LMFDB - Embedded newform 378.8.a.p.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [378,8,Mod(1,378)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(378, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 8, names="a")

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

chi := DirichletCharacter("378.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 378 = 2 \cdot 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	378.a (trivial)

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:	yes
Analytic conductor:	$118.081539633$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 4051x^{2} - 71519x + 526930 $ x^4 - x^3 - 4051x^2 - 71519x + 526930
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2}\cdot 3^{6} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-27.8481$ of defining polynomial
Character	$\chi$	$=$	378.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-8.00000 q^{2} +64.0000 q^{4} -54.6225 q^{5} -343.000 q^{7} -512.000 q^{8} +O(q^{10})$	\(q-8.00000 q^{2} +64.0000 q^{4} -54.6225 q^{5} -343.000 q^{7} -512.000 q^{8} +436.980 q^{10} -4876.93 q^{11} -6223.88 q^{13} +2744.00 q^{14} +4096.00 q^{16} -21519.5 q^{17} -7110.08 q^{19} -3495.84 q^{20} +39015.4 q^{22} -19282.1 q^{23} -75141.4 q^{25} +49791.1 q^{26} -21952.0 q^{28} -67658.9 q^{29} -15172.2 q^{31} -32768.0 q^{32} +172156. q^{34} +18735.5 q^{35} +426622. q^{37} +56880.6 q^{38} +27966.7 q^{40} -121201. q^{41} -521047. q^{43} -312124. q^{44} +154257. q^{46} +374597. q^{47} +117649. q^{49} +601131. q^{50} -398329. q^{52} -879416. q^{53} +266390. q^{55} +175616. q^{56} +541271. q^{58} -1.06375e6 q^{59} +1.19667e6 q^{61} +121378. q^{62} +262144. q^{64} +339964. q^{65} +3.58644e6 q^{67} -1.37725e6 q^{68} -149884. q^{70} -2.49300e6 q^{71} -2.00068e6 q^{73} -3.41298e6 q^{74} -455045. q^{76} +1.67279e6 q^{77} -1.16695e6 q^{79} -223734. q^{80} +969606. q^{82} -7.54648e6 q^{83} +1.17545e6 q^{85} +4.16838e6 q^{86} +2.49699e6 q^{88} +7.67375e6 q^{89} +2.13479e6 q^{91} -1.23405e6 q^{92} -2.99677e6 q^{94} +388370. q^{95} +3.36001e6 q^{97} -941192. q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 32 q^{2} + 256 q^{4} + 100 q^{5} - 1372 q^{7} - 2048 q^{8}+O(q^{10}) $ 4 * q - 32 * q^2 + 256 * q^4 + 100 * q^5 - 1372 * q^7 - 2048 * q^8	\( 4 q - 32 q^{2} + 256 q^{4} + 100 q^{5} - 1372 q^{7} - 2048 q^{8} - 800 q^{10} + 6446 q^{11} + 6594 q^{13} + 10976 q^{14} + 16384 q^{16} + 22184 q^{17} - 15730 q^{19} + 6400 q^{20} - 51568 q^{22} - 107176 q^{23} + 25790 q^{25} - 52752 q^{26} - 87808 q^{28} - 163844 q^{29} + 35424 q^{31} - 131072 q^{32} - 177472 q^{34} - 34300 q^{35} + 371678 q^{37} + 125840 q^{38} - 51200 q^{40} + 79212 q^{41} + 1186652 q^{43} + 412544 q^{44} + 857408 q^{46} - 952110 q^{47} + 470596 q^{49} - 206320 q^{50} + 422016 q^{52} - 155082 q^{53} + 3728930 q^{55} + 702464 q^{56} + 1310752 q^{58} + 1193848 q^{59} + 129102 q^{61} - 283392 q^{62} + 1048576 q^{64} - 9519330 q^{65} - 2555300 q^{67} + 1419776 q^{68} + 274400 q^{70} - 2975872 q^{71} + 6428020 q^{73} - 2973424 q^{74} - 1006720 q^{76} - 2210978 q^{77} + 6843240 q^{79} + 409600 q^{80} - 633696 q^{82} - 11642074 q^{83} - 1650490 q^{85} - 9493216 q^{86} - 3300352 q^{88} - 21148548 q^{89} - 2261742 q^{91} - 6859264 q^{92} + 7616880 q^{94} - 8711770 q^{95} - 5112362 q^{97} - 3764768 q^{98}+O(q^{100}) \) 4 * q - 32 * q^2 + 256 * q^4 + 100 * q^5 - 1372 * q^7 - 2048 * q^8 - 800 * q^10 + 6446 * q^11 + 6594 * q^13 + 10976 * q^14 + 16384 * q^16 + 22184 * q^17 - 15730 * q^19 + 6400 * q^20 - 51568 * q^22 - 107176 * q^23 + 25790 * q^25 - 52752 * q^26 - 87808 * q^28 - 163844 * q^29 + 35424 * q^31 - 131072 * q^32 - 177472 * q^34 - 34300 * q^35 + 371678 * q^37 + 125840 * q^38 - 51200 * q^40 + 79212 * q^41 + 1186652 * q^43 + 412544 * q^44 + 857408 * q^46 - 952110 * q^47 + 470596 * q^49 - 206320 * q^50 + 422016 * q^52 - 155082 * q^53 + 3728930 * q^55 + 702464 * q^56 + 1310752 * q^58 + 1193848 * q^59 + 129102 * q^61 - 283392 * q^62 + 1048576 * q^64 - 9519330 * q^65 - 2555300 * q^67 + 1419776 * q^68 + 274400 * q^70 - 2975872 * q^71 + 6428020 * q^73 - 2973424 * q^74 - 1006720 * q^76 - 2210978 * q^77 + 6843240 * q^79 + 409600 * q^80 - 633696 * q^82 - 11642074 * q^83 - 1650490 * q^85 - 9493216 * q^86 - 3300352 * q^88 - 21148548 * q^89 - 2261742 * q^91 - 6859264 * q^92 + 7616880 * q^94 - 8711770 * q^95 - 5112362 * q^97 - 3764768 * q^98

Display 100 coefficients 10 coefficients

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 $

$2$	−8.00000	−0.707107
$2$	−8.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	64.0000	0.500000
$5$	−54.6225	−0.195424	−0.0977118	−	0.995215i	$-0.531152\pi$
$5$	−54.6225	−0.195424	−0.0977118	+	0.995215i	$0.531152\pi$
$6$	0	0
$7$	−343.000	−0.377964
$7$	−343.000	−0.377964
$8$	−512.000	−0.353553
$9$	0	0
$10$	436.980	0.138185
$11$	−4876.93	−1.10477	−0.552385	−	0.833589i	$-0.686282\pi$
$11$	−4876.93	−1.10477	−0.552385	+	0.833589i	$0.686282\pi$
$12$	0	0
$13$	−6223.88	−0.785706	−0.392853	−	0.919601i	$-0.628512\pi$
$13$	−6223.88	−0.785706	−0.392853	+	0.919601i	$0.628512\pi$
$14$	2744.00	0.267261
$15$	0	0
$16$	4096.00	0.250000
$17$	−21519.5	−1.06233	−0.531167	−	0.847267i	$-0.678246\pi$
$17$	−21519.5	−1.06233	−0.531167	+	0.847267i	$0.678246\pi$
$18$	0	0
$19$	−7110.08	−0.237814	−0.118907	−	0.992905i	$-0.537939\pi$
$19$	−7110.08	−0.237814	−0.118907	+	0.992905i	$0.537939\pi$
$20$	−3495.84	−0.0977118
$21$	0	0
$22$	39015.4	0.781191
$23$	−19282.1	−0.330451	−0.165225	−	0.986256i	$-0.552835\pi$
$23$	−19282.1	−0.330451	−0.165225	+	0.986256i	$0.552835\pi$
$24$	0	0
$25$	−75141.4	−0.961810
$26$	49791.1	0.555578
$27$	0	0
$28$	−21952.0	−0.188982
$29$	−67658.9	−0.515148	−0.257574	−	0.966259i	$-0.582923\pi$
