi$
$269$	−886839.	−0.747247	−0.373624	+	0.927580i	$0.621885\pi$
$270$	0	0
$271$	−376955.	−0.311793	−0.155897	−	0.987773i	$-0.549827\pi$
$271$	−376955.	−0.311793	−0.155897	+	0.987773i	$0.549827\pi$
$272$	542441.i	0.444559i
$273$	− 38852.9i	− 0.0315513i
$274$	−185688.	−0.149420
$275$	0	0
$276$	−468613.	−0.370289
$277$	880363.i	0.689386i	0.938716	+	0.344693i	$0.112017\pi$
$277$	880363.i	0.689386i	−0.938716	+	0.344693i	$0.887983\pi$
$278$	− 1.89831e6i	− 1.47317i
$279$	461231.	0.354738
$280$	0	0
$281$	2.08492e6	1.57515	0.787577	−	0.616217i	$-0.211335\pi$
$281$	2.08492e6	1.57515	0.787577	+	0.616217i	$0.211335\pi$
$282$	− 815689.i	− 0.610804i
$283$	692401.i	0.513915i	0.966423	+	0.256958i	$0.0827201\pi$
$283$	692401.i	0.513915i	−0.966423	+	0.256958i	$0.917280\pi$
$284$	708238.	0.521055
$285$	0	0
$286$	−30700.2	−0.0221935
$287$	− 710689.i	− 0.509301i
$288$	− 512905.i	− 0.364381i
$289$	354186.	0.249452
$290$	0	0
$291$	−85134.1	−0.0589347
$292$	− 271405.i	− 0.186278i
$293$	2.27624e6i	1.54899i	0.632580	+	0.774495i	$0.281996\pi$
$293$	2.27624e6i	1.54899i	−0.632580	+	0.774495i	$0.718004\pi$
$294$	115248.	0.0777616
$295$	0	0
$296$	1.87730e6	1.24539
$297$	− 358421.i	− 0.235777i
$298$	− 294354.i	− 0.192013i
$299$	−288703.	−0.186755
$300$	0	0
$301$	482371.	0.306877
$302$	− 1.72093e6i	− 1.08579i
$303$	− 1.32395e6i	− 0.828448i
$304$	−16645.1	−0.0103300
$305$	0	0
$306$	−611730.	−0.373470
$307$	2.65187e6i	1.60586i	0.596076	+	0.802928i	$0.296726\pi$
$307$	2.65187e6i	1.60586i	−0.596076	+	0.802928i	$0.703274\pi$
$308$	50869.7i	0.0305550i
$309$	−811825.	−0.483689
$310$	0	0
$311$	2.57626e6	1.51039	0.755195	−	0.655501i	$-0.227542\pi$
$311$	2.57626e6	1.51039	0.755195	+	0.655501i	$0.227542\pi$
$312$	156175.i	0.0908293i
$313$	2.51549e6i	1.45131i	0.688057	+	0.725657i	$0.258464\pi$
$313$	2.51549e6i	1.45131i	−0.688057	+	0.725657i	$0.741536\pi$
$314$	1.27572e6	0.730182
$315$	0	0
$316$	−518295.	−0.291984
$317$	2.30397e6i	1.28774i	0.765135	+	0.643870i	$0.222672\pi$
$317$	2.30397e6i	1.28774i	−0.765135	+	0.643870i	$0.777328\pi$
$318$	1.42181e6i	0.788452i
$319$	78472.9	0.0431760
$320$	0	0
$321$	−1.43324e6	−0.776349
$322$	− 856369.i	− 0.460279i
$323$	− 32700.7i	− 0.0174402i
$324$	−116592.	−0.0617031
$325$	0	0
$326$	−1.73355e6	−0.903427
$327$	− 2.11347e6i	− 1.09302i
$328$	2.85672e6i	1.46617i
$329$	−832683.	−0.424121
$330$	0	0
$331$	697305.	0.349827	0.174913	−	0.984584i	$-0.444035\pi$
$331$	697305.	0.349827	0.174913	+	0.984584i	$0.444035\pi$
$332$	448385.i	0.223258i
$333$	1.24649e6i	0.615999i
$334$	1.76096e6	0.863739
$335$	0	0
$336$	−272754.	−0.131802
$337$	1.14848e6i	0.550868i	0.961320	+	0.275434i	$0.0888216\pi$
$337$	1.14848e6i	0.550868i	−0.961320	+	0.275434i	$0.911178\pi$
$338$	− 1.65699e6i	− 0.788911i
$339$	1.29842e6	0.613642
$340$	0	0
$341$	319241.	0.148673
$342$	− 18771.3i	− 0.00867817i
$343$	− 117649.i	− 0.0539949i
$344$	−1.93896e6	−0.883433
$345$	0	0
$346$	−1.76575e6	−0.792937
$347$	2.43716e6i	1.08658i	0.839546	+	0.543289i	$0.182821\pi$
$347$	2.43716e6i	1.08658i	−0.839546	+	0.543289i	$0.817179\pi$
$348$	− 105325.i	− 0.0466213i
$349$	−896801.	−0.394124	−0.197062	−	0.980391i	$-0.563140\pi$
$349$	−896801.	−0.394124	−0.197062	+	0.980391i	$0.563140\pi$
$350$	0	0
$351$	−296377.	−0.128403
$352$	− 355007.i	− 0.152715i
$353$	4.33422e6i	1.85129i	0.378396	+	0.925644i	$0.376476\pi$
$353$	4.33422e6i	1.85129i	−0.378396	+	0.925644i	$0.623524\pi$
$354$	2.43347e6	1.03209
$355$	0	0
$356$	480444.	0.200917
$357$	− 535849.i	− 0.222521i
$358$	− 678817.i	− 0.279927i
$359$	3.59167e6	1.47082	0.735411	−	0.677621i	$-0.236989\pi$
$359$	3.59167e6	1.47082	0.735411	+	0.677621i	$0.236989\pi$
$360$	0	0
$361$	−2.47510e6	−0.999595
$362$	3.72014e6i	1.49207i
$363$	1.61928e6i	0.644992i
$364$	42063.9	0.0166401
$365$	0	0
$366$	140226.	0.0547175
$367$	− 361791.i	− 0.140215i	−0.997539	−	0.0701073i	$-0.977666\pi$
$367$	− 361791.i	− 0.140215i	0.997539	−	0.0701073i	$-0.0223342\pi$
$368$	2.02674e6i	0.780152i
$369$	−1.89682e6	−0.725202
$370$	0	0
$371$	1.45144e6	0.547473
$372$	− 428481.i	− 0.160537i
