LMFDB - Embedded newform 88.6.a.c.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [88,6,Mod(1,88)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(88, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("88.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 88 = 2^{3} \cdot 11 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	88.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:	$14.1137761435$
Analytic rank:	$1$
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} - 2x^{3} - 158x^{2} - 78x + 2316 $ x^4 - 2x^3 - 158x^2 - 78*x + 2316
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-4.56967$ of defining polynomial
Character	$\chi$	$=$	88.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+13.8972 q^{3} -4.29272 q^{5} -204.438 q^{7} -49.8668 q^{9} -121.000 q^{11} -939.867 q^{13} -59.6569 q^{15} +741.838 q^{17} +1868.56 q^{19} -2841.12 q^{21} -1932.30 q^{23} -3106.57 q^{25} -4070.04 q^{27} +7684.78 q^{29} -8497.26 q^{31} -1681.57 q^{33} +877.593 q^{35} -136.803 q^{37} -13061.6 q^{39} +103.129 q^{41} -2837.74 q^{43} +214.064 q^{45} -17669.0 q^{47} +24987.8 q^{49} +10309.5 q^{51} +1282.99 q^{53} +519.419 q^{55} +25967.9 q^{57} +28047.9 q^{59} +44146.9 q^{61} +10194.6 q^{63} +4034.58 q^{65} -21350.8 q^{67} -26853.6 q^{69} +49741.1 q^{71} +59439.4 q^{73} -43172.8 q^{75} +24737.0 q^{77} -81188.9 q^{79} -44444.7 q^{81} -58038.4 q^{83} -3184.50 q^{85} +106797. q^{87} -14792.3 q^{89} +192144. q^{91} -118088. q^{93} -8021.21 q^{95} +40704.5 q^{97} +6033.88 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 13 q^{3} - 19 q^{5} + 58 q^{7} + 191 q^{9} - 484 q^{11} - 1266 q^{13} - 2173 q^{15} - 504 q^{17} - 4016 q^{19} - 2902 q^{21} - 7837 q^{23} - 1845 q^{25} - 16531 q^{27} - 4114 q^{29} - 6989 q^{31}+ \cdots - 23111 q^{99}+O(q^{100}) $ 4 * q - 13 * q^3 - 19 * q^5 + 58 * q^7 + 191 * q^9 - 484 * q^11 - 1266 * q^13 - 2173 * q^15 - 504 * q^17 - 4016 * q^19 - 2902 * q^21 - 7837 * q^23 - 1845 * q^25 - 16531 * q^27 - 4114 * q^29 - 6989 * q^31 + 1573 * q^33 - 18886 * q^35 - 1997 * q^37 - 6528 * q^39 + 1278 * q^41 - 1354 * q^43 + 35048 * q^45 - 29152 * q^47 + 40464 * q^49 - 29370 * q^51 + 46060 * q^53 + 2299 * q^55 + 80276 * q^57 + 13569 * q^59 + 15778 * q^61 + 27956 * q^63 + 83964 * q^65 - 11395 * q^67 + 108327 * q^69 - 16615 * q^71 + 78546 * q^73 + 48352 * q^75 - 7018 * q^77 + 25958 * q^79 + 197252 * q^81 - 43882 * q^83 + 75834 * q^85 + 42816 * q^87 - 31785 * q^89 - 105024 * q^91 - 269505 * q^93 - 43924 * q^95 - 17011 * q^97 - 23111 * q^99

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 $

$2$	0	0
$2$	0	0
$3$	13.8972	0.891508	0.445754	−	0.895155i	$-0.352936\pi$
$3$	13.8972	0.891508	0.445754	+	0.895155i	$0.352936\pi$
$4$	0	0
$5$	−4.29272	−0.0767904	−0.0383952	−	0.999263i	$-0.512225\pi$
$5$	−4.29272	−0.0767904	−0.0383952	+	0.999263i	$0.512225\pi$
$6$	0	0
$7$	−204.438	−1.57694	−0.788471	−	0.615072i	$-0.789127\pi$
$7$	−204.438	−1.57694	−0.788471	+	0.615072i	$0.789127\pi$
$8$	0	0
$9$	−49.8668	−0.205213
$10$	0	0
$11$	−121.000	−0.301511
$11$	−121.000	−0.301511
$12$	0	0
$13$	−939.867	−1.54244	−0.771220	−	0.636569i	$-0.780353\pi$
$13$	−939.867	−1.54244	−0.771220	+	0.636569i	$0.780353\pi$
$14$	0	0
$15$	−59.6569	−0.0684593
$16$	0	0
$17$	741.838	0.622568	0.311284	−	0.950317i	$-0.399241\pi$
$17$	741.838	0.622568	0.311284	+	0.950317i	$0.399241\pi$
$18$	0	0
$19$	1868.56	1.18747	0.593736	−	0.804660i	$-0.297652\pi$
$19$	1868.56	1.18747	0.593736	+	0.804660i	$0.297652\pi$
$20$	0	0
$21$	−2841.12	−1.40586
$22$	0	0
$23$	−1932.30	−0.761648	−0.380824	−	0.924648i	$-0.624360\pi$
$23$	−1932.30	−0.761648	−0.380824	+	0.924648i	$0.624360\pi$
$24$	0	0
$25$	−3106.57	−0.994103
$26$	0	0
$27$	−4070.04	−1.07446
$28$	0	0
$29$	7684.78	1.69682	0.848411	−	0.529337i	$-0.177559\pi$
$29$	7684.78	1.69682	0.848411	+	0.529337i	$0.177559\pi$
$30$	0	0
$31$	−8497.26	−1.58809	−0.794044	−	0.607860i	$-0.792028\pi$
$31$	−8497.26	−1.58809	−0.794044	+	0.607860i	$0.792028\pi$
$32$	0	0
$33$	−1681.57	−0.268800
$34$	0	0
$35$	877.593	0.121094
$36$	0	0
$37$	−136.803	−0.0164283	−0.00821414	−	0.999966i	$-0.502615\pi$
$37$	−136.803	−0.0164283	−0.00821414	+	0.999966i	$0.502615\pi$
$38$	0	0
$39$	−13061.6	−1.37510
$40$	0	0
$41$	103.129	0.00958124	0.00479062	−	0.999989i	$-0.498475\pi$
$41$	103.129	0.00958124	0.00479062	+	0.999989i	$0.498475\pi$
$42$	0	0
$43$	−2837.74	−0.234046	−0.117023	−	0.993129i	$-0.537335\pi$
$43$	−2837.74	−0.234046	−0.117023	+	0.993129i	$0.537335\pi$
$44$	0	0
$45$	214.064	0.0157584
$46$	0	0
$47$	−17669.0	−1.16672	−0.583360	−	0.812214i	$-0.698262\pi$
$47$	−17669.0	−1.16672	−0.583360	+	0.812214i	$0.698262\pi$
$48$	0	0
$49$	24987.8	1.48675
$50$	0	0
$51$	10309.5	0.555025
$52$	0	0
$53$	1282.99	0.0627386	0.0313693	−	0.999508i	$-0.490013\pi$
$53$	1282.99	0.0627386	0.0313693	+	0.999508i	$0.490013\pi$
$54$	0	0
$55$	519.419	0.0231532
$56$	0	0
$57$	25967.9	1.05864