$29$	−67658.9	−0.515148	−0.257574	+	0.966259i	$0.582923\pi$
$30$	0	0
$31$	−15172.2	−0.0914710	−0.0457355	−	0.998954i	$-0.514563\pi$
$31$	−15172.2	−0.0914710	−0.0457355	+	0.998954i	$0.514563\pi$
$32$	−32768.0	−0.176777
$33$	0	0
$34$	172156.	0.751184
$35$	18735.5	0.0738632
$36$	0	0
$37$	426622.	1.38464	0.692321	−	0.721589i	$-0.256588\pi$
$37$	426622.	1.38464	0.692321	+	0.721589i	$0.256588\pi$
$38$	56880.6	0.168160
$39$	0	0
$40$	27966.7	0.0690927
$41$	−121201.	−0.274639	−0.137319	−	0.990527i	$-0.543849\pi$
$41$	−121201.	−0.274639	−0.137319	+	0.990527i	$0.543849\pi$
$42$	0	0
$43$	−521047.	−0.999396	−0.499698	−	0.866200i	$-0.666556\pi$
$43$	−521047.	−0.999396	−0.499698	+	0.866200i	$0.666556\pi$
$44$	−312124.	−0.552385
$45$	0	0
$46$	154257.	0.233664
$47$	374597.	0.526286	0.263143	−	0.964757i	$-0.415241\pi$
$47$	374597.	0.526286	0.263143	+	0.964757i	$0.415241\pi$
$48$	0	0
$49$	117649.	0.142857
$50$	601131.	0.680102
$51$	0	0
$52$	−398329.	−0.392853
$53$	−879416.	−0.811389	−0.405694	−	0.914009i	$-0.632970\pi$
$53$	−879416.	−0.811389	−0.405694	+	0.914009i	$0.632970\pi$
$54$	0	0
$55$	266390.	0.215898
$56$	175616.	0.133631
$57$	0	0
$58$	541271.	0.364265
$59$	−1.06375e6	−0.674307	−0.337154	−	0.941450i	$-0.609464\pi$
$59$	−1.06375e6	−0.674307	−0.337154	+	0.941450i	$0.609464\pi$
$60$	0	0
$61$	1.19667e6	0.675024	0.337512	−	0.941321i	$-0.390415\pi$
$61$	1.19667e6	0.675024	0.337512	+	0.941321i	$0.390415\pi$
$62$	121378.	0.0646798
$63$	0	0
$64$	262144.	0.125000
$65$	339964.	0.153545
$66$	0	0
$67$	3.58644e6	1.45681	0.728403	−	0.685149i	$-0.240263\pi$
$67$	3.58644e6	1.45681	0.728403	+	0.685149i	$0.240263\pi$
$68$	−1.37725e6	−0.531167
$69$	0	0
$70$	−149884.	−0.0522291
$71$	−2.49300e6	−0.826643	−0.413321	−	0.910585i	$-0.635631\pi$
$71$	−2.49300e6	−0.826643	−0.413321	+	0.910585i	$0.635631\pi$
$72$	0	0
$73$	−2.00068e6	−0.601932	−0.300966	−	0.953635i	$-0.597309\pi$
$73$	−2.00068e6	−0.601932	−0.300966	+	0.953635i	$0.597309\pi$
$74$	−3.41298e6	−0.979090
$75$	0	0
$76$	−455045.	−0.118907
$77$	1.67279e6	0.417564
$78$	0	0
$79$	−1.16695e6	−0.266293	−0.133146	−	0.991096i	$-0.542508\pi$
$79$	−1.16695e6	−0.266293	−0.133146	+	0.991096i	$0.542508\pi$
$80$	−223734.	−0.0488559
$81$	0	0
$82$	969606.	0.194199
$83$	−7.54648e6	−1.44868	−0.724338	−	0.689445i	$-0.757854\pi$
$83$	−7.54648e6	−1.44868	−0.724338	+	0.689445i	$0.757854\pi$
$84$	0	0
$85$	1.17545e6	0.207605
$86$	4.16838e6	0.706679
$87$	0	0
$88$	2.49699e6	0.390595
$89$	7.67375e6	1.15383	0.576916	−	0.816803i	$-0.304256\pi$
$89$	7.67375e6	1.15383	0.576916	+	0.816803i	$0.304256\pi$
$90$	0	0
$91$	2.13479e6	0.296969
$92$	−1.23405e6	−0.165225
$93$	0	0
$94$	−2.99677e6	−0.372140
$95$	388370.	0.0464744
$96$	0	0
$97$	3.36001e6	0.373800	0.186900	−	0.982379i	$-0.440156\pi$
$97$	3.36001e6	0.373800	0.186900	+	0.982379i	$0.440156\pi$
$98$	−941192.	−0.101015
$99$	0	0
$100$	−4.80905e6	−0.480905
$101$	−1.15294e7	−1.11348	−0.556739	−	0.830687i	$-0.687948\pi$
$101$	−1.15294e7	−1.11348	−0.556739	+	0.830687i	$0.687948\pi$
$102$	0	0
$103$	−1.11729e7	−1.00748	−0.503738	−	0.863856i	$-0.668042\pi$
$103$	−1.11729e7	−1.00748	−0.503738	+	0.863856i	$0.668042\pi$
$104$	3.18663e6	0.277789
$105$	0	0
$106$	7.03533e6	0.573738
$107$	−3.42954e6	−0.270641	−0.135320	−	0.990802i	$-0.543206\pi$
$107$	−3.42954e6	−0.270641	−0.135320	+	0.990802i	$0.543206\pi$
$108$	0	0
$109$	−1.88945e6	−0.139747	−0.0698736	−	0.997556i	$-0.522260\pi$
$109$	−1.88945e6	−0.139747	−0.0698736	+	0.997556i	$0.522260\pi$
$110$	−2.13112e6	−0.152663
$111$	0	0
$112$	−1.40493e6	−0.0944911
$113$	−1.41109e7	−0.919982	−0.459991	−	0.887924i	$-0.652147\pi$
$113$	−1.41109e7	−0.919982	−0.459991	+	0.887924i	$0.652147\pi$
$114$	0	0
$115$	1.05324e6	0.0645779
$116$	−4.33017e6	−0.257574
$117$	0	0
$118$	8.51001e6	0.476807
$119$	7.38119e6	0.401525
$120$	0	0
$121$	4.29728e6	0.220518
$122$	−9.57335e6	−0.477314
$123$	0	0
$124$	−971023.	−0.0457355
$125$	8.37180e6	0.383384
$126$	0	0
$127$	3.09430e7	1.34045	0.670223	−	0.742160i	$-0.266198\pi$
$127$	3.09430e7	1.34045	0.670223	+	0.742160i	$0.266198\pi$
$128$	−2.09715e6	−0.0883883
$129$	0	0
$130$	−2.71972e6	−0.108573
$131$	−2.12709e7	−0.826679	−0.413340	−	0.910577i	$-0.635638\pi$
$131$	−2.12709e7	−0.826679	−0.413340	+	0.910577i	$0.635638\pi$
$132$	0	0
$133$	2.43876e6	0.0898851
$134$	−2.86915e7	−1.03012
$135$	0	0
$136$	1.10180e7	0.375592
$137$	−2.72502e7	−0.905416	−0.452708	−	0.891659i	$-0.649542\pi$
$137$	−2.72502e7	−0.905416	−0.452708	+	0.891659i	$0.649542\pi$
$138$	0	0
$139$	2.83364e7	0.894939	0.447469	−	0.894299i	$-0.352325\pi$
$139$	2.83364e7	0.894939	0.447469	+	0.894299i	$0.352325\pi$
$140$	1.19907e6	0.0369316
$141$	0	0
$142$	1.99440e7	0.584525
$143$	3.03535e7	0.868024
$144$	0	0
$145$	3.69570e6	0.100672
$146$	1.60054e7	0.425630
$147$	0	0
$148$	2.73038e7	0.692321
$149$	2.69055e7	0.666329	0.333164	−	0.942869i	$-0.391884\pi$
$149$	2.69055e7	0.666329	0.333164	+	0.942869i	$0.391884\pi$
$150$	0	0
$151$	4.85034e7	1.14644	0.573221	−	0.819400i	$-0.305693\pi$
$151$	4.85034e7	1.14644	0.573221	+	0.819400i	$0.305693\pi$
$152$	3.64036e6	0.0840798
$153$	0	0
$154$	−1.33823e7	−0.295262
$155$	828746.	0.0178756
$156$	0	0
$157$	5.78740e7	1.19353	0.596767	−	0.802414i	$-0.296452\pi$
$157$	5.78740e7	1.19353	0.596767	+	0.802414i	$0.296452\pi$
$158$	9.33563e6	0.188297
$159$	0	0
$160$	1.78987e6	0.0345463
$161$	6.61376e6	0.124899
$162$	0	0
$163$	7.36007e6	0.133114	0.0665572	−	0.997783i	$-0.478799\pi$
$163$	7.36007e6	0.133114	0.0665572	+	0.997783i	$0.478799\pi$
$164$	−7.75685e6	−0.137319
$165$	0	0
$166$	6.03719e7	1.02437
$167$	6.95449e7	1.15547	0.577733	−	0.816225i	$-0.303937\pi$
$167$	6.95449e7	1.15547	0.577733	+	0.816225i	$0.303937\pi$
$168$	0	0
$169$	−2.40118e7	−0.382667