$373$	− 3.08325e6i	− 1.14746i	−0.819045	−	0.573729i	$-0.805496\pi$
$373$	− 3.08325e6i	− 1.14746i	0.819045	−	0.573729i	$-0.194504\pi$
$374$	−423409.	−0.156524
$375$	0	0
$376$	3.34710e6	1.22095
$377$	− 64888.9i	− 0.0235135i
$378$	− 879131.i	− 0.316464i
$379$	5.04227e6	1.80313	0.901567	−	0.432639i	$-0.142418\pi$
$379$	5.04227e6	1.80313	0.901567	+	0.432639i	$0.142418\pi$
$380$	0	0
$381$	−424225.	−0.149721
$382$	− 1.29871e6i	− 0.455357i
$383$	− 3.11928e6i	− 1.08657i	−0.839548	−	0.543285i	$-0.817180\pi$
$383$	− 3.11928e6i	− 1.08657i	0.839548	−	0.543285i	$-0.182820\pi$
$384$	330622.	0.114420
$385$	0	0
$386$	4.50201e6	1.53794
$387$	− 1.28744e6i	− 0.436968i
$388$	− 92169.9i	− 0.0310821i
$389$	1.64560e6	0.551378	0.275689	−	0.961247i	$-0.411094\pi$
$389$	1.64560e6	0.551378	0.275689	+	0.961247i	$0.411094\pi$
$390$	0	0
$391$	−3.98171e6	−1.31713
$392$	472908.i	0.155440i
$393$	− 2.24541e6i	− 0.733355i
$394$	173130.	0.0561865
$395$	0	0
$396$	135770.	0.0435078
$397$	− 3.72727e6i	− 1.18690i	−0.804870	−	0.593451i	$-0.797765\pi$
$397$	− 3.72727e6i	− 1.18690i	0.804870	−	0.593451i	$-0.202235\pi$
$398$	3.30993e6i	1.04740i
$399$	16442.8	0.00517063
$400$	0	0
$401$	4.84820e6	1.50564	0.752818	−	0.658229i	$-0.228694\pi$
$401$	4.84820e6	1.50564	0.752818	+	0.658229i	$0.228694\pi$
$402$	− 1.97214e6i	− 0.608656i
$403$	− 263979.i	− 0.0809667i
$404$	1.43337e6	0.436923
$405$	0	0
$406$	192477.	0.0579515
$407$	862761.i	0.258169i
$408$	2.15393e6i	0.640591i
$409$	3.03126e6	0.896014	0.448007	−	0.894030i	$-0.352134\pi$
$409$	3.03126e6	0.896014	0.448007	+	0.894030i	$0.352134\pi$
$410$	0	0
$411$	−434123.	−0.126768
$412$	− 878918.i	− 0.255097i
$413$	− 2.48417e6i	− 0.716649i
$414$	−2.28563e6	−0.655399
$415$	0	0
$416$	−293554.	−0.0831676
$417$	− 4.43808e6i	− 1.24984i
$418$	− 12992.5i	− 0.00363708i
$419$	−1.66905e6	−0.464444	−0.232222	−	0.972663i	$-0.574600\pi$
$419$	−1.66905e6	−0.464444	−0.232222	+	0.972663i	$0.574600\pi$
$420$	0	0
$421$	−1.76031e6	−0.484043	−0.242022	−	0.970271i	$-0.577810\pi$
$421$	−1.76031e6	−0.484043	−0.242022	+	0.970271i	$0.577810\pi$
$422$	− 880335.i	− 0.240639i
$423$	2.22242e6i	0.603913i
$424$	−5.83427e6	−1.57606
$425$	0	0
$426$	−2.96415e6	−0.791364
$427$	− 143148.i	− 0.0379939i
$428$	− 1.55169e6i	− 0.409445i
$429$	−71774.4	−0.0188290
$430$	0	0
$431$	−648832.	−0.168244	−0.0841219	−	0.996455i	$-0.526809\pi$
$431$	−648832.	−0.168244	−0.0841219	+	0.996455i	$0.526809\pi$
$432$	2.08061e6i	0.536392i
$433$	− 360993.i	− 0.0925293i	−0.998929	−	0.0462646i	$-0.985268\pi$
$433$	− 360993.i	− 0.0925293i	0.998929	−	0.0462646i	$-0.0147318\pi$
$434$	783031.	0.199551
$435$	0	0
$436$	2.28814e6	0.576455
$437$	− 122181.i	− 0.0306055i
$438$	1.13590e6i	0.282913i
$439$	−1.07021e6	−0.265038	−0.132519	−	0.991180i	$-0.542307\pi$
$439$	−1.07021e6	−0.265038	−0.132519	+	0.991180i	$0.542307\pi$
$440$	0	0
$441$	−314003.	−0.0768843
$442$	350115.i	0.0852423i
$443$	− 797145.i	− 0.192987i	−0.995334	−	0.0964935i	$-0.969237\pi$
$443$	− 797145.i	− 0.192987i	0.995334	−	0.0964935i	$-0.0307627\pi$
$444$	1.15799e6	0.278770
$445$	0	0
$446$	5.14594e6	1.22498
$447$	− 688175.i	− 0.162903i
$448$	− 1.69468e6i	− 0.398927i
$449$	−6.40103e6	−1.49842	−0.749211	−	0.662331i	$-0.769567\pi$
$449$	−6.40103e6	−1.49842	−0.749211	+	0.662331i	$0.769567\pi$
$450$	0	0
$451$	−1.31288e6	−0.303937
$452$	1.40572e6i	0.323634i
$453$	− 4.02338e6i	− 0.921181i
$454$	651605.	0.148370
$455$	0	0
$456$	−66094.4	−0.0148851
$457$	− 5.91843e6i	− 1.32561i	−0.748791	−	0.662806i	$-0.769366\pi$
$457$	− 5.91843e6i	− 1.32561i	0.748791	−	0.662806i	$-0.230634\pi$
$458$	− 17365.5i	− 0.00386834i
$459$	−4.08755e6	−0.905588
$460$	0	0
$461$	−8.84337e6	−1.93805	−0.969026	−	0.246960i	$-0.920569\pi$
$461$	−8.84337e6	−1.93805	−0.969026	+	0.246960i	$0.920569\pi$
$462$	− 212902.i	− 0.0464060i
$463$	− 6.76858e6i	− 1.46739i	−0.679479	−	0.733695i	$-0.737794\pi$
$463$	− 6.76858e6i	− 1.46739i	0.679479	−	0.733695i	$-0.262206\pi$
$464$	−455531.	−0.0982252
$465$	0	0
$466$	−6.14416e6	−1.31068
$467$	1.04740e6i	0.222239i	0.993807	+	0.111120i	$0.0354437\pi$