$58$	0	0
$59$	28047.9	1.04899	0.524495	−	0.851414i	$-0.324254\pi$
$59$	28047.9	1.04899	0.524495	+	0.851414i	$0.324254\pi$
$60$	0	0
$61$	44146.9	1.51906	0.759532	−	0.650470i	$-0.225428\pi$
$61$	44146.9	1.51906	0.759532	+	0.650470i	$0.225428\pi$
$62$	0	0
$63$	10194.6	0.323609
$64$	0	0
$65$	4034.58	0.118445
$66$	0	0
$67$	−21350.8	−0.581069	−0.290535	−	0.956865i	$-0.593833\pi$
$67$	−21350.8	−0.581069	−0.290535	+	0.956865i	$0.593833\pi$
$68$	0	0
$69$	−26853.6	−0.679015
$70$	0	0
$71$	49741.1	1.17103	0.585517	−	0.810660i	$-0.300891\pi$
$71$	49741.1	1.17103	0.585517	+	0.810660i	$0.300891\pi$
$72$	0	0
$73$	59439.4	1.30547	0.652736	−	0.757586i	$-0.273621\pi$
$73$	59439.4	1.30547	0.652736	+	0.757586i	$0.273621\pi$
$74$	0	0
$75$	−43172.8	−0.886251
$76$	0	0
$77$	24737.0	0.475466
$78$	0	0
$79$	−81188.9	−1.46362	−0.731811	−	0.681508i	$-0.761324\pi$
$79$	−81188.9	−1.46362	−0.731811	+	0.681508i	$0.761324\pi$
$80$	0	0
$81$	−44444.7	−0.752675
$82$	0	0
$83$	−58038.4	−0.924741	−0.462370	−	0.886687i	$-0.653001\pi$
$83$	−58038.4	−0.924741	−0.462370	+	0.886687i	$0.653001\pi$
$84$	0	0
$85$	−3184.50	−0.0478073
$86$	0	0
$87$	106797.	1.51273
$88$	0	0
$89$	−14792.3	−0.197953	−0.0989763	−	0.995090i	$-0.531557\pi$
$89$	−14792.3	−0.197953	−0.0989763	+	0.995090i	$0.531557\pi$
$90$	0	0
$91$	192144.	2.43234
$92$	0	0
$93$	−118088.	−1.41579
$94$	0	0
$95$	−8021.21	−0.0911866
$96$	0	0
$97$	40704.5	0.439251	0.219626	−	0.975584i	$-0.429516\pi$
$97$	40704.5	0.439251	0.219626	+	0.975584i	$0.429516\pi$
$98$	0	0
$99$	6033.88	0.0618741
$100$	0	0
$101$	−125801.	−1.22710	−0.613552	−	0.789654i	$-0.710260\pi$
$101$	−125801.	−1.22710	−0.613552	+	0.789654i	$0.710260\pi$
$102$	0	0
$103$	−169673.	−1.57587	−0.787936	−	0.615757i	$-0.788850\pi$
$103$	−169673.	−1.57587	−0.787936	+	0.615757i	$0.788850\pi$
$104$	0	0
$105$	12196.1	0.107956
$106$	0	0
$107$	−74755.7	−0.631226	−0.315613	−	0.948888i	$-0.602210\pi$
$107$	−74755.7	−0.631226	−0.315613	+	0.948888i	$0.602210\pi$
$108$	0	0
$109$	123083.	0.992276	0.496138	−	0.868244i	$-0.334751\pi$
$109$	123083.	0.992276	0.496138	+	0.868244i	$0.334751\pi$
$110$	0	0
$111$	−1901.19	−0.0146459
$112$	0	0
$113$	−146460.	−1.07900	−0.539502	−	0.841985i	$-0.681387\pi$
$113$	−146460.	−1.07900	−0.539502	+	0.841985i	$0.681387\pi$
$114$	0	0
$115$	8294.80	0.0584873
$116$	0	0
$117$	46868.2	0.316529
$118$	0	0
$119$	−151660.	−0.981754
$120$	0	0
$121$	14641.0	0.0909091
$122$	0	0
$123$	1433.21	0.00854176
$124$	0	0
$125$	26750.4	0.153128
$126$	0	0
$127$	85789.4	0.471981	0.235990	−	0.971755i	$-0.424167\pi$
$127$	85789.4	0.471981	0.235990	+	0.971755i	$0.424167\pi$
$128$	0	0
$129$	−39436.8	−0.208654
$130$	0	0
$131$	98589.9	0.501943	0.250971	−	0.967995i	$-0.419250\pi$
$131$	98589.9	0.501943	0.250971	+	0.967995i	$0.419250\pi$
$132$	0	0
$133$	−382005.	−1.87258
$134$	0	0
$135$	17471.5	0.0825080
$136$	0	0
$137$	140094.	0.637703	0.318852	−	0.947805i	$-0.396703\pi$
$137$	140094.	0.637703	0.318852	+	0.947805i	$0.396703\pi$
$138$	0	0
$139$	−180709.	−0.793308	−0.396654	−	0.917968i	$-0.629829\pi$
$139$	−180709.	−0.793308	−0.396654	+	0.917968i	$0.629829\pi$
$140$	0	0
$141$	−245550.	−1.04014
$142$	0	0
$143$	113724.	0.465063
$144$	0	0
$145$	−32988.6	−0.130300
$146$	0	0
$147$	347261.	1.32545
$148$	0	0
$149$	−418243.	−1.54334	−0.771672	−	0.636021i	$-0.780579\pi$
$149$	−418243.	−1.54334	−0.771672	+	0.636021i	$0.780579\pi$
$150$	0	0
$151$	149557.	0.533782	0.266891	−	0.963727i	$-0.414004\pi$
$151$	149557.	0.533782	0.266891	+	0.963727i	$0.414004\pi$
$152$	0	0
$153$	−36993.1	−0.127759
$154$	0	0
$155$	36476.3	0.121950
$156$	0	0
$157$	−181195.	−0.586676	−0.293338	−	0.956009i	$-0.594766\pi$
$157$	−181195.	−0.586676	−0.293338	+	0.956009i	$0.594766\pi$
$158$	0	0
$159$	17830.1	0.0559320
$160$	0	0
$161$	395034.	1.20107
$162$	0	0
$163$	420310.	1.23908	0.619542	−	0.784963i	$-0.287318\pi$
$163$	420310.	1.23908	0.619542	+	0.784963i	$0.287318\pi$
$164$	0	0
$165$	7218.48	0.0206413
$166$	0	0
$167$	90313.7	0.250589	0.125295	−	0.992120i	$-0.460012\pi$
$167$	90313.7	0.250589	0.125295	+	0.992120i	$0.460012\pi$
$168$	0	0
$169$	512057.	1.37912
$170$	0	0
$171$	−93179.2	−0.243685
$172$	0	0
$173$	394997.	1.00341	0.501705	−	0.865039i	$-0.332706\pi$
$173$	394997.	1.00341	0.501705	+	0.865039i	$0.332706\pi$
$174$	0	0
$175$	635100.	1.56764
$176$	0	0
$177$	389789.	0.935182
$178$	0	0
$179$	−519681.	−1.21228	−0.606142	−	0.795357i	$-0.707283\pi$
$179$	−519681.	−1.21228	−0.606142	+	0.795357i	$0.707283\pi$
$180$	0	0
$181$	−365931.	−0.830237	−0.415119	−	0.909767i	$-0.636260\pi$
$181$	−365931.	−0.830237	−0.415119	+	0.909767i	$0.636260\pi$
$182$	0	0
$183$	613520.	1.35426
$184$	0	0
$185$	587.258	0.00126153
$186$	0	0
$187$	−89762.4	−0.187711
$188$	0	0
$189$	832069.	1.69436
$190$	0	0
$191$	160811.	0.318957	0.159478	−	0.987201i	$-0.449019\pi$
$191$	160811.	0.318957	0.159478	+	0.987201i	$0.449019\pi$
$192$	0	0