$170$	−9.40360e6	−0.146799
$171$	0	0
$172$	−3.33470e7	−0.499698
$173$	5.03380e7	0.739153	0.369577	−	0.929200i	$-0.379503\pi$
$173$	5.03380e7	0.739153	0.369577	+	0.929200i	$0.379503\pi$
$174$	0	0
$175$	2.57735e7	0.363530
$176$	−1.99759e7	−0.276193
$177$	0	0
$178$	−6.13900e7	−0.815883
$179$	−2.04870e7	−0.266989	−0.133494	−	0.991050i	$-0.542620\pi$
$179$	−2.04870e7	−0.266989	−0.133494	+	0.991050i	$0.542620\pi$
$180$	0	0
$181$	7.45972e6	0.0935078	0.0467539	−	0.998906i	$-0.485112\pi$
$181$	7.45972e6	0.0935078	0.0467539	+	0.998906i	$0.485112\pi$
$182$	−1.70783e7	−0.209989
$183$	0	0
$184$	9.87244e6	0.116832
$185$	−2.33032e7	−0.270592
$186$	0	0
$187$	1.04949e8	1.17364
$188$	2.39742e7	0.263143
$189$	0	0
$190$	−3.10696e6	−0.0328623
$191$	−7.23735e7	−0.751559	−0.375779	−	0.926709i	$-0.622625\pi$
$191$	−7.23735e7	−0.751559	−0.375779	+	0.926709i	$0.622625\pi$
$192$	0	0
$193$	1.51484e8	1.51676	0.758380	−	0.651812i	$-0.225991\pi$
$193$	1.51484e8	1.51676	0.758380	+	0.651812i	$0.225991\pi$
$194$	−2.68801e7	−0.264317
$195$	0	0
$196$	7.52954e6	0.0714286
$197$	9.53636e6	0.0888691	0.0444346	−	0.999012i	$-0.485851\pi$
$197$	9.53636e6	0.0888691	0.0444346	+	0.999012i	$0.485851\pi$
$198$	0	0
$199$	2.04414e8	1.83876	0.919378	−	0.393375i	$-0.128693\pi$
$199$	2.04414e8	1.83876	0.919378	+	0.393375i	$0.128693\pi$
$200$	3.84724e7	0.340051
$201$	0	0
$202$	9.22351e7	0.787348
$203$	2.32070e7	0.194708
$204$	0	0
$205$	6.62030e6	0.0536709
$206$	8.93830e7	0.712393
$207$	0	0
$208$	−2.54930e7	−0.196426
$209$	3.46753e7	0.262729
$210$	0	0
$211$	1.03468e8	0.758258	0.379129	−	0.925344i	$-0.376224\pi$
$211$	1.03468e8	0.758258	0.379129	+	0.925344i	$0.376224\pi$
$212$	−5.62826e7	−0.405694
$213$	0	0
$214$	2.74364e7	0.191372
$215$	2.84609e7	0.195305
$216$	0	0
$217$	5.20408e6	0.0345728
$218$	1.51156e7	0.0988162
$219$	0	0
$220$	1.70490e7	0.107949
$221$	1.33935e8	0.834682
$222$	0	0
$223$	−1.44604e8	−0.873200	−0.436600	−	0.899656i	$-0.643818\pi$
$223$	−1.44604e8	−0.873200	−0.436600	+	0.899656i	$0.643818\pi$
$224$	1.12394e7	0.0668153
$225$	0	0
$226$	1.12887e8	0.650525
$227$	2.43618e8	1.38235	0.691176	−	0.722687i	$-0.257093\pi$
$227$	2.43618e8	1.38235	0.691176	+	0.722687i	$0.257093\pi$
$228$	0	0
$229$	3.51662e6	0.0193509	0.00967546	−	0.999953i	$-0.496920\pi$
$229$	3.51662e6	0.0193509	0.00967546	+	0.999953i	$0.496920\pi$
$230$	−8.42590e6	−0.0456635
$231$	0	0
$232$	3.46414e7	0.182132
$233$	2.03341e8	1.05313	0.526563	−	0.850136i	$-0.323481\pi$
$233$	2.03341e8	1.05313	0.526563	+	0.850136i	$0.323481\pi$
$234$	0	0
$235$	−2.04614e7	−0.102849
$236$	−6.80801e7	−0.337154
$237$	0	0
$238$	−5.90495e7	−0.283921
$239$	3.32740e8	1.57657	0.788284	−	0.615312i	$-0.210970\pi$
$239$	3.32740e8	1.57657	0.788284	+	0.615312i	$0.210970\pi$
$240$	0	0
$241$	4.77330e7	0.219664	0.109832	−	0.993950i	$-0.464969\pi$
$241$	4.77330e7	0.219664	0.109832	+	0.993950i	$0.464969\pi$
$242$	−3.43782e7	−0.155930
$243$	0	0
$244$	7.65868e7	0.337512
$245$	−6.42629e6	−0.0279176
$246$	0	0
$247$	4.42523e7	0.186851
$248$	7.76818e6	0.0323399
$249$	0	0
$250$	−6.69744e7	−0.271093
$251$	−3.32721e8	−1.32807	−0.664037	−	0.747699i	$-0.731158\pi$
$251$	−3.32721e8	−1.32807	−0.664037	+	0.747699i	$0.731158\pi$
$252$	0	0
$253$	9.40375e7	0.365072
$254$	−2.47544e8	−0.947839
$255$	0	0
$256$	1.67772e7	0.0625000
$257$	−1.47355e8	−0.541500	−0.270750	−	0.962650i	$-0.587272\pi$
$257$	−1.47355e8	−0.541500	−0.270750	+	0.962650i	$0.587272\pi$
$258$	0	0
$259$	−1.46331e8	−0.523346
$260$	2.17577e7	0.0767727
$261$	0	0
$262$	1.70167e8	0.584550
$263$	1.75903e8	0.596249	0.298125	−	0.954527i	$-0.403639\pi$
$263$	1.75903e8	0.596249	0.298125	+	0.954527i	$0.403639\pi$
$264$	0	0
$265$	4.80359e7	0.158564
$266$	−1.95101e7	−0.0635584
$267$	0	0
$268$	2.29532e8	0.728403
$269$	−2.84105e8	−0.889909	−0.444954	−	0.895553i	$-0.646780\pi$
$269$	−2.84105e8	−0.889909	−0.444954	+	0.895553i	$0.646780\pi$
$270$	0	0
$271$	2.54945e7	0.0778134	0.0389067	−	0.999243i	$-0.487612\pi$
$271$	2.54945e7	0.0778134	0.0389067	+	0.999243i	$0.487612\pi$
$272$	−8.81439e7	−0.265584
$273$	0	0
$274$	2.18002e8	0.640226
$275$	3.66459e8	1.06258
$276$	0	0
$277$	1.15506e8	0.326532	0.163266	−	0.986582i	$-0.447797\pi$
$277$	1.15506e8	0.326532	0.163266	+	0.986582i	$0.447797\pi$
$278$	−2.26691e8	−0.632817
$279$	0	0
$280$	−9.59259e6	−0.0261146
$281$	1.00901e8	0.271283	0.135642	−	0.990758i	$-0.456690\pi$
$281$	1.00901e8	0.271283	0.135642	+	0.990758i	$0.456690\pi$
$282$	0	0
$283$	8.65941e7	0.227110	0.113555	−	0.993532i	$-0.463776\pi$
$283$	8.65941e7	0.227110	0.113555	+	0.993532i	$0.463776\pi$
$284$	−1.59552e8	−0.413321
$285$	0	0
$286$	−2.42828e8	−0.613786
$287$	4.15719e7	0.103804
$288$	0	0
$289$	5.27509e7	0.128554
$290$	−2.95656e7	−0.0711859
$291$	0	0
$292$	−1.28043e8	−0.300966
$293$	−1.82281e8	−0.423356	−0.211678	−	0.977339i	$-0.567893\pi$
$293$	−1.82281e8	−0.423356	−0.211678	+	0.977339i	$0.567893\pi$
$294$	0	0
$295$	5.81048e7	0.131776
$296$	−2.18431e8	−0.489545
$297$	0	0
$298$	−2.15244e8	−0.471165
$299$	1.20010e8	0.259637
$300$	0	0
$301$	1.78719e8	0.377736
$302$	−3.88027e8	−0.810658
$303$	0	0
$304$	−2.91229e7	−0.0594534
$305$	−6.53651e7	−0.131916
$306$	0	0
$307$	−3.43383e8	−0.677320	−0.338660	−	0.940909i	$-0.609974\pi$
$307$	−3.43383e8	−0.677320	−0.338660	+	0.940909i	$0.609974\pi$
$308$	1.07058e8	0.208782
$309$	0	0
$310$	−6.62997e6	−0.0126400
$311$	6.10123e8	1.15015	0.575077	−	0.818099i	$-0.304972\pi$
$311$	6.10123e8	1.15015	0.575077	+	0.818099i	$0.304972\pi$
$312$	0	0
$313$	−6.59850e7	−0.121630	−0.0608149	−	0.998149i	$-0.519370\pi$
$313$	−6.59850e7	−0.121630	−0.0608149	+	0.998149i	$0.519370\pi$
$314$	−4.62992e8	−0.843956
$315$	0	0
$316$	−7.46851e7	−0.133146
$317$	−6.08186e7	−0.107233	−0.0536166	−	0.998562i	$-0.517075\pi$