$467$	1.04740e6i	0.222239i	−0.993807	+	0.111120i	$0.964556\pi$
$468$	− 112268.i	− 0.0236941i
$469$	−2.01322e6	−0.422630
$470$	0	0
$471$	2.98252e6	0.619486
$472$	9.98551e6i	2.06308i
$473$	− 891101.i	− 0.183136i
$474$	2.16919e6	0.443457
$475$	0	0
$476$	580134.	0.117358
$477$	− 3.87386e6i	− 0.779556i
$478$	− 2.16757e6i	− 0.433913i
$479$	9.37725e6	1.86740	0.933699	−	0.358060i	$-0.116562\pi$
$479$	9.37725e6	1.86740	0.933699	+	0.358060i	$0.116562\pi$
$480$	0	0
$481$	713414.	0.140598
$482$	5.93950e6i	1.16448i
$483$	− 2.00212e6i	− 0.390500i
$484$	−1.75310e6	−0.340168
$485$	0	0
$486$	−3.87181e6	−0.743572
$487$	2.88960e6i	0.552098i	0.961144	+	0.276049i	$0.0890251\pi$
$487$	2.88960e6i	0.552098i	−0.961144	+	0.276049i	$0.910975\pi$
$488$	575404.i	0.109376i
$489$	−4.05290e6	−0.766467
$490$	0	0
$491$	−1.04845e6	−0.196265	−0.0981324	−	0.995173i	$-0.531287\pi$
$491$	−1.04845e6	−0.196265	−0.0981324	+	0.995173i	$0.531287\pi$
$492$	1.76213e6i	0.328191i
$493$	− 894930.i	− 0.165833i
$494$	−10743.5	−0.00198074
$495$	0	0
$496$	−1.85318e6	−0.338230
$497$	3.02590e6i	0.549495i
$498$	− 1.87660e6i	− 0.339077i
$499$	−6.38524e6	−1.14796	−0.573979	−	0.818870i	$-0.694601\pi$
$499$	−6.38524e6	−1.14796	−0.573979	+	0.818870i	$0.694601\pi$
$500$	0	0
$501$	4.11696e6	0.732795
$502$	− 1.54110e6i	− 0.272942i
$503$	1.35497e6i	0.238787i	0.992847	+	0.119394i	$0.0380950\pi$
$503$	1.35497e6i	0.238787i	−0.992847	+	0.119394i	$0.961905\pi$
$504$	1.26218e6	0.221333
$505$	0	0
$506$	−1.58200e6	−0.274682
$507$	− 3.87390e6i	− 0.669312i
$508$	− 459285.i	− 0.0789628i
$509$	−6.43410e6	−1.10076	−0.550381	−	0.834914i	$-0.685518\pi$
$509$	−6.43410e6	−1.10076	−0.550381	+	0.834914i	$0.685518\pi$
$510$	0	0
$511$	1.15956e6	0.196445
$512$	5.37268e6i	0.905767i
$513$	− 125429.i	− 0.0210428i
$514$	266610.	0.0445111
$515$	0	0
$516$	−1.19603e6	−0.197750
$517$	1.53825e6i	0.253104i
$518$	2.11617e6i	0.346519i
$519$	−4.12817e6	−0.672727
$520$	0	0
$521$	6.16552e6	0.995120	0.497560	−	0.867430i	$-0.334229\pi$
$521$	6.16552e6	0.995120	0.497560	+	0.867430i	$0.334229\pi$
$522$	− 513719.i	− 0.0825181i
$523$	7.12400e6i	1.13886i	0.822040	+	0.569429i	$0.192836\pi$
$523$	7.12400e6i	1.13886i	−0.822040	+	0.569429i	$0.807164\pi$
$524$	2.43098e6	0.386770
$525$	0	0
$526$	−6256.55	−0.000985985 0
$527$	− 3.64072e6i	− 0.571033i
$528$	503869.i	0.0786562i
$529$	−8.44071e6	−1.31141
$530$	0	0
$531$	−6.63021e6	−1.02045
$532$	17801.7i	0.00272699i
$533$	1.08562e6i	0.165523i
$534$	−2.01077e6	−0.305148
$535$	0	0
$536$	8.09246e6	1.21666
$537$	− 1.58702e6i	− 0.237490i
$538$	4.01838e6i	0.598544i
$539$	−217337.	−0.0322227
$540$	0	0
$541$	−9.51129e6	−1.39716	−0.698580	−	0.715532i	$-0.746184\pi$
$541$	−9.51129e6	−1.39716	−0.698580	+	0.715532i	$0.746184\pi$
$542$	1.70803e6i	0.249746i
$543$	8.69737e6i	1.26587i
$544$	−4.04862e6	−0.586556
$545$	0	0
$546$	−176048.	−0.0252725
$547$	− 1.48406e6i	− 0.212071i	−0.994362	−	0.106036i	$-0.966184\pi$
$547$	− 1.48406e6i	− 0.212071i	0.994362	−	0.106036i	$-0.0338158\pi$
$548$	− 470001.i	− 0.0668570i
$549$	−382058.	−0.0541002
$550$	0	0
$551$	27461.4	0.00385340
$552$	8.04782e6i	1.12417i
$553$	− 2.21438e6i	− 0.307921i
$554$	3.98904e6	0.552197
$555$	0	0
$556$	4.80486e6	0.659164
$557$	7.72192e6i	1.05460i	0.849680	+	0.527299i	$0.176795\pi$
$557$	7.72192e6i	1.05460i	−0.849680	+	0.527299i	$0.823205\pi$
$558$	− 2.08990e6i	− 0.284144i
$559$	−736847.	−0.0997351
$560$	0	0
$561$	−989894.	−0.132795
$562$	− 9.44703e6i	− 1.26169i
$563$	3.41747e6i	0.454395i	0.973849	+	0.227198i	$0.0729563\pi$
$563$	3.41747e6i	0.454395i	−0.973849	+	0.227198i	$0.927044\pi$
$564$	2.06461e6	0.273301
$565$	0	0
$566$	3.13736e6	0.411645
$567$	− 498133.i	− 0.0650710i
$568$	− 1.21631e7i	− 1.58188i
$569$	2.42117e6	0.313505	0.156752	−	0.987638i	$-0.449898\pi$
$569$	2.42117e6	0.313505	0.156752	+	0.987638i	$0.449898\pi$
$570$	0	0
$571$	2.31941e6	0.297705	0.148853	−	0.988859i	$-0.452442\pi$
$571$	2.31941e6	0.297705	0.148853	+	0.988859i	$0.452442\pi$
$572$	− 77706.2i	− 0.00993037i
$573$	− 3.03626e6i	− 0.386325i