$193$	124698.	0.240972	0.120486	−	0.992715i	$-0.461555\pi$
$193$	124698.	0.240972	0.120486	+	0.992715i	$0.461555\pi$
$194$	0	0
$195$	56069.6	0.105594
$196$	0	0
$197$	−784680.	−1.44054	−0.720272	−	0.693691i	$-0.755983\pi$
$197$	−784680.	−1.44054	−0.720272	+	0.693691i	$0.755983\pi$
$198$	0	0
$199$	−584647.	−1.04655	−0.523277	−	0.852163i	$-0.675291\pi$
$199$	−584647.	−1.04655	−0.523277	+	0.852163i	$0.675291\pi$
$200$	0	0
$201$	−296718.	−0.518028
$202$	0	0
$203$	−1.57106e6	−2.67579
$204$	0	0
$205$	−442.704	−0.000735748 0
$206$	0	0
$207$	96357.4	0.156300
$208$	0	0
$209$	−226096.	−0.358037
$210$	0	0
$211$	−376077.	−0.581529	−0.290764	−	0.956795i	$-0.593910\pi$
$211$	−376077.	−0.581529	−0.290764	+	0.956795i	$0.593910\pi$
$212$	0	0
$213$	691264.	1.04399
$214$	0	0
$215$	12181.6	0.0179725
$216$	0	0
$217$	1.73716e6	2.50432
$218$	0	0
$219$	826044.	1.16384
$220$	0	0
$221$	−697229.	−0.960274
$222$	0	0
$223$	−186863.	−0.251629	−0.125814	−	0.992054i	$-0.540154\pi$
$223$	−186863.	−0.251629	−0.125814	+	0.992054i	$0.540154\pi$
$224$	0	0
$225$	154915.	0.204003
$226$	0	0
$227$	−1.35845e6	−1.74977	−0.874883	−	0.484334i	$-0.839062\pi$
$227$	−1.35845e6	−1.74977	−0.874883	+	0.484334i	$0.839062\pi$
$228$	0	0
$229$	−165090.	−0.208033	−0.104016	−	0.994576i	$-0.533169\pi$
$229$	−165090.	−0.208033	−0.104016	+	0.994576i	$0.533169\pi$
$230$	0	0
$231$	343775.	0.423882
$232$	0	0
$233$	−972832.	−1.17395	−0.586973	−	0.809607i	$-0.699681\pi$
$233$	−972832.	−1.17395	−0.586973	+	0.809607i	$0.699681\pi$
$234$	0	0
$235$	75847.9	0.0895929
$236$	0	0
$237$	−1.12830e6	−1.30483
$238$	0	0
$239$	−1.59700e6	−1.80846	−0.904230	−	0.427045i	$-0.859555\pi$
$239$	−1.59700e6	−1.80846	−0.904230	+	0.427045i	$0.859555\pi$
$240$	0	0
$241$	1.66500e6	1.84660	0.923299	−	0.384082i	$-0.125482\pi$
$241$	1.66500e6	1.84660	0.923299	+	0.384082i	$0.125482\pi$
$242$	0	0
$243$	371361.	0.403442
$244$	0	0
$245$	−107265.	−0.114168
$246$	0	0
$247$	−1.75620e6	−1.83160
$248$	0	0
$249$	−806573.	−0.824414
$250$	0	0
$251$	344631.	0.345279	0.172639	−	0.984985i	$-0.444771\pi$
$251$	344631.	0.345279	0.172639	+	0.984985i	$0.444771\pi$
$252$	0	0
$253$	233808.	0.229645
$254$	0	0
$255$	−44255.8	−0.0426206
$256$	0	0
$257$	1.60526e6	1.51605	0.758023	−	0.652227i	$-0.226165\pi$
$257$	1.60526e6	1.51605	0.758023	+	0.652227i	$0.226165\pi$
$258$	0	0
$259$	27967.7	0.0259065
$260$	0	0
$261$	−383215.	−0.348210
$262$	0	0
$263$	794473.	0.708255	0.354128	−	0.935197i	$-0.384778\pi$
$263$	794473.	0.708255	0.354128	+	0.935197i	$0.384778\pi$
$264$	0	0
$265$	−5507.53	−0.00481773
$266$	0	0
$267$	−205572.	−0.176476
$268$	0	0
$269$	−1.59318e6	−1.34241	−0.671203	−	0.741274i	$-0.734222\pi$
$269$	−1.59318e6	−1.34241	−0.671203	+	0.741274i	$0.734222\pi$
$270$	0	0
$271$	−978012.	−0.808949	−0.404474	−	0.914549i	$-0.632546\pi$
$271$	−978012.	−0.808949	−0.404474	+	0.914549i	$0.632546\pi$
$272$	0	0
$273$	2.67027e6	2.16845
$274$	0	0
$275$	375895.	0.299733
$276$	0	0
$277$	247857.	0.194089	0.0970447	−	0.995280i	$-0.469061\pi$
$277$	247857.	0.194089	0.0970447	+	0.995280i	$0.469061\pi$
$278$	0	0
$279$	423731.	0.325897
$280$	0	0
$281$	−2.56291e6	−1.93628	−0.968140	−	0.250410i	$-0.919435\pi$
$281$	−2.56291e6	−1.93628	−0.968140	+	0.250410i	$0.919435\pi$
$282$	0	0
$283$	751020.	0.557424	0.278712	−	0.960375i	$-0.410093\pi$
$283$	751020.	0.557424	0.278712	+	0.960375i	$0.410093\pi$
$284$	0	0
$285$	−111473.	−0.0812936
$286$	0	0
$287$	−21083.5	−0.0151091
$288$	0	0
$289$	−869533.	−0.612409
$290$	0	0
$291$	565680.	0.391596
$292$	0	0
$293$	−361215.	−0.245808	−0.122904	−	0.992419i	$-0.539221\pi$
$293$	−361215.	−0.245808	−0.122904	+	0.992419i	$0.539221\pi$
$294$	0	0
$295$	−120402.	−0.0805523
$296$	0	0
$297$	492475.	0.323961
$298$	0	0
$299$	1.81610e6	1.17480
$300$	0	0
$301$	580141.	0.369078
$302$	0	0
$303$	−1.74829e6	−1.09397
$304$	0	0
$305$	−189510.	−0.116650
$306$	0	0
$307$	2.26761e6	1.37316	0.686581	−	0.727054i	$-0.259111\pi$
$307$	2.26761e6	1.37316	0.686581	+	0.727054i	$0.259111\pi$
$308$	0	0
$309$	−2.35799e6	−1.40490
$310$	0	0
$311$	1.47991e6	0.867628	0.433814	−	0.901002i	$-0.357168\pi$
$311$	1.47991e6	0.867628	0.433814	+	0.901002i	$0.357168\pi$
$312$	0	0
$313$	−1.51012e6	−0.871268	−0.435634	−	0.900124i	$-0.643476\pi$
$313$	−1.51012e6	−0.871268	−0.435634	+	0.900124i	$0.643476\pi$
$314$	0	0
$315$	−43762.7	−0.0248501
$316$	0	0
$317$	−583893.	−0.326351	−0.163176	−	0.986597i	$-0.552174\pi$
$317$	−583893.	−0.326351	−0.163176	+	0.986597i	$0.552174\pi$
$318$	0	0
$319$	−929859.	−0.511611
$320$	0	0
$321$	−1.03890e6	−0.562743
$322$	0	0
$323$	1.38617e6	0.739283
$324$	0	0
$325$	2.91977e6	1.53334
$326$	0	0
$327$	1.71052e6	0.884622
$328$	0	0
$329$	3.61220e6	1.83985
$330$	0	0
$331$	534027.	0.267913	0.133956	−	0.990987i	$-0.457232\pi$
$331$	534027.	0.267913	0.133956	+	0.990987i	$0.457232\pi$
$332$	0	0
$333$	6821.94	0.00337130
$334$	0	0
$335$	91653.0	0.0446205
$336$	0	0