$317$	−6.08186e7	−0.107233	−0.0536166	+	0.998562i	$0.517075\pi$
$318$	0	0
$319$	3.29968e8	0.569120
$320$	−1.43190e7	−0.0244279
$321$	0	0
$322$	−5.29101e7	−0.0883167
$323$	1.53005e8	0.252638
$324$	0	0
$325$	4.67671e8	0.755699
$326$	−5.88805e7	−0.0941261
$327$	0	0
$328$	6.20548e7	0.0970995
$329$	−1.28487e8	−0.198917
$330$	0	0
$331$	−1.28950e9	−1.95444	−0.977222	−	0.212222i	$-0.931930\pi$
$331$	−1.28950e9	−1.95444	−0.977222	+	0.212222i	$0.931930\pi$
$332$	−4.82975e8	−0.724338
$333$	0	0
$334$	−5.56359e8	−0.817039
$335$	−1.95901e8	−0.284694
$336$	0	0
$337$	1.28266e9	1.82560	0.912800	−	0.408408i	$-0.133916\pi$
$337$	1.28266e9	1.82560	0.912800	+	0.408408i	$0.133916\pi$
$338$	1.92094e8	0.270586
$339$	0	0
$340$	7.52288e7	0.103803
$341$	7.39939e7	0.101055
$342$	0	0
$343$	−4.03536e7	−0.0539949
$344$	2.66776e8	0.353340
$345$	0	0
$346$	−4.02704e8	−0.522660
$347$	2.45054e8	0.314854	0.157427	−	0.987531i	$-0.449680\pi$
$347$	2.45054e8	0.314854	0.157427	+	0.987531i	$0.449680\pi$
$348$	0	0
$349$	3.32721e7	0.0418978	0.0209489	−	0.999781i	$-0.493331\pi$
$349$	3.32721e7	0.0418978	0.0209489	+	0.999781i	$0.493331\pi$
$350$	−2.06188e8	−0.257054
$351$	0	0
$352$	1.59807e8	0.195298
$353$	6.08648e8	0.736470	0.368235	−	0.929733i	$-0.379962\pi$
$353$	6.08648e8	0.736470	0.368235	+	0.929733i	$0.379962\pi$
$354$	0	0
$355$	1.36174e8	0.161545
$356$	4.91120e8	0.576916
$357$	0	0
$358$	1.63896e8	0.188790
$359$	−1.64857e9	−1.88052	−0.940260	−	0.340458i	$-0.889418\pi$
$359$	−1.64857e9	−1.88052	−0.940260	+	0.340458i	$0.889418\pi$
$360$	0	0
$361$	−8.43319e8	−0.943445
$362$	−5.96778e7	−0.0661200
$363$	0	0
$364$	1.36627e8	0.148484
$365$	1.09282e8	0.117632
$366$	0	0
$367$	−1.33484e9	−1.40960	−0.704802	−	0.709404i	$-0.748964\pi$
$367$	−1.33484e9	−1.40960	−0.704802	+	0.709404i	$0.748964\pi$
$368$	−7.89795e7	−0.0826127
$369$	0	0
$370$	1.86426e8	0.191337
$371$	3.01640e8	0.306676
$372$	0	0
$373$	−5.52949e8	−0.551701	−0.275851	−	0.961200i	$-0.588960\pi$
$373$	−5.52949e8	−0.551701	−0.275851	+	0.961200i	$0.588960\pi$
$374$	−8.39593e8	−0.829886
$375$	0	0
$376$	−1.91794e8	−0.186070
$377$	4.21101e8	0.404755
$378$	0	0
$379$	−1.32582e9	−1.25097	−0.625484	−	0.780237i	$-0.715099\pi$
$379$	−1.32582e9	−1.25097	−0.625484	+	0.780237i	$0.715099\pi$
$380$	2.48557e7	0.0232372
$381$	0	0
$382$	5.78988e8	0.531432
$383$	6.70594e8	0.609907	0.304954	−	0.952367i	$-0.401359\pi$
$383$	6.70594e8	0.609907	0.304954	+	0.952367i	$0.401359\pi$
$384$	0	0
$385$	−9.13719e7	−0.0816018
$386$	−1.21187e9	−1.07251
$387$	0	0
$388$	2.15041e8	0.186900
$389$	8.45061e8	0.727888	0.363944	−	0.931421i	$-0.381430\pi$
$389$	8.45061e8	0.727888	0.363944	+	0.931421i	$0.381430\pi$
$390$	0	0
$391$	4.14941e8	0.351049
$392$	−6.02363e7	−0.0505076
$393$	0	0
$394$	−7.62909e7	−0.0628400
$395$	6.37420e7	0.0520398
$396$	0	0
$397$	−1.09248e8	−0.0876287	−0.0438143	−	0.999040i	$-0.513951\pi$
$397$	−1.09248e8	−0.0876287	−0.0438143	+	0.999040i	$0.513951\pi$
$398$	−1.63531e9	−1.30020
$399$	0	0
$400$	−3.07779e8	−0.240452
$401$	2.12954e8	0.164923	0.0824615	−	0.996594i	$-0.473722\pi$
$401$	2.12954e8	0.164923	0.0824615	+	0.996594i	$0.473722\pi$
$402$	0	0
$403$	9.44302e7	0.0718693
$404$	−7.37881e8	−0.556739
$405$	0	0
$406$	−1.85656e8	−0.137679
$407$	−2.08061e9	−1.52971
$408$	0	0
$409$	−2.01464e9	−1.45602	−0.728008	−	0.685569i	$-0.759554\pi$
$409$	−2.01464e9	−1.45602	−0.728008	+	0.685569i	$0.759554\pi$
$410$	−5.29624e7	−0.0379511
$411$	0	0
$412$	−7.15064e8	−0.503738
$413$	3.64867e8	0.254864
$414$	0	0
$415$	4.12208e8	0.283105
$416$	2.03944e8	0.138894
$417$	0	0
$418$	−2.77403e8	−0.185778
$419$	1.54148e9	1.02374	0.511870	−	0.859063i	$-0.328953\pi$
$419$	1.54148e9	1.02374	0.511870	+	0.859063i	$0.328953\pi$
$420$	0	0
$421$	2.50008e8	0.163293	0.0816463	−	0.996661i	$-0.473982\pi$
$421$	2.50008e8	0.163293	0.0816463	+	0.996661i	$0.473982\pi$
$422$	−8.27744e8	−0.536170
$423$	0	0
$424$	4.50261e8	0.286869
$425$	1.61701e9	1.02176
$426$	0	0
$427$	−4.10457e8	−0.255135
$428$	−2.19491e8	−0.135320
$429$	0	0
$430$	−2.27687e8	−0.138102
$431$	1.11874e9	0.673070	0.336535	−	0.941671i	$-0.390745\pi$
$431$	1.11874e9	0.673070	0.336535	+	0.941671i	$0.390745\pi$
$432$	0	0
$433$	−3.22641e9	−1.90991	−0.954953	−	0.296758i	$-0.904095\pi$
$433$	−3.22641e9	−1.90991	−0.954953	+	0.296758i	$0.904095\pi$
$434$	−4.16326e7	−0.0244467
$435$	0	0
$436$	−1.20925e8	−0.0698736
$437$	1.37097e8	0.0785857
$438$	0	0
$439$	−1.62773e9	−0.918239	−0.459120	−	0.888374i	$-0.651835\pi$
$439$	−1.62773e9	−0.918239	−0.459120	+	0.888374i	$0.651835\pi$
$440$	−1.36392e8	−0.0763315
$441$	0	0
$442$	−1.07148e9	−0.590209
$443$	2.86632e8	0.156643	0.0783217	−	0.996928i	$-0.475044\pi$
$443$	2.86632e8	0.156643	0.0783217	+	0.996928i	$0.475044\pi$
$444$	0	0
$445$	−4.19160e8	−0.225486
$446$	1.15683e9	0.617446
$447$	0	0
$448$	−8.99154e7	−0.0472456
$449$	−2.33699e9	−1.21841	−0.609206	−	0.793012i	$-0.708512\pi$
$449$	−2.33699e9	−1.21841	−0.609206	+	0.793012i	$0.708512\pi$
$450$	0	0
$451$	5.91088e8	0.303413
$452$	−9.03096e8	−0.459991
$453$	0	0
$454$	−1.94894e9	−0.977470
$455$	−1.16608e8	−0.0580347
$456$	0	0
$457$	−6.92110e8	−0.339210	−0.169605	−	0.985512i	$-0.554249\pi$
$457$	−6.92110e8	−0.339210	−0.169605	+	0.985512i	$0.554249\pi$
$458$	−2.81330e7	−0.0136832
$459$	0	0
$460$	6.74072e7	0.0322889
$461$	3.00297e9	1.42757	0.713784	−	0.700365i	$-0.246980\pi$
$461$	3.00297e9	1.42757	0.713784	+	0.700365i	$0.246980\pi$
$462$	0	0
$463$	−5.27708e7	−0.0247093	−0.0123546	−	0.999924i	$-0.503933\pi$
$463$	−5.27708e7	−0.0247093	−0.0123546	+	0.999924i	$0.503933\pi$
$464$	−2.77131e8	−0.128787
$465$	0	0
$466$	−1.62673e9	−0.744672
$467$	6.98361e8	0.317300	0.158650	−	0.987335i	$-0.449286\pi$
$467$	6.98361e8	0.317300	0.158650	+	0.987335i	$0.449286\pi$