$574$	−3.22022e6	−0.407949
$575$	0	0
$576$	−4.52307e6	−0.568038
$577$	− 2.56645e6i	− 0.320917i	−0.987043	−	0.160459i	$-0.948703\pi$
$577$	− 2.56645e6i	− 0.320917i	0.987043	−	0.160459i	$-0.0512973\pi$
$578$	− 1.60486e6i	− 0.199811i
$579$	1.05253e7	1.30478
$580$	0	0
$581$	−1.91570e6	−0.235443
$582$	385753.i	0.0472066i
$583$	− 2.68129e6i	− 0.326718i
$584$	−4.66103e6	−0.565522
$585$	0	0
$586$	1.03139e7	1.24074
$587$	− 2.09851e6i	− 0.251371i	−0.992070	−	0.125685i	$-0.959887\pi$
$587$	− 2.09851e6i	− 0.251371i	0.992070	−	0.125685i	$-0.0401130\pi$
$588$	291708.i	0.0347940i
$589$	111718.	0.0132688
$590$	0	0
$591$	404763.	0.0476686
$592$	− 5.00829e6i	− 0.587334i
$593$	− 5.11831e6i	− 0.597709i	−0.954299	−	0.298854i	$-0.903396\pi$
$593$	− 5.11831e6i	− 0.597709i	0.954299	−	0.298854i	$-0.0966045\pi$
$594$	−1.62405e6	−0.188857
$595$	0	0
$596$	745048.	0.0859150
$597$	7.73833e6i	0.888610i
$598$	1.30815e6i	0.149591i
$599$	2.61775e6	0.298099	0.149049	−	0.988830i	$-0.452379\pi$
$599$	2.61775e6	0.298099	0.149049	+	0.988830i	$0.452379\pi$
$600$	0	0
$601$	9.49925e6	1.07276	0.536381	−	0.843976i	$-0.319791\pi$
$601$	9.49925e6	1.07276	0.536381	+	0.843976i	$0.319791\pi$
$602$	− 2.18568e6i	− 0.245808i
$603$	5.37326e6i	0.601789i
$604$	4.35589e6	0.485830
$605$	0	0
$606$	−5.99899e6	−0.663586
$607$	− 6.07366e6i	− 0.669081i	−0.942381	−	0.334541i	$-0.891419\pi$
$607$	− 6.07366e6i	− 0.669081i	0.942381	−	0.334541i	$-0.108581\pi$
$608$	− 124234.i	− 0.0136295i
$609$	449996.	0.0491660
$610$	0	0
$611$	1.27197e6	0.137839
$612$	− 1.54837e6i	− 0.167107i
$613$	− 9.20907e6i	− 0.989839i	−0.868939	−	0.494920i	$-0.835198\pi$
$613$	− 9.20907e6i	− 0.989839i	0.868939	−	0.494920i	$-0.164802\pi$
$614$	1.20160e7	1.28629
$615$	0	0
$616$	873621.	0.0927622
$617$	− 1.37224e7i	− 1.45117i	−0.688133	−	0.725585i	$-0.741569\pi$
$617$	− 1.37224e7i	− 1.45117i	0.688133	−	0.725585i	$-0.258431\pi$
$618$	3.67848e6i	0.387434i
$619$	6.80356e6	0.713690	0.356845	−	0.934164i	$-0.383852\pi$
$619$	6.80356e6	0.713690	0.356845	+	0.934164i	$0.383852\pi$
$620$	0	0
$621$	−1.52725e7	−1.58921
$622$	− 1.16734e7i	− 1.20982i
$623$	2.05266e6i	0.211884i
$624$	416647.	0.0428358
$625$	0	0
$626$	1.13980e7	1.16250
$627$	− 30375.4i	− 0.00308570i
$628$	3.22901e6i	0.326716i
$629$	9.83921e6	0.991593
$630$	0	0
$631$	−2.80897e6	−0.280850	−0.140425	−	0.990091i	$-0.544847\pi$
$631$	−2.80897e6	−0.280850	−0.140425	+	0.990091i	$0.544847\pi$
$632$	8.90105e6i	0.886438i
$633$	− 2.05815e6i	− 0.204158i
$634$	1.04396e7	1.03148
$635$	0	0
$636$	−3.59880e6	−0.352788
$637$	179715.i	0.0175484i
$638$	− 355571.i	− 0.0345839i
$639$	8.07608e6	0.782435
$640$	0	0
$641$	−3.83494e6	−0.368649	−0.184325	−	0.982865i	$-0.559010\pi$
$641$	−3.83494e6	−0.368649	−0.184325	+	0.982865i	$0.559010\pi$
$642$	6.49420e6i	0.621854i
$643$	1.59562e7i	1.52195i	0.648779	+	0.760977i	$0.275280\pi$
$643$	1.59562e7i	1.52195i	−0.648779	+	0.760977i	$0.724720\pi$
$644$	2.16758e6	0.205949
$645$	0	0
$646$	−148171.	−0.0139695
$647$	7.48025e6i	0.702514i	0.936279	+	0.351257i	$0.114246\pi$
$647$	7.48025e6i	0.702514i	−0.936279	+	0.351257i	$0.885754\pi$
$648$	2.00232e6i	0.187325i
$649$	−4.58910e6	−0.427677
$650$	0	0
$651$	1.83066e6	0.169299
$652$	− 4.38785e6i	− 0.404234i
$653$	2.62102e6i	0.240540i	0.992741	+	0.120270i	$0.0383760\pi$
$653$	2.62102e6i	0.240540i	−0.992741	+	0.120270i	$0.961624\pi$
$654$	−9.57641e6	−0.875504
$655$	0	0
$656$	7.62121e6	0.691456
$657$	− 3.09485e6i	− 0.279721i
$658$	3.77299e6i	0.339720i
$659$	8.01276e6	0.718734	0.359367	−	0.933196i	$-0.382993\pi$
$659$	8.01276e6	0.718734	0.359367	+	0.933196i	$0.382993\pi$
$660$	0	0
$661$	−7.21439e6	−0.642238	−0.321119	−	0.947039i	$-0.604059\pi$
$661$	−7.21439e6	−0.642238	−0.321119	+	0.947039i	$0.604059\pi$
$662$	− 3.15958e6i	− 0.280210i
$663$	818539.i	0.0723195i
$664$	7.70044e6	0.677790
$665$	0	0
$666$	5.64803e6	0.493414
$667$	− 3.34377e6i	− 0.291019i
$668$	4.45721e6i	0.386475i
$669$	1.20308e7	1.03927
$670$	0	0
$671$	−264442.	−0.0226738
$672$	− 2.03576e6i	− 0.173901i
