LMFDB - Embedded newform 275.6.a.e.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [275,6,Mod(1,275)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(275, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("275.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 275 = 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	275.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:	$44.1055504486$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 129x^{3} + 45x^{2} + 2924x - 5216 $ x^5 - x^4 - 129x^3 + 45x^2 + 2924*x - 5216
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 55)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-6.57133$ of defining polynomial
Character	$\chi$	$=$	275.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-8.57133 q^{2} -16.0464 q^{3} +41.4676 q^{4} +137.539 q^{6} -3.71474 q^{7} -81.1502 q^{8} +14.4878 q^{9} +O(q^{10})$	\(q-8.57133 q^{2} -16.0464 q^{3} +41.4676 q^{4} +137.539 q^{6} -3.71474 q^{7} -81.1502 q^{8} +14.4878 q^{9} -121.000 q^{11} -665.407 q^{12} -94.0133 q^{13} +31.8402 q^{14} -631.400 q^{16} -196.903 q^{17} -124.180 q^{18} -425.349 q^{19} +59.6082 q^{21} +1037.13 q^{22} -518.668 q^{23} +1302.17 q^{24} +805.819 q^{26} +3666.80 q^{27} -154.041 q^{28} -378.739 q^{29} +5098.68 q^{31} +8008.74 q^{32} +1941.62 q^{33} +1687.72 q^{34} +600.775 q^{36} +4418.94 q^{37} +3645.81 q^{38} +1508.58 q^{39} +12225.3 q^{41} -510.922 q^{42} +12069.9 q^{43} -5017.58 q^{44} +4445.67 q^{46} -1450.03 q^{47} +10131.7 q^{48} -16793.2 q^{49} +3159.59 q^{51} -3898.51 q^{52} +24797.4 q^{53} -31429.4 q^{54} +301.452 q^{56} +6825.33 q^{57} +3246.30 q^{58} -22302.7 q^{59} +52941.4 q^{61} -43702.5 q^{62} -53.8183 q^{63} -48440.7 q^{64} -16642.2 q^{66} -65036.7 q^{67} -8165.11 q^{68} +8322.76 q^{69} +60745.5 q^{71} -1175.69 q^{72} -54856.0 q^{73} -37876.1 q^{74} -17638.2 q^{76} +449.483 q^{77} -12930.5 q^{78} +40298.7 q^{79} -62359.6 q^{81} -104787. q^{82} -74192.7 q^{83} +2471.81 q^{84} -103455. q^{86} +6077.41 q^{87} +9819.17 q^{88} +551.359 q^{89} +349.235 q^{91} -21507.9 q^{92} -81815.7 q^{93} +12428.6 q^{94} -128512. q^{96} -82992.9 q^{97} +143940. q^{98} -1753.02 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 9 q^{2} + 115 q^{4} - 237 q^{6} - 70 q^{7} - 753 q^{8} + 1059 q^{9}+O(q^{10}) $ 5 * q - 9 * q^2 + 115 * q^4 - 237 * q^6 - 70 * q^7 - 753 * q^8 + 1059 * q^9	\( 5 q - 9 q^{2} + 115 q^{4} - 237 q^{6} - 70 q^{7} - 753 q^{8} + 1059 q^{9} - 605 q^{11} + 1605 q^{12} - 1498 q^{13} + 2113 q^{14} + 4883 q^{16} - 3874 q^{17} - 5838 q^{18} + 882 q^{19} + 1092 q^{21} + 1089 q^{22} + 5344 q^{23} - 13119 q^{24} - 4478 q^{26} + 2160 q^{27} - 12565 q^{28} + 5318 q^{29} - 7916 q^{31} - 21385 q^{32} - 18605 q^{34} + 5628 q^{36} + 1788 q^{37} + 34421 q^{38} - 29760 q^{39} + 5854 q^{41} + 46725 q^{42} + 4364 q^{43} - 13915 q^{44} - 33834 q^{46} - 46452 q^{47} + 127545 q^{48} - 34217 q^{49} + 19842 q^{51} - 3222 q^{52} - 4412 q^{53} - 86535 q^{54} + 115575 q^{56} - 137160 q^{57} + 58221 q^{58} + 17896 q^{59} - 35930 q^{61} + 19627 q^{62} - 100980 q^{63} + 14779 q^{64} + 28677 q^{66} - 73136 q^{67} + 83409 q^{68} + 34296 q^{69} + 43612 q^{71} - 372276 q^{72} - 142306 q^{73} - 95609 q^{74} - 6617 q^{76} + 8470 q^{77} - 15750 q^{78} - 46504 q^{79} + 79101 q^{81} - 175798 q^{82} - 81604 q^{83} - 532533 q^{84} - 101788 q^{86} - 219750 q^{87} + 91113 q^{88} + 8664 q^{89} - 203380 q^{91} - 251174 q^{92} + 46470 q^{93} - 71458 q^{94} - 925479 q^{96} + 22230 q^{97} - 59962 q^{98} - 128139 q^{99}+O(q^{100}) \) 5 * q - 9 * q^2 + 115 * q^4 - 237 * q^6 - 70 * q^7 - 753 * q^8 + 1059 * q^9 - 605 * q^11 + 1605 * q^12 - 1498 * q^13 + 2113 * q^14 + 4883 * q^16 - 3874 * q^17 - 5838 * q^18 + 882 * q^19 + 1092 * q^21 + 1089 * q^22 + 5344 * q^23 - 13119 * q^24 - 4478 * q^26 + 2160 * q^27 - 12565 * q^28 + 5318 * q^29 - 7916 * q^31 - 21385 * q^32 - 18605 * q^34 + 5628 * q^36 + 1788 * q^37 + 34421 * q^38 - 29760 * q^39 + 5854 * q^41 + 46725 * q^42 + 4364 * q^43 - 13915 * q^44 - 33834 * q^46 - 46452 * q^47 + 127545 * q^48 - 34217 * q^49 + 19842 * q^51 - 3222 * q^52 - 4412 * q^53 - 86535 * q^54 + 115575 * q^56 - 137160 * q^57 + 58221 * q^58 + 17896 * q^59 - 35930 * q^61 + 19627 * q^62 - 100980 * q^63 + 14779 * q^64 + 28677 * q^66 - 73136 * q^67 + 83409 * q^68 + 34296 * q^69 + 43612 * q^71 - 372276 * q^72 - 142306 * q^73 - 95609 * q^74 - 6617 * q^76 + 8470 * q^77 - 15750 * q^78 - 46504 * q^79 + 79101 * q^81 - 175798 * q^82 - 81604 * q^83 - 532533 * q^84 - 101788 * q^86 - 219750 * q^87 + 91113 * q^88 + 8664 * q^89 - 203380 * q^91 - 251174 * q^92 + 46470 * q^93 - 71458 * q^94 - 925479 * q^96 + 22230 * q^97 - 59962 * q^98 - 128139 * q^99

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 $

$2$	−8.57133	−1.51521	−0.757605	−	0.652713i	$-0.773631\pi$
$2$	−8.57133	−1.51521	−0.757605	+	0.652713i	$0.773631\pi$
$3$	−16.0464	−1.02938	−0.514689	−	0.857377i	$-0.672093\pi$
$3$	−16.0464	−1.02938	−0.514689	+	0.857377i	$0.672093\pi$
$4$	41.4676	1.29586
$5$	0	0
$5$	0	0
$6$	137.539	1.55973
$7$	−3.71474	−0.0286538	−0.0143269	−	0.999897i	$-0.504561\pi$
$7$	−3.71474	−0.0286538	−0.0143269	+	0.999897i	$0.504561\pi$
$8$	−81.1502	−0.448296
$9$	14.4878	0.0596205
$10$	0	0
$11$	−121.000	−0.301511
$11$	−121.000	−0.301511
$12$	−665.407	−1.33393
$13$	−94.0133	−0.154288	−0.0771438	−	0.997020i	$-0.524580\pi$
$13$	−94.0133	−0.154288	−0.0771438	+	0.997020i	$0.524580\pi$
$14$	31.8402	0.0434166
$15$	0	0
$16$	−631.400	−0.616601
$17$	−196.903	−0.165246	−0.0826229	−	0.996581i	$-0.526330\pi$
$17$	−196.903	−0.165246	−0.0826229	+	0.996581i	$0.526330\pi$
$18$	−124.180	−0.0903377
$19$	−425.349	−0.270310	−0.135155	−	0.990824i	$-0.543153\pi$
$19$	−425.349	−0.270310	−0.135155	+	0.990824i	$0.543153\pi$
$20$	0	0
$21$	59.6082	0.0294957
$22$	1037.13	0.456853
$23$	−518.668	−0.204442	−0.102221	−	0.994762i	$-0.532595\pi$
$23$	−518.668	−0.204442	−0.102221	+	0.994762i	$0.532595\pi$
$24$	1302.17	0.461466
$25$	0	0
$26$	805.819	0.233778
$27$	3666.80	0.968007
$28$	−154.041	−0.0371315