$468$	0	0
$469$	−1.23015e9	−0.550621
$470$	1.63691e8	0.0727249
$471$	0	0
$472$	5.44640e8	0.238404
$473$	2.54111e9	1.10410
$474$	0	0
$475$	5.34261e8	0.228731
$476$	4.72396e8	0.200762
$477$	0	0
$478$	−2.66192e9	−1.11480
$479$	1.95368e9	0.812231	0.406115	−	0.913822i	$-0.366883\pi$
$479$	1.95368e9	0.812231	0.406115	+	0.913822i	$0.366883\pi$
$480$	0	0
$481$	−2.65525e9	−1.08792
$482$	−3.81864e8	−0.155326
$483$	0	0
$484$	2.75026e8	0.110259
$485$	−1.83532e8	−0.0730494
$486$	0	0
$487$	2.70361e9	1.06070	0.530349	−	0.847779i	$-0.322061\pi$
$487$	2.70361e9	1.06070	0.530349	+	0.847779i	$0.322061\pi$
$488$	−6.12694e8	−0.238657
$489$	0	0
$490$	5.14103e7	0.0197408
$491$	4.11747e9	1.56980	0.784902	−	0.619619i	$-0.212713\pi$
$491$	4.11747e9	1.56980	0.784902	+	0.619619i	$0.212713\pi$
$492$	0	0
$493$	1.45599e9	0.547259
$494$	−3.54018e8	−0.132124
$495$	0	0
$496$	−6.21455e7	−0.0228678
$497$	8.55099e8	0.312442
$498$	0	0
$499$	−1.06407e9	−0.383371	−0.191686	−	0.981456i	$-0.561395\pi$
$499$	−1.06407e9	−0.383371	−0.191686	+	0.981456i	$0.561395\pi$
$500$	5.35795e8	0.191692
$501$	0	0
$502$	2.66177e9	0.939091
$503$	−3.92965e9	−1.37678	−0.688392	−	0.725339i	$-0.741683\pi$
$503$	−3.92965e9	−1.37678	−0.688392	+	0.725339i	$0.741683\pi$
$504$	0	0
$505$	6.29765e8	0.217600
$506$	−7.52300e8	−0.258145
$507$	0	0
$508$	1.98035e9	0.670223
$509$	3.61544e9	1.21520	0.607601	−	0.794242i	$-0.292132\pi$
$509$	3.61544e9	1.21520	0.607601	+	0.794242i	$0.292132\pi$
$510$	0	0
$511$	6.86233e8	0.227509
$512$	−1.34218e8	−0.0441942
$513$	0	0
$514$	1.17884e9	0.382898
$515$	6.10291e8	0.196885
$516$	0	0
$517$	−1.82688e9	−0.581425
$518$	1.17065e9	0.370061
$519$	0	0
$520$	−1.74062e8	−0.0542865
$521$	−6.32520e7	−0.0195949	−0.00979743	−	0.999952i	$-0.503119\pi$
$521$	−6.32520e7	−0.0195949	−0.00979743	+	0.999952i	$0.503119\pi$
$522$	0	0
$523$	−1.14761e8	−0.0350784	−0.0175392	−	0.999846i	$-0.505583\pi$
$523$	−1.14761e8	−0.0350784	−0.0175392	+	0.999846i	$0.505583\pi$
$524$	−1.36134e9	−0.413340
$525$	0	0
$526$	−1.40722e9	−0.421612
$527$	3.26499e8	0.0971728
$528$	0	0
$529$	−3.03303e9	−0.890802
$530$	−3.84288e8	−0.112122
$531$	0	0
$532$	1.56080e8	0.0449425
$533$	7.54340e8	0.215785
$534$	0	0
$535$	1.87330e8	0.0528896
$536$	−1.83626e9	−0.515059
$537$	0	0
$538$	2.27284e9	0.629261
$539$	−5.73766e8	−0.157824
$540$	0	0
$541$	4.29211e9	1.16542	0.582708	−	0.812682i	$-0.301993\pi$
$541$	4.29211e9	1.16542	0.582708	+	0.812682i	$0.301993\pi$
$542$	−2.03956e8	−0.0550224
$543$	0	0
$544$	7.05151e8	0.187796
$545$	1.03207e8	0.0273099
$546$	0	0
$547$	−1.10905e9	−0.289732	−0.144866	−	0.989451i	$-0.546275\pi$
$547$	−1.10905e9	−0.289732	−0.144866	+	0.989451i	$0.546275\pi$
$548$	−1.74401e9	−0.452708
$549$	0	0
$550$	−2.93167e9	−0.751357
$551$	4.81060e8	0.122509
$552$	0	0
$553$	4.00265e8	0.100649
$554$	−9.24049e8	−0.230893
$555$	0	0
$556$	1.81353e9	0.447469
$557$	−5.32743e9	−1.30624	−0.653122	−	0.757252i	$-0.726541\pi$
$557$	−5.32743e9	−1.30624	−0.653122	+	0.757252i	$0.726541\pi$
$558$	0	0
$559$	3.24294e9	0.785231
$560$	7.67407e7	0.0184658
$561$	0	0
$562$	−8.07208e8	−0.191826
$563$	5.11161e9	1.20720	0.603599	−	0.797288i	$-0.293733\pi$
$563$	5.11161e9	1.20720	0.603599	+	0.797288i	$0.293733\pi$
$564$	0	0
$565$	7.70772e8	0.179786
$566$	−6.92753e8	−0.160591
$567$	0	0
$568$	1.27642e9	0.292262
$569$	−4.96214e9	−1.12921	−0.564606	−	0.825360i	$-0.690972\pi$
$569$	−4.96214e9	−1.12921	−0.564606	+	0.825360i	$0.690972\pi$
$570$	0	0
$571$	−1.94091e8	−0.0436294	−0.0218147	−	0.999762i	$-0.506944\pi$
$571$	−1.94091e8	−0.0436294	−0.0218147	+	0.999762i	$0.506944\pi$
$572$	1.94262e9	0.434012
$573$	0	0
$574$	−3.32575e8	−0.0734003
$575$	1.44888e9	0.317831
$576$	0	0
$577$	−2.65966e9	−0.576383	−0.288191	−	0.957573i	$-0.593054\pi$
$577$	−2.65966e9	−0.576383	−0.288191	+	0.957573i	$0.593054\pi$
$578$	−4.22007e8	−0.0909017
$579$	0	0
$580$	2.36525e8	0.0503360
$581$	2.58844e9	0.547548
$582$	0	0
$583$	4.28885e9	0.896398
$584$	1.02435e9	0.212815
$585$	0	0
$586$	1.45825e9	0.299358
$587$	8.46766e8	0.172794	0.0863972	−	0.996261i	$-0.472465\pi$
$587$	8.46766e8	0.172794	0.0863972	+	0.996261i	$0.472465\pi$
$588$	0	0
$589$	1.07876e8	0.0217531
$590$	−4.64838e8	−0.0931794
$591$	0	0
$592$	1.74745e9	0.346161
$593$	−1.74406e9	−0.343456	−0.171728	−	0.985144i	$-0.554935\pi$
$593$	−1.74406e9	−0.343456	−0.171728	+	0.985144i	$0.554935\pi$
$594$	0	0
$595$	−4.03180e8	−0.0784674
$596$	1.72195e9	0.333164
$597$	0	0
$598$	−9.60077e8	−0.183591
$599$	6.51168e9	1.23794	0.618969	−	0.785415i	$-0.287551\pi$
$599$	6.51168e9	1.23794	0.618969	+	0.785415i	$0.287551\pi$
$600$	0	0
$601$	9.00575e8	0.169223	0.0846115	−	0.996414i	$-0.473035\pi$
$601$	9.00575e8	0.169223	0.0846115	+	0.996414i	$0.473035\pi$
$602$	−1.42975e9	−0.267100
$603$	0	0
$604$	3.10421e9	0.573221
$605$	−2.34728e8	−0.0430945
$606$	0	0
$607$	4.66484e8	0.0846597	0.0423298	−	0.999104i	$-0.486522\pi$
$607$	4.66484e8	0.0846597	0.0423298	+	0.999104i	$0.486522\pi$
$608$	2.32983e8	0.0420399
$609$	0	0
$610$	5.22920e8	0.0932784
$611$	−2.33145e9	−0.413505
$612$	0	0
$613$	1.02926e10	1.80474	0.902368	−	0.430966i	$-0.141827\pi$
$613$	1.02926e10	1.80474	0.902368	+	0.430966i	$0.141827\pi$
$614$	2.74706e9	0.478938
$615$	0	0
$616$	−8.56467e8	−0.147631
$617$	6.57345e9	1.12667	0.563333	−	0.826230i	$-0.309519\pi$
$617$	6.57345e9	1.12667	0.563333	+	0.826230i	$0.309519\pi$
$618$	0	0
$619$	−3.71722e9	−0.629943	−0.314971	−	0.949101i	$-0.601995\pi$
$619$	−3.71722e9	−0.629943	−0.314971	+	0.949101i	$0.601995\pi$
$620$	5.30397e7	0.00893779
$621$	0	0
$622$	−4.88098e9	−0.813281
$623$	−2.63210e9	−0.436108
$624$	0	0
$625$	5.41313e9	0.886887
$626$	5.27880e8	0.0860052
$627$	0	0
$628$	3.70394e9	0.596767
$629$	−9.18071e9	−1.47095