$673$	1.43323e7i	1.21977i	0.792488	+	0.609887i	$0.208785\pi$
$673$	1.43323e7i	1.21977i	−0.792488	+	0.609887i	$0.791215\pi$
$674$	5.20390e6	0.441244
$675$	0	0
$676$	4.19406e6	0.352994
$677$	4.94781e6i	0.414898i	0.978246	+	0.207449i	$0.0665161\pi$
$677$	4.94781e6i	0.414898i	−0.978246	+	0.207449i	$0.933484\pi$
$678$	− 5.88330e6i	− 0.491526i
$679$	393790.	0.0327786
$680$	0	0
$681$	1.52340e6	0.125877
$682$	− 1.44652e6i	− 0.119087i
$683$	7.77331e6i	0.637608i	0.947821	+	0.318804i	$0.103281\pi$
$683$	7.77331e6i	0.637608i	−0.947821	+	0.318804i	$0.896719\pi$
$684$	47512.5	0.00388300
$685$	0	0
$686$	−533083.	−0.0432498
$687$	− 40599.1i	− 0.00328190i
$688$	5.17280e6i	0.416634i
$689$	−2.21715e6	−0.177929
$690$	0	0
$691$	−7.69151e6	−0.612797	−0.306398	−	0.951903i	$-0.599124\pi$
$691$	−7.69151e6	−0.612797	−0.306398	+	0.951903i	$0.599124\pi$
$692$	− 4.46934e6i	− 0.354795i
$693$	580070.i	0.0458825i
$694$	1.10431e7	0.870347
$695$	0	0
$696$	−1.80883e6	−0.141538
$697$	1.49725e7i	1.16738i
$698$	4.06352e6i	0.315692i
$699$	−1.43645e7	−1.11198
$700$	0	0
$701$	1.53139e7	1.17704	0.588521	−	0.808482i	$-0.299711\pi$
$701$	1.53139e7	1.17704	0.588521	+	0.808482i	$0.299711\pi$
$702$	1.34292e6i	0.102851i
$703$	301921.i	0.0230412i
$704$	−3.13065e6	−0.238069
$705$	0	0
$706$	1.96389e7	1.48288
$707$	6.12397e6i	0.460771i
$708$	6.15944e6i	0.461804i
$709$	−1.84381e7	−1.37753	−0.688764	−	0.724986i	$-0.741846\pi$
$709$	−1.84381e7	−1.37753	−0.688764	+	0.724986i	$0.741846\pi$
$710$	0	0
$711$	−5.91015e6	−0.438454
$712$	− 8.25100e6i	− 0.609968i
$713$	− 1.36030e7i	− 1.00210i
$714$	−2.42800e6	−0.178239
$715$	0	0
$716$	1.71817e6	0.125252
$717$	− 5.06758e6i	− 0.368132i
$718$	− 1.62743e7i	− 1.17813i
$719$	99951.2	0.00721051	0.00360525	−	0.999994i	$-0.498852\pi$
$719$	99951.2	0.00721051	0.00360525	+	0.999994i	$0.498852\pi$
$720$	0	0
$721$	3.75512e6	0.269021
$722$	1.12150e7i	0.800673i
$723$	1.38860e7i	0.987945i
$724$	−9.41616e6	−0.667617
$725$	0	0
$726$	7.33715e6	0.516637
$727$	− 1.25198e7i	− 0.878538i	−0.898356	−	0.439269i	$-0.855238\pi$
$727$	− 1.25198e7i	− 0.878538i	0.898356	−	0.439269i	$-0.144762\pi$
$728$	− 722393.i	− 0.0505179i
$729$	−1.15223e7	−0.803007
$730$	0	0
$731$	−1.01624e7	−0.703401
$732$	354931.i	0.0244831i
$733$	− 1.72341e7i	− 1.18476i	−0.805660	−	0.592378i	$-0.798189\pi$
$733$	− 1.72341e7i	− 1.18476i	0.805660	−	0.592378i	$-0.201811\pi$
$734$	−1.63932e6	−0.112312
$735$	0	0
$736$	−1.51270e7	−1.02934
$737$	3.71910e6i	0.252214i
$738$	8.59472e6i	0.580886i
$739$	1.83842e7	1.23832	0.619160	−	0.785265i	$-0.287473\pi$
$739$	1.83842e7	1.23832	0.619160	+	0.785265i	$0.287473\pi$
$740$	0	0
$741$	−25117.3	−0.00168046
$742$	− 6.57664e6i	− 0.438525i
$743$	2.79507e7i	1.85746i	0.370752	+	0.928732i	$0.379100\pi$
$743$	2.79507e7i	1.85746i	−0.370752	+	0.928732i	$0.620900\pi$
$744$	−7.35861e6	−0.487375
$745$	0	0
$746$	−1.39706e7	−0.919111
$747$	5.11296e6i	0.335252i
$748$	− 1.07170e6i	− 0.0700358i
$749$	6.62950e6	0.431794
$750$	0	0
$751$	1.70136e7	1.10077	0.550383	−	0.834912i	$-0.314482\pi$
$751$	1.70136e7	1.10077	0.550383	+	0.834912i	$0.314482\pi$
$752$	− 8.92944e6i	− 0.575810i
$753$	− 3.60295e6i	− 0.231564i
$754$	−294020.	−0.0188342
$755$	0	0
$756$	2.22519e6	0.141600
$757$	1.91600e7i	1.21522i	0.794234	+	0.607611i	$0.207872\pi$
$757$	1.91600e7i	1.21522i	−0.794234	+	0.607611i	$0.792128\pi$
$758$	− 2.28472e7i	− 1.44431i
$759$	−3.69858e6	−0.233040
$760$	0	0
$761$	−2.06404e7	−1.29198	−0.645992	−	0.763344i	$-0.723556\pi$
$761$	−2.06404e7	−1.29198	−0.645992	+	0.763344i	$0.723556\pi$
$762$	1.92222e6i	0.119926i
$763$	9.77592e6i	0.607919i
$764$	3.28719e6	0.203747
$765$	0	0
$766$	−1.41339e7	−0.870341
$767$	3.79471e6i	0.232911i
$768$	1.02259e7i	0.625605i
$769$	−1.11748e7	−0.681432	−0.340716	−	0.940166i	$-0.610669\pi$
$769$	−1.11748e7	−0.681432	−0.340716	+	0.940166i	$0.610669\pi$
$770$	0	0
$771$	623311.	0.0377632
$772$	1.13952e7i	0.688141i
$773$	− 3.07555e7i	− 1.85129i	−0.378394	−	0.925645i	$-0.623523\pi$
$773$	− 3.07555e7i	− 1.85129i	0.378394	−	0.925645i	$-0.376477\pi$
$774$	−5.83355e6	−0.350010
$775$	0	0