$337$	2.75088e6	1.31946	0.659731	−	0.751502i	$-0.270670\pi$
$337$	2.75088e6	1.31946	0.659731	+	0.751502i	$0.270670\pi$
$338$	0	0
$339$	−2.03539e6	−0.961940
$340$	0	0
$341$	1.02817e6	0.478827
$342$	0	0
$343$	−1.67246e6	−0.767572
$344$	0	0
$345$	115275.	0.0521419
$346$	0	0
$347$	241141.	0.107510	0.0537548	−	0.998554i	$-0.482881\pi$
$347$	241141.	0.107510	0.0537548	+	0.998554i	$0.482881\pi$
$348$	0	0
$349$	3.58337e6	1.57481	0.787405	−	0.616436i	$-0.211424\pi$
$349$	3.58337e6	1.57481	0.787405	+	0.616436i	$0.211424\pi$
$350$	0	0
$351$	3.82530e6	1.65729
$352$	0	0
$353$	−1.84061e6	−0.786184	−0.393092	−	0.919499i	$-0.628595\pi$
$353$	−1.84061e6	−0.786184	−0.393092	+	0.919499i	$0.628595\pi$
$354$	0	0
$355$	−213525.	−0.0899243
$356$	0	0
$357$	−2.10765e6	−0.875242
$358$	0	0
$359$	1.78372e6	0.730451	0.365226	−	0.930919i	$-0.380992\pi$
$359$	1.78372e6	0.730451	0.365226	+	0.930919i	$0.380992\pi$
$360$	0	0
$361$	1.01543e6	0.410092
$362$	0	0
$363$	203469.	0.0810462
$364$	0	0
$365$	−255157.	−0.100248
$366$	0	0
$367$	4.33788e6	1.68117	0.840587	−	0.541677i	$-0.182211\pi$
$367$	4.33788e6	1.68117	0.840587	+	0.541677i	$0.182211\pi$
$368$	0	0
$369$	−5142.72	−0.00196620
$370$	0	0
$371$	−262292.	−0.0989352
$372$	0	0
$373$	−5.12472e6	−1.90721	−0.953604	−	0.301063i	$-0.902658\pi$
$373$	−5.12472e6	−1.90721	−0.953604	+	0.301063i	$0.902658\pi$
$374$	0	0
$375$	371756.	0.136515
$376$	0	0
$377$	−7.22267e6	−2.61725
$378$	0	0
$379$	−4.19347e6	−1.49960	−0.749801	−	0.661664i	$-0.769851\pi$
$379$	−4.19347e6	−1.49960	−0.749801	+	0.661664i	$0.769851\pi$
$380$	0	0
$381$	1.19224e6	0.420775
$382$	0	0
$383$	−2.60071e6	−0.905930	−0.452965	−	0.891528i	$-0.649634\pi$
$383$	−2.60071e6	−0.905930	−0.452965	+	0.891528i	$0.649634\pi$
$384$	0	0
$385$	−106189.	−0.0365112
$386$	0	0
$387$	141509.	0.0480294
$388$	0	0
$389$	−563376.	−0.188766	−0.0943832	−	0.995536i	$-0.530088\pi$
$389$	−563376.	−0.188766	−0.0943832	+	0.995536i	$0.530088\pi$
$390$	0	0
$391$	−1.43345e6	−0.474178
$392$	0	0
$393$	1.37013e6	0.447486
$394$	0	0
$395$	348521.	0.112392
$396$	0	0
$397$	−1.13156e6	−0.360331	−0.180165	−	0.983636i	$-0.557663\pi$
$397$	−1.13156e6	−0.360331	−0.180165	+	0.983636i	$0.557663\pi$
$398$	0	0
$399$	−5.30881e6	−1.66942
$400$	0	0
$401$	3.54308e6	1.10032	0.550161	−	0.835058i	$-0.314566\pi$
$401$	3.54308e6	1.10032	0.550161	+	0.835058i	$0.314566\pi$
$402$	0	0
$403$	7.98630e6	2.44953
$404$	0	0
$405$	190788.	0.0577982
$406$	0	0
$407$	16553.2	0.00495331
$408$	0	0
$409$	−4.23104e6	−1.25066	−0.625330	−	0.780360i	$-0.715036\pi$
$409$	−4.23104e6	−1.25066	−0.625330	+	0.780360i	$0.715036\pi$
$410$	0	0
$411$	1.94692e6	0.568518
$412$	0	0
$413$	−5.73406e6	−1.65420
$414$	0	0
$415$	249142.	0.0710112
$416$	0	0
$417$	−2.51135e6	−0.707241
$418$	0	0
$419$	−5.85784e6	−1.63006	−0.815028	−	0.579422i	$-0.803278\pi$
$419$	−5.85784e6	−1.63006	−0.815028	+	0.579422i	$0.803278\pi$
$420$	0	0
$421$	6.65403e6	1.82970	0.914850	−	0.403794i	$-0.132309\pi$
$421$	6.65403e6	1.82970	0.914850	+	0.403794i	$0.132309\pi$
$422$	0	0
$423$	881095.	0.239426
$424$	0	0
$425$	−2.30457e6	−0.618897
$426$	0	0
$427$	−9.02529e6	−2.39548
$428$	0	0
$429$	1.58045e6	0.414607
$430$	0	0
$431$	6.54588e6	1.69736	0.848682	−	0.528904i	$-0.177397\pi$
$431$	6.54588e6	1.69736	0.848682	+	0.528904i	$0.177397\pi$
$432$	0	0
$433$	−4.67682e6	−1.19876	−0.599379	−	0.800466i	$-0.704586\pi$
$433$	−4.67682e6	−1.19876	−0.599379	+	0.800466i	$0.704586\pi$
$434$	0	0
$435$	−458450.	−0.116163
$436$	0	0
$437$	−3.61062e6	−0.904436
$438$	0	0
$439$	2.67406e6	0.662231	0.331115	−	0.943590i	$-0.392575\pi$
$439$	2.67406e6	0.662231	0.331115	+	0.943590i	$0.392575\pi$
$440$	0	0
$441$	−1.24606e6	−0.305100
$442$	0	0
$443$	−2.40974e6	−0.583393	−0.291697	−	0.956511i	$-0.594220\pi$
$443$	−2.40974e6	−0.583393	−0.291697	+	0.956511i	$0.594220\pi$
$444$	0	0
$445$	63499.2	0.0152009
$446$	0	0
$447$	−5.81242e6	−1.37590
$448$	0	0
$449$	6.56118e6	1.53591	0.767956	−	0.640503i	$-0.221274\pi$
$449$	6.56118e6	1.53591	0.767956	+	0.640503i	$0.221274\pi$
$450$	0	0
$451$	−12478.6	−0.00288885
$452$	0	0
$453$	2.07843e6	0.475871
$454$	0	0
$455$	−824821.	−0.186780
$456$	0	0
$457$	177486.	0.0397533	0.0198767	−	0.999802i	$-0.493673\pi$
$457$	177486.	0.0397533	0.0198767	+	0.999802i	$0.493673\pi$
$458$	0	0
$459$	−3.01931e6	−0.668923
$460$	0	0
$461$	7.95359e6	1.74305	0.871527	−	0.490347i	$-0.163130\pi$
$461$	7.95359e6	1.74305	0.871527	+	0.490347i	$0.163130\pi$
$462$	0	0
$463$	−741086.	−0.160663	−0.0803315	−	0.996768i	$-0.525598\pi$
$463$	−741086.	−0.160663	−0.0803315	+	0.996768i	$0.525598\pi$
$464$	0	0
$465$	506920.	0.108719
$466$	0	0
$467$	5.12831e6	1.08813	0.544066	−	0.839042i	$-0.316884\pi$
$467$	5.12831e6	1.08813	0.544066	+	0.839042i	$0.316884\pi$
$468$	0	0
$469$	4.36491e6	0.916312
$470$	0	0
$471$	−2.51812e6	−0.523026
$472$	0	0
$473$	343367.	0.0705676
$474$	0	0
$475$	−5.80483e6	−1.18047
$476$	0	0
$477$	−63978.8	−0.0128748