$630$	0	0
$631$	−7.08019e9	−1.12187	−0.560934	−	0.827860i	$-0.689558\pi$
$631$	−7.08019e9	−1.12187	−0.560934	+	0.827860i	$0.689558\pi$
$632$	5.97481e8	0.0941486
$633$	0	0
$634$	4.86549e8	0.0758253
$635$	−1.69019e9	−0.261955
$636$	0	0
$637$	−7.32234e8	−0.112244
$638$	−2.63974e9	−0.402429
$639$	0	0
$640$	1.14552e8	0.0172732
$641$	−1.25546e9	−0.188278	−0.0941388	−	0.995559i	$-0.530010\pi$
$641$	−1.25546e9	−0.188278	−0.0941388	+	0.995559i	$0.530010\pi$
$642$	0	0
$643$	3.34406e9	0.496062	0.248031	−	0.968752i	$-0.420216\pi$
$643$	3.34406e9	0.496062	0.248031	+	0.968752i	$0.420216\pi$
$644$	4.23281e8	0.0624493
$645$	0	0
$646$	−1.22404e9	−0.178642
$647$	1.13016e8	0.0164050	0.00820248	−	0.999966i	$-0.497389\pi$
$647$	1.13016e8	0.0164050	0.00820248	+	0.999966i	$0.497389\pi$
$648$	0	0
$649$	5.18784e9	0.744955
$650$	−3.74137e9	−0.534360
$651$	0	0
$652$	4.71044e8	0.0665572
$653$	−1.41007e10	−1.98173	−0.990863	−	0.134876i	$-0.956937\pi$
$653$	−1.41007e10	−1.98173	−0.990863	+	0.134876i	$0.956937\pi$
$654$	0	0
$655$	1.16187e9	0.161553
$656$	−4.96438e8	−0.0686597
$657$	0	0
$658$	1.02789e9	0.140656
$659$	−2.92109e9	−0.397599	−0.198800	−	0.980040i	$-0.563704\pi$
$659$	−2.92109e9	−0.397599	−0.198800	+	0.980040i	$0.563704\pi$
$660$	0	0
$661$	−1.05718e9	−0.142379	−0.0711894	−	0.997463i	$-0.522679\pi$
$661$	−1.05718e9	−0.142379	−0.0711894	+	0.997463i	$0.522679\pi$
$662$	1.03160e10	1.38200
$663$	0	0
$664$	3.86380e9	0.512184
$665$	−1.33211e8	−0.0175657
$666$	0	0
$667$	1.30461e9	0.170231
$668$	4.45087e9	0.577733
$669$	0	0
$670$	1.56720e9	0.201309
$671$	−5.83607e9	−0.745747
$672$	0	0
$673$	1.74723e9	0.220952	0.110476	−	0.993879i	$-0.464762\pi$
$673$	1.74723e9	0.220952	0.110476	+	0.993879i	$0.464762\pi$
$674$	−1.02612e10	−1.29089
$675$	0	0
$676$	−1.53675e9	−0.191333
$677$	1.40297e9	0.173776	0.0868878	−	0.996218i	$-0.472308\pi$
$677$	1.40297e9	0.173776	0.0868878	+	0.996218i	$0.472308\pi$
$678$	0	0
$679$	−1.15248e9	−0.141283
$680$	−6.01831e8	−0.0733995
$681$	0	0
$682$	−5.91951e8	−0.0714563
$683$	−1.13805e10	−1.36675	−0.683373	−	0.730069i	$-0.739488\pi$
$683$	−1.13805e10	−1.36675	−0.683373	+	0.730069i	$0.739488\pi$
$684$	0	0
$685$	1.48848e9	0.176940
$686$	3.22829e8	0.0381802
$687$	0	0
$688$	−2.13421e9	−0.249849
$689$	5.47339e9	0.637513
$690$	0	0
$691$	1.54508e10	1.78147	0.890734	−	0.454524i	$-0.150191\pi$
$691$	1.54508e10	1.78147	0.890734	+	0.454524i	$0.150191\pi$
$692$	3.22163e9	0.369577
$693$	0	0
$694$	−1.96043e9	−0.222635
$695$	−1.54781e9	−0.174892
$696$	0	0
$697$	2.60818e9	0.291758
$698$	−2.66177e8	−0.0296262
$699$	0	0
$700$	1.64950e9	0.181765
$701$	−9.89620e9	−1.08506	−0.542532	−	0.840035i	$-0.682534\pi$
$701$	−9.89620e9	−1.08506	−0.542532	+	0.840035i	$0.682534\pi$
$702$	0	0
$703$	−3.03332e9	−0.329287
$704$	−1.27846e9	−0.138096
$705$	0	0
$706$	−4.86919e9	−0.520763
$707$	3.95458e9	0.420855
$708$	0	0
$709$	4.16857e9	0.439264	0.219632	−	0.975583i	$-0.429514\pi$
$709$	4.16857e9	0.439264	0.219632	+	0.975583i	$0.429514\pi$
$710$	−1.08939e9	−0.114230
$711$	0	0
$712$	−3.92896e9	−0.407941
$713$	2.92552e8	0.0302267
$714$	0	0
$715$	−1.65798e9	−0.169632
$716$	−1.31117e9	−0.133494
$717$	0	0
$718$	1.31886e10	1.32973
$719$	−1.28659e9	−0.129089	−0.0645446	−	0.997915i	$-0.520559\pi$
$719$	−1.28659e9	−0.129089	−0.0645446	+	0.997915i	$0.520559\pi$
$720$	0	0
$721$	3.83230e9	0.380790
$722$	6.74655e9	0.667116
$723$	0	0
$724$	4.77422e8	0.0467539
$725$	5.08398e9	0.495474
$726$	0	0
$727$	−2.57738e9	−0.248776	−0.124388	−	0.992234i	$-0.539697\pi$
$727$	−2.57738e9	−0.248776	−0.124388	+	0.992234i	$0.539697\pi$
$728$	−1.09301e9	−0.104994
$729$	0	0
$730$	−8.74257e8	−0.0831781
$731$	1.12127e10	1.06169
$732$	0	0
$733$	3.86930e9	0.362884	0.181442	−	0.983402i	$-0.441924\pi$
$733$	3.86930e9	0.362884	0.181442	+	0.983402i	$0.441924\pi$
$734$	1.06787e10	0.996740
$735$	0	0
$736$	6.31836e8	0.0584160
$737$	−1.74908e10	−1.60944
$738$	0	0
$739$	1.73788e10	1.58403	0.792016	−	0.610501i	$-0.209032\pi$
$739$	1.73788e10	1.58403	0.792016	+	0.610501i	$0.209032\pi$
$740$	−1.49140e9	−0.135296
$741$	0	0
$742$	−2.41312e9	−0.216853
$743$	−6.09852e9	−0.545461	−0.272730	−	0.962090i	$-0.587927\pi$
$743$	−6.09852e9	−0.545461	−0.272730	+	0.962090i	$0.587927\pi$
$744$	0	0
$745$	−1.46964e9	−0.130216
$746$	4.42359e9	0.390112
$747$	0	0
$748$	6.71675e9	0.586818
$749$	1.17633e9	0.102293
$750$	0	0
$751$	−3.39867e9	−0.292799	−0.146399	−	0.989226i	$-0.546768\pi$
$751$	−3.39867e9	−0.292799	−0.146399	+	0.989226i	$0.546768\pi$
$752$	1.53435e9	0.131571
$753$	0	0
$754$	−3.36881e9	−0.286205
$755$	−2.64938e9	−0.224042
$756$	0	0
$757$	1.90249e10	1.59399	0.796995	−	0.603986i	$-0.206422\pi$
$757$	1.90249e10	1.59399	0.796995	+	0.603986i	$0.206422\pi$
$758$	1.06065e10	0.884568
$759$	0	0
$760$	−1.98846e8	−0.0164312
$761$	6.26428e9	0.515258	0.257629	−	0.966244i	$-0.417059\pi$
$761$	6.26428e9	0.515258	0.257629	+	0.966244i	$0.417059\pi$
$762$	0	0
$763$	6.48082e8	0.0528195
$764$	−4.63191e9	−0.375779
$765$	0	0
$766$	−5.36475e9	−0.431270
$767$	6.62066e9	0.529807
$768$	0	0
$769$	−1.42387e10	−1.12909	−0.564545	−	0.825402i	$-0.690949\pi$
$769$	−1.42387e10	−1.12909	−0.564545	+	0.825402i	$0.690949\pi$
$770$	7.30975e8	0.0577012
$771$	0	0
$772$	9.69500e9	0.758380
$773$	−1.01066e10	−0.787007	−0.393503	−	0.919323i	$-0.628737\pi$
$773$	−1.01066e10	−0.787007	−0.393503	+	0.919323i	$0.628737\pi$
$774$	0	0
$775$	1.14006e9	0.0879777
$776$	−1.72033e9	−0.132158
$777$	0	0
$778$	−6.76049e9	−0.514694
$779$	8.61747e8	0.0653128
$780$	0	0
$781$	1.21582e10	0.913251
$782$	−3.31953e9	−0.248229
$783$	0	0
$784$	4.81890e8	0.0357143
$785$	−3.16123e9	−0.233245
$786$	0	0
$787$	3.95662e9	0.289343	0.144671	−	0.989480i	$-0.453788\pi$
$787$	3.95662e9	0.289343	0.144671	+	0.989480i	$0.453788\pi$