$776$	−1.58290e6	−0.0943624
$777$	4.94743e6i	0.293986i
$778$	− 7.45641e6i	− 0.441653i
$779$	−459440.	−0.0271259
$780$	0	0
$781$	5.58986e6	0.327924
$782$	1.80417e7i	1.05502i
$783$	− 3.43264e6i	− 0.200089i
$784$	1.26163e6	0.0733065
$785$	0	0
$786$	−1.01742e7	−0.587416
$787$	− 3.52357e6i	− 0.202790i	−0.994846	−	0.101395i	$-0.967669\pi$
$787$	− 3.52357e6i	− 0.202790i	0.994846	−	0.101395i	$-0.0323305\pi$
$788$	438214.i	0.0251403i
$789$	−14627.3	−0.000836509 0
$790$	0	0
$791$	−6.00587e6	−0.341299
$792$	− 2.33168e6i	− 0.132086i
$793$	218666.i	0.0123480i
$794$	−1.68887e7	−0.950706
$795$	0	0
$796$	−8.37786e6	−0.468652
$797$	3.57469e6i	0.199339i	0.995021	+	0.0996695i	$0.0317785\pi$
$797$	3.57469e6i	0.199339i	−0.995021	+	0.0996695i	$0.968221\pi$
$798$	− 74504.5i	− 0.00414167i
$799$	1.75426e7	0.972139
$800$	0	0
$801$	5.47853e6	0.301705
$802$	− 2.19678e7i	− 1.20601i
$803$	− 2.14210e6i	− 0.117233i
$804$	4.99173e6	0.272340
$805$	0	0
$806$	−1.19612e6	−0.0648542
$807$	9.39463e6i	0.507804i
$808$	− 2.46163e7i	− 1.32646i
$809$	−3.38466e7	−1.81821	−0.909104	−	0.416569i	$-0.863233\pi$
$809$	−3.38466e7	−1.81821	−0.909104	+	0.416569i	$0.863233\pi$
$810$	0	0
$811$	3.21058e7	1.71408	0.857042	−	0.515247i	$-0.172300\pi$
$811$	3.21058e7	1.71408	0.857042	+	0.515247i	$0.172300\pi$
$812$	487185.i	0.0259301i
$813$	3.99323e6i	0.211884i
$814$	3.90928e6	0.206793
$815$	0	0
$816$	5.74628e6	0.302108
$817$	− 311839.i	− 0.0163446i
$818$	− 1.37350e7i	− 0.717706i
$819$	479657.	0.0249874
$820$	0	0
$821$	3.04076e6	0.157443	0.0787216	−	0.996897i	$-0.474916\pi$
$821$	3.04076e6	0.157443	0.0787216	+	0.996897i	$0.474916\pi$
$822$	1.96707e6i	0.101541i
$823$	3.72147e6i	0.191520i	0.995404	+	0.0957601i	$0.0305282\pi$
$823$	3.72147e6i	0.191520i	−0.995404	+	0.0957601i	$0.969472\pi$
$824$	−1.50943e7	−0.774452
$825$	0	0
$826$	−1.12561e7	−0.574034
$827$	8.77895e6i	0.446354i	0.974778	+	0.223177i	$0.0716427\pi$
$827$	8.77895e6i	0.446354i	−0.974778	+	0.223177i	$0.928357\pi$
$828$	− 5.78523e6i	− 0.293255i
$829$	−6.61553e6	−0.334332	−0.167166	−	0.985929i	$-0.553462\pi$
$829$	−6.61553e6	−0.334332	−0.167166	+	0.985929i	$0.553462\pi$
$830$	0	0
$831$	9.32602e6	0.468483
$832$	2.58872e6i	0.129651i
$833$	2.47858e6i	0.123763i
$834$	−2.01095e7	−1.00112
$835$	0	0
$836$	32885.8	0.00162739
$837$	− 1.39646e7i	− 0.688991i
$838$	7.56267e6i	0.372019i
$839$	4.67110e6	0.229094	0.114547	−	0.993418i	$-0.463458\pi$
$839$	4.67110e6	0.229094	0.114547	+	0.993418i	$0.463458\pi$
$840$	0	0
$841$	−1.97596e7	−0.963359
$842$	7.97619e6i	0.387718i
$843$	− 2.20863e7i	− 1.07042i
$844$	2.22824e6	0.107673
$845$	0	0
$846$	1.00701e7	0.483733
$847$	− 7.49000e6i	− 0.358735i
$848$	1.55648e7i	0.743280i
$849$	7.33487e6	0.349239
$850$	0	0
$851$	3.67627e7	1.74014
$852$	− 7.50264e6i	− 0.354091i
$853$	− 3.66146e7i	− 1.72299i	−0.507768	−	0.861494i	$-0.669529\pi$
$853$	− 3.66146e7i	− 1.72299i	0.507768	−	0.861494i	$-0.330471\pi$
$854$	−648620.	−0.0304331
$855$	0	0
$856$	−2.66483e7	−1.24304
$857$	− 1.15520e7i	− 0.537288i	−0.963240	−	0.268644i	$-0.913425\pi$
$857$	− 1.15520e7i	− 0.537288i	0.963240	−	0.268644i	$-0.0865755\pi$
$858$	325219.i	0.0150820i
$859$	−3.53878e7	−1.63633	−0.818165	−	0.574983i	$-0.805009\pi$
$859$	−3.53878e7	−1.63633	−0.818165	+	0.574983i	$0.805009\pi$
$860$	0	0
$861$	−7.52860e6	−0.346104
$862$	2.93994e6i	0.134763i
$863$	1.07433e7i	0.491034i	0.969392	+	0.245517i	$0.0789577\pi$
$863$	1.07433e7i	0.491034i	−0.969392	+	0.245517i	$0.921042\pi$
$864$	−1.55291e7	−0.707720
$865$	0	0
$866$	−1.63571e6	−0.0741158
$867$	− 3.75203e6i	− 0.169519i
$868$	1.98195e6i	0.0892881i
$869$	−4.09071e6	−0.183759
$870$	0	0
$871$	3.07531e6	0.137355
$872$	− 3.92958e7i	− 1.75007i
$873$	− 1.05102e6i	− 0.0466740i
$874$	−553618.	−0.0245150
$875$	0	0
$876$	−2.87510e6	−0.126588
$877$	− 4.58293e6i	− 0.201208i	−0.994927	−	0.100604i	$-0.967923\pi$
$877$	− 4.58293e6i	− 0.201208i	0.994927	−	0.100604i	$-0.0320775\pi$
$878$	4.84927e6i	0.212295i
$879$	2.41131e7	1.05264
$880$	0	0
$881$	1.68070e7	0.729543	0.364772	−	0.931097i	$-0.381147\pi$
