LMFDB - Embedded newform 2240.2.h.e.671.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2240,2,Mod(671,2240)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2240, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1, 0, 1]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("2240.671");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2240 = 2^{6} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2240.h (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$17.8864900528$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			671.3
Character	$\chi$	$=$	2240.671
Dual form			2240.2.h.e.671.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.237691i q^{3} -1.00000 q^{5} +(-2.63997 - 0.174795i) q^{7} +2.94350 q^{9} +O(q^{10})$	$q-0.237691i q^{3} -1.00000 q^{5} +(-2.63997 - 0.174795i) q^{7} +2.94350 q^{9} -4.69677 q^{11} +2.77882 q^{13} +0.237691i q^{15} +3.55805i q^{17} +6.72232i q^{19} +(-0.0415472 + 0.627498i) q^{21} -8.45582i q^{23} +1.00000 q^{25} -1.41272i q^{27} -5.80792i q^{29} +6.92430 q^{31} +1.11638i q^{33} +(2.63997 + 0.174795i) q^{35} -0.171482i q^{37} -0.660500i q^{39} -5.71397i q^{41} +3.26264 q^{43} -2.94350 q^{45} -2.83304 q^{47} +(6.93889 + 0.922907i) q^{49} +0.845718 q^{51} -7.09578i q^{53} +4.69677 q^{55} +1.59784 q^{57} -4.79554i q^{59} -9.63085 q^{61} +(-7.77076 - 0.514509i) q^{63} -2.77882 q^{65} -9.99904 q^{67} -2.00988 q^{69} -7.49472i q^{71} -8.35078i q^{73} -0.237691i q^{75} +(12.3993 + 0.820971i) q^{77} -5.05279i q^{79} +8.49472 q^{81} -9.53742i q^{83} -3.55805i q^{85} -1.38049 q^{87} +5.75795i q^{89} +(-7.33599 - 0.485723i) q^{91} -1.64585i q^{93} -6.72232i q^{95} -12.7637i q^{97} -13.8249 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q - 24 q^{5} - 36 q^{9}+O(q^{10}) $ 24 * q - 24 * q^5 - 36 * q^9	$ 24 q - 24 q^{5} - 36 q^{9} - 4 q^{13} - 8 q^{21} + 24 q^{25} + 36 q^{45} + 24 q^{57} - 56 q^{61} + 4 q^{65} - 96 q^{69} + 12 q^{77} + 24 q^{81}+O(q^{100}) $ 24 * q - 24 * q^5 - 36 * q^9 - 4 * q^13 - 8 * q^21 + 24 * q^25 + 36 * q^45 + 24 * q^57 - 56 * q^61 + 4 * q^65 - 96 * q^69 + 12 * q^77 + 24 * q^81

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2240\mathbb{Z}\right)^\times$.

$n$	$897$	$1471$	$1541$	$1921$
$\chi(n)$	$1$	$-1$	$-1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$		−	0.237691i		−	0.137231i	−0.997643	−	0.0686156i	$-0.978142\pi$
$3$		−	0.237691i		−	0.137231i	0.997643	−	0.0686156i	$-0.0218582\pi$
$4$		0			0
$5$	−1.00000			−0.447214
$5$	−1.00000			−0.447214
$6$		0			0
$7$	−2.63997	−	0.174795i	−0.997815	−	0.0660663i
$7$	−2.63997	−	0.174795i	−0.997815	−	0.0660663i
$8$		0			0
$9$	2.94350			0.981168
$10$		0			0
$11$	−4.69677			−1.41613			−0.708064	−	0.706148i	$-0.750431\pi$
$11$	−4.69677			−1.41613			−0.708064	+	0.706148i	$0.750431\pi$
$12$		0			0
$13$	2.77882			0.770705			0.385352	−	0.922769i	$-0.374080\pi$
$13$	2.77882			0.770705			0.385352	+	0.922769i	$0.374080\pi$
$14$		0			0
$15$			0.237691i			0.0613716i
$16$		0			0
$17$			3.55805i			0.862955i	0.902124	+	0.431477i	$0.142008\pi$
$17$			3.55805i			0.862955i	−0.902124	+	0.431477i	$0.857992\pi$
$18$		0			0
$19$			6.72232i			1.54221i	0.636711	+	0.771103i	$0.280295\pi$
$19$			6.72232i			1.54221i	−0.636711	+	0.771103i	$0.719705\pi$
$20$		0			0
$21$	−0.0415472	+	0.627498i	−0.00906635	+	0.136931i
$22$		0			0
$23$		−	8.45582i		−	1.76316i	−0.472034	−	0.881581i	$-0.656480\pi$
$23$		−	8.45582i		−	1.76316i	0.472034	−	0.881581i	$-0.343520\pi$
$24$		0			0
$25$	1.00000			0.200000
$26$		0			0
$27$		−	1.41272i		−	0.271878i
$28$		0			0
$29$		−	5.80792i		−	1.07850i	−0.842145	−	0.539252i	$-0.818707\pi$
$29$		−	5.80792i		−	1.07850i	0.842145	−	0.539252i	$-0.181293\pi$
$30$		0			0
$31$	6.92430			1.24364			0.621820	−	0.783160i	$-0.286393\pi$
$31$	6.92430			1.24364			0.621820	+	0.783160i	$0.286393\pi$
$32$		0			0
$33$			1.11638i			0.194337i
$34$		0			0
$35$	2.63997	+	0.174795i	0.446237	+	0.0295457i
$36$		0			0
$37$		−	0.171482i		−	0.0281915i	−0.999901	−	0.0140958i	$-0.995513\pi$
$37$		−	0.171482i		−	0.0281915i	0.999901	−	0.0140958i	$-0.00448697\pi$
$38$		0			0
$39$		−	0.660500i		−	0.105765i
$40$		0			0
$41$		−	5.71397i		−	0.892372i	−0.894940	−	0.446186i	$-0.852782\pi$
$41$		−	5.71397i		−	0.892372i	0.894940	−	0.446186i	$-0.147218\pi$
$42$		0			0
$43$	3.26264			0.497549			0.248774	−	0.968561i	$-0.419972\pi$
$43$	3.26264			0.497549			0.248774	+	0.968561i	$0.419972\pi$
$44$		0			0
$45$	−2.94350			−0.438791
$46$		0			0
$47$	−2.83304			−0.413242			−0.206621	−	0.978421i	$-0.566247\pi$
$47$	−2.83304			−0.413242			−0.206621	+	0.978421i	$0.566247\pi$
$48$		0			0
$49$	6.93889	+	0.922907i	0.991270	+	0.131844i
$50$		0			0
$51$	0.845718			0.118424
$52$		0			0
$53$		−	7.09578i		−	0.974681i	−0.873212	−	0.487340i	$-0.837967\pi$
$53$		−	7.09578i		−	0.974681i	0.873212	−	0.487340i	$-0.162033\pi$
$54$		0			0
$55$	4.69677			0.633312
$56$		0			0
$57$	1.59784			0.211639
$58$		0			0
$59$		−	4.79554i		−	0.624326i	−0.950029	−	0.312163i	$-0.898947\pi$
$59$		−	4.79554i		−	0.624326i	0.950029	−	0.312163i	$-0.101053\pi$
$60$		0			0
$61$	−9.63085			−1.23310			−0.616552	−	0.787314i	$-0.711471\pi$
$61$	−9.63085			−1.23310			−0.616552	+	0.787314i	$0.711471\pi$
$62$		0			0
$63$	−7.77076	−	0.514509i	−0.979024	−	0.0648221i
$64$		0			0
$65$	−2.77882			−0.344670
$66$		0			0
$67$	−9.99904			−1.22158			−0.610789	−	0.791794i	$-0.709147\pi$
$67$	−9.99904			−1.22158			−0.610789	+	0.791794i	$0.709147\pi$
$68$		0			0
$69$	−2.00988			−0.241961
$70$		0			0
$71$		−	7.49472i		−	0.889459i	−0.895665	−	0.444730i	$-0.853300\pi$