$788$	6.10327e8	0.0444346
$789$	0	0
$790$	−5.09936e8	−0.0367977
$791$	4.84003e9	0.347720
$792$	0	0
$793$	−7.44793e9	−0.530370
$794$	8.73983e8	0.0619628
$795$	0	0
$796$	1.30825e10	0.919378
$797$	−5.24101e9	−0.366700	−0.183350	−	0.983048i	$-0.558694\pi$
$797$	−5.24101e9	−0.366700	−0.183350	+	0.983048i	$0.558694\pi$
$798$	0	0
$799$	−8.06114e9	−0.559091
$800$	2.46223e9	0.170026
$801$	0	0
$802$	−1.70364e9	−0.116618
$803$	9.75717e9	0.664996
$804$	0	0
$805$	−3.61260e8	−0.0244081
$806$	−7.55442e8	−0.0508193
$807$	0	0
$808$	5.90305e9	0.393674
$809$	1.41795e10	0.941544	0.470772	−	0.882255i	$-0.343975\pi$
$809$	1.41795e10	0.941544	0.470772	+	0.882255i	$0.343975\pi$
$810$	0	0
$811$	2.11422e10	1.39180	0.695899	−	0.718139i	$-0.255006\pi$
$811$	2.11422e10	1.39180	0.695899	+	0.718139i	$0.255006\pi$
$812$	1.48525e9	0.0973538
$813$	0	0
$814$	1.66449e10	1.08167
$815$	−4.02026e8	−0.0260137
$816$	0	0
$817$	3.70469e9	0.237670
$818$	1.61171e10	1.02956
$819$	0	0
$820$	4.23699e8	0.0268354
$821$	−4.59683e9	−0.289906	−0.144953	−	0.989439i	$-0.546303\pi$
$821$	−4.59683e9	−0.289906	−0.144953	+	0.989439i	$0.546303\pi$
$822$	0	0
$823$	1.94286e9	0.121491	0.0607453	−	0.998153i	$-0.480652\pi$
$823$	1.94286e9	0.121491	0.0607453	+	0.998153i	$0.480652\pi$
$824$	5.72051e9	0.356197
$825$	0	0
$826$	−2.91893e9	−0.180216
$827$	−7.90499e9	−0.485995	−0.242998	−	0.970027i	$-0.578131\pi$
$827$	−7.90499e9	−0.485995	−0.242998	+	0.970027i	$0.578131\pi$
$828$	0	0
$829$	−1.91855e10	−1.16959	−0.584793	−	0.811182i	$-0.698824\pi$
$829$	−1.91855e10	−1.16959	−0.584793	+	0.811182i	$0.698824\pi$
$830$	−3.29766e9	−0.200186
$831$	0	0
$832$	−1.63155e9	−0.0982132
$833$	−2.53175e9	−0.151762
$834$	0	0
$835$	−3.79872e9	−0.225805
$836$	2.21922e9	0.131365
$837$	0	0
$838$	−1.23319e10	−0.723893
$839$	−5.71006e9	−0.333791	−0.166895	−	0.985975i	$-0.553374\pi$
$839$	−5.71006e9	−0.333791	−0.166895	+	0.985975i	$0.553374\pi$
$840$	0	0
$841$	−1.26721e10	−0.734623
$842$	−2.00006e9	−0.115465
$843$	0	0
$844$	6.62195e9	0.379129
$845$	1.31158e9	0.0747821
$846$	0	0
$847$	−1.47397e9	−0.0833481
$848$	−3.60209e9	−0.202847
$849$	0	0
$850$	−1.29360e10	−0.722496
$851$	−8.22618e9	−0.457556
$852$	0	0
$853$	−2.37284e10	−1.30902	−0.654512	−	0.756052i	$-0.727126\pi$
$853$	−2.37284e10	−1.30902	−0.654512	+	0.756052i	$0.727126\pi$
$854$	3.28366e9	0.180408
$855$	0	0
$856$	1.75593e9	0.0956859
$857$	−1.74012e9	−0.0944379	−0.0472189	−	0.998885i	$-0.515036\pi$
$857$	−1.74012e9	−0.0944379	−0.0472189	+	0.998885i	$0.515036\pi$
$858$	0	0
$859$	1.32629e10	0.713940	0.356970	−	0.934116i	$-0.383810\pi$
$859$	1.32629e10	0.713940	0.356970	+	0.934116i	$0.383810\pi$
$860$	1.82150e9	0.0976527
$861$	0	0
$862$	−8.94995e9	−0.475932
$863$	4.37161e9	0.231528	0.115764	−	0.993277i	$-0.463068\pi$
$863$	4.37161e9	0.231528	0.115764	+	0.993277i	$0.463068\pi$
$864$	0	0
$865$	−2.74959e9	−0.144448
$866$	2.58113e10	1.35051
$867$	0	0
$868$	3.33061e8	0.0172864
$869$	5.69115e9	0.294192
$870$	0	0
$871$	−2.23216e10	−1.14462
$872$	9.67399e8	0.0494081
$873$	0	0
$874$	−1.09678e9	−0.0555685
$875$	−2.87153e9	−0.144905
$876$	0	0
$877$	6.99690e9	0.350273	0.175137	−	0.984544i	$-0.443963\pi$
$877$	6.99690e9	0.350273	0.175137	+	0.984544i	$0.443963\pi$
$878$	1.30218e10	0.649293
$879$	0	0
$880$	1.09113e9	0.0539746
$881$	−3.55447e10	−1.75129	−0.875646	−	0.482953i	$-0.839564\pi$
$881$	−3.55447e10	−1.75129	−0.875646	+	0.482953i	$0.839564\pi$
$882$	0	0
$883$	−3.13427e10	−1.53205	−0.766027	−	0.642808i	$-0.777769\pi$
$883$	−3.13427e10	−1.53205	−0.766027	+	0.642808i	$0.777769\pi$
$884$	8.57184e9	0.417341
$885$	0	0
$886$	−2.29306e9	−0.110764
$887$	3.34580e10	1.60978	0.804892	−	0.593421i	$-0.202223\pi$
$887$	3.34580e10	1.60978	0.804892	+	0.593421i	$0.202223\pi$
$888$	0	0
$889$	−1.06134e10	−0.506641
$890$	3.35328e9	0.159443
$891$	0	0
$892$	−9.25467e9	−0.436600
$893$	−2.66341e9	−0.125158
$894$	0	0
$895$	1.11905e9	0.0521759
$896$	7.19323e8	0.0334077
$897$	0	0
$898$	1.86959e10	0.861548
$899$	1.02654e9	0.0471211
$900$	0	0
$901$	1.89246e10	0.861966
$902$	−4.72870e9	−0.214545
$903$	0	0
$904$	7.22477e9	0.325263
$905$	−4.07469e8	−0.0182736
$906$	0	0
$907$	−9.00067e9	−0.400543	−0.200272	−	0.979740i	$-0.564182\pi$
$907$	−9.00067e9	−0.400543	−0.200272	+	0.979740i	$0.564182\pi$
$908$	1.55915e10	0.691176
$909$	0	0
$910$	9.32862e8	0.0410367
$911$	−2.29482e10	−1.00562	−0.502811	−	0.864397i	$-0.667701\pi$
$911$	−2.29482e10	−1.00562	−0.502811	+	0.864397i	$0.667701\pi$
$912$	0	0
$913$	3.68037e10	1.60045
$914$	5.53688e9	0.239858
$915$	0	0
$916$	2.25064e8	0.00967546
$917$	7.29593e9	0.312455
$918$	0	0
$919$	−4.02376e9	−0.171012	−0.0855061	−	0.996338i	$-0.527251\pi$
$919$	−4.02376e9	−0.171012	−0.0855061	+	0.996338i	$0.527251\pi$
$920$	−5.39257e8	−0.0228317
$921$	0	0
$922$	−2.40237e10	−1.00944
$923$	1.55161e10	0.649498
$924$	0	0
$925$	−3.20570e10	−1.33176
$926$	4.22166e8	0.0174721
$927$	0	0
$928$	2.21705e9	0.0910661
$929$	−3.66147e10	−1.49830	−0.749152	−	0.662398i	$-0.769539\pi$
$929$	−3.66147e10	−1.49830	−0.749152	+	0.662398i	$0.769539\pi$
$930$	0	0
$931$	−8.36493e8	−0.0339734
$932$	1.30139e10	0.526563
$933$	0	0
$934$	−5.58688e9	−0.224365
$935$	−5.73259e9	−0.229356
$936$	0	0
$937$	−2.13032e10	−0.845971	−0.422985	−	0.906136i	$-0.639018\pi$
$937$	−2.13032e10	−0.845971	−0.422985	+	0.906136i	$0.639018\pi$
$938$	9.84120e9	0.389348
$939$	0	0
$940$	−1.30953e9	−0.0514243
$941$	−3.58329e10	−1.40190	−0.700952	−	0.713209i	$-0.747241\pi$
$941$	−3.58329e10	−1.40190	−0.700952	+	0.713209i	$0.747241\pi$
$942$	0	0
$943$	2.33701e9	0.0907546
$944$	−4.35712e9	−0.168577
$945$	0	0
$946$	−2.03289e10	−0.780719
$947$	−8.36114e9	−0.319919	−0.159960	−	0.987124i	$-0.551136\pi$