$29$	−378.739	−0.0836268	−0.0418134	−	0.999125i	$-0.513313\pi$
$29$	−378.739	−0.0836268	−0.0418134	+	0.999125i	$0.513313\pi$
$30$	0	0
$31$	5098.68	0.952914	0.476457	−	0.879198i	$-0.341921\pi$
$31$	5098.68	0.952914	0.476457	+	0.879198i	$0.341921\pi$
$32$	8008.74	1.38258
$33$	1941.62	0.310369
$34$	1687.72	0.250382
$35$	0	0
$36$	600.775	0.0772601
$37$	4418.94	0.530656	0.265328	−	0.964158i	$-0.414520\pi$
$37$	4418.94	0.530656	0.265328	+	0.964158i	$0.414520\pi$
$38$	3645.81	0.409576
$39$	1508.58	0.158820
$40$	0	0
$41$	12225.3	1.13579	0.567896	−	0.823101i	$-0.307758\pi$
$41$	12225.3	1.13579	0.567896	+	0.823101i	$0.307758\pi$
$42$	−510.922	−0.0446921
$43$	12069.9	0.995477	0.497738	−	0.867327i	$-0.334164\pi$
$43$	12069.9	0.995477	0.497738	+	0.867327i	$0.334164\pi$
$44$	−5017.58	−0.390718
$45$	0	0
$46$	4445.67	0.309772
$47$	−1450.03	−0.0957483	−0.0478741	−	0.998853i	$-0.515245\pi$
$47$	−1450.03	−0.0957483	−0.0478741	+	0.998853i	$0.515245\pi$
$48$	10131.7	0.634716
$49$	−16793.2	−0.999179
$50$	0	0
$51$	3159.59	0.170101
$52$	−3898.51	−0.199936
$53$	24797.4	1.21259	0.606297	−	0.795238i	$-0.292654\pi$
$53$	24797.4	1.21259	0.606297	+	0.795238i	$0.292654\pi$
$54$	−31429.4	−1.46673
$55$	0	0
$56$	301.452	0.0128454
$57$	6825.33	0.278251
$58$	3246.30	0.126712
$59$	−22302.7	−0.834119	−0.417060	−	0.908879i	$-0.636939\pi$
$59$	−22302.7	−0.834119	−0.417060	+	0.908879i	$0.636939\pi$
$60$	0	0
$61$	52941.4	1.82167	0.910837	−	0.412766i	$-0.135437\pi$
$61$	52941.4	1.82167	0.910837	+	0.412766i	$0.135437\pi$
$62$	−43702.5	−1.44387
$63$	−53.8183	−0.00170836
$64$	−48440.7	−1.47829
$65$	0	0
$66$	−16642.2	−0.470275
$67$	−65036.7	−1.76999	−0.884997	−	0.465597i	$-0.845839\pi$
$67$	−65036.7	−1.76999	−0.884997	+	0.465597i	$0.845839\pi$
$68$	−8165.11	−0.214136
$69$	8322.76	0.210448
$70$	0	0
$71$	60745.5	1.43011	0.715053	−	0.699071i	$-0.246403\pi$
$71$	60745.5	1.43011	0.715053	+	0.699071i	$0.246403\pi$
$72$	−1175.69	−0.0267276
$73$	−54856.0	−1.20481	−0.602403	−	0.798192i	$-0.705790\pi$
$73$	−54856.0	−1.20481	−0.602403	+	0.798192i	$0.705790\pi$
$74$	−37876.1	−0.804056
$75$	0	0
$76$	−17638.2	−0.350284
$77$	449.483	0.00863946
$78$	−12930.5	−0.240646
$79$	40298.7	0.726479	0.363240	−	0.931696i	$-0.381671\pi$
$79$	40298.7	0.726479	0.363240	+	0.931696i	$0.381671\pi$
$80$	0	0
$81$	−62359.6	−1.05607
$82$	−104787.	−1.72096
$83$	−74192.7	−1.18213	−0.591066	−	0.806623i	$-0.701293\pi$
$83$	−74192.7	−1.18213	−0.591066	+	0.806623i	$0.701293\pi$
$84$	2471.81	0.0382223
$85$	0	0
$86$	−103455.	−1.50836
$87$	6077.41	0.0860836
$88$	9819.17	0.135166
$89$	551.359	0.00737836	0.00368918	−	0.999993i	$-0.498826\pi$
$89$	551.359	0.00737836	0.00368918	+	0.999993i	$0.498826\pi$
$90$	0	0
$91$	349.235	0.00442093
$92$	−21507.9	−0.264929
$93$	−81815.7	−0.980910
$94$	12428.6	0.145079
$95$	0	0
$96$	−128512.	−1.42319
$97$	−82992.9	−0.895594	−0.447797	−	0.894135i	$-0.647791\pi$
$97$	−82992.9	−0.895594	−0.447797	+	0.894135i	$0.647791\pi$
$98$	143940.	1.51397
$99$	−1753.02	−0.0179763
$100$	0	0
$101$	−16805.0	−0.163921	−0.0819605	−	0.996636i	$-0.526118\pi$
$101$	−16805.0	−0.163921	−0.0819605	+	0.996636i	$0.526118\pi$
$102$	−27081.9	−0.257738
$103$	16517.4	0.153409	0.0767043	−	0.997054i	$-0.475560\pi$
$103$	16517.4	0.153409	0.0767043	+	0.997054i	$0.475560\pi$
$104$	7629.20	0.0691665
$105$	0	0
$106$	−212546.	−1.83734
$107$	127625.	1.07765	0.538825	−	0.842418i	$-0.318868\pi$
$107$	127625.	1.07765	0.538825	+	0.842418i	$0.318868\pi$
$108$	152054.	1.25440
$109$	−155192.	−1.25113	−0.625567	−	0.780171i	$-0.715132\pi$
$109$	−155192.	−1.25113	−0.625567	+	0.780171i	$0.715132\pi$
$110$	0	0
$111$	−70908.1	−0.546246
$112$	2345.48	0.0176680
$113$	64553.7	0.475582	0.237791	−	0.971316i	$-0.423577\pi$
$113$	64553.7	0.475582	0.237791	+	0.971316i	$0.423577\pi$
$114$	−58502.1	−0.421609
$115$	0	0
$116$	−15705.4	−0.108369
$117$	−1362.05	−0.00919871
$118$	191164.	1.26387
$119$	731.444	0.00473493
$120$	0	0
$121$	14641.0	0.0909091
$122$	−453778.	−2.76022
$123$	−196172.	−1.16916
$124$	211430.	1.23485
$125$	0	0
$126$	461.294	0.00258852
$127$	−153990.	−0.847193	−0.423596	−	0.905851i	$-0.639232\pi$
$127$	−153990.	−0.847193	−0.423596	+	0.905851i	$0.639232\pi$
$128$	158922.	0.857350
$129$	−193678.	−1.02472
$130$	0	0
$131$	134029.	0.682372	0.341186	−	0.939996i	$-0.389171\pi$
$131$	134029.	0.682372	0.341186	+	0.939996i	$0.389171\pi$
$132$	80514.3	0.402196
$133$	1580.06	0.00774541
$134$	557451.	2.68191
$135$	0	0
$136$	15978.7	0.0740790
$137$	−100996.	−0.459730	−0.229865	−	0.973223i	$-0.573828\pi$
$137$	−100996.	−0.459730	−0.229865	+	0.973223i	$0.573828\pi$
$138$	−71337.1	−0.318873
$139$	−298270.	−1.30940	−0.654700	−	0.755889i	$-0.727205\pi$
$139$	−298270.	−1.30940	−0.654700	+	0.755889i	$0.727205\pi$
$140$	0	0
$141$	23267.7	0.0985613
$142$	−520669.	−2.16691
$143$	11375.6	0.0465195
$144$	−9147.59	−0.0367621
$145$	0	0
$146$	470189.	1.82554
$147$	269471.	1.02853
$148$	183243.	0.687658
$149$	169128.	0.624093	0.312047	−	0.950067i	$-0.398986\pi$
$149$	169128.	0.624093	0.312047	+	0.950067i	$0.398986\pi$
$150$	0	0
$151$	−161845.	−0.577641	−0.288821	−	0.957383i	$-0.593263\pi$
$151$	−161845.	−0.577641	−0.288821	+	0.957383i	$0.593263\pi$
$152$	34517.2	0.121179
$153$	−2852.69	−0.00985205
$154$	−3852.67	−0.0130906
$155$	0	0
$156$	62557.2	0.205810
$157$	421890.	1.36600	0.683000	−	0.730419i	$-0.260675\pi$
$157$	421890.	1.36600	0.683000	+	0.730419i	$0.260675\pi$
$158$	−345413.	−1.10077
$159$	−397909.	−1.24822
$160$	0	0
$161$	1926.71	0.00585804
$162$	534505.	1.60016
$163$	380794.	1.12259	0.561295	−	0.827616i	$-0.310303\pi$
$163$	380794.	1.12259	0.561295	+	0.827616i	$0.310303\pi$
$164$	506953.	1.47183
$165$	0	0
$166$	635930.	1.79118
$167$	652565.	1.81064	0.905321	−	0.424727i	$-0.139630\pi$
$167$	652565.	1.81064	0.905321	+	0.424727i	$0.139630\pi$
$168$	−4837.22	−0.0132228
$169$	−362454.	−0.976195
$170$	0	0
$171$	−6162.37	−0.0161160
$172$	500509.	1.29000
$173$	−541735.	−1.37617	−0.688084	−	0.725631i	$-0.741548\pi$
$173$	−541735.	−1.37617	−0.688084	+	0.725631i	$0.741548\pi$