$881$	1.68070e7	0.729543	0.364772	+	0.931097i	$0.381147\pi$
$882$	1.42279e6i	0.0615842i
$883$	− 5.28260e6i	− 0.228006i	−0.993480	−	0.114003i	$-0.963633\pi$
$883$	− 5.28260e6i	− 0.228006i	0.993480	−	0.114003i	$-0.0363673\pi$
$884$	−886186.	−0.0381412
$885$	0	0
$886$	−3.61197e6	−0.154582
$887$	− 4.43107e7i	− 1.89104i	−0.325569	−	0.945518i	$-0.605556\pi$
$887$	− 4.43107e7i	− 1.89104i	0.325569	−	0.945518i	$-0.394444\pi$
$888$	− 1.98870e7i	− 0.846322i
$889$	1.96226e6	0.0832728
$890$	0	0
$891$	−920219.	−0.0388326
$892$	1.30250e7i	0.548109i
$893$	538305.i	0.0225892i
$894$	−3.11821e6	−0.130485
$895$	0	0
$896$	−1.52930e6	−0.0636389
$897$	3.05834e6i	0.126913i
$898$	2.90039e7i	1.20023i
$899$	3.05741e6	0.126170
$900$	0	0
$901$	−3.05783e7	−1.25488
$902$	5.94883e6i	0.243453i
$903$	− 5.10994e6i	− 0.208543i
$904$	2.41415e7	0.982524
$905$	0	0
$906$	−1.82304e7	−0.737865
$907$	3.95608e7i	1.59679i	0.602135	+	0.798394i	$0.294317\pi$
$907$	3.95608e7i	1.59679i	−0.602135	+	0.798394i	$0.705683\pi$
$908$	1.64930e6i	0.0663872i
$909$	1.63448e7	0.656099
$910$	0	0
$911$	−2.40982e7	−0.962030	−0.481015	−	0.876712i	$-0.659732\pi$
$911$	−2.40982e7	−0.962030	−0.481015	+	0.876712i	$0.659732\pi$
$912$	176328.i	0.00701994i
$913$	3.53894e6i	0.140506i
$914$	−2.68172e7	−1.06181
$915$	0	0
$916$	43954.4	0.00173087
$917$	1.03862e7i	0.407881i
$918$	1.85212e7i	0.725375i
$919$	2.24660e7	0.877480	0.438740	−	0.898614i	$-0.355425\pi$
$919$	2.24660e7	0.877480	0.438740	+	0.898614i	$0.355425\pi$
$920$	0	0
$921$	2.80923e7	1.09129
$922$	4.00704e7i	1.55238i
$923$	− 4.62223e6i	− 0.178586i
$924$	538882.	0.0207641
$925$	0	0
$926$	−3.06693e7	−1.17538
$927$	− 1.00224e7i	− 0.383063i
$928$	− 3.39995e6i	− 0.129599i
$929$	1.16280e7	0.442043	0.221021	−	0.975269i	$-0.429061\pi$
$929$	1.16280e7	0.442043	0.221021	+	0.975269i	$0.429061\pi$
$930$	0	0
$931$	−76056.7	−0.00287583
$932$	− 1.55517e7i	− 0.586458i
$933$	− 2.72913e7i	− 1.02641i
$934$	4.74591e6	0.178013
$935$	0	0
$936$	−1.92806e6	−0.0719333
$937$	2.40956e7i	0.896580i	0.893888	+	0.448290i	$0.147967\pi$
$937$	2.40956e7i	0.896580i	−0.893888	+	0.448290i	$0.852033\pi$
$938$	9.12218e6i	0.338525i
$939$	2.66475e7	0.986264
$940$	0	0
$941$	−567590.	−0.0208959	−0.0104479	−	0.999945i	$-0.503326\pi$
$941$	−567590.	−0.0208959	−0.0104479	+	0.999945i	$0.503326\pi$
$942$	− 1.35142e7i	− 0.496207i
$943$	5.59425e7i	2.04863i
$944$	2.66395e7	0.972962
$945$	0	0
$946$	−4.03769e6	−0.146692
$947$	− 9.04501e6i	− 0.327743i	−0.986482	−	0.163872i	$-0.947602\pi$
$947$	− 9.04501e6i	− 0.327743i	0.986482	−	0.163872i	$-0.0523983\pi$
$948$	5.49050e6i	0.198423i
$949$	−1.77129e6	−0.0638446
$950$	0	0
$951$	2.44068e7	0.875105
$952$	− 9.96305e6i	− 0.356287i
$953$	− 1.48053e7i	− 0.528063i	−0.964514	−	0.264032i	$-0.914948\pi$
$953$	− 1.48053e7i	− 0.528063i	0.964514	−	0.264032i	$-0.0850524\pi$
$954$	−1.75529e7	−0.624423
$955$	0	0
$956$	5.48639e6	0.194152
$957$	− 831294.i	− 0.0293410i
$958$	− 4.24895e7i	− 1.49578i
$959$	2.00805e6	0.0705062
$960$	0	0
$961$	−1.61911e7	−0.565546
$962$	− 3.23257e6i	− 0.112619i
$963$	− 1.76940e7i	− 0.614838i
$964$	−1.50336e7	−0.521041
$965$	0	0
$966$	−9.07185e6	−0.312790
$967$	2.02304e7i	0.695726i	0.937545	+	0.347863i	$0.113093\pi$
$967$	2.02304e7i	0.695726i	−0.937545	+	0.347863i	$0.886907\pi$
$968$	3.01072e7i	1.03272i
$969$	−346411.	−0.0118517
$970$	0	0
$971$	−255216.	−0.00868679	−0.00434340	−	0.999991i	$-0.501383\pi$
$971$	−255216.	−0.00868679	−0.00434340	+	0.999991i	$0.501383\pi$
$972$	− 9.80004e6i	− 0.332707i
$973$	2.05284e7i	0.695142i
$974$	1.30932e7	0.442229
$975$	0	0
$976$	1.53507e6	0.0515827
$977$	− 3.15610e7i	− 1.05783i	−0.848675	−	0.528914i	$-0.822599\pi$
$977$	− 3.15610e7i	− 1.05783i	0.848675	−	0.528914i	$-0.177401\pi$
$978$	1.83642e7i	0.613938i
$979$	3.79196e6	0.126447
$980$	0	0
$981$	2.60917e7	0.865627
$982$	4.75064e6i	0.157208i
$983$	− 5.24054e7i	− 1.72978i	−0.501958	−	0.864892i	$-0.667387\pi$
$983$	− 5.24054e7i	− 1.72978i	0.501958	−	0.864892i	$-0.332613\pi$
$984$	3.02624e7	0.996357
$985$	0	0
$986$	−4.05504e6	−0.132832