$947$	−8.36114e9	−0.319919	−0.159960	+	0.987124i	$0.551136\pi$
$948$	0	0
$949$	1.24520e10	0.472941
$950$	−4.27409e9	−0.161738
$951$	0	0
$952$	−3.77917e9	−0.141960
$953$	2.16103e10	0.808789	0.404395	−	0.914585i	$-0.367482\pi$
$953$	2.16103e10	0.808789	0.404395	+	0.914585i	$0.367482\pi$
$954$	0	0
$955$	3.95323e9	0.146872
$956$	2.12954e10	0.788284
$957$	0	0
$958$	−1.56294e10	−0.574334
$959$	9.34683e9	0.342215
$960$	0	0
$961$	−2.72824e10	−0.991633
$962$	2.12420e10	0.769276
$963$	0	0
$964$	3.05491e9	0.109832
$965$	−8.27446e9	−0.296411
$966$	0	0
$967$	−4.42197e9	−0.157262	−0.0786308	−	0.996904i	$-0.525055\pi$
$967$	−4.42197e9	−0.157262	−0.0786308	+	0.996904i	$0.525055\pi$
$968$	−2.20021e9	−0.0779650
$969$	0	0
$970$	1.46826e9	0.0516537
$971$	6.23091e9	0.218416	0.109208	−	0.994019i	$-0.465169\pi$
$971$	6.23091e9	0.218416	0.109208	+	0.994019i	$0.465169\pi$
$972$	0	0
$973$	−9.71940e9	−0.338255
$974$	−2.16288e10	−0.750027
$975$	0	0
$976$	4.90155e9	0.168756
$977$	1.29962e10	0.445846	0.222923	−	0.974836i	$-0.428440\pi$
$977$	1.29962e10	0.445846	0.222923	+	0.974836i	$0.428440\pi$
$978$	0	0
$979$	−3.74244e10	−1.27472
$980$	−4.11282e8	−0.0139588
$981$	0	0
$982$	−3.29398e10	−1.11002
$983$	4.60389e10	1.54592	0.772961	−	0.634454i	$-0.218775\pi$
$983$	4.60389e10	1.54592	0.772961	+	0.634454i	$0.218775\pi$
$984$	0	0
$985$	−5.20900e8	−0.0173671
$986$	−1.16479e10	−0.386971
$987$	0	0
$988$	2.83215e9	0.0934257
$989$	1.00469e10	0.330251
$990$	0	0
$991$	2.28769e10	0.746687	0.373343	−	0.927693i	$-0.378211\pi$
$991$	2.28769e10	0.746687	0.373343	+	0.927693i	$0.378211\pi$
$992$	4.97164e8	0.0161699
$993$	0	0
$994$	−6.84079e9	−0.220930
$995$	−1.11656e10	−0.359336
$996$	0	0
$997$	−3.62930e10	−1.15982	−0.579910	−	0.814681i	$-0.696912\pi$
$997$	−3.62930e10	−1.15982	−0.579910	+	0.814681i	$0.696912\pi$
$998$	8.51258e9	0.271084
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	378.8.a.p.1.2	✓	4
3.2	odd	2		378.8.a.q.1.3	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
378.8.a.p.1.2	✓	4	1.1	even	1	trivial
378.8.a.q.1.3	yes	4	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 378 = 2 \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	378.a (trivial)

\(f(q)\)	\(=\)	\(q-8.00000 q^{2} +64.0000 q^{4} -54.6225 q^{5} -343.000 q^{7} -512.000 q^{8} +O(q^{10})\)	\(q-8.00000 q^{2} +64.0000 q^{4} -54.6225 q^{5} -343.000 q^{7} -512.000 q^{8} +436.980 q^{10} -4876.93 q^{11} -6223.88 q^{13} +2744.00 q^{14} +4096.00 q^{16} -21519.5 q^{17} -7110.08 q^{19} -3495.84 q^{20} +39015.4 q^{22} -19282.1 q^{23} -75141.4 q^{25} +49791.1 q^{26} -21952.0 q^{28} -67658.9 q^{29} -15172.2 q^{31} -32768.0 q^{32} +172156. q^{34} +18735.5 q^{35} +426622. q^{37} +56880.6 q^{38} +27966.7 q^{40} -121201. q^{41} -521047. q^{43} -312124. q^{44} +154257. q^{46} +374597. q^{47} +117649. q^{49} +601131. q^{50} -398329. q^{52} -879416. q^{53} +266390. q^{55} +175616. q^{56} +541271. q^{58} -1.06375e6 q^{59} +1.19667e6 q^{61} +121378. q^{62} +262144. q^{64} +339964. q^{65} +3.58644e6 q^{67} -1.37725e6 q^{68} -149884. q^{70} -2.49300e6 q^{71} -2.00068e6 q^{73} -3.41298e6 q^{74} -455045. q^{76} +1.67279e6 q^{77} -1.16695e6 q^{79} -223734. q^{80} +969606. q^{82} -7.54648e6 q^{83} +1.17545e6 q^{85} +4.16838e6 q^{86} +2.49699e6 q^{88} +7.67375e6 q^{89} +2.13479e6 q^{91} -1.23405e6 q^{92} -2.99677e6 q^{94} +388370. q^{95} +3.36001e6 q^{97} -941192. q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 32 q^{2} + 256 q^{4} + 100 q^{5} - 1372 q^{7} - 2048 q^{8}+O(q^{10}) \) 4 * q - 32 * q^2 + 256 * q^4 + 100 * q^5 - 1372 * q^7 - 2048 * q^8	\( 4 q - 32 q^{2} + 256 q^{4} + 100 q^{5} - 1372 q^{7} - 2048 q^{8} - 800 q^{10} + 6446 q^{11} + 6594 q^{13} + 10976 q^{14} + 16384 q^{16} + 22184 q^{17} - 15730 q^{19} + 6400 q^{20} - 51568 q^{22} - 107176 q^{23} + 25790 q^{25} - 52752 q^{26} - 87808 q^{28} - 163844 q^{29} + 35424 q^{31} - 131072 q^{32} - 177472 q^{34} - 34300 q^{35} + 371678 q^{37} + 125840 q^{38} - 51200 q^{40} + 79212 q^{41} + 1186652 q^{43} + 412544 q^{44} + 857408 q^{46} - 952110 q^{47} + 470596 q^{49} - 206320 q^{50} + 422016 q^{52} - 155082 q^{53} + 3728930 q^{55} + 702464 q^{56} + 1310752 q^{58} + 1193848 q^{59} + 129102 q^{61} - 283392 q^{62} + 1048576 q^{64} - 9519330 q^{65} - 2555300 q^{67} + 1419776 q^{68} + 274400 q^{70} - 2975872 q^{71} + 6428020 q^{73} - 2973424 q^{74} - 1006720 q^{76} - 2210978 q^{77} + 6843240 q^{79} + 409600 q^{80} - 633696 q^{82} - 11642074 q^{83} - 1650490 q^{85} - 9493216 q^{86} - 3300352 q^{88} - 21148548 q^{89} - 2261742 q^{91} - 6859264 q^{92} + 7616880 q^{94} - 8711770 q^{95} - 5112362 q^{97} - 3764768 q^{98}+O(q^{100}) \) 4 * q - 32 * q^2 + 256 * q^4 + 100 * q^5 - 1372 * q^7 - 2048 * q^8 - 800 * q^10 + 6446 * q^11 + 6594 * q^13 + 10976 * q^14 + 16384 * q^16 + 22184 * q^17 - 15730 * q^19 + 6400 * q^20 - 51568 * q^22 - 107176 * q^23 + 25790 * q^25 - 52752 * q^26 - 87808 * q^28 - 163844 * q^29 + 35424 * q^31 - 131072 * q^32 - 177472 * q^34 - 34300 * q^35 + 371678 * q^37 + 125840 * q^38 - 51200 * q^40 + 79212 * q^41 + 1186652 * q^43 + 412544 * q^44 + 857408 * q^46 - 952110 * q^47 + 470596 * q^49 - 206320 * q^50 + 422016 * q^52 - 155082 * q^53 + 3728930 * q^55 + 702464 * q^56 + 1310752 * q^58 + 1193848 * q^59 + 129102 * q^61 - 283392 * q^62 + 1048576 * q^64 - 9519330 * q^65 - 2555300 * q^67 + 1419776 * q^68 + 274400 * q^70 - 2975872 * q^71 + 6428020 * q^73 - 2973424 * q^74 - 1006720 * q^76 - 2210978 * q^77 + 6843240 * q^79 + 409600 * q^80 - 633696 * q^82 - 11642074 * q^83 - 1650490 * q^85 - 9493216 * q^86 - 3300352 * q^88 - 21148548 * q^89 - 2261742 * q^91 - 6859264 * q^92 + 7616880 * q^94 - 8711770 * q^95 - 5112362 * q^97 - 3764768 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more