$174$	−52091.5	−0.130435
$175$	0	0
$176$	76399.3	0.185912
$177$	357879.	0.858625
$178$	−4725.88	−0.0111798
$179$	185595.	0.432945	0.216472	−	0.976289i	$-0.430545\pi$
$179$	185595.	0.432945	0.216472	+	0.976289i	$0.430545\pi$
$180$	0	0
$181$	−194784.	−0.441933	−0.220967	−	0.975281i	$-0.570921\pi$
$181$	−194784.	−0.441933	−0.220967	+	0.975281i	$0.570921\pi$
$182$	−2993.40	−0.00669865
$183$	−849520.	−1.87519
$184$	42090.0	0.0916504
$185$	0	0
$186$	701269.	1.48629
$187$	23825.3	0.0498235
$188$	−60129.1	−0.124077
$189$	−13621.2	−0.0277371
$190$	0	0
$191$	71148.2	0.141117	0.0705586	−	0.997508i	$-0.477522\pi$
$191$	71148.2	0.141117	0.0705586	+	0.997508i	$0.477522\pi$
$192$	777300.	1.52172
$193$	−627831.	−1.21325	−0.606624	−	0.794989i	$-0.707476\pi$
$193$	−627831.	−1.21325	−0.606624	+	0.794989i	$0.707476\pi$
$194$	711359.	1.35701
$195$	0	0
$196$	−696374.	−1.29480
$197$	−802033.	−1.47240	−0.736201	−	0.676763i	$-0.763382\pi$
$197$	−802033.	−1.47240	−0.736201	+	0.676763i	$0.763382\pi$
$198$	15025.7	0.0272378
$199$	−1.02974e6	−1.84329	−0.921647	−	0.388030i	$-0.873156\pi$
$199$	−1.02974e6	−1.84329	−0.921647	+	0.388030i	$0.873156\pi$
$200$	0	0
$201$	1.04361e6	1.82199
$202$	144041.	0.248375
$203$	1406.92	0.00239623
$204$	131021.	0.220427
$205$	0	0
$206$	−141576.	−0.232446
$207$	−7514.35	−0.0121889
$208$	59360.0	0.0951339
$209$	51467.2	0.0815014
$210$	0	0
$211$	−642370.	−0.993297	−0.496648	−	0.867952i	$-0.665436\pi$
$211$	−642370.	−0.993297	−0.496648	+	0.867952i	$0.665436\pi$
$212$	1.02829e6	1.57136
$213$	−974748.	−1.47212
$214$	−1.09392e6	−1.63287
$215$	0	0
$216$	−297562.	−0.433953
$217$	−18940.3	−0.0273047
$218$	1.33020e6	1.89573
$219$	880243.	1.24020
$220$	0	0
$221$	18511.5	0.0254954
$222$	607777.	0.827678
$223$	1.29531e6	1.74427	0.872133	−	0.489269i	$-0.162736\pi$
$223$	1.29531e6	1.74427	0.872133	+	0.489269i	$0.162736\pi$
$224$	−29750.3	−0.0396161
$225$	0	0
$226$	−553311.	−0.720607
$227$	−51821.2	−0.0667487	−0.0333743	−	0.999443i	$-0.510625\pi$
$227$	−51821.2	−0.0667487	−0.0333743	+	0.999443i	$0.510625\pi$
$228$	283030.	0.360575
$229$	−543679.	−0.685100	−0.342550	−	0.939500i	$-0.611291\pi$
$229$	−543679.	−0.685100	−0.342550	+	0.939500i	$0.611291\pi$
$230$	0	0
$231$	−7212.60	−0.00889328
$232$	30734.8	0.0374895
$233$	−567463.	−0.684775	−0.342387	−	0.939559i	$-0.611236\pi$
$233$	−567463.	−0.684775	−0.342387	+	0.939559i	$0.611236\pi$
$234$	11674.5	0.0139380
$235$	0	0
$236$	−924842.	−1.08090
$237$	−646650.	−0.747822
$238$	−6269.44	−0.00717442
$239$	244565.	0.276948	0.138474	−	0.990366i	$-0.455780\pi$
$239$	244565.	0.276948	0.138474	+	0.990366i	$0.455780\pi$
$240$	0	0
$241$	1.25296e6	1.38961	0.694806	−	0.719197i	$-0.255490\pi$
$241$	1.25296e6	1.38961	0.694806	+	0.719197i	$0.255490\pi$
$242$	−125493.	−0.137746
$243$	109616.	0.119085
$244$	2.19535e6	2.36064
$245$	0	0
$246$	1.68145e6	1.77152
$247$	39988.5	0.0417054
$248$	−413759.	−0.427188
$249$	1.19053e6	1.21686
$250$	0	0
$251$	−1.28427e6	−1.28668	−0.643340	−	0.765581i	$-0.722452\pi$
$251$	−1.28427e6	−1.28668	−0.643340	+	0.765581i	$0.722452\pi$
$252$	−2231.72	−0.00221380
$253$	62758.8	0.0616415
$254$	1.31990e6	1.28368
$255$	0	0
$256$	187934.	0.179228
$257$	−190501.	−0.179914	−0.0899569	−	0.995946i	$-0.528673\pi$
$257$	−190501.	−0.179914	−0.0899569	+	0.995946i	$0.528673\pi$
$258$	1.66008e6	1.55267
$259$	−16415.2	−0.0152053
$260$	0	0
$261$	−5487.10	−0.00498587
$262$	−1.14881e6	−1.03394
$263$	631347.	0.562832	0.281416	−	0.959586i	$-0.409196\pi$
$263$	631347.	0.562832	0.281416	+	0.959586i	$0.409196\pi$
$264$	−157563.	−0.139137
$265$	0	0
$266$	−13543.2	−0.0117359
$267$	−8847.35	−0.00759512
$268$	−2.69692e6	−2.29367
$269$	1.32229e6	1.11415	0.557077	−	0.830461i	$-0.311923\pi$
$269$	1.32229e6	1.11415	0.557077	+	0.830461i	$0.311923\pi$
$270$	0	0
$271$	−2.26680e6	−1.87495	−0.937475	−	0.348053i	$-0.886843\pi$
$271$	−2.26680e6	−1.87495	−0.937475	+	0.348053i	$0.886843\pi$
$272$	124325.	0.101891
$273$	−5603.97	−0.00455081
$274$	865669.	0.696588
$275$	0	0
$276$	345125.	0.272712
$277$	−1.21741e6	−0.953315	−0.476658	−	0.879089i	$-0.658152\pi$
$277$	−1.21741e6	−0.953315	−0.476658	+	0.879089i	$0.658152\pi$
$278$	2.55657e6	1.98402
$279$	73868.7	0.0568133
$280$	0	0
$281$	−1.94463e6	−1.46917	−0.734584	−	0.678518i	$-0.762623\pi$
$281$	−1.94463e6	−1.46917	−0.734584	+	0.678518i	$0.762623\pi$
$282$	−199435.	−0.149341
$283$	−1.61891e6	−1.20159	−0.600796	−	0.799402i	$-0.705150\pi$
$283$	−1.61891e6	−1.20159	−0.600796	+	0.799402i	$0.705150\pi$
$284$	2.51897e6	1.85322
$285$	0	0
$286$	−97504.1	−0.0704868
$287$	−45413.6	−0.0325448
$288$	116029.	0.0824300
$289$	−1.38109e6	−0.972694
$290$	0	0
$291$	1.33174e6	0.921906
$292$	−2.27475e6	−1.56126
$293$	−232000.	−0.157877	−0.0789384	−	0.996879i	$-0.525153\pi$
$293$	−232000.	−0.157877	−0.0789384	+	0.996879i	$0.525153\pi$
$294$	−2.30972e6	−1.55845
$295$	0	0
$296$	−358598.	−0.237891
$297$	−443683.	−0.291865
$298$	−1.44965e6	−0.945633
$299$	48761.7	0.0315428
$300$	0	0
$301$	−44836.4	−0.0285242
$302$	1.38723e6	0.875248
$303$	269660.	0.168737
$304$	268565.	0.166673
$305$	0	0
$306$	24451.4	0.0149279
$307$	83735.6	0.0507066	0.0253533	−	0.999679i	$-0.491929\pi$
$307$	83735.6	0.0507066	0.0253533	+	0.999679i	$0.491929\pi$
$308$	18639.0	0.0111956
$309$	−265046.	−0.157915
$310$	0	0
$311$	467964.	0.274354	0.137177	−	0.990547i	$-0.456197\pi$
$311$	467964.	0.274354	0.137177	+	0.990547i	$0.456197\pi$
$312$	−122421.	−0.0711985
$313$	−2.81546e6	−1.62439	−0.812193	−	0.583389i	$-0.801726\pi$
$313$	−2.81546e6	−1.62439	−0.812193	+	0.583389i	$0.801726\pi$
$314$	−3.61616e6	−2.06978
$315$	0	0
$316$	1.67109e6	0.941418
$317$	26046.1	0.0145578	0.00727889	−	0.999974i	$-0.497683\pi$
$317$	26046.1	0.0145578	0.00727889	+	0.999974i	$0.497683\pi$
$318$	3.41061e6	1.89132
$319$	45827.5	0.0252144
$320$	0	0
$321$	−2.04793e6	−1.10931
$322$	−16514.5	−0.00887617
$323$	83752.6	0.0446675
$324$	−2.58591e6	−1.36852
$325$	0	0
$326$	−3.26391e6	−1.70096
$327$	2.49028e6	1.28789
$328$	−992082.	−0.509170
$329$	5386.46	0.00274356