$478$	0	0
$479$	4.98194e6	0.992111	0.496055	−	0.868291i	$-0.334781\pi$
$479$	4.98194e6	0.992111	0.496055	+	0.868291i	$0.334781\pi$
$480$	0	0
$481$	128577.	0.0253396
$482$	0	0
$483$	5.48988e6	1.07077
$484$	0	0
$485$	−174733.	−0.0337303
$486$	0	0
$487$	−1.40032e6	−0.267550	−0.133775	−	0.991012i	$-0.542710\pi$
$487$	−1.40032e6	−0.267550	−0.133775	+	0.991012i	$0.542710\pi$
$488$	0	0
$489$	5.84115e6	1.10465
$490$	0	0
$491$	1.63264e6	0.305623	0.152811	−	0.988255i	$-0.451167\pi$
$491$	1.63264e6	0.305623	0.152811	+	0.988255i	$0.451167\pi$
$492$	0	0
$493$	5.70086e6	1.05639
$494$	0	0
$495$	−25901.7	−0.00475134
$496$	0	0
$497$	−1.01690e7	−1.84665
$498$	0	0
$499$	−3.05894e6	−0.549945	−0.274973	−	0.961452i	$-0.588669\pi$
$499$	−3.05894e6	−0.549945	−0.274973	+	0.961452i	$0.588669\pi$
$500$	0	0
$501$	1.25511e6	0.223402
$502$	0	0
$503$	−1.01326e6	−0.178566	−0.0892831	−	0.996006i	$-0.528458\pi$
$503$	−1.01326e6	−0.178566	−0.0892831	+	0.996006i	$0.528458\pi$
$504$	0	0
$505$	540029.	0.0942299
$506$	0	0
$507$	7.11618e6	1.22950
$508$	0	0
$509$	9.88684e6	1.69147	0.845733	−	0.533607i	$-0.179164\pi$
$509$	9.88684e6	1.69147	0.845733	+	0.533607i	$0.179164\pi$
$510$	0	0
$511$	−1.21517e7	−2.05865
$512$	0	0
$513$	−7.60512e6	−1.27589
$514$	0	0
$515$	728360.	0.121012
$516$	0	0
$517$	2.13795e6	0.351779
$518$	0	0
$519$	5.48936e6	0.894548
$520$	0	0
$521$	5.99108e6	0.966965	0.483482	−	0.875354i	$-0.339372\pi$
$521$	5.99108e6	0.966965	0.483482	+	0.875354i	$0.339372\pi$
$522$	0	0
$523$	1.35910e6	0.217269	0.108634	−	0.994082i	$-0.465352\pi$
$523$	1.35910e6	0.217269	0.108634	+	0.994082i	$0.465352\pi$
$524$	0	0
$525$	8.82614e6	1.39757
$526$	0	0
$527$	−6.30359e6	−0.988693
$528$	0	0
$529$	−2.70257e6	−0.419893
$530$	0	0
$531$	−1.39866e6	−0.215266
$532$	0	0
$533$	−96927.7	−0.0147785
$534$	0	0
$535$	320905.	0.0484721
$536$	0	0
$537$	−7.22213e6	−1.08076
$538$	0	0
$539$	−3.02352e6	−0.448271
$540$	0	0
$541$	−219465.	−0.0322382	−0.0161191	−	0.999870i	$-0.505131\pi$
$541$	−219465.	−0.0322382	−0.0161191	+	0.999870i	$0.505131\pi$
$542$	0	0
$543$	−5.08542e6	−0.740163
$544$	0	0
$545$	−528361.	−0.0761973
$546$	0	0
$547$	−9.22935e6	−1.31887	−0.659436	−	0.751760i	$-0.729205\pi$
$547$	−9.22935e6	−1.31887	−0.659436	+	0.751760i	$0.729205\pi$
$548$	0	0
$549$	−2.20146e6	−0.311732
$550$	0	0
$551$	1.43595e7	2.01493
$552$	0	0
$553$	1.65981e7	2.30805
$554$	0	0
$555$	8161.26	0.00112467
$556$	0	0
$557$	−1.29470e6	−0.176819	−0.0884096	−	0.996084i	$-0.528178\pi$
$557$	−1.29470e6	−0.176819	−0.0884096	+	0.996084i	$0.528178\pi$
$558$	0	0
$559$	2.66710e6	0.361002
$560$	0	0
$561$	−1.24745e6	−0.167346
$562$	0	0
$563$	3.39881e6	0.451915	0.225957	−	0.974137i	$-0.427449\pi$
$563$	3.39881e6	0.451915	0.225957	+	0.974137i	$0.427449\pi$
$564$	0	0
$565$	628711.	0.0828571
$566$	0	0
$567$	9.08617e6	1.18692
$568$	0	0
$569$	9.92266e6	1.28484	0.642418	−	0.766355i	$-0.277931\pi$
$569$	9.92266e6	1.28484	0.642418	+	0.766355i	$0.277931\pi$
$570$	0	0
$571$	−2.11476e6	−0.271438	−0.135719	−	0.990747i	$-0.543334\pi$
$571$	−2.11476e6	−0.271438	−0.135719	+	0.990747i	$0.543334\pi$
$572$	0	0
$573$	2.23482e6	0.284352
$574$	0	0
$575$	6.00282e6	0.757157
$576$	0	0
$577$	1.41413e6	0.176828	0.0884140	−	0.996084i	$-0.471820\pi$
$577$	1.41413e6	0.176828	0.0884140	+	0.996084i	$0.471820\pi$
$578$	0	0
$579$	1.73296e6	0.214829
$580$	0	0
$581$	1.18652e7	1.45826
$582$	0	0
$583$	−155242.	−0.0189164
$584$	0	0
$585$	−201192.	−0.0243064
$586$	0	0
$587$	−1.82594e6	−0.218722	−0.109361	−	0.994002i	$-0.534880\pi$
$587$	−1.82594e6	−0.218722	−0.109361	+	0.994002i	$0.534880\pi$
$588$	0	0
$589$	−1.58777e7	−1.88581
$590$	0	0
$591$	−1.09049e7	−1.28426
$592$	0	0
$593$	−1.00356e7	−1.17194	−0.585971	−	0.810332i	$-0.699287\pi$
$593$	−1.00356e7	−1.17194	−0.585971	+	0.810332i	$0.699287\pi$
$594$	0	0
$595$	651032.	0.0753893
$596$	0	0
$597$	−8.12498e6	−0.933011
$598$	0	0
$599$	6.95642e6	0.792170	0.396085	−	0.918214i	$-0.370369\pi$
$599$	6.95642e6	0.792170	0.396085	+	0.918214i	$0.370369\pi$
$600$	0	0
$601$	−1.09278e7	−1.23409	−0.617043	−	0.786929i	$-0.711670\pi$
$601$	−1.09278e7	−1.23409	−0.617043	+	0.786929i	$0.711670\pi$
$602$	0	0
$603$	1.06470e6	0.119243
$604$	0	0
$605$	−62849.6	−0.00698095
$606$	0	0
$607$	845095.	0.0930966	0.0465483	−	0.998916i	$-0.485178\pi$
$607$	845095.	0.0930966	0.0465483	+	0.998916i	$0.485178\pi$
$608$	0	0
$609$	−2.18334e7	−2.38549
$610$	0	0
$611$	1.66065e7	1.79959
$612$	0	0
$613$	−2.16942e6	−0.233180	−0.116590	−	0.993180i	$-0.537196\pi$
$613$	−2.16942e6	−0.233180	−0.116590	+	0.993180i	$0.537196\pi$
$614$	0	0
$615$	−6152.37	−0.000655925 0
$616$	0	0
$617$	−1.29619e7	−1.37074	−0.685372	−	0.728193i	$-0.740361\pi$
$617$	−1.29619e7	−1.37074	−0.685372	+	0.728193i	$0.740361\pi$
$618$	0	0
$619$	−1.11194e7	−1.16642	−0.583209	−	0.812322i	$-0.698203\pi$
$619$	−1.11194e7	−1.16642	−0.583209	+	0.812322i	$0.698203\pi$
$620$	0	0
$621$	7.86452e6	0.818358