$71$		−	7.49472i		−	0.889459i	0.895665	−	0.444730i	$-0.146700\pi$
$72$		0			0
$73$		−	8.35078i		−	0.977385i	−0.872456	−	0.488692i	$-0.837474\pi$
$73$		−	8.35078i		−	0.977385i	0.872456	−	0.488692i	$-0.162526\pi$
$74$		0			0
$75$		−	0.237691i		−	0.0274462i
$76$		0			0
$77$	12.3993	+	0.820971i	1.41303	+	0.0935583i
$78$		0			0
$79$		−	5.05279i		−	0.568483i	−0.958753	−	0.284242i	$-0.908258\pi$
$79$		−	5.05279i		−	0.568483i	0.958753	−	0.284242i	$-0.0917418\pi$
$80$		0			0
$81$	8.49472			0.943858
$82$		0			0
$83$		−	9.53742i		−	1.04687i	−0.852066	−	0.523434i	$-0.824651\pi$
$83$		−	9.53742i		−	1.04687i	0.852066	−	0.523434i	$-0.175349\pi$
$84$		0			0
$85$		−	3.55805i		−	0.385925i
$86$		0			0
$87$	−1.38049			−0.148004
$88$		0			0
$89$			5.75795i			0.610341i	0.952298	+	0.305171i	$0.0987135\pi$
$89$			5.75795i			0.610341i	−0.952298	+	0.305171i	$0.901286\pi$
$90$		0			0
$91$	−7.33599	−	0.485723i	−0.769021	−	0.0509176i
$92$		0			0
$93$		−	1.64585i		−	0.170666i
$94$		0			0
$95$		−	6.72232i		−	0.689695i
$96$		0			0
$97$		−	12.7637i		−	1.29596i	−0.761659	−	0.647978i	$-0.775615\pi$
$97$		−	12.7637i		−	1.29596i	0.761659	−	0.647978i	$-0.224385\pi$
$98$		0			0
$99$	−13.8249			−1.38946
$100$		0			0
$101$	2.61152			0.259856			0.129928	−	0.991523i	$-0.458525\pi$
$101$	2.61152			0.259856			0.129928	+	0.991523i	$0.458525\pi$
$102$		0			0
$103$	7.95941			0.784264			0.392132	−	0.919909i	$-0.371738\pi$
$103$	7.95941			0.784264			0.392132	+	0.919909i	$0.371738\pi$
$104$		0			0
$105$	0.0415472	−	0.627498i	0.00405460	−	0.0612375i
$106$		0			0
$107$	0.217879			0.0210632			0.0105316	−	0.999945i	$-0.496648\pi$
$107$	0.217879			0.0210632			0.0105316	+	0.999945i	$0.496648\pi$
$108$		0			0
$109$			13.9465i			1.33583i	0.744238	+	0.667914i	$0.232813\pi$
$109$			13.9465i			1.33583i	−0.744238	+	0.667914i	$0.767187\pi$
$110$		0			0
$111$	−0.0407599			−0.00386876
$112$		0			0
$113$	14.6924			1.38215			0.691073	−	0.722785i	$-0.257138\pi$
$113$	14.6924			1.38215			0.691073	+	0.722785i	$0.257138\pi$
$114$		0			0
$115$			8.45582i			0.788510i
$116$		0			0
$117$	8.17945			0.756191
$118$		0			0
$119$	0.621930	−	9.39316i	0.0570122	−	0.861070i
$120$		0			0
$121$	11.0596			1.00542
$122$		0			0
$123$	−1.35816			−0.122461
$124$		0			0
$125$	−1.00000			−0.0894427
$126$		0			0
$127$		−	12.6197i		−	1.11981i	−0.828555	−	0.559907i	$-0.810837\pi$
$127$		−	12.6197i		−	1.11981i	0.828555	−	0.559907i	$-0.189163\pi$
$128$		0			0
$129$		−	0.775502i		−	0.0682792i
$130$		0			0
$131$			8.57631i			0.749316i	0.927163	+	0.374658i	$0.122240\pi$
$131$			8.57631i			0.749316i	−0.927163	+	0.374658i	$0.877760\pi$
$132$		0			0
$133$	1.17503	−	17.7467i	0.101888	−	1.53884i
$134$		0			0
$135$			1.41272i			0.121587i
$136$		0			0
$137$	21.2941			1.81927			0.909637	−	0.415404i	$-0.136360\pi$
$137$	21.2941			1.81927			0.909637	+	0.415404i	$0.136360\pi$
$138$		0			0
$139$		−	11.6407i		−	0.987354i	−0.869645	−	0.493677i	$-0.835653\pi$
$139$		−	11.6407i		−	0.987354i	0.869645	−	0.493677i	$-0.164347\pi$
$140$		0			0
$141$			0.673389i			0.0567096i
$142$		0			0
$143$	−13.0514			−1.09142
$144$		0			0
$145$			5.80792i			0.482321i
$146$		0			0
$147$	0.219367	−	1.64931i	0.0180931	−	0.136033i
$148$		0			0
$149$		−	17.9197i		−	1.46804i	−0.679129	−	0.734019i	$-0.737642\pi$
$149$		−	17.9197i		−	1.46804i	0.679129	−	0.734019i	$-0.262358\pi$
$150$		0			0
$151$			6.13504i			0.499262i	0.968341	+	0.249631i	$0.0803094\pi$
$151$			6.13504i			0.499262i	−0.968341	+	0.249631i	$0.919691\pi$
$152$		0			0
$153$			10.4731i			0.846703i
$154$		0			0
$155$	−6.92430			−0.556173
$156$		0			0
$157$	−3.33383			−0.266069			−0.133034	−	0.991111i	$-0.542472\pi$
$157$	−3.33383			−0.266069			−0.133034	+	0.991111i	$0.542472\pi$
$158$		0			0
$159$	−1.68661			−0.133757
$160$		0			0
$161$	−1.47804	+	22.3231i	−0.116486	+	1.75931i
$162$		0			0
$163$	−2.24745			−0.176034			−0.0880169	−	0.996119i	$-0.528053\pi$
$163$	−2.24745			−0.176034			−0.0880169	+	0.996119i	$0.528053\pi$
$164$		0			0
$165$		−	1.11638i		−	0.0869101i
$166$		0			0
$167$	−2.19756			−0.170052			−0.0850261	−	0.996379i	$-0.527097\pi$
$167$	−2.19756			−0.170052			−0.0850261	+	0.996379i	$0.527097\pi$
$168$		0			0
$169$	−5.27818			−0.406014
$170$		0			0
$171$			19.7872i			1.51316i
$172$		0			0
$173$	−3.25420			−0.247412			−0.123706	−	0.992319i	$-0.539478\pi$
$173$	−3.25420			−0.247412			−0.123706	+	0.992319i	$0.539478\pi$
$174$		0			0
$175$	−2.63997	−	0.174795i	−0.199563	−	0.0132133i
$176$		0			0
$177$	−1.13986			−0.0856769
$178$		0			0
$179$	15.1239			1.13041			0.565207	−	0.824949i	$-0.308796\pi$
$179$	15.1239			1.13041			0.565207	+	0.824949i	$0.308796\pi$
$180$		0			0
$181$	−20.9063			−1.55396			−0.776978	−	0.629528i	$-0.783248\pi$
$181$	−20.9063			−1.55396			−0.776978	+	0.629528i	$0.783248\pi$
$182$		0			0
$183$			2.28917i			0.169220i
$184$		0			0
$185$			0.171482i			0.0126076i
$186$		0			0
$187$		−	16.7113i		−	1.22205i
$188$		0			0
$189$	−0.246936	+	3.72954i	−0.0179620	+	0.271284i
$190$		0			0
$191$		−	5.75197i		−	0.416198i	−0.978108	−	0.208099i	$-0.933272\pi$
$191$		−	5.75197i		−	0.416198i	0.978108	−	0.208099i	$-0.0667276\pi$
$192$		0			0
$193$	−13.9932			−1.00726			−0.503628	−	0.863921i	$-0.668002\pi$
$193$	−13.9932			−1.00726			−0.503628	+	0.863921i	$0.668002\pi$
$194$		0			0
$195$			0.660500i			0.0472994i
$196$		0			0
$197$		−	12.1902i		−	0.868518i	−0.900788	−	0.434259i	$-0.857010\pi$
$197$		−	12.1902i		−	0.868518i	0.900788	−	0.434259i	$-0.142990\pi$
$198$		0			0