$330$	0	0
$331$	−3.14034e6	−1.57546	−0.787729	−	0.616022i	$-0.788743\pi$
$331$	−3.14034e6	−1.57546	−0.787729	+	0.616022i	$0.788743\pi$
$332$	−3.07660e6	−1.53188
$333$	64020.6	0.0316380
$334$	−5.59335e6	−2.74351
$335$	0	0
$336$	−37636.6	−0.0181871
$337$	−1.01090e6	−0.484880	−0.242440	−	0.970166i	$-0.577948\pi$
$337$	−1.01090e6	−0.484880	−0.242440	+	0.970166i	$0.577948\pi$
$338$	3.10672e6	1.47914
$339$	−1.03586e6	−0.489554
$340$	0	0
$341$	−616941.	−0.287315
$342$	52819.7	0.0244191
$343$	124816.	0.0572842
$344$	−979472.	−0.446268
$345$	0	0
$346$	4.64339e6	2.08518
$347$	−1.73452e6	−0.773315	−0.386657	−	0.922223i	$-0.626370\pi$
$347$	−1.73452e6	−0.773315	−0.386657	+	0.922223i	$0.626370\pi$
$348$	252016.	0.111553
$349$	−839301.	−0.368854	−0.184427	−	0.982846i	$-0.559043\pi$
$349$	−839301.	−0.368854	−0.184427	+	0.982846i	$0.559043\pi$
$350$	0	0
$351$	−344728.	−0.149351
$352$	−969057.	−0.416862
$353$	−1.53771e6	−0.656808	−0.328404	−	0.944537i	$-0.606511\pi$
$353$	−1.53771e6	−0.656808	−0.328404	+	0.944537i	$0.606511\pi$
$354$	−3.06750e6	−1.30100
$355$	0	0
$356$	22863.6	0.00956135
$357$	−11737.1	−0.00487404
$358$	−1.59079e6	−0.656003
$359$	3.58199e6	1.46686	0.733430	−	0.679765i	$-0.237918\pi$
$359$	3.58199e6	1.46686	0.733430	+	0.679765i	$0.237918\pi$
$360$	0	0
$361$	−2.29518e6	−0.926933
$362$	1.66956e6	0.669622
$363$	−234936.	−0.0935799
$364$	14481.9	0.00572893
$365$	0	0
$366$	7.28151e6	2.84131
$367$	2.65471e6	1.02885	0.514424	−	0.857536i	$-0.328006\pi$
$367$	2.65471e6	1.02885	0.514424	+	0.857536i	$0.328006\pi$
$368$	327487.	0.126059
$369$	177117.	0.0677165
$370$	0	0
$371$	−92115.6	−0.0347455
$372$	−3.39270e6	−1.27113
$373$	−1.09038e6	−0.405793	−0.202897	−	0.979200i	$-0.565036\pi$
$373$	−1.09038e6	−0.405793	−0.202897	+	0.979200i	$0.565036\pi$
$374$	−204214.	−0.0754931
$375$	0	0
$376$	117670.	0.0429236
$377$	35606.5	0.0129026
$378$	116752.	0.0420276
$379$	2.62200e6	0.937637	0.468819	−	0.883295i	$-0.344680\pi$
$379$	2.62200e6	0.937637	0.468819	+	0.883295i	$0.344680\pi$
$380$	0	0
$381$	2.47098e6	0.872082
$382$	−609834.	−0.213822
$383$	2.67839e6	0.932990	0.466495	−	0.884524i	$-0.345516\pi$
$383$	2.67839e6	0.932990	0.466495	+	0.884524i	$0.345516\pi$
$384$	−2.55012e6	−0.882538
$385$	0	0
$386$	5.38134e6	1.83833
$387$	174866.	0.0593509
$388$	−3.44152e6	−1.16057
$389$	−3.20686e6	−1.07450	−0.537250	−	0.843423i	$-0.680537\pi$
$389$	−3.20686e6	−1.07450	−0.537250	+	0.843423i	$0.680537\pi$
$390$	0	0
$391$	102127.	0.0337832
$392$	1.36277e6	0.447928
$393$	−2.15069e6	−0.702420
$394$	6.87448e6	2.23100
$395$	0	0
$396$	−72693.7	−0.0232948
$397$	4.87414e6	1.55211	0.776053	−	0.630668i	$-0.217219\pi$
$397$	4.87414e6	1.55211	0.776053	+	0.630668i	$0.217219\pi$
$398$	8.82623e6	2.79298
$399$	−25354.3	−0.00797296
$400$	0	0
$401$	3.60536e6	1.11966	0.559831	−	0.828606i	$-0.310866\pi$
$401$	3.60536e6	1.11966	0.559831	+	0.828606i	$0.310866\pi$
$402$	−8.94510e6	−2.76070
$403$	−479344.	−0.147023
$404$	−696863.	−0.212419
$405$	0	0
$406$	−12059.1	−0.00363079
$407$	−534691.	−0.159999
$408$	−256402.	−0.0762554
$409$	−1.61929e6	−0.478649	−0.239325	−	0.970940i	$-0.576926\pi$
$409$	−1.61929e6	−0.478649	−0.239325	+	0.970940i	$0.576926\pi$
$410$	0	0
$411$	1.62062e6	0.473236
$412$	684939.	0.198797
$413$	82848.8	0.0239007
$414$	64408.0	0.0184688
$415$	0	0
$416$	−752928.	−0.213314
$417$	4.78617e6	1.34787
$418$	−441142.	−0.123492
$419$	3.67267e6	1.02199	0.510995	−	0.859583i	$-0.329277\pi$
$419$	3.67267e6	1.02199	0.510995	+	0.859583i	$0.329277\pi$
$420$	0	0
$421$	3.38719e6	0.931395	0.465697	−	0.884944i	$-0.345804\pi$
$421$	3.38719e6	0.931395	0.465697	+	0.884944i	$0.345804\pi$
$422$	5.50596e6	1.50505
$423$	−21007.7	−0.00570857
$424$	−2.01231e6	−0.543601
$425$	0	0
$426$	8.35488e6	2.23057
$427$	−196663.	−0.0521980
$428$	5.29233e6	1.39649
$429$	−182538.	−0.0478861
$430$	0	0
$431$	−4.13106e6	−1.07120	−0.535598	−	0.844473i	$-0.679914\pi$
$431$	−4.13106e6	−1.07120	−0.535598	+	0.844473i	$0.679914\pi$
$432$	−2.31522e6	−0.596874
$433$	5.97405e6	1.53126	0.765631	−	0.643280i	$-0.222427\pi$
$433$	5.97405e6	1.53126	0.765631	+	0.643280i	$0.222427\pi$
$434$	162343.	0.0413723
$435$	0	0
$436$	−6.43545e6	−1.62130
$437$	220615.	0.0552626
$438$	−7.54485e6	−1.87917
$439$	−3.38748e6	−0.838911	−0.419456	−	0.907776i	$-0.637779\pi$
$439$	−3.38748e6	−0.838911	−0.419456	+	0.907776i	$0.637779\pi$
$440$	0	0
$441$	−243296.	−0.0595716
$442$	−158668.	−0.0386309
$443$	3.50641e6	0.848893	0.424447	−	0.905453i	$-0.360469\pi$
$443$	3.50641e6	0.848893	0.424447	+	0.905453i	$0.360469\pi$
$444$	−2.94039e6	−0.707861
$445$	0	0
$446$	−1.11026e7	−2.64293
$447$	−2.71390e6	−0.642428
$448$	179945.	0.0423588
$449$	7.19124e6	1.68340	0.841701	−	0.539943i	$-0.181554\pi$
$449$	7.19124e6	1.68340	0.841701	+	0.539943i	$0.181554\pi$
$450$	0	0
$451$	−1.47926e6	−0.342454
$452$	2.67689e6	0.616289
$453$	2.59704e6	0.594612
$454$	444176.	0.101138
$455$	0	0
$456$	−553877.	−0.124739
$457$	−6.32089e6	−1.41575	−0.707877	−	0.706336i	$-0.750347\pi$
$457$	−6.32089e6	−1.41575	−0.707877	+	0.706336i	$0.750347\pi$
$458$	4.66005e6	1.03807
$459$	−722006.	−0.159959
$460$	0	0
$461$	−5.31660e6	−1.16515	−0.582574	−	0.812777i	$-0.697955\pi$
$461$	−5.31660e6	−1.16515	−0.582574	+	0.812777i	$0.697955\pi$
$462$	61821.5	0.0134752
$463$	−2.17724e6	−0.472013	−0.236007	−	0.971751i	$-0.575839\pi$
$463$	−2.17724e6	−0.472013	−0.236007	+	0.971751i	$0.575839\pi$
$464$	239136.	0.0515644
$465$	0	0
$466$	4.86391e6	1.03758
$467$	−67604.5	−0.0143444	−0.00717222	−	0.999974i	$-0.502283\pi$
$467$	−67604.5	−0.0143444	−0.00717222	+	0.999974i	$0.502283\pi$
$468$	−56480.8	−0.0119203
$469$	241594.	0.0507171
$470$	0	0
$471$	−6.76983e6	−1.40613
$472$	1.80987e6	0.373932
$473$	−1.46045e6	−0.300148
$474$	5.54265e6	1.13311
$475$	0	0
$476$	30331.2	0.00613582
$477$	359259.	0.0722956
$478$	−2.09624e6	−0.419635
$479$	2.70352e6	0.538382	0.269191	−	0.963087i	$-0.413244\pi$
$479$	2.70352e6	0.538382	0.269191	+	0.963087i	$0.413244\pi$
$480$	0	0
$481$	−415439.	−0.0818737
$482$	−1.07395e7	−2.10556