$622$	0	0
$623$	3.02411e6	0.312160
$624$	0	0
$625$	9.59321e6	0.982344
$626$	0	0
$627$	−3.14211e6	−0.319193
$628$	0	0
$629$	−101486.	−0.0102277
$630$	0	0
$631$	4.91924e6	0.491841	0.245921	−	0.969290i	$-0.420910\pi$
$631$	4.91924e6	0.491841	0.245921	+	0.969290i	$0.420910\pi$
$632$	0	0
$633$	−5.22644e6	−0.518438
$634$	0	0
$635$	−368269.	−0.0362436
$636$	0	0
$637$	−2.34852e7	−2.29322
$638$	0	0
$639$	−2.48043e6	−0.240312
$640$	0	0
$641$	−1.65493e7	−1.59087	−0.795434	−	0.606040i	$-0.792757\pi$
$641$	−1.65493e7	−1.59087	−0.795434	+	0.606040i	$0.792757\pi$
$642$	0	0
$643$	−572421.	−0.0545994	−0.0272997	−	0.999627i	$-0.508691\pi$
$643$	−572421.	−0.0545994	−0.0272997	+	0.999627i	$0.508691\pi$
$644$	0	0
$645$	169291.	0.0160226
$646$	0	0
$647$	−1.32996e7	−1.24905	−0.624525	−	0.781005i	$-0.714707\pi$
$647$	−1.32996e7	−1.24905	−0.624525	+	0.781005i	$0.714707\pi$
$648$	0	0
$649$	−3.39380e6	−0.316282
$650$	0	0
$651$	2.41417e7	2.23263
$652$	0	0
$653$	1.05394e7	0.967234	0.483617	−	0.875280i	$-0.339323\pi$
$653$	1.05394e7	0.967234	0.483617	+	0.875280i	$0.339323\pi$
$654$	0	0
$655$	−423218.	−0.0385444
$656$	0	0
$657$	−2.96405e6	−0.267900
$658$	0	0
$659$	−8.04359e6	−0.721500	−0.360750	−	0.932663i	$-0.617479\pi$
$659$	−8.04359e6	−0.721500	−0.360750	+	0.932663i	$0.617479\pi$
$660$	0	0
$661$	1.74407e7	1.55261	0.776303	−	0.630360i	$-0.217093\pi$
$661$	1.74407e7	1.55261	0.776303	+	0.630360i	$0.217093\pi$
$662$	0	0
$663$	−9.68956e6	−0.856092
$664$	0	0
$665$	1.63984e6	0.143796
$666$	0	0
$667$	−1.48493e7	−1.29238
$668$	0	0
$669$	−2.59688e6	−0.224329
$670$	0	0
$671$	−5.34178e6	−0.458015
$672$	0	0
$673$	9.89064e6	0.841757	0.420878	−	0.907117i	$-0.361722\pi$
$673$	9.89064e6	0.841757	0.420878	+	0.907117i	$0.361722\pi$
$674$	0	0
$675$	1.26439e7	1.06812
$676$	0	0
$677$	6.20931e6	0.520681	0.260341	−	0.965517i	$-0.416165\pi$
$677$	6.20931e6	0.520681	0.260341	+	0.965517i	$0.416165\pi$
$678$	0	0
$679$	−8.32154e6	−0.692674
$680$	0	0
$681$	−1.88787e7	−1.55993
$682$	0	0
$683$	1.18068e7	0.968461	0.484230	−	0.874940i	$-0.339100\pi$
$683$	1.18068e7	0.968461	0.484230	+	0.874940i	$0.339100\pi$
$684$	0	0
$685$	−601384.	−0.0489695
$686$	0	0
$687$	−2.29429e6	−0.185463
$688$	0	0
$689$	−1.20584e6	−0.0967705
$690$	0	0
$691$	−1.28250e7	−1.02179	−0.510896	−	0.859643i	$-0.670686\pi$
$691$	−1.28250e7	−1.02179	−0.510896	+	0.859643i	$0.670686\pi$
$692$	0	0
$693$	−1.23355e6	−0.0975718
$694$	0	0
$695$	775731.	0.0609185
$696$	0	0
$697$	76505.1	0.00596498
$698$	0	0
$699$	−1.35197e7	−1.04658
$700$	0	0
$701$	−5.99809e6	−0.461018	−0.230509	−	0.973070i	$-0.574039\pi$
$701$	−5.99809e6	−0.461018	−0.230509	+	0.973070i	$0.574039\pi$
$702$	0	0
$703$	−255626.	−0.0195081
$704$	0	0
$705$	1.05408e6	0.0798728
$706$	0	0
$707$	2.57185e7	1.93507
$708$	0	0
$709$	1.33200e7	0.995150	0.497575	−	0.867421i	$-0.334224\pi$
$709$	1.33200e7	0.995150	0.497575	+	0.867421i	$0.334224\pi$
$710$	0	0
$711$	4.04863e6	0.300354
$712$	0	0
$713$	1.64192e7	1.20956
$714$	0	0
$715$	−488184.	−0.0357124
$716$	0	0
$717$	−2.21938e7	−1.61226
$718$	0	0
$719$	−5.94742e6	−0.429049	−0.214524	−	0.976719i	$-0.568820\pi$
$719$	−5.94742e6	−0.429049	−0.214524	+	0.976719i	$0.568820\pi$
$720$	0	0
$721$	3.46876e7	2.48506
$722$	0	0
$723$	2.31389e7	1.64626
$724$	0	0
$725$	−2.38733e7	−1.68682
$726$	0	0
$727$	7.06271e6	0.495605	0.247802	−	0.968811i	$-0.420292\pi$
$727$	7.06271e6	0.495605	0.247802	+	0.968811i	$0.420292\pi$
$728$	0	0
$729$	1.59610e7	1.11235
$730$	0	0
$731$	−2.10515e6	−0.145710
$732$	0	0
$733$	1.01810e7	0.699891	0.349945	−	0.936770i	$-0.386200\pi$
$733$	1.01810e7	0.699891	0.349945	+	0.936770i	$0.386200\pi$
$734$	0	0
$735$	−1.49069e6	−0.101782
$736$	0	0
$737$	2.58345e6	0.175199
$738$	0	0
$739$	8.90680e6	0.599944	0.299972	−	0.953948i	$-0.403023\pi$
$739$	8.90680e6	0.599944	0.299972	+	0.953948i	$0.403023\pi$
$740$	0	0
$741$	−2.44063e7	−1.63289
$742$	0	0
$743$	3.70751e6	0.246383	0.123191	−	0.992383i	$-0.460687\pi$
$743$	3.70751e6	0.246383	0.123191	+	0.992383i	$0.460687\pi$
$744$	0	0
$745$	1.79540e6	0.118514
$746$	0	0
$747$	2.89419e6	0.189769
$748$	0	0
$749$	1.52829e7	0.995408
$750$	0	0
$751$	2.30940e7	1.49417	0.747085	−	0.664729i	$-0.231453\pi$
$751$	2.30940e7	1.49417	0.747085	+	0.664729i	$0.231453\pi$
$752$	0	0
$753$	4.78941e6	0.307819
$754$	0	0
$755$	−642005.	−0.0409894
$756$	0	0
$757$	1.25262e7	0.794474	0.397237	−	0.917716i	$-0.369969\pi$
$757$	1.25262e7	0.794474	0.397237	+	0.917716i	$0.369969\pi$
$758$	0	0
$759$	3.24928e6	0.204731
$760$	0	0
$761$	1.66360e7	1.04133	0.520663	−	0.853762i	$-0.325685\pi$
$761$	1.66360e7	1.04133	0.520663	+	0.853762i	$0.325685\pi$
$762$	0	0
$763$	−2.51628e7	−1.56476
$764$	0	0
$765$	158801.	0.00981068
$766$	0	0
$767$	−2.63613e7	−1.61800
$768$	0	0
$769$	−4.17179e6	−0.254394	−0.127197	−	0.991877i	$-0.540598\pi$
$769$	−4.17179e6	−0.254394	−0.127197	+	0.991877i	$0.540598\pi$
$770$	0	0
$771$	2.23087e7	1.35157