$199$	−17.9685			−1.27375			−0.636877	−	0.770965i	$-0.719774\pi$
$199$	−17.9685			−1.27375			−0.636877	+	0.770965i	$0.719774\pi$
$200$		0			0
$201$			2.37668i			0.167638i
$202$		0			0
$203$	−1.01520	+	15.3327i	−0.0712527	+	1.07615i
$204$		0			0
$205$			5.71397i			0.399081i
$206$		0			0
$207$		−	24.8897i		−	1.72996i
$208$		0			0
$209$		−	31.5732i		−	2.18396i
$210$		0			0
$211$	11.8169			0.813508			0.406754	−	0.913538i	$-0.366661\pi$
$211$	11.8169			0.813508			0.406754	+	0.913538i	$0.366661\pi$
$212$		0			0
$213$	−1.78143			−0.122062
$214$		0			0
$215$	−3.26264			−0.222511
$216$		0			0
$217$	−18.2800	−	1.21033i	−1.24092	−	0.0821627i
$218$		0			0
$219$	−1.98491			−0.134128
$220$		0			0
$221$			9.88718i			0.665083i
$222$		0			0
$223$	26.9064			1.80178			0.900892	−	0.434044i	$-0.142914\pi$
$223$	26.9064			1.80178			0.900892	+	0.434044i	$0.142914\pi$
$224$		0			0
$225$	2.94350			0.196234
$226$		0			0
$227$		−	19.4994i		−	1.29422i	−0.762397	−	0.647110i	$-0.775977\pi$
$227$		−	19.4994i		−	1.29422i	0.762397	−	0.647110i	$-0.224023\pi$
$228$		0			0
$229$	16.0562			1.06102			0.530510	−	0.847679i	$-0.322000\pi$
$229$	16.0562			1.06102			0.530510	+	0.847679i	$0.322000\pi$
$230$		0			0
$231$	0.195138	−	2.94721i	0.0128391	−	0.193912i
$232$		0			0
$233$	5.18783			0.339866			0.169933	−	0.985456i	$-0.445645\pi$
$233$	5.18783			0.339866			0.169933	+	0.985456i	$0.445645\pi$
$234$		0			0
$235$	2.83304			0.184807
$236$		0			0
$237$	−1.20100			−0.0780136
$238$		0			0
$239$			17.3426i			1.12180i	0.827883	+	0.560900i	$0.189545\pi$
$239$			17.3426i			1.12180i	−0.827883	+	0.560900i	$0.810455\pi$
$240$		0			0
$241$			26.3029i			1.69432i	0.531341	+	0.847158i	$0.321688\pi$
$241$			26.3029i			1.69432i	−0.531341	+	0.847158i	$0.678312\pi$
$242$		0			0
$243$		−	6.25728i		−	0.401404i
$244$		0			0
$245$	−6.93889	−	0.922907i	−0.443310	−	0.0589624i
$246$		0			0
$247$			18.6801i			1.18859i
$248$		0			0
$249$	−2.26696			−0.143663
$250$		0			0
$251$			31.3756i			1.98041i	0.139616	+	0.990206i	$0.455413\pi$
$251$			31.3756i			1.98041i	−0.139616	+	0.990206i	$0.544587\pi$
$252$		0			0
$253$			39.7150i			2.49686i
$254$		0			0
$255$	−0.845718			−0.0529609
$256$		0			0
$257$		−	25.4959i		−	1.59039i	−0.606353	−	0.795195i	$-0.707368\pi$
$257$		−	25.4959i		−	1.59039i	0.606353	−	0.795195i	$-0.292632\pi$
$258$		0			0
$259$	−0.0299743	+	0.452709i	−0.00186251	+	0.0281300i
$260$		0			0
$261$		−	17.0956i		−	1.05819i
$262$		0			0
$263$			4.84565i			0.298796i	0.988777	+	0.149398i	$0.0477335\pi$
$263$			4.84565i			0.298796i	−0.988777	+	0.149398i	$0.952266\pi$
$264$		0			0
$265$			7.09578i			0.435890i
$266$		0			0
$267$	1.36861			0.0837578
$268$		0			0
$269$	25.4301			1.55050			0.775251	−	0.631654i	$-0.217624\pi$
$269$	25.4301			1.55050			0.775251	+	0.631654i	$0.217624\pi$
$270$		0			0
$271$	7.31193			0.444168			0.222084	−	0.975028i	$-0.428714\pi$
$271$	7.31193			0.444168			0.222084	+	0.975028i	$0.428714\pi$
$272$		0			0
$273$	−0.115452	+	1.74370i	−0.00698748	+	0.105534i
$274$		0			0
$275$	−4.69677			−0.283226
$276$		0			0
$277$		−	13.4332i		−	0.807121i	−0.914953	−	0.403560i	$-0.867773\pi$
$277$		−	13.4332i		−	0.807121i	0.914953	−	0.403560i	$-0.132227\pi$
$278$		0			0
$279$	20.3817			1.22022
$280$		0			0
$281$	−10.5575			−0.629806			−0.314903	−	0.949124i	$-0.601972\pi$
$281$	−10.5575			−0.629806			−0.314903	+	0.949124i	$0.601972\pi$
$282$		0			0
$283$			8.29530i			0.493104i	0.969130	+	0.246552i	$0.0792977\pi$
$283$			8.29530i			0.493104i	−0.969130	+	0.246552i	$0.920702\pi$
$284$		0			0
$285$	−1.59784			−0.0946477
$286$		0			0
$287$	−0.998773	+	15.0847i	−0.0589557	+	0.890422i
$288$		0			0
$289$	4.34025			0.255309
$290$		0			0
$291$	−3.03382			−0.177845
$292$		0			0
$293$	−0.858796			−0.0501714			−0.0250857	−	0.999685i	$-0.507986\pi$
$293$	−0.858796			−0.0501714			−0.0250857	+	0.999685i	$0.507986\pi$
$294$		0			0
$295$			4.79554i			0.279207i
$296$		0			0
$297$			6.63521i			0.385014i
$298$		0			0
$299$		−	23.4972i		−	1.35888i
$300$		0			0
$301$	−8.61329	−	0.570294i	−0.496462	−	0.0328712i
$302$		0			0
$303$		−	0.620734i		−	0.0356603i
$304$		0			0
$305$	9.63085			0.551461
$306$		0			0
$307$		−	22.1449i		−	1.26387i	−0.775019	−	0.631937i	$-0.782260\pi$
$307$		−	22.1449i		−	1.26387i	0.775019	−	0.631937i	$-0.217740\pi$
$308$		0			0
$309$		−	1.89188i		−	0.107625i
$310$		0			0
$311$	4.87944			0.276688			0.138344	−	0.990384i	$-0.455822\pi$
$311$	4.87944			0.276688			0.138344	+	0.990384i	$0.455822\pi$
$312$		0			0
$313$		−	27.6236i		−	1.56138i	−0.624921	−	0.780688i	$-0.714869\pi$
$313$		−	27.6236i		−	1.56138i	0.624921	−	0.780688i	$-0.285131\pi$
$314$		0			0
$315$	7.77076	+	0.514509i	0.437833	+	0.0289893i
$316$		0			0
$317$			29.9581i			1.68261i	0.540558	+	0.841307i	$0.318213\pi$
$317$			29.9581i			1.68261i	−0.540558	+	0.841307i	$0.681787\pi$
$318$		0			0
$319$			27.2784i			1.52730i
$320$		0			0
$321$		−	0.0517880i		−	0.00289052i
$322$		0			0
$323$	−23.9184			−1.33085
$324$		0			0
$325$	2.77882			0.154141
$326$		0			0
$327$	3.31495			0.183317
$328$		0			0
$329$	7.47915	+	0.495201i	0.412339	+	0.0273013i
$330$		0			0
$331$	18.4351			1.01328			0.506642	−	0.862157i	$-0.330887\pi$
$331$	18.4351			1.01328			0.506642	+	0.862157i	$0.330887\pi$
$332$		0			0
$333$		−	0.504759i		−	0.0276606i
$334$		0			0
$335$	9.99904			0.546306
$336$		0			0
$337$	−10.4810			−0.570939			−0.285469	−	0.958388i	$-0.592149\pi$
$337$	−10.4810			−0.570939			−0.285469	+	0.958388i	$0.592149\pi$
$338$		0			0
$339$		−	3.49226i		−	0.189674i
$340$		0			0