$483$	−30916.9	−0.00603014
$484$	607128.	0.117806
$485$	0	0
$486$	−939554.	−0.180439
$487$	9.64211e6	1.84225	0.921127	−	0.389262i	$-0.127270\pi$
$487$	9.64211e6	1.84225	0.921127	+	0.389262i	$0.127270\pi$
$488$	−4.29620e6	−0.816649
$489$	−6.11039e6	−1.15557
$490$	0	0
$491$	6.66155e6	1.24701	0.623507	−	0.781818i	$-0.285707\pi$
$491$	6.66155e6	1.24701	0.623507	+	0.781818i	$0.285707\pi$
$492$	−8.13478e6	−1.51507
$493$	74575.0	0.0138190
$494$	−342754.	−0.0631925
$495$	0	0
$496$	−3.21931e6	−0.587568
$497$	−225653.	−0.0409780
$498$	−1.02044e7	−1.84380
$499$	4.19798e6	0.754725	0.377362	−	0.926066i	$-0.376831\pi$
$499$	4.19798e6	0.754725	0.377362	+	0.926066i	$0.376831\pi$
$500$	0	0
$501$	−1.04713e7	−1.86384
$502$	1.10079e7	1.94959
$503$	4.23018e6	0.745486	0.372743	−	0.927935i	$-0.378417\pi$
$503$	4.23018e6	0.745486	0.372743	+	0.927935i	$0.378417\pi$
$504$	4367.37	0.000765850 0
$505$	0	0
$506$	−537926.	−0.0933999
$507$	5.81610e6	1.00487
$508$	−6.38559e6	−1.09785
$509$	−5.17751e6	−0.885782	−0.442891	−	0.896575i	$-0.646047\pi$
$509$	−5.17751e6	−0.885782	−0.442891	+	0.896575i	$0.646047\pi$
$510$	0	0
$511$	203776.	0.0345223
$512$	−6.69634e6	−1.12892
$513$	−1.55967e6	−0.261661
$514$	1.63285e6	0.272607
$515$	0	0
$516$	−8.03137e6	−1.32790
$517$	175453.	0.0288692
$518$	140700.	0.0230393
$519$	8.69291e6	1.41660
$520$	0	0
$521$	7.76284e6	1.25293	0.626465	−	0.779450i	$-0.284501\pi$
$521$	7.76284e6	1.25293	0.626465	+	0.779450i	$0.284501\pi$
$522$	47031.7	0.00755465
$523$	−3.35346e6	−0.536092	−0.268046	−	0.963406i	$-0.586378\pi$
$523$	−3.35346e6	−0.536092	−0.268046	+	0.963406i	$0.586378\pi$
$524$	5.55788e6	0.884262
$525$	0	0
$526$	−5.41148e6	−0.852808
$527$	−1.00395e6	−0.157465
$528$	−1.22594e6	−0.191374
$529$	−6.16733e6	−0.958204
$530$	0	0
$531$	−323117.	−0.0497306
$532$	65521.3	0.0100370
$533$	−1.14934e6	−0.175238
$534$	75833.5	0.0115082
$535$	0	0
$536$	5.27774e6	0.793480
$537$	−2.97813e6	−0.445664
$538$	−1.13338e7	−1.68818
$539$	2.03198e6	0.301264
$540$	0	0
$541$	2.87042e6	0.421650	0.210825	−	0.977524i	$-0.432385\pi$
$541$	2.87042e6	0.421650	0.210825	+	0.977524i	$0.432385\pi$
$542$	1.94295e7	2.84094
$543$	3.12559e6	0.454917
$544$	−1.57695e6	−0.228465
$545$	0	0
$546$	48033.5	0.00689544
$547$	4.40734e6	0.629808	0.314904	−	0.949124i	$-0.398028\pi$
$547$	4.40734e6	0.629808	0.314904	+	0.949124i	$0.398028\pi$
$548$	−4.18806e6	−0.595747
$549$	767004.	0.108609
$550$	0	0
$551$	161096.	0.0226051
$552$	−675394.	−0.0943429
$553$	−149699.	−0.0208164
$554$	1.04348e7	1.44447
$555$	0	0
$556$	−1.23685e7	−1.69680
$557$	−1.15892e7	−1.58276	−0.791379	−	0.611325i	$-0.790637\pi$
$557$	−1.15892e7	−1.58276	−0.791379	+	0.611325i	$0.790637\pi$
$558$	−633153.	−0.0860841
$559$	−1.13473e6	−0.153590
$560$	0	0
$561$	−382311.	−0.0512873
$562$	1.66681e7	2.22610
$563$	2.32310e6	0.308885	0.154443	−	0.988002i	$-0.450642\pi$
$563$	2.32310e6	0.308885	0.154443	+	0.988002i	$0.450642\pi$
$564$	964858.	0.127722
$565$	0	0
$566$	1.38762e7	1.82067
$567$	231650.	0.0302603
$568$	−4.92951e6	−0.641110
$569$	−1.40597e6	−0.182052	−0.0910260	−	0.995849i	$-0.529015\pi$
$569$	−1.40597e6	−0.182052	−0.0910260	+	0.995849i	$0.529015\pi$
$570$	0	0
$571$	−5.52531e6	−0.709196	−0.354598	−	0.935019i	$-0.615382\pi$
$571$	−5.52531e6	−0.709196	−0.354598	+	0.935019i	$0.615382\pi$
$572$	471720.	0.0602829
$573$	−1.14167e6	−0.145263
$574$	389255.	0.0493122
$575$	0	0
$576$	−701799.	−0.0881367
$577$	−3.12727e6	−0.391044	−0.195522	−	0.980699i	$-0.562640\pi$
$577$	−3.12727e6	−0.391044	−0.195522	+	0.980699i	$0.562640\pi$
$578$	1.18377e7	1.47384
$579$	1.00744e7	1.24889
$580$	0	0
$581$	275606.	0.0338726
$582$	−1.14148e7	−1.39688
$583$	−3.00048e6	−0.365611
$584$	4.45158e6	0.540110
$585$	0	0
$586$	1.98854e6	0.239216
$587$	−7.45555e6	−0.893068	−0.446534	−	0.894767i	$-0.647342\pi$
$587$	−7.45555e6	−0.893068	−0.446534	+	0.894767i	$0.647342\pi$
$588$	1.11743e7	1.33284
$589$	−2.16872e6	−0.257582
$590$	0	0
$591$	1.28698e7	1.51566
$592$	−2.79011e6	−0.327203
$593$	−7.46375e6	−0.871607	−0.435803	−	0.900042i	$-0.643536\pi$
$593$	−7.46375e6	−0.871607	−0.435803	+	0.900042i	$0.643536\pi$
$594$	3.80295e6	0.442237
$595$	0	0
$596$	7.01333e6	0.808740
$597$	1.65236e7	1.89745
$598$	−417952.	−0.0477940
$599$	314190.	0.0357787	0.0178894	−	0.999840i	$-0.494305\pi$
$599$	314190.	0.0357787	0.0178894	+	0.999840i	$0.494305\pi$
$600$	0	0
$601$	1.42821e7	1.61290	0.806450	−	0.591303i	$-0.201386\pi$
$601$	1.42821e7	1.61290	0.806450	+	0.591303i	$0.201386\pi$
$602$	384307.	0.0432202
$603$	−942239.	−0.105528
$604$	−6.71135e6	−0.748544
$605$	0	0
$606$	−2.31134e6	−0.255672
$607$	188944.	0.0208142	0.0104071	−	0.999946i	$-0.496687\pi$
$607$	188944.	0.0208142	0.0104071	+	0.999946i	$0.496687\pi$
$608$	−3.40651e6	−0.373724
$609$	−22576.0	−0.00246663
$610$	0	0
$611$	136322.	0.0147728
$612$	−118294.	−0.0127669
$613$	−3.63784e6	−0.391014	−0.195507	−	0.980702i	$-0.562635\pi$
$613$	−3.63784e6	−0.391014	−0.195507	+	0.980702i	$0.562635\pi$
$614$	−717725.	−0.0768312
$615$	0	0
$616$	−36475.6	−0.00387303
$617$	−6.34013e6	−0.670479	−0.335240	−	0.942133i	$-0.608817\pi$
$617$	−6.34013e6	−0.670479	−0.335240	+	0.942133i	$0.608817\pi$
$618$	2.27179e6	0.239275
$619$	−1.42152e6	−0.149117	−0.0745583	−	0.997217i	$-0.523755\pi$
$619$	−1.42152e6	−0.149117	−0.0745583	+	0.997217i	$0.523755\pi$
$620$	0	0
$621$	−1.90185e6	−0.197901
$622$	−4.01107e6	−0.415704
$623$	−2048.15	−0.000211418 0
$624$	−952515.	−0.0979288
$625$	0	0
$626$	2.41323e7	2.46129
$627$	−825865.	−0.0838958
$628$	1.74948e7	1.77015
$629$	−870103.	−0.0876888
$630$	0	0
$631$	1.68033e7	1.68005	0.840025	−	0.542547i	$-0.182540\pi$
$631$	1.68033e7	1.68005	0.840025	+	0.542547i	$0.182540\pi$
$632$	−3.27025e6	−0.325678
$633$	1.03077e7	1.02248
$634$	−223250.	−0.0220581
$635$	0	0
$636$	−1.65003e7	−1.61752
$637$	1.57878e6	0.154161
$638$	−392802.	−0.0382052
$639$	880068.	0.0852637
$640$	0	0
$641$	−3.74097e6	−0.359616	−0.179808	−	0.983702i	$-0.557548\pi$
$641$	−3.74097e6	−0.359616	−0.179808	+	0.983702i	$0.557548\pi$
$642$	1.75535e7	1.68084