$987$	8.82093e6i	0.288218i
$988$	− 27193.1i	0 0.000886270i
$989$	−3.79702e7	−1.23439
$990$	0	0
$991$	3.73103e7	1.20683	0.603413	−	0.797429i	$-0.293807\pi$
$991$	3.73103e7	1.20683	0.603413	+	0.797429i	$0.293807\pi$
$992$	− 1.38316e7i	− 0.446264i
$993$	− 7.38682e6i	− 0.237730i
$994$	1.37108e7	0.440145
$995$	0	0
$996$	4.74992e6	0.151718
$997$	2.27918e7i	0.726174i	0.931755	+	0.363087i	$0.118277\pi$
$997$	2.27918e7i	0.726174i	−0.931755	+	0.363087i	$0.881723\pi$
$998$	2.89323e7i	0.919512i
$999$	3.77398e7	1.19643

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	175.6.b.d.99.1		4
5.2	odd	4		35.6.a.b.1.2	✓	2
5.3	odd	4		175.6.a.d.1.1		2
5.4	even	2	inner	175.6.b.d.99.4		4
15.2	even	4		315.6.a.c.1.1		2
20.7	even	4		560.6.a.l.1.2		2
35.27	even	4		245.6.a.c.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
35.6.a.b.1.2	✓	2	5.2	odd	4
175.6.a.d.1.1		2	5.3	odd	4
175.6.b.d.99.1		4	1.1	even	1	trivial
175.6.b.d.99.4		4	5.4	even	2	inner
245.6.a.c.1.2		2	35.27	even	4
315.6.a.c.1.1		2	15.2	even	4
560.6.a.l.1.2		2	20.7	even	4

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:	\( N \)	\(=\)	\( 175 = 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	175.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-4.53113i q^{2} -10.5934i q^{3} +11.4689 q^{4} -48.0000 q^{6} +49.0000i q^{7} -196.963i q^{8} +130.780 q^{9} +O(q^{10})\)	\(q-4.53113i q^{2} -10.5934i q^{3} +11.4689 q^{4} -48.0000 q^{6} +49.0000i q^{7} -196.963i q^{8} +130.780 q^{9} +90.5195 q^{11} -121.494i q^{12} -74.8502i q^{13} +222.025 q^{14} -525.461 q^{16} -1032.31i q^{17} -592.582i q^{18} +31.6771 q^{19} +519.076 q^{21} -410.156i q^{22} -3857.08i q^{23} -2086.51 q^{24} -339.156 q^{26} -3959.60i q^{27} +561.975i q^{28} +866.917 q^{29} +3526.76 q^{31} -3921.89i q^{32} -958.908i q^{33} -4677.54 q^{34} +1499.90 q^{36} +9531.22i q^{37} -143.533i q^{38} -792.917 q^{39} -14503.9 q^{41} -2352.00i q^{42} -9844.30i q^{43} +1038.16 q^{44} -17476.9 q^{46} +16993.5i q^{47} +5566.41i q^{48} -2401.00 q^{49} -10935.7 q^{51} -858.447i q^{52} -29621.1i q^{53} -17941.4 q^{54} +9651.19 q^{56} -335.568i q^{57} -3928.11i q^{58} -50697.4 q^{59} -2921.38 q^{61} -15980.2i q^{62} +6408.23i q^{63} -34585.3 q^{64} -4344.94 q^{66} +41086.2i q^{67} -11839.5i q^{68} -40859.5 q^{69} +61753.1 q^{71} -25758.9i q^{72} -23664.5i q^{73} +43187.2 q^{74} +363.300 q^{76} +4435.46i q^{77} +3592.81i q^{78} -45191.5 q^{79} -10166.0 q^{81} +65718.8i q^{82} +39095.9i q^{83} +5953.22 q^{84} -44605.8 q^{86} -9183.58i q^{87} -17829.0i q^{88} +41891.1 q^{89} +3667.66 q^{91} -44236.3i q^{92} -37360.4i q^{93} +76999.9 q^{94} -41546.1 q^{96} -8036.53i q^{97} +10879.2i q^{98} +11838.2 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 62 q^{4} - 192 q^{6} + 378 q^{9}+O(q^{10}) \) 4 * q + 62 * q^4 - 192 * q^6 + 378 * q^9	\( 4 q + 62 q^{4} - 192 q^{6} + 378 q^{9} - 1202 q^{11} + 98 q^{14} - 1086 q^{16} - 1260 q^{19} - 294 q^{21} - 9120 q^{24} + 2868 q^{26} - 11770 q^{29} - 792 q^{31} - 16356 q^{34} + 5274 q^{36} + 12066 q^{39} + 3548 q^{41} - 24936 q^{44} - 59072 q^{46} - 9604 q^{49} - 48822 q^{51} - 64800 q^{54} + 1470 q^{56} - 119200 q^{59} - 103692 q^{61} - 110978 q^{64} + 57696 q^{66} - 174564 q^{69} + 161488 q^{71} + 90804 q^{74} - 25120 q^{76} + 103590 q^{79} - 103356 q^{81} - 14112 q^{84} + 32808 q^{86} + 75300 q^{89} + 56546 q^{91} + 123664 q^{94} - 239712 q^{96} - 56844 q^{99}+O(q^{100}) \) 4 * q + 62 * q^4 - 192 * q^6 + 378 * q^9 - 1202 * q^11 + 98 * q^14 - 1086 * q^16 - 1260 * q^19 - 294 * q^21 - 9120 * q^24 + 2868 * q^26 - 11770 * q^29 - 792 * q^31 - 16356 * q^34 + 5274 * q^36 + 12066 * q^39 + 3548 * q^41 - 24936 * q^44 - 59072 * q^46 - 9604 * q^49 - 48822 * q^51 - 64800 * q^54 + 1470 * q^56 - 119200 * q^59 - 103692 * q^61 - 110978 * q^64 + 57696 * q^66 - 174564 * q^69 + 161488 * q^71 + 90804 * q^74 - 25120 * q^76 + 103590 * q^79 - 103356 * q^81 - 14112 * q^84 + 32808 * q^86 + 75300 * q^89 + 56546 * q^91 + 123664 * q^94 - 239712 * q^96 - 56844 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more