$341$	−32.5218			−1.76116
$342$		0			0
$343$	−18.1572	−	3.64933i	−0.980394	−	0.197045i
$344$		0			0
$345$	2.00988			0.108208
$346$		0			0
$347$	−15.9120			−0.854203			−0.427102	−	0.904204i	$-0.640465\pi$
$347$	−15.9120			−0.854203			−0.427102	+	0.904204i	$0.640465\pi$
$348$		0			0
$349$	−8.26239			−0.442276			−0.221138	−	0.975243i	$-0.570977\pi$
$349$	−8.26239			−0.442276			−0.221138	+	0.975243i	$0.570977\pi$
$350$		0			0
$351$		−	3.92569i		−	0.209538i
$352$		0			0
$353$			26.9601i			1.43494i	0.696589	+	0.717470i	$0.254700\pi$
$353$			26.9601i			1.43494i	−0.696589	+	0.717470i	$0.745300\pi$
$354$		0			0
$355$			7.49472i			0.397778i
$356$		0			0
$357$	−2.23267	−	0.147827i	−0.118166	−	0.00782385i
$358$		0			0
$359$		−	17.8000i		−	0.939448i	−0.882813	−	0.469724i	$-0.844353\pi$
$359$		−	17.8000i		−	0.939448i	0.882813	−	0.469724i	$-0.155647\pi$
$360$		0			0
$361$	−26.1896			−1.37840
$362$		0			0
$363$		−	2.62877i		−	0.137975i
$364$		0			0
$365$			8.35078i			0.437100i
$366$		0			0
$367$	−28.2711			−1.47574			−0.737869	−	0.674944i	$-0.764168\pi$
$367$	−28.2711			−1.47574			−0.737869	+	0.674944i	$0.764168\pi$
$368$		0			0
$369$		−	16.8191i		−	0.875566i
$370$		0			0
$371$	−1.24031	+	18.7327i	−0.0643935	+	0.972551i
$372$		0			0
$373$			11.6874i			0.605150i	0.953126	+	0.302575i	$0.0978462\pi$
$373$			11.6874i			0.605150i	−0.953126	+	0.302575i	$0.902154\pi$
$374$		0			0
$375$			0.237691i			0.0122743i
$376$		0			0
$377$		−	16.1391i		−	0.831208i
$378$		0			0
$379$	8.53868			0.438603			0.219301	−	0.975657i	$-0.429622\pi$
$379$	8.53868			0.438603			0.219301	+	0.975657i	$0.429622\pi$
$380$		0			0
$381$	−2.99958			−0.153673
$382$		0			0
$383$	6.77864			0.346372			0.173186	−	0.984889i	$-0.444594\pi$
$383$	6.77864			0.346372			0.173186	+	0.984889i	$0.444594\pi$
$384$		0			0
$385$	−12.3993	−	0.820971i	−0.631928	−	0.0418406i
$386$		0			0
$387$	9.60360			0.488179
$388$		0			0
$389$		−	4.94055i		−	0.250496i	−0.992125	−	0.125248i	$-0.960027\pi$
$389$		−	4.94055i		−	0.250496i	0.992125	−	0.125248i	$-0.0399726\pi$
$390$		0			0
$391$	30.0863			1.52153
$392$		0			0
$393$	2.03851			0.102829
$394$		0			0
$395$			5.05279i			0.254233i
$396$		0			0
$397$	31.5497			1.58343			0.791717	−	0.610889i	$-0.209188\pi$
$397$	31.5497			1.58343			0.791717	+	0.610889i	$0.209188\pi$
$398$		0			0
$399$	−4.21824	−	0.279294i	−0.211176	−	0.0139822i
$400$		0			0
$401$	−26.1468			−1.30571			−0.652853	−	0.757484i	$-0.726428\pi$
$401$	−26.1468			−1.30571			−0.652853	+	0.757484i	$0.726428\pi$
$402$		0			0
$403$	19.2414			0.958480
$404$		0			0
$405$	−8.49472			−0.422106
$406$		0			0
$407$			0.805413i			0.0399228i
$408$		0			0
$409$			28.8672i			1.42739i	0.700456	+	0.713696i	$0.252980\pi$
$409$			28.8672i			1.42739i	−0.700456	+	0.713696i	$0.747020\pi$
$410$		0			0
$411$		−	5.06141i		−	0.249661i
$412$		0			0
$413$	−0.838236	+	12.6601i	−0.0412469	+	0.622962i
$414$		0			0
$415$			9.53742i			0.468173i
$416$		0			0
$417$	−2.76690			−0.135496
$418$		0			0
$419$		−	18.1741i		−	0.887865i	−0.896060	−	0.443933i	$-0.853583\pi$
$419$		−	18.1741i		−	0.887865i	0.896060	−	0.443933i	$-0.146417\pi$
$420$		0			0
$421$		−	20.8669i		−	1.01699i	−0.861065	−	0.508494i	$-0.830202\pi$
$421$		−	20.8669i		−	1.01699i	0.861065	−	0.508494i	$-0.169798\pi$
$422$		0			0
$423$	−8.33906			−0.405459
$424$		0			0
$425$			3.55805i			0.172591i
$426$		0			0
$427$	25.4252	+	1.68342i	1.23041	+	0.0814666i
$428$		0			0
$429$			3.10222i			0.149776i
$430$		0			0
$431$			8.24061i			0.396936i	0.980107	+	0.198468i	$0.0635967\pi$
$431$			8.24061i			0.396936i	−0.980107	+	0.198468i	$0.936403\pi$
$432$		0			0
$433$		−	17.1319i		−	0.823308i	−0.911340	−	0.411654i	$-0.864951\pi$
$433$		−	17.1319i		−	0.823308i	0.911340	−	0.411654i	$-0.135049\pi$
$434$		0			0
$435$	1.38049			0.0661895
$436$		0			0
$437$	56.8427			2.71916
$438$		0			0
$439$	−24.4443			−1.16666			−0.583331	−	0.812234i	$-0.698251\pi$
$439$	−24.4443			−1.16666			−0.583331	+	0.812234i	$0.698251\pi$
$440$		0			0
$441$	20.4247	+	2.71658i	0.972603	+	0.129361i
$442$		0			0
$443$	4.13069			0.196255			0.0981276	−	0.995174i	$-0.468715\pi$
$443$	4.13069			0.196255			0.0981276	+	0.995174i	$0.468715\pi$
$444$		0			0
$445$		−	5.75795i		−	0.272953i
$446$		0			0
$447$	−4.25936			−0.201461
$448$		0			0
$449$	−6.99796			−0.330254			−0.165127	−	0.986272i	$-0.552803\pi$
$449$	−6.99796			−0.330254			−0.165127	+	0.986272i	$0.552803\pi$
$450$		0			0
$451$			26.8372i			1.26371i
$452$		0			0
$453$	1.45824			0.0685143
$454$		0			0
$455$	7.33599	+	0.485723i	0.343917	+	0.0227710i
$456$		0			0
$457$	−34.0073			−1.59080			−0.795398	−	0.606088i	$-0.792738\pi$
$457$	−34.0073			−1.59080			−0.795398	+	0.606088i	$0.792738\pi$
$458$		0			0
$459$	5.02653			0.234618
$460$		0			0
$461$	−36.5119			−1.70053			−0.850265	−	0.526354i	$-0.823559\pi$
$461$	−36.5119			−1.70053			−0.850265	+	0.526354i	$0.823559\pi$
$462$		0			0
$463$			36.9093i			1.71532i	0.514218	+	0.857659i	$0.328082\pi$
$463$			36.9093i			1.71532i	−0.514218	+	0.857659i	$0.671918\pi$
$464$		0			0
$465$			1.64585i			0.0763243i
$466$		0			0
$467$			6.57990i			0.304481i	0.988343	+	0.152241i	$0.0486489\pi$
$467$			6.57990i			0.304481i	−0.988343	+	0.152241i	$0.951351\pi$
$468$		0			0
$469$	26.3972	+	1.74778i	1.21891	+	0.0807051i
$470$		0			0
$471$			0.792423i			0.0365129i
$472$		0			0
$473$	−15.3239			−0.704593
$474$		0			0
$475$			6.72232i			0.308441i
$476$		0			0
$477$		−	20.8865i		−	0.956325i
$478$		0			0
$479$	−23.2509			−1.06236			−0.531180	−	0.847259i	$-0.678251\pi$