$643$	5.88753e6	0.561573	0.280786	−	0.959770i	$-0.409405\pi$
$643$	5.88753e6	0.561573	0.280786	+	0.959770i	$0.409405\pi$
$644$	79896.3	0.00759122
$645$	0	0
$646$	−717871.	−0.0676807
$647$	−5.19398e6	−0.487797	−0.243899	−	0.969801i	$-0.578426\pi$
$647$	−5.19398e6	−0.487797	−0.243899	+	0.969801i	$0.578426\pi$
$648$	5.06050e6	0.473430
$649$	2.69863e6	0.251496
$650$	0	0
$651$	303924.	0.0281068
$652$	1.57906e7	1.45472
$653$	5.73999e6	0.526779	0.263390	−	0.964690i	$-0.415160\pi$
$653$	5.73999e6	0.526779	0.263390	+	0.964690i	$0.415160\pi$
$654$	−2.13450e7	−1.95143
$655$	0	0
$656$	−7.71902e6	−0.700330
$657$	−794743.	−0.0718312
$658$	−46169.1	−0.00415707
$659$	−1.18769e7	−1.06534	−0.532671	−	0.846323i	$-0.678812\pi$
$659$	−1.18769e7	−1.06534	−0.532671	+	0.846323i	$0.678812\pi$
$660$	0	0
$661$	−1.10514e7	−0.983813	−0.491907	−	0.870648i	$-0.663700\pi$
$661$	−1.10514e7	−0.983813	−0.491907	+	0.870648i	$0.663700\pi$
$662$	2.69169e7	2.38715
$663$	−297044.	−0.0262444
$664$	6.02075e6	0.529945
$665$	0	0
$666$	−548742.	−0.0479383
$667$	196440.	0.0170968
$668$	2.70603e7	2.34635
$669$	−2.07851e7	−1.79551
$670$	0	0
$671$	−6.40591e6	−0.549255
$672$	477387.	0.0407800
$673$	−1.81997e7	−1.54891	−0.774454	−	0.632631i	$-0.781975\pi$
$673$	−1.81997e7	−1.54891	−0.774454	+	0.632631i	$0.781975\pi$
$674$	8.66477e6	0.734695
$675$	0	0
$676$	−1.50301e7	−1.26502
$677$	1.94815e6	0.163362	0.0816810	−	0.996659i	$-0.473971\pi$
$677$	1.94815e6	0.163362	0.0816810	+	0.996659i	$0.473971\pi$
$678$	8.87866e6	0.741777
$679$	308297.	0.0256622
$680$	0	0
$681$	831545.	0.0687097
$682$	5.28800e6	0.435342
$683$	−2.17374e7	−1.78302	−0.891510	−	0.453001i	$-0.850354\pi$
$683$	−2.17374e7	−1.78302	−0.891510	+	0.453001i	$0.850354\pi$
$684$	−255539.	−0.0208841
$685$	0	0
$686$	−1.06984e6	−0.0867976
$687$	8.72411e6	0.705228
$688$	−7.62090e6	−0.613812
$689$	−2.33128e6	−0.187088
$690$	0	0
$691$	−1.99747e6	−0.159142	−0.0795711	−	0.996829i	$-0.525355\pi$
$691$	−1.99747e6	−0.159142	−0.0795711	+	0.996829i	$0.525355\pi$
$692$	−2.24645e7	−1.78333
$693$	6512.02	0.000515089 0
$694$	1.48672e7	1.17174
$695$	0	0
$696$	−493183.	−0.0385909
$697$	−2.40719e6	−0.187685
$698$	7.19393e6	0.558891
$699$	9.10575e6	0.704893
$700$	0	0
$701$	6.76258e6	0.519778	0.259889	−	0.965639i	$-0.416314\pi$
$701$	6.76258e6	0.519778	0.259889	+	0.965639i	$0.416314\pi$
$702$	2.95478e6	0.226299
$703$	−1.87959e6	−0.143442
$704$	5.86133e6	0.445722
$705$	0	0
$706$	1.31802e7	0.995203
$707$	62426.1	0.00469697
$708$	1.48404e7	1.11266
$709$	−1.31058e7	−0.979148	−0.489574	−	0.871962i	$-0.662848\pi$
$709$	−1.31058e7	−0.979148	−0.489574	+	0.871962i	$0.662848\pi$
$710$	0	0
$711$	583839.	0.0433131
$712$	−44742.9	−0.00330769
$713$	−2.64452e6	−0.194816
$714$	100602.	0.00738519
$715$	0	0
$716$	7.69617e6	0.561037
$717$	−3.92439e6	−0.285085
$718$	−3.07024e7	−2.22260
$719$	1.24991e7	0.901691	0.450845	−	0.892602i	$-0.351123\pi$
$719$	1.24991e7	0.901691	0.450845	+	0.892602i	$0.351123\pi$
$720$	0	0
$721$	−61357.9	−0.00439574
$722$	1.96727e7	1.40450
$723$	−2.01055e7	−1.43044
$724$	−8.07723e6	−0.572685
$725$	0	0
$726$	2.01371e6	0.141793
$727$	−1.55739e7	−1.09285	−0.546424	−	0.837508i	$-0.684011\pi$
$727$	−1.55739e7	−1.09285	−0.546424	+	0.837508i	$0.684011\pi$
$728$	−28340.5	−0.00198189
$729$	1.33944e7	0.933482
$730$	0	0
$731$	−2.37660e6	−0.164498
$732$	−3.52276e7	−2.42999
$733$	−926307.	−0.0636789	−0.0318394	−	0.999493i	$-0.510137\pi$
$733$	−926307.	−0.0636789	−0.0318394	+	0.999493i	$0.510137\pi$
$734$	−2.27544e7	−1.55892
$735$	0	0
$736$	−4.15387e6	−0.282656
$737$	7.86944e6	0.533673
$738$	−1.51813e6	−0.102605
$739$	3.44944e6	0.232347	0.116174	−	0.993229i	$-0.462937\pi$
$739$	3.44944e6	0.232347	0.116174	+	0.993229i	$0.462937\pi$
$740$	0	0
$741$	−641672.	−0.0429307
$742$	789553.	0.0526468
$743$	4.52822e6	0.300923	0.150462	−	0.988616i	$-0.451924\pi$
$743$	4.52822e6	0.300923	0.150462	+	0.988616i	$0.451924\pi$
$744$	6.63936e6	0.439738
$745$	0	0
$746$	9.34598e6	0.614862
$747$	−1.07489e6	−0.0704794
$748$	987979.	0.0645645
$749$	−474095.	−0.0308788
$750$	0	0
$751$	1.53285e7	0.991745	0.495873	−	0.868395i	$-0.334848\pi$
$751$	1.53285e7	0.991745	0.495873	+	0.868395i	$0.334848\pi$
$752$	915545.	0.0590385
$753$	2.06079e7	1.32448
$754$	−305195.	−0.0195501
$755$	0	0
$756$	−564839.	−0.0359435
$757$	−1.46683e7	−0.930339	−0.465169	−	0.885222i	$-0.654007\pi$
$757$	−1.46683e7	−0.930339	−0.465169	+	0.885222i	$0.654007\pi$
$758$	−2.24740e7	−1.42072
$759$	−1.00705e6	−0.0634525
$760$	0	0
$761$	−6.76195e6	−0.423263	−0.211632	−	0.977350i	$-0.567878\pi$
$761$	−6.76195e6	−0.423263	−0.211632	+	0.977350i	$0.567878\pi$
$762$	−2.11796e7	−1.32139
$763$	576498.	0.0358498
$764$	2.95035e6	0.182869
$765$	0	0
$766$	−2.29574e7	−1.41368
$767$	2.09675e6	0.128694
$768$	−3.01567e6	−0.184493
$769$	−2.65717e7	−1.62033	−0.810166	−	0.586201i	$-0.800623\pi$
$769$	−2.65717e7	−1.62033	−0.810166	+	0.586201i	$0.800623\pi$
$770$	0	0
$771$	3.05686e6	0.185199
$772$	−2.60347e7	−1.57220
$773$	1.30745e7	0.787000	0.393500	−	0.919325i	$-0.371264\pi$
$773$	1.30745e7	0.787000	0.393500	+	0.919325i	$0.371264\pi$
$774$	−1.49883e6	−0.0899291
$775$	0	0
$776$	6.73489e6	0.401491
$777$	263405.	0.0156521
$778$	2.74871e7	1.62809
$779$	−5.20000e6	−0.307015
$780$	0	0
$781$	−7.35020e6	−0.431193
$782$	−875367.	−0.0511886
$783$	−1.38876e6	−0.0809513
$784$	1.06032e7	0.616095
$785$	0	0
$786$	1.84343e7	1.06431
$787$	−3.73173e6	−0.214770	−0.107385	−	0.994218i	$-0.534248\pi$
$787$	−3.73173e6	−0.214770	−0.107385	+	0.994218i	$0.534248\pi$
$788$	−3.32584e7	−1.90803
$789$	−1.01309e7	−0.579367
$790$	0	0
$791$	−239800.	−0.0136272
$792$	142258.	0.00805869
$793$	−4.97720e6	−0.281062
$794$	−4.17778e7	−2.35177
$795$	0	0
$796$	−4.27008e7	−2.38866
$797$	−2.53825e7	−1.41543	−0.707714	−	0.706499i	$-0.750273\pi$
$797$	−2.53825e7	−1.41543	−0.707714	+	0.706499i	$0.750273\pi$
$798$	217320.	0.0120807
$799$	285515.	0.0158220
$800$	0	0
$801$	7987.98	0.000439902 0
$802$	−3.09027e7	−1.69653
$803$	6.63758e6	0.363263
$804$	4.32759e7	2.36106