$772$	0	0
$773$	6.18595e6	0.372355	0.186178	−	0.982516i	$-0.440390\pi$
$773$	6.18595e6	0.372355	0.186178	+	0.982516i	$0.440390\pi$
$774$	0	0
$775$	2.63974e7	1.57872
$776$	0	0
$777$	388674.	0.0230958
$778$	0	0
$779$	192703.	0.0113775
$780$	0	0
$781$	−6.01868e6	−0.353080
$782$	0	0
$783$	−3.12774e7	−1.82316
$784$	0	0
$785$	777821.	0.0450511
$786$	0	0
$787$	1.37144e7	0.789294	0.394647	−	0.918833i	$-0.370867\pi$
$787$	1.37144e7	0.789294	0.394647	+	0.918833i	$0.370867\pi$
$788$	0	0
$789$	1.10410e7	0.631415
$790$	0	0
$791$	2.99419e7	1.70153
$792$	0	0
$793$	−4.14922e7	−2.34306
$794$	0	0
$795$	−76539.4	−0.00429504
$796$	0	0
$797$	−4.09307e6	−0.228246	−0.114123	−	0.993467i	$-0.536406\pi$
$797$	−4.09307e6	−0.228246	−0.114123	+	0.993467i	$0.536406\pi$
$798$	0	0
$799$	−1.31075e7	−0.726362
$800$	0	0
$801$	737645.	0.0406225
$802$	0	0
$803$	−7.19217e6	−0.393615
$804$	0	0
$805$	−1.69577e6	−0.0922310
$806$	0	0
$807$	−2.21408e7	−1.19677
$808$	0	0
$809$	3.09632e6	0.166332	0.0831658	−	0.996536i	$-0.473497\pi$
$809$	3.09632e6	0.166332	0.0831658	+	0.996536i	$0.473497\pi$
$810$	0	0
$811$	−1.78715e7	−0.954133	−0.477066	−	0.878867i	$-0.658300\pi$
$811$	−1.78715e7	−0.954133	−0.477066	+	0.878867i	$0.658300\pi$
$812$	0	0
$813$	−1.35917e7	−0.721185
$814$	0	0
$815$	−1.80427e6	−0.0951499
$816$	0	0
$817$	−5.30250e6	−0.277924
$818$	0	0
$819$	−9.58162e6	−0.499148
$820$	0	0
$821$	2.63991e7	1.36689	0.683443	−	0.730004i	$-0.260482\pi$
$821$	2.63991e7	1.36689	0.683443	+	0.730004i	$0.260482\pi$
$822$	0	0
$823$	−4.12330e6	−0.212200	−0.106100	−	0.994355i	$-0.533836\pi$
$823$	−4.12330e6	−0.212200	−0.106100	+	0.994355i	$0.533836\pi$
$824$	0	0
$825$	5.22391e6	0.267215
$826$	0	0
$827$	−1.10990e7	−0.564314	−0.282157	−	0.959368i	$-0.591050\pi$
$827$	−1.10990e7	−0.564314	−0.282157	+	0.959368i	$0.591050\pi$
$828$	0	0
$829$	−7.60846e6	−0.384513	−0.192256	−	0.981345i	$-0.561580\pi$
$829$	−7.60846e6	−0.384513	−0.192256	+	0.981345i	$0.561580\pi$
$830$	0	0
$831$	3.44453e6	0.173032
$832$	0	0
$833$	1.85369e7	0.925601
$834$	0	0
$835$	−387691.	−0.0192429
$836$	0	0
$837$	3.45842e7	1.70633
$838$	0	0
$839$	2.64407e7	1.29678	0.648392	−	0.761306i	$-0.275442\pi$
$839$	2.64407e7	1.29678	0.648392	+	0.761306i	$0.275442\pi$
$840$	0	0
$841$	3.85447e7	1.87921
$842$	0	0
$843$	−3.56174e7	−1.72621
$844$	0	0
$845$	−2.19812e6	−0.105903
$846$	0	0
$847$	−2.99317e6	−0.143358
$848$	0	0
$849$	1.04371e7	0.496948
$850$	0	0
$851$	264345.	0.0125126
$852$	0	0
$853$	−2.20001e7	−1.03527	−0.517633	−	0.855602i	$-0.673187\pi$
$853$	−2.20001e7	−1.03527	−0.517633	+	0.855602i	$0.673187\pi$
$854$	0	0
$855$	399992.	0.0187127
$856$	0	0
$857$	1.08709e7	0.505605	0.252803	−	0.967518i	$-0.418648\pi$
$857$	1.08709e7	0.505605	0.252803	+	0.967518i	$0.418648\pi$
$858$	0	0
$859$	−1.63430e7	−0.755699	−0.377849	−	0.925867i	$-0.623336\pi$
$859$	−1.63430e7	−0.755699	−0.377849	+	0.925867i	$0.623336\pi$
$860$	0	0
$861$	−293002.	−0.0134699
$862$	0	0
$863$	2.31095e7	1.05624	0.528121	−	0.849169i	$-0.322897\pi$
$863$	2.31095e7	1.05624	0.528121	+	0.849169i	$0.322897\pi$
$864$	0	0
$865$	−1.69561e6	−0.0770522
$866$	0	0
$867$	−1.20841e7	−0.545968
$868$	0	0
$869$	9.82385e6	0.441298
$870$	0	0
$871$	2.00669e7	0.896264
$872$	0	0
$873$	−2.02980e6	−0.0901402
$874$	0	0
$875$	−5.46878e6	−0.241474
$876$	0	0
$877$	−4.30032e7	−1.88800	−0.944000	−	0.329944i	$-0.892970\pi$
$877$	−4.30032e7	−1.88800	−0.944000	+	0.329944i	$0.892970\pi$
$878$	0	0
$879$	−5.01989e6	−0.219140
$880$	0	0
$881$	−1.80461e7	−0.783326	−0.391663	−	0.920109i	$-0.628100\pi$
$881$	−1.80461e7	−0.783326	−0.391663	+	0.920109i	$0.628100\pi$
$882$	0	0
$883$	2.02867e6	0.0875607	0.0437803	−	0.999041i	$-0.486060\pi$
$883$	2.02867e6	0.0875607	0.0437803	+	0.999041i	$0.486060\pi$
$884$	0	0
$885$	−1.67325e6	−0.0718131
$886$	0	0
$887$	−1.67196e7	−0.713537	−0.356768	−	0.934193i	$-0.616121\pi$
$887$	−1.67196e7	−0.713537	−0.356768	+	0.934193i	$0.616121\pi$
$888$	0	0
$889$	−1.75386e7	−0.744286
$890$	0	0
$891$	5.37781e6	0.226940
$892$	0	0
$893$	−3.30156e7	−1.38545
$894$	0	0
$895$	2.23084e6	0.0930917
$896$	0	0
$897$	2.52388e7	1.04734
$898$	0	0
$899$	−6.52996e7	−2.69471
$900$	0	0
$901$	951774.	0.0390591
$902$	0	0
$903$	8.06236e6	0.329036
$904$	0	0
$905$	1.57084e6	0.0637543
$906$	0	0
$907$	1.13533e7	0.458250	0.229125	−	0.973397i	$-0.426414\pi$
$907$	1.13533e7	0.458250	0.229125	+	0.973397i	$0.426414\pi$
$908$	0	0
$909$	6.27331e6	0.251818
$910$	0	0
$911$	−2.72350e7	−1.08725	−0.543627	−	0.839327i	$-0.682949\pi$
$911$	−2.72350e7	−1.08725	−0.543627	+	0.839327i	$0.682949\pi$
$912$	0	0
$913$	7.02264e6	0.278820
$914$	0	0
$915$	−2.63367e6	−0.103994
$916$	0	0
$917$	−2.01555e7	−0.791535
$918$	0	0
$919$	1.65016e7	0.644522	0.322261	−	0.946651i	$-0.395557\pi$
$919$	1.65016e7	0.644522	0.322261	+	0.946651i	$0.395557\pi$
$920$	0	0
$921$	3.15135e7	1.22418
$922$	0	0
$923$	−4.67501e7	−1.80625
$924$	0	0
$925$	424989.	0.0163314