$479$	−23.2509			−1.06236			−0.531180	+	0.847259i	$0.678251\pi$
$480$		0			0
$481$		−	0.476518i		−	0.0217274i
$482$		0			0
$483$	5.30601	+	0.351316i	0.241432	+	0.0159854i
$484$		0			0
$485$			12.7637i			0.579569i
$486$		0			0
$487$			23.8122i			1.07903i	0.841975	+	0.539517i	$0.181393\pi$
$487$			23.8122i			1.07903i	−0.841975	+	0.539517i	$0.818607\pi$
$488$		0			0
$489$			0.534199i			0.0241573i
$490$		0			0
$491$	20.7348			0.935750			0.467875	−	0.883795i	$-0.345020\pi$
$491$	20.7348			0.935750			0.467875	+	0.883795i	$0.345020\pi$
$492$		0			0
$493$	20.6649			0.930700
$494$		0			0
$495$	13.8249			0.621385
$496$		0			0
$497$	−1.31004	+	19.7858i	−0.0587633	+	0.887516i
$498$		0			0
$499$	12.5588			0.562207			0.281103	−	0.959677i	$-0.409300\pi$
$499$	12.5588			0.562207			0.281103	+	0.959677i	$0.409300\pi$
$500$		0			0
$501$			0.522341i			0.0233365i
$502$		0			0
$503$	−5.71884			−0.254991			−0.127495	−	0.991839i	$-0.540694\pi$
$503$	−5.71884			−0.254991			−0.127495	+	0.991839i	$0.540694\pi$
$504$		0			0
$505$	−2.61152			−0.116211
$506$		0			0
$507$			1.25458i			0.0557178i
$508$		0			0
$509$	2.13629			0.0946894			0.0473447	−	0.998879i	$-0.484924\pi$
$509$	2.13629			0.0946894			0.0473447	+	0.998879i	$0.484924\pi$
$510$		0			0
$511$	−1.45967	+	22.0458i	−0.0645722	+	0.975249i
$512$		0			0
$513$	9.49675			0.419292
$514$		0			0
$515$	−7.95941			−0.350734
$516$		0			0
$517$	13.3061			0.585203
$518$		0			0
$519$			0.773495i			0.0339526i
$520$		0			0
$521$			11.9780i			0.524765i	0.964964	+	0.262383i	$0.0845083\pi$
$521$			11.9780i			0.524765i	−0.964964	+	0.262383i	$0.915492\pi$
$522$		0			0
$523$		−	27.5958i		−	1.20668i	−0.797483	−	0.603341i	$-0.793836\pi$
$523$		−	27.5958i		−	1.20668i	0.797483	−	0.603341i	$-0.206164\pi$
$524$		0			0
$525$	−0.0415472	+	0.627498i	−0.00181327	+	0.0273863i
$526$		0			0
$527$			24.6370i			1.07321i
$528$		0			0
$529$	−48.5009			−2.10874
$530$		0			0
$531$		−	14.1157i		−	0.612568i
$532$		0			0
$533$		−	15.8781i		−	0.687755i
$534$		0			0
$535$	−0.217879			−0.00941974
$536$		0			0
$537$		−	3.59482i		−	0.155128i
$538$		0			0
$539$	−32.5904	−	4.33468i	−1.40377	−	0.186708i
$540$		0			0
$541$			43.8250i			1.88418i	0.335357	+	0.942091i	$0.391143\pi$
$541$			43.8250i			1.88418i	−0.335357	+	0.942091i	$0.608857\pi$
$542$		0			0
$543$			4.96925i			0.213251i
$544$		0			0
$545$		−	13.9465i		−	0.597401i
$546$		0			0
$547$	44.5179			1.90345			0.951725	−	0.306952i	$-0.0993091\pi$
$547$	44.5179			1.90345			0.951725	+	0.306952i	$0.0993091\pi$
$548$		0			0
$549$	−28.3484			−1.20988
$550$		0			0
$551$	39.0427			1.66327
$552$		0			0
$553$	−0.883202	+	13.3392i	−0.0375576	+	0.567241i
$554$		0			0
$555$	0.0407599			0.00173016
$556$		0			0
$557$			24.4288i			1.03508i	0.855658	+	0.517541i	$0.173152\pi$
$557$			24.4288i			1.03508i	−0.855658	+	0.517541i	$0.826848\pi$
$558$		0			0
$559$	9.06629			0.383463
$560$		0			0
$561$	−3.97214			−0.167704
$562$		0			0
$563$		−	0.730916i		−	0.0308045i	−0.999881	−	0.0154022i	$-0.995097\pi$
$563$		−	0.730916i		−	0.0308045i	0.999881	−	0.0154022i	$-0.00490287\pi$
$564$		0			0
$565$	−14.6924			−0.618115
$566$		0			0
$567$	−22.4258	−	1.48483i	−0.941795	−	0.0623572i
$568$		0			0
$569$	−12.9007			−0.540825			−0.270412	−	0.962745i	$-0.587160\pi$
$569$	−12.9007			−0.540825			−0.270412	+	0.962745i	$0.587160\pi$
$570$		0			0
$571$	6.90455			0.288946			0.144473	−	0.989509i	$-0.453851\pi$
$571$	6.90455			0.288946			0.144473	+	0.989509i	$0.453851\pi$
$572$		0			0
$573$	−1.36719			−0.0571153
$574$		0			0
$575$		−	8.45582i		−	0.352632i
$576$		0			0
$577$		−	0.449436i		−	0.0187103i	−0.999956	−	0.00935514i	$-0.997022\pi$
$577$		−	0.449436i		−	0.0187103i	0.999956	−	0.00935514i	$-0.00297788\pi$
$578$		0			0
$579$			3.32607i			0.138227i
$580$		0			0
$581$	−1.66709	+	25.1785i	−0.0691626	+	1.04458i
$582$		0			0
$583$			33.3272i			1.38027i
$584$		0			0
$585$	−8.17945			−0.338179
$586$		0			0
$587$		−	45.3127i		−	1.87026i	−0.354309	−	0.935128i	$-0.615284\pi$
$587$		−	45.3127i		−	1.87026i	0.354309	−	0.935128i	$-0.384716\pi$
$588$		0			0
$589$			46.5473i			1.91795i
$590$		0			0
$591$	−2.89751			−0.119188
$592$		0			0
$593$			19.7345i			0.810400i	0.914228	+	0.405200i	$0.132798\pi$
$593$			19.7345i			0.810400i	−0.914228	+	0.405200i	$0.867202\pi$
$594$		0			0
$595$	−0.621930	+	9.39316i	−0.0254966	+	0.385082i
$596$		0			0
$597$			4.27096i			0.174799i
$598$		0			0
$599$		−	0.997402i		−	0.0407528i	−0.999792	−	0.0203764i	$-0.993514\pi$
$599$		−	0.997402i		−	0.0407528i	0.999792	−	0.0203764i	$-0.00648645\pi$
$600$		0			0
$601$		−	18.3186i		−	0.747231i	−0.927584	−	0.373616i	$-0.878118\pi$
$601$		−	18.3186i		−	0.747231i	0.927584	−	0.373616i	$-0.121882\pi$
$602$		0			0
$603$	−29.4322			−1.19857
$604$		0			0
$605$	−11.0596			−0.449637
$606$		0			0
$607$	−2.88415			−0.117064			−0.0585320	−	0.998286i	$-0.518642\pi$
$607$	−2.88415			−0.117064			−0.0585320	+	0.998286i	$0.518642\pi$
$608$		0			0
$609$	3.64446	+	0.241303i	0.147681	+	0.00977809i
$610$		0			0
$611$	−7.87250			−0.318487
$612$		0			0
$613$			4.70796i			0.190153i	0.995470	+	0.0950764i	$0.0303095\pi$
$613$			4.70796i			0.190153i	−0.995470	+	0.0950764i	$0.969690\pi$
$614$		0			0
$615$	1.35816			0.0547663
$616$		0			0
$617$	−11.0431			−0.444578			−0.222289	−	0.974981i	$-0.571353\pi$
$617$	−11.0431			−0.444578			−0.222289	+	0.974981i	$0.571353\pi$
$618$		0			0
$619$			44.6125i			1.79313i	0.442914	+	0.896564i	$0.353945\pi$
$619$			44.6125i			1.79313i	−0.442914	+	0.896564i	$0.646055\pi$
$620$		0			0