$805$	0	0
$806$	4.10862e6	0.222771
$807$	−2.12180e7	−1.14689
$808$	1.36373e6	0.0734851
$809$	−2.24328e7	−1.20507	−0.602534	−	0.798093i	$-0.705842\pi$
$809$	−2.24328e7	−1.20507	−0.602534	+	0.798093i	$0.705842\pi$
$810$	0	0
$811$	−2.48963e7	−1.32918	−0.664588	−	0.747210i	$-0.731393\pi$
$811$	−2.48963e7	−1.32918	−0.664588	+	0.747210i	$0.731393\pi$
$812$	58341.5	0.00310519
$813$	3.63740e7	1.93003
$814$	4.58301e6	0.242432
$815$	0	0
$816$	−1.99497e6	−0.104884
$817$	−5.13390e6	−0.269087
$818$	1.38795e7	0.725255
$819$	5059.64	0.000263578 0
$820$	0	0
$821$	−4.11968e6	−0.213307	−0.106654	−	0.994296i	$-0.534014\pi$
$821$	−4.11968e6	−0.213307	−0.106654	+	0.994296i	$0.534014\pi$
$822$	−1.38909e7	−0.717052
$823$	1.14381e7	0.588648	0.294324	−	0.955706i	$-0.404905\pi$
$823$	1.14381e7	0.588648	0.294324	+	0.955706i	$0.404905\pi$
$824$	−1.34039e6	−0.0687724
$825$	0	0
$826$	−710124.	−0.0362146
$827$	−3.28841e6	−0.167195	−0.0835974	−	0.996500i	$-0.526641\pi$
$827$	−3.28841e6	−0.167195	−0.0835974	+	0.996500i	$0.526641\pi$
$828$	−311602.	−0.0157952
$829$	−8.29845e6	−0.419383	−0.209691	−	0.977768i	$-0.567246\pi$
$829$	−8.29845e6	−0.419383	−0.209691	+	0.977768i	$0.567246\pi$
$830$	0	0
$831$	1.95350e7	0.981322
$832$	4.55407e6	0.228082
$833$	3.30664e6	0.165110
$834$	−4.10238e7	−2.04231
$835$	0	0
$836$	2.13422e6	0.105615
$837$	1.86959e7	0.922428
$838$	−3.14797e7	−1.54853
$839$	−1.43568e7	−0.704130	−0.352065	−	0.935976i	$-0.614520\pi$
$839$	−1.43568e7	−0.704130	−0.352065	+	0.935976i	$0.614520\pi$
$840$	0	0
$841$	−2.03677e7	−0.993007
$842$	−2.90327e7	−1.41126
$843$	3.12044e7	1.51233
$844$	−2.66376e7	−1.28718
$845$	0	0
$846$	180064.	0.00864968
$847$	−54387.5	−0.00260489
$848$	−1.56570e7	−0.747687
$849$	2.59778e7	1.23689
$850$	0	0
$851$	−2.29196e6	−0.108488
$852$	−4.04205e7	−1.90767
$853$	2.78233e7	1.30929	0.654646	−	0.755935i	$-0.272818\pi$
$853$	2.78233e7	1.30929	0.654646	+	0.755935i	$0.272818\pi$
$854$	1.68567e6	0.0790909
$855$	0	0
$856$	−1.03568e7	−0.483106
$857$	−1.42355e7	−0.662096	−0.331048	−	0.943614i	$-0.607402\pi$
$857$	−1.42355e7	−0.662096	−0.331048	+	0.943614i	$0.607402\pi$
$858$	1.56459e6	0.0725576
$859$	4.22812e6	0.195508	0.0977540	−	0.995211i	$-0.468834\pi$
$859$	4.22812e6	0.195508	0.0977540	+	0.995211i	$0.468834\pi$
$860$	0	0
$861$	728726.	0.0335009
$862$	3.54087e7	1.62309
$863$	−4.11087e7	−1.87891	−0.939456	−	0.342670i	$-0.888669\pi$
$863$	−4.11087e7	−1.87891	−0.939456	+	0.342670i	$0.888669\pi$
$864$	2.93665e7	1.33834
$865$	0	0
$866$	−5.12056e7	−2.32018
$867$	2.21615e7	1.00127
$868$	−785408.	−0.0353831
$869$	−4.87614e6	−0.219042
$870$	0	0
$871$	6.11432e6	0.273088
$872$	1.25939e7	0.560878
$873$	−1.20238e6	−0.0533958
$874$	−1.89096e6	−0.0837344
$875$	0	0
$876$	3.65016e7	1.60713
$877$	1.50841e6	0.0662247	0.0331124	−	0.999452i	$-0.489458\pi$
$877$	1.50841e6	0.0662247	0.0331124	+	0.999452i	$0.489458\pi$
$878$	2.90352e7	1.27113
$879$	3.72276e6	0.162515
$880$	0	0
$881$	3.81167e7	1.65454	0.827268	−	0.561808i	$-0.189894\pi$
$881$	3.81167e7	1.65454	0.827268	+	0.561808i	$0.189894\pi$
$882$	2.08537e6	0.0902635
$883$	2.07323e7	0.894842	0.447421	−	0.894323i	$-0.352343\pi$
$883$	2.07323e7	0.894842	0.447421	+	0.894323i	$0.352343\pi$
$884$	767629.	0.0330385
$885$	0	0
$886$	−3.00546e7	−1.28625
$887$	2.93790e7	1.25380	0.626900	−	0.779100i	$-0.284323\pi$
$887$	2.93790e7	1.25380	0.626900	+	0.779100i	$0.284323\pi$
$888$	5.75421e6	0.244880
$889$	572031.	0.0242753
$890$	0	0
$891$	7.54552e6	0.318416
$892$	5.37136e7	2.26033
$893$	616767.	0.0258817
$894$	2.32617e7	0.973414
$895$	0	0
$896$	−590352.	−0.0245664
$897$	−782451.	−0.0324695
$898$	−6.16385e7	−2.55071
$899$	−1.93107e6	−0.0796892
$900$	0	0
$901$	−4.88268e6	−0.200376
$902$	1.26792e7	0.518890
$903$	719463.	0.0293622
$904$	−5.23855e6	−0.213201
$905$	0	0
$906$	−2.22601e7	−0.900962
$907$	−2.30519e7	−0.930441	−0.465220	−	0.885195i	$-0.654025\pi$
$907$	−2.30519e7	−0.930441	−0.465220	+	0.885195i	$0.654025\pi$
$908$	−2.14890e6	−0.0864972
$909$	−243467.	−0.00977306
$910$	0	0
$911$	2.54013e7	1.01405	0.507025	−	0.861931i	$-0.330745\pi$
$911$	2.54013e7	1.01405	0.507025	+	0.861931i	$0.330745\pi$
$912$	−4.30951e6	−0.171570
$913$	8.97732e6	0.356426
$914$	5.41784e7	2.14516
$915$	0	0
$916$	−2.25451e7	−0.887797
$917$	−497883.	−0.0195526
$918$	6.18855e6	0.242372
$919$	250887.	0.00979919	0.00489959	−	0.999988i	$-0.498440\pi$
$919$	250887.	0.00979919	0.00489959	+	0.999988i	$0.498440\pi$
$920$	0	0
$921$	−1.34366e6	−0.0521963
$922$	4.55703e7	1.76545
$923$	−5.71088e6	−0.220648
$924$	−299089.	−0.0115245
$925$	0	0
$926$	1.86619e7	0.715200
$927$	239301.	0.00914630
$928$	−3.03322e6	−0.115620
$929$	2.87097e7	1.09141	0.545707	−	0.837976i	$-0.316261\pi$
$929$	2.87097e7	1.09141	0.545707	+	0.837976i	$0.316261\pi$
$930$	0	0
$931$	7.14297e6	0.270088
$932$	−2.35313e7	−0.887375
$933$	−7.50915e6	−0.282414
$934$	579460.	0.0217348
$935$	0	0
$936$	110530.	0.00412374
$937$	−4.58299e7	−1.70530	−0.852649	−	0.522484i	$-0.825005\pi$
$937$	−4.58299e7	−1.70530	−0.852649	+	0.522484i	$0.825005\pi$
$938$	−2.07078e6	−0.0768471
$939$	4.51781e7	1.67211
$940$	0	0
$941$	1.33338e7	0.490887	0.245443	−	0.969411i	$-0.421066\pi$
$941$	1.33338e7	0.490887	0.245443	+	0.969411i	$0.421066\pi$
$942$	5.80264e7	2.13058
$943$	−6.34085e6	−0.232203
$944$	1.40819e7	0.514319
$945$	0	0
$946$	1.25180e7	0.454787
$947$	3.58301e7	1.29829	0.649147	−	0.760663i	$-0.275126\pi$
$947$	3.58301e7	1.29829	0.649147	+	0.760663i	$0.275126\pi$
$948$	−2.68150e7	−0.969076
$949$	5.15720e6	0.185887
$950$	0	0
$951$	−417947.	−0.0149855
$952$	−59356.8	−0.00212265
$953$	−1.71351e7	−0.611158	−0.305579	−	0.952167i	$-0.598850\pi$
$953$	−1.71351e7	−0.611158	−0.305579	+	0.952167i	$0.598850\pi$
$954$	−3.07933e6	−0.109543
$955$	0	0
$956$	1.01415e7	0.358887
$957$	−735367.	−0.0259552
$958$	−2.31727e7	−0.815761
$959$	375173.	0.0131730
$960$	0	0
$961$	−2.63256e6	−0.0919540
$962$	3.56086e6	0.124056
$963$	1.84901e6	0.0642501
$964$	5.19572e7	1.80075
$965$	0	0
$966$	264999.	0.00913694
$967$	5.18927e7	1.78460	0.892298	−	0.451448i	$-0.149092\pi$