$926$	0	0
$927$	8.46107e6	0.323389
$928$	0	0
$929$	−1.68977e7	−0.642373	−0.321186	−	0.947016i	$-0.604082\pi$
$929$	−1.68977e7	−0.642373	−0.321186	+	0.947016i	$0.604082\pi$
$930$	0	0
$931$	4.66912e7	1.76547
$932$	0	0
$933$	2.05666e7	0.773497
$934$	0	0
$935$	385325.	0.0144144
$936$	0	0
$937$	1.25638e7	0.467490	0.233745	−	0.972298i	$-0.424902\pi$
$937$	1.25638e7	0.467490	0.233745	+	0.972298i	$0.424902\pi$
$938$	0	0
$939$	−2.09866e7	−0.776743
$940$	0	0
$941$	−2.78645e7	−1.02583	−0.512917	−	0.858438i	$-0.671435\pi$
$941$	−2.78645e7	−1.02583	−0.512917	+	0.858438i	$0.671435\pi$
$942$	0	0
$943$	−199276.	−0.00729753
$944$	0	0
$945$	−3.57184e6	−0.130110
$946$	0	0
$947$	1.30308e7	0.472169	0.236085	−	0.971732i	$-0.424136\pi$
$947$	1.30308e7	0.472169	0.236085	+	0.971732i	$0.424136\pi$
$948$	0	0
$949$	−5.58652e7	−2.01361
$950$	0	0
$951$	−8.11451e6	−0.290945
$952$	0	0
$953$	1.15678e7	0.412589	0.206295	−	0.978490i	$-0.433859\pi$
$953$	1.15678e7	0.412589	0.206295	+	0.978490i	$0.433859\pi$
$954$	0	0
$955$	−690315.	−0.0244928
$956$	0	0
$957$	−1.29225e7	−0.456106
$958$	0	0
$959$	−2.86405e7	−1.00562
$960$	0	0
$961$	4.35743e7	1.52203
$962$	0	0
$963$	3.72783e6	0.129536
$964$	0	0
$965$	−535294.	−0.0185044
$966$	0	0
$967$	1.48070e7	0.509214	0.254607	−	0.967045i	$-0.418054\pi$
$967$	1.48070e7	0.509214	0.254607	+	0.967045i	$0.418054\pi$
$968$	0	0
$969$	1.92639e7	0.659077
$970$	0	0
$971$	−4.54956e7	−1.54854	−0.774269	−	0.632857i	$-0.781882\pi$
$971$	−4.54956e7	−1.54854	−0.774269	+	0.632857i	$0.781882\pi$
$972$	0	0
$973$	3.69437e7	1.25100
$974$	0	0
$975$	4.05767e7	1.36699
$976$	0	0
$977$	−3.59948e7	−1.20643	−0.603216	−	0.797578i	$-0.706114\pi$
$977$	−3.59948e7	−1.20643	−0.603216	+	0.797578i	$0.706114\pi$
$978$	0	0
$979$	1.78987e6	0.0596850
$980$	0	0
$981$	−6.13776e6	−0.203628
$982$	0	0
$983$	−4.41236e7	−1.45642	−0.728210	−	0.685354i	$-0.759648\pi$
$983$	−4.41236e7	−1.45642	−0.728210	+	0.685354i	$0.759648\pi$
$984$	0	0
$985$	3.36841e6	0.110620
$986$	0	0
$987$	5.01996e7	1.64024
$988$	0	0
$989$	5.48336e6	0.178261
$990$	0	0
$991$	−3.80658e7	−1.23126	−0.615631	−	0.788034i	$-0.711099\pi$
$991$	−3.80658e7	−1.23126	−0.615631	+	0.788034i	$0.711099\pi$
$992$	0	0
$993$	7.42151e6	0.238847
$994$	0	0
$995$	2.50972e6	0.0803653
$996$	0	0
$997$	−1.03251e6	−0.0328971	−0.0164485	−	0.999865i	$-0.505236\pi$
$997$	−1.03251e6	−0.0328971	−0.0164485	+	0.999865i	$0.505236\pi$
$998$	0	0
$999$	556795.	0.0176515

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	88.6.a.c.1.4	✓	4
3.2	odd	2		792.6.a.l.1.3		4
4.3	odd	2		176.6.a.l.1.1		4
8.3	odd	2		704.6.a.u.1.4		4
8.5	even	2		704.6.a.x.1.1		4
11.10	odd	2		968.6.a.d.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
88.6.a.c.1.4	✓	4	1.1	even	1	trivial
176.6.a.l.1.1		4	4.3	odd	2
704.6.a.u.1.4		4	8.3	odd	2
704.6.a.x.1.1		4	8.5	even	2
792.6.a.l.1.3		4	3.2	odd	2
968.6.a.d.1.4		4	11.10	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:	\( N \)	\(=\)	\( 88 = 2^{3} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	88.a (trivial)

\(f(q)\)	\(=\)	\(q+13.8972 q^{3} -4.29272 q^{5} -204.438 q^{7} -49.8668 q^{9} -121.000 q^{11} -939.867 q^{13} -59.6569 q^{15} +741.838 q^{17} +1868.56 q^{19} -2841.12 q^{21} -1932.30 q^{23} -3106.57 q^{25} -4070.04 q^{27} +7684.78 q^{29} -8497.26 q^{31} -1681.57 q^{33} +877.593 q^{35} -136.803 q^{37} -13061.6 q^{39} +103.129 q^{41} -2837.74 q^{43} +214.064 q^{45} -17669.0 q^{47} +24987.8 q^{49} +10309.5 q^{51} +1282.99 q^{53} +519.419 q^{55} +25967.9 q^{57} +28047.9 q^{59} +44146.9 q^{61} +10194.6 q^{63} +4034.58 q^{65} -21350.8 q^{67} -26853.6 q^{69} +49741.1 q^{71} +59439.4 q^{73} -43172.8 q^{75} +24737.0 q^{77} -81188.9 q^{79} -44444.7 q^{81} -58038.4 q^{83} -3184.50 q^{85} +106797. q^{87} -14792.3 q^{89} +192144. q^{91} -118088. q^{93} -8021.21 q^{95} +40704.5 q^{97} +6033.88 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 13 q^{3} - 19 q^{5} + 58 q^{7} + 191 q^{9} - 484 q^{11} - 1266 q^{13} - 2173 q^{15} - 504 q^{17} - 4016 q^{19} - 2902 q^{21} - 7837 q^{23} - 1845 q^{25} - 16531 q^{27} - 4114 q^{29} - 6989 q^{31}+ \cdots - 23111 q^{99}+O(q^{100}) \) 4 * q - 13 * q^3 - 19 * q^5 + 58 * q^7 + 191 * q^9 - 484 * q^11 - 1266 * q^13 - 2173 * q^15 - 504 * q^17 - 4016 * q^19 - 2902 * q^21 - 7837 * q^23 - 1845 * q^25 - 16531 * q^27 - 4114 * q^29 - 6989 * q^31 + 1573 * q^33 - 18886 * q^35 - 1997 * q^37 - 6528 * q^39 + 1278 * q^41 - 1354 * q^43 + 35048 * q^45 - 29152 * q^47 + 40464 * q^49 - 29370 * q^51 + 46060 * q^53 + 2299 * q^55 + 80276 * q^57 + 13569 * q^59 + 15778 * q^61 + 27956 * q^63 + 83964 * q^65 - 11395 * q^67 + 108327 * q^69 - 16615 * q^71 + 78546 * q^73 + 48352 * q^75 - 7018 * q^77 + 25958 * q^79 + 197252 * q^81 - 43882 * q^83 + 75834 * q^85 + 42816 * q^87 - 31785 * q^89 - 105024 * q^91 - 269505 * q^93 - 43924 * q^95 - 17011 * q^97 - 23111 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more