$621$	−11.9457			−0.479364
$622$		0			0
$623$	1.00646	−	15.2008i	0.0403230	−	0.609008i
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$	−7.50466			−0.299707
$628$		0			0
$629$	0.610144			0.0243280
$630$		0			0
$631$			21.1799i			0.843157i	0.906792	+	0.421578i	$0.138524\pi$
$631$			21.1799i			0.843157i	−0.906792	+	0.421578i	$0.861476\pi$
$632$		0			0
$633$		−	2.80877i		−	0.111639i
$634$		0			0
$635$			12.6197i			0.500796i
$636$		0			0
$637$	19.2819	+	2.56459i	0.763977	+	0.101613i
$638$		0			0
$639$		−	22.0607i		−	0.872709i
$640$		0			0
$641$	37.2700			1.47208			0.736039	−	0.676939i	$-0.236694\pi$
$641$	37.2700			1.47208			0.736039	+	0.676939i	$0.236694\pi$
$642$		0			0
$643$		−	4.33442i		−	0.170933i	−0.996341	−	0.0854664i	$-0.972762\pi$
$643$		−	4.33442i		−	0.170933i	0.996341	−	0.0854664i	$-0.0272380\pi$
$644$		0			0
$645$			0.775502i			0.0305354i
$646$		0			0
$647$	−40.4334			−1.58960			−0.794801	−	0.606871i	$-0.792425\pi$
$647$	−40.4334			−1.58960			−0.794801	+	0.606871i	$0.792425\pi$
$648$		0			0
$649$			22.5235i			0.884125i
$650$		0			0
$651$	−0.287686	+	4.34498i	−0.0112753	+	0.170293i
$652$		0			0
$653$		−	35.9486i		−	1.40678i	−0.710805	−	0.703389i	$-0.751669\pi$
$653$		−	35.9486i		−	1.40678i	0.710805	−	0.703389i	$-0.248331\pi$
$654$		0			0
$655$		−	8.57631i		−	0.335104i
$656$		0			0
$657$		−	24.5805i		−	0.958978i
$658$		0			0
$659$	15.0569			0.586535			0.293267	−	0.956030i	$-0.405257\pi$
$659$	15.0569			0.586535			0.293267	+	0.956030i	$0.405257\pi$
$660$		0			0
$661$	−34.7595			−1.35199			−0.675995	−	0.736907i	$-0.736286\pi$
$661$	−34.7595			−1.35199			−0.675995	+	0.736907i	$0.736286\pi$
$662$		0			0
$663$	2.35010			0.0912702
$664$		0			0
$665$	−1.17503	+	17.7467i	−0.0455656	+	0.688189i
$666$		0			0
$667$	−49.1107			−1.90158
$668$		0			0
$669$		−	6.39541i		−	0.247261i
$670$		0			0
$671$	45.2339			1.74623
$672$		0			0
$673$	−37.5179			−1.44621			−0.723105	−	0.690739i	$-0.757286\pi$
$673$	−37.5179			−1.44621			−0.723105	+	0.690739i	$0.757286\pi$
$674$		0			0
$675$		−	1.41272i		−	0.0543756i
$676$		0			0
$677$	−40.0920			−1.54086			−0.770431	−	0.637524i	$-0.779959\pi$
$677$	−40.0920			−1.54086			−0.770431	+	0.637524i	$0.779959\pi$
$678$		0			0
$679$	−2.23103	+	33.6957i	−0.0856190	+	1.29312i
$680$		0			0
$681$	−4.63484			−0.177607
$682$		0			0
$683$	44.8609			1.71656			0.858278	−	0.513186i	$-0.171535\pi$
$683$	44.8609			1.71656			0.858278	+	0.513186i	$0.171535\pi$
$684$		0			0
$685$	−21.2941			−0.813604
$686$		0			0
$687$		−	3.81641i		−	0.145605i
$688$		0			0
$689$		−	19.7179i		−	0.751191i
$690$		0			0
$691$			39.6466i			1.50823i	0.656744	+	0.754113i	$0.271933\pi$
$691$			39.6466i			1.50823i	−0.656744	+	0.754113i	$0.728067\pi$
$692$		0			0
$693$	36.4975	+	2.41653i	1.38642	+	0.0917964i
$694$		0			0
$695$			11.6407i			0.441558i
$696$		0			0
$697$	20.3306			0.770077
$698$		0			0
$699$		−	1.23310i		−	0.0466402i
$700$		0			0
$701$		−	14.1895i		−	0.535929i	−0.963429	−	0.267965i	$-0.913649\pi$
$701$		−	14.1895i		−	0.535929i	0.963429	−	0.267965i	$-0.0863510\pi$
$702$		0			0
$703$	1.15276			0.0434772
$704$		0			0
$705$		−	0.673389i		−	0.0253613i
$706$		0			0
$707$	−6.89433	−	0.456480i	−0.259288	−	0.0171677i
$708$		0			0
$709$		−	21.0785i		−	0.791621i	−0.918332	−	0.395810i	$-0.870464\pi$
$709$		−	21.0785i		−	0.791621i	0.918332	−	0.395810i	$-0.129536\pi$
$710$		0			0
$711$		−	14.8729i		−	0.557777i
$712$		0			0
$713$		−	58.5507i		−	2.19274i
$714$		0			0
$715$	13.0514			0.488096
$716$		0			0
$717$	4.12219			0.153946
$718$		0			0
$719$	37.6604			1.40449			0.702247	−	0.711933i	$-0.252180\pi$
$719$	37.6604			1.40449			0.702247	+	0.711933i	$0.252180\pi$
$720$		0			0
$721$	−21.0126	−	1.39127i	−0.782551	−	0.0518134i
$722$		0			0
$723$	6.25196			0.232513
$724$		0			0
$725$		−	5.80792i		−	0.215701i
$726$		0			0
$727$	38.8467			1.44075			0.720373	−	0.693587i	$-0.243971\pi$
$727$	38.8467			1.44075			0.720373	+	0.693587i	$0.243971\pi$
$728$		0			0
$729$	23.9969			0.888772
$730$		0			0
$731$			11.6087i			0.429362i
$732$		0			0
$733$	−7.81498			−0.288653			−0.144326	−	0.989530i	$-0.546102\pi$
$733$	−7.81498			−0.288653			−0.144326	+	0.989530i	$0.546102\pi$
$734$		0			0
$735$	−0.219367	+	1.64931i	−0.00809147	+	0.0608359i
$736$		0			0
$737$	46.9632			1.72991
$738$		0			0
$739$	−9.09188			−0.334450			−0.167225	−	0.985919i	$-0.553481\pi$
$739$	−9.09188			−0.334450			−0.167225	+	0.985919i	$0.553481\pi$
$740$		0			0
$741$	4.44009			0.163111
$742$		0			0
$743$		−	25.4240i		−	0.932716i	−0.884596	−	0.466358i	$-0.845566\pi$
$743$		−	25.4240i		−	0.932716i	0.884596	−	0.466358i	$-0.154434\pi$
$744$		0			0
$745$			17.9197i			0.656527i
$746$		0			0
$747$		−	28.0734i		−	1.02715i
$748$		0			0
$749$	−0.575195	−	0.0380842i	−0.0210172	−	0.00139157i
$750$		0			0
$751$		−	10.1548i		−	0.370553i	−0.982686	−	0.185277i	$-0.940682\pi$
$751$		−	10.1548i		−	0.370553i	0.982686	−	0.185277i	$-0.0593182\pi$
$752$		0			0
$753$	7.45771			0.271774
$754$		0			0
$755$		−	6.13504i		−	0.223277i
$756$		0			0
$757$		−	13.0574i		−	0.474578i	−0.971439	−	0.237289i	$-0.923741\pi$
$757$		−	13.0574i		−	0.474578i	0.971439	−	0.237289i	$-0.0762588\pi$
$758$		0			0
$759$	9.43991			0.342647
$760$		0			0
$761$			20.4698i			0.742028i	0.928627	+	0.371014i	$0.120990\pi$
$761$			20.4698i			0.742028i	−0.928627	+	0.371014i	$0.879010\pi$
$762$		0			0
$763$	2.43777	−	36.8182i	0.0882532	−	1.33291i
$764$		0			0
$765$		−	10.4731i		−	0.378657i
$766$		0			0
$767$		−	13.3259i		−	0.481171i
$768$		0			0