$967$	5.18927e7	1.78460	0.892298	+	0.451448i	$0.149092\pi$
$968$	−1.18812e6	−0.0407542
$969$	−1.34393e6	−0.0459798
$970$	0	0
$971$	−8.70846e6	−0.296410	−0.148205	−	0.988957i	$-0.547350\pi$
$971$	−8.70846e6	−0.296410	−0.148205	+	0.988957i	$0.547350\pi$
$972$	4.54551e6	0.154318
$973$	1.10799e6	0.0375193
$974$	−8.26456e7	−2.79140
$975$	0	0
$976$	−3.34272e7	−1.12325
$977$	−2.09303e7	−0.701519	−0.350760	−	0.936466i	$-0.614077\pi$
$977$	−2.09303e7	−0.701519	−0.350760	+	0.936466i	$0.614077\pi$
$978$	5.23741e7	1.75093
$979$	−66714.5	−0.00222466
$980$	0	0
$981$	−2.24839e6	−0.0745933
$982$	−5.70983e7	−1.88949
$983$	−1.93101e7	−0.637382	−0.318691	−	0.947859i	$-0.603243\pi$
$983$	−1.93101e7	−0.637382	−0.318691	+	0.947859i	$0.603243\pi$
$984$	1.59194e7	0.524129
$985$	0	0
$986$	−639207.	−0.0209387
$987$	−86433.5	−0.00282416
$988$	1.65823e6	0.0540445
$989$	−6.26025e6	−0.203517
$990$	0	0
$991$	1.51318e7	0.489449	0.244725	−	0.969593i	$-0.421302\pi$
$991$	1.51318e7	0.489449	0.244725	+	0.969593i	$0.421302\pi$
$992$	4.08340e7	1.31748
$993$	5.03913e7	1.62174
$994$	1.93415e6	0.0620903
$995$	0	0
$996$	4.93684e7	1.57689
$997$	4.47054e7	1.42437	0.712184	−	0.701993i	$-0.247706\pi$
$997$	4.47054e7	1.42437	0.712184	+	0.701993i	$0.247706\pi$
$998$	−3.59822e7	−1.14357
$999$	1.62034e7	0.513679

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	275.6.a.e.1.2		5
5.2	odd	4		275.6.b.e.199.3		10
5.3	odd	4		275.6.b.e.199.8		10
5.4	even	2		55.6.a.c.1.4	✓	5
15.14	odd	2		495.6.a.h.1.2		5
20.19	odd	2		880.6.a.r.1.2		5
55.54	odd	2		605.6.a.d.1.2		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
55.6.a.c.1.4	✓	5	5.4	even	2
275.6.a.e.1.2		5	1.1	even	1	trivial
275.6.b.e.199.3		10	5.2	odd	4
275.6.b.e.199.8		10	5.3	odd	4
495.6.a.h.1.2		5	15.14	odd	2
605.6.a.d.1.2		5	55.54	odd	2
880.6.a.r.1.2		5	20.19	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:	\( N \)	\(=\)	\( 275 = 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	275.a (trivial)

\(f(q)\)	\(=\)	\(q-8.57133 q^{2} -16.0464 q^{3} +41.4676 q^{4} +137.539 q^{6} -3.71474 q^{7} -81.1502 q^{8} +14.4878 q^{9} +O(q^{10})\)	\(q-8.57133 q^{2} -16.0464 q^{3} +41.4676 q^{4} +137.539 q^{6} -3.71474 q^{7} -81.1502 q^{8} +14.4878 q^{9} -121.000 q^{11} -665.407 q^{12} -94.0133 q^{13} +31.8402 q^{14} -631.400 q^{16} -196.903 q^{17} -124.180 q^{18} -425.349 q^{19} +59.6082 q^{21} +1037.13 q^{22} -518.668 q^{23} +1302.17 q^{24} +805.819 q^{26} +3666.80 q^{27} -154.041 q^{28} -378.739 q^{29} +5098.68 q^{31} +8008.74 q^{32} +1941.62 q^{33} +1687.72 q^{34} +600.775 q^{36} +4418.94 q^{37} +3645.81 q^{38} +1508.58 q^{39} +12225.3 q^{41} -510.922 q^{42} +12069.9 q^{43} -5017.58 q^{44} +4445.67 q^{46} -1450.03 q^{47} +10131.7 q^{48} -16793.2 q^{49} +3159.59 q^{51} -3898.51 q^{52} +24797.4 q^{53} -31429.4 q^{54} +301.452 q^{56} +6825.33 q^{57} +3246.30 q^{58} -22302.7 q^{59} +52941.4 q^{61} -43702.5 q^{62} -53.8183 q^{63} -48440.7 q^{64} -16642.2 q^{66} -65036.7 q^{67} -8165.11 q^{68} +8322.76 q^{69} +60745.5 q^{71} -1175.69 q^{72} -54856.0 q^{73} -37876.1 q^{74} -17638.2 q^{76} +449.483 q^{77} -12930.5 q^{78} +40298.7 q^{79} -62359.6 q^{81} -104787. q^{82} -74192.7 q^{83} +2471.81 q^{84} -103455. q^{86} +6077.41 q^{87} +9819.17 q^{88} +551.359 q^{89} +349.235 q^{91} -21507.9 q^{92} -81815.7 q^{93} +12428.6 q^{94} -128512. q^{96} -82992.9 q^{97} +143940. q^{98} -1753.02 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 9 q^{2} + 115 q^{4} - 237 q^{6} - 70 q^{7} - 753 q^{8} + 1059 q^{9}+O(q^{10}) \) 5 * q - 9 * q^2 + 115 * q^4 - 237 * q^6 - 70 * q^7 - 753 * q^8 + 1059 * q^9	\( 5 q - 9 q^{2} + 115 q^{4} - 237 q^{6} - 70 q^{7} - 753 q^{8} + 1059 q^{9} - 605 q^{11} + 1605 q^{12} - 1498 q^{13} + 2113 q^{14} + 4883 q^{16} - 3874 q^{17} - 5838 q^{18} + 882 q^{19} + 1092 q^{21} + 1089 q^{22} + 5344 q^{23} - 13119 q^{24} - 4478 q^{26} + 2160 q^{27} - 12565 q^{28} + 5318 q^{29} - 7916 q^{31} - 21385 q^{32} - 18605 q^{34} + 5628 q^{36} + 1788 q^{37} + 34421 q^{38} - 29760 q^{39} + 5854 q^{41} + 46725 q^{42} + 4364 q^{43} - 13915 q^{44} - 33834 q^{46} - 46452 q^{47} + 127545 q^{48} - 34217 q^{49} + 19842 q^{51} - 3222 q^{52} - 4412 q^{53} - 86535 q^{54} + 115575 q^{56} - 137160 q^{57} + 58221 q^{58} + 17896 q^{59} - 35930 q^{61} + 19627 q^{62} - 100980 q^{63} + 14779 q^{64} + 28677 q^{66} - 73136 q^{67} + 83409 q^{68} + 34296 q^{69} + 43612 q^{71} - 372276 q^{72} - 142306 q^{73} - 95609 q^{74} - 6617 q^{76} + 8470 q^{77} - 15750 q^{78} - 46504 q^{79} + 79101 q^{81} - 175798 q^{82} - 81604 q^{83} - 532533 q^{84} - 101788 q^{86} - 219750 q^{87} + 91113 q^{88} + 8664 q^{89} - 203380 q^{91} - 251174 q^{92} + 46470 q^{93} - 71458 q^{94} - 925479 q^{96} + 22230 q^{97} - 59962 q^{98} - 128139 q^{99}+O(q^{100}) \) 5 * q - 9 * q^2 + 115 * q^4 - 237 * q^6 - 70 * q^7 - 753 * q^8 + 1059 * q^9 - 605 * q^11 + 1605 * q^12 - 1498 * q^13 + 2113 * q^14 + 4883 * q^16 - 3874 * q^17 - 5838 * q^18 + 882 * q^19 + 1092 * q^21 + 1089 * q^22 + 5344 * q^23 - 13119 * q^24 - 4478 * q^26 + 2160 * q^27 - 12565 * q^28 + 5318 * q^29 - 7916 * q^31 - 21385 * q^32 - 18605 * q^34 + 5628 * q^36 + 1788 * q^37 + 34421 * q^38 - 29760 * q^39 + 5854 * q^41 + 46725 * q^42 + 4364 * q^43 - 13915 * q^44 - 33834 * q^46 - 46452 * q^47 + 127545 * q^48 - 34217 * q^49 + 19842 * q^51 - 3222 * q^52 - 4412 * q^53 - 86535 * q^54 + 115575 * q^56 - 137160 * q^57 + 58221 * q^58 + 17896 * q^59 - 35930 * q^61 + 19627 * q^62 - 100980 * q^63 + 14779 * q^64 + 28677 * q^66 - 73136 * q^67 + 83409 * q^68 + 34296 * q^69 + 43612 * q^71 - 372276 * q^72 - 142306 * q^73 - 95609 * q^74 - 6617 * q^76 + 8470 * q^77 - 15750 * q^78 - 46504 * q^79 + 79101 * q^81 - 175798 * q^82 - 81604 * q^83 - 532533 * q^84 - 101788 * q^86 - 219750 * q^87 + 91113 * q^88 + 8664 * q^89 - 203380 * q^91 - 251174 * q^92 + 46470 * q^93 - 71458 * q^94 - 925479 * q^96 + 22230 * q^97 - 59962 * q^98 - 128139 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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