$769$		−	27.0520i		−	0.975519i	−0.872978	−	0.487759i	$-0.837814\pi$
$769$		−	27.0520i		−	0.975519i	0.872978	−	0.487759i	$-0.162186\pi$
$770$		0			0
$771$	−6.06015			−0.218251
$772$		0			0
$773$	1.40240			0.0504408			0.0252204	−	0.999682i	$-0.491971\pi$
$773$	1.40240			0.0504408			0.0252204	+	0.999682i	$0.491971\pi$
$774$		0			0
$775$	6.92430			0.248728
$776$		0			0
$777$	0.107605	+	0.00712462i	0.00386031	+	0.000255594i
$778$		0			0
$779$	38.4111			1.37622
$780$		0			0
$781$			35.2009i			1.25959i
$782$		0			0
$783$	−8.20496			−0.293221
$784$		0			0
$785$	3.33383			0.118990
$786$		0			0
$787$		−	23.8331i		−	0.849557i	−0.905297	−	0.424778i	$-0.860352\pi$
$787$		−	23.8331i		−	0.849557i	0.905297	−	0.424778i	$-0.139648\pi$
$788$		0			0
$789$	1.15177			0.0410041
$790$		0			0
$791$	−38.7876	−	2.56816i	−1.37913	−	0.0913133i
$792$		0			0
$793$	−26.7624			−0.950359
$794$		0			0
$795$	1.68661			0.0598177
$796$		0			0
$797$	15.2266			0.539354			0.269677	−	0.962951i	$-0.413083\pi$
$797$	15.2266			0.539354			0.269677	+	0.962951i	$0.413083\pi$
$798$		0			0
$799$		−	10.0801i		−	0.356609i
$800$		0			0
$801$			16.9485i			0.598847i
$802$		0			0
$803$			39.2217i			1.38410i
$804$		0			0
$805$	1.47804	−	22.3231i	0.0520939	−	0.786787i
$806$		0			0
$807$		−	6.04452i		−	0.212777i
$808$		0			0
$809$	5.89511			0.207261			0.103631	−	0.994616i	$-0.466954\pi$
$809$	5.89511			0.207261			0.103631	+	0.994616i	$0.466954\pi$
$810$		0			0
$811$		−	12.8325i		−	0.450609i	−0.974288	−	0.225304i	$-0.927662\pi$
$811$		−	12.8325i		−	0.450609i	0.974288	−	0.225304i	$-0.0723377\pi$
$812$		0			0
$813$		−	1.73798i		−	0.0609537i
$814$		0			0
$815$	2.24745			0.0787247
$816$		0			0
$817$			21.9325i			0.767322i
$818$		0			0
$819$	−21.5935	−	1.42973i	−0.754539	−	0.0499587i
$820$		0			0
$821$		−	45.4206i		−	1.58519i	−0.609749	−	0.792595i	$-0.708730\pi$
$821$		−	45.4206i		−	1.58519i	0.609749	−	0.792595i	$-0.291270\pi$
$822$		0			0
$823$		−	24.3742i		−	0.849630i	−0.905280	−	0.424815i	$-0.860339\pi$
$823$		−	24.3742i		−	0.849630i	0.905280	−	0.424815i	$-0.139661\pi$
$824$		0			0
$825$			1.11638i			0.0388674i
$826$		0			0
$827$	−6.59791			−0.229432			−0.114716	−	0.993398i	$-0.536596\pi$
$827$	−6.59791			−0.229432			−0.114716	+	0.993398i	$0.536596\pi$
$828$		0			0
$829$	25.4531			0.884024			0.442012	−	0.897009i	$-0.354265\pi$
$829$	25.4531			0.884024			0.442012	+	0.897009i	$0.354265\pi$
$830$		0			0
$831$	−3.19295			−0.110762
$832$		0			0
$833$	−3.28375	+	24.6890i	−0.113775	+	0.855422i
$834$		0			0
$835$	2.19756			0.0760497
$836$		0			0
$837$		−	9.78209i		−	0.338118i
$838$		0			0
$839$	5.67643			0.195972			0.0979860	−	0.995188i	$-0.468760\pi$
$839$	5.67643			0.195972			0.0979860	+	0.995188i	$0.468760\pi$
$840$		0			0
$841$	−4.73193			−0.163170
$842$		0			0
$843$			2.50942i			0.0864290i
$844$		0			0
$845$	5.27818			0.181575
$846$		0			0
$847$	−29.1971	−	1.93316i	−1.00322	−	0.0664243i
$848$		0			0
$849$	1.97172			0.0676693
$850$		0			0
$851$	−1.45003			−0.0497062
$852$		0			0
$853$	−31.7516			−1.08716			−0.543578	−	0.839359i	$-0.682931\pi$
$853$	−31.7516			−1.08716			−0.543578	+	0.839359i	$0.682931\pi$
$854$		0			0
$855$		−	19.7872i		−	0.676707i
$856$		0			0
$857$		−	23.1227i		−	0.789857i	−0.918712	−	0.394928i	$-0.870769\pi$
$857$		−	23.1227i		−	0.789857i	0.918712	−	0.394928i	$-0.129231\pi$
$858$		0			0
$859$		−	3.25293i		−	0.110989i	−0.998459	−	0.0554943i	$-0.982327\pi$
$859$		−	3.25293i		−	0.110989i	0.998459	−	0.0554943i	$-0.0176735\pi$
$860$		0			0
$861$	3.58550	+	0.237400i	0.122194	+	0.00809056i
$862$		0			0
$863$		−	33.5622i		−	1.14247i	−0.820786	−	0.571236i	$-0.806464\pi$
$863$		−	33.5622i		−	1.14247i	0.820786	−	0.571236i	$-0.193536\pi$
$864$		0			0
$865$	3.25420			0.110646
$866$		0			0
$867$		−	1.03164i		−	0.0350363i
$868$		0			0
$869$			23.7318i			0.805045i
$870$		0			0
$871$	−27.7855			−0.941475
$872$		0			0
$873$		−	37.5699i		−	1.27155i
$874$		0			0
$875$	2.63997	+	0.174795i	0.0892473	+	0.00590915i
$876$		0			0
$877$			37.9085i			1.28008i	0.768342	+	0.640039i	$0.221082\pi$
$877$			37.9085i			1.28008i	−0.768342	+	0.640039i	$0.778918\pi$
$878$		0			0
$879$			0.204128i			0.00688508i
$880$		0			0
$881$			35.2003i			1.18593i	0.805229	+	0.592963i	$0.202042\pi$
$881$			35.2003i			1.18593i	−0.805229	+	0.592963i	$0.797958\pi$
$882$		0			0
$883$	−4.51832			−0.152054			−0.0760268	−	0.997106i	$-0.524223\pi$
$883$	−4.51832			−0.152054			−0.0760268	+	0.997106i	$0.524223\pi$
$884$		0			0
$885$	1.13986			0.0383159
$886$		0			0
$887$	−13.1235			−0.440644			−0.220322	−	0.975427i	$-0.570711\pi$
$887$	−13.1235			−0.440644			−0.220322	+	0.975427i	$0.570711\pi$
$888$		0			0
$889$	−2.20585	+	33.3155i	−0.0739820	+	1.11737i
$890$		0			0
$891$	−39.8977			−1.33662
$892$		0			0
$893$		−	19.0446i		−	0.637303i
$894$		0			0
$895$	−15.1239			−0.505537
$896$		0			0
$897$	−5.58507			−0.186480
$898$		0			0
$899$		−	40.2158i		−	1.34127i
$900$		0			0
$901$	25.2472			0.841105
$902$		0			0
$903$	−0.135554	+	2.04730i	−0.00451095	+	0.0681300i
$904$		0			0
$905$	20.9063			0.694950
$906$		0			0
$907$	28.4945			0.946145			0.473072	−	0.881024i	$-0.343145\pi$
$907$	28.4945			0.946145			0.473072	+	0.881024i	$0.343145\pi$
$908$		0			0
$909$	7.68700			0.254962
$910$		0			0
$911$		−	8.54564i		−	0.283130i	−0.989929	−	0.141565i	$-0.954787\pi$
$911$		−	8.54564i		−	0.283130i	0.989929	−	0.141565i	$-0.0452134\pi$
$912$		0			0
$913$			44.7950i			1.48250i
$914$		0			0
$915$		−	2.28917i		−	0.0756776i
$916$		0			0
$917$	1.49910	−	22.6412i	0.0495045	−	0.747679i
$918$		0			0
$919$			45.6546i			1.50600i	0.658018	+	0.753002i	$0.271395\pi$
$919$			45.6546i			1.50600i	−0.658018	+	0.753002i	$0.728605\pi$
$920$		0			0
$921$	−5.26364			−0.173443
$922$		0			0
$923$		−	20.8264i		−	0.685511i
$924$		0			0
$925$		−	0.171482i		−	0.00563831i
$926$		0			0
$927$	23.4285			0.769495
$928$		0			0
$929$		−	32.1038i		−	1.05329i	−0.850085	−	0.526645i	$-0.823450\pi$
$929$		−	32.1038i		−	1.05329i	0.850085	−	0.526645i	$-0.176550\pi$
$930$		0			0
$931$	−6.20408	+	46.6455i	−0.203330	+	1.52874i
$932$		0			0
$933$		−	1.15980i		−	0.0379702i
$934$		0			0
$935$			16.7113i			0.546520i
$936$		0			0
$937$		−	45.6950i		−	1.49279i	−0.665503	−	0.746395i	$-0.731783\pi$
$937$		−	45.6950i		−	1.49279i	0.665503	−	0.746395i	$-0.268217\pi$
$938$		0			0
$939$	−6.56588			−0.214269
$940$		0			0
$941$	−19.0939			−0.622442			−0.311221	−	0.950338i	$-0.600738\pi$
$941$	−19.0939			−0.622442			−0.311221	+	0.950338i	$0.600738\pi$
$942$		0			0
$943$	−48.3163			−1.57340
$944$		0			0
$945$	0.246936	−	3.72954i	0.00803283	−	0.121322i
$946$		0			0
$947$	−46.0057			−1.49498			−0.747492	−	0.664271i	$-0.768742\pi$
$947$	−46.0057			−1.49498			−0.747492	+	0.664271i	$0.768742\pi$
$948$		0			0
$949$		−	23.2053i		−	0.753275i
$950$		0			0
$951$	7.12078			0.230907
$952$		0			0
$953$	16.1643			0.523612			0.261806	−	0.965121i	$-0.415682\pi$
$953$	16.1643			0.523612			0.261806	+	0.965121i	$0.415682\pi$
$954$		0			0
$955$			5.75197i			0.186129i
$956$		0			0
$957$	6.48385			0.209593
$958$		0			0
$959$	−56.2157	−	3.72209i	−1.81530	−	0.120193i
$960$		0			0
$961$	16.9459			0.546643
$962$		0			0
$963$	0.641328			0.0206665
$964$		0			0
$965$	13.9932			0.450458
$966$		0			0
$967$		−	14.1944i		−	0.456462i	−0.973607	−	0.228231i	$-0.926706\pi$
$967$		−	14.1944i		−	0.456462i	0.973607	−	0.228231i	$-0.0732942\pi$
$968$		0			0
$969$			5.68519i			0.182635i
$970$		0			0
$971$			33.2911i			1.06836i	0.845370	+	0.534181i	$0.179380\pi$
$971$			33.2911i			1.06836i	−0.845370	+	0.534181i	$0.820620\pi$
$972$		0			0
$973$	−2.03474	+	30.7312i	−0.0652308	+	0.985196i
$974$		0			0
$975$		−	0.660500i		−	0.0211529i
$976$		0			0
$977$	−16.9401			−0.541962			−0.270981	−	0.962585i	$-0.587348\pi$
$977$	−16.9401			−0.541962			−0.270981	+	0.962585i	$0.587348\pi$
$978$		0			0
$979$		−	27.0437i		−	0.864322i
$980$		0			0
$981$			41.0514i			1.31067i
$982$		0			0
$983$	28.4007			0.905843			0.452921	−	0.891551i	$-0.350382\pi$
$983$	28.4007			0.905843			0.452921	+	0.891551i	$0.350382\pi$
$984$		0			0
$985$			12.1902i			0.388413i
$986$		0			0
$987$	0.117705	−	1.77773i	0.00374659	−	0.0565857i
$988$		0			0
$989$		−	27.5883i		−	0.877258i
$990$		0			0
$991$			13.8298i			0.439319i	0.975577	+	0.219659i	$0.0704946\pi$
$991$			13.8298i			0.439319i	−0.975577	+	0.219659i	$0.929505\pi$
$992$		0			0
$993$		−	4.38186i		−	0.139054i
$994$		0			0
$995$	17.9685			0.569640
$996$		0			0
$997$	16.2256			0.513871			0.256936	−	0.966429i	$-0.417287\pi$
$997$	16.2256			0.513871			0.256936	+	0.966429i	$0.417287\pi$
$998$		0			0
$999$	−0.242257			−0.00766466

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2240.2.h.e.671.3	✓	24
4.3	odd	2	inner	2240.2.h.e.671.22	yes	24
7.6	odd	2		2240.2.h.f.671.22	yes	24
8.3	odd	2		2240.2.h.f.671.21	yes	24
8.5	even	2		2240.2.h.f.671.4	yes	24
28.27	even	2		2240.2.h.f.671.3	yes	24
56.13	odd	2	inner	2240.2.h.e.671.21	yes	24
56.27	even	2	inner	2240.2.h.e.671.4	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2240.2.h.e.671.3	✓	24	1.1	even	1	trivial
2240.2.h.e.671.4	yes	24	56.27	even	2	inner
2240.2.h.e.671.21	yes	24	56.13	odd	2	inner
2240.2.h.e.671.22	yes	24	4.3	odd	2	inner
2240.2.h.f.671.3	yes	24	28.27	even	2
2240.2.h.f.671.4	yes	24	8.5	even	2
2240.2.h.f.671.21	yes	24	8.3	odd	2
2240.2.h.f.671.22	yes	24	7.6	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:	\( N \)	\(=\)	\( 2240 = 2^{6} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2240.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-0.237691i q^{3} -1.00000 q^{5} +(-2.63997 - 0.174795i) q^{7} +2.94350 q^{9} +O(q^{10})\)	\(q-0.237691i q^{3} -1.00000 q^{5} +(-2.63997 - 0.174795i) q^{7} +2.94350 q^{9} -4.69677 q^{11} +2.77882 q^{13} +0.237691i q^{15} +3.55805i q^{17} +6.72232i q^{19} +(-0.0415472 + 0.627498i) q^{21} -8.45582i q^{23} +1.00000 q^{25} -1.41272i q^{27} -5.80792i q^{29} +6.92430 q^{31} +1.11638i q^{33} +(2.63997 + 0.174795i) q^{35} -0.171482i q^{37} -0.660500i q^{39} -5.71397i q^{41} +3.26264 q^{43} -2.94350 q^{45} -2.83304 q^{47} +(6.93889 + 0.922907i) q^{49} +0.845718 q^{51} -7.09578i q^{53} +4.69677 q^{55} +1.59784 q^{57} -4.79554i q^{59} -9.63085 q^{61} +(-7.77076 - 0.514509i) q^{63} -2.77882 q^{65} -9.99904 q^{67} -2.00988 q^{69} -7.49472i q^{71} -8.35078i q^{73} -0.237691i q^{75} +(12.3993 + 0.820971i) q^{77} -5.05279i q^{79} +8.49472 q^{81} -9.53742i q^{83} -3.55805i q^{85} -1.38049 q^{87} +5.75795i q^{89} +(-7.33599 - 0.485723i) q^{91} -1.64585i q^{93} -6.72232i q^{95} -12.7637i q^{97} -13.8249 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 24 q^{5} - 36 q^{9}+O(q^{10}) \) 24 * q - 24 * q^5 - 36 * q^9	\( 24 q - 24 q^{5} - 36 q^{9} - 4 q^{13} - 8 q^{21} + 24 q^{25} + 36 q^{45} + 24 q^{57} - 56 q^{61} + 4 q^{65} - 96 q^{69} + 12 q^{77} + 24 q^{81}+O(q^{100}) \) 24 * q - 24 * q^5 - 36 * q^9 - 4 * q^13 - 8 * q^21 + 24 * q^25 + 36 * q^45 + 24 * q^57 - 56 * q^61 + 4 * q^65 - 96 * q^69 + 12 * q^77 + 24 * q^